Commun. Math. Phys. 303, 1–30 (2011) Digital Object Identifier (DOI) 10.1007/s00220-010-1179-4
Communications in
Mathematical Physics
Emergent Spacetime from Modular Motives Rolf Schimmrigk Indiana University South Bend, 1700 Mishawaka Ave., South Bend, IN 46634, USA. E-mail:
[email protected] Received: 28 April 2009 / Accepted: 4 October 2010 Published online: 25 February 2011 – © Springer-Verlag 2011
Abstract: The program of constructing spacetime geometry from string theoretic modular forms is extended to Calabi-Yau varieties of dimensions three and four, as well as higher rank motives. Modular forms on the worldsheet can be constructed from the geometry of spacetime by computing the L–functions associated to omega motives of Calabi-Yau varieties, generated by their holomorphic n−forms via Galois representations. The modular forms that emerge in this way are related to characters of the underlying rational conformal field theory. The converse problem of constructing space from string theory proceeds in the class of diagonal theories by determining the motives associated to modular forms in the category of pure motives with complex multiplication. The emerging picture suggests that the L–function can be viewed as defining a map between the geometric category of motives and the category of conformal field theories on the worldsheet. Contents 1. 2.
3.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Modularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Modular forms from affine Lie algebras . . . . . . . . . . . . 2.2 Modular forms from Größencharacters . . . . . . . . . . . . . 2.3 Rankin-Selberg products of modular forms . . . . . . . . . . . 2.4 Complex multiplication modular forms . . . . . . . . . . . . . −Motives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Grothendieck motives . . . . . . . . . . . . . . . . . . . . . . 3.2 −motives for manifolds of Calabi-Yau and special Fano type 3.3 L–function of −motives . . . . . . . . . . . . . . . . . . . 3.4 String theoretic modularity and automorphy . . . . . . . . . . 3.5 Grothendieck motives for Brieskorn-Pham varieties . . . . . . 3.6 −motives for CY and SF BP hypersurfaces . . . . . . . . .
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3.7 Lower weight motives from higher dimensional varieties . . . . L–Functions for Calabi-Yau Threefolds and Fourfolds . . . . . . . . L–Functions Via Jacobi Sums . . . . . . . . . . . . . . . . . . . . . Modularity Results for Rank two Motives in Dimension One and Two A Nonextremal K3 Surface X 212 ⊂ P(2,3,3,4) . . . . . . . . . . . . . Modular Motives of the Calabi-Yau Threefold X 36 ⊂ P(1,1,1,1,2) . . . 8.1 The modular −motive of X 36 . . . . . . . . . . . . . . . . . . 8.2 Lower weight modular motives of X 36 . . . . . . . . . . . . . . 9. The K3 Fibration Hypersurface X 312 ⊂ P(2,2,2,3,3) . . . . . . . . . . 9.1 The −motive of X 312 . . . . . . . . . . . . . . . . . . . . . . 9.2 Lower weight modular forms of X 312 . . . . . . . . . . . . . . . 10. A Modular Calabi-Yau Fourfold X 46 ⊂ P5 . . . . . . . . . . . . . . . 11. Emergent Space from Characters and Modular Forms . . . . . . . . 12. Further Considerations . . . . . . . . . . . . . . . . . . . . . . . . .
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1. Introduction The present paper continues the program of applying methods from arithmetic geometry to the problem of understanding how spacetime emerges in string theory. The goal is to construct a direct relation between the physics on the worldsheet and the geometry of the extra dimensions. One way to formulate this question is by asking whether it is possible to explicitly determine the structure of the compact dimensions from the building blocks of the two-dimensional worldsheet theory. In this general, but vague, form the problem of constructing an emergent geometry in string theory could have been formulated more than thirty years ago. The reason that it was not can probably be traced to both the lack of a concrete framework, and the lack of useful tools. The framework of the heterotic string of the 1980s, in combination with the web of dualities between different string models discovered in the 1990s, motivates a more concrete version of this problem, which aims at the relation between Calabi-Yau varieties and worldsheet physics given by exactly solvable conformal field theories. Both Calabi-Yau varieties and conformal field theories define rich structures, raising a number of problems which have not been addressed in the past. The key ingredient of the program pursued here is the modular invariance of the theory. From a spacetime physics perspective it is initially somewhat surprising that this feature of string theory should turn out to provide a useful tool for the understanding of its geometric consequences, because it is the modular invariance of the two-dimensional theory that would appear to be most difficult to explain from a geometric perspective. By now there exists a fair amount of evidence that shows that methods from arithmetic geometry provide promising tools for this problem, at least in lower dimensions. The purpose of the present paper is threefold. First, to describe the notion of the −motive for arbitrary Calabi-Yau varieties (and more generally, Fano manifolds of special type), independent of any specific framework. Second, to extend the string modularity results obtained previously for the -motive of diagonal Calabi-Yau manifolds in complex dimensions one and two to spaces of dimension three and four. Finally, to extend, in the process, previous results for rank two motives to higher rank motives. The basic problem of establishing modularity results for motivic L–functions in dimensions larger than one is made difficult by the fact that no generalization of the elliptic modularity theorem [1–3] is known, even conjecturally. This makes already
Emergent Spacetime from Modular Motives
3
the first step, the construction of modular forms from algebraic varieties, nontrivial. There exists, however, an extensive web of conjectures, the Langlands program, that suggests that in higher dimensions the Hasse-Weil L–functions of geometric structures have modular properties in a generalized sense. It is expected in particular that associated to each cohomology group is an automorphic representation, leading to an automorphic form. The class of automorphic L–functions contains a special type of object, the standard L–function [4], which generalizes the notion of a Hecke L–function. Modularity of Hecke L–functions is known by virtue of their analytic continuation and their functional equations [5,6]. Langlands’ vision thus extends results obtained by Artin and Hecke. While Artin considered representations ρ of the Galois group Gal(K /Q) of a number field K to define L–functions L(ρ, s), Hecke had previously introduced L–functions based on certain characters χ associated to number fields (called Größencharaktere by Hecke, also called algebraic Hecke characters), whose structure was motivated by an attempt to establish modularity of the L–function. It turns out that these a priori different concepts lead to the same object in the sense that Artin’s L–functions and Hecke’s L–functions agree [7,8]. Langlands’ conjectures involve a generalization of the Artin-Hecke framework to GL(n). More precisely, the connection between geometry and arithmetic can be made because representations of the Galois group can be constructed by considering the −adic cohomology as a representation space. This strategy has proven difficult to implement for higher n in general, and in particular in the context of obtaining a string theoretic interpretation of geometric modular forms beyond the case of elliptic curves and rigid Calabi-Yau varieties. For varieties of higher dimensions the results obtained so far indicate that it is more important to identify irreducible pieces of low rank in the cohomology groups, and to consider the L–functions associated to these subspaces. The difficulty here is that at present there exists no general framework that provides guidance for the necessary decomposition of the full cohomology groups. Nevertheless, the Langlands program suggests that modularity, and more generally automorphy, are phenomena that transcend the framework of elliptic curves, and one can ask the question whether the methods described in [9–11] to establish modularity relations between elliptic curves and conformal field theories can be generalized to higher dimensional varieties. Results in this direction have been obtained for extremal K3 surfaces of Brieskorn-Pham type in ref. [12]. In the present paper the string modularity results obtained previously for diagonal Calabi-Yau manifolds in complex dimensions one and two are extended to all higher dimensions that are of physical relevance. The idea is to consider particular subgroups of the intermediate cohomology group of a variety, defined by the representation of a Galois group associated to the manifold. For spaces of Calabi-Yau and special Fano type there exists at least one nontrivial orbit, defined by the holomorphic n−form in the case of Calabi-Yau spaces, and the corresponding cohomology group in the case of special Fano manifolds. This orbit will be called the omega motive M (X ), and its L–function will be noted by L(M (X ), s) = L (X, s). The precise definition of these objects and their concrete implementation will be given in Sect. 2. The strategy developed here is completely general and can be applied to any Calabi-Yau variety, as well as Fano varieties of special type. In later sections this framework will be applied to Calabi-Yau varieties of Brieskorn-Pham type, the class of varieties for which Gepner [13] originally discovered a relation between the spectra of a certain type of conformal field theory and the cohomology of the manifolds. For such models the −motive, as well as the other submotives, can be determined explicitly.
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A simplifying characteristic of extremal K3 surfaces of Brieskorn-Pham type analyzed in [12] is that their −motive is of rank two. This is not the case in general, hence the problem arises of extending the string theoretic analysis from extremal K3 surfaces to K3 motives of higher rank. An example that leads to a motive of rank four is defined by the surface X 212 = (z 0 : z 1 : z 2 : z 3 ) ∈ P(2,3,3,4) z 06 + z 14 + z 24 + z 33 = 0 . (1) It is shown in Sect. 7 that the L–function L (X 212 , s) of the −motive of X 212 is obtained from those of the diagonal elliptic curves E 4 ⊂ P(1,1,2) and E 6 ⊂ P(1,2,3) . It was shown in [11] that the L–functions of E 4 and E 6 have a string theoretic structure, i.e. the modular forms f E 4 ∈ S2 (0 (64)) and f E 6 ∈ S2 (0 (144)) are determined by the Hecke indefinite theta series that appear in the conformal field theory on the string worldsheet. The concrete relations are summarized in Theorem 6 in Sect. 6. This shows that L (X 212 , s) is string theoretic in the same sense, i.e. its L–function is determined by modular forms that arise in the underlying string model on the worldsheet. More precisely, the following result will be shown. Theorem 1. The L–series L (X 212 , s) of the rank four −motive of the K3 surface X 212 is given by the Rankin-Selberg product of the L–functions of the modular forms f E 4 ∈ S2 (0 (64)) and f E 6 ∈ S2 (0 (144)) of the elliptic curves E 4 and E 6 , (2) L (X 212 , s) = L f E 4 ⊗ f E 6 , s . The precise meaning of the tensor product of modular forms will become clear below. Previous work on the construction of space via modular forms arising in the conformal field theory on the worldsheet was restricted to elliptic curves and K3 surfaces. To lift this restriction, consider first the Calabi-Yau threefolds X 36 = {z 06 + z 16 + z 26 + z 36 + z 43 = 0} ⊂ P(1,1,1,1,2) , X 312 = {z 06 + z 16 + z 26 + z 34 + z 44 = 0} ⊂ P(2,2,2,3,3) .
(3)
It is shown in Sect. 8 that the rank two −motive of X 36 leads to a string theoretic modular form of weight four, and that the remaining part of the intermediate cohomology leads to modular forms of weight two. The −motive of X 312 is of rank four and it is shown in Sect. 9 that its L–function L (X 312 , s) is determined by the weight two modular form f E 4 of the elliptic curve E 4 and the weight three cusp form f X 6A ∈ S3 (1 (27)) of the 2
−motive of the diagonal K3 surface X 26A ⊂ P(1,1,1,3) , whose string theoretic interpretation was established in [12]. More precisely, the following results will be shown. Theorem 2. 1) The inverse Mellin transform of the −motivic L (X 36 , s) of the threefold X 36 is a cusp form f (q) ∈ S4 (0 (108)). The L–function of the intermediate cohomology group H 3 (X 36 ) decomposes into modular factors as L( f i ⊗ χi , s), (4) L(H 3 (X 36 ), s) = L( f , s) · i
where f i ∈ S2 (0 (Ni )) with Ni = 27, 144, 432, and the χi are twist characters.
Emergent Spacetime from Modular Motives
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2) The −motivic L–series of X 312 is given by the Rankin-Selberg product of the modular forms associated to E 4 ⊂ P(1,1,2) and the extremal weighted Fermat surface X 26A ⊂ P(1,1,1,3) , L (X 312 , s) = L( f E 4 ⊗ f X 6A , s). 2
(5)
The L–series of the remaining part of the intermediate cohomology H 3 (X 312 ) decomposes into a product of factors that include L–series of modular forms of weight two and levels N = 27, 64, 144, 432, possibly including a twist. The results concerning the rank four motives in Theorems 1 and 2 can be viewed from a somewhat different perspective. A result shown by Ramakrishnan in ref. [14] implies that the Rankin-Selberg L–function associated to any pair of elliptic modular cusp forms comes from an automorphic form. Hence the fact that the L–functions of the −motives of both X 212 and X 312 are given in terms of the Rankin-Selberg convolution shows that these motives are automorphic in the sense that there exists an automorphic representation whose L–function agrees with that of the motive. In F-theory Calabi-Yau fourfolds are of interest, and as a final example in this paper the modularity of the −motive of the degree six fourfold
5 X 46 = (z 0 : · · · : z 5 ) ∈ P5 z i6 = 0 (6) i=0
is shown in Sect. 10. Theorem 3. The inverse Mellin transform of the L–function of the −motive of X 46 is of the form f X 46 , q = f 27 (q) ⊗ χ3 , (7) where f 27 (q) is a cusp Hecke eigenform of weight w = 5 and level N = 27, and χ3 is the Legendre character. There exists an Größencharacter ψ27 with congruence ideal 4 , s) ⊗ χ . m = (3) such that the motivic L–series is given by L (X 46 , s) = L(ψ27 3 The basic question raised by these results is whether the −motive is string automorphic in general. More generally one may view the L–function as a link between the geometry of spacetime and the physics of the worldsheet. One way to make this idea more explicit is by viewing L as a functor from the category of Fano varieties of special type (or rather their motives) to the category of superconformal field theories. The evidence obtained so far supports this perspective for a physical interpretation of L–functions. The notion e.g. of composing motives then translates into a corresponding composition of conformal field theories. A concrete example is the motivic tensor structure which maps into a tensor structure for conformal field theories. The basic tensor structure of motives is described in L–function terms by the Rankin-Selberg convolution L( f 1 ⊗ f 2 , s) of the modular forms f i associated to the modular motives Mi , and also leads to the symmetric square of modular forms. Denote by χ = i i ⊗ χ twisted products constructed from modular forms i on the string worldsheet. The motivic L– functions L(M(X ), s) that emerge from Calabi-Yau varieties and special Fano varieties can be expressed in terms of string modular L–functions of the type
L χ , s , L Symr χ , s , L 1χ1 ⊗ 2χ2 , s , (8)
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where L(Symr f, s) describes the L–function associated to a symmetric tensor product of a modular form f . The emerging picture thus indicates that L–functions of automorphic forms facilitate a structural map between the space of Calabi-Yau varieties (and their generalizations) and the space of certain conformal field theories. The paper is organized as follows. Section 2 briefly introduces the necessary modular theoretic background. Section 3 describes the notion of a Grothendieck motive and introduces the general concept of −motives for arbitrary Calabi-Yau manifolds, as well as for the class of Fano varieties of special type. This provides the framework for the relation between the geometry of spacetime and physics on the string worldsheet. Sections 4 and 5 describe the basic structure of the L–functions for Calabi-Yau surfaces and threefolds, derived from Artin’s zeta function. To make this paper more self-contained, Sect. 6 briefly reviews the results for modular motives of rank two that appear as building blocks for the examples of higher dimension. More details can be found in [11,12]. Sections 7 through 10 contain the string theoretic modularity analysis of the higher dimensional varieties and higher rank motives, leading to the results of Theorem 1, 2, 3. Section 11 shows how the converse problem can be approached in the context of modular motives, and Sect. 12 ends the paper with some final remarks. 2. Modularity In order to make the paper more self-contained the following paragraphs briefly summarize the types of modular forms that eventually are reflected in the geometry of weighted hypersurfaces of Calabi-Yau and special Fano type. The affine Lie algebraic forms introduced by Kac and Peterson provide the structures on the worldsheet, while certain types of modular Hecke L–series will arise from the arithmetic of the geometry. 2.1. Modular forms from affine Lie algebras. The simplest class of N = 2 supersymmetric exactly solvable theories is built in terms of the affine SU(2) theory as a coset model SU(2)k ⊗ U(1)2 . (9) U(1)k+2,diag Coset theories G/H lead to central charges of the form cG − c H , hence the supersymmetric affine theory at level k still has central charge ck = 3k/(k + 2). The spectrum of anomalous dimensions k,q,s and U(1)−charges Q k,q,s of the primary fields k,q,s at level k is given by ( + 2) − q 2 s 2 + , 4(k + 2) 8 (10) s q k Q ,q,s = − + , k+2 2 where ∈ {0, 1, . . . , k}, + q + s ∈ 2Z, and |q − s| ≤ . Associated to the primary fields are characters defined as k,q,s =
c
k 2πiτ (L 0 − 24 ) 2πi J0 χ,q,s (τ, z, u) = e−2πiu tr Hq,s e e k = c,q+4 j−s (τ )θ2q+(4 j−s)(k+2),2k(k+2) (τ, z, u),
(11)
to a definite fermion number (mod where the trace is to be taken over a projection Hq,s 2) of a highest weight representation of the (right-moving) N = 2 algebra with highest weight vector determined by the primary field. The expression of the rhs in terms of the string functions [15],
Emergent Spacetime from Modular Motives
k c,m (τ ) =
7
k,m (τ ) η3 (τ )
,
(12)
where η(τ ) is the Dedekind eta function, and k,m (τ ) are the Hecke indefinite modular forms 2 2 sign(x)e2πiτ ((k+2)x −ky ) (13) k,m (τ ) = −|x|
+1 m ∈Z2 + 2(k+2) , 2k
and theta functions θn,m (τ, z, u) = e−2πimu
e2πim
2 τ +2πiz
(14)
n ∈Z+ 2m
is useful because it follows from this representation that the modular behavior of the N = 2 characters decomposes into a product of the affine SU(2) structure in the index and into -function behavior in the charge and sector index. It follows from the coset construction that the essential ingredient in the conformal field theory is the SU(2) affine theory. The issue of understanding the emergence of spacetime in string theory can now be reformulated in a more concrete way as the problem of relating string theoretic modular forms to geometric ones. It is apparent from the results of [9–12] that more important than the string functions are the associated SU(2) theta functions k,m (τ ). These indefinite Hecke forms are associated to quadratic number fields determined by the level of the affine theory. They are modular forms of weight 1 and cannot, therefore, be identified with geometric modular forms. However, products of these forms can lead to interesting motives, and it was shown in the above references that appropriate products lead to modular forms associated to Calabi-Yau varieties. 2.2. Modular forms from Größencharacters. The modularity of the L–series determined in this paper follows from the fact that they can be interpreted in terms of Hecke L–series associated to Größencharacters, defined by Jacobi sums. A Größencharacter can be associated to any number field. While some of the Jacobi sums encountered in this paper take values in higher degree cyclotomic fields, it will be sufficient for the modularity proofs that follow to focus on the special case of imaginary quadratic fields. Hecke’s modularity results (see e.g. [16,17]) have been extended in work by Shimura [18] and Ribet [19], and introductions can be found e.g. in Miyake [20] and Iwaniec [21]. In the brief summary of the necessary background that follows the notation of Ribet is adopted for the most part. √ Let K = Q( −D) be an imaginary quadratic field, σ : K −→ C an embedding, m ⊂ O K an integral ideal in the ring of algebraic integers, and w > 1 a positive integer. Denote further by mod× m the multiplicative congruence modulo m and by (z) the principal ideal of z ∈ K . Definition. A Größencharacter is a homomorphism ψ : Im(K ) → C× from the fractional ideals of K prime to the integral ideal m such that ψ((z)) = σ (z)w−1 , for all z ∈ K × such that z ≡ 1(mod× m). The type of behavior of ψ on the principal ideals is called the infinity type.
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The Hecke L–series of the character ψ is defined by 1 . L(ψ, s) = ψ(p) p∈Spec O K 1 − Nps
(15)
The modularity of the corresponding q−series f (ψ, q) = n an q n associated to the L–series via the inverse Mellin transform is characterized by a Nebentypus character defined in terms of the Dirichlet character ϕ D associated to K and a second character λ defined mod Nm by λ(a) =
ψ((a)) , σ (a)w−1
a ∈ Z.
(16)
The Nebentypus character is given by the product = λϕ D of these two characters. Modularity of the L–series follows from the results of Hecke and Shimura, adapted here following the formulation of Ribet [19]. Denote by Nm the norm of the ideal m, and by D K the discriminant of K . Theorem 4. Let ψ be a Größencharacter of the imaginary quadratic field K with infinity type σ w−1 . Define the coefficients cn as
ψ(a)q Na =:
(a,m)=1
∞
cn q n .
(17)
n=1
a integral
Then there exists a unique newform f (q) = |D K |Nm, and character = λϕ D such that
∞
n=1 an q
n
of weight w, level N =
a p = c p ∀ p |/ DNm.
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Of particular importance in this paper are √ √ Größencharacters √ associated to the Gauss field Q( −1) and the Eisenstein field Q( −3). For Q( −1) consider prime ideals p = (z p) and define the character ψ32 by setting ψ32 (p) = z p,
(19)
where the generator z p is determined by the congruence relation z p ≡ 1(mod (2 + 2i))
(20)
for the congruence √ ideal m = (2 + 2i). For the field Q( −3) two characters ψ N , N = 27, 36 and their twists will appear. The congruence ideals here are given by m27 = (3), m36 = 2 + 4ξ3 ,
(21)
where ξ3 = e2πi/3 , leading to the characters ψ N (p) = z p,
(22)
where the generator is determined uniquely by the congruence relations z p ≡ 1(mod m N ).
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Emergent Spacetime from Modular Motives
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2.3. Rankin-Selberg products of modular forms. An important ingredient in the analysis of higher dimensional varieties is the fact that the L–series of motives carrying higher dimensional representations of the Galois group can sometimes be expressed in terms of L–functions of lower rank motives. This leads to Rankin-Selberg products of L–functions. This construction is quite general, but will be applied here only to products of L–series that are associated to Hecke eigenforms. If the L–series of two modular forms f, g of arbitrary weight and arbitrary levels are given by L( f, s) = an n −s , n
L(g, s) =
bn n −s ,
(24)
n
it is natural to consider the naive Rankin-Selberg L–series associated to f, g as L( f × g, s) = an bn n −s . n
It turns out that a slight modification of this product has better properties, and is more appropriate for geometric constructions. If f ∈ Sw (0 (N )) and g ∈ Sv (0 (N )) are cusp forms with characters and λ, respectively, the modified Rankin-Selberg product is defined as L( f ⊗ g, s) = L N (λ, 2s + 2 − (w + v))L( f × g, s),
(25)
where L N (χ , s) is the truncated Dirichlet L–series defined by the condition that χ (n) = 0 if (n, N ) > 1 [23]. Hecke showed that such forms f, g have Euler products given by −1 L( f, s) = 1 − α p p −s 1 − β p p −s , p
L(g, s) =
1 − γ p p −s
−1 1 − δ p p −s ,
(26)
p
where α p + β p = a p , γ p + δ p = b p and α p β p = p w−1 , γ p δ p = p v−1 . It can be shown that the modified Rankin-Selberg product has the Euler product L( f ⊗ g, s) −1 = 1 − α p γ p p −s 1 − α p δ p p −s 1 − β p γ p p −s 1 − β p δ p p −s .
(27)
p
The tensor notation is at this point formal, but it will become clear that this product is indicative of a representation theoretic tensor product. Furthermore, it also describes the L–series of the tensor product M f ⊗ Mg of motives M f , Mg associated to the modular forms f, g via the constructions of Deligne [24], Jannsen [25] and Scholl [26], L( f ⊗ g, s) = L(M f ⊗ Mg , s). The motivic tensor product will be described below. The Rankin-Selberg products which will appear below involve modular forms of weight two and three, leading to rank four motives on Calabi-Yau varieties of dimension two and three.
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2.4. Complex multiplication modular forms. A special class of modular forms that is relevant in this paper are forms which are sparse in the sense that a particular subset of the coefficients a p of their Fourier expansion f (q) = n an q n vanish. A conceptual way to formulate this idea was introduced by Ribet [19]. A modular form f (q) is called to be of complex √ multiplication (CM) type if there exists an imaginary quadratic number field K = Q( −d) such that the coefficients a p vanish for those rational primes p which are inert in K . It follows from this definition that any such form can be described by the inverse Mellin transform of an L–series associated to a Hecke Größencharacter, which is the view originally adopted by Hecke, and also Shimura. Modular forms with complex multiplication are more transparent than general forms, in particular in the context of their associated geometry. This will become important further below in the construction of the Calabi-Yau motives from the conformal field theory on the worldsheet. 3. −Motives The results in the present and previous papers show that it is useful for modularity to consider the Galois orbit in the cohomology defined by the holomorphic n−form of a Calabi-Yau variety. This Galois orbit defines a geometric substructure of the manifold, called here the −motive, which provides an example of a Grothendieck motives in this manifolds. Similar orbits can be considered in the context of so-called special Fano varieties considered in [27–29], whose modularity properties were analyzed in the context of mirror pairs of rigid Calabi-Yau varieties in [30]. The aim of the present section is to provide the geometric framework for the construction of the motivic structure from the conformal field theory. The circle of ideas that is concerned with the relations between characters, modular forms, and motives extends beyond the class of weighted Fermat varieties, and it is useful to formulate the constructions in its most general context. The outline of this section is to first describe the concept of Grothendieck motives, also called pure motives, then to define the notion of −motives in complete generality, and finally to consider the −motive in detail for weighted Fermat hypersurfaces, i.e. of Brieskorn-Pham type.
3.1. Grothendieck motives. The idea to construct varieties directly from the conformal field theory on the worldsheet without the intermediary of Landau-Ginzburg theories or sigma models becomes more complex as the number of dimensions increases. Mirror symmetry and other dualities show that it should not be expected that any particular model on the worldsheet should lead to a unique variety. Rather, one should view manifolds as objects which can be built from irreducible geometric structures. This physical expectation is compatible with Grothendieck’s notion of motives. The original idea for the existence of motives arose from a plethora of cohomology theories Grothendieck was led to during his pursuit of the Weil conjectures [31–33]. There are several ways to think of motives as structures that support these various cohomology theories, such as Betti, de Rham, étale, crystalline cohomology groups, etc., and to view these cohomology groups as realization of motives. Grothendieck’s vision of motives as basic building blocks that support universal structures is based on the notion of correspondences. This is an old concept that goes back to Klein and Hurwitz in the late 19th century. The idea is to define a relation between two varieties by considering an algebraic cycle class on their product. In order to do so
Emergent Spacetime from Modular Motives
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an algebraic cycle is defined as a finite linear combination of irreducible subvarieties Vα ⊂ X of codimension r of a variety X . The set of all these algebraic cycles defines a group
r n α Vα Vα ⊂ X . (28) Z = α
This group is too large to be useful, hence one considers equivalence relations between its elements. There are a variety of such equivalence relations, resulting in quite different structures. The most common of these are rational, homological, and numerical equivalence. A description of these can be found in [34], but for the following it will not be important which of these is chosen. Given any of these equivalences one considers the group of equivalence classes Ar (X ) = Z r (X )/ ∼
(29)
of algebraic cycles to define the group of algebraic correspondences of degree r between manifolds X, Y of equal dimension d as Corrr (X, Y ) = Ad+r (X × Y ).
(30)
Correspondences can be composed f · g, leading to the notion of a projector p such that p · p = p. The first step in the construction of Grothendieck motives is the definition of an effective motive, obtained by considering a pair defined by a variety and a projector M = (X, p), where p is a projector in the ring of algebraic correspondences of degree zero, p ∈ Corr 0 (X, X ). Maps between such objects are of the form [35] Hom((X, p), (Y, q)) = q ◦ Corr 0 (X, Y )Q ◦ p.
(31)
This formulation of morphisms between effective motives is equivalent to the original view of Grothendieck described in [33]. It is important for physical applications of motives to enlarge the class of effective motives by introducing twists of effective motives by powers of the inverse Lefschetz motive L. This is an effective motive defined as L = (P1 , 1 − Z ), where Z is the cycle class Z ∈ A1 (P1 × P1 ) defined by the cycle P1 × pt. It is possible to tensor effective motives M by L and its inverse. Combining the notions of effective motives and the Lefschetz motive leads to the concept of a Grothendieck motive. Definition. A Grothendieck motive is a triple M(m) = (X, p, m), where M = (X, p) is an effective motive, and m ∈ Z. M(m) = (M, m) is the m−fold Tate twist of M. If N (n) = (Y, q, n) is another motive morphisms are defined as Hom(M(m), N (n)) := p ◦ Corr n−m (X, Y ) ◦ q.
(32)
The tensor product of two Grothendieck motives Mi = (X i , pi , m i ), i = 1, 2 is defined as M1 ⊗ M2 = (X 1 × X 2 , p1 ⊕ p2 , m 1 + m 2 ).
(33)
A discussion of the virtues and disadvantages of the various realizations in terms of specific equivalence relations can be found in [35,36], building on earlier references, such as [31–33]. A more detailed discussion of motives can be found in [37].
12
R. Schimmrigk
3.2. −motives for manifolds of Calabi-Yau and special Fano type. In the context of the emergent geometry problem via string theoretic modular forms it is necessary to consider L–functions associated to motives of low rank, not of the full cohomology groups of a variety. For higher genus curves and higher dimensional varieties the experimental evidence [9–12] suggests that the relevant physical information is encoded in subspaces of the cohomology. A possible strategy therefore is to consider the factorization of L– functions and to ask whether modular forms arise from the emerging pieces, and if so, whether these modular forms admit a string theoretic Kac-Moody interpretation. This section formulates a general strategy, valid for any Calabi-Yau variety X of dimension dimC X = n, for decomposing the L–function of its intermediate cohomology H n (X ), into pieces which lead to L–functions with integral coefficients. These L–functions then have the potential, when modular, to admit a factorization into KacMoody theoretic modular forms along the lines discussed in [9–11] for elliptic curves and higher genus curves. The basic strategy outlined below is a generalization of the method described in [12], which was based on Jacobi sums associated to hypersurfaces embedded in weighted projective spaces. The idea is to consider an orbit in the cohomology which is generated by the holomorphic n−form ∈ H n,0 (X ) via the action of the Galois group Gal(K X /Q) of a number field K X that is determined by the arithmetic properties of the variety as dictated by the Weil conjectures [38], proven by Grothendieck [39] and Deligne [40]. The resulting orbit of this action turns out to define a motive in the sense of Grothendieck, as will be shown further below. To see how the group structure appears in full generality it is necessary to briefly describe the arithmetic structure of arbitrary Calabi-Yau varieties. For a general smooth algebraic variety X defined over the rational integers the reduction mod q leads to the congruence zeta function of X/Fq , tr # X/Fq r Z (X/Fq , t) ≡ exp . (34) r r ∈N
Here the sum is over all finite extensions Fq r of Fq of degree r = Fq r : Fq . By definition Z (X/Fq , t) ∈ 1 + Q[[t]], but the expansion can be shown to be integer valued by writing Euler product. The main virtue of Z (X/Fq , t) is that the numbers it as an N pr = # X/F pr show a simple behavior, as a result of which the zeta function can be shown to be a rational function. 1) The first step is to consider the rational form of Artin’s congruence zeta function. This leads to a link between the purely arithmetic geometric objects Nr, p = #(X/F pr ) for all primes p and the cohomology of the variety. This was first shown for curves by F.K. Schmidt in the thirties in letters to Hasse [41–43]. Further experience by Hasse, Weil, and others led to the conjecture that this phenomenon is more general, culminating in the cohomological part of the Weil conjecture. According to Weil [38] and Grothendieck [39] Z (X/F p , t) is a rational function which can be written as n 2 j−1 (t) j=1 P p Z (X/F p , t) = n , (35) 2j j=0 P p (t) where n P 0p (t) = 1 − t, P 2n p (t) = 1 − p t,
(36)
Emergent Spacetime from Modular Motives
13
and for 1 ≤ j ≤ 2n − 1, j
deg P p (t) = b j (X ),
(37) j
where b j (X ) denotes the j th Betti number of the variety, b j (X ) = dim HdR (X ). The rationality of the zeta function was first shown by Dwork [44] by adelic methods. j The resulting building blocks given by the polynomials P p (t) associated to the full cohomology group H j (X ) are not useful in the present context, leading to L–series whose Mellin transforms in general cannot directly be identified with string theoretic modular forms of the type considered in [9–12,30] and in the present paper. The idea instead is to decompose these objects further, which leads to the factorization of the polynomials P ip (t). 2) The most difficult part of the Weil conjectures is concerned with the nature of the factorization of the polynomials b j
j P p (t)
=
j 1 − γi ( p)t .
(38)
i=1
Experience with Jacobi and Gauss sums in the context of diagonal weighted proj jective varieties indicates that the inverse eigenvalues γi ( p) are algebraic integers that satisfy the Riemann hypothesis j (39) γi ( p) = p j/2 , ∀i. This part of the Weil conjectures was finally proven by Deligne [40]. j 3) The algebraic nature of the inverse roots of the polynomials P p (t) can be used to define a number field K X associated to the intermediate cohomology of the variety of complex dimension n as K X = Q γin |i = 1, . . . , bn . The field K X is separable and therefore one can consider orbits within the cohomology with respect to the embedding monomorphisms. For n−dimensional Calabi-Yau varieties one can, in particular, consider the orbits O associated to the holomorphic n−forms ∈ H n,0 (X ), while for special Fano manifolds of charge Q one can consider ∈ H n−(Q−1),(Q−1) (X ). The orbits of these forms generated by the embeddings of the field K X define a projection p on the intermediate cohomology which can be used to define a motive. Definition. The −motive of varieties of Calabi-Yau or special Fano type is defined as the motive M = (X, p , Q) which cohomologically is represented as the orbit of the class ∈ H n−(Q−1),(Q−1) (X ) with respect to the Galois group Gal(K X /Q). 3.3. L–function of −motives. Given the −motive M one can combine the local factors of the zeta functions, leading to the motivic L–function of the variety . L (X, s) := L(M , s) =
1
−s P p (p ) p
,
(40)
14
R. Schimmrigk
n−(Q−1),(Q−1) (X ). where P p (t) are the polynomials described by the orbit of ∈ H . The symbol = is the common notation to indicate that the bad primes are neglected in the discussion. Denoting the Weil number corresponding to in (38) by γ , the polynomial can be expressed for the good primes p as P (1 − σ (γ )t). (41) p (t) = σ ∈Hom(K ,C)
The σ −orbits defined via the embedding monomorphisms define traces of the number field K X , which implies that the corresponding −motives have L–functions L (X, s) with integral coefficients. 3.4. String theoretic modularity and automorphy. With the above structures in place we can ask in full generality for any Calabi-Yau variety the following Questions. When is the L–function L (X, s) of the −motive M of a Calabi-Yau variety modular? Further, if it is modular, can L (X, s) be expressed in terms of string theoretic forms associated to the Kac-Moody algebras that arise in the conformal field theories on the worldsheet? Modularity of L (X, s) here is understood to include the usual linear operations on modular forms. More generally, this question can be raised for the class of motives associated to Fano varieties of special type. It has been shown in refs. [9–12] that the answer to this question is affirmative at least sometimes in lower dimensions, and generalizations to higher dimensions will be established below. In such cases the L–function can be viewed as a map that takes motives and turns them into conformal field theoretic objects. This framework therefore leads to the following picture. Conjecture. The L–function defines a map from the category of Calabi-Yau motives (more general special Fano type motives) to the category of N = 2 supersymmetric conformal field theories. These ideas can be generalized to the context of automorphic motives via automorphic representations. The Langlands program leads to the expectation that every pure motive is automorphic, and at this level of speculation the question becomes whether the L–functions of the irreducible automorphic motives admit a string theoretic interpretation. 3.5. Grothendieck motives for Brieskorn-Pham varieties. For weighted projective hypersurfaces of Brieskorn-Pham type the motives that emerge can be described concretely in terms of the Jacobi sums that reflect the cohomology of these varieties. Consider a diagonal hypersurface X nd of degree d and dimension n in a weighted n+2 projective space with weights (k0 , . . . , kn+1 ) ∈ N . With di = d/ki one can consider the group G dn = i (μdi ), where μdi is the cyclic group with generator ξdi = e2πi/di . This group acts on the projective space as n+1 gz = (ξda00 z 0 , . . . , ξdan+1 z n+1 )
(42)
for a vector a = (a0 , . . . , an+1 ) ∈ Zn+2 . Motives of Fermat type can be defined via projectors that are associated to the characters of the symmetry group. Denote the dual dn and associate to a ∈ G dn a projector pa as group of G dn by G 1 pa = d a(g)−1 g, (43) |G n | d g∈G n
Emergent Spacetime from Modular Motives
15
where the character defined by a is given by a(g) =
n+1
ξdaii .
(44)
i=0
Combining the projectors pa within a Gal(Q(μd )/Q)−orbit then leads to pa . pO :=
(45)
a∈O
The projectors pO can now be regarded as algebraic cycles on X × X with rational coefficients by considering their associated graphs. By abuse of notation, effective motives of Fermat type can therefore be defined as objects that are determined by the Galois orbits O as (X nd , pO ). Including Tate twists then leads to Grothendieck motives, MO := (X nd , pO , m).
(46)
3.6. −motives for CY and SF BP hypersurfaces. For varieties of Calabi-Yau and special Fano type there exists a particularly important cohomology group which can be parametrized explicitly. For Calabi-Yau manifolds of complex dimension n these forms take values in H n,0 (X ), while for special Fano varieties of complex dimension n and charge Q one has more generally ∈ H n−(Q−1),(Q−1) (X ), where Q ∈ N. The −motive associated to the Galois orbit of of a variety of charge Q will be denoted by Q
M = (X nd , p , Q − 1).
(47)
1. For Q = 1 one recovers the Calabi-Yau case, which will be denoted by M = M
3.7. Lower weight motives from higher dimensional varieties. The −motive is not the only motive of Calabi-Yau and special Fano varieties that can lead to interesting modular forms. Instead, one can construct Grothendieck motives MO (X ) associated to Galois orbits O that do not arise from the −form, but come from motives represented by subgroups of the remainder of the intermediate cohomology group. The associated L–series L(MO (X ), s) may, or may not lead to interesting modular forms. In the case they do in the examples below, the resulting modular forms can be described as determined by Tate twists of forms L(MO (X ), s) = L( f O , s − 1), where f O (q) is a modular associated to the motive MO (X ). In general one would expect that these lower weight motives lead to Tate twists of automorphic forms of lower dimensional varieties that are embedded in the higher-dimensional manifolds. 4. L–Functions for Calabi-Yau Threefolds and Fourfolds In this paper the focus is on Calabi-Yau manifolds in complex dimensions two, three and four. String theoretic examples of rank two and dimension one and two are considered in [9–12]. Combining these examples with the varieties discussed in the present paper covers all physically interesting dimensions in string theory, M-theory, and F-theory.
16
R. Schimmrigk
Calabi-Yau threefolds with finite fundamental group lead to zeta functions of the form Z (X/F p , t) = with
P 3p (t)
(48)
(1 − t)P 2p (t)P 4p (t)(1 − p 3 t)
deg P 3p (t) = 2 + 2h (2,1) , deg P 2p (t) = h (1,1) .
This follows from the fact that for Calabi-Yau threefolds without a torus factor we have b1 = 0. For Calabi-Yau threefolds with h 1,1 = 1 the zeta function reduces to Z X/F p , t =
P 3p (t) (1 − t)(1 − pt)(1 − p 2 t)(1 − p 3 t)
,
(49)
and therefore becomes particularly simple. The coefficients βi3 ( p) of the polynomial P 3p (t)
=
3 (X ) b
βi3 ( p)t i
(50)
j=0
are related to the cardinalities of the variety via the expansion 1 2 2 Z (X/F p , t) = 1 + N1, p t + (N1, p + N2, p )t 2 1 1 1 3 3 4 N3, p + N1, p N2, p + N1, + p t + O(t ) 3 2 6
(51)
as β13 ( p) = N1, p − (1 + p + p 2 + p 3 ), 1 2 2 3 β23 ( p) = (N1, p + N2, p ) − N1, p (1 + p + p + p ) 2 +(1 + p + p 2 + p 3 )2 + (1 + p + p 2 ) + p 2 (1 + p) + p 4 (1 + p + p 2 ),
(52)
etc. This procedure is useful because the knowledge of a finite number of terms in the L–function determines it uniquely. In dimension four the cohomology of Calabi-Yau varieties is more complicated, leading to zeta functions Z (X 4 /F p , t) =
P 3p (t)P 5p (t) (1 − t)P 2p (t)P 4p (t)P 6p (t)(1 − p 4 t)
(53)
for spaces without torus factors, but the procedure is the same as above. For smooth hypersurfaces the cohomology groups except of degree given by the dimension are either trivial or inherited from the ambient space, leading to the intermediate L–function 1 (54) L(X, s) = 4 P p ( p −s ) p as the only nontrivial factor.
Emergent Spacetime from Modular Motives
17
5. L–Functions Via Jacobi Sums For the class of hypersurfaces of Brieskorn-Pham type it is possible to gain insight into the precise structure of the L–function by using a result of Weil [38] which expresses the cardinalities of the variety in terms of Jacobi sums of finite fields. In this context there the L–function of the −motive of such weighted Fermat hypersurfaces can be made explicit. For any degree vector d = (d0 , . . . , dn+1 ) and for any prime p define the numbers ei = (di , p − 1) and the set
p,d n+2 An = (α0 , . . . , αn+1 ) ∈ Q 0 < αi < 1, ei αi ≡ 0(mod 1), αi ≡ 0(mod 1) . i
(55) Theorem 5. The number of solutions of the smooth projective variety
n+1 di n+1 X n = (z 0 : · · · : z n+1 ) ∈ P bi z i = 0
(56)
i=0
over the finite field F p is given by N p (X n ) = 1 + p + p 2 + · · · + p n +
j p (α)
χ¯ αi (bi ),
(57)
d α∈An
where j p (α) =
1 p−1
χα0 (u 0 ) · · · χαn+1 (u n+1 ).
(58)
u i ∈F p
u 0 +···+u n+1 =0
With these Jacobi sums jq (α) one defines the polynomials 1/ f n n/2 |n| n i f P p (t) = (1 − p t) χ¯ αi (b )t 1 − (−1) j p f (α) and the associated L–function L (n) (X, s) =
(59)
i
d α∈An
p
1 . P np ( p −s )
(60)
Here |n| = 1 if n is even and |n| = 0 if n is odd. A slight modification of this result is useful even in the case of smooth weighted projective varieties because it can be used to compute the factor of the zeta function coming from the invariant part of the cohomology, when viewing these spaces as quotient varieties of projective spaces. The Jacobi-sum formulation allows to write the L–function of the −motive M of weighted Fermat hypersurfaces in a more explicit way. Define the vector α =
kn+1 k0 corresponding to the holomorphic s−form, and denote its Galois orbit d ,..., d d
by O ⊂ An . Then L (X, s) =
−1/ f 1 − (−1)n j p f (α) p − f s . p α∈O
(61)
18
R. Schimmrigk
6. Modularity Results for Rank two Motives in Dimension One and Two In the examples of higher rank motives considered in Theorems 1 and 2 the geometry of the varieties can be constructed from that of lower-dimensional manifolds. This leads to an induced string theoretic modular structure for the higher rank motive by using known modularity results for the building blocks involved. To make this paper more self-contained this section briefly summarizes the necessary ingredients obtained in [11] for the diagonal weighted elliptic curves and in [12] for the diagonal weighted extremal K3 surfaces. More details can be found in these references. The simplest possible framework in which the problem of an emergent spacetime can be raised is for compactifications on tori, in particular Brieskorn-Pham curves E d of degree d embedded in the weighted projective plane. The elliptic curves E d are defined over the rational number Q, and therefore modular in lieu of the Shimura-Taniyama conjecture, proven in complete generality in ref. [3], based on Wiles’ paradigmatic results [1,2]. This theorem says that any elliptic curve over the rational numbers is modular in the sense that the inverse Mellin transform of the Hasse-Weil L–function is a modular form of weight two for some congruence subgroup 0 (N ). This raises the question whether the modular forms derived from these Brieskorn-Pham curves are related in some way to the characters of the conjectured underlying conformal field theory models. The conformal field theory on the string worldsheet is fairly involved, and a priori there are a number of different modular forms that could play a role in the geometric construction of the varieties. The first string theoretic modularity result showed that the modular form associated to the cubic Fermat curve E 3 ⊂ P2 factors into a product of SU(2)−modular forms that arise from the characters of the underlying world sheet [9]. More precisely, the worldsheet modular forms that encode the structure of the compact spacetime geometry are Hecke indefinite theta series associated to Kac-Moody theoretic string functions introduced by Kac and Peterson. For the remaining two elliptic weighted Fermat curves this relation requires a modification involving a twist character that is physically motivated by the number field generated by the quantum dimensions of the string model [10,11]. The modular forms that emerge from the weighted Fermat curves provide a string theoretic interpretation of the Hasse-Weil L–function in terms of the exactly solvable Gepner models at central charge c = 3. Ref. [11] furthermore identifies the criteria that lead to the derivation of these elliptic curves from the conformal field theory itself, with no a priori input from the geometry. The class of elliptic Brieskorn-Pham curves is given by E 3 = (z 0 : z 1 : z 2 ) ∈ P2 | z 03 + z 13 + z 23 = 0 , E 4 = (z 0 : z 1 : z 2 ) ∈ P(1,1,2) | z 04 + z 14 + z 22 = 0 , (62) E 6 = (z 0 : z 1 : z 2 ) ∈ P(1,2,3) | z 06 + z 13 + z 22 = 0 . The modular forms associated to these curves are cusp forms of weight two with respect to congruence groups of level N , 0 (N ) ⊂ SL(2, Z), defined by a b ∗ ∗ a b (mod N ) . (63) ≡ ∈ SL2 (Z) 0 (N ) = 0 ∗ c d c d Recalling the string theoretic Hecke indefinite theta series k,m (τ ) defined in Sect. 2 k (τ ), the geometric modular forms in terms of the Kac-Peterson string functions c,m
Emergent Spacetime from Modular Motives
19
decompose as f (E d , q) =
kii ,m i (q ai ) ⊗ χd ,
(64)
i
where ai ∈ N, and χd (·) = worldsheet theta series
d ·
is the Legendre symbol. More precisely, given the
11,1 (τ ) = η2 (τ ), 21,1 (τ ) = η(τ )η(2τ ),
(65)
the elliptic modular forms factorize in the following way [11]. Theorem 6. The inverse Mellin transforms f (E d , q) of the Hasse-Weil L–functions L HW (E d , s) of the curves E d , i = 3, 4, 6 are modular forms f (E d , q) ∈ S2 (0 (N )), with N = 27, 64, 144 respectively. These cusp forms factor as f (E 3 , q) = 11,1 (q 3 )11,1 (q 9 ), f (E 4 , q) = 21,1 (q 4 )2 ⊗ χ2 , f (E , q) = 6
11,1 (q 6 )2
(66)
⊗ χ3 .
Consider next the class of extremal K3 surfaces that can be constructed as weighted Fermat varieties X 24 = (z 0 : · · · : z 3 ) ∈ P3 z 04 + z 14 + z 24 + z 34 = 0 , X 26A = (z 0 : · · · : z 3 ) ∈ P(1,1,1,3) z 06 + z 16 + z 26 + z 32 = 0 , (67) X 26B = (z 0 : · · · : z 3 ) ∈ P(1,1,2,2) z 06 + z 16 + z 23 + z 33 = 0 . For the remainder of this section let X 2 be any of these three varieties. All three K3 surfaces X 2 lead to an −motive that is modular and admits a string theoretic interpretation. Denote by M(X ) ⊂ H 2 (X 2 ) the cohomological realization of a motive M, with M (X ) the motive associated to the holomorphic 2-form, and let L (X, s) = L(M (X ), s) be the associated L–series, with f (X, q) denoting the inverse Mellin transform of L (X, s). The main result of [12] shows that the modular forms determined by the −motives of these extremal K3 surfaces are determined by the string theoretic modular forms that enter in Theorem 6. Theorem 7. Let M ⊂ H 2 (X 2 ) be the irreducible representation of Gal(Q(μd )/Q) associated to the holomorphic 2−form ∈ H 2,0 (X 2 ) for any of the K3 surfaces X 2 in (67). Then the q−series f (X 2 , q) of the L–function L (X 2 , s) are modular forms given, respectively, by f X 24 , q = η6 (q 4 ), f X 26A , q = ϑ q 3 η2 q 3 η2 (q 9 ), (68) 6B 3 2 3 6 f X 2 , q = η q η (q ) ⊗ χ3 .
20
R. Schimmrigk
These functions are cusp forms of weight three with respect to 0 (N ) with levels 16, 27 and 48, respectively. For X 24 and X 26A the L–functions can be written as 2 ,s , (69) L X 24 , s = L ψ64 6A 2 L X 2 , s = L ψ27 , s , (70) where ψd with d = 27, 64 are Größencharacters associated to cusp forms f d (q) of weight two and levels 64 and 27, respectively, given by the elliptic forms f 4 τ = f E 4 , q = η 2 q 4 η 2 q 8 ⊗ χ2 , (71) f 6A τ = f E 3 , q = η2 q 3 η2 q 9 . 2 ⊗ χ , s), leading to the cusp For X 26B the L–series is given by L (X 26B , s) = L(ψ144 3 form of level 144,
f 6B (τ ) = f (E 6 , q) = η4 (q 6 ) ⊗ χ3 .
(72)
Several of these results will enter in the discussion below of higher dimensional varieties. 7. A Nonextremal K3 Surface X 212 ⊂ P(2,3,3,4) String theoretic modularity of the class of extremal K3 surfaces of Brieskorn-Pham type has been established in [12]. Extremal K3 surfaces are special in the sense that their Picard number is maximal, i.e. ρ = 20, and the resulting motive has rank two. In this section the results of [12] are extended to a nonextremal K3 surface with a rank four −motive. Consider the Brieskorn-Pham hypersurface (1) of degree twelve in weighted projective space P(2,3,3,4) . The Galois group of the cyclotomic field Q(μ12 ) has order four, hence the −motive has rank four. The four Jacobi sums which parametrize this motive are given by j p (σ α ), σ ∈ Gal(Q(μ12 )/Q), i.e. j p (α) with
α∈
5 1 1 2 1 1 1 1 , , , , , , , , 6 4 4 3 6 4 4 3
(73)
(74)
and their conjugates α¯ = 1 − α, where 1 denotes the unit vector. The computation of these sums leads to the L–function of the −motive 12 30 20 14 140 60 . L (X 212 , s) = 1 − s + s − s − s + s + s + · · · . 13 25 37 49 61 73
(75)
Insight into the structure of this L–function can be obtained by noting that the surface X 212 can be constructed as a quotient of a product of elliptic curves E 1 × E 2 /ι, where ι is an involution on the product. More precisely, the elliptic curves are given by 4 = x02 − (x14 + x24 = 0) ⊂ P(2,1,1) , (76) E−
Emergent Spacetime from Modular Motives
21
4 and E 6 Table 1. Coefficient comparison of the surface X 212 and the curves E −
pf ap f
13 (E 4 )
b p f (E 6 )
c p f X 212
25
37
49
61
73
−6
−1
2
−7
10
−6
2
−5
−10
9
14
−10
−12
30
−20
−14
140
60
and the degree six curve E 6 ⊂ P(1,2,3) . The twist map [48,49] (see also [50,51])
: P(2,1,1) × P(3,1,2) −→ P(3,3,2,4)
(77)
1/3 2/3 1/2 1/2 ((x0 , x1 , x2 ), (y0 , y1 , y2 )) → y0 x1 , y0 x2 , x0 y1 , x0 y2 .
(78)
is defined by
4 × E 6 and leads to the K3 surface of degree twelve X 12 . This maps the product E − 2 4 , s) = L(E 4 , s) = an (E 4 )n −s and L(E 6 , s) = The coefficients of L(E − n 6 −s of the Hasse-Weil L–functions of the elliptic curves E 4 and E 6 , respecn bn (E )n tively, can be obtained by expanding the results of Theorem 6. Multiplying the coefficients a p (E 4 ) and b p (E 6 ) leads to
a p (E 4 )b p (E 6 ) = c p (X 212 ).
(79)
For low primes the results are collected in Table 1. The inverse Mellin transform of the L–function of the two building blocks E 4 and 6 E of the surface X 212 are given in terms of string theoretic theta functions as described in Theorem 6. The modular forms of E 4 and E 6 are both of complex multiplication type, leading to an interpretation of the L–series in terms of Hecke’s Größencharacters. More precisely, they are described by twists by Legendre symbols χn , n = 2, 3 of the characters ψ32 and ψ36 , as described in [11], L(E 4 , s) = L(ψ32 ⊗ χ2 , s), L(E 6 , s) = L(ψ36 ⊗ χ3 , s).
(80)
√ Here the characters√ψ32 and ψ36 are associated to the Gauss field Q( −1) and the Eisenstein field Q( −3) respectively, as described in §2. The factorization of the coefficients c p (X 212 ) suggests that the rank four −motive M (X 212 ) is the tensor product of the elliptic motives of E 4 and E 6 , but a priori leaves open the precise nature of the L–function product. This can be determined by combining the iterative reduction of Jacobi sums introduced by Weil [52] in his discussion of cyclotomic fields with the results of the elliptic curves E 4 and E 6 given in Sect. 6. Applying Weil’s reduction to the Jacobi sums associated to the −motive of the surface X 212 shows that they can be expressed for the relevant primes p = 1 + 12n, n ∈ N, completely in terms of the Jacobi sums of E 4 and E 6 . Furthermore, the resulting local factors of the L–function L (X 212 , s) are precisely those of the modified Rankin-Selberg product considered in Sect. 2, applied to the modular forms f 64 ∈ S2 (0 (64)) and f 144 ∈ S2 (0 (144)) of the curves E 4 and E 6 respectively, L X 212 , s = L M f64 ⊗ M f144 , s , where M f Ni are the elliptic motives of f Ni .
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R. Schimmrigk
The string theoretic interpretation of the Hasse-Weil L–series of E 4 and E 6 described in Theorem 6 therefore induces a string interpretation of the modular blocks of the K3 surface X 212 . The CM property of these forms will allow a systematic discussion in Sect. 11 of the reverse construction of emergent space from the modular forms of the worldsheet field theory. 8. Modular Motives of the Calabi-Yau Threefold X 36 ⊂ P(1,1,1,1,2) In this section string theoretic modularity is proven for the −motive of the diagonal weighted Calabi-Yau threefold X 36 , extending the lower-dimensional results of [11,12]. 8.1. The modular −motive of X 36 . Consider the Calabi-Yau variety X 36 defined as the double cover branched over the degree six Fermat surface in projective threespace P3 . This manifold can be viewed as a smooth degree six hypersurface of Brieskorn-Pham type in the weighted projective fourspace P(1,1,1,1,2) as in (3). The Galois group of Q(μ6 ) has order two, hence the −motive M (X 36 ) has rank two. The Jacobi sums associated √ to this motive take values in the imaginary quadratic field K = Q( −3), and therefore define a Größencharacter of the type described in Sect. 2. The results of Hecke and Shimura thus imply that the L–series of the motive M of X 36 is modular. Computing sufficiently many coefficients of this L–function 17 89 107 125 308 433 . (81) L (X 36 , s) = 1 + s + s + s − s + s − s + · · · , 7 13 19 25 31 37 therefore determines the resulting modular form uniquely. It follows from (81) that this motivic L–function is the Mellin transform of a modular form of weight 4 and level N = 108, f ∈ S4 (0 (108)). This form cannot be written as a product or a quotient of Dedekind eta-functions [53,54], but the fact that it admits complex multiplication by the imaginary quadratic field K implies that L (X 36 , s) can be written as the Hecke L–series of a twisted Größencharacter associated to the field K . Such characters were considered in [11,12] in the context of the elliptic Brieskorn-Pham curve E 3 ⊂ P2 and E 6 ⊂ P(1,2,3) , as well as modular K3 surfaces. The Hasse-Weil L–series L(E 3 , s) is a Hecke series for the character ψ27 considered in §2, which can be written as [9], L(E 3 , s) = L(ψ27 , s) = L 11,1 (q 3 )11,1 (q 9 ), s . (82) The character ψ27 turns out to be the fundamental building block of the L–series of the −motive of X 36 . For higher dimensional varieties it is possible to generalize such relations by considering powers of the Größencharacter in order to obtain higher weight modular forms, as 3 leads to a modular form described above in Hecke’s theorem. In the present case ψ27 which can be written in terms of the Dedekind eta function η(q) as η8 (q 3 ) ∈ S4 (0 (9)), which also appears as a motivic form in a number of geometries different from X 36 . In order to obtain the motivic L–series L (X 36 , s) computed above it is necessary to introduce a twist character. This can be chosen to be the cubic residue power symbol, denoted here by 2 (3) χ2 ( p) := , (83) p 3
Emergent Spacetime from Modular Motives
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where p| p and the congruence ideal is chosen to be m = (3). With this character the Hecke interpretation of the motivic L–function of X 36 takes the form (3)
3 L (X 36 , s) = L(ψ27 ⊗ (χ2 )2 , s).
(84)
The inverse Mellin transform f (X 36 , q) of this L–series . f (X 36 , q) = q + 17q 7 + 89q 13 + 107q 19 − 125q 25 + 308q 31 − 433q 37 + · · · (85) is therefore of complex multiplication type in the sense of Ribet [19], and the string theoretic nature of the character ψ27 implies the same for the weight four modular form of X 36 . 8.2. Lower weight modular motives of X 36 . The degree six Calabi-Yau hypersurface X 36 provides an example of the phenomenon noted in Sect. 3 that the intermediate cohomology can lead to modular motives beyond the −motive. For X 36 the motives are of rank two, given by the Galois group Gal(Q(μ6 )/Q) with certain multiplicities and twists. Modulo these twists and multiplicities the group H 2,1 (X ) ⊕ H 1,2 (X ) leads to three different types of modular motives of weight two and rank two, denoted in the following by M A ∈ {MI , MII , MIII }. The L–series L(M A , s) that result from these motives have coefficients a Ap that are all divisible by the prime p. By introducing the twisted coefficients a pA = a Ap / p, these L–series lead to modular forms of weight two f A ∈ S2 (0 (N A )), where the level N A is determined by the motive M A , L M A (X 36 ), s = L( f A , s − 1). The modular forms are of levels N A = 27, 144, 432, the first two given by the curves E 3 , E 6 described in Theorem 6, f I (q) = f (E 3 , q), f II (q) = f (E 6 , q),
(86)
while the level NIII = 432 form is given by . f III (q) = q − 5q 7 − 7q 13 + q 19 + 4q 31 + · · · , It follows that the L–series L(H 3 (X 36 ), s) of the intermediate cohomology group decomposes into modular pieces in the sense that each factor arises from a modular form L(H 3 (X 36 ), s) = L( f , s) L( f i ⊗ χi , s − 1)ai , i
where ai ∈ N, f ∈ S4 (0 (108)) is as determined above, the f i (q) are modular forms of weight two and levels Ni = 27, 144, 432, and χi is a Legendre character (which can be trivial). The Jacobi sums corresponding to the motives MI , MII , MII are listed in Table 2, together with the level N A of the corresponding modular form f A ∈ S2 (0 (N A )). The weight two modular forms f A that emerge from X 36 have a natural geometric interpretation. The threefold X 36 contains divisors given by degree six Fermat curves 6
C 6 ⊂ P2 and C ⊂ P(1,1,2) , obtained from the original hypersurface via intersections
24
R. Schimmrigk Table 2. The Jacobi sums of X 36 that lead to non-isogenic modular motives
Type
Jacobi sum j p 13 , 13 , 13 , j p 16 , 16 , 21 , j p 16 , 16 , 23 ,
I II III
1, 3 5, 6 2, 3
Level N
2 3
1 3
1 3
27 144 432
with coordinate hyperplanes. These curves are of genus ten and can be shown to decompose into ten elliptic factors of three different types E I = E 3 , E II = E 6 , and E III an elliptic curve of conductor 432. Hence its L–function factors as L(C 6 , s) = L(E I , s)L(E II , s)6 L(E III , s)3 , 6
taking into account their multiplicities. The curve C is of genus four and leads to the same modular forms, with different multiplicities. These results establish and make precise the structure of X 36 described in Theorem 2. 9. The K3 Fibration Hypersurface X 312 ⊂ P(2,2,2,3,3) The diagonal hypersurface X 312 given in Eq. (3) is the second manifold that extends previous results to three dimensions. It differs in character from the degree six hypersurface considered in the previous section because its motive is of higher rank. 9.1. The −motive of X 312 . The Galois group of X 312 is of order four, leading to an −motive of rank four. The Jacobi sums that parametrize this motive are given by j p 16 , 16 , 16 , 41 , 41 , j p 16 , 16 , 16 , 43 , 43 and their complex conjugates. The computation of these sums for low p f leads to the L–series
6 150 94 497 1210 582 . L X 312 , s = 1 + s − s + s − s − + s + ··· . (87) 13 25 37 49 61s 73 The structure of the −motivic L–series of X 312 can be understood by noting that the threefold is a K3 fibration with typical fiber X 26A given in (67). The interpretation of L (X 312 , s) in terms of the fibration is also useful because it makes the complex multiplication structure of the associated modular form transparent. The threefold can be constructed explicitly as the quotient of a product of a torus E and a K3 surface E × K3/ι,
(88)
where ι is an involution acting on the product. More precisely, the elliptic curve is the 4 considered in the discussion of X 12 , and the K3 surface is weighted quartic curve E − 2 6A X 2 . Applying the twist map of [48,49] gives the map
: P(2,1,1) × P(3,1,1,1) −→ P(3,3,2,2,2) defined by ((x0 , x1 , x2 ), (y0 , y1 , y2 , y3 )) →
(89)
1/3 1/3 1/2 1/2 1/2 y0 x1 , y0 x2 , x0 y1 , x0 y2 , x0 y3 . (90)
4 × X 6A to threefold X 12 . This maps the product E − 2 3
Emergent Spacetime from Modular Motives
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Table 3. Coefficient comparison for the threefold X 312 pf a p f (E 4 ) b p f (X 26A ) c p f (X 312 )
13 −6 −1 6
25 −1 25 −150
37 2 47 94
49 −7 120 −994
61 10 −121 −1210
73 −6 −97 582
The fibration structure suggests that the L–function of the threefold X 312 can be understood in terms of those of its building blocks. The L–function of the K3 fiber of this threefold was determined in Theorem 7 to be given by the Mellin transform of the cusp form f (X 26A , q) of weight w = 3 and level N = 27, and the L–function of the quartic curve is given by f (E 4 , q) ∈ S2 (0 (64)) according to Theorem 6 [11]. The geometric structure of X 312 suggests that a comparison of the coefficients of the L–function of the −motive of X 312 with those of its building blocks E 4 and X 26A should lead to a composite structure. Table 3 illustrates that this is indeed the case. The coefficients a p (E 4 ) arise from the L–series of the quartic E 4 , while the surface expansion b p (X 26A ) is that of (68) in Theorem 7, which leads to the expansion . f (X 26A , q) = q − 13q 7 − q 13 + 11q 19 + 25q 25 − 46q 31 + 47q 37 −22q 43 + 120q 49 − 121q 61 − 109q 67 − 97q 73 + · · · .
(91)
It follows that the products a p (E 4 )b p (X 26A ) agree with the expansion coefficients c p (X 312 ) of the threefold X 312 . The factorization of the −motivic L–series L (X 312 , s) also shows that its building blocks have complex multiplication. For the curve E 4 this was already discussed above in the context of the K3 surface X 212 [11]. For the K3 surface X 26A it was shown in [12] that the modular form of Theorem 7 is the Mellin √ transform of the Hecke L–series of a 2 of the Eisenstein field Q( −3), Größencharacter ψ27 2 ,s , (92) L X 26A , s = L ψ27 where the character ψ27 has been defined in Sect. 2. The motivic L–series of both building blocks of X 312 therefore are of complex multiplication type. The proof that the precise relation between the −motivic L–function L (X 312 , s) and its building blocks is again given by the Rankin-Selberg convolution proceeds in the same way in the case of the K3 surface X 212 ⊂ P(2,3,3,4) discussed in Sect. 7. By using Weil’s dimensional reduction of the Jacobi sums it becomes clear that the only nontrivial ingredients at q = 1 + 12n are the lower-dimensional Jacobi sums of E 4 and X 26A , and these combine in the local factors of the L–function of X 312 in precisely the way required by the Rankin-Selberg product. Combining this result with the theorem of Ramakrishnan [14] shows that the −motive of X 312 is automorphic as well. 9.2. Lower weight modular forms of X 312 . Similar to the degree six threefold X 36 the cohomology H 2,1 ⊕ H 1,2 of the degree twelve hypersurface X 312 leads to modular forms of weight two. There are again Jacobi sums that lead to precisely the same modular forms f A (q), A = I, II, III of weight two and levels N A = 27, 144, 432, possibly including a twist, as for X 36 . This is expected because X 312 contains the plane Fermat
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R. Schimmrigk
curve C 6 ⊂ P2 already encountered in X 36 . There is a further L–series of a rank two motive that is determined by the quartic elliptic weighted Fermat curve E 4 ⊂ P(1,1,2) considered in Theorem 6, L IV X 312 , s = L E 4 , s − 1 .
10. A Modular Calabi-Yau Fourfold X 46 ⊂ P5 Calabi-Yau varieties of complex dimension four are useful in the context of F-theory in four dimensions and M-theory in three dimensions. In this section it is shown that the −motive of the fourfold of degree six in projective fivespace P5 defined by the Brieskorn-Pham hypersurface (6) is modular. The Galois orbit is of length two, and the motivic L–function is described by com puting the Jacobi sums j p (α ) with α = 16 , . . . , 16 . The resulting L–function is given by
71 337 601 625 194 529 . L X 46 , s = 1 − s − s + s + s − s − s + · · · . 7 13 19 25 31 37
(93)
The associated q−expansion f (X 46 , q) differs from that of a newform of weight five and level 27, f 27 (q) = q + 71q 7 − 337q 13 − 601q 19 + 194q 31 − 529q 37 + · · · ,
(94)
only in signs. These signs can be adjusted to the quadratic character χ3 defined by the Legendre symbol 3 , χ3 ( p) = p
(95)
f X 46 , q = f 27 (q) ⊗ χ3 .
(96)
leading to
This modular form can also be described as a Hecke √ L–series associated to the character ψ27 associated to the Eisenstein field K = Q( −3) and defined in Sect. 2 in (21). More precisely, the twisted Hecke L–series agrees with that of the −motive of X 46 , 4 L X 46 , s = L(ψ27 , s) ⊗ χ3 .
(97)
Hence the motivic L–function is again a purely algebraic object and its fundamental structure is determined by the L–series of a Hecke indefinite theta series, as noted in (82).
Emergent Spacetime from Modular Motives
27
11. Emergent Space from Characters and Modular Forms In the discussion so far the goal was to formulate a general framework of −motives of varieties of Calabi-Yau type, and more generally, of special Fano type, in the context of Grothendieck’s framework of motives, and to test the conjecture that these motives are string modular in the sense that it is possible to identify modular forms on the worldsheet whose Mellin transform agrees with the L–function of the resulting motives. The problem of constructing spacetime geometry from fundamental string input involves the inverse problem of this strategy. The aim of the present section is to address this “emergent space” problem. As already mentioned earlier, the idea here must be to obtain a construction of the motivic pieces of the compact varieties from the basic conformal field theoretic modular forms. There are several ways to think about these objects and the following remarks describe how these various constructions, which a priori are independent, fit together in the context of the −motive of Brieskorn-Pham type varieties. The main simplifying observation in the present context of weighted Fermat hypersurfaces is that all the motivic L–functions L (X, s) that have been encountered so far in this program [9–12,55,56] lead to modular forms which are of complex multiplication type [19] (see [22] for geometric aspects of CM). The structure of such forms has been described in detail in [11,12,57]. The important point in the present context is that modular −motives of CM type are algebraic objects whose L–functions are given by the Hecke L–series of a Größencharacter (possibly including a twist). This construction of Größencharacters from geometry can be inverted, and it is known how to construct motives directly from the characters. It is in particular possible to construct a Grothendieck motive of the form Mχ = (Aχ , pχ ), where Aχ is an abelian variety associated to the character χ by the theorem of Casselman [58], where pχ is a projector associated to χ . This leads to an apparent problem of riches, because given a modular form Sw (0 (N )) it is possible to construct a Grothendieck motive M f = (X f , p f ) by considering the cohomology of an associated Kuga-Sato variety X f , as shown by Deligne [24], Jannsen [25], and Scholl [26]. Combining the abelian motives and the Kuga-Sato motives with those of the −motives thus leads to three a priori different motivic constructions associated to the Größencharacters encountered here and in the earlier papers. It turns out that the motives Mχ , M f and M all are isomorphic because they arise from the same CM modular form. This follows because motives associated to CM modular forms have CM, and one can generalize Faltings result that L–series characterize abelian varieties up to isogeny to CM motives, as shown e.g. by Anderson [59]. In the case of varieties of Brieskorn-Pham type it is possible to make the relation between abelian varieties and −motives more concrete by noting that the cohomology of Fermat varieties has a well-known inductive structure, which was first noted by Shioda-Katsura [60] in the context of Fermat varieties (see also Deligne [61]). This inductive structure allows to reduce the cohomology of higher-dimensional varieties in terms of the cohomology of algebraic curves (modulo Tate twists). Hence the basic building blocks are abelian varieties derived from the Jacobians of these curves. It follows from results of Gross and Rohrlich [62] that these Jacobians factor into simple abelian varieties and that these abelian factors have complex multiplication. The final step in the construction is provided by the fact that the L–function of the abelian variety attached to χ by Casselman’s result is given by the conjugates of the L–function of the Größencharacter.
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12. Further Considerations The goal of the program continued in this paper is to investigate the relation between the geometry of spacetime and the physics of the worldsheet by analyzing in some depth the connection between the modular symmetry encoded in exact models on the worldsheet and the modular symmetries that emerge from the nontrivial arithmetic structure of spacetime. The techniques introduced for this purpose provide a stronger, and more precise, alternative to the framework of Landau-Ginzburg theories and σ −models. The latter in particular presupposes the concept of an ambient space in which the string propagates, a notion that should emerge as a derived concept in a fundamental theory. The focus of the results obtained in previous work and the present paper has been on the class of diagonal models, related to Gepner’s construction. It would be interesting to extend these considerations to the more general class of hypersurfaces that are related to the Kazama-Suzuki models [63]. Of particular interest in that class are certain ‘irreducible’ models which are not tensor products, hence a single conformal field theoretic quotient suffices to saturate the necessary central charge. Such models exist for both K3 surfaces and Calabi-Yau threefolds, and establishing modularity in the sense described here would be a starting point for the exploration of modular points in the moduli space of nondiagonal varieties. Results in this direction would illuminate relations between different conformal field theories. A second open problem is the analysis of families of varieties with respect to their modular properties. First steps in this direction have been taken in refs. [64–66], where the zeta functions for particular one-parameter families of Calabi-Yau threefolds are computed. It is of interest to understand how the modular behavior of these families is related to deformations along marginal directions of the associated conformal field theory. Deforming away from the weighted Fermat point in Brieskorn-Pham type families will in general change the structure of the motive, but the reciprocity conjecture of Langlands implies that automorphy is expected to occur for different fibers in such families. Of particular interest is the modular behavior of singular fibers in families of Calabi-Yau varieties. It is known from work of the 1980s that the moduli spaces of different Calabi-Yau spaces are connected via singular varieties [67,68]. This raises the question what the precise structure is of the conjectured family of modular and automorphic forms at such singular fibers in a given family. The results of such a modularity analysis should prove useful for a deeper understanding of the conformal field theoretic behavior of phase transitions between Calabi-Yau varieties, a problem that has proven challenging in the past. Progress in this direction will be described in [69]. Acknowedgements. It is a pleasure to thank Monika Lynker for conversations, and Rob Myers for raising the point of string dualities in the context of the motivic picture. Part of this work was completed while the author was supported as a Scholar at the Kavli Institute for Theoretical Physics in Santa Barbara. The work at the KITP was supported in part by the NSF under Grant No. PHY05-51164. The author is also grateful to the National Science Foundation for support provided under Grant No. PHY09-69875. This work was furthermore supported in part by Faculty Research Grants from Indiana University South Bend. Thanks are due to the referee, whose comments and questions led to an improved presentation of this paper.
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40. Deligne, P.: La conjecture de Weil I. Publ. Math. IHES 43, 273–307 (1974) 41. Schmidt, F.K.: Analytische Zahlentheorie in Körpern der Charakteristik p. Math. Z. 33, 1–32 (1931) 42. Hasse, H.: Beweis des Analogons der Riemannschen Vermutung für die Artinschen und F.K. Schmidtschen Kongruenzzetafunktionen in gewissen elliptischen Fällen. Vorläufige Mitteilung. Nachrichten V.d. Gesellschaft d. Wiss. Zu Göttingen I 42, 253–262 (1933) 43. Hasse, H.: Über die Kongruenzzetafunktionen. Unter Benutzung von Mitteilungen von Prof. Dr. F.K. Schmidt Und Prof. Dr. E. Artin. S. Ber. Preuß. Akad. Wiss. H. 17, 250–263 (1934) 44. Dwork, B.: On the rationality of the zeta function of an algebraic variety. Amer. J. Math. 82, 631–648 (1960) 45. Shioda, T.: Some observations on Jacobi sums. In: Galois Representations and Arithmetic Geometry, ed., Y. Ihara, Advanced Studies in Pure Mathematics 12, Tokyo: Math. Soc. Japan, 1987 46. Gouvea, F., Yui, N.: Arithmetic of diagonal hypersurfaces over finite fields. London: London Math. Soc., 1995 47. Kadir, S., Yui, N.: Motives and mirror symmetry for Calabi-Yau orbifolds. In: Modular Forms and String Duality. Fields Inst. Commun. Providence, PI: Amer. Math. Soc. Felds. Dist. 54, 2008, pp. 3–46 48. Hunt, B., Schimmrigk, R.: Heterotic gauge structure of type II K3 fibrations. Phys. Lett. B381, 427–436 (1996) 49. Hunt, B., Schimmrigk, R.: K3 Fibered Calabi-Yau threefolds I: The twist map. Int. J. Math. 10, 833–866 (1999) 50. Voisin, C.: Miroirs ét involutions sur les surfaces K 3. Journées de Géométrie Algébrique d’Orsay, Astérisque 218, 273–323 (1993) 51. Borcea, C.: K3 Surfaces with involutions and mirror pairs of Calabi-Yau manifolds. In: Mirror Symmetry, II, AMS/IP Stud. Adv. Math. 1, Providence, RI: Amer. Math. Soc., 1997, pp. 717–743 52. Weil, A.: Jacobi sums as “Grössencharaktere”. Trans. Amer. Math. Soc. 73, 487–495 (1952) 53. Dummit, D., Kisilvesky, H., McKay, J.: Multiplicative properties of η−functions. Contemp. Math. 45, 89–98 (1985) 54. Martin, Y.: Multiplicative η−Quotients. Trans. Amer. Math. Soc. 348, 4825–4856 (1996) 55. Schimmrigk, R.: Calabi-Yau arithmetic and rational conformal field theories. J. Geom. Phys. 44, 555–569 (2003) 56. Schimmrigk, R.: A modularity test of elliptic mirror symmetry Phys. Lett. B655, 84–89 (2007) 57. Lynker, M., Schimmrigk, R., Stewart, S.: Complex multiplication of exactly solvable Calabi-Yau varieties. Nucl. Phys. B700, 463–489 (2004) 58. Shimura, G.: Introduction to the arithmetic theory of automorphic functions. Princeton, NJ: Princeton UP, 1971 59. Anderson, G.W.: Cyclotomy and an extension of the Taniyama group. Compositio Math. 57, 153–217 (1986) 60. Shioda, T., Katsura, T.: On Fermat varieties. Tohoku Math. J. 31, 97–115 (1979) 61. Deligne, P.: Hodge cycles on abelian varieties. In: Hodge Cycles, Motives, and Shimura Varieties, eds., P. Deligne, J.S. Milne, A. Ogus, K.-y. Shih, LNM 900, Berlin-Heidelberg-New York: Springer Verlag, 1982 62. Gross, B.: On the periods of abelian integrals and a formula of Chowla and Selberg. Invent. Math. 45, 193–211, (with an appendix by D.E. Rohrlich) 63. Kazama, Y., Suzuki, H.: New N = 2 superconformal field theories and superstring compactification. Nucl. Phys. B321, 232 (1989) 64. Candelas, P., de la Ossa, X., Rodriguez-Villegas, F.: Calabi-Yau manifolds over finite fields I. http://arXiv. org/abs/hep-th/0012233/v1, 2000 65. Candelas, P., de la Ossa, X., Rodriguez-Villegas, F.: Calabi-Yau manifolds over finite fields II. In: CalabiYau Varieties and Mirror Symmetry. Fields Inst. Commun., 38, Providence, PI: Amer. Math. Soc., 2003, pp. 121–157 66. Kadir, S.N.: The arithmetic of Calabi-Yau manifolds and mirror symmetry. Oxford Univ. Dphil thesis, 2004, available at http://arXiv.org/abshep-th/0409202v1, 2004 Arithmetic mirror symmetry for a two-parameter family of Calabi-Yau manifolds. In: Mirror Symmetry V, AMS/IP Stud. Adv. Math. 38. Providence, PI: Amer. Math. Soc., 2006, pp. 35–86 67. Candelas, P., Dale, A., Lütken, C.A., Schimmrigk, R.: Complete intersection Calabi-Yau manifolds. Nucl. Phys. B298, 493–525 (1988) 68. Candelas, P., Green, P., Hübsch, T.: Rolling among Calabi-Yau vacua. Nucl. Phys. B330, 49–102 (1990) 69. Kadir, S.N., Lynker, M., Schimmrigk, R.: Work in progress Communicated by A. Kapustin
Commun. Math. Phys. 303, 31–71 (2011) Digital Object Identifier (DOI) 10.1007/s00220-011-1192-2
Communications in
Mathematical Physics
An Infinite Class of Extremal Horizons in Higher Dimensions Hari K. Kunduri1 , James Lucietti2 1 Theoretical Physics Institute, Department of Physics, University of Alberta, Edmonton,
AB T6G 2J1, Canada. E-mail:
[email protected]
2 Theoretical Physics, Blackett Laboratory, Imperial College London, London SW7 2AZ, UK.
E-mail:
[email protected] Received: 12 March 2010 / Accepted: 19 September 2010 Published online: 5 February 2011 – © Springer-Verlag 2011
Abstract: We present a new class of near-horizon geometries which solve Einstein’s vacuum equations, including a negative cosmological constant, in all even dimensions greater than four. Spatial sections of the horizon are inhomogeneous S 2 -bundles over any compact Kähler-Einstein manifold. For a given base, the solutions are parameterised by one continuous parameter (the angular momentum) and an integer which determines the topology of the horizon. In six dimensions the horizon topology is either S 2 × S 2 or CP2 #CP2 . In higher dimensions the S 2 -bundles are always non-trivial, and for a fixed base, give an infinite number of distinct horizon topologies. Furthermore, depending on the choice of base we can get examples of near-horizon geometries with a single rotational symmetry (the minimal dimension for this is eight). All of our horizon geometries are consistent with all known topology and symmetry constraints for the horizons of asymptotically flat or globally Anti de Sitter extremal black holes. Contents 1. 2. 3.
4.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary of Six Dimensional Ricci Flat Near-Horizon Geometry Construction of Near-Horizon Geometries . . . . . . . . . . . . 3.1 Near-horizon equations . . . . . . . . . . . . . . . . . . . . 3.2 A class of near-horizon geometries in even dimensions . . . 3.2.1 Summary of solutions. . . . . . . . . . . . . . . . . . . Global Analysis of Horizon Geometries . . . . . . . . . . . . . . 4.1 Inhomogeneous S 2n . . . . . . . . . . . . . . . . . . . . . . 4.2 Inhomogeneous S 2 bundles over Kähler-Einstein spaces . . . 4.2.1 The topology of H. . . . . . . . . . . . . . . . . . . . . 2 4.2.2 Four dimensional horizons: S 2 × S 2 and CP2 #CP . . . . 4.2.3 Examples in all even dimensions. . . . . . . . . . . . .
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Area and Angular Momentum Formulas . . . . . . . . . 5.1 Area . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Angular momentum . . . . . . . . . . . . . . . . . . 5.3 Area versus angular momentum curves . . . . . . . . 6. Which Horizon Geometries Arise from New Black Holes? 6.1 Black holes with R × U (1)[(D−1)/2] symmetry . . . . 6.2 Black holes with R × U (1) symmetry . . . . . . . . 6.3 Associated (boosted) black strings . . . . . . . . . . 7. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . A. Local Properties of Horizon Metrics . . . . . . . . . . . . A.1 Curvature calculations . . . . . . . . . . . . . . . . . A.2 Conformal Kähler structure . . . . . . . . . . . . . . B. Topology of Hm, p . . . . . . . . . . . . . . . . . . . . . C. Inhomogeneous CPn Horizon with Conical Singularity . . D. Computation of “Internal” Angular Momenta on K . . . . D.1 K = CPn−1 . . . . . . . . . . . . . . . . . . . . . . D.2 Toric K . . . . . . . . . . . . . . . . . . . . . . . .
H. K. Kunduri, J. Lucietti
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1. Introduction One of the classic results of four dimensional General Relativity is Hawking’s horizon topology theorem [1,2]. This states that spatial sections of the event horizon of an asymptotically flat black hole solution to Einstein’s equations must be homeomorphic to S 2 . This theorem is a key ingredient to the black hole uniqueness theorem, being the first logical step required to prove such a classification theorem for black holes. For a variety of reasons, mainly stemming from String Theory and AdS/CFT (see e.g. [3] for a clear account and references), the study of black hole solutions to higher dimensional General Relativity has recently attracted a great deal of attention. The classification of black hole solutions in higher than four dimensions is a difficult open problem. However, it is possible to extend some of the ingredients which were used for D = 4 to higher dimensions. For example, Hawking’s horizon topology theorem uses the two dimensional Gauss-Bonnet theorem in a crucial way and thus does not generalise straightforwardly. Nevertheless, Galloway and Schoen [4] have established a generalisation which constrains the horizon topology of asymptotically flat black holes in the following manner: spatial sections of the event horizon, H (which are D −2 dimensional orientable and closed manifolds), must be positive Yamabe type1 . For D = 5, so H is three dimensional, this constraint is strong enough to allow only S 3 (and quotients) and S 1 × S 2 (and connected sums of these). In fact, explicit asymptotically flat black hole solutions are known for both of these topology types [5–7]. In D ≥ 6, so dim H ≥ 4, the complete list of positive Yamabe type manifolds is not known. However, it is clearly less of a constraint than in D = 5. There is in fact another way of constraining horizon topologies, as noted in [8]. Suppose we have a black hole solution which is asymptotically flat or globally Anti de Sitter2 (AdS). Now consider a spacelike hypersurface which intersects the future 1 A compact manifold is positive Yamabe type if and only if it admits a positive scalar curvature metric. 2 By asymptotically globally AdS we mean the conformal boundary is R × S D−2 . We will not consider
asymptotically locally AdS spacetimes, i.e. with conformal boundary R × X for more general X . For black holes with these asymptotics, H would have to be cobordant to X . Recall that another important case in the context of AdS/CFT is X = R D−2 in which case the known “black hole” solutions have H = R D−2 .
Infinite Class of Extremal Horizons in Higher Dimensions
33
event horizon and conformal future infinity. The D − 1 dimensional manifold has a boundary which is the disjoint union of H and S D−2 (the sphere at infinity), i.e. it defines a cobordism between H and S D−2 . In fact since the manifolds in question all have an orientation induced from the spacetime orientation, then H and S D−2 must be oriented cobordant. It is a standard result that two closed manifolds are (oriented) cobordant if and only if their corresponding Stiefel-Whitney and Pontryagin numbers are equal [9,10]. Since these numbers all vanish for spheres we deduce that H must have vanishing Stiefel-Whitney and Pontryagin numbers. It is worth noting that topological censorship requires to be simply connected [2,11]. However, for dim H ≥ 3 this provides no extra constraint on the topology of H because given any oriented cobordism there must always exist a simply connected oriented cobordism [12]. In fact for dim H = 3 all the Stiefel-Whitney and Pontryagin numbers trivially vanish for any H, and thus the existence of such cobordisms provides no constraint. However, for dim H ≥ 4 the existence of a cobordism to a sphere does give non-trivial constraints on the topology, which is different to the positive Yamabe constraint. For example, CP2 is positive Yamabe but has non vanishing Stiefel-Whitney and Pontryagin numbers, whereas T 4 is zero Yamabe type but has vanishing Stiefel-Whitney and Pontryagin numbers. Therefore for D > 5 the existence of such cobordisms provides a refinement of allowed horizon topologies for asymptotically flat and AdS black holes. Now, as is well known, finding, and let alone classifying, black hole solutions is a difficult task. In this paper we are motivated by the question: what horizon topologies are actually realised by asymptotically flat and globally AdS black hole solutions in D > 5?3 As we have discussed above a necessary condition is they are orientedcobordant to a sphere (or equivalently have vanishing Stiefel-Whitney and Pontryagin numbers), and, at least in the asymptotically flat case, positive Yamabe type. But are these conditions sufficient? This is a fundamental open problem towards the classification of higher dimensional black holes. Interestingly, for extremal black holes, one can show that the full spacetime Einstein equations imply the metric induced on H satisfies an equation which depends only on intrinsic data on H. Thus in a precise sense the Einstein equations on the horizon can be decoupled and solved separately. This is in fact intimately related to the existence of the so called near-horizon limit of the full black hole metric [8,13,14]. Therefore, by studying the horizon equation one can learn about the possible horizon geometries and topologies for H, without finding the full black hole metric. This is the approach we will take. It is worth emphasising that this method can allow one to rule out possible black horizon topologies, but not prove their existence (since given a near-horizon geometry there need not be a corresponding black hole solution). To get some insight into what one might expect, consider possible near-horizon geometries in Einstein-Maxwell theory. The reason for doing this is that it is easy to construct some simple static examples, i.e. the direct product AdS2 × H, where the metric on H is positive Einstein, with a Maxwell field proportional to the volume form on AdS2 . For simplicity consider D = 6, and thus in this example H is a positive Einstein closed 4-manifold. The classification problem for such spaces is a famous open problem in differential geometry. Only a few explicit examples of such Einstein spaces are known [15]: S 4 with the round metric, CP2 with the Fubini-Study metric, S 2 × S 2 with the standard product metric and CP2 #CP2 (i.e. CP2 with 1-point blown up) with the Page metric [16] (which is cohomogeneity-1 and conformally Kähler). Existence of Einstein 3 Note that this question is still open for D = 5 black holes.
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H. K. Kunduri, J. Lucietti
metrics has been established for CP2 #kCP2 with 2 ≤ k ≤ 8.4 This provides us with a host of Einstein metrics which can be used to give near-horizon geometries of exotic horizon topology. Out of this list only S 4 , S 2 × S 2 and CP2 #CP2 are actually cobordant5 to S 4 . However, in view of the uniqueness theorem for static black holes with an electric field [17] only the S 4 case is expected to arise as a limit of a static asymptotically flat black hole (although note this theorem has only been proved for non-extremal black holes). Nevertheless, one might expect that rotating black holes (stationary and non-static case) could have S 2 × S 2 or CP2 #CP2 horizon topologies. This would be analogous to black rings in D = 5 which must be rotating. In this paper we construct an infinite family of vacuum (and also Einstein) near-horizon geometries in D = 2n + 2 ≥ 6 which have non-spherical horizon topology. Spatial sections of the horizon H are smooth inhomogeneous 2n-dimensional S 2 -bundles over a compact Kähler-Einstein manifold K (the Einstein metric on K has positive curvature). Our ansatz is inspired by the local form of the horizon metric of certain even dimensional extremal Myers-Perry black holes (which can be written as fibrations over CPn−1 ), as well as certain Einstein metrics on complex line bundles over K [18] (which include Page’s metric [16]). For fixed K , our solutions are parameterised by a continuous parameter L > 0 and an integer m > p which specifies the topology of H (where p is an integer associated to the Kähler-Einstein base called the Fano index). The isometry group of our near-horizon geometries is S O(2, 1) × U (1) × G, where G is the isometry group of K . Note that by construction, the local form of our solutions (with K = CPn−1 ) also contain the near-horizon geometry of the extremal Myers-Perry-(AdS) black holes (which have H = S 2n ) with all angular momenta equal as a special case. In D = 6 (so n = 2), these solutions give smooth cohomogeneity-1 horizon geometries for H = S 2 × S 2 (if m is even) and H = CP2 #CP2 (if m is odd). Note that these horizon metrics are not the Einstein ones for S 2 × S 2 or CP2 #CP2 discussed above in the context of static near-horizon geometries in Einstein-Maxwell – in particular our S 2 × S 2 metric is not even a product metric. As discussed above both of these manifolds are cobordant to S 4 and are positive Yamabe and therefore candidates as horizons of black holes. Therefore we will discuss the possibility that there are asymptotically flat or globally AdS vacuum black hole solutions with such horizon topologies, and that the solution we have is the near-horizon limit of an extremal black hole of this kind. In higher dimensions (i.e. n > 2), the new near-horizon geometries we find are all non-trivial H-bundles over AdS2 , with H itself always a non-trivial S 2 -bundle over any compact Kähler-Einstein space. Strikingly, for a fixed Kähler-Einstein base, the topology of the horizon (i.e. the S 2 -bundle over K ) is different for each value of the integer m. Therefore, we have an infinite discrete class of horizon topologies (in contrast to the n = 2 case above). Furthermore, as we explain later, any S 2 -bundle over a compact manifold is guaranteed to be cobordant to a sphere, and any S 2 -bundle over a compact base with positive Ricci curvature must be of positive Yamabe type. Therefore, all our horizon topologies are cobordant to S 2n and positive Yamabe type, and are thus all consistent with the topological restrictions discussed above for the horizons of asymptotically flat and globally AdS black holes. If the Kähler-Einstein base space K is toric (i.e. admits U (1)n−1 isometry) then these near-horizon geometries have S O(2, 1) × U (1)n isometry where n = [(D−1)/2] 4 The k = 2 case is conformally Kähler whereas the rest are Kähler. 5 This can been seen from the fact that CP2 is the generator of the oriented cobordism group in four
dimensions, which is in fact isomorphic to Z.
Infinite Class of Extremal Horizons in Higher Dimensions
35
is the rank of S O(D − 1). Interestingly, if one chooses the Kähler-Einstein space to have no isometries,6 we get examples of near-horizon geometries with isometry exactly S O(2, 1) × U (1), i.e. just one rotational isometry U (1) and no more. This is interesting as it has been conjectured that in view of the higher dimensional version of the rigidity theorem [19,20], there should be stationary black hole solutions with R × U (1) symmetry [8]. Therefore we will discuss the interesting possibility that our near-horizon geometries are near-horizon limits of extremal black holes with this minimal amount of rotational symmetry. It is worth emphasising that our solutions possess no more abelian rotational symmetry than is allowed for asymptotically flat or globally AdS spacetimes: U (1)[(D−1)]/2 (and this is saturated when K is toric). Indeed this was the motivation for focusing on the class of near-horizon geometries considered in this paper. However, they do not constitute the most general possibility with U (1)[(D−1)/2] rotational symmetry. Indeed the classification of near-horizon geometries with U (1)[(D−1)]/2 symmetry is an interesting open problem out of reach with current methods. However, when the Kähler-Einstein space is chosen to be homogeneous our near-horizon geometries are cohomogeneity-1. It is then plausible (at least for K = CPn−1 ) that our solutions are the most general cohomogeneity-1 near-horizon geometries with a maximal abelian isometry group U (1)[(D−1)/2] , although we have not proved this. The analogous classification problem in D = 4 [21–23] and D = 5 [23] has been solved (including a Maxwell field and a cosmological constant in D = 4 [22–24]) – i.e. the classification of near-horizon geometries with U (1) and U (1)2 rotational symmetry respectively. Crucially, these near-horizon classifications have been used recently to prove uniqueness theorems for extremal Kerr [25–27] and Kerr-Newman [25,27] as well as a D = 5 generalisation [26]. For D > 5 there is another possible generalisation of these 4d and 5d problems. That is, the classification of near-horizon geometries with U (1) D−3 rotational symmetry. This has also been solved [28], however it is worth emphasising that for D > 5 such near-horizon geometries cannot be near-horizon limits of asymptotically flat black holes since they have too many commuting rotational isometries – instead they would arise as near-horizon limits of Kaluza-Klein (KK) black holes which are uniform in the KK direction. The organisation of this paper is as follows. In Sect. 2 we present the simplest example of our solutions, a Ricci-flat near-horizon geometry in six dimensions with horizon 2 topology either S 2 × S 2 or CP2 #CP . This short summary is intended for readers who wish to avoid our analysis in detail. Section 3 presents the derivation of our new nearhorizon solutions. The global analysis of the resulting horizon geometries is given in Sect. 4. Section 5 presents a derivation of the physical properties of these solutions. Finally, in Sect. 6 we gather the preceding results and consider the possibility that these near-horizon geometries extend to extremal, asymptotically flat or asymptotically globally AdS black hole solutions. We conclude with a discussion. Some useful technical results are collected in the Appendices. 2. Summary of Six Dimensional Ricci Flat Near-Horizon Geometry In this section we will summarise the six dimensional Ricci flat near-horizon geometry we have found. We have also constructed analogous solutions with a negative 6 The minimal dimension for such near-horizon geometries is 8 (i.e. n = 3 so dim K = 4). Explicit examples for dim K = 4 are given by the del Pezzo surfaces d Pk for 4 ≤ k ≤ 8, i.e. CP2 with k points blown up in general positions.
36
H. K. Kunduri, J. Lucietti
cosmological constant and also in any even dimension. However, the Ricci flat six dimensional case is the simplest and thus a good example to illustrate our more general class of solutions. We will not give any derivations in this section, however we will make the presentation self-contained. In the subsequent sections we will provide complete derivations of the general class in all even dimensions including a cosmological constant. The D = 6 vacuum solution takes the explicit form ⎡ ⎤ 2 2 2 2 4r L (1 − x )d x ⎦ ds 2 = (ξm + x 2 ) ⎣− 2 dv 2 + 2dvdr + 4x 2 L (4 − m 2 x 2 ) ξm − 3m 2 ⎡ 2 4x
2 (4 − m 2 x 2 ) ξm − 3m 2 1 dφ + cos θ dχ + 2 +L 2 ⎣ ξ r dv m (ξm + x 2 )(1 − x 2 ) 2 ⎤ 1 + (1 − x 2 )(dθ 2 + sin2 θ dχ 2 )⎦ , (1) 4 where 4 ξm = 3
3−
4 m2 4 + m2
,
(2)
where L > 0 and m > 2 is an integer. The coordinate ranges are −2/m ≤ x ≤ 2/m, 0 ≤ θ ≤ π , φ ∼ φ + 2π/m, χ ∼ χ + 2π and (v, r ) can take any value (the horizon is at r = 0). Cross sections of the horizon H, are homeomorphic to S 2 × S 2 if m is even 2 or CP2 #CP if m is odd. Note that the only other way the local form of the horizon metric can be extended to a smooth metric on a compact manifold is if m = 2 and φ ∼ φ + 2π , which gives H = S 4 and corresponds to the near-horizon limit of extremal 6d Myers-Perry with all angular momenta equal. The area and Komar angular momentum, defined with respect to the rotational Killing field m −1 ∂φ (since this has orbits with canonical period 2π ), are:
4 3− 2 , m √
4 πL 4 ξm J =± 1+ 2 , 2G m m
A(H) =
8π 2 L 4 3m 2
(3) (4)
and therefore A(H) = 4π G m ξm |J |.
(5)
It is interesting to compare, for fixed J , the area of the new horizons H to the spherical topology case (79). This can be expressed as
3 − m42 A(H) = m 3ξ = 2m , (6) m A(S 4 ) 4 + m2
Infinite Class of Extremal Horizons in Higher Dimensions
37
and therefore 2<
√ A(H) < 2 3, A(S 4 )
(7)
where the inequalities follow from the fact that (6) is a monototically increasing function of m and m > 2. In Sect. 6 we discuss the possibility that such near-horizon geometries arise as nearhorizon limits of yet to be found asymptotically flat extremal black hole solutions. 3. Construction of Near-Horizon Geometries 3.1. Near-horizon equations. We will assume that the event horizon of a stationary extremal black hole solution must be a Killing horizon of a Killing vector field V . In a neighbourhood of such a Killing horizon we can always introduce Gaussian null coordinates [20] (v, r, x A ) such that V = ∂/∂v, the horizon is at r = 0 and x A are coordinates on a spatial section of the horizon H (which of course is D − 2 dimensional). We will assume that H is an oriented compact manifold without boundary. Near the extremal Killing horizon, the space-time metric in these coordinates reads ds 2 = r 2 F(r, x)dv 2 + 2dvdr + 2r h A (r, x)dvd x A + γ AB (r, x)d x A d x B .
(8)
The near-horizon limit [8,14] is obtained by taking the limit v → v/ , r → r and
→ 0. The resulting metric is ds 2 = r 2 F(x)dv 2 + 2dvdr + 2r h A (x)dvd x a + γ AB (x)d x A d x B ,
(9)
where F, h A , γ AB are a function, a one-form, and a Riemannian metric respectively, defined on H. In this paper we will be interested in finding near-horizon geometry solutions to Einstein’s vacuum equations Rμν = gμν . We will be mainly focused on ≤ 0. One can prove (see e.g. [23]) that these spacetime equations for a near-horizon geometry are in fact equivalent to the following set of equations on H: R AB =
1 h A h B − ∇(A h B) + γ AB 2
(10)
with the function F determined by F=
1 1 h A h A − ∇ A h A + , 2 2
(11)
where R AB and ∇ are the Ricci tensor and the covariant derivative of the metric γ AB . In particular, (10) is the AB component of the Einstein equations, (11) is the vr component, all written covariantly on H. It can be shown that the rest of the Einstein equations are satisfied as a consequence of the above set of equations. Before moving on we note that static near-horizon geometries of this kind (which are equivalent to dh = 0) have been classified in [13]. It was found that for ≤ 0 the only solution is F = , h A = 0 and R AB = γ AB . In this paper we will focus on the non-static case.
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3.2. A class of near-horizon geometries in even dimensions. We will consider (2n + 2)dimensional near-horizon geometries (so dim H = 2n) of the form γ AB d x A d x B = L 2 A(ρ)2 g¯ ab d x¯ a d x¯ b + dρ 2 + B(ρ)2 (dφ + σ )2 , h A d x A = (ρ)−1 [k(ρ)B(ρ)2 (dφ + σ ) − (ρ)dρ],
(12) (13)
where g¯ ab is a (2n − 2)-dimensional Kähler-Einstein metric on a base manifold K , normalised to R¯ ab = 2n g¯ ab , with Kähler form J = 21 dσ and x¯ a some set of coordinates on K . As we will see later, compactness of H requires the coordinates ρ ∈ [ρ1 , ρ2 ], φ ∼ φ + φ and that K be compact. The constant L is taken to have dimensions of length and is introduced for later convenience. The isometry group of γ AB is G × U (1), where G is the isometry group of g¯ ab and the U (1) is generated by the Killing field ∂/∂φ. The 1-form h is chosen to respect the G × U (1) symmetry too, so the total near-horizon geometry also has this symmetry. In this parameterisation there is a scaling freedom, (ρ, A, B, k, L) → (sρ, s A, s B, s −2 k, s −1 L),
(14)
where s = 0 is a constant, which leaves the near-horizon data invariant. Let us explain our motivation for studying this class of near-horizon geometries. One reason comes from choosing the base space K to be a homogeneous space. Then our ansatz is in fact the most general horizon geometry with G × U (1) isometry group whose principle orbits are U (1) bundles over a homogeneous base K (even if K is not Kähler-Einstein). The (ρ, φ) are coordinates valid on the principal orbits. Furthermore in this case h is also the most general 1-form invariant under G × U (1) provided there is a unique – up to homothety – homogeneous Einstein metric on K [29] (in the Kähler case there is in fact a unique homogeneous Kähler-Einstein metric [15]). Note that if K is homogeneous both the horizon and near-horizon geometries are cohomogeneity-1. The case of most interest to us when K is homogeneous is K = CPn−1 , which has a unique Kähler-Einstein metric g¯ ab given by the Fubini-Study metric. This is indeed a homogeneous metric on CPn−1 with G = SU (n), in which case the isometry group of the horizon geometry is SU (n) × U (1) with cohomogeneity-1 orbits. This class of near-horizon geometries includes those of the extremal Myers-Perry black holes with all angular momenta equal [29]. If instead we choose other Kähler-Einstein spaces K with less symmetry these metrics are no longer cohomogeneity-1—we will also be interested in this possibility (although then the above is not necessarily the most general near-horizon geometry with such symmetries). Another important point is that if K is toric (i.e. has U (1)n−1 symmetry) then the maximal abelian subgroup of G × U (1) is U (1)[(D−1)]/2 which is the maximal abelian subgroup of the rotation group S O(D − 1) for asymptotically flat or globally AdS spacetimes. We will now turn to solving Eq. (10). It is convenient to introduce a set of vielbeins7 e A for the metric γ AB : e0 = Ldρ,
ea = L Ae¯a ,
e2n−1 = L B(dφ + σ ),
(15)
7 To avoid a proliferation of indices we will use the same symbols for coordinate and vielbein indices.
Infinite Class of Extremal Horizons in Higher Dimensions
39
where a = 1, . . . 2n − 2 and e¯a are vielbeins for g¯ = e¯a e¯a . The Ricci tensor of (12) in this basis is diagonal with 2(n − 1)A 1 B , (16) R00 = 2 − − L A B 2
B B 1 A B R2n−1 2n−1 = 2 − + 2(n − 1) , (17) − L B A4 AB (2n − 3)A2 2n A 2B 2 A B δab Rab = − − − − . (18) A2 A A2 A4 AB L 2 Details of the calculation are given in Appendix A.1. The source term S AB ≡ 21 h A h B − ∇(A h B) + γ AB in the vielbein basis reads S00
1 = 2 L
2 − 2 2
+ ,
S0 2n−1 = −
Bk L 2
1 k 2 B 2 B S2n−1 2n−1 = 2 + , + L 2 2 B (19) A δab Sab = + δab . A L2
It immediately follows that the 0 2n − 1 component of Eq. (10) implies k(ρ) = κ a constant. We will assume κ = 0, otherwise the near-horizon geometry is static. For solving the rest of the equations it is convenient to define a new coordinate x by x (ρ) = B(ρ). Note that the coordinate is valid wherever B = 0 (which occurs on the principal orbits), and is defined up to the scaling freedom x → s 2 x (inherited from (14)), as well as x → −x and x → x + const . We will use these to simplify the solution. For later reference it is worth noting that (d x)2 = B 2 /L 2 . From now on we will be treating everything as a function of x and will denote d/d x = . Firstly, if one subtracts the 2n − 1 2n − 1 component of Eq. (10) from the 0 0 component of Eq. (10) one gets
1 κ2 1 2 2(n − 1) A + 3 = 2 − − . (20) − A A 2 We will determine the most general solution for which the left and right sides of the above equation vanish separately. It is not obvious that there must be solutions of this form, but as we shall see there are; it is also not necessarily the case that all solutions must be of this form.8 It is a guess inspired by the form of 4d near-horizon geometries [23,24] and even dimensional inhomogeneous Einstein spaces [18]. Therefore we should emphasise that we have not necessarily classified all solutions of the form (12) and (13). Setting the RHS of (20) to zero gives an equation for which is identical to that which occurs for 4d near-horizon geometries [23,24]. Its general solution is (x) =
κ 2 β(x − x0 )2 + , β 4
(21)
8 In fact, in the = 0 case, it is easy to show there are solutions not in this class, e.g. there is a solution √ 2 B = n A2 and = |κ|A2 /(2 n − 1), where A is constant. If K = CPn−1 this is the near-horizon geometry
of the direct product of an odd dimensional extremal Myers-Perry black hole with all angular momenta equal, and a line.
40
H. K. Kunduri, J. Lucietti
where β > 0 and x0 are integration constants. For later convenience we will introduce a positive constant ξ > 0 defined by κ2 =
β2ξ , 4
(22)
in terms of which (x) =
β [ξ + (x − x0 )2 ]. 4
(23)
Setting the LHS of (20) to zero gives A + A−3 = 0,
(24)
which in fact has two different families of solutions. Notice that (24) implies A is a non-constant function, a fact we will use below. We have thus determined the functions (x), A(x). Before solving for A explicitly, we will now use (24) to simplify the other field equations. It is clear now that the ab component of (10) is a first order equation for the remaining function B. In fact it is convenient to introduce the function ˜ P(x) ≡ B 2 A2(n−1) . Then the ab component of (10) can be simplified, using (24), resulting in d A2(n−1) (2n − λA2 ) P˜ , = d x A A A2 A 2
(25)
(26)
where λ ≡ L 2 is dimensionless. It is now sufficient to impose one other component of the field equations, say the 00 component. We find that the 00 component of (10) is (without using (24))
2 2(n − 1)A 2(n − 1)A 2(n − 1)A + + + P˜ − P˜ + P˜ A A A2 = −2λ A2(n−1) . Using (24) this can be rewritten as
1 d 2(n − 1)A P˜ P˜ 2 2 d d + AA − A A A A d x d x A A A d x A A
˜ A A 1 P 2 + + − A = −2λ A2(n−1) . 2 A A A Now, substituting (26) into (28) results in many cancellations, and simplifies to
1 1 2 + = 0. − A A A A2
(27)
(28)
(29)
Infinite Class of Extremal Horizons in Higher Dimensions
41
To summarise, we have shown that if we impose the ansatz (24), then the field equations are equivalent to solving (26), (29) with (23). Thus, this system of equations is overdetermined and it is not obvious there exist solutions. In fact as we shall now see, one class of solutions to (24) leads to a full solution for this system. First note that (24) implies A 2 − A−2 = const . In fact this constant must be nonzero, otherwise Eq. (29) implies = 0 which is inconsistent with (23). Therefore we can write A 2 − A−2 = − α −1 for some constant α > 0 and = ±1. Integrating9 one gets A2 = (α − α −1 x 2 ), where we have used the translation freedom in the definition of x √ to fix the integration constant. Furthermore, using the scaling freedom (14) (with s = α) the solution can be written as A2 = (1 − x 2 ).
(30)
Now, Eq. (29), using (23) and (30), is satisfied if and only if x0 = 0. Therefore is simply =
β (ξ + x 2 ). 4
Finally, substituting (30) and (31) into (26) gives
P(x) d (ξ + x 2 )(1 − x 2 )n−1 [λ (1 − x 2 ) − 2n] , = dx x x2
(31)
(32)
where for convenience we have defined ˜ P(x) ≡ 4β −1 n P(x).
(33)
To summarise, we have found a solution to the near-horizon equations (10) of the form (12) and (13), with A2 and given by (30) and (31) respectively, and B 2 determined up to a first order ODE for P (32) where P is defined by (33) and (25). To integrate the ODE for P(x) (32) explicitly it is convenient to define the polynomial n l − (n − l)ξ n (−1)l u l , Q n (u, ξ ) ≡ Cl (34) n(2l − 1) l=0
which satisfies d dx
Q n (x 2 , ξ ) x
=−
(ξ + x 2 )(1 − x 2 )n−1 . x2
It is worth noting for later reference that √ (n − 1)! π (−n+ 12 ) (−n+ 21 ) xC (x) + ξ C (x) , Q n (x 2 , ξ ) = 2n−1 2n−2 2(n + 21 )
(35)
(36)
(α)
where Cn (x) are the Gegenbauer polynomials. It then follows that the general solution to (32) is: P(x) = 2n Q n (x 2 , ξ ) − λQ n+1 x 2 , ξ + cx, (37) 9 Note that the other solution A2 = α to the first order equation A 2 − A−2 = − α −1 does not solve the original second order equation A + A−3 = 0 and thus must be discarded.
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H. K. Kunduri, J. Lucietti
where c is an integration constant. This explicit expression will be useful for later analysis. We have thus determined the horizon data γ AB , h A in the coordinate system (x, φ, x¯ a ). We may now evaluate the remaining near-horizon data, namely the function F given by (11). Using the explicit form of A(x) and (x), together with (32) (i.e. we do not need the explicit expression for P(x)), we find it can be written as: F=
A0 κ 2 B 2 + 2 2 L
(38)
for a constant A0 given by
2 β β
nβ κ + = A0 = − 2 + [−2n + λ(1 + ξ )] , 2L β 4 4L 2
(39)
where in the second equality we have used (22). The significance of this particular form for F is revealed by changing the radial variable in the full near-horizon geometry to r → r which allows one to write it as ds 2 = (x)[A0 r 2 dv 2 + 2dvdr ] +L
2
κr dv 2 dx2 2 + B(x) dφ + σ + 2 A(x) g¯ ab d x¯ d x¯ + . B(x)2 L 2
a
b
(40)
This form of the near-horizon geometry has manifest S O(2, 1) × G × U (1) symmetry as guaranteed by the theorem proved in [29]. For our solution we have (x) =
β (ξ + x 2 ), 4
A(x)2 = (1 − x 2 ),
B(x)2 =
P(x) (1 −
x 2 )n−1 (ξ
+ x 2) (41)
with A0 given by (39), κ given by (22) and P(x) by (37). This solution has the scaling symmetry (β, v) → (Kβ, K −1 v), where K > 0. This allows one to set the constant β to any desired value (i.e. it is a redundant parameter). It is worth pointing out that if one analytically continues the AdS2 → S 2 we have an Einstein metric which falls in the general class derived in [30,31]. 3.2.1. Summary of solutions. We have derived a set of near-horizon geometries of the form (9), and satisfying Rμν = gμν in D = 2n + 2 dimensions for n ≥ 2, given by (ξ + x 2 )(1 − x 2 )n−1 d x 2 P(x) + γ AB d x A d x B = L 2 (dφ + σ )2 2 P(x) (ξ + x )(1 − x 2 )n−1 (42) +(1 − x 2 )g¯ ab d x¯ a d x¯ b , √ 2 ξ P(x) 2x h Ad x A = ± (dφ + σ ) − d x, (43) 2 2 2 n−1 (ξ + x ) (1 − x ) ξ + x2 −2n + λ(1 + ξ ) 4ξ P(x) F= + 2 , (44) 2 2 L (ξ + x ) L (1 − x 2 )n−1 (ξ + x 2 )3 where P(x) is a polynomial given by (37) and λ ≡ L 2 . The metric g¯ ab is any (2n − 2)dimensional Kähler-Einstein metric such that R¯ ab = 2n g¯ ab and J = dσ/2 is the Kähler
Infinite Class of Extremal Horizons in Higher Dimensions
43
form. For fixed choice of the Kähler-Einstein structure (g, ¯ J ), the solution is parameterised by the constants (ξ, c, L), where ξ, L > 0, c is the integration constant occurring in P(x) and = ±1. Note that the parameter β has cancelled from the solutions as it is redundant (see above). Therefore this constitutes a three parameter family of metrics. It is worth noting that although they are valid for n ≥ 2, if one sets n = 1 and formally g¯ ab = 0, σ = 0, then the solution is locally isometric to the near-horizon limit of extremal Kerr-NUT-(AdS4 ) [23]. These solutions may thus be regarded as a generalisation to all even dimensions. It is worth pointing out that these solutions are valid for any . 4. Global Analysis of Horizon Geometries In this section we will derive the conditions necessary for extending our local horizon metric to a complete Riemannian metric on a compact smooth manifold H with no boundary (i.e. a cross section of the horizon). We will only analyse the case ≤ 0. First note that since we require γ AB to be a positive definite metric we must have A2 ≥ 0 (with a possible equality only at isolated points) and therefore either = 1 and x 2 ≤ 1 or = −1 with x 2 ≥ 1. For = +1 we must have P(x) ≥ 0, whereas for = −1 we must have (−1)n P(x) ≥ 0, in other words n P(x) ≥ 0. We first consider potential singularities in the metric as one varies x. Inspecting the metric we see that these can only occur at x = ±1, x = ±∞ or the roots of P(x). Following the terminology of [18] we refer to these points as endpoints. A complete manifold requires x1 ≤ x ≤ x2 , where x1 < x2 are two adjacent endpoints and that the singularities at these endpoints are removable by coordinate transformations. Compactness further requires that x1 , x2 are finite endpoints (i.e the metric distance between points in x1 < x < x2 and the endpoints is finite). This leaves a number of possibilities: the endpoints can be either at ±1 or at simple zeros of P(x). Note that regularity of the metric requires that if either x1 or x2 are equal to ±1 then P(x) must also vanish at these points (otherwise, for example, the norm of ∂/∂φ diverges at these endpoints). Therefore in all cases we must have x1 ≤ x ≤ x2 with P(x1 ) = P(x2 ) = 0 and n P(x) > 0 for x1 < x < x2 . Note that this implies n P (x1 ) > 0 and n P (x2 ) < 0. For later use it is convenient to note the identity P (xi ) =
(ξ + xi2 )(1 − xi2 )n−1 [λ (1 − xi2 ) − 2n] , xi
(45)
which can be derived using (32) and P(xi ) = 0. It is worth noting that necessarily xi = 0, since all roots of P are non-zero.10 The possibilities for the endpoints are listed in Table 1 below. Note that in Case IIa (Case IIb), the inequality x2 < 1(x1 > −1) follows by the assumptions that x2 = 1(x1 = 1) and x2 (x1 ) is an adjacent endpoint to −1(1). Also note that Case IIa and IIb, Case IIIa and IIIb, and Case Va and Vb, can be mapped into each other using the freedom in the definition of x → −x. Therefore without loss of generality we refer to these cases as II, III, V respectively, and we need only consider the five cases I-V. In fact, Case III and V cannot occur. This is easy to see as follows. Simply note that in both cases = −1 and thus without loss of generality one has a root x2 > 1 such that sgn P (x2 ) = (−1)n+1 (i.e. Case IIIb and Case Va). However Eq. (45) implies 10 This follows from the fact that P(0) = (2n − λ )ξ . For = 1 we see that P(0) > 0. For = −1 we see that P(0) = 0 unless λ = −2n, and thus for simplicity we assume λ = −2n.
44
H. K. Kunduri, J. Lucietti
Table 1. Endpoints Case x1 x2
I −1 1
IIa −1 <1
IIb > −1 1
IIIa < −1 −1
IIIb 1 >1
IV > −1 <1
Va >1 >1
Vb < −1 < −1
sgn P (x2 ) = (−1)n and therefore we have a contradiction. In Appendix C we show Case II cannot be made smooth with compact topology. This leaves Case I and Case IV, and as we will show these can be made smooth with compact topology, and end up homeomorphic to S 2n and S 2 -bundles over compact Kähler-Einstein manifolds K , respectively. Before moving on we note that the horizon metric has conical singularities at the roots xi of P(x) as long as xi2 = 1. Removal of the conical singularity at x = xi is equivalent to n−1 (1 − x 2 )n−1
x i i (46) = 4π φ = 4π(ξ + xi2 ) , 2 |P (xi )| λ (1 − xi ) − 2n where the second equality follows from the identity (45). 4.1. Inhomogeneous S 2n . In this section we analyse Case I listed in Table 1. Note that since in this case −1 ≤ x ≤ 1 we must have = 1. From the explicit expression for P(x) given in Eq. (37) we see that P(1) = 0 = P(−1) implies c = 0 and therefore P(x) is an even function. The constraint P(1) = 0 now provides one linear equation for the parameter ξ . One can solve this by performing the various binomial sums involved, to get an explicit value for ξ given by ξ = ξ∗ ≡
2n + 1 − λ . (2n + 1)(2n − 1 − λ)
(47)
Substituting this value back into P(x) and simplifying gives, after some work,11 λ(1 − x 2 ) . (48) P(x) = (1 − x 2 )n 1 + ξ∗ − 2n + 1 It is worth noting that for λ = 0 we have ξ∗ = 1/(2n − 1), and the solution is much easier to obtain using the identity Q n (u, 1/(2n − 1)) = (1 − u)n /(2n − 1). Putting all this together gives the following horizon metric: ⎡ (ξ∗ + x 2 )d x 2 γ AB d x A d x B = L 2 ⎣ 2) (1 − x 2 ) 1 + ξ∗ − λ(1−x 2n+1 ⎤ λ(1−x 2 ) 2 (1 − x ) 1 + ξ∗ − 2n+1 + (dφ + σ )2 + (1 − x 2 )g¯ ab d x¯ a d x¯ b ⎦ , ξ∗ + x 2 (49) where ξ∗ is given by (47). 11 In fact it is easier to go back and solve (32).
Infinite Class of Extremal Horizons in Higher Dimensions
45
The above metric is smooth and invertible for −1 < x < 1. We must now check regularity at x = ±1. Set x = ±(1 − η2 ) and expanding for small η one gets (50) γ AB d x A d x B = 2L 2 dη2 + η2 ((dφ + σ )2 + g¯ ab d x¯ a d x¯ b ) + · · · , where . . . signify terms higher order in the η expansion. Smoothness at η = 0 requires that φ = 2π and that g¯ be the Fubini-Study metric on K = CPn−1 . The horizon metric then looks like the origin of R2n near x = ±1. The horizon topology in this case is then H = S 2n . To summarise, this case gives a 1-parameter (given by L) family of inhomogeneous horizon geometries on H = S 2n . In fact these near-horizon geometries are isometric to the near-horizon limits of the extremal Myers-Perry-(AdS) in 2n + 2 dimensions with all angular momenta equal. For λ = 0 these near-horizon limits were calculated in [29] and it is easy to check that they are the same by setting x = cos θ and L2 =
2na 2 . 2n − 1
(51)
It can be checked that our solution for < 0 is isometric to the near-horizon geometry of extremal Myers-Perry-AdS in 2n + 2 dimensions with all angular momenta equal, although we will not give the details here. 4.2. Inhomogeneous S 2 bundles over Kähler-Einstein spaces. In this section we analyse Case IV listed in Table 1. This corresponds to the generic case when the endpoints are −1 < x1 < x2 < 1 and thus = 1. This implies that A2 = (1 − x 2 ) > 0 for all x1 ≤ x ≤ x2 and P(x) > 0 for x1 < x < x2 . It follows that we must have P (x1 ) > 0 and P (x2 ) < 0. The horizon metric is thus smooth and invertible for x1 < x < x2 with potential conical singularities at x = x1 , x2 . From (46) we see that simultaneous removal of these singularities implies −
λ(1 − x12 ) − 2n λ(1 − x22 ) − 2n = . x1 x2
(52)
For λ = 0 this immediately implies that x1 = −x2 . For λ < 0 notice that f (x) = [λ(1 − x 2 ) − 2n]/x is a monotonically increasing function for all x > 0. Therefore the regularity condition f (x2 ) = − f (x1 ) implies x1 = −x2 . From the form of P(x) the condition x1 = −x2 implies c = 0. The period of φ is given by (46) and is simply φ =
4π x2 . 2n − λ(1 − x22 )
(53)
This means that for a fixed point on the base K the (x, φ) part of the metric is smooth and of S 2 topology. Compactness of H then clearly requires K to be compact. So far we have only considered regularity of the horizon metrics when one varies x. In this case (in contrast to the other cases considered) we have not constrained the Kähler-Einstein space (K , J, g). ¯ We must also impose that dφ + σ is independent of the coordinate chart on K used. This implies that 2J C over any 2-cycle C in K must be an integer multiple of φ, and hence C 2J and C 2J for any two 2-cycles C, C must be rationally related. This constraint is automatically satisfied for Kähler-Einstein manifolds since the first Chern class of its tangent bundle, which is an integral class, is
46
H. K. Kunduri, J. Lucietti
given by c1 (K ) = [ρ/2π ¯ ] = (n/2π )[2J ] (ρ¯ is the Ricci form of g). ¯ Let p be the Fano index of the Kähler-Einstein base K , which by definition is the largest positive integer such that p −1 c1 (K ) is an integral class. Now consider a set of 2-cycles i ⊂ K which form a representative basis of the free part of H2 (K , Z). It follows that c1 (K ) = n i p (54) i
for a set of integers n i ∈ Z such that gcd(n i ) = 1; note that without loss of generality we can always take n i to be non-negative. It follows that 2π p , (55) 2J = n i n i and thus for any integral 2-cycle C = i ci i we have 2π p 2π p 2J = ≥ , (56) c n i i n n C i
where the last equality follows from the fact one can always find integers ci such that 12 It follows that the minimum absolute value of c n = 1. i i i C 2J over all possible 2-cycles is then simply 2π p/n. Therefore the period of φ must satisfy mφ =
2π p n
(57)
for some positive integer m. In fact for any compact Kähler-Einstein manifold p ≤ n with equality if and only if K = CPn−1 . Equating the two expressions (53) and (57) for φ thus gives the following quantisation condition m=
p[2n − λ(1 − x22 )] . 2nx2
(58)
Note that this is particularly simple for λ = 0 which gives x2 = p/m. The existence of smooth metrics in this case thus boils down to proving that the (even) polynomial P(x) = 2n Q n (x 2 , ξ ) − λQ n+1 (x 2 , ξ ) must have a smallest positive root x2 < 1, such that x2 satisfies (58) for some integer m. If this can be achieved then the horizon metric is a smooth inhomogeneous metric on a compact manifold which is a fibre bundle (possibly trivial) over K with S 2 fibre. To prove the existence of x2 first note the identities: √ n! π (2n − 1 − λ)(ξ − ξ∗ ) , (59) P(0) = (2n − λ)ξ, P(1) = (n + 21 ) where ξ∗ is given by (47). For λ ≤ 0 it is clear that P(0) > 0. Furthermore if ξ < ξ∗ then P(1) < 0. It follows that, for λ ≤ 0, if ξ < ξ∗ then there must exist a root x0 of P(x) in the interval 0 < x < 1. It remains to show that x0 is the smallest positive root, i.e. P(x) > 0 for −x0 < x < x0 , so that x2 = x0 . Fortunately this is easy to prove. Suppose that x0 is not the smallest positive root. It follows that there must exist a root 12 This is a basic generalisation of a number theory result called Bézout’s identity.
Infinite Class of Extremal Horizons in Higher Dimensions
47
x− ≤ x0 < 1 such that P (x− ) ≥ 0. However from the identity (45) we see that any root 0 < xi < 1 must have P (xi ) < 0. This is a contradiction and hence x0 must be the smallest positive root as required. Therefore we have proved that a sufficient condition for the existence of x2 is ξ < ξ∗ . We can also prove the condition ξ < ξ∗ is necessary as follows. First note that ξ = ξ∗ was analysed earlier and has x2 = 1 and thus we discard in this case. Now assume ξ > ξ∗ so that P(1) > 0. Therefore, either P(x) > 0 for all 0 ≤ x ≤ 1 or there exists a root 0 < x− < 1 such that P (x− ) ≥ 0. As argued above this is a contradiction and therefore we have shown that if ξ ≥ ξ∗ then P(x) > 0 for all 0 < x < 1, i.e. x2 does not exist. To summarise, in the λ ≤ 0 case we have shown that a root 0 < x2 < 1, such that P(x) > 0 for x 2 ≤ x22 , exists if and only if ξ < ξ∗ . Finally it remains to show that the quantisation condition (58) can always be satisfied for some positive integer m. This can be established as follows. First note that for λ ≤ 0 the function [2n − λ(1 − x 2 )]/x is monotonically decreasing, and takes the value +∞ at x = 0 and 2n at x = 1. Therefore (58) has solutions if and only if m is any integer satisfying m > p.
(60)
It is worth noting that one can solve (58) explicitly for x2 = x2 (m, p) (e.g. for λ = 0 it is just x2 = p/m) which in turn gives ξ = ξ(m, p) via the identity, (−n+ 21 ) (−n− 21 ) λ x2 C2n−1 (x2 ) − 2n+1 C2n+1 (x2 ) (61) ξ =− (−n+ 1 ) −n− 1 λ C2n−2 2 (x2 ) − 2n+1 C2n 2 (x2 ) which follows from P(x2 ) = 0 and (36). In summary, we have smooth horizon metrics which are S 2 -bundles over a compact Kähler-Einstein space K if and only if ξ < ξ∗ and (58) are satisfied. For a given base K (which gives p), these metrics are parameterised by one continuous parameter L > 0 and an integer m satisfying the bound (60). 4.2.1. The topology of H. The topology of our bundles depends on the integers m, p and thus we will refer to these horizons by Hm, p . In Appendix B we derive some of the basic invariants of these spaces, which we summarise at the end of this section. Recall though that the two topological restrictions we are interested in are: is Hm, p cobordant to S 2n ? Is Hm, p positive Yamabe type? Fortunately these are easy to deal with as follows. First, observe that any S 2 -bundle over an oriented compact manifold is oriented cobordant to a sphere. This can be seen as follows. Let S 2 → H → K be any S 2 -bundle over a compact manifold K , with dim H = N . The structure group of such bundles is S O(3). One can construct an associated ball bundle over K . That is, one replaces the fibres S 2 with the 3-ball B3 (so ∂ B3 ∼ = S 2 ) and then constructs the associated bundle using the same S O(3) transition functions acting on B3 ⊂ R3 . Call this new N + 1 dimensional (oriented) manifold X so that by construction ∂ X = H. It is then clear that a (oriented) cobordism between H and S N exists. One simply cuts a sufficiently small (N + 1)-ball B N +1 from the interior of X so that the remaining manifold has boundaries H and S N as required. Explicitly, the cobordism is given by = X \B N +1 . Therefore we immediately deduce that our Hm, p are always (oriented) cobordant to S 2n for any m > p and n ≥ 2. Next, observe that any S 2 -bundle over a compact manifold K , such that the base admits a positive Ricci curvature metric, must also admit a positive Ricci curvature
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H. K. Kunduri, J. Lucietti
metric [32]. Since by assumption our K is compact and Einstein with positive scalar curvature, we immediately deduce that our Hm, p must admit a positive Ricci curvature metric. It follows that Hm, p are all of positive Yamabe type for any m > p and n ≥ 2. Summary of topology. We will now summarise the key topological properties of the 2n-dimensional manifolds Hm, p , where m, p are positive integers such that m > p. • For n = 2 it is either homeomorphic to the trivial bundle S 2 × S 2 (if m is even) or the non-trivial S 2 -bundle over S 2 which is CP2 #CP2 (if m is odd). For n ≥ 3 it is always a non-trivial S 2 -bundle over a compact positive Kähler-Einstein manifold K , where p is its Fano index; furthermore different m give different topologies. • They are simply connected for all n ≥ 2. • They are spin if and only if m + p is even, for all n ≥ 2. • The Euler characteristic is 2χ (K ) for any n ≥ 2 and all m. • They are oriented cobordant to S 2n for all n ≥ 2 and all m. • They are positive Yamabe type for all n ≥ 2 and all m. The derivation of the n = 2 case is discussed in the next section and the rest are discussed in detail in Appendix B. 2
4.2.2. Four dimensional horizons: S 2 × S 2 and CP2 #CP . In this section we consider the n = 2 case explicitly. We note that the only Kähler-Einstein space in this case is S 2 with (note the normalisations) cos θ 1 2 dθ + sin2 θ dχ 2 , σ = g¯ ab d x¯ a d x¯ b = dχ (62) 4 2 and χ ∼ χ +2π . Since there is only one 2-cycle, S 2 itself, the invariant S 2 2J = 2π , i.e. p = 2. Therefore, for every integer m > 2 we have an explicit smooth inhomogeneous metric on S 2 bundles over S 2 . As is well known [33], up to homeomorphism, there are only two types of S 2 -bundles over S 2 since they are classified by π1 (S O(3)) = Z2 . One is the trivial bundle S 2 × S 2 and the other is a non-trivial bundle which has the same topology as CP2 #CP2 . In our case these arise depending on whether m is even or odd respectively. This fact can be deduced from our analysis as follows (see [34] for a simple argument). Since p = 2 we note that Hm,2 is spin if and only if m is even. Since S 2 × S 2 is spin and CP2 #CP2 is not spin, it immediately follows that m even corresponds to the former and m odd to the latter. It is worth pointing out that the λ = 0 solution is particularly simple. Then P(x) =
4 4 x + 4(ξ − 1)x 2 + 4ξ. 3
(63)
The existence of a positive root x2 < 1 occurs if and only if ξ < 1/3 and explicitly is given by 3 4ξ 2 2 x2 = 1 − ξ − (1 − ξ ) − . (64) 2 3 Since we must have ξ > 0 we see that 0 < x2 < 1 is uniquely parameterised by 0 < ξ < 1/3 (the function x2 (ξ ) is a monotonically increasing function on the domain
Infinite Class of Extremal Horizons in Higher Dimensions
[0, 1/3] with range [0, 1]). The quantisation condition is simply x2 = one to give a simple explicit expression for ξ : 4 4 3 − m2 ξ = ξm ≡ , 3 4 + m2
49 2 m,
which allows
(65)
where recall the integer m > 2. Note that in this parameterisation, the quartic (63) is simply:
4x 2 . (66) P(x) = (4 − m 2 x 2 ) ξm − 3m 2 In Sect. 2 we give the explicit form of the full near-horizon geometry in this simple case. 4.2.3. Examples in all even dimensions. We now discuss some interesting examples in dimensions n > 2. K = CPn−1 . Here we consider the Kähler base to be K = CPn−1 which has a unique Einstein metric given by the Fubini-Study metric, and this has p = n. It is a homogeneous metric with SU (n) isometry. Therefore, the horizon and near-horizon geometries are both cohomogeneity-1. The resultant near-horizon geometries then possess an isometry group S O(2, 1) × SU (n) × U (1). This class is the natural generalisation of the n = 2 case. Recall that for n = 2 we showed H is homeomorphic to the trivial bundle S 2 × S 2 (if m is even) or the nontrivial bundle CP2 #CP2 (if m is odd). However, for n ≥ 3 the possible topologies of our solutions are very different from the n = 2 case, despite the similarly of the local forms of the metric. In particular our Hm, p is never homeomorphic to the trivial bundle or CPn #CPn . To see this observe that for n ≥ 3, different m must give different topologies (as we show in Appendix B), and m = 0 and m = 1 correspond to the trivial bundle and CPn #CPn respectively [18]. However regularity of our bundles requires m > n, and thus these two topologies are immediately ruled out as claimed. dim H = 6. This corresponds to the case n = 3, which corresponds to D = 8 dimensional near-horizon geometries. The Kähler-Einstein space K in this case is 4 dimensional and spatial sections of the horizon are six dimensional S 2 -bundles over K . In fact, all Kähler-Einstein metrics on complex 2-manifolds (K , J ) with positive curvature have been classified [35]. These occur exactly on CP2 , CP1 × CP1 , or the del Pezzo surfaces d Pk = CP2 #kCP2 for 3 ≤ k ≤ 8 (i.e. CP2 blown up at k points in a general position), and the Kähler-Einstein metric is uniquely determined by (K , J ). Each of these provides us with a near-horizon geometry of the form derived earlier. Let us now consider the isometry groups of these Kähler-Einstein metrics. In fact CP2 , CP1 ×CP1 and d P3 are all toric manifolds (i.e. admit an effective U (1)2 -action), and their associated Kähler-Einstein metrics13 must be invariant under U (1)2 . Further, the CP2 and CP1 ×CP1 cases have enhanced symmetry of SU (3) and SU (2)2 respectively. Recall our corresponding near-horizon geometries have isometry groups S O(2, 1)×U (1)× G, where G is the isometry group of K , and thus the cases CP2 and CP1 × CP1 are both cohomogeneity-1. On the other hand, d Pk for 4 ≤ k ≤ 8 are not toric manifolds and their Kähler-Einstein metrics generically have no continuous isometries. Therefore, choosing 13 Note that this metric on d P is only known numerically [36]. 3
50
H. K. Kunduri, J. Lucietti
K = d Pk for 4 ≤ k ≤ 8 gives us examples of near-horizon geometries with only U (1) rotational isometry, or in total S O(2, 1) × U (1). Interestingly, for k ≥ 5 these solutions possess 2k − 8 extra continuous parameters, corresponding to the complex structure moduli of d Pk (these correspond to the positions of the k blow up points). dim H ≥ 8. This corresponds to n > 3 and is qualitatively similar to the n = 3 case just described, although there is no analogous classification of possible compact positive curvature Kähler-Einstein manifolds available yet. We can give some explicit examples though. If we choose the Kähler-Einstein space to be homogeneous then the horizon, and near-horizon geometry, is cohomogeneity-1. The most symmetric case is K = CPn−1 which we have already considered above. Another homogeneous possibility is to take ×(n−1) ∼ ×(n−1) K = (CP1 ) . The resultant near-horizon geometry then possesses = (S 2 ) an isometry group S O(2, 1) × S O(3)×(n−1) × U (1). More generally consider the case when K is a toric manifold so the isometry group is U (1)n−1 . It follows that the rotational isometry group of the associated near-horizon geometries is U (1)n . These horizon and near-horizon geometries are generically cohomogeneity-n. A special case of course includes the homogeneous examples above. Finally the most extreme case occurs for Kähler-Einstein metrics that possess no symmetries whatsoever (generalising the higher del Pezzo surfaces). In this case the S 2 bundle over K has only one U (1) symmetry and the associated near-horizon geometry has isometry S O(2, 1) × U (1).
5. Area and Angular Momentum Formulas In this section we will discuss various physical properties of the near-horizon geometries we have derived. Recall that, for any compact Kähler-Einstein manifold K , we found a family of compact horizon geometries parameterised by one positive real number L (which sets the scale of the horizon) and an integer m ≥ p, where p is the Fano index of K . Our horizons Hm, p have their topology determined by the integers (m, p). For m = p one must have K = CPn−1 and so p = n, and the horizons have S 2n topology and are isometric to those of extremal Myers-Perry with all angular momenta equal. For m > p the topology of the horizon is of an S 2 -bundle over any K . Recall the coordinate ranges are −x2 ≤ x ≤ x2 and φ ∼ φ + φ, where x2 is the smallest positive root of p P(x) and depends on the integer m. Note that φ = 2π nm for m > p and φ = 2π for m = p. We first give the volume form on the horizon
=
√ γ d x 1 ∧ · · · ∧ d x 2n = L 2n (1 − x 2 )n−1 d x ∧ dφ ∧ ¯ ,
(67)
where we have chosen an orientation and ¯ is the volume form associated to the Kähler-Einstein space K .
5.1. Area. The area of a cross section of the horizon is A(Hm, p ) =
H
= −L 2n φ vol(K )
√ (n − 1)! π (n +
1 2)
(−n+ 1 )
C2n−1 2 (x2 ),
(68)
Infinite Class of Extremal Horizons in Higher Dimensions
where vol(K ) =
K
n−1 l=0
51
¯ and we have used n−1
√ (−1)l 2x 2l+1 (n − 1)! π (−n+ 21 ) C Cl (x), =− 2l + 1 (n + 21 ) 2n−1
(69)
(α)
where Cn (x) is a Gegenbauer polynomial. The H = S 2n case, ξ = ξ∗ , corresponds to x2 = 1, φ = 2π , vol(CPn−1 ) = (−n+ 1 )
π n−1 /(n − 1)!, and using C2n−1 2 (1) = −1, gives14 1
A(S ) = 2n
2π n+ 2 L 2n (n + 21 )
= A2n L 2n ,
(70)
where A2n is the volume of a unit round sphere. Note that for λ = 0 this agrees with the Myers-Perry value [29] upon using (51) (as it should). 5.2. Angular momentum. The Komar angular momentum of the near-horizon geometry is given by [29] 1 √ Ji ≡ J [m i ] = γ h · mi , (71) 16π G H where m i is a rotational Killing field. For our near-horizon geometry the available rotational Killing fields are mφ ≡
φ ∂ , 2π ∂φ
m i = m¯ i
(72)
for 1 ≤ i ≤ d − 1, where d ≤ n depends on K , where m¯ i are the commuting Killing fields of K . Note that if K is toric then d = n. We have defined m φ such that its orbits are canonically normalised with period 2π . The angular momentum associated to m φ evaluates to √ L 2n (φ)2 vol(K ) ξ x2 2P(x) d x. (73) Jφ = ± 2 2 32π 2 G −x2 (ξ + x ) The integral for Jφ can be done by parts using (32), resulting in x2 2P(x) 2 2 2 2n Q d x = − (x , −1) − λQ (x , −1) , n−1 n 2 2 2 2 x2 −x2 (ξ + x )
(74)
where Q n (u, ξ ) is the polynomial (34). For the case at hand Q n (x 2 , −1) =
n n Cl (−1)l x 2l l=0
2l − 1
=−
n!( 23 ) (n +
3 2)
(−n− 21 )
C2n
(x),
(75)
n+1 14 The area of a unit round S n is given by A = 2π 2 / ( n+1 ) (note that there is a typo in [29]). The n 2 n−1 volume of CP with the Fubini-Study metric normalised as in this paper, can be deduced from the volume of an S 2n−1 with unit round metric, A2n−1 , using the fact it can be written as a Hopf fibration. Explicitly A2n−1 = φ vol(CPn−1 ).
52
H. K. Kunduri, J. Lucietti (α)
where Cn (x) is a Gegenbauer polynomial. Putting all this together gives √ L 2n (φ)2 vol(K )A2n n ξ λ (−n+ 21 ) (−n− 21 ) C2n−2 (x2 ) − C (x2 ) , (76) Jφ = ± 16π 2 G A2n−1 x2 2n + 1 2n where An is the area of a unit round n−sphere as above. In the case when K = CPn−1 we have checked explicitly, in Appendix D, that the angular momenta associated to the internal Killing fields m¯ i of CPn−1 are J¯i = 0
(77)
for 1 ≤ i ≤ n − 1. It also follows that these angular momenta vanish for direct products of lower dimensional complex projective spaces. In fact in Appendix D we also show that for general toric K these internal angular momenta must vanish as well (one does not need the explicit metric for this calculation). (−n− 21 )
For the H = S 2n case the formulas for Jφ can be simplified using C2n 2n + 1, resulting in
(1) =
A2n L 2n ξ∗ n (2n − 1 − λ) . (78) 8π G √ When λ = 0 this gives Jφ = ±A2n L 2n n 2n − 1/(8π G) which agrees with [29] upon using (51). Jφ = ±
5.3. Area versus angular momentum curves. Let us first consider the simplest case of H = S 2n corresponding to the near-horizon geometry of an extremal Myers-Perry(-AdS) black hole. For λ = 0, using the formula for the area of the horizon and the angular momentum we find 8π G A(S 2n ) = √ |Jφ |. n 2n − 1
(79)
For λ < 0 one can also write Jφ as a function of the horizon area:
|Jφ | =
2n 1/n 2 ) 2n − A(S −1 A2n n A(S 2n ) 8π G
2n + 1
.
(80)
From this expression it is easy to show that, for fixed Jφ , A(S 2n ) decreases monotonically as one makes more negative starting from = 0. This makes intuitive sense, as turning on a negative cosmological constant makes gravity more attractive. These area versus angular momentum curves can also be written for our new nearhorizon geometry. For λ = 0 it reads: ⎡ A(Hm, p ) =
8π G ⎣ − √ x2 ξ
(−n+ 1 )
x2 C2n−1 2 (x2 ) (−n+ 1 ) C2n−2 2 (x2 )
⎤
√ ⎦ |Jφ | = 8π G ξ |Jφ |, x2
(81)
Infinite Class of Extremal Horizons in Higher Dimensions
53
where the second equality follows from the identity: (−n+ 1 )
ξ =−
x2 C2n−1 2 (x2 ) (−n+ 1 )
C2n−2 2 (x2 )
,
(82)
which is a consequence of P(x2 ) = 0 and (36). For λ < 0 one can write down an analogous curve for Hm, p although it is not so revealing. We can compare the curve for our new near-horizon geometry to the spherical topology case. For λ = 0 we find that, at fixed Jφ , (2n − 1)C (−n+ 21 ) (x ) A(Hm, p ) n 2 2n−1 = . (83) ξ(2n − 1) = n − (−n+ 12 ) A(S 2n ) x2 x2 C2n−2 (x2 ) We have explicitly checked that (83) is a monotonically decreasing function of x2 in the interval 0 < x2 < 1 for low values of n. Assuming this is the case for all n (as seems reasonable), it follows that n<
√ A(Hm, p ) < n 2n − 1. 2n A(S )
(84)
Thus in particular, for fixed Jφ , the area of Hm, p is always larger than that of the spherical topology case. It is worth emphasising that if there are new black hole solutions corresponding to our near-horizon geometries, the canonical rotational Killing field m φ need not correspond to the same combination of rotational Killing fields at asymptotic infinity as it does for the Myers-Perry solution (see next section). Therefore, the above comparison of the area at fixed Jφ may not be meaningful outside the context of this class of near-horizon geometries. 6. Which Horizon Geometries Arise from New Black Holes? In this section we will investigate the possibility that the D = 2n + 2 ≥ 6 dimensional near-horizon geometries we have found are in fact the near-horizon limits of yet to be known, stationary extremal black hole solutions to Rμν = gμν for ≤ 0. We will focus on asymptotically flat black holes ( = 0) and asymptotically globally AdS black holes ( < 0).15 Near spatial infinity, asymptotically these black hole spacetimes would look like
R 2 d R2 2 ds ∼ − 1 − dt 2 + + R 2 ds 2 (S 2n ), (85) 2 D−1 1 − R D−1 ds 2 (S 2n ) =
n
dμ2I + μ2I (dψ I )2
(86)
I =1
as R → ∞, where R is some radial coordinate, nI=1 μ2I = 1 are the latitude coordinates on S 2n and ψ I ∼ ψ I + 2π . Note that the Killing fields ψ I = ∂/∂ψ I are the 15 We remark that our near-horizon geometries can be obtained as “near-horizon” limits of spacetimes with Taub-NUT like asymptotics. These spacetimes can be obtained by analytically continuing certain Einstein metrics in [31]. However, due to their asymptotics, they necessarily have closed time like curves everywhere.
54
H. K. Kunduri, J. Lucietti
standard generators of the Cartan subgroup U (1)n ⊂ S O(2n + 1). There are a number of constraints on the symmetries and topologies of the horizons of such black holes which we now recall. In both of these cases, the maximal rotational isometry group is S O(2n + 1) and its maximal abelian subgroup is U (1)n . Furthermore, for a rotating black hole, the rigidity theorem [19,20] guarantees the existence of at least one U (1) isometry (although this has only been proved for non-extremal black holes).16 Therefore black holes of this kind have an isometry group whose abelian subgroup is R × U (1)d such that 1 ≤ d ≤ n (as shown above though, asymptotically they do have the maximal abelian symmetry R × U (1)n ). Our near-horizon geometries are D = 2n + 2 dimensional spacetimes which satisfy Rμν = gμν for ≤ 0. Their isometry groups are S O(2, 1)×G ×U (1), where G is the isometry group of K . The S O(2, 1) component is typical for near-horizon geometries and is guaranteed by general theorems regarding symmetry enhancement [14,29]. The isometry group of spatial sections of the horizon H is (by assumption) G × U (1) and thus depends on the choice of K . If K is toric then the maximal abelian subgroup of G is U (1)n−1 and thus in this case the total abelian isometry group of the near-horizon geometries is R × U (1)n . If K has a metric with no isometries then the total abelian isometry group of the near-horizon geometry is R × U (1). For asymptotically flat, or globally AdS black holes the exterior to the black hole defines a cobordism from H to S 2n . As discussed earlier all our horizons geometries are guaranteed to be cobordant to a sphere since they are S 2 -bundles over a compact manifold. Furthermore for asymptotically flat black holes the horizon must be positive Yamabe type, which is also the case for our horizon geometries. Therefore in all even dimensions greater than four we have found examples of nearhorizon geometries such that the spatial sections of the horizon are not of spherical topology, but still consistent with all known symmetry and topology constraints required for asymptotically flat and globally AdS black holes. It is therefore natural to speculate whether these are the near-horizon geometries of yet to be found extremal black holes in such spacetimes. We now expand on some examples in more detail. 6.1. Black holes with R × U (1)[(D−1)/2] symmetry. For this discussion we will focus on the class of near-horizon geometries with the Kähler-Einstein base K = CPn−1 for n ≥ 2. Recall that for n = 2 then H is either the trivial bundle S 2 × S 2 (for m even) 2 or the non-trivial S 2 -bundle over S 2 (for m odd) which is homeomorphic to CP2 #CP . For n ≥ 3 then H is always a non-trivial S 2 -bundle over CPn−1 and for different m they have different topology (in this case m > n). The rotational symmetry of these near-horizon geometries is SU (n) × U (1). As discussed above the abelian subgroup of this is U (1)n , the maximal abelian rotational symmetry group possible for asymptotically flat or globally AdS black holes in 2n + 2 dimensions. These isometries are associated to angular momenta Ji for i = 1, . . . , n. We calculated these with respect to the Killing fields on the horizon (72) and found Jφ = 0 (76), and that the “internal” angular momenta J¯i = 0 for 1 ≤ i ≤ n − 1 (see Appendix D). It is natural to expect that the putative black hole solutions would have the same rotational symmetry as the horizon geometry, and thus a total of R × SU (n) × U (1). For example, the H = S 2n case we derived earlier which also has SU (n) × U (1) rotational 16 See [37] for partial results on the extremal case.
Infinite Class of Extremal Horizons in Higher Dimensions
55
symmetry, arises from the extremal Myers-Perry-(AdS) black hole with equal angular momenta which does have the enhanced isometry group R × SU (n) × U (1). However, for our new horizon geometries Hm,n will in fact now argue that this cannot be the case. Suppose that the full black hole solution does have a global SU (n) × U (1) isometry with orbits which are U (1) bundles over CPn−1 . This gives a natural way of identifying the rotational Killing fields on the horizon (72), in terms of those of the total rotational symmetry S O(2n + 1) at infinity. Simply write the round S 2n in (85) in terms of a round S 2n−1 , and in turn, the round S 2n−1 as a Hopf fibration over CPn−1 , i.e. ds 2 (S 2n ) = dθ 2 + sin2 θ [(dφ + σ )2 + g¯ ],
(87)
where g¯ is the Fubini-Study metric (normalised by Ric(g¯ ) = 2n g¯ ) with Kähler form J = dσ /2 and of course φ = 2π . It is then easy to show that in terms of the standard set of rotational Killing fields ψ I for 1 ≤ I ≤ n of S 2n defined in (86), we have ∂ = ψI , ∂φ n
m¯ i = ψn − ψi
(88)
I =1
for 1 ≤ i ≤ n − 1, where m¯ i are the U (1)n−1 generators on CPn−1 . Now, due to the assumption of the global SU (n) × U (1) isometry, it is most natural to identify the data (φ , σ , g¯ ) in this sphere at infinity with the corresponding data on the horizon (φ, σ, g) ¯ (since we are choosing K = CPn−1 ). However, since φ = 2π/m = φ , and (g, ¯ J) and (g¯ , J ) are both normalised in the same way, we find a contradiction. Therefore this argument implies, surprisingly, that there can be no asymptotically flat or globally AdS black hole with a global SU (n) × U (1) rotational isometry and horizon geometry Hm,n . This is unfortunate as it means we can say less about any potential black hole solutions with such horizons. For example, the above identification would have allowed one to deduce the angular momenta as viewed from infinity are all equal. Also, the problem of determining such black hole solutions would have been more tractable, since the black hole metrics would have been cohomogeneity-2 and thus one could cast the problem on the 2 dimensional orbit space Mˆ = M/[R × SU (n) × U (1)] (analogous to the Weyl solutions case). In the absence of any other symmetry, there is no natural way to identify the data on the horizon (φ, σ, g) ¯ with the data at infinity (φ, σ , g¯ ). However, note that one has the same number of commuting Killing fields on the horizon and the sphere at infinity (this is true for any toric K ). Therefore, a natural expectation for the symmetries of the hypothetical black hole solutions is that they have the same commuting rotational symmetries as the near-horizon geometry, i.e. a global isometry group R × U (1)n . This then allows one to identify the commuting rotational Killing fields in the most general possible way: namely, the two sets of commuting Killing fields (∂φ , m¯ i ) and (∂φ , m¯ i ) are necessarily related by some constant matrix in S L(n, Z). Since we do not know the explicit form of this constant matrix, we cannot deduce what the angular momenta would be from the point of view of asymptotic infinity; in particular, they need not have all angular momenta equal. Finally, it is worth noting that in the generic toric K case, Hm, p has isometry group U (1)n and thus one might expect any corresponding full black hole solutions to also have R ×U (1)n symmetry. This would then allow one to identify the commuting Killing fields on the horizon and at infinity via some constant matrix, as in the K = CPn−1 case just discussed.
56
H. K. Kunduri, J. Lucietti
6.2. Black holes with R × U (1) symmetry. In this section we consider the class of near-horizon geometries we have derived where the Kähler-Einstein base K is chosen so that it has no isometries at all. Such spaces are known to exist in four and higher dimensions. Therefore the associated near-horizon geometries in this case start in eight dimensions. The near-horizon geometries in this case would have total isometry group S O(2, 1) ×U (1). If they arose as near-horizon limits of an extremal black hole, it would necessarily have only R × U (1) symmetry. In this case the near-horizon geometry only has one angular momentum, corresponding to the U (1) generator ∂/∂φ, and as we showed this is necessarily non-zero. Since the horizon has less abelian symmetry than asymptotic infinity there is no natural way to identify the U (1) generators and therefore we cannot guess what angular momenta these black holes would carry relative to infinity. Let us now mention some explicit examples. The simplest case for which we can have such symmetry is n = 3, i.e. 4d Kähler-Einstein space K . The only non-toric possibilities are the complex del Pezzo surfaces d Pk for 4 ≤ k ≤ 8. Therefore we have established the existence of examples of horizon geometries which are fully consistent with the black hole horizon topology theorems, but also have only one U (1) rotational symmetry. As mentioned earlier, the d Pk for k ≥ 5 have a moduli space of complex structures of dimension 2k − 8, corresponding to the freedom in placing the blowup points. Hence the associated near-horizon geometries possess these additional continuous parameters. These parameters could in turn constitute further continuous ‘hair’ for the putative black holes. For n > 3 the Kähler-Einstein space K is 2n − 2 ≥ 6 dimensional. Much less is known about the classification of such spaces. However, presumably it is the case there are examples with no isometries, as well as continuous moduli. Therefore, in all even dimensions D ≥ 8 we have candidate near-horizon geometries for new asymptotically flat or globally AdS extremal black holes with just R × U (1) isometry. 6.3. Associated (boosted) black strings. In this section we point out that given our Ricci-flat near-horizon geometries in 2n + 2 dimensions with horizon section H, we may trivially construct Ricci-flat near-horizon geometries in 2n + 3 dimensions with horizon section S 1 × H. This construction is analogous to that used in [29] where, for every extremal MyersPerry black hole in 2n + 2 dimensions, a boosted extremal black string was constructed in 2n + 3 dimensions. Its near-horizon geometry then provides an example of one with H = S 1 × S 2n and U (1)n+1 rotational isometry. This is the correct maximal abelian rotational symmetry for an asymptotically flat black hole in 2n + 3 dimensions. This led [29] to conjecture that a subset of these solutions could be the near-horizon limit of the yet to be found extremal asymptotically flat black rings in odd dimensions (in 5d one can explicitly check this is true as solutions are known [14]). In the present case, obviously we do not have the full even dimensional black hole solution (i.e. the analogue of the Myers-Perry solution with our horizon Hm, p ) or for that matter know whether it actually exists. Nevertheless, one may still construct the near-horizon geometry of the corresponding boosted black string as follows. Consider the direct product of our Ricci flat solutions with a line dz 2 . Now consider the “boost” φ → φ + sβ z
z → cβ z,
(89)
where sβ ≡ sinh β, cβ ≡ cosh β and are constants. The resultant near-horizon geometry is
Infinite Class of Extremal Horizons in Higher Dimensions
57
dx2 A(x)2 g¯ ab d x¯ a d x¯ b + B(x)2
2 κr dv +B(x)2 dφ + σ + sβ dz + + cβ2 dz 2 L2
ds = (x)[A0 r dv + 2dvdr ] + L 2
2
2
2
(90)
which is Ricci flat in 2n + 3 dimensions. One can perform a global analysis of the horizon metrics and the result are smooth compact horizon geometries with topology of Hm, p × S 1 , where Hm, p is identical to the 2n dimensional horizons we have derived and z must be periodic on the S 1 . Thus we have a family with four continuous parameters (L , β, , z) and an integer m > p, although it is expected that would be related to L if a black hole exists (see below). As a consequence of our construction, the isometry group of these near-horizon geometries is S O(2, 1) × G × U (1)2 . Note that S 1 × Hm, p is cobordant to S 2n+1 and positive Yamabe type17 . For K = CP1 for example, we have D = 7 smooth cohomogeneity-1 near-horizon geometries with H = S 1 × S 2 × S 2 , abelian symmetry U (1)2 , which are cobordant to S 5 and positive Yamabe type. The constants β and introduced above have the following significance. Suppose that there exists an asymptotically flat extremal black hole with a horizon geometry given by our Hm, p and consider the associated black string obtained by adding a line dz 2 . Now boost (t, z) → (cβ t − sβ z, cβ z − sβ t), where t is the asymptotically flat time coordinate of the lower dimensional black hole. Upon taking the near-horizon limit of this black string one will end up with a near-horizon geometry that is related to that of the black hole as above where is the angular velocity of the black hole in the φ direction (it is not possible to calculate from knowledge of the near-horizon geometry alone [29]). Note that these hypothetical black strings would have standard KK asymptotics (i.e. R1,2n+1 × S 1 ). Generically such strings possess a tension. Typically one can choose the boost β = βcrit such that the tension vanishes. In this case, one might expect new asymptotically flat black holes in R1,2n+2 to exist with horizon topology S 1 × Hm, p . Physically, this is achievable because the straight string can be “bent” into a ring in such a way as to have Minkowski asymptotics while inputting no energy. But since near-horizon geometries are independent of the asymptotic geometry, the near-horizon geometry of strings with β = βcrit is expected to be the same as such higher dimensional “black rings”. As mentioned above, this is precisely what occurs for extremal black rings and the Kerr black string in D = 5 [14]. It is natural to expect a similar phenomenon to occur here and so we have candidate near-horizon geometries for asymptotically flat extremal black holes in 2n + 3 dimensions with horizon topology S 1 × Hm, p . 7. Discussion In this paper, we have investigated the space of allowed extremal black hole horizon topologies, for vacuum general relativity, including a negative cosmological constant, in even dimensions greater than four. Our strategy was based on the observation that for such black holes, Einstein’s equations can be decoupled and solved on the horizon H alone. We have constructed an infinite class of (non-static) vacuum near-horizon geometries in D = 2n + 2 for n ≥ 2 (see Sect. 2 for the n = 2 Ricci flat case and (42) 17 In fact S 1 × M n is cobordant to S n+1 for any closed manifold M n . This is simply because it is the boundary of the n + 2 dimensional manifold D × M n , where D is a disk. Also S 1 × M n is positive Yamabe type if M n is.
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for the general local form of the solutions). A global analysis of the local horizon metrics reveals that the topology of H is either an S 2 -bundle over any compact KählerEinstein manifold K , or S 2n . Regularity implies that the S 2 -bundle topology solutions are characterised, for a given base K , by a continuous parameter (essentially the angular momentum) and a positive integer m satisfying some bound determined by the topology of K . The spherical topology solutions are simply the near-horizon geometries of extremal Myers-Perry-(AdS) with all angular momenta equal. We emphasise that all our new near-horizon geometries have horizons of non-spherical topology (with the precise topology determined by the choice of K and the integer m). First, consider the simplest case n = 2. Then for our new near-horizon solutions, H 2 must have topology S 2 × S 2 or CP2 #CP . We have analogous near-horizon geometries in higher dimensions n > 2, where K can be any (2n − 2)-dimensional Kähler-Einstein space. In contrast to the n = 2 case, the horizon H is always a non-trivial S 2 -bundle over K , and furthermore, for a fixed base K , one has an infinite number (countable) of distinct horizon topologies. As explained in the Introduction, the horizon topology of asymptotically flat or globally AdS black holes is constrained by the existence of a cobordism from H to a sphere, and the generalisation of Hawking’s topology theorem (positive Yamabe). In fact all our horizon topologies automatically satisfy these restrictions. We emphasise that all these conclusions are equally valid with and without a negative cosmological constant. Therefore, our work raises the possibility that there exist extremal, asymptotically flat or globally AdS black holes with precisely these non-spherical horizon geometries. Our work leaves a number of open questions. Most interestingly, of course, is whether there are in fact asymptotically flat or AdS extremal black holes with near-horizon limits given by our new near-horizon geometries. If this is the case then is reasonable to expect non-extremal and non-vacuum generalisations as well. These black holes would have to be rotating and thus constructing them would be a difficult task. An important feature of our analysis is that it yields, in addition to the topology, the explicit horizon geometries for the proposed extremal black holes. Although asymptotic information is lost in the near-horizon limit (e.g. angular velocities, mass), we can still compute the conserved angular momenta and area from the near-horizon geometry alone. In fact for our near-horizon geometries we find only one independent Komar angular momentum J . Interestingly, in the pure vacuum case ( = 0) we find that for fixed J , the area of the non-spherical horizons is always more than that of the corresponding Myers-Perry horizons. However, for the corresponding candidate nonspherical extremal black holes, we cannot say how this angular momentum J would be distributed at asymptotic infinity, and thus this comparison should be taken with caution. Another interesting point is that for fixed angular momentum J , we can have more than one near-horizon solution. In particular, in D = 6 for fixed J , we can have an infinite number of near-horizon geometries with H = S 2 × S 2 (even m) but also an infinite number of near-horizon geometries with H = CP2 #CP2 (odd m). Thus we see two phenomena: discrete hair for a fixed topology, and non-uniqueness of the near-horizon geometry for fixed J . On the other hand, in higher (even) dimensions D ≥ 8, we do not have the discrete hair for fixed topology, although we do have infinite non-uniqueness of the near-horizon geometry for fixed J (since we have an infinite number of distinct possible horizon topologies with the same J ). It should be emphasised though, that since we do not have the full black hole solutions available (or even know they exist), we can
Infinite Class of Extremal Horizons in Higher Dimensions
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not compute the mass and thus we cannot say anything about the corresponding black hole (non-)uniqueness problem. One particularly interesting example of a near-horizon geometry arises when we choose the Kähler-Einstein base to have no isometries (e.g. the del Pezzo surfaces d Pk for 4 ≤ k ≤ 8). The resulting near-horizon geometry then has just S O(2, 1) × U (1) symmetry. If it arises as the near-horizon limit of an extremal black hole, the black hole would have at most R × U (1) symmetry. This is the minimum symmetry requirement for a rotating black hole (all known solutions have more symmetries), and in fact it has been conjectured that black hole solutions with this symmetry should exist [8]. It is worth noting that evidence for asymptotically flat rotating black holes with this minimal symmetry has been found recently using a variety of different approaches [38,39]. In particular, the analysis of linearized gravitational perturbations of odd dimensional D ≥ 9 Myers-Perry black holes with equal angular momenta suggests a new branch of solutions with R × U (1) symmetry [39].18 Such black holes have an upper bound on their angular momenta which is saturated when they are extremal, and the instability sets in for angular momenta larger than some lower bound, which of course includes the extremal limit. However, this potential new branch of solutions would still have spherical horizon topology. More approachable open problems involve the analysis of near-horizon geometries. In this paper we found a class of solutions within some general ansatz (i.e. near-horizons of the form (12) and (13)). An interesting question is whether we have in fact found all solutions within this ansatz or not. We note that one can in fact classify compact Einstein spaces of the same form as our horizon geometries and the possible topologies one finds are S 2n , CPn and a finite discrete family of S 2 -bundles over Kähler-Einstein base K of the same form as ours (although we have an infinite discrete family for fixed K ) [18]. It is thus striking that a regular H = CPn case does not arise in our solutions (we did find an example with a conical singularity though). Therefore it is possible that there are other solutions within this class which would give such horizon topologies. We note that for n odd CPn is cobordant to S 2n and therefore consistent with all known restrictions on asymptotically flat or AdS black holes. In this paper we have focussed our attention on the classification problem of higherdimensional black holes. Since we included a negative cosmological constant, our results have potential applications to AdS/CFT. Most obviously, the existence of non-spherical horizon topology black holes which are asymptotic to global AdS is currently unknown. These would correspond to some interesting phase of the dual gauge theory on R × S 3 . Various attempts at finding AdS black rings have so far failed.19 In particular the existence of supersymmetric AdS5 black rings with R × U (1)2 symmetry has been ruled out [41,42], suggesting that the known spherical horizon topology solutions [44,45] may in fact be the most general ones. However in this paper we have found non-supersymmetric non-spherical topology (and non-black ring-like) examples. For instance, in 6d we have explicit S 2 × S 2 topology extremal horizon metrics. It is thus natural to wonder whether these are in fact near-horizon geometries of yet to be found extremal AdS black holes. If so, then it seems reasonable non-extremal generalisations should exist. Such objects would correspond to novel thermal phases of the dual gauge theory. 18 It is worth noting that previously [40], for the odd dimensional D ≥ 7 Myers-Perry-AdS with equal angular momenta, an instability was found (which sets in for sufficiently large rotation and includes the extremal limit), and was conjectured to have an endpoint which is a non-rotating black hole with even less symmetry, i.e. just stationary but not axisymmetric. 19 Although see [43] for some approximate constructions, see also [23] for an approximate near-horizon geometry.
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Acknowledgements. We would like to thank Jerome Gauntlett, Don Page, James Sparks, Dan Waldram and Toby Wiseman for useful conversations. We would especially like to thank Jerome Gauntlett for comments on a draft of our paper, and James Sparks for valuable correspondence on topology. HK is supported by a fellowship from the Pacific Institute of Mathematical Sciences and NSERC. JL is supported by an EPSRC career acceleration fellowship.
A. Local Properties of Horizon Metrics We consider 2n-dimensional Riemannian manifolds (H, γ ) (spatial sections of the horizon) endowed with metrics of the form: (91) γ AB d x A d x B = L 2 dρ 2 + A(ρ)2 g¯ ab d x¯ a d x¯ b + B(ρ)2 (dφ + σ )2 , where g¯ is a Kähler metric on the 2n − 2 dimensional Kähler-Einstein ‘base’ space K with Kähler form J¯ = 21 dσ and curvature normalised such that Ric(g) ¯ = 2n g. ¯ Lower Latin indices a, b, c, . . . take values from 1, . . . , 2n − 2. Henceforth we adopt the convention that all quantities defined on the base are “barred”, unless otherwise stated. Note that L is a parameter with dimensions length. We introduce a vielbein basis {e A : A ∈ {0, a, 2n − 1}} defined by e0 = Ldρ,
ea = L Ae¯a ,
e2n−1 = L B(dφ + A)
(92)
¯ Note we also sometimes use the coordinate with e¯a a vielbein basis for (K , g). d x = Bdρ. A.1. Curvature calculations. In this subsection we denote ρ-derivatives by d/dρ = . The spin connection, defined by de A = −ω A B ∧ e B , is readily computed: ω0a = −
A a e , LA
ωa 2n−1 = −
ω0 2n−1 = −
B ¯ b Jab e . L A2
B 2n−1 e , LB
ωab = ω¯ ab −
B ¯ 2n−1 , Jab e L A2 (93)
The curvature two-form AB = dω AB +ω AC ∧ωC B has the following non-vanishing components:
A 0 1 B A B a 2n−1 b ¯ − , e J ∧ e + ∧ e (94) e − ab L2 A A2 A B
1 B A B 0 B 0 2n−1 = 2 − (95) e ∧ e2n−1 + 2 − ea ∧ eb J¯ab , L B A A B
B2 B A 1 A B B ¯ 0 Jab e ∧ eb + a 2n−1 = 2 − (96) − δac ec ∧ e2n−1 , 2 4 L A A B A AB
1 2B A B2 ¯ ¯ B2 ¯ ¯ A2 a b B 0 2n−1 ¯ c d J J J J J = 2 ∧ e + e ∧ e δ δ − − − − e ab ac bd ab cd c d L A2 A B A2 A4 A4 ¯ + ab . (97) 0a =
ab
Note that we have used the fact ∇¯ J¯ = 0 and J¯ab J¯bc = −δ ac to simplify the above expression. Finally, the Riemann curvature is read off from AB = 21 R ABC D eC ∧ e D .
Infinite Class of Extremal Horizons in Higher Dimensions
61
A calculation yields the following independent components (for clarity we will label the 2n − 1 basis component simply as φ below):
A B ¯ A B B − R0a0b = − 2 δab , R0aφb = 2 2 Jab , R0φ0φ = − 2 , L A L A A B L B
2 B A 2B B A B δab R0φab = 2 2 − − , (98) J¯ab , Raφbφ = L A A B A4 AB L2 ¯
1 Rabcd A2 B2 ¯ ¯ Rabcd = 2 . Ja[c Jd]b − J¯ab J¯cd + 2 − δ δ + a[c d]b 2 2 4 L A A A From here it is straightforward to calculate the Ricci tensor (16). A.2. Conformal Kähler structure. In this subsection we denote x-derivatives by d/d x = . Define the two-forms J± ≡ L 2 A2 J¯ ± e0 ∧ e2n−1 = L 2 [A2 J¯ ± d x ∧ (dφ + σ )]
(99)
which satisfy J± AB J± BC = −δ BA (i.e. they are almost complex structures). One can show that
2 2A ∓ 2 d x ∧ J± . (100) d J± = A A Since J¯ is the Kähler form on the base K , the volume form ¯ = J¯n−1 /(n − 1)! and thus J¯n = 0. It follows that J±n = ±n(L 2 A2 )n−1 J¯n−1 ∧ e0 ∧ e2n−1 = ±n! ,
(101)
where is the volume form of (H, γ ). Now, since K is an n − 1-dimensional complex manifold there is an (n − 1, 0) form on K such that J¯n−1 = i n−1 (−1)(n−1)(n−2)/2 ∧ ¯ where ¯ is the complex conjugate of . This allows one to write J±n = , n n(n−1)/2 ¯ ± , where ± ∧ i (−1) n 2 2 n−1 ± = (L A ) 2 ∧ e0 ± ie2n−1 . (102) 2 One can check that d± =
(An−1 B) d x ∧ ± . An−1 B
(103)
From these local properties it follows that H is a complex manifold with complex structure specified by ± , and furthermore (H, γ ) is conformally Kähler (see [46] for a similar argument). In fact from (102) we may read off a set of complex coordinates for H. First let z i be a ¯ where k is the Kähler set of complex coordinates for K . Then note that σ = −i(∂ − ∂)k, potential of K . However since ∂k is a (1, 0)-form on K we must have ∧ ∂k = 0. This allows us to write n 2 2 n−1 ± = (L A ) 2 B ∧ dw ± , (104) 2
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where w ± = ±[iφ − k(z i , z¯ i )] +
x
dx . B2
(105)
Thus Z I = (w ± , z i ) are two different sets of complex coordinates on H. Now let us consider our explicit solution. Using A2 = (1 − x 2 ) we get d J± = ∓
2 d x ∧ J± , 1∓x
(106)
and therefore Jˆ± =
J± (1 ∓ x)2
(107)
is closed. The 2-form Jˆ± is a Kähler form and the associated Kähler metric is ± = γˆ AB
γ AB . (1 ∓ x)2
(108)
It is useful to have an expression for the Ricci form of this Kähler metric. Recall that in general for a Kähler metric gμ¯ν , in some set of complex coordinates z μ , the Ricci form is ρ = −i∂ ∂¯ log det(gμ¯ν ) = db,
b=
i ¯ log det(gμ¯ν ), (∂ − ∂) 2
(109)
¯ Therewhere the expression in terms of b follows from the identity ∂ ∂¯ = − 21 d(∂ − ∂). fore one simply needs det gμ¯ν in a set of complex coordinates. For the case at hand, using (104) one can calculate (and thus det γ from = ±i n det(γ Z I Z¯ J ) dw ± ∧ d w¯ ± ∧ dz 1 ∧ d z¯ 1 · · · ∧ dz n−1 ∧ d z¯ n−1 ) in the coordinates Z I = (w ± , z i ). Taking care of the conformal rescaling it follows that det(γˆZ±I Z¯ j ) = and therefore
(L 2 A2 )n−1 B 2 det(g¯i j¯ ), 2(1 ∓ x)2n
2(n−1) B 2 ∂ i A ¯ b± = b¯ + log dw ± ± (∂ − ∂)k 2 ∂w ± (1 ∓ x)2n i ∂ A2(n−1) B 2 ± ¯ − , log d w¯ ∓ (∂ − ∂)k 2 ∂ w¯ ± (1 ∓ x)2n
(110)
(111)
± where ρˆ ± = db± and ρ¯ = d b¯ are the Ricci forms of γˆ AB and g¯ ab respectively. Note that to perform this calculation one must take care of the fact that by definition the complex coordinates w ± are shifted by a function of (z i , z¯ i ) (i.e. k): while this does not change the ∂/∂w ± derivatives it does shift
∂ ∂ ∂ ∂ . (112) → ± ∂ k + i ∂z i ∂z i ∂w ± ∂ w¯ ±
Infinite Class of Extremal Horizons in Higher Dimensions
63
Noting that ∂ B2 ∂ i ∂ = ∓ , ± ∂w 2 ∂x 2 ∂φ
(113)
the final two terms of (111) can then be written back in our real coordinates: B2 d A2(n−1) B 2 ± ¯ b =b∓ (dφ + σ ). log 2 dx (1 ∓ x)2n
(114)
Since the base is Kähler-Einstein ρ¯ = 2n J¯ and therefore locally b¯ = nσ up to an exact 1-form – we will choose this to be such that b¯ = n(dφ + σ ) and therefore B2 d A2(n−1) B 2 ± (dφ + σ ). (115) b = n∓ log 2 dx (1 ∓ x)2n Using our explicit form for B 2 finally gives the Ricci form of (H, γˆ ± ) ρˆ ± = d [ f (x)(dφ + σ )] , where f (x) = n ∓
1 2(1 − x 2 )n−1 (ξ + x 2 )
P −
2n P 2x P ± 2 (ξ + x ) 1 ∓ x
(116)
.
(117)
B. Topology of Hm, p In the main text we established that our horizon metrics extend smoothly onto compact Hm, p , and that the resulting Hm, p is the total space of an S 2 -bundle over a compact Kähler-Einstein manifold K . Note that the structure group for such bundles is at most S O(3). In this section we will discuss the topology of these bundles in more detail. To avoid clutter we will refer to Hm, p simply as H. One of the most fundamental topological invariants of a manifold is the fundamental group. Since the fibre of the bundle H over K is S 2 , and S 2 is simply connected, it follows that π1 (H) ∼ = π1 (K ).20 Furthermore since any closed Kähler manifold with positive definite Ricci tensor must be simply connected [48] we must have π1 (K ) = 0 (recall K is positive Einstein). Therefore π1 (H) = 0.
(118)
From this we immediately deduce that H1 (H) = 0 and by Poincaré duality the free part of H2n−1 (H) = 0. In fact we can easily deduce the whole cohomology ring H ∗ (H) as follows. First note that there is a closed global 2-form J+ /(1 − x)2 on H, see Eq. (99), whose restriction to each fiber S 2 generates the cohomology of the fiber. Then, by the Leray-Hirsch theorem [47], the cohomology of H is H (H) = H (K ) ⊗ H (S 2 ).
(119)
20 For any fibre bundle F → E → B, where F is the fibre, E the total space and B the base, exactness of its homotopy sequence · · · → π1 (F) → π1 (E) → π1 (B) → π0 (F) → · · · , implies that if π0 (F) = π1 (F) = 0, then π1 (E) is isomorphic π1 (B), see e.g. [33,47].
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This allows us to deduce H k (H, R) =H k (K , R) ⊕ H k−2 (K , R) for k ≥ 2. Now, recall 2n that the Euler characteristic χ (H) = i=0 (−1)i bi (H), where bi (H) = dim H i (H, R) are the Betti numbers of H. Since bk (H) = bk (K ) + bk−2 (K ) for 2 ≤ k ≤ 2n − 2, and b1 = b2n−1 = 0 from above, we deduce χ (H) = 2χ (K )
(120)
for any choice of K . For later use we now construct an explicit basis for the second integral homology groups H2 (H, Z) in terms of that of H2 (K , Z), following [46]. Since π1 (K ) = 0 we have H 2 (K , Z) ∼ = Zr for some r (i.e. the torsion vanishes), and therefore we can find 2-cycles i for i = 1, . . . , r such that the homology classes [i ] form a basis for the free part of H2 (K , Z). Now define a submanifold ∼ = S 2 of H corresponding to the fibre of H at some fixed point on K . Also define the global section s : K → H, of π : H → K , by the property that each base point in K is mapped to the pole x = x2 of the fibre S 2 . Therefore {, si } forms a representative basis for the free part of H2 (H, Z). Observe that we have H 2 (H, Z) = Z ⊕ H 2 (K , Z) which agrees with the general result derived above. Next we note that H is a complex manifold – this is proved in Appendix (A.2). We can introduce complex coordinates (W, z i ), where W = exp(2π w+ /φ) with x dx w+ = iφ − k(z, z¯ ) + (121) B2 and z i a set of complex coordinates inherited from K , where k is the Kähler potential of K . Note that W is well defined everywhere except at the pole x = x2 . Furthermore W = 0 at the other pole x = x1 = −x2 . To see these x facts2one simply has to note that B 2 ∼ ( const )2 (x22 − x 2 ) as x 2 → x22 and thus d x/B → ∞ as x → x2 and x d x/B 2 → −∞ as x → −x2 . Thus W is a coordinate on the CP1 ∼ = S 2 fibre of Hm, p which covers everywhere except the x = x2 pole. Hence 1/W is a coordinate on the fibre which covers everywhere except the other pole x = x1 . Using the complex structure we can now relate our bundle H to a standard one. Let L denote the canonical line bundle over K (i.e. the holomorphic line bundle of (n − 1, 0) forms over K ). By adding a point to each fibre C one has an associated CP1 ∼ = S2 bundle over K , such that the U (1) subgroup of transitions functions of L acts isometrically on CP1 . For this canonical S 2 bundle (φ)c = 2π/n (see e.g. [46]). In our case we have φ = ( p/m)(φ)c . Therefore the coordinate on the CP1 fibre of Hm, p m/ p can be written as W = Wc , where Wc = exp(2π w+ /(φ)c ) is a coordinate on the 1 CP fibre of L. It follows that our S 2 bundles may be thought of as being associated ⊗m to the (m/ p)th power of the canonical line bundle, i.e. L p . Once again the association involves simply adding a point to each fibre C and taking the same U (1) transition function such that they act isometrically on the S 2 fibre. These bundles are always well defined since L1/ p is always well defined (e.g. for CPn−1 this is the tautological bundle). An easy way to see this is from Eq. (54). The line bundle L1/ p may be defined by < c1 (L1/ p ), [i ] >= p −1 < c1 (L), [i ] >= − p −1 < c1 (K ), [i ] >= −n i ∈ Z. As in [46], we may therefore write Hm, p ∼ = Lm/ p ⊗U (1) CP1 , which means take the same U (1) transition functions as for Lm/ p and use them to construct an associated CP1 ∼ = S2 bundle with the U (1) acting isometrically on the fibre. It is also worth noting that our bundle can be written as a projectivised bundle.
Infinite Class of Extremal Horizons in Higher Dimensions
65
We are interested in determining the conditions for H to be a spin manifold. Recall this requires one to compute the second Stiefel-Whitney class w2 (H). Since T H is a complex vector bundle we can use the fact that w2 (H) is the mod 2 reduction of the first Chern class c1 (H). Now we remark that our (H, γ ) is a conformally Kähler manifold – this property is shown in the Appendix A.2. For definiteness we will choose the Kähler metric to be γˆ AB = (1−x)−2 γ AB (see Appendix A.2). Then, since the Chern class c1 (H) (i.e. the first Chern class of the complex tangent bundle T H) is a topological invariant, it is in fact easier to calculate this using the Kähler metric γˆ AB . This is a standard result for Kähler manifolds and is given by c1 (H) = [ρ/2π ˆ ] where ρˆ is the Ricci form of γˆ AB . The Ricci form ρˆ is given by (116). We can now evaluate the first Chern class evaluated on our basis of the free part of H2 (H, Z): 1 φ c1 (H), [] = ( f (x2 ) − f (x1 )) = 2, ρˆ = 2π 2π 1 f (x2 ) c1 (H), s∗ [i ] = ρˆ = 2J = n i (m + p), 2π si 2π i
(122) (123)
where the second equality follows from the form of f (x) (117), x1 = −x2 , the identity f (±x2 ) = n ±
2π φ
(124)
(which can be derived using (45) and (53)), Eq. (55) and (57). Note that n i ∈ Z are the same integers as in (54) and satisfy gcd(n i ) = 1. We are now in a position to deduce the second Stiefel-Whitney class using w2 (H) = c1 (H) mod 2. This means that evaluating w2 (H) on H2 (H, Z) gives a set of integers which are the same mod 2 as evaluating c1 (H) on H2 (H, Z). Therefore we deduce that w2 (H) is trivial, and hence H is a spin manifold, if and only if m + p is even. Note that this does not depend on whether w2 (K ) is trivial which occurs if and only if p is even (see (54)). Finally, we show that for n ≥ 3 the bundle S 2 → H → K is never trivial. This is in contrast to the case n = 2, which as we saw is the trivial bundle S 2 × S 2 if m is even. To prove this we will use the fact that S 2 -bundles over a compact manifold are partially classified by the first Pontryagin class of the associated R3 -bundle (constructed with the same S O(3) transition functions). Explicitly, Hm, p may be thought of as the unit sphere bundle in V3 = IR ⊕ L−m/ p , where IR is the trivial real line bundle over K . Therefore we need p1 (V3 ) = −c2 (V3 ⊗ C) ∈ H 4 (K , Z), where V3 ⊗ C = IC ⊕ L−m/ p ⊕ Lm/ p and IC is the trivial complex line bundle over K . The total Chern class c(V3 ⊗C) = c(L−m/ p )c(Lm/ p ) = (1−c1 (Lm/ p ))(1+c1 (Lm/ p )) = 1−c1 (Lm/ p )2 . It follows that p1 (V3 ) = m 2 c1 (L)2 / p 2 = m 2 c1 (K )2 / p 2 . Since K is Kähler-Einstein, c1 (K ) = n[J ]/π , and thus finally we have p1 (V3 ) =
m 2n2 [J ]2 . π 2 p2
(125)
This immediately implies that p1 (V3 ) = 0 (since for our solutions m = 0) and therefore all our bundles are non-trivial – of course this argument only works for n ≥ 3. In fact we may go further and define a topological invariant for any S 2 -bundle over a compact
66
H. K. Kunduri, J. Lucietti
manifold, by the scalar quantity21 ρ(H) ≡
p1 (V3 ) ∧ J n−3 = K
m 2 nn! vol(K ), p2 π 2
(126)
which of course is only defined for n ≥ 3. This number is an invariant of our bundles Hm, p – in particular any two S 2 -bundles over K are not homeomorphic if the invariant ρ just defined is different. We deduce that for different m > 0 the manifolds Hm, p are not homeomorphic. This is in marked contrast to the n = 2 case. Thus since we can have m > p we in fact have an infinite discrete family of topologies (this is in contrast to the compact Einstein spaces in [18] which have m < p). C. Inhomogeneous CPn Horizon with Conical Singularity Here we perform the global analysis of Case II in Table 1, and wlog take −1 < x1 ≤ x ≤ 1 with P(x1 ) = 0 and P(1) = 0 and P(x) > 0 for x1 < x < 1, so we must have
= 1. This form of P(x) clearly requires P (x1 ) > 0 and comparing to Eq. (45) implies x1 < 0. One can solve the constraint P(1) = 0 to get √ n! π(2n − 1 − λ) c = c∗ = − (ξ − ξ∗ ), (127) (n + 21 ) where ξ∗ is given by (47). Substituting back gives22 P(x) = (1 − x)n R(x),
(128)
where R(x) is a polynomial of order n + 2 which we do not need explicitly for our analysis below. However, we do require the polynomial R(x) to have a root −1 < x1 < 0 such that R(x) > 0 for x1 < x < 1. The horizon metric in this case is (ξ + x 2 )(1 + x)n−1 d x 2 (1 − x)R(x) 2 + (1 − x 2 )g¯ . + γ AB d x A d x B = L 2 (dφ + σ ) (1 − x)R(x) (ξ + x 2 )(1 + x)n−1
(129) Let us now examine regularity of this metric which is clearly non-degenerate and smooth for x1 < x < 1. Near x = 1 set x = 1 − η2 and expand for small η. To leading order one gets ¯ , (130) γ AB d x A d x B ∼ 2L 2 dη2 + η2 ((dφ + σ )2 + g) where we have used the identity23 R(1) = 2n (1 + ξ ). Therefore smoothness at η = 0 requires φ = 2π and g¯ to be the Fubini-Study metric on CPn−1 . At the other endpoint 21 We thank James Sparks for pointing this out. 22 Note that this form for P(x) is guaranteed by (32) which allows one to show that if P(1) = 0 then P (m) (1) = 0 for 1 ≤ m ≤ n − 1. 23 This may be obtained by differentiating (128) n times, evaluating the result at x = 1, and comparing to the expression obtained from computing d n P(x)/d x n directly from (32).
Infinite Class of Extremal Horizons in Higher Dimensions
67
x1 we have A2 > 0 and thus we have a bolt. Smoothness requires the conical singularity at x = x1 to be removed. The condition for this is (46) (recall x1 < 0 in this case) φ =
4π |x1 | , 2n − λ(1 − x12 )
(131)
and therefore since we have already shown φ = 2π , smoothness requires |x1 | = n −
λ (1 − x12 ). 2
(132)
Recall that the root x1 must satisfy −1 < x1 < 0. Therefore for λ ≤ 0 the regularity condition (132) implies x1 ≤ −n which is a contradiction. Therefore for λ ≤ 0 the above metric necessarily is singular (either at x = 1 of x = x1 ). It can be thought of as a metric on CPn with a conical singularity at the bolt. It is possible that for λ > 0 this metric can be made smooth – to prove this one needs to show that R(x) has a root x1 such that −1 < x1 < 1, R(x) > 0 for x1 < x ≤ 1 and (132) is satisfied. Then the horizon metric would be a smooth and inhomogeneous metric on CPn . We will not pursue this here. D. Computation of “Internal” Angular Momenta on K In this section we compute the Komar integral (71) associated to the U (1) Killing vector fields m¯ i = ∂/∂φ i for i = 1, . . . , n − 1 for K toric. Explicitly this is √ L 2n φ ξ x2 P(x) ¯ Ji = ± d x (σ · m¯ i ) , ¯ (133) 2 2 8π G −x2 (ξ + x ) K where ¯ is the volume form on K . The calculation thus reduces to evaluation of the integrals Ii ≡ (σ · m¯ i ) ¯ . (134) K
First observe that this integral is actually well defined despite σ not being a globally defined object on K . To see this recall the Kähler form J = 21 dσ and Lm¯ i J = 0. It is therefore possible to choose a gauge for σ such that Lm¯ i σ = 0. Any residual gauge transformations σ → σ +dλ must satisfy Lm¯ i dλ = 0 which is equivalent to the function m¯ i · dλ = const . However this constant must vanish since m¯ i each vanish somewhere on K . This proves that Ii is gauge invariant and thus well defined. We will now do an explicit calculation for K = CPn−1 and also give a more general argument for toric K . D.1. K = CPn−1 . The calculation thus reduces to evaluation of the integrals Ii ≡ (σ · m¯ i ) ¯ . CPn−1
(135)
As noted above Ii is gauge invariant. Therefore it may be computed by choosing an open covering of CPn−1 and working in a gauge there σ is smooth in each open set. Now recall the standard open cover of CPn−1 which consists of n patches Uk = {Z k = 0} with k = 1, . . . , n and Z k the usual homogeneous coordinates. In each patch
68
H. K. Kunduri, J. Lucietti (k)
Uk we can introduce inhomogeneous coordinates z i = Z i /Z k for i = k. Let us work (n) in one patch, say Un , and set z i = z i for i = 1, . . . , n − 1. The Fubini-Study metric in such a patch is given by 2 g¯ ab d x¯ a d x¯ b = dn−1 =
z¯ i z j dz i d z¯ j dz i d z¯ i − , f f2
(136)
where f = 1 + z i z¯ i and i, j = 1, . . . , n − 1 and we are summing over repeated indices (this metric satisfies Ric(g) ¯ = 2n g). ¯ The Kähler form is i i i dσ , σ = z d z¯ − z¯ i dz i . (137) J= 2 2f There are a number of ways to introduce real coordinates in order to identify the rotai tional Killing fields m¯ i . The simplest for our purposes is to set z i = ri eiφ . The vector fields m¯ i = ∂/∂φ i have closed orbits with period 2π and generate the U (1)n−1 isometry subgroup, and 0 ≤ ri ≤ ∞. Then (136) becomes ⎡ 2 n−1 2 ⎤ n−1 n−1 1 1 2 dn−1 = dri2 + ri2 (dφ i )2 − 2 ⎣ ri dri + ri2 dφ i ⎦ , (138) f f i=1
where f = 1 +
n−1
2 i=1 ri .
i=1
i=1
It follows that
!n−1 g¯ =
i=1 ri fn
.
(139)
We also have σ =
n−1 1 2 ri dφi . f
(140)
i=1
Note this is in a gauge which ensures σ is smooth at ri = 0. Putting this together gives ! ∞ ∞ ∞ ri3 n−1 j=1, j =i r j (n) n−1 (σ · m¯ i ) ¯ = (2π ) dr1 dr2 . . . drn−1 Ii ≡ f n+1 Un 0 0 0 (2π )n−1 ∞ ri3 dri π n−1 (141) = n−3 = 2 n! 0 (1 + ri2 )3 n! by repeated integration. (k) Now, it is clear that the analogous integrals Ii in the patches Uk for k = 1, . . . , n −1 give the same value. However, it is not the case that adding all these together gives Ii , since the overlaps Ui ∩ U j = 0. In fact there is a trick to avoid this complication. Instead one can work in a gauge which is singular in every patch, in such a way that performing the integral in any patch gives the correct total answer. For the case at hand this gauge is given by n−1 ri2 1 σ = − (142) dφi , f n i=1
Infinite Class of Extremal Horizons in Higher Dimensions
which gives (n)
Ii = Ii
−
69
1
¯ = 0, n CPn−1
(143)
π where in the last equality we have used vol(CPn−1 ) = (n−1)! . Hence as expected, the conserved angular momenta associated with the ‘internal’ rotational Killing fields on CPn−1 vanish. n−1
D.2. Toric K . In this section we generalise the calculation of the previous section to cover the general case of when K is a toric manifold. As we will see, we do not actually need the explicit metric in order to calculate the internal angular momenta, just the toric data. We will employ well known constructions of toric symplectic geometry. First recall that for a 2(n − 1) dimensional toric Kähler manifold one may introduce symplectic coordinates (x i , φ i ) for i = 1, . . . n − 1 such that the metric is g¯ ab d x¯ a d x¯ b = G i j (x)d x i d x j + G i j (x)dφ i dφ j ,
(144)
where G i j is the matrix inverse of G i j , and G i j = ∂ 2 g/∂ x i ∂ x j , where g is called the symplectic potential. The symplectic form is the Kähler form which is J = d x i ∧ dφ i ,
(145)
so in the language of symplectic geometry these are Darboux coordinates. The coordinates ranges are φ i ∼ φ i + 2π , so that the Killing vector fields m¯ i = ∂/∂φ i generate the toric symmetry U (1)n−1 . By a classic result the x i are coordinates which lie in a so called Delzant polytope . This is a subset of Rn−1 defined by the intersection of a set of linear inequalities = {x : (va · x + λa ) ≥ 0, ∀a}, where a labels the faces of the polytope and va is the normal vector to each face such that va form a basis for Zn−1 . Note that symplectic coordinates are not unique. In particular x i → M i j x j , where M ∈ G L(n − 1, Z) and x i → x i + k i are both freedoms. The polytope is then invariant under a subgroup of these transformations. As in the previous section the calculation of the internal angular momenta reduces to Ii = (σ · m¯ i ) ¯ . (146) K
In symplectic coordinates we can always choose a gauge such that σ = 2(x i + ci )dφ i for some constants √ g¯ = 1. Therefore
ci .
(147)
Also note that the volume form in these coordinates is trivial so
Ii = 2(2π )n−1
(x i + ci ) d x 1 · · · d x n−1 .
(148)
It is then clear that we can always pick a gauge such that Ii = 0 for all 1 ≤ i ≤ n − 1, i.e. ci = −vol()−1 x i d x 1 · · · d x n−1 . By the same reasoning as in the previous section, since Ii is gauge invariant, if it vanishes in every coordinate patch then it must vanish everywhere. Thus we deduce that for general toric K the internal angular momenta vanish.
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References 1. Hawking, S.W.: Black holes in general relativity. Commun. Math. Phys. 25, 152 (1972) 2. Chrusciel, P.T., Wald, R.M.: On The Topology Of Stationary Black Holes. Class. Quant. Grav. 11, L147 (1994) 3. Emparan, R., Reall, H.S.: Black Holes in Higher Dimensions. Living Rev. Rel. 11, 6 (2008) 4. Galloway, G.J., Schoen, R.: A generalization of Hawking’s black hole topology theorem to higher dimensions. Commun. Math. Phys. 266, 571 (2006) 5. Myers, R.C., Perry, M.J.: Black Holes In Higher Dimensional Space-Times. Annal. Phys. 172, 304 (1986) 6. Emparan, R., Reall, H.S.: A rotating black ring in five dimensions. Phys. Rev. Lett. 88, 101101 (2002) 7. Pomeransky, A.A., Sen’kov, R. A.: Black ring with two angular momenta. http://arxiv.org/abs/hep-th/ 0612005v1, 2006 8. Reall, H.S.: Higher dimensional black holes and supersymmetry. Phys. Rev. D 68, 024024 (2003) [Erratum-ibid. D 70, 089902 (2004)] 9. Milnor, J.W., Stasheff, J.D.: Characteristic classes. Princeton, NJ: Princeton University Press 1974 10. Wall, C.T.C.: Determination of the cobordism ring. Ann. Math. Second Series 72, 292–311 (1960) 11. Galloway, G.J., Schleich, K., Witt, D.M., Woolgar, E.: Topological Censorship and Higher Genus Black Holes. Phys. Rev. D 60, 104039 (1999) 12. Milnor, J.: A Procedure for killing homotopy groups of differentiable manifolds. Proc. Sympos. Pure Math. Vol. III Providence, RI: Amer. Math. Soc., 1961, pp. 39–55 13. Chrusciel, P.T., Reall, H.S., Tod, P.: On non-existence of static vacuum black holes with degenerate components of the event horizon. Class. Quant. Grav. 23, 549 (2006) 14. Kunduri, H.K., Lucietti, J., Reall, H.S.: Near-horizon symmetries of extremal black holes. Class. Quant. Grav. 24, 4169 (2007) 15. Besse, A.L.: Einstein Manifolds. Berlin-Heidelberg-New York: Springer-Verlag, 2nd edition, 1987 16. Page, D.N.: A Compact Rotating Gravitational Instanton. Phys. Lett. B 79, 235 (1978) 17. Gibbons, G.W., Ida, D., Shiromizu, T.: Uniqueness of (dilatonic) charged black holes and black p-branes in higher dimensions. Phys. Rev. D 66, 044010 (2002) 18. Page, D.N., Pope, C.N.: Inhomogeneous Einstein Metrics On Complex Line Bundles. Class. Quant. Grav. 4, 213 (1987) 19. Hollands, S., Ishibashi, A., Wald, R.M.: A Higher Dimensional Stationary Rotating Black Hole Must be Axisymmetric. Commun. Math. Phys. 271, 699 (2007) 20. Moncrief, V., Isenberg, J.: Symmetries of Higher Dimensional Black Holes. Class. Quant. Grav. 25, 195015 (2008) 21. Hajicek, P.: Three remarks on axisymmetric stationary horizons. Commun.Math. Phys. 36, 305–320 (1974) 22. Lewandowski, J., Pawlowski, T.: Extremal Isolated Horizons: A Local Uniqueness Theorem. Class. Quant. Grav. 20, 587 (2003) 23. Kunduri, H.K., Lucietti, J.: A classification of near-horizon geometries of extremal vacuum black holes. J. Math. Phys. 50, 082502 (2009) 24. Kunduri, H.K., Lucietti, J.: Uniqueness of near-horizon geometries of rotating extremal AdS(4) black holes. Class. Quant. Grav. 26, 055019 (2009) 25. Amsel, A.J., Horowitz, G.T., Marolf, D., Roberts, M.M.: Uniqueness of Extremal Kerr and Kerr-Newman Black Holes. Phys. Rev. D 81, 024033 (2010) 26. Figueras, P., Lucietti, J.: On the uniqueness of extremal vacuum black holes. Class. Quant. Grav. 27, 095001 (2010) 27. Chrusciel, P.T., Nguyen, L.: A uniqueness theorem for degenerate Kerr-Newman black holes. http://arciv. org/abs/1002.1737v1 [gr-qc], (2010) 28. Hollands, S., Ishibashi, A.: All vacuum near horizon geometries in D-dimensions with (D − 3) Commuting Rotational Symmetries. Ann. H. Poincaré 10, 1537–1557 (2010) 29. Figueras, P., Kunduri, H.K., Lucietti, J., Rangamani, M.: Extremal vacuum black holes in higher dimensions. Phys. Rev. D 78, 044042 (2008) 30. Mann, R.B., Stelea, C.: New multiply nutty spacetimes. Phys. Lett. B 634, 448 (2006) 31. Houri, T., Oota, T., Yasui, Y.: Generalized Kerr-NUT-de Sitter metrics in all dimensions. Phys. Lett. B 666, 391 (2008) 32. Nash, J.: Positive Ricci Curvature on Fibre Bundles. J. Diff. Geom. 14, 241–254 (1979) 33. Stenrod, N.: The topology of fibre bundles. Princeton, NJ: Princeton University Press 1951 34. Gauntlett, J.P., Martelli, D., Sparks, J., Waldram, D.: Sasaki-Einstein metrics on S(2) x S(3). Adv. Theor. Math. Phys. 8, 711 (2004) 35. Tian, G.: On Calabis conjecture for complex surfaces with positive first Chern class. Invent. Math. 101, 101–172 (1990)
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36. Doran, C., Headrick, M., Herzog, C.P., Kantor, J., Wiseman, T.: Numerical Kaehler-Einstein metric on the third del Pezzo. Commun. Math. Phys. 282, 357 (2008) 37. Hollands, S., Ishibashi, A.: On the ‘Stationary Implies Axisymmetric’ Theorem for Extremal Black Holes in Higher Dimensions. Commun. Math. Phys. 291, 403 (2009) 38. Emparan, R., Harmark, T., Niarchos, V., Obers, N.A.: New Horizons for Black Holes and Branes. http:// arxiv.org/abs/0912.2352v3 [hep-th], 2010 39. Dias, O.J.C., Figueras, P., Monteiro, R., Reall, H.S., Santos, J.E.: An instability of higher-dimensional rotating black holes. JHEP 1005, 076 (2010) 40. Kunduri, H.K., Lucietti, J., Reall, H.S.: Gravitational perturbations of higher dimensional rotating black holes: Tensor Perturbations. Phys. Rev. D 74, 084021 (2006) 41. Kunduri, H.K., Lucietti, J., Reall, H.S.: Do supersymmetric anti-de Sitter black rings exist? JHEP 0702, 026 (2007) 42. Kunduri, H.K., Lucietti, J.: Near-horizon geometries of supersymmetric AdS(5) black holes. JHEP 0712, 015 (2007) 43. Caldarelli, M.M., Emparan, R., Rodriguez, M.J.: Black Rings in (Anti)-deSitter space. JHEP 0811, 011 (2008) 44. Chong, Z.W., Cvetic, M., Lu, H., Pope, C.N.: General non-extremal rotating black holes in minimal five-dimensional gauged supergravity. Phys. Rev. Lett. 95, 161301 (2005) 45. Kunduri, H.K., Lucietti, J., Reall, H.S.: Supersymmetric multi-charge AdS(5) black holes. JHEP 0604, 036 (2006) 46. Gauntlett, J.P., Martelli, D., Sparks, J.F., Waldram, D.: A new infinite class of Sasaki-Einstein manifolds. Adv. Theor. Math. Phys. 8, 987 (2006) 47. Bott, R., Tu, L.: Differential Forms in Algebraic Topology. Berlin-Heidelberg-New York: Springer-Verlag 1982 48. Kobayashi, S.: On Compact Kähler Manifolds with Positive Definite Ricci tensor. Ann. Math. Second Series 74, 570–574 (1961) Communicated by P.T. Chru´sciel
Commun. Math. Phys. 303, 73–87 (2011) Digital Object Identifier (DOI) 10.1007/s00220-011-1198-9
Communications in
Mathematical Physics
Completely Integrable Hamiltonian Systems with Weak Lyapunov Instability or Isochrony Gaetano Zampieri Università di Verona, Dipartimento di Informatica, Strada Le Grazie, 15, I-37134 Verona, Italy. E-mail:
[email protected] Received: 15 March 2010 / Accepted: 19 October 2010 Published online: 2 February 2011 – © Springer-Verlag 2011
Dedicated to Angelo Barone Netto for his 75th birthday Abstract: The aim of this paper is to introduce a class of Hamiltonian autonomous systems in dimension 4 which are completely integrable and their dynamics is described in all details. They have an equilibrium point which is stable for some rare elements of the class, and unstable in most cases. Anyhow, it is linearly stable (all orbits of the linearized system are bounded) and no motion is asymptotic in the past, namely no nonconstant solution has the equilibrium as limit point as time goes to minus infinity. In the unstable cases, there is a sequence of initial data which converges to the equilibrium point whose corresponding solutions are unbounded and the motion is slow. So instability is quite weak and perhaps no such explicit examples of instability are known in the literature. The stable cases are also interesting since the level sets of the 2 first integrals independent and in involution keep being non-compact and stability is related to the isochronous periodicity of all orbits near the equilibrium point and the existence of a further first integral. Hopefully, these superintegrable Hamiltonian systems will deserve further research. 1. Introduction Let us have a quick look of our main results, the details and the proofs are in the paper. We introduce the Hamiltonian system in R4 , ⎧ q˙ = p2 ⎪ ⎨ 1 q˙2 = p1 (1.1) ⎪ ⎩ p˙ 1 = −g (q1 ) q2 p˙ 2 = −g(q1 ), where g is a C 1 function near 0 and satisfies g(0) = 0 and g (0) > 0. The Hamiltonian function for the system is H (q, p) = p1 p2 + g(q1 ) q2 , Supported by the PRIN 2007 directed by Fabio Zanolin.
(1.2)
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G. Zampieri
and another first integral is K (q, p) =
p22 + V (q1 ), V (q1 ) = 2
q1
g(s) ds.
(1.3)
0
The q1 , p2 -plane is invariant and the origin is a center on it, namely an open neighborhood C of (0, 0) in this plane is the union of periodic orbits which enclose (0, 0). We restrict the phase space to M = {(q1 , q2 , p1 , p2 ) ∈ R4 : (q1 , p2 ) ∈ C, (q2 , p1 ) ∈ R2 }, so we have a global center in the q1 , p2 -plane. The Hamiltonian vector fields ∇ H (q, p) in (1.1) and ∇ K (q, p) are both complete on M (with skew-symmetric matrix, see (4.4)). The functions H, K are in involution. The vectors ∇ H (q, p), ∇ K (q, p) are linearly independent at each point of the set N := {(q, p) ∈ M : (q1 , p2 ) = (0, 0)} which is invariant for both the Hamiltonian vector fields ∇ H and ∇ K . A nonempty component ⊂ N of a level set of H, K is diffeomorphic to S1 × R and there are coordinates φ mod 2π and z on S1 × R such that the differential equations defined by the vector field ∇ H on take the form φ˙ = ω,
z˙ = v
(ω, v = const).
(1.4)
If (x0 , 0) belongs to the projection of on the q1 , p2 -plane and T (x0 ) is the period of the first component of any integral curve of ∇ H on , then v = 0 if and only if T (x0 ) = 0. A necessary and sufficient condition for the stability of the origin of (1.1) is that all periodic orbits in a neighborhood of (0, 0) in the q1 , p2 -plane have the same period so the center is (locally) isochronous. In this case all orbits of (1.1) in R4 , with (q1 , p2 ) near (0, 0), are periodic and have the same period. Moreover, a further first integral appears as we shall see using a result of Barone and Cesar [4]. This happens for instance for the following two functions (see (5.7) and (5.10)): g(x) = 1 − √
1 1+x
,
g(x) = 1 + x −
1 . (1 + x)3
(1.5)
The corresponding Hamiltonian systems are superintegrable, see Fassò [6], ‘Maximally superintegrable systems’ p. 110, and the Definition at p. 106. In Sect. 5 we show other explicit examples of our superintegrable isochronous systems, more precisely we find all those which have a third quadratic in the momenta first integral. Isochronous systems had been recently studied starting from powerful ideas of Calogero by himself and collaborators, see [3], Françoise [7], and the references therein. There do not seem to be immediate connections of our systems with Calogero’s techniques; however, perhaps the question deserves further work. In Sect. 2 there are results on 2 dimensional isochronous centers taken from Zampieri [12]. We reproduce here the proofs to be self-contained. We relate the functions g, giving isochronous centers, with the even functions and so isochrony, and Lyapunov stability for (1.1), is a rare phenomenon among the class (1.1). Isochronous centers were first studied by Urabe [10] with a different approach. In [12] a new characterization by means of the so called ‘involutions’ h was found which permits to construct the isochronous centers. One of the referees points out that the survey [5] shows my Theorem 2.1 as a classical result and asks to clarify the paternity. Indeed a 1998 preprint is quoted (see Theorem 7.3 in [5]) instead of the 1989 paper [12]. However, Chavarriga & Sabatini’s [5] is a good survey on general isochronous centers, not necessarily of the form considered in this paper (among the vast literature on this field see [9]).
Weak Instability or Isochrony
75
Generally we have instability of the equilibrium for (1.1) and it is very easy to show explicit functions for it (see Sect. 2 Corollary 2.3 and below), for instance g(x) = sin x,
g(x) = αx + βx 2 + γ x 3 ,
(1.6)
with α > 0, (β, γ ) = (0, 0). This kind of instability is quite weak since all orbits of the linearization of the system (1.1) at the origin in R4 are bounded and there are no asymptotic motions to the equilibrium, namely no non-constant solutions with the origin as limit point as t → −∞. In the present paper, instability without asymptotic motions is called weak instability. In our Hamiltonian systems, instability occurs because there is a sequence of initial data which converges to the origin whose corresponding solutions are unbounded and the motion is slow, indeed (1.4) shows that the coordinate z is an affine function of time. Perhaps no such explicit examples of instability are known in the literature. The famous Cherry Hamiltonian system has a linearly stable equilibrium point which is Lyapunov unstable, however it has an asymptotic motion, see [11] p. 412. 2. Isochronous Oscillations Let us start from the following equation: x¨ = −g(x),
g(0) = 0, g (0) > 0,
(2.1)
where g is continuous in a neighborhood of 0 in R and differentiable at 0 (in Sect. 1 the function g was C 1 , while in the present Sect. 2 the existence of g (0) is enough). This o.d.e. has the first integral of energy x x˙ 2 + V (x), V (x) = G(x, x) ˙ = g(s) ds. (2.2) 2 0 By means of this first integral, we easily see that each Cauchy problem for (2.1) has a unique solution if g(x) = 0 only at x = 0. The potential energy V is a C 1 function and there exists V (0) = g (0) > 0 . We can restrict the domain of g to an open interval J 0 such that V is strictly increasing on J ∩ R+ , strictly decreasing on J ∩ R− , and for each point x ∈ J there is a unique point h(x) ∈ J with V (h(x)) = V (x),
sgn(h(x)) = − sgn(x),
where sgn(x) is the sign of x. We check at once that the function u(x) := sgn(x) 2V (x) is a C 1 diffeomorphism onto the image I = u(J ) which is a symmetric interval u (0) = V (0), u ∈ Diff 1 (J ; I ), 0 ∈ J, 0 ∈ I,
y ∈ I ⇒ −y ∈ I.
(2.3)
(2.4)
(2.5)
Moreover, since h(x) = u −1 (−u(x)) we have that h is also a diffeomorphism: h ∈ Diff 1 (J ; J ), h(0) = 0,
h (h(x)) = x,
h (0) = −1.
(2.6)
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We call h the involution associated with V . Remark that the graph of h is symmetric with respect to the diagonal which intersects at the origin; indeed (x, h(x)) has (h(x), x) as the symmetric point and this coincides with the point (h(x), h(h(x))) of the graph. Theorem 2.1 (Determining isochronous centers). Let V be a C 1 function near 0 in R with V (0) = V (0) = 0, assume there exists V (0) > 0, let J be an open interval as above and let h ∈ Diff 1 (J ; J ) be the involution associated with V . Then all orbits of x¨ = −V (x) which intersect the J interval of the x-axis in the x, x-plane, ˙ are periodic and enclose (0, 0). Moreover, they all have the same period if and only if V (x) =
V (0) (x − h(x))2 , 8
x ∈ J.
(2.7)
In this case we say that the origin is an isochronous center for x¨ = −V (x) . Formula (2.7) corresponds to formula (6.2) in [12], the proof is included in the proof of Proposition 1 in [12] as a particular case. Proof. By composition with the inverse of the diffeomorphism (2.4), the first integral (2.2) gives 2 G u −1 (y), x˙ = x˙ 2 + y 2 . The first part of the thesis follows at once. Now, consider a periodic orbit in the x, x-plane ˙ which intersects the x-axis at x0 ∈ J , x0 > 0, then h(x0 ) < 0 is the other intersection. By the energy conservation we get the period of the orbit as x0 dx . (2.8) T (x0 ) = 2 √ 2 (V (x0 ) − V (x)) h(x0 ) If y0 ∈ I, y0 > 0, x0 = u −1 (y0 ), then h(x0 ) = u −1 (−y0 ), and the period y0 −1 u −1 (y0 ) u (r ) dr dx
=2 T (y0 ) := 2 √ −1 2 (V (x0 ) − V (x)) −y0 u (−y0 ) y02 − r 2 y0 u −1 (r ) + u −1 (−r ) dr
= 2 . 0 y02 − r 2 The change of integration variable s = arcsin(r/y0 ) gives for y0 ∈ I, y0 > 0, π/2 u −1 (y0 sin s) + u −1 (−y0 sin s) ds. T (y0 ) = 2
(2.9)
(2.10)
0
So
lim T (y0 ) = 2π u −1 (0) = 2π/ V (0).
y0 →0+
(2.11)
All orbits have the same period if and only if T (y0 ) equals this limit value for all y0 ∈ I, y0 > 0. By (2.9) this is equivalent to the following condition for all z ∈ I, z > 0: z u −1 (r ) + u −1 (−r ) dr 2π =2 . (2.12) √ √ V (0) z2 − r 2 0
Weak Instability or Isochrony
77
Fig. 1. Changing the order of integration
We multiply both sides by z/ y 2 − z 2 and we integrate from 0 to y ∈ I, y > 0, y π z dz √ V (0) 0 y2 − z2 y z u −1 (r ) + u −1 (−r ) dr z dz = . (2.13) √ z2 − r 2 y2 − z2 0 0 By the change of the integration order, see Fig. 1, we get
y y πy z −1 −1 = dz u (r ) + u (−r ) dr, (2.14) √ √ V (0) y2 − z2 z2 − r 2 0 r =π/2
√
2y = u −1 (y) − u −1 (−y), V (0)
y ∈ I, y > 0.
(2.15)
We deduced this condition from (2.12), and now we see at once that it implies (2.12). By plugging y = u(x) into (2.15) we have 2 u(x) = x − h(x), √ V (0) This condition is equivalent to (2.7).
x ∈ J, x > 0.
(2.16)
In [12], Sect. 6, there is the construction we are going to show and also another one (see from formula (6.3)). From the theorem we easily get the following result. Corollary 2.2 (Constructing isochronous centers). Let h : J → J be a C 1 function on an open interval J ⊆ R containing 0 which satisfies the conditions in (2.6). Let ω > 0 and define V (x) =
ω2 (x − h(x))2 , 8
x ∈ J.
(2.17)
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Then there exists V (0) = ω2 and all orbits of x¨ = −V (x) which intersect the J interval of the x-axis in the x, x-plane, ˙ are periodic and have the same period 2π/ω. To get h as above, we can just consider an arbitrary even C 1 function on a (symmetric) open interval which vanishes at 0, then a π/4 clockwise rotation of its graph gives a curve containing an arc y = h(x) which satisfies (2.6) with J open interval. √ For instance, starting from x → x 2 / 2 we can (even explicitly) calculate h(x) = 1 + x −
√ 1 + 4x,
V (x) =
2 √ ω2 −1 + 1 + 4x . 8
(2.18)
Notice that the original quadratic function is defined on the whole R; √ to get a function after rotation we throw out an unbounded arc and obtain (x, 1 + x − 1 + 4x) with x ≥ −1/4; finally, to satisfy (2.6) with J open interval, we must restrict x to (−1/4, 3/4) or to suitable smaller open intervals. Finally, a simple example is also h(x) = −
x , 1+x
V (x) =
ω2 2 x 8
2+x 1+x
2 .
(2.19)
Now, let us see some necessary conditions to have constant period which can be used to check whether a given function V , or its derivative g, locally gives an isochronous center or not. In the sequel V has as many derivatives as necessary. We saw how h is related to an even function. The even functions have vanishing odd derivatives at 0, so it will not be a surprise to see that the derivatives of the involution h are not arbitrary. Consider the relation h(h(x)) = x, perform 2 derivations and calculate at 0 taking into account h(0) = 0 and h (0) = −1. In this way we get an identity which shows that h (0) can take any value. By 3 derivations and calculating at 0 we get h (3) (0) = −3h (0)2 /2. Going further we see that the even derivatives are free while the odd ones are uniquely determined by the preceding derivatives. Indeed, differentiating n times we get h (n) (h(x))h (x)n + · · · + h (h(x))h (n) (x) = 0, which is true for n = 2 and is proved at once by induction for all n ≥ 2, where the dots represent omitted terms which do not contain the highest order derivative h (n) . So calculating at x = 0 we have h (n) (0)(−1)n +· · ·− h (n) (0) = 0 and for even n the n th order derivative disappears while for odd n we can solve for h (n) (0) in terms of the lower order derivatives. The first 5 terms in Taylor’s formula are given by h(x) = −x + a x 2 − a 2 x 3 + b x 4 + 2a 4 − 3ab x 5 + o(x 5 ), (2.20) where a, b are free parameters. Next, formula (2.7) shows that the derivatives of the isochronous V at 0 are constrained to obey some conditions. By means of (2.20) we can calculate V (4) (0) and V (6) (0) by the lower order derivatives and get 2 necessary conditions. Of course we can go forward to infinite conditions. Corollary 2.3 (Necessary conditions). Let V admit V (6) (0) and satisfy V (0) = V (0) = 0, V (0) > 0. Moreover, let the origin be an isochronous center for x¨ = −V (x) . Then V (4) (0) =
5V (3) (0)2 7V (3) (0)V (5) (0) 140V (3) (0)4 (6) (0) = . , V − 3V (0) V (0) 9V (0)3
(2.21)
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To illustrate the necessary conditions let us use it to prove the well known lack of isochronism of the simple pendulum x¨ + sin x = 0, V (4) (0) = sin(3) (0) = −1 = 0 =
5V (3) (0)2 5 sin(2) (0)2 = . 3 sin (0) 3V (0)
(2.22)
Another example is α 2 β 3 γ 4 x + x + x (α > 0) (2.23) 2 3 4 for which the second condition in (2.21) implies β = 0, and the first condition with β = 0 gives γ = 0 too. We could also prove the above formulas in (2.21) using the approach of Barone, Cesar and Gorni [2], where the first 2 derivatives of the period function T (x0 ) at 0 are computed with a procedure which carries on to higher-order derivatives. With some regularity on V they find the following formulas for V (0) = 1:
π 5 (3) 2 (4) V (0) − V (0) . T (0) = 0, T (0) = (2.24) 4 3 V (x) =
3. The Dynamics in the Lagrangian Framework In this section we deal with the following 4-dimensional system defined by the Lagrangian function L: L(x, y, x, ˙ y˙ ) = x˙ y˙ − g(x) y, x¨ = −g(x),
y¨ = −g (x)y,
g(0) = 0, g (0) > 0, (3.1)
where g ∈ C 1 near 0 in R. This system of differential equations has two first integrals G(x, x) ˙ and F(x, y, x, ˙ y˙ ), x x˙ 2 + V (x), V (x) = G(x, x) ˙ = g(s) ds, (3.2) 2 0 F(x, y, x, ˙ y˙ ) = y˙ x˙ + g(x)y. The first differential equation (3.1) separates and its dynamics was studied in the previous section. We restrict our attention to an open interval J as in Theorem 2.1 and to the orbits in the x, x-plane ˙ which intersect J ; their union is an open neighborhood C of (0, 0). Let us fix x0 ∈ J, x0 > 0, and denote by t → X (t, x0 ) the periodic solution of x¨ = −g(x) with (x0 , 0) as initial condition at time 0. Next, we plug X (t, x0 ) into the second differential equation in (3.1) and get the linear equation with periodic coefficient, Hill’s equation, y¨ = −g (X (t, x0 )) y.
(3.3)
The partial derivatives of X (t, x0 ) give 2 independent solutions of (3.3) as one sees by derivation of the first equation for X (t, x0 ), in particular ∂2 ∂ X ∂ ∂2 X ∂ (t, x ) = (t, x0 ) = (−g (X (t, x0 ))) 0 2 2 ∂t ∂ x0 ∂ x0 ∂t ∂ x0 ∂X = −g (X (t, x0 )) (t, x0 ). ∂ x0
(3.4)
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Let us define φ(t) =
∂X (t, x0 ), ∂ x0
ψ(t) = −
1 ∂X (t, x0 ), g(x0 ) ∂t
(3.5)
so that for all t, ˙ ˙ ˙ φ(t) − φ(t) ˙ ψ(t) = 1. φ(0) = 1, φ(0) = 0, ψ(0) = 0, ψ(0) = 1, ψ(t) (3.6) Let us denote by T (x0 ), briefly τ , the period of t → X (t, x0 ), then t → φ(t + τ ) is also a solution of (3.3), so a linear combination of φ and ψ, and ˙ ) ψ(t) = φ(t) + φ(τ ˙ ) ψ(t), φ(t + τ ) = φ(τ ) φ(t) + φ(τ
(3.7)
where we used φ(τ ) = 1 which comes from the last equality in (3.6) with t = τ if we ˙ ) = 1. Taking the derivative, calculating at t = τ , and notice that ψ(τ ) = 0 and ψ(τ ˙ taking into account ψ(nτ ) = 1, we have ˙ + τ ) = φ(t) ˙ + φ(τ ˙ ) ψ(t) ˙ φ(t
⇒
˙ ˙ ˙ ), φ((n + 1)τ ) = φ(nτ ) + φ(τ
(3.8)
for all n ∈ Z. By induction we have ˙ ˙ ), φ(nτ ) = n φ(τ ∀n ∈ Z. (3.9) ˙ ) = 0 then φ, φ˙ is periodic, otherwise it is unbounded. A necessary and sufficient If φ(τ ˙ ) = 0 namely condition to have the former case is φ(τ ∂2 X (T (x0 ), x0 ) = 0. ∂t∂ x0
(3.10)
To go ahead we need the differentiability of the period function Proposition 3.1. The period T (x0 ) of t → X (t, x0 ) is a C 1 function on {x0 ∈ J : x0 > 0}. Proof. By formula (2.10), for x0 ∈ J, x0 > 0, we have π/2 u −1 (u(x0 ) sin s) + u −1 (−u(x0 ) sin s) ds. T (x0 ) = 2
(3.11)
0
The function u is a C 1 diffeomorphism as in the previous √ section. In the stronger hypothesis of this section, g ∈ C 1 , we have that u(x) = V (x) for x > 0 is C 2 as well as V = g and its inverse u −1 (y) for y > 0. This proves the result. Now, we can differentiate ∂t X (T (x0 ), x0 ) = 0 with respect to x0 , ∂2 X ∂2 X (T (x0 ), x0 ) T (x0 ) + (T (x0 ), x0 ) = 0. 2 ∂t ∂ x0 ∂t
(3.12)
We just recall that (∂ 2 X/∂t 2 ) (T (x0 ), x0 ) = −g(x0 ) < 0 for x0 > 0, and have that condition (3.10) is equivalent to T (x0 ) = 0.
(3.13)
This is the condition in order φ to be periodic. Since the linear combination of φ and ψ gives the general solution, we have just proved the following:
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y
x
Fig. 2. Projection of an unbounded orbit
Proposition 3.2. Let us fix x0 ∈ J with x0 > 0, then the solutions to the T (x0 )-periodic equation (3.2) are either all T (x0 )-periodic, or all unbounded but those proportional to t → ∂t X (t, x0 ). A necessary and sufficient condition for the former case is (3.13). In Fig. 2 the curve (x(t), y(t)) = (X (t, x0 ), φ(t)), namely the solution to the system of differential equations (3.1) with x(0) = x0 > 0, y(0) = 1, x(0) ˙ = y˙ (0) = 0, in a non-periodic case. The initial point (x0 , 1) and the point (h(x0 ), g(x0 )/g(h(x0 ))) are crossed at any period (as one can see from (3.6) at t = τ/2). Now, notice that the origin in R4 is an unstable equilibrium for (3.1) if and only if we can find x0 > 0 arbitrarily close to 0 such that equation (3.2) has unbounded solutions. So by Proposition 3.2 we have Theorem 3.3 (Stability and weak instability). Let g be a C 1 function near 0 in R with g(0) = 0, g (0) > 0, then the origin in R4 is a stable equilibrium for the system (3.1), if and only if the origin in R2 is a locally isochronous center for x¨ = −g(x), in this case all orbits of (3.1) with (x, x) ˙ near (0, 0) are periodic and have the same period. If the equilibrium is unstable, then there is a sequence of initial data which converges to the origin whose corresponding solutions are unbounded. The instability is weak, by this we mean that there are no asymptotic motions to the equilibrium, namely no non-constant solutions which have the origin as limit point as t → −∞. We already proved the first part of the statement; for the last, let us only remark that the distance of an orbit from the origin in R4 is greater than the distance of the projection in the x, x-plane ˙ which is strictly positive unless the projection is (0, 0) and in this case the second differential equation in (3.1) gives the harmonic oscillator y¨ = −g (0)y. An explicit function g to give such an example is sin(x) (see (2.22)). However, as we know from Sect. 2, almost all functions g as in the statement of the theorem, give instability.
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Examples of the rare functions which give stable equilibria are (see (2.18) and (2.19))
1 1 , g (0) = 1, 1− √ (3.14) g(x) = 2 1 + 4x
1 1 , g (0) = 1. 1+x − g(x) = (3.15) 4 (1 + x)3 For all these functions which give stable equilibria, at least whenever g ∈ C 2 , the system (3.1) admits a Lyapunov function which is a further first integral and it is positive definite near the origin. This additional first integral is smooth in a neighborhood of the origin in R4 and at least continuous at the origin as proved by Barone and Cesar in [4]. In the following statement C ⊆ R2 is the set defined below formula (3.2). Theorem 3.4 (Lyapunov functions). Let g : J → R be a C 2 function on the open interval J ⊆ R with 0 ∈ J, g(0) = 0, g (0) > 0. Suppose that all orbits of x¨ = −g(x) which intersect the ˙ are periodic and have the J interval of the x-axis in the x, x-plane, same period 2π/ g (0), so the origin in R4 is a stable equilibrium for the system (3.1). Then there exists E(x, y, x, ˙ y˙ ) = a(x, x) ˙ y˙ 2 + b(x, x)y ˙ y˙ + c(x, x)y ˙ 2 , a, b, c : C → R,
(3.16)
a continuous and positive definite function, which is a (global) first integral for the system (3.1). The first integral is obtained by means of the functions in (3.5) and a suitable inverse function, see [4] for details. Proposition 3.5 (Eigenvalues). Let g be a C 1 function near 0 in R with g(0) = 4 0, g (0) > 0, then the linearization of the system (3.1) at the origin in R has the double eigenvalues ±i g (0). Let us remark that we can easily construct unstable cases with some x0 at which (3.13) holds and so the corresponding orbits are all periodic. If we want the period function composed with u −1 to be for instance T (y0 ) = 2π(1 − y02 + y04 ), whose derivative √ vanishes at y0 = 1/ 2 we integrate
y0 zT (z) 2 8 5
(3.17) y0 . dz = 2π y0 − y03 + 3 15 0 y02 − z 2 So we get the relation u
−1
(y0 ) − u
−1
2 3 8 5 y . (−y0 ) = 2 y0 − y0 + 3 15 0
(3.18)
The symmetric choice u −1 (−y0 ) = −u −1 (y0 ) gives u −1 (y0 ) = y0 −
2 3 8 5 y + y . 3 0 15 0
(3.19)
By inversion we get u(x), then V (x) = u(x)2 /2 and finally g(x) = V (x). There is an isolate x0 at which (3.13) holds. We can also imagine more complicated examples where these x0 accumulate at 0.
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Finally, let us mention that most of the results in Sect. 3 were found in [13] by a different (more complicated) proof which included other differential equations in a family for which the present system was a particular case. However, in [13] I had not realized the Lagrangian character of the present case which is very important to our purposes and will be exploited in the next section. 4. Complete Integrability In this section we write q = (q1 , q2 ) = (x, y). The Legendre transformation takes the Lagrangian system (3.1) into H (q, p) = p1 p2 + g(q1 ) q2 , ∂H ∂H (q, p) = p2 , q˙2 = (q, p) = p1 , q˙1 = ∂ p1 ∂ p2 ∂H ∂H p˙ 1 = − (q, p) = −g (q1 ) q2 , p˙ 2 = − (q, p) = −g(q1 ). ∂q1 ∂q2
(4.1)
The Hamiltonian function corresponds to the first integral F while the first integral G becomes q1 p2 K (q, p) = 2 + V (q1 ), V (q1 ) = g(s) ds. (4.2) 2 0 Our phase space is M = {(q1 , q2 , p1 , p2 ) ∈ R4 : (q1 , p2 ) ∈ C, (q2 , p1 ) ∈ R2 },
(4.3)
R2
where C ⊆ is an open set mentioned in Sect. 3 between formulas (3.2) and (3.3) so to have a global center in the q1 , p2 -plane. The Hamiltonian vector fields ⎛ ⎞⎛ ⎞ ⎛ ⎞ 0 0 1 0 ∂q1 H (q, p) p2 0 0 1 ⎟ ⎜ ∂q2 H (q, p) ⎟ ⎜ p1 ⎜ 0 ⎟ ∇ H (q, p) = ⎝ = , (4.4) −1 0 0 0 ⎠ ⎝ ∂ p1 H (q, p) ⎠ ⎝ −g (q1 )q2 ⎠ 0 −1 0 0 ∂ p2 H (q, p) −g(q1 ) ⎛ ⎞⎛ ⎞ ⎞ ⎛ 0 0 1 0 ∂q1 K (q, p) 0 0 0 1 ⎟ ⎜ ∂q2 K (q, p) ⎟ ⎜ p2 ⎟ ⎜ 0 ∇ K (q, p) = ⎝ , (4.5) = −1 0 0 0 ⎠ ⎝ ∂ p1 K (q, p) ⎠ ⎝ −g(q1 ) ⎠ 0 −1 0 0 ∂ p2 K (q, p) 0 are both complete on M, namely their integral curves are all defined on the whole R. Indeed, this is clear for (4.4) since we have a global center on the q1 , p2 -plane and linear equations in q2 , p1 ; for (4.5) we simply integrate and recall (4.3) q1 (t) = q1 (0),
q2 (t) = q2 (0) + p2 (0) t,
p1 (t) = p1 (0) − g(q1 (0)) t,
p2 (t) = p2 (0).
(4.6)
The functions H, K are in involution, indeed their Poisson brackets vanish: ∂H ∂K ∂K ∂H · − · ∂q ∂ p ∂q ∂ p
p2 0 g(q1 ) g (q1 )q2 · · − = 0. = p2 0 g(q1 ) p1
{H, K } =
(4.7)
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The vectors ∇ H (q, p), ∇ K (q, p) are linearly independent at each point of the set N := {(q, p) ∈ M : (q1 , p2 ) = (0, 0)}
(4.8)
which is invariant for both the Hamiltonian vector fields ∇ H and ∇ K . Indeed, the condition α∇ H (q, p) + β∇ K (q, p) = 0 implies αg(q1 ) = 0 and αp2 = 0, so α = 0 since (q1 , p2 ) = (0, 0); then it also implies βg(q1 ) = 0 and βp2 = 0 so β = 0. Let ⊂ N be a nonempty component of a level set of (H, K ), then we see at once that it is not compact since it contains the unbounded curve p1 → (q1 , c/g(q1 ), p1 , 0), where c is the value of H and V (q1 ) = 0 is the one of K . By a well known theorem (see Theorem 3, Chap. 4, in Arnold, Kozlov and Neishtadt [1]), is diffeomorphic to S1 × R. More precisely Theorem 4.1 (Complete integrability). Let g be a C 1 function near 0 in R with g(0) = 0, g (0) > 0. The functions H, K : M → R, in (4.1) and (4.2), are in involution on M, dim M = 4, and the Hamiltonian vector fields ∇ H and ∇ K are complete on M. The set N in (4.8) is invariant for ∇ H and ∇ K and H, K are independent on N . Let ⊂ N be a nonempty component of a level set of (H, K ), then it is diffeomorphic to S1 × R and there are coordinates φ mod 2π and z on S1 × R such that the differential equations defined by the vector field ∇ H on take the form φ˙ = ω,
z˙ = v
(ω, v = const).
(4.9)
If (x0 , 0) belongs to the projection of on the q1 , p2 -plane and T (x0 ) is the period of the first component of any integral curve of ∇ H on , then ω = 2π/T (x0 ) and v = 0 if and only if T (x0 ) = 0. We already proved everything in the statement of the theorem but the last sentence which is a straight consequence of T (x0 ) = 0 being the necessary and sufficient condition for the solutions on to be all periodic (see Proposition 3.2), and (4.9) gives periodic solutions if and only if v = 0. For the isochronous systems, namely whenever the period function is constant and all integral curves of ∇ H in M have the same period, we know from Theorem 3.4 that a further first integral exists which is quadratic in q2 , p1 . Of course we do not expect this first integral to be in involution with K . In the next section we classify the cases where a third first integral quadratic in the momenta p1 , p2 exits and we arrive at explicit formulas for it. They are all isochronous. 5. Explicit Superintegrable Systems Let g be a C 1 function near 0 in R with g(0) = 0, g (0) > 0. Suppose that the system (4.1) has a first integral W of the form A(q1 , q2 ) p12 + B(q1 , q2 ) p1 p2 + C(q1 , q2 ) p22 + U (q1 , q2 ).
(5.1)
Then the equation {H, W } = 0 gives a cubic polynomial in p1 , p2 . The coefficients vanish if and only if ∂2 A = 0, ∂1 C = 0, ∂1 A + ∂2 B = 0, ∂1 B + ∂2 C = 0, ∂1 U (q1 , q2 ) = 2C(q1 , q2 )g(q1 ) + B(q1 , q2 )g (q1 )q2 , ∂2 U (q1 , q2 ) = 2 A(q1 , q2 )g (q1 )q2 + B(q1 , q2 )g(q1 ).
(5.2) (5.3)
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The conditions in (5.2) are the same as those in (3.2.2) in Hietarinta [8], where the celebrated Darboux problem is treated (only the order of the coordinates is changed since q˙1 = p2 , q˙2 = p1 ). They have the following solution where a, b1 , b2 , c1 , c2 , c3 ∈ R are arbitrary: A(q1 , q2 ) = aq12 + b1 q1 + c1 , B(q1 , q2 ) = −2aq1 q2 − b1 q2 − b2 q1 + c3 , C(q1 , q2 ) = aq22 + b2 q2 + c2 .
(5.4)
Plugging (5.4) in (5.3) we get the following condition for the integrability of U : 3(b2 + 2aq2 )g(q1 ) = 3(b1 + 2aq1 )q2 g (q1 ) + 2(c1 + q1 (b1 + aq1 ))q2 g (q1 ). (5.5) For q2 = 0 we get 3b2 g(q1 ) = 0 which implies b2 = 0, and back to (5.5) we have 6ag(q1 ) = 3(b1 + 2aq1 )g (q1 ) + 2(c1 + q1 (b1 + aq1 ))g (q1 ).
(5.6)
For a = 0, g(0) = 0 and g (0) = ω2 , this equation gives ω2 g(q1 ) = λ
1 1− √ 1 + 2λq1
,
(5.7)
where λ = b1 /(2c1 ) (for c1 = 0 there are no solutions). The values λ = 2 and ω = 1 give (3.14). The first integral (5.1) for this g is easily obtained with b1 = 2λc1 , a = 0, b2 = 0, c1 p12 (1 + 2λq1 ) + p1 p2 (c3 − 2λc1 q2 ) + c2 p22 + c2
2 ω2 −1 + 1 + 2λq 1 λ2
+ c1 ω2 q22 + q2 (c3 − 2λc1 q2 ) g(q1 ) + d,
(5.8)
where d ∈ R. For c1 = c3 = d = 0, c2 = 1/2, we have the first integral K we already know, while H corresponds to the choice c1 = c2 = d = 0, c3 = 1. For c1 = c2 = 1, c3 = d = 0 we have the first integral p12 (1 + 2λq1 ) − 2λq2 p1 p2 + p22 + 2V (q1 ) + ω2 − 2λg(q1 ) q22 .
(5.9)
This function is positive definite near the origin of R4 as we check at once. This is why we have Lyapunov stability of the equilibrium and compact orbits. We dealt with the particular case a = 0. In the sequel a = 0 and we fix a = 1 dividing the first integral by a. The coefficient of g in Eq. (5.6) is a quadratic function in q1 with discriminant b12 − 4c1 . Let us consider first the case where this vanishes, namely c1 = b12 /4. If b1 = 0 then (5.6) with g(0) = 0, g (0) = ω2 , gives the trivial function g(x) = ω2 x, while for λ := 2b1 = 0 we get the following solution with g(0) = 0, g (0) = ω2 : g(q1 ) =
ω2 4
λ + q1 −
λ4 (λ + q1 )3
.
(5.10)
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Notice that this formula includes (3.15) for ω = 1 and λ = 1. The first integral (5.1), which corresponds to (5.10), with c3 = 0, c2 = 1, and which vanishes at the origin of R4 , is p12 (λ + q1 )2 − 2 p1 p2 (λ + q1 )q2 + p22 (1 + q22 ) 2 λ2 ω 2 ω2 2 4 1 + 4q2 . − + (λ + q1 ) + λ 4 (λ + q1 )2 2
(5.11)
We can check that at the origin of R4 its gradient vanishes and the Hessian matrix is the diagonal matrix 2(ω2 , λ2 ω2 , λ2 , 1), so we have again a (positive definite near the origin) Lyapunov function for all values of the parameter λ = 0. Now we are ready to deal with a = 1 and b12 − 4c1 = 0, we also consider c1 = 0 since c1 = 0 gives no solutions. Then the solution of Eq. (5.6) with g(0) = 0 and g (0) = ω2 > 0 is b12 − 4c1 − 2 (b1 + 2q1 )2 2c1 ω2 2 g(q1 ) = 2 (b1 + 4c1 )(b1 + 2q1 ) + b1 √ . (5.12) 1 + q1 (b1 + q1 )/c1 (b1 − 4c1 )2 The first integral (5.1) with c3 = 0, c2 = c1 , divided by c1 , and which vanishes at the origin of R4 , is 1 ( p1 q1 − p2 q2 ) ( p1 (b1 + q1 ) − p2 q2 ) c1 ω2 2 2 2 2 4 2 + 2 8b b (b c + 4c + 4c + q )q + 16c − b 1 1 1 1 1 1 1 1 1 1 q2 (b1 − 4c1 )2 2b1 ω2 (b1 + 2q1 ) −4c1 (c1 + (b1 + q1 )q1 ) + b12 − 4c1 q22 + 2 . (5.13) √ 1 + q1 (b1 + q1 )/c1 (b1 − 4c1 )2
p12 + p22 +
We can check that at the origin of R4 its gradient vanishes and the Hessian matrix is the diagonal matrix 2(ω2 , ω2 , 1, 1), so we have a positive definite function near the origin; this yields Lyapunov stability of the equilibrium and compact orbits. Acknowledgements. I thank the referees for the thorough, constructive and helpful comments and suggestions on the manuscript. The picture in Sect. 3 was made using the application Mathematica by Wolfram Research Inc. by means of the package CurvesGraphics6 by Gianluca Gorni.
References 1. Arnold, V.I., Kozlov, V.V., Neishtadt, A.I.: Mathematical aspects of classical and celestial mechanics. Encyclopaedia of Mathematical Sciences 3. Dynamical Systems III. Berlin-Heidelberg-New York: Springer-Verlag, 1988 2. Barone-Netto, A., Cesar, M.O., Gorni, G.: A computational method for the stability of a class of mechanical systems. J. Diff. Eqs. 184, 1–19 (2002) 3. Calogero, F.: Isochronous systems. Oxford: Oxford University Press, 2008 4. Cesar, M.O., Barone-Netto, A.: The existence of Liapunov functions for some non-conservative positional mechanical systems. J. Diff. Eqs. 91, 235–244 (1991) 5. Chavarriga, J., Sabatini, M.: A survey of isochronous systems. Qual. Theory Dyn. Syst. 1, 1–79 (1999) 6. Fassò, F.: Superintegrable Hamiltonian systems: geometry and perturbations. Acta Appl. Math. 87, 93–121 (2005)
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7. Françoise, J.P.: Isochronous systems and perturbation theory. J. Nonlinear Math. Phys. 12(Supp. 1), 315–326 (2005) 8. Hietarinta, J.: Direct methods for the search of the second invariant. Phys. Reps. 147, 87–154 (1987) 9. Mardeši´c, P., Moser-Jauslin, L., Rousseau, C.: Darboux linearization and isochronous centers with a rational first integral. J. Diff. Eqs. 134, 216–268 (1997) 10. Urabe, M.: Potential forces which yield periodic motions of fixed period. J. Math. Mech. 10, 569–578 (1961) 11. Whittaker, E.T.: A Treatise on the Analytical Dynamics of Particles and Rigid Bodies. Fourth ed. New York: Dover Publications, 1944 12. Zampieri, G.: On the periodic oscillations of x¨ = g(x). J. Diff. Eqs. 78, 74–88 (1989) 13. Zampieri, G.: Solving a collection of free coexistence-like problems in stability. Rend. Sem. Mat. Univ. Padova 81, 95–106 (1989) Communicated by G. Gallavotti
Commun. Math. Phys. 303, 89–125 (2011) Digital Object Identifier (DOI) 10.1007/s00220-011-1193-1
Communications in
Mathematical Physics
Global Smooth Ion Dynamics in the Euler-Poisson System Yan Guo1 , Benoit Pausader2 1 Division of Applied Mathematics, Brown University, Providence, RI 02912, USA.
E-mail:
[email protected]
2 Mathematics Department, Brown University, Providence, RI 02912, USA.
E-mail:
[email protected] Received: 17 March 2010 / Accepted: 16 September 2010 Published online: 13 February 2011 – © Springer-Verlag 2011
Abstract: A fundamental two-fluid model for describing dynamics of a plasma is the Euler-Poisson system, in which compressible ion and electron fluids interact with their self-consistent electrostatic force. Global smooth electron dynamics were constructed in Guo (Commun Math Phys 195:249–265, 1998) due to dispersive effect of the electric field. In this paper, we construct global smooth irrotational solutions with small amplitude for ion dynamics in the Euler-Poisson system. Contents 1. 2.
Introduction . . . . . . . . . . . . . Method and Preliminary Results . . . 2.1 Presentation of the method . . . 2.2 Notations and preliminary results 3. Linear Decay . . . . . . . . . . . . . 4. Normal Form Transformation . . . . 5. The L 2 -Type Norm . . . . . . . . . 5.1 The energy estimate . . . . . . . 5.2 The H˙ −1 -norm . . . . . . . . . 6. Bilinear Multiplier Theorem . . . . . 6.1 A general multiplier theorem . . 6.2 Multiplier analysis . . . . . . . 7. The L 10 Bound and End of the Proof 7.1 Estimating the L 10 bound . . . . 7.2 End of the proof . . . . . . . . . References . . . . . . . . . . . . . . . . .
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89 92 92 94 96 100 104 105 106 107 107 109 120 120 124 124
1. Introduction The “two-fluid” models in plasma physics describe dynamics of two separate compressible fluids of ions and electrons interacting with their self-consistent electromagnetic
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field. Many famous nonlinear dispersive PDE, such as Zakharov’s equation, nonlinear Schrödinger equations, as well as KdV equations, can be formally derived from “twofluid” models under various asymptotic limits. In the absence of the magnetic effects, the fundamental two-fluid model for describing the dynamics of a plasma is given by the following Euler-Poisson system ∂t n ± + ∇ · (n ± v ± ) = 0 n ± m ± (∂t v± + v± · ∇v± ) + T± ∇n ± = ∓en ± ∇φ φ = 4π e(n − − n + ).
(1.1)
Here n ± are the ion (+) and electron density (−), v± are the ion (+) and electron (−) velocity, m ± are the masses of the ions (+) and electrons (−), T± are their effective temperatures, and e is the charge of an electron. The self-consistent electric field −∇φ satisfies the Poisson equation. The Euler-Poisson system describes rich dynamics of a plasma. Indeed, even at the linearized level, there are electron waves, ion acoustic waves in the Euler-Poisson system. Despite its importance, there have been few mathematical studies of its global solutions in 3D. This stems from the fact that the Euler-Poisson system belongs to the general class of hyperbolic conservation laws with zero dissipation, for which no general mathematical framework for construction of global in time solutions exists in 3D. In fact, as expected [10], solutions of the Euler-Poisson system with large amplitude in general will develop shocks. However, unlike the pure Euler equations, shock formation for solutions of the EulerPoisson system with small amplitude has remained open. In Guo [9], the first author studied a simplified model of the Euler-Poisson system for an electron fluid: ∂t n − + ∇ · (n − v − ) = 0 n − m − (∂t v− + v− · ∇v− ) + T− ∇n − = en − ∇φ, φ = 4π e(n − − n 0 ).
(1.2)
In this model, the ions are treated as immobile and only form a constant charged background n 0 . Surprisingly, it was observed [9] that the linearized Euler-Poisson system for the electron fluid is the Klein-Gordon equation, due to plasma oscillations created by the electric field φ. In this case, the dispersion relation reads ω(ξ ) 1 + |ξ |2 . Such a “Klein-Gordon” effect led to construction of smooth irrotational electron dynamics with small amplitude for all time in three space dimension. This is in stark contrast to the pure Euler equations for neutral fluids where the dispersion relation reads ω(ξ ) |ξ |, in which shock waves can develop even for small smooth initial data (see Sideris [23]). It is the dispersive effect of the electric field that enhances the linear decay rate and prevents shock formation. The natural open question remains: does such a dispersive effect exist generally? If so, can it prevent shock formation for the general Euler-Poisson system (1.1)? In the current paper, we make another contribution towards answering this question. We consider another (opposite) asymptotic limit of the original Euler-Poisson system (1.1) for the ion dynamics. It is well-known that mm−+ << 1 in all physical situations. By
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letting the electron mass m − go to zero, we formally obtain T− ∇n − = en − ∇φ and the famous Boltzmann relation eφ n − = n 0 exp (1.3) T− for the electron density (n 0 is a constant). Such an important relation (1.3) can also be verified through arguments from kinetic theory, see Cordier and Grenier [3]. We then obtain the well-known ion dynamic equations as ∂t n + + ∇ · (n + v+ ) = 0, n + m + (∂t v+ + v+ · ∇v+ ) = −T+ ∇n + − n + e∇φ, eφ − n+ . φ = 4π e n 0 exp T−
(1.4)
We also assume that curl(v(0)) = 0.
(1.5)
It is standard that the condition (1.5) is preserved by the flow. As a matter of fact, nonzero vorticity leads to creation of a non-vanishing magnetic field, which is omitted in the Euler-Poisson system but retained in a more general Euler-Maxwell system [1]. The linear dispersion relation for (1.4) behaves like p(ξ ) ≡ |ξ |
2 + |ξ |2 ≡ |ξ |q(|ξ |), 1 + |ξ |2
(1.6)
which is much closer to the wave dispersion ω(ξ ) = |ξ | than to the Klein-Gordon one, ω(ξ ) = 1 + |ξ |2 (in particular note that this dispersion relation behaves near 0 as in the Schrödinger case, whereas in (1.6) p remains very similar to the wave case). Intuitively, one might expect formation of singularity for (1.4) as in the pure Euler equations. Nevertheless, we demonstrate that small smooth irrotational flows exist globally in time, and there is no shock formation. Without loss of generality, we study the global behavior of irrotational perturbations of the uniform state [n + , v+ ] = [n 0 + ρ, v]. We use two important norms defined as follows: u(x)Y = |∇|−1 u H 2k+1 + u k+ 12 , 10 , W 5 9 16 k −1 k+ 21 u(t, x) X = sup |∇| (1 − ) u(t) L 2 + (1 + t) 15 (1 − ) 2 u(t) L 10
(1.7)
t
for k ≥ 5. Here, we keep k as a parameter to emphasize the fact that smoother initial data lead to smoother solutions. Hidden in the X -norm is a statement about preservation of regularity of (ρ, v). Our main result is the following
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Theorem 1.1. Let x ∈ R3 . Let m + , T± , e, n 0 be any given positive constants. There exists ε0 > 0 such that any initial perturbation (n 0 + ρ0 , v0 ) satisfying ∇ × v0 = 0 and ρ0 Y + v0 Y ≤ ε0 leads to a global solution (n 0 + ρ, v) of (1.4) with ρ X + v X ≤ 2ε0 . 16
In particular, the perturbations ρ and v decay in L ∞ as t − 15 . Together with the earlier result in Guo [9] this shows that global smooth potential flows with small velocity exist for two opposite scaling limits of (1.1). This is a strong and exciting indication that shock waves of small amplitude should be absent for the full Euler-Poisson system (1.1), at least in certain physical regimes. Our method developed in this paper should be useful in the future study of (1.1). There have been a lot of mathematical studies of various aspects of the EulerPoisson system for a plasma. Texier [26,27] studied the Euler-Maxwell system and its approximation by the Zakharov equations. Wang and Wang [29] constructed large BV radially symmetric solutions outside the origin. In Liu and Tadmor [16,17], threshold for singularity formation has been studied for the Euler-Poisson system with T± = 0 in one and two dimensions. In Feldman, Ha and Slemrod [4,5], the plasma sheath problem of the Euler-Poisson system was investigated. In Peng and Wang [20], the Euler-Poisson system is derived from the Euler-Maxwell system with a magnetic field. Quasi-neutral limit in the Euler-Poisson system was studied in Cordier and Grenier [3] as well as in Peng and Wang [21]. When n + is replaced by a doping profile and a momentum relaxation is present, the Euler-Poisson system describes electron dynamics in a semiconductor device. There has been much more mathematical study of such a model, for which we only refer to Chen, Jerome and Wang [1] and the references therein. This paper is organized as follows: in Sect. 3 we study the relevant linear dispersive equation. In Sect. 4, we introduce our normal form transformation. In Sect. 5, we get an estimate on the L 2 -part of the X norm using the energy method. In Sect. 6 we state and prove the relevant multiplier estimate we need in order to control our bilinear terms. Finally, in Sect. 7, we control the high integrability part of the norm, and finish the analysis to obtain global solutions with small initial data in Theorem 1.1. 2. Method and Preliminary Results 2.1. Presentation of the method. For notational simplicity, we let n 0 = e = T+ = T− = m + = 4π = 1 in (1.4) throughout the paper. Even though the ion dynamics system (1.4) is the most natural system to further understand the dispersive effects in the full Euler-Poisson system (1.1), it has remained an open problem to construct global smooth solutions until now due to much more challenging mathematical difficulties than those in the case of the electron Euler-Poisson equation (1.3) studied by Guo [9]. The first difficulty is to understand the time decay rate of the linearized ion dynamics equation: ∂tt ρ − ρ − (− + 1)−1 ρ = 0, whose solutions are given by the operator e±i p(|∇|)t with p given by (1.6). Unlike the linearized electron equations studied in [9], there is no direct study of the linear decay of such a system. Only recently [11], the time-decay rate for general dispersive equations
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has been carried out in detail with asymptotic conditions near low frequency |ξ | = 0 and high frequency |ξ | = ∞. Interestingly, any phase p(ξ ) which is not exactly the phase function of the wave equation p(ξ ) = |ξ | commands a decay rate better than 1 t . We are able to employ this result together with a stationary phase analysis near the 1 inflection point of p(ξ ) to obtain a decay rate of t 4/3 , which is between the wave and the Klein-Gordon equations. A consequence of the linear estimates of Sect. 3 is that eit p(|∇|) α0 X α0 Y . The main mathematical difficulty in this paper stems from bootstrapping the linear decay into a construction of global solutions to the nonlinear problem. Based on Germain, Masmoudi and Shatah [6–8], Gustafson, Nakanishi and Tsai [12,13], Shatah [22], we follow a new set-up for normal form transformation in [6,13,22]. Using that ∇ × v ≡ 0, we can introduce a pair of complex-valued new unknowns: α1 = ρ −
i R−1 v, q(|∇|)
and
α2 = ρ +
i R−1 v, q(|∇|)
(2.1)
for q defined in (1.6), where R = ∇|∇|−1 stands for the Riesz transform, and v = ∇ψ, by (1.5) R−1 v ≡ |∇|ψ. After the normal form transformation (4.14), it suffices for us to control m(ξ, η) α(ξ ˆ − η)α(η)dη ˆ α(t) ˆ R3 1 (ξ, η) t m(ξ, η)m(η, ζ ) + α(ξ ˆ − η)α(η ˆ − ζ )α(ζ ˆ )dηdζ ds, (2.2) ei(t−s) p(|ξ |)
1 (ξ, η) 0 R6 where m denotes a generic multiplier given by (4.12). Here
1 = p(|ξ |) − p(|ξ − η|) − p(|η|). In the Klein-Gordon case, the phase is bounded away from zero so there is no singularity. However, for 1 , there is a significant zero set when |ξ − η||η| = 0, (see Lemma 6.3) and there is no “null form” structure to cancel with the multiplier m. We first observe that m(ξ, η)m(η, ζ ) |ξ ||η|. We then make use of such a structure to form a locally bounded multiplier M1 =
|ξ ||ξ − η||η| 1.
1 (ξ, η)
ˆ −η) This process introduces a singular term α(ξ |ξ −η| , which will be controlled in a separate −1 fashion by the H norm in our norm · X . We believe that including this H −1 control in the norm should work equally well for equations with nonlinearity which has perfect spatial derivatives. Even though M1 is locally bounded, it is very difficult to employ the classical bilinear estimates such as Coifman-Meyer Theorem [2] to control (2.2). This is due to the anisotropic nature of M1 since |η| can be very small with respect to |ξ − η|. Instead, we make use of a very recent multiplier estimate by Gustafson, Nakanishi and Tsai [13].
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It is important to use L 10 -norms as a proxy for the L ∞ -norm for which our degenerp ate multipliers are not well-suited (we need an L -norm with p < 12). The optimal 5/4−ε 5/4−ε ∞ ∞ ˙ ∩ L η H˙ ξ is crucial in applying Sobolev regularity for M1 ∈ L ξ Hη such an estimate to obtain L 10 decay, and its proof is particularly delicate for small frequencies. We split the phase space and make a careful interplay between angles and the lengths of ξ, η, ξ − η. We also make use of Littlewood-Paley decomposition and interpolation to obtain a sharp Sobolev estimate for M1 . On the other hand, to reduce the requirement of the number in our norm of derivatives X, we also need to show a 3/2−ε ˙ η3/2−ε ∩ L ∞ ˙ H H for large frequencies. stronger estimate M1 ∈ L ∞ η ξ ξ 2.2. Notations and preliminary results. We work in dimension n = 3, although we state some results in arbitrary dimension n. We introduce a = 1 + a 2 . We write A B to signify that there exists a constant C such that A ≤ C B. We write A B if A B A. Our phases and some multipliers are radial functions, and in some cases we might abuse notations and write, for a radial function f, f (x) = f (|x|). Our multipliers are estimated using the homogeneous Sobolev norm defined for 0 ≤ s < n/2 by f H˙ s = |∇|s f L 2 , where |∇| is defined by F (|∇| f ) (ξ ) = |ξ | fˆ(ξ ). We will also use the Littlewood-Paley multipliers PN defined for dyadic numbers N ∈ 2Z by ξ Fξ g, PN g = Fξ−1 ϕ (2.3) N where ϕ ∈ Cc∞ (Rn ) is such that ∀ξ = 0,
ϕ
N ∈2Z
ξ N
= 1.
An important estimate on these Littlewood-Paley multipliers is the Bernstein inequality: 1
1
PN f L r N n( l − r ) PN f L l , |∇|±s PN f L l s N ±s PN f L l s N ±s f L l
(2.4)
for all s ≥ 0, and all 1 ≤ l ≤ r ≤ ∞, independently of f, N , and p, where |∇|s is the classical fractional differentiation operator. We will also need the two following product estimates: Lemma 2.1. Let τ be a multi-index of length |τ | and γ < τ , then for all u ∈ Cc∞ (Rn ) and all small δ > 0, there holds that D τ −γ u D γ ∂ j u L 2 δ uW 1+δ,∞ u H |τ | uW 2,10 u H |τ | .
Euler-Poisson: Ion Equation
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Proof. Without loss of generality, we may assume that |γ | + 1, |τ | − |γ | ≥ 2, otherwise Hölder’s inequality gives the result. We use a simple paradifferential decomposition, in other words, we write ⎛ ⎞ ⎠ PM D τ −γ u PN D γ ∂ j u D τ −γ u D γ ∂ j u = ⎝ + + M/N ≤1/16
M∼N
N /M≤1/16
= R + T1 + T2 where M and N are dyadic numbers. We first estimate R. Using Bernstein properties and in particular the fact that PN u L ∞ min(1, N −1−δ )uW 1+δ,∞ , we deduce from the Cauchy Schwartz inequality that PM D τ −γ u PN D γ ∂ j u L 2 R L 2 M∼N
M |τ |−|γ | PM u L 2 M |γ |+1 PN u L ∞
M∼N
M
2|τ |
PM u2L 2
1 2
M
1 2
M
2
PM u2L ∞
M
u H |τ | uW 1+δ,∞ . Independently, we estimate T1 . Using that if 16Mi ≤ Ni , i = 1, 2 then PM1 f PN1 g, PM2 h PN2 k L 2 ×L 2 = 0 unless N1 ≤ 4N2 ≤ 16N1 (intersection of the Fourier support), and letting f = D τ −γ u, g = D γ ∂ j u, we get PN1 f PM1 g, PN2 f PM2 g L 2 ×L 2 T1 2L 2 N1 ∼N2 ,16Mi ≤Ni
u2W 1,∞
u2W 1,∞
PN1 D τ −γ u L 2 PN2 D τ −γ u L 2 (M1 M2 )|γ |
N1 ∼N2 ,16Mi ≤Ni
|γ |
|γ |
N1 PN1 D τ −γ u L 2 N2 PN2 D τ −γ u L 2
N1 ∼N2 2 uW 1,∞ u2H |τ | ,
and T2 is treated exactly in the same way.
We also need the following “tame” product estimate (see e.g. Tao [25]) Lemma 2.2. For 1 < p < ∞, s ≥ 0, uvW s, p u L ∞ vW s, p + uW s, p v L ∞ for u and v in L ∞ ∩ W s, p .
(2.5)
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3. Linear Decay In this section, we investigate the decay of linear solutions of the linearized equation ∂tt ρ − ρ − (− + 1)−1 ρ = 0.
(3.1)
These solutions can be expressed in terms of the initial data and of one “half-wave” operator Tt = eit p(|∇|) for p defined in (1.6) that we now study. Our main result in this section is the following 5
Proposition 3.1. For any δ > 0, for any f ∈ W 2 +δ,1 , there holds that 4 3 eit p(|∇|) f L ∞ δ |t|− 3 + |t|− 2 f 5 +δ,1 W2
(3.2)
for all t = 0. Besides, we have the L 10 -decay estimate 16
eit p(|∇|) f L 10 (1 + |t|)− 15 f
W
12 , 10 5 9
(3.3)
uniformly in t, f . More precise estimates are derived below. The rest of the section in devoted to a proof of (3.2) and (3.3). For most of this section, we study the dispersive features of our operator in general dimension n. Proposition 3.1 is a consequence of the particular case n = 3. Direct computations give that 1 1 , p (r ) = 1 + r2 + 1 + r2 (1 + r 2 )(2 + r 2 ) r r 4 − 2r 2 − 6 p (r ) = (3.4) 3 and, (1 + r 2 ) (1 + r 2 )(2 + r 2 ) 2 r 2 r 4 − 2r 2 − 6 (8r 2 + 13) 5r 4 − 6r 2 − 6 − . p (r ) = 5 3 7 5 (1 + r 2 ) 2 (2 + r 2 ) 2 (1 + r 2 ) 2 (2 + r 2 ) 2 We note that p (r ) has one unique positive root at √ r = r0 = 1 + 7.
(3.5)
In order to state our first result, we define a frequency localization function around the inflection point r0 . Let ψr0 ∈ C ∞ (R) be a smooth function such that 0 ≤ ψr0 ≤ 1, ψr0 (r0 + r ) = 1 when |r | ≤ ε and ψr0 (r0 + r ) = 0 when |r | ≥ 2ε. Lemma 3.1. For all time t = 0, and all f ∈ L 1 , eit p(|∇|) ψr0 (|∇|) f L ∞ n,ε (1 + |t|)−
n−1 1 2 −3
f L 1 holds.
(3.6)
Euler-Poisson: Ion Equation
97
Proof. We note that eit p(|∇|) ψr0 (|∇|) f (x)∞ = ||F −1 {eit p(|ξ |) ψr0 (|ξ |) fˆ(ξ )}||∞ = ||F −1 {eit p(|ξ |) ψr0 (|ξ |)} ∗ f (x)||∞ ≤ ||F −1 {eit p(|ξ |) ψr0 (|ξ |)}||∞ || f || L 1 . Since ψr0 is chosen to be spherically symmetric, it is well-known that ∞ F −1 {eit p(|ξ |) ψr0 (|ξ |)}(x) = 2π eit p(r ) ψr0 (r ) J˜n−2 (r |x|)r n−1 dr 2 0 ∞ eit p(r ) ψr0 (r ) J˜n−2 (r |x|)r n−1 dr, = 2π 2
0
where for all n ≥ 2, J˜n−2 (s) ≡ s −
n−2 2
2
J n−2 (s) = Re eis Z (s) = eis Z (s) − e−is Z¯ (s). 2
(3.7)
Here Z (s) is a smooth function satisfying (cf. John [14]) that for all k ≥ 0 and all s, |∂ k Z (s)| n,k (1 + s)−
n−1 2 −k
.
(3.8)
We first estimate e−ir |x| Z¯ (r |x|). Changing the variable r → r + r0 , and letting (r ) = (r0 + r )n−1 ψr0 (r0 + r ), we get ∞ I˜1 = ei(t p(r )−r |x|) ψ(r )Z (r |x|)r n−1 dr 0
=
2ε
−2ε
ei(t p(r0 +r )−(r0 +r )|x|) (r )Z ((r0 + r )|x|)dr,
and a first crude estimate allows us to conclude that | I˜1 | n,ε 1
(3.9)
which takes care of the small times |t| 1. Thus, we now assume that t > 1. We consider the phase (r, |x|, t) = p(r ) − r
|x| . t
By (3.4), we directly compute that p (r0 ) = 0 and p (r0 ) = Case 1. Suppose that |x| ≥
1 4
4r04 − 4r02 5
3
(1 + r02 ) 2 (2 + r02 ) 2
= 0.
p (r0 )t. Then, since |r | ≤ 2ε,
|∂r3 (r, |x|, t)| = | p (r )| >
1 | p (r0 )|, 2
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Y. Guo, B. Pausader
if ε > 0 is chosen sufficiently small, and using (3.8), by the Van der Corput Lemma (see e.g. Stein [24]), we get that 1 − k |Z ((r0 + r )|x|)∂ (r )| + |x| sup |(r )Z ((r0 + r )|x|)| | I˜1 | ε |t| 3 sup |r |≤2ε
k∈{0,1},|r |≤2ε
ε |t|
− 31
ε |t|
− 31 − n−1 2
|x|
− n−1 2
.
(3.10)
Case 2. Suppose now that |x| ≤
1 4
p (r0 )t. Then
|∂r (r, |x|, t)| = | p (r0 )| −
|x| ≥ | p (r0 )|/2 t
and therefore, using the nonstationary phase and the fact that Z has all derivatives bounded, we obtain that | I˜1 | |t|− 2 . n
(3.11)
The estimation of eir |x| Z (r |x|) is easier. Proceeding as above, we introduce I˜2 =
2ε −2ε
ei(t p(r0 +r )+(r0 +r )|x|) (r )Z ((r0 + r )|x|)dr.
But the phase in I˜2 satisfies |∂r 2 (r, |x|, t)| = |∂r
p(r ) + r
|x| | ≥ | p (r )| 1, t
and we can conclude as in Case 2 above to get | I˜2 | |t|− 2 . n
Now this, (3.9), (3.10) and (3.11) prove (3.6).
Now that we have dealt with the degeneracies at r0 , the other degeneracies at 0 and ∞ are more easily dealt with at the price of losing derivatives. To isolate these regions, we introduce two smooth cut-off functions. We let ψ0 and ψ∞ such that 0 ≤ ψ0 + ψ∞ ≤ 1, ψ0 is supported on (−r0 + ε, r0 − ε), ψ∞ is supported on {|x| ≥ r0 + ε} and ψ0 + ψr0 + ψ∞ = 1.
(3.12)
We first treat the case of small frequencies. We note that since r0 is the only positive root of p , p (r ) = 0 for either r ∈ (r0 + ε, ∞) or r ∈ (0, r0 − ε). Therefore we can apply Theorem 1 from Guo, Peng and Wang [11], case (a) and (b) respectively, to obtain with (3.4): Lemma 3.2. For all f ∈ L 1 , n
eit p(|∇|) ψ0 (|∇|) f L ∞ n,ε (1 + |t|)− 2 |∇|
n−2 2
f L 1 holds.
(3.13)
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99
Lemma 3.3. For all f ∈ L 1 , n
eit p(|∇|) ψ∞ (|∇|) f L ∞ n,ε |t|− 2 |∇|
n+2 2
f B0
1,1
holds.
(3.14)
Finally, from Lemma 3.1, 3.2, 3.3, we can prove Proposition 3.1. Proof of Proposition 3.1. We take n = 3 here. Equation (3.2) follows directly from Lemma 3.1, (3.13) and (3.14). In order to get (3.3), we interpolate between the isometric property eit p(|∇|) P f L 2 = P f L 2 for P a Fourier projector and the various L ∞ estimates. Interpolating with Lemma 3.1 gives that 16
eit p(|∇|) ψr0 (|∇|) f L 10 |t|− 15 f
L
10 9
.
Interpolating with (3.13) gives 6
eit p(|∇|) ψ0 (|∇|) f L 10 |t|− 5 f
L
10 9
.
Finally, interpolating with (3.14) and using the inclusions of Besov spaces, 0 ⊂ L 10 and L B10,2
10 9
⊂ B 010 ,2 , 9
and Bernstein estimates (2.4), we get that eit p(|∇|) ψ∞ (|∇|) f 2L 10
N ≥1
eit p(|∇|) ψ∞ (|∇|)PN f 2L 10 12
|t|− 5
N 4 PN f 2 10 L
N ≥1 12
|t|− 5 f 2
W 2,
10 9
9
.
Since for small time t ≤ 1, we also have that eit p(|∇|) f L 10 eit p(|∇|) f
6
H5
f
6
H5
f
W
12 , 10 5 9
and since f = ψr0 (|∇|) f + ψ0 (|∇|) f + ψ∞ (|∇|) f , this ends the proof.
,
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4. Normal Form Transformation In this section, we derive the normal form transformation for α j . Isolating linear, quadratic and higher order terms, we can rewrite the normalized Euler-Poisson system (1.4) as follows: ∂t ρ + div(v) + div(ρv) = 0, 2 ∂t v + ∇ρ + ∇φ + (v · ∇)v − ∇ ρ2 = −∇ ln(1 + ρ) − ρ + 2 2 ρ = (1 − )φ + φ2 + eφ − 1 − φ − φ2 .
ρ2 2
(4.1) ,
(4.2) (4.3)
For small ρ, the last line defines an inverse operator ρ → φ(ρ). We further expand this inverse operator up to third order as 2 1 φ(ρ) ≡ (1 − )−1 ρ − (1 − )−1 (1 − )−1 ρ + R(ρ), (4.4) 2 where R satisfies good properties. We note that since ∇ ×v = 0 there exists a function ψ 2 such that v = ∇ψ and consequently, (v · ∇)v = ∇ |v|2 . In terms of the velocity potential ψ, we can rewrite the above system as 0 ρ ρ + ∂t ψ ψ (1 − )−1 + 1 0 −∇ · (ρ∇ψ) 2 = 1 . (4.5) −1 (1 − )−1 ρ 2 − R(ρ) − ln(1 + ρ) + ρ − |∇ψ| 2 (1 − ) 2 1 1 We denote the pair of eigenfunctions of the linear part as , = ± p(|∇|) ± q(|∇|) i|∇| i|∇|2 i −1 (−1) i ∇ and recall α j = ρ + (−1) q(|∇|) R v ≡ ρ + q(|∇|) |∇|ψ, where R = |∇| stands for the Riesz transform. We can diagonalize the matrix as: 1 i|∇| 1 1 0 −i p(|∇|) 0 2 − 2q(|∇|) = . q(|∇|) i|∇| 1 0 i p(|∇|) − q(|∇|) (1 − )−1 + 1 0 i|∇| i|∇| 2 2q(|∇|) j
j
∇ Now, with α j given in (2.1), using that R−1 ∇ = |∇|R−1 |∇| = |∇|, and
div(v) = div(
∇ −1 R v) = −|∇|R−1 v, |∇|
we diagonalize the matrix and rewrite (4.5) in terms of α as (∂t + (−1) j i p(|∇|))α j = Q j (α) + N j ,
(4.6)
where Q 2 = Q¯ 1 , and N2 = N¯ 1 such that the quadratic term Q j and the cubic term N j take the form: 2 j i|∇| −1 −1 2 2 (1 − ) Q j = −div(ρv) + (−1) (1 − ) ρ + ρ − |v| 2q(|∇|) (4.7) ρ2 j i|∇| N j = (−1) − ln(1 + ρ) + ρ − − R(ρ) . q(|∇|) 2
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The most important step is to study the linear profiles ω j (t) = e(−1)
j it p(|∇|)
α j (t),
so that its temporal derivatives are of at least quadratic order: ∂t ω j = e(−1) 2 and v = Plugging ρ = α1 +α 2 transform of Q j (α) as
j i p(|∇|)t
{Q j (α) + N j }.
∇ p(|∇|) α1 −α2 2i
(4.8)
into Q j , we now compute the Fourier
Qˆ j (α)(t, ξ ) i ξ ·η = q(η)(αˆ 1 (ξ − η) + αˆ 2 (ξ − η))(αˆ 1 (η) − αˆ 2 (η)) 3 4 |η| R i|ξ | 1 1 1 +(−1) j (αˆ 1 (ξ − η) + αˆ 2 (ξ − η))(αˆ 1 (η) + αˆ 2 (η)) 1+ 8q(ξ ) ξ 2 ξ − η 2 η 2 (ξ − η) η · q(ξ − η)q(η)(αˆ 1 (ξ − η) − αˆ 2 (ξ − η))(αˆ 1 (η) − αˆ 2 (η)) dη + |ξ − η| |η| j ≡ m rl (ξ, η)αˆ r (ξ − η)αˆ l (η) dη. (4.9) R3
We now integrate (4.6) to get t j+1 j+1 αˆ j (t) = e(−1) i p(|ξ |)t αˆ j (0) + e(−1) i p(|ξ |)(t−s) Qˆ j (α)(s)ds 0 t (−1) j+1 i p(|ξ |)(t−s) ˆ N j (α)(s)ds + e 0 t j+1 j+1 = e(−1) i p(|ξ |)t αˆ j (0) + e(−1) i p(|ξ |)(t−s) Nˆ j (α)(s)ds 0 t j+1 j j + e(−1) i p(|ξ |)t e(−1) i p(|ξ |)s m rl αˆ r (ξ − η)αˆ l (η)(s)dsdη. 0
R3
The crucial step is to replace αˆ j (s) = e(−1) then takes the form ˆ j (α) ≡ e(−1) j+1 i p(|ξ |)t
2 t r,l=1 0
R3
j+1 i p(|ξ |)s
j
(4.10)
ωˆ j (s) in the third term, which
j
m rl eis rl ωˆ r (ξ − η)ωˆ l (η)dηds.
(4.11)
Here ωˆ 1 (ξ ) = eit p(ξ ) αˆ 1 (ξ ) and ωˆ 2 (ξ ) = e−it p(ξ ) αˆ 2 (ξ ) = ωˆ 1 (ξ ), j
rl (ξ, η) = (−1) j p(ξ ) − (−1)r p(ξ − η) − (−1)l p(η), and j
j
j
j
m rl (ξ, η) = |ξ |n 1rl (ξ )n 2rl (ξ − η)n 3rl (η),
(4.12)
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is a factorable multiplier defined in (4.9), where the n olk are either smooth functions or a x product of a smooth function with the angle function x → |x| . More specifically, there are only four different phases of the following types:
1 (ξ, ξ
2 (ξ, ξ
3 (ξ, ξ
4 (ξ, ξ
− η, η) = − η, η) = − η, η) = − η, η) =
p(ξ ) − p(ξ − η) − p(η), p(ξ ) + p(ξ − η) + p(η), p(ξ ) − p(ξ − η) + p(η), p(ξ ) + p(ξ − η) − p(η).
(4.13)
Integrating by parts in s in the integral in , and making use of the fact that ∂t ωˆ is at least quadratic by (4.8), we obtain from (4.11) that1 ˆ j (α)(t, ξ ) t j m rl is j e rl ωˆ r (ξ − η)ωˆ l (η)dη = 3 i j R r,l=1 rl 0 j 2 t m (ξ, η) is j +2 i rl j e rl ωˆ r (ξ − η)∂t ωˆ l (η)dηds 3
rl r,l=1 0 R j 2 m rl =i ωˆ (0, ξ − η)ωˆ l (0, η)dη j r 3 r,l=1 R rl j 2 m rl j + e(−1) it p(ξ ) αˆ r (t, ξ − η)αˆ l (t, η)dη 3 i j R r,l=1 rl j 2 t im rl (ξ, η) is j l +2 e rl ωˆ r (ξ − η)eis(−1) p(|η|) Qˆ r (α)(η)dηds j 3
rl r,l=1 0 R j 2 t im rl is j l +2 e rl ωˆ r (ξ − η)eis(−1) p(η) Nˆ l (η)dηds. j 3 r,l=1 0 R rl
e(−1)
j i p(|ξ |)t
2
We then change back to ωˆ r (s) = e(−1) i p(|ξ |)s αˆ r (s), and using (4.7), we write j ˆ j (α) e(−1) i p(|ξ |)t αˆ j (t) + B t j ˆ = αˆ j (0) + B j (α(0)) + e(−1) i p(|ξ |)s Nˆ j (α)(s)ds r +1
0
t + 0
R3
e
(−1) j i p(|ξ |)s
j
im lk (ξ, η) j
lk
where the normal form transformation is 2 FB j (α)(ξ ) =
3 r,l=1 R
αˆ r (ξ − η)hˆ l (α(η))(s)dsdη,
(4.14)
αˆ r (ξ − η)αˆ l (η)dη
(4.15)
j
m rl j
i rl
1 For notational simplicity, we do not distinguish m j (ξ, η) and m j (ξ, ξ − η). lr rl
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and hˆ l (α) ≡
R3
m r1 l1 (η, ζ )αˆ r1 (η − ζ )αˆ l1 (ζ )dζ + Nˆ l
(4.16)
is the associated nonlinearity. We next show that h(α) behaves like a quadratic term in α. Lemma 4.1. Assuming that α has small X -norm, we have that 16
|∇|−1 h(α(t)) H 2k + |∇|−1 N H 2k (1 + t)− 15 α2X holds.
(4.17)
Proof. When h is a product of α’s, this follows directly from the Sobolev embedding W 1,10 ⊂ L ∞ . Note that h and N both are derivatives of a nonlinear term. When h = N , we see from (4.7) that, except for the term involving R, a similar proof works. For the terms involving R, it suffices to prove that 16 (1 + t) 15 R(ρ)W k+2,10 + R(ρ) H 2(k+1) α3X .
(4.18)
2
Letting E(x) = e x − 1 − x − x2 , we plug (4.4) into (4.3) and expand ϕ 2 /2 to get that 2 R2 1 R+ (1 − )−1 ρ − (1 − )−1 (1 − )−1 ρ 2 2 2 1 +E((1 − )−1 ρ − (1 − )−1 (1 − )−1 ρ + R) 2 2 1 −1 (1 − ) ρ (1 − )−1 (1 − )−1 ρ = 2 2 2 1 −1 −1 (1 − ) ρ − (1 − ) . 8
(1 − )R +
(4.19)
In order to solve (4.19), we define the following iterative scheme. For ρ sufficiently small in X -norm, we let R0 = 0 2 1 R2 −1 −1 −1 Rn − n (1 − ) ρ (1 − )Rn+1 = − (1 − ) ρ − (1 − ) 2 2 2 1 −E((1 − )−1 ρ − (1 − )−1 (1 − )−1 ρ + Rn ) 2 2 1 2 2 1 − (1−)−1 (1 − )−1 ρ . + (1 − )−1 ρ (1 − )−1 (1−)−1 ρ 2 8 We see that, if s > 3/2 and 2 ≤ l < ∞, using the tame estimate (2.5),
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Rn+1 W s+2,l ρ L ∞ Rn W s,l + ρW s−2,l Rn L ∞ + Rn L ∞ Rn W s,l +ρ L ∞ ρ2W s−2,l + ρ3W s−2,l 3 +C (ρ L ∞ + Rn L ∞ ) Rn W s,l + ρW s−2,l , (4.20) and, assuming that sup Rn L ∞ + ρ L ∞ ≤ 2, n
we also see that Rn+1 − Rn H 2 ρ L ∞ + sup Rn L ∞ Rn − Rn−1 L 2 . n
Hence, if ρ X < 1 is sufficiently small, 16
(1 + t) 15 Rn W s+2,10 + Rn H 2(s+1) ρ3X 1 holds. for all 0 ≤ s ≤ k and that (Rn )n is a Cauchy sequence in H 2 , hence converges to a unique limit R = R(ρ), the given function which solves (4.19) and satisfies (4.18). Let l = 2 in (4.20). Using that L ∞ ⊂ W 2,10 , one recovers from (4.20) that for all n, 16
(1 + t) 15 Rn (t) H 2k+2 α3X .
Passing to the limit in n, we finish the proof of (4.17).
5. The L 2 -Type Norm In this section, we get control on the first part of the X -norm, namely, we control the L 2 -based norms as follows Proposition 5.1. Let α correspond to a solution of (1.4) by (2.1), then if α has small X -norm, 3
α H˙ −1 ∩H 2k α(0)Y + α X2 holds. The remainder of this section is devoted to the proof of Proposition 5.1. We first control the high derivatives and then the H˙ −1 -norm.
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5.1. The energy estimate. In this subsection, we use energy methods to control high derivatives of the solution in L 2 , assuming a control on the X -norm, and most notably integrability of the solution in L 10 -norms. In order to prove this, we rewrite (1.4) into the symmetrized form ∂t u + A j (u)∂ j u = (0, −∇φ), where u = (ln(1 + ρ), v1 , v2 , v3 ), Aj =
v j e Tj e j v j I3
(5.1)
.
Now, for a multi-index τ , we take derivatives of (5.1) τ times and take the scalar product with D τ u to get 1 d D τ u2L 2 = −(A j (u)D τ ∂ j u, D τ u) L 2 ×L 2 2 dt − cγ (D τ −γ [A j (u)]D γ ∂ j (u), D τ u) − (∇ D τ φ, D τ v) L 2 ×L 2 γ <τ
div(v) L ∞ D τ u2L 2 +
γ <τ
cγ (D τ −γ [A j (u)]D γ ∂ j (u), D τ u) L 2 ×L 2
+|(D τ φ, D τ div(v)) L 2 ×L 2 |. Besides, using (4.1) and (4.4), one sees that ˜ (D τ φ, D τ div(v)) = (D τ (1 − )−1 ρ, D τ div(v)) − (∇ D τ R(ρ), D τ v) τ −1 τ τ −1 = −(D (1 − ) ρ, D ∂t ρ) − (D (1 − ) ρ, D τ div(ρv)) ˜ −(∇ D τ R(ρ), D τ v) with
2 1 ˜ R(ρ) = (1 − )−1 (1 − )−1 ρ − R(ρ) 2 and R given in (4.4). Now, using Lemma 2.1, we remark that for all γ < τ , D τ −γ u D γ ∂ j u L 2 uW 2,10 u H |τ | holds,
and combining this with (4.18), we obtain 1 1 d τ 2 D u L 2 + (1 − )− 2 D τ ρ2L 2 uW 2,10 u2H |τ | + R H |τ |+1 u H |τ | 2 dt 16
(1 + t)− 15 u3X as long as τ ≤ 2k and that α X is sufficiently small. Finally, integrating this in time and remarking that ρ = Re(α) and v = −q(|∇|)RIm(α), we obtain that u2H τ u(0)2H τ + u3X ,
(5.2)
τ provided that τ ≤ 2k. Since control of ln(1 + ρ) in the L ∞ t Hx -norm gives control of ρ τ -norm, this gives us the global bound on the derivatives we needed. in the L ∞ H t x
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5.2. The H˙ −1 -norm. In this subsection, we control the H˙ −1 norm of the solution, which is the other L 2 component of the X -norm. We use (4.10) and we first deal with the quadratic terms Q j (α), whose contribution can be written as a finite sum of terms like (recall that α1 = α 2 ) t m ˆ − η)α(η)dηds, ˆ ei(t−s) p(ξ ) |ξ | α(ξ I = F −1 |ξ | 0 R3 where, from (4.12) we see that one can write m = |ξ |n 1 (ξ )n 2 (ξ − η)n 3 (η) with n i (ζ ) =
ζ ˜ ) |ζ | n(ζ
or n i (ζ ) = n(ζ ˜ ) for n˜ an S 0 -symbol. In particular, n i (|∇|) f L r f L r
for 1 < r < ∞. We use a standard energy estimate and the inclusion W 1,10 ⊂ L ∞ to get t I m α(ξ ˆ − η)α(η)dηds ˆ L 2 L 2 ds 3 |ξ | 0 R |ξ | t n 2 (ξ − η)α(ξ ˆ − η) n 3 (η)α(η) ˆ dη L 2 ds 3 R 0 t (n 2 (|∇|)α) (n 3 (|∇|)α) L 2 ds 0 t n 2 (|∇|)α L ∞ 2 n 3 (|∇|)α(s) L ∞ ds x t Lx 0 t 1 α X (1 − ) 2 n 3 (|∇|)α(s) L 10 ds x 0 t t 1 ds 2 α X (1 − ) 2 α(s) L 10 α X 16 x 0 0 (1 + s) 15 α2X . Next we control the contribution of the cubic term N by using the fact that eit p(|∇|) is a unitary operator and (4.17): t t |∇|−1 e−i(t−s) p(|∇|) N (s)ds L 2 |∇|−1 N L 2 ds 0 0 t ds α2X α2X . 16 0 (1 + s) 15 Combining the two above estimates gives that |∇|
−1
α L ∞ 2 |∇| t Lx
−1
α0 L 2
I (ξ ) + 2 + |ξ | L
α0 Y + α2X , so that we control the first part in the X -norm.
t 0
e−i(t−s) p(|ξ |) ˆ N (s)ds L 2 |ξ | (5.3)
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6. Bilinear Multiplier Theorem 6.1. A general multiplier theorem. In order to control the last part of the norm, we need to deal with bilinear terms in (4.15), (4.16) which involve convolution with a singular symbol. Note that since p(0) = 0, the symbol is quite singular on the whole parameter space and especially near (ξ, η) = (0, 0). In particular, we cannot use the traditional Coifman-Meyer multiplier theorem [2], or a more refined version as in Muscalu, Pipher, Tao and Thiele [18,19] since in all these cases, the multipliers need to satisfy some homogeneity conditions. In order to overcome this we use estimates inspired from Gustafson, Nakanishi and Tsai [13] that we present now. Although most of the results in this subsection are essentially contained in Gustafson, Nakanishi and Tsai [13], for selfcontainedness, we give a direct proof. We introduce the following multiplier norm: η mMs,b = PN m(ξ, η) L b H˙ s , (6.1) ξ,η
η
ξ
N ∈2Z
and we let Msξ,η = Ms,∞ ξ,η , which will be the norm that we mostly use. To a multiplier m, we associate the bilinear “pseudo-product” operator B[ f, g] = Fξ−1 m(ξ, η) fˆ(ξ − η)g(η)dη. ˆ (6.2) R3
Our goal in this section is to obtain robust estimates on B. Lemma 6.1. If m L ∞ H˙ ηs−ε + m L ∞ H˙ s+ε < ∞, then the Msξ,η -norm of m is finite. η
ξ
ξ
Proof. Indeed, we have that η
PN m(ξ, η) L ∞ H˙ s ≤ min(N −ε m L ∞ H˙ s+ε , N ε m L ∞ H˙ ηs−ε ), η η ξ ξ ξ ⎛ ⎞ η ⎠ P η m(ξ, η) ∞ ˙ s PN m(ξ, η) L ∞ H˙ s ⎝ + L H N N
ξ
η
N ≤1
N ≤1
ξ
N ≥1
N ε m L ∞ H˙ ηs−ε + ξ
so that
η
N −ε m L ∞ H˙ s+ε < +∞.
N ≥1
ξ
η
Theorem 6.1. Suppose that 0 ≤ s ≤ n/2 and mMs,∞ = mMsη,ξ < ∞, then η,ξ
B[ f, g]
L l1
mMsη,ξ f L l2 g L 2 ,
(6.3)
for l1 , l2 satisfying 2 ≤ l1 , l2 ≤
2n n − 2s
and
s 1 1 + =1− . l1 l2 n
(6.4)
Remark 6.1. Actually, by changing coordinates (ξ, η) to (ξ, ζ = ξ −η), we could replace the norm Msξ,η by min mMsξ,η , mMsξ,ζ .
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Theorem 6.1 follows by duality from the following estimate which is an adaptation of an estimate from Gustafson, Nakanishi and Tsai [13]. Lemma 6.2. Let 0 ≤ s ≤ n/2, 2 ≤ l1 , l2 , l3 ≤
2n n−2s ,
then
B[ f, g] L l1 mMs,b f L l2 g L l3 ξ,η
for all f ∈ L l2 , g ∈ L l3 , where
1 b
+
1 l1
= 21 , l12 +
1 l3
(6.5)
= 1 − ns . η
Proof. We consider m with a finite Ms,b ξ,η norm. Let Fx denote the Fourier transform from x → η. By definition, we have fˆ(η)g(ξ ˆ − η) = Fxη F yξ f (x + y)g(y).
(6.6)
We introduce a function φ ∈ Cc∞ (Rn ) such that φϕ = ϕ, and we let m N (ξ, η) = η PN m(ξ, η) so that Fηz m N (ξ, η) = φ( Nz )Fηz m N (ξ, η). Using first Parseval’s equality in x, then in η and then in ξ , we see that ˆ ) fˆ(η)g(ξ B[ f, g](x)h(x)d x = m N (ξ, η)h(ξ ˆ − η)dηdξ Rn
= =
R
2n
R
n
R
n
ˆ ) h(ξ ˆ ) h(ξ
R
n
m N (ξ, η) Fxη F yξ f (x + y)g(y) dηdξ Fηx m N (ξ, η) F yξ f (x + y)g(y) dηdξ
R x ˆ ) = φ( )Fηx m N (ξ, η) F yξ f (x + y)g(y) d xdξ h(ξ n n N R R x ˆ ) = Fηx m N (ξ, η) F yξ φ( ) f (x + y)g(y) d xdξ. h(ξ N Rn Rn n
We then use Cauchy-Schwarz’s inequality for the inner integral for x, and then use the Hölder inequality with a1 + b1 = 21 to get ˆ )|Fηx m N (ξ, η) L 2 (ξ )F yξ φ x ( f (x + y)g(y)) L 2 (ξ )dξ |h(ξ x x N Rn x ˆ L a m N (ξ, η) b 2 F yξ φ ≤ h ( f (x + y)g(y)) L 2 L ξ (L η ) ξ x,ξ xN f (x + y)g(y) L 2x,y , ≤ h L a m N (ξ, η) L b (L 2 ) φ η x ξ N where we have used the Hausdroff-Young Inequality for a > 2, and Parseval’s Equality in η for the second factor, as well as Parseval’s Equality in ξ for the third factor. Finally, since φ( Nx ) L n/s N s , we employ the Hardy-Littlewood-Young Inequality with ns + l12 + l13 = 1 to get that φ(
x ) f (x + y)g(y) L 2x,y N s f L l1 g L l2 . N
Combining N s with ||m(ξ, η)|| L b (L 2 ) and by (6.4), we complete the proof. ξ
η
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109
In order to prove Theorem 6.1, it suffices to remark that ˆ )dξ B[ f, g](x)h(x)d x = Fxξ B[ f, g](ξ )h(ξ Rn
R
= =
n
R2n Rn
ˆ )g(ξ fˆ(η)m(ξ, η)h(ξ ˆ − η)dξ dη f (x)B ∗ [h, g](x)d ¯ x.
Applying (6.5) with l1 = 2 to the bilinear operator B ∗ corresponding to the multiplier m∗ (ξ, η) = m(η, ξ ), we get the theorem. 6.2. Multiplier analysis. The control of the L 10 norm is the main mathematical difficulty in this paper. In this subsection, we prove the relevant estimate to apply Theorem 6.1 to the multipliers that appear in our analysis. Lemma 6.3. Let a = b + c ∈ R3 , and let |c| ≤ min{|a|, |b|}, then | p(a) − p(b) − p(c)| |c| {1 − cos[c, a] + 1 − cos[b, a]} |a||b||c| + , (1 + |a||b|)(1 + |c|2 )
(6.7)
where [·, ·] denote the angle between two vectors. Proof. We first note that if |b| ≥ |a|, then p(b) ≥ p(a) and | p(a) − p(b) − p(c)| ≥ p(c) |c|, |a||b| ≤ 1. and the lemma follows since 1+|a||b| We assume |b| ≤ |a|. We remark that, as written in (1.6), p(r ) = rq(r ), where √ 1 ≤ q(r ) ≤ q(0) = 2 and
q (r ) = − q (0)
r (1 + r 2 )2
2+r 2 1+r 2
∼r →∞ −
1 , r3
1 = 0, q (0) = − √ . 2
(6.8)
From this, we get that p(a) − p(b) − p(c) ≤ [|a| − |b| − |c|] q(a) − |b| (q(b) − q(a)) − |c| (q(c) − q(a)) .
(6.9)
From (6.8), we see that q is decreasing and hence each term is non positive. Since |a| = |b| cos[b, a] + |c| cos[c, a], the first term on the right hand side of (6.9) gives the first term on the right hand side in (6.7). |a||b||c| Now we bound (1+|a||b|)(1+|c| 2 ) . Notice first that if cos[c, a] ≤ 9/10, then the last term is bounded by |c|(1 − cos[c, a]) and the lemma is clearly valid. So we can assume that c and a are almost collinear with cos[c, a] ≥ 9/10. In which case, from the geometry of the triangle a, b, c, we get that |a| ≥ 4/3|c| and |a| − |c| ∼ |b| ∼ |a|.
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It suffices to show that, in this case, |a||b||c| |c| (q(a) − q(c)) . (1 + |a||b|)(1 + |c|2 ) We separate three cases: |a| small, |c| large and the rest. Using (6.8), we see that there exists δ > 0 such that −s ≤ q (s) ≤ − 2s for 0 ≤ s ≤ δ. Consequently, if |a| ≤ δ, we get that |c|(q(a) − q(c)) |a| |a|2 − |c|2 −|a||c|(|a| − |c|) −|a||b||c|. = |c| q (s)ds ≤ −|c| 4 |c| On the other hand, since q (r ) ∼ −r −3 at ∞, we see that |a| |a| ds |c|2 − |a|2 |a| − |c| q (s)ds ∼∞ − = − , q(a) − q(c) = 3 2 |c|2 s 2|a| |a||c|2 |c| |c| so that if |c| ≥ δ −1 is sufficiently large, we get that |c|(q(a) − q(c)) −
1 . |c|
Finally, in the last case |a| ≥ δ, |c| ≤ δ −1 and |a| = |c| + (|a| − |c|) ≥ |c| + δ/2. Therefore, |c|+δ/2 |a| δ q (s)ds q (s)ds − max q , δ 4 [ 4δ ,δ −1 +δ] |c| |c|+ 4 and we recover the last term once again.
In the remaining part of this section, we consider the triangle with vertices ξ, η, ξ −η. Definition 6.1. We let θ be the angle between ξ and η (0 ≤ θ ≤ π ), γ the angle between ξ and ξ − η (0 ≤ γ ≤ π ) and we let the angle between η and η − ξ by π − β such that β = γ + θ. We note that sin
β 2
≤
β 2
and sin
β 2
β for 0 ≤ β ≤ π so that
β β2. 2 We now obtain general bounds on the multipliers that arise in our analysis. We first focus on the multiplier associated with the phase 1 . In the end, in Sect. 7, we recover the bounds on the other multipliers using symmetry. 1 − cos β = 2 sin2
Lemma 6.4. For all ξ, η ∈ R3 we have the following estimates on 1 : |η| |∂ξ 1 (ξ, η)| + | sin γ |, max{|ξ − η|, |ξ |} min{|ξ − η|, |ξ |} 2 |ξ | + | sin β|, |∂η 1 (ξ, η)| max{|ξ − η|, |η|} min{|ξ − η|, |η|} 2 |η| , |ξ 1 (ξ, η)| |ξ − η||ξ | 1 . |η 1 (ξ, η)| min(|ξ − η|, |η|)
(6.10) (6.11) (6.12) (6.13)
Euler-Poisson: Ion Equation
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Proof. Recall 1 = p(ξ ) − p(ξ − η) − p(η). We compute ∇ξ 1 = p (ξ ) ξ − p (ξ − η) ξ − η |ξ | |ξ − η| ξ ξ − η ≤ p (ξ ) − p (ξ − η) + | p (ξ )| − |ξ | |ξ − η| γ p (ξ ) − p (ξ − η) + 2 sin . 2 We now claim that p (ξ ) − p (ξ − η)
|η| . max{|ξ − η|, |ξ |} min{|ξ − η|, |ξ |} 2
(6.14)
In fact, if max(|ξ |, |ξ − η|) ≤ 20, from (3.4), using the crude bound | p (s)| 1, we obtain that
|ξ |
|ξ −η|
p (s)ds ||ξ | − |ξ − η|| |η|
|η| . max{|ξ − η|, |ξ |} min{|ξ − η|, |ξ |} 2
Therefore, we only need to consider the case max{|ξ |, |ξ − η|} ≥ 20. Then, if min{|ξ |, |ξ − η|} ≤ 10, we get that |η| max{|ξ |, |ξ − η|} and the right-hand side of (6.10) is of order 1 and the claim is valid. Finally, if min{|ξ |, |ξ − η|} ≥ 10, from (3.4), p (r ) ∼ r13 as r → ∞, and we conclude that claim since |ξ | max{|ξ |,|ξ −η|} 1 p (s)ds dr |ξ −η| min{|ξ |,|ξ −η|} r 3 1 1 − 2 min{|ξ |, |ξ − η|} max{|ξ |, |ξ − η|}2 ||ξ | − |ξ − η|| (|ξ | + |ξ − η|) |η| = |ξ |2 |ξ − η|2 min{|ξ |, |ξ − η|}2 max{|ξ |, |ξ − η|} |η| . max{|ξ − η|, |ξ |} min{|ξ − η|, |ξ |} 2
Similarly, as in the proof of (6.14), ∇η 1 = p (ξ − η) ξ − η − p (η) η |ξ − η| |η| η ξ − η ≤ p (η) − p (ξ − η) + | p (η)| − |η| |ξ − η| β ≤ p (η) − p (ξ − η) + 2 sin 2 |ξ | + | sin β|. max{|ξ − η|, |η|} min{|ξ − η|, |η|} 2
(6.15)
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Using (6.14) and the fact that p (3) (r ) ∼ r −4 as r → +∞ we now compute as in the proof of (6.15) that |ξ 1 | = |p(ξ ) − p(ξ − η)| p (ξ ) p (ξ − η) = p (ξ ) − p (ξ − η) + 2 − |ξ | |ξ − η| |η| | p (ξ ) − p (ξ − η)| + 3 max{|ξ − η|, |ξ |} min{|ξ − η|, |ξ |} |ξ | 1 1 | p (ξ − η)| + − |ξ | |ξ − η|
|η| max{|ξ − η|, |ξ |} min{|ξ − η|, |ξ |} 3 |η| |η| + + . max{|ξ − η|, |ξ |} min{|ξ − η|, |ξ |} 2 |ξ | |ξ − η||ξ |
Finally, we also get that η 1 = −η p(ξ − η) − η p(η) = − p (ξ − η) + p (η) −
2 2 p (ξ − η) + p (η) |ξ − η| |η| 1 1 1 1 + . + + 1 + |ξ − η|3 1 + |η|3 |η| |ξ − η|
This ends the proof.
Proposition 6.1. Define M1 = then if f is either
χ
1
ξ −η 2
or
χ
|ξ ||ξ − η||η| ,
1 ξ − η 2λ η 2λ
for any cutoff function χ with support in
1
η 2
= {max{|ξ |, |ξ − η|, |η|} 1},
(6.16)
we have that, for any ε > 0, ||M1 || || f M1 ||
L∞ η
5 −ε Hξ4 3 −ε
Hξ2 L∞ η
+ ||M1 ||
+ || f M1 ||
L∞ ξ
5 −ε Hη4 3 −ε
Hη2 L∞ ξ
ε 1 f or λ >
9 , 8
(6.17)
ε 1 f or λ > 1.
(6.18)
Proof of Proposition 6.1. In order to prove this proposition, we split R3 into a union of |ξ | three regions: {|ξ | < 21 |η|}, {|η| < 21 |ξ |} and { 13 < |η| < 3}. Before we start, we remark that, in the triangle defined by ξ, η and ξ − η, we have that (see Definition 6.1) |ξ | |η| |η − ξ | = = . sin θ sin β sin γ
Euler-Poisson: Ion Equation
113
! Case 1. T he r egion 1 = |ξ | < 21 |η| . In this case, |ξ − η| ≥ |η| − |ξ | > |ξ |, so |ξ | has the smallest size. We also deduce that |ξ − η| |η| and consequently, since p(ξ ) ≤ p(η), | 1 (ξ, η)| = | p(ξ ) − p(ξ − η) − p(η)| max{|η|, |ξ − η|}. We note that since p is bounded, |∇ξ,η 1 | 1 and from Lemma 6.4, we obtain that ∇ξ,η 1 1 1 = − ∇ξ,η ,
1
21 {|η| + |ξ − η|}2 2 ξ 1 = − ξ 1 + 2 |∇ξ 1 |
1
21
31 1 1 1 1 + + {|η| + |ξ − η|}2 {1 + |ξ |3 } |ξ | {|η| + |ξ − η|}3 1 , |ξ |3 2 η 1 = − η 1 + 2 |∇η 1 |
2
3 1
1
1
1 1 1 1 + . {|η| + |ξ − η|}2 |η| {|η| + |ξ − η|}3 |η|3 1 |η| . Recall the definition of ϕ from (2.3) and denote g = |ξ ||ξ −η||η| 2|ξ | ξ ϕ N ϕ˜ |η| , for some ϕ˜ ∈ Cc∞ . A direct computation yields ξ −η 2λ η 2λ
Note that
1
1
1 |ξ ||ξ − η||η| 1|ξ |N ,|ξ |≤|η| , |ξ | ξ − η 2λ η 2λ 1 |ξ ||ξ − η||η| |∂ξ2 g| 1|ξ |N ,|ξ |≤|η| , |ξ |2 ξ − η 2λ η 2λ
|∇ξ g|
so that
1 M1 ϕ ξ ϕ˜ 2|ξ | 1|ξ |N ,|ξ |≤|η| , and N |η| N 4λ−2 ξ M1 ϕ ξ ϕ˜ 2|ξ | = ξ 1 g + 2∇ξ 1 · ∇ξ g + 1 ξ g N |η|
1
1
1 1 2 1|ξ |N ,|ξ |≤|η| . N N 4λ−2
We thus have that 2|ξ | ξ N 3/2 ϕ˜ || L 2 ||M1 ϕ , ξ N |η| N 4λ−2 2|ξ | ξ 1 ϕ˜ L 2 1/2 . ξ M1 ϕ N |η| N N 4λ−2
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Interpolating between the above estimates, we get that for any ε > 0 and any fixed fixed η, " " " " 2|ξ | ξ "M1 ϕ˜ 2|ξ | " ϕ˜ H˙ σ M 1 ϕ " ξ |η| " H˙ σ N |η| N ξ σ 2|ξ | 1− σ2 2|ξ | ξ ξ ϕ˜ L 2 ξ M1 ϕ ϕ˜ L2 2 ||M1 ϕ N |η| N |η| N
N 23 −σ , N 4λ−2 N
which is summable in N for λ > 1 and 0 ≤ σ < 23 . The same proof (switching ξ to η) # η | works for M ||M1 ϕ( M )ϕ( ˜ 2|ξ |η| )|| H˙ ησ . Both (6.17) and (6.18) are valid in this case if λ > 1 and σ = 3/2 − ε. Case 2. I n the r egion 2 = {|η| ≤ 21 |ξ |}. We note that |η| is the smallest, and |ξ −η| |ξ |. We first claim that |ξ |2 ≡ |η| θ 2 + d 2 . (6.19) | 1 | |η| θ 2 + 2 2 η ξ In fact, if |ξ | is not the largest, then we know that | 1 | ≥ |η| and the claim is clearly valid. If |ξ | is the largest, then θ is the angle between ξ and η (see Definition 6.1) and 1 − cos θ θ 2 . Therefore we deduce (6.19) from Lemma 6.3. We note from Lemma 6.4 that in this case, |η| + | sin γ |, 1 + |ξ |3 |η| . |ξ 1 (ξ, η)| |ξ |2
|∇ξ 1 (ξ, η)|
∇ξ 1
1 ξ 1
1
sin γ |η|
=
sin β |ξ | ,
the inequality above and (6.19), we can obtain that ∇ 1 sin β ξ 1 = − 2 + , 1 |η|{θ 2 + d 2 }2 (1 + |ξ |3 ) |η||ξ |{θ 2 + d 2 }2 |∇ξ 1 |2 ξ 1 = − +2
21
31
Besides, using that
|η| + sin2 γ 1 1 (1+|ξ |3 )2 + |η|{θ 2 + d 2 }2 |ξ |2 |η|3 {θ 2 + d 2 }3 1 1 + |η|{θ 2 + d 2 }2 |ξ |2 (1 + |ξ |3 )2 |η|{θ 2 + d 2 }3 sin2 β + . |η||ξ |2 {θ 2 + d 2 }3 2
Now, for fixed η, denote g=
|ξ ||ξ − η||η| ϕ ξ − η 2λ η 2λ
ξ N
2|η| ϕ˜ , |ξ |
(6.20)
Euler-Poisson: Ion Equation
115
since |ξ | N and |η| ≤ |ξ |, direct computation yields 1 |ξ ||ξ − η||η| 1|ξ |N ,|η|≤|ξ | , |ξ | ξ − η 2λ η 2λ 1 |ξ ||ξ − η||η| |∂ξ2 g| 1|ξ |N ,|η|≤|ξ | . |ξ |2 ξ − η 2λ η 2λ
|∇ξ g|
Therefore, we have that |ξ |2 M1 ϕ ξ ϕ˜ 2|ξ | 1|ξ |N ,|η|≤|ξ | , N |η| N 2λ η 2λ (θ 2 + d 2 ) and by (6.20) we also have that ξ M1 ϕ ξ ϕ˜ 2|ξ | N |η| 1 1 1 = ξ g + 2∇ξ · ∇ξ g + ξ g
1
1
1 2 |ξ | 1|ξ |N ,|η|≤|ξ | 1 1 sin2 β , + + N 2λ η 2λ {θ 2 + d 2 }2 |ξ |2 (1 + |ξ |3 )2 {θ 2 + d 2 }3 |ξ |2 {θ 2 + d 2 }3 η where we have used the fact that θ 2 + d 2 1. By using |η| as the north pole, we thus compute: 2 |ξ |2 sin θ N4 M1 ϕ ξ ϕ˜ 2|ξ | dξ d|ξ |dθ 4λ 4λ 2 2 2 N |η| N η |ξ |N (θ + d ) N6 θ dθ dθ = d|ξ | + 3 N 4λ η 4λ |ξ |∼N θ≤d d 4 θ≥d θ
N7 N 4λ η 4λ
1 N5 . 2 4λ−2 d N η 4λ−2
Next, since β = θ + γ and γ , β θ (see Definition 6.1) and d
(6.21) N η N ,
we have
2 ξ M1 ϕ ξ ϕ˜ 2|ξ | dξ N |η| 6 1 N 1 sin4 β + + θ dθ d|ξ | N 4λ η 4λ |ξ |N {θ 2 + d 2 }4 N 4 N 12 {θ 2 + d 2 }6 N 4 {θ 2 + d 2 }6 θ dθ dθ N6 × + 8 4 7 4 N 4λ η 4λ |ξ |N θ≤d d N θ≥d θ N $ % 1 θ dθ dθ θ 5 dθ dθ 1 + 4 + + + d|ξ | 11 12 7 N 12 θ≤d d 12 N θ≥d θ θ≤d d θ≥d θ N3 N7 N3 + + N 4λ η 4λ d 6 N 4λ η 4λ N 12 d 10 N 4λ η 4λ d 6 1 , 4λ−6 N η 4λ−6 N 3
(6.22)
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where we have used the fact
1+|η|2 1+N 2
1. By interpolation between (6.21) and (6.22),
2|ξ | ϕ˜ H˙ σ M1 ϕ ξ |η| σ 2|ξ | 1− σ2 2|ξ | ξ ξ ϕ˜ L 2 ξ M1 ϕ ϕ˜ L2 2 M1 ϕ N |η| N |η| 5 σ η σ N σ N 2 (1− 2 ) N 2λ−1 η 2λ−1 N 43 σ
ξ N
5
η σ N σ N 2 −2σ N 2λ−1 η 2λ−1
as |η| N . By taking σ = 45 −ε, this is summable for N for 4λ−2 > 25 . This concludes (6.17). On the other hand, in (see (6.16) we have |ξ | |ξ − η| ≥ 1 so that N 1 and we deduce that this is summable for N ≥ 1, σ = 3/2 − ε when λ > 1. This concludes (6.18). We now turn to the η derivatives. Using again Lemma 6.4, we have that |∇η 1 (ξ, η)|
|ξ | 1 + | sin β|, and |η 1 (ξ, η)| , 2 η ξ |η|
and therefore by (6.19), |ξ | sin β ∇η 1 = − ∇η 1 + 2 2 , 2 2 2 2 2 2 1 |η| {θ + d } ξ η
1 |η| {θ + d 2 }2 2 η 1 |∇
| 1 η 1 = − η +2
1
21
31 1 |ξ |2 1 2 + + sin β . |η|3 {θ 2 + d 2 }2 |η|3 {θ 2 + d 2 }3 ξ 2 η 4 Define g by g=
|ξ ||ξ − η||η| η ϕ˜ ϕ ξ − η 2λ η 2λ M
2|η| . |ξ |
Since |η| M and |η| |ξ |, a direct computation yields 1 |ξ ||ξ − η||η| 1|η|M,|η|≤|ξ | , |η| ξ − η 2λ η 2λ 1 |ξ ||ξ − η||η| 1|η|M,|η|≤|ξ | . |∂η2 g| |η|2 ξ − η 2λ η 2λ
|∇η g|
Euler-Poisson: Ion Equation
117
Hence, since sin β sin θ by Definition 6.1, (6.19) and the fact that θ 2 + d 2 1, η M1 ϕ η ϕ˜ 2|ξ | M |η| 1 1 1 = η g + 2∇η · ∇η g + η g
1
1
1
|ξ |2
M 2 {θ 2
1|η|M,|η|≤|ξ | + d 2 }2 M 2λ ξ 2λ
|ξ |2 + 2 2 M {θ + d 2 }3 M 2λ ξ 2λ By using
ξ |ξ |
as the north pole, and d
|ξ |2 2 + sin θ 1|η|M,|η|≤|ξ | . ξ 2 η 4
|ξ | M ξ ,
we thus compute that
η 2|ξ | ϕ˜ |2 dη M |η| |η|∼M sin θ |ξ |4 dηdθ M 4λ ξ 4λ |η|M θ 2 + d 2 2 θ dθ dθ |ξ |4 M 2 d|η| + 3 M 4λ ξ 4λ |η|M θ≤d d 4 θ≥d θ
|M1 ϕ
|ξ |4 M 3 1 |ξ |2 M 3 . 4λ 4λ 2 4λ−2 M ξ d M ξ 4λ−2
(6.23)
Next, since β = θ + γ and γ , β θ , and we have that |η|∼M
2 η M1 ϕ η ϕ˜ 2|ξ | dη M |η| 1
M 2 M 4λ ξ 4λ
|ξ |4 |ξ |8 |ξ |4 sin4 θ θ dθ d|η| + + 2 2 4 {θ 2 + d 2 }6 ξ 4 η 8 {θ 2 + d 2 }6 |η|M {θ + d } |ξ |4 θ dθ dθ 2 + 8 7 M M 4λ ξ 4λ |η|M θ≤d d θ≥d θ $ % θ dθ dθ θ 5 dθ dθ |ξ |4 + + 4 + + d|η| 11 12 7 ξ M 8 θ≤d d 12 θ≥d θ θ≤d d θ≥d θ
×
|ξ |4 |ξ |8 |ξ |4 + + M M 4λ ξ 4λ d 6 M M 4λ+8 ξ 4λ+4 d 10 M M 4λ ξ 4λ d 6 1 , 4λ−6 M M ξ 4λ−6 |ξ |2
(6.24)
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Y. Guo, B. Pausader
where d
|ξ | M ξ .
Interpolating between (6.23) and (6.24), we obtain
η 2|ξ | η 2|ξ | 1− σ ϕ˜ H˙ σ M1 ϕ ϕ˜ L 2 2 η M1 ϕ η M |η| M |η| η 2|ξ | σ ϕ˜ L2 2 × M1 ϕ M |η| 21 − σ4 σ 4 |ξ |2 M 3 1 4λ−2 4λ−2 4λ−6 4λ−6 2 M ξ M M ξ |ξ | 3
M 2 −σ . M 2λ−1−σ ξ 2λ−1−σ |ξ |σ −1 By taking σ = 45 − ε, this is summable in M if 2λ − 1 − σ > 0 and we conclude (6.17). On the other hand, in , we have |ξ | ≥ 1 so that by taking σ = 23 − ε, this is summable if λ > 1 with f = 1 1 or 1 1 . ξ −η 2
η 2
|ξ | Case 3. T he r egion 3 = 13 < |η| < 3 . In this region, we have |ξ − η| ≤ 4 min(|ξ |, |η|), |ξ | |η| are of the order of the longest side and sin γ sin β. Therefore we claim that | 1 | |ξ − η|(γ 2 + β 2 +
|ξ |2 + |η|2 ) ≡ |ξ − η|(γ 2 + β 2 + d12 ). (1 + |ξ |2 + |η|2 ) ξ − η 2 (6.25)
The above lower bound is trivial if |ξ | is not the largest by (6.7). If |ξ | is the largest and |ξ − η| is not the smallest, then ξ, η, ξ − η are all comparable so that γ θ π − β and from (6.7), | 1 | θ 2 |η| +
|η| |ξ − η|(γ 2 + β 2 + d12 ). 1 + |η|2
Finally, when |ξ − η| is the smallest, this follows from (6.7). Moreover, from (6.10) and (6.11), |∂ξ,η 1 (ξ, η)|
|η| 1 , + | sin γ |, |ξ,η 1 (ξ, η)| 2 η ξ − η |ξ − η|
(6.26)
and therefore, ∇ξ,η 1 1 ∇ξ,η = −
1
21 |η| | sin γ | + |ξ − η|2 (β 2 + γ 2 + d12 )2 η ξ − η 2 |ξ − η|2 (β 2 + γ 2 + d12 )2 |∇ξ,η 1 |2 ξ,η 1 = − +2
21
31
ξ,η 1
1
Euler-Poisson: Ion Equation
119
1 |η|2 + |ξ |2 + |ξ − η|3 (β 2 + γ 2 + d12 )2 |ξ − η|3 (β 2 + γ 2 + d12 )3 η 2 ξ − η 4 +
sin2 γ + sin2 β . |ξ − η|3 (β 2 + γ 2 + d12 )3
For fixed η and a dyadic number N , denote g=
|ξ ||ξ − η||η| ϕ ξ − η 2λ η 2λ
ξ −η |η| ϕ . N |ξ |
As before, direct computation yields |ξ ||ξ − η||η| 1 1|ξ −η|N ,|ξ ||η| , |ξ − η| ξ − η 2λ η 2λ |ξ ||ξ − η||η| 1 2 g| 1|ξ −η|N ,|ξ ||η| . |∂ξ,η |ξ − η|2 ξ − η 2λ η 2λ
|∇ξ,η g|
Therefore, 2 M1 ϕ ξ − η ϕ |η| |η| 1|ξ −η|N ,|ξ ||η| , N |ξ | N 2λ η 2λ β 2 + d12 and ξ M1 ϕ ξ − η ϕ |η| N |ξ | 1 1 1 |ξ g + 2∇ξ · ∇ξ g + ξ g|
1
1
1 $ % |η|2 |η|4 |η|2 sin2 β 1|ξ −η|N ,|ξ ||η| + . 2 + 3 3 N 2 N 2λ η 2λ β 2 + d12 β 2 + d12 η 2 ξ − η 4 β 2 + d12 η By using − |η| as the north pole, and d1
|η| η N ,
we thus compute from (6.25):
2 M1 ϕ ξ − η ϕ |η| dξ N |ξ | sin β |η|4 N 2 d|ξ |dβ N 4λ η 4λ |ξ −η|N (β 2 + d12 )2 $ % |η|4 N 3 βdβ dβ = + 4 3 N 4λ η 4λ β≤d1 d1 β≥d β
|η|4 N 3 1 |η|2 N 3 . N 4λ η 4λ d12 N 4λ−2 η 4λ−2
(6.27)
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Y. Guo, B. Pausader
Next, we have that
2 |η|4 ξ M1 ϕ ξ − η ϕ |η| dξ N |ξ | N 4λ η 4λ N 4 |ξ −η|N |ξ −η|∼N $ % 1 |η|4 β4 × + + d(ξ − η) (β 2 + d12 )4 (β 2 + d12 )6 η 4 N 8 (β 2 + d12 )6 $ % βdβ dβ |η|4 N 2 d|ξ − η| + 8 7 N 4λ η 4λ N 4 |ξ −η|N β≤d1 d1 β≥d1 β $ % $ % βdβ dβ β 5 dβ dβ |η|4 + + + 4 + 12 12 11 7 η N 8 β≤d1 d1 β≥d1 β β≤d1 d1 β≥d1 β |η|4 |η|8 |η|4 + + N 4λ η 4λ N d16 N 4λ+8 η 4λ+4 N d110 N 4λ N η 4λ d16 1 , N 4λ−6 η 4λ−6 |η|2 N
since d1
|η| η N .
(6.28)
Interpolating between (6.27) and (6.28), we have
ξ −η |η| ϕ || H˙ σ ξ N |ξ | σ |η| |η| ξ −η ξ −η 1− σ ϕ || L 2 2 ||ξ M1 ϕ ϕ || L2 2 ||M1 ϕ N |ξ | N |ξ | 21 − σ4 σ 4 1 |η|2 N 3 N 4λ−2 η 4λ−2 N 4λ−6 η 4λ−6 |η|2 N
||M1 ϕ
3
N 2 −σ 2λ−1−σ N η 2λ−1−σ |η|σ −1
as N |η|. By taking σ = 45 − ε, this is summable for N when 4λ − 2 > 25 , hence we deduce (6.17). On the other hand, in , we know that |η| |ξ | ≥ 1. Hence for 1 1 3 f = 1 or 1 , we can take σ = 2 − ε and still get a convergent series. Equa ξ −η 2
η 2
tion (6.18) therefore follows. The η derivatives can be controlled similarly since we had the same control. 7. The L 10 Bound and End of the Proof 7.1. Estimating the L 10 bound. Using the results of Sect. 6, we can now estimate the last part of the X norm. Proposition 7.1. Let α be a solution of (4.14), then 16
sup(1 + t) 15 α(t)W k,10 α0 Y + α2X . t≥0
(7.1)
Euler-Poisson: Ion Equation
121
Proof. We use Theorem 6.1 and Proposition 6.1 to control the nonlinear terms appearing in (4.14). Our strategy is first to establish the proposition for 1 , and then we use symmetry to conclude all the other cases. We first deal with the cubic terms using (4.17) as follows. We let
A(ξ − η) =
n 2 (ξ − η) n 3 (η) 2λ ξ − η 2λ and B(η) = η . |ξ − η| |η|
We note that A and B are smooth and obey symbol like estimates. We first apply Proposition 3.1 to get that, for a typical term,
k
(1 − ) 2 F −1
0 t
t
t
0
0
(1 + t − s) 15 1 16
0
1 16
0
t
im(ξ, η) ˆ ei(t−s) p(|ξ |) α(s, ˆ ξ − η)h(α)(s, η)dsdη L 10
1 R3 |ξ | −1 k ˆ Fξ n 2 (ξ − η)α(ξ ˆ − η)n 3 (η)h(η)dη 12 , 10 ds n 1 (ξ ) ξ W 5 9 R 3 1 12 ˆ Fξ−1 ξ k+ 5 M1 A(ξ − η)α(ξ ˆ − η)B(η)h(η) 10 ds
t
(1 + t − s) 15 1 16
A(|∇|)α
16
α
(1 + t − s) 15 1 (1 + t − s) 15
L
R3
k+ 12 H 5
k+2λ+ 7 5 H˙ −1 ∩H
9
B(|∇|)h L l2 + A(|∇|)α L l2 B(|∇|)h |∇|−1 h H 3 + α H˙ −1 ∩H 2 |∇|−1 h
k+ 12 H 5
ds
k+2λ+ 12 5 H
ds
16
(1 + t)− 15 α3X
since k ≥ 2λ + 75 . Here we have applied Lemma 6.1 around s = 45 − ε, Proposition 6.1 60 > 2. To finish the analysis of and Theorem 6.1 with l1 = 10, b = ∞, and l2 = 29+20ε the cubic term, we also need to control the cubic term pre-normal form in (4.14). We use the fact that eit p(|∇|) is a unitary operator on H k and (4.17) to get that
F −1
0
t
ei(t−s) p(ξ ) Nˆ 1 (α)(s)dsW k,10
t
0
0
t
ei(t−s) p(|∇|) N1 (α)(s) H k+2 ds |∇|−1 N1 (α)(s) H k+3 ds 16
(1 + t)− 15 α2X . To estimate the integrated term B in (4.14), we need to separate the regions. First we control the integrated part when all the terms are small, M = max(|ξ |, |ξ − η|, |η|) < 3.
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6 To do this, we first note that Sobolev’s embedding L 10 ⊂ H˙ 5 , and the fact that for 6 bounded ξ, |ξ | 5 |ξ | |ξ − η| + |η| to get2
B(α(t), α(t)) L 10 6 ˆ ξ − η) α(t, ˆ η) |ξ ||ξ − η|||η| m α(t, dη L 2 Fξ−1 |ξ | 5 i 1 |ξ | |ξ − η| |η| R3 ˆ η) η 2λ α(t, dη L 2 M1 n 1 (ξ )n 2 (ξ − η)n 3 (η) ξ − η 2λ α(t, ˆ ξ − η) 3 |η| R n 2 (|∇|)α L 10 |∇|−1 n 3 (|∇|)α L 2 α L 10 α X ,
where we have applied Proposition 6.1 for M1 for s = 45 − ε, Lemma 6.1, and Lemma 60 > 2. 6.2 with l1 = 2, b = ∞, l2 = 10 and l3 = 29+20ε Next we deal with the case when one of the frequencies is large, M > 1. This can happen in two cases. First, if |ξ | ≤ 1, M > 1. In this case, we have |η| |ξ − η| ≥ 1. We bound the L 10 norm by the L 2 norm via Sobolev’s inequality (with bounded ξ ) to get B L 10 Fξ−1 Fξ−1 Fξ−1
R3
R3
m(ξ, η) χ α(t, ˆ ξ − η)α(t, ˆ η)dη L 10 i 1 1
n 1 (ξ ) f M j n 2 (ξ −η)
R3
1
ˆ ξ − η) ˆ η) ξ − η 2λ+4 α(t, η 2λ+ 4 α(t, n 3 (η) dη L 10 |ξ − η| |η| 1
1
ˆ ξ − η) ˆ η) ξ − η 2λ+ 4 α(t, η 2λ+ 4 α(t, n 3 (η) dη L 2 |ξ − η| |η| α X αW 3,10 .
f M j n 2 (ξ − η)
α H 2 n 3 (|∇|)α 2, 3 W ε
We have applied Proposition 6.1 with s = 23 − ε, Lemma 6.1 around s = 23 − ε and Lemma 6.2 with s = 23 − ε, b = ∞ and l1 = 2. This concludes the estimates in the region {|ξ | ≤ 1} ∩ {M ≥ 1}. The other case is included in the region = {|η| ≤ 2|ξ − η|, |ξ | ≥ 1/2} ∪ {|η| > |ξ − η|, |ξ | ≥ 1/2} and leads to the worst loss in derivatives (whereas the region when all frequencies are small leads to the loss of smoothness of the multiplier and hence to the loss of decay in time). In the case |η| ≤ 2|ξ − η|, we choose f = χ 1 in Proposition 6.1. We apply η 2
Lemma 6.1 to deduce that 6
|ξ |k+ 5
k+ 65
ξ − η
3
−ε
2 M1 f ∈ Mξ,η .
2 Here we forget the difference between n and n and treat the terms as symmetric. 2 3
Euler-Poisson: Ion Equation
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Hence |∇|k B(α, α) L 10 $
% 1 ˆ −η) ˆ ξ −η 2λ α(ξ η 2λ+2 α(η) n 3 (η) dη L 10 f M1 n 2 (ξ −η) |ξ −η| |η| R3 6 6 1 |ξ |k+ 5 ˆ −η) ξ −η k+2λ+5 α(ξ η 2λ+2 −1 n α(η)dη ˆ F M f n (ξ −η) (η) 1 2 3 L2 6 |ξ −η| |η| R3 ξ −η k+ 5 Fξ−1
|ξ |k n 1 (ξ )
6
|ξ |k+ 5 ξ −
6 η k+ 5
(1 − )2 k+ 11 5 +2δ n 2 (|∇|)α l n 3 (|∇|)α L l2 L3 |∇|
M1 f
|∇| 3 2 −ε Mξ,η 2
11 (1 − ) |∇|k+ 5 +2δ α L l2 α L l3 . |∇|
We have applied Lemma 6.2 with 3 − ε, 2
s=
1 1 7 17 11 4 + ε and − ε. = = l2 25 15 l3 50 5
Now, using Bernstein estimates (2.4), we compute that P≤1
(1 − )2 α L l2 N −1 PN α L l2 |∇| N ≤1
N ≤1
N ≤1
N
−1
N
3
1 1 l4 − l2
PN α L l4
1−σ σ N ε N −1 PN α L 2 PN ασL 10 α1−σ X α L 10 ,
for σ =
3 1 19 4 1 2σ − 2ε, and = + ε = − 10 l4 2 5 50 5
while for the high frequencies, we have that P≥1
3 3 (1 − )2 α L l2 (1 − ) 2 ασL 10 (1 − ) 2 α1−σ L l5 |∇|
ασW 3,10 α1−σ X for 5 + 40 1 21 ε . = 14 20 l5 1+ 7 ε
Independently, we have that 11
k
k
11
1
|∇|k+ 5 +2δ α L l3 (1 − ) 2 α1−σ (1 − ) 2 +( 10 +2δ) σ ασL 2 L 10 α1−σ ασX , W k,10 provided that k > 11/(5σ ) =
22 3
+ ε.
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χ In the case |η| > |ξ − η| we proceed similarly with f = ξ −η 1/2 . We therefore conclude the proposition for 1 . We now have completed the proof for j = 1. To establish (7.1) for j = 1, we note that the proposition is clearly valid for 2 because the proof in Case 1 shows that Proposition 6.1 is also valid in this easier case (indeed, | 2 | max(|ξ |, |ξ − η|, |η|)). For 4 , we note 4 (ξ, η) = − 1 (η, ξ ), and repeat the same proof in light of Proposition 6.1. Finally, for 3 (ξ, η) = − 1 (ξ − η, ξ ), we make a change of integration variable η → ξ − η in the integrations in both the cubic terms and B and get back to the previous case. We thus conclude the proof.
7.2. End of the proof. Now, we are ready to finish the proof of Theorem 1.1. Proof of Theorem 1.1. The existence of a local regular solution β ∈ C(0, T ∗ ), X ) follows from the standard method of Kato [15]. Combining Proposition 5.1 and Proposition 7.1, we obtain that β X α(0)Y + β2X , so that if α(0)Y is sufficiently small, we get a global bound on the X -norm of the solution, which implies that T ∗ = ∞ and gives a global bound on the X -norm of ρ and v. This ends the proof. Acknowledgments. Y. Guo’s research is supported in part by DMS-0530862 and a Chinese NSF grant. The authors thank Jingjun Zhang for pointing out typos in the original manuscript.
References 1. Chen, G.Q., Jerome, J.W., Wang, D.: Compressible Euler-Maxwell equations. Proceedings of the Fifth International Workshop on Mathematical Aspects of Fluid and Plasma Dynamics (Maui, HI, 1998). Transport Theory Statist. Phys. 29(3–5), 311–331, (2000) 2. Coifman, R., Meyer, Y.: Commutateurs d’intégrales singulières et opérateurs multilinéaires. Ann. Inst. Fourier (Grenoble) 28(3, xi), 177–202 (1978) 3. Cordier, S., Grenier, E.: Quasineutral limit of an Euler-Poisson system arising from plasma physics. Comm. Part. Diff. Eqs. 25(5–6), 1099–1113 (2000) 4. Feldman, M., Ha, S.-Y., Slemrod, M.: Self-similar isothermal irrotational motion for the Euler, EulerPoisson systems and the formation of the plasma sheath. J. Hyp. Diff. Eq. 3(2), 233–246 (2006) 5. Feldman, M., Ha, S.-Y., Slemrod, M.: A geometric level-set formulation of a plasma-sheath interface. Arch. Ratn. Mech. Anal. 178(1), 81–123 (2005) 6. Germain, P., Masmoudi, N., Shatah, J.: Global solutions for 3D quadratic Schrödinger equations. Int. Math. Res. Not., 2009(3), 414–432 (2009) 7. Germain, P., Masmoudi, N., Shatah, J.: Global solutions for the gravity water waves equation in dimension 3. Preprint, available at http://arxiv.org/abs/1001.5158v1 [math.AP], 2010 8. Germain, P., Masmoudi, N., Shatah, J.: Global solutions for 2D quadratic Schrödinger equations. Preprint, available at http://arxiv.org/abs/0906.5343v1 [math.Ap], 2009 9. Guo, Y.: Smooth irrotational Flows in the large to the Euler-Poisson system in R 3+1 . Commun. Math. Phys. 195, 249–265 (1998) 10. Guo, Y., Tahvildar-Zadeh, A.S.: Formation of singularities in relativistic fluid dynamics and in spherically symmetric plasma dynamics. In: Nonlinear partial differential equations (Evanston, IL, 1998), Contemp. Math., 238, Providence, RI: Amer. Math. Soc., 1999, pp. 151–161 11. Guo, Z., Peng, L., Wang, B.: Decay estimates for a class of wave equations. J. Funct. Anal. 254(6), 1642– 1660 (2008) 12. Gustafson, S., Nakanishi, K., Tsai, T.P.: Global dispersive solutions for the Gross-Pitaevskii equation in two and three dimensions. Ann. IHP 8(7), 1303–1331 (2007)
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13. Gustafson, S., Nakanishi, K., Tsai, T.P.: Scattering theory for the Gross-Pitaevskii equation in three dimensions. Commun. Contemp. Math. 11(4), 657–707 (2009) 14. John, F.: Plane Waves and Spherical Means, Applied to Partial Differential Equations. J. Appl. Math. Mech. 62(7), 285–356 (1982) 15. Kato, T.: The Cauchy problem for quasilinear symmetric systems. Arch. Rat. Mech. Anal. 58, 181–205 (1975) 16. Liu, H., Tadmor, E.: Critical thresholds in 2D restricted Euler-Poisson equations. SIAM J. Appl. Math. 63(6), 1889–1910 (2003) (electronic) 17. Liu, H., Tadmor, E.: Spectral dynamics of the velocity gradient field in restricted flows. Commun Math. Phys. 228(3), 435–466 (2002) 18. Muscalu, C.: Paraproducts with flag singularities. I. A case study. Rev. Mat. Iberoam. 23(2), 705–742 (2007) 19. Muscalu, C., Pipher, J., Tao, T., Thiele, C.: Multi-parameter paraproducts. Rev. Mat. Iberoam. 22(3), 963–976 (2006) 20. Peng, Y., Wang, S.: Convergence of compressible Euler-Maxwell equations to compressible EulerPoisson equations. Chin. Ann. Math. Ser. B 28(5), 583–602 (2007) 21. Peng, Y., Wang, Y.-G.: Boundary layers and quasi-neutral limit in steady state Euler-Poisson equations for potential flows. Nonlinearity 17(3), 835–849 (2004) 22. Shatah, J.: Normal forms and quadratic nonlinear Klein-Gordon equations. Comm. Pure Appl. Math. 38(5), 685–696 (1985) 23. Sideris, T.: Formation of singularities in three-dimensional compressible fluids. Commun. Math. Phys. 101, 475–485 (1985) 24. Stein, E.: Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals. Volume 43 of Princeton Mathematical Series. Princeton, NJ: Princeton University Press 1993, With the assistance of Timothy S. Murphy, Monographs in Harmonic Analysis, III 25. Tao, T.: Nonlinear dispersive equations, local and global analysis. CBMS. Regional Conference Series in Mathematics, 106. Published for the Conference Board of the Mathematical Science, Washington, DC; Providence, RI: Amer. Math. Soc., 2006 26. Texier, B.: WKB asymptotics for the Euler-Maxwell equations. Asymptot. Anal. 42(3-4), 211–250 (2005) 27. Texier, B.: Derivation of the Zakharov equations. Arch. Ration. Mech. Anal. 184(1), 121–183 (2007) 28. Wang, D.: Global solution to the equations of viscous gas flows. Proc. Roy. Soc. Edinburgh Sect. A 131(2), 437–449 (2001) 29. Wang, D., Wang, Z.: Large BV solutions to the compressible isothermal Euler-Poisson equations with spherical symmetry. Nonlinearity 19(8), 1985–2004 (2006) Communicated by P. Constantin
Commun. Math. Phys. 303, 127–148 (2011) Digital Object Identifier (DOI) 10.1007/s00220-011-1195-z
Communications in
Mathematical Physics
Topological Geon Black Holes in Einstein-Yang-Mills Theory George T. Kottanattu, Jorma Louko School of Mathematical Sciences, University of Nottingham, Nottingham NG7 2RD, UK. E-mail:
[email protected];
[email protected] Received: 24 March 2010 / Accepted: 19 September 2010 Published online: 10 February 2011 – © Springer-Verlag 2011
Abstract: We construct topological geon quotients of two families of Einstein-YangMills black holes. For Künzle’s static, spherically symmetric SU(n) black holes with n > 2, a geon quotient exists but generically requires promoting charge conjugation into a gauge symmetry. For Kleihaus and Kunz’s static, axially symmetric SU(2) black holes a geon quotient exists without gauging charge conjugation, and the parity of the gauge field winding number determines whether the geon gauge bundle is trivial. The geon’s gauge bundle structure is expected to have an imprint in the Hawking-Unruh effect for quantum fields that couple to the background gauge field. 1. Introduction Given a stationary black hole spacetime with a bifurcate Killing horizon, it may be possible to construct from it a time-orientable quotient spacetime in which the two exterior regions separated by the Killing horizon become identified. If the quotient is asymptotically flat, its spatial geometry is that of a compact manifold minus a point, with the omitted point at an asymptotically flat infinity. This makes the quotient a topological geon in the sense introduced by Sorkin [1], as motivated by the earlier work in [2–4]. The showcase example is the RP3 geon [5–8], formed as a Z2 quotient of Kruskal. There exist also quotients in which the infinity is only asymptotically locally flat, and others in which the infinity is asymptotically anti-de Sitter or asymptotically locally anti-de Sitter [9–12]. In this paper we shall understand topological geons to encompass all of these cases, the characteristic property being that the infinity consists of only one component. Topological geon black holes of the kind described above are unlikely to be created in an astrophysical star collapse, as their formation from conventional initial data would require a change in the spatial topology. However, they provide an arena for the Hawking-Unruh effect in a setting where the black hole is eternal and has nonvanishing surface gravity but thermality for a quantum field cannot arise by the usual procedure
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of tracing over a causally disconnected exterior [13]. There is still thermality, in the usual Hawking temperature, but only for a limited set of observations, and the non-thermal correlations bear an imprint of the absence of the causally disconnected exterior [9–12,14,15]. In a sense, the Hawking-Unruh effect on a topological geon black hole reveals to an exterior observer features of the geometry that are classically hidden behind the horizons. A recent review can be found in [16]. When the black hole has a Maxwell field, it may be necessary to include charge conjugation in the map with which the black hole gauge bundle is quotiented into the geon gauge bundle [12]. This happens for example for the Reissner-Nordström hole, both with electric and magnetic charge; it also happens for the higher-dimensional ReissnerNordström hole with electric charge in any dimension and with magnetic charge in even dimensions. Maxwell’s theory on the geon must then incorporate charge conjugation as a gauge symmetry, rather than just as a global symmetry: technically, the gauge group is no longer U(1) SO(2) but Z2 U(1) O(2), where the nontrivial element of Z2 acts on U(1) by complex conjugation [17]. The presence of the charge conjugation in the quotienting map can further be verified to leave its imprint in the Hawking-Unruh effect for a quantum field that couples to the background Maxwell field [18,19]. By contrast, spherically symmetric Einstein-SU(2) black holes admit a geon quotient without the inclusion of charge conjugation in the quotienting map, and the geon’s gauge bundle is in fact trivial [12]. The purpose of this paper is to construct two new families of Einstein-Yang-Mills geon black holes. We shall specifically examine whether charge conjugation needs to be promoted into a gauge symmetry when taking the geon quotient. We take the gauge group to be SU(n) with n ≥ 2, a choice motivated physically by the appearance of these groups in particle physics and mathematically by their amenability to a unified treatment. We shall work with pure Einstein-Yang-Mills, but we note that these gauge groups, and the definition of spherical symmetry in terms of SU(2) rather than SO(3) [20,21], provide opportunities for extensions that include spinor as well as scalar fields. In Sects. 2 and 3 we consider the static, spherically symmetric Einstein-SU(n) black holes of Künzle [22] and their generalisations to a negative cosmological constant [23,24]. The case n = 2 was covered in [12] as discussed above. For n > 2 we show that a geon quotient exists and generically requires including charge conjugation in the quotienting map: the enlarged gauge group is Z2 SU(n), where the nontrivial element of Z2 acts on SU(n) by complex conjugation. A quotient without charge conjugation is possible only for certain special field configurations, of which we give a complete list, and we show that the geon gauge bundle is then trivial. In Sects. 4 and 5 we consider the static, axially symmetric Einstein-SU(2) black holes of Kleihaus and Kunz [25,26]. We show that all these holes admit a geon quotient without gauging charge conjugation. When the winding number of the gauge field configuration is odd, the geon gauge bundle is trivial; this includes as a special case the spherically symmetric geon discussed in [12]. When the winding number is even, the geon gauge bundle is nontrivial. Section 6 summarises the results and discusses their relevance for the Hawking-Unruh effect. The metric signature is (−+++). Sections 2 and 3 use the convention of an antihermitian gauge field, common in mathematical literature. Sections 4 and 5 use the convention of a hermitian gauge field, common in physics literature. Homotopies are assumed smooth, without loss of generality [27].
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2. Spherically Symmetric SU(n) Black Holes In this section we review the relevant properties of the static, spherically symmetric SU(n) Einstein-Yang-Mills black holes of Künzle [22] and their generalisations to a negative cosmological constant [23,24]. We also give explicit Kruskal-like coordinates that extend these solutions across the Killing horizon. We assume n > 2 when not explicitly mentioned otherwise, although most of the formulas hold also for n = 2. 2.1. Exterior ansatz. Given a group action on the base space of a principal bundle, the notion of a symmetric gauge field can be formulated as invariance of the connection under an appropriate group action on the total space. Concretely, let P be a principal bundle with base manifold M, projection π : P → M and structure group G with Lie algebra g. Let H be a group and φ : H × M → M its action on M. We say that the connection form ω ∈ 1 (P, g) is H -symmetric if the following three conditions hold: 1. For each h ∈ H , there is a h ∈ Aut(P) such that π ◦h = φh ◦π with Id H = Id P ; 2. ∗h ω = ω for all h ∈ H ; 3. The map H → Aut(P) given by h → h is a group homomorphism. This is essentially the definition adopted in [28]. Condition 3 is known to have undesirably restrictive consequences in some situations, such as when H is a translation group [29], but for our applications the definition will be satisfactory. We take M to be a static, spherically symmetric spacetime and G to be SU(n). In this subsection we specify M by a coordinate-based ansatz. The ansatz does not cover all regions of the Kruskal-type black and white whole spacetimes that will be introduced in Subsect. 2.2, but we shall see that the ensuing gauge field will remain spherically symmetric when appropriately continued beyond the coordinate singularities. The metric ansatz is ds 2 = −N e−2δ dt 2 + N −1 dr 2 + r 2 (dθ 2 + sin2 θ dφ 2 ),
(2.1)
where the functions N and δ depend only on the coordinate r and we assume N > 0 and r > 0. The factor dθ 2 + sin2 θ dφ 2 is recognised as the metric on unit S 2 , where (θ, φ) are standard angle coordinates with coordinate singularities at θ = 0 and θ = π . It is evident from the ansatz that the metric has an SO(3) isometry whose orbits are spacelike with topology S 2 and area 4πr 2 . We refer to this SO(3) isometry as spherical symmetry. The metric (2.1) is static, with the timelike Killing vector ∂t that is orthogonal to the hypersurfaces of constant t. We refer to the coordinates (t, r, θ, φ) as Schwarzschild-like coordinates and to r as the area-radius. A systematic derivation of the ansatz (2.1) from the assumptions of spherical symmetry and staticity is given in [20]. The ansatz does not cover static, spherically symmetric spacetimes in which the area of the SO(3) orbits is constant (see Sect. 2 in [20], Sect. 15.4 in [30] or Exercise 32.1 in [31]), but this special case does not occur within the black hole spacetimes in which we are interested. The SO(3) action on M induces an action of the covering group SU(2), by the double cover map SU(2) → SU(2)/{± Id} SO(3). Following [20,21], we adopt SU(2) as the group H of spherical symmetry in the above definition of a spherically symmetric connection. We shall now recall the resulting classification of these configurations and their description in an adapted Lie algebra basis [20]. The first part of the argument consists of determining all SU(n) principal bundles that admit an SU(2) action of the required kind. For spacetimes that are regularly foliated
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by the SO(3) orbits, as is the case in (2.1), this amounts to classifying all SU(n) principal bundles over S 2 . The classification relies on presenting S 2 as the quotient space SU(2)/U(1) of the base space and analysing the action of the isotropy subgroup U(1) ⊂ SU(2) on the total space of the bundle. The result is that, up to isomorphisms, the bundles are in one-to-one correspondence with the conjugacy classes of group homomorphisms from U(1) to SU(n) [28]. A convenient unique representative from each conjugacy class is the map λ : U(1) → SU(n), z → diag(z k1 , . . . , z kn ), where the n integers k1 , . . . , kn satisfy k1 ≥ k2 ≥ · · · ≥ kn and sum to zero. It follows that the equivalence classes of the SU(n) principal bundles can be uniquely indexed by sets of n integers {k1 , . . . , kn } that sum to zero and are ordered so that k1 ≥ k2 ≥ · · · ≥ kn . For the second part of the argument, one fixes the bundle and an SU(2) action on it and considers all connections that are invariant under this action. Let the map λ : U(1) → SU(n) be as defined above, and let λ : su(2) → su(n) denote the derivative of λ at the identity. A theorem of Wang [32] then states that the invariant connections are in one-to-one correspondence with the set of linear maps : su(2) → su(n) satisfying the conditions (X ) = λ (X ), ◦ ad z = adλ(z) ◦ ,
(2.2a) (2.2b)
for all X ∈ u(1) and z ∈ U(1), where U(1) is again the isotropy subgroup of the SU(2) action. The curvature F of these connections takes the form F( X˜ , Y˜ ) = [(X ), (Y )] − ([X, Y ]) ,
(2.3)
where X, Y ∈ su(2) and X˜ , Y˜ are the corresponding vector fields induced by the SU(2) action on the total space. We adopt for su(2) the basis τl := − 2i σl , l = 1, 2, 3, where σl are the Pauli matrices, 0 1 0 −i 1 0 , σ2 = , σ3 = (2.4) σ1 = 1 0 i 0 0 −1 We write l := (τl ), l = 1, 2, 3, and we may without loss of generality choose the z 0 isotropy subgroup U(1) to be embedded in SU(2) as z → 0 z −1 . From (2.2a) it then follows that 3 = − 2i diag (k1 , . . . , kn ). The infinitesimal version of (2.2b) reads ([τ3 , τl ]) = [3 , l ] , i = 1, 2, which implies that 1 and 2 can be written as 1 i C − C H , 2 = − C + C H , 1 = 2 2
(2.5)
(2.6)
where C is a strictly upper triangular complex n×n matrix, C H is its hermitian conjugate, and Ci j = 0 if and only if ki = k j + 2. Evaluating (2.3) on the su(2) basis τl shows that the only non-vanishing component of the curvature form is F(τ˜1 , τ˜2 ) = [1 , 2 ] − 3 . As the base space S 2 is two-dimensional, the curvature form must be proportional to the spherically symmetric volume form sin θ dθ ∧ dφ. The curvature form on S 2 must hence take the form F = ([1 , 2 ] − 3 ) sin θ dθ ∧ dφ.
(2.7)
Topological Geon Black Holes in Einstein-Yang-Mills Theory
131
A corresponding connection form is Aˆ := 1 dθ + (2 sin θ + 3 cos θ ) dφ.
(2.8)
Finally, the connection form A on the four-dimensional spacetime (2.1) can be decomposed as ˆ A = A˜ + A,
(2.9)
where Aˆ is as in (2.8) but the components of the matrix C in (2.6) are allowed to depend on the coordinates (t, r ). The remaining part A˜ is an su(n)-valued one-form on the twodimensional spacetime obtained by dropping the angles from (2.1), invariant under the adjoint action of the subgroup λ([U(1)]) [28]. In what follows we consider only the case [22–24] where the set of n integers is {k1 , . . . , kn } = {n − 1, n − 3, n − 5, . . . , −n + 3, −n + 1}. The connection form is taken to have a vanishing Coulomb component, At = 0, and one can then choose the gauge so that also the radial component Ar is zero. This means that we consider purely magnetic configurations of the form A = 1 dθ + (2 sin θ + 3 cos θ ) dφ, where the traceless antihermitian matrices 1 , 2 and 3 are given by ⎞ ⎛ 0 w1 ⎟ ⎜ −w1 0 w2 ⎟ 1⎜ −w2 0 w3 ⎟ ⎜ 1 = ⎜ ⎟, . . . . . . . . . ⎟ ⎜ 2 ⎝ −wn−2 0 wn−1 ⎠ −wn−1 0 ⎞ ⎛ 0 w1 ⎟ ⎜ w1 0 w2 ⎟ i ⎜ w2 0 w3 ⎟ ⎜ 2 = − ⎜ ⎟ ... ... ... ⎟ 2⎜ ⎝ wn−2 0 wn−1 ⎠ wn−1 0 ⎞ ⎛ n−1 ⎟ ⎜ n−3 ⎟ i ⎜ n−5 ⎟ ⎜ 3 = − ⎜ ⎟, ... ⎟ 2⎜ ⎠ ⎝ −n + 3 −n + 1
(2.10)
(2.11a)
(2.11b)
(2.11c)
and the real-valued functions w j , j = 1, . . . , n − 1, depend only on the coordinate r . We end the subsection with three comments. First, the one-form (2.10) has a Dirac string singularity as θ → 0 and θ → π [33]. The regularity of the curvature form (2.7) shows that this singularity is a gauge artefact. As the triviality of the fundamental group of SU(n) implies that SU(n) principal bundles over two-spheres are trivial [34,35], the one-form (2.10) must therefore be a local representative of a connection one-form in the trivial SU(n) bundle over the spacetime. We shall explicitly remove the Dirac string singularity in Subsect. 3.3.
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Second, the ansatz (2.10) has a residual gauge freedom in that a gauge transformation by
⎛
⎞
eikπ/n diag ⎝ −1, −1, . . . , −1, 1, 1, . . . , 1 ⎠ ∈ SU(n)
k
(2.12)
n−k
leaves w j invariant for j = k but changes the sign of wk [22]. We shall use this gauge freedom to simplify the special geon configurations that will be found in Subsect. 3.2. Third, we note that the embedding of the isotropy subgroup U(1) ⊂ SU(2) in SU(n) maps exp(2π τ3 ) = − IdSU(2) to exp (2π 3 ), which for our configurations (2.11) equals IdSU(n) for odd n and − IdSU(n) for even n. A gauge transformation by exp (2π 3 ) hence leaves the ansatz (2.10) invariant, and by the discussion in Subsect. 3.3 the same holds also in a globally regular gauge in which the Dirac string singularities of (2.10) have been removed. For the gauge configurations that we consider, the spherical symmetry action of SU(2) hence projects to a spherical symmetry action of SU(2)/{± Id} SO(3). 2.2. Nondegenerate Killing horizon: Kruskal-like extension. The metric (2.1) and the connection form (2.10) give an ansatz that can be inserted in the Einstein-Yang-Mills field equations. We are interested in spacetimes that have a nondegenerate Killing horizon at r = rh > 0, where N (rh ) = 0 and N (rh ) > 0, the prime indicating derivative with respect to r . Initial data for integrating the field equations from r = rh towards increasing r then consists of rh , δ(rh ) and w j (rh ), j = 1, . . . , n − 1. Local solutions in some neighbourhood of the horizon exist under a weak regularity restriction on w j (rh ) [22,23]. Not all of these local solutions extend to an asymptotically flat (for a vanishing cosmological constant) or asymptotically anti–de Sitter (for a negative cosmological constant) infinity at r → ∞, but for those that do, the solution is a static region of a nondegenerate black hole spacetime. Numerical results are given in [22–24,36]. To extend the metric across the Killing horizon, we start in the exterior region and define the Kruskal-like coordinates (U, V, θ, φ) by r δ(r ) e U := − exp −α t − dr , (2.13a) N (r ) r0 r eδ(r ) V := exp α t + dr , (2.13b) r0 N (r ) where α := 21 N (rh )e−δ(rh ) and the constant r0 is chosen so that the product U V , r eδ(r ) dr , (2.14) U V = − exp 2α r0 N (r ) has the Taylor expansion r − rh 1 N
(rh )
2 UV = − 1 + δ (rh ) − (r − rh ) + O (r − rh ) rh 2 N (rh )
(2.15)
as r → rh . It follows that in the exterior we have U < 0 and V > 0, and the Killing horizon is at U V → 0− . Whether U V is bounded below depends on the asymptotic behaviour of the metric at large r , but this will not affect what follows.
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The metric in the coordinates (U, V, θ, φ) reads ds 2 =
1 N (r )e−2δ(r ) dU dV + r 2 (dθ 2 + sin2 θ dφ 2 ), α2 UV
(2.16)
where r is a function of U V via (2.14). Inverting (2.15) as 1 N
(rh ) r − rh 2 r , = −U V 1 + δ (rh ) − U V + O V (U ) h rh 2 N (rh )
(2.17)
we find that the metric (2.16) has the near-horizon expansion 4rh N
(rh )
2 1 + 3δ r dU dV (r ) − U V + O V (U ) h h N (rh ) N (rh ) + r 2 dθ 2 + sin2 θ dφ 2 , (2.18)
ds 2 = −
which is regular across U V = 0. The metric can hence be extended from the original, ‘right-hand-side’ exterior to the black hole interior where U > 0 and V > 0, to the white hole interior where U < 0 and V < 0 and to the ‘left-hand-side’ exterior where U > 0 and V < 0. If the functions N (r ) and δ(r ) are smooth at r = rh , it further follows that the metric in the Kruskal coordinates is smooth at the horizon. Whether U V has an upper limit in the black and white hole regions, and whether there are further Killing horizons past these regions, will not affect what follows. The extension of the gauge potential across the horizon is given by (2.10) and (2.11), with w j = w j (r (U V )). The extension is regular since w j (rh ) are part of the boundary data for the exterior solution, and the extension is smooth if w j (r ) are smooth at r = rh . The resulting Kruskal-like extension is spherically symmetric, with the orbits of the SO(3) isometry being spacelike and having topology S 2 . As the exterior gauge potential ansatz (2.10) does not have terms proportional to dt or dr , and as the coefficients depend only on r , the gauge field on the Kruskal-like extension is spherically symmetric in the same sense as in the exterior.
3. Geon Quotient of the Spherically Symmetric SU(n) Black Hole We wish to take a geon quotient of the Kruskal-like SU(n) black hole of Sect. 2. For the spacetime manifold this is a straightforward adaptation of the procedure with which the Kruskal manifold is quotiented into the RP3 geon [5–8], and we shall review the requisite notions in Subsect. 3.1. The new issues arise with including in the quotient the principal bundle in which the gauge field lives. These issues will be addressed in Subsects. 3.2–3.5. For presentational simplicity, we take the gauge group of the black hole bundle to be SU(n) for odd n and SU(n)/{± Id} for even n. We denote this gauge group by G. We write equations in G as matrix equations in the defining matrix representation, understanding for even n the matrices to be defined up to overall sign. Proceeding with the gauge group SU(n) for all n would yield the same end results but we shall see in Subsect.3.3 that our choice of G will shorten the technical steps.
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3.1. Geon quotient of the spacetime manifold. Let M denote the spacetime manifold of the Kruskal-like extension, with the metric constructed in Subsect. 2.2. M is covered by the Kruskal-like coordinates (U, V, θ, φ), with the usual coordinate singularities of the angle coordinates at θ = 0 and θ = π . In addition to the three Killing vectors of the SO(3) isometry, M has the Killing vector ξ := V ∂V − U ∂U , which is timelike in the right and left exterior quadrants where U V < 0, spacelike in the black and white hole quadrants where U V > 0 and null on the bifurcate Killing horizon U V = 0. In the right exterior quadrant where V > 0 and U < 0, covered by the metric (2.1), ξ = α −1 ∂t . M has topology R2 × S 2 and is foliated by spacelike hypersurfaces of topology R × S 2 S 3 \{two points}, each omitted point being at a spatial infinity. The product U V may be bounded below by some negative constant, depending on the nature of the spatial infinities [12], and it may be bounded above by some positive constant, depending on the properties of the spacetime in the black and white hole regions. The possible existence of such bounds will not affect what follows. Consider now the map J : M → M; (U, V, θ, φ) → (V, U, π − θ, φ + π ),
(3.1)
where the action on the angle coordinates is recognised as the S 2 antipodal map and is understood in this sense at the coordinate singularities. J is an involutive isometry without fixed points, and it preserves both space and time orientation. The quotient M := M/{Id, J } is therefore a time and M is foliated 3 space orientable spacetime. 3 by spacelike hypersurfaces of topology S \{two points} /Z2 RP \{point}, with the omitted point being at a spatial infinity. As recalled in Sect. 1, these properties make M a topological geon spacetime, in the asymptotically flat case in the sense of Sorkin [1–4] and in the asymptotically anti-de Sitter case in the generalised sense of [9–12]. As the quotienting identifies the two exterior regions of M, M is an eternal black and white hole spacetime, with a single exterior region that is isometric to one exterior region of M. We may hence refer to M as a topological geon black hole. The conformal diagram depends on the character of the spatial infinity and on the structure of the black hole interior: representative samples may be found in [8,9,14,16]. We end with two observations on the isometries of M . First, as the SO(3) action on M commutes with J , there is an induced SO(3) action on M , with two-dimensional spacelike orbits. The generic orbits have again topology S 2 , but the special orbits that come from the U = V subset of M have topology RP2 . We shall regard M as a spherically symmetric spacetime despite these exceptional orbits. Second, J changes the sign of the Killing vector ξ . The isometries generated by ξ on M do therefore not induce an isometry on M : while such isometries exist within the exterior region of M , they cannot be extended past the horizon. The ramifications of this phenomenon for the Hawking-Unruh effect on related Einstein(-Maxwell) topological geon black holes have been investigated in [9–11,14,15,18,19]. 3.2. Special gauge field configurations: geon quotient of the principal bundle. We now embark on the task of examining whether the spacetime quotient M → M can be extended to the principal bundle in which the gauge field lives. Let Aext denote the gauge potential (2.10) on M, Aext := 1 dθ + (2 sin θ + 3 cos θ ) dφ.
(3.2)
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We need to examine whether there is a bundle map that projects to J on M and leaves the gauge field invariant. In terms of the gauge potential Aext (3.2), this amounts to J asking whether J maps Aext to a gauge-equivalent gauge potential. Denoting by Aext the pull-back of Aext by J , we thus seek a gauge function : M → G, such that a J back to A , gauge transformation by maps Aext ext J Aext −1 + d−1 = Aext .
(3.3)
J = −1 dθ + (2 sin θ − 3 cos θ ) dφ. Aext
(3.4)
From (3.1) we find
As neither (3.2) nor (3.4) involves dU or dV , we may assume to depend only on the angular coordinates (θ, φ). Equation (3.3) is then equivalent to the pair − 1 −1 + ∂θ −1 = 1 , (2 sin θ − 3 cos θ )−1 + ∂φ −1 = 2 sin θ + 3 cos θ.
(3.5a) (3.5b)
To find a necessary condition for a solution to (3.5) to exist, we consider the field J . These can be computed from strengths of Aext and Aext 1 F(X, Y ) = d A(X, Y ) + [A(X ), A(Y )], 2
(3.6)
with the result Fext = ∂U 1 dU ∧ dθ + ∂V 1 dV ∧ dθ + ∂U 2 sin θ dU ∧ dφ + ∂V 2 sin θ dV ∧ dφ + ([1 , 2 ] − 3 ) sin θ dθ ∧ dφ, J Fext = −∂U 1 dU ∧ dθ − ∂V 1 dV ∧ dθ + ∂U 2 sin θ dU ∧ dφ + ∂V 2 sin θ dV ∧ dφ − ([1 , 2 ] − 3 ) sin θ dθ ∧ dφ.
(3.7a) (3.7b)
From (3.3) it follows that these field strengths are related by J −1 = Fext . Fext
(3.8)
Inserting (3.7) in (3.8) and using the fact that only depends on the angular coordinates, (3.8) reduces to 1 −1 = −1 , 2 −1 = 2 , ([1 , 2 ] − 3 )−1 = −([1 , 2 ] − 3 ).
(3.9a) (3.9b) (3.9c)
Simplifying (3.9c) with the help of (3.9a) and (3.9b) shows that the set (3.9) is equivalent to 1 −1 = −1 , 2 −1 = 2 , 3 −1 = −3 . The set (3.10) is hence a necessary condition for (3.5) to hold.
(3.10a) (3.10b) (3.10c)
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To analyse (3.10), observe first from (2.11c) that 3 and −3 are diagonal and their diagonal elements appear in the reverse order, ⎞ ⎛ −n + 1 ⎟ ⎜ −n + 3 ⎟ i ⎜ −n + 5 ⎟ ⎜ (3.11) − 3 = − ⎜ ⎟. · · · ⎟ ⎜ 2 ⎠ ⎝ n−3 n−1 Identity (3.10c) thus implies that has the form ⎛ ⎜ ⎜ = (−i)n−1 ⎜ ⎝ αn
···
αn−1
α1
α2
where α j are complex numbers with unit magnitude and −1
⎞ ⎟ ⎟ ⎟, ⎠
n
j=1 α j
(3.12)
= 1.
T
= , where the overline denotes Consider then (3.10a) and (3.10b). Using complex conjugation and T transposition, we find ⎛
0 ⎜ α2 α 1 wn−1 1⎜ 1 −1 = ⎜ 2⎜ ⎝
−α1 α 2 wn−1 0 ···
⎞ −α2 α 3 wn−2 ··· αn−1 α n−2 w2
··· 0 αn α n−1 w1
⎟ ⎟ ⎟. ⎟ −αn−1 α n w1 ⎠ 0
(3.13) By (2.11a) and (3.13), (3.10a) reduces to the set α1 α 2 wn−1 = w1 = α2 α 1 wn−1 , α2 α 3 wn−2 = w2 = α3 α 2 wn−2 , .. . αn−2 α n−1 w2 = wn−2 = αn−1 α n−2 w2 , αn−1 α n w1 = wn−1 = αn α n−1 w1 ,
(3.14)
and it can be similarly verified that also (3.10b) reduces to (3.14). As n > 2 by assumption, alphas satisfying (3.14) do not exist for generic gauge field configurations. There is however a special class of gauge field configurations for which such alphas exist. If w j is vanishing, the j th line of (3.14) requires wn− j to vanish. If w j is nonvanishing, the j th line of (3.14) implies wn− j = j w j and α j+1 = j α j , where j ∈ {−1, +1}, and if n is even, n/2 = 1. A necessary condition for the alphas to exist is therefore that the gauge potential functions satisfy wn− j = j w j for all j, with j ∈ {−1, +1} and j = n− j . Note that this condition is compatible with the radial evolution equation for the gauge potential functions [22]. As observed in Subsect. 2.1, the sign of each w j can be independently changed by a gauge transformation. The gauge can therefore be chosen so that the necessary condition for the alphas to exist reads wn− j = w j , ∀ j.
(3.15)
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Now, when (3.15) holds, (3.14) is solved by α j = 1 ∀ j, and (3.12) then takes the form ⎞ ⎛ 1 1 ⎟ ⎜ ⎟ ⎜ ··· (3.16) = (−i)n−1 ⎜ ⎟. ⎠ ⎝ 1 1 The condition (3.15) is hence also sufficient for a solution to (3.10) to exist, and as (3.16) is independent of the angles, this also provides a solution to (3.5). We summarise. The necessary and sufficient condition for a geon quotient with the gauge group G to exist is (3.15), up to gauge transformations. When (3.15) holds, the gauge transformation that compensates for J in the quotienting bundle map is given by (3.16). We note in passing that for n = 2 the only gauge potential function is w1 and Eqs. (3.14) have the solution α1 = α2 = 1. This yields the purely magnetic special case of the SU(2) geon described in [12].
3.3. Triviality of the black hole bundle. Up to now we have been working in a gauge in which the gauge potential Aext (3.2) has Dirac string singularities at θ = 0 and θ = π . As noted at the end of Subsect. 2.1, the gauge bundle over the Kruskal-like spacetime M is trivial, and a globally regular gauge on M must hence exist. In this subsection we transform Aext into a globally regular gauge. This will be used in Subsect. 3.4 to analyse the gauge bundle over the geon spacetime. To begin, observe that gauge transformations by the functions N := diag e−i(n−1)φ/2 , e−i(n−3)φ/2 , . . . , e−i(−n+3)φ/2 , e−i(−n+1)φ/2 , (3.17a) S := diag ei(n−1)φ/2 , ei(n−3)φ/2 , . . . , ei(−n+3)φ/2 , ei(−n+1)φ/2 , (3.17b) make the gauge potential Aext (3.2) regular everywhere except respectively at θ = π and θ = 0. This is the step where taking the gauge group to be SU(n)/{± Id} for even n shortens the discussion, as the expressions (3.17) are not single-valued in SU(n) for even n. It therefore suffices to find a gauge function H : S 2 \ ({θ = 0} ∪ {θ = π }) that agrees with N in some punctured neighbourhood of θ = 0, agrees with S in some punctured neighbourhood of θ = π , and interpolates in between: a transformation by H puts Aext (3.2) into a globally regular gauge. We shall show that such gauge functions exist. Let first n be odd. The formulas (3.17) for N and S define two paths in G = SU(n), with path parameter φ ∈ [0, 2π ]. These paths are closed, starting and ending at the identity. As the fundamental group of SU(n) is trivial [37], these paths are homotopic, and any homotopy between them, with θ as the homotopy parameter (for example with π/2 ≤ θ ≤ 3π/4), provides the interpolation we need. Let then n be even. The formulas (3.17) for N and S again define two closed paths in G = SU(n)/{± Id}, starting and ending at the identity, with path parameter φ ∈ [0, 2π ]. When these paths are lifted from G to its double cover SU(n), formulas (3.17) show that each lift starts at Id ∈ SU(n) and ends at − Id ∈ SU(n). As the fundamental group of SU(n) is trivial, these two lifts are homotopic to each other in SU(n), and
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this homotopy in SU(n) projects down into a homotopy between the original closed paths in G = SU(n)/{± Id}. Hence the homotopy between the closed paths in G provides again the interpolation we need. Finally, note that for even n a connection in the trivial SU(n)/{± Id} bundle lifts into a connection in the trivial SU(n) bundle. Using the gauge group SU(n)/{± Id} instead of SU(n) for even n is hence just a presentational convenience.
3.4. Triviality of the geon bundle for the configurations of Subsect. 3.2. In this subsection we show that the geons of Subsect. 3.2 have a trivial gauge bundle. We showed in Subsect. 3.3 that the black hole bundle P is trivial and we can realise it as P := M × G. In this realisation, the geon bundle P is the quotient of P by the Z2 group of bundle automorphisms whose nontrivial element K takes the form K : M × G → M × G; (x, h) → J (x), h · (x)−1 ,
(3.18)
where : M → G is the gauge function that compensates for J in a globally regular gauge. The G-multiplication denoted by a dot is matrix multiplication for odd n and matrix multiplication up to overall sign for even n. We shall work in the globally regular gauge that is obtained from the gauge (3.15) by the procedure of Subsect. 3.3. In this gauge we have (x) = H (x)[H (J (x))]−1
(3.19)
for 0 < θ < π , where is given by (3.16) and H was defined in Subsect. 3.3. It follows from (3.16) and (3.17) that takes a constant value in sufficiently small punctured neighbourhoods of θ = 0 and θ = π . is therefore well defined on M, by (3.19) for 0 < θ < π and by continuity at θ = 0 and θ = π . Recall that a principal bundle is trivial iff it admits a global section. The geon bundle P admits a global section iff P admits a global section σ that is invariant under K . By (3.18), this invariance condition reads σ (J (x)) = σ (x) · (x)−1 , ∀x ∈ M.
(3.20)
As the gauge potential depends on U and V only through the combination U V , it suffices to consider the condition (3.20) on the two-sphere at U = V = 0. It further suffices to consider (3.20) on the equator θ = π/2 of the two-sphere. To see this, let γ and eq denote the respective restrictions of σ and to the equator. The restriction of (3.20) to the equator then reads γ (φ + π ) = γ (φ) · eq (φ)−1 .
(3.21)
If σ exists, it defines a solution to (3.21) by restriction. Conversely, suppose that a solution to (3.21) exists. We can view γ equivalently as a G-valued function on S 1 or as a closed path in G, denoted by the same letter and given by γ : [0, 2π ] → G; φ → γ (φ). When viewed as a closed path, γ is contractible. For odd n this follows because G = SU(n) has a trivial fundamental group. For even n the fundamental group of G = SU(n)/{± Id} is Z2 , but γ is contractible by the observation made in the last paragraph of Subsect. 3.3, or alternatively by the explicit construction of γ below. Given γ , we can define σ for
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0 ≤ θ ≤ π/2 by an arbitrary contraction of γ into a trivial path at θ = 0. Defining σ for π/2 < θ ≤ π by (3.20) then gives the desired σ . What hence remains is to show that a solution to (3.21) exists. We now proceed to construct such a solution. Let Heq denote the restriction of H to the equator. From (3.19) we have eq (φ) = −1 Heq (φ) Heq (φ + π ) . Defining γ˜ (φ) := γ (φ) · Heq (φ + π ),
(3.22)
the condition (3.21) can be rearranged into γ˜ (φ + π ) = γ˜ (φ) · −1 .
(3.23)
Without loss of generality, we may set γ˜ (0) = Id; then γ˜ (π ) = −1 . Since −1 is special unitary, it can be diagonalised by −1 = U DU −1 ,
(3.24)
where U is unitary and D is a diagonal special unitary matrix whose diagonal elements are the eigenvalues of −1 . We need to analyse these eigenvalues. Let n be odd. A recursive evaluation of the determinant shows that −1 − λ Id = (n−1)/2 −(λ2 − 1) λ − (−1)(n−1)/2 . The eigenvalues of −1 are hence ±1, and the ˆ multiplicity of −1 is even. We now define the φ-dependent matrix D(φ) by replacing an arbitrarily-chosen half of the −1s in D by eiφ and the other half by e−iφ . It is immediate ˆ ˆ ˆ ) = D. Given D, ˆ we define that D(φ) ∈ G, Dˆ has period 2π, D(0) = Id and D(π −1 −1 −1 −1 ˆ ˆ ˆ = U D(φ)DU −1 = γ˜ (φ) := U D(φ)U . Then γ˜ (φ) · = U D(φ)U U DU −1 −1 ˆ ˆ ˆ U D(φ) D(π )U = U D(φ + π )U = γ˜ (φ + π ), so that γ˜ satisfies (3.23) and γ satisfies (3.21). n/2 Let then n be even. Proceeding as above, we find −1 − λ Id = (λ2 + 1) . The ˆ eigenvalues of −1 are hence ±i, each with multiplicity n/2. We now define D(φ) by replacing in D the eigenvalues i by eiφ/2 and the eigenvalues −i by e−iφ/2 . Then ˆ ˆ ) = D, and although Dˆ is not 2π -periodic as an SU(n) matrix, it is as a D(0) = Id, D(π −1 , the conditions (3.23) ˆ G = SU(n)/{± Id} matrix. Defining again γ˜ (φ) := U D(φ)U and (3.21) can be verified as for odd n. Finally, for even n, we verify explicitly the claim that the path γ : [0, 2π ] → G; φ → γ (φ) constructed above is contractible in G. Without loss of generality, the gauge function H can be chosen to equal N (3.17a) on the equator. In this gauge it is transparent that the lift of Heq into SU(n) is periodic in φ with period 4π and changes sign after a translation in φ by 2π . From (3.22) it follows that the lift of γ to SU(n) is a closed path in SU(n), and the contraction of this lift in SU(n) projects down to a contraction of γ in G. This completes the proof of triviality of the geon bundle.
3.5. Generic gauge field configurations: geon quotient with gauged charge conjugation. We saw in Subsect. 3.2 that a geon quotient with gauge group G does not exist for generic gauge field configurations. A similar obstacle for the Maxwell gauge field in the Reissner-Nordström black hole [5] can be overcome by promoting U(1) charge conjugation
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from a global symmetry into a gauge symmetry [12]. In this subsection we show that a similar gauging of charge conjugation works also for the SU(n) black holes at hand. In the abelian case, the usual Maxwell gauge group U(1) SO(2) is enlarged into O(2) Z2 SO(2) Z2 U(1). In the Z2 U(1) representation, the group multiplication law reads (a1 , u 1 ) · (a2 , u 2 ) = a1 a2 , u 1 ρa1 (u 2 ) , (3.25) where ai ∈ Z2 , u i ∈ U(1), and ρ : Z2 → Aut (U(1)) , a → ρa , is the group homomorphism for which the nontrivial element of Z2 acts on U(1) by complex conjugation. Writing Z2 {0, 1}, where the identity element is 0, the explicit formula for ρ is ρ0 (u) = u, ρ1 (u) = u.
(3.26a) (3.26b)
In the nonabelian case at hand, the original gauge group G is SU(n) for odd n and SU(n)/{± Id} for even n. We enlarge G to G enl := Z2 G by (3.25) and (3.26). The group multiplication table of G enl reads (0, u 1 ) · (0, u 2 ) = (0, u 1 u 2 ), (0, u 1 ) · (1, u 2 ) = (1, u 1 u 2 ), (1, u 1 ) · (0, u 2 ) = (1, u 1 u 2 ), (1, u 1 ) · (1, u 2 ) = (0, u 1 u 2 ).
(3.27)
:= (a, ) : If is a gauge function with values in G, it follows that the gauge function M → G enl transforms the gauge potential by −1 −1 A −1 + d −1 = A−1 + d−1 if = (0, ), A → (3.28) ¯ = (1, ). A + d if . The conditions (3.10) To find a geon, we follow Subsect. 3.2 with replaced by are replaced by 1 −1 = −1 , −1 = 2 , 2 −1 = −3 . 3
(3.29a) (3.29b) (3.29c)
= (1, ), where It follows from (2.11) and (3.28) that the set (3.29) is solved by = diag i −n+1 , i −n+3 , . . . , i n−3 , i n−1 = (−i)n−1 diag 1, −1, 1, −1, . . . , (−1)n−1 . (3.30) Hence the black hole bundle now admits a geon quotient without restrictions on the gauge field configuration. If desired, the geon quotient can be described as in Subsect. 3.4, by adopting in the trivial black hole bundle M × G enl a globally regular gauge. Now, however, the geon bundle is not trivial, since the gauge transformation part of the bundle map is in the disconnected component of G enl .
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4. Axially Symmetric SU(2) Black Holes In this section we first review the static, axially symmetric Einstein-SU(2) black holes discovered by Kleihaus and Kunz [25,26]. For a generalisation to a negative cosmological constant, see [38]. We then give Kruskal-like coordinates that extend the spacetime across the horizon.
4.1. The exterior solution of Kleihaus and Kunz. A static, axially symmetric metric can be written in the isotropic coordinates (t, r, θ, φ) as ds 2 = − f dt 2 +
m 2 m 2 2 l 2 2 dr + r dθ + r sin θ dφ 2 , f f f
(4.1)
where the positive functions f, m and l depend only on r and θ . Here θ and φ are the usual angular coordinates on the (topological) S 2 , with coordinate singularities at θ = 0 and θ = π ; for regularity of the spacetime at these coordinate singularities, we need l/m → 1 as θ → 0 and as θ → π . The spacetime is static, with the timelike hypersurface-orthogonal Killing vector ∂t . The Killing vector of axial symmetry is ∂φ , with the symmetry axis at θ = 0 and θ = π . The ansatz for the gauge potential is A=
1 n τφ (H1 dr + (1 − H2 ) r dθ ) − n τrn H3 + τθn (1 − H4 ) r sin θ dφ , (4.2) 2er
where e is the coupling constant, the functions Hi depend only on r and θ , and τrn := sin θ cos nφ τ x + sin θ sin nφ τ y + cos θ τ z , τθn := cos θ cos nφ τ x + cos θ sin nφ τ y − sin θ τ z , τφn := − sin nφ τ x + cos nφ τ y ,
(4.3a) (4.3b) (4.3c)
where n is a positive integer and, to conform to the notation of [25,26], τ x , τ y and τ z denote respectively the Pauli matrices σ1 , σ2 and σ3 (2.4). This ansatz is purely magnetic, with no term proportional to dt. The ansatz is static, containing no dependence on t, and it is axially symmetric, in thesense that the rotation φ → φ + α can be undone by a gauge transformation with exp −in(α/2)τ z . With a 2π rotation in φ, the ansatz undergoes a 2π n rotation in su(2): we hence refer to n as the winding number. Finally, we require both the metric and the gauge field to be invariant, in an appropriate sense, under the north-south reflection θ → π − θ . For the metric the sense is that of isometry, implying that f, m and l are even under θ → π − θ . For the gauge field the sense is [26] that H1 and H3 and are odd and H2 and H4 are even under θ → π − θ . We are interested in solutions to the Einstein-SU(2) field equations with a nondegenerate Killing horizon of the Killing vector ∂t at r = rh > 0. The boundary conditions at the horizon and at the symmetry axis and the integration of the field equations into the exterior region r > rh were discussed in [25,26,39–41], and numerical evidence was found that solutions exist, including solutions that have an asymptotically flat infinity at r → ∞. The defining properties of the nondegenerate horizon in the isotropic coordinates of the anzatz (4.1) are f (rh , θ ) = 0 = f (rh , θ ) and f
(rh , θ ) > 0, where the prime indicates derivative with respect to r . Working in the dimensionless variable
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δ := (r/rh −1), it follows that the near-horizon Taylor expansions of the metric functions and the gauge potential functions begin 1 2 2 3 f (δ, θ ) = f 2 (θ )δ 1 − δ + δ F(θ ) + O(δ ) , (4.4a) 24 1 (4.4b) m(δ, θ ) = m 2 (θ )δ 2 1 − 3δ + δ 2 M(θ ) + O(δ 3 ) , 24 1 (4.4c) l(δ, θ ) = l2 (θ )δ 2 1 − 3δ + δ 2 L(θ ) + O(δ 3 ) , 12 δ H11 (θ ) + O(δ 3 ), H1 (δ, θ ) = δ 1 − (4.5a) 2 1 H2 (δ, θ ) = H20 (θ ) + δ 2 H21 (θ ) + O(δ 3 ), (4.5b) 4 1 H3 (δ, θ ) = H30 (θ ) + δ 2 H31 (θ ) + O(δ 3 ), (4.5c) 8 1 H4 (δ, θ ) = H40 (θ ) + δ 2 H41 (θ ) + O(δ 3 ), (4.5d) 8 where the O-terms may depend on θ and the field equations yield various relations among the coefficient functions [26]. One of these relations is 1 dm 2 2 d f2 − = 0, m 2 dθ f 2 dθ
(4.6)
from which it follows that f 22 /m 2 is independent of θ , implying that the horizon has constant surface gravity [26]. The gauge potential can further be chosen regular everywhere, including θ = 0 and θ = π [40,41]. The SU(2) bundle is thus trivial and the gauge potential is expressed in a globally regular gauge. In the special case n = 1 the field equations imply that l = m, H1 = H3 = 0, H2 = H4 and all the metric and gauge potential functions are independent of θ . The metric and the gauge field are then spherically symmetric, and the solution reduces to that of [42,43]. 4.2. Kruskal-like extension. A complication with finding Kruskal-like coordinates that cover a neighbourhood of the full bifurcate Killing horizon is that the null geodesics with constant φ generically have nontrivial evolution in both r and θ . However, because of the discrete isometry θ → π − θ , the submanifold at θ = π/2 is totally geodesic, and Kruskal-like coordinates that extend this submanifold across the horizon can be found as in the spherically symmetric case of Subsect. 2.2. We shall show that the Kruskal-like coordinates adapted to the θ = π/2 submanifold can be extended to other values of θ to give a C 0 extension across the horizon. This C 0 extension will suffice for taking the geon quotient in Sect. 5. We start at r > rh and define the coordinates (U, V, θ, φ) by r √ m(r, π/2) dr , (4.7a) U := − exp −α t − f (r, π/2) r0 r √ m(r, π/2) dr , (4.7b) V := exp α t + f (r, π/2) r0
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where α :=
f 2 (π/2) , √ rh m 2 (π/2)
and r0 is chosen so that r √ 1 1 m(r, π/2) 2 dr = ln δ − δ + O(δ ) f (r, π/2) α 2 r0
(4.8)
(4.9)
as r → rh . The region r > rh is at U < 0 and V > 0, and the Killing horizon is at U V → 0− . The metric in the coordinates (U, V, θ, φ) reads m(r, θ ) f (r, π/2)2 1 1 2 + 1 dU dV ds = f (r, θ ) 2 2α U V f (r, θ )2 m(r, π/2) 1 m(r, θ ) f (r, π/2)2 1 2 2 2 2 + f (r, θ ) 2 − 1 V dU + U dV 4α (U V )2 f (r, θ )2 m(r, π/2) m(r, θ ) 2 2 l(r, θ ) 2 2 r dθ + r sin θ dφ 2 , (4.10) + f (r, θ ) f (r, θ ) where r is a function of U V by
r √ m(r, π/2) dr U V = − exp 2α f (r, π/2) r0 = −δ 2 1 − δ + O(δ 2 ) , r → rh .
(4.11a) (4.11b)
Inverting the near-horizon expansion (4.11b) and substituting in (4.10) yields 1 1 f (θ ) [1 + O(U V )] dU dV + f (θ ) 2 2 2 2 α 96α √ −U V V 2 dU 2 + U 2 dV 2 × M(θ ) − M(π/2) − 2F(θ ) + 2F(π/2) + O
ds 2 = −
+
1 l2 (θ ) f 2 (θ ) [1 + O(U V )] dθ 2 + [1 + O(U V )] sin2 θ dφ 2 . 2 α f 2 (θ )
(4.12)
Similarly, the near-horizon expansion of the gauge potential (4.2) reads 1 1 A= τ n − (1 + O(U V )) H11 (θ )(V dU + U dV ) 2e φ 2 + (1 − H20 (θ ) + O(U V )) dθ ! −n τrn (H30 (θ ) + O(U V )) + τθn (1 − H40 + O(U V ))] sin θ dφ . (4.13) The components of the metric (4.12) and the gauge potential (4.13) are well defined at the horizon, U V → 0− , but the of the metric are not guaranteed to be √ components differentiable because of the O −U V error term. Our Kruskal coordinates therefore give a C 0 extension of the spacetime into a neighbourhood of the bifurcate Killing horizon, but they are not sufficiently regular for discussing the field equations across the horizon. Coordinates that allow a smooth extension are discussed in [44,45], but at the expense of rendering the discrete isometry that we wish to utilise less transparent. We shall work with the above C 0 extension.
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5. Geon Quotient of the Axially Symmetric SU(2) Black Hole In this section we show that the Kruskal-like SU(2) black hole of Sect. 4 has a geon quotient. As in Sect. 3, quotienting the spacetime manifold proceeds as taking the RP3 geon quotient of Kruskal [5–8] and the issues of interest to us arise with quotienting the principal bundle in which the gauge field lives. 5.1. Spacetime quotient. Let M denote the spacetime manifold of the Kruskal-like (C 0 ) extension covered by the coordinates (U, V, θ, φ), with the coordinate singularities at θ = 0 and θ = π understood to be handled as in Sect. 3. The map J defined by (3.1) is an involutive isometry without fixed points, preserving both space and time orientation and mapping the two exterior regions of M to each other. The quotient spacetime M := M/{Id, J } is hence a time and space orientable black and white hole spacetime, its single exterior region is isometric to one exterior region of M, and it is foliated by spacelike hypersurfaces of topology RP3 \{point} with the omitted point being at an asymptotically flat spatial infinity. M is hence a topological geon in the sense of Sorkin [1–4] and we may refer to it as a topological geon black hole. The isometries of M may be discussed as in Subsect. 3.1. In particular, the Killing vector ∂φ of M is invariant under J and there is hence an induced U(1) isometry group on M , with subtleties at the orbits coming from the subset of M where U = V and θ = π/2. We shall regard M as an axially symmetric spacetime despite these exceptional orbits. The isometry properties associated with the Killing vector V ∂V − U ∂U of M are as in Subsect. 3.1. 5.2. Principal bundle quotient. Let Aext denote the gauge potential on M, given in the right-hand-side exterior by (4.2) and having the near-horizon form (4.13). We need to investigate whether there exists a bundle map that projects to J on M and leaves the gauge potential invariant. As in Sect. 3, this reduces to examining whether Aext is invariant under J up to a gauge transformation. From the evenness of the gauge potential functions H2 and H4 and the oddness of the gauge potential functions H1 and H3 under θ → π − θ , and from the properties of the matrices (4.3) under J , it follows that the cases of odd and even n require separate treatment. Let first n be odd. Aext is then clearly invariant under J , and the bundle map can be chosen to be K odd : M × SU(2) → M × SU(2); (U, V, θ, φ, h) → (V, U, π − θ, φ + π, h).
(5.1)
K odd is involutive, and the quotient bundle is the trivial SU(2) bundle over Mg := M/{Id, J }. As the gauge potential is invariant under a gauge transformation by − Id ∈ SU(2), the geon bundle can be alternatively taken to be the trivial SO(3) SU(2)/{± Id} bundle over Mg . J denote the pull-back of A Let then n be even, and let Aext ext by J . In the right-handJ takes the form side exterior covered by the coordinates (t, r, θ, φ), Aext J = Aext
1 " x (τ sin nφ − τ y cos nφ) [H1 dr + (1 − H2 )r dθ ] 2er
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145
−n (−τ x sin θ cos nφ − τ y sin θ sin nφ + τ z cos θ )H3
# +(−τ x cos θ cos nφ − τ y cos θ sin nφ − τ z sin θ ) (1 − H4 ) r sin θ dφ . (5.2) J and A Comparison with (4.2) shows that Aext ext do not coincide. They are however x y z taken to each other by (τ , τ , τ ) → (−τ x , −τ y , τ z ), which is a gauge transformation: defining π i 0 z ∈ SU(2), (5.3) g0 := exp i τ = 0 −i 2
we have J Aext = g0 Aext g0−1 ,
(5.4)
and (5.4) is a gauge transformation because the inhomogeneous term involving dg0 vanishes. The bundle map can thus be chosen to be K ev : M × SU(2) → M × SU(2); (U, V, θ, φ, h) → V, U, π − θ, φ + π, h · g0−1 ,
(5.5)
where the dot denotes matrix multiplication in SU(2). K ev generates the cyclic 2 , K 3 }, and the geon bundle is the quotient group of order four, ¯ := {Id, K ev , K ev ev 2}⊂ ¯ ¯ identifies points in M×SU(2) (M × SU(2)) /. As the normal subgroup {Id, K ev by the position-independent gauge transformation by g02 = − Id ∈ SU(2), and as this gauge transformation leaves the gauge potential invariant, the geon bundle can be equivalently presented as a Z2 quotient of the trivial SO(3) SU(2)/{± Id} bundle over M. Explicitly, we may realise the projection $ SU(2) → SO(3), g → g, ˆ in the defining i τ j . Note that gˆ = diag(−1, −1, 1). matrix representations so that gτ i g −1 = g ˆ 0 j j The involutive bundle map then reads Kˆ ev : M × SO(3) → M × SO(3); ˆ → V, U, π − θ, φ + π, hˆ · gˆ −1 , (U, V, θ, φ, h) 0
(5.6)
where the dot denotes matrix multiplication in SO(3). The geon bundle for even & trivial. To see this, we view the geon bundle as the % n is not ˆ quotient (M × SO(3)) / Id, K ev . Suppose this bundle is trivial. Proceeding as in the discussion of Subsect. 3.4 leading to (3.21), we see that there then exists a continuous 2π -periodic function γ : R → SO(3) such that γ (φ + π ) = γ (φ) · gˆ 0−1
(5.7)
and the closed path γ0 : [0, 2π ] → SO(3); φ → γ (φ) is contractible. We may assume without loss of generality that γ (0) = Id ∈ SO(3). The condition (5.7) then implies that γ0 is homotopic to the path γ1 : [0, 2π ] → SO(3); φ → γ1 (φ), where ⎛ ⎞ cos φ − sin φ 0 cos φ 0⎠. γ1 (φ) := ⎝ sin φ (5.8) 0 0 1 But as the lift of γ1 to SU(2) is not closed, γ1 is not contractible, and hence neither is γ0 . This is a contradiction and implies that the assumed triviality of the geon bundle cannot hold.
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6. Conclusions We have shown that the static, spherically symmetric SU(n) black hole solutions of Künzle [20,22] and the static, axially symmetric SU(2) black hole solutions of Kleihaus and Kunz [25,26] admit topological geon quotients. These constructions extend the family of known non-abelian Einstein-Yang-Mills geon-type black holes from the static, spherically symmetric SU(2) geon-type black hole [12] to include geons with a more general Yang-Mills gauge group and to geons with less symmetry. For Künzle’s static, spherically symmetric SU(n) black holes with n > 2, we showed that a geon quotient generically requires an extension of the gauge group from SU(n) to Z2 SU(n), where the nontrivial element of Z2 acts on SU(n) by complex conjugation. This means that the SU(n) charge conjugation must be treated as a gauge symmetry, rather than just as a global symmetry. This gauging is very similar to the U(1) charge conjugation gauging that is necessary for taking a geon quotient of the Reissner-Nordström black hole [12]. By contrast, static, spherically symmetric SU(2) black holes were known to admit a geon quotient without the need to gauge the SU(2) charge conjugation [12], and we showed that the same holds for the static, axially symmetric SU(2) black holes of Kleihaus and Kunz [25,26]. In the cases where gauging the charge conjugation is not required, we showed that the geons built from Künzle’s black holes have a trivial gauge bundle, whereas those built from the black holes of Kleihaus and Kunz have a trivial (respectively nontrivial) gauge bundle for odd (even) winding number of the gauge field configuration. We have not investigated whether this phenomenon reflects some deeper geometric property. Our results on the axially symmetric solutions have a technical limitation in that the extension across the Killing horizon was C 0 but was not guaranteed to be differentiable. We suspect that this limitation is an artefact of a non-optimal coordinate choice and the results continue to hold within extensions of higher differentiability. It should be possible to examine this question with the techniques of Rácz and Wald [44,45]. The topological geon black holes that we have found should provide an interesting arena for investigating the Hawking-Uhruh effect for quantum fields coupled to the background Yang-Mills field. How does the geon’s charge show up in the Hawking-Unruh effect, compared with the Hawking-Unruh effect on the conventional Kruskal-like extension? In particular, does the Hawking-Unruh effect feel the gauging of SU(n) charge conjugation, as it does feel the gauging of U(1) charge conjugation [18,19]? When the charge conjugation is not gauged, does the Hawking-Unruh effect feel the triviality versus nontriviality of the geon’s gauge bundle? A technically simple test field with which to address these questions might be a multiplet of charged scalars minimally coupled to the Yang-Mills field. A more interesting case might be a neutrino multiplet, for which the additional issue of inequivalent spin structures arises [15]. Acknowledgements. We thank Martin Edjvet, Yakov Shnir, Elizabeth Winstanley and especially John Barrett for helpful discussions. We also thank Burkhard Kleihaus and two anonymous referees for helpful comments on the manuscript. GTK was supported in part by the Sunburst Fund of ETH (Switzerland). JL was supported in part by STFC (UK) Rolling Grant PP/D507358/1.
References 1. Sorkin, R.D.: Introduction to topological geons. In: Topological Properties and Global Structure of Space-time: Proceedings of the NATO Advanced Study Institute on Topological Properties and Global Structure of Space-time, Erice, Italy, 12–22 May 1985, edited by P. G. Bergmann, V. de Sabbata, New York: Plenum Press, 1986, pp. 249–270
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Sorkin, R.: The quantum electromagnetic field in multiply connected space. J. Phys. A 12, 403 (1979) Friedman, J.L., Sorkin, R.D.: Spin 1/2 from gravity. Phys. Rev. Lett. 44, 1100 (1980) Friedman, J.L., Sorkin, R.D.: Half integral spin from quantum gravity. Gen. Rel. Grav. 14, 615 (1982) Misner, C.W., Wheeler, J.A.: Classical physics as geometry: Gravitation, electromagnetism, unquantized charge, and mass as properties of curved empty space. Annals Phys. (N.Y.) 2, 525, (1957); Reprinted in: J. A. Wheeler, Geometrodynamics. New York: Academic, 1962 Giulini, D.: 3-manifolds in canonical quantum gravity. Ph.D. Thesis, University of Cambridge, 1990 Giulini, D.: Two-body interaction energies in classical general relativity. In: Relativistic Astrophysics and Cosmology, Proceedings of the Tenth Seminar, Potsdam, October 21–26 1991, edited by Gottlöber, S., Mücket, J.P., Müller V. Singapore: World Scientific, 1992, pp. 333–338 Friedman, J.L., Schleich, K., Witt, D.M.: Topological censorship. Phys. Rev. Lett. 71, 1486 (1993) [Erratum-ibid. 75, 1872 (1995)] Louko, J., Marolf, D.: Single-exterior black holes and the AdS-CFT conjecture. Phys. Rev. D 59, 066002 (1999) Louko, J., Marolf, D., Ross, S.F.: On geodesic propagators and black hole holography. Phys. Rev. D 62, 044041 (2000) Maldacena, J.M.: Eternal black holes in Anti-de-Sitter. JHEP 0304, 021 (2003) Louko, J., Mann, R.B., Marolf, D.: Geons with spin and charge. Class. Quant. Grav. 22, 1451 (2005) Birrell, N.D., Davies, P.C.W.: Quantum fields in curved space. Cambridge: Cambridge University Press, 1984 Louko, J., Marolf, D.: Inextendible Schwarzschild black hole with a single exterior: how thermal is the Hawking radiation? Phys. Rev. D 58, 024007 (1998) Langlois, P.: Hawking radiation for Dirac spinors on the RP3 geon. Phys. Rev. D 70, 104008 (2004) [Erratum-ibid. D 72, 129902 (2005)] Louko, J.: Geon black holes and quantum field theory. J. Phys. Conf. Ser. 222, 012038 (2010) Kiskis, J.E.: Disconnected gauge groups and the global violation of charge conservation. Phys. Rev. D 17, 3196 (1978) Bruschi, D.E., Louko, J.:Charged Unruh effect on geon spacetimes. http://arXiv./orglabs/1003.1297v1 [gr-qc], 2010 talk given by D. E. Bruschi at the 12th Marcel Grossmann meeting, Paris, France, 12–18 July 2009 Bruschi, D.E., Louko, J.: In preparation Künzle, H.P.: SU(n) Einstein Yang-Mills fields with spherical symmetry. Class. Quant. Grav. 8, 2283 (1991) Bartnik, R.: The structure of spherically symmetric su(n) Yang-Mills fields. J. Math. Phys. 38, 3623 (1997) Künzle, H.P.: Analysis of the static spherically symmetric SU(n) Einstein Yang-Mills equations. Commun. Math. Phys. 162, 371 (1994) Baxter, J.E., Helbling, M., Winstanley, E.: Soliton and black hole solutions of su(N) Einstein-Yang-Mills theory in anti-de Sitter space. Phys. Rev. D 76, 104017 (2007) Baxter, J.E., Helbling, M., Winstanley, E.: Abundant stable gauge field hair for black holes in anti-de Sitter space. Phys. Rev. Lett. 100, 011301 (2008) Kleihaus, B., Kunz, J.: Static black hole solutions with axial symmetry. Phys. Rev. Lett. 79, 1595 (1997) Kleihaus, B., Kunz, J.: Static axially symmetric Einstein-Yang-Mills-dilaton solutions. II: Black hole solutions. Phys. Rev. D 57, 6138 (1998) Conlon, L.: Differentiable manifolds. 2nd edition, Boston: Birkhauser, 2001 Harnad, J.P., Vinet, L., Shnider, S.: Group actions on principal bundles and invariance conditions for gauge fields. J. Math. Phys. 21, 2719 (1980) Molelekoa, M.: Symmetries of gauge fields. J. Math. Phys. 26, 192 (1985) Stephani, H., Kramer, D., MacCallum, M., Hoenselaers, C., Herlt, E.: Exact Solutions of Einstein’s Field Equations. 2nd edition. Cambridge: Cambridge University Press, 2003 Misner, C.W., Thorne, K.S., Wheeler, J.A.: Gravitation. San Francisco: Freeman, 1973 Wang, H.-C.: On invariant connections over a principal fibre bundle. Nagoya Math. J. 13, 1 (1958) Volkov, M.S., Gal’tsov, D.V.: Gravitating non-abelian solitons and black holes with Yang-Mills fields. Phys. Rept. 319, 1 (1999) Steenrod, N.: The topology of fibre bundles. Princeton: Princeton University Press, 1951 Naber G.L.: Topology, geometry and gauge fields: foundations. New York: Springer, 1997 Kleihaus, B., Kunz, J., Sood, A.: Charged SU(N) Einstein-Yang-Mills black holes. Phys. Lett. B 418, 284 (1998) Nakahara, M.: Geometry, topology and physics. 2nd edition., Bristol: IOP Publishing, 2003 Radu, E., Winstanley, E.: Static axially symmetric solutions of Einstein-Yang-Mills equations with a negative cosmological constant: Black hole solutions. Phys. Rev. D 70, 084023 (2004)
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39. Kleihaus, B., Kunz, J.: Static axially symmetric Einstein Yang-Mills-dilaton solutions. I: Regular solutions. Phys. Rev. D 57, 834 (1998) 40. Kleihaus, B.: On the regularity of static axially symmetric solutions in SU(2) Yang-Mills dilaton theory. Phys. Rev. D 59, 125001 (1999) 41. Kleihaus, B., Kunz, J.: Comment on ‘Singularities in axially symmetric solutions of Einstein-Yang-Mills and related theories, by L. Hannibal’, arXiv:hep-th/9903235 42. Bizon, P.: Colored black holes. Phys. Rev. Lett. 64, 2844 (1990) 43. Künzle, H.P., Masood-ul-Alam, A.K.M.: Spherically symmetric static SU(2) Einstein-Yang-Mills fields. J. Math. Phys. 31, 928 (1990) 44. Rácz, I., Wald, R.M.: Extension of space-times with Killing horizon. Class. Quant. Grav. 9, 2643 (1992) 45. Rácz, I., Wald, R.M.: Global extensions of space-times describing asymptotic final states of black holes. Class. Quant. Grav. 13, 539 (1996) Communicated by P.T. Chru´sciel
Commun. Math. Phys. 303, 149–173 (2011) Digital Object Identifier (DOI) 10.1007/s00220-011-1199-8
Communications in
Mathematical Physics
Supersymmetric QCD and Noncommutative Geometry Thijs van den Broek1,2 , Walter D. van Suijlekom1 1 Institute for Mathematics, Astrophysics and Particle Physics, Faculty of Science,
Radboud University Nijmegen, Heyendaalseweg 135, 6525 AJ Nijmegen, The Netherlands. E-mail:
[email protected];
[email protected] 2 National Institute for Subatomic Physics, Science Park 105, 1098 XG Amsterdam, The Netherlands Received: 30 March 2010 / Accepted: 4 October 2010 Published online: 10 February 2011 – © The Author(s) 2011. This article is published with open access at Springerlink.com
Abstract: We derive supersymmetric quantum chromodynamics from a noncommutative manifold, using the spectral action principle of Chamseddine and Connes. After a review of the Einstein–Yang–Mills system in noncommutative geometry, we establish in full detail that it possesses supersymmetry. This noncommutative model is then extended to give a theory of quarks, squarks, gluons and gluinos by constructing a suitable noncommutative spin manifold (i.e. a spectral triple). The particles are found at their natural place in a spectral triple: the quarks and gluinos as fermions in the Hilbert space, the gluons and squarks as the (bosonic) inner fluctuations of a (generalized) Dirac operator by the algebra of matrix-valued functions on a manifold. The spectral action principle applied to this spectral triple gives the Lagrangian of supersymmetric QCD, including supersymmetry breaking (negative) mass terms for the squarks. We find that these results are in good agreement with the physics literature. 1. Introduction Over the last few decades, noncommutative geometry [9] has proven to be very successful in deriving models in high-energy physics from geometrical principles. This started with the particle theories studied by Connes and Lott from a noncommutative perspective [12], culminating in the work of Chamseddine and Connes [3,4]. Therein, the full Standard Model of high-energy physics —including the Higgs field— was derived from a noncommutative manifold, through the so-called spectral action principle. This principle puts gauge theories such as the Standard Model on the same geometrical footing as Einstein’s general theory of relativity by deriving a Lagrangian from a noncommutative spacetime. For more details, see eg. Sect. 2 below. More recently, in [7] (see also [13]) this noncommutative model was enhanced to also include massive neutrinos while solving a technical issue (i.e. ‘fermion doubling’) at the same time. Ever since the early models introduced by Connes and Lott, there has been interest in the connection between noncommutative geometry and supersymmetry. An early
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instance of this subject is found in [15,16], and also [1,18]. However, this was all before the elegant spectral action principle was introduced, in particular the last article needed to turn the noncommutative algebra of coordinates into a superalgebra. Throughout the present paper, the algebra of noncommutative coordinates are M N (C)-valued functions on spacetime, that is, A = C ∞ (M, M N (C)) (possibly with N = 3). In the paradigm of noncommutative geometry, the gauge group consists of special unitary elements in this algebra; in this case SU (A) = C ∞ (M, SU (N )). The supersymmetric gauge theories we will derive thus have SU (N ) as a gauge group. Then, as is intended, the supersymmetry will manifest itself as a transformation between bosonic and fermionic degrees of freedom; this was suggested in [4]. The natural place for the fermionic degrees of freedom is in the Hilbert space of spinors. As we will see below, the bosonic degrees of freedom are generated naturally by a generalized Dirac operator; this is very similar to the origin of the Higgs boson through the finite Dirac operator in the noncommutative description of the Standard Model [7]. This article is organized as follows. We start by giving a short overview of the spectral action in noncommutative geometry, since it is the main technique exploited in this article. In Sect. 4 we demonstrate that the Einstein–Yang–Mills system as derived from a noncommutative manifold —which we recall in Sect. 3— is actually supersymmetric. More precisely, it is N = 1 supersymmetric SU (N ) Yang–Mills theory, minimally coupled to gravity. Section 5 forms the main part of this article, we define a noncommutative manifold on which the spectral action gives the Lagrangian of supersymmetric quantum chromodynamics (QCD). Besides a quark and a gluon we recognize their superpartners: the squark and the gluino. The squark appears naturally as the finite part of the inner fluctuations of the noncommutative manifold, besides the gluons as the continuous part. We discuss the several terms that appear in the spectral action and find that they coincide with the usual dynamics and interaction terms between gluons, gluinos, quarks and squarks that appear in the physics literature. In addition, we find supersymmetry breaking (negative) mass terms for the squarks. 2. Preliminaries In [4], Chamseddine and Connes introduced the spectral action principle as a powerful device to derive (potentially physical) Lagrangians from a noncommutative spin manifold. For convenience, we will start by quickly recalling their setup and approach. The basic device in noncommutative geometry [9] is a spectral triple (A, H, D) consisting of a ∗-algebra A of bounded operators in a Hilbert space H, and an unbounded self-adjoint operator D in H, such that 1. The commutator [D, a] is a bounded operator; 2. The resolvent (i + D)−1 of D is a compact operator. One may further enrich this set of data by a self-adjoint operator γ on H that commutes with all elements in A and is such that γ 2 = 1 (grading), and an anti-unitary operator J on H (reality) such that the following hold: a, J b J −1 = 0; ∀a, b ∈ A. (2.1) [D, a], J b J −1 = 0, These conditions are called the first-order condition and the commutant property, respectively. The following ±-signs for the commutation relations between J, γ and D:
Supersymmetric QCD and Noncommutative Geometry KO-dimension 0 2 4 6
J2 = + − − +
J D = D J + + + +
151 J γ = γ J + − + −
determine the so-called KO-dimension of the real spectral triple. The notion of a real spectral triple generalizes Riemannian spin geometry to the noncommutative world. In fact, there exists a reconstruction theorem [10,11] which states that if the algebra A in (A, H, D) is commutative, then the spectral triple is of the form C ∞ (M), L 2 (M, S), ∂/ , canonically associated to a Riemannian spin manifold M. Here S → M is a spinor bundle and ∂/ is the corresponding Dirac operator. 2.1. Inner fluctuations. Rather than isomorphisms of algebras, a natural notion of equivalence for noncommutative (C ∗ -)algebras is Morita equivalence [25]. Given a spectral triple (A, H, D) and an algebra B that is Morita equivalent to A, one can define [10] a spectral triple (B, H , D ) on B. Interestingly, upon taking B to be A, this leads to a whole family of spectral triples (A, H, D A ), where D A := D + A with A ∈ 1D (A) self-adjoint with 1 (A) := ai [D, bi ] : ai , bi ∈ A . (2.2) i
The bounded operators A are generally referred to as the inner fluctuations of D and may be interpreted as gauge fields. When considering a real spectral triple (A, H, D; J ), we have the additional restriction that the real structure J of the spectral triple (A, H , D ; J ) on the Morita equivalent algebra should be compatible with the relation J D = D J . Upon taking B to be A again in such a case, the resulting spectral triple is of the form (A, H, D A ; J ), but now with D A := D + A + J A J ∗ .
(2.3)
Note that in the commutative case these inner fluctuations vanish. The gauge group is defined to be U (A) := {u ∈ A : uu ∗ = u ∗ u = 1}. It acts on elements ψ in the Hilbert space via ψ → u J u J −1 ψ. This induces an action on D A as D A → u D A u ∗ . Consequently, the inner fluctuations transform as A → Au := u Au ∗ + u[D, u ∗ ].
(2.4)
In the presence of a determinant on A, we can restrict U (A) to SU (A) for which in addition the determinant is equal to the identity. 2.2. The spectral action. The above suggests that a (real) spectral triple defines a gauge theory, with the gauge fields arising as the inner fluctuations of the Dirac operator and the gauge group is given by the unitary elements in the algebra. It is thus natural to seek for gauge invariant functionals of A ∈ 1D (A) and the so-called spectral action [4] is the most natural. Let (A, H, D; J ) be a real spectral triple. Given the above operator D A , a cut-off scale and some positive, even function f one can define (cf. [4,10]) the gauge invariant spectral action: Sb [A] := Tr f (D A /).
(2.5)
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The cut-off parameter is used to obtain an asymptotic series for the spectral action; the physically relevant terms then appear with a positive power of as a coefficient. Besides this bosonic action, one can define a fermionic action in terms of ψ ∈ H and A ∈ 1D (A): S f [A, ψ] := ψ, D A ψ .
(2.6)
It was shown in [4] that for a suitable choice of a spectral triple the spectral action equals the full Standard Model Lagrangian, including the Higgs boson. More recently, in [7] (see also [13]) this was enhanced to also include massive neutrinos while solving a technical issue (i.e. ‘fermion doubling’ as pointed out in [20]) at the same time. We will not further go into details but refer to the mentioned literature instead. For convenience, we end this section by recalling some results on heat kernel expansions and Seeley–DeWitt coefficients; these will be useful later on; for more details we refer to [14]. If V is a vector bundle on a compact Riemannian manifold (M, g) and if P : C ∞ (V ) → C ∞ (V ) is a second-order elliptic differential operator of the form (2.7) P = − g μν ∂μ ∂ν + K μ ∂μ + L with K μ , L ∈ Γ (End(V )), then there exist a unique connection ∇ and an endomorphism E on V such that P = ∇∇ ∗ − E.
(2.8)
, where Explicitly, we write locally ∇μ = ∂μ + ωμ = ωμ
1 ν gμν K ν + gμν g ρσ Γρσ . 2
(2.9)
and L we find E ∈ Γ (End(V )) and compute for the curvature Using this ωμ μν of ∇: ρ E := L − g μν ∂ν (ωμ ) − g μν ωμ ων + g μν ωρ Γμν ;
μν :=
∂μ (ων ) − ∂ν (ωμ ) − [ωμ , ων ].
(2.10a) (2.10b)
In this situation we can make an asymptotic expansion (as t → 0) of the trace of the operator e−t P in powers of t: √ Tr e−t P ∼ t (n−m)/2 an (P), an (P) := an (x, P) gd m x, (2.11) n≥0
M
where m is the dimension of M and the coefficients an (x, P) are called the Seeley– DeWitt coefficients. It turns out [14, Ch 4.8] that an (x, P) = 0 for n odd and that the first three even coefficients are given by a0 (x, P) = (4π )−m/2 Tr(id); a2 (x, P) = (4π )−m/2 Tr(−R/6 id + E);
1 μ a4 (x, P) = (4π )−m/2 Tr −12R;μ + 5R 2 − 2Rμν R μν 360
(2.12a) (2.12b)
μ +2Rμνρσ R μνρσ − 60R E + 180E 2 + 60E ;μ + 30μν μν , (2.12c)
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μ
where R;μ := ∇ μ ∇μ R and the same for E. In all cases that we will consider, the manμ
μ
ifold will be taken without boundary so that the terms E ;μ , R;μ vanish by Stokes’ Theorem. This can be used in the computation of the spectral action as follows. Assume that the inner fluctuations give rise to an operator D A for which D 2A is of the form (2.7) on some vector bundle V on a compact Riemannian manifold M. Then, on writing f as a Laplace transform, we obtain −t D 2A /2 f (D A /) = g(t)e ˜ dt. t>0
In the case of a four-dimensional manifold the dominant terms of the expansion are found with Eq. (2.11) to be Tr f (D A /) = 24 f 4 a0 D 2A + 22 f 2 a2 D 2A + a4 D 2A f (0) + O −2 , (2.13) where the f k are moments of the function f : ∞ f (w)w k−1 dw; f k :=
(k > 0).
0
3. The Einstein–Yang–Mills System A spectral triple that will serve as the starting point for many of the subsequent considerations is the one that results in the Einstein–Yang–Mills system; it was introduced and studied in [4] (cf. also [13, Sect. 11.4]). From now on, M will denote a compact four-dimensional Riemannian spin manifold (with metric g). We take our spectral triple to be the tensor product of the canonical one on (M, g), and the finite spectral triple (M N (C), M N (C), 0): A = C ∞ (M) ⊗ M N (C) C ∞ (M, M N (C)), H = L 2 (M, S) ⊗ M N (C), D = ∂/ M ⊗ id, with ∂/ M = iγ μ ∇μS , where the representation of M N (C) on M N (C) is by left multiplication. We make the spectral triple real by defining J : H → H by J (s ⊗ T ) := J M s ⊗ T ∗ , s ⊗ T ∈ H,
(3.1)
where J M is the real structure on L 2 (M, S) (i.e. charge conjugation) and T ∗ is the adjoint of the matrix T . The inner fluctuations (2.3) of this Dirac operator are seen to be of the form A + J A J ∗ = γ μ ad(Aμ ),
(3.2)
where ad(Aμ )T := [Aμ , T ], T ∈ M N (C) and the minus sign giving rise to this commutator comes from the identity −1 ∗ = JM γ μ JM = −γ μ . JM γ μ JM
(3.3)
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The local expression for the fluctuated Dirac operator is then D A = ieaμ γ a [(∂μ + ωμ ) ⊗ id + id ⊗ Aμ ],
(3.4)
where ωμ is the spin connection and Aμ := −i ad Aμ is skew-Hermitian due to the self-adjointness of Aμ . From the demand of self-adjointness of D A , it follows that A is a u(N )-valued oneform. Now, U (N ) is not a simple group but U (N ) SU (N ) U (1) resulting in u(N ) u(1) ⊕ su(N ) for the corresponding Lie algebras. But since A + J A J −1 is in the adjoint representation [cf. (3.2)] of U (N ), we retain only a traceless object. The symmetry group of the fluctuations is therefore effectively SU (N ). Proposition 1. The square D 2A of the operator given in (3.4) is of the form − g μν ∂μ ∂ν + K μ ∂μ + L (cf. (2.7)) with K μ = 2ωμ − Γ μ ⊗ id + 2 id ⊗ Aμ ,
1 μ μ μ L = ∂ ωμ + ω ωμ − Γ ωμ + R ⊗ id + id ⊗ ∂ μ Aμ + Aμ Aμ 4 1 +2 ωμ ⊗ Aμ − Γ μ ⊗ Aμ − γ μ γ ν ⊗ Fμν , 2 ν g μλ and F where Γ ν = Γμλ μν is the curvature of the connection Aμ :
Fμν := ∂μ Aν − ∂ν Aμ + [Aμ , Aν ].
(3.5)
ωμ
With this we can both determine (and consequently μν ) and E [cf. (2.9), (2.10b) and (2.10a) respectively] uniquely: ωμ = ωμ ⊗ id + id ⊗ Aμ , 1 1 E = R ⊗ id − γ μ γ ν ⊗ Fμν , 4 2 1 ab μν = Rμν γab ⊗ id + id ⊗ Fμν . 4
We have shown that the fluctuated Dirac operator D A meets the demands needed to apply the heat kernel expansion of the spectral action, as sketched at the end of the previous section. Now for the first three coefficients appearing in (2.13) we have the following expressions: N2 √ 4 a0 (D 2A ) = g d x, (3.6) 2 4π M N2 √ a2 (D 2A ) = R g d4 x, (3.7) 2 48π M 1 N2 5R 2 − 8Rμν R μν − 7Rμνρσ R μνρσ a4 (D 2A ) = 2 16π 360 M 1 √ Tr(Fμν Fμν ) g d4 x, (3.8) − 2 24π M where N 2 originates from Tr M N (C) id. Inserting these expressions into (2.13) then results in
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Theorem 2 (Chamseddine–Connes [4]). The bosonic action for the inner fluctuations of the spectral triple (C ∞ (M, M N (C)), L 2 (M, S) ⊗ M N (C), ∂/ ⊗ 1) is given by Sb [A] ≡ Tr f (D A /) =
1 √ Lb (g, A) g d4 x + O(−2 ), 4π 2 M
with Lagrangian N2 f (0)N 2 2 f 2 2 R + 5R − 8Rμν R μν − 7Rμνρσ R μνρσ 6 1440 f (0) − Tr Fμν Fμν . 6
Lb (g, A) = 2 f 4 4 N 2 +
This expression contains both the Einstein–Hilbert action of General Relativity and the Yang–Mills action of a SU (N )-gauge field. Since the term ψ, D A ψ accounts for the fermionic propagator and interactions of the fermion ψ with the gauge field, the sum S[A, ψ] := Sb [A] + S f [A, ψ] = Tr f (D A /) + ψ, D A ψ
gives the full action of the Einstein–Yang–Mills system plus terms of order −2 . The gauge group SU (A) = C ∞ (M, SU (N )) acts on the gauge potential A and on ψ in the adjoint representation. 4. Supersymmetry in the Einstein–Yang–Mills System We would like to obtain a realization of supersymmetry for the Einstein–Yang–Mills system, as considered in the previous section, in the framework of noncommutative geometry. The possibility of such a supersymmetry was suggested in [4]. We work this out in full detail and give the supersymmetry transformation establishing this symmetry between the fermionic and bosonic fields. In trying to do so, we immediately stumble upon the problem that bosonic and fermionic fields do not have the same number of degrees of freedom, as is required for supersymmetry. Indeed, both in the spinorial as in the finite part the fermionic degrees of freedom exceed those of the bosons: by requiring self-adjointness and unimodularity, the finite part of the bosons was seen to be su(N )-valued one-forms. The finite part of the fermions, on the other hand, is an element of M N (C). On top of that, a spinor ψ(x) has eight real (four complex) degrees of freedom whereas the continuous part of the gauge potential has only four: Aμ , μ = 1, . . . , 4. We will solve these two problems one by one in the subsequent subsections.
4.1. Majorana and Weyl fermions. The basic fermionic constituents of most supersymmetric theories are Majorana fermions; particles that are invariant under charge conjugation. However, in this Euclidean set up, we have J 2 = −1 with which only ψ(x) = 0 could be Majorana. Indeed (massless) Majorana fermions do not exist in a 4 dimensional Euclidean space, as was pointed out by Schwinger [26] already in 1959. An alternative way to correctly reduce the number of degrees of freedom is to restrict the input of the inner product to eigenspaces H± of γ . To this end Chamseddine, Connes and Marcolli [7] propose as a fermionic action 21 J ψ, D A ψ instead of ψ, D A ψ . This would be of no avail to us, since this allows for such a restriction only when J γ = γ J ,
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in accordance with the classification of [5,6]. In our case, it would automatically yield J ψ1 , Dψ2 = − J ψ1 , Dψ2 for all ψ1 , ψ2 ∈ H. Different but similar solutions of this problem were given by Van Nieuwenhuizen and Waldron [22,23] and Nicolai [21]. To obtain a Lagrangian in Euclidean space, whose Green functions are analytic continuations of their Minkowskian counterpart, Van Nieuwenhuizen and Waldron propose the following. Starting from a Lagrangian for a single Weyl fermion, they define a Wick rotation on the spinors themselves. When applying this, one is obliged to drop the Minkowskian reality constraint ψ := ψ † γ 0 —rotating ψ and ψ separately. The result is then a Lagrangian containing Weyl spinors χ and ψ of opposite chirality. Since the system still contains two fermionic variables (χ and ψ instead of ψ and ψ) the path integral is insensitive to such a rotation. The hermiticity of the action is lost due to this procedure. This is not a problem however “since hermiticity is primarily needed for unitarity, and unitarity only makes sense in a theory with time”, as it is put in [23]. The solution is thus to take as the fermionic part of the action (ψ ∈ H+ , χ ∈ H− ),
S f [A, ψ, χ ] := χ , D A ψ ;
(4.1)
which is the Euclidean counterpart of the action for ψ and ψ in Minkowskian space. 4.2. Unimodularity for fermions. The reduction from M N (C) to su(N ) takes place in two steps; first from M N (C) to u(N ) and second from u(N ) to su(N ). For the first part we simply use the fact that the M N (C) is the complexification of u(N ) : M N (C) C ⊗R u(N ). For the full Hilbert space H this implies already that H = L 2 (M, S) ⊗C M N (C) L 2 (M, S) ⊗R u(N ). We obtain the reduction from u(N ) to su(N ) by splitting any fermion into a trace and a traceless part: = Tr ψ + ψ ∈ L 2 (M, S) ⊗ (u(1) ⊕ su(N )) . ψ Inserting this expression into the inner product, we get = Tr χ + χ , D A Tr ψ + Tr χ χ , D A ψ , D A Tr ψ , D A ψ + χ , D A ψ
= Tr χ , D Tr ψ + χ , D A ψ , where we have used that for λ ∈ u(1) and X, X 1 , X 2 ∈ su(N ) that [X, λ] = 0 = Tr(X λ). So the two different parts decouple and the trace-part lacks any gauge interactions; it describes a totally free fermion. We therefore discard it from the theory. 4.3. Supersymmetry transformations. After the preparations done in the two previous subsections, the Einstein–Yang–Mills system is at least suited for supersymmetry. What is left is actually proving that the system is supersymmetric. Thus, consider the action S[A, ψ, χ ] = Sb [A] + S f [A, ψ, χ ] in terms of the two traceless Weyl spinors ψ and χ and the SU (N )-gauge field A. We conveniently write the fermionic action, √ S f [A, ψ, χ ] = χ , D A ψ = Tr F (χ , D A ψ) gd 4 x, M
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in terms of a Hermitian pairing (. , .) : Γ ∞ (S) × Γ ∞ (S) → C ∞ (M) and a trace Tr F over the finite part. In order to see whether this system exhibits supersymmetry, we will define δA ∈ B(H) and δψ ∈ H+ , δχ ∈ H− , where the expressions for δA, δψ and δχ contain their respective superpartners— under which the action is invariant: d S[A + tδ A, ψ + tδψ, χ + tδχ ] δS[A, ψ, χ ] := = 0. (4.2) dt t=0 From here on ± will denote a pair of γ 5 eigenspinors that are singlets of the gauge group and vanish covariantly: ∇μS ± = 0. Definition 3. For A ∈ 1D (A), ψ ∈ H+ , χ ∈ H− , we define δ A ∈ B(H), δψ ∈ H+ , δχ ∈ H− by δ A = δ1 A + δ2 A, δψ := c3 F+
where and
δ1 A := γ μ c1 (− , γμ ψ), δ2 A := γ μ c2 (χ , γμ + ), δχ := c4 F− ,
a ⊗T ,F where F ≡ γ μ γ ν Fμν a μν = ∂μ Aν − ∂ν Aμ + [Aμ , Aν ] and c1 . . . c4 ∈ R.
The constants c1 . . . c4 are yet to be determined. Note that the expression for δ A from the above definition is not self-adjoint anymore. This is directly related to the action not being real as was mentioned before. In fact, one could take the real part of the fermionic action, allowing a self-adjoint transformation of the form A → A + δ A + δ A∗ . Proposition 4. With the definitions given above, we have for the fermionic part of the action, δS f [A, ψ, χ ] = −2ic4 − , Fμν γ μ D ν ψ − 2ic3 Fμν γ μ D ν χ , + , where Dμ = ∇μS + Aμ is the covariant derivative. Proof. We apply the above supersymmetry transformations to the fermionic part of the action to obtain d χ + tδχ , D A+tδ A (ψ + tδψ) δS f [A, ψ, χ ] = dt t=0 (4.3) = c4 F− , D A ψ + χ , δAψ + c3 χ , D A F+ . Let us look at the terms on the right hand side one by one. Writing out F, using the self-adjointness of D A and the identity μ α ∇νS γ μ = −Γνα γ
twice, we get for the first term c4 F− , D A ψ = −c4 i Γσμα γ σ γ α , γ ν Fμν − , ψ + c4 i γ σ γ μ γ ν Dσ Fμν − , ψ .
(4.4)
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Using the identity γ μ γ ν γ σ = g μν γ σ − g σ ν γ μ + g μσ γ ν − i μνσ λ γ 5 γλ μ
μ
and the symmetry Γνλ = Γλν of the Christoffel symbols, the first term on the RHS of (4.4) is seen to vanish whereas the second term now reads
(F− , D A ψ) = i g σ μ γ ν − g σ ν γ μ + g μν γ σ − i σ μνλ γ 5 γλ Dσ Fμν − , ψ . (4.5) The first two terms of (4.5) add up by the antisymmetry of Fμν , whereas the third term vanishes for that very reason. The fourth term vanishes in view of the Bianchi identity: [Dμ Fνσ + Dσ Fμν + Dν Fσ μ ](x) = 0 ∀x ∈ M. We are thus left with: c4 F− , D A ψ = 2c4 i(γ ν D μ Fμν − , ψ). By exactly the same reasoning we can rewrite the third term of (4.3). Now we are still left with the second term of (4.3), which yields for each point x ∈ M: Tr F (χ , δAψ)(x) = f abc (χ a , γ μ ψ c )[c1 (− , γμ ψ) + c2 (χ , γμ + )](x).
(4.6)
Both terms are seen to vanish separately using the antisymmetry of f abc and a Fierz transformation (see Appendix A for details). Adding the results for the first and third terms of (4.3) yields the expression: δS f [A, ψ, χ ] = 2c4 i γ ν D μ Fμν − , ψ + 2c3 i χ , γ ν D μ Fμν + . That covered the fermionic part of the action. For brevity we will from here on solely focus on the part of δ A featuring ψ (i.e. δ1 A), mentioning only that the other term can be handled analogously. Regarding the bosonic part we can see that after performing the supersymmetry transformation Proposition 5. The square of the operator D A+tδ1 A with δ1 A given in Definition 3 is of the form in (2.7): μ D 2A+tδ1 A = − gμν ∂ μ ∂ ν + K ∂μ + L with K μ and L given in terms of the K μ and L of Proposition 1 as μ
K = K μ − 2c1 ⊗ ad(− , γ μ ψ)t, L = L − c1 γ ν γ μ ⊗ ad(− , γμ Dν ψ)t − 2c1 ⊗ ad(− , γ ν ψ)[(ων − Γν ) ⊗ 1 + Aν t] + O(t 2 ). Proof. We will explicitly calculate D 2A+tδ1 A = [D + A + tδ1 A + J (A + tδ1 A)J ∗ ]2 = D 2A + i{D A , δ1 A}t + O(t 2 ), where δ1 A + J δ1 A J = iδ1 A.
(4.7)
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Then, the second term on the right hand side of (4.7) reads
c1 γ μ ⊗ ad(− , γμ ψ) {D A , δ1 A} = iγ ν ∇νS ⊗ id + Aν + ic1 γ μ ⊗ ad(− , γμ ψ) γ ν (∇νS ⊗ id + Aν ) = ic1 γ μ γ ν (∂μ ⊗ 1 + Aμ ) ad(− , γν ψ) −ic1 γ μ γ ν ad(− , γν ψ)(∂μ ⊗ 1 + Aμ ) ν μ λ +2ic1 ad(− , γ μ ψ)(∇μS + Aμ ) − ic1 Γμλ γ γ ad(− , γν ψ),
(4.8) where the Christoffel symbol stems from interchanging the spin connection ∇μS with a gamma matrix. Using that − vanishes covariantly we have
λ (− , γλ ψ) + (− , γν ψ)∂μ . ∂μ (− , γν ψ) = − , γν ∇μS ψ − Γμν μ
Inserting this expression into (4.8) and using the definition Γ μ = g νλ Γνλ , we receive for {D A , δ1 A}: {D A , δ1 A} = ic1 γ μ γ ν ad(− , γν Dμ ψ) + 2ic1 ad(− , γ ν ψ)[(∂μ + ωμ −Γμ ) ⊗ 1 + Aμ ]. Plugging this into (4.7) yields the desired form of K μ and L . We are thus allowed to perform a heat kernel expansion (2.11) for D A+tδ1 A to see to what extent each of the coefficients an (D 2A ) (for n = 0, 2, 4) is invariant under supersymmetry. The objects that are of interest to us are
d 2 d δan D 2A := an D A+tδ1 A an D 2A + {δ1 A, D A }t + O t 2 , = dt dt t=0 t=0 (4.9) the first of which are given by (2.12a), (2.12b) and (2.12c). The E and μν appearing in these formulas are of course different than before, but still related to K μ and L (as given above) in the same way; by (2.10a) and (2.9). Short calculations show that the changes of K μ to K μ and L to L have the following effect on the variable E and μν :
E = E + c1 γ μ γ ν ⊗ ad(− , γν Dμ ψ)t − c1 id ⊗ ad(− , γ μ Dμ ψ)t + O t 2 ,
μν = μν + c1 id ⊗ [ad(− , γν Dμ ψ) − c1 ad(− , γμ Dν ψ)]t + O t 2 . Having found these particular expressions, we are ready to determine (4.9). Proposition 6. The Seeley–DeWitt coefficients a0 (D 2A ) and a2 (D 2A ) are invariant under the supersymmetry transformation A → A+tδ1 A given in Definition 3, whereas a4 (D 2A ) transforms as d 2 c1 c2 μν a4 D A+tδ1 A − , F μν γν Dμ ψ + F γ ν D μ χ , + . = 2 2 dt 6π 6π t=0
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Proof. The first coefficient a0 D 2A+tδ1 A is trivial: the identity does not transform under supersymmetry. Ignoring for the moment the term linear in χ for the second Seeley– DeWitt coefficient, there is only one contribution [see (2.12b)]: d = c1 i Tr[γ μ γ ν ⊗ ad(− , γν Dμ ψ)id ⊗ ad(− , γ μ Dμ ψ)] = 0. (4.10) Tr(E ) dt t=0 For the third coefficient (2.12c) there are three terms of interest:
1 d 2 μν √ 4 Tr 6E + + 2R E gd x δa4 D 2A = μν , (4.11) 2 192π dt M t=0 where the last one vanishes by the same reasoning as employed above. For the first term we use that E = 41 R ⊗ 1 − 21 γ μ γ ν ⊗ Fμν and obtain d dt
Tr E
2√
M
gd x 4
t=0
1 = −2 c1 2
M
λ σ μ ν Tr γ γ γ γ ⊗ Fλσ ad(− , γν Dμ ψ)
√ 4 −γ γ ⊗ Fμν ad(− , γ λ Dλ ψ) gd x λν σ μ λμ νσ = −4c1 δ δ − δ δ √ × Tr[Fλσ ad(− , γν Dμ ψ)] gd 4 x M √ = 8c1 Tr Fμν ad(− , γν Dμ ψ) gd 4 x M = −8N c1 − , F μν γμ Dν ψ , μ ν
where at various points we have used that F is antisymmetric. For the second term in ab γ ⊗ 1, (4.11) we have with μν = 1 ⊗ Fμν + 41 Rμν ab 2 √ Tr μν gd 4 x M t=0 1 ab √ 4 = 2c1 Tr 1 ⊗ Fμν + Rμν γab ⊗ 1][id ⊗ ad(− , γ[ν Dμ] ψ)] gd x 4 M √ =16c1 Tr Fμν ad(− , γ ν D μ ψ) gd 4 x = −16N c1 − , Fμν γ μ D ν ψ .
d dt
M
We thus get for (4.11):
c1 c1 μν μν δa4 D 2A = − (48 + 16)N , F γ D ψ = − , F γ D ψ . − μ ν − μ ν 192π 2 3π 2 As was said before, repeating the calculations for δ2 A gives a similar result involving χ . Thus, although a0 (D 2A ) (proportional to 4 ) and a2 (D 2A ) (proportional to 2 ) are supersymmetry invariants, a4 (D 2A ) transforms to an expression that equals the one of the fermionic action (cf. Proposition 4) by the right choice of coefficients. This means that
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Theorem 7. The action S[A, ψ, χ ] = Sb [A] + S f [A, ψ, χ ] (with S f defined in (4.1)) of the Einstein–Yang–Mills system is invariant under supersymmetry for at least all positive powers of , provided c4 = −
ic1 f (0) 6π 2
and
c3 = −
ic2 f (0) . 6π 2
Though these results are encouraging, it is still somewhat unsatisfactory that we had to resort to a heat kernel expansion; a question whether or not the full spectral action is supersymmetry invariant remains to be answered. As was noted by Chamseddine in [2], noncommutative geometry treats bosons (spectral action) and fermions (inner product) on a different footing. Hence, any attempt (such as [27]) that combines both the inner product and the spectral action into a single expression is well worth studying from the perspective of supersymmetry. 5. Supersymmetric QCD In this section, we consider a supersymmetric version of QCD —the theory of quarks and gluons. For that we will be regarding only one of three generations of particles and we leave all leptons and electroweak gauge bosons out. 5.1. The finite spectral triple. If we want any chance of finding supersymmetry, we need to enlarge the finite part of the Hilbert space such that it contains not only the quarks and antiquarks, but the gluinos1 —the supersymmetric partners of the gluons and therefore fermions— as well. Moreover, in order to keep the gauge group to be SU (3) the algebra in our spectral triple should be M3 (C). Definition 8. The finite spectral triple (A F , H F , D F ) is defined by – A F := M3 (C). – H F := C3 ⊕ M3 (C) ⊕ C3 , carrying the following representation of A F : π(m)(q, g, q ) = (mq, mg, q ). – D F is defined as the matrix:
⎛
⎞ 0 d 0 D F := ⎝ d ∗ 0 e∗ ⎠ , 0 e 0
with d : M3 (C) → C3 and e : M3 (C) → C3 arbitrary linear maps. Note that this approach is not entirely in the same spirit as [7,13], where the Hilbert space is defined as an A bimodule. This would have required us to take A = C ⊕ M3 (C) but would have led both to a different gauge group as to an extra —unwanted— copy of C in the Hilbert space. Starting out with a simple algebra requires this slightly different approach, but entails that, strictly speaking, (5.1) is not an algebra representation. We do not consider this to be a problem however, since it does not occur in more realistic situations (i.e. those with a non-simple algebra). 1 We will postpone the (partial) justification of this terminology until Proposition 13.
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The conditions of a spectral triple are trivially satisfied; we would like to define a real structure J F on it as well. Our candidate is J F (q1 , g, q2 ) := (q2 , g ∗ , q1 ) ∈ C3 ⊕ M3 (C) ⊕ C3 .
(5.1)
This form of J F , as with the representation of the algebra, is as expected: on the two copies of C3 it is —up to interchanging these two copies— the same as in the noncommutative description of the Standard Model [7]; on M3 (C) it is the same as in the Einstein–Yang–Mills system. Lemma 9. With J F as above, the requirement D F J F = J F D F uniquely determines e in terms of d: e(g) = d(g ∗ )
∀g ∈ M3 (C).
Proof. This follows from a direct computation of D F J F and J F D F acting on an element in H F . Let us check the other conditions for a real spectral triple (of KO-dimension 0). We compute for the opposite representation π ◦ (m) = J F π(m ∗ )J F∗ : π ◦ (m)(q1 , g, q2 ) := (q1 , gm, m t q2 ).
(5.2)
One easily checks that π(m) commutes with π ◦ (m ) for any m, m ∈ M3 (C), thus fulfilling the second condition in Eq. (2.1). In order to satisfy the first (i.e. the first-order condition), we make the following choice of D F in terms of a map d : M3 (C) → C3 of the form: d(g) = gv
∀g ∈ M3 (C)
(5.3)
for a fixed v ∈ C3 . This definition for d corresponds to d ∗ (q) = qv t (considered as the 3 × 3 matrix (qv t )i j = qi v j ) for the adjoint of d and by Lemma 9 to
e(g) = g t v, e∗ (q) = vq t ; g ∈ M3 (C), q ∈ C3 for the map e and its adjoint. Proposition 10. Given the representations of the algebra (5.1) and (5.2), the finite Dirac operator with d and e as above, satisfies the order one condition (2.1). Consequently, the set of data (A F , D F , H F , J F ) defines a finite real spectral triple of KO-dimension 0. Proof. Writing out (2.1) and using that π(m) = 1 on antiquarks and π ◦ (m) = 1 on quarks gives a number of simultaneous demands: d(mg) = md(g), e(gm) = m t e(g),
d ∗ (mq) = md ∗ q, e∗ (m t q ) = e∗ (q )m
∀m ∈ m, g ∈ M3 (C), q, q ∈ C3 .
These are easily seen to be met for the given representations and maps d and e. As a preparation for the next section, we determine the inner fluctuations of the finite Dirac operator, as well as the (finite) gauge group and its action.
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Lemma 11. The inner fluctuations D F + A F + J F A∗F J F∗ with A F ∈ D F (A F ) of the finite Dirac operator D F are parametrized by a vector q˜ ∈ C3 as ⎛ ⎞ 0 A 0 q ∗ ∗ ∗ ∗ Aq 0 B ⎠ D q := D F + A F + J F A F J F = g3 ⎝ q 0 B 0 q with g3 the QCD-coupling constant and ∗ A q , q (q) = q
A q, q (g) = g
∗ B (q) q
t B q. q (g) = g
t
= qq , t
Proof. We have A F := i π(m i )[D F , π(n i )] [cf. (2.2)] which, applied to an element (q1 , g, q2 ) ∈ H F , gives
0, m i [1 − n i ]vq2t , g t (n i∗ − 1)v) . (5.4) π(m i )[D F , π(n i )](q1 , g, q2 ) = i
i
For the other part, J F A∗F J F∗ , we compute J F (π(m i )[D F , π(n j )])∗ J F∗ = − J F [D F , π(n i∗ )]π(m i∗ )J F i
i
=−
[D F , π ◦ (n i )]π ◦ (m i ),
i
where we have used that D F J F = J F D F . We therefore get
t gm i (1 − n i )v, q1 (n i∗ − 1)v , 0 . J F A∗F J F (q1 , g, q2 ) =
(5.5)
i
Requiring A F to be self-adjoint yields the demand n i∗ − 1 = i m i (1 − n i ),
(5.6)
i
for the elements of the algebra. Defining q := g3−1 [1 + m i (1 − n i )]v = g3−1 n i∗ v, i
(5.7)
i
and adding the expressions (5.4) and (5.5) for A F and J F A∗F J F∗ respectively to that of D F , we get
t q , q1 q + q q2t , g t q , D F + A F + J F A∗F J F∗ (q1 , g, q2 ) = g3 g which is of the desired form. We can visually represent the finite spectral triple and the inner fluctuations of its Dirac operator by means of a Krajewski diagram [17] as depicted in Fig. 1.
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Fig. 1. Krajewski diagram of super-QCD. The nodes represent the representation of C ⊕ M3 (C) horizontally and of C◦ ⊕ M3 (C)◦ vertically. The dotted line indicates how the squark and antisquark appear as components in the (finite) Dirac operator. Note that we do not actually have a copy of C in the algebra, but see the discussion in the beginning of this subsection
5.2. The product geometry and its inner fluctuations. We next consider the product of the canonical spectral triple (C ∞ (M), L 2 (M, S), ∂/ M ) associated to a four-dimensional Riemannian spin manifold M, with the above spectral triple (A F , H F , D F ). Explicitly, we have A = C ∞ (M, M3 (C)),
H = L 2 (M, S) ⊗ C3 ⊕ M3 (C) ⊕ C3 , D = ∂/ M ⊗ id + γ5 ⊗ D F . The grading γ = γ5 ⊗ 1 and real structure J = J M ⊗ J F give the resulting real spectral triple KO-dimension 4. We will write a generic element in the Hilbert space as ψ = (ψq , ψg , ψq¯ ), according to the above direct sum decomposition. For the bosons, we derive from Eq. (2.2) and Lemma 11 that Proposition 12. The inner fluctuations D → D A = D + A + J A J ∗ with A ∈ 1D (A) are parametrized by an SU (3)-gauge potential Aμ (x)(μ = 1, . . . , 4) and a C3 -valued function q (x)(x ∈ M). Explicitly, we have with A = iγ μ Aμ : D A = ∂/ ⊗ 1 + A + γ5 ⊗ Dq˜ with Dq˜ as in Lemma 11. q as the squark and anti-squark, respectively. As before, A will We will identify q and be the gluon, and ψg the gluino. This terminology is justified by the action of the gauge group on these fields: Proposition 13. The gauge group SU (A) = C ∞ (M, SU (3)) acts on the squarks and quarks in the defining representation, on the gluinos in the adjoint representation, and on the gluon as a SU (3)-gauge field, i.e. for u ∈ SU (A): q˜ → u q; ˜
ψq → uψq ;
ψg → uψg u ∗ ;
Aμ → uAμ u ∗ + u∂μ u ∗ .
Proof. For a real spectral triple, the gauge group SU (A) acts on the Hilbert space in the adjoint representation Ad(u) := π(u)π ◦ (u ∗ ) = u J u J ∗ . A direct computation shows that on an element in Hilbert space: Ad(u)(ψq , ψg , ψq¯ ) = (uψq , uψg u ∗ , uψq¯ ).
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Next, we look at how D q transforms: ∗ ∗ ∗ t Ad(u)D q Ad(u )(ψq , ψg , ψq¯ ) = Ad(u)D q (u ψq , u ψg u, u ψq¯ )
t = Ad(u) (u ∗ ψg u) q , (u ∗ ψq ) q + q (u t ψq¯ )t , (u ∗ ψg u)t q
t q , u(u ∗ ψq ) q u ∗ + u q (u t ψq¯ )t u ∗ , u(u ∗ ψg u)t q = u(u ∗ ψg u)
q , ψq (u q ) t + u q ψqt¯ , ψgt u q , = ψg u
which corresponds to applying Du q to (ψq , ψg , ψq¯ ). Last, we check how the gluons transform. For instance, when applied to a gluino ψg : Ad(u)(∂μ + Aμ ) Ad(u ∗ )ψg = π(u)π ◦ (u ∗ )(∂μ (u ∗ ψg u) + [Aμ , u ∗ ψg u]) = Ad(u)ψg + ∂μ ψg + ψg (∂μ u)u ∗ + u[Aμ , u ∗ ψg u]u ∗ = ∂μ ψg + ad(u Aμ u ∗ + u[∂μ , u ∗ ])ψg = ∂μ + Auμ ψg , with Au as in (2.4). Similar statements hold when acting on ψq and ψq¯ , respectively. 5.3. The spectral action. Having found an expression for the inner fluctuations of the product geometry, we now determine the corresponding spectral and fermionic action. Let us abbreviate D (1,0) = ∂/ ⊗1+A ≡ iγ μ Dμ to write for the fluctuated Dirac operator: D A = D (1,0) + γ 5 ⊗ D q,
(5.8)
Before we compute the spectral action, we will first prove some useful lemmas. Lemma 14. For the square of D A we have 2
2 Tr D 2A = Tr D (1,0) + Tr D q with
2 2 Tr D q |2 . q = 12g3 | Proof. The cross term in the square of D A equals γ 5 [D (1,0) , 1 ⊗ D q ], which vanishes upon taking the trace. For the square of the finite part we find
2 2 ∗ 2 ∗ = 2g + 2g Tr A A Tr B B Tr D q q q 3 3 q . q ∗ If we apply this first operator on the right hand side on a quark q we find that A q A q = diag | q |2 . With a similar calculation for B q , we arrive at the result.
Lemma 15. For the fourth power of the finite Dirac operator D q we have 4 4 Tr D q |4 . q = 16g3 |
(5.9)
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Proof. The calculation bears strong resemblance with the previous lemma, the main difference lies in additional cross terms. Lemma 16. For the commutator between the continuous D (1,0) = iγ μ Dμ and finite Dirac operators we have Dμ , D q (ψq , g, ψq¯ ) = D(∂μ +g3 Aμ ) q (ψq , g, ψq¯ ). Proof. We use that Dμ acts on the Hilbert space as: Dμ ψq , ψg , ψq¯ = ∂μ + g3 Aμ ψq , ∂μ + g3 Aμ ψg , ∂μ + g3 Aμ ψq¯ . Thus we get from applying the commutator (whilst putting g3 = 1 for simplicity):
t t q , ψq ∂μ q − (ψq q )Aμ Dμ , Dq˜ (ψq , g, ψq¯ ) = g(∂μ + Aμ ) + (∂μ + Aμ ) q ψqt¯ , g t (∂μ − Atμ ) q t q , ψq (∂μ + Aμ ) q = g(∂μ + Aμ )
+ (∂μ + Aμ ) q ψqt¯ , g t (∂μ + Aμ ) q , where we have frequently used that A∗μ = −Aμ . We will proceed —as in Sect. 3— by making an expansion in powers of D 2A . We first determine the endomorphism E defined by D 2A = ∇ ∗ ∇ − E . Here ∇ is the connection defined by A on the tensor product of the spinor bundle by the trivial bundle with fiber C3 ⊕ M3 (C) ⊕ C3 . With respect to the Einstein–Yang–Mills system, we are simply adding the term γ5 ⊗ Dq˜ to D (1,0) ; this is easily seen to leave μν unchanged, and having the following effect on E: − E → −E = −E − iγ 5 γ μ [Dμ , Dq˜ ] + Dq2˜ ,
(5.10)
compared to E = 14 R ⊗ id − 21 γ μ γ ν ⊗ Fμν prior to adding squarks and quarks. The minus sign giving rise to the commutator comes from interchanging γ μ and γ 5 . The term ωμ then drops from the expression, leaving the commutator of Dμ := ∂μ + g3 Aμ with Dq˜ . Theorem 17. The spectral action Sb [A] for the inner fluctuations of the spectral triple (A, H, D) introduced above is given by the spectral action Sb for the Einstein–Yang– Mills system (cf. Theorem 2) plus additional terms of the form f (0) 2 6 f2 Sb [A] = Sb [A] + 8g3 | q (x)|2 + g32 q (x)|4 + 6|Dμ q (x)|2 − 2 g32 2 | 2 π 4π M √ −3Rg32 | q (x)|2 g d4 x.
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Proof. From (2.12b) we see that the contributions to the Lagrangian of O 2 come from Tr(E ). Since the trace of the second term of (5.10) vanishes, we are left with 2
q |2 , Tr(E ) = Tr(E) − 4 Tr D (0,1) = Tr(E) − 48g32 | by virtue of Lemma 14. Since μν is unaltered, all extra terms we have on O(0 ) result from Tr(R E ) and Tr(E 2 ) [see (2.12c)]. For the first we have 2
q |2 , (5.11) Tr(R E ) = Tr(R E) − 4R Tr D (0,1) = Tr(R E) − 48g32 R| whereas the second gives
2 2 2 1 Tr(E 2 ) = Tr(E 2 ) + Tr iγ 5 γ μ [Dμ , D + Tr[ Dq˜ ]2 − Tr[R ⊗ Dq˜ ] q]
2 2 μ = Tr E + 4 Tr [Dμ , Dq˜ ][D , Dq˜ ] + 4 Tr Dq4˜ − 2R Tr Dq2˜ (5.12) . In the first step we have used that terms of the Clifford algebra proportional to γ μ γ ν (μ < ν), γ 5 γ μ and 1 are orthogonal, and we consequently only retain the squares of the terms in (5.10) plus one cross-term. Now for the second and the last terms of (5.12) we can use Lemmas 15 and 16 with which the former becomes 2 q |2 . Tr([Dμ , Dq˜ ][D μ , Dq˜ ]) = Tr D(∂μ +g3 Aμ ) q D(∂μ +g3 Aμ ) q = 12g3 |(∂μ + g3 Aμ ) (5.13)
Taking the expansion of the spectral action (2.13), with the coefficients taken from (2.12b) and (2.12c) we get the following extra contributions: order 2 : order 0 :
2 6 4 Tr Dq˜ = − 2 f 2 g32 | q |2 , π (4π )
1 1 2 2 −60 −48g f (0) R| q | 3 (4π )2 360
2 q | + 64| q |4 − 24R| q |2 , +180 4 · 12| ∂μ + g3 Aμ
−2 f 2
1
2
which ends the proof. In order to have manifest supersymmetry with the fermionic action S f [A, ψq , ψg ] we have to reduce once more the degrees of freedom for the spinor ψg . This is completely analogous to what happens in the Einstein–Yang–Mills system in Sect. 4: we replace the M3 (C)-valued Dirac spinor ψg by two su(3)-valued Weyl spinors ψg and χg of opposite chirality. For the spinorial part of the quark and antiquark we can do a similar thing, but to prevent terms from vanishing automatically we have to interchange the left and right handed parts as compared to the gluino. Theorem 18. The fermionic action for the triple (A, H, D) is given by S f [A, ψq , χq , ψg , χg , ψq¯ , χq¯ ] ≡ (ψq , χg , ψq¯ ), D A (χq , ψg , χq¯ ) ¯ q¯
= ψq , (∂/ + A)χq + χg , (∂/ + A)ψg + ψq¯ , (∂/ + A)χ t ¯˜ χq¯ + ψq , ψ t q¯˜ . + ψq , ψg q˜ − χg q, ˜ χq − χg q, g
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5.4. Discussion. In summary, we have added the superpartners of the QCD-particles (squarks and gluinos) to the theory, in conformity to the ‘paradigm’ of NCG: fermions are elements of the Hilbert space, whereas bosons arise as inner fluctuations of a Dirac operator. Having quarks, gluinos and anti-quarks as the fermionic constituents, the freedom to choose the (finite part of the) Dirac operator was seen to be very little. On top of that, this construction led to the fact that these superpartners are in the right representation of the gauge group. A computation of the spectral action and the fermionic action then led to additional terms over the supersymmetric Einstein–Yang–Mills system considered in Sect. 4. We will now interpret these additional terms. Note that for supersymmetry at least the number of degrees of freedom need to be the same. For that, the finite part of the gluinos has to be reduced from M3 (C) to su(3) —a problem that was dealt with in Sect. 4. As far as the quarks and squarks are concerned, we have not addressed the apparent discrepancy between degrees of freedom yet. Indeed, the squarks are described by a C3 -valued function, whereas quarks are described by a C3 -valued Dirac spinor, i.e. a mismatch of a factor of 4. This is due to the fact that we have ignored isospin, something that will await another time. We next compare the above results (Proposition 17 and 18) with that of the Minimally Supersymmetric Standard Model (MSSM); Kraml [19] and Chung et al. [8] provide lengthy expositions on the MSSM. In the latter, the various MSSM-interactions are conveniently listed in the Appendix. We first switch to flat Euclidean space by taking ωμ = 0 and R = 0 and working on a local chart of M. For each of the interactions that appear we will at the same time make the switch from the current notation to the one more common in physics and translate the (relevant pieces of the) Lagrangian as found in the literature to this context. First, the free part of the action Sb + S f in Proposition 17 and 18 coincides with the usual kinetic terms for the quark, squark, gluon and gluino. Note the additional coupling of the squark to the scalar curvature of M. –
Squark-quark-gluino. The quark is described by ψq = ψqi ⊗ ei ∈ L 2 (M, S) ⊗ C3 , the antiquark by ψq¯ = ψqi¯ ⊗ ei ∈ L 2 (M, S) ⊗ C3 , and the gluino by a pair of su(3)-valued Weyl spinors ψg = ψga ⊗ Ta and χg = χga ⊗ Ta . The finite part of the Dirac operator gives in the fermionic action the term:
–
(ψq , χg , ψq¯ ), (γ 5 ⊗ Dq˜ )(χq , ψg .χq¯ )
= (ψq , χg , ψq¯ ), (γ5 ψg q , γ5 (χq qt + q χqt¯ ), γ5 ψgt q)
i q k − χga , γ 5 χqk = g3 (Ta )ik ψqi , γ 5 ψga q
i q k + ψqk¯ , γ5 ψga − χga , γ 5 χqi¯ q .
Here the transpose t refers to the finite index only and (·, ·) is the hermitian structure in the spinor bundle (i.e. summation over spinor indices). Note that an interaction such as (ψqi , γ 5 ψga ) actually only involves the positive chirality part (with respect to γ5 ) of ψqi in accordance with [8,19]. As in the Einstein–Yang–Mills system, we get a gluon-gluino-gluino interaction from the continuous part of D A in the fermionic action:
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χg , ig3 γ μ Aμ ψg = ig3 χgc , γ μ Aaμ ψgb Tr (Tc [Ta , Tb ])
= ig3 f abc χgc , γ μ ψgb Aaμ .
–
Similarly, we have the usual quark-quark-gluon interaction, which reads (ψq ig3 γ μ Aμ ψq ). From (5.13) we can extract a squark-squark-gluon term, that is of the form i i −g32 g3 Aμ q i ∂ μ q − g32 ∂μ q g3 Aμ q i i j = −g33 Aaμ (Ta )i j q j ∂μ q − g33 ∂μ ( q q )i Aaμ Ta i j j i j i = g33 Aaμ (Ta )i j q ∂μ q − g33 ∂ μ q q .
–
Equation (5.13) provides us a squark-squark-gluon-gluon term as well: i g32 g3 Aμ g3 Aμ q q i = −g34 Aμ a Abμ (Tb Ta )i j qi qj 1 1 = − Aμa Aμa qi q i − dabc Aμ a Abμ (T c )i j qi qj. 6 2 In going to the last line, we have used the identity Tb Ta =
–
1 1 δab id3 + (i f bac + dbac )T c , 6 2
where the term with f abc vanishes since Aμ a Abμ is symmetric upon interchanging a and b. Finally, there is a four squark self-interaction q (x)|4 = g34 q (x)i q (x)i q (x) j q (x) j , g34 | originating from the third term of the display in Theorem 17.
To summarize, all results are in perfect agreement with the literature, in the sense that all interactions are present and their form is precisely the same. In three terms that we compared however, we were off by two powers of the coupling constants and a sign. However, it is precisely these ‘erroneous’ terms of the Lagrangian that are accompanied by a factor f (0), in which we can absorb this excess of coupling constants. The minus sign is unresolved still, since f has to be a positive function. There is one other unresolved issue: the constants appearing in our results do not in all cases match those of the literature. However, to properly address all these issues, one has to wait for a description of the full MSSM in terms of a noncommutative manifold —since that is what we are comparing our model with here— taking also into account isospin and hypercharge. This is part of future research. One observation that we cannot refrain from doing is that the sum Sb + S f of the actions in Theorem 17 and 18 is not supersymmetric. In fact, there appear terms proportional to | q |2 as allowed in soft supersymmetry breaking (see for instance [8] and references therein). If it were not for these terms, the action would be invariant under the transformations as laid out in Definition 3 together with
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δψq ∼ D A q − ,
δχq ∼ D A q + ,
q − , δψq¯ ∼ D A
δχq¯ ∼ D A q + ,
δ q ∼ (+ , ψq ) + (χq , − ),
δ q ∼ (+ , ψq¯ ) + (χq¯ , − ),
where ∼ means modulo a multiplicative constant. The ‘mass’ terms then transforms like 4 2 δ d x| q| ∼ d4 x q [(+ , ψq¯ ) + (χq¯ , − )] + [(+ , ψq ) + (χq , − )] q , M
M
which, given the action and transformation laws of the fields, cannot be cancelled. Though for now these mass terms are accompanied by a minus sign —much like in the case of the Higgs mechanism— we consider the presence of these terms as a merit of the above model, leaving the question open whether a description of the spontaneous supersymmetry breaking mechanism responsible for these soft-breaking terms can be found within noncommutative geometry. Of course, a search for such a mechanism is motivated by the derivation of the Higgs spontaneous gauge symmetry breaking mechanism from a noncommutative manifold in the case of the Standard Model. Possibly, one of the noncommutative manifolds that appear in the classification of [6] will describe the supersymmetric theory with spontaneous supersymmetry breaking mechanism. Open Access This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
A. Fierz Identities The topic of this section are (Euclidean) Fierz identities. For these identities in a Minkowskian background, we refer to eg. [24]. Definition 19 (Orthonormal Clifford basis). Let Cl(V ) be the Clifford algebra over a vector space V of dimension n. Then γ K := γk1 · · · γkr for all strictly ordered sets K = {k1 < . . . < kr } ⊆ {1, · · · , n} form a basis for Cl(V ). If γ K is as above, we denote with γ K the element γ k1 · · · γ kr . The basis spanned by the γ K is said to be orthonormal if Tr γ K γ L = nn K δ K L ∀ K , L. Here n K := (−1)r (r −1)/2 , where r denotes the cardinality of the set K and with δ K L we mean δK L =
1 if K = L . 0 else
(A.1)
Example 20. Take V = R4 and let Cl(4, 0) be the Euclidean Clifford algebra, that is, with signature (+ + + +). Its basis are the sixteen matrices 1, γμ γμ γν γμ γν γλ γ1 γ2 γ3 γ4 =: γ5 .
μ<ν μ<ν<λ
(4 elements), (6 elements), (4 elements),
We can identify γ1 γ3 γ4 = γ2 γ5 , γ1 γ2 γ3 = γ4 γ5 , γ1 γ2 γ4 = −γ3 γ5 , γ2 γ3 γ4 = −γ1 γ5 , establishing a connection with the basis most commonly used by physicists.
(A.2)
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Lemma 21 (Completeness relation). If the basis of the Clifford algebra is orthonormal, it satisfies the following completeness relation: 1 L c nL γ (A.3) (γ L )a b = δa c δd b . d n L
Proof. Since the γ K form a basis, we can write any element Γ of the Clifford algebra as Γ = m K γ K m K ∈ C, (A.4) K
where the sum runs over all (strictly ordered) sets. By multiplying both sides with γ L and taking the trace we find the expression for the coefficient m L to be: 1 n L Tr Γ γ L . n Applying this result in particular to Γ = γ K , and writing matrix indices explicitly, (A.4) yields
c 1 n L (γ K )cd γ L (γ K )a b = (γ L )a b , d n mL =
L
for which (A.3) is required. Theorem 22 ((Generalized) Fierz identity). If for any two strictly ordered sets K , L there exists a third strictly ordered set M and c ∈ N such that γ K γ L = c γ M , we have the four-spinor identity: 1 ψ1 , γ K ψ2 ψ3 , γ K ψ4 = − C K L ψ3 , γ L ψ2 ψ1 , γ L ψ4 , C K L ∈ N, n L
(A.5) for any ψ1 , . . . , ψ4 in the n-dimensional spin representation of the Clifford algebra. Here we denote by ., . the inner product on the spinor representation. f Proof. We start by multiplying the completeness relation (A.3) with (γ K )ce γ K b yielding
f f 1 = (γ K )a e γ K (γ L γ K )d e γ L γ K a , d n L or
f M f 1 e = , (A.6) (γ K )a e γ K M C K M (γ M )d γ a d n by the assumption made. Here we have accommodated the proportionality constants in a matrix C K L . Now we have to contract the above expression with the four spinors ψ1a , ψ2 e , ψ3d and ψ4 f . But, remembering that they are Grassmann variables — i.e. their components anticommute— we get one minus sign on the left hand side of (A.6) from interchanging ψ1 and ψ3 . Hence we arrive at the result. Now how do we compute the constants C K L ? Just multiply (A.6) again by a (γ L )ed γ L f , yielding:
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1 Tr γ K γ L γ K γ L = C K M Tr γ M γ L Tr γ M γ L . n M
(A.7)
On the other hand, we have γ K γ L γK = f K L γ L
f K L ∈ N (no sum over L),
(A.8)
using the anticommutator repeatedly2 . Putting (A.8) into (A.7) we get: L f K L Tr(γ L γ L ) = n C K M δ LM δ M M
or CK L = nL fK L , since
(A.9)
Tr γ L γ L = (−1)r (r −1)/2 n,
by orthonormality. Corollary A1 (Fierz identity). We work out one example of particular interest to us. Consider again Cl(4, 0)(n = 4) with the basis as in Example 20. As can readily be checked, this basis satisfies the requirement for Theorem 22. The spinors we will contract with, are the four Weyl spinors: χ , − ∈ S − , ψ1 , ψ2 ∈ S + . We start with determining the numbers f 1r , r = 0, . . . , 4 defined by γ μ γ L γμ = f 1r γ L (see above) where r is the cardinality of L. We find the recursive relation γ μ 1γμ = n · 1 ≡ f 10 1, γ μ γν γμ = 2γν − γ μ γμ γν = (2 − f 10 )γ ν ≡ f 11 γ ν , ... γ μ (γ ν1 · · · γ νn )γμ = [2(−1)n−1 − f 1(n−1) ]γ ν1 · · · γ νn ≡ f 1n γ ν1 · · · γ νn (n ≤ 4), which gives f 10 = 4,
f 11 = −2,
f 12 = 0,
f 13 = 2,
f 14 = −4,
and consequently, using (A.9) C10 = 4, C11 = −2, C12 = 0, C13 = −2, C14 = −4. Now applying (A.5) yields
1 χ , γ μ ψ1 − , γμ ψ2 = − C11 − , γ μ ψ1 χ , γμ ψ2 4 1 − C13 − , γ μ γ ν γ λ ψ1 χ , γμ γν γλ ψ2 , 4
since only terms with an odd number of γ -matrices survive due to the different chirality of the spinors. Identifying the terms with three γ -matrices with ±γ μ γ5 as in (A.2), we get 2 For example: γ μ γ λ γ = (2 − dim V )γ λ ∀ λ ∈ {1, 2, . . . , dim V }. μ
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1 1 χ , γ μ ψ1 − , γμ ψ2 = − , γ μ ψ1 χ , γμ ψ2 + − , γ μ ψ1 χ , γμ ψ2 2 2 = − , γ μ ψ1 χ , γμ ψ2 . (A.10)
References 1. Chamseddine, A.H.: Connection between space-time supersymmetry and noncommutative geometry. Phys. Lett. B332, 349–357 (1994) 2. Chamseddine, A.H.: Remarks on the spectral action principle. Phys. Lett. B436, 84–90 (1998) 3. Chamseddine, A.H., Connes, A.: Universal formula for noncommutative geometry actions: Unifications of gravity and the standard model. Phys. Rev. Lett. 77, 4868–4871 (1996) 4. Chamseddine, A.H., Connes, A.: The spectral action principle. Commun. Math. Phys. 186, 731–750 (1997) 5. Chamseddine, A.H., Connes, A.: Conceptual Explanation for the Algebra in the Noncommutative Approach to the Standard Model. Phys. Rev. Lett. 99, 191601 (2007) 6. Chamseddine, A.H., Connes, A.: Why the Standard Model. J. Geom. Phys. 58, 38–47 (2008) 7. Chamseddine, A.H., Connes, A., Marcolli, M.: Gravity and the standard model with neutrino mixing. Adv. Theor. Math. Phys. 11, 991–1089 (2007) 8. Chung, D.J.H., et al.: The soft supersymmetry-breaking Lagrangian: Theory and applications. Phys. Rept. 407, 1–203 (2005) 9. Connes, A.: Noncommutative Geometry. San Diego: Academic Press, 1994 10. Connes, A.: Gravity coupled with matter and the foundation of non-commutative geometry. Commun. Math. Phys. 182, 155–176 (1996) 11. Connes, A.: On the spectral characterization of manifolds. Preprint, 2009 12. Connes, A., Lott, J.: Particle models and noncommutative geometry. Nucl. Phys. Proc. Suppl. 18B, 29–47 (1991) 13. Connes, A., Marcolli, M.: Noncommutative Geometry, Quantum Fields and Motives. Providence, RI: Amer. Math. Soc., 2008 14. Gilkey, P.B.: Invariance theory, the heat equation, and the Atiyah-Singer index theorem. Volume 11 of Mathematics Lecture Series. Wilmington, DE: Publish or Perish Inc., 1984 15. Hussain, F., Thompson, G.: Noncommutative geometry and supersymmetry. Phys. Lett. B260, 359–364 (1991) 16. Hussain, F., Thompson, G.: Noncommutative geometry and supersymmetry. 2. Phys. Lett. B265, 307–310 (1991) 17. Krajewski, T.: Classification of finite spectral triples. J. Geom. Phys. 28, 1–30 (1998) 18. Kalau, W., Walze, M.: Supersymmetry and noncommutative geometry. J. Geom. Phys. 22, 77–102 (1997) 19. Kraml, S.: Stop and sbottom phenomenology in the MSSM. PhD thesis, TU Wien, http://arxiv.org/abs/ hep-ph/9903257v1, 1999 20. Lizzi, F., Mangano, G., Miele, G., Sparano, G.: Fermion Hilbert space and fermion doubling in the noncommutative geometry approach to gauge theories. Phys. Rev. D55, 6357–6366 (1997) 21. Nicolai, H.: A Possible constructive approach to (super − φ 3 )4 . 1. Euclidean formulation of the model. Nucl. Phys. B140, 294 (1978) 22. van Nieuwenhuizen, P., Waldron, A.: On Euclidean spinors and Wick rotations. Phys. Lett. B389, 29–36 (1996) 23. van Nieuwenhuizen, P., Waldron, A.: A continuous Wick rotation for spinor fields and supersymmetry in Euclidean space. http://arxiv.org./abs/hep-th/9611043v1, 1996 24. Nieves, J.F., Pal, P.B.: Generalized Fierz identities. Am. J. Phys. 72, 1100–1108 (2004) 25. Rieffel, M.A.: Morita equivalence for C ∗ -algebras and W ∗ -algebras. J. Pure Appl. Algebra 5, 51–96 (1974) 26. Schwinger, J.: Euclidean Quantum Electrodynamics. Phys. Rev. 115, 721–731 (1959) 27. Sitarz, A.: Spectral action and neutrino mass. Europhys. Lett. 86, 10007 (2009) Communicated by A. Connes
Commun. Math. Phys. 303, 175–211 (2011) Digital Object Identifier (DOI) 10.1007/s00220-010-1166-9
Communications in
Mathematical Physics
Aharonov-Bohm Effect and High-Velocity Estimates of Solutions to the Schrödinger Equation Miguel Ballesteros1 , Ricardo Weder2, 1 Institut für Mathematik, Johannes Gutenberg-Universität, Staudingerweg 9, 55099 Mainz, Germany.
E-mail:
[email protected]
2 Departamento de Métodos Matemáticos y Numéricos, Instituto de Investigaciones en Matemáticas
Aplicadas y en Sistemas, Universidad Nacional Autónoma de México, Apartado Postal 20-726, México, DF 01000, México. E-mail:
[email protected] Received: 4 April 2010 / Accepted: 8 June 2010 Published online: 14 December 2010 – © Springer-Verlag 2010
To Mario Castagnino on the occasion of his 75th birthday Abstract: The Aharonov-Bohm effect is a fundamental issue in physics that has been extensively studied in the literature and is discussed in most of the textbooks in quantum mechanics. The issues at stake are what are the fundamental electromagnetic quantities in quantum physics, if magnetic fields can act at a distance on charged particles and if the magnetic potentials have a real physical significance. The Aharonov-Bohm effect is a very controversial issue. From the experimental side the issues were settled by the remarkable experiments of Tonomura et al. (Phys Rev Lett 48:1443–1446, 1982; Phys Rev Lett 56:792–795, 1986) with toroidal magnets that gave a strong experimental evidence of the physical existence of the Aharonov-Bohm effect, and by the recent experiment of Caprez et al. (Phys Rev Lett 99:210401, 2007) that shows that the results of the Tonomura et al. experiments can not be explained by the action of a force. Aharonov and Bohm (Phys Rev 115:485-491, 1959) proposed an Ansatz for the solution to the Schrödinger equation in simply connected regions of space where there are no electromagnetic fields. It consists of multiplying the free evolution by the Dirac magnetic factor. The Aharonov-Bohm Ansatz predicts the results of the experiments of Tonomura et al. and of Caprez et al. Recently in Ballesteros and Weder (Math Phys 50:122108, 2009) we gave the first rigorous proof that the Aharonov-Bohm Ansatz is a good approximation to the exact solution for toroidal magnets under the conditions of the experiments of Tonomura et al. We provided a rigorous, simple, quantitative, error bound for the difference in norm between the exact solution and the Aharonov-Bohm Ansatz. In this paper we prove that these results do not depend on the particular geometry of the magnets and on the velocities of the incoming electrons used on the experiments, and on the gaussian shape of the wave packets used to obtain our quantitative error bound. We consider a general class of magnets that are a finite union of handlebodies. Each handlebody is diffeomorphic to a torus or a ball, and some of them can be patched though the boundary. We formulate the Aharonov-Bohm Ansatz that is appropriate to Research partially supported by CONACYT under Project CB-2008-01-99100. Fellow, Sistema Nacional de Investigadores.
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this general case and we prove that the exact solution to the Schrödinger equation is given by the Aharonov-Bohm Ansatz up to an error bound in norm that is uniform in time and that decays as a constant divided by v ρ , 0 < ρ < 1, with v the velocity. The results of Tonomura et al., of Caprez et al., our previous results and the results of this paper give a firm experimental and theoretical basis to the existence of the Aharonov-Bohm effect and to its quantum nature. Namely, that magnetic fields act at a distance on charged particles, and that this action at a distance is carried by the circulation of the magnetic potential which gives a real physical significance to magnetic potentials. 1. Introduction In classical physics the dynamics of a charged particle in the presence of a magnetic field is completely described by Newton’s equation with the Lorentz force, F = qv × B, where B is the magnetic field, q is the charge of the particle and v its velocity. Newton’s equation implies that in classical physics the magnetic field acts locally. If a particle propagates in a region where the magnetic field is zero the Lorentz force is zero and the trajectory of the particle is a straight line. The dynamics of a classical particle is not affected by magnetic fields that are located in regions of space that are not accessible to the particle. The action at a distance of magnetic fields on charged particles is not possible in classical electrodynamics. Furthermore, the relevant physical quantity is the magnetic field. The magnetic potentials have no physical meaning, they are just a convenient mathematical tool. In quantum physics this changes in a dramatic way. Quantum mechanics is a Hamiltonian theory where the dynamics of a charged particle in the presence of a magnetic field is governed by the equation of Schrödinger that can not be formulated directly in terms of the magnetic field; it requires the introduction of a magnetic potential. This makes the action at a distance of magnetic fields possible, since in a region of space with non-trivial topology, like the exterior of a torus, the magnetic potential has to be different from zero if there is a magnetic flux inside the torus, even if the magnetic field is identically zero outside. The reason is quite simple: if the magnetic potential is zero outside the torus it follows from Stoke’s theorem that the magnetic flux inside has to be zero. Aharonov and Bohm observed [3] that this implies that in quantum physics the magnetic flux inside the torus can act at a distance in a charged particle outside the torus, in spite of the fact that the magnetic field is identically zero along the trajectory of the particle and, furthermore, that the action of the magnetic field is carried over by the magnetic potential, which gives a real physical significance to the magnetic potentials. The possibility that magnetic fields can act at a distance on charged particles and that the magnetic potentials can have a physical significance is such a strong departure from the physical intuition coming from classical physics that it is no wonder that the Aharonov-Bohm effect was, and still is, a very controversial issue. In fact, the experimental verification of the Aharonov-Bohm effect constitutes a test of the validity of the theory of quantum mechanics itself. For a review of the literature up to 1989 see [20] and [22]. In particular, in [22] there is a detailed discussion of the large controversy -involving over three hundred papers- concerning the existence of the Aharonov-Bohm effect. For a recent update of this controversy see [28,31]. In their seminal paper Aharonov and Bohm [3] proposed an experiment to verify their theoretical prediction. They suggested to use a thin straight solenoid. They supposed that the magnetic field was confined to the solenoid. They suggested to send a coherent electron wave packet towards the solenoid and to split it in two parts, each one going through one side of the solenoid, and to bring both parts together behind the
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solenoid in order to create an interference pattern due to the difference in phase in the wave function of each part, produced by the magnetic field inside the solenoid. In fact, the existence of this interference pattern was first predicted by Franz [12,13]. There is a very large literature for the case of a solenoid, both theoretical and experimental. The theoretical analysis is reduced to a two dimensional problem after making the assumption that the solenoid is infinite. Of course, it is experimentally impossible to have an infinite solenoid. It has to be finite, and the magnetic field has to leak outside. The leakage of the magnetic field was a highly controversial point. Actually, if we assume that the magnetic field outside the finite solenoid can be neglected, there is no AharonovBohm effect at all because, if this is true, the exterior of the finite solenoid is a simply connected region of space without magnetic field where the magnetic potential can be gauged away to zero. In order to circumvent this issue it was suggested to use a toroidal magnet, since it can contain a magnetic field inside without a leak. The experiments with toroidal magnets were carried out by Tonomura et al. [21,29,30]. In these remarkable experiments they split a coherent electron wave packet into two parts. One traveled inside the hole of the magnet and the other outside the magnet. They brought both parts together behind the magnet and they measured the phase shift produced by the magnetic flux enclosed in the magnet, giving a strong evidence of the existence of the AharonovBohm effect. The Tonomura et al. experiments [21,29,30] are widely considered as the only convincing experimental evidence of the existence of the Aharonov-Bohm effect. After the fundamental experiments of Tonomura et al. [21,29,30] the existence of the Aharonov-Bohm effect was largely accepted and the controversy shifted into the interpretation of the results of the Tonomura et al. experiments. It was claimed that the outcome of the experiments could be explained by the action of some force acting on the electron that travels through the hole of the magnet. See, for example, [6,16] and the references quoted there. Such a force would accelerate the electron and it would produce a time delay. In a recent crucial experiment Caprez et al. [8] found that the time delay is zero, thus experimentally excluding the explanation of the results of the Tonomura et al. experiments by the action of a force. Aharonov and Bohm [3] proposed an Ansatz for the solution to the Schrödinger equation in simply connected regions of space where there are no electromagnetic fields. The Aharonov-Bohm Ansatz consists of multiplying the free evolution by the Dirac magnetic factor [10] (see Definition 4.2 in Sect. 4). The Aharonov-Bohm Ansatz predicts the interference fringes observed by Tonomura et al. [21,29,30] and it also predicts the absence of acceleration observed in the Caprez et al. [8] experiments, because in the Aharonov-Bohm Ansatz the electron is not accelerated since it propagates following the free evolution, with the wave function multiplied by a phase. As the experimental issues have already been settled by Tonomura et al. [21,29,30] and by Caprez et al. [8], the whole controversy can now be summarized in a single mathematical question: is the Aharonov-Bohm Ansatz a good approximation to the exact solution to the Schrödinger equation for toroidal magnets and under the conditions of the experiments of Tonomura et al.? Of course, there have been numerous attempts to give an answer to this question. Several Ansätze have been provided for the solution to the Schrödinger equation and for the scattering matrix, without giving error bound estimates for the difference, respectively, between the exact solution and the exact scattering matrix, and the Ansätze. Most of these works are qualitative, although some of them give numerical values for their Ansätze. Methods like Fraunhöfer diffraction, first-order Born and high-energy approximations, Feynman path integrals and the Kirchhoff method in optics were used to propose the Ansätze. For a review of the literature up to 1989 see [20] and [22] and for
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Fig. 1. The magnet K = ∪ Lj=1 K j ⊂ R3 , where K j are handlebodies , for every j ∈ {1, . . . , L}. The exterior domain, := R3 \K .The curves γk , k = 1, 2, . . . m are a basis of the first singular homology group of K and the curves γˆk , k = 1, 2, . . . m are a basis of the first singular homology group of
a recent update see [4,5]. The lack of any definite rigorous result on the validity of the Aharonov-Bohm Ansatz is perhaps the reason why this controversy lasted for so many years. It is only very recently that this situation has changed. In our paper [5] we gave the first rigorous proof that the Ansatz of Aharonov-Bohm is a good approximation to the exact solution of the Schrödinger equation. We provided, for the first time, a rigorous quantitative mathematical analysis of the Aharonov-Bohm effect with toroidal magnets under the conditions of the experiments of Tonomura et al. [21,29,30]. We assumed that the incoming free electron is represented by a gaussian wave packet, which from the physical point of view is a reasonable assumption. We provided a rigorous, simple, quantitative, error bound for the difference in norm between the exact solution and the approximate solution given by the Aharonov-Bohm Ansatz. Our error bound is uniform in time. We also proved that on the gaussian asymptotic state the scattering operator is given by a constant phase shift, up to a quantitative error bound, that we provided. Actually, the error bound is the same in the cases of the exact solution and the scattering operator. As mentioned above, the results of [5] were proven under the experimental conditions of Tonomura et al., in particular for the magnets and for the velocities of the incoming electrons considered in [21,29,30]. This was necessary to obtain rigorous quantitative results that can be compared with the experiments. This raises the question if the experimental results of [21,29,30] and the rigorous mathematical results of [5] depend or not on the particular geometry of the magnets, on the velocities of the incoming electrons used in the experiments, and on the gaussian shape of the wave packets. In this paper we give a general answer to this question. We assume that the magnet K is a compact submanifold of R3 . Moreover, K = ∪ Lj=1 K j where K j , 1 ≤ j ≤ L are the connected components of K . We suppose that the K j are handlebodies. For a precise definition of handle bodies see [4]. In intuitive terms, K is the union of a finite number of bodies diffeomorphic to tori or to balls. Some of them can be patched through the boundary. See Fig. 1. For the Aharonov-Bohm Ansatz to be valid it is necessary that, to a good approximation, the electron does not interact with the magnet K , because if the electron hits K it will
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be reflected and the solution can not be the free evolution modified with a phase. This is true no matter how big the velocity is. Actually, in the case of the infinite solenoid with non-zero cross section this can be seen in the explicit solution [27]. We dealt with this issue in [5] requiring that the variance of the gaussian state be small in order that the interaction with the magnet be small. In this paper we consider a general class of incoming asymptotic states with the property that under the free classical evolution they do not hit K . The intuition is that for high velocity the exact quantum mechanical evolution is close to the free quantum mechanical evolution and that as the free quantum mechanical evolution is concentrated on the classical trajectories, we can expect that, in the leading order for high velocity, we do not see the influence of K and that only the influence of the magnetic flux inside K shows up in the form of a phase, as predicted by the Aharonov-Bohm Ansatz. In our general case K has several holes and the parts of the wave packet that travel through different holes acquire different phases. For this reason we decompose our electron wave packet into the parts that travel through the different holes of K and we formulate the Aharonov-Bohm Ansatz for each one of them. We prove that the exact solution to the Schrödinger equation is given by the Aharonov-Bohm Ansatz up to an error bound in norm that is uniform in time and that decays as a constant divided by v ρ , 0 < ρ < 1, with v the velocity. In our bound the direction of the velocity is kept fixed as its absolute value goes to infinity. The results of this paper complement the results of our previous paper [4] where we proved that for the same class of incoming high-velocity asymptotic states the scattering operator is given by multiplication by a constant phase shift, as predicted by the Aharonov-Bohm Ansatz. Our results here, that are obtained with the help of results from [4], prove in a qualitative way that the Ansatz of Aharonov-Bohm is a good approximation to the exact solution of the Schrödinger equation for high velocity for a very general class of magnets K and of incoming asymptotic states, proving that the experimental results of Tonomura et al. [21,29,30] and of Caprez et al. [8] and the rigorous mathematical results of [5] hold in general and that they do not depend on the particular geometry of the magnets, on the velocities of the incoming electrons used in the experiments, and on the gaussian shape of the wave packets. Summing up, the experiments of Tonomura et al. [21,29,30] give a strong evidence of the existence of the interference fringes predicted by Franz [12,13] and by Aharonov and Bohm [3]. The experiment of Caprez et al. [8] verifies that the interference fringes are not due to a force acting on the electron, and the results [4,5] and in this paper rigorously prove that quantum mechanics theoretically predicts the observations of these experiments in a extremely precise quantitative way under the experimental conditions in [5] and in a qualitative way for general magnets and incoming asymptotic states on [4] and in this paper. These results give a firm experimental and theoretical basis to the existence of the Aharonov-Bohm effect [3] and to its quantum nature. Namely, that magnetic fields act at a distance on charged particles, even if they are identically zero in the space accessible to the particles, and that this action at a distance is carried by the circulation of the magnetic potential, what gives magnetic potentials a real physical significance. The results of this paper, as well as the ones of [4,5] and of [19,34] where the Aharonov-Bohm effect in the case of solenoids contained inside infinite cylinders with arbitrary cross section was rigorously studied, are proven using the method introduced in [11] to estimate the high-velocity limit of solutions to the Schrödinger equation and of the scattering operator.
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The paper is organized as follows. In Sect. 2 we state preliminary results that we need. In Sect. 3 we obtain estimates in norm for the leading order at high velocity of the exact solution to the Schrödinger equation in the case where besides the magnetic flux inside K there are a magnetic field and an electric potential outside K . Our estimates are uniform in time. These results are of independent interest and they go beyond the Aharonov-Bohm effect. The main results of this section are Theorems 3.9 and 3.10 and Sect. 3.2 where the physical interpretation of our estimates is given. In Sect. 4 we consider the Aharonov-Bohm effect and we prove our estimates that show that the Aharonov-Bohm Ansatz is a good approximation to the exact solution to the Schrödinger equation. The main results are Theorems 4.12, 4.13 and 4.14. In the Appendix we prove a result that we need, namely the triviality of the first group of singular homology of the sets where electrons that travel through different holes are located. Let us mention some related rigorous results on the Aharonov-Bohm effect. For further references see [4,5] and [34]. In [17], a semi-classical analysis of the AharonovBohm effect in bound-states in two dimensions is given. The papers [25,26,35], and [36] study the scattering matrix for potentials of Aharonov-Bohm type in the whole space. Finally some words about our notations and definitions. We denote by C any finite positive constant whose value is not specified. For any x ∈ R3 , x = 0, we denote, xˆ := x/|x|. For any v ∈ R3 we designate, v := |v|. By B R (x) we denote the open ball of center x and radius R. B R (0) is denoted by B R . For any set O we denote by F(x ∈ O) the operator of multiplication by the characteristic function of O. By · we denote the norm in L 2 (), where := R3 \K . The norm of L 2 R3 is denoted by · L 2 (R3 ) . For any open set, O, we denote by Hs (O), s = 1, 2, . . . the Sobolev spaces [1] and by Hs,0 (O) the closure of C0∞ (O) in the norm of Hs (O). By B(O) we designate the Banach space of all bounded operators on L 2 (O). We use notions of homology and cohomology as defined, for example, in [7,9,14,15], and [33]. In particular, for a set O ⊂ R3 we denote by H1 (O; R) the first group of sin1 (O) the first de Rham gular homology with coefficients in R, [7] p. 47, and by Hde R cohomology class of O [33]. We define the Fourier transform as a unitary operator on L 2 R3 as follows: ˆ p) := Fφ( p) := φ(
1 e−i p·x φ(x) d x. (2π )3/2 R3
We define functions of the operator p := −i∇ by Fourier transform, ˆ p) ∈ L 2 R3 , f (p)φ := F ∗ f ( p)Fφ, D( f (p)) := φ ∈ L 2 R3 : f ( p) φ( for every measurable function f . 2. Preliminary Results We study the propagation of a non-relativistic particle -an electron for example- outside a bounded magnet, K , in three dimensions, i.e. the electron propagates in the exterior domain := R3 \K . We asssume that inside K there is a magnetic field that produces a magnetic flux. We suppose, furthermore, that in there are an electric potential V and a magnetic field B. This is a more general situation than the one of the Aharonov-Bohm effect.
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2.1. The magnet K . We assume that the magnet K is a compact submanifold of R3 . Moreover, K = ∪ Lj=1 K j , where K j , 1 ≤ j ≤ L are the connected components of K . We suppose that the K j are handlebodies. For a precise definition of handlebodies see [4] were we study in detail the homology and the cohomology of K and . In intuitive terms, K is the union of a finite number of bodies diffeomorphic to tori or to balls. Some of them can be patched through the boundary. See Fig. 1. 2.2. The magnetic field and the electric potential. In the following assumptions we summarize the conditions on the magnetic field and the electric potential that we use (see [4]). We denote by the self-adjoint realization of the Laplacian in L 2 R3 with domain H2 R3 . Below we assume that V is - bounded with relative bound zero. By this we mean that the extension of V to R3 by zero is − with relative bound bounded zero. Using an extension operator from H2 () to H2 R3 [32] we prove that this is 2 equivalent to require that V is bounded from H2 () into 3 L () with relative bound 2 zero. We denote by · B(R3 ) the operator norm in L R . Assumption 2.1. We assume that the magnetic field, B, is a real-valued, bounded 2- form in , that is continuous in a neighborhood of ∂ K , and furthermore, 1. B is closed: d B| ≡ divB = 0. 2. There are no magnetic monopoles in K : B = 0, j ∈ {1, 2, . . . , L}. (2.1) ∂K j
3. |B(x)| ≤ C(1 + |x|)−μ , for some μ > 2.
(2.2)
4. d ∗ B| ≡ curl B is bounded and, |curl B| ≤ C(1 + |x|)−μ .
(2.3)
5. The electric potential, V , is a real-valued function, it is −bounded, with relative bound zero and ≤ C(1 + r )−α , for some α > 1. (2.4) F(|x| ≥ r )V (− + I )−1 B (R 3 ) Condition (2.1) means that the total contribution of magnetic monopoles inside each component K j of the magnet is 0. In a formal way we can use Stokes theorem to conclude that B = 0 ⇐⇒ div B = 0, j ∈ {1, 2, . . . , L}. ∂K j
Kj
As div B is the density of magnetic charge, ∂ K j B is the total magnetic charge inside K j , and our condition (2.1) means that the total magnetic charge inside K j is zero. This condition is fulfilled if there is no magnetic monopole inside K j , j ∈ {1, 2, . . . , L}. Furthermore, condition (2.4) is equivalent to the following assumption [24] (2.5) ≤ C(1 + r )−α , for some α > 1. V (− + I )−1 F(|x| ≥ r ) B (R 3 ) Condition (2.4) has a clear intuitive meaning, it is a condition on the decay of V at infinity. However, in the proofs below we use the equivalent statement (2.5).
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m 2.3. The magnetic potentials. Let γˆ j j=1 be the closed curves defined in Eq. (2.6) of [4] (see Fig. 1). We prove in Corollary 2.4 of [4] that the equivalence classes of these curves are a basis of the first singular homology group of . We mintroduce below a function that gives the magnetic flux across surfaces that have γˆ j j=1 as their boundaries. m Definition 2.2. The flux, , is a function : γˆ j j=1 → R. We now define a class of magnetic potentials with a given flux modulo 2π . Definition 2.3. Let B be a closed 2− form that satisfies Assumption 2.1. We denote by A ,2π (B) the set of all continuous 1-forms, A, in that satisfy. 1. |A(x)| ≤ C
2.
1 , 1 + |x|
(2.6)
A(x) · xˆ ≤ C(1 + |x|)−βl , βl > 1, where xˆ := x/|x|.
(2.7)
A = (γˆj ) + 2π n j (A), n j (A) ∈ Z,
(2.8)
γˆj
j ∈ {1, 2, . . . , m}.
3. d A| ≡ curl A = B| .
(2.9)
Furthermore, we say that two potentials, A, A˜ ∈ A ,2π (B) have the same fluxes if ˜ j ∈ {1, 2, . . . , m}. A= A, (2.10) γˆj
γˆj
Moreover, we say that A ∈ A ,2π (B) is short range if |A(x)| ≤ C
1 , β > 1. (1 + |x|)β
(2.11)
We denote by A ,2π,SR (B) the set of all potentials in A ,2π (B) that are short range. m The definition of the flux depends on the particular choice of the curves γˆ j j=1 . However, the class A ,2π (B) is independent of this particular choice. In fact it can be equivalently defined taking any other basis
of the first singular homology group in . See [4]. By Stoke’s theorem the circulation γˆ j A of a potential A ∈ A ,2π (B) represents the flux of the magnetic field B in any surface whose boundary is γˆ j , j = 1, 2, . . . , m. As the magnetic field is a priori known outside the magnet, it is natural to specify the magnetic potentials fixing fluxes of the magnetic field in surfaces inside the magnet taking the circulation of A in closed curves in the boundary of K . We prove in [4] that this gives the same class of potentials. We find, however, that it is technically more convenient to work with closed curves in that define a basis of the first singular homology group. Note that in [4] we use the same symbol to denote a larger class of magnetic potentials where (2.7) is only required to hold in the L 1 sense. Here we assume that it holds in the pointwise sense to obtain precise error bounds.
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In Theorem 3.7 of [4] we construct the Coulomb potential, AC , that belongs to A ,2π,SR (B) with n j (A) = 0, j ∈ {1, 2, . . . , m}. For this purpose condition (2.1) is essential. In Lemma 3.8 of [4] we prove that for any A, A˜ ∈ A ,2π (B) with the same fluxes there is a gauge transformation between them. Namely, that there is a C 1 0− form λ in such that, A˜ − A = dλ.
(2.12)
Moreover, we can take λ(x) := C(x0 ,x) A˜ − A , where x0 is any fixed point in and C(x0 , x) is any curve from x0 to x contained in . Furthermore, λ∞ (x) := limr →∞ λ(r x) exists and it is continuous in R3 \{0} and homogeneous of order zero, i.e. λ∞ (r x) = λ∞ (x), r > 0, x ∈ R3 \{0}. Moreover, |λ∞ (x) − λ(x)| ≤
∞
|x|
b(|x|), for some b(r ) ∈ L 1 (0, ∞),
and |λ∞ (x + y) − λ∞ (x)| ≤ C|y|, ∀x : |x| = 1, and ∀y : |y| < 1/2.
(2.13)
2.4. The Hamiltonian. Let us denote p := −i∇. The Schrödinger equation for an electron in with electric potential V and magnetic field B is given by i
∂ 1 q φ= (P − A)2 + q V, ∂t 2M c
(2.14)
where is Planck’s constant, P := p is the momentum operator, c is the speed of light, M and q are, respectively, the mass and the charge of the electron and A a magnetic potential with curlA = B. To simplify the notation we multiply both sides of (2.13) by 1 and we write Schrödinger’s equation as follows: i
1 ∂ φ= (p − A)2 φ + V φ, ∂t 2m
(2.15)
with m := M/, A = qc A and V := q V. Note that since we write Schrödinger’s equation in this form our Hamiltonian below is the physical Hamiltonian divided by . We fix the flux modulo 2π by taking A ∈ A ,2π , where B := qc B. Note that this corresponds to fixing the circulations of A modulo qc 2π , or equivalently, to fixing the fluxes of the magnetic field B modulo qc 2π . We define the quadratic form, h 0 (φ, ψ) :=
1 (pφ, pψ), D(h 0 ) := H1,0 (). 2m
(2.16)
−1 D , where D is the Laplacian The associated positive operator in L 2 () [18,23] is 2m with Dirichlet boundary condition on ∂. Note that the functions in H1,0 (O) vanish in −1 the trace sense in the boundary of O. We define H (0, 0) := 2m D . By elliptic regularity [2], D(H (0, 0)) = H2 () ∩ H1,0 ().
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For any A ∈ A ,2π (B) we define, 1 ((p − A)φ, (p − A)ψ) 2m 1 1 = h 0 (φ, ψ) + (−(pφ, Aψ) − (Aφ, pψ)) + (Aφ, Aψ), 2m 2m D(h A ) = H1,0 ().
h A (φ, ψ) :=
(2.17)
1 1 As the quadratic form − 2m ((pφ, Aψ) + (Aφ, pψ)) + 2m (Aφ, Aψ) is h 0 -bounded with relative bound zero, h A is closed and positive. We denote by H (A, 0) the associated positive self-adjoint operator [18,23]. H (A, 0) is the Hamiltonian with magnetic potential A. As the electric potential V is h 0 -bounded with relative bound zero it follows [18,23] that the quadratic form,
h A,V (φ, ψ) := h A (φ, ψ) + (V φ, ψ), D(h A,V ) = H1,0 (),
(2.18)
is closed and bounded from below. The associated operator, H (A, V ), is self-adjoint and bounded from below. H (A, V ) is the Hamiltonian with magnetic potential A and electric potential V . 1 Suppose that div A is bounded. In this case the operator 2m −2 A · p − (p · A) + A2 1 is H (0, 0) bounded with relative bound zero and we have that H (0, 0) − 2m (2 A · p + 1 (p · A)) + 2m A2 is self-adjoint on the domain of H (0, 0) and since also V is H (0, 0) bounded with relative bound zero we have that, 1 1 2 H (A, V ) = H (0, 0) − (2 A · p + (p · A)) + A + V, (2.19) 2m 2m D(H (A, V )) = H2 () ∩ H1,0 (). We define the Hamiltonian H (A, V ) in L 2 () with Dirichlet boundary condition at ∂, i.e. ψ = 0 for x ∈ ∂. This is the standard boundary condition that corresponds to an impenetrable magnet K . It implies that the probability that the electron is at the boundary of the magnet is zero. Note that the Dirichlet boundary condition is invariant under gauge transformations. In the case of the impenetrable magnet the existence of the Aharonov-Bohm effect is more striking, because in this situation there is zero interaction of the electron with the magnetic field inside the magnet. Note, however, that once a magnetic potential is chosen the particular self-adjoint boundary condition taken at ∂ does not play an essential role in our calculations. Furthermore, our results hold also for a penetrable magnet where the interacting Schrödinger equation is defined in all space. Actually, this latercase is slightly simpler because we do not need to work with two Hilbert spaces, L 2 R3 for the free evolution, and L 2 () for the interacting evolution, which simplifies the proofs. We prove in Theorem 4.1 of [4] that if A, A˜ ∈ A ,2π (B) ˜ V ) are unitarily equivalent and we give explicitly the Hamiltonians H (A, V ) and H ( A, the unitary operator that relates them. 2.5. The wave and scattering operators. Let J be the identification operator from L 2 R3 onto L 2 () given by multiplication by the characteristic function of . The free Hamiltonian is the self-adjoint operator, 1 2 p , H0 := (2.20) 2m 3 with domain H2 R .
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The wave operators are defined as follows: W± (A, V ) := s- lim eit H (A,V ) J e−it H0 , t→±∞
(2.21)
provided that the strong limits exist. We prove in [4] that if Assumption 2.1 holds the wave operators exist and are partially isometric for every A ∈ A ,2π (B) and that J can be replaced by the operator of multiplication by any function χ ∈ C ∞ R3 that satisfies χ (x) = 0 in a neighborhood of K and χ (x) = 1 for x ∈ R3 \B R , where K ⊂ B R . Furthermore, the wave operators satisfy the intertwining relations, eit H (A,V ) W± (A, V ) = W± (A, V ) eit H0 .
(2.22)
Moreover, if A, A˜ ∈ A ,2π (B) and they have the same fluxes (for the case where the fluxes are not equal see [4]) ˜ V = eiλ(x) W± (A, V ) e−iλ∞ (±p) . W± A, (2.23) The scattering operator is defined as S(A, V ) := W+∗ (A, V ) W− (A, V ). If A, A˜ ∈ A ,2π (B) [4], ˜ V = eiλ∞ (p) S(A, V ) e−iλ∞ (−p) , S A,
˜ A ∈ A ,2π (B). A,
(2.24)
(2.25)
If A, A˜ ∈ A ,2π,SR (B) (or more generally if A − A˜ satisfies (2.11)) λ∞ is constant ˜ V = S(A, V ). That is to say, the scattering operator is uniquely and by (2.25) S A, defined by K , B, V and the flux modulo 2π , if we restrict the potentials to be of short range. 3. Uniform Estimates We first prepare some results that we need. In Theorem 3.2 of [4] we proved that B has an extension to a closed 2-form in R3 . Below we use the same symbol, B, for this closed extension. Furthermore, in Theorem 3.7 of [4] we constructed the Coulomb potential, AC ∈ A ,2π,SR (B), that actually has the fluxes (2.8) with n j (A) = 0, j ∈ {1, 2, . . . , m}. In fact, AC extends to a continuous 1-form in R3 , that we denote by the same symbol, AC , such that div AC n-times differentiable with n arbitrary and with support contained in K . See the proof of Lemma 5.6 of [4]. For any potential A ∈ A ,2π (B) we can construct a Coulomb potential AC with the same fluxes as A. As mentioned above (see (2.12)), by Lemma 3.8 of [4] there is a C 1 0− form λ such that A = AC + dλ.
(3.1)
Note that λ has an extension to a C 1 0− form in R3 ( Theorem 4.22, p.311 [32] ) that we denote by the same symbol, λ. Then, Eq. (3.1) defines an extension of A to a continuous one form in R3 that we denote by the same symbol, A. Furthermore, the gauge ˜ A and λ to R3 . transformation formula (2.12) holds for the extensions of A,
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We define for v ∈ R3 \0,
t
(ˆv × B)(x + τ vˆ )dτ,
(3.2)
vˆ · A(x + τ vˆ )dτ, −∞ ≤ t ≤ ∞, t b(x, t) := A(x + t vˆ ) + (ˆv × B)(x + τ vˆ )dτ.
(3.3)
η(x, t) :=
0 t
L A,ˆv (t) :=
0
(3.4)
0
For f : R3 × R → R3 with ft (x) := f(x, t) ∈ L 1loc (R3 , R3 ) we define, 1 χ (x) −p · f(x, t) − f(x, t) · p + (f(x, t))2 , (3.5) f (x, t) := 2m where χ ∈ C ∞ R3 satisfies χ (x) = 0 for x in a neighborhood of K , χ (x) = 1, x ∈ {x : |x| ≥ R} with R such that K ⊂ B R . It follows by Fourier transform that under translation in configuration or momentum space generated, respectively, by p and x we obtain eip·vt f (x) e−ip·vt = f (x + vt), e−imv·x f (p) eimv·x = f (p + mv),
(3.6) (3.7)
and, in particular, e−imv·x e−it H0 eimv·x = e−imv
2 t/2
e−ip·vt e−it H0 .
(3.8)
We define [34], H1 :=
1 −imv·x 1 e H0 eimv·x , H2 := e−imv·x H (A, V ) eimv·x . v v
(3.9)
We need the following lemma from [34]. Lemma 3.1. For any f ∈ C0∞ (Bη ) and for any j = 1, 2, . . . there is a constant C j such that mv F |x − z| > |z| e−i vz H0 f p − F (|x| ≤ |z|/8) ≤ C j (1 + |z|)− j , √ 4 v 3 B (R ) (3.10) for v := |v| > (8η/m)2 . Proof. Corollary 2.2 of [34] with Q = 0. Note that the proof in three dimensions is the same as the one in two dimensions given in [34]. Lemma 3.2. Let g ∈ C0∞ R3 satisfy, g( p) = 1, | p| < m/16 and g( p) = 0, | p| ≥ m/8. Suppose that V satisfies (2.4) or, equivalently, (2.5). Then, for any compact set D ⊂ R3 there is a constant C such that −i z H p −α 1g V e φ (3.11) √ 2 3 ≤ C(1 + |z|) φH2 (R3 ) , v L (R )
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for all v > 1, z ∈ R and all φ ∈ H2 R3 with support in D. Furthermore, if V ∈ L ∞ R3 and for some z ∈ R, −α 3 V (x)F(|x − z vˆ | ≤ |z/4|) B(R3 ) ≤ C(1 + |z|) , ∀x ∈ R ,
(3.12)
then, there is a constant C1 such that −i z H p 1g V e φ √ v
L2
( ) R3
≤ C1 (1 + |z|)−α φ L 2 (R3 ) ,
(3.13)
for all v > 1 and all φ ∈ L 2 R3 with support in D. The constant C1 depends only on V L ∞ and on C. Proof. By (3.8), −i z H p 1g V e √ φ 2 3 v L (R ) p − mv −1 −i vz H0 ˆ V (− + 1) F(|x| ≤ |z|/8) F(|x − z v | > |z|/4) e g ≤ √ v B (R 3 ) ×φH2 (R3 ) + C V (− + 1)−1 F(|x − z vˆ | ≤ |z|/4) φH2 (R3 ) B (R 3 ) (3.14) F(|x| > |z|/8)(− + 1)φ L 2 (R3 ) . + g L ∞ V (− + 1)−1 B (R 3 ) Equation (3.11) follows from (2.5, 3.10, 3.14) and using that as φ has compact support in D, F(|x| > |z|/8)(− + 1)φ L 2 (R3 ) ≤ Cl (1 + |z|)−l (1 + |x|)l ( + 1)φ
L 2 (R 3 )
≤ Cl (1 + |z|)−l φH2 (R3 ) . −1 Equation (3.12) is proven in the same way, 3 but as the regularization (− + 1) is not 2 needed we obtain the norm of φ in L R .
With g as in Lemma 3.2 we denote, √ ϕ˜ := g(p/ v) ϕ, v > 0.
(3.15)
By Fourier transform we prove that ϕ˜ − ϕ L 2 (R3 ) ≤
C ϕH2 (R3 ) . 1+v
(3.16)
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3.1. High-velocity solutions to the Schrödinger equation. At the time of emission, i.e., as t → −∞, the electron wave packet is far away from K and it does not interact with it, therefore, it can be parametrised with kinematical variables and it can be assumed that it follows the free evolution, i
∂ φ(x, t) = H0 φ(x, t), x ∈ R3 , t ∈ R, ∂t
(3.17)
where H0 is the free Hamiltonian (2.20). We represent the emitted electron wave packet by the free evolution of an asymptotic state with velocity v, (3.18) ϕv := eimv·x ϕ, ϕ ∈ L 2 R3 . Recall that in the momentum representation eimv·x is a translation operator by the vector mv, which implies that the asymptotic state (3.18) is centered at the classical momentum mv in the momentum representation, ϕˆv ( p) = ϕ( ˆ p − mv). Then, the electron wave packet is represented at the time of emission by the following incoming wave packet that is a solution to the free Schrödinger equation (3.17), ψv,0 := e−it H0 ϕv .
(3.19)
The (exact) electron wave packet, ψv (x, t), satisfies the interacting Schrödinger equation (2.15) for all times and as t → −∞ it has to approach the incoming wave packet, i.e., lim ψv − J ψv,0 = 0. t→−∞
Hence, we have to solve the interacting Schrödinger equation (2.15) with initial conditions at minus infinity. This is accomplished with wave operator W− . In fact, we have that ψv = e−it H (A,V ) W− (A, V ) ϕv ,
(3.20)
because, as e−it H (A,V ) is unitary, lim e−it H (A,V ) W− ϕv − J e−it H0 ϕv = 0. t→−∞
Moreover, lim e−it H (A,V ) W− ϕv − J e−it H0 ϕv,+ = 0, where ϕv,+ := W+∗ W− ϕv . (3.21) t→∞
This means that -as to be expected- for large positive times, when the exact electron wave packet is far away from K , it behaves as the outgoing solution to the free Schrödinger equation (3.17), e−it H0 ϕv,+ ,
(3.22)
where the Cauchy data at t = 0 of the incoming and the outgoing wave packets (3.18, 3.22) are related by the scattering operator, ϕv,+ = S(A, V ) ϕv .
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In order to see the Aharonov-Bohm effect we need to separate the effect of K as a rigid body from that of the magnetic flux inside K . For this purpose we need asymptotic states that have negligible interaction with K for all times. This is possible if the velocity is high enough, as we explain below. For any v = 0 we denote, vˆ := {x ∈ : x + τ vˆ ∈ , ∀τ ∈ R}.
(3.23)
Let us consider asymptotic states (3.18), where ϕ has compact support contained in vˆ . For the discussion below it is better to parametrise the free evolution of ϕv by the distance z = vt rather than by the time t. At distance z the state is given by z
e−i v H0 ϕv = eimv·x e−i
mzv 2
z
e−i v H0 e−ip·z vˆ ϕ,
(3.24)
where we used (3.8). Note that e−ip·z vˆ is a translation in straight lines along the classical free evolution, e−ip·z vˆ ϕ (x) = ϕ(x − z vˆ ). (3.25) z
The term e−i v H0 gives rise to the quantum-mechanical spreading of the wave packet. For high velocities this term is one order of magnitude smaller than the classical translation, and if we neglect it we get that, z mzv e−i v H0 ϕv (x) ≈ ei 2 ϕv (x − z vˆ ), for large v. (3.26) We see that, in this approximation, for high velocities our asymptotic state evolves along mzv the classical trajectory, modulo the global phase factor ei 2 that plays no role. The key issue is that the support of our incoming wave packet remains in v for all distances, or for all times, and in consequence it has no interaction with K . We can expect that for high velocities the exact solution, ψv (3.20), to the interacting Schrödinger equation (2.15) is close to the incoming wave packet ψv,0 and that, in consequence, it also has negligible interaction with K , provided, of course, that the support of ϕ is contained in v . Below we give a rigorous ground for this heuristic picture proving that in the leading order ψv is not influenced by K and that it only contains information on the potential A. We define, W±,v (A, V ) := e−imv·x W± (A, V ) eimv·x = s- lim ei z H2 (A,V ) J e−i z H1 . (3.27) z→±∞
Lemma 3.3. Let 0 be a compact subset of vˆ , v ∈ R3 \0. Then, for all A ∈ A ,2π (B) and all χ ∈ C ∞ R3 that satisfies χ (x) = 0 for x in a neighborhood of K , χ (x) = 1, for x ∈ {x : x = y + τ vˆ , y ∈ 0 , τ ∈ R} ∪ {x : |x| ≥ R} with R such that K ⊂ B R , there is a constant C such that z z −i v H (A,V ) W± (A, V ) ϕv − χ e−i L A,ˆv (±∞) e−i v H0 ϕv e C (1 + (1 ∓ sign(z))|z|) ϕH2 (R3 ) , v for all z ∈ R and all ϕ ∈ H2 R3 with support contained in 0 . ≤
(3.28)
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Proof. By (3.16) it is enough to prove the lemma for ϕ. ˜ We first give the proof for a potential A ∈ A ,2π (B) that satisfies |A(x)| + |divA(x)| ≤ C(1 + |x|)−β1 , β1 > 1,
(3.29)
for example, for the Coulomb potential. By the intertwining relations (2.22), z
e−i v
z
H (A,V )
W± (A, V ) ϕ˜v − χ e−i L A,ˆv (±∞) e−i v H0 ϕ˜v = eimv·x s- lim eit H2 χ (x)e−it H1 − χ (x)e−i L A,ˆv (t) e−i z H1 ϕ. ˜ t→±∞
(3.30)
Denote
P(t, τ, z) := ei(τ −z)H2 i H2 e−i L A,ˆv (t−(τ −z)) χ (x) −e−i L A,ˆv (t−(τ −z)) χ (x) H1 − vˆ · A(x + (t − (τ − z))ˆv) e−iτ H1 ϕ. ˜ (3.31)
Then, by Duhamel’s formula - see Eq. (5.26) of [4] and [34] t+z dτ P(t, τ, z). eit H2 χ (x)e−it H1 − χ (x)e−i L A,ˆv (t) ϕ˜ =
(3.32)
z
We have that (see Eqs. (5.29–5.32) of [4] and [34]) P(t, τ, z) = T1 + T2 + T3 ,
(3.33)
where T1 :=
1 i(τ −z)H2 −i L A,ˆv (t−(τ −z)) e ie ˜ (b (x, t − (τ − z)) + χ V (x)) e−iτ H1 ϕ, v (3.34)
1 i(τ −z)H2 −i L A,ˆv (t−(τ −z)) e ie {−(χ ) + 2(pχ ) · p T2 := 2mv −2b(x, t − (τ − z)) · (pχ )}e−iτ H1 ϕ, ˜ −iτ H i(τ −z)H2 −i L A,ˆv (t−(τ −z)) 1 ϕ. (pχ ) · vˆ e ie ˜ T3 := e
(3.35) (3.36)
Note that
η(x, t − (τ − z)) F(|x − τ vˆ | ≤ |τ/4|) ≤ C(1 + |τ |)−μ+1 , if t + z ≥ 0 and τ ∈ [0, t + z] or if t + z ≤ 0 and τ ∈ [t + z, 0]. (3.37) Furthermore, since ∇ · (ˆv × B) = −ˆv · curl B,
p · η(x, t − (τ − z))F(|x − τ vˆ | ≤ |τ/4|) ≤ C(1 + τ )−μ+1 , if t + z ≥ 0 and τ ∈ [0, t + z] or if t + z ≤ 0 and τ ∈ [t + z, 0]. (3.38) We give the proof for W+ (A, V ). The case of W− (A, V ) follows in the same way. Since we have to take the limit t → ∞ in (3.30), we can assume that t > 2|z|. Let us estimate t+z T1 dτ . z
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We consider first the terms in b that do not contain A. For example the term, −1 I1 := mv
t+z
dτ ei(τ −z)H2 ie−i L A,ˆv (t−(τ −z)) χ (x) η(x, t − (z − τ )) · pe−iτ H1 ϕ. ˜
z
We have that 0 1 dτ η(x, t − (z − τ )) · pe−iτ H1 ϕ˜ mv z t+z 1 + dτ η(x, t − (z − τ )) · pe−iτ H1 ϕ˜ mv 0 C ≤ (1 + (1 − sign(z))|z|) ϕ ˜ H 1 (R 3 ) , v
I1 ≤
where we used (3.13) and (3.37). Let us now estimate a term in b that contains A. For example, I2 :=
−1 mv
t+z
dτ ei(τ −z)H2 ie−i L A,ˆv (t−(τ −z)) χ (x) A(x +(t − (z − τ ))ˆv) · pe−iτ H1 ϕ. ˜
z
Since, z ≤ τ ≤ t + z and t ≥ 2|z|, we have that |τ | ≤ t + z. Then, for |x − τ vˆ | ≤ |τ |/4, we have that |x + (t − (τ − z))ˆv)| ≥ |t + z| − |τ |/4 ≥ 3|τ |/4. Then by (3.13, 3.29), I2 ≤
C ϕ ˜ H 1 (R 3 ) . v
The remaining terms in T1 are estimated in the same way, using (3.11) in the term containing χ V . In this way we prove that
t+z z
C ˜ H 2 (R 3 ) . T1 ≤ v (1 + (1 − sign(z))|z|) ϕ
(3.39)
In the same way we prove that
t+z z
C ˜ H 1 (R 3 ) . T2 ≤ v ϕ
(3.40)
Moreover, by Eq. (5.37) of [4] (see also the proof of Lemma 2.4 of [34]),
t+z z
C T3 (τ ) ≤ v ϕH2 (R3 ) .
(3.41)
Note that it is in the proof of (3.41) that the condition χ (x) = 1, for x ∈ {x : x = y + τ vˆ , y ∈ 0 , τ ∈ R} is used. Equation (3.28) follows from (3.33–3.36) and (3.39–3.41). Let us now consider the case of A ∈ A ,2π (B). We take A˜ ∈ A ,2π (B) that satisfies (3.29) and has the same fluxes as A. Let λ be as in (2.12). We give the proof
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for W+ (A, V ). The case of W− (A, V ) is similar. By the gauge transformation formula (2.23), z z −i v H (A,V ) W+ (A, V ) ϕv − χ e−i L A,ˆv (∞) e−i v H0 ϕv e −iλ(x) −i z H A,V ˜ v ˜ V eiλ∞ (p) ϕv = e e W+ A, −i L A,ˆ v) −iλ(x) −i vz H0 ˜ v (∞) iλ∞ (ˆ −χ e e e e ϕv ≤
C (1 + (1 − sign(z))|z|) ϕH2 (R3 ) + C eiλ∞ (p) − eiλ∞ (ˆv) ϕv 2 3 . (3.42) L (R ) v
But, by (2.13), (3.7) and since λ∞ is homogenous of degree zero, C iλ∞ (p) ϕH1 (R3 ) . − eiλ∞ (ˆv) ϕv 2 3 ≤ e L (R ) v Equation (3.28) follows from (3.42, 3.43).
(3.43)
Lemma 3.4. Suppose that A ∈ A ,2π (B). Then, there is a constant C such that z −i L A,ˆv (±∞) − 1 e−i v H0 ϕv 2 3 e L (R ) 1 ϕH2 (R3 ) , for ± z > 0, (3.44) ≤ C (1 + |z|)−βl +1 + v and all ϕ ∈ H2 R3 . Proof. By (3.16) it is enough to prove the lemma for ϕ. ˜ We give the proof in the + case. The − case follows in the same way. By (3.7, 3.9) we have that z −i L A,ˆv (∞) − 1 e−i v H0 ϕ˜v 2 3 e L (R ) ∞ −i z H1 ˆ ˆ (A · v )(x + τ v ) dτ e ϕ ˜ ≤ (3.45) 2 3 . 0 L (R ) Furthermore, denoting x = x vˆ + x⊥ , where x is the component of x parallel to vˆ and x⊥ is the component of x perpendicular to vˆ , it follows from (3.6) that ∞ F(|x − z vˆ | < |z|/4) ˆ ˆ (A · v )(x + τ v ) dτ 0 B (R 3 ) ∞ ip·x⊥ −ip·x⊥ (A · vˆ )(τ vˆ ) dτ e = F(|x − z vˆ | < |z|/4)e x B (R 3 ) ∞ (A · vˆ )(τ vˆ ) dτ = F(|x − z| < |z|/4) x B (R 3 ) ∞
(A · vˆ )(τ vˆ ) dτ ≤ C (1 + z)−βl +1 . (3.46) ≤ 3z/4
The lemma follows from (3.45, 3.46) and Lemma 3.2.
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Lemma 3.5. Let 0 be a compact subset of vˆ , v ∈ R3 \0. Then, for all A ∈ A ,2π (B), there is a constant C such that z z −i v H (A,V ) W± (A, V ) ϕv − e−i v H0 ϕv e 1 ≤C (3.47) + (1 + |z|)−βl +1 ϕH2 (R3 ) , for ± z > 0, v and all ϕ ∈ H2 R3 with support contained in 0 . Proof. The lemma follows from Lemmata 3.3, 3.4, (3.16) and since by Lemma 3.2, z (3.48) ˜ L 2 (R3 ) , l = 1, 2, . . . . (1 − χ )e−i v H0 ϕ˜v ≤ Cl (1 + |z|)−l ϕ Lemma 3.6. Let 0 be a compact subset of vˆ , v ∈ R3 \0. Then, for all A ∈ A ,2π (B) with divA ∈ L 2loc () there is a constant C such that z
∞ z −i v H (A,V ) W− (A, V ) ϕv − e−i v H0 ei −∞ A·ˆv(x+τ vˆ ) dτ ϕv e 1 ≤C (3.49) + (1 + z)−βl +1 ϕH2 (R3 ) , for z ≥ 0, v and all ϕ ∈ H2 R3 with support contained in 0 . Proof. First note that ∞ −∞
A · vˆ (x + τ vˆ ) dτ = L A,ˆv (∞) − L A,ˆv (−∞).
By Eqs. (5.19) and (5.42) of [4], C ϕH2 (R3 ) . (3.50) W− (A, V ) ϕv − W+ (A, V ) ei L A,ˆv (∞)−i L A,ˆv (−∞) ϕv ≤ v Then
∞ z 1 −i vz H (A,V ) W− (A, V ) ϕv − e−i v H0 ei −∞ A·ˆv(x+τ vˆ ) dτ ϕv ≤ C ϕH2 (R3 ) e v z z + e−i v H (A,V ) W+ (A, V ) ei L A,ˆv (∞)−i L A,ˆv (−∞) ϕv − e−i v H0 ei L A,ˆv (∞)−i L A,ˆv (−∞) ϕv 1 1 ≤ C ϕH2 (R3 ) + C + (1 + z)−βl +1 ϕH2 (R3 ) , for z > 0, (3.51) v v
where we used Lemma 3.5 and Eq. (5.42) of [4].
Lemma 3.7. For all A ∈ A ,2π (B) with divA ∈ L 2loc () there is a constant C such that, ∀z ∈ R, ∞
∞ z |z| i −∞ A·ˆv(x+τ vˆ ) dτ −i vz H0 e ϕv − e−i v H0 ei −∞ A·ˆv(x+τ vˆ ) dτ ϕv ≤ C ϕH2 (R3 ) , e v (3.52) 3 for all ϕ ∈ H2 R .
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Proof. By (3.9), ∞
∞ z z N := ei −∞ A·ˆv(x+τ vˆ ) dτ e−i v H0 ϕv − e−i v H0 ei −∞ A·ˆv(x+τ vˆ ) dτ ϕv ∞
∞ = ei −∞ A·ˆv(x+τ vˆ ) dτ e−i z H1 ϕ − e−i z H1 ei −∞ A·ˆv(x+τ vˆ ) dτ ϕ .
(3.53)
Moreover by (3.8), ∞ z |z| i −∞ A·ˆv(x+τ vˆ ) dτ −i z H1 ϕH2 (R3 ) . e − e−i(zp·ˆv+mvz/2) ϕ ≤ H0 ϕ ≤ C e v v (3.54) Furthermore, by (3.6), ei
∞
−∞
A·ˆv(x+τ vˆ ) dτ
e−i(zp·ˆv+mvz/2) ϕ = e−i(zp·ˆv+mvz/2) ei
∞
−∞
A·ˆv(x+τ vˆ ) dτ
ϕ. (3.55)
Then by (3.53, 3.54, 3.55),
∞
∞ |z| N ≤ e−i(zp·ˆv+mvz/2) ei −∞ A·ˆv(x+τ vˆ ) dτ ϕ − e−i z H1 ei −∞ A·ˆv(x+τ vˆ ) dτ ϕ + C ϕH2 (R3 ) v
∞ |z| |z| ≤C ϕH2 (R3 ) + e−i z H1 − e−i(zp·ˆv+mvz/2) ei −∞ A·ˆv(x+τ vˆ ) dτ ϕ ≤ C ϕH2 (R3 ) , v v
(3.56) where we used Eq. (5.42) of [4]. Lemma 3.8. Let 0 be a compact subset of vˆ , v ∈ R3 \0. Then, for all A ∈ A ,2π (B) with divA ∈ L 2loc () there is a constant C such that ∀ z ≥ Z ≥ 0, z z−Z Z −i v H (A,V ) W− (A, V ) ϕv − e−i v H0 e−i L A,ˆv (−∞) e−i v H0 ϕv e 1 Z −βl +1 ≤C + (1 + Z ) ϕH2 (R3 ) , (3.57) + v v and all ϕ ∈ H2 R3 with support contained in 0 . Proof. The lemma follows from Lemmata 3.4, 3.6 and 3.7 and Eq. (5.42) of [4].
We summarize the results that we have obtained in the following theorem. Theorem 3.9. Let 0 be a compact subset of vˆ , v ∈ R3 \0. Then, for all A ∈ A ,2π (B) there is a constant C such that the following estimates hold for all ϕ ∈ H2 R3 with support contained in 0 : 1. For all Z ≥ 0 and all z ≤ Z , z z −i v H (A,V ) W− (A, V ) ϕv − e−i L A,ˆv (−∞) e−i v e
If furthermore, divA ∈ L 2loc (),
H0
C ϕv ≤ (1 + Z ) ϕH2 (R3 ) . v (3.58)
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195
2. For all Z ≥ 0 and all z ≥ Z , z −i v e
z−Z
H (A,V )
Z
W− (A, V ) ϕv − e−i v H0 e−i L A,ˆv (−∞) e−i v 1 Z −βl +1 ≤C + (1 + Z ) ϕH2 (R3 ) . + v v
H0
ϕv (3.59)
3. For all z ≥ 0, z −i v e
z
H (A,V )
W− (A, V ) ϕv − e−i v H0 ei 1 −βl +1 ≤C ϕH2 (R3 ) . + (1 + z) v
∞
−∞
A·ˆv(x+τ vˆ ) dτ
ϕv (3.60)
Proof. The theorem follows from Eq. (3.48) and Lemmata 3.3, 3.6 and 3.8.
Theorem 3.10. Let 0 be a compact subset of vˆ , v ∈ R3 \0. Then, for all A ∈ A ,2π (B) there is a constant C such that the following estimates hold for all ϕ ∈ H2 R3 with support contained in 0 : 1. For all z ≤ v 1/βl , z −i v e
H (A,V )
z
W− (A, V ) ϕv − e−i L A,ˆv (−∞) e−i v
H0
ϕv ≤
C ϕH2 (R3 ) . v 1−1/βl (3.61)
If furthermore, divA ∈ L 2loc (), 2. For all z ≥ v 1/βl , −i z e v ≤
H (A,V )
W− (A, V ) ϕv − e−i
z−v 1/βl v
H0 −i L A,ˆv (−∞) −i v
e
e
1/βl v
H0
C ϕH2 (R3 ) . v 1−1/βl
ϕv (3.62)
3. For all z ≥ v 1/βl , z −i v e ≤
H (A,V )
z
W− (A, V ) ϕv − e−i v
C ϕH2 (R3 ) . v 1−1/βl
H0 i
e
∞
−∞
A·ˆv(x+τ vˆ ) dτ
ϕv (3.63)
Proof. In Theorem 3.9 we take Z = v ρ , 0 < ρ < 1. The error terms are of the form, 1/v, 1/v ρ(βl −1) and 1/v 1−ρ . As for v ≥ 1 the error 1/v is smaller than 1/v 1−ρ , we only have to consider 1/v ρ(βl −1) and 1/v 1−ρ . Looking to these errors as a function of ρ we see that the point where the smallest exponent is bigger is the point of intersection of the lines 1 − ρ and ρ(βl − 1), i.e., 1 − ρ = ρ(βl − 1). Hence we take, ρ = 1/βl . The theorem follows from Theorem 3.9.
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3.2. Physical interpretation. In Theorems 3.9 and 3.10 we give the leading order for high-velocity of the solution to the Schrödinger equation. In Eq. (3.58) we give the leading order when the electron is incoming and interacting. We see that as the solution propagates towards the magnet, and it crosses it, it picks up a phase. In Eqs. (3.59, 3.60) we give two different expressions for the leading order when the electron is outgoing, i.e. after it leaves the magnet. The distance Z separates the incoming and interacting region from the outgoing one. In Eq. (3.59) we see that the leading order for the outgoing electron at distance z consists of the incoming and interacting leading order taken as the initial data at distance Z followed by the free evolution during distance z − Z . Finally, in Eq. (3.60) we give another representation of the leading order of the outgoing electron. Recall that the Cauchy data of the outgoing solution is given by Sϕv , with S the scattering operator. Furthermore (see Theorem 5.7 of [4]), up to an error of order
∞ 1/v, Sϕv = ei −∞ A·ˆv(x+τ vˆ ) dτ ϕv . Then, Eq. (3.60) expresses the leading order when the electron is outgoing as the free evolution applied to the Cauchy data of the outgoing solution. Note that scattering theory and Theorem 5.7 of [4] tell us that, up to an error ∞ z of order 1/v, the interacting solution tends to e−i v H0 ei −∞ A·ˆv(x+τ vˆ ) dτ ϕv at t → ∞. Equation (3.60) is more precise. It actually gives us an estimate of the error bound for large distances. Note that the leading orders for the outgoing electron given in Eqs. (3.59, 3.60) are close to each other for high velocity. It follows from Lemmata 3.4 and 3.7 that for z ∈ R, Z ≥ 0, z−Z
∞ z −i v H0 −i L A,ˆv (−∞) −i Zv H0 e e ϕv − e−i v H0 ei −∞ A·ˆv(x+τ vˆ ) dτ ϕv e 1 Z ≤C + (1 + Z )−βl +1 + ϕH2 (R3 ) . v v
(3.64)
In Eqs. (3.61, 3.62, 3.63) we optimize the error bounds taking the transition distance as Z = v 1/βl and we obtain high-velocity estimates that are uniform, respectively, for z ≤ v 1/βl , and z ≥ v 1/βl . Furthermore, taking Z = v 1/βl in (3.64) we obtain −i z−v1/βl e v ≤
H0 −i L A,ˆv (−∞) −i v
e
e
1/βl v
H0
ϕv − e
−i vz H0 i
e
∞
−∞
A·ˆv(x+τ vˆ ) dτ
C ϕH2 (R3 ) , z ∈ R. v 1−1/βl
ϕv (3.65)
In the transition region around Z the different expressions that we have obtained for the leading order are close to each other, as we show in the next sub-subsection. 3.2.1. The transition region We estimate the difference between the leading orders in Theorems 3.9 and 3.10 in the transition region z ∈ [Z /L , Z L], Z , L > 1. It follows from Lemmata 3.4, 3.7 and from Eq. ( 5.42) of [4] that for z ∈ [Z /L , Z L],
∞ z −i L A,ˆv (−∞) −i vz H0 e ϕv − e−i v H0 ei −∞ A·ˆv(x+τ vˆ ) dτ ϕv e 1 + ZL ≤ C (1 + Z /L)−βl +1 + ϕH2 (R3 ) . v
(3.66)
Aharonov-Bohm Effect and Solutions to the Schrödinger Equation
In the same way we prove that for z ∈ [Z /L , Z L], v > 1, z−Z Z −i L A,ˆv (−∞) −i vz H0 e ϕv − e−i v H0 e−i L A,ˆv (−∞) e−i v H0 ϕv e 1 + ZL ϕH2 (R3 ) . ≤ C (1 + Z /L)−βl +1 + v 1/β l Taking as in Theorem 3.10, Z = v 1/βl , we obtain that for z ∈ v L , Lv 1/βl ,
∞ z −i L A,ˆv (−∞) −i vz H0 e ϕv − e−i v H0 ei −∞ A·ˆv(x+τ vˆ ) dτ ϕv e 1 ≤ C L βl −1 + 1 + L ϕH2 (R3 ) , 1−1/βl v z−Z Z −i L A,ˆv (−∞) −i vz H0 e ϕv − e−i v H0 e−i L A,ˆv (−∞) e−i v H0 ϕv e 1 ≤ C L βl −1 + 1 + L ϕH2 (R3 ) . 1−1/β l v
197
(3.67)
(3.68)
(3.69)
3.3. Final formulae. Summing up, we have proven in Theorems 3.9 and 3.10 that the leading order for high velocity of the exact solution to the Schrödinger equation, ψv = e−it H (A,V ) W− (A, V ) ϕv , that behaves as, ψv,0 := e−it H0 ϕv , when t → −∞, is given by the following approximate solution to the Schrödinger equation: z e−i L A,ˆv (−∞) e−i v H0 ϕv , z = vt ≤ Z ≥ 0, (3.70) ψv,App (x, z) := z−Z Z e−i v H0 e−i L A,ˆv (−∞) e−i v H0 ϕv , z = vt ≥ Z , and, equivalently, by the approximate solution, z e−i L A,ˆv (−∞) e−i v H0 ϕv , z = vt ≤ Z ≥ 0,
φv,App (x, z) := ∞ z e−i v H0 ei −∞ A·ˆv(x+τ vˆ ) dτ ϕv , z = vt ≥ Z .
(3.71)
4. The Aharonov-Bohm Effect We will consider now the case where the magnetic field, B, outside K is zero but with a non-trivial magnetic flux, , inside K . For the moment we also suppose that the electric potential, V , outside K is zero, but this actually is not essential as the electric potential gives rise to a lower order effect for high velocity. This situation corresponds to the Aharonov-Bohm effect [3] and in particular to the experiments of Tonomura et al. [21,29,30] with toroidal magnets that are widely considered as the only convincing experimental verification of the Aharonov-Bohm effect. The physical interpretation of the results of the Tonomura et al. experiments is based on the validity of the Ansatz of Aharonov-Bohm [3] that is an approximate solution to the Schrödinger equation. Aharonov-Bohm propose a solution to the Schrödinger equation when, to a good aproximation, the electron stays in a simply connected region of space, C (more precisely in a region with trivial first group of singular homology), where the electromagnetic field is zero. Aharonov-Bohm point out that in this region the magnetic potential is the gradient of a scalar function, λ(x), and that the solution can be found by means of a change of gauge from the free evolution. The chosen scalar function depends on the simply connected region and it is only defined there. We now state the Aharonov-Bohm Ansatz in a precise way.
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Definition 4.1. Aharonov-Bohm Ansatz with Initial Condition at Time Zero. Let A be a magnetic potential with curl A = 0, defined in a region C that is simply connected, or more precisely with trivial first group of singular homology. Let A = ∇λ(x), for some scalar function λ. Let φ be the initial data at time zero of a solution to the Schrödinger equation that stays in C for all times, to a good approximation. Then, the change of gauge formula ([3], p. 487), e−it H (A) φ ≈ φ AB (x, t) := eiλ(x) e−it H0 e−iλ(x) φ
(4.1)
holds. To be more precise, in (4.1) we denote by λ(x) an extension of λ(x) to a function defined in R3 . Note that if the initial state at t = 0 is taken as e−iλ(x) φ, the Aharonov-Bohm Ansatz is the multiplication of the free solution by the Dirac magnetic factor eiλ(x) [10]. Equation (4.1) is formulated when the initial conditions are taken at time zero. We now find the appropriate Aharonov-Bohm Ansatz for the high-velocity solution ψv = e−it H (A,V ) W− (A, V ) ϕv , that satisfies the initial condition at time −∞, lim ψv − J ψv,0 = 0, t→−∞
(4.2)
(4.3)
where ψv,0 is the free incoming wave packet that represents the electron at the time of emission, ψv,0 := e−it H0 ϕv .
(4.4)
We have to find the initial state at time zero in (4.1) in order that the initial condition at time −∞ is satisfied. We take φ = eiλ(x) e−iλ∞ (−p) ϕv , where λ∞ (x) := limr →∞ λ(r x). We have that eiλ(x) e−it H0 e−iλ(x) φ = e−it H0 eiλ(x+(p/m)t) e−iλ∞ (−p) ϕv . But as λ∞ is homogeneous of order zero, s − lim eiλ∞ (x+(p/m)t) = eiλ∞ (−p) . t→−∞
Then
lim eiλ(x) e−it H0 e−iλ(x) φ − e−it H0 ϕv = 0.
t→−∞
Furthermore, for the high-velocity state ϕv and large v we have that e−iλ∞ (−p) ϕv ≈ e−iλ∞ (−ˆv) ϕv .
(4.5)
For this statement see the proof of Theorem 5.7 of [4]. It follows that the Aharonov-Bohm Ansatz for ψv is given by ψv (x, t) ≈ eiλ(x) e−it H0 e−iλ∞ (−ˆv) ϕv . We prove below that without loss of generality we can assume that the potential A has compact support in B R and λ∞ (−ˆv) = 0. In this case the Aharonov-Bohm Ansatz for high-velocity solutions with initial data at time −∞ is given by the following definition.
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199
Definition 4.2. Aharonov-Bohm Ansatz with Initial condition at Time Minus Infinite. Let A be a magnetic potential with curl A = 0, defined in a region C with trivial first group of singular homology. Let A = ∇λ(x) for some scalar function λ with λ∞ (−ˆv) = 0 t for some unit vector vˆ . Let ψv (x, t) := e−i H (A) W− (A, V ) ϕv be the solution to the Schrödinger equation that behaves like ψv,0 := e−it H0 ϕv when time goes to minus infinity. We suppose that ψv is approximately localized for all times in C. Then, the following change of gauge formula holds: ψv ≈ ψ AB,v (x, t) := eiλ(x) e−it H0 ϕv .
(4.6)
Observe that, again, the Aharonov-Bohm Ansatz is the multiplication of the free solution by the Dirac magnetic factor eiλ(x) [10]. Note that for the validity of the Aharonov-Bohm Ansatz it is necessary that the electron stays in the simply connected region C (disjoint from the magnet) and that it is not directed towards the magnet K (it does not hit it). In fact, if the electron hits K it will be reflected no matter how big the velocity is, and then, it will not follow the free evolution multiplied by a phase, as is the case in the Aharonov-Bohm Ansatz. This can be seen, for example, in the case of a solenoid contained inside an infinite cylinder, that has explicit solution [27]. See for example Eq. (4.22) of [27] that gives the phase shifts in the case with Dirichlet boundary condition, that shows that the scattering from the cylinder is always present and that it appears in the leading order together with the contribution of the magnetic flux inside the cylinder. In fact, the magnet K amounts to an infinite electric potential. Observe, however, that, as we prove below, a finite potential V that satisfies (2.4) produces a lower order term and, hence, it does not affect the validity of the Aharonov-Bohm Ansatz for high velocity. Recall that the set vˆ (3.23) corresponds to trajectories that do not hit the magnet under the classical free evolution. Since for high velocities the electron follows the quantum free evolution and as the quantum free evolution is concentrated along the classical trajectories, it is natural to require that when the electron is inside B R it is actually in vˆ ∩ B R , in such a way that as it crosses the region where the magnet is located it does so through the holes of K that are in vˆ or that it crosses outside of the holes of K . In general, vˆ crosses several holes of K , and if two electrons cross different holes of K there can be no simply connected region that contains both of them for all times. In order to make the idea above precise we have first to decompose vˆ on its components that cross the same holes of K . This was accomplished in [4] as follows. Recall that K ⊂ B R . For any x ∈ R3 and any unit vector vˆ in R3 , we denote L(x, vˆ ) := x + Rˆv, and we give to L(x, vˆ ) the orientation of vˆ . Suppose that L(x, vˆ ) ⊂ , and L(x, vˆ ) ∩ B R = ∅. We denote by c(x, vˆ ) the curve consisting of the segment L(x, vˆ ) ∩ B R and an arc on ∂ B R that connects the points L(x, vˆ ) ∩ ∂ B R . We orient c(x, vˆ ) in such a way that the segment of straight line has the orientation of vˆ . See Fig. 2. Definition 4.3. A line L(x, vˆ ) ⊂ goes through holes of K if L(x, vˆ ) ∩ B R = ∅ and [c(x, vˆ )] H1 (;R) = 0. Otherwise we say that L(x, vˆ ) does not go through holes of K . Note that this characterization of lines that go or do not go through holes of K is independent of the R that was used in the definition. This follows from the homotopic invariance of homology. See Theorem 11.2, p. 59 of [14].
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M. Ballesteros, R. Weder
Fig. 2. The curves c(x, vˆ )
In an intuitive sense [c(x, vˆ )] H1 (;R) = 0 means that c(x, vˆ ) is the boundary of a surface (actually of a chain) that is contained in and then it can not go through holes of K . Obviously, as K ⊂ B R , if L(x, vˆ ) ∩ B R = ∅ the line L(x, vˆ ) can not go through holes of K . ˆ ⊂ that go through holes of K go through Definition 4.4. Two lines L(x, vˆ ), L(y, w) ˆ H1 (;R) . Furthermore, we say that the the same holes if [c(x, vˆ )] H1 (;R) = ±[c(y, w)] ˆ H1 (;R) . lines go through the holes in the same direction if [c(x, vˆ )] H1 (;R) = [c(y, w)] Remark 4.5. If (x, vˆ ) ∈ × S2 , there are neighborhoods Bx ⊂ R3 , Bvˆ ⊂ S2 such that ˆ ∈ Bx × Bvˆ then, the following is true: if L(x, vˆ ) (x, vˆ ) ∈ Bx × Bvˆ , and if (y, w) ˆ does not go through holes of does not go through holes of K , then, also L(y, w) ˆ goes through the same holes K . If L(x, vˆ ) goes through holes of K , then, L(y, w) and in the same direction. This follows from the homotopic invariance of homology, Theorem 11.2, p. 59 of [14]. Definition 4.6. For any vˆ ∈ S2 we denote by vˆ ,out the set of points x ∈ vˆ such that L(x, vˆ ) does not go through holes of K . We call this set the region without holes of vˆ . The holes of vˆ is the set vˆ ,in := vˆ \vˆ ,out . We define the following equivalence relation on vˆ ,in . We say that x Rvˆ y if and only if L(x, vˆ ) and L(y, vˆ ) go through the same holes and in the same direction. By [x] we designate the classes of equivalence under Rvˆ . We denote by vˆ ,h h∈I the partition of vˆ ,in given by this equivalence relation. It is defined as follows. I := {[x]}x∈vˆ ,in .
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201
Given h ∈ I there is x ∈ vˆ ,in such that h = [x]. We denote vˆ ,h := {y ∈ vˆ ,in : y Rvˆ x}. Then vˆ ,in = ∪h∈I vˆ ,h , vˆ ,h 1 ∩ vˆ ,h 2 = ∅, h 1 = h 2 . We call vˆ ,h the subset of vˆ that goes through the holes h of K in the direction of vˆ . Note that {vˆ ,h }h∈I ∪ {vˆ ,out }
(4.7)
is an disjoint open cover of vˆ . We visualize the dynamics of the electrons that travel through the holes of K in vˆ ,h as follows. For large negative times the incoming electron wave packet is in , far away from K . As time increases the electron travels towards K and it reaches the region where K is located, let us say that it is inside B R . At these times the electron has to be in vˆ ,h in order to cross B R through the holes of K in vˆ ,h . After crossing the holes it travels again away from K towards spatial infinity in . This means that the classical trajectories have to be in the following domain: Ch := \ B R ∪ Pvˆ ∪ B R ∩ vˆ ,h , (4.8) where Pvˆ is the plane orthogonal to vˆ that passes through zero, Pvˆ := x ∈ R3 : x · vˆ = 0 .
(4.9)
Note that we take away from Ch the part of Pvˆ that does not intersect v,h in order that the only way that the electron in Ch can classically cross the plane Pvˆ is through vˆ ,h . In a similar way, the classical trajectories of the electrons that do not cross any hole of K have to be on the set Cout := \B R ∪ B R ∩ vˆ ,out . (4.10) In Corollary 5.9 in the Appendix we prove that the first group of singular homology with coefficients in R of Ch , H1 (Ch ; R), h ∈ I, and of Cout , H1 (Cout ; R) are trivial. We actually prove that the first de Rham cohomology class of Ch and of Cout are trivial by explicitly constructing a function λ such that A = ∇λ for any magnetic potential A with curl A = 0, or in differential geometric language by constructively proving that any closed one form is exact. Then, the triviality of the first group of singular homology with coefficients in R of Ch and of Cout follows from de Rham’s theorem (Theorem 4.17 p. 154 of [33]). Let x0 be a fixed point with x0 · vˆ < −R. We define A, h ∈ I, where C h is any differentiable path from x0 to x in Ch , λh (x) := Ch
(4.11) and λout (x) :=
C out
A, where C out is any differentiable path from x0 to x in Cout . (4.12)
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Since H1 (Ch ; R), h ∈ I and H1 (Cout ; R) are trivial, λh , h ∈ I and λout do not depend in the particular curve form x0 to x that we take, respectively, in Ch , h ∈ I and Cout . Furthermore, they are differentiable and ∇λh (x) = A(x), x ∈ Ch , h ∈ I and ∇λout (x) = A(x), x ∈ Cout . Before we prove the validity of the Aharonov-Bohm Ansatz we prepare some simple the complement of results on the free evolution that we need. Below we denote by O any set O ⊂ R3 . Lemma 4.7. We denote
C−,h := x ∈ \B R : x · vˆ < 0 ∪ vˆ ,h , h ∈ I, C−,out := x ∈ \B R : x · vˆ < 0 ∪ vˆ ,out .
(4.13)
Then, for any l = 0, 1, . . . and any compact set 0 ⊂ vˆ,h , h ∈ I, there is a constant Cl such that ∀Z ≥ 0, ∀z ∈ (−∞, Z ], and for all ϕ ∈ H2 R3 with support in 0 , 1+ Z −i vz H0 −l (1 + Z ) ϕH2 (R3 ) . (4.14) e ϕ + ≤ C χ v 2 3 l C−,h L (R ) v Furthermore, for any l = 0, 1, . . . and any compact set 0 ⊂ vˆ ,out , there is a constant Cl such that ∀Z ≥ 0, ∀z ∈ (−∞, Z ], and for all ϕ ∈ H2 R3 with support in 0 , z 1+ Z ϕH2 (R3 ) . (4.15) χC e−i v H0 ϕv 2 3 ≤ Cl (1 + Z )−l + −,out v L (R ) Proof. We give the proof of (4.14). Equation (4.15) follows in the same way. 1.
Suppose that z ≤ min (− 43 R, −Z ). By (3.16) it is enough to prove (4.14) for ϕ. ˜ The estimate follows from (3.9) and Lemma 3.2, observing that χ (x) = C−,h (x)F(|x − z vˆ | > |z|/4). χ C
2.
e−i zp·ˆv ϕ = 0, it follows from (3.8) that Suppose that z ∈ [−Z , Z ]. Since, χ C−,h −i vz H0 −i z H1 −i zp·ˆv −i zmv/2 e ϕ 2 3 e ϕ − e e = χ χ v C−,h C−,h L 2 (R 3 ) L (R ) Z (4.16) ≤ C ϕH2 (R3 ) . v If Z ≤ 43 R it remains to consider z ∈ − 43 R, −Z . In this case we just say that −i vz H0 e ϕ 2 3 ≤ ϕ L 2 (R3 ) ≤ Cl (1 + Z )−l ϕ L 2 (R3 ) . (4.17) χ v C−,h L (R )
−,h
3.
Lemma 4.8. We denote
C+0 := x ∈ \B R : x · vˆ > 0 .
(4.18)
Then, for any l = 0, 1, . . . there is a constant Cl such that ∀Z ≥ 0, ∀z ≥ Z , and for all ϕ ∈ H2 R3 , 1 −i vz H0 −l ϕH2 (R3 ) . (4.19) ϕv 2 3 ≤ Cl (1 + Z ) + χC0 e + L (R ) v
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Proof. If Z ≥ 43 R we prove (4.19) as in item 1 of the proof of Lemma 4.7, observing that χC+ (x) = χC+ (x)F(|x − z vˆ | > |z|/4). If Z ≤ 43 R it remains to consider z ∈ Z , 43 R , but in this case (4.19) follows as in item 3 of the proof of Lemma 4.7. Corollary 4.9. For any l = 0, 1, . . . and any compact set 0 ⊂ vˆ ,h , h ∈ I there is a constant Cl such that ∀Z ≥ 0, ∀z ∈ R, and for all ϕ ∈ H2 R3 with support in 0 , z 1+ Z ϕH2 (R3 ) . (4.20) χCh e−i v H0 ϕv 2 3 ≤ Cl (1 + Z )−l + L (R ) v Furthermore, for any l = 0, 1, . . . and any compact set 0 ⊂ vˆ ,out there is a constant Cl such that ∀Z ≥ 0, ∀z ∈ R, and for all ϕ ∈ H2 R3 with support in 0 , 1+ Z −i vz H0 −l ϕH2 (R3 ) . (4.21) e ϕv 2 3 ≤ Cl (1 + Z ) + χ Cout L (R ) v Proof. Note that since \B R ∩ vˆ ,h ⊂ \ B R ∪ Pvˆ , h ∈ I, we have that C−,h ⊂ Ch , h ∈ I. Moreover, C−,out ⊂ Cout , and C+0 ⊂ Ch ∩ Cout . Hence, the corollary follows from Lemma 4.7 when z ≤ Z and from Lemma 4.8 when z ≥ Z. Definition 4.10. We designate by A ,2π (0) the set of all potentials A ∈ A ,2π (B) that satisfy curl A = B = 0. Remark 4.11. For any A ∈ A ,2π (0) ∩ C l , R3 , l = 1, 2, . . ., there is a A˜ ∈ A ,2π (0) ∩ C l , R3 with the same flux as A and with support A˜ ⊂ B R . To prove this statement we take any x0 ∈ \B R and let ε > 0 be so small that K ⊂ B R−ε . We define λ(x) := A, for x ∈ \B R−ε , C(xo ,x)
where C(x 0 , x) is any differentiable path from x0 to x contained in \B R−ε . Then, λ ∈ C l+1 \B R−ε . We denote by λ any extension of λ to R3 such that λ ∈ C l+1 R3 [32]. We define, ˜ A(x) := A(x) − ∇λ(x), x ∈ . l Then, A˜ ∈ A ,2π (0) ∩ C , R3 , l = 1, 2, . . ., the flux of A˜ is the same as the one of A and support A˜ ⊂ B R . Note that if B = 0 the Coulomb potential AC ∈ C ∞ , R3 (see Theorem 3.7 of [4]). Doing the gauge transformation above we see that for every l = 1, 2, . . . there is a potential in A ,2π (0) ∩ C l , R3 with compact support in B R .
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By Remark 4.11 we can use the freedom of taking a gauge transformation to assume that A ∈ A ,2π (0) ∩ C 1 , R3 and that supportA ⊂ B R , which we do from now on. Theorem 4.12. For any l = 0, 1, . . . and any compact set 0 ⊂ vˆ ,h , h ∈ I, there is a constant Cl such that ∀Z ≥ 0, ∀z ∈ R, and for all ϕ ∈ H2 R3 with support in 0 , z z −i v H (A,V ) W− (A, V ) ϕv − eiλh χCh e−i v H0 ϕv 2 3 e L (R ) 1 + Z ϕH2 (R3 ) . (4.22) ≤ Cl (1 + Z )−l + v Furthermore, for any l = 0, 1, . . . and any compact set 0 ⊂ vˆ ,out , there is a constant Cl such that ∀Z ≥ 0, ∀z ∈ R, and for all ϕ ∈ H2 R3 with support in 0 , z z −i v H (A,V ) W− (A, V ) ϕv − eiλout χCout e−i v H0 ϕv 2 3 e L (R ) 1+ Z ϕH2 (R3 ) . (4.23) ≤ Cl (1 + Z )−l + v Proof. We first consider the case z ≤ Z . In this case the theorem follows from Lemmata 4.7, 4.8, Corollary 4.9, and (3.58) observing that since support A ⊂ B R , −L A,ˆv (−∞) = λh (x), x ∈ C−,h , h ∈ I, −L A,ˆv (−∞) = λout (x), x ∈ C−,out . For z ≥ Z we use (3.60), Lemma 4.8 and Corollary 4.9. For this purpose note that ∞ A(x + τ vˆ ) · vˆ dτ = A, for x ∈ vˆ ,h , h ∈ I, −∞ c(x,ˆv) ∞ A(x + τ vˆ ) · vˆ dτ = 0, for x ∈ out . −∞
Moreover, recall that (see Definition 7.10 of [4]) A, x ∈ vˆ ,h , h ∈ I, Fh := c(x,ˆv)
and that Fh is constant for all x ∈ vˆ ,h . Fh is the magnetic flux over any surface (or a chain) in R3 whose boundary is c(x, vˆ ). In other words, it is the flux associated to the holes of K in vˆ ,h . Furthermore, we have that Fh = λh (x), x ∈ C+0 ,
(4.24)
which completes the proof for z ≥ Z , h ∈ I. For the case vˆ ,out and z ≥ Z we observe that λout (x) = 0, for x ∈ C+0 .
(4.25)
We now state our main results on the validity of the Aharonov-Bohm Ansatz.
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Theorem 4.13. For any 1 > δ > 0 and any compact set 0 ⊂ vˆ ,h , h ∈ I, there is a constant Cδ such that ∀t ∈ R and for all ϕ ∈ H2 R3 with support in 0 , −it H (A,V ) W− (A, V ) ϕv − eiλh χCh e−it H0 ϕv e
L2
( ) R3
≤
Cδ ϕH2 (R3 ) . (4.26) v 1−δ
Furthermore, for any 1 > δ > 0 and any compact set 0 ⊂ vˆ ,out , there is a constant 3 Cδ such that ∀t ∈ R and for all ϕ ∈ H2 R with support in 0 , −it H (A,V ) W− (A, V ) ϕv − eiλout χCout e−it H0 ϕv e
L 2 (R 3 )
≤
Cδ ϕH2 (R3 ) . v 1−δ (4.27)
Proof. We take in Theorem 4.12, Z = v 1/(1+l) and t = z/v. Then, for v > 1, v1 (1+ Z ) ≤ 1 1 1 and (1 + Z )−l ≤ v 1−1/(1+l) . The theorem follows taking 1+l ≤ δ. 2 v 1−1/(1+l) Let us take any ϕ ∈ H2 R3 with compact support in vˆ . Then, since (4.7) is a disjoint open cover of vˆ , ϕ=
ϕh + ϕout ,
(4.28)
h∈I
where ϕh , ϕout ∈ H2 R3 , ϕh has compact support in vˆ ,h , h ∈ I, and ϕout has compact support in vˆ ,out . The sum is finite because ϕ has compact support. We denote ϕv := eimv·x ϕ, ϕv,h := eimv·x ϕh , h ∈ I, ϕv,out := eimv·x ϕout .
(4.29)
ψ AB,v,h := χCh eiλh e−it H0 ϕv,h , h ∈ I, ψ AB,v,out := χCout eiλout e−it H0 ϕv,out , ψ AB,v := ψ AB,v,h + ψ AB,v,out .
(4.30)
We define
(4.31)
h∈I
Equation (4.31) gives the Aharonov-Bohm Ansatz in the domain ∪h∈I Ch ∪ Cout that has non-trivial first group of singular homology as the sum of the Aharonov-Bohm Ansätze in each of the components, Ch , h ∈ I, Cout that have trivial first group of singular homology. As we already mentioned, for the Ansatz of Aharonov-Bohm to be valid, it is necessary that the electron does not hit the magnet. Otherwise, the electron will be reflected and the Ansatz cannot be an approximate solution because it consists of the free evolution multiplied by a phase in configuration space. Hence, the wave function that represents such an electron has to have its support approximately contained for all times in the domain ∪h∈I Ch ∪ Cout . In the next theorem we prove that the Ansatz of Aharonov-Bohm is actually valid on the biggest domain where it can be valid, ∪h∈I Ch ∪ Cout , and, in this way, we provide an approximate solution for all times for every electron that does not hit the magnet.
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Theorem 4.14. The Validity of the Aharonov-Bohm Ansatz. For any 1 > δ > 0 and any compact set 0 ⊂ vˆ , there is a constant Cδ such that ∀t ∈ R and for all ϕ ∈ H2 R3 with support in 0 the solution to the Schrödinger equation e−it H (A,V ) W− (A, V ) ϕv that behaves as e−it H0 ϕv as t → −∞ is given at time t by the Aharonov-Bohm Ansatz, ψ AB,v , up to the following error: −it H (A,V ) W− (A, V ) ϕv − ψ AB,v e
L 2 (R 3 )
≤
Cδ ϕH2 (R3 ) . v 1−δ
Proof. The theorem follows from Theorem 4.13 and Eqs. (4.28 to 4.31).
(4.32)
Note that by (4.24, 4.25) behind the magnet in C+0 , ψ AB,v,h := χCh ei Fh e−it H0 ϕv,h , h ∈ I, x ∈ C+0 ,
(4.33)
and that ψ AB,v,out := χCout e−it H0 ϕv,out ,
x ∈ C+0 .
(4.34)
As mentioned in the Introduction the phase shifts ei Fh were measured in the experiments of Tonomura et al. [21,29,30] and, furthermore, since the Aharonov-Bohm Ansatz is free evolution, up to a phase, the electron is not accelerated, which explains the results of the experiment of Caprez et al. [8]. Hence, Theorem 4.14 rigorously proves that quantum mechanics predicts the results of the experiments of Tonomura et al. and of Caprez et al.. Acknowledgements. This research was partially done while M. Ballesteros was at Departamento de Métodos Matemáticos y Numéricos. Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas. Universidad Nacional Autónoma de México.
5. Appendix In this Appendix we prove that the first group of singular homology with coefficients in R of Ch and of Cout are trivial. The sets Ch and Cout are defined, respectively, in (4.8) and (4.10). We denote C+ := x ∈ R3 \B R : x · vˆ > 0 , C− := x ∈ R3 \B R : x · vˆ < 0 , (5.1) 0 the interior of C . Recall that P is defined in (4.9). Then, and by C± ± vˆ
0 Ch = C − ∪ C+0 ∪ B R ∩ vˆ ,h , 0 Cout = C− ∪ C+0 ∪ B R ∩ vˆ ,out ∪ Pvˆ \B R .
(5.2) (5.3)
We first prepare several results that we need. Below we denote by A any continuously differentiable vector field defined, respectively, in Ch , h ∈ I, and in Cout , with curl A = 0. Let x0 be a fixed point with x0 < −R. For any x ∈ B R we denote, respectively by xin , xout the intersection of the line {x + τ vˆ , τ ∈ R} with ∂ B R such that xin · vˆ < 0, xout · vˆ > 0. For any h ∈ I, let x h be a fixed point in vˆ ,h ∩ B R , and let x out be a fixed point in vˆ ,out ∩ B R .
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x any differentiable path in C that goes Remark 5.1. For every x ∈ C− we denote by C− − from x0 to x and we define λ− (x) := A. (5.4) x C−
Since C− is simply connected the line integral in (5.4) does not depend on the x that we choose. Then, for x ∈ C 0 , λ (x) is differentiable particular curve C− − − and ∇λ− (x) = A(x). For any x, y ∈ R3 we denote by [x, y] the straight line that goes from x to y. Remark 5.2. For every x ∈ B R ∩ vˆ we denote by C0x the differentiable path consisting xin followed by the segment [xin , x] and we define for every x ∈ B R ∩ vˆ , of a path C− A. (5.5) λ0 (x) := C0x
xin By Remark 5.1 the line integral in (5.5) does not depend on the particular curve C− that we choose. Then, for x ∈ B R ∩ vˆ , λ0 (x) is differentiable and ∇λ0 (x) = A(x). To prove this statement we observe that for each x ∈ B R ∩ vˆ , there is ε > 0 such that Bε (x) ⊂ B R ∩ vˆ . The set Cs := {C− ∪ (Bε (x) + Rˆv)} is simply connected and, furthermore, λ0 (x) = C A, where C is any differentiable path contained in Cs that goes from x0 to x. x a differentiable path Remark 5.3. For every x ∈ C+ and any h ∈ I we denote by C h,+ xh
h , x h ] and of a differconsisting of any curve C−in followed from the straight line [xin out x h ,x
h to x. The differentiable path C x entiable path C+out in C+ that goes from xout out,+ is h defined in the same way, but replacing x by x out . We define A, x ∈ C+ , h ∈ I, (5.6) λ+h (x) := x Ch,+
and
λout + (x)
:=
x Cout,+
A, x ∈ C+ .
(5.7) xh
x h ,x
Since C± are simple connected, λ+h does not depend on the particular paths C−in , C+out x out
x out ,x
in out that we choose and, λout that + does not depend on the particular paths C − , C + h out 0 we choose. It follows that λ+ and λ+ are continuously differentiable in C+ and that ∇λ+h (x) = A(x), ∇λout + (x) = A(x).
Remark 5.4. λ+h , h ∈ I does not depend on the particular x h ∈ vˆ ,h that we choose. To prove this statement let us take any y ∈ vˆ ,h ∩ B R and let the differentiable path x be defined as C x but with y instead of x h . Let γ be any differentiable path from C y,+ h,+ x to x0 contained in \B R . Let C be the closed oriented differentiable path consisting x , from x to x, followed from γ . C is defined in the same way, but with C x of C h,+ 0 y y,+ x . Let D be an arc on ∂ B from x h to x h and let G be a differentiable instead of C h,+ R out in
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x h ,x
path consisting of C−in followed of D, C+out and γ . Since R3 \B R is simply connected, we have that A = 0, G
and then
A=
A.
h c xin ,ˆv
C
We prove in the same way that
A=
Cy
A.
c(yin ,ˆv)
h , vˆ H (;R) = [c(yin , vˆ )] H1 (;R) , Furthermore, since x h , y ∈ vˆ ,h , we have that c xin 1 and then, by Stoke’s theorem, A, A= h ,ˆ c xin v
c(yin ,ˆv)
which proves that
A=
A,
C
Cy
and then
λ+h (x) :=
x Ch,+
A=
x C y,+
A.
out ∈ Remark 5.5. λout vˆ ,out that we choose. + does not depend on the particular x h out by x . Furthermore, as in this case This is proven as in Remark 5.4, replacing x out c xin , vˆ H (;R) = 0, 1
A = 0, C
and then,
λout + (x) =
γ
A,
(5.8)
where γ is any differentiable path from x0 to x contained in \B R . Definition 5.6. For all h ∈ I we define λh ⎧ ⎪ ⎨ λ− (x), h λ (x) := λ0 (x), ⎪ ⎩ λh (x), +
: Ch → R as follows: if x ∈ C− , if x ∈ vˆ ,h ∩ B R , if x ∈ C+ .
(5.9)
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Furthermore, we define λout : Cout → R as, ⎧ λ− (x), if x ∈ C− , ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ λ0 (x), if x ∈ vˆ ,out ∩ B R , (x), if x ∈ C+ , (5.10) λout (x) := λout
+ ⎪ ⎪ ⎪ A, if x ∈ Pvˆ \B R , where γ is any differentiable path from x0 to x ⎪ ⎪ ⎩ γ contained in \B R . Lemma 5.7. The functions λh , h ∈ I and λout are continuously differentiable and ∇λh (x) = A(x), x ∈ Ch , h ∈ I and ∇λout (x) = A(x), x ∈ C out . Proof. We first consider λh , h ∈ I. By Remarks 5.1, 5.2 and 5.3 λh (x) is continuously 0 ∪ C0 ∪ differentiable and ∇λh (x) = A(x) for x ∈ C− vˆ ,h ∩ B R . It follows from (5.2) + that it only remains to prove the result for x ∈ vˆ ,h ∩ ∂ B R . Let ε > 0 be such that, Bε (x) ⊂ vˆ ,h (see Remark 4.5). The set 0 ∪ (Bε (x) + Rˆv) ∪ C+0 C p,h := C− is simply connected and by Remark 5.4 λh (y) = A,
y ∈ C p,h ,
C
where C is any differentiable path from x0 to y that is contained in C p,h . It follows that λh (x) is differentiable for x ∈ vˆ ,h ∩ ∂ B R and that ∇λh (x) = A(x). out Let us now lemma holds for x ∈ consider λ . By Remarks 5.1, 5.2 and 5.3 the 0 0 C− ∪ C+ ∪ vˆ ,out ∩ B R . Furthermore, by the definition of λout and (5.8) it also holds for x ∈ Pvˆ \B R . By (5.3) it only remains to consider the case of x ∈ ∂ B R ∩ vˆ ,out . Take ε > 0 such that K ⊂ B R−ε . Then, since R3 \B R−ε is a simply connected set where curl A = 0 we have that for x ∈ Cout \B R−ε , out A, λ (x) = γ
where γ is any differentiable path from x0 to x contained in R3 \B R−ε . This implies that λout (x) is continuously differentiable with ∇λout (x) = A(x) for x ∈ C out \B R−ε , and in particular for x ∈ ∂ B R ∩ vˆ .out . 1 (C ), h ∈ I, and H 1 (C ) Lemma 5.8. The first de Rham cohomology groups Hde R h de R out are trivial.
Proof. In differential geometric terms Lemma 5.7 means that every closed 1-differential form in Ch , h ∈ I, and in Cout is exact, which proves the lemma. Corollary 5.9. The first groups of singular homology H1 (Ch ; R), h ∈ I and H1 (Cout ; R) are trivial. Proof. The corollary follows from Lemma 5.8 and de Rham’s theorem (Theorem 4.17 p. 154 of [33]).
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References 1. Adams, R.A., Fournier, J.J.F.: Sobolev Spaces. Oxford: Amsterdam Academic Press, 2003 2. Agmon, S.: Lectures on Elliptic Boundary Value Problems. Princeton, NJ: D. Van Nostrand, 1965 3. Aharonov, Y., Bohm, D.: Significance of electromagnetic potentials in the quantum theory. Phys. Rev. 115, 485–491 (1959) 4. Ballesteros, M., Weder, R.: High-velocity estimates for the scattering operator and Aharonov-Bohm effect in three dimensions. Commun. Math. Phys. 285, 345–398 (2009) 5. Ballesteros, M., Weder, R.: The Aharonov-Bohm effect and Tonomura et al. experiments: Rigorous results. J. Math. Phys. 50, 122108 (2009) (54 pp) 6. Boyer, T.H.: Darwin-Lagrangian analysis for the interaction of a point charge and a magnet: considerations related to the controversy regarding the Aharonov-Bohm and the Aharonov-Casher phase shifts. J. Phys. A: Math. Gen. 39, 3455–3477 (2006) 7. Bredon, G.E.: Topology and Geometry. New York: Springer-Verlag, 1993 8. Caprez, A., Barwick, B., Batelaan, H.: Macroscopic test of the Aharonov-Bohm effect. Phys. Rev. Lett. 99, 210401 (2007) (4 pp.) 9. de Rham, G.: Differentiable Manifolds. Berlin: Springer-Verlag, 1984 10. Dirac, P.: Quantized singularities in the electromagnetic field. Proc. R. Soc. A 133, 60–72 (1931) 11. Enss, V., Weder, R.: The geometrical approach to multidimensional inverse scattering. J. Math. Phys. 36, 3902–3921 (1995) 12. Franz, W.: Elektroneninterferenzen im Magnetfeld. Verh. D. Phys. Ges. (3) 20, Nr.2, 65–66 (1939) 13. Franz, W.: Elektroneninterferenzen im Magnetfeld. Physikalische Berichte 21, 686 (1940) 14. Greenberg, M.J., Harper, J.R.: Algebraic Topology, A First Course. New York: Addison-Wesley, 1981 15. Hatcher, A.: Algebraic Topology. Cambridge: Cambridge University Press, 2002 16. Hegerfeldt, G.C., Neumann, J.T.: The Aharonov–Bohm effect: the role of tunneling and associated forces. J. Phys. A: Math. Theor. 41, 155305 (2008) (11pp) 17. Helffer, B.: Effet d’Aharonov-Bohm sur un état borné de l’équation de Schrödinger. (French) [The Aharonov-Bohm effect on a bound state of the Schrödinger equation], Commun. Math. Phys. 119, 315–329 (1988) 18. Kato, T.: Perturbation Theory for Linear Operators. Second Edition, Berlin: Springer-Verlag, 1976 19. Nicoleau, F.: An inverse scattering problem with the Aharonov-Bohm effect. J. Math. Phys. 41, 5223–5237 (2000) 20. Olariu, S., Popescu, I.I.: The quantum effects of electromagnetic fluxes. Rev. Modern. Phys. 57, 339–436 (1985) 21. Osakabe, N., Matsuda, T., Kawasaki, T., Endo, J., Tonomura, A., Yano, S., Yamada, H.: Experimental confirmation of Aharonov-Bohm effect using a toroidal magnetic field confined by a superconductor. Phys. Rev. A 34, 815–822 (1986) 22. Peshkin, M., Tonomura, A.: The Aharonov-Bohm Effect. Lecture Notes in Phys. 340, Berlin: Springer, 1989 23. Reed, M., Simon, B.: Methods of Modern Mathematical Physics. II. Fourier Analysis. Self-Adjointness. New York: Academic Press, 1975 24. Reed, M., Simon, B.: Methods of Modern Mathematical Physics. III. Scattering Theory. New York: Academic Press, 1979 25. Roux, Ph.: Scattering by a toroidal coil. J. Phys. A: Math. Gen. 36, 5293–5304 (2003) 26. Roux, Ph., Yafaev, D.: On the mathematical theory of the Aharonov-Bohm effect. J. Phys. A: Math. Gen. 35, 7481–7492 (2002) 27. Ruijsenaars, S.N.M.: The Aharonov-Bohm effect and scattering theory. Ann. Phys. (New York) 146, 1–34 (1983) 28. Tonomura, A.: Direct observation of hitherto unobservable quantum phenomena by using electrons. Proc. Natl. Acad. Sci. U.S.A. 102, 14952–14959 (2005) 29. Tonomura, A., Matsuda, T., Suzuki, R., Fukuhara, A., Osakabe, N., Umezaki, H., Endo, J., Shinagawa, K., Sugita, Y., Fujiwara, H.: Observation of Aharonov-Bohm effect by electron holography. Phys. Rev. Lett. 48, 1443–1446 (1982) 30. Tonomura, A., Osakabe, N., Matsuda, T., Kawasaki, T., Endo, J., Yano, S., Yamada, H.: Evidence for Aharonov-Bohm effect with magnetic field completely shielded from electron wave. Phys. Rev. Lett. 56, 792–795 (1986) 31. Tonomura, A., Nori, F.: Disturbance without the force. Nature 452–20, 298–299 (2008) 32. Triebel, H.: Interpolation Theory, Function Spaces, Differential Operators. Amsterdam: North-Holland, 1978 33. Warner, F.W.: Foundations of Differentiable Manifolds. Berlin: Springer-Verlag, 1983 34. Weder, R.: The Aharonov-Bohm effect and time-dependent inverse scattering theory. Inverse Problems 18, 1041–1056 (2002)
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35. Yafaev, D.R.: Scattering matrix for magnetic potentials with Coulomb decay at infinity. Integral Equations Operator Theory 47, 217–249 (2003) 36. Yafaev, D.R.: Scattering by magnetic fields. St. Petersburg Math. J. 17, 875–895 (2006) Communicated by I.M. Sigal
Commun. Math. Phys. 303, 213–232 (2011) Digital Object Identifier (DOI) 10.1007/s00220-010-1133-5
Communications in
Mathematical Physics
An Algebraic Construction of Boundary Quantum Field Theory Roberto Longo1, , Edward Witten2 1 Dipartimento di Matematica, Università di Roma “Tor Vergata”, Via della Ricerca Scientifica, 1,
I-00133 Roma, Italy. E-mail:
[email protected]
2 Institute for Advanced Study, School of Natural Sciences, Einstein Drive, Princeton, NJ 08540, USA.
E-mail:
[email protected] Received: 5 April 2010 / Accepted: 5 May 2010 Published online: 21 September 2010 – © Springer-Verlag 2010
Abstract: We build up local, time translation covariant Boundary Quantum Field Theory nets of von Neumann algebras AV on the Minkowski half-plane M+ starting with a local conformal net A of von Neumann algebras on R and an element V of a unitary semigroup E(A) associated with A. The case V = 1 reduces to the net A+ considered by Rehren and one of the authors; if the vacuum character of A is summable, AV is locally isomorphic to A+ . We discuss the structure of the semigroup E(A). By using a one-particle version of Borchers theorem and standard subspace analysis, we provide an abstract analog of the Beurling-Lax theorem that allows us to describe, in particular, all unitaries on the one-particle Hilbert space whose second quantization promotion belongs to E(A(0) ) with A(0) the U (1)-current net. Each such unitary is attached to a scattering function or, more generally, to a symmetric inner function. We then obtain families of models via any Buchholz-Mack-Todorov extension of A(0) . A further family of models comes from the Ising model.
1. Introduction As is known Conformal Quantum Field Theory is playing a crucial role in several research areas, both in Physics and in Mathematics. Boundary Quantum Field Theory, related to Conformal Field Theory, is also receiving increasing attention. In recent years, the Operator Algebraic approach to Conformal Field Theory has provided a simple, model independent description of Boundary Conformal Field Theory on the Minkowski half-plane M+ = {t, x : x > 0} [12,13]. The purpose of this paper is to give a general Operator Algebraic method to build up new Boundary Quantum Field Theory models on M+ . We shall obtain local, Boundary Supported in part by the ERC Advanced Grant 227458 OACFT “Operator Algebras and Conformal Field Theory”, PRIN-MIUR, GNAMPA-INDAM and EU network “Noncommutative Geometry” MRTN-CT-20060031962.
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QFT nets of von Neumann algebras on M+ that are not conformally covariant but only time translation covariant. Motivation for such a construction comes from the papers [10,15,16] where one needs a description of the space of all possible Boundary QFT’s in two dimensions compatible with a given theory in bulk. There however a general framework for Boundary QFT was missing and sample computations were given for certain second quantization unitaries V as in the following. Let us then explain our basic set up. Let A be a local Möbius covariant net of von Neumann algebras on the real line; so we have a von Neumann algebra A(I ) on a fixed Hilbert space H associated with every interval I of R satisfying natural properties: isotony, locality, Möbius covariance with positive energy and vacuum vector (see Appendix B). We identify the real line with the time-axis of the 2-dimensional Minkowski spacetime M. Suppose that V is a unitary on H commuting with the time translation unitary group, such that V A(I+ )V ∗ commutes with A(I− ) whenever I− , I+ are intervals of R and I+ is contained in the future of I− . Then we can define a local, time translation covariant net AV of von Neumann algebras on the half-plane M+ by setting1 AV (O) ≡ A(I− ) ∨ V A(I+ )V ∗ . Here O = I− × I+ is the double cone (rectangle) of M+ given by O ≡ {t, x : x ± t ∈ I± }. The unitaries V as above (that we renormalize for V to be vacuum preserving) form a semigroup that we denote by E(A). The case V = 1 has been studied in [12] and gives a Möbius covariant net A+ on M+ . So a local, Möbius covariant net A and an element V of the semigroup E(A) give rise to a Boundary QFT net AV on the half-plane. Furthermore, if the split property holds for the local Möbius covariant net A (in particular if the vacuum character is summable) the net AV is locally isomorphic to the net A+ on M+ . Our first problem in this paper is to analyze the structure of E(A). We begin by considering the case A is the net A(0) generated by the U (1)-current and second quantization unitaries, i.e. the unitary on the Fock space one obtains by promoting unitaries on the one-particle Hilbert space. It turns out we are to consider the semigroup E(H, T ) of unitaries V on the one-particle Hilbert space, commuting with the translation oneparameter unitary group T , such that V H ⊂ H, where H is the standard real Hilbert subspace associated with the positive half-line (see [11]). By using a one-particle version of Borchers theorem and the standard subspace analysis in [11] we obtain a complete characterization of these unitaries: V ∈ E(H, T ) if and only if V = ϕ(Q) with ϕ the boundary value of a symmetric inner function on the strip Sπ ≡ {z : 0 < z < π } and Q is the logarithm of the one-particle energy operator P, the generator of T . For instance, T¯ (t) ≡ e−it (1/P) gives a one-parameter unitary semigroup in E(H, T ), the only one with negative generator. The inner function structure is well known in Complex Analysis and we collect in Appendix A the basic facts needed in this paper. The above result also characterizes the closed real subspaces K ⊂ H such that T (t)K ⊂ K , t ≥ 0, and so is an abstract analog of (a real version of) the Beurling-Lax theorem [2,8] characterizing the Hilbert subspaces of H ∞ (S∞ ) mapped into themselves by positive translations in Fourier transform, with S∞ the upper complex plane. Now symmetric inner functions S2 on the strip Sπ with the further symmetry S2 (−q) = S2 (q) are called scattering functions and appear in low dimensional Quantum Field 1 If M , M are von Neumannn algebras on the same Hilbert space, M ∨M denotes the von Neumann 1 2 1 2 algebra generated by M1 and M2 .
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Theory (see [17]); in particular every scattering function will give here a local Boundary QFT net on M+ . One may wonder whether our construction is related to Lechner models on the 2-dimensional Minkowski spacetime associated with a scattering function [9], yet at the moment there is no link between the two constructions. Our work continues with the construction of local Boundary QFT models associated with other local conformal nets A on R. We consider any Buchholz-Mack-Todorov local extension A of A(0) (coset models S O(4N )1 /S O(2N )2 ) [6]: every unitary V ∈ E(A(0) ), obtained by second quantization of a unitary V0 = ϕ(Q) ∈ E(H ) as above, extends to a unitary V˜ ∈ E(A), provided ϕ is non-singular in zero. We so obtain other infinite families of local, translation covariant Boundary QFT nets of von Neumann algebras on M+ . A further family of local, translation covariant Boundary QFT nets of von Neumann algebras comes from the Ising model. Also in this case every such model is associated with a symmetric inner function. 2. Endomorphisms of Standard Subspaces We first recall some basic properties of standard subspaces, we refer to [11] for more details. Let H be a complex Hilbert space and H ⊂ H a real linear subspace. The symplectic complement H of H is the real Hilbert subspace H ≡ {ξ ∈ H : (ξ, η) = 0 ∀η ∈ H } so H is the closure of H . A closed real linear subspace H is called cyclic if H +i H is dense in H and separating if H ∩ i H = {0}. H is cyclic if and only if H is separating. A standard subspace H of H is a closed, real linear subspace of H which is both cyclic and separating. Thus a closed linear subspace H is standard iff H is standard. Let H be a standard subspace of H. Define the anti-linear operator S ≡ S H : D(S) ⊂ H → H, where D(S) ≡ H + i H , S : ξ + iη → ξ − iη, ξ, η ∈ H. As H is standard, S is well-defined and densely defined, and clearly S 2 = 1| D(S) . S is ∗ = S . Let a closed operator and indeed its adjoint is given by S H H S = J 1/2 be the polar decomposition of S. Then J is an anti-unitary involution, namely J is antilinear with J = J ∗ = J −1 , and ≡ S ∗ S is a positive, non-singular selfadjoint linear operator with J J = −1 . and J are called the modular operator and the modular conjugation of H . The content of the following relations is the real Hilbert subspace (much easier) analog of the fundamental Tomita-Takesaki theorem for von Neumann algebras: it H = H,
J H = H ,
for all t ∈ R. With a ∈ (0, ∞] we denote by Sa the strip of the complex plane {z ∈ C : 0 < z < a} (so S∞ is the upper plane). Lemma 2.1. Let H be a standard subspace of the Hilbert space H and V ∈ B(H) a bounded linear operator on H. The following are equivalent:
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• VH ⊂ H • J V J 1/2 ⊂ 1/2 V • The map s ∈ R → V (s) ≡ −is V is extends to a bounded weakly continuous function on the closed strip S1/2 , analytic in S1/2 , such that V (i/2) = J V J .
Proof. See [1,11].
Note that, if the equivalent properties of Lemma 2.1 hold, then V (s + i/2) = J V (s)J . Indeed V (s + i/2) = i z V −i z |z=s+i/2 = i z V (s)−i z |z=i/2 = J V (s)J ,
(1)
where we have applied Lemma 2.1 to the unitary V (s). Let H be a standard subspace of the Hilbert space H and assume that there exists a one parameter unitary group T (t) = eit P on H such that • T (t)H ⊂ H for all t ≥ 0 • P > 0. We refer to a pair (H, T ) with H and T as above as a standard pair (of the Hilbert space H). The following theorem is the one-particle analog of Borchers theorem for von Neumann algebras [3]. Theorem 2.2. Let (H, T ) be a standard pair as above. The following commutation relations hold for all t, s ∈ R: is T (t)−is = T (e−2π s t), J T (t)J = T (−t). Proof. See [11].
(2) (3)
Note that (2) gives a positive energy unitary representation of the translation-dilation group on R that we denote by L (L is usually called the “ax +b” group): T (t) is the unitary corresponding to the translation x → x + t on R and is is the unitary corresponding to the dilation x → e−2π s x. We shall say that the standard pair (H, T ) is non-degenerate if the kernel of P is {0}. Now there exists only one irreducible unitary representation of the group L with strictly positive energy, up to unitary equivalence (log P and log satisfies the canonical commutation relations). Therefore, if the standard pair (H, T ) is non-degenerate, the associated representation of L is a multiple of the unique irreducible one and (H, T ) is irreducible iff the associated unitary representation of L is irreducible. Let the standard pair (H, T ) be non-degenerate and (non-zero) irreducible. We can then identify (up to unitary equivalence) H with L 2 (R, dq), Q ≡ log P with the operator of multiplication by q on L 2 (R, dq) and −is by the translation by 2π s on this function space: eit Q : f (q) → eitq f (q),
−is : f (q) → f (q + 2π s).
(4)
In this representation J can be identified with the complex conjugation J f = f¯ and f ∈ H iff f admits an analytic continuation on the strip Sπ such that f (· + a) ∈ L 2 for every a ∈ (0, π ) with boundary values satisfying f (q + iπ ) = f¯(q). We now describe the endomorphisms of the standard pair (H, T ), namely the semigroup E(H, T ) of unitaries V of H commuting with T such that V H ⊂ H (sometimes
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abbreviated E(H )). We denote by P the generator of T and begin with the irreducible case. With a > 0, we denote by H∞ (Sa ) the space of bounded analytic functions on the strip Sa . If ψ ∈ H∞ (Sa ) then the limit limε→0+ ψ(q +iε) exists for almost all q ∈ R and defines a function in L ∞ (R, dq) that determines ψ (and similarly on the line z = ia if a < ∞). Theorem 2.3. Assume the standard pair (H, T ) of H to be irreducible and let V be a bounded linear operator on H. The following are equivalent: (i) V commutes with T and V H ⊂ H ; (ii) V = ψ(Q), where Q ≡ log P and ψ ∈ L ∞ (R, dq) is the boundary value of a ¯ function in H∞ (Sπ ) such that ψ(q + iπ ) = ψ(q), for almost all q ∈ R. In this case V is unitary, i.e. V ∈ E(H ), iff |ψ(q)| = 1 for almost all q ∈ R, namely ψ is an inner function on Sπ , see Appendix A.2 Proof. (i) ⇒ (ii): With and J the modular operator and the modular conjugation of H we have the commutation relations (2, 3). As the standard pair (H, V ) is assumed to be irreducible, the associated positive energy unitary representation of L is irreducible. Therefore the von Neumann algebra generated by {T (t) : t ∈ R} is maximal abelian in B(H). As V commutes with T , setting Q ≡ log P we have V = ψ(Q) for some Borel complex function ψ on R. By (2, 3) we then have −is ψ(Q)is = ψ(Q + 2π s), ¯ J ψ(Q)J = ψ(Q)
(5) (6)
As V H ⊂ H , by Lemma 2.1 and Eq. (1) the function V (s) ≡ −is ψ(Q)is = ψ(Q + 2π s) extends to a bounded continuous function on the strip S1/2 , analytic in S1/2 , and ¯ V (s + i/2) = J V (s)J = J ψ(Q + 2π s)J = ψ(Q + 2π s) = V (s)∗ . We now identify H with L 2 (R, dq) with Q ≡ log P and as in (4). Then V is identified with the multiplication operator Mψ : f ∈ L 2 (R, dq) → ψ f ∈ L 2 (R, dq). It then follows by Lemma A.4 that ψ is the boundary value of a function ψ ∈ H∞ (Sπ ) ¯ and ψ(q + iπ ) = ψ(q) for almost all q ∈ R. (ii) ⇒ (i): Conversely, let V = ψ(Q), where ψ is the boundary value of a function ¯ in H∞ (Sπ ) with ψ(q + iπ ) = ψ(q) for almost all q ∈ R. Then clearly V commutes with T , Eqs. (5,6) hold, the function V (s) = −is ψ(Q)is is the boundary value of a bounded continuous function on S1/2 , analytic on S1/2 , and V (i/2) = ψ(Q + iπ ) = ¯ ψ(Q) = J ψ(Q)J = J V J , so V H ⊂ H by Lemma 2.1. Clearly V is unitary iff |ψ(q)| = 1 for almost all q ∈ R. If (H, T ) is reducible (and non-degenerate so Q = log P is defined) the proof of (ii) ⇒ (i) in Th. 2.3 remains valid, so the implication still holds true. Note that E(A, T ) is commutative if (H, T ) is irreducible: it is isomorphic to the semigroup of inner functions. It will be useful to formulate Th. 2.3 in terms of functions of P and we do this in the unitary case. 2 In the scattering context the variable q is usually denoted by θ , the rapidity.
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Corollary 2.4. Let (H, T ) be an irreducible standard pair and V a unitary on H. Then V ∈ E(H, T ) iff V = ϕ(P) with ϕ the boundary value of a symmetric inner function on S∞ such that ϕ(− p) = ϕ( ¯ p), p ≥ 0. Proof. Easy consequence of the conformal identification of S∞ and Sπ by the logarithm function. We now describe the Lie algebra of E(H, T ), i.e. the generators of the one-parameter semigroups of unitaries in E(H, T ). Corollary 2.5. Let (H, T ) be a standard pair of the Hilbert space H and P the generator of T . Let V (s) = eis A be a one-parameter unitary group of H. Then V (s) ∈ E(H, T ) for all s ≥ 0 if A = f (P), where f : R → R is an odd function f (− p) = − f ( p) that admits an analytic continuation in the upper plane S∞ with f (z) ≥ 0. Conversely, if (H, T ) is irreducible, every unitary one-parameter group V (s) on H such that V (s) ∈ E(H, T ) for all s ≥ 0 has the form V (s) = eis f (P) with f as above. The proof of Corollary 2.5 follows from the analysis in Sect. A of the semigroup of inner functions; we shall write the explicit form of f and of the inner functions ψ in Theorem 2.3. Example. If (H, T ) is a non-degenerate standard pair, the self-adjoint operator − P1 belongs to the Lie algebra of E(H, T ), namely e−it (1/P) H ⊂ H , t ≥ 0, with P the generator of T . We may now describe the reducible case. Let ( H˜ , T˜ ) be a non-zero, non-degenerate stan˜ Since, up to unitary equivalence, there exists only one dard pair on the Hilbert space H. non-zero, non-degenerate irreducible standard pair (H, T ), the pair ( H˜ , T˜ ) is the direct sum of copies (H, T ). In other words we may write H˜ = nk=1 Hk , H˜ = nk=1 Hk , n T˜ = k=1 Tk , for some finite or infinite n, where every Hilbert space Hn is identified with the same Hilbert space H and each pair (Hk , Tk ) is identified with (H, T ). With this identification we have: Theorem 2.6. A unitary V˜ belongs to E( H˜ , T˜ ) if and only if V˜ is an n × n matrix (Vhk ) with entries in B(H) such that Vhk = ϕhk (P). Here ϕhk : R → C are complex Borel functions such that (ϕhk ( p)) is a unitary matrix for almost every p > 0, each ϕhk is the boundary value of a function in H(S∞ ) and is symmetric, i.e. ϕ¯ hk ( p) = ϕhk (− p). Proof. Assume that V˜ belongs to E( H˜ , T˜ ). We may write H˜ = H ⊗ 2 and T˜ = T ⊗ 1; here 2 is the Hilbert space of n-tuples (finite n) or of countable square summable sequences (n = ∞). As V˜ commutes with T˜ , V˜ belongs to the von Neumann algebra {T } ⊗ B( 2 ) which coincides with {T } ⊗ B( 2 ) because T generates a maximal abelian von Neumann algebra. Therefore V˜ = (Vhk ), where Vhk = ϕhk (P) for some complex functions ϕhk : R → C and (ϕhk ( p)) is a unitary matrix for (almost) every p because V˜ is unitary. ˜ = ⊗ 1 and J˜ = J ⊗ 1 are constant diagonal matrices, therefore by Now Eqs. (2, 3) we have is ϕhk (P)−is = ϕhk (e−2π s P), J ϕhk (P)J = ϕ¯ hk (P).
(7) (8)
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˜ 1/2 V˜ ⊃ J˜ V˜ J˜ ˜ 1/2 so By Lemma 2.1 we have (ϕhk (−P)) = −is ϕhk (P)is |i=1/2 = J˜ V˜ J˜ = (J ϕhk (P)J ) = (ϕ¯ hk (P)) . Therefore ϕ¯ hk ( p) = ϕhk (− p). Clearly the matrix operator norm ||ϕ(z)|| is bounded, indeed ||ϕ(z)|| ≤ 1. We may now reverse the above proof to get the converse statement. We only have to check that if each ϕhk (z) is bounded then ||ϕ(z)|| ≤ 1. If n is finite this is true because each ϕhk (z) is bounded iff ||ϕ(z)|| is bounded and in this case ||ϕ(z)|| ≤ 1 by the maximum modulus principle. If n = ∞ we then note that the operator norm of each finite corner matrix must be bounded by 1 so ||ϕ(z)|| ≤ 1 also in this case. We note the following proposition: when combined with Cor. 2.4 or Thm. 2.6, it gives an abstract, (real) analog of the Beurling-Lax theorem [2,8], see also [14]. Proposition 2.7. Let (H, T ) be a non-degenerate standard pair of the Hilbert space H. A standard subspace K ⊂ H satisfies T (t)K ⊂ K for t ≥ 0 if and only if K = V H for some V ∈ E(H, T ). In particular, if (H, T ) is irreducible, K = ϕ(P)H with P the generator of T and ϕ a symmetric inner function on S∞ . Proof. Let U H and U K be the representations of L associated with (H, T ) and (K , T ). By assumptions U H and U K agree on the translation one-parameter group, in particular (K , T ) is non-degenerate too. Moreover U H and U K have the same multiplicity because this is also the multiplicity of the abelian von Neumann algebra generated by T . Therefore U H and U K are unitarily equivalent and indeed also the associated anti-unitary representations of the group generated by L and the reflection x → −x are unitarily equivalent, namely there exists a unitary V ∈ B(H) such that U K (g) = V U H (g)V ∗ , g ∈ L,
V JK V ∗ = J H ,
and in particular V commutes with T (t). Then V S H V ∗ = V J H H V ∗ = JK K = SK , 1/2
1/2
hence V H = K and we conclude that V ∈ E(H, T ). The converse statement that if V ∈ E(H, T ) then K ≡ V H satisfies T (t)K ⊂ K for t ≥ 0 is obvious. In the irreducible case K = ϕ(P)H by Cor. 2.4. Note that the unitary V in Prop. 2.7 is not unique (but in the irreducible case V is unique up to a sign). On the other hand the unitary ≡ JK J H is a canonical unitary associated with the inclusion K ⊂ H and commutes with T because J H T (t)J H = T (−t) and JK T (t)JK = T (−t), so ∈ E(H, T ). Clearly = V JH V ∗ JH . In the irreducible case V = ϕ(P) for some symmetric inner function ϕ so J H V J H = V ∗ and = V 2 . We now consider the von Neumann algebraic case.
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Corollary 2.8. Let M be a von Neumann algebra on a Hilbert space H with a cyclic and separating vector and T (t) = eit P a one-parameter unitary group on H, with positive generator P, such that T (t)M T (−t) ⊂ M for all t ≥ 0. Suppose that the kernel of P is C . If V is a unitary on H commuting with T such that V M V ∗ ⊂ M, then V |H0 = (ϕhk (P0 )), where (ϕhk ( p)) is a matrix of functions as in Theorem 2.6. Here H0 is the orthogonal complement of in H and P0 = P|H0 . Proof. With Msa the self-adjoint part of M, the closed linear subspace H ≡ Msa is a standard subspace of H and V H ⊂ H . Thus (H0 , T ) is a non-degenerate standard pair of H0 , where H0 = H R and T0 (t) ≡ T (t)| H0 . By Theorem 2.6 we then have V |H0 = (ϕhk (P0 )) with (ϕhk ) a matrix of functions as in that theorem. 3. Constructing Boundary QFT on the Half-plane In this section we introduce the unitary semigroup E(A) associated with a local Möbius covariant net A. By generalizing the construction in [12], each element in E(A) produces a Boundary QFT net of local algebras on the half-plane M+ ≡ {t, x ∈ R2 : x > 0}. 3.1. The semigroup E(A). Let A be a local Möbius covariant net of von Neumann algebras on R (see Appendix B); so we have an isotonous map that associates a von Neumann algebra A(I ) on a fixed Hilbert space H to every interval or half-line I of R. A is local namely A(I1 ) and A(I2 ) commute if I1 and I2 are disjoint intervals. Denote by T the one-parameter unitary translation group on H. Then T (t)A(I ) T (−t) = A(I + t), T has positive generator P and T (t) = , where is the vacuum vector, the unique (up to a phase) T -invariant vector. By the Reeh-Schlieder theorem is cyclic and separating for A(I ) for every fixed interval or half-line I . Lemma 3.1. Let V be a unitary on H commuting with T . The following are equivalent: (i) V A(I2 )V ∗ commutes with A(I1 ) for all intervals I1 , I2 of R such that I2 > I1 (I2 is contained in the future of I1 ). (ii) V A(a, ∞)V ∗ ⊂ A(a, ∞) for every a ∈ R. (iii) V A(0, ∞)V ∗ ⊂ A(0, ∞). Proof. Clearly (ii) ⇔ (iii) by translation covariance as V commutes with T . Moreover (ii) ⇒ (i) because V A(I2 )V ∗ ⊂ V A( I˜2 )V ∗ ⊂ A( I˜2 ), where I˜2 is the smallest right half-line containing I2 . Finally, assuming (i), by additivity we have that V A(0, ∞)V ∗ commutes with A(−∞, 0), so (iii) follows by duality: V A(0, ∞)V ∗ ⊂ A(−∞, 0) = A(0, ∞). Note that a unitary V in the above Lemma 3.1 fixes up to a phase as it commutes with T . We will assume that indeed V = . Let A be a local Möbius covariant net of von Neumann algebras on R on the Hilbert space H. The unitaries V on H satisfying the equivalent conditions in Lemma 3.1, normalized with V = , form a semigroup that we denote by E(A) (or E(A, T )).
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Note that E(A, T ) ⊂ E(H, T ), where H ≡ A(0, ∞)sa . As a consequence of Corollary 2.8 every unitary V in E(A) must have the form V |H0 = (ϕhk (P0 )) there described on the orthogonal complement H0 of . Examples of unitaries V in E(A) are easily obtained by taking either V to implement an internal symmetry (first kind gauge group element), namely V A(I )V ∗ = A(I ) for all intervals I , or by taking V = T (t) a translation unitary with t ≥ 0. We give now an example of V in E(A) not of this form. Example. (See Sect. 8.2 of [5].) Let O be a double cone in the Minkowski spacetime Rd+1 with d odd. We denote here by A(O) the local von Neumann algebra associated with O by the d + 1-dimensional scalar, massless, free field. With I an interval of the time-axis {x ≡ t, x1 , . . . xd : x1 = · · · = xd = 0} we set A0 (I ) ≡ A(O I ), where O I is the double cone I ⊂ Rd+1 , the causal envelope of I . Then A0 is a local translation covariant net on R. (Indeed A0 extends to a local Möbius covariant net on S 1 .) With U the translation unitary group of A, the translation unitary group of A0 is T (t) = U (t, 0, . . . , 0). Let V ≡ U (x) be the unitary corresponding to a positive time-like or light-like translation vector x = t, x1 , . . . xd for A, thus t 2 ≥ dk=1 xk2 . Then V ∈ E(A0 , T ). Indeed V A0 (0, ∞)V ∗ = V A(V+ )V ∗ = A(V+ + x) ⊂ A(V+ ) = A0 (0, ∞), where V+ denotes the forward light cone. The net A0 is described as follows: A0 =
∞
Nd (k + 1)A(k) ,
k=0
where A(k) is the local Möbius covariant net on S 1 associated with the k th -derivative of the U (1)-current and the multiplicity factor Nd k + d−1 is the dimension of the space 2 d of harmonic spherical functions of degree k on R . Before further considerations we characterize the unitaries in E(A) implementing internal symmetries. Proposition 3.2. Let A be a local Möbius covariant net of von Neumann algebras on R and U the associated unitary representation of the Möbius group. Then V ∈ E(A) commutes with U if and only if V implements an internal symmetry of A. Proof. We know that if V implements an internal symmetry then V commutes with U as a consequence of the Bisognano-Wichmann property, see [11]. Conversely if V commutes with U then V A(I )V ∗ ⊂ A(I ) for every interval I of S 1 because the Möbius group acts transitively on open intervals of S 1 ; in particular also V A(I )V ∗ ⊂ A(I ), thus V A(I )V ∗ ⊃ A(I ) by Haag duality, namely V implements an internal symmetry. 3.2. Translation covariant Boundary QFT. Consider now the 2-dimensional Minkowski spacetime M. Let I1 , I2 be intervals of time-axis such that I2 > I1 and let O = I1 × I2 be the double cone (rectangle) of M+ associated with I1 , I2 , namely a point t, x belongs to O iff x − t ∈ I1 and x + t ∈ I2 .
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We shall say that a double cone O = I1 × I2 of M+ is proper if it has positive distance from the time axis x = 0, namely if the closures of I1 and I2 have empty intersection. We shall denote by K+ the set of proper double cones of M+ . A local, (time) translation covariant Boundary QFT net of von Neumann algebras on M+ on a Hilbert space H is a triple (B+ , U, ), where • B+ is a isotonous map O ∈ K+ → B+ (O) ⊂ B(H), where B+ (O) is a von Neumann algebra on H; • U is a one-parameter group on H with positive generator P such that U (t)B+ (O)U (−t) = B+ (O + t, 0), O ∈ K+ , t ∈ R; • ∈ H is a unit vector such that C are the U -invariant vectors and is cyclic and separating for B+ (O) for each fixed O ∈ K+ ; • B+ (O1 ) and B+ (O2 ) commute if O1 , O2 ∈ K+ are spacelike separated. 3.3. A construction by an element of the semigroup E(A). Let now A be a local, Möbius covariant net of Neumann algebras on the time-axis R of M+ . With V a unitary in E(A) we set AV (O) ≡ A(I1 ) ∨ V A(I2 )V ∗ , where I1 , I2 are intervals of time-axis such that I2 > I1 and O = I1 × I2 . Proposition 3.3. AV is a local, translation covariant Boundary QFT net of von Neumann algebras on M+ . Proof. Isotony of AV is obvious. Locality means that AV (O1 ) commutes elementwise with AV (O2 ) if the double cone O2 = I3 × I4 is contained in the spacelike complement of the double cone O1 = I1 × I2 . Say O2 is contained in the right spacelike complement of O1 . Then I4 > I2 > I1 > I3 . Now V A(I4 )V ∗ commutes with V A(I2 )V ∗ by the locality of A and with A(I1 ) because V ∈ E(A); analogously A(I3 ) commutes with A(I1 ) by locality and with V A(I2 )V because V ∈ E(A). Therefore A(I3 ) ∨ V A(I4 )V ∗ and A(I1 ) ∨ V A(I2 )V ∗ commute. Finally translation covariance with respect to T follows at once because V commutes with T by assumptions. If V = 1 the net AV is the net A+ in [12]. Corollary 3.4. Let V1 , V2 ∈ E(A). The following are equivalent: (i) AV1 = AV2 ; (ii) V2 = V1 V with V implementing an internal symmetry of A; (iii) V1 A(0, ∞)V1∗ = V2 A(0, ∞)V2∗ . Proof. (iii) ⇔ (ii) follows by Lemma 3.1 and (ii) ⇒ (i) is immediate. (i) ⇒ (iii): note that the von Neumann algebra Vi A(−∞, 0)Vi∗ is generate by the von Neumann algebras AVi (O) as O = I1 × I2 ∈ K+ varies with I1 , I2 ⊂ (−∞, 0); therefore AV1 = AV2 ⇒ V1 A(−∞, 0)V1∗ = V2 A(−∞, 0)V2∗ ⇒ V1 A(0, ∞)V1∗ = V2 A(0, ∞)V2∗ by duality.
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So we have constructed a map: local Möb-covariant net A on R & V ∈ E(A)
→
BQFT net AV on M+
and, given A, the map V ∈ E(A) → AV is one-to-one modulo internal symmetries. We shall say that two nets B1 , B2 on M+ , acting on the Hilbert spaces H1 and H2 , are locally isomorphic if for every proper double cone O ∈ K+ there is an isomorphism O : B1 (O) → B2 (O) such that O˜ |B1 (O) = O if O, O˜ ∈ K+ , O ⊂ O˜ and U2 (t)O (X )U2 (−t) = O+t (U1 (t)XU1 (−t)),
X ∈ B1 (O) ,
with U1 and U2 the corresponding time translation unitary groups on H1 and H2 . Proposition 3.5. Let A be a local Möbius covariant net of Neumann algebras on R with the split property. If V is a unitary in E(A) the net AV is locally isomorphic to A+ . Proof. Let I2 > I1 be intervals with disjoint closures and O = I1 × I2 . Let I˜2 be the smallest right half-line containing I2 . By the split property there is a natural isomorphism : A(I1 ) ∨ A( I˜2 ) → A(I1 ) ⊗ A( I˜2 ) with (ab) = a ⊗ b for a ∈ A(I1 ), b ∈ A( I˜2 ). Then the commutative diagram A+ (O) ⊂ A(I1 ) ∨ A( I˜2 ) ⏐ ⏐ O
−−−−→
A(I1 ) ⊗ A( I˜2 ) ⏐ ⏐
id⊗AdV
AV (O) ⊂ A(I1 ) ∨ V A( I˜2 )V ∗ −−−−→ A(I1 ) ⊗ V A( I˜2 )V ∗
defines a natural isomorphism O : A+ (O) → AV (O) and the family {O : O ∈ K+ } has the required consistency properties. As an immediate consequence, if Vt is a one-parameter semigroup of unitaries in E(A), the family AVt gives a deformation of the conformal net A+ on M+ with translation covariant nets on M+ that are locally isomorphic to A+ . Let again A(0) be the Möbius covariant net on R associated with the U (1)-current. In other words A(0) is generated by the U (1)-current j, (0) j (x) f (x)dx : supp f ⊂ I , A (I ) = W ( f ) ≡ exp −i and similarly A(k) by the net generated by the k-derivative of j. Then A(k) is the net obtained by second quantization of the irreducible, positive energy representation U (k+1) of Möbius group with lowest weight k + 1 or, equivalently, A(k) is the net associated with the irreducible Möbius covariant net of standard subspaces of the one-particle Hilbert space associated with U (k+1) , see [7]. With V0 a unitary on the one-particle Hilbert space H0 we denote by (V0 ) the second quantization promotion to the Bosonic Fock space over H0 . We shall refer to a unitary of the form (V0 ) as a second quantization unitary.
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Theorem 3.6. A second quantization unitary (V0 ) belongs to E(A(k) ) if and only if V0 = ϕ(P (k) ). Here P (k) is the generator of the translation unitary group on the oneparticle Hilbert space H0 and ϕ : [0, ∞) → C is the boundary value of a symmetric inner function on S∞ as in Corollary 2.4. Proof. With H (k) (0, ∞) the standard subspace of H0 associated with (0, ∞), the von Neumann algebra A(0, ∞) is generated by the Weyl unitaries W (h) as h varies in H (k) (0, ∞) (see [11]). As (V0 )W (h) (V0 )∗ = W (V0 h), we immediately see that V0 ∈ E(H (k) (0, ∞)) ⇒ (V0 ) ∈ E(A(k) ). The converse implication follows because W (h) ∈ A(0, ∞) if and only if h ∈ H (k) (0, ∞) (e.g. by Haag duality). Note that (V0 ) belongs to a one-parameter semigroup of E(A(k) ) if ϕ is a singular inner (k) function (see Cor. A.2) so it provides a deformation of the net A+ ; this is not the case if ϕ is a Blaschke product. 3.4. Families of models. We now construct elements of E(A) with A a local extension of the U (1)-current net A(0) ; so we get further families of local, translation covariant Boundary QFT nets of von Neumann algebras on M+ . For convenience we regard A(0) as a net on R. The local extensions of A(0) are classified in [6]. Such an extension A is the crossed product of A(0) by a localized automorphism β. Recall that β acts on Weyl unitaries by β (W (h)) = e−i
(x)h(x)dx
W (h)
for every localized element h of the one-particle space, say h ∈ S(R) and h has zero integral, where S(R) denotes the Schwartz real function space, see [6,7]. In other words β is associated with the action on the U (1)-current j (x) → j (x) + (x) A(0)
and A is generated by and a unitary U implementing β, see below. Here ∈ S(R) and the sector class of β (i.e. the class of β modulo inner auto1 morphisms) is determined by the charge g ≡ 2π (x)dx. β is inner iff the charge of is zero x and in this case β = AdW (L), where L is the primitive of , namely L(x) = −∞ (s)ds. For the extension A to be local the spin N = 21 g 2 , given by the Sugawara construction, is to be an integer. We take with support in (0, ∞) so that β is localized in (0, ∞) and, in particular, β gives rise to an automorphism of the von Neumann algebra A(0) (0, ∞). As said, A(0) acts on the Bose Fock space on the one particle Hilbert space H and H carries the irreducible unitary representation U (1) of the Möbius group with lowest weight 1. Therefore we may identify H with the Hilbert space K1 = L 2 (R+ , pd p) with the known lowest weight 1 unitary representation of the Möbius group; S(R) embeds into ∞K1 (thus in H) by Fourier transform and the scalar product determined by ( f, g) = 0 p fˆ( p)g( ˆ p)d p, f, g ∈ S(R), see [7]. Let H (0, ∞) be the standard real Hilbert subspace of H associated with (0, ∞). Then, in the configuration space, a function h on Rbelongs to H (0, ∞) if it is real, ∞ ˆ p)|2 d p < ∞. supph ⊂ [0, ∞) and its Fourier transform hˆ satisfies 0 | p||h( Let ϕ be a symmetric inner function ϕ on S∞ and set V0 = ϕ(P) with P the positive generator of the time-translation unitary one-parameter group on the one-particle Hilbert
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space. By Cor. 2.4 the unitary V0 belongs to E(H (0, ∞)). So V ≡ (V0 ) ∈ E(A(0) ) and we denote by η the endomorphism of A(0) (0, ∞) implemented by V . 2 is locally integrable in zero (Hölder continuity at We shall assume that |ϕ( p)−1| | p| 0). Also, as ϕ(0) = ±1, replacing ϕ by −ϕ if necessary we may and do assume that ϕ(0) = 1. We formally set 1 = V0 ; rigorously 1 is defined to be the function onR whose ˆ ˆ ˆ Fourier transform is ϕ( p) ( p). Clearly (0) = 1 (0), namely 1 (x)dx = (x)dx; moreover 1 is real because is real and ϕ is symmetric. Note that the support of 1 is contained in [0, ∞) by the Paley-Wiener theorem as ϕ ∈ H (S∞ ). So − 1 has zero charge and belongs to H (0, ∞). In the following β is the localized automorphism of A(0) associated with . Denote by L 1 the primitive of 1 . Note that the primitive L − L 1 of − 1 belongs to H (0, ∞), indeed
∞
∞ |1 − ϕ( p)|2 ˆ 2 ˆ ˆ | ( p)|2 d p < ∞. | p|| L( p) − L 1 ( p)| d p = | p| 0 0 Lemma 3.7. On A(0) (0, ∞) we have η · β = Adz · β · η, where the unitary z belongs to A(0) (0, ∞), indeed z = W (L − L 1 ). Proof. For every h ∈ H (0, ∞) we have η · β(W (h)) = η e−i (x)h(x)dx W (h) = e−i (x)h(x)dx W (V0 h) and Adz · β · η(W (h)) = Adz · β(W (V0 h)) = e−i = e−i =e
−i
1 (x)V0 h(x)dx (x)h(x)dx
W (V0 h)
(x)V0 h(x)dx Adz (W (V0 h)) −i V0 (x)V0 h(x)dx =e W (V0 h)
W (V0 h).
With A a local extension of A(0) , the von Neumann algebra A(0, ∞) is generated by A(0) (0, ∞) and a unitary U implementing β, namely β(a) = U aU ∗ , a ∈ A(0) (0, ∞); finite sums nk=−n ak U k , ak ∈ A(0) (0, ∞), are dense in A(0, ∞). Proposition 3.8. η extends to a vacuum preserving endomorphism η˜ of A(0, ∞) determined by η(U ˜ ) = zU with z as in Lemma 3.7. Proof. Let N be the subalgebra of A(0, ∞) of finite sums { k ak U k } with ak ∈ A(0) (0, ∞). It is immediate to check that the map η˜ 0 η˜ 0 : ak U k → η(ak )(zU )k , ak ∈ A(0) (0, ∞) , k
k
is an endomorphism of N . η˜ 0 preserves the vacuum conditional expectation k ak U k → a0 , thus the vacuum state. Moreover η˜ 0 (N ) is cyclic on the vacuum vector because
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η˜ 0 (N ) contains the closure of η(A(0) (0, ∞)) = V A(0) (0, ∞) , namely the Hilbert space of A(0) , and is U -invariant because U ∈ η˜ 0 (N ). Then we have a unitary V˜ determined by V˜ X = η˜ 0 (X ) ,
X ∈N ,
(9)
that implements η˜ 0 . Therefore η˜ = AdV˜ is a normal extension of η˜ 0 to A(0) (0, ∞). Proposition 3.9. The unitary V˜ defined by Eq. (9) belongs to E(A). Proof. By construction V implements the endomorphism η˜ of A(0) (0, ∞) and V˜ = . We only need to show that V˜ commutes with the translation unitary group T of A, namely ˜ t ≥ 0, on A(0) (0, ∞) with τt ≡ AdT (t). Since V ∈ E(A(0) ), we have η˜ · τt = τt · η, η · τt = τt · η on A(0) (0, ∞) so it suffices to show that ητt (U ) = τt η(U ). We have τt (U ) = u ∗t U,
(10)
where u t is a unitary τ -cocycle Adu t · τt · β = β · τt , actually u t = W (L − L t ), where t (x) ≡ (x − t) and L t is the primitive of t . Therefore ητ ˜ t (U ) = τt η(U ˜ ) means η(u ∗t )zU = τt (z)u ∗t U and we need to show that zu t = η(u t )τt (z). Indeed we have zu t = W (L 1 − L)W (L − L t ) = W (L 1 − L 1t )W (L 1t − L t ) = W (V0 (L − L t ))W (L 1t − L t ) = η(u t )τt (z) , where L 1t is the primitive of 1t , so the proof is complete. Corollary 3.10. Let ϕ be a symmetric inner function on S∞ which is Hölder continuous at 0 as above with ϕ(0) = 0, and N ∈ N be an integer. There is a local, translation covariant Boundary QFT net of von Neumann algebras on M+ associated with ϕ and N . Proof. Given N ∈ N, the extension A N of the U (1)-current net with charge g such that 1 2 ˜ 2 g = N is local [6] and ϕ determines an element V ∈ E(A N ) as above. Hence we have a Boundary QFT net by the above construction. Recall for example the structure of the net A N (cf. [6]): A1 is associated with the level 1 su(2)-Kac-Moody algebra with central charge 1, A2 is the Bose subnet of the free complex Fermi field net, A3 appears in the Z4 -parafermion current algebra analyzed by Zamolodchikov and Fateev, and in general A N is a coset model S O(4N )1 /S O(2N )2 . 3.4.1. Case of the Ising model. One further family of local Boundary QFT nets comes by considering the Ising model conformal net AIsing on R, namely the Virasoro net with central charge c = 1/2. AIsing is the fixed point net of F under the Z2 gauge group action, where F is the twisted-local net of von Neumann algebras on R generated by a real Fermi field. The one-particle Hilbert space of F in H carries the irreducible unitary spin 1/2 representation of the double cover of the Möbius group and F acts on the Fermi Fock space over H. With T the translation unitary group on H, the standard subspace H of H associated with (0, ∞) is the one associated with the unique irreducible, non-zero standard pair (H, T ).
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With P the generator of T , then every symmetric inner function ϕ on S∞ gives a unitary V0 = ϕ(P) on H mapping H into itself. The Fermi second quantization V of V0 then satisfies V F(0, ∞)V ∗ ⊂ F(0, ∞) and commutes with translations. Moreover V commutes with the Z2 gauge group unitary (the Fermi second quantization of −1) so it restricts to a unitary V− on the AIsing Hilbert subspace and V− AIsing (0, ∞)V−∗ ⊂ AIsing (0, ∞) namely V− ∈ E(AIsing ). By applying our construction we conclude: Proposition 3.11. Given any symmetric inner function ϕ on S∞ , there is a local, translation covariant Boundary QFT associated with AIsing as above. Acknowledgements. The first named author is grateful to K.-H. Rehren for comments.
Appendix A. One-Parameter Semigroups of Inner Functions We recall and comment on basic facts about inner functions, see [14]. Consider the disk D ≡ {z ∈ C : |z| < 1} and the Hardy space H∞ (D) of bounded analytic functions on D. Every ϕ ∈ H∞ (D) has a radial limit f ∗ (eiθ ) ≡ limr →1− ϕ(r eiθ ) almost everywhere with respect to the Lebesgue measure of ∂D and defines a function ϕ ∗ ∈ L ∞ (∂D, dθ ), where ∂D is the boundary of D. As ||ϕ ∗ ||∞ = sup{ϕ(z) : z ∈ D} by the maximum modulus principle, and in particular ϕ ∗ determines ϕ, we may identify H∞ (D) with a Banach subspace of L ∞ (∂D, dθ ). We shall then denote ϕ ∗ by the same symbol ϕ if no confusion arises. Given a sequence of elements an ∈ D such that ∞ n=1 (1 − |an |) < ∞ , the function B(z) ≡
∞
Ban (z)
n=1 z−a is called the Blaschke product. Here Ba (z) is the Blaschke factor |a| a 1−az ¯ if a = 0 and B0 (z) ≡ z. This product converges uniformly on compact subsets of the D, and thus B is a holomorphic function on the disk. Moreover |B(z)| ≤ 1 for z ∈ D. An inner function ϕ on D is a function ϕ ∈ H∞ (D) such that |ϕ(z)| = 1 for almost all z ∈ ∂D.3 A Blaschke product is an inner function. Indeed, up to a phase, Ba (z) is the only inner function with a simple zero in a (thus the Möbius transformation mapping a to 0) and B(z) the only inner function on D that has zeros exactly at {an }, with multiplicity. If an inner function ϕ has no zeros on D, then ϕ is called a singular inner function. ϕ is an inner function if and only if π iθ e +z iθ dμ(e ϕ(z) = α B(z) exp − ) , (11) iθ −π e − z
where μ is a positive, finite, Lebesgue singular measure on ∂D, B(z) is a Blaschke product and α is a constant with |α| = 1. The decomposition is unique. Note that all the zeros of ϕ come from the Blaschke product so ϕ is singular if and only if B is the identity. 3 Every function in H∞ (D) factorizes into the product of an inner function and an outer function [14]. We don’t need this fact here.
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Note that the inner functions form a (multiplicative) semigroup and the singular inner functions form a sub-semigroup. We now consider a one-parameter semigroup {ϕt , t ≥ 0} of inner functions. Namely ϕt is an inner function for every t ≥ 0, and ϕt+s = ϕt ϕs . Clearly ϕ0 = 1. We require that the map t ∈ [0, ∞) → ϕt∗ is weak∗ continuous in L ∞ (∂D, dθ ). This is equivalent to the weak operator continuity (hence to the strong operator continuity) of the one-parameter unitary group Mϕt∗ on L 2 (∂D, dθ ), where Mϕt∗ is the multiplication operator by ϕt∗ on L 2 (∂D, dθ ). Proposition A.1. Let ϕt be a one-parameter semigroup of inner functions on D. Then: a) ϕt (z) → 1 as t → 0 uniformly on compact subsets of D, b) every ϕt is singular, c) ϕt (z) = eit f (z) where f is a analytic function on D with f (z) ≥ 0 such that the radial limit function of f on ∂D exists almost everywhere and is real. Proof. a) By the weak∗ continuity of ϕt we have ϕt (z)g(z)dz −→ ∂D
∂D
g(z)dz
(12)
as t → 0, for all g ∈ L 1 (∂D, dθ ). Let z 0 ∈ D. Since ϕt ∈ H∞ (D) the value ϕt (z 0 ) is given by the Cauchy integral ϕt (z) 1 ϕt (z 0 ) = dz 2πi ∂ D z − z 0 1 1 so, choosing g(z) ≡ 2πi z−z 0 in (12), we see that ϕt (z 0 ) → 1 as t → 0. As the family of analytic functions {ϕt : t > 0} is bounded, hence normal, the convergence is indeed uniform on compact subsets of D. b) Fix z 0 ∈ D and suppose z 0 is a zero of some ϕt . Let t0 ≡ inf{t > 0 : ϕt (z 0 ) = 0}. Since ϕt (z 0 ) → 1 as t → 0 we have t0 > 0. Write now t = ns with s ∈ (0, t0 ) and n an integer. Then ϕt (z 0 ) = ϕns (z 0 ) = ϕs (z 0 )n = 0, so we conclude that ϕt never vanishes in D for every t > 0. c) For a fixed z ∈ D, the map t → ϕt (z) is a one-parameter semigroup of complex numbers with modulus less than one, therefore ϕt (z) = eit f (z) for a complex number f (z) such that f (z) ≥ 0. Now, by point a), on any given compact subset of D, we have |ϕt (z) − 1| < 1 for a sufficiently small t > 0; thus it f (z) = log ϕt (z) showing that f is an analytic function on D. This also shows that f (z) has a real radial limit to almost all points of ∂D.
We shall say that ϕ ∈ H∞ (D) is symmetric if ϕ(z) = ϕ(¯ ¯ z ) for all z ∈ D, thus iff ϕ is real on the interval (−1, 1) (ϕ is real analytic). Of course ϕ is symmetric iff the equality ϕ(z) = ϕ(¯ ¯ z ) holds almost everywhere on the boundary ∂D. Note that a Blaschke factor Ba is symmetric iff a is real, thus a Blaschke product is symmetric iff the non-real zeros come in pairs, with multiplicity. We now determine all semigroups of inner functions. Corollary A.2. Every one-parameter semigroup of inner functions ϕt on D is given by π iθ e +z iθ ) , (13) ϕt (z) = eitλ exp −t dμ(e iθ −π e − z
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where μ is a positive, finite measure on ∂D which is singular with respect to the Lebesgue measure and λ ∈ R is a constant. Conversely, given a finite, positive, Lebesgue singular measure μ and a constant λ ∈ R, formula (13) defines a one-parameter semigroup of inner functions on D. All functions ϕt are symmetric if and only if λ = 0 and μ(eiθ ) = μ(e−iθ ). Proof. Let ϕ be a semigroup of inner functions ϕt on D. By point c) in Prop. A.1 every ϕt is singular for every t ≥ 0. By formula (11) we then have π iθ e +z iθ ϕt (z) = α(t) exp − dμt (e ) , iθ −π e − z where α(t) is a complex number of modulus one and μt is a Lebesgue singular measure. Clearly α is a semigroup, so α(t) = eitλ for a real constant λ. By comparing the above expression with the formula ϕt = eit f given by point c) in Prop. A.1, we see that μt t is a constant, namely μt = tμ for a Lebesgue singular measure μ as desired. The rest is immediate. Therefore if ϕ is a symmetric inner function then: ϕ belongs to a one-parameter semigroup of symmetric inner functions ⇔ ϕ is singular. 1+z Set h(z) ≡ i 1−z . We now use the conformal maps h and log to identify D with S∞ and with Sπ as follows h
log
D −→ S∞ −→ Sπ . With this identification we shall carry the above notions to S∞ and Sπ . In particular given a function ϕ ∈ H∞ (Sπ ) (resp. ϕ ∈ H∞ (S∞ )) we shall say that ϕ is symmetric iff ϕ(q + iπ ) = ϕ(q) ¯ (resp. ϕ(−q) = ϕ(q)) ¯ for almost all q ∈ R; and ϕ is inner if |ϕ(q)| = |ϕ(q + iπ )| = 1 for almost all q ∈ R (resp. |ϕ(q)| = 1 for almost all q > 0). Note that by Eq. (11) every inner function ϕ on S∞ can be uniquely written as +∞ 1 + ps dm(s) . (14) ϕ( p) = B( p) exp −i −∞ p − s Here m is a measure on R ∪ {∞} singular with respect to the Lebesgue measure (the point at infinity can have positive measure). A factor in the Blasckhe product B here have the form p−α p+α with α ≥ 0 and is symmetric iff α = 0. Clearly ϕ is symmetric iff the Blasckhe factors corresponding to α and −α, ¯ with α = 0, appear in pairs (with the same multiplicity) and m(s) = m(−s). Corollary A.3. Let ϕt , t ≥ 0, be a semigroup of symmetric inner functions on S∞ . Then ϕt (z) = exp(it f (z)), where f is holomorphic on S∞ with f (z) ≥ 0. For almost all p ∈ R the limit f ∗ ( p) = limε→0+ f ( p + iε) exists almost everywhere and is real with f ∗ (− p) = − f ∗ ( p). For p ≥ 0 we have:
+∞ p dν(λ), (15) f ∗ ( p) = cp + 2 λ − p2 0
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where c ≥ 0 is a constant and (1 + λ2 )−1 ν(λ) is a finite, positive measure on [0, +∞) which is singular with respect to the Lebesgue measure. Conversely every function f ∗ on [0, ∞) given by the right hand side of (15) is the boundary value of a function f analytic in S∞ and ϕt ≡ eit f , t ≥ 0, is a one-parameter semigroup of symmetric inner functions on S∞ . Proof. This is a consequence of Cor. A.2 and formula (14). With ψ ∈ L ∞ (R, dq), denote by Mψ the operator of multiplication by ψ on L 2 (R, dq). Set also ψs (q) = ψ(q + s). Lemma A.4. Let ψ ∈ L ∞ (R, dq). The operator-valued map s ∈ R → V (s) ≡ Mψs ∈ B(L 2 (R, dq)) extends to a bounded weakly continuous function on the strip Sa , a > 0, analytic in Sa , such that V (s + ia) = V (s)∗ if and only if ψ is the boundary value of a ¯ function in H∞ (Sa ) such that ψ(s + ia) = ψ(a) for almost s ∈ R. Proof. Suppose that s ∈ R → V (s) ∈ B(L 2 (R, dq)) extends to a bounded weakly continuous function on the strip Sa , analytic in Sa , and V (s + ia) = V (s)∗ . Then for every g ∈ L 1 (R, dq) the map
s ∈ R → (g1 , V (s)g2 ) = ψ(q + s)g(q)dq is the boundary value of a function Vg in H∞ (Sa ) such that Vg (s +ia) = Vg¯∗ . Here g1 , g2 are L 2 -functions with g1 g¯ 2 = g. For a fixed u ∈ (0, a) the map g → Vg (iu) is a linear functional on L ∞(R, dq) which is weak∗ continuous by the maximum modulus theorem. Thus Vg (iu) = ψiu (q)g(q)dq with ψiu a L ∞ -function. Setting ψ(z) = ψiu (q) with ¯ z = q + iu one can then show that ψ is a function in H∞ (Sa ) and ψ(s + ia) = ψ(a). The converse statement is easily verified. By a scattering function S2 we shall mean a symmetric inner function on Sπ which is continuous on Sπ with the additional symmetry S2 (−¯z ) = S2 (z) (cf. [9]). Let ϕ be an inner function on Sπ which is continuous on Sπ . Viewed as a function on D, ϕ has only two possible singularities at 1 and −1; if it is further singular, then by Eq. (13) z+1 z−1 ϕ(z) = exp c1 − c2 z−1 1+z for some constants c2 ≥ 0, c2 ≥ 0 and ϕ is a scattering function iff c1 = c2 . Corollary A.5. Let ϕt be a one-parameter semigroup of symmetric inner functions on S∞ and let f be its generator, i.e. ϕt (z) = eit f (z) . The following are equivalent: • f is holomorphic in C with at most one singularity in 0, • f (z) = c1 z − c2 1z for some constants c2 ≥ 0, c2 ≥ 0, • viewed as a function on Sπ , ϕt is continuous up to the boundary for every t ≥ 0. In particular ϕt is a scattering function for every t ≥ 0 iff f (z) = c(z − 1z ) with c ≥ 0. Proof. Immediate by the above discussion.
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B. Möbius Covariant Nets of von Neumann Algebras The reader may find in [11] the basic properties of local, Möbius covariant net of von Neumann algebras on R. Here we recall the definition. A net A of von Neumann algebras on S 1 is a map I → A(I ) from I, the set of open, non-empty, non-dense intervals of S 1 , to the set of von Neumann algebras on a (fixed) Hilbert space H that verifies the following isotony property: 1. Isotony: If I1 , I2 are intervals and I1 ⊂ I2 , then A(I1 ) ⊂ A(I2 ). The net A is said to be Möbius covariant if the following properties 2,3 and 4 are satisfied: 2. Mobius ¨ Invariance There is a strongly continuous unitary representation U of G on H such that U (g)A(I )U (g)∗ = A(g I ) , g ∈ G, I ∈ I. Here G denotes the Möbius group (isomorphic to P S L(2, R)) that naturally acts on S 1 . 3. Positivity of the Energy: U is a positive energy representation. 4. Existence and Uniqueness of the Vacuum: There exists a unique (up to a phase) unit U -invariant vector (vacuum vector) and is cyclic for the von Neumann algebra ∨ I ∈I A(I ) The net A is said to be local if the following property holds: 5. Locality: If I1 and I2 are disjoint intervals, the von Neumann algebras A(I1 ) and A(I2 ) commute: A(I1 ) ⊂ A(I2 ) . One of the main consequence of the axioms is Haag duality, namely A(I ) = A(I ) for every interval I ∈ I, see [11]. A local Möbius covariant net on R is the restriction of a local Möbius covariant net on S 1 to R = S 1 {−1} (identification by the stereographic map). We say that the split property holds for a local Möbius covariant net A on S 1 if A(I1 ) ∨ A(I2 ) is naturally isomorphic with A(I1 ) ⊗ A(I2 ) when I1 , I2 are intervals with disjoint closures. (If A is non-local one requires that the inclusion A(I1 ) ⊂ A(I2 ) has an intermediate type I factor.) This very general property holds in particular if Tr(e−β L 0 ) < ∞ for all β > 0, where L 0 is the conformal Hamiltonian, see [5].
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References 1. Araki, H., Zsido, L.: Extension of the structure theorem of Borchers and its application to half-sided modular inclusions. Rev. Math. Phys. 17, 491–543 (2005) 2. Beurling, A.: On two problems concerning linear transformations in Hilbert space. Acta Math. 81, 239– 255 (1949) 3. Borchers, H.-J.: The CPT Theorem in two-dimensional theories of local observables. Commun. Math. Phys. 143, 315 (1992) 4. Brunetti, R., Guido, D., Longo, R.: Modular localization and Wigner particles. Rev. Math. Phys. 14, 759– 785 (2002) 5. Buchholz, D., D’Antoni, C., Longo, R.: Nuclearity and thermal states in Conformal Field Theory. Commun. Math. Phys. 270, 267–293 (2007) 6. Buchholz, D., Mack, G., Todorov, I.: The current algebra on the circle as a germ of local field theories. Nucl. Phys. B (Proceedings Supplement) 5, 20–56 (1988) 7. Guido, D., Longo, R., Wiesbrock, H.-W.: Extensions of conformal nets and superselection structure. Commun. Math. Phys. 192, 217–244 (1998) 8. Lax, P.D.: Translation invariant subspaces. Acta Math. 101, 163–178 (1959) 9. Lechner, G.: Polarization-free quantum fields and interaction. Lett. Math. Phys. 64, 137–154 (2003) 10. Li, K., Witten, E.: Role of short distance behavior in off-shell Open-String Field Theory. Phys. Rev. D 48, 853–860 (1993) 11. Longo, R.: Lectures on Conformal Nets. Preliminary lecture notes that are available at http://www.mat. uniroma2.it/~longo. The first part is published as follows: Longo, R.: Real Hilbert subspaces, modular theory, S L(2, R) and CFT. In: “Von Neumann algebras in Sibiu”, Theta Series Adv. Math. 10, Bucharest: Theta, 2008, pp. 33–91 12. Longo, R., Rehren, K.H.: Local fields in boundary CFT. Rev. Math. Phys. 16, 909–960 (2004) 13. Longo, R., Rehren, K.H.: How to remove the boundary in CFT, an operator algebraic procedure. Commun. Math. Phys. 285, 1165–1182 (2009) 14. Rudin, W.: Real and Complex Analysis. New York: McGraw-Hill, 1970 15. Witten, E.: Some computations in background independent Open-String Field Theory. Phys. Rev. D 47, 3405–3410 (1993) 16. Witten, E.: Quantum background independence in String Theory. http://arXiv.org/abs/hep-th/9306122v1, 1993 17. Zamolodchikov, A.: Factorized S-matrices as the exact solutions of certain relativistic quantum field theory models. Ann. Phys. 120, 253–291 (1979) Communicated by Y. Kawahigashi
Commun. Math. Phys. 303, 233–260 (2011) Digital Object Identifier (DOI) 10.1007/s00220-011-1203-3
Communications in
Mathematical Physics
Decorrelation Estimates for the Eigenlevels of the Discrete Anderson Model in the Localized Regime Frédéric Klopp1,2, 1 LAGA, U.M.R. 7539 C.N.R.S, Institut Galilée, Université Paris-Nord, 99 Avenue J.-B. Clément,
93430 Villetaneuse, France. E-mail:
[email protected]
2 Institut Universitaire de France, Paris, France
Received: 6 April 2010 / Accepted: 18 October 2010 Published online: 12 February 2011 – © Springer-Verlag 2011
Abstract: The purpose of the present work is to establish decorrelation estimates for the eigenvalues of the discrete Anderson model localized near two distinct energies inside the localization region. In dimension one, we prove these estimates at all energies. In higher dimensions, the energies are required to be sufficiently far apart from each other. As a consequence of these decorrelation estimates, we obtain the independence of the limits of the local level statistics at two distinct energies. Résumé: Dans ce travail, nous établissons des inégalités de décorrélation pour les valeurs propres proches de deux énergies distinctes. En dimension 1, nous démontrons que ces inégalités sont vraies quel que soit le choix de ces deux énergies. En dimension supérieure, il nous faut supposer que les deux énergies sont suffisamment éloignées l’une de l’autre. Comme conséquence de ces inégalités de décorrélation, nous démontrons que les limites des statistiques locales des valeurs propres sont indépendantes pour deux énergies distinctes. 1. Introduction On 2 (Zd ), consider the random Anderson model Hω = − + Vω , where − is the free discrete Laplace operator u m for u = (u n )n∈Zd ∈ 2 (Zd ) (−u)n =
(1.1)
|m−n|=1
and Vω is the random potential (Vω u)n = ωn u n
for u = (u n )n∈Zd ∈ 2 (Zd ).
The author is supported by the grant ANR-08-BLAN-0261-01.
(1.2)
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We assume that the random variables (ωn )n∈Zd are independent identically distributed and that their common distribution admits a compactly supported bounded density, say g. It is then well known (see e.g. [12]) that • let := [−2d, 2d]+supp g and S− and S+ be the infimum and supremum of ; for almost every ω = (ωn )n∈Zd , the spectrum of Hω is equal to ; • for some S− < s− ≤ s+ < S+ , the intervals I− = [S− , s− ) and I+ = (s+ , S+ ] are contained in the region of localization for Hω , i.e. the region of where the finite volume fractional moment criteria of [1] are verified for restrictions of Hω to sufficiently large cubes (see also Proposition 2.1). In particular, I := I− ∪ I+ contains only pure point spectrum associated to exponentially decaying eigenfunctions; for the precise meaning of the region of localization, we refer to Sect. 2.1.2; if the disorder is sufficiently large or if the dimension d = 1 then one can pick I = ; • there exists a bounded density of states, say λ → ν(E), such that, for any continuous function ϕ : R → R, one has R
ϕ(E)ν(E)d E = E(δ0 , ϕ(Hω )δ0 ).
(1.3)
Here, and in the sequel, E(·) denotes the expectation with respect to the random parameters, and P(·) the probability measure they induce. Let N be the integrated density of states of Hω i.e. N is the distribution function of the measure ν(E)d E. The function ν is only defined E-almost everywhere. In the sequel, when we speak of ν(E) for some E, we mean that the non decreasing function N is differentiable at E and that ν(E) is its derivative at E. 1.1. The results. For L ∈ N, let = L = [−L , L]d be a large box and | | := # = (2L + 1)d be its cardinality. Let Hω ( ) be the operator Hω restricted to with periodic boundary conditions. The notation | | → +∞ is a shorthand for considering = L in the limit L → +∞. Let us denote the eigenvalues of Hω ( ) ordered increasingly and repeated according to multiplicity by E 1 (ω, ) ≤ E 2 (ω, ) ≤ · · · ≤ E | | (ω, ). Let E be an energy in I such that ν(E) > 0. The local level statistics near E is the point process defined by (ξ, E, ω, ) =
| |
δξn (E,ω, ) (ξ ),
(1.4)
n=1
where ξn (E, ω, ) = | | ν(E) (E n (ω, ) − E), 1 ≤ n ≤ | |.
(1.5)
One of the most striking results describing the localization regime for the Anderson model is Theorem 1.1 ([15]). Assume that E ∈ I such that ν(E) > 0. When | | → +∞, the point process (·, E, ω, ) converges weakly to a Poisson process on R with intensity
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1, i.e. for (U j )1≤ j≤J , U j ⊂ R bounded measurable and U j ∩ U j = ∅ if j = j and (k j )1≤ j≤J ∈ N J , one has ⎫⎞ ⎛⎧ #{ j; ξn (E, ω, ) ∈ U1 } = k1 ⎪ ⎪ J ⎬ ⎨ kj ⎟ ⎜ .. .. −|U j | |U j | → . P ⎝ ω; e ⎠ . . ⎪ →Zd ⎪ kj! ⎭ ⎩ j=1 #{ j; ξn (E, ω, ) ∈ U J } = k J An analogue of Theorem 1.1 was first proved in [17] for a different one-dimensional random operator. Once Theorem 1.1 is known, a natural question arises: • for E = E , are the limits of (ξ, E, ω, ) and (ξ, E , ω, ) stochastically independent? This question has arisen and has been answered for other types of random operators like random matrices (see e.g. [14]); in this case, the local statistics are not Poissonian. For the Anderson model, this question has been open (see e.g. [16,19]) and to the best of our knowledge, the present paper is the first to bring an answer. The conjecture is also open for the continuous Anderson model and random CMV matrices where the local statistics have also been proved to be Poissonian (see e.g. [4,7,19,20]). The main result of the present paper is Theorem 1.2. Assume that the dimension d = 1. Pick E ∈ I and E ∈ I such that E = E , ν(E) > 0 and ν(E ) > 0. When | | → +∞, the point processes (E, ω, ) and (E , ω, ), defined in (1.4), converge weakly respectively to two independent Poisson processes on R with intensity 1. That is, for (U j )1≤ j≤J , U j ⊂ R bounded measurable and U j ∩ U j = ∅ if j = j and (k j )1≤ j≤J ∈ N J and (U j )1≤ j≤J , U j ⊂ R
bounded measurable and U j ∩ U j = ∅ if j = j and (k j )1≤ j≤J ∈ N J one has ⎫⎞ ⎛⎧ #{ j; ξn (E, ω, ) ∈ U1 } = k1 ⎪ ⎪ ⎪ ⎪ ⎪ .. .. ⎪ ⎟ ⎪ ⎜⎪ ⎪ ⎪ ⎟ ⎪ ⎜⎪ . . ⎪ ⎪ ⎬⎟ ⎜⎨ #{ j; ξn (E, ω, ) ∈ U J } = k J ⎟ ⎜ P ⎜ ω; ⎟ , ω, ) ∈ U } = k #{ j; ξ (E ⎜⎪ ⎪ n 1 1 ⎪⎟ ⎟ ⎪ ⎜⎪ ⎪ ⎪ ⎪ .. .. ⎪ ⎠ ⎝⎪ ⎪ ⎪ . . ⎪ ⎪ ⎩ ⎭ #{ j; ξn (E , ω, ) ∈ U J } = k J →
→Zd
J j=1
e
−|U j | |U j |
kj
kj!
.
J j=1
e−|U j |
|U j |k j . k j !
(1.6)
When d ≥ 2, we also prove Theorem 1.3. Assume that d is arbitrary. Pick E ∈ I and E ∈ I such that |E − E | > 2d, ν(E) > 0 and ν(E ) > 0. When | | → +∞, the point processes (E, ω, ) and (E , ω, ), defined in (1.4), converge weakly respectively to two independent Poisson processes on R with intensity 1. In Sect. 3, we show that Theorems 1.2 and 1.3 follow from Theorem 1.1 and the decorrelation estimates that we present now. They are the main technical results of the present paper.
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Lemma 1.1. Assume d = 1 and pick β ∈ (1/2, 1). For α ∈ (0, 1) and {E, E } ⊂ I s.t. E = E , for any c > 0, there exists C > 0 such that, for L ≥ 3 and cL α ≤ ≤ L α /c, one has σ (Hω ( )) ∩ (E + L −d (−1, 1)) = ∅, β P (1.7) ≤ C(/L)2d e(log L) . σ (Hω ( )) ∩ (E + L −d (−1, 1)) = ∅ This lemma shows that, up to sub-polynomial errors, the probability to obtain simultaneously an eigenvalue near E and another one near E is bounded by the product of the estimates given for each of these events by Wegner’s estimate (see Sect. 2.1.1). In this sense, (1.7) is similar to Minami’s estimate for two distinct energies. Lemma 1.1 proves a result conjectured in [16,19] in dimension 1. In arbitrary dimension, we prove (1.7), actually a somewhat stronger estimate, only when the two energies E and E are sufficiently far apart. Lemma 1.2. Assume d is arbitrary. Pick β ∈ (1/2, 1). For α ∈ (0, 1) and {E, E } ⊂ I s.t. |E − E | > 2d, for any c > 0, there exists C > 0 such that, for L ≥ 3 and cL α ≤ ≤ L α /c, one has P
σ (Hω ( )) ∩ (E + L −d (−1, 1)) = ∅, σ (Hω ( )) ∩ (E + L −d (−1, 1)) = ∅
≤ C(/L)2d (log L)C .
(1.8)
This e.g. proves the independence of the processes for energies in opposite edges of the almost sure spectrum. The estimate (1.8) in Lemma 1.2 is somewhat stronger than (1.7); one can obtain an analogous estimate in dimension 1 if one restricts oneself to energies E and E such that E − E does not belong to some set of measure 0 (see Lemma 2.11 in Remark 2.2 at the end of Sect. 2.3). Remark 1.1. As the proof of Theorems 1.2 and 1.3 shows, the estimates (1.7) are (1.8) are stronger than what it needed. It suffices to show that the probabilities in (1.7) are (1.8) are o((/L)d ). In [7] (see also [8]), the authors provide another proof of Theorems 1.1 and of Theorems 1.2 and 1.3 under the assumption that the probabilities in (1.7) are (1.8) are o((/L)d ). The analysis done in [8] deals with both discrete and continuous models. It yields a stronger version of Theorem 1.1 and Theorems 1.2 and 1.3 in essentially the same step. Whereas in the proof of Lemma 1.1, we explicitly use the fact that Hω = H0 + Vω where H0 is the free Laplace operator (1.1), the proof we give of Lemma 1.2 still works if H0 is any convolution matrix with exponentially decaying off diagonal coefficients if one replaces the condition |E − E | > 2d with the condition |E − E | > sup σ (H0 ) − inf σ (H0 ). 2. Proof of the Decorrelation Estimates Before starting with the proofs of Lemma 1.1 and 1.2, let us recall additional properties for the discrete Anderson model known to be true under the assumptions we made on the distribution of the random potential.
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2.1. Some facts on the discrete Anderson model. Basic estimates on the distribution of the eigenvalues of the Anderson model are the Wegner and Minami estimates. 2.1.1. The Wegner and Minami estimates. One has Theorem 2.1 ([22]). There exists C > 0 such that, for J ⊂ R, and , a cube in Zd , one has E [tr(1 J (Hω ( )))] ≤ C|J | | |,
(2.1)
where • Hω ( )) is the operator Hω restricted to with periodic boundary conditions, • 1 J (H ) is the spectral projector of the operator H on the energy interval J . We refer to [10,13,21] for simple proofs and more details on the Wegner estimate. Another crucial estimate is the Minami estimate. Theorem 2.2 ([2,5,9,15]). There exists C > 0 such that, for J ⊂ K , and , a cube in Zd , one has E [tr(1 J (Hω ( ))) · (tr(1 K (Hω ( ))) − 1)] ≤ C|J | |K | | |2 .
(2.2)
For J = K , the estimate (2.2) was proved in [2,5,9,15]; for J = K , it can be found in [5]. In their nature, (1.7) or (1.8) and (2.2) are quite similar: the Minami estimate can be interpreted as a decorrelation estimate for close together eigenvalues. It can be used to obtain the counterparts of Theorems 1.2 and 1.3 when E and E tend to each other as | | → +∞ (see [7]). Note that the Minami estimate (2.2) has been proved for the discrete Anderson model on intervals I irrelevant of the spectral type of Hω in I . Our proof of the decorrelation estimates (1.7) and (1.8) makes use of the fact that I lies in the localized region. 2.1.2. The localized regime. Let us now give a precise description of what we mean with the region of localization or the localized regime. We prove Proposition 2.1. Recall that I = I+ ∪ I− is the region of where the finite volume fractional moment criteria of [1] for Hω ( ) are verified for sufficiently large. Then, (Loc) There exists ν > 0 such that, for any p > 0, there exists q > 0 and L 0 > 0 such that, for L ≥ L 0 , with probability larger than 1 − L − p , if (1) ϕn,ω is a normalized eigenvector of Hω ( L ) associated to an energy E n,ω ∈ I , (2) xn,ω ∈ L is a maximum of x → |ϕn,ω (x)| in L , then, for x ∈ L , one has |ϕn,ω (x)| ≤ L q e−ν|x−xn,ω | .
(2.3)
The point xn,ω is called a localization center for ϕn,ω or E n,ω . Note that, by Minami’s estimate, the eigenvalues of Hω ( ) are almost surely simple. Thus, we can associate a localization center to an eigenvalue as is done in Proposition 2.1. In its spirit, this result is not new (see e.g. [1,6,7]). We state it in a form convenient for our purpose. We prove Proposition 2.1 in Sect. 4
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2.2. The proof of Lemmas 1.1 and 1.2. The basic idea of the proof is to show that, when ω varies, two eigenvalues of Hω ( ) cannot vary in a synchronous manner, or, put in another way, locally in ω, if E(ω) and E (ω) denote the two eigenvalues under consideration, for some γ and γ , the mapping (ωγ , ωγ ) → (E(ω), E (ω)) is a local diffeomorphism when all the other random variables, that is (ωα )α∈{γ ,γ } , are fixed. As we are in the localized regime, we will exploit this by noting that eigenvalues of Hω ( ) can only depend significantly on (log L)d random variables, i.e. we can study what happens in cubes that are of side-length log L while the energy interval where we want to control things are of size L −d . This is the essence of Lemma 2.1 below. This lemma is proved under the general assumptions (2.1), (2.2) and (Loc). In particular, it is valid for if one replaces the discrete Laplacian with any convolution matrix with exponentially decaying off diagonal coefficients. The second step consists in analyzing the mapping (ωγ , ωγ ) → (E(ω), E (ω)) on these smaller cubes. The main technical result is Lemma 2.4 that shows that, under the conditions of Lemmas 1.1 and 1.2, with a large probability, eigenvalues away from each other cannot move synchronously as functions of the random variables. Of course, this will not be correct for all random models: constructing artificial degeneracies, one can easily come up with random models where this is not the case. Lemmas 1.1 and 1.2 will be proved in essentially the same way; the only difference will be in Lemma 2.4 that controls the joint dependence of two distinct eigenvalues on the random variables. Let JL = E + L −d [−1, 1] and JL = E + L −d [−1, 1]. Pick L sufficiently large so that JL ⊂ I and JL ⊂ I are contained in I where (Loc) holds true. Pick cL α ≤ ≤ L α /c, where c > 0 is fixed. By (2.2), we know that P #[σ (Hω ( )) ∩ JL ] ≥ 2 or #[σ (Hω ( )) ∩ JL ] ≥ 2 ≤ C(/L)2d where #[·] denotes the cardinality of ·. So if we define P0 = P #[σ (Hω ( )) ∩ JL ] = 1, #[σ (Hω ( )) ∩ JL ] = 1 , it suffices to show that
P0 ≤ C(/L)2d ·
β
e(log L) (log L)C
if the dimension d = 1, if the dimension d > 1.
(2.4)
First, using the assumption (Loc), we are going to reduce the proof of (2.4) to the proof of a similar estimate where the cube will be replaced by a much smaller cube, a cube of side length of order log L. We prove Lemma 2.1. There exists C > 0 such that, for L sufficiently large, ˜ d P1 , P0 ≤ C(/L)2d + C(/) where ˜ = C log L and P1 := P(#[σ (Hω ( ˜)) ∩ J˜L ] ≥ 1) and #[σ (Hω ( ˜)) ∩ J˜L ] ≥ 1), where J˜L = E + L −d (−2, 2) and J˜L = E + L −d (−2, 2). Proof of Lemma 2.1. Fix C > 0 large so that e−Cγ log L/2 ≤ L −2d−q , where q and γ are given by assumption (Loc) where we choose p = d. Let 0 be the set of probability 1 − L − p , where (1) and (2) in assumption (Loc) are satisfied. Define ˜ = C log L. We prove
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Lemma 2.2. There exists a covering of by cubes, say = ∪γ ∈ [γ + ˜], such that ˜ d , and, if ω ∈ 0 is such that Hω ( ) has exactly one eigenvalue in JL and # (/) exactly one eigenvalue in JL , then (1) either, there exists γ and γ such that γ + ˜ ∩ γ + ˜ = ∅ and • Hω (γ + ˜) has exactly one e.v. in J˜L , • Hω (γ + ˜) has exactly one e.v. in J˜L . (2) or Hω ( ˜(γ )) has exactly one e.v. in J˜L and exactly one e.v. in J˜ . 5
L
We postpone the proof of Lemma 2.2 to complete that of Lemma 2.1. Using the estimate on P(0 ), the independence of Hω (γ + ˜) and Hω (γ + ˜) when alternative (1) is the case in Lemma 2.2, Wegner’s estimate (2.1) and the fact the random variables are identically distributed, we compute σ (Hω ( 3˜(0))) ∩ J˜L = ∅ P0 ≤ L σ (Hω ( 3˜(0))) ∩ J˜L = ∅ ˜ 2d P(#[σ (Hω ( ˜(0))) ∩ J˜L ] ≥ 1)P(#[σ (Hω ( ˜(0))) ∩ J˜L ] ≥ 1) + C(/) −2d 2d ˜ 2d d 2d ˜ ˜ ˜ d P1 , ≤ CL + C(/) (/L) + C(/) P1 ≤ C(/L) + C(/) −2d
˜ dP + C(/)
˜ This completes the proof of where P1 is defined in Lemma 2.1 for 5˜ replaced with . Lemma 2.1. ˜ d ∩ , consider the cubes (γ + ˜) ˜ d Proof of Lemma 2.2. For γ ∈ Z γ ∈Z ∩ . They cover . Recall that we are taking periodic boundary conditions. If the localization centers associated to the two eigenvalues of Hω ( ) assumed to be respectively in J˜L ˜ d such and J˜L are at a distance less than 3˜ from one another, then we can find γ ∈ Z that both localization centers belong γ + 4˜ (for ˜ = C log L and C > 0 sufficiently large). Thus, by the localization property (Loc), we are in case (2). ˜ d such that each ˜ we can find γ ∈ Z ˜ d and γ ∈ Z If the distance is larger than 3, + of the cubes γ + /2 and γ contains exactly one of the localization centers and ˜ ˜ /2 ˜ (γ + /2 ˜ ) ∩ (γ + /2 ˜ ) = ∅. So for = C log L and C > 0 sufficiently large, by the localization property (Loc), we are in case (1). This completes the proof of Lemma 2.2. We now proceed with the proof of (2.4). Therefore, by Lemma 2.1, it suffices to prove that P1 , defined in Lemma 2.1, satisfies, for some C > 0, ˜β e if the dimension d = 1, 2d ˜ P1 ≤ C(/L) · (2.5) ˜C if the dimension d > 1. ˜ Let (E j (ω, )) ˜ d be the eigenvalues of Hω ( ˜ ) ordered in an increasing way 1≤ j≤(2+1) and repeated according to multiplicity. Assume that ω → E(ω) is the only eigenvalue of Hω ( ˜) in JL . In this case, by standard perturbation theory arguments (see e.g. [11,18]), we know that (1) E(ω) being simple, ω → E(ω) is real analytic, and if ω → ϕ(ω) = (ϕ(ω; γ ))γ ∈ ˜ denotes the associated normalized real eigenvector, it is also real analytic in ω;
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(2) one has ∂ωγ E(ω) = ϕ 2 (ω; γ ) ≥ 0 which, in particular, implies that ∇ω E(ω)1 = 1;
(2.6)
(3) the Hessian of E is given by Hessω E(ω) = ((h γβ ))γ ,β , where • h γ ,β = −2Re(Hω ( ˜) − E(ω))−1 ψγ (ω), ψβ (ω), • ψγ = ϕ(ω; γ )(ω)δγ • (ω) is the orthogonal projector on the orthogonal to ϕ(ω). We prove Lemma 2.3. There exists C > 0 such that Hessω (E(ω))∞ →1 ≤
C . dist (E(ω), σ (Hω ( ˜)) \ {E(ω)})
˜ d random Proof of Lemma 2.3. First, note that, by definition, Hω ( ˜) depends on (2+1) d d ˜ ˜ variables so that Hessω E(ω) is a (2+1) ×(2+1) matrix. Hence,for a = (aγ )γ ∈ ˜ ∈ C ˜ and b = (bγ )γ ∈ ˜ ∈ C ˜ , we compute Hessω E a, b = −2(Hω ( ˜) − E(ω))−1 ψa , ψb , where
⎛ ψa = (ω) ⎝
⎞ aγ |δγ δγ |⎠ ϕ(ω) =
γ ∈ ˜
aγ ϕ(ω; γ )(ω)δγ .
γ ∈ ˜
Hence, ψa 2 ≤ Ca∞ and, for some C > 0, Hessω (E(ω))∞ →1 ≤
C . dist (E(ω), σ (Hω ( ˜)) \ {E(ω)})
This completes the proof of Lemma 2.3. Note that, using (2.2), Lemma 2.3 yields, for ε ∈ (4L −d , 1), σ (Hω ( ˜)) ∩ J˜L = {E(ω)} ≤ Cε˜ 2d L −d . P ω; Hessω (E(ω))∞ →1 ≥ ε−1 Hence, for ε ∈ (4L −d , 1), one has P1 ≤ Cε˜2d L −d + Pε ,
(2.7)
Pε = P(0 (ε))
(2.8)
⎫ ⎪ σ (Hω ( ˜)) ∩ J˜L = {E(ω)} ⎪ ⎬ {E(ω)} = σ (Hω ( ˜)) ∩ (E − Cε, E + Cε), . 0 (ε) = ω; ⎪ ⎪ σ (Hω ( ˜)) ∩ J˜L = {E (ω)} ⎪ ⎪ ⎩ ⎭ {E (ω)} = σ (Hω ( ˜)) ∩ (E − Cε, E + Cε)
(2.9)
where
and
⎧ ⎪ ⎪ ⎨
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We will now estimate Pε . The basic idea is to prove that the eigenvalues E(ω) and E (ω) depend effectively on at least two independent random variables. A simple way to guarantee this is to ensure that their gradients with respect to ω are not co-linear. In the present case, the gradients have non negative components and their 1 -norm is 1; hence, it suffices to prove that they are different to ensure that they are not co-linear. We prove Lemma 2.4. Let L ≥ 1. For the discrete Anderson model, one has (1) in any dimension d: for E > 2d, if the random variables (ωγ )γ ∈ L are bounded by K , for E j (ω) and E k (ω), simple eigenvalues of Hω ( L ) such that |E k (ω) − E j (ω)| ≥ E, one has ∇ω (E j (ω) − E k (ω))2 ≥
E − 2d (2L + 1)−d/2 ; K
(2.10)
(2) in dimension 1: fix E < E and β > 1/2; let P denote the probability that there exists E j (ω) and E k (ω), simple eigenvalues of Hω ( L ) such that |E k (ω) − E| + β |E j (ω) − E | ≤ e−L and such that β
∇ω (E j (ω) − E k (ω))1 ≤ e−L ;
(2.11)
then, there exists c > 0 such that P ≤ e−cL . 2β
We postpone the proof of Lemma 2.4 for a while to estimate Pε . Set ˜β e− if the dimension d = 1, λ = λ L = E−2d −d/2 ˜ if the dimension d > 1. K
(2.12)
(2.13)
For γ and γ in ˜, define γ ,γ 0,β (ε) = 0 (ε) ∩ ω; |Jγ ,γ (E(ω), E (ω))| ≥ λ
(2.14)
where Jγ ,γ (E(ω), E (ω)) is the Jacobian of the mapping (ωγ , ωγ ) → (E(ω), E (ω)) i.e. ∂ωγ E(ω) ∂ωγ E(ω) . Jγ ,γ (E(ω), E (ω)) = ∂ωγ E (ω) ∂ωγ E (ω) In Sect. 2.4, we prove Lemma 2.5. Pick (u, v) ∈ (R+ )2n such that u1 = v1 = 1. Then u j u k 2 ≥ 1 u − v2 . max 1 j=k v j vk 4n 5 We apply Lemma 2.4 with L = ˜ and Lemma 2.5 to obtain that γ ,γ Pε ≤ P(0,β (ε)) + Pr , γ =γ
(2.15)
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F. Klopp
where ˜2β
(1) in dimension 1, we have Pr ≤ C ˜2d e−c for any 1/2 < β < β; thus, for L sufficiently large, as ˜ ≥ c log L and β > 1/2, we have Pr ≤ L −2d .
(2.16)
(2) in dimension d, as by assumption E > 2d, one has Pr = 0, thus, (2.16) still holds. In the sequel, we will write ω = (ωγ , ωγ , ωγ ,γ ), where ωγ ,γ = (ωβ )β∈{γ ,γ } . γ ,γ
To estimate P(0,β (ε)), we use Lemma 2.6. Pick ε = L −d λ−3 . For any ωγ ,γ , if there exists (ωγ0 , ωγ0 ) ∈ R2 such γ ,γ
that (ωγ0 , ωγ0 , ωγ ,γ ) ∈ 0,β (ε), then, for (ωγ , ωγ ) ∈ R2 such that |(ωγ , ωγ ) − (ωγ0 , ω0 )|∞ ≥ L −d λ−2 , one has (E j (ω), E j (ω)) ∈ J˜L × J˜ . γ
L
Recall that g is the density of the random variables (ωγ )γ ; it is assumed to be bounded γ ,γ
and compactly supported. Hence, the probability P(0,β (ε)) is estimated as follows
γ ,γ
P(0,β (ε)) = Eγ ,γ ≤ Eγ ,γ
R
1 2
γ ,γ 0,β (ε)
(ω)g(ωγ )g(ωγ )dωγ dωγ
|(ωγ ,ωγ )−(ωγ0 ,ωγ0 )|∞
g(ωγ )g(ωγ )dωγ dωγ
≤ C L −2d λ−4 ,
(2.17)
where Eγ ,γ denotes the expectation with respect to all the random variables except ωγ and ωγ . Summing (2.17) over (γ , γ ) ∈ 2˜ , using (2.15) and (2.16), we obtain
Pε ≤ C L −2d λ−4 . We now plug this into (2.7) and use the fact that ε = L −d λ−3 to complete the proof of (2.5). This completes the proofs of Lemmas 1.1 and 1.2. Proof of Lemma 2.6. Recall that, for any γ , ωγ → E j (ω) and ωγ → E j (ω) are non decreasing. Hence, to prove Lemma 2.6, it suffices to prove that, for |(ωγ , ωγ ) − (ωγ0 , ωγ0 )|∞ = L −d λ−2 , one has (E j (ω), E j (ω)) ∈ J˜L × J˜L . Let Sβ denote the square Sβ = {|(ωγ , ωγ ) − (ωγ0 , ωγ0 )|∞ ≤ L −d λ−2 }. Recall that ε = L −d λ−3 . Pick ωγ ,γ γ ,γ
such that there exists (ωγ0 , ωγ0 ) ∈ R2 for which one has (ωγ0 , ωγ0 , ωγ ,γ ) ∈ 0,β (ε). To shorten the notations, in the sequel, we write only the variables (ωγ , ωγ ) as ωγ ,γ stays fixed throughout the proof; e.g. we write E((ωγ , ωγ )) instead of E((ωγ , ωγ , ωγ ,γ )). Consider the mapping (ωγ , ωγ ) → ϕ(ωγ , ωγ ) := (E(ω), E (ω)). We will show that ϕ defines an analytic diffeomorphism form Sβ to ϕ(Sβ ). By (2.14) and (2.9), the
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γ ,γ
definitions of 0,β (ε) and 0 (ε), we know that σ (H(ω0 ,ω0 ) ( ˜)) ∩ (E − Cε, E + Cε) = {E(ω)} ⊂ (E − C L −d , E + C L −d ), γ
γ
γ
γ
γ
γ
γ
γ
σ (H(ω0 ,ω0 ) ( ˜)) ∩ [(E − Cε, E − Cε/2) ∪ (E + Cε/2, E + Cε)] = ∅,
σ (H(ω0 ,ω0 ) ( ˜)) ∩ (E − Cε, E + Cε) = {E (ω)} ⊂ (E − C L −d , E + C L −d ), σ (H(ω0 ,ω0 ) ( ˜)) ∩ [(E − Cε, E − Cε/2) ∪ (E + Cε/2, E + Cε)] = ∅.
By (2.6), as L −d λ−2 ≤ λε, for (ωγ , ωγ ) ∈ Sβ , one has σ (Hω ( ˜)) ∩ (E − Cε/2, E + Cε/2) = {E(ω)} ⊂ (E − Cε/4, E + Cε/4), σ (Hω ( ˜)) ∩ [(E − Cε/2, E − Cε/4) ∪ (E + Cε/4, E + Cε/2)] = ∅, σ (Hω ( ˜)) ∩ (E − Cε/2, E + Cε/2) = {E(ω)} ⊂ (E − Cε/4, E + Cε/4), σ (Hω ( ˜)) ∩ [(E − Cε/2, E − Cε/4) ∪ (E + Cε/4, E + Cε/2)] = ∅. Hence, by Lemma 2.3, for (ωγ , ωγ ) ∈ Sβ , one has Hessω (E(ω))∞ →1 + Hessω (E (ω))∞ →1 ≤ Cε−1 ≤ C L d λ3 . By (2.6) and the Fundamental Theorem of Calculus, for (ωγ , ωγ ) ∈ Sβ , we get that, ∇ϕ(ωγ , ωγ ) − ∇ϕ(ωγ0 , ωγ0 ) ≤ Hessω (E(ω))∞ →1 + Hessω (E (ω))∞ →1 L −d λ−1 ≤ Cλ2 . (2.18) Let us show that ϕ is one-to-one on the square Sβ . Using (2.18), we compute ϕ(ω , ω ) − ϕ(ωγ , ωγ ) − ∇ϕ(ω0 , ω0 ) · ωγ − ωγ ≤ λ2 ωγ − ωγ γ γ γ γ ω − ωγ ω − ωγ γ
γ
γ ,γ
As (ωγ0 , ωγ0 , ωγ ,γ ) ∈ 0,β (ε), we have Jac ϕ(ωγ0 , ωγ0 ) ≥ λ. Hence, for ˜ large, we have
1 ωγ − ωγ ϕ(ωγ , ωγ ) − ϕ(ωγ , ωγ ) ≥ λ 2 ωγ − ωγ
so ϕ is one-to-one. The estimate (2.18) yields |Jac ϕ(ωγ , ωγ ) − Jac ϕ(ωγ0 , ωγ0 )| ≤ λ2 . γ ,γ
As (ωγ0 , ωγ0 , ωγ ,γ ) ∈ 0,β (ε), for L sufficiently large, this implies that ∀(ωγ , ωγ ) ∈ Sβ , |Jγ ,γ (E(ω), E (ω))| ≥
1 λ. 2
(2.19)
The Local Inversion Theorem then guarantees that ϕ is an analytic diffeomorphism from Sβ onto ϕ(Sβ ). By (2.19), the Jacobian matrix of its inverse is bounded by C ˜β for some
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F. Klopp
C > 0 independent of L. Hence, if for some |(ωγ , ωγ ) − (ωγ0 , ωγ0 )|∞ = L −d λ−2 , one has (E(ω), E (ω)) ∈ J˜L × J˜ , then L
L
−d −2
λ
= |(ωγ , ωγ ) − (ωγ0 , ωγ0 )|∞ = |ϕ −1 (E(ω), E (ω)) − ϕ −1 (E, E )|∞ ≤ C L −d λ−1 ,
which is absurd when L → +∞ as λ = λ L → 0 (see (2.13)). This completes the proof of Lemma 2.6. 2.3. Proof of Lemma 2.4. A fundamental difference between points (1) and (2) in Lemma 2.4 is that to prove point (2), we will need the fact that H0 is the discrete Laplacian. In the proof of point (1), we can take H0 to be any convolution matrix with exponentially decaying off diagonal coefficients if one replaces the condition |E − E | > 2d with the condition |E − E | > sup σ (H0 ) − inf σ (H0 ). As it is simpler, we start with the proof of point (1). 2.3.1. The proof of point (1). Let E j (ω) and E k (ω) be simple eigenvalues of Hω ( L ) such that |E k (ω) − E j (ω)| ≥ E > 2d. Then, ω → E j (ω) and ω → E k (ω) are real analytic functions. Let ω → ϕ j (ω) and ω → ϕk (ω) be normalized eigenvectors associated respectively to E j (ω) and E k (ω). Differentiating the eigenvalue equation in ω, one computes ω · ∇ω (E j (ω) − E k (ω)) = Vω ϕ j (ω), ϕ j (ω) − Vω ϕk (ω), ϕk (ω) = E j (ω) − E k (ω) + −ϕk (ω), ϕk (ω) − −ϕ j (ω), ϕ j (ω). As 0 ≤ − ≤ 2d and as ϕ j (ω) and ϕk (ω) are normalized, we get that E − 2d ≤ |E j (ω) − E k (ω)| − 2d ≤ |ω · ∇ω (E j (ω) − E k (ω))|. Hence, as the random variables (ωγ )γ ∈ are bounded, the Cauchy Schwartz inequality yields ∇ω (E j (ω) − E k (ω))2 ≥
E − 2d (2L + 1)−d/2 , K
which completes the proof of (2.10). 2.3.2. The proof of point (2). Let us now assume d = 1. Fix E < E . Pick E j (ω) and β E k (ω), simple eigenvalues of Hω ( L ) such that |E k (ω) − E| + |E j (ω) − E | ≤ e−L . Then, ω → E j (ω) and ω → E k (ω) are real analytic functions. Let ω → ϕ j (ω) and ω → ϕ k (ω) be normalized eigenvectors associated respectively to E j (ω) and E k (ω). One computes ∇ω E j (ω) = ([ϕ j (ω; γ )]2 )γ ∈ L and ∇ω E k (ω) = ([ϕ k (ω; γ )]2 )γ ∈ L . Hence, if β
e−L ≥ ∇ω (E j (ω) − E k (ω))1 = |ϕ j (ω; γ ) − ϕ k (ω; γ )| · |ϕ j (ω; γ ) + ϕ k (ω; γ )|
(2.20)
γ ∈ L
as ∇ω E j (ω) = ∇ω E k (ω) = 1, there exists a partition of L = {−L , . . . , L}, say P ⊂ L and Q ⊂ L such that P ∪ Q = L and P ∩ Q = ∅ and such that
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245
β
• for γ ∈ P, |ϕ j (ω; γ ) − ϕ k (ω; γ )| ≤ e−L /2 ; β • for γ ∈ Q, |ϕ j (ω; γ ) + ϕ k (ω; γ )| ≤ e−L /2 . Introduce the orthogonal projectors P and Q defined by P= |γ γ | and Q = |γ γ |. γ ∈P
One has Pϕ j − Pϕ k 2 ≤
√
γ ∈Q
L e−L
β /2
and
Qϕ j + Qϕ k 2 ≤
√
L e−L
β /2
.
Clearly Pϕ j 2 + Qϕ j 2 = ϕ j 2 = 1. As ϕ j , ϕ k = 0, one has k j k 0 = (P + Q)ϕ j , (P + Q)ϕ k = Pϕ j , Pϕ ! + Qϕ , Qϕ √ β L e−L /2 . = Pϕ j 2 − Qϕ j 2 + O
Hence Pϕ j 2 =
√ √ 1 1 β β + O( L e−L /2 ) and Qϕ j 2 = + O( L e−L /2 ). 2 2
This implies that P = ∅ and Q = ∅. We set h − = and E k (ω) yield
Pϕ j
−
Pϕ k
and h + =
Qϕ j
+
Qϕ k .
(2.21)
The eigenvalue equations for E j (ω)
(− + Wω )ϕ j = E(ω)ϕ j and (− + Wω )ϕ k = −E(ω)ϕ k , where E(ω) = (E j (ω) − E k (ω))/2, Wω = Vω − E(ω),
E(ω) = (E j (ω) + E k (ω))/2.
To simplify the notation, from now on, we write u = ϕ j ; then, one has ϕ k = √ β Pu − Qu + O( L e−L /2 ). This yields (− + Wω )(Pu + Qu) = E(ω)(Pu + Qu), (− + Wω )(Pu − Qu − h − + h + ) = −E(ω)(Pu − Qu − h − + h + ) that is
(− + Wω )(Pu) = E(ω)Qu − h, (− + Wω )(Qu) = E(ω)Pu + h
where h := (− + Wω − E(ω))(h − − h + )/2. As P Wω Q = 0, this can also be written as "[−(PQ + QP) − E] u = # h1, (2.22) −(PP + QQ) + Vω − E u = h 2 , where h 1 := (P − Q)h + (E(ω) − E)u, h 2 := (Q − P)h + (E(ω) − E)u, E = (E − E)/2, E = (E + E )/2.
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By our assumption on E j (ω) and E k (ω), we know that √ β β β |E(ω) − E| ≤ 2e−L , |E(ω) − E| ≤ e−L , h ≤ C L e−L /2 . Hence, we get that √ β h 1 + h 2 ≤ C L e−L /2 .
(2.23)
So the above equations imply that √ β • E is at a distance at most L e−L /2 to the spectrum of the deterministic operator −(PQ + QP), • u is close to being in the eigenspace associated to the eigenvalues close to E, • finally, u is close to being in the kernel of the random operator −(PP + QQ) + Vω − E. The first conditions will be used to describe u. The last condition will be interpreted as a condition determining the random variables ωγ for sites γ such that |u γ | is not too small. We will show that the number of these sites is of the size of the volume of the cube L ; so, the probability that the second equation in (2.22) is satisfied should be very small. To proceed, we first study the operator −PQ − QP. As we consider periodic boundary conditions, we compute − PQ − QP = (|γ + 1γ | + |γ γ + 1|) γ ∈∂ P
+
(|γ + 1γ | + |γ γ + 1|),
(2.24)
γ ∈∂ Q
where ∂P = {γ ∈ P; γ + 1 ∈ Q} ⊂ P and ∂Q = {γ ∈ Q; γ + 1 ∈ P} ⊂ Q. By (2.21), we know that ∂P = ∅ and ∂Q = ∅. We first note that ∂P ∩ ∂Q = ∅. Here, as we are considering the operators with periodic boundary conditions on L , we identify L with Z/LZ. For A ⊂ L we define A +1 = { p +1; p ∈ A} to be the shift by one of A. By definition, (∂P +1) ⊂ Q and (∂Q+1) ⊂ P. Hence, (∂P +1)∩∂P = ∅ and (∂Q+1)∩∂Q = ∅. Consider the set C := ∂P ∪ ∂Q. We can partition it into its “connected components” i.e. 0 C can be written as a disjoint union of intervals of integers, say C = ∪ll=1 Clc . Then, by the definition of ∂P and ∂Q, for l = l , one has, Clc ∩ Clc = Clc ∩ (Clc + 1) = ∅.
(2.25)
Define Cl = Clc ∪ (Clc + 1). Equation (2.25) implies that, for l = l , Cl ∩ Cl = ∅.
(2.26)
0 Cl ∪ll=1
= L . The representation (2.24) then implies that the Note that one may have following block decomposition. − PQ − QP = − where Cl is the projector Cl =
$
l0 l=1
γ ∈C j
|γ γ |.
Cl Cl ,
(2.27)
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247
Note that, by (2.26), the projectors Cl and Cl are orthogonal to each other for l = l . So the spectrum of the operator −PQ − QP is given by the union of the spectra of (Cl Cl )1≤l≤l0 . Each of these operators is the Dirichlet Laplacian on an interval of length #Cl . Its spectral decomposition can be computed explicitly. We will use some facts from this decomposition that we state now. Lemma 2.7. On a segment of length n, the Dirichlet Laplacian n i.e. the n × n matrix ⎛0 1 0 ··· ··· 0⎞ ⎟ .. ⎟ . .⎟ 0 ⎟ ⎟ .. .. ⎟ . . 1 0⎟ ⎟ .. ⎠ . 1 0 1 0 ··· ··· 0 1 0
⎜1 0 ⎜ ⎜ ⎜0 1 n = ⎜ ⎜ .. ⎜. 0 ⎜. ⎝. .
1
0 .. .
..
satisfies • its eigenvalues are simple and are given by (2 cos(kπ/(n + 1)))1≤k≤n ; • for k ∈ {1, . . . , n}, the eigenspace associated to 2 cos(kπ/(n + 1)) is generated by the vector (sin[k jπ/(n + 1)])1≤ j≤n . Moreover, there exists K 1 > 0 such that, for any n ≥ 1, one has 1 kπ k π ≥ − 2 cos . inf 2 cos n+1 n+1 K1n2 1≤k
(2.28)
Proof of Lemma 2.7. The first statement follows immediately from the identity k( j + 1)π k( j − 1)π kπ k jπ sin + sin = 2 cos sin . n+1 n+1 n+1 n+1 The estimate (2.28) is an immediate consequence of kπ kπ (k + k )π (k − k )π cos − cos = −2 sin sin . n+1 n+1 2(n + 1) 2(n + 1) This completes the proof of Lemma 2.7
We now solve the first equation in (2.22) that describes the u solution to this equation. Lemma 2.8. Let u be a solution to (2.22) such that u = 1. Then, for L sufficiently large, one has l0 β Cl u ≤ e−L /3 (2.29) u − l=1
where, if for 1 ≤ l ≤ l0 , we write Cl = {γl− , . . . , γl+ } (nl = γl+ − γl− + 1), then, • either there exists a unique kl ∈ {1, . . . , nl } satisfying 1 2 cos kl π − E < nl + 1 K1n2
(2.30)
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F. Klopp
and αl ∈ R such that Cl u − αl u l ≤ e−L where u lγ =
⎧ ⎨
sin
⎩
0
kl (γ −γl− +1)π nl +1
β /3
(2.31)
if γ ∈ Cl , if γ ∈ Cl .
• there exists no kl ∈ {1, . . . , nl } satisfying (2.30) then Cl u ≤ e−L
β /3
.
Proof of Lemma 2.8. By Lemma 2.7, the spacing between consecutive eigenvalues of $0 −Cl Cl is bounded below by 1/(K 1 n 2 ). Let C ⊥ = 1 − ll=1 Cl . Hence, u = $l0 ⊥ u, the terms in this sum being two by two orthogonal to each other. C u + C l=1 l As E > 0, the first equation in (2.22) then yields √ √ β β ∀1 ≤l ≤l0 , − Cl Cl u − E Cl u ≤ C L e−L /2 and C ⊥ u ≤ C L e−L /2 . (2.32) Write Cl = {γl− , γl− + 1, . . . , γl+ } where one may have γl− = γl+ . We assume that the − . By the characterization of the spectrum of (Cl )1≤l≤l0 are ordered so that γl+ < γl+1 −Cl Cl , • if 2 cos(kl π/(n + 1)) is an eigenvalue of −Cl Cl closer to E than a distance L −2 /4K 1 (by the remark made above, such an eigenvalue is unique), then, for some αl real, one has Cl u − αl u l ≤ C L 5/2 e−L
β /2
.
• if there is no such eigenvalue, then Cl u ≤ C L 5/2 e−L
β /2
.
(2.33)
This completes the proof of Lemma 2.8. We now prove that |u γ | cannot be really small for too many γ . Lemma 2.9. There exists c > 0 such that, for L sufficiently large, β
(1) either #C ≥ L/3 and, for γ ∈ C, |u γ | ≥ e−L /6 , (2) or l0 ≥ 2cL β and there exists l ∗ ∈ {1, . . . , l0 } such that, for |l − l ∗ | ≤ cL β and β γ ∈ Cl , one has |u γ | ≥ e−L /6 . Proof of Lemma 2.9. To prove Lemma 2.9, we compare the values of u on Cl and Cl+1 , that is, the vectors Cl u and Cl+1 u given by Lemma 2.8. First, notice that up to an error β of size at most e−L /3 , u on Cl is determined by its coefficient u γl+ , or equivalently, by its coefficient u γ − ; in particular as sin(kl π/(nl + 1)) = (−1)kl −1 sin(kl nl π/(nl + 1)), l the representations (2.29) and (2.31) yields β (2.34) |u γ − | − |u γl+ | + u γl+ − αl sin(kl π/(nl + 1)) ≤ Ce−L /3 . l
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% Notice also that, as 2 ≤ nl ≤ 2L + 1 is fixed, for ρ ∗ :=
nl 2
+
1 2
cos
β sup Cl u − ρl |αl | ≤ Ce−L /3 .
2kl π nl +1
! , one has (2.35)
1≤l≤l0
To compare the values of u on Cl and Cl+1 , we use the second equation of (2.22) or, equivalently, the eigenvalue equation for u that reads (see (2.22)) (− + Vω − E)u = E u + e, √ β where e = h 1 + h 2 (see (2.22)); hence, e ≤ C L e−L /2 .
(2.36)
− We will discuss three cases depending on how far γl+ and γl+1 are from one another:
− − − 1 < γl+1 : as {γl+ + (1) If dist(Cl , Cl+1 ) ≥ 3, that is, if γl+ < γl+ + 1 < γl+1 β − 0 1, . . . , γl+1 − 1} ∩ [∪ll=1 Cl ] = ∅, by (2.29), we know that |u n | ≤ e−L /3 for − − 1}. The eigenvalue equation (2.36) at the points γl+ + 1 n ∈ {γl+ + 1, . . . , γl+1 − and γl+1 − 1 then tells us that
|u γl+ | + |u γ − | ≤ Ce−L
β /3
l+1
.
Thus, by (2.34) and (2.35) Cl u + Cl+1 u ≤ Ce−L
β /4
.
(2.37)
− − 0 (2) If dist(Cl , Cl+1 ) = 2, that is, if γl+ < γl+ + 1 = γl+1 − 1 < γl+1 : as γl+ + 1 ∈ ∪ll=1 Cl , β by (2.29), we know that |u γl+ +1 | ≤ e−L /3 . Hence, in the same way as above, the eigenvalue equation (2.36) at the point γl+ + 1 tells us that
|u γl+ + u γ − | ≤ Ce−L l+1
β /3
.
Thus, by (2.34) and (2.35) | Cl u − Cl+1 u | ≤ Ce−L γl+
(3) If dist(Cl , Cl+1 ) = 1, that is, if the decomposition (2.27) yield
+1 =
− γl+1 :
β /4
.
(2.38)
then, the first equation in (2.22) and
|u γl+ −1 − E u γl+ | + |u γ − +1 − E u γ − | ≤ Ce−L l+1
β /3
l+1
.
− The eigenvalue equation (2.36) at the points γl+ and γl+1 yields
|u γl+ −1 + u γ − + (ωγl+ − E − E)u γl+ | l+1
+|u γl+ + u γ − +1 + (ωγ − − E − E)u γ − | ≤ Ce−L l+1
l+1
β /3
l+1
.
Summing these two equations, we obtain |u γ − + (ωγl+ − E)u γl+ | + |u γl+ + (ωγ − − E)u γ − | ≤ Ce−L l+1
l+1
l+1
β /3
.
Then, as the random variables (ωn )n∈Z are bounded, using (2.34) and (2.35), there exists C > 1 such that 1 β β (2.39) (Cl u − Ce−L /4 ) ≤ Cl+1 u ≤ C(Cl u + e−L /4 ). C
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Notice that (2.38) and (2.37) also imply that (2.39) (at the expense of possibly changing the constant C) also holds in case (1) and case (2). Hence, for 1 ≤ l, l ≤ l0 , we have
C −|l −l| Cl u − C |l −l| e−L
β /4
≤ Cl u ≤ C |l −l| Cl u + C |l −l| e−L
β /4
. (2.40)
If case (1) in the above alternative never holds, i.e. if for 1 ≤ l ≤ l0 , one has β dist(Cl , Cl+1 ) ≤ 2,then, one has #C ≥ L/3. We know that Cu = 1 + O(e−L /3 ). ∗ So, for L sufficiently large, there exists 1 ≤ l ≤ l0 such that & √ Cl ∗ u ≥ (2 0 )−1 ≥ (4 L)−1 . Hence, by (2.40), either of two things occur β
˜ β for some c˜ > 0; thus, • for some l, one has Cl u ≤ e−L /5 , then |l − l ∗ | ≥ cL β β ∗ β ˜ ; and for some 0 < c < c, ˜ for |l − l | ≤ cL , one has Cl u ≥ e−L /5 . l0 ≥ 2cL β • for 1 ≤ l ≤ l0 , one has Cl u ≥ e−L /5 ; then, case (1) never occurs, thus, by the observation made above, #C ≥ L/3. Finally, notice that, by (2.35), (2.34) and the form of u l (see Lemma 2.8), Cl u ≥ e−L β implies that |u n | ≥ e−L /6 for n ∈ Cl . This completes the proof of Lemma 2.9.
β /5
We now show that our characterization of u, a solution of (2.22), imposes very restrictive conditions on the random variables (ωγ )−L≤γ ≤L . If γ is inside one of the connected components of C, say Cl , that is, if {γ − 1, γ , γ + 1} ⊂ Cl , then, by the first equation in (2.22), we know that |u γ +1 + u γ −1 − Eu γ | ≤ Ce−L
β /3
.
Plugging this into (2.36), the eigenvalue equation for u, we get |(ωγ − E)u γ | ≤ e−L
β /4
.
Hence, if γ belongs to one of the (Cl )l singled out in Lemma 2.9, the lower bound for |u γ | given in Lemma 2.9 yields |ωγ − E| ≤ Ce−L
β /12
.
(2.41)
Note that, if nl > 2, at least nl − 2 points γ in Cl satisfy {γ − 1, γ , γ + 1} ⊂ Cl . On the other hand, if nl = 2, then, the approximate eigenvalue equation on Cl reads √ ωγ − − E 1 u γl+ ≤ C L e−L β /2 . l − u + 1 ωγl − E γl So, if Cl u ≥ e−L
β /6
, one has
|1 − (ωγ − − E)(ωγl+ − E)| ≤ Ce−L l
β /4
.
(2.42)
Hence, by Lemma 2.9, we see that the random variables must satisfy at least cL β distinct condition of the type (2.41) or (2.42). As the random variables are supposed to be independent, identically distributed with a bounded density, these condition imply 2β that (2.20) can occur with a given partition P and Q with a probability at most, e−cL for some c > 0. As the total number of partitions is bounded by 2 L and as β > 1/2, we obtain that, P, the probability that (2.20) holds, is bounded by (2.12). This completes the proof of Lemma 2.4.
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Remark 2.1. The estimate (2.12) can be improved as, actually, not all partitions are allowed as we saw in the course of the proof. Moreover, it is sufficient to assume that the distribution function of the random variables be Hölder continuous for the method to work. Remark 2.2. We now present a natural weaker analogue of point (2) in Lemma 2.4. Fix ρ > 0 and define E Lc =
L '
σ (−Cl Cl ) + [−L −ρ , L −ρ ],
l=0
then, for ρ > 3, one has |E Lc | ≤ 2L 2−ρ , thus, ( ' c E L = 0. n≥1 L≥n Define the set of total measure ⎛ E = R \ ⎝
( '
⎞ E Lc ⎠ .
M≥1 L≥M
Hence, if E − E = E ∈ E, for L sufficiently large, as inf dist(E, σ (−Cl Cl )) ≥ L −ρ ,
1≤l≤L
by the decomposition (2.27), a solution u to the first equation in (2.22) must satisfy u ≤ L −(ν−ρ) if h 1 ≤ L −ν . Hence, we obtain Lemma 2.10. Fix ν > 4. For the discrete Anderson model in dimension 1, for E − E ∈ E, for L sufficiently large, if E j (ω) and E k (ω) are simple eigenvalues of Hω ( L ) such that |E k (ω) − E| + |E j (ω) − E | ≤ L −ν , then ∇ω (E j (ω) − E k (ω))1 ≥ L −ν . This can then be used as Lemma 2.4 is used in the proof of Lemma 1.1 to prove the following variant of the decorrelation estimates in dimension 1 Lemma 2.11. Assume d = 1. For α ∈ (0, 1) and E − E ∈ E s.t. {E, E } ⊂ I , for any c > 0, there exists C > 0 such that, for L ≥ 3 and cL α ≤ ≤ L α /c, one has P
σ (Hω ( )) ∩ (E + L −d (−1, 1)) = ∅, σ (Hω ( )) ∩ (E + L −d (−1, 1)) = ∅
≤ C(/L)2d (log L)C .
Comparing with Lemma 1.1, we improved the bound on the probability at the expense of reducing the set of validity in (E, E ).
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F. Klopp
2.4. Proof of Lemma 2.5. Pick (u, v) ∈ (R+ )2n such that u1 = v1 = 1. At the expense of exchanging u and v, we may assume that v2 ≥ u2 . Write u = αv + v ⊥ , where v, v ⊥ = 0. Note that, as all the coefficient of both u and v are non negative, v ⊥ = 0 is equivalent u = v. Let us now assume u = v, that is v ⊥ = 0. One computes u22 = α 2 v22 + v ⊥ 22 and u − v22 = (α − 1)2 v22 + v ⊥ 22 .
(2.43)
Moreover, as all the coefficients of v are non negative, v ⊥ admits at least one negative coefficient. As all the coefficients of u are non negative, the decomposition u = αv + v ⊥ implies that α > 0. The first equation in (2.43) and the condition v2 ≥ u2 then imply α ∈ (0, 1). Combining this with u = αv + v ⊥ and u1 = v1 = 1 yields 0 < 1 − α ≤ v ⊥ 1 . Hence, by the second equation in (2.43) and the Cauchy-Schwartz inequality, we get √ 1 √ u − v1 ≤ u − v2 ≤ v2 v ⊥ 1 + v ⊥ 2 ≤ 2 nv ⊥ 2 . n For any ( j, k), one has
(2.44)
u j u k v⊥ v⊥ k . = j v j vk v j vk
As v, v ⊥ = 0, one computes ! u u 2 2 ⊥ ⊥ j k = (v j vk⊥ )2 + (vk v ⊥ j ) − 2v j vk vk v j v j vk j,k j,k ⎛ ⎞ ⎞ ⎛ 2 ⊥ 2 ⊥ ⊥ = 2⎝ vj⎠ (vk ) − 2 ⎝ vjvj ⎠ vk vk j
= 2v22 v ⊥ 22 ≥
k
j
k
1 u − v21 . 2n 3
Thus, u u 2 1 max j k ≥ 5 u − v21 j=k v j vk 2n which completes the proof of Lemma 2.5. 3. The Proofs of Theorems 1.2 and 1.3 In [7], the authors extensively study the distribution of the energy levels of random systems in the localized phase. Their results apply also to the discrete Anderson model; in particular, they provide a proof of Theorems 1.2 and 1.3 once the decorrelation estimates obtained in Lemmas 1.1 and 1.2 are known. We provide an alternate proof. The proof in [7] relies on a construction that also proves Theorem 1.1 (actually a stronger uniform result). Here, we only prove Theorems 1.2 and 1.3 independently of the values of the limits in Theorem 1.1.
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The localization centers of Proposition 2.1 are not defined uniquely. One can easily check that, under the assumptions of Proposition 2.1, all the localization centers for a given eigenvalue or eigenfunction are contained in a disk of radius at most C log L (for some C > 0). To define a unique localization center, we order the centers lexicographically and let the localization center associated to the eigenvalue or eigenfunction be the largest one (i.e. the one most upper left in dimension 2). We prove Lemma 3.1. Pick α ∈ (0, 1) and c > 0. Let ν be defined by (Loc). Assume = (L) satisfies cL α ≤ ≤ L α /c. If (Loc) (see Proposition 2.1) is satisfied then, for any p > 0 and ε > 0, there exists L 0 > 0 such that, for L ≥ L 0 , with probability larger than 1 − L−p: (1) If (E j )1≤ j≤J ∈ I J are eigenvalues of Hω ( L ) with localization center in γ + , then the operator Hω (γ + (1+ε) ) has J eigenvalues, say ( E˜ j )1≤ j≤J , with localization center in γ + (1+ε/2) and such that sup1≤ j≤J |E j − E˜ j | ≤ e−νε/4 . (2) If (E j )1≤ j≤J ∈ I J are eigenvalues of Hω (γ + (1+ε) ) with localization center in γ + , then the operator Hω ( L ) has J eigenvalues, say ( E˜ j )1≤ j≤J , with localization center in γ + (1+ε/2) and such that sup1≤ j≤J |E j − E˜ j | ≤ e−νε/4 . (3) If (E j )1≤ j≤J ∈ I J are eigenvalues of Hω (γ + (1+ε) ) with localization center in γ + ( (1+ε/2) \ ), then there exists (β j )1≤ j≤J such that, for 1 ≤ j ≤ J , one has " # ε d • β j ∈ 16 Z ∩ γ + ( (1+ε/2) \ ) , • the operator Hω (β j + ε/4 ) has an eigenvalue, say E˜ j , satisfying |E j − E˜ j | ≤ e−νε/8 . The number ν > 0 is given by (Loc). Similar results can be found in [7]. Proof. With probability at least 1 − L − p , the conclusions of Proposition 2.1 hold which we assume from now on. To prove (1), let (ϕ j )1≤ j≤J be normalized eigenfunctions associated to (E j )1≤ j≤J . Then, setting ϕ˜ j = 1γ + (1+ε) ϕ j and using (2.3) from (Loc) and the assumption that the localization center are in γ + , one obtains !! ϕ˜ j , ϕ˜k 2 (γ + (1+ε) ) 1≤ j≤J − Id ≤ J 2 e−νε/4 , 1≤k≤J sup 1γ +( (1+ε) \ (1+ε/2) ) ϕ˜ j 2 (γ + (1+ε) ) ≤ e−νε/6 ,
1≤ j≤J
sup (Hω (γ + (1+ε) ) − E j )ϕ˜ j 2 (γ + (1+ε) ) ≤ e−νε/4 .
1≤ j≤J
This immediately yields (1) for L sufficiently large as • J ≤ (2L + 1)d and cL α ≤ ≤ L α /c, • at a localization center, the modulus of an eigenfunction is at least of order L −d/2 . Point (2) is proved in the same way. We omit further details. ε d To prove (3), we set ϕ˜ j = 1β j + ε/4 ϕ j , where β j is the point in 16 Z closest to the localization center of ϕ j . The conclusion then follows from the same reasoning as above. This completes the proof of Lemma 3.1.
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F. Klopp
Pick (U j )1≤ j≤J , (k j )1≤ j≤J ∈ N J , (U j )1≤ j≤J and (k j )1≤ j≤J ∈ N J as in Theorem 1.2. To prove Theorems 1.2 and 1.3, it suffices to prove (1.6) for (U j )1≤ j≤J and (U j )1≤ j≤J non-empty compact intervals which we assume from now on. Pick L and such that (2L + 1) = (2 + 1)(2 + 1), cL α ≤) ≤ L α /c for some α ∈ (0, 1) and c > 0. Pick ε > 0 small. Partition L = |γ |≤ (γ ), where (γ ) = (2 + 1)γ + . For ⊂ and U ⊂ R, consider the random variables ⎧ ⎨ 1 if Hω ( ) has at least one eigenvalue in X (E, U, , ) := E + (ν(E)| |)−1 U with localization center in , ⎩ 0 if not; if = , we write X (E, U, ) := X (E, U, , ), and
(E, U ) :=
X (E, U, , (γ )), (E, U, ) :=
|γ |≤
X (E, U, (γ )).
|γ |≤
We prove Lemma 3.2. ⎛⎧ ⎫⎞ ⎛⎧ #{ j; ξn (E, ω, ) ∈ U1 } = k1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ .. .. ⎜⎪ ⎜⎪ ⎪ ⎪ ⎟ ⎪ ⎪ ⎜ ⎜⎪ ⎪ ⎪ ⎪ ⎟ . . ⎪ ⎜⎪ ⎜⎪ ⎪ ⎪ ⎪ ⎟ ⎬ ⎨ ⎨ ⎜ ⎜ ⎟ P ⎜ ω; #{ j; ξn (E, ω, ) ∈ U J } = k J ⎟ − P ⎜ ω; ⎜⎪ ⎜⎪ ⎟ #{ j; ξn (E , ω, ) ∈ U1 } = k1 ⎪ ⎪ ⎜⎪ ⎜⎪ ⎟ ⎪ ⎪ ⎜⎪ ⎜⎪ ⎪ ⎪ ⎟ . . ⎪ ⎪ ⎪ ⎝⎪ ⎪ . . ⎝⎪ ⎠ ⎪ ⎪ ⎪ . . ⎪ ⎪ ⎪ ⎭ ⎩ ⎩ #{ j; ξn (E , ω, ) ∈ U J } = k J
⎛⎧ #{ j; ξn (E, ω, ) ∈ U1 } = k1 ⎪ ⎨ ⎜ .. .. P ⎝ ω; . . ⎪ ⎩ #{ j; ξn (E, ω, ) ∈ U J } = k J
⎫ ⎞ (E, U1 , ) = k1 ⎪ ⎪ ⎪ .. .. ⎪ ⎟ ⎪ ⎪ ⎟ . . ⎪ ⎪ ⎬⎟ ⎟ (E, U J , ) = k J ⎟ → 0, ⎟ L→+∞ (E , U1 , ) = k1 ⎪ ⎪ ⎟ ⎪ ⎪ ⎟ .. .. ⎪ ⎪ ⎠ ⎪ . . ⎪ ⎭ (E , U J , ) = k J
⎫⎞ ⎫ ⎞ ⎛⎧ (E, U1 , ) = k1 , ⎪ ⎪ ⎪ ⎨ ⎬ ⎬ ⎜ ⎟ ⎟ . . .. .. ⎠ − P ⎝ ω; ⎠ → 0, ⎪ ⎪ ⎪ ⎩ ⎭ ⎭ L→+∞ (E, U J , ) = k J
and ⎛⎧ ⎫⎞ ⎫⎞ ⎛⎧ (E , U1 , ) = k1 , ⎪ #{ j; ξn (E , ω, ) ∈ U1 } = k1 ⎪ ⎪ ⎪ ⎨ ⎬ ⎨ ⎬ ⎜ ⎟ ⎜ ⎟ .. .. .. .. − P ω; P ⎝ ω; ⎠ ⎝ ⎠ → 0. . . . . ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ ⎩ ⎭ L→+∞ (E , U J , ) = k J #{ j; ξn (E , ω, ) ∈ U J } = k J
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255
Proof. We first prove Lemma 3.3. For any p > 0 and ε > 0, there exists C > 0 such that, for U a compact interval and L sufficiently large, one has P ({ω; #{n; ξn (E, ω, ) ∈ U } = (E, U )}) ≤ Cd L −d (|U | + 1)2 + L − p , (3.1) and P ({ω; (E, U ) = (E, U, )}) ≤ L − p + C ε |U |. (3.2) ) Proof of Lemma 3.3. As = |γ |≤ (γ ) and these sets are two by two disjoint, the quantities #{n; ξn (E, ω, ) ∈ U } and (E, U ) differ if and only if, for some |γ | ≤ , Hω ( ) has at least two eigenvalues in E + (ν(E)| |)−1 U with localization center in (γ ). By Lemma 3.1, this implies that, except on a set of probability at most L − p , Hω ((2 + 1)γ + 2 ) has at least two eigenvalues in U + [−e−ν/4 , e−ν/4 ]. Thus, by Minami’s estimate (2.2), this happens with a probability at most C2d L −2d (|U | + 1)2 + L − p . Summing this estimate over all the possible γ ’s, we complete the proof of (3.1). The proof of (3.2) is split into two steps. Define (E, U, ε) = X (E, U, (2 + 1)γ + (1+ε) , (2 + 1)γ + (1−ε) ). |γ |≤
Then, we successively prove P ({ω; (E, U ) = (E, U, ε)}) ≤ L − p + C ε |U |
(3.3)
P ({ω; (E, U, ε) = (E, U, )}) ≤ L − p + C ε |U |
(3.4)
and
which implies (3.2). To prove (3.3), we note that, by Lemma 3.1, except on a set of probability at most L − p , (E, U ) and (E, U, ε) differ if and only if, for some |γ | ≤ , one has (1) either σ (Hω ( )) ∩ δU˜ = ∅, (2) or σ (Hω ((2 + 1)γ + (1+ε) )) ∩ δU˜ = ∅, (3) or Hω ((2 + 1)γ + (1+ε) ) has an eigenvalue in U˜ with a localization center in the cube (2 + 1)γ + ( (1+ε) \ (1−ε) ) where U˜ = E + (ν(E)| |)−1 U + e−νε/8 [−1, 1] and δU˜ = U˜ \ (E + (ν(E)| |)−1 U ). The probability of alternatives (1) and (2) is estimated using the Wegner estimate (2.1). It is bounded by 2L d e−νε/8 ≤ L − p for L sufficiently large. By point (3) of Lemma 3.1, except on a set of probability at most L − p , alternative (3) implies that, for some β ∈ γ +( (1+ε/2) \ ), the operator Hω (β + ε/4 ) has an eigenvalue in E +(ν(E)| |)−1 U + e−νε/8 [−1, 1]. The number of possible β’s is bounded by Cεd ε−d −d = Cε1−d . Using Wegner’s estimate (2.1) and summing over the possible β’s, this probability is bounded by Cε1−d (ε/L)d |U | + L − p ≤ Cε(/L)d |U | + L − p . Finally, we sum this over all possible γ ’s to obtain that the probability that alternative (3) holds for some γ is bounded by Cε|U | + L − p . This yields (3.3). To prove (3.4), the reasoning is similar. By Lemma 3.1, except on a set of probability at most L − p , (E, U, ) and (E, U, ε) differ if and only if, for some |γ | ≤ , one has
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F. Klopp
(1) either σ (Hω ((2 + 1)γ + (1+ε) )) ∩ U˜ = ∅, (2) or σ (Hω ( (γ ))) ∩ U˜ = ∅, (3) or Hω ((2 + 1)γ + (1+ε) ) has an eigenvalue in U˜ with localization center in the cube (2 + 1)γ + ( (1+ε) \ (1−ε) ). (4) or Hω ( (γ )) has an eigenvalue in U˜ with localization center in (2 + 1)γ + ( \ (1−ε) ). Following the same steps as in the proof of (3.3), we obtain (3.4). We omit further details. This completes the proof of Lemma 3.3. As ε > 0 can be chosen arbitrarily small and J and J are finite and fixed, Lemma 3.3 clearly implies Lemma 3.2. In view of Theorem 1.1 and Lemma 3.2, to prove (1.6), it suffices to prove that, in the limit L → +∞, the difference between the following quantities vanishes (E, U1 , ) = k1 , . . . , (E, U J , ) = k J P ω; , (E , U1 , ) = k1 , . . . , (E , U J , ) = k J
(3.5)
⎫⎞ ⎛⎧ ⎫⎞ ⎛⎧ (E , U1 , ) = k1 , ⎪ (E, U1 , ) = k1 , ⎪ ⎪ ⎪ ⎨ ⎬ ⎨ ⎬ ⎜ ⎟ ⎜ ⎟ .. .. .. .. P ⎝ ω; ⎠ P ⎝ ω; ⎠. . . . . ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ ⎩ (E , U J , ) = k J ⎭ (E, U J , ) = k J
(3.6)
and
Both terms in (3.5) and (3.6) define probability measures on N J +J . By Theorems 1.1 and Lemma 3.2, we know that the limit of the term in (3.6) also defines a probability measure on N J +J . Thus, by standard results on the convergence of probability measures (see e.g. [3]), the difference of (3.5) and (3.6) vanishes in the limit L → +∞ if and only if, for any (t j )1≤ j≤J and (t j )1≤ j ≤J real, in the limit L → +∞, the following quantity vanishes $ $ − Jj=1 t j (E,U j ,)− Jj =1 t j (E ,U j ,) E e ! $J $J − j =1 t j (E ,U j ,) . −E e− j=1 t j (E,U j ,) E e Note that, as the sets ( (γ ))|γ |≤ are two by two disjoint and translates of each other, for a fixed U , the random variables (X (E, U, (γ ))|γ |≤ are i.i.d. Bernoulli random variables. Thus, $ $ − Jj=1 t j (E,U j ,)− Jj =1 t j (E ,U j ,) E e − $ J t X (E,U , (γ ))−$ J t X (E ,U , (γ )) j j j=1 j j =1 j . = E e |γ |≤
Decorrelation Estimates for Localized Discrete Anderson Model Eigenlevels
257
The Minami estimate (2.2) and the decorrelation estimates (1.7) and (1.8) of Lemmas 1.1 and 1.2 guarantee that, for any ρ ∈ (0, 1), one has, for some C > 0 independent of γ , sup ˜ 1≤ j< j≤J
+
P
sup
X (E, U j , (γ )) = 1 X (E, U j˜ , (γ )) = 1 X (E , U j , (γ )) = 1 P X (E , U , (γ )) = 1 j˜
1≤ j < j˜ ≤J
+ sup P 1≤ j≤J 1≤ j ≤J
X (E, U j , (γ )) = 1 X (E , U j , (γ )) = 1
≤C
d(1+ρ) . L
(3.7)
Using this, we compute E e
−
$J
j=1 t j
$ X (E,U j , (γ ))− Jj =1 t j X (E ,U j , (γ ))
=1+
J
(e−t j − 1) · P(X (E, U j , (γ )) = 1)
j =1
+
J
(e
−t j
j =1
d(1+ρ) − 1) · P(X (E , U j , (γ )) = 1) + O . L
(3.8)
Here, the term O((/L)d(1+ρ) ) is uniform in γ . On the other hand, one has ! E e−t j X (E,U j , (γ )) = 1 + (e−t j − 1) · P(X (E, U j , (γ )) = 1), E e
−t j X (E ,U j , (γ ))
!
(3.9) = 1 + (e
−t j
− 1) · P(X (E, U j , (γ )) = 1).
By the Wegner estimate (2.1), we know that " # sup P(X (E, U j , (γ )) = 1) + P(X (E , U j , (γ )) = 1) ≤ C
1≤ j≤J 1≤ j ≤J
d . (3.10) L
Thus, by (3.8), we have $ $ − Jj=1 t j X (E,U j , (γ ))− Jj =1 t j X (E ,U j , (γ )) E e =
J j=1
E e
−t j X (E,U j , (γ ))
J ! j =1
E e
−t j X (E ,U j , (γ ))
!* !d(1+ρ) + −1 . 1 + O L
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In the same way, one proves E e−
$J
j=1 t j
X (E,U j , (γ ))
!
=
J
E e−t j X (E,U j , (γ ))
!, 1+O
L −1
d(1+ρ) !-
,
j=1
E e
−
$J
t j =1 j
X (E ,U j , (γ ))
J
=
E e−t j X (E ,U j , (γ ))
!,
(3.11) d(1+ρ) !1 + O L −1 .
j =1
As #{|γ | ≤ } ≤ C(L−1 )d , we obtain that $ $ − Jj=1 t j (E,U j ,)− Jj =1 t j (E ,U j ,) E e * ! $J !dρ + $ − j =1 t j (E ,U j ,) − Jj=1 t j (E,U j ,) −1 E e =E e 1 + O L . (3.12) Finally, note that (3.9), (3.10) and (3.11) imply that, for any (t j )1≤ j≤J and (t j )1≤ j ≤J , one has * + $ ! $ − Jj =1 t j (E ,U j ,) − Jj=1 t j (E,U j ,) < +∞. sup E e +E e L≥1
Hence, by (3.12), as cL α ≤ ≤ L α /c for some α ∈ (0, 1), we obtain that $ $ − Jj=1 t j (E,U j ,)− Jj =1 t j (E ,U j ,) E e ! $J $J − j =1 t j (E ,U j ,) → 0. −E e− j=1 t j (E,U j ,) E e L→+∞
This completes the proof of Theorems 1.2 and 1.3. Remark 3.1. The basic idea we used here is to split the cube into smaller two-bytwo disjoint cubes ( γ ())γ in such a way that, up to exponentially small errors, the eigenvalues of Hω ( ) can be represented as eigenvalues for Hω ( γ ()) and that they are independent of each other. In [7] (see also [8] for a review of the results), this idea is exploited thoroughly to study the eigenvalue statistics for random operators in the localized regime. 4. Proof of Proposition 2.1 Let I be a compact subset of the region of localization i.e. the region of where the finite volume fractional moment criteria of [1] for Hω ( ) are verified for sufficiently large. Then, by (A.6) of [1], we know that there exists α > 0 such that, for any F ⊂ I, ∀(x, y) ⊂ 2 , one has E(|μω, |(F)) ≤ Ce−α|x−y| , x,y
(4.1)
x,y
where μω, denotes the spectral measures of Hω ( ) associated to the vector δx and δ y . In particular, if F contains a single eigenvalue of Hω ( ), say E, that is simple and associated to the normalized eigenvector, say, ϕ then x,y
|μω, |(F) = |ϕ(x)| |ϕ(y)|.
(4.2)
Decorrelation Estimates for Localized Discrete Anderson Model Eigenlevels
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Pick ε and δ positive such that ε| |2 = δ/K for some large K to be chosen below. Then, partition I = ∪1≤n≤M In into intervals (In )n of length ε. By Minami’s estimate, one has P({ω; ∃n s.t. In contains 2 e.v. of Hω ( )}) ≤ Cδ|I |/K ≤ δ/2 if C|I |/K ≤ 1/2. Pick K so that this be satisfied. We now apply (4.1) to F = In for 1 ≤ n ≤ M and sum the results for s < α to get s|x−y| x,y ∀y ∈ , E e |μω, |(In ) ≤ C|I |ε−1 . n x∈
Hence, by Markov’s inequality, ⎛ ⎞ ˜ C| | |I | x,y ⎠ ≤ δ/2. P⎝ es|x−y| |μω, |(In ) ≥ δε 2 n (x,y)∈
Thus, using the relation between δ and ε, with a probability larger than 1 − δ, we know that (1) each interval In contains at most a single eigenvalue, say, E n associated to the normalized eigenfunction, say, ϕn ; (2) by (4.2), one has ∀(x, y) ∈ 2 , |ϕn (x)| |ϕn (y)| ≤
C| |3 e−s|x−y| . δ2
As ϕn is normalized, if xn is a maximum of x → |ϕn (x)|, one has |ϕn (xn )| ≥ | |−1/2 , thus, ∀x ∈ , |ϕn (x)| ≤ C| |7/2 δ −2 e−s|x−xn | . This yields Proposition 2.1 if one picks δ = L − p when = L . References 1. Aizenman, M., Schenker, J.H., Friedrich, R.M., Dirk, H.: Finite-volume fractional-moment criteria for Anderson localization. Comm. Math. Phys. 224(1), 219–253 (2001) 2. Bellissard, J.V., Hislop, P.D., Stolz, G.: Correlation estimates in the Anderson model. J. Stat. Phys. 129(4), 649–662 (2007) 3. Billingsley, P.: Convergence of probability measures. Wiley Series in Probability and Statistics: Probability and Statistics, second edition. New York: John Wiley & Sons Inc., 1999 4. Combes, J.-M., Germinet, F., Klein, A.: Poisson statistics for eigenvalues of continuum random Schrödinger operators. Preprint, available at http://arxiv.org/abs/0807.0455v1 [math.ph], (2009) 5. Combes, J.-M., Germinet, F., Klein, A.: Generalized eigenvalue-counting estimates for the Anderson model. J. Stat. Phys. 135(2), 201–216 (2009) 6. Germinet, F., Klein, A.: New characterizations of the region of complete localization for random Schrödinger operators. J. Stat. Phys. 122(1), 73–94 (2006) 7. Germinet, F., Klopp, F.: Spectral statistics for random Schrödinger operators in the localized regime. In progress, available at http://arxiv.org/abs/1011.1832v1 [math.sp], (2010) 8. Germinet, F., Klopp, F.: Spectral statistics for the discrete Anderson model in the localized regime. http://arxiv.org/abs/1006.4427v1 [math.sp], (2010)
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9. Graf, G.M., Vaghi, A.: A remark on the estimate of a determinant by Minami. Lett. Math. Phys. 79(1), 17–22 (2007) 10. Hislop, P.D.: Lectures on random Schrödinger operators. In: Fourth Summer School in Analysis and Mathematical Physics, Volume 476 of Contemp. Math., Providence, RI: Amer. Math. Soc., 2008, pp. 41–131 11. Kato, T.: Perturbation theory for linear operators. Classics in Mathematics. Berlin: Springer-Verlag, 1995. Reprint of the 1980 edition 12. Kirsch, W.: An invitation to random Schrödinger operators. In: Random Schrödinger operators, Vol. 25 of Panor. Synthèses. Paris: Soc. Math. France, 2008. With an appendix by Frédéric Klopp, pp. 1–119 13. Kirsch, W., Metzger, B.: The integrated density of states for random Schrödinger operators. In: Spectral theory and mathematical physics: a Festschrift in honor of Barry Simon’s 60th birthday, Volume 76 of Proc. Sympos. Pure Math., Providence, RI: Amer. Math. Soc., 2007, pp. 649–696 14. Mehta, M.L.: Random matrices. Volume 142 of Pure and Applied Mathematics (Amsterdam). Amsterdam: Elsevier/Academic Press, third edition. 2004 15. Minami, N.: Local fluctuation of the spectrum of a multidimensional Anderson tight binding model. Comm. Math. Phys. 177(3), 709–725 (1996) 16. Minami, N.: Energy level statistics for random operators. In: Götze, F., Kirsch, W., Klopp, F., Kriecherbauer, T. (eds.) Disordered systems: random Schrödinger operators and random matrices, Volume 5 of Oberwolfach Reports, 2008, pp. 842–844 17. Molchanov, S.: The local structure of the spectrum of a random one-dimensional Schrödinger operator. Trudy Sem. Petrovsk. 8, 195–210 (1982) 18. Reed, M., Simon, B.: Methods of modern mathematical physics. IV. Analysis of operators. New York: Academic Press [Harcourt Brace Jovanovich Publishers], 1978 19. Simon, B.: Fine structure of the zeros of orthogonal polynomials. I. A tale of two pictures. Electron. Trans. Numer. Anal. 25, 328–368 (electronic), (2006) 20. Stoiciu, M.: The statistical distribution of the zeros of random paraorthogonal polynomials on the unit circle. J. Approx. Theory 139(1-2), 29–64 (2006) 21. Veseli´c, I.: Existence and regularity properties of the integrated density of states of random Schrödinger operators. Volume 1917 of Lecture Notes in Mathematics. Berlin: Springer-Verlag, 2008 22. Wegner, F.: Bounds on the density of states in disordered systems. Z. Phys. B 44(1–2), 9–15 (1981) Communicated by B. Simon
Commun. Math. Phys. 303, 261–288 (2011) Digital Object Identifier (DOI) 10.1007/s00220-011-1190-4
Communications in
Mathematical Physics
Analytic Sharp Fronts for the Surface Quasi-Geostrophic Equation Charles Fefferman1 , José L. Rodrigo2 1 Mathematics Department, Princeton University, Princeton, NJ 08544, USA.
E-mail:
[email protected]
2 Mathematics Department, Warwick University, Coventry CV4 7AL, UK.
E-mail:
[email protected] Received: 9 April 2010 / Accepted: 31 July 2010 Published online: 11 February 2011 – © Springer-Verlag 2011
Abstract: We study the evolution of sharp fronts for the Surface Quasi-Geostrophic equation in the context of analytic functions. We showed that, even though the equation contains operators of order higher than 1, by carefully studying the evolution of the second derivatives it can be adapted to fit an abstract version of the Cauchy-Kowaleski Theorem.
1. Introduction In this paper we study the existence of analytic sharp fronts for the Surface QuasiGeostrophic (SQG) equation. SQG is given by the equations ∂θ + u · ∇θ = 0, ∂t u = −(−)−1/2 ∇ ⊥ θ,
(1) (2)
and has been the subject of extensive research in recent years for its connections with 3D Euler. We refer the reader to [1] and [2] for more details. A sharp front for SQG corresponds to the evolution of an initial data given by the characteristic function of an open set (with sufficiently regular boundary). It is easy to see that the solution remains the characteristic function of an evolving set, and so the problem reduces to the contour dynamics problem obtained by considering the evolution of the boundary of the patch. We refer the reader to [3,4,9,10 and 6] for more details. For simplicity we will the consider periodic case in which the boundary is a graph. That is we take θ of the form 1 y ≥ ϕ(x, t) θ (x, y, t) = (3) 0 y < ϕ(x, t),
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where ϕ(x, t) is periodic in x of period 1. In [9] the following equation is derived for the evolution of the sharp front in this periodic setting ∂ϕ ∂ϕ ¯ t) ∂ϕ ∂ x (x, t) − ∂ x¯ ( x, χ (x − x, ¯ ϕ(x, t) − ϕ(x, ¯ t))d x¯ (x, t) = 2 + (ϕ(x, t) − ϕ( x, ∂t [(x − x) ¯ ¯ t))2 ]1/2 R/Z ∂ϕ ∂ϕ (x, t) − (x, ¯ t) η(x − x, ¯ ϕ(x, t) − ϕ(x, ¯ t))d x. ¯ (4) + ∂ x¯ R/Z ∂ x In addition local existence results are obtained for smooth initial data [9]. In [6] a similar equation is obtained for the case in which the curve is closed (and hence no longer a graph) and local existence results were obtained in Sobolev space. The purpose of this article is to study the existence of analytic solutions of (4). The motivation is two-fold. Analytic solutions are important in the study of almost sharp fronts, as described in [3], in the upcoming work of the authors [5]. An additional motivation is of theoretical nature. In particular studying whether the system can fit in the Cauchy-Kowaleski scheme even though the right hand side of (4) is of order higher than one. We recall that in [9] the author showed that the most singular term in the right hand side of the equation is a nonlinear, nonlocal version of the operator given by the multiplier i k log |k|, which in principle does not fit in the standard Cauchy-Kowaleski machinery. We will show, by studying the evolution of ϕ (x, t) (where the primes represent derivatives with respect to x) that the new system fits the scheme of the abstract version of the Cauchy-Kowaleski Theorem of Sammartino and Caflisch [11,12]. 2. Adapting the Equation In order to simplify the presentation we consider the equation ∂ϕ ∂ϕ ¯ t) ∂ϕ ∂ x (x, t) − ∂ x¯ ( x, (x, t) = χ (x − x)d ¯ x, ¯ 2 ∂t ¯ + (ϕ(x, t) − ϕ(x, ¯ t))2 ]1/2 R/Z [(x − x)
(5)
with ϕ(x, 0) = ϕ0 (x) an analytic initial data. Remark 1. Equation (5) arises from (4) by making the following simplifications: – We ignore the correction term η as, in the context of analytic functions, it can be handled by any version of the Cauchy-Kowaleski Theorem. – We take the cut-off function χ to be an analytic function of x − x¯ alone (notice that analyticity is not an additional constraint in this case). – We also assume that χ (0) = 1 and χ (± 21 ) = 0 to some fixed order. In addition, and to simplify the notation we will write ϕ(x), suppressing the explicit t dependence. Also, we will use a prime to denote partial derivatives with respect to the space variables. Finally we will use (x, x) ¯ or simply to denote (x − x) ¯ 2 + (ϕ(x, t) − ϕ(x, ¯ t))2 .
Analytic Sharp Fronts for the Surface Quasi-Geostrophic Equation
263
The main result we will obtain is the following Theorem 1. Given an analytic, periodic (of period 1) function ϕ0 (x) there exists T > 0 and a unique solution of Eq. (5) in C([0, T ); H k (R)) (for any k ≥ 1) that is analytic with respect to the space variable. Additional time regularity can, of course, be read directly from the equation. The main tool in the proof will be a version of the Cauchy-Kowaleski Theorem due to Sammartino and Caflisch. In general these arguments are restricted to equations with a right-hand side of first order. Notice that operator on the right hand side of (5) is of order greater than 1, as it can be seen by observing that the linearization will correspond to the Fourier multiplier given by i k log |k|. To recast (5) in the setting of Cauchy-Kowaleski we differentiate the equation twice and study the evolution equations for ϕ, ϕx and ϕx x , as independent unknowns. In terms on these new unknowns the system will be recast as an operator of order 1. The following results will be useful in calculating a suitable expression for the evolution equations of ϕ, ϕx and ϕx x . Lemma 1. Given a smooth function f (x, x) ¯ and under the assumptions above on χ the derivative of sgn(x − x) ¯ f (x, x)χ ¯ (x − x)d ¯ x¯ R/Z
is given by d sgn(x − x) ¯ f (x, x)χ ¯ (x − x)d ¯ x¯ d x R/Z d f (x, x)χ ¯ (x − x) ¯ d x. ¯ = 2 f (x, x) + sgn(x − x) ¯ dx R/Z
(6)
Lemma 2. We will use the following Taylor expansions with remainder: 1 ϕ(x) ¯ = ϕ(x) + ϕ (x)(x¯ − x) + ϕ ((1 − τ )x¯ + τ x)τ dτ (x¯ − x)2 , 0
1 ϕ(x) ¯ = ϕ(x) + ϕ (x)(x¯ − x) + ϕ (x)(x¯ − x)2 2 1 1 + ϕ ((1 − τ )x¯ + τ x)τ 2 dτ (x¯ − x)3 2 0
and ϕ (x) ¯ = ϕ (x) + ϕ (x)(x¯ − x) +
1
ϕ ((1 − τ )x¯ + τ x)τ dτ (x¯ − x)2 .
0
To simplify the notation when we use the above Taylor expansions we define, for a, b ∈ N, 1 Ia,b = Ia,b (x, x, ¯ t) = ϕ (a) ((1 − τ )x¯ + τ x, t)τ b dτ. (7) 0
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C. Fefferman, J. L. Rodrigo
As indicated above we are going to study the evolution of ϕ and ϕ . Differentiating Eq. (5) with respect to x and using the lemma above we obtain ϕt (x) = 2
ϕ (x)
+
1/ [1 + (ϕ (x))2 ] 2
ϕ (x) R/Z
1/2
χ (x − x) ¯
¯ − x) ¯ + (ϕ(x) − ϕ(x))ϕ ¯ (x)] [ϕ (x) − ϕ (x)][(x χ (x − x)d ¯ x¯ − 3/ 2 ϕ (x) − ϕ (x) ¯ + χ (x − x)d ¯ x, ¯ 1/2 R/Z
(8)
and so an additional differentiation yields ϕt (x) = 2
ϕ (x) 1/
−2
ϕ (x)(ϕ (x))2 3/
[1 + (ϕ (x))2 ] 2 [1 + (ϕ (x))2 ] 2 ϕ (x)−ϕ (x) ¯ ϕ(x)−ϕ(x) ¯ 1 + ϕ (x) x−x¯ x−x¯ ϕ (x) + lim+ 2 2 1/2 − 2 3/2 x→x ¯ x) ¯ x) ¯ (x − x) ¯ 1 + ϕ(x)−ϕ( (x − x) ¯ 1 + ϕ(x)−ϕ( x−x¯ x−x¯
ϕ (x) ¯ sgn(x − x)∂ ¯ x sgn(x − x) χ (x − x) ¯ + 1/ 2 R/Z ¯ − x) ¯ + (ϕ(x) − ϕ(x))ϕ ¯ (x)] [ϕ (x) − ϕ (x)][(x χ (x − x) ¯ d x¯ − 3/2
ϕ (x) − ϕ (x) ¯ + ∂x χ (x − x) ¯ d x. ¯ (9) 1/ 2 R/Z
Notice that this last term is zero. Using Taylor’s Theorem we will rewrite Eqs. (5), (8) and (9). In the case of (5) we obtain −ϕ (x) − I3,1 (x¯ − x) ϕt (x, t) = sgn(x¯ − x)χ (x¯ − x)d x. ¯ (10) 2 1/ R/Z [1 + (ϕ (x) + I2,1 ( x¯ − x)) ] 2 In order to rewrite Eq. (8) for ϕt we start by considering the integrand of the most singular expression. We have ϕ (x)
1/2
−
[ϕ (x) − ϕ (x)][(x ¯ − x) ¯ + (ϕ(x) − ϕ(x))ϕ ¯ (x)] 3/
= (x − x) ¯
2
= (x − x) ¯ 2 = (x − x) ¯ 2 = (x¯ − x)3
2 2 x) ¯ − ϕ (x) 1 + ϕ(x)−ϕ( x−x¯
ϕ (x)−ϕ (x) ¯ x−x¯
1+
ϕ(x)−ϕ(x) ¯ ϕ (x) x−x¯
3/
ϕ (x)[1 + (ϕ (x) +
2 I2,1 (x¯ − x))2 ] − [ϕ (x) + I3,1 (x¯ − x)][1 + (ϕ (x) + I2,1 (x¯ − x))ϕ (x)] 3/
ϕ (x)ϕ (x)I ϕ (x)ϕ (x)I
2,1 ( x¯
2,1
−
2 ( x¯ x) + ϕ (x)I2,1
− [1 + (ϕ (x))2 ]I |x −
−
2 − [1 + (ϕ (x))2 ]I3,1 (x¯ − x) − ϕ (x)I2,1 I3,1 (x¯ − x)2
x)2
3/2
2 3,1 + ϕ (x)I2,1 ( x¯ − x) − ϕ (x)I2,1 I3,1 ( x¯ 3/ 3 2 x| ¯ [1 + (ϕ (x) + I2,1 (x¯ − x)) ] 2
− x)
,
Analytic Sharp Fronts for the Surface Quasi-Geostrophic Equation
265
and so Eq. (8) becomes ϕ (x)
ϕt (x) = 2 +
1/
[1 + (ϕ (x))2 ] 2 2 ( x¯ − x) − ϕ (x)I ϕ (x)ϕ (x)I2,1 − [1 + (ϕ (x))2 ]I3,1 + ϕ (x)I2,1 2,1 I3,1 ( x¯ − x)
R /Z
× sgn(x¯ − x)χ(x − x)d ¯ x¯ +
[1 + (ϕ (x) + I2,1 (x¯ − x))2 ]3/2 ϕ (x) − ϕ (x) ¯ 1/2
R /Z
χ (x − x)d ¯ x. ¯
(11)
A simple use of Taylor’s formula shows that the limit in Eq. (9) equals −
ϕ (x)(ϕ (x))2 [1 + (ϕ (x))2 ]
3/2
+
ϕ (x) [1 + (ϕ (x))2 ]
1/2
,
and using this expression we can rewrite Eq. (9) for the evolution of ϕ (x) as follows: ϕ (x)
ϕt (x) = 3
1/
−3
ϕ (x)(ϕ (x))2 3/
[1 + (ϕ (x))2 ] 2 [1 + (ϕ (x))2 ] 2
χ (x − x) ¯ ϕ (x)2 − 2ϕ (x) (x − x) + ¯ + (ϕ(x) − ϕ(x))ϕ ¯ (x)] 5/2 −[ϕ (x) − ϕ (x)][1 ¯ + (ϕ (x))2 +(ϕ(x) − ϕ(x))ϕ ¯ (x)] 2 ¯ − x) ¯ + (ϕ(x) − ϕ(x))ϕ ¯ (x)] d x¯ +3[ϕ (x) − ϕ (x)][(x
+
ϕ (x) R/Z
1/2
−
¯ − x) ¯ + (ϕ(x) − ϕ(x))ϕ ¯ (x)] [ϕ (x) − ϕ (x)][(x χ (x − x)d ¯ x. ¯ 3/ 2 (12)
We want to systematically apply the Taylor’s expansions of Lemma 2 to simplify (2). We will show the details for the first term, leaving the details of the rest to the interested reader. We have ϕ(x) − ϕ(x) ¯ 2 2 ϕ (x)2 = (x¯ − x)4 ϕ (x) 1 + x − x¯ 2 2 = (x¯ − x)4 ϕ (x) 1 + ϕ (x) + I2,1 (x¯ − x) 2 2 = (x¯ − x)4 ϕ (x) 1 + (ϕ (x))2 + 2ϕ (x)I2,1 (x¯ − x) + I2,1 (x¯ − x)2 = (x¯ − x)4 ϕ (x) [1 + (ϕ (x))2 ]2 + 4(1 + (ϕ (x))2 )ϕ (x)I2,1 (x¯ − x) 2 2 + 2[1 + (ϕ (x))2 ] I2,1 (x¯ − x)2 + 4(ϕ (x))2 I2,1 (x¯ − x)2 3 4 3 4 + 4ϕ (x) I2,1 (x¯ − x) + I2,1 (x¯ − x) .
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Since we will need the following expression later we also include the details for the expression multiplying the curly brackets in (2), ignoring χ : 1
5/2
=
1 1 1 1 + 5/ 5 5 2 |x¯ − x| [1 + (ϕ (x))2 ] |x¯ − x| [1 + (ϕ (x))2 ]5/2 1 × 2 5/2 − 1 . (x)( x−x) 2 ¯ (x−x) ¯ 1 + 2ϕ I I + [1+(ϕ (x))2 ] 2,1 [1+(ϕ (x))2 ] 2,1
(13)
We can rewrite the integrand of the most singular term in (2) as χ (x¯ − x) ϕ (x)[1 + (ϕ (x))2 ]2 (x¯ − x)4 − 2[1 + (ϕ (x))2 ]2 I3,1 (x¯ − x)4 5/ 2 +( power s o f (x¯ − x) o f degr ee 5 or higher ) =
χ (x¯ − x)
5/2
2 2 4 [1 + (ϕ (x)) ] (x¯ − x) [ϕ (x) − 2I3,1 ] + other ter ms... .
We can rewrite equation as χ (x¯ − x) ϕt (x) = + [1 + (ϕ (x))2 ]2 (x¯ − x)4 ϕ (x) − 2I3,1 d x¯ + U1 (ϕ, x, t), 5/2 R/Z (14) where U1 can be written as 3
ϕ (x)
−3
ϕ (x)(ϕ (x))2
+
ϕ (x)
1/ 3/ 2 R/Z (1 + (ϕ (x) + I2,1 ( x¯ − x))2 ) [1 + (ϕ (x))2 ] 2 [1 + (ϕ (x))2 ] 2 ¯ [ϕ (x) + I3,1 (x¯ − x)][1 + (ϕ (x) + I2,1 (x¯ − x))ϕ (x)] χ (x − x) d x¯ − 3/ 2 |x¯ − x| (1 + (ϕ (x) + I2,1 (x¯ − x))2 ) (x¯ − x)5 + χ (x¯ − x) 5/ 2 R/Z ¯ × a polynomial in ϕ(x), ϕ (x), ϕ (x), ϕ (x), (x¯ − x), and I2,1 , I3,1 , I3,2 d x. 1/
(15) Now, using (13) we have χ (x¯ − x) ϕt (x) = + ϕ (x) − 2I3,1 d x¯ 1/ 2 R/ | x¯ − x|[1 + (ϕ (x))2 ] Z χ (x¯ − x)(x¯ − x)4 [1 + (ϕ (x))2 ]2 ϕ (x) − 2I3,1 + R/Z
×
1
5/2
−
1 |x¯ − x|5 [1 + (ϕ (x))2 ]
d x¯ + U1 (x, t). 5/2
(16)
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267
We use the following lemma to deal with the most singular term: Lemma 3.
ϕ (z) − ϕ (x) 1 ϕ (x) − 2I3,1 d x¯ = − χ (z − x)dz |x¯ − x| |z − x| R/Z R/Z x+ 1 x+ 1 x+ 1 2 2 χ (x 2 χ (x ¯ − x) ¯ − x) + (ϕ (z) − ϕ (x)) (z − x) d x¯ − 2 d x¯ dz (x¯ − x)2 (x¯ − x) x z z z x z χ ( x¯ − x) χ (x¯ − x) d x¯ dz. (ϕ (z) − ϕ (x)) (z − x) d x¯ − 2 + 2 x− 21 x− 21 ( x¯ − x) x− 21 ( x¯ − x) χ (x¯ − x)
Proof. Notice that since 2 −2
0
1
R/Z 0
1
ϕ ((1 − τ )x¯ + τ x) − ϕ (x) τ dτ d x¯ |x¯ − x| x 1 1 ··· − 2 ··· .
χ (x¯ − x)
x+ 21
= −2
τ dτ = 1 we can rewrite the left-hand side as
x
x− 21
0
0
For the first of the two double integrals we consider the change of variables z = (1 − τ )x¯ + τ x and x¯ = x. ¯ Then we have dτ d x¯ = (x¯ − x)−1 dzd x¯ and so
x+ 21
−2 x
1
0 x+ 21
=2
x+ 12
· · · = −2 x
x¯ x
x+ 21
(ϕ (z) − ϕ (x))
x x+ 21
z − x¯ 1 ϕ (z) − ϕ (x) χ (x¯ − x) dzd x¯ |x¯ − x| x − x¯ x¯ − x χ (x¯ − x)
z x+ 21
z − x¯ d xdz ¯ (x¯ − x)3
z−x x − x¯ + χ (x¯ − x) d xdz ¯ 3 ( x ¯ − x) ( x ¯ − x)3 x z x+ 1 x+1 2 2 χ (x¯ − x) d χ (x¯ − x) − + (z − x) = (ϕ (z)−ϕ (x)) (z − x) d xdz ¯ 2 d x¯ (x¯ − x) (x¯ − x)2 x z x+ 1 x+ 1 χ (x¯ − x) χ (x¯ − x) 2 2 d − d xdz ¯ +2 (ϕ (z) − ϕ (x)) d x¯ (x¯ − x) (x¯ − x) x z x+ 1 x+ 1 2 2 χ (x ¯ − x) χ (z − x) = (ϕ (z) − ϕ (x)) (z − x) + (z − x) d x¯ dz (z − x)2 (x¯ − x)2 x z x+ 1 x+ 1 2 2 χ (x ¯ − x) χ (z − x) −2 d x¯ dz + (ϕ (z) − ϕ (x)) − 2 (z − x) (x¯ − x) x z x+ 1 2 ϕ (z) − ϕ (x) =− χ (z − x)dz (z − x) x x+ 1 x+ 1 x+ 1 2 2 χ (x 2 χ (x ¯ − x) ¯ − x) + d x ¯ dz. (ϕ (z)−ϕ (x)) (z − x) d x ¯ −2 (x¯ − x)2 (x¯ − x) x z z
=2
(ϕ (z) − ϕ (x))
χ (x¯ − x)
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As for the second integral (notice that dτ d x¯ = (x − x) ¯ −1 dzd x), ¯ x 1 x x ϕ (z) − ϕ (x) z − x¯ 1 −2 · · · = −2 χ (x¯ − x) dzd x¯ 1 1 | x ¯ − x| x − x¯ x − x¯ x− 2 0 x− 2 x¯ z x z − x¯ (ϕ (z) − ϕ (x)) χ (x¯ − x) d xdz ¯ = −2 (x − x) ¯ 3 x− 21 x− 12 x z z−x x − x¯ = −2 (ϕ (z) − ϕ (x)) χ (x¯ − x) + χ (x¯ − x) d xdz ¯ 3 1 1 (x − x) ¯ (x − x) ¯ 3 x− 2 x− 2 z x χ (x¯ − x) d χ (x¯ − x) − +(z − x) (ϕ (z)−ϕ (x)) (z − x) d xdz ¯ = d x¯ (x¯ − x)2 (x¯ − x)2 x− 21 x− 12 z x d χ (x¯ − x) χ (x¯ − x) − (ϕ (z) − ϕ (x)) +2 d xdz ¯ (x¯ − x) (x¯ − x) x− 21 x− 21 d x¯ z x χ (x¯ − x) χ (z − x) (ϕ (z) − ϕ (x)) − (z − x) + (z − x) d x ¯ dz = 2 (z − x)2 x− 21 x− 21 ( x¯ − x) z x χ (x¯ − x) χ (z − x) −2 d x¯ dz (ϕ (z) − ϕ (x)) 2 + (z − x) x− 21 x− 21 ( x¯ − x) x ϕ (z) − ϕ (x) = χ (z − x)dz (z − x) x− 21 x z z χ (x¯ − x) χ (x¯ − x) d x¯ dz, + (ϕ (z)−ϕ (x)) (z − x) d x¯ − 2 2 x− 21 x− 21 ( x¯ − x) x− 21 ( x¯ − x) and so the integral becomes ϕ (z) − ϕ (x) =− χ (z − x)dz |z − x| R/Z x+ 1 x+ 1 x+ 1 2 2 χ (x 2 χ (x ¯ − x) ¯ − x) + d x ¯ dz (ϕ (z)−ϕ (x)) (z − x) d x ¯ −2 (x¯ − x)2 (x¯ − x) x z z x z z χ (x¯ − x) χ (x¯ − x) d x ¯ dz. + (ϕ (z)−ϕ (x)) (z − x) d x ¯ − 2 2 x− 21 x− 21 ( x¯ − x) x− 21 ( x¯ − x)
We remark that x+ 1 x+ 1 x+ 1 2 2 χ (x 2 ¯ − x) + (ϕ (z) − ϕ (x)) (z − x) d x¯ − 2 2 (x¯ − x) x z z x z z ( x¯ − x) χ + (ϕ (z) − ϕ (x)) (z − x) d x¯ − 2 2 x− 21 x− 21 ( x¯ − x) x− 21
χ (x¯ − x) d x¯ dz (x¯ − x) χ (x¯ − x) d x¯ dz (x¯ − x) (17)
can be written as
R/Z
(ϕ (z) − ϕ (x))K 1 (z, x)dz,
Analytic Sharp Fronts for the Surface Quasi-Geostrophic Equation
269
where K 1 is continuous, and moreover smooth outside z = x. Actually it can be written as the sum of a smooth function in x and z and |z − x| times a smooth funcation in x and z. Notice that since we have taken χ to be even we have
x+ 21 x
χ (x¯ − x) d x¯ = (x¯ − x)
x x− 21
χ (x¯ − x) d x, ¯ (x¯ − x)
and this shows that the expressions in the square brackets in (17) agree when z = x. Using this lemma we can rewrite (16) as follows: ϕ (z) − ϕ (x) 1 χ (z − x)dz ϕt (x) = − 1/ |z − x| [1 + (ϕ (x))2 ] 2 R/Z 1 + (ϕ (z) − ϕ (x))K 1 (x, z)dz 1/ [1 + (ϕ (x))2 ] 2 R/Z + χ (x¯ − x)(x¯ − x)4 [1 + (ϕ (x))2 ]2 ϕ (x) − 2I3,1 R/Z
1 1 d x¯ + U1 (x, t). × 5/ − 5/ 2 |x¯ − x|5 [1 + (ϕ (x))2 ] 2
We obtain the equation
ϕ (z) − ϕ (x) χ (z − x)dz |z − x| [1 + (ϕ (x))2 ] R/Z + U1 (x, t) + U2 (x, t), 1
ϕt (x) = −
1/2
where U2 is implicitly defined by the equality above. Notice that U2 can be rewritten as 1 (ϕ (z) − ϕ (x))K 1 (x, z)dz U2 = 1/ [1 + (ϕ (x))2 ] 2 R/Z 2 2 χ (x¯ − x) ϕ (x) − 2I3,1 + [1 + (ϕ (x)) ] ×
R/Z
1 1 1 d x. ¯ − 5/ 2 5/2 |x¯ − x| [1 + (ϕ (x))2 ] 2 ϕ(x)−ϕ(x) ¯ 1+ x−x¯
(18)
In order to make the main term simpler we will introduce new coordinates, based on −1/ arch length, as this will make the term [1 + (ϕ (z))2 ] 2 disappear. Before we change coordinates into arc-length we make one additional algebraic manipulation and introduce new unknowns. We rewrite the equation as ϕ (z) 1 ϕ (x) ϕt (x) = − − χ (z − x)dz 1/2 1/2 2 2 |z − x| [1 + (ϕ (x)) ] R/Z [1 + (ϕ (z)) ] 1 1 1 χ (z − x)dz ϕ (z) − + 1/2 1/2 2 2 |z − x| [1 + (ϕ (z)) ] [1 + (ϕ (x)) ] R/Z + U1 (x, t) + U2 (x, t),
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which we can rewrite as ϕt (x) = − R/Z
ϕ (z) 1/ [1 + (ϕ (z))2 ] 2
−
ϕ (x) 1/ [1 + (ϕ (x))2 ] 2
1 χ (z − x)dz |z − x|
+ U1 (x, t) + U2 (x, t) + U3 (x, t),
(19)
where U3 =
R/Z
ϕ (z)
1 [1 + (ϕ (z))2 ]
1/2
−
1 [1 + (ϕ (x))2 ]
1/2
1 χ (z − x)dz. (20) |z − x|
We will prove the existence of analytic sharp fronts by studying a system involving Eqs. (10), (11) and (19) (to which we will eventually add an equation for the length of the curve) where we want to consider the functions ϕ, ϕ and ϕ as independent unknowns. We introduce the following functions ¯¯ ¯¯ t) := ϕ (x, t). f¯¯(x, t) := ϕ(x, t), g(x, t) := ϕ (x, t), h(x,
(21)
We want to rewrite the system formed by Eqs. (10), (11) and (19) in terms of the new unknowns in (21). We obtain (we keep the notation Ia,b as it makes no explicit mention to ϕ or f¯¯) f¯¯t (x, t) = g¯¯ t (x) = 2
¯¯ h(x) + I3,1 (x¯ − x) sgn(x − x)χ ¯ (x¯ − x)d x¯ ¯ [1 + (g(x) ¯ + I2,1 (x¯ − x))2 ]1/2 ¯¯ h(x)
(22)
2] 2 ¯¯ [1 + (g(x)) ¯¯ ¯¯ 2 2 ]I ¯¯ ¯¯ g(x) ¯¯ h(x)I 2,1 − [1 + (g(x)) 3,1 + h(x)I 2,1 I3,1 ( x¯ − x) 2,1 ( x¯ − x) − g(x)I + [1 + (g(x) ¯¯ + I2,1 (x¯ − x))2 ]3/2 R/Z ¯¯ ¯¯ x) χ (x − x) ¯ g(x) − g( ¯ d x¯ (23) ×sgn(x¯ − x)χ (x − x)d ¯ x¯ + 1/ 2 ] 2 | x¯ − x| ¯ [1 + ( g(x) ¯ + I ( x ¯ − x)) R/Z 2,1 1/
and finally 1 h¯¯ (x) h¯¯ (z) χ (z − x)dz − 1/2 1/2 2 2 ¯ ¯ |z − x| ¯ ] [1 + (g(x)) ¯ ] R/Z [1 + (g(z)) ¯¯ g, ¯¯ x, t) + U ( f, ¯ ¯¯ h, ¯¯ x, t) + U ( f, ¯ ¯¯ h, ¯¯ x, t). ¯¯ h, + U1 ( f, (24) 2 ¯ g, 3 ¯ g,
h¯¯ t (x) = −
We need to keep careful track of the expressions for Ui . From (15) we have U1 = 3
h¯¯ (x)
−3
¯¯ 2 g(x)( ¯¯ h(x))
+
¯¯ h(x)
1/ ¯¯ 2 ]1/2 2 ]3/2 ¯¯ ¯¯ + I2,1 (x¯ − x))2 ) 2 R/Z (1 + (g(x) [1 + (g(x)) [1 + (g(x)) ¯¯ χ (x − x) ¯¯ ¯¯ + I2,1 (x¯ − x))g(x)] ¯ [h(x) + I3,1 (x¯ − x)][1 + (g(x) d x¯ − 3/ 2 2 ¯ | x ¯ − x| (1 + (g(x) ¯ + I2,1 (x¯ − x)) )
Analytic Sharp Fronts for the Surface Quasi-Geostrophic Equation
+
sgn(x¯ − x) R/Z
¯¯ (1 + (g(x) + I2,1 (x¯ − x))2 )
5/2
χ (x¯ − x)
¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ × pol in f (x), g(x), ¯ h(x), h (x), (x¯ − x), and I2,1 , I3,1 , I3,2 d x.
As for U2 , (18) becomes
1
U2 =
2] ¯¯ [1 + (g(x))
1/2
2 2 ¯¯ + [1 + (g(x)) ]
R/Z
R/Z
(25)
(h¯¯ (z) − h¯¯ (x))K 1 (x, z)dz χ (x¯ − x) h¯¯ (x) − 2I3,1
1 1 1 × d x. ¯ 5/2 − 2 ]5/2 ¯¯ |x¯ − x| [1 + (g(x)) ¯¯ 1 + [g(x) + I2,1 (x¯ − x)]2
271
(26)
As for U3 (20) becomes 1 1 1 ¯ ¯ χ (z − x)dz. (27) U3 = − h (z) 1/2 1/2 2 2 ¯ ¯ |z − x| [1 + (g(z)) ¯ ] [1 + (g(x)) ¯ ] R/Z 1 1/ We will incorporate the evolution equation for the length L(t) = 0 [1+(ϕ (y, t))2 ] 2 dy of the curve to our system (as it will appear in the change of coordinates involving arc length). We have L (t) = =
1
ϕ (x)ϕt (x)
1/ 0 [1 + (ϕ (x))2 ] 2 ϕ (x)
dx
1
1 ϕ (x) ϕ (x) 2 d x + 1/ 1/ 1/ 0 [1 + (ϕ (x))2 ] 2 [1 + (ϕ (x))2 ] 2 0 [1 + (ϕ (x))2 ] 2 ϕ (x)ϕ (x)I − [1 + (ϕ (x))2 ]I + ϕ (x)I 2 (x¯ − x) − ϕ (x)I I (x¯ − x) 2,1 3,1 2,1 3,1 2,1 × [1 + (ϕ (x) + I2,1 (x¯ − x))2 ]3/2 R/Z
× sgn(x¯ − x)χ (x − x)d ¯ xd ¯ x 1 ϕ (x) ϕ (x) − ϕ (x) ¯ + χ (x − x)d ¯ xd ¯ x, 1/2 1/ 2 2 R/Z 0 [1 + (ϕ (x)) ]
(28)
which in terms of the new unknowns becomes L (t) = 2
1
×
1 ¯¯ ¯¯ ¯¯ g(x) g(x) h(x) d x + 1/ 1/ 2] 2 2 ]1/2 ¯¯ 2] 2 ¯¯ ¯¯ 0 [1 + (g(x)) 0 [1 + (g(x)) [1 + (g(x)) ¯¯ 2 2 ]I ¯¯ ¯¯ ¯¯ ¯¯ h(x)I g(x) 2,1 − [1 + (g(x)) 3,1 + h(x)I 2,1 I3,1 ( x¯ − x) 2,1 ( x¯ − x) − g(x)I R/Z
¯¯ [1 + (g(x) + I2,1 (x¯ − x))2 ]3/2
× sgn(x¯ − x)χ (x − x)d ¯ xd ¯ x 1 ¯¯ ¯¯ ¯¯ x) χ (x − x) ¯ g(x) g(x) − g( ¯ d xd ¯ x. + 1/2 1/ 2 ¯ 2 |x¯ − x| ¯¯ ¯ ] 2 R/Z [1 + (g(x) 0 [1 + (g(x)) + I2,1 (x¯ − x)) ]
(29)
We will rewrite the 4 equations of our system (22), (23), (24) and (29) in new coordinates to simplify the most singular term in (24). We introduce a renormalized
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C. Fefferman, J. L. Rodrigo
arc length, that is we divide by the total length, to keep the period of the new functions constant in time. We define (we use g¯¯ and ϕ interchangeably) new coordinates s and τ given by 1 s := R(x, t) = L(t)
x
[1 + (ϕ (y, t))2 ] dy τ := t. 1/2
0
Notice that ∂ 1 ∂ ∂R ∂ = ∂x = , ∂x ∂s ∂ x ∂s ∂s
∂x ∂ ∂ ∂R ∂ ∂ ∂ = − ∂∂τx + = + , ∂t ∂s ∂τ ∂t ∂s ∂τ ∂s
and so ∂ L(t) ∂ = . 1/2 2 ∂s ∂ x [1 + (ϕ (x)) ] We now consider dz =
L(t) d s¯ 1/ [1+(ϕ (z))2 ] 2
dz |z−x| .
We need to write it in terms of the new variables. Notice that
, and so
L(t) d s¯ 1 dz = 1/ 2 |z − x| [1 + (ϕ (z(¯s )))2 ] |z(¯s , t) − z(s, t)| 1 1 1 L(t) d s¯ + − = 1/ [1 + (ϕ (z(¯s )))2 ] 2 ∂z |s − s¯ | |z(¯s , τ ) − z(s, τ )| ∂z |s − s¯ | ∂ s¯ ∂ s¯ d s¯ 1 L(t) d s¯ 1 . = + − |s − s¯ | [1 + (ϕ (z(¯s )))2 ]1/2 |z(¯s , τ ) − z(s, τ )| ∂z |s − s¯ | ∂ s¯ Also, we write χ (z − x) = χ˜ (s − s¯ ) + χ (z − x) − χ˜ (s − s¯ ). We introduce new unknowns f¯(s, τ ) = f¯¯(x(s, τ ), τ )
¯¯ g(s, ¯ τ ) = g(x(s, τ ), τ )
¯¯ ¯ τ ) = h(x(s, h(s, τ ), τ )
and Ja,b (s, s¯ , τ ) = Ia,b (x(s, τ ), x(¯ ¯ s , τ ), τ ). We can rewrite the system formed by Eqs. (22), (23), (24) and (29) as ∂R ¯ f¯τ (s, τ ) + f s (s, τ ) = ∂t
¯ + J3,1 (x(¯ ¯ s ) − x(s)) h(s) [1 + (g(s) ¯ + J2,1 (x(¯ ¯ s ) − x(s)))2 ]1/2 L(t)d s¯ × sgn(s − s¯ )χ (x(¯ ¯ s ) − x(s)) , [1 + (g(¯ ¯ s ))2 ]1/2
(30)
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273
where we have used sgn(x(s) − x(¯ ¯ s )) = sgn(s − s¯ ). Equation (23) becomes g¯ τ (s, τ ) + +L(t)
¯ h(s) ∂R g¯ s (s, τ ) = 2 ∂t 2 ]1/2 [1 + (g(s)) ¯ 2 ¯ s ) − x(s)) − g(s)J 2 ]J ¯ ¯ g(s) ¯ h(s)J ¯ ¯ ¯ s ) − x(s)) 2,1 − [1 + (g(s)) 3,1 + h(s)J 2,1 J3,1 ( x(¯ 2,1 ( x(¯
[1 + (g(s) ¯ + J2,1 (x(¯ ¯ s ) − x(s)))2 ]3/2
R/Z
× sgn(¯s − s) +L(t)
R/Z
χ(x(s) − x(¯ ¯ s ))d s¯ [1 + (g(¯ ¯ s ))2 ]1/2 g(s) ¯ − g(¯ ¯ s) 1/2
[1 + (g(s) ¯ + J2,1 (x(¯ ¯ s ) − x(s)))2 ]
χ (x(s) − x(¯ ¯ s )) d s¯ , |x(¯ ¯ s ) − x(s)| [1 + (g(¯ ¯ s ))2 ]1/2
(31)
(24) becomes ∂R ¯ h¯ τ (s, τ ) + h s (s, τ ) ∂t 1 =− (h¯ (¯s ) − h¯ (s))[χ(s ˜ − s¯ ) + χ (z(¯s ) − z(s)) − χ˜ (s − s¯ )] L(τ ) R/Z 1 L(t) 1 1 d s¯ + − × 1/ |s − s¯ | [1 + (g(¯ ¯ s ))2 ] 2 |z(¯s , τ ) − z(s, τ )| ∂z |s − s¯ | ∂ s¯ + U¯ 1 + U¯ 2 + U¯ 3 ,
(32)
which we can rewrite as 1 ¯h τ (s, τ ) = − 1 d s¯ + U¯ 1 + U¯ 2 + U¯ 3 + U¯ 4 , (h¯ (¯s ) − h¯ (s))χ(s ˜ − s¯ ) L(τ ) R/Z |¯s − s| (33) with ∂R ¯ U¯ 4 = − h s (s, τ ) ∂t (h¯ (¯s ) − h¯ (s)) − −
R/Z
1 [1 + (g(¯ ¯ s ))2 ]
1/2
1 1 χ˜ (s − s¯ )d s¯ − |z(¯s , τ ) − z(s, τ )| ∂z |s − s¯ | ∂ s¯
(h¯ (¯s ) − h¯ (s))[χ (z(¯s ) − z(s)) − χ(s ˜ − s¯ )]
R/Z
d s¯ 1 . 1/ |z(¯s , t) − z(s, t)| [1 + (g(¯ ¯ s )))2 ] 2 (34)
Finally for L, (29) becomes,
×
1 ¯ g(s) ¯ h(s) g(s) ¯ 2 ds + (L(τ )) 3/ 2] 2] 2 [1 + (g(s)) ¯ [1 + ( g(s)) ¯ 0 0 2 ( x(¯ 2 ]J ¯ ¯ g(s) ¯ h(s)J ¯ ¯ s ) − x(s)) − g(s)J ¯ ¯ s ) − x(s)) 2,1 − [1 + (g(s)) 3,1 + h(s)J 2,1 J3,1 ( x(¯
L (τ ) = 2L(t)
1
2,1
[1 + (g(s) ¯ + J2,1 (x(¯ ¯ s ) − x(s)))2 ]3/2 1 1 g(s) ¯ 2 × sgn(¯s − s)χ (x(s) − x(¯ ¯ s )) d s ¯ ds + (L(τ )) 1/ 2] [1 + (g(s)) ¯ [1 + (g(¯ ¯ s ))2 ] 2 0 1 g(s) ¯ − g(¯ ¯ s) χ (x(s) − x(¯ ¯ s )) × d s¯ ds. 1/2 2 | x(¯ ¯ s ) − x(s)| [1 + (g(¯ ¯ s ))2 ]1/2 ¯ + J2,1 (x(¯ ¯ s ) − x(s))) ] R/Z [1 + (g(s) R /Z
(35)
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The final transformation that we make in the equation is defining new unknowns f, g, h and l as f¯(s, τ ) = f (s, τ ) + f¯0 (s), where f¯0 (s) = f¯(s, 0), g(s, ¯ τ ) = g(s, τ ) + g¯ 0 (s), where g¯ 0 (s) = g(s, ¯ 0), ¯ 0), ¯ τ ) = h(s, τ ) + h¯ 0 (s), where h¯ 0 (s) = h(s, h(s, L(t) = l(t) + L(0). The system of equations becomes ∂R ∂ R ¯ f s (s, τ ) + f (s) ∂t ∂t 0 h(s) + h¯ 0 (s) + J3,1 (x(¯ ¯ s ) − x(s)) = [l(τ ) + L 0 ] sgn(s − s¯ )χ (x(¯ ¯ s) ¯ s ) − x(s)))2 ]1/2 R/Z [1 + (g(s) + g¯ 0 (s) + J2,1 ( x(¯ d s¯ −x(s)) , (36) [1 + (g(¯s ) + g¯ 0 (¯s ))2 ]1/2 ∂R ∂R h(s) + h¯ 0 (s) gs (s, τ ) + g0 (s) = 2 gτ (s, τ ) + 1/ ∂t ∂t [1 + (g(s) + g¯ 0 (s))2 ] 2 (g(s) + g¯ 0 (s))(h(s) + h¯ 0 (s))J2,1 − [1 + (g(s) + g¯ 0 (s))2 ]J3,1 +[l(τ ) + L 0 ] [1 + (g(s) + g¯ 0 (s) + J2,1 (x(¯ ¯ s ) − x(s)))2 ]3/2 R/Z 2 ( x(¯ ¯ s ) − x(s)) − (g(s) + g¯ 0 (s))J2,1 J3,1 (x(¯ ¯ s ) − x(s)) (h(s) + h¯ 0 (s))J2,1 + [1 + (g(s) + g¯ 0 (s) + J2,1 (x(¯ ¯ s ) − x(s)))2 ]3/2 χ (x(s) − x(¯ ¯ s ))d s¯ × sgn(¯s − s) [1 + (g(¯s ) + g¯ 0 (¯s ))2 ]1/2 (g(s) + g¯ 0 (s)) − (g(¯s ) + g¯ 0 (¯s )) χ (x(s) − x(¯ ¯ s )) + [l(τ ) + L 0 ] 1/ 2 2 | x(¯ ¯ s ) − x(s)| ¯ s ) − x(s))) ] R/Z [1 + (g(s) + g¯ 0 (s) + J2,1 ( x(¯ f τ (s, τ ) +
d s¯ , [1 + (g(¯s ) + g¯ 0 (¯s ))2 ]1/2 1 1 d s¯ (h (¯s , τ ) − h (s, τ ))χ˜ (s − s¯ ) h τ (s, τ ) = − l(τ ) + L 0 R/Z |¯s − s| ×
+ U¯ 1 + U¯ 2 + U¯ 3 + U¯ 4 + U¯ 5 ,
where U¯ 5 = −
1 1 d s¯ , (h¯ (¯s ) − h¯ 0 (s))χ˜ (s − s¯ ) l(τ ) + L 0 R/Z 0 |¯s − s|
and finally
(g(s) + g¯ 0 (s))(h(s) + h¯ 0 (s)) ds [1 + (g(s) + g¯ 0 (s))2 ]3/2 0 1 (g(s) + g¯ 0 (s)) + [l(t) + L 0 ]2 2 0 [1 + (g(s) + g¯ 0 (s)) ]
l (τ ) = 2[l(t) + L 0 ]
1
(37)
(38)
(39)
Analytic Sharp Fronts for the Surface Quasi-Geostrophic Equation
275
(g(s) + g¯ 0 (s))(h(s) + h¯ 0 (s))J2,1 − [1 + (g(s) + g¯ 0 (s))2 ]J3,1 [1 + ((g(s) + g¯ 0 (s)) + J2,1 (x(¯ ¯ s ) − x(s)))2 ]3/2 R/Z ¯ s ) − x(s)) − (g(s) + g¯ 0 (s))J2,1 J3,1 (x(¯ ¯ s ) − x(s)) (h(s) + h¯ 0 (s))J 2 (x(¯
× +
2,1
[1 + ((g(s) + g¯ 0 (s)) + J2,1 (x(¯ ¯ s ) − x(s)))2 ]3/2 d s¯ ds × sgn(¯s − s)χ (x(s) − x(¯ ¯ s )) [1 + (g(¯s ) + g¯ 0 (¯s ))2 ]1/2 1 (g(s) + g¯ 0 (s)) + [l(t) + L 0 ]2 2 0 [1 + (g(s) + g¯ 0 (s)) ] (g(s) + g¯ 0 (s)) − (g(¯s ) + g¯ 0 (¯s )) × 1/ ¯ s ) − x(s)))2 ] 2 R/Z [1 + (g(s) + g¯ 0 (s) + J2,1 ( x(¯ d s¯ ¯ s )) χ (x(s) − x(¯ × ds. |x(¯ ¯ s ) − x(s)| [1 + (g(¯s ) + g¯ 0 (¯s ))2 ]1/2
(40)
In order to adapt the system to the version of Cauchy-Kowaleski that we will use we rewrite the system as τ ∂R ∂ R ¯ − f (s, τ ) = f s (s, t¯) − f (s) ∂ t¯ ∂ t¯ 0 0 h(s) + h¯ 0 (s) + J3,1 (x(¯ ¯ s ) − x(s)) ¯ + [l(t ) + L 0 ] ¯ s ) − x(s)))2 ]1/2 R/Z [1 + (g(s) + g¯ 0 (s) + J2,1 ( x(¯ d s¯ d t¯, (41) × sgn(s − s¯ )χ (x(¯ ¯ s ) − x(s)) [1 + (g(¯s ) + g¯ 0 (¯s ))2 ]1/2 τ ∂R ∂R h(s) + h¯ 0 (s) g(s, τ ) = − gs (s, t¯) − g0 (s) + 2 1/ ∂ t¯ ∂ t¯ 0 [1 + (g(s) + g¯ 0 (s))2 ] 2 (g(s) + g¯ 0 (s))(h(s) + h¯ 0 (s))J2,1 − [1 + (g(s) + g¯ 0 (s))2 ]J3,1 + [l(t¯) + L 0 ] [1 + (g(s) + g¯ 0 (s)) + J2,1 (x(¯ ¯ s ) − x(s)))2 ]3/2 R/Z 2 ( x(¯ ¯ s ) − x(s)) − (g(s) + g¯ 0 (s))J2,1 J3,1 (x(¯ ¯ s ) − x(s)) (h(s) + h¯ 0 (s))J2,1 + [1 + (g(s) + g¯ 0 (s) + J2,1 (x(¯ ¯ s ) − x(s)))2 ]3/2 χ (x(s) − x(¯ ¯ s ))d s¯ × sgn(¯s − s) [1 + (g(¯s ) + g¯ 0 (¯s ))2 ]1/2 (g(s) + g¯ 0 (s)) − (g(¯s ) + g¯ 0 (¯s )) + [l(t¯) + L 0 ] 1/ ¯ s ) − x(s)))2 ] 2 R/Z [1 + (g(s) + g¯ 0 (s) + J2,1 ( x(¯ d s¯ ¯ s )) χ (x(s) − x(¯ d t¯, (42) × |x(¯ ¯ s ) − x(s)| [1 + (g(¯s ) + g¯ 0 (¯s ))2 ]1/2 τ h(s, τ ) = eim(k)a(τ −t¯) (U¯ 1 (t¯) + U¯ 2 (t¯) + U¯ 3 (t¯) + U¯ 4 (t¯) + U¯ 5 (t¯) )d t¯, (43) 0
where a(t) := 0
t
1 dr, l(r ) + L 0
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and m(k) ∼ k ln|k|. τ l(τ ) = 2[l(t¯) + L 0 ] 0
1 0
(g(s) + g¯ 0 (s))(h(s) + h¯ 0 (s)) ds [1 + (g(s) + g¯ 0 (s))2 ]3/2
(g(s) + g¯ 0 (s)) + [l(t¯) + L 0 ]2 [1 + (g(s) + g¯ 0 (s))2 ] 0 (g(s) + g¯ 0 (s))(h(s) + h¯ 0 (s))J2,1 − [1 + (g(s) + g¯ 0 (s))2 ]J3,1 × [1 + ((g(s) + g¯ 0 (s)) + J2,1 (x(¯ ¯ s ) − x(s)))2 ]3/2 R/Z 2 ( x(¯ ¯ s ) − x(s)) − (g(s) + g¯ 0 (s))J2,1 J3,1 (x(¯ ¯ s ) − x(s)) (h(s) + h¯ 0 (s))J2,1 + [1 + ((g(s) + g¯ 0 (s)) + J2,1 (x(¯ ¯ s ) − x(s)))2 ]3/2 d s¯ ds × sgn(¯s − s)χ (x(s) − x(¯ ¯ s )) [1 + (g(¯s ) + g¯ 0 (¯s ))2 ]1/2 1 (g(s) + g¯ 0 (s)) + [l(t¯) + L 0 ]2 [1 + (g(s) + g¯ 0 (s))2 ] 0 (g(s) + g¯ 0 (s)) − (g(¯s ) + g¯ 0 (¯s )) × 1/ ¯ s ) − x(s)))2 ] 2 R/Z [1 + ((g(s) + g¯ 0 (s)) + J2,1 ( x(¯ d s¯ ¯ s )) χ (x(s) − x(¯ × ds d t¯. (44) |x(¯ ¯ s ) − x(s)| [1 + (g(¯s ) + g¯ 0 (¯s ))2 ]1/2 1
We want to show that the right hand sides of Eqs. (41)–(44) can be written as an analytic function whose arguments only involve f, g, h, l, f , g , h , l with no higher derivatives of the unknowns being involved. Clearly right hand sides of that form would satisfy the Cauchy estimates required by the Cauchy-Kowaleski Theorem. We start by considering the terms U¯ 1 , . . . , U¯ 5 . We have 1 ¯ d s¯ . (45) (h¯ 0 (¯s ) − h¯ 0 (s))χ(s ˜ − s¯ ) U5 (s) = − |¯ s − s| R/Z The expression for U4 becomes ∂R (h (s, τ ) + h¯ 0 (s)) − U¯ 4 = − (h (¯s , τ ) + h¯ 0 (¯s ) − (h (s, τ ) + h¯ 0 (s))) ∂τ R /Z 1 1 1 χ(s ˜ − s¯ )d s¯ − × 1/ ∂z [1 + (g(¯s ) + g¯ 0 (¯s ))2 ] 2 |z(¯s , τ ) − z(s, τ )| |s − s¯ |
−
R /Z
∂ s¯
(h (¯s , τ ) + h¯ 0 (¯s ) − (h (s, τ ) + h¯ 0 (s)))[χ (z(¯s ) − z(s))
−χ(s ˜ − s¯ )]
d s¯ 1 . |z(¯s , t) − z(s, t)| [1 + (g(¯s ) + g¯ 0 (¯s ))2 ]1/2
We need to rewrite 1 1 − |z(¯s , τ ) − z(s, τ )| ∂z |s − s¯ | ∂ s¯ as an analytic function of the arguments described before.
(46)
Analytic Sharp Fronts for the Surface Quasi-Geostrophic Equation
Recall that z(s) = L(t)
s
1 d s¯¯ , 0 ([1+(g( ¯ s¯¯ ))2 ])1/2
1 sgn(¯s − s) s¯ L(t) =
1 L(t)
=
1 L(t)
=
1 L(t)
1
277
and so we have 1
−
1 1 (¯s − s) d s¯¯ s [1+(g( [1+(g(¯ ¯ s ))2 ]1/2 ¯ s¯¯ ))2 ]1/2 s¯ 1 1 − [1+(g(¯ d s¯¯ sgn(¯s − s) − s [1+(g( ¯ s ))2 ]1/2 ¯ s¯¯ ))2 ]1/2 s¯ 1 1 s¯ − s d s¯¯ [1+(g(¯ s [1+(g( ¯ s ))2 ]1/2 ¯ s¯¯ ))2 ]1/2 s¯ s¯ g(˜ ¯ s )g¯ (˜s ) d s¯¯ sgn(¯s − s) s s¯¯ [1+(g(˜ ¯ s ))2 ]3/2 s¯ 1 1 s¯ − s d s¯¯ [1+(g(¯ s [1+(g( ¯ s ))2 ]1/2 ¯ s¯¯ ))2 ]1/2 s¯ ¯ s )g¯ (˜s ) ¯ s − s) g(˜ 3/ d s¯ s (˜ sgn(¯s − s) [1+(g(˜ ¯ s ))2 ] 2
s¯ − s
s¯
1 1 d s¯¯ s [1+(g( ¯ s¯¯ ))2 ]1/2 [1+(g(¯ ¯ s ))2 ]1/2
,
and taking s˜ = (1 − ρ)s + ρ s¯ = s + ρ(¯s − s) (which leads to s˜ − s = ρ(¯s − s) and d s˜ = dρ(¯s − s)) we obtain 1
g((1−ρ)s+ρ ¯ s¯ )g¯ ((1−ρ)s+ρ s¯ ) dρ(¯s − s) 1 sgn(¯s − s) 0 ρ(¯s − s) [1+(g((1−ρ)s+ρ ¯ s¯ ))2 ]3/2 = 1 1 1 L(t) s¯ − s dρ(¯s − s) 2 1/
=
1 sgn(¯s L(t)
0 [1+(g((1−ρ)s+ρ [1+(g(¯ ¯ s )) ] ¯ s¯ ))2 ]1/2 1 g((1−ρ)s+ρ ¯ s¯ )g¯ ((1−ρ)s+ρ s¯ ) dρ 0 ρ [1+(g((1−ρ)s+ρ ¯ s¯ ))2 ]3/2 − s) 1 . 1 1 dρ 1/ 2 1/ 0 [1+(g((1−ρ)s+ρ [1+(g(¯ ¯ s )) ] 2 ¯ s¯ ))2 ] 2
2
Using this expression U4 becomes ∂R 1 (h (s, τ ) + h¯ 0 (s)) − (h (¯s , τ ) + h¯ 0 (¯s ) − (h (s, τ ) + h¯ 0 (s))) U¯ 4 = − ∂τ L(t) R/Z 1 [g((1−ρ)s+ρ s¯ )+g¯0 ((1−ρ)s+ρ s¯ )][g ((1−ρ)s+ρ s¯ )−g¯ ((1−ρ)s+ρ s¯ )] dρ 0 ρ [1+(g((1−ρ)s+ρ s¯ )+g¯ 0 ((1−ρ)s+ρ s¯ ))2 ]3/2 χ(s ˜ − s¯ )d s¯ × sgn(¯s − s) 1 1 0 [1+(g((1−ρ)s+ρ s¯ )+g¯ ((1−ρ)s+ρ s¯ ))2 ]1/2 dρ 0 ¯ ¯ − (h (¯s , τ ) + h 0 (¯s ) − (h (s, τ ) + h 0 (s)))[χ (z(¯s ) − z(s)) R/Z
−χ(s ˜ − s¯ )]
d s¯ 1 . |z(¯s , t) − z(s, t)| [1 + (g(¯s ) + g¯ 0 (¯s ))2 ]1/2
We still need to consider ∂R −L (τ ) = 2 ∂τ L (τ )
x 0
∂R ∂τ
(47)
in the above expression. We have
1 [1 + (ϕ (y, t)) ] dy + L(τ )
2
1/2
0
x
ϕ (y)ϕt (y) [1 + (ϕ (y, t))2 ]
1/2
dy,
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C. Fefferman, J. L. Rodrigo
and so in terms of f¯ and g¯ it becomes (the last integral in the RHS above is L when x = 1, and so we can use expression (35)) ∂R −L (τ ) = 2 ∂τ L (τ )
s
[1 + (g¯ (¯s , t))2 ]
L(τ )
1/2
0
1/2
[1 + (g(¯ ¯ s , t))2 ]
d s¯
s s ¯ ) 1 1 g(r ¯ )h(r g(r ¯ ) 2 dr + 2L(t) (L(τ )) 2 ]3/2 L(τ ) L(τ ) [1 + (g(r ¯ ))2 ] [1 + ( g(r ¯ )) 0 0 2 2 ¯ ¯ g(r ¯ )h(r )J2,1 − [1 + (g(r ¯ )) ]J3,1 + h(r )J2,1 (x(¯ ¯ s ) − x(r )) − g(r ¯ )J2,1 J3,1 (x(¯ ¯ s ) − x(r )) × [1 + (g(r ¯ ) + J2,1 (x(¯ ¯ s ) − x(r )))2 ]3/2 R/Z s g(r ¯ ) 1 1 2 ×sgn(¯s − r )χ(x(r ) − x(¯ ¯ s )) d s ¯ dr + (L(τ )) L(τ ) [1 + (g(r ¯ ))2 ] [1 + (g(¯ ¯ s ))2 ]1/2 0 1 g(r ¯ ) − g(¯ ¯ s) χ (x(r ) − x(¯ ¯ s )) × d s¯ dr, (48) 1/ ¯ ) − x(¯s )| [1 + (g(¯ ¯ s ))2 ]1/2 ¯ ) + J2,1 (x(¯ ¯ s ) − x(r )))2 ] 2 |x(r R/Z [1 + ( g(r +
which in terms of g and g¯ 0 becomes
(g(r ) + g¯ 0 (r ))(h(r ) + h¯ 0 (r )) dr [1 + (g(r ) + g¯ 0 (r ))2 ]3/2 0 s (g(r ) + g¯ 0 (r )) + [l(τ ) + L 0 ] [1 + (g(r ) + g¯ 0 (r ))2 ] 0 (g(r ) + g¯ 0 (r ))(h(r ) + h¯ 0 (r ))J2,1 − [1 + (g(r ) + g¯ 0 (r ))2 ]J3,1 × [1 + ((g(r ) + g¯ 0 (r )) + J2,1 (x(¯ ¯ s ) − x(r )))2 ]3/2 R /Z
∂R −L (τ ) = s+2 ∂τ L(τ )
+
s
2 ( x(¯ (h(r ) + h¯ 0 (r ))J2,1 ¯ s ) − x(r )) − (g(r ) + g¯ 0 (r ))J2,1 J3,1 (x(¯ ¯ s ) − x(r ))
[1 + ((g(r ) + g¯ 0 (s)) + J2,1 (x(¯ ¯ s ) − x(r )))2 ]3/2 d s¯ × sgn(¯s − r )χ(x(r ) − x(¯ ¯ s )) dr [1 + (g(¯s ) + g¯ 0 (¯s ))2 ]1/2 s (g(r ) + g¯ 0 (r )) (g(r ) + g¯ 0 (r )) − (g(¯s ) + g¯ 0 (¯s )) + [l(τ ) + L 0 ] 1/ 2] [1 + (g(r ) + g ¯ (r )) 0 ¯ s ) − x(r )))2 ] 2 0 R/Z [1 + ((g(r ) + g¯ 0 (s)) + J2,1 ( x(¯
×
¯ s )) d s¯ χ (x(r ) − x(¯ dr. |x(r ) − x(¯ ¯ s )| [1 + (g(¯s ) + g¯ 0 (¯s ))2 ]1/2
As for U¯ 3 we have 1 1 1 ¯ ¯ χ (z(¯s ) − z(s))d s¯ . − U3 = h (¯s ) 1/2 1/2 2 2 |z(¯s ) − z(s)| [1 + (g(¯ ¯ s )) ] [1 + (g(s)) ¯ ] We need to rewrite
1
1/ [1 + (g(¯ ¯ s ))2 ] 2
−
1 2 ]1/2 [1 + (g(s)) ¯ 1/
=
1 |z(¯s ) − z(s)| 1/2
2 ] 2 − [1 + (g(¯ ¯ ¯ s ))2 ] 1 [1 + (g(s)) 1/ 2 ]1/2 [1 + (g(¯ L(t) [1 + (g(s)) ¯ ¯ s ))2 ] 2
1
s¯
s¯ ¯ s¯¯)g¯ (s¯¯) d s¯¯ − s g( 1 [1+(g( ¯ s¯¯ ))2 ]1/2 = 1/ 2 ]1/2 [1 + (g(¯ L(t) [1 + (g(s)) ¯ ¯ s ))2 ] 2 s¯
1 d s¯¯ s [1+(g( ¯ s¯¯ ))2 ]1/2
1
1 d s¯¯ s [1+(g( ¯ s¯¯ ))2 ]1/2
,
Analytic Sharp Fronts for the Surface Quasi-Geostrophic Equation
279
and taking s¯¯ = (1 − ρ)s + ρ s¯ = s + ρ(¯s − s) (which leads to s¯¯ − s = ρ(¯s − s) and d s¯¯ = dρ(¯s − s)) we obtain 1 ¯ s¯ )g¯ ((1−ρ)s+ρ s¯ ) dρ(¯s − s) − 0 g((1−ρ)s+ρ 1 [1+(g((1−ρ)s+ρ ¯ s¯ ))2 ]1/2 = 1/ 1/ 1 1 2 ] 2 [1 + (g(¯ L(t) [1 + (g(s)) ¯ ¯ s ))2 ] 2 0 [1+(g((1−ρ)s+ρ dρ(¯s − s) ¯ s¯ ))2 ]1/2 1 g((1−ρ)s+ρ ¯ s¯ )g¯ ((1−ρ)s+ρ s¯ ) dρ 1 − 0 [1+(g((1−ρ)s+ρ 1 ¯ s¯ ))2 ]1/2 = . 1 1/2 1/2 1 2 2 L(t) [1 + (g(s)) ¯ ] [1 + (g(¯ ¯ s )) ] 2 1/ dρ 0 [1+(g((1−ρ)s+ρ ¯ s¯ )) ]
2
Using this expression U3 becomes
1 − U¯ 3 = [h (¯s ) + h¯ 0 (¯s )] L(t) 1 × 1 1
1 0
[g((1−ρ)s+ρ s¯ )+g¯ 0 ((1−ρ)s+ρ s¯ )][g ((1−ρ)s+ρ s¯ )+g¯ 0 ((1−ρ)s+ρ s¯ )] [1+(g((1−ρ)s+ρ s¯ )+g¯ 0 ((1−ρ)s+ρ s¯ ))2 ]1/2 1/2
1/2
[1 + (g(s) + g¯ 0 (s))2 ] [1 + (g(¯s ) + g¯ 0 (¯s ))2 ]
0 [1+(g((1−ρ)s+ρ s¯ )+g¯ 0 ((1−ρ)s+ρ s¯ ))2 ]1/2
dρ
χ (z(¯s ) − z(s))d s¯ .
As for U¯ 2 we have U¯ 2 =
1 2] [1 + (g(s)) ¯
1/
[1 + (g(¯ ¯ s ))2 ] 2 ¯ h (¯s ) L(τ ) R/Z (
1/2 1/
2] 2 L(τ ) [1 + (g(s)) ¯ d s¯ − h¯ (s))K 1 (x(s), z(¯s )) 1/ L(τ ) [1 + (g(¯ ¯ s ))2 ] 2 [1 + (g(s)) 2 ]1/2 ¯ 2 2 ] χ (x(¯ ¯ s ) − x(s)) +[1 + (g(s)) ¯ h¯ (s) − 2J3,1 L(τ ) R/Z 1 1 × 5/2 |x(¯ ¯ s ) − x(s)| 2 1 + [g(s) ¯ + J2,1 (x(¯ ¯ s ) − x(s))] L(τ ) 1 − d s¯ . 1/ 5/2 2 [1 + (g(¯ ¯ s ))2 ] 2 [1 + (g(s)) ¯ ]
And so we need to rewrite 1 1 1 − 5/2 2 ]5/2 |x(¯ ¯ s ) − x(s)| [1 + (g(s)) ¯ 1 + [g(s) ¯ + J2,1 (x(¯ ¯ s ) − x(s))]2
=
1 |x(¯ ¯ s ) − x(s)|
2] [1 + (g(s)) ¯
5/2
5/2 − 1 + [g(s) ¯ + J2,1 (x(¯ ¯ s ) − x(s))]2
2 ]5/2 [1 + (g(s)) ¯
5/2 . 2 1 + [g(s) ¯ + J2,1 (x(¯ ¯ s ) − x(s))]
dρ
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C. Fefferman, J. L. Rodrigo
Define f (τ ) =
5/2 1 1 + [g(s) ¯ + J2,1 τ ]2 , and use f (0) − f (τ ) = − 0 f (ρτ )dρτ
to obtain 1 |x(¯ ¯ s ) − x(s)| ¯ + J2,1 ρ(x(¯ ¯ s ) − x(s)))2 )3/2 [g(s) ¯ + J2,1 ρ(x(¯ ¯ s ) − x(s))]J2,1 dρ(x(¯ ¯ s ) − x(s)) −3 01 (1 + (g(s) × 5/2 2 ]5/2 1 + [g(s) ¯ + J2,1 (x(¯ [1 + (g(s)) ¯ ¯ s ) − x(s))]2
=
¯ + J2,1 ρ(x(¯ ¯ s ) − x(s)))2 )3/2 [g(s) ¯ + J2,1 ρ(x(¯ ¯ s ) − x(s))]J2,1 dρ −3 01 (1 + (g(s) = sgn(¯s − s). 5/2 5/2 2 2 1 + [g(s) ¯ + J2,1 (x(¯ [1 + (g(s)) ¯ ] ¯ s ) − x(s))]
Using this expression U2 becomes U¯ 2 =
1 1/2
[1 + (g(s) + g¯ 0 (s))2 ]
R/Z
1/2 [1 + (g(¯s ) + g¯ 0 (¯s ))2 ] [h (¯s ) + h¯ 0 (¯s )]
−[1 + (g(s) + g¯ 0 (s)) ] [h (s) + h¯ 0 (s)] × K 1 (x(s), z(¯s )) 2
1/2
+[1 + (g(s) + g¯ 0 (s))2 ]2
R/Z
1 1/2
[1 + (g(¯s ) + g¯ 0 (¯s ))2 ]
d s¯
1/2 χ(x(¯ ¯ s ) − x(s)) [1 + (g(s) + g¯ 0 (s))2 ] [h (s) + h¯ 0 (s)]
−2(l(τ ) + L 0 )J3,1 1 −3 0 (1 + (g(s) + g¯ 0 (s)+ J2,1 ρ(x(¯ ¯ s ) − x(s)))2 )3/2 [g(s)+ g¯ 0 (s) + J2,1 ρ(x(¯ ¯ s )−x(s))]J2,1 dρ × 5/2 5/2
[1 + (g(s) + g¯ 0 (s))2 ] ×
sgn(¯s − s) 1/2
[1 + (g(¯s ) + g¯ 0 (¯s ))2 ]
1 + [g(s) ¯ + J2,1 (x(¯ ¯ s ) − x(s))]2
d s¯ .
As for U¯ 1 (25) becomes U¯ 1 =
2 ¯ 3 ¯ g(s)( ¯ h(s)) h (s) − 3 3/ L(τ ) [1 + (g(¯ ¯ s ))2 ] 2
¯ h(s) + 1/ ¯ + J2,1 (x(¯ ¯ s ) − x(s)))2 ) 2 R/Z (1 + (g(s) ¯ + I3,1 (x(¯ ¯ s ) − x(s))][1 + (g(s) ¯ + I2,1 (x(¯ ¯ s ) − x(s)))g(s)] ¯ [h(s) − 3/ (1 + (g(s) ¯ + I2,1 (x(¯ ¯ s ) − x(s)))2 ) 2 L(τ ) ¯ s )) χ (x(s) − x(¯ × d s¯ 1/ |x(¯ ¯ s ) − x(s)| [1 + (g(¯ ¯ s ))2 ] 2 sgn(¯s − s) χ (x(¯ ¯ s ) − x(s)) + 5/ ¯ + J2,1 (x(¯ ¯ s ) − x(s)))2 ] 2 R/Z [1 + (g(s) 2 ]1/2 [1 + (g(s)) ¯ ¯ ¯ s) × pol in f¯(s), g(s), ¯ h(s), h¯¯ (s), (x(¯ L(τ ) L(τ ) −x(s)), and J2,1 , J3,1 , J3,2 d s¯ , 1/ [1 + (g(¯ ¯ s ))2 ] 2
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Analytic Sharp Fronts for the Surface Quasi-Geostrophic Equation
281
which leads to U¯ 1 =
3 [g(s) + g¯ 0 (s)][h(s) + h¯ 0 (s)]2 [h (s) + h¯ 0 (s)] − 3 3/ l(t) + L(0) [1 + (g(s) + g¯ 0 (s))2 ] 2
h(s) + h¯ 0 (s) + 1/ ¯ s ) − x(s)))2 ) 2 R/Z (1 + (g(s) + g¯ 0 (s) + J2,1 ( x(¯ [h(s) + h¯ 0 (s) + I3,1 (x(¯ ¯ s ) − x(s))][1 + (g(s) + g¯ 0 (s) + I2,1 (x(¯ ¯ s ) − x(s)))(g(s) + g¯ 0 (s))] − 3/ (1 + (g(s) + g¯ 0 (s) + I2,1 (x(¯ ¯ s ) − x(s)))2 ) 2 χ (x(s) − x(¯ l(t) + L(0) ¯ s )) × d s¯ |x(¯ ¯ s ) − x(s)| [1 + (g(¯s ) + g¯ 0 (¯s ))2 ]1/2 sgn(¯s − s) + χ (x(¯ ¯ s ) − x(s)) 5/ ¯ s ) − x(s)))2 ])2 ] 2 R/Z [1 + (g(s) + g¯ 0 (s) + J2,1 ( x(¯ 1/ [1 + (g(s) + g¯ 0 (s))2 ] 2 × pol in [ f (¯s ) + f¯0 (¯s )], [g(s) + g¯ 0 (s)], [h(s) + h¯ 0 (s)], [h (s) l(t) + L(0) l(t) + L(0) + h¯ 0 (s)], (x(¯ ¯ s ) − x(s)), and J2,1 , J3,1 , J3,2 d s¯ . 1/ [1 + (g(¯s ) + g¯ 0 (¯s ))2 ] 2
In order to complete the task of checking that the right hand sides of (41)–(44) have the correct analytic structure, the final point to consider is Ja,b . Recall that we have 1 Ia,b = Ia,b (x, x, ¯ t) = ϕ (a) ((1 − ρ)x¯ + ρx, t)ρ b dρ, 0
Ja,b (s, s¯ , τ ) = Ia,b (x(s, τ ), x(¯ ¯ s , τ ), τ ). In the rewriting of these expressions we will need to consider expressing f¯ (s), ¯ f (s), f¯ (s), g¯ (s) and g¯ (s) in terms of expressions that involve at most one derivative. Recall that ∂s =
L(t) L(t) ∂x = ∂ . 2 )1/2 x (1 + (ϕ (x))2 )1/2 (1 + (g(s)) ¯
We obtain L(t) L(t) ∂x f¯¯ = g(s), ¯ 1/ 2 2 )1/2 2 (1 + (ϕ (x)) ) (1 + (g(s)) ¯ L(t) ¯ g¯ (s) = h(s), 2 )1/2 (1 + (g(s)) ¯ L(t)g(s) ¯ g¯ (s) L(t) g(s) ¯ + g¯ (s) f¯ (s) = ∂s f¯ (s) = − 2 ]3/2 2 ]1/2 [1 + (g(s)) ¯ [1 + (g(s)) ¯ 2 h(s) ¯ ¯ (L(t))2 ¯ (L(t))2 (g(s)) + =− h(s), 2 ]2 2] [1 + (g(s)) ¯ [1 + (g(s)) ¯ L(t)g(s) ¯ g¯ (s) ¯ L(t) h(s) + h¯ (s) g¯ (s) = ∂s g¯ (s) = − 3/ 2 2 ]1/2 2 [1 + (g(s)) ¯ ] [1 + (g(s)) ¯ 2 g(s) ¯ ¯ L(t) (L(t))2 (h(s)) + h¯ (s), =− 2 2 2 ]1/2 [1 + (g(s)) ¯ ] [1 + (g(s)) ¯ f¯ (s) =
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and finally (a simple calculation shows) ¯ g¯ (s)h(s) 4(L(t))2 g(s) (L(t))2 + h¯ (s). f¯ (s) = − 2 ]3 2 ]2 [1 + (g(s)) ¯ [1 + (g(s)) ¯ Other recurrent expressions that will be needed are considered now:
s¯
1 d s¯¯ [1 + (g( ¯ s¯¯ ))2 ]1/2 1 1 = L(t) dρ(¯s − s), ¯ − ρ)s + ρ s¯ ))2 ]1/2 0 [1 + (g((1
x(¯ ¯ s ) − x(s) = L(t)
s
where we have taken s¯¯ = s + ρ(¯s − s) = ρ s¯ + (1 − ρ)s, yielding d s¯¯ = dρ(¯s − s). Also 1 ∂x = L(t) , 2 ]1/2 ∂s [1 + (g(s)) ¯ and hence ∂x (¯s − s) = L(t) ∂s
s¯
1 1 d s¯¯ − L(t) (¯s − s) 1/ 2 ]1/2 2 ¯ [1 + (g(s)) ¯ ¯ s¯ )) ] 2 s [1 + (g( s¯ 1 1 ¯¯ − d s = L(t) 1/ 2] 2 [1 + (g(s)) ¯ ¯ s¯¯ ))2 ]1/2 s [1 + (g( s¯ s¯ s¯¯ g(˜ ¯ s )g¯ (˜s ) g(˜ ¯ s )g¯ (˜s ) ¯¯ = −L(t) − d s ˜ d s (s − s˜ ) d s˜ = L(t) 3/ 2 [1 + (g(˜ ¯ s )) ] 2 [1 + (g(˜ ¯ s ))2 ]3/2 s s s 1 g(ρ ¯ s¯ + (1 − ρ)s)g¯ (ρ s¯ + (1 − ρ)s) = −L(t) (1 − ρ) dρ(¯s − s)2 . 2 ]3/2 [1 + ( g((1 ¯ − ρ)s + ρ s ¯ )) 0
x(¯ ¯ s ) − x(s)−
Also ∂2x L(t)g(s) ¯ g¯ (s) =− , 2 2 ]3/2 ∂s [1 + (g(s)) ¯ and so ∂x 1 ∂2x x(¯ ¯ s ) − x(s) − (¯s − s) − (¯s − s)2 ∂s 2 ∂s 2 s¯ 1 1 = L(t) d s¯¯ − L(t) (¯s − s) 1/ 2 ]1/2 2 ¯ 2 [1 + ( g(s)) ¯ ¯ s¯ )) ] s [1 + (g( L(t)g(s) ¯ g¯ (s) 1 − (¯s − s)2 − 2 ]3/2 2 [1 + (g(s)) ¯ s¯ 1 1 g(s) ¯ g¯ (s) = L(t) − + (s¯¯ − s)d s¯¯, 2 ]1/2 2 ]1/2 [1 + (g(s)) ¯ [1 + (g(s)) ¯ ¯ s¯¯ ))2 ]1/2 s [1 + (g(
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which by Taylor’s Theorem becomes
1 ∂2 (s¯¯ − s˜ )d s˜ d s¯¯ 2 [1 + (g(˜ ¯ s ))2 ]1/2 s ∂ s˜ s s¯ 2 1 1 ∂ = L(t) (¯s − s˜ )2 d s˜ 2 [1 + (g(˜ ¯ s ))2 ]1/2 2 s ∂ s˜ s¯ (g¯ (˜s ))2 g(˜ ¯ s )L(t)h¯ (˜s ) (g(˜ ¯ s ))2 (g¯ (˜s ))2 1 + − 3 = L(t) (¯s − s)2 d s¯ ¯ s ))2 ]2 [1 + (g(˜ ¯ s ))2 ]3/2 [1 + (g(˜ [1 + (g(˜ ¯ s ))2 ]5/2 2 s 1 (g¯ (˜s ))2 g(˜ ¯ s )L(t)h¯ (˜s ) (g(˜ ¯ s ))2 (g¯ (˜s ))2 1 (1−ρ)2 dρ(¯s −s)3 . = L(t) + −3 ¯ s ))2 ]2 [1 + (g(˜ ¯ s ))2 ]3/2 [1 + (g(˜ [1 + (g(˜ ¯ s ))2 ]5/2 2 0
= L(t)
s¯
s¯¯
We start by considering I2,1 . It is originally defined by Taylor’s formula, ϕ(x) ¯ = ϕ(x) + ϕ (x)(x¯ − x) + I2,1 (x¯ − x)2 , which in terms of f¯, g, ¯ . . . becomes ¯ s ) − x(s))2 . f¯(x) ¯ = f¯(x) + g(s)( ¯ x(¯ ¯ s ) − x(s)) + J2,1 (x(¯ Also, Taylor-expanding f (s) directly we obtain 1 ¯ ¯ ¯ f¯ ((1 − ρ)¯s + ρs)ρdρ(¯s − s)2 . f (¯s ) = f (s) + f (s)(¯s − s) + 0
And so ∂x ∂x (¯s − s) + g(s)[x(¯ ¯ s ) − x(s) − (¯s − s)] + J2,1 (x(¯ ¯ s ) − x(s))2 ∂s ∂s 1 = f¯ (s)(¯s − s) + f¯ ((1 − ρ)¯s + ρs)ρdρ(¯s − s)2 ,
g(s) ¯
0
which yields ∂x 1 (¯s − s)] g(s)[ ¯ x(¯ ¯ s ) − x(s) − 2 (x(¯ ¯ s ) − x(s)) ∂s 1 + f¯ ((1 − ρ)¯s + ρs)ρdρ(¯s − s)2 0 1 = ¯ 2 − g(s)L(t) 1 1 L(t) 0 dρ(¯s − s) [1+(g((1−ρ)s+ρ ¯ s¯ ))2 ]1/2 1 g(ρ ¯ s¯ + (1 − ρ)s)g¯ (ρ s¯ + (1 − ρ)s) × (1 − ρ) dρ [1 + (g((1 ¯ − ρ)s + ρ s¯ ))2 ]3/2 0 1 ¯ ¯ − ρ)¯s + ρs))2 h((1 − ρ)¯s + ρs) (L(t))2 (g((1 − + 2 ]2 [1 + ( g((1 ¯ − ρ)s + ρ s ¯ )) 0 2 (L(t)) ¯ + h((1 − ρ)¯s + ρs)dρ . [1 + (g((1 ¯ − ρ)s + ρ s¯ ))2 ]
J2,1 =
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The next term to consider is J3,1 . We have ϕ (x) ¯ = ϕ (x) + ϕ (x)(x¯ − x) + I3,1 (x¯ − x)2 , which in terms of f¯, . . . becomes ¯ ¯ s ) − x(s))2 , g(¯ ¯ s ) = g(s) ¯ + h(s)( x(¯ ¯ s ) − x(s)) + J3,1 (x(¯ ∂x ¯ ¯ ∂ x + h(s)( x(¯ ¯ s ) − x(s) − ) + J3,1 (x(¯ ¯ s ) − x(s))2 . g(¯ ¯ s ) = g(s) ¯ + h(s) ∂s ∂s Taylor for g¯ yields g(¯ ¯ s ) = g(s) ¯ + g¯ (s)(¯s − s) +
1
g¯ ((1 − ρ)¯s + ρs)ρdρ(¯s − s)2 ,
0
and so J3,1
∂x 1 ¯ − h(s) x(¯ ¯ s ) − x(s) − = (x(¯ ¯ s ) − x(s))2 ∂s 1 g¯ ((1 − ρ)¯s + ρs)ρdρ(¯s − s)2 + 0 1 × ¯ 2 − g(s)L(t) 1 1 L(t) 0 dρ(¯ s − s) [1+(g((1−ρ)s+ρ ¯ s¯ ))2 ]1/2 1 g(ρ ¯ s¯ + (1 − ρ)s)g¯ (ρ s¯ + (1 − ρ)s) × (1 − ρ) dρ [1 + (g((1 ¯ − ρ)s + ρ s¯ ))2 ]3/2 0 1 ¯ s¯ + (1 − ρ)s))2 g(ρ ¯ s¯ + (1 − ρ)s) (L(t))2 (h(ρ − + [1 + ( g((1 ¯ − ρ)s + ρ s¯ ))2 ]2 0 L(t) ¯ + h (ρ s¯ + (1 − ρ)s)dρ . [1 + (g((1 ¯ − ρ)s + ρ s¯ ))2 ]1/2
The last term that we need to consider is I3,2 1 1 ϕ(x) ¯ = ϕ(x) + ϕ (x)(x¯ − x) + ϕ (x)(x¯ − x)2 + I3,2 (x¯ − x)3 , 2 2 which becomes 1 1¯ ¯ s ) − x(s))3 , f¯(¯s ) = f¯(s) + g(s)( ¯ x(¯ ¯ s ) − x(s)) + h(s)( x(¯ ¯ s ) − x(s))2 + J3,2 (x(¯ 2 2 2 1 ∂2x ∂x 1 ¯ ∂x 2 (¯s − s) + (¯ s − s) f¯(¯s ) = f¯(s) + g(s)( ¯ (¯ s − s) )) + h(s) ∂s 2 ∂s 2 2 ∂s 2 ∂x 1∂ x +g(s)( ¯ x(¯ ¯ s ) − x(s) − (¯s − s)2 ) (¯s − s) − ∂s 2 ∂s 2 2 ∂x 1¯ (¯s − s) + h(s) (x(¯ ¯ s ) − x(s))2 − 2 ∂s 1 + J3,2 (x(¯ ¯ s ) − x(s))3 . 2
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On the other hand Taylor for f¯ yields 1 1 f¯(¯s ) = f¯(s) + f¯ (s)(¯s − s) + f¯ (s)(¯s − s)2 + 2 2
1
f¯ (ρ s¯ + (1 − ρ)s)ρ 2 dρ.
0
This yields ∂x 2 1 ∂2x − g(s)( ¯ x(¯ ¯ s ) − x(s) − (¯s − s)2 ) (¯s − s) − 3 (x(¯ ¯ s ) − x(s)) ∂s 2 ∂s 2 2 1 1 ∂x 1¯ 2 2 ¯ (¯s − s) + f (ρ s¯ + (1 − ρ)s)ρ dρ − h(s) (x(¯ ¯ s ) − x(s)) − 2 ∂s 2 0 ∂x 1 ∂2x 2 − g(s)( ¯ x(¯ ¯ s ) − x(s) − (¯s − s) − (¯s − s)2 ) = 3 (x(¯ ¯ s ) − x(s)) ∂s 2 ∂s 2 ∂x 1¯ ∂x ∂x (¯s − s))2 + 2 (¯s − s) x(¯ ¯ s ) − x(s) − (¯s − s) − h(s) (x(¯ ¯ s ) − x(s) − 2 ∂s ∂s ∂s 1 1 ¯ + f (ρ s¯ + (1 − ρ)s)ρ 2 dρ , 2 0
J3,2 =
and so we obtain
2
3 1 L(t) 0 dρ (¯s − s)3 [1+(g((1−ρ)s+ρ ¯ s¯ ))2 ]1/2 1 (g¯ (˜s ))2 g(˜ ¯ s )L(t)h¯ (˜s ) × − g(s) ¯ L(t) + ¯ s ))2 ]2 [1 + (g(˜ ¯ s ))2 ]3/2 [1 + (g(˜ 0 (g(˜ ¯ s ))2 (g¯ (˜s ))2 1 (1 − ρ)2 dρ (x¯ − x)3 −3 [1 + (g(˜ ¯ s ))2 ]5/2 2 1 2 1¯ g(ρ ¯ s¯ + (1 − ρ)s)g¯ (ρ s¯ + (1 − ρ)s) 2 − h(s) − L(t) (1 − ρ) dρ(¯ s − s) 2 [1 + (g((1 ¯ − ρ)s + ρ s¯ ))2 ]3/2 0 1 L(t) ¯ −h(s) (¯s − s) − L(t) (1 − ρ) 2 ]1/2 [1 + (g(s)) ¯ 0 g(ρ ¯ s¯ + (1 − ρ)s)g¯ (ρ s¯ + (1 − ρ)s) 2 dρ(¯ s − s) × [1 + (g((1 ¯ − ρ)s + ρ s¯ ))2 ]3/2 1 ¯ s¯ + (1 − ρ)s) ¯ s¯ + (1 − ρ)s)g¯ (ρ s¯ + (1 − ρ)s)h(ρ 4(L(t))2 g(ρ 1 − + 2 3 2 0 [1 + (g((1 ¯ − ρ)s + ρ s¯ )) ] 2 (L(t)) ¯ (ρ s¯ + (1 − ρ)s)ρ 2 dρ(¯s − s)3 . + h [1 + (g((1 ¯ − ρ)s + ρ s¯ ))2 ]2 1
This completes the analysis of the system of 4 equations to which we will apply the Cauchy-Kowaleski Theorem. In the next section we will briefly describe the abstract scheme and check that the system formed by (41)–(44) fits that scheme.
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3. Cauchy-Kowaleski Theorem We begin with some definitions. Definition 1. A Banach scale {X ρ , 0 < ρ < ρ0 } with norms ρ is a collections of Banach spaces such that X ρ ⊂ X ρ with ρ ≤ ρ whenever ρ ≤ ρ ≤ ρ0 . Definition 2. Given τ > 0, 0 < ρ ≤ ρ0 and R > 0 we define: 1.
X ρ,τ to be the set of all functions u(t) from [0, τ ] to X ρ endowed with the norm u ρ,τ = sup u(t) ρ . 0≤t≤τ
2. Yρ,β,τ is the set of functions u(t) from [0, T ] to X ρ with the norm u(t) ρ,β,τ = sup u(t) ρ−βτ . 0≤t≤τ
3. We will denote by X ρ,τ (R) and Yρ,β,τ (R) the balls of radius R in X ρ,τ and Yρ,β,τ respectively. Theorem 2 (Sammartino–Caflisch). Suppose that there exist R > 0, T > 0 and β0 > 0 such that for 0 < t ≤ T the following holds: 1. For every 0 < ρ < ρ < ρ0 − β0 T and every u ∈ X ρ,T (R) the function F(t, u) : [0, T ] → X ρ is continuous. 2. For every 0 < ρ ≤ ρ0 −β0 T the function F(t, 0) : [0, T ] → X ρ,T (R) is continuous in [0, T ] and F(t, 0) ρ0 −β0 T ≤ R0 < R. 3. For every 0 < ρ < ρ(s) ≤ ρ0 − β0 s and every u 1 , u 2 ∈ Yρ0 ,β0 ,T (T ) we have t u 1 − u 2 ρ(s) ds. F(t, u 1 ) − F(t, u 2 ) ρ ≤ C ρ(s) − ρ 0 Then there exist β > β0 and T ∗ > 0 such that u + F(t, u) = 0 has a unique solution in Yρ0 ,β,T ∗ . We begin by defining the spaces of functions. We will be complexifying only the space variable. We look at functions in C, periodic in x. Definition 3. Given l ∈ N and ρ > 0 we say that a function f (x) is in K l,ρ if and only if – f is periodic in x and analytic in |x| < ρ. – For every |x| < ρ, ∂xα f ∈ L 2 for every fixed x, that is, as a function of the real part only. – The norm in f l,ρ is finite, where f l,ρ := sup ∂xα f (· + x) L 2 (R/ ) . α≤l |x|<ρ
Z
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The following two lemmas about analytic functions will be used below to prove the main theorem. Lemma 4. Suppose that (z 1 , . . . , z n ) is analytic with {(z 1 , . . . , z n ) ∈ Cn : |z 1 | ≤ C A1 , . . . , |z n | ≤ C An } ⊂⊂ U . Then the map ( f 1 , . . . , f n ) → ( f 1 , . . . , fl ) (where f i belongs to the Banach space under consideration (be precise) ) is Lip(1) on {( f 1 , . . . , f n ) ∈ Cl : f 1 ≤ C A1 , . . . , f n ≤ C An }, with Lip(1)-norm bounded a priori by and Ai . Lemma 5. Define
F(x) :=
( f 1 (x + τ1 (ρ)), . . . , f n (x + τn (ρ)))dμ(ρ),
where f i are in the Banach space under consideration and satisfy f i ≤ C Ai , and we assume that all τi (ρ) are smooth real functions, is as in the previous lemma and ( , dμ) is a probability measure. Then the map ( f 1 , . . . , f n ) → F(x) is Lip(1), with Lip(1)-norm a priori bounded by and Ai . In order to prove Theorem 1 we start by defining the spaces appearing in the Theorem of Sammartino and Caflisch. We define X k,ρ = H k,ρ × H k,ρ × H k,ρ × R. We rewrite the system of equations in the form ⎛ ⎞ ⎛ ⎞ f F1 (t, f, g, h, l) ⎜ g ⎟ ⎜ F2 (t, f, g, h, l) ⎟ ⎝ h ⎠ = ⎝ F (t, f, g, h, l) ⎠ , 3 l F4 (t, f, g, h, l) where F1 , F2 , F3 and F4 are given by the right hand sides of Eqs. (41), (42), (43) and (44). Conditions 1 and 2 of the theorem are easily checked. We leave the details to the interested reader. We concentrate on the more complicated Cauchy estimate 3. By simple inspection (using the careful rewriting of the right hand sides at the end of the previous section) it is straightforward to check that F1 , F2 and F4 can be written as analytic functions of the arguments f, g, h, l, f g , h , l and integrals of the form 1 ( f 1 (x + τ (ρ)), . . . , f n (x + τn (ρ)))dρ,
(50)
0
where is analytic and the functions f i can only be taken from the list f, g, h f , g , h . Lemmas 4 and 5 show that F1 , F2 and F4 satisfy the Lipschitz estimate required by the Cauchy-Kowaleski Theorem.
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As for F3 we notice that the equation for h is given by (38) which can be rewritten as 1 1 d s¯ h τ (s, τ ) + (h (¯s , τ ) − h (s, τ ))χ˜ (s − s¯ ) l(τ ) + L 0 R/Z |¯s − s| = U¯ 1 + U¯ 2 + U¯ 3 + U¯ 4 + U¯ 5 ,
which using notation from pseudo-differential operators can be rewritten as (∂t +
1 im(k))h = U¯ 1 + U¯ 2 + U¯ 3 + U¯ 4 + U¯ 5 , l(τ ) + L 0
and so h = (∂t +
1 im(k))−1 (U¯ 1 + U¯ 2 + U¯ 3 + U¯ 4 + U¯ 5 ), l(τ ) + L 0
which has the integral representation form given in (43). Notice that the operator (∂t +
1 im(k))−1 l(τ ) + L 0
preserves all L 2 -based Sobolev norms in the spatial variable. This is a simple consequence of the fact that the multiplier is purely imaginary (since m(k) is real), and can also be seen given the integral representation in (43) where the exponent of e is purely imaginary (notice that t has not been complexified). Since U1 , . . . , U5 can be written as analytic functions of the arguments described in (50) we obtain the required Lipschitz estimate concluding the proof. References 1. Constantin, P., Majda, A., Tabak, E.: Singular front formation in a model for quasigesotrophic flow. Phys. Fluids 6(1), 9–11 (1994) 2. Constantin, P., Majda, A., Tabak, E.: Formation of strong fronts in the 2 − D quasigeostrophic thermal active scalar. Nonlinearity 7(6), 1495–1533 (1994) 3. Córdoba, D., Fefferman, C., Rodrigo, J.: Almost sharp fronts for the surface Quasi-Geostrophic equation. PNAS 101(9), 2487–2491 (2004) 4. Córdoba, D., Fontelos, M.A., Mancho, A.M., Rodrigo, J.: Evidence of singularities for a family of contour dynamics equations. PNAS 102(17), 5949–5952 (2005) 5. Fefferman, C., Rodrigo, J.: On the limit of almost sharp fronts for the Surface Quasi-Geostrophic equation. In preparation. 6. Gancedo, F.: Existence for the α-patch model and the QG sharp front in Sobolev spaces. Adv. Math. 217(6), 2569–2598 (2008) 7. Majda, A., Bertozzi, A.: Vorticity and incompressible flow. Cambridge Texts in Applied Mathematics 27, Cambridge: Cambridge Univ. Press, 2002 8. Madja, A., Tabak, E.: A two-dimensional model for quasigeostrophic flow: comparison with the twodimensional Euler flow. Physisa D 98(2-4), 515–522 (1996) 9. Rodrigo, J.: The vortex patch problem for the Quasi-Geostrophic equation. PNAS 101(9), 2484–2486 (2004) 10. Rodrigo, J.: On the evolution of sharp fronts for the quasi-geostrophic equation. Comm. Pure Appl. Math. 58(6), 821–866 (2005) 11. Sammartino, M., Caflisch, R.E.: Zero Viscosity Limit for Analytic Solutions of the Navier-Stokes Equation on a Half-Space I. Existence for Euler and Prandtl equations. Commun. Math. Phys. 192, 433–461 (1998) 12. Sammartino, M., Caflisch, R.E.: Zero Viscosity Limit for Analytic Solutions of the Navier-Stokes Equation on a Half-Space II. Construction of Navier-Stokes Solution. Commun. Math. Phys. 192, 463– 491 (1998) Communicated by P. Constantin
Commun. Math. Phys. 303, 289–300 (2011) Digital Object Identifier (DOI) 10.1007/s00220-011-1197-x
Communications in
Mathematical Physics
On Vorticity Directions near Singularities for the Navier-Stokes Flows with Infinite Energy Yoshikazu Giga1 , Hideyuki Miura2 1 Graduate School of Mathematical Sciences, University of Tokyo, Komaba 3-8-1, Meguro,
Tokyo 153-8914, Japan. E-mail:
[email protected]
2 Department of Mathematics, Osaka University, Machikaneyamacho 1-1, Toyonaka,
Osaka 560-0043, Japan. E-mail:
[email protected] Received: 7 April 2010 / Accepted: 16 September 2010 Published online: 12 February 2011 – © Springer-Verlag 2011
Abstract: We give a geometric nonblow-up criterion on the direction of the vorticity for the three dimensional Navier-Stokes flow whose initial data is just bounded and may have infinite energy. We prove that under a restriction on behavior in time (type I condition) the solution does not blow up if the vorticity direction is uniformly continuous at the place where the vorticity magnitude is large. This improves the regularity condition for the vorticity direction first introduced by P. Constantin and C. Fefferman (1993) for finite energy weak solution. Our method is based on a simple blow-up argument which says that the situation looks like two-dimensional under continuity of the vorticity direction. We also discuss boundary value problems. 1. Introduction There is a large number of work on regularity criteria or nonblow-up criteria for the three dimensional Navier-Stokes flow started by Ohyama [O] and Serrin [Se]; see also [Le,L]. Most of them are analytic in the sense that boundedness for some quantity is assumed. In 1993, P. Constantin and C. Fefferman [CF] gave a geometric condition for the vorticity direction ζ = ω/|ω|, where ω is the vorticity. It says that if the vorticity is Lipschitz continuous in space uniformly in time in the region where vorticity is large, then the Leray-Hopf type weak solution is regular. Since then there are several improvement [BB,B1,B2,B3,C,CKL,Gr,GrR,GrZ,Z]. However, most of these works discuss a solution having a bounded kinetic energy. In this paper we consider a smooth mild solution of the Navier-Stokes equations in R3 × (0, T ), where a solution is just bounded in the space direction and may not have a finite energy. We shall give a condition for the vorticity direction so that the solution can be extended smoothly across t = T . Roughly speaking our result reads: the solution does not blow up at t = T if the blow-up is type I and the vorticity direction is uniformly continuous in space variables. For regularity assumptions on the vorticity direction our result improves the existing one. However, we are forced to assume the growth of the
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L ∞ -norm of the solution is bounded by a self-similar rate (T − t)−1/2 which is called the type I rate. All previous results are proved by integral estimates while ours is by a blow-up argument. It was first used by De Giorgi [DG] (see also [Giu, Theorem 8.1]) for the study of the minimal surfaces. The first author [G] applied the same kind of argument for semilinear parabolic equations to derive a global uniform bound (see also Giga-Kohn [GK] for the derivation of the blow-up rate). The blow-up argument also plays important roles for the analysis of the singularities for geometric flows like the harmonic map heat flow [St] and the Ricci flow [H]. Very recently, Koch, Nadirashvili, Seregin and Sverak [KNSS] applied it to show that type I axisymmetric Navier-Stokes flows must be regular; see [SS] for a local version and also [CSYT,CSTY] for different proofs. We also note that the blow-up argument is effectively studied to derive several estimates for semilinear equations [PQS]. Our blow-up argument is not only simple but also clarifies the structure of the problem. The continuous alignment assumption on the vorticity directions eventually implies that the blow-up limit is a two dimensional flow. It is easy to guess that the solution is regular. Our argument justifies physical intuition. We give an explicit form of our results. We consider the Navier-Stokes equations u t − u + (u, ∇)u + ∇ p = 0 in × (−1, 0), div u = 0 in × (−1, 0),
(NS)
where is either R3 or a half space R+3 . We assume that the solution u is smooth and ||u||∞ (t) = sup |u(x, t)| x∈
is bounded for all t ∈ (−1, 0). Unfortunately, the solution is not unique even if we fix an initial data for = R3 if one allows a solution with infinite energy. In fact, u(x, t) = g(t), p = −g (t) · x always solve the equation. So as in [GIM] we only consider a mild solution, i.e., solution satisfying an integral equation, which is equivalent to require that p = (−)−1 ∇ · (u ⊗ u) in some sense. (Such a relation is automatic for decaying solutions.) For the initial value problem there is a unique local in time mild solution for = R3 [GIM] and = R+3 [So,BJ] with the zero Dirichlet condition. The time t = 0 is considered as a possible blow-up time. We say the blow-up is type I if ||u||∞ (t) ≤ C0 (−t)−1/2
(I)
with some C0 > 0 independent of t ∈ (−1, 0). We consider a type I mild solution which is a mild solution satisfying (I ). According to the estimate of existence time for the mild solution evolving from bounded initial data [GIM], we have u∞ (t) ≥ ε0 (−t)−1/2 with some ε0 > 0 if t = 0 is the blow-up time. Thus if C0 is small, then the solution cannot blow up at t = 0. We are now in position to state one of our main results. Theorem 1.1. Let u be a type I mild solution of (NS) for R3 ×(−1, 0). For a given d > 0 let η be a modulus such that |ζ (x, t) − ζ (y, t)| ≤ η(|x − y|) for all x, y ∈ d (t), t ∈ (−1, 0),
(CA)
where d (t) = {x ∈ R3 | |ω(x, t)| > d} and ζ = ω/|ω|, ω = curl u. Then u does not blow up at t = 0.
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Remark 1.2. Here by a modulus we simply mean that η is a nondecreasing continuous function defined in [0, ∞) with η(0) = 0. For a uniformly continuous function f in Rn one defines a modulus of continuity by m(σ ) = sup{| f (x) − f (y)| | |x − y| ≤ σ }; this is of course, a modulus in our sense. Such m is moreover subadditive. Conversely if η is a modulus and subadditive, then η is a modulus of continuity of some f , see [Ku]. Remark 1.3. In [CF] η is taken η(σ ) = Aσ in (CA), where A is a positive constant. In [BB] η is taken η(σ ) = Aσ 1/2 . In [B1] η is taken η(σ ) = Aσ β for some β ∈ (0, 1/2) 0 3 under the assumption that −1 ωr2 (t)dt is finite for r = β+1 . These authors considered weak solutions and did not assume that the blow-up is type I. See also [C,GrR] and [Z] for related results. Remark 1.4. The continuous alignment assumption (CA) can be relaxed as follows: |x − y| |ζ (x, t) − ζ (y, t)| ≤ η(o(1) √ ) for x, y ∈ d (t) and t ∈ (−1, 0) −t
(C A )
as t ↑ 0. We will prove the theorem under the condition (C A ) in Sect. 2.1. We can replace these continuous alignment assumptions by a condition for ∇ζ . See Corollary 2.6. Our blow-up argument can be also applied for a local regularity criterion away from the boundary, which is regarded as a local version of Theorem 1.1. In order to prove the local regularity criterion, we will use a compactness theorem for a sequence of suitable weak solutions by Seregin and Sverak [SS]. For this reason, we have to assume local energy is finite uniformly in (−1, 0). It should be noted that local energy is allowed to diverge as t tends to zero in Theorem 1.1. See Sect. 2.2. A regularity criterion near the boundary turns out to work for the slip boundary condition by a simlar blow-up argument. A key step for the blow-up argument is to establish a Liouville type result. For the slip boundary value problem we may still use the Liouville type theorem for the flow without the boundary condition since our continuous alignment condition (CA) implies that the vorticity of the blow-up limit is orthogonal to the boundary. We give an explicit statement in Sect. 3 when is a half space. A corresponding result to [CF] for the slip boundary condition is discussed in [B2], where η(σ ) = Aσ 1/2 in (CA). In the case of the zero Dirichlet condition (u = 0) we need to study the Liouville type problem with the boundary. However, we do not know the answer since the vorticity on the boundary is not well controlled. In [B3] a regularity criterion is given for the → Dirichlet problem by assuming further that the boundary integral ∂|ω|2 /∂ − n is small − → in some sense, where n is the normal of the boundary. We do not know whether such an extra assumption is necessary. We shall discuss the Dirichlet problem in Sect. 3. 2. Blow-up Analysis 2.1. Proof of Theorem 1.1. To prove the main theorem it suffices to prove that lim sup ||u||∞ (τ ) < ∞ t↑0 −1≤τ ≤t
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for example by [GIM]. Assume the contrary so that lim sup ||u||∞ (τ ) = ∞. t↑0 −1≤τ ≤t
Then there is a sequence {tk }∞ k=1 with tk ↑ 0 such that sup
−1≤τ ≤tk
||u||∞ (τ ) = ||u||∞ (tk ) := Mk ↑ ∞
as k → ∞. We take xk such that |u(xk , tk )| ≥ Mk − 1. We rescale the solution by u k (x, t) = λk u(xk + λk x, tk + λ2k t) with λk = 1/Mk . Clearly, we know |u k (0, 0)| ≥ 1 − 1/Mk , and the uniform bound for derivatives of the rescaled velocity is given as |u k (x, t)| ≤ 1 in R3 × (−Mk2 , 0]. By the scaling invariance properties of (NS), u k is a mild solution of (NS) in R3 × (−Mk2 , 0). The parabolic regularity theory (see [GIM,GS,MS]) also implies the uniform bound for the rescaled velocity, j
||∂t u k || L ∞ R3 ×(−M 2 /2,0) + ||∇ j u k || L ∞ R3 ×(−M 2 /2,0) ≤ C j . k
k
(2.1)
Note that [GS] discussed the case of decaying initial data but their argument easily extends to L ∞ case. Thus we can find a subsequence (still denoted u k , ωk ) which converges to bounded continuous functions u and ω locally uniformly in R3 × (−∞, 0]. Moreover, u is a mild solution for (NS) in R3 × (−∞, 0), since u k ⊗ u k converges to u ⊗ u ∗-weakly in L ∞ . The limit function (u, ω) solves the vorticity equation: ωt − ω + (u, ∇)ω − (ω, ∇)u = 0. Moreover, u and ω have bounds |u| ≤ 1, |ω| ≤ C in R3 × (−∞, 0] and |u(0, 0)| = 1. Under the type I condition we will show that the backward global solution ω is nontrivial. Proposition 2.1. Assume that u is a type I mild solution of (NS) in R3 × (−1, 0). Then ω ≡ 0 in R3 × (−∞, 0]. Proof. Suppose that ω ≡ 0 so that curl ω = 0. Since −u = curl ω by div u = 0, the Liouville theorem for harmonic functions yields u is constant in space. Moreover since u is a mild solution, u is also constant in time. Then the condition |u(0, 0)| = 1 implies ||u||∞ (t) = 1 for all t < 0. From the type I assumption it follows that ||u k ||∞ (t) ≤ C0 Mk−1 (|tk | + Mk−2 |t|)−1/2 = C0 (Mk2 |tk | + |t|)−1/2 ≤ C0 (−t)−1/2 , which yields ||u||∞ (t) ≤ C0 (−t)−1/2 . This contradicts the fact ||u||∞ (t) ≡ 1.
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We shall show that the continuous alignment condition (CA) or more generally (C A ) implies ω ≡ 0. Proposition 2.2. Assume that u is a mild solution of (NS) in R3 × (−1, 0) which is bounded in R3 ×(−1, −δ) for all δ > 0. If ζ satisfies (C A ), then ω ≡ 0 in R3 ×(−∞, 0]. These two propositions imply the main theorem (Theorem 1.1). Proof of Proposition 2.2. We argue by contradiction. Suppose that ω ≡ 0. Since ω is continuous in R3 × (−∞, 0], the set U = {(x, t) ∈ R3 × (−∞, 0] | ω(x, t) = 0} is a nonempty open set. Let K be a compact subset of U . Then there exists δ > 0 and k0 ∈ N such that |ωk (x, t)| > δ on K for k ≥ k0 . Thus ζk := ωk /|ωk | is well-defined in K for k ≥ k0 . For sufficiently large k, say k ≥ k1 for some k1 (≥ k0 ) we have δ > Mk−2 d, where d is the constant in the definition of d (t). For k ≥ k1 we have ⎞ ⎛ |x − y| ⎠ |ζk (x, t) − ζk (y, t)| ≤ η ⎝o(1) Mk |tk | + Mk−2 |t| ⎛ ⎞ |x − y| ⎠. ≤ η ⎝o(1) 2 Mk |tk | + |t| By the local existence theorem for a bounded mild solution [GIM], we have a bound ε0 such that Mk2 |tk | > ε0 > 0 for all k > k1 . Thus the difference is estimated as
−1/2 |ζk (x, t) − ζk (y, t)| ≤ η o(1)ε0 |x − y| . The right-hand side converges to zero as tk ↑ 0. We thus observe that ζk tends to ζ with ζ = ω/|ω| uniformly in K and |ζ (x, t) − ζ (y, t)| = 0 for x, y ∈ K . Since K is an arbitrary compact set in U , we have ω(x, t) = |ω(x, t)|ζ 0 (t), where ζ 0 (t) is a vector in R3 depending only on t. By rotation we may assume that ω(x, t0 ) = (0, 0, ω3 (x, t0 )) for a given time t0 . If we assume (CA) instead of (CA ), we need not invoke the local existence theorem for a bounded mild solution in [GIM] to conclude that ζ is spatially constant.
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We would like to prove that ζ 0 is also independent of time. At t = t0 , since (curl ω)3 = 0, we have −u 3 = (curl ω)3 = 0 in R3 . By the Liouville theorem for harmonic functions, this implies that u 3 is spatially constant. Since 0 = ω1 = ∂u 3 /∂ x2 − ∂u 2 /∂ x3 , 0 = ω2 = ∂u 1 /∂ x3 − ∂u 3 /∂ x1 , we observe that u 1 and u 2 are independent of x3 . Thus the flow is two-dimensional at t = t0 . Once the flow is x3 -independent and u 3 is constant, by the unique local existence theorem of a bounded mild solution, the solution stays two-dimensional (independent of x3 ) for t ≥ t0 . One may take t0 arbitrary, so we conclude that ζ 0 is independent of t. We thus show that u = (u 1 , u 2 , c) and ω = (0, 0, ω3 ) solve the two-dimensional vorticity equation ω3t − ω3 + (u, ∇)ω3 = 0 in R2 × (−∞, 0).
(2.2)
However, the following lemma which we may call a Liouville type theorem implies that ω ≡ 0 since we know that ω and u are bounded and smooth in R2 × (−∞, 0] and that curl u = ω and div u = 0. Lemma 2.3. Assume that bounded smooth functions u = (u 1 , u 2 ) and ω3 solve (2.2), curl u = ω3 and div u = 0. Then ω3 ≡ 0. Corollary 2.4. Assume that u is a bounded smooth mild solution of (NS) in R2 ×(−∞, 0). Then u must be constant in space-time. Remark 2.5. (i) Corollary 2.4 follows immediately from Lemma 2.3 since ω is bounded in R2 × (−∞, 0) (cf. [GIM]) and satisfies the vorticity equation. (If ω ≡ 0, as discussed above, u must be spatially constant which implies that it is a space-time constant mild solution.) (ii) Corollary 2.4 is already proved in [KNSS] by using the stability of the strong maximum principle. We shall give a shorter proof for Lemma 2.3 for completeness. Proof of Lemma 2.3. We may assume that u and ω3 are defined in (−∞, 0] as smooth bounded functions by shifting in time as u ε (x, t) := u(x, t−ε), ωε,3 (x, t) := ω3 (x, t−ε) for ε > 0. (If we are able to prove that ωε,3 ≡ 0 for t ≤ 0, for sufficiently small ε > 0 this implies ω3 ≡ 0 in R2 × (−∞, 0)). Suppose that L = ||ω3 || L ∞ (R2 ×(−∞,0]) > 0. Then there exists a sequence of points (xk , tk ) ∈ R2 × (−∞, 0] satisfying ω3 (xk , tk ) → L (or − L). Let u (k) (x, t) := u(x + (k) xk , t + tk ) and ω3 (x, t) := ω3 (x + xk , t + tk ). Then there exist u, ˜ ω˜ 3 such that u (k) (k) and ω3 subsequently converge to u˜ and ω˜ 3 respectively (in locally uniform sense) and u, ˜ ω˜ 3 satisfy the two-dimensional vorticity equations in R2 × (−∞, 0). (Here we have invoked L ∞ theory of the Navier-Stokes equations [GIM] or the vorticity equations (cf. [GGS])). By the choice of (xk , tk ) we have ω˜ 3 (0, 0) = L (or − L) and |ω˜ 3 | ≤ L in R2 × (−∞, 0). Therefore the strong maximum principle implies that ω˜ 3 ≡ L(or − L). Since −u˜ = (∂ ω˜ 3 /∂ x2 , −∂ ω˜ 3 /∂ x1 ) = 0, the Liouville theorem for harmonic functions yields that u˜ is spatially constant. Thus ω˜ 3 ≡ 0 which contradicts the fact that |ω˜ 3 | ≡ L > 0.
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The continuous alignment assumption (CA ) can be replaced by the condition for ∇ζ as follows: Corollary 2.6. Let u be a type I mild solution of (NS) in R3 × (−1, 0). For a given d > 0 assume that 0 ∇ζ 2L ∞ (d (t)) (t)dt < +∞, −1
where d (t) is defined as Theorem 1.1. Then u does not blow up at t = 0. Remark 2.7. This assumption implies that ∇ζ is identically zero. Hence ζ turns out to be a constant vector as in the proof of Proposition 2.2. Thus the proof is reduced to one of Theorem 1.1. A similar criterion is established for solutions of the Euler equations by [CFM]. They assumed that the above type integral bound for ∇ζ in a bunch of trajectories. Moreover, they assumed that such neighborhood is large enough to capture the local intensification of the vorticity. Under these assumptions they proved nonblow-up of solutions. Remark 2.8. In [BB], regularity of a weak solution of (NS) is established under the assumption that 0 ∇ζ aL b ( (t)) (t)dt < +∞ −1
d
for 2/a + 3/b = 1/2. This assumption is not scaling invariant while ours is scaling invariant. It is easy to generalize our assumption in this form with 2/a + 3/b = 1 and 2 ≤ a < ∞. 2.2. Local regularity. We will give a remark on a local version of Theorem 1.1. For the purpose, we introduce the notion of suitable weak solutions following [Lin,SS], which is originally introduced by [CKN]. Definition 2.9. Let B(r ) be a ball centered at the origin with radius r > 0 and Q = B(1) × (−1, 0) be a unit parabolic cylinder in R3 × (−1, 0). We say the pair u and p is a suitable weak solution of (NS) in Q if the following the assumptions are satisfied u ∈ L ∞ (−1, 0; L 2 (B(1))) ∩ L 2 (−1, 0; W 1,2 (B(1))), p ∈ L 3/2 (−1, 0; L 3/2 (B(1))), and (u, p) satisfies (NS) in the sense of distributions and the local energy inequality t ϕ(x, t)|u(x, t)|2 d x + 2 ϕ|∇u|2 d xds B(1)
≤
t
−1
−1
B(1)
{|u|2 (ϕ + ∂t ϕ) + u · ∇ϕ(|u|2 + 2 p)}d xds B(1)
for all nonnegative functions ϕ ∈ C0∞ (Q) and almost all t ∈ (−1, 0). We now state our main result in this subsection.
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Theorem 2.10. Let (u, p) be a suitable weak solution for (NS) in Q safistying u L ∞ (B(r0 )) (t) ≤ C0 (−t)−1/2 with some positive constants r0 ∈ (0, 1) and C0 > 0 independent of t ∈ (−1, 0). For a given d > 0 let η be a modulus such that |ζ (x, t) − ζ (y, t)| ≤ η(|x − y|)
for all x, y ∈ d (t) t ∈ (−1, 0),
(D)
where d (t) = {x ∈ B(r0 ); |ω(x, t)| > d}. Then u is regular at (x, t) = (0, 0). Remark 2.11. We do not prove Theorem 2.10 because it is almost parallel as the proof of Theorem 1.1. The only difference is the compactness argument which justifies convergence of sequences of the rescaled local solutions. For the purpose we invoke a result in Seregin and Sverak [SS, Theorem 2.8], which says that the rescaled solutions constructed by the same way as in Sect. 2.1 are Hölder continuous locally uniformly in R3 × (−∞, 0], and then the subsequence converges to some backward global mild solution u locally uniformly in R3 × (−∞, 0]. We notice that since local existence theory in general domains is not known, it is not easy to weaken the assumption of the vorticity directions as (C A ) in the local case. 3. Effect of Boundary We now consider the boundary value problem for (NS) when is the half space R+3 . If one imposes the slip boundary condition for example − → → → n T (u)− τ = 0 and u · − n = 0, → → where − n is the normal (0, 0, −1) and − τ is a tangential vector on ∂R+3 , then we have ∂u 1 ∂u 2 = = u 3 = 0 on {x3 = 0}. ∂ x3 ∂ x3
(3.1)
This in particular implies that ω1 = ω2 = 0 on {x3 = 0},
(3.2)
where ω = (ω1 , ω2 , ω3 ). Here T (u) is the stress tensor defined by
∂u i ∂u j Ti j (u) = ( + ) − pδi j , 1 ≤ i, j ≤ 3. ∂ x j ∂ xi Arguing in the same way to prove Thorem 1.1 we have Theorem 3.1. Let u be a type I mild solution of (NS) with = R+3 , where the slip boundary condition (3.1) is imposed. For a given d > 0 let η be a modulus satisfying the continuous alignment condition (CA) for ζ . Then u does not blow up at t = 0.
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The L ∞ -theory of the half space with the slip boundary condition is not explicit in the literature. However, since the Stokes operator with this boundary condition is the same as the heat operator with the same boundary condition. We are able to establish L ∞ theory in the same as the whole space case. The blow-up limit (u, ω) should fulfill either the boundary value problem or the whole space problem. It depends on the behavior of dist (xk , ∂R+3 )Mk . If lim sup dist (xk , ∂R+3 )Mk = +∞, k→∞
then the problem is reduced to the whole space case cf. Sect. 2. We may assume that lim dist (xk , ∂R+3 )Mk < +∞
k→∞
by taking a subsequence. In this case the limit u becomes a bounded backward global mild solution of (NS) in the half space with the slip boundary condition (cf. [G]). Since ω1 = ω2 = 0 on the boundary by (3.2), the continuous alignment condition (CA) implies that ω = (0, 0, ω3 ). The proof of Theorem 3.1 is completed by the following Liouville type theorem. Lemma 3.2. Assume that the pair u = (u 1 , u 2 , u 3 ) and ω = (0, 0, ω3 ) is a bounded backward global mild solution of ω3t − ω3 + (u, ∇)ω3 − ω3 ∂x3 u 3 = 0 in R+3 × (−∞, 0)
(3.3)
with u 3 = 0 on {x3 = 0}, where curl u = ω, div u = 0. Then ω = 0. Proof. We first observe that (curl ω)3 = 0. Since ω3 solves −u 3 = (curl ω)3 in R+3 and u 3 is bounded with u 3 = 0 on {x3 = 0}, a classical Liouville theorem implies that u 3 is constant therefore u 3 ≡ 0. Since 0 = ω1 = ∂u 3 /∂ x2 − ∂u 2 /∂ x3 , 0 = ω2 = ∂u 1 /∂ x3 − ∂u 3 /∂ x1 , we conclude that u 1 , u 2 are independent of x3 so that ω3 is independent of x3 , ω3t − ω3 + (u, ∇)ω3 = 0 in R2 × (−∞, 0) with parameter x3 . Thus the problem is reduced to the Liouville problem for the whole space. Remark 3.3. Even if the domain has a curved boundary by blow-up argument the limit u is expected to solve the problem with flat boundary as in [G]. However, one should establish necessary L ∞ theory for curved boundary case. We now consider the Dirichlet boundary condition u 1 = u 2 = u 3 = 0 on {x3 = 0}. We argue in the same way as for the slip boundary condition. The case we should discuss is the case: lim sup dist(xk , ∂R+3 )Mk < +∞. k→∞
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In this case the limit u becomes a bounded backward global mild solution of (NS) in the half space with the Dirichlet boundary condition. Since ω3 = 0 by u 1 = u 2 = 0 on the boundary, (CA) implies that ω = (ω1 , 0, 0) by rotating coordinates in x1 , x2 space. Arguing in the same way as Lemma 3.2 we conclude that the problem is twodimensional, i.e., u 1 = 0, u 2 , u 3 is independent of x1 . If the following Liouville type problem is solved, we are able to establish a regularity criterion on the vorticity direction for the Dirichlet problem similar to Theorem 3.1. Problem 3.4. Assume that u = (0, u 2 , u 3 ) and ω = (ω1 , 0, 0) are bounded smooth (up to the boundary) solutions of ω1t − ω1 + (u, ∇)ω1 = 0 in R+2 × (−∞, 0), where ω1 , u 2 , u 3 is independent of x1 and R+2 = {(x2 , x3 )|x3 > 0}. Assume that curl u = ω and div u = 0. Assume furthermore that u = 0 on the boundary ∂R+2 × (−∞, 0). Are there any (nontrivial) solutions other than u ≡ 0? Remark 3.5. If u decays rapidly enough as t → −∞ and |x| → ∞, by the standard energy inequality we know u must be zero. We do not know the decay of u and ω. If we apply the similar argument to prove the Liouville type theorem for the whole space, we are able to prove the following decay estimate: Proposition 3.6. Under the assumption of Problem 3.4, we have lim
sup
|m|→∞ x2 ∈R,|x3 |≥m,t<0
|ω1 (x2 , x3 , t)| = 0.
Proof. Suppose that there exist positive constants m 0 > 0 and α > 0 and a sequence (xk , tk ) ∈ R+2 × (−∞, 0) such that xk,3 ↑ +∞ and |ω1 (xk , tk )| ≥ α. (k)
Let u (x, t) := u(x + xk , t + tk ) and ω(k) (x, t) := ω(x + xk , t + tk ). By a standard compactness argument there exists the limit (u, ˜ ω) ˜ (by taking an appropriate subsequence of (u (k) , ω(k) )) which is a classical solution of the vorticity equation in the whole plane for t < 0. Indeed, since u (k) and ω(k) are bounded, a standard parabolic theory [LSU] implies that ω(k) is locally bounded in W p2,1 for any p > 1. This already implies that ˜ ω) ˜ in a distribution sense. Since u · ∇ω = ∇ · (u ω), this (u (k) , ω(k) ) converges to (u, observation yields that u˜ and ω˜ solve the vorticity equations in the whole plane for t < 0. It is a classical solution by a regularity theory of parabolic and the Poisson equations for u. Since W p2,1 is compactly embedded in a Hölder space locally ([LSU], Chap. II, Lemma 3.3), we may conclude that ω(k) converges to its limit at least uniformly. This implies that |ω(0, ˜ 0)| ≥ α, and so ω˜ is not identically zero. Therefore we get a nontrivial bounded solution for the vorticity equation in the whole plane for t < 0. This contradicts the Liouville type theorem Lemma 2.3. Remark 3.7. If we assume in addition that → (||∂ωtan /∂ − n ||∞ /||u||2∞ ) (t) → 0 → n = 0 at the limit problem. If we have such as t ↑ 0 other than (CA), then we get ∂ω /∂ − 1
a condition for our Liouville problem, the similar argument to solve the Liouville problem for the whole space yields nonexistence of the nontrivial solution since ω1 cannot attain its maximum on the boundary. Here ωtan denotes the tangential component of ω.
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Acknowledgements. The authors are grateful to Professor Tai-Peng Tsai for many helpful comments. The authors are grateful to Professor Shige-Toshi Kuroda for his useful comments about the modulus. The authors are grateful to Dr. Yuko Nagase for informative remarks about papers of De Giorgi. The research of Y. Giga was partly supported by the Grant-in-Aid for Scientific Research, No. 21224001, No. 20654017, the Japan Society of the Promotion of Science (JSPS). The research of H. Miura was partly supported by the Grant-inAid for Scientific Research, No. 20431497, JSPS, and the Sumitomo Foundation Grant, No. 090885. Much of the work of H. Miura was done while he was at the University of Tokyo during 2008-2009. Its hospitality is gratefully acknowledged as well as the support from global COE, “The research and training center for new development in mathematics” supported by JSPS.
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[H] [ISS] [KNSS] [Ku] [L] [Le] [LSU] [Lin] [MS] [NRS] [O] [PQS] [SS] [Se] [So] [St] [Z]
Y. Giga, H. Miura
Hamilton, R.: The formation of singularities in the Ricci flow. Surv. Diff. Geom. 2, 7–136 (1995) Iskauriaza, L., Serëgin, G.A., Shverak, V.: L3,∞ -solutions of Navier-Stokes equations and backward uniqueness. Russ. Math. Surv. 58, 211–250 (2003) Koch, G., Nadirashvili, N., Seregin, G., Sverak, V.: Liouville theorems for the Navier-Stokes equations and applications. Acta Math. 203, 83–105 (2009) Kuroda, S.-T.: Diagonalization modulo norm ideals; spectral method and modulus of continuity. RIMS KˆoKyˆuroku Bessatsu, 16, 101–126 (2010) Ladyzhenskaya, O.A.: The mathematical theory of viscous incompressible flow. New York: Gordon and Breach Science Publishers, 1969 Leray, J.: Sur le mouvement d’un liquide visqueux emplissant l’espace. Acta Math. 63, 193– 248 (1934) Ladyzhenskaya, O.A., Solonnikov, V.A., Uralt’seva, N.N.: Linear and quasilinear equations of parabolic type, Moscow 1967; English translation: Providence, RI: Amer. Math. Soc., 1968 Lin, F.-H.: A new proof of the Caffarelli-Kohn-Nirenberg theorem. Comm. Pure Appl. Math. 51, 241–257 (1998) Miura, H., Sawada, O.: On the regularizing rate estimates of Koch-Tataru’s solution to the Navier-Stokes equations. Asymptotic Anal. 49, 1–15 (2006) Necas, J., Ruzicka, M., Sverak, V.: On Leray’s self-similar solutions of the Navier-Stokes equations. Acta Math. 176, 283–294 (1996) Ohyama, T.: Interior regularity of weak solutions to the Navier-Stokes equation. Proc. Japan Acad. 36, 273–277 (1960) Polácik, P., Quittner, P., Souplet, P.: Singularity and decay estimates in superlinear problems via Liouville-type theorems. II: parabolic equations. Indiana Univ. Math. J. 56, 879–908 (2007) Seregin, G., Sverak, V.: On type I singularities of the local axi-symmetric solutions of the NavierStokes equations. Comm. Part. Diff. Eqs. 34, 171–201 (2009) Serrin, J.: On the interior regularity of weak solutions of the Navier-Stokes equations. Arch. Rat. Mech. Anal. 9, 187–195 (1962) Solonnikov, V.A.: On nonstationary Stokes problem and Navier-Stokes problem in a half-space with initial data nondecreasing at infinity, function theory and applications. J. Math. Sci. 114, 726– 1740 (2003) Struwe, M.: Geometric evolution problems, Nonlinear partial differential equations in differential geometry (Park City, UT, 1992), IAS/Park City Math. Ser., 2, Providence, RI: Amer. Math. Soc., 1996, pp. 257–339 Zhou, Y.: A new regularity criterion for the Navier-Stokes equations in terms of the direction of vorticity. Monatsh. Math. 144, 251–257 (2005)
Communicated by P. Constantin
Commun. Math. Phys. 303, 301–316 (2011) Digital Object Identifier (DOI) 10.1007/s00220-011-1196-y
Communications in
Mathematical Physics
Global Structure of Quaternion Polynomial Differential Equations Xiang Zhang Department of Mathematics, Shanghai Jiaotong University, Shanghai 200240, P. R. China. E-mail:
[email protected] Received: 9 April 2010 / Accepted: 9 September 2010 Published online: 26 January 2011 – © Springer-Verlag 2011
Abstract: In this paper we mainly study the global structure of the quaternion Bernoulli equations q˙ = aq + bq n for q ∈ H, the quaternion field and also some other form of cubic quaternion differential equations. By using the Liouvillian theorem of integrability and the topological characterization of 2–dimensional torus: orientable compact connected surface of genus one, we prove that the quaternion Bernoulli equations may have invariant tori, which possesses a full Lebesgue measure subset of H. Moreover, if n = 2 all the invariant tori are full of periodic orbits; if n = 3 there are infinitely many invariant tori fulfilling periodic orbits and also infinitely many invariant ones fulfilling dense orbits.
1. Introduction and Main Results The dynamics of ordinary differential equations in R or C has been intensively studied from many different points of view. While because of the noncommutativity of the quaternion algebra, the study on quaternion differential equations becomes very difficult and much more involved, and the results in this field are very few. In recent years because of their application in quantum and fluid mechanics, see e.g. [2,3,12,16,17,29,30], the study on the dynamics of quaternion differential equations has been attracting more interesting. In 2006 Campos and Mawhin [10] initiated the study on the existence of periodic solutions of one–dimensional first order periodic quaternion differential equations. Wilczy´nski [31] continued this study and paid more attention to the existence of two periodic solutions of quaternion Riccati equations. Our work in [15] presented a study on the global structure of the quaternion autonomous homogeneous differential equations, The author is partially supported by NNSF of China grant 10831003 and Shanghai Pujiang Program grant 09PJD013.
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q˙ = aq n ,
q ∈ H,
(1)
where a ∈ H is a parameter. Recall that H is the quaternion field. In this paper we will study the global dynamics of the quaternion Bernoulli equations q˙ = bq + aq n ,
(2)
with a, b, q ∈ H, 2 ≤ n ∈ N and also of the third order equation q˙ = a(q − c0 )(q + c0 )q,
(3)
with a ∈ H and c0 ∈ R. The quaternion Bernoulli equation (2) consists of linear terms and homogenous nonlinearities of degree n. We note that real planar polynomial vector fields generalizing the linear systems with homogeneous nonlinearities have been extensively studied from different points of view, for instance limit cycles, centers, phase portraits and integrability, see e.g. [14,18,23,25]. Some famous three dimensional real differential systems exhibiting chaotic phenomena, for instance the Lorenz system, Rabinovich systems and Rikitake systems and so on, also have this form, which consist of linear terms and homogeneous nonlinearity of degree 2. To our knowledge, the dynamics of the quaternion equations of form (2) with a, b = 0 has never been studied. We note that for either a = 0 or b = 0, Eq. (2) is in fact Eq. (1), and it has been studied in [15]. Equation (2) with a, b = 0 can be written as q˙ = a(cq − q n ),
(4)
with a, c ∈ H not zero. Our first result is the following. Theorem 1. For the quaternion differential Eq. (4) with c ∈ R not zero, the following statements hold: (a) Assume that a + a = 0 and a − a = 0. (a1 ) The phase space R4 , i.e., H, is foliated by invariant planes of (4), which all pass through the origin. (a2 ) On each invariant plane, there are n singularities: one is the origin and √ the others are located on the circle centered at the origin with the radius n−1 |c|, denoted by Sc . All non–trivial orbits are heteroclinic, and connect the origin and one of the singularities on Sc except the following 2(n − 1) ones: there are exactly n −1 heteroclinic orbits connecting the origin and infinity, and also n −1 ones connecting each one of the singularities on Sc and infinity. (b) Assume that a 2 − a 2 = 0. (b1 ) Each orbit of system (4) starting on the branch of P := {q n−1 + q n−1 − c = 0} is heteroclinic connecting the origin and one of the singularities given by q n−1 = c, which are located in two consecutive regions; limited by the branches of P. (b2 ) There exists at least one orbit in each connected region limited by the branches of P, which connects infinity and one of the singularities of (4). (c) Assume that a + a = 0 and a − a = 0. (c1 ) The hypersurfaces P are invariant, on which all orbits are nontrivial and located in two dimensional invariant algebraic varieties.
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(c2 ) The invariant set R4 \{P} is foliated by one invariant plane, two 2–dimensional invariant algebraic varieties and 2–dimensional invariant tori. The invariant plane is foliated by n isochronous centers with n separatrices going to infinity. One of the algebraic varieties is full of singularities and the other fulfills periodic orbits with a center and finitely many heteroclinic orbits. A nontrivial orbit is an orbit which is not a singularity. An algebraic variety is a subset of R4 formed by the common zeros of finitely many polynomials. In statement (c2 ) of Theorem 1 we do not study the dynamics of Eq. (4) on the invariant tori. In fact, the next theorem shows that the dynamics on the invariant tori depend on the degree n of the equations. Now we study the dynamics of Eq. (4) on the invariant tori appearing in statement (c2 ) of the last theorem for n = 2, 3. For larger n, we have no methods to tackle it. The difficulty is the parametrization of the invariant tori as we will see in the proof of the following results. Theorem 2. For the 2–dimensional invariant tori stated in (c2 ) of Theorem 1 the following statements hold: (a) n = 2. Each torus is full of periodic orbits. (b) n = 3. Among the tori there are infinitely many ones with fulfilled periodic orbits and also infinitely many ones with fulfilled dense orbits. The above results are on Eq. (4) with c ∈ R. We now study the equation with c ∈ H\R. For general a ∈ H and 2 < n ∈ N, we have no method to deal with it. The next result is on Eq. (4) with 0 = a ∈ R and n = 2. Theorem 3. For Eqs. (4) with a ∈ R nonzero, n = 2 and c − c = 0, set L = c0 q0 + c1 q1 + c2 q2 + c3 q3 − c02 + c12 + c22 + c32 /2, the following statements hold. (a) If c + c = 0, all the orbits of system (4) starting on the hyperplane L = 0 are heteroclinic and spirally approach the singularities O = (0, 0, 0, 0) and S = (c0 , c1 , c2 , c3 ). There are other two heteroclinic orbits connecting infinity and either S or O. (b) If c + c = 0, the hyperplane L = 0 is invariant. The invariant set R4 \{L = 0} is foliated by one invariant plane foliated by two period annuli, one invariant sphere fulfilling periodic orbits, and 2–dimensional invariant tori. We remark that the case c − c = 0 was studied in Theorems 1 and 2. Finally we study the cubic quaternion differential equation (3). Without loss of generality we assume c0 > 0. Theorem 4. Consider the cubic equation (3) with c0 > 0 and a ∈ H nonzero. Set L = q02 − q12 − q22 − q32 − c02 /2 and denote by L + and L − the two sheets of the √ √ generalized hyperboloid of L = 0 corresponding to q0 ≥ c0 / 2 and q0 ≤ −c0 / 2, respectively. The following statements hold: (a) If a +a = 0, any orbit starting on L + (r esp.L − ) is heteroclinic connecting the singularities O = (0, 0, 0, 0) and S+ = (c0 , 0, 0, 0) (resp. O and S− = (−c0 , 0, 0, 0)). (b) If a + a = 0, the hyperboloid L = 0 is invariant under the flow of (3). The invariant subset R4 \{L = 0} is foliated by periodic orbits and 2–dimensional invariant tori. Of the invariant tori, there are infinitely many ones fulfilling periodic orbits and also infinitely many ones fulfilling dense orbits.
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From Theorems 2 and 4 we conjecture that for quaternion polynomial differential equations of degree larger than 2, if the equations have invariant tori, then there are infinitely many ones fulfilling periodic orbits and also infinitely many ones fulfilling dense orbits. We remark that in the proof of the existence of invariant tori, we will use both the Liouvillian theorem of integrability and also the topological characterization of torus. In the case that the mentioned equations have two functionally independent first integrals but they are not Liouvillian integrable, we prove the existence of invariant tori by showing that the connected parts of the intersection of the level sets of the two first integrals are orientable compact smooth surfaces of genus one. In this paper, as a by product of our results we find some new class of integrable systems. The problem on searching integrable differential equations, including the integrable Hamiltonian systems, has a long history. It can be traced back to Poincaré and Darboux, and even earlier. In recent years Calogero has done a series of researches in this direction, see for instance [6–8,21] and the reference therein. The paper is organized as follows. In the next section we recall some basic facts on quaternion which will be used later on. In Sect. 3 we will prove our main results. The last section is the Appendix presenting the results for linear quaternion equations. 2. Basic Preliminaries In this section for readers’ convenience we recall some basic facts on quaternion algebra (see e.g., [13,19,20]), which will be used later on. Quaternions are a non–commutative extension of complex numbers, which are defined as the field H = {q = q0 + q1 i + q2 j + q3 k; q0 , q1 , q2 , q3 ∈ R}, with i, j, k satisfying i 2 = j 2 = k 2 = −1, i j = − ji = k. For a, b ∈ H, their addition and multiplication are defined respectively as a + b = (a0 + b0 ) + (a1 + b1 )i + (a2 + b2 ) j + (a3 + b3 )k, ab = (a0 b0 − a1 b1 − a2 b2 − a3 b3 ) + (a1 b0 + a0 b1 − a3 b2 + a2 b3 )i + (a2 b0 + a3 b1 + a0 b2 − a1 b3 ) j + (a3 b0 − a2 b1 + a1 b2 + a0 b3 )k. Obviously a, b ∈ H commute if and only if the vectors (a1 , a2 , a3 ) and (b1 , b2 , b3 ) are parallel in R3 . For a ∈ H, its conjugate is a = a0 − a1 i − a2 j − a3 k. Then we have ab = b a, ab + b a = ba + a b and aa = a02 + a12 + a22 + a32 . The last equality implies that (a1 i + a2 j + a3 k)2 = − a12 + a22 + a32 . For any a ∈ H nonzero, a/(aa) is its unique inverse, denoted by a −1 . Moreover, it is easy to check that the elements in H satisfy the law of association and distribution under the action of the addition and multiplication. Mostly we will use the quaternion structures to prove our results. But sometimes it is not enough in the proof, we need to write the quaternion differential equations in components. Considering one–dimensional quaternion ordinary differential equations, q˙ =
dq = f (q, q ), dt
q ∈ H,
(5)
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where f (q, q ) is an H–valued function in the variables q and q. Set f (q, q ) = f 0 (q ∗ ) + f 1 (q ∗ )i + f 2 (q ∗ ) j + f 3 (q ∗ )k, where q = q0 + q1 i + q2 j + q3 k and q ∗ = (q0 , q1 , q2 , q3 ) ∈ R4 . Then Eq. (5) can be written in an equivalent way as q˙s = f s (q0 , q1 , q2 , q3 )
for s = 0, 1, 2, 3.
The last paragraph shows that a one–dimensional quaternion ordinary differential equation is in fact equivalent to a system of four–dimensional real ordinary differential equations. It is well known that the dynamics of higher dimensional real differential systems is usually very difficult to study. Sometimes the existence of suitable invariants is very useful in the study. The first integral and invariant algebraic hypersurface are two important invariants. A real valued differentiable function H (q, q ) is a first integral of (5) if the derivative of H with respect to the time t along the solutions of (5) is identically zero. An invariant algebraic hypersurface of (5) is defined by the vanishing set of a real polynomial F(q, q ) satisfying d F(q, q ) = K (q, q )F(q, q ), dt (5) with the cofactor K (q, q ) a real polynomial. In this paper the most difficult part is the search of invariant algebraic hypersurfaces and of first integrals. Having them we can obtain the dynamics of the equations with the help of qualitative methods. This idea can be found in the study of the Lorenz system [27], of the Rabinovich system [9] and of the Einstein-Yang-Mills Equations [26] and so on. 3. Proof of the Main Results 3.1. Proof of Theorem 1. Statement (a). Under the assumption of the theorem we assume without loss of generality that a = 1, and set c = c0 ∈ R. Then system (4) can be written in q˙ = c0 q −
qn − qn qn + qn − (q1 i + q2 j + q3 k), 2 q −q
(6)
for q − q = 0, where we have used the fact that q − q = 2(q1 i + q2 j + q3 k) and qn =
qn + qn qn − qn + (q1 i + q2 j + q3 k). 2 q −q
Obviously, q n + q n and (q n − q n )/(q − q ) are real. Furthermore using the Darboux theory of integrability we can check easily that H2 =
q2 , q1
H3 =
q3 , q1
are two first integrals of Eq. (6), which follows from the facts that q1 = 0, q2 = 0 and q3 = 0 are three invariant algebraic hyperplanes with the same cofactor c0 − (q n − q n )/(q − q ). For more information on the Darboux theory of integrability, see for instance [24,28].
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We remark that the Darboux theory of integrability was developed for polynomial vector fields in Cn and Rn . Here we can use this theory in the non–commutative field H, because the mentioned invariant algebraic hyperplanes and their cofactors are all real. Generally, if a polynomial differential equation in H has its invariant algebraic hypersurfaces all real, we can apply the Darboux theory of integrability by using these hypersurfaces. The existence of the two functionally independent first integrals shows that the R4 space is foliated by invariant planes given by {H2 = h 2 } ∩ {H3 = h 3 } with h 2 , h 3 ∈ R ∪ {∞}. This proves statement (a1 ). We now prove statement (a2 ), that is, study the dynamics of Eq. (6) on each invariant plane. For any h 2 , h 3 ∈ R, restricted to each invariant plane P23 := {H2 = h 2 } ∩ {H3 = h 3 } Eq. (6) becomes s n −2 q0n−2s , q˙0 = c0 q0 − 2s s=0 [(n+1)/2] s−1 n −2 q0n−2s+1 , q˙1 = c0 q1 − 2s − 1 [n/2]
(7)
s=1
where [·] denotes the integer part function, 2 = q12 + q22 + q32 = q12 1 + h 22 + h 23 , and we have used the binormal expansion q = n
[n/2] s=0
+
n 2s
s −2 q0n−2s
[(n+1)/2] s=1
n 2s − 1
s−1 −2 q0n−2s+1 (q1 i + q2 j + q3 k),
and the fact that (q1 i + q2 j + q3 k)2 = −2 . For
studying the dynamics of Eq. (7) we transfer it to the complex field. Set z =
q0 + q1 1 + h 22 + h 23 i. Then Eq. (7) can be written in a one dimensional complex differential equation z˙ = c0 z − z n .
(8) δπ Clearly, this last equation has n singularities in C : z 0 = 0 and z k = |c0 | exp i n−1 + 2(k−1)π for k = 1, . . . , n − 1, where δ = 0 if c0 > 0 or δ = 1 if c0 < 0. These sinn−1 gularities are all nodes (see e.g. [4]), and z 0 = 0 has different stability than the other n − 1 ones. By introducing the polar coordinates z = r eiθ we can prove that Eq. (8) has exactly n − 1 heteroclinic orbits connecting the origin and infinity, and the unique heteroclinic orbit connecting each z k for k = 1, . . . , n − 1, and the infinity. All the other orbits are heteroclinic and connect the origin and one of the z k s. This proves statement (a2 ), and consequently statement (a). For proving statements (b) and (c), we note that for any a ∈ H there exists a c ∈ H
such that cac−1 = a0 +
√ n−1
a12 + a22 + a32 i. Moreover Eq. (4) with c0 ∈ R can be trans-
formed to p˙ = cac−1 (c0 p − p n ) by the change of variables p = cqc−1 . So in what
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follows we assume without loss of generality that a = a0 + a1 i. Set H=
(qq )n−1 q n−1 + q n−1 − c0
,
S = q n−1 + q n−1 − c0 .
We claim that the derivatives of H and S along Eq. (4) are d H = (n − 1)(a + a)(c0 − H )H, dt (4) d S n−1 n−1 n−1 n−1 c + c = (n − 1) aq − q − q q a . 0 0 dt
(9) (10)
(4)
Indeed, d q n−1 + q n−1 dt
= (4)
n−2 q l qq ˙ n−2−l + q n−2−l q˙ q l l=0
n−2 q l ac0 q n−1−l + q n−1−l c0 a q l − q l aq 2n−2−l − q 2n−2−l a q l = l=0
= (n − 1) c0 aq n−1 + q n−1 a − aq 2n−2 + q 2n−2 a .
In the last equality we have used the fact that q l aq k + q k a q l = aq k+l + q k+l a. This proves Eq. (10). Using Eq. (10) and the fact that qq and q n−1 + q n−1 − c0 are real, we can prove easily Eq. (9). This proves the claim. Restricted to the hypersurface P := {S = 0}, Eq. (10) becomes d S = (n − 1)(a + a)(qq )n−1 . (11) dt (4),P For convenience of the following proof, we write Eq. (4) in a system: qn + qn qn − qn − a1 c0 − q1 , q˙0 = a0 c0 q0 − 2 q −q qn − qn qn + qn q1 + a1 c0 q0 − , q˙1 = a0 c0 − q −q 2 qn − qn q˙2 = (a0 q2 − a1 q3 ) c0 − , q −q qn − qn q˙3 = (a1 q2 + a0 q3 ) c0 − . q −q
(12)
Statement (b). By the assumption we can assume that a0 = 1. From (11) we get that if an orbit of (4) passes through P, it should intersect P transversally. So each region limited by the branches of P is either positively or negatively invariant.
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Since S + c0 is a homogeneous polynomial in q ∗ = (q1 , q2 , q3 , q4 ) ∈ R4 , it follows that each branch of P is either a hyperplane or a generalized hyperboloid. From the expression of H it follows that each branch of the level hypersurfaces H = h for h ∈ R (if it exist) is compact. Obviously the level set H = 0 is the origin, and the level set H = c0 consists of the roots of q n−1 = c0 , because H = c0 is equivalent to (q n−1 − c0 )( q n−1 − c0 ) = 0. In fact, these level sets are exactly formed by the singularities. In addition the compact hypersurfaces H = h approach P when h → ±∞. The facts from the last paragraph and Eq. (9) imply that each orbit starting on P will finally approach two singularities, which are located in two consecutive regions limited by P. Furthermore, since the function H has different signs in the two consecutive regions limited by P, it follows from the continuation of H in each region limited by P that any heteroclinic orbit should go to the level set H = 0, i.e. the origin. This proves statement (b1 ). As a by product of the last results we get that for 0 < h < c0 the level set H = h is empty. Statement (b2 ) follows from the proof of statement (b1 ), especially the fact that the orbits starting on two consecutive branches of P either all get into or all go out of the region limited by the two branches. Statement (c). The assumption means that a0 = 0 and a1 = 0. Without loss of generality we take a1 = 1. Set F = q22 + q32 . Then F is a first integral of (4), which follows easily from (12) with a0 = 0. Moreover, we get from (9) that H is also a first integral of (4), which is functionally independent with F. From (11) it follows that each branch of the hypersurface P is invariant. We first study the dynamics of (4) on P. For n = 2 the level set P is a hyperplane, on which all orbits are parallel straight lines. For n > 2 some easy calculations show that (q n − q n )/(q − q ) = q n−1 + q n−1 . This verifies that system (12) on P has no singularities. Moreover each orbit on P is located on a cylinder F = f > 0 and rotates strictly along the cylinder. This proves (c1 ). By some direct calculations and using the equality iq n + q n i = L n−1 (iq + qi), we get that 2(n − 1)(qq )n−2 ∇H = 2 q n−1 + q n−1 − c0 n q + qn − c0 q0 , (L n−1 − c0 )q1 , (L n−1 − c0 )q2 , (L n−1 − c0 )q3 . × 2 So, in the invariant space R4 \ {P}, the critical points of (H, F) form the invariant plane S1 := {q2 = 0} ∩ {q3 = 0}, the invariant varieties S2 := {(q n + q n )/2 − c0 q0 = 0} ∩ {L n−1 = c0 } and S3 := {(q n + q n )/2 − c0 q0 = 0} ∩ {q1 = 0}, where L n−1 = (q n − q n )(q − q ). We get from (12) that the invariant variety S2 is full of singularities and that the invariant variety S3 is full of periodic orbits with a center and the heteroclinic orbits connecting the singularities on L n−1 = c0 .
Global Structure of Quaternion Differential Equations
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On the invariant plane S1 , system (12) is simply q˙0 = −c0 q1 +
[(n+1)/2] s=1
q˙1 = c0 q0 −
[n/2] s=0
n 2s
n 2s − 1
−q12
s
−q12
s−1
q0n−2s+1 ,
q0n−2s .
Taking z = q0 + iq1 , the last equation can be written in z˙ = i c0 z − z n .
(13)
Clearly Eq. (13) √ has n singularities: one is at the origin and the others are located on the circle |z| = n−1 |c0 |. Applying Theorem 2.1 of [4] to these singularities we get that the origin is an isochronous center with the period 2π/|c0 |, and the other singularities are also isochronous centers with the common period 2π/((n − 1)|c0 |). Furthermore, the periodic orbits surrounding the origin have different orientation than the ones around the other singularities. Hence we have obtained the dynamics of Eq. (13), and consequently that of Eq. (12) on the critical sets. For all regular values (h, f ) of (H, F), the intersection Mh, f = {H = h} ∩ {F = f } is a two dimensional compact invariant manifold, because Mh, f does not contain singularities and the intersection is transversal. We claim that the connected submanifolds of Mh, f are all invariant tori. Indeed, system (12) can be written in a Hamiltonian system with the Hamiltonian H under the Poisson bracket {·, ·} defined by {P, Q} = ∇ P M(q)∇ Q, where P, Q are two arbitrary smooth functions in R 4 and ⎛ n−1 2 0 −1 n−1 q +q − c0 ⎜ 1 0 M(q) = ⎝0 0 2(n − 1)(qq )n−2 0 0
⎞ 0 0 0 0 ⎟ . 0 −1 ⎠ 1 0
Furthermore the first integrals H and F are in involution under the Poisson bracket. Then the claim follows from the classic Liouvillian theorem on integrability. For more information on Poisson structures and Liouvillian integrability, see for instance [1,5]. This proves statement (c). We complete the proof of the theorem. 3.2. Proof of Theorem 2. As in the proof of statement (c) of Theorem 1 we take a = i, c0 > 0 and use the notations given there. Statement (a). Equation (12) with n = 2 becomes q˙0 = (2q0 − c0 )q1 ,
q˙1 = c0 q0 − q02 + q12 + q22 + q32 ,
q˙2 = (2q0 − c0 )q3 ,
q˙3 = −(2q0 − c0 )q2 .
Now P := {q0 = c0 /2},
H=
q02 + q12 + q22 + q32 , 2q0 − c0
F = q22 + q32 .
(14)
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Recall that P is an invariant hyperplane and H and F are two functionally independent first integrals. In the invariant space R4 \{P}, we have 2q0 − c0 = 0. For all regular values (h, f ) with f > 0 and either h > c0 or h < 0, we will prove that the invariant torus Mh, f = {H = h} ∩ {F = f } is full of periodic orbits. Taking the change of coordinates z = q0 − c0 /2 + i q1 , q2 = r cos θ and q3 = r sin θ , Eqs. (14) become c02 2 2 z˙ = −i z + i r + , r˙ = 0, θ˙ = −2 Re(z), (15) 4 where Re(z) denotes the real part of z. Equations (15) have the solutions r (t) = r, (z 0 + R + (z 0 − R) exp (−2Rt i)) R , z(t) = z 0 + R − (z 0 − R) exp (−2Rt i) t θ (t) = θ0 + 2 Re z(s)ds 0 1 z 0 + R − (z 0 − R) exp(−2Rti) ln = θ0 + 2Rt + Re i 2R z 0 + R − (z 0 − R) exp(−2Rti) , = θ0 + 2Rt + Arg 2R
with r ∈ (0, ∞) and R = r 2 + c02 /4. Clearly, z(t) is a periodic function of period π/R in t. Moreover, the third part in the summation of the last equality of θ (t) is also a periodic function of period π/R in t. These show that q2 and q3 are periodic functions of period π/R in t, and consequently the orbits on the invariant tori are all periodic. As a by product of the above proof, we get that with the expansion of the tori their periods become smaller and smaller. This proves statement (a). Statement (b). Equation (12) with n = 3 is q˙0 = − c0 − 3q02 + q12 + q22 + q32 q1 , q˙1 = c0 − q02 + 3q12 + 3q22 + 3q32 q0 , (16) q˙2 = − c0 − 3q02 + q12 + q22 + q32 q3 , q˙3 = c0 − 3q02 + q12 + q22 + q32 q2 . Now
P := 2 q02 − q12 − q22 − q32 − c0 ,
is an invariant generalized hyperboloid, and 2 2 q0 + q12 + q22 + q32 H= 2 , 2 q0 − q12 − q22 − q32 − c0
F = q22 + q32 ,
are two functionally independent first integrals. For each regular value (h, f ) of (H, F), we study the dynamics on the invariant tori Mh, f . Since f > 0, the generalized cylinder F = f is parametrized by q2 = f cos θ, q3 = f sin θ.
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Restricted to the F = f , the hypersurface H = h with f 2 + (2 f + c0 )h < 0 can be parametrized by q0 =
G(ϕ, h) cos ϕ, q1 =
G(ϕ, h) sin ϕ,
with G(ϕ, h) = h cos 2ϕ − f +
h 2 cos2 2ϕ − 2 f h cos 2ϕ − 2 f h − c0 h.
Note that here we study only those tori Mh, f with f > 0 and f 2 + (2 f + c0 )h < 0. They are probably the simplest ones which can be parametrized. On the above mentioned invariant torus Mh, f , system (16) is written: h˙ = 0, θ˙ = c0 + f − G(ϕ, h)(2 cos 2ϕ + 1),
(17)
r˙ = 0, ϕ˙ = c0 − G(ϕ, h) cos 2ϕ + f cos 2ϕ + 2 f. Set
A(ϕ, h) =
h 2 cos2 2ϕ − 2 f h cos 2ϕ − 2 f h − c0 h,
B(ϕ, h) = c0 + f − G(ϕ, h)(2 cos 2ϕ + 1). Then c0 − G(ϕ, h) cos 2ϕ + f cos 2ϕ + 2 f =
AB . A + 2h cos2 ϕ
If there is a periodic orbit on Mh, f , we assume that its smallest positive period is 2mπ in ϕ and 2nπ in θ for m, n ∈ N. We get from (17) that
2mπ
0
2nπ 2h cos2 ϕ 1+ dϕ = dθ. A 0
The last equality can be written in 2(n − m)π = m
2π
0
h(1 + cos2 ψ) h 2 cos2 ψ − 2 f h cos ψ − (2 f + c0 )h
dψ.
We can check easily that I (h) :=
1 2π
0
2π
h(1 + cos2 ψ) h 2 cos2 ψ − 2 f h cos ψ − (2 f + c0 )h
dψ,
is analytic in h with f 2 + (2 f + c0 )h < 0, and I (h) ≡ 0 in any open subset of R. The last claim implies that I (h) is a locally open mapping. So, for any given f > 0 there exist infinitely many h such that Mh, f is full of periodic orbits, and also infinitely many h for which Mh, f has dense orbits. This proves statement (b). We complete the proof of the theorem.
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3.3. Proof of Theorem 3. Working in a similar way to the proof of Theorem 1, we only need to study Eq. (4) with c = c0 + ic1 . Writing Eq. (4) in a system gives q˙0 = c0 q0 − c1 q1 − q02 + q12 + q22 + q32 , q˙1 = c1 q0 + (c0 − 2q0 )q1 , q˙2 = (c0 − 2q0 )q2 − c1 q3 , q˙3 = c1 q2 + (c0 − 2q0 )q3 . 2 Recall that L = c0 q0 + c1 q1 − c0 + c12 /2. (a) Restricted to the hyperplane L = 0 we have d L = 4c0 q02 + q12 + q22 + q32 . dt
(18)
(18)
So every orbit intersects the hyperplane L = 0 transversally. Obviously L = 0 is orthogonal to the line connecting the singularities O and S, and have the same distance to O and S. Set H=
q02 + q12 + q22 + q32 , 2c0 q0 + 2c1 q1 − K 0
where K 0 = c02 + c12 . We have −2c0 q02 + q12 + q22 + q32 B d H = , dt (18) (2c0 q0 + 2c1 q1 − K 0 )2
(19)
(20)
where B = (q0 −c0 )2 +(q1 −c1 )2 +q22 +q32 . The level set H = h is empty
if 0 < h < 1, and is a ball, denoted by Bh , centered at (hc0 , hc1 , 0, 0) with the radius c02 + c12 (h 2 − h) if h > 1 or h < 0. We can check easily that the balls Bh with h > 1 (resp. h < 0) contain the singularity S (resp. O) in their interiors and are located in L > 0 (resp. L < 0). Furthermore, it is easy to prove that when h 1 (resp. h 0) the ball Bh shrinks to the singularity S (resp. O), and that when h ∞ (resp. h −∞) the ball Bh expands and approaches the hyperplane L = 0. From the derivative of H and the property of the ball Bh , it follows that each orbit starting on L = 0 will be heteroclinic connecting the two singularities S and O. Moreover we get from the last two equations of (18) that these orbits spirally approach S and O. These last proofs imply that except those orbits being heteroclinic to S and O, there are two other ones: one is heteroclinic to S and infinity, and another is heteroclinic to O and infinity. This proves statement (a). Statement (b). Since c0 = 0, we get from (20) that the function H defined in (19) is a first integral of system (18). Furthermore we can prove that 2 2 q0 + q12 + q22 + q32 F= , (2q1 − c1 )2 + 4q22 + 4q32 is also a first integral of system (18). In addition, L = 0, i.e. 2q1 = c1 is invariant. Some calculations show that H and F are functionally independent, and that the critical points are {q2 = 0} ∩ {q3 = 0} and {q0 = 0} ∩ −a1 q1 + q12 + q22 + q32 = 0 . The corresponding values of (H, F) are (h, 0) and (h, f ) with f > 0 and √ √ critical h = c1 f / 2 f − c1 . On the invariant plane {q2 = 0} ∩ {q3 = 0}, there are
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two by the invariant line L = 0. On the invariant sphere period annuli separated q0 = 0} ∩ {−a1 q1 + q12 + q22 + q32 = 0 all orbits are periodic. This last claim follows from the fact that system (18) restricted to the sphere has the first integral q22 + q32 . √ √ For any h > 1 or h < 0 and f > 0 with h = c1 f / 2 f − c1 , the values (h, f ) are regular for (H, F). Since the hypersurfaces H = h and F = f are compact and intersect transversally, their intersections denoted by Mh, f should be two dimensional compact invariant manifolds (if they exist). We claim that the connected parts of Mh, f are 2–dimensional invariant tori. We now prove the claim. For doing so, it suffices to show that Mh, f is orientable and has genus 1. Associated to the 2–field, ∂H ∂F ∂H ∂F ∂ ∂ ∂H ∂F ∂H ∂F ∂ ∂ ∇ H ∧ ∇ F = ∂q − ∧ + − ∂q1 ∂q0 ∂q0 ∂q1 ∂q2 ∂q0 ∂q0 ∧ ∂q2 0 ∂q1 ∂q0 ∂q2 ∂H ∂F ∂H ∂F ∂ ∂H ∂F ∂ + ∂q − ∂q ∧ ∂ + ∂ H ∂ F − ∂q ∧ ∂ 3 ∂q0 2 ∂q1 ∂q0 ∂q3 ∂q1 ∂q2 ∂q1 ∂q2 0 ∂q3 ∂H ∂F ∂H ∂F ∂ ∂ ∂H ∂F ∂H ∂F ∂ ∂ + ∂q − ∂q ∂q1 ∧ ∂q3 + ∂q2 ∂q3 − ∂q3 ∂q2 ∂q2 ∧ ∂q3 , 1 ∂q3 3 ∂q1 the dual 2–form is ∂H ∂F ∂H ∂F ∂H ∂F ∂H ∂F ω = ∂q dq dq0 dq2 − dq − − 0 1 ∂q ∂q ∂q ∂q ∂q ∂q ∂q 3 2 3 1 2 3 1 3 ∂H ∂F ∂H ∂F ∂H ∂F ∂H ∂F + ∂q1 ∂q2 − ∂q2 ∂q1 dq0 dq3 + ∂q0 ∂q3 − ∂q3 ∂q0 dq1 dq2 ∂H ∂F ∂H ∂F ∂H ∂F ∂H ∂F dq dq2 dq3 . − ∂q − dq + − 1 3 ∂q ∂q ∂q ∂q ∂q ∂q ∂q 0 2 2 0 0 1 1 0 Recall that ∇ denotes the gradient of a smooth function. Since the fields ∇ H and ∇ F are linearly independent on Mh, f , the two form ω is non–zero on Mh, f . Hence Mh, f is orientable, see e.g. [1, Sect. 2.5] and also [11]. Denote by Xh, f the restriction of the vector field defined by (12) to Mh, f . Since the vector field Xh, f has no singularities, applying the Poincaré–Hopf formula to the manifold Mh, f we get 0 = ind(Xh,h 1 ) = χ (Mh,h 1 ) = 2 − 2g, where ind(Xh, f ) denotes the sum of the indices of the singularities of Xh, f on Mh, f , and χ (Mh, f ) and g are the Euler characteristic and the genus of the surface Mh, f , respectively. This shows that the genus of Mh, f is one. It is well–known that an orientable compact connected surface of genus one is a torus, see e.g., [22, Sect. X] for more details. We complete the proof of statement (b) and consequently the proof of the theorem. 3.4. Proof of Theorem 4. Recall that L = q02 − q12 − q22 − q32 − c02 /2. For simplifying the notations we denote by H+ and H− the subset of R4 with L√> 0 and L < 0 respec√ tively, and by H++ and H+− the two parts of H+ with q0 > c0 / 2 and q0 < −c0 / 2, respectively. Working in a similar way to the proof of Theorem 1, we assume without loss of generality that a = a0 + a1 i. Eq. (3) is equivalent to the system q˙0 = a1 q1 A − a0 q0 B,
q˙1 = −a0 q1 A − a1 q0 B,
q˙2 = −(a0 q2 − a1 q3 )A,
q˙3 = −(a1 q2 + a0 q3 )A,
(21)
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where A = c02 − 3q02 + q12 + q22 + q32 and B = c02 − q02 + 3 q12 + q22 + q32 . It is easy to check that system (21) has the three finite singularities O, S+ and S− . Furthermore restricted to L = 0 the derivative of L along the solutions of system (21) with respect to the time t is 2 d L 2 2 2q = −2a − c /2 . (22) 0 0 0 dt (21)
Statement (a). We consider the case a0 < 0. The proof of the case a0 > 0 follows from the same arguments as that of a0 < 0. By (22) we get that if an orbit intersects L = 0, it should transversally pass through it. Moreover the orbits meeting L = 0 will go from H− to H+ as the time increases. Set 2 (23) H = q02 + q12 + q22 + q32 / q02 − q12 − q22 − q32 − c02 /2 . We can check that for h ∈ (2c02 , ∞) the hypersurface E h := {H = h} has two branches which are located in H++ and H+− respectively, and that for h ∈ (−∞, 0) the hypersurface E h has a unique branch which is located in H− . Moreover we can check that for h ∈ (2c02 , ∞) ∪ (−∞, 0) the hypersurface E h is compact and contains one of the three singularities in its interior. When h → ±∞ the hypersurface E h approaches the hyperboloid L = 0. Now we can verify that 2 2 q0 + q12 + q22 + q32 d H = 8a0 N (24) 2 , dt (21) c02 − 2 q02 − q12 − q22 − q32 2 where N = c04 − 2c02 q02 − q12 − q22 − q32 + q02 + q12 + q22 + q32 . Since outside S1 and S2 we have N > 0, it follows that the subsets H+ (resp. H− ) are positively (resp. negatively) invariant by the flow of the system. Furthermore, all orbits starting in H++ (resp. H+− ) will approach S+ (resp. S− ) when t → ∞. All orbits starting in H− will go to √ √O when t → −∞. So all orbits starting on L = 0 with q0 > c0 / 2 (resp. q0 < −c0 / 2) will be heteroclinic to O and S+ (resp. to O and S− ). This proves statement (a). Statement (b). Equations (22) and (24) show that the hyperboloid L = 0 is invariant and that H is a first integral of system (21). Moreover we can prove that F = q22 + q32 , is also a first integral of (21), and that H and F are functionally independent. Recall that F = f is a 3–dimensional cylinder when f > 0 and is a plane when f = 0. Working in a similarway to the proof of statement (b) of Theorem 3 we can prove that for h ∈ (−∞, 0) ∪ 2c02 , ∞ the intersections E h ∩ {F = f } are either formed by periodic orbits for (h, f ) being critical values or 2–dimensional invariant tori for (h, f ) being regular values. Using the same methods as those given in the proof of statement (b) of Theorem 2 we can prove that of the invariant tori there are infinitely many ones fulfilling periodic orbits and also infinitely many ones fulfilling dense orbits. This proves statement (b) and consequently the theorem. Acknowledgements. The author thanks Professors Armengol Gasull and Jaume Llibre for their discussion and comments on part of results given in the first version of this paper. I appreciate the referees for their excellent comments and suggestions, which improved our paper both in the mathematics and in its expression.
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4. Appendix: The Linear Case For the homogeneous linear differential equations q˙ = aq + qb,
(25)
with a, b ∈ H nonzero, taking H = qq we have d H = (a + a + b + b)(qq ). dt (25) Moreover its equivalent 4–dimensional linear differential system has at the origin the four eigenvalues
2 2 (a + a + b + b)/2 ± (a − a) ± (b − b) /2. So the dynamics of (25) follows easily from these eigenvalues. For the homogeneous linear differential equations q˙ = aq + qb,
(26)
with a, b ∈ H nonzero, its equivalent 4–dimensional linear differential system has the four eigenvalues (a − b + a − b)/2 ±
(a + b − a + b)2 /2, (a + a)/2 ± (a − a)2 − bb /2.
Then the dynamics of (26) follows easily from these eigenvalues. For the homogeneous linear equations q˙ = aq + bq,
(27)
with a, b ∈ H nonzero, its equivalent 4–dimensional linear differential system has the four eigenvalues (a − b + a − b)/2 ±
(a − b − a − b)2 /2, (a + a)/2 ± (a − a)2 − bb /2.
Then its dynamics follows also from these eigenvalues. For the non–homogeneous linear quaternion differential equations q˙ = b + aq,
q˙ = b + qa,
(28)
with a, b ∈ H nonzero, they can be transformed to homogeneous ones via the change of variables p = q + a −1 b or p = q + ba −1 . So their dynamics can be obtained from Theorem 2 of [15].
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References 1. Abraham, R., Marsden, J.E.: Foundations of Mechanics. 2nd Ed., Redwood City, CA: Addison–Wesley, 1987 2. Adler, S.L.: Quaternionic quantum field theory. Commun. Math. Phys. 104, 611–656 (1986) 3. Adler, S.L.: Quaternionic Quantum Mechanics and Quantum Fields. New York: Oxford University Press, 1995 4. Álvarez, M.J., Gasull, A., Prohens, R.: Configurations of critical points in complex polynomial differential equations. Nonlin. Anal. 71, 923–934 (2009) 5. Arnold, V.I.: Mathematial Methods of Classical Mechanics. New York: Springer-Verlag, 1978 6. Bruschi, M., Calogero, F.: Integrable systems of quartic oscillators. II. Phys. Lett. A 327, 320–326 (2004) 7. Calogero, F., Degasperis, A.: New integrable PDEs of boomeronic type. J. Phys. A 39, 8349–8376 (2006) 8. Calogero, F., Degasperis, A.: New integrable equations of nonlinear Schrödinger type. Stud. Appl. Math 113, 91–137 (2004) 9. Chen, C., Cao, J., Zhang, X.: The topological structure of the Rabinovich system having an invariant algebraic surface. Nonlinearity 21, 211–220 (2008) 10. Campos, J., Mawhin, J.: Periodic solutions of quaternionic-values ordinary differential equations. Ann. di Mat. 185, S109–S127 (2006) 11. Cima, A., Gasull, A., Mañosa, V.: Some properties of the k–dimensional Lyness’s map. J. Phys. A: Math. Theor. 41, 285205 (2008) 12. Finkelstein, D., Jauch, J.M., Schiminovich, S., Speiser, D.: Foundations of quaternion quantum mechanics. J. Math. Phys. 3, 207–220 (1962) 13. Frobenius, F.G.: Ueber lineare Substitutionen und bilineare Formen. J. Reine Angew. Math 84, 1–63 (1878) 14. Gasull, A., Llibre, J., Mãosa, V., Mãosas, F.: The focus-centre problem for a type of degenerate system. Nonlinearity 13, 699–729 (2000) 15. Gasull, A., Llibre, J., Zhang, X.: One–dimensional quaternion homogeneous polynomial differential equations. J. Math. Phys. 50, 082705 (2009) 16. Gibbon, J.D.: A quaternionic structure in the three–dimensional Euler and ideal magneto–hydrodynamics equation. Physica D 166, 17–28 (2002) 17. Gibbon, J.D., Holm, D.D., Kerr, R.M., Roulstone, I.: Quaternions and particle dynamics in the Euler fluid equations. Nonlinearity 19, 1969–1983 (2006) 18. Giné, J., Llibre, J.: Integrability and algebraic limit cycles for polynomial differential systems with homogeneous nonlinearities. J. Diff. Eqs. 197, 147–161 (2004) 19. Hamilton, S.W.R.: Lectures on Quaternions. Royal Irish Academy, Dublin: Hodges and Smith, 1853 20. Hanson, A.J.: Visualizing Quaternions. San Francisco: Elsevier, 2006 21. Iona, S., Calogero, F.: Integrable systems of quartic oscillators in ordinary (three-dimensional) space. J. Phys. A 35, 3091–3098 (2002) 22. Lang, S.: Differential and Riemannian Manifolds. New York: Springer-Verlag, 1995 23. Li, C., Li, W., Llibre, J., Zhang, Z.: On the limit cycles of polynomial differential systems with homogeneous nonlinearities. Proc. Edinburgh Math. Soc. 43(2), 529–543 (2000) 24. Llibre, J.: Handbook of Differential Equations. Amsterdam: Elsevier/North–Holland, 2004, pp. 437–532 25. Llibre, J., Valls, C.: Classification of the centers, their cyclicity and isochronicity for a class of polynomial differential systems generalizing the linear systems with cubic homogeneous nonlinearities. J. Diff. Eqs. 246, 2192–2204 (2009) 26. Llibre, J., Yu, J.: On the periodic orbits of the static, spherically symmetric Einstein-Yang-Mills equations. Commun. Math. Phys. 286, 277–281 (2009) 27. Llibre, J., Zhang, X.: Invariant algebraic surfaces of the Lorenz systems. J. Math. Phys. 43, 1622–1645 (2002) 28. Llibre, J., Zhang, X.: Darboux theory of integrability in C n taking into account the multiplicity. J. Diff. Eqs. 246, 541–551 (2009) 29. Roubtsov, V.N., Roulstone, I.: Examples of quaternionic and K¨ahler structures in Hamiltonian models of nearly geostrophic flow. J. Phys. A: Math. Gen. 30, L63–L68 (1997) 30. Roubtsov, V.N., Roulstone, I.: Holomorphic structures in hydrodynamical models of nearly geostrophic flow. Proc. R. Soc. London A 457, 1519–1531 (2001) 31. Wilczynski, P.: Quaternionic–valued ordinary differential equations. The Riccati equation. J. Diff. Eqs. 247, 2163–2187 (2009) Communicated by G. Gallavotti
Commun. Math. Phys. 303, 317–330 (2011) Digital Object Identifier (DOI) 10.1007/s00220-011-1194-0
Communications in
Mathematical Physics
QP-Structures of Degree 3 and 4D Topological Field Theory Noriaki Ikeda1 , Kyousuke Uchino2 1 Maskawa Institute for Science and Culture, Kyoto Sangyo University, Kyoto 603-8555, Japan.
E-mail:
[email protected]
2 Department of Mathematics, Tokyo University of Science, Wakamiya 26, Shinjyuku, Tokyo, Japan.
E-mail:
[email protected] Received: 13 April 2010 / Accepted: 25 August 2010 Published online: 11 February 2011 – © Springer-Verlag 2011
Abstract: A BV algebra and a QP-structure of the degree 3 is formulated. A QP-structure of degree 3 gives rise to Lie algebroids up to homotopy and its algebraic and geometric structure is analyzed. A new algebroid is constructed, which derives a new topological field theory in 4 dimensions by the AKSZ construction. 1. Introduction A BV algebra and a QP-structure has been motivated by the structure of the BatalinVilkovisky formalism of a gauge theory [1,2] and is its mathematical formulation [3,4]. In case of a topological field theory of Schwarz type, a BV formalism has been reformulated to the AKSZ formulation, which is a clear construction using geometry of a graded manifold [5,6]. Application to higher n + 1 dimensions has been formulated and new topological field theories in higher dimensions have been founded by applying this construction [7–9]. In n = 1, a classical QP-structure is equivalent to a Poisson structure on a manifold M and is also a Lie algebroid on T ∗ M from the explicit construction. This is the construction of a Poisson structure by the Schouten-Nijenhuis bracket in a classical limit. The topological field theory in two dimensions constructed by the AKSZ formulation [6] is the Poisson sigma model [10–12] and the quantization of this model on disc derives the Kontsevich formula of the deformation quantization on a Poisson manifold [14,15]. In n = 2, a classical QP-structure is a Courant algebroid [16–19,32]. The topological field theory derived in three dimensions is the Courant sigma model [20–23]. However structures for higher n, more than 2, have not been understood enough apart from BF theories. In this paper, we analyze the n = 3 case. A QP-structure of degree 3 leads us to a new type of algebroid, which is called a Lie algebroid up to homotopy. The notion of this algebroid is defined as a homotopy deformation of a Lie algebroid satisfying some integrability conditions. We will prove that a QP-structure of degree 3 on a N-manifold
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(nonnegatively graded manifold) is equivalent to a Lie algebroid up to homotopy. This QP-structure defines a new natural 4-dimensional topological field theory via the AKSZ construction. The paper is organized as follows. In Sect. 2, a BV algebra and a QP-structure of degree 3 are formulated. In Sect. 3, a QP-structure of degree 3 is constructed and analyzed. In Sect. 4, examples of QP-structures of degree 3 are listed. In Sect. 5, the AKSZ construction of a topological field theory in four dimensions is formulated and examples are listed.1 2. QP-Manifolds and BV Algebras 2.1. Classical QP-manifold. Definition 2.1. A graded manifold M is by definition a sheaf of a graded commutative algebra over an ordinary smooth manifold M. In the following, we assume the degrees are nonnegative. The structure sheaf of M is locally isomorphic to a graded commutative algebra C ∞ (U ) ⊗ S(V ), where U is an ordinary local chart of M, S(V ) is the polynomial algebra over V and where V := i≥1 Vi is a graded vector space such that the dimension of Vi is finite for each i. For example, when V = V1 , M is a vector bundle whose fiber is V1∗ : the dual space of V1 . Definition 2.2. A graded manifold (M, ω) equipped with a graded symplectic structure ω of degree n is called a P-manifold of degree n. In the next section, we will study a concrete P-manifold of degree 3. The structure sheaf C ∞ (M) of a P-manifold becomes a graded Poisson algebra. The Poisson bracket is defined in the usual manner, {F, G} = (−1)|F|+1 ι X F ι X G ω,
(2.1)
C ∞ (M), |F|
where F, G ∈ is the degree of F and X F := {F, −} is the Hamiltonian vector field of F. We recall the basic properties of the Poisson bracket, {F, G} = −(−1)(|F|−n)(|G|−n) {G, F}, {F, G H } = {F, G}H + (−1)(|F|−n)|G| G{F, H }, {F, {G, H }} = {{F, G}, H } + (−1)(|F|−n)(|G|−n) {G, {F, H }}, where n is the degree of the symplectic structure and F, G, H ∈ C ∞ (M). We remark that the degree of the Poisson bracket is −n. Definition 2.3. Let (M, ω) be a P-manifold of degree n. A function ∈ C ∞ (M) of degree n + 1 is called a Q-structure, if it is a solution of the classical master equation, {, } = 0.
(2.2)
The triple (M, ω, ) is called a QP-manifold. We define an operator Q := {, −}, which is called a homological vector field. From (2.2) we have the cocycle condition, Q 2 = 0, which says that the homological vector field is a coboundary operator on C ∞ (M) and defines a cohomology called the classical BRST cohomology. 1 Very recently, Grützmann’s paper appears which has overlaps with our paper [24].
QP-Structures with Degree 3 and 4D Topological Field Theory
319
2.2. Quantum QP-manifold. Definition 2.4. A graded manifold is called a quantum BV-algebra if it has an odd Laplace operator , which is a linear operator on C ∞ (M) satisfying 2 = 0, and the graded Poisson bracket is given by {F, G} = (−1)|F| (F G) − (−1)|F| (F)G − F(G),
(2.3)
where F, G ∈ C ∞ (M). If n is odd, a P-manifold (M, ω) has the odd Poisson bracket. If an odd P-manifold (M, ω) has a volume form ρ, one can define an odd Laplace operator (see [25]): F :=
1 (−1)|F| divρ X F . 2
Here a divergence divρ is a map from a space of vector fields on M to C ∞ (M) and is defined by divρ X Fdv = − X (F)dv, M
M
for a vector field X on M. The pair (M, ) is called a quantum P-structure. An odd Laplace operator has degree −n. Definition 2.5. A function ∈ C ∞ (M) with degree n + 1 is called a quantum Q-structure, if it satisfies a quantum master equation i
(e ) = 0,
(2.4)
where is a formal parameter. The triple (M, , ) is called a quantum QP-manifold. From the definition of an odd Laplace operator, Eq. (2.4) is equivalent to {, } − 2i = 0.
(2.5)
If we take the limit of → 0 in (2.5), which is called a classical limit, the classical master equation {, } = 0 is derived. Since 2 = 0, is also a coboundary operator. i This defines a quantum BRST cohomology. Let O = Oe ∈ C ∞ (M) be a cocycle i with respect to . The cocycle condition (O ) = (Oe ) = 0 is equivalent to {, O} − iO = 0.
(2.6)
The solutions of (2.6) are called observables in physics. In the classical limit, (2.6) is {, O} = QO = 0. O reduces to an element of a classical BRST cohomology. 3. Structures and Homotopy Algebroids In this section, we construct and analyze a classical QP-structure of degree 3 explicitly.
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3.1. P-structures. Let E → M be a vector bundle over an ordinary smooth manifold M. The shifted bundle E[1] → M is a graded manifold whose fiber space has degree +1. We consider the shifted cotangent bundle M := T ∗ [3]E[1]. It is a P-manifold of degree 3 over M, T ∗ [3]E[1] → M2 → E[1] → M, where M2 is a certain graded manifold.2 The structure sheaf C ∞ (M) of M is decomposed into the homogeneous subspaces, C ∞ (M) = C i (M), i≥0
where C i (M) is the space of functions of degree i. In particular, C 0 (M) = C ∞ (M): the algebra of smooth functions on the base manifold and C 1 (M) = E ∗ : the space of sections of the dual bundle of E. Let us denote by (x, q, p, ξ ) a canonical (Darboux) coordinate on M, where x is a smooth coordinate on M, q is a fiber coordinate on E[1] → M, (ξ, p) is the momentum coordinate on T ∗ [3]E[1] for (x, q). The degrees of the variables (x, q, p, ξ ) are respectively (0, 1, 2, 3). Two directions of counting the degree of functions on T ∗ [3]E[1] are introduced. Roughly speaking, these are the fiber direction and the base direction. Definition 3.1 (Bi-degree, see also Remark 3.3.3 in [18]). Consider a monomial ξ i p j q k on a local chart (U ; x, q, p, ξ ) of M, of which the total degree is 3i +2 j +k. The bidegree of the monomial is, by definition, (2(i + j), i + k). This definition is invariant under the natural coordinate transformation, xi = xi (x1 , x2 , ..., xdim(M) ), qi = ti j q j , j
pi
=
ti−1 j pj,
j
ξi
=
∂t −1 ∂tkl −1 jl ξj + ( tlk + t ) p j qk , ∂ xi ∂ xi ∂ xi l j
∂x j j
jkl
where t is a transition function. Since T ∗ [3]E[1] is covered by the natural coordinates, the bidegree is globally well-defined (See also Remark 3.2 below.) The space C n (M) is uniquely decomposed into the homogeneous subspaces with respect to the bidegree, C n (M) = C 2i, j (M). 2i+ j=n
Since C 2,0 (M) = E and C 0,2 (M) = ∧2 E ∗ , we have C 2 (M) = E ⊕ ∧2 E ∗ . 2 In fact, M is E[1] ⊕ E ∗ [2], which is derived from the result in the previous sentence of Remark 3.2. 2
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Remark 3.2. The P-manifold T ∗ [3]E[1] is regarded as a shifted manifold of T ∗ [2]E[1]. The structure sheaf is also a shifted sheaf of the one on T ∗ [2]E[1]. In particular, the space C 2i, j is the shifted space of C i, j on T ∗ [2]E[1]. For the canonical coordinate on M, the symplectic structure has the following form: ω = δx i δξ i + δq a δ pa , and the associated Poisson bracket has the following expression: → → → → ← − − ← − − ← − − ← − − ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ G−F G + F G − F G, {F, G} = F i ∂ x ∂ξ i ∂ξ i ∂ x i ∂q a ∂ pa ∂ pa ∂q a − →
← −
∂ ∂ and ∂φ are the right and left differentiations, respecwhere F, G ∈ C ∞ (M) and ∂φ tively. Note that the degree of the symplectic structure is +3 and the one of the Poisson bracket is −3. The bidegree of the Poisson bracket is (−2, −1), that is,
{(2i, j), (2k, l)} = (2(i + k) − 2, j + l − 1), where (2i, j)... are functions with the bidgree (2i, j). 3.2. Q-structures. We consider a (classical) Q-structure, , on the P-manifold. It is required that has degree 4. That is, ∈ C 4 (M). Because C 4 (M) = C 4,0 (M) ⊕ C 2,2 (M) ⊕ C 0,4 (M), the Q-structure is uniquely decomposed into = θ2 + θ13 + θ4 , where the bidegrees of the substructures are (4, 0), (2, 2) and (0, 4), respectively. In the canonical coordinate, is the following polynomial: = f 1 i a (x)ξ i q a + +
1 ab 1 f 2 (x) pa p b + f 3 a bc (x) pa q b q c 2 2
1 f 4abcd (x)q a q b q c q d , 4!
(3.7)
and the substructures are 1 ab f 2 (x) pa p b , 2 1 θ13 = f 1 i a (x)ξ i q a + f 3 a bc (x) pa q b q c , 2 1 a b c d θ4 = f 4abcd (x)q q q q , 4! θ2 =
where f 1 − f 4 are structure functions on M. By counting the bidegree, one can easily prove that the classical master equation {, } = 0 is equivalent to the following three identities: {θ13 , θ2 } = 0, 1 {θ13 , θ13 } + {θ2 , θ4 } = 0, 2 {θ13 , θ4 } = 0.
(3.8) (3.9) (3.10)
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The conditions (3.8), (3.9) and (3.10) are equivalent to f 1 i b f 2 ba = 0, ∂ f2 + f 2 da f 3 b cd + f 2 db f 3 a cd ∂xk i ∂ f 1i a k ∂f 1 b − f + f 1 i c f 3 c ab f 1k b 1 a ∂xk ∂xk ∂ f 3 a bc] + f 2 ae f 4bcde − f 3 a e[b f 3 e cd] f 1 k [d ∂xk ∂ f 4bcde] f 1 k [a + f 3 f [ab f 4cde] f ∂xk f 1k c
ab
(3.11)
= 0,
(3.12)
= 0,
(3.13)
= 0,
(3.14)
= 0,
(3.15)
where [b c d · · · ] is a skewsymmetrization with respect to indices b, c, d, . . ., etc.
3.3. Lie algebroid up to homotopy. In this section we study an algebraic structure associated with the QP-structure in 3.1 and 3.2. Definition 3.3. Let Q = θ2 + θ13 + θ4 be a Q-structure on T ∗ [3]E[1], where (θ2 , θ13 , θ4 ) is the unique decomposition of . We call the quadruple (E; θ2 , θ13 , θ4 ) a Lie algebroid up to homotopy, or in shorthand, Lie algebroid u.t.h. We should study the algebraic properties of the Lie algebroid up to homotopy. Let us define a bracket product by [e1 , e2 ] := {{θ13 , e1 }, e2 },
(3.16)
where e1 , e2 ∈ E. By the bidegree counting, E is closed under this bracket. The bracket is not necessarily a Lie bracket, but it is still skewsymmetric: [e1 , e2 ] = {{θ13 , e1 }, e2 }, = {θ13 , {e1 , e2 }} + {e1 , {θ13 , e2 }}, = −{{θ13 , e2 }, e1 } = −[e2 , e1 ], where {e1 , e2 } = 0 is used. A bundle map ρ : E → T M which is called an anchor map is defined by the following identity: ρ(e)( f ) := {{θ13 , e}, f }, where f ∈ C ∞ (M). The bracket and the anchor map satisfy the algebroid conditions (A0) and (A1) below: (A0) ρ[e1 , e2 ] = [ρ(e1 ), ρ(e2 )], (A1) [e1 , f e2 ] = f [e1 , e2 ] + ρ(e1 )( f )e2 ,
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where the bracket [ρ(e1 ), ρ(e2 )] is the usual Lie bracket on T M. The bracket (3.16) does not satisfy the Jacobi identity in general. So we should study its Jacobi anomaly, which characterizes the algebraic structure of the Lie algebroid u.t.h. The structures θ13 , θ2 and θ4 define the three operations: • δ(−) := {θ13 , −}: a de Rham type derivation on ∧· E ∗ , • (α1 , α2 ) := {{θ2 , α1 }, α2 }: a symmetric pairing on E ∗ , where α1 , α2 ∈ E ∗ , • (e1 , e2 , e3 , e4 ) := {{{{{θ4 , e1 }, e2 }, e3 }, e4 }: a 4-form on E. Remark that δδ = 0 in general. Because the degree of the pairing is −2, it is C ∞ (M)valued. The pairing induces a symmetric bundle map ∂ : E ∗ → E which is defined by the equation, (α1 , α2 ) = ∂α1 , α2 , where − , − is the canonical pairing of the duality of E and E ∗ . Since α , e = {α, e}, we have ∂α = −{θ2 , α}. By direct computation, we obtain 1 {{{{θ13 , θ13 }, e1 }, e2 }, e3 } = [[e1 , e2 ], e3 ] + (cyclic permutations), 2 and {{{{θ2 , θ4 }, e1 }, e2 }, e3 } = −∂(e1 , e2 , e3 ). From Eq. (3.9), we get an explicit formula of the Jacobi anomaly, (A2) [[e1 , e2 ], e3 ] + (cyclic permutations) = ∂(e1 , e2 , e3 ). In a similar way, we obtain the following identities: (A3) ρ∂ = 0, (A4) ρ(e)(α1 , α2 ) = (Le α1 , α2 ) + (α1 , Le α2 ), (A5) δ = 0, where Le (−) := {{θ13 , e}, −} is the Lie type derivation which acts on E ∗ . Axioms (A3) and (A4) are induced from Eq. (3.8) and (A5) is from Eq. (3.10). The fundamental relations (3.11)–(3.15) correspond to Axioms (A1)–(A5):3 Thus, the notion of the Lie algebroid up to homotopy is characterized by the algebraic properties (A1)–(A5). One concludes that The classical algebra associated with the QP-manifold (T ∗ [3]E[1], ) is the space of sections of the vector bundle E with the operations ([·, ·], ρ, ∂, ) satisfying (A1)–(A5). In the next section, we will study some special examples of Lie algebroid u.t.h.s. Remark 3.4. If the pairing is nondegenerate, then the bundle map ∂ is bijective and then from (A3) we have ρ = 0. Remark 3.5 (Higher Courant-Dorfman brackets). We define a bracket on C ∞ (M) by [−, −]C D := {{, −}, −}, which is called a Courant-Dorfman (CD) bracket. It is well-known that [, ]C D is a Loday bracket ([26]). Since the degree of the CD-bracket is −2, the total space of degree i ≤ 2, C 2 (M) ⊕ C 1 (M) ⊕ C 0 (M), 3 Actually, Axiom (A0) depends on (A1) and (A2).
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is closed under the CD-bracket, in particular, the top space C 2 (M) = (E ⊕ ∧2 E ∗ ) is a subalgebra. If θ2 = 0, the CD-bracket on E ⊕ ∧2 E ∗ has the following form, [e1 + β1 , e2 + β2 ]C D = [e1 , e2 ] + Le1 β2 − i e2 δβ1 + (e1 , e2 ), where β1 , β2 ∈ ∧2 E ∗ . This CD-bracket is regarded as a higher analogue of CourantDofman’s original bracket (cf. [16,17]). We refer the reader to Hagiwara [27] and Sheng [28] for the detailed study of the higher CD-brackets.
4. Examples and Twisting Transformations 4.1. The cases of θ2 = θ4 = 0. In this case, the bracket (3.16) satisfies (A0), (A1) and the Jacobi identity. Therefore, the bundle E → M becomes a Lie algebroid: Definition 4.1 ([29]). A Lie algebroid over a manifold M is a vector bundle E → M with a Lie algebra structure on the space of the sections (E) defined by the bracket [e1 , e2 ] for e1 , e2 ∈ (E) and an anchor map ρ : E → T M satisfying (A0) and (A1) above. We take {ea } as a local basis of E and let a local expression of an anchor map be ρ(ea ) = f i 1a (x) ∂∂x i and a Lie bracket be [eb , ec ] = f 3 a bc (x)ea . The Q-structure associated with the Lie algebroid E is defined as a function on T ∗ [3]E[1], := θ13 := f 1 i a (x)ξ i q a +
1 a f 3 bc (x) pa q b q c , 2
which is globally well-defined. Conversely, if we consider := θ13 , the classical master equation induces the Lie algebroid structure on E. Let us consider the case that the bundle is a vector space on a point. A Lie algebroid over a point g → { pt} is a Lie algebra g. The P-manifold over g → { pt} is isomorphic to g∗ [2] ⊕ g[1] and the structure sheaf is the polynomial algebra over g[2] ⊕ g∗ [1], C ∞ (M) = S(g) ⊗
·
g∗ .
The bidegree is defined by the natural manner,
C 2i, j (M) = S i (g) ⊗
j
g∗ .
The Q-structure associated with the Lie bracket on g is θ13 =
1 a 1 f bc pa q b q c ∼ = f a bc pa ⊗ (q b ∧ q c ), 2 2
where p· ∈ g, q· ∈ g∗ and f a bc is the structure constant of the Lie algebra.
(4.17)
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4.2. The cases of θ2 = 0 and θ4 = 0. In this case, the bracket induced by θ13 still satisfies the Jacobi identity. We assume that g is semi-simple. Then the dual space g∗ has a metric, (·, ·) K −1 , which is the inverse of the Killing form on g. The metric inherits the following invariant condition from the Killing form: (L p q1 , q2 ) K −1 + (q1 , L p q2 ) K −1 = 0,
(4.18)
where L p (−) is the canonical coadjoint action of g to g∗ . Equation (4.18) is a linear version of (A4). Thus, we obtain a Q-structure, := k ab pa p b +
1 a f bc pa q b q c , 2
(4.19)
where k ab pa p b := (·, ·) K −1 .
4.3. Non Lie algebra example. We consider the cases that the Jacobi identity is broken. Let (g, [·, ·], (·, ·) K ) be a vector space (not necessarily Lie algebra) equipped with a skewsymmetric bracket [·, ·] and an invariant metric (·, ·) K . The metric induces a bijection K : g → g∗ which is defined by the identity, ( p1 , p2 ) K = K p1 , p2 . We define a map from g∗ to g by ∂ := K −1 and define a 4-form by, ( p1 , p2 , p3 , p4 ) := ([[ p1 , p2 ], p3 ] + cyclic permutations, p4 ) K . Remark 4.2. The 4-form above is considered to be a higher analogue of the Cartan 3-form ([ p1 , p2 ], p3 ) K . Axioms (A0)–(A4) obviously hold on g. We check (A5). It suffices to show (3.10). Let us denote by {−, p1 , p2 , . . . , pn } the n-fold bracket {. . . {{−, p1 }, p2 }, . . . , pn }. We already have (3.8) and (3.9). From {θ13 , {θ13 , θ13 }} = 0 and (3.9), we have {θ13 , {θ2 , θ4 }} = 0. Since {θ13 , θ2 } = 0, this is equal to {θ2 , {θ3 , θ4 }} = 0 up to sign. This gives {{θ2 , {θ3 , θ4 }}, p1 , . . . , p5 } = 0 for any p1 , . . . , p5 . From {θ2 , p} = 0, we have {θ2 , {{θ3 , θ4 }, p1 , . . . , p5 }} = 0. Since K −1 = −{θ2 , −} is bijective, we get {{θ3 , θ4 }, p1 , . . . , p5 } = 0, which yields the desired relation {θ3 , θ4 } = 0. Proposition 4.3. The triple (g, ∂, ) is a Lie algebra(oid) up to homotopy.
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4.4. Twisting by 3-form and the cases of θ2 = 0 and θ4 = 0. We introduce the notion of twisting transformation by 3-form before studying the cases of θ2 = 0. Given a Q-structure and a 3-form φ ∈ C 0,3 (M), there exists the second Q-structure which is defined by the canonical transformation, φ := exp(X φ )(),
(4.20)
where X φ := {φ, −} is the Hamiltonian vector field of φ. The transformation (4.20) is called a twisting by 3-form, or simply twisting. By a direct computation, we obtain φ
θ2 = θ2 , φ
θ13 = θ13 − {θ2 , φ}, 1 φ θ4 = θ4 − {θ13 , φ} + {{θ2 , φ}, φ}, 2 φ
φ
φ
where φ = θ2 + θ13 + θ4 and X φi≥3 () = 0. The twisting by 3-form defines an equivalence relation on the Q-structures. We notice that θ2 is an invariant for the twisting. If θ2 = 0, then θ13 is an invariant and φ
θ4 = θ4 − δφ, where δφ = {θ13 , φ}. This leads us to Proposition 4.4. The class of Q-structures which have no θ2 is classified into Hd4R ( · E ∗ , δ) by the twisting by 3-form. 5. AKSZ Construction of Topological Field Theory in 4 Dimensions 5.1. General theory. In this section, we consider the AKSZ construction of a topological field theory in 4 dimensions. For a graded manifold N , let N |0 be the degree zero part. Let X be a manifold in 4 dimensions and M be a manifold in d dimensions. Let (X , D) be a differential graded (dg) manifold X with a D-invariant nondegenerate measure μ, such that X |0 = X , where D is a differential on X . (M, ω, ) is a QP-manifold of degree 3 and M|0 = M. A degree deg(−) on X is called the form degree and a degree gh(−) on M is called the ghost number.4 Let Map(X , M) be a space of smooth maps from X to M. | − | = deg(−) + gh(−) is the degree on Map(X , M) and called the total degree. A QP-structure on Map(X , M) is constructed from the above data. Since Diff(X ) × Diff(M) naturally acts on Map(X , M), D and Q induce homoˇ logical vector fields on Map(X , M), Dˆ and Q. Two maps are introduced. An evaluation map ev : X × MX −→ M is defined as ev : (z, ) −→ (z), ∈ MX . : • (X
where z ∈ X and A chain map μ∗ × M) −→ • (M) is defined as μ∗ F = X μF, where F ∈ • (X × M) and X μ is an integration on X by the D-invariant measure μ. It is an usual integral for the even degree parts and the Berezin integral for the odd degree parts. 4 The ghost number gh(−) is the degree | − | on M in Sect. 2.
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A (classical) P-structure on Map(X , M) is defined as follows: Definition 5.1. For a graded symplectic form ω on M, a graded symplectic form ω on Map(X , M) is defined as ω := μ∗ ev∗ ω. We can confirm that ω satisfies the definition of a graded symplectic form because μ∗ ev∗ preserves nondegeneracy and closedness. Thus ω is a P-structure on Map(X , M) and induces a graded Poisson bracket {−, −} on Map(X , M). Since |μ∗ ev∗ | = −4, |ω| = −1 and {−, −} on Map(X , M) has the degree one and an odd Poisson bracket. Next we define a Q-structure S on Map(X , M). S is called a BV action and consists of two parts S = S0 + S1 . S0 is constructed as follows: Let ω be the odd symplectic form on M. We take a fundamental form ϑ such that ω = −dϑ and define S0 := ι Dˆ μ∗ ev∗ ϑ. |S0 | = 0 because μ∗ ev∗ has degree −4. S1 is constructed as follows: We take a Q-structure on M and define S1 := μ∗ ev∗ . S1 also has degree 0. We can prove that S is a Q-structure on Map(X , M), since {, } = 0, ⇐⇒ {S, S} = 0
(5.21)
from the definition of S0 and S1 . A quantum version is ˆ S ) = 0, (e ) = 0 ⇐⇒ (e i
i
(5.22)
ˆ is an odd Laplace operator on Map(X , M). The infinitesimal form of the right where ˆ = 0, which is called a quantum master equation. 5 hand side in (5.22) is {S, S} − 2iS The following theorem has been confirmed [5]: Theorem 5.2. If X is a dg manifold and M is a QP-manifold, the graded manifold Map(X , M) has a QP-structure. Definition 5.3. A topological field theory in 4 dimensions is a triple (X , M, S), where X is a dg manifold with dim X |0 = 4, M is a QP-manifold with the degree 3, and S is a BV action with the total degree 0. In order to interpret this theory as a ‘physical’ topological field theory, we must take X = T [1]X . Then we can confirm that a QP-structure on Map(X , M) is equivalent to the AKSZ formulation of a topological field theory [6,13]. We set X = T [1]X from now. In ‘physics’, a quantum field theory is constructed by quantizing a classical field theory. First we consider a Q-structure {·, ·} and a classical P-structure S such that {S, S} = 0. ˆ and confirm that Next we define a quantum P-structure ˜ S ) = 0. (e i
Finally we calculate a partition function Z=
i
L
e S,
on a Lagrangian submanifold L ⊂ Map(X , M). Quantization is not discussed in this paper. 5 Discussion for an odd Laplace operator is too naive. In general, the quantum master equation has an obstruction expressed by the modular class [30]. We must regularize an odd Laplace operator and a quantum BV action.
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5.2. Local coordinate expression and examples. A general theory in the previous subsection is applied to the local coordinate expression in Sect. 3.1 and a known topological field theory in 4 dimensions is obtained as a special case and a new nontrival topological field theory is constructed. Let us take a manifold X in 4 dimensions and a manifold M in d dimensions. Let E[1] be a graded vector bundle on M. We take X = T [1]X and M = T ∗ [3]E[1]. Let (σ μ , θ μ ) be a local coordinate on T [1]X . σ μ is a local coordinate on the base manifold X and θ μ is one on the fiber of T [1]X , respectively. Let x i be a smooth map x i : X −→ M and ξ i be a section of T ∗ [1]X ⊗ x ∗ (T ∗ [3]M), q a be a section of T ∗ [1]X ⊗ x ∗ (E[1]) and pa be a section of T ∗ [1]X ⊗ x ∗ (T ∗ [3]E x [1]). These are called superfields. The exterior derivative d is taken as a differential D on X . From d, a differential d = θ μ ∂σ∂ μ on X is induced. Then a BV action S has the following expression: S = S0 + S1 , S0 = μ (ξ i d x i − pa dq a ), X 1 1 S1 = μ ( f 1 i a (x)ξ i q a + f 2 ab (x) pa pb + f 3 a bc (x) pa q b q c 2 2 X 1 a b c d + f 4abcd (x)q q q q ). 4! Nonabelian BF theory. Let be a Q-structure (4.17) for a Lie algebra g. ξ i d x i = 0, since M = { pt}. If we define a curvature F a = dq a − 21 f a bc q b q c , a Q-structure is μ(− pa F a ), S= X
which is equivalent to a BV formalism for a nonabelian BF theory in 4 dimensions. Topological Yang-Mills theory. We take a nondegenerate Killing form (·, ·) K for a Lie algebra g and consider the Q-structure (4.19). A topological field theory constructed from (4.19) is S= μ (− pa F a + k ab pa pb ). X
This is equivalent to a topological Yang-Mills theory, 1 μ kab F a F b , S=− 4 X if we delete pa by the equations of motion. Nonassociative BF theory. Let us take a non Lie algebra (g, [·, ·], (·, ·) K ) in Sect. 4.3. If we take M = { pt} and M = g∗ [2] ⊕ g[1], (g, [·, ·], (·, ·) K ) leads a QP-structure with degree 3. In the canonical basis, it is expressed as f 1 i a (x) = 0, f 2 ab (x) = K ab , −1 f e f 3 a bc (x) = f a bc , f 4abcd (x) = K ae f [b f
f
cd] ,
where K ab = ( pa , pb ) is nondegenerate and [ pa , pb ] = f c ab pc is a nonassociative bracket and does not satisfy the Jacobi identity. The AKSZ construction derives a new
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nontrivial topological field theory in 4 dimensions. A BV action S has the following expression: 1 1 1 −1 e S= μ(− pa dq a + K ab pa pb + f a bc pa q b q c + K ae f f [b f f cd] q a q b q c q d ) 2 2 4! X 1 1 −1 e =− μ(K ab F a F b + K ae f f [b f f cd] q a q b q c q d ). 4 X 3! It is easily confirmed that {S, S} = 0. Topological 3-brane on Spin(7)-structure. Let (M, ) be an 8-dimensional Spin(7)manifold. Here is a Spin(7)4-form, which satisfies d = 0 and the selfdual condition = ∗. A Spin(7) structure is defined as the subgroup of G L(8) to preserve . The Q-structure on (T M, ) is given by = ξ i qi +
1 i jkl (x)q i q j q k q l . 4!
(5.23)
The BV action S for (5.23) defines the same theory as the topological 3-brane analyzed in [31]. 6. Conclusions and Discussion We have defined a BV algebra and a QP-structure of degree 3. A QP-structure of degree 3 has been constructed explicitly and a Lie algebroid u.t.h. has been defined as its algebraic and geometric structure. A general theory of the AKSZ construction of a topological field theory has been expressed and a new topological field theory in four dimensions has been constructed from a QP-structure. Quantization of this theory and analysis of a Lie algebroid u.t.h. will shed light on a super Poisson geometry and a quantum field theory. They are future problems. Acknowledgements. The first author (N.I.) would like to thank the Maskawa Institute for Science and Culture, Kyoto Sangyo University for hospitality. We would like to thank to referees for their useful advice. The authors would like to thank Klaus Bering, Maxim Grigoriev, Camille Laurent-Gengoux, Yvette KosmannSchwarzbach, Kirill Mackenzie, Dmitry Roytenberg, Alexei Sharapov, Thomas Strobl and Theodore Voronov for their comments and discussions.
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14. 15. 16. 17.
Kontsevich, M.: Lett. Math. Phys. 66, 157 (2003) Cattaneo, A.S., Felder, G.: Commun. Math. Phys. 212, 591 (2000) Courant, T.: Trans. A. M. S 319, 631 (1990) Liu, Z.J., Weinstein, A., Xu, P.: Dirac structures and Poisson homogeneous spaces. http://arxiv.org/abs/ dg-ga/9611001v1, 1996 Roytenberg, D.: Quasi-Lie bialgebroids and Twisted Poisson manifolds. Lett. Math. Phys. 61, 123–137 (2002) Roytenberg, D.: On the structure of graded symplectic supermanifolds and Courant algebroids. Contemp. Math. Vol. 315, Providence, RI: Amer. Math. Soc., 2002 Ikeda, N.: Int. J. Mod. Phys. A 18, 2689 (2003) Ikeda, N.: JHEP 0210, 076 (2002) Hofman, C., Park, J.S.: Topological Open Membranes. http://arxiv.org/abs/[hep-th/0209148]v1, 2002 Roytenberg, D.: Lett. Math. Phys. 79, 143 (2007) Grützmann, M.: H-twisted Lie algebroids. J. Geom. Phys. 61, 476–484 (2011) Khudaverdian, H.O.M.: Commun. Math. Phys. 247, 353 (2004) Kosmann-Schwarzbach, Y.: Lett. Math. Phys. 69, 61 (2004) Hagiwara, Y.: J. Phys. A: Math. Gen. 35, 1263 (2002) Sheng, Y.: On higher-order Courant Brackets. http://arxiv.org/abs/1003.1350v1 [math.DG], 2010 Mackenzie, K.: Lie Groupoids and Lie Algebroids in Differential Geometry, LMS Lecture Note Series 124, Cambridge: Cambridge U. Press, 1987 Lyakhovich, S.L., Sharapov, A.A.: Nucl. Phys. B 703, 419 (2004) Bonelli, G., Zabzine, M.: JHEP 0509, 015 (2005) Roytenberg, D.: Courantalgebroids, derived brackets and even symplectic supermanifolds, http://arxiv. org/abs/math.DG/9910078
18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32.
Communicated by Y. Kawahigashi
Commun. Math. Phys. 303, 331–359 (2011) Digital Object Identifier (DOI) 10.1007/s00220-011-1206-0
Communications in
Mathematical Physics
Quasiclassical Lian-Zuckerman Homotopy Algebras, Courant Algebroids and Gauge Theory Anton M. Zeitlin Department of Mathematics, Yale University, 442 Dunham Lab, 10 Hillhouse Ave., New Haven, CT 06511, USA. E-mail:
[email protected] URL: http://math.yale.edu/az84; http://www.ipme.ru/zam.html Received: 27 April 2010 / Accepted: 18 October 2010 Published online: 15 February 2011 – © Springer-Verlag 2011
To Gregg J. Zuckerman for his 60th birthday Abstract: We define a quasiclassical limit of the Lian-Zuckerman homotopy BV algebra (quasiclassical LZ algebra) on the subcomplex, corresponding to “light modes”, i.e. the elements of zero conformal weight, of the semi-infinite (BRST) cohomology complex of the Virasoro algebra associated with vertex operator algebra (VOA) with a formal parameter. We also construct a certain deformation of the BRST differential parametrized by a constant two-component tensor, such that it leads to the deformation of the A∞ -subalgebra of the quasiclassical LZ algebra. Altogether this gives a functor the category of VOA with a formal parameter to the category of A∞ -algebras. The associated generalized Maurer-Cartan equation gives the analogue of the Yang-Mills equation for a wide class of VOAs. Applying this construction to an example of VOA generated by β-γ systems, we find a remarkable relation between the Courant algebroid and the homotopy algebra of the Yang-Mills theory.
1. Introduction The relation between the dynamics of two-dimensional world and D-dimensional field theory is in the very heart of String Theory. An important problem is to find how the classical nonlinear equations of motion of gauge theory and gravity emerge from such two-dimensional dynamics. In the early days of string theory, the solutions of the linearized equations of motion modulo gauge symmetry were identified with the semi-infinite cohomology classes of the Virasoro algebra for a certain Virasoro module. On the other hand, in the same time period the nonlinear equations of motion, e.g. Yang-Mills and Einstein equations with extra fields, have been derived as relations coming from the conformal invariance condition for two-dimensional sigma models [5,6,8,9,29]. These relations are usually written as β(φ{ν} , h) = 0, where β(φ{ν} , h) = i∈N h i βi (φ{ν} ) is some function of all the fields φν , which are present in the sigma-model, and h is a formal parameter having a meaning
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of the Planck constant. The equation β1 (φ{ν} ) = 0 (β1 is known as a 1-loop β-function) is equivalent to the classical nonlinear field equations. The definition of β was and is still a mystery from the mathematical point of view. However, it was suggested by some authors in the 1980s that the algebraic version of the equation β(φ{ν} , h) = 0 should be something like the (generalized) Maurer-Cartan equation. Soon after that, the open and the closed String Field Theories (SFT) have been constructed [36,49]. The equations of motion of String Field Theory, which are supposed to contain nonlinear field-theoretic equations of motion, had the form of Maurer-Cartan equations for certain homotopy algebras: associative algebras for open strings and strong homotopy Lie algebras for closed ones. However, it was very hard to derive these equations from SFT [3,7], moreover, the relation of these Maurer-Cartan equations to the β-function vanishing condition was also not clear. It is worth noting several attempts to establish such a relation [1,33]. At the same time, in the 1980s, the Vertex Operator Algebra (VOA) theory was constructed (see e.g. [10,12]): a mathematical theory describing two-dimensional conformally invariant models, which are of special importance in string theory. It was observed by B.H. Lian and G.J. Zuckerman [21], that a special class of VOAs, the topological VOAs (TVOAs), possesses a homotopy algebra. This algebra turned out to be a homotopy Batalin-Vilkovisky (BV) algebra. It was also conjectured in [20] that there are “higher homotopies” for this algebra, such that it can be extended to the object which the authors of [20] called G ∞ -algebra (see also [13,18]), which first appeared in [15] (see also [35]). In a recent article [14], it was also conjectured and proven (for a certain class of TVOA) that there exists a BV∞ -algebra, which is the extension of Lian-Zuckerman homotopy algebra. One of the important classes of TVOAs is the semi-infinite cohomology complex (or simply BRST complex) associated with certain VOAs. Taking into account the physical motivation, in order to construct the β-function and classical field equations, one has to construct a functor from the category of vertex operator algebras to the category of homotopy algebras. A natural candidate for this functor is the one provided by the construction of LZ algebra on the BRST complex of the Virasoro algebra. We show by means of several examples, that it is enough to consider a certain quasi-isomorphic subcomplex of this BRST complex, the so-called light modes, which are annihilated by the L 0 Virasoro mode and therefore generate a subalgebra in the LZ algebra. When conformal weights in the VOA are bounded from below (in this paper we assume them to be bounded by zero), the complex of light modes is easy to work with. In this paper we will consider only this light mode complex. We define what we call the quasiclassical limit of the LZ homotopy BV algebra, which leads to a certain “truncation” of higher homotopies. This means that it contains A∞ - and L ∞ -algebras, such that all polylinear operations vanish starting from the quadrilinear ones. We conjecture that it is actually the BV∞ -algebra, motivated by the results of [13,14]. In the β-γ example, this gives a homotopy BV algebra associated to The Courant algebroid on the sum of tangent and cotangent bundles. The corresponding L ∞ -algebra is the one constructed by D. Roytenberg and A. Weinstein [32]. Then we introduce a deformation of the BRST differential, which we call a f lat backgr ound deformation, which corresponds to the Abelian vertex subalgebra and involves a constant two-component tensor. We prove that this deformation can be continued to the deformation of the homotopy commutative A∞ subalgebra of the LZ
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algebra. This gives us a functor from the category of VOAs with a chosen Abelian subalgebra to the category of A∞ -algebras parametrized by this constant tensor. In the beta-gamma example, this deformed A∞ -algebra turns out to be the A∞ -algebra of the Yang-Mills theory with external fields. The structure of the paper is as follows. In Sect. 2 we set up notations and recall the basic facts concerning Lian-Zuckerman homotopy BV algebra. We also give a short reminder about A∞ -algebras. In Sect. 3, we define the quasiclassical limit of the LZ algebra of light modes. We prove explicitly that the quasiclassical limit of LZ product extends to the A∞ -algebra. In Sec. 4 we define what we call a flat background deformation of the quasiclassical LZ algebra. This is a deformation of the differential, such that in general only the homotopy commutative A∞ -subalgebra of the homotopy BV algebra can be deformed in such a way to satisfy all necessary conditions with the deformed differential. In Sect. 5 we apply the constructions we introduced earlier to the VOA generated by a family of β-γ systems. In particular, we obtain that the quasiclassical LZ algebra corresponds to the BV algebra, which in conformal weight 1 reproduces the Courant algebroid. After that, we show that the flat background deformation in a certain case leads to the Yang-Mills theory with matter fields, which yields a remarkable relation between Courant/Dorfman brackets and gauge theory. In the last section we outline some of the numerous possible directions continuing the studies started in this paper. 2. Lian-Zuckerman Homotopy BV Algebra 2.1. Notation and conventions. Throughout the paper we will work with vertex operator algebras (VOA), using physics notation. Therefore the elements of the VOA’s vector space will be referred to as states and A(z) denotes the vertex operator Y (A, z) (see e.g. [10,12]), corresponding to the state A. To simplify the calculations, we introduce a special notation for certain operator product coefficients. Namely if A and B are the elements of the VOA, then we denote A(z)B A, B ≡ Resz (z A(z)B), [A, B] ≡ Resz (A(z)B), AB ≡ Resz . (1) z 2.2. Topological VOA and the Lian-Zuckerman homotopy BV algebra. Topological vertex operator algebra (TVOA) is a vertex superalgebra (see e.g.[10]) that has an additional odd operator Q which makes the graded vector space of VOA a chain complex, such that the Virasoro element L(z) is Q-exact. The formal definition is as follows (see e.g. [20] for more details). Definition 2.1. Let V be a Z-graded vertex operator superalgebra, such that V = ⊕i V i = ⊕i,μ V i [μ], where i represents grading of V with respect to conformal weight and μ represents fermionic grading of V i . We call V a topological vertex operator algebra (TVOA) if there exist four elements: J ∈ V 1 [1], b ∈ V 2 [−1], F ∈ V 1 [0], L ∈ V 2 [0], such that (2) [Q, G(z)] = L(z), Q 2 = 0, G 20 = 0, where Q = J0 and G(z) = n bn z −n−2 , J (z) = n Jn z −n−1 , L(z) = n Ln z −n−2 , F(z) = n Fn z −n−1 . Here L(z) is the Virasoro element of V ; the operators F0 , L0 are
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diagonalizable, commute with each other and their egenvalues coincide with fermionic grading and conformal weight correspondingly. A natural example of such object, which will be crucial in the following, is the semiinfinite cohomology (or simply BRST) complex for the Virasoro algebra [11] of some VOA with central charge equal to 26. The necessary setup for the construction of the semi-infinite complex (for more details, see [11] or [22]) is the VOA , obtained from the following super—Heisenberg algebra: {bn , cm } = δn+m,0 , n, m ∈ Z.
(3)
One can construct the space of as a Fock module: = {b−n 1 . . . b−n k c−m 1 . . . c−m l 1, n 1 , . . . , n k > 0, m 1 , . . . , m l > 0; ck 1 = 0, k 2; bk 1 = 0, k −1}. Then one can define two fields: b(z) = bm z −m−2 ,
c(z) =
m
cn z −n+1 ,
(4)
(5)
n
which according to the commutation relations between modes have the following operator product: b(z)c(w) ∼
1 . z−w
(6)
The Virasoro element is given by the following expression: L (z) = 2 : ∂b(z)c(z) : + : b(z)∂c(z) :,
(7)
such that b(z) has conformal weight 2, and c(z) has conformal weight −1. Here, as 1 usual, :: stand for normal ordered product, e.g. b(z)c(w) = z−w + : b(z)c(w) : (for more details see e.g. [10], Sect. 2.2). Now let V be a VOA with the Virasoro element L(z). Let us consider the tensor product V ⊗ . Then we have the following proposition. Proposition 2.1 [11]. If V is a VOA with the central charge of Virasoro algebra equal to 26, then V ⊗ is a topological vertex algebra, where 3 J (z) = c(z)L(z)+ : c(z)∂c(z)b(z) : + ∂ 2 c(z), G(z) = b(z), 2 F(z) =: c(z)b(z) :, L(z) = L(z) + L (z).
(8)
The operator Q = J0 is traditionally called the B RST operator and the eigenvalue of F0 , i.e. fermionic grading is usually called the ghost number . Lian and Zuckerman observed that each TVOA possesses a rich algebraic structure. They have shown the following. One can define two operations which are cochain maps with respect to Q: a1 (z)a2 (−1)|a1 | μ(a1 , a2 ) = Resz (9) , {a1 , a2 } = dz(b−1 a1 )(z)a2 . z 2πi These operations satisfy the following relations:
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Proposition 2.2 [21]. The operation μ is homotopy commutative and homotopy associative: Qμ(a1 , a2 ) = μ(Qa1 , a2 ) + (−1)|a1 | μ(a1 , Qa2 ), μ(a1 , a2 ) − (−1)|a1 ||a2 | μ(a2 , a1 ) = Qm(a1 , a2 ) + m(Qa1 , a2 ) + (−1)|a1 | m(a1 , Qa2 ), Qn(a1 , a2 , a3 ) + n(Qa1 , a2 , a3 ) + (−1) + (−1)
|a1 |+|a2 |
|a1 |
(10)
n(a1 , Qa2 , a3 )
n(a1 , a2 , Qa3 ) = μ(μ(a1 , a2 ), a3 ) − μ(a1 , μ(a2 , a3 )),
where m(a1 , a2 ) =
(−1)i i≥0
i +1
Resw Resz−w (z − w)i w −i−1 b−1 (a1 (z − w)a2 )(w)1,
1 Resz Resw wi z −i−1 (b−1 a1 )(z)a2 (w)a3 n(a1 , a2 , a3 ) = i +1 i≥0
+(−1)|a1 ||a2 |
(11)
1 Resw Resz z i w −i−1 (b−1 a2 )(w)a1 (z)a3 . i +1 i≥0
The operation {·, ·} measures the failure of b0 to be a derivation of μ. In other words we have the following proposition: Proposition 2.3 [21]. The operations μ and {·, ·} are related in the following way: {a1 , a2 } = b0 μ(a1 , a2 ) − μ(b0 a1 , a2 ) − (−1)|a1 | μ(a1 , b0 a2 ).
(12)
Moreover, the bracket satisfies the relations of a homotopy Gerstenhaber algebra. Proposition 2.4 [21]. The operations μ and {·, ·} satisfy the relations: {a1 , a2 } + (−1)(|a1 |−1)(|a2 |−1) {a2 , a1 } = (−1)|a1 |−1 (Qm (a1 , a2 ) − m (Qa1 , a2 ) − (−1)|a2 | m (a1 , Qa2 )), {a1 , μ(a2 , a3 )} = μ({a1 , a2 }, a3 ) + (−1)(|a1 |−1)||a2 | μ(a2 , {a1 , a3 }), {μ(a1 , a2 ), a3 } − μ(a1 , {a2 , a3 }) − (−1)(|a3 |−1)|a2 | μ({a1 , a3 }, a2 ) |a1 |+|a2 |−1
= (−1)
(13)
(Qn (a1 , a2 , a3 ) − n (Qa1 , a2 , a3 )
|a1 |
− (−1)
n (a1 , Qa2 , a3 ) − (−1)|a1 |+|a2 | n (a1 , a2 , Qa3 ),
{{a1 , a2 }, a3 } − {a1 , {a2 , a3 }} + (−1)(|a1 |−1)(|a2 |−1) {a2 , {a1 , a3 }} = 0, where m , n are some bilinear and trilinear operations on the TVOA (see p. 621 of [21]). In fact, the operations m and n are constructed from μ, m and n. However, we will not need explicit expressions in the following. The relations (10)–(13) by definition mean that μ(·, ·) and {·, ·} generate the homotopy BV algebra. In the following we will refer to the concrete homotopy BV algebra generated by μ and {, } as the L Z algebra. We note here, that in this article when we say “homotopy” (associative, Lie, Gerstenhaber, BV…) algebra it means that we have a certain set of operations endowed with just a first level of homotopies. This provides the structure of a strict (associative,
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Lie, Gerstenhaber, BV…) algebra structure on the cohomology, as in [21]. When we talk about ∞-algebras, we mean that there is a full set of higher homotopies (see below the explicit description of A∞ -algebras) satisfying some relations. Therefore ∞-algebra is a homotopy algebra, but the inverse is not true in general. Lian and Zuckerman made a conjecture, that the LZ algebra can be extended to G ∞ algebra (see e.g. [20]). It was recently proved [13,14] that for a certain class of TVOAs, that this algebra is in fact the BV∞ -algebra [14]. As we discussed above, this means that there are “higher homotopies”, i.e. in the general case nonzero multilinear operations which satisfy the higher associativity/Jacobi/Leibniz relations. Here we will not discuss this complicated object in detail. However, this conjecture implies that there exists a homotopy algebra, which “extends” the relations between μ and n only. Such an algebra is called an A∞ -algebra, and we will discuss the precise definition in the next subsection.
2.3. Short reminder of A∞ -algebras. The A∞ -algebra is a generalization of a differential graded associative algebra. Namely, consider a graded vector space V with a differential Q. Consider the multilinear operations μi : V ⊗i → V of the degree 2 − i, such that μ1 = Q. Definition 2.2 (see e.g. [27]). The space V is an A∞ -algebra if the operations μn satisfy bilinear identity: n−1 (−1)i Mi ◦ Mn−i+1 = 0
(14)
i=1
on V ⊗n , where Ms acts on V ⊗m for any m ≥ s as the sum of all possible operators of l m−s−l the form 1⊗ ⊗ μs ⊗ 1⊗ taken with appropriate signs. In other words, Ms =
n−s l m−s−l (−1)l(s+1) 1⊗ ⊗ μs ⊗ 1⊗ .
(15)
l=0
Let us write several relations which are satisfied by Q, μ1 , μ2 , μ3 : Q 2 = 0, Qμ2 (a1 , a2 ) = μ2 (Qa1 , a2 ) + (−1)|a1 | μ2 (a1 , Qa2 ), Qμ3 (a1 , a2 , a3 ) + μ3 (Qa1 , a2 , a3 ) + (−1)|a1 | μ3 (a1 , Qa2 , a3 )
(16)
+ (−1)|a1 |+|a2 | μ3 (a1 , a2 , Qa3 ) = μ2 (μ2 (a1 , a2 ), a3 ) − μ2 (a1 , μ2 (a2 , a3 )). In such a way we see that if μn = 0, n ≥ 3, then we have just a differential graded associative algebra (DGA). If the operations μn vanish for all n > k, such A∞ -algebras are sometimes called A(k) -algebras [34], so e.g. DGA is the A(2) algebra. We observe that putting μ2 ≡ μ and μ3 = n, these relations are manifestly the same as the ones relating Q, μ and n. The Lian-Zuckerman conjecture states that there are “higher homotopies” μn , n > 3 satisfying the relations (14). It is well known that the relations (14) can be encoded into one equation ∂ 2 = 0 [27]. To see this one can apply the desuspension operation (the operation which shifts
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the grading s −1 : Vn → (s −1 V )n−1 ) to μn . In such a way we can define operations of ⊗n degree 1: μ˜ n = sμn (s −1 ) . More explicitly, μ˜ n (s −1 a1 , . . . , s −1 an ) = (−1)s(a) s −1 μn (a1 , . . . , an ),
(17)
such that s(a) = (1 − n)|a1 | + (2 − n)|a2 | + . . . + |an−1 |. The relations between μ˜ n operations can be summarized in the following simple equations: n
M˜ i ◦ M˜ n+1−i = 0
(18)
i=1
on V ⊗n , where each M˜ s acts on V ⊗m (for m ≥ s) as the sum of all operators 1⊗l ⊗ μ˜ s ⊗ 1⊗k , such that l + s + k = m. Combining them into one operator ∂ = n M˜ n , acting on a space ⊕k V ⊗k the relations (14) can be summarized in one equation ∂ 2 = 0. An important object in the theory of A∞ -algebras is the generalized Maurer-Cartan (GMC) equation. Let us pick X ∈ V of degree 1. Then the equation QX + μn (X, . . . , X ) = 0 (19) n≥2
is called the generalized Maurer-Cartan equation on X . It is worth mentioning that it is well defined in general only on nilpotent elements, i.e. such that μn (X, . . . , X ) = 0 for n > k. This will not be a problem in the following, because the only A∞ algebras we will consider in this article, are A(3) -algebras and therefore GMC equation will be well defined for all elements of degree 1. The Generalized Maurer-Cartan equation is known to have the following infinitesimal symmetry: X → X + (Qα + (−1)n−k μn (X, . . . , α, . . . , X )), (20) n≥2,k
where is infinitesimal, α is an element of degree 0 and k means the position of α in μn . 3. Light Modes and the Quasiclassical Limit 3.1. Vertex operator algebras with a formal parameter. In this article we will be interested in computing quasiclassical limits, therefore we need to consider the vertex operator algebras depending on a formal parameter. Namely, we consider the VOAs on the spaces of the form V = ⊕n∈Z≥0 V n , where n stands for the grading with respect to conformal weight, and V n = V n [h −1 , h], where V n are some vector spaces and h is a formal parameter. Let us denote V = ⊕n∈Z≥0 V n . Then, we require the operator products to meet the following conditions: i) For any state A the associated vertex operator A(z) = n An z −n−1 is such that its Fourier modes An ∈ EndC[h,h −1 ] (V), i.e. they commute with the natural action of C[h, h −1 ] on V. ii) Let A, B ∈ V and A(z) = n An z −n−1 , then An B ∈ hV [h], n ≥ 0. Moreover, we put the following conditions on the Virasoro action:
(21)
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iii) V is invariant under the action of the operator L −1 . iv) Let A ∈ V be a state of conformal weight 1, then L 1 A ∈ hV .1 As a consequence of properties i) − iv), we observe that letting h = 1 we obtain a VOA structure on V . Let us consider some simple examples of such parameter-dependent VOAs. Examples. i) Heisenberg VOA. We consider the Fock space for the Heisenberg algebra, [an , am ] = hnδn,−m ,
(22)
i.e. the space Fa [h −1 , h], where Fa = {a−n 1 . . . a−n k |0, n 1 , . . . , n k > 0; an |0 = 0, n ≥ 0}. The space Fa is a VOA, such that on the operator product expansion language these commutation relations are summarized as follows: a(z)a(w) ∼
h , (z − w)2
(23)
where a(z) = n an z −n−1 . The Virasoro element is given by the operator L(z) = 1 2 h : a(z) :, where dots stand for standard normal ordering in the Fock space. ii) β-γ system. This is the most interesting example for us, because in this case there is a nontrivial subspace of the VOA of conformal weight equal to zero. For all the details we refer to the paper [26]. Let us consider the Heisenberg algebra of the following kind: [γn , βm ] = hδn,−m .
(24)
The construction of the corresponding VOA space Fβγ [h −1 , h] is as follows. Let us consider again the Fock space for the Heisenberg algebra (24): F˜βγ = {β−n 1 . . . .β−n k γ−m 1 . . . γ−m l |0, n 1 , . . . , n k > 0, m 1 , . . . , m l > 0; βn |0 = 0, n ≥ 0, γn |0 = 0, n > 0}. In other words, F˜βγ = C[β−n , γ−m ]n>0,m≥0 . The space F˜βγ [h −1 , h] already carries an algebraic structure of a VOA, but one can proceed further ˆ where A = C[γ0 ] and Aˆ = C[[γ0 ]] or any and construct the space Fβγ = F˜βγ ⊗ A A, other function field containing C[γ0 ]. From [26] one can prove that Fβγ [h −1 , h] is a VOA with a formal parameter, such that the commutation relations (24) can be expressed via the operator product: γ (z)β(w) ∼
h , β(z)β(w) ∼ 0, γ (z)γ (w) ∼ 0, z−w
(25)
where β(z) = n βn z −n−1 is a quantum field of conformal dimension 1 and γ (z) = −n is a quantum field of conformal dimension 0. The Virasoro element is given n γn z by L β,γ (z) = − h1 : β∂γ (z) :, where dots stand for normal ordering. One can easily continue this list of examples. In fact, there is a quite large class of VOAs, which can be obtained from the VOAs with a formal parameter by letting this parameter be equal to 1. 1 This condition, in principle, can be relaxed, by letting L A ∈ hV [h], and (with some modifications) one 1 can apply the constructions of the article to this case. However, being motivated by certain concrete examples we keep the more restrictive form of condition iv).
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3.2. The light modes. Let us consider the space V ⊗ , where is the b-c ghost VOA defined in Subsect. 2.2. When the central charge of the Virasoro algebra of V is equal to 26, this space is the semi-infinite cohomology complex of the Virasoro algebra. However, there is a subspace of V ⊗ which is a complex with respect to the BRST operator regardless of the value of the central charge. Definition 3.1. We call the subspace of V ⊗ which is annihilated by L0 (i.e. space of states of conformal weight zero) the space of light modes. Since the conformal weights in the VOA are greater than zero, the following proposition holds. Proposition 3.1. (i) The space of light modes FL0 is linearly spanned by the elements which correspond to the operators: ˜ u(z), c(z)A(z), ∂c(z)a(z), c(z)∂c(z) A(z), 2 2 c(z)∂ c(z)a(z), ˜ c(z)∂c(z)∂ c(z)u(z). ˜
(26)
Here u, u, ˜ a, a˜ ∈ V are of conformal weight 0 and A, A˜ ∈ V are of conformal weight 1. (ii) The space FL0 is a chain complex, quasi-isomorphic to the semi-infinite complex when the central charge is equal to 26. The differential acts on FL0 in the following way (we recall that the grading is given by the ghost number): 1 1 V V E 66 D 66 66 6 66 L −1 666 66 66 1 L L −1 662 1 666 66 6 66 1 66 − 66 2 L 1 66 / V0 V0 V0 V0
(27)
id
where V i (i = 0, 1) is the space of the elements of V of conformal dimension i. Proof. The first part of the proposition can be proven by comparing the conformal weights of differential polynomials of operator c(z) and the operators in vertex algebra V in such a way that the total conformal weight is zero. Part ii) can be obtained using the following formulas: 1 Q A(z) = ∂c(z)A(z) + c(z)∂ A(z) + ∂ 2 c(z)L 1 A(z), 2
Qc(z) = c(z)∂c(z), (28)
where u ∈ V 0 , A ∈ V 1 . Really, using (28), we get: Qu(z) = c(z)∂u(z), Q(c(z)A(z)) = −c(z)∂ 2 c(z)L 1 A(z), Q(∂c(z)a(z)) = c(z)∂ 2 c(z)a(z) + c(z)∂c(z)∂a(z), (29) 1 2 2 ˜ ˜ ˜ = 0. Q(c(z)∂ c(z)a(z)) Q(c(z)∂c(z) A(z)) = c(z)∂c(z)∂ c(z)L 1 A(z), 2 Proposition 3.1 is proved. Another important observation is about the operator b0 .
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Corollary 3.1. The space FL0 is invariant under the action of the operator b0 , [Q, b0 ] = 0 on FL0 and b0 has a trivial cohomology. The explicit form of the action of b0 is: V1 o V0 o
id
−id
V1 V0 o
V0
(30) −id
V0
Important remark about notation. In order to simplify some of the calculations, in the following we will sometimes write instead of the element of the complex FL0 the corresponding state of V, i.e. instead of the element corresponding to c A we will just write A, ˜ Since according to our notation the states without tilde correspond or A˜ instead of c∂c A. to tensor products with elements of ghost number 0 and 1, and the states marked by tilde correspond to the tensor product with elements of ghost number 2 and 3, this should never lead to confusion. We note here that the operations μ and {·, ·} act invariantly on the space of light modes, therefore the light modes form a homotopy subalgebra of LZ algebra. Moreover, from Proposition 3.1, and Corollary 3.1, we see that the Lian-Zuckerman algebra on light modes is determined by means of operator product expansion of the elements of V of conformal dimensions 0 and 1. For example, explicit expressions for the bilinear operation μ is collected in the table: μ(a1 , a2 ) = H H a1 u1 a2 HHH u2 u1u2 A2
u 1 A2
v2 A˜ 2 v˜2 u˜ 2
u 1 u˜ 2 u 1 A˜ 2 u 1 u˜ 2 u 1 u˜ 2
(31)
A1
v1
A˜ 1
v˜1
A1 u 2 +[A1 , u 2 ] −[A1 , A2 ]+ 1 2 A1 , A2 A1 v2 1 ˜ 2 A 1, A 2 [A1 , v˜2 ] 0
v1 u 2
A˜ 1 u 2
v˜1 u 2
−v1 A2 −[v˜1 , A2 ] 0 [v1 , A˜ 2 ] −v1 v˜2 0
1 ˜ 2 A1 ,
A2
−[ A˜ 1 , v2 ] 0 0 0
u˜ 1 u˜ 1 u 2
[v˜1 , A2 ]
0
−v˜1 v2 0 0 0
0 0 0 0
Let us denote by FL+0 the space of light modes, which belong to V [h] ⊗ . The Lian-Zuckerman algebraic operations act on this space invariantly since V [h] is a vertex algebra. Let us denote FL+0 (1) the Lian-Zuckerman algebra on FL+0 when h = 1. Below we construct the embedding of FL+0 (1) into FL+0 as a chain complex. The following proposition holds: Proposition 3.2. Let F be the subspace of FL+0 , linearly spanned by the elements corresponding to the vertex operators: ˜ u(z), c(z)A(z), h∂c(z)v(z), hc∂c(z) A(z), 2 2 2 hc(z)∂ c(z)v(z), ˜ h c(z)∂c(z)∂ c(z)u(z), ˜ where u, u, ˜ v, v, ˜ A, A˜ ∈ V . Then the following two statements hold:
(32)
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i) The complex F is a BRST subcomplex of FL+0 isomorphic to FL+0 (1). Moreover, it is a subcomplex with respect to the operator h −1 b0 . ii) The Lian-Zuckerman operations act as follows on F: μ : Fi ⊗ F j → Fi+ j [h], m : Fi ⊗ F j → Fi+ j−1 [h], n : Fi ⊗ F j ⊗ Fk → Fi+ j+k−1 [h], {Fi , F j } → hFi+ j−1 [h],
(33)
where Fi is a subspace of F of ghost number i.
Therefore, one can consider the expansions μ(·, ·) = μ0 (·, ·) + O(h), m(·, ·)h = m 0 (·, ·) + O(h), n(·, ·, ·)h = n 0 (·, ·, ·) + O(h), {·, ·} = h{·, ·}0 + O(h 2 ). Then we have a theorem, which follows from Proposition 3.2.
Theorem 3.1. The operations μ0 (·, ·), m 0 (·, ·), {·, ·}0 , n 0 (·, ·, ·) and the differentials Q and h −1 b0 , defined on the space of the light modes of V ( i.e. FL0 (1)), satisfy the relations of the homotopy BV algebra.
We will call the resulting algebra on F the quasiclassical Lian-Zuckerman algebra. 2 Further we need explicit expressions for all bilinear operations and their homotopies. We use the following notation. Let a1 , a2 0 = lim h→0 h −1 a1 , a2 and [a1 , a2 ]0 = lim h→0 h −1 [a1 , a2 ], where a1 , a2 ∈ V . Below, the expression a1 a2 means the normal ordered product of the two elements of V in the h → 0 limit. Then we can express the bilinear operations μ0 (·, ·) and {·, ·}0 via the following tables: μ0 (a1 , a2 )= HH a1 u1 a2 HHH u2 u1u2 A2
u 1 A2
v2 A˜ 2 v˜2 u˜ 2
u 1 u˜ 2 u 1 A˜ 2 u 1 u˜ 2 u 1 u˜ 2
A1
v1
A˜ 1
v˜1
A1 u 2 +[A1 , u 2 ]0 −[A1 , A2 ]0 − 1 2 A1 , A2 0 A1 v2 1 ˜ 2 A1 , A2 0 [A1 , v˜2 ]0 0
v1 u 2
A˜ 1 u 2
v˜1 u 2
−v1 A2
1 ˜ 2 A1 ,
0 0 −v1 v˜2 0
0 0 0 0
A2 0
u˜ 1 u˜ 1 u 2
[v˜1 , A2 ]0
0
−v˜1 v2 0 0 0
0 0 0 0
2 Let us stress here, that the quasiclassical limit we consider, is different from the standard limit that takes vertex algebra to Poisson vertex algebra. If one applies such limit to our construction, one gets only the subalgebra of the homotopy Lie algebra, which is a part of the homotopy BV algebra.
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{a1 , a2 }0 =
HH a1 a2 HH H
u1
A1
v1
A˜ 1
v˜1
u2 A2
0 0
−[A1 , u 2 ]0 −[A1 , A2 ]0
0 0
0 [v˜1 , A2 ]0
0 −[u˜ 1 , A2 ]0
v2 A˜ 2
0 0 0 −[A1 , u˜ 2 ]0
−[A1 , v2 ]0 −[A1 , A˜ 2 ]0 −[A1 , v˜2 ]0 0
0 0 0 0
[ A˜ 1 , u 2 ]0 −[ A˜ 1 , A2 ]0 − 21 A˜ 1 , A2 0 0 A˜ 1 , A˜ 2 0 −[ A˜ 1 , v˜2 ]0 0
0 [v˜1 , A˜ 2 ]0 0 0
0 0 0 0
v˜2 u˜ 2
u˜ 1
In the tables above we keep the same notation as in (31) except for the fact that u i vi , u˜ i , v˜i and Ai , A˜ i are associated with the elements from V 0 and V 1 correspondingly. The bilinear operation m is nonzero only if both its arguments…belong to F1 : m 0 (A1 , A2 ) = −A1 , A2 0 .
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The expression n 0 (a1 , a2 , a3 ) is nonzero only when all three elements belong to F1 or one of the first two belongs to F2 and the other lies in F1 : n 0 (A1 , A2 , A3 ) = A2 A1 , A3 0 − A1 A2 , A3 0 , n 0 (A1 , v, ˜ A2 ) = n 0 (v, ˜ A1 , A2 ) = −vA ˜ 1 , A2 0 .
(35)
We believe that the quasiclassical Lian-Zuckerman homotopy BV algebra is actually a BV∞ -algebra from [14], motivated by the results of [13,14]. Below we prove an important part of this conjecture, which is needed in the paper. Theorem 3.2. The homotopy associative algebra with the operations Q, μ0 , n 0 is an A∞ -algebra where the higher homotopies vanish starting from the tetralinear one. In other words, Q, μ0 , n 0 generate the A(3) algebra. Proof. In order to prove this statement, it is enough to show that the relations: (−1)n a1 μ0 (a1 , n 0 (a2 , a3 , a4 )) + μ0 (n 0 (a1 , a2 , a3 ), a4 ) = n 0 (μ0 (a1 , a2 ), a3 , a4 ) − n 0 (a1 , μ0 (a2 , a3 ), a4 ) + n 0 (a1 , a2 , μ0 (a3 , a4 )). (36) hold. The last relation is trivial. Let us prove the first one in the most nontrivial case, when ai = Ai ∈ F1 : (−1)n A1 μ0 (A1 , n 0 (A2 , A3 , A4 )) + μ0 (n 0 (A1 , A2 , A3 ), A4 ) 1 1 = − A1 , A3 0 A2 , A4 0 + A1 , A2 0 A3 , A4 0 2 2 1 1 + A2 , A4 0 A1 , A3 0 − A1 , A4 0 A2 , A3 0 2 2 1 1 = A1 , A2 0 A3 , A4 0 − A1 , A4 0 A2 , A3 0 2 2 = n 0 (μ0 (a1 , a2 ), a3 , a4 ) − n 0 (a1 , μ0 (a2 , a3 ), a4 ) + n 0 (a1 , a2 , μ0 (a3 , a4 )). (37) We leave for the reader to establish these relations in all other situations.
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3.3. Remarks on quasiclassical limits and Vertex/Courant algebroids. In the end of this section we want to draw attention to an important notion, introduced in [16] called the ver tex algebr oid. It basically reflects the relations between the operator products of the elements of conformal dimension 0 and 1 and the operator L −1 in the vertex algebra (the authors of [16] do not require the presence of the Virasoro element). As one can see from the explicit construction of the operations, those relations are included in the relations of the LZ algebra (10). At the same time, for the LZ algebra one needs extra relations, involving the L 1 operator, which are absent in the definition of vertex algebroid (see e.g. [4,16,25]). We also indicate here that the quasiclassical limit of the LZ algebra is not equivalent to the usual quasiclassical limit, which truncates the vertex algebra into the Poisson algebra, and the vertex algebroid into the Courant algebroid (see e.g. [4]). In this limit the L 1 action vanishes, partly destroying the differential and almost all structures in the homotopy BV algebra, which we obtained, leaving only the subalgebra of the L ∞ algebra corresponding to the states of ghost number 0 and 1. This is, in fact, the usual L ∞ -algebra of the Courant algebroid [32]. In the same way the authors of [16] build a functor assigning to the vertex algebra the vertex algebroid and therefore the L ∞ -algebra, LZ construction yields a functor which assigns to the VOA the homotopy BV algebra. Considering VOA with a parameter, we see that we have a functor which maps each such VOA in the quasiclassical limit of the LZ algebra, which, as we have seen are highly truncated homotopy BV algebra (see Theorem 3.2). As a direct consequence of Theorem 3.2 we have a functor from the category of VOA with a parameter into the category of A∞ -algebras. One might wonder if this construction of the classical limit could be modified to be applied to other TVOA, e.g. the chiral de Rham complex [26] in order to obtain some nontrivial homotopy BV structure. However, the space of light modes there is just spanned by differential forms and the operation μ is given by the wedge product, while the analog of b0 acts as a derivation of μ. Therefore the bracket {·, ·} vanishes and the whole LZ algebra is just the differential graded algebra of differential forms. There is, in fact, a nontrivial modification of the LZ algebra in this case, which is worth mentioning here. It basically corresponds to the interchanging of the roles of b(z) and J (z) operators (see Definition 2.1), such that the bracket is constructed by means of Q, but not b0 . Reducing it to a certain subcomplex of b0 one can construct a BV algebra of polyvector fields on a manifold [25]. This might lead to the chiral version of the Barannikov-Kontsevich results [2]. 4. Deformation of Lian-Zuckerman Homotopy Commutative Algebra 4.1. Flat background deformation: the case of general TVOA. Let V be a TVOA. Let us consider the set of elements { f i }0 , where f i ∈ V 0 [1] are primary elements (i.e. they correspond to the highest weight vectors of the Virasoro algebra) of conformal dimension 0 and of fermionic degree 1, such that b0 f i = 0,
Q f i = 0, μ( f i , f j ) = 0 ∀i, j.
(38)
An immediate consequence is that { f i , f j }0 = 0. We will call the operator Rη =
i, j
ηi j μ( f i , { f j , ·}),
(39)
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where ηi j is some constant matrix, the flat background deformation of Q. First of all we show that our definition is consistent, i.e. the following proposition holds: Proposition 4.1. The operator R η obeys the properties: (R η )2 = 0, [Q, R η ] = 0.
(40)
Proof. The second relation is an immediate consequence of the fact that Q is a derivation of both μ(·, ·) and {·, ·}0 . Let us prove the first one. In order to do that let us write in detail (R η )2 a for some a ∈ V : (R η )2 a =
ηi j μ( f i , { f j , ηkl μ( f k , { fl , a})})
i, j,k,l
=
ηi j ηkl μ( f i μ( f k , { f j { fl , a}0 }0 )
i, j,k,l
=
ηi j ηkl (μ(μ( f i , f k ), { f j { fl , a}}) + (Qn + n Q)( f i , f k , { f j { fl , a}})).
i, j,k,l
(41) Since { f j , { fl , a}} = { fl , { f j , a}}, (R η )2 a = 0. Remark. Geometric/physical meaning of the flat metric deformation. It can been shown η that the operator R i j has the natural geometric meaning. Let us consider the operator (0) φ (z, z¯ ) = i, j η f i (z) f j (¯z ), where z¯ is the complex conjugated variable for z, and the operator-valued differential forms φ (2) = i, j ηi j [b−1 , f i (z)][b−1 , f j (¯z )], φ (1) = d z¯ i, j ηi j [b−1 , f i (z)] f j (¯z ) − dz i, j ηi j f i (z)[b−1 , f j (¯z )], such that the following descent equations are satisfied: Qφ (2) = dφ (1) ,
Qφ (1) = dφ (0) ,
Qφ (0) = 0.
(42)
Let us consider the operator R˜ η which acts on the elements of TVOA as follows: R˜ η a = P0
φ (1) a,
(43)
C ,0
where C ,0 is the contour around zero of the radius and P0 is the projection on the
0 -component. Counting the powers of , one can see that R˜ η = R η . Physically, it means the following. Suppose the TVOA V is described by the action (2) S0 . Let us(2)consider its perturbation by the operator φ , i.e. the perturbed action is Sφ(1)= is given by J B,φ = J B (z)dz + φ , S0 − φ . Therefore, the deformed B RST current i.e. the deformed charge has the form Q φ = J B,φ , but it must involve some regularization, as it usually happens in the quantum theory. The regularization is given by the projection on the -independent part. After examples are considered, it will be quite clear why we call the perturbation of the form φ (2) = i, j ηi j [b−1 , f i (z)][b−1 , f j (¯z )] by “flat background”.
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4.2. Deformation of the quasiclassical LZ algebra of light modes. Let V be a VOA and V ⊗ is the corresponding BRST complex. Then the following proposition holds. Proposition 4.2. The flat background deformation of the BRST operator is determined by Abelian vertex subalgebras of V generated by the primary elements {si } of conformal weight 1, such that si (z)s j (w) ∼ 0, ∀i, j, where si (z) =
n sn,i z
−n−1 ,
(44)
in other words, [sn,i , sm, j ] = 0 for any m, n.
Proof. Let us consider all possible primary fields of conformal weight 0 and of ghost number 1 in V ⊗ . They have the form cs(z) and ∂cu(z), where s(z), u(z) are primary fields of conformal dimensions 1 and 0 correspondingly. The condition that such field should be annihilated by b0 leaves us only the quantum fields of the form cs(z). Therefore, the flat background deformation of the BRST operator is determined by the sets of primary fields si of conformal weight 1. Finally, condition (44) can be obtained from the fact that μ( f i , f j ) = 0, where f i is the state corresponding to the quantum field csi (z). Therefore, one can introduce the elements f i from FL0 corresponding to si (z) of ghost number 1 of conformal weight 0. In the case with parameter, let us define the η operator Rh ≡ h −1 i, j ηi j μ( f i , { f j , ·}). Then we have a proposition. η
η 2
Proposition 4.3. i) The operator Rh commutes with the BRST operator and (Rh ) = 0. It acts on FL0 as follows:
/ V1 VE 15 D 5 5 55 555 5 Lˆ −1 5 555 555 1 Lˆ ˆL −1 552 1 5 55 555 55 1 ˆ 55 − 55 2 L 1 55 / V0 V0 V 0 − / V 0
(45)
where = h −1 i, j ηi j s0,i s0, j , Lˆ 1 · = −h −1 i, j ηi j si , s0, j ·, Lˆ −1 · = −h −1 i, j ηi j si s0, j ·. η
ii) On the complex F the operator Rh acts in such a way: η
Rh : F → F[h].
(46)
η η The quasiclassical version of Rh , i.e. the operator R0 = i, j ηi j μ0 ( f i , { f j , ·}0 ), where f i = csi and si ∈ V , acts on F invariantly and commutes with Q on F. A natural question is whether one can deform the quasiclassical LZ algebra in such η a way that all the relations will be satisfied with the differential Q η = Q + R0 . The answer is only partly positive. Namely, the following theorem holds.
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Theorem 4.1. i) There exist a flat metric deformation of the homotopy associative subalgebra of the quasiclassical LZ homotopy BV algebra, i.e. there exist η-deformed η η η multilinear maps μ0 , m 0 , n 0 on F, which together with Q η satisfy the relations of η η the homotopy associative algebra. Moreover, m 0 ≡ m 0 and n 0 ≡ n 0 . η η η ii) The homotopy associative algebra on F with operations Q , μ0 , n 0 is an A∞ -algebra, such that all multilinear operations vanish starting from the tetralinear one. Proof. For simplicity of calculations in this proof we assume the Einstein summation convention, i.e. when an index variable appears twice in a single term, once in an upper (superscript) and once in a lower (subscript) position, it implies that we are summing over all of its possible values. In the beginning, let us show how the bilinear operation is deformed. In order to do η that let us find an obstacle for R0 to be a derivation of μ0 : η
R0 μ0 (a1 , a2 ) = μ0 ( f i , μ0 ({ f j , a1 }0 , a2 ))ηi j + μ0 ( f i , μ0 (a1 , { f j , a2 }0 ))ηi j = μ0 (μ0 ( f i , { f j , a1 }0 ), a2 ) + μ0 (μ0 ( f i , a1 ), { f j , a2 }0 )ηi j −(Qn 0 + n 0 Q)( f i , { f j , a1 }0 , a2 )ηi j − (Qn 0 + n 0 Q)( f i , a1 , { f j , a2 }0 )ηi j = μ0 (μ0 ( f i , { f j , a1 }0 ), a2 ) + (−1)|a1 | μ0 (μ0 (a1 , f i ), { f j , a2 }0 ) −(Qn 0 + n 0 Q)( f i , { f j , a1 }0 , a2 )ηi j − (Qn 0 + n 0 Q)( f i , a1 , { f j , a2 }0 )ηi j +(Qr0 + r0 Q)( f i , a1 , { f j , a2 }0 )ηi j = μ0 (μ0 ( f i , { f j , a1 }0 ), a2 )ηi j + (−1)|a1 | μ0 (a1 , μ0 ( f i , [ f j , a2 ]))ηi j +(Qr0 + r0 Q)( f i , a1 , { f j , a2 }0 )ηi j η
η
η
η
= μ0 (R0 a1 , a2 ) + (−1)|a1 | μ0 (a1 , R0 a2 ) − (Qν0 + ν0 Q)(a1 , a2 ),
(47)
where r0 (a1 , a2 ) = m 0 ( f i , a1 ){ f j , a2 }0 ηi j and η
ν0 (a1 , a2 ) = n 0 ( f i , { f j , a1 }0 , a2 )ηi j − m 0 ( f i , a1 ){ f j , a2 }0 ηi j .
(48)
η
The explicit expression for ν0 is given in the following table: η
ν0 (a1 , a2 )= H H a1 a2 HHH u2 A2 v2 A˜ 2 v˜2 u˜ 2
u1
A1
v1
A˜ 1
v˜1
u˜ 1
0 0 0 0 0 0
ηi j si , A1 0 { f j , u 2 }0 ν η (A1 , A2 )
0 0 0 0 0 0
0 0 0 0 0 0
0 −ηi j si , A2 0 { f j , v˜1 }0 0 0 0 0
0 0 0 0 0 0
0 0 −ηi j si , A1 0 { f j , v˜2 }0 0
where η
ν0 (A1 , A2 ) = −si , A1 0 { f j , A2 }0 ηi j − si { f j , A1 }0 , A2 0 ηi j + { f j , A1 }0 si , A2 0 ηi j .
(49)
Quasiclassical LZ Homotopy Algebras, Courant Algebroids and Gauge Theory η
η
347
η
We see that [R0 , μ0 ] + [Q, ν η ] = 0 and if [R0 , ν0 ] = 0, then Q η is a derivation of η η the bilinear operation μ0 = μ0 + ν0 . One can show it explicitly. We check the most nontrivial situations, i.e. when a1 ∈ F1 , a2 ∈ F1 , and also cases when a2 ∈ F0 , a1 ∈ F1 and a1 ∈ F0 , a2 ∈ F1 . η η η So, let a1 = u ∈ F0 and a2 = A ∈ F1 . Then ν0 (u, A) = 0 and ν0 (u, R0 A) = 0. At the same time η
η
η
ν0 (R0 u, A) = ν0 (si { f j , u}0 , A)ηi j = (−sk si { fl , { f j , u}0 }0 , A0 + si { f k , { f j , u}0 }0 sl , A0 )ηi j ηkl = 0. (50) Now we check the case a2 = u ∈ F0 , a1 = A ∈ F1 . The bilinear operation between these two elements is nontrivial, moreover: η η
R0 ν0 (A, u) = ηi j ηkl sk { fl , si , A0 { f j , u}0 }0 . η
η
(51)
η
According to the definition of ν0 , ν0 (R0 A, u) = 0. At the same time η
(−1)|A| ν η (A, R0 u) = sk , A0 { fl , si { f j , u}0 }0 ηi j ηkl + si { f j , A}0 , sk { fl , u}0 0 ηi j ηkl .
(52)
η
Therefore, R0 is a derivation in this case also. Let us consider the last possibility, when a1 = A1 ∈ F1 and a2 = A2 ∈ F1 . 1 sk , { fl , si , A2 0 { f j , A1 }0 }0 0 ηi j ηi j ηkl 2 1 − sk , { fl , si , A1 0 { f j , A2 }0 }0 0 ηi j ηkl , 2 1 η η ν0 (R0 A1 , A2 ) = si , A2 0 { f j , sk , { fl , A1 }0 0 }0 ηi j ηkl , 2 1 η η −ν0 (A1 , R0 A2 ) = − si , A1 0 { f j , sk , { fl , A2 }0 }0 0 ηi j ηkl . 2 η η
R0 ν0 (A1 , A2 ) =
η η
η
(53)
η
Comparing the terms above, we find that R0 ν0 (A1 , A2 ) = ν0 (R0 A1 , A2 ) − η η η ν0 (A1 , R0 A2 ), i.e. R0 is a derivation. η The next step is to show that μ0 satisfies the homotopy commutativity relation. From η the table for ν0 we see that η
η
η η
η
η
η
η
ν0 (a1 , a2 ) − ν0 (a2 , a1 ) = R0 ν0 (a1 , a2 ) + ν0 (R0 a1 , a2 ) + (−1)n a1 ν0 (a1 , R0 a2 ). (54) η
η
Therefore, m 0 ≡ m 0 . The last statement does mean that μ0 satisfies the homotopy associativity relation. We will show this in the case when all the arguments belong to
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A. M. Zeitlin
F1 , leaving to check the other combinations to the reader. Let A1 , A2 , A3 ∈ F1 , then 1 (A3 , { f j , A1 }0 0 si , A2 0 ηi j 2 − f i , A1 0 { f j , A2 }0 , A3 0 ηi j − si , A3 0 { f j , A1 }0 , A2 0 ηi j , 1 η η ν0 (μ0 (A1 , A2 ), A3 ) = ν0 (− A1 , A2 0 , A3 ) 2 1 = ηi j f i , A3 0 { f j , A1 , A2 0 }0 , 2 1 η η ν0 (A1 , μ0 (A2 , A3 )) = ν0 (A1 , − A2 , A3 0 ) 2 1 ij = η { f j , A2 , A3 0 }0 si , A1 0 , 2 1 η μ0 (A1 , ν0 (A2 , A3 )) = (A1 , { f j , A2 }0 0 si , A3 0 ηi j 2 −si ,A2 0 { f j , A3 }0 , A1 0 ηi j −si , A1 0 { f j , A2 }0 , A3 0 ηi j ). (55) η
μ0 (ν0 (A1 , A2 ), A3 ) =
Therefore, η
η
μ0 (ν η (A1 , A2 ), A3 ) + ν0 (μ0 (A1 , A2 ), A3 ) − ν0 (A1 , μ0 (A2 , A3 )) −μ0 (A1 , ν η (A2 , A3 )) 1 1 = ηi j { f j , A3 , A1 0 }0 si , A2 0 − ηi j { f j , A2 , A3 0 }0 si , A1 0 . 2 2
(56)
On the other hand, η
η
η
η
R0 n 0 (A1 , A2 , A3 ) + n 0 (R0 A1 , A2 , A3 ) − n 0 (A1 , R0 A2 , A3 ) + n(A1 , A2 , R0 A3 ) 1 1 = ηi j { f j , A3 , A1 0 si , A2 0 }0 − ηi j { f j , A2 , A3 0 si , A1 0 }0 2 2 1 1 ij + A2 , A3 0 si , { f j , A1 }0 0 η − A1 , A3 0 si , { f j , A2 }0 0 ηi j 2 2 η = μ0 (ν η (A1 , A2 ), A3 ) + ν0 (μ0 (A1 , A2 ), A3 ) η
η
−ν0 (A1 , μ0 (A2 , A3 )) − μ0 (A1 , ν0 (A2 , A3 )).
(57)
η
Hence, μ0 is associative up to homotopy provided by n. Hence, we proved i). In order to prove ii) we just note that ν η doesn’t contribute to the higher associativity η η relation involving μ0 , n 0 . Then ii) follows from the proof of the similar statement for Q, μ0 , n 0 . Since we have an A∞ -algebra it is natural to consider the associated generalized Maurer-Cartan equation. However, due to the properties of the operations μ0 , n 0 , the resulting equation coincides with the linear one. In order to get around this, we multiply the A∞ -algebra with some noncommutative associative algebra S. The resulting object is also from the category of A∞ -algebras, but there will be no homotopy commutativity in general. Let us choose S = U (g), the universal enveloping algebra of some Lie algebra g. In this case it is possible (as we will see on several examples below) to find the relation with gauge theory.
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η
Definition 4.1. Consider the A∞ -algebra on F ⊗ U (g) generated by Q η , μ0 , n 0 . Consider the Maurer-Cartan element ∈ F1 ⊗ g. We will refer to the Maurer-Cartan equation η
η
Q η + μ0 (, ) + n 0 (, , ) = 0
(58)
as the Yang-Mills equation associated to the VOA with a formal parameter V and Lie algebra g. We will refer to the infinitesimal symmetries of the generalized Maurer-Cartan equations, which are η
η
→ + (Q η u + μ0 (, u) − μ0 (u, )),
(59)
where u ∈ F0 ⊗ g, as gauge symmetries. In the next section we will show that this equation and its gauge symmetries are actually equivalent to the system of Yang-Mills equations with matter fields and their gauge symmetries for certain vertex algebras. In the end of this section we write the explicit expression for the Yang-Mills equation associated with VOA V and the Lie algebra g, since it is needed in the following. The Maurer-Cartan element for F ⊗ U (g) has the form = A + v, where A, v ∈ F1 ⊗ g are the elements corresponding to the states of conformal weights 1, 0 correspondingly. The equation for v is as follows: v=
1 1 1 L 1 A + ηi j si , {s j , A}0 0 + A, A0 . 2 2 2
(60)
Then the equation for A is: 2 A + L −1 L 1 A + +
η sk {sl , L 1 A +
+A(L 1 A) + A +
ηi j L −1 si , {s j , A}0 0
i, j kl
k,l
ηi j si , {s j , A}0 0 }0
i, j
ηi j si , {s j , A}0 0 − (L 1 A)A
i, j
η si , {s j , A}0 0 A + 2[A, A] − Aad , Aad 0 A = 0, ij
(61)
i, j
where Aad ∈ F1 ⊗ End(g) stands for the element A ∈ F1 ⊗ g with the Lie algebra elements are considered in the adjoint representation. The gauge symmetries of this equation can be written as follows: A → A + (L −1 u + ηi j si {s j , u}0 + Au − u A), where u ∈ F0 ⊗ g.
(62)
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5. Beta-Gamma Systems, Courant Algebroid and Yang-Mills Theory 5.1. A toy model: Heisenberg VOA. Let us start with the simplest nontrivial vertex algebra with a formal parameter, i.e. we consider the vertex algebra V (a, g)[h, h −1 ], generated by quantum fields a i (z) (i = 1, . . . , D) with operator products a i (z)a j (w) ∼
hg i j , z−w
(63)
where g i j is a symmetric matrix. In the case of this VOA the elements of zero conformal weight are just constants and the fields of conformal dimension 1 have the form Ai a i (z), where Ai are constant. This example is very special because of the following two facts presented in Proposition 5.1: Proposition 5.1. i) The quasiclassical limit of the LZ algebra of light modes on V (a, g)[h, h −1 ] is isomorphic to the LZ algebra of light modes on V (a, g). ii) Any flat background deformation of BRST operator is trivial for V (a, g). Proof. To prove i) it is enough to note that the terms corresponding to higher powers in h usually correspond to the multiple “contractions” of the quantum fields. In the case of V (a, g) this does not happen, since all the elements from the light mode complex are at most linear in a i (z). Expressing a i (z) = n ani z −n−1 , we find that a0i annihilates each element in V (a, g) for every i. Therefore, ii) is also proven. The next proposition is about the Yang-Mills equation on V (a, g). Proposition 5.2. The Yang-Mills equation for V (a, g) with Lie-algebra g is equivalent to the system of equations g i j [Ai , [A j , Ak ]] = 0 (64) i, j
for certain Ai ∈ g (i = 1, . . . , D) One can see that these equations coincide with Yang-Mills equations with a flat metric g i j and constant gauge fields Ai . However, this example was a toy model for us: it was too degenerate. In order to have less trivial example, one should “enrich” the space of fields of conformal weight zero. 5.2. BV extension of the Courant algebroid. Let us consider a family of β-γ systems by quantum pi (z), X i (z), (i = 1, . . . , D), where pi (z) = generated fields −n−1 i i −n−1 , X (z) = n X n z are quantum fields of conformal dimensions 1 n pi,n z and 0 correspondingly, and the operator products are X i (z) p j (w) ∼
hδi, j , z−w
X i (z)X j (w) ∼ 0,
pi (z) p j (w) ∼ 0
(65)
and the Virasoro element is given by the formula L(z) = −
1 pi ∂ X i . h i
(66)
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D F The space of the VOA is given by F p,X [h −1 , h], F p,X ≡ ⊗i=1 pi ,X i (see Sec. 3.1). For definiteness, from now on let us assume that the space of zero conformal weight of F p,X is given by the formal powers series in X 0i . The operators from the VOA F p,X of conformal dimensions 0 and 1 have the form u(z) = u(X )(z), A(z) = : Ai (X )(z) pi (z) :,
B(z) =
i
(67)
B j (X )(z)∂ X (z), j
j
where u(X ), Ai (X ), B j (X ) are considered as power series in X i . Let M be the formal scheme Spf(C[[X 01 , . . . , X 0D ]]). Then the states u, A, B can be identified with sections of O M , T M, T ∗ M correspondingly. Now let us consider the semi-infinite complex associated with F p,X . The BRST operator acts as follows:
OM
where divA =
TM L TMK K L LLLL21 div KKKK− 21 div KKK LLL KK% LL& id / O OM O M L M KKK LLL KKK LLL KK L d LLL d KKK % &
i ∂i A
T ∗M
OM o
T ∗M
i . The action of the BV operator b 0
TM o id
−id
−id
is given by the diagram below: (69)
TM OM o
OM T ∗M o
(68)
−id
OM
T ∗M
It is useful to write down the explicit values for the operation μ in the LZ algebra on the complex F: μ0 (a1 , a2 )= H H a1 X˜1 u1 X1 v1 v˜1 u˜ 1 a2 HHH X˜1 u 2 u2 u1u2 X1 u 2 v1 u 2 v˜1 u 2 u˜ 1 u 2 −L X1 u 2 X2 u 1 X2 (X1 , X2 ) D + −v1 X2 − 21 X˜1 , X2 L X2 v˜1 0 1 X , X 1 2 2 v2 u 1 u˜ 2 v2 X1 0 0 −v˜1 v2 0 X˜2 u 1 X˜2 − 21 X1 , X˜2 0 0 0 0 v˜2 u 1 u˜ 2 L X1 v˜2 −v1 v˜2 0 0 0 u˜ 2 u 1 u˜ 2 0 0 0 0 0
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˜ i , B˜ i ), such that Ai , A ˜ i ∈ T M and Bi , B˜ i ∈ where Xi , X˜i stand for pairs (Ai , Bi ), (A T ∗ M. The expression (X1 , X2 ) D stands for the Dorfman bracket (see e.g. [23,31]) defined as ((A1 , B1 ), (A2 , B2 )) D = (L A1 A2 , L A1 B2 − i A2 dB1 ).
(70)
The pairing ·, · between twoμ elements (A1 , μB1 ) and (A2 , B2 ) is defined as follows: (A1 , B1 ), (A2 , B2 ) = μ (A1 B2,μ + B1,μ A2 ), i.e. it is a natural pairing between the elements of T M and T ∗ M. The operation L X u for X = (A, B) denotes the Lie derivative with respect to the vector field A. This suggests that there is a certain relation between the Courant algebroid [23] on T M ⊕ T ∗ M and the quasiclassical LZ algebra for β-γ systems. Namely the following proposition holds: Proposition 5.3. The homotopy BV algebra generated by μ0 and {, }0 on F contains the Courant algebroid structure on T M ⊕ T ∗ M, i.e. {, }0 being restricted to T M ⊕ T ∗ M coincides with the Dorfman bracket, μ0 : F0 ⊗ F1 → F1 is a multiplication of function on an element of T M ⊕ T ∗ M, and the pairing between two elements of F1 is given by the operator product coefficient ·, ·. In fact, this relation between such “short” homotopy BV algebras and Courant brackets can be extended to the large class of Courant algebroids (we will discuss this and related questions in [47]). In the following we will call this homotopy BV algebra the BV double of Courant algebroid. 5.3. Deformation of the homotopy commutative algebra and Yang-Mills equations. In the previous subsection we showed that the quasiclassical Lian-Zuckerman algebra associated with a family of β-γ systems gives a homotopy BV algebra extending the Courant algebroid. In this section we consider the flat background deformation of this algebra according to considerations of the previous section. First of all we will pick the Abelian subalgebra of the operators of conformal dimension 1 in the beta-gamma VOA. We will consider the VOA subalgebra generated by pi (z), (i = 1, . . . , D). Moreover, in this subsection we assume that the deformation matrix ηi j is symmetric. Then the deformation of the BRST operator is given in the following proposition. η Proposition 5.4. Let f i (z) = cpi (z). Then R0 = ı, j ηi j μ0 ( f i , { f j , ·}0 ) acts on F as follows:
OM
/ TM TM 9 r8 s r s r s dˆ ss dˆ rr s r s r s r sss − rrr
/ OM / OM 8 OM 1 r s9 1
r s
r s − d iv d iv 2 2 rr ss rrr sss r s r s r s / T ∗M T ∗M
(71)
ij ij ∗ ˆ j
where = i, j η ∂i ∂ j , divB = i, j η ∂i B j (here B ∈ T M) and (du) = ij i η ∂i u (here u ∈ O M ).
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Let us also assume that the matrix η{i j} is invertible, such that η{i j} is the inverse matrix. This yields the following proposition. Proposition 5.5. The complex (F, Q η ) is isomorphic to the following complex (G, Q η ) which decomposes into a direct sum of three subcomplexes: 0
/ 1 (M)
∗d∗d
/ 1 (M)
0
/ 1 (M)
/ 1 (M)
/0
0
/ 0 (M)
id
/ 0 (M)
/0
/ 0 (M)
d
∗d∗
/ 0 (M)
/ 0 (72)
where 0 (M) ≡ O M and 1 (M) ≡ T ∗ M and the Hodge star operator ∗ is constructed via the metric corresponding to the invertible and symmetric matrix η{i j} . Proof. The embedding can be constructed in the following way. Let us denote the three subcomplexes above as (Gi· , Q η ) (i = 1, 2, 3). We construct the following emebeddings: id
id
f1
g1
f2
g2
f3
g3
G01 − → F0 , G31 − → F3 , → F1 , G11 − → F1 , G11 −
(73)
→ F1 , G12 − → F1 , G12 − → F1 , G13 − → F1 , G13 − such that f i , gi are explicitly given by: ˜ = B˜ + B˜ ∗ , ˆ f 1 (B) = B + B∗ − divB, g1 (B) ˜ = B˜ − B˜ ∗ , f 2 (B) = B − B∗ , g2 (B)
(74)
f 3 ≡ id, g3 (v) ˜ = v˜ − (d v˜ + dˆ v), ˜ where B∗ ∈ T M such that B ∗ i = ηi j B j . Combining f i , gi and other maps into the map of complexes, one can see that this is an isomorphism. It can be observed that the cohomology of the complex (F, Q η ) in degree 1 is equivalent to the solutions of Maxwell equations and scalar field equations ∗ d ∗ dA = 0, = 0,
(75)
modulo gauge transformations A → A + du. Now we recall that we have the structure of the homotopy associative and homotopy commutative algebra on (F, Q η ), which is in fact the A∞ -algebra. Let us consider the tensor product of (F, Q η ) with U (g), where g is some Lie algebra and find out what the Yang-Mills equations are for the β-γ VOA. The answer is given in the proposition below.
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Proposition 5.6. The Yang-Mills equation for the VOA F p,X and the Lie algebra g is equivalent to the following system of equations ηi j [∇i , [∇ j , ∇k ]] = ηi j [[∇k , i ], j ], i, j
i, j
η [∇i , [∇ j , k ]] = ij
i, j
ηi j [i , [ j , k ]],
(76)
i, j
where i = Bi − j A j ηi j , Ai = Bi + j A j ηi j and ∇i = ∂i + Ai . Here Bi are the components of B ∈ T ∗ M ⊗ g and Ai are the components of A ∈ T M ⊗ g. The gauge symmetries correspond to the following transformation of fields: Ai → Ai + (∂i u + [Ai , u]), i → i + [i , u].
(77)
Proof. To prove this statement we just need to substitute the generic Maurer-Cartan element, which is the sum of (A, B, v) ∈ (T M ⊕ T ∗ M ⊕ O M ) ⊗ g. Substituting it in (61), we obtain Eqs. (76). The equation for v is as follows: 1 1 v=− ( ∂i Ai + ηi j ∂i B j ) − (Ai B i + B i Ai ). 2 2 i
ij
(78)
i
Substituting appropriate elements in (62), we obtain gauge symmetries (77). This statement is very close to the particular results obtained in relation to the “original” logarithmic open string vertex algebra [39–42]. Equations (76) are the Yang-Mills equations in the presence of D scalar fields i (see also [46]). 5.4. Smooth manifold case. All the considerations we had above were applied only to the case of the flat metric and a standard volume form on D-dimensional (pseudo-) Euclidean space. Here we will give some statements about the case of a general smooth manifold M. First of all we generalize the BV double of the Courant algebroid to the case of a D-dimensional smooth manifold M with a volume form , such that in local coordinates = eφ(X ) d X 1 ∧ . . . . ∧ d X D . We have the following proposition. Proposition 5.7. Let us consider the quasiclassical LZ algebra for the VOA F p,X , such that the Virasoro element is given by L φ (z) = −
1 pi ∂ X i + ∂ 2 φ(X ). h
(79)
i
Then there exist a BV algebra defined on the sections of certain bundles of the manifold M, such that in local coordinates it is given by this quasiclassical LZ algebra. Proof. The shift in ∂ 2 φ(X ) of the Virasoro element changes the action of the differential in (68) in such a way that the div operation is changed by divA = i (∂i Ai +∂i φ Ai ). This is a local coordinate expression for the operator invariant under the coordinate change. As one can see, the other operations are already written in the covariant form, therefore, we have a BV algebra defined globally on the sections of appropriate bundles on M.
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The next question is whether the deformed homotopy commutative algebra can be generalized to the case of the manifold M with the metric g = gi j d X i d X j . The answer is positive and the following proposition holds: Proposition 5.8. There exists an A∞ -algebra on the complex (72) for the smooth Riemannian manifold (M, g), such that in the case of M = R D and g i j = ηi j it coincides with the A∞ -algebra, generated by Q η , μη , n η . Proof. Similar A∞ -algebra for the general manifold M on the sum of upper and lower complexes from (72) was considered in [42]. In the same way one can construct it on the full complex F. We leave the technical details to the reader. For a compact manifold M, one can write the action for the Yang-Mills theory interacting with 1-forms, corresponding to this A∞ -algebra: 1 1 1 (− F ∧ ∗F + dA ∧ ∗dA + dA ∗ ∧ ∗dA ∗ S= 4 4 4 M 1 1 + F ∧ ∗( ∧ ) − ( ∧ ) ∧ ∗( ∧ )), (80) 2 4 where F = dA + A ∧ A is a curvature for A and dA = d + A is a covariant derivative. It is not clear, however, how to derive the Yang-Mills equations on the smooth manifold M with some metric gi j from a the point of view of VOA, like we did before in the flat case (see some suggestions in the last section). However, the following statement is still true: Proposition 5.9. Let us consider the A∞ -algebra on the manifold M with the metric g from Proposition 5.8. Let us introduce a formal parameter into g, such that g i j → g i j (t) = tg i j . Then taking the limit t → 0 we recover √ the A∞ -subalgebra of the BV Courant algebroid on M with a volume form = g(X )d X 1 ∧ . . . ∧ d X N , where g(X ) = det (g{i j} (X )). Proof. In the flat √ case this result is obvious. For nonconstant g i j we need to watch that terms containing g(t) will not blow up in the t → 0 limit. This never happens, since they always enter the expressions in the form ∂i log(g(t)) ≡ ∂i log(g). 6. Some Remarks, Open Problems and Conjectures 6.1. Physical interpretation: beta-functions and (deformed) LZ algebras. In this paper we have shown that the correspondence between VOA and A∞ -algebras we constructed using the quasiclassical limit of the Lian-Zuckerman homotopy algebra of light modes allows us to write down an analogue of the Yang-Mills equation for the general VOA with a formal parameter. It is known that Yang-Mills equations show up as a 1-loop beta-function for the open string theory. Therefore, one can give the pure algebraic meaning to the 1-loop beta function in the generic case. In such a way the Maurer-Cartan element is associated with a perturbation term. One can think of that as follows. Let the VOA correspond to some CFT with the action S0 on the half-plane H + . Then the background term may be interpreted as a perturbation of an action of the form flat 1 ij z ) as we already have seen. The perturbation corresponding to + i, j η si (z)s j (¯ H h the Maurer-Cartan element has the form ∂ H + b−1 . Therefore, the complete action
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of the theory (sigma-model) for which we write the conformal invariance condition has the form (for the β-γ example such action was considered in [46]): ⎛ ⎞ 1 ij ⎝ ⎠ dz ∧ d z¯ η si (z)s j (¯z ) + dz(b−1 ). (81) S = S0 + H+ h ∂ H+ i, j
In the case of non-Abelian Lie algebra (when ∈ F ⊗ g) one cannot just add the last term to the action. One has to insert a trace of the path ordered exponential of b−1 in the partition function. One of the open mathematical questions, which is motivated by this paper, is to prove partly the Lian-Zuckerman conjecture, i.e. reconstruct completely the LZ algebra of light modes, and to compare the corresponding Maurer-Cartan equations with the known expressions for the beta-functions. In particular, we partly know the explicit form of the Maurer-Cartan equations in certain nontrivial situations, i.e. in the standard open string case, the low derivative terms should correspond to the equation in the Born-Infeld theory. One of the ways to construct this homotopy algebra is to treat it as a deformation of the quasiclassical one. However, there can be several such deformations. If so, it is also interesting to know what it means from the physical point of view, i.e. from the point of view of perturbation theory. 6.2. B-field and the deformations of the BV homotopy algebra. In the example, studied in Subsect. 5.2, corresponding to the vertex algebra, generated by β-γ systems for simplicity we considered the deformation via the symmetric matrix ηi j . In the case of general η, the antisymmetric part is related to the so-called Kalb-Ramond field which is a necessary ingredient in string theory. It is interesting to write the Yang-Mills equations and the corresponding action in this instance. Another interesting structure arises when the matrix ηi j becomes purely antisymmetric. We state here a proposition to which we will return in the subsequent publications. Proposition 6.1. Let ηi j be antisymmetric. Then there exists a homotopy BV algebra η η η on F such that it is generated by Q η , μ0 , b˜0 = h −1 b0 , {·, ·}0 , such that {a1 , a2 }0 = η η η b˜0 μ0 (a1 , a2 ) − μ0 (b˜0 a1 , a2 ) − (−1)n a1 μ0 (a1 , b˜0 a2 ). We note that if ηi j is not antisymmetric, such a BV algebra does not exist, and thei j obstacle corresponds to the generalization of the Laplace operator · = i, j η {si , {s j , ·}. As the simplest example, let us consider the β-γ system. In this case the deformation of the corresponding homotopy BV algebra via antisymmetric tensor (related to the Abelian vertex subalgebra generated by p{i} ) leads to a certain deformation of the BV double of the Courant algebroid via the antisymmetric bivector field. One can check that the corresponding deformation of the Courant algebroid coincides with the one considered in [31] (see also [17] for the physical insight). It appears that a necessary condition on the bivector field η for this deformation to hold on the general smooth manifold M was that [η, η] S = 0, where [·, ·] is a Schouten bracket. It is known that the Courant algebroid admits also another deformation (or a twist), via the 3-form (see e.g. [31]) which was introduced by Severa. It is interesting how to incorporate this deformation in the VOA formalism we studied in this paper. One of the ways to do that is to consider instead of the η-deformation based on the Abelian vertex algebra generated by pi ,
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another deformation based on the abelian vertex algebra generated by vertex operators of the type ωk (X )∂ X k . 6.3. Yang-Mills equations for β-γ systems and T-duality. Looking at the Yang-Mills equations in the β-γ example, a physicist may be interested in what is the meaning of the “matter” 1-form fields in the equations. The answer is as follows. If you consider the conformal field theory, corresponding to open strings in dimension 2D on the torus, it appears to be a logarithmic (see e.g. [28]) VOA, and the Lian-Zuckerman construction does not work there. In order to get rid of logarithms we considered another VOA on this space, corresponding to β-γ systems, with the deformed BRST operator. However, even with this deformation we didn’t recover the original open string theory, but the one, where half of dimensions is T -duali zed. We expect also the relation of our considerations to the so-called “Double Field Theory” introduced by Hull and Zwiebach [19], where the similar structures, like the Courant algebroid appear in the context of T-duality in String Field Theory.
6.4. Nontrivial metric and a B-field. Einstein equations. Another question one can ask, being motivated by the example with the β-γ system is about the meaning of the YangMills A∞ -algebra on a manifold with nontrivial metric and a B-field from the VOA point of view. So far, we have the construction only with a flat metric. We propose the following solution to this problem. We conjecture that there should be an L ∞ -algebra action on the LZ homotopy commutative algebra, in such a way that the flat background deformation can be treated as a deformation related to the Abelian L ∞ -subalgebra. From the form of the η-deformation, one may suggest that this L ∞ -algebra comes from a tensor product of two LZ algebras. There is an indication of the existence of such L ∞ -algebra in [24,43–45]. The Maurer-Cartan equation for this homotopy Lie algebra should be equivalent to the Einstein equations with external fields in the case of β-γ VOA. We will address this question in [48]. Acknowledgements. I am indebted to I.B. Frenkel, M.M. Kapranov and M. Movshev for valuable discussions. I am very grateful to J. Stasheff for reading the draft version of this article and his comments. I would like to thank the referees for their valuable comments and remarks. I am grateful to the organizers of Simons Workshop’09, where this work was partly done.
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Commun. Math. Phys. 303, 361–383 (2011) Digital Object Identifier (DOI) 10.1007/s00220-010-1170-0
Communications in
Mathematical Physics
Global Well-Posedness for the 2D Micro-Macro Models in the Bounded Domain Yongzhong Sun1 , Zhifei Zhang2 1 Department of Mathematics, Nanjing University, Nanjing 210093, P. R. China.
E-mail:
[email protected]
2 School of Mathematical Sciences, Peking University, Beijing 100871, P. R. China.
E-mail:
[email protected] Received: 28 April 2010 / Accepted: 31 July 2010 Published online: 20 November 2010 – © Springer-Verlag 2010
Abstract: In this paper, we establish new a priori estimates for the coupled 2D Navier-Stokes equations and Fokker-Planck equation. As its applications, we prove the global existence of smooth solutions for the coupled 2D micro-macro models for polymeric fluids in the bounded domain. 1. Introduction In this paper, we study the coupled micro-macro models for polymeric fluids. These models play an important role in applied physics, chemistry and biology. At the level of polymeric fluid, we get a system coupling the fluid and the polymers. The motion of the fluid is described by the Navier-Stokes equations with an elastic stress which reflects the microscopic contribution of the polymer molecules to the overall macroscopic flow fields. The evolution of polymer density is described by the Fokker-Planck equation with a drift term depending on the spatial gradient of the velocity. Mathematically, the system reads ⎧ ⎪ ⎨ u t + u · ∇u + ∇ p = u + ∇ · τ in (0, T ) × , ∇ · u = 0 in (0, T ) × , (1.1) ⎪ ⎩ f + u · ∇ f = f − ∇ · ∇uq f − ∇ U f in (0, T ) × × D, t q q q where the polymer stress τ is given by ∇q U ⊗ q f dq, τ= D
U (|q|2 )
is the potential. We may refer to [2,10] for more physical backand U (q) = grounds. The known mathematical results for micro-macro models of polymeric fluids are usually limited to the small-time existence and uniqueness of strong solutions of the
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corresponding PDE systems. We refer to [14,15,17,18] for the Oldroyd B model. We refer to [12,19,20,34] for the FENE Dumbbell model. Concerning the general coupled PDE systems, some preliminary studies were made in the earlier work [33]. In recent works [24,25], Lin, Liu and Zhang studied the global existence of smooth solutions to micro-macro models of polymeric fluids near equilibrium. We also refer to [5,22,26,27] for more recent works in this direction. Lions and Masmoudi [29,30] also proved the global existence of weak solutions for some micro-macro models, see [1] for the models with a technical smoothing. Recently, in the case of = R2 or T2 , the authors [7] proved the global existence of smooth solutions to a coupled nonlinear Fokker-Planck and Navier-Stokes system when the convection velocity u in the Fokker-Planck equation is replaced by a sort of time averaged one. Later this assumption was removed by Constantin and Masmoudi [8]. At the same time, Lin, Zhang and Zhang [28] independently proved the global regularity for the 2D co-rotational FENE model, see [32] for the dumbbell model. The purpose of this paper is to generalize these global results in 2D to the case of bounded domain. Let us recall two key technical points in the proof of [8,28]. The first point is a priori estimate of 2D Navier-Stokes equations given by Chemin and Masmoudi [4]. More precisely, let us consider the 2D Navier-Stokes equations with an external force u t + u · ∇u − u = −∇ p + f, where the external force term f ∈ L 2T H −1 ∩ L 1T C −1 . Then it is proved that the solution u is bounded in L 1T C 1 , which is defined by the Littlewood-Paley localization operators. Note that this space is slightly weaker than the space L 1T (Li p). Thus, the priori bound of u is not enough to propagate the regularity of the density distribution f in the x direction. Thus, we need to introduce the second key technical device, i.e, so called estimates on losing derivatives in [3], which enables us to show that u is actually bounded in L 1T (Li p). It should be pointed out that the Littlewood-Paley theory, Bony’s decomposition and the explicit form of the Stokes semigroup e−t A play the key roles in order to use the above two technical devices. It seems difficult to adapt them to the case of bounded domain. Very recently, Constantin and Seregin [9] gave a new proof for the 2D Smoluchowski model. However, their proof still works for the case of = R2 or T2 . To introduce our ideas, let us look at a simplified model of (1.1):
u t − u + ∇ p = ∇ · τ, ∇ · u = 0 in (0, T ) × , (1.2) τt + u · ∇τ = ∇uτ in (0, T ) × . Here is a bounded smooth domain and we impose the Dirichlet boundary condition for the velocity u. Let us assume that τ ∈ L ∞ ((0, T ) × ) for the moment, which can be obtained if ∇u in the second equation of (1.2) is replaced by its asymmetric part. Let def
v = A−1 ∇ · τ be a solution of Stokes system
−v(t) + ∇π(t) = ∇ · τ (t), ∇ · v(t) = 0 in , v(t, x) = 0 on ∂. def
And we denote w = u − v. Then the new unknown w satisfies the following equations ⎧ −1 ⎪ ⎨ ∂t w − w + ∇ p = A (u · ∇τ − ∇uτ ) in (0, T ) × , ∇ · w = 0 in (0, T ) × , ⎪ ⎩ w(t, x) = 0 on (0, T ) × ∂.
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Then by using the classical results concerning the stationary Stokes system, the energy estimates enable us to obtain the uniform bound of w in L 2T H 2 () . Roughly speaking, this bound implies u ∈ L 2T H 2 + L 2T C 1 . Obviously, the obtained estimate is better than that of Chemin and Masmoudi [4]. Here we use the structure of the system. Unfortunately, this estimate is still not enough to propagate the regularity of τ . Fortunately, because of better estimate of u, we don’t need to use the second device: losing derivative estimates to prove u ∈ L 1T (Li p). Instead, we just use the following logarithmic Sobolev inequality: ∇ A−1 (∇ · F) L ∞ () ≤ C(1 + F L ∞ () ) ln(e + ∇ F L q () ). Here A is the Stokes operator. The other parts of this paper are organized as follows. In Sect. 2, we introduce some classical results about the Stokes system. Section 3 is devoted to the proof of new a priori estimates for the coupled 2D Navier-Stokes equations and Fokker-Planck equation. In Sect. 4, we apply this estimate to prove the global existence of smooth solutions for some polymer fluids in the bounded domain. Finally, we present some basic energy estimates in an appendix. Let us conclude this section by introducing some notations. We denote by · L p () the norm of L p (), and W k,q () the Sobolev space whose norm is defined by def ∂ α u L q () . uW k,q () = |α|≤k
k,q
And we denote H k () = W k,2 (), W0 () the closure of smooth functions with compact support in the norm of W k,q (). 2. Preliminaries In this section, we introduce some preliminary results for the Stokes system. Assume that is a bounded smooth domain in R N . Let us first consider the stationary Stokes system
−U + ∇ P = f, ∇ · U = 0 in , (2.1) U (x) = 0 on ∂. Proposition 2.1. Let q ∈ (1, ∞). If f ∈ L q (), then there exists a unique solution 1,q U ∈ W0 ∩ W 2,q of the Stokes equations (2.1) such that U W 2,q () + ∇ P L q () ≤ C f L q () . If f = ∇ · F with F ∈
L q (),
(2.2)
then
U W 1,q () ≤ CF L q () .
(2.3)
1,q
Finally, if f = ∇ · F with Fi j = ∂k Hikj and Hikj ∈ W0 () for i, j, k = 1, . . . , N , then U L q () ≤ CH L q () . Here the constant C depends on q, .
(2.4)
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Proof. The inequalities (2.2) and (2.3) are classical and can be found in [13,21]. Before the proof of (2.4), some explanations are needed. In the case of = R N , the inequality (2.4) is followed by representing U = P(R j Rk Hikj ), where R = (R1 , . . . , R N ) is the Riesz transform and P = I d + R R· is the projection on L q (R N ) N into the divergence free vector fields. In the case of bounded domain, the inequality (2.4) is in general not true if there is no vanishing boundary condition imposed on H . We give a proof of (2.4) by a duality argument. Let g ∈ L p () N with 1p + q1 = 1 and let V solve the following Stokes system
−V + ∇ = g, ∇ · V = 0 in , V (x) = 0 on ∂. In the following calculations, integrating by part is verified by the vanishing boundary condition of U, V and G. On the one hand, (−U + ∇ P) · V d x = U · (−V )d x = U · (−V + ∇)d x = U · gd x.
On the other hand, (−U + ∇ P) · V d x = (∇ · ∇ · H ) · V d x =− ∇ · H : ∇V d x = H : ∇ ⊗ ∇V d x.
Hence, U · gd x = H : ∇ ⊗ ∇V d x ≤ H L q () ∇ 2 V L p () .
We conclude the proof of (2.4) by using (2.2).
Proposition 2.2. Let f = ∇ · F with F ∈ W 1,q () for q ∈ (N , ∞). Then the solution U of (2.1) satisfies (2.5) ∇U L ∞ () ≤ C 1 + F L ∞ () ln e + ∇ F L q () . Proof. We use the Green matrix G(x, y) = (G i j (x, y)) N ×N to represent U by U (x) = G(x, z)∇ · F(z)dz = − ∇z G(x, z)F(z)dz.
The following estimates for G can be found in [21]: |∇G(x, z)| ≤
C C , |∇ 2 G(x, z)| ≤ , |x − z| N −1 |x − z| N
with C depending only on N and .
(2.6)
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Denote d = diam(). Note that U (x) = G(x, z)∇z · F(z)dz = G(x, z)∇z · (F(z) − F(x))dz = ∇z G(x, z)(F(x) − F(z))dz.
Let δ ∈ (0, d) be a real number to be determined later and write ∇x U (x) = ∇x ∇z G(x, z)(F(x) − F(z))dz = + ∇x ∇z G(x, z)(F(x) − F(z))dz. {z∈:|x−z|≥δ}
{z∈:|x−z|<δ}
Using (2.6), we find that 1 |F(z)− F(x)| dz +C dz N |x −z| N {z∈:|x−z|≥δ} |x −z| {z∈:|x−z|<δ} d δ dr dr + C[F]α, ≤ CF L ∞ () 1−α r r δ 0 d d = CF L ∞ () ln + C[F]α, δ α ≤ CF L ∞ () ln + C∇ F L q () δ α . δ δ
|∇x U (x)| ≤ CF L ∞ ()
Here α = 1 − Nq > 0 and the Hölder seminorm [F]α, = supx = y∈ |F(x)−F(y)| |x−y|α and we used the fact [F]α, ≤ C∇ F L q () by Sobolev embedding. By choosing 1 δ = d(e + ∇ F L q () )− α , we conclude the proof of (2.5).
Next we consider the non-stationary Stokes system
∂t U − U + ∇ P = f, ∇ · U = 0 in (0, T ) × , U (t, x) = 0 on (0, T ) × ∂, U (0, x) = U0 (x) in .
(2.7)
We have the following classical result from [16]. 1− 1p , p
Proposition 2.3. Let p, q ∈ (1, ∞). Assume that U0 ∈ Dq Then there holds
, f ∈ L p (0, T ; L q ()).
∂t U L p (L q ()) + ∇ 2 U L p (L q ()) + ∇ P L p (L q ()) T T T ≤ C f L p (L q ()) + U0 T
If f = ∇ · F, then we have
1− 1p , p
;
(2.8)
Dq
∇U L p (L q ()) ≤ C F L p (L q ()) + U0 T
Here the constant C depends on p, q, .
T
1 − 1 ,p p
Dq2
.
(2.9)
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Remark 2.4. The space Dqα,s is defined as follows Dqα,s =
def
v ∈ L qσ () : v Dqα,s = v L q ()
+∞
+
t
1−α
Ae
−t A
0 q
dt vsL q ()
1 s
t
< +∞ ,
def
where A is the Stokes operator and L σ () = {v ∈ L q () N : div v = 0 in , v · n = 0 on ∂}. The following fact will be used: if 0 < α < k2 − N2 ( 1p − q1 ) for k = 1, 2, then v Dqα,s ≤ CvW k, p () for 1 < p ≤ q < ∞. We give a proof for k = 1. Recall the decay estimates for the Stokes semigroup in the bounded domain: As e−t A v L q () ≤ Ct
−s− N2
1 1 p−q
v L p () ,
for any 1 < p ≤ q < ∞ and s ∈ [0, 1], see [16]. Thus we have 1 1 dt −1+ 21 −α− N2 1p − q1 s 1−α −t A s ≤ C∇v L p () t t Ae v L q () dt ≤ CvW 1, p () t 0 0 for α < 21 − N 1p − q1 and for α > 0, +∞ +∞ dt −1− α+ N2 1p − q1 s 1−α −t A s s ≤ Cv L p () t Ae v L q () t dt ≤ CvsL p () . t 1 1 3. New a Priori Estimates for Coupled 2D Navier-Stokes Equations and Fokker-Planck Equation In this section, we consider the coupled 2D Navier-Stokes equations and Fokker-Planck equation in a general form: ⎧ in (0, T ) × , ⎨ ∂t u + u · ∇u − u + ∇ p = ∇ · τ, ∇ · u = 0, ∂t τ + u · ∇τ = Q(∇u, g), in (0, T ) × , (3.1) ⎩ u(t, x) = 0 on (0, T ) × ∂, u(0, x) = u 0 (x), τ (0, x) = τ0 (x), x ∈ . Here u = (u 1 , u 2 ) is the velocity field, p is the scalar pressure and the stress tensor τ = (τi j ) is a 2 × 2 symmetric matrix. In general, the matrix Q(∇u, g) is a quadratic form in (∇u, g) with L ∞ coefficients. More precisely, Q(∇u, g) = B(∇u, g) + A1 (∇u) + A2 (g), where B(∇u, g) is a bilinear form in (∇u, g), A1 and A2 are linear forms of ∇u and g respectively. Thus, |B(∇u, g)| ≤ C(|∇u|2 + |g|2 ), |A1 (∇u)| ≤ C|∇u|, |A2 (g)| ≤ C|g|.
(3.2)
The goal of this section is to establish new a priori estimates for the system (3.1) which will play an important role in the proof of global well-posedness for some 2D micro-macro models in the bounded domain.
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Theorem 3.1. Let be a bounded smooth domain in R2 . Let u 0 ∈ H01 () ∩ H 2 () 4q with div u 0 = 0, τ0 ∈ H 1 () and g ∈ L 4 (0, T ; L p ()) for p = 4−q and q ∈ [2, 4). Assume that (u, τ ) is a strong solution of (3.1) in [0, T ). Then there holds (3.3) w L 2 (W 2,q ()) ≤ C T, u 0 H 2 () , τ0 H 1 () , (τ, g) L 4 (L p ()) , T
T
def
where w = u − A−1 ∇ · τ . Remark 3.2. In the case of = R2 or T2 , we can also obtain similar estimates for w. In fact, the proof will be easier. We leave it to the interested readers. def
Proof. We denote v = A−1 ∇ · τ . Then by the definition of A, v satisfies
−v(t) + ∇π(t) = ∇ · τ (t), ∇ · v(t) = 0 in , v(t, x) = 0 on ∂, for each t ∈ [0, T ). Thus, the first equation of (3.1) can be rewritten as ∂t u + u · ∇u − w + ∇ p = 0, for some pressure p. On the other hand, taking A−1 ∇· for the equation of τ to obtain ∂t v + A−1 ∇ · (u · ∇τ ) = A−1 ∇ · (Q(∇u, g)). Thus, the new unknown w satisfies the following initial-boundary value problem: ⎧ ⎨ ∂t w − w + ∇ p = F in (0, T ) × , ∇ · w = 0 in (0, T ) × , (3.4) ⎩ w(t, x) = 0 on (0, T ) × ∂, w(0, x) = w0 (x), x ∈ . Here w0 (x) = u 0 (x) − v0 (x), v0 = A−1 ∇ · τ0 and F = −u · ∇u − A−1 ∇ · (u · ∇τ ) + A−1 ∇ · Q(∇u, g). def
Step 1. Estimate of ∇ 2 w L 2 (L 2 ()) . Firstly, the standard energy estimate gives T
u2L ∞ (L 2 ()) T
+ ∇u2L 2 (L 2 ()) ≤ τ 2L 2 (L 2 ()) + u 0 2L 2 () . T
(3.5)
T
Multiplying the first equation of (3.4) by ∂t w, then integrating over to obtain 1 d 1 1 |∇w(t)|2 d x + |∂t w(t)|2 d x ≤ F2L 2 () + ∂t w(t)2L 2 () . 2 dt 2 2 Thus we have d dt
|∇w(t)| d x + 2
|∂t w(t)|2 d x ≤ F2L 2 () .
(3.6)
In the following arguments, we will frequently use the Gagliardo-Nirenberg (GN) inequality: 1
1− 1
q f L 2q () ≤ C f Lq 2 () ∇ f L 2 () , q ∈ [1, ∞).
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By the GN inequality and Proposition 2.1, we have u · ∇u2L 2 () ≤ 2 u · ∇w2L 2 () + u · ∇v2L 2 () ≤ 2 u2L 4 () ∇w2L 4 () + u2L 4 () ∇v2L 4 () ≤ Cu L 2 () ∇u L 2 () ∇w L 2 () ∇ 2 w L 2 () + u L 2 () ∇u L 2 () τ 2L 4 () ≤ Cε u2L 2 () ∇u2L 2 () ∇w2L 2 () + ε∇ 2 w2L 2 () + u2L 2 () ∇u2L 2 () + τ 4L 4 () . Note that by ∇ · u = 0, u · ∇τ = ∇ · (uτ ) and uτ = 0 on ∂, hence by Proposition 2.1 and the GN inequality, A−1 ∇ · (u · ∇τ )2L 2 () ≤ Cuτ 2L 2 () ≤ Cu2L 4 () τ 2L 4 () ≤ Cu L 2 () ∇u L 2 () τ 2L 4 () ≤ C u2L 2 () ∇u2L 2 () + τ 4L 4 () . Using Proposition 2.1, (3.2) and the GN inequality, we have A−1 ∇ · Q(∇u, g)2L 2 () ≤ CQ(∇u, g)2L 2 () ≤ C g2L 2 () + ∇u2L 2 () + B(∇w, g)2L 2 () + B(∇v, g)2L 2 () ≤ C (τ, g)2L 2 () + ∇w2L 2 () + ∇w2L 4 () g2L 4 () + ∇v2L 4 () g2L 4 () ≤ C(1 + (τ, g)4L 4 () ) + Cε (1 + g4L 4 () )∇w2L 2 () + ε∇ 2 w2L 2 () . We denote def
ψ(t) = 1 + u(t)2L 2 () ∇u(t)2L 2 () + (τ, g)(t)4L 4 () . Substituting these estimates into the right hand side of (3.6), we obtain d |∇w(t)|2 d x + |∂t w(t)|2 d x dt ≤ Cε ψ(t)∇w(t)2L 2 () + Cψ(t) + 2ε∇ 2 w(t)2L 2 () .
(3.7)
On the other hand, from Proposition 2.1 and the above estimates, we also have ∇ 2 w(t)2L 2 () ≤ C ∂t w2L 2 () + F2L 2 () ≤ C∂t w2L 2 () + Cε ψ(t)∇w(t)2L 2 () + Cψ(t) + ε∇ 2 w(t)2L 2 () . Choosing ε =
1 2
in the above inequality to obtain
∇ 2 w(t)2L 2 () ≤ C∂t w2L 2 () + Cψ(t) 1 + ∇w(t)2L 2 () .
1 Inserting (3.8) into (3.7) and taking ε = 2C in (3.7), we deduce that d |∇w(t)|2 d x + |∂t w(t)|2 d x ≤ Cψ(t) 1 + ∇w(t)2L 2 () , dt
(3.8)
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from which and Gronwall’s inequality, we infer that ∇w2L ∞ (L 2 ()) + ∂t w2L 2 (L 2 ()) ≤ 1 + ∇w0 2L 2 () exp Cψ L 1 (0,T ) . T
T
Hence by (3.8), ∇ 2 w2L 2 (L 2 ()) ≤ 1 + ∇w0 2L 2 () exp(Cψ L 1 (0,T ) ). T
Note that by (3.5), ψ L 1 (0,T ) =
T
0
1 + u(t)2L 2 () ∇u(t)2L 2 () + τ (t)4L 4 () + g(t)4L 4 () dt
2 ≤ T + u 0 2L 2 () + τ 2L 2 (L 2 ()) + (τ, g)4L 4 (L 4 ()) . T
T
Thus, we obtain ∇w2L ∞ (L 2 ()) + ∂t w2L 2 (L 2 ()) + ∇ 2 w2L 2 (L 2 ()) T T T def ≤ C T, u 0 H 1 () , τ0 L 2 () , τ L 4 (L 4 ()) , g L 4 (L 4 ()) = F. T
T
(3.9)
On the other hand, we get by Proposition 2.1 that ∇u L 2 () ≤ ∇w L 2 () + Cτ L 2 () , from which and (3.9), we infer that ∇u L 4 (L 2 ()) ≤ C T, u 0 H 1 () , τ0 L 2 () , τ L 4 (L 4 ()) , g L 4 (L 4 ()) . T
T
T
(3.10)
Step 2. Estimate of ∇ 2 w L 2 (L q ()) . Applying Proposition 2.3 to (3.4), we find that T
∇ 2 w L 2 (L q ()) ≤ C F L 2 (L q ()) + w0 H 2 () . T
T
(3.11)
Here we have used Remark 2.4. In the following, we will estimate F L 2 (L q ()) . From T the GN inequality, it follows that u · ∇w L q () ≤ u L p () ∇w L 4 () 1
1
≤ Cu H 1 () ∇w L2 2 () ∇ 2 w L2 2 () ≤ C u2H 1 () + ∇w L 2 () ∇ 2 w L 2 () , and by Proposition 2.1, u · ∇v L q () ≤ C u2H 1 () + τ 2L 4 () , which together with (3.9) and (3.10) imply that u · ∇u L 2 (L q ()) ≤ F. T
(3.12)
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Using the fact that uτ |∂ = 0 and Proposition 2.1, we have A−1 ∇ · (u · ∇τ ) L 2 (L q ()) ≤ Cuτ L 2 (L q ()) ≤ Cu L 4 (L p ()) τ L 4 (L 4 ()) T
T
≤
Cu2L 4 (H 1 ()) T
T
T
+ τ 2L 4 (L 4 ()) T
≤ F.
(3.13)
Finally, by Proposition 2.1, Sobolev embedding and (3.9), we obtain A−1 ∇ · Q(∇u, g) L 2 (L q ()) ≤ CQ(∇u, g) L 2 (L q ()) T T ≤ C g L 2 (L q ()) + ∇u L 2 (L q ()) T T +B(∇w, g) L 2 (L q ()) + B(∇v, g) L 2 (L q ()) T T ≤ C (g, τ ) L 2 (L q ()) + ∇w L 2 (L q ()) + ∇w L 4 (L 4 ()) g L 4 (L p ()) . T T T T +∇v L 4 (L 4 ()) τ L 4 (L p ()) T T 2 ≤ F + C τ L 4 (L p ()) + g2L 4 (L p ()) . (3.14) T
T
Summing up (3.11)–(3.14), we conclude the proof of (3.3).
4. Applications In the following subsections, we will apply new a priori estimates obtained in Sect. 3 to some 2D polymer models.
4.1. Oldroyd B model. We consider the Oldroyd B model which is one of the basic macroscopic models for visco-elastic fluids. This system reads ⎧ ⎨ ∂t u + u · ∇u − u + ∇ p = ∇ · τ, ∇ · u = 0, ∂t τ + u · ∇τ = Q(∇u, τ ), in (0, T ) × , ⎩ u(0, x) = u 0 (x), τ (0, x) = τ0 (x), x ∈ ,
in (0, T ) × , (4.1)
where Q(∇u, τ ) = −aτ + bD(u) + W (u)τ − τ W (u) + c(D(u)τ + τ D(u)), with the constants a, b, c ≥ 0 and D(u) = 21 ((∂i u j + ∂ j u i ))2×2 is the symmetric part of ∇u and the anti-symmetric part W (u) = ∇u − D(u). If is a bounded domain, we also impose the Dirichlet boundary condition for the velocity u. The local existence and uniqueness of the strong solution of (4.1) were studied in [14,15,17,18]. In [29], Lions and Masmoudi proved the global existence of the weak solution under assumption c = 0. Chemin and Masmoudi [4] proved the local and global well-posedness in critical Besov spaces. Moreover, in the case of = R2 , they also obtained the following blow-up criterion of smooth solution.
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Theorem 4.1. Assume that (u, τ ) is a smooth solution of (4.1) with u ∈ C([0, T ); H s (R2 )) ∩ L 2 (0, T ; H s+1 (R2 )), τ ∈ C([0, T ); H s (R2 )) for s > 1. Let T ∗ be a maximal existence time of the solution. Then we have the following necessary condition for the blow-up: T∗ τ (t)2L 2 + τ (t) L ∞ dt = +∞. T ∗ < +∞ ⇒ 0
The proof of Theorem 4.1 is based on the a priori estimate of 2D Navier-Stokes equations and the losing derivative estimates. Recently, Lei, Masmoudi and Zhou [22] gave a new proof which avoids using the losing derivative estimates, but still works for = R2 or T2 . Based on new a priori estimate obtained in Sect. 3, we generalize Theorem 4.1 to the case of bounded domain. Theorem 4.2. Let be a bounded smooth domain in R2 , u 0 ∈ D(A) and τ0 ∈ H 2 (). Assume that (u, τ ) is a strong solution of (4.1) with u ∈ C([0, T ); D(A)) ∩ L 2 (0, T ; H 3 ()), τ ∈ C([0, T ); H 2 ()). Let T ∗ be a maximal existence time of the solution. Then we have the following necessary condition for the blow-up: T∗ τ (t)4L ∞ () dt = +∞. T ∗ < +∞ ⇒ 0
Here and hereafter, we denote D(A) by the domain of the Stokes operator A. Remark 4.3. In the case of c = 0, the global existence of the weak solution of (4.1) is proved by Lions and Masmoudi [29]. However the global existence of the smooth solution is still open, although we can show that τ is bounded in L p () for any p < ∞. Proof. We will prove it by contradiction argument. Let us assume T ∗ < +∞ and
T∗ 0
τ (t)4L ∞ () dt < +∞.
(4.2)
In what follows, we will prove that the solution (u, τ ) can be extended after t = T ∗ . Step 1. Estimate of ∇u L 1 (L ∞ ()) . Let w = u − A−1 ∇ · τ . Thanks to (4.2) and TheoT rem 3.1, we have w L 2 (W 2,q ()) ≤ C T, u 0 H 2 () , τ0 H 1 () , τ L 4 (L ∞ ()) , (4.3) T
T
for any q ∈ [2, 4) and T ≤
T ∗.
By Sobolev embedding, we have
∇u L 1 (L ∞ ()) ≤ C∇u L 1 (W 1,q ()) , T
T
and by Proposition 2.1, ∇u L 1 (W 1,q ()) ≤ w L 1 (W 2,q ()) + Cτ L 1 (W 1,q ()) . T
T
T
(4.4)
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Thus it suffices to estimate τ L 1 (W 1,q ()) . Taking ∇ for the equation of τ , and T
multiplying the resulting equation by |∇τ |q−2 ∇τ , then integrating over , we obtain 1 d q q |∇τ (t)| d x ≤ |∇u(t)||∇τ (t))| d x + |∇ Q(∇u, τ )||∇τ (t))|q−1 d x q dt q
q−1
≤ ∇u L ∞ () ∇τ L q () + ∇ Q(∇u, τ ) L q () ∇τ L q () . Thanks to Sobolev embedding, we find that for q > 2, ∇u L ∞ () ≤ ∇w L ∞ () + ∇ A−1 ∇ · τ L ∞ () ≤ CwW 2,q () + ∇ A−1 ∇ · τ L ∞ () , and by Proposition 2.2, ∇ A−1 ∇ · τ L ∞ () ≤ Cτ (t) L ∞ () ln(e + ∇τ (t)) L q () ). Thus, we obtain ∇u L ∞ () ≤ CwW 2,q () + Cτ (t) L ∞ () ln(e + ∇τ (t) L q () ). On the other hand, we get by Proposition 2.1 that ∇ Q(∇u, τ ) L q () ≤ C(1 + τ L ∞ () )(wW 2,q () + ∇τ L q () ). Summing up the above estimates to obtain d ∇τ (t) L q () ≤ Cη(t)(1 + ∇τ (t) L q () ) ln(e + ∇τ (t) L q () ), dt
(4.5)
where η(t) = 1 + τ (t) L ∞ () 1 + w(t)W 2,q () . From (4.2) and (4.3), we find that η(t) ∈ L 1 (0, T ). Then Gronwall’s inequality ensures that q ∇τ L ∞ ≤ C T, u , τ , τ 2 2 4 ∞ 0 H () 0 H () L (L ()) , T (L ()) T
which together with (4.4) gives ∇u L 1 (L ∞ ()) ≤ C T, u 0 H 2 () , τ0 H 2 () , τ L 4 (L ∞ ()) . T
T
(4.6)
Step 2. Energy estimate. Energy estimates will ensure that for any t ≤ T ≤ T ∗ , (u, τ )(t) H 2 () + u(t) L 2 (H 3 ()) ≤ C T, (u 0 , τ0 ) H 2 () , ∇u L 1 (L ∞ ()) , T
T
(4.7) which together with (4.6) implies that the solution (u, τ ) can be extended after t = T ∗ . Since the proof of (4.7) is very long, we will present it in the appendix.
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4.2. Smoluchowski equation. We consider the system
∂t u + u · ∇u − u + ∇ p = ∇ · τ, ∇ · u = 0, in (0, T ) × , ∂t f + u · ∇ f + ∇g · (G(u, f ) f ) − g f = 0, in (0, T ) × × M.
(4.8)
Here M is a smooth compact Riemannian manifold without boundary, ∇g and g are covariant derivative and Laplace-Beltrami operator on M respectively. We denote ij K (m, q) f (t, x, q)dq, W = cl (m)∂ j u i , G(u, f ) = ∇g U + W, U (t, x, m) = M
(4.9) ij
with K and cl , i, j = 1, 2; l = 1, 2, . . . , n are smooth functions defined on M independent of t, x. The tensor stress τ takes the form (1) (2) γi j (m) f (t, x, m)dm + γi j (m 1 , m 2 ) f (t, x, m 1 ) f (t, x, m 2 )dm, τi j = M
M
M
(4.10) (k)
where γi j (m) are smooth functions on M for i, j, k = 1, 2. We supplement (4.8) with the initial-boundary conditions: u(0, x) = u 0 (x),
f (0, x, m) = f 0 (x, m), u(t, x) = 0 on ∂.
(4.11)
In the case of = R2 or T2 , the global regularity of (4.8) was established by Constantin and Masmoudi [8], see also [9] for a different proof. In this subsection, we generalize their result to the case of bounded domain. Theorem 4.4. Let be a bounded smooth domain in R2 . Assume that u 0 (x) ∈ D(A), f 0 (x, m) ≥ 0, f 0 (x, m) ∈ H 2 (; H −s (M)) ∩ L ∞ (; L 1 (M)) for some s > 2. Then there exists a unique global solution (u, f ) to the system (4.8) such that u ∈ C([0, +∞); D(A)) ∩ L 2 (0, T ; H 3 ()), f ∈ C([0, +∞); H 2 (; H −s (M))), for any T < +∞. Before the proof of Theorem 4.4, let us recall some basic facts about the system (4.8). The Fokker-Planck equation satisfies the weak maximum principle, therefore f 0 (x, m) ≥ 0 ⇒ f (t, x, m) ≥ 0. Integrating the second equation of (4.8) on M to obtain f (t, x, m)dm + u · ∇ f (t, x, m)dm = 0, ∂t M
M
and hence,
f (t, x, m)dm + C
|τ (t, x)| ≤ C
2 f (t, x, m)dm
M
≤
M C f 0 L ∞ (;L 1 (M)) + C f 0 2L ∞ (;L 1 (M)) .
(4.12)
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Moreover, we define def
N 2 (t, x) =
|Rs ∇x f (t, x, m)|2 dm,
s
Rs = (1 − g )− 2 , def
M
then there hold (see [6]) |∇τ (t, x)| ≤ C N (t, x), ∂t N + u · ∇ N ≤ C(1 + |∇u|)N + C|∇ 2 u|.
(4.13) (4.14)
In order to apply Theorem 3.1 to the system (4.8), we need to derive the equation for τ . (1) For the sake of simplicity, we assume τi j (t, x) = M γi j (m) f (t, x, m)dm. Multiplying (1)
the second equation of (4.8) by γi j and integrating on M, we find that (1) (1) ∂t τi j + u · ∇τi j = ∇g γi j (m) · ∇g U + g γi j (m) f (t, x, m)dm M ij + ∂ j u i (t, x) ∇g γi(1) j (m) · c (m)dm.
(4.15)
M
Proof of Theorem 4.4. As the local existence of solutions to the system (4.8) follows from Galerkin’s method and the a priori estimates (for example, the energy estimate (4.16)) for the approximate solutions, here we only need to prove that the maximal existence time T ∗ of the solution can be taken to +∞. By following the proof in the appendix, we can obtain u(t) H 2 () + u(t) L 2 (H 3 ()) + f (t) H 2 (;H −s (M)) T ≤ C T, u 0 H 2 () , f 0 H 2 (;H −s (M)) , ∇u L 1 (L ∞ ()) , T
(4.16) (4.17)
for any t ≤ T ≤ T ∗ . Thus, to prove T ∗ = +∞, it suffices to show that ∇u L 1 (L ∞ ()) < +∞, T
T ∗.
for any T ≤ Fix q ∈ (2, 4). Thanks to (4.12), (4.15) and Theorem 3.1, we have w L 2 (W 2,q ()) ≤ C T, u 0 H 2 () , f 0 H 1 (;H −s (M)) , f 0 L ∞ (;L 1 (M)) , (4.18) T
where w = u − A−1 ∇ · τ . On the other hand, multiplying (4.14) by N (t)q−1 and integrating the resulting euqtion on , we obtain d q q q−1 N (t) L q () ≤ C 1 + ∇u(t) L ∞ () N (t) L q () + ∇ 2 u(t) L q () N (t) L q () . dt Exactly as in the proof of Theorem 4.1, we have ∇u L ∞ () ≤ CwW 2,q () + Cτ (t) L ∞ () ln(e + ∇τ (t)) L q () ). And by Proposition 2.1 and (4.13), we have ∇ 2 u(t) L q () ≤ ∇ 2 w(t) L q () + ∇τ (t) L q () ≤ ∇ 2 w(t) L q () + CN (t) L q () .
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Thus, we deduce that d q N (t) L q () dt q q ≤ C 1 + wW 2,q () + τ (t) L ∞ () ln(e + N (t) L q () )N (t) L q () , from which and Gronwall’s inequality, we infer that t q q N (t) L q () ≤ N (0) L q () exp exp C (1 + w(t )W 2,q () + τ (t ) L ∞ () )dt 0 ≤ C T, u 0 H 2 () , f 0 H 2 (;H −s (M)) , f 0 L ∞ (;L 1 (M)) . Here we used (4.12) and (4.18) in the last inequality. Now we get by Sobolev embedding and Proposition 2.1 that ∇u L 1 (L ∞ ()) ≤ C∇u L 1 (W 1,q ()) ≤ C(w L 1 (W 2,q ()) + τ L 1 (W 1,q ()) ) T T T T ≤ C T, u 0 H 2 () , f 0 H 2 (;H −s (M)) , f 0 L ∞ (;L 1 (M)) . This completes the proof of Theorem 4.4.
4.3. Corotational FENE model. We consider the co-rotational FENE Dumbbell model, which models polymers by nonlinear springs, and which takes into account the finite extensibility of the polymer chains. Mathematically, this system reads ⎧ ⎪ ⎨ u t + u · ∇u − u + ∇ p = ∇ · τ, in (0, T ) × , ∇ · u = 0, in (0, T ) × , (4.19) ⎪ ⎩ f + u · ∇ f + ∇ · (S(u)q f ) = 1 f + 1 ∇ · (∇ U f ), in (0, T ) × × D, t q q 2 q 2 q with S(u), the potential U (s) and the extra-stress tensor τ being given by 2 |q| b |q|2 ∇u − ∇u t , U = − log 1 − , τ= ∇q U ⊗ q f dq, S(u) = 2 2 2 b D (4.20) √ √ where D = B(0, b), the ball with center 0 and radius b. We should point out that in general S(u) = ∇u in (4.20), and in the simpler co-rotational case, S(u) is given by the anti-symmetric part of ∇u. In the case when S(u) = ∇u and the last equation of (4.19) is replaced by a stochastic PDE, one may find interesting studies in [19] for FENE models with b > 2 and sometime b > 6, in [12] , an additional polynomial force term is added. See also discussions in [34] for FENE models with b > 76. Recently, Lin, Zhang and Zhang [28] proved the global existence of smooth solution to the system (4.19) with b > 12 in the case of = R2 , see [31] for b > 0. In this subsection, we generalize the result of [28] to the case of bounded domain. We assume f satisfies the natural flux boundary condition: 1 1 q ∇q U f + ∇q f − S(u)q f · |∂ D = 0. (4.21) 2 2 |q|
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In order to get round the singularity of ∇q U in (4.21), as in [11], we introduce the following transformation:
f (t, x, q) = e−
U
|q|2 2 2
g(t, x, q),
so that as long as b > 4 and g|∂ D = 0, the flux boundary condition (4.21) is automatically satisfied. Furthermore, the system (4.19) is now reduced to the following new system for (u, g): ⎧ ⎨ u t + u · ∇u − u + ∇ p = ∇ · τ, (4.22) ∇ · u = 0, ⎩ gt + u · ∇g + ∇q · (S(u)qg) = 21 q g + 41 q U − 21 |∇q U |2 g, with the extra-stress tensor τ given by
∇q U ⊗ qe−
τ=
U
|q|2 2 2
g(t, x, q)dq,
(4.23)
D
together with the initial boundary conditions: u|t=0 = u 0 , g|t=0 = g0 , u|∂ = 0, g|∂ D = 0.
(4.24)
Now we state our main result as follows. Theorem 4.5. Let be a bounded smooth domain in R2 and b > 12. Assume that u 0 ∈ D(A), g0 ∈ H 2 (; H01 (D)). Then the system (4.22)–(4.24) has a unique global solution (u, g) such that for any T > 0, there hold u ∈ C([0, T ); D(A)) ∩ L 2 ((0, T ); H 3 ()), g ∈ L 2 ((0, T ); H 2 (; L 2 (D))). g ∈ C [0, T ); H 2 (; H01 (D)) , q g, (b − |q|2 )2 Before the proof of Theorem 4.5, let us recall some basic facts about the system (4.22). Firstly, multiplying the third equation of (4.22) by g , then integrating the resulting equation over D to obtain 1 1 2 2 ∂t |g| dq + u · ∇ |g| dq + ∇q · (S(u)qg)gdq 2 2 D D D 1 1 1 q U − |∇q U |2 g 2 dq. |∇q g|2 dq + =− 2 D 4 D 2 Note that ∇ · u = 0 implies that ∇q · (S(u)q) = 0, we get by integrating by parts that 1 1 ∇q · (S(u)qg)gdq = − (S(u)q) · ∇q g 2 dq = divq (S(u)q)g 2 dq = 0. 2 D 2 D D
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On the other hand, integration by parts also gives 1 1 1 1 1 q U − |∇q U |2 g 2 dq = − − |∇q g|2 dq + |∇q g + ∇q U g|2 dq. 2 D 4 D 2 2 D 2 Therefore, we obtain 1 |g|2 dq + u · ∇ |g|2 dq + |∇q g + ∇q U g|2 dq = 0. ∂t 2 D D D In particular, this implies that 2 |g| dq + u · ∇ |g|2 dq ≤ 0, ∂t D
D
from which and ∇ · u = 0, we deduce |g(t, x, q)|2 dq
L ∞ ()
D
2 ≤ |g0 (x, q)| dq D
On the other hand, thanks to (4.20) and (4.23), we have ∇q U =
L ∞ ()
bq , b−|q|2
|τ (t, x)| ≤ C
(4.25) and
1
|g(t, x, q)|dq ≤ C
|g(t, x, q)| dq 2
D
.
2
.
D
The latter estimate together with (4.25) ensures that τ (t, x) L ∞ ()
1 2 2 ≤C |g0 (x, q)| dq D
.
(4.26)
L ∞ ()
Secondly, if b > 4 and g ∈ H01 (D), we have (Lemma 3.2 in [28])
1 |∇q g + ∇q U g|2 dq ≥ 2 D
|∇q g| dq − C
|g|2 dq.
2
D
(4.27)
D
Finally, we derive the equation of the tensor τ . Multiplying the equation of g in (4.22) by ∇q U ⊗ qe
−
U
|q|2 2 2
, then integrating the resulting equation over D, we obtain
∇q U ⊗ qe−
τt + u · ∇τ = −
U
|q|2 2 2
D
∇q · (S(u)qg)dq
1 1 − 2 q U − |∇q U | gdq − ∇q U ⊗ qe 4 D 2 2 |q| U 2 def + ∇q U ⊗ qe− 2 q gdq = Q (∇u, g ),
D
U
|q|2 2 2
(4.28)
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where g is some function of (t, x) depending on g. Due to b > 12 and (4.25), we can get by integration by parts that |Q(∇u, g )| ≤ C(1 + |∇u|) |g(t, x, q)|dq D
1
≤ C(1 + |∇u|)
|g0 (x, q)| dq 2
2
.
(4.29)
D
Now we are in position to prove Theorem 4.5. Proof of Theorem 4.5. As in the proof of Theorem 4.4, here we only need to prove that the maximal existence time T ∗ of the solution can be taken to +∞. By using the energy estimates (see Theorem 3.1 in [28] and the appendix), it can be shown that u(t) H 2 () + u(t) L 2 (H 3 ()) + g(t) H 2 (;H 1 (D)) + q g(t) L 2 (H 2 (;L 2 (D))) T T ≤ C T, u 0 H 2 () , g0 H 2 (;H 1 (D)) , ∇u L 2 (L ∞ ()) , T
for any t ≤ T ≤ T ∗ . Thus, to prove T ∗ = +∞, it suffices to show that ∇u L 2 (L ∞ ()) < +∞, T
for any T ≤ T ∗ . Fix q ∈ (2, 4). Thanks to (4.28), (4.29) and Theorem 3.1, we have w L 2 (W 2,q ()) ≤ C T, u 0 H 2 () , g0 H 1 (;L 2 (D)) , g0 L ∞ (;L 2 (D)) , T
(4.30)
where w = u − A−1 ∇ · τ . While by Sobolev embedding and Proposition 2.1, we have ∇u L 2 (L ∞ ()) ≤ C∇u L 2 (W 1,q ()) ≤ C w L 2 (W 2,q ()) + τ L 2 (W 1,q ()) . T
T
T
T
Thus, due to (4.30), it remains to estimate τ L 2 (W 1,q ()) . Note that T
1
|∇g(t, x, q)| dq
|∇τ (t, x)| ≤ C
2
2
1
def
= C N (t, x) 2 .
D
Therefore, we need to derive a differential inequality for N (t, x). Taking ∇ for the equation of g in (4.22), and multiplying the resulting equation by ∇g, then integrating over D, we get by integrating by parts that 1 ∂t N + u · ∇ N + |∇q ∇g + ∇q U ∇g|2 dq 2 D 2 2 2 ≤ C|∇u|N + C|∇ u| |g(t, x, q)| dq + |∇q ∇g|2 dq, D
D
which together with (4.25) and (4.27) implies that ∂t N + u · ∇ N ≤ C|∇u|N + C(1 + |∇ 2 u|2 ).
(4.31)
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With (4.31), exactly as in the proof of Theorem 4.4, we can obtain q d ≤ C 1 + w2W 2,q () + τ (t) L ∞ () N (t) 2 q dt L 2 () q q × ln e + N (t) 2 q N (t) 2 q + C, L 2 ()
L 2 ()
from which and Gronwall’s inequality, we infer that ∇τ L q () ≤ CN (t) q ≤ C T, u 0 H 2 () , g0 H 2 (;H 1 (D)) . L 2 ()
This completes the proof of Theorem 4.5.
Acknowledgements. This work was done while Yongzhong Sun and Zhifei Zhang were visiting the Morningside Center of Mathematics in CAS. We would like to thank the center for hospitality and support. Yongzhong Sun is supported by NSF of China under Grant No. 10771097, 10931007. Zhifei Zhang is supported by NSF of China under Grant 10990013, 11071007, and SRF for ROCS, SEM.
5. Appendix Let be a bounded smooth domain in R2 . We consider the system ⎧ in (0, T ) × , ⎨ ∂t u + u · ∇u − u + ∇ p = ∇ · τ, ∇ · u = 0, ∂t τ + u · ∇τ = Q(∇u, τ ), in (0, T ) × , (5.1) ⎩ u(t, x) = 0 on (0, T ) × ∂, (u(0, x), τ (0, x)) = (u 0 (x), τ0 (x)) x ∈ . For simplicity, here we assume that Q(∇u, τ ) = ∇uτ . The following proposition can be easily generalized to the general form Q(∇u, τ ). Proposition 5.1. Let u 0 ∈ D(A), τ0 ∈ H 2 (). Assume that (u, τ ) is a strong solution of (5.1) in [0, T ). Then there holds (u, τ )(t) H 2 () + u(t) L 2 (H 3 ()) ≤ C T, (u 0 , τ0 ) H 2 () , ∇u L 1 (L ∞ ()) , T
T
for any t ≤ T . Proof. We divide the proof into several steps. Step 1. L 2 energy estimate. The following estimate is classical: d u(t)2L 2 () + τ (t)2L 2 () + ∇u(t) L 2 () dt ≤ (1 + ∇u L ∞ () )τ (t)2L 2 () . From the equation of τ , it is easy to show that t ∇u(t ) L ∞ () τ (t ) L ∞ () dt . τ (t) L ∞ () ≤ τ0 L ∞ () + 0
Then Gronwall’s inequality ensures that τ (t) L ∞ () ≤ τ0 L ∞ () exp
0
t
∇u(t ) L ∞ () dt ,
(5.2)
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from which and (5.2), we get by Gronwall’s inequality that (u, τ )(t) L 2 () + u(t) L 2 (H 1 ()) + τ (t) L ∞ () T ≤ C T, (u 0 , τ0 ) L 2 () , τ0 L ∞ () , ∇u L 1 (L ∞ ()) .
(5.3)
T
Step 2. H 1 energy estimate. Taking ∇ for the equation of τ , multiplying the resulting equation by ∇τ , then integrating over , we get by integration by parts that d ∇τ 2L 2 () = (−∇u · ∇τ + ∇ Q(∇u, τ ))∇τ d x dt ≤ 2∇u L ∞ () ∇τ 2L 2 () + τ L ∞ () ∇ 2 u L 2 () ∇τ L 2 () . (5.4) Multiplying the equation of u by Au, then integrating on , we have d 2 2 ∇u L 2 () + Au L 2 () = (−u · ∇u + ∇ · τ )Aud x dt 1 1 2 ≤ C u L2 2 () ∇u L 2 () ∇ u L2 2 () + ∇τ L 2 () Au L 2 () ,
(5.5)
where we used the inequality 1
1
u · ∇u L 2 () ≤ u L 4 () ∇u L 4 () ≤ Cu L2 2 () ∇u L 2 () ∇ 2 u L2 2 () . From (5.4) and (5.5), we infer that d ∇u(t)2L 2 () + ∇τ (t)2L 2 () + Au(t)2L 2 () dt ≤ C 1 + τ 2L ∞ () + ∇u L ∞ () + u2L 2 () ∇u2L 2 () × ∇u(t)2L 2 () + ∇τ (t)2L 2 () , from which and (5.3), we get by Gronwall’s inequality that (u, τ )(t) H 1 () + Au(t) L 2 (L 2 ()) T ≤ C T, (u 0 , τ0 ) H 1 () , τ0 L ∞ () , ∇u L 1 (L ∞ ()) . T
On the other hand, we also have d ∇u2L 2 () + u t 2L 2 () = dt
(−u · ∇u + ∇ · τ )u t d x,
from which and (5.6), it is easy to show that u t L 2 (L 2 ()) ≤ C T, (u 0 , τ0 ) H 1 () , τ0 L ∞ () , ∇u L 1 (L ∞ ()) . T
(5.6)
T
(5.7)
Step 3. H 2 energy estimate. Taking ∇ 2 for the equation of τ , multiplying the resulting equation by ∇ 2 τ , then integrating over , we get d ∇ 2 τ 2L 2 () = −∇ 2 (u · ∇τ ) + ∇ 2 Q(∇u, τ ) ∇ 2 τ d x. dt
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Note that u|∂ = 0 and ∇ · u = 0, we have ∇ 2 (u · ∇τ ) − u · ∇∇ 2 τ ∇ 2 τ d x ∇ 2 (u · ∇τ )∇ 2 τ d x =
≤ ∇u L ∞ () ∇
2
τ 2L 2 ()
1
1
1
3
+ C∇τ L2 2 () ∇ 2 u L2 2 () ∇ 3 u L2 2 () ∇ 2 τ L2 2 () ,
where we used the inequality ∇ 2 u∇τ L 2 () ≤ ∇ 2 u L 4 () ∇τ L 4 () 1
1
1
1
≤ C∇ 2 u L2 2 () ∇ 3 u L2 2 () ∇τ L2 2 () ∇ 2 τ L2 2 () . Similarly, we have ∇ 2 Q(∇u, τ )∇ 2 τ d x ≤ τ L ∞ () ∇ 3 u L 2 () ∇ 2 τ L 2 ()
1
1
1
3
+ ∇u L ∞ () ∇ 2 τ 2L 2 () + C∇τ L2 2 () ∇ 2 u L2 2 () ∇ 3 u L2 2 () ∇ 2 τ L2 2 () . Consequently, we obtain d ∇ 2 τ 2L 2 () ≤ ε∇ 3 u2L 2 () dt 2
2
+ C(∇u L ∞ () + τ 2L ∞ () + ∇τ L3 2 () ∇ 2 u L3 2 () )∇ 2 τ 2L 2 () .
(5.8)
On the other hand, similar to the proof of Step 2, we can obtain d u t 2L 2 () + ∇u t 2L 2 () ≤ C∇u L ∞ () u t 2L 2 () + τt 2L 2 () , dt and for τt , d τt 2L 2 () ≤ τ L ∞ () τt L 2 () ∇u t L 2 () + ∇u L ∞ () τt 2L 2 () dt 1
1
1
1
+ C∇τ L2 2 () ∇ 2 τ L2 2 () u t L2 2 () ∇u t L2 2 () τt L 2 () . Thus we have d (u t 2L 2 () + τt 2L 2 () ) + ∇u t 2L 2 () dt 2
2
≤ C(1 + ∇u L ∞ () + τ 2L ∞ () + ∇τ L3 2 () u t L3 2 () )h(t),
(5.9)
where def
h(t) = u t 2L 2 () + τt 2L 2 () + ∇ 2 τ 2L 2 () . Now we rewrite the first equation of (5.1) as − u + ∇ p = −u t − u · ∇u + ∇ · τ.
(5.10)
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Then we get by (77) in [21] that u H 3 () ≤ C(u t H 1 () + u · ∇u H 1 () + τ H 2 () ). By the GN inequality, we have u · ∇u H 1 ()
1 1 2 2 ≤ Cu H 1 () u H 2 () + u H 2 () u H 3 () .
Hence,
u H 3 () ≤ C u t H 1 () + u2H 1 () u H 2 () + u H 1 () u H 2 () + τ H 2 () .
Substituting it into (5.8) to obtain d ∇ 2 τ 2L 2 () ≤ Cε u t 2H 1 () + u4H 1 () + u2H 1 () u2H 2 () + τ 2H 2 () dt 2 2 + C ∇u L ∞ () + τ 2L ∞ () + ∇τ L3 2 () ∇ 2 u L3 2 () ∇ 2 τ 2L 2 () , from which and (5.9), we infer that d h(t) ≤ ψ1 (t) + ψ2 (t)h(t), dt for some ψ1 (t), ψ2 (t) ∈ L 1 (0, T ) by (5.6) and (5.7). And from the equations, we have (u t (0), τt (0)) L 2 () ≤ C(u 0 H 2 () , τ0 H 2 () ). Now Gronwall’s inequality ensures that (u t (t), τt (t)) L 2 () + τ (t) H 2 () ≤ C T, (u 0 , τ0 ) H 2 () , ∇u L 1 (L ∞ ()) . T
(5.11) With (5.6) and (5.11), it is easy to deduce from (5.10) and Proposition 2.1 that u(t) H 2 () ≤ C T, (u 0 , τ0 ) H 2 () , ∇u L 1 (L ∞ ()) . T
This completes the proof of Proposition 5.1.
References 1. Barrett, J.W., Schwab, C., Süli, E.: Existence of global weak solutions for some polymeric flow models. Math. Models Methods Appl. Sci. 15, 939–983 (2005) 2. Bird, R.B., Curtis, C.F., Armstrong, R.C., Hassager, O.: Dynamics of Polymeric Liquids, Volume 2: Kinetic Theory. New York: Weiley Interscience, 1987 3. Bahouri, H., Chemin, J.Y.: Équations de transport relatives á des champs de vecteurs non-lipschitziens et mécanique des fluides. Arch. Rational Mech. Anal. 127, 159–181 (1994) 4. Chemin, J.Y., Masmoudi, N.: About lifespan of regular solutions of equations related to viscoelastic fluid. SIAM. J. Math. Anal. 33, 84–112 (2001) 5. Chen, Y., Zhang, P.: The global existence of small solutions to the incompressible viscoelastic fluid system in 2 and 3 space dimensions. Comm. Part. Diff. Eq. 31, 1793–1810 (2006) 6. Constantin, P.: Nonlinear Fokker-Planck Navier-Stokes systems. Commun. Math. Sci. 3, 531–544 (2005)
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7. Constantin, P., Fefferman, C., Titi, E., Zarnescu, A.: Regularity for coupled two-dimensional nonlinear Fokker-Planck and Navier-Stokes systems. Comm. Math. Phys. 270, 789–811 (2007) 8. Constantin, P., Masmoudi, N.: Global well-posdness for a Smoluchowski equation coupled with NavierStokes equations in 2D. Comm. Math. Phys. 278, 179–191 (2008) 9. Constantin, P., Seregin, G.: Global Regulatity of Solutions of Coupled Navier-Stokes Equations and Nolinear Fokker-Planck Equations. Discrete Contin. Dyn. Syst. 26, 1185–1196 (2010) 10. Doi, M., Edwards, S.F.: The Theory of Polymer Dynamics. Oxford: Oxford Science Publication, 1986 11. Du, Q., Liu, C., Yu, P.: FENE dumbbell model and its several linear and nonlinear closure approximations. Multiscale Model. Simul. 4, 709–731 (2005) 12. Weinan, E., Li, T.J. Zhang, P.W.: Well-posedness for the dumbbell model of polymeric fluids. Comm. Math. Phys. 248, 409–427 (2004) 13. Galdi, G.P.: An introduction to the mathematical theory of Navier-Stokes equations, Vol. 1, Springer Tracts in Natural Philosophy, Vol. 38, Berlin-Heidelberg-New York: Springer, 1994 14. Fernández-Cara, E., Guillén, F., Ortega, R.R.: Some theoretical results concerning non-Newtonian fluids of the Oldroyd kind. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4)26, 1–29 (1998) 15. Fernández-Cara, E., Guillén, F., Ortega, R.R.: Mathematical modeling and analysis of viscoelastic fluids of the Oldroyd kind. Handbook of numerical analysis, Vol. 8, Amsterdam: North-Holland, 2002, pp. 543–661 16. Giga, Y., Sohr, H.: Abstract L p estimates for the Cauchy problem with applications to the Navier-Stokes equations in exterior domains. J. Funct. Anal. 102, 72–94 (1991) 17. Guillopé, C., Saut, J.C.: Existence results for the flow of viscoelastic fluids with a differential constitutive law. Nonlinear Anal. 15, 849–869 (1990) 18. Guillopé, C., Saut, J.C.: Global existence and one-dimensional nonlinear stability of shearing motions of viscoelastic fluids of Oldroyd type. RAIRO Modél. Math. Anal. Numér. 24, 369–401 (1990) 19. Jourdain, B., Leliévre, T., Le Bris, C.: Existence of solution for a micro-macro model of polymeric fluid: the FENE model. J. Funct. Anal. 209, 162–193 (2004) 20. Jourdain, B., Leliévre, T., Le Bris, C., Otto, F.: Long-time asymptotics of a multiscale model for polymeric fluid flows. Arch. Ration. Mech. Anal. 181, 97–148 (2006) 21. Ladyzhenskaya, O.A.: The Mathematical Theory of Viscous Incompressible Flow. London: Gordon and Breach, 1969 22. Lei, Z., Liu, C., Zhou, Y.: Global solutions for incompressible viscoelastic fluids. Arch. Rat. Mech. Anal. 188, 371–398 (2008) 23. Lei, Z., Masmoudi, N., Zhou, Y.: Remarks on the Blowup criteria for Oldroyd models. Jour. Diff. Eqs. 248, 328–341 (2009) 24. Lin, F., Liu, C., Zhang, P.: On hydrodynamics of viscoelastic fluids. Comm. Pure Appl. Math. 58, 1437– 1471 (2005) 25. Lin, F., Liu, C., Zhang, P.: On a Micro-Macro Model for Polymeric Fluids near Equilibrium. Comm. Pure Appl. Math. 60, 838–866 (2007) 26. Lin, F., Zhang, P.: The FENE dumbell model near equilibrium. Acta Math. Sin. (Engl. Ser.) 24, 529– 538 (2008) 27. Lin, F., Zhang, P.: On the initial-boundary value problem of the incompressible viscoelastic fluid system. Comm. Pure Appl. Math. 61, 539–558 (2008) 28. Lin, F., Zhang, P., Zhang, Z.: On the global existence of smooth solution to the 2-D FENE dumbbell model. Comm. Math. Phys. 277, 531–553 (2008) 29. Lions, P.L., Masmoudi, N.: Global solutions for some Oldroyd models of non-Newtonian flows. Chinese Ann. Math. Ser. B 21, 131–146 (2000) 30. Lions, P.L., Masmoudi, N.: Global existence of weak solutions to some micro-macro models. C. R. Math. Acad. Sci. Paris 345, 15–20 (2007) 31. Masmoudi, N.: Well-posedness for the FENE dumbbell model of polymeric flows. Comm. Pure Appl. Math. 61, 1685–1714 (2008) 32. Masmoudi, N., Zhang, P., Zhang, Z.: Global well-posedness for 2D polymeric fluid models and growth estimate. Phys. D 237, 1663–1675 (2008) 33. Renardy, M.: An existence theorem for model equations resulting from kinetic theories of polymer solutions. SIAM J. Math. Anal. 22, 313–327 (1991) 34. Zhang, H., Zhang, P.W.: Local existence for the FENE-dumbbell model of polymeric fluids. Arch. Ration. Mech. Anal. 181, 373–400 (2006) Communicated by P. Constantin
Commun. Math. Phys. 303, 385–420 (2011) Digital Object Identifier (DOI) 10.1007/s00220-011-1200-6
Communications in
Mathematical Physics
On the Extension of Stringlike Localised Sectors in 2+1 Dimensions Pieter Naaijkens Radboud University Nijmegen, Institute for Mathematics, Astrophysics and Particle Physics, Heyendaalseweg 135, 6525 AJ Nijmegen, The Netherlands. E-mail:
[email protected] Received: 30 April 2010 / Accepted: 9 September 2010 Published online: 2 February 2011 – © The Author(s) 2011. This article is published with open access at Springerlink.com
Abstract: In the framework of algebraic quantum field theory, we study the category A BF of stringlike localised representations of a net of observables O → A(O) in three dimensions. It is shown that compactly localised (DHR) representations give rise to a A non-trivial centre of A BF with respect to the braiding. This implies that BF cannot be modular when non-trivial DHR sectors exist. Modular tensor categories, however, are important for topological quantum computing. For this reason, we discuss a method to remove this obstruction to modularity. Indeed, the obstruction can be removed by passing from the observable net A(O) to the Doplicher-Roberts field net F(O). It is then shown that sectors of A can be extended to sectors of the field net that commute with the action of the corresponding symmetry group. Moreover, all such sectors are extensions of sectors of A. Finally, the category F BF of sectors of F is studied by investigating the relation with the categorical crossed product of A BF by the subcategory of DHR representations. Under appropriate conditions, this completely determines the category F BF . 1. Introduction The study of superselection sectors and particle statistics has been a long-standing subject in algebraic quantum field theory [26]. Superselection sectors can be described as representations of a local net O → A(O) of observables. The physically relevant representations are selected by a certain selection criterion. A superselection sector, then, is a (unitary) equivalence class of representations satisfying this criterion. These representations can be shown to have the structure of a tensor category resembling the category of representations of a compact group. In this category, one can define a braiding, closely related to the statistics of sectors. It is well known that for the compactly localised representations first considered by Doplicher, Haag and Roberts, the braiding is in fact symmetric in spacetimes of dimension three or higher [20]. However, if one considers the weaker condition of localisation
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in some “fattening string” extending to spacelike infinity, the braiding is non-symmetric for spacetimes of dimension 3 or less [24]. Buchholz and Fredenhagen have shown that for massive particle states, this localisation condition holds [7]. The category of such stringlike localised representations in three dimensions automatically satisfies most of the axioms of a modular tensor category [2,56]. This class of tensor categories plays a prominent role in the theory of topological quantum computation, see e.g. [22,23,31,32,46]. A good review can be found in [45]. This is one of the reasons why modular tensor categories are interesting, providing a reason to investigate if we can obtain modular tensor categories from algebraic quantum field theory. Another part of the motivation is provided by related constructions and results in e.g. [30,41,48], where the extension of compactly localised representations in d = 1 + 1 is discussed. First, we give a brief overview of the basics of algebraic quantum field theory (AQFT), also called local quantum physics. The leading idea in AQFT is that local algebras of observables encode all relevant information of a given physical theory. For each double cone O in Minkowski space M3 there is an associated unital C ∗ -algebra A(O) of observables, which are said to be localised in O. This assignment of observable algebras should satisfy the following properties: (i) Isotony: If O1 ⊂ O2 , then A(O1 ) ⊂ A(O2 ). We assume the inclusions are injective unital ∗-homomorphisms. (ii) Locality: If O1 is spacelike separated from O2 , then the associated local observable algebras commute. (iii) Translation covariance: There is a strongly continuous action x → βx of the translation group M3 on the local algebras, such that βx (A(O)) = A(O + x). To avoid the trivial case we assume in addition that for each double cone O the algebra A(O) contains an element that is not a multiple of the identity. Note that the set of double cones in M3 is directed by inclusion. The inductive limit of this net in the category of C ∗ -algebras is denoted by A and is called the quasi-local algebra. By means of a specific faithful irreducible representation π0 : A → B(H0 ), typically the vacuum representation, A is represented as a net of bounded operators on a Hilbert space H0 . It is then natural to consider π0 (A(O)) for each O, where the prime denotes the commutant. This leads to a net of von Neumann algebras, which we will again denote by A(O). This net turns out to be more convenient to work with, and thus we will from now on assume that A(O) is a von Neumann algebra for each O. The algebra A again will be the norm closure of the union of these local (von Neumann) algebras. Note that A is not a von Neumann algebra in general. The vacuum representation π0 must satisfy a few additional conditions. It should be covariant under translations, say with a strongly continuous group of unitaries U0 (x), x ∈ M3 . There is a unique (up to a phase) vacuum vector such that U0 (x) = for all x. Moreover, the spectrum condition for the generators of translations should hold: the joint spectrum of the generators of the translations should be contained in the forward lightcone V + . For details and motivations see e.g. [8]. Buchholz and Fredenhagen provide a construction that, given a massive single particle representation, produces a corresponding vacuum representation π0 satisfying these criteria [7]. A superselection sector is then a unitary equivalence class of representations of A satisfying a certain (physically motivated) selection criterion. For example, Buchholz and Fredenhagen were led to consider stringlike localised sectors [7]. The category of these representations, denoted by A BF , has a very rich structure. An essential ingredient in the analysis of this structure is the axiom of Haag duality, which strengthens locality.
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If S is some unbounded region of spacetime, the C ∗ -algebra A(S ) is defined by A(S ) =
A(O)
·
,
O⊂S
where the closure in norm is taken and the union is taken over all double cones contained in S . Suppose S is any connected causally complete region, that is, S = (S ) , where the prime denotes taking the causal complement. Haag duality then is the condition that π0 A S = π0 (A(S )) . (1.1) Here the prime in S denotes taking the causal complement, whereas the other primes stand for the commutant. We will only need this duality relation in the case where S is either a double cone or a spacelike cone. Haag duality has been proven for free fields [1], but to the knowledge of the author no result is known (in d = 2 + 1) for interacting fields. Every representation in A BF can be described as an endomorphism of some algebra ASa containing A as a subalgebra. The category A BF then can be equipped with a tensor product defined by composition of such endomorphisms. As mentioned before, a particularly interesting feature is that it is in fact a braided tensor category. In three dimensions, the DHR sectors, which are localised in bounded regions, form a degenerate tensor subcategory of A BF with respect to the braiding: the braiding with objects from this subcategory reduces to a symmetry. By a result of Rehren, this implies that the category A BF cannot be modular [47,48]. The basic idea now is to pass to the field net F, as constructed by Doplicher and Roberts [18]. The field net is a net of algebras that generate the different superselection sectors by acting on the vacuum. It is endowed with an action of a compact group G of symmetries (sometimes called the gauge group). The observables are precisely those elements of the field algebra that are invariant under the action of this symmetry group. At the end of the 1980s, Doplicher and Roberts solved a long-standing problem in algebraic quantum field theory, namely how to construct the group G and the corresponding field net from the observable algebra [18]. Their investigations led to a new duality theory for compact groups [17], on which we will elaborate below. It is important to note however that these constructions only work if all sectors have permutation statistics. In the braided case, instead of a group one expects an object with a (quasi-)Hopf algebra-like structure, see for example [49,55], or even a more general notion of symmetry [33]. In the special case where A has no fermionic DHR sectors, we can interpret O → F(O) as a new AQFT. Conti, Doplicher and Roberts have shown that the field net does not have any non-trivial representations satisfying the DHR criterion any more [9]. The theory F is an extension of A, in the sense that any stringlike localised representation of A can be extended to a representation of F with the same localisation properties. This A extension factors through the categorical crossed product A BF DHR of [40]. Under certain conditions, this crossed product is in fact equivalent, in the categorical sense, to the category F BF . This makes it possible to understand the latter completely in terms of the original theory O → A(O). To summarise, the obstruction for modularity is removed by passing from a theory A to a new theory F that extends A in a systematic way. Although some constructions in this paper are motivated by results in d = 1+1, there are also some notable differences with the case d = 2+1 considered in the present work.
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In d = 2 + 1, passing from a net F to the fixpoint theory A = FG with respect to the action of some group G introduces DHR sectors, which are automatically degenerate in d = 2 + 1. In d = 1 + 1, DHR sectors also appear when passing to the fixpoint net. In this case, however, they are never degenerate, at least not if the symmetry group G is finite and the theory is “completely rational” [30]. In that situation there appear automatically “twisted” sectors which prevent degeneracy of the new DHR sectors in the fixpoint theory [41]. The paper is organised as follows. In Sect. 2, the basic structure of stringlike localised sectors in three dimensions is recalled. The next section is concerned with the construction of the field net F, and it is shown that this can be interpreted as a new AQFT without DHR sectors. Section 4 then discusses how stringlike localised sectors of our original theory A can be extended to the new theory F. Section 5 deals with the reverse problem of restricting sectors that are invariant under the action of the symmetry group, using results from the theory of non-abelian cohomology. In the last part of the paper, it is investigated how these results are related to the purely mathematical theory of crossed products of braided tensor categories by symmetric subcategories. This gives a better understanding of the sectors of the new theory in terms of those of the old theory. In particular, conditions are given under which all sectors of F are related to the sectors of A. In the last section, the main results are summarised and some open problems are indicated. Some terminology regarding category theory and algebraic quantum field theory, which will be used throughout the article, is recollected in an Appendix. 2. Stringlike Localised Sectors In algebraic quantum field theory a superselection criterion identifies the physically relevant representations of the observable algebra. Usually one selects those representations π that cannot be distinguished from the vacuum representation π0 in the spacelike complement of some causally complete region. The selection criterion used by Doplicher, Haag and Roberts (DHR) requires that the relevant representations π satisfy, for each double cone O, π A O ∼ (2.1) = π0 A O . That is, π is unitarily equivalent to the vacuum representation when restricted to observables in the causal complement of an arbitrary double cone. The structure of the DHR superselection sectors is well understood, see e.g. [26,27] for reviews. A DHR represen1 ∼ tation is of the form π = π0 ◦ ρ, where ρ is an endomorphism of A that acts trivially on A O for some O. Such an endomorphism is said to be localised in O. Furthermore, ρ there is a morphism ρ is transportable, in the sense that for any double cone O localised in O, unitarily equivalent to ρ. Localised transportable endomorphisms can be regarded as objects of a braided tensor category. However, the criterion (2.1) is too narrow for many physical applications. For example, consider the case of an electrically charged particle. Then, by Gauss’ theorem, it is possible to measure the electric flux through a surface at arbitrary large distance. This implies that the presence of an electric charge can be detected at arbitrarily large distances, i.e., there is no double cone O such that the state cannot be distinguished from the vacuum in the spacelike complement of this O. See [6] for a discussion of states in QED. 1 All (endo)morphisms and representations are assumed to be unital and to preserve the ∗ -operation, unless stated otherwise.
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This is why Buchholz and Fredenhagen consider a more general selection criterion [7], namely (2.2) π A C ∼ = π0 A C , for each spacelike cone C in the following sense: Definition 2.1. A spacelike cone is a set C = x + λ>0 λ · O, for some double cone O not containing the origin, and x ∈ Md . Moreover, we demand that C is causally complete2 , i.e., C = C . Such a spacelike cone can be visualised as a semi-infinite string that becomes thicker and thicker when moving towards spacelike infinity. Since again this criterion means that such representations cannot be distinguished from the vacuum in the spacelike complement of a spacelike cone, such representations are called localisable in cones. We will call the equivalence class of such a representation a BF sector, and call a representative a BF representation. Buchholz and Fredenhagen show that in a relativistic quantum field theory massive single-particle representations always have such localisation properties. Roughly speaking, a massive representation is a representation that is covariant under translation (covariance under the full Poincaré group is not required). Moreover, the joint spectrum of the generators of the translations is bounded away from zero and contains an isolated mass shell, separated by a gap from the rest of the spectrum. There are several methods to study the superselection structure of charges localised in spacelike cones (also called “topological charges”). Recall that we identified π0 (A) with A. Contrary to the case of DHR sectors, BF sectors cannot be described in terms of endomorphisms of the quasi-local algebra A. Instead, the representations map cone algebras A(C ) to weak closures of the algebra, that is, η(A(C )) ⊂ A(C ) if η is localised in a spacelike cone C ⊂ C . For double cones O there is the inclusion A(O) ⊂ A (recall that the local algebras are assumed to be von Neumann algebras), but for spacelike cones in general the weak closure A(C ) is not contained in A. This implies that BF representations do not map A into A, as is the case in the DHR situation, but into some larger algebra. This situation is rather inconvenient, but fortunately this problem can be solved by introducing an auxiliary algebra [7]. The BF representations can be extended to proper endomorphisms of this auxiliary algebra. At the end of this section we comment on some other approaches. To motivate the introduction of the auxiliary algebra, consider a BF representation π and spacelike cone C . By the selection criterion (2.2) there is a unitary V such that π0 (A) = V π(A)V ∗ for all A ∈ A C . Consider the equivalent representation η(A) = V π(A)V ∗ , A ∈ A. It follows that η(A) = A for all A ∈ A C . By localisation and locality it follows that η(AB) = η(A)B = Bη(A) for all A ∈ A C and B ∈ A C , where C ⊃ C is a spacelike cone. Therefore, invoking Haag duality (1.1) for spacelike cones we have η A C ⊂ A C . 2 Buchholz and Fredenhagen do not demand that C is causally complete [7]. However, in view of our definition of Haag duality, it is more natural to consider only causally complete spacelike cones. See the Appendix to [18] for an alternative, but equivalent, definition.
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Definition 2.2. A representation η of A is a BF representation localised in C if it sat isfies the selection criterion (2.2) and η(A) = A for all A ∈ A C . This is denoted by η ∈ A BF (C ). From now on, fix a spacelike cone C . We will consider the category A BF (C ) of BF representations localised in C and intertwiners3 as morphisms. Note that the objects of the category are still transportable, i.e., if η ∈ A BF (C ) and if C is an arbitrary spacelike cone, there is a unitary equivalent representation (that might not be an object of A BF (C )) that is localised in C. This restriction to a fixed spacelike cone is for technical reasons only. As will be demonstrated below, for two spacelike cones C1 and C2 , the corresponding categories A BF (Ci ) are equivalent as braided tensor categories. In the remainder of this section, the structure of this category is described. The reader unfamiliar with these constructions is advised to keep in mind the category of finite-dimensional unitary representations of a compact group, which shares many of its features with the category of BF representations. There is, however, one notable difference: the representation category of a compact group is always symmetric, whereas the category of BF representations in d = 2 + 1 is interesting precisely because it is braided, but in general not symmetric. We now come to the construction of the auxiliary algebra. One starts by choosing an auxiliary spacelike cone Sa . This can be interpreted as a “forbidden” direction. From now on this auxiliary cone will be fixed. It should be noted that the results will not depend onthe specific choice of Sa . After fixing Sa we can consider the family of algebras A (Sa + x) , for x ∈ M3 . This set is partially ordered by x ≤ y ⇔ Sa + x ⊃ Sa + y and is directed, i.e., each pair of elements in this poset has an upper bound. Hence it is possible to consider the C ∗ -inductive limit (here the norm closure of the union of algebras) ASa =
A ((Sa + x) )
·
⊂ B(H0 ).
x∈M3
Clearly for every x ∈ M3 , we have ASa = ASa +x . The point is then that BF representations can be extended to endomorphisms of the auxiliary algebra. After the introduction of this auxiliary algebra, the structure of the superselection sectors can be studied with essentially the same methods as in the case of compactly localised (DHR) sectors, see e.g. [26,27]. For the convenience of the reader and to establish our notation, the main features and constructions are outlined below. The results are phrased in terms of tensor C ∗ -categories. See [17,36,38] for an overview of the relevant notions. Lemma 2.3. Let η be a BF representation. Then η has a unique extension ηSa to ASa that agrees with η on A and is weakly continuous on A (Sa + x) for each x ∈ M3 . If η is localised in C ⊂ (Sa + x) for some x ∈ M3 , then ηSa is an endomorphism of ASa . In the latter case we have η1Sa ◦ η2Sa = η2Sa ◦ η1Sa if the localisation regions of η1 and η2 are spacelike separated. Proof. We give a sketch of the proof; for the full proof see Lemma 4.1 and Proposition 4.3 of [7]. By the superselection criterion it is possible to find a unitary V in B(H0 ) 3 Recall that for two representations η and η of an algebra A, an intertwiner T from η to η is an operator 1 2 1 2 such that for all A ∈ A, T η1 (A) = η2 (A)T .
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such that η(A) = V AV ∗ for A ∈ A (Sa + x) . This representation can be extended uniquely to the weak closure A (Sa + x) . Obviously, this extension is weakly continuous. This leads to an extension ηSa of η. By Haag duality the localisation of η implies, in particular, that the unitaries V can be chosen in the auxiliary algebra, so that ηSa is an endomorphism of this auxiliary algebra. The final statement of the lemma can be checked for A ∈ A. We then invoke weak continuity to arrive at the desired conclusion. With this result, the analysis of the structure of the BF representations proceeds analogously to the DHR case: one just extends the representations to ASa as appropriate. In particular, it is possible to compose endomorphisms, which can be interpreted as composition of charges. Definition 2.4. Let ηi ∈ A BF (C ) (i = 1, 2), with C spacelike to Sa + x for some x. Define a tensor product on A BF (C )) by η1 ⊗ η2 = η1Sa ◦ η2 , and if Ti ∈ HomA(ηi , σi ) for i = 1, 2, by T1 ⊗ T2 = T1 η1Sa (T2 ) = σ1Sa (T2 )T1 . It can be shown that η1 ⊗η2 ∈ A BF (C ) and that η1 ⊗η2 is independent of the specific choice of auxiliary cone. Moreover if ηi ∼ η2 . See Sect. 4 of ηi , then η1 ⊗ η2 ∼ η1 ⊗ = = [7] for proofs. To proceed, an additional property is necessary, namely Borchers’ Property B for spacelike cones. Property B. Let E ∈ A C be a non-zero projection. Then, for any spacelike cone C ⊃ C , where the bar denotes closure in M3 , there is an isometry W ∈ A C such that W W ∗ = E. In fact, this property follows from the spectrum condition and locality [4], or [10] for a more recent exposition. Note that the assumption of weak additivity is not necessary, since this is automatically satisfied for algebras of observables localised in spacelike cones. Moreover, if the A(C ) are Type III factors Property B is satisfied automatically and one can even choose W ∈ A(C ) . Theorem 2.5. The category A BF (C ) has subobjects (notation: η1 ≺ η2 ), direct sums η1 ⊕ η2 , and can be endowed with a tensor product η1 ⊗ η2 . Proof. The first two properties can be derived using Property B. First, consider η ∈ a spacelike cone C ⊃ C . By PropA BF (C ) and a projection P ∈ EndA(η). Consider erty B there exists an isometry W ∈ A C such that P = W W ∗ . Define σ (−) = W ∗ η(−)W . Note that W ∈ HomA(σ, η). By duality and the localisation of η, it follows that σ is localised in C. Moreover, since η is localisable in cones it is easy to exhibit unitary charge transporters of σ , hence σ ∈ A BF (C ). By transportability it is possible to find a unitarily equivalent σ localised in C . It follows that σ ≺ η. For the existence of direct sums, consider η1 , η2 ∈ A . Using again Property B it BF is possible to find isometries V1 , V2 ∈ A C such that V1 V1∗ + V2 V2∗ = I (consider
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projections P = 0, I and I − P). Define η(−) = V1 η1 (−)V1∗ + V2 η2 (−)V2∗ . Then η is localised in C and localisable in cones. Using the same argument as above, an equivalent η localised in C can be found. This is the direct sum η = η1 ⊕ η2 , unique up to isomorphism. To see this, suppose η (−) = W1 η1 (−)W1∗ + W2 η2 (−)W2∗ . Then U := V1 W1∗ + V2 W2∗ is a unitary intertwiner from η to η . Similarly, it is not hard to see that if η ∼ = η , then η is a direct sum of η1 and η2 as well. The tensor product was already defined in Definition 2.4. With these definitions it is straightforward to verify that ⊗ defines a bifunctor on the category, and turns A BF (C ) into a strict monoidal category, with monoidal unit ι, given by the identity endomorphism of A. Now that a tensor product has been defined on the category A BF (C ), the next step is to look for a braiding. The braiding is intimately related to the statistics of a sector. It gives rise to representations of the braid group, or of the symmetric group if the braiding is symmetric, describing the interchange of identical particles. These notions were first studied in the context of algebraic quantum field theory by Doplicher, Haag and Roberts [13,14]. Braid statistics have been studied, for example, in [20]. The constructions below are essentially the same as in these original papers, and have merely been adapted to the case at hand. A convenient technical tool when dealing with BF representations is that of an interpolating sequence of spacelike cones. This can be used, e.g., to show that a certain construction is independent of the specific choice of spacelike cones, or to choose charge transporters in the auxiliary algebra. Definition 2.6. Let C1 and C2 be spacelike cones in Sa . An interpolating sequence between C1 and C2 , is a set of spacelike cones C1 , . . . Cn , each contained in (Sa + xi ) for some xi ∈ M3 , such that C1 = C1 , Cn = C2 , and for each i we have either Ci ⊂ Ci+1 or Ci+1 ⊂ Ci . With this definition it is possible to prove the following result: Lemma 2.7. Let η ∈ A BF (C1 ). For any spacelike cone C2 ⊂ Sa there is an equivalent representation η∼ = η localised in C2 , such that a unitary intertwiner V in ASa can be found.
Proof. Choose an interpolating sequence Ci between C1 and C2 . Set η1 = η. We then define a sequence of unitarily equivalent representations ηi+1 ∼ ηi+1 = ηi , such that Vi = ηi Vi . Since either Ci+1 ⊂ Ci or Ci ⊂ Ci+1 , it follows by Haag duality that either Vi ∈ A(Ci ) or Vi ∈ A(Ci+1 ) , hence Vi ∈ ASa . But then Vn−1 · · · V1 is a unitary intertwiner between η := ηn , and because ASa is an algebra, it follows that V := Vn−1 · · · V1 ∈ ASa . A braiding on the category relates the objects η1 ⊗ η2 and η2 ⊗ η1 . In this case it is a unitary operator εη1 ,η2 that intertwines the representations η1 ⊗ η2 and η2 ⊗ η1 . A particular example is the statistics operator εη,η that describes the statistics of a sector. To define the braiding εη1 ,η2 between η1 ⊗ η2 and η2 ⊗ η1 , with ηi ∈ A BF (C ), first choose two spacelike cones C1 and C2 . Both spacelike cones should lie in the causal complement of Sa + x for some x and should lie spacelike with respect to each other, i.e. C1 ⊂ C2 . By transportability there are BF-representations ηi ∼ = ηi localised in Ci . These morphisms are called spectator morphisms. Moreover, by Lemma 2.7 the corresponding
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unitary intertwiners V1 ∈ HomA(η1 , η1 ) and V2 can be chosen to be in ASa . After these choices have been made, one can define the braiding by εη1 ,η2 = (V2 ⊗ V1 )∗ ◦ (V1 ⊗ V2 ). It follows that εη1 ,η2 is a unitary in HomA(η1 ⊗ η2 , η2 ⊗ η1 ). A standard argument using interpolating sequences of spacelike cones shows that the definition of εη1 ,η2 is independent of the specific choice of intertwiners and localisation regions, up to the relative position of C1 and C2 , in the following sense. Definition 2.8. Suppose we have a spacelike cone C in the causal complement of Sa . If we rotate the spatial coordinates counter-clockwise, at some point it will fail to be spacelike to Sa . Now suppose we have two spacelike separated cones C1 and C2 . We define an orientation C1 < C2 if and only if we can move C1 by translation and rotating counter-clockwise to Sa while remaining in the spacelike complement of C2 . Note that for any two spacelike separated cones, there is always precisely one cone for which this is possible. We will always choose C2 < C1 to define the braiding εη1 ,η2 . One can then show that εη1 ,η2 is natural, in the categorical sense, in both the first and second variable. Moreover, εη1 ,η2 satisfies the braid relations. The verification becomes straightforward if one chooses the spacelike cones Ci in the definition in a convenient way, so as to be able to make use of the localisation properties of the endomorphisms. See [27] for the way this works in the DHR case. Theorem 2.9. The category A BF (C ) is a strict braided tensor category, where the braiding is given by εη1 ,η2 . The appearance of braid (but not symmetric) statistics is due to the fact that in 2+1 dimensions the manifold of spacelike directions is not simply connected, unlike the situation in higher dimensions. See Sect. 2 of [43] for a clarification of this point. Finally, there is the categorical notion of a conjugate object. In this setting, conjugates can be interpreted as “anti-particles”, and are closely related to the statistics of a sector. To each BF representation η a dimension d(η) and phase ωη are associated. For bosons (resp. fermions) the phase is +1 (resp. −1), but in A BF (C ) these are not the only possibilities (for d = 2 + 1). There are several ways to introduce these parameters. The traditional way is to introduce a left inverse [13,14]. Longo discovered a connection between the Jones index of an inclusion of factors and the dimension [34,35]. Finally, one can define the dimension, and twist (or phase), in a general categorical setting [36], see also [38]. The dimension d(η) takes values in [1, ∞]. If d(η) < ∞, one says that η has finite statistics. Restricted to objects of finite dimension, the dimension function satisfies the following identies: d (η) = d(η), d(η1 ⊗ η2 ) = d(η1 )d(η2 ), d(η1 ⊕ η2 ) = d(η1 ) + d(η2 ). Here η is a conjugate representation of η (see the Appendix). From now on, we will consider only categories where all objects have finite dimension, i.e., we leave out any sectors with infinite statistics the observable net may admit. Objects with finite dimension are precisely those for which there is a conjugate (or “anti-particle”). To avoid cumbersome notation, the category of all BF representations with finite statistical dimension will also be denoted by A BF (C ).
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Under weak additional assumptions, Guido and Longo showed that the DHR sectors with finite statistics are Poincaré covariant with positive energy [25], in particular they are covariant under translations as well. Hence under their assumptions, the set of finite DHR sectors coincides with the set of Poincaré covariant finite sectors with positive energy. Moreover, Buchholz and Fredenhagen show that massive irreducible single particle representations automatically have finite statistics [7]. They also show that all representations of interest for particle physics are indeed described by (direct sums of) representations with finite statistics. One may therefore argue that restricting to sectors of finite dimension is not too restrictive from the point of view of physics. Finally, we would like to mention that Mund recently proved a version of the spinstatistics theorem for massive particles obeying braid group statistics [43]. The restriction to sectors with finite statistics implies that the category A BF (C ) is semi-simple, i.e. that every representation can be decomposed into a direct sum of irreducibles. Indeed, let η ∈ A BF (C ). If η is not irreducible there is a non-trivial projection E ∈ EndA(η). By the existence of subobjects, one has η = η1 ⊕ η2 for some η 1 , η 2 ∈ A BF (C ). Semi-simplicity now follows, since d(η) = d(η1 ) + d(η2 ) and the dimension function d takes values in [1, ∞), since we restricted to objects of finite dimension. The results so far can be summarised by the following theorem. ∗ Theorem 2.10. The category A BF (C ) is a braided tensor C -category. That is it has duals (or conjugates), direct sums, subobjects, a braiding and a positive ∗-operation. The Hom-sets are Banach spaces, such that T ◦ S ≤ ST and S ∗ ◦ S = S2 for all morphisms S, T (whenever the composition is defined). Moreover, the tensor unit ι is irreducible: HomA(ι, ι) C.
It then follows automatically that the Hom-sets are finite-dimensional vector spaces [36]. In the case of interest here, the ∗-operation and norm are inherited from the observable algebra. One question that remains to be answered is to which extent the category A BF (C ) depends on the choice of C . It turns out that in fact for any two choices C1 , C2 the resulting categories are equivalent as tensor categories, cf. [18, Theorem 4.11]. Proposition 2.11. Let C1 and C2 be two spacelike cones. Then the categories A BF (C1 ) and A (C ) are equivalent as braided tensor categories. BF 2 Proof. We give a sketch of the proof; the details are left to the reader. One first proves the result in the case C1 ⊂ C2 . This gives rise to a full and faithful inclusion of categoA ries A BF (C1 ) ⊂ BF (C2 ). Clearly this inclusion is braided. In addition, the inclusion is essentially surjective, since for each representation localised in C2 one can find a unitary equivalent representation localised in C1 . Hence, the inclusion is in fact an equivalence of categories, hence an equivalence of braided tensor categories [53]. To prove the full result, one uses an argument with interpolating sequences of spacelike cones. Thus the BF representations form a braided tensor category. However, if there are DHR localised sectors, the braiding has a “trivial” part. Indeed, the DHR sectors form a symmetric subcategory of A BF (C ). But more importantly, the DHR sectors are degenerate objects with respect to the braiding. That is, they have trivial braiding with any object of A BF (C ), in a sense made precise below. In such a situation, one says that the category has a non-trivial centre [40].
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Fig. 1. This figure shows why the braiding is degenerate for compactly localised endomorphisms. The compactly localised (dashed lines) endomorphism can move from one side of the spacelike cone to the other, keeping it in the causal complement of the auxiliary cone (shaded region) and spacelike cone C (solid lines) at all times
Definition 2.12. The centre of a braided category is the full subcategory of degenerate objects. That is, it consists of all objects ρ such that ερ,η ◦ εη,ρ = Iη⊗ρ for all objects η. A non-trivial centre is an obstruction to modularity, since by a result of Rehren the existence of (non-trivial) degenerate sectors implies that the so-called S-matrix (in the sense of Verlinde [57]) is not invertible [47]. To make this situation more precise, we study the properties of the DHR sectors within A BF (C ). Definition 2.13. Let S be either a double cone or a spacelike cone. We write A DHR (S ) for the category of DHR localised sectors whose localisation region lies in S . Note that ρ ∈ DHR (C ) in particular is also an element of BF (C ), so the constructions in the first part of this section go through without change. For example, the tensor product of ρ1 and ρ2 in DHR (C ) is again in DHR (C ). Since objects from DHR (C ) can be localised in bounded regions of spacetime, one can say even more about them: A Lemma 2.14. Let η ∈ A BF (C ) and ρ ∈ DHR (O) for some double cone O ⊂ Sa . Then the DHR sectors are degenerate with respect to the braiding, i.e.,
ερ,η ◦ εη,ρ = Iη⊗ρ . Proof. The basic idea is depicted in Fig. 1. Because ρ is localised in a bounded region, there is more freedom in the choice of localisation cones of the spectator morphisms. In particular, it is possible to “flip” the cones, that is, if ρˆ is localised in some spacelike cone C, it is possible to find a spacelike cone Cpointing in the opposite direction, such that ρˆ is localised in C. Using this, it is not difficult to see that the braiding ερ,η does not depend on the orientation of the spacelike cones of the spectator morphisms. It follows −1 , which proves the result. that ερ,η = εη,ρ To conclude this section we briefly comment on other methods to describe the superselection structure of charges localised in spacelike cones. Doplicher and Roberts take
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a different approach in [18], which does not need the auxiliary algebra. This method, however, works only in spacetimes of dimension at least 4 and would need adaptation to the d = 2 + 1 case we are interested in. In the approach of both Buchholz & Fredenhagen and of Doplicher & Roberts, only representations localised in a fixed spacelike cone C can be considered. A related approach by Fröhlich and Gabbiani [24], which also uses the auxiliary algebra, does not require one to fix a spacelike cone. Instead, they consider two coordinate patches, and show that it is possible to pass from one to the other in a “smooth” way. Finally, it is possible to use the so-called universal algebra, introduced by Fredenhagen [19], see also [42]. This has the advantage that we do not have to choose an auxiliary cone. On the other hand, there are drawbacks, for example the universal algebra is not simple and the vacuum representation is not faithful [21]. In the end, each method gives the same result, so the choice of method only matters for the technical details. 3. The Field Net In this section we consider the field net of the observable algebras with respect to the DHR sectors. In other words, the field operators by construction only generate the DHR sectors. This is possible since the DHR sectors have permutation statistics in 2+1 dimensions. At the end of this section we discuss an alternative, more abstract construction of the field net, that turns out to be helpful in the applications we have in mind. For the convenience of the reader we first recall the definition of a field net [18]. We specialise to the case of interest here: that of a complete, normal field net without fermionic sectors. Definition 3.1. Let (π0 , H0 ) be a vacuum representation of the net O → A(O). A complete normal field net (π, G, F) is a representation (π, H) of A and a net O → F(O) of von Neumann algebras acting on H, such that (i) H0 ⊂ H; (ii) π0 is a subrepresentation of π ; (iii) there is a compact group G of unitaries on H leaving H0 pointwise fixed, inducing an action αg = Ad g; (iv) for each g ∈ G, αg is an automorphism of F(O) with fixed-point algebra π(A(O)); (v) the inductive limit F of the local algebras F(O) is irreducible; (vi) the Hilbert space H0 is cyclic for F(O); (vii) if O1 and O2 are spacelike separated double cones, F(O1 ) and F(O2 ) commute; (viii) every irreducible DHR representation with finite statistics is included as a subrepresentation of π . In the presence of fermionic sectors, item (vii) has to be modified to graded commutativity. Doplicher and Roberts show that such a field net exists and is unique up to a suitable notion of equivalence. The main point for us is that (at least in the purely bosonic case) this field net can be interpreted as an algebraic quantum field theory in its own right. The proof of this fact will be given below, after some preparatory results on harmonic analysis on the field net. Definition 3.2. Let ξ be a finite-dimensional continuous unitary representation of a group G as in Definition 3.1. A set of operators X 1 , . . . X d , where d = dim ξ , is said to
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be a multiplet transforming according to ξ if αg (X i ) =
d
ξ
u ji (g)X j ,
j=1 ξ
where u ji (g) are the matrix coefficients of ξ . An operator X is said to transform irreducibly according to ξ , or to be an irreducible tensor, if it is part of a multiplet transforming according to an irreducible representation ξ . Irreducible tensors can be obtained by averaging over the symmetry group G, and their span is weakly dense in the field algebra, see e.g. [15, Sect. 2]. Recall that for each irreducible DHR endomorphism ρ there is a Hilbert space Hρ in the field net transforming according to some irrep ξ of G. That is, Hρ is a closed linear subspace of F such that ψ1∗ ψ2 ∈ CI for all ψ1 , ψ2 ∈ Hρ . The space Hρ is precisely the set of operators ψ in F such that ψ A = ρ(A)ψ for all A ∈ A, and α Hρ = ξ . Moreover, there is a basis of Hρ that is a multiplet transforming according to ξ . Irreducible tensors may then be decomposed into a G-invariant part and an operator in Hρ , in the following sense: Lemma 3.3. Let B ⊂ B(H) be a ∗ -algebra, such that F(O) ⊂ B for some double cone O. Suppose that X transforms irreducibly under the action of G, that is, is contained in a finite dimensional Hilbert space transforming according to an irrep of G. Then there is a B ∈ B ∩ G and a ψ ∈ Hρ ⊂ F(O) such that X = Bψ, where ψ transforms according to the same irreducible representation as X . This decomposition is not unique, but depends on the specific choice of Hρ . Proof. Complete X to a multiplet X 1 , . . . X d . Without loss of generality, assume X = X 1 . Let ξ denote the representation according to which X transforms. Since the field net has full spectrum, there is a Hilbert space Hρ in F(O), such that Hρ transforms according to ξ . Note that the equivalence class of ρ corresponds to the class of the representation ξ ξ . If u ji are the matrix coefficients describing the transformation of the multiplet, it is ξ possible to choose an orthonormal basis ψi of Hρ such that αg (ψi ) = dj=1 u ji (g)ψ j . Now define B=
d
X i ψi∗ .
i=1
Since ξ is a unitary representation, it follows that αg (B) = B, i.e. B ∈ B∩G . Moreover, taking ψ = ψ1 , it follows that Bψ = X 1 = X . Now that we have the field net F at hand, it is possible to construct an auxiliary algebra with respect to F, analogous to the one defined in terms of the algebra of observables A. Hence we define · FSa = (F ((Sa + x) )) , x∈M3
where the closure in norm is taken.
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Since the observable net embeds into the field net, one expects the auxiliary algebra of the observable net to embed into the auxiliary algebra of the field net. The next lemma demonstrates that this is indeed the case. Lemma 3.4. Let (π, G, F) be a complete normal field net for (A, ω0 ). Then the representation (π, H) of A can be uniquely extended to a faithful representation π Sa : S A a → B(H) that is weakly continuous on A (Sa + x) . for the set of equivalence classes of irreducible representations of the Proof. Write G group G. The representation (π, H), viewed as a representation of A, is a direct sum ⊕ξ ∈Gˆ dξ πξ , where each πξ is a DHR representation [18]. We will extend each πξ to a
representation πξSa of ASa , and set π Sa = ⊕ξ ∈Gˆ dξ πξSa . So consider such a representation πξ . By Lemma 2.3, πξ has a unique weakly continuous extension. In fact, since πξ is localised in a bounded region, it follows in particular that πξSa is an endomorphism of ASa , viewed as a subalgebra of B(H). To see that π Sa is faithful, construct a left inverse ϕ of π Sa , as in [7].
This result makes it possible to identify ASa with the subalgebra π Sa ASa of B(H). When there is no risk of confusion, we will sometimes identify A ∈ ASa with its image π Sa (A). It is fruitful to investigate the relationship between the auxiliary algebra and the action of the symmetry group. Just as the observable net consists of precisely those operators that are fixed by the G-action on the field net, the same is true for the auxiliary algebras.
Lemma 3.5. Let (π, H, F, G) be a normal field net. Then: (i) For each spacelike cone, F(C ) ∩ G = π(A C ) . G
= π Sa ASa . (ii) The fixpoint algebra is given by FSa = G and A C is a subalgebra of A, it is obvious Proof. (i) First of all, since π(A) that π(A C ) ⊆ G . From relative locality, π(A C ) ⊆ F(C ) . By taking double commutants, π(A C ) ⊆ F(C ) . Note that for each double cone O, H0 is cyclic for F(O), hence also for F(C ). This implies that an element T ∈ F(C ) ∩ G is uniquely determined by its restriction to H0 . Furthermore, H0 is an invariant subspace for T , since T ∈ G . We have F(C ) ∩ G ⊆ π(A(C )) , so if E 0 denotes the projection onto H0 ⊂ H, it follows that T |H0 ∈ π(A(C )) E 0 = π0 (A(C )) = π0 A C . The last step follows by Haag duality for spacelike cones in the vacuum representation. (ii) Note that αg extends to B(H), where H is the Hilbert space on which F acts irreducibly. Using the Haar measure of G, one can define a conditional expectation E : F → A by E(A) = αg (A)dg. G
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It then follows that ⎛ ⎞·
· E FSa = E ⎝ F(Sa + x) ⎠ = E (F(Sa + x) ) , x∈M3
x∈M3
where [12]. Now by part (i) it follows that we usedthat E is weak- and norm-continuous E F(Sa + x) = π Sa A(Sa + x) , see also [12, Lemma 3.2]. Therefore,
· π Sa (A(Sa + x) ) = π Sa ASa , E FSa = x
which proves the claim.
With the aid of these lemmas it is possible to prove the main result of this section: without fermionic sectors, the field net can be interpreted as an AQFT in its own right, but one without non-trivial DHR sectors. Theorem 3.6. Assume that O → A(O) satisfies the following conditions: (i) there are at most countably many DHR sectors; (ii) there are no fermionic DHR sectors; (iii) each DHR sector with finite statistics is covariant under translations satisfying the spectrum condition. Then the field net O → F(O) satisfies the axioms of an algebraic QFT, i.e. it is a local, translation covariant net satisfying Haag duality and the spectrum condition, hence it also has Property B for spacelike cones. The complete normal field net admits only the trivial DHR representation. Proof. Isotony follows, since the field net is, in particular, a net. Since we assumed the absence of fermionic sectors, twisted duality for the field net reduces to Haag duality for double cones. Thus only the questions of translation covariance and duality for spacelike cones remain. The covariance properties follow from the results in Sect. 6 of [18], and the assumption that we only have translation covariant sectors. In fact, one can show in this case that the representation π of F is translation covariant. The generators of translations again satisfy the spectrum condition and the vacuum vector is invariant under the action of the translation group [18, Sect. 6]. By the same reasoning as before, Property B follows. Toprove duality for spacelike cones, consider such a cone C . First, note that by locality F C ⊂ F(C ) . Let F ∈ F(C ) transform irreducibly under the action of G. But then by Lemma 3.3, F = Bψ, where B ∈ F(C ) ∩G and ψ ∈ Hρ . Applying Lemma 3.5 gives B ∈ π(A C ) and, since Hρ ⊂ F C , one obtains F ∈ F C . The irreduc ible tensors form a dense subset, which allows us to conclude F C = F(C ) . Taking commutants then proves Haag duality. For the last assertion, note that the observable net is embedded in the field net. More precisely, we have an inclusion of subsystems A ⊂ F. By [9, Theorem 4.7], every DHR representation of the field net F with finite statistics is a direct sum of representations with finite statistics. Moreover, these sectors are labelled by the equivalence classes of irreducible representations of a compact group L, such that F(A) L = B (see also [9, Theorem 4.1]). But in this case, B = F(A) = F, hence L is the trivial group and the only irreducible DHR sector is the vacuum sector.
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Let us briefly comment on the assumptions of Theorem 3.6. The first condition is a technical one, needed for the results in [9] and Corollary 5.4 below. By construction of the set the field net, DHR sectors are in 1-1 correspondence with irreps of G, hence G, of irreps of G, is also countable. The second condition implies that the field net satisfies ordinary locality, as opposed to twisted locality. The final condition is needed to lift the translation covariance of A to the field net. As mentioned before, by weak additional assumptions on A, it follows automatically that every DHR sector with finite statistics is translation covariant. Therefore, the conditions appear not to be unreasonably restrictive. From now on, we will assume that A satisfies all assumptions in the theorem. Roughly speaking, Doplicher and Roberts construct the field net as a crossed product of the observable algebras by a semigroup of endomorphisms. As mentioned before, this construction is intimately related to the theory of representations of compact groups. It is therefore not surprising that an alternative construction, based on results on the category of representations of compact groups, exists. Indeed, based on an unpublished manuscript of Roberts and on Deligne’s embedding theorem [11], Halvorson and Müger describe such a construction [27,38], which is of a more algebraic nature compared to the original analytic approach. Since the algebraic formulation is easier to work with in the present case, the rest of this section will be used to outline the main features of this approach and to fix the notation. The results in Sect. 2 state that the DHR representations form a symmetric tensor (C ∗ )-category. By Deligne’s embedding theorem, this gives rise to a faithful symmetric tensor ∗ -functor E : A DHR → SH f , the category of finite-dimensional (super) Hilbert spaces. The embedding theorem also gives a compact supergroup (G, k) of natural monoidal transformations of E, and an equivalence of categories such that A DHR is equivalent to Rep f (G, k). All monoidal categories and functors are assumed to be strict, unless noted otherwise. The “super” structure gives a Z2 -grading on the Hilbert spaces, corresponding to the action of a central element k ∈ G such that k 2 = e. Since we assumed that all DHR sectors are bosonic, we can forget about the super structure. The group G from the embedding theorem will be the symmetry group. The embedding functor E associates to each DHR endomorphism ρ a Hilbert space E(ρ). Using this embedding functor E, we first construct a field algebra F0 . We cite the definition: Definition 3.7. The field algebra F0 consists of triples (A, ρ, ψ), where A ∈ A, ρ ∈ DHR , and ψ ∈ E(ρ), modulo the equivalence relation (AT, ρ, ψ) ≡ A, ρ , E(T )ψ , for T an intertwiner from ρ to ρ . For λ ∈ C, we have E(λ idρ ) = λ id E(ρ) , hence (λA, ρ, ψ) = (A, ρ, λψ). In particular, it follows that any element with ψ = 0, is the zero element of the algebra. One then proceeds by defining a complex-linear structure on this algebra, a multiplication, as well as an involutive ∗ -operation. The multiplication is defined by (A1 , ρ1 , ψ1 )(A2 , ρ2 , ψ2 ) = (A1 ρ1 (A2 ), ρ1 ⊗ ρ2 , ψ1 ⊗ ψ2 ). The definition of the ∗-operation is a bit more involved. First, if H and H are two Hilbert spaces and S : H ⊗ H → C is a bounded linear map, one can define an anti- linear map J S : H → H . This map is defined by setting (J S)ψ, ψ = S ψ ⊗ ψ for all ψ ∈ H, ψ ∈ H . The brackets denote the inner product on H . If ρ is a DHR endomorphism, choose a conjugate (see the Appendix) ρ, R, R . The ∗-operation is
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∗ then defined by (A, ρ, ψ)∗ = R ∗ ρ(A)∗ , ρ, J E R ψ . For a verification that this is well-defined and indeed defines a ∗-algebra, see [27]. Note that this construction is purely algebraic, for instance, there is no norm defined on F0 . The algebra A can be identified with the subalgebra {(A, ι, 1) : A ∈ A} of F0 , and E(ρ) can be identified with the subspace {(I, ρ, ψ) : ψ ∈ E(ρ)}.4 The compact group G associated with the embedding functor E gives rise to an action on F0 . Recall that the elements of G are monoidal natural transformations of the functor E. If g ∈ G, write gρ for the component at ρ. The action of G on F0 is then defined by αg (A, ρ, ψ) = (A, ρ, gρ ψ),
A ∈ A, ψ ∈ E(ρ).
This is in fact a group isomorphism g → αg into Aut A(F0 ), the group of automorphisms of F0 that leave A pointwise fixed. Finally, for a double cone O, it is possible to define the local ∗ -subalgebra F0 (O) of F0 , consisting of elements (A, ρ, ψ), with A ∈ A(O), ψ ∈ E(ρ), and ρ localized in O. To construct the field net, a faithful, G-invariant positive linear projection (in fact, a conditional expectation) m : F0 → A is defined. If ω0 is the vacuum state of A, the GNS construction on the state ω0 ◦ m is used to create a representation (π, H) of F0 . The local algebras are then defined by F(O) = π(F0 (O)) . As usual, the algebra F is defined to be the norm closure of the union of all local algebras. Since m is G-invariant, the action of αg is implemented on H by unitaries U (g). In other words, π(αg (F)) = U (g)π(F)U (g)∗ for g ∈ G and F ∈ F0 . This action can be extended to F in an obvious way. With these definitions, (π, G, F) is a complete normal field net for (A, ω0 ) with local commutation relations. In fact, any complete normal field net for A is equivalent to the field net constructed here. The final technical lemma concerns field operators. In the field net there are field operators, which can be interpreted as operators creating the DHR charges from the vacuum state. That is, for a DHR endomorphism ρ there are ∈ F such that ρ(A) = A, with A ∈ A. It is convenient in calculations to know how this works on the auxiliary algebras. Lemma 3.8. Let ρ be an endomorphism of A localised in a double cone O, and take ψ ∈ E(ρ). Then
π Sa ρ Sa (A) π(I, ρ, ψ) = π(I, ρ, ψ)π Sa (A), (3.1) for all A ∈ ASa . Proof. Note that for A ∈ A, the equality holds basically by construction of the field . Then there is a net (in the sense of topology) + x) net. Now suppose A ∈ A (S a Aλ → A in A (Sa + x) that converges weakly to A. Equation (3.1) holds for Aλ by the previous remark. The result now follows by weak continuity of the extensions and of separate weak continuity of multiplication. 4. Extension to the Field Net Our next goal is to understand the BF-superselection structure of F, including the way it is related to that of A. Now that we have established how the auxiliary algebra is included 4 These Hilbert spaces E(ρ) play the same role as the Hilbert spaces H in [18]. ρ
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in the field net, a natural question is how BF representations of A can be extended to BF representations of F. This section is devoted to this problem. At the end of the section we comment on alternative approaches. A If η ∈ F BF (C ) is an extension of η ∈ BF (C ), it follows that η(A) = αg ◦ η(A) = η(A) = η ◦ αg (A) αg ◦ for all A ∈ A. The next theorem gives a characterisation of extensions such that αg ◦ η(F) = η ◦ αg (F) for all F ∈ π(F0 ). Such extensions are in 1-1 correspondence with certain families of unitaries Wρ (η) in ASa . A proof of this result for extensions of automorphisms was given in [16, Thm. 8.2]. Later, the result of Doplicher and Roberts was adapted to endomorphisms [39]. The explicit description of the field net allows us to verify this construction, without invoking e.g. universality properties as in the original proof. The first step is to show that we can define an extension on the subalgebra π(F0 ) of F. We will then extend this to the algebra F. Proposition 4.1. Let η be a representation of A. Then representations η of π(F0 ) that extend η and commute with αg are in one-to-one correspondence with mappings A Sa satisfying (ρ, η) → Wρ (η) from A DHR × BF (C ) to unitaries in A Wρ (η) ∈ HomA(ρ ⊗ η, η ⊗ ρ), Wρ (η)(T ⊗ Iη ) = (Iη ⊗ T )Wρ (η), T ∈ HomA ρ, ρ , Wρ⊗ρ (η) = (Wρ (η) ⊗ Iρ )(Iρ ⊗ Wρ (η)), Wρ η ⊗ η = Iη ⊗ Wρ η (Wρ (η) ⊗ Iη ).
(4.1) (4.2) (4.3) (4.4)
The extension is determined by η(π(A, ρ, ψ)) = π Sa (ηSa (A)Wρ (η))π(I, ρ, ψ). (4.5) Moreover, if S ∈ HomA η, η satisfies SWρ (η) = Wρ η ρ Sa (S) for all ρ ∈ A DHR ˆ ηˆ . (that is, Wρ (η) is natural in η), then π Sa (S) ∈ HomF0 η,
Proof. To avoid cumbersome notation, π Sa ASa will be identified with ASa in the proof. First, assume η is a representation of F that commutes with the G-action. Lemma 3.5 implies that η restricts to a representation of ASa , which we will denote by η. For ρ ∈ A DHR , write i = π(I, ρ, ψi ), where ψi is an orthonormal basis of E(ρ). Define Wρ (η) =
d
η(i )i∗ .
i=1
This definition is independent of the chosen basis of E(ρ). The i generate a Hilbert space with support I , [27, Prop. 270], from which it follows that Wρ (η) is unitary. The Hilbert space E(ρ) transforms according to some irreducible representation. Since η commutes with the G-action, it is easy to verify that αg (Wρ (η)) = Wρ (η). By Lemma 3.5(ii), Wρ (η) is a unitary in ASa . Note that Wι (η) = I , since η is unital. Also note that for ψ ∈ E(ρ), it follows that Wρ (η)π(I, ρ, ψ) = η(π(I, ρ, ψ)). Because
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(4.5) is in particular a ∗-endomorphism (see below for a verification) and F0 is generated by elements of this form, we see that η can indeed be defined as in (4.5). It remains to verify properties (4.1)–(4.4). The verification of these properties is quite straightforward. We give a proof of (4.2) and leave the rest to the reader. So, let T ∈ HomA ρ, ρ . Note that T ∈ A by Haag duality for double cones. Then
η(π(T, ρ, ψi ))π(I, ρ, ψi )∗ =
i
η(π(I, ρ , E(T )ψi ))π(I, ρ, ψi )∗
i
=
π Sa (Wρ (η))π(I, ρ , E(T )ψi )π(I, ρ, ψi )∗
i
= π Sa (Wρ (η))π(T, ι, 1). This is Eq. (4.2). In the second line Eq. (4.5) has been used. As for the converse, we have to show that Eq. (4.5) indeed defines a ∗-representation of π(F0 ) that extends η. For (A, ρ, ψ) ∈ F0 , define η(π(A, ˆ ρ, ψ)) as in Eq. (4.5). Note that (4.3) together with the unitarity of Wι (η) imply that Wι (η) = I . Considering the embedding of A into F0 (by A → (A, ι, 1)), it follows that η(π(A, ˆ ι, 1)) = π Sa (η(A)). This shows that we can view ηˆ as an extension of η. To check that η is well-defined, suppose (AT, ρ, ψ) = (A, ρ , E(T )ψ), with T intertwining ρ and ρ . A simple computation, using π Sa (T ) = π(T ), and the fact that π is well-defined, shows that well-definedness of ηˆ boils down to the identity η(A)Wρ (η)T = η(AT )Wρ (η), which in turn is easily verified using the properties of Wρ (η). η is multiplicative, consider F = (A, ρ, ψ) and F = In order to show that A , ρ , ψ as elements of F0 . Then: ηˆ π(F)π F = ηˆ π Aρ A , ρ ⊗ ρ , ψ ⊗ ψ = π Sa η Aρ A Wρ⊗ρ (η)π I, ρ ⊗ ρ , ψ ⊗ ψ .
(4.6)
On the other hand, η(π(F)) ˆ ηˆ π F = π Sa (η(A)Wρ (η))π(I, ρ, ψ)π Sa η A Wρ (η) π I, ρ , ψ . An application of Lemma 3.8 reduces the right hand side to
π Sa η(A)Wρ (η)ρ Sa η A Wρ (η) π I, ρ ⊗ ρ , ψ ⊗ ψ . Then one should note that Wρ (η) intertwines ρ Sa ◦ η and ηSa ◦ ρ, and use the fact that ρ is an endomorphism of A, so that ηSa ρ A = η ρ A . By using (4.3), one then obtains Eq. (4.6), so η preserves multiplication. To check that ηˆ is a ∗ -homomorphism, we have to show η (π(F)∗ ) = η(π(F))∗ . Since ηˆ preserves multiplication, it is enough to show this for (A, ι, 1) and (I, ρ, ψ) ∈ F0 . The first case is easy: ηˆ π(A, ι, 1)∗ = ηˆ π A∗ , ι, 1 = π Sa η A∗ π(I, ι, 1) = π Sa (η(A))∗ ,
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since η and π Sa are ∗ -homomorphisms. To check the remaining case, let ρ, R, R be a conjugate. Then, R ∗ ∈ HomA (ρ ⊗ ρ, ι), so we have ∗ η R ∗ Wρ (η) = Wι (η)R ∗ Wρ⊗ρ (η)∗ Wρ (η) = R ∗ Wρ (η)∗ Wρ⊗ρ (η) = R ∗ ρ Sa Wρ (η)∗ ,
(4.7)
where the properties of Wρ (η) have been used in each step. Recall the anti-linear map J used in the definition of the ∗-operation on F0 . Then, by definition of η,
∗ ηˆ π(I, ρ, ψ)∗ = ηˆ π R ∗ , ρ, J E R ψ
∗ = π Sa η R ∗ Wρ (η) π I, ρ, J E R ψ . Substitute Eq. (4.7) and apply Lemma 3.8. Together with the fact that π Sa agrees with π on A, this gives
∗
η π(I, ρ, ψ)∗ = π Sa R ∗ ρ S Wρ (η)∗ π I, ρ, J E R ψ
∗ = π R ∗ , ι, 1 π I, ρ, J E R ψ π Sa Wρ (η)∗ = π(I, ρ, ψ)∗ π Sa (Wρ (η))∗ = η(π(1, ρ, ψ))∗ , which concludes the proof that η is a representation. To prove that η commutes with the G-action, consider (A, ρ, ψ) ∈ F0 , and let g ∈ G. Then η(π(A, ρ, gρ ψ)) = π Sa (η(A)Wρ (η)π(I, ρ, gρ ψ). η(αg π(A, ρ, ψ)) = On the other hand, αg is implemented by U (g), so we have η(π(A, ρ, gρ )) = U (g)π Sa (η(A)Wρ (η))π(I, ρ, ψ)U (g)∗ αg ◦ = U (g)π Sa (η(A)Wρ (η))U (g)∗ π(I, ρ, gρ ψ). η commutes with the From this it follows that if π Sa (η(A)Wρ (η)) is G-invariant, then S a action of G. Since η(A)Wρ (η) ∈ A this is nothing but Lemma 3.5(ii). Finally, let S ∈ HomA η, η be an intertwiner, and F = (A, ρ, ψ) ∈ F0 . Then η(π(F)) = π Sa (Sη(A)Wρ (η))π(1, ρψ) π Sa (S) = π Sa η (A)SWρ (η) π(I, ρ, ψ)
= π Sa η (A)Wρ η ρ Sa (S) π(I, ρ, ψ) = η (π(F))π Sa (S), Sa where in the last line Lemma 3.8 has been used. Hence we see that π (S) ∈ HomF0 η, η , completing the proof.
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It should be noted that conditions (4.1)–(4.4) are very similar to the conditions on a braiding, in particular the braiding ερ,η satisfies these conditions. The only difference is that Wρ (η) need only be defined for ρ a DHR endomorphism and η a BF endomorphism. The construction above gives an extension of representations of A to F. To verify if these extensions are BF representations one should look at the localisation properties of the extension. The next lemma gives a necessary and sufficient condition for the extension of a localised representation to be cone localised again. Lemma 4.2. Consider the notation and assumptions of Proposition 4.1. If η is localised in C , its extension η is localised in C if and only if Wρ (η) = I for each ρ ∈ A DHR localised spacelike to C . Here, η is called localised in C if it acts trivially on all F ∈ π(F0 (O)) for O ⊂ C . Proof. The localisation properties follow from the localisation of η. If F ∈ F0 (O) for some double cone O ⊂ C , it is of the form F = (A, ρ, ψ), with A ∈ A(O) and ρ localised in O. But η acts trivially on such A, and Wρ (η) = I . Hence η(π(A, ρ, ψ)) = π(A, ρ, ψ). For the converse, suppose that ρ ∈ A DHR is localised spacelike to C . Choose an orthonormal basis ψi of E(ρ). Then π(I, ρ, ψi ) ∈ π(F0 (O)) for O ⊂ C . Hence η(π(I, ρ, ψi )) = π Sa (Wρ (η))π(I, ρ, ψi ) = π(I, ρ, ψi ). We multiply on the right by π(I, ρ, ψi )∗ and sum over i. Since E(ρ) has support I , it follows that π Sa (Wρ (η)) is the identity. As a consequence of these results, we can canonically extend BF representations of A to BF representations of F. This way of extending representations was first pointed out by Rehren [48], where the author sketches a proof in the case of compactly localised sectors. Theorem 4.3. Every BF representation η of A can be extended to a BF representation of F that commutes with the G-action. This extension is unique. Proof. One readily verifies that Wρ (η) = ερ,η has the properties required in Proposition 4.1. Moreover, Wρ (η) = I if ρ is localised spacelike to η. Hence there is a ∗ -representation η of π(F0 ) extending η. If η is localised in C , Lemma 4.2 shows that η is localised in the same region. If C is another spacelike cone, by transportability of η there is a unitarily equivalent η localised in C. By Proposition 4.1, this lifts to a unitary equivalence of η and η , since the condition stated on S is nothing but naturality of ερ,η in η. This shows transportability of the extension. We now have a representation defined on the algebra π(F0 ). To extend this representation to F, we first show that it can be extended to the local algebras F(O) = π(F0 (O)) . Consider a double cone O. If O is spacelike to C , localisation implies η(π(F)) = π(F) for all π(F) ∈ π(F0 (O)). In this case it is clear that this extends to the weak closure F(O). Now suppose O is not spacelike to C . Then by the argument above, there is a unitary V such that η(π(F)) = V ∗ η(π(F))V which is localised spacelike to O. In other words, η(π(F)) = V π(F)V ∗ , by localisation of η. The right hand side is weakly continuous, hence we can extend η to F(O) for every O. But the argument also shows that η is in fact an isometry, since V π(F)V ∗ = π(F). The union of the local algebras is norm dense in F, hence by continuity η extends uniquely to a representation of F. Finally, we show that the extension is unique. Suppose that we have another localised extension that commutes with the action of G. Proposition 4.1 then asserts the existence
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of a family Wρ (η). We show Wρ (η) = ερ,η . First of all, suppose ρ ∈ A DHR is localised spacelike to the localisation of η. Then, by Lemma 4.2, Wρ (η) = I . But this is equal to ερ,η , since ρ is degenerate. Now consider an arbitrary ρ ∈ A DHR . Choose a unitary equivalent ρ localised spacelike to the localisation of η, with corresponding unitary T . Then, (T ⊗ Iη ) = (Iη ⊗ T )Wρ (η), (T ⊗ Iη ) = (Iη ⊗ T )ερ,η , where the first equation follows from (4.2), and the second follows from naturality with respect to ρ of the braiding. Since T is a unitary, it follows that Wρ (η) = ερ,η . Remark 4.4. (i) Localisation properties are used to show that η can be extended to a representation of F. By applying the results of [16], as in [39], it can be proved that in fact every extension (whether it is cone localised or not) as in Proposition 4.1 can be defined on the whole of F. (ii) Denote the canonical extension by (η) or η. It turns out that : η → η is in fact a faithful, but not full, tensor functor. These and other categorical aspects are discussed in Sect. 6. Let us briefly comment on other approaches to the problem of extending representations. Firstly one could use techniques from the theory of subfactors. For this to work A(C ) ⊂ F(C ) needs to be an inclusion of factors. Moreover, the Jones index of this inclusion should be finite. In this case the machinery of α-induction and σ -restriction can be applied [3]. In the present situation, however, it is not clear if these requirements are satisfied. Another approach that can be used in the DHR setting is Roberts’ theory of localised cocycles [51,52], see also [9]. It is not immediately clear, however, if this can be modified to apply to the case of BF sectors. For one, the set of all double cones is directed, unlike the set of all spacelike cones. 5. Non-abelian Cohomology and Restriction to the Observable Algebra In the previous section, extension of BF representations of the observable algebra to the field algebra was discussed. Here we investigate the other direction: does every BF representation of the field algebra that commutes with the group action come from such an extension? This is a first step in understanding the category F BF (C ). In answering this question, one encounters problems of a cohomological nature in a natural way. For convenience of the reader we recall the notion of an α-1-cocycle and an α-2-cocycle in a von Neumann algebra M; for the complete definition see [54]. A Borel map v : G → U(M) is an α-1-cocycle if it satisfies the identity v(gh) = αg (v(h))v(g); a map w : G × G → U(M) is an α-2-cocycle if w(gh, k)w(g, h) = w(g, hk)αg (w(h, k)). It is possible to define a coboundary map ∂. For example, a 1-cocycle v(g) is a coboundary if there is a unitary w ∈ M such that v(g) = αg (w)w ∗ . A 2-cocycle w(g, h) is a coboundary if there is a Borel map ψ : G → U(M) such that w(g, h) = αg (ψ(h))ψ(g)ψ(gh)∗ .
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It turns out that each cocycle taking values in F(C ) is in fact a coboundary in a bigger algebra F C ⊃ F(C ) . This is essentially due to the field net having full G-spectrum, which allows to use the construction of Sutherland to construct a coboundary [54]. Before proving this result, we first recall some notions regarding Hilbert spaces in von Neumann algebras [50]. Definition 5.1. Let M be a von Neumann algebra. A Hilbert space in M is a norm closed linear subspace H , such that a ∈ H implies a ∗ a ∈ CI and x ∈ M, xa = 0 for all a ∈ H implies x = 0. An inner product is then defined by (a, b)I = a ∗ b. One can check that this indeed defines a Hilbert space. If {Vi }i∈J is an orthonormal basis for H Vi Vi∗ are , the operators ∗ (mutually orthogonal) projections, the Vi are isometries, and i∈J Vi Vi = I . Certain operators x ∈ M can be identified with operators in B(H ). More generally, if H1 and H2 are two Hilbert spaces in M, write (H1 , H2 ) = x ∈ M : ψ2∗ xψ1 ∈ CI, ψ1 ∈ H1 , ψ2 ∈ H2 . These operators are in 1-1 correspondence with operators in B(H1 , H2 ), see [50, Lemma 2.3]. For x ∈ (H1 , H2 ), write L(x) for the corresponding linear operator in B(H1 , H2 ). In this case, (ψ1 , L(x)ψ2 )I = ψ1∗ xψ2 . With these preparations we can prove the triviality of cocycles. Theorem 5.2. Assume G is second countable. Let v(g1 , . . . , gn ) be a unitary α-ncocycle in F(C ) . Then there is a spacelike cone C ⊃ C such that v is a coboun dary in F C . Proof. Pick a double cone O ⊂ C , such that there is a spacelike cone C ⊃ C ∪O. Note that this is always possible. Since the field net has full spectrum, for each irreducible representation ξ of G, there is a Hilbert space in F(O), transforming according to this representation. That is, there are isometries ψi , i = 1, . . . , d spanning a Hilbert space Hξ in F(O), such that αg (ψi ) =
d
ξ
u ji (g)ψ j ,
j=1 ξ
where u ji (g) are the matrix coefficients of ξ . The left regular action λ(g) on L 2 (G) decomposes as a direct sum of irreducible representations. By the Peter-Weyl theorem the Hilbert space L 2 (G) decomposes as [28] L 2 (G) = dξ Hξ , (5.1) ξ ∈G
where dξ is the dimension of the representation ξ . For each irreducible representation ξ , the algebra F(O) contains a Hilbert space Hξ (as in Definition 5.1), transforming according to the corresponding representation. The group G is second countable, hence the number of irreducible representations is at most countable [28]. Since A(O) is a properly infinite von Neumann algebra acting on a separable Hilbert space, it is possible to find a countable family of isometries Vi such that Vi∗ V j = δi, j I and i Vi Vi∗ = I .
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Moreover, they are invariant under the action of G. These isometries enable us to construct an image of the direct sum decomposition (5.1) of L 2 (G) in F(O) as follows. counted with multiplicities. For each i choose an First choose an enumeration ξi of G, orthonormal basis ψ j of Hξi where j = 1, . . . , dξi . Then ei j = Vi ψ j Vi∗ forms an orthonormal basis of a Hilbert space in F(O). This Hilbert space will be denoted by L 2F(G). If T : L 2F(G) → L 2 (G) denotes the corresponding isomorphism of Hilbert spaces, the above remarks imply that T (αg (ψ)) = λ(g)T (ψ) for all ψ ∈ L 2F(G).
Note that the action αg induces an action on B L 2F(G) . To see what effect this has on the corresponding operators in B L 2 (G) , consider the following calculation, where −, − is the inner product of L 2 (G), x ∈ B L 2F(G) , and g ∈ G: T (ψ1 ), L(x)T (ψ2 )I = ψ1∗ xψ2 = αg ψ1∗ αg (x)αg (ψ2 ) = (αg (ψ1 ), L(αg (x))αg (ψ2 ))I = λ(g)T (ψ1 ), L(αg (x))λ(g)T (ψ2 )I = T (ψ1 ), λ(g)∗ L(αg (x))λ(g)T (ψ2 )I. In other words, L(αg (x)) = λ(g)L(x)λ(g)∗ = Ad λ(g)L(x), since the left regular representation is unitary. The situation canbe summarised as follows: there is a copy of L 2 (G) in F(O), as well as a copy of B L 2 (G) . Moreover, the action αg of G acts as Ad λ(g) on these operators. We are now in a position to apply Proposition 2.5.1 from [54]. Define an injective representation π : F(C ) ⊗ B L 2 (G) → F C by π(x ⊗ y) = x F −1 (y). Note that this is indeeda representation, since F(C ) commutes with 2 F(O). Endow the algebra F(C ) ⊗ B L (G) with the action βg of G defined by βg = αg ⊗ Ad λ(g). It follows that for each g ∈ G, π(βg (x ⊗ y)) = αg (π(x ⊗ y)). By Proposition 2.1.5 of [54] v(g1 , . . . gn ) ⊗ I is a β-coboundary. But since v(g1 , . . . gn ) = π(v(g1 , . . . gn )⊗ I ) and αg ◦π = π ◦βg , it follows that v(g1 , . . . gn ) is an α-coboundary in F C . Remark 5.3. The DHR sectors of A are in one-to-one correspondence with irreducible representations of the group G. Hence under the assumption already made in Theorem 3.6, it follows that G is indeed second countable. With this theorem we are able to prove the main result of this section, namely that every BF representation of F that commutes with the G-action comes from the extension of a representation of A. Corollary 5.4. Let η ∈ F BF (C ), such that αg ◦η = η ◦αg for all g ∈ G. Then η restricts S ASa = η. to a BF sector η A a of the observable net. Moreover, η Proof. Since the representation η commutes with the action of G, by Lemma 3.5(ii) it restricts to an endomorphism of ASa . It is clear that this restriction is localised in C as well. To prove transportability, proceed in a similar way as in [41, Prop. 3.5]. Suppose C is another spacelike cone. For simplicity we assume it is spacelike to Sa . In the general case, one has to apply an argument as in the proof of Proposition 2.11. Pick a spacelike cone C ⊂ C such that there is a double cone C ⊃ O ⊂ C . By Lemma 2.7
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and transportability, there is a unitary V ∈ FSa such that η = Ad V ◦ η is localised in C. Now consider g η := αg ◦ η ◦ αg−1 . Since η is G-invariant, αg (V ) ∈ HomF(η, g η). Because αg leaves F C globally invariant, g η is also localised in C. Define an η). By Haag duality, v(g) ∈ F C . Moreη, g α-1-cocycle v(g) = αg (V )V ∗ ∈ HomF ( over g → v(g) is strongly continuous. By Theorem 5.2 there is a unitary W ∈ F C such that v(g) = αg (W )W ∗ . Define η := Ad W ∗ ◦ η. It is easy to see that η is localised ∗ ∗ η). Moreover, by definition αg (V )V = αg (W )W ∗ , in C and that W V ∈ HomF (η, ∗ from which it follows that αg (W V ) = W ∗ V for all g ∈ G. Hence W ∗ V is in ASa , and η ASa . is the desired intertwiner from η ASa to Since extensions commuting with G are unique by Theorem 4.3, the last statement is obvious. 6. Categorical Crossed Products The results in the previous section give a complete understanding of all G-invariant BF representations of F BF (C ). Indeed, these are all of the form (η) for some BF representation η of A. Recall that this extension functor is defined by (η) = η, and by (S) = π Sa (S) for intertwiners S (see Proposition 4.1). In fact, this extension preserves all relevant properties of the category A BF (C ). F Proposition 6.1. The functor : A BF (C ) → BF (C ) is a strict braided monoidal func∼ tor. It also preserves direct sums: (η1 ⊕ η2 ) = (η1 ) ⊕ (η2 ). Finally, d((η)) = d(η).
Proof. Functoriality of is immediate. Note that (ι) is just the identity endomorphism of F, hence it preserves the tensor unit. We verify (η1 ⊗ η2 ) = (η1 ) ⊗ (η2 ) on a dense subalgebra. Consider F = (A, ρ, ψ) ∈ F0 . Then the extension of the tensor product is given by
Sa η η1Sa η2 (A)ερ,η1 ⊗η2 π(1, ρ, ψ). (6.1) 1 ⊗ η2 (π(F)) = π Note that by definition, η1 (π(A, ι, 1)) = π Sa (η1 (A)) for all A ∈ A. Passing to the
unique weakly continuous
extension, and taking weak limits, it follows that Sa Sa S S a a η1 (A) for all A ∈ ASa . We then calculate η1 π (A) = π
(η1 ⊗ η2 ) (π(F)) = η1 Sa π Sa η2 (A)ερ,η2 π(I, ρ, ψ)
= η1 Sa π Sa η2 (A)ερ,η2 π Sa ερ,η1 π(I, ρ, ψ)
= π Sa η1Sa η2 (A)ερ,η2 ερ,η1 π(I, ρ, ψ). By the braid equations (cf. conditions (4.2)–(4.4)), the last line is equal to Eq. (6.1). For η 1 , η 2 ∈ A = ε(η1 ),(η2 ) . This follows from uniqueness of (C ), note that ε η ,η 1 2 BF the braiding of F (C ), and by noticing that the functor sends spectator morphisms BF used in the definition of εη1 ,η2 to spectator morphisms for (η1 ) and (η2 ).
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To prove that preserves direct sums, assume η1 ⊕ η2 = Ad V1 ◦ η1 + Ad V2 ◦ η2 . It is then not hard to show that for F ∈ F0 , (η1 ⊕ η2 )(π(F)) = (V1 )(η1 )(π(F)) V1∗ + (V2 )(η1 )(π(F)) V2∗ . The right hand side is just the direct sum (η1 ) ⊕ (η 2 ). Finally, for the last statement one can show that if η, R, R is a standard conjugate for η, then (η) , (R), R is a standard conjugate for (η), and this determines the dimension. Details can be found in [38, Prop. 344]. Using some harmonic analysis, the intertwiners between two extensions can be described explicitly. Proposition 6.2. For γ ∈ A DHR , write Hγ for the Hilbert space in F generated by π(I, γ , ψ), ψ ∈ E(γ ). Then for η1 , η2 ∈ A BF (C ), Sa HomF((η1 ), (η2 )) = spani∈G (HomA(γi ⊗ η1 , η2 )) Hγi , π
(6.2)
where γi ∈ A DHR corresponds to the irrep i. Moreover, we can choose each γi to be localised in a double cone Oi ⊂ C . Proof. Consider T ∈ HomA(γ ⊗η1 , η2 ) and = π(I, γ , ψ) ∈ Hγ . By Proposition 4.1, T lifts to an intertwiner π Sa (T ) from γ ⊗ η1 to η2 , hence ⊗ η1 π A, ρ, ψ . η2 π A, ρ, ψ π Sa (T ) = π Sa (T )γ Since the DHR morphisms form a symmetric category and E is a symmetric symmetry E(γ ),E(ρ) , it follows functor, that is, it maps εγ ,ρ to the canonical that π I, ρ, ψ π(I, γ , ψ) = π(εγ ,ρ , γ , ψ)π I, ρ, ψ . Using the braid equations, we then have
π Sa γ Sa η1 (A)ερ,γ ⊗η1 π I, ρ, ψ = π Sa γ Sa η1 (A)ερ,γ ⊗η1 εγ ,ρ π I, ρ, ψ
= π Sa γ Sa η1 (A)ερ,η1 π I, ρ, ψ .
∗ -tensor
An application of Lemma 3.8 then shows that π Sa (T ) ∈ HomF((η1 ), (η2 )). For the other direction, note that since (η1 ) and (η2 ) are G-invariant extensions, it follows that HomF((η1 ), (η2 )) is stable under the action of G. Since the Hom-sets are finite-dimensional vector spaces, it is clear that in this case they are generated linearly by irreducible tensors under G. So let T1 , . . . Tn be some multiplet in HomF((η1 ), (η2 )) transforming according to the representation ξ . By the proof of Lemma 3.3 there is a G-invariant X such that Ti = X i , where the i ∈ Hγ form an orthonormal basis for E(γ ). Moreover, γ is localised in some O ⊂ C and transforms according to ξ . Since Ti ∈ HomF((η1 ), (η2 )), we have, with F = (A, ι, 1) ∈ F0 ,
X i η1 (π(F)) = η2 (π(F))X i = X π Sa γ Sa (η1 (A)) i , where the last identity follows by applying Lemma 3.8 to the first term in the equa d tion. Now, multiply on the right by i∗ , and sum over i. Since i=1 i i∗ = I by [27, Prop. 270], this leads to
(6.3) X π Sa γ Sa η1 (A) = π Sa (η2 (A))X.
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By Lemma 3.5(ii) there is a T ∈ ASa such that π Sa (T ) = X , and by Eq. (6.3) and faithfulness of π Sa , we have T ∈ HomA(γ ⊗ η1 , η2 ). Corollary 6.3. The tensor functor is an embedding (i.e. faithful and injective on objects), but not full. Proof. It follows from Corollary 5.4 that is injective on objects. Since π Sa is a faithful representation, Proposition 4.1 implies is faithful. The preceding proposition implies that it is not full. Indeed, the image of HomA(η1 , η2 ) under the functor is π Sa (HomA(η1 , η2 )), which in general is a proper subset of HomF((η1 ), (η2 )) as given by Eq. (6.2). Inspired by the results of Doplicher and Roberts, Müger formulated a categorical version of the field net construction [40]. In a different context, a similar construction is due to Brugières [5]. In both approaches, modular categories are obtained by getting rid of a non-trivial centre. Here we investigate this in the present situation, cf. [41]. We follow the approach of [40], since it also works when the symmetric subcategory has infinitely many isomorphism classes of irreducible objects. Let us recall the basic ideas used in this construction. Suppose C is a braided tensor C ∗ -category and S is a full symmetric subcategory. By the Doplicher-Roberts theorem [17], there is a unique compact group G and an equivalence of categories E : S → Rep f (G). In the case at hand, C is the category A BF (C ) and S is the symmetric subcatA 5 egory DHR (C ). The group G will be the symmetry group, and E is the functor used in Sect. 3. choose a corresponding γk ∈ S First a category C 0 S is defined. For each k ∈ G, such that Hk = E(γk ) transforms according to k. The category C 0 S is the category with the same objects as C, but with Hom-sets HomC0 S (ρ, σ ) = ⊕k∈G Hom C (γk ⊗ ρ, σ ) ⊗ Hk , where the usual tensor product of vector spaces over C is used. One can then define a composition of arrows, a ∗-operation, conjugates, direct sums and in the case at hand, where the objects of S are degenerate, a braiding. Since the details are quite involved, we refer to the original paper [40]. The category C 0 S already has most of the desired structure. One property, however, is missing: in general it is not closed under subobjects. To remedy this, a closure construction is defined. This closure is denoted by C S. It is called the crossed product of C by S. The basic idea is to add a corresponding (sub)object for each projection in HomC0 S (η, η). To make this precise: the category C S has pairs (η, P) as objects, where η ∈ C and P = P 2 = P ∗ ∈ HomC0 S (η, η). The morphisms are given by HomCS ((η1 , P1 ), (η2 , P2 )) = {T ∈ HomC0 S (η1 , η2 ) | T = T ◦ P1 = P2 ◦ T }, which is just P2 ◦ HomC0 S (η1 , η2 ) ◦ P1 . Composition is as in C 0 S. Because P is a projection, id(η,P) = P. The tensor product can be defined by (η1 , P1 ) ⊗ (η2 , P2 ) = (η1 ⊗ η2 , P1 ⊗ P2 ), and the same as in C 0 S on morphisms. One can then show that C S is a braided tensor C ∗ -category with conjugates, direct sums and subobjects. The 5 Note that in the construction of the field net, the subcategory A DHR was used, without the localisation in C . Using transportability, however, it is easy to see that one might as well choose A DHR (C ), since this category is equivalent to A . DHR
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category C is embedded into the crossed product C S by a tensor functor ι : C → C S, defined by η → (η, idη ) and HomC (η1 , η2 ) T → T ⊗ . Here is a unit vector in the Hilbert space transforming according to the trivial representation of G. Like the functor , ι is a embedding functor that is not full. The following proposition clarifies the relation between the crossed product A A BF (C ) DHR (C ) and the BF representations of the field net F. F Proposition 6.4. The extension functor : A BF (C ) → BF (C ) factors through the A A A canonical inclusion functor ι : BF (C ) → BF (C ) DHR (C ). That is, there is a F A braided tensor functor H : A BF (C ) DHR (C ) → BF (C ) such that the diagram
A BF (C )
ι-
H
A A BF (C ) DHR (C )
HH H j H
H
? F BF (C )
commutes. Moreover, H is full and faithful. A Proof. First define H on the category A BF (C ) 0 DHR (C ). Clearly, for objects η we must set H (η) = (η). In view of Proposition 6.2, it is natural to set for the morphisms H (T ⊗ ψk ) = π Sa (T )π(I, γk , ψk ), where T ∈ HomA(γk ⊗ ρ, σ ), ψk ∈ E(γk ), and and extend by linearity. It is not very difficult, although quite tedious, to verify k ∈ G, F A that H defines a strict braided monoidal functor from A BF (C ) 0 DHR (C ) to BF (C ). It is clear that H is faithful, and by Proposition 6.2 it is full. A To define H on the closure A BF (C ) DHR (C ), consider one of its objects (η, P). 2 ∗ By definition, P = P = P ∈ HomA (C )0 A (C ) (η, η). It follows that H (P) as BF DHR defined above is a projection in HomF((η), (η)). By localisation of H (η) and Haag duality it follows that H (P) ∈ F(C ) . Consider a spacelike cone C such that C ⊂ C. Then by Property B there is an isometry W ∈ F C such that W W ∗ = H (P). Now define H (η, P)(·) = W ∗ η(·)W . This defines a ∗-representation of F that is localised in C, due to localisation properties of E. Using transportability, an equivalent representation localised in C can be obtained, in a similar way as done in Sect. 2. Again it can be verified that H is a braided monoidal functor. It is clearly faithful, and by Proposition 6.2 and the definition of the Hom-sets in the crossed product, it is also full. Note that H is not a strict tensor functor, but only a strong one. This is due to the arbitrary choices one has to make in finding the isometry W , which is merely unique up to unitary equivalence. A A Finally, A BF (C ) is embedded in BF (C ) DHR (C ) by η → (η, I ). Hence H ◦ ι(η) = H ((η, I )) = η, thus H ◦ ι = .
7. Essential Surjectivity of H One of our goals is to understand the category F BF (C ) in terms of the original AQFT O → A(O). The functor is not full, so it cannot provide a complete answer to this question. The functor H , however, is full and faithful. Moreover, we have an explicit description of the crossed product in terms of our original net of observables A(O). Since a tensor functor is an equivalence of tensor categories if and only if it is an equivalence
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of categories [53], it is enough to show that H is an equivalence of categories. By the previous section H is full and faithful, hence only essential surjectivity has to be shown. In this section this question is investigated. The first observation is that this is related to a property of the extension functor . Proposition 7.1. The functor H is essentially surjective if and only if is dominant. That is, for each irreducible η ∈ F η) for some η ∈ A BF (C ), η ≺ ( BF (C ). Proof. Suppose first that H is essentially surjective. Then for an irreducible object η∼ η ∈ F = H η , P . But by construction BF (C ), thereis some η , P suchthat H, of evidently H η , P is a subobject of η . Since η ∼ = H η , P , also η ≺ η . F Conversely, suppose η is dominant. Let η ∈ BF (C ) be irreducible, andsuppose is such that η ≺ η . Then there is acorresponding isometry W ∈ HomF η, η . Hence W W ∗ is a projection in EndF η , η . Proposition 6.2 shows that this in Hom A projection comes from a corresponding projection P η ,η , BF (C )0 A DHR (C ) F ∼ and we see that η = H η , P . The result follows because (C ) is semi-simple. BF
In the remainder of this section, we comment on the question of finding conditions such that is dominant. In the case of finite G this problem has been solved in [39]. Given an irreducible sector of the field net, one can use the full G-spectrum of the field net to construct a direct sum that is G-invariant and contains η. This construction works in the present case of BF sectors as well. By Corollary 5.4 it follows that this direct sum comes from extending a representation of the observable net. A straightforward attempt to generalise this to arbitrary compact groups would be to replace the (finite) direct sum by a countable direct sum or even a direct integral. However, apart from convergence problems one might encounter, there is another issue: since the dimension d(η) is strictly positive, and is additive under taking direct sums, this leads to a sector with infinite dimension. Hence it is not an element of our category F BF (C ). Let us first recall how the group G acts on the sectors, or more precisely, on equivalence classes of localised representations. g g Lemma 7.2. Let η ∈ F BF (C ). Then G acts on equivalence classes [η] by [η] = [ η] = [αg ◦ η ◦ αg−1 ].
Proof. This obviously defines an action. This action is well-defined: suppose η1 (−) = V η2 (−)V ∗ for some unitary V . Then g η2 (−) = αg ◦ η2 ◦ αg−1 (−) = αg (V η1 ◦ αg−1 (−)V ∗ ) = αg (V )αg ◦ η1 αg−1 (−)αg (V ∗ ), hence g η1 ∼ = g η2 . The previous observations suggest that if there is any hope to construct a G-invariant direct sum of a sector of the field net, the action of G on this sector should not be too “wild”, in the sense that there should only be a finite number of mutually inequivalent sectors under the action of G. This is indeed a necessary condition, as will be shown below. This behaviour is described by the stabiliser subgroup. Definition 7.3. Suppose η ∈ F BF (C ). The stabiliser subgroup G η is defined by G η = {g ∈ G | g η ∼ = η}. By Lemma 7.2 this is well-defined. Moreover, the index [G : G η ] is finite if and only if there are only finitely many equivalence classes under the action of G. Note that G η is a closed subgroup of G, hence compact. The condition that the index be finite is necessary for finding a G-invariant dominating representation.
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Lemma 7.4. Suppose η ≺ η for η ∈ F η commutes with the action of G. BF (C ), where Then [G : G η ] < ∞. Proof. Assume for simplicity that η is irreducible; the general case readily follows. Decompose η = ⊕i∈I ηi , where I is some finite set. Then there is an i ∈ I such that ηi ∼ η = η for all g ∈ G, it follows η. Because g = η, since η ≺ that for every g ∈ G there is some j ∈ I such that g ηi ∼ = η j . As g runs over G, g ηi runs over all equivalence classes g [η]. It follows that there are at most |I | such equivalence classes, or by the remark above: [G : G η ] ≤ |I |. Our next goal is to construct a BF representation η that commutes with the action of G, such that η ≺ η. In other words: η is a direct summand of η. Observe that it is enough to consider only summands ηi ∼ = gi η for some gi ∈ G. Now assume that [G : G η ] is finite. Then there is a finite dimensional representation of G, permuting a basis of the space spanned by the left cosets G/G η . Write [g] for the coset of g ∈ G. Pick a representative gi of each coset. Since the field net has full G-spectrum, it is possible to find isometries V[gi ] such that αg V[gi ] = V[ggi ] and the following relations hold: V[g∗ i ] V[g j ] = δi, j I, V[gi ] V[g∗ i ] = I. [gi ]∈G/G η
Now if g ∈ G, there is a g j and a h j ∈ G η such that ggi = g j h j . Moreover, multiplication on the left induces a permutation on the cosets, hence also of the representatives gi . Let η be such that η ≺ η. Consider η(−) = [gi ]∈G/G η V[gi ] gi η(−)V[g∗ i ] . Then for g ∈ G,
∗ g η(−) = αg V[gi ] ggi η(−)αg V[gi ] = V[gi ] gi h i η(−) V[g∗ i ] , [gi ]∈G/G η
[gi ]∈G/G η
η to commute with the G-action, it is sufficient that h η = η where h i is as above. So for for all h ∈ G η . The existence of such a η is also necessary. To find such an η, by semi-simplicity of F BF (C ) it is enough to consider an irreducible η. We will do this in the rest of this section. By definition, for each g ∈ G η there is a unitary v(g) such that g η(−) = v(g)η(−)v(g)∗ . By considering gh η = g h η and using that η is irreducible, it follows that v(gh) = c(g, h)αg (v(h))v(g), g, h ∈ G η , where c(g, h) is a complex number of modulus one. In fact, it is not difficult to show that c(g, h) is a 2-cocycle, with equivalence class [c] ∈ H 2 (G η , T). The cohomology class does not depend on the specific choice of unitaries v(g) and is the same for each η ∼ = η. Hence (G η , [c]) can be seen as an invariant of the sector. If [c] is the trivial cohomology class, v(g) is in fact an α-one-cocycle and we can construct an η ∼ = η that commutes with the action of G η , just as in the proof of Corollary 5.4. The following observation, which amounts to the fact that the direct sum is independent of the chosen basis, turns out to be convenient. irreducible. Consider two direct sums of copies of Lemma 7.5. Let η ∈ F BF (C ) be n n ∗ η, η = i=1 Vi η(−)Vi∗ and η = i=1 W η = η if and only if there is i . Then i η(−)W n a unitary n × n matrix λ such that Wi = i=1 λ ji V j .
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Proof. (⇒) Define λi j = Vi∗ W j , then λi j ∈ EndF(η) ∼ = C, by irreducibility of η. By a straightforward calculation one easily verifies that λ is indeed a unitary matrix, and n Wi = i=n λ ji V j . (⇐) Easy calculation. n Now suppose we have a direct sum η(A) = i=1 Vi η(A)Vi∗ . An easy calculation then shows that for g ∈ G η : g
η(−) =
n
αg (Vi )v(g)η(−)v(g)∗ αg Vi∗ ,
i=1
where the v(g) are unitaries as above. Because v(g) is unitary, it follows that αg (Vi )v(g) is a basis of HomF (η, g η). This space has a Hilbert space structure, defining an inner product by V, W I = W ∗ V for V, W ∈ Hom (η, g η). Combining this with the previous observations, we find the following necessary and sufficient criterion. Proposition 7.6. There is a G-equivariant (i.e., commuting with the action of G) dominating sector η η if and only if the following conditions hold: (i) the stabiliser group G η has finite index in G, i.e. [G : G η ] < ∞, (ii) there is a finite-dimensional non-trivial Hilbert space H in F such that αg (V )v(g) ∈ H for all V ∈ H and g ∈ G η . We end this section with a few remarks. First of all, the author unfortunately does not know of any physical interpretation of the conditions in the proposition. Furthermore it seems to be difficult to verify these conditions. However, the proposition generalises the situation where G is finite. In this case, the conditions are trivially satisfied. If one can show that the cocycle c(g, h) is trivial (as a cocycle in H 2 (G η , T)), it follows by Theorem 5.2 that there is a unitary w such that v(g) = αg (w)w ∗ . Condition (ii) is then satisfied by taking the one-dimensional Hilbert space spanned by w. Using Theorem 5.2 one can show that c(g, h) is trivial as a cocycle in the field net, which, however, is not sufficient here. As a final remark, suppose that condition (ii) is satisfied. It follows that there is a Hilbert space in F carrying a projective unitary representation. Indeed, choose an orthonormal basis Vi of H. Then for g ∈ G η , αg (Vi )v(g) is a new basis for H. Write λ(g) for the unitary transformation that implements the basis change. It follows that λ(gh) = c(g, h)λ(g)λ(h). 8. Conclusions and Open Problems It would be desirable to arrive at a modular category starting from an AQFT in three dimensions, for example because of their relevance to topological quantum computing. In this paper some steps in this direction are taken. In particular, the category of stringlike localised or BF representations has many of the properties of a modular category. The existence of DHR sectors, which cannot be ruled out a priori, is shown to be an obstruction for modularity. To remove this obstruction, the original theory A is extended to the field net F, which can be seen as a new AQFT without DHR sectors. The relation between those theories is partially made clear, in particular by the crossed product construction of Sect. 6. There is, however, one point that is not fully understood, namely the question whether the sectors in the new theory F can be completely described by the
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sectors of the theory A. This is the case if for example G is finite, or the conditions of Proposition 7.6 hold for each BF sector of F. In this case, the sectors of F are completely A determined by the crossed product A BF DHR (C ). Although one major obstruction for modularity has now been removed, this is not enough to conclude that F BF (C ) is modular. In particular, there may be degenerate BF (but not DHR) sectors of F. The other condition is that there should be only finitely many equivalence classes of BF representations of F. In case the functor H of Sect. 7 is indeed an equivalence, both properties are determined by the crossed product, and hence ultimately by A BF (C ). In particular, in this situation, absence of (C ) is equivalent to the absence of degenerate objects in degenerate sectors in F BF A A BF (C ) DHR (C ). This is essentially because H is a braided functor, which makes it possible to transfer the degeneracy condition of the braiding from one category to A the other. The absence of degenerate objects of A BF (C ) DHR (C ) is equivalent to the absence of degenerate BF sectors (that are not DHR) of A, since by [40] the crossed product has trivial centre if and only if A DHR (C ) is equal to the centre of A BF (C ). The finiteness condition would follow by counting arguments from finiteness of A BF (C ). We give a list of some open problems and questions. (i) In view of the remarks above, it would be interesting to understand the set of BF (that are not DHR) sectors of A. In particular, are there conditions that imply that this set is finite, or does not contain any degenerate sectors? As for the latter: in the DHR case a condition for this was given in [39]. Perhaps this condition might be adapted to the case of BF sectors. It should be noted that both conditions (i.e. non-degeneracy and finiteness) are completely understood in the case of conformal field theory on the circle, in terms of an index of certain subfactors [30]. That method, however, cannot obviously be adapted to the case we are interested in, among other reasons because we have no condition for factoriality of the relevant algebras of observables. However, it would be interesting to know if there is an analogue of the condition of “complete rationality” that ensures modularity. (ii) It would be desirable to have a physical interpretation for the conditions given in Sect. 7. This might give some hints on how to prove these conditions in concrete theories. (iii) One of our assumptions was the absence of fermionic DHR sectors of A. It would be interesting to see what can still be done if this assumption is dropped. In this case, the field net does not satisfy locality, but only twisted locality. Thus one would lose the interpretation of F as an AQFT in the sense that it should only consist of observables commuting at spacelike distances. (iv) Can the techniques be useful in describing quantum spin systems? Such systems are more appropriate for topological quantum computing than relativistic quantum field theories, see e.g. [31]. There is some evidence that points in this direction [44]. In particular, it can be shown that in Kitaev’s Z2 model on the plane, single excitations can be described by automorphisms of an observable algebra. These automorphisms fulfill a selection criterion similar to the BF criterion. Moreover, they are localised and transportable, and using the methods here, one can explicitly calculate the statistics of these excitations. The results are consistent with Kitaev’s results [31]. Although this simple model is by no means sufficient for quantum computing, it might be possible to extend the methods to more interesting models.
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Acknowledgements. This research is funded by NWO grant no. 613.000.608, which is gratefully acknowledged. I would also like to thank Michael Müger for valuable discussions and suggestions, and Klaas Landsman for a critical reading of the manuscript. Open Access This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
Appendix A In this appendix we collect some of the terminology regarding (tensor) categories and notions of superselection theory that will be used throughout the article. Due to lack of space, we restrict to the essentials. In particular, the categorical concepts can be defined much more generally than necessary for our purposes. For the essentials of category theory, details can be found in the book by Mac Lane [37]. For the structure of categories appearing in algebraic quantum field theory, see [38]. Modular categories are described in [2]. An overview of superselection theory can be found in the book by Haag [26].
A.1. Superselection theory. A sector is a unitary equivalence class of representations (satisfying some selection criterion such as the DHR or BF criterion) of the observable algebra. Representations satisfying the BF or DHR criterion can be described by localised and transportable endomorphisms of the observable algebra. Sometimes we will identify such an endomorphism ρ with its sector, i.e., all unitary equivalent localised endomorphisms. These endomorphisms are the objects of a category, with intertwiners as morphisms. An intertwiner from η1 to η2 (and hence a morphism in Hom(η1 , η2 )) is an operator T such that T η1 (A) = η2 (A)T for all observables A. There is a natural tensor product ⊗ (defined by composition of endomorphisms) on this category. Another important concept is that of a conjugate sector. A conjugate of a DHR or BF representation can be interpreted as an “anti-charge”. Formally, a conjugate for a BF (or DHR) representation ρ is a triple (ρ, R, R), where ρ is a BF (resp. DHR) representation. The operators R, R are intertwiners satisfying R ∈ Hom (ι, ρ ⊗ ρ), with ι the trivial endomorphism, and R ∈ Hom (ι, ρ ⊗ ρ) such that ∗
R ρ(R) = I,
R ∗ ρ R = I,
where I is the unit of the observable algebra. If a conjugate exists, one can always choose ∗ a standard conjugate. A conjugate ρ, R, R is called standard if R ∗ ρ(S)R = R S R for all S ∈ Hom(ρ, ρ). The conditions for a conjugate imply that ρ ⊗ ρ (and ρ ⊗ ρ) contain a copy of the vacuum sector. A conjugate exists if and only if the sector has finite (statistics) dimension. The latter is then given by d(ρ)I = R ∗ R with R standard. Conjugates are intimately related to the statistics of a sector. It should be noted that conjugates can be defined in a much more general categorical setting, e.g. [36]. In the category of BF representations, a braiding ερ,η ∈ Hom(ρ ⊗ η, η ⊗ ρ) for every pair of objects ρ, η can be defined. A sector is called degenerate if, roughly speaking, it has trivial braiding with all objects. More precisely, ρ is degenerate if and only if ερ,η ◦ εη,ρ = I for all objects η. If this holds for a particular representative of a sector, it holds for all representatives of the sector. An object of the form ι ⊕ · · · ⊕ ι is always degenerate. Degenerate sectors of this form are called trivial.
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A.2. Category theory. Let F : C → D be a functor. Then, for every pair of objects ρ, σ in C, there is a map Fρ,σ : Hom(ρ, σ ) → Hom(F(ρ), F(σ )) defined by S → F(S) for S ∈ Hom(ρ, σ ). The functor F is called faithful, if Fρ,σ is injective for each pair of objects ρ, σ . Likewise, if it is surjective for all pairs, it is called full. Note that a faithful functor is not necessarily injective on objects, that is, it might happen that F(ρ) = F(σ ) for distinct objects ρ and σ of C. A faithful functor that is also injective on objects, is called an embedding.6 In particular, subcategories give rise to embedding functors. A subcategory of a category C is a category that contains a collection of the objects and morphisms of C. A subcategory is called full if it has the same morphisms as the bigger category, hence in that case it is completely determined by specifying its objects. Finally, a functor F : C → D is called an equivalence of categories if it is full, faithful and essentially surjective, which means that for each object D of D, there is an object C of C such that F(C) is isomorphic to D. From a categorical perspective, equivalent categories are “essentially the same”. Certain categories admit a tensor (or monoidal) product ⊗. That is, one can form tensor products of objects and morphisms. In a tensor category there is a tensor unit ι, such that ρ ∼ = ι⊗ρ ∼ = ρ ⊗ ι, where ∼ = means isomorphic in the category. Associativity is described by natural isomorphisms αρ,σ,τ : ρ ⊗ (σ ⊗ τ ) → (ρ ⊗ σ ) ⊗ τ satisfying certain coherence conditions. A tensor category is called strict if the associativity morphisms reduce to the identity, and ρ ⊗ ι = ι ⊗ ρ = ρ for all objects ρ. The categories encountered in this paper are all strict. Every tensor category is monoidally equivalent to a strict tensor category. That is, there is a tensor functor between the two categories, that is also an equivalence of categories. A tensor functor is a functor F together with natural isomorphisms F(ρ ⊗ σ ) → F(ρ) ⊗ F(σ ), and similarly for the tensor unit. Again, the functor is called strict if the isomorphisms are all identities. Even between strict tensor categories, however, it might be necessary to consider non-strict tensor functors. In case both categories have a braid C ing, a braided tensor functor F : C → D is a functor such that F ερ,σ = εD F(ρ),F(σ ) (or C a suitably modified condition if the categories are not strict), where ερ,σ is the braiding of C. The category of stringlike localised representations is called modular, if it has only finitely many equivalence classes of representations and the centre (with respect to the braiding) is trivial. The latter condition is the statement that if ερ,η ◦ εη,ρ = I for each object ρ, then η = ι ⊕ · · · ⊕ ι, i.e., it is a direct sum of trivial endomorphisms. A modular category satisfies additional axioms (for example the existence of duals or conjugates), but these are automatically satisfied by the category of BF representations. The nondegeneracy condition is equivalent to Turaev’s condition on a modular category [56], which is stated in terms of invertibility of a certain matrix S, by a result of Rehren [47]. References 1. Araki, H.: von Neumann algebras of local observables for free scalar field. J. Math. Phys. 5, 1–13 (1964) 2. Bakalov, B., Kirillov, A. Jr.: Lectures on tensor categories and modular functors. Volume 21 of University Lecture Series. Providence, RI: Amer. Math. Soc., 2001 3. Böckenhauer, J., Evans, D.E.: Modular invariants, graphs and α-induction for nets of subfactors. I. Commun. Math. Phys. 197(2), 361–386 (1998) 4. Borchers, H.-J.: A remark on a theorem of B. Misra. Commun. Math. Phys. 4, 315–323 (1967) 5. Bruguières, A.: Catégories prémodulaires, modularisations et invariants des variétés de dimension 3. Math. Ann. 316(2), 215–236 (2000) 6 Note, however, that for some authors an embedding functor is only a faithful functor.
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6. Buchholz, D.: The physical state space of quantum electrodynamics. Commun. Math. Phys. 85(1), 49–71 (1982) 7. Buchholz, D., Fredenhagen, K.: Locality and the structure of particle states. Commun. Math. Phys. 84(1), 1–54 (1982) 8. Buchholz, D., Haag, R.: The quest for understanding in relativistic quantum physics. J. Math. Phys. 41(6), 3674–3697 (2000) 9. Conti, R., Doplicher, S., Roberts, J.E.: Superselection theory for subsystems. Commun. Math. Phys. 218(2), 263–281 (2001) 10. D’Antoni, C.: Technical properties of the quasi-local algebra. In: Kastler [29], pp. 248–258 11. Deligne, P.: Catégories tannakiennes. In: The Grothendieck Festschrift, Vol. II, Volume 87 of Progr. Math., Boston, MA: Birkhäuser Boston, 1990, pp. 111–195 12. Doplicher, S., Haag, R., Roberts, J.E.: Fields, observables and gauge transformations. I. Commun. Math. Phys. 13, 1–23 (1969) 13. Doplicher, S., Haag, R., Roberts, J.E.: Local observables and particle statistics. I. Commun. Math. Phys. 23, 199–230 (1971) 14. Doplicher, S., Haag, R., Roberts, J.E.: Local observables and particle statistics. II. Commun. Math. Phys. 35, 49–85 (1974) 15. Doplicher, S., Roberts, J.E.: Fields, statistics and non-abelian gauge groups. Commun. Math. Phys. 28, 331–348 (1972) 16. Doplicher, S., Roberts, J.E.: Endomorphisms of C ∗ -algebras, cross products and duality for compact groups. Ann. Math. (2) 130(1), 75–119 (1989) 17. Doplicher, S., Roberts, J.E.: A new duality theory for compact groups. Invent. Math. 98(1), 157–218 (1989) 18. Doplicher, S., Roberts, J.E.: Why there is a field algebra with a compact gauge group describing the superselection structure in particle physics. Commun. Math. Phys. 131(1), 51–107 (1990) 19. Fredenhagen, K.: Generalizations of the theory of superselection sectors. In: Kastler [29], pp. 379–387 20. Fredenhagen, K., Rehren, K.-H., Schroer, B.: Superselection sectors with braid group statistics and exchange algebras. I. General theory. Commun. Math. Phys. 125(2), 201–226 (1989) 21. Fredenhagen, K., Rehren, K.-H., Schroer, B.: Superselection sectors with braid group statistics and exchange algebras. II. Geometric aspects and conformal covariance. Rev. Math. Phys. 4(Special Issue), 113–157 (1992) 22. Freedman, M.H., Kitaev, A., Larsen, M.J., Wang, Z.: Topological quantum computation. From: Mathematical challenges of the 21st century (Los Angeles, CA, 2000) Bull. Amer. Math. Soc. (N.S.), 40(1), 31–38 (2003) 23. Freedman, M.H., Larsen, M., Wang, Z.: A modular functor which is universal for quantum computation. Commun. Math. Phys. 227(3), 605–622 (2002) 24. Fröhlich, J., Gabbiani, F.: Braid statistics in local quantum theory. Rev. Math. Phys. 2(3), 251–353 (1990) 25. Guido, D., Longo, R.: Relativistic invariance and charge conjugation in quantum field theory. Commun. Math. Phys. 148(3), 521–551 (1992) 26. Haag, R.: Local quantum physics: Fields, particles, algebras. Texts and Monographs in Physics. Berlin: Springer-Verlag, Second edition, 1996 27. Halvorson, H.: Algebraic quantum field theory. In: Butterfield, J., Earman, J. (eds.) Philosophy of Physics, London: Elsevier, 2006, pp. 731–922 28. Hewitt, E., Ross, K.A.: Abstract harmonic analysis. Vol. II: Structure and analysis for compact groups. Analysis on locally compact Abelian groups. Die Grundlehren der mathematischen Wissenschaften, Band 152. New York: Springer-Verlag, 1970 29. Kastler, D. (ed.): The algebraic theory of superselection sectors: Introduction and recent results, River Edge, NJ: World Scientific Publishing Co. Inc., 1990 30. Kawahigashi, Y., Longo, R., Müger, M.: Multi-interval subfactors and modularity of representations in conformal field theory. Commun. Math. Phys. 219(3), 631–669 (2001) 31. Kitaev, A.: Fault-tolerant quantum computation by anyons. Ann. Phys. 303(1), 2–30 (2003) 32. Kitaev, A.: Anyons in an exactly solved model and beyond. Ann. Phys 321(1), 2–111 (2006) 33. Kowalzig, N.: Hopf Algebroids and Their Cyclic Theory. PhD thesis, Universiteit van Amsterdam and Universiteit Utrecht, 2009 34. Longo, R.: Index of subfactors and statistics of quantum fields. I. Commun. Math. Phys. 126(2), 217–247 (1989) 35. Longo, R.: Index of subfactors and statistics of quantum fields. II. Correspondences, braid group statistics and Jones polynomial. Commun. Math. Phys. 130(2), 285–309 (1990) 36. Longo, R., Roberts, J.E.: A theory of dimension. K -Theory 11(2), 103–159 (1997) 37. Mac Lane, S.: Categories for the working mathematician, Volume 5 of Graduate Texts in Mathematics. New York: Springer-Verlag, second edition, 1998 38. Müger, M.: Abstract duality for symmetric tensor ∗-categories. Appendix to [27]
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39. Müger, M.: On charged fields with group symmetry and degeneracies of Verlinde’s matrix S. Ann. Inst. H. Poincaré Phys. Théor. 71(4), 359–394 (1999) 40. Müger, M.: Galois theory for braided tensor categories and the modular closure. Adv. Math. 150(2), 151– 201 (2000) 41. Müger, M.: Conformal orbifold theories and braided crossed G-categories. Commun. Math. Phys. 260(3), 727–762 (2005) 42. Mund, J.: Borchers’ commutation relations for sectors with braid group statistics in low dimensions. Ann. Henri Poincaré 10(1), 19–34 (2009) 43. Mund, J.: The spin-statistics theorem for anyons and plektons in d = 2+1. Commun. Math. Phys. 286(3), 1159–1180 (2009) 44. Naaijkens, P.: Localized endomorphisms in Kitaev’s toric code on the plane. Preprint arXiv:1012.3857 45. Nayak, C., Simon, S.H., Stern, A., Freedman, M., Das Sarma, S.: Non-abelian anyons and topological quantum computation. Rev. Mod. Phys. 80(3), 1083–1159 (2008) 46. Panangaden, P., Paquette, É.: A categorical presentation of quantum computation with anyons. In: Coecke, B. (ed.), New structures for Physics, Lecture Notes in Physics. Berlin-Heidelberg-New York: Springer (2011) 47. Rehren, K.-H.: Braid group statistics and their superselection rules. In: Kastler [29], pp. 333–355 48. Rehren, K.-H.: Markov traces as characters for local algebras. From Recent advances in field theory (Annecy-le-Vieux, 1990), Nucl. Phys. B Proc. Suppl., 18B, 259–268 (1991) 49. Rehren, K.-H.: Field operators for anyons and plektons. Commun. Math. Phys. 145(1), 123–148 (1992) 50. Roberts, J.E.: Cross products of von Neumann algebras by group duals. In: Symposia Mathematica, Volume XX, London: Academic Press, 1976, pp. 335–363 51. Roberts, J.E.: Local cohomology and superselection structure. Commun. Math. Phys. 51(2), 107–119 (1976) 52. Roberts, J.E.: Lectures on algebraic quantum field theory. In: Kastler [29], pp. 1–112 53. Saavedra Rivano, N.: Catégories Tannakiennes. Lecture Notes in Mathematics, Vol. 265. Berlin: SpringerVerlag, 1972 54. Sutherland, C.E.: Cohomology and extensions of von Neumann algebras. II. Publ. Res. Inst. Math. Sci. 16(1), 135–174 (1980) 55. Szlachányi, K., Vecsernyés, P.: Quantum symmetry and braid group statistics in G-spin models. Commun. Math. Phys. 156(1), 127–168 (1993) 56. Turaev, V.G.: Quantum invariants of knots and 3-manifolds. Volume 18 of de Gruyter Studies in Mathematics. Berlin: Walter de Gruyter & Co., 1994 57. Verlinde, E.: Fusion rules and modular transformations in 2D conformal field theory. Nucl. Phys. B 300(3), 360–376 (1988) Communicated by Y. Kawahigashi
Commun. Math. Phys. 303, 421–449 (2011) Digital Object Identifier (DOI) 10.1007/s00220-010-1171-z
Communications in
Mathematical Physics
An Isoperimetric Inequality for Fundamental Tones of Free Plates L. M. Chasman Knox College, Galesburg, IL 61401, U.S.A. E-mail:
[email protected] Received: 6 May 2010 / Accepted: 27 July 2010 Published online: 27 November 2010 – © Springer-Verlag 2010
Abstract: We establish an isoperimetric inequality for the fundamental tone (first nonzero eigenvalue) of the free plate of a given area, proving the ball is maximal. Given τ > 0, the free plate eigenvalues ω and eigenfunctions u are determined by the equation u − τ u = ωu together with certain natural boundary conditions. The boundary conditions are complicated but arise naturally from the plate Rayleigh quotient, which contains a Hessian squared term |D 2 u|2 . We adapt Weinberger’s method from the corresponding free membrane problem, taking the fundamental modes of the unit ball as trial functions. These solutions are a linear combination of Bessel and modified Bessel functions. 1. Introduction Laplacian and bi-Laplace operators are used to model many physical situations, with their eigenvalues representing quantities such as energy and frequency. The eigenvalues μ of the Neumann Laplacian on a region determine the frequencies of vibration of a free membrane with that shape. If ∗ is the ball of same volume as , then we have μ1 () ≤ μ1 (∗ )
with equality if and only if is a ball.
First conjectured by Kornhauser and Stakgold [13], this isoperimetric inequality was proved for simply connected domains in R2 by Szeg˝o [26,28] and extended to all domains and dimensions by Weinberger [32]. The main goal of this paper is to establish the analogous result for the eigenvalues ω the free plate under tension. That is, of all regions with the same volume, we have the bound ω1 () ≤ ω1 (∗ )
with equality if and only if is a ball.
Our proof relies on the variational characterization of eigenvalues with suitable trial functions. Similar to Weinberger’s approach for the free membrane, we take our trial
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functions to be extensions of the fundamental mode of the unit ball. However, because the plate equation is fourth order, finding the trial functions and establishing the appropriate monotonicities is significantly more complicated than in the membrane case. The eigenmodes of the unit ball are identified in our companion paper [6], where we also establish several properties of these functions that are used in the proof of the isoperimetric inequality. If the reader is satisfied with a numerical demonstration, the needed properties of ultraspherical Bessel functions can be verified in any given dimension using Mathematica or Maple. The boundary conditions of the free plate are not imposed, but instead arise naturally from the Rayleigh quotient. It is therefore extremely important that we begin with the correct Rayleigh quotient for the plate, which includes a Hessian term. These natural boundary conditions have long been known in the case d = 2 (see, eg, [33]); we include in this paper their derivation for all dimensions. The fundamental tone of the ball is extremal for other physically meaningful plate boundary conditions. In particular, the ball provides a lower bound for the clamped plate eigenvalues [3,18,19,29]. The methods used by Talenti, Nadirashvilli, Ashbaugh and Benguria to prove the clamped plate isoperimetric inequality are quite different than those for the free plate and membrane and only establish the bound in dimensions 2 and 3. The problem remains open for dimensions four and higher, with a partial result by Ashbaugh and Laugesen [4]. For an overview of work on the clamped plate problem, see [10, Chap. 11, p. 169–174] and [12, p. 105–116]. Other plate boundary conditions include the simply supported plate, hinged plate, and Neumann boundary conditions. Plate problems are fourth-order and generally more difficult than their second-order membrane counterparts, because the theory of the bi-Laplace operator is not as well understood as the theory of the Laplacian. Verchota recently established the solvability of the biharmonic Neumann problem [31]; these boundary conditions arise from the zero-tension plate and allow consideration of Poisson’s ratio, a measure of a material property that we take to be zero for our free plate. Supported plate work includes Payne [23] and Licar and Warner [15], who examine domain dependence of plate eigenvalues. It would be natural to conjecture an isoperimetric inequality for the simply supported plate, although there does not seem to be any work on this problem to the best of our knowledge. Work with hinged plates includes Nazarov and Sweers [20]. Other notable mathematical work on plates includes Kawohl, Levine, and Velte [11], who investigated the sums of low eigenvalues for the clamped plate under tension and compression, and Payne [23], who considered both vibrating and buckling free and clamped plates and established inequalities bounding plate eigenvalues by their (free or fixed) membrane counterparts. For a broad survey of results, see [1,5]. 2. Formulating the Problem We now develop the mathematical formulation of the free plate isoperimetric problem. Let be a smoothly bounded region in Rd , d ≥ 2, and fix a parameter τ > 0. The “plate” Rayleigh quotient is |D 2 u|2 + τ |Du|2 d x . (1) Q[u] = 2 |u| d x Here |D 2 u| = ( jk u 2x j xk )1/2 is the Hilbert-Schmidt norm of the Hessian matrix D 2 u of u, and Du denotes the gradient vector.
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Physically, when d = 2 the region is the shape of a homogeneous, isotropic plate. The parameter τ represents the ratio of lateral tension to flexural rigidity of the plate; for brevity we refer to τ as the tension parameter. Positive τ corresponds to a plate under tension, while taking τ negative would give us a plate under compression. The function u describes a transverse vibrational mode of the plate, and the Rayleigh quotient Q[u] gives the bending energy of the plate. From the Rayleigh quotient (1), we will derive the partial differential equation and boundary conditions governing the vibrational modes of a free plate. The critical points of (1) are the eigenstates for the plate satisfying the free boundary conditions and the critical values are the corresponding eigenvalues. The equation is: u − τ u = ωu,
(2)
where ω is the eigenvalue, with the natural (i.e., unconstrained or “free”) boundary conditions on ∂: ∂ 2u = 0, ∂n 2 ∂(u) ∂u − div∂ P∂ (D 2 u)n − = 0. V u := τ ∂n ∂n Mu :=
(3) (4)
Here n is the outward unit normal to the boundary and div∂ and grad∂ are the surface divergence and gradient. The operator P∂ projects onto the space tangent to ∂. We will prove in a later section that the spectrum of the Rayleigh quotient Q is discrete, consisting entirely of eigenvalues with finite multiplicity: 0 = ω0 < ω1 ≤ ω2 ≤ · · · → ∞. We also have a complete L 2 -orthonormal set of eigenfunctions u 0 ≡ const, u 1 , u 2 , and so forth. We call u 1 the fundamental mode and the eigenvalue ω1 the fundamental tone; the latter can be expressed using the Rayleigh-Ritz variational formula: ω1 () = min{Q[u] : u ∈ H (), 2
u d x = 0}.
In general, the k th eigenvalue is the minimum of Q[u] over the space of all functions u L 2 -orthogonal to the eigenfunctions u 0 , u 1 , .. . , u k−1 . Because u 0 is the constant function, the condition u ⊥ u 0 can be written u d x = 0. Note that in the limiting case τ = 0, the first d + 1 eigenvalues of are trivial because Q[u] = 0 for all linear functions u. We therefore need the tension parameter τ to be positive in order to have a nontrivial inequality. The eigenvalue equation (2) can also be obtained by separating the plate wave equation φtt = −φ + τ φ, √ by the separation φ(x, t) = u(x) cos( ωt). The eigenvalue ω is therefore the square of the frequency of vibration of the plate.
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3. Main Result The main goal of this paper is to prove an isoperimetric inequality for the fundamental tone of the free plate under tension. Let ∗ denote the ball with the same volume as our region . Theorem 1. For all smoothly bounded regions of a fixed volume, the fundamental tone of the free plate with a given positive tension is maximal for a ball. That is, if τ > 0 then the first nonzero eigenvalue ω of u − τ u = ωu subject to the natural boundary conditions (3) and (4) satisfies ω1 () ≤ ω1 (∗ ),
with equality if and only if is a ball.
(5)
The proof of Theorem 1 will proceed from a series of lemmas, following roughly this outline. A more detailed summary follows. – Section 8 – Defining the trial functions, showing concavity of the radial part of the trial function, and evaluating the Rayleigh quotient – Section 9 – Proving partial monotonicity of the Rayleigh quotient – Section 10 – Establishing rescaling and rearrangement results, and proving the theorem. Adapting Weinberger’s approach for the membrane [32], we construct in Lemma 5 trial functions with radial part ρ matching the radial part of the fundamental mode of the ball. We follow by proving in Lemma 6 a concavity property of ρ that will be needed later on. We next bound the eigenvalue ω by a quotient of integrals over our region , both of whose integrands are radial functions (Lemma 7). These integrands will be shown to have a “partial monotonicity”. The denominator’s integrand is increasing by Lemma 8 and the numerator’s integrand satisfies a decreasing partial monotonicity condition by Lemma 9. The proof of Lemma 9 becomes rather involved and so is contained in its own section and broken into two cases, Lemma 10 for large τ values, and Lemma 11 for small values of τ . The latter in turn requires some facts about particular polynomials, proved in Lemmas 12 and 13. We then exploit partial monotonicity to see that the quotient of integrals is bounded above by the quotient of the same integrals taken over ∗ , by Lemma 14. Finally, we conclude that the quotient of integrals on ∗ is in fact equal to the eigenvalue ω∗ of the unit ball. From there we deduce the theorem. 4. Existence of the Spectrum and Regularity of Solutions Our first task is to investigate the spectrum of the fourth-order operator associated with our Rayleigh quotient Q in (1). In this section we show the spectrum is entirely discrete, with an associated weak eigenbasis. We will then establish regularity of the eigenfunctions up to the boundary and derive the natural boundary conditions. For this section only we will allow τ to be any real number. We continue to require ⊂ Rd to be smoothly bounded unless otherwise stated. The existence of the spectrum. We consider the sesquilinear form d a(u, v) = u xi x j vxi x j + τ (Du · Dv) d x i, j=1
in L 2 () with form domain H 2 (). Note the plate Rayleigh quotient Q can be written in terms of a, with Q[u] = a(u, u)/ u 2L 2 .
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Proposition 1. The spectrum of the operator 2 − τ associated with the form a(·, ·) consists entirely of isolated eigenvalues of finite multiplicity ω0 ≤ ω1 ≤ ω2 ≤ · · · → ∞. There exists an associated set of real-valued weak eigenfunctions which is an orthonormal basis for L 2 (). Proof. By Cauchy-Schwarz, the form a(·, ·) is bounded on H 2 (), and so is continuous. We will show the quadratic form a(u, u) is coercive; that is, for some positive constants c1 and c2 a(u, u) + c1 u 2 ≥ c2 u 2H 2 () . By the boundedness of a on H 2 , this is equivalent to showing the norm associated with a, u a2 = a(u, u) + c1 u 2 is equivalent to · 2H 2 () , and hence a is closed on H 2 (). Because is smoothly
bounded, H 2 () and L 2 () can be extended to H 2 (Rd ) and L 2 (Rd ) respectively. The space H 2 (Rd ) is compactly embedded in L 2 (Rd ). Then by a standard result (see e.g., Cor. 7.D [25, p. 78]), the form a has a set of weak eigenfunctions which is an orthonormal basis for L 2 (), and the corresponding eigenvalues are of finite multiplicity and satisfy ω1 ≤ ω2 ≤ · · · ≤ ωn → ∞ as n → ∞.
(6)
For τ > 0, coercivity of the form a is easily proved: a(u, u) + τ u 2 ≥ D 2 u 2 + τ Du 2 + τ u 2 ≥ min(τ, 1) u 2H 2 , where all unlabeled norms are L 2 norms on . To prove coercivity when τ ≤ 0, we must somehow arrive at a positive constant in front of the |Du|2 term. We cannot use Poincaré’s inequality on the |D 2 u| term as this will introduce terms involving the average value of Du. Instead, we will exploit an interpolation inequality. By Theorem 7.28 of [9, p. 173], we have that for any index 1 ≤ j ≤ n and any ε > 0, ∂x j u 2L 2 () ≤ ε u 2H 2 () + Cε−1 u 2L 2 ()
(7)
with C = C() a constant. Replacing ε by ε/d and summing over j, we see
1 C D 2 u 2L 2 ≥ − 1 Du 2L 2 − 2 + 1 u 2L 2 . ε ε Fix δ ∈ (0, 1). Let K > 0. Then a(u, u) + K u 2L 2 = D 2 u 2L 2 − |τ | Du 2L 2 + K u 2L 2
δ Cδ − δ − |τ | Du 2L 2 + K − 2 − δ u 2L 2 ≥ (1 − δ) D 2 u 2L 2 + ε ε
Cδ δ ≥ min 1 − δ, − δ − |τ |, K − 2 − δ u H 2 . ε ε We can choose our ε small and our K large so that the minimum is positive, which proves coercivity. For example, for δ = 1/2, we need to take ε < 1/(1 + 2|τ |) and K > 21 (C + 1 + 2|τ |). Thus a is coercive for all τ .
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Now suppose u is a weak eigenfunction corresponding to eigenvalue ω. Because ω is real-valued, by taking the complex conjugate of the weak eigenvalue equation we see that u is also a weak eigenfunction with the same eigenvalue. Thus the real and imaginary parts of u are both eigenfunctions associated with ω, and we may choose our eigenfunctions to be real-valued. Note that for any bounded region and all real values of τ , the constant function solves the weak eigenvalue equation with eigenvalue zero. For all nonnegative values of τ , the Rayleigh quotient is nonnegative for all functions and so 0 = w0 ≤ w1 ≤ · · · ≤ wk ≤ . . .. When τ = 0, the coordinate functions x1 , . . . , xd are also solutions with eigenvalue zero, and so the lowest eigenvalue is at least d + 1-fold degenerate. Taking instead τ > 0, the Raleigh quotient shows that the fundamental tone ω1 is positive, and so we have: 0 = ω0 < ω1 ≤ ω2 ≤ · · · ≤ ωn → ∞ as n → ∞. Regularity. We aim to establish regularity of the weak eigenfunctions by appealing to interior and boundary regularity theory for elliptic operators. Proposition 2. For any τ ∈ R and smoothly bounded , the weak eigenfunctions of − τ are smooth on . Proof. Let u be a weak eigenfunction of A with associated eigenvalue ω; by Proposition 1 we have u ∈ D(a) = H 2 (). Then by a theorem in [21, p 668], we have u ∈ H k () for every positive integer k. Thus we have u ∈ H k () for all k ∈ Z+ , and so u ∈ C ∞ (). Regularity on the boundary follows from global interior regularity and the Trace Theorem (see, for example, [30, Prop 4.3, p. 286 and Prop 4.5, p. 287.]). Thus we have u ∈ C ∞ (), as desired. 5. The Natural Boundary Conditions In this section, our goal is to derive the form of the natural boundary conditions necessarily satisfied by all eigenfunctions. Consider the weak eigenvalue equation for eigenfunction u with eigenvalue ω and some test function φ ∈ Cc∞ (): u xi x j φxi x j + τ Du · Dφ − ωuφ d x. i, j
Because the eigenfunction u is smooth, we may use integration by parts to move most of the derivatives on φ to u; this gives us a volume integral and two surface integrals that must vanish for all φ. The natural boundary conditions are rather complicated in higher dimensions, and so we state the two-dimensional case first. The boundary conditions in this case have been known for some time: see, for example, [33] Proposition 3 (Two dimensions). For ⊂ R2 , the natural boundary conditions for eigenfunctions of the free plate under tension have the form ∂ 2u = 0, ∂n 2
2 ∂(u) ∂ ∂ u ∂u ∂u − − − K (s) = 0, V u := τ ∂n ∂n ∂s ∂s∂n ∂s Mu :=
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where n denotes the outward unit normal derivative, s the arclength, and K the curvature of ∂. We also look at one example of the natural boundary conditions for a region with corners. Notice that an additional condition arises at the corners! Proposition 4 (Rectangular region in two dimensions). When ⊂ R2 is a rectangular region with edges parallel to the coordinate axes, the natural boundary conditions for eigenfunctions of the free plate under tension have the form ∂ 2u = 0 at each edge, ∂n 2 ∂ 3u ∂(u) ∂u − 2 − = 0 on each edge, τ ∂n ∂s ∂n ∂n u x y = 0 at each corner, where n and s indicate the normal and tangent directions. Finally, we state the natural boundary conditions for a smoothly-bounded region in higher dimensions: Proposition 5 (General). For any smoothly bounded , the natural boundary conditions for eigenfunctions of the free plate under tension have the form ∂ 2u = 0 on ∂, ∂n 2 ∂u ∂u − div∂ P∂ (D 2 u)n − = 0 on ∂, V u := τ ∂n ∂n where n denotes the normal derivative and div∂ is the surface divergence. The projection P∂ projects a vector v at a point x on ∂ into the tangent space of ∂ at x. Mu :=
Proof (Proof of Proposition 5). Our eigenfunctions u are smooth on by Proposition 2 and satisfy the weak eigenvalue equation a(u, φ)−ω(u, φ) L 2 () = 0 for all φ ∈ H 2 (). That is, ⎞ ⎛ d ⎝ u xi x j φxi x j + τ Dφ · Du − ωuφ ⎠ d x = 0.
i, j=1
As in the derivation of the natural boundary conditions for the membrane, we will make much use of integration by parts. Let n denote the outward unit normal to the surface ∂. To simplify our calculations, we consider each term separately. The gradient term only needs one use of integration by parts: ∂u dS − Du · Dφ d x = φ φ(u) d x. ∂ ∂n The Hessian term becomes: u xi x j φxi x j d x i, j
=
∂
∂(u) 2 Dφ · (D u)n − φ d S + (2 u)φ d x, ∂n
after integrating by parts twice.
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We wish to transform the term involving Dφ in the above surface integral using integration by parts. Because we are on ∂, we must treat the normal and tangential components separately. We can then use the Divergence Theorem for integration on ∂. We note that the surface gradient grad∂ equals D − n∂n when applied to a function (like φ) that is defined on a neighborhood of the boundary. Thus grad∂ φ gives the tangential part of the Euclidean gradient vector. Hence, Dφ · (D 2 u)n d S ∂ 2
∂φ ∂ u n = + grad∂ φ · n 2 + P∂ (D 2 u)n dS ∂n ∂n ∂ ∂φ ∂ 2 u = + grad∂ φ, P∂ (D 2 u)n dS 2 ∂ ∂ ∂n ∂n ∂φ ∂ 2 u 2 = P (D − φ div u)n d S, ∂ ∂ 2 ∂ ∂n ∂n by the Divergence Theorem on the surface ∂. Here ·, ·∂ denotes the inner product on the tangent space to ∂. Recall P∂ projects a vector at a point x on ∂ onto the tangent space of ∂ at x. Thus for u an eigenfunction associated with eigenvalue ω, we see 0= φ 2 u − τ u − ωu d x
∂u ∂φ ∂ 2 u ∂u 2 − − div∂ P∂ (D u)n + +φ τ d S. 2 ∂n ∂n ∂ ∂n ∂n As in the membrane case, this identity must hold for all φ ∈ H 2 (). If we take any compactly supported φ, then the volume integral must vanish; because φ is arbitrary, we must therefore have 2 u − τ u − ωu = 0 everywhere. Similarly, the terms multiplied by φ and ∂φ/∂n must vanish on the boundary. Collecting these results, we obtain the eigenvalue equation (2) and natural boundary conditions of Proposition 5. Proof (Proof of Proposition 3). Here d = 2; take rectangular coordinates (x, y). We parametrize ∂ by arclength s and define coordinates (n, s), with n the normal distance from ∂, taken to be positive outside . Write n(s) ˆ and tˆ(s) for the outward unit normal and unit tangent vectors to the boundary. Then P∂ f 1 nˆ + f 2 tˆ = f 2 tˆ and the operators div∂ and grad∂ both simply take the derivative with respect to arclength s. That is, for a scalar function f (s), and taking t (s) to be the tangent vector to the surface, we have grad∂ f (s) = f (s)
and
div∂ ( f (s)tˆ(s)) = f (s),
and so we may write ∂ T 2 t (D u)n. div∂ P∂ (D 2 u)n = ∂s The tangent line to ∂ at the point (0, s) in our new coordinates forms an angle α = α(s) with the x-axis (see [33, p. 230]); the curvature of ∂ is given by K (s) = α (s). Then
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in rectangular coordinates, the unit tangent vector is (cos α, sin α), and the outward unit normal is (sin α, − cos α). Thus we have ∂ T 2 t (D u)s = ∂s sin α cos α(u x x − u yy ) + (sin2 α − cos2 α)u x y . ∂s By [33, p. 233], on ∂ under our change of coordinates, we have u x x = u nn sin2 α + u ss cos2 α + 2u ns sin α cos α + K u n cos2 α − 2K u s sin α cos α, u yy = u nn cos2 α + u ss sin2 α − 2u ns sin α cos α + K u n sin2 α + 2K u s sin α cos α, u x y = −u nn cos α sin α + u ss cos α sin α + u ns (sin2 α − cos2 α) +K u n cos α sin α − K u s (sin2 α − cos2 α). So after simplification, ∂ T 2 ∂ [n (D u)t] = ∂t ∂s
∂ 2u ∂u − K (s) ∂s∂n ∂s
.
This together with the results of Proposition 5 yields the form of V u given in Proposition 3. Mu is unchanged, and so this completes the proof. Proof (Proof of Proposition 4). Our previous findings do not completely apply because ∂ has corners, although our argument proceeds similarly. For convenience of notation, we will take to be the square [0, 1]2 . The Hessian term gives us a condition at the corners. In particular, after integrating by parts twice, we have: u x x φx x + 2u x y φx y + u yy φ yy d A = φ u x x x x + 2u x x yy + u yyyy d A
x=1 + u x x φx − u x x x φ + u x y φ y − u x yy φ dy 0 x=0 1 y=1 u yy φ y − u yyy φ + u x y φx − u x x y φ d x. +
1
0
y=0
Since
1 0
y=1 u x y φ y dy = u x y φ − y=0
1
u x yy φ dy
0
and
1 0
x=1 u x y φx d x = u x y φ − x=0
1 0
u x x y φ d x,
430
we obtain
L. M. Chasman
u x x φx x + 2u x y φx y + u yy φ yy d A = φ u x x x x + 2u x x yy + u yyyy d A
x=1 u x x φx − φ(2u x yy + u x x x ) dy 0 x=0 1 y=1 u yy φ y − φ(2u x x y + u yyy ) dx + 0 y=0 x=1 y=1 + 2u x y x=0 .
+
1
y=0
Because the Divergence Theorem does apply to regions with piecewise-smooth boundaries, the gradient term is the same as in the smooth-boundary case. The final term above is the only term that depends only on the behavior of u and φ at the corners; arguing as before, we obtain the eigenvalue equation and natural boundary conditions, with the additional condition 1 0 = u x y φ 1x=0 . y=0
That is, we must have u x y = 0 at the corners. Example: natural boundary conditions on the ball. When is a ball, we can simplify the general boundary conditions. Proposition 6 (Ball). The natural boundary conditions in the case = Bd (R), the ball of radius R, are Mu := u rr = 0 at r = R, 1 u − (u)r = 0 at r = R. V u := τ u r − 2 S u r − r r
(8) (9)
Proof. When is a ball, the normal vector to the surface at a point x is n = x/R. Then the i th component of (D 2 u)n is given by d
u xi x j
j=1
and can be rewritten as
xj R
⎞ ⎛ d 1 ∂ ⎝ u x j x j − u ⎠. R ∂ xi j=1
Therefore, u x (D 2 u)n = D Du · − . R R
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Then the projection P∂ takes the tangential component of the above gradient vector, and so u x . P∂ (D 2 u)n = grad∂ Bd (R) Du · − R R We know div∂ grad∂ = ∂ by definition. For the ball of radius R, we have ∂ Bd (R) = 1 . The operator S is the spherical Laplacian, consisting of the angular part of the R2 S ∂ d−1 ∂ 1 Laplacian. It satisfies the identity = ∂r 2 + r ∂r + r 2 S . Thus u x div∂ P∂ (D 2 u)n = ∂ Bd (R) Du · − R R u , = ∂ Bd (R) u r − R 2
by noting that Du · n = ∂u/∂r . The boundary conditions of Proposition 5 then simplify to (8) and (9), as desired. The one-dimensional case. The one-dimensional analog of the free plate is the free rod, represented by an interval I = [a, b] on the real line. We include its boundary conditions for the sake of completeness. We may derive the natural boundary conditions from the weak eigenvalue equation as before. We obtain as boundary conditions u ab = 0, b τ u − u a = 0, and the eigenvalue equation u − τ u = ωu. Note that these are in fact the one-dimensional analogues of the boundary conditions and eigenvalue equation obtained in Proposition 5. The computations are straightforward integrations by parts and thus omitted. The fundamental tone of the free plate in dimensions d ≥ 2 had simple angular dependence; the fundamental tone of the free rod under tension can be proved to be an odd function. See [7, Chap. 7]. Note that we do not have an isoperimetric inequality for the free rod, because all connected domains of the same area are now intervals of the same length, and are identical up to translation. 6. The Fundamental Tone as a Function of Tension Fix the smoothly bounded domain . We will estimate how the fundamental tone ω1 = ω1 (τ ) depends on the tension parameter τ , establishing bounds used in the proof of Theorem 1. We will also examine the behavior of ω1 in the extreme case as τ → ∞. First we note that the Rayleigh quotient (1) is linear and increasing asa function of τ . Our eigenvalue ω1 (τ ) is the infimum of Q[u] over u ∈ H 2 () with u d x = 0, and thus ω1 (τ ) is itself a concave, increasing function of τ . Next, we will prove ω1 (τ )/τ is bounded above and below for all τ > 0. Recall μ1 is the fundamental tone of the free membrane.
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Fig. 1. The fundamental tone of the disk (solid curve) together with the linear bounds from Lemma 1 (dashed lines)
Lemma 1. For all τ ≥ 0 we have ||d , ¯ 2 dx |x − x|
τ μ1 ≤ ω1 (τ ) ≤ τ where x¯ =
(10)
x d x/|| is the center of mass of . In particular, when is the unit ball, τ μ1 ≤ ω1 (τ ) ≤ τ (d + 2).
(11)
Furthermore, the upper bounds in (10) and (11) hold for all τ ∈ R. These bounds are illustrated in Fig. 1. Proof. To establish the upper bound, take the coordinate functions as trial functions: u k = xk − x¯k , for k = 1, . . . , d. Note u k d x = 0 by definition of center of mass, so the u k are valid trial functions. All second derivatives of the u k are zero, so we have 2 τ |Du k | d x 1 dx 2 . ω1 (τ ) ≤ Q[u k ] = =τ 2 (x k − x¯ k ) d x uk d x Clearing the denominator and summing over all indices k, we obtain ω1 (τ ) |x − x| ¯ 2 d x ≤ τ ||d,
which is the desired upper bound. When is the unit ball, note |x|2 d x = ||d/(d+2). Now we treat the lower bound. Let u ∈ H 2 () with u d x = 0. Then Q[u] ≥
τ
|Du|2 d x ≥ τ μ1 2 u dx
by the variational characterization of μ1 . Taking the infimum over all trial functions u for the plate yields ω1 (τ ) ≥ τ μ1 .
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Fig. 2. The fundamental tone of the disk (solid curve) together with the upper bound of Lemma 2 (top dashed line) and the lower bound of Lemma 1 (bottom dashed line)
Note that Payne [22] proved linear bounds for eigenvalues of the clamped plate under tension. Kawohl, Levine, and Velte [11] investigated the sums of the first d eigenvalues as functions of parameters for the clamped plate under tension and compression. (See Fig. 2) We can also prove another linear upper bound on ω1 , which is just a constant plus the lower bound in Lemma 1. Lemma 2. For all τ ∈ R, ω1 ≤ C() + τ μ1 , where the value C() =
|D
2 v|2 d x
v
2 dx
is given explicitly in terms of the fundamental mode v of the free membrane on . Proof. Let v be a fundamental mode of the membrane with v = −μ1 v and v d x = 0; the membrane boundary condition is ∂u/∂n = 0 on ∂. Then by the variational characterization of eigenvalues, ω1 (τ ) ≤ Q[v] = C() + τ Q M [v] = C() + τ μ1 , as desired. Infinite tension limit. A plate behaves like a membrane as the flexural rigidity tends to zero, that is, as τ = (tension/flexural rigidity) tends to infinity. For the fundamental tone, that means: Corollary 1. For the fundamental tone of the free plate, ω1 (τ ) → μ1 τ
as τ → ∞.
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Proof. By Lemmas 1 and 2, we have μ1 ≤
ω1 (τ ) C ≤ μ1 + . τ τ
Let τ → ∞. The eigenfunctions should converge as τ → ∞ to the eigenfunctions of the free membrane problem. Proving this for all eigenfunctions seems to require a singular perturbation approach, which has been carried out for the clamped plate in [8], but we will not need any such facts for our work. For the convergence of the fundamental tone of the clamped plate to the first fixed membrane eigenvalue, see [11]. Remark. In the limit as τ → 0, we find a relationship between ω1 (τ )/τ and the scalar moment of inertia of the region ; see [7]. 7. Summary of Bessel Function Facts The radial part of the fundamental tone of the unit ball is a linear combination of Bessel functions. Before we begin constructing our trial functions, we need to gather some results established in the companion paper [6]. The ultraspherical Bessel functions jl (z) of the first kind are defined in terms of the Bessel functions of the first kind, Jν (z), as follows: jl (z) = z −s Js+l (z), d −2 , with s = 2 and solve the ultraspherical Bessel equation z 2 w + (d − 1)zw + z 2 − l(l + d − 2) w = 0.
(12)
Note that this notation suppresses the dependence of the jl functions on the dimension d; we assume dimension d ≥ 2 is fixed. Ultraspherical modified Bessel functions il (z) of the first kind are defined analogously, with il (z) = z −s Is+l (z) solving the modified ultraspherical Bessel equation z 2 w + (d − 1)zw − z 2 + l(l + d − 2) w = 0.
(13)
Ultraspherical Bessel functions satisfy the following recurrence relations: d − 2 + 2l jl (z) = jl−1 (z) + jl+1 (z), z l jl (z) = jl (z) − jl+1 (z), z d − 2 + 2l il (z) = il−1 (z) − il+1 (z), z l il (z) = il (z) + il+1 (z). z
(14) (15) (16) (17)
An Isoperimetric Inequality for Fundamental Tones of Free Plates
435
Ultraspherical Bessel functions and their derivatives may be expressed by converging series. The first few terms in these expansions may be used to bound the Bessel functions and their derivatives; we will need such bounds for the second derivatives of j1 and i 1 . Let dk denote the coefficients of the series expansion for i 1 (z), so that j1 (z) =
∞ ∞ (−1)k dk z 2k−1 and i 1 (z) = dk z 2k−1 k=1
k=1
by the series expansions of jl and il in [6, p. 5], where dk =
2k + 1 21−2k−d/2 . (k − 1)! (k + 1 + d/2)
Lemma 3 [6, Lemma 10], We have the following bounds: −d1 z + d2 z 3 ≥ j1 (z) for all z ∈ 0, 3(d + 2)/(d + 5) , √ 6 d1 z + d2 z 3 ≥ i 1 (z) for all z ∈ 0, 3 . 5 While proofs are provided in the companion paper, these bounds and those listed below in Lemma 4 can also all be demonstrated numerically in Mathematica or Maple for any given dimension. We will also be using additional facts about the signs of certain Bessel functions and derivatives. These were proven in [6] and are collected below. We write a∞ for the first nontrivial zero of jl . Lemma 4 [6, Lemmas 5 through 9]. We have the following: 1. 2. 3. 4.
For l = 1, . . . , 5, we have jl > 0 on (0, a∞ ]. We have j1 > 0 on (0, a∞ ). We have j2 > 0 on (0, a∞ ]. We have j1 < 0 on (0, a∞ ].
5. We have jl(4) > 0 on (0, a∞ ]. We are now ready to begin proving Theorem 1. 8. Trial Functions In this and the next sections, we establish the lemmas which allow us to prove Theorem 1: Among all regions of a fixed volume, when τ > 0 the fundamental tone of the free plate is maximal for a ball. That is, ω1 () ≤ ω1 (∗ ),
with equality if and only if is a ball.
(18)
For simplicity, as we prepare to prove Theorem 1, we will write ω instead of ω1 for the fundamental tone of the free plate with shape ; the fundamental tone of the unit ball will be denoted by ω∗ . When dependence on the region and the tension τ need be made explicit, we write ω(τ, ) for the fundamental tone and Q τ, for the Rayleigh quotient. The tension parameter τ > 0 throughout the remainder of this paper.
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We begin with the assumption that our domain has the same volume as the unit ball Bd ; this is justified by the scaling argument in Lemma 15. In [6, Theorem 2], we identified the fundamental mode of the unit ball, written in spherical coordinates as: u 1 = R(r )Y1 := ( j1 (ar ) + γ i 1 (br )) Y1 ,
(19)
where Y1 is any of the d spherical harmonics of order 1, j1 (z) and i 1 (z) are ultraspherical Bessel functions, and a and b are positive constants satisfying the conditions a 2 b2 = ω and b2 − a 2 = τ . Note that we may take our spherical harmonics Y1 to be the xi /r for i = 1 . . . , d, where xi is the i th coordinate function. Finally, recall that we took a∞ to be the first nontrivial zero of jl (z); by the proof of [6, Theorem 3], we have a < a∞ . Note that the fundamental tone of the free membrane 2 . with shape Bd is given by μ∗1 = a∞ We are now able to choose our trial functions. Inspired by Weinberger’s proof for the membrane [32], we choose appropriate trial functions from the fundamental modes of the unit ball. In the following lemmas, we take R(r ) = j1 (ar ) + γ i 1 (br ) to be the radial part of the fundamental mode of the unit ball. Recall a and b are positive constants determined by τ and the boundary conditions, as in the proof of Theorem 3 in [6]. The constant γ is positive and determined by a, τ , and the boundary conditions as follows: γ :=
−a 2 j1 (a) > 0. b2 i 1 (b)
(20)
Recall also that R(r ) > 0 on (0, 1] and R (1) > 0. Lemma 5 (Trial functions). Let the radial function ρ be given by the function R, extended linearly. That is, R(r ) when 0 ≤ r ≤ 1, ρ(r ) = R(1) + (r − 1)R (1) when r ≥ 1. After translating suitably, the functions u k = xk ρ(r )/r , for k = 1, . . . , d, are valid trial functions for the fundamental tone. Proof. To be valid trial functions, the u k must be in H 2 (). Because is bounded in Rd , the only possible issue would be a singularity at the origin. The series expansions given in [6, p. 5] for j1 and i 1 give us that R(r )/r approaches a constant as r → 0. Thus, u k ∈ H 2 () as desired. The trial functions must also be perpendicular to the constant function, and so we will need ρ(r )xk dx = 0 for k = 1, . . . , d. r We use the Brouwer Fixed Point Theorem to translate our region so that the above conditions are guaranteed; here again we follow Weinberger [32]. Write x = (x1 , . . . , xd ) and consider the vector field ρ(|x − v|) (x − v) d x. X (v) = |x − v|
An Isoperimetric Inequality for Fundamental Tones of Free Plates
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The vector field X is continuous by construction. For any vector v along the boundary of the convex hull of , the vector field X (v) is inward-pointing, because ρ ≥ 0 and the entire region lies in a half-space to one side of v. Thus by the Brouwer Fixed Point Theorem, our vector field X vanishes at some v in the convex hull of . If we first translate by v, then we have X (0) = ρ(r )x/r d x = 0. This gives us u k d x = 0, as desired. We will need one further fact about our radial function ρ. Lemma 6 (Concavity). The function ρ (r ) ≤ 0 for r ∈ [0, 1], with equality only at the endpoints. Proof. First note that on [0, 1], the function ρ ≡ R. We see R (r ) = a 2 j1 (ar ) + γ b2 i 1 (br ), which is zero at r = 0 because the individual Bessel derivatives vanish there, by the series expansions for the Bessel jl (z) and il (z) in [6]. At r = 1, the function R vanishes because of the boundary condition Mu = 0. The fourth derivative of R is given by R (r ) = a 4 j1 (ar ) + γ b4 i 1 (br ). Because all derivatives of i 1 (z) are positive when z ≥ 0, the second term above is positive on (0, ∞). Lemma 9 in [6] states that j1 (z) is positive on (0, a∞ ]. Thus R (r ) > 0 on (0, 1], and so R (r ) is a convex function on [0, 1]. Since R = 0 at r = 0 and r = 1, the function R must be negative on the interior of the interval [0, 1]. We now bound our fundamental tone above by a quotient of integrals whose integrands are radial functions. The numerator will be quite complicated, so we write N [ρ] := (ρ )2 +
3(d − 1) τ (d − 1) 2 (ρ − r ρ )2 + τ (ρ )2 + ρ . r4 r2
We will also need the following calculus facts: Fact [7, Appendix]. We have the sums d
|u k |2 = ρ 2 ,
k=1 d k=1 d k=1
|Du k |2 =
d −1 2 ρ + (ρ )2 , r2
|D 2 u k |2 = (ρ )2 +
3(d − 1) (ρ − r ρ )2 . r4
We may now use the trial functions to bound our fundamental tone by a quotient of integrals.
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Lemma 7 (Using the trial functions). For any , translated as in Lemma 5, we have
N [ρ] d x 2 ρ dx
ω≤
(21)
with equality if = ∗ . Proof. For u k defined as in Lemma 5, we have ω ≤ Q[u k ] =
|D
+ τ |Du k |2 d x , 2 |u k | d x
2u
k|
2
from the Rayleigh-Ritz characterization. We have equality when = ∗ because the u k are the eigenfunctions for the ball associated with the fundamental tone; see [6]. Multiplying both sides by |u k |2 d x and summing over all k, we obtain ω
d k=1
|u k | d x ≤ 2
d k=1
|D u k | + τ 2
2
d
|Du k |2 d x
(22)
k=1
again with equality if = ∗ . By these sums in Fact 8, we see inequality (22) becomes ω
ρ2 d x ≤
3(d − 1) τ (d − 1) 2 2 2 (ρ )2 + d x, (ρ − r ρ ) + τ (ρ ) + ρ r4 r2
once more with equality if is the ball ∗ . Dividing both sides by (21).
ρ
2 d x,
we obtain
9. Partial Monotonicity of the Integrands We want to show the quotient (21) in Lemma 7 has a sort of monotonicity with respect to the region , and so we examine the integrands of the numerator and denominator separately. The case of the denominator is much simpler; the partial monotonicity of the integrand of the numerator is much more difficult, and requires several lemmas. We begin with the denominator. Lemma 8 (Monotonicity in the denominator). The function ρ(r )2 is strictly increasing. Proof. Differentiating, we see
ρ (r ) =
j1 (ar ) + γ i 1 (br )
when 0 ≤ r ≤ 1,
R (1)
when r ≥ 1.
Obviously i 1 (br ) ≥ 0. Because we have a < a∞ from the proof of Theorem 3 in [6], the function j1 (ar ) is positive on [0, 1]. Thus ρ (r ) is positive everywhere, and ρ (and therefore ρ 2 ) is an increasing function.
An Isoperimetric Inequality for Fundamental Tones of Free Plates
439
We do not need to prove the integrand of the numerator is strictly decreasing; a weaker “partial monotonicity” condition is sufficient. We will say a function F is partially monotonic for if it satisfies F(x) > F(y)
for all x ∈
and y ∈ .
(23)
This desired partial monotonicity in the numerator is established by Lemmas 9 through 13. The proofs of these lemmas rely on properties of the trial function ρ and ball eigenvalue ω∗ , made explicit below: √ 1. ρ(r ) = j1 (ar ) + γ i 1 (br ) with 0 < a < a∞ , b = a 2 + τ , and 0 < γ < 1 determined by the natural boundary conditions. 2. γ ≥ −a 2 j1 (a)/b2 i 1 (b) (we in fact have equality). 3. ρ ≤ 0 for all values of r under our consideration. 4. ρ − r ρ ≥ 0 for all values of r under our consideration. 5. (d + 2)τ > ω∗ > τ d. Properties 1, 2, 3, and 5 have already been proven; property 4 is established in the proof of Lemma 9, below. Lemma 9 (Partial monotonicity in the numerator). The function
3(d − 1) ρ2 2 2 2 N [ρ] = (ρ ) + (ρ − r ρ ) + τ (ρ ) + (d − 1) 2 r4 r satisfies condition (23) for the unit ball. Proof. Given that ρ < 0 on (0, 1) and equals zero elsewhere by Lemma 6, the function (ρ )2 satisfies condition (23) for the unit ball. The derivative of the function τ (ρ )2 with respect to r is 2τ ρ ρ , and hence negative on (0, r ) and zero everywhere else. Thus τ (ρ )2 is a decreasing function of r . It remains to show that the remaining term h(r ) =
3(ρ − r ρ )2 ρ2 + τ r4 r2
is also a decreasing function of r . Differentiating, we see
−2 6 h (r ) = 3 (ρ − r ρ ) 2 (ρ − r ρ ) + 3ρ + τρ . r r Now, ρ − r ρ = 0 at r = 0 and d (ρ − r ρ ) = −r ρ , dr so by Lemma 6, (ρ − r ρ ) is positive on (0, ∞) and vanishes at zero. Thus in order for h(r ) to be decreasing, we must have 6 (ρ − r ρ ) + 3ρ + τρ > 0. r2
(24)
Let r ρ := ρ − (d − 1)r −2 (ρ − r ρ ). Recall from the Bessel equations (12) and (13) that r j1 (ar ) = −a 2 j1 (ar ) and r i 1 (br ) = b2 i 1 (br ).
(25)
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Then on the interval [0, 1],
6 6 d −1 (ρ − r ρ ) + 3ρ + τρ = 2 (ρ − r ρ ) + 3 r ρ + (ρ − r ρ ) + τρ r2 r r2 3(d + 1) = (ρ − r ρ ) + 3 −a 2 j1 (ar ) + γ b2 i 1 (br ) + τρ, 2 r
with the last equality by (25). Considering the first term of the last line above, we see by (15) and (17), 1 1 (ρ − r ρ ) = 2 (ar j2 (ar ) − br γ i 2 (br )) r2 r 1 2 = a ( j1 (ar ) + j3 (ar )) + γ b2 (i 3 (br ) − i 1 (br )) d +2 with the second equality by (14), (16), and simplifying. Therefore our quantity of interest in (24) can be bounded below in terms of jl ’s and il ’s: 6 (ρ − r ρ ) + 3ρ + τρ r2
3a 2 3a 2 (d + 1) j1 (ar ) + j3 (ar ) = τ− d +2 d +2
3b2 3b2 (d + 1) i 1 (br ) + γ i 3 (br ). +γ τ + d +2 d +2 ≥
3a 2 (d + 1) 3b2 (d + 1) j3 (ar ) + γ i 3 (br ) d +2 d +2
3b2 3a 2 +γ τ + j1 (ar ) + τ− d +2 d +2
with the inequality by jl (ar ) ≤ il (ar ) ≤ il (br ), since τ > 0 and so a < b. The function i 3 is everywhere nonnegative and the constant γ is positive. We have a < a∞ , so ar < a∞ on [0, 1] and hence the functions j3 (ar ) and j1 (ar ) are positive on [0, 1] by Lemma 1 of [6]. The remaining factor is positive for all τ > 0 by Lemmas 10 and 11 (to follow), thus establishing (24) and completing the proof. We establish the positivity of the remaining factor first for those τ values such that τ > 9/(d + 5); the proof for smaller τ values is more complicated and is treated in another lemma. Lemma 10 (Large τ ). We have τ−
3a 2 >0 d +2
(26)
for all τ > 9/(d + 5). Proof. We use the bounds we established for ω(τ ) in Sect. 4. 2 . Lemma 1 Recall that the first free membrane eigenvalue for the ball is μ∗1 = a∞ and Proposition of Lorch and Szego ([17], but see the statement in [6, Prop. 4]) together
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441
give (d + 2)τ > ω∗ > τ d. Because ω∗ = a 4 + a 2 τ [6, Prop 2], we obtain inequalities relating τ and a: a4 a4 > τ > , d − a2 d + 2 − a2
(27)
with the upper bound holding only if a 2 < d. Using the lower bound, we see τ−
3a 2 3a 2 a4 − > 2 d +2 d +2−a d +2 a 4 (d + 5) − 3a 2 (d + 2) , = (d + 2)(d + 2 − a 2 )
which is nonnegative whenever a 2 ≥ 3(d + 2)/(d + 5). When a 2 < 3(d + 2)/(d + 5), we have τ−
3a 2 9 >τ− > 0, d +2 d +5
by our choice of τ . Lemma 11 (Small τ ). We have τ−
3a 2 +γ d +2
3b2 τ+ >0 d +2
(28)
for all 0 < τ ≤ 9/(d + 5). Proof. The proof will proceed as follows. We restate the desired inequality (28) as a condition on γ , (30). We then use properties of Bessel functions to establish a lower bound on γ in terms of a rational function of a; we then show this function satisfies (30). We will need to treat the cases of d ≥ 3 and d = 2 separately, because the two-dimensional case requires better bounds than we can derive for general d. First note that b2 = a 2 + τ , so the inequality (28) is equivalent to τ>
3a 2 (1 − γ ) . (d + 5)γ + (d + 2)
(29)
Using the lower bound on τ in (27), we see that the above will hold if γ ≥
3(d + 2) − a 2 (d + 5) =: γ ∗ . (3 + a 2 )(d + 2)
(30)
We need only show that (30) holds for all 0 < τ ≤ 9/(d + 5). We will use Taylor polynomial estimates to bound γ below by a rational function. From Lemma 6 of [6], we have j1 (z) ≤ −d1 z + d2 z 3 on 0, 3(d + 2)/(d + 5) , √ 6 i 1 (z) ≤ d1 z + d2 z 3 on 0, 3 . 5
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L. M. Chasman
These bounds apply to z = ar and z = br respectively, when r ∈ [0, 1], as we show below by obtaining bounds on a 2 and b2 . To derive our bound on a 2 , we note that the lower bound of (27) together with our assumption τ ≤ 9/(d + 5) implies a4 9 , < d + 2 − a2 d +5 so that (d + 5)a 4 + 9a 2 − 9(d + 2) < 0. The left-hand side is increasing with respect to a 2 and equals zero when a 2 = 3(d + 2)/(d + 5). Hence a 2 < 3(d + 2)/(d + 5) and the bound on j1 (z) holds for z = ar when τ ≤ 9/(d + 5). We use these to obtain a further bound: 9 d +5 a4 9 > − d + 2 − a2 d +5
3(d + 2) a2 + 3 2 a − , = d + 2 − a2 d +5
0≥τ−
and so we have a 2 < d. To bound b2 , we use b2 = a 2 + τ and obtain b2 = a 2 + τ ≤
9 3(d + 2) + = 3, d +5 d +5
and so b2 ≤ 3. We also have, from (27), da 2 (d + 2)a 2 2 > b > , d − a2 d + 2 − a2
(31)
with the upper bound holding in this regime because a 2 < d. We also need the following binomial estimate: 3 1 − x < (1 − x)3/2 2
for 0 < x < 1.
Using these bounds, we see γ =
−a 2 j1 (a) b2 i 1 (b)
a 2 (d1 a − d2 a 3 ) b2 (d1 b + (6/5)d2 b3 ) a 3 (d1 − d2 a 2 ) ≥ 3/2 da 2 da 2 (d1 + (6/5)d2 d−a 2) d−a 2
≥
by definition (20) by Lemma 6 of [6] by (31)
(32)
An Isoperimetric Inequality for Fundamental Tones of Free Plates
=
d − a2 d
3/2
(d − a 2 )(1 − c1 a 2 ) (d − a 2 + (6/5)c1 da 2 )
3a 2 (d − a 2 )(6(d + 4) − 5a 2 ) ≥ 1− 2d (6d(d + 4) − 24a 2 )
writing c1 = d2 /d1 =
443
5 6(d + 4)
by (32),
noting that a 2 /d < 1 and a 2 < 3(d + 2)/(d + 5). Thus we have γ − γ ∗ ≥ 0 if
3a 2 (d − a 2 )(6(d + 4) − 5a 2 ) 3(d + 2) − a 2 (d + 5) 1− ≥ 0, − 2d 6d(d + 4) − 24a 2 (3 + a 2 )(d + 2) or, clearing the denominators and writing x = a 2 , if (2d − 3x)(d − x) (6(d + 4) − 5x) (3 + x)(d + 2) −2d (6d(d + 4) − 24x) (3(d + 2) − x(d + 5)) ≥ 0. The above polynomial is fourth degree in each of d and x and has the root x = 0; because we are only interested in its behavior for x ∈ (0, 3(d + 2)/(d + 5)), we may divide by x and work to show the resulting polynomial P(x, d) = 24d 4 + 60d 3 − 120d 2 − 432d − 40d 3 x − 119d 2 x − 6d x + 432x + 43d 2 x 2 + 113d x 2 + 54x 2 − 15d x 3 − 30x 3 is nonnegative for x ∈ (0, 3(d + 2)/(d + 5). This claim is addressed in Lemma 12 for d ≥ 3. For d = 2, the function P(x, 2) is negative on most of our interval of interest [0, 12/7], and so we must improve our lower bound on γ . The derivation follows that of inequality (27) in the proof of Lemma 10, as follows. 2 τ , where a ≈ 1.84118 is the first zero of J (z). By PropoBy Lemma 1, ω∗ > a∞ ∞ 1 ∗ sition 2 of [6] we have ω = a 4 + a 2 τ , giving us τ≤
a4 . 2 − a2 a∞
Using b2 = a 2 + τ , we obtain also a bound on b2 : b2 ≤
2 a2 a∞ . 2 − a2 a∞
Proceeding as before, we deduce 2
(a∞ − a 2 )(36 − 5a 2 ) 3a 2 γ ≥ 1− 2 2 + (6a 2 − 36)a 2 2a∞ 36a∞ ∞ with the last again from (32). So γ − γ ∗ ≥ 0 if 2
(a∞ − a 2 )(36 − 5a 2 ) 3a 2 12 − 7a 2 1− 2 − ≥0 2 + (6a 2 − 36)a 2 2a∞ 36a∞ 12 + 4a 2 ∞
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L. M. Chasman
or, setting x = a 2 , if the fourth degree polynomial
3x 2 − x)(36 − 5x)(12 + 4x) Q(x) = 1 − 2 (a∞ 2a∞ 2 2 + (6a∞ − 36)x (12 − 7x) − 36a∞ is positive on [0, 12/7]. This positivity follows from Lemma 13, completing our proof. The next two lemmas regarding the polynomials P and Q allow us to complete the proof of Lemma 11. Lemma 12. The polynomial P(x, d) = 24d 4 + 60d 3 − 120d 2 − 432d − 40d 3 x − 119d 2 x − 6d x + 432x + 43d 2 x 2 + 113d x 2 + 54x 2 − 15d x 3 − 30x 3 is nonnegative for all x ∈ (0, 3(d + 2)/(d + 5)) and integers d ≥ 3. Proof. First note that 3(d + 2)/(d + 5) < 3. We bound P below on the interval x ∈ [0, 3] by taking x = 3 in terms with negative coefficients and taking x = 0 in terms with positive coefficients, obtaining P(x, d) ≥ 24d 4 − 60d 3 − 477d 2 − 855d − 810 =: g(d). The highest order term is 24d 4 , and so g is ultimately positive and increasing in d. Note also that g (d) = 96d 3 − 180d 2 − 954d − 855, g (d) = 288d 2 − 360d − 954. The function g (d) is a quadratic polynomial with positive leading coefficient and roots at d ≈ −1.30 and 2.55; thus g (d) is increasing for all d ≥ 3. We see that g (5) = 1875, so g is increasing for all d ≥ 5. Finally, g(7) = 6876, so for all d ≥ 7 we have g(d) > 0 and hence P(x, d) > 0 for all d ≥ 7 and x ∈ [0, 3]. For d = 3, 4, 5, 6, we look at the polynomials Pd (x) = P(x, d) directly to show that Pd (x) > 0 on [0, 3(d + 2)/(d + 5)]. Each Pd is a cubic polynomial in x; its first derivative Pd (x) is quadratic and so the critical points of Pd (x) can all be found exactly. For d = 4, 5, and 6, direct calculations show Pd < 0 on [0, 3(d + 2)/(d + 5)] and Pd (3(d + 2)/(d + 5)) > 0, so Pd (x) > 0 on [0, 3(d + 2)/(d + 5)]. For d = 3, our interval of interest is [0, 15/8]. We have a critical point c ≈ 1.4 ∈ [0, 15/8], with P3 < 0 on [0, c] and P3 > 0 on [c, 15/8]. The critical value P3 (c) ≈ 79 is positive, so P3 (x) > 0 on the desired interval [0, 15/8]. Lemma 13. The polynomial
3x 2 Q(x) = 1 − 2 (a∞ − x)(36 − 5x)(12 + 4x) 2a∞ 2 2 + (6a∞ − 36)x (12 − 7x) − 36a∞ is positive on [0, 12/7]. Proof. As in previous cases, x = 0 is a root of this polynomial, so we examine g(x) := Q(x)/x. The derivative g (x) is a quadratic polynomial, so its roots can be found exactly. We see that g has a critical point c ≈ 1.4 in [0, 12/7], with g < 0 on [0, c] and g > 0 on [c, 12/7]. The critical value g(c) ≈ 177.8 is positive, so g > 0 on [0, 12/7].
An Isoperimetric Inequality for Fundamental Tones of Free Plates
445
10. Proof of the Isoperimetric Inequality Now that we have established the desired monotonicity of our quotient, we need two more lemmas before we can prove the isoperimetric inequality for the free plate under tension. Our next lemma is a simple observation about integrals of monotone and partially monotone functions, which is a special case of more general rearrangement inequalities (see [16, Chap. 3]). Lemma 14. For any radial function F(r ) that satisfies the partial monotonicity condition (23) for ∗ ,
F dx ≤
∗
F dx
with equality if and only if = ∗ . For any strictly increasing radial function F(r ),
F dx ≥
∗
F dx
with equality if and only = ∗ . Proof. Note that || = |∗ | with |\∗ | = |∗ \|. Suppose F satisfies (23) for ∗ . The result follows from decomposing the domain:
F dx =
∩∗
F dx +
≤ ≤ ≤
∩∗ ∩∗ ∩∗ ∗
F dx
F d x + sup |F(x)|| \ ∗ | x∈\∗
F dx +
inf
x∈∗ \
|F(x)||∗ \ |
since F satisfies (23).
F dx +
=
\∗
∗ \
F dx
F d x.
Note that if |\∗ | > 0, either the second inequality or the third is strict by the strict inequality in (23). If F is increasing, then apply the first part of the lemma to the function −F. The final lemma describes how the eigenvalues change with the dilation of the region, and is used in the proof of the theorem to show we need only consider with volume equal to that of the unit ball. We will use the notation s := {x ∈ Rd : x/s ∈ } for s > 0. Lemma 15 (Scaling). For all s > 0, we have ω(τ, ) = s 4 ω(s −2 τ, s).
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Proof. For any u ∈ H 2 () with u d x = 0, let u(x) ˜ = u(x/s). Then u˜ is a valid trial function on s and so |D 2 u| ˜ 2 + s −2 τ |D u| ˜ 2 dx ˜ = s Q s −2 τ,s [u] 2 s u˜ d x −2 2 |s (D u)(x/s)|2 + s −2 τ |s −1 (Du)(x/s)|2 d x = s 2 s u(x/s) d x s −4+d |D 2 u|2 + τ |Du|2 dy taking y = x/s, = s d u 2 dy = s −4 Q τ, [u].
Now the lemma follows from the variational characterization of the fundamental tone. We can now prove our main result. Proof (Proof of Theorem 1). Once we have established inequality (18) for all regions of volume equal to that of the unit ball and all τ > 0, we obtain (18) for regions of arbitrary volume, since ω(τ, ) = s 4 ω(s −2 τ, s) ≤ s 4 ω(s −2 τ, s∗ ) = ω(τ, ∗ ), for all s > 0 by Lemma 15. Thus it suffices to prove the theorem for with volume equal to that of the unit ball, so that ∗ is the unit ball. We may also translate as in Lemma 5, which leaves the fundamental tone unchanged. Then, N [ρ] d x ω≤ by Lemma 7 ρ2 d x ∗ N [ρ] d x by Lemmas 8, 9, and 14 ≤ 2 ∗ ρ d x = ω∗ ,
by applying the equality condition in Lemma 7. Finally, if equality holds, then must be a ball, by the equality statement in Lemma 14. 11. Further Directions The isoperimetric problem for the free plate considered in this paper can be generalized in several different directions: considering the case where the material property Poisson’s ratio is nonzero, investigating a stronger inequality involving the harmonic mean of eigenvalues, and considering the problem on curved spaces. Poisson’s Ratio. One generalization of the free plate problem is to account for Poisson’s Ratio, a property of the material of the plate that describes how a rectangle of the material stretches or shrinks in one direction when stretched along the perpendicular direction. Our Rayleigh quotient and work so far all hold for a material where Poisson’s Ratio is zero. Most real-world materials have σ ∈ [0, 1/2], although there exist some materials with negative Poisson’s Ratio.
An Isoperimetric Inequality for Fundamental Tones of Free Plates
447
We will assume σ ∈ [0, 1) in order to be assured of coercivity of the generalized Rayleigh quotient, given by (1 − σ )|D 2 u|2 + σ (u)2 + τ |Du|2 d x Q[u] = . (33) 2 |u| d x This quotient reduces to our previous quotient (1) when σ = 0. Following our earlier derivation, we obtain the same eigenvalue equation u − τ u = ωu, along with new natural boundary conditions on ∂, which reduce to the old ones when σ = 0, ∂ 2u (1 − σ ) 2 + σ u = 0, ∂n ∂ ∂u ∂u 2 = 0. − (1 − σ )div∂ P∂ (D u)n − τ ∂n ∂n ∂ The generalization to nonzero σ does not change the eigenvalue equation and hence the general form of solutions is preserved. However, the change in the Rayleigh quotient affects the proof of Theorem 3 of [6], which identified the fundamental mode of the ball. This in turn affects the proof of the isoperimetric inequality in this paper. We can no longer complete the square in the Rayleigh quotient as in [6, Theorem 3] to show the fundamental mode of the ball corresponds to l = 0 or 1, although for some values of σ we can adapt the proof to show the lowest eigenvalue corresponding to l = 1 is lower than that for l = 0. Harmonic mean of low eigenvalues. In two dimensions, Szeg˝o was able to prove a stronger statement of the Szeg˝o-Weinberger inequality using conformal mappings [26,28]. Specifically, he proved that the sum of reciprocals 1 1 + μ1 μ2 is minimal for a disk. In other words, the harmonic mean of μ1 and μ2 is maximal for the disk. Our investigation in [7, Chap. 3] with the moment of inertia suggests a similar result for the free plate, since the moment of inertia is minimal for a ball. That is, for the free plate, we conjecture d 1 1 1 ≥ . d ωi () ω1 (∗ ) i=1
Curved spaces. We have taken our region to be in Euclidean space Rd , but we could consider the same eigenvalue problem on a region in spaces of constant curvature: the sphere and hyperbolic space. Other eigenvalue inequalities have been proven in these spaces [1]. In particular, the Szeg˝o-Weinberger inequality was proved for domains on the sphere by Ashbaugh and Benguria [2]. Another direction of generalization would be Hersch-type bounds for metrics on the whole sphere or torus; see [14].
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Acknowledgements. I am grateful to the University of Illinois Department of Mathematics and the Research Board for support during my graduate studies, and the National Science Foundation for graduate student support under grants DMS-0140481 (Laugesen) and DMS-0803120 (Hundertmark) and DMS 99-83160 (VIGRE), and the University of Illinois Department of Mathematics for travel support to attend the 2007 Sectional meeting of the AMS in New York. I would also like to thank the Mathematisches Forschungsinstitut Oberwolfach for travel support to attend the workshop on Low Eigenvalues of Laplace and Schrödinger Operators in 2009. Finally, I would like to thank my advisor Richard Laugesen for his support and guidance throughout my time as his student and for his assistance with refining this paper.
References 1. Ashbaugh, M.S.: Isoperimetric and universal inequalities for eigenvalues. In: Spectral theory and geometry (Edinburgh, 1998), London Math. Soc. Lecture Note Ser., 273, Cambridge: Cambridge Univ. Press, 1999, pp. 95–139 2. Ashbaugh, M.S., Benguria, R.: Sharp upper bound to the first nonzero Neumann eigenvalue for bounded domains in spaces of constant curvature. J. London Math. Soc. (2) 52(2), 402–416 (1995) 3. Ashbaugh, M.S., Benguria, R.: On Rayleigh’s conjecture for the clamped plate and its generalization to three dimensions. Duke Math. J. 78, 1–17 (1995) 4. Ashbaugh, M.S., Laugesen, R.S.: Fundamental tones and buckling loads of clamped plates. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 23(2), 383–402 (1996) 5. Bandle, C.: Isoperimetric Inequalities and Applications. Pitman Advanced Publishing Program, BostonLondon-Melbourne, Pitman (1980) 6. Chasman, L.M.: The fundamental tone of the free circular plate. http://arxiv.org/abs/1004.3316v1 [math.AP], 2010 Applicable Analysis (to appear) 7. Chasman, L.M.: Isoperimetric problem for eigenvalues of free plates. Ph.D thesis, University of Illinois at Urbana-Champaign, 2009. http://arxiv.org/abs/1004.0016v1 [math.SP], 2010 8. de Groen P.P.N. (1979) Singular perturbations of spectra. Asymptotic analysis, Lecture Notes in Math. 711. Springer, Berlin, pp. 9–32 9. Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Berlin: SpringerVerlag, 2001 (Reprint of 1998 edition) 10. Henrot A.: Extremum problems for eigenvalues of elliptic operators. Frontiers in Mathematics. Birkhäuser Verlag, Basel, 2006 11. Kawohl, B., Levine, H.A., Velte, W.: Buckling eigenvalues for a clamped plate embedded in an elastic medium and related questions. SIAM J. Math. Anal. 24(2), 327–340 (1993) 12. Kesavan, S.: Symmetrization and Applications. World Scientific, Singapore (2006) 13. Kornhauser, E.T., Stakgold, I.: A variational theorem for ∇ 2 u + λu = 0 and its application. J. Math. Phys. 31, 45–54 (1952) 14. Hersch, J.: Quatre propriétés isopérimétriques de membranes sphériques homogènes. C. R. Acad. Sci. Paris Ser. A-B 270, A1645–A1648 (1970) 15. Licari, H.P., Warner, H.: Domain dependence of eigenvalues of vibrating plates. SIAM J. Appl. Math. 24(3), 383–395 (1973) 16. Lieb, E.H., Loss, M.: Analysis. Second edition. Graduate Studies in Mathematics, 14. Providence, RI: Amer. Math. Soc. 2001 17. Lorch, L., Szego, P.: Bounds and monotonicities for the zeros of derivatives of ultraspherical Bessel functions. SIAM J. Math. Anal. 25(2), 549–554 (1994) 18. Nadirashvili, N.S.: New isoperimetric inequalities in mathematical physics. In: Partial differential equations of elliptic type (Cortona, 1992), Sympos. Math. XXXV, Cambridge: Cambridge Univ. Press, 1994, pp. 197–203 19. Nadirashvili, N.S.: Rayleigh’s conjecture on the principal frequency of the clamped plate. Arch. Rat. Mech. Anal 129, 1–10 (1995) 20. Nazarov, S.A., Sweers, G.: A hinged plate equation and iterated Dirichlet Laplace operator on domains with concave corners. J. Diff. Eqs. 233(1), 151–180 (2007) 21. Nirenberg, L.: Remarks on strongly elliptic partial differential equations. Comm. Pure Appl. Math. 8, 649– 675 (1955) 22. Payne, L.E.: New isoperimetric inequalities for eigenvalues and other physical quantities. Comm. Pure Appl. Math. 9, 531–542 (1956) 23. Payne, L.E.: Inequalities for eigenvalues of supported and free plates. Quart. Appl. Math. 16, 111–120 (1958) 24. Rayleigh, J.W.S.: The theory of sound. New York: Dover Pub, 1945, Re-publication of the 1894/96 edition 25. Showalter, R.E.: Hilbert space methods for partial differential equations. Monographs and Studies in Mathematics, Vol. 1. London-San Francisco-Melbourne: Pitman, 1977
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26. Szeg˝o, G.: On membranes and plates. Proc. Nat. Acad. Sci. 36, 210–216 (1950) 27. Szeg˝o, G.: Inequalities for certain eigenvalues of a membrane of given area. J. Rat. Mech. Anal. 3, 343– 356 (1954) 28. Szeg˝o, G.: Note to my paper “On membranes and plates”. Proc. Nat. Acad. Sci. (USA) 44, 314–316 (1958) 29. Talenti, G.: On the first eigenvalue of the clamped plate. Ann. Mat. Pura Appl. (Ser. 4) 129, 265–280 (1981) 30. Taylor, M.E.: Partial Differential Equations. I. Basic Theory. Applied Mathematical Sciences, 115. New York: Springer-Verlag, 1996 31. Verchota, G.C.: The biharmonic Neumann problem in Lipschitz domains. Acta Math. 194(2), 217– 279 (2005) 32. Weinberger, H.F.: An isoperimetric inequality for the N -dimensional free membrane problem. J. Rat. Mech. Anal. 5, 633–636 (1956) 33. Weinstock, R.: Calculus of Variations. New York: Dover, 1974 (Reprint of 1952 edition) Communicated by B. Simon
Commun. Math. Phys. 303, 451–508 (2011) Digital Object Identifier (DOI) 10.1007/s00220-011-1202-4
Communications in
Mathematical Physics
The Transition to a Giant Vortex Phase in a Fast Rotating Bose-Einstein Condensate M. Correggi1 , N. Rougerie2 , J. Yngvason3,4 1 2 3 4
CIRM, Fondazione Bruno Kessler, Via Sommarive 14, 38123 Trento, Italy Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, 4 place Jussieu, 75005 Paris, France Erwin Schrödinger Institute for Mathematical Physics, Boltzmanngasse 9, 1090 Vienna, Austria Fakultät für Physik, Universität Wien, Boltzmanngasse 5, 1090 Vienna, Austria. E-mail:
[email protected]
Received: 6 May 2010 / Accepted: 9 September 2010 Published online: 17 February 2011 – © Springer-Verlag 2011
Abstract: We study the Gross-Pitaevskii (GP) energy functional for a fast rotating Bose-Einstein condensate on the unit disc in two dimensions. Writing the coupling parameter as 1/ε2 we consider the asymptotic regime ε → 0 with the angular velocity proportional to (ε2 | log ε|)−1 . We prove that if = 0 (ε2 | log ε|)−1 and 0 > 2(3π )−1 then a minimizer of the GP energy functional has no zeros in an annulus at the boundary of the disc that contains the bulk of the mass. The vorticity resides in a complementary ‘hole’ around the center where the density is vanishingly small. Moreover, we prove a lower bound to the ground state energy that matches, up to small errors, the upper bound obtained from an optimal giant vortex trial function, and also that the winding number of a GP minimizer around the disc is in accord with the phase of this trial function. Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . 2. Useful Estimates . . . . . . . . . . . . . . . . . . 3. Reduction to an Auxiliary Problem on an Annulus 4. Estimates of the Reduced Energy . . . . . . . . . 5. Energy Asymptotics and Absence of Vortices . . . Appendix A . . . . . . . . . . . . . . . . . . . . . . .
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451 462 474 478 499 502
1. Introduction An ultracold Bose gas in a magneto-optical trap exhibits remarkable phenomena when the trap is set in rotational motion. In the ground state the gas is a superfluid and responds to the rotation by the creation of quantized vortices whose number increases with the angular velocity. The literature on this subject is vast but there exist some excellent reviews [A,Co,Fe1]. The mathematical analysis is usually carried out in the framework of the (time independent) Gross-Pitaevskii (GP) equation that has been derived from the
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many-body quantum mechanical Hamiltonian in [LSY] for the non-rotating case and in [LS] for a rotating system at fixed coupling constant and rotational velocity. When these parameters tend to infinity the leading order approximation was established in [BCPY] but it is still an open problem to derive the full GP equation rigorously in this regime. The present paper, however, is not concerned with the many-body problem and the GP description will be assumed throughout. When studying gases in rapid rotation a distinction has to be made between the case of harmonic traps, where the confining external potential increases quadratically with the distance from the rotation axis, and anharmonic traps where the confinement is stronger. In the former case there is a limiting angular velocity beyond which the centrifugal forces drive the gas out of the trap. In the latter case the angular velocity can in principle be as large as one pleases [Fe2]. Using variational arguments it was predicted in [FB] that at sufficiently large angular velocities in such traps a phase transition changing the character of the wave function takes place: Vortices disappear from the bulk of the density and the vorticity resides in a ‘giant vortex’ in the central region of the trap where the density is very low. This phenomenon was also discussed and studied numerically in both earlier and later works [FJS,FZ,KB,KF,KTU], but proving theorems that firmly establish it mathematically has remained quite challenging. The emergence of a giant vortex state at large angular velocity with the interaction parameter fixed was proved in [R1] for a quadratic plus quartic trap potential. The present paper is concerned with the combined effects of a large angular velocity and a large interaction parameter on the distribution of vorticity in an anharmonic trap. In particular, we shall establish rigorous estimates on the relation between the interaction strength and the angular velocity required for creating a giant vortex. The physical regime of interest and hence the mathematical problem treated is rather different from that in [R1] but a common feature is that the annular shape of the condensate is created by the centrifugal forces at fast rotation. By contrast, the paper [AAB] focuses on a situation where, even at slow rotation speeds, the condensate has a fixed annular shape due to the choice of the trapping potential. The basic methodology of the present paper is related to that of [AAB] and [IM1,IM2] but with important novel aspects that will become apparent in the sequel. As in the papers [CDY1,CY], that deal mainly with rotational velocities below the giant vortex transition, the mathematical model we consider is that of a two-dimensional ‘flat’ trap. The angular velocity vector points in the direction orthogonal to the plane and the energy functional in the rotating frame of reference is1 (1.1) E GP [] := dr |∇|2 − 2 · ∗ L + ε−2 ||4 , B
where we have denoted the physical angular velocity by 2, the integral domain is the unit disc B = {r ∈ R2 : r ≤ 1}, and L = −i r ∧ ∇ is the angular momentum operator. It is also useful to write the functional in the form E GP [] = (1.2) dr |(∇ − i A) |2 − 2 r 2 ||2 + ε−2 ||4 , B
where the vector potential A is given by A := ∧ r = r eϑ . 1 The notation “a := b” means that a is by definition equal to b.
(1.3)
Transition to Giant Vortex in Fast Rotating BE Condensates
453
Here (r, ϑ) are two-dimensional polar coordinates and eϑ a unit vector in the angular direction. The complex valued is normalized in L 2 (B) and the ground state energy is defined as E GP :=
inf
2 =1
E GP [].
(1.4)
The minimization in (1.4) leads to Neumann boundary conditions on ∂B. Alternatively we could have imposed Dirichlet boundary conditions, or considered the case of a homogeneous trapping potential as in [CDY2]. Our general strategy applies to these cases too, but with nontrivial modifications and new aspects that are dealt with in separate papers [CPRY1,CPRY2]. We denote by GP a minimizer of (1.2), i.e., any normalized function such that GP E [ GP ] = E GP . The minimizer is in general not unique because vortices can break the rotational symmetry [CDY1, Prop. 2.2] but any minimizer satisfies the variational equation (GP equation) 2 (1.5) − GP − 2 · L GP + 2ε−2 GP GP = μGP GP , with Neumann boundary conditions and the chemical potential 4 1 dr GP . μGP := E GP + 2 ε B
(1.6)
The subsequent analysis concerns the asymptotic behavior of GP and E GP as ε → 0 with tending to ∞ in a definite way. As discussed in [CDY1] the centrifugal term −2 r 2 ||2 in (1.2) creates for ε−1 a ‘hole’ around the center where the density | GP |2 is exponentially small while the mass is concentrated in an annulus of thickness ∼ ε at the boundary. Moreover, by establishing upper and lower bounds on E GP , it was shown in [CY] that in the asymptotic parameter regime | log ε| (ε2 | log ε|)−1
(1.7)
the ground state energy is to subleading order correctly reproduced by the energy of a trial function exhibiting a lattice of vortices reaching all the way to the boundary of the disc. It can even be shown that in the whole regime (1.7) the vorticity of a true minimizer GP in the annulus is uniformly distributed2 with density /π . 1.1. Main results. In this paper we investigate the case =
0 2 ε | log ε|
(1.8)
with 0 a fixed constant and prove that for 0 sufficiently large a phase transition takes place: Vortices disappear from the bulk of the density and all vorticity is contained in a hole where the density is low. 2 Cf. [CY, Theorem 3.3] and [CPRY1, Theorem 1.1].
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To state this result precisely, we first recall from [CDY1] that the leading order in the asymptotic expansion of the GP ground state energy is given by the minimization of the ‘Thomas-Fermi’ (TF) functional obtained from E GP by simply neglecting the kinetic term: (1.9) dr −2 r 2 ρ + ε−2 ρ 2 , E TF [ρ] := B
where ρ plays the role of the density ||2 . The minimizing normalized density, denoted by ρ TF , can be computed explicitly (see Appendix). For > 2π −1/2 ε−1 it vanishes for r < Rh with √ 1 − Rh2 = (2/ π )(ε)−1 ∼ −1 (1.10) 0 ε| log ε|, while for Rh ≤ r ≤ 1 it is given by (ε)2 2 (r − Rh2 ). (1.11) 2 Our proof of the disappearance of vortices is based on estimates that require the TF density to be sufficiently large. For this reason we consider an annulus ρ TF (r ) =
A˜ := {r : Rh + ε| log ε|−1 ≤ r ≤ 1},
(1.12)
where ρ TF (r ) 0 ε−1 | log ε|−3 . Our result on the disappearance of vortices from the essential support of the density is as follows: Theorem 1.1 (Absence of vortices in the bulk). If is given by (1.8) with 0 > 2(3π )−1 , ˜ then GP does not have any zero in A˜ for ε > 0 small enough. More precisely, for r ∈ A, | log ε|3 GP (1.13) | (r)|2 − ρ TF (r ) ≤ C 7/8 ρ TF (r ). ε Remark 1.1 (Bulk of the condensate). The annulus A˜ contains the bulk of the mass. Indeed, because A˜ ρ TF = 1 − o(1) and GP is normalized, (1.13) implies that also GP 2 A˜ | | = 1 − o(1). The prosof of Theorem 1.1 is based on precise energy estimates that involve a comparison of the energy of the restriction of GP to A˜ with the energy of a giant vortex trial function. By the latter we mean a function of the form ˆ (r) = f (r) exp(i ϑ)
(1.14)
ˆ ∈ N. It is convenient to write3 with f real valued and ˆ = [] − ω
(1.15)
ˆ with ω ∈ Z and use ω as a label for the -dependent quantities in the sequel. We define a functional of f by ˆ EˆωGP [ f ] : = E GP [ f exp(i ϑ)] ˆ 2 r −2 − 2) ˆ f 2 + ε−2 f 4 . = dr |∇ f |2 + ( B
3 We use the notation ‘[ · ]’ for the integer part of a real number.
(1.16)
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Minimizing the functional EˆωGP [ f ] (see Proposition 2.3), that is convex in f 2 , over L 2 ˆ we obtain an energy denoted by Eˆ ωGP and a minimizer that normalized f for fixed , is rotationally symmetric, without a zero for r > 0, and unique up to sign.4 We denote the unique positive minimizer by gω . If 0 ≤ ω ≤ Cε−1 the corresponding density gω2 is close to ρ TF (see Remark 1.3 below). It is clear that ˆ = Eˆ ωGP E GP ≤ E GP [gω exp(i ϑ)]
(1.17)
for any value of ω and hence E GP ≤ Eˆ GP := inf Eˆ ωGP . ω∈Z
(1.18)
Our second main result is a lower bound that matches (1.18) up to small errors (see Remark 1.4 below) and an estimate of the phase that optimizes Eˆ ωGP . Theorem 1.2 (Ground state energy). For 0 > 2(3π )−1 and ε > 0 small enough the ground state energy is | log ε|3/2 E GP = Eˆ GP − O 1/2 . (1.19) ε (log | log ε|)2 Moreover Eˆ GP = Eˆ ωGP , with ωopt ∈ N satisfying opt
2 ωopt = √ 1 + O(| log ε|−1/2 ) . 3 πε
(1.20)
Because GP does not have any zeros in the annulus A˜ the winding number (degree) of GP around the unit disc is well defined. Our third main result is that the giant vortex phase [] − ωopt is, up to possible small errors, equal to this winding number. Theorem 1.3 (Degree of a minimizer). For 0 > 2(3π )−1 and ε > 0 small enough, the winding number of GP is deg { GP , ∂B} = [] − ωopt (1 + o(1))
(1.21)
with ωopt as in (1.20). The following remarks are intended to elucidate the phase difference ωopt and to justify the claim that the remainder in (1.19) is, indeed, a small correction to Eˆ GP . Remark 1.2 (Giant vortex phase). For a fixed trial function f the energy (1.16) is minimal for −1 ˆ = + O(1). (1.22) dr r −2 f 2 4 Instead of requiring f to be real we could have required f to be radial in which case the minimizer is unique up to a constant phase.
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If f 2 is close to ρ TF and thus essentially concentrated in an annulus of width ∼ (ε) = O(ε| log ε|), it follows that ˆ = O(ε−1 ). 0 < ( − )
(1.23)
√ The leading term 2/(3 π )ε−1 in (1.20) can, indeed, be computed from (1.22) with 2 TF f = ρ , or more simply, for f 2 (r ) increasing linearly from 0 at r = Rh to its maximal value at r = 1. It is worth noting that if f 2 (r ) were constant in the interval [Rh , 1], √ ˆ therefore the leading term for ω according to (1.22) would be 2/( π ε) and the phase ˆ is closer to because approximately equal to (/π ) × area of the hole. The optimal of the inhomogeneity of the density in the annulus. Remark 1.3 (Giant vortex density functional). The functional (1.16) can also be written EˆωGP [ f ] = (1.24) dr |∇ f |2 − 2 r 2 f 2 + Bω2 f 2 + ε−2 f 4 B
with ˆ Bω (r ) := r − /r.
(1.25)
If ω = O(ε−1 ), then Bω (r ) = O(ε−1 ) close to the boundary of the disc. Neglecting the term Bω2 f 2 as well as the gradient term in (1.24) in comparison with the centrifugal and interaction terms leads to the TF functional evaluated at f 2 . This makes plausible the assertion above that gω2 is close to ρ TF if ω = O(ε−1 ) and this will indeed be proved in Sect. 2. Remark 1.4 (Composition of the ground state energy). If one drops the gradient term in (1.24) but retains the term with Bω2 , one obtains a modified (and ω-dependent) TF functional EˆωTF [ρ] := (1.26) dr −2 r 2 ρ + Bω2 ρ + ε−2 ρ 2 . B
Its minimizer and minimizing energy Eˆ ωTF can be computed √ explicitly (see the Appendix) and one sees that the energy is minimal for ω = 2/(3 πε)(1 + o(1)). Denoting the minimal value by Eˆ TF , we have Eˆ TF = E TF + O(ε−2 ),
(1.27)
while √ E TF = −2 − 4/(3 π )ε−1 = O(ε−4 | log ε|−2 ) + O(ε−3 | log ε|−1 ). (1.28) The term O(ε−2 ) in (1.27) is the angular kinetic energy corresponding to the third term in (1.24). The difference between Eˆ GP and Eˆ TF is the radial kinetic energy of the order (ε)2 | log ε| = O(ε−2 | log ε|−1 ). The remainder in (1.19) is thus much smaller than all terms in Eˆ GP . It is also smaller than the energy a vortex in A˜ would have, which is | log ε| × density = O(ε−1 ).
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1.2. Proof strategy. We now explain the general strategy for the proof of the main results. The first step is an energy splitting as in [LM]. We write a generic, normalized wave function as (r) = gω (r )w(r) with a complex valued function w. The identity dr |∇|2 = dr (−gω )gω |w|2 + dr gω2 |∇w|2 (1.29) B
B
B
holds by partial integration and the boundary condition on gω . Using the variational ˆ ˆ as in (1.15), we obtain a equation for gω and writing w(r) = exp(i ϑ)v(r) and splitting of the energy functional: E GP [] = Eˆ ωGP + Eω [v] with Eω [v] := where
(1.30)
g4 dr gω2 |∇v|2 − 2gω2 B ω · (iv, ∇v) + ω2 (1 − |v|2 )2 , ε B
B ω (r ) := Bω (r )eϑ = r − ([] − ω) r −1 eϑ ,
(1.31)
(1.32)
and we have used the notation (iv, ∇v) :=
1 i(v∇v ∗ − v ∗ ∇v). 2
(1.33)
Since the divergence of the two-dimensional vector field gω2 B ω vanishes we can write 2gω2 B ω = ∇ ⊥ Fω
(1.34)
with a scalar potential Fω and the dual gradient ∇ ⊥ = (−∂ y , ∂x ). Stokes theorem then gives Eω [v] = dr gω2 |∇v|2 + Fω curl(iv, ∇v) B
−
∂B
dσ Fω (iv, ∂τ v) +
B
dr
gω4 (1 − |v|2 )2 . ε2
(1.35)
Here and in the rest of the paper ‘curl (iv, ∇v)’ stands for the 3-component of ∇ ∧ (iv, ∇v), ∂τ is the tangential derivative and dσ the Lebesgue measure on the circle, i.e, given a ball B R of radius R centered at the origin, ∂τ := R −1 ∂ϑ and dσ := Rdϑ on ∂B R . ˆ The giant vortex trial function gω exp(i ϑ) gives an upper bound to the energy and hence we see from (1.30) that Eω [u] ≤ 0,
(1.36)
where u is the remaining factor of GP after the giant vortex trial function has been ˆ u. The proof of Theorem 1.1 is based on a lower extracted, e.g., GP = gω exp(i ϑ) bound on Eω [u] that, for 0 large enough, would be positive and hence contradict (1.36) ˜ The main steps are as follows: if u had zeros in A.
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˜ so that – Concentration of the density: There is an annulus A, slightly larger than A, the supremum of | GP |2 over B\A is5 O(ε∞ ). The same holds for gω2 provided ω ≤ Cε−1 . This implies that for v = u the integrations in (1.31) can be restricted to A up to errors that are O(ε∞ ). – Optimization of the phase: Choice of ω = O(ε−1 ) so that Fω , chosen to vanish on the inner boundary of A, is small on ∂B. – Boundary estimate: Using the variational equations for GP and gω and the smallness of Fω on the boundary it is proved that the boundary term in (1.35) is small. This step uses the Neumann boundary conditions on ∂B in a strong way and its extension to the case of Dirichlet boundary conditions is a nontrivial open problem.6 – Vortex balls: Isolation of the possible zeros of u in ‘vortex balls’ and an estimate of the first term in (1.35) from below in terms of the infimum of gω2 on the vortex balls and the winding numbers of u around the centers. For the construction of vortex balls a suitable upper bound on the last term in (1.35) is essential. – Jacobian estimate: Approximation of the integral of Fω against the vorticity curl(iu, ∇u) by a sum of the values of Fω at the centers of the vortex balls multiplied by the corresponding winding numbers. – Gradient estimate: Estimates on ∇u leading to a lower bound on the last term in (1.35) that, for 0 large enough, excludes zeros of u. A key point to notice is that the vortex ball construction in combination with the jacobian estimate leads to a cost function defined as Hω (r ) :=
1 2 g (r )| log ε| + Fω (r ). 2 ω
(1.37)
The first term is, to leading approximation, the kinetic energy of a vortex of unit strength at radius r , the second term is the gain due to the potential energy of the vortex in the field 2gω2 B ω . If Hω is negative at some point, the energy may be lowered by inserting a vortex at this point. A positive value of Hω means that the cost of the kinetic energy outweighs the gain. Note that Hω depends on 0 through Fω ; in fact it is not difficult to see that7 |Fω (r )| ≤
C 2 g (r )| log ε|. 0 ω
(1.38)
Hence, if 0 is large enough, the cost function is positive everywhere on the annulus and vortices are energetically unfavorable. An upper bound, 2(3π )−1 , on the critical value for 0 is computed in the Appendix. This upper bound is in fact optimal, as demonstrated in [R2]. The proof requires additional ingredients. The construction of vortex balls is a technique introduced independently by Jerrard [J1] and Sandier [Sa] in Ginzburg-Landau (GL) theory, whereas the jacobian estimate originates from the work [JS]. Both techniques are described in details in the monograph [SS]. This method has been applied in [AAB] and [IM1,IM2] to functionals that at first sight look exactly like (1.31). There is, however, an essential difference: While in [AAB] and [IM1,IM2] the size of the relevant integral domain is fixed, the weight function gω2 5 We use the symbol O(ε ∞ ) to denote a remainder which is smaller than any positive power of ε, e.g., exponentially small. 6 This problem has recently been solved [CPRY1]. 7 Here, as in the rest of the paper, C stands for a finite, positive constant whose value may vary from one formula to another.
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is in our case concentrated in an annulus whose width tends to zero as ε → 0. This shrinking of the width of the annulus and the large gradient of gω2 are the main reasons for the complications encountered in the proof of our main theorems. To appreciate the difficulty the following consideration is helpful. As already noted, the concentration of gω2 on the annulus A˜ of width O(ε| log ε|) implies that Bω = O(ε−1 ) on this annulus. From (1.36) and the Cauchy-Schwarz inequality applied to the second term in (1.31), one obtains the bound A˜
dr
gω4 C (1 − |u|2 )2 ≤ 2 . ε2 ε
(1.39)
˜ the estimate (1.39) is not sufficient Now, although gω4 ∼ (ε)2 ∼ ε−2 | log ε|−2 on A, ˜ In fact, (1.39) is compatible with the vanishing for the construction of vortex balls on A. ˜ while for a of u on a ball of radius O(ε| log ε|), i.e., comparable to the width of A, construction of vortex balls one must be able to isolate the possible zeros of u in balls of much smaller radius. The solution of this problem, elaborated in Sect. 4, involves a division of the annulus into cells of area ∼ (ε)2 with the upshot that a local version of (1.39) holds in every cell. The local version, dr Cell
gω4 C | log ε| (1 − |u|2 )2 ≤ 2 × (number of of cells)−1 ∼ ε2 ε ε
(1.40)
means that, in the cell, u can only vanish in a region of area O(ε3 | log ε|3 ) that is much smaller than the area of the cell, i.e., ∼ ε2 | log ε|2 . Hence the vortex ball technique applies in the cells where (1.40), or a sufficiently close approximation to it, holds. The gist of the proof of a global lower bound, that eventually leads to Theorems 1.1 and 1.2, is a stepwise increase in the number of cells where an estimate close to (1.40) is valid until all potential zeros in the annulus are included in vortex balls. The proof of Theorem 1.3 involves in addition an estimate on the winding number of u. In the sketch of the proof strategy above we have for simplicity deviated slightly from the actual procedure that will be followed in the sequel. Namely, for technical reasons, we find it necessary to restrict the considerations to a problem on the annulus before the splitting of the GP energy functional as in (1.30). This means that, instead of the function gωopt and the optimal phase ωopt defined above, we shall work with corresponding quantities for a functional like (1.24) but with the integration restricted to A. The main reason for this complication is lack of precise information about the behavior of GP in a neighborhood of the origin. In fact, the distribution of the ‘giant vorticity’ of GP in the hole is unknown. In particular it is not known whether GP vanishes at the origin ˆ like gω , which has a zero there of order . 1.3. Heuristic considerations. A heuristic argument for the giant vortex transition, based on an analogy with an electrostatic problem, has been given in [CY] and goes as follows: Writing a wave function as (r) = |(r)| exp(iϕ(r)) with a real phase ϕ the kinetic energy term in (1.2) is (1.41) dr |∇|||2 + ||2 |∇ϕ − A|2 B
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with A(r) = r eϑ . We focus on the second term and consider the situation where the phase ϕ contains, besides the giant vortex, also a possible contribution from a vortex of unit degree at a point r 0 in the annulus of thickness ∼ (ε)−1 . The phase can thus be written ˆ + arg(r − r 0 ), ϕ(r) := ϑ
(1.42)
where arg(x, y) = arctan(y/x). √ The modulus || vanishes at r 0 and is small in a disc (‘vortex core’) of radius ∼ ε/ ∼ ε3/2 | log ε|1/2 around r 0 . The phase arg(r − r 0 ) can be regarded as the imaginary part of the complex logarithm if we identify R2 with C. The corresponding real part, i.e., conjugate harmonic function, is log |r − r 0 | which is the two-dimensional electrostatic potential of a unit point charge localized in r 0 . Likeˆ is ˆ log r , i.e., the electrostatic potential wise, the conjugate harmonic function of ϑ ˆ of a charge placed at the origin. By the Cauchy-Riemann equations for the complex logarithm we can write ∇ϕ = ∂r ϕ er + r −1 ∂ϑ ϕ eϑ = ∂r χ eϑ − r −1 ∂ϑ χ er
(1.43)
ˆ log r + log |r − r 0 |. χ (r) :=
(1.44)
with
After a rotation by π/2, i.e., replacement of eϑ by er at every point, the vector potential also has an electrostatic interpretation: r er is the electrostatic field of a uniform charge distribution with charge density /π . Employing (1.43) we now have |∇ϕ − A|2 = |∇χ − r er |2
(1.45)
and E(r) := ∇χ − r er is the electric field generated by the point charges and the uniform background. We can now apply Newton’s theorem to argue that the effect of the giant vortex, i.e., the first term in (1.44), is to neutralize in the annulus the field generated by the uniform ˆ is chosen to charge distribution in the ‘hole’, i.e., the second term in (1.45), provided match the area of the hole times the uniform charge density. √ By the same argument the point charge at r 0 neutralizes, outside a disc of radius ∼ 1/ , the effect of one unit of the continuous charge. The total charge of the continuous distribution in the annulus is Q ∼ (ε)−1 ∼ ε−1 . Inserting the vortex reduces the effective charge by one unit so the corresponding energy gain is ∼ ε−1 . On the other hand, the energy associated with the electrostatic field from the point charge outside the vortex core (that is cut off by the modulus of the wave function) is ∼ ||2 | log ε| ∼ (ε)| log ε|. The condition for the cost outweighing the gain is ε| log ε| ε−1 , i.e.,
1 , ε2 | log ε|
(1.46)
marking the transition to the giant vortex phase. It is also instructive to consider a version of the functional (1.31) where the variables have been scaled so that the width of the annulus becomes O(1). We define −1 : = (ε)−1 = −1 ˇ := gω (s), 0 ε| log ε|, s := r, g(s) ˇ v(s) ˇ := v(s), B(s) := B ω (s).
(1.47)
Transition to Giant Vortex in Fast Rotating BE Condensates
Then (1.31) is equal to gˇ 2 2 ˇ v] ˇ · (i v, ˇ 2 − 2B E[ ˇ := ds gˇ 2 |∇ v| ˇ ∇ v) ˇ + 2 (1 − |v| ˇ 2 )2 ε Bˇ
461
(1.48)
with Bˇ := B−1 a ball of radius −1 = 0 ε−1 | log ε|−1 . For the sake of a heuristic consideration we now assume that gω2 is constant, ∼ −1 , on the annulus of width and zero otherwise. We define a new small parameter εˇ := ε−1/2 ∼ ε1/2 | log ε|−1/2
(1.49)
ˇ ∼ ε−1 ∼ −1 (| log εˇ | + O(log | log εˇ |)). | B| 0
(1.50)
and note that We are thus led to consider the functional 1 2 2 2 ˇ ds |∇ v| ˇ − 2 B · (i v, ˇ ∇ v) ˇ + 2 (1 − |v| ˇ ) εˇ Aˇ
(1.51)
on an annulus Aˇ of width O(1) and an effective vector potential of strength O(−1 0 | log εˇ |). This is reminiscent of the situation considered in [AAB] and [IM1,IM2] where the domain is fixed and the rotational velocity is proportional to the logarithm of the small parameter. Moreover, increasing 0 decreases the coefficient in front of the logarithm. Hence for large 0 this coefficient is small and if the analysis of [AAB] and [IM1,IM2] would apply, one could conclude that vortices are absent. This reduction of the problem to known results is, however, too simplistic because the annulus Aˇ is not fixed: Although its width stays constant, its diameter and hence the area increases as (ˇε | log εˇ |)−2 . A new ingredient is needed, and in our approach this is the division of the annulus into cells as mentioned in the previous subsection. In the scaled version (1.48) of the energy functional these cells are (essentially) of fixed size. When writing the actual proofs we prefer to use the original unscaled variables but the picture provided by the scaling is still helpful, in particular for comparison with [AAB] and [IM1,IM2]. Remark 1.5 (Alternative approach). After the submission of this paper we learned [J3] of a possible alternative approach to prove the absence of vortices in the bulk, relying on [J2, Lemma 8] (see also [AJR, Lemma 4.1]). This method could replace some of the arguments in Sect. 4 and lead to a shorter proof of our Theorem 1.1 but is likely to yield worse remainder terms in the energy (Theorem 1.2). We also stress that the tools developed in Sect. 4 of the present paper are an essential input in the paper [R2], where it is proved that vortices do appear in the bulk if 0 < 2(3π )−1 . For this Lemma 8 in [J2] would not be sufficient. 1.4. Organization of the paper. The paper is organized as follows. In Sect. 2 we gather some definitions and notation that are to be used in the rest of the paper. We also prove useful estimates on the matter densities, both for the actual GP minimizer GP and for the ‘giant vortex profiles’ gω . In Sect. 3 we introduce the auxiliary problem that we are going to study on the annulus A and prove that it indeed captures the main energetic features of the full problem. Section 4 is devoted to the study of the auxiliary problem via tools from the Ginzburg-Landau theory. We conclude the proof of our main results in Sect. 5. Finally an Appendix gathers important facts about the TF functionals that we use in our analysis, as well as the analysis of the cost function that leads to an upper bound on the critical speed.
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2. Useful Estimates In this section we state some useful estimates which are going to be used in the rest of the paper. Some of them are simple consequences of energy considerations and in particular energy upper bounds, whereas others depend in a crucial way on the variational equations solved by GP , etc., and apply therefore only to (global) minimizers. In the first part of the section we investigate the properties of any GP minimizer: Starting from simple energy estimates, we prove the concentration of GP on an annulus of width O(ε| log ε|) close to the boundary of the trap and in particular its exponential smallness inside the hole. This is the major result about GP . The second part of the section is devoted to the analysis of the densities associated with the giant vortex energy. We shall introduce first some notation and then, exploiting energy considerations, discuss the properties of those minimizers: Most of them are very similar to the ones proven for GP , as, e.g., the exponential smallness, but we also need other general properties as, for instance, an a priori bound on the gradient of the densities and an estimate of their difference from the TF density ρ TF .
2.1. Estimates for GP minimizers. We briefly recall the main notation: Given the GP energy functional E GP [] defined in (1.2), we denote by E GP its infimum over L 2 -normalized wave functions. Any GP minimizer is denoted by GP and solves the variational equation (1.5). The TF functional E TF [ρ] was introduced in (1.9) and E TF and ρ TF stand for its ground state energy and density respectively (see also the Appendix). GP , denotes a restriction Any further label to the functionals E GP and E TF , as, e.g., ED of the integration in the functional to the domain D. The same convention is used for the corresponding ground state energies and minimizers. Finally we use the notation B(r, ) for a ball of radius centered at r, whereas B R is a ball with radius R centered at the origin. The starting point is a simple GP energy upper bound proven for instance in [CY, Prop. 4.2], i.e., E GP ≤ E TF + | log ε|(1 + o(1)) ≤ E TF + Cε−2 .
(2.1)
A straightforward consequence of such an upper bound is that the L 2 norm of GP is concentrated in the support ATF := {r : Rh ≤ r ≤ 1} of the TF minimizer. At the same time the bound implies a useful upper bound on | GP |: Proposition 2.1 (Preliminary estimates for GP ). As ε → 0, GP 2 ≤ O(1), | | − ρ TF 2 L (B ) GP 2 ≤ ρ TF ∞ L (B )
(2.2) L ∞ (B )
1 + O( ε| log ε|) .
(2.3)
Proof. In order to prove the first statement it is sufficient to use the fact that ρ TF is the positive part of the function (ε2 /2)(μTF + 2 r 2 ) together with the normalization of ρ TF
Transition to Giant Vortex in Fast Rotating BE Condensates
and GP and estimate 4 2 2
dr | GP |2 − ρ TF = GP + ρ TF − 2 dr | GP |2 ρ TF 4 2 B B GP 4 TF 2 ≤ + ρ − ε2 μTF − ε2 2 dr r 2 | GP |2 4 2 B
= ε2 E TF | GP |2 − E TF
≤ ε2 E GP − E TF ≤ C,
463
(2.4)
by (2.1). The proof of the second inequality is similar to the proof of Lemma 5.1 in [CY] and involves the variational equation (1.5). We define ρ GP := | GP |2 .
(2.5)
¯ it has to be The crucial point is that at any maximum point of ρ GP in the closed ball B, ρ GP ≤ 0. This is trivially true in the open ball B but can be extended to the boundary thanks to Neumann boundary conditions: Since ∂r GP = 0 at the boundary ∂B, which implies ∂r ρ GP = 0 there, ρ GP can have a maximum at r 0 ∈ ∂B only if ∇ρ GP (r 0 ) = 0 and therefore ρ GP (r 0 ) ≤ 0. From the variational equation (1.5) solved by GP , one can estimate
− ρ GP ≤ 4ε−2 ε2 μGP + ε2 2 − 2ρ GP ρ GP , (2.6) by using the properties 2 ∗ ∗ − ρ GP = − GP GP − GP GP − 2 ∇ GP , 2 2 GP ∗ 2 · L GP ≤ ∇ GP + 2 r 2 GP .
(2.7)
Now since ρ GP ≤ 0 at any maximum point of ρ GP , one immediately has GP 2 ≤ (ε2 μGP + ε2 2 )/2 ≤ ρ TF (1) + Cε2 |μGP − μTF |. ∞
On the other hand
ρ TF (1) = ρ TF
∞
= Cε = Cε−1 | log ε|−1 ,
(2.8)
and the difference between the chemical potentials (see (1.6) and (A.2) for the definitions) can be estimated as follows 1/2 GP 2 GP GP TF GP TF −2 TF |μ − μ | ≤ E − E + ε ρ + | | − ρ TF ∞ 2 ∞ 1/2
1/2 1 + ε3 | log ε||μGP − μTF | ≤ Cε−2 1 + ρ TF ∞
≤ Cε−5/2 | log ε|−1/2 1 + ε3/2 | log ε|1/2 |μGP − μTF |1/2 ,
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which yields |μGP − μTF | ≤ Cε−5/2 | log ε|−1/2 ,
(2.9)
and thus the result. A consequence of the L 2 estimate in (2.2) is that the L 2 norm of GP is concentrated inside the support of ρ TF or, in other words, the mass of GP inside the hole B Rh is small. Indeed one can easily realize that (2.2) implies GP 2 ≤ O( ε| log ε|), (2.10) 2 L (B R h )
since, due to the normalization of ρ TF and its support in ATF , 2 2 1 = GP 2 + GP 2 TF L (B R h ) L (A ) 2 2 = GP 2 + dr GP − ρ TF + 1 L (B R h )
2 ≥ GP 2
L (B R h )
ATF
+ 1 − O( ε| log ε|)),
where in the last step the Cauchy-Schwarz inequality and the size O(ε| log ε|) of ATF is used. This simple estimate can be refined (see, e.g., [CDY1, Prop. 2.5]) and one can actually show that GP is exponentially small in ε inside B Rh . The next proposition is devoted to the proof of two pointwise inequalities of this kind: Proposition 2.2 (Exponential smallness of GP inside the hole). As ε → 0 and for any r ∈ B, 1 − r2 GP 2 −1 −1 . (2.11) (r) ≤ Cε | log ε| exp − 1 − Rh2 Moreover there exists a strictly positive constant c such that for any O(ε7/6 ) ≤ r ≤ Rh − O(ε7/6 ), c GP 2 (2.12) (r) ≤ Cε−1 | log ε|−1 exp − 1/6 . ε Remark 2.1 (Comparison between (2.11) and (2.12)). At first sight the pointwise estimates in (2.11) and (2.12) look very similar. In fact, since 1 − Rh2 = O(ε| log ε|), one can easily realize that the first one yields a much better upper bound than the latter as soon as 1 − r 2 O(ε5/6 ), i.e., in particular for 1 − r 2 = O(1). The main drawback of the first inequality is however that it becomes much weaker and even useless if one gets closer to the radius Rh . More precisely as soon as 1 − r 2 O(ε) the bound is no longer exponentially small in ε. On the opposite the second inequality has no r dependence and a worse coefficient in the exponential function but it holds true and yields some exponential smallness up to a distance of order ε7/6 from the boundary ∂B Rh . Because of this fact the second inequality will be crucial in the reduction of the original problem to another one on an annulus close to the support of the TF minimizer. The first inequality,
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on the contrary, will be useful in a region far from the boundary of the hole ∂B Rh and close to the origin, where the second bound provides a worse estimate. Note also that the factor in front of the exponential in both (2.11) and (2.12) is essentially given by the sup of | GP |2 over the domain B. In the case of the first inequality this is actually needed in order to make it meaningful on the whole of B. Proof. The starting point of the proof of (2.11) is the inequality (2.6) together with the estimate (2.9), which yield
1 1 − ρ GP ≤ 4ε−2 ρ˜ TF (r ) + Cε− 2 | log ε|− 2 − ρ GP ρ GP , (2.13) where we have set ρ˜ TF (r ) :=
ε2 TF μ + 2 r 2 , 2
(2.14)
which coincides with the TF density ρ TF inside ATF and is negative everywhere else. More precisely, for any r such that r 2 ≤ Rh2 − ε, one has − ρ GP + 2ε−3 | log ε|−2 ρ GP ≤ 0,
(2.15)
since in that region 1
1
ρ˜ TF (r ) + Cε− 2 | log ε|− 2 ≤ −ε3 2 (1 − o(1)) ≤ − On the other hand the function
r2 − 1 W (r ) := exp 1 − Rh2 − ε
1 . 2ε| log ε|2
,
satisfies for any r ≤ 1, −W + 2ε−3 | log ε|−2 W ≥
C 2 −1 −ε| log ε| − r W ≥ 0. + ε ε2 | log ε|2
If we then multiply W by ρ GP ∞ we get a supersolution to (2.15), so that by the maximum principle (see, e.g., [E, Theorem 1 at p. 508]) GP 2 GP 2 (r) ≤ W (r ), ∞
for any r ≤ Rh − ε. However W (r ) is an increasing function and W ( Rh2 − ε) = ρ GP ∞ by construction, which implies that the estimate trivially holds true for any r ∈ B. The estimate (2.11) is then a consequence of (2.3) and the inequality 1 − Rh2 = ε| log ε| ε. Concerning the proof of the second inequality (2.12), we first notice that (2.13) implies that, for any r ∈ B Rh such that r ≤ Rh − O(ε7/6 ), − ρ GP ≤ −Cε−17/6 | log ε|−2 ρ GP ,
(2.16)
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since in that region ρ˜ TF (r ) =
C ε2 2 2 r − Rh2 ≤ − 5/6 . 2 ε | log ε|2
(2.17)
For any r 0 such that r0 ≤ Rh − O(ε7/6 ) we can thus consider the annulus I = {r ∈ B : r ∈ [r0 − , r0 + ]} for some = O(ε7/6 ), such that r0 + ≤ Rh − O(ε7/6 ), i.e., the outer boundary of the annulus is still at a distance of order ε7/6 from ∂B Rh , and it is straightforward to verify that the function 2 − (r − r0 )2 U (r ) := ρ GP exp , ∞ ε5/2
(2.18)
satisfies − U = −
2 4(r − r0 )2 2 2(r − r0 ) 1 + + + U (r ) U (r ) ≥ −C ε5 ε5/2 ε5/2 r ε5 ε5/2
≥ −Cε−8/3 U (r ) −Cε−17/6 | log ε|−2 U (r ),
(2.19)
where we have used the fact that |r − r0 | ≤ inside I and = O(ε7/6 ). Denoting now V (r) = ρ GP (r) − U (r ), we aim at proving that V ≤ 0, but one has, for any r ∈ I, −V (r) ≤ −Cε−17/6 | log ε|−2 V (r). Now at any maximum point of V in the interior of I it must be V ≤ 0, which implies V ≤ 0 because of the above inequality. Hence it remains only to prove that V ≤ 0 at boundary ∂I since the function might have a positive maximum there. However by construction V ≤ 0 at ∂I since U (r0 − ) = U (r0 + ) = ρ GP ∞ , and thus V (r) is negative everywhere. In particular ρ GP (r) ≤ U (r ) for all r such that r = r0 and = O(ε7/6 ), which yields (2.12). 2.2. The giant vortex densities. We now discuss some basic properties of the energy functional (1.16) together with those of an analogous functional where the integration is restricted to an annulus A = {r : R< ≤ r ≤ 1} ⊂ B with R< < Rh and |R< − Rh | ε| log ε|. A precise choice for R< will be made at the beginning of Sect. 3 (see (3.2)) but the results contained in this section apply to any R< satisfying the above conditions. As indicated at the end of Sect. 1.2 the restricted functional is for technical reasons actually more useful for the proof of the main results than the original functional. It is defined in analogy with (1.16) and (1.24) for real valued functions f on A by GP 2 2 −2 2 2 −2 4 EˆA [ f ] := dr |∇ f | + ([] − ω) r f − 2([] − ω) f + ε f ,ω A (2.20) dr |∇ f |2 − 2 r 2 f 2 + Bω2 f 2 + ε−2 f 4 , = A
with Bω (r ) given by (1.25). We use A as a subindex to label quantities associated with the annulus and thus denote the infimum of the functional (2.20) by
Transition to Giant Vortex in Fast Rotating BE Condensates GP ˆ GP Eˆ A ,ω := inf EA,ω [ f ] f 2 =1
467
(2.21)
while Eˆ ωGP stands for the corresponding infimum of (1.16). We shall use the short-hand notation EˆGP , Eˆ GP , g , etc., for quantities related either to (1.16) or (2.20), e.g., a statement about g is meant to apply to both gω and gA,ω . Proposition 2.3 (Minimization of EˆGP ). For any ω ∈ Z such that |ω| ≤ O(ε−1 ), the ground state energies Eˆ GP satisfy the estimates E TF ≤ Eˆ ωTF ≤ Eˆ GP ≤ Eˆ ωTF + O(ε−2 | log ε|−1 ) ≤ E TF + O(ε−2 ).
(2.22)
The minimizers gω and gA,ω exist, are radially symmetric and unique up to a global sign, which can be chosen so that they are given by positive functions solving the variational equation − g +
([] − ω)2 g − 2([] − ω)g + 2ε−2 g3 = μˆ GP g , r2
(2.23)
2 where the chemical potentials μˆ GP ˆ GP are fixed by the L normalization of g , i.e., μ = 4 GP −2 ˆ E + ε g 4 . In addition g are smooth and increasing and satisfy Neumann conditions at the boundary ∂B, i.e., ∂r g (1) = 0; gA,ω satisfies an identical condition at the inner boundary as well, i.e., ∂r gA,ω (R< ) = 0.
Proof. The lower bounds to the ground state energies are simply obtained by neglecting positive terms (the kinetic energies) in the functionals: In the case of EˆωGP (the other case is identical), one has EˆωGP [ f ] ≥ Eˆ TF [ f 2 ] ≥ Eˆ TF ≥ E TF , where we refer to the Appendix for the simple proof of the last inequality. The upper bound can be easily obtained by testing the functionals on suitable regularizations of ρˆωTF (see (A.6) for its definition and [CY], where such regularizations are performed on ρ TF ): The main correction to the energy is due to the radial kinetic energy of the regularization and one can easily realize that this energy can be made of order O(ε−2 | log ε|−2 ) times | log ε|. Indeed, ρˆωTF is a monotone function going from 0 to O(ε−1/2 | log ε|−1/2 ) in an interval of size O(ε| log ε|) and if it were smooth its kinetic energy would hence be O(ε−2 | log ε|−2 ). The exctra factor | log ε| is due to the regularization close to r = Rh . The last inequality in (2.22) is also discussed in the Appendix but it is basically due to the estimate |Bω (r)| ≤ r −1 − r + C|ω| + O(1) ≤ Cε−1 , (2.24) for any ω such that |ω| ≤ O(ε−1 ) and r ∈ A. We remark that the restriction of the integration to the annulus A has no effect because the support of the trial function can be assumed to be contained inside the support of ρˆωTF , i.e., the region where r ≥ Rˆ ω . On the other hand by (A.9), Rˆ ω ≥ Rh − O(ε2 | log ε|2 ), for any ω ∈ Z such that |ω| ≤ Cε−1 , which implies that the support of ρˆωTF is contained inside A.
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Existence and uniqueness of the minimizers trivially follow from strict convexity of the functionals with respect to the density f 2 , which can be made clearer by writing them as √ √ 2 EˆGP [ ρ] = dr ∇ ρ + ([] − ω)2 r −2 ρ + ε−2 ρ 2 − 2([] − ω), (2.25) where ρ = f 2 and we have used the L 2 normalization of f . Similarly the radial symmetry of g can be proven by averaging over the angular variable and exploiting the convexity of the functional. All the other properties of g , including the variational equations (2.23), are trivial consequences of the minimization: Positivity can be proven by noticing that the minimizers g are actually ground states of suitable one-dimensional Schrödinger operators and therefore cannot vanish except at the origin (see, e.g., [LL, Theorem 11.8]). Smoothness follows from (2.23) by a simple bootstrap argument, etc. The only property which requires a brief discussion is the monotonicity and we state it in a separate lemma. Lemma 2.1 (Monotonicity of the density). Let ρ(r ) ≥ 0 be the L 1 -normalized minimizer of √ 1 d ρ 2 a 2 + ρ + bρ dr r dr r2 R in H 1 (B\B R ) ∩ L 2 (B\B R ) with a, b > 0 and 0 ≤ R < 1. Then ρ is monotonously increasing in r . Proof. After a transformation of variables r 2 → s and considering ρ as a function of s, the functional takes the form 1 √ 2 d ρ a 2 + ρ + bρ , ds s ds s R2 and the normalization condition is
1 R2
ds ρ = const.
Suppose the assertion is false. Then ρ has a maximum at s = s1 for some R ≤ s1 < 1 and a local minimum at some s2 with s1 < s2 ≤ 1. For 0 < < ρ(s1 ) − ρ(s2 ) consider the set I = {s < s2 : ρ(s1 ) − ≤ ρ(s) ≤ ρ(s1 )}. Then, because ρ is continuous, (ε) := ds ρ(s) I
is strictly positive and () → 0 as → 0. Likewise, for δ > 0 we consider Jδ = {s > s1 : ρ(s2 ) ≤ ρ(s) ≤ ρ(s2 ) + δ} and the function (δ) := ds ρ(s) Jδ
that has the same properties as .
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Since both and are continuous, strictly positive and tend to zero when their argu¯ ments tend to zero, there exist ¯ , δ¯ > 0 such that ρ(s2 )+ δ¯ < ρ(s1 )− ¯ and (¯ ) = (δ): ¯ Because ρ is continuous one can, for any given δ > 0, always find an ¯ > 0 such that the ¯ holds and ε¯ → 0 as δ¯ → 0. The inequality ρ(s2 ) + δ¯ < ρ(s1 ) − ¯ equality (¯ ) = (δ) ¯ and hence also ¯ , are small enough. is fulfilled if δ, We now define a new function ρ¯ by putting ⎧ ⎨ ρ(s1 ) − ¯ , if s ∈ I¯ , ¯ if s ∈ Jδ¯ , ρ(s) ¯ := ρ(s2 ) + δ, ⎩ ρ(s), otherwise. We compute the energy of the new density ρ. ¯ Note that it belongs by definition to the ¯ minimization domain and it is normalized in L 1 because (¯ ) = (δ). The kinetic energy of ρ¯ vanishes in the intervals I¯ and Jδ¯ and the potential term for ρ¯ is strictly smaller than for ρ because 1/s is strictly decreasing and the value of ρ¯ on I¯ is larger than on Jδ¯ . Finally 1 1 ds ρ¯ 2 < ds ρ 2 R2
R2
because when modifying ρ to define ρ, ¯ mass is moved from I¯ to Jδ , where the density is lower. Altogether the functional evaluated on ρ¯ is strictly smaller than on ρ and this contradicts the assumption that ρ is a minimizer. Next we compare the densities g2 with the TF density and prove exponential smallness in the hole. The analogue of Proposition 2.1 is Proposition 2.4 (Preliminary estimates for g ). As ε → 0 and for any ω ∈ Z such that |ω| ≤ O(ε−1 ),
TF 2 2 ρ ∞ g − ρ TF 2 g 1 + O( = O(1), ≤ ε| log ε| . (2.26) ∞ L (B ) L (B ) L (B ) Proof. The proof of the L 2 estimate is exactly the same as for (2.2). The only difference occurs in the energy remainder on the r.h.s. of (2.4) which can now be bounded by ε2 ( Eˆ GP − E TF ) ≤ C because of (2.22). Here the condition on ω, i.e., |ω| ≤ Cε−1 , is used as in Proposition 2.3. The sup estimate can be proven by applying the same argument used to prove (2.3) to the variational equations (2.23) solved by g , exploiting as well Neumann boundary conditions: Indeed thanks to the monotonicity of the potential ([] − ω)2 r −2 , one obtains the inequality
2 2 2 2 2 −g2 ≤ 4ε−2 ε2 μˆ GP + 2ε ([] − ω) − ε ([] − ω) − 2g g , which as in the proof of Proposition 2.1 implies
1 2 GP g 2∞ ≤ ε μˆ + 2ε2 ([] − ω) − ε2 ([] − ω)2 . 2 The difference between the chemical potential can be estimated as in (2.9): GP μˆ − μTF ≤ O(ε−5/2 | log ε|−1/2 ), so that g 2∞ ≤ ρ TF (1)(1 + o(1)).
(2.27)
(2.28)
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As for GP the above estimates imply the exponential smallness of the densities g inside the hole. Moreover the L 2 estimate (2.2) can be refined to a pointwise estimate inside ATF and one can actually prove that there is a region where the difference between g2 and ρ TF is much smaller than ρ TF : Proposition 2.5 (Exponential smallness of g inside the hole). As ε → 0 and for any r ∈ B, 1 − r2 2 −1 −1 g (r ) ≤ Cε | log ε| exp − . (2.29) 1 − Rh2 Moreover there exists a strictly positive constant c such that for any r ≤ Rh − O(ε7/6 ), c (2.30) g2 (r ) ≤ Cε−1 | log ε|−1 exp − 1/6 . ε Proof. The proof can be easily reduced to the proof of Proposition 2.2 by using (2.28). Proposition 2.6 (Pointwise estimate for g ). As ε → 0 and for any ω ∈ Z with |ω| ≤ O(ε−1 ), 2 (2.31) g (r ) − ρ TF (r ) ≤ Cε2 | log ε|2 (r 2 − Rh2 )−3/2 ρ TF (r ) for any r ∈ ATF such that r ≥ Rh + O(ε3/2 | log ε|2 ). Proof. The proof is similar to the proof of Proposition 1 in [AAB] but for the sake of completeness we bring here all the details. The variational equation (2.23) can be rewritten in the following form: 2 2 ρ ˜ (r ) − g g , ε2
(2.32)
1 2 GP ε μˆ + ε2 2 r 2 − ε2 Bω2 (r ) . 2
(2.33)
− g = where the function ρ˜ is given by ρ˜ (r ) :=
Moreover by (2.24) and (2.28), ρ˜ − ρ TF ∞ TF ≤ ε2 μˆ GP − μTF + O(1) ≤ O(ε−1/2 | log ε|−1/2 ). (2.34) L (A ) On the other hand, for any r ∈ ATF such that r − Rh ≥ O(ε3/2 | log ε|2 ), ρ TF (r ) ≥ O(ε−1/2 ), and therefore ρ˜ (r ) ≥ ρ TF (r )(1 − C| log ε|−1/2 ) > Cε−1/2 ,
(2.35)
and in particular ρ˜ is strictly positive for such r, which is crucial in order to apply the maximum principle. The pointwise estimates are indeed proven by providing local super- and subsolutions to (2.32).
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For the upper bound, we consider an interval [r0 −δ, r0 +δ], where Rh +Cε3/2 | log ε|2 + δ < r0 < 1 − δ with δ 1, and the function ρ˜ (1) −1 W (r ) : = ρ˜ (r0 + δ) coth coth ρ˜ (r0 + δ)
−1 2 δ − |r − r0 | 2ρ˜ (r0 + δ) . (2.36) + (3δε) One has (see [AS, Proof of Prop. 2.1]) for any r ∈ [r0 − δ, r0 + δ],
2 2 − W ≥ 2 ρ˜ (r0 + δ) − W 2 W ≥ 2 ρ˜ (r ) − W 2 W, ε ε
(2.37)
where we have used the fact that ρ˜ (r ) is an increasing function of r . Moreover at the boundary of the interval W (r0 − δ) = W (r0 + δ) = ρ˜ (1) which is not smaller than g thanks to the upper bound (2.27), which reads g2 ≤ ρ˜ (1). Therefore W (r ) is a supersolution to (2.32) in the interval [r0 − δ, r0 + δ] and by the maximum principle δ g (r0 ) ≤ W (r0 ) = ρ˜ (r0 + δ) coth 2ρ˜ (r0 + δ) , (2.38) 3ε where we have used the fact that coth(x) is a non-increasing function. By the explicit expression (2.33) and the inequality (2.35), one has |ρ˜ (r0 + δ) − ρ˜ (r0 )| ≤ Cε2 2 δ,
ρ˜ (r0 ) ≥ ρ TF (r0 ) 1 − O(| log ε|−1/2 ≥ Cε2 2 (r02 − Rh2 ), so that (2.38) becomes g (r0 ) ≤
Cδ ρ˜ (r0 ) 1 + 2 r0 − Rh2
coth
δ 2ρ˜ (r0 + δ) . 3ε
(2.39) (2.40)
(2.41)
When the argument of coth tends to ∞, i.e., δ ρ˜ (r0 + δ) 1, ε we can bound coth(x) =
1 + e−2x ≤ (1 + Ce−2x ), 1 − e−2x
and obtain the inequality
g (r0 ) ≤ ρ˜ (r0 ) 1 +
Cδ 2 r0 − Rh2
2δ 1 + C exp − 2ρ˜ (r0 + δ) . 3ε
(2.42)
By (2.40) the second error term on the r.h.s. of the above expression is bounded from above by exp −Cδ r02 − Rh2 ,
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so that by taking
−1/2 , δ = Cε2 | log ε|2 r02 − Rh2
(2.43)
such an error can be made smaller than any power of ε, since one can choose the constant coefficient in δ arbitrarily large. With such a choice the other error term becomes
−3/2 > Cε1/2 | log ε|1/2 , Cε2 | log ε|2 r02 − Rh2 since by definition r 2 − Rh2 ≤ 1 − Rh2 ≤ O(ε| log ε|). Hence the second factor on the r.h.s. of (2.42) can be absorbed in the above remainder. Moreover for any r ≥ Rh , δ ≤ ε5/2 | log ε|1/2 ε3/2 and one can extend the estimate to any r ≥ Rh + O(ε3/2 | log ε|2 ). For the same reason the estimate applies also to the region [1 − 2δ, 1]: There one can use (2.27) and the fact that ρ˜ (1) − ρ˜ (1 − 2δ) ≤ C2 ε2 δ ≤ Cε1/2 , which is much smaller than the error term above. The final estimate is then
−3/2 , (2.44) g (r ) ≤ ρ˜ (r ) 1 + Cε2 | log ε|2 r 2 − Rh2 for any Rh + Cε3/2 | log ε|2 ≤ r ≤ 1. The next step is the replacement of ρ˜ with ρ TF by means of (2.34), which yields an additional remainder given by
−1 ρ TF (r )−1 ρ˜ (r ) − ρ TF (r ) ≤ C(ε)−2 r 2 − Rh2 ρ˜ − ρ TF ∞ TF L (A )
−1 ≤ Cε2 | log ε|2 r 2 − Rh2 , (2.45) which can however be absorbed in the error term in (2.41), since r 2 − Rh2 ≤ ε| log ε|. In order to prove a corresponding lower bound, we fix some r0 and δ as before, i.e., such that Rh + Cε3/2 | log ε|2 + δ < r0 < 1 − δ and δ 1. Since ρ˜ is an increasing function of r and g is positive 2 (2.46) − g ≥ 2 ρ˜ (r0 − δ) − g2 g . ε Then we denote by h(r ) the function solving for r ∈ B,
−h = ε˜ −2 1 − h 2 h, with Dirichlet boundary condition h(1) = 0 and ε˜ → 0. In [Se] it was proven that h satisfies the bound 1 − r2 1 − c exp − ≤ h(r ) ≤ 1. 2˜ε If we now set ˜ ) := h(r
ρ˜ (r0 − δ) h
ε ε˜ := , δ 2ρ˜ (r0 − δ)
|r − r0 | , δ
(2.47) (2.48)
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then h˜ solves in [r0 − δ, r0 + δ] the equation −h˜ =
2 ˜ ρ˜ (r0 − δ) − h˜ 2 h, 2 ε
with Dirichlet conditions at the boundary r = r0 ± δ. Thanks to (2.46), g is a superso˜ ) inside the lution for the same problem, so that by the maximum principle g (r ) ≥ h(r interval and in particular ˜ 0) ≥ g (r0 ) ≥ h(r
1 , ρ˜ (r0 − δ) 1 − c exp − 2˜ε
(2.49)
for any Rh + Cε3/2 | log ε|2 + δ < r0 < 1 − δ. Note that by choosing δ as in (2.43) the remainder in the above expression can be made smaller than any power of ε. However the estimate of ρ˜ (r − δ) in terms of ρ˜ (r ) provides the same remainder as in the upper bound proof, i.e., g (r ) ≥
−3/2 . ρ˜ (r ) 1 − Cε2 | log ε|2 r 2 − Rh2
The extension of the estimate to the whole region [Rh +O(ε3/2 | log ε|2 ), 1] as well as the replacement of ρ˜ with ρ TF can be done exactly as in the upper bound and the remainders included in the above error term. We conclude the section with an useful estimate of the gradient of the profiles g : Proposition 2.7 (Gradient estimate for g ). As ε → 0 and for any R > Rh + O(ε3/2 | log ε|2 ), one has
−1/4 ∇g L ∞ (B\B R ) ≤ Cε−1 R 2 − Rh2 g L ∞ (B) ,
(2.50)
which inside A˜ = {r : Rh + ε| log ε|−1 ≤ r ≤ 1} becomes ∇g L ∞ (A˜ ) ≤ Cε−7/4 | log ε|−3/4 .
(2.51)
Proof. We first exploit the fact that any g is radial to rewrite (2.23) as
− g − r −1 g + ([] − ω)2 r −2 g + 2ε−2 g3 = μˆ GP + 2[ − ω] g (2.52) and take the L ∞ norm inside B\B R , for some R > Rh + O(ε3/2 | log ε|2 ), of both sides to obtain (see (2.33)) g
L ∞ (B\B R )
≤ C r −1 g ∞ + Cε−2 (ρ˜ − g2 )g ∞ L (B\B R ) L (B\B R )
−1/2 −2 2 2 g L ∞ (B) , R − Rh ≤ C g L ∞ (B\B ) + ε R
(2.53)
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by (2.34) and the pointwise estimate (2.31): ρ˜ − g 2
L ∞ (B\B R )
≤ ρ TF − g2 L ∞ (B\B ) + O(ε−1/2 | log ε|−1/2 ) R −1/2 −1/2 ≤ C R 2 − Rh2 + O(ε−1/2 | log ε|−1/2 ) ≤ C R 2 − Rh2 ,
since R 2 − Rh2 ≤ Cε| log ε|. On the other hand by the Gagliardo-Nirenberg inequality (see, e.g., [N, Theorem at p. 125]) 1/2 1/2 g ∞ L (B\B R ) ≤ C g L ∞ (B\B R ) g L ∞ (B)
−1/4 1/2 1/2 −1 2 2 g L ∞ (B) + g L ∞ (B\B ) g L ∞ (B) , R − Rh ≤C ε R
which implies the result. 3. Reduction to an Auxiliary Problem on an Annulus The first step towards the proof of the main results is the reduction of the original GP energy functional to an analogous functional on a suitable annulus. The main ingredients of such a reduction are, on the one hand, the exponential smallness of the GP minimizers proven in the last section (see, e.g., (2.11), (2.12), (2.29), etc.), which intuitively implies that almost all the L 2 mass and therefore the energy are concentrated in a annulus close to the boundary, and on the other a decoupling of the GP energy functional, which allows the extraction of the giant vortex energies introduced in (1.16). The second part of the section is devoted to the discussion of the optimal giant vortex phase: Whereas both the energy splitting and upper bound hold true for any reasonable phase ω, they are useful only for specific choices of the phase. As we are going to see, the optimal phase ω0 can be defined as the minimizer of a suitable coupled problem in the annulus. We also discuss the existence of the analogous minimizer associated with the original problem in the ball B, i.e., the phase ωopt occurring in Theorems 1.2 and 1.3, as well as some relevant properties of it. 3.1. Energy decoupling. Before stating the main result of this section, we first recall and introduce some notation. The main object of the reduction is an annulus A := {r : R< ≤ r ≤ 1},
(3.1)
where the inner radius R< < Rh has to be chosen in a proper way so that two conditions are simultaneously fulfilled: The radius R< should be sufficiently far from Rh in such a way that the exponential smallness proven in Propositions 2.2 applies inside the complement of A. At the same time, R< must not be too far from Rh ; in fact we shall in Sect. 4 need that |R< − Rh | ε| log ε|−1 (see (4.89) and the use of (4.22) in the proof of Proposition 4.1). All these conditions are satisfied if one chooses R< := Rh − ε8/7 .
(3.2)
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The apparently strange power of ε occurring in the above expression is essentially motivated by the pointwise estimates (2.12) and (2.30), which hold true for any r ≤ Rh − O(ε7/6 ): In the definition of the inner radius of the annulus we could have chosen any R< satisfying such a condition in addition to |R< − Rh | ε| log ε|−1 . In particular any remainder given by a power of ε smaller than 7/6 but larger than 1 would have been all right. However for the sake of simplicity we make an explicit choice among the allowed remainders and pick ε8/7 . The auxiliary problem on the annulus is associated with the estimate of the energy functional
2 2 2 −2 2 2 |∇v| , (3.3) EA,ω [v] := 1 − |v| dr gA − 2B · ∇v) + ε g ω (iv, ,ω A,ω A
where B ω is defined in (1.32). According to the convention used in the rest of the paper, Eω without the label A stands for the same energy as above but with the integral extended to the whole of B. A key ingredient in the proof of the GP energy asymptotics is a lower estimate of (3.3). In fact in the next proposition, which is the main result proven in this section, we GP when v is suitably show that the energy (3.3) provides a lower bound to E GP − Eˆ A ,ω linked to GP : Proposition 3.1 (Reduction to an annulus). For any ω ∈ Z such that |ω| ≤ O(ε−1 ) and for ε sufficiently small GP ∞ GP GP ∞ Eˆ A ≤ Eˆ A ,ω + EA,ω [u ω ] − O(ε ) ≤ E ,ω + O(ε ),
(3.4)
where the function u ω ∈ H 1 (A) is given by the decomposition GP (r) =: gA,ω (r )u ω (r) exp {i([] − ω)ϑ} .
(3.5)
Proof. The proof of (3.4) is done by proving suitable upper and lower bounds to the GP ground state energy. The upper bound is obtained by testing E GP on a trial function of the form g(r ˜ ) exp{i([] − ω)ϑ}, where g˜ is an appropriate regularization of gA,ω , and using the definition (2.20). Since gA,ω does not vanish at R< , it is not in the minimization domain of E GP and one has to regularize it at the boundary ∂B R< , e.g., taking ⎧ if r ∈ A, ⎨ gA,ω (r ) if R< − εn ≤ r ≤ R< , g(r ˜ ) := c ε−n gA,ω (R< )(R< − εn − r ) (3.6) ⎩0 if r ≤ R< − εn , where c is a normalization constant and n some arbitrary power greater than 0. Thanks to the exponential smallness (2.30), which implies gA,ω (R< ) = O(ε∞ ), the normalization constant is c = 1 − O(ε∞ ) and the energy of g˜ in [R< − εn , R< ] is exponentially small as well. The upper bound trivially follows. The lower bound is mainly a consequence of a classical result of energy decoupling, which has been already used in different contexts, e.g., in [LM]. The starting point is however a reduction to the annulus A of the GP energy functional: Exploiting the exponential smallness (2.12) of GP , it is very easy to show that
GP GP − O(ε∞ ), (3.7) E GP = E GP GP ≥ EA
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GP to denote the restriction of the integration to A. where we have used the symbol EA Note that in the above inequality one can avoid the estimate of the gradient of GP by simply rewriting the functional as in (1.2) and neglecting all the positive terms given by the integration over B\A; the only negative term (centrifugal energy) can then be estimated by means of (2.12). Thanks to the exponential smallness, one also has that the L 2 mass of GP outside A is very small, i.e., GP 2 ≥ 1 − O(ε∞ ). (3.8) L (A)
Now inside A the decomposition (3.5) is well defined and therefore one can calculate
2 GP GP = EA,ω [u ω ] + EA dr |u ω |2 ∇gA,ω + gA,ω ∇gA,ω · ∇ |u ω |2 A
2 2 2 2 + ([] − ω)2 r −2 gA ,ω |u ω | − 2 ([] − ω) gA,ω |u ω |
4 2 + ε−2 gA . ,ω 2 |u ω | − 1
A simple integration by parts then yields 2 2 | |u dr gA,ω ∇gA,ω · ∇ ω = − dr |u ω |2 ∇gA,ω + gA,ω |u ω |2 gA,ω , A
A
since the boundary terms vanish because of the Neumann conditions satisfied by gA,ω at the boundaries ∂B R< and ∂B. Then one can replace in the above expression gA,ω by means of the variational equation (2.23) and the result is
GP 2 2 −2 4 |u | GP = EA,ω [u ω ] + μˆ GP EA dr g − ε dr gA A,ω A,ω ω ,ω A
A
GP 2 4 2 = EA,ω [u ω ] + μˆ GP − ε−2 gA,ω L 4 (A) A,ω L (A) GP ∞ ≥ EA,ω [u ω ] + Eˆ A ,ω − O(ε ),
(3.9)
by (3.8) and the definition of the chemical potential μˆ GP A,ω (see Proposition 2.3). 3.2. Optimal phases and densities. The idea behind the decomposition (3.5) is that, after the extraction from GP of a density gω and a giant vortex phase, i.e., the phase factor exp{i([]−ω)ϑ}, what is left is a function u ω , which contains all the remaining vorticity in GP . Therefore in the giant vortex regime, one would like to prove that |u ω | ∼ 1 inside a suitable annulus at the boundary of the trap, and the key tool to proving such a behavior is a detailed analysis of the reduced energy EA,ω . However, in order to prove such a result, both the phase and the associated density gA,ω have to be chosen in an appropriate way: The result stated in Proposition 3.1 is basically independent of ω, i.e., it applies to any reasonable giant vortex phase ω. NevGP ertheless the leading order term in the GP energy asymptotics is given in (3.4) by Eˆ A ,ω which depends in crucial way on ω and it is clear that in order to extract some delicate information about u ω like the absence of zeros, the estimate (upper bound) of the reduced energy EA,ω [u ω ] through (3.4) has to be very precise. This leads to the definition of a
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giant vortex optimal phase (associated with the annulus A), which is nothing but the GP with respect to ω. minimizer of the energy Eˆ A ,ω GP as In the next proposition we show that there exists at least one ω0 minimizing Eˆ A ,ω well as some properties which are going to be crucial in the rest of the proof. Proposition 3.2 (Optimal phase ω0 and density gA,ω0 ). For every ε > 0 there exists an GP . It satisfies ω0 ∈ Z minimizing Eˆ A ,ω
2 ω0 = √ 1 + O(| log ε|−1/2 ) . 3 πε
GP satisfies the bound Moreover the minimizer gA,ω0 of EˆA ,ω0 [] − ω0 2 − = O(1). dr gA ,ω0 r2 A
(3.10)
(3.11)
Proof. The existence of a minimizing ω0 ∈ Z can be easily proven by noticing that GP = +∞ since by (2.20), limω→±∞ Eˆ A ,ω GP 2 Eˆ A ,ω ≥ ([] − ω) − 2 ([] − ω),
which implies that, for given ε > 0, only finitely many ω ∈ Z can minimize the energy. The main ingredient for the proof of (3.10) is the energy bound (2.22), which implies, for any ω ∈ Z such that |ω| ≤ Cε−1 , GP −2 −1 ˆ GP ˆ TF (3.12) Eˆ ωTF0 ≤ Eˆ A ,ω0 ≤ E A,ω ≤ E ω + O(ε | log ε| ), √ by definition of ω0 . Choosing now ω = [2/(3 π ε)] in the r.h.s. of the above expression and using (A.10) for Eˆ ωTF , we get the inequality 2 2 2 TF E + ω0 − √ + − 2 ([] − ) ≤ Eˆ ωTF0 9π ε2 3 πε 2 ≤ E TF + − 2 ([] − ) + Cε−2 | log ε|−1, (3.13) 9π ε2 which yields the result. We now prove (3.11). A simple estimate yields
! ! GP −2 2 ˆ GP gA,ω ± 2 EˆA gA g = E dr − − ω r ([] ) 0 A ,ω 0 0 ,ω0 ±1 A,ω0 ,ω0 A
2 2 −2 ˆ GP + dr r −2 gA dr − ([] − ω0 ) r −2 gA ,ω0 ≤ E A,ω0 ± 2 ,ω0 + R< . (3.14)
A
Now suppose that
A
dr − ([] − ω0 ) r −2 g 2 > 2 , A,ω0 3 A
then, since R< = 1 − o(1), (3.14) would imply that there exists some ω¯ ∈ Z equal to ω0 ± 1 such that ! GP ˆ GP ˆ GP Eˆ A ,ω¯ ≤ EA,ω¯ gA,ω0 < E A,ω0 ,
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GP . Note that the above argument implies which contradicts the fact that ω0 minimizes Eˆ A ,ω that the constant on the r.h.s. of (3.11) is actually smaller than 1, i.e.,
dr − ([] − ω0 ) r −2 g 2 < 1. (3.15) A,ω0
A
As anticipated above and in the Introduction, there is another giant vortex phase that naturally emerges in the study of the above functionals, i.e., the one associated with the energy in the whole ball B: More precisely it is defined as the minimizer of Eˆ ωGP := inf EˆωGP [ f ] f 2 =1
with respect to ω ∈ Z and denoted by ωopt : Proposition 3.3 (Optimal phase ωopt ). For every ε > 0 there exists an ωopt ∈ Z minimizing Eˆ ωGP . It satisfies
2 (3.16) 1 + O(| log ε|−1/2 ) . ωopt = √ 3 πε Proof. The proof is basically identical to the one of (3.10): For instance it is sufficient to replace (3.12) with the corresponding version in B (see (2.22)). 4. Estimates of the Reduced Energy In this section we study the auxiliary problem introduced in Sect. 3. From now on we shall simplify notation by dropping some subscripts: gA,ω0 =: g, also
and
[] − ω0 eϑ , B(r ) := B ω0 (r ) = r − r
(4.1)
2 g4 2 . E[v] := dr g |∇v| − 2g B · (iv, ∇v) + 2 1 − |v| ε A
2
2
2
The following energy is crucial in our analysis:
2 g4 F[v] := . dr g 2 |∇v|2 + 2 1 − |v|2 ε A
(4.2)
We recall that u := u ω0 is defined by GP (r) =: g(r )u(r) exp {i([] − ω0 )ϑ}
(4.3)
and that from Proposition 3.1 we have E[u] ≤ O(ε∞ ).
(4.4)
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We also need to define a reduced annulus A˜ := {r : R> ≤ r ≤ 1}
(4.5)
R> := Rh + ε| log ε|−1 .
(4.6)
with
An important point is that from (2.31) we have the lower bound g 2 (r ) ≥
C ε| log ε|3
˜ on A.
(4.7)
The main result of this section is Proposition 4.1 (Bounds for the reduced energies). Let u be defined by (4.3). If 0 > 2(3π )−1 we have for ε small enough, F[u] ≤ C
| log ε|5/2
,
(4.8)
| log ε|3/2 . ε1/2 log | log ε|
(4.9)
ε1/2 (log | log ε|)2
E[u] ≥ −C
Since the proof of Proposition 4.1 is rather involved we sketch the main ideas before going into the details. It is useful to introduce a potential function F defined as follows: r r 1 F(r ) := 2 =2 ds g 2 (s) s − ([] − ω0 ) ds g 2 (s)B(s) · eϑ . (4.10) s R< R< We have ∇ ⊥ F = 2g 2 B,
F(R< ) = 0,
(4.11)
i.e., F is the “primitive” of 2g 2 B vanishing at R< . We refer to Subsect. 4.1 for further properties of F. Integrating by parts we have dr g 2 |∇u|2 − 2g 2 B.(iu, ∇u) = dr g 2 |∇u|2 + Fcurl(iu, ∇u) A A dσ F(1)(iu, ∂τ u) (4.12) − ∂B
and thus the energy E[u] can be rewritten as follows E[u] = dr g 2 |∇u|2 + dr Fcurl(iu, ∇u) A
A
−
∂B
dσ F(1)(iu, ∂τ u) +
A
dr
2 g4 2 1 − |u| . ε2
(4.13)
It is in this form that the energy is best bounded from below. The boundary term (third term in (4.13)) is estimated in the following way: The property of ω0 given in (3.11) implies that F(1) = O(1). We combine this fact with an estimate of the circulation of u on the boundary of the unit ball that we provide in
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Subsect. 4.2. Proving this estimate requires to derive a PDE satisfied by u and use it in much the same way as in the proof of the Pohozaev identity [P]. The first two terms can be estimated in terms of the vorticity of u. Indeed, suppose that |u| ∼ 1 except in some balls (that we identify with vortices) whose radii are much " # smaller than the width of A. Let us denote these balls by B(a j , t) j∈J with J ⊂ N and t ε| log ε|, i.e., much smaller than the width of the annulus. Then by Stokes theorem, if the degree of u around a j is d j (we systematically neglect remainder terms in this sketch) $ dr Fcurl(iu, ∇u) 2π F(a j )d j . (4.14) A
j∈J
Minimizing the sum of the first and the last term with respect to t yields t ∝ ε3/2 | log ε|1/2 (see also [CY, p. 6] for heuristics about the optimal size of the vortex core), which implies an estimate of the form $ ε| log ε| dr g 2 |∇u|2 2πg 2 (a j )|d j | log t A j∈J $ πg 2 (a j )|d j || log ε|. (4.15) j∈J
In Subsects. 4.3 and 4.4 we give a rigorous version of this heuristic analysis. The main tools were originally introduced in the context of Ginzburg-Landau (GL) theory (we refer to [BBH2,SS] and references therein). The method of growth and merging of vortex balls introduced independently by Sandier [Sa] and Jerrard [J1] provides a lower bound of the form (4.15) (see Subsect. 4.3). On the other hand, the jacobian estimate of Jerrard and Soner [JS] allow to deduce in Subsect. 4.4 that $ curl(iu, ∇u) 2π d j δ(r − a j ), (4.16) j∈J
i.e., the vorticity measure of u is close to a sum of Dirac masses. This implies an estimate of the form (4.14). Having performed this analysis, the role of the critical velocity becomes clear: We have essentially (note that the boundary term is negligible in a first approach) $ 1 2 E[u] g (a j )| log ε| + F(a j ) . 2π |d j | (4.17) 2 j∈J
If 0 > 2(3π )−1 the sum in the parenthesis is positive for any a j in the bulk (see the Appendix), which means that vortices become energetically unfavorable. So far we have assumed that the zeros of u were isolated in small ‘vortex’ balls. One can show that this is indeed the case by exploiting upper bounds to F[u] (this energy controls, via the co-area formula the size of the set where |u| is not close to 1). We derive a first bound from our a priori estimate on E[u] (4.4) in Subsect. 4.1. We emphasize however that this first bound is not strong enough to construct vortex balls in the whole annulus A. To get around this point we split in Subsect. 4.3 the annulus into cells and distinguish between two type of cells. In ‘good’ cells we have the proper control and perform locally the vortex balls construction. On the other hand we are able to show
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that there are relatively few ‘bad’ cells where the construction is not possible. Thus the analysis in the good cells allows to get lower bounds for E[u]. These bounds can in turn be used to improve our control on F[u] and reduce the number of bad cells. The analysis can then be repeated, but now on a larger set. Finally we see that there are no bad cells. Such an induction process is ‘hidden’ at the end of the proof of Proposition 4.1 in Subsect. 4.5 (see in particular the discussion after (4.101)). 4.1. Preliminaries. We first give some elementary estimates on B and F that we need in our analysis: Lemma 4.1 (Useful properties of B and F). Let B and F be defined in (4.1) and (4.10) respectively. We have B L ∞ (A) ≤ Cε−1 ,
|F(1)| = 2
1 R<
F L ∞ (A) ≤ Cε
−1
∇ F L ∞ (A) ≤ Cε 2 ds g (s)B(s) · eϑ ≤ C.
−2
(4.18)
, | log ε|
(4.19) −1
,
(4.20) (4.21)
Moreover there is a constant C such that |r − R< | 2 C |F(r )| ≤ C min g (r ), 1 + 2 |r − 1| ε ε | log ε|
(4.22)
for any r ∈ A. Proof. We have
B(r ) = ( − [])r + [](r − r −1 ) + ω0 r −1 eϑ
so (4.18) is a consequence of |ω0 | ≤ Cε−1 (see (3.10)) and the fact that |A| ∝ ε| log ε| (see (3.1) and (3.2)) and ∝ ε−2 | log ε|−1 . We obtain (4.20) from (2.26), (4.11) and (4.18). On the other hand F(R< ) = 0, which implies that (4.19) follows from (4.20) and |A| ∝ ε| log ε|. Moreover (4.21) is exactly the same as (3.11) in Proposition 3.2. To prove (4.22) we prove that |F(r )| is smaller than both terms on the right-hand side. From Eq. (4.10) we have r r 2 −1 |F(r )| ≤ 2 B L ∞ (A) ds g (s) ≤ Cε ds g 2 (s) R<
but
g2
is increasing so
r R<
R<
ds g 2 (s) ≤ g 2 (r )|r − R< |.
We deduce that |F(r )| is smaller than the first term on the right-hand side of (4.22). On the other hand, using (4.20) and (4.21) we have immediately |F(r )| ≤ C(1 + ε−2 | log ε|−1 |r − 1|) for some finite constant C. Thus (4.22) is proven.
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We now prove the energy bounds that are the starting point for the vortex balls construction in Subsect. 4.3. Lemma 4.2 (Preliminary energy bounds). We have, for ε > 0 small enough, C , ε2 C E[u] ≥ − 2 . ε
F[u] ≤
Proof. We have, using (4.18) together with the normalization of GP , 1 2 2 2 2 dr g B · (iu, ∇u) ≤ dr g |∇u| + 2 dr g 2 B 2 |u|2 2 A A A 1 C ≤ dr g 2 |∇u|2 + 2 . 2 A ε
(4.23) (4.24)
(4.25)
Equation (4.24) immediately follows. We obtain (4.23) from (4.4): F[u] = E[u] + 2 dr g 2 B · (iu, ∇u) ≤ O(ε∞ ) + 2 dr g 2 B · (iu, ∇u) A A 1 C 1 C 2 2 ≤ dr g |∇u| + 2 ≤ F[u] + 2 . (4.26) 2 A ε 2 ε Subtracting 21 F[u] from both sides of the last inequality we get (4.23). 4.2. Equation for u and boundary estimate. We first derive from the equations for GP and g an equation satisfied by u: Lemma 4.3 (Equation for u). Let u be defined by (4.3). We have on A − ∇(g 2 ∇u) − 2ig 2 B · ∇u + 2 where λ ∈ R satisfies the estimate |λ| ≤ C |E[u]| +
g4 2 |u| − 1 u = λg 2 u, ε2
1 1/2 . F[u] ε3/2 | log ε|1/2
(4.27)
(4.28)
Moreover u satisfies the boundary condition ∂r u = 0 on ∂B.
(4.29)
Proof. In this proof we use the short hand notation ˆ = e ˆ r := ([] − ω0 ) er .
(4.30)
The equation for u is a consequence of the variational equation (1.5) satisfied by GP , 2 (4.31) − GP − 2 · L GP + 2ε−2 GP GP = μGP GP ,
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and that solved by g (see (2.23)),
ˆ − g + r − r
2 g − 2 r 2 + 2ε−2 g 3 = μˆ GP g.
(4.32)
Direct computations starting from (4.3) show that − and
GP
ˆ2 2 ˆ ˆ = −ug − gu − 2∇u · ∇g + 2 gu + 2 g · Lu ei ϑ r r
ˆ ˆ − 2 · L GP = −2g · Lu − 2gu ei ϑ .
(4.33)
(4.34)
Plugging the equation for g in (4.33) we obtain 2 3 ˆ GP GP ˆ ˆ − = −gu − 2∇u · ∇g + μˆ gu + 2gu − 2 g u + 2g · Lu ei ϑ . ε (4.35) Next, combining (4.31), (4.34) and (4.35), we have − gu − 2∇u · ∇g +
2 3 2 2 ˆ |u| g − 1 u + 2 g · Lu − 2g · Lu = λgu, (4.36) ε2 r
where λ = μGP − μˆ GP .
(4.37)
There only remains to multiply (4.36) by g and reorganize the terms to obtain (4.27). The Neumann condition (4.29) is a straightforward consequence of the corresponding boundary conditions for g and GP . We now turn to the proof of (4.28). Multiplying (4.27) by u ∗ and integrating over A, we obtain, recalling (3.8), |λ| 1 − O(ε∞ ) ≤ |λ|
A
dr | GP |2 = E[u] +
A
dr
g4 4 |u| − 1 . ε2
(4.38)
Using Cauchy-Schwarz inequality
2 1/2
g 4
2 1/2 1 g4 4 2 2 2 2 ≤ |u| |u| g − 1 − 1 |u| + g 2 2 2 A ε A ε A ε
so (4.28) follows using the upper bounds (2.3) and (2.26) on g 2 |u|2 = | GP |2 and g 2 respectively together with |A| ∝ ε| log ε|. We remark that the equation satisfied by u is exactly of the form that we would have obtained if u had been a minimizer of E under a mass constraint for gu. We are now able to estimate the circulation of u at the boundary of the unit ball:
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Lemma 4.4 (Boundary estimate). We have, for ε > 0 small enough,
2 g4 C C 2 2 dσ g |∂τ u| + dσ 2 1 − |u|2 ≤ 3/2 F[u] + |E[u]|. (4.39) 1/2 ε ε | log ε| ε ∂B ∂B Proof. The main idea is a Pohozaev-like trick, like in [BBH2, Theorem 3.2]. We multiply both sides of (4.27) by (r − R> ) er · ∇u ∗ , then integrate over A˜ ⊂ A (see (4.5) for ˜ and take the real part. Recall that 1 − R> ∝ ε| log ε|. The first term the definition of A) is given by
1 − dr (r − R> ) er · ∇u ∗ ∇ g 2 ∇u + complex conjugate 2 A˜ R> R> 1 |∇u|2 + g 2 |er · ∇u|2 + g 2 (r − R> ) er · ∇|∇u|2 dr g 2 1 − = r r 2 A˜ 1 dσ g 2 (1 − R> ) |∂τ u|2 = 2 ∂B R> 2 g 2 |er · ∇u|2 − |∇u|2 , (4.40) − dr g (r − R> ) er · ∇g |∇u|2 + 2r A˜ where we have integrated by parts twice, remembering the Neumann boundary conditions for g and u on ∂B. The third term yields 1 ∗ 4 2 dr − R · ∇u g (|u| − 1)u + complex conjugate e (r ) > r ε2 A˜
2
2 1 1 dr g 4 (r − R> ) er ·∇ 1−|u|2 = 2 dσ (1− R> ) g 4 1−|u|2 = 2 2ε A˜ 2ε ∂ B
2
2 1 R> . g 4 1 − |u|2 + 2g 3 (r − R> ) er · ∇g 1 − |u|2 − 2 dr 1− ε A˜ 2r (4.41) Altogether we thus obtain
2 1 g4 dσ g 2 |∂τ u|2 + 2 1 − |u|2 (1 − R> ) 2 ε ∂ B R> 2 g 2 |er · ∇u|2 − |∇u|2 = dr g (r − R> ) er · ∇g |∇u|2 + 2r A˜
2 1 R> g 4 1 − |u|2 dr 1− +2ig 2 (r − R> ) er · ∇u ∗ B · ∇u + 2 ε A˜ 2r
2 λ + +2g 3 (r − R> ) er · ∇g 1 − |u|2 dr g 2 (r − R> ) er · ∇|u|2 . 2 A˜ (4.42) We then estimate the moduli of the terms of the r.h.s.. Obviously g4
2 R> R> 2 ≤ F[u]. g 2 |er · ∇u|2 − |∇u|2 + 2 1 − 1 − |u|2 dr 2r ε 2r A˜ (4.43)
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Using (4.6) and (4.18) we obtain dr g 2 (r − R> ) er · ∇u ∗ B · ∇u ≤ C| log ε| dr g 2 |∇u|2 ≤ C| log ε|F[u], A˜
A˜
(4.44) whereas, using the normalization of GP , (4.23) and (4.28), λ 2 2 2 ∗ 2 ˜ dr g (r − R> ) er · ∇|u| = λ ˜ dr g u (r − R> ) er · ∇u A A 1/2 1/2 F[u]1/2 2 2 2 2 ≤ Cε| log ε| |E[u]| + 3/2 dr g |u| dr g |∇u| ε | log ε|1/2 A˜ A˜ ≤ C| log ε| |E[u]| +
C| log ε|1/2 F[u]. ε1/2
Combining (2.51) with (4.7), we obtain
2 1 3 2 dr g · ∇g 1 − |u| − R e (r ) > r ε2 ˜ A C| log ε|9/4 1 4 C| log ε|9/4 2 2 ≤ dr g (1 − |u| ) ≤ F[u]. ε1/4 ε2 ε1/4 A˜ By similar arguments 9/4 dr g (r − R> ) er · ∇g |∇u|2 ≤ C| log ε| F[u]. ˜ ε1/4 A
(4.45)
(4.46)
(4.47)
Gathering equations (4.42) to (4.47), we have
2 1 g4 dσ g 2 |∂τ u|2 + 2 1 − |u|2 (1 − R> ) 2 ε ∂B ≤ C| log ε| |E[u]| +
C| log ε|1/2 F[u], ε1/2
and there only remains to divide by 1 − R> ∝ ε| log ε| to get the result. 4.3. Cell decomposition and vortex ball construction. In this subsection we aim at constructing vortex balls for u. Namely, we want to construct a collection of balls whose radii are much smaller than the width of A and that cover the set where |u| is not close to 1. The usual of way of performing this task is to exploit bounds on F[u]. Unfortunately, the bound (4.23) is not sufficient for our purpose. It only implies that the area of the set where u can possibly vanish is of order ε2 | log ε|2 , whereas the vortex balls method requires to cover it by balls of radii much smaller than the width of A, which is O(ε| log ε|). However, the estimates of Lemma 4.2 are definitely not optimal and can be improved by using a procedure of local vortex balls construction. The idea is the following: If the bound (4.23) could be localized (in a sense made clear below), we could construct vortex balls. As this is not the case we split the annulus A into regions where the bound can be localized and therefore vortex balls can be constructed as usual, and regions where this is not the case.
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Definition 4.1 (Good and bad cells). We cover A with (almost rectangular) cells of side length Cε| log ε|, using a corresponding division of the angular variable. We denote by N ∝ ε−1 | log ε|−1 the total number of cells and label the cells as An , n ∈ {1, . . . , N }. Let 0 ≤ α < 21 be a parameter to be fixed later on. – We say that An is an α-good cell if
2 | log ε| g4 ≤ dr g 2 |∇u|2 + 2 1 − |u|2 (4.48) ε−α . ε ε An We denote by NαG the number of α-good cells and G Sα the (good) set they cover. – We say that An is an α-bad cell if
2 | log ε| g4 > dr g 2 |∇u|2 + 2 1 − |u|2 (4.49) ε−α . ε ε An We denote by NαB the number of α-bad cells and B Sα the (bad) set they cover. Note that the annulus A has a width ∝ ε| log ε| (which implies that N ∝ ε−1 | log ε|−1 ) so that we are actually dividing it into bad cells where there is much more energy than what would be expected from the localization of the bound (4.23) (namely Cε−2 ∝ ε−1 | log ε|) and regions (good cells) of reasonably small energy. A first consequence of this is, neglecting the good cells and using (4.23), NαB ≤
ε C εα F[u] ≤ εα N , | log ε| ε| log ε|
(4.50)
i.e., there are very few α-bad cells. A consequence of the refined bound (4.8) that we are aiming at is that there are actually no α-bad cells at all. We now construct the vortex balls in the good set. The proof is merely sketched because it is an adaptation of well-established methods (see [SS] and references therein). Note that the construction is possible only in the subdomain A˜ where the density is large enough. Proposition 4.2 (Vortex ball construction in the good set). Let 0 ≤ α < 21 . There is a certain ε0 so that, for ε ≤ ε0 there exists a finite collection {Bi }i∈I := {B(ai , i )}i∈I of disjoint balls with centers ai and radii i such that % 1. r ∈ G Sα ∩ A˜ : ||u| − 1| > | log ε|−1 ⊂ i∈I Bi , & 2. for any α-good cell An , i,Bi ∩An =∅ i = ε| log ε|−5 . Setting di := deg{u, ∂Bi }, if Bi ⊂ A˜ ∩ G Sα , and di = 0 otherwise, we have the lower bounds 1 log |log ε| 2 2 2 . (4.51) − α |di |g (ai ) |log ε| 1 − C dr g |∇u| ≥ 2π |log ε| 2 Bi ˜ where u can possibly vanish by Proof. We begin by covering the subset of G Sα ∩ A, balls of much smaller radii than those announced in the proposition. Let An be an α-good cell. We have, using (4.7) and (4.48),
2 1 2 1 − |u| dr |∇u|2 + 3 ε | log ε|3 An ∩A˜
2 g4 3 2 2 2 ≤ C| log ε|4 ε−α . ≤ Cε| log ε| dr g |∇u| + 2 1 − |u| ε An
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Fig. 1. Initial collection of vortex balls
Then, using the Cauchy-Schwarz inequality, |∇|u|| 1 − |u|2 dr 3/2 ≤ C| log ε|4 ε−α , ε | log ε|3/2 An ∩A˜ and the coarea formula implies
1 − t 2 1 ˜ : |u|(r) = t ≤ C| log ε|4 ε−α , r ∈ A dt 3/2 H ∩ A n ε | log ε|3/2 t∈R+ where H1 stands for the one-dimensional Hausdorff measure. We argue as in [SS, Props. 4.4 and 4.8] to deduce from this that the set {r ∈ An ∩ A˜ : ||u| − 1| > | log ε|−1 } can be covered by a finite number of disjoint balls {B( a˜ j , ˜ j )} j∈J with & ˜ j ≤ Cε3/2−α | log ε|13/2 . Doing likewise on each α-good cell, we get a colj∈J lection {B( a˜ i , ˜ i )}i∈I , covering {r ∈ G Sα ∩ A˜ : ||u| − 1| > | log ε|−1 } and satisfying $ ˜ i ≤ Cε3/2−α | log ε|13/2 , for any α-good cell An . (4.52) i∈I,B( a˜ i ,˜ i )∩An =∅
The balls in this collection may overlap because we have constructed them locally in the cells. However, by merging the balls that intersect as in [SS, Lemma 4.1], we can construct a finite collection of disjoint balls (still denoted by B( a˜ i , ˜ i )) satisfying the same bounds on their radii and still covering {r ∈ G Sα ∩ A˜ : ||u| − 1| > | log ε|−1 }. The collection B( a˜ i , ˜ i ) is represented in Fig. 1. The gray regions are those where u could vanish, i.e., bad cells and vortex balls. The dashed circle is the inner boundary ˜ of A. We now let the balls in our initial collection grow and merge using the method described in [SS, Sect. 4.2], adding lower bounds on the conformal annuli constructed.
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Fig. 2. Growth and merging process
In brief, the idea is to introduce a process parametrized by a variable t (interpreted as ‘time’) so that, as t increases, the balls will grow all with the same dilation factor. There are two phases in the process: When balls grow independently their radii get multiplied by some factor, say m(t), and we add lower bounds to the kinetic energy of u on the annuli between the initial and grown balls. The first time two (or more) balls touch we merge them into larger balls (see [SS, Lemma 4.1]), and continue with the growing phase. Fig. 2 shows this process. The lower bounds on the annuli are obtained by integrating the kinetic energy over circles centered at the balls’ centers ai during the growth phases: B( ai ,m(t)i )\B( ai ,i )
dr |∇u|2 ≥ 2π di2 log (m(t)) 1 − | log ε|−1 ,
(4.53)
where di := deg{u, ∂B(ai , i )}. The main point in this computation is that ||u| − 1| < | log ε|−1 (in particular u cannot vanish) between the circles ∂B(ai , i ) and ∂B(ai , m(t)i ). Thus the degree of u is di on any circle ∂B(ai , ) with i < < m(t)i . To add the lower bounds obtained during the different growth phases we use that if two balls, say B(a p , p ) and B(aq , q ) merge into a new ball B(as , s ) one always has d 2p + dq2 ≥ |d p | + |dq | ≥ |d p + dq | = |ds |. This accounts for the fact that we get a factor |di | in (4.51), whereas the factor in (4.53) is di2 . We stop the process when we have a collection of disjoint balls {Bi }i∈I := {B(ai , i )}i∈I satisfying condition 2 of Proposition 4.2 (Property 1 is satisfied by construction, since it holds true for the initial family of balls), i.e., the balls are large enough, so that we have 1 log |log ε| 2 , (4.54) − α |di | |log ε| 1 − C dr |∇u| ≥ 2π |log ε| 2 B( ai ,i )
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Fig. 3. Final configuration of cells and vortex balls
˜ and di = 0 otherwise. The final conwhere di := deg{u, ∂Bi }, if Bi ⊂ G Sα ∩ A, figuration is drawn in Fig. 3. Note that the radii of the final vortex balls are still much smaller than the diameter of the cells. This is important for (4.55) below and the jacobian estimate in the next section.
ε| comes from the logarithm of The logarithmic factor 21 − α |log ε| 1 − C log|log |log ε| the dilation factor of the collections of balls, i.e., & i &i∈I ≥ Cεα−1/2 | log ε|−23/2 . ˜ i∈ I˜ i On the other hand, using (2.31) and the fact that |∇ρ TF | ≤ Cε−2 | log ε|−2 (see (1.11)), we have min g 2 (r ) − g 2 (ai ) r ∈Bi 1/2 3/2 TF TF TF ≤ C ε | log ε| ρ ∞ + min ρ (r ) − ρ (ai ) L (A) r ∈Bi
≤ C ε−1/2 | log ε|1/2 + ε−2 | log ε|−2 i ≤ Cε−1 | log ε|−7 ≤ C| log ε|−4 g 2 (ai ),
(4.55)
because ai ∈ A˜ and i ≤ Cε| log ε|−5 . We conclude from (4.54) and (4.55) that the lower bound (4.51) holds on each ball we have constructed by bounding below g 2 with its minimum on the ball Bi . The error min r ∈Bi g 2 (r) − g 2 (ai ) can then be absorbed into the g 2 (ai ) log | log ε| term.
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4.4. Jacobian estimate. We now turn to the jacobian estimate. With the vortex balls that we have constructed, it is to be expected that in the α-good set the vorticity measure of u will be close to a sum of Dirac masses curl(iu, ∇u)
$
2π di δ(r − ai ),
i∈I
where δ(r − ai ) stands for the Dirac mass at ai . Indeed, outside of the balls |u| 1, so curl(iu, ∇u) 0, and the balls have very small radii compared to the size of A. Proposition 4.3 gives a rigorous statement of this fact. Note that we can quantify the & difference between curl(iu, ∇u) and i 2π di δ(r − ai ) in terms of the energy only in the domain A˜ where the density is large enough. Proposition 4.3 (Jacobian estimate). Let 0 ≤ α < function with compact support
1 2
and φ be any piecewise-C 1 test
supp(φ) ⊂ A˜ ∩ G Sα . Let {Bi }i∈I := {B(ai , i )}i∈I be a disjoint collection of balls as in Proposition 4.2. Setting di := deg{u, ∂Bi }, if Bi ⊂ A˜ ∩ G Sα , and di = 0 otherwise, one has $ 2π di φ(ai ) − dr φ curl(iu, ∇u) ≤ C ∇φ L ∞ (G Sα ) ε2 | log ε|−2 F[u]. ˜ G Sα ∩A i∈I
(4.56) Proof. We argue as in [SS, Chap. 6]. We first introduce a function ξ : R+ → R+ as follows: ξ(x) = 2x, if x ∈ [0, 1/2], and ξ(x) = 1, if x ∈ [1/2, +∞[. This function satisfies – ξ(t) ≤ 2t and ξ (t) ≤ 2, – |ξ(t) − t| ≤ |1 − t| and |ξ(t) − 1| ≤ |1 − t|, – ξ(t)2 − t 2 ≤ 3t |1 − t|, and we define w as a regularization of u (in the sense that |w| = 1 and therefore curl(iw, ∇w) = 0 when |u| is far enough from 0): w= Remark that we have (iw, ∇w) =
ξ(|u|) u. |u|
|w|2 (iu, ∇u) |u|2
(4.57)
and this has a meaning even if u van-
ishes. By integrating by parts on G Sα ∩ A˜ and using the assumptions on φ, one has G Sα ∩A˜
dr [curl(iu, ∇u) − curl(iw, ∇w)] φ
=−
G Sα ∩A˜
dr [(iu, ∇u) − (iw, ∇w)] ∇ ⊥ φ.
(4.58)
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Now,
491
dσ [(iu, ∇u) − (iw, ∇w)] ∇ φ G Sα ∩A˜ ≤ ∇φ L ∞ (G Sα ) dr |(iu, ∇u) − (iw, ∇w)| A˜ 2 |u| − |w|2 |∇u| dr ≤ ∇φ L ∞ (G Sα ) |u| A˜ ⊥
≤ ∇φ L ∞ (G Sα ) 1 − |u| L 2 (A˜ ) ∇u L 2 (A˜ ) ≤ C ∇φ L ∞ (G Sα ) ε5/2 | log ε|9/2 F[u], (4.59) using the properties of ξ , the Cauchy-Schwarz inequality, the definition (4.2) of F[u], 2 g 2 ≥ Cε−1 | log ε|−3 on A˜ and (1 − |u|)2 ≤ 1 − |u|2 . We now evaluate $ dr curl(iw, ∇w)φ = dr curl(iw, ∇w)φ, (4.60) G Sα ∩A˜
Bi ∩A˜
i∈I
which follows from the fact that ||u| − 1| < | log ε|−1 outside ∪i∈I Bi , so that |w| = 1 and curl(iw, ∇w) = 0 outside ∪i∈I Bi . If Bi ⊂ A˜ ∩ G Sα , we have dr |curl(iw, ∇w)| |φ(r) − φ(ai )| ≤ C ∇φ L ∞ (Bi ) i dr |∇u|2 (4.61) Bi
and
Bi
Bi
dr curl(iw, ∇w) = deg{u, ∂Bi } = 2π di
(4.62)
˜ Sα , then Bi ∩∂ A˜ ∩ G Sα = ∅ by definition of w and di . On the other hand, if Bi A∩G and thus ≤ dr curl(iw, ∇w)φ dr |φ||∇w|2 α ˜ ˜ Bi ∩A Bi ∩A∩G S 2 ≤4 dr |φ||∇u| ≤ C ∇φ L ∞ (Bi ) i dr |∇u|2 , (4.63) Bi ∩A˜ ∩G S α
Bi ∩A˜ ∩G Sα
because |∇w| ≤ 2|∇u| and φ is supported in the interior of A˜ ∩ G Sα . Gathering Eqs. (4.58) to (4.63), we obtain $ di φ(ai ) − dr φ curl(iu, ∇u) ˜ G Sα ∩A i∈I $ 5/2 9/2 2 ∇φ L ∞ (Bi ) i ≤ C ∇φ L ∞ (G Sα ) ε | log ε| F[u] + dr |∇u| , i
Bi ∩A˜ ∩G Sα
(4.64)
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and the result follows because, using i ≤ ε| log ε|−5 and (4.7), $ ∇φ L ∞ (Bi ) i dr |∇u|2 i
Bi ∩A˜ ∩G Sα
≤ C ∇φ L ∞ (G Sα ) ε2 | log ε|−2
$ i
≤ C ∇φ L ∞ (G Sα ) ε | log ε| 2
−2
Bi ∩A˜ ∩G Sα
dr g 2 |∇u|2
F[u].
(4.65) &
Note that (4.56) is equivalent to saying that the norm of curl(iu, ∇u)− i di δ(r −ai ) in (Cc1 (A˜ ∩ G Sα ))∗ , i.e., the dual space of Cc1 (A˜ ∩ G Sα ), is controlled by the energy. 4.5. Completion of the proof of Proposition 4.1. We now complete the proof of Proposition 4.1, collecting the estimates of the preceding subsections. We want to avoid any unwanted boundary term when performing integrations by parts in the proof below. Indeed, our radial frontiers between the good set and the bad set are somewhat artificial and have no physical interpretation. Therefore it is difficult to estimate integrals on these boundaries. To get around this point we need to introduce an azimuthal partition of unity on the annulus in order to ‘smooth’ the radial boundaries appearing in our construction. This requires new definitions: Definition 4.2 (Pleasant and unpleasant cells). Recall the covering of the annulus A by cells An , n ∈ {1, . . . , N }. We say that An is – an α-pleasant cell if An and its two neighbors are good cells. We denote P Sα the union of all α-pleasant cells and NαP their number, – an α-unpleasant cell if either An is a bad cell, or An is a good cell but its two neighbors are bad cells. We denote U P Sα the union of all α-unpleasant cells and NαUP their number, – an α-average cell if An is a good cell but exactly one of its neighbors is not. We denote ASα the union of all α-average cells and NαA their number. Remark that one obviously has, recalling (4.50), 3 B N N 2 α
(4.66)
NαA ≤ 2NαB N .
(4.67)
NαUP ≤ and
The average cells will play the role of transition layers between the pleasant set, where we will use the tools of Subsects. 4.3 and 4.4, and the unpleasant set, where we have little information and therefore have to rely on more basic estimates (like those we used in the proof of Lemma 4.2). To make this precise we now introduce the azimuthal partition of unity we have announced. Let us label U P Sαl , l ∈ {1, . . . , L}, and P Sαm , m ∈ {1, . . . , M}, the connected components of the α-unpleasant set and α-pleasant set respectively. We construct azimuthal
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positive functions, bounded independently of ε, denoted by χlU and χmP (the labels U and P stand for “pleasant set” and “unpleasant set”) so that χlU := 1
on U P Sαl ,
χlU := 0
on P Sαm , ∀m ∈ {1, . . . , M} , and on U P Sαl , ∀l = l,
χmP := 1
on P Sαm ,
χmP $ $ χmP + χlU m l
:= 0
on
= 1
on A.
(4.68)
U P Sαl ,
∀l ∈ {1, . . . , L} , and on
P Sαm ,
∀m = m,
It is important to note that each function so defined varies from 0 to 1 in an average cell. A crucial consequence of this is that we can take functions satisfying C C U , ∇χmP ≤ , (4.69) ∇χl ≤ ε| log ε| ε| log ε| because the side length of a cell is ∝ ε| log ε|. For example one can choose this partition of unity to be constituted of piecewise affine functions of the angle. We will use the short-hand notation χin :=
M $
χmP ,
(4.70)
χlU .
(4.71)
m=1
χout :=
L $ l=1
The subscripts ‘in’ and ‘out’ refer to ‘in the pleasant set’ and ‘out of the pleasant set’ respectively. We would like to use the jacobian estimate of Proposition 4.3 with φ = χin F, whose support is not included in A˜ but only in A (moreover it does not vanish on ∂B). We will + and R − as thus need a radial partition of unity: We introduce two radii Rcut cut + := 1 − ε| log ε|−1 , Rcut − Rcut
:= R> + ε| log ε|
−1
(4.72) .
(4.73)
Let ξin (r ) and ξout (r ) be two positive radial functions satisfying ξin (r ) ξin (r ) ξout (r ) ξout (r ) ξin + ξout
:= 1 := 0 := 1 := 0 = 1
− + for Rcut ≤ r ≤ Rcut , for R< ≤ r ≤ R> and for r = 1, for R< ≤ r ≤ R> , − + for Rcut ≤ r ≤ Rcut , on A.
(4.74)
For example ξin and ξout can be defined as piecewise affine functions of the radius. Moreover, because of (4.72) and (4.73), we can impose |∇ξin | ≤
C| log ε| C| log ε| , |∇ξout | ≤ . ε ε
˜ and ‘outside of A’ ˜ respectively. The subscripts ‘in’ and ‘out’ refer to ‘inside A’
(4.75)
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In the sequel {Bi }i∈I := {B(ai , i )}i∈I is a collection of disjoint balls as in Proposition 4.2. For the sake of simplicity we label B j , j ∈ J ⊂ I , the balls such that B j ⊂ A˜ ∩ G Sα . Proof (Proposition 4.1). Recall the properties of F (4.11). By integration by parts, we have dr g 2 |∇u|2 − 2g 2 B · (iu, ∇u) A dr g 2 |∇u|2 + Fcurl(iu, ∇u) − dσ F(1)(iu, ∂τ u) (4.76) = A
∂B
and we are going to evaluate the three terms using our previous results. We begin with a lower bound on the kinetic term in (4.76), using Proposition 4.2. We introduce a parameter γ to be fixed later in the proof and estimate $ dr g 2 |∇u|2 ≥ (1 − γ ) dr ξin g 2 |∇u|2 A
Bj
j∈J
+ (1 − γ )
A
dr ξout g 2 |∇u|2 + γ
dr g 2 |∇u|2 .
A
(4.77)
Using the lower bound (4.51), we have dr ξin g 2 |∇u|2 Bj
≥ ξin (a j )
Bj
dr g 2 |∇u|2 +
inf ξin (r ) − ξin (a j )
r ∈B j
Bj
dr g 2 |∇u|2
1 log |log ε| 2 − α |d j |ξin (a j )g (a j ) |log ε| 1 − C ≥ 2π |log ε| 2 C − dr g 2 |∇u|2 . | log ε|4 B j
(4.78)
The estimate of inf B j ξin − ξin (a j ) is a consequence of (4.75) combined with j ≤ Cε| log ε|−5 . We now compute dr F curl(iu, ∇u) = dr [ξin χin F curl(iu, ∇u) A
A
+ ξout χin F curl(iu, ∇u) + χout F curl(iu, ∇u)] .
(4.79)
We can use Proposition 4.3 to estimate the first term because ξin χin F is a piecewise-C 1 function with support included in A˜ ∩ G Sα . We obtain $ dr ξin χin F curl(iu, ∇u) ≥ 2π d j F(a j )ξin (a j )χin (a j ) A
j∈J
− C ∇(ξin χin F) L ∞ (G Sα ) ε2 | log ε|−2 F[u].
(4.80)
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Now, using (4.19), (4.20), (4.69) and (4.75), we have ∇(ξin χin F) L ∞ (A) ≤
C| log ε| , ε2
so that $ dr ξin χin F curl(iu, ∇u) ≥ 2π d j F(a j )ξin (a j )χin (a j ) − C| log ε|−1 F[u]. A
j∈J
(4.81) The second term in the r.h.s. of (4.79) is simply bounded below as follows: dr ξout χin F curl(iu, ∇u) ≥ − dr ξout |F||∇u|2 . A
A
(4.82)
We now estimate the third term in the r.h.s. of (4.79): We integrate by parts back to get dr χout F curl(iu, ∇u) ≥ − dr ∇ ⊥ (χout F) · (iu, ∇u) A A −C dσ |F(1)| |(iu, ∂τ u)| , (4.83) ∂B
but
dr ∇ ⊥ (χout F) · (iu, ∇u) = dr F∇ ⊥ (χout ) · (iu, ∇u) + 2χout g 2 B · (iu, ∇u) ,
A
A
(4.84)
and the second term can be bounded using the same computations as in the proof of Lemma 4.2: 2 dr χout g 2 B · (iu, ∇u) ≤ δ dr χout g 2 |∇u|2 + Cδ −1 dr χout g 2 B 2 |u|2 , A
A
A
where δ is a parameter to be fixed later. For the first term in (4.84) we use (4.22): ⊥ dr ∇ ⊥ (χout )F · (iu, ∇u) ≤ Cε−1 ∇ dr χ |r − R< | g 2 |u||∇u| out A A C 2 2 dr g |∇u| + 2 dr g 2 |u|2 . ≤δ δε {∇χout =0} {∇χout =0} The second inequality uses |1 − R< | ∝ ε| log ε| and (4.69). We inject the preceding computations in (4.83), taking into account that B ≤ Cε−1 . We also note that we have χout = 0 and/or ∇χout = 0 only in the unpleasant set and the average set, so dr χout F curl(iu, ∇u) ≥ −Cδ dr g 2 |∇u|2 A U P Sα ∪ASα C − 2 dr g 2 |u|2 − C dσ |F(1)| |(iu, ∂τ u)| . (4.85) δε U P Sα ∪ASα ∂B
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We gather Eqs. (4.77), (4.78), (4.79), (4.81), (4.82) and (4.85) to obtain (recalling that |χin | ≤ 1) $ dr g 2 |∇u|2 + F curl(iu, ∇u) ≥ 2π ξin (a j )|d j | A
j∈J
1 log |log ε| − |F(a j )| · (1 − γ ) − α g 2 (a j ) |log ε| 1 − C |log ε| 2 dr ξout g 2 |∇u|2 − dr ξout |F||∇u|2 − C dσ |F(1)| |(iu, ∂τ u)| + (1 − γ ) ∂B A A C + (γ − δ) dr g 2 |∇u|2 − 2 dr g 2 |u|2 − C| log ε|−1 F[u]. (4.86) δε A U P Sα ∪ASα
We now choose the parameters in (4.86) as follows: γ = 2δ =
log | log ε| log | log ε| , α = α˜ , | log ε| | log ε|
(4.87)
where α˜ is a large enough constant (see below). This choice allows to bound the terms in (4.86) from below: Indeed, if 0 > 2(3π )−1 , we have from Proposition A.2, 1 2 C g (a j ) |log ε| − |F(a j )| ≥ 2 ε| log ε|2 for any a j ∈ A˜ and thus 1 log |log ε| − |F(a j )| − α g 2 (a j ) |log ε| 1 − C (1 − γ ) |log ε| 2 log |log ε| C 1 − |F(a j )| ≥ > 0, ≥ g 2 (a j ) |log ε| 1 − C 2 | log ε| ε| log ε|2
(4.88)
where we have used (2.26). On the other hand, by the definition of ξout , for any r ∈ supp(ξout ), we have either |r − R< | ≤ Cε| log ε|−1
(4.89)
|r − 1| ≤ Cε| log ε|−1 .
(4.90)
or
Therefore, using (4.22), we have in the first case |F(r )| ≤ C
g 2 (r ) . | log ε|
In the second case (4.22) yields |F(r )| ≤
C , ε| log ε|2
but (2.31) shows that, if r satisfies (4.90), g 2 (r ) ≥
C . ε| log ε|
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We conclude that g 2 (r ) ≥ C| log ε||F(r )| |F(r )| for any r ∈ supp(ξout ) and thus dr ξout g 2 |∇u|2 − dr ξout |F||∇u|2 ≥ 0. (1 − γ ) A
A
(4.91)
(4.92)
Finally we have from (4.86), (4.88) and (4.92), dr g 2 |∇u|2 + F curl(iu, ∇u) A | log ε| log |log ε| dr g 2 |∇u|2 − 2 dr g 2 |u|2 ≥C |log ε| ε log |log ε| U P Sα ∪ASα A − dσ |F(1)| |(iu, ∂τ u)| − C| log ε|−1 F[u] . (4.93) ∂B
Adding
2 g4 2 dr 2 1 − |u| − dσ F(1)(iu, ∂τ u) ε A ∂B
to both sides of (4.93) and using (4.76), we get the lower bound log |log ε| | log ε| E[u] ≥ C dr g 2 |u|2 F[u] − 2 |log ε| ε log |log ε| U P Sα ∪ASα − dσ F(1)(iu, ∂τ u) , ∂B
(4.94)
valid for ε small enough and 0 > 2(3π )−1 . But g 2 |u|2 = | GP |2 ≤ Cε−1 | log ε|−1 , whereas the side length of a cell is O(ε| log ε|), thus
C |U P Sα ∪ ASα | ≤ Cε| log ε| NαUP + NαA . (4.95) dr g 2 |u|2 ≤ ε| log ε| U P Sα ∪ASα Using (4.50), (4.66) and (4.67), we deduce dr g 2 |u|2 ≤ Cε| log ε|NαB ≤ Cε2 εα F[u]. U P Sα ∪ASα
(4.96)
On the other hand (2.31) implies that g 2 (1) ≥ Cε−1 | log ε|−1 . Combining this fact with the upper bound (2.3) yields |u| ≤ C
on ∂B
and thus, using (4.21) and Cauchy-Schwarz inequality, 1/2 2 ≤C dσ F(1)(iu, ∂ u) dσ |∂ u| . τ τ ∂B
∂B
(4.97)
(4.98)
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Using Lemma 4.4 we conclude | log ε|1/4 1/2 1/2 1/2 ≤ C | log ε| . dσ F(1)(iu, ∂ u) |E[u]| + F[u] τ ε1/4 ∂B
(4.99)
Combining (4.94), (4.96) and (4.99), we have log |log ε| | log ε| α F[u] − E[u] ≥ C ε F[u] |log ε| log | log ε| −| log ε|1/2 |E[u]|1/2 −
| log ε|1/4 1/2 . F[u] ε1/4
(4.100)
Recall the choice of α in (4.87): We choose now a constant α˜ > 2. Then | log ε| α | log ε|1−α˜ log |log ε| , ε = |log ε| log | log ε| log | log ε| and thus there exists a finite constant c such that log |log ε| | log ε|1/4 1/2 , O(ε∞ ) ≥ E[u] ≥ c F[u] F[u] − | log ε|1/2 |E[u]|1/2 − |log ε| ε1/4 (4.101) where the upper bound comes from (4.4). Since the sign of E[u] is not known, we might have two possible cases: If E[u] ≥ 0, (4.4) implies that |E[u]| ≤ O(ε∞ ), which can be plugged in (4.101) yielding log |log ε| | log ε|1/4 1/2 , (4.102) O(ε∞ ) ≥ E[u] ≥ c F[u] F[u] − |log ε| ε1/4 This implies F[u] ≤ C
| log ε|5/2 ε1/2 log | log ε|2
which concludes the proof of Proposition 4.1, if E[u] ≥ 0. On the opposite, if E[u] < 0, either log |log ε| | log ε|1/4 1/2 1/2 1/2 , 0 ≥ E[u] + c| log ε| |E[u]| ≥ c F[u] F[u] − |log ε| ε1/4 which implies the result as before, or |E[u]| ≤ C| log ε|, which gives log |log ε| | log ε|1/4 3/2 1/2 C| log ε| ≥ c , F[u] F[u] − |log ε| ε1/4 and thus again (4.8) and (4.9). As already noted, the end of the proof could be formulated as an induction. Plugging the estimates of Lemma 4.2 in (4.100) and using the upper bound on E[u] would yield improved estimates of F[u] and |E[u]|, thus reducing the number of bad cells and improving the boundary estimate. The process could then be repeated a large number of times, proving that there are no bad cells at all. The second term in the lower bound (4.100) would then vanish and the process stop when the first term would reach the order of magnitude of the last one (coming from the boundary estimate), thus giving the results of Proposition 4.1.
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5. Energy Asymptotics and Absence of Vortices In this section we conclude the proofs of our main results. The proof of the energy asymptotics is a straightforward combination of the results of Sects. 3 and 4: Proof (Theorem 1.2). From Proposition 3.1 we have, using the simplified notation of Sect. 4, GP ∞ E GP ≥ Eˆ A ,ω0 + E[u] − O(ε ),
which reduces to | log ε|3/2 GP E GP ≥ Eˆ A − C ,ω0 ε1/2 log | log ε| thanks to (4.9). Using a regularization of gA,ω0 as a trial function for the functional EˆωGP 0 GP ≥ Eˆ GP − O(ε ∞ ) and thus as in the proof of Proposition 3.1, we get Eˆ A ω ,ω0 0 E GP ≥ Eˆ ωGP −C 0
| log ε|3/2 . ε1/2 log | log ε|
We conclude the proof of the lower bound recalling that, by definition, Eˆ GP = inf Eˆ ωGP ≤ Eˆ ωGP . 0 ω
For the upper bound , E GP ≤ Eˆ GP = Eˆ ωGP opt we simply use gωopt (r ) exp{i([] − ωopt )ϑ} as a trial function for E GP . The proof of the absence of vortices requires an additional ingredient: Lemma 5.1 (Estimate for the gradient of u ω0 ). Recall the definition of u ω0 in (3.5). There is a finite constant C such that ∇u ω 0
L ∞ (A˜ )
≤C
| log ε|3/2 . ε3/2
(5.1)
Proof. We use the short-hand notation defined at the beginning of Sect. 4 (in particular u = u ω0 ). Recall the variational equation (4.27)
2 (5.2) − ∇ g 2 ∇u − 2ig 2 B · ∇u + 2 g 4 |u|2 − 1 u = λg 2 u. ε From this equation we get the pointwise estimate
|∇g| 1 |u| ≤ C |∇u| + |B||∇u| + 2 g 2 |u|2 − 1 u + |λ| |u| , g ε
(5.3)
holding true on A. Recalling that | GP | ≤ ε−1/2 | log ε|−1/2 and g≥
C ε1/2 | log ε|3/2
˜ on A,
(5.4)
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we have ˜ |u| ≤ C| log ε| on A.
(5.5)
| log ε|2 1 2 2 ≤ C |u| − 1 u . g ε2 ε3
(5.6)
Thus, using (2.26),
On the other hand, estimating the chemical potential with (4.28) and plugging in the results of Proposition 4.1, we have C| log ε|7/4 | log ε|2 1 1/2 ≤ |λu| ≤ C|u| |E[u]| + 3/2 F[u] (5.7) ε | log ε|1/2 ε7/4 log | log ε| ε3 ˜ on A. Combining (2.26), (2.51), (4.7) and (4.18) with (5.6) and (5.7), we deduce from (5.3), | log ε|3/4 | log ε|2 u L ∞ (A˜ ) ≤ C ∇u . + L ∞ (A˜ ) ε3 ε5/4 From the Gagliardo-Nirenberg inequality [N, Theorem p. 125], we deduce | log ε|3/4 | log ε|2 1/2 1/2 u L ∞ (A˜ ) ≤ C u u . + L ∞ (A˜ ) L ∞ (A˜ ) ε3 ε5/4
(5.8)
Inserting (5.5), we conclude u L ∞ (A˜ ) ≤ C
| log ε|2 , ε3
and we get (5.1) by using (5.5) and the Gagliardo-Nirenberg inequality again. We are now in position to complete the Proof (Theorem 1.1). The proof relies on a combination of (4.8) and (5.1), as in [BBH1]. Suppose that at some point r 0 such that 1 Rh + ε| log ε|−1 ≤ r0 ≤ 1, 2 we have ||u(r 0 )| − 1| ≥ ε1/8 | log ε|3 . Then, using (5.1), there is a constant C such that, for any r ∈ B0 with B0 := B(r 0 , Cε13/8 | log ε|3/2 ), we have ||u(r)| − 1| ≥ This implies (recall (4.7)) B0
dr
1 1/8 ε | log ε|3 . 2
2 C| log ε|3 g4 2 1 − |u| ≥ , ε2 ε1/2
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and thus (note that by the initial condition on r0 , B0 ⊂ A) F[u] ≥
C| log ε|3 , ε1/2
(5.9)
which is a contradiction with (4.8). We have thus proven that (recall (5.5)) | log ε|3 GP 2 (5.10) | | − g 2 ≤ g 2 |u|2 − 1 ≤ C 7/8 ε # " on A˜ = Rh + ε| log ε|−1 ≤ r ≤ 1 . The result then follows by a combination of (2.31) and (5.10). Remark 5.1 (Absence of vortices in a larger domain) . By direct inspection of the proof of Theorem 1.1, one can easily realize that we could have proven the main result, i.e., ˜ i.e., there is some freedom in the the absence of vortices, in a domain larger than A, choice of the bulk of the condensate. More precisely the choice of a larger domain would have implied a worse lower bound on g 2 via (2.31) and in turn a worse remainder in (5.10), but at the same time this would have allowed the extension of the no vortex result up to a distance of order ε| log ε|−a from Rh for some power a > 1. We have however chosen to state the main result in A˜ for the sake of simplicity. The proof of the result about the degree of GP is a corollary of the main result proven above: Proof (Theorem 1.3). We first note that the pointwise estimate in (5.10) implies that GP does not vanish on ∂B, so that its degree is indeed well defined. We then compute GP | GP | 2π deg{ GP , ∂B} = −i dσ ∂ GP τ | GP | ∂B u i([]−ω0 )ϑ −i([]−ω0 )ϑ |u| e ∂τ e = −i dσ u |u| ∂B u |u| ∂τ . (5.11) = 2π ([] − ω0 ) − i dσ u |u| ∂B Then
|u| ∂τ dσ u ∂B
u ≤ dσ |u| ∂B
∂τ u ≤ C dσ |∂τ u| , |u| ∂B
(5.12)
where we have used that |u| is bounded below by a constant on ∂B. Finally, combining (4.39) and the results of Proposition 4.1, we obtain (recall that g 2 ≥ Cε−1 | log ε|−1 on ∂B) C| log ε|3 dσ |∂τ u|2 ≤ . (5.13) ε(log | log ε|)2 ∂B Using the Cauchy-Schwarz inequality, we thus conclude from (5.11), (5.12) and (5.13) that
deg{ GP , ∂B} = [] − ω0 + O ε−1/2 | log ε|3/2 (log | log ε|)−1 . (5.14)
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There only remains to recall that (see (3.10)) 2 ω0 = √ + O(ε−1 | log ε|−1/2 ) 3 πε and that an identical estimate applies to ωopt (see (3.16)). Remark 5.2 (Degree of a GP minimizer) . According to (5.14), we could have stated the result (1.21) about the degree of GP in terms of ω0 , i.e., the optimal giant vortex phase when the minimization problem is restricted to the annulus A, instead of ωopt , i.e., the optimal giant vortex phase in the whole of B. Moreover the remainder in (5.14), i.e., O(ε−1/2 | log ε|3/2 (log | log ε|)−1 ), is much better than the one contained in the final result (1.21), i.e., O(ε−1 | log ε|−1/2 ), which is inherited from (3.10) and (3.16). Note however that the latter remainder is the best precision√to which one can estimate the giant vortex phase in terms of the explicit quantity 2/(3 π)ε−1 . For this reason and the fact that ωopt occurs more naturally in the analysis, we have used it in (1.21). Acknowledgements. MC and NR gratefully acknowledge the hospitality of the Erwin Schrödinger Institute (ESI). JY acknowledges the hospitality of the Institute for Mathematical Sciences (IMS) at the National University of Singapore. MC is partially supported by a grant Progetto Giovani GNFM and NR by Région Ile-de-France through a PhD grant. NR thanks Sylvia Serfaty for helpful discussions.
Appendix A In this Appendix we discuss some useful properties of the TF-like functionals involved in the analysis as well as the critical angular velocity c for the emergence of the giant vortex phase.
A.1. The TF functionals. We start by considering the TF functional defined in (1.9): E TF [ρ] := dr −2 r 2 ρ + ε−2 ρ 2 . B
Its minimizer over positive functions in L 1 (B) is unique and is given by the radial density ρ TF (r ) :=
1 2 TF ε2 2 2 ε μ + ε2 2 r 2 = r − Rh2 , + + 2 2
(A.1)
where [ · ]+ stands for the positive part and the chemical potential is fixed by normalizing ρ TF in L 1 (B), i.e., 2 μTF := E TF + ε−2 ρ TF 2 = −2 Rh2 .
(A.2)
√ Note that, if ≥ 2/( π ε), the TF minimizer is a compactly supported function, since it vanishes outside ATF , i.e., for r ≤ Rh , where 2 . (A.3) Rh := 1 − √ π ε
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The corresponding ground state energy can be explicitly evaluated and is given by 4 TF 2 E = − 1 − √ . (A.4) 3 π ε By (A.3) and (A.4) the annulus ATF has a shrinking width of order ε| log ε| and the leading order term in the ground state energy asymptotics is −2 , which is due to the convergence of ρ TF to a delta function supported at the boundary of the trap. In other sections of the paper we often consider the restrictions of the functionals TF ). However in the case of the to domains D strictly contained inside B (denoted by ED TF since all the TF functional there is no need to make a distinction between E TF and ED ground state properties are basically independent of the integration domain, provided ATF ⊂ D. Another important TF-like functional is defined in (1.26) and includes the giant vortex energy contribution, i.e., EˆωTF [ρ] := dr −2 r 2 ρ + Bω2 (r )ρ + ε−2 ρ 2 B (A.5) dr ([] − ω)2 r −2 ρ + ε−2 ρ 2 − 2[ − ω], = B
where the potential B ω is defined in (1.25), ω ∈ Z and we have used the normalization in L 1 (B) of the density in the last term. The minimization is essentially the same as for (1.9): The normalized minimizer is ρˆωTF (r ) :=
ε2 TF μˆ ω − ([] − ω)2 r −2 , + 2
(A.6)
and the normalization condition becomes 1 − Rˆ ω2 2 + log Rˆ ω2 = , π ε2 ([] − ω)2 Rˆ ω2
(A.7)
([] − ω)2 . Rˆ ω2 := μˆ TF ω
(A.8)
where we have denoted
With such a definition the minimizer (A.6) can be rewritten in a form very close to the TF minimizer (A.1), i.e., ρ˜ωTF (r ) =
ε2 ([] − ω)2 2 r − Rˆ ω2 . + 2 Rˆ ω2 r 2
In order to make a comparison it would then be useful to evaluate the radius Rˆ ω but Eq. (A.7) has no explicit solution. However, since the right-hand side of (A.7) tends to zero as ε → 0, we can expand the left-hand side assuming Rˆ ω−2 = 1 + δ for some δ 1: 1 2 1 3 2 δ − δ + O(δ 4 ) = , 2 3 π ε2 ([] − ω)2
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which yields 1 −1 = δ Rˆ ω2
2 2 1 −3 −3 = √ + O(ε ) . 1+ √ + π ε([] − ω) 3 π ε([] − ω) 9π ε2 ([] − ω)2
We thus have 2 4 + O(ε−3 −3 ) − Rˆ ω2 = 1 − √ 2 π ε([] − ω) 3π ε ([] − ω)2 2ω 4 − + O(ε3 | log ε|3 ), = Rh2 + √ 3π ε2 2 π ε2
(A.9)
and whether Rˆ ω is larger or smaller than Rh depends in a crucial way on the phase ω: In particular in the case of the giant vortex phase ω0 (see Proposition 3.2), the sum of the two last terms in the above expression vanishes to the leading order (see (3.10)), i.e., it is much smaller than O(ε2 | log ε|2 ). The ground state energy Eˆ TF is easy to compute:
2 π ε2 ([] − ω)4 ˆ −2 Rω − 1 Eˆ ωTF = −2 ([] − ω) + 4 4 4ω 2 2 2 = − + √ − 2 ([] − ) + O(ε−2 | log ε|−2 ), +ω − √ + 3 πε 3 π ε 3π ε2 and, assuming that |ω| ≤ O(ε−1 ), one can easily recognize that the leading term and the first remainder coincide with (A.4), i.e., the energy Eˆ ωTF is equal to E TF up to second order corrections: 2 2 2 Eˆ ωTF = E TF + ω − √ + − 2 ([] − ) + O(ε−2 | log ε|−2 ). (A.10) 9π ε2 3 πε This formula implies that Eˆ ωTF is minimized by a phase which is given up to corrections of order ε−1 | log ε|−1 by 2 ωTF := √ , 3 πε
(A.11)
and the same is true for the giant vortex phases ω0 (see Proposition 3.2 and (3.10)) and ωopt (see Proposition 3.3 and (3.16)). A.2. The critical angular velocity and the vortex energy. The last part of this Appendix is devoted to the study of the critical velocity c , which is defined as the angular velocity at which vortices disappear from the bulk of the condensate. To estimate this velocity, according to the discussion in Sect. 4, we have to compare the vortex energy cost 21 g 2 (r )| log ε| and the vortex energy gain |F(r )| (see (4.10)): This leads to the definition of the function 1 2 (A.12) H (r ) := gA (r )| log ε| − Fω0 (r ) , 2 ,ω0
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which yields the overall energy contribution of a vortex at a radius r inside the bulk: If H is positive in some region, then a vortex is energetically unfavorable there, and, if this holds true in the whole of the bulk, the condensate is in the giant vortex phase. Before studying the behavior of the above function H , it is however convenient to obtain an explicit approximative value for the critical velocity and to this purpose we replace the density g 2 with ρ TF and study the function 1 (A.13) H TF (r ) := | log ε|ρ TF (r ) − F TF (r ) , 2 where the cost function F TF is explicitly given by r TF F (r ) := 2 ds B ωTF (s) · eϑ ρ TF (s) Rh r
2 2 ds s − [] − ωTF s −1 (s 2 − Rh2 ), = ε
(A.14)
Rh
with ωTF defined in (A.11). In order to investigate the behavior of the infimum of H TF inside the bulk, it is convenient to rescale the quantities and set z := ε(r 2 − Rh2 ),
(A.15) √ so that z varies on a scale of order one, i.e., more precisely z ∈ [0, 2/ π ] (see (A.3)). With such a choice the gain function can be easily estimated: 2 2
−1
ε2 2 r −Rh TF dt t (t + Rh2 ) − [] + ωTF t + Rh2 F (r ) = 2 0 −1 r 2 −R 2 2 2 h ε 4 2 = dt t t − √ + O(1) 1− √ +t 2 3 πε π ε 0 z 1 4 2 s −1 = ds s s − √ + O(ε) 1− √ + 2ε 0 3 π π ε ε z 2 1 4 z2 z−√ = ds s s − √ + O(| log ε|) = + O(| log ε|), 2ε 0 6ε 3 π π where we have used the approximation [1 − O((ε)−1 )]−1 = 1 + O((ε)−1 ). Applying the same rescaling to the energy cost function, we thus obtain H TF (r ) := where
z H˜ TF (z) , 12ε
2 TF ˜ H (z) = 30 − 2z z − √ + O(ε| log ε|) π 2 = 30 − 2z √ − z − O(ε| log ε|), π
(A.16)
(A.17)
√ since z ≤ 2/ π by the definition of the scaling. Now it is very easy to see that H˜ TF (z) ≥ 30 − 2π −1 − O(ε| log ε|).
(A.18)
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The above considerations lead to the following Proposition A.1 (TF vortex energy). For any 0 > 2(3π )−1 and ε small enough, there exists a finite constant C such that H TF (r ) ≥ Cε−1 | log ε|−2 > 0 for any r such that r ≥ Rh + ε| log ε|−1 . Proof. It is sufficient to collect (A.16), (A.18) and recall the rescaling (A.15). We now go back to the original function H and show that the above property holds true as well, i.e., c = 2(3π )−1 ε−2 | log ε|−1 is the critical velocity for the disappearance of vortices from the bulk of the condensate: Proposition A.2 (Critical angular velocity). If 0 > 2(3π )−1 and ε is small enough, there exists a finite constant C such that H (r ) ≥ Cε−1 | log ε|−2 > 0 for any r such that r ≥ Rh + ε| log ε|−1 . Proof. The result basically follows from what is proven about H TF : We are going to show that (A.19) sup H TF (r ) − H (r ) ≤ Cε−1 | log ε|−1 . r ∈A˜
In order to prove the above inequality, we use the estimates (2.31) and (2.26) to get 2 1/2 7/2 TF ρ ≤ Cε (r ) | log ε| sup ρ TF (r ) − gA ≤ Cε−1/2 | log ε|5/2 , (A.20) ,ω0 ∞
r ∈A˜
and
sup F TF (r ) − Fω0 (r )
r ∈A˜
≤2
R> R<
2 2 2 TF ds Bω0 (s) gA,ω0 (s) + Cε ω0 − ω
2 (r ) +2 sup ρ TF (r ) − gA ,ω0 r ∈A˜
1 Rh
1 Rh
ds s −1 (s 2 − Rh2 )
ds Bω0 (s)
2 3 2 −1/2 ≤ C ε−1 |R> − R< | gA | log ε|7/2 ,ω0 (R> ) + ε | log ε| + ε
≤ C | log ε|−1 ρ TF (R> ) + ε−1 | log ε|−1 ≤ Cε−1 | log ε|−1 ,
(A.21)
where we have used (A.20), the monotonicity of gA,ω0 (r ) (see Proposition 2.3) and the estimate (3.10). Hence one obtains (A.19) and the final result follows from Proposition A.1 if r ≥ Rh + ε| log ε|1/2 . Indeed, using (A.15), (A.16) and (A.18) we have H TF (r ) ≥ Cε−1 | log ε|−1/2 in this case.
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On the other hand, if Rh + ε| log ε|−1 ≤ r ≤ Rh + ε| log ε|1/2 , it follows from (4.22) that 2 −2 H (r ) ≥ 21 | log ε|gA ,ω0 (r )(1 − C| log ε| ) ≥ C
1 , ε| log ε|2
where the last inequality comes from (4.7). References [A] [AAB] [AJR] [AS] [BBH1] [BBH2] [BCPY] [Co] [CDY1] [CDY2] [CPRY1] [CPRY2] [CY] [E] [Fe1] [Fe2] [FJS] [FB] [FZ] [IM1] [IM2] [J1] [J2] [J3] [JS] [KTU] [KB]
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Communicated by I.M. Sigal
Commun. Math. Phys. 303, 509–554 (2011) Digital Object Identifier (DOI) 10.1007/s00220-011-1204-2
Communications in
Mathematical Physics
Quantum Diffusion and Eigenfunction Delocalization in a Random Band Matrix Model László Erd˝os1, , Antti Knowles2, 1 Institute of Mathematics, University of Munich, Theresienstr. 39, D-80333 Munich, Germany.
E-mail:
[email protected]
2 Department of Mathematics, Harvard University, Cambridge, MA 02138, USA.
E-mail:
[email protected];
[email protected] Received: 14 May 2010 / Accepted: 30 September 2010 Published online: 11 February 2011 – © Springer-Verlag 2011
Abstract: We consider Hermitian and symmetric random band matrices H in d 1 dimensions. The matrix elements Hx y , indexed by x, y ∈ ⊂ Zd , are independent, uniformly distributed random variables if |x − y| is less than the band width W , and zero otherwise. We prove that the time evolution of a quantum particle subject to the Hamiltonian H is diffusive on time scales t W d/3 . We also show that the localization length of the eigenvectors of H is larger than a factor W d/6 times the band width. All results are uniform in the size || of the matrix. 1. Introduction The general formulation of the universality conjecture for disordered systems states that there are two distinctive regimes depending on the energy and the disorder strength. In the strong disorder regime, the eigenfunctions are localized and the local spectral statistics are Poisson. In the weak disorder regime, the eigenfunctions are delocalized and the local statistics coincide with those of a Gaussian matrix ensemble. Random band matrices are natural intermediate models to study eigenvalue statistics and quantum propagation in disordered systems as they interpolate between Wigner matrices and random Schrödinger operators. Wigner matrix ensembles represent meanfield models without spatial structure, where the quantum transition rates between any two sites are i.i.d. random variables with zero expectation. In the celebrated Anderson model [5], only a random on-site potential V is present in addition to a short range deterministic hopping (Laplacian) on a graph that is typically a regular box in Zd . For the Anderson model, a fundamental open question is to establish the metalinsulator transition, i.e. to show that in d 3 dimensions the eigenfunctions of −+λV are delocalized for small disorder λ. The localization regime at large disorder or near the Partially supported by SFB-TR 12 Grant of the German Research Council. Partially supported by U.S. National Science Foundation Grant DMS 08–04279.
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spectral edges has been well understood by Fröhlich and Spencer with the multiscale technique [29,30], and later by Aizenman and Molchanov by the fractional moment method [3]; many other works have since contributed to this field. In particular, it has been established that the local eigenvalue statistics are Poisson [38] and that the eigenfunctions are exponentially localized with an upper bound on the localization length that diverges as the energy parameter approaches the presumed phase transition point [15,43]. The progress in the delocalization regime has been much slower. For the Bethe lattice, corresponding to the infinite-dimensional case, delocalization has been established in [4,27,35]. In finite dimensions only partial results are available. The existence of an absolutely continuous spectrum (i.e. extended states) has been shown for a rapidly decaying potential, corresponding to a scattering regime [8,10,39]. Diffusion has been established for a heavy quantum particle immersed in a phonon field in d 4 dimensions [28]. For the original Anderson Hamiltonian with a small coupling constant λ, the eigenfunctions have a localization length of at least λ−2 (see [9]). The time and space scale λ−2 corresponds to the kinetic regime where the quantum evolution can be modelled by a linear Boltzmann equation [24,45]. Beyond this time scale the dynamics is diffusive. This has been established in the scaling limit λ → 0 up to time scales t ∼ λ−2−κ with an explicit κ > 0 in [18–20]. There are no rigorous results on the local spectral statistics of the Anderson model, but it is conjectured – and supported by numerous arguments in the physics literature, especially by supersymmetric methods (see [14]) – that the local correlation function of the eigenvalues of the finite volume Anderson model follows the GOE statistics in the thermodynamic limit. Due to their mean-field character, Wigner matrices are simpler to study than the Anderson model and they are always in the delocalization regime. The complete delocalization of the eigenvectors was proved in [21]. The local spectral statistics in the bulk are universal, i.e. they follow the statistics of the corresponding Gaussian ensemble (GOE, GUE, GSE), depending on the symmetry type of the matrix (see [37] for explicit formulas). For an arbitrary single entry distribution, bulk universality has been proved recently in [17,22,23] for all symmetry classes. A different proof was given in [46] for the Hermitian case. Random band matrices H = {Hx y }x,y∈ represent systems on a large finite graph with a metric. The matrix elements between two sites, x and y, are independent random variables with a variance σx2y := E|Hx y |2 depending on the distance between the two sites. The variance typically decays with the distance on a characteristic length scale W , called the band width of H . This terminology comes from the simplest one-dimensional model where the graph is a path on N vertices, labelled by = {1, 2, . . . , N }, and the matrix elements Hx y vanish if |x − y| W . If W = N and all variances are equal, we recover the usual Wigner matrix. The case W = O(1) is a one-dimensional Andersontype model with random hoppings at bounded range. Higher-dimensional models are obtained if the graph is a box in Zd . For more general random band matrices and for a systematic presentation, see [44]. Since the one-dimensional Anderson-type models are always in the localization regime, varying the band width W offers a possibility to test the localization-delocalization transition between an Anderson-type model and the Wigner ensemble. Numerical simulations and theoretical arguments based on supersymmetric methods [31] suggest that the local eigenvalue statistics change from Poisson, for W N 1/2 , to GOE (or GUE), for W N 1/2 . The eigenvectors are expected to have a localization length of order W 2 . In particular the eigenvectors are fully delocalized for W N 1/2 . In two
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dimensions the localization length is expected to be exponentially large in W ; see [1]. In accordance with the extended states conjecture for the Anderson model, the localization length is expected to be macroscopic, ∼ N , independently of the band width in d 3 dimensions. Extending the techniques of the rigorous proofs for Anderson localization, Schenker has recently proved the upper bound W 8 for the localization length in d = 1 dimensions [40]. In this paper we prove a counterpart of this result from the side of delocalization. More precisely, we show a lower bound W 1+d/6 for the eigenvectors of d-dimensional band matrices with uniformly distributed entries. We remark that the lower bound W was proved recently in [25] for very general band matrices. On the spectral side, we mention that, apart from the semicircle law (see [2,25,33] for d = 1 and [11] for d = 3), the question of bulk universality of local spectral statistics for band matrices is mathematically open even for d = 1. In the spirit of the general conjecture, one expects GUE/GOE statistics in the bulk for the delocalization regime, W N 1/2 . The GUE/GOE statistics have recently been established [25] for a class of generalized Wigner matrices, where the variances of different matrix elements are not necessarily identical, but are of comparable size, i.e. E|Hx y |2 ∼ E|Hx y |2 ; in particular, the band width is still macroscopic (W ∼ N ). Supersymmetric methods offer a very attractive approach to study the delocalization transition in band matrices, but the rigorous control of the functional integrals away from the saddle points is difficult and it has been performed only for the density of states [11]. Effective models that emerge near the saddle points can be more accessible to rigorous mathematics. Recently Disertori, Spencer and Zirnbauer studied a related statistical mechanics model that is expected to reflect the Anderson localization and delocalization transition for real symmetric band matrices. They proved a quasi-diffusive estimate for the two-point correlation functions in a three dimensional supersymmetric hyperbolic nonlinear sigma model at low temperatures [13]. Localization was also established in the same model at high temperatures [12]. We also mention that band matrices are not the only possible interpolating models to mimic the metal-insulator transition. Other examples include the Anderson model with a spatially decaying potential [8,34] and a quasi one-dimensional model with a weak on-site potential for which a transition in the sense of local spectral statistics has been established in [6,47]. A natural approach to study the delocalization regime is to show that the quantum time evolution is diffusive on large scales. We normalize the matrix entries so that the rate of quantum jumps is of order one. The typical distance of a single jump is the band width√ W . If the jumps were independent, the typical distance travelled in time t would be W t. Using the argument of [9], we show that a typical localization length √is incompatible with a diffusion on spatial scales larger than . Thus we obtain W t, provided that the diffusion approximation can be justified up to time t. The main result of this paper is that the quantum dynamics of the d-dimensional band matrix is given by a superposition of heat kernels up to time scales t W d/3 . Although diffusion is expected to hold up to time t ∼ W 2 for d = 1 and up to any time for d 3 (assuming the thermodynamic limit has been taken), our method can follow the quantum dynamics only up to t W d/3 . The threshold exponent d/3 originates in technical estimates on certain Feynman graphs; going beyond the exponent d/3 would require a further resummation of certain four-legged subdiagrams (see Sect. 11). Finally, we remark that our method also yields a bound on the largest eigenvalue of a band matrix; see Theorem 3.4 in the forthcoming paper [16] for details.
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2. The Setup Let the dimension d 1 be fixed and consider the d-dimensional lattice Zd equipped with the Euclidean norm |·|Zd (any other norm would also do). We index points of Zd with x, y, z, . . . . Let W > 1 denote a large parameter (the band width) and define M ≡ M(W ) := {x ∈ Zd : 1 |x|Zd W }, the number of points at distance at most W from the origin. In the following we tacitly make use of the obvious relation M ∼ C W d . For notational convenience, we use both W and M in the following. In order to avoid dealing with the infinite lattice directly, we restrict the problem to a finite periodic lattice N of linear size N . More precisely, for N ∈ N, we set N := {−[N /2], . . . , N − 1 − [N /2]}d ⊂ Zd , a cube with side length N centred around the origin. Here [·] denotes integer part. We regard N as periodic, i.e. we equip it with periodic addition and the periodic distance |x| := inf{|x + N ν|Zd : ν ∈ Zd }. Unless otherwise stated, all summations x are understood to mean x∈ N . We consider random matrices H ω ≡ H whose entries Hx y are indexed by x, y ∈ N . Here ω denotes the running element in probability space. The large parameter of the model is the band width W . We shall always assume that N W M 1/6 . Under this condition all our results hold uniformly in N . We assume that H is either Hermitian or symmetric. The entries Hx y satisfying 1 |x − y| W are i.i.d. (with the obvious restriction that Hyx = Hx y ). In the Hermitian case they are uniformly distributed on a circle of appropriate radius in the complex plane, Hx y ∼ √
1 M −1
Unif(S1 ),
1 |x − y| W.
(2.1a)
In the symmetric case they are Bernoulli random variables, 1 1 −1 = P Hx y = √ = , 1 |x − y| W. (2.1b) P Hx y = √ 2 M −1 M −1 If |x − y| ∈ / [1, W ], then Hx y = 0. An important consequence of our assumptions (2.1a) and (2.1b) is |Hx y |2 =
1 1(1 |x − y| W ). M −1
(2.2)
We remark that the assumption that the matrix entries have the special form (2.1a) or (2.1b) is not necessary for our results to hold. We make it here because it greatly simplifies our proof. The reason for this is that, as observed by Feldheim and Sodin [26,42], the condition (2.2) allows one to obtain a simple algebraic expression for the nonbacktracking powers of H ; see Lemma 5.2. In the forthcoming paper [16] we extend our results to random matrix ensembles in which the matrix elements Hx y are allowed to have a general distribution (and thus in
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particular a genuinely random absolute value); moreover their variances E|Hx y |2 are given by a general profile on the scale W in x − y (as opposed to the step function profile in (2.2)). Under these assumptions, the algebraic identity of Lemma 5.2 is no longer exact, and needs to be amended with additional random terms. The resulting graphical expansion is considerably more involved than in the case (2.2), and its control requires essential new ideas. However, the fundamental mechanism underlying quantum diffusion for band matrices is already apparent in the special case (2.2) discussed in this paper. Let α ∈ A := {1, . . . , | N |} index the orthonormal basis {ψαω }α∈A of eigenvectors of the matrix H ω , i.e. H ω ψαω = λωα ψαω , where λωα ∈ R. The normalization of the matrix elements is chosen in such a way that the typical eigenvalue of the matrix is of order one: 1 2 1 1 M E Tr H 2 = E . Eλα = |Hx y |2 = |A| α |A| |A| x,y M −1 3. Scaling and Results The central quantity of our analysis is 2 (t, x) := E δx , e−it H/2 δ0 , where δx ∈ 2 ( N ) denotes the standard basis vector, defined by (δx ) y = δx y . The factor 1/2 is a convenient normalization since, by a standard result of random matrix theory, the spectrum of H/2 is asymptotically equal to the unit interval [−1, 1]. The function (t, x) describes the ensemble average of the quantum transition probability of a particle starting from position 0 ending up at position x after time t. Note that x (t, x) = 1 for any t ∈ R. Heuristically, the particle performs a series of random jumps of size W . The typical number of jumps in time t = O(1) is of order one. Indeed, by first order perturbation theory, the small-times probability distribution for 1 |x| W is given by (t, x) ∼ E | δx , (1 − it H/2)δ0 |2 =
1 t2 t2 E |Hx0 |2 = , 4 4 M −1
up to higher order terms in t. Thus x =0 (t, x) is an O(1) quantity, separated away from zero, indicating that the distance from the origin is of O(W ) for times t ∼ O(1). In time t the particle performs O(t) jumps of size O(W ). We expect that the jumps are approximately independent and the trajectory is a random walk consisting of O(t) steps with size O(W ) each. Thus, the typical distance from the origin is of order t 1/2 W . We rescale time and space (t, x) → (T, X ) so as to make the macroscopic quantities T and X of order one, i.e. we set t = ηT,
x = η1/2 W X,
where W and η are two large parameters. Ideally, one would like to study the long time limit η → ∞ for a fixed W . In this case, however, we know that the dynamics cannot be diffusive for d = 1. Indeed, as explained in the Introduction, it is expected that the motion cannot be diffusive for distances larger than W 2 ; this has in fact been proved [40] for distances larger than W 8 . Thus we have to consider a scaling limit where η and W
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are related and they tend simultaneously to infinity. To that end we choose an exponent κ > 0 and set η ≡ η(W ) := W dκ . Our first main result establishes that (t, x) behaves diffusively up to time scales t = O(W dκ ) if κ < 1/3. Theorem 3.1 (Quantum diffusion). Let 0 < κ < 1/3 be fixed. Then for any T0 > 0 and any continuous bounded function ϕ ∈ Cb (Rd ) we have
x dκ = W T, x ϕ dX L(T, X ) ϕ(X ), (3.1) lim W →∞ W 1+dκ/2 Rd x∈ N
uniformly in N W 1+d/6 and 0 T T0 . Here 1 λ2 4 L(T, X ) := dλ √ G(λT, X ), π 1 − λ2 0 and G is the heat kernel G(T, X ) :=
d +2 2π T
d/2
d+2
e− 2T
|X |2
,
(3.2)
Remark 3.2. The factor d + 2 arises from a random walk in d dimensions with steps in the unit ball. If B is a random variable uniformly distributed in the d-dimensional unit ball, the covariance matrix of B is (d + 2)−1 1. This result can be interpreted as follows. The limiting dynamics at macroscopic time T is not given by a single heat kernel, but by a weighted superposition of heat kernels at times λT , for 0 λ 1. The factor λ expresses a delay arising from backtracking paths, in which the quantum particle “wastes time” by retracing its steps. If the particle is not backtracking, it is moving according to diffusive dynamics. The backtracking paths correspond to two-legged subdiagrams, and have the interpretation of a self-energy renormalization in the language of diagrammatic perturbation theory. Thus, out of the total macroscopic time T during which the particle moves, a fraction λ of T is spent moving diffusively, and a fraction (1 − λ) of T backtracking. Theorem 3.1 gives an explicit 2 expression for the probability density f (λ) = π4 √ λ 2 1(0 λ 1) for the particle to 1−λ move during a fraction λ of T . Our proof precisely exhibits this phenomenon. As explained in Sect. 4, the proof is based on an expansion of the quantum time evolution in terms of nonbacktracking paths. At time t = W dκ T , this expansion yields a weighted superposition of paths of lengths n = 1, . . . , [t] (higher values of n are strongly suppressed). Here n is the number of nonbacktracking steps, i.e. the number of steps that contribute to the effective motion of the particle. The difference [t] − n is the number of steps that the particle spends backtracking. Our expansion (or, more precisely, its leading order ladder terms) shows that the weight of a path of n nonbacktracking steps is given by |αn (t)|2 , where αn (t) is the Chebyshev transform of the propagator e−itξ in ξ ; see (5.3). The probability density f arises from this microscopic picture by setting n = [λt]. Then we have, as proved in Proposition 8.5 below, t|α[λt] (t)|2 → f (λ) weakly as t → ∞. Our second main result shows that the eigenvectors of H have a typical localization length larger than W 1+dκ/2 , for any κ < 1/3. For x ∈ N and > 0 we define the characteristic function Px, projecting onto the complement of an -neighbourhood of x,
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Px, (y) := 1(|y − x| ). Let ε > 0 and define the random subset Aωε, ⊂ A of eigenvectors through
ω ω ω Aε, := α ∈ A : |ψα (x)| Px, ψα < ε . x
Aωε,
The set contains, in particular, all eigenvectors that are exponentially localized in balls of radius O( ); see Corollary 3.4 below for a more general and precise statement. Theorem 3.3 (Delocalization). Let ε > 0 and 0 < κ < 1/3. Then A 1+dκ/2 √ ε,W lim sup E 2 ε, |A| W →∞ uniformly in N W 1+d/6 . Theorem 3.3 implies that the fraction of eigenvectors subexponentially localized on scales W 1+κd/2 converges to zero in probability. Corollary 3.4. For fixed γ > 0 and K > 0 define the random subset of eigenvectors
|x − u| γ ω ω 2 B := α ∈ A : ∃ u ∈ N : |ψα (x)| exp K . (3.3) x Then for 0 < κ < 1/3 we have lim E
W →∞
|BW 1+κd/2 | = 0, |A|
uniformly in N W 1+d/6 . 4. Main Ideas of the Proof We need to compute the expectation of the squared matrix elements of the unitary time evolution e−it H/2 . A natural starting point is the power series expansion e−it H/2 = n n 0 (−it H/2) /n!. Unfortunately, the resulting series is unstable for t → ∞, as is manifested by the large cancellations in the sum E| δx , e
−it H/2
in−n t n+n n EHxny Hyx δ y | = . n+n n!n ! 2 2
(4.1)
n,n
This can be seen as follows. The expectation n =E Hx x1 Hx1 x2 . . . Hxn−1 y Hyyn −1 . . . Hy1 x EHxny Hyx
(4.2)
x1 ,...xn−1 y1 ,... yn −1
is traditionally represented graphically by drawing the labels x, x1 , x2 , . . . , y1 , x as vertices of a path, and by identifying vertices whose labels are identical. Since the matrix elements are centred (i.e. EHx y = 0 for all x, y), each edge must be traveled at
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least twice in any path that yields a nonzero contribution to (4.2). It is well known that the leading order contribution to (4.2) is given by the so-called fully backtracking paths. A fully backtracking path is a path generated by successively applying the transformation a → aba to the trivial path x. A typical fully backtracking path may be thought of as a tree with double edges. It is not hard to see that, after summing over y, each fully backtracking path yields a contribution of order 1 to (4.2). Also, the number of fully backtracking paths is of order 4n+n , so that the expectation (4.2) is of order 4n+n . In particular, this implies that the main contribution to (4.1) comes from terms satisfying n + n ∼ t. Moreover, the series (4.1) is unstable in the sense that the sum of the absolute values of its summands behaves like e4t as t → ∞. The large terms in (4.1) systematically cancel each other out similarly to the twolegged subdiagram renormalization in perturbative field theory. In perturbative renormalization, these cancellations are exploited by introducing appropriately adjusted fictitious counter-terms. In the current problem, however, we make use of the Chebyshev transformation, which removes the contribution of all backtracking paths in one step. The key observation is that, if Un denotes the n th Chebyshev polynomial of the second kind, then Un (H/2) can be expressed in terms of nonbacktracking paths. A nonbacktracking path is a path which contains no subpath of the form aba. Thus the strongest instabilities in (4.1) can be removed if e−it H/2 is expanded into a series of Chebyshev polynomials. This idea appeared first in [7] and has recently been exploited in [26,42] to prove, among other things, the edge-universality for band matrices. In [42] it is also stated that the same method can be used to prove delocalization of the edge eigenvectors if W N 5/6 , i.e. to get the bound W 6/5 on the localization length . Our estimate gives a slightly weaker bound, W 7/6 , for this special case, but it applies to bulk eigenvectors as well as higher dimensions. After the Chebyshev transform, we need to compute expectations E Hx x1 Hx1 x2 . . . Hxn−1 y Hyyn −1 . . . Hy1 x , (4.3) x1 ,...xn−1 y1 ,...yn −1
where the summations are restricted to nonbacktracking paths. As above, since EHab = 0, every matrix element must appear at least twice in the non-trivial terms of (4.3). Taking the expectation effectively introduces a pairing, or more generally a lumping, of the factors, which can be conveniently represented by Feynman diagrams. The main contribution comes from the so-called ladder diagrams, corresponding to n = n and xi = yi . The contribution of these diagrams can be explicitly computed, and showed to behave diffusively. More precisely: Since we express nonbacktracking powers of H as Chebyshev polynomials in H/2, the contribution of each graph to the propagator e−it H/2 carries a weight equal to the Chebyshev transform αn (t) of e−itξ in ξ . We shall show that αn (t) is given essentially by a Bessel function of the first kind. In order to identify the limiting behaviour of the ladder we therefore need to analyse a diagrams, probability distribution on N of the form |αn (t)|2 n∈N for large t (Sect. 8). The main work consists of proving that the non-ladder diagrams are negligible. Similarly to the basic idea of [18–20], the non-ladder diagrams are classified according to their combinatorial complexity. The large number of complex diagrams is offset by their small value, expressed in terms of powers of W . Conversely, diagrams containing large pieces of ladder subdiagrams have a relatively large contribution but their number is small. More precisely, focusing only on the pairing diagrams in the Hermitian case, it is easy to see that ladder subdiagrams are marginal for power counting. We define the skeleton
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of a graph by collapsing parallel ladder rungs (called bridges) into a single rung. We show that the value of a skeleton diagram is given by a negative power of M ∼ C W d that is proportional to the size of the skeleton diagram. This is how the dimension d enters our estimate. We then sum up all possible ladder subdiagrams corresponding to a given skeleton. Although the ladder subdiagrams do not yield additional W -powers, they represent classical random walks for which dispersive bounds are available, rendering them summable. The restriction t W d/3 comes from summing up the skeleton diagrams. In Sect. 11 we present a critical skeleton that shows that this restriction is necessary without further resummation or a more refined classification of complex graphs. 5. The Path Expansion We start by writing the expansion of e−it H/2 in terms of nonbacktracking paths by using the Chebyshev transform. 5.1. The Chebyshev transform of e−itξ . The Chebyshev transform αk (t) of e−itξ is defined by e−itξ =
∞
αk (t) Uk (ξ ).
k=0
Here Uk denotes the Chebyshev polynomial of the second kind, defined through Uk (cos θ ) =
sin(k + 1)θ sin θ
(5.1)
for k = 0, 1, 2, . . .. The Chebyshev polynomials satisfy the orthogonality relation 2 1 dξ 1 − ξ 2 Uk (ξ ) Ul (ξ ) = δkl . π −1 Therefore the coefficients αk (t) are given by 2 1 dξ 1 − ξ 2 e−itξ Uk (ξ ). αk (t) = π −1
(5.2)
The coefficient αk (t) can be evaluated explicitly using the standard identities (see [32]) Uk (ξ ) =
ξ Tk+1 (ξ ) − Tk+2 (ξ ) , Tk+2 (ξ ) − 2ξ Tk+1 (ξ ) + Tk (ξ ) = 0, 1 − ξ2 2 1 T2l (ξ ) cos(tξ ) dξ = 2(−1)l J2l (t), π −1 1 − ξ2 2 1 T2l+1 (ξ ) sin(tξ ) dξ = 2(−1)l J2l+1 (t), 2 π −1 1−ξ 2(k + 1) Jk+1 (t). Jk (t) + Jk+2 (t) = t
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Here Tk denotes the Chebyshev polynomial of the first kind and Jk the Bessel function of the first kind; they are defined through 1 π Tk (cos θ ) := cos(kθ ), Jk (t) := dθ cos(t sin θ − tθ ). π 0 If k = 2l is even we may therefore compute 2 1 α2l (t) = dξ 1 − ξ 2 cos(tξ ) Uk (ξ ) π −1 ξ Tk+1 (ξ ) − Tk+2 (ξ ) 2 1 dξ 1 − ξ 2 cos(tξ ) = π −1 1 − ξ2 1 2 Tk (ξ ) − Tk+2 (ξ ) = dξ cos(tξ ) = (−1)l [J2l (t) + J2l+2 (t)] 2 π −1 2 1−ξ 2l + 1 = 2(−1)l J2l+1 (t). t If k = 2l + 1 is odd a similar calculation yields α2l+1 (t) = −2i(−1)l
2l + 2 J2l+2 (t). t
Thus we have the following result. Lemma 5.1. We have that e−itξ =
αk (t) Uk (ξ ),
k
where αk (t) = 2(−i)k
k+1 Jk+1 (t). t
Also, for all t ∈ R we have the identity |αk (t)|2 = 1,
(5.3)
(5.4)
k 0
as follows from the orthonormality of the Chebyshev polynomials. 5.2. Expansion in terms of nonbacktracking paths. For n = 0, 1, 2, . . . , let H (n) denote the n th nonbacktracking power of H . It is defined by := Hx(n) Hx0 x1 · · · Hxn−1 xn , 0 ,x n x1 ,...,xn−1
where means sum under the restriction xi = xi+2 for i = 0, . . . , n − 2. We call this restriction the nonbacktracking condition. The following key observation is due to Bai and Yin [7].
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Lemma 5.2. The nonbacktracking powers of H satisfy H (0) = 1,
H (1) = H,
H (2) = H 2 −
M 1, M −1
as well as the recursion relation H (n) = H H (n−1) − H (n−2)
(n 3).
(5.5)
Proof. For the convenience of the reader we give the simple proof. The cases n = 0, 1, 2 are easily checked. Moreover,
H H (n−1)
x0 xn
=
n−2
1(xi = xi+2 ) Hx0 x1 · · · Hxn−1 xn
x1 ,...,xn−1 i=1 n−2
=
1(xi = xi+2 ) Hx0 x1 · · · Hxn−1 xn
x1 ,...,xn−1 i=0
+
1(x0 = x2 )
x1 ,...,xn−1
n−2
1(xi = xi+2 ) Hx0 x1 · · · Hxn−1 xn
i=1
= (H (n) )x0 ,xn
+
1(x0 = x4 )
x3 ,...,xn−1
×
n−2
1(xi = xi+2 )Hx0 x3 Hx3 x4 · · · Hxn−1 xn
i=3
1(x1 = x3 )|Hx0 x1 |2
x1
= (H
(n)
)x0 ,xn + (H (n−2) )x0 xn .
Notice that in the last step we used (2.2).
Feldheim and Sodin have observed [26,42] that (5.5) is reminiscent of the recursion n (ξ ) := relation for the Chebyshev polynomials of the second kind. Let us abbreviate U Un (ξ/2). Then we have (see e.g. [32]) 0 (ξ ) = 1, U
1 (ξ ) = ξ, U
2 (ξ ) = ξ 2 − 1, U
and for n 2, n−1 (ξ ) − U n−2 (ξ ). n (ξ ) = ξ U U Comparing this to Lemma 5.2, we get, following [26,42], n (H ) − H (n) = U
1 Un−2 (H ). M −1
n (H ) yields Solving for U n (H ) = U
k 0
1 H (n−2k) , (M − 1)k
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Fig. 1. The graphical representation of paths of vertices
with the convention that H (n) = 0 for n < 0. Therefore Lemma 5.1 yields e−it H/2 =
n (H ) = αn (t) U
n 0
H (m)
m 0
αm+2k (t) . (M − 1)k
k 0
We have proved the following result. Lemma 5.3. We have that e−it H/2 =
am (t)H (m) ,
m 0
where am (t) :=
αm+2k (t) . (M − 1)k
k 0
6. Graphical Representation For ease of presentation, we assume throughout the proof of Theorem 3.1 (Sects. 6–8) that we are in the Hermitian case (2.1a). How to extend our arguments to cover the symmetric case (2.1b) is described in Sect. 9. Using Lemma 5.3 we get (t, x) =
n,n 0
(n)
(n )
an (t)an (t) E H0x Hx0 . (n)
(n )
Expanding in nonbacktracking paths yields a graphical expansion. Let us write H0x Hx0 as a sum over paths x0 , x1 , . . . , xn+n −1 , x0 , where x0 = 0 and xn = x. Such a path is graphically represented as a loop of n + n vertices belonging to the set Vn,n := {0, . . . , n + n − 1}; see Fig. 1. Vertices i ∈ Vn,n satisfying the nonbacktracking condition (i.e. xi−1 = xi+1 ) are drawn using black dots; other vertices are drawn using white dots. There are n + n oriented edges e0 , . . . , en+n −1 defined by ei := (i, i + 1) (here, and in the following, Vn,n is taken to be periodic). We denote by En,n := {e0 , . . . , en+n −1 } the set of edges. In Fig. 1 the edges are oriented clockwise. Each vertex has an outgoing and an incoming edge, and each edge e has an initial vertex a(e) and final vertex b(e). Moreover, we order the edges using their initial vertices.
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Each vertex i ∈ Vn,n carries a label xi ∈ N . The labels x = (x0 , . . . , xn+n −1 ) are summed over under the restriction Q x (x) = 1, where Q x (x) := δ0x0 δxn x
n+n −1
1(1 |xi − xi+1 | W )
i=0
n−2
1(xi = xi+2 )
i=0
n+n −2
1(xi = xi+2 ).
i=n
The two last products implement the nonbacktracking condition. We define the unordered pair of labels corresponding to the edge e through x (e) := {xa(e) , xb(e) }. Next, to each configuration of labels x = (x0 , . . . , xn+n −1 ) we assign a lumping = (x) of the set of edges En,n . Here a lumping means a partition of En,n or, equivalently, an equivalence relation on En,n . We use the notation = {γ }γ ∈ , where γ ∈ is lump of , i.e. an equivalence class. The lumping = (x) associated with the labels x is defined according to the rule that e and e are in the same lump γ ∈ if and only if x (e) = x (e ). Let Gn,n denote the set of lumpings of En,n obtained in this manner. Thus we may write (n) (n ) E H0x Hx0 = Vx (). ∈Gn,n
Here Vx () =
∗
Q x (x) E Hx0 x1 · · · Hxn+n −1 x0 ,
x
where the summation is restricted to label configurations yielding the lumping . Next, observe that the expectation of a monomial y,z (Hyz )ν yz is nonzero if and only if ν yz = νzy for all y, z (here we only use that the law of the matrix entries is invariant under rotations of the complex plane). In particular, Vx () vanishes if one lump γ ∈ is of odd size. Defining the subset Gn,n ⊂ Gn,n of lumpings whose lumps are of even size, we find that (n) (n ) Hx0 = Vx (). E H0x ∈Gn,n
We summarize the key properties of Gn,n . Lemma 6.1. Let ∈ Gn,n . Then each lump γ ∈ is of even size. Moreover, any two edges e, e ∈ γ in the same lump γ are separated by either at least two edges or a vertex in {0, n} (nonbacktracking property). Next, we give an explicit expression for Vx (). We start by assigning to each lump γ ∈ an unordered pair of labels γ . Then we pick a partition πγ of γ into two subsets of equal size. Abbreviate these families as = {γ }γ ∈ and π = {πγ }γ ∈ . Thus we get ⎞ ⎛ ⎝ Vx () = Q x (x) x (γ , πγ )⎠ x
⎛
×⎝
π
γ =γ
γ ∈
⎞
1(γ = γ )⎠ EHx0 x1 · · · Hxn+n −1 x0 .
(6.1)
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Fig. 2. A pairing of edges
Here, for each γ ∈ , γ ranges over all unordered pairs of labels and πγ ranges over all partitions of γ into two subsets of equal size; x (γ , πγ ) is the indicator function of the following event: For all e ∈ γ we have that x (e) = γ , and e, e ∈ γ belong to the same subset of πγ ⇒ xa(e) = xa(e ) , xb(e) = xb(e ) , e, e ∈ γ belong to different subsets of πγ ⇒ xa(e) = xb(e ) , xb(e) = xa(e ) . This definition of x (γ , πγ ) has the following interpretation. All edges in γ (corresponding to matrix elements) have the same unordered pair of labels (and hence represent copies of the same random variable Hyz or its complex conjugate). Moreover, each random variable Hyz must appear as many times as its complex conjugate; random variables indexed by two edges e, e ∈ γ are identical if e, e belong to the same subset of πγ , and each other’s complex conjugates if e, e belong to different subsets of πγ . Note that the expectation in (6.1) is equal to 1 , (M − 1)n¯
(6.2)
where n¯ := n+n 2 . In particular, Vx () 0. An important subset of lumpings of En,n is the set of pairings, Pn,n ⊂ Gn,n , which contains all lumpings satisfying |γ | = 2 for all γ ∈ . We call two-element lumps σ ∈ Pn,n bridges. Given a pairing ∈ Pn,n , we say that e and e are bridged (in ) if there is a σ ∈ such that σ = {e, e }. Bridges are represented graphically by drawing a line, for each {e, e } ∈ , from the edge e to e ; see Fig. 2. Thus a pairing ∈ Pn,n is the edge set of a graph whose vertex set is En,n . If is a pairing, each bridge σ ∈ has a unique partition πσ of its edges, so that the expression (6.1) for Vx () may be rewritten in the simpler form ⎛ ⎞ Vx () = Q x (x) ⎝ 1(xa(e) = xb(e ) )1(xb(e) = xa(e ) )⎠ x
⎛
×⎝
{e,e }∈
⎞
1(x (e) = x (e ))⎠
σ =σ e∈σ e ∈σ
1 . (M − 1)n¯
(6.3)
The main contribution to the expansion is given by the ladder pairing L n ∈ Pn,n . It is defined as L n := {{e0 , e2n−1 }, {e1 , e2n−2 }, . . . , {en−1 , en }} . The ladder is represented graphically in Fig. 3.
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Fig. 3. The ladder pairing
7. The Non-ladder Lumpings In this section we estimate the contribution of the non-ladder lumpings and show that it ∗ ⊂ G denote the set of non-ladder lumpings, vanishes in the limit W → ∞. Let Gn,n n,n ∗ := G if n = n and G ∗ := G ∗ ∗ i.e. Gn,n n,n \{L n }. Similarly, let Pn,n := Pn,n ∩ Gn,n n,n n,n denote the set of non-ladder pairings. We shall prove the following result. Proposition 7.1. Let 0 < κ < 1/3 and pick a β satisfying 0 < β < 2/3 − 2κ. Then there is a constant C such that
|an (ηT ) an (ηT )|
Vx ()
∗ ∈Gn,n
x n,n 0
C , W dβ
for W larger than some W0 (T, κ) and N W 1+d/6 . The rest of this section is devoted to the proof of Proposition 7.1.
7.1. Controlling the non-pairings. Replacing the expectation in (6.1) with (6.2) we get Vx () =
Q x (x)
x
π
⎛ ⎝
⎞⎛ x (γ , πγ )⎠ ⎝
⎞ 1(γ = γ )⎠
γ =γ
γ ∈
1 . (M − 1)n¯
∗ in terms of a sum over all We start by estimating the sum over all lumpings ∈ Gn,n ∗ pairings ∈ Pn,n . Let us define
Rx () :=
Q x (x)
⎛ ⎝
π
x
⎞ x (γ , πγ )⎠
γ ∈
Lemma 7.2. For all n, n ∈ N we have ∗ ∈Gn,n
Vx ()
∗ ∈Pn,n
Rx ().
1 . (M − 1)n¯
(7.1)
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Fig. 4. Two parallel bridges
Proof. Let γ and πγ be given for each γ ∈ . For each γ , pick any pairing γ of γ that is compatible with πγ in the sense that, for each bridge σ ∈ γ , the two edges of σ belong to different subsets of πγ . If n = n , we additionally require that not all γ ’s are subsets of the Ladder L n (such a choice is always possible). Next, set σ = γ for all σ ∈ γ . Note that each bridge σ carries a unique partition πσ . It is then easy to see that for any pairing γ as above, we have x (σ , πσ ). x (γ , πγ ) σ ∈γ
Thus, by partitioning each γ ∈ into bridges, we see that each term in ∈G ∗ Vx () n,n is bounded by a corresponding term in ∈P ∗ Rx (). In fact, there is an overcounting n,n arising from the different ways of partitioning γ into bridges. Because of Lemma 7.2 we may restrict ourselves to pairings. We estimate Rx (). If is a pairing we may write, just like (6.3), the expression (7.1) ∗ ∈Pn,n in the simpler form ⎛ ⎞ 1 Q x (x) ⎝ 1(xa(e) = xb(e ) )1(xb(e) = xa(e ) )⎠ . (7.2) Rx () = (M − 1)n¯ x {e,e }∈
7.2. Collapsing of parallel bridges. Let us introduce the set P ∗n,n , defined as the set ∗ is a proper subset of P ∗ of all non-ladder pairings of En,n . Clearly, Pn,n n,n (due to the ∗ ). nonbacktracking condition of Lemma 6.1 which is imposed on pairings in Pn,n Let n, n 0 and ∈ P ∗n,n . For any i, j, we say that the two bridges {ei , e j } and / {0, n}; see Fig. 4. Two parallel bridges may {ei+1 , e j−1 } of are parallel if i + 1, j ∈ be collapsed to obtain a new pairing of a smaller set of edges, in which the parallel bridges are replaced by a single bridge. More precisely: We obtain ∈ P ∗m,m from ∈ P ∗n,n by removing the vertices i + 1 and j, by creating the edges (i, i + 2) and ( j − 1, j + 1), and by bridging them. Finally, we rename the vertices using the increasing integers 0, 1, 2, . . . , n + n − 3; by definition, the new name of the vertex n is m, and m is defined through m + m + 2 = n + n . The converse operation of collapsing bridges, expanding bridges, is self-explanatory.
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Fig. 5. Collapsing parallel bridges to obtain the skeleton pairing
In the next lemma we iterate the above procedure → until all parallel bridges have been collapsed. ∗ . Then there exist m n, m n , and a pairing S() ∈ Lemma 7.3. Let ∈ Pn,n ∗ G m,m containing no parallel bridges, such that may be obtained from S() by successively expanding bridges. This defines S() uniquely.
Proof. Successively collapse all parallel bridges in ; see Fig. 5. The result is clearly independent of the order in which this is done. We call the pairing = S() the skeleton of . The set of skeleton pairings of the edges Em,m is denoted by ∗ ∗ ∗ Sm,m S() : ∈ Pn,n := ∩ P m,m . n,n 0 ∗ ∗ Note that Sm,m is in general not a subset of Pm,m . The following lemma summarizes ∗ the key properties of Sm,m . ∗ Lemma 7.4. (i) Each ∈ Sm,m contains no parallel bridges. ∗ (ii) Let ∈ Sm,m and σ = {e, e } ∈ . Then e, e are adjacent only if e ∩ e ∈ {0, m}. ∗ (iii) If m¯ := m+m 2 = 1, then Sm,m = ∅.
Proof. Statement (i) follows immediately from the definition of S(). Statement (ii) is ∗ , i.e. Lemma 6.1. To a consequence of the nonbacktracking property of pairings in Pn,n ∗ ∗ . If = S() see this, let ∈ Sm,m be of the form = S() for some ∈ Pn,n contains a bridge {e, e } consisting of two consecutive edges e, e , then must also contain a bridge { f, f } consisting of two consecutive edges f, f . If e ∩ e ∈ / {0, m}, then f ∩ f ∈ / {0, n}, in contradiction to Lemma 6.1. Statement (iii) is an immediate ∗ . consequence of (ii) and the requirement that L 1 ∈ / S1,1 7.3. Contribution of parallel bridges. For given n and n , we estimate ∈P ∗ Rx () n,n by summing over skeleton pairings , followed by summing over all possible ways of expanding the bridges of . ∗ We observe that a pairing ∈ Pn,n is uniquely determined by its skeleton = ∗ S() ∈ Sm,m for some positive integers m, m as well as a family = { σ }σ ∈ satis¯ where σ encodes the number of parallel bridges that were collapsed to fying | | = n,
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L. Erd˝os, A. Knowles
Fig. 6. Summing up parallel bridges
form the bridge σ . Here σ 1 is a positive integer and | | := σ ∈ σ . Let G () denote the pairing obtained from by expanding the bridge σ into σ parallel bridges, for each σ ∈ . Thus may be recovered from its skeleton through = G () for a unique family . For given p ∈ N, the sum over all pairings satisfying || = p therefore becomes Rx () = Rx (G ()). (7.3) ∗ n+n =2 p ∈Pn,n
∗ m+m 2 p ∈Sm,m :| |= p
Next, we define and estimate the contribution to Rx () of a set of parallel bridges. Let 1, and two labels y, z be given. Then we define D (y, z) :=
x0 ,...,x
δx0 y δx z
−1
1(1 |xi − xi+1 | W ).
i=0
Thus, D (y, z) is equal to the number of paths of length from y to z, whereby each step takes values in {x : 1 |x| W }. (We could also have included the nonbacktracking restriction in the definition of D , but this is not needed as we only want an upper bound on Rx ()). Graphically, D corresponds to the contribution of parallel bridges; see Fig. 6. We need the following straightforward properties of D . Lemma 7.5. Let ∈ N. Then for each y we have D (y, z) = M . z
Moreover, for each y and z we have D (y, z) M −1 , as well as D (y, z)
C d/2
M −1 +
C M Nd
for some constant C. Proof. The first two statements are obvious. The last follows from a standard local central limit theorem; see for instance the proof in [42].
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Fig. 7. Construction of the orbit [i] of the vertex i ∗ . We observe that the product in (7.2) may be 7.4. Orbits of vertices. Fix ∈ Pn,n interpreted as an indicator function that fixes labels along paths of vertices. To this end, we define a map τ ≡ τ on the vertex set Vn,n . Start with a vertex i ∈ Vn,n . Let e be the outgoing edge of i (i.e. e = (i, i + 1)), and e the edge bridged by to e. Then we define τ i as the final vertex of e (i.e. e = (τ i − 1, τ i)). Thus the product in (7.2) may be rewritten as 1(xa(e) = xb(e ) )1(xb(e) = xa(e ) ) = δxi xτ i . {e,e }∈
i∈Vn,n
Starting from any vertex i ∈ Vn,n we construct a path (i, τ i, τ 2 i, . . . ). In this fashion the set of vertices is partitioned into orbits of τ ; see Fig. 7. Let [i] ⊂ Vn,n denote the orbit of the vertex i ∈ Vn,n . ∗ Next, let = S() ∈ Sm,m be the skeleton pairing of , and let the family be defined through = G (). The map τ ≡ τ on the skeleton pairing is defined exactly as for above. In order to sum over all labels x = (x0 , . . . , xn+n −1 ) in the expression for Rx (G ()), we split the set of labels x into two parts: labels of vertices between two parallel bridges, and labels associated with vertices of . In order to make this precise, we need the following definitions. Let Z () be the set of orbits of . It contains the distinguished orbits [0] and [m], which receive the labels 0 and x respectively. (Note that we may have [0] = [m], in which case x must be 0.) We assign a label yζ to each orbit ζ ∈ Z (), and define the family y := {yζ }ζ ∈Z () . Each bridge σ ∈ “sits between two orbits” ζ1 (σ ) and ζ2 (σ ). More precisely, let e = (i, i + 1) ∈ σ be the smaller edge of σ . Then we set ζ1 (σ ) := [i] and ζ2 (σ ) := [i + 1]. (Note that using the larger edge of σ in this definition would simply exchange ζ1 (σ ) and σ2 (σ ); this is of no consequence for the following.) ∗ , , and = G () ∈ P ∗ we have Lemma 7.6. For given ∈ Sm,m n,n
Rx ()
1 1(0 = y )1(x = y ) D σ (yζ1 (σ ) , yζ2 (σ ) ). [0] [m] (M − 1)n¯ y
(7.4)
σ ∈
Proof. The left-hand side of (7.4) is given by the expression (7.2). The summation over all xi ’s between parallel bridges of is contained in the factors D , and the summation over all the remaining xi ’s is replaced by the sum over y . We relaxed the nonbacktracking condition in Q x (x) to obtain an upper bound. ∗ ∗ Next, let Z ∗ () := Z ()\{[0]} and define L() := |Z ()|. The set Z () is the set of orbits whose label is summed over in x Rx (). The following lemma gives an
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Fig. 8. Left: a skeleton pairing giving rise to 5 orbits indexed by Z () = {1, . . . , 5}; the bridges in T , for one possible choice of T , are drawn using thick lines. Right: the corresponding multigraph on the vertex set Z (); the edges in φ(T ) are drawn using thick lines
upper bound on L(). It states, roughly, that the number of orbits (or free labels) is bounded by 2m/3; ¯ we refer to it as the 2/3 rule. Compare this bound with the trivial bound L() m, ¯ which would be sharp if were allowed to have parallel bridges. ∗ . Then L() Lemma 7.7 (The 2/3 Rule). Let ∈ Sm,m
2m¯ 3
+ 13 .
Proof. Let Z () := Z ()\{[0], [m]}. We show that every orbit ζ ∈ Z () consists of at least 3 vertices. Let i ∈ Vm,m belong to ζ ∈ Z (). Then, by Lemma 7.4 (ii), we have that τ i = i. By assumption, τ i ∈ / {0, m}. Hence τ 2 i = i, for otherwise would have two parallel bridges, in contradiction to Lemma 7.4 (i). Therefore the orbit of τ contains at least 3 vertices. Note that there are orbits containing exactly 3 vertices, as depicted in Fig. 7. The total number of vertices of not including the vertices 0 and m is 2m¯ − 2, so that we get 3|Z ()| 2m¯ − 2. The claim follows from the bound |Z ∗ ()| |Z ()| + 1. ∗ , = S() ∈ 7.5. Bound on Rx (). As in the previous subsection, we fix ∈ Pn,n ∗ Sm,m , and satisfying = G (). We start by observing that the product in (7.4) may be rewritten in terms of a multigraph () on the vertex set Z (). Each factor D σ (yζ1 (σ ) , yζ2 (σ ) ) yields an edge connecting the orbits ζ1 and ζ2 . In other words, there is a one-to-one map, which we denote by φ, between bridges of and edges of (); each bridge σ ∈ gives rise to an edge φ(σ ) of () connecting ζ1 (σ ) and ζ2 (σ ). See Fig. 8 for an example of such a multigraph.
Lemma 7.8. There is a subset of bridges T ⊂ of size |T | = L(), such that, in the subgraph of () with the edge set φ(T ), each orbit ζ ∈ Z ∗ () is connected to [0]. Proof. Starting from ζ0 = [0], we construct a sequence of orbits ζ0 , ζ1 , . . . , ζ L() , and a sequence of bridges σ1 , . . . , σ L() , with the property that for all k = 1, . . . , L() there is a k < k such that ζk and ζk are connected by φ(σk ). Assume that ζ0 , . . . , ζk−1 have already been constructed. Let i be the smallest vertex of Vm,m \(ζ0 ∪ · · · ∪ ζk−1 ). Then we set ζk = [i]. By construction, the vertex i − 1
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belongs to an orbit ζk for some k < k. Set σk to be the bridge containing {i − 1, i}. Hence, by definition of (), we see that ζk and ζk are connected by φ(σk ). The set T is given by {σ1 , . . . , σ L() }. Because |T | = L(), the subgraph of () with the edge set φ(T ) is a tree that connects all orbits in Z ∗ () to [0]. Let us call this tree T (). Its root is [0]. Next, we observe that |\T | 1.
(7.5)
Indeed, using Lemma 7.7 and m¯ 2 we find m¯ 1 1 − . (7.6) 3 3 3 We now estimate (7.4) as follows. Each factor indexed by σ ∈ \T is estimated by sup y,z D σ (y, z). As it turns out, we need to exploit the heat kernel decay for at least one bridge in \T . Pick a bridge σ¯ ∈ \T . (By (7.5) there is such a bridge.) Using Lemma 7.5, we estimate |\T | = m¯ − L()
sup D σ (y, z) M σ −1 if σ ∈ \(T ∪ {σ¯ }),
(7.7a)
y,z
sup D σ (y, z) y,z
C d/2 σ
M σ −1 +
C σ M if σ = σ¯ . Nd
Since N W M 1/6 and M ∼ C W d we find C d/2
σ¯
M
σ¯ −1
C + d M σ¯ C M σ¯ −1 N
1 1/2
σ¯
(7.7b)
1 + 1/6 , M
where we replaced d with 1 to obtain an upper bound. Thus we get 1 C 1 Rx () + (M − 1)n¯ 1/2 M 1/6 x σ¯ × M σ −1 1(0 = y[0] ) D σ (yζ1 (σ ) , yζ2 (σ ) ). σ ∈\T
y
σ ∈T
We perform the summation over y by starting at the leaves of T () and moving towards the root [0]. Each vertex ζ of T () carries a label yζ . Let us choose a leaf ζ of T (), and denote by ζ the parent of ζ in T (). Let σ ∈ be the (unique) bridge such that φ(σ ) connects ζ and ζ . Then summation over yζ yields the factor D σ (yζ , yζ ) = M σ , (7.8) yζ
by Lemma 7.5. Continuing in this manner until we reach the root, we find 1 C 1 Rx () + 1/6 M σ −1 M σ 1/2 n ¯ (M − 1) M σ¯ x σ ∈T σ ∈\T n¯ 1 1 M 1 =C + 1/6 . 1/2 |\T | M −1 M M σ¯
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Now (7.6) implies |\T |
m¯ 1 − , 3 3
so that
Rx () C
x
M M −1
n¯
1 1/2
σ¯
1 + 1/6 M
M 1/3 . ¯ M m/3
(7.9)
Notice that (7.9) results from an 1 - ∞ -summation procedure, where the 1 -bound (7.8) was used for propagators associated with bridges in T , and the ∞ -bound (7.7) for propagators associated with bridges in \T . The bound (7.7a) is a simple power counting bound; the bound (7.7b), improved by the heat kernel decay, is used only for one bridge. Note that in the original setup (2.1) each row and column of H contains M nonzero entries Hx y , whose positions are determined by the condition 1 |x − y| W . If we removed this last condition and only required that each row and column contain M nonzero entries in arbitrary locations off the diagonal, then all bounds relying solely on power counting would remain valid. In particular, (7.9) would be valid without the factor −1/2 σ¯ + M −1/6 , which results from the heat kernel decay associated with the special band structure.
7.6. Sum over pairings. We may now estimate p. Let first
p, m, m
0 and ∈
∗ . Sm,m
Rx (G ()) C
:| |= p x
n+n =2 p
∗ ∈Pn,n
x
Rx () for fixed
Then (7.9) yields
M M −1
p
M 1/3 ¯ M m/3
:| |= p
1 1/2
σ¯
1 + 1/6 M
.
The sum on the right-hand side is equal to
1 +···+ m¯ = p
1 1/2
1
1 + 1/6 M
1 + 1 = 1/2 M 1/6 1 1 =1 2 +···+ m¯ = p− 1 p 1 p − 1 − 1 1 + 1/6 1/2 M m¯ − 2 1 =1 1 ¯ p m−1 1 1 . + C p 1/2 M 1/6 (m¯ − 2)! p− m+1 ¯
1
Next, we note that ∗ |Sm,m ¯ − 1)(2m¯ − 3) · · · 3 · 1 2m¯ m¯ !. | (2m
This expresses the fact that the first edge of can be bridged with at most (2m¯ − 1) edges, the next remaining edge with at most (2m¯ − 3) edges, and so on. Therefore (7.3)
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531
and Lemma 7.4 (iii) yield
∗ n+n =2 p ∈Pn,n
Rx () C
m¯
2 m! ¯
4m+m 2 p
x
× C
1
+
p 1/2 1/3
M p
×
1 M 1/6
M 1/3 p
p
¯ M 1/3 p m−1 ¯ M m/3 (m¯ − 2)!
p M 1 p 1/2 M 1/6 M −1 p m¯ m¯ 2 2m¯ M 1/3 1
+
4m+m 2 p
M M −1
1 p 1/2
+
1 M 1/6
M M −1
p p C p m¯ . M 1/3 m=2 ¯
Thus, Lemma 7.2 yields
h n,n
n+n =2 p
M 1/3 p
1 p 1/2
1 + 1/6 M
M M −1
p p r =2
Cp M 1/3
r ,
(7.10)
where we abbreviated h n,n :=
∗ ∈Gn,n
Vx ().
x
7.7. Conclusion of the proof. In this subsection we complete the proof of Proposition 7.1 by showing that the error E W :=
|an (ηT ) an (ηT )| h n,n
(7.11)
n,n
satisfies E W = o(1) as W → ∞, uniformly in N W 1+d/6 . We begin by deriving bounds on the coefficients an (t). Lemma 7.9.
(i) We have
|an (t)|2 = 1 + O(M −1 ),
(7.12)
n 0
uniformly in t ∈ R. (ii) We have |an (t)| C
tn . n!
(7.13)
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Proof. We start with (i). Write
|an (t)|2 =
n 0
αn+2k (t) αn+2k (t) . (M − 1)k+k
n 0 k,k 0
The term k = k = 0 yields 1 by (5.4). The rest is equal, by (5.4), to |αn+2k (t)|2 αn+2k (t) αn+2k (t) (M − 1)k+k (M − 1)k+k
n 0 k+k >0
k+k >0 n 0
k+k >0
1 −1 ). = O(M (M − 1)k+k
In order to prove (ii), we use the integral representation (see [32]) t Jn (t) = √
n
1
2
π n+
1 2
−1
1
dλ eitλ (1 − λ2 )n− 2 .
Therefore t n+1 n+1 2 |αn (t)| 2 √ t π n+
3 2
π tn . 2 n!
(7.14)
Moreover, (5.4) yields |αn (t)| 1.
(7.15)
We use the estimate |an (t)|
|αn+2k (t)| k 0
(M − 1)k
.
Let us first consider the case t n. Then it is easy to see that with (7.14) this yields |an (t)|
k 0
If t > n we have
tn n!
t n+2k (n+2k)!
tn n! .
Together
tn 1 t n+2k 1 tn . C (M − 1)k (n + 2k)! n! (M − 1)k n! k 0
C. Thus the bound (7.15) yields |an (t)|
k 0
C tn . C (M − 1)k n!
Quantum Diffusion and Eigenfunction Delocalization in a Random Band Matrix Model n+n 2
Using the new variables p := n¯ = (7.11), EW
p
and q :=
n−n 2
533
we find from the definition
|a p+q (ηT )a p−q (ηT )| h p+q, p−q .
p 0 q=− p
Next, we observe that Lemma 7.9 (ii) implies that terms corresponding to n, n t = ηT ∼ C M κ T are strongly suppressed. Thus we introduce a cutoff at p = M μ , where κ < μ < 13 . Let us first consider the terms p M μ . We need to estimate μ
EW
:=
p M
|a p+q (ηT )a p−q (ηT )| h p+q, p−q
p=0 q=− p
⎛
⎞1/2 ⎛ μ ⎞1/2 p p Mμ M 2 2 a p+q (ηT )a p−q (ηT ) ⎠ ⎝ h p+q, p−q ⎠ ⎝ p=0 q=− p
⎛
C⎝
p=0 q=− p
⎞1/2
μ
p M
(h p+q, p−q )2 ⎠
,
p=0 q=− p
where we used Lemma 7.9 (i). Thus,
EW
2
μ
C
M p=0
⎣
p
⎤2 h p+q, p−q ⎦
q=− p
p &2 p M 1/3 1 M C p m¯ 1 + , p p 1/2 M 1/6 M −1 M 1/3 Mμ
%
⎡
m=2 ¯
p=2
by (7.10). For p M μ and W large enough, the term in the square brackets is bounded by p 2 1 1 M 1/3 1 p 1/2 C C + C 1/6 . p 1/2 1/6 1/3 1/3 1 p p M M M M 1− M
2
C M μ−1/3 . Thus we find E W Let us now consider the case p > M μ , i.e. estimate > := EW
p a p+q (ηT )a p−q (ηT ) h p+q, p−q . p>M μ q=− p
By (7.13) and the elementary inequality |a p+q (t)a p−q (t)| C
p! ( p−q)!
( p+q)! p!
we have
t2p t2p C . ( p + q)!( p − q)! p! p!
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L. Erd˝os, A. Knowles
This gives > C EW
p (ηT )2 p h p+q, p−q p! p! q=− p μ
p>M
C
p p (ηT )2 p M 1/3 1 M C p m¯ 1 + p! p! p p 1/2 M 1/6 M −1 M 1/3 μ m=2 ¯
p>M
by (7.10). Setting η ∼ C M κ yields > EW
p−2 (C M κ T )2 p C p m¯ p! p! M 1/3 m=0 ¯ p>M μ C M κ T 2 p C M 2κ T 2 p + p pM 1/3 p>M μ p>M μ
p 2p C M 2κ−1/3−μ T 2 C M κ−μ T + p>M μ
p>M μ
C M κ−μ T
2M μ
Mμ + C M 2κ−1/3−μ T .
Choosing μ = 1/3 − β (where, we recall, 0 < β < 2/3 − 2κ) completes the proof of Proposition 7.1. 8. The Ladder Pairings In this section we analyse the contribution of the ladder pairings, n 0 |an (ηT )|2 Vx (L n ), and complete the proof of Theorem 3.1. (Recall that η := W dκ is the time scale.) Recalling the expression (6.3), and noting that in the case of the ladder the variables x0 , . . . , xn determine the value of all variables x0 , . . . , x2n−1 , we readily find Vx (L n ) =
n−1 1 δ δ 1(1 |xi+1 − xi | W ) 0x x x n 0 (M − 1)n n+1 i=0
x∈ N
×
n−2 i=0
1(xi = xi+2 )
1 {xi , xi+1 } = {x j , x j+1 } .
(8.1)
0i< j n−1
Throughout this section we assume that η = W dκ for some κ < 1/3. We perform a series of steps to simplify the expression (8.1). In a first step, we get rid of the last product. Lemma 8.1. Under the assumptions of Proposition 7.1 we have |an (ηT )|2 Vx (L n ) = |an (ηT )|2 Vx1 (n) − E x1 , n 0
n 0
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535
where Vx1 (n) :=
n−1 n−2 1 δ δ 1(1 |x − x | W ) 1(xi = xi+2 ) 0x x x i+1 i n 0 (M − 1)n n+1 i=0
x∈ N
i=0
and
|E x1 |
x
C . W dβ
Proof. For each x = (x0 , . . . , xn ) ∈ n+1 P:{0,...,n−1} P (x), where N we write 1 = the sum ranges over all partitions P of the set {0, . . . , n − 1}, and P (x) is the indicator function P (x) :=
0i< j n−1
' 1 {xi , xi+1 } = {x j , x j+1 } 1 {xi , xi+1 } = {x j , x j+1 }
if i and j belong to the same lump of P if i and jbelong to different lumps of P.
Notice that if P = P0 := {{0}, . . . , {n − 1}}, then P (x) is the last product of (8.1). Let us define E x1 (n) :=
n−1 1 δ δ (1 |xi+1 − xi | W ) 0x x x n 0 n (M − 1) n+1 i=0
x∈ N
×
n−2
1(xi = xi+2 )
P (x).
P = P0
i=0
Thus, by definition, we have E x1 =
|an (ηT )|2 E x1 (n).
n 0
Next, we estimate x |E x1 |. We begin by observing that each partition P of {0, . . . , ∗ n − 1} uniquely defines a partition (P) ( ∈ Gn,n . Indeed, each lump p ∈ P gives rise to the lump γ ∈ (P) defined by γ = i∈ p {ei , e2n−1−i }. In particular, (P) = (P ) if P = P . We now claim that n−1 1 δ δ 1(1 |xi+1 − xi | W ) 0x x x n 0 n (M − 1) n+1 x∈ N
×
n−2
i=0
1(xi = xi+2 ) P (x) Vx ((P)).
i=0
This can be directly read off (6.1); there is in fact an overcounting arising from the summation over π . Thus we find |E x1 | |an (ηT )|2 Vx ((P)) |an (ηT )|2 Vx (). x
x n 0
P:{0,...,n−1} P = P0
Invoking Proposition 7.1 completes the proof.
x n 0
∗ ∈Gn,n
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L. Erd˝os, A. Knowles
In a second step, we get rid of the second to last product in (8.1), i.e. the nonbacktracking condition. Lemma 8.2. For any T 0 we have |an (ηT )|2 Vx1 (n) = |an (ηT )|2 Vx2 (n) − E x2 , n 0
n 0
where Vx2 (n) :=
n−1 1 δ δ 1(1 |xi+1 − xi | W ) 0x x x n 0 (M − 1)n n+1 i=0
x∈ N
and |E x2 | x
C W 2d/3
.
Proof. We find |E x2 | = |an (ηT )|2 n 0
x
n−1 1 δ 1(1 |xi+1 − xi | W ) 0x 0 (M − 1)n n+1 i=0
x∈ N
%
× 1−
n−2
&
1(xi = xi+2 ) .
i=0
The expression in the square brackets is equal to 1−
n−2
n−2 (−1)k+1 (1 − 1(xi = xi+2 )) =
i=0
k=1
k
1(xi j = xi j +2 ).
0i 1 <···
Therefore summing over x yields x
|E x2 |
n−2 n−2 1 M n−k |an (ηT )| k (M − 1)n k=1 n 0 & n % 1 n−2 M 2 1+ = |an (ηT )| −1 . M −1 M
2
n 0
We introduce a cutoff at n = M 1/3 . The part n M 1/3 is bounded by & M % 1/3 −2/3
1 M M C 2 1+ |an (ηT )| − 1 C eM − 1 2/3 , M −1 M M n 0
by Lemma 7.9 (i). The part n > M 1/3 is estimated using Lemma 7.9 (ii), exactly as in > in Sect. 7.7. the estimate of E W We summarize what we have proved so far.
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537
Lemma 8.3. Under the assumptions of Proposition 7.1 we have |an (ηT )|2 Vx (L n ) = |αn (ηT )|2 Px (n) + E x , n 0
n 0
where Px (n) :=
n−1 1 δ δ 1(1 |xi+1 − xi | W ) 0x x x n 0 Mn n+1 i=0
x∈ N
and C |E x | dβ . W x Proof. The claim follows from Lemmas 8.1 and 8.2, combined with an argument identical to the proof of Lemma 7.9 (i) that allows us to replace |an (t)|2 with |αn (t)|2 . We 1 1 1/3 , exactly as in replaced the factor (M−1) n with M n by introducing a cutoff at n = M the proof of Lemma 8.2. The expression Px (n) is the (normalized) number of paths in Zd of length n from 0 to any point in the set x + N Zd , whereby each step takes values in {y : 1 |y| W }. In a third step, we use the central limit theorem to replace Px (n) with a Gaussian. Recall the definition of the heat kernel d + 2 d/2 − d+2 |X |2 G(T, X ) = e 2T . 2π T Lemma 8.4. Let ϕ ∈ Cb (Rd ) and T 0. Then we have
x = dX G(T, X )ϕ(X ), lim Px ([ηT ]) ϕ W →∞ W 1+dκ/2 x
(8.2)
where [·] denotes the integer part. x (n) denote the normalized number of paths in Zd of length n from 0 to x, Proof. Let P whereby each step takes values in {y : 1 |y| W }. Then we have
x x x+ν N ([ηT ]) ϕ = Px ([ηT ]) ϕ P 1+dκ/2 1+dκ/2 W W x∈ N x∈ N ν∈Zd π(x) x ([ηT ]) ϕ , P = 1+dκ/2 W d x∈Z
where π(x) is defined through π(x) ∈ N and x − π(x) ∈ N Zd . Define the sequence 1 of i.i.d. random variables A1 , A2 , . . . whose law is M a∈Zd 1(1 |a| W ) δa , where δa denotes the point measure at a. Then we have π A1 + · · · A[ηT ] π(x) = Eϕ . (8.3) Px ([ηT ]) ϕ W 1+dκ/2 W 1+dκ/2 d x∈Z
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L. Erd˝os, A. Knowles
Next, we introduce the partition 1 = 1 A1 + · · · + A[ηT ] < N /2 + 1 A1 + · · · + A[ηT ] N /2 in the expectation in (8.3). The second resulting term is bounded by ϕ∞ P A1 + · · · + A[ηT ] N /2 . This vanishes in the limit W → ∞ by the central limit theorem, since W √N[ηT ] → ∞ by assumption. The first term resulting from the partition is A1 + · · · A[ηT ] A1 + · · · A[ηT ] Eϕ 1 A1 + · · · + A[ηT ] < N /2 = E ϕ + o(1), W 1+dκ/2 W 1+dκ/2 by the same argument as above. Therefore we get
B1 + · · · + B[ηT ] x = E ϕ + o(1), Px ([ηT ]) ϕ √ W 1+dκ/2 [ηT ] x∈ N
√
[ηT ] Ai √ where Bi := W η . The covariance matrix of Bi is follows by the central limit theorem.
T d+2 1 + o(1),
and the claim (8.2)
In a fourth and final step, we replace the probability distribution |αn (t)|2 with its asymptotic distribution. For the following we fix some test function ϕ ∈ Cb (Rd ). Testing against ϕ in Lemma 8.3 yields
x x = |an (ηT )|2 Vx (L n ) ϕ |αn (ηT )|2 Px (n) ϕ 1+dκ/2 1+κd/2 W W x x n 0 n 0 ϕ∞ +O . (8.4) W dβ While the distribution |αn (t)|2 has no limit as t → ∞, it turns out that the rescaled distribution, f t (λ) := t |α[tλ] (t)|2 , converges weakly to f (λ) :=
4 λ2 1(0 λ 1). √ π 1 − λ2
In order to prove this, we consider the integrated distribution λ Ft (λ) := dξ f t (ξ ). 0
We now show that Ft (λ) converges pointwise to F(λ) = of the functions f t , f, Ft , F.
)λ 0
f . See Fig. 9 for a graph
Quantum Diffusion and Eigenfunction Delocalization in a Random Band Matrix Model
10
1
8
0.8
6
0.6
4
0.4
2
0.2
0
0.2
0.4
0.6
0.8
0
1
0
0.2
0.4
0.6
539
0.8
1
Fig. 9. The functions f t (λ), f (λ) (left) and Ft (λ), F(λ) (right). Here we chose t = 150
Proposition 8.5. The pointwise limit F(λ) := lim Ft (λ) t→∞
exists for all λ 0 and satisfies
λ
F(λ) =
dξ 0
ξ2 4 2 arcsin λ − λ 1 − λ2 (λ ∈ [0, 1]), (8.5a) = π 1 − ξ2 π F(λ) = 1 (λ > 1). (8.5b)
Proof. See Appendix A. In order to conclude the proof of Theorem 3.1, we need the following result. Proposition 8.6. Let T 0. Then lim
W →∞
|an (ηT )|
n 0
2
x
Vx (L n ) ϕ
x W 1+dκ/2
∞
=
dλ f (λ)
dX G(λT, X ) ϕ(X ).
0
Indeed, Theorem 3.1 is an immediate consequence of Propositions 7.1 and 8.6. The rest of this section is devoted to the proof of Proposition 8.6. We begin by observing that the family of probability measures defined by the densities { f t }t 0 is tight, so that we may cut out values of λ in the range [0, δ) ∪ (1 − δ, ∞). Lemma 8.7. Let ε > 0. Then there is a δ > 0 and a t0 0 such that F(δ) + 1 − F(1 − δ)
ε ϕ∞
Ft (δ) + 1 − Ft (1 − δ)
ε ϕ∞
and
for all t t0 .
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L. Erd˝os, A. Knowles
Proof. By Proposition 8.5 we have that F(δ) + 1 − F(1 − δ) → 0
(8.6)
as δ → 0. Choose δ > 0 small enough that the left-hand side of (8.6) is bounded by ε 2ϕ∞ . Moreover, Proposition 8.5 also implies that there is a t0 such that Ft (δ) + 1 − Ft (1 − δ) F(δ) + 1 − F(1 − δ) +
ε 2ϕ∞
for all t t0 . Now by (8.4), Proposition 8.6 will follow if we can show
x |αn (ηT )|2 Px (n) ϕ 1+dκ/2 W x n 0 ∞ = dλ f (λ) dX G(λT, X ) ϕ(X ) + o(1), 0
i.e.
∞
0
dλ f ηT (λ)
x
∞
=
dλ f (λ)
Px ([ηT λ]) ϕ
x
W 1+dκ/2
dX G(λT, X ) ϕ(X ) + o(1).
(8.7)
Lemma 8.7 implies that in order to prove (8.7) it suffices to prove 1−δ
x dλ f ηT (λ) Px ([ηT λ]) ϕ W 1+dκ/2 δ x 1−δ = dλ f (λ) dX G(λT, X ) ϕ(X ) + o(1),
(8.8)
0
δ
for every δ > 0. ) Next, note that, by Lemma 8.4, the sum on the left-hand side of (8.8) converges to dX G(λT, X ) ϕ(X ) for each λ ∈ [δ, 1 − δ]. In order to invoke the dominated convergence theorem, we need an integrable bound on f t (λ). Lemma 8.8. Let δ > 0. Then there is a C > 0 such that f t (λ) C for all λ ∈ [δ, 1 − δ] and t large enough. Proof. From Lemma 5.1 we get α[tλ] (t)2 t C J[tλ]+1 (t)2 t. We estimate this using the following result due to Krasikov (see [36], Theorem 2). Setting μ := (2ν + 1)(2ν + 3) and assuming that ν > −1/2 and t > μ + μ2/3 /2, we have the bound |Jν (t)|2
4 4t 2 − (2ν + 1)(2ν + 5) . π (4t 2 − μ)3/2 − μ
Setting ν = [tλ] + 1 yields |J[tλ]+1 (t)|2 completes the proof.
C t
for λ ∈ (δ, 1 − δ) and t large enough. This
Quantum Diffusion and Eigenfunction Delocalization in a Random Band Matrix Model
541
By Lemmas 8.8 and 8.4, it is enough to prove that 1−δ dλ f ηT (λ) dX G(λT, X ) ϕ(X ) δ
=
1−δ δ
dλ f (λ)
Let us abbreviate
dX G(λT, X ) ϕ(X ) + o(1).
(8.9)
g(λ) :=
dX G(λT ) ϕ(X ).
The proof of Proposition 8.6 is therefore completed by the following result. Lemma 8.9. Let δ > 0. Then 1−δ lim dλ f t (λ)g(λ) = t→∞ δ
1−δ δ
dλ f (λ)g(λ).
Proof. The proof is a simple integration by parts. It is easy to check that on [δ, 1 − δ] the function g is smooth and its derivative is bounded. We find 1−δ 1−δ dλ f t (λ)g(λ) = dλ Ft (λ)g(λ) δ
δ
=−
1−δ δ
dλ Ft (λ)g (λ) + Ft (1 − δ)g(1 − δ) − Ft (δ)g(δ).
Proposition 8.5 and dominated convergence yield the claim. 9. Symmetric Matrices In this section we describe how to extend the argument of Sects. 6–8 to the symmetric case (2.1b). While in the Hermitian case (2.1a) we had EHx y Hyx =
1 , M −1
EHx y Hx y = 0,
we now have EHx y Hyx = EHx y Hx y =
1 . M −1
(9.1)
Since the distribution of Hx y is symmetric, Lemma 6.1 also holds in the symmetric case. However, (9.1) implies that there is no restriction on the order of the labels associated with an edge. Thus we replace (6.1) with ⎞⎛ ⎞ ⎛ 1 ⎝ Vx () = Q x (x) x (γ )⎠ ⎝ 1(γ = γ )⎠ , (9.2) (M − 1)n¯ x
γ ∈
where x (γ ) :=
γ =γ
e∈γ
1(x (e) = γ ).
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L. Erd˝os, A. Knowles
Fig. 10. Two antiparallel twisted bridges. Compare to Fig. 4 ∗ as the set of lumpings G without the complete ladder Next, we define the set Gn,n n,n and the complete antiladder (see its definition below). It is easy to see that the analogue of Lemma 7.2 holds with ⎞ ⎛ 1 ⎝ Rx () := Q x (x) x (γ )⎠ . (M − 1)n¯ x
γ ∈
∗ . We have that It therefore suffices to estimate the contribution of pairings ∈ Pn,n
Rx () =
x
×
Q x (x)
1 (M − 1)n¯
1(xa(e) = xb(e ) )1(xb(e) = xa(e ) ) {e,e }∈
+ 1(xa(e) = xa(e ) )1(xb(e) = xb(e ) ) .
(9.3)
Thus, the graphical representation of pairings has to be modified as follows. Each bridge σ ∈ carries a tag, straight or twisted, which arises from multiplying out the product in (9.3). Twisted bridges are graphically represented with dashed lines. In order to find a good notion of combinatorial complexity of pairings, we define antiparallel bridges as follows. Two bridges {ei , e j } and {ei+1 , e j+1 } are antiparallel if i + 1, j + 1 ∈ / {0, n}; see Fig. 10. An antiladder is a sequence of bridges such that two consecutive bridges are antiparallel. It is easy to see that, in addition to ladders whose rungs are straight bridges, antiladders whose rungs are twisted bridges have a leading order contribution. The skeleton = S() of the pairing is obtained from by the following procedure. A pair of parallel straight bridges is collapsed to form a single straight bridge. A pair of antiparallel twisted bridges is collapsed to form a single twisted bridge. This is repeated until no parallel straight bridges or antiparallel twisted bridges remain. The resulting pairing is the skeleton = S(); see Fig. 11. Thus we see that Lemma 7.3 holds. Moreover, Lemma 7.4 holds, provided that (i) is replaced with ∗ (i’) Each ∈ Sm,m contains no parallel straight bridges and no antiparallel twisted bridges.
Crucially, Lemma 7.7 remains valid for such tagged skeletons. This can be easily seen using the orbit construction of the proof of Lemma 7.7, combined with (i’).
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543
Fig. 11. The construction of the tagged skeleton graph
Next, we associate a factor D (y, z) with each bridge σ ∈ . If σ is straight, this is done exactly as in Sect. 7.4. If σ is twisted, this association follows immediately from the definition of the antiladder. Thus we find that Lemma 7.6 holds. The rest of the analysis in Sect. 7 carries over almost verbatim; the only required modification is the summation over 2m¯ tag configurations of the bridges of . The resulting factor 2m¯ is immaterial. Finally, the complete ladder pairing yields (3.1). The complete antiladder is subleading, as its contribution vanishes unless x = 0. 10. Delocalization: Proofs of Theorem 3.3 and Corollary 3.4 In this section we show how to derive Theorem 3.3 from Theorem 3.1, and derive Corollary 3.4 as a consequence. Proof of Theorem 3.3. We use an argument due to Chen [9] showing that diffusive motion implies delocalization of the vast majority of eigenvectors. Recall that Px, (y) := 1(|y −x| ) is the characteristic function of the complement (in N ) of the -neighborhood of x. Also, Aωε, , defined by
ω ω ω Aε, = α ∈ A : |ψα (x)| Px, ψα < ε , x
is the set of eigenvectors localized on a scale up to an error of ε. By diagonalizing H ω , λωα |ψαω ψαω |, Hω = α∈A
we have *2 * * * * * −itλω ω ω α ψ (x) ψ * Px, e * α α * * α∈A *2 * * * * 1 * ω −itλα ω ω* * 1+ P e ψ (x) ψ x, α* α * ζ * * α∈Aω ε, * *2 * * * * ω −itλα ω ω* + (1 + ζ ) * P e ψ (x) ψ x, α* , α * *α∈A\Aω *
*2 * ω * * *Px, e−it H δx * =
ε,
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L. Erd˝os, A. Knowles
for any ζ > 0. Next, we observe that the norm in the first term may be bounded by 1: *2 * *2 * * * * * * * * * ω −itλα ω ω* ω* ω * * Px, e ψα (x) ψα * * ψα (x) ψα * * *α∈Aω * * *α∈Aω ε, ε, = |ψαω (x)|2 |ψαω (x)|2 = 1. α∈Aω ε,
α∈A
Thus we get
* * * * *2 * * * 1 ω ω * * −it H −itλα ω ω* * δx * 1 + Px, e ψα (x) ψα * *Px, e * ζ * * α∈Aω ε, * *2 * * * * ω* ω * +(1 + ζ ) * ψα (x) ψα * *α∈A\Aω * ε, 1 1+ |ψαω (x)| Px, ψαω + (1 + ζ ) ζ ω α∈Aε,
α∈A\Aω ε,
|ψαω (x)|2 .
Averaging over x ∈ N yields *2 1 ω 1 * 1 * −it H ω * δx * 1 + |ψα (x)| Px, ψαω *Px, e |A| x ζ |A| ω x α∈Aε,
+(1 + ζ )
1 |A|
|ψαω (x)|2 .
x α∈A\Aω ε,
|A\Aωε, | 1 ε + (1 + ζ ) , 1+ ζ |A| by definition of Aωε, . Therefore |A\Aωε, | |A|
*2 ε 1 1 * ω * * * Px, e−it H δx * − . 1 + ζ |A| x ζ
Taking the expectation yields E
* *2 ε *2 ε |A\Aε, | 1 1 * 1 * * * * E E * P0, e−it H δ0 * − , * Px, e−it H δx * − = |A| 1 + ζ |A| x ζ 1+ζ ζ (10.1)
by translation invariance. Note that this estimate holds uniformly in t. Next, pick a continuous function ϕ(X ) that is equal to 0 if |X | 1 and 1 if |X | 2. Recalling that (t, x) = | δx , e−it H/2 δ0 |2 , we find *2 * x
* * (t, x). 1(|x| W 1+dκ/2 ) (t, x) ϕ E *P0,W 1+dκ/2 e−it H/2 δ0 * = W 1+dκ/2 x x
Quantum Diffusion and Eigenfunction Delocalization in a Random Band Matrix Model
545
Now choose an exponent κ˜ satisfying κ < κ˜ < 1/3 and set t = W d κ˜ . Thus, *2 * x d κ˜ * * ˜ ϕ W d/2(κ−κ) E * P0,W 1+dκ/2 e−iW H/2 δ0 * (W d κ˜ , x). 1+κd/2 ˜ W x Since we have
˜ X =1 lim ϕ W d/2(κ−κ)
W →∞
for X = 0 and L(1, X ) is continuous at X = 0, a simple limiting argument shows that Theorem 3.1 implies x d/2(κ−κ) ˜ d κ˜ (W , x) = dX L(1, X ) = 1. ϕ W lim ˜ W →∞ W 1+d κ/2 x We have hence proved that
*2 * d κ˜ * * lim E *P0,W 1+dκ/2 e−iW H/2 δ0 * = 1.
W →∞
Plugging this into (10.1) yields lim inf E Setting ζ =
√
W →∞
|A\Aε,W 1+dκ/2 | 1 ε − . |A| 1+ζ ζ
ε completes the proof.
Proof of Corollary 3.4. Pick an intermediate exponent κ satisfying κ < κ < 1/3 and abbreviate := W 1+dκ/2 ,
:= W 1+dκ /2 .
Let α ∈ Bω and let u ∈ N be as in (3.3). Then we find by Cauchy–Schwarz, 2 * * ω ω * |ψα (x)| * Px, ψα x
+ ' * |x − u| γ |x − u| γ * ω *2 * Px, exp − ψα x x + ' |x − u| γ |ψαω (y)|2 K exp − |x−y| + ' |x − u| γ |x − y| γ −δ( / )γ |ψαω (y)|2 , Ke exp − +δ |ψαω (x)|2 exp
|x−y|
where δ > 0 is some small constant to be chosen later. Using (a + b)γ (2a)γ + (2b)γ we find 2 ' * * |x − u| γ ω ω* −δ( / )γ * |ψα (x)| Px, Ke exp − ψα x x,y γ + |x − u| |y − u| γ |ψαω (y)|2 . +δ 2 +δ 2
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L. Erd˝os, A. Knowles
Fig. 12. A critical skeleton pairing k . The label of each vertex is indicated next to its vertex
Choosing δ < 2−γ therefore yields 2 * * γ ω ω* 2 * |ψα (x)| Px, C d K 2 e−δ( / ) =: εW . ψα x
. Then Corollary 3.4 follows from lim W →∞ εW = We have thus proved that Bω ⊂ Aωε , W 0 and Theorem 3.3. 11. Critical Pairings In this section we give an example family of pairings which are critical in the sense that they saturate the 2/3 rule (Lemma 7.7). This implies that extending our results beyond time scales of order W d/3 requires either a further resummation of pairings or a more refined classification of graphs in terms of their deviation from the 2/3 rule. Let k 1 and consider the skeleton pairing k defined in Fig. 12. It is a critical pairing in the sense that all orbits not containing the vertices 0, m consist of 3 vertices. It is easy to see that for k we have m¯ = 6k + 1,
L(k ) = 4k + 1.
In particular, the 2/3 rule of Lemma 7.7 is saturated. Moreover, if k satisfies σ 2 for all σ ∈ k then the associated pairing := G k (k ) has a nonzero contribution Vx () ∼ Rx () ≈ M −2k (here, and in the following, we ignore any powers of W with exponent of order one). Indeed, it is easy to check that under the condition σ 2 for all σ the above satisfies all nonbacktracking conditions. (In fact, it suffices to require that σ¯ 2, where σ¯ is the bridge drawn as a vertical line in Fig. 12.) As shown in Sect. 7 (see (7.13)), the coefficients an (t) essentially vanish if n > (1 + o(1))t. Setting t = M κ thus means restricting the summation to n, n M κ . Assume, to begin with, that we adopt the strategy of Sect. 7 in estimating the contribution of each graph, i.e. we use the 2/3 rule for each skeleton pairing and the 1 - ∞ -type estimates from Lemma 7.5 on the edges of the associated multigraph. We show that the
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547
sum of the contributions of the skeleton pairings k diverges if κ > 1/3. Indeed, noting that n, n M κ implies n¯ M κ , we find that the contribution of all k ’s is p/6 1 M 2k k=1
1,
(11.1)
1 +···+ 6k = p
where p = M κ and the sum over i is restricted to i 2 for all i. Here we only sum over the pairing of maximal n¯ = p (so as to obtain a lower bound), and set 6k + 1 ≈ 6k. Now (11.1) is equal to p/6 p/6 p − 6k C(M κ − 6k) 6k 1 ∼ , 6k M 2k k M 1/3 k=1
k=1
which diverges as W → ∞ if κ > 1/3. Hence a control of the error term at time scales κ larger than 1/3 would require further resummation of such critical pairings. In the estimates of the preceding paragraph we did not make full use of the heat kernel decay associated with each skeleton bridge. For simplicity, the following discussion is restricted to d = 1 (it may be easily extended to higher dimensions; in fact some estimates are better in higher dimensions). Using Lemma 7.5, we may improve (11.1) to p/6 1 M 2k k=1
√
1 +···+ 6k = p
1 ; 1 · · · 2k
(11.2)
this is a simple consequence of the heat kernel bound of Lemma 7.5 and the fact that each six-block of k contains two bridges in k \(k )T for which we may apply the bound (7.7b) (in which we drop the unimportant second term for simplicity). Now (11.2) is bounded by 6k p/6 p/6 C M 5κ/6 1 k p ∼ p , (11.3) 4k M 2k k 2/3 M 1/3 k=1
k=1
which is summable for κ < 2/5. Note, however, that the factor k −6k from (11.1) has been replaced with the larger factor k −4k . Recall that the factor k −6k is used to cancel the combinatorics m! ¯ ∼ k 6k arising from the summation over all skeletons. In the present example this small factor is not needed, as the family {k } is small. It is clear, however, that a systematic application of this approach requires a more refined classification of skeletons in terms of how much they deviate from the 2/3 rule. One expects that the number of skeletons saturating the 2/3 rule is small, and that they are therefore amenable to estimates of type (11.3). Conversely, most of the m! ¯ skeletons are expected to deviate strongly from the 2/3 rule, so that their greater number is compensated by their small individual contributions. Finally, we mention that the upper bound (7.7), used in the 1 - ∞ -type estimates above, neglects the spatial decay of the heat kernel, i.e. that D (x, y) ∼ −1/2 e−(x−y)
2 /
(11.4)
for |x − y| N . Thus a correct lower bound on the contribution of each skeleton graph should have taken into account this additional decay as well. A somewhat lengthier
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calculation shows that with the asymptotics (11.4) the estimate (11.2) may be improved to p/6 1 M 2k k=1
k
1 +···+ 6k = p j=1
1 ×, , ( j) ( j) ( j) ( j) ( j) ( j) ( j) ( j) ( j) ( j) ( j) ( j) ( j) ( j) 2 4 + 2 6 + 3 4 + 3 6 + 4 5 + 4 6 + 5 6 ( j) where we abbreviated i := 6( j−1)+i . It is not hard to see that the resulting bound is the same as (11.3), with a smaller constant C. In other words, the gain obtained from the spatial decay of the heat kernel is immaterial, and the 1 - ∞ -estimates cannot be improved. In conclusion: Our estimates rely on an indiscriminate application of the 2/3 rule to all skeleton pairings; going beyond time scales of order W d/3 would require either (i) a refined classification of the skeleton pairings in terms of how much they deviate from the 2/3 rule, combined with a systematic use of the bound (7.7b) on all bridges in \T ; or (ii) a further resummation of graphs in order to exploit cancellations. The approach (i) can be expected to reach at most times of order W 2/5 for d = 1.
Acknowledgements. The problem of diffusion for random band matrices originated from several discussions with H.T. Yau and J. Yin. The authors are especially grateful to J. Yin for various insights and for pointing out an improvement in the counting of the skeleton diagrams.
A. Proof of Proposition 8.5 Note first that F is monotone nondecreasing and satisfies 0 F(λ) 1, as follows from (5.4). Hence it is enough to prove (8.5a) for λ ∈ (0, 1). t , defined by For the following it is convenient to replace Ft with F t (λ) := F
[λt] [tλ + 1] 2 . |αn (t)| = Ft t n=0
t (λ) = o(1) as t → ∞. By Lemma 8.8 we have Ft (λ) − F Thus let λ ∈ (0, 1) be fixed. From (5.2) we find t (λ) = F
[λt] 2 π n=0
[λt] 2 = π n=0
1 −1 π
dξ
1 − ξ2 U
n (ξ ) e
2
−itξ
dθ sin θ sin[(n + 1)θ ] e
0
where we used (5.1). Thus, π π t (λ) = 1 F dθ dθ sin θ sin θ eit (cos θ−cos θ ) π2 0 0
2 ,
−it cos θ
Quantum Diffusion and Eigenfunction Delocalization in a Random Band Matrix Model
%
549
ei([λt]+1)(θ+θ ) − 1 e−i([λt]+1)(θ+θ ) − 1 ei([λt]+1)(θ−θ ) − 1 + − e−i(θ+θ ) − 1 ei(θ+θ ) − 1 e−i(θ−θ ) − 1 & e−i([λt]+1)(θ−θ ) − 1 − . (A.1) ei(θ−θ ) − 1
×
We now claim that the limit t → ∞ of the first two terms of (A.1) vanish by a stationary phase argument. Let us write the first term of (A.1) as Rt1 + Rt2 , where Rt1 :=
1 π2
π
π
dθ 0
dθ sin θ sin θ eit (cos θ−cos θ )
0
ei([λt]+1)(θ+θ ) e−i(θ+θ ) − 1
π π sin θ sin θ ei(1−{λt})(θ+θ ) it (cos θ−cos θ +λθ+λθ ) 1 e dθ dθ = 2 π e−i(θ+θ ) − 1 π 0 π 0 =: dθ dθ at (θ, θ ) eitφ(θ,θ ) , 0
0
where {ξ } := ξ − [ξ ] ∈ [0, 1). One readily finds the bounds inf
θ,θ ∈[0,π ]
|∇φ(θ, θ )| λ,
sup
|∇ 2 φ(θ, θ )| < ∞, sup
θ,θ ∈[0,π ]
t
sup
θ,θ ∈[0,π ]
|∇at (θ, θ )| < ∞.
A standard stationary phase argument therefore yields limt→∞ Rt1 = 0. Similarly, we find π π sin θ sin θ 1 2 dθ dθ eit (cos θ−cos θ ) −i(θ+θ ) . Rt = − 2 π 0 0 -e ./ − 10 =:b(θ,θ )
As above, the functions b and ∇b are bounded on [0, π ]2 . The phase cos θ − cos θ has four stationary points, (0, 0), (0, π ), (π, 0), (π, π ), all of them nondegenerate. Therefore a standard stationary phase argument implies that Rt2 = O(t −1/2 ). (Note that the stationary points lie on the boundary of the integration domain. This is not a problem, however, as the usual stationary phase argument may be applied in combination with the )∞ 2 identity 0 dx eit x = O(t −1/2 ).) Similarly, one shows that the second term of (A.1) vanishes as t → ∞. Next, as we have just shown, we have t (λ) = F t0 (λ) + F t+ (λ) + F t− (λ) + o(1) F for t → ∞, where π 1 1 1 π 0 it (cos θ−cos θ ) Ft (λ) := 2 + , dθ dθ sin θ sin θ e π 0 e−i(θ−θ ) − 1 ei(θ−θ ) − 1 0 π π ±i([λt]+1)(θ−θ ) it (cos θ−cos θ ) e t± (λ) := − 1 , dθ P dθ sin θ sin θ e F π2 0 e∓i(θ−θ ) − 1 0 t0 (λ) = o(1). Indeed, the expression where P denotes principal value. We now show that F 0 t (λ) is equal to −1. Exactly as above we therefore in square brackets in the definition of F t0 (λ) = O(t −1/2 ). conclude that F
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t+ (λ). In a first step, we replace the factor Next, let us consider F 1 −i(θ−θ ) .
1 e−i(θ−θ ) −1
with
The error is
−
1 π2
π
π
dθ sin θ sin θ eit (cos θ−cos θ ) ei([λt]+1)(θ−θ ) 0 0 1 1 , − × −i(θ−θ ) e − 1 −i(θ − θ ) dθ P
which vanishes in the limit t → ∞ by the above saddle point argument (the expression in the square brackets is an entire analytic function, and the phase cos θ − cos θ + λθ − λθ has the four nondegenerate saddle points defined by sin θ = sin θ = λ). In a second step, we choose a scale 2/5 < ε < 1/2 and introduce a cutoff in |θ − θ | at t −ε . Thus we have t+ (λ) + o(1) F π π i([λt]+1)(θ−θ ) 1 e = 2 , dθ P dθ 1(|θ − θ | t −ε ) sin θ sin θ eit (cos θ−cos θ ) π 0 i(θ − θ ) 0 π π i([λt]+1)(θ−θ ) 1 e + 2 dθ dθ 1(|θ − θ | > t −ε ) sin θ sin θ eit (cos θ−cos θ ) . π 0 i(θ − θ ) 0 (A.2) Let us abbreviate Dt := {(θ, θ ) ∈ [0, π ]2 : |θ − θ | > t −ε }. The second term on the right-hand side of (A.2) is equal to 1 π2
dθ dθ eit (cos θ−cos θ +λθ−λθ ) Dt
dθ dθ eitφ(θ,θ )
=: Dt
sin θ sin θ ei(1−{λt})(θ−θ ) i(θ − θ )
at (θ, θ ) . θ − θ
(A.3)
In the domain Dt the phase φ has two stationary points defined by sin θ = sin θ = λ and θ = θ . For all (θ, θ ) not in some fixed neighbourhood of these stationary points and satisfying |θ − θ | > t −ε , we have the bound |∇φ(θ, θ )| Ct −ε , for some constant C > 0 depending on λ, and large enough t. Thus a standard saddle point analysis shows that (A.3) is of the order t −1/2 + t 2ε−1 = o(1). In a third step, we analyse the first term on the right-hand side of (A.2). We introduce the new coordinates u=
θ + θ , 2
v = θ − θ ,
Quantum Diffusion and Eigenfunction Delocalization in a Random Band Matrix Model
and write
t+ (λ) + o(1) = 1 F π2
π
π
dθ P
0
551
dθ 1(|θ − θ | t −ε )
0 it (cos θ−cos θ ) e
i([λt]+1)(θ−θ )
× sin θ sin θ e i(θ − θ ) at,u π v
1 v
sin u − = 2 du P dv sin u + π 0 2 2 −at,u v
×ei(1−{λt})v
eit (λv−2 sin u sin 2 ) , iv
where at,u := min{t −ε , 2u, 2(π − u)}. v
Now we replace the factor eit (λv−2 sin u sin 2 ) with eitv(λ−sin u) . The resulting error is at,u π v
v
1 sin u − Rt := 2 du P dv sin u + π 0 2 2 −at,u v
×ei(1−{λt})v eitv(λ−sin u)
eit sin u (v−2 sin 2 ) − 1 . iv
(A.4)
It is easy to check that, for v ∈ [−at,u , at,u ], we have eit sin u (v−2 sin v2 ) − 1 5 1 Ct 1− 2 ε √ . iv |v| Therefore
π
|Rt | C
2π
du 0
0
5 1 dv t 1− 2 ε √ = o(1). |v|
Thus we may write t+ (λ) + o(1) = 1 F π2
0
π
du P -
at,u −at,u
v
v i(1−{λt})v eitv(λ−sin u) dv sin u + . sin u − e 2 2 iv ./ 0 =:It (u)
In a fourth step, we analyse It (u) using contour integration. Abbreviate b := λ−sin u. Let us assume that u satisfies b = 0. Then, setting z = |b|tv, we find |b|tat,u z eiz sgn b z z sin u − ei(1−{λt}) |b|t . It (u) = P dz sin u + 2|b|t 2|b|t iz −|b|tat,u Let us consider the case b > 0. Using the identity P
1 1 = iπ δ(v) + v v − i0
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and Cauchy’s theorem, we find z i(1−{λt}) z eiz z
bt It (u) = π sin2 (u) + dz sin u + sin u − e , 2bt 2bt iz γ where γ is the arc {btat,u (cos ϕ, sin ϕ) : ϕ ∈ [0, π ]}. The absolute value of the integral is bounded by π dϕ eat,u sin ϕ e−btat,u sin ϕ , 0
which is bounded uniformly in t and b = 0, and vanishes in the limit t → ∞ for all b = 0. The case b < 0 is treated in the same way. In summary, we have, for each u satisfying sin u = λ, that |It (u)| C,
lim It (u) = π sin2 (u) sgn(λ − sin u).
t→∞
Hence by dominated convergence we get π t+ (λ) = 1 lim F du sin2 (u) sgn(λ − sin u). t→∞ π 0 A similar (in fact easier) analysis yields π t− (λ) = 1 lim F du sin2 (u) sgn(λ + sin u). t→∞ π 0 Therefore we get t (λ) = 2 lim F t→∞ π
0
π
4 du sin (u) 1(sin u < λ) = π
2
0
λ
dξ
ξ2 1 − ξ2
.
This completes the proof of Proposition 8.5. References 1. Abrahams, E., Anderson, P.W., Licciardello, D.C., Ramakrishnan, T.V.: Scaling theory of localization: absence of quantum diffusion in two dimensions. Phys. Rev. Lett. 42, 673–676 (1979) 2. Anderson, G., Zeitouni, O.: A CLT for a band matrix model. Probab. Theory Related Fields 134(2), 283–338 (2006) 3. Aizenman, M., Molchanov, S.: Localization at large disorder and at extreme energies: an elementary derivation. Commun. Math. Phys. 157, 245–278 (1993) 4. Aizenman, M., Sims, R., Warzel, S.: Absolutely continuous spectra of quantum tree graphs with weak disorder. Commun. Math. Phys. 264(2), 371–389 (2006) 5. Anderson, P.: Absences of diffusion in certain random lattices. Phys. Rev. 109, 1492–1505 (1958) 6. Bachmann, S., De Roeck, W.: From the Anderson model on a strip to the DMPK equation and random matrix theory. J. Stat. Phys. 139, 541–564 (2010) 7. Bai, Z.D., Yin, Y.Q.: Limit of the smallest eigenvalue of a large dimensional sample covariance matrix. Ann. Probab. 21(3), 1275–1294 (1993) 8. Bourgain, J.: Random lattice Schrödinger operators with decaying potential: some higher dimensional phenomena. Lecture Notes in Mathematics, Vol. 1807, Berlin-Heidelberg-New York: Springer, 2003, pp. 70–99 9. Chen, T.: Localization lengths and Boltzmann limit for the Anderson model at small disorders in dimension 3. J. Stat. Phys. 120(1–2), 279–337 (2005)
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10. Denisov, S.A.: Absolutely continuous spectrum of multidimensional Schrödinger operator. Int. Math. Res. Not. 2004(74), 3963–3982 (2004) 11. Disertori, M., Pinson, H., Spencer, T.: Density of states for random band matrices. Commun. Math. Phys. 232, 83–124 (2002) 12. Disertori, M., Spencer, T.: Anderson localization for a supersymmetric sigma model. http://arxiv.org/abs/ 0910.3325v1 [math-ph], 2009 13. Disertori, M., Spencer, T., Zirnbauer, M.: Quasi-diffusion in a 3D Supersymmetric Hyperbolic Sigma Model. Commun. Math. Phys. 300, 435–486 (2010) 14. Efetov, K.B.: Supersymmetry in Disorder and Chaos. Cambridge: Cambridge University Press, 1997 15. Elgart, A.: Lifshitz tails and localization in the three-dimensional Anderson model. Duke Math. J. 146(2), 331–360 (2009) 16. Erd˝os, L., Knowles, A.: Quantum diffusion and delocalization for band matrices with general distribution. http://arxiv.org/abs/1005.1838v3 [math-ph], 2010 17. Erd˝os, L., Péché, G., Ramírez, J., Schlein, B., Yau, H.-T.: Bulk universality for Wigner matrices. Comm. Pure Appl. Math. 63(7), 895–925 (2010) 18. Erd˝os, L., Salmhofer, M., Yau, H.-T.: Quantum diffusion of the random Schrödinger evolution in the scaling limit. Acta Math. 200(2), 211–277 (2008) 19. Erd˝os, L., Salmhofer, M., Yau, H.-T.: Quantum diffusion of the random Schrödinger evolution in the scaling limit II. The recollision diagrams. Commun. Math. Phys. 271, 1–53 (2007) 20. Erd˝os, L., Salmhofer, M., Yau, H.-T.: Quantum diffusion for the Anderson model in scaling limit. Ann. Inst. H. Poincare 8(4), 621–685 (2007) 21. Erd˝os, L., Schlein, B., Yau, H.-T.: Local semicircle law and complete delocalization for Wigner random matrices. Comm. Math. Phys. 287, 641–655 (2009) 22. Erd˝os, L., Schlein, B., Yau, H.-T.: Universality of Random Matrices and Local Relaxation Flow. http:// arxiv.org/abs/0907.5605v4 [math-ph], 2010 23. Erd˝os, L., Schlein, B., Yau, H.-T., Yin, J.: The local relaxation flow approach to universality of the local statistics for random matrices. http://arxiv.org/abs/0911.3687v5 [math-ph], 2010 24. Erd˝os, L., Yau, H.-T.: Linear Boltzmann equation as the weak coupling limit of the random Schrödinger equation. Commun. Pure Appl. Math. LIII, 667–735 (2000) 25. Erd˝os, L., Yau, H.-T., Yin, J.: Bulk universality for generalized Wigner matrices. http://arxiv.org/abs/ 1001.3453v6 [math-ph], 2010 26. Feldheim, O., Sodin, S.: A universality result for the smallest eigenvalues of certain sample covariance matrices. http://arxiv.org/abs/0812.1961v4 [math-ph], 2009 27. Froese, R., Hasler, D., Spitzer, W.: Transfer matrices, hyperbolic geometry and absolutely continuous spectrum for some discrete Schrödinger operators on graphs. J. Funct. Anal. 230(1), 184–221 (2006) 28. Fröhlich, J., de Roeck, W.: Diffusion of a massive quantum particle coupled to a quasi-free thermal medium in dimension d 4. http://arxiv.org/abs/0906.5178v3 [math-ph], 2010 29. Fröhlich, J., Spencer, T.: Absence of diffusion in the Anderson tight binding model for large disorder or low energy. Commun. Math. Phys. 88, 151–184 (1983) 30. Fröhlich, J., Martinelli, F., Scoppola, E., Spencer, T.: Constructive proof of localization in the Anderson tight binding model. Commun. Math. Phys. 101(1), 21–46 (1985) 31. Fyodorov, Y.V., Mirlin, A.D.: Scaling properties of localization in random band matrices: a σ -model approach. Phys. Rev. Lett. 67, 2405–2409 (1991) 32. Gradshteyn, I.S., Ryzhik, I.M.: Tables of integrals, series, and products. New York: Academic Press, 2007 33. Guionnet, A.: Large deviation upper bounds and central limit theorems for band matrices. Ann. Inst. H. Poincaré Probab. Stat. 38, 341–384 (2002) 34. Kirsch, W., Krishna, M., Obermeit, J.: Anderson model with decaying randomness: mobility edge. Math. Z. 235, 421–433 (2000) 35. Klein, A.: Absolutely continuous spectrum in the Anderson model on the Bethe lattice. Math. Res. Lett. 1, 399–407 (1994) 36. Krasikov, I.: Uniform bounds for Bessel functions. J. Appl. Anal. 12(1), 83–91 (2006) 37. Mehta, M.L.: Random Matrices. New York: Academic Press, 1991 38. Minami, N.: Local fluctuation of the spectrum of a multidimensional Anderson tight binding model. Commun. Math. Phys. 177(3), 709–725 (1996) 39. Rodnianski, I., Schlag, W.: Classical and quantum scattering for a class of long range random potentials. Int. Math. Res. Not. 5, 243–300 (2003) 40. Schenker, J.: Eigenvector localization for random band matrices with power law band width. Commun. Math. Phys. 290, 1065–1097 (2009) 41. Schlag, W., Shubin, C., Wolff, T.: Frequency concentration and location lengths for the Anderson model at small disorders. J. Anal. Math. 88, 173–220 (2002)
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42. Sodin, S.: The spectral edge of some random band matrices. http://arxiv.org/abs/0906.4047v4 [math-ph], 2010 43. Spencer, T.: Lifshitz tails and localization. Preprint, 1993 44. Spencer, T.: Random banded and sparse matrices (Chapter 23). To appear in “Oxford Handbook of Random Matrix Theory”, edited by G. Akemann, J. Baik, P. Di Francesco, Oxford Univ. Press, 2010 45. Spohn, H.: Derivation of the transport equation for electrons moving through random impurities. J. Stat. Phys. 17(6), 385–412 (1977) 46. Tao, T., Vu, V.: Random matrices: Universality of the local eigenvalue statistics. http://arxiv/abs/0906. 0510v10 [math.PR], 2010, to appear Acta Math. 47. Valkó, B., Virág, B.: Random Schrödinger operators on long boxes, noise explosion and the GOE. http:// arxiv.org/abs/0912.0097v2 [math.PR], 2009 48. Wigner, E.: Characteristic vectors of bordered matrices with infinite dimensions. Ann. of Math. 62, 548– 564 (1955) Communicated by H.-T. Yau
Commun. Math. Phys. 303, 555–594 (2011) Digital Object Identifier (DOI) 10.1007/s00220-011-1216-y
Communications in
Mathematical Physics
Factorization and Dilation Problems for Completely Positive Maps on von Neumann Algebras Uffe Haagerup , Magdalena Musat Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, 2100 Copenhagen Ø, Denmark. E-mail:
[email protected];
[email protected] Received: 12 September 2010 / Accepted: 14 December 2010 Published online: 11 March 2011 – © Springer-Verlag 2011
Abstract: We study factorization and dilation properties of Markov maps between von Neumann algebras equipped with normal faithful states, i.e., completely positive unital maps which preserve the given states and also intertwine their automorphism groups. The starting point for our investigation has been the question of existence of non-factorizable Markov maps, as formulated by C. Anantharaman-Delaroche. We provide simple examples of non-factorizable Markov maps on Mn (C) for all n ≥ 3, as well as an example of a one-parameter semigroup (T (t))t≥0 of Markov maps on M4 (C) such that T (t) fails to be factorizable for all small values of t > 0. As applications, we solve in the negative an open problem in quantum information theory concerning an asymptotic version of the quantum Birkhoff conjecture, as well as we sharpen the existing lower bound estimate for the best constant in the noncommutative little Grothendieck inequality. 1. Introduction Motivated by the study of ergodic actions of free groups on noncommutative spaces, C. Anantharaman-Delaroche investigated in [2] the noncommutative analogue of G.-C. Rota’s “Alternierende Verfahren” theorem from classical probability, asserting that if T is a measure-preserving Markov operator on the probability space (, μ), then for every p ≥ 1 and f ∈ L p (, μ), the sequence T n (T ∗ )n ( f ) converges almost everywhere, as n → ∞. In the noncommutative setting, the probability space is replaced by a von Neumann algebra M, equipped with a normal faithful state φ, and T is now a unital, completely positive map on M such that φ ◦ T = φ. However, in this setting the existence of the adjoint map T ∗ is not automatic. It turns out (see more precise references below) that it is equivalent to the fact that T commutes with the modular automorphism Supported by the ERC Advanced Grant no. OAFPG 247321, and partially supported by the Danish Natural Science Research Council and the Danish National Research Foundation. Partially supported by the National Science Foundation, DMS-0703869, and by the Danish Natural Science Research Council and the Danish National Research Foundation.
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group of φ. This motivated the following definition considered in [2] (cf. Definition 2.6), where we have chosen to follow the slightly modified notation from [31]: Definition 1.1. Let (M, φ) and (N , ψ) be von Neumann algebras equipped with normal faithful states φ and ψ, respectively. A linear map T : M → N is called a (φ, ψ)-Markov map if (1) (2) (3) (4)
T is completely positive. T is unital. ψ ◦ T = φ. ψ ψ φ φ T ◦ σt = σt ◦ T , for all t ∈ R, where (σt )t∈R and (σt )t∈R denote the automorphism groups of the states φ and ψ, respectively. In particular, when (M, φ) = (N , ψ), we say that T is a φ-Markov map.
Note that a linear map T : M → N satisfying conditions (1)–(3) above is automatically normal. If, moreover, condition (4) is satisfied, then it was proved in [1] (see also Lemma 2.5 in [2]) that there exists a unique completely positive, unital map T ∗ : N → M such that φ(T ∗ (y)x) = ψ(yT (x)), x ∈ M, y ∈ N . It is easy to show that
T∗
(1.1)
is a (ψ, φ)-Markov map.
Remark 1.2. A special case of interest is the one of a (φ, ψ)-Markov map J : M → N which is a ∗-monomorphism. In this case J (M) is a σ ψ -invariant sub-von Neumann algebra of N . Hence, by [34], there is a unique ψ-preserving normal faithful conditional expectation E J (M) of N onto J (M), and thus the adjoint J ∗ is given by J ∗ = J −1 ◦E J (M) . C. Anantharaman-Delaroche proved in [2] that the noncommutative analogue of Rota’s theorem holds for Markov maps which are factorizable in the following sense (cf. Def. 6.2 in [2]): Definition 1.3. A (φ, ψ)-Markov map T : M → N is called factorizable if there exists a von Neumann algebra P equipped with a faithful normal state χ , and ∗-monomorphisms J0 : M → P and J1 : N → P such that J0 is (φ, χ )-Markov and J1 is (ψ, χ )-Markov, satisfying, moreover, T = J0∗ ◦ J1 . Remark 1.4. (a) Note that if both φ and ψ are tracial states on M and N , respectively, and T : M → N is factorizable, then the factorization can be chosen through a von Neumann algebra with a faithful normal tracial state, as well. This can be achieved by replacing (P, χ ) whose existence is ensured by the definition of factorizability by (Pχ , χ| Pχ ), where Pχ denotes the centralizer of the state χ , since J0 (M) ⊆ Pχ and J1 (N ) ⊆ Pχ . (b) The class of factorizable (φ, ψ)-Markov maps is known to be closed under composition, the adjoint operation, taking convex combinations and w ∗ -limits (see Prop. 2 in [31]). C. Anantharaman-Delaroche raised in [2] the question whether all Markov maps are factorizable. This was the starting point of investigation for our paper. The class of maps which are known to be factorizable includes all Markov maps between abelian von Neumann algebras (as it was explained in [2], Remark 6.3 (a)), the trace-preserving Markov maps on M2 (C) (due to a result of B. Kümmerer from [21]), as well as Schur
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multipliers associated to positive semi-definite real matrices having diagonal entries all equal to 1 (as shown by E. Ricard in [31]). It is therefore natural to further study the problem of factorizability of τn -Markov maps on Mn (C), for n ≥ 3, where τn denotes the unique normalized trace on the n × n complex matrices. In Sect. 2 we give a general characterization of factorizable τn -Markov maps on Mn (C), as well as a characterization of those τn -Markov maps which lie in the convex hull of ∗-automorphisms of Mn (C). We also discuss the case of Schur multipliers. As an application, we construct in Sect. 3 several examples of non-factorizable τn -Markov maps on Mn (C), n ≥ 3 (cf. Examples 3.1 and 3.2), an example of a factorizable Schur multiplier on (M6 (C), τ6 ) which does not lie in the convex hull of ∗-automorphisms of M6 (C) (cf. Example 3.3), as well as an example of a one-parameter semigroup (T (t))t≥0 of τ4 -Markov maps on M4 (C) such that T (t) fails to be factorizable for all small values of t > 0 (see Theorem 3.4). This latter example is to be contrasted with a result of B. Kümmerer and H. Maasen from [24], asserting that if (T (t))t≥0 is a one-parameter semigroup of τn -Markov maps on Mn (C), n ≥ 1, such that each T (t) is self-adjoint, then T (t) is factorizable, for all t ≥ 0. We have been informed of recent work of M. Junge, E. Ricard and D. Shlyakhtenko, where they have generalized Kümmerer and Maasen’s result to the case of a strongly continuous one-parameter semigroup of self-adjoint Markov maps on an arbitrary finite von Neumann algebra. This result has been independently obtained by Y. Dabrowski (see the preprint [9]). In Sect. 4 we discuss the connection between the notion of factorizability and Kümmerer’s notions of dilation, respectively, of Markov dilation, that he introduced in [22]. The starting point for our analysis was a private communication by C. Koestler [20], who informed us in the Spring of 2008 that for a φ-Markov map on a von Neumann algebra M, factorizability is equivalent to the existence of a dilation (in the sense of [22]), and that Kümmerer in his unpublished Habilitationsschrift [23] had constructed examples of τn -Markov maps on Mn (C), n ≥ 3, having no dilations, and hence being non-factorizable. The equivalence between factorizability and the existence of a dilation is based on an inductive limit argument also from Kümmerer’s unpublished work [23]. In Theorem 4.4 and its proof, we provide the details of the argument communicated to us by C. Koestler. Moreover, we show that the existence of a dilation is equivalent to the–seemingly stronger–condition of existence of a Markov dilation in the sense of Kümmerer. Section 5 is devoted to the study of the so-called Rota dilation property of a Markov map, introduced by M. Junge, C. LeMerdy and Q. Xu in [16]. This notion has proven to be very fruitful for the development of semigroup theory in the noncommutative setting, and applications to noncommutative L p -spaces and noncommutative harmonic analysis (see, e.g., [15,26]). The Rota dilation property of a Markov map implies its factorizability, but it is more restrictive, as it forces the map to be self-adjoint. As a consequence of Theorem 6.6 in [2], the square of any factorizable self-adjoint Markov map has the Rota dilation property. Our main result in this section is that there exists a self-adjoint τn -Markov map T on Mn (C), for some positive integer n, such that T 2 does not have the Rota dilation property (see Theorem 5.4), and therefore the analogue of Rota’s classical dilation theorem for Markov operators does not hold, in general, in the noncommutative setting. The existence of non-factorizable Markov maps turned out to have an interesting application to an open problem in quantum information theory, known as the asymptotic quantum Birkhoff conjecture. The conjecture, originating in joint work of A. Winter, J. A. Smolin and F. Verstraete (cf. [33]), asserts that if T : Mn (C) → Mn (C) is a
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τn -Markov map, n ≥ 1, then T satisfies the following asymptotic quantum Birkhoff property: k k lim dcb T, conv(Aut( Mn (C))) = 0. k→∞
i=1
i=1
We would like to thank V. Paulsen for bringing this problem to our attention. In Sect. 6 we solve the conjecture in the negative (see Theorem 6.1)), by showing that every non-factorizable τn -Markov map on Mn (C), n ≥ 3, fails the above asymptotic quantum Birkhoff property. Finally, in Sect. 7, as an application of some of the techniques developed in the previous sections, we prove that the best constant in the noncommutative little Grothendieck inequality (cf. [30] and [14]) is strictly greater than 1, thus sharpening the existing lower bound estimate for it. 2. Factorizability of τn -Markov maps on Mn (C) Let P be a von Neumann algebra and n a positive integer. Recall (see, e.g., [18] (Vol. II, Sect. 6.6) that a family ( f i j )1≤i, j≤n of elements of P is a set of matrix units in P if the following conditions are satisfied: f i j f kl = δ jk f il , 1 ≤ i, j, k, l ≤ n, f i∗j = f ji , 1 ≤ n f ii = 1 P . If this is the case, then F := Span{ f i j : 1 ≤ i, j ≤ n} is a i, j ≤ n and i=1 ∗-subalgebra of P isomorphic to Mn (C) and 1 P ∈ F. The following result is well-known, but we include a proof for the convenience of the reader. Lemma 2.1. Let P be a von Neumann algebra, n a positive integer, and ( f i j )1≤i, j≤n , (gi j )1≤i, j≤n two sets of matrix units in P. Then there exists a unitary operator u ∈ P such that u f i j u ∗ = gi j , 1 ≤ i, j ≤ n. Proof. By hypothesis, ( f ii )1≤i≤n and (gii)1≤i≤n are two sets of pairwise orthogonal n n projections in P with i=1 f ii = 1 P = i=1 gii , satisfying, moreover, f 11 ∼ f 22 ∼ · · · ∼ f nn and g11 ∼ g22 ∼ · · · ∼ gnn , respectively, where ∼ denotes the relation of equivalence of projections. By, e.g., [18] (Vol. II, Ex. 6.9.14), it follows that f 11 ∼ g11 , ∗ ∗ i.e., there n exists a partial isometry v ∈ P such that v v = f 11 and vv = g11 . Set now u := i=1 gi1 v f 1i . It is elementary to check that u is a unitary in P. Moreover, for 1 ≤ k, l ≤ n, u f kl u ∗ = i,n j=1 gi1 v f 1i f kl f j1 v ∗ g1 j = gk1 v f 11 v ∗ g1l = gk1 g11 g1l = gkl , which proves the result. By a result of M.-D. Choi (see [7]), a linear map T : Mn (C) → Mn (C) is completely positive if and only if T can be written in the form Tx =
d
ai∗ xai , x ∈ Mn (C),
(2.1)
i=1
for some a1 , . . . , ad ∈ Mn (C). The condition that T is unital is then equivalent to d ∗ 1n , while the condition that T is trace-preserving, i.e., τn ◦ T = τn , is i=1 ai ai = d equivalent to i=1 ai ai∗ = 1n . Here 1n denotes the identity matrix in Mn (C).
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Theorem 2.2. Let T : Mn (C) → Mn (C) be a τn -Markov map, written in the form (2.1). where a1 , . . . , ad ∈ Mn (C) are chosen to be linearly independent and satisfy d d ∗ ∗ i=1 ai ai = i=1 ai ai = 1n . Then the following conditions are equivalent: (i) T is factorizable. (ii) There exists a finite von Neumann algebra N equipped with a normal faithful tracial state τ N and a unitary operator u ∈ Mn (N ) = Mn (C) ⊗ N such that T x = (id Mn (C) ⊗ τ N )(u ∗ (x ⊗ 1 N )u) , x ∈ Mn (C).
(2.2)
(iii) There exists a finite von Neumann algebra N equipped with a normal faithful d tracial state τ N and v1 , . . . , vd ∈ N such that u := i=1 ai ⊗ vi is a unitary operator in Mn (C) ⊗ N and τ N (vi∗ v j ) = δi j , 1 ≤ i, j ≤ d. Proof. We first show that (i) ⇒ (ii). Assume that T is factorizable. By Remark 1.4 (a), there exists a finite von Neumann algebra P with a normal faithful tracial state τ P and two unital ∗-monomorphisms α, β : Mn (C) → P such that T = β ∗ ◦ α. Note that α and β are automatically (τn , τ P )-Markov maps, since τn is the unique normalized trace on Mn (C). Let {ei j }1≤i, j≤n be the standard matrix units in Mn (C) and set f i j := α(ei j ), respectively, gi j := β(ei j ), for all 1 ≤ i, j ≤ n. Choose now a unitary operator u ∈ P as in Lemma 2.1. Then β(x) = uα(x)u ∗ , for all x ∈ Mn (C). Equivalently, α(x) = u ∗ β(x)u, for all x ∈ Mn (C). Consider now the relative commutant N := (β(Mn (C))) ∩ P = {gi j : 1 ≤ i, j ≤ n} ∩ P, and let τ N be the restriction of τ P to N . Since the map i,n j=1 ei j ⊗xi j → i,n j=1 gi j xi j , where xi j ∈ N , 1 ≤ i, j ≤ n, defines a ∗-isomorphism of Mn (C) ⊗ N onto P (see, e.g., [18], Vol. II, Sect. 6.6.), we can make the identifications P = Mn (C)⊗ N , τ P = τn ⊗τ N and β(x) = x ⊗ 1 N , x ∈ Mn (C). This implies that α(x) = u ∗ (x ⊗ 1 N )u, x ∈ Mn (C). Since T = β ∗ ◦ α = β −1 ◦ Eβ(Mn (C)) ◦ α (see Remark 1.2), then T x ⊗ 1 N = E Mn (C)⊗1 N (u ∗ (x ⊗ 1 N )u), x ∈ Mn (C), where E Mn (C)⊗1 N is the unique τ P = τn ⊗ τ N -preserving conditional expectation of Mn (C) ⊗ N onto Mn (C) ⊗ 1 N . Then (2.2) follows and the implication is proved. Conversely, assume that (ii) holds. Define maps α, β : Mn (C) → Mn (C) ⊗ N by α(x) := u ∗ (x ⊗ 1 N )u, respectively, β(x) := x ⊗ 1 N , for all x ∈ Mn (C). Then α and β are (τn , τn ⊗ τ N )-Markov ∗-monomorphisms of Mn (C) into Mn (C) ⊗ N satisfying T = β ∗ ◦ α, which proves that T is factorizable. Next we prove the implication (ii) ⇒ (iii). Assume that (ii) holds and choose a von Neumann algebra N with a normal faithful tracial state τ N and a unitary operator u ∈ Mn (C) ⊗ N satisfying (2.2). Since a1 , . . . , ad ∈ Mn (C) are linearly independent, we can extend the set {a1 , . . . , ad } to an algebraic basis {a1 , . . . , an 2 } for Mn (C). Then n 2 ai ⊗ vi , where v1 , . . . , vn 2 ∈ N . By (2.1) u has a representation of the form u = i=1 and (2.2) we deduce that ⎞ ⎛ 2 d n n2 ∗ ∗ ∗ ⎠ ⎝ ai xai = (id Mn (C) ⊗ τ N ) ai xa j ⊗ vi v j = τ N (vi∗ v j )ai∗ xa j . i=1
i, j=1
i, j=1
(2.3)
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For a ∈ Mn (C) we let L a and Ra denote, respectively, the operators of left and right multiplicationby a on Mn (C), i.e., L a x = ax, Ra x = xa, x ∈ Mn (C). It is well-known that the map rk=1 ak ⊗ bk → rk=1 L ak ⊗ Rbk defines a vector space isomorphism of Mn (C) ⊗ Mn (C) onto B(Mn (C)). By (2.3), d
2
ai∗
⊗ ai =
i=1
n
τ N (vi∗ v j )ai∗ ⊗ a j .
(2.4)
i, j=1
Moreover, the set {ai∗ ⊗ a j : 1 ≤ i, j ≤ n 2 } is an algebraic basis for Mn (C) ⊗ Mn (C), so in particular, this set is linearly independent. Therefore (2.4) implies that if 1 ≤ i, j ≤ d δ τ N (vi∗ v j ) = i j 0 else. In particular, τ N (vi∗ vi ) = 0 for i > d, which by the faithfulness of τ N implies that d vi = 0, for all i > d. Hence u = i=1 ai ⊗ vi , which proves (iii). It remains to prove that (iii) implies (ii). Choose (N , τ N ) and operators v1 , . . . , vd ∈ d ai ⊗vi is a unitary operator in Mn (C)⊗ N and τ (vi∗ v j ) = N as in (iii). Then u := i=1 δi j , for 1 ≤ i, j ≤ d. Thus T (x) =
d i=1
ai∗ xai =
n
τ N (vi∗ v j )ai∗ xa j = (id Mn (C) ⊗ τ N )(u ∗ (x ⊗ 1 N )u),
i, j=1
which gives (ii) and the proof is complete. Corollary 2.3. Let T : Mn (C) → Mn (C) be a τn -Markov map of the form (2.1), where d d a1 , . . . , ad ∈ Mn (C) and i=1 ai∗ ai = i=1 ai ai∗ = 1n . If d ≥ 2 and the set {ai∗ a j : 1 ≤ i, j ≤ d} is linearly independent, then T is not factorizable. Proof. Assume that T is factorizable. Since the linear independence of the set {ai∗ a j : 1 ≤ i, j ≤ d} implies that the set {ai : 1 ≤ i ≤ d} is linearly independent, as well, Theorem 2.2 applies. Hence, by the equivalence (ii) ⇔ (iii) therein, there exists a finite von Neumann algebra N with a normal, faithful tracial state τ N and operators d ai ⊗ vi ∈ Mn (C) ⊗ N = Mn (N ) is a unitary v1 , . . . , vd ∈ N such that u := i=1 operator and τ N (ai∗ a j ) = δi j , for 1 ≤ i, j ≤ d. Then d i, j=1
ai∗ a j
⊗ (vi∗ v j −δi j 1 N ) = u ∗ u
−
d
ai∗ ai
⊗1 N = 1 Mn (N ) −1n ⊗ 1 N = 0 Mn (N ) .
i=1
By the linear independence of the set {ai∗ a j : 1 ≤ i, j ≤ d} it follows that for every functional φ ∈ N ∗ , we have φ(vi∗ v j − δi j 1 N ) = 0, for all 1 ≤ i, j ≤ d. Hence vi∗ v j = δi j 1 N , 1 ≤ i, j ≤ d. Since d ≥ 2, we infer in particular that v1∗ v2 = 0 N and v1∗ v1 = v2∗ v2 = 1 N . The latter condition ensures that v1 and v2 are unitary operators, since N is a finite von Neumann algebra. But this leads to a contradiction with the fact that v1∗ v2 = 0 N . This proves that T is not factorizable.
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Let Aut(M) denote the set of ∗-automorphisms of a von Neumann algebra M. If M = Mn (C), for some n ∈ N, then Aut(Mn (C)) = {ad(u) : u ∈ U(n)}, where ad(u)x = uxu ∗ , x ∈ Mn (C), and U(n) denotes the unitary group of Mn (C). Since B(Mn (C)) is finite dimensional, the convex hull of Aut(Mn (C)), denoted by conv(Aut(Mn (C)), is closed in the norm topology on B(Mn (C)). Further, let us denote by FM(Mn (C)) the set of factorizable τn -Markov maps on Mn (C). By Remark 1.4 (b), conv(Aut(Mn (C)) ⊆ FM(Mn (C)).
(2.5)
Note that for n = 2 the two sets above are equal, as they are further equal to the set of τ2 -Markov maps on M2 (C), as shown by Kümmerer in [21]. Proposition 2.4. Let T : Mn (C) → Mn (C) be a τn -Markov map written in the d form (2.1), where a1 , . . . , ad ∈ Mn (C) are linearly independent and i=1 ai∗ ai = d ∗ i=1 ai ai = 1n . Then the following conditions are equivalent: (1) T ∈ conv(Aut(Mn (C)). (2) T satisfies condition (ii) of Theorem 2.2 with N abelian. (3) T satisfies condition (iii) of Theorem 2.2 with N abelian. Proof. We first show that (1) ⇒ (2). Assume that T ∈ conv(Aut(Mn (C)). Then there exist u 1 , . . . , u s ∈ U(n) and positive real numbers c1 , . . . , cs with sum equal to 1, for some positive integer s, so that
Tx =
s
ci u i∗ xu i , x ∈ Mn (C).
i=1
Next, consider the abelian von Neumann algebra N := l ∞ ({1, . . . , s}) with faiths ful tracial state τ N given by τ N (a) := i=1 ci ai , a = (a1 , . . . , as ) ∈ N . Set ∞ u := (u 1 , . . . , u s ) ∈ l ({1, . . . , s}, Mn (C)) = Mn (C) ⊗ N . Then u is unitary and relation (2.2) is satisfied. We now show that (2) ⇒ (1). Assume that (2) holds, i.e., there exists an abelian von Neumann algebra N with a normal faithful tracial state τ N and a unitary opera denote the spectrum of N (i.e., the tor u ∈ Mn (N ) such that (2.2) is satisfied. Let N is compact in the set of non-trivial multiplicative linear functionals on N ). Then N ) and τ N corresponds to a regular Borel probability measure μ w∗ -topology, N C( N . By identifying N with C( N ), we have u ∈ Mn (C( N )) = C( N , Mn (C)) and on N Tx =
N
u(t)∗ xu(t)dμ(t), x ∈ Mn (C).
). Since conv(Aut(Mn (C)) Thus T lies in the norm-closure of conv(ad(u(t)∗ ) : t ∈ N is a closed set in B(Mn (C)), condition (1) follows. The implication (2) ⇒ (3) follows immediately from the proof of the corresponding implication (ii) ⇒ (iii) in Theorem 2.2.
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Corollary 2.5. Let T : Mn (C) → Mn (C) be a τn -Markov map of the form (2.1), where d a1 , . . . , ad ∈ Mn (C) are self-adjoint, i=1 ai2 = 1n and ai a j = a j ai , for all 1 ≤ i, j ≤ d. Then the following hold: (a) T is factorizable. (b) If d ≥ 3 and the set {ai a j : 1 ≤ i ≤ j ≤ d} is linearly independent, then T ∈ / conv(Aut(M)). Proof. The proof of (a) is inspired by the proof of Theorem 3 in [31]. Let N be the CARalgebra over a d-dimensional Hilbert space H with orthonormal basis e1 , . . . , ed , and let a(ei ), 1 ≤ i ≤ d be the corresponding annihilation operators. Then N M2d (C) and the operators defined by vi := a(ei ) + a(ei )∗ , 1 ≤ i ≤ d form a set of anti-commuting self-adjoint unitaries (see, e.g., [6]). Set now u := d i=1 ai ⊗ vi ∈ Mn (C) ⊗ N . Then u is unitary since u∗u =
d
ai a j ⊗ vi v j =
i, j=1
=
1 2
d 1 (ai a j + a j ai ) ⊗ vi v j 2 i, j=1
d
ai a j ⊗ (vi v j + v j vi ) =
i, j=1
d
ai2 ⊗ 1 N = 1 Mn (N ) .
i=1
Moreover, τ N (vi∗ v j ) = τ N ((vi v j + v j vi )/2) = δi j , 1 ≤ i, j ≤ d. Hence, by the implication (iii) ⇒ (i) of Theorem 2.2, we deduce that T is factorizable. We now prove (b). Assume that d ≥ 3 and that {ai a j : 1 ≤ i ≤ j ≤ d} is linearly independent. In particular, the set {ai : 1 ≤ i ≤ d} is linearly independent. If T ∈ conv(Aut(Mn (C)), then by Proposition 2.4, there exists an abelian von Neumann algebra N with normal faithful tracial state τ N and operators v1 , . . . , vd ∈ N such d that the operator u := i=1 ai ⊗ vi ∈ Mn (C) ⊗ N = Mn (N ) is unitary. There fore, 1 Mn (N ) = u ∗ u = i,d j=1 ai a j ⊗ vi∗ v j . Using the fact that ai a j = a j ai , for all d 1 ≤ i, j ≤ d, and that i=1 ai2 = 1n , we infer that d
ai2 ⊗ (vi∗ vi − 1 N ) +
i=1
ai a j ⊗ (vi∗ v j + v ∗j vi ) = 0 Mn (N ) .
1≤i< j≤d
By the linear independence of the set {ai a j : 1 ≤ i ≤ j ≤ d}, it follows that vi∗ vi = 1 N , for 1 ≤ i ≤ d, and, respectively, that vi∗ v j + v ∗j vi = 0 N , for 1 ≤ i < j ≤ d. Since ), we deduce that N C( N , 1 ≤ i ≤ d, |vi (t)| = 1, t ∈ N and, respectively, , 1 ≤ i = j ≤ d. Re(vi (t)v j (t)) = 0, t ∈ N Since d ≥ 3, it follows that v1 (t)v2 (t), v2 (t)v3 (t) and v3 (t)v1 (t) are purely imaginary . Hence the product of these numbers is also purely complex numbers, for all t ∈ N imaginary. On the other hand, this product equals |v1 (t)|2 |v2 (t)|2 |v3 (t)|2 = 1, which gives rise to a contradiction. We conclude that T ∈ / conv(Aut(M)).
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We now discuss the case of Schur multipliers. The following fact is probably well-known, but we include a proof for completeness. Proposition 2.6. Let B = (bi j )i,n j=1 ∈ Mn (C) and let TB : Mn (C) → Mn (C) be its corresponding Schur multiplier, i.e., TB (x) = (bi j xi j )i,n j=1 , for all x = (xi j )i,n j=1 ∈ Mn (C). The following conditions are equivalent: (1) TB is positive. (2) TB is completely positive. (3) There exist diagonal matrices a1 , . . . , ad ∈ Mn (C) such that TB (x) =
d
ai∗ xai , x ∈ Mn (C).
(2.6)
i=1
(4) There exist linearly independent diagonal matrices a1 , . . . , ad ∈ Mn (C) such that (2.6) holds. (5) B is a positive semi-definite matrix, i.e., B = B ∗ and all eigenvalues of B are non-negative. Proof. The series of implications (4) ⇒ (3) ⇒ (2) ⇒ (1) is trivial, so we only have to prove that (1) ⇒ (5) ⇒ (4). Assume that TB is positive, i.e., TB ((Mn (C))+ ) = (Mn (C))+ , where (Mn (C))+ denotes the set of positive semi-definite n × n complex matrices. Clearly, the matrix x0 whose entries are all equal to 1 belongs to (Mn (C))+ , and therefore B = TB (x0 ) ∈ (Mn (C))+ . This shows that (1) ⇒ (5). To prove (5) ⇒ (4), assume that B = B ∗ with non-negative eigenvalues. Then B = C ∗ DC, where C is unitary and D = diag(λ1 , . . . , λn ) is a diagonal matrix whose diagonal entries are the eigenvalues of B repeated according to multiplicity. In particular, λi ≥ 0, for 1 ≤ i ≤ n. Let d := rank(B) = rank(D). We may assume that λ1 , . . . , λd > 0 and λd+1 = · · · = λ√n = 0. For any 1 ≤ i ≤ d consider now the diagonal n × n matrix given by ai := λi diag(ci1 , . . . cin ), where (ci1 , . . . , cin ) is the i th row of C. Then a1 , . . . , ad are linearly independent and one checks easily that (2.6) holds. Let B ∈ Mn (C). By Proposition 2.6, the Schur multiplier TB associated to the matrix B is a τn -Markov map if and only if B is positive semi-definite and b11 = b22 = · · · = bnn = 1, because the latter condition is equivalent to having TB (1n ) = 1n and τn ◦ TB = τn . Remark 2.7. In [31] E. Ricard proved that if B = (bi j )i,n j=1 ∈ Mn (R) is a positive semi-definite matrix whose diagonal entries are all equal to 1, then the associated Schur multiplier TB is always factorizable. This result can also be obtained from Corollary 2.5 (a). Indeed, under the above hypotheses, B = C t DC, where C is an orthogonal matrix and D = diag{d1 , . . . , dn } is a diagonal matrix with λi ≥ 0. Let d := rank(D). Then, following the proof of the implication (5) ⇒ (4) in Proposition 2.6, we deduce that d TB (x) = i=1 ai∗ xai , for all x ∈ Mn (C), where a1 , . . . , ad are linearly independent diagonal matrices with ai = ai∗ , 1 ≤ i ≤ d (since the entries of C are real numbers). d d Moreover, i=1 ai2 = i=1 ai∗ ai = TB (1n ) = 1n , and ai a j = a j ai , for 1 ≤ i, j ≤ d. It then follows from Corollary 2.5 (a) that TB is factorizable. We end this section with a general characterization of factorizable Schur multipliers, which turns out to be useful for applications.
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U. Haagerup, M. Musat
Proposition 2.8. Let B = (bi j )i,n j=1 be a positive semi-definite n × n complex matrix having all diagonal entries equal to 1. Then the associated Schur multiplier TB is a factorizable τn -Markov map if and only if there exists a finite von Neumann algebra N with normal faithful tracial state τ N and unitaries u 1 , . . . , u n ∈ N such that bi j = τ N (u i∗ u j ), 1 ≤ i, j ≤ n.
(2.7)
Proof. Assume that TB is factorizable. Then by Theorem 2.2, there exists a finite von Neumann algebra N with normal faithful tracial state τ N , and a unitary u ∈ Mn (N ) = Mn (C) ⊗ N such that TB (x) = (id Mn (C) ⊗ τ N )(u ∗ (x ⊗ 1 N )u) , x ∈ Mn (C).
(2.8)
It follows that τn (yTB (x)) = (τn ⊗ τ N )((y ⊗ 1 N )u ∗ (x ⊗ 1 N )u), x, y ∈ Mn (C). (2.9) n Let (e jk )1≤ j,k≤n be the standard matrix units in Mn (C). Then u = i,k=1 e jk ⊗ u jk , n ∗ ∗ where u jk ∈ N , 1 ≤ j, k ≤ n, and u = i,k=1 ek j ⊗ u jk . Consider now j, k ∈ {1, . . . , n}, j = k. By applying (2.9) to x = e j j and y = ekk , we get τn (ekk TB (e j j )) = b j j τn (ekk e j j ) = 0. Therefore, 0 = (τn ⊗ τ N )((ekk ⊗ 1 N )u ∗ (e j j ⊗ 1 N )u) = (τn ⊗ τ N )((ekk ⊗ 1 N )u ∗ (e j j ⊗ 1 N )u(ekk ⊗ 1 N )) = (τn ⊗ τ N )(ekk ⊗ u ∗jk u jk ) = (1/n)τ N (u ∗jk u jk ). By the faithfulness of τ N , u jk = 0 N for j = k. Thus u = 1 ≤ j, k ≤ n we then get
n
j=1 e j j
⊗ u j j . For
b jk = b jk nτn (ek j e jk ) = nτn (ek j TB (e jk )) = n(τn ⊗ τ N )((ek j ⊗ 1 N )u ∗ (e jk ⊗ 1 N )u) = n(τn ⊗ τ N )(ekk ⊗ u ∗j j u kk ) = τ N (u ∗j j u kk ). Hence (2.7) holds with u j = u j j , for 1 ≤ j ≤ n. Conversely, if (2.7) holds for a set of n unitaries u 1 , . . . , u n in a finite von Neumann algebra N with normal, faithful, tracial state τ N , then the operator u := nj=1 e j j ⊗u j is a unitary in Mn (C) ⊗ N and one checks easily that (2.8) holds. Hence, by Theorem 2.2, TB is factorizable. 3. Examples We begin by exhibiting an example of a non-factorizable τ3 -Markov maps on M3 (C). Example 3.1. Set ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 1 ⎝0 0 0 ⎠ 1 ⎝ 0 0 1⎠ 1 ⎝ 0 −1 0 ⎠ 0 0 −1 , a2 = √ 0 0 0 , a3 = √ 1 0 0 . a1 = √ 2 0 1 0 2 −1 0 0 2 0 0 0
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3 3 Then i=1 ai∗ ai = i=1 ai ai∗ = 13 , and hence the operator T defined by T x := 3 ∗ i=1 ai xai , for all x ∈ Mn (C) is a τ3 -Markov map. If T were factorizable, then by the implication (i) ⇒ (iii) in Theorem 2.2, there would exist a finite von Neumann algebra N with a normal faithful tracial state τ N and elements v1 , v2 , v3 ∈ N such that the operator ⎞ ⎛ 3 1 ⎝ 0 −v3 v2 ⎠ v3 0 −v1 u := ai ⊗ vi = √ 2 −v2 v1 0 i=1 is unitary, but as observed in [13] (pp. 282–283), this is impossible. Indeed, since u ∗ u = 1 N , we have v1∗ v1 + v2∗ v2 = v2∗ v2 + v3∗ v3 = v3∗ v3 + v1∗ v1 = 2 1 N
(3.1)
v1∗ v2 = v2∗ v3 = v3∗ v1 = 0 N .
(3.2)
and, respectively,
Note that (3.1) implies that v1∗ v1 = v2∗ v2 = v3∗ v3 = 1 N , and since N is finite, it follows that v1 , v2 and v3 are unitary operators, which contradicts (3.2). This shows that T is not factorizable. Alternatively, one can check that {ai∗ a j : 1 ≤ i, j ≤ 3} is a linearly independent set in M3 (C) and then use Corollary 2.3 to prove that T is not factorizable. We now present some concrete examples of non-factorizable Schur multipliers. Example 3.2. Following an example constructed in [8], for 0 ≤ s ≤ 1 set √ √ √ ⎞ ⎛ ⎛ ⎞ s s s 0 0 0 0 √1 s s ⎟ ⎜ s s ⎜0 1 ω ω⎟ B(s) := ⎝ √ ⎠ + (1 − s)⎝ 0 ω 1 ω ⎠, s s s s √ 0 ω ω 1 s s s s √ where ω := ei2π /3 = −1/2 + i 3/2 and ω is the complex conjugate of ω. Note that B(s) is positive semi-definite, since B(s) = x1 (s)∗ x1 (s) + x2 (s)∗ x2 (s), √ √ √ √ where x1 (s) = (1, s, s, s) and x2 (s) = 1 − s(0, 1, ω, ω). Moreover, b11 = b22 = b33 = b44 = 1. Thus TB(s) is a τ4 -Markov map for all 0 ≤ s ≤ 1. We claim that for 0 < s < 1, the map TB(s) is not factorizable. To prove this, we will use Corollary 2.3. Let 0 < s < 1 and observe that TB(s) (x) = a1 (s)∗ xa1 (s) + a2 (s)∗ xa2 (s), x ∈ M4 (C), where a1 (s) and a2 (s) are the diagonal 4 × 4 matrices √ √ √ √ a1 (s) := diag(1, s, s, s), a2 (s) := 1 − s diag(0, 1, ω, ω). (s)∗ a1 (s) = diag(1, s, It is elementary to check that the following four matrices: a1√ ∗ ∗ s, s), a2 (s) a2 (s) √ = (1 − s) diag(0, 1, 1, 1), a1 (s) a2 (s) = s diag(0, 1, ω, ω) and a2 (s)∗ a1 (s) = s diag(0, 1, ω, ω) are linearly independent. Hence, by the above-mentioned corollary, TB(s) is not factorizable.
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Note that for s = 1/3, B(s) has a particularly simple form, namely, √ √ ⎞ √ ⎛ 1√ 1/ 3 1/√ 3 1/ √3 ⎜ 1/ 3 1√ i/ 3 −i/√ 3 ⎟ ⎟. √ B(1/3) = ⎜ ⎝ 1/ 3 −i/ 3 1√ i/ 3 ⎠ √ √ 1/ 3 i/ 3 −i/ 3 1 The above 4 × 4 matrix examples can easily be generalized to examples of nonfactorizable Schur multipliers on (Mn (C), τn ), for all n ≥ 4, by setting B(s) := x1 (s)∗ x1 (s) + x2 (s)∗ x2 (s), where 0 < s < 1 and x1 , x2 are the column vectors in Cn given by √ √ √ √ x1 (s) := (1, s, s, . . . , s), x2 (s) := 1 − s(0, 1, ρ, ρ 2 , . . . , ρ n−2 ), where ρ = ei 2π /(n−1) . Further, let a1 (s) and a2 (s) be the corresponding n × n diagonal matrices. Then the linear independence of the set {(a j (s))∗ ak (s) : 1 ≤ j, k ≤ 2} follows from the computation ⎛ ⎞ 1 1 1 det ⎝ 1 ρ ρ 2 ⎠ = ρ¯ 2 (ρ + 1)(ρ − 1)3 = 0, 1 ρ¯ ρ¯ 2 where ρ¯ is the conjugate of ρ. Then, an application of Corollary 2.3 shows that the Schur multiplier TB(s) is not factorizable. √ Example 3.3. Let β = 1/ 5 and set ⎞ ⎛ 1 β β β β β ⎜β 1 β −β −β −β ⎟ ⎟ ⎜ 1 β −β −β ⎟ ⎜β β . B := ⎜ 1 β −β ⎟ ⎟ ⎜ β −β β ⎠ ⎝ β −β −β β 1 β β β −β −β β 1 We claim that TB is a factorizable τ6 -Markov map on M6 (C), but TB ∈conv(Aut(M / √ 6 (C))). To prove this, observe first that since cos(2π/5) = cos(8π /5) = (−1 + 5)/4 and √ cos(4π /5) = cos(6π /5) = (−1 − 5)/4, then B can be written in the form B = x1∗ x1 + x2∗ x2 + x3∗ x3 , √ √ √ √ √ √ where x1 := (1, 1/ 5, 1/ 5, 1/ 5, 1/ 5, 1/ 5), x2 := 2/5(0, 1, ei2π/5 , ei4π/5 , √ ei6π/5 , ei8π/5 ) and x3 := x2 = 2/5(0, 1, e−i2π/5 , e−i4π/5 , e−i6π/5 , e−i8π/5 ). Hence B is positive semi-definite. By Remark 2.7, TB is a factorizable τ6 -Markov map on M6 (C). Moreover, TB (x) =
3 i=1
bi∗ xbi , x ∈ M6 (C),
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√ √ √ √ √ √ where b1 := diag (1, 1/ 5, 1/ 5, 1/ 5, 1/ 5, 1/ 5), b2 := 2/5 diag(0, 1, ei2π/5 , ei4π/5 , ei6π/5 , ei8π/5 ) and b3 := b2 ∗ . Set now 1 1 a1 := b1 , a2 := √ (b2 + b3 ), a3 := √ (b2 − b3 ). 2 i 2 3 Then TB (x) = i=1 ai xai , for all x ∈ M6 (C). Note that a1 , a2 , a3 are commuting 3 ai2 = 16 . Thus, if we knew that the set self-adjoint (diagonal) matrices with i=1 {ai a j : 1 ≤ i ≤ j ≤ 3} is linearly independent, then, by Corollary 2.5 (b) we could conclude that T ∈ / conv(Aut(Mn (C))). Note that the linear independence of the above set is equivalent to the linear independence of the set {bi b j : 1 ≤ i ≤ j ≤ 3}.
(3.3)
Set γ := ei2π /5 . Then the following relations hold: √ 2 1 diag (0, 1, γ , γ 2 , γ 3 , γ 4 ), b12 = diag (5, 1, 1, 1, 1, 1) , b1 b2 = 5√ 5 2 b1 b3 = diag (0, 1, γ 4 , γ 8 , γ 12 , γ 16 ), 5 2 b22 = diag 0, 1, γ 2 , γ 4 , γ 6 , γ 8 , 5 2 2 b2 b3 = diag (0, 1, 1, 1, 1, 1), b32 = diag 0, 1, γ 3 , γ 6 , γ 9 , γ 12 . 5 5 Now let
⎛
2π kl H := exp i 5 0≤k,l≤4
1 ⎜1 ⎜ =⎜ ⎜1 ⎝1 1
1 γ γ2 γ3 γ4
(3.4)
⎞ 1 1 1 γ2 γ3 γ4 ⎟ ⎟ γ4 γ6 γ8 ⎟ ⎟. γ 6 γ 9 γ 12 ⎠ γ 8 γ 12 γ 16
Then H is a complex 5 × 5 Hadamard matrix, i.e., |h i, j |2 = 1, for all 0 ≤ i, j ≤ 4 and H ∗ H = H H ∗ = 5 15 . It follows that the rows of H are linearly independent. This fact, combined with the relations (3.4), shows that the set in (3.3) is linearly independent. Hence the assertion is proved. Theorem 3.4. Let L = (L jk )1≤ j,k≤4 be the 4 × 4 complex matrix given by ⎛
⎞ 0 1/2 1/2 1/2 0 1 − ω 1 − ω⎟ ⎜ 1/2 L := ⎝ , 1/2 1 − ω 0 1 − ω⎠ 1/2 1 − ω 1 − ω 0
(3.5)
where ω := ei2π /3 and ω is the complex conjugate of ω. Let (C(t))t≥0 denote the one-parameter family of 4 × 4 complex matrices C(t) := (e−t L jk )1≤ j,k≤4 , t ≥ 0.
(3.6)
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U. Haagerup, M. Musat
Then the corresponding Schur multipliers Tt := TC(t) , t ≥ 0
(3.7)
form a continuous one-parameter semigroup of τ4 -Markov maps on M4 (C) starting at T (0) = id M4 (C) . Moreover, there exists t0 > 0 such that T (t) is not factorizable for any 0 < t < t0 . Proof. Let ((B(s))1≤s≤1 be the positive semi-definite 4 × 4 complex matrices considered in Example 3.2. In particular, the matrix B(1) has all entries equal to 1. Note that the matrix L given by (3.5) is the first derivative of B(s) at s = 1, i.e., L=
B(s) − B(1) d B(s) . = lim ds |s=1 s1 s−1
Since B(s) is positive semi-definite for all 0 ≤ s ≤ 1, we have that 4
(B(s) − B(1))c j c¯k ≤ 0,
j,k=1
whenever c1 , . . . , c4 ∈ C and c1 +, · · · + c4 = 0. This implies that 4j,k=1 L jk c j c¯k ≥ 0, i.e., L is a conditionally negative definite matrix. By Schoenberg’s theorem (see, e.g., [3]), the matrices C(t), t ≥ 0 given by (3.6) are all positive semi-definite. Moreover, since L 11 = L 22 = L 33 = L 44 = 0, we also have C(t)11 = C(t)22 = C(t)33 = C(t)44 = 0, for all t ≥ 0. Hence the Schur multipliers T (t) = TC(t) , t ≥ 0 are all τ4 -Markov maps. Clearly, the family (T (t))t≥0 forms a continuous one-parameter semigroup of Schur multipliers starting at T (0) = id M4 (C) . Now set F := {t > 0 : TC(t) is factorizable}. We will show that there exists t0 > 0 such that F ∩ (0, t0 ) = ∅. By Proposition 2.8, for each t ∈ F we can find a finite von Neumann algebra N (t) with normal faithful tracial state τ N (t) and four unitary operators u 1 (t), . . . , u 4 (t) ∈ N (t) such that C(t) jk = τ N (t) (u j (t)∗ u k (t)), 1 ≤ j, k ≤ 4. Since ω + ω + 1 = 0 and ω2 = ω, we can express u 1 (t), . . . , u 4 (t) in the form u 1 (t) = x(t) + w(t), u 2 (t) = x(t) + y(t) + z(t), u 3 (t) = x(t) + ωy(t) + ωz(t), u 4 (t) = x(t) + ωy(t) + ωz(t), where x(t) := (u 2 (t) + u 3 (t) + u 4 (t))/3, y(t) := (u 2 (t) + ωu 3 (t) + ωu 4 (t))/3, z(t) := (u 2 (t) + ωu 3 (t) + ωu 4 (t))/3, w(t) = x(t) − u 1 (t)). Note that for all t ∈ F, x(t) ≤ 1, y(t) ≤ 1, z(t) ≤ 1.
(3.8)
y(t)2 ≤ 2z(t)2 + w(t)2 .
(3.9)
We prove next that For this, observe first that x(t)∗ y(t) + y(t)∗ z(t) + z(t)∗ x(t) = 0, which implies that x(t)∗ y(t)2 ≤ y(t)∗ z(t)2 + z(t)∗ x(t)2 ≤ y(t)z(t)2 + z(t)2 x(t) ≤ 2z(t)2 , wherein we used (3.8).
(3.10)
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569
Now recall that x(t) = u 1 (t) + w(t), where u 1 (t) is a unitary. Thus, by (3.10), y(t)2 = u 1 (t)∗ y(t)2 = x(t)∗ y(t) − w(t)∗ y(t)2 ≤ x(t)∗ y(t)2 + w(t)2 y(t)2 ≤ 2z(t)2 + w(t)2 , which proves (3.9). Next, observe that the 2-norms of y(t), z(t) and w(t) can be expressed in terms of the entries of the matrix L, because for c := (c1 , c2 , c3 , c4 ) ∈ C4 and t ∈ F we have 2 4 4 4 ∗ c u (t) = c ¯ c τ (u (t) u (t)) = c¯j ck e−t L jk = f (c, t), j j j k N j k j=1 j,k=1 j,k=1 2
where the function t → f (c, t) is actually defined for all t ≥ 0 and satisfies ⎛ ⎞ 4 2 f (c, t) = |c1 + c2 + c3 + c4 | − ⎝ c¯j ck L jk ⎠ t + O(t 2 ), as t 0,
(3.11)
j,k=1
in Landau’s O-notation. Consider now the three special cases where c := (c1 , c2 , c3 , c4 ) ∈ C4 is equal to (0, 1/3, ω/3, ω/3), (0, 1/3, ω/3, ω/3) and (−1, 1/3, 1/3, 1/3), respectively. For t ≥ 0, let us denote the function f (c, t) in (3.11) by g(t), h(t) and k(t), respectively, in each of the corresponding cases. Then, g(t) = y(t)22 , h(t) = z(t)22 , k(t) = z(t)22 ,
t ∈ F.
(3.12)
as t 0.
(3.13)
Moreover, by (3.11), g(t) = t + O(t 2 ), h(t) = O(t 2 ), k(t) = O(t 2 ),
Assume now that inf(F) = 0. Then there exists a sequence (tn )n≥1 in F such that tn → 0 as n → ∞. By (3.9) and (3.12) we have g(tn )1/2 ≤ 2h(tn )1/2 + k(tn )1/2 , n ≥ 1. 1/2
3/2
However, by (3.13), 2h(tn )1/2 + k(tn )1/2 = O(tn ), while g(tn )1/2 = tn + O(tn ), both for large enough n. This gives rise to a contradiction. Hence inf(F) > 0, i.e., there exists t0 > 0 such that (0, t0 ) ∩ F = ∅. The proof is complete. Remark 3.5. The above theorem is to be contrasted with a result of Kümmerer and Maassen (cf. [24]), showing that if (T (t))t≥0 is a one-parameter semigroup of τn -Markov maps on Mn (C) satisfying T (t)∗ = T (t), for all t ≥ 0, then T (t) ∈ conv(Aut(Mn (C))), for all t ≥ 0. In particular, T (t) is factorizable, for all t ≥ 0. In very recent work, Junge, Ricard and Shlyakhtenko [17] have generalized Kümmerer and Maassen’s result to the case of a strongly continuous one-parameter semigroup (T (t))t≥0 of self-adjoint Markov maps on an arbitrary von Neumann algebra with a faithful, normal tracial state by proving that also in this case T (t) is factorizable, for all t ≥ 0. This result has been independently obtained by Y. Dabrowski (see [9]).
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Remark 3.6. In the recent preprint [10], K. Dykema and K. Juschenko have indirectly exhibited an example of a τ4 -Markov map on M4 (C) which is not factorizable. More precisely, for every n ≥ 1 they considered the sets Fn , defined as the closure of the union over k ≥ 1 of sets of n × n complex matrices (bi j )1≤i, j≤n such that bi j = τk (u i u ∗j ), where u 1 , . . . , u n ∈ U(k), respectively, Gn , consisting of all n × n complex matrices (bi j )1≤i, j≤n such that bi j = τ M (u i u ∗j ), where u 1 , . . . , u n are unitaries in some von Neumann algebra M equipped with normal faithful tracial state τ M (where M varies). By a refinement of Kirchberg’s deep results from [19], they concluded that Connes’ embedding problem whether every I I1 -factor with separable predual embeds in the ultrapower of the hyperfinite I I1 factor has an affirmative answer, if and only if Fn = Gn , for all n ≥ 1. Further, they pointed out that Fn ⊆ Gn ⊆ n , for all n ≥ 1, where n is the set of n × n (complex) correlation matrices, i.e., positive semi-definite matrices whose entries on the main diagonal are all equal to 1. A natural question to consider is whether Fn = n , for all n ≥ 1. One of the results of [10] is that the answer to this question is negative, as soon as n ≥ 4. More precisely, Dykema and Juschenko showed that G4 has no extreme points of rank 2, while there are extreme points of rank 2 in 4 . Hence G4 = 4 . In view of Proposition 2.8 above, any element of 4 \G4 is an example of a non-factorizable τ4 -Markov map on M4 (C). 4. Kummerer’s Notions of Dilation and Their Connection to Factorizability The following definitions are due to Kümmerer (see [22], Defs. 2.1.1 and 2.2.4, respectively): Definition 4.1. Let M be a von Neumann algebra with a normal faithful state φ and let T : M → M be a φ-Markov map. A dilation of T is a quadruple (N , ψ, α, ι), where N is a von Neumann algebra with a normal faithful state ψ, α ∈ Aut(N , ψ), i.e., α is an automorphism of N leaving ψ invariant and ι : M → N is a (φ, ψ)-Markov ∗-monomorphism, satisfying T n = ι∗ ◦ α n ◦ ι, n ≥ 1.
(4.14)
Furthermore, we say that (N , ψ, α, ι) is a dilation of T of order 1 if (4.14) holds for n = 1 but not necessarily for n ≥ 2. Definition 4.2. A dilation (N , ψ, α, ι) of a φ-Markov map T : M → M is called a Markov dilation if P{0} (x) = P(−∞,0] (x), x ∈ ( α k ◦ ι(M)) , (4.15) k≥0
where for I ⊆ Z, P I denotes (cf. Lemma 2.1.3 of [22]) the unique ψ-preserving normal k ◦ ι(M) . faithful conditional expectation of N onto its subalgebra α k∈I Remark 4.3. The condition (4.15) is equivalent to P{0} P[0,∞) = P(−∞,0] P[0,∞) . Clearly, P[0,∞) P{0} = P{0} , and since both P{0} and P[0,∞) extend uniquely to self-adjoint projections on L 2 (N , ψ), it also follows that P{0} P[0,∞) = P{0} . Hence (4.15) is further equivalent to P{0} = P(−∞,0] P[0,∞) .
(4.16)
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571
In the Spring of 2008, C. Koestler informed us in private correspondence that he was aware of the fact that for a given Markov map, the existence of a dilation (in the sense of Definition 4.1) is actually equivalent to C. Anantharaman-Delaroche’s condition of factorizability of the map. The proof relies on construction of the inductive limit of von Neumann algebras naturally associated to a Markov ∗-monomorphism, studied by Kümmerer in his unpublished Habilitationsschrift [23]. We are very grateful to C. Koestler for sharing all this information with us and for kindly providing us with a copy of [23]. We were further able to show that the existence of a dilation for a given Markov map is equivalent to the existence of a Markov dilation (in the sense of Definition 4.2) for it. For completeness, we collect together all these equivalent statements in the following theorem. Theorem 4.4. Let M be a von Neumann algebra with normal faithful state φ and let T : M → M be a φ-Markov map. The following statements are equivalent: (1) T is factorizable. (2) T has a dilation. (3) T has a Markov dilation. For convenience, we include the details of the above-mentioned inductive limit construction for von Neumann algebras, that will be used in the proof of Theorem 4.4. Lemma 4.5. Suppose that for each positive integer k, we are given a von Neumann algebra Mk with a normal faithful state ψk and a unital ∗-monomorphism βk : Mk → Mk+1 such that ψk+1 ◦ βk = ψk , satisfying, moreover, ψk+1
σt
ψ
◦ βk = βk ◦ σt k , t ∈ R.
(4.17)
Then, there exists a von Neumann algebra M with a normal faithful state ψ and unital ∗ψ monomorphisms μk : Mk → M such that μk+1 ◦ βk = μk , ψ ◦ μk = ψk and σt ◦ μk = ∞ ψk μk ◦ σt , t ∈ R, for all k ≥ 1, and such that k=1 μk (Mk ) is weakly dense in M. Moreover, if we are given another von Neumann algebra N with a normal faithful state ϕ and (normal) ∗-monomorphisms λk : Mk → N such that λk+1 ◦ βk = λk and ϕ ◦ λk = ψk for all k ≥ 1, then there exists a unique ∗-monomorphism λ : M → N such that λ ◦ μk = λk for all k ≥ 1 and ϕ ◦ λ = ψ. Proof. As a first step towards the existence of M, let M∞ be the C ∗ -algebra inductive limit of the sequence M1 → M2 → M3 → · · ·. This is a C ∗ -algebra equipped with ∗ -monomorphisms μk : Mk → M∞ (which are unital when the connectingmappings βk all are unital) satisfying μk+1 ◦ βk = μk for all k ≥ 1, and which contains ∞ μk (Mk ) k=1 k on k ◦ as a norm-dense sub-algebra. The states ψ μk (Mk ), defined by ψ μk = ψk , are k+1 to k , and so they extend to a coherent, i.e., the restriction of ψ μk (Mk ) is equal to ψ on M∞ . Let M be the weak closure of M∞ in the GNS-representation of M∞ state ψ , and let μk : Mk → M be the composition of with respect to the state ψ μk with the extends to a normal state ψ on M. It is inclusion mapping of M∞ into M. The state ψ clear that μk+1 ◦ βk = μk , ψ ◦ μk = ψk ,
k ≥ 1.
We prove next that ψ is faithful. For simplicity, we will now identify Mk with μk (Mk ), k ≥ 1. Then we have the inclusions M1 ⊆ M2 ⊆ M3 ⊆ · · · and ∞ k=1 Mk
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is weakly dense in M. Moreover, these inclusions extend to isometric embeddings 2 of the corresponding GNS Hilbert spaces ∞for 2(Mk , ψk ), k ≥ 1, i.e., 2L (M1 , ψ1 ) ⊆ 2 2 L (M2 , ψ2 ) ⊆ L (M3 , ψ3 ) ⊆ · · · and k=1 L (Mk , ψk ) is dense in L (M, ψ). After ψ these identifications, for every k ≥ 1, the condition σt k+1 ◦ β = β ◦ σtσk , t ∈ R, is equivalent to ψk+1
σt
ψ
(x) = σt k (x), x ∈ Mk , t ∈ R.
(4.18)
Hence, by [34] there exist unique normal faithful conditional expectations Ek : Mk+1 → Mk such that ψk ◦ Ek = ψk+1 , k ≥ 1. Also from [34] it follows that the isometric modular conjugation operator Jψk on L 2 (Mk , ψk ) is the restriction of Jψk+1 to L 2 (Mk , ψk ). Hence, there exists a conjugate-linear isometric involution J on L 2 (M, ψ) which extends all the Jk ’s. Since, moreover, Jψk πψk (Mk )Jψk = πψk (Mk ) , for all k ≥ 1, it follows that J πψ (M)J ⊆ πψ (M) .
(4.19)
Let ξψ be the cyclic vector in L 2 (M, ψ) corresponding to the unit operator 1 in M. Then J ξψ = ξψ , and therefore by (4.19), ξψ is also cyclic for πψ (M) . Hence ξψ is separating for πψ (M), which proves that ψ is faithful. Finally, in order to prove that ψ ψ σt ◦μk = μk ◦σt k , t ∈ R, k ≥ 1, we have to show that under the above identifications, ψ
ψ
σt (x) = σt k (x), x ∈ Mk , t ∈ R, k ≥ 1.
(4.20)
By (4.18), together with the fact that ψk = ψk+1 | M , k ≥ 1, it follows that the modular ψ
k
automorphism groups (σt k )t∈R , k ≥ 1 have a unique extension to a strongly continuous one-parameter group of automorphisms (σt )t∈R on M. Moreover, ψ satisfies the KMS ψ condition with respect to (σt )t∈R , since each ψk is a (σt k )t∈R -KMS state on Mk (see ψ Theorem 1.2, Chap. VIII, in [35]). Therefore σt = σt , t ∈ R, which proves (4.20). To the second part of the lemma we can without loss of generality assume prove ∞ that λ k ∞ k=1 (Mk ) is weakly dense in N (otherwise replace N by the weak closure of k=1 λk (Mk )). Consider the GNS-representations of M and N on Hilbert spaces H and H with respect to the normal faithful states ψ and ϕ, respectively. Then there exist cyclic and separating vectors ξ ∈ H and ξ ∈ H for M and N , respectively, such that ψ(x) = xξ, ξ , ϕ(y) = yξ , ξ ,
x ∈ M, y ∈ N .
By the universal property of the C ∗ -algebra inductive limit, there is a ∗-monomorphism λ : M∞ → N satisfying λ ◦ μk = λk for all k ≥ 1. Observe that ϕ ◦ λ(x) = ψ(x), for all x ∈ M∞ . It follows that the map u 0 : M∞ ξ → H defined by u 0 xξ = λ(x)ξ , x ∈ M∞ , is isometric and has dense range in H . Hence it extends to a unitary u : H → H . We see that uxu ∗ ξ = λ(x)ξ , and hence that uxu ∗ = λ(x), for all x ∈ M∞ . The map λ : M → N defined by λ(x) = uxu ∗ , for x ∈ M, has the desired properties. Remark 4.6. We would like to draw the reader’s attention to the subtle fact that condition on the C ∗ -algebra inductive (4.17) is crucial for guaranteeing that the canonical state ψ limit M∞ extends to a faithful state ψ on the von Neumann algebra M, obtained via the . GNS representation of M∞ with respect to ψ
Factorization and Dilation Problems
573
The von Neumann algebra (M, ψ) is said to be the von Neumann algebra inductive limit of the sequence (M1 , ψ1 )
β1
/ (M2 , ψ2 )
β2
/ (M3 , ψ3 )
β3
/ ··· .
The following result is a reformulation of Proposition 2.1.7 in [23]. Proposition 4.7. Let N be a von Neumann algebra with a normal faithful state ψ, and let β : N → N be a ψ-Markov ∗-monomorphism. Then there exists a von Neumann with a normal faithful state ψ and , a (ψ, ψ )-Markov embedding ι : N → N algebra N ∗ ˜ an α ∈ Aut( N ) for which ψ ◦ α = ψ such that β = ι ◦ α ◦ ι. , ψ ) be the von Neumann algebra inductive limit of the sequence Proof. Let ( N (N , ψ)
β
/ (N , ψ)
β
/ (N , ψ)
β
/ ··· ,
be the associated ∗-monomorphism from the k th and for every k ≥ 1, let μk : N → N copy of N into N . By the second part of Lemma 4.5 applied to the ∗-monomorphisms given by λk = μk ◦ β, there exists a ∗-monomorphism α on N such that λk : N → N α ◦ μk = μk ◦ β, for all k ≥ 1. It follows that ∞ k=1
μk (N ) =
∞ k=1
μk+1 ◦ β(N ) =
∞
α ◦ μk+1 (N ) ⊆ Im(α).
k=1
As ∞ k=1 μk (N ) is dense in N and the image of α is a von Neumann subalgebra of N , this shows that α is onto, and hence an automorphism. to be μ1 . Then α ◦ ι = ι ◦ β. Moreover, by Lemma 4.5, ι is a Take ι : N → N ψ-Markov map. Since ◦ α ◦ μk = ψ ◦ μk ◦ β = ψ ◦ β = ψ = ψ ◦ μk , k ≥ 1, ψ ◦ α and ψ coincide on ∞ ψ k≥1 μk (N ). Therefore, ψ = ψ ◦ α. The existence of the ∗ ∗ → N follows from Remark 1.2. As ι ◦ ι is the identity on N we get adjoint map ι : N that β = ι∗ ◦ α ◦ ι using the previously obtained identity ι ◦ β = α ◦ ι. This completes the proof. Proof of Theorem 4.4. The implication (3) ⇒ (2) is trivial. Also, the implication (2) ⇒ (1) follows immediately, since if (N , ψ, α, ι) is a dilation of T , then T = ι∗ ◦ (α ◦ ι) is a factorization of T through (N , ψ) in the sense of Definition 1.3, because α ∈ Aut(N , ψ) ψ ψ implies that α ◦ σt = σt ◦ α, t ∈ R. Next we prove that (1) ⇒ (3). Assume that T is factorizable. Then, by Theorem 6.6 in [2], there exists a von Neumann algebra N with a normal faithful state ψ, a ψ-Markov normal ∗-endomorphism β : N → N and a (φ, ψ)-Markov ∗-monomorphism j : M → N such that T n = j ∗ ◦ β n ◦ j, for all n ≥ 1. By Proposition 4.7, we ˜ α, ι) of β, where α ∈ Aut( N˜ , ψ). ˜ We may (and will) consider can find a dilation ( N˜ , ψ, ˜ N as a subalgebra of N˜ . In this way, ι is just the inclusion map, ι∗ is the ψ-preserving normal faithful conditional expectation of N˜ onto N , and β = α| N . Then it is clear that , ψ , α, with j := ι ◦ j, the quadruple ( N j) is a dilation of T (which actually proves (1) ⇒ (2)). To complete the proof of the implication (1) ⇒ (3), we will show that by
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U. Haagerup, M. Musat
, ψ , α, the construction of β from [2], the quadruple ( N j) becomes a Markov dilation of T . For J ⊆ {n ∈ Z : n ≥ 0}, let E J denote the unique ψ-preserving conditional expectation of N onto its subalgebra B J := ( k∈J β k ◦ j (M)) . Then, by condition (6.2) in Theorem 6.6 of [2], we get E[0,n+k] ◦ β k = β k ◦ E[0,n] , n, k ≥ 0.
(4.21)
Moreover, with B and (Bn , φn ), n ≥ 0 defined as in the proof of the above-mentioned theorem, we have Bn = B[0,n] and N = B = B[0,∞) . Hence E[0,n] = EBn , the unique ψ-invariant conditional expectation of N onto Bn , and E[0,∞) = E N = id N . Set now , be the unique extension H := L 2 (N , ψ), Hn := L 2 (Bn , φn ), n ≥ 0, and let V := β 2 of β to an isometry on L (N , ψ). By (4.21), PHn+k V k = V k PHn , n, k ≥ 0, where PK ∈ B(H ) denotes the orthogonal projection onto a closed subspace K of H . Hence, PHn+k PV k (H ) = PHn+k V k (V ∗ )k = V k PHn (V ∗ )k = PV k (Hn ) , n, k ≥ 0. (4.22) By the definition, it is clear that β k (B J ) = B J +k , for all k ≥ 0 and all J ⊆ {n ∈ Z : n ≥ 0}. In particular, β k (N ) = β k (B[0,∞) ) = B[k,∞) , β k (Bn ) = B[k,k+n] ,
n, k ≥ 0.
Thus, by restricting (4.22) to N ⊆ L 2 (N , ψ), we get E[0,n+k] E[k,∞) = E[k,k+n] , for all n, k ≥ 0. In particular, we have E[0,k] E[k,∞) = E{k} , k ≥ 0.
(4.23)
Since j = ι ◦ j, we have from Proposition 4.7 that ˜ α k ◦ j(M) = β k ◦ j (M), k ≥ 0. -preserving conditional expectaHence, by composing (4.23) from the right with the ψ ∗ ˜ tion ι of N onto N , we get (following the notation set forth in Definition 4.2) that P[0,k] P[k,∞) = P{k} , k ≥ 0.
(4.24)
Note that for every I ⊆ Z, by the definition of P I one has α n P I α −n = P I +n , for all n ∈ Z. Hence, from (4.24) we get that P[−n,0] P[0,∞) = P{0} , n ≥ 0. In the limit as n → ∞, this yields P(−∞,0] P[0,∞) = P{0} , ˜ is a Markov dilation of T . ˜ α, j) a condition which, by Remark 4.3, ensures that ( N˜ , ψ,
Factorization and Dilation Problems
575
Note that by the proof of Theorem 4.4 it follows that a φ-Markov map T : M → M admits a dilation if and only if it has a dilation of order 1 (see Definition 4.1), since in order to show that (2) ⇒ (1) above we have only used the existence of a dilation of order 1 for the given map. Remark 4.8. Kümmerer constructed in [23] examples of τn -Markov maps on Mn (C), n ≥ 3, having no dilation, as follows: ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 0 0 0 0 1 0 0 0 1 (1) Let a1 = √1 ⎝ 1 0 0 ⎠ , a2 = √1 ⎝ 0 0 1 ⎠, and a3 = √1 ⎝ 0 0 0 ⎠. 2 2 2 0 1 0 0 0 0 1 0 0 3 ∗ Then the map given by T x := i=1 ai xai , x ∈ M3 (C), is a τ3 -Markov map having no dilation. √ √ = diag(1, 1/ 2, 1/ 2, 0, (2) Let n ≥ 4 and consider the n × n√diagonal matrices a 1 √ . . . , 0) and a2 = diag(0, 1/ 2, i/ 2, 1, . . . , 1). Then the map given by 2 ai∗ xai , x ∈ Mn (C), is a τn -Markov Schur multiplier with no dilation. T x := i=1 In view of Koestler’s communication, these are all examples of non-factorizable Markov maps. In [23], Kümmerer also constructed an example of a τ6 -Markov Schur multiplier on M6 (C) which admits a dilation, hence it is factorizable, but does not lie in conv(Aut(M6 (C))). However, he did not consider the one-parameter semigroup case. We were also informed by B. V. R. Bhat and A. Skalski [4] that, unaware of Kümmerer’s examples and their connection with Anantharaman-Delaroche’s problem, as well as of our already existing work, they have also constructed examples of a nonfactorizable τ3 -Markov map on M3 (C), respectively, of a τ4 -Markov Schur multiplier on M4 (C) which is not factorizable. 5. The Noncommutative Rota Dilation Property The following was introduced in [16] (see Definition 10.2 therein): Definition 5.1. Let M be a von Neumann algebra equipped with a normalized (normal and faithful) trace τ . We say that a bounded operator T : M → M satisfies the Rota dilation property if there exists a von Neumann algebra N equipped with a normalized (normal and faithful) trace τ N , a normal unital faithful ∗-representation π : M → N which preserves the traces (i.e., τ N ◦ π = τ ), and a decreasing sequence (Nm )m≥1 of von Neumann subalgebras of N such that T m = Q ◦ Em ◦ π, m ≥ 1. Here Em denotes the canonical (trace-preserving) conditional expectation of N onto Nm , and Q : N → M is the conditional expectation associated to π , that is, Q := π ∗ = π −1 ◦ Eπ(M) , where Eπ(M) is the trace-preserving conditional expectation of N onto π(M). Remark 5.2. If T : M → M has the Rota dilation property, then T is completely positive, unital and trace-preserving. Since in the tracial setting condition (4) in Definition 1.1 is trivially satisfied, it follows that T is automatically a τ -Markov map. Moreover, since E1 (viewed as an operator from N into N ) can be written as E1 = j1∗ ◦ j1 , where j1 : N1 → N is the inclusion map, then T = Q ◦ E1 ◦ π = ( j1∗ ◦ π )∗ ◦ ( j1∗ ◦ π ).
(5.25)
576
U. Haagerup, M. Musat
Hence, T is positive as an operator on the pre-Hilbert space M with inner product x, y := τ (y ∗ x), x, y ∈ M. This also implies that T = T ∗ , where T ∗ : M → M is the adjoint of T in the sense of (1.1). (This observation is also stated in [16] (cf. Remark 10.3 therein)). Furthermore, equalities (5.25) show that the Rota dilation property implies factorizability of T , in view of Remark 1.4 (b). The following is a consequence of Theorem 6.6 in [2]: Theorem 5.3. If T : M → M is a factorizable τ -Markov map with T = T ∗ , then T 2 has the Rota dilation property. Proof. Since T is factorizable, Theorem 6.6 in [2] ensures the existence of a von Neumann algebra N with a normal faithful state ψ, a normal unital endomorphism β : N → N which is ψ-Markov and a normal unital ∗-homomorphism J0 : M → N which is (φ, ψ)-Markov such that, if we set Jn := β n ◦ J0 and E[n denotes the condi tional expectation of N onto its von Neumann subalgebra generated by k≥n Jk (M) for all n ≥ 0, while E0] is the conditional expectation of N onto J0 (M), then E0] ◦ Jn = J0 ◦ T n , n ≥ 1, ∗ n E[n ◦ J0 = Jn ◦ (T ) , n ≥ 1. T∗
(5.26)
It follows that J0 ◦ T n ◦ (T ∗ )n = E0] ◦ Jn ◦ (T ∗ )n = E0] ◦ E[n ◦ J0 , n ≥ 1. Since = T , we infer that (T 2 )n = T n ◦ (T ∗ )n = J0−1 ◦ E0] ◦ E[n ◦ J0 = J0∗ ◦ E[n ◦ J0 , n ≥ 1.
(5.27)
Observing that (E[n )n≥1 is a sequence of conditional expectations with decreasing ranges, (5.27) shows that T 2 has the Rota dilation property. Note that if M is abelian and T : M → M is factorizable, then, following the construction in [2], one can choose an abelian dilation N for T . Therefore, Theorem 5.3 is a noncommutative analogue of Rota’s classical dilation theorem for Markov operators. The next result shows that the factorizability condition cannot be removed from the hypothesis of Theorem 5.3, thus the Rota dilation theorem does not hold in general in the noncommutative setting. Theorem 5.4. There exists a τn -Markov map T : Mn (C) → Mn (C), for some n ≥ 1, such that T = T ∗ , but T 2 is not factorizable. In particular, T 2 does not have the Rota dilation property. To prove the theorem, we start with the following Lemma 5.5. Let n, d ∈ N and consider a1 , . . . , ad ∈ Mn (C) to be self-adjoint with d 2 i=1 ai = 1n . Set T (x) :=
d
ai xai , x ∈ Mn (C).
i=1
Suppose that the following conditions hold: (i) ai2 a j = a j ai2 , 1 ≤ i, j ≤ d. (ii) A := {ai a j : 1 ≤ i, j ≤ d} is linearly independent. 6 B is linearly independent, where (iii) B := ∪i=1 i
(5.28)
Factorization and Dilation Problems
577
B1 := {ai a j ak al : 1 ≤ i = j = k = l ≤ d}, B2 := {ai a j ak2 : 1 ≤ i = j = k = k ≤ d}, B3 := {ai3 a j : 1 ≤ i = j ≤ d}, B4 := {ai a 3j : 1 ≤ i = j ≤ d}, B5 := {ai2 a 2j : 1 ≤ i < j ≤ d}, B6 := {ai4 : 1 ≤ i ≤ d}. Furthermore, it is assumed that B is the disjoint union of the sets Bi , 1 ≤ i ≤ 6, and that the elements listed in each Bi are distinct. Assume further that (iv) d ≥ 5. Then T is a self-adjoint τn -Markov map, for which T 2 is not factorizable. Proof. We have T 2 x = i,d j=1 (ai a j )∗ x(ai a j ), for all x ∈ Mn (C). It is clear that T 2 is a τn -Markov map, for which Theorem 2.2 can be applied, due to condition (ii). Hence, if T 2 were factorizable, it would then follow that there exists a finite von Neumann algebra N with a normal faithful tracial state τ N , and a unitary u ∈ Mn (N ) such that T 2 x ⊗ 1 N = (id Mn (C) ⊗ τ N )(u ∗ (x ⊗ 1 N )u), x ∈ Mn (C). Moreover, since T 2 is self-adjoint, an easy argument shows that u can be chosen 0 u∗ ∈ to be self-adjoint. Namely, one can replace N by M2 (N ) and u by u˜ := u 0 M2 (Mn (N )) = Mn (M2 (N )). Moreover, Theorem 2.2 ensures that u is of the form u=
d
ai a j ⊗ vi j ,
(5.29)
τ N (vi∗j vkl ) = δik δ jl .
(5.30)
i, j=1
where vi j ∈ N , for all 1 ≤ i, j ≤ d, and
By (i), the elements vi j , 1 ≤ i, j ≤ d are uniquely determined from (5.29). Since, moreover, u = u ∗ , we deduce that vi∗j = v ji , 1 ≤ i, j ≤ d.
(5.31)
Now, by condition (i), we obtain the following set of relations for all i = j = k = i: ai a j ak2 = ai ak2 a j = ak2 ai a j ,
(5.32)
and, respectively, for all i = j, ai3 a j = ai a j ai2 , ai a 3j = a 2j ai a j , ai2 a 2j = ai a 2j ai = a 2j ai2 = a j ai2 a j .
(5.33)
These conditions imply that every matrix of the form ai a j ak al , 1 ≤ i, j, k, l ≤ d 6 B . Moreover, two elements of the form occurs precisely once in the set B = ∪i=1 i b = ai a j ak al , b = ai a j ak al , where (i, j, k, l) = (i , j , k , l ) are equal if and only if one of the four cases listed in (5.32) and (5.33) holds. Furthermore, since u 2 = u ∗ u = 1 Mn (N ) , we have 1 Mn (N ) =
d i, j,k,l
ai a j ak al ⊗ vi j vkl .
(5.34)
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U. Haagerup, M. Musat
By applying id Mn (C) ⊗ τ N on both sides of (5.34), we get 1n = i,d j,k,l=1 τ N (vi j vkl ) d ai a j ak al , and therefore 0 Mn (N ) = i, j,k,l=1 ai a j ak al ⊗ (vi j vkl − τ N (vi j vkl )1 N ). By (5.30) and (5.31), this can further be reduced to d
0 Mn (N ) =
ai a j ak al ⊗ (vi j vkl − δil δ jk 1 N ).
(5.35)
i, j,k,l=1
Using the remark following (5.33), Eq. (5.35) can be rewritten as 0 Mn (N ) = b∈B b⊗wb , where wb ∈ N . Since B is a linearly independent set, this implies that wb = 0 N , for all b ∈ B. Hence, if b ∈ B1 , i.e., b = ai a j ak al , where i = j = k = l, we infer that 0 N = wb = vi j vkl − δil δ jk 1 N , which implies that vi j vkl = 0 N , i = j = k = l.
(5.36)
Similarly, if b ∈ B6 , i.e., b = ai4 , for some 1 ≤ i ≤ d, then the same argument applies, and we obtain 0 N = wb = vii2 − δii2 1 N , i.e., vii2 = 1 N , 1 ≤ i ≤ d.
(5.37)
On the other hand, if b ∈ B2 , then by (5.33) it follows that b = ai a j ak2 = ai ak2 a j = 2 + v v + v2 v − ai a j ak2 , for some 1 ≤ i = j = k = i ≤ d, and therefore wb = vi j vkk ik k j kk i j (δik δ jk + δi j δkk + δk j δki )1 N . Hence 2 2 + vik vk j + vkk vi j = 0 N , 1 ≤ i = j = k = i ≤ d. vi j vkk
(5.38)
Similarly, using wb = 0 N for all b ∈ Bm , where 3 ≤ m ≤ 5, we obtain that the following relations hold for all 1 ≤ i = j ≤ d:
vii v j j
vii vi j + vi j vii = 0 N , vi j v j j + v j j vi j = 0 N , + vi j v ji + v j j vii + v ji vi j = 21 N .
(5.39) (5.40) (5.41)
vi∗j
Now, recall that = v ji , 1 ≤ i, j ≤ d, so by (5.37), we deduce that {vii , 1 ≤ i ≤ d} is a set of self-adjoint unitaries. Thus, by (5.41) we have vi j vi∗j + vi∗j vi j = 21 N − vii v j j − v j j vii ≤ 4, 1 ≤ i = j ≤ d, which implies that vi j ≤ 2, for all 1 ≤ i = j ≤ d. Now, for every 1 ≤ j ≤ d set p j := i= j s(vi∗j vi j ), where s(vi∗j vi j ) denotes the support projection of vi∗j vi j . By (5.36) it follows that p j and pk are orthogonal projections, whenever 1 ≤ j = k ≤ d, and hence d
τ N ( p j ) ≤ τ N (1) = 1.
(5.42)
j=1
On the other hand, by (5.30), τ N (vi∗j vi j ) = 1, for all 1 ≤ i, j ≤ d. Moreover, for i = j, vi∗j vi j ≤ vi∗j vi j p j ≤ 4 p j . Thus τ N ( p j ) ≥ (τ N (vi∗j vi j ))/4 = 1/4, for all 1 ≤ i = j ≤ d. This implies that d j=1 τ N ( p j ) ≥ d/4, and since d ≥ 5, this contradicts (5.42). The proof is complete.
Factorization and Dilation Problems
579
The condition d ≥ 5 is essential in the statement of Lemma 5.5 above, as it can be seen from the following remark. Remark 5.6. Assume that T : Mn (C) → is of the form (5.28), where d Mn (C) ai2 = 1n . If d ≤ 4, then T 2 is faca1 , . . . , ad ∈ Mn (C) are self-adjoint with i=1 torizable. Proof. We can assume without loss of generality that d = 4 (otherwise add zero-terms). Set 4
u :=
ai a j ⊗ (2ei j − δi j 14 ),
i, j=1
where (ei j )1≤i, j≤4 are the standard matrix units in M4 (C). Then u = u ∗ ∈ Mn (C) ⊗ M4 (C) = M4n (C). We first show that u is a unitary. We have u ∗ u = uu ∗ = u 2 =
4
ai a j ak al ⊗ (2ei j − δi j 14 )(2ekl − δkl 14 )
i, j,k,l=1
=
4
ai a j ak al ⊗ (4δ jk eil − 2δi j ekl − 2δkl ei j + δi j δkl 14 )
i, j,k,l=1
= 4s1 − 2s2 − 2s3 + s4 , 4 4 2 2 where s1 := i, j,l=1 ai a j al ⊗ eil = i,l=1 ai al ⊗ eil , s2 := i,k,l=1 ai ak al ⊗ 4 4 4 2 ekl = ak al ⊗ ekl , s3 := i, j,k=1 ai a j ak ⊗ ei j = i, j=1 ai a j ⊗ ei j and 4 k,l=1 2 2 d s4 := i, j=1 ai a j ⊗ 14 = 14n . We have repeatedly used the fact that i=1 ai2 = 1n . Hence s1 = s2 = s3 , and therefore 4
u ∗ u = uu ∗ = s4 = 14n . Next we prove that T 2 is factorizable by showing that E Mn (C)⊗14 (u ∗ (x ⊗ 14 )u) = T 2 x, x ∈ Mn (C). Since u = u ∗ , the left hand side above becomes E Mn (C)⊗14 (u ∗ (x ⊗ 14 )u) ⎞ ⎛ 4 = E Mn (C)⊗14 ⎝ ai a j ⊗ (2ei j − δi j 14 )(x ⊗ 14 )(ak al ⊗ (2ekl − δkl 14 ))⎠ i, j,k,l=1
=
4
ai a j xak al τ4 ((2ei j − δi j 1)(2ekl − δkl I ))
i, j,k,l=1
=
4
ai a j xa j ai = T 2 x,
i, j=1
wherein we have used the fact that τ4 ((2ei j − δi j 14 )(2ekl − δkl 14 )) = (4/4)δil δ jk − (2/4)δi j δkl − (2/4)δi j δkl + δi j δkl = δil δ jk . The proof is complete.
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Lemma 5.7. Let d ≥ 5, b1 , . . . , bd be self-adjoint matrices in Mm (C), and u 1 , . . . , u d be self-adjoint unitary matrices in Mr (C), where m and r are positive integers. Assume that the following conditions hold: d 2 (a) i=1 bi = 1m . (b) bi b j = b j bi , for all 1 ≤ i, j ≤ d. (c) bi b j bk bl = 0m , for all 1 ≤ i, j, k, l ≤ d. (d) For every 1 ≤ i = j ≤ d, the set {bi b j bk2 : 1 ≤ k ≤ d} is linearly independent in Mm (C). (e) The set {bi2 b2j : 1 ≤ i < j ≤ d} is linearly independent in Mm (C). (f) The set {1r } ∪ {u i u j : 1 ≤ i = j ≤ d} ∪ {u i u j u k u l : 1 ≤ i = j = k = l ≤ d} is linearly independent in Mr (C). Then ai := bi ⊗ u i , 1 ≤ i ≤ d are self-adjoint matrices in Mmr (C) = Mm (C) ⊗ Mr (C) d ai2 = 1mr , as well as the conditions (i)–(iv) in Lemma 5.5 with which satisfy i=1 n = mr . Proof. Note first that conditions (a) and (b), together with the fact that u i2 = 1r , for all 1 ≤ i ≤ d, imply that dj=1 a 2j = 1mr and ai2 a j = a j ai2 , 1 ≤ i, j ≤ d, i.e., condition (i) in Lemma 5.5 is satisfied. Further, the set A := {ai a j : 1 ≤ i, j ≤ d} is equal to {bi2 ⊗ 1r : 1 ≤ i ≤ d} ∪ {bi b j ⊗ u i u j : 1 ≤ i = j ≤ d}. By (e), the set {1r } ∪ {u i u j : 1 ≤ i = j ≤ d} is linearly independent. Hence A is linearly independent if and only if bi b j = 0m , whenever 1 ≤ i = j ≤ d. The linear independence of b12 , . . . , bd2 follows from (d), and by (c) we get that bi b j = 0m , for all 1 ≤ i, j ≤ d. This proves condition (ii) in Lemma 5.5. Next, consider the set B := B1 ∪ . . . ∪ B6 , where B1 , . . . , B6 are defined as in (iii) in the above mentioned lemma. Since u i2 = 1r , for all 1 ≤ i ≤ d, the sets B1 , . . . , B6 can be rewritten as: B1 = {bi b j bk bl ⊗ u i u j u k u l : 1 ≤ i = j = k = l ≤ d}, B2 = {bi b j bk2 ⊗ u i u j : 1 ≤ i = j = k = i ≤ d}, B3 = {bi3 b j ⊗ u i u j : 1 ≤ i = j ≤ d}, B4 = {bi b3j ⊗ u i u j : 1 ≤ i = j ≤ d}, B5 = {bi2 b2j ⊗ 1l : 1 ≤ i < j ≤ d} and B6 = {bi4 ⊗ 1l : 1 ≤ i ≤ d}. By (e), B is a linearly independent set if and only if the following three conditions hold: (1) bi b j bk bl = 0m , whenever 1 ≤ i = j = k = l ≤ d. (2) For every 1 ≤ i = j ≤ d, the set {bi b j bk2 : 1 ≤ k ≤ d, k = i, k = j} ∪ {bi3 b j : 1 ≤ i, j ≤ d} ∪ {bi b3j : 1 ≤ i, j ≤ d} is linearly independent. (3) The set {bi2 b2j : 1 ≤ i < j ≤ d} ∪ {bi4 : 1 ≤ i ≤ d} is linearly independent. Clearly, (c) implies (1), (e) implies (3), and by (b), condition (d) implies (2). Hence (iii) in Lemma 5.5 holds, and since d ≥ 5, condition (iv) holds, as well, thus completing the proof. Proof of Theorem 5.4. It remains to be proved that for d ≥ 5, there exist positive integers m, r and matrices b1 , . . . , bd ∈ Mm (C), u 1 , . . . , u d ∈ Mr (C) satisfying the hypotheses of Lemma 5.7. Let S d−1 = S(Rd ) denote the unit sphere in Rd , and let φ1 , . . . , φd : S d−1 → R be the coordinate functions. It is not difficult to check that these functions in C(S d−1 ) satisfy conditions (a) − (e) in Lemma 5.7. Conditions (a) − (c) are, indeed, obvious.
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2 . Hence, the linear independence of To prove (e), note that φd2 = 1 − φ12 − · · · − φd−1 the set {φi2 φ 2j : 1 ≤ i < j ≤ d} is equivalent to the linear independence of the set of polynomials
P := {xi2 x 2j : 1 ≤ i ≤ j ≤ d − 1} ∪ {xi2 : 1 ≤ i ≤ d − 1} ∪ {1} in C(B(Rd−1 )), where B(Rd−1 ) is the closed unit ball in Rd−1 . But P is clearly a linearly independent set, because if a polynomial in Rd−1 vanishes in a neighborhood of 0, then all its coefficients are 0. This shows that φ1 , . . . , φd satisfy (e). The same method gives that {φ12 , . . . , φd2 } is a linearly independent set, and since for 1 ≤ i = j ≤ d, the set {x ∈ S d−1 : φi (x)φ j (x) = 0} is dense in S d−1 , it follows that also condition (d) holds for φ1 , . . . , φd . Next we show that (a)−(d) hold for the restriction of (φ1 , . . . , φd ) to some finite subset of S d−1 . For this, assume that (e) fails for the restriction of (φ1 , . . . , φd ) to any finite subset F of S d−1 . Then, for each such F, we can find coefficients ciFj , 1 ≤ i ≤ j ≤ d, not all equal to zero, such that ciFj φi2 (x)φ 2j (x) = 0, x ∈ F. 1≤i≤ j≤d
Moreover, we can assume that 1≤i≤ j≤d |ciFj |2 = 1. Take now a weak∗ -limit point c = (ci j )1≤i≤ j≤d of the net ((ciFj )1≤i≤ j≤d ) F , where the finite subsets F ⊆ S d−1 are ordered by inclusion. Then ci j φi2 (x)φ 2j (x) = 0, x ∈ S d−1 , 1≤i≤ j≤d
and not all coefficients ci j above vanish. This contradicts the fact that φ1 , . . . , φd satisfy (e). Using this type of argument, it is easy to see that one can choose a finite subset F of S d−1 such that not only (e), but also (d) and (c) hold for the restrictions of φ1 , . . . , φd to F. Of course, conditions (a) and (b) also hold for these restrictions. Set now m := |F|, and let F = { p1 , . . . , pm }. Then the diagonal matrices bi := diag{φi ( p1 ), . . . , φi ( pm )}, 1 ≤ i ≤ d in Mm (C) satisfy (a) − (d). It remains to be proved that we can find self-adjoint unitaries u 1 , . . . , u d in some matrix algebra Mr (C) satisfying ( f ). For this, consider the free product group G := Z2 ∗. . .∗Z2 (d copies), and let g1 , . . . , gd be its generators. Then gi2 = 1, for all 1 ≤ i ≤ d and gi1 gi2 . . . gis = 0, whenever s is a positive integer and i 1 = i 2 = · · · = i s . Since G is residually finite (cf. [11]), by passing to a quotient of G we can find a finite group generated by γ1 , . . . , γd such that γi2 = 1, for all 1 ≤ i ≤ d, and γi1 γi2 . . . γis = 1, whenever 1 ≤ s ≤ d and i 1 = i 2 = · · · = i s . This implies that the group elements listed in the set {1} ∪ {γi γ j : 1 ≤ i = j ≤ d} ∪ {γi γ j γk γl : 1 ≤ i = j = k = l ≤ d} are all distinct. Set now r := || and let u 1 , . . . , u d be the ranges of γ1 , . . . , γd by the left regular representation λ of in B(l 2 ()) Mr (C). Then u 1 , . . . , u d are selfadjoint unitaries. Moreover, the set {λ (γ ) : γ ∈ } is linearly independent, because λ (γ )δe = δγ , where δγ ∈ l 2 () is defined by δγ (γ ) = 1, if γ = γ and δγ (γ ) = 0, else. Hence u 1 , . . . , u d satisfy ( f ) and the proof is complete.
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Remark 5.8. The above proof does not provide explicit numbers m and r for which b1 , . . . , bd and u 1 , . . . u d can be realized, but it is easy to find lower bounds. For d = 5, the set in (e) has 15 linearly independent elements, and since the bi ’s can be simultaneously diagonalized, it follows that m ≥ 15. Also, for d = 5, the set in ( f ) has 1 + 5 × 4 + 5 × 43 = 341 linearly independent elements in Mr (C). Hence r 2 ≥ 341, which implies that r ≥ 19. Altogether, we conclude that n := mr ≥ 15 × 19 = 285. We will end this section by giving a characterization of those τ M -Markov maps S : M → M which admit a Rota dilation. Theorem 5.9. Let M be a finite von Neumann algebra with normal, faithful, tracial state τ M , and let S : M → M be a linear operator. Then the following statements are equivalent: (i) S satisfies the Rota dilation property. (ii) S = T ∗ T , for some factorizable (τ M , τ N )-Markov map T : M → N taking values in a von Neumann algebra N with a normal, faithful, tracial state τ N . Proof. The implication (i) ⇒ (ii) follows immediately from Remark 5.2 (see (5.25) therein). We now prove that (ii) ⇒ (i). Suppose that there exists a factorizable (τ M , τ N )Markov map T : M → N , where N is a von Neumann algebra with a normal, faithful, tracial state τ N such that S = T ∗ T. Since T is factorizable, it follows by Remark 1.4 (a) that there exists a finite von Neumann algebra P with a normal, faithful tracial state τ P such that T = β ∗ ◦ α, where α : M → P and β : N → P are unital ∗-monomorphisms satisfying τ M = τ P ◦ α and τ N = τ P ◦ β. Consider now the von Neumann algebras M ⊕ N and P ⊕ P, equipped with the normal, faithful, tracial states defined by τ M⊕N := (τ M ⊕ τ N )/2 and τ P⊕P := (τ P ⊕ τ P )/2, respectively. Further, define an operator T˜ on M ⊕ N by T˜ (x, y) := (T ∗ y, T x), x ∈ M, y ∈ N . Then T˜ is a τ M⊕N -Markov map on M ⊕ N and T˜ ∗ = T˜ . Moreover, T˜ is factorizable, since T˜ = δ ∗ ◦ γ , where δ, γ : M ⊕ N → P ⊕ P are the ∗-monomorphisms given by γ (x, y) := (α(x), β(y)), respectively, δ(x, y) := (β(y), α(x)), x ∈ M, y ∈ N , and τ M⊕N = (τ P⊕P ) ◦ γ = (τ P⊕P ) ◦ δ. Hence, by Theorem 5.3, T˜ 2 has a Rota dilation, i.e., there exists a finite von Neumann algebra Q with a normal, faithful, tracial state τ Q , a unital ∗-monomorphism i : M ⊕ N → Q for which τ M⊕N = τ Q ◦ i, and a decreasing sequence (Q n )n≥1 of von Neumann subalgebras of Q such that T˜ 2n = i ∗ ◦ E Q n ◦ i, n ≥ 1, where E Q n is the unique τ Q -preserving normal conditional expectation of Q onto Q n . Note that T˜ 2n (x, y) = ((T ∗ T )n x, (T T ∗ )n y), (x, y) ∈ M ⊕ N . In particular, T˜ 2n (1 M , 0 N ) = (1 M , 0 N ). Set e := i((1 M , 0 N )). Then e is a projection in Q. We will show next that e ∈ Q n , for all n ≥ 1. For simplicity of notation, set
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z := (1 M , 0 N ) and w := 1 M⊕N − z. For all n ≥ 1, w ∗ T˜ 2n (z) = w ∗ z = 0 M⊕N , and therefore 0 = T˜ 2n (z), w L 2 (M⊕N ) = = = =
(i ∗ ◦ E Q n ◦ i)(z), w L 2 (M⊕N ) E Q n (i(z)), i(w) L 2 (Q) τ Q ((1 Q − e)E Q n (e)) τ Q ((1 Q − e)E Q n (1 Q − e)).
Since τ Q is faithful and E Q n (e) ≥ 0, it follows that E Q n (e) ∈ eQe. Similarly, we obtain that E Q n (1 − e) ∈ (1 Q − e)Q(1 Q − e). Since E Q n (e) − e = (1 Q − e) − E Q n (1 Q − e), we deduce that E Q n (e) − e ∈ eQe ∩ (1 Q − e)Q(1 Q − e) = {0 Q }, n ≥ 1, which proves the claim. Further, note that τ Q (e) = τ M⊕N ((1 N , 0 N )) = 1/2. Set R := eQe, τ R := 2(τ Q )| R , and for x ∈ M, let j (x) := i(x, 0 N ). Then it is easy to check that R is a von Neumann algebra with normal, faithful tracial state τ R and that the map j : M → R above defined is a unital ∗-monomorphism for which τ M = j ◦ τ R . Moreover, for all n ≥ 1, (T ∗ T )n = j ∗ ◦ E Rn ◦ j, n ≥ 1, where Rn := eQ n e, n ≥ 1, form a decreasing sequence of von Neumann subalgebras of R, and E Rn := (E Q n )| R is the unique τ R -preserving conditional expectation of R onto Rn . It follows by the definition that S = T ∗ T has a Rota dilation. Note that from the proof of Theorem 5.9 it follows right-away that in order for a linear map T : M → M to satisfy the Rota dilation property, it suffices that it satisfies the conditions set forth in Definition 5.1 for m = 1, only.
6. On the Asymptotic Quantum Birkhoff Conjecture In 1946 G. Birkhoff [5] proved that every doubly stochastic matrix is a convex combination of permutation matrices. Note that if one considers the abelian von Neumann algebra D := l ∞ ({1, 2, . . . , n}) with trace given by τ ({i}) = 1/n, 1 ≤ i ≤ n, then the positive unital trace-preserving maps on D are those linear operators on D which are given by doubly stochastic n × n matrices. Since every automorphism of D is given by a permutation of {1, 2, . . . , n}, this led naturally to the question whether Birkhoff’s classical result extends to the quantum setting. The statement that every completely positive, unital trace-preserving map T : (Mn (C), τn ) → (Mn (C), τn ) lies in conv(Aut(Mn (C))) turned out to be false for n ≥ 3. For the case n ≥ 4, this was shown by Kümmerer and Maasen (cf. [24]), while the case n = 3 was settled by Kümmerer in [23] (see Remark 4.8). In [25], Landau and Streater gave a more elementary proof of Kümmerer and Maasen’s result, and also constructed another counterexample to the quantum Birkhoff conjecture in the case n = 3.
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Recently, V. Paulsen brought to our attention the following asymptotic version of the quantum Birkhoff conjecture, listed as Problem 30 on R. Werner’s web page of open problems in quantum information theory (see [36]): The asymptotic quantum Birkhoff conjecture. Let n ≥ 1. If T : Mn (C) → Mn (C) is a τn -Markov map, then T satisfies the following asymptotic quantum Birkhoff property: k k T, conv(Aut( Mn (C))) = 0. (6.43) lim dcb k→∞
i=1
i=1
As mentioned In the introduction, this conjecture originates in joint work of A. Winter, J. A. Smolin and F. Verstraete. The main results obtained in [33] motivated its formulation. We would like to thank M.-B. Ruskai for kindly providing us with the Report of the workshop on Operator structures in quantum information theory that took place at BIRS, February 11–16, 2007, where A. Winter discussed the conjecture, as well as for pointing out related very recent work of C. Mendl and M. Wolf (cf. [27]). Using the existence of non-factorizable Markov maps, we prove the following: Theorem 6.1. For every n ≥ 3, there exist τn -Markov maps on Mn (C) which do not satisfy the asymptotic quantum Birkhoff property (6.43). Proof. We will show that any non-factorizable τn -Markov map on Mn (C) does not satisfy (6.43). Such maps do exist for every n ≥ 3, as was shown in Sect. 3. The key point in our argument is to prove that, given any τn -Markov map T : Mn (C) → Mn (C), n ≥ 3, its cb-distance to the set FM(Mn (C)) of factorizable Markov maps on Mn (C) does not decrease by applying successive tensor powers. More precisely, we will show that k k T, FM Mn (C) ≥ dcb (T, FM(Mn (C))), k ≥ 1. (6.44) dcb i=1
i=1
k k Then, since conv(Aut( i=1 Mn (C))) ⊂ FM i=1 Mn (C) , for all k ≥ 1 (as pointed out in (2.5)), the desired conclusion will follow immediately, using the fact that the set of factorizable maps on Mn (C) is closed in the norm-topology, cf. Remark 1.4 (b). (Note that in our concrete finite-dimensional setting, this latter fact can also be obtained directly from Theorem 2.2 using a simple ultraproduct argument.) Now, in order to prove (6.44), we show that given m, l ≥ 3, then for any τm -Markov map T on Mm (C) and any τl -Markov map S on Ml (C), we have dcb (T ⊗ S, FM(Mm (C) ⊗ Ml (C))) ≥ dcb (T, FM(Mm (C))).
(6.45)
Let ι : Mm (C) → Mm (C) ⊗ Ml (C) be defined by ι(x) := x ⊗ 1l , for all x ∈ Mm (C). Note that ι is a (τm , τm ⊗ τl )-Markov map and a ∗-monomorphism. Its adjoint map ι∗ : Mm (C) ⊗ Ml (C) → Mm (C) is given by ι∗ (z) = (1m ⊗ τl )(z), for all z ∈ Mm (C) ⊗ Ml (C). It is easily checked that ι∗ (T ⊗ S)ι = T . Since ιcb = ι∗ cb = 1, we then obtain dcb (T ⊗ S, FM(Mm (C) ⊗ Ml (C))) ≥ dcb (T, ι∗ FM(Mm (C) ⊗ Ml (C)))ι). (6.46) It follows from the definition of factorizability (cf. Definition 1.3) that ι∗ FM (Mm (C) ⊗ Ml (C))ι ⊂ FM(Mm (C)). Together with (6.46), this completes the proof of (6.45), which, in turn, yields (6.44).
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It is now a natural question whether every factorizable τn -Markov map on Mn (C) does satisfy the asymptotic quantum Birkhoff property, for all n ≥ 3. It appears that this question has a close connection to Connes’ embedding problem, which is known to be equivalent to a number of different fundamental problems in operator algebras (for references, see, e.g., Ozawa’s excellent survey paper [28]). Theorem 6.2. If for any n ≥ 3, every factorizable τn -Markov map on Mn (C) satisfies the asymptotic quantum Birkhoff property, then Connes’ embedding problem has a positive answer. Proof. Assume by contradiction that Connes’s embedding problem has a negative answer. Then, by Dykema and Juschenko’s results from [10] (as explained in Remark 3.6), there exists a positive integer n such that Gn \Fn = ∅. Choose B = (blt )1≤l,t≤n ∈ Gn \Fn . It follows that the associated Schur multiplier TB is factorizable. We will prove that TB does not satisfy the asymptotic quantum Birkhoff property. Suppose by contradiction that TB does satisfy (6.43). For every positive integer k, let ιk : Mn (C) → Mn k (C) be the map defined by ιk (x) := x ⊗ 1n ⊗ · · · ⊗ 1n , for all x ∈ Mn (C). Then TB = ι∗k ◦ T ⊗k ◦ ιk , for all k ≥ 1, and we deduce that lim dcb (TB , conv(ι∗k ◦ Aut(Mn k (C)) ◦ ιk )) = 0.
k→∞
For k ≥ 1, choose operators Tk ∈ conv(ι∗k ◦ Aut(Mn k (C)) ◦ ιk ) such that lim TB − Tk cb = 0.
k→∞
(6.47)
sk (k) ∗ (k) Each Tk is of the form Tk = i=1 ci ιk ◦ ad(u i ) ◦ ιk , for some positive integer sk , sk (k) (k) (k) ci = 1. unitaries u i ∈ U(n k ) and positive numbers ci , 1 ≤ i ≤ sk , with i=1 (k) (k) sk M k (C) := Ak . Equip Ak with the Set u k := (u 1 , . . . , u sk ). Then u k ∈ sk (k) i=1 n trace given by τ ((a1 , . . . , ask )) := i=1 ci τn k (ai ), for all (a1 , . . . , ask ) ∈ Ak . Finally, define jk : Mn (C) → Ak by jk (x) := (ιk (x), . . . , ιk (x)) (sk terms), for all x ∈ Mn (C). It sk (k) can be checked that the adjoint jk∗ of jk is given by jk∗ ((a1 , . . . , ask )) = ι∗k ( i=1 ci ai ), for all (a1 , . . . , ask ) ∈ Ak . Then Tk can be rewritten as Tk = jk∗ ◦ ad(u k ) ◦ jk .
(6.48)
Since Ak admits a τk -preserving embedding into the hyperfinite II1 -factor R, equipped with its trace τ R , we can view jk as a unital embedding of Mn (C) into R. Respectively, we can view u k as a unitary in R. By taking ultraproducts, and using (6.47) we obtain that TB = j ∗ ◦ ad(u) ◦ j, where u is a unitary in the ultrapower R ω of R and j : Mn (C) → R ω is a unital embedding. More precisely, if πω denotes the quotient map of l ∞ (R) onto R ω , then u = πω (u 1 , u 1 , . . .) ∈ R ω and the map j is given by j (x) = πω ( j1 (x), j2 (x), . . .), for all x ∈ Mn (C). Further, by using the identification R ω = j (Mn (C)) ⊗ ( j (Mn (C)) ∩ R ω ) Mn ( j (Mn (C)) ∩ R ω ), we obtain from the proof of Proposition 2.8 applied to the factorizable Schur multiplier TB that u = diag(v1 , . . . , vn ), where vi ∈ j (Mn (C)) ∩ R ω ⊂ R ω , for all 1 ≤ i ≤ n,
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and that the (l, t)th entry blt of B is given by blt = τ R ω (vl∗ vt ), for all 1 ≤ l, t ≤ n. Here τ R ω denotes the trace on R ω . By a standard ultraproduct argument, for every 1 ≤ i ≤ n, we can find a sequence (m) (1) (2) (vi )m≥1 of unitaries in R representing vi , i.e., vi = πω (vi , vi , . . .) ∈ R ω , and we conclude that (m) ∗ (m) = τ R ω (vl∗ vt ), 1 ≤ l, t ≤ n. vt (6.49) lim τ R vl ω
Now let ε > 0. We deduce from (6.49), that there exists a positive integer m such that (m) ∗ (m) − τ R ω (vl∗ vt ) < ε/2, 1 ≤ l, t ≤ n. vt τ R vl s.o.t
Using the fact that R = ∪k Dk , where D1 ⊆ D2 ⊆ · · · are unital finite dimensional factors, Dk M2k (C), it follows from the proof of Corollary 5.3.7 in Vol. I of [18] that every unitary operator in R can be approximated in · 2 -norm by unitaries from ∪∞ k=1 U(Dk ). Hence, we can find a positive integer k and w1 , w2 , . . . , wn ∈ U(Dk ) such that τ R (w ∗ wt ) − τ R ω (v ∗ vt ) < ε, 1 ≤ l, t ≤ n. l l This implies that the n × n matrix B belongs to the closure of Fn . Since Fn is already a closed set, this shows that B ∈ Fn , which contradicts the assumption on B. Therefore, the proof is complete. Remark 6.3. After this paper was submitted for publication we discovered that, in fact, there is no direct connection between the asymptotic quantum Birkhoff property and the Connes embedding problem. More precisely, for a large class of τn -Markov maps T on Mn (C), n ≥ 3, including the one in Example 3.3 above, T is factorizable (even through R ω ), but T does not satisfy (6.43). However, it turns out that Connes’ embedding problem has a positive answer if and only if the following equality holds for every n ≥ 3 and every factorizable τn -Markov map T on Mn (C): lim dcb (T ⊗ Sk , conv(Aut(Mn (C) ⊗ Mk (C)))) = 0,
k→∞
where Sk is the completely depolarizing channel on Mk (C), i.e., Sk (x) = τk (x)1k , for all x ∈ Mk (C). The proofs of the statements in this remark will appear in a forthcoming paper. 7. On the Best Constant in the Noncommutative Little Grothendieck Inequality Let O H (I ) denote Pisier’s operator Hilbert space based on l 2 (I ), for a given index set I . Further, let A be a C∗ -algebra and T : A → O H (I ) a completely bounded map. Then, by the refinement of the second part of Corollary 3.4 of [30] obtained in [14], there exist states f 1 , f 2 on A such that √ (7.50) T x ≤ 2 T cb f 1 (x x ∗ )1/4 f 2 (x ∗ x)1/4 , x ∈ A. Definition 7.1. For a completely bounded map T : A → O H (I ) we denote by C(T ) the smallest constant C > 0 for which there exist states f 1 , f 2 on A such that T x ≤ C f 1 (x x ∗ )1/4 f 2 (x ∗ x)1/4 , x ∈ A.
(7.51)
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The existence of a smallest constant C(T ) as above follows from a simple compactness argument using the fact that the set Q(A) := { f ∈ A∗+ : f ≤ 1} is w∗ -compact. From [30] we know that T cb ≤ C(T ). Hence, by (7.50), we infer that √ (7.52) T cb ≤ C(T ) ≤ 2 T cb . In the following we √ will prove that for suitable choices of A, I and T, C(T ) > T cb , i.e., the constant 2 in (7.50) cannot be reduced to 1. It would be interesting to know what is the best constant C0 in the noncommutative little Grothendieck inequality (7.50), i.e., what is the smallest constant C0 for which C(T ) ≤ C0 T cb , for arbitrary choices of A, I and T . Theorem 7.2. There exist linear maps T1 : M3 (C) → O H ({1, 2, 3}) and T2 : l ∞ ({1, . . . , 4}) → O H ({1, 2}) such that Ti cb < C(Ti ), i = 1, 2. In particular, the best constant C0 in the noncommutative little Grothendieck inequality (7.50) is strictly larger than 1. The key result that will be used in the proof of the above theorem is the following: Theorem 7.3. Let (A, τ ) be a finite dimensional (unital) C∗ -algebra with a faithful trace τ . Furthermore, let d be a positive integer and let a1 , . . . , ad be elements in A satisfying (i) τ (ai∗ a j ) = δi j , 1 ≤ i, j ≤ d, d d ∗ ∗ (ii) i=1 ai ai = i=1 ai ai = d1 A . Consider the map T : A → O H (d) := O H ({1 . . . , d}) given by (iii) T x := (τ (a1∗ x), . . . , τ (ad∗ x)), a ∈ A. Then C(T ) = 1. If, furthermore, (iv) d ≥ 2, (v) {ai∗ a j : 1 ≤ i, j ≤ d} is linearly independent, then T cb < 1. We will first prove a number of intermediate results. Lemma 7.4. Let A, τ, a1 , . . . , ad and T : A → O H (d) be as in Theorem 7.3 (i), (ii) and (iii). Then C(T ) = 1. Proof. By (i), a1 , . . . , ad is an orthonormal set in L 2 (A, τ ). Moreover, τ (a1∗ x), . . . , τ (ad∗ x) are the coordinates of the orthogonal projection P of x ∈ A = L 2 (A, τ ) onto E := span{a1 . . . , ad } with respect to the basis {a1 , . . . , ad }. Thus T x =
d
|τ (ai∗ x)|2 = P x2 ≤ x2 = τ (x ∗ x)1/2 = τ (x ∗ x)1/4 τ (x ∗ x)1/4 .
i=1
Hence C(T ) ≤ 1. Conversely, assume that T x ≤ K f 1 (x x ∗ )1/4 f 2 (x ∗ x)1/4 , x ∈ A,
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for a constant K > 0 and states f 1 , f 2 on A. By (i) and (ii), it follows that for every 1 ≤ i ≤ d, T ai = (0, . . . , 1, 0, . . . , 0), where the number 1 is at the i th coordinate. Therefore, 1 = T ai 2 ≤ K 2 f 1 (ai ai ∗ )1/2 f 2 (ai ∗ ai )1/2 . By the Cauchy-Schwarz inequality and (ii), we infer that d=
d
T ai 2 ≤ K 2
2
i=1
f 1 (ai ai∗ )
1/2 2
i=1
1/2 f 2 (ai∗ ai )
= K 2 d.
i=1
Hence K ≥ 1, which proves that C(T ) ≥ 1, and the conclusion follows.
Lemma 7.5. Let A, τ, a1 , . . . , ad be as in Theorem 7.3 (i) and (ii), set r := dim(A) and choose ad+1 , . . . , ar such that the set {a1 , . . . , ar } is an orthonormal basis for A. Let B be a unital C∗ -algebra. Then every element u ∈ A ⊗ B has a unique representation of the form u=
r
ai ⊗ u i ,
(7.53)
i=1
where u i ∈ B, 1 ≤ i ≤ d. Moreover, if u is unitary, then r
u i∗ u i =
i=1
r
u i u i∗ = 1 B .
(7.54)
i=1
Proof. Existence and uniqueness of u 1 , . . . , u r ∈ B in (7.53) is obvious. To prove (7.54), note that if u ∗ u = uu ∗ = 1 A⊗B , then by (i), 1 B = (τ A ⊗ id B )(u ∗ u) =
d
τ A (ai∗ a j )u i∗ u j =
i, j=1
d
u i∗ u i ,
i=1
and similarly, by the trace property of τ A and (i), 1 B = (τ A ⊗ id B )(uu ∗ ) =
d i, j=1
which completes the proof.
τ A (ai a ∗j )u i u ∗j =
d
u i u i∗ ,
i=1
Lemma 7.6. Let A, τ, a1 , . . . , ad and T be as in Theorem 7.3 (i), (ii) and (iii), and choose ad+1 , . . . , ar as in Lemma 7.5. Assume further that T cb = 1. Then the following statements hold:
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589
(a) For every ε > 0, there exists a unital C∗ -algebra B(ε), a Hilbert B(ε)-bimodule H(ε), elements u 1 , . . . , u r ∈ B(ε) and unit vectors ξ, η ∈ H(ε) such that the operator u := ri=1 ai ⊗ u i ∈ A ⊗ B(ε) is unitary and the following inequalities are satisfied: d i=1 d
r
u i ξ − ηu i 2 +
(u i ξ 2 + ηu i 2 ) < ε,
(7.55)
(u i∗ η2 + ξ u i∗ 2 ) < ε.
(7.56)
i=d+1 r
u i∗ η − ξ u i∗ 2 +
i=1
i=d+1
(b) There exist a unital C∗ -algebra B, a Hilbert bimodule H,elements u 1 , . . . , u r ∈ B and unit vectors ξ, η ∈ H such that the operator u := ri=1 ai ⊗ u i ∈ A ⊗ B is unitary and the following identities are satisfied: u i ξ = ηu i , u i∗ η = ξ u i∗ , 1 ≤ i ≤ d, u i ξ = ηu i = u i∗ η = ξ u i∗ = 0, d + 1 ≤ i ≤ r.
(7.57) (7.58)
Proof. Let ε > 0. Since T cb = 1, there exists a positive integer n such that T ⊗ id Mn (C) cb > 1 − ε/4. Since dim(A) < ∞, the unit ball of Mn (A) = A ⊗ Mn (C) is the convex hull of its unitary operators. Hence there exists a unitary operator u ∈ A⊗ Mn (C) such that (T ⊗ id Mn (C) )(u) Mn (O H ) > 1 − ε/4. r By Lemma 7.5, u has the form u = i=1 ai ⊗ u i for a unique set of elements u 1 , . . . , u r ∈ Mn (C) satisfying r
u i∗ u i =
i=1
By condition (iii) in Theorem 7.3, T (ai ) =
r
u i u i∗ = 1n .
(7.59)
i=1
ei , 1 ≤ i ≤ d 0, d + 1 ≤ i ≤ r,
where ei is the i th vector in the standard unit vector basis of l 2 ({1, . . . , d}) = O H (d). Hence (T ⊗ id Mn (C) )(u) =
d
ei ⊗ u i .
i=1
It then follows (cf. [29]) that d u i ⊗ u¯i i=1
2 d = ei ⊗ u i i=1 Mn (C)⊗Mn (C) Mn (O H ) = (T ⊗ id Mn (C) )(u) M (O H ) n ε 2 ε > 1− >1− . 4 2
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U. Haagerup, M. Musat
We can identify Mn (C) ⊗ Mn (C) isometrically with the bounded operators on H S(n) = L 2 (Mn (C), Tr), where Tr denotes the standard non-normalized trace on Mn (C), by let¯ = aξ b∗ , for all ξ ∈ H S(n). In particular, ting a ⊗ b¯ correspond to L a Rb∗ , i.e., (a ⊗ b)ξ d r (u i ⊗ u i )ξ = u i ξ u i∗ , ξ ∈ H S(n). i=1
i=1
Let H S(n)sa denote the self-adjoint part of H S(n). Since H S(n) = H S(n)sa +i H S(n)sa , d u ⊗ u i ∈ B(H S(n)) leaves H S(n)sa invariant, one checks easily and the operator i=1 d i d that the norm of i=1 u i ⊗ u i is the same as the norm of i=1 u i ⊗ u i restricted to H S(n)sa . By compactness of the unit ball in H S(n)sa we deduce that there exist vectors ξ, η ∈ H S(n)sa such that ξ 2 = η2 = 1,
(7.60)
satisfying, moreover, d d ε ∗ = u ξ u , η u ⊗ u i i i > 1 − . i 2 i=1 i=1 H S(n) d ∗ Furthermore, since is a real number, we infer that i=1 u i ξ u i , η H S(n)
d u i ξ, ηu i =
d
i=1
u i ξ u i∗ , η
i=1
H S(n)
ε >1− , 2
(7.61)
by replacing η with −η, if needed. Hence, d
d u i ξ 22 + ηu i 22 − u i ξ − ηu i 22 = 2Re u i ξ, ηu i H S(n)
i=1
i=1 d =2 u i ξ u i∗ , η H S(n) > 2 − ε. (7.62) i=1
Moreover, by (7.59) and (7.60), r
u i ξ 22 +
i=1
r
ηu i 22 = 2.
i=1
Subtracting (7.63) from (7.62), we get d
u i ξ − ηu i 22 +
i=1
i=d+1
Furthermore, since ξ = d i=1
r u i ξ 22 + ηu i 22 < ε.
ξ∗
and η = η∗ , we get by taking adjoints that
u i∗ ξ − ηu i∗ 22 +
r u i ξ 22 + ηu i 22 < ε. i=d+1
This proves (a) with B(ε) = Mn (C) and H(ε) = H S(n).
(7.63)
Factorization and Dilation Problems
591
We now prove (b). Given a positive integer n, let εn := 1/n 2 and set Bn := B(εn ) and Hn := H(εn ). Then Bn is a unital C∗ -algebra and Hn is a Bn -Hilbert bimodule. (n) (n) , η(n) ∈ H(n) Moreover, there exist elements u (n) 1 , . . . , u r ∈ Bn and unit vectors ξ (n) such that the operator u (n) := ri=1 ai ⊗ u i ∈ A ⊗ Bn is unitary and the following inequalities hold: (n) ∗ (n) (n) (n) ∗ 1≤i ≤d ηn − ξn u i < 1/n, u i ξn − ηn u i < 1/n, u i respectively, (n) ∗ (n) ∗ (n) ξn < 1/n, ηn u i < 1/n, u i ηn < 1/n, ui (n) ∗ d + 1 ≤ i ≤ r. ξn u i < 1/n, Now (b) follows from (a) by a standard ultraproduct construction (see, e.g., [12]). Lemma 7.7. Let A, τ, a1 , . . . , ad and T be as in Theorem 7.3 (i), (ii) and (iii), and assume that T cb = 1. Then the following statements hold: (a) There exist a finite von Neumann algebra N with a normal, faithful tracial state τ N , a projection p ∈ N and elements v1 , . . . , vd ∈ (1 − p)N p such that the operator d v := i=1 ai ⊗ vi ∈ A ⊗ N is a partial isometry satisfying v ∗ v = 1 A ⊗ p, vv ∗ = 1 A ⊗ (1 − p). (b) There exists a finite von Neumann algebra P with a normal, faithful tracial state τ P d and elements w1 , . . . , wd ∈ P such that the operator w := i=1 ai ⊗ wi ∈ A ⊗ P is unitary. Proof. Let r := dim(A) and choose a1 , . . . , ar ∈ A as in Lemma 7.5. Further, let B, H, u 1 , . . . , u r and ξ, η be as in Lemma 7.6 (b). In particular, the operator u := r i=1 ai ⊗ u i is a unitary in A ⊗ B. Note that M2 (H) is an M2 (B)-bimodule by standard matrix multiplication, and M2 (H) is a Hilbert space with norm 2 σ11 σ12 2 = σi j 2 , σi j ∈ H, 1 ≤ i, j ≤ 2. σ21 σ22 M (H ) 2
i, j=1
ξ 0 0 0 ∈ M2 (H) and si := ∈ M2 (B), for all 1 ≤ i ≤ r . Then Set ζ := 0 η ui 0 ζ = 1, and by (7.57) and (7.58) it follows that
√1 2
Further, set e :=
1B 0
si ζ = ζ si , si∗ ζ = ζ si∗ , 1 ≤ i ≤ d, si ζ = si∗ ζ = 0, d + 1 ≤ i ≤ r. 0 ∈ M2 (B). Then e is a projection and by (7.54), 0 r i=1
si∗ si = e,
r i=1
si si∗ = 1 M2 (B) − e.
(7.64) (7.65)
(7.66)
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Next, set s :=
r
i=1 ai
⊗ si . Since u is a unitary operator, it follows that
s ∗ s = 1 A ⊗ e, ss ∗ = 1 A ⊗ (1 M2 (B) − e).
(7.67)
Let C denote the C∗ -algebra generated by s1 , . . . , sd in M2 (B). By (7.66), both e and 1 M2 (B) − e belong to C, and hence 1 M2 (B) ∈ C. Moreover, by (7.64) and (7.65), cζ = ζ c, c ∈ C.
(7.68)
Define now a state φ on C by φ(c) := cζ, ζ M2 (H) , for all c ∈ C. Note that φ is tracial, since φ(c∗ c) = cζ, cζ M2 (H) = ζ c, ζ c M2 (H) = ζ cc∗ , ζ M2 (H) = cc∗ ζ, ζ M2 (H) = φ(cc∗ ), c ∈ C. Let (πφ , Hφ , ξφ ) be the GNS-representation of C with respect to φ. Then N := πφ (C) is a finite von Neumann algebra with normal, faithful tracial state τ N given by τ N (x) := xξφ , ξφ Hφ , for all x ∈ N . Moreover, φ(c) = τ N (πφ (c)), for all c ∈ C. Now set vi := πφ (si ), for all 1 ≤ i ≤ r . By (7.65), it follows that τ N (vi∗ vi ) = φ(si∗ si ) = 0, for all d + 1 ≤ i ≤ r , and hence vi = 0, d + 1 ≤ i ≤ r.
(7.69)
Next set p := πφ (e). Then p is a projection in N . By (7.66) and (7.69) we infer that d
vi∗ vi = p,
i=1
d
vi vi∗ = 1 N − p.
(7.70)
i=1
d Finally, set v := (id A ⊗ πφ )(s) = ri=1 ai ⊗ vi = i=1 ai ⊗ vi . Then, by (7.67) it ∗ ∗ follows that v v = 1 A ⊗ p and vv = 1 A ⊗ (1 N − p). This proves part (a). To prove (b), we note first that by (7.70), p and 1 N − p have the same central valued trace, and therefore they are equivalent (as projections in N ) (see, e.g., [18] (Vol. II, Chap. 8)). In particular, τ N ( p) = τ N (1 N − p) = 1/2. Choose now t ∈ N such that t ∗ t = p and tt ∗ = 1 N − p, and set wi := t ∗ vi , for all 1 ≤ i ≤ d. Then d i=1
and the operator w :=
d
i=1 ai
wi∗ wi =
d
wi wi∗ = p,
i=1
⊗ wi satisfies
w ∗ w = ww ∗ = 1 A ⊗ p. Hence (b) follows from (a) by setting P := pN p and defining τ P (x) := 2τ N (x), for all x ∈ P. Proof of Theorem 7.3. By (7.52) and Lemma 7.4 we have that T cb ≤ C(T ) = 1. If we assume by contradiction that T cb = 1, then by Lemma 7.7 (b), there exist a finite von Neumann algebra P with a normal, faithful tracial state τ P and elements d ai ⊗ wi is unitary in A ⊗ P. By the w1 , . . . , wd ∈ P such that the operator w := i=1 hypothesis of Theorem 7.3 (cf. (iv) and (v)), the additional assumptions that d ≥ 2 and the set {ai∗ a j : 1 ≤ i, j ≤ d} is linearly independent do hold. Therefore, we can proceed
Factorization and Dilation Problems
593
almost as in the proof of Corollary 2.3. Namely, we have w ∗ w = Therefore, using (ii) we deduce that ∗
0 A⊗P = w w − 1 A ⊗ 1 P =
d i, j=1
ai∗ a j
d
∗ i, j=1 ai a j
⊗ wi∗ w j .
1 ∗ ⊗ wi w j − δi j 1 P . d
Hence, by (v) we conclude that wi∗ wi =
1 δi j 1 P , 1 ≤ i, j ≤ d. d
√ √ This implies that dw1 and dw2 are two isometries in the finite von Neumann algebra P, having orthogonal ranges. This is impossible. Therefore T cb < 1 and the proof is complete. Proof of Theorem 7.2. (1) Consider A = M3 (C), τ = τ3 , d = 3, and let a1 , a2 , a3 ∈ M3 (C) be given by ! ⎛0 0 0 ⎞ ! ⎛ 0 0 1⎞ ! ⎛ 0 −1 0 ⎞ 3⎝ 3⎝ 3⎝ 0 0 −1 ⎠ , a2 = 0 0 0 ⎠ , a3 = 1 0 0⎠. a1 = 2 0 1 0 2 −1 0 0 2 0 0 0 Define T1 : M3 (C) → O H (3) by T1 (x) := (τ (a1∗ x), τ (a2∗ x), τ (a3∗ x)), x ∈ M3 (C). (2) Consider A = l ∞ ({1, . . . , 4}), τ (c) = (c1 + · · · + c4 )/4, c = (c1 , . . . , c4 ) ∈ A, d = 2, and let a1 , a2 ∈ A be given by √ √ √ √ √ √ √ √ √ √ a1 := ( 2, 2/ 3, 2/ 3, 2/ 3), a2 := (0, 2/ 3, (2/ 3)ω, (2/ 3)ω), ¯ where ω := ei2π /3 , and ω¯ is its complex conjugate. Define T2 : l ∞ ({1, . . . , 4}) → O H (2) by T2 (x) := (τ (a1∗ x), τ (a2∗ x)), x ∈ l ∞ ({1, . . . , 4}). In each of the cases (1) and (2) it is easily checked that conditions (i), (ii), (iv) and (v) in the hypothesis of Theorem 7.3 are verified. Hence, the maps Ti (defined above according to condition (iii) of Theorem 7.3) will satisfy C(Ti ) = 1 > Ti cb , for i = 1, 2. Note that a1 , a2 , a3 in case (1) are scalar multiples of the matrices considered in Example 3.1, while a1 , a2 in case (2) correspond, up to a scalar factor, to the diagonal 4 × 4 matrices in Example 3.2 with s = 1/3. References 1. Accardi, L., Cecchini, C.: Conditional expectations in von Neumann algebras and a theorem of Takesaki. J. Funct. Analysis 45, 245–273 (1982) 2. Anantharaman-Delaroche, C.: On ergodic theorems for free group actions on noncommutative spaces. Probab. Theory Rel. Fields 135, 520–546 (2006) 3. Berg, C., Christensen, J.P.R., Ressel, P.: Harmonic Analysis on Semigroups. New York: Springer Verlag, 1984 4. Bhat, B.V.R., Skalski, A.: Personal communication. 2008 5. Birkhoff, G.: Three observations on linear algebra. Univ. Nac. Tucuan, Revista A 5, 147–151 (1946)
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6. Brattelli, O., Robinson, D. W.: Operator Algebras and Quantum Statistical Mechanics II. New YorK: Springer Verlag, 1981 7. Choi, M.-D.: Completely positive linear maps on complex matrices. Lin. Alg. Applic. 10, 285–290 (1975) 8. Christensen, J.P.R., Vesterstrøm, J.: A note on extreme positive definite matrices. Math. Ann 244, 65–68 (1979) 9. Dabrowski, Y.: A non-commutative path space approach to stationary free stochastic differential equations. http://arxiv.org/abs/1006.4351v2 [math.OA], 2010 10. Dykema, K., Juschenko, K.: Matrices of unitary moments. http://arxiv.org/abs/0901.0288v2 [math.OA], 2009 11. Gruenberg, K.W.: Residual properties of infinite soluble groups. Proc. London Math. Soc. 3(7), 29–62 (1957) 12. Haagerup, U.: The injective factors of type IIIλ , 0 < λ < 1. Pacific J. Math. 137(2), 265–310 (1989) 13. Haagerup, U., Itoh, T.: Grothendieck type norms for bilinear forms on C ∗ -algebras. J. Operator Theory 34, 263–283 (1995) 14. Haagerup, U., Musat, M.: The Effros-Ruan conjecture for bilinear maps on C ∗ -algebras. Invent. Math. 174(1), 139–163 (2008) 15. Junge, M., Mei, T.: Noncommutative Riesz transforms–a probabilistic approach. Amer. J. Math. 132(3), 611–680 (2010) 16. Junge, M., Le Merdy, C., Xu, Q.: H ∞ functional calculus and square functions on noncommutative L p -spaces. Astérisque No. 305, parts: Soc. Math. de France, 2006 17. Junge, M., Ricard, E., Shlyakhtenko, D.: In preparation 18. Kadison, R.V., Ringrose, J.R.: Fundamentals of the Theory of Operator Algebras I, II, London-Newyork: Academic Press, 1986 19. Kirchberg, E.: On nonsemisplit extensions, tensor products and exactness of group C ∗ -algebras. Invent. Math. 112(3), 449–489 (1993) 20. Koestler, C.: Personal communication, 2008 21. Kümmerer, B.: Markov dilations on the 2 × 2 matrices. In: Operator algebras and their connections with topology and ergodic theory (Busteni, 1983), Lect. Notes Math. 1132, Berlin: Springer, 1985, pp. 312–323 22. Kümmerer, B.: Markov dilations on W∗ -algebras. J. Funct. Analysis 63, 139–177 (1985) 23. Kümmerer, B.: Construction and structure of Markov dilations on W∗ -algebras, Habilitationsschrift, Tübingen, 1986 24. Kümmerer, B., Maasen, H.: The essentially commutative dilations of dynamical semigroups on Mn . Commun. Math. Phys. 109, 1–22 (1987) 25. Landau, L.J., Streater, R.F.: On Birkhoff’s theorem for doubly stochastic completely positive maps on matrix algebras. Lin. Alg. Applic. 193, 107–127 (1993) 26. Mei, T.: Tent spaces associated with semigroups of operators. J. Funct. Anal 255, 3356–3406 (2008) 27. Mendl, C.B., Wolf, M.M.: Unital quantum channels - convex structure and revivals of Birkhoff’s theorem. Commun. Math. Phys. 289, 1057–1096 (2009) 28. Ozawa, N.: About the QWEP conjecture. Internat. J. Math. 15, 501–530 (2004) 29. Pisier, G.: The Operator Hilbert Space O H , Complex Interpolation and Tensor Norms. Mem. Amer. Math. Soc. Number 585, Vol. 122, Providence, RI: Amer. math. Soc., 1996 30. Pisier, G., Shlyakhtenko, D.: Grothendieck’s theorem for operator spaces. Invent. Math. 150, 185–217 (2002) 31. Ricard, E.: A Markov dilation for self-adjoint Schur multipliers. Proc. Amer. Math. Soc. 136(12), 4365–4372 (2008) 32. Rota, G.-C.: An Alternierende Verfahren for general positive operators. Bull. Amer. Math. Soc. 68 (1962), 95–102; Commun. Math. Phys. 106(1), 91–103, 1986 33. Smolin, J.A., Verstraete, F., Winter, A.: Entanglement of assistance and multipartite state distillation. Phys. Rev. A 72, 052317 (2005) 34. Takesaki, M.: Conditional expectations in von Neumann algebras. J. Funct. Anal. 9, 306–321 (1972) 35. Takesaki, M.: Theory of Operator Algebras II. New-York: Springer-Verlag, 2003 36. Werner’s R.: open problems in quantum information theory web site http://www.imaph.tu-bs.de/qi/ problems/problems.html., also available at http://arxiv.org/abs/quant-ph/0504166v1, 2005 Communicated by M.B. Ruskai
Commun. Math. Phys. 303, 595–612 (2011) Digital Object Identifier (DOI) 10.1007/s00220-011-1229-6
Communications in
Mathematical Physics
Quivers, Quasi-Quantum Groups and Finite Tensor Categories Hua-Lin Huang1 , Gongxiang Liu2 , Yu Ye3 1 School of Mathematics, Shandong University, Jinan, Shandong 250100, China. E-mail:
[email protected] 2 Department of Mathematics, Nanjing University, Nanjing, Jiangsu 210093, China. E-mail:
[email protected] 3 Department of Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, China.
E-mail:
[email protected] Received: 18 June 2009 / Accepted: 1 November 2010 Published online: 27 March 2011 – © Springer-Verlag 2011
Abstract: We study finite quasi-quantum groups in their quiver setting developed recently by the first author. We obtain a classification of finite-dimensional pointed Majid algebras of finite corepresentation type, or equivalently a classification of elementary quasi-Hopf algebras of finite representation type, over the field of complex numbers. By the Tannaka-Krein duality principle, this provides a classification of the finite tensor categories in which every simple object has Frobenius-Perron dimension 1 and there are finitely many indecomposable objects up to isomorphism. Some interesting information of these finite tensor categories is given by making use of the quiver representation theory.
1. Introduction This paper is devoted to the classification of finite quasi-quantum groups and the associated representation theory, whence the classification of finite tensor categories [19], within the quiver setting developed recently in [32,33]. The notion of quasi-Hopf algebras was introduced by Drinfeld [14] in connection with the Knizhnik-Zamolodchikov system of equations from conformal field theory. The definition of quasi-Hopf algebras is not selfdual, so there is a dual notion which is called the Majid algebra after Shnider-Sternberg [45]. In accordance with Drinfeld’s philosophy of quantum groups [13], we understand both of these mutually dual algebraic structures in the framework of quasi-quantum groups. We focus on finite-dimensional pointed Majid algebras, or equivalently elementary quasi-Hopf algebras. Within this restriction, we can take full advantage of the quiver techniques to tackle the problems of classification and representation theory. Recall that by pointed it is meant that the simple subcoalgebras of the underlying coalgebras are one-dimensional. Dually, by elementary it is meant that the underlying algebras are finite-dimensional and their simple modules are one-dimensional.
596
H.-L. Huang, G. Liu, Y. Ye
Quivers are oriented diagrams consisting of vertices and arrows. Due to Gabriel [22,23], in the early 1970’s quivers and their representations became widespread first in the representation theory of associative algebras. Nowadays these notions show up in various areas of mathematics and physics. What we rely on is still their combinatorial behavior and hence very handy applications in the study of algebraic structures and representation theory. In connection with Hopf algebras and quantum groups, Hopf quivers [12] and covering quivers [29] were introduced, see also [7,10,11,28,46] for related works. For the quiver setting of the broader class of quasi-quantum groups, it turns out that there is nothing new other than the Hopf quivers. In principle, it is shown in [32,33] that pointed Majid algebras and elementary quasi-Hopf algebras can be constructed on Hopf quivers exhaustively with a help of the projective representation theory of groups and a proper deformation theory. According to the well-known Tannaka-Krein duality principle (see for instance [6,42]), finite quasi-quantum groups are deeply related to finite tensor categories. More precisely, the representation categories of finite-dimensional quasi-Hopf algebras and the corepresentation categories of finite-dimensional Majid algebras are finite tensor categories; conversely, finite tensor categories with some mild conditions are obtained in this way. In recent years finite tensor categories and finite quasi-quantum groups have been intensively studied by Etingof, Gelaki, Nikshych, Ostrik, and many other authors. In [18], the fusion categories, that is, the semisimple finite tensor categories, are investigated in depth and a number of general properties are obtained. A systematic study of not necessarily semisimple finite tensor categories initiated in [19], and some classification results were obtained in [15–17,25] through concrete constructions of elementary quasi-Hopf algebras. The aim of this paper is to classify finite-dimensional pointed Majid algebras of finite corepresentation type, or equivalently elementary quasi-Hopf algebras of finite representation type. The obvious motivation for this, from a mathematical point of view, is two-fold. On the one hand, the algebras of finite representation type are very important in the representation theory of associative algebras. The study of such algebras has been a central theme in the area all along. Given an interesting class of algebras, one is always tempted to classify those of finite representation type. On the other hand, their associated representation categories are the finite tensor categories in which there are only finitely many non-isomorphic indecomposable objects. Such tensor categories of finite type are simplest after the semisimple ones. It is natural to pay prior attention to these finite tensor categories with very good finiteness property. We also expect that the obtained classification results are useful in physics and mathematical physics. The logarithmic conformal field theories [20,21,30] naturally motivated the research of non-semisimple finite tensor categories [19]. It seems of interest to investigate the associated rational logarithmic conformal field theories which lead to our tensor categories of finite type (see [20,21]). Another possible application could be the use of tensor categories in topological quantum computation (see [1]). A complete classification result like ours might provide a suitable palate for model building. Among the tensor categories, the braided ones (see [37,42]) are probably of most interest. It turns out that the methods and obtained results of the present paper can be applied to this class of tensor categories. In particular, the first two authors give in a subsequent paper [35] a classification of braided tensor categories of finite type. Moreover, with help of the “quasi” version of the quantum double theory developed by Majid [41,43], one can obtain an interesting class of quasitriangular quasi-Hopf algebras, whence braided tensor
Quasi-Quantum Groups and Tensor Categories
597
categories, from those studied in this paper. It is reasonable to view these quasitriangular quasi-Hopf algebras as the “quasi” analogue of the small quantum sl2 [40] and consider their possible applications in quantum integrable systems, low-dimensional topology and nonassociative geometry. The paper is organized as follows. In Sect. 2 we recall the quiver setting of quasiquantum groups. Sect. 3 is devoted to the classification of finite-dimensional pointed Majid algebras of corepresentation type. In Sect. 4 we investigate finite tensor categories by making use of quiver representation theory. Throughout, we work over the field C of complex numbers for simplicity. For the convenience of the exposition, we deal mainly with pointed Majid algebras and mention briefly the situation of elementary quasi-Hopf algebras. About general background knowledge, the reader is referred to [2,3] for quivers and representation theory of algebras, to [37,42,45] for quasi-quantum groups, and to [4,6] for tensor categories. 2. Hopf Quivers and Quasi-Quantum Groups In this section we recall the quiver framework of quasi-quantum groups for the convenience of the reader. 2.1. Hopf quivers. A quiver is a quadruple Q = (Q 0 , Q 1 , s, t), where Q 0 is the set of vertices, Q 1 is the set of arrows, and s, t : Q 1 −→ Q 0 are two maps assigning respectively the source and the target for each arrow. A path of length l ≥ 1 in the quiver Q is a finitely ordered sequence of l arrows al · · · a1 such that s(ai+1 ) = t (ai ) for 1 ≤ i ≤ l − 1. By convention a vertex is said to be a trivial path of length 0. For a quiver Q, the associated path coalgebra CQ is the C-space spanned by the set of paths with counit and comultiplication maps defined by ε(g) = 1, (g) = g ⊗ g for each g ∈ Q 0 , and for each nontrivial path p = an · · · a1 , ε( p) = 0, (an · · · a1 ) = p ⊗ s(a1 ) +
n−1
an · · · ai+1 ⊗ ai · · · a1 + t (an ) ⊗ p.
i=1
The length of paths gives a natural gradation to the path coalgebra. Let Q n denote the set of paths of length n in Q, then CQ = ⊕n≥0 CQ n and (CQ n ) ⊆ ⊕n=i+ j CQ i ⊗ CQ j . Clearly CQ is pointed with the set of group-like G(CQ) = Q 0 , and has the following coradical filtration: CQ 0 ⊆ CQ 0 ⊕ CQ 1 ⊆ CQ 0 ⊕ CQ 1 ⊕ CQ 2 ⊆ · · · . Hence CQ is coradically graded. The path coalgebras can be presented as cotensor coalgebras, so they are cofree in the category of pointed coalgebras and enjoy a universal mapping property. According to [12], a quiver Q is said to be a Hopf quiver if the corresponding path coalgebra CQ admits a graded Hopf algebra structure. Hopf quivers can be determined by ramification data of groups. Let G be a group and denote its set of conjugacy classes by C. A ramification datum R of the group G is a formal sum C∈C RC C of conjugacy classes with coefficients in N = {0, 1, 2, · · · }. The corresponding Hopf quiver Q = Q(G, R) is defined as follows: the set of vertices Q 0 is G, and for each x ∈ G and c ∈ C, there are RC arrows going from x to cx. It is clear by definition that Q(G, R)
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is connected if and only if the the set {c ∈ C|C ∈ C with RC = 0} generates the group G. For a given Hopf quiver Q, the set of graded Hopf structures on CQ is in one-to-one correspondence with the set of CQ 0 -Hopf bimodule structures on CQ 1 . 2.2. Quasi-quantum groups. We recall explicitly the definitions about Majid algebras only. Those about quasi-Hopf algebras can be written out in a dual manner. A dual quasi-bialgebra, or Majid bialgebra for short, is a coalgebra (H, , ε) equipped with a compatible quasi-algebra structure. Namely, there exist two coalgebra homomorphisms M : H ⊗ H −→ H, a ⊗ b → ab, μ : k −→ H, λ → λ1 H and a convolution-invertible map : H ⊗3 −→ k called reassociator, such that for all a, b, c, d ∈ H the following equalities hold: a1 (b1 c1 )(a2 , b2 , c2 ) = (a1 , b1 , c1 )(a2 b2 )c2 , 1 H a = a = a1 H , (a1 , b1 , c1 d1 )(a2 b2 , c2 , d2 ) = (b1 , c1 , d1 )(a1 , b2 c2 , d2 )(a2 , b3 , c3 ), (a, 1 H , b) = ε(a)ε(b).
(2.1) (2.2) (2.3) (2.4)
Here and below we use the Sweedler sigma notation (a) = a1 ⊗ a2 for the coproduct and a1 ⊗ a2 ⊗ · · · ⊗ an+1 for the result of the n-iterated application of on a.H is called a Majid algebra if, moreover, there exist a coalgebra antimorphism S : H −→ H and two functionals α, β : H −→ k such that for all a ∈ H, S(a1 )α(a2 )a3 = α(a)1 H , a1 β(a2 )S(a3 ) = β(a)1 H , (a1 , S(a3 ), a5 )β(a2 )α(a4 ) = −1 (S(a1 ), a3 , S(a5 ))α(a2 )β(a4 ) = ε(a).
(2.5) (2.6)
Assume that H is a Majid algebra with reassociator . A linear space M is called an H -Majid bimodule, if M is an H -bicomodule with structure maps (δ L , δ R ), and there are two H -bicomodule morphisms, ρ L : H ⊗ M −→ M, h ⊗ m → h.m, ρ R : M ⊗ H −→ M, m ⊗ h → m.h, such that for all g, h ∈ H, m ∈ M, the following equalities hold: 1 H .m = g1 .(h 1 .m 0 )(g2 , h 2 , m 1 ) = m 0 .(g1 h 1 )(m 1 , g2 , h 2 ) = g1 .(m 0 .h 1 )(g2 , m 1 , h 2 ) =
m = m.1 H , (g1 , h 1 , m −1 )(g2 h 2 ).m 0 , (m −1 , g1 , h 1 )(m 0 .g2 ).h 2 , (g1 , m −1 , h 1 )(g2 .m 0 ).h 2 ,
(2.7) (2.8) (2.9) (2.10)
where we use the Sweedler notation δ L (m) = m −1 ⊗ m 0 , δ R (m) = m 0 ⊗ m 1 for comodule structure maps. A Majid algebra H is said to be pointed, if the underlying coalgebra is pointed. For a given pointed Majid algebra (H, , ε, M, μ, , S, α, β), let {Hn }n≥0 be its coradical
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filtration, and gr H = H0 ⊕ H1 /H0 ⊕ H2 /H1 ⊕· · · the corresponding coradically graded coalgebra. Then gr H has an induced graded Majid algebra structure. The corresponding ¯ c) ¯ c¯ ∈ gr H unless they graded reassociator gr satisfies gr (a, ¯ b, ¯ = 0 for all a, ¯ b, all lie in H0 . Similar condition holds for gr α and gr β. In particular, H0 is a sub-Majid algebra and turns out to be the group algebra kG of the group G = G(H ), the set of group-like elements of H. 2.3. Quiver setting for quasi-quantum groups. It is shown in [32] that the path coalgebra CQ admits a graded Majid algebra structure if and only if the quiver Q is a Hopf quiver. Moreover, for a given Hopf quiver Q = Q(G, R), if we fix a Majid algebra structure on CQ 0 = (CG, ) with quasi-antipode (S, α, β), then the set of graded Majid algebra structures on CQ with CQ 0 = (CG, , S, α, β) is in one-to-one correspondence with the set of (CG, )-Majid bimodule structures on CQ 1 . According to [14], by transforming the quasi-antipode (S, α, β) via convolution invertible functionals in HomC (CG, C), one obtains all the graded Majid algebra structures on CQ with CQ 0 = (CG, ) and an arbitrary quasi-antipode. The category of Majid bimodules over a general group with an arbitrary 3-cocycle is characterized in [33] by the admissible collections of projective representations. Let G be a group and a 3-cocycle on G. Denote by C the set of conjugacy classes of G and ˜ C be a by Z C the centralizer of one of the elements, say gC , in the class C ∈ C. Let 2-cocycle on Z C defined by ˜ C (e, f ) =
(e, f, gC )(e f, f −1 , e−1 )(e, f gC , f −1 ) . (e f gC , f −1 , e−1 )(e, f, f −1 )
(2.11)
Then the category of (CG, )-Majid bimodules is equivalent to the product of categories ˜ ˜ ˜ C∈C (CZ C , C )−rep, where (CZ C , C )−rep is the category of projective C -repre˜ sentations, or equivalently the left module category of the twisted group algebra CC Z C (see [36]). Thanks to the Gabriel type theorem in [32], for an arbitrary pointed Majid algebra H, its graded version gr H can be realized uniquely as a large sub-Majid algebra of some graded Majid algebra structure on a Hopf quiver. By “large” it is meant the sub-Majid algebra contains the set of vertices and arrows of the Hopf quiver. Therefore, in principle all pointed Majid algebras are able to be constructed on Hopf quivers. The classification project can be carried out in two steps. The first step is to classify large sub-Majid algebras of those on path coalgebras. This gives a classification of graded pointed Majid algebras. The second step is to perform a suitable deformation process to get general pointed Majid algebras from the graded ones. 2.4. Multiplication formula for quiver Majid algebras. In order to construct graded Majid algebras on Hopf quivers, we need to compute the product of paths. It is shown in [32] that the multiplication formula can be given via the quantum shuffle product [44]. Suppose that Q is a Hopf quiver with a necessary CQ 0 -Majid bimodule structure on CQ 1 . Let p ∈ Q l be a path. An n-thin split of it is a sequence ( p1 , . . . , pn ) of vertices and arrows such that the concatenation pn · · · p1 is exactly p. These n-thin splits are in one-to-one correspondence with the n-sequences of (n − l) 0’s and l 1’s. Denote the set of such sequences by Dln . Clearly |Dln | = nl . For d = (d1 , · · · , dn ) ∈ Dln , the corresponding n-thin split is written as dp = ((dp)1 , · · · , (dp)n ), in which (dp)i is a
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vertex if di = 0 and an arrow if di = 1. Let α = am · · · a1 and β = bn · · · b1 be paths of m+n and d¯ ∈ D m+n be the complement sequence length m and n respectively. Let d ∈ Dm n which is obtained from d by replacing each 0 by 1 and each 1 by 0. Define an element ¯ m+n ] · · · [(dα)1 .(dβ) ¯ 1] (α · β)d = [(dα)m+n .(dβ) ¯ i ] is understood as the action of the CQ 0 -Majid bimodule in CQ m+n , where [(dα)i .(dβ) on CQ 1 and these terms in different brackets are put together by the cotensor product, or equivalently concatenation. In terms of these notations, the formula of the product of α and β is given as follows: α·β = (α · β)d . (2.12) m+n d∈Dm
We should remark that, for general Majid algebras, the product is not associative. So the order must be concerned for the product of more than two terms.
Convention 2.1. For an arbitrary path p and an integer n ≥ 3, let p n denote the prodn−2
uct (· · · ( p · p) · · · · ) · p calculating from the left side. For consistency, when n < 3, we
still use the notation p n although there is no risk of associative problem. Similarly we
use the notation p n for the product calculating from the right side. 3. Pointed Majid Algebras of Finite Corepresentation Type In this section we give an explicit classification of finite-dimensional graded pointed Majid algebras of finite corepresentation type. We start by fixing the Hopf quivers on which such Majid algebras live. Then we calculate all the possible Majid bimodules for the construction. Finally we provide the classification by making use of quiver techniques.
3.1. Determination of Hopf quivers. Recall that a finite-dimensional algebra is defined to be of finite representation type if the number of the isomorphism classes of indecomposable finite-dimensional modules is finite. A finite-dimensional coalgebra C is said to be of finite corepresentation type if the dual algebra C ∗ is of finite representation type. Since finite-dimensional coalgebras and finite-dimensional algebras are dual to each other, we apply the known results of algebras to the coalgebra setting without explanation. It is well-known that the module category of a finite-dimensional elementary algebra, or the comodule category of a finite-dimensional pointed coalgebra, can be visualized as the representation category of the corresponding bound quiver (see [2,9]). Hence the quiver presentation of algebras or coalgebras can provide important information for their representation or corepresentation type. This is the starting point of our classification. Lemma 3.1. Let C = C be a finite-dimensional pointed coalgebra and assume that its bound quiver Q is a connected Hopf quiver. Then C is of finite corepresentation type if and only if its bound quiver Q = Q(Zn , g), where Zn = g|g n = 1 for some integer n ≥ 1.
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Proof. “⇒” We denote the vertex group of the quiver Q by G and assume that Q = Q(G, R) for some ramification datum R = C∈C RC C of G. Note that a Hopf quiver is very symmetric, its shape is completely determined once the arrows with source 1, the unit element of G, are known. For our Hopf quiver Q, we claim that the number of arrows starting from 1 is 1. Assume otherwise there are at least two arrows with source 1. Then the Hopf quiver Q either contains the Kronecker quiver /
◦
/◦
as a sub-quiver, or contains a sub-quiver of the form go
1
/h
for some g = h. For the latter case, by the definition of Hopf quivers, in Q there are also arrows as the following: · · · ←− h −2 g 2 −→ h −1 g 2 ←− h −1 g −→ g. Since G is a finite group, there is a positive integer N such that h −N +1 g N or h −N g N = 1. It follows that the quiver Q contains a sub-quiver whose underlying graph is a cycle and whose paths are of length less than 2. By the Gabriel type theorem for pointed coalgebras (see e.g. [8,9]), the bound quiver Q of C contains either of the above two quivers as a sub-quiver, hence the corepresentation category of C contains either of their representation categories as a sub-category. According to Gabriel’s famous classification of quivers of finite representation type [22], both of these two quivers admit infinitely many finite-dimensional indecomposable representations. On the other hand, since the quiver Q is assumed to be connected, so the number of arrows starting from 1 can only be 1. Assume that 1 −→ g is the unique arrow of Q with source 1. By the definition of Hopf quiver, the element g itself must constitute a conjugacy class and generate the group G since Q is connected. Therefore, such a quiver can only be of the following form: jjj41TTTTTTT jjjj TTTT j j j TTTT jj j j j TT* j g ··· ··· o g n−1 o where the set of vertices {1, g, · · · · · · , g n−1 } constitutes a cyclic group Zn of order n and Q = Q(Zn , g). “⇐” Assume that Q = Q(Zn , g) is the bound quiver of C, then C is a large subd−1 coalgebra of CQ. Denote CQ(d) := ⊕i=0 CQ i , the d-truncated sub-coalgebra. Since C is finite-dimensional, C is a sub-coalgebra of CQ(d) for some d. It is well-known that CQ(d) is of finite corepresentation type [2,3] and thus so is C. Assume that M is a finite-dimensional pointed Majid algebra and the corresponding Hopf quiver is Q. To avoid the trivial case, we assume that Q contains at least one arrow. This excludes the situation for M being cosemisimple. According to the quasi-Hopf analogue of the Cartier-Gabriel decomposition theorem in [32], we may assume without loss of generality that the quiver Q is connected. This is equivalent to saying that the underlying coalgebra of M is connected.
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Corollary 3.2. Keep the previous assumption. If M is of finite corepresentation type then the set of group-like G(M) = Zn and Q = Q(Zn , g) for some integer n ≥ 2. Proof. By Lemma 3.1, Q = Q(Zn , g) for some n ≥ 1. We claim the case n = 1 would not occur. Otherwise, the quiver Q consists of one vertex and one loop. By the multiplication formula (2.12), it is easy to see that the sub-Majid algebra generated by the loop is the shuffle algebra in one variable (see e.g. [37]) which is of course infinite-dimensional. By the Gabriel type theorem, this must be contained in gr M. This implies that gr M, hence M, is infinite-dimensional. Contradiction. 3.2. Twisted group algebras. From now on we let Z n denote the Hopf quiver Q(Zn , g), which is known as the basic cycle of length n. In order to classify graded Majid algebras on Z n we need to classify (CZn , )-Majid bimodule structures on CZ 1n for an arbitrary 3-cocycle on Zn , which can be reduced to the classification of one-dimensional modules over some twisted group algebra Cσ Zn by [33]. Firstly we recall the nontrivial 3-cocycles on Zn . It is well-known that H 3 (Zn , C∗ ) ∼ = Zn , so there are n mutually non-cohomologous 3-cocycles. We give a list after Lemmas 3.3 and 3.4 of [25]. Let q be a primitive root of unity of order n. For any integer i ∈ N, we denote by i the remainder of division of i by n. A list of 3-cocycles on Zn are
s (g i , g j , g k ) = qsi( j+k−( j+k) )/n
(3.1)
for all 0 ≤ s, j, k ≤ n − 1. Obviously, s is trivial (i.e., cohomologous to a 3-coboundary) if and only if s = 0. For a 3-cocycle s , we define a 2-cocycle σs on Zn by (2.11) as follows: σs (g i , g j ) :=
s (g i , g j , g)s (g i+ j , g − j , g −i )s (g i , g j+1 , g − j ) s (g i+ j+1 , g − j , g −i )s (g i , g j , g − j )
for all 0 ≤ i, j ≤ n − 1. Consider the associated twisted group algebra Cσs Zn . We denote the multiplication in Cσs Zn by “*”. Thus g ∗ g = σs (g, g)g 2 =
s (g, g, g)s (g 2 , g n−1 , g n−1 )s (g, g 2 , g n−1 ) 2 g = q−s g 2 . s (g 3 , g n−1 , g n−1 )s (g, g, g n−1 )
i
Denote g ∗ · · · ∗ g by g ∗i , then we have Lemma 3.3. In Cσs Zn , we have g i = q(i−1)s g ∗i for 1 ≤ i ≤ n. Proof. Induction on i, g ∗ g i = q(i−1)s g ∗(i+1) = σs (g, g i )g i+1 s (g, g i , g)s (g i+1 , g n−i , g n−1 )s (g, g i+1 , g n−i ) i+1 g = s (g i+2 , g n−i , g n−1 )s (g, g i , g n−i ) = q−s g i+1 . This implies that g i+1 = qis g ∗(i+1) .
(3.2)
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Now we consider one-dimensional modules of Cσs Zn . Let the one-dimensional space V = CX be a Cσs Zn -module with action given by g X = λX
(3.3)
for some λ ∈ C. Then we have Lemma 3.4. λn = qs . Proof. Indeed, λn X = g ∗n X = q−(n−1)s g n X = qs X .
So λ = q s for some q which is an n th root of q. Thus q is a primitive root of unity of order n 2 when s = 0. Therefore when s = 0, the set of one-dimensional Cσs Zn modules is in one-to-one correspondence with the set of n th roots of q. When s = 0, we have Cσs Zn = CZn and the set of one-dimensional Zn -module is in one-to-one correspondence with the set of n th roots of unity. 3.3. Computation of Majid bimodules. Here and below, let X i denote the arrow g i−1 −→ g i of the Hopf quiver Z n for 1 ≤ i ≤ n. For convenience the subscript of X i is read modulo n in some circumstances. By Theorem 3.3 of [33], the (CZn , s )Majid bimodule structures on CZ 1n can be obtained by extending the Cσs Zn -module structures on the one-dimensional space CX 1 . Recall that, for an arbitrary group G and a 3-cocycle , a (kG, )-Majid bimodule M is simultaneously a kG-bicomodule and a quasi kG-bimodule such that the quasi-module structure maps are kG-bicomodule morphisms. Assume that M = g,h∈G g M h is the decomposition into isotypic components, where g
M h = {m ∈ M | δ L (m) = g ⊗ m, δ R (m) = m ⊗ h}.
Here we use (δ L , δ R ) to denote the bicomodule structure maps. Now, the quasi-associativity of the quasi-actions given in (2.8)–(2.10) can be simplified in the following form: (e, f, g) (e f ).m, (e, f, h) (h, e, f ) m.(e f ), (m.e). f = (g, e, f ) (e, h, f ) (e.m). f = e.(m. f ), (e, g, f ) e.( f.m) =
(3.4) (3.5) (3.6)
for all e, f, g, h ∈ G and m ∈ g M h . These equalities will be used freely. A (kG, )-Majid bimodule can be associated to an admissible collection of projective modules as follows. We still let 1 denote the unit element, and let C denote the set of conjugacy classes of the group G. For each C ∈ C, let Z C denote the centralizer of one ˜ C the corresponding 2-cocycle ˜ g(C) on ZC as defined in element in C, say g(C), and g(C) 1 ˜ (2.11), and MC = M the C -representation of Z C given by h m = (h.m).h −1 , ∀ h ∈ Z C , m ∈ MC .
(3.7)
Then (MC )C∈C is called the corresponding admissible collections of projective representations of M. Conversely, given an admissible collection of projective representations,
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one can extend it by a twisted version of induced representation to a Majid bimodule, see [33] for detail. This provides the category equivalence mentioned previously in Subsect. 2.3. Now let’s get back to our situation of the cyclic group Zn . Let q be an n th root of q and fix an Cσs Zn -action on CX 1 by g X 1 = q s X 1.
(3.8)
We extend this Cσs Zn -module to an (CZn , s )-Majid bimodule on CZ 1n . The bicomod-
ule structure is defined according to the quiver structure, namely, for 1 ≤ i ≤ n, δ L (X i ) = g i ⊗ X i , δ R (X i ) = X i ⊗ g i−1 .
(3.9)
For the quasi-bimodule, there is no harm to assume that g.X i = X i+1 for 1 ≤ i ≤ n − 1. With this, we have Lemma 3.5. The following equations: g.X i = X i+1 (1 ≤ i ≤ n − 1), g.X n = qs X 1 , X i .g = q−s q −s X i+1 (1 ≤ i ≤ n),
(3.10) (3.11) (3.12)
define a quasi-CZn -bimodule on CZ 1n and make it a (CZn , s )-Majid bimodule together with the bicomodule structure defined by (3.9). Proof. Inductively, we have X i = g i−1 .X 1 ,
for 1 ≤ i ≤ n.
Thus g.X n = g.(g n−1 .X 1 ) =
s (g, g n−1 , g) n g .X 1 = qs X 1 . s (g, g n−1 , 1)
We proceed to determine the right quasi-action. On the one hand, g n−1 X 1 = q(n−2)s g ∗n−1 X 1 = q−2s q (n−1)s X 1 = q−s q −s X 1 . On the other hand, by the relation between “” and the quasi-actions “.” given by (3.7), we have g n−1 X 1 = (g n−1 .X 1 ).g = X n .g. Thus X n .g = q−s q −s X 1 . Now assume that X n−1 .g = cX n for some c ∈ C. Thus X n .g = (g.X n−1 ).g =
s (g, g n−2 , g) g.(X n−1 .g) = cq−s g.X n = cX 1 s (g, g n−1 , g)
and X n .g = q−s q −s X 1 . Therefore, c = q−s q −s . Inductively, assume that X i .g = q−s q −s X i+1 for some i ≤ n − 1. Suppose that X i−1 .g = cX i for some c ∈ C, then q−s q −s X i+1 = X i .g = (g.X i−1 ).g = g.(X i−1 .g) = cg.X i = cX i+1 . Thus c = q−s q −s . It is straightforward to verify that the quasi-bimodule structure maps are bicomodule morphisms. This completes the proof.
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3.4. Graded Majid algebras on Z n . In this subsection we calculate the graded Majid algebra on the quiver Z n associated to the (CZn , )-Majid bimodule given in Lemma 3.5. Firstly we need to fix some notations. For any ∈ C, define l =1 + + · · · + l−1 (l+m)! and l! = 1 · · · l. The Gaussian binomial coefficient is defined by l+m l := l! m! . In the path coalgebra CZ n , let pil denote the path starting from g i with length l. The index “i” is read modulo n when there is no risk of confusion. We keep the (CZn , s )Majid bimodule on CZ 1n as in Lemma 3.5 and consider the associated graded Majid algebra as given in [32]. Let CZ n (s, q) denote the resulting Majid algebra on the path coalgebra CZ n and “.” denote its multiplication. Note that the quasi-antipode (S, α, β) of CZ n (s, q) satisfies S(g) = g −1 , α(g) = 1, β(g) = 1/s (g, g −1 , g) for all g ∈ Zn . Lemma 3.6. For any natural number l, we have
X 1l = l!q−s q −s p0l ,
X 1l = qsl (l−l )/n l!q−s q −s p0l .
Proof. By induction on l. Firstly, we have X 1 · X 1 = [g.X 1 ][X 1 .1] + [X 1 .g][1.X 1 ] = (1 + q−s q −s )X 2 X 1 = 2q−s q −s X 2 X 1 .
Assume l = an + i with 0 ≤ i ≤ n − 1 and X 1l−1 = (l − 1)!q−s q −s p0l−1 , then
X 1l = X 1l−1 · X 1 = (l − 1)!q−s q −s p0l−1 · X 1 = (l − 1)!q−s q −s ([g i .X 1 ][X i .1] · · · [X 1 .1] + · · · + [X i .g] · · · [X 1 .g][1.X 1 ]) = (l − 1)!q−s q −s (1 + q−s q −s + · · · + q−(l−1)s q −(l−1)s ) p0l = l!q−s q −s p0l . Similarly, for l = an + i as above we have
X 1l+1 = X 1 · X 1l = qsia l!q−s q −s X 1 · p0l = qsia l!q−s q −s ((q−s q −s )i + · · · (q−s q −s )1 + 1 +qs n q−s q −s + · · · + qsa n q−s q −s ) p0l+1 = qsia l!q−s q −s qsa (l + 1)q−s q −s p0l+1 = qs(i+1)a (l + 1)!q−s q −s p0l+1 . Lemma 3.7. For all 0 ≤ i, j ≤ n − 1 and all non-negative integers a, b, ⎧ ⎨ −(a+1)(i+ j)s (a+b)n+(i+ j)
X1 if i + j > n − 1 q bn+ j an+i X1 · X1 = ⎩ −a(i+ j)s (a+b)n+(i+ j) X1 if i + j ≤ n − 1. q
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bn+ j
Proof. Consider the multiplication of X 1an+i−1 , X 1 and X 1 . By the quasi-associativity axiom (2.1) of Majid algebras, we have the following equality:
bn+ j
X 1an+i · X 1
=
(1, 1, 1) bn+ j+1 an+i−1 X · X . 1 (g i−1 , g, g j ) 1
By this, it follows that the coefficient is trivial unless j = n − 1. Thus if i + j > n − 1, by making use of the previous equation iteratively we have
bn+ j
X 1an+i · X 1
an+(i+ j+1−n)
= X1
· X 1bn+n−1
= q−(i+ j)s X 1
an+(i+ j−n)
(b+1)n
· X1
(a+b)n+ j
= q−a(i+ j)s X 1i · X 1 = q−a(i+ j)s X 1
(b+1)n+ j
= q−(i+ j)s X 1(a−1)n+i · X 1 = ···
i+ j+1−n
= q(−a−1)(i+ j)s X 1
· X 1(a+b)n+n−1
(a+b+1)n
i+ j−n
· X1
(a+b)n+(i+ j)
= q−(a+1)(i+ j)s X 1
.
The case of i + j ≤ n − 1 can be proved in the same manner. With these preparations, now we can give the product formula. Proposition 3.8. For all non-negative integers l, m, we have l m sl (m−m )/n l + m p0 · p0 = q p0l+m . l q−s q −s Proof. Assume that l = an + i, m = bn + j with 0 ≤ i, j ≤ n − 1. We only prove the formula when i + j ≤ n − 1 since the case i + j > n − 1 can be proved similarly. Indeed, by Lemmas 3.6 and 3.7, we have p0l · p0m = =
1 l!q−s q −s
1 qs jb m!
q−s q −s
q−s jb q−a(i+ j)s l!q−s q −s m!q−s q −s
X 1l · X 1m
X 1l+m
q−s jb q−a(i+ j)s s(i+ j)(a+b) q (l + m)!q−s q −s p0l+m l!q−s q −s m!q−s q −s l +m = qsib p0l+m . l −s −s q q =
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Observe that
g i · p0l = qsi(l−l )/n pil ,
p0l · g i = q−sil q −sil pil .
(3.13)
Then the previous product formula can be extended to general paths as follows. We leave the proof to the interested reader. Corollary 3.9. For all 0 ≤ i, j ≤ n − 1 and all non-negative integers l, m, we have in CZ n (s, q) the following multiplication formula: l +m l+m pil · p mj = q−s jl q −s jl qs(i+l )[m+ j−(m+ j) ]/n pi+ j . l −s −s q q We remark that the set {CZ n (0, q)|q n = 1} ∪ {CZ n (s, q)|1 ≤ s ≤ n − 1, q is a primitive root of unity of order n 2 } gives an explicit classification of graded Majid algebras on the path coalgebra CZ n by neglecting the minor difference of the quasi-antipodes [14]. In particular, when s = 0, the set {CZ n (0, q)|q n = 1} gives a classification of graded Hopf algebras on CZ n , which recovers the result in [10]; when s = 0, the set {CZ n (s, q)|1 ≤ s ≤ n − 1, q is a primitive root of unity of order n 2 } gives a classification of non-trivial graded Majid algebras (i.e., not gauge equivalent to Hopf algebras) on CZ n by [33]. 3.5. Finite sub-Majid algebra on Z n . In this subsection we investigate the possible finite-dimensional graded large sub-Majid algebras of CZ n (s, q). Proposition 3.10. There is a unique finite-dimensional graded large sub-Majid algebra of CZ n (s, q). In addition, such unique Majid algebra is generated by g and X 1 . Proof. By definition, the smallest graded large sub Majid algebra of CZ n (s, q) is the one generated by the set of vertices and arrows, which can be given by g and X 1 clearly. Assume that the multiplicative order of q−s q −s is d. Then by Lemmas 3.5 and 3.6, the quasi-algebra generated by g and X 1 has {g i · X 1l |0 ≤ i ≤ n − 1, 0 ≤ l ≤ d − 1} as a basis. Thus by (3.13) the set { pil |0 ≤ i ≤ n − 1, 0 ≤ l ≤ d − 1} is also a basis. Clearly the space spanned by these paths is closed under the coproduct of the path coalgebra, and also closed under the counit and the quasi-antipode of CZ n (s, q). Therefore it is an nd-dimensional graded large sub-Majid algebra. We denote this graded Majid algebra by M(n, s, q). It remains to prove that any graded large sub-Majid algebra of CZ n (s, q) is infinitedimensional if it strictly contains M(n, s, q). Assume that M is such a Majid algebra. Then consider a nontrivial homogeneous space of degree l ≥ d in M. Note that such a space must be spanned by some paths of length l. Since there is only one path of length l with fixed source and target, it follows by the axioms of coalgebras that there are paths of length l that lie in H. Then by (3.13) all the paths of length l lie in H. Consider the coproduct of these paths, it follows that all the paths of length d must lie in H. Now by
making use of Proposition 3.8 with induction, we have ( p0d )m = m! p0md . It follows that p0md ∈ H for all m ≥ 0. That means H must be infinite-dimensional. We are done. Now we can conclude that the set {M(n, 0, q)|q n = 1} ∪ {M(n, s, q)|1 ≤ s ≤ n−1, q is a primitive root of unity of order n 2 } provides a complete classification of the finite-dimensional graded large sub Majid algebra on the Hopf quiver Z n . Note that the
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set {M(n, 0, q)|q n = 1} are the usual Hopf algebras, called generalized Taft algebras in [34]. This gives a classification of finite-dimensional graded Hopf algebras on the quiver Z n . While the set {M(n, s, q)|1 ≤ s ≤ n − 1, q is a primitive root of unity of order n 2 } gives a classification of finite-dimensional graded non-trivial Majid algebras on Z n . It is also worthy to remark that the underlying coalgebras of these Majid algebras are truncated subcoalgebras of CZ n , namely they have a basis consisting of all the paths of length smaller than some fixed integer d ≥ 2. In particular, they are monomial in the sense of [7]. The set {M(n, 0, q)|q n = 1} ∪ {M(n, s, q)|1 ≤ s ≤ n − 1, q is a primitive root of unity of order n 2 } also gives a classification of connected monomial graded Majid algebras, which contains the classification result of monomial Hopf algebras in [7]. Corollary 3.11. Let CZ n (d) denote the truncated sub coalgebra of CZ n spanned by the paths of length smaller than some fixed integer d ≥ 2. Then CZ n (d) admits a graded n2 Majid algebra structure if and only if d|n, or d = (s,n 2 ) for some 1 ≤ s ≤ n − 1. In this n case, the Majid algebra on CZ (d) is gauge equivalent to a Hopf algebra if and only if d|n. 3.6. Classification results. Now we are ready to give the main result. In the following we always assume that, for a pointed Majid algebra M with set of group-like elements G(M), its quasi-antipode (S, α, β) satisfies S(g) = g −1 , α(g) = 1, β(g) = 1/(g, g −1 , g) for all g ∈ G(M). The observation of Drinfeld [14] guarantees that this assumption is harmless. Theorem 3.12. Suppose that M = C is a finite-dimensional graded pointed Majid algebra with connected underlying coalgebra. If M is of finite corepresentation type, then M∼ = M(n, s, q) for some positive integers n ≥ 2 and 0 ≤ s ≤ n − 1, and q is an n th root of unity if s = 0, or some primitive root of unity of order n 2 if s = 0. Proof. By the assumption, the corresponding Hopf quiver of M is Z n for some n ≥ 2 according to Corollary 3.2. Now by the Gabriel type theorem for Majid algebra and Subsect. 3.4, M can be viewed as a graded large sub-Majid algebra of some CZ n (s, q). By Proposition 3.10, there is only one possible finite-dimensional graded large sub-Majid algebra M(n, s, q). So M can only be one of the M(n, s, q). On the other hand, since the M(n, s, q) are truncated sub-coalgebras of CZn , so they are of finite corepresentation type as mentioned in Lemma 3.1. This completes the proof. With the help of this theorem, in the following we deal with a not necessarily graded situation by making use of deformation theory [26] and a deep result from geometric methods of representation theory [24]. Without loss of generality, let M be a finitedimensional pointed Majid algebras with connected underlying coalgebra, that is, its bound quiver is connected. As before, we exclude the trivial case for M = C. Corollary 3.13. Keep the above assumption. Then M is of finite corepresentation type if and only if gr M ∼ = M(n, s, q) for some appropriate n, s, q. Proof. First suppose that M is of finite corepresentation type. Consider its underlying coalgebra and apply the Gabriel type theorem. Since M and gr M share the same quiver, so there is a unique Hopf quiver Q such that M can be viewed as a large sub coalgebra
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of the path coalgebra CQ. Now Lemma 3.1 applies, that is, when M is of corepresentation type, then the quiver Q can only be Z n for some integer n ≥ 2. According to Proposition 3.10, the graded version gr M of M is isomorphic to some M(n, s, q). Conversely, suppose gr M ∼ = M(n, s, q). Note that M is a deformation of gr M (it is also said that gr M is a degeneration of M). Then that the Majid algebra M(n, s, q) is of finite corepresentation type implies that so is M, according to the famous theorem of Gabriel [24] which says that finite representation type is open. To complete the classification of non-graded connected pointed Majid algebras of finite corepresentation type, it suffices to calculate all the deformations of M(n, s, q). Note that finite-dimensional Majid algebras are co-Frobenius (i.e., the dual algebra is Frobenius) according to [5,31], then by the same argument as in Sect. 2 of [7] we can conclude that the underlying coalgebra of a connected pointed Majid algebra of finite corepresentation type is isomorphic to a truncated sub coalgebra of CZ n . It follows that one only needs to calculate the coalgebra-preserving deformation of M(n, s, q), which is a quasi-analogue of the preferred deformation of Hopf algebras [27].
3.7. Some remarks. We conclude this section with some remarks. (1) The preferred deformations for M(n, 0, q) were explicitly given in [7]. For s = 0, it seems that the preferred deformations for M(n, s, q) are much more complicated since the deformation of reassociators gets involved. We leave this problem for future work. (2) For a not necessarily connected situation, the underlying coalgebra of a finite-dimensional pointed Majid algebra is a direct sum of finite copies of some CZ n (d) and the Majid algebras is a crossed product of some deformation of M(n, s, q) with a group twisted by a three cocycle [32]. (3) Our classification of finite-dimensional pointed Majid algebras of finite corepresentation type contains the corresponding classification result for pointed Hopf algebras, which was given in [39] by different method. (4) A standard dualization process gives parallel classification results for elementary quasi-Hopf algebras of finite representation type. Some of the dual of M(n, s, q) appeared in previous works of Etingof and Gelaki [15–17,25].
4. Tensor Categories of Finite Type As an application, we will classify a class of tensor categories of finite type in this section. In addition, some information of these finite tensor categories are given by making use of quiver representation theory.
4.1. Finite tensor categories. By a tensor category we mean an abelian rigid monoidal category over C in which the neutral object 1 is simple. A tensor category C is said to be finite if (1) C has finitely many simple objects, (2) any object has finite length, and (3) any simple object admits a projective cover.
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For a finite tensor category C , we denote its Grothendieck ring by Gr (C ). It is a free abelian group of finite rank, whose basis S is the set of isomorphism classes of simple objects in C . Let X ∈ C , then its Frobenius-Perron dimension d+ (X ) is defined to be the largest non-negative real eigenvalue of the matrix of left multiplication in the Grothendieck ring by X under the basis S. For more knowledge about finite tensor categories, see [6,18,19] and references therein. The following result of Etingof and Ostrik [19] provides the close relation between finite tensor categories and finite quasi-quantum groups. Lemma 4.1 (Etingof-Ostrik). For a finite tensor category C , it is tensor equivalent to the representation category Rep H of a finite-dimensional quasi-Hopf algebra H if and only if the Frobenius-Perron dimensions of objects in C are integers. Since Majid algebras are dual of quasi-Hopf algebras, so a finite tensor category C is tensor equivalent to the corepresentation category Corep M of a finite-dimensional Majid algebra M if and only if the Frobenius-Perron dimensions of objects in C are integers. 4.2. Tensor categories of finite type. The simplest finite tensor categories are of course the semisimple ones. After the semisimple situation, the simplest ones are those having finitely many isomorphism classes of indecomposable objects in view of the KrullRemak-Schmidt property of finite tensor categories. A finite tensor category C with this property must be equivalent to the representation category Rep A of a finite-dimensional algebra A of finite representation type, or the corepresentation category Corep C of a finite-dimensional coalgebra C of finite corepresentation type. Inspired by this, such tensor categories are said to be of finite type. By Lemma 4.1, the classification of tensor categories of finite type whose objects have integer Frobenius-Perron dimensions is equivalent to the classification of finite-dimensional quasi-Hopf algebras of finite representation type, or finite-dimensional Majid algebras of finite corepresentation type. Now the results of Sec. 3 can be applied to classify some class of such tensor categories. 4.3. Some classification results. Let C be a finite tensor category and assume that C = Corep C as an abelian category for a finite-dimensional coalgebra C. We say that C is connected if the coalgebra C is connected, that is, the dual algebra C ∗ of C is indecomposable [3]. To avoid the trivial case, in the following we always assume that C has at least two simple objects. Theorem 4.2. Assume that C is a tensor category of finite type. If every simple object of C has Frobenius-Perron dimension 1, then as an abelian category C is equivalent n2 to the direct product of finite copies of Corep CZ n (d) with d|n, or d = (s,n 2 ) for some 1 ≤ s ≤ n − 1. Proof. By the assumption, first of all we have C = Corep C as an abelian category for some finite-dimensional pointed coalgebra C. By the property of Frobenius-Perron dimension, the fact that every simple object of C has Frobenius-Perron dimension 1 implies all objects of C have integer Frobenius-Perron dimensions. Then C must be a pointed Majid algebra of finite corepresentation type by Lemma 4.1. Now the theorem follows immediately from Subsects. 3.6 and 3.7.
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The corepresentation category Corep CZ n (d) is well understood, see for example [2,3] and also [34] for its Auslander-Reiten quiver. In particular, CZ n (d) is Nakayama [2,3]. Therefore, we have Corollary 4.3. Assume that C is a tensor category of finite type and that every simple object of C has Frobenius-Perron dimension 1, then every indecomposable object X ∈ C is uniserial, that is, the set of sub objects of X is totally ordered by inclusion. Combining Corollary 3.13 and Lemma 4.1 we have Corollary 4.4. Assume that C is a connected tensor category of finite type. If every simple object of C has Frobenius-Perron dimension 1, then as a tensor category C is equivalent to Corep M for some pointed Majid algebras M with gr M ∼ = M(n, s, q). Acknowledgements. The authors are very grateful to the referee for the valuable comments and suggestions which improved the exposition. The research was supported by the NSFC grants (10601052, 10801069, 10971206), the SDNSF grant (2009ZRA01128), and the IIFSDU grant (2010TS021). The second author was also supported by Japan Society for the Promotion of Science under the item “JSPS Postdoctoral Fellowship for Foreign Researchers” and Grant-in-Aid for Foreign JSPS Fellow.
References 1. Abramsky, S., Duncan, R.: Mathematical Structures in Computer Science 16(3), 469–489 (2006) 2. Assem, I. , Simson, D., Skowronski, A.: Elements of the Representation Theory of Associative Algebras. Vol. 1. Techniques of Representation Theory. London Mathematical Society Student Texts, 65. Cambridge: Cambridge University Press, 2006 3. Auslander, M., Reiten, I., Smalø, S.O.: Representation Theory of Artin Algebras. Cambridge Studies in Adv. Math. 36. Cambridge: Cambridge Univ. Press, 1995 4. Bakalov, B., Kirilov, A. Jr.: Lectures on tensor categories and modular funtors. Providence, RI: Amer. Math. Soc., 2001 5. Bulacu, D., Caenepeel, S.: Integrals for (dual) quasi-Hopf algebras. Applications. J. Algebra 266(2), 552–583 (2003) 6. Calaque, D., Etingof, P.: Lectures on tensor categories. IRMA Lect. in Math. Theor. Phys. 12, 1–38 (2008) 7. Chen, X.-W. Huang H.-L., Ye, Y., Zhang, P.: Monomial Hopf algebras. J. Algebra 275, 212–232 (2004) 8. Chen, X.-W., Huang, H.-L., Zhang, P.: Dual Gabriel theorem with applications. Sci. in China Series A Math. 49(1), 9–26 (2006) 9. Chin, W.: A brief introduction to coalgebra representation theory. Lecture Notes in Pure and Appl. Math. Vol. 237, New York: Dekker, 2004, pp. 109–131 10. Cibils, C.: A quiver quantum group. Commun. Math. Phys. 157, 459–477 (1993) 11. Cibils, C., Rosso, M.: Algèbres des chemins quantiques. Adv. Math. 125, 171–199 (1997) 12. Cibils, C., Rosso, M.: Hopf quivers. J. Algebra 254, 241–251 (2002) 13. Drinfeld, V.G.: Quantum groups. Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986), Providence, RI: Amer. Math. Soc., 1987, pp. 798–820 14. Drinfeld, V.G.: Quasi-Hopf algebras. Leningrad Math. J. 1, 1419–1457 (1990) 15. Etingof, P., Gelaki, S.: Finite dimensional quasi-Hopf algebras with radical of codimension 2. Math. Res. Lett. 11, 685–696 (2004) 16. Etingof, P., Gelaki, S.: On radically graded finite-dimensional quasi-Hopf algebras. Mosc. Math. J. 5(2), 371–378 (2005) 17. Etingof, P., Gelaki, S.: Liftings of graded quasi-Hopf algebras with radical of prime codimension. J. Pure Appl. Algebra 205(2), 310–322 (2006) 18. Etingof, P., Nikshych, D., Ostrik, V.: On fusion categories. Ann. of Math. 162, 581–642 (2005) 19. Etingof, P., Ostrik, V.: Finite tensor categories. Moscow Math. J. 4, 627–654 (2004) 20. Gaberdiel, M.R.: An algebraic approach to logarithmic conformal field theory. Int. J. Mod. Phys. A 18(25), 4593–4638 (2003) 21. Gaberdiel, M.R., Kausch, H.G.: A rational logarithmic conformal field theory. Phys. Lett. B 386, 131–137 (1996) 22. Gabriel, P.: Unzerlegbare Darstellungen I. Manus. Math. 6, 71–103 (1972)
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23. Gabriel, P.: Indecomposable representations II. In: Symposia Mathematica, Vol. XI (Convegno di Algebra Commutativa, INDAM, Rome, 1971), London: Academic Press, 1973, pp. 81–104 24. Gabriel, P.: Finite representation type is open. In: Representations of Algebras, Lecture Notes in Math., Vol. 488, Berlin-Heidelberg-New York: Springer Verlag, 1975, pp. 132–155 25. Gelaki, S.: Basic quasi-Hopf algebras of dimension n 3 . J. Pure Appl. Algebra 198(1–3), 165–174 (2005) 26. Gerstenhaer, M.: On the deformation of rings and algebras. Ann. of Math. 79(2), 59–103 (1964) 27. Gerstenhaber, M., Schack, S.D.: Bialgebra cohomology, deformations. and quantum groups. Proc. Nat. Acad. Sci. 87, 478–481 (1990) 28. Green, E.L.: Constructing quantum groups and Hopf algebras from coverings. J. Algebra 176, 12–33 (1995) 29. Green, E.L., Solberg, Ø.: Basic Hopf algebras and quantum groups. Math. Z. 229, 45–76 (1998) 30. Gurarie, V.: Logarithmic operators in conformal field theory. Nucl. Phys. B 410, 535–549 (1993) 31. Hausser, F., Nill, F.: Integral theory for quasi-Hopf algebras. http://arxiv.org/abs/math./9904164.v2 [math.QA], 1999 32. Huang, H.-L.: Quiver approaches to quasi-Hopf algebras. J. Math. Phys. 50(4), 043501-1–043501-9 (2009) 33. Huang, H.-L.: From projective representations to quasi-quantum groups. http://arxiv.org/0903.1472.v1 [math.QA], 2009 34. Huang, H.-L., Chen, H.-X., Zhang, P.: Generalized Taft algebras. Algebra Colloq. 11(03), 313–320 (2004) 35. Huang, H.-L., Liu, G.: On coquasitriangular pointed Majid algebras. http://arxiv.org/labs/1002.0518.v1 [math.QA], 2010 36. Karpilovsky, G.: Projecive Representations of Finite Groups. New York: Marcel Dekker, 1985 37. Kassel, C.: Quantum Groups. Graduate Texts in Math. 155, New York: Springer-Verlag, 1995 38. Liu, G.: Classification of finite dimensional basic Hopf algebras according to their representation type. Contemp. Math. 478, 103–124 (2009) 39. Liu, G., Li, F.: Pointed Hopf algebras of finite corepresentation type and their classifications. Proceedings of AMS 135(3), 649–657 (2007) 40. Lusztig, G.: Finite dimensional Hopf algebras arising from quantized universal enveloping algebras. J. AMS 3, 257–296 (1990) 41. Majid, S.: Representations, duals and quantum doubles of monoidal categories. Rend. Circ. Mat. Palermo (2) Suppl. No. 26, 197–206 (1991) 42. Majid, S.: Foundations of Quantum Group Theory. Cambridge: Cambridge University Press, 1995 43. Majid, S.: Quantum double for quasi-Hopf algebras. Lett. Math. Phys. 45(1), 1–9 (1998) 44. Rosso, M.: Quantum groups and quantum shuffles. Invent. Math. 133, 399–416 (1998) 45. Shnider, S., Sternberg, S.: Quantum Groups: From Coalgebras to Drinfeld Algebras. Boston, MA: International Press Inc., 1993 46. Van Oystaeyen, F., Zhang, P.: Quiver Hopf algebras. J. Algebra 280(2), 577–589 (2004) Communicated by A. Connes
Commun. Math. Phys. 303, 613–707 (2011) Digital Object Identifier (DOI) 10.1007/s00220-011-1222-0
Communications in
Mathematical Physics
Diffusion of a Massive Quantum Particle Coupled to a Quasi-Free Thermal Medium W. De Roeck1 , J. Fröhlich2 1 Institut für Theoretische Physik, Universität Heidelberg, D 69120 Heidelberg, Germany.
E-mail:
[email protected]
2 Institute for Theoretical Physics, ETH Zürich, CH-8093 Zürich, Switzerland
Received: 19 September 2009 / Accepted: 2 December 2010 Published online: 22 March 2011 – © Springer-Verlag 2011
Abstract: We consider a heavy quantum particle with an internal degree of freedom moving on the d-dimensional lattice Zd (e.g., a heavy atom with finitely many internal states). The particle is coupled to a thermal medium (bath) consisting of free relativistic bosons (photons or Goldstone modes) through an interaction of strength λ linear in creation and annihilation operators. The mass of the quantum particle is assumed to be of order λ−2 , and we assume that the internal degree of freedom is coupled “effectively” to the thermal medium. We prove that the motion of the quantum particle is diffusive in d ≥ 4 and for λ small enough. 1. Introduction 1.1. Diffusion. Diffusion and Brownian motion are central phenomena in the theory of transport processes and nonequilibrium statistical physics in general. One can think of the diffusion of a tracer particle in interacting particle systems, the diffusion of energy in coupled oscillator chains, and many other examples. From a heuristic point of view, diffusion is rather well-understood in most of these examples. It can often be successfully described by some Markovian approximation, e.g. the Boltzmann equation or Fokker-Planck equation, depending on the example under study. In fact, this has been the strategy of Einstein in his ground breaking work of 1905, in which he modeled diffusion as a random walk. However, up to this date, there is no rigorous derivation of diffusion from classical Hamiltonian mechanics or unitary quantum mechanics, except for some special chaotic systems; see Sect. 1.3.1. Such a derivation ought to allow us, for example, to prove that the motion of a tracer particle that interacts with its environment is diffusive at large times. In other words, one would like to prove a central limit theorem for the position of such a particle. In recent years, some promising steps towards this goal have been taken. We provide a brief review of previous results in Sect. 1.3. In the present paper, we rigorously exhibit
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diffusion for a quantum particle weakly coupled to a thermal reservoir. However, our method is restricted to spatial dimension d ≥ 4. 1.2. Informal description of the model and main results. We consider a quantum particle hopping on the lattice Zd , and interacting with a reservoir of bosons (photons or phonons) at temperature β −1 > 0. In the present section, we describe the system in a way that is appropriate at zero temperature, but is formal when β < ∞. The total Hilbert space, H , of the coupled system is a tensor product of the particle space, HS , with a reservoir space, HR . Thus H := HS ⊗ HR .
(1.1)
The particle space HS is given by l 2 (Zd ) ⊗ S , where the Hilbert space S describes the internal degrees of freedom of the particle, e.g., a (pseudo-)spin or dipole moment, and the particle Hamiltonian is given by the sum of the kinetic energy and the energy of the internal degrees of freedom HS := HS,kin ⊗ 1 + 1 ⊗ HS,spin .
(1.2)
The kinetic energy is chosen to be small in comparison with the interaction energy, and this is made manifest in its definition by a factor λ2 , where λ is the coupling strength between the particle and the reservoir (to be introduced below). Hence we set HS,kin = λ2 ε(P),
(1.3)
where the function ε is the dispersion law of the particle and P is the lattice-momentum operator. The most natural choice is to take ε(P) to be (minus) the discrete lattice Laplacian, −. The energy of states of the internal degree of freedom is to a large extent arbitrary HS,spin := Y,
for some Hermitian matrix Y,
(1.4)
the main requirement being that Y not be equal to a multiple of the identity. The reservoir is described by a free boson field; creation and annihilation operators creating/annihilating bosons with momentum q ∈ Rd are written as aq∗ , aq , respectively. They satisfy the canonical commutation relations [aq# , aq# ] = 0,
[aq , aq∗ ] = δ(q − q ),
(1.5)
where a # stands for either a or a ∗ . The energy of a reservoir mode q is given by the dispersion law ω(q) ≥ 0. To describe the coupling of the particle to the reservoir, we introduce a Hermitian matrix W on S and we write X for the position operator on l 2 (Zd ). The total Hamiltonian of the system is taken to be Hλ := HS + dq ω(q)aq∗ aq d R (1.6) +λ dq eiq·X ⊗ W ⊗ φ(q)aq + e−iq·X ⊗ W ⊗ φ(q)aq∗ , Rd
acting on HS ⊗ HR . The function φ(q) is a form factor and λ ∈ R is the coupling strength. We write HS instead of HS ⊗ 1, etc.
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We introduce three important assumptions: 1) The kinetic energy is small w.r.t. the coupling term in the Hamiltonian, as has already been indicated by the inclusion of λ2 in the definition of HS,kin . Physically, this means that the particle is heavy. 2) We require a linear dispersion law for the reservoir modes, ω(q) ≡ |q|, in order to have good decay estimates at low speed. This means that the reservoir consists of photons, phonons or Goldstone modes of a Bose-Einstein condensate. 3) We assume that the amplitude of the wave front of a reservoir excitation (located on the light cone) has integrable (in time) decay. This is satisfied if the dimension of space1 is at least 4. Additional assumptions will concern the smoothness of the form factor φ and the “effectiveness” of the coupling to the heat bath (e.g., the interaction between the internal degrees of freedom and the reservoir, described by the matrix W , should not vanish.) β The initial state, ρR , of the reservoir is chosen to be an equilibrium state at tem−1 perature β > 0. For mathematical details on the construction of infinite reservoirs, see e.g. [2,4,11]. The initial state of the whole system, consisting of the particle and β the reservoir, is a product state ρS ⊗ ρR , with ρS a density matrix for the particle that will be specified later. The time-evolved density matrix of the particle (‘subsystem’) is called ρS,t and is obtained by “tracing out the reservoir degrees of freedom” after the time-evolution has acted on the initial state during a time t, i.e., formally, β ρS,t := Tr HR e−it Hλ ρS ⊗ ρR eit Hλ ,
(1.7)
where Tr HR is the partial trace over HR . We warn the reader that the above formula does not make sense mathematically for an infinitely extended reservoir, since the reservoir β state ρR is not a density matrix on HR . This is a consequence of the fact that the reservoir is described from the start in the thermodynamic limit and, hence, the reservoir modes form a continuum. Nevertheless, the LHS of formula (1.7) can be given a meaning in the thermodynamic limit. The density matrix ρS,t obviously depends on the coupling strength λ, but we do not indicate this explicitly. We also drop the subscript S and we simply write ρt , instead of ρS,t , in what follows. We will often represent ρt as a B(S )-valued kernel on Zd × Zd : ρt (x L , x R ) ∈ B(S ),
x L , x R ∈ Zd .
(1.8)
Although this is not necessary for many of our results, we require the initial state of the particle to be exponentially localized near the origin of the lattice, i.e.,
ρt (x L , x R ) B (S ) ≤ Ce−δ |x L | e−δ |x R | ,
for some constants C, δ > 0.
(1.9)
Our first result concerns the diffusion of the position of the particle. 1 Since the integrability in time is only needed for reservoir excitations, we can in principle also treat models in which the particle is 3-dimensional, but the reservoir is effectively 4-dimensional.
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1.2.1. Diffusion. We define the probability density μt (x) := Tr S ρt (x, x),
(1.10)
where Tr S denotes the partial trace over the internal degrees of freedom. The number μt (x) is the probability to find the particle at site x after time t. By diffusion, we mean that, for large t, μt (x) ∼
1 2π t
d/2 (det D)
−1/2
1 x −1 x , exp − √ ·D √ 2 t t
(1.11)
where the diffusion tensor D ≡ Dλ is a strictly positive matrix with real entries; actually, if the particle dispersion law ε is invariant under lattice rotations, then the tensor D is isotropic and hence a scalar. The magnitude of D is inferred from the following reasoning: The particle undergoes collisions with the reservoir modes. Let tm be the mean time between two collisions, and let vm be the mean speed of the particle (the direction of the particle velocity is assumed to be random). Then the mean free path is vm × tm and the central limit theorem suggests that the particle diffuses with diffusion constant D∼
(vm × tm )2 . tm
(1.12)
The mean time tm is of order tm ∼ λ−2 since the interaction with the reservoir contributes only in second order. The mean velocity vm is of order vm ∼ λ2 because of the factor λ2 in the definition of the kinetic energy. Hence D ∼ λ2 . We now move towards quantifying (1.11). Let us fix a time t. Since μt (x) is a probability measure, one can think of xt as a random variable with Prob(xt = x) := μt (x).
(1.13)
The claim that the random variable √xtt converges in distribution, as t ∞, to a Gaussian random variable with mean 0 and variance D is called a Central Limit Theorem (CLT). It is equivalent to pointwise convergence of the characteristic function, i.e., x∈Zd
e
− √i x·q t
1
μt (x) −→ e− 2 q·Dq , for all q ∈ Rd , t ∞
(1.14)
and it is this statement which is our main result, Theorem 3.1. A stronger version of the convergence in (1.14) (also included in Theorem 3.1) implies that the rescaled moments of μt converge. For example, for i, j = 1, . . . , d, 1 1 Tr [ρt X i ] = xi μt (x) −→ 0, t ∞ t t x
1 1 Tr ρt X i X j = xi x j μt (x) −→ Di, j . t ∞ t t x
(1.15) (1.16)
In the form as stated here, these results are expected only if one assumes that the model has space inversion symmetry, which is assumed throughout.
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1.2.2. Equipartition. Our second result concerns the asymptotic expectation value of the kinetic energy of the particle and the internal degrees of freedom. The equipartition theorem suggests that the energy of all degrees of freedom of the particle, the translational and internal degrees of freedom, thermalizes at the temperature β −1 of the heat bath. We will establish this property up to a correction that is small in the coupling strength λ. This is acceptable, since the interaction effectively modifies the Gibbs state of the particle. We prove that, for all bounded functions F, 1 2 (1.17) Tr ρt F HS,kin −→ dk F λ2 ε(k) e−βλ ε(k) + o |λ|0 , t ∞ Z Td 1 Tr ρt F HS,spin −→ (1.18) F(e)e−βe + o |λ|0 , as λ 0, t ∞ Z e∈spY
where Z , Z are normalization constants and the sum
e∈spY ranges over all eigen2 ε(k) −βλ e can be replaced by 1 (as
values of the Hamiltonian Y . We note that the factor in Theorem 3.2) since we anyhow allow a correction term that is small in λ and the function ε(k) is bounded. For this reason, one could say that, for very small values of λ, the translational degrees of freedom thermalize at infinite temperature (β = 0). 1.2.3. Decoherence. By decoherence we mean that off-diagonal elements ρt (x, y) of the density matrix ρt in the position representation fall off rapidly as a function of |x − y|. Of course, this property can only hold at large enough times when the effect of the reservoir on the particle has destroyed all initial long-distance coherence, i.e., after a time of order λ−2 . Thus, there is a decoherence length 1/γdch and a decay rate g such that ρt (x L , x R ) B (S ) ≤ Ce−γdch |x L −x R | + C e−λ
2 gt
,
as t ∞
(1.19)
for some constants C, C . The magnitude of the inverse decoherence length γdch is determined as follows: The time the reservoir needs to destroy coherence is of the order of the mean free time tm , while the time that is needed for coherence to be built up over a distance 1/γdch is given by (γdch × vm )−1 , where vm is the mean velocity of the particle. Equating these two times yields γdch ∼ (tm × vm )−1 ,
(1.20)
and hence, recalling that tm ∼ λ−2 and vm ∼ λ2 , as argued in Sect. 1.2.1, we find that γdch does not scale with λ. 1.3. Related results and discussion. 1.3.1. Classical mechanics. Diffusion has been established for the two-dimensional finite horizon billiard in [7]. In that setup, a point particle travels in a periodic, planar array of fixed hard-core scatterers. The finite-horizon condition refers to the fact that the particle cannot move further than a fixed distance without hitting an obstacle. Knauf [29] replaced the hard-core scatterers by a planar lattice of attractive Coulombic potentials, i.e., the potential is V (x) = − j∈Z2 |x−1 j| . In that case, the motion of the particle can be mapped to the free motion on a manifold with strictly negative curvature, and one can again prove diffusion.
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Recently, a different approach was taken in [5]: Interpreted freely, the model in [5] consists of a d = 3 lattice of confined particles that interact locally with chaotic maps such that the energy of the particles is preserved but their momenta are randomized. Neighboring particles can exchange energy via collisions and one proves diffusive behaviour of the energy profile. 1.3.2. Quantum mechanics for extended systems. The earliest result for extended quantum systems that we are aware of, [32], treats a quantum particle interacting with a time-dependent random potential that has no memory (the time-correlation function is δ(t)). Recently, this was generalized in [27] to the case of time-dependent random potentials where the time-dependence is given by a Markov process with a gap (hence, the free time-correlation function of the environment is exponentially decaying). In [15], we treated a quantum particle interacting with independent heat reservoirs at each lattice site. This model also has an exponentially decaying free reservoir time-correlation function and as such, it is very similar to [27]. Notice also that, in spirit, the model with independent heat baths is comparable to the model of [5], but, in practice, it is easier since quantum mechanics is linear! The most serious shortcoming of these results is the fact that the assumption of exponential decay of the correlation function in time is unrealistic. In the model of the present paper, the space-time correlation function, called ψ(x, t) in what follows, is the correlation function of freely-evolving excitations in the reservoir, created by interaction with the particle. Since momentum is conserved locally, these excitations cannot decay exponentially in time t, uniformly in x. For example, if the dispersion law of the reservoir modes is linear, then ψ(x, t) is a solution of the linear wave equation. In d = 3, it behaves qualitatively as ψ(x, t) ∼
1 δ(c|t| − |x|), with c the propagation speed of the reservoir modes. |x| (1.21)
In higher dimensions, one has better dispersive estimates, namely supx |ψ(x, t)| ≤ d−1
O t − 2 (under certain conditions), and this is the reason why, for the time being, our approach is restricted to d ≥ 4. In the Anderson model, the analogue of the correlation function does not decay at all, since the potentials are fixed in time. Indeed, the Anderson model is different from our particle-reservoir model: diffusion is only expected to occur for small values of the coupling strength, whereas the particle gets trapped (Anderson localization) at large coupling. Finally, we mention a recent and exciting development: in [17], the existence of a delocalized phase in three dimensions is proven for a supersymmetric model which is interpreted as a toy version of the Anderson model. 1.3.3. Quantum mechanics for confined systems. The theory of confined quantum systems, i.e., multi-level atoms, in contact with quasi-free thermal reservoirs has been intensively studied in the last decade, e.g. by [3,13,26]. In this setup, one proves approach to equilibrium for the multi-level atom. Although at first sight, this problem is different from ours (there is no analogue of diffusion), the techniques are quite similar and we were mainly inspired by these results. However, an important difference is that, due to its confinement, the multi-level atom experiences a free reservoir correlation function with better decay properties than that of our model. For example, in [26], the free reservoir correlation function is actually exponentially decaying.
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1.3.4. Scaling limits. Up to now, most of the rigorous results on diffusion starting from deterministic dynamics are formulated in a scaling limit. This means that one does not fix one dynamical system and study its behaviour in the long-time limit, but, rather, one compares a family of dynamical systems at different times. The precise definition of the scaling limit differs from model to model, but, in general, one scales time, space and the coupling strength (and possibly also the initial state) such that the Markovian approximation to the dynamics becomes exact. In our model the natural scaling limit is the so-called weak coupling limit: one introduces the macroscopic time τ := λ2 t and one takes the limit λ 0, t ∞ while keeping τ fixed. In that limit, the dynamics of the particle becomes Markovian in τ (as if the heat bath had no memory) and it is described by a Lindblad evolution. The long-time behavior of this Lindblad evolution is diffusive. This is explained in detail in Sect. 4. One may say that, in this scaling limit, the heuristic reasoning employed in the previous sections to deduce the λ-dependence of the diffusion constant and the decoherence length becomes exact. The same scaling is known very well in the theory of confined open quantum systems as it gives rise to the Pauli master equation. This was first made precise in [10]. If we had set up the model with a kinetic energy of O(1) (instead of O(λ2 )), then one should also rescale space by introducing the macroscopic space-coordinate χ := λ2 x. The reason for this additional rescaling is that, between two collisions, a particle with mass of order 1 moves during a time of order λ−2 , and hence it travels a distance of order λ−2 . The resulting scaling limit x → λ−2 x, t → λ−2 t, λ 0
(1.22)
is often called the kinetic limit. In the kinetic limit the dynamics of the particle is described by a linear Boltzmann equation (LBE) in the variables (χ , τ ). The convergence of the particle dynamics to the LBE has been proven in [18] for a quantum particle coupled to a heat bath, and in [21] for a quantum particle coupled to a random potential (Anderson model). The long-time, large-distance limit of the Boltzmann equation is the heat equation, which suggests that one should be able to derive the heat equation directly in the limiting regime corresponding to x → λ−(2+κ) x, t → λ−(2+2κ) t, λ 0,
for some κ > 0.
(1.23)
This was accomplished in [19,20] for the Anderson model. An analogous result was obtained in [30] for a classical particle moving in a random force field. 1.3.5. Limitations to our result. Two striking features of our model are the large mass, of order λ−2 , and the internal degrees of freedom described by the Hamiltonian HS,spin = Y . Physically speaking, these choices are of course not necessary for diffusion, they just make our task of proving it easier. Let us explain why this is so. First of all, once the mass is chosen to be of order λ−2 , the internal degrees of freedom are necessary to make the model diffusive in second order perturbation theory. Without the internal degrees of freedom, it would be ballistic. This is explained in Sect. 4.2; in particular, it can be deduced immediately from conservation of momentum and energy for the processes in Fig. 3. Note also that the dependence on λ is chosen such that the kinetic term HS,kin = λ2 ε(P) is comparable to the particle-reservoir interaction in second order of perturbation theory (both are of order λ2 ). The large mass ensures that the position of the particle remains well-defined for a time of order λ−2 , which permits us to sum up Feynman diagrams in real space.
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Further, we note that our result requires an analyticity assumption on the form factor φ, see Assumption 2.3. This assumption ensures that the free reservoir correlation function ψ(x, t) is exponentially decaying for small x, even though it has slow decay on the lightcone, as explained in Sect. 1.3.2.
1.4. Outline of the paper. The model is introduced in Sect. 2 and the results are stated in Sect. 3. In Sect. 4, we describe the Markovian approximation to our model. This approximation provides most of the intuition and it is a key ingredient of the proofs. Sect. 5 describes the main ideas of the proof, which is contained in the remaining Sects. 6–9 and the four Appendices A-D. 2. The Model After fixing conventions in Sect. 2.1, we introduce the model. Sect. 2.2 describes the particle, while Sect. 2.3 deals with the reservoir. In Sect. 2.4, we couple the particle to the reservoir, and we define the reduced particle dynamics Zt . Section 2.5 introduces the fiber decomposition. 2.1. Conventions and notation. Given a Hilbert space E , we use the standard notation 1 ≤ p ≤ ∞, (2.1) B p (E ) := S ∈ B(E ), Tr (S ∗ S) p/2 < ∞ , with B∞ (E ) ≡ B(E ) the bounded operators on E , and 1/ p S p := Tr (S ∗ S) p/2 , S := S ∞ .
(2.2)
For bounded operators acting on B p (E ), i.e. elements of B(B p (E )), we use in general the calligraphic font: V, W, T , . . .. An operator X ∈ B(E ) determines an operator ad(X ) ∈ B(B p (E )) by ad(X )S := [X, S] = X S − S X,
S ∈ B p (E ).
(2.3)
The norm of operators in B(B p (E )) is defined by W :=
W(S) p . S p S∈B p (E ) sup
(2.4)
We will mainly work with Hilbert-Schmidt operators ( p = 2) and, unless mentioned otherwise, the notation W will refer to this case. For vectors υ ∈ Cd , we let Re υ, Im υ denote the vectors (Re υ1 , . . . , Re υd ) and (Im υ1 , . . . , Im √ υd ), respectively. The scalar product on Cd is written as υ · υ and the norm as |υ| := υ · υ. The scalar product on a general Hilbert space E is written as ·, ·, or, occasionally, as ·, ·E . All scalar products are defined to be linear in the second argument and anti-linear in the first one. We use the physicist’s notation |ϕϕ |
for the rank-1 operator in B(E ) acting as ϕ → ϕ , ϕ ϕ.
(2.5)
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We write s (E ) for the symmetric (bosonic) Fock space over the Hilbert space E and we refer to [11] for definitions and discussion. If ω is a self-adjoint operator on E , then its (self-adjoint) second quantization, ds (ω), is defined by ds (ω)Sym(ϕ1 ⊗ . . . ⊗ ϕn ) :=
n
Sym(ϕ1 ⊗ . . . ⊗ ωϕi ⊗ . . . ⊗ ϕn ),
(2.6)
i=1
where Sym projects on the symmetric subspace of ⊗n E and ϕ1 , . . . , ϕn ∈ E . We use C, C to denote constants whose precise value can change from equation to equation. 2.2. The particle. We choose a finite-dimensional Hilbert space S , which can be thought of as the state space of some internal degrees of freedom of the particle, such as spin or a dipole moment. The total Hilbert space of the particle is given by HS := l 2 (Zd , S ) = l 2 (Zd ) ⊗ S (the subscript S refers to ‘system’, as is customary in system-reservoir models). We define the position operators, X j , on HS by (X j ϕ)(x) = x j ϕ(x),
x ∈ Zd , ϕ ∈ l 2 (Zd , S ),
j = 1, . . . , d.
(2.7)
In what follows, we will almost always drop the component index j and write X ≡ (X j ) to denote the vector-valued position operator. We will often consider the space HS in its dual representation, i.e. as L 2 (Td , S ), where Td is the d-dimensional torus (momentum space), which is identified with L 2 ([−π, π ]d , S ). We formally define the ‘momentum’ operator P as multiplication by k ∈ Td , i.e., Pϕ(k) = kϕ(k), k ∈ [−π, π ]d , ϕ ∈ L 2 (Td , S ).
(2.8)
Although P is well-defined as a bounded operator, it does not correspond to a continuous function on Td , and it is not true that [X j , P j ] = −i. Throughout the paper, we will only use operators F(P) where F is a function on Td that is extended periodically to Rd . We choose such a periodic function, ε, of P to determine the dispersion law of the particle. The kinetic energy of our particle is given by λ2 ε(P), where λ is a small parameter, i.e., the ‘mass’ of the particle is of order λ−2 . The energy of the internal degrees of freedom is given by a self-adjoint operator Y ∈ B(S ), acting on HS as (Y ϕ)(k) = Y (ϕ(k)). The Hamiltonian of the particle is HS := λ2 ε(P) ⊗ 1 + 1 ⊗ Y.
(2.9)
As in Sect. 1, we will mostly write ε(P) instead of ε(P) ⊗ 1 and Y instead of 1 ⊗ Y . Our first assumption ensures that the Hamiltonian HS = Y + λ2 ε(P) has good regularity properties. Assumption 2.1 (Analyticity of the particle dynamics). The function ε, defined originally on Td , extends to an analytic function in a neighborhood of the complex multistrip of width δε > 0. That is, when viewed as a periodic function on Rd , ε is analytic (and bounded) in a neighborhood of (R + i[−δε , δε ])d . Moreover, ε is symmetric with respect to space inversion, i.e., ε(k) = ε(−k).
(2.10)
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Furthermore, we assume there is no υ ∈ Rd such that the function k → υ · ∇ε(k) vanishes identically and that ε does not have a smaller periodicity than that of Td , i.e., we assume that ε(k) = ε(z + k) for all k ∈ Td
⇔
z ∈ (2π Z)d .
(2.11)
d The most natural choice for ε is ε(k) = i=1 2(1 − cos(ki )), which corresponds to −ε(P) being the lattice Laplacian. As already indicated in Sect. 1.2.1, the symmetry assumption (2.10) is necessary to exclude an asymptotic drift of the particle. By a simple Paley-Wiener argument, Assumption 2.1 implies that one has exponential propagation estimates for the evolution generated by the operator ε(P). Indeed, from the relation iν·X −itε(P) −iν·X −itε(P+ν) e e e = e ≤ eqε (|Im ν|)|t| ,
for |Im ν| ≤ δε
(2.12)
for γ ≤ δε ,
(2.13)
with qε (γ ) := sup|Im p|≤γ |Im ε( p)|, one obtains −itε(P) (x L , x R ) e
S
≤ e−γ |x L −x R | eqε (γ )|t| ,
where we write S(x L , x R ) for a B(S )-valued ‘matrix element’ of S ∈ B(HS ).
2.3. The reservoirs. 2.3.1. The reservoir space. We introduce a one-particle reservoir space h = L 2 (Rd ) and a positive one-particle Hamiltonian ω ≥ 0. The coordinate q ∈ Rd should be thought of as a momentum coordinate, and ω acts by multiplication with a function ω(q), (ωϕ)(q) = ω(q)ϕ(q).
(2.14)
In other words, ω is the dispersion law of the reservoir particles. The full reservoir Hilbert space, HR , is the symmetric Fock space (see Sect. 2.1 or [11]) over the one-particle space h, HR := s (h).
(2.15)
The reservoir Hamiltonian, HR , acting on HR , is then the second quanitzation of ω, HR := ds (ω) =
Rd
dq ω(q)aq∗ aq ,
with the creation/annihilation operators aq∗ , aq to be introduced below.
(2.16)
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2.3.2. The system-reservoir coupling. The coupling between system and reservoir is assumed to be translation invariant. We choose a ‘form factor’ φ ∈ L 2 (Rd ) and a self-adjoint operator W = W ∗ ∈B(S ) with W ≤1, and we define the interaction Hamiltonian HSR by HSR
on HS ⊗ HR , dq eiq·X ⊗ W ⊗ φ(q)aq + e−iq·X ⊗ W ⊗ φ(q)aq∗
:=
(2.17) aq , aq∗
are the creation/annihilation operators (actually, operator-valued distribuwhere tions) on h satisfying the canonical commutation relations (CCR) aq , aq∗ = δ(q − q ), (2.18) aq# , aq# = 0, with a # standing for either a or a ∗ . We also introduce the smeared creation/annihilation operators a ∗ (ϕ) := dq ϕ(q)aq∗ , a(ϕ) := dq ϕ(q)aq , ϕ ∈ L 2 (Rd ). (2.19) Rd
Rd
In what follows we will specify our assumptions on HSR , but we already mention that we need [W, Y ] = 0 for the internal degrees of freedom to be coupled effectively to the field. 2.3.3. Thermal states. Next, we put some tools in place to describe the positive temperature state of the reservoir. We introduce the density operator −1 Tβ = eβω − 1 on h = L 2 (Rd ). (2.20) Let C be the ∗ algebra consisting of polynomials in the creation and annihilation operaβ tors a(ϕ), a ∗ (ϕ ) with ϕ, ϕ ∈ h. We define ρR as a quasi-free state defined on C. It is a linear functional on C, fully specified by the following properties: 1) Gauge-invariance
β β ρR a ∗ (ϕ) = ρR [a(ϕ)] = 0. (2.21) 2) The choice of the two-particle correlation function β
β
ρR a ∗ (ϕ)a(ϕ ) ρR a ∗ (ϕ)a ∗ (ϕ ) 0 ϕ |Tβ ϕ , =
β β 0 ϕ|(1 + Tβ )ϕ ρ a(ϕ)a(ϕ ) ρ a(ϕ)a ∗ (ϕ ) R
R
(2.22) 3)
β The state ρR
is quasifree. This means that the higher correlation functions are related to the two-particle correlation function via Wick’s theorem β β ρR a # (ϕ1 ) . . . a # (ϕ2n ) = ρR a # (ϕi )a # (ϕ j ) , (2.23) π ∈Pn (i, j)∈π
a∗
where stands for either or a, and Pn is the set of pairings π , partitions of {1, . . . , 2n} into n pairs (r, s). By convention, we fix the order within the pairs such that r < s. The reason why it suffices to specify the state on C has been explained in many places, see e.g. [4,11,24]. a#
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2.3.4. Assumptions on the reservoir. Next, we state our main assumption restricting the type of reservoir and the dimensionality of space. Assumption 2.2 (Relativistic reservoir and d ≥ 4). We assume that the dimension of space d ≥ 4.
(2.24)
Further, we assume the dispersion law of the reservoir particles to be linear ω(q) := |q|.
(2.25)
For simplicity, we will assume that the form factor φ is rotationally symmetric and we write φ(q) ≡ φ(|q|),
q ∈ Rd .
We define the “effective squared form factor” as 1 2 −βω |φ(|ω|)| ˆ ψ(ω) := |ω|(d−1) 1−e1 |φ(|ω|)|2 e−βω −1
(2.26)
ω≥0 ω<0
,
(2.27)
where we are abusing the notation by letting ω denote a variable in R. Previously, ω was the energy operator on the one-particle Hilbert space and as such, it could assume only ˆ positive values. Indeed, at positive temperature, the function ψ(ω) plays a similar role as |φ(|ω|)|2 at zero-temperature: It describes the intensity of the coupling to the reservoir modes of frequency ω. Modes with ω < 0 appear only at positive temperature and they ˆ
correspond physically to “holes”. One checks that ˆψ(ω) = eβω , which is Einstein’s ψ(−ω) emission-absorption law (i.e. detailed balance). This particle-hole point of view can be incorporated into the formalism by the Araki-Woods representation, see e.g. [4,11,24]. ˆ The next assumption restricts the “effective squared form factor” ψ. Assumption 2.3 (Analytic form factor). Let the form factor be rotation-symmetric ˆ φ(q) ≡ φ(|q|), as in (2.26), and let ψˆ be defined as in (2.27). We assume that ψ(0) =0 ˆ and that the function ω → ψ(ω) has an analytic extension to a neighborhood of the strip R + i[δR , δR ], for some δR > 0, such that ˆ dω|ψ(ω)| < ∞. (2.28) sup −δR ≤χ ≤δR R+iχ
We note that Assumption 2.3 is satisfied (in d ≥ 4) if one chooses: 1 φ(|q|) := √ ϑ(|q|) |q|
(2.29)
with ϑ a function on R with ϑ(−ω) = ϑ(ω) and analytic in the strip of width δR , and ˆ such that (2.28) holds with |ϑ(ω)|2 substituted for |ψ(ω)|. The motivation for Assumptions 2.2 and 2.3 will become clear in Sect. 5.1, where we discuss the reservoir space-time correlation function ψ(x, t).
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The last assumption is a Fermi Golden Rule condition that ensures that the spin degrees of freedom are effectively coupled to the reservoir. To state it, we need the following operators 1e (Y )W 1e (Y ), a ∈ sp(ad(Y )). (2.30) Wa := e, e ∈ spY e − e = a
Note that the variable a labels the Bohr-frequencies of the internal degrees of freedom of the particle. Assumption 2.4 (Fermi Golden Rule). Recall the function ψˆ as defined in (2.27). The set of matrices ˆ BW := ψ(a)W (2.31) a , a ∈ sp(ad(Y )) ⊂ B(S ) generates the complete algebra B(S ). This means that any S ∈ B(S ) which commutes with all operators in BW is necessarily a multiple of the identity. We also require the following non-degeneracy condition: • Every eigenvalue of Y is nondegenerate (multiplicity 1). • For all eigenvalues e, e , e , e of Y such that e = e , we have ⇒ e = e and e = e . e − e = e − e
(2.32)
This condition implies in particular that all eigenvalues of ad(Y ) are nondegenerate, except for the eigenvalue 0, whose multiplicity is given by dim S . The strict nondegeneracy condition on Y , in contrast to the condition on BW , is not crucial to our technique of proof, but it allows us to be more concrete in some stages of the calculation. In particular, the matrices Wa=0 , introduced above in (2.30), can be rewritten as Wa = e , W e × |e e|,
(2.33)
where e, e are the unique eigenvalues s.t. e − e = a = 0, and we have denoted the corresponding eigenvectors by the same symbols e, e (cfr. (2.5)). The condition that BW generates the complete algebra, can then be rephrased as follows: Consider an undirected graph with vertex set spY and let the vertices e and e be connected by an edge if and only if ˆ − e)|e, W e |2 = 0 ψ(e
(2.34)
(note that this condition is indeed symmetric in e, e , as long as β < ∞). Then Assumption 2.4 is satisfied if and only if this graph is connected. Assumptions of the type above have their origin in a criterion for ergodicity of quantum master equations due to [23,33], that is the noncommutative analogue of the PerronFrobenius theorem. In our analysis, too, Assumption 2.4 is used to ensure that the Markovian semigroup t (to be introduced in Sect. 4) has good ergodic properties. This can be seen in Sect. C.1.1 in Appendix C.
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2.4. The dynamics of the coupled system. Consider the Hilbert space H := HS ⊗ HR . The Hamiltonian Hλ (with coupling constant λ) on H is (formally) given by Hλ := HS + HR + λHSR .
(2.35)
If the following condition is satisfied: φ, ω−1 φh < ∞,
(2.36)
then HSR is a relatively bounded perturbation of HS + HR and hence Hλ is a self-adjoint operator. One easily checks that (2.36) is implied by Assumptions 2.2 and 2.3. For the purposes of our analysis, it is important to understand the dynamics of the coupled system at positive temperature. To this end, we introduce the reduced dynamics of the quantum particle. β By a slight abuse of notation, we use ρR to denote the conditional expectation B(HS ) ⊗ C → B(HS ), given by β
β
ρR [S ⊗ R] = SρR [R],
(2.37)
β
where ρR (R) is defined by (2.21–2.22–2.23) for R ∈ C, i.e. a polynomial in creation and annihilation operators. Formally, the reduced dynamics in the Heisenberg picture is given by β Zt (S) := ρR eit Hλ (S ⊗ 1) e−it Hλ .
(2.38)
However, this definition does not make sense a priori, since eit Hλ (S ⊗ 1) e−it Hλ ∈ / B(HS ) ⊗ C in general. A mathematically precise definition of Zt is the subject of the upcoming Lemma 2.5. β Since both the initial reservoir state ρR and the Hamiltonian Hλ are translationinvariant, we expect that the reduced evolution Zt is also translation invariant in the sense that Tz Zt T−z = Zt ,
where (Tz S)(x L , x R ) := S(x L + z, x R + z).
(2.39)
By the requirement ε(k) = ε(−k) in Assumption 2.1 and the requirement that φ(q) = φ(−q) in Assumption 2.3, the Hamiltonian Hλ is also invariant with respect to spaceinversion x → −x, or, equivalently, k → −k. Since the initial reservoir state is also invariant with respect to space inversion (this follows from the fact that ω(q) = ω(−q)), we expect that T E Zt T E = Zt ,
where (T E S)(x L , x R ) := S(−x L , −x R ).
(2.40)
Finally, the unitarity of the microscopic time-evolution implies that T J Zt T J = Zt ,
where (T J S)(x L , x R ) := S ∗ (x R , x L ),
where the ∗ in S ∗ (·, ·) is the Hermitian conjugation on B(S ).
(2.41)
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Lemma 2.5. Assume Assumptions 2.1, 2.2 and 2.3, and let H0 := HS + HR , The Lie-Schwinger series n Zt (S) := (iλ)
HSR (t) := eit H0 HSR e−it H0 .
0 ≤ t1 ≤ · · · ≤ tn ≤ t
n∈N
(2.42)
dt1 · · · dtn
β ×ρR ad(HSR (t1 ))ad(HSR (t2 )) · · · ad(HSR (tn )) eitad(H0 ) (S ⊗ 1) (2.43) is well-defined for all λ, t ∈ R, that is, the RHS is a norm convergent family of operators and Zt∗ has the following properties: 1) 2) 3) 4) 5) 6) 7)
Zt (1) = 1. Tz Zt T−z = Zt with Tz as defined in (2.39). T E Zt T E = Zt with T E as defined in (2.40). T J Zt T J = Zt with T J as defined in (2.41). Zt (S) ∞ ≤ S ∞ . Zt (S) ≥ 0 for S ≥ 0 For S ∈ B2 (HS ), the map S → Zt (S) is continuous in t in the Hilbert-Schmidt norm · 2 .
These properties of Zt should not come as a surprise, they hold true trivially if one β pretends that the initial reservoir state ρR is a density matrix and Zt is obtained by taking the partial trace over the reservoir space, as in (1.7). One can prove this lemma, under much less restrictive conditions than the stated assumptions, by estimates on the RHS. For this purpose, the estimates given in the present paper amply suffice. However, one can also define the system-reservoir dynamics as a dynamical system on a von Neumann algebra through the Araki-Woods representation and this is the usual approach in the mathematical physics literature, see e.g. [3,11,13,24,26]. We also define Zt : B1 (HS ) → B1 (HS ), the reduced dynamics in the Schrödinger representation, by duality
Tr SZt (S ) = Tr Zt (S)S . (2.44) Physically, Zt is the reduced dynamics on observables of the system and Zt is the reduced dynamics on states. 2.5. Translation invariance and the fiber decomposition. In this section, we introduce concepts and notation that will prove useful in the analysis of the reduced evolution Zt . These concepts will be used in Sect. 3.2. However, Sect. 3.1, which contains the main results, can be understood without the concepts introduced in the present section. Consider the space of Hilbert-Schmidt operators B2 (HS ) ∼ B2 (l 2 (Zd ) ⊗ S ) ∼ L 2 (Td × Td , B2 (S ), dk L dk R ) and define ˆ L , k R ) := S(k
x L ,x R
S(x L , x R )e−i(x L k L −x R k R ) ,
(2.45)
S ∈ B2 (l 2 (Zd ) ⊗ S ).
∈Zd
(2.46)
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Note the asymmetric normalization of the Fourier transform, which serves to eliminate factors of 2π in the bulk of the paper. In what follows, we will write S for Sˆ to keep the notation simple, since the arguments x ↔ k will indicate whether we are dealing with ˆ To deal conveniently with the translation invariance of our model, we change S or S. variables, see also Fig. 1. k=
kL + kR , 2
p = kL − k R ,
k, p ∈ Td ,
(2.47)
and, for a.e. p ∈ Td , we obtain a function S p ∈ L 2 (Td , B2 (S )) by putting (S p )(k) := S(k +
p p , k − ). 2 2
(2.48)
This follows from the fact that the Hilbert space B2 (HS ) ∼ L 2 (Td × Td , B2 (S ), dk L dk R ) can be represented as a direct integral ⊕ ⊕ p B2 (HS ) = dp G , S= d p Sp, (2.49) Td
Td
where each ‘fiber space’ G p is naturally identified with G ≡ L 2 (Td , B2 (S )). Elements of G will often be denoted by ξ, ξ and the scalar product is dk Tr S [ξ ∗ (k)ξ (k)] (2.50) ξ, ξ G := Td
with Tr S the trace over the space of internal degrees of freedom S . Let Tz , z ∈ Zd be the lattice translation defined in (2.39). In momentum space, (Tz S) p = e−i pz S p ,
S ∈ B2 (HS ).
(2.51)
β
Since Hλ and ρR are translation invariant, it follows that T−z Zt Tz = Zt .
(2.52)
Let W ∈ B(B2 (HS )) be translation invariant in the sense that T−z WTz = W (cf. (2.52)). Then it follows that, in the representation (2.49), W acts diagonally in p, i.e. (W S) p depends only on S p and we define W p by (W S) p = W p S p ,
S p ∈ G , W p ∈ B(G ).
(2.53)
For the sake of clarity, we give an explicit expression for W p . Define the kernel Wx L ,x R ;x L ,x R by (W S)(x L , x R ) =
x L ,x R ∈Zd
Wx L ,x R ;x L ,x R S(x L , x R ),
x L , x R ∈ Zd .
(2.54)
Translation invariance is expressed by Wx L ,x R ;x L ,x R = Wx L +z,x R +z;x L +z,x R +z ,
z ∈ Zd ,
(2.55)
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Fig. 1. The thick black square [−π, π ] × [−π, π ] is the momentum space Td × Td (drawn here for d = 1), with k L , k R ∈ Td . After changing variables to (k, p) ∈ Td × Td , the momentum space is transformed into the gray rectangle. One sees that the four triangles which lie inside the square but outside the rectangle, are identified with the four triangles inside the rectangle but outside the square
and, as an integral kernel, W p ∈ B(L 2 (Td , B2 (S ))) is given by p W p (k , k) = eik(x L −x R )−ik (x L −x R ) e−i 2 ((x L +x R )−(x L +x R )) Wx L ,x R ;x L ,x R . x R , x L , x R ∈ Zd xL = 0
(2.56) To avoid confusion with other subscripts we will often write {S} p instead of S p
and
{W} p instead of W p .
(2.57)
We also introduce the following transformations. For ν ∈ Td , let Uν be the unitary operator acting on the fiber spaces G as (Uν ξ )(k) = ξ(k + ν),
ξ ∈ G.
(2.58)
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Next, let κ = (κ L , κ R ) ∈ Cd × Cd and define the operators Jκ by 1
1
(Jκ S)(x L , x R ) := ei 2 κ L ·x L S(x L , x R )e−i 2 κ R ·x R .
(2.59)
Note that Jκ is unbounded if κ ∈ / Rd × Rd . The relation between the operators Jκ and the fiber decomposition is given by the relation {Jκ WJ−κ } p = U− κ L +κ R {W} 4
p−
κ L −κ R 2
U κ L +κ R ,
(2.60)
4
as follows from (2.56) and the definition (2.59). From (2.56) and (2.60), we check that p
is conjugate to
ν
is conjugate to
1 (x L + x R ) − (x L + x R ) , 2 (x L − x R ) − (x L − x R ).
(2.61) (2.62)
We state an important lemma on the fiber decomposition. Lemma 2.6. Let S ∈ B1 (L 2 (Td , S )). Then, S p is well-defined, for every p, as a function in L 1 (Td , B2 (S )) and Tr Jκ S = e−i px S(x, x) = 1, S p G , x∈Zd
with p = −
κL − κ R and κ = (κ L , κ R ), 2
(2.63)
where 1 stands for the constant function on Td with value 1 ∈ B(S ). If, moreover, Jκ S is a Hilbert-Schmidt operator for |Im κ L ,R | ≤ δ , then the function Td → G :
p → S p ,
(2.64)
as defined in (2.48), is well-defined for all p ∈ Td and has a bounded-analytic extension to the strip |Im p| < δ . The first statement of the lemma follows from the singular-value decomposition for trace-class operators. In fact, the correct statement asserts that one can choose S p such that (2.63) holds. Indeed, one can change the value of the kernel S(k L , k R ) on the line k L − k R = p without changing the operator S, and hence S p in (2.63) can not be defined via (2.48) in general, if the only condition on S is S ∈ B1 . The second statement of Lemma 2.6 is the well-known relation between exponential decay of functions and analyticity of their Fourier transforms. Since we will always demand the initial density matrix ρ0 to be such that Jκ ρ0 2 is finite for κ in a complex domain, we will mainly need the second statement of Lemma 2.6. By employing Lemma 2.6 and the properties of Zt∗ listed in Lemma 2.5, it is easy to show that the function k → {Zt ρ0 }0 (k) ∈ B(S ) takes values in the positive matrices on S and is normalized, i.e., dk Tr S [{Zt ρ0 }0 (k)] = 1, {Zt ρ0 }0 G = 1.
(2.65)
(2.66)
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Further, the space-inversion symmetry and self-adjointness of the density matrix (the third and fourth property in Lemma 2.5 ) imply that E {Zt } p E = {Zt }− p ,
where (Eξ )(k) := ξ(−k), for ξ ∈ G ,
J {Zt } p J = {Zt }− p ,
where (J ξ )(k) := (ξ(k)) , for ξ ∈ G ,
∗
(2.67) (2.68)
∗
where the on ξ(k) is the Hermitian conjugation on B(S ). 3. Results In this section, we describe our main results. In Sect. 3.1, we state the results in a direct way, emphasizing the physical phenomena. In Sect. 3.2, we describe more general statements that imply all the results stated in Sect. 3.1. 3.1. Diffusion, decoherence and equipartition. We choose the initial state of the particle to be a density matrix ρ ∈ B1 (HS ) satisfying ρ > 0,
Tr[ρ] = 1
Jκ ρ 2 < ∞,
(3.1)
for κ in some neighborhood of 0 ∈ Cd × Cd . The condition Jκ ρ 2 < ∞ reflects the fact that, at time t = 0, the particle is exponentially localized near the origin. Our results describe the time-evolved density matrix ρt := Zt ρ. Note that ρt depends on λ, too. First, we state that the particle exhibits diffusive motion. Define the probability density μt ≡ μλt , depending on the initial state ρ ∈ B1 (HS ), by μt (x) := Tr S [ρt (x, x)] .
(3.2)
It is easy to see that μt (x) ≥ 0,
μt (x) = Tr[ρt ] = 1.
(3.3)
x∈Zd
The following theorem states that the family of probability densities μt (·) converges in distribution and in the sense of moments to a Gaussian, after rescaling space as x → √x t . Theorem 3.1 (Diffusion). Assume Assumptions 2.1, 2.2, 2.3 and 2.4. Let the initial state ρ satisfy condition (3.1) and let μt be as defined in (3.2). There is a positive constant λ0 such that, for 0 < |λ| ≤ λ0 , 1 − √i q·x μt (x)e t −→ e− 2 q·Dλ q (3.4) t ∞
x∈Zd
with the diffusion matrix Dλ given by Dλ = λ2 (Dr w + o(λ)),
(3.5)
where Dr w is the diffusion matrix of the Markovian approximation to our model, to be defined in Sect. 4. Both Dλ and Dr w are strictly positive matrices (i.e., all eigenvalues are strictly positive) with real entries.
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The convergence of μt (·) to a Gaussian also holds in the sense of moments: For any natural number ∈ N, we have ⎛ ⎞ 1 i √ − q·x ⎝ − 2 q·Dλ q t ⎠ (∇q ) e μt (x)e −→ (∇ ) . (3.6) q t ∞ q=0 x∈Zd q=0
In particular, for = 2, this means that 1 xi x j μt (x) −→ (Dλ )i, j . t ∞ t d
(3.7)
x∈Z
Our next result describes the asymptotic ‘state’ of the particle. Not all observables reach a stationary value as t ∞. For example, as stated in Theorem 3.1, the position diffuses. The asymptotic state applies to the internal degrees of freedom of the particle and to functions of its momentum. Hence, we look at observables of the form F(P) ⊗ A,
F = F ∈ L ∞ (Td ),
A = A∗ ∈ B(S ),
(3.8)
with P = P ⊗ 1 the lattice momentum operator defined in Sect. 2.2. Such observables can be represented as elements of the Hilbert space L 2 (Td ) ⊗ B2 (S ) ∼ L 2 (Td , B2 (S ))=G (recall that S is finite-dimensional) by the obvious mapping F(P) ⊗ A → F ⊗ A L ∞ (Td )
(3.9)
L 2 (Td ). Consequently,
since ⊂ the asymptotic state is not described by a density matrix on HS , but by a functional on the Hilbert space G . This functional is called eq ξ eq ≡ ξλ (‘eq’ for equilibrium) and we identify it with an element of G . The asymptotic expectation value of F ⊗ A is given by
dk F(k) Tr S ξ eq (k)A . (3.10) F ⊗ A, ξ eq G = Td
We also state a result on decoherence: Equation (3.13) expresses that the off-diagonal elements of ρt in position representation are exponentially damped in the distance from the diagonal. Note that this is not in contradiction with Theorem 3.1 as the latter speaks about diagonal elements of ρt . Theorem 3.2 (Equipartition and decoherence). Assume Assumptions 2.1, 2.2, 2.3 and 2.4. Let the same conditions on the coupling constant λ and the initial state ρ be satisfied as in Theorem 3.1. Let A, F be as defined above. Then Tr[ρt (F(P) ⊗ A)] = F ⊗ A, ξ eq G + O(e−gλ t ), 2
for some decay rate g > 0. The function ξ eq (k) =
ξ eq
1 −βY e + o(|λ|0 ), Z (β)
≡
eq ξλ
t ∞
(3.11)
∈ G is given by
for all k ∈ Td ,
λ0
(3.12)
with the normalization constant Z (β) := (2π )d Tr(e−βY ). Further, there is a decoherence length (γdch )−1 > 0 such that ρt (x, y) B (S ) ≤ Ce−γdch |x−y| + O(e−gλ t ), 2
t ∞.
(3.13)
In particular, Theorem 3.2 implies that the inverse decoherence length γdch remains strictly positive as λ 0. Theorems 3.1 and 3.2 are derived from more general statements in the next section.
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3.2. Asymptotic form of the reduced evolution. In the following theorem, we present a more general statement about the asymptotic form of the reduced evolution Zt . The two previous results, Theorems 3.1 and 3.2, are in fact immediate consequences of this more general statement. As argued in Sect. 2.5, the operator Zt is translation invariant and hence it can be decomposed along the fibers, ⊕ {Zt } p ∈ B(G ). d p {Zt } p , (3.14) Zt = Td
The next result, Theorem 3.3, lists some long-time properties of the operators {Zt } and Uν {Zt } p U−ν with Uν as defined in (2.58). To fix the domains of the parameters p and ν, we define Dlow := p ∈ Td + iTd , ν ∈ Td + iTd |Re p| < p ∗ , |Im p| < δ, |Im ν| < δ , (3.15) Dhigh := p ∈ Td + iTd , ν ∈ Td + iTd |Re p| > p ∗ /2, |Im p| < δ, |Im ν| < δ , (3.16) depending on some positive constants p ∗ , δ > 0. Theorem 3.3 (Asymptotic form of reduced evolution). Assume Assumptions 2.1, 2.2, 2.3 and 2.4, and let the same conditions on the coupling constant λ and the initial state ρ be satisfied as in Theorem 3.1. Then there are positive constants p ∗ > 0 and δ > 0, determining the sets Dlow , Dhigh above, such that the following properties hold: 1) For small fibers p, i.e., such that ( p, 0) ∈ Dlow , there are rank-1 operators P( p, λ), bounded operators R low (t, p, λ) and numbers f ( p, λ), analytic in p on Dlow and satisfying Uν P( p, λ)U−ν < C,
(3.17)
sup Uν R low (t, p, λ)U−ν < C,
(3.18)
sup ( p,ν)∈Dlow
sup ( p,ν)∈Dlow
t≥0
such that {Zt } p = e f ( p,λ)t P( p, λ) + R low (t, p, λ)e−(λ
2 glow )t
(3.19)
for a positive rate glow > 0. 2) For large fibers p, i.e., such that ( p, 0) ∈ Dhigh , there are bounded operators R high (t, p, λ), analytic in p on Dhigh and satisfying sup ( p,ν)∈Dhigh
sup Uν R high (t, p, λ)U−ν = O(1),
λ0
(3.20)
t≥0
and {Zt } p = R high (t, p, λ)e−(λ
2 g high )t
for some positive rate g high > 0.
,
t ∞
(3.21)
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Fig. 2. The singular points of {R(z)} p as a function of the fiber momentum p. Above the irregular black line, the only singular points are given by f ( p, λ), in every small fiber p. Below the irregular black lines, we have no control
3) The function f ( p, λ) and rank-1 operator P( p, λ) satisfy sup f ( p, λ) − λ2 fr w ( p) = o(|λ|2 ),
(3.22)
( p,0)∈Dlow
sup ( p,ν)∈Dlow
Uν P( p, λ)U−ν − Uν Pr w ( p)U−ν = o(|λ|0 ),
λ 0,
(3.23)
where the function fr w ( p) and the projection operator Pr w ( p) are defined in Sect. 4. The main conclusion of this theorem is presented in Fig. 2. Let R(z) be the Laplace transform of the reduced evolution Zt and {R(z)} p its fiber decomposition, i.e., ⊕ −t z R(z) := dt e Zt and R(z) = d p {R(z)} p . (3.24) R+
Td
The figure shows the singular points, z = f ( p, λ), of {R(z)} p . Those singular points determine the large time asymptotics. If we had not integrated out the reservoirs, i.e., if Zt were the unitary dynamics, then one could identify f ( p, λ) with resonances of the generator of Zt . The proof of Theorem 3.3 forms the bulk of the present paper. 3.3. Connection between Theorem 3.3 and the Results in Section 3.1. In this section, we show how to derive Theorems 3.1 and 3.2 from Theorem 3.3. Since P( p, λ) is a rank-1 operator, we can write for some ξ( p, λ), ξ˜ ( p, λ) ∈ G (3.25) P( p, λ) = |ξ( p, λ) ξ˜ ( p, λ) ,
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635
using the notation introduced in (2.5). We derive a bound on the eigenvectors ξ( p, λ) and ξ˜ ( p, λ), analytically continued in the coordinate k. This bound follows from the analyticity and uniform boundedness of Uν P( p, λ)U−ν on Dlow and straightforward symmetry arguments. Lemma 3.4. The vectors Uν ξ( p, λ) and Uν ξ˜ ( p, λ) can be chosen bounded-analytic on Dlow . In other words, the operator P( p, λ) has a kernel P( p, λ)(k, k ) = ξ( p, λ)(k) ξ˜ ( p, λ)(k ) ,
(3.26)
which is bounded-analytic in both k and k in the domain |Im k|, |Im k | < δ. Note that for fixed k, k , the RHS of (3.26) belongs to B(B2 (S )). Proof. Since P( p, λ) is the dominant contribution to {Zt } for large t, the properties (2.67- 2.68) imply that (J E)P( p, λ)(J E) = P( p, λ)
(3.27)
(note that J E is an anti-unitary involution). Consequently, the eigenvectors ξ( p, λ) and ξ˜ ( p, λ) can be chosen such that J Eξ( p, λ) = ξ( p, λ) and J E ξ˜ ( p, λ) = ξ˜ ( p, λ). Then Uν ξ( p, λ) = Uν J Eξ( p, λ) = J EU−ν ξ( p, λ) = U−ν ξ( p, λ) . (3.28) Since Uν = eν∇k , we have also ξ( p, λ) ≤ 2 cosh(Im ν(i∇k ))ξ( p, λ) = (UiIm ν + U−iIm ν )ξ( p, λ) ≤ 2 Uν ξ ,
for any ν ∈ Cd .
(3.29)
The same relation holds for ξ˜ ( p, λ) and hence none of the factors on the RHS of Uν P( p, λ)U−ν B (G ) = Uν ξ( p, λ) Uν ξ˜ ( p, λ)
B (G )
= Uν ξ( p, λ) G Uν ξ˜ ( p, λ) G
(3.30)
can become small as ν varies. The lemma now follows from the uniform boundedness of Uν P( p, λ)U−ν . For p = 0, the vectors ξ( p, λ) and ξ˜ ( p, λ) play a distinguished role, and we rename them as eq
ξ eq = ξλ := ξ( p = 0, λ),
eq ξ˜ eq = ξ˜λ := ξ˜ ( p = 0, λ),
(3.31)
Note that ξ eq was already referred to in Theorem 3.2. By exploiting symmetry and positivity properties of the reduced evolution Zt , we can infer some further properties of the function f ( p, λ) and the operator P( p, λ).
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Proposition 3.5. The function f ( p, λ), defined for all p with ( p, 0) ∈ Dlow , has a negative real part, Re f ( p, λ) ≤ 0, and satisfies the following properties: f ( p = 0, λ) = 0, and ∇ p f ( p, λ) p=0 = 0. (3.32) The Hessian Dλ := (∇ p )2 f ( p, λ) p=0
has real entries and is strictly positive. The functions ξ eq and ξ˜ eq can be chosen such that
ξ˜ eq = 1, ξ eq (k) ≥ 0, dk Tr S ξ eq (k) = 1, ξ eq = 1, Td
(3.33)
(3.34)
where 1 ∈ G is the constant function on Td with value 1 ∈ B2 (S ). Moreover, it satisfies the space inversion symmetry (ξ eq ) (k) = (ξ eq ) (−k). The fact that f ( p = 0, λ) = 0, ξ˜ eq = 1 and (3.34) follow in a straightforward way from (2.66) and the asymptotic form (3.19). The symmetry property ξ eq (k) = ξ eq (−k) and ∇ p f ( p, λ) p=0 = 0 follow from (2.67) and (3.19). The fact that Dλ has real entries follows from f ( p, λ) = f (− p, λ) which in turn follows from the reality of the probabilities μt (x) and the convergence (3.4). To derive the strict positivity of Dλ , we use the claim (in Proposition 4.2) that Dr w , the Hessian of fr w ( p) at p = 0, is strictly positive. By the convergence (3.22) and the analyticity of fr w ( p), it follows that |Dλ − λ2 Dr w | 0 as λ 0. Indeed, if a sequence of analytic functions is uniformly bounded on some open set and converges pointwise on that set, then all derivatives converge as well. 3.3.1. Diffusion. We outline the derivation of Theorem 3.1. Let p be such that ( p, 0) ∈ Dlow . We calculate the logarithm of the characteristic function: log e−i px μt (x) x
= log
e−i px Tr S ρt (x, x)
x
= log1, {ρt } p = log1, {Zt } p {ρ0 } p 2 low = log e f ( p,λ)t 1, P( p, λ) {ρ0 } p + e−λ g t 1, R low (t, p, λ) {ρ0 } p 2 low = log e f ( p,λ)t 1, P( p, λ) {ρ0 } p + e−(λ g − f ( p,λ))t C 1 {ρ0 } p 2 low = f ( p, λ)t + log 1, P( p, λ) {ρ0 } p + e−(λ g − f ( p,λ))t C 1 {ρ0 } p , (3.35) where the scalar product ·, · and · refer to the Hilbert space G . The second equality follows from Lemma 2.6, the fourth from (3.19) and the fifth from (3.18). The second term between brackets in the last line vanishes as t ∞, for | p| small enough, such that λ2 glow − f ( p, λ) > 0. To conclude the calculation, we need to check that the expression in log (·) does not vanish. We note that 1, P( p = 0, λ) {ρ0 }0 = 1, ξ eq ξ˜ eq , {ρ0 }0 = 1
(3.36)
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as follows from the fact that ξ˜ eq = 1 and the normalization of ξ eq in (3.34). Hence, for p in a complex neighborhood of 0, the expression 1, P( p, λ) {ρ0 } p is bounded away from 0 by analyticity in p. Consequently, 1 e−i px μt (x) = f ( p, λ). log t ∞ t x lim
(3.37)
Next, we remark that, for i p real, the LHS of (3.37) is a large deviation generating function for the family of probability densities (μt (·))t∈R+ . A classical result [6] in large deviation theory states that the analyticity of the large deviation generating function in a neighborhood of 0 implies a central limit theorem for the variable √x t , both in distribution, see (3.4), as in the sense of moments, see (3.6). 3.3.2. Equipartition. To derive the result on equipartition in Theorem 3.2, we consider F, A as in (3.8). Since ρt (F(P) ⊗ A) is a trace-class operator, Lemma 2.6 implies that Tr [(F(P) ⊗ A)ρt ] = 1, {(F(P) ⊗ A)ρt }0 G = F ⊗ A, {ρt }0 G ,
(3.38)
where, as in (3.10), F ⊗ A stands for the function k → F(k)A in L 2 (Td , B2 (S )). Using Theorem 3.3 for the fiber p = 0, we obtain F ⊗ A, {ρt }0 = e f (0,λ)t F ⊗ A, P( p = 0, λ){ρ0 }0 + e−(λ
2 glow )t
F ⊗ A, R low (t, p = 0, λ){ρ0 }0
= F ⊗ A, ξ eq + Ce−(λ
2 glow )t
F ⊗ A G {ρ0 }0 G .
(3.39)
To obtain the second equality, we have used the uniform boundedness of the operators R low (t, p = 0, λ) (Statement 1) of Theorem 3.3), the fact that f ( p = 0, λ) = 0 (Proposition 3.5) and the identities P( p = 0, λ){ρ0 }0 = ξ˜ eq , {ρ0 }0 ξ eq = 1, {ρ0 }0 ξ eq = ξ eq .
(3.40)
Hence, from (3.39), we obtain the asymptotic expression (3.11) by choosing g ≤ glow . 3.3.3. Decoherence. In this section, we derive the bound (3.13) in Theorem 3.2. We decompose ρt as follows, using Theorem 3.3: ⊕ d p {ρt } p (3.41) ρt := Td ⊕ ⊕ 2 2 low = d p eλ f ( p,λ)t P( p, λ) {ρ0 } p +e−λ g t d p R low (t, p, λ) {ρ0 } p ∗ ∗ | p|≤ p | p|≤ p ! " ! " =:A1
+e
−λ2 g high t
=:A2
(3.42) ⊕
| p|> p ∗
d p R high (t, p, λ) {ρ0 } p . ! " =:A3
(3.43)
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The terms A2 and A3 are bounded by 2 A2,3 2 ≤ C d p {ρ0 } p 2G = C ρ0 22 ≤ C ρ0 21 , Td
(3.44)
where the first inequality follows from the bounds (3.18) and (3.20). Hence, for our purposes, it suffices to consider the first term A1 . To calculate the operator A1 in position representation, we use the kernel expression (3.26) for P( p, λ) to obtain A1 (x L , x R ) p = d pei 2 ·(x L +x R ) e f ( p,λ)t ξ˜ ( p, λ), {ρ0 } p | p|≤ p ∗
Td
dk ξ( p, λ)(k)eik·(x L −x R ) . (3.45)
We now shift the path of integration (in k) into the complex plane, using that the function ξ( p, λ)(·) is bounded-analytic in a strip of width δ. This yields exponential decay in (x L − x R ). Using also that Re f ( p, λ) ≤ 0, for | p| ≤ p ∗ (see Proposition 3.5), we obtain the bound A1 (x L , x R ) B2 (S ) ≤ Ce−γ |x L −x R | ,
for γ < δ.
(3.46)
Combining the bounds on A1 and A2 , A3 , we obtain for γ < δ (3.47) ρt (x L , x R ) B2 (S ) ≤ Ce−γ |x L −x R | + C e−(λ g)t , low high . The fact that this bound is valid for any γ < δ, confirms the with g := min g , g claim that the inverse decoherence length γdch can be chosen uniformly in λ as λ 0. 2
4. The Markov Approximation For small coupling strength λ and times of order λ−2 , one can approximate the reduced evolution Zt by a “quantum Markov semigroup” t which is of the form t = et
−iad(Y )+λ2 M
,
(4.1)
where Y = 1 ⊗ Y is the Hamiltonian of the internal degrees of freedom, and M is a Lindblad generator, see e.g. [1]. Lindblad generators, and especially the semigroups they generate, have received a lot of attention lately in quantum information theory. The operator M has the additional property of being translation-invariant. Translation-invariant Lindbladians have been classified in [25] and, recently, studied in a physical context; see [34] for a review. In Sect. 4.1, we construct M and we state its relation with Zt . We also describe heuristically how M emerges from time-dependent perturbation theory in λ as a lowest order approximation to Zt . In Sect. 4.2, we discuss the momentum representation of M (the derivation of this representation is however deferred to Appendix C), and we recognise that the evolution equation generated by M is a mixture of a linear Boltzmann equation for the translational degrees of freedom and a Pauli master equation for the internal degrees of freedom. In Sect. 4.3, we discuss spectral properties of M, which are largely proven in Appendix C. Finally, in Sect. 4.3.1, we derive bounds on the long-time behavior of t ρ, for any density matrix ρ ∈ B1 (HS ).
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4.1. Construction of the semigroup. First, we define the operator Lˆ (t) on B(HS ): β Lˆ (t)(S) = −ρR ad(HSR ) eitad(Y +HR ) ad(HSR )(S ⊗ 1) . (4.2) β
This definition makes sense since the conditional expectation ρR is applied to an element of B(HS ) ⊗ C, see Sect. 2.4. Then we consider the Laplace transform of Lˆ (t), i.e. L (z) = dt e−t z Lˆ (t), Re z > 0, (4.3) R+
and, finally, we let L(z) be the dual operator to L (z), acting on B1 (HS ), see (2.44). Then the operator M is obtained from L by “spectral averaging” and adding the “Hamiltonian” term −iad(ε(P)): M := −iad(ε(P)) + 1a (ad(Y ))L(−ia)1a (ad(Y )). (4.4) a∈sp(ad(Y ))
For now, this definition is formal, since it involves (4.3) with Re z = 0. The following proposition provides a careful definition of M and collects some basic properties of the semigroup evolution t . Proposition 4.1. Assume Assumptions 2.1, 2.2 and 2.3. Then, the operators L(z), defined above, can be continued from Re z > 0 to a continuous function in the region Re z ≥ 0 and sup Jκ L(z)J−κ < ∞,
for κ ∈ Cd × Cd .
(4.5)
Re z≥0
(In fact, Jκ L(z)J−κ = L(z).) The operator M, as defined in (4.4), is bounded both on B1 (HS ) and B2 (HS ). Recall the constants qε (γ ), γ > 0, defined in Assumption 2.1. Then Jκ MJ−κ − M ≤ qε (|Im κ L |) + qε (|Im κ R |),
|Im κ L ,R | ≤ δε ,
(4.6)
where the norm · refers to the operator norm on B(B2 (HS )). The family of operators t , defined in (4.1), 2 t = et (−iad(Y )+λ M) ,
t ∈ R+
(4.7)
is a “quantum dynamical semigroup”. This means:2 i) t1 t2 = t1 +t2 ii) t ρ ≥ 0 iii) Tr t ρ = Tr ρ
for all t1 , t2 ≥ 0 (semigroup property), for any 0 ≤ ρ ∈ B1 (HS ) (positivity preservation), (4.8) for any 0 ≤ ρ ∈ B1 (HS ) (trace preservation)
We postpone the proof of this proposition to Appendix C. 2 Most authors include “complete positivity” as a property of quantum dynamical semigroups, see e.g. [1]. Although the operators t satisfy complete positivity, we do not stress this since it is not important for our analysis.
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4.1.1. Motivation of the semigroup t . The connection of the semigroup t with the reduced evolution Zt is that, for any T < ∞, sup 0
Zt − t B (B2 (HS )) = o(λ0 ),
λ 0.
(4.9)
Results in the spirit of (4.9) have been advocated by [35] and first proven, for confined (i.e. with no translational degrees of freedom) systems, in [10]. They go under the name “weak coupling limit” and they have given rise to extended mathematical studies, see e.g. [12,31]. In our model, (4.9) will be implied by our proofs but we will not state it explicitly in the form given above. In fact, statements like (4.9) can be proven under much weaker assumptions than those in our model; see [16] for a proof which holds in all dimensions d > 1. Here, we restrict ourselves to a short and heuristic sketch of the way the Lindblad generator M emerges from the full dynamics. First, we consider the Lie-Schwinger series (2.43) in the interaction picture with respect to the free internal degrees of freedom, i.e. we consider Zt e−itad(Y ) instead of Zt . Keeping only terms up to second order in λ in (2.43) and substituting our definition for Lˆ (t) we obtain Zt e−itad(Y ) = 1 + iλ2 tad(ε(P)) 2 +λ dt1 dt2 ei(t−t2 )ad(Y ) Lˆ (t2 − t1 )e−i(t−t1 )ad(Y ) + O(λ4 ), (4.10) 0
where we have also used [ε(P), Y ] = 0 to simplify the second term on the RHS. It is useful to rewrite the third term by splitting t2 = t1 + (t2 − t1 ) and to insert the spectral decomposition of unity corresponding to the operator ad(Y ): we get t−t1 t 2 i(t−t1 )(a−a ) −iua ˆ λ L (u) 1a (ad(Y )). dt1 e 1a (ad(Y )) du e 0
a,a ∈sp(ad(Y )) 0
(4.11) Next, we analyze the RHS of (4.10) for long times; we choose t = λ−2 t, and we argue that, in the limit λ 0, it reduces to 1 + iad(ε(P))t + 1a (ad(Y ))L (ia)1a (ad(Y )) t. (4.12) a
This limit can be straightforwardly justified if the function t → L∗ (t) is (norm-)integrable, which will follow from Assumptions 2.2 and 2.3 in our case. Indeed, if t → L∗ (t) # λ−2 t−t1 #∞ is integrable, then the integral 0 du . . . in (4.11) converges to 0 du . . . for fixed t1 , as λ → 0. This yields the Laplace transform L (ia). The restriction a = a appears then because λ−2 t 2 λ dt1 ei(t−t1 )(a−a ) →λ→0 δa,a t (4.13) 0
and one finishes the argument by invoking dominated convergence. Comparing with (4.4) and using that (1a (ad(Y ))) = 1−a (ad(Y )), one checks that (4.12) is equal to 1 + tM , with M the dual to M. This is the beginning of a series defining the semi group etM (we got only the first two terms because we kept only terms of order λ0 and 2 λ in the original Lie-Schwinger series).
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4.2. Momentum space representation of M. In this section, we give an explicit and intuitive expression for the operator M. As M is translation covariant, i.e., Tz MT−z = M, as in (2.52), we have the fiber decomposition, ⊕ M= d p Mp, (4.14) Td
where the notation is as introduced in Sect. 2.5. We describe M p explicitly as an operator on G . It is of the form p p −ε k− ξ(k) + (N ξ )(k), ξ ∈ G , M p ξ (k) = −i[ϒ, ξ(k)] − i ε k + 2 2 (4.15) where ε is the dispersion law of the particle, see Sect. 2.2, and ϒ is a self-adjoint matrix in B(S ) whose only relevant property is that it commutes with Y , i.e., [Y, ϒ] = 0. Physically, it describes the Lamb-shift of the internal degrees of freedom due to the coupling to the reservoir and its explicit form is given in Appendix C. The operator N is given, for ξ ∈ C(Td , B2 (S )), by (N ξ )(k) = dk ra (k , k)Wa ξ(k )Wa∗ a∈sp(ad(Y ))
1 − ra (k, k ) ξ(k)Wa∗ Wa + Wa∗ Wa ξ(k) 2
(4.16)
with the (singular) jump rates ra (k, k ) := 2π
Rd
dq |φ(q)|
2
1 δ(ω(q) − a)δTd (k − k − q) 1−e−βω(q) 1 δ(ω(q) + a)δTd (k − k + q) eβω(q) −1
a ≥ 0, a < 0,
(4.17)
where φ is the form-factor, see Sect. 2.3, and δTd (·) is a sum of Dirac delta distributions on the torus; δTd (·) := δ(· − q0 ). (4.18) q0 =0+(2π Z)d
Note that ra (·, ·) vanishes at a = 0, due to the fact that the ‘effective squared form factor’ ˆ vanishes at 0, see Assumption 2.3. ψ(·) Equation (4.15) is most easily checked starting from the expressions for M in Sect. C.1. In particular, the three terms in (4.16) correspond to the fiber decompositions of the operators (ρ), − 21 (1)ρ, − 21 ρ (1) in (C.7), and the first two terms on the RHS of (4.15) correspond to the commutator with ϒ and ε(P) in (C.7). We already stated that M is translation-invariant, hence it commutes with ad(P). However, the operator M also commutes with ad(Y ), as can be easily checked starting from the expressions (4.15) and (4.16) and employing the definitions of Wa in (2.30) and the fact that [Y, ϒ] = 0. We can therefore construct the double decomposition ⊕ M= ⊕ d p M p,a , (4.19) a∈sp(ad(Y )) Td
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where M p,a := 1a (ad(Y ))M p 1a (ad(Y )).
(4.20)
To proceed, we make use of our strong nondegeneracy condition in Assumption 2.4. Indeed, the operators M p,a act on functions ξ ∈ G that satisfy the constraint ξ(k) = 1a (ad(Y ))ξ(k) = 1e (Y )ξ(k)1e (Y ),
ξ ∈ G ∼ L 2 (Td , B2 (S )). (4.21)
e,e ∈spY,e−e =a
Due to the non-degeneracy assumption, the sum on the RHS contains only one non-zero term for a = 0, i.e., there are unique eigenvalues e, e such that a = e − e . Let us denote the eigenvector in the space S of the operator Y with eigenvalue e by e as well (cfr. the discussion following Assumption 2.4), then this unique term in (4.21) can be written as 1e (Y )ξ(k)1e (Y ) = e, ξ(k)e |ee |, e − e = a. (4.22) It follows that the matrix valued function ξ(k) satisfying (4.21) can be identified with the C-valued function ϕ(k) ≡ e, ξ(k)e S .
(4.23)
For a = 0, a function ξ(k) satisfying (4.21) is necessarily diagonal in the basis of eigenvectors of Y . In that case, we can identify ξ with ϕ(k, e) ≡ e, ρ(k, k)eS .
(4.24)
Hence, we can identify M p,a=0 with an operator on L 2 (Td ) and M p,0 with an operator on L 2 (Td × spY ). A careful analysis of these operators is performed in Appendix C. Here, we discuss the operator M0,0 because it is crucial for understanding our model. 4.2.1. The markov generator M0,0 . Let us choose ϕ ∈ C(Td × spY ). Then, by the formulas given above, the operator M0,0 acts as M0,0 ϕ(k, e) := r (k , e ; k, e)ϕ(k , e ) − r (k, e; k , e )ϕ(k, e) , dk Td
(4.25)
e ∈spY
where r (k, e; k , e ) are (singular) transition rates given explicitly by r (k, e; k , e ) := re−e (k, k )|e , W e|2 .
(4.26)
In formula (4.25), one recognizes the structure of a Markov generator, acting on densities of absolutely continuous probability measures (hence L 1 -functions) on Td × spY . The numbers j (e, k) := dk r (k, e; k , e ) (4.27) Td
e ∈spY
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643
(b)
Fig. 3. The processes contributing to the gain term (the first term on the RHS in (4.25)) of the operator M0,0 . Emission corresponds to e > e and absorption to e < e
are processes. Let ϕ 1 # called escape rates in the context of Markov is a positive measure, we get dk |ϕ(k, e)|. Using that r (k, e; k , e )dk e∈spY Td
=
M0,0 ϕ 1 ≤ 2 sup j (e, k) ϕ 1 ,
(4.28)
e,k
which implies that M0,0 is bounded on L 1 (Td × spY ). In particular, this means that M0,0 is a bonafide Markov generator (i.e. it generates a strongly continuous (in our case even norm-continuous) semigroup) and et M0,0 ϕ is a probability density for all t ≥ 0. Physically speaking, the probability density ϕ is read off from the diagonal part of the density matrix ρ, see (4.24). We note that the transition rates r (k, e; k , e ) satisfy the ‘detailed balance property’ at inverse temperature β for the internal energy levels e, e , and at infinite temperature for the momenta k, k :
r (k, e; k , e ) = eβ(e−e )r (k, e ; k, e).
(4.29)
Physically, we would expect overall detailed balance at inverse temperature β, i.e.
r (k, e; k , e ) = eβ(E(e,k)−E(e ,k ))r (k, e ; k, e),
(4.30)
where the energy E(k, e) should depend on both e and k. To understand why E does not depend on k in (4.29), we recall that the kinetic energy of the particle is assumed to be of order λ2 ; hence, the total energy is e + λ2 ε(k) which reduces to e in zeroth order in λ. One can associate an intuitive picture with the operator M0,0 . It describes the stochastic evolution of a particle with momentum k and energy e. The state of the particle changes from (k, e) to (k , e ) by emitting and absorbing reservoir particles with momentum q and energy ω(q), such that total momentum and total energy (which does not include any contribution from k, k ) are conserved, see Fig. 3. It is clear from the collision rules in Fig. 3 that, in the absence of internal degrees of freedom, the particle can only emit or absorb bosons with momentum q = 0, and hence it cannot change its momentum. This means that without the internal degrees of freedom, the semigroup t would not exhibit any diffusive motion. This is indeed the reason why we introduced these internal degrees of freedom.
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4.3. Asymptotic properties of the semigroup. The following Proposition 4.2 states some spectral results on the Lindblad operator M and its restriction to momentum fibers M p ∈ B(G ). These results are stated in a way that mirrors, as closely as possible, the statements of Theorem 3.3. These results are useful for two purposes. First of all, they show that our main physical results, Theorems 3.1 and 3.2, hold true if one replaces the reduced evolution Zt by the semigroup t (see the remark following Proposition 4.2). Second, a bound which follows directly from Proposition 4.2 will be a crucial ingredient in the proof of our main result Theorem 3.3. This bound is stated in (4.55) in Sect. 4.3.1. We introduce the following sets (cf. (3.15–3.16)): d d d d ∗ Dlow r w := p ∈ T + iT , ν ∈ T + iT |Re p| < pr w , |Im p| ≤ δr w , |Im ν| ≤ δr w , (4.31) 1 high Dr w := p ∈ Td + iTd , ν ∈ Td + iTd |Re p| > pr∗w , 2
(4.32) |Im p| ≤ δr w , |Im ν| ≤ δr w , depending on positive parameters pr∗w > 0 and δr w > 0. The subscript ‘r w’ stands for ‘random walk’ and it will always be used for objects related to t . Proposition 4.2. Assume Assumptions 2.1, 2.2, 2.3 and 2.4. There are positive conhigh stants pr∗w > 0 and δr w > 0, determining Dlow r w , Dr w above, such that the following properties hold: 1) For small fibers p, i.e., such that ( p, ν) ∈ Dlow r w , the operator Uν M p U−ν is bounded and has a simple eigenvalue fr w ( p), independent of ν, sp(Uν M p U−ν ) = { fr w ( p)} ∪ p,ν .
(4.33)
The eigenvalue fr w ( p) is elevated above the rest of the spectrum, uniformly in p, i.e., there is a positive grlow w > 0 such that sup ( p,ν)∈Dlow rw
Re p,ν < −grlow w <
inf
( p,ν)∈Dlow rw
Re fr w ( p) ≤ 0.
(4.34)
The one-dimensional spectral projector Uν Pr w ( p)U−ν corresponding to the eigenvalue fr w ( p), is uniformly bounded: sup ( p,ν)∈Dlow
Uν Pr w ( p)U−ν ≤ C.
(4.35)
high
2) For large fibers p, i.e., such that ( p, 0) ∈ Dr w , the operator Uν M p U−ν is bounded and its spectrum lies entirely below the real axis, i.e., sup high
( p,ν)∈Dr w
high Re sp Uν M p U−ν < −gr w ,
high
for some gr w > 0.
(4.36)
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Fig. 4. The spectrum of M p as a function of the fiber momentum p. Above the irregular black line, the only spectrum consists of the isolated eigenvalue fr w ( p), in every small fiber p. Below the irregular black lines, we have no control
3) The function fr w ( p), defined for all p such that ( p, 0) ∈ Dlow r w , has a negative real part, Re fr w ( p) ≤ 0, and satisfies and ∇ p fr w ( p)) p=0 = 0. (4.37) fr w ( p = 0) = 0, The Hessian Dr w := (∇ p )2 fr w ( p) p=0
has real entries and is strictly positive.
(4.38)
The spectral projector Pr w ( p = 0) is given by eq eq Pr w ( p = 0) = |ξ˜r w ξr w |,
(4.39)
with eq ξ˜r w (k) = 1B (S ) ,
and
eq
ξr w (k) =
e−βY 1 , d (2π ) Tr(e−βY )
k ∈ Td .
(4.40)
The conclusion of Proposition 4.2 is sketched in Fig. 4. The proof of this proposition is very analogous to the proof in [8] (which, however, does not consider internal degrees of freedom). For completeness, we reproduce the proof in Appendix C. From Proposition 4.2, one can derive that the semigroup et (−iad(Y )+M) exhibits diffusion, decoherence and equipartition. This follows by analogous reasoning as in Sects. 3.3.1, 3.3.2 and 3.3.3, but starting from Proposition 4.2 instead of Theorem 3.3. The matrix Dr w is the diffusion constant, the inverse decoherence length has to be choeq sen smaller than δr w and the function ξr w is the ‘equilibrium state’. We do not state these properties explicitly as they are not necessary for the proof of our main results.
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4.3.1. Bound on t in position representation. By virtue of Proposition 4.2, we can write {t } p = Pr w ( p)eλ
fr w ( p)t
2
high
{t } p = Rr w ( p, t)e
−λ + Rrlow w (t, p)e
2 glow t rw
high −λ2 gr w t
,
( p, ν) ∈ Dlow rw ,
high
, ( p, ν) ∈ Dr w ,
(4.41) (4.42)
high
with Pr w ( p) as defined above, satisfying (4.35), and the operators Rrlow w , Rr w satisfying sup ( p,ν)∈Dlow rw
sup Uν Rrlow w (t, p)U−ν < C,
(4.43)
t≥0
high
sup high ( p,ν)∈Dr w
sup Uν Rr w (t, p)U−ν < C.
(4.44)
t≥0
The appearance of the factor λ2 is due to the fact that λ2 multiplies M in the definition of the semigroup t . Next, we derive estimates on t (see e.g. the bound (4.55) below) starting from (4.41– 4.42) and (4.43–4.44), without using explicitly the semigroup property of t . This is important since in the proof of Lemma 6.2, we will carry out an analogous derivation for objects which are not semigroups. We choose κ = (κ L , κ R ) ∈ Cd × Cd such that Re κ L = Re κ R = 0 and we calculate, using relation (2.60), ⊕ d p Uν {t } p+p U−ν , Jκ t J−κ = Td
with p :=
κ + κR κL − κ R ν := L , 2 4
(4.45)
where we use the analyticity in ( p, ν), see (4.35) and (4.43–4.44). Recall that {} p acts on G p ∼ G ∼ L 2 (Td , B(S )). Our choice for κ ensures that p and ν are purely imaginary. Next, we split the integration over p ∈ Td into small fibers (| p| < pr∗w ) and large fibers (| p| ≥ pr∗w ) by defining I low := p + p | p ∈ Td , | p| < pr∗w , (4.46) I high := p + p | p ∈ Td , | p| ≥ pr∗w . Using the relations (4.41) and (4.42), we obtain Jκ t J−κ =
⊕
⊕ 2 2 low d p eλ fr w ( p)t Uν Pr w ( p)U−ν + e−λ gr w t d p Uν Rrlow w ( p, t)U−ν I low I low ! " ! " =:B1
+
2 high e−λ gr w t
⊕
I high
=:B2
high
d p Uν Rr w ( p, t)U−ν . ! "
(4.47)
=:B3
We establish decay properties of the operators B1,2,3 in position representation. For B2 and B3 we proceed as follows. Recall the duality (2.61–2.62). By varying p and ν,
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and using the bounds (4.43–4.44), we obtain γ
(B2,3 )x L ,x R ;x L ,x R S ≤ Ce− 2 |(x L +x R )−(x L +x R )| e−γ |(x L −x R )−(x L −x R )|
(4.48)
for any γ < δr w . For B1 , we need a better bound, which is attained by exploiting the fact that Pr w ( p) is a rank-1 operator with a kernel of the form (recall the notation of (2.5)) Pr w ( p)(k, k ) = (ξr w ( p)) (k) (ξ˜r w ( p))(k ) , for some ξr w ( p), ξ˜r w ( p) ∈ G ,
(4.49)
where both ξr w ( p), ξ˜r w ( p) are bounded-analytic functions of k, k , respectively, in a strip of width δr w . This follows from boundedness and analyticity of Pr w ( p) by the same reasoning as in Lemma 3.4. By the definition of B1 in (4.47) and (2.56), (2.60), p 2 (B1 )x L ,x R ;x L ,x R = d p eλ fr w ( p)t e−i 2 ((x L +x R )−(x L +x R )) (4.50) low I × dk dk e−ik(x L −x R )+ik (x L −x R ) Pr w ( p)(k, k ). (4.51) Td +ν
Td +ν
By the analyticity of Pr w ( p)(·, ·) in k, k , p, we derive, for γ < δr w , γ
(B1 )x L ,x R ;x L ,x R S ≤ Cerr w (γ ,λ)t e− 2 |(x L +x R )−(x L +x R )| e−γ |x L −x R | e−γ |x L −x R | , (4.52)
where the function rr w (γ , λ) is given by rr w (γ , λ) := λ2
sup
|Im p|≤γ ,|Re p|≤ pr∗w
max (Re fr w ( p), 0) .
(4.53)
Note that rr w (γ , λ) := O(λ2 )O(γ 2 ),
λ 0, γ 0,
(4.54)
as follows from Re fr w ( p) ≤ 0 for p ∈ R. The bound (4.54) will be used to argue that the exponential blowup in time, given by err w (γ ,λ)t can be compensated by the decay 2 e−λ gr w t by choosing γ small enough, see Sect. 5. Putting the bounds on B1,2,3 together, we arrive at γ
(t )x L ,x R ;x L ,x R ≤ Cerr w (γ ,λ)t e− 2 |(x L +x R )−(x L +x R )| e−γ |x L −x R | e−γ |x L −x R | + C e−λ
2g t rw
γ
e− 2 |(x L +x R )−(x L +x R )| e−γ |(x L −x R )−(x L −x R )| (4.55)
high
for γ < δr w and with the rate gr w := min (grlow w , gr w ). The bound (4.55) is the main result of the present section and it will be used in a crucial way in the proofs. The importance of this bound is explained in Sect. 5.4. For completeness, we note that a bound like 2q
(t )x L ,x R ;x L ,x R ≤ e2λ
ε (γ )t
e−γ |x L −x L | e−γ |x R −x R |
(4.56)
can be derived simply from the fact that Jκ MJκ is bounded for complex κ, see (4.6), since κL κR
is conjugate to is conjugate to
(x L − x L ), (x R − x R ).
(4.57) (4.58)
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5. Strategy of the Proofs In this section, we outline our strategy for proving the results in Sect. 3. We start by introducing and analyzing the space-time reservoir correlation function ψ(x, t). Then we introduce a perturbation expansion for the reduced evolution Zt (which involves the reservoir correlation function). Afterwards, we describe and motivate the temporal cutoff that we will put into the expansion. Finally, a plan of the proof is given.
5.1. Reservoir correlation function. A quantity that will play an important role in our analysis is the free reservoir correlation function ψ(x, t), which we define next. Let x ISR (5.1) := dq φ(q)eiq·x 1x ⊗ aq + φ(q)e−iq·x 1x ⊗ aq∗ , where 1x = 1x (X ) is the projection on HS , acting as (1x ϕ)(x ) = δx,x ϕ(x) for ϕ ∈ x is the part of the system-reservoir coupling that acts at site l 2 (Zd , S ). The operator ISR x after setting the matrix W ∈ B(S ) equal to 1 (recall that the matrix W describes the coupling of the internal degrees of freedom to the reservoir). We also define the timeevolved interaction term, with the time-evolution given by the free reservoir dynamics x x −it HR ISR (t) := eit HR ISR e .
(5.2)
The reservoir correlation function ψ is then defined as β x 0 ψ(x, t) := ρR ISR (t)ISR (0) , = φ x , Tβ eitω φ + φ x , (1 + Tβ )e−itω φh h iωt ˆ = dωψ(ω)e ds eiωs·x , R
Sd−1
(5.3)
where (φ x )(q) := eiq·x φ(q) and Sd−1 is the d − 1-dimensional hypersphere of unit radius. The ‘effective squared form factor’ ψˆ was introduced in (2.27), and the density operator Tβ in Sect. 2.3.3). Assumptions 2.2 and 2.3 imply certain properties of the correlation function that will be primary ingredients of the proofs. We state these properties as lemmata. In fact, one could treat these properties as the very assumptions of our paper, since, in practice, Assumptions 2.2 and 2.3 will only be used to guarantee these properties, Lemmata 5.1 and 5.2. The straightforward proofs of Lemmata 5.1 and 5.2 are postponed to Appendix A. The following lemma states that the free reservoir has exponential decay in t whenever |x|/t is smaller than some speed v∗ . Lemma 5.1 (Exponential decay at ‘subluminal’ speed). Assume Assumptions 2.2 and 2.3. Then there are positive constants v∗ > 0, gR > 0 such that |ψ(x, t)| ≤ C exp (−gR |t|), if
|x| ≤ v∗ , t
for some constant C.
(5.4)
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Property (5.4) is satisfied if the reservoir is ‘relativistic’, i.e., if the dispersion law ω(q) of the reservoir particles is linear in the momentum q, temperature β −1 is positive and the form factor φ satisfies the infrared regularity condition that q → |φ(q)|2 |q| is analytic in a strip around the real axis. The speed v ∗ has to be chosen strictly smaller than the propagation speed of the reservoir modes given by the slope of ω. In fact, the decay rate gR vanishes when v ∗ approaches the propagation speed of the reservoir modes. Lemma 5.1 does not depend on the fact that the dimension d ≥ 4. Lemma 5.2 (Time-integrable correlations). Assume Assumptions 2.2 and 2.3. Then dt sup |ψ(x, t)| < ∞. (5.5) R+
x∈Zd
This property is satisfied for non-relativistic reservoirs, with ω(q) ∝ |q|2 , in d ≥ 3 and for relativistic reservoirs, with ω(q) ∝ |q|, in d ≥ 4, provided that we choose the coupling to be sufficiently regular in the infrared.
5.2. The Dyson expansion. In this section, we set up a convenient notation to handle the Dyson expansion introduced in Lemma 2.5. We define the group Ut on B(HS ) by Ut S := e−it HS Seit HS ,
S ∈ B(HS ),
(5.6)
and the operators Ix,l , with x ∈ Zd and l ∈ {L , R} (L , R stand for ‘left’ and ‘right’), as if l = L −i (1x ⊗ W )S (5.7) Ix,l S := S ∈ B(HS ), i S(1x ⊗ W ) if l = R where the operators 1x ≡ 1x (X ) are projections on a lattice site x ∈ Zd , as used in Sect. 5.1. We write (ti , xi , li ), i = 1, . . . , 2n to denote 2n triples in R × Zd × {L , R} and we assume them to be ordered by the time coordinates, i.e., ti < ti+1 . We evaluate the Lie-Schwinger series (2.43) using the properties (2.21–2.22–2.23), and we arrive at 2n 2n Vt (ti , xi , li )i=1 , Zt = dt1 . . . dt2n ζ π, (ti , xi , li )i=1 n∈Z+ π ∈Pn
0
(5.8) where π ∈ Pn are pairings, as in (2.23), and we define 2n := Ut−t2n Ix2n ,l2n . . . Ix2 ,l2 Ut2 −t1 Ix1 ,l1 Ut1 Vt (ti , xi , li )i=1 with Ut as in (5.6) and
ζ
2n π, (ti , xi , li )i=1
⎧ ⎪ ⎪ ψ (xs ⎨ ψ (xs 2 := λ ⎪ ψ (xs ⎪ ⎩ (r,s)∈π ψ (xs
− x r , ts − x r , ts − x r , ts − x r , ts
− tr ) − tr ) − tr ) − tr )
lr lr lr lr
(5.9)
= ls = L = ls = R (5.10) = L , ls = R = R, ls = L
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Fig. 5. Graphical representation of a term contributing to the RHS of (5.8) with π = {(1, 3), (2, 4), (5, 8), (6, 10), (7, 11), (9, 12)} ∈ P6 . The times ti correspond to the position of the points on the horizontal axis
with the correlation function ψ as defined in (5.3). We recall the convention r < s for each element of a pairing π . For n = 0, the integral in (5.8) is meant to be equal to Ut . In Sect. 7, we will introduce some combinatorial concepts to deal with the pairings π ∈ Pn that are used in (5.8). 2n ) ∈ P × ([0, t] × Zd × For convenience, we will replace the variables (π, (ti , xi , li )i=1 n 2n {L , R}) by a single variable σ that carries the same information (Fig. 5). Starting from this graphical representation, we can reconstruct the corresponding term in (5.8) - an operator on B2 (HS ))- as follows: • To each straight line between the points (ti , xi , li ) and (t j , x j , l j ), one associates the operators e±i(t j −ti )HS , with ± being − for li = l j = L and + for li = l j = R. • To each point (ti , xi , li ), one associates the operator λ2 Ixi ,li , defined in (5.7). • To each curved line between the points (tr , xr , lr ) and (ts , xs , ls ), with r < s, we associate the factor ψ # (xs − xr , ts − tr ) with ψ # being ψ or ψ, depending on lr , ls , as prescribed in (5.10). Rules like these are commonly called “Feynman rules” by physicists. 5.3. The cut-off model. In our model, the space-time correlation function ψ(x, t) does not decay exponentially in time, uniformly in space, i.e, there is no g > 0
such that
sup |ψ(x, t)| ≤ Ce−g|t| .
(5.11)
x∈Zd
The impossibility of choosing the form factor φ or any other model parameter such that one has exponential decay is a fundamental consequence of local momentum conservation, as explained in Sect. 1.3. However, if the correlation function ψ(x, t) did decay exponentially, we could set up a perturbation expansion for Zt around the Markovian limit t . Such a scheme was implemented in [15], building on an expansion introduced in [14]. In the present section, we modify our model by introducing a cutoff time τ into the correlation function ψ(x, t). More concretely, we modify the perturbation expansion for Zt by replacing ψ(x, t),
−→
1|t|≤τ ψ(x, t).
(5.12)
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The cutoff time τ will be chosen as a function of λ satisfying τ (λ) → ∞,
τ (λ)λ → 0,
as λ 0.
(5.13)
However, we will take care to keep τ explicit in the estimates, until Sect. 8 where the τ -dependence will often be hidden in generic constants c(γ , λ), c (γ , λ). With the cutoff in place, the correlation function ψ(x, t) decays exponentially, uniformly in x ∈ Zd , i.e., obviously, |t|
sup 1|t|≤τ |ψ(x, t)| ≤ Ce− τ .
(5.14)
x∈Zd
The modified reduced dynamics obtained in this way will be called Ztτ . That is, 2n 2n Vt (ti , xi , li )i=1 Ztτ = dt1 . . . dt2n ζτ π, (ti , xi , li )i=1 n∈Z+ π ∈Pn
0
(5.15) with ⎛ ⎞ 2n 2n ζτ π, (ti , xi , li )i=1 := ⎝ . 1|ts −tr |≤τ ⎠ ζ π, (ti , xi , li )i=1
(5.16)
(r,s)∈π
If τ is chosen to be independent of λ then one can analyze Ztτ by the technique deployed in [15]. It turns out that for a λ-dependent τ , one can still analyze the cutoff model by the same techniques as long as λ2 τ (λ) 0 as λ 0, which is satisfied by our choice (5.13). The analysis of Ztτ is outlined in Lemma 6.1, in Sect. 6.1. The main conclusion of the treatment of the cutoff model is that The cutoff reduced dynamics Ztτ is ‘close’ to the semigroup t .
(5.17)
This conclusion is partially embodied in Lemma 6.2. For the sake of this explanatory chapter, one can identify Ztτ with t . The reason why it is useful to treat the cutoff model first, is that we will perform a renormalization step, effectively replacing the free evolution Ut in the expansion (5.8) by the cutoff reduced dynamics Ztτ . The benefit of such a replacement is explained in Sect. 5.4.
5.4. Exponential decay for the renormalized correlation function. 5.4.1. The joint system-reservoir correlation function. We recall that the free reservoir correlation function ψ(x, t) does not decay exponentially in t, uniformly in x. This was mentioned already in Sect. 5.3 and it motivated the introduction of the temporal cutoff τ . In the perturbation expansion for the reduced evolution Zt , the correlation function ψ(x, t) models the propagation of reservoir modes over a space-time ‘distance’ (x, t)
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and it occurs together with terms describing the propagation of the particle. Let us look at the lowest-order terms in the expansion of Zt , introduced in Sect. 5.2 above; Zt = λ
2 0
dt2 dt1
ψ # (x2 − x1 , t2 − t1 ) Ut−t2 Ix2 ,l2 Ut2 −t1 Ix1 ,l1 Ut1
x1 ,x2 ,l1 ,l2
+ higher orders in λ
(5.18)
with ψ # being ψ or ψ, as prescribed by the rules in (5.10). It is natural to ask whether the ‘joint correlation function’ ψ # (x2 − x1 , t2 − t1 ) Ix2 ,l2 Ut2 −t1 Ix1 ,l1
(5.19)
has better decay properties than ψ(x, t) by itself. In particular, we ask whether (5.19) is exponentially decaying in t2 − t1 , uniformly in x2 − x1 . This turns out to be the case only if l1 = l2 since in that case, the question essentially amounts to bounding 2 |ψ(x2 − x1 , t2 − t1 )| × e±i(t2 −t1 )λ ε(P) (x1 , x2 ) .
(5.20)
The expression (5.20) has exponential decay in time because 1 • For speed xt22 −x −t1 greater than some v > 0, we estimate 2 ±i(t2 −t1 )λ2 ε(P) (x1 , x2 ) ≤ e−(γ v−λ qε (γ ))|t2 −t1 | , e
for 0 < γ ≤ δε
(5.21)
with δε , qε (·) as in (2.12) and Assumption 2.1. Hence, for fixed v, one can choose γ so as to make on the RHS of (5.21) negative, for λ small enough. the exponent x2 −x1 • For speeds t2 −t1 smaller than v ∗ > 0, the reservoir correlation function ψ(x2 − x1 , t2 − t1 ) decays with rate gR , as asserted in Lemma 5.1 with v ∗ as defined therein. When l1 = l2 in (5.19), there is no decay at all from Ut2 −t1 , in other words, the ‘matrix element’
Ut2 −t1
x L ,x R ;x L ,x R
(5.22)
is obviously not decaying in the variables x L − x R or x R − x L , since it is a function of x L − x L and x R − x R only. Hence, for l2 = l1 , the joint correlation function (5.19) has as poor decay properties as the reservoir correlation function ψ(x, t). The situation is summarized in the following table:
l1 = l2 No exp. decay
joint S − R correlation fct. |x|/t > v ∗ |x|/t ≤ v ∗ l1 = l2 l1 , l2 arbitrary exp. decay from exp. decay from e±it HS (0, x) ψ(x, t)
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5.4.2. Renormalized joint correlation function. The bad decay property of the joint correlation function (5.19) suggests to perform a renormalization step, replacing the free propagator Ut by the cutoff reduced dynamics Ztτ , for which (5.19) has exponential decay when l1 = l2 . The cutoff reduced dynamics Ztτ was introduced in Sect. 5.3, where we argued that it is well approximated by the Markov semigroup t . Hence, we replace the group Ut by the semigroup t in (5.19), thus obtaining a ‘renormalized joint system-reservoir correlation function’. We then check that the so-defined renormalized correlation function has exponential decay in time, uniformly in space: For λ small enough, −tλ2 gr w ψ(x − x , t2 − t1 ) × , t2 −t1 x ,x ;x ,x ≤ e l1 l2 for l1 , l2 ∈ {L , R}
L
R
L
R
(5.23)
with the decay rate gr w as in (4.55). To verify (5.23), we assume for concreteness that l1 = L and l2 = R, and we estimate by the triangle inequality x − x ≤ 1 x − x + 1 x + x − x + x + 1 x − x . (5.24) L R L R L R R L R L 2 2 2 We note that the three terms on the RHS of (5.24) correspond (up to factors 21 ) to the three spatial arguments multiplying γ in the first line of (4.55). By (5.24), at least one of these terms is larger than 13 x R − x L . Hence we dominate (4.55) by replacing that particular term by 13 x R − x L . Setting all other spatial arguments in (4.55) equal to zero, we obtain γ 2 (t )x L ,x R ;x L ,x R ≤ Cerr w (γ ,λ)t e− 6 |x R −x L | + C e−(λ gr w )t . (5.25) Assuming that x R − x L ≥ v ∗ |t2 − t1 | and using that rr w (γ , λ) = O(γ 2 )O(λ2 ), see (4.54), we choose γ such that the first term of (5.25) decays exponentially in t2 − t1 with a rate or order 1. Hence, at high speed (≥ v ∗ ) (5.23) is satisfied. At low speed (≤ v ∗ ), 2 (5.23) holds by the exponential decay of ψ and the bound t ≤ Ce O(λ )t , which is easily derived from (4.55). For l1 = l2 , we can apply the same reasoning, and hence (5.23) is proven in general. However, in the case l1 = l2 , the proof is actually simpler. We can follow the same strategy as used for bounding (5.20), but replacing the propagation estimate (2.12) for Ut by the analogous estimate (4.6) for t . Indeed, the exponential decay in the case l1 = l2 was already present without the coupling to the reservoir, as explained in Sect. 5.4.1, whereas the decay in the case l1 = l2 is a nontrivial consequence of the decoherence induced by the reservoir.
renormalized S − R correlation fct. |x|/t > v ∗ |x|/t ≤ v ∗ l1 = l2 l1 = l2 l1 , l2 arbitrary exp. decay from exp. decay from exp. decay from e±it HS (0, x) ψ(x, t) decoherence of t Along the same line, we note that the decay rate in (5.23) cannot be made greater than O(λ2 ), since the effect of the reservoir manifests itself only after a time O(λ−2 ). This should be contrasted with the decay rate for (5.20), which can be chosen to be independent of λ.
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5.5. The renormalized model. We have argued in the previous section that it makes sense to evaluate the perturbation expansion (5.8) in two steps by introducing a cutoff τ for the temporal arguments of the correlation function ζ . The resulting cutoff reduced evolution Ztτ was described in Sect. 5.3. By reordering the perturbation expansion, we are able to rewrite the reduced evolution Zt approximatively as Zt ≈ λ2
τ I ψ # (x2 − x1 , t2 − t1 ) Zt−t Zτ I Zτ 0 < t1 < t2 < t dt2 dt1 2 x 2 ,l2 t2 −t1 x 1 ,l1 t1 x1 ,x2 ,l1 ,l2 |t2 − t1 | > τ
+ higher orders in λ,
(5.26)
where the restriction that t2 − t1 > τ reflects the fact that the short diagrams have been resummed. Note that it is somewhat misleading to call the remainder of the perturbation series ‘higher order in λ’, since τ will be λ-dependent, too. The main tools in dealing with the renormalized model are 1) The exponential decay of the renormalized joint correlation function, as outlined in Sect. 5.4. This property holds true thanks to the decoherence in the Markov semigroup t and the exponential decay for low (‘subluminal’) speed of the bare reservoir correlation function. The latter is a consequence of the fact that the dispersion law of the reservoir modes is linear (see Lemma 5.1). The necessity of the exponential decay of the renormalized joint correlation function for the final analysis will become apparent in Lemma 9.4. 2) The integrability in time of the correlation function, uniformly in space, as stated in Lemma 5.2. This property allows us to sum up all subleading diagrams in the renormalized model. This will be made more explicit in Sect. 9.2, in particular in Lemma 9.3. The most convenient description of the renormalized model will be reached at the end of Sect. 8 and the beginning of Sect. 9, where a representation in the spirit of (5.26) is discussed. The treatment of the renormalized model is contained in Sect. 9. 5.6. Plan of the proofs. In Sect. 6, we present the analysis of the cutoff reduced dynamics Ztτ and the full reduced dynamics Zt , starting from bounds that are obtained in later sections. The main ingredient of this analysis is spectral perturbation theory, contained in Appendix B. In Sect. 7, we introduce Feynman diagrams and we use them to derive convenient expressions for the cutoff reduced dynamics Ztτ and the full reduced dynamics Zt . We will distinguish between long and short diagrams. The cutoff reduced dynamics contains only short diagrams. Section 8 contains the analysis of the short diagrams. In particular, we prove the bounds on Ztτ , which were used in Sect. 6. In Sect. 9, we deal with the long diagrams. In particular, we prove the bounds on Zt from Sect. 6. At the end of the paper, in Sect. 9.4, we collect the most important constants and parameters of our analysis. A flow chart of the proofs is presented in Fig. 6. 6. Large Time Analysis of the Reduced Evolution Z t and the Cutoff Reduced Evolution Z tτ In this section, we analyze the evolution operators Zt and Ztτ starting from bounds on their Laplace transforms
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Fig. 6. A flow chart of the proofs. The arrows mean “implies”. The arrows pointing to arrows specify the proof of the implication
τ
R (z) := and
R+
R(z) :=
R+
dt e−t z Ztτ
(6.1)
dt e−t z Zt .
(6.2)
These bounds are proven by diagrammatic expansions in Sects. 7, 8 and 9. However, the present section is written in such a way that one can ignore these diagrammatic expansions and consider the bounds on Rτ (z) and R(z) as an abstract starting point. Our results, Lemma 6.2, and Theorem 3.3, follow from these bounds by an application of the inverse Laplace transform and spectral perturbation theory. For convenience, these tools are collected in Lemma B.1 in Appendix B. 6.1. Analysis of Ztτ . Our main tool in the study of Rτ (z) is Lemma 6.1 below. Loosely speaking, the important consequence of this lemma is the fact that we can represent the Laplace transform Rτ , defined in (6.1), as −1 Rτ (z) = z − −iad(Y ) + λ2 M + Aτ (z) , (6.3) where the operator Aτ (z) is “small” wrt. λ2 M, in a sense specified by the theorem. Note that if we set Aτ (z) = 0, then the RHS of (6.3) is the Laplace transform of the Markov semigroup t . This is consistent with the claim that Ztτ is ‘close to’ t .
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The subscripts ‘ld’ and ‘ex’, introduced below, stand for “ladder” and “excitations”, respectively. These subscripts will acquire an intuitive meaning in Sect. 7 when the diagrammatic representation of the expansion is introduced. The (sub)superscript τ indicates the dependence on the cutoff τ , but sometimes we will also use the (sub)superscript c. This will be done for quantities that are designed for the cut-off model but that do not necessarily change when τ is varied. Lemma 6.1 can be stated for any τ , but, as announced, it will be used for a λ-dependent τ . Lemma 6.1. For λ small enough, there are operators Rτex (z) and Rτld (z) in B(B2 (HS )), depending on λ and τ , satisfying the following properties: 1) For Re z sufficiently large, the integral in (6.1) converges absolutely in B(B2 (HS )) and Rτ (z) = (z − (−iad(HS ) + Rτld (z) + Rτex (z)))−1 .
(6.4)
1 2) The operators Rτld (z), Rτex (z) are analytic in z in the domain Re z > − 2τ . Moreover, there is a positive constant δ1 > 0 such that Jκ Rτex (z)J−κ = O(λ2 )O(λ2 τ ), λ2 τ 0, λ 0 . sup Jκ Rτld (z)J−κ ≤ λ2 C |Im κ |≤δ1 ,Re z>− 1 L ,R
2τ
(6.5) 3) Recall the operator L(z), introduced in Sect. 4.1. It satisfies sup Jκ Rτld (z) − λ2 L(z) J−κ |Im κ L ,R |≤δ1 ,Re z≥0 +∞ 2
≤λ C
τ
dt sup |ψ(x, t)| + λ4 τ C .
(6.6)
x
The proof of this lemma is given in Sect. 8. From Lemma 6.1, one can deduce, by spectral methods, that Ztτ inherits some of the properties of the Markovian dynamics t . Instead of stating explicitly all possible results about Ztτ , we restrict our attention to Lemma 6.2, in particular, to the bound (6.7). This bound is the analogue of the bound (4.55) for the semigroup dynamics t , and it will be used heavily in the analysis of Zt in Sect. 8. Lemma 6.2. Let the cutoff reduced evolution Ztτ be as defined in Sect. 5.3, with the cutoff time τ = τ (λ) satisfying (5.13). Then there are positive numbers δc > 0, λc > 0 and gc > 0 such that, for 0 < |λ| < λc and γ ≤ δc , τ 1 rτ (γ ,λ)t − γ2 |(x L +x R )−(x L +x R )| −γ |x L −x R | −γ |x L −x R | ≤ cZ e e e e Zt x ,x ;x ,x L
R
L
R
B2 (S )
2 −λ e + cZ
2g t c
γ
e− 2 |(x L +x R )−(x L +x R )| e−γ |(x L −x R )−(x L −x R )| , (6.7)
1 , c2 > 0, and with for constants cZ Z
rτ (γ , λ) = O(λ2 )O(γ 2 ) + o(λ2 ) where the bound o(λ2 ) is uniform for γ ≤ δc .
λ 0, γ 0,
(6.8)
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The constants δc > 0 and decay rate gc > 0 are in general smaller than the analogues δr w and gr w in the bound (4.55). Proof. We apply Lemma B.1 in Appendix B with := λ2 and ( ) V (t, ) := Uν Ztτ p U−ν , ( ) ( τ ) A1 (z, ) := Uν Rτex (z) p + Rld (z) p − λ2 i {ad(ε(P))} p U−ν , N := Uν {M} p U−ν , B := −Uν {ad(Y )} p U−ν = −ad(Y ),
(6.9) (6.10) (6.11) (6.12)
and ( p, ν) ∈ Dlow with c
1 low Dlow min(δ := D ∩ |Im p| ≤ min(δ , δ ), |Im ν| ≤ , δ ) . 1 ε 1 ε c rw 2
(6.13)
The set Dlow r w has been defined before Proposition 4.2, the bound on p, ν involving δ1 ensures that we can convert the domain of analyticity in the variable κ in Lemma 6.1 into a domain of analyticity in the variables ( p, ν), via the relation (2.60). Similarly, the bound on p, ν involving δε ensures that sup ( p,ν)∈Dlow c
Uν {ad(ε(P))} p U−ν ≤ C
(6.14)
as a consequence of the bound on Jκ ad(ε(P))J−κ provided by Assumption 2.1 and Eq. (2.12). We now check, step by step, the conditions of Lemma B.1. First, the continuity of V (t, ) and the bound (B.1) follow from Lemma 2.5 and Statement 1) of Lemma 6.1. The relation (B.4) is Statement 1) of Lemma 6.1 Condition 1) of Lemma B.1 is trivially satisfied since Y is a Hermitian matrix on a finite-dimensional space. To check Condition 2) of Lemma B.1, we choose g A as g A = 2grlow w and we will actually show that the bound (B.6), which is required in the region Re z > −λ2 g A , holds in the region Re z > −1/(2τ ), as long as λ2 τ is small enough. By Cauchy’s formula, this implies that * A1 (z , λ) 1 ∂ A1 (z, λ) = dz = O(λ2 τ ), for Re z > −λ2 g A , (6.15) ∂z 2πi Cz (z − z )2 with Cz a circle of radius O(1/τ ) centered at z. Hence (B.7) follows. To check (B.6), we use the bound (6.14) for ad(ε(P)). The boundedness of the other terms in A1 (z, λ) follows immediately from (6.5). Condition 3) contains conditions on the spectrum of M p that are satisfied thanks to Proposition 4.2. It remains to check (B.8). By the bound on Rτex (z) in (6.5), it suffices to check that, for any a ∈ sp(ad(Y )), τ 1a (ad(Y ))Jκ λ2 M− −λ2 iad(ε(P))+Rld (−ia) J−κ 1a (ad(Y ))=o(λ2 ), as λ→0. (6.16) This follows by the estimate in (6.6) and the relation between M and L in (4.4). Note that we used that τ (λ) → ∞ as λ → 0 to get o(λ2 ) from the estimate (6.6).
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Hence, we can apply Lemma B.1 and we obtain a number f τ ( p, λ), a rank-one projector Pτ ( p, λ) and a family of operators Rτlow (t, p, λ) such that ( τ) 2 low (6.17) Zt p = e fτ ( p,λ)t Pτ ( p, λ) + e−(λ gc )t Rτlow (t, p, λ) for some gclow > 0 (which can be chosen arbitrarily close to grlow w by taking |λ| small enough), and such that Uν Pτ ( p, λ)U−ν ≤ C,
(6.18)
sup Uν Rτlow ( p, t)U−ν ≤ C.
(6.19)
sup ( p,ν)∈Dlow c
sup ( p,ν)∈Dlow c
t≥0
The above reasoning applies to small fibers, since we use the spectral analysis of Proposition 4.2. We now establish a simpler result about the cut-off reduced evolution ( τ) Zt p , for large fibers. Let
1 high high Dc := Dr w ∩ |Im p| ≤ min(δ1 , δε ), |Im ν| ≤ min(δ1 , δε ) . (6.20) 2 high
Although for ( p, ν) ∈ Dc , we cannot apply Lemma B.1, we can still apply Lemma B.2 to conclude that, for λ small enough, the singularities of {Rτ (z)} p in the domain, say, high
Re z > −2λ2 gr w lie at a distance o(λ2 ) from spM p . One can then easily prove that high {Rτ (z)} p is bounded-analytic in a domain of the form Re z > −λ2 gr w + o(λ2 ) and hence ( high
with a rate gc enough) and
)
Ztτ p
=
high − λ2 gc t high Rτ ( p, t)e
(6.21) high
> 0 (which can be chosen arbitrarily close to gr w by making λ small sup high ( p,ν)∈Dc
sup Uν Rτhigh (t, p, λ)U−ν ≤ C.
(6.22)
t≥0
high
and Dc are Finally, we note that one can easily find a constant δc such that Dlow c of the form (4.31) and (4.32) with the parameters δc instead of δr w (the parameter pr∗w does not need to be readjusted). With the information on Ztτ obtained above, we are able to prove the bound (6.7) by the same reasoning as we employed in the lines following Proposition 4.2 to derive the bound (4.55). The function rτ (γ , λ) in the statement of Lemma 6.2 is determined as rτ (γ , λ) :=
sup p∈Td ,| p|≤γ
max (Re f τ ( p, λ), 0),
(6.23)
and the bound (6.8) follows by (4.54) and f τ ( p, λ) − λ2 f ( p) = o(λ2 ),
λ 0,
(6.24)
which follows from (B.13) in Lemma B.1. The decay rate gc is chosen as gc := high min(gclow , gc ). This concludes the proof of Lemma 6.2.
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We close this section with two remarks which are however not necessary for an understanding of the further stages of the proofs. Remark 6.3. As apparent from the bound (6.6), one cannot take τ ≡ const, since in that case, this bound becomes O(λ2 ) instead of o(λ2 ). This would mean that there is a difference of O(λ2 ) between Ztτ and t , whereas the important terms in t are themselves of O(λ2 ). This is however not an essential point: as one can see from the classification of diagrams in Sect. 7, one could easily modify the definition of the cutoff model such that Ztτ is close to t even at τ ≡ const. This can be achieved by performing the cutoff on the non-ladder diagrams only, which is a notion introduced in Sect. 7. The true reason why τ must diverge to ∞ when λ 0 will become clear in the proof of Lemma 6.5, in Sect. 9.3. Remark 6.4. One is tempted to say that any claim that is made about Zt in Sect. 3 could be stated for Ztτ as well. While this is correct for Theorem 3.3, it fails for Proposition 3.5. The reason is that the identity f ( p = 0, λ) = 0 follows from the fact that Zt conserves the trace of density matrices, as it is the reduced dynamics of a unitary evolution. This is not true for Ztτ , and hence we cannot prove (or even expect) that f τ ( p = 0, λ) = 0. 6.2. Spectral analysis of Zt . In this section, we state Lemma 6.5, the τ = ∞ analogue of Lemma 6.1. This lemma leads to our main result, Theorem 3.3, via reasoning that is almost identical to the one that led from Lemma 6.1 to Lemma 6.2. Essentially (and analogously to Lemma 6.1), Lemma 6.5 states that the Laplace transform R(z), defined in (6.2), is of the form R(z) = (z − (−iad(Y ) + λ2 M + A(z)))−1 , where A(z) is ‘small’ w.r.t.
(6.25)
λ2 M.
Lemma 6.5. There is an operator Rex (z) ∈ B(B2 (HS )), depending on λ and satisfying the following properties, for λ small enough: 1) For Re z sufficiently large, the integral in (6.2) converges absolutely in B(B2 (HS )) and R(z) = (Rτ (z)−1 − Rex (z))−1 ,
(6.26)
Rτ (z)
was introduced in (6.1) and τ = τ (λ) was defined in (5.13). where 2) There are positive constants δex , gex such that the operator Rex (z) is analytic in z in the domain Re z > −λ2 gex and sup |Im κ L ,R |≤δex ,Re z>−λ2 gex
Jκ Rex (z)J−κ = o(λ2 ),
λ 0.
(6.27)
The proof of Lemma 6.5 is contained in Sect. 8. Starting from Lemma 6.5, we can prove our main result, Theorem 3.3, by the spectral analysis outlined in Appendix B. Proof of Theorem 3.3. We apply Lemma B.1 with := λ2 and V (t, ) := Uν {Zt } p U−ν , ( ) A1 (z, ) := Uν {Rex (z)} p + Rτex (z) p ( τ ) + Rld (z) p − λ2 i {ad(ε(P))} p U−ν , N := Uν {M} p U−ν , B := −Uν {ad(Y )} p U−ν = −ad(Y ),
(6.28)
(6.29) (6.30) (6.31)
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1 min(δ ∩ |Im p| ≤ min(δ , δ ), |Im ν| ≤ , δ ) . ( p, ν) ∈ Dlow ex ε ex ε rw 2
(6.32)
Hence, the only difference with the relations (6.9–6.10–6.11–6.12) is that we have added the term {Rex (z)} p in (6.29), we consider Zt instead of Ztτ in (6.28), and we replace δ1 by δex in (6.32). This means that we can copy the proof of Lemma 6.2, except that, in addition, we have to check the bounds (B.6) and (B.7) for the term Rex (z). We choose g A := 21 gex . Then the bound (B.6) follows from (6.27), and (B.7) follows since, by the Cauchy integral formula and (6.27), ∂ Uν {Rex (z)} p U−ν ∂z Re z≥− 12 λ2 gex −1 = Re z − (−λ2 gex ) o(λ2 ) = o(|λ|0 ), sup
λ 0,
(6.33)
the same argument as in (6.15), but with a circle radius of the order of where we use Re z − (−λ2 gex ) . This application of the Cauchy integral formula is the reason for the factor 21 into the definition of g A . Lemma B.1 yields the function f ( p, λ), the rank-one projector P( p, λ) and the operator R low (t, p, λ) required in the small fiber statements of Theorem 3.3. For
1 high ( p, ν) ∈ Dr w ∩ |Im p| ≤ min(δex , δε ), |Im ν| ≤ min(δex , δε ) , (6.34) 2 we can again apply Lemma B.2 to derive the large fiber statements of Theorem 3.3. As in the proof of Lemma 6.2, we can again choose parameters δ, p ∗ such that domains Dlow , Dhigh as defined in (3.15–3.16), are included in the domains for ( p, ν) specified by (6.32) and (6.34). 7. Feynman Diagrams In this section, we introduce the expansion of the reduced evolution Zt and the cutoff reduced evolution Ztτ in amplitudes labelled by Feynman diagrams. These expansions will be the main tool in the proofs of Lemmata 6.1 and 6.5. We start by introducing a notation for the Dyson expansion of Zt which is more convenient than that of Sect. 5.2. 7.1. Diagrams σ . Consider a pair of elements in I × Zd × {L , R} with I ⊂ R+ a closed interval whose elements should be thought of as times. The smaller time of the pair is called u and the larger time is called v, and we require that u = v, i.e. u < v. The set of pairs satisfying this constraint is called 1I . We define nI as the set of n pairs of elements in I × Zd × {L , R} such that no two times coincide. That is, each σ ∈ nI consists of n pairs whose time-coordinates are parametrized by (u i , vi ), for i = 1, . . . , n, and with the convention that u i < vi and u i < u i+1 . The elements σ are called diagrams. As announced in Sect. 5.2, there is a one-to-one mapping between, on the one hand, a set of 2n with t < t triples (ti (σ ), xi (σ ), li (σ ))i=1 i i+1 and ti ∈ I , together with a pairing π ∈ Pn , and, on the other hand, a diagram σ ∈ nI as defined above.
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Fig. 7. A diagram σ ∈ I with |σ | = 5. Its time coordinates are shown explicitly. Note that the parametrization by u i , vi encodes the combinatorial structure (the way the times are connected by pairings), whereas the ti are ordered. We consistenly draw the long pairings (see later) above the time-axis and the short ones below
To construct this mapping, proceed as follows: Choose from the pairing π the pair (r, s) for which r = 1 and set u 1 = tr , v1 = ts . The pair ((tr , xr , lr ), (ts , xs , ls )) becomes the first pair in the diagram σ . Then choose the pair (r , s ) ∈ π such that r = min{{1, 2, . . . , 2n}\{r, s}}.
(7.1)
Set u 2 = tr , v2 = ts . The pair ((tr , xr , lr ), (ts , xs , ls )) becomes the second pair of σ . Repeat this until one has n pairs, each time picking the pair whose r is the smallest of the remaining integers. The mapping is easily visualized in a picture, see Fig. 7. We also use the notation t(σ ), x(σ ), l(σ ) to denote the ‘coordinates’ of the diagram σ . Here, t(σ ), x(σ ), l(σ ) are 2n-tuples of elements in I, Zd , {L , R}, respectively, and such that the i th components of these 2n-tuples constitutes the i th triple (ti (σ ), xi (σ ), li (σ )). Note that the time-coordinates t ≡ t1 (σ ), . . . , t2n (σ ) can also be defined as the ordered set of times containing the elements {u i , vi , i = 1, . . . , n}. Evidently, the triples 2n do not fix a diagram uniquely since the combinatorial structure (ti (σ ), xi (σ ), li (σ ))i=1 that is encoded in π is missing. That combinatorial structure is now encoded in the way the time coordinates t(σ ) are partitioned into pairs (u i , vi ), see also Fig. 7. We drop the superscript n to denote the union over all n ≥ 1, i.e. I
:=
+
n I,
(7.2)
n≥1
and we write |σ | = n to denote that σ ∈ We define the domain of a diagram as Domσ :=
n +
n. I
[u i , vi ],
for σ ∈
n I.
(7.3)
i=1
We call a diagram σ ∈ I irreducible (notation: ir) whenever its domain is a connected set (hence an interval). In other words, σ is irreducible whenever there are no two (sub)diagrams σ1 , σ2 ∈ I such that σ = σ1 ∪ σ2 ,
and
Domσ1 ∩ Domσ2 = ∅,
(7.4)
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Fig. 8. The left figure shows an irreducible diagram σ in the interval I = [I− , I+ ] with |σ | = 7. This diagram is not minimally irreducible. The right figure shows a minimally irreducible subdiagram. In this case, there is only one such minimally irreducible subdiagram, but this need not always be the case
where the union refers to a union of pairs of elements in I × Zd × {L , R}. For any σ ∈ I that is not irreducible, we can thus find a unique (up to the order) sequence of (sub) diagrams σ1 , . . . , σm such that σ1 , . . . , σm
σ = σ1 ∪ . . . ∪ σm .
are irreducible and
(7.5)
We fix the order of σ1 , . . . , σm by requiring that max Domσi ≤ min Domσi+1 and we call the sequence (σ1 , . . . , σm ) obtained in this way the decomposition of σ into irreducible components. We let nI (ir) ⊂ nI stand for the set of irreducible diagrams σ (with n pairs) that satisfy Domσ = I , that is, u 1 = t1 (σ ) = min I and maxi ti (σ ) = maxi vi = max I . A diagram σ ∈ I (ir) is called minimally irreducible in the interval I whenever it has the following property: For any subdiagram σ ⊂ σ , the diagram σ \σ does not belong to I (ir). Intuitively, this means that either the diagram σ contains a boundary point of I as one of its time-coordinates, or the diagram σ \σ is not irreducible. The set of minimally irreducible diagrams (with n pairs) is denoted by nI (mir). See Figs. 7 and 8 for a graphical representation of the diagrams. Since, up to now, most definitions depend solely on the time-coordinates, we only indicate the time-coordinates in the pictures. In the terminology introduced below, we draw equivalence classes of diagrams [σ ] rather than the diagrams σ themselves. A diagram σ in I for which each pair of time coordinates (u, v) satisfies |v −u| ≥ τ , or |v − u| ≤ τ , is called long, or short, respectively. The set of all long/small diagrams with n pairs is denoted by nI (> τ ) / nI (< τ ). Note that nI (> τ ) ∪ nI (< τ ) is strictly smaller than nI whenever n > 1. In addition to the sets nI (ir), nI (mir), nI (> τ ), we will sometimes use more than one specification (adj) to denote a subset of I or nI , and we will drop the superscript n to denote the union over all |σ |, as in (7.3), for example, n I (<
τ, ir),
I (>
τ, mir)
(7.6)
are the sets of short irreducible diagrams with |σ | = n and long minimally irreducible diagrams, respectively. On the set nI , we define the “Lebesgue measure” dσ by
n I
dσ F(σ ) :=
where I = [I− , I+ ].
I−
du 1 · · · du n
u i
dv1 · · · dvn
x(σ ),l(σ )
F(σ ), (7.7)
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Since nI (ir) is a zero-measure subset of nI , the definition of the measure dσ on n (ir) has to be modified in an obvious way: For all continuous (in the time coordinates I t(σ )) functions F on nI , we set
n (ir) I
dσ F(σ ) =
n I
dσ δ(max t(σ ) − I+ )δ(min t(σ ) − I− )F(σ ),
(7.8)
where the Dirac distributions δ(I+ − ·) and δ(I− − ·) are a priori ambiguous since I− , I+ are the boundary points of the interval I . They are defined as δ(· − I+ ) := lim δ(· − s), s →< I +
δ(· − I− ) := lim δ(· − s). s →> I−
We extend the definition of the measure dσ also to dσ F(σ ) := I (adj)
n≥1
I
and the various
n (adj) I
(7.9)
I (adj) by setting
dσ Fn (σ ),
(7.10)
where Fn is the restriction to nI (adj) of a function F on I (adj). We will often encounter functions of σ that are independent of the coordinates x(σ ), l(σ ) and that must be integrated only over t(σ ) and summed over |σ |. To deal elegantly with such situations, we let [σ ] stand for an equivalence class of diagrams that is obtained by dropping the x, l-coordinates. That is |σ | = |σ | [σ ] = [σ ] ⇔ . (7.11) u i (σ ) = u i (σ ), vi (σ ) = vi (σ ), for all i = 1, . . . , |σ | The set of such equivalence classes is denoted by !T I (the symbol !T can be thought of as a projection onto the time coordinates) and we naturally extend the definition to !T I (adj) where adj can again stand for ir, mir, > τ , < τ . The integration over equivalence classes of diagrams is defined as above in (7.7) and (7.8), but with x(σ ),l(σ ) omitted, i.e., such that for all functions F˜ on I (adj): ˜ ), ˜ ). (7.12) d[σ ]F([σ ]) = dσ F(σ with F([σ ]) = F(σ !T
I (adj)
I (adj)
x(σ ),l(σ )
Lemma 7.1 contains the main application of this construction. It is in fact a simple L 1 − L ∞ -bound. Lemma 7.1. Let F and G be positive, continuous functions on I . Then ⎤ , -⎡ dσ F(σ )G(σ ) ≤ d[σ ] sup G(σ ) ⎣sup F(σ )⎦ , I (adj)
!T
I (adj)
x(σ ),l(σ )
t(σ ) x(σ ),l(σ )
(7.13) where it is understood that the sum and sup over x(σ ), l(σ ) are performed while keeping |σ | and t(σ ) fixed. In (7.13), the sum/sup, over x(σ ), l(σ ) is in fact a shorthand notation for the sum/sup over all σ such that [σ ] = [σ ] for a given σ . Hence, supx(σ ),l(σ ) G(σ ) is a function of [σ ] only, as required. The supremum supt(σ ) is over I 2|σ | , with |σ | fixed. Hence, the second factor on the RHS of (7.13) is in fact a function of |σ | only. Proof. We start from the explicit expressions in (7.7) or (7.8), and we use a L 1 − L ∞ inequality: first for the sum over x(σ ), l(σ ) and then for the integration over u i , vi .
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2n ) 7.1.1. Representation of the reduced evolution Zt . Recall the operators Vt ((ti , xi , li )i=1 defined in (5.9). Since, by the above discussion, there is a one-to-one correspondence 2|σ | between a diagram σ and (π, (ti , xi , li )i=1 ), where π ∈ P|σ | and ti < ti+1 , we can write 2|σ | 2|σ | Vt (σ ) instead of Vt ((ti , xi , li )i=1 ) and ζ (σ ) instead of ζ (π, (ti , xi , li )i=1 ), i.e. 2|σ | (7.14) Vt (σ ) := Vt (ti (σ ), xi (σ ), li (σ ))i=1
and
⎧ ψ(x − x, v − u) l = l = L ⎪ ⎪ ⎨ ψ(x − x, v − u) l = l = R ζ (σ ) := λ2 . ⎪ ψ(x − x, v − u) l = L , l = R ⎪ ⎩ ((u,x,l),(v,x ,l ))∈σ ψ(x − x, v − u) l = R, l = L
(7.15)
As a slight generalization of the operators Vt (σ ), we also define V I (σ ) for a closed interval I := [I− , I+ ] by V I (σ ) := U I+ −t2n Ix2n ,l2n . . . Ix2 ,l2 Ut2 −t1 Ix1 ,l1 Ut1 −I− ,
for σ such that Domσ ⊂ I. (7.16)
The only difference with Vt (σ ) is in the time-arguments ‘t’ of Ut at the beginning and the end of the expression. With this new notation, Vt (σ ) = V[0,t] (σ ). Next, we state the representation of the reduced evolution Zt as an integral over diagrams Zt = Ut + dσ ζ (σ )V[0,t] (σ ). (7.17) [0,t]
Similarly, the cutoff dynamics Ztτ is represented as τ dσ ζ (σ )V[0,t] (σ ). Zt = Ut + [0,t] (<τ )
(7.18)
Formulas (7.17) and (7.18) are immediate consequences of (5.8) and (5.15), respectively. We use the notion of irreducible diagrams σ to decompose the operators V[0,t] (σ ) into products and to derive a new representation, (7.23), for Zt and Ztτ . Let (σ1 , . . . , σ p ) be the decomposition of a diagram σ ∈ [0,t] into irreducible components. Define the times s1 , . . . , s2 p to be the boundaries of the domains of the irreducible components, i.e., [s2i−1 , s2i ] = Domσi , for i = 1, . . . , p. Then V I (σ ) = U I+ −s2 p V[s2 p−1 ,s2 p ] (σ p ) Us2 p−1 −s2 p−2 . . . Us3 −s2 V[s1 ,s2 ] (σ1 ) Us1 −I− ,
(7.19)
as can be checked from (7.15–7.16). Here, the essential observation is that all time coordinates of σi are smaller than those of σi+1 . We introduce τ,ir ir Zt := dσ ζ (σ )V[0,t] (σ ), Zt := dσ ζ (σ )V[0,t] (σ ), (7.20) [0,t] (ir)
[0,t] (<τ,ir)
and we remark that the definitions in (7.20) allow for a shift of time on the RHS, that is dσ ζ (σ )V I (σ ), for any I = [s, s + t], s ∈ R, (7.21) Ztir = I (ir)
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and similarly for Ztτ,ir . By this time-translation invariance, the factorization property (7.19) and the factorization property of the correlation function in (7.15), i.e., ζ (σ1 ∪ . . . ∪ σ p ) =
p
ζ (σi ),
(7.22)
i=1
we can rewrite the expression (7.17) as Zt = ds1 . . . dsm Ut−sm Zsirm −sm−1 . . . Us3 −s2 Zsir2 −s1 Us1 , m∈2Z+
0≤s1 ≤···≤sm ≤t
(7.23) where the term on the RHS corresponding to m = 0 is understood to be equal to Ut . The idea behind (7.23) is that, instead of summing over all diagrams, we sum over all sequences of irreducible diagrams. An analogous formula holds with Zt and Ztir replaced by Ztτ and Ztτ,ir . 7.2. Ladder diagrams and excitations. We are ready to identify the operators Rτex (z) and Rτld (z), whose existence was postulated in Lemma 6.1 and the operator Rex (z), which was postulated in Lemma 6.5. The Laplace transform, R(z), of Zt has been introduced in (6.2). We calculate R(z) starting from (7.23), R(z) = dt e−t z Zt (7.24) R+ 3m 2 (z + iad(HS ))−1 dt e−t z Ztir (z + iad(HS ))−1 (7.25) = R+
m≥0
= (z + iad(HS ) − Rir (z))−1 ,
with Rir (z) :=
R+
dt e−t z Ztir . (7.26)
# The second equality follows by R+ dte−t z Ut = (z + iad(HS ))−1 for Re z > 0. The third equality follows by summing the geometric series. An identical computation yields Rτ (z) = (z + iad(HS ) − Rτir (z))−1 , with Rτir (z) := dt e−t z Ztτ,ir . (7.27) R+
The definition of Rτex (z) and Rτld (z) relies on the following splitting of Rτir (z): Rτld (z) := dt e−t z ζ (σ )1|σ |=1 V[0,t] (σ ), (7.28) Rτex (z) :=
R+ R+
dt e−t z
[0,t] (<τ ,ir)
[0,t] (<τ ,ir)
ζ (σ )1|σ |≥2 V[0,t] (σ ).
(7.29)
The subscripts refer to ‘ladder’- and ‘excitation’-diagrams. The name ‘ladder’ originates from the graphical representation of diagrams whose irreducible components consist of one pair (it is standard in condensed matter theory). Since obviously Rτex (z) + Rτld (z) = Rτir (z), the relation (7.27) implies Statement (1) of Lemma 6.1.
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In the model without cutoff, we do not disentangle ladder and excitation diagrams, since every diagram that contains a long pairing, is considered an excitation. We can thus define Rex (z) := Rir (z) − Rτir (z).
(7.30)
We will come up with a more constructive representation of Rex (z) in formula (7.33). 7.2.1. The reduced evolution as a double integral over long and short diagrams. We develop a new representation of Ztir by fixing the long diagrams, i.e., those in [0,t] (> τ ), and integrating the short ones. We define the conditional cutoff dynamics, Ct (σl ), depending on a long diagram σl ∈ [0,t] (> τ ), as follows: Ct (σl ) = 1σl ∈ [0,t] (>τ,ir) V[0,t] (σl ) + dσ ζ (σ )V[0,t] (σ ∪ σl ). (< τ ) [0,t]
σl ∪ σ ∈
[0,t] (ir)
(7.31) In words, Ct (σl ) contains contributions of short diagrams σ ∈ t (< τ ) such that σl ∪ σ is irreducible in the interval [0, t]. Hence, if σl is itself irreducible in the interval [0, t], then there is a term without any short diagrams; this is the first term in (7.31). In general, σl need not be irreducible. Note that the constraint on σ (in the domain of the integral) in the second term of (7.31) depends crucially on the nature of σl . In particular, if Domσl does not contain the boundary points 0 or t, then σ has to contain 0 or t, and this introduces one or two delta functions into the constraint on σ . To relate Ct (σl ) to Ztir , we must explicitly add those σl that contain one or both of the times 0 and t. This is visible in the following formula, which follows from (7.31) and the definition of Ztir in (7.20):
τ,ir ir Zt − Zt = dσl ζ (σl )Ct (σl ) [1 + δ(t1 (σl )] 1 + δ(t2|σl | (σl ) − t) . (7.32) [0,t] (>τ )
We must subtract Ztτ,ir on the LHS, since all contributions to the RHS involve at least one long diagram. The δ-functions on the RHS are defined as in (7.9). The following formula is an obvious consequence of (7.30) and (7.32): Rex (z) −t z = dt e R+
[0,t] (>τ )
dσl ζ (σl )Ct (σl ) [1 + δ(t1 (σl )] 1 + δ(t2|σl | (σl ) − t) . (7.33)
All of Sect. 8 will be devoted to proving good bounds on Rex (z), as claimed in Lemma 6.5. 7.3. Decomposition of the conditional cutoff dynamics Ct (σl ). Our next step is to decompose the conditional cutoff dynamics Ct (σl ), as defined in (7.31), into components. Since Ct (σl ) is defined as an integral over short diagrams σ , we can achieve this by classifying the short diagrams σ that contribute to this integral. The idea is to look at the irreducible components of σ whose domain contains one or more of the time-coordinates of σl (In our final formula, (7.45), these domains correspond to the intervals [ski , skf ]). The irreducible components whose domain does not contain any of the time coordinates of σl can be resummed right away, and they do not play a role in our classification (this corresponds to the operators Ztτ in (7.45). We outline the abstract decomposition procedure in Sect. 7.3.1, and we present an example (with figures) in Sect. 7.3.2.
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7.3.1. Vertices and vertex partitions. Consider a long diagram σl ∈ [0,t] (> τ ) with |σl | = n and time-coordinates t(σl ) = (t1 , . . . , t2n ). With this diagram, we will associate different vertex partitions L. First, we define vertices. A vertex l is determined by a label, bare or dressed, and a vertex set S(l), given by S(l) = {t j , t j+1 . . . , t j+m−1 },
for some 1 ≤ j < j + m − 1 ≤ 2n.
(7.34)
Hence, the vertex set is a a subset of the times {t1 (σl ), . . . , t2n (σl )}. Moreover, a vertex l with |S(l)| > 1 is always dressed. Hence, if l is bare then S(l) is necessarily a singleton, i.e., S(l) = {t j } for some j. A vertex partition L compatible with σl (notation: L ∼ σl ) is a collection of vertices l1 , . . . , lm such that • The vertex sets S(l1 ), . . . , S(l p ) form a partition of {t1 (σl ), . . . , t2n (σl )}. By convention, we always number the vertices in a vertex partition such that the elements of S(lk ) are smaller than those of S(lk+1 ). The number, p, of vertices in a vertex partition is called the cardinality of the vertex partition and is denoted by |L|. • Any two consecutive times t j , t j+1 such that [t j , t j+1 ] ⊂ Domσl , belong to the vertex set S(lk ) of one of the vertices lk . Such a vertex lk is necessarily dressed since its vertex set contains at least two elements. • If t1 = 0, then S(l1 ) = {t1 } and l1 is bare. If t1 > 0, then S(l1 ) # t1 and l1 is dressed. • If t2n = t, then S(lm ) = {t2n } and lm is bare. If t2n < t, then S(lm ) # t2n and lm is dressed. The idea is to split Ct (σl ) =
Ct (σl , L),
(7.35)
L∼σl
where the sum is over all L compatible with σl and Ct (σl , L) contains the contributions of all short pairings σ that match the vertex partition L; ⎧ ⎨ ∀ dressed lk : ∃! irr. component σ j ⊂ σ such that S(lk ) ⊂ Domσ j . σ matches L ⇔ and S(lk ) ∩ Domσ j = ∅ for all k = k ⎩ ∀ bare lk : S(lk ) ∩ Domσ = ∅
(7.36) For the sake of completeness, we define the operators Ct (σl , L), below, but, in Sect. 7.3.3, we will provide a more constructive expression for them. First, assume that the vertex partition L contains at least one dressed vertex. Then Ct (σl , L) is defined by restricting the second integral in (7.31) to those σ that match L, dσ ζ (σ )V[0,t] (σ ∪ σl )1σ matches L. (7.37) Ct (σl , L) := (< τ ) [0,t]
σl ∪ σ ∈
[0,t] (ir)
Next, we assume that the vertex partition L contains only bare vertices. If σl is irreducible in the interval [0, t], i.e., σl ∈ [0,t] (> τ, ir), then the vertex partition with only bare vertices is compatible with σl . If σl ∈ / [0,t] (> τ, ir), then this vertex partition is not compatible with σl . Hence, we assume that σl ∈ [0,t] (> τ, ir) and we define Ct (σl , L) := V[0,t] (σl ) +
[0,t] (<τ )
dσ ζ (σ )V[0,t] (σ ∪ σl ) × 1({t1 ,...,t2|σ | }∩Domσ =∅) . l
(7.38)
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Fig. 9. A long diagram with time coordinates as in (7.39)
The second term is the same as in (7.37) (but specialized to the partition with only bare vertices) and the first term is a contribution without any short diagrams. This first term equals the first term (on the RHS) of (7.31). In Sect. 7.3.2, we give examples of vertex partitions that are intended to render the above concepts more intuitive. 3 (> τ ) which 7.3.2. Examples of vertex partitions. We choose a long diagram σl ∈ [0,t] 3 consists of three pairs such that the time coordinates (u i , vi )i=1 are ordered as
t2 t3 t4 t5 t6 t1 . 0 < u 1 < u 2 < v1 < u 3 < v2 < v3 = t
(7.39)
Hence, σl is irreducible in the interval [u 1 , t], but not in the interval [0, t], at least not if t1 = 0 (Fig. 9). Below we display three diagrams σ ∈ [0,t] (< τ ) satisfying the condition σl ∪ σ ∈ [0,t] (ir). To assign to each of those diagrams a vertex partition, we proceed as follows. Starting on the left, we look at the time-coordinates t(σl ) and we check whether these times are ‘bridged’ by a short pairing, i.e., whether they belong to the domain of a short diagram. If this is the case then such a time belongs to the vertex set of a dressed vertex. The vertex set of this vertex is the set of all time-coordinates that are connected to this point by short pairings. If this is not the case, i.e., if a time-coordinate of σl is not ‘bridged’ by any short pairing, than such a point constitutes a bare vertex, whose vertex set is just this one point. Actually, for the first time-coordinate (in our case u 1 ), this is particularly simple. Either the first time-coordinate is not equal to 0, in which case it has to be ‘connected’ by short pairings to 0 (indeed, if this were not the case, then σl ∪ σ cannot be irreducible in [0, t]), or the first time-coordinate is equal to 0, in which case it cannot be connected by short pairings to the second coordinate, because then the first time-coordinate of the short diagram would have to be 0 as well, which is a zero measure event (for this reason, we have excluded this case in the definition of the diagrams in Sect. 7.1). In our example u 1 = 0, and one checks that, in all three choices of σ , there are short diagrams connecting u 1 and 0. Let us determine the vertices in the three displayed figures vertex partition 1 l1 {t1 } dressed l2 {t2 } bare l3 {t3 } bare l4 {t4 } dressed l5 {t5 } bare l6 {t6 } bare
vertex partition 2 l1 l2 l3 l4
{t1 } {t2 } {t3 , t4 , t5 } {t6 }
dressed bare dressed bare
vertex partition 3 l1 l2 l3 l4 l5
{t1 } {t2 , t3 } {t4 } {t5 } {t6 }
dressed dressed bare bare bare
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In the example displayed above, it is also very easy to determine which vertex partitions L are compatible with σl (L ∼ σl ). Apart from the fact that the vertex sets S(lk ) of the vertices in L have to form a partition of {t1 , . . . , t6 }, we need that l1 is dressed and l|L| (the last vertex in the partition) is bare. To each vertex lk in the above examples, we can associate time coordinates ski and skf as the boundary times of the domains of irreducible diagrams bridging the times in the vertex. Eventually, we intend to fix a vertex partition and associated time coordinates s i and s f and to integrate over all short diagrams that are irreducible in the interval [s i , s f ]. This integration gives rise to the vertex operators, see Eq. (7.42). To illustrate this, we zoom in on a part of a long diagram, shown in Fig. 11. A formal definition is given in the next section. 7.3.3. Abstract definition of the vertex operator. Let L be a vertex partition compatible with σl , with vertices lk , k = 1, . . . , |L|. In what follows, we focus on one particular vertex lk which we assume first to be dressed. The vertex lk is assumed to have a vertex set S(lk ) = {t j , t j+1 , . . . , t j+m−1 }. This means in particular that the time-coordinate t j−1 belongs to the vertex set of the vertex lk−1 (unless j = 1) and the time-coordinate t j+m belongs to the vertex set of the vertex lk+1 (unless j + m − 1 = 2|σl |). We fix an initial time ski and final time skf such that t j−1 ≤ ski ≤ t j ≤ t j+m−1 ≤ skf ≤ t j+m ,
(7.40)
where it is understood that t j−1 = 0 if j = 1 and t j+m = t if j +m −1 = 2|σl |. The vertex operator B(lk , ski , skf ) is defined by summing the contributions of all σ ∈ [s i ,s f ] (< τ , ir). k
k
To write a formula for the vertex operator B(lk , ski , skf ), we need to relabel the timecoordinates of σl and σ ∈ [s i ,s f ] (< τ , ir). k k Consider the m triples (ti (σl ), xi (σl ), li (σl )), for i = j, . . . , j +m −1, i.e., a subset of the 2|σl | triples determined by the long diagram σl , and the 2|σ | triples ti (σ ), xi (σ ), li (σ ) with i = 1, . . . , 2|σ | determined by σ ∈ [s i ,s f ] (< τ , ir). We now define the triples m+2|σ |
(ti , xi , li )i
k
k
) of the union of triples by time-ordering (i.e. such that ti ≤ ti+1 2|σ |
j+m−1
(ti (σ ), xi (σ ), li (σ ))i=1 and (ti (σl ), xi (σl ), li (σl ))i= j
.
The vertex operator B(lk , ski , skf ) is then defined as follows: m+2|σ | , B lk , ski , skf := dσ ζ (σ )V[s i ,s f ] ti , xi , li i=1 (<τ ,ir) ski ,skf
k
k
(7.41)
(7.42)
where the dependence of the integrand on σ is implicit in the above definition of the triples (ti , xi , li ). The double primes in the coordinates (ti , xi , li ) are supposed to render the comparison with later formulas easier. We now treat the simple case in which the vertex lk is bare. In that case, there is a j such that S(lk ) = {t j } and the vertex operator is simply defined as ski = skf = t j . (7.43) B lk , ski , skf := Ix j ,l j , Hence, in this case, the vertex time-coordinates ski , skf are dummy coordinates, see also Fig. 10.
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Fig. 10. The picture shows three different choices of short diagrams σ ∈ [0,t] (< τ ). Recall that short diagrams are drawn below the horizontal (time) axis. In each picture, we show the resulting vertex partition by listing the vertices l1 , l2 , . . .. The dressed vertices are denoted by a horizontal bar whose endpoints represent the vertex time-coordinates ski , skf . The bare vertices are denoted by a short vertical line whose position represents the (dummy) vertex time coordinates ski = skf = t j . The time-coordinates of the bare vertices are not shown since they coincide with time-coordinates of long pairings. For example, in the bottom picture, l1 , l2 are dressed and l3 , l4 , l5 are bare
7.3.4. The operator Ct (σl , L) as an integral over time-coordinates of vertex operators. We are ready to give a constructive formula for Ct (σl , L), as announced in Sect. 7.3.1. First, we define the integration measure over the vertex time-coordinates ski , skf : Ds Ds := i
f
k = 1, . . . , |L| lk dressed
dski dskf
i δ s1 l1 dressed × 1 l1 bare
4 δ s|fL| − t l|L| dressed × . 1 l|L| bare
(7.44)
To understand this formula, we observe that only non-dummy vertex time coordinates need to be integrated over. A dummy vertex time coordinate is a time coordinate whose value is a-priori fixed by σl and the vertex partition L. The non-dummy times are the time coordinates of the dressed vertices, except at the temporal boundaries 0, t, where such a time coordinate is also a dummy coordinate. The terms between {·}-brackets in
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Fig. 11. A part of a long diagram σl ∈ [0,t] (> τ ) is shown, suggesting a dressed vertex l with vertex set S(l) = {t j , . . . , t j+4 }. The end points of the pairings that are ‘floating’ in the air are immaterial to this vertex, as long as they land on the time-axis outside the interval [s i , s f ]. The vertex operator B(l, s i , s f ) is obtained by integrating all diagrams in [s i ,s f ] (< τ , ir)
formula (7.44) take care of this . Finally, the formula for Ct (σl , L) is Ct (σl , L) = 0 < s i < s f < t Ds i Ds f B l|L| , s|iL| , s|fL| Zsτi k
k
skf < ski for k > k . . . Zsτi −s f B l2 , s2i , s2f Zsτi −s f B l1 , s1i , s1f , 3 2 2 1
f |L| −s|L|−1
(7.45)
where the indices k, k correspond to vertices lk , lk : only the time-coordinates of dressed vertices are integrated over, even though all vertices appear on the RHS. This formula can be checked from the definition (7.37) and the explicit expressions for the vertex operators B(·; ·, ·) above. The cutoff reduced dynamics Ztτ in (7.45) appears by summing the small diagrams between the vertices, using formula (7.18). 8. The Sum Over “Small” Diagrams In this section, we establish two results. First, we analyze the cutoff-dynamics Ztτ . The main bound is stated in Lemma 8.3, and a proof of Lemma 6.1 (concerning the Laplace transform of Ztτ ) is outlined immediately after Lemma 8.3. Second, we resum the small subdiagrams within a general irreducible diagram: Recall that the conditional cutoff dynamics Ct (σl ) is defined as the sum over all irreducible diagrams in [0, t] containing the long diagram σl . In Lemma 8.6, we obtain a description of Ct (σl ) that does not involve any small diagrams. In this sense, we have performed a blocking procedure, getting rid of information on time-scales smaller than τ . Since this section uses parameters and constants that were introduced earlier in the paper, we encourage the reader to consult the overview tables in Sect. 9.4. 8.1. Generic constants. In Sects. 8 and 9, we will state bounds that will depend in a crucial way on the parameters λ, γ and τ . The parameter γ is a momentum-like variable used to bound matrix elements in position representation, see below in Sect. 8.3. It appeared first in Sect. 4.3.1. To simplify the presentation, we introduce the following notation and conventions: • We write c(γ ) for functions of γ ≥ 0 with the property that γ → c(γ ) is decreasing, and c(γ ) is finite, except, possibly, at γ = 0. It is understood that c(γ ) is independent of λ.
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• We write c(γ , λ) for functions of γ ≥ 0 and λ ∈ R that have the asymptotics c(γ , λ) = o(γ 0 )O(λ2 ) + o(λ2 ),
γ → 0, λ → 0.
(8.1)
• We write c (γ , λ) for functions of γ ≥ 0 and λ ∈ R that have the asymptotics c (γ , λ) = o(γ 0 )O(λ2 ) + c(γ )o(λ2 ),
γ → 0, λ → 0.
(8.2)
• The cutoff time τ = τ (λ) is treated as an implicit function of λ, satisfying (5.13). In particular, c(γ , λ) and c (γ , λ) can depend on τ .
8.2. Bounds in the sense of matrix elements. In Sect. 2.5 , we introduced the kernel notation Ax L ,x R ;x L ,x R , for operators A on B2 (l 2 (Zd , S )); Ax L ,x R ;x L ,x R is an element of B(B2 (S )) such that S, AS =
x L ,x R ;x L ,x R
S(x L , x R ), Ax L ,x R ;x L ,x R S (x L , x R )B2 (S ) .
(8.3)
First, we introduce a notion that allows us to bound operators A by their ‘matrix elements’ Ax L ,x R ;x L ,x R . Definition 1. Let A and A˜ be operators on B2 (l 2 (Zd , S )) and B2 (l 2 (Zd )), respectively. We say that A˜ dominates A in ‘the sense of matrix elements’, denoted by ˜ A ≤ A,
(8.4)
Ax L ,x R ;x L ,x R B (B2 (S )) ≤ A˜ x L ,x R ;x L ,x R .
(8.5)
m.e.
iff
Note that, if A is an operator on B2 (l 2 (Zd )), the inequality A ≤m.e. A˜ literally means that the absolute values of the matrix elements of A are smaller than the matrix elements ˜ We will need the following implication: of A. A ≤ A˜ m.e.
⇒
˜ A ≤ A .
(8.6)
Indeed, for any S ∈ B2 (l 2 (Zd ) ⊗ S ) ∼ l 2 (Zd × Zd , B2 (S )), we construct ˜ L , x R ) := S(x L , x R ) B (S ) , S(x 2
(8.7)
˜ l 2 (Zd ×Zd ) = S l 2 (Zd ×Zd ,B (S )) and such that S 2 ˜ A˜ S˜ |S, AS | ≤ S, from which (8.6) follows.
(8.8)
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8.3. Bounding operators. We introduce operators on B2 (l 2 (Zd )) that will be used as upper bounds ‘in the sense of matrix elements’, as defined above. These bounding operators will depend on the coupling constant λ, the conjugation parameter γ > 0 and the 1 , c2 be as defined cutoff time τ = τ (λ). Let the function rτ (γ , λ) and the constants cZ Z in Lemma 6.2 and, in addition, let rε (γ , λ) := 2λ2 qε (2γ ),
for 2γ ≤ δε ,
(8.9)
with qε (·) and δε as in Assumption 2.1. We define (8.10) I˜ x,l := δx ,x δx ,x δl=L δx L =x + δl=R δx R =x , L R L R x ,x ;x ,x L R L R τ,γ 1 rτ (γ ,λ)t − γ2 |(x L +x R )−(x L +x R )| −γ |x L −x R | −γ |x L −x R | Z˜t := cZ e e e e x L ,x R ;x L ,x R
γ U˜t
2 −λ e + cZ
2g t c
x L ,x R ;x L ,x R
γ
γ
e− 2 |(x L +x R )−(x L +x R )| e−γ |(x L −x R )−(x L −x R )| , (8.11)
:= erε (γ ,λ)t e− 2 |(x L +x R )−(x L +x R )| e−γ |(x L −x R )−(x L −x R )| .
(8.12)
In order for definitions (8.11, 8.12) to make sense, λ and γ > 0 have to be sufficiently small, such that the functions rε (γ , λ) and rτ (γ , λ) are well-defined. In particular, we need conditions on λ and γ such that Lemma 6.2 applies. τ,γ γ The operators I˜ z,l , Z˜t , U˜t inherit their notation from the operators they are designed to bound, as we have the following inequalities, for λ, γ small enough: Ix,l ≤ I˜ x,l ,
(8.13)
m.e.
τ,γ Ztτ ≤ Z˜t ,
(8.14)
γ Ut ≤ U˜t .
(8.15)
m.e. m.e.
The first inequality is obvious from the definition of Ix,l in (5.7) and the fact that W B (S ) ≤ 1. Indeed, I˜ x,l can be obtained from Ix,l by replacing W by 1. The second inequality is the result of Lemma 6.2 and the third inequality follows from the bounds following Assumption 2.1. τ,γ γ We start by stating obvious rules to multiply the operators Z˜t and U˜t . Lemma 8.1. For λ, γ small enough, the following bounds hold (with c(γ ) and c(γ , λ) as defined in Sect. 8.1) n si , • For all sequences of times s1 , . . . , sn with t = i=1 γ
γ γ γ U˜sn . . . U˜s2 U˜s1 ≤ [c(γ )]n−1 ec(γ ,λ)t U˜t 2 , m.e.
(8.16)
γ
τ, τ,γ τ,γ τ,γ Z˜sn . . . Z˜s2 Z˜s1 ≤ [c(γ )]n−1 ec(γ ,λ)t Z˜t 2 . m.e.
(8.17)
• For all times s < t, γ
s τ, τ,γ γ Z˜t−s U˜s ≤ c(γ )e 2τ ec(γ ,λ)t Z˜t 2 ,
m.e.
(8.18)
γ
t−s τ, γ τ,γ U˜t−s Z˜s ≤ c(γ )e 2τ ec(γ ,λ)t Z˜t 2 .
m.e.
(8.19)
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Proof. Inequalities (8.16) and (8.17) are immediate consequences of the fact that γ γ e−γ |x−x1 | e−γ |x−x2 | ≤ e− 2 |x1 −x2 | e− 2 |x| , for any γ > 0. (8.20) x∈Zd
x∈Zd
To derive inequalities (8.18) and (8.19), we use (8.20) and we dominate exponential t 2 factors e O(λ )t on the RHS by e 2τ , using that τ λ2 → 0 as λ 0. Lemma 8.2, below, shows how the bounds of Lemma 8.1 are used to integrate over diagrams. This lemma will be used repeatedly in the next sections, and, since it is a crucial step, we treat the following simple example in detail: We attempt to bound the expression γ γ γ F := dt1 dt2 ζ (σ ) U˜s f −t I˜ x2 ,l2 U˜t2 −t1 I˜ x1 ,l1 U˜t −s i (8.21) 2 1 i f s
in “the sense of matrix elements”, with σ being the diagram in ordered pair ((t1 , x1 , l1 ), (t2 , x2 , l2 )). We proceed as follows:
1 [s i ,s f ]
consisting of the
1) We bound ζ (σ ) by supx1 ,x2 ,l1 ,l2 |ζ (σ )|. Note that the latter expression is a function of t2 − t1 only. 2) Since the only dependence on x1 , x2 , l1 , l2 is in the operators I˜ xi ,li , we perform the sum xi ,li I˜ xi ,li = 1, for i = 1, 2. 3) Since the operators I˜ xi ,li have disappeared, we can bound γ
f i γ γ γ U˜s f −t U˜t2 −t1 U˜t −s i ≤ [c(γ )]2 ec(γ ,λ)|s −s | U˜s2f −s i 2
1
(8.22)
m.e.
using Lemma 8.1. Thus γ
f i F ≤ U˜s2f −s i ec(γ ,λ)|s −s |
m.e.
s i
1
<s f
dt1 dt2 [c(γ )]2
sup
x1 ,x2 ,l1 ,l2
|ζ (σ )|.
(8.23)
Note that supx1 ,x2 ,l1 ,l2 |ζ (σ )| = supx |ψ(x, t2 − t1 )| because |σ | = 1. The short derivation above can be considered to be an application of Lemma 7.1, as we illustrate by writing Fx
L ,x R ;x L ,x R
=
1 [s i ,s f ]
dσ G(σ )F(σ ),
with G(σ ) = ζ (σ ), F(σ ) := A˜ x
L ,x R ;x L ,x R
, (8.24)
and hence (8.23) follows from Lemma 7.1 after applying (8.22). Lemma 8.2 is a generalization of the bound (8.23) above. m such that Lemma 8.2. Fix an interval I = [s i , s f ] and a set of m triples (ti , xi , li )i=1 ti ∈ I and ti < ti+1 . For any σ ∈ I (ir), we define the set of n := m + 2|σ | triples m+2|σ | ) of the union of triples (ti , xi , li )i=1 by time-ordering (i.e., such that ti ≤ ti+1 m 2|σ | and (8.25) ti , xi , li i=1 , (ti (σ ), xi (σ ), li (σ ))i=1 .
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Then
675
γ γ γ dσ |ζ (σ )| I˜ xn ,ln U˜t −t . . . U˜t −t I˜ x2 ,l2 U˜t −t I˜ x1 ,l1 n n−1 3 2 2 1 I (ir)
≤
m.e.
ec(γ ,λ)|I |
!T
γ
d[σ ] [c(γ )]2|σ |
I (ir) γ 2 tm −tm−1
× U˜s2f −t I˜ xm ,lm U˜ m
sup
x(σ ),l(σ )
γ
|ζ (σ )| γ
. . . I˜ x2 ,l2 U˜t 2 −t I˜ x1 ,l1 U˜t 2 −s i . 2
1
(8.26)
1
γ τ,γ Moreover, the statement remains true if one replaces U˜t → Z˜t on the LHS and γ γ τ, U˜t 2 → Z˜t 2 on the RHS of (8.26).
Proof. The proof is a copy of the proof of the the bound (8.23). The steps are 1) 2) 3) 4)
Dominate |ζ (σ )| by supx(σ ),l(σ ) |ζ (σ )|. Sum over x(σ ), l(σ ) by using xi ,li I˜ xi ,li = 1. γ τ,γ Multiply the operators U˜t or Z˜t , using the bound (8.16) or (8.17). Interpret the remaining sum over |σ | and integration over t(σ ) as an integration over equivalence classes [σ ].
8.4. Bound on short pairings and proof of Lemma 6.1. We recall that the crucial result in Lemma 6.1 (see Statement 2 therein) is the bound Jκ Rτex (z)J−κ = dt e−t z R+
[0,t] (<τ ,ir)
dσ 1|σ |≥2 Jκ V[0,t] (σ )J−κ = O(λ2 )O(λ2 τ ), (8.27)
1 uniformly for Re z ≥ − 2τ and for |Im κ| small enough. In the first step of the proof of (8.27), we sum over the x(σ ), l(σ )- coordinates of the diagrams in (8.27). The strategy for doing this has been outlined in Sects. 8.2 and 8.3.
Lemma 8.3. For λ, γ smalll enough, dσ 1|σ |≥2 ζ (σ )V[0,t] (σ ) [0,t] (<τ ,ir)
≤ e
m.e.
c(γ ,λ)t
γ U˜t
!T
[0,t] (<τ ,ir)
d[σ ] c(γ )|σ | 1|σ |≥2
sup
x(σ ),l(σ )
|ζ (σ )|.
(8.28)
γ Proof. In the definition of V I (σ ), see e.g. (7.16), we bound Ix,l by I˜ x,l and Ut by U˜t . Then, we use the bound (8.26) with m = 0 to obtain (8.28). Note that, since m = 0, the m+2|σ | 2|σ | set of triples (ti , xi , li )i=1 is equal to the set of triples (ti (σ ), xi (σ ), li (σ ))i=1 . Note also that we use (8.26) with I (< τ, ir) instead of I (ir), and with the restriction to |σ | ≥ 2. However, this does not change the validity of (8.26), as one easily checks.
To appreciate how Lemma 8.3 relates to the bound (8.27), we note already that the bound (8.28) remains true if one puts left and right hand sides between Jκ · J−κ for purely imaginary κ (for general κ the matrix elements can become negative, which is not allowed by our definition of ≤m.e. ).
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In the second step of the proof, we estimate the Laplace transform of the integral over equivalence classes [σ ] appearing on the RHS of (8.28). This estimate uses three important facts 1) The correlation functions in (8.28) decay exponentially with rate 1/τ , due to the cutoff. 2) The diagrams are restricted to |σ | ≥ 2, they are therefore subleading with respect to a diagram with |σ | = 1. 3) We allow the estimate to depend on γ in a non-uniform way. Indeed, γ will be fixed in the last step of the argument. Concretely, we show that, for 0 < a ≤
1 τ
R+
dt e
and for λ small enough (depending on γ )
at [0,t] (<τ ,ir)
d[σ ] 1|σ |≥2 [c(γ )]
= O(λ2 )O(λ2 τ )c(γ ),
2|σ |
sup
x(σ ),l(σ )
|ζ (σ )|
λ 0, λ2 τ 0.
(8.29)
To verify (8.29), we set k(t) := λ2 c(γ ) 1|t|≤τ sup |ψ(x, t)|
(8.30)
x
and we calculate, by exploiting the cutoff τ in the definition of k(·), 1
e τ t k 1 < λ2 c(γ ),
te
1 τ + k 1
t
k 1 = τ O(λ2 )c(γ ). (8.31) #∞ The norm · 1 refers to the variable t, i.e., h 1 = 0 dt|h(t)|. Hence (8.29) follows from the bound (D.4) in Lemma D.1, in Appendix D, after using that τ O(λ2 ) < C, as λ 0, and choosing λ small enough. In the third step of the proof, we fix γ . By using the explicit form (8.12) and the relation (2.60), we check that Jκ U˜tγ J−κ ≤ ec(γ ,λ)t , for any |Im κ L ,R | < γ . (8.32) x L ,x R ;x L ,x R
Next, we make use of the following general fact that can be easily checked (e.g., by the Cauchy-Schwarz inequality): If, for some γ > 0 and C < ∞, (Jκ AJ−κ )x
L ,x R ;x L ,x R
≤ C,
uniformly for κ L ,R s.t. |Im κ L ,R | ≤ γ , (8.33)
then A ≤ c(γ ),
(8.34)
where the norms refer to the operator norm on B(B2 (l 2 (Zd , S ))), as in Definition 1. Hence, from (8.32) we get sup
|Im κ L ,R |<γ /2
γ
J−κ U˜t Jκ ≤ c(γ )ec(γ ,λ)t .
(8.35)
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By the first equality in (8.27) and Lemma 8.3, Jκ Rex (z)J−κ γ ≤ dt e−tRe z Jκ U˜t J−κ ec(γ ,λ)t m.e. R+
[0,t] (<τ ,ir)
d[σ ] 1|σ |≥2 [c(γ )]2|σ |
sup
x(σ ),l(σ )
|ζ (σ )|. (8.36)
We combine (8.36) and (8.35) with (8.29), setting a ≡ max (−Re z, 0) + c(γ , λ)
(8.37)
1 for λ small enough such that c(γ , λ) ≤ 2τ and such that (8.29) applies. At this point, the parameter γ has been fixed and this choice determines the maximal value of |Im κ|. This concludes the proof of the bound in (8.27). The other statements of Lemma 6.1 are proven below. τ (z) (Statement 2) follows by a drastically Proof of Lemma 6.1. The claim about Rld simplified version of the above argument for Rτex (z). To establish the convergence claim in Statement 1) of Lemma 6.1, it suffices, by (7.27), to check that Ztτ,ir ≤ eCt for some constant C. This has been established in the τ (z) + Rτ (z) is the Laplace transform of Z τ,ir . proof of Statement 2), above, since Rld t ex The identity (6.4) was established in Sect. 7.2. τ (z). To check Statement 3), we employ expression (C.1) for L(z) and (7.28) for Rld The latter differs from L(z) in that it is the Laplace transform of a quanitity with a cutoff 2 at t = τ and in the fact that it includes the propagator Ut = ei(ad(Y )+λ ad(ε))t , whereas L(z) includes only ei(ad(Y ))t . We observe that Jκ Rτld (z) − λ2 L(z) J−κ ∞ ≤ 4λ2 (8.38) dt sup |ψ(x, t)| Jκ e−iad(Y )t J−κ x τ τ 2 + 4λ2 dt sup |ψ(x, t)| Jκ e−iad(Y )t eiλ ad(ε(P))t − 1 J−κ , (8.39) 0
x
where the factors ‘4’ originate from the sum over l1 , l2 and we use that Re z ≥ 0. In the first term on the RHS, Jκ e−iad(Y )t J−κ = 1, since Y commutes with the position operator X . The second term is bounded by τ 2 2 4λ ds sup |ψ(x, s)| × sup λ2 t Jκ ad(ε(P))eiλ ad(ε(P))t J−κ ≤ τ λ4 C, 0
x
t≤τ
(8.40) where we have used Lemma 5.2 and the bound (2.12).
8.5. Bound on the vertex operators B(l, s i , s f ). In this section, we prove a bound on the ‘dressed vertex operators’, which were introduced in Sect. 7.3.3. Since such ‘dressed vertex operators’ contain an irreducible short diagram in the interval [s i , s f ], we obtain a bound that is exponentially decaying in |s f − s i |. In (8.41), this exponential decay resides in the function w(·) and it is made explicit through the calculation in (8.45).
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The proof of the next lemma parallels the proof of Lemma 8.3 above. Consider m m and let l be a (dressed) vertex with vertex set S(l) = {t , . . . , t }. triples (ti , xi , li )i=1 m 1 i f Let s , s be vertex time-coordinates associated to l, i.e., such that s i < t1 and s f > tm . Lemma 8.4. For λ, γ small enough, the following bound holds: γ γ B(l, s i , s f ) ≤ w(s f − s i ) U˜s f −t I˜ xm ,lm U˜t −t m.e.
m
m
m−1
γ γ . . . U˜t −t I˜ x1 ,l1 U˜t −s i , 2
1
1
sup
|ζ (σ )|.
(8.41)
where w(s − s ) := e f
i
λ2 C |s f −s i |
!T
[s i ,s f ] (<τ ,ir)
d[σ ] C |σ |
x(σ ),l(σ )
(8.42)
The RHS of (8.42) indeed depends only on s f − s i , since the correlation function ζ (σ ) depends only on differences of the time-coordinates of σ . The function w(·) depends on the coupling strength λ via the correlation function ζ (σ ), see 7.15). Proof. Starting from the definition of the vertex operator B(l, s i , s f ) given in (7.42), we γ bound the operators Ix,l , Ut by I˜ x,l , U˜t and we apply Lemma 8.2 to obtain B(l, s i , s f ) ≤ ec(γ ,λ)|s
f −s i |
!T
m.e.
γ
[s i ,s f ] (<τ ,ir) γ 2 tm −tm−1
× U˜s2f −t I˜ xm ,lm U˜ m
d[σ ] [c(γ )]2|σ | γ
sup
x(σ ),l(σ )
|ζ (σ )|
γ
. . . U˜t 2 −t I˜ x1 ,l1 U˜t 2 −s i . 2
1
(8.43)
1
γ
From the definition of U˜t in (8.12), we see that γ γ U˜t 1 ≤ ec(γ ,λ)t U˜t 2 ,
for γ2 < γ1 .
(8.44)
We dominate the RHS of (8.43) by fixing γ /2 = γ1 and applying (8.44) for any γ2 ≤ γ1 . This yields (8.41), with the constant C in (8.42) given by fixing γ = γ1 in c(γ ). One sees that the maximal value we can choose for γ1 is γ1 = 41 δε , with δε as in Assumption 2.1. For later use, we note here that, for λ sufficiently small and with w(t) defined in (8.42): R+
dt |t|w(t)e
t 2τ
≤ τC
t
R+
dtw(t)e τ
≤ O(λ2 τ 2 ),
λ 0, λ2 τ 0,
(8.45)
where the second inequality follows by the bound (D.3) in Lemma D.1, with k(t) := λ2 C sup |ψ(x, t)|1t≤τ x
and
a :=
for λ such that λ2 C < 1/τ with C as in the exponent of (8.42).
1 τ
(8.46)
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8.6. Bound on the conditional cutoff dynamics Ct (σl ). In this section, we state bounds on Ct (σl , L) and Ct (σl ), defined in Sects. 7.2.1 and 7.3.1, respectively. Our bounds will follow in a straightforward way from Lemma 8.4 and formula (7.45), which we repeat here for convenience: Ct (σl , L) = 0 < s i < s f < t Ds i Ds f B l|L| , s|iL| , s|fL| Zsτf −s i k
|L|
k
skf < ski for k > k . . . Zsτf −s i B l2 , s2i , s2f Zsτf −s i B l1 , s1i , s1f , 3 2 2 1
|L|−1
(8.47)
By inserting the bound from Lemma 8.4 in (8.47), we obtain a bound on Ct (σl , L) depending on the vertex time-coordinates s i , s f . In the next bound, Lemma 8.5, we simply integrate out these coordinates. To describe the result, it is convenient to introduce some tailor-made notation. Let the times (t1 , . . . , t2n ) be the time-coordinates of σl . We will now specify the effective dynamics between each of those times, depending on the vertex partition L. • If the times ti and ti+1 belong to the vertex set of the same vertex, then γ γ H˜ ti+1 ,ti := G˜ti+1 −ti ,
t γ γ with G˜t := e− 3τ U˜t .
(8.48)
• If the times ti and ti+1 belong to different vertices, then γ τ,γ H˜ ti+1 ,ti := Z˜ti+1 −ti .
(8.49)
The idea of this distinction is clear: within a dressed vertex, we get additional decay t γ from the short diagrams; this is the origin of the exponential decay e− 3τ in G˜t . Between τ the vertices, we encounter the cutoff reduced evolution Zt , as already visible in (8.47). Moreover, we get an additional small factor for each dressed vertex. To make this explicit, we define |L|dressed := #{dressed lk } (= number of dressed vertices in the vertex partition L).
(8.50)
γ Lemma 8.5. Let the operators H˜ ti ,ti+1 be defined as above, depending on the diagram σl and the vertex partition L. Then, for λ, γ small enough, |L|dressed γ γ G˜t−t I˜ x2n ,l2n H˜ t ,t Ct (σl , L) ≤ (|λ|τ )2 c(γ ) m.e.
2n
γ γ γ . . . H˜ t3 −t2 I˜ x2 ,l2 H˜ t2 −t1 I˜ x1 ,l1 G˜t1 .
2n 2n−1
(8.51)
γ Note that between the times 0 and t1 , we always (for each vertex partition) put G˜t1 . γ This is because either t1 = 0, in which case G˜t1 = 1, or t1 belongs to a dressed vertex whose initial time coordinate, s1i , is fixed to be s1i = 0. The same remark applies between the times tn and t (Fig. 12).
Proof. The proof starts from the representation of Ct (σl , L) in (8.47) and the bound for the vertex operators B(lk , ski , skf ) given in Lemma 8.4. Then we integrate out the ski , skf -coordinates for the dressed vertices lk . The main tool in doing so is the fast decay of the function w(·), as follows from (8.45).
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Fig. 12. Consider the long diagram σl ∈ [0,t] (>τ ) with |σ | = 4 shown above. In the upper figure, we show a short diagram σ such that σl ∪ σ is irreducible in [0, t]. The corresponding vertex partition L = {l1 , . . . , l6 } is indicated by vertical lines for the bare vertices l1 , l4 , l5 and horizontal bars for the dressed vertices l2 , l3 , l6 . In the picture below we suggest the representation that emerges after applying Lemma 8.5: There are no ˜ The vertex time coordinates any more. The time-coordinates of the long diagrams correspond to operators I. intervals between time-coordinates of the long diagrams correspond to operators Z˜ τ,γ or G˜ γ . The intervals corresponding to G˜ γ are those which in the upper picture belong entirely to the domain of a short diagram
We consider a simple example. Take t1 = 0 and t2n = t and let |L| = 1, i.e. there is one vertex l. It follows that l is dressed and S(l) = {t2 , . . . , t2n−1 }. In this case, formula (7.45) reads Ct (σl , L) =
0 < s i < t2 t2n−1 < s f < t
τ i f Zsτi −t Ix1 ,l1 , (8.52) ds i ds f Ix2n ,l2n Zt−s f B l, s , s 1
and the bound in Lemma 8.4 is γ B(l, s i , s f ) ≤ w s f − s i U˜s f −t m.e.
2n−1
γ × I˜ x2n−1 ,l2n−1 U˜t2n−1 −t2n−2
!
A˜ γ
γ
γ
. . . U˜t3 −t2 I˜ x2 ,l2 × U˜t −s i , 2 "
(8.53)
where the operator A˜ γ is defined as the ‘interior part’ of the vertex operator. The sole property of A˜ γ that is relevant for the present argument is that γ
A˜ γ ≤ ec(γ ,λ)t A˜ 2 , m.e.
(8.54)
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γ
as follows from the definition of U˜t and the bound (8.16). From (8.53), (8.54) and (8.18, 8.19) , we obtain s f −s i (t2n−1 −t2 ) i f f i e 2τ (c(γ ))2 ds ds w s − s Ct (σl , L) ≤ e− 3τ i 0 < s < t2 m.e. t2n−1 < s f < t
τ, γ2 t2n −t2n−1
γ
τ, A˜ 2 Z˜t2 −t2 1 I˜ x1 ,l1 , (8.55) where we have used the decomposition s f − s i = s f − t2n−1 + (t2n−1 − t2 ) + t2 − s i and we have chosen λ, γ small enough such that c(γ , λ) < 1/(6τ ) in (8.54). By a change of integration variables, we find that s f −s i t i f f i 2τ ds ds w s − s e ≤ dt |t|w(t)e 2τ , (8.56) 0 < si < t
× I˜ x2n ,l2n Z˜
γ
2
R+
t2n−1 < s f < t
and we note that this bound remains valid if, in the integration domain on the LHS, we replaced 0 by a smaller number, or t2n by a larger number. Hence, by the bound (8.45), we obtain Ct (σl , L) ≤ [c(γ )]2 (|λ|τ )2 e−
(t2n−1 −t2 ) 3τ
m.e.
τ,γ
τ,γ
Ix2n ,l2n Z˜t2n −t2n−1 A˜ γ Z˜t2 −t1 I˜ x1 ,l1 ,
(8.57)
where the constant that originates from the RHS of the bound (8.45) has been absorbed in c(γ ). The bound (8.57) is indeed (8.51) for our special choice of L in which |L|dressed =1. To obtain the general bound, one repeats the above calculation for each dressed vertex. These calculations can be performed completely independently of each other, as is visible from the remark below (8.56). In Lemma 8.5, the bound depends on L through H˜ γ , see (8.48) and (8.49). The next step is to sum over L. First, we weaken our bound in (8.51) to be valid for all L, such that the sum over L amounts to counting all possible L ∼ σl . By “weakening the bound”, γ τ,γ we mean that we bound some of the operators G˜t by Z˜t . This can always be done, since, for λ small enough, γ τ,γ G˜t ≤ Z˜t m.e.
(8.58)
γ γ τ,γ with Gt as in (8.48) (in fact, G˜t is smaller than the second term of Z˜t , see (8.11)). Let σ1 , . . . , σm be the decomposition of σl into irreducible components and let s2i−1 , s2i be the boundaries of the domain of σi . These times si should not be confused with the vertex time-coordinates s i , s f that were employed in an earlier stage of our analysis. In particular, the times s2i−1 , s2i , i = 1, . . . m, are a subset of the times ti , i = 1, . . . , 2n. The central remark is that
For any i, the times s2i , s2i+1 belong to the same vertex for all vertex partitions L ∼ σl . Indeed, since the interval [s2i , s2i+1 ] is not in the domain of σl , it must be in the domain of any short diagram contributing to Ct (σl ), or, in other words, any vertex partition L ∼ σl must contain a vertex whose vertex set contains both s2i , s2i+1 . Consequently, γ the operators H˜ s2i ,s2i+1 in (8.51) are always (i.e., for each compatible vertex partition) γ γ equal to G˜s2i ,s2i+1 , and we will not replace them. However, we replace all other H˜ t j ,t j+1 ,
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Fig. 13. Consider the long diagram σl ∈ [0,t] (> τ ) with |σ | = 4 shown above. It has two irreducible components with domains [s1 , s2 ] and [s3 , s4 ]. In the upper figures, two different vertex partitions (compatible with σl ) are shown together with their respective bounds, obtained in Lemma 8.5. These bounds are represented by the operators G˜ γ and Z˜ τ,γ , as in Fig. 12, except for the fact that we omit the operators I˜ corresponding to the time coordinates of σl . In the lower figure, we show the (weaker) bound that gives rise to Lemma 8.6. To establish this weaker bound, we replace the G˜ γ that are ‘bridged’ by the long diagram by Z˜ τ,γ
i.e. those with the property that the times t j , t j+1 are in the domain of the same irreducible τ,γ component of σl , by Z˜t j+1 −t j . This procedure is illustrated in Fig. 13. After this replacement, the operator part of the resulting expression is independent of L, and we can perform the sum over L ∼ σl by estimating |L|dressed (|λ|τ )2 c(γ ) ≤ (|λ|τ )2v(σl ) c(γ )|σl | , for |λ|τ ≤ 1 (8.59) L∼σl
with v(σl ) := min |L|dressed .
(8.60)
#{L ∼ σl } ≤ 42|σl |−1 .
(8.61)
L∼σl
To obtain (8.59), one uses that 22|σl |−1
is the number of ways to partition the time-coordinates into vertex sets. Indeed, The extra factor 2 for each vertex takes into account the choice bare/dressed. We have thus arrived at the following lemma Lemma 8.6. Let s2i−2 , s2i be the boundaries of the domain of σi , the i th irreducible component of σl . Then, for λ, γ small enough, γ γ γ Ct (σl ) ≤ (|λ|τ )2v(σl ) G˜t−s2m E˜ γ (σm )G˜s2m−1 −s2m−2 E˜ γ (σm−1 ) . . . E˜ γ (σ1 )G˜s1 , m.e.
(8.62)
where, for an irreducible diagram σ with |σ | = p, τ,γ τ,γ E˜ γ (σ ) := [c(γ )]|σ | I˜ x2 p (σ ),l2 p (σ ) Z˜t2 p (σ )−t2 p−1 (σ ) . . . Z˜t2 (σ )−t1 (σ ) I˜ x1 (σ ),l1 (σ )
(8.63)
with v(σl ) as defined above. γ Note that v(σl ) is actually the number of factors G˜u in the expression (8.62) for which γ u = 0 (u can be zero only for the rightmost and leftmost G˜u ). Or, alternatively,
v(σl ) = #{irreducible components in σl } − 1 + 1t2n =t + 1t1 =0 .
(8.64)
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8.7. Bounds on Rex (z) in terms of E˜ γ (σ ). To realize why the bound (8.62) in Lemma 8.6 is useful, we recall that our aim is to calculate Rex (z), given by (see (7.33)) Rex (z) −t z = dt e R+
[0,t] (>τ )
dσ ζ (σ )Ct (σ ) [1 + δ(t1 (σl )] 1 + δ(t2|σl | (σl ) − t) . (8.65)
We calculate Rex (z) by replacing the integral over diagrams by an integral over sequences of irreducible diagrams, as we did in (7.23), i.e., n dσ . . . = dσ j . . . . (8.66) [0,t] (>τ )
n≥1 0≤s1 <···<s2n ≤t j=1
[s2 j−1 ,s2 j ] (>τ,ir)
Using the bound (8.62), we obtain, with the shorthand zr := Re z, n Rex (z) ≤ 1 + RG˜ (zr ) RE˜ (zr ) RG˜ (zr )RE˜ (zr ) 1 + RG˜ (zr ) m.e.
n≥0
−1 1 + RG˜ (zr ) , (8.67) = 1 + RG˜ (zr ) RE˜ (zr ) 1 − RG˜ (zr )RE˜ (zr ) where, for Re z large enough,
γ RG˜ (z) := (τ |λ|)2 dt e−t z G˜t + R −t z RE˜ (z) := dt e dσ ζ (σ )E˜ γ (σ ). R+
[0,t] (>τ,ir)
(8.68) (8.69)
γ
Since G˜t is known explicitly, the only task that remains is to study RE˜ (z). This study is undertaken in Sect. 9. 9. The Renormalized Model In this section, we prove Lemma 6.5, thereby concluding the proof of our main result, Theorem 3.3. We briefly recall the logic of our proof. As announced in Sect. 5.4.1, we analyze Ztir and Zt through a renormalized perturbation series, where the short diagrams have already been resummed. However, we do not study the Laplace transform of irreducible diagrams (defined in (7.20)) Rir (z) = dt e−t z Ztir = dt e−t z dσ ζ (σ )Vt (σ ), (9.1) R+
R+
[0,t] (ir)
directly, but rather the Laplace transform of irreducible renormalized diagrams (defined in Lemma 8.6 and Sect. 8.7) dt e−t z dσ ζ (σ )E˜ γ (σ ). (9.2) RE˜ (z) = R+
[0,t] (>τ,ir)
Although the quantities (9.2) and (9.1) are not equal, we will argue below (in the proof of Lemma 6.5 starting from Lemma 9.1) that good bounds on RE˜ (z) yield good bounds on Rir (z) and hence also on R(z). The reason that the expression (9.1) itself cannot
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be bounded by an integral over long irreducible diagrams, is the fact that an irreducible diagram in the interval [0, t] does not necessarily contain an irreducible long subdiagram in the interval [0, t]. Indeed, Lemma 8.6 decomposes the domain of an irreducible diagram into domains of long, irreducible subdiagrams and intermediate intervals. These γ remaining intervals give rise to operators G˜t , which are easily dealt with, as we will see below, in the proof of Lemma 6.5, since they originate from short diagrams and therefore have good decay properties. Nevertheless, we clearly see the similarity between (9.1) and (9.2). To highlight this similarity, we write the inverse Laplace transform of RE˜ (z): For r > 0 large enough, we have 1 dz et z RE˜ (z) 2πi r +iR τ,γ τ,γ = dσ c(γ )|σ | ζ (σ ) I˜ x2n ,l2n Z˜t2n −t2n−1 . . . I˜ x2 ,l2 Z˜t2 −t1 I˜ x1 ,l1 , (9.3) [0,t] (>τ,ir)
where t, x, l are the coordinates of σ and, since σ is irreducible in [0, t], t1 = 0 and t2n = t. The inverse Laplace transform of Rir (z), i.e. Ztir , is ir dσ ζ (σ )Ix2n ,l2n Ut2n −t2n−1 . . . Ix2 ,l2 Ut2 −t1 Ix1 ,l1 , (9.4) Zt = [0,t] (ir)
where t, x, l have the same meaning as above. Thus, the perturbation series in (9.3) is indeed a renormalized version of (9.4). The diagrams are constrained to be long, and the τ,γ short diagrams have been absorbed into the ‘dressed free propagator’ Z˜t . This point τ,γ of view has also been stressed in Sect. 5.5. Observe, however, that Zt depends on the positive parameter γ , whereas there is no such dependence in (9.1). The following lemma is our main result on RE˜ (z). Lemma 9.1. Recall that RE˜ (z) depends on γ , because E˜ γ (·) does. One can choose γ such that there are positive constants gex > 0 and δex > 0, such that sup |Im κ R,L |≤δex ,Re z≥−λ2 gex
Jκ RE˜ (z)J−κ = o(λ2 ),
as λ 0.
(9.5)
The main tools in the proof of Lemma 9.1 will be the exponential decay of the ‘renormalized correlation function’, which follows from the bounds on Ztτ stated in Lemma 6.2, and the strategy for integrating over diagrams presented in Lemma 8.2. With Lemma 9.1 at hand, the proof of Lemma 6.5 is immediate. Proof of Lemma 6.5. We only need to prove Statement 2) since Statement 1) will follow by a remark analogous to that in the proof of Statement 1) of Lemma 6.1. Clearly, for λ small enough, as λ 0. (9.6) sup Jκ RG˜ (z)J−κ ≤ O(λ), 1 |Im κ R,L |≤γ ,Re z≥− 4τ
γ This follows from the properties of U˜t , see e.g. the proof of Lemma 6.1, and the defiγ nition of G˜t , see (8.48). Next, we remark that
Jκ Rex (z)J−κ
−1 ≤ 1 + Jκ RG˜ (zr )J−κ 2 Jκ RE˜ (zr )J−κ 1 − Jκ RG˜ (zr )RE˜ (zr )J−κ (9.7)
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685
with zr = Re z. This follows from the bound (8.67), the fact that Jκ J−κ = 1, and the implication (8.6) (which allows to pass from ‘≤m.e. ’ to an inequality between norms). Hence, Statement 2) follows by plugging the bounds of Lemma 9.1 and (9.6) into the the RHS of (9.7).
9.1. Bound on the renormalized correlation function. In this section, we prove Lemma 9.2, which establishes (as its first claim) the property (5.23) with t replaced by Ztτ . Indeed, in Sect. 6.1, we argued that Ztτ is very close to t , and this was made explicit in Lemma 6.2. Let h(t) := λ ch sup 2
x∈Zd
e−(1/2)gR t |x|/t ≤ v ∗ |ψ(x, t)| |x|/t ≥ v ∗
(9.8)
with the velocity v ∗ and decay rate gR as in Lemma 5.1 and the constant ch chosen such that λ2 sup |ψ(x, t)| ≤ h(t),
for t > τ,
x
(9.9)
Lemma 5.1 ensures that such a choice is possible. Lemma 9.2. There are positive constants δr > 0 and gr > 0 such that, for all γ < δr , λ small enough, and κ ≡ (κ L , κ R ) satisfying |Im κ R,L | ≤ δ2r , λ |ψ 2
for
xl2
˜tτ,γ J−κ J Z − xl 1 , t | × κ x
l1 , l2 ∈ {L , R}
L ,x R ;x L ,x R
≤ h(t)e−λ2 gr t , (9.10)
and Jκ Z˜tτ,γ J−κ x
L ,x R ;x L ,x R
≤ Cec(γ ,λ)t .
(9.11)
This lemma is derived from the bound (6.7) in Lemma 6.2 in a way that is completely analogous to the proof of (5.23) starting from (4.55), as outlined in Sect. 5.4.1. The only difference is that in Lemma 9.2, we allow for a small blowup in space given by the multiplication operator Jκ . For future use, we also define h τ (t) := 1|t|≥τ h(t),
(9.12)
and we note that h τ 1 :=
R+
h τ (t) = o(λ2 ),
since h 1 < ∞ and τ (λ) → ∞ as λ 0.
as λ 0,
(9.13)
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9.2. Sum over non-minimally iducible diagrams. In a first step towards performing the integral in (9.2), we reduce the integral over irreducible diagrams to an integral over minimally irreducible diagrams. Indeed, since any diagram that is irreducible in I has a minimally irreducible (in I ) subdiagram, we have, for any positive function F, dσ F(σ ) ≤ dσ F(σ ) + dσ F(σ ∪ σ ) . (9.14) I (ir,>τ )
I (mir,>τ )
I (>τ )
The first term between brackets on the RHS corresponds to the minimally irreducible diagrams on the LHS. The second term contains the integration over ‘additional’ diagrams σ . The integration over these diagrams is unconstrained since σ ∪σ is irreducible in I for any σ , provided that σ is irreducible in I . This is also explained and used in Appendix D: see (D.5) and (D.6). Lemma 9.3 shows that such an integration over unconstrained long diagrams yields a factor exp{c (γ , λ)|I |}, with the generic constant c (γ , λ) as introduced in Sect. 8.1. Lemma 9.3. For λ, γ small enough, dσ E˜ γ (σ )|ζ (σ )| [0,t] (>τ,ir)
≤ e
c (γ ,λ)t
γ
[0,t] (>τ,mir)
m.e.
dσ |ζ (σ )|E˜ 2 (σ ).
(9.15)
Proof. By formula (9.14) (applied in the case where F(σ ) is a matrix element of the operator |ζ (σ )|E˜ γ (σ )), we have that γ ˜ dσ E (σ )|ζ (σ )| ≤ dσ E˜ γ (σ )|ζ (σ )| [0,t] (>τ,ir)
+
[0,t] (>τ,mir)
dσ
[0,t] (>τ,mir)
m.e.
[0,t] (>τ )
dσ E˜ γ (σ ∪ σ )|ζ (σ ∪ σ )|.
(9.16)
First, we bound [0,t] (>τ )
dσ E˜ γ (σ ∪ σ )|ζ (σ ∪ σ )|
(9.17)
with σ fixed. To perform the integral over σ in (9.17), we recall that E˜ γ (·) consists of τ,γ γ products of the operators I˜ xi ,li and Zti+1 −ti . Hence, by Lemma 8.2 with U˜t replaced τ,γ τ,γ by Z˜t , we can sum over the x, l-coordinates of σ and multiply the Z˜ti+1 −ti operators using the bound (8.17). This yields γ d[σ ]c(γ )|σ | sup |ζ (σ )|. (9.18) (9.17) ≤ E˜t 2 (σ )|ζ (σ )| ec(γ ,λ)t !T
m.e.
t (>τ )
x(σ ),l(σ )
The integral on the RHS of (9.18) is estimated as d[σ ]c(γ )|σ | sup |ζ (σ )| ≤ ec(γ ) h τ 1 t − 1 !T
t (>τ )
x(σ ),l(σ )
(9.19)
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Fig. 14. The decomposition of a minimally irreducible diagram σ into two ‘ladder diagrams’ σ1 and σ2 . In the upper figure, one can easily check that any point on the (horizontal) time-axis is bridged by at most two pairings
with h τ as defined in (9.12). This follows from the bound (D.8) (integral over unconstrained diagrams) in Appendix D. To bound the first term in (9.16), we dominate γ
E˜ γ (σ ) ≤ ec (γ ,λ)t E˜ 2 (σ ).
m.e.
(9.20)
The lemma follows by inserting the bounds (9.20) and (9.18, 9.19) in (9.16) and using that c(γ ) h τ 1 = c (γ , λ), since h τ 1 = o(λ2 ). 9.3. Sum over minimally irreducible diagrams. In this section, we perform the integral dσ |ζ (σ )|E˜ γ (σ ) (9.21) t (>τ,mir)
that appears on the RHS of the bound in Lemma 9.3 (upon replacing γ → γ /2), and we prove that it is exponentially decaying in time with decay rate O(λ2 ), for well-chosen γ and λ small enough, depending on γ . It is in this place that we use the decay property of the renormalized correlation function that was stated in Lemma 9.2. The key idea is the following. If σ is a long diagram with |σ | = 1 consisting of the 2 , then two triples (ti , xi , li )i=1 τ,γ |ζ (σ )|E˜ γ (σ ) = c(γ )|ψ # (x2 − x1 , t2 − t1 ) | I˜ x2 ,l2 Z˜t2 −t1 I˜ x1 ,l1 .
(9.22)
In this case, we can obviously use Lemma 9.2 to deduce exponential decay in t2 − t1 of (9.22), uniformly in x1 , x2 , l1 , l2 . In general, there is of course more than one pairing in τ,γ an irreducible diagram and so one has to ‘split’ the decay coming from I˜ x2 ,l2 Z˜t2 −t1 I˜ x1 ,l1 between the different pairings, thus weakening the decay by a factor which can be as high as |σ |. However, since we are considering minimally irreducible pairings, there are at most two pairings bridging any given time t , see Fig. 14. Hence, one can attempt to split the τ,γ decay from I˜ x2 ,l2 Z˜t2 −t1 I˜ x1 ,l1 in half. This can be done and it is described in Lemma 9.4. For σ ∈ [0,t] , we define the function Hτ (σ ) =
|σ |
h τ (v j − u j )
(9.23)
j=1
with h τ as in (9.12). Note that Hτ (σ ) depends only on the equivalence class [σ ], and hence we can write Hτ ([σ ]) := Hτ (σ ).
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Lemma 9.4. Let the positive constants gr and δr be defined as in Lemma 9.2 and choose γ < δr . Let κ = (κ L , κ R ) such that |Im κ L ,R | ≤ γ /8 and fix a long irreducible diagram class [σ ] ∈ !T [0,t] (> τ, mir). Then γ |σ | c(γ ,λ)t − 21 λ2 gr t |ζ (σ )|Jκ E˜ (σ )J−κ e Hτ ([σ ]). (9.24) ≤ c(γ ) e x(σ ),l(σ ) Note that the operator between · depends on the equivalence class [σ ] only, due to the sum over x(σ ), l(σ ). Proof. For concreteness, we assume that |σ | = n is even; the argument for |σ | odd is analogous. We can find σ1 and σ2 such that σ1 ∪ σ2 = σ, |σ1 | = |σ2 | = n/2 and σ1 and σ2 are ladder diagrams, i.e. their decompositions into irreducible components consists of singletons. To be more concrete, the time-pairs of σ1 are (t1 , t3 ), (t4 , t7 ), (t8 , t11 ) . . . (t2n−4 , t2n−1 ),
(9.25)
(t2 , t5 ), (t6 , t9 ), (t10 , t13 ) . . . (t2n−2 , t2n ).
(9.26)
and those of σ2 are The possibility of making such a decomposition is a consequence of the structure of minimally irreducible diagrams, as illustrated in Fig. 14. We now estimate the LHS of (9.24) in two ways. In our first estimate, we take the supremum over the x, l-coordinates of σ2 and we keep those of σ1 . In the second estimate, the roles of σ1 and σ2 are reversed. We estimate
|ζ (σ )|E˜ γ (σ )
x(σ ),l(σ )
= [c(γ )]|σ |
x(σ2 ),l(σ2 )
x(σ1 ),l(σ1 )
x(σ1 ),l(σ1 )
τ,γ τ,γ |ζ (σ1 )| I˜ x2n ,l2n Zt2n −t2n−1 . . . I˜ x2 ,l2 Zt2 −t1 I˜ x1 ,l1
≤ [c(γ )]|σ |ec(γ ,λ)t [c(γ )]|σ2 | m.e. ×
|ζ (σ2 )|
sup
x(σ2 ),l(σ2 )
|ζ (σ2 )|
γ
γ
γ
τ, τ, 2 ˜ τ, 2 ˜ ˜ |ζ (σ1 )|Z˜t2n 2−t2n−1 I˜ x2n−1 ,l2n−1 Z˜t2n−1 −t2n−4 . . . Ix3 ,l3 Zt3 −t1 Ix1 ,l1 ) . (9.27)
The equality follows from the definition of E˜ γ (σ ). To obtain the inequality on the third line, we perform the sum over x(σ2 ), l(σ2 ) by the same procedure that was used to obtain (9.18), i.e., by using Lemma 8.2. Since |ζ (σ1 )| factorizes into a function of the pairs in σ1 , the operator part in the last line of (9.27) is a product of two types of terms, namely; τ, γ |ψ (x4i+3 − x4i , t4i+3 − t4i )| I˜ x4i+3 ,l4i+3 Z˜t4i+32 −t4i I˜ x4i ,l4i λ2 (9.28) x4i , x4i+3 l4i , l4i+3
for i = 1, . . . , n/2−1 (and an analogous term where we replace 4i → 1 and 4i +3 → 3, corresponding to the first pair in σ1 ) and γ
τ, 2 Z˜t4i −t 4i−1
for i = 1, . . . , n/2.
(9.29)
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We note that Lemma 9.2 provides a bound on the matrix elements of these expressions. In particular, we use (9.10) to bound (9.28) and (9.11) to bound (9.29). We obtain, for |Im κ L ,R | ≤ γ4 , (Jκ (9.28) J−κ )x
L ,x R ;x L ,x R
(Jκ (9.29) J−κ )x
L ,x R ;x L ,x R
≤ h τ (t4i+3 − t4i ) e−λ
2 g (t r 4i+3 −t4i )
,
≤ ec(γ ,λ)(t4i −t4i−1 ) .
(9.30) (9.31)
By the relation stated in (8.34) and the line following it, we can convert these bounds on the kernels into bounds on the operator norms, yielding, for |Im κ L ,R | ≤ 18 γ , Jκ (9.28) J−κ ≤ c(γ )h τ (t2i+1 − t2i−1 ) e−λ
2 g (t r 4i+3 −t4i ),
(9.32)
Jκ (9.29) J−κ ≤ c(γ )e
(9.33)
c(γ ,λ)(t4i −t4i−1 )
,
and hence, by multiplying these bounds for the operators appearing in (9.27) and using that h τ (t3 − t1 )
n/2−1
h τ (t4i+3 − t4i ) ≤ Hτ (σ1 ),
i=1
sup
x(σ2 ),l(σ2 )
|ζ (σ2 )| ≤ Hτ (σ2 ),
(9.34)
(see (9.9)), we arrive at Jκ (9.27) J−κ ≤ c(γ )|σ | Hτ (σ )e−λ
2 g |Domσ | r 1
ec(γ ,λ)t .
(9.35)
The claim of the lemma now follows by applying the same bound with the roles of σ1 and σ2 swapped, taking the geometric mean of the two bounds and noting that [0, t] ≤ |Domσ1 | + |Domσ2 |.
(9.36)
Next, we use Lemmata 9.3 and 9.4 to prove Lemma 9.1. By these two lemmas, the integral over renormalized irreducible diagrams is reduced to an integral over minimally irreducible equivalence classes [σ ]. Each equivalence class [σ ] essentially contributes c(γ )|σ | Hτ (σ ) to the integral. Since Hτ (σ ) is not exponentially decaying in Domσ , the Laplace transform of Hτ (σ ) cannot be continued to negative Re z. However, the 21 factor e−λ 2 gr t in Lemma 9.4 enables us to do such a continuation since the factors c (γ , λ), c(γ , λ) from Lemmata 9.3 and 9.4 can be made smaller than λ2 21 gr by first choosing γ small enough, and then adjusting λ. Proof of Lemma 9.1. We choose γ small enough, as required in the conditions of Lemmata 9.3 and 9.4, and we estimate, for |Im κ L ,R | ≤ γ /8,
γ (γ ,λ)t γ c ˜ ˜ dσ |ζ (σ )|Jκ E (σ )J−κ ≤ e dσ |ζ (σ )|Jκ E 2 (σ )J−κ [0,t] (>τ,mir) [0,t] (>τ,ir) γ ˜ 2 ≤ ec (γ ,λ)t E d[σ ] |ζ (σ )| J (σ )J κ −κ !T [0,t] (>τ,mir) x(σ ),l(σ ) 1 2 1 2 ≤ e(c (γ ,λ)− 4 λ gr )t d[σ ] c(γ )|σ | e− 4 λ gr t Hτ (σ ). (9.37) !T
[0,t] (>τ,mir)
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The first inequality is Lemma 9.3, the second inequality uses the definition of the measure dσ , and the third inequality follows from Lemma 9.4. We will now estimate the Laplace transform of the integral in the last line of (9.37). To prove Lemma 9.1, we fix gex := 18 gr and we show that one can choose γ such that, for λ small enough and Re z ≥ −λ2 gex ,
R+
dt e
−t z
e
c (γ ,λ)− 14 λ2 gr t
1 2 gr t
!T
[0,t] (>τ,mir)
d[σ ]c(γ )|σ | e− 4 λ
as λ 0.
Hτ (σ ) = o(λ2 ), (9.38)
Of course, the choice of γ will have to depend on the specific value of c (γ , λ). To show (9.38), we first choose γ such that, for λ small enough, we have 1 1 c (γ , λ) − λ2 gr ≤ − λ2 gr 4 8 with c (γ , λ) as in (9.38).
(9.39)
c (γ ,λ)− 14 λ2 gr t −zt e e
by 1 in (9.38). Next, Consequently, we can dominate the factor we note that, if σ is minimally irreducible in the interval [0, t], then |σ |
|vi − u i | ≤ 2t,
(9.40)
i=1
where (u i , vi ) are the pairs of time-coordinates associated to σ . This follows from the observation that each point in the interval [0, t] is bridged by at most two pairings of σ , see also Fig. 14. Consequently, we find that 1 2 gr t
Hτ (σ )e− 4 λ
≤
|σ |
1 2 gr |v j −u j |
h τ (v j − u j )e− 8 λ
.
(9.41)
j=1 c (γ ,λ)− 14 λ2 gr t −zt We estimate the LHS of (9.38), with e e
replaced by 1, by invoking
(D.2) in Lemma D.1, with 1 2 gr t
k(t) := c(γ )e− 8 λ
h τ (t)
and
a := 0. 1 2 gr t
Indeed, by using h τ 1 = o(λ2 ) and the exponential decay e− 8 λ k 1 = c(γ )o(λ2 ), tk 1 = c(γ )o(|λ|0 ), Therefore, the bound (D.2) yields (9.38).
as λ 0,
(9.42) , we obtain that (9.43) (9.44)
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9.4. List of important parameters. We list some constants and functions that we use throughout the paper. We start with the different decay rates; in the third column we indicate where the constant appears first. By “full model”, we mean: “the model without cutoff”. gR 1 τ
=
|λ|1/2
λ2 gr λ2 gr w λ2 gc λ2 gex λ2 g
bare reservoir correlation fct. (for subluminal speed)
Lemma 5.1
bare reservoir correlation fct. in the cut-off model
Section 5.3
renormalized joint S − R correlation function
Lemma 9.2
Markov semigroup
Proposition 4.2
cut-off model
Lemma 6.2
excitations in the full model
Lemma 9.1
full model
Theorem 3.2
Additionally, the rates gr w , gc , g come with a superscript low, high indicating that the gap refers to small, large fibers p, respectively. The following constants restrict the values of complex deformation parameters, in particular the parameter κ in Jκ , as defined in 2.59. δε δR δ δr w δex δr
particle dispersion law reservoir dispersion law full model Markov semigroup excitations in the full model renormalized S − R correlation fct.
Assumption 2.1 Assumption 2.3 Theorem 3.3 Proposition 4.2 Lemma 6.5 Lemma 9.2
The following functions of γ , λ appear as blowup-rates in exponential bounds. rr w (γ , λ) rε (γ , λ) rτ (γ , λ)
Markov semigroup γ U˜t
τγ Z˜t
Lemma 8.1 Section 8.3 Lemma 6.2
In the final Sects. 8 and 9, these rates are represented by the generic constants c(γ , λ), c (γ , λ), introduced in Sect. 8.1. Acknowledgements. W. De Roeck thanks J. Bricmont and H. Spohn for helpful discussions and suggestions, and for pointing out several references. He has also greatly benefited from collaboration with J. Clark and C. Maes. In particular, the results described in Sect. 4 and Appendix C were essentially obtained in [8]. After the first version was submitted, some inaccuracies were pointed out by L. Erdös, A. Knowles, H.-T. Yau and J. Yin. Most importantly, the remarks of an anonymous referee allowed for serious improvements in the presentation of the proof. At the time when this work was completed, W.D.R. was a postdoctoral fellow of the Flemish research fund ‘FWO-Vlaanderen’. The support of this institution is gratefully acknowledged.
A. Appendix: The Reservoir Correlation Function In this appendix, we study the reservoir correlation function ψ(x, t) and we prove Lemmata 5.1 and 5.2. Recall the definition of the “effective squared form factor” ψˆ
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in (2.27). It is related to ψ(x, t) by, see (5.3), itω iωs·x ˆ ψ(x, t) = dω ds ψ(ω)e e . R
(A.1)
Sd−1
From this expression, one understands that ψ(x, t) cannot have exponential decay in t, uniformly in x. One also sees that, for x fixed, there is exponential decay provided that ˆ is analytic in a strip around R. ψ(·) Consider q(·) such that ψ(x, t) = ds q(t + s · x). (A.2) Sd−1
By Assumption 2.3, there is a δR > 0 such that q(t) decays as Ce−δR |t| . Choosing v ∗ = 21 , we obtain, for |x|/|t| ≤ v ∗ , |ψ(x, t)| ≤ e−gR |t| C,
with gR :=
2 δR , 3
(A.3)
which proves Lemma 5.1. From now on, we assume that |x|/|t| > v ∗ . We remark that (A.2) can be rewritten, after an explicit calculation, as ψ(x, t) =
1 −1
dη q(t + η|x|)a(η),
d−3 2 a(η) := Area Sd−2 1 − η2 .
(A.4)
ˆ ˆ we By Assumption 2.3, in particular the condition ψ(0) = 0 and the analyticity of ψ, ˆ
deduce that ψ(ω) ω is analytic in a strip around R, as well. Its Fourier transform, Q, is an ˆ ∈ L 1 ) whose derivative equals q. exponentially decaying C 1 -function (since ψ(ω) Hence, by partial integration, the fact that a(η) |−1 = a(η)|1 = 0 for d > 3, and the change of variables ζ = |x|η, we obtain 1 ψ(x, t) = − 3/2 |x|
|x|
1 dζ Q(t + ζ ) 1/2 a |x| −|x|
ζ , |x|
Q = q.
(A.5)
Here, Q and a stand for the derivatives of Q and a. We evaluate this integral by splitting it into the regions − |x| ≤ ζ ≤ −|x| + 1,
−|x| + 1 ≤ ζ ≤ |x| − 1,
|x| − 1 ≤ ζ ≤ |x|. (A.6)
ζ ) ∞ (we In the second region, we dominate the integral (A.5) by Q 1 × |x|11/2 a ( |x| assume here that |x| ≥ 1, otherwise the decay in t has been proven above). In the first ζ and third region, we dominate the integral (A.5) by Q ∞ × |x|11/2 a ( |x| ) 1 . Using the ∗ explicit form of a and the fact that |x|/|t| > v , we conclude
sup |ψ(x, t)| ≤ C(1 + |t|)−3/2 , x
for d ≥ 4,
(A.7)
which implies Lemma 5.2. Obviously, dispersive estimates like (A.7) can be derived in much greater generality, see e.g. [22].
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B. Appendix: Spectral Perturbation Theory Let ∈ R be a small parameter and consider a continuous function R+ # t → V (t, ), taking values in a Banach space, and such that sup e−tm V (t, ) < ∞,
for some m > 0.
(B.1)
t≥0
The Laplace transform A(z, ) :=
R+
dte−t z V (t, )
(B.2)
is well-defined for Re z > m and it follows (by the inverse Laplace transform) that 1 V (t, ) = dz ezt A(z, ), with → := m + iR for any m > m, (B.3) 2π i → where the integral is meant in the sense of improper Riemann integrals. We will state assumptions that allow to continue A(z, ) downwards in the complex plane, i.e., to Re z ≤ m and to obtain bounds on V (t, ). Lemma B.1. For Re z large enough, let A(z, ) := (z − iB − A1 (z, ))−1
(B.4)
and assume the following conditions: 1) B is bounded and its spectrum consists of finitely many semisimple eigenvalues on the real axis, that is B= b1b (B), (B.5) b∈spB
where 1b (B) is the spectral projection corresponding to the eigenvalue b. For concreteness, we assume that 0 ∈ spB. 2) For small enough, the operator-valued function z → A1 (z, ) is analytic in the domain Re z > − g A and A1 (z, ) = O( ),
sup
0,
(B.6)
Re z>− g A
sup Re z>− g A
∂ A1 (z, ) = o(| |0 ), ∂z
0,
(B.7)
3) There are bounded operators Nb , for b ∈ spB, acting on the spectral subspaces Ran1b (B) and such that, for all b ∈ spB, Nb − 1b (B)A1 (ib, )1b (B) = o( ),
0.
(B.8)
Consider the operator N := ⊕ Nb , b∈spB
with [B, N ] = 0,
(B.9)
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and assume that N has a simple eigenvalue f N such that spN = { f N } ∪ N
sup Re N ≤ −g N
and
(B.10)
for some gap g N > 0. We also require that Re f N > −g N ,
Re f N > −g A .
(B.11)
The eigenvalue f N is necessarily an eigenvalue of Nb for some b ∈ spB. For concreteness (and to match with our applications), we assume that it is an eigenvalue of N0 . Then, there is an 0 > 0 such that, for | | ≤ 0 , there is a number f ( ), a rank-one operator P( ), bounded operators R(t, ) and a decay rate g > 0, such that V (t, ) = P( )e f ( )t + R(t, )e− gt
(B.12)
with f ( ) − f N = o( ), P( ) − 1 f N (N ) = o(| |0 ), sup R(t, ) = O(| |0 ),
t∈R+
(B.13) (B.14) as | | 0,
(B.15)
with 1 f N (N ) the spectral projection of N associated to the eigenvalue f N . The decay rate g can be chosen arbitrarily close to min{g N , g A } by making 0 small enough. In particular, one can choose g and 0 such that Re f ( ) > − g for all | | ≤ 0 . If, in addition N and A1 depend analytically on a parameter α in a complex domain D ⊂ C, such that (B.6)–(B.7)–(B.8)–(B.10)–(B.11) hold uniformly in α ∈ D, then (B.12) holds with f, P and R analytic in α and the estimates (B.13)–(B.14)–(B.15) are satisfied uniformly in α ∈ D. Lemma B.1 follows in a straightforward way from spectral perturbation theory of discrete spectra. For completeness, we give a proof below, using freely some well-known results that can be found in, e.g., [28]. Lemma B.2. The singular points of A(z, ) in the domain Re z ≥ − g A lie within a distance of o( ) of the spectrum of iB + N (provided that there are any singular points at all). Proof. Standard perturbation theory implies that the spectrum of the operator iB + A1 (z, ),
for Re z ≥ − g A
(B.16)
lies at a distance O( ) from the spectrum of iB. Here and in what follows, the estimates in powers of are uniform for Re z ≥ − g A . Let 10b ≡ 1b (B) be the spectral projections of B on the eigenvalue b. As long as is small enough, there is an invertible operator U ≡ U ( , z) satisfying U − 1 = O( ) and such that the projections 1b := U 10b U −1 ,
b ∈ spB
(B.17)
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are spectral projections of the operator (B.16) associated to the spectral patch originating from the eigenvalue b at = 0 (see Chap. 2.4 of [28] for an explicit construction of U ). It follows that the spectral problem for (B.16) is equivalent to the spectral problem for ib10b + Nb + Aex,b (z, ) , U −1 1b (iB + A1 (z, )) 1b U = (B.18) b
b
where Aex,b (z, ) := 10b U −1 (iB)U 10b − ib10b , + 10b U −1 Nb U 10b − Nb , + 10b U −1 (A1 (ib, ) − Nb ) U 10b , + 10b U −1 (A1 (z, ) − A1 (ib, )) U 10b ,
(O( 2 )), (O( 2 )), (B.19) (o( )), (|z − ib|o(| |0 )).
The estimates in powers of are obtained by using U − 1 = O( ), the property 1b U = U 10b and the bounds (B.6)–(B.7)–(B.8). When z is chosen at a distance O( ) from ib, then all terms in (B.19) are o( ). The claim of Lemma B.2 now follows by simple perturbation theory applied to the RHS of (B.18). Lemma B.3. The function A(z) has exactly one singularity at a distance o( ) from f N . This singularity is called f ≡ f ( ). The corresponding residue P ≡ P( ) is a rank-one operator satisfying P − 1 N ( f N ) = o | |0 , 0. (B.20) Proof. By Lemma B.2, there can be at most one singularity. We prove below that there is at least one. By the reasoning in the proof of Lemma B.2 and the fact that the eigenvector corresponding to f N belongs to Ran10b=0 (see condition 3) of Lemma B.1), it suffices to study the singularities of the function −1 z → z − N0 + Aex,0 (z, ) . (B.21) Let the contour f ≡ f ( ) be a circle with center f N and radius r for some r > 0. Clearly, for r small enough, the entire spectrum of N0 lies outside the contour f , except for the eigenvalue f N . The contour integral of (z − N )−1 along f equals the spectral projection corresponding to f N . We estimate * −1 (B.22) dz z − N0 − Aex,0 (z, ) − (z − N0 )−1 f * −1 = dz (z − N0 )−1 Aex,0 (z, ) z − N0 − Aex,0 (z, ) (B.23) f * 2 = dz −2 c(r ) o( ), as 0 with f
c(r ) :=
sup
|z− f N |=r
(z − N0 )−1 .
(B.24)
The last estimate holds in norm and it follows from the bound Aex,0 (z, ) = o( ), see (B.19). The expression (B.24) is o(1), as 0, since the circumference of the contour f is 2πr . From the fact that the contour integral of (B.21) does not vanish, we conclude that A(z) has at least one singularity inside f .
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The claim about the residue is most easily seen in an abstract setting. Let F(z) be an operator-valued analytic function in some open domain containing 0, and such that 0 ∈ spF(0) is an isolated eigenvalue. We have the Taylor expansion F(z) =
zn Fn , n!
Fn := F (n) (0),
0 ∈ spF0 .
(B.25)
n≥0
If F1 − 1 is small enough, then also F1−1 F0 has 0 as an isolated eigenvalue. We denote the corresponding spectral projection by 10 (F1−1 F0 ) and we calculate the residue (res) at 0, res(F(z)−1 ) = res (F0 + z F1 )−1 −1 F1−1 = 10 F1−1 F0 F1−1 . (B.26) = res F1−1 F0 + z The last expression is clearly a rank-one operator. In the case at hand, F1−1 = 1 + o(| |0 ), as 0, which yields (B.20). We proceed to the proof of Lemma B.1. First, we choose the rate g such that f N < g < min{g A , g N } and we fix the contours f and → (see also Fig. 15); • The contour f is as described in Lemma B.3, with r < |g − f N |. In particular, for small , it encircles the point f but no other singular points of A(z). • The contour → is given by → := − g + iR. By Lemma B.2, we know that for small , there are no singularities of A(z) in the region Re z > − g except for the point z ≡ f . Hence, we can deform contours as follows: 1 V (t, ) = dz ezt A(z, ) (B.27) 2π i → * 1 1 = dz ezt A(z, ) + dz ezt A(z, ). (B.28) 2π i f 2π i → The first term in (B.28) yields et f P. The second term of (B.28) is split as follows: dz ezt A(z, ) (B.29) → dz ezt (z − iB − N )−1 (B.30) = → dz ezt (z − iB − N )−1 (A1 (z, ) − N ) A(z, ). (B.31) + →
The term (B.30) equals et
iB+ 1 N (N )N
= O e− g N t ,
t ∞,
(B.32)
since the contour → can be closed in the lower half-plane to enclose the spectrum of iB + N minus the eigenvalue f N , i.e., the set N .
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Fig. 15. The (rotated) complex plane. The black dots indicate the spectrum of iB + N (which need not be discrete). The upper dot is the eigenvalue f N . In the picture, we have assumed that the spectrum of B consists of 3 semisimple eigenvalues: 0, b1 , b−1 . The gray patches contain the possible singularities of the function A(z) above the irregular black line. These singularities lie at o( ) from the spectrum of iB + N . Below the irregular black line, i.e., in the region Re z < − g A , we have no control since A1 (z, ) ceases to be analytic in that region (hence we have also not drawn a patch around b−1 ). The integration contours → , → and f are drawn as dashed lines
The integrand of (B.31) decays as |z|−2 for z ∞ , since for a bounded operator M, (z − M)−1 = O
1 , |z|
|z| ∞.
(B.33)
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Using that A1 (z, ) = O( ), it is now easy to establish that the integral in (B.31) is O(1), as 0. One extracts etRe z from the integration (B.31) to get the bound O(e− gt ). Together with (B.32), this proves Lemma B.1. C. Appendix: Construction and Analysis of the Lindblad Generator M The operator M was introduced at the beginning of Sect. 4. We provide a more explicit construction and we prove Propositions 4.1 and 4.2. C.1. Construction of M. First, we note that by using the notions introduced in Sect. 5.2, the operator L(z), defined in Sect. 4.1, can be expressed as L(z) = dt e−t z ψ # (x2 − x1 , t) Ix2 ,l2 e−iad(Y )t Ix1 ,l1 , (C.1) R+
x1 ,x2 ,l1 ,l2
where ψ # equals ψ or ψ, depending on l1 , l2 , according to the rules in (5.10). In words, λ2 L(z) contains the terms of order λ2 in the Lie-Schwinger series of Lemma 2.5. Next, we define some auxiliary objects: ∗ ϒ := Im Wa Wa dt ψ(0, t)eiat , (C.2) a∈sp(ad(Y ))
$(ρ) :=
x L ,x R ∈Zd a∈sp(ad(Y ))
R+
R
dt eiat ψ(x L − x R , t)
∗ × 1x L ⊗ Wa ρ 1x R ⊗ Wa .
(C.3)
The operator ϒ = ϒ ∗ ∈ B(S ) was already referred to in Sect. 4. From the above expression and the definition of Wa in (2.30), we check immediately that [Y, ϒ] = 0. Further, we can rewrite (C.3) as ˆ $(ρ) = 2π ds ψ(a)V (s, a)ρV ∗ (s, a) (C.4) a∈sp(ad(Y ))
ˆ as in (2.27) and (5.3), and with ψ(·) V (s, a) :=
Sd−1
eias·x 1x ⊗ Wa .
(C.5)
x∈Zd
The expression (C.4) is essentially the Kraus decomposition of $, see [1], and hence it shows that $ is a completely positive map. This means in particular that for ρ ≥ 0, $(ρ) 1 = Tr $(ρ), and hence, by using (C.3), 2 Tr S [($ρ)(x, x)] ≤ dim S dt|ψ(0, t)| W ρ 1 , (C.6) $ρ 1 = x
where the finiteness of the factor between brackets on the RHS is implied by Lemma 5.2. Since any trace class operator can be written as a linear combination of four positive trace class operators, it follows that $ is bounded on B1 (HS ) and by a similar calculation, one can check that $ is also bounded on B2 (HS ).
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We are now ready to verify that 1 ∗ $ (1)ρ + ρ$ ∗ (1) . (C.7) 2 Indeed, this is checked most conveniently starting from (4.4) and employing (C.1). The terms with l1 = l2 give rise to $(ρ), while the terms with l1 = l2 give rise to −i[ϒ, ρ] and − 21 ($ ∗ (1)ρ +ρ$ ∗ (1)). Moreover, by the boundedness and complete positivity of $ and the representation (C.7), it follows that M is of Lindblad type, see e.g. [1]. Starting from (C.4), one can derive the momentum space representation of M given in Sect. 4.2. For example, by expressing V (s, a) in momentum representation, one obtains ∗ ˆ $(ρ)(k L , k R ) = 2π ds ψ(a)W a ρ(k L + sa, k R + sa)Wa , (C.8) M(ρ) = −i[ε(P) + ϒ, ρ] + $(ρ) −
a∈sp(ad(Y ))
Sd−1
which gives rise to the first term of (4.16). C.1.1. Proof of Proposition 4.1. By the integrability in time of the correlation function ψ(x, t), as stated in Lemma 5.2, the expression (C.1) implies immediately that L(z) can be continued continuously to z ∈ R. This proves (4.5). The boundedness of M on B2 (HS ) and B1 (HS ) follows from the boundedness of $, which was explained above. The complete positivity of the map $ and the canonical form (C.7) imply that M is a Lindblad generator, see e.g. [1]. Consequently, −iad(Y ) + λ2 M is also a Lindblad generator and the semigroup t is positivity-preserving and trace-preserving. To check (4.6), we note that
Jκ MJ−κ − M = −i Jκ ad(ε(P))J−κ − ad(ε(P)) , (C.9) and hence (4.6) follows immediately from Assumption 2.1. C.2. Spectral analysis and proof of Proposition 4.2. The claims of Proposition 4.2 require a spectral analysis which we present now. We recall the decomposition ⊕ ⊕ M= d p Mp = dp ⊕ M p,a , (C.10) Td
Td
a∈sp(ad(Y ))
and we keep in mind that Proposition 4.2 treats M p as an operator on the Hilbert space G ∼ L 2 (Td , B2 (S )). C.2.1. Explicit representation of M p,0 . By exploiting the nondegeneracy condition in Assumption 2.4, we can identify M p,a=0 , for each p with an operator on L 2 (Td × spY ). This was explained in Sect. 4.2. We introduce explicitly gain, loss and kinetic operators; G, L and K p , acting on L 2 (Td × spY ), by dk r k , e ; k, e ϕ(k , e ), (C.11) Gϕ(k, e) := e ∈spY
Lϕ(k, e) := −
Td
e ∈spY
Td
dk r k, e; k , e ϕ(k, e),
(C.12)
p p −ε k− ϕ(k, e), ϕ ∈ L 2 Td × spY . (C.13) K p ϕ(k, e) := i ε k + 2 2
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The kinetic operator K p models the free flight of the particle between collisions. The operators L and K p act by multiplication in the variables k, e. The expression for M p,0 , given in Sect. 4.2 (in particular in (4.15)), can be rewritten as M p,0 = G + L + K p .
(C.14)
We define the similarity transformation A → Aˆ := e 2 βY Ae− 2 βY , 1
1
for A ∈ B L 2 Td × spY ,
(C.15)
where we have slightly abused the notation by writing Y to denote a multiplication operator on spY , i.e., Y ϕ(k, e) = eϕ(k, e). Since L and K p act by multiplication, we have Lˆ = L and Kˆ p = K p . The usefulness of this similarity transformation resides in the ˆ 0,0 , are self-adjoint on L 2 (Td × spY ). ˆ and hence also M fact that G, C.2.2. Explicit representation of M p,a = 0 . To write an explicit expression for M p,a=0 , we first define the operators (acting on B(S )) ada (ϒ) = 1a (ad(Y ))ad(ϒ)1a (ad(Y )), a ∈ sp(ad(Y )),
(C.16)
which satisfy ad0 (ϒ) = 0 and ad(ϒ) = ⊕a ada (ϒ) since [ϒ, Y ] = 0. Due to the nondegeneracy condition in Assumption 2.4, both 1a (ad(Y )) and ada (ϒ) are rank-one operators and we identify ada (ϒ) with a number ϒa , such that ada (ϒ) = ϒa 1a (ad(Y )). In fact, as already remarked in Sect. 4.2, the operator M p,a=0 itself acts as the rankone operator 1a (ad(Y )) on B(S ) and hence we identify it with an operator on L 2 (Td ) (which is also called M p,a=0 here): p p −ε k− ϕ(k) M p,a=0 ϕ(k) = −i ϒa + ε k + 2 2 1 − j (k, e) + j (k, e ) ϕ(k), ϕ ∈ L 2 (Td ), 2
(C.17)
where j (k, e) are the escape rates introduced in (4.27) and e, e are determined by a = e − e . To check (C.17), one starts from (4.15) and one uses • The fact that r (k , e ; k, e) vanishes for e = e, as remarked following (4.17). • The definition of the matrices Wa in (2.30) and the escape rates j (·, ·) in (4.27). • The definition ϕ(k) ≡ e, ξ(k)e S in (4.23). In particular, the last term on the RHS of (C.17) appears because Td
6 5 dk re −e (k, k ) e , Wa Wa∗ ξ(k) + ξ(k)Wa Wa∗ e S = j (k, e) + j (k, e ) ϕ(k).
Hence, M p,a=0 acts by multiplication in the variable k.
(C.18)
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C.2.3. Analysis of M0,0 . We already established that M0,0 is a bounded Markov generator on L 1 (Td × spY ). This implies that Re sp L 1 M0,0 ≤ 0.
(C.19)
ˆ 0,0 is no longer a Markov generator, but its spectrum is identical to The operator M ˆ 0,0 , since e± 12 βY is bounded and invertible. The loss operator L is a multiplication M operator and its spectrum is found to be (see (4.27)) (C.20) sp(L) = − j (e, k) | e ∈ spY, k ∈ Td . It is important to note that the escape rates j (e, k) are bounded away from 0; this is a consequence of Assumption 2.4 and more concretely, of the fact that for each e, there is a e such that (2.34) holds. Hence, we have sp(L) < 0.
(C.21) L 2.
e, e ,
Next, we argue that G is a compact operator on Indeed, for fixed the kernel r (k , e ; k, e) depends only on k ≡ k − k and, hence, its Fourier transform acts on l 2 (Zd ) by multiplication with the function d(k)eik·x r (0, e ; k, e). (C.22) Zd # x → Td From the explicit expression for r (k , e ; k, e), one checks that the function (C.22) decays
at infinity if the dimension d > 1 (recall that d ≥ 4 by Assumption 2.2). Hence G is compact. Given the compactness of G, Weyl’s theorem ensures that the self-adjoint operators ˆ M0,0 and Lˆ have the same essential spectrum. ˆ 0,0 has an eigenvalue 0, corresponding to the eigenBy inspection, we check that M β
vector ϕˆ eq (k, e) ≡ e− 2 e . Note that the corresponding right eigenvector of M0,0 is the (unnormalized) Gibbs state ϕ eq (k, e) ≡ e−βe and the corresponding left eigenvector is the constant function, since indeed β
β
ϕˆ eq = e 2 Y ϕ eq , ϕˆ eq = e− 2 Y 1Td ×spY .
(C.23)
ˆ 0,0 on L 1 (note that ˆ 0,0 on L 2 has to be an eigenvalue of M Since any eigenvalue of M 2 d 1 d L (T ×spY ) ⊂ L (T ×spY )), the relation (C.19) implies that there are no eigenvalues with strictly positive real part. We now exploit a Perron-Frobenius argument to argue that the eigenvalue 0 is simple and that it is the only eigenvalue on the real axis. See e.g. Theorem 13.3.6 in [9] for a version of the Perron-Frobenius theorem that establishes this in our case, proˆ vided that the semigroup et M0,0 is irreducible, i.e., that for any nonnegative functions 1 d ϕ ∈ L (T × spY ), the inclusion ˆ (C.24) Supp et M0,0 ϕ ⊂ Supp(ϕ), (Supp stands for ‘support’) implies that either Supp(ϕ) = Td × spY or ϕ = 0. This irreducibility criterion is easily checked starting from Assumption 2.4, in particular its rephrasing in terms of a connected ˆ graph. Theorem 13.3.6 yields that the eigenvalue 1 of et M0,0 is simple, which implies ˆ 0,0 is simple. To exclude purely imaginary eigenvalues ib of that the eigenvalue 0 of M ˆ 0,0 , we apply this theorem for t such that eibt = 1. M
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ˆ p,0 as follows. C.2.4. Analysis of M p,0 and M p,a . We investigate the spectrum of M By the same reasoning as in Sect. C.2.3, any spectrum with real part greater than (the ˆp negative number) sup spL consists of eigenvalues of finite multiplicity. Assume that M has an eigenvalue m p with (right) eigenvector) ϕˆ p . Then ˆ p,0 ϕˆ p Re m p ϕˆ p , ϕˆ p = Re ϕˆ p , M ˆ 0,0 ϕˆ p . = Re ϕˆ p , K p ϕˆ p + Re ϕˆ p , M
(C.25) (C.26)
The first term in (C.26) vanishes because the multiplication operator K p is purely imaginary. The second term can only become positive if ϕˆ = ϕˆ eq , with ϕˆ eq the eigenvector of ˆ 0,0 corresponding to the eigenvalue 0. This means that either the eigenvalue m p has M ˆ 0,0 with eigenvalue 0. strictly negative real part, or the vector ϕˆ eq is an eigenvector of M eq In the latter case, ϕˆ must also be an eigenvector of K p with eigenvalue 0, which can only hold if ε k + 2p − ε k − 2p = 0 for all k. This is however excluded by the condition (2.11) in Assumption 2.1. We conclude that for all p ∈ Td \{0}, we have Re spM p < 0. By compactness of Td and the lower semicontinuity of the spectrum, we deduce hence that sup Re spM p,0 = c(I0 ) < 0, for any neighborhood I0 of 0.
(C.27)
Td \I0
For a = 0, the operator M p,a=0 is a multiplication operator in k and 1 Re spM p,a ≤ − inf j (k, e) < 0, independently of p, 2 k,e
(C.28)
as follows by (C.17) and the fact that j (k, e) is bounded away from 0. C.2.5. Proof of Proposition 4.2 We summarize the results of Sects. C.2.3 and C.2.4. For a = 0, the real part of the spectrum of the operators M p,a is strictly negative, uniformly in p, see (C.28). The real part of the spectrum of M p,0 is strictly negative, uniformly in p except for a neighborhood of 0 ∈ Td . The operator M0,0 has a simple eigenvalue at 0 with corresponding eigenvector ϕ eq , as defined in Sect. C.2.3. The rest of the spectrum of M0,0 is separated from the eigenvalue by a gap. Since M p = ⊕a M p,a , and using the uniform bound (C.28), we obtain immediately that the operator M0 has a simple eigenvalue at 0 with corresponding eigenvector ξ eq , ξ eq := ϕ eq ⊕ 0 ⊕ · ·!· ⊕ 0",
(C.29)
a=0
separated from the rest of the spectrum of M0 by a gap. By the analyticity in κ, see (4.6), and the correspondance between κ and ( p, ν), as stated in (2.60), we can apply analytic perturbation theory in p to the family of operators M p . We conclude that for p in a neighborhood of 0, the operator M p has a simple eigenvalue, which we call fr w ( p), that is separated by a gap from the rest of the spectrum. We also obtain that the corresponding eigenvector is analytic in p and ν. In this way we have derived all claims of Proposition 4.2, except for the symmetry ∇ p fr w ( p) = 0 and the strict postive-definiteness of the matrix (∇ p )2 fr w ( p). These two claims will be proven in Sect. C.2.6. We note that the function fr w ( p), which we defined above as the simple and isolated eigenvalue of M p with maximal real part, is also a simple and isolated eigenvalue of M p,0 with maximal real part.
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C.2.6. Strict positivity of the diffusion constant. By the remark at the end of Sect. C.2.5 ˆ p,0 = spM p,0 , we view fr w ( p) as the eigenvalue of M ˆ p,0 that and the fact that spM reduces to 0 for p = 0. ˆ 0,0 + K p and we define the operator-valued vector V := ˆ p,0 = M We recall that M ∇ p K p p=0 (note that V is in fact a vector of operators). The first order shift of the eigenvalue is given by ∇ p fr w ( p) :=
1 ϕˆ eq , V ϕˆ eq = 0. ϕˆ eq , ϕˆ eq
(C.30)
To check that (C.30) indeed vanishes, we use that ϕˆ eq is symmetric under the transformation k → −k (in fact, it is independent of k) while V is anti-symmetric under k → −k (this follows from the symmetry ε(k) = ε(−k) in Assumption 2.1). The second order shift is then given by 2 1 ˆ −1 V ϕˆ eq Dr w := ∇ p fr w ( p) = − 5 eq eq 6 ϕˆ eq , V M 0,0 ϕˆ , ϕˆ 1 (C.31) + 5 eq eq 6 ϕˆ eq , (∇ p )2 K p ϕˆ eq , ϕˆ , ϕˆ where the first term on the RHS of (C.31) is well-defined since V ϕˆ eq is orthogonal to the 0-spectral subspace of M0,0 , by (C.30). The second term vanishes because (∇ p )2 K p = 0, as can again be checked explicitly. Let υ ∈ Rd and Vυ := υ · V (recall that V is a vector). Then, by (C.31), υ · Dr w υ = −
1 ˆ −1 Vυ ϕˆ eq . ϕˆ eq , Vυ M 0,0 ϕˆ eq , ϕˆ eq
(C.32)
ˆ 0,0 , we see Upon using the spectral theorem and the gap for the self-adjoint operator M that the RHS of the last expression is positive and it can only vanish if ⎡ ⎤ 0 = Vυ ϕˆ eq 2 = ⎣ e−βe ⎦ dk|υ · ∇ε(k)|2 , (C.33) e∈spY
which is however excluded by Assumption 2.1. The strict positive-definiteness of the diffusion constant Dr w is hence proven. D. Appendix: Combinatorics In this appendix, we show how to integrate over irreducible equivalence classes of diagrams. In other words, we assume that the x, l-coordinates have already been summed over (or a supremum over them has been taken) and we carry out the remaining integration over the time-coordinates t and the diagram size |σ |. We first define a function of diagrams, K (σ ), that depends only on the equivalence class [σ ]. Let k be a positive function on R+ and put K (σ ) :=
|σ |
k(vi − u i ),
(D.1)
i=1
where (u i , vi ) are the pairs of times in the diagram σ . In the applications, the function k will be (a multiple of) supx |ψ(x, t)|, sometimes restricted to t < τ or t > τ .
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# Lemma D.1. Let a ≥ 0 and assume that teat k 1 = R+ dt teat k(t) < 1, then 1 dt eat d[σ ]K (σ ) ≤ eat k 1 . at k + 1 − te 1 R !T [0,t] (mir) ˜ k < 1 with a ˜ := a + k 1 , then If in addition, teat 1 ˜ dt eat d[σ ] K (σ ) ≤ 2 eat k 1
R+
dt eat
R+
!T
!T
[0,t] (ir)
[0,t] (ir)
˜ d[σ ]1|σ |≥2 K (σ ) ≤ 2 eat k 1
(D.2)
1 , ˜ k 1 − teat 1
(D.3)
˜ k teat 1 . ˜ k 1 − teat 1
(D.4)
Proof. First, we note that for each irreducible diagram σ ∈ [0,t] (ir), we can find a subdiagram σ ⊂ σ such that σ is minimally irreducible in [0, t], i.e., σ ∈ [0,t] (mir). Note that the choice of subdiagram σ is not necessarily unique. Conversely, given a minimally irreducible diagram σ ∈ [0,t] (mir), we can add any diagram σ ∈ [0,t] to σ , thereby creating a new irreducible diagram σ := σ ∪ σ ∈ [0,t] (ir). By these considerations, we easily deduce d[σ ] 1|σ |≥2 K (σ ) !T
[0,t] (ir)
≤
!T
+
[0,t] (mir)
d[σ ]1|σ |≥2 K (σ )
!T
[0,t] (mir)
1+
d[σ ]1|σ |=1 K (σ )
!T
d[σ ]K (σ ) [0,t]
!T
(D.5)
d[σ ]K (σ ) .
(D.6)
[0,t]
The 1+· in (D.5) covers the case in which the diagram σ was itself minimally irreducible, and hence no diagrams σ are added to σ . In (D.6), one always has to add at least one pair to σ , since |σ | ≥ 2 but |σ | = 1. In fact, the equivalence classes in the inequality could be dropped, i.e., one can omit the projections !T and replace d[σ ], d[σ ], d[σ ] by dσ, dσ , dσ , respectively. We recall that if a diagram σ with |σ | = 1 is irreducible (or minimally irreducible) in the interval I , then its time-coordinates are fixed to be the boundaries of I ; i.e., there is only one equivalence class of such diagrams: d[σ ]1|σ |=1 K (σ ) = d[σ ]1|σ |=1 K (σ ) = k(t). (D.7) !T
[0,t] (ir)
!T
[0,t] (mir)
The unconstrained integral over all (equivalence classes of) diagrams, that appears in (D.5) and (D.6), can be performed as follows: !T
≤
d[σ ]K (σ ) = [0,t]
n≥1 0
n≥1 0
du 1 . . . du n
du 1 . . . du n ( k 1 )n =
vi >u i
⎛ dv1 . . . dvn ⎝
n
k(vi − u i )⎠
i=1
tn ( k 1 )n = et k 1 − 1. n!
n≥1
⎞
(D.8)
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Next, we perform the integral over (equivalence classes of) minimally irreducible diagrams. For σ ∈ [0,t] (mir) with |σ | = n > 1, the relative order of the times u i , vi is fixed as follows: 0 = u 1 ≤ u 2 ≤ v1 ≤ u 3 ≤ v2 ≤ u 4 ≤ · · · ≤ vn−2 ≤ u n ≤ vn−1 ≤ vn = t. (D.9) We have hence R+
=
dt eat
!T
[0,t] (mir)
dσ K (σ )1|σ |=n
v1 ∞ dv1 k(v1 − u 1 )ea(v1 −u 1 ) du 2 dv2 . . . 0 0 v1 vn−4 ∞ ... du n−2 dvn−2 ∞
... ×
vn−5 vn−2
vn−3 vn−1 vn−2
du n−1 du n
∞
vn−3 ∞
vn−2
vn−1
dvn−1 ea(vn−1 −vn−2 ) k(vn−1 − u n−1 )
dvn ea(vn −vn−1 ) k(vn − u n ).
(D.10)
First we extend the domain of integration of u n from [vn−2 , vn−1 ] to (−∞, vn−1 ] and we estimate the integrals over the variables u n and vn by ∞ vn−1 du n dvn ea(vn −vn−1 ) k(vn − u n ) ≤ teat k 1 . (D.11) −∞
vn−1
Next, we perform the integration over u n−1 , vn−1 in the same way, we continue the procedure until only the variable v1 is left (note that u 1 = 0 is fixed). The integral over v1 gives eat k 1 . This yields the bound LHS of D.10 ≤ eat k 1 × teat k n−1 1 .
(D.12)
We are ready to evaluate the Laplace transform of (D.5)–(D.6). Using (D.8), we bound 1+ d[σ ]K (σ ) ≤ et k 1 , ! T I d[σ ]K (σ ) ≤ et k 1 − 1 ≤ t k 1 et k 1 . (D.13) !T
I
Combining this with (D.7) and (D.12), and summing over n ≥ 2, we obtain dt eat d[σ ]1|σ |≥2 K (σ ) !T
˜ k 1 ≤ eat
[0,t] (ir)
˜ k teat 1 ˜ + k 1 teat k 1 , ˜ k 1 − teat 1
(D.14)
where the two terms on the RHS correspond to (D.5) and (D.6), respectively. This ends the proof of (D.4). The bound in (D.3) follows by adding eat k 1 , which is the contribution of |σ | = 1 (see (D.7), to (D.4). The bound (D.2) is proven by summing (D.12) over n ≥ 1.
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References 1. Alicki, R., Fannes, M.: Quantum Dynamical Systems. Oxford: Oxford University Press, 2001 2. Araki, H., Woods, E.J.: Representations of the canonical commutation relations describing a nonrelativistic infinite free Bose gas. J. Math. Phys. 4, 637 (1963) 3. Bach, V., Fröhlich, J., Sigal, I.: Return to equilibrium. J. Math. Phys. 41, 3985 (2000) 4. Brattelli, O., Robinson, D.W.: Operator Algebras and Quantum Statistical Mechanics: 2. Berlin: SpringerVerlag, 2nd edition, 1996 5. Bricmont, J., Kupiainen, A.: Diffusion in Energy Conserving Coupled Maps, http://arxiv.org/abs/1102. 3831 6. Bryc, W.: A remark on the connection between the large deviation principle and the central limit theorem. Stat. and Prob. Lett. 18, 253–256 (1993) 7. Bunimovich, L.A., Sinai, Ya.G.: Statistical properties of Lorentz gas with periodic configuration of scatterers. Commun Math. Phys. 78, 479–497 (1981) 8. Clark, J., De Roeck,W., Maes, C.: Diffusive behaviour from a quantum master equation. http://arXiv.org/ abs0812.2858v3 [math-ph], 2008 9. Davies, E.B.: Linear Operators and their spectra. Cambridge: Cambridge University Press, 2007 10. Davies, E.B.: Markovian master equations. Commun. Math. Phys. 39, 91–110 (1974) 11. Derezi´nski, J.: Introduction to Representations of Canonical Commutation and Anticommutation Relations. Volume 695 of Lecture Notes in Physics. Berlin: Springer-Verlag 2006 12. Derezi´nski, J., Früboes, R.: Fermi golden rule and open quantum systems. In: Attal, S., Joye, A., Pillet, C.-A. (eds.) Lecture notes Grenoble Summer School on Open Quantum Systems. Lecture Notes in Mathematics. Vol. 118. Berlin: Springer, 2003, pp. 67–116 13. Derezi´nski, J., Jakši´c, V.: Return to equilibrium for Pauli-Fierz systems. Ann. H. Poincaré 4, 739–793 (2003) 14. De Roeck, W.: Large deviation generating function for currents in the Pauli-Fierz model. Rev. Math. Phys. 21(4), 549–585 (2009) 15. De Roeck, W., Fröhlich, J., Pizzo, A.: Quantum Brownian motion in a simple model system. Commun. Math. Phys. 293(2), 361–398 (2010) 16. De Roeck, W., Spehner, D.: Derivation of the quantum master equation for massive tracer particles. In preparation 17. Disertori, M., Spencer, T., Zirnbauer, M.: Quasi-diffusion in a 3d supersymmetric hyperbolic sigma model. Commun. Math. Phys. 300, 435–486 (2010) 18. Erdös, L.: Linear Boltzmann equation as the long time dynamics of an electron weakly coupled to a phonon field. J. Stat. Phys. 107(85), 1043–1127 (2002) 19. Erdös, L., Salmhofer, M., Yau, H.-T.: Quantum diffusion of the random Schrödinger evolution in the scaling limit ii. the recollision diagrams. Commun. Math. Phys 271, 1–53 (2007) 20. Erdös, L., Salmhofer, M., Yau, H.-T.: Quantum diffusion of the random Schrödinger evolution in the scaling limit i. the non-recollision diagrams. Acta Mathematica 200, 211–277 (2008) 21. Erdös, L., Yau, H.-T.: Linear Boltzmann equation as the weak coupling limit of a random Schrödinger equation. Commun. Pure Appl. Math. 53(6), 667–735 (2000) 22. Evans, L.C.: Partial Differential Equations (Graduate Studies in Mathematics, V. 19) GSM/19. Providence, RI: Amer. Math. Soc., 1998 23. Frigerio, A.: Stationary states of quantum dynamical semigroups. Commun. Math. Phys. 63(3), 269–276 (1978) 24. Fröhlich, J., Merkli, M.: Another return of ‘return to equilibrium’. Commun. Math. Phys. 251, 235–262 (2004) 25. Holevo, A.S.: A note on covariant dynamical semigroups. Rep. Math. Phys. 32, 211–216 (1992) 26. Jakši´c, V., Pillet, C.-A.: On a model for quantum friction. iii: Ergodic properties of the spin-boson system. Commun. Math. Phys. 178, 627–651 (1996) 27. Kang, Y., Schenker, J.: Diffusion of wave packets in a Markov random potential. J. Stat. Phys. 134, 1005– 1022 (2008) 28. Kato, T.: Perturbation Theory for Linear Operators. Berlin: Springer, second edition, 1976 29. Knauf, A.: Ergodic and topological properties of coulombic periodic potentials. Commun. Math. Phys. 110(1), 89–112 (1987) 30. Komorowski, T., Ryzhik, L.: Diffusion in a weakly random hamiltonian flow. Commun. Math. Phys 263, 277–323 (2006) 31. Lebowitz, J., Spohn, H.: Irreversible thermodynamics for quantum systems weakly coupled to thermal reservoirs. Adv. Chem. Phys. 39, 109–142 (1978) 32. Ovchinnikov, A.A., Erikhman, N.S.: Motion of a quantum particle in a stochastic medium. Sov. Phys.JETP 40, 733–737 (1975)
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33. Spohn, H.: An algebraic condition for the approach to equilibrium of an open N-level system. Lett. Math. Phys. 2, 33–38 (1977) 34. Vacchini, B., Hornberger, K.: Quantum linear boltzmann equation. Phys. Rept. 478, 71–120 (2009) 35. Van Hove, L.: Quantum-mechanical perturbations giving rise to a statistical transport equation. Physica 21, 517–540 (1955) Communicated by H.-T. Yau
Commun. Math. Phys. 303, 709–720 (2011) Digital Object Identifier (DOI) 10.1007/s00220-011-1234-9
Communications in
Mathematical Physics
Loop-Erasure of Planar Brownian Motion Dapeng Zhan Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA. E-mail:
[email protected] Received: 18 January 2010 / Accepted: 26 November 2010 Published online: 27 March 2011 – © Springer-Verlag 2011
Abstract: We use a coupling technique to prove that there exists a loop-erasure of the time-reversal of a planar Brownian motion stopped on exiting a simply connected domain, and that the loop-erased curve is a radial SLE2 curve. This result extends to Brownian motions and Brownian excursions under certain conditioning in a finitely connected plane domain, and the loop-erased curve is a continuous LERW curve. 1. Introduction In this paper we will derive the existence of a loop-erasure of the time-reversal of a planar Brownian motion up to some finite stopping time. It is well-known that simple random walks on a regular lattice such as δZ2 converge to planar Brownian motions as the mesh δ → 0. The loop-erasure of a simple random walk is called a loop-erased random walk (LERW). Lawler, Schramm, and Werner proved [8] that the LERW on the discrete approximation of a simply connected domain converges to the SchrammLoewner evolution (SLE) [4] with parameter κ = 2, i.e., SLE2 , when the mesh tends to 0. So it is reasonable to conjecture that a planar Brownian motion in a simply connected domain a.s. has a unique (up to equivalence) loop-erasure, which is an SLE2 curve. In this paper we will prove the existence. The uniqueness is still open to the author. In addition, we expect that there exists a deterministic algorithm to erase the loops on the Brownian motion. This is also not solved in this paper. The result in this paper extends naturally to finitely connected domains. For simplicity, we will only deal with simply connected domains, and work on the time-reversal of the Brownian motion. From [1], the Hausdorff dimension of SLE2 curve is 5/4. The result of this paper implies that the percolation dimension ([2]) of a planar Brownian motion (the minimal Hausdorff dimension of a subpath of a Brownian path) is no more than 5/4. This value is strictly less than the boundary dimension of planar Brownian motion, which is equal to 4/3 ([5,6]). Supported by NSF grant 0963733.
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In [7], Lawler, Schramm, and Werner proved that, by adding Brownian bubbles to a chordal SLE2 curve and filling the holes, one obtains a Brownian excursion in a simply connected domain from one boundary point to another boundary point with holes filled in. Their result gives an evidence that a loop-erasure of a planar Brownian motion should exist. We will use the coupling technique introduced in [14] to prove the existence of the loop-erasure. The coupling technique is used to create a coupling of a conditional planar Brownian motion with a radial SLE2 curve in a simply connected domain such that, for every t in the definition domain of the radial SLE2 curve, say β, the first hitting point of the planar Brownian motion at the set β[0, t] is the tip point: β(t). Corollary 2.1 will then be applied. 2. Preliminary In [3], the loop-erasure of a finite path on a graph is defined as follows. Let X = (X 0 , X 1 , . . . , X n ) be a finite path. Let w(0) = 0 and τ = 0. If X w(τ ) = X n , and let w(τ + 1) = sup{k : X k = X w(τ ) } + 1, let the value of τ be incremented by 1, and repeat this process; if X w(τ ) = X n , stop. In the end, we get integer numbers τ ≥ 0 and 0 = w(0) < w(1) < · · · < w(τ ). Then the lattice path Yk = X w(k) , 0 ≤ k ≤ τ , is called the loop-erasure of X . It is easy to see that every vertex of Y that lies on X, Y is a simple lattice path, and has the same initial and final vertices as X . From the definition, it is clear that a path Y = (Y0 , Y1 , . . . , Yτ ) is the loop-erasure of another path X = (X 0 , X 1 , . . . , X n ) if and only if there is an increasing function w : {0, 1, . . . , τ } → {0, 1, . . . , n} such that w(0) = 0, Yk = X w(k) for 0 ≤ k ≤ τ, Yτ = X n , and for 0 ≤ k ≤ τ − 1, the path (Y0 , Y1 , . . . , Yk ) is disjoint from the path (X w(k+1) , X w(k+1)+1 , . . . , X n ). From this observation, we may extend the definition of loop-erasure to (continuous) curves. Definition 2.1. We say a continuous curve Y (t), c ≤ t ≤ d, is a loop-erasure of another continuous curve X (t), a ≤ t ≤ b, if Y (c) = X (a), Y (d) = X (b), and there is an increasing function w from [c, d] into [a, b] such that Y (t) = X (w(t)) for c ≤ t ≤ d, and for any t1 < t2 ∈ [c, d], Y [c, t1 ] ∩ X [w(t2 ), b] = ∅. It is easy to see that Y must be a simple curve. In fact, we have an equivalent definition. Lemma 2.1. A simple curve Y (t), c ≤ t ≤ d, is a loop-erasure of a curve X (t), a ≤ t ≤ b, if and only if Y (c) = X (a), Y (d) = X (b), and for any T ∈ (c, d), the biggest s ∈ [a, b] such that X (s) ∈ Y [c, T ] satisfies that X (s) = Y (T ). Proof. First, suppose Y is a loop-erasure of X , and let w be as in the definition. Fix T ∈ [c, d). For any t ∈ (T, d], we have Y [c, T ] ∩ X [w(t), d] = ∅. Thus, X (s) ∈ Y [c, T ] if s > w+ (T ), where w+ (T ) is the right-hand limit of w at T . On the other hand, for any t ∈ (T, d], we have Y (t) = X (w(t)). By letting t → T + , we conclude that Y (T ) = X (w+ (T )). So the biggest s such that X (s) ∈ Y [c, T ] is w+ (T ), and X (w+ (T )) = Y (T ). Now we prove the other direction. For c ≤ t ≤ d, let w(t) be the biggest s ∈ [a, b] such that X (s) ∈ Y [c, t]. Then w is an increasing function, and from the assumption, Y (t) = X (w(t)) for c < t < d. The equality holds for t = c since in that case Y [c, t] is a single point Y (c). It also holds for t = d because X (b) = Y (d) implies that w(d) = b, and so we have Y (d) = X (b) = X (w(d)). Since Y is simple, so w is strictly increasing. For t1 < t2 ∈ [c, d], we have w(t2 ) > w(t1 ), so from the definition of w(t1 ) we have X [w(t2 ), b] ∩ Y [c, t1 ] = ∅.
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Corollary 2.1. A simple curve Y (t), c ≤ t ≤ d, is a loop-erasure of the time-reversal of a curve X (t), a ≤ t ≤ b, if and only if Y (c) = X (b), Y (d) = X (a), and for any T ∈ (c, d), the first s such that X (s) ∈ Y [c, T ] satisfies that X (s) = Y (T ). Two loop-erasures of a curve X are called equivalent if they have the same image. Given a curve X (t), a ≤ t ≤ b, there may exist more than one loop-erasure, which are not equivalent. For example, in the compact space S obtained by adding +∞ and −∞ to the strip {z ∈ C : 0 ≤ Im z ≤ 1}, there is a curve, which starts from −∞, ends at +∞, and travels through the line segments in the following order: . . . , [n, n + 1], [n + 1, n + i], [n + i, n + 1 + i], [n + 1 + i, n + 1], [n + 1, n + 2], . . ., where n ∈ Z. Such curve has at least two loop-erasures: one has image R ∪ {+∞, −∞}, the other has image (R + i) ∪ {+∞, −∞}, which can not be equivalent. 3. Planar Brownian Motion in Simply Connected Domains We identify R2 with the complex plane C, and use the convention that a standard real Brownian motion starts from 0, and has variance t at time t for t ≥ 0, and that a standard complex Brownian motion is a complex valued random process whose real part and imaginary part are two independent standard real Brownian motions. Suppose BC (t) is a standard complex Brownian motion, and D C is a simply connected domain containing 0. Let τ = τ D be the first time that BC (t) ∈ D. Then τ is an a.s. finite stopping time. We will focus on the loop-erasures of the time-reversal of BC (t), 0 ≤ t ≤ τ . From the remarks in Sect. 7, we will see that BC (t), 0 ≤ t ≤ τ , itself has a loop-erasure, which is a disc SLE2 curve. The following is the main theorem in this paper. Theorem 3.1. Almost surely there is a loop-erasure of the time-reversal of BC (t), 0 ≤ t ≤ τ , which is a radial SLE2 curve that grows in D towards 0 from a random boundary point of D, whose distribution is the harmonic measure in D seen from 0. From the Riemann Mapping Theorem and conformal invariance (up to time-change) of complex Brownian motion [9], SLE, and harmonic measure, we suffice to consider ρ the special case that D = D := {z ∈ C : |z| < 1}. For ρ ∈ T, let BC (t) be BC (t) ρ conditioned to exit D at ρ. The explicit definition of BC (t) will be given below. From ρ the two lemmas below in this section and the rotation symmetry of both BC and radial SLE in D from ρ to 0, to prove Theorem 3.1, we suffice to show the following theorem. 1 (t), 0 ≤ Theorem 3.2. Almost surely there is a loop-erasure of the time-reversal of BC t ≤ τ1 , which is a radial SLE2 curve that grows in D from 1 towards 0. 1+z Let P(z) = Re 1−z and Pρ (z) = P(z/ρ) for ρ ∈ T := {z ∈ C : |z| = 1}. Then Pρ is harmonic and positive in D; vanishes on T except at ρ; and Pρ (0) = 1. We call Pρ the normalized (by its value at 0) Poisson kernel in D with the pole at ρ. Let δρ (t), 0 ≤ t < τρ , be a complex valued function that solves the ODE
δρ (t) =
2∂z Pρ (δρ (t) + BC (t)) , δρ (0) = 0; Pρ (δρ (t) + BC (t))
and suppose that the solution can not be extended beyond τρ . Here 2∂z = ∂x + i∂ y . Let ρ ρ BC (t) = BC (t) + δρ (t), 0 ≤ t < τρ . Then BC (t) starts from 0 and satisfies the SDE ρ
d BC (t) = d BC (t) +
ρ
2∂z Pρ (BC ) ρ
Pρ (BC )
dt, 0 ≤ t < τρ .
(3.1)
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This is a complex SDE, and its real stochastic part and imaginary stochastic part are two independent standard Brownian motions. If f is an analytic function, then from Itô’s ρ formula [9] for real valued functions, the process f (BC (t)) satisfies the complex SDE: ρ
ρ
ρ
d f (BC (t)) = | f (BC (t))|d BC (t) + f (BC (t)) where BC (t) :=
ρ
2∂z Pρ (BC ) ρ
Pρ (BC )
dt,
(3.2)
f (BC (t)) B (t) ρ | f (BC (t))| C
has the same distribution as BC (t). There is no drift term coming from the second derivatives of f because f ≡ 0. ρ The process BC (t) satisfies rotation symmetry, which means that for a ∈ T, ρ aρ (Ra (BC (t))) has the same distribution as (BC (t)), where Ra (z) := az. This follows easily from (3.2) with f = Ra . Note that for any z ∈ C, a ρ
2∂z Pρ (z) Pρ (z)
=
2∂z Paρ (az) Paρ (az) .
There is no compact set K ⊂ D such that BC (t) ∈ K for 0 ≤ t < τρ . For otherwise, the solution δρ (t) could be extended beyond τρ . The next two lemmas give the reason ρ why BC (t) is viewed as BC (t) conditioned to exit D at ρ. Lemma 3.1. Let ν denote the distribution of (BC (t) : 0 ≤ t < τ ). For every ρ ∈ T, ρ let μ(ρ, ·) denote the distribution of (BC (t) : 0 ≤ t < τρ ). Then ν = T μ(ρ, ·)dλ(ρ), where λ is the uniform probability measure on T. Proof. From Itô’s formula, the process Mρ (t) := Pρ (BC (t)), 0 ≤ t < τ , is a positive local martingale. So if σ is any Jordan curve in D surrounding 0, and τσ is the first time that BC (t) visits σ , then E [Mρ (τσ )] = Mρ (0) = 1. From the Girsanov Theorem, it is ρ easy to check that the distribution of (BC (t) : 0 ≤ t < τσ ) is absolutely continuous w.r.t. that of (BC (t) : 0 ≤ t < τσ ), and the Radon-Nikodym derivative is Mρ (τσ ). We are considering probability measures on the space of curves γ (t), 0 ≤ t < T , in D, started from 0. Let (Ft ) denote the natural filtration generated by the curves. For each n ∈ N, let τn denote the first time when |γ (t)| ≥ 1 − 1/n. Then each τn is an (Ft )-stopping time, and the whole sigma-algebra F is generated by the union n∈N Fτn . For each n ∈ N and ρ ∈ T, μ(ρ, ·) is absolutely continuous w.r.t. ν on Fτn , and the Radon-Nikodym derivative is Pρ (BC (τn )). We have that ρ → Pρ (BC (τn )) is continuous, and T Pρ (BC (τn ))dλ(ρ) = 1. Thus, ν = T μ(ρ, ·)dλ(ρ) on Fτn . Finally, since F is the σ -algebra generated by the union n∈N Fτn , which is an algebra, so the proof is finished by the Monotone Class Theorem.
ρ
Lemma 3.2. Almost surely limt→τρ− BC (t) = ρ. Proof. Let Wρ (z) =
ρ+z ρ−z ,
which maps D conformally onto the right half plane
{Re z > 0}, and maps ρ to ∞. We have Pρ = Re Wρ , so 2∂z Pρ = Wρ . Let Z ρ (t) = ρ ρ Wρ (BC (t)). Then Pρ (BC (t)) = Re Z ρ (t). From (3.2), there is another standard complex Brownian motion BC (t) such that Z ρ (t) satisfies the SDE: ρ
BC (t) + d Z ρ (t) = |Wρ (BC (t))|d
ρ
|Wρ (BC (t))|2 Re Z ρ (t)
dt, 0 ≤ t < τρ .
t ρ Let u ρ (t) = 0 |Wρ (BC (s))|2 ds, 0 ≤ t < τρ . Then u ρ is continuous and increasing, and maps [0, τρ ) onto [0, Sρ ) for some Sρ ∈ (0, ∞]. Let Z ρu (t) = Z ρ (u −1 ρ (t)), 0 ≤ t < Sρ .
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Then there is another standard complex Brownian motion BC (t) such that Z ρu (t) satisfies the SDE: d Z ρu (t) = d BC (t) +
1 dt, 0 ≤ t < Sρ . Re Z ρu (t)
(3.3)
ρ
Since the curve BC (t), 0 ≤ t < τρ , is not contained in any compact subset of D, so Z ρu (t), 0 ≤ t < Sρ , is not contained in any compact subset of {Re z > 0}. Thus, Sρ = ∞. From (3.3), Re Z ρu is a Bessel process of dimension 3 started from 1. Since |Z ρu (t)| ≥ Re Z ρu (t), and Sρ = ∞, so a.s. limt→∞ |Z ρu (t)| = ∞. Since lim z→∞ Wρ−1 (z) = ρ, so we derive the conclusion.
4. Schramm-Loewner Evolution Schramm-Loewner evolution (SLE) was introduced by Oded Schramm [11] to study the scaling limits of the 2-dimensional statistical lattice model at criticality, where the conformal invariance property appears in the limit. It is very successful in giving mathematical proofs of the conjectures proposed by physicists. The definition of SLE combines Loewner’s differential equation with a stochastic input. For the completeness of this paper, we now give a brief introduction of radial SLE, which is one of the major versions of SLE. The reader may refer to [10] and [4] for more properties of SLE. √ Let B(t) be a standard real Brownian motion. Let κ > 0 be a parameter. Let ξ(t) = κ B(t), t ≥ 0. The following differential equation is called the radial Loewner equation driven by ξ . ∂t gt (z) = gt (z)
eiξ(t) + gt (z) , g0 (z) = z. eiξ(t) − gt (z)
(4.1)
It turns out that there is a decreasing family of domains (Dt : 0 ≤ t < ∞) with D0 = D and 0 ∈ Dt for all t ≥ 0, such that each gt is defined on Dt , maps Dt conformally onto D, and satisfies gt (0) = 0 and gt (0) = et . Moreover, almost surely β(t) :=
lim
Dz→eiξ(t)
gt−1 (z)
(4.2)
exists for 0 ≤ t < ∞, and β(t), 0 ≤ t < ∞, is a continuous curve in D with β(0) = 1 and limt→∞ β(t) = 0. Such β is called a standard radial SLEκ curve. The radial SLEκ curve in a general simply connected domain which grows from a boundary point to an interior point is defined as the image of such β under a conformal map from D onto this domain, which takes 1 and 0 to the initial and end points, respectively. If κ ∈ (0, 4], β is a simple curve, intersects T only at its initial point, and for each t ≥ 0, Dt = D\β((0, t]); if κ > 4, β is no longer a simple curve, and for each t ≥ 0, Dt is the connected component of D\β((0, t]) which contains 0. In this paper we are mostly interested in the case κ = 2, so β is a simple curve. There is an interesting local martingale associated with radial SLE2 , which was used to prove the convergence of LERW to SLE2 [8]. Recall that Peiξ(t) is the normalized Poisson kernel in D with the pole at eiξ(t) . Since gt−1 maps D conformally onto Dt = D\β(0, t], fixes 0, and has continuous extension to D, which maps eiξ(t) to β(t), so Q t := Peiξ(t) ◦ gt is the normalized (Q t (0) = 1) Poisson kernel in Dt with the pole at β(t). We have the following proposition.
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Proposition 4.1. Let κ = 2. Then for any z ∈ D, (Q t (z) : 0 ≤ t < Tz ) is a local martingale, where Tz ∈ (0, ∞] is such that [0, Tz ) is the maximal interval with z ∈ Dt for t ∈ [0, Tz ). 5. Local Martingale in Two Time Variables 1 (t), 0 ≤ t < τ , Theorem 3.2 will be proved by constructing a coupling of the process BC 1 with a standard radial SLE2 curve β(t), 0 ≤ t < ∞, such that conditioned on β up to 1 before hitting β[0, T ] is a complex Brownian a finite stopping time T , the part of BC motion in D\β[0, T ] conditioned to hit β(T ). In this section, we will first construct a local coupling. 1 (t ), 0 ≤ t < τ , First we suppose that the conditional complex Brownian motion BC 1 1 1 and the standard radial SLE2 curve β(t2 ), 0 ≤ t2 < ∞, are independent. This is a trivial √ coupling of the above two processes. Let ξ(t2 ) = 2B(t2 ) be the driving function of β, and let gt denote the radial Loewner maps. Let (Ft11 ) and (Ft22 ) be the natural filtrations 1 (t ) and (ξ(t )), respectively. Then (β(t )) and (g ) are (F 2 )-adapted. generated by BC 1 2 2 t2 t2 Let 1 D = {(t1 , t2 ) ∈ [0, τ1 ) × [0, ∞) : BC [0, t1 ] ∩ β[0, t2 ] = ∅}.
For every t2 ∈ [0, ∞), let T1 (t2 ) be the maximal number such that (t1 , t2 ) ∈ D for t1 ∈ [0, T1 (t2 )); for every t1 ∈ [0, τ1 ), let T2 (t1 ) be the maximal number such that (t1 , t2 ) ∈ D for t2 ∈ [0, T2 (t1 )). If t¯2 < ∞ is an (Ft22 )-stopping time, then T1 (t¯2 ) is an (Ft11 × Ft¯2 )-stopping time; if t¯1 < τ1 is an (Ft11 )-stopping time, then T2 (t¯1 ) is an 2 (Ft¯1 × Ft22 )-stopping time. 1 Let Q t2 = Peiξ(t2 ) ◦ gt2 be as in Proposition 4.1. Since g0 = id and ξ(0) = 0, so 1+z . Define M on D such that Q 0 (z) = P1 (z) = 1−z M(t1 , t2 ) =
1 (t )) Q t2 (BC 1 1 (t )) Q 0 (BC 1
.
1 (0) = 0 and Q (0) ≡ 1, so It is clear that M(t1 , 0) = 1 for any 0 ≤ t1 < τ1 . Since BC t2 M(0, t2 ) = 1 for any 0 ≤ t2 < ∞.
Lemma 5.1. (a) For any (Ft11 )-stopping time t¯1 < τ1 , M(t¯1 , t2 ), 0 ≤ t2 < T2 (t¯1 ), is an (Ft¯1 × Ft22 )-local martingale. (b) For any (Ft22 )-stopping time t¯2 < ∞, M(t1 , t¯2 ), 0 ≤ 1 t1 < T1 (t¯2 ), is an (Ft11 × Ft¯2 )-local martingale. 2
Proof. (a) This part follows immediately from Proposition 4.1. 1 (t )). Recall (b) Let f t¯2 = Q t¯2 /Q 0 . Then f t¯2 is Ft¯2 -measurable, and M(t1 , t¯2 ) = f t¯2 (BC 1 2 that Q 0 = P1 . From (3.1) (ρ = 1) and Itô’s formula, we see that M(t1 , t¯2 ), 0 ≤ t1 < T1 (t¯2 ), satisfies the (Ft11 × Ft¯2 )-adapted SDE: 2
1 1 (t1 ))d BC (t1 )] + Re[2∂z f t¯2 (BC (t1 )) d1 M(t1 , t¯2 ) = Re[2∂z f t¯2 (BC
1 1 + f t¯2 (BC (t1 ))dt1 . 2
1) 2∂z Q 0 (BC 1) Q 0 (BC
]dt1
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We have f t¯2 Q 0 = Q t¯2 , and both Q 0 and Q t¯2 are harmonic. So 0 = Q t¯2 = 4∂z ∂z ( f t¯2 Q 0 ) = f t¯2 Q 0 + Q 0 f t¯2 + 4∂z f t¯2 ∂z Q 0 + 4∂z f t¯2 ∂z Q 0 = Q 0 f t¯2 + 8 Re[∂z f t¯2 ∂z Q 0 ]. So we have 1 (t1 ))d BC (t1 )]. d1 M(t1 , t¯2 ) = Re[2∂z f t¯2 (BC
Thus, M(t1 , t¯2 ), 0 ≤ t1 < T1 (t¯2 ), is an (Ft11 × Ft¯2 )-local martingale. 2
(5.1)
Let J denote the set of Jordan curves in D\{0} that surround 0. For every σ ∈ J, let 1 (t ) hits σ ; let T 2 be the first time that β(t ) hits σ . Then Tσ1 be the first time that BC 1 2 σ j j Tσ is an (Ft j )-stopping time, j = 1, 2. Let JP denote the set of (σ1 , σ2 ) ∈ J2 such that σ1 ∩ σ2 = ∅, and σ2 surrounds σ1 . Then for any (σ1 , σ2 ) ∈ JP, [0, Tσ11 ] × [0, Tσ22 ] ⊂ D. Lemma 5.2. For any (σ1 , σ2 ) ∈ JP, | ln(M)| is bounded on [0, Tσ11 ] × [0, Tσ22 ] by a constant depending only on σ1 and σ2 . Proof. Fix (σ1 , σ2 ) ∈ JP. In this proof, a uniform constant means a constant depending only on σ1 and σ2 ; and we say a variable is uniformly bounded if its absolute value 1 (t )). Since M(t , t ) = is bounded by a uniform constant. Let N (t1 , t2 ) = Q t2 (BC 1 1 2 N (t1 , t2 )/N (t1 , 0), so it suffices to show that ln(N ) is uniformly bounded on [0, Tσ11 ] × [0, Tσ22 ]. Fix t1 ∈ [0, Tσ11 ] and t2 ∈ [0, Tσ22 ]. Let E σ j denote the domain bounded by σ j , j = 1, 2. Let = E σ2 \E σ1 and t2 = Dt2 \E σ1 for t2 ∈ [0, Tσ22 ]. Recall that Dt2 = D\β((0, t2 ]). Let m and m t2 denote the moduli of the above doubly connected domains, respectively. Then m is a uniform constant, and m ≤ m t2 . Since gt2 maps Dt2 conformally onto D, so it maps t2 onto D\gt2 (E σ1 ), which must have modulus m t2 ≥ m. Since 0 ∈ E σ1 and gt2 (0) = 0, so 0 ∈ gt2 (E σ1 ). There is uniform constant rm ∈ (0, 1) such that the modulus of D\[0, rm ] equals m. It is known that, for connected compact sets K ⊂ D with 0 ∈ K and the modulus of D\K being at least m, the maximum of r (K ) := supz∈K |z| is attained when K = [0, rm ]. Now gt2 (E σ1 ) satisfies the 1 (t ) ∈ E , so |g (B 1 (t ))| ≤ r . property of K , so gt2 (E σ1 ) ⊂ {|z| ≤ rm }. Since BC 1 σ1 t2 m C 1 1+z 1 1 Since N (t1 , t2 ) = Q t2 (BC (t1 )) = P(gt2 (BC (t1 ))/eiξ2 (t2 ) ), where P(z) = Re 1−z , so 1−rm 1+rm 1+rm
1+rm ≤ N (t1 , t2 ) ≤ 1−rm . Thus, | ln(N )| ≤ ln( 1−rm ), which is a uniform constant. The stochastic process M(t1 , t2 ) valued at a certain pair of times (T1 , T2 ) will be used as a Radon-Nikodym derivative to weight some simple probability distribution to get a somehow complicated distribution. Here are the details. Fix (σ1 , σ2 ) ∈ JP. Let μ denote 1 (t ), 0 ≤ t < τ , with β(t ), 0 ≤ t < ∞, which are indethe joint distribution of BC 1 1 1 2 2 pendent of each other. From Lemma 5.1 and Lemma 5.2, we have M(Tσ11 , Tσ22 )dμ = M(0, 0) = 1. Define νσ1 ,σ2 such that dνσ1 ,σ2 /dμ = M(Tσ11 , Tσ22 ). Then νσ1 ,σ2 is also a probability measure. Now suppose the joint distribution of the above two random curves is νσ1 ,σ2 instead of μ. Since M = 1 when either t1 or t2 equals 0, so the marginal distributions of νσ1 ,σ2 agree with those of μ. Thus, νσ1 ,σ2 is also a coupling measure of 1 (t ), 0 ≤ t < τ , with β(t ), 0 ≤ t < ∞. We now look at the behavior of the BC 1 1 1 2 2 1 (t ), 0 ≤ t ≤ T 1 , and β(t ), 0 ≤ t ≤ T 2 . Fix any (F 2 )-stopping sub-curves BC 1 1 2 2 σ1 σ2 t2 time t¯2 ≤ Tσ22 . From (3.1), (5.1), and the Girsanov Theorem, under νσ1 ,σ2 , there is an
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1 (t ), 0 ≤ t ≤ T 1 , (Ft11 × Ft¯2 )-standard complex Brownian motion BC (t1 ) such that BC 1 1 σ1 2 2 1 satisfies the (Ft1 × Ft¯ )-adapted SDE: 2
1 (t1 ) = d BC (t1 ) + d BC
= d BC (t1 ) +
1) 2∂z P1 (BC
P1
(B 1 )
1) 2∂z f t¯2 (BC
dt1 +
1) f t¯2 (BC
C
1) 2∂z Q t¯2 (BC 1) Q t¯2 (BC
dt1
dt1 ,
(5.2)
where the second equality holds because P1 f t¯2 = Q 0 f t¯2 = Q t¯2 . 6. Coupling Measures Let M be as in the last section. Then we have the following proposition. Proposition 6.1. For any finite collection (σ1m , σ2m ), 1 ≤ m ≤ n, in JP, there is an a.s. continuous stochastic process M∗ defined on [0, ∞]2 , which satisfies the following properties: (i) M∗ = M on [0, Tσ1m ] × [0, Tσ2m ], 1 ≤ m ≤ n; 1 2 (ii) M∗ (t, 0) = M∗ (0, t) = 1 for any t ∈ [0, ∞]; (iii) There are constants C2 > C1 > 0 depending only on (σ1m , σ2m ), 1 ≤ m ≤ n, such that C1 ≤ M∗ (t1 , t2 ) ≤ C2 on [0, ∞]2 ; (iv) For any (Ft22 )-stopping time t¯2 , M∗ (t1 , t¯2 ), 0 ≤ t1 ≤ ∞, is an (Ft11 × Ft¯2 )2 martingale; 1 1 2 (v) For any (Ft1 )-stopping time t¯1 , M∗ (t¯1 , t2 ), 0 ≤ t2 ≤ ∞, is an (Ft¯ × Ft2 )1 martingale. For the proof, we may first define M∗ on [0, ∞]×{0}∪{0}×[0, ∞]∪ nm=1 [0, Tσ1m ]× 1
[0, Tσ2m ] by (i) and (ii), and then extend M∗ to [0, ∞]2 in such a way that: if R is a rectan2
gle obtained by dividing [0, ∞]2 using the lines {t1 = Tσ1m } and {t2 = Tσ2m }, 1 ≤ m ≤ n, 1
2
and R is not contained in any [0, Tσ1m ] × [0, Tσ2m ], then there are functions f 1R (t1 ) and 1
2
f 2R (t2 ) such that M∗ (t1 , t2 ) = f 1R (t1 ) f 2R (t2 ) on R. Such M∗ is well constructed, and is unique. Property (iii) follows from Lemma 5.2. Property (iv) and (v) follow from the local martingale property of M. The reader may refer to [14] (Theorem 6.1) for the explicit formula of M∗ and a detailed proof of a similar proposition. Let JP∗ be the set of (σ1 , σ2 ) ∈ JP such that both σ1 and σ2 are polygonal curves whose vertices have rational coordinates. Then JP∗ is countable. Let (σ1m , σ2m ), m ∈ N, be an enumeration of JP∗ . For each n ∈ N, let M∗n be the M∗ given by the above proposition for (σ1m , σ2m ), 1 ≤ m ≤ n, in the above enumeration. Let the probability μ be as in the last section. For each n ∈ N, define ν n such that dν n = M∗n (∞, ∞)dμ. From the property of M∗ , M∗n (∞, ∞)dμ = M∗n (0, 0) = 1, so ν n is a probability measure. Since M∗n = 1 when either t1 or t2 equals 0, so ν n is also a coupling measure 1 (t ), 0 ≤ t < τ , with β(t ), 0 ≤ t < ∞. of BC 1 1 1 2 2 Fix any m ∈ N. If n ≥ m, from the martingale property of M∗n , we have E [M∗n (∞, ∞)|FT1 1 × FT2 2 ] = M∗n (Tσ1m , Tσ2m ) = M(Tσ1m , Tσ2m ). σ1m
σ2m
1
2
1
2
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Thus, on FT1 1 × FT2 2 , ν n equals νσ1m ,σ2m defined in the last section. We want to conσ1m
σ2m
1 (t ), 0 ≤ t < τ , with β(t ), 0 ≤ t < ∞, such struct a coupling measure ν ∞ of BC 1 1 1 2 2 that for any m ∈ N, ν ∞ equals νσ1m ,σ2m on FT1 1 × FT2 2 . Such ν ∞ could be defined as σ1m
(ν n )
σ2m
in some suitable topology as follows. a subsequential weak limit of 1 to [0, τ ] such that B 1 (τ ) = 1, and Let C := ∪T ∈[0,∞] C([0, T ], D). Extend BC 1 C 1 1 and β are random elements extend β to [0, ∞] such that β(∞) = 0. Then both BC in C. Let μ1 and μ2 be their distributions, respectively. We view them as probability measures on C, where the σ -algebra is generated by the events {T ≥ a, f (a) ∈ A}, where 0 ≤ a ≤ ∞. So μ = μ1 × μ2 is a probability measure on C × C. Let denote the space of nonempty compact subsets of [0, ∞] × D endowed with Hausdorff metric. Then is a compact metric space. Define G : C → such that G( f ) is the graph of f . Then G is a one-to-one map. Let IG = G(C). One may check that G and G −1 (defined on IG ) are both measurable. This is also true for G × G and G −1 × G −1 . For n ∈ N, ν¯ n := (G × G)∗ (ν n ) is a probability measure on 2 . Since 2 is compact, so (¯ν n ) has a subsequence (¯ν n k ) that converges weakly to some probability measure ν¯ ∞ on × . Let ν nj k and ν ∞ j , j = 1, 2, denote the marginal distributions nk n ∞ k of ν and ν . Then for j = 1, 2, ν¯ j → ν¯ ∞ j weakly. For n ∈ N and j = 1, 2, since ∞ is supported by I 2 . ν nj = μ j , ν¯ nj = G ∗ (μ j ). Thus, ν¯ ∞ = G (μ ∗ j ), j = 1, 2. So ν¯ j G Let ν ∞ = (G −1 × G −1 )∗ (¯ν ∞ ) be a probability measure on C 2 . For j = 1, 2, we have −1 ∞ ν∞ ν∞ j = (G )∗ (¯ j ) = μ j . So ν is also a coupling measure of μ1 and μ2 . It remains to check that for any m ∈ N, ν ∞ equals νσ1m ,σ2m on FT1 1 × FT2 2 . For σ1m
σ2m
any σ ∈ J, define a truncate map Pσ from C onto itself such that Pσ ( f ) is the restriction of f to [0, τσ ], where τσ is the first time that f (t) ∈ σ . Fix m ∈ N. Then ν ∞ equals νσ1m ,σ2m on FT1 1 × FT2 2 iff σ1m
σ2m
(Pσ1m × Pσ2m )∗ (ν ∞ ) = (Pσ1m × Pσ2m )∗ (νσ1m ,σ2m ).
(6.1)
From an earlier observation, (6.1) holds if ν ∞ is replaced by ν n with n ≥ m. Let D : ( f, g) → ( f, g, f, g) be a diagonal map from C 2 to C 4 . For n ∈ N, let λ¯ n = [((G ◦ Pσ1m ) × (G ◦ Pσ2m )) × (G × G)]∗ ◦ D∗ (ν n ). Then λ¯ n is a probability measure on 4 = 2 × 2 . It is a coupling of (G × G)∗ ◦ (Pσ1m × Pσ2m )∗ (ν n ) and (G × G)∗ (ν n ), and is supported by G := {(F1 , F2 , F3 , F4 ) ∈ 4 : (0, 0) ∈ F1 ⊂ F3 , (0, 1) ∈ F2 ⊂ F4 }. 1 (0) = 0 and β(0) = 1. Here we use the facts that BC n 4 Since is a compact space, the sequence (λ¯ n k ) has a subsequence, say (λ¯ k j ), which ∞ 2 2 ∞ converges weakly to a probability measure λ¯ on × . Then λ¯ is also supported by ¯∞ ¯∞ G. Let λ¯ ∞ 1 and λ2 be the marginal distributions of λ on the first two variables and the n last two variables, respectively. Then we have (G × G)∗ ◦ (Pσ1m × Pσ2m )∗ (ν k j ) → λ¯ ∞ 1 nk j n ∞ replaced by ν k j if n ≥ m, so and (G × G)∗ (ν ) → λ¯ ∞ . Since (6.1) holds with ν k j 2 λ¯ ∞ = (G × G)∗ ◦ (Pσ m × Pσ m )∗ (νσ m ,σ m ). Since (G × G)∗ (ν n k ) → (G × G)∗ (ν ∞ ), so 1
1
2
1
2
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∞ ∞ ∞ ∞ −1 × G −1 × G −1 × G −1 ) (λ ∞ λ¯ ∞ ∗ ¯ ), and let λ1 and λ2 2 = (G × G)∗ (ν ). Let λ = (G ∞ ∞ ∞ be its marginal distributions. Then we have λ1 = (Pσ1m × Pσ2m )∗ (νσ1m ,σ2m ) and λ2 = ν . Since λ¯ ∞ is supported by G, so λ∞ is supported by the set of ( f 1 , f 2 , f 3 , f 4 ) ∈ C 4 such that f 1 and f 2 are subcurves of f 3 and f 4 , respectively. From the property of (Pσ1m × Pσ2m )∗ (νσ1m ,σ2m ), we can further conclude that λ∞ is supported by {( f 1 , f 2 , f 3 , f 4 ) ∈ C 4 : f 1 = Pσ1m ( f 3 ), f 2 = Pσ2m ( f 4 )}. So we obtain (6.1). The reader may refer to Lemma 4.1 in [15] for a more detailed argument. Now for each m ∈ N, ν ∞ = νσ1m ,σ2m on FT1 1 × FT2 2 . Let t¯2 be an (Ft22 )-stopping σ1m
time with
σ2m
t¯2 ≤ Tσ2m . From the discussion at the end of the last section, we see that 2 ≤ t1 ≤ Tσ1m , satisfies (5.2) for some (Ft11 × Ft¯2 )-standard complex Brownian 2 1
1 (t ), 0 BC 1 motion BC (t1 ). Fix t2 ∈ (0, ∞). For n ∈ N, define
Rn = sup{Tσ1m : 1 ≤ m ≤ n, Tσ2m ≥ t2 }. 1
Fix n ∈ N. Then for any 1 ≤ m ≤ n, if t2 ≤
2
Tσ2m , 2
1 (t ), 0 ≤ t ≤ T 1 , satisfies then BC 1 1 σm 1
1 (t ), 0 ≤ t ≤ R , should also satisfy (5.2). (5.2). So BC 1 1 n 1 (t ) is disjoint From the definition, T1 (t2 ) is the maximal number such that BC 1 from β[0, t2 ] for 0 ≤ t1 < T1 (t2 ). It is easy to check that T1 (t2 ) = supn∈N Rn . 1 (t ), 0 ≤ t < T (t ), should also satisfy (5.2). Let W (z) = Thus, BC 1 1 1 2 t2
eiξ(t2 ) +gt2 (z) . eiξ(t2 ) −gt2 (z)
Then Q t2 = Re Wt2 ; Wt2 maps Dt2 conformally onto the right half plane, and maps β(t2 ) to ∞. The argument in the proof of Lemma 3.2 can be used here to show that 1 (t ) = β(t ). Thus, B 1 (T (t )) = β(t ). In fact, we may view a.s. limt1 →T1 (t2 ) BC 1 2 2 C 1 2 1 BC (t1 ), 0 ≤ t1 < T1 (t2 ), as the complex Brownian motion BC (t) conditioned to leave Dt2 at β(t2 ). This result holds for every t2 ∈ (0, ∞). So a.s. for every t2 ∈ Q ∩ (0, ∞), 1 (T (t )) = β(t ). we have BC 1 2 2 From the definition, it is clear that T1 as a function of t2 is decreasing, and 1 [0, T (t )) is disjoint from β[0, t ] for any t ∈ [0, ∞). For any a ∈ R, it is easy BC 1 2 2 2 to check that {t2 : T1 (t2 ) > a} is an open subset of [0, ∞). So t2 → T1 (t2 ) is right1 and β are continuous, and Q ∩ (0, ∞) is dense in (0, ∞), continuous. Since both BC 1 (T (t )) = β(t ). From Corollary 2.1, we see that β is a so a.s. for any t2 ∈ (0, ∞), BC 1 2 2 1. loop-erasure of the reversal of BC 7. Some Remarks 1. One can prove that, under the new coupling measure ν ∞ , for any (Ft11 )-stopping time t¯1 < τ1 , the curve β(t2 ), 0 ≤ t2 < T2 (t¯1 ), is a radial SLE2 curve in D started 1 (t¯ ), which stops on hitting B 1 [0, t¯ ]. In general, β may not visit from 1 aimed at BC 2 1 C 1 BC (t¯2 ). 2. Theorem 3.2 can be extended to finitely connected plane domains. Let γ1 be a Brownian motion started from an interior point z 0 in a finitely connected domain D, stopped on exiting D, and conditioned to hit ∂ D at z 1 . The process satisfies SDE (3.1) with Pρ replaced by the Poisson kernel function in D with the pole at z 2 . Then the time-reversal of γ1 has a loop-erasure, which is a continuous LERW in D growing from z 1 to z 0 ([13]).
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3. Let γ2 be the Brownian excursion in D from one boundary point z 1 to another boundary point z 2 . The process starts from z 1 , and after the initial time, it becomes a Brownian motion in D conditioned to exit D at z 2 . We can conclude that the time-reversal of γ2 has a loop-erasure, which is a continuous LERW in D from z 2 to z 1 . For the proof, we may use the coupling technique to construct a coupling of γ2 with a continuous LERW β in D from z 2 to z 1 such that conditioned on the part of β up to a finite stopping time T , the part of γ2 before hitting β[0, T ] is a Brownian excursion in D\β[0, T ] from z 1 to β(T ). It is well known that the time-reversal of γ2 is the Brownian excursion in D from z 2 to z 1 . So γ2 itself has a loop-erasure, which is a continuous LERW in D from z 1 to z 2 . Especially, if D is simply connected, then the Brownian excursion from z 1 to z 2 has a loop-erasure, which is a chordal SLE2 curve in D from z 1 to z 2 . 4. Let γ3 be the Brownian motion in D started from an interior point z 0 and conditioned to hit another interior point z 3 . The process satisfies SDE (3.1) with Pρ replaced by G D (z 1 , ·), where G D (·, ·) is the Green function in D. Using the coupling technique, we can conclude that the time-reversal of γ3 has a loop-erasure, which is a continuous LERW in D from z 3 to z 0 ([16]). It is well known that the time-reversal of γ3 is the Brownian motion in D started from z 3 conditioned to hit z 0 . So γ3 itself has a loop-erasure, which is a continuous LERW in D from z 0 to z 3 . 5. Let γ4 be a Brownian excursion in D started from a boundary point z 1 and conditioned to hit an interior point z 0 . The process starts from z 1 , and after the initial time, it becomes the Brownian motion in D conditioned to hit z 0 . We can conclude that its time-reversal has a loop-erasure, which is a continuous LERW in D from z 0 to z 1 . It is well known that γ4 is the time-reversal of the γ1 in Remark 2. So γ1 has a loop-erasure, which is a continuous LERW in D from z 0 to z 1 ; and γ4 has a loop-erasure, which is a continuous LERW in D from z 1 to z 0 . In particular, if D is a simply connected domain, a continuous LERW from an interior point z 0 to a random boundary point with harmonic measure distribution is a disc SLE2 curve ([12]) in D started from z 0 . So we conclude that the BC (t), 0 ≤ t ≤ τ , in Theorem 3.1 has a loop-erasure, which is a disc SLE2 curve in D started from z 0 . Acknowledgements. I would like to thank the referees for providing constructive comments and help in improving the contents of this paper.
References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.
Beffara, V.: The dimension of the SLE curves. Ann. Probab. 36(4), 1421–1452 (2008) Burdzy, K.: Percolation dimension of fractrals. J. Math. Anal. Appl. 145, 282–288 (1990) Lawler, G.F.: Intersection of random walks. Boston: Birkhäuser, 1991 Lawler, G.F.: Conformally Invariant Processes in the Plane. Providence, RI: Amer. Math. Soc., 2005 Lawler, G.F., Schramm, O., Werner, W.: Values of Brownian intersection exponents I: Half-plane exponents. Acta Math. 187(2), 237–273 (2001) Lawler, G.F.: Oded Schramm and Wendelin Werner. Values of Brownian intersection exponents II: Plane exponents. Acta Math. 187, 275–308 (2001) Lawler, G.F.: Oded Schramm and Wendelin Werner. Conformal restriction: the chordal case. J. Amer. Math. Soc. 16(4), 917–955 (2003) Lawler, G.F.: Oded Schramm and Wendelin Werner. Conformal invariance of planar loop-erased random walks and uniform spanning trees. Ann. Probab. 32(1B), 939–995 (2004) Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion. Berlin: Springer, 1991 Rohde, S., Schramm, O.: Basic properties of SLE. Ann. of Math. 161(2), 883–924 (2005) Schramm, O.: Scaling limits of loop-erased random walks and uniform spanning trees. Israel J. Math. 118, 221–288 (2000)
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12. Zhan, D.: Stochastic Loewner evolution in doubly connected domains. Probab. Theory Related Fields 129(3), 340–380 (2004) 13. Zhan, D.: The Scaling Limits of Planar LERW in Finitely Connected Domains. Ann. Probab. 36(2), 467–529 (2008) 14. Zhan, D.: Reversibility of chordal SLE. Ann. Probab. 36(4), 1472–1494 (2008) 15. Zhan, D.: Duality of chordal SLE. Inven. Math. 174(2), 309–353 (2008) 16. Zhan, D.: Continuous LERW started from interior points. Stoch. Proc. Appl. 120, 1267–1316 (2010) Communicated by S. Smirnov
Commun. Math. Phys. 303, 721–759 (2011) Digital Object Identifier (DOI) 10.1007/s00220-011-1223-z
Communications in
Mathematical Physics
On the Integral Geometry of Liouville Billiard Tables G. Popov1 , P. Topalov2 1 Laboratoire de Mathématiques Jean Leray, CNRS: UMR 6629, Université de Nantes,
2, rue de la Houssinière, BP 92208, 44072 Nantes Cedex 03, France
2 Department of Mathematics, Northeastern University, 360 Huntington Avenue,
Boston, MA 02115, USA. E-mail:
[email protected] Received: 17 February 2010 / Accepted: 12 September 2010 Published online: 28 March 2011 – © Springer-Verlag 2011
Abstract: The notion of a Radon transform is introduced for completely integrable billiard tables. In the case of Liouville billiard tables of dimension 3 we prove that the Radon transform is one-to-one on the space of continuous functions K on the boundary which are invariant with respect to the corresponding group of symmetries. We prove also that the frequency map associated with a class of Liouville billiard tables is non-degenerate. This allows us to obtain spectral rigidity of the corresponding Laplace-Beltrami operator with Robin boundary conditions.
1. Introduction This paper is concerned with the integral geometry and the spectral rigidity of Liouville billiard tables. By a billiard table we mean a smooth compact connected Riemannian manifold (X, g) of dimension n ≥ 2 with a non-empty boundary := ∂ X . The elastic reflection of geodesics at determines continuous curves on X called billiard trajectories as well as a discontinuous dynamical system on T ∗ X – the “billiard flow” – that generalizes the geodesic flow on closed manifolds without boundary. The billiard flow on T ∗ X induces a discrete dynamical system in the open coball bundle B ∗ of given by the corresponding billiard ball map B and its iterates. The map B is defined in an open subset of B ∗ = {ξ ∈ T ∗ : ξ g < 1}, where ξ g denotes the norm induced by the Riemannian metric g on the corresponding cotangent plane and it can be considered as a discrete Lagrangian system as in [9,11,15]. The orbits of B can be obtained by a variational principal and they can be viewed as “discrete geodesics” of the corresponding Lagrangian. In this context, periodic orbits of B can be considered as “discrete closed geodesics”. Let μ be a positive continuous function on B ∗ . Denote by π∗ K the pull-back of the continuous function K ∈ C() with respect to the projection π : T ∗ → . We are interested in the following problems.
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Problem A. Let K be a continuous function on such that the mean value of the product π∗ K · μ is zero on any periodic orbit of the billiard ball map B. Does it imply K ≡ 0? The mapping assigning to any periodic orbit γ = {0 , 1 , . . . , m−1 } ⊂ B ∗ of the m−1 ∗ map B the mean value (1/m) j=0 π K · μ ( j ) of the function π∗ K · μ on γ can be viewed as a discrete analogue of the Radon transform, considering the periodic orbits of the billiard ball map as discrete closed geodesics. Problem A has a positive answer for any ball in the Euclidean space Rn centered at the origin if μ = 1 and K is even. In fact, approximating the great circles on the sphere by closed billiard trajectories of the billiard table we obtain from the hypothesis in Problem A that the integral of K over any great circle is zero. Since K is even, by Funk’s theorem we obtain K ≡ 0 ([3, Theorem 4.53]). The case of general Riemannian manifold is much more complicated. Denote by π X : T ∗ X → X the natural projection of the cotangent bundle T ∗ X onto X . Let S ∗ X | = {ξ ∈ T ∗ X : π X (ξ ) ∈ , ξ g = 1} be the restriction of the unit co-sphere bundle to . There are two natural choices for the function μ we are concerned with, namely, μ ≡ 1 or μ(ξ ) = π + (ξ ), n g −1 , ξ ∈ B ∗ , where ·, · is the standard pairing between vectors and covectors, n g is the inward unit normal to at x = π (ξ ), and π + : B ∗ → S ∗ X | assigns to any ξ ∈ Tx∗ with norm ξ g < 1 the unit outgoing covector the restriction of which to Tx coincides with ξ . Recall that a covector based on x is outgoing if its value on n g (x) is non-negative. The latter choice of μ is related to the wave-trace formula for manifolds with boundary obtained by V. Guillemin and R. Melrose [4,5]. It appears also in the iso-spectral invariants of the Robin boundary problem for the Laplace-Beltrami operator obtained in [12]. From now on we fix the positive function μ ∈ C(B ∗ ) by μ ≡ 1 , or by μ(ξ ) = π + (ξ ), n g −1 , ξ ∈ B ∗ .
(1.1)
For that choice of μ, it will be shown that Problem A has a positive solution for a class of Liouville billiard tables of classical type. A Liouville billiard table (L.B.T.) of dimension n ≥ 2, is a completely integrable billiard table (X, g) (the notion of complete integrability will be recalled in Sect. 2) admitting n functionally independent and Poisson commuting integrals of the billiard flow on T ∗ X which are quadratic forms in the impulses. A L.B.T. can be viewed as a 2n−1 -folded branched covering of a disk-like domain in Rn by the cylinder Tn−1 × [−N , N ], where T = R/Z and N > 0. Liouville billiard tables of dimension two are defined in [10] and in any dimension n ≥ 2 in [11], where the integrability of the billiard ball map is shown via the geodesic equivalence principal. Here we write explicitly first integrals of the billiard flow and show that it is completely integrable (see Sect. 3.1). An important subclass of L.B.T.s are the Liouville billiard tables of classical type having an additional symmetry and for which the boundary is strictly geodesically convex (with respect to the outward normal −n g ). It turns out that the group of isometries of a L.B.T. of classical type is isomorphic to (Z/2Z)n . Moreover, the group of isometries of (X, g) induces a group of isometries G on which is isomorphic to (Z/2Z)n . An important example of a L.B.T. of classical type is the interior of the n-axial ellipsoid equipped with the Euclidean metric. More generally, there is a non-trivial two-parameter family of L.B.T.s of classical type of constant scalar curvature κ having the same broken geodesics (considered as non-parameterized curves) as the ellipsoid [11, Theorem 3]. This family includes the ellipsoid (κ = 0), a L.B.T. on the sphere (κ = 1) and a L.B.T. in the hyperbolic space (κ = −1). Theorem 1. Let (X, g), dim X = 3, be an analytic L.B.T. of classical type. Suppose that there is at least one non-periodic geodesic on the boundary . Choose μ as in (1.1).
On the Integral Geometry of Liouville Billiard Tables
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Let K ∈ C() be invariant with respect to the group of isometries G ∼ = (Z/2Z)3 of ∗ the boundary and such that the mean value of π K · μ on any periodic orbit of the billiard ball map is zero. Then K ≡ 0. In particular, Problem A has a positive solution for ellipsoidal billiard tables in R3 with μ ≡ 1 as well as for μ(ξ ) = π + (ξ ), n g −1 , for any K ∈ C() which is invariant under the reflections with respect to the coordinate planes Ox y , O yz , and Ox z . More generally, Theorem 3 can be applied for any L.B.T. of the family described in [11, Theorem 3]. The condition that the boundary contains at least one non-closed geodesic will become clear after the discussion of Problem C. As it was mentioned above the map assigning to each periodic orbit of the billiard ball map B the mean value of π∗ K · μ on it can be considered as a discrete analogue of the Radon transform. Another version of the Radon transform can be defined as follows. Denote by F the family of all Lagrangian tori ⊂ B ∗ which are invariant with respect to some exponent B m , m ≥ 1, of the billiard ball map B, i.e. B m ( ) ⊆ . For any continuous function K on we denote by R K ,μ ( ) the mean value of the integral of π∗ K · μ on ∈ F with respect to the Leray form (see Sect. 2). The mapping → R K ,μ ( ), ∈ F, will be called a Radon transform of K as well. Problem B. Let K be a continuous function on which is invariant with respect to the group of isometries G. Does the relation R K ,μ ≡ 0 imply K ≡ 0? The main result of the paper is the following theorem, which gives a positive answer to Problem B for L.B.T.s. Theorem 2. Let (X, g), dim X = 3, be a Liouville billiard table of classical type. Fix μ by (1.1). If K ∈ C() is invariant under the group of symmetries G of and R K ,μ ( ) = 0 for any ∈ F, then K ≡ 0. We point out that L.B.T.s of classical type are smooth by construction but they are not supposed to be analytic. A similar result has been obtained for the ellipse in [4] and more generally for L.B.T.s of classical type in dimension n = 2 in [10] and [12]. It is always interesting to find a smaller set of data for which the Radon transform is one-to-one. In the case n = 2 the proof is done by analyticity, and we need to know the values of the Radon transform R K ,μ ( ) only on a family of invariant circles { j } j∈N approaching the boundary S ∗ of B ∗ . The case n = 3 is more complicated, since the argument using analyticity does not work any more. Nevertheless, we can restrict the Radon transform to data “close” to the boundary in the following sense: It will be shown in Sect. 3.3 that any L.B.T. of classical type of dimension 3 admits four not necessarily connected charts U j , 1 ≤ j ≤ 4, of action-angle variables in B ∗ . Two of them, say U1 and U2 , have the property that any unparameterized geodesic in S ∗ can be obtained as a limit of orbits of B lying either in U1 or in U2 (then the corresponding broken geodesics approximate geodesics of the boundary). Moreover, in any connected component of U1 and U2 there is such a sequence of orbits of B, while U3 and U4 do not enjoy this property. In other words, the charts U1 and U2 can be characterized by the property that there is a family of “whispering gallery rays” issuing from any of their connected components. For this reason the two cases j = 1, 2 will be referred as to boundary cases. Denote by Fb the set of all ∈ F lying either in U1 or in U2 . We will show in Theorem 4.1 that the restriction of the Radon transform R K ,μ on Fb determines uniquely K .
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As an application we prove spectral rigidity of the Robin boundary problem for Liouville billiard tables. Given a real-valued function K ∈ C(, R), we consider the “positive” Laplace-Beltrami operator on X with domain ∂u 2 D := u ∈ H (X ) : | = K u| , ∂n g where H 2 (X ) is the Sobolev space, and n g (x), x ∈ , is the inward unit normal to with respect to the metric g. We denote this operator by g,K . It is a selfadjoint operator in L 2 (X ) with discrete spectrum Spec g,K := {λ1 ≤ λ2 ≤ · · · }, where each eigenvalue λ = λ j is repeated according to its multiplicity, and it solves the spectral problem
u = λu in X, (1.2) ∂u | = K u| . ∂n g Let [0, 1] t → K t ∈ C ∞ (, R) be a continuous family of smooth real-valued functions on . To simplify the notations we denote by t the corresponding operators g,K t . This family is said to be isospectral if ∀ t ∈ [0, 1], Spec ( t ) = Spec ( 0 ).
(1.3)
We consider here a weaker notion of isospectrality which has been introduced in [12]. Fix two positive constants c and d>1/2, and consider the union of infinitely many disjoint intervals (H1 ) I := ∪∞ k=1 [ak , bk ], 0 < a1 < b1 < · · · < ak < bk < · · ·, such that lim ak = lim bk = +∞,
k→∞
ak+1 − bk ≥
k→∞ cbk−d
lim (bk − ak ) = 0, and
k→∞
for any k ≥ 1.
We impose the following “weak isospectral assumption”: (H2 ) There is a > 0 such that ∀ t ∈ [0, 1], Spec ( t ) ∩ [a, +∞) ⊂ I, where I is given by (H1 ). Using the asymptotics of the eigenvalues λ j as j → ∞ we have shown in [12] that the conditions (H1 )–(H2 ) are “natural” for any d > n/2 (n = dim X ), which means that the usual isospectral assumption (1.3) implies (H1 )–(H2 ) for any such d and any c > 0. Theorem 3. Let (X, g) be a 3-dimensional analytic Liouville billiard table of classical type such that the boundary has at least one non-periodic geodesic. Let [0, 1] t → K t ∈ C ∞ (, R) be a continuous family of real-valued functions on satisfying the isospectral conditions (H1 )–(H2 ). Suppose that K 0 and K 1 are invariant with respect to the group of symmetries G = (Z/2Z)3 of . Then K 0 ≡ K 1 .
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A similar result has been proved in [12] for smooth 2-dimensional billiard tables. The idea of the proof of Theorem 3 is as follows. Fix the continuous function μ by μ(ξ ) = π + (ξ ), n g −1 . First, using [12, Theorem 1.1] we obtain that R K 1 ,μ ( ) = R K 0 ,μ ( ),
(1.4)
for any Liouville torus of a frequency vector satisfying a suitable Diophantine condition.1 Next, we prove that the union of such tori is dense in the union of the two charts U j , j = 1, 2, of “action-angle” coordinates in B ∗ , which implies (1.4) for any torus ∈ Fb . Now the claim follows from Theorem 4.1. In the same way we prove Theorem 1. First we obtain that R K ,μ ( ) = 0 for a set of “rational tori” . Then we prove that the union of these tori is dense in U1 ∪ U2 , and we apply Theorem 4.1. We point out that the proof of Theorem 3 presented in Sect. 6 requires only finite smoothness of K t (see Theorem 6.1). An important ingredient in the proof of both theorems is the density of the corresponding families of invariant tori in U j , j = 1, 2. This follows from the non-degeneracy of the frequency map for Liouville billiard tables of classical type studied in Sect. 5. Recall that in any chart U j of action-angle coordinates the frequency map assigns to any value of the momentum map the frequency vector of the minimal power B m : U j → U j , m ≥ 1, that leaves invariant the corresponding Liouville tori ⊂ U j . The frequency map is said to be non-degenerate in U j if its Hessian with respect to the action variables is non-degenerate in a dense subset of U j . We are interested in the following problem: Problem C. Is the frequency map non-degenerate in any chart of action-angle coordinates? We prove in Theorem 5.1 that this is true in the charts U j , j = 1, 2, for any analytic L.B.T. of classical type for which the boundary admits at least one non-closed geodesic. The 3-axial ellipsoid and more generally any billiard table of the two-parameter family of L.B.T.s of classical type of constant scalar curvature described in [11, Theorem 3] has these properties. The non-degeneracy of the frequency map appears also as a hypothesis in the Kolmogorov-Arnold-Moser theorem. In particular, Theorem 5.1 allows us to apply the KAM theorem for the billiard ball maps associated with small perturbations of the L.B.T.s in [11, Theorem 3]. It is a difficult problem to prove that the frequency map of a specific completely integrable system is non-degenerate. The non-degeneracy of the frequency map of completely integrable Hamiltonian systems has been systematically investigated in [7]. The main idea in [7] is to investigate the system at the singularities of the momentum map. In our case we reduce the system at the boundary S ∗ of B ∗ . To our best knowledge this problem has not been rigorously studied for completely integrable billiard tables even in the case of the billiard table associated with the interior of the ellipsoid. The article is organized as follows. In Sect. 2 we recall certain facts about the billiard ball map and define a Radon transform for completely integrable billiard tables. Section 3 is concerned with the construction of L.B.T.s. First we consider a cylinder C = Tω1 × Tω2 × [−N , N ], where Tl = R/lZ for l > 0 and N > 0 and define a “metric” g and two Poisson commuting quadratic with respect to the impulses integrals I1 and I2 of g in C. The non-negative quadratic form g is degenerate at a submanifold 1 Recall that in action-angle coordinates on , the billiard ball map B m : → , m ≥ 1, is a shift by a constant vector that is the frequency vector of .
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S of C. To make g a Riemannian metric we consider its push-forward on the quotient σ : C → C˜ of C with respect to the group generated by two commuting involutions σ1 and σ2 whose fix point set is just S. The main result in this section is Proposition 3.3 which provides C˜ with a differentiable structure such that the push-forwards g˜ := σ∗ g, I˜1 := σ∗ I1 and I˜2 := σ∗ I2 are smooth forms, g˜ is a Riemannian metric on C˜ and I˜1 and I˜2 are Poisson commuting integrals of g. ˜ In Sect. 3.3 we write an explicit parameterization of the regular tori by means of the values of the momentum map corresponding to the integrals I˜1 and I˜2 . The injectivity of the Radon transform is investigated in Sect. 4 using the Stone-Weierstrass theorem and some properties of the classical Legendre polynomials. The non-degeneracy of the frequency map of an analytic L.B.T. is investigated in Sect. 5. The proof of Theorem 1 and Theorem 3 is given in Sect. 6. In the Appendix we investigate the frequency map and the action-angle coordinates of completely integrable billiard tables and derive a formula for the frequency vectors of B m . 2. Invariant Manifolds, Leray Form, and Radon Transform In the present section we define the Radon transform for integrable billiard tables. First we recall the definition of the billiard ball map B associated to a billiard table (X, g), dim X = n, with boundary . Denote by H ∈ C ∞ (T ∗ X, R) the Hamiltonian corresponding to the Riemannian metric g on X via the Legendre transformation and set S ∗ X := {ξ ∈ T ∗ X : H (ξ ) = 1}, S ∗ X | := {ξ ∈ S ∗ X : π X (ξ ) ∈ }, ∗ S± X | := {ξ ∈ S ∗ X | : ± ξ, n g > 0}, n g being the inward unit normal to . Denote by r : T ∗ X | → T ∗ X | the “reflection” at the boundary given by r : v → w, where w|Ty = v|Ty and w, n g + v, n g = 0. Obviously r : S ∗ X | → S ∗ X | . Take u ∈ S+∗ X | ⊂ T ∗ X and consider the integral curve γ (t; u) of the Hamiltonian vector field X H on T ∗ X starting at u. If it intersects transversally S ∗ X | at a time t1 > 0 and lies entirely in the interior of S ∗ X for t ∈ (0, t1 ), ∗ X | . The set O ⊆ S ∗ X | of all such u is open in S ∗ X | . we set B0 (u) := γ (t1 , u) ∈ S− + + The billiard ball map is defined by B := r ◦ B0 : O → S+∗ X | . Denote by B ∗ := {ξ ∈ T ∗ : H (ξ ) < 1} the (open) coball bundle of . The natural projection π+ : S+∗ X | → B ∗ assigning to each u ∈ S ∗ X | the covector u|Tx ∈ B ∗ admits a smooth inverse map π + : B ∗ → S+∗ X | . The map B := π+ ◦ B ◦ π + is defined in the open subset π+ (O) of the coball bundle of and it is a smooth symplectic map, i.e. it preserves the canonical symplectic two-form ω = dp ∧ d x on B ∗ . The map B will be called a billiard ball map as well. From now on we assume that the billiard ball map B : B ∗ → B ∗ is globally defined and completely integrable. By definition,2 the complete integrability of the billiard ball map of (X, g) means that there exist n − 1 invariant with respect to B smooth functions F1 , . . . , Fn−1 on B ∗ which are functionally independent and in involution with respect to the canonical Poisson bracket on T ∗ , i.e. {Fi , F j } = 0,
1 ≤ i, j ≤ n − 1.
2 This is one of the many definitions of complete integrability of the billiard ball map.
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The functions F1 , . . . , Fn−1 are said to be functionally independent in B ∗ if the form d F1 ∧. . .∧d Fn−1 does not vanish almost everywhere. A function f on B ∗ is said to be invariant with respect to the billiard ball map B if B ∗ f = f . The invariant functions with respect to the billiard ball map are called also integrals. In particular, as F1 , . . . , Fn−1 are integrals, then any non-empty level set L c := {ξ ∈ B ∗ : F1 (ξ ) = c1 , . . . , Fn−1 (ξ ) = cn−1 }, c = (c1 , . . . , cn−1 ) ∈ Rn−1 , is invariant with respect to the billiard ball map B : B ∗ → B ∗ . By the ArnoldLiouville theorem any regular compact component c of L c is diffeomorphic to the (n − 1)-dimensional torus Tn−1 and there exists a tubular neighborhood of c in B ∗ symplectically diffeomorphic to Drn−1 × Tn−1 that is supplied with the canonical symplectic n−1 := {J = (J , . . . , J n−1 : |J | < r } structure n−1 1 n−1 ) ∈ R k=1 d Jk ∧ dθk . Here Dr n−1 for some r > 0, θ = (θ1 , . . . , θn−1 ) are the periodic coordinates on T , and | · | is the Euclidean norm in Rn−1 . The coordinates (J, θ ) are called action-angle coordinates of the billiard ball map. Recall that c is regular if the (n − 1)-form d F1 ∧ . . . ∧ d Fn−1 does not vanish at the points of c . Any regular torus c is a Lagrangian submanifold of B ∗ and it is also called a Liouville torus. Assume that the Liouville torus c is invariant with respect to B m for some m ≥ 1, i.e. B m ( c ) = c . Let αc be a (n − 1)-form defined in a tubular neighborhood of c in B ∗ so that ωn−1 := ω ∧ . . . ∧ ω = αc ∧ d F1 ∧ . . . ∧ d Fn−1 .
(2.1)
It follows from (2.1) that the restriction λc := αc | c of αc to c is uniquely defined. The form λc is a volume form on c which is called the Leray form. As B preserves both the symplectic structure ω and the functions F1 , . . . , Fn−1 , one obtains from (2.1) that the restriction of B m to c preserves λc . Fix a positive continuous function μ on B ∗ and denote by F the set of all Liouville tori. For any continuous function K on the mapping R K ,μ : F → R, given by 1 (π ∗ K )μ λc , (2.2) R K ,μ ( c ) := λc ( c ) c is called a Radon transform of K . It is easy to see that the Radon transform does not depend on the different choices made in the definition of the Leray form. Remark 2.1. An alternative definition of the Radon transform would be m−1
1 K ,μ ( c ) := (B ∗ ) j (π∗ K )μ λc , R λc ( c ) c
(2.3)
j=0
where m ≥ 1 is the minimal power of B that leaves c invariant, i.e., B m ( c ) = c . Note that (2.3) appears as a spectral invariant of (1.2) in [12]. We show in Sect. 5 that for L.B.T. of classical type m = 1 in the charts U1 and U2 . In particular, (2.2) and (2.3) coincide in this case. There is another notion of complete integrability which is related to the “billiard flow” of the billiard table (X, g) (cf. Definition 7.2). We reformulate Definition 7.2 in terms of the cotangent bundle T ∗ X : A billiard table is completely integrable if there exist n smooth functions H1 , . . . , Hn−1 , Hn = H in a neighborhood U of S ∗ X in T ∗ X with the following properties:
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G. Popov, P. Topalov
(i) the functions H j are in involution in U with respect to the canonical Poisson bracket on T ∗ X , i.e. {Hi , H j } = 0, 1 ≤ i, j ≤ n, (ii) H1 , . . . , Hn are functionally independent in U , (iii) r ∗ H j = H j in U | for 1 ≤ j ≤ n. The properties (i) and (iii) imply that H j is invariant with respect to the billiard flow in U for any 1 ≤ j ≤ n. In particular, the functions F j = H j ◦ π + , 1 ≤ j ≤ n − 1, are integrals of the billiard ball map B. As H1 , . . . , Hn are functionally independent in U the billiard ball map is completely integrable if, for example, the integrals H j are homogeneous functions with respect to the standard action of R∗ := R\0 on the fibers of T ∗ X \0. In this way we see that the billiard ball map of a completely integrable billiard table is completely integrable if the integrals are homogeneous functions on the fibers of T ∗ X \0. Definition 2.2. A billiard table (X, g) with a completely integrable billiard ball map will be called R-rigid with respect to the density μ if Problem B has a positive solution. 3. Liouville Billiard Tables 3.1. Construction of Liouville billiard tables. In this section we describe a class of 3-dimensional completely integrable billiard tables called Liouville billiard tables. The interior of an ellipsoid is a particular case of a Liouville billiard table – see § 3.2 below as well as § 5.3 in [11] for the general construction of Liouville billiard tables of arbitrary dimension, where the integrability of the billiard ball map was deduced from geodesically equivalence principle. Here we write explicitly integrals of the billiard flow of a Liouville billiard table which are quadratic forms in momenta, and hence, homogeneous functions of degree 2 on the fibers of T ∗ X \0. For any N > 0 and any ωk > 0(k = 1, 2) consider the cylinder C := {(θ1 , θ2 , θ3 )} ∼ = Tω1 × Tω2 × [−N , N ], Tl := R/l Z, where θ1 and θ2 are periodic coordinates with minimal periods ω1 and ω2 respectively and θ3 takes its values in the closed interval [−N , N ]. Define the involutions σ1 , σ2 : C → C of the cylinder C by σ1 : (θ1 , θ2 , θ3 ) → (−θ1 ,
ω2 − θ2 , θ3 ) 2
(3.1)
and σ2 : (θ1 , θ2 , θ3 ) → (θ1 , −θ2 , −θ3 ).
(3.2)
As the commutator [σ1 , σ2 ] vanishes one can define the action of the Abelian group A := Z2 ⊕ Z2 on C by (α, θ ) → α · θ := (σ1α1 ◦ σ2α2 )(θ ), where α = (α1 , α2 ) ∈ A and θ ∈ C. Consider the equivalence relation ∼A on C defined as follows: The points p, q ∈ C are equivalent p ∼A q iff they belong to the same orbit of A (i.e., there is α ∈ A such that α · p = q). Denote by C˜ the topological quotient C/ ∼A of C with respect to the action of A and let σ : C → C˜
(3.3)
be the corresponding projection. A point p ∈ C is called a regular point of the projection (3.3) iff it is not a fixed point of the action for any 0 = α ∈ A. The points in C that
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are not regular will be called singular or branched points of the projection σ . The set of singular points is given by S := S1 S2 , where ω2 ω1 ω2 S1 := θ1 ≡ 0 mod , θ2 ≡ mod , θ3 : −N ≤ θ3 ≤ N 2 4 2 and S2 :=
ω2 , θ3 = 0 : θ1 ∈ Tω1 . θ1 , θ2 ≡ 0 mod 2
The set S1 ⊂ C has four connected components homeomorphic to the unit interval [0, 1] ⊂ R while S2 ⊂ C has two connected components homeomorphic to T. Lemma 3.1. The space C˜ is homeomorphic to the unit disk D3 in R3 . The map σ : C→C˜ ˜ is a 4-folded branched covering of C. Remark 3.2. The image S˜1 of S1 under the projection σ :C → C˜ is homeomorphic to the disjoint union of two unit intervals and the image S˜2 of S2 is homeomorphic to T. Proof of Lemma 3.1. First consider the action of the involution σ1 on the cylinder C = Tω1 × Tω2 × [−N , N ]. For any value c ∈ [−N , N ] the involution σ1 is acting on the 2-torus T2c := Tω1 × Tω2 × {θ3 = c} by
ω2 σ1 (c) : (θ1 , θ2 ) → −θ1 , − θ2 . 2 The involution σ1 (c) : T2c → T2c has four fixed points and it is easy to see that the topological quotient S2 (c) of T2c with respect to the orbits of the action of σ1 (c) is homeomorphic to the 2-sphere S2 := {x ∈ R3 : |x|2 = 1}. Hence, C1 := (C/ ∼σ1 ) ∼ = {(S2 (c), c) : c ∈ [−N , N ]}.
(3.4)
Under the identification (3.4), the involution σ2 : C1 → C1 becomes S2 × [−N , N ] → S2 × [−N , N ], (x1 , x2 , x3 ; c) → (x1 , x2 , −x3 ; −c). ∼ T, The fixed points of this involution form a submanifold, {(x1 , x2 , 0; 0) : x12 +x22 = 1} = 3 ∼ ∼ ˜ and the corresponding quotient is homeomorphic to D , hence, C = (C1 / ∼σ2 ) = D3 . In what follows we will define a differential structure D on C˜ and a smooth ˜ D) ∼ Riemannian metrics g˜ on the manifold X := (C, = D3 such that the billiard table (X, g) ˜ becomes completely integrable. The branched covering σ : C → C˜ defined above will play an important role in our construction. To this end choose three realvalued C ∞ -smooth functions ϕ1 , ϕ2 : R → R and ϕ3 : [−N , N ] → R satisfying the following properties:
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(A1 ) ϕk (k = 1, 2, 3) is an even function depending only on the variable θk and ϕ1 (θ1 ) ≥ ϕ2 (θ2 ) ≥ 0 ≥ ϕ3 (θ3 ); ϕk (k = 1, 2) is periodic with period ωk ; ω
2 − θ2 , ϕ2 satisfies the additional symmetry ϕ2 (θ2 ) = ϕ2 2 (A2 ) νk := min ϕk = max ϕk+1 (k = 1, 2) and ν1 > ν2 = 0; ωk ; for any k ∈ {1, 2}, ϕk (θk ) = νk iff θk ≡ 0 mod 2 ω2 ω2 ϕ2 (θ2 ) = ν1 iff θ2 ≡ mod ; 4 2 ϕ3 (θ3 ) = ν2 iff θ3 = 0; (A3 ) compatibility conditions: (2l) (2l) (1) for k ∈ {1, 2}, ϕk (0) = ϕk (ωk /2) > 0 and ϕ1 (0) = ϕ1 (ω1 /2);3 (2l) (2l) (2l) (2l) (2) for any l ≥ 0, ϕ1 (0) = (−1)l ϕ2 (ω2 /4) and ϕ2 (0) = (−1)l ϕ3 (0). Consider the following quadratic forms on T C (quadratic on any fiber Tθ C): dg 2 := 1 dθ12 + 2 dθ22 + 3 dθ32
(3.5)
and d I12 := (ϕ2 + ϕ3 )1 dθ12 + (ϕ1 + ϕ3 )2 dθ22 + (ϕ1 + ϕ2 )3 dθ32 , d I22 := (ϕ2 ϕ3 )1 dθ12 + (ϕ2 ϕ3 )2 dθ22 + (ϕ2 ϕ3 )3 dθ32 ,
(3.6)
where 1 := (ϕ1 − ϕ2 )(ϕ1 − ϕ3 ), 2 := (ϕ1 − ϕ2 )(ϕ2 − ϕ3 ), and 3 := (ϕ1 − ϕ3 )(ϕ2 − ϕ3 ). We say also that the forms above are quadratic forms on C. Notice that dg 2 is degenerate, it vanishes on S. Proposition 3.3. Assume that the functions ϕ1 , ϕ2 , ϕ3 satisfy (A1 ) ÷ (A3 ). Then there exists a differential structure D on C˜ such that the projection σ : C → C˜ is smooth and σ is a local diffeomorphism at the regular points. The push-forwards g˜ := σ∗ g and I˜k := σ∗ Ik (k = 1, 2) are smooth quadratic forms and g˜ is a Riemannian metric ˜ D). In addition, the billiard table (X, g) on X := (C, ˜ is completely integrable and the quadratic forms I˜1 , I˜2 , and I˜3 (ξ ) := g(ξ, ˜ ξ )/2, considered as functions on T X are functionally independent and Poisson commuting4 integrals of the billiard flow of g. ˜ Proof of Proposition 3.3. Consider the set S = S1 S2 ⊂ C of branched points of the ˜ Take a point p = (θ 0 , θ 0 , θ 0 ) ∈ S1 and assume for example covering σ : C → C. 1 2 3 ω2 0 0 that θ1 = 0, θ2 = 4 and θ30 ∈ [−N , N ]. Define a new chart V1 = {(x1 , x2 , x3 )} in a neighborhood of p by xk := θk − θk0 , k = 1, 2, and x3 := θ3 , where |xk | < ωk /8 for k = 1, 2 and |x3 | ≤ N . In this chart p = (0, 0, θ30 ) and dg 2 = Q 1 d x12 + Q 2 d x22 + Q 3 d x32 , d I12 d I12
= =
(φ2 + φ3 )Q 1 d x12 + (φ1 + φ3 )Q 2 d x22 + (φ1 + φ2 )Q 3 d x32 , (φ2 φ3 )Q 1 d x12 + (φ1 φ3 )Q 2 d x22 + (φ1 φ2 )Q 3 d x32 ,
(3.7) (3.8) (3.9)
3 Item (A )(i) in [11] has to be written similarly. 5 4 The canonical symplectic structure on T ∗ X induces a symplectic structure on T X by identifying vectors
and covectors by means of the Riemannian metric g. ˜
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where Q 1 := (φ1 − φ2 )(φ1 − φ3 ), Q 2 := (φ1 − φ2 )(φ2 − φ3 ), Q 3 := (φ1 − φ3 ) (φ2 − φ3 ), φk (xk ) = ϕk (θk0 + xk ), k = 1, 2, and φ3 (x3 ) = ϕ3 (x3 ). Note that V1 is a tubular neighborhood of the chosen component of S1 and it does not intersect the other components of S. It follows from (A1 ) ÷ (A3 ) that the functions φ1 , φ2 , and φ3 are smooth and have the following properties in V1 : (L 1 ) φk is even and depends only on the variable xk ; (L 2 ) φ1 > φ3 and φ2 > φ3 ; (L 3 ) φ1 and φ2 satisfy: (i) φ1 > ν1 if x1 = 0 and φ1 (0) = ν1 , φ1 (0) > 0; (ii) φ2 < ν1 if x2 = 0 and φ2 (0) = ν1 , φ2 (0) < 0; (iii) φ1(2l) (0) = (−1)l φ2(2l) (0) for any l ≥ 0. In the new coordinates, the involution σ1 |V1 becomes σ1 |V1 :(x1 , x2 , x3 ) →(−x1 , −x2 , x3 ). In order to define a differential structure in a neighborhood of σ ( p) in C˜ consider the mapping 1 : V1 → Im 1 := W1 , 1 : (x1 , x2 , x3 ) → (y1 = x12 − x22 , y2 = 2x1 x2 , y3 = x3 ).
(3.10)
By Lemma 3.4 below the push-forwards g| ˜ W1 := 1∗ (g|V1 ), I˜1 |W1 := 1∗ (I1 |V1 ), and ˜ W1 is positive definite. I˜2 |W1 : = 1∗ (I2 |V1 ) are smooth quadratic forms on W1 and g| Since 1 ◦ (σ1 |V1 ) = 1 and σ2 (V1 ) ∩ V1 = ∅ we can identify σ |V1 with 1 and get a ˜ In a similar way we construct a differential structure in the neighborhood of σ ( p) ∈ C. tubular neighborhood V2 of the component θ10 = ω2 /2, θ20 = ω42 and θ30 ∈ [−N , N ] of S1 together with a mapping 2 : V2 → W2 such that the push-forward of g|V2 , I1 |V2 , and I2 |V2 are smooth quadratic forms on W2 . Consider also the tubular neighborhoods V3 := σ2 (V1 ) and V4 := σ2 (V2 ) of the other two components of S1 in C˜ together with the mappings 3 := 1 ◦ (σ2 |V3 ) : V3 → W1 and 4 := 2 ◦ (σ2 |V4 ) : V4 → W2 .
(3.11)
For j = 3, 4 one has j ◦ (σ1 |V j ) = j , and therefore we can identify j with σ |V j . As the quadratic forms (3.5) and (3.6) are invariant with respect to σ2 we obtain from (3.11) that 3∗ (g|V3 ) = g| ˜ W1 , 4∗ (g|V4 ) = g| ˜ W2 , 3∗ (I j |V3 ) = I˜j |W1 , and ˜ 4∗ (I j |V4 ) = I j |W2 , j = 1, 2. In particular, the mappings σ |V3 : V3 → W1 and σ |V4 : V4 → W2 and the push-forward of (3.5) and (3.6) with respect to them are smooth. Arguing similarly we treat the case p ∈ S2 and construct a coordinate chart W3 ˜ of S˜2 = σ (S2 ) in C. Covering the image of the branched points of σ by the charts W1 , W2 , and W3 we get ˜ S˜1 S˜2 ) consists of regular a differential structure on 3j=1 W j ⊃ S˜1 S˜2 . As the set C\( points of σ we can induce a differential structure on it from the differential structure of the cylinder C. The union of these two differential structures is compatible and defines a ˜ Denote by X the smooth manifold X = (C, ˜ D). It follows differential structure D on C. from (A1 ) that the forms (3.5) and (3.6) on C are invariant under the involutions (3.1) and (3.2). In particular, the push-forwards g˜ := σ∗ g, I˜1 := σ∗ I1 , and I˜2 := σ∗ I2 are smooth quadratic forms on X \( S˜1 S˜2 ). Moreover, we have seen that the push-forwards g, ˜ I˜1 , and I˜2 are smooth quadratic forms on W j , and that g˜ is a Riemannian metric in W j for any j ∈ {1, 2, 3}. Hence, the push-forwards g, ˜ I˜1 , and I˜2 are smooth quadratic forms on X and g˜ is a Riemannian metric. We will show that I1 and I2 are integrals
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of the billiard flow of the metric g on C\S. Indeed, applying the Legendre transformation pk = k θ˙k , k = 1, 2, 3 (which is well defined only on C\S) and dropping for simplicity the factor 21 in the Hamiltonian we get ⎧ ⎪ H = ⎪ ⎨ I1 = ⎪ ⎪ ⎩ I2 =
1 1 1 2 2 2 1 p 1 + 2 p 2 + 3 p 3 , ϕ2 +ϕ3 2 ϕ1 +ϕ3 2 ϕ1 +ϕ2 2 1 p 1 + 2 p 2 + 3 p 3 , (ϕ2 ϕ3 ) 2 (ϕ1 ϕ3 ) 2 (ϕ1 ϕ2 ) 2 1 p 1 + 2 p 2 + 3 p 3 ,
(3.12)
which can be rewritten in Stäkel form (cf. [14,13, § 2]) ⎧ ⎪ p 2 = ϕ12 H − ϕ1 I1 + I2 , ⎪ ⎨ 1 p22 = −ϕ22 H + ϕ2 I1 − I2 , . ⎪ ⎪ ⎩ 2 p3 = ϕ32 H − ϕ3 I1 + I2 .
(3.13)
In particular, the functions H, I1 , and I2 Poisson commute with respect to the canonical symplectic form ω := dp1 ∧ dθ1 + dp2 ∧ dθ2 + dp3 ∧ dθ3 on the cotangent bundle T ∗ (C\S) (see for example [13, Prop. 1]). Moreover, the forms I1 and I2 are invariant with respect to the reflection map at the boundary ρ : (T C)|∂C → (T C)|∂C given by ρ (θ1 , θ2 , ±N , θ˙1 , θ˙2 , θ˙3 ) −→ (θ1 , θ2 , ±N , θ˙1 , θ˙2 , −θ˙3 ).
Hence I˜1 and I˜2 are Poisson commuting integrals of the billiard flow of the metric g˜ on X \σ (S). As σ (S) is a 1-dimensional submanifold in the 3-manifold X we get that I˜1 and ˜ A direct comI˜2 are Poisson commuting integrals of the billiard flow of the metric g. putation shows that H, I1 and I2 in (3.12) are functionally independent on T ∗ (C\S). Hence, H˜ , I˜1 and I˜2 are functionally independent on T ∗ (X \σ (S)). Lemma 3.4. The quadratic forms g| ˜ W1 = 1∗ (g|V1 ), I˜1 |W1 = 1∗ (I1 |V1 ), and I˜2 |W1 = 1∗ (I2 |V1 ) are smooth and g| ˜ W1 is positive definite. Proof of Lemma 3.4. A direct computation involving (3.10) shows that d g˜ 2 |(W1 \ S˜1 ) = g˜ 11 dy12 + 2g˜ 12 dy1 dy2 + g˜ 22 dy22 + g˜ 33 dy32 , where g˜ 11
1 = 4
φ 1 − φ2 x12 + x22
(φ1 − φ3 )x12 + (φ2 − φ3 )x22
x12 + x22
,
g˜ 12
1 = 4
φ 1 − φ2 x12 + x22
2 x1 x2 , (3.14)
and g˜ 22
1 = 4
φ 1 − φ2 x12 + x22
(φ1 − φ3 )x22 + (φ2 − φ3 )x12 x12 + x22
, g˜ 33 = (φ1 − φ3 )(φ2 − φ3 ). (3.15)
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Let A := φ1 ∂∂x1 ⊗ d x1 + φ2 ∂∂x2 ⊗ d x2 + φ1 ∂∂x3 ⊗ d x3 ∈ C ∞ (T V1 ⊗ T ∗ V1 ). A similar computation as above shows that ∂ ˜ ∂ ˜ ∂ ˜ ∂ ˜ A| (W1 \ S˜1 ) = A11 ∂ y ⊗ dy1 + A12 ∂ y ⊗ dy2 + A21 ∂ y ⊗ dy1 + φ3 ∂ y ⊗ dy3 , 1 1 2 3 where A˜ 11 =
φ1 x12 + φ2 x22 x12
+
x22
,
φ1 − φ2 A˜ 21 = 2 x1 x2 , x1 + x22
φ1 − φ2 A˜ 12 = 2 x1 x2 , x1 + x22 A˜ 22 =
φ1 x22 + φ2 x12 x12 + x22
(3.16)
.
Consider the tensor field A as a section in Hom (T ∗ V1 , T ∗ V1 ). Then we have det(A + c)g((A + c)−1 ξ, ξ ) = c2 g(ξ, ξ ) + cI1 (ξ, ξ ) + I2 (ξ, ξ )
(3.17)
for any c > − max0≤θ1 ≤ω1 ϕ1 (θ1 ). We will show that the coefficients (3.14), (3.15), and (3.16), when re-expressed in terms of the variables (y1 , y2 , y3 ), are smooth in W1 . Then the statement of the lemma will follow from the relation (3.17) and the properties of the Vandermonde determinant. Consider, for example, the function (x1 , x2 ) :=
φ1 (x1 ) − φ2 (x2 ) , (x1 , x2 ) = (0, 0). x12 + x22
Fix m ∈ N, m ≥ 1. Using (L 3 ) and the Taylor formula with an integral reminder term we get φ1 (x1 ) =
m
ak x12k + x12m+1 S1,2m+1 (x1 ),
k=0 m
φ2 (x2 ) = (−1)k ak x22k + x22m+1 S2,2m+1 (x2 ), k=0
where S j,2m+1 , j = 1, 2, are smooth functions in a neighborhood of 0. Lemma 3.5 below implies that (x1 , x2 ) =
m−1
k (y1 , y2 ) + S2m+1 (x1 , x2 ) for (x1 , x2 ) = (0, 0),
k=0
where k (y1 , y2 ) := Pk (y1 , y2 ) for k-odd and k (y1 , y2 ) := Rk (y1 , y2 ) for k-even are homogeneous polynomials of degree 2k with respect to (y1 , y2 ), and S2m+1 (x1 , x2 ) :=
x12m+1 S1,2m+1 (x1 ) − x22m+1 S2,2m+1 (x2 ) x12 + x22
, (x1 , x2 ) = (0, 0).
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Consider the directional derivatives
∂ ∂ ∂ 1 x and = − x 1 2 ∂ y1 ∂ x1 ∂ x2 2(x12 + x22 ) ∂ ∂ ∂ 1 x2 . := = + x1 ∂ y2 ∂ x1 ∂ x2 2(x12 + x22 )
∂ y1 := ∂ y2 We have
lim
(x1 ,x2 )→(0,0)
∂ yα1 ∂ yβ2 S2m+1 (x1 , x2 ) = 0
for α + β ≤ m. Hence, can be extended by continuity to a C ∞ -smooth function in the variables (y1 , y2 ) in a neighborhood of (0, 0) and its Taylor series is ∞ k=0 k (y1 , y2 ). In the case when φ1 and φ2 are real analytic the power series ∞ k=0 k (y1 , y2 ) is uniformly convergent in a neighborhood of (0, 0). Arguing similarly we obtain that the coefficients (3.14)–(3.16) are C ∞ -smooth in the variables (y1 , y2 ) when φ1 and φ2 are smooth and real analytic if φ1 and φ2 are real analytic. Moreover, by Taylor’s formula φ1 (x1 ) = ν1 + a1 x12 + o(x12 ) as x1 → 0 and φ2 (x2 ) = ν1 − a1 x22 + o(x22 ) as x2 → 0 that together with (3.14) and (3.15) implies g˜ 11 = a1 (ν1 −φ3 )+o(1), g˜ 12 = o(1), and g˜ 22 = a1 (ν1 −φ3 )+o(1) as y → ((0, 0, y30 )). Hence, d g˜ 2 |(W1 \ S˜1 ) can be extended by continuity to (0, 0, y30 ) ∈ S˜1 and by (L 2 ) the extension is positive definite. This completes the proof of the lemma. Lemma 3.5. For any m ≥ 2, x12m
−
x22m
+
x22m
=
x12m
=
(x12 + x22 ) Pm−1 (y1 , y2 ), m − even, m − odd, Q m (y1 , y2 ), (x12 + x22 ) Rm−1 (y1 , y2 ), m − odd, m − even, Nm (y1 , y2 ),
where Pm , Q m , Rm , and Nm are polynomials of y1 and y2 of degree m. Proof of Lemma 3.5. Introduce the complex variables z := x1 +i x2 and w := y1 +iy2 and note that w = z 2 . Then, for any m ≥ 2, x12m ±x22m = (z+¯z )2m ± (−1)m (z − z¯ )2m /22m . Finally, using Newton’s binomial formula one concludes the lemma. Following [10] we impose the following additional assumptions on the functions ϕk : (A4 ) ϕ1 (θ1 ) = ϕ1 (ω1 /2 − θ1 ), (A5 ) for any k ∈ {1, 2} the derivative ϕk (θk ) > 0 on (0, ωk /4) and ϕ3 (θ3 ) < 0 on (0, N ]. The condition ϕ3 (N ) < 0 means that the boundary of X is locally geodesically convex. Definition 3.6. The billiard table (X, g) ˜ in Proposition 3.3 is called a Liouville billiard table (L.B.T.). Liouville billiard tables satisfying conditions (A4 ) and (A5 ) are called Liouville billiard tables of classical type. In the case when ϕ1 , ϕ2 , and ϕ3 are real analytic, the billiard table is called analytic L.B.T.
On the Integral Geometry of Liouville Billiard Tables
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The involutions, (θ1 , θ2 , θ3 ) → (−θ1 , θ2 , θ3 ),
ω 1 (θ1 , θ2 , θ3 ) → − θ1 , θ2 , θ3 , 2 (θ1 , θ2 , θ3 ) → (θ1 , −θ2 , θ3 ),
(3.18)
induce a group of isometries G(X )=G(X, g) ˜ on X which is isomorphic to the direct sum G(X ) ∼ = Z2 ⊕ Z2 ⊕ Z2 . Remark 3.7. The action of G(X ) on (X, g) ˜ is an analog of the action of the group Z2 ⊕ Z2 ⊕ Z2 in the interior of the ellipsoid in R3 = {(x, y, z)} generated by the reflections with respect to the coordinate planes Ox y , O yz and Ox z . (2l)
(2l)
Remark 3.8. The compatibility conditions ϕ1 (0) = ϕ1 (ω1 /2), l = 0, 1, . . ., in (A3 ) follow from (A4 ) for L.B.T.s of classical type. 3.2. Ellipsoidal billiard tables. Denote by R3 the Euclidean space R3 = {(x1 , x2 , x3 )} supplied with the standard Euclidean metric dg02 := d x12 + d x22 + d x32 . A class of L.B.T.s in R3 depending on 3 real parameters b1 > b2 > b3 can be obtained using the mapping: ⎧ 3 ⎪ x = (b1 − λ2 ) bb11 −λ ⎪ 1 ⎪ −b3 cos φ1 ⎨ √ 0 : x2 = b2 − b3 sin φ1 cos φ2 , ⎪ ⎪ ⎪ ⎩ x3 = b3 −λ1 φ3 sin φ2 b3 −b1 where λk := bk+1 + (bk − bk+1 ) sin2 φk (k = 1, 2), λ3 := b3 − φ32 , φk (k = 1, 2) are periodic coordinates with period 2π , and −N ≤ φ3 ≤ N . The mapping 0 : T2 × [−N , N ] → R3 gives a 4-folded branched covering of an ellipsoidal domain X in R3 and (X, g0 ) is a L.B.T. of classical type – for details see § 5 in [11]. More generally, the two-parameter family of billiard tables (M 3 , gα,β ) of constant scalar curvature κ in [11, Theorem 3] consists of L.B.T.s of classical type according to § 5.4 in [11]. The boundary of any billiard table of the family is geodesically equivalent to the ellipsoid. In particular, it has non-periodic geodesics and satisfies the hypothesis of Theorem 1 and Theorem 3. This family contains the ellipsoid (κ = 0) and L.B.T.s of both positive and negative scalar curvature that are realized on the standard sphere and on the hyperbolic space respectively. 3.3. Parameterization of the Lagrangian tori. The aim of this section is to obtain charts of action-angle coordinates for L.B.T.s of classical type and to parameterize the corresponding Liouville tori. Recall that a L.B.T. (X, g) ˜ is obtained as a quotient space of the cylinder C = {(θ1 (mod ω1 ), θ2 (mod ω2 ), θ3 )} ∼ = Tω1 × Tω2 × [−N , N ] with respect to the group action of A = Z2 ⊕Z2 as described in Sect. 3.1. By Proposition 3.3, the projection σ : C → X is smooth and invariant with respect to the group action
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of A on C. Moreover, the push-forwards of the quadratic forms (3.6) with respect to the projection σ : C → X are integrals of the billiard flow on (X, g). ˜ The boundary ∂C of C has two connected components defined by θ3 = ±N and we set T2N := {(θ1 (mod ω1 ), θ2 (mod ω2 ), θ3 = N )}. By construction the restriction σ |T2 of the projection σ : C → X to T2N is a double N branched covering of the boundary = ∂ X . Denote Cr := C\S and introduce on T ∗ C the coordinates {(θ1 , θ2 , θ3 ; p1 , p2 , p3 )}, where p1 , p2 , and p3 are the conjugated impulses. The Legendre transformation corresponding to the Lagrangian L g (ξ ) := g(ξ, ξ )/2, ξ ∈ T Cr , transforms the Lagrangian and the integrals (3.6) to the functions H, I1 and I2 on T ∗ Cr given by (3.12).5 Set Q 1 := T ∗ Cr |T2 = {η = (θ, p) ∈ T ∗ Cr : θ3 = N }. N
The restriction ω˜ 1 of the symplectic two-form ω = dp1 ∧ dθ1 + dp2 ∧ dθ2 + dp3 ∧ dθ3 to Q 1 is ω˜ 1 := ω| Q 1 = dp1 ∧ dθ1 + dp2 ∧ dθ2 . This form is degenerate and its kernel Kerω˜ 1 is spanned on the vector field ∂∂p3 . Denote by Q the isoenergy surface Q := {η ∈ T ∗ Cr : H (η) = 1}, and consider the set Q 2 := Q ∩ Q 1 . It is clear that Q 2 is diffeomorphic to the restriction of the unit cosphere bundle Sg∗ Cr of Cr to the torus T2N . The set Q +2 := {η = (θ, p) ∈ Q 2 : p3 < 0}
can be identified with the set S+∗ Cr |T2 of all η in Sg∗ Cr |T2 such that η, n g > 0, where n g N
N
denotes the inward unit normal to T2N \S1 . Moreover, the open coball bundle Bg∗ (T2N \S1 ) can be identified with p12 p22 ∗ 2 + (3.19) |θ3 =N < 1 , (θ1 , θ2 ; p1 , p2 ) ∈ T (T \S ) : 1 2 where S := {(θ1 ≡ 0 (mod → Q +2 given by
ω1 2 ), θ2
≡
ω2 4
(mod
ω2 2 )}. Consider the map
R : Bg∗ (T2N \S1 )
p2 p2 R : (θ1 , θ2 ; p1 , p2 ) → (θ1 , θ2 ; p1 , p2 , p3 ) where p3 = − 3 1 − 1 − 2 . 1 2
The coball bundle Bg∗ (T2N \S1 ) can be considered as a phase space of the billiard ball map B : B ∗ (\σ (S)) → B ∗ (\σ (S)) via the branched double covering σ |T2 : T2N → . N In this setting the map R can be identified with π + . We have also ω˜ 2 := R ∗ ω˜ 1 = dp1 ∧ dθ1 + dp2 ∧ dθ2 . Moreover, the functions I1 := R ∗ I1 and I2 := R ∗ I2 are functionally independent integrals of B in Bg∗ (T2N \S1 ). 5 For simplicity we drop the factor 1 in the Hamiltonian function. 2
On the Integral Geometry of Liouville Billiard Tables
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In the coordinates {(θ1 , θ2 ; p1 , p2 )} the integrals I1 and I2 become (cf. (3.12)) p12 p2 − (ϕ2 − ν3 ) 2 , 1 2 p12 p2 I2 = ϕ1 ϕ2 − ϕ2 (ϕ1 − ν3 ) − ϕ1 (ϕ2 − ν3 ) 2 , 1 2 I1 = (ϕ1 + ϕ2 ) − (ϕ1 − ν3 )
(3.20) (3.21)
where ν3 := ϕ3 (N ) < ν2 = 0 in view of (A1 ) and (A2 ). In order to describe the invariant manifolds of the billiard ball map B we choose real constants h 1 and h 2 and consider the level set L˜ h := {I1 = h 1 , I2 = h 2 } ⊂ Bg∗ (T2N \S1 ), h = (h 1 , h 2 ). Consider the quadratic polynomial, P(t) := t 2 − h 1 t + h 2 = (t − κ1 )(t − κ2 ),
(3.22)
where κ1 and κ2 are the roots of P and h 1 = κ1 +κ2 , h 2 = κ1 κ2 . If (θ1 , θ2 ; p1 , p2 ) ∈ L˜ h , it follows from (3.13) that P(ϕ1 (θ1 )) = ϕ12 (θ1 ) − h 1 ϕ1 (θ1 ) + h 2 = p12 ≥ 0 , −P(ϕ2 (θ2 )) =
−ϕ22 (θ2 ) + h 1 ϕ2 (θ2 ) −
h2 =
p22
≥ 0,
(3.23) (3.24)
and P(ϕ3 (N )) = P(ν3 ) = ν32 − h 1 ν3 + h 2 ≥ 0.
(3.25)
Then the set L˜ h is non-empty if and only if there is a point (θ1 , θ2 ) ∈ (R/ω1 Z)×(R/ω2 Z) such that the inequalities (3.23), (3.24), and (3.25) are satisfied. In particular, it follows from (3.23) and (3.24) that the roots κ1 ≤ κ2 are real, hence, D := h 21 − 4h 2 ≥ 0. Moreover, (A1 ) ÷ (A2 ) imply ⎧ ⎪ ⎨ ν1 ≤ ϕ1 (θ1 ) ≤ ν0 , (3.26) 0 = ν2 ≤ ϕ2 (θ2 ) ≤ ν1 , ⎪ ⎩ ν3 = ϕ3 (N ) < ν2 = 0, where ν0 := max ϕ1 . Then by inspection one sees that only the following four cases can occur: ν3 ≤ κ1 ≤ ν2 = 0 and 0 = ν2 ≤ κ2 ≤ ν1 ; ν3 ≤ κ1 ≤ ν2 = 0 and ν1 ≤ κ2 ≤ ν0 ; 0 = ν2 ≤ κ1 ≤ κ2 ≤ ν1 ; 0 = ν2 ≤ κ1 ≤ ν1 and ν1 ≤ κ2 ≤ ν0 . Consider the union U˜ 1 of all L˜ h in B ∗ (T2N \S1 ) such that (A) with strict inequalities holds for the corresponding (κ1 , κ2 ). We will see below that any L˜ h in U˜ 1 is a disjoint union of Liouville tori. In the same way we define U˜ 2 corresponding to (B), U˜ 3 corresponding to (C) and U˜ 4 corresponding to (D). Denote U j := σ∗ (U˜ j ) ⊂ B ∗ , j = 1, 2, 3, 4, where σ∗ is the push-forward of covectors corresponding to σ : C → X . (A) (B) (C) (D)
Definition 3.9. We refer to cases (A) and (B) as boundary cases and denote Fb := U1 ∪ U2 .
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Remark 3.10. We will see in Sect. 5 that the billiard trajectories in T ∗ X issuing from U1 ∪ U2 “approximate” the geodesics on the boundary . We are going to parameterize the invariant tori belonging to the level set L˜ h . To that end we need the inverse functions of ϕ1 |[0,ω1 /4] and ϕ2 |[0,ω2 /4] . According to (A1 ) ÷ (A5 ) the function ϕ1 has the following properties. It is a periodic function of period ω1 /2, ϕ1 (ω1 /4 + θ1 ) = ϕ1 (ω1 /4 − θ1 ) for any θ1 , the map ϕ1 : [0, ω1 /4] −→ [ν1 , ν0 ] is a homeomorphism, ϕ1 (θ1 ) > 0 in the interval (0, ω1 /4), and the critical points of ϕ1 at θ1 = 0 and θ1 = ω1 /4 are non degenerate. Denote by f 1 : [ν1 , ν0 ] −→ [0, ω1 /4] the inverse map of ϕ1 |[0,ω1 /4] . Then f 1 is smooth in (ν1 , ν0 ), f 1 > 0 in that interval, and √ f 1 (x1 ) = F1+ ( x1 − ν1 ) as x1 → ν1 + 0, (3.27) √ f 1 (x1 ) = F1− ( ν0 − x1 ) as x1 → ν0 − 0, where F1∓ are smooth functions in a neighborhood of 0, and F1+ (0) = 0, F1− (0) = ω1 /4, (F1+ ) (0) = 2ϕ1 (0)−1 and (F1− ) (0) = − −2ϕ1 (ω1 /4)−1 .
(3.28)
The function ϕ2 |[0,ω2 /4] has the same properties, and we denote by f 2 : [0, ν1 ] −→ [0, ω2 /4] its inverse function. Then f 2 is smooth in (0, ν1 ) and f 2 > 0 in that interval, and √ as x2 → 0 + 0, f 2 (x2 ) = F2+ ( x2 ) (3.29) √ f 2 (x2 ) = F2− ( ν1 − x2 ) as x2 → ν1 − 0, where F2∓ are smooth functions in a neighborhood of 0 and F2+ (0) = 0, F2− (0) = ω2 /4, (F2+ ) (0) = (F2− ) (0) = − −2ϕ2 (ω2 /4)−1 .
2ϕ2 (0)−1 , (3.30)
Assume that L˜ h ⊂ U˜ 1 . We have ν3 < κ1 < 0 and 0 < κ2 < ν1 . It follows from (3.23)– (k) (3.24) and (3.26) that L˜ h consists of four connected components Th (1 ≤ k ≤ 4) which (k) are diffeomorphic to T2 . Moreover, the image of each Th with respect to the bundle projection T ∗ T2N → T2N coincides with one of the annuli
Ah := {0 ≤ θ1 ≤ ω1 ; − f 2 (κ2 ) ≤ θ2 ≤ f 2 (κ2 )} and
Ah := {0 ≤ θ1 ≤ ω1 ; ω2 /2 − f 2 (κ2 ) ≤ θ2 ≤ ω2 /2 + f 2 (κ2 )}. Assume that the tori Th(1) and Th(2) are projected onto Ah and similarly, Th(3) and Th(4) are projected onto Ah . As the map σ |T2 : T2N → is invariant with respect to the N involution
ı : (θ1 , θ2 ) → (−θ1 , ω2 /2 − θ2 ),
On the Integral Geometry of Liouville Billiard Tables
(1)
(2)
739 (3)
(4)
and ı(Ah ) = Ah , the pairs (Th , Th ) and (Th , Th ) correspond to the same pair of (1) (2) invariant tori in T ∗ , which we identify with (Th , Th ). It follows from (3.23), (3.24) (1) and (3.26) that the map r1 2 : Ah → Th defined by
r1 2 (θ1 , θ2 ) −→ θ1 , θ2 ; 1 ϕ1 (θ1 )2 − h 1 ϕ1 (θ1 ) + h 2 , 2 −ϕ2 (θ2 )2 + h 1 ϕ2 (θ2 ) − h 2 , (3.31) (1)
gives a parametrization of the torus Th for 1 = 1 and 2 = ±1. In the same way, (2) taking 1 = −1 and 2 = ±1 we parametrize Th . In the same way one treats the cases (B), (C) and (D). In particular, one gets that U˜ 1 , U˜ 2 , and U˜ 3 have 4 connected components while U˜ 4 has 8 connected components. Similarly, U1 , U2 , and U3 have 2 connected components and U4 has 4 connected components. 4. R-Rigidity We are going to prove that Liouville billiard tables of classical type are R-rigid with respect to the densities μ defined by (1.1). Theorem 4.1. Let (X, g) ˜ be a Liouville billiard table of classical type and let K ∈ C(, R) be invariant with respect to the action of the group G(X ) on . Suppose that R K ,μ ( ) = 0 for any Liouville torus ⊂ Fb . Then K ≡ 0. Proof of Theorem 4.1. First, consider the case when μ ≡ 1. Denote the pull-back of K under the projection σ |T2 : T2N → by K, K ∈ C(T2N ). Let h ∈ Fb be a N
Liouville torus and let Th be a connected component of (σ |T2 )∗ h ⊂ B ∗ (T2N ), where N h = (h 1 , h 2 ) are the values of the integrals I1 and I2 on Th . Then we have −1 (λh (Th )) Kλh = 2R K ,1 ( h ) = 0, Th
where λh is the corresponding Leray’s form on Th . Note that K is invariant under the involution (θ1 , θ2 ) → (−θ1 , ω22 − θ2 ) since σ |T2 is invariant under the involution (3.1) N for θ3 = N . Recall that the group G(X ) ∼ = Z2 ⊕ Z2 ⊕ Z2 defined by (3.18) acts by isometries on X and on its boundary . Since K is invariant under this action, the function K(θ1 , θ2 ) is invariant with respect to the involutions (θ1 , θ2 ) → (−θ1 , θ2 ), (θ1 , θ2 ) → (ω1 /2 − θ1 , θ2 )
(4.1)
(θ1 , θ2 ) → (θ1 , −θ2 ), (θ1 , θ2 ) → (θ1 , ω2 /2 − θ2 ).
(4.2)
and
From now on we consider K ∈ C(T2N , R) which is invariant with respect to the involutions (4.1) and (4.2) and such that for any Kλh = 0 ∀ Th ∈ Fb . (4.3) Th
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First, take h = (h 1 , h 2 ) and assume, for example, that Th ⊂ U˜ 1 . We shall give an explicit formula for the Leray form on the connected components of L˜ h , using the parameter (1) ization obtained in Sect. 3.3. Set Th := Th , and let Th+ be the “half torus” r11 (Ah ), where the map r11 is defined by (3.31). Consider the set
Ah (δ) := {0 ≤ θ1 ≤ ω1 ; − f (κ2 ) + δ ≤ θ2 ≤ f (κ2 ) − δ}, where δ > 0 is sufficiently small. It follows from (3.31) that the functions (θ1 , θ2 , I1 , I2 ) give a coordinate chart in a neighborhood of the branch Th+ (δ) := r11 (Ah (δ)) ⊆ Th+ . We will compute the Leray form on it. In the coordinates {(θ1 , θ2 ; p1 , p2 )} on B ∗ T2N we have ω˜ 2 ∧ ω˜ 2 = 2 d p1 ∧ dθ1 ∧ d p2 ∧ dθ2 = 2 d( ϕ1 (θ1 )2 − I1 ϕ1 (θ1 ) + I2 ) ∧ dθ1 ∧ d( −ϕ2 (θ2 )2 + I1 ϕ2 (θ2 ) − I2 ) ∧ dθ2 =−
(ϕ1 (θ1 ) − ϕ2 (θ2 )) dθ1 ∧ dθ2 1 ∧ dI1 ∧ dI2 . 2 ϕ1 (θ1 )2 − I1 ϕ1 (θ1 ) + I2 −ϕ2 (θ2 )2 + I1 ϕ2 (θ2 ) − I2
In particular, letting δ → 0 + 0 we see that the Leray form on Th+ can be identified with λh :=
(ϕ1 (θ1 ) − ϕ2 (θ2 )) dθ1 ∧ dθ2 . ϕ1 (θ1 )2 − h 1 ϕ1 (θ1 ) + h 2 −ϕ2 (θ2 )2 + h 1 ϕ2 (θ2 ) − h 2
(4.4)
We have
f 2 (κ2 ) ω1
K λh = 2 Th
− f 2 (κ2 ) 0 f 2 (κ2) ω1 /4
= 16
K(θ1 , θ2 )(ϕ1 (θ1 ) − ϕ2 (θ2 )) dθ1 dθ2 √ √ (ϕ1 (θ1 ) − κ1 )(ϕ1 (θ1 ) − κ2 ) (ϕ2 (θ2 ) − κ1 )(κ2 − ϕ2 (θ2 ))
0
0
K(θ1 , θ2 )(ϕ1 (θ1 ) − ϕ2 (θ2 )) dθ1 dθ2 , √ √ (ϕ1 (θ1 ) − κ1 )(ϕ1 (θ1 ) − κ2 ) (ϕ2 (θ2 ) − κ1 )(κ2 − ϕ2 (θ2 ))
as the functions K, ϕ1 , and ϕ2 are invariant with respect to the involutions (4.1) and (4.2). Set K˜ (θ1 , θ2 ) := K(θ1 , θ2 )(ϕ1 (θ1 ) − ϕ2 (θ2 )) and denote f 2 (κ2) ω1 /4
M A (κ1 , κ2 ) := 0
0
K˜ (θ1 , θ2 ) dθ1 dθ2 , √ √ (ϕ1 (θ1 ) − κ1 )(ϕ1 (θ1 ) − κ2 ) (ϕ2 (θ2 ) − κ1 )(κ2 − ϕ2 (θ2 )) (4.5)
where the subscript in M A refers to the case (A). Then (4.3) implies M A (κ1 , κ2 ) = 0 for any κ1 ∈ (−ν3 , 0) and any κ2 ∈ (0, ν1 ).
Remark 4.2. Note that for any fixed κ2 ∈ (0, ν1 ) the function κ1 → M A (κ1 , κ2 ) can be extended to an analytic (possibly multivalued) function on C\([0, κ2 ][ν1 , ν0 ]). Since it vanishes for κ1 ∈ (−ν3 , 0) we obtain that M A (κ2 , κ1 ) ≡ 0, ∀κ1 ∈ C\([0, κ2 ] [ν1 , ν0 ]) and ∀κ2 ∈ (0, ν1 ).
On the Integral Geometry of Liouville Billiard Tables
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Set K˜ 1 (x1 , x2 ) := K˜ ( f 1 (x1 ), f 2 (x2 )) f 1 (x1 ) f 2 (x2 ). It follows from (3.27) and (3.29) that K˜ 1 ∈ L 1 ((ν1 , ν0 )×(0, ν1 )). More precisely, (3.27) and (3.29) imply Lemma 4.3. We have K˜ ( f 1 (x1 ), f 2 (x2 ))F(x1 , x2 ) K˜ 1 (x1 , x2 ) = √ , √ √ √ x1 − ν1 ν0 − x1 x2 ν1 − x2
(4.6)
where the function (x1 , x2 ) → K ( f 1 (x1 ), f 2 (x2 )) is continuous on [ν1 , ν0 ] × [0, ν1 ], the function F ∈ C([ν1 , ν0 ] × [0, ν1 ]) does not dependent on K˜ and F > 0. Passing to the variables x1 = ϕ1 (θ1 ) and x2 = ϕ2 (θ2 ) in (4.5) we get M A (κ1 , κ2 ) =
ν0
κ2
√
ν1 0
K˜ 1 (x1 , x2 ) d x2 d x1 ≡0 √ (x1 − κ1 )(x2 − κ1 ) (x1 − κ2 )(κ2 − x2 )
(4.7)
for any κ1 ∈ (−∞, 0) and any κ2 ∈ (0, ν1 ). Consider now the case (B). Arguing in the same way we obtain M B (κ1 , κ2 ) :=
1 16
Kλh = Th
ν1
ν0
√
κ2
0
K˜ 1 (x1 , x2 ) d x1 d x2 ≡0 √ (x1 − κ1 )(x2 − κ1 ) (x1 − κ2 )(κ2 − x2 ) (4.8)
for any κ1 ∈ (−∞, 0) and κ2 ∈ (ν1 , ν0 ). In the same way one obtains: Case (C). For any 0 < κ1 < κ2 < ν1 , MC (κ1 , κ2 ) :=
Kλh = Th
κ2
ν0
κ1 ν1
√
K˜ 1 (x1 , x2 ) d x1 d x2 . (4.9) √ (x1 − κ1 )(x2 − κ1 ) (x1 − κ2 )(κ2 − x2 )
Case (D). For any κ1 ∈ (0, ν1 ) and κ2 ∈ (ν1 , ν0 ), M D (κ1 , κ2 ) :=
Kλh = Th
ν1
ν0
κ1 κ2
√
K˜ 1 (x1 , x2 ) d x1 d x2 . √ (x1 − κ1 )(x2 − κ1 ) (x1 − κ2 )(κ2 − x2 ) (4.10)
Remark 4.4. In what follows we will not use the identities (4.9) and (4.10). Now, we argue as follows: Take a continuous function χ on the interval [0, ν1 ] and consider the mean ν1 M A (ϕ; κ1 ) := M A (κ1 , κ2 )χ (κ2 ) dκ2 , 0
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where M A (κ1 , κ2 ) is given by (4.7). In view of Lemma 4.3, we can apply Fubini’s theorem to the following integral: ν1 κ2 ν0 K˜ 1 (x1 , x2 )χ (κ2 ) d x1 d x2 0 ≡ M A (ϕ; κ1 ) = dκ2 √ √ (x1 − κ1 )(x2 − κ1 ) (x1 − κ2 )(κ2 − x2 ) 0 0 ν1 ν1 ν1 ν0 χ (κ2 ) dκ2 K˜ 1 (x1 , x2 ) = (4.11) d x1 d x2 √ √ 0 ν1 (x 1 − κ1 )(x 2 − κ1 ) x2 (x 1 − κ2 )(κ2 − x 2 ) for any κ1 ∈ (−∞, 0) and any χ ∈ C([0, ν1 ]). Similarly, consider the mean ν0 M B (ϕ; κ1 ) := M B (κ1 , κ2 )χ (κ2 ) dκ2 , ν1
where χ is a continuous function on the interval [ν1 , ν0 ]. We obtain as above ν0 ν1
K˜ 1 (x1 , x2 )χ (κ2 ) d x1 d x2 dκ2 √ √ (x1 − κ1 )(x2 − κ1 ) (x1 − κ2 )(κ2 − x2 ) ν1 0 κ2 x1 ν1 ν0 χ (κ2 ) dκ2 K˜ 1 (x1 , x2 ) = d x1 d x2 (4.12) √ √ 0 ν1 (x 1 − κ1 )(x 2 − κ1 ) ν1 (x 1 − κ2 )(κ2 − x 2 )
0 ≡ M B (ϕ; κ1 ) =
ν1
for any κ1 ∈ (−∞, 0). Finally, combining (4.11) and (4.12) we obtain for any χ ∈ C([0, ν0 ]) and any κ1 ∈ (−∞, 0) the equality ν1 0
ν0 ν1
√
x1 χ (κ2 ) dκ2 K˜ 1 (x1 , x2 ) d x1 d x2 ≡ 0. √ (x1 − κ1 )(x2 − κ1 ) x2 (x1 − κ2 )(κ2 − x2 )
(4.13)
In particular, for any k ≥ 0 and for any κ1 ∈ (−∞, 0), ν1 0
K˜ 1 (x1 , x2 ) Rk (x1 , x2 ) d x1 d x2 ≡ 0, √ ν1 (x 1 − κ1 )(x 2 − κ1 ) ν0
(4.14)
where Rk (x1 , x2 ) :=
x1
√
x2
z k dz = (x1 − z)(z − x2 )
1
0
(x2 + t (x1 − x2 ))k dt. √ t (1 − t)
(4.15)
Recall that the Legendre polynomials Pk , k ≥ 0, can be generated by the power series expansion, (1 − 2ωz + z 2 )−1/2 =
∞
Pk (ω) z k ,
(4.16)
k=0
which√is convergent for small z. For 0 < x2 ≤ x1 we set s1 := (x1 + x2 )/2 and s2 := x1 x2 . Lemma 4.5. For any k ≥ 0 and for any 0 < x2 ≤ x1 , Rk (x1 , x2 ) = π s2k Pk (s1 /s2 ).
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Proof. For any given values of x1 and x2 , 0 < x2 ≤ x1 , consider the power series in z, I (z) :=
∞
Rk (x1 , x2 ) z k .
k=0
There exists 0 < r < ∞ sufficiently small such that the power series converges for |z| ≤ r and 1 1 1 t dt. I (z) = t 1 − z(x + t (x − x )) 1 − t 2 1 2 0 t Using the substitution, s = 1−t we get I (z) = √
π , (1 − zx1 )(1 − zx2 )
and by (4.16) we obtain I (z) = π
∞
s2k Pk (s1 /s2 ) z k ,
k=0
which proves the lemma.
Note that the function (x1 , x2 ) → Q(x1 , x2 , κ1 ) := √
K˜ 1 (x1 , x2 ) (x1 − κ1 )(x2 − κ1 )
belongs to L 1 ([ν1 , ν0 ] × [0, ν1 ]) in view of Lemma 4.3, and it depends analytically on κ1 ∈ (−∞, 0). Consider the power series expansion ∞
1 −j = Q j (x1 , x2 )κ1 , √ (x1 − κ1 )(x2 − κ1 ) k=1
(4.17)
where (x1 , x2 ) ∈ [ν1 , ν0 ] × [0, ν1 ] and κ1 < 0. Now (4.16) implies Q j (x1 , x2 ) = j−1 −s2 P j−1 (s1 /s2 ) for any j ≥ 1. Using Lemma 4.5, (4.14) and (4.17) we obtain that for any k, j ≥ 0, ν1 ν0 k+ j (4.18) K˜ 1 (x1 , x2 )s2 Pk (s1 /s2 )P j (s1 /s2 ) d x1 d x2 = 0. 0
ν1
Let k and m be non-negative integers such that 2k ≤ m and let d be the integer part of m/2. We have the following relation due to Adams (see [1,16, Chap. XV, Legendre functions, Miscellaneous Examples, Ex. 11]), Pk (z)Pm−k (z) =
k
r =0
m ck,r Pm−2r (z) =
d
r =0
m ck,r Pm−2r (z),
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where for any 0 ≤ r ≤ d, m ck,r :=
with
Ak−r Ar Am−k−r 2m − 4r + 1 , . Am−r 2m − 2r + 1
⎧ 1.3.5...(2k−1) ⎪ , k ≥ 1, ⎨ k Ak := 1, k = 0, ⎪ ⎩ Ak = 0, k ≤ −1.
m )d Hence, for any given m ≥ 0 we obtain a (d + 1) × (d + 1) matrix (ck,r k,r =0 which is triangular (all the elements over the diagonal vanish) and with non-vanishing diagonal elements. This together with (4.18) (take m = j + k) implies that for any m ≥ 0, 0 ≤ 2r ≤ m, ν1 ν0 K˜ 1 (x1 , x2 )s2m Pm−2r (s1 /s2 ) d x1 d x2 = 0. ν1
0
On the other hand, for any m ≥ 0 the monomial z m can be written as a linear combination of the Legendre polynomials Pm−2r (z), 0 ≤ 2r ≤ m, and we get ν1 ν0 (4.19) K˜ 1 (x1 , x2 )s1m−2r s22r d x1 d x2 = 0. ν1
0
Consider the set of monomials M = {s1m−2r s22r : r, m ∈ Z, 0 ≤ 2r ≤ m}. Obviously M is closed under multiplication, 1 ∈ M, and it separates the points (x1 , x2 ) of the compact [ν1 , ν0 ] × [0, ν1 ], since s1 , s22 ∈ M and 0 ≤ x2 ≤ x1 are the unique solutions of x 2 − 2s1 x + s22 = 0. The Stone-Weierstrass theorem implies that the vector space Span(M) of all finite linear combinations of monomials of M is dense in C([ν1 , ν0 ] × [0, ν1 ]). Choose ψ ∈ C([ν1 , ν0 ] × [0, ν1 ]). Then for any ε > 0 there is P ∈ Span(M) such that P − ψC([ν1 ,ν0 ]×[0,ν1 ]) < ε. Now (4.19) implies ν1 0
≤
ν0
K˜ 1 (x1 , x2 )ψ(x1 , x2 )d x1 d x2 ν1 ν1 ν0
| K˜ 1 (x1 , x2 )||ψ(x1 , x2 ) − P(x1 , x2 )|d x1 d x2
ν1
0
≤ ε K˜ 1 L 1 ((ν1 ,ν0 )×(0,ν1 )) . Hence, 0
ν1 ν0 ν1
K˜ 1 (x1 , x2 )ψ(x1 , x2 )d x1 d x2 = 0
for any ψ ∈ C([ν1 , ν0 ] × [0, ν1 ]) which implies K˜ 1 ≡ 0 on that compact. In particular, K ≡ 0, and hence K ≡ 0. This completes the proof when μ ≡ 1.
On the Integral Geometry of Liouville Billiard Tables
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Now, consider the case when μ(ξ ) = π + (ξ ), n g −1 . Assume that ξ ∈ h , where h is a Liouville torus in Fb and h = (h 1 , h 2 ) are the values of the integrals I˜1 and I˜2 on h . Let h ⊂ U1 . Using the mapping (3.31), we introduce coordinates {(θ1 , θ2 )} on the (1) (1) (1) (2) “half" tori Th+ = r11 (Ah ) and Th− = r1,−1 (Ah ) of Th as well as on Th+ = r−11 (Ah ) (2)
(2)
and Th− = r−1,−1 (Ah ) of Th . Similarly, we parametrize the Liouville tori h in U2 . Lemma 4.6. In coordinates {(θ1 , θ2 )} on h ⊂ Fb , μ(ξ ) = π + (ξ ), n g −1 is given by μ(θ1 , θ2 ) =
√ (ϕ1 (θ1 ) − ν3 )(ϕ1 (θ1 ) − ν3 ) , √ (κ1 − ν3 )(κ2 − ν3 )
(4.20)
where κ1 = h 1 + h 1 and κ2 = h 1 h 2 . Proof of Lemma 4.6. Fix h = (h 1 , h 2 ) so that h ⊂ Fb . It follows from (3.5) that ∂ 1 ng = − √ . 3 ∂θ3 On the other hand, the third equation in (3.13) shows that p3 = − ν32 − h 1 ν3 + h 2 = − (κ1 − ν3 )(κ2 − ν3 ). Hence, π + (ξ ), n g =
√
√ (κ1 − ν3 )(κ2 − ν3 )/ 3 .
Remark 4.7. The statement of Lemma 4.6 holds also for any h ∈ F not necessarily in Fb . Note that the denominator in (4.20) is a positive constant on Th and the numerator is independent of h 1 and h 2 and does not vanish. The relation (4.3) with K˜ (θ1 , θ2 ) = K(θ1 , θ1 )(ϕ1 (θ1 ) − ϕ2 (θ2 )) (ϕ1 (θ1 ) − ν3 )(ϕ1 (θ1 ) − ν3 ) implies that the expression (4.5) vanishes. In particular, (4.7) and (4.8) hold. Finally, arguing in the same way as in the case μ ≡ 1 one concludes that K ≡ 0.
5. Non-Degeneracy of the Frequency Map In this section we investigate the non-degeneracy of the frequency map of Liouville billiard tables of classical type. Theorem 5.1. Let (X, g) be an analytic 3-dimensional Liouville billiard table of classical type. Suppose that there is at least one non-periodic geodesic on . Then the frequency map is non-degenerate in the union U1 ∪ U2 corresponding to the boundary cases (A) and (B).
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Proof. As in Sect. 3.3 we introduce coordinates {(θ1 , θ2 , θ3 ; p1 , p2 , p3 )} on the cotangent bundle T ∗ C, where p1 , p2 , p3 are the conjugate variables to θ1 , θ2 , and θ3 . Solving the system of equations (3.12) with respect to p12 , p22 , and p32 , where H = 1, I1 = h 1 , and I2 = h 2 are given values of the integrals, we get ⎧ 2 2 ⎪ ⎨ p1 = ϕ1 − h 1 ϕ1 + h 2 (5.1) p22 = −(ϕ22 − h 1 ϕ2 + h 2 ) ⎪ ⎩ 2 p3 = ϕ32 − h 1 ϕ3 + h 2 . In particular, it follows from (5.1) that the invariant set Th := {H = 1, I1 = h 1 , I2 = h 2 } ⊂ T ∗ C
(5.2)
is non-empty if and only if the quadratic polynomial P(t) = t 2 − h 1 t + h 2 has real roots κ1 ≤ κ2 (i.e., D = h 21 − 4h 2 ≥ 0). As in Sect. 4 we obtain four cases related to the position of the roots κ1 and κ2 with respect to the constants ν3 < ν2 = 0 < ν1 < ν0 , namely, (A) (B) (C) (D)
ν3 ≤ κ1 ≤ ν2 = 0 and 0 ≤ κ2 ≤ ν1 ; ν3 ≤ κ1 ≤ 0 and ν1 ≤ κ2 ≤ ν0 ; 0 ≤ κ1 ≤ κ2 ≤ ν1 ; 0 ≤ κ1 ≤ ν1 and ν1 ≤ κ2 ≤ ν0 .
Recall that ν3 = min ϕ3 , ν2 = max ϕ3 = min ϕ2 = 0, ν1 = max ϕ2 = min ϕ1 , and ν0 = max ϕ1 . In what follows we consider κ1 and κ2 as new parameters (constants of motion)6 that parametrize the invariant set (5.2). We first consider Case (A) where ν3 ≤ κ1 ≤ ν2 = 0 and 0 = ν2 ≤ κ2 ≤ ν1 . It follows from (5.1) that the impulses p1 , p2 , and p3 are real-valued if and only if ⎧ ⎪ ⎨ ν1 ≤ ϕ1 (θ1 ) ≤ ν0 (5.3) 0 ≤ ϕ2 (θ2 ) ≤ κ2 ⎪ ⎩ ν3 ≤ ϕ3 (θ3 ) ≤ κ1 . Hence, the projection of the invariant set (5.2) onto the base C is described by the following inequalities: 0 ≤ θ1 ≤ ω1 ; − f 2 (κ2 ) ≤ θ2 ≤ f 2 (κ2 ) or
ω2 ω2 ≤ θ2 ≤ f 2 (κ2 ) + ; 2 2 − N ≤ θ3 ≤ − f 3 (κ1 ),
− f 2 (κ2 ) +
f 3 (κ1 ) ≤ θ3 ≤ N or
where f 2 is the inverse of ϕ2 |[0, ω2 /4] and f 3 is the inverse of ϕ3 |[0,N ] . These inequalities give four rectangular boxes in C that project onto an unique set in C˜ via the projection (3.3). Consider, for example, the rectangular box Bh given by ⎧ ⎪ ⎨ 0 ≤ θ1 ≤ ω1 ; Bh : − f 2 (κ2 ) ≤ θ2 ≤ f 2 (κ2 ); (5.4) ⎪ ⎩ f 3 (κ1 ) ≤ θ3 ≤ N . 6 h = κ + κ and h = κ κ . 1 1 2 2 1 2
On the Integral Geometry of Liouville Billiard Tables
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For any given θ ∈ Bh we obtain from (5.1) that p1 (θ ) = 1 (ϕ1 (θ1 ) − κ1 )(ϕ1 (θ1 ) − κ2 ), p2 (θ ) = 2 (ϕ2 (θ2 ) − κ1 )(κ2 − ϕ2 (θ2 )), p3 (θ ) = 3 (κ1 − ϕ3 (θ3 ))(κ2 − ϕ3 (θ3 )), where k = ±1. Then the mapping r+ : Bh → T ∗ C, (θ1 , θ2 , θ3 ) → (θ1 , θ2 , θ3 ; p1 (θ ), p2 (θ ), p3 (θ )), where 1 = 1, 2 = ±1, and 3 = ±1, parametrizes one of the two connected components of the subset Th := { H˜ = 1, I˜1 = h 1 , I˜2 = h 2 } ⊂ T ∗ X . Assume that the strict inequalities ν3 < κ1 < ν2 = 0 and 0 < κ2 < ν1 hold. Remark 5.2. The component of Th parametrized by r+ is diffeomorphic to T2 × [0, 1] and its intersection with the boundary T ∗ X | of T ∗ X has two components which can be identified with the two components of the image of the slice {0 ≤ θ1 ≤ ω1 , − f 2 (κ2 ) ≤ θ2 ≤ f 2 (κ2 ), θ3 = N } of Bh with respect to r+ with 3 = 1 and 3 = −1 respectively. In particular, the impulse p3 takes constant values of different sign on them. Moreover, the reflection map r : T ∗ X | → T ∗ X | is given by (θ1 , θ2 ; p1 , p2 , p3 ) → (θ1 , θ2 ; p1 , p2 , − p3 ). Hence, the reflection map interchanges these two components, and by Lemma 7.4 (c), m = 1 (cf. Remark 2.1). Similarly, we get m = 1 in Case (B). Now we compute the generalized actions of the billiard flow corresponding (see (7.8), Appendix), ω1 1 J1 (κ1 , κ2 ) = (ϕ1 (θ1 ) − κ1 )(ϕ1 (θ1 ) − κ2 ) dθ1 2π 0 ν0 2 = (x1 − κ1 )(x1 − κ2 ) ρ1 (x1 ) d x1 , π ν1 f2 (κ2 ) 1 J2 (κ1 , κ2 ) = (ϕ2 (θ2 ) − κ1 )(κ2 − ϕ2 (θ2 )) dθ2 π − f2 (κ2 ) 2 κ2 = (x2 − κ1 )(κ2 − x2 ) ρ2 (x2 ) d x2 , π 0 1 N J3 (κ1 , κ2 ) = (κ1 − ϕ3 (θ3 ))(κ2 − ϕ3 (θ3 )) dθ3 π f3 (κ1 ) 1 κ1 = (κ1 − x3 )(κ2 − x3 ) ρ3 (x3 ) d x3 , π ν3
to Th
where ρ1 (x1 ) := f 1 (x1 ) > 0, ρ2 (x2 ) := f 2 (x2 ) > 0 and ρ3 (x3 ) := − f 3 (x3 ) > 0
(5.5)
(5.6)
(5.7)
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are analytic functions in the intervals (ν1 , ν0 ), (0, ν1 ) and (ν3 , 0), respectively, and f 1 and f 2 satisfy (3.27) and (3.29). Notice that the functions F1± and F2± in (3.27) and (3.29) √ √ are smooth and even analytic in a neighborhood of 0, and f 3 (x3 ) = −x3 F3 ( −x3 ), where F3 is analytic in a neighborhood of [ν3 , 0]. By the assumption (A5 ), ρ3 (x3 ) is smooth at x3 = ν3 . In particular, we obtain Remark 5.3. The functions ρ1 , ρ2 and ρ3 are analytic in the intervals (ν1 , ν0 ), (0, ν1 ) and (ν3 , 0), respectively, and √ √ G − ( x1 − ν1 ) G + ( ν0 − x1 ) ρ1 (x1 ) = 1√ as x1 →ν1 + 0 and ρ1 (x1 ) = 1√ as x1 →ν0 − 0, x1 − ν1 ν0 − x1 √ √ G− G +2 ( ν1 − x2 ) 2 ( x2 ) as x2 → 0 + 0 and ρ2 (x2 ) = √ as x2 → ν1 − 0, and ρ2 (x2 ) = √ x2 ν1 − x2 √ G 3 ( −x3 ) ρ3 (x3 ) = √ as x3 → 0 − 0, −x3 ± where G ± 1 and G 2 are analytic in a neighborhood of 0, and G 3 is analytic in a neighborhood of [ν3 , 0]. Moreover, G 3 (0) > 0 and by (3.28) and (3.30) we have G +1 (0) = 2ϕ1 (0)−1 , G − (0) = − −2ϕ1 (ω1 /4)−1 , 1 −1 G +2 (0) = 2ϕ2 (0)−1 , G − 2 (0) = − −2ϕ2 (ω2 /4) .
As a corollary we obtain Lemma 5.4. The functions J1 , J2 , and J3 are analytic in (κ1 , κ2 ) ∈ (ν3 , 0) × (0, ν1 ). Proof of Lemma 5.4. The function J1 is obviously analytic in that domain. Fix a ∈ (0, ν1 ) and take 0 < δ " 1 such that ρ2 (z) is holomorphic in the disc D2δ (a) := {|z − a| < 2δ} ⊂ C. Then write a−δ κ2 √ √ J2 (κ1 , κ2 ) = κ2 − x2 f (x2 , κ1 ) d x2 + κ2 − x2 f (x2 , κ1 ) d x2 , a−δ
0
where f (x2 , κ1 ) =
1√ x2 − κ1 ρ2 (x2 ) π
is analytic in (0, ν1 ) × (ν3 , 0). Then the first integral defines an analytic function in (κ1 , κ2 ) ∈ (ν3 , 0)×(a−δ/2, a+δ/2). Consider now the second one. We expand f (x2 , κ1 ) in Taylor series with respect to x2 at x2 = κ2 . Then integrating with respect to x2 and using j Cauchy inequalities for d fj (κ2 , κ1 ), where (κ2 , κ1 ) ∈ Dδ/2 (a) × (ν3 + δ, −δ), we obtain d x2
that the second integral defines an analytic function in (κ1 , κ2 ) ∈ (ν3 + δ, −δ) × Dδ/2 (a). In the same way we prove that J3 is analytic in (ν3 , 0) × (0, ν1 ). In order to obtain suitable formulas for the frequencies of the billiard ball map we proceed as in the Appendix. Denote by H(J1 , J1 , J3 ) the Hamiltonian of the billiard flow expressed in the corresponding action-angle coordinates. Then for any κ1 and κ2 such that ν3 < κ1 < 0 and 0 < κ2 < ν1 , one has H(J1 (κ1 , κ2 ), J1 (κ1 , κ2 ), J3 (κ1 , κ2 )) ≡ 1.
(5.8)
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Differentiating (5.8) with respect to κ1 and κ2 we get that the frequencies 1 and 2 of the billiard ball map satisfy ⎡ ⎤ ! ∂ J ∂ J ∂ J " 1 1 2 3 ⎥ ∂κ1 ∂κ1 ∂κ1 ⎢ ⎣ 2 ⎦ = 0. ∂J ∂J ∂J 1
2
∂κ2
∂κ2
3
∂κ2
2π
and therefore (cf. formula (7.7) in the Appendix) ! ∂J ∂J "! " ! ∂J " 1 2 3 1 ∂κ1 ∂κ1 ∂κ1 = −2π . ∂ J3 ∂ J1 ∂ J2 2 ∂κ ∂κ ∂κ 2
(5.9)
2
2
The latter relation and the formulas for the actions (5.5)–(5.7) lead to the following formulas for the frequencies: 1 (κ1 , κ2 ) = π where
A(κ1 , κ2 ) B(κ1 , κ2 ) and 2 (κ1 , κ2 ) = π , D(κ1 , κ2 ) D(κ1 , κ2 )
κ1
κ2
κ2
(5.10)
(x2 − x3 )ρ2 (x2 )ρ3 (x3 ) d x2 d x3 , (x2 − κ1 )(κ2 − x2 )(κ1 − x3 )(κ2 − x3 ) ν 0 3κ1 ν0 (x1 − x3 )ρ1 (x1 )ρ3 (x3 ) d x1 d x3 B(κ1 , κ2 ) := , √ (x1 − κ1 )(x1 − κ2 )(κ1 − x3 )(κ2 − x3 ) ν3 ν1 A(κ1 , κ2 ) :=
and
D(κ1 , κ2 ) := 0
ν0
ν1
√
√
(x1 − x2 )ρ1 (x1 )ρ2 (x2 ) d x1 d x2 . (x1 − κ1 )(x1 − κ2 )(x2 − κ1 )(κ2 − x2 )
It follows from Lemma 5.4 that A, B and D are analytic functions in (κ1 , κ2 ) ∈ (ν3 , 0)× (0, ν1 ). Moreover, D = 0 in that domain, which implies that 1 and 2 are analytic in (κ1 , κ2 ) ∈ (ν3 , 0) × (0, ν1 ). Denote by J the Jacobian of the frequency map (κ1 , κ2 ) → ( 1 (κ1 , κ2 ), 2 (κ1 , κ2 )), J (κ1 , κ2 ) :=
π 2 Aκ1 D − ADκ1 Aκ2 D − ADκ2 ∂( 1 , 2 ) = 4 ∂(κ1 , κ2 ) D Bκ1 D − B Dκ1 Bκ2 D − B Dκ2 .
(5.11)
Since J (κ1 , κ2 ) is analytic in (κ1 , κ2 ) ∈ (ν3 , 0) × (0, ν1 ), either J (κ1 , κ2 ) = 0 in an open dense subset of (ν3 , 0) × (0, ν1 ) or J (κ1 , κ2 ) = 0 for any (κ1 , κ2 ) ∈ (ν3 , 0) × (0, ν1 ).
(5.12)
We are going to compute the limit of J (κ1 , κ2 ) as k1 → ν3 + 0. To do this we will need the following auxiliary lemma. Lemma 5.5. Let f (x, κ) be a function on (a, b) × (a, b) such that f and its partial are continuous and bounded on (a, b) × (a, b). derivatives f x , f κ and f xκ exist ) κand f (x,κ) Consider the function F(κ) := a √ d x. Then κ−x √ (a) F(κ) = 2 f (a, κ) κ − a + O(|κ − a|3/2 ),
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(b) F (κ) =
f (a,κ) √ κ−a
√ + O( κ − a ),
where the estimates above are uniform in κ ∈ (a, b). Proof of Lemma 5.5. An integration by parts leads to F(κ) = −2
κ−0
√
√ f (x, κ) d κ − x = 2 f (a, κ) κ − a + 2
a
κ
√ f x (x, κ) κ − x d x
a
(5.13) that together with the boundedness of f x proves (a). Differentiating (5.13) with respect to κ and using the boundedness of f x , f κ , and f xκ , we prove (b). The expression for A(κ1 , κ2 ) can be rewritten in the form κ1 f (x3 , κ1 ; κ2 ) A(κ1 , κ2 ) = d x3 , √ κ1 − x 3 ν3 where ρ3 (x3 ) f (x3 , κ1 ; κ2 ) := √ κ2 − x 3
0
κ2 (x
2 − x 3 )ρ2 (x 2 ) d x 2 . √ (x2 − κ1 )(κ2 − x2 )
3 ,κ1 ;κ2 ) satisfy the condiFor any given κ2 ∈ (0, ν1 ) the functions f (x3 , κ1 ; κ2 ) and ∂ f (x∂κ 2 tions of lemma 5.5 (with x ≡ x3 , κ = κ1 , a = ν3 < 0 < b < 0) in view of Remark 5.3. Applying the lemma we get κ2 √ √ x2 − ν3 ρ2 (x2 ) d x2 √ ρ3 (ν3 ) A(κ1 , κ2 ) = 2 √ κ1 − ν3 + o( κ1 − ν3 ), √ κ2 − ν3 0 κ2 − x 2 (5.14) * κ2 √ √ √ ∂ A(κ1 , κ2 ) x2 − ν3 ρ2 (x2 ) d x2 ρ3 (ν3 ) = √ κ1 − ν3 + o(1/ κ1 − ν3 ), √ ∂κ1 κ2 − ν3 0 κ2 − x 2 (5.15)
and ∂ A(κ1 , κ2 ) ∂ =2 ∂κ2 ∂κ2
ρ3 (ν3 ) √ κ2 − ν3
κ2 √ x
0
2
− ν3 ρ2 (x2 ) d x2 √ κ2 − x 2
√ √ κ1 − ν3 + o( κ1 − ν3 ), (5.16)
as κ1 → ν3 + 0. In the same way one obtains ν0 √ √ x1 − ν3 ρ1 (x1 ) d x1 √ ρ3 (ν3 ) B(κ1 , κ2 ) = 2 √ κ1 − ν3 + o( κ1 − ν3 ), √ κ2 − ν3 ν1 x 1 − κ2 (5.17) * ν0 √ √ √ ∂ B(κ1 , κ2 ) x1 − ν3 ρ1 (x1 ) d x1 ρ3 (ν3 ) = √ κ1 − ν3 + o(1/ κ1 − ν3 ), √ ∂κ1 κ2 − ν3 ν1 x 1 − κ2 (5.18)
On the Integral Geometry of Liouville Billiard Tables
and ∂ B(κ1 , κ2 ) ∂ =2 ∂κ2 ∂κ2
ρ3 (ν3 ) √ κ2 − ν3
ν0 √ x
ν1
1
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− ν3 ρ1 (x1 ) d x1 √ x 1 − κ2
√ √ κ1 − ν3 + o( κ1 − ν3 ), (5.19)
as κ1 → ν3 + 0. Note also that for any κ2 ∈ (0, ν1 ), D(κ1 , κ2 ) is a continuous (even real-analytic) function with respect to κ1 on the whole interval (−∞, 0). Consider the limit δ(κ2 ) := limκ1 →ν3 +0 π −2 D 3 J (κ1 , κ2 ) for κ2 ∈ (0, ν1 ). It follows from (5.11) and (5.14)–(5.19) that D κ1 D κ2 (−ABκ2 + Aκ2 B) + (ABκ1 − Aκ1 B) D D = (Aκ1 Bκ2 − Bκ1 Aκ2 ) + o(1) ⎛ ) ν √x −ν ρ (x ) d x ⎞ 0 1 √3 1 1 1 κ2 √ 2 ρ3 (ν3 ) 2 x2 − ν3 ρ2 (x2 ) d x2 ∂ ⎝ ν1 x1 −κ2 ⎠ + o(1) √ =2 √ √ ρ (x ) d x2 ∂κ2 ) κ2 x2 −ν κ2 − ν3 κ2 − x2 0 √3 2 2
π −2 D 3 J = (Aκ1 Bκ2 − Bκ1 Aκ2 ) +
κ2 −x2
0
as κ1 → ν3 + 0. Hence,
ρ3 (ν3 ) δ(κ2 ) = 2 √ κ2 − ν3
2
κ2 0
√
x2 − ν3 ρ2 (x2 ) d x2 √ κ2 − x 2
2
⎛) ν 0 ∂ ⎝ ν1 ∂κ2 ) κ2 0
√
x1 −ν3 ρ1 (x1 ) d x1 √ x1 −κ2 √ x2 −ν3 ρ2 (x2 ) d x2 √ κ2 −x2
⎞ ⎠.
(5.20) Suppose that (5.12) holds. Then δ(κ2 ) = 0 for any κ2 ∈ (0, ν1 ) and it follows from (5.20) that there is a constant C = 0 such that ν0 √ κ2 √ x1 − ν3 ρ1 (x1 ) d x1 x2 − ν3 ρ2 (x2 ) d x2 =C (5.21) √ √ x − κ κ2 − x 2 1 2 ν1 0 for any κ2 ∈ (0, ν1 ). Lemma 5.6. Let (X, g) ˜ be a Liouville billiard table of classical type. Then the geodesic flow of the restriction l˜ = g| ˜ of the Riemannian metric g˜ to the boundary is completely integrable. A functionally independent with l˜ integral of the geodesic flow of l˜ is given by the restriction I˜ = I˜2 | of I˜2 to and the level set {l˜ = 1, I˜ = κ} is non empty if and only if κ ∈ [0, ν0 ]. In action-angle coordinates the rotation function corresponding to the Liouville torus T˜κ := {l˜ = 1, I˜ = κ}7 for κ ∈ (0, ν1 ) is * κ √ ν0 √ x1 − ν3 ρ1 (x1 ) d x1 x2 − ν3 ρ2 (x2 ) d x2 ρ(κ) = 2 . (5.22) √ √ x1 − κ κ − x2 ν1 0 Proof of Lemma 5.6. It follows from the construction of the Liouville billiard tables that the mapping σ |T2 : T2N → is a double branched covering of the boundary , where N
T2N = {(θ1 (mod ω1 ), θ2 (mod ω2 ), θ3 = N )} ⊂ C. 7 This set has two connected components that correspond to two Liouville tori with the same rotation function.
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In the coordinates {(θ1 , θ2 )} on T2N we get the following expressions for the metric l = (σ |T2 )∗l˜ and the integral I = (σ |T2 )∗ I˜, N
N
dl 2 = (ϕ1 − ϕ2 ) (ϕ1 − ν3 ) dθ12 + (ϕ2 − ν3 ) dθ22 ,
d I 2 = (ϕ1 − ϕ2 ) ϕ2 (ϕ1 − ν3 ) dθ12 + ϕ1 (ϕ2 − ν3 ) dθ22 .
Applying the Legendre transformation corresponding to l we obtain the following system of equations for the level set Tκ := {l = 1, I = κ}, p12 p22 1 L= + = 1, ϕ1 − ϕ2 ϕ1 − ν3 ϕ2 − ν3 p12 p22 1 + ϕ1 I = ϕ2 = κ, ϕ1 − ϕ2 ϕ1 − ν3 ϕ2 − ν3 that leads to the following expression of the impulses on Tκ : p1 (θ1 )2 = (ϕ1 (θ1 ) − ν3 ) (ϕ1 (θ1 ) − κ) ≥ 0, p2 (θ2 )2 = (ϕ2 (θ2 ) − ν3 ) (κ − ϕ2 (θ2 )) ≥ 0.
(5.23) (5.24)
In particular, Tκ = ∅ if and only if κ ∈ [0, ν0 ]. Hence, the projection of Tκ into the base T2N is given by the union of the sets Aκ := {(θ1 , θ2 ) : 0 ≤ θ1 ≤ ω1 , − f 2 (κ) ≤ θ2 ≤ f 2 (κ)} and Aκ := {(θ1 , θ2 ) : 0 ≤ θ1 ≤ ω1 , − f 2 (κ) + ω2 /2 ≤ θ2 ≤ f 2 (κ) + ω2 /2}. As the sets Aκ and Aκ have the same image under the projection σ |T2 : T2N → we N restrict our attention only to the set Aκ . It follows from (5.23)–(5.24) that the mapping r+ : Aκ → T ∗ T2N , (θ1 , θ2 ) → (θ1 , θ2 ; (ϕ1 (θ1 ) − ν3 ) (ϕ1 (θ1 ) − κ), ± (ϕ2 (θ2 ) − ν3 ) (κ − ϕ2 (θ2 )) ), parametrizes one of the two connected components of the set T˜κ = {l˜ = 1, I˜ = κ} ⊂ T ∗ X . By Liouville-Arnold formula we get the following formulas for the corresponding actions: 2 ω1 J1 (κ) = (ϕ1 (θ1 ) − ν3 ) (ϕ1 (θ1 ) − κ) dθ1 π 0 ν0 8 = (x1 − ν3 ) (x1 − κ)ρ1 (x1 ) d x1 , (5.25) π ν1 f2 (κ) 2 J2 (κ) = (ϕ2 (θ2 ) − ν3 ) (κ − ϕ2 (θ2 )) dθ2 π − f2 (κ) 4 κ (x2 − ν3 ) (κ − x2 )ρ2 (x2 ) d x2 . = (5.26) π 0
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In the corresponding action-angle coordinates the Hamiltonian L becomes L = L 0 (J1 , J2 ), where L 0 is smooth, and the frequency vector
ω of the invariant torus ∂ L0 ∂ L0 {J1 = c1 , J2 = c2 } is ω = − ∂ J1 (c1 , c2 ), ∂ J1 (c1 , c2 ) . Then, differentiating the relation L 0 (J1 (κ), J2 (κ)) ≡ 1 with respect to κ ∈ (0, ν1 ) we get (5.22).
We need the following technical lemma. Lemma 5.7. Let m < 0 < M be real constants, F1 ∈ C 1 ([0, M]), and F2 ∈ C 1 ([m, 0]). Then √ M √ F1 ( t) (5.27) dt = −2F1 (0) log −α + O(1) √√ t t −α 0 and √ α √ F2 ( −t) (5.28) dt = 2F2 (0) log −α + O(1) √ √ −t α − t m as α → 0 − 0. Proof. We have √ √M M √ F1 ( t) 1 dt = 2F1 (0) du + O(1) = −2F1 (0) log −α + O(1). √√ √ t t −α u2 − α 0 0 The proof of (5.28) is similar and we omit it. Lemma 5.7 can be applied to the two integrals in (5.22) using Remark 5.3. In this way we obtain ν0 √ √ √ x1 − ν3 ρ1 (x1 ) d x1 = −2G +1 (0) ν1 − ν3 log ν1 − κ + O(1) √ x1 − κ ν1 and
κ 0
√
√ √ x2 − ν3 ρ2 (x2 ) d x2 = 2G − √ 2 (0) ν1 − ν3 log ν1 − κ + O(1). κ − x2
On the other hand, Remark 5.3 and assumption (A3 ), (2), in Sect. 3.1 imply −1 = − 2ϕ (0)−1 = −G + (0), G− 1 2 (0) = − −2ϕ2 (ω2 /4) 1 and by (5.22) we obtain ρ(κ) → 2 as κ → ν1 − 0. As by (5.21), ρ ≡ const we conclude that ρ ≡ 2 on the interval (0, ν1 ). The latter implies that all the geodesics of lying on a torus T˜κ with κ ∈ (0, ν1 ) (see Lemma 5.6) are periodic. Using the analyticity of the billiard table and considering the Poincaré map in a tubular neighborhood of the “hyperbolic” level set {l = 1, I = ν1 } we obtain that any geodesics of corresponding to some κ ∈ [ν1 , ν2 ) is periodic as well. As the level sets {l = 1, I = 0} and {l = 1, I = ν0 } consists of periodic geodesics we see that all the geodesics on are periodic. Hence, the assumption that the Jacobian J of the frequency map vanishes in an open subset of (κ1 , κ2 ) ∈ (ν3 , 0) × (0, ν1 ) implies that all the geodesics of are periodic. Case (B) can be studied by the same argument.
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6. Proof of Theorem 1 and Theorem 3 In this section we prove Theorem 3 and Theorem 1 formulated in the Introduction. Let (X, g) be a 3-dimensional analytic Liouville billiard table of classical type such that := ∂ X admits at least one non-closed geodesic. We will prove a more general result than Theorem 3 which requires only finite smoothness of K t . Namely, fix d > 1/2 and > 4[2d] + 11, where [2d] is the integer part of 2d and d is the exponent in (H1 ). Denote by C (, R) the corresponding class of Hölder continuous functions. Theorem 6.1. Let [0, 1] t → K t be a continuous curve in C (, R) and suppose that it satisfies (H1 ) and (H2 ), where and d are fixed as above. If K 0 and K 1 are invariant with respect to the group of symmetries G, then K 0 ≡ K 1 . Proof. Given α > 0 and τ > 2 we denote by τα the set of all frequencies ( 1 , 2 ) ∈ R2 satisfying the Diophantine condition: α For any (k1 , k2 , k3 ) ∈ Z3 , (k1 , k2 ) = (0, 0) : | 1 k1 + 2 k2 + k3 | ≥ . (|k1 | + |k2 |)τ Note that the set τ := ∪α>0 τα is of full lebesgue measure in R2 for any τ > 2 fixed (cf. [8, Prop. 9.9]). Then it follows from Theorem 5.1 that the subset of U1 ∪ U2 filled by invariant tori with frequencies in τ is dense in U1 ∪ U2 . Take 0 < τ − 2 " 1 so that > ([2d] + 1)(τ + 2) + 7. Then we apply [12, Theorem 1.1] for any in that family. By Remark 5.2 we have R K 0 ,μ ( ) = R K t ,μ ( )
(6.1)
for any t ∈ [0, 1] and for any torus with frequency in τ , where μ = π + (ξ ), n g −1 . By continuity we obtain (6.1) for any Liouville torus lying in the part U1 ∪ U2 of B ∗ corresponding to the boundary cases. Finally, Theorem 6.1 follows from (6.1) and Theorem 4.1. Proof of Theorem 1. Let (X, g) be a 3-dimensional analytic Liouville billiard table of classical type and let μ = 1 or μ = π + (ξ ), n g −1 . Assume that K ∈ C(, R) is invariant with respect to the group of symmetries G = (Z/2Z)3 of and let the mean value of μ · K on any periodic orbit of the billiard ball map be zero. It follows from Theorem 5.1 that the set filled by Liouville tori of the billiard ball map with frequency vectors := ( 1 , 2 ) ∈ Q×Q is dense in the part of B ∗ corresponding to boundary cases. Let be such a rational torus. In action-angle coordinates, ∼ = R2 /Z2 .There exists N ∈ N and two relatively prime numbers p, q ∈ Z such that ≡ Np , Nq (mod Z2 ). Hence, there is an affine change of coordinates on R2 /Z2 such that ≡ (1/N , 0) (mod Z2 ). Denote D := {(x, y) : 0 ≤ x < 1/N , 0 ≤ y ≤ 1}. Using the invariance of and of the Leray form on with respect to B N we obtain,
N N 1 1
∗ k ∗ k (μ · K ) λ = (B ) (μ · K ) λ = (B ) (μ · K ) λ = 0, (6.2) N D D N k=1
k=1
N
as by assumption the mean k=1 (B ∗ )k (μ · K ) vanishes. Using the density of rational tori in boundary cases, equality (6.2), and Theorem 4.1 we see that K ≡ 0.
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7. Appendix: Frequencies of Integrable Billiard Tables In this Appendix we collect the necessary facts used for the computation of the frequency map in Sect. 5. Our main task is to derive formula (7.9) for the frequencies of the billiard ball map. Let (X, g), n = dim X ≥ 2, be a billiard table with non-empty locally convex boundary . Consider the reflection map at the boundary, ρ : T X | → T X | ,
ξ → ξ − 2g(ξ, n g )n g ,
(7.1)
where T X | := {ξ ∈ T X : π(ξ ) ∈ } is the restriction of the tangent bundle to , π : T X → X is the natural projection onto the base, and n g is the inward unit normal to the boundary. The restriction ρ is an involution on T X | the set of fixed point of which coincides with T ⊆ T X | . Note that ρ preserves the values of of the Hamiltonian Hg (ξ ) := 21 g(ξ, ξ ) and when restricted to the unit spherical bundle Sg X | := {ξ ∈ T X | : ξ g = 1} it coincides with the mapping r : → considered in Sect. 2 if we identify vectors and covectors with the help of the Legendre transform, F L g : T X → T ∗ X, ξ → g(ξ, ·). More generally, the notions and mappings considered in Sect. 2 have their analogs on T X via the Legendre transform. Denote by αg the Liouville 1-form on T X given by αg (v)(·) := g(v, dv π(·)), where v ∈ T X and (·) stands for an arbitrary element of Tv (T X ). Note that the differential ωg := dαg of the 1-form αg corresponds to the symplectic form dp∧d x on the cotangent bundle T ∗ X via the Legendre transform. Lemma 7.1. The reflection map ρ : T X | → T X | satisfies the following properties: (a) the reflection ρ preserves the restriction of the Liouville form αg to T X | ; (b) the reflection ρ preserves the values of the Hamiltonian Hg (ξ ) = 21 g(ξ, ξ ); (c) in the case when (X, g) ˜ is a Liouville billiard table the reflection ρ corresponding to the Riemannian metric g˜ preserves the values of the pairwise commuting integrals I˜k (k = 1, 2) of the billiard flow (cf. Proposition 3.3). Proof of Lemma 7.1. (a) Let t → v(t) be a smooth curve in T X | defined in an open neighborhood of t = 0 such that v(0) = v ∈ Tx X, x ∈ , and v(0) ˙ = ! ∈ Tv (T X | ). One has d αg (ρ(v))(dv ρ(!)) = g(ρ(v), dρ(v) π ◦ dv ρ(!)) = g ρ(v), π(ρ(v(t))|t=0 ) dt d d = g v − 2g(v, n g )n g , π(v(t))|t=0 = g v, π(v(t))|t=0 dt dt (7.2) = αg (v)(!), d where we have used that π ◦ ρ = π and that dt π(v(t))|t=0 ∈ Tx is orthogonal to n g . This proves statement (a). The proof of (b) is straightforward and we omit it. Statement (c) was established in the proof of Proposition 3.3.
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Now we will describe a special variant of the symplectic gluing procedure introduced by Lazutkin in [8, § 4]. The main idea is to identify parts of the boundary ∂(T X ) of the configuration space T X of the billiard flow in order to “eliminate" the reflections and obtain a new “glued” configuration space together with a smooth billiard flow on it. Note that the glued configurations space becomes a smooth symplectic manifold so that the billiard flow is a smooth Hamiltonian system on it. Divide the boundary of T X into three parts: ∂(T X ) = T X | = T − X | T + X | T , where T ± X | := {ξ ∈ T X : ±g(ξ, n g ) > 0} and T is assumed naturally embedded into T X | . Note that ρ|T − X : T − X | → T + X |
(7.3)
is a diffeomorphism and the elements of T ⊆ T X | are fixed points of ρ. Now, using (7.3) we identify the points ξ − ∈ T − X and ρ(ξ − ) ∈ T + X of the boundary of T X \T ρ / and obtain a new glued space T X that we supply with the factor topology so that the ρ / projection πρ : T X \T → T X , ⎧ ⎪ / ∂(T X ) ⎨ ξ if ξ ∈ T X \T ξ → {ξ, ρ(ξ )} if ξ ∈ T − X | , ⎪ ⎩ −1 {ρ (ξ ), ξ } if ξ ∈ T + X | is continuous. Definition 7.2. The billiard flow of (X, g) is called completely integrable if there exist n functionally independent integrals Q 1 , . . . , Q n ≡ Hg ∈ C ∞ (T X, R) of the billiard flow such that ∀ 1 ≤ k, l ≤ n, {Q k , Q l } = 0, and ∀ 1 ≤ k ≤ n∀ ξ ∈ T X | , Q k (ρ(ξ )) = Q k (ξ ). Assume that the billiard flow on T X is completely integrable. Denote by X g the Hamiltonian vector field on T X with Hamiltonian Hg . The following proposition follows from Lemma 7.1 (a), (b), and is a special case of the symplectic gluing developed in [8, § 4]. ρ
Proposition 7.3. There exists a smooth differentiable structure on T X , a symplecρ ρ ∞ ˜ tic form ω˜ g on T X , and functions Q k ∈ C (T X , R)(1 ≤ k ≤ n), such that the projection TX πρ : T X \T →
ρ
(7.4)
is smooth, πρ∗ (ω˜ g ) = ωg , and πρ∗ ( Q˜ k ) = Q k for any 1 ≤ k ≤ n. In particular, the Hamiltonian vector field X˜ g corresponding to H˜ g := Q˜ n is completely integrable in ρ T X and (πρ )∗ (X g ) = X˜ g . Denote ρ / T := πρ (T ± X | ) ⊂ T X .
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Note that T is a disjoint union of connected non-intersecting embedded hypersurfaces ρ / in T X that are transversal to the Hamiltonian vector field X˜ g . Let c = (c1 , . . . , cn ) be a regular value of the “momentum” map ρ / M:T X → Rn ,
ξ → ( Q˜ 1 (ξ ), . . . , Q˜ n (ξ )),
and let T˜c be a connected component of the level set M −1 (c). The compactness of X implies that T˜c is compact. By the Liouville-Arnold theorem T˜c is diffeomorphic to the n dimensional torus Tn and one can introduce action-angle coordinates in a tubular ρ / neighborhood of T˜c in T X (see [2]). Assume that T˜c ∩ T = ∅ is a non-empty compact set. In this case we will call T˜c glued Liouville torus. As X˜ g is tangent to T˜c and transversal to T the submanifolds T˜c and T intersect transversally. Hence, T˜c ∩ T is a disjoint union of finitely many compact embedded submanifolds in T˜c . Denote by m ≥ 1 the number of the connected components of T˜c ∩ T . The proof of the following lemma is straightforward and we omit it. Lemma 7.4. (a) The connected components of T˜c ∩ T are diffeomorphic to Tn−1 ; (b) The closure of any of the connected components of T˜c \T is diffeomorphic to [0, 1] × Tn−1 . The n-torus T˜c is obtained by a “cyclic" gluing together of all m ≥ 1 copies of [0, 1] × Tn−1 along their boundaries; (c) Let Sc be a connected component of T˜c ∩ T and let c = p+ (πρ−1 (Sc )), where p+ : T X | → T denotes the orthogonal projection ξ → ξ − g(n g , ξ ) n g onto T . Then the number m ≥ 1 of the connected components of T˜c ∩ T is the minimal power of the billiard ball map B that leaves c invariant, i.e., B m ( c ) = c .8 Choose a component Sc of T˜c ∩ T and a basis of cycles γ˜1 , . . . , γ˜n−1 of its homology group as well as a transversal cycle γ˜n in T˜c so that γ˜1 , . . . , γ˜n is a basis of the homology group of T˜c . Let { J˜1 , . . . , J˜n ; θ˜1 (mod 2π ), . . . , θ˜n (mod 2π )} be action-angle coordinates in a tubular neighborhood of the glued Liouville torus T˜c that corresponds to the cycles γ˜1 , . . . , γ˜n , i.e., ∀ 1 ≤ k ≤ n, 1 α˜ g , J˜k = 2π γ˜k ρ / where α˜ g := (πρ )∗ (αg ) is the push-forward of the Liouville form αg onto T X . In the action-angle coordinates,
∂ ∂ X˜ g = η1 ( J˜) + · · · + ηn ( J˜) , ˜ ∂ θ1 ∂ θ˜n where ηk ( J˜1 , . . . , J˜n ) :=
g ∂H ( J˜1 , . . . , J˜n ), 1 ≤ k ≤ n. ∂ J˜k
8 Note that we identify vectors and covectors via the Riemannian metric g.
(7.5)
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It follows from the choice of the cycles γ˜1 , . . . , γ˜n that Sc is a section of the bundle, (θ˜1 , . . . , θ˜n ) → (θ˜1 , . . . , θ˜n−1 ).
Tn−1 × T → Tn−1 ,
As Sc is transversal to X˜ g one concludes that ηn ( J˜) = 0. It follows from our construction that the billiard ball map B m is conjugated to the following diffeomorphism of the n − 1-dimensional torus {(θ˜1 (mod 2π ), . . . , θ˜n−1 (mod 2π ))}, θ˜k → θ˜k + 2π
ηk ( J˜) , 1 ≤ k ≤ n − 1. ηn ( J˜)
Parameterizing the glued Liouville tori with fixed energy { H˜ g = 1} by the values of the integrals Q˜ = ( Q˜ 1 , . . . , Q˜ n−1 ) we obtain the following mapping for the frequencies of B m : ( Q˜ 1 , . . . , Q˜ n−1 ) → ( k ( Q˜ 1 , . . . , Q˜ n−1 ))1≤k≤n−1 := 2π
ηk (J ( Q˜ 1 , . . . , Q˜ n−1 , 1)) ηn (J ( Q˜ 1 , . . . , Q˜ n−1 , 1))
, 1≤k≤n−1
(7.6) where ηk ( J˜1 , . . . , J˜n ) is defined by (7.5). Finally, by partial differentiation of the identity, g ( J˜( Q˜ 1 , . . . , Q˜ n−1 , 1)) ≡ 1, H one gets that the frequency vector := ( 1 , . . . , n−1 )T satisfies the linear relation A = −2π b, where A(Q) := and ∀1 ≤ k ≤ n,
∂ Jl ∂ Q k (Q, 1) 1≤k,l≤n−1 ,
(7.7)
∂ Jn b(Q) := ( ∂∂QJn1 (Q, 1), . . . , ∂ Q (Q, 1))T , n−1
1 Jk := 2π
γk
αg ,
(7.8)
where γk is the connected component of πρ−1 (γ˜k ) lying in T + X | and Q := (Q 1 , . . . , Q n−1 ). The functions Jk (1 ≤ k ≤ n) will be called generalized actions of the billiard flow. Using that ηn = 0 one can prove that A(Q) is non-degenerate. Hence, (Q) = −2π A(Q)−1 b(Q),
(7.9)
where Q 1 , . . . , Q n−1 are the integrals of the billiard flow in a tubular neighborhood of the invariant set πρ−1 (T˜c ) of the billiard flow.
On the Integral Geometry of Liouville Billiard Tables
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References 1. Adams, J.: Expression of the product of any two Legendre’s coefficients by means of Legendre’s coefficients. Proc. R. Soc. Lond. 27, 63–71 (1878) 2. Arnold, V.: Mathematical methods of classical mechanics. NY: Springer-Verlag, 1989 3. Besse, A.: Manifolds all of whose geodesics are closed. Berlin-New York: Springer-Verlag, 1978 4. Guillemin, V., Melrose, R.: An inverse spectral result for elliptical regions in R2 . Adv. in Math. 32, 128–148 (1979) 5. Guillemin, V., Melrose, R.: The Poisson summation formula for manifolds with boundary. Adv. in Math. 32, 204–232 (1979) 6. Kiyohara, K.: Two classes of Riemannian manifolds whose geodesic flows are integrable. Memoirs of the AMS 130, Number 619 (1997) 7. Knörrer, H.: Singular fibers of the momentum mapping for integrable Hamiltonian systems. J. Reine Angew. Math. 355, 67–107 (1985) 8. V. Lazutkin: KAM theory and semiclassical approximations to eigenfunctions. Berlin: Springer-Verlag, 1993 9. Moser, J., Veselov, A.: Discrete versions of some integrable systems and factorization of matrix polynomials. Commun. Math. Phys. 139, 217–243 (1991) 10. Popov, G., Topalov, P.: Liouville billiard tables and an inverse spectral result. Ergod. Th. & Dynam. Sys. 23, 225–248 (2003) 11. Popov, G., Topalov, P.: Discrete analog of the projective equivalence and integrable billiard tables. Ergod. Th. & Dynam. Sys. 28, 1657–1684 (2008) 12. Popov, G., Topalov, P.: Invariants of isospectral deformations and spectral rigidity. http://arxiv.org/abs/ 0906.0449v1 [math.SP], 2 Jun 2009 13. Topalov, P.: Integrability criterion of geodesical equivalence. Hierarchies, Acta Appl. Math. 59(3), 271– 298 (1999) 14. Perelomov, A.: Integrable Systems of Classical Mechanics and Lie Algebras. Basel: Birkhäuser-Verlag, 1990 15. Tabachnikov, S.: Billiards. In: Panoramas et Syntheses, Paris: Societe Mathematique de France, 1995 16. Whittaker, E., Watson, G.: A course of modern analysis. Cambridge: Cambridge University Press, 1927 Communicated by S. Zelditch
Commun. Math. Phys. 303, 761–784 (2011) Digital Object Identifier (DOI) 10.1007/s00220-011-1224-y
Communications in
Mathematical Physics
Inverse Scattering at Fixed Energy on Surfaces with Euclidean Ends Colin Guillarmou1 , Mikko Salo2 , Leo Tzou3 1 DMA, U.M.R. 8553 CNRS, Ecole Normale Supérieure, 45 rue d’Ulm, F 75230 Paris Cedex 05, France.
E-mail:
[email protected]
2 Department of Mathematics, University of Helsinki, PO Box 68, 00014 Helsinki, Finland.
E-mail:
[email protected]
3 Department of Mathematics, Stanford University, Stanford, CA 94305, USA.
E-mail:
[email protected] Received: 6 April 2010 / Accepted: 16 November 2010 Published online: 23 March 2011 – © Springer-Verlag 2011
Abstract: On a fixed Riemann surface (M0 , g0 ) with N Euclidean ends and genus g, we show that, under a topological condition, the scattering matrix SV (λ) at frequency λ > 0 j for the operator + V determines the potential V if V ∈ C 1,α (M0 )∩e−γ d(·,z 0 ) L ∞ (M0 ) for all γ > 0 and for some j ∈ {1, 2}, where d(z, z 0 ) denotes the distance from z to a fixed point z 0 ∈ M0 . The topological condition is given by N ≥ max(2g + 1, 2) for j = 1 and by N ≥ g + 1 if j = 2. In R2 this implies that the operator SV (λ) determines 2 any C 1,α potential V such that V (z) = O(e−γ |z| ) for all γ > 0. 1. Introduction The purpose of this paper is to prove an inverse scattering result at fixed frequency λ > 0 in dimension 2. The typical question one can ask is to show that the scattering matrix SV (λ) for the Schrödinger operator + V determines the potential. This is known to be false if V is only assumed to be Schwartz, by the example of Grinevich-Novikov [6], but it is also known to be true for exponentially decaying potentials (i.e. V ∈ e−γ |z| L ∞ (R2 ) for some γ > 0) with norm smaller than a constant depending on the frequency λ, see Novikov [16]. For other partial results we refer to [2,11,20–22]. The determinacy of V from SV (λ) when V is compactly supported, without any smallness assumption on the norm, follows from the recent work of Bukhgeim [1] on the inverse boundary problem after a standard reduction to the Dirichlet-to-Neumann operator on a large sphere (see [26] for this reduction). In dimensions n ≥ 3, it is proved in Novikov [17] (see also [3] for the case of magnetic Schrödinger operators) that the scattering matrix at a fixed frequency λ determines an exponentially decaying potential. When V is compactly supported this also follows directly from the result by Sylvester-Uhlmann [23] on the inverse boundary problem, by reducing to the Dirichlet-to-Neumann operator on a large sphere. Melrose [15] gave a direct proof of the last result based on the methods of [23], and this proof was extended to exponentially decaying potentials in [27] and to the magnetic case in [18].
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In the geometric scattering setting, [12,13] reconstruct the asymptotic expansion of a potential or metrics from the scattering operator at fixed frequency on asymptotically Euclidean/hyperbolic manifolds. Further results of this type are given in [28,29]. The method for proving the determinacy of V from SV (λ) in [15,27] is based on the construction of complex geometric optics solutions u(z) = eρ.z (1 + r (ρ, z)) of ( + V − λ2 )u = 0with ρ ∈ Cn , z ∈ Rn , and the density of the oscillating scattering solutions u sc (z) = S n−1 V (λ, z, ω) f (ω)dω within those complex geometric optics 1
solutions, where V (λ, z, ω) = eiλω.z + e−iλω.z |z|− 2 (n−1) a(λ, z, ω) are the perturbed plane wave solutions (here ω ∈ S n−1 and a ∈ L ∞ ). Unlike when n ≥ 3, the problem in dimension 2 is that the set of complex geometrical optics solutions of this type is not large enough to show that the Fourier transform of V1 − V2 is 0. The real novelty in the recent work of Bukhgeim [1] in dimension 2 is the construction of new complex geometric optics solutions (at least on a bounded domain ⊂ C) of ( + Vi )u i = 0 of the form u 1 = e/ h (1 + r1 (h)) and u 2 = e−/ h (1 + r2 (h)) with 0 < h 1, where is a holomorphic function in C with a unique non-degenerate critical point at a fixed z 0 ∈ C (for instance (z) = (z − z 0 )2 ), and ||r j (h)|| L p is small as h → 0 for p > 1. These solutions allow to use the stationary phase at z 0 to get (V1 − V2 )u 1 u 2 = C(V1 (z 0 ) − V2 (z 0 ))h + o(h), C = 0
as h → 0 and thus, if the Dirichlet-to-Neumann operators on ∂ are the same, then V1 (z 0 ) = V2 (z 0 ). One of the problems to extend this to inverse scattering is that a holomorphic function in C with a non-degenerate critical point needs to grow at least quadratically at infinity, which would somehow force consideration of potentials V having Gaussian decay. On the other hand, if we allow the function to be meromorphic with simple poles, then we can construct such functions, having a single critical point at any given point p, for instance by considering (z) = (z − p)2 /z. Of course, with such we then need to work on C\{0}, which is conformal to a surface with no hole but with 2 Euclidean ends, and has linear growth in the ends. In general, on a surface with genus g and N Euclidean ends, we can use the Riemann-Roch theorem to construct holomorphic functions with linear or quadratic growth in the ends, the dimension of the space of such functions depending on g, N . In the present work, we apply this idea to obtain an inverse scattering result for g0 + V on a fixed Riemann surface (M0 , g0 ) with Euclidean ends, under some topological condition on M0 and some decay condition on V . Theorem 1.1. Let (M0 , g0 ) be a non-compact Riemann surface with genus g and N ends isometric to R2 \{|z| ≤ 1} with metric |dz|2 . Let V1 and V2 be two potentials in C 1,α (M0 ) with α > 0, and such that SV1 (λ) = SV2 (λ) for some λ > 0. Let d(z, z 0 ) denote the distance between z and a fixed point z 0 ∈ M0 . (i) If N ≥ max(2g + 1, 2) and Vi ∈ e−γ d(·,z 0 ) L ∞ (M0 ) for all γ > 0, then V1 = V2 . 2 (ii) If N ≥ g + 1 and Vi ∈ e−γ d(·,z 0 ) L ∞ (M0 ) for all γ > 0, then V1 = V2 . In R2 , where g = 0 and N = 1, we have an immediate corollary: Corollary 1.2. Let λ > 0 and let V1 , V2 ∈ C 1,α (R2 ) ∩ e−γ |z| L ∞ (R2 ) for all γ > 0. If the scattering matrices satisfy SV1 (λ) = SV2 (λ), then V1 = V2 . 2
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This is an improvement on the result of Bukhgeim [1] which shows identifiability for compactly supported functions, and in a certain sense on the result of Novikov [16] since it is assumed there that the potential has to be of small L ∞ norm. Some techniques used here are borrowed from previous works of the first and last authors [7,8], however it is worth to notice that we need new ingredients in the present case. The first point is to get a sharp description of Carleman weights in terms of the topology of the surface through the Riemann-Roch theorem. The set of possible weights in the present case (linearly or quadratically growing holomorphic functions on the surface) is only finite dimensional, and therefore much smaller than the sets of functions used in [7,8]. Another novelty in this work is the construction of complex geometric optics using a Carleman estimate for − λ2 on a Riemann surface with Euclidean ends, with a linearly growing holomorphic phase: as compared to the Carleman estimate in [7], this introduces non-trivial difficulties at infinity and, as far as we know, this estimate is new on its own and could have other applications. In general, the usual method in scattering theory for proving existence of complex geometric optics goes through integral kernels or Fourier analysis on Rn . We also prove, for a potential V with Gaussian decay, a density result for scattering solutions in the set of solutions of − λ2 + V with Gaussian growths using a certain type of Paley-Wiener theorem. This did not seem to be written down in the literature. The structure of the paper is as follows. In Sect. 2 we employ the Riemann-Roch theorem and a transversality argument to construct Morse holomorphic functions on (M0 , g0 ) with linear or quadratic growth in the ends. Section 3 considers Carleman estimates with harmonic weights on (M0 , g0 ), where suitable convexification and weights at the ends are required since the surface is non-compact. Complex geometrical optics solutions are constructed in Sect. 4. Section 5 discusses direct scattering theory on surfaces with Euclidean ends and contains the proof that scattering solutions are dense in the set of suitable solutions, and Sect. 6 gives the proof of Theorem 1.1. Finally, there is an Appendix discussing a Paley-Wiener type result for functions with Gaussian decay which is needed to prove density of scattering solutions. 2. Holomorphic Morse Functions on a Surface with Euclidean Ends 2.1. Riemann surfaces with Euclidean ends. Let (M0 , g0 ) be a non-compact connected smooth Riemannian surface with N ends E 1 , . . . , E N which are Euclidean, i.e. isometric to C\{|z| ≤ 1} with metric |dz|2 . By using a complex inversion z → 1/z, each end is also isometric to a pointed disk E i {|z| ≤ 1, z = 0}
with metric
|dz|2 , |z|4
thus conformal to the Euclidean metric on the pointed disk. The surface M0 can then be compactified by adding the points corresponding to z = 0 in each pointed disk corresponding to an end E i , we obtain a closed Riemann surface M with a natural complex structure induced by that of M0 , or equivalently a smooth conformal class on M induced by that of M0 . Another way of thinking is to say that M0 is the closed Riemann surface M with N points e1 , . . . , e N removed. The Riemann surface M has holomorphic charts z α : Uα → C and we will denote by z 1 , . . . z N the complex coordinates corresponding to the ends of M0 , or equivalently to the neighbourhoods of the points ei . The Hodge star operator acts on the cotangent bundle T ∗ M, its eigenvalues are ±i and the respec∗ M := ker( + iId) and T ∗ M := ker( − iId) are sub-bundles of tive eigenspaces T1,0 0,1
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∗ M ⊕ T∗ M the complexified cotangent bundle CT ∗ M and the splitting CT ∗ M = T1,0 0,1 holds as complex vector spaces. Since is conformally invariant on 1-forms on M, the complex structure depends only on the conformal class of g. In holomorphic coordinates z = x + i y in a chart Uα , one has (ud x + vdy) = −vd x + udy and ∗ ∗ T1,0 M|Uα Cdz, T0,1 M|Uα Cd z¯ ,
where dz = d x + idy and d z¯ = d x − idy. We define the natural projections induced by the splitting of CT ∗ M, ∗ ∗ π1,0 : CT ∗ M → T1,0 M, π0,1 : CT ∗ M → T0,1 M.
The exterior derivative d defines the de Rham complex 0 → 0 → 1 → 2 → 0, where k := k T ∗ M denotes the real bundle of k-forms on M. Let us denote C k the complexification of k , then the ∂ and ∂¯ operators can be defined as differential ∗ M) and ∂¯ : C ∞ (M) → C ∞ (M, T ∗ M) by operators ∂ : C ∞ (M) → C ∞ (M, T1,0 0,1 ∂ f := π1,0 d f, ∂¯ f := π0,1 d f, they satisfy d = ∂ + ∂¯ and are expressed in holomorphic coordinates by ∂ f = ∂z f dz, ∂¯ f = ∂z¯ f d z¯ , with ∂z := 21 (∂x − i∂ y ) and ∂z¯ := 21 (∂x + i∂ y ). Similarly, one can define the ∂ and ∂¯ operators mapping sections of C 1 to sections of C 2 by setting ¯ 1,0 + ω0,1 ) := dω1,0 ∂(ω1,0 + ω0,1 ) := dω0,1 , ∂(ω ∗ M and ω ∗ if ω0,1 ∈ T0,1 1,0 ∈ T1,0 M. In coordinates this is simply
¯ ¯ ∧ dz. ∂(udz + vd z¯ ) = ∂v ∧ d z¯ , ∂(udz + vd z¯ ) = ∂u ∗ M, there If g is a metric on M whose conformal class induces the complex structure T1,0 is a natural operator, the Laplacian acting on functions and defined by
¯ f = d ∗ d, f := −2i ∂∂ where d ∗ is the adjoint of d through the metric g and is the Hodge star operator mapping
2 to 0 and induced by g as well. 2.2. Holomorphic functions. We are going to construct Carleman weights given by holomorphic functions on M0 which grow at most linearly or quadratically in the ends. We will use the Riemann-Roch theorem, following ideas of [7], however, the difference in the present case is that we have very little freedom to construct these holomorphic functions, simply because there is just a finite dimensional space of such functions by Riemann-Roch. For the convenience of the reader, and to fix notations, we recall the usual Riemann-Roch index theorem (see Farkas-Kra [5] for more details). A divisor D on M is an element D = (( p1 , n 1 ), . . . , ( pk , n k )) ∈ (M × Z)k ,
where k ∈ N
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k which will also be denoted D = i=1 pin i or D = p∈M p α( p) , where α( p) = 0 for all p except α( pi ) = n i . The inverse divisor of D is defined to be D −1 := p∈M p −α( p) k and the degree of the divisor D is defined by deg(D) := i=1 ni = α( p). A p∈M ord( p) non-zero meromorphic function on M is said to have divisor D if ( f ) := p∈M p is equal to D, where ord( p) denotes the order of p as a pole or zero of f (with positive sign convention for zeros). Notice that in this case we have deg( f ) = 0. For divisors D = p∈M p α ( p) and D = p∈M p α( p) , we say that D ≥ D if α ( p) ≥ α( p) for all p ∈ M. The same exact notions apply for meromorphic 1-forms on M. Then we define for a divisor D, r (D) := dim({ f meromorphic function on M; ( f ) ≥ D} ∪ {0}), i(D) := dim({u meromorphic 1 form on M; (u) ≥ D} ∪ {0}). The Riemann-Roch theorem states the following identity: for any divisor D on the closed Riemann surface M of genus g, r (D −1 ) = i(D) + deg(D) − g + 1.
(1)
Notice also that for any divisor D with deg(D) > 0, one has r (D) = 0 since deg( f ) = 0 for all f meromorphic. By [5, Th. p70], let D be a divisor, then for any non-zero meromorphic 1-form ω on M, one has i(D) = r (D(ω)−1 ),
(2)
which is thus independent of ω. For instance, if D = 1, we know that the only holomorphic function on M is 1 and one has 1 = r (1) = r ((ω)−1 )− g +1 and thus r ((ω)−1 ) = g if ω is a non-zero meromorphic 1 form. Now if D = (ω), we obtain again from (1), g = r ((ω)−1 ) = 2 − g + deg((ω)), which gives deg((ω)) = 2(g − 1) for any non-zero meromorphic 1-form ω. In particular, if D is a divisor such that deg(D) > 2(g − 1), then we get deg(D(ω)−1 ) = deg(D) − 2(g − 1) > 0 and thus i(D) = r (D(ω)−1 ) = 0, which implies by (1), deg(D) > 2(g − 1) ⇒ r (D −1 ) = deg(D) − g + 1 ≥ g.
(3)
Now we deduce Lemma 2.1. Let e1 , . . . , e N be distinct points on a closed Riemann surface M with genus g, and let z 0 be another point of M\{e1 , . . . , e N }. If N ≥ max(2g + 1, 2), the following hold true: (i) there exists a meromorphic function f on M with at most simple poles, all contained in {e1 , . . . , e N }, such that ∂ f (z 0 ) = 0, (ii) there exists a meromorphic function h on M with at most simple poles, all contained in {e1 , . . . , e N }, such that z 0 is a zero of order at least 2 of h.
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Proof. Let first g ≥ 1, so that N ≥ 2g + 1. By the discussion before the lemma, we know that there are at least g + 2 linearly independent (over C) meromorphic functions f 0 , . . . , f g+1 on M with at most simple poles, all contained in {e1 , . . . , e2g+1 }. Without loss of generality, one can set f 0 = 1 and by linear combinations we can assume that −j f 1 (z 0 ) = · · · = f g+1 (z 0 ) = 0. Now consider the divisor D j = e1 . . . e2g+1 z 0 for j = 1, 2, with degree deg(D j ) = 2g + 1 − j, then by the Riemann-Roch formula (more precisely (3)) r (D −1 j ) = g + 2 − j. Thus, since r (D1−1 ) > r (D2−1 ) = g and using the assumption that g ≥ 1, we deduce that there is a function in span( f 1 , . . . , f g+1 ) which has a zero of order 2 at z 0 and a function which has a zero of order exactly 1 at z 0 . The same method clearly works if g = 0 by taking two points e1 , e2 instead of just e1 . If we allow double poles instead of simple poles, the proof of Lemma 2.1 shows Lemma 2.2. Let e1 , . . . , e N be distinct points on a closed Riemann surface M with genus g, and let z 0 be another point of M\{e1 , . . . , e N }. If N ≥ g + 1, then the following hold true: (i) there exists a meromorphic function f on M with at most double poles, all contained in {e1 , . . . , e N }, such that ∂ f (z 0 ) = 0, (ii) there exists a meromorphic function h on M with at most double poles, all contained in {e1 , . . . , e N }, such that z 0 is a zero of order at least 2 of h. 2.3. Morse holomorphic functions with prescribed critical points. We follow in this section the arguments used in [7] to construct holomorphic functions with non-degenerate critical points (i.e. Morse holomorphic functions) on the surface M0 with genus g and N ends, such that these functions have at most linear growth (resp. quadratic growth) in the ends if N ≥ max(2g + 1, 2) (resp. if N ≥ g + 1). We let H be the complex vector space spanned by the meromorphic functions on M with divisors larger or equal to e1−1 . . . e−1 N (resp. by e1−2 . . . e−2 N ) if we work with functions having linear growth (resp. quadratic growth), where e1 , . . . e N ∈ M are points corresponding to the ends of M0 as explained in Sect. 2. Note that H is a complex vector space of complex dimension greater or equal −2 −2 to N − g + 1 (resp. 2N − g + 1) for the e1−1 . . . e−1 N divisor (resp. the e1 . . . e N divisor). We will also consider the real vector space H spanned by the real parts and imaginary parts of functions in H, this is a real vector space which admits a Lebesgue measure. We now prove the following Lemma 2.3. The set of functions u ∈ H which are not Morse in M0 has measure 0 in H , in particular its complement is dense in H . Proof. We use an argument very similar to that used by Uhlenbeck [25]. We start by defining m : M0 × H → T ∗ M0 by ( p, u) → ( p, du( p)) ∈ T p∗ M0 . This is clearly a smooth map, linear in the second variable, moreover m u := m(., u) = (·, du(·)) is smooth on M0 . The map u is a Morse function if and only if m u is transverse to the zero section, denoted T0∗ M0 , of T ∗ M0 , i.e. if Image(D p m u ) + Tm u ( p) (T0∗ M0 ) = Tm u ( p) (T ∗ M0 ), ∀ p ∈ M0 such that m u ( p) = ( p, 0).
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This is equivalent to the fact that the Hessian of u at critical points is non-degenerate (see for instance Lemma 2.8 of [25]). We recall the following transversality result, the proof of which is contained in [25, Th.2] by replacing the Sard-Smale theorem by the usual finite dimensional Sard theorem: Theorem 2.4. Let m : X × H → W be a C k map, let X, W be smooth manifolds and H be a finite dimensional vector space. If W ⊂ W is a submanifold such that k > max(1, dim X − dim W + dim W ), then the transversality of the map m to W implies that the complement of the set {u ∈ H ; m u is transverse to W } in H has Lebesgue measure 0. We want to apply this result with X := M0 , W := T ∗ M0 and W := T0∗ M0 , and with the map m as defined above. We have thus proved our lemma if one can show that m is transverse to W . Let ( p, u) be such that m( p, u) = ( p, 0) ∈ W . Then identifying T( p,0) (T ∗ M0 ) with T p M0 ⊕ T p∗ M0 , one has Dm ( p,u) (z, v) = (z, dv( p) + Hess p (u)z), where Hess p (u) is the Hessian of u at the point p, viewed as a linear map from T p M0 to T p∗ M0 (note that this is different from the covariant Hessian defined by the LeviCivita connection). To prove that m is transverse to W we need to show that (z, v) → (z, dv( p) + Hess p (u)z) is onto from T p M0 ⊕ H to T p M0 ⊕ T p∗ M0 , which is realized if the map v → dv( p) from H to T p∗ M0 is onto. But from Lemma 2.1, we know that there exists a meromorphic function f with real part v = Re( f ) ∈ H such that v( p) = 0 and dv( p) = 0 as an element of T p∗ M0 . We can then take v1 := v and v2 := Im( f ), which are functions of H such that dv1 ( p) and dv2 ( p) are linearly independent in T p∗ M0 by the Cauchy-Riemann equation ∂¯ f = 0. This shows our claim and ends the proof by using Theorem 2.4. In particular, by the Cauchy-Riemann equation, this lemma implies that the set of Morse functions in H is dense in H. We deduce Proposition 2.1. There exists a dense set of points p in M0 such that there exists a Morse holomorphic function f ∈ H on M0 which has a critical point at p. Proof. Let p be a point of M0 and let u be a holomorphic function with a zero of order at least 2 at p, the existence is ensured by Lemma 2.1. Let B( p, η) be any small ball of radius η > 0 near p, then by Lemma 2.3, for any > 0, we can approach u by a holomorphic Morse function u ∈ H which is at distance less than of u in a fixed norm on the finite dimensional space H. Rouché’s theorem for ∂z u and ∂z u (which are viewed as functions locally near p) implies that ∂z u has at least one zero of order exactly 1 in B( p, η) if is chosen small enough. Thus there is a Morse function in H with a critical point arbitrarily close to p. Remark 2.5. In the case where the surface M has genus 0 and N ends, we have an explicit formula for the function in Proposition 2.1: indeed M0 is conformal to C\{e1 , . . . , e N −1 } for some ei ∈ C - i.e. the Riemann sphere minus N points - then the function f (z) = (z − z 0 )2 /(z − e1 ) with z 0 ∈ {e1 , . . . , e N −1 } has z 0 for unique critical point in C\{e1 , . . . , e N −1 } and it is non-degenerate. We end this section by the following lemmas which will be used for the amplitude of the complex geometric optics solutions but not for the phase.
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Lemma 2.6. For any p0 , p1 , . . . pn ∈ M0 some points of M0 and L ∈ N, then there exists a function a(z) holomorphic on M0 which vanishes to order L at all p j for j = 1, . . . , n and such that a( p0 ) = 0. Moreover a(z) can be chosen to have at most polynomial growth in the ends, i.e. |a(z)| ≤ C|z| J for some J ∈ N. Proof. It suffices to find on M some meromorphic function with divisor greater or equal L L to D := e1−J . . . e−J N p1 . . . pn but not greater or equal to Dp0 and this is insured by the Riemann-Roch theorem as long as J N − n L ≥ 2g since then r (D) = −g + 1 + J N − n L and r (Dp0 ) = −g + J N − n L. Lemma 2.7. Let { p0 , p1 , .., pn } ⊂ M0 be a set of n + 1 disjoint points. Let c0 , c1 , . . . , c K ∈ C, L ∈ N, and let z be a complex coordinate near p0 such that p0 = {z = 0}. Then there exists a holomorphic function f on M0 with zeros of order at least L at each p j , such that f (z) = c0 + c1 z + ... + c K z K + O(|z| K +1 ) in the coordinate z. Moreover f can be chosen so that there is J ∈ N such that, in the ends, |∂z f (z)| = O(|z| J ) for all ∈ N0 . Proof. The proof goes along the same lines as in Lemma 2.6. By induction on K and linear combinations, it suffices to prove it for c0 = · · · = c K −1 = 0. As in the proof of Lemma 2.6, if J is taken large enough, there exists a function with divisor greater K −1 L or equal to D := e1−J . . . e−J p1 . . . pnL but not greater or equal to Dp0 . Then it N p0 suffices to multiply this function by c K times the inverse of the coefficient of z K in its Taylor expansion at z = 0. 2.4. Laplacian on weighted spaces. Let x be a smooth positive function on M0 , which is equal to |z|−1 for |z| > r0 in the ends E i {z ∈ C; |z| > 1}, where r0 is a large fixed number. We now show that the Laplacian g0 on a surface with Euclidean ends has a / N positive. right inverse on the weighted spaces x −J L 2 (M0 ) for J ∈ Lemma 2.8. For any J > −1 which is not an integer, there exists a continuous operator G mapping x −J L 2 (M0 ) to x −J −2 L 2 (M0 ) such that g0 G = Id. Proof. Let gb := x 2 g0 be a metric conformal to g0 . The metric gb in the ends can be written gb = d x 2 /x 2 + dθ S21 by using radial coordinates x = |z|−1 , θ = z/|z| ∈ S 1 ; this is thus a b-metric in the sense of Melrose [14], giving the surface a geometry of surface with cylindrical ends. Let us define for m ∈ N0 , Hbm (M0 ) := {u ∈ L 2 (M0 ; dvolgb ); (x∂x ) j ∂θk u ∈ L 2 (M0 ; dvolgb ) for all j + k ≤ m}. The Laplacian has the form gb = −(x∂x )2 + S 1 in the ends, and the indicial roots of gb in the sense of Sect. 5.2 of [14] are given by the complex numbers λ such that x −iλ gb x iλ is not invertible as an operator acting on the circle Sθ1 . Thus the indicial roots are the solutions of λ2 + k 2 = 0, where k 2 runs over the eigenvalues of S 1 , that is, k ∈ Z. The roots are simple at ±ik ∈ iZ\{0} and 0 is a double root. In Theorem 5.60 of [14], Melrose proves that gb is Fredholm on x a Hb2 (M0 ) if and only if −a is not the imaginary part of some indicial root, that is here a ∈ Z. For J > 0, the kernel of gb on the space x J Hb2 (M0 ) is clearly trivial by an energy estimate. Thus gb : x −J Hb0 (M0 ) → x −J Hb−2 (M0 ) is surjective for J > 0 and J ∈ Z, and the same then holds for gb : x −J Hb2 (M0 ) → x −J Hb0 (M0 ) by elliptic regularity.
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Now we can use Proposition 5.64 of [14], which asserts, for all positive J ∈ Z, the existence of a pseudodifferential operator G b mapping continuously x −J Hb0 (M0 ) to x −J Hb2 (M0 ) such that gb G b = Id. Thus if we set G = G b x −2 , we have g0 G = Id and G maps continuously x −J +1 L 2 (M0 ) to x −J −1 L 2 (M0 ) (note that L 2 (M0 ) = x Hb0 (M0 )). 3. Carleman Estimate for Harmonic Weights with Critical Points 3.1. The linear weight case. In this section, we prove a Carleman estimate using harmonic weights with non-degenerate critical points, in a way similar to [7]. Here however we need to work on a non-compact surface and with weighted spaces. We first consider a Morse holomorphic function ∈ H obtained from Proposition 2.1 with the condition that has linear growth in the ends, which corresponds to the case where V ∈ e−γ /x L ∞ (M0 ) for all γ > 0. The Carleman weight will be the harmonic function ϕ := Re(). We let x be a positive smooth function on M0 such that x = |z|−1 in the complex charts {z ∈ C; |z| > 1} E i covering the end E i . It is always possible to choose this chart so that ϕ has no critical point in E i . By the discussions in Sect. 2, the weight is asymptotic to a linear complex function at infinity: there exist q j ∈ C such that (z) = q0 z + q1 + O(1/|z|), |z| → ∞, and therefore its real part ϕ satisfies ϕ(z) = r0 .(x, y) + r1 + O(|z|−1 ) for some r0 ∈ R2 , r1 ∈ R if z = x + i y. In some sense, the Carleman estimate we shall prove can be compared to the more usual estimates with radial weights near infinity [9]. Let δ ∈ (0, 1) be small and let us take ϕ0 ∈ x −α L 2 (M0 ) a solution of g0 ϕ0 = x 2−δ , a solution exists by Proposition 2.8 if α > 1 + δ. Actually, by using Proposition 5.61 of [14], if we choose α < 2, then it is easy to see that ϕ0 is smooth on M0 and has polyhomogeneous expansion as |z| → ∞, with leading asymptotic in the end E i given by ϕ0 = −x −δ /δ 2 + ci log(x) + di + O(x) for some ci , di which are smooth functions in S 1 . For > 0 small, we define the convexified weight ϕ := ϕ − h ϕ0 . If h is small enough, ϕ has no critical points in the end E i . We recall from the proof of Proposition 3.1 in [7] the following estimate which is valid in any compact set K ⊂ M0 : for all w ∈ C0∞ (K ), we have C 1 1 1 w2L 2 + 2 w|dϕ|2L 2 + 2 w|dϕ |2L 2 + dw2L 2 (K ) h h h ≤ eϕ / h g e−ϕ / h w2L 2 ,
(4)
where C depends on K but not on h and . So for functions supported in the end E i , it clearly suffices to obtain a Carleman estimate in E i R2 \{|z| ≤ 1} by using the Euclidean coordinate z of the end. Proposition 3.1. Let δ ∈ (0, 1), and ϕ as above, then there exists C > 0 such that for all h > 0 small enough, and all u ∈ C0∞ (E i ), h 2 ||eϕ / h ( − λ2 )e−ϕ / h u||2L 2 ≥
δ δ C (||x 1− 2 u||2L 2 + h 2 ||x 1− 2 du||2L 2 ).
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Proof. In this proof, we shall use the semi-classical pseudo-differential calculus on R2 with Weyl quantization and we use the notations of [4]. In particular the classes of symbols which we use are defined by α −k m S k (ξ m ) := {a ∈ C ∞ (R2 , T ∗ R2 ); ∀α ∈ N4 ∃Cα > 0, |∂(x,ξ ) a(x, ξ )| ≤ C α h ξ } 1
for k, m ∈ R (recall that ξ := (1 + |ξ |2 ) 2 ). The metric g0 can be extended to R2 to be the Euclidean metric and we shall denote by the flat positive Laplacian on R2 . Let us write P := g0 − λ2 , then the operator Ph := h 2 eϕ / h Pe−ϕ / h is given by Ph = h 2 − |dϕ |2 + 2h∇ϕ .∇ − hϕ − h 2 λ2 , following the notation of [4, Chap. 4.3], it is a semiclassical operator in S 0 (ξ 2 ) with semiclassical full Weyl symbol σ (Ph ) := |ξ |2 − |dϕ |2 − h 2 λ2 + 2idϕ , ξ = a + ib. We can define A := (Ph + Ph∗ )/2 = h 2 − |dϕ |2 − h 2 λ2 and B := (Ph − Ph∗ )/2i = −2i h∇ϕ .∇ + i hϕ which have respective semiclassical full symbols a and b, i.e. A = Oph (a) and B = Oph (b) for the Weyl quantization. Notice that A, B are symmetric operators, thus for all u ∈ C0∞ (E i ), ||(A + i B)u||2 = (A2 + B 2 + i[A, B])u, u.
(5)
It is easy to check that the operator i h −1 [A, B] is a semiclassical differential operator in S 0 (ξ 2 ) with full semiclassical symbol {a, b}(ξ ) = 4(D 2 ϕ (dϕ , dϕ ) + D 2 ϕ (ξ, ξ )).
(6)
Let us now decompose the Hessian of ϕ in the basis (dϕ , θ ), where θ is a covector orthogonal to dϕ and of norm |dϕ |. This yields coordinates ξ = ξ0 dϕ + ξ1 θ (recall that |dϕ | > 0 in |z| > 1 if h/ is small) and there exist smooth functions M, N , K so that D 2 ϕ (ξ, ξ ) = |dϕ |2 (Mξ02 + N ξ12 + 2K ξ0 ξ1 ). Notice that ϕ has a polyhomogeneous expansion at infinity of the form ϕ (z) = γ .z +
h rδ h + c1 log(r ) + c2 + c3r −1 + O((1 + )r −2 ), δ2
where r = |z|, ω = z/r, γ = (γ1 , γ2 ) ∈ R2 and ci are some smooth functions on S 1 bounded uniformly by C(1 + h/) < 2C if h/ < 1 is small enough; in particular β
dϕ = γ1 dz 1 + γ2 dz 2 + O(r −1+δ ), ∂zα ∂z¯ ϕ (z) = O(r −2+δ ) for all α + β ≥ 2, where the remainders are uniform in h/ < 1, which implies that M, N , K ∈ r −2+δ L ∞ (E i ). Then one can write {a, b} = 4|dϕ |2 (M + Mξ02 + N ξ12 + 2K ξ0 ξ1 ) = 4(N (a + h 2 λ2 ) + ((M − N )ξ0 + 2K ξ1 )b/2 + (N + M)|dϕ |2 ),
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and since M + N = Tr(D 2 ϕ ) = −ϕ = hϕ0 /, we obtain h {a, b} = 4|dϕ |2 (c(z)(a + h 2 λ2 ) + (z, ξ )b + r −2+δ ), N (M − N )ξ0 + 2K ξ1 c(z) = , (z, ξ ) = . |dϕ |2 2|dϕ |2
(7)
Now we take a smooth extension of |dϕ |2 , a(z, ξ ), (z, ξ ) and r to z ∈ R2 , this can be done for instance by extending r as a smooth positive function on R2 and then extending dϕ to a smooth non-vanishing 1-form on R2 (this can be easily arranged by considering (1 − η)dϕ + Aηd x for some η ∈ C0∞ (B(0, 1)) and A > 0 well chosen) and extending dϕ0 smoothly on R2 so that |dϕ |2 is smooth positive (for h/ small) and polynomials in h/ and a, are of the same form as in {|z| > 1}. Let us define the symbol and quantized differential operator on R2 , e := 4|dϕ |2 (c(z)(a + h 2 λ2 ) + (z, ξ )b),
E := Oph (e),
and write δ
h 2 (A + B 2 ), δ δ 1 F := h −1r 1− 2 (i h −1 [A, B] − E)r 1− 2 + (A2 + B 2 ). δ
δ
δ
i h −1r 1− 2 [A, B]r 1− 2 = h F + r 1− 2 Er 1− 2 − with
(8)
We deduce from (6) and (7) the following Lemma 3.2. The operator F is a semiclassical differential operator in the class S 0 (ξ 4 ) with semiclassical principal symbol σ (F)(ξ ) =
4|dϕ|2 1 4 + (|ξ |2 − |dϕ|2 )2 + (ξ, dϕ)2 .
By the semiclassical Gårding estimate, we obtain Corollary 3.3. The operator F of Lemma 3.2 is such that there is a constant C so that Fu, u ≥
C (||u||2L 2 + h 2 ||du||2L 2 ).
Proof. It suffices to use that σ (F)(ξ ) ≥ semiclassical Gårding estimate. δ
C (1
+ |ξ |4 ) for some C > 0 and use the δ
δ
δ
So by writing i[A, B]u, u = ir 1− 2 [A, B]r 1− 2 r −1+ 2 u, r −1+ 2 u in (5) and using (8) and Corollary 3.3, we obtain that there exists C > 0 uniform in h/ < 1 such that for all u ∈ C0∞ (E i ), ||Ph u||2L 2 ≥ (A2 + B 2 )u, u + −
δ δ Ch 2 (||r −1+ 2 u||2L 2 + h 2 ||r −1+ 2 du||2L 2 ) + hEu, u
δ δ h2 (||A(r −1+ 2 u)||2L 2 + ||B(r −1+ 2 u)||2L 2 ).
(9)
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δ
δ
δ
We observe that h −1 [A, r −1+ 2 ]r 1+ 2 ∈ S 0 (ξ ) and h −1 [B, r −1+ 2 ]r 1+ 2 ∈ h S 0 (1), and thus δ
δ
δ
||A(r −1+ 2 u)||2L 2 + ||B(r −1+ 2 u)||2L 2 ≤ C (||Au||2L 2 + ||Bu||2L 2 + h 2 ||r −1+ 2 u||2L 2 δ
+h 4 ||r −1+ 2 du||2L 2 ) for some C > 0 uniform in h/ < 1. Taking h small, this implies with (9) that there exists a new constant C > 0 such that δ δ 1 Ch 2 (||r −1+ 2 u||2L 2 + h 2 ||r −1+ 2 du||2L 2 ) + hEu, u. ||Ph u||2L 2 ≥ (A2 + B 2 )u, u + 2 (10) It remains to deal with hEu, u: we first write E = 4|dϕ |2 (c(z)(A+h 2 λ2 )+Oph ()B)+ δ δ hr −1+ 2 Sr −1+ 2 , where S is a semiclassical differential operator in the class S 0 (ξ ) by the decay estimates on c(z), (z, ξ ) as z → ∞, then by Cauchy-Schwartz (and with L := Oph ()) δ
δ
δ
|h Eu, u| ≤ Ch(||Au|| L 2 + h 2 ||r −1+ 2 u|| L 2 + h||Sr −1+ 2 u|| L 2 )||r −1+ 2 u|| L 2 + Ch||Bu|| L 2 ||Lu|| L 2 δ δ 1 1 ≤ ||Au||2L 2 + h 2 ||Sr −1+ 2 u||2L 2 + Ch 2 ||r −1+ 2 u||2L 2 + ||Bu||2L 2 + Ch 2 ||Lu||2L 2 , 4 4
where C is a constant independent of h, but may change from line to line. Now we δ observe that Lr 1− 2 and S are in S 0 (ξ ) and thus δ
δ
δ
||Sr −1+ 2 u||2L 2 + ||Lu||2L 2 ≤ C(||r −1+ 2 u||2L 2 + h 2 ||r −1+ 2 du||2L 2 ), which by (10) implies that there exists C > 0 such that for all h > 0 with small enough δ δ Ch 2 (||r −1+ 2 u||2L 2 + h 2 ||r −1+ 2 du||2L 2 ) for all u ∈ C0∞ (E i ). The proof is complete.
||Ph u||2L 2 ≥
Combining now Proposition 3.1 and (4), we obtain Proposition 3.4. Let (M0 , g0 ) be a Riemann surface with Euclidean ends with x a boundary defining function of the radial compactification M 0 and let ϕ = ϕ − h ϕ0 , where ϕ is a harmonic function with non-degenerate critical points and linear growth on M0 δ and ϕ0 satisfies g0 ϕ0 = x 2−δ as above. Then for all V ∈ x 1− 2 L ∞ (M0 ) there exists an h 0 > 0, 0 and C > 0 such that for all 0 < h < h 0 , h < 0 and u ∈ C0∞ (M0 ), we have δ δ 1 1− δ 2 1 x 2 u L 2 + 2 x 1− 2 u|dϕ|2L 2 + x 1− 2 du2L 2 h h ≤ Ceϕ / h (g + V − λ2 )e−ϕ / h u2L 2 . (11) Proof. As in the proof of Proposition 3.1 in [7], by taking small enough, we see that the combination of (4) and Proposition 3.1 shows that for any w ∈ C0∞ (M0 ), C 1 1− δ 2 1 1− δ 1 1− δ 2 2 1− 2δ 2 2 2 2 x w L 2 + 2 x w|dϕ| L 2 + 2 x w|dϕ | L 2 + x dw L 2 h h h ϕ
ϕ
≤ e h ( − λ2 )e− h w2L 2 , which ends the proof.
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3.2. The quadratic weight case for surfaces. In this section, ϕ has quadratic growth at 2 infinity, which corresponds to the case where V ∈ e−γ /x L ∞ for all γ > 0. The proof when ϕ has quadratic growth at infinity is even simpler than the linear growth case. We define ϕ0 ∈ x −2 L ∞ to be a solution of g0 ϕ0 = 1; this is possible by Lemma 2.8 and one easily obtains from Proposition 5.61 of [14] that ϕ0 = −x −2 /4 + O(x −1 ) as x → 0. We let ϕ := ϕ − h ϕ0 which satisfies g0 ϕ / h = −1/. If K ⊂ M0 is a compact set, the Carleman estimate (4) in K is satisfied by Proposition 3.1 of [7]; it then remains to get the estimate in the ends E 1 , . . . , E N . But the exact same proof as in Lemma 3.1 and Lemma 3.2 of [7] gives directly that for any w ∈ C0∞ (E i ), C 1 1 1 2 2 2 2 w L 2 + 2 w|dϕ| L 2 + 2 w|dϕ | L 2 + dw L 2 ≤ eϕ / h g0 e−ϕ / h w2L 2 h h h (12) for some C > 0 independent of , h, and it suffices to glue the estimates in K and in the ends E i as in Proposition 3.1 of [7], to obtain (12) for any w ∈ C0∞ (M0 ). Then by using the triangle inequality, ||eϕ / h (g0 + V − λ2 )e−ϕ / h u|| L 2 ≤ ||eϕ / h g0 e−ϕ / h u|| L 2 + C||u|| L 2 for some C depending on λ, ||V || L ∞ , we see that the V − λ2 term can be absorbed by the left-hand side of (12) and we finally deduce Proposition 3.5. Let (M0 , g0 ) be a Riemann surface with Euclidean ends and let ϕ = ϕ − h ϕ0 where ϕ is a harmonic function with non-degenerate critical points and quadratic growth on M0 and ϕ0 satisfies g0 ϕ0 = 1 with ϕ0 ∈ x −2 L ∞ (M0 ). Then for all V ∈ L ∞ there exists an h 0 > 0, 0 and C > 0 such that for all 0 < h < h 0 , h < 0 and u ∈ C0∞ (M0 ), 1 C 1 2 2 2 u L 2 + 2 ||u|dϕ||| L 2 + du L 2 ≤ eϕ / h (g0 + V − λ2 )e−ϕ / h u2L 2 . h h The main difference with the linear weight case is that one can use a convexification which has quadratic growth at infinity which allows to absorb the λ2 term, while it was not the case for the linearly growing weights. 4. Complex Geometric Optics on a Riemann Surface with Euclidean Ends As in [1,7,10], the method for identifying the potential at a point p is to construct complex geometric optic solutions depending on a small parameter h > 0, with phase a Morse holomorphic function with a non-degenerate critical point at p, and then to apply the stationary phase method. Here, in addition, we need the phase to be of linear growth at infinity if V ∈ e−γ /x L ∞ for all γ > 0 while the phase has to be of quadratic growth 2 at infinity if V ∈ e−γ /x L ∞ for all γ > 0. We shall now assume that M0 is a non-compact surface with genus g with N ends equipped with a metric g0 which is Euclidean in the ends, and V is a C 1,α function in M0 . Moreover, if V ∈ e−γ /x L ∞ for all γ > 0, we ask that N ≥ max(2g + 1, 2) while 2 if V ∈ e−γ /x L ∞ for all γ > 0, we assume that N ≥ g + 1. As above, let us use a smooth positive function x which is equal to 1 in a large compact set of M0 and is equal
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to x = |z|−1 in the regions |z| > r0 of the ends E i {z ∈ C; |z| > 1}, where r0 is a fixed large number. This function is a boundary defining function of the radial compactification of M0 in the sense of Melrose [14]. To construct the complex geometric optics solutions, we will need to work with the weighted spaces x −α L 2 (M0 ), where α ∈ R+ . Let H be the finite dimensional complex vector space defined in the beginning of Sect. 2.3. Choose p ∈ M0 such that there exists a Morse holomorphic function = ϕ + iψ ∈ H on M0 , with a critical point at p; there is a dense set of such points by Proposition 2.1. The purpose of this section is to construct solutions u on M0 of ( − λ2 + V )u = 0 of the form u = e/ h (a + r1 + r2 )
(13)
for h > 0 small, where a ∈ x −J +1 L 2 with J ∈ R+ \N is a holomorphic function on M0 , obtained by Lemma 2.6, such that a( p) = 0 and a vanishing to order L (for some fixed large L) at all other critical points of , and finally r1 , r2 will be remainder terms which are small as h → 0 and have particular properties near the critical points of . More precisely, eϕ0 / r2 will be a o L 2 (h), and r1 will be a Ox −J L 2 (h), but with an explicit expression, which can be used to obtain sufficient information in order to apply the stationary phase method. 4.1. Construction of r1 . We want to construct r1 = Ox −J L 2 (h) which satisfies e−/ h (g0 − λ2 + V )e/ h (a + r1 ) = Ox −J L 2 (h) for some large J ∈ R+ \N so that a ∈ x −J +1 L 2 . Let G be the operator of Lemma 2.8, mapping continuously x −J +1 L 2 (M0 ) to −J ¯ x −1 L 2 (M0 ). Then clearly ∂∂G = 2i −1 when acting on x −J +1 L 2 , here −1 is the inverse of mapping functions to 2-forms. First, we will search for r1 satisfying e−2iψ/ h ∂e2iψ/ h r1 = −∂G(a(V − λ2 )) + ω + Ox −J H 1 (h)
(14)
with ω ∈ x −J L 2 (M0 ) a holomorphic 1-form on M0 and r1 x −J L 2 = O(h). Indeed, using the fact that is holomorphic we have ¯ ¯ ¯ −/ h ∂e/ h = −2i ∂e ¯ − h (−) ∂e h (−) e−/ h g0 e/ h = −2i ∂e ¯ −2iψ/ h ∂e2iψ/ h = −2i ∂e 1
1
and applying −2i ∂¯ to (14), this gives e−/ h (g0 + V )e/ h r1 = −a(V − λ2 ) + Ox −J L 2 (h). Writing −∂G(a(V − λ2 )) =: c(z)dz in local complex coordinates, c(z) is C 2,α by elliptic regularity and we have 2i∂z¯ c(z) = a(V − λ2 ), therefore ∂z ∂z¯ c( p ) = ∂z¯2 c( p ) = 0 at each critical point p = p by construction of the function a. Therefore, we deduce that at each critical point p = p, c(z) has Taylor series expansion 2j=0 c j z j + O(|z|2+α ). That is, all the lower order terms of the Taylor expansion of c(z) around p are polynomials of z only. By Lemma 2.7, and possibly by taking J larger, there exists a holomorphic function f ∈ x −J L 2 such that ω := ∂ f has Taylor expansion equal to that of ∂G(a(V − λ2 ))
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at all critical points p = p of . We deduce that, if b := −∂G(a(V −λ2 ))+ω = b(z)dz, we have |∂z¯m ∂z b(z)| = O(|z|2+α−−m ),
for + m ≤ 2, at critical points p = p,
|b(z)| = O(|z|),
if p = p.
(15)
Now, we let χ1 ∈ C0∞ (M0 ) be a cutoff function supported in a small neighbourhood U p of the critical point p and identically 1 near p, and χ ∈ C0∞ (M0 ) is defined similarly with χ = 1 on the support of χ1 . We will construct r1 to be a sum r1 = r11 + hr12 , where r11 is a compactly supported approximate solution of (14) near the critical point p of and r12 is a correction term supported away from p. We define locally in complex coordinates centered at p and containing the support of χ , r11 := χ e−2iψ/ h R(e2iψ/ h χ1 b), (16) 1 −1 ∞ ¯ where R f (z) := −(2πi) R2 z¯ −ξ¯ f d ξ ∧ dξ for f ∈ L compactly supported is the classical Cauchy operator inverting locally ∂z (r11 is extended by 0 outside the neighbourhood of p). The function r11 is in C 3,α (M0 ) and we have e−2iψ/ h ∂(e2iψ/ h r11 ) = χ1 (−∂G(a(V − λ2 )) + ω) + η with η := e−2iψ/ h R(e2iψ/ h χ1 b)∂χ .
(17)
We then construct r12 by observing that b vanishes to order 2 + α at critical points of other than p (from (15)), and ∂χ = 0 in a neighbourhood of any critical point of ψ, so we can find r12 satisfying 2ir12 ∂ψ = (1 − χ1 )b.
(18) ∗ M T1,0 0
This is possible since both ∂ψ and the right-hand side are valued in and ∂ψ has 2,α finitely many isolated zeroes on M0 : r12 is then a function which is in C (M0 \P), where P := { p1 , . . . , pn } is the set of critical points other than p, it extends to a function in C 1,α (M0 ) and it satisfies in local complex coordinates z at each p j , β γ
|∂z¯ ∂z r12 (z)| ≤ C|z|1+α−β−γ , β + γ ≤ 2 by using also the fact that ∂ψ can be locally considered as a smooth function with a zero of order 1 at each p j . Moreover b ∈ x −J H 2 (M0 ), thus r1 ∈ x −J H 2 (M0 ) and we have e−2iψ/ h ∂(e2iψ/ h r1 ) = b + h∂r12 + η = −∂G(a(V − λ2 )) + ω + h∂r12 + η. Lemma 4.1. The following estimates hold true: ||η|| H 2 (M0 ) = O(| log h|), η H 1 (M0 ) ≤ O(h| log h|), ||x J ∂r12 || H 1 (M0 ) = O(1), r12 )|| L 2 = o(h), ||x J r1 || L 2 = O(h), ||x J (r1 − h where r12 solves 2i r12 ∂ψ = b. Proof. The proof is exactly the same as the proof of Lemma 4.2 in [8], except that one needs to add the weight x J to have bounded integrals. As a direct consequence, we have Corollary 4.2. With r1 = r11 + hr12 , there exists J > 0 such that ||e−/ h (g0 − λ2 + V )e/ h (a + r1 )||x −J L 2 (M0 ) = O(h| log h|).
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4.2. Construction of r2 . In this section, we complete the construction of the complex geometric optic solutions. We deal with the general case of surfaces and we shall show the following Proposition 4.1. If ϕ0 is the subharmonic function constructed in Sect. 3, then for small enough there exist solutions to (g0 − λ2 + V )u = 0 of the form u = e/ h (a + r1 + r2 ) with r1 = r11 + hr12 constructed in the previous section and r2 ∈ e−ϕ0 / L 2 satisfying eϕ0 / r2 L 2 ≤ Ch 3/2 | log h|. This is a consequence of the following lemma (which follows from the Carleman estimate obtained in Sect. 3 above) δ
Lemma 4.3. Let δ ∈ (0, 1), V ∈ x 1− 2 L ∞ (M0 ), and ϕ = ϕ − h ϕ0 be a weight with linear growth at infinity as in Proposition 3.4. For all f ∈ L 2 (M0 ) and all h > 0 small enough, there exists a solution v ∈ L 2 (M0 ) to the equation δ
e−ϕ / h (g − λ2 + V )eϕ / h v = x 1− 2 f
(19)
satisfying 1
v L 2 (M0 ) ≤ Ch 2 f L 2 (M0 ) . If ϕ has quadratic growth at infinity, the same result is true when V ∈ L ∞ (M0 ) but δ x 1− 2 f can be replaced by f ∈ L 2 in (19). Proof. The proof is based on a duality argument. Let Ph := eϕ / h (g − λ2 + V )e−ϕ / h δ and for all h > 0 the real vector space A := {u ∈ x −1+ 2 H 1 (M0 ); Ph u ∈ L 2 (M0 )} equipped with the real scalar product (u, w)A := Ph u, Ph w L 2 . By the Carleman estimate of Proposition 3.4, the space A is a Hilbert space equipped with the scalar product above if h < h 0 , and thus the linear functional L : w → 1 1− 2δ f w dvolg0 on A is continuous with norm bounded by Ch 2 || f || L 2 by PropM0 x osition 3.4, and by Riesz theorem there is an element u ∈ A such that (., u)A = L and with norm bounded by the norm of L. It remains to take v := Ph u which solves δ Ph∗ v = x 1− 2 f , where Ph∗ = e−ϕ / h (g − λ2 + V )eϕ / h is the adjoint of Ph and v satisfies the desired norm estimate. The proof when the weight ϕ has quadratic growth at infinity is the same, but improves slightly due to the Carleman estimate of Proposition 3.5. Proof of Proposition 4.1. We first solve the equation
δ δ ( + V − λ2 )eϕ / h r2 = x 1− 2 eϕ / h x −1+ 2 e−ϕ / h ( + V − λ2 )e/ h (a + r1 ) by using Lemma 4.3 and the fact that for J large, there is C > 0 such that for all h < h 0 , δ
||x −1+ 2 e−ϕ / h ( + V − λ2 )e/ h (a + r1 )|| L 2 ≤ C||x J e−/ h ( − λ2 + V )e/ h (a + r1 )|| L 2 ,
since x −J −1 eϕ0 / ∈ L ∞ (M0 ) for all J (recall that ϕ0 ∼ −x −δ /δ 2 as x → 0). But now the right-hand side is bounded by O(h| log h|) according to Corollary 4.2, therefore we set r2 := −e−iψ/ h−ϕ0 / r2 which satisfies (g0 − λ2 + V )e/ h (a + r1 + r2 ) = 0 and, by Lemma 4.3, the norm estimate ||eϕ0 / r2 || L 2 ≤ O(h 3/2 | log h|).
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5. Scattering on Surfaces with Euclidean Ends Let (M0 , g0 ) be a surface with Euclidean ends and V ∈ e−γ /x L ∞ (M0 ) for some γ . The scattering theory in this setting is described for instance in Melrose [15]; here we will follow this presentation (see also Sect. 3 in Uhlmann-Vasy [27] for the Rn case). First, using standard methods in scattering theory, we define the resolvent on the continuous spectrum as follows Lemma 5.1. The resolvent R V (λ) := (g0 + V − λ2 )−1 admits a meromorphic extension from {Im(λ) < 0} to {Im(λ) ≤ A, Re(λ) = 0}, as a family of operators mapping e−γ /x L 2 (M0 ) to eγ /x L 2 (M0 ) for any γ > A. Moreover, for λ ∈ R\{0} not a pole, R V (λ) maps continuously x α L 2 to x −α L 2 for any α > 1/2. Proof. The statement is known for V = 0 and M0 = R2 by using the explicit formula of the resolvent convolution kernel on R2 in terms of Hankel functions (see for instance [15]), we shall denote R0 (λ) this continued resolvent. More precisely, for all A > 0, the operator R0 (λ) continues analytically from {Im(λ) < 0} to {Im(λ) ≤ A, Re(λ) = 0} as a family of bounded operators mapping e−γ /x L 2 to eγ /x L 2 for any γ > A. Now we can set χ ∈ C0∞ (M0 ) such that 1 − χ is supported in the ends E i , and let χ0 , χ1 ∈ C0∞ (M0 ) such that (1 − χ0 ) = 1 on the support of (1 − χ ) and χ1 = 1 on the support of χ . Let λ0 ∈ −iR+ with iλ0 0, then the resolvent R V (λ0 ) is well defined from L 2 (M0 ) to H 2 (M0 ) since the Laplacian is essentially self-adjoint [24, Prop. 8.2.4], and we have a parametrix E(λ) := (1 − χ0 )R0 (λ)(1 − χ ) + χ1 R V (λ0 )χ which satisfies (g0 − λ2 + V )E(λ) = 1 + K (λ), K (λ) := ([g0 , χ1 ] − (λ2 − λ20 )χ1 )R V (λ0 )χ −[g0 , χ0 ]R0 (λ)(1 − χ ) + V E(λ), where here we use the notation R0 (λ) for an integral kernel on M0 , which in the charts {z ∈ R2 ; |z| > 1} corresponding to the ends E 1 , . . . E N , is given by the integral kernel of (R2 − λ2 )−1 . Using the explicit expression of the convolution kernel of R0 (λ) in the ends (see for instance Sect. 1.5 of [15]) and the decay assumption on V , it is direct to see that for Im(λ) < A, Re(λ) = 0, the map λ → K (λ) is a compact analytic family of bounded operators from e−γ /x L 2 to e−γ /x L 2 for any γ > A. Moreover 1 + K (λ0 ) is invertible since ||K (λ0 )|| L 2 →L 2 ≤ 1/2 if iλ0 is large enough. Then by analytic Fredholm theory, the resolvent R V (λ) has a meromorphic extension to Im(λ) < A, Re(λ) = 0 as a bounded operator from e−γ /x L 2 to eγ /x L 2 if γ > A, given by R V (λ) = E(λ)(1 + K (λ))−1 . Now (1+ K (λ))−1 = 1+ Q(λ) for some Q(λ) = −K (λ)(1+ K (λ))−1 mapping e−γ /x L 2 to itself for any γ > A, which proves the mapping properties of R V (λ) on exponential weighted spaces. For the mapping properties on {Re(λ) = 0}, a similar argument works.
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A corollary of this lemma is the mapping property Corollary 5.2. For λ ∈ R\{0} not a pole of R V (λ), and f ∈ e−γ /x L ∞ for some γ > 0, then there exists v ∈ C ∞ (∂ M 0 ) such that 1
R V (λ) f − x 2 e−iλ/x v ∈ L 2 . Proof. Using the expression R V (λ) = E(λ)(1 + Q(λ)) of the proof of Lemma 5.1, it suffices to know the mapping property of E(λ) on e−γ /x L 2 , but since outside a compact set (i.e. in the ends) E(λ) is given by the free resolvent on R2 ; this amounts to proving the statement in R2 , which is well-known: for instance, this is proved for f ∈ C0∞ (R2 ) in Sect. 1.7 [15] but the proof extends easily to f ∈ e−γ /x L ∞ (R2 ) since the only used assumption on f for applying a stationary phase argument is actually that the Fourier transform fˆ(z) has a holomorphic extension in a complex neighbourhood of R2 . We also have a boundary pairing, the proof of which is exactly the same as [15, Lemma 2.2] (see also Proposition 3.1 of [27]). Lemma 5.3. For λ > 0 and V ∈ e−γ /x L ∞ (M0 ), if u ± ∈ x −α L 2 (M0 ) for some α > 1/2 and (g0 − λ2 + V )u ± ∈ x α L 2 (M0 ) with 1
1
1
1
u + − x 2 eiλ/x f ++ − x 2 e−iλ/x f +− ∈ L 2 , u − − x 2 eiλ/x f −+ − x 2 e−iλ/x f −− ∈ L 2 for some f ±± ∈ C ∞ (∂ M 0 ), then
u + , (g0 + V − λ2 )u − − (g0 + V − λ2 )u + , u − = 2iλ
∂ M0
( f ++ f −+ − f +− f −− ),
N S 1 is induced by the metric x 2 g| where the volume form on ∂ M 0 i=1 T ∂ M0 .
As a corollary, the same exact arguments as in Secs. 2.2 to 2.5 in [15] show1 Corollary 5.4. The operator R V (λ) is well defined for any λ ∈ R\{0} as a bounded operator from x α L 2 to x −α L 2 if α > 1/2. In R2 there is a Poisson operator P0 (λ) mapping C ∞ (S 1 ) to x −α L 2 (R2 ) for α > 1/2, which satisfies that for any f + ∈ C ∞ (S 1 ) there exists f − ∈ C ∞ (S 1 ) such that 1
1
P0 (λ) f + − x 2 eiλ/x f + − x 2 e−iλ/x f − ∈ L 2 , ( − λ2 )P0 (λ) f + = 0. We can therefore define in our case a similar Poisson operator PV (λ) mapping C ∞ (∂ M 0 ) to x −α L 2 for α > 1/2, by PV (λ) f + := (1 − χ )P0 (λ) f + − R V (λ)(g0 + V − λ2 )(1 − χ )P0 (λ) f + ,
(20)
where 1 − χ ∈ C ∞ (M0 ) equals 1 in the ends E i and P0 (λ) denotes here the Schwartz kernel of the Poisson operator on R2 pulled back to each of the Euclidean ends E i of M0 in the obvious way. Then, since (g0 + V −λ2 )(1−χ )P0 (λ) f + ∈ e−γ /x L 2 for all γ > 0, it suffices to use Corollaries 5.2 and 5.4 to see that it defines an analytic Poisson operator 1 In [15], a unique continuation is used for Schwartz solutions of (+ V −λ2 )u = 0 when V is a compactly supported potential on Rn , but the same result is also true in our setting, it just suffices to use a standard local Carleman estimate at infinity to obtain the unique continuation [9].
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PV (λ) on λ ∈ R\{0} satisfying that for all f + ∈ C ∞ (∂ M0 ), there exists f − ∈ C ∞ (∂ M 0 ) such that 1
1
PV (λ) f + − x 2 eiλ/x f + − x 2 e−iλ/x f − ∈ L 2 , ( + V − λ2 )PV (λ) f + = 0. (21) Moreover, it is easily seen to be the unique solution of (21): indeed, if two such solu1 tions exist then the difference is a solution u with asymptotic x 2 e−iλ/x f − + L 2 for some f − ∈ C ∞ (∂ M 0 ), but applying Lemma 5.3 with u − = u + = u shows that f − = 0, thus u ∈ L 2 , which implies u = 0 by Corollary 5.4. Definition 5.5. The scattering matrix SV (λ) : C ∞ (∂ M 0 ) → C ∞ (∂ M 0 ) for λ ∈ R\{0} is defined to be the map SV (λ) f + := f − , where f − is given by the asymptotic 1
1
PV (λ) f + = x 2 eiλ/x f + + x 2 e−iλ/x f − + g, with g ∈ L 2 . We remark that, using Lemma 5.3 and the uniqueness of the Poisson operator, one easily deduces for λ ∈ R\{0}, SV (λ)∗ = SV (−λ) = SV (λ)−1 ,
(22)
where the scalar product on L 2 (∂ M 0 ) is induced by the metric x 2 g0 |T ∂ M 0 . We can now state a density result similar to Proposition 3.3 of [27]: Proposition 5.6. If V ∈ e−γ0 /x L ∞ (M0 ) (resp. V ∈ e−γ0 /x L ∞ (M0 )) for some γ0 > 0, and λ ∈ R\{0}, then for any 0 < γ < γ < γ0 the set 2
F := {PV (λ) f + ; f + ∈ C ∞ (∂ M 0 )} is dense in the null space of g0 + V −λ2 in eγ /x L 2 (M0 ) for the topology of eγ 2 2 (resp. in eγ /x L 2 (M0 ) for the topology of eγ /x L 2 (M0 )).
/x
L 2 (M0 )
Proof. First assume V ∈ e−γ0 /x L ∞ (M0 ). Let w ∈ e−γ /x L 2 be orthogonal to F, and set u − := R V (λ)w and u + = PV (λ) f ++ for some f ++ ∈ C ∞ (∂ M 0 ). Then, define 1 f −− ∈ C ∞ (∂ M 0 ) by R V (λ)w − x 2 e−iλ/x f −− ∈ L 2 , and from Lemma 5.3 we obtain f +− , f −− = 0 since w, PV (λ) f ++ = 0 by assumption. Since f +− = SV (λ) f ++ is arbitrary, then f −− = 0 and u − ∈ L 2 . In particular, from the parametrix constructed in the proof of Lemma 5.1, R V (λ)w − (1 − χ0 )R0 (λ)(1 − χ )(1 + Q(λ))w ∈ L 2
with (1 + Q(λ))w ∈ e−γ /x L 2 . Since in each end, R0 (λ) is the integral kernel of the free resolvent of the Euclidean Laplacian on R2 and (1 − χ0 ) and (1 − χ ) are supported in the ends, we can view the term (1 − χ0 )R0 (λ)(1 − χ )(1 + Q(λ))w as a disjoint sum (over the ends) of functions on R2 of the form 1 ei zξ (ξ 2 − λ2 − i0)−1 fˆ(ξ )dξ, (23) (1 − χ0 (z)) (2π )2 R2
where in each end E i , f = (1 − χ )(1 + Q(λ))w ∈ e−γ /x L 2 (E i ) can be considered as a function in e−γ |z| L 2 (R2 ). By the Paley-Wiener theorem, fˆ is holomorphic in a strip U = {|Im(ξ )| < γ } with bound supη≤γ || fˆ(· + iη)|| L 2 (R2 ) < ∞ for all γ < γ , so the fact that (23) is in L 2 implies that fˆ vanishes at the real sphere {ξ ∈ R2 ; ξ 2 = λ2 },
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and thus there exists h holomorphic in U such that fˆ(ξ ) = (ξ 2 − λ2 )h(ξ ) (see e.g. the proof of Lemma 2.5 in [18]), and satisfying the same types of L 2 estimates as fˆ in U on lines Im(ξ ) = cst. By the Paley-Wiener theorem again, we deduce that (23) is in e−γ |z| L 2 and thus R V (λ)w ∈ e−γ /x L 2 (M0 ) for any γ < γ . Then if v ∈ eγ /x L 2 (M0 ) and (g0 + V − λ2 )v = 0, one has by integration by parts 0 = R V (λ)w, (g0 + V − λ2 )v = w, v, which ends the proof in the case V ∈ e−γ0 /x L ∞ (M0 ). The quadratic decay case V ∈ 2 e−γ0 /x L ∞ (M0 ) is exactly similar but instead of Paley-Wiener theorem, we use Corollary 2 2 2 2 2 7.3 and the inclusions e−γ /x L 2 ⊂ e−γ /x L 1 ∩e−γ /x L 2 and e−γ /x L ∞ ⊂ e−γ /x L 2 for all γ < γ < γ . 6. Identifying the Potential 6.1. The case of a surface. On a Riemann surface (M0 , g0 ) with N Euclidean ends and genus g, we assume that V1 , V2 ∈ C 1,α (M0 ) are two real valued potentials such that the respective scattering operators SV1 (λ) and SV2 (λ) agree for a fixed λ > 0. We also assume that for all γ > 0, −γ /x ∞ L (M0 ) if N ≥ max(2g + 1, 2) e V1 , V2 ∈ 2 e−γ /x L ∞ (M0 ) if N ≥ g + 1. By considering the asymptotics of u 1 := PV1 (λ) f 1 and PV2 (−λ) f 2 for f i ∈ C ∞ (∂ M 0 ) we easily have by integration by parts that (V1 − V2 )u 1 u 2 dvolg0 = −2iλ SV1 (λ) f 1 . f 2 − f 1 .SV2 (−λ) f 2 M0 ∂ M0 = −2iλ (SV1 (λ) − SV2 (λ)) f 1 . f 2 = 0 (24) ∂ M0
by using (22). From Proposition 5.6, this implies by density that, if V ∈ e−γ /x L ∞ (resp. 2 V ∈ e−γ /x L ∞ ) for all γ > 0, then for all solutions u i of (g0 + Vi − λ2 )u i = 0 in 2 eγ /x L 2 (M0 ) (resp. u i ∈ eγ /x L 2 (M0 )) for some γ > 0, we have (V1 − V2 )u 1 u 2 dvolg0 = 0. (25) M0
We shall now use our complex geometric optics solutions as special solutions in 2 the weighted space e−γ / hx L 2 (M0 ) (resp. e−γ / hx L 2 (M0 )) for some γ > 0 if 2 V ∈ e−γ /x L ∞ (resp. V ∈ e−γ /x L ∞ ) for all γ > 0. Let p ∈ M0 be such that, using Proposition 2.1, we can choose a holomorphic Morse function = ϕ + iψ with linear or quadratic growth on M0 (depending on the topological assumption), with a critical point at p. Then for the complex geometric optics solutions u 1 , u 2 with phase constructed in Sect. 4, the identity (25) holds true. We will then deduce Proposition 6.1. Let λ ∈ (0, ∞) and assume that SV1 (λ) = SV2 (λ), then V1 ( p) = V2 ( p).
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Proof. Let u 1 and u 2 be solutions on M0 to (g + V j − λ2 )u j = 0 constructed in Sect. 4 with phase for u 1 and − for u 2 , thus of the form u 1 = e/ h (a + r11 + r21 ), u 2 = e−/ h (a + r12 + r22 ). We have the identity M0
u 1 (V1 − V2 )u 2 dvolg0 = 0.
Then by using the estimates in Lemma 4.1 and Proposition 4.1 we have, as h → 0,
e
2iψ/ h
M0
|a| (V1 − V2 ) dvolg0 + h 2
M0
1 2 )(V − V ) dvol + o(h) = 0, e2iψ/ h (a r12 + a r12 1 2 g0
where r12 ∈ L ∞ (M0 ) are defined in Lemma 4.1, with the superscript j referring to the j solution for the potential V j ; in particular these functions r12 are independent of h. By splitting Vi (·) = (Vi (·)− Vi ( p))+ Vi ( p) and using the C 1,α regularity assumption on Vi , one can use the stationary phase for the Vi ( p) term and integration by parts to gain a power of h for the Vi (·) − Vi ( p) term (see the proof of Lemma 5.4 in [8] for details) to deduce j
M0
e2iψ/ h |a|2 (V1 − V2 ) dvolg0 = Ch(V1 ( p) − V2 ( p)) + o(h)
for some C = 0. Therefore, Ch(V1 ( p) − V2 ( p)) + h M0
1 2 )(V − V ) dvol = o(h). e2iψ/ h (a r12 + a r12 1 2 g0
Now to deal with the middle terms, it suffices to apply a Riemann-Lebesgue type argument like Lemma 5.3 of [8] to deduce that it is a o(h). The argument is simply to approximate the amplitude in the L 1 (M0 ) norm by a smooth compactly supported function and then use stationary phase to deal with the smooth function. We have thus proved that V1 ( p) = V2 ( p) by taking h → 0. Acknowledgements. M.S. is supported partly by the Academy of Finland. C.G. is supported by ANR grant ANR-09-JCJC-0099-01, and is grateful to the Math. Dept. of Helsinki where part of this work was done. L.T is partially supported by NSF Grant No. DMS-0807502. We thank the referee for a careful reading and comments.
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7. Appendix To obtain mapping properties of the resolvent of R2 acting on functions with Gaussian decay, we shall give two lemmas on Fourier transforms of functions with Gaussian decay. Lemma 7.1. Let f (z) ∈ e−γ |z| L 2 (R2 ) for some γ > 0. Then the Fourier transform fˆ(ξ ) extends analytically to C2 and for all ξ, η ∈ R2 , 2
|| fˆ(ξ + iη)|| L 2 (R2 ,dξ ) ≤ 2π e
|η|2 4γ
||eγ |z| f || L 2 (R2 ) . 2
If f (z) ∈ e−γ |z| L 1 (R2 ) for some γ > 0 then 2
sup | fˆ(ξ + iη)| ≤ e
ξ ∈R2
|η|2 4γ
||eγ |z| f || L 1 (R2 ) . 2
Proof. The first statement is clear. For the bound, we write |η|2 η 2 |η|2 η 2 −γ |z− 2γ | γ |z|2 −γ |z− 2γ | γ |z|2 e−iξ.z e e f (z)dz = e 4γ Fz→ξ (e e f (z)). fˆ(ξ + iη) = e 4γ R2
−γ |z−
η 2 |
2γ e γ |z| f (z) is in L 2 (R2 , dz) and its norm is bounded uniBut the function e 2 γ |z| formly by ||e f || L 2 , thus it suffices to use the Plancherel theorem to obtain the desired bound. The L ∞ bound is similar. 2
Lemma 7.2. Let F(ξ + iη) be a complex analytic function on R2 + iR2 = C2 such that there is C > 0 and γ > 0 with ||F(ξ + iη)|| L 2 (R2 ,dξ ) ≤ Ce
|η|2 4γ
and sup |F(ξ + iη)| ≤ Ce ξ ∈R2
|η|2 4γ
.
F(ξ ) −γ |z| L ∞ (R2 ). If F vanishes on the real submanifold {|ξ |2 = λ2 }, then Fξ−1 →z ( |ξ |2 −λ2 ) ∈ e 2
Proof. First by analyticity of F, one has that F vanishes on the complex hypersurface Mλ := {ζ ∈ C2 ; ζ.ζ = λ2 } (see for instance the proof of Lemma 2.5 of [18]), and in particular G(ζ ) = F(ζ )/(ζ.ζ − λ2 ) is an analytic function on C2 . We will first prove that for each η ∈ R2 , G(ξ + iη) ∈ L 1 (R2 , dξ ) ∩ L ∞ (R2 , dξ ) and ||G(ξ + iη)|| L 1 (R2 ,dξ ) ≤ Ce
|η|2 4γ
.
(26)
If |η| ≤ 2 we choose the disc B := {ξ ∈ R2 ; |ξ |2 < 2(4 + λ2 )} and let ζ := ξ + iη. Then ||G(ξ + iη)|| L 1 (B,dξ ) and ||(ζ.ζ − λ2 )−1 || L 2 (R2 \B,dξ ) are uniformly bounded for |η| ≤ 2, and we obtain by Cauchy-Schwarz that (26) holds for |η| ≤ 2. For the case |η| > 2 we define Uη := {ξ ∈ R2 ; |ζ.ζ − λ2 | > |η|} and note that sup ||(ζ.ζ − λ2 )−1 || L 1 (R2 \Uη ,dξ ) < ∞,
|η|>2
sup ||(ζ.ζ − λ2 )−1 || L 2 (Uη ,dξ ) < ∞.
|η|>2
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These results follow by decomposing the integration sets to parts where one can change coordinates ξ1 + iξ2 to ξ˜1 + i ξ˜2 := ζ.ζ − λ2 , and by evaluating simple integrals. Then (26) follows from Cauchy-Schwarz and the estimates for F. Let η = 2γ z; we use a contour deformation from R2 to 2iγ z + R2 in C2 , ei z.ξ G(ξ )dξ = ei z.(ξ +2iγ z) G(ξ + 2iγ z)dξ, R2
R2
which is justified by the fact that G(ξ + iη) ∈ L 1 (R2 × K , dξ dη) for any compact set K in R2 by the uniform bound (26). Now using (26) again shows that 2 i z.ξ e G(ξ )dξ ≤ Ce−γ |z| , R2
which ends the proof.
Corollary 7.3. Let f (z) ∈ e−γ |z| L 2 (R2 ) ∩ e−γ |z| L 1 (R2 ) for some γ > 0. Assume that its Fourier transform fˆ(ξ ) vanishes on the sphere {|ξ | = |λ|}, then one has
fˆ(ξ ) 2 −1 Fξ →z ∈ e−γ |z| L ∞ (R2 ). 2 2 |ξ | − λ 2
2
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Commun. Math. Phys. 303, 785–824 (2011) Digital Object Identifier (DOI) 10.1007/s00220-011-1205-1
Communications in
Mathematical Physics
Concentration of Measure for Quantum States with a Fixed Expectation Value Markus P. Müller1,2 , David Gross3,4 , Jens Eisert2,5 1 Institute of Mathematics, Technical University of Berlin, 10623 Berlin, Germany 2 Institute of Physics and Astronomy, University of Potsdam, 14476 Potsdam, Germany.
E-mail:
[email protected]
3 Institute for Theoretical Physics, Leibniz University Hannover, 30167 Hannover, Germany 4 Institute for Theoretical Physics, ETH Zürich, 8093 Zürich, Switzerland.
E-mail:
[email protected]
5 Institute for Advanced Study Berlin, 14193 Berlin, Germany. E-mail:
[email protected]
Received: 20 April 2010 / Accepted: 24 October 2010 Published online: 8 March 2011 – © Springer-Verlag 2011
Abstract: Given some observable H on a finite-dimensional quantum system, we investigate the typical properties of random state vectors |ψ that have a fixed expectation value ψ|H |ψ = E with respect to H . Under some conditions on the spectrum, we prove that this manifold of quantum states shows a concentration of measure phenomenon: any continuous function on this set is almost everywhere close to its mean. We also give a method to estimate the corresponding expectation values analytically, and we prove a formula for the typical reduced density matrix in the case that H is a sum of local observables. We discuss the implications of our results as new proof tools in quantum information theory and to study phenomena in quantum statistical mechanics. As a by-product, we derive a method to sample the resulting distribution numerically, which generalizes the well-known Gaussian method to draw random states from the sphere. 1. Introduction The term concentration of measure phenomenon refers to the observation that in many high-dimensional spaces “continuous functions are almost everywhere close to their mean”. A well-known illustration is the fact that on a high-dimensional sphere “most points lie close to the equator”. In other words, the values of the coordinate functions concentrate about 0, their mean. On the sphere, the effect exists not only for coordinate functions, but for any Lipschitz-continuous function. The result—known as Lévy’s Lemma—has surprisingly many applications in both mathematics and physics (see below). Our main contribution is a “Lévy’s Lemma”-type concentration of measure theorem for the set of quantum states with fixed expectation value. More concretely, suppose that we are given any observable H = H † with eigenvalues {E k }nk=1 on Cn . In the following, we will often call H a “Hamiltonian” and E k the “energy levels”, but this is not the only possible physical interpretation. We fix some arbitrary value E, and we are interested in
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the set of pure quantum states with fixed expectation value E, i.e., M E := |ψ ∈ Cn | ψ|H |ψ = E and ψ = 1 . Our main Theorem 1 shows that, subject to conditions on the spectrum of H , any continuous function on M E concentrates about its mean. The motivation for the approach taken here is two-fold.
1.1. Motivation 1: The probabilistic method. Beyond being a geometric curiosity, the concentration of measure effect is a crucial ingredient to an extremely versatile proof technique: the probabilistic method [1]. Recall the basic idea. Assume, by way of example, one wants to ascertain the existence of a state vector ψ on n qubits, such that ψ is “highly entangled” with respect to any bipartition of the n systems into two sets. The problem seems daunting: There are exponentially many ways of dividing the composite system into two parts. For any bipartition, we need to make a statement about the entropy of the eigenvalue distribution of the reduced density matrix – a highly non-trivial function. Lastly, in any natural parametrization of the set of state vectors, a change of any of the parameters will affect the vast majority of the constraints simultaneously. Given these difficulties, it is an amazing fact that the probabilistic method reduces the problem above to a simple lemma with a schematic proof (detailed, e.g., in Ref. [2,3]). Neither the non-trivial nature of the entropy function, nor the details of the tensor product space from which the vectors are drawn enters the proof. Only extremely coarse information – the Lipschitz constant of the entropy and the concentration properties of the unit sphere – are needed. Consequently, proofs based on concentration properties are now common specifically in quantum information theory. Examples include the investigation of “generic entanglement” [3], random coding arguments to assess optimum rates in quantum communication protocols, state merging [5], the celebrated counterexample to the additivity conjecture in quantum information theory [6], or the resource character of quantum states for measurement-based computing [7,8]. The tremendous reduction of complexity afforded by the probabilistic method motivates our desire to prove measure concentration for other naturally occurring spaces, besides the sphere. For the set of “states under a constraint”, Theorem 1 achieves that goal and opens up the possibility of applying randomized arguments in this setting.
1.2. Motivation 2: Statistical mechanics. The second motivation draws from notions of quantum statistical mechanics [13,9–12,14–18]. The predictions of statistical mechanics are based on ensemble averages, yet in practice prove to apply already to single instances of thermodynamical systems. This phenomenon needs to be explained. It becomes at least plausible if there is a measure concentration effect on the ensemble under consideration. Concentration implies that any observable will give values close to the ensemble mean for almost every state in the ensemble. This will in particular happen at almost every point on a sufficiently generic trajectory through the ensemble. Thus there may be an “apparent relaxation” [10,11,15–18] even in systems not in a global equilibrium state. Recently, several authors realized that it is particularly simple to state a precise quantitative version of this intuition for ensembles consisting of random vectors drawn from some subspace [10,18]. However, in the context of statistical mechanics, it may be more
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natural to consider sets of states with prescribed energy expectation value, rather than elements of some linear subspace. Indeed, such “mean energy ensembles” have been studied before [19–22]. Thus, it is natural to ask whether the concentration results for linear spaces translate to mean energy ensembles. We present both positive and negative results on this problem. Since the mean energy ensemble and its properties depend on the spectrum of the chosen observable H , so does the degree of measure concentration. For many spectra typically encountered in large many-body systems, our main theorem yields trivial bounds. As explained in Sect. 3, this is partly a consequence of the fact that “Lévy’s Lemma-type” exponential concentration simply does not exist for such systems. However, for families of Hamiltonians with, for example, constant spectral radius, we do get meaningful concentration inequalities. Therefore, the methods presented in this paper are expected to have a range of applicability complementary to other approaches. The question whether weakened concentration properties can be proven for more general many-body systems under energy constraints remains an interesting problem (see Sect. 3). 2. Main Results and Overview As stated above, we will analyze the set M E := |ψ ∈ Cn | ψ|H |ψ = E and ψ = 1 for some observable H and expectation value E. The set of all pure quantum states is a complex sphere in Cn ; equivalently, we can view it as the unit sphere S 2n−1 in R2n . The obvious geometric volume measure on S 2n−1 corresponds to the unitarily invariant measure on the pure quantum states [3]. As we will see below, the set M E is a submanifold of the sphere (and thus of R2n ); hence it carries a natural volume measure as well, namely the “Hausdorff measure” [23] that it inherits from the surrounding Euclidean space R2n . Normalizing it, we get a natural probability measure on M E . Our first main theorem can be understood as an analog of Lévy’s Lemma [24] for the manifold M E . It says that the measure on M E is strongly concentrated, in the sense that the values of Lipschitz-continuous functions are very close to their mean on almost all points of M E . In some sense, almost all quantum states with fixed expectation value behave “typically”. To understand the theorem, note that M E is invariant with respect to energy shifts of the form E := E + s, H := H + s1, such that the new eigenvalues are E k := E k + s. We then have M E = M E , i.e., the manifold of states does not change (only its description does). We call a function f : M E → R λ-Lipschitz if it satisfies | f (x) − f (y)| ≤ λx − y, where · denotes the Euclidean norm in Cn . Theorem 1 (Concentration of measure). Let H = H † be any observable on Cn , with eigenvalues {E k }nk=1 , E min := mink E k , E max := maxk E k , and arithmetic mean E A := 1 k E k . Let E > E min be any value which is not too close to the arithmetic mean, i.e., n E ≤ EA −
π(E max − E min ) . √ 2(n − 1)
Suppose we draw a normalized state vector |ψ ∈ Cn randomly under the constraint ψ|H |ψ = E, i.e., |ψ ∈ M E is a random state according to the natural distribution described above. Then, if f : M E → R is any λ-Lipschitz function, we have 3 −cn Prob | f (ψ) − f¯| > λt ≤ a · n 2 e
1 t− 4n
2
√ +2ε n
,
(1)
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where f¯ is the median of f on M E , and the constants a, c and ε can be determined in the following way: > 0 and – Shift the energies by some offset s (as described above) such that E min ε 1 E H E = 1 + 1+ √ (2) n n
with ε > 0, where E H denotes the harmonic mean energy. The offset may be chosen arbitrarily subject only to the constraint that the constant a below is positive. /(32E ) and – Compute c = 3E min E Q
:=
1 −2 Ek n k
− 1
2
,
a=
2 3040E max
E
2
1−
E 2 ε2 E 2Q
−1 .
The theorem involves an energy offset s, shifting all energy levels to E k := E k + s. The idea is to choose this shift such that E ≈ E H , i.e. such that the energy in question becomes close to the harmonic mean energy (we show in Lemma 11 below that this is always possible). Specifically, the theorem demands that E becomes a bit larger than E H , resulting in a constant ε > 0 defined in Eq. (2). The theorem does not specify s uniquely – there is some freedom for optimizing over the different possible choices of s. However, there is the constraint that a > 0, which prevents us from choosing too small values of ε (indeed, a > 0 is equivalent to ε > E /E Q ). On the other hand, ε should not be too large, because it appears in the exponent in Eq. (1). To apply the theorem, it is often useful to know the value of the median f¯. Our second main theorem gives an approximation of f¯ in the limit n → ∞: Theorem 2 (Estimation of the median f¯). With the notation from Theorem 1, let N be the full ellipsoid 1 N := z ∈ Cn | z|H |z ≤ E 1 + , 2n and let f : N → R be any λ N -Lipschitz function. Then, the median f¯ of f on the energy manifold M E satisfies 21 f¯ − E N f ≤ λ N 3 + 15 E · O n − 21 , 8n E min 1 √ 3 where O n − 2 := ε/ n + ln 2an 2 /(2n). We proceed by discussing a simple example. Suppose we have a bipartite Hilbert space A ⊗ B with dimensions |A| = 3 and large, but arbitrary |B|, and the Hamiltonian ⎛ ⎞ 1 H = ⎝ 2 ⎠ ⊗ 1 B =: H A ⊗ 1 B . (3) 3 We fix the arbitrary energy value E = 23 , and draw a state |ψ ∈ A ⊗ B randomly under the constraint ψ|H |ψ = E, which is Tr(ψ A H A ) = E. What does Theorem 1
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tell us about concentration of measure for this manifold of quantum states? To have all positive eigenvalues, our offset s must be s > −1, and the shifted harmonic mean energy becomes −1 1 1 1 E H (s) = 3 . + + 1+s 2+s 3+s The offset s (equivalently, the constant ε) is not specified uniquely by Theorem 1; we try to find a good choice by fixing ε independently of n. After some trial-and-error, ε = 2 turns out to be a good choice (other values work as well, but not ε = 1). The next task is to estimate the shift s which results from our choice of ε = 2; it is determined by the equation 3 1 2 E H (s), +s = 1+ 1+ √ 2 n n where n = 3|B|. It is difficult to solve this equation directly, but it is easy to see that a √ solution close to (−4 + 7)/3 ≈ −.45 exists for large n. This fact helps to gain a rough estimate of s which is sufficient to prove strong concentration of measure: denote the difference of the left- and right-hand side by f n (s), then f n − 21 > 0 for all n ≥ 8193. Since f n is decreasing, we get f n (x) > 0 for all x ∈ (−1, − 21 ], hence s > − 21 . The constant c = c(s) =
3(1 + s) 32 23 + s
3 /E ≤ 5 . On the other , and similarly E max is increasing in s, hence c ≥ c − 21 = 64 2 hand, since f n (0) < 0 for all n ∈ N, we have s < 0. Hence we have to consider the expression E 2 /(ε2 E 2Q ) only in the relevant interval s ∈ (− 21 , 0), where it is decreasing and thus upper-bounded by 259 675 . Consequently, a is positive and satisfies a < 30830. Substituting these expressions into Theorem 1, we get the following result: Example 1. Drawing random pure state vectors |ψ under the constraint ψ|H |ψ = 23 , where H is the observable defined in Eq. (3), we get the concentration of measure result
Prob | f (ψ) − f¯| > λt ≤ 30830 n e 3 2
2 √ 3 1 − 64 n t− 4n +4 n
for every λ-Lipschitz function f and all n ≥ 8193. It is clear that the amount of measure concentration that we get from Theorem 1 depends sensitively on the spectrum of the Hamiltonian H . In particular, not all natural Hamiltonians yield a non-trivial concentration result. For example, in Sect. 3, we show that for a sequence of m non-interacting spins, Theorem 1 does not give a useful concentration result in the sense that the corresponding concentration constant c in (1) will be very close to zero. However, we will also prove that this is not a failure of our method, but reflects the fact that there simply is no concentration in that case, at least no concentration which is exponential in the dimension. In the “thermodynamic limit” of large dimensions n, the condition on the energy E in Theorem 1 becomes E min < E < E A . From a statistical physics point of view, E min is the ground state energy of “temperature zero”, while E A corresponds to the “infinite temperature” energy. Hence, the condition on E can be interpreted as a “finite
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temperature” condition. However, this condition is no restriction: if one is interested in concentration of measure for E A < E < E max , then the simple substitution H → −H and E → −E will make Theorem 1 applicable in this case as well. In the situation of Example 1 above, with Hamiltonian (3) on the bipartite Hilbert space A ⊗ B with fixed |A| and large |B|, we may ask what the reduced density matrix ψ A typically looks like. It is well-known [3] that for quantum states without constraints, the reduced density matrix is typically close to the maximally mixed state. To estimate the typical reduced in our case, we may consider the Lipschitz-continuous func state tions f i, j (ψ) := ψ A i, j , that is, the matrix elements of the reduced state. Theorem 2 gives a way to estimate these matrix elements by integration over some ellipsoid. Instead of doing this calculation directly, we give a general theorem below which gives the typical reduced density matrix in the more general case that the global Hamiltonian H can be written H = H A + HB , i.e., if it describes two systems without interaction. The Hamiltonian (3) corresponds to the special case H B = 0. In this case, we get: Example 2. Random state vectors |ψ under the constraint ψ|H |ψ = 23 , where H is the observable defined in (3), typically have a reduced density matrix ψ A close to √ ⎞ ⎛ 0√ 0 5+ 7 1 ⎝ ρc = 0 2(4 − 7) 0 √ ⎠. 12 0 0 −1 + 7 Specifically, we have for all t > 0 and n ≥ 8193 2 √ √ 3 − 3 n t− 1 59 +4 n A 64 4n ≤ 369960 n 2 e Prob ψ − ρc > 3 8 t + √ . 4 2 n This example is a special case of our third main theorem: Theorem 3 (Typical reduced density matrix). Let H = H A + H B be an observable in the Hilbert space A ⊗ B := C|A| ⊗ C|B| of dimension n = |A| · |B| with |A|, |B| ≥ 2. |A| |B| Denote the eigenvalues of H A and H B by {E iA }i=1 and {E Bj } j=1 , and the eigenvalues of H by E kl := E kA + ElB respectively. Suppose that the assumptions of Theorem 1 hold, and adopt the notation from there, in particular, E k := E k + s with the energy offset s specified there. Then, the reduced density matrix ψ A := Tr B |ψψ| of random pure state vectors |ψ ∈ A ⊗ B under the constraint ψ|H |ψ = E satisfies 2 √ √ 3 −cn t− 1 +2ε n A 4n 2 Prob ψ − ρc > 8|A|(t + δ) ≤ |A|(|A| + 1)an e 2
for all t > 0, where the “canonical” matrix ρc is given by ⎛ |B| E 0 ... 0 k=1 E 1k ⎜ .. |B| E 1 ⎜ ⎜ . 0 1 + 2n k=1 E 2k ⎜ ρc = . . n+1 ⎜ ⎜ .. .. ⎝ |B| E 0 ... k=1 E
|A|k
⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
Concentration of Measure for Quantum States with a Fixed Expectation Value
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and the constant δ equals 21 1 21 3 1 E E −2 1+ + 15 δ= , O n E min n 8n E min 1 where O n − 2 =
√ε n
3 ln 2an 2
+
2n
.
The “canonical” matrix ρc given above does not immediately have a useful physical interpretation. In particular, it is not in general proportional to exp(−β H A ), i.e., it is not necessarily the Gibbs state corresponding to H A as one might have expected based on intuition from statistical mechanics (we discuss this point in more detail in Sect. 3 below). Note also that ρc is not exactly normalized, but it is close to being normalized (i.e., Tr ρc ≈ 1 in large dimensions n, which follows from E ≈ E H ). To illustrate the use of Theorem 3, we give a proof of Example 2. We use the notation and intermediate results from the proof of Example 1. The energy shift s depends on √ −4+ 7 the dimension n, i.e., s = s(n), and we have limn→∞ s(n) = 3 . More in detail, if we use Mathematica to compute the Taylor expansion of s(n) at n = ∞, we find the inequality √ √ √ −4 + 7 4(35 + 16 7) −4 + 7 − (4) < s(n) < √ 3 3 63 n for all n ∈ N. The typical reduced density matrix that Theorem 3 supplies depends on n. It is ⎛ 1 ⎞ 3 0 0 1 1+s(n) + s(n) 1 + n⎜ ⎟ 1 2n 2 0 ⎠, ρc(n) := · ⎝ 0 2+s(n) n+1 3 1 0 0 3+s(n) which tends to the matrix ρc from the statement of Example 2 as n → ∞. We can use (n)
Eq. (4) to bound the difference between ρc while
3 2 +s
2+s
and
3 2 +s
3+s
and ρc . Using that
are increasing, a standard calculation yields 4 (n) ρc − ρc ≤ √ 2 n
3 2 +s 1+s
is decreasing in s,
(5)
for all n ≥ 829. Similar calculations can be used to bound the constant δ from above: 58 δ< √ 4 n for all n ≥ 550. Thus, according to Theorem 3,
2 √ √ 3 − 3 n t− 1 58 +4 n A (n) 64 4n ≤ 369960 n 2 e Prob ψ − ρc > 3 8 t + √ 4 2 n √ √ 58 59 √4 < 3 8 · √ holds for all n ≥ 8193. The estimate (5) together with 3 8 · √ 4n + 4n n proves the claim in Example 2. An interesting aspect of Theorem 3 is that the typical reduced density matrix does not maximize the entropy locally (if so, it would be the Gibbs state corresponding to H A ). This is expected to have applications in quantum information theory in situations where random bipartite states with non-maximal entanglement are considered. It will be shown in Sect. 4 below that the reduced density matrices maximize a different functional instead which is related to the determinant.
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3. Implications for Statistical Mechanics Recently, the concentration of measure phenomenon has attracted a considerable amount of attention in the context of quantum statistical mechanics. Consider some ensemble of quantum states, such as the set of all pure quantum states in a certain subspace of the global Hilbert space. The subspace might be given, for example, by the span of all eigenvectors corresponding to an energy in some small interval [E − E, E + E] with respect to a given Hamiltonian H . Suppose we are given a single, particular, random state vector |ψ from this subspace. What properties will this pure state have? At first one might be tempted to think that there is very little knowledge available on the properties of the state, and that most properties should turn out to be random. However, the concentration of measure phenomenon shows that this is not the case – almost all the possible state vectors |ψ will have many properties in common. Several authors [13,9–12] have recently argued that this property of measure concentration may help to better understand certain foundational issues of statistical mechanics. Conceptually, statistical mechanics aims to predict outcomes of measurements on systems even in the case that we have only very limited knowledge about the system (say, we only know a few macroscopic variables). Ensemble averages are employed to make predictions, and the predictions agree very well with experiment even in single instances of the system. Concentration of measure is then viewed as a possible theoretical explanation of some aspects of this phenomenon. As a paradigmatic example, Ref. [10] considers the situation of a bipartite Hilbert space H S ⊗ H E , consisting of system S and environment E. Then the setting is investigated where the set of physically accessible quantum states is a subspace H R ⊂ H S ⊗ H E (for example, a spectral window subspace as explained above). If we are given an unknown random global quantum state vector |ψ ∈ H R , then what does the state typically look like for the system S alone? In this case, the postulate of equal apriori probabilities from statistical physics suggests to use the maximally mixed state on H R , that is 1/ dim(H R ), as an ensemble description. Then, taking the partial trace over the environment will yield a state S = Tr E 1/ dim(H R ) which may be used by observers in S to predict measurement outcomes. According to Ref. [10], concentration of measure in the subspace H R proves that almost all quantum state vectors |ψ ∈ H R have the property that the corresponding reduced state ψ S := Tr E |ψψ| is very close to S . That is, ψ S ≈ S with overwhelming probability. It is then argued in Ref. [10] that this result explains why using the ensemble average S is in good agreement with experiment even in the case of a single instance of the physical system. While Ref. [10] considers a very general situation that does not allow for specifying directly what the “typical” state S looks like, Ref. [9] make additional assumptions that allow to specify S in more detail. That is, if the Hamiltonian H is H = HS + H E (that is, there is no or negligible interaction between system and environment), if the restriction H R is given by a spectral energy window, and if the bath’s spectral density scales exponentially, then S ∼ exp(−β HS ) for some suitable β > 0. That is, under these standard assumptions from statistical mechanics, Ref. [9] argues that the typical reduced state is a Gibbs state. These results raise an immediate question: what happens if the restriction is not given by a subspace? In statistical mechanics, one often considers the situation that an observer
Concentration of Measure for Quantum States with a Fixed Expectation Value
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“knows the total energy of the system”, but does not know the exact microscopic state. In all the papers previously mentioned, the intuitive notion of “knowing the energy” has been translated to the technical statement of “knowing with certainty that the quantum state is supported in a spectral subspace”. Obviously, there is at least one natural alternative: knowing the energy might also be read as “knowing the energy expectation value ψ|H |ψ of |ψ”. In fact, this possibility has been proposed by several authors [19–21] as a possible alternative definition of the “quantum microcanonical ensemble” (we call it “mean energy ensemble”). The results in this paper give some information on the applicability of the mean energy ensemble in statistical mechanics: 1. As a positive result, we know from Theorem 1 that the concentration of measure phenomenon occurs for the mean energy ensemble as well, if that theorem is applicable to the particular Hamiltonian H and energy value E that is considered. In this case, many interesting results from Refs. [13,9–12] carry over to the mean energy ensemble. 2. On the other hand, the ensemble does not seem to reproduce well-known properties of statistical mechanics, such as the occurrence of the Gibbs state (cf. Theorem 3). Hence it seems to describe rather exotic physical situations. There is an intuitive reason why the mean energy ensemble behaves exotically: calculations involving Theorem 2 suggest that the k th energy level |E k typically has an occupation proportional to |ψ|E k |2 ∼ 1/E k , where E k is the corresponding shifted energy value (cf. Theorem 1); this is also visible in the form of the typical reduced density matrix ρc in Theorem 3. In particular, typical state vectors |ψ “spread out” a lot on the small energy levels. This produces a “Schrödinger cat state” which is in a coherent superposition of many different energy states. Such states are not normally observed in statistical physics. However, Point 2 does not completely rule out the mean energy ensemble as a description of actual physics, due to the following fact: 3. Our result is tailor-made for systems with the property that the corresponding mean energy ensemble concentrates exponentially in the dimension n, similarly as in Lévy’s Lemma (corresponding to inequality (6) below with κ(n) = const.). However, many natural many-body systems do not have such a concentration property, which is why Theorems 1, 2 and 3 do not apply in those cases. As an example, we will now show that the mean energy ensemble does not concentrate exponentially in the case of m non-interacting spins. Let the energy levels of each spin m be 0 and +1. The total Hilbert space has m dimension n = 2 . If k is an integer between 0 and m, then the energy level k is k -fold degenerate. Moreover, suppose that we are interested in the energy value E = αm, where α ∈ (0, 21 ). To see if Theorem 1 is applicable, we determine a rough estimate of the energy shift s that has to be employed such that E ≈ E H . We thus have to find s > 0 such that E H
= 2
−m
m
m 1 k k+s
−1
!
≈ αm + s;
k=0
this is only possible if the (k = 0)-term gives a significant contribution. The conclusion is that s must be very small, that is, of the order s ≈ αm2−m (and this conclusion is confirmed by more elaborate large deviations arguments similar to those discussed below).
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But then, the constant c in the exponent in (1) is approximately c=
3E min 3 3 αm2−m 3 1 · = · 2−m = · ≈ 32E 32 αm 32 32 n
such that Theorem 1 does not give any measure concentration at all. It turns out, however, that this result is not a failure of our method, but reflects the fact that there simply is no concentration of measure which is exponential in the dimension n in this case. The precise statement below makes use of the binary entropy function H (γ ) = −γ log2 γ − (1 − γ ) log2 (1 − γ ) defined for γ ∈ [0, 1] (c.f. [25]). Example 3 (Non-interacting spins: no exp. concentration). Suppose we have m noninteracting spins as explained above, and fix the energy value E = αm with α ∈ (0, 21 ). Consider a hypothetical concentration of measure inequality 2 Prob{| f − f¯| > t} ≤ b · e−κ(n)t
(6)
for all 1-Lipschitz functions f . Here, f¯ denotes either the mean or the median of f, n = 2m is the dimension of the system and b > 0 a fixed constant. If such an inequality is to hold, then necessarily κ(n) = o n H (β)
(7)
for any β > α. In particular, the optimal exponent in (7) is strictly smaller than one if α = 21 and goes to zero as α → 0. Proof. Let γ be such that α < γ < 21 . Let L=
E[|ψi |2 ],
{i | E i <γ m}
R=
E[|ψi |2 ].
{i | E i ≥γ m}
Normalization gives L + R = 1 from which we get αm = E =
i
⇒ L ≥1−
E[|ψi |2 ] E i ≥ γ m R = γ m(1 − L) α . γ
(8)
We can also bound L using inequality (6). To this end, consider the 1-Lipschitz function |ψ → ψi , that is, the real part of the i th component of |ψ in the Hamiltonian’s eigenbasis. Due to the invariance of the energy manifold with respect to reflections |ψ → −|ψ, both expectation value and median of this function equal zero. If hypothesis (6) is true, then it follows by squaring that Prob{(ψi )2 > u} ≤ b · e−κ(n) u ,
Concentration of Measure for Quantum States with a Fixed Expectation Value
and thus
E[(ψi )2 ] =
∞
u=0
795
u ∂u Prob{(ψi )2 ≤ u} du ∞
=− u ∂u Prob{(ψi )2 > u} du u=0 ∞ = Prob{(ψi )2 > u} du u=0 ∞ b ≤ b · e−κ(n)u du = , κ(n) u=0
(9)
having used integration by parts. An analogous inequality obviously holds for the imaginary part E[(ψi )2 ]. From basic information theory (e.g., Chap. 11 in Ref. [25]), we borrow the fact that the number of terms in the definition of L is upper-bounded by |{i | E i < γ m}| ≤ 2m H (γ )+log2 m .
(10)
Combining (8), (9) and (10): α 2b m(H (γ )+ 1 log2 m) m 2 ≥ L ≥1− κ(n) γ 2b H (γ )+ 1 log2 m H (β) m n = o n ⇒ κ(n) ≤ 1 − γα for every β > γ .
It is highly plausible that similar results hold for the mean energy ensemble of many other many-body systems: the best possible rate of concentration (such as the right-hand side in (6)) is not exp(−cnt 2 ), but at most exp(−cn p t 2 ) with energy-dependent exponent p < 1. Deciding whether this upper bound can be achieved remains an interesting open problem. Indeed, we end the present section by sketching a possible route for tackling this question. The proof of Example 3 uses the coordinate functions ψ → ψi as an example for continuous functions without strong concentration properties. We conjecture that this is already the worst case, i.e. that no function with Lipschitz-constant equal to one “concentrates less” than the “most-spread out” of the coordinate functions. A strongly simplified version of this conjecture is easily made precise. We restrict attention to real spaces and linear functions
ψ → ci ψi i
of the state vectors. Here, the ψi ’s are the coefficients of ψ in the eigenbasis of the Hamiltonian and the ci ’s are arbitrary coefficients subject to the normalization con straint i ci2 = 1. The latter constraint ensures that the Lipschitz constant of the linear funtion is one. If the vectors ψ are drawn from an energy ensemble M E , we have the elementary estimate
ci ψi = ci c j E[ψi ψ j ] = ci2 E[ψi2 ] ≤ max E[ψi2 ] = max Var[ψi ], Var i
i, j
i
i
i
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M. P. Müller, D. Gross, J. Eisert
having made use of the fact that M E is invariant under the transformation which takes ψi → −ψi for some i while leaving the other coordinates fixed. Upper bounds on the variance of a random variable are sufficient to establish simple concentration estimates (by means of Chebychev’s inequality). From that point of view, we have proven that coordinate functions show the least concentration among all linear functions, according to methods based on second moments alone. Generalizing this observation to more general functions and higher moments would allow us to restrict attention to the eigenbasis of the Hamiltonian, thus “taking the non-commutativity out of the problem”. Conceivably, this would constitute a relatively tractable path to a more complete understanding of concentration in typical many-body systems. 4. Invitation: A Simple Hamiltonian As a preparation for the proof in the next section, we give a particularly simple example of a Hamiltonian which admits a more direct proof of concentration of measure; in the meantime, we will also see that the “harmonic mean energy” from Theorem 1 appears naturally. Consider a bipartite quantum system on a Hilbert space H = H A ⊗ H B , with dimensions |A| := dim H A and |B| := dim H B . We may assume without restriction that |A| ≤ |B|. We are interested in the manifold of state vectors |ψ ∈ H with average energy ψ|H |ψ = E, where H = H † is some Hamiltonian on H. What happens if we draw a state |ψ from that submanifold at random? Instead of studying this question in full generality (which we will do in Sect. 5), we start with the simple example H = H A ⊗ 1, where H A = H A† is an observable on H A alone. This is a special case of a more general bipartite Hamiltonian H = H A + H B = H A ⊗ 1 + 1 ⊗ H B without interaction as studied in Theorem 3. A nice consequence is that ψ|H |ψ = E ⇔ Tr(ψ A H A ) = E, that is, the constraint depends on the reduced density matrix ψ A := Tr B |ψψ| alone. The unitarily invariant measure ν on the pure quantum states on the global Hilbert space H induces in a natural way a measure ν A on the density matrices in H A : if S is some measurable subset of the density matrices, then ν A (S) := ν ({|ψ | Tr B |ψψ| ∈ S}) = ν Tr −1 (S) . B Formally, the measure ν A is the pushforward measure [26] of the unitarily invariant measure ν on the pure states of H A ⊗H B with respect to the map Tr B ; that is, ν A = (Tr B )∗ (ν). Due to the simple form of the Hamiltonian H , we may calculate probabilities with respect to ν A . The probability density distribution corresponding to ν A is invariant with respect to unitaries on H A , but it depends on the eigenvalues t = (t1 , . . . , t|A| ) of the reduced density matrix ψ A . The relation is [2,27]: ⎛ ⎞ |A| |A|
|B|−|A| dν A (t) = z −1 δ ⎝1 − ti ⎠ ti (ti − t j )2 dt, (11) i=1
i=1
i< j≤|A|
where z is the normalization constant and dt = dt1 . . . dt|A| .
Concentration of Measure for Quantum States with a Fixed Expectation Value
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We will now study the simplest case |A| = 2, where E1 0 HA = , 0 E2 and we may assume without loss of generality that E 1 > E 2 . We fix some arbitrary energy value E between E 1 and E 2 . Since the average energy E A := 21 (E 1 + E 2 ) is the energy of an “infinite temperature” Gibbs state, we additionally assume that E < E A , which is no restriction, but saves us some distinction of cases. Thus, E 2 < E < E A < E 1 . The state space is now the Bloch ball; that is, the unit ball in R3 . The distribution dν A has been calculated for the case |A| = 2 already by Hall [28]. In ordinary spherical coordinates, it can be written dν A (r ) = ρ|B| (r )r 2 dr dϕ dθ, where ρ|B| (r ) = c B (1 − r 2 )|B|−2 , and c B is some normalization constant. It can be derived from Eq. (11) by using that points r in the Bloch ball correspond to density matrices with eigenvalues t1 = 21 (1 + r ) and t2 = 21 (1 − r ). If |B| = 2, this measure becomes the usual Euclidean measure in the Bloch ball, corresponding to the Hilbert-Schmidt measure [27]. In general, ν A (S) = ρ|B| (r )r 2 dr dϕ dθ = ρ|B| (|x|)d x. S
S
We write ρ B (x) := ρ|B| (|x|) for x ∈ R3 . It is easy to see (and we will show this below) that the subset of mixed states ψ A in the Bloch ball with Tr(ψ A H A ) = E is the intersection of a plane with the Bloch ball; that is, a disc K E . To determine probabilities, we need to compute the area μ(X ) of two-dimensional subsets X ⊂ K E . In light of the density ρ B introduced above, it is tempting to use the term X ρ B (x) d x as the measure of X . The normalized version is a probability measure: Prob(X ) :=
ρ B (x) d x . K E ρ B (x) d x X
(12)
It turns out that this measure agrees with the normalized geometric volume measure Prob that we use elsewhere in this paper (for example in Theorem 1). We discuss this fact in detail below after the proof of Proposition 1. Using this identity and deferring its justification to below, we get a concentration of measure result: Proposition 1. The reduced density matrix ψ A := |ψψ| concentrates exponentially on the canonical state E−E 2 0 E −E 1 2 ρc := , E 1 −E 0 E 1 −E 2 that is, " ! (|B| − 1)(E 1 − E 2 )2 Prob ψ A − ρc 1 ≥ ε ≤ exp −ε2 4(E 1 − E)(E − E 2 ) for all ε ≥ 0.
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M. P. Müller, D. Gross, J. Eisert
Proof. It follows from the condition Tr(ψ A H A ) = E that ψ A must have diagonal elements d1 :=
E − E2 E1 − E , d2 := , E1 − E2 E1 − E2
where d1 < d2 . By Schur’s Theorem [29], the eigenvalues t1 and t2 of ψ A must majorize the diagonal elements, that is, (d2 , d1 ) ≺ (t2 , t1 ), hence t2 = ψ A ∞ ≥ d2 . Thus, among all states ψ A satisfying the energy condition, the canonical state ρc is the “most mixed” one, in the sense that its radius in the Bloch ball is the smallest possible. Geometrically, this has the following interpretation. Translating the condition Tr(ψ A H A ) = E to the Bloch ball representation, that is, representing density matrices ψ A by vectors r ∈ R3 , we get ⎞ ⎛ 0 ⎠ = 2(E − E A ). 0 r ·⎝ E1 − E2 This defines a plane with rz =
2(E − E A ) <0 E1 − E2
in R3 ; the intersection of that plane with the Bloch ball gives the set of density matrices that fulfill the energy condition. This set is a disc K E , with the vector representation of ρc at its center. Since the trace distance · 1 on the density matrices corresponds to the Euclidean distance in the Bloch ball, the set of states ψ A satisfying the energy condition with ψ A − ρc 1 ≥ ε corresponds to an annulus in K E with inner radius ε; we denote this annulus by K E (ε). Hence Prob ψ A − ρc 1 ≥ ε =
x∈K E (ε) ρ B (x)d x x∈K E (0) ρ B (x)d x
.
By elementary integration, we get
x∈K E (ε)
√1−rz2
2π
ρ B (x)d x =
dϕ 0
= 2π c B =
ε
√1−rz2 ε
⎛
⎞ s cos ϕ ds · sρ B ⎝ s sin ϕ ⎠ rz
s(1 − s 2 − r z2 )|B|−2 ds
|B|−1 π cB 1 − r z2 − ε2 . |B| − 1
Since K E = K E (0), this yields
|B|−1 |B|−1 1 − r z2 − ε2 ε2 = 1− 1 − r z2 1 − r z2 ε2 . ≤ exp −(|B| − 1) 1 − r z2
Prob ψ A − ρc 1 ≥ ε =
Concentration of Measure for Quantum States with a Fixed Expectation Value
799
Substituting 1 − r z2 = proves the claim.
4(E 1 − E)(E − E 2 ) (E 1 − E 2 )2
In this paper, the measure that we are interested in is the Hausdorff volume measure μ M E on the energy manifold M E . To compute the correct probabilities when restricting to the Bloch ball by partial trace, we need to invoke the pushforward of μ M E with respect to Tr B . So is the probability measure Prob that we have defined above in Eq. (12) equal to this pushforward measure, i.e., does ?
Prob = (Tr B )∗ (μ M E ) hold? Fortunately the answer is “yes”, but this is not directly obvious. First we observe that Prob may be interpreted as an “energy shell measure” in the following sense. The submanifold measure from Eq. (12) can be given by a limit, in a spirit similar to the definition of the “Minkowski content” (cf. Ref. [23]): For X ⊂ K E , denote by Uδ− (X ) the set of matrices ψ A that are δ-close to X , and have an energy expectation value of Tr(ψ A H A ) < E. This is half of the δ-neighborhood of X . Then, up to a normalization constant, Prob(X ) ∼ ρ B (x)d x X 1 = lim ρ B (x)d x δ→0 δ Uδ− (X ) 1 = lim ν A Uδ− (X ) δ→0 δ 1 − = lim ν Tr −1 (U (X )) . δ B δ→0 δ But Uδ− (K E ) is a “slice” of the Bloch ball in between two parallel planes. The set consists of those matrices ψ A determined by the inequality E − ε < Tr(ψ A H A ) < E, where ε > 0 is some energy difference corresponding to δ. In particular, − Tr −1 B Uδ (K E ) = {|ψ | ψ|H |ψ ∈ (E − ε, E)}. This is an energy shell. Hence it is basically the uniform distribution on this energy shell on the pure states which, by taking the partial trace and the limit ε → 0, generates our probability measure Prob. In the general case of arbitrarily many different energy levels {E k }nk=1 , this energy shell has different “widths” at different points x ∈ M E ; in the limit δ → 0, this width is proportional to PS ∇ E(x)−1 , where PS denotes the projection onto the sphere’s tangent space at x. Hence the corresponding “energy shell measure” does not in general equal the geometric (Hausdorff) measure Prob used elsewhere in this paper, which arises from an analogous limit procedure, but starting with the uniform distribution in an ε-neighborhood of M E . However, here we are in a very special situation: we only have two different energy levels E 1 and E 2 which are highly degenerate. In this case, it turns out that PS ∇ E(x) is constant along M E . To simplify the argument, we double the dimensions and work in real space Rn ; the case n = 2|A| |B| applies to Proposition 1 above.
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Lemma 1. Consider the real energy manifold n # $ n
M E := x ∈ Rn E k xk2 = E, xk2 = 1 k=1
k=1
in the special case that there are only two different energy levels, i.e. there exist E1 , E2 such that E 1 = E 2 = · · · = E m = E1 , and E m+1 = · · · = E n = E2 . Furthermore, assume that E1 < E < E2 . Let PS denote the projection onto the tangent space of the unit sphere, and let E(x) := nk=1 E k xk2 . Then, PS ∇ E(x) is constant on M E . Proof. Direct calculation yields PS ∇ E(x)2 = 4E 2 (x) − 4E(x)2 , where E 2 (x) :=
n
E k2 xk2 . Since m m
2 2 E(x) = xk E1 + 1 − xk E2
k=1
k=1
k=1
2 equals E on all of M E , it follows that m k=1 x k is constant on M E . But then, m m
2 2 2 2 E (x) = xk E1 + 1 − xk E22 k=1
is also constant on all of M E .
k=1
Hence, in our case of only two different energy levels, the energy shell has constant width everywhere, such that the measure defined in Eq. (12) indeed agrees with the geometric measure that we use elsewhere in the paper. Proposition 1 shows that a typical reduced density matrix is close to ρc ; that is, it is diagonal in H A -basis, and it can be written ρc = e−β H A /Z with Z := Tr(e−β H A ) and some appropriate “inverse temperature” β > 0. Hence it is a Gibbs state. This result suggests that the local state always concentrates on a Gibbs state when H = H A ⊗ 1, also in the more general case |A| = dim H A ≥ 3. But this guess is false, as we have already shown in Theorem 3 and Example 2 – the local density matrix always commutes with H A , but it is not in general the corresponding Gibbs state. In light of the calculation above, it is now easy to give an intuitive explanation for this fact. In analogy to the previous proposition for |A| = 2, we expect that the distribution of eigenvalues as given in Eq. (11) will dominate the concentration of measure to some local “canonical” state. It also seems reasonable to assume (and can be verified numerically) that the (ti − t j )2 -terms do not contribute much for large |B|, and that the other terms exponential in |B| dominate. But these terms are |A|
|B|−|A|
ti
= (det ψ A )|B|−|A| ,
i=1
and so we conclude that the reduced density matrix should concentrate on the one ψ A which maximized the previous expression. This suggests the following conjecture:
Concentration of Measure for Quantum States with a Fixed Expectation Value
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Conjecture 1. Suppose that H = H A ⊗ 1 is a Hamiltonian on a bipartite Hilbert space H = H A ⊗ H B with fixed |A| := dim H A and varying |B| := dim H B . Then typical quantum states |ψ ∈ H with fixed mean energy ψ|H |ψ = E have the property that ψ A := Tr B |ψψ| concentrates exponentially on the determinant maximizer ρc := arg max det ψ A | Tr(ψ A H A ) = E , given the maximizer is unique. We will now see that the prediction of this conjecture is consistent with Theorem 3. To this end, we now compute the determinant maximizer ρc explicitly. First of all, it is easy to see that ρc must be diagonal if written in the basis of H A , i.e., [ρc , H A ] = 0: |A| Suppose σ is any density matrix with Tr(σ H A ) = E, and let {σk,k }k=1 denote the diagonal elements. Then Tr(σ H A ) = k σk,k E k = E, where E k denotes the eigenvalues of H A . Now if σ˜ is the matrix with diagonal elements σk,k and other entries zero, then Tr(σ˜ H A ) = E is still true, but by the Hadamard determinant theorem [29]
det σ ≤
|A|
σk,k = det σ˜ .
k=1
Hence the maximizer is diagonal; itremains to determine its diagonal elements (λ1 , . . . , λ|A| ). We have to maximize k ln λk subject to the constraints k λk = 1 and k λk E k = E. It turns out to be difficult to do that directly, so we drop the normalization condition k λk = 1 for the moment and solve the resulting equation ∂
∂
ln λk − λ λk E k = 0, ∂λi ∂λi k
k
E 1 where λ is the Lagrange multiplier. This gives λ = |A| E , and λi = |A| · E i . Since this distribution is not automatically normalized, we can use the freedom to shift the energy levels by some offset s ∈ R, i.e., E k := E k + s, E := E + s. The resulting distribution E · E1 is normalized, i.e., i λi = 1, if and only if λi := |A| i
⎛
⎞−1 |A|
1 1 ⎠ =: E H , E = ⎝ |A| E k k=1
that is, if the offset is shifted such that the new energy value E equals the harmonic mean energy E H . We have thus reproduced the canonical density matrix ρc of Theorem 3. Moreover, the calculation above shows that the occurrence of the harmonic mean energy is very natural and not a technical artifact of our proof. (In Sect. 6, we give a method for numerical sampling of the energy manifold, and there, the harmonic mean energy will appear in a natural way as well.)
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5. Proof of Main Theorem Before proving the main theorems, we fix some notation that will be useful for the proof. Following the lines of Gromov [26], we define a metric measure space X to be a separable complete metric space with a finite Borel measure μ, i.e., X = (X, dist, μ). (In fact, one could more generally consider Polish spaces with a σ –finite Borel measure, as described by Gromov.) In this paper, we will only consider the two cases that X is the energy manifold M E , or a full ellipsoid N , both equipped with the obvious geometric measure and the metric dist that is induced by the surrounding Euclidean space R2n Cn (for more details, see Sect. 5). We denote the (n − 1)-sphere by S n−1 , and the n-ball is denoted B n , i.e., S n−1 := x ∈ Rn | x = 1 , B n := x ∈ Rn | x ≤ 1 . The symbol ∂ is used for the topological boundary, for example, ∂ B n = S n−1 . We denote the k-dimensional (Hausdorff) volume measure on k-dimensional submanifolds M ⊂ Rn by μk . In case that μk (M) < ∞, we write μ M for the normalized measure on M, i.e., μ M (X ) :=
μk (X ) μk (M)
for Borel subsets X ⊂ M. Sometimes we consider subsets that are not actually submanifolds, but are turned into submanifolds in an obvious way. For example, the Ball B n is itself a metric measure space, but not a submanifold, since it is not open. However, its interior is a submanifold of Rn , and n
μn (B n ) =
π2 , 1 + n2
while μ B n (B n ) = 1. Expectation values with respect to μ M are denoted E M . Given some Hamiltonian H = H † on Cm , we would like to prove concentration of measure for the “mean energy ensemble” M E := ψ ∈ Cm | ψ|H |ψ = E, ψ = 1 . To relate our discussion to real-valued geometry, we work instead in Rn with n = 2m. That is, doubling all energy eigenvalues of H to get the energy levels E 1 , . . . , E n , and slightly abusing notation, we can write n # $ n
M E = x ∈ Rn xk2 E k = E, xk2 = 1 . k=1
k=1
Geometrically, M E is the intersection of the unit sphere with an ellipsoid (given by the energy condition). The action of shifting all energies by some offset (as postulated in Theorem 1) alters the ellipsoid, but not the ellipsoid’s intersection with the sphere; it leaves M E invariant. It is interesting to note that the corresponding full ellipsoid’s volume turns out to be minimal exactly if the energy shift is tuned such that E = E H , i.e., for the harmonic mean energy shift close to the one which is postulated in Theorem 1.
Concentration of Measure for Quantum States with a Fixed Expectation Value
803
We would like to prove measure concentration for the algebraic variety M E in ordinary Euclidean space. Introducing a function f : Rn → R2 via n T n
f (x) := xk2 , xk2 E k , k=1
k=1
we can write ME = f
−1
1 . E
If E is any energy value with mink E k < E < maxk E k and E = E k for all k, then the differential d f has full rank on all of M E , such that M E is a proper submanifold of Rn of codimension 2. If E = E k for some k (but E = mink E k and E = maxk E k ), the eigenvectors corresponding to this energy value are singular points of f . Still, removing those eigenvectors from M E , we get a valid submanifold M˜ E of Rn of codimension 2. Since M˜ E and M E agree up to a set of measure zero, we will drop the tilde in the following, and simply write M E for the manifold with eigenvectors removed. If we treat M E as a metric measure space, we include the eigenvectors in its definition to have a complete metric space. As a submanifold of Rn , the sets M E carry a natural geometric volume measure. Since every M E is a compact submanifold, it makes sense to talk about the normalized measure μ M E on M E , and to ask whether this measure exhibits a concentration of measure phenomenon. Our main proof strategy to answer this question in the positive is due to Gromov [26]. To explain this strategy, we introduce the notion of a “typical submanifold”. Suppose that N is a metric measure space, and M ⊂ N is a subset (say, a submanifold) which is itself a metric measure space. Then we say that “M is typical in N ” if M has small codimension, and if a small neighborhood of M covers almost all of N . That is, μ N (Uε (M)) ≈ 1 already for small ε. For example, given an n-dimensional sphere N = S n , any equator M (which is itself an (n − 1)-dimensional sphere) is typical in N ; this is just Lévy’s Lemma. On the other hand, a polar cap with angle θ is only typical in S n if θ ≈ π2 . Gromov’s idea can now be explained as follows: If N is a metric measure space that shows concentration of measure, and if M is a typical submanifold of N , then M shows measure concentration as well: it “inherits” concentration of measure from N . The intuitive reason why this idea works is as follows. Consider the behavior of Lipschitzcontinuous functions on M ⊂ N . By continuity, those functions do not change much if we turn to an ε-neighborhood of M; but then, since M is assumed to be typical in N , we already obtain almost all of N , and so the behavior of the functions on M will be similar to that on N – in particular, expectation values will be similar, and the concentration of measure phenomenon will occur in M if it occurs in N . However, Gromov seems to explore this idea in his book only for the case that N is a sphere. In our case, we have to find a submanifold N ⊂ Rn which is itself subject to the concentration of measure phenomenon, such that M E ⊂ N holds and such that M E is typical in N . A first obvious guess is to use the sphere itself; clearly, M E is a subset of the sphere S n−1 ⊂ Rn , and we have concentration of measure on the sphere by Lévy’s Lemma. However, M E can only be typical in S n−1 if the energy E is close to the “typical”
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M. P. Müller, D. Gross, J. Eisert
value E S n−1 nk=1 xk2 E k , which turns out to be the mean value n1 (E 1 + E 2 + · · · + E n ). Since we definitely want to consider different energies far away from the mean energy, N = S n−1 is not a useful choice. Instead, it will turn out that we can choose N to be a full ellipsoid with appropriate equatorial radii (in fact, N will be the slightly enlarged full energy ellipsoid for an appropriate energy shift). Our main tool from integral geometry is the Crofton formula [30,31]. It expresses the volume of a submanifold of Rn in terms of the average volume of intersection of that manifold with random hyperplanes. Lemma 2 (The Crofton formula [30]). Let M be a q-dimensional submanifold of Rn . Consider the invariant measure d L r on the planes of dimension r in Rn . If r + q ≥ n, μr +q−n (M ∩ L r ) d L r = σ (q, r, n)μq (M), Lr
where σ (q, r, n) =
%n Or +q−n · i=n−r Oi %r Oq · i=0 Oi
, and On :=
n+1
2π 2 n+1 2
denotes the surface area of the
n-sphere. Note that the Crofton formula is formulated in Ref. [30, (14.69)] only for the case that M is a compact submanifold, but the proof remains valid also in the case that M is not compact. This observation is also expressed in Ref. [31]. For the details of the definition of the invariant measure d L R , see Ref. [30]. In short, the Lie group M of all motions (translations and rotations) in Rn possesses the closed subgroup Hr of motions leaving a fixed r -dimensional plane L r0 invariant. Then, there is a one-to-one correspondence between the set of r -planes in Rn and the homogeneous space M/Hr . Since both M and Hr are unimodular, M/Hr possesses an invariant density d L r which can then be interpreted as a density on the r -planes in Rn . Note that this measure is defined only up to some multiplicative constant – different authors use different normalizations (cf. Ref. [32]), which give different constants σ˜ (q, r, n) instead of σ (q, r, n) in Lemma 2. However, for fixed l and n, we always have σ (k, l, n) σ˜ (k, l, n) = , σ (k , l, n) σ˜ (k , l, n) and it is only those ratios that are relevant for the calculations. If r + q = n, then μr +q−n (X ) = μ0 (X ) equals the number of points in the set X . A useful possible expression for the constants is [32] q+1 r +1 2 2 . σ˜ (q, r, n) = q+r −n+1 n+1 2 2 The Crofton formula will be useful in the following lemma, which gives a lower bound on the measure of ε-neighborhoods of subsets of M E . Given any subset X ⊂ Rn (say, any curve), directly estimating μn (Uε (X )) seems difficult – if the curve intersects itself many times, the neighborhood can be almost arbitrarily small. On the other hand, the Crofton formula takes into account how “meandering” subsets X are, by counting the number of intersections with hyperplanes.
Concentration of Measure for Quantum States with a Fixed Expectation Value
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Lemma 3 (Measure of neighborhood). For every open subset X ⊂ M E ⊂ Rn with n ≥ 3 and ε > 0, we have μn (Uε (X )) ≥
π ε2 μn−2 (X ), 4(n − 1)
where Uε (X ) denotes the open ε-neighborhood in Rn of X . Proof. If L 2 is any 2-dimensional plane in Rn , then M E ∩ L 2 = (∂ N ∩ L 2 ) ∩ (S n ∩ L 2 ), where ∂ N is the surface of an ellipsoid. If the intersection is not empty, then S n ∩ L 2 is a circle, and ∂ N ∩ L 2 is a compact quadratic curve, hence an ellipse. By the Crofton formula, μ0 (M E ∩ L 2 ) d L 2 = σ (n − 2, 2, n)μn−2 (M E ) < ∞, L2
so the set of planes L 2 with μ0 (M E ∩ L 2 ) = #(M E ∩ L 2 ) = ∞ has measure zero, and we can ignore them. Let now L X be the set of all planes L 2 such that X ∩ L 2 is a finite non-empty set. Since a circle and an ellipse in the plane intersect in at most four points if they are not equal, we have #(X ∩ L 2 ) ≤ 4 for all L 2 ∈ L X . Hence, the Crofton formula yields μ0 (X ∩ L 2 ) d L 2 σ (n − 2, 2, n)μn−2 (X ) = L 2 = #(X ∩ L 2 ) d L 2 LX ≤4 d L 2. LX
Using Crofton’s formula for the n-dimensional submanifold Uε (X ), we get on the other hand μ2 (Uε (X ) ∩ L 2 ) d L 2 σ (n, 2, n)μn (Uε (X )) = L 2 ≥ μ2 (Uε (X ) ∩ L 2 ) d L 2 LX ≥ π ε2 d L 2, LX
since every plane that intersects X intersects Uε (X ) in at least a disc of radius ε. Combining both inequalities, the claim follows. To deal with concentration of measure on submanifolds, we need to introduce some additional notions of the theory of measure concentration; they all can be found in the book by Gromov [26], and also, e.g., in Ref. [33] with a few errors corrected. Let (Y, ν) be a metric measure space with 0 < m := ν(Y ) < ∞, then the “partial diameter” diam is defined by diam(ν, m − κ) := inf{diam(Y0 ) | Y0 ⊂ Y, ν(Y0 ) ≥ m − κ},
806
M. P. Müller, D. Gross, J. Eisert
that is, the smallest diameter of any Borel subset with measure larger than m − κ. If (X, μ) is a metric measure space with m := μ(X ) < ∞, the observable diameter ObsDiam is defined as ObsDiam(X, κ) := sup diam(μ ◦ f −1 , m − κ) | f : X → R is 1-Lipschitz , where μ ◦ f −1 is the push-forward measure on f (X ) ⊂ R. In the special case that μ = μ X , i.e., if μ is normalized such that m = 1, this definition implies ObsDiam(X, κ) ≤ D ⇔
∀ f : X → R 1-Lipschitz ∀ε > 0∃Y0 ⊂ f (X ) ⊂ R with μ X f −1 (Y0 ) ≥ 1 − κ and diam(Y0 ) ≤ D + ε. (13)
This shows that small ObsDiam amounts to a large amount of measure concentration – in fact, one can conversely infer that ObsDiam(X, κ) ≤ D for D > 0, 0 < κ < 21 implies that all λ-Lipschitz maps f : X → R satisfy μ X {x ∈ X : | f (x) − m X f | ≥ λD} ≤ κ,
(14)
where m X f is the median of f on X , see Ref. [26] and, for a proof, Ref. [34, Lemma 2.3]. (Replacing κ by 2κ on the right-hand side removes the restriction κ < 21 .) Another useful notion is the separation distance Sep(X ; κ0 , . . . , κ N ) := sup{δ | ∃X i ⊂ X : μ(X i ) ≥ κi , dist(X i , X j ) ≥ δ, X i open}. In Ref. [26], arbitrary Borel sets are allowed in the definition of the separation distance; here, we only use open sets, since they are submanifolds and hence subject to the Crofton formula. In the most important case of two parameters, the equation Sep(X, κ, κ) = D implies that for every ε > 0 there are open subsets X 1 , X 2 ⊂ X with μ(X 1 ) ≥ κ and μ(X 2 ) ≥ κ such that dist(X 1 , X 2 ) ≥ D − ε. In the following, we need an inequality relating separation distance and observable diameter. It has first been stated in Ref. [26], and a small lapse has been corrected in Ref. [33]. Since the notation of Ref. [33] differs significantly from the notation used here, we give a proof in order to keep the presentation self-contained. Lemma 4 (Observable diameter). For every metric measure space X , and for every κ > κ > 0, ObsDiam(X, 2κ) ≤ Sep(X ; κ, κ) ≤ ObsDiam(X, κ ) holds. Proof. Suppose that ObsDiam(X ; κ ) ≤ D. Let X 1 ⊂ X be an open set with μ X (X 1 ) ≥ κ, and define a function f : X → R via f (x) := dist(x, X 1 ) = inf dist(x, y), y∈X 1
where d denotes the metric on X . It follows from the triangle inequality that f is continuous with Lipschitz constant 1. Let ε > 0. According to (13), there is a subset Y0 ⊂ R with diam(Y0 ) ≤ D +ε such that μ X {x ∈ X | f (x) ∈ Y0 } ≥ 1−κ . Since 1−κ +κ > 1, it follows that X 1 ∩ f −1 (Y0 ) = ∅, hence we have 0 ∈ Y0 . Similarly, if X 2 ⊂ X is another
Concentration of Measure for Quantum States with a Fixed Expectation Value
807
open set with μ X (X 2 ) ≥ κ, it follows that X 2 ∩ f −1 (Y0 ) = ∅, so there is some x ∈ X 2 with f (x) ∈ Y0 . Thus, dist(X 1 , X 2 ) = inf f (x) ≤ sup y = diam(Y0 ) ≤ D + ε. x∈X 2
y∈Y0
Since ε > 0 was arbitrary, it follows that dist(X 1 , X 2 ) ≤ D, and thus Sep(X ; κ, κ) ≤ D. This proves the second inequality. To prove the first inequality, suppose that ObsDiam(X, 2κ) ≥ D. This means that there exists a 1-Lipschitz function f : X → R such that for all Y0 ⊂ R with μ X {x ∈ X | f (x) ∈ Y0 } ≥ 1 − 2κ, diam(Y0 ) ≥ D holds. Clearly, the function M(a) := μ X f −1 ((−∞, a]) is increasing in a ∈ R. Let a0 := inf{a ∈ R | M(a) ≥ κ}, then μ X f −1 ((−∞, a0 ]) ∪ f −1 ((a0 , a]) ≥ κ for all a > a0 . Since every finite Borel measure on a Polish space is regular [35, Ulam Theorem], it follows that μ X f −1 ((−∞, a0 ]) ≥ κ. Similarly, define N (b) := μ X f −1 ([b, ∞)) , and let b0 := sup{b ∈ R | N (b) ≥ κ}. An analogous argument shows that μ X f −1 ([b0 , ∞)) ≥ κ. Moreover, μ X ( f −1 ((a, b))) ≥ 1 − 2κ if a < a0 and b > b0 . Due to the regularity of μ X , we conclude that μ X f −1 (Y0 ) ≥ 1 − 2κ for Y0 := [a0 , b0 ]. Setting Y1 := (−∞, a0 ] and Y2 := [b0 , ∞), we get dist(Y1 , Y2 ) ≥ diam(Y0 ) ≥ D. Letting X i := f −1 (Yi ) for i = 1, 2, it follows from the Lipschitz continuity of f that dist(X 1 , X 2 ) ≥ D, and μ X (X i ) ≥ κ. Now if ε > 0 is arbitrary, the open sets X˜ i := Uε (X i ) have measure μ X ( X˜ i ) ≥ κ, and dist( X˜ 1 , X˜ 2 ) ≥ D − 2ε. This proves that Sep(X ; κ, κ) ≥ D. Now we can formulate a lemma on how M E inherits measure concentration from surrounding bodies: Lemma 5 (Measure concentration on M E from that of surrounding body). Let ε > 0, and let N ⊂ Rn be a metric measure space such that Uε (M E ) ⊂ N . Then, for all κ > 0, μn−2 (M E )π ε2 κ . ObsDiam(M E , 2κ) ≤ 2ε + ObsDiam N , μn (N )4(n − 1) Proof. Abbreviate X := M E . Suppose that Sep(X ; κ, κ) = D, that is, for every δ > 0, there are open subsets X 1 , X 2 ⊂ X such that μ X (X 1 ) ≥ κ and μ X (X 2 ) ≥ κ and dist(X 1 , X 2 ) ≥ D − δ. Let X˜ i := Uε (X i ) ⊂ N . Using Lemma 3, we get 1 π ε2 μn ( X˜ i ) ≥ μn−2 (X i ) μ N ( X˜ i ) = μn (N ) μn (N ) 4(n − 1) π ε2 1 μ X (X i )μn−2 (X ) = μn (N ) 4(n − 1) μn−2 (X ) π ε2 κ =: κε . ≥ μn (N ) 4(n − 1)
808
M. P. Müller, D. Gross, J. Eisert
Since also dist( X˜ 1 , X˜ 2 ) ≥ D − δ − 2ε, we get Sep(N ; κε , κε ) ≥ D − 2ε. All in all, we have shown that Sep(X ; κ, κ) ≤ 2ε + Sep(N ; κε , κε ). Lemma 4 yields the chain of inequalities ObsDiam(X, 2κ) ≤ Sep(X ; κ, κ) ≤ Sep(N ; κε , κε ) + 2ε ≤ ObsDiam(N , κε ) + 2ε for every κε < κε . Our goal in the following will thus be to find a good n-dimensional body N with (M E ) small observable diameter and Uε (M E ) ⊂ N such that the ratio μn−2 μn (N ) appearing in the previous lemma is not too small. To this end, note that μn−2 (M E ) = μn−2 (∂(S n−1 ∩ N E )), n 2 where N E = x ∈ Rn | k=1 E k x k ≤ E is the full energy ellipsoid. We would like to relate μn−2 (∂(S n−1 ∩ N E )) to μn−1 (S n−1 ∩ N E ). For this purpose, the following isoperimetric inequality will be useful. Lemma 6 (An isoperimetric inequality). Let n ≥ 3, and let B ⊂ S n−1 ⊂ Rn be a Borel subset which covers at most half of the sphere, i.e., μ S n−1 (B) ≤ 21 . Then, μn−2 (∂ B) 1√ > n. μn−1 (B) 2 Proof. We use the isoperimetric inequality on the sphere [24,36, App. I]: Among all Borel sets in S n−1 with fixed volume, the minimal volume of the boundary is assumed by a round ball. Thus, let Ctn−1 ⊂ S n−1 be a polar cap (i.e., round ball) with corresponding angle 0 < t ≤ π2 such that μn−1 (Ctn−1 ) = μn−1 (B). By the isoperimetric inequality, μn−2 (∂Ctn−1 ) μn−2 (S n−2 ) · sinn−2 t μn−2 (∂ B) ≥ = t μn−1 (B) μn−1 (Ctn−1 ) μn−2 (S n−2 ) · 0 sinn−2 θ dθ =
sinn−2 t t 0
sinn−2 θ dθ
=: f n (t). n−1
For t ∈ [0, π/2), let h n (t) := (n − 2) 0 sinn−2 θ dθ − sincos t t , then h n (t) = − sinn−2 t (1+tan2 t) ≤ 0, so h n is decreasing. Since h n (0) = 0, it follows that h n (t) ≤ 0 for all t ∈ [0, π/2). Multiplying the corresponding equation with the non-negative expression cos tt gives for 0 < t < π2 , t
sin t
0 ...
(n − 2)
cos t − sin t
sinn−2 t t 0
sinn−2 θ dθ
≤ 0.
Concentration of Measure for Quantum States with a Fixed Expectation Value
809
But the left-hand side is exactly (ln f n (t)) , such that ln f n (t) is decreasing, and hence f n (t) is decreasing, so π 2 n2 =√ f n (t) ≥ f n , 2 π n−1 2 and this expression is larger than
1√ 2 n
if n≥3 (it grows asymptotically like
& √ 2 π
n).
To deal with ellipsoids, we need some results on expectation values of certain functions. Lemma 7 (Ellipsoidal expectation values). Let N ⊂ Rn be the full ellipsoid with equatorial radii {ak }nk=1 , ak > 0. Then we have the following expectation values with respect to the geometric measure in N : a12 + a22 + · · · + an2 , n+2 n 2 2 2 nk=1 ak4 + k=1 ak 4 . E N x = (n + 2)(n + 4)
E N x2 =
Proof. First, we use a linear transformation to reduce the expectation value calculations to integrals on the ball B n = {x ∈ Rn : x ≤ 1}. That is, let (x) := diag(a1 , . . . , an )x for x ∈ Rn , then by the transformation formula for integrals, we have f (x)d x = f ((x))| det D(x)|d x, N
Bn
where det D(x) = a1 a2 . . . an . The only other non-trivial ingredients are the spherical integrals μn (B n ) , x12 d x = n n+2 B 3μn (B n ) , (15) x14 d x = (n + 2)(n + 4) Bn μn (B n ) . x12 x22 d x = (n + 2)(n + 4) Bn These formulas can be proved directly by applying hyperspherical coordinates, cf. Ref. [37]. Now we are ready to estimate the crucial expression soids N .
μn−2 (M E ) μn (N )
for surrounding ellip-
Lemma 8 (Ratio of M E and the surrounding ellipsoid). Let n ≥ 3, and E H :=
n 1 1 n Ek k=1
−1
810
M. P. Müller, D. Gross, J. Eisert
be the harmonic mean energy corresponding to the energy levels E k > 0, k = 1, . . . , n. Suppose that n+2 (1 + δ)E H n with some δ > 0 such that the energy manifold M E is not empty. Moreover, suppose that E is less than the median of k E k xk2 on the sphere S n−1 . Let N be a full ellipsoid with equatorial radii {ak }nk=1 , ak > 0, i.e., n # $ x2 k n N := x ∈ R ≤1 . ak2 E=
k=1
Then, we have 1 3 n n 2
E 1 n2 1 μn−2 (M E ) 2E 2 > · 1− 2 . n μn (N ) 2 (1 + 2δ) 2 − 1 δ (n + 2)(n + 4) ak2 E k E2 k=1 k=1 k Proof. According to the isoperimetric inequality from Lemma 6, we have (with N E the energy ellipsoid as defined directly above that lemma), √ n μn−2 (M E ) = μn−2 (∂(S n−1 ∩ N E )) > μn−1 (S n−1 ∩ N E ). 2 Hence it remains to lower-bound μn−1 (S n−1 ∩ N E ); this is exactly the Haar measure probability Prob {ψ|H |ψ ≤ E}. In principle, this probability can be computed exactly: using the volume-preserving map from the unit sphere to the probability simplex [38], this probability equals the ratio of the volumes of the two bodies that originate from intersecting the simplex with a hyperplane. This ratio has been computed in Ref. [39], % k and the result is k:E k ≤E i=k EE−E for E ≤ E max . Unfortunately, the result is only i −E k valid in the non-degenerate case; moreover, despite its simple form, it is hard to estimate that value in a way which is useful in the current calculation. Thus, we instead use a different approach which is based on geometry of the ellipsoid. Let E(x) := nk=1 E k xk2 , and let Srn−1 be the sphere of radius r in Rn . First we show the inequality n−1 μn−1 (Sλr ∩ N E ) ≥ λn−1 μn−1 (Srn−1 ∩ N E )
(16)
Srn−1 ∩ N E
such that x = r and for any λ ∈ (0, 1). This can be seen as follows: let x ∈ n−1 2 ∩ NE . E(x) ≤ E. Then, λx = λx and E(λx) = λ E(x) < E, such that x ∈ Sλr n−1 n−1 n−1 Hence λ(Sr ∩ N E ) ⊂ Sλr ∩ N E , and so μn−1 (Sλr ∩ N E ) is lower-bounded by μn−1 (λ(Srn−1 ∩ N E )), which equals λn−1 μn−1 (Srn−1 ∩ N E ). n be the set of vectors in Rn with norm between a and b. With the help of Let Ba,b Ineq. (16), we get √1+2δ n√ μn (N E ∩ B1, )= μn−1 (Srn−1 ∩ N E ) dr 1+2δ
1
≤
√
1+2δ
r n−1 μn−1 (S n−1 ∩ N E ) dr
1 n
= μn−1 (S
n−1
(1 + 2δ) 2 − 1 . ∩ NE ) · n
Concentration of Measure for Quantum States with a Fixed Expectation Value
811
Thus, we have reduced the problem to finding a lower bound on μn (N E ∩ B1,√1+2δ ). Indeed, applying Lemma 7 to the assumptions of this lemma, we see that the expectation value of x2 with respect to the geometric measure in N E is exactly E N E x2 = 1 + δ, so we can indeed expect that much of the weight of N E is contained in B n √ prove this, we use the Chebyshev inequality. Let σ 2
be the variance of x2
1, 1+2δ
. To
with respect
to the geometric measure on N E , then it is easy to see that 2 σ 2 = E N E x4 − E N E x2 ≤
1 2E 2 , (n + 2)(n + 4) E2 k=1 k n
and the probability that a point in N E is not contained in B n √ σ2 δ2
1, 1+2δ
is upper-bounded by
. Hence
μn (N E ∩
n√ B1, ) 1+2δ
1 2E 2 ≥ 1− 2 δ (n + 2)(n + 4) E2 k=1 k n
μn (N E ).
The claim follows from substituting explicit expressions for μn (N E ) and μn (N ).
Since we want to show that our energy submanifold M E inherits measure concentration from an ellipsoid, we first have to prove concentration of measure for ellipsoids: Lemma 9 (Measure concentration for ellipsoids). Let N ⊂ Rn be the full ellipsoid with equatorial radii {ak }nk=1 , ak > 0. Then, for every κ > 0, we have ' 4a 4 ObsDiam(N , κ) ≤ √ ln , κ n where a := maxk ak . Proof. In accordance with Refs. [24,40], for a (convex) body K ⊂ Rn with surface S := ∂ K , we say that K is strictly convex if for every ε > 0 there exists some δ = δ(ε) > 0 such that x, y ∈ S and x − y ≥ ε implies (x + y)/2 ∈ (1 − δ)K . The unit ball B n is strictly convex, and one can choose 1 ε2 2 ε2 δ(ε) = 1 − 1 − ≥ 4 8 for 0 ≤ ε ≤ 2. According to Ref. [24, p. 37], if A ⊂ B n is any measurable subset with ε-neighborhood Aε , we have μ B n (Aε ) ≥ 1 −
1 2 · e−nε /4 . μ (A) Bn
Now let L = diag(a1 , . . . , an ) be the linear map which maps the ball B n onto the ellipsoid N . Since L is linear, it preserves the geometric measure; that is, for every
812
M. P. Müller, D. Gross, J. Eisert
measurable subset A ⊂ B n , we have μ N (L(A)) = μ B n (A). Now let B ⊂ N be any measurable subset. We claim that ! " −1 L L B ε ⊂ Bε . a
To prove this, let y∈L
! L
−1
" B
ε a
,
and let x := L −1 y. It follows that there is some x ∈ L −1 (B) such that x − x ≤ aε . Let y := L x ∈ B, then y − y = L(x − x ) ≤ L · x − x ≤ ε since L = a, so y ∈ Bε . Thus ! " = μBn L −1 B ε μ N (Bε ) ≥ μ N L L −1 B ε a
a
ε 2 1 ≥ 1− · e−n ( a ) /4 −1 μ B n (L B) 1 2 2 · e−nε /(4a ) . = 1− μ N (B)
Let now f : N → R be any 1-Lipschitz function, let m f be the median of f on N , and let A := {x ∈ N | f (x) ≤ m f } such that μ N (A) = 21 . It follows that μ N {x ∈ N | f (x) > m f + ε} ≤ μ N {x ∈ N | x ∈ Aε } = 1 − μ N (Aε ) 1 2 2 · e−nε /(4a ) . ≤ μ N (A) Repeating the calculation for f (x) − m f − ε and applying the union bound yields μ N {| f (x) − m f | > ε} ≤ 4 · e−nε
2 /(4a 2 )
=: κ.
(17)
Due to the characterization of the observable diameter as given in (13), we obtain ObsDiam(N , κ) ≤ 2ε, and the claim follows from expressing ε in terms of κ. As a last technical lemma, we need a comparison between the mean and the median on the (n − 1)-dimensional sphere. We suspect that the statement is well-known, but we have been unable to locate a proof in the literature. Lemma 10 (Mean vs. median on the sphere). Let f : S n−1 → R be any function that is λ-Lipschitz with respect to the Euclidean distance measure, inherited from the surrounding space Rn . Moreover, let E S n−1 f denote the expectation value, and m S n−1 f denote the median of f on S n−1 . Then, E S n−1 f − m S n−1 f ≤ √λπ . 2 n−2
Concentration of Measure for Quantum States with a Fixed Expectation Value
813
Proof. Without loss of generality, we may assume that λ = 1. We use Lévy’s Lemma on the sphere [36]. Normally, it is formulated for the geodesic distance ρ instead of the Euclidean distance d; but since d ≤ ρ, it follows that f must be 1-Lipschitz also for the geodesic distance. Since two arbitrary points on the sphere always have geodesic distance less than or equal to π , it is clear that | f (x) − m S n−1 f | ≤ π for all x ∈ S n−1 . Let ε > 0. Lévy’s Lemma states that ' π −ε2 (n−2)/2 ·e . μ S n−1 | f (x) − m S n−1 f | > ε ≤ 2 Abbreviating m := m S n−1 f and μ := μ S n−1 , we get E S n−1 f − m = |E S n−1 ( f − m)| ≤ E S n−1 | f − m| ∞ =− ε∂ε μ S n−1 {| f − m| > ε} dε ∞0 μ S n−1 {| f − m| > ε} dε = 0 ' ∞ π π 2 ≤ . e−ε (n−2)/2 dε = √ 2 0 2 n−2 This proves the claim.
Now we are ready to prove concentration of measure for the manifold that arises from intersecting a sphere with an ellipsoid in Euclidean space. Theorem 4 (Concentration of measure for M E , R-version). Let n ≥ 3, and {E k }nk=1 any set of positive energy levels with arithmetic mean E A := n1 nk=1 E k , harmonic mean −1 E H := n1 nk=1 E1k , maximum E max := maxk E k , minimum E min := mink E k , and −2 1 n 1 E Q := n k=1 E 2 . Suppose that E is any energy value which satisfies k 2 – E = 1 + n 1 + √εn E H for some ε > 0, – E ≤ EA −
π(E max √ −E min ) . n−2
Then, for every λ-Lipschitz function f : M E → R with median f¯, we have for the normalized geometric measure μ ≡ μ M E ,
3
2 n2 −n β E max e μ | f − f¯| > t ≤ 2 E 2 1 − ε2E 2 E2
!
3E min 64E
t 1 λ − 2n
2 "
√ +ε n
Q
whenever the denominator √ on the right-hand side is positive. The constant β > 0 can be chosen as β = 2048 e < 1075. π Moreover, the median f¯ of f on M E can be estimated as follows. Let N be the full ellipsoid of points x ∈ Rn with nk=1 E k xk2 ≤ E 1 + n1 , and let λ N be the Lipschitz constant of f in N . Then 1 21 3 E − f¯ − E N f ≤ λ N +b ·O n 2 , 4n E min
814
M. P. Müller, D. Gross, J. Eisert
& where the constant b > 0 can be chosen as b = 8 1 + 23 < 15, and 3 1 2 2βn 2 E max ε 1 . O n − 2 = √ + ln n n E 2 1 − 2E 2 2 2 ε E¯
Proof. Let δ := √εn . We may suppose that not all energy levels are equal, i.e., there otherwise, there is nothing to prove. Define the funcexist k and l such that E k = El ; n ˜ := E min + tion E : R → R by E(x) := nk=1 E k xk2 , and E˜ : Rn → R by E(x) n 2 2 ˜ ˜ (E − E )x , then E = E . Moreover, ∇ E(x) ≤ 4(E − E min )2 n−1 n−1 k min max S S k=1 k for all x ≤ 1, which proves that the Lipschitz constant of E S n−1 with respect to the Euclidean distance in Rn is upper-bounded by 2(E max − E min ). Since E A = E S n−1 E(x), the third condition together with Lemma 10 ensures that E is less than or equal to the median of E(x) on the sphere. For every E , define the energy ellipsoid N E via N E := {x ∈ Rn | E(x) ≤ E }. Suppose that x ∈ Uε (M E ), so there is some y ∈ M E with x − y < ε, hence |E(x) − E(y)| ≤
max
z∈Uε (M E )
∇ E(z) · x − y ≤ 2E max (1 + ε)ε.
It follows that E(x) ≤ E + 4E max ε if 0 < ε ≤ 1. Consequently, Uε (M E ) ⊂ N E+4εE max , and N := N E+4εE max will be the surrounding body of M E that we use when applying Lemma 5. We arbitrarily fix the value ε :=
E 4n E max
which turns out to be an almost optimal choice (clearly, 0 < ε ≤ 1). Hence N = N
E 1+ n1
is a full ellipsoid with equatorial radii ak =
E Ek
1 2 1 1+ . n
Using that ((1 + 2δ)n/2 − 1)−1 ≥ e−nδ and a few more easy simplifications, we get by applying Lemma 8, n 1 −2 μn−2 (M E ) 1 3 −nδ 2E 2 1+ > n2e . 1− 2 2 μn (N ) 2 n nδ E Q By Lemma 9 and 1 +
1 n
≤ 1 + 13 , we have measure concentration in N :
ObsDiam(N , κ) ≤ 8
E 1 4 · ln n 3E min κ
1 2
.
Then Lemma 5 yields measure concentration in M E – applying that lemma, using the pre − n vious inequalities together with the fact that E ≤ E max and that 1 + n1 2 is decreasing
Concentration of Measure for Quantum States with a Fixed Expectation Value
and hence lower-bounded by limn→∞
815
n − 21 1 1 + n1 = e− 2 , we get
( E 1 +8 ObsDiam(M E , 2κ) ≤ 2n 3E min 21 3 2 1 2E 2 1 512n 2 E max + − ln 1 − 2 2 × δ+ . ln n π E 2κ 2 nδ E Q Then the first claimed inequality follows from the characterization of the observable diameter as given in (14). To prove the second claim, suppose that f : N → R is any (λ N = 1)-Lipschitz function, and define for ξ > 0, X ξ := x ∈ M E : | f (x) − m M E f | ≤ λξ . We already know that μ M E (X ξ ) is large. Lemma 3 yields μn (Uε (X ξ )) π ε2 μn−2 (X ξ ) ≥ μn (N ) 4(n − 1) μn (N ) 2 μn−2 (M E ) πε μ M E (X ξ ) · =: P. = 4(n − 1) μn (N )
μ N (Uε (X ξ )) =
We know that we have measure concentration in N ; setting a := min ak = k
E E min
1 1+ n
1 2
and using Eq. (17) of Lemma 9, we get for all C > 0, μ N {x ∈ N : | f (x) − m N f | > C} ≤ 4 · e−nC
2 /(4a 2 )
,
and using [36, App. V.4], μ N {x ∈ N : | f (x) − E N f | > C} ≤ 8 · e−nC
2 /(32a 2 λ2 )
.
Set now C := a
8 32 ln n P
1 2
,
then μ N {| f (x) − E N f | ≤ C} > 1 − P. Thus, Uε (X ξ ) ∩ {x ∈ N : | f (x) − E N f | ≤ C} = ∅, and if x is any element of that intersection, | f (x)−E N f | ≤ C and | f (x)−m M E f | ≤ ξ +ε holds, such that m M f − E N f ≤ a E
32 8 ln n P
1 2
+ ξ + ε.
(18)
816
M. P. Müller, D. Gross, J. Eisert
Now we specialize ξ by setting ξ :=
1 +8 2n
1 E α 2 δ+ , E min n
where 3 2 2E 2 1 4096n 2 E max α := − ln 1 − 2 2 + ln . 2 π E2 nδ E Q This ξ is chosen such that μ M E (X ξ ) ≥ 21 . The assertion of the theorem is then proved by substituting all the previously established inequalities into (18). Consider now the assumptions given in Theorem 1, but denote the complex dimension by n. ˜ Define ε˜ := √ε , double all energy eigenvalues, and substitute this into Theorem 4. 2 After dropping all tildes, this substitution yields the statements of Theorem 1 and 2. The proof of Theorem 3 can now be given as follows: Proof of Theorem 3. Every |ψ ∈ A ⊗ B can be written
|ψ = ψ jk | j ⊗ |k, j,k
where H A | j = E Aj | j and H B |k = E kB |k. We embed all vectors in real space R2n by introducing coordinates x jk and y jk such that ψ jk =: x jk + i y jk . First we apply Theorem 2 to estimate the matrix elements of ψ A := Tr B |ψψ|. Embedding the ellipsoid N from Theorem 2 into R2n , we get a real ellipsoid with equatorial radii 1 1 /E jk ) 2 , where each equatorial radius appears twice, namely for the coordi(E 1 + 2n nate axes x jk and y jk . Let v, w ∈ {x, y}, then a transformation to spherical coordinates yields 1 E N (vab wcd ) = vab wcd dz μ2n (N ) N 1 E 1 + 2n = vab wcd dz E ) 21 B 2n μ2n (B 2n )(E ab cd ⎧ if (a, b) = (c, d) or v = w ⎨0 = E 1+ 2n1 ⎩ otherwise, E (2n+2) ab
where we have used Eq. (15) and the equation
Bn
x1 x2 d x = 0. Hence
1 δ p,q E 1 + 2n ¯ . E N ψ pk ψqk = E N (x pk xqk + y pk yqk ) + i(y pk xqk − x pk yqk ) = E pk (n + 1)
Concentration of Measure for Quantum States with a Fixed Expectation Value
Since p|ψ A |q =
|B|
¯
k=1 ψ pk ψqk ,
817
this yields
|B| 1
δ p,q 1 + 2n E E N p|ψ |q = . n+1 E pk A
k=1
To bound the Lipschitz constants, we compute gradients: the result for the real part is ⎧ 2 ⎨ |B| x 2 + x 2 + y 2 + y 2 ψ2 if p = q qj pj q j pj j=1 ≤ ∇ p|ψ A |q = |B| 2 2 4ψ2 if p = q, ⎩ x +y 4 j=1
pj
pj
and for the imaginary part, we get # 2 |B| 2 + y2 + x 2 + x 2 y ψ2 if p = q A pj q j q j pj j=1 p|ψ ≤ |q = ∇ 0 if p = q. 0 Consider the functions r pq (ψ) := p|ψ A |q and i pq (ψ) := p|ψ A |q. All corresponding Lipschitz constants λ N in the ellipsoid N satisfy λN ≤ 2
E E min
1 1+ n
21
,
since this square root denotes the largest equatorial radius, which is an upper bound to ψ. It follows from Theorem 2 that the median of both functions satisfies |¯r pq − (ρc ) p,q | |i¯ pq − (ρc ) p,q |
≤2 ,
E E min
21 1 3 1+ + 15on , n 8n -. / =:δ
where on =
E E min
3
ε ln(2an 2 ) √ + 2n n
21 .
By the triangle inequality, we have r pq (ψ) − (ρc ) p,q > t + 2δ ⇒ r pq (ψ) − r¯ pq > t i pq (ψ) − (ρc ) p,q > t + 2δ ⇒ i pq (ψ) − i¯ pq > t for every t ≥ 0. Since the Lipschitz constants of r pq and i pq in the sphere (and thus in M E ) satisfy λ ≤ 2, it follows from Theorem 1 that Prob r pq (ψ) − (ρc ) p,q > 2t + 2δ ≤ η, and similarly for i pq , where we used the abbreviation √ 3 1 2 η := a · n 2 exp −cn t − + 2ε n . 4n
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Combining the two inequalities for the real and the imaginary part, we get √ Prob p|ψ A |q − (ρc ) p,q > 2(2t + 2δ) ≤ 2η. By the√probability that there exist indices p and q such that theA union bound, p|ψ |q − (ρc ) p,q > 2(2t + 2δ) is upper-bounded by |A|(|A| + 1)η. If this is not the case, i.e., if no such indices exist, then 2 2 A p|ψ A |q − (ρc ) p,q ≤ 2|A|2 (2t + 2δ)2 . ψ − ρc = 2
Thus,
p,q
2 Prob ψ A − ρc ≤ 8|A|2 (t + δ)2 ≥ 1 − |A|(|A| + 1)η. 2
This proves the claim.
As stated in the Introduction, we now give a proof that it is always possible to shift the energy offset such that the energy E in question becomes (close to) the harmonic mean energy. Lemma 11 (Harmonic mean and energy shifts). Suppose we are given energy levels {E k }nk=1 and an energy E between the smallest and the arithmetic mean energy E A , that is min E k ≤ E < E A := k
n 1
Ek . n k=1
Then there exists E ∈ R such that the harmonic mean of the energies {E k + E}nk=1 equals E + E, and all energies are non–negative: E k + E ≥ 0. Moreover, E is unique unless all energies are equal. Proof. Denote the harmonic mean of n energy values E 1 , . . . , E n by E H {E k }nk=1 := −1 n 1 1 ; similarly, we use E A {E k }nk=1 := n1 nk=1 E k to emphasize the depenk=1 n Ek dence of the arithmetic mean E A on the energy values. Let E( E) := E H {E k + E}nk=1 − E which defines a continuous function. We may assume without restriction that E 1 = mini E i and E n = maxi E i . If E = −E 1 , then E( E) = E H {E k − E 1 }nk=1 + E 1 = E H {0, . . .} + E 1 = E 1 . It remains to show that lim E→∞ E( E) = E A , then by continuity there must be some E ≥ −E 1 such that E( E) = E. The limit identity we would like to show is equivalent to lim E H {E k + E}nk=1 − E A {E k + E}nk=1 = 0. E→∞
We apply an inequality given by Furuta [41]: E A {E k + E}nk=1 ≥ E H {E k + E}nk=1 4(E 1 + E)(E n + E) ≥ E A {E k + E}nk=1 . (E 1 + E n + 2 E)2 , -. / →1 for E→∞
Concentration of Measure for Quantum States with a Fixed Expectation Value
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This proves existence of some E such that E( E) = E. In order to see uniqueness, T note that the Cauchy-Schwarz inequality for two vectors v = n1 , . . . , n1 and a = 2 (a1 , . . . , an )T , i.e. |v, a|2 ≤ v, va, a implies that n1 nk=1 ak ≤ n1 nk=1 ak2 , and we have equality if and only if a1 = a2 = · · · = an . The derivative of the function E turns out to be n n 2 1
1 1 1 −1 E (x) = / n (E k + x)2 n Ek + x k=1
k=1
which must thus be strictly positive unless all energies are equal.
6. Approximate Sampling of the Manifold There is a well-known method [42,43] to pick random points from the surface of a hypersphere S n−1 ⊂ Rn : Generate X 1 , X 2 , . . . , X n random real numbers, distributed independently identically according to the normal distribution with density proportional 1 to exp(−nx 2 /2). Then, normalize the resulting vector: For r := (X 12 + X 22 + · · · + X n2 ) 2 , the point (X 1 , X 2 , . . . , X n )T /r is uniformly distributed on the unit hypersphere. If the uniform distribution on the hypersphere shall be sampled only approximately, then the normalization is not necessary: we have the norm expectation value EX 2 = 1 and the variance VarX 2 = n2 , such that the distribution of the vectors X themselves closely resembles the uniform distribution on the sphere in high dimensions n. This way, expectation values of functions f : Rn → R with respect to the uniform distribution on the sphere can be estimated numerically to good accuracy (assuming that f is slowly varying and not growing too fast at infinity). This has the quantum interpretation (if n is even) of drawing random pure states in Cn/2 . It turns out that a simple modification of this algorithm yields approximate sampling of the energy manifold M E , or rather of its measure μ M E = Prob that we use in this paper. We describe the algorithm below. In contrast to the rest of this paper, we do not give explicit error bounds in this case, because the necessary calculations are straightforward but very lengthy, and the resulting error bounds depend sensitively on the assumptions on the regularity of the functions f that are considered. However, we discuss a rough estimate of the error at the end of this section. Algorithm 5 (Approximate sampling of M E ). Suppose we are given an observable H = H † on Cn with eigenvalues {E k }nk=1 and an energy value E such that Theorem 1 applies and proves sufficient concentration of measure. Then, the uniform (Hausdorff) measure on the manifold of quantum states |ψ with ψ|H |ψ = E can be numerically sampled in the following way: 1. Find an energy shift s ∈ R such that H := H + s1 ≥ 0, and such that the harmonic mean of the new energy levels E k := E k + s equals E := E + s, i.e., n −1 1 1 E H := = E . n E k k=1
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2. Generate all real and imaginary parts ψk and ψk of |ψ (in the eigenbasis of H ) independently according to the Gaussian distribution proportional to E exp −n k x 2 . E If H = 1 and E = 1 (which means that we have a void constraint), this algorithm reduces to the well-known sphere point picking algorithm as a special case (note that the real dimension is 2n, which cancels a factor 21 in the exponent). If H is not proportional to the identity, then the entries of the random vector |ψ are independently, but not identically distributed. Note that a similar “Gaussian approximation” has been used in Ref. [21] right from the start in the analysis of the mean energy ensemble (without error bounds). Without using the results in this paper, direct calculation shows that the distribution generated by the algorithm above satisfies Eψ2 =
E =1 E H
(explaining the choice of the energy shift) and Eψ|H |ψ = E . The corresponding 2 variances are Varψ|H |ψ = En and n 1 E 2 ; Varψ = 2 n E k 2
k=1
this expression is also present in Theorem 1, where it is called E 2 /(n E 2Q ) and assumed to be small (the factor n is absorbed into ε there). Thus, the algorithm above produces points close to the energy manifold M E with high probability. But does it approximate the uniform distribution on M E ? Since physics mainly involves computing expectation values of observables, we are interested in a weak form of approximation where we say that two measure μ and ν on Cn ∼ R2n (or on submanifolds) are close, i.e., μ ≈ ν, if Eμ f ≈ Eν f for all real functions f : Cn → R that satisfy certain regularity conditions (such as Lipschitz continuity and polynomial growth at infinity). For example, the uniform measure on the sphere μ S 2n−1 and in the ball μ B 2n are close if n is large: Since μ B 2n {|ψ | ψ < 1 − ε} = (1 − ε)2n ≤ exp(−2nε), most of the points in the ball are close to the surface. As a consequence, a simple calculation shows that expectation values of λ-Lipschitz functions f : B 2n → R satisfy E B 2n f − E S 2n−1 f ≤
λ . 2n + 1
Are the uniform measure μ M E and the resulting Gaussian measure from Algorithm 5 close in this sense? The answer is yes, and the results in this paper give a simple geometric explanation for this fact, which is schematically depicted in Fig. 1:
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Fig. 1. Geometric caricature why Algorithm 5 samples the measure μ M E on the energy manifold to good approximation. Point a) denotes the well-known sphere point picking algorithm, and e) is for Algorithm 5. See the text below for an explanation how the one leads to the other
a) As explained at the beginning of this section, it is well-known how to pick random points approximately from the uniform distribution on the sphere S 2n−1 : choose real and imaginary parts randomly, distributed independently identically according to a Gaussian distribution proportional to exp(−nx 2 ). b) We have just seen that the uniform distribution in the ball B 2n and on the sphere S 2n−1 are close to each other. Hence the algorithm from a) also samples the uniform distribution in the ball to good approximation. c) Let N be the full ellipsoid with equatorial radii 1 n {ak }nk=1 := (E /E k ) 2 k=1
H ,
in the directions of the eigenvectors of i.e., N := z ∈ C | z|H |z ≤ E . Then the ball B 2n and the ellipsoid N are related by a linear transformation L : Cn → Cn which preserves the normalized geometric volume measure: L := diag(a1 , . . . , an ), then N = L B (2n) and μ B 2n = L ∗ (μ N ). Sampling the ball B 2n , and then applying the linear transformation L, is the same as sampling the full ellipsoid N . Writing y = L x, the components yk of vectors after 1 the transformation are related to the components x K before by yk = (E /E k ) 2 xk . Hence ⎛ ⎛ ⎞2 ⎞ 21 E E k 2 ⎜ ⎟ k ⎠ . = exp −n exp −nxk2 = exp ⎝−n ⎝ y y ⎠ k E E k Thus, we have shown that Algorithm 5 samples the full ellipsoid N to good approximation. d) The full ellipsoid N is close to the full ellipsoid N from Theorem 2. There, it was shown that the uniform volume measure in N is close to the uniform measure μ M E on the energy manifold M E . Hence the uniform volume measure in N is close to μME . e) In summary, the measure produced by Algorithm 5 is close to μ M E as claimed – assuming that the underlying Hamiltonian is in the range of applicability of Theorem 1 and 2.
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The sampling algorithm gives a simple method for a numerical check of identities such as the form of the reduced density matrix in Example 2. It is interesting to note that even though the typical reduced density matrices are not Gibbs states (cf. Theorem 3), the distribution involved in Algorithm 5 involves the Boltzmann-like term exp(−cE k ), where c > 0 is constant, and E k denotes the k th energy level. How good is the approximation given by Algorithm 5? As explained above, the algorithm is meant to approximate expectation values of functions f : Rn → R on the energy manifold with respect to the measure μ M E . Thus, we would like to estimate the expression |E f − E M E f |, where E denotes the expectation value with respect to the Gaussian measure used in the algorithm. As a lower bound on that error (for some f ), recall that 2 E 2 Varψ2 = E ψ2 − 1 = . n E 2Q 2 The function f (ψ) := ψ2 − 1 is Lipschitz continuous, and the upper bound on the √ Lipschitz constant ∇ f (ψ) = 4ψ(1 − ψ2 ) ≤ 8/(3 3) on the unit ball does not grow with n. Assume for simplicity that E and E Q are constant in n (like in Example 1 and Example 2). Then |E f − E M E f | =
E 2 E 2Q
·
1 , n
which shows that we have to expect at least an error of the order 1/n even for functions f and Hamiltonians H that behave very regularly. A rough upper bound on the error can be given by adding the error contributions of steps a), b), c), and d) in Algorithm 5. It seems that the dominant contribution comes from step d) – a corresponding error estimate is given in Theorem 2. It is roughly of the order n −1/4 , again assuming that the energies and the Lipschitz constant are constant in n. 7. Conclusions In this work, we have established the notion of concentration of measure for quantum states with a fixed expectation value. The results that we established constitute on the one hand a new proof tool to assess properties of quantum states with the probabilistic method. Such a proof tool is expected to be helpful in a number of contexts, e.g., when sharpening counterexamples to additivity by enforcing a strong “conspiracy” by means of a suitable Hilbert Schmidt constraint, adding to the portfolio of techniques available related to the idea of a probabilistic method. On the other hand, in this work we are in the position to introduce concentration of measure ideas to notions from quantum statistical mechanics, specifically to the mean energy ensemble, and link this physically meaningful ensemble to ideas of typicality. Obviously, a constraint of the type introduced here could as well relate to settings where the particle number is held constant, so is expected to be applicable to a quite wide range of physical settings. It is also the hope that methods similar to the ones established here help assessing questions of typicality in the context of quantum dynamics and addressing key open problems in the theory of relaxation [15–18] of non-equilibrium complex quantum systems.
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Acknowledgements. We would like to thank C. Gogolin and R. Seiler for discussions. This work has been supported by the EU (COMPAS, CORNER, QESSENCE, MINOS) and the EURYI. It was done partially while DG was visiting the Institute for Mathematical Sciences, National University of Singapore in 2010. The visit was supported by the Institute.
References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33.
Alon, N., Spencer, J.H.: The probabilistic method. Newyork: Wiley 2000 Lloyd, S., Pagels, H.: Complexity as thermodynamic depth. Ann. Phys. 188, 186 (1988) Hayden, P., Leung, D., Winter, A.: Aspects of generic entanglement. Commun. Math. Phys. 265, 95 (2006) Hayden, P., Leung, D.W., Shor, P.W., Winter, A.: Randomizing quantum states: Constructions and applications. Commun. Math. Phys. 250, 371 (2004) Horodecki, M., Oppenheim, J., Winter, A.: Quantum information can be negative. Nature 436, 673 (2005) Hastings, M.B.: A counterexample to additivity of minimum output entropy. Nature Phys. 5, 255 (2009) Gross, D., Flammia, S.T., Eisert, J.: Most quantum states are too entangled to be useful as computational resources. Phys. Rev. Lett. 102, 190501 (2009) Bremner, M.J., Mora, C., Winter, A.: Are random pure states useful for quantum computation. Phys. Rev. Lett. 102, 190502 (2009) Goldstein, S., Lebowitz, J.L., Tumulka, R., Zanghi, N.: Canonical typicality. Phys. Rev. Lett. 96, 050403 (2006) Popescu, S., Short, A.J., Winter, A.: Entanglement and the foundations of statistical mechanics. Nature Phys. 2, 754 (2006) Reimann, P.: Foundation of statistical mechanics under experimentally realistic conditions. Phys. Rev. Lett. 101, 190403 (2008) Gogolin, C.: Einselection without pointer states. Phys. Rev. E 81, 051127 (2010) Srednicki, M.: Chaos and quantum thermalization. Phys. Rev. E 50, 888 (1994) Garnerone, S., de Oliveira, T.R., Zanardi, P.: Typicality in random matrix product states. Phys. Rev. A 81, 032336 (2010) Kollath, C., Läuchli, A., Altman, E.: Quench dynamics and non equilibrium phase diagram of the BoseHubbard model. Phys. Rev. B 74, 174508 (2006) Rigol, M., Dunjko, V., Yurovsky, V., Olshanii, M.: Relaxation in a completely integrable many-body quantum system: An ab initio study of the dynamics of the highly excited states of lattice hard-core bosons. Phys. Rev. Lett. 98, 050405 (2007) Cramer, M., Dawson, C.M., Eisert, J., Osborne, T.J.: Exact relaxation in a class of non-equilibrium quantum lattice systems. Phys. Rev. Lett. 100, 030602 (2008) Linden, N., Popescu, S., Short, A.J., Winter, A.: On the speed of fluctuations around thermodynamic equilibrium. http://arXiv.org/abs/0907.1267v1 [quant-ph], 2009 Brody, D.C., Hook, D.W., Hughston, L.P.: Quantum phase transitions without thermodynamic limits. Proc. R. Soc. A 463, 2021 (2007) Bender, C.M., Brody, D.C., Hook, D.W.: Solvable model of quantum microcanonical states. J. Phys. A 38, L607 (2005) Fresch, B., Moro, G.J.: Typicality in ensembles of quantum states: Monte Carlo sampling versus analytical approximations. J. Phys. Chem. A 113, 14502 (2009) Jiang, Z., Chen, Q.: Understanding Statistical Mechanics from a Quantum Point of View. In preparation Federer, H.: Geometric measure theory. Berlin-Heidelberg-New York: Springer-Verlag, 1969 Ledoux, M.: The concentration of measure phenomenon. Mathematical Surveys and Monographs 89, Providence, RI: Amer. Math. Soc., 2001 Cover, T.M., Thomas, J.M.: Elements of information theory, Second Edition. New York: Wiley, 2006 Gromov, M.: Metric structures for Riemannian and Non-Riemannian spaces. Modern Birkhäuser Classics, Basel-Boston: Birkhäuser, 2007 Zyckowski, K., Sommers, H.-J.: Induced measures in the space of mixed quantum states. J. Phys. A 34(35), 7111 (2001) Hall, M.: Random quantum correlations and density operator distributions. Phys. Lett. A 242, 123 (1998) Bhatia, R.: Matrix analysis. Berlin-Heidelberg-New York: Springer, 1997 Santaló, L.A.: Integral geometry and geometric probability. Reading, MA: Addison-Wesley, 1972 Tasaki, H.: Geometry of reflective submanifolds in Riemannian symmetric spaces. J. Math. Soc. Japan 58(1), 275–297 (2006) Schneider, R., Weil, W.: Stochastic and integral geometry. Reading, MA: Springer, 2008 Funano, K.: Concentration of 1-Lipschitz Maps into an infinite dimensional p -ball with the q -distance function. Proc. Amer. Math. Soc. 137, 2407 (2009)
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34. Funano, K.: Observable concentration of mm-spaces into nonpositively curved manifolds. Geometriae Dedicata 127, 49 (2007) 35. Elstrodt, J.: Maß–und Integrationstheorie. Reading, MA: Springer, 1996 36. Milman, V.D., Schechtman, G.: Asymptotic theory of finite dimensional normed spaces. Lecture Notes in Mathematics 1200. Reading, MA: Springer, 2001 37. Blumenson, L.E.: A derivation of n-dimensional spherical coordinates. American Mathematical Monthly 67(1), 63 (1960) 38. Bengtsson, I., Zyczkowski, K.: Geometry of quantum states - an introduction to quantum entanglement. Cambridge: Cambridge University Press, 2006 39. Dempster, A.P., Kleyle, R.M.: Distributions determined by cutting a simplex with hyperplanes. Ann. Math. Stat. 39(5), 1473 (1968) 40. Barvinok, : Measure concentration in optimization. Reading, MA: Springer, 2007 41. Furuta, T.: Short proof that the arithmetic mean is greater than the harmonic mean and its reverse inequality. Math Ineq and Appl. 8(4), 751 (2005) 42. Müller, M.E.: A note on a method for generating points uniformly on N -dimensional spheres. Comm. Assoc. Comput. Mach. 2, 19 (1959) 43. Marsaglia, G.: Choosing a point from the surface of a sphere. The Annals of Mathematical Statistics 43(2), 645 (1972) Communicated by M.B. Ruskai
Commun. Math. Phys. 303, 825–844 (2011) Digital Object Identifier (DOI) 10.1007/s00220-011-1209-x
Communications in
Mathematical Physics
C2 -Cofiniteness of the 2-Cycle Permutation Orbifold Models of Minimal Virasoro Vertex Operator Algebras Toshiyuki Abe Graduate School of Science and Engineering, Ehime University, 2-5, Bunkyocho, Matsuyama, Ehime 790-8577, Japan. E-mail:
[email protected] Received: 30 April 2010 / Accepted: 26 October 2010 Published online: 11 February 2011 – © Springer-Verlag 2011
Abstract: In this article, we give a sufficient and necessary condition for the = (V ⊗ V )σ for a C2 -cofinite vertex operator algebra V and C2 -cofiniteness of V the 2-cycle permutation σ of V ⊗ V . As an application, we show that the 2-cycle permutation orbifold model of the simple Virasoro vertex operator algebra L(c, 0) of minimal central charge c is C2 -cofinite.
1. Introduction Given a vertex operator algebra V , the tensor product V ⊗k of k-copies of V as a C-vector space canonically has a vertex operator algebra structure (see [FHL]). Each permutation σ of tensor factors gives rise to an automorphism of V ⊗k of finite order, and the fixed point set (V ⊗k )σ = {u ∈ V ⊗k | σ (u) = u} becomes a vertex operator subalgebra called a σ -permutation orbifold model. In this article, we consider the 2-transposition σ of V ⊗ V for a C2 -cofinite vertex operator algebra V , and study the C2 -cofiniteness condition of (V ⊗ V )σ . We give a sufficient and necessary condition for the C2 -cofiniteness of the σ -permutation orbifold model, and we show that (L(c, 0) ⊗ L(c, 0))σ is C2 -cofinite 2 with coprime integers p, q ≥ 2. for any minimal central charge c = c p,q = 1 − 6 ( p−q) pq The C2 -cofiniteness condition requires that V / a(−2) b a, b ∈ V C is finite dimensional, where a(−2) b denotes the −2-product of ordered pair (a, b) for a, b ∈ V . A vertex operator algebra satisfying this condition is called C2 -cofinite. The C2 -cofiniteness condition is one of the most important properties in the vertex operator algebra theory, and once a vertex operator algebra satisfies this condition, its modules have a lot of remarkable features, for example, modular invariance of (extended) trace functions and the existence of fusion products (see [H,M1,M2,M3,M4] and references therein). In spite of its importance, a verification of the C2 -cofiniteness condition is a very difficult task in general, and This work was supported by Grant-in-Aid for Young Scientists (B) 20740017.
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some conjectures for the C2 -cofiniteness have still remained open. One of them is that if V is a C2 -cofinite simple vertex operator algebra and G is a finite automorphism group, then the fixed point vertex operator subalgebra V G := {a ∈ V | g(a) = a, ∀g ∈ G} is C2 -cofinite. This conjecture has been checked only for some examples such as lattice vertex operator algebras with a certain involution ([Yam]), and no definitive idea to prove this conjecture has been found. The permutation orbifold theory has been studied by physicists two decades ago (see [KS,FKS], see also [Ban2]). Its systematic study has been started with the papers [BHS] for cyclic permutations, and was generalized to the general symmetric group in [Ban1]. The techniques are effectively applied to the study of elliptic genera in terms of the second quantized string theory in [DMVV] and also applied to the study of the kernels of the modular representations in [Ban3] for example. On the other hand, the remarkable paper [BDM] seems to be known as a unique paper in which permutation orbifold models are studied from the point of view of the vertex operator algebra theory. The aim of this article is to start the study of structure theory of permutation orbifold models in the theory of vertex operator algebras, and especially to show the C2 -cofiniteness condition is valid for permutation orbifold models of C2 -cofinite vertex operator algebras. As the very first step, we consider 2-cycle permutation orbifold models of C2 -cofinite vertex operator algebras. We could not show that an arbitrary permutation orbifold model is C2 -cofinite without any condition except a simpleness and C2 -cofiniteness of the based vertex operator algebra. But we give a sufficient and necessary condition for this assertion. Let V be a vertex operator algebra. Then the 2-cycle permutation orbifold model = (V ⊗ V )σ is linearly spanned by vectors φ(a, b) = a ⊗ b + b ⊗ a for a, b ∈ V . V We then have an identity φ(a(−n) u, v) ≡ −φ(u, a(−n) v)
) mod C2 (V
(1.1)
for a, u, v ∈ V and n ≥ 2, where a(−n) b denotes the −n-product. This identity plays an important role in our arguments. Suppose that V is C2 -cofinite and assume that V is strongly generated by a finite set T . Then by using the identity (1.1) and other facts, we can prove that if V is C2 -cofinite, then the finiteness of the dimension of the subspace ) a, b ∈ T, n > 0 φ(a(−n) b, 1) + C2 (V C /C2 (V ) is equivalent to the C2 -cofinitenss of V . in V For the case V is the simple Virasoro vertex operator algebra L(c, 0) of a central charge c, V is strongly generated by the Virasoro vector ω. It is also known that V is C2 -cofinite if and only if c is a minimal central charge c p,q with relatively prime integers p, q ≥ 2 (see [Ar] for example). Therefore under the assumption that c is a /C2 (V ) spanned by φ(ω(−n) ω, 1) + C2 (V ) minimal central charge, if the subspace of V is C2 -cofinite. We in fact show that for n ∈ Z>0 is finite dimensional then V ) φ(ω(−n) ω, 1) ∈ C2 (V
(1.2)
for n ≥ 30 regardless of the central charge c by direct computations. As a result we find is C2 -cofinite in the case c is a minimal central charge. that V This article is organized as follows. In Sect. 2 we recall some notions and results from representation theory of vertex operator algebras, and give some identities which are used in the later sections. In Sect. 3, we prove (1.1) and some useful lemmas. In Sect. 4, we give when V is C2 -cofinite. a sufficient and necessary condition for the C2 -cofiniteness of V
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Section 5 is devoted to calculations concerning with the Virasoro vectors. We have some numerical results with the help of a computer and show that (L(c, 0) ⊗ L(c, 0))σ is C2 -cofinite for a minimal central charge c. Section 6 is an appendix where we give a list of results of computations which is necessary in Sect. 5 and show that (1.2) is valid for n ≥ 30. 2. Preliminaries A vertex operator algebra is a quadruplet (V, ·(n) ·, 1, ω) consisting of a Z≥0 -graded C-vector space V = ∞ d=0 Vd , bilinear maps V × V (a, b) → a(n) b ∈ V associated to each integer n ∈ Z, and two distinguished vectors 1 ∈ V0 called the vacuum vector and ω ∈ V2 called the Virasoro vector. For its axioms, refer to [MN] for example. We write down some identities from the axiom of the vertex operator algebras which we need in this article: Associativity formula. ∞ m (−1)i (a(m−i) b(n+i) u − (−1)m b(m+n−i) a(i) u) (a(m) b)(n) u = i
(2.1)
i=0
for a, b, u ∈ V and m, n ∈ Z. Commutativity formula. [a(m) , b(n) ]u =
∞ m (a(i) b)(m+n−i) u i
(2.2)
i=0
for a, b, u ∈ V and m, n ∈ Z. The vacuum vector satisfies that a(i) 1 = 0 for i ∈ Z≥0 , 1(n) a = δn,−1 a. As for the Virasoro vector ω, if we set L n = ω(n+1) then m + 1 cV id V , [L m , L n ] = (m − n)L m+n + δm+n,0 (2.3) 3 2 (L −1 a)(n) = −na(n−1) , (2.4) L 0 a = da for d ∈ Z≥0 and a ∈ Vd . (2.5) Throughout this article, we assume that V is of CFT type, that is, V0 = C1. We give an identity deduced from the associativity formula. Lemma 2.1. For a, b, u ∈ V and m, n ∈ Z>0 , (a(−m) b(−n) 1)(−1) u = a(−m) b(−n) u ∞ + (αm,n;i a(−m−n−i) b(i) u + αn,m;i b(−m−n−i) a(i) u), i=0
where αm,n;i =
m + n − 1 + i (−1)n−1 (m + n − 1)! (m − 1)!(n − 1)! i n+i
for m, n ∈ Z>0 and i ∈ Z≥0 .
(2.6)
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Proof. By the Associativity formula, we have (a(−m) b(−n) 1)(−1) u ∞ −m (−1)i a(−m−i) (b(−n) 1)(−1−i) u = i i=0 ∞ −m (−1)i+m (b(−n) 1)(−m−1−i) a(i) u − i i=0 ∞ m − 1 + i −1 + i (−1)n−1 a(−m−i) b(−n+i) u = m−1 n−1 i=0 ∞ m − 1 + i −m − 1 − i (−1)m+n−1 b(−m−n−i) a(i) u. − m−1 n−1 Since
−1+i n−1
(2.7)
i=0
= 0 if 1 ≤ i ≤ n − 1, the first summation in the right hand side is equal to
a(−m) b(−n) u +
∞ m +n−1+i n−1+i (−1)n−1 a(−m−n−i) b(i) u. m−1 n−1 i=0
We see that m +n−1+i n−1+i (m + n − 1 + i)!(−1)n−1 (−1)n−1 = = αm,n;i . m−1 n−1 (m − 1)!(n − 1)!i!(n + i)
On the other hand, it follows from −m−1−i = m+n−1+i (−1)n−1 that the second n−1 n−1 summation of the right hand side in (2.7) is equal to ∞ m −1+i m +n−1+i (−1)m−1 b(−m−n−1−i) a(i) u m−1 n−1 i=0
=
∞
αn,m;i b(−m−n−1−i) a(i) u.
i=0
Hence we have the lemma.
We next recall the definition of the C2 -cofiniteness condition and related results following [Z]. A vertex operator algebra V is said to satisfy the C2 -cofiniteness condition or to be C2 -cofinite if the subspace C2 (V ) := a(−2) b a, b ∈ V C is finite codimensional in V . We denote by · : V → V /C2 (V ) the canonical projection. The quotient space V /C2 (V ) becomes a Poisson algebra with multiplication and Lie bracket a · b = a(−1) b, [a, b] = a(0) b for a, b ∈ V respectively. In this article we do not use its Lie algebra structure. A vertex operator algebra V is called strongly generated by a subset T ⊂ V if V is 1 r spanned by 1 and vectors of the form a(−n · · · a(−n 1 with r ≥ 1, a i ∈ T and n i ≥ 1. r) 1)
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Proposition 2.2. If V is strongly generated by a set T , then V /C2 (V ) is generated by T as a commutative associative algebra with unit 1. Proposition 2.2 can be proved by the facts that for a i ∈ T and n i ∈ Z>0 , 1 r a(−1) . . . a(−1) 1 = a 1 · · · · · ar ,
and 1 r a(−n . . . a(−n 1 = 0, r) 1)
if some n i > 1. The following theorem is one of the important properties of the subspace C2 (V ). Theorem 2.3 ([GN]). Let V be a vertex operator algebra of CFT type, and S a subset of V such that V = C1 ⊕ SC ⊕ C2 (V ). Then V is spanned by 1 and vectors of the 1 r form a(−n . . . a(−n 1 satisfying r ≥ 1, a i ∈ S, n 1 > n 2 > · · · > nr ≥ 1. r) 1) By Theorem 2.3, any C2 -cofinite vertex operator algebra has a finite subset T by which V is strongly generated. 3. 2-Cyclic Permutation Orbifold Models Let V be a vertex operator algebra. Then V ⊗ V becomes a vertex operator algebra with Virasoro vector ω ⊗ 1 + 1 ⊗ ω and vacuum vector 1 ⊗ 1. The n th product of a ⊗ b and u ⊗ v for a, b, u, v ∈ V, n ∈ Z is given by (a ⊗ b)(n) (u ⊗ v) =
(a(i) u) ⊗ (b(n−1−i) v). i∈Z
In particular we have C2 (V ⊗ V ) = C2 (V ) ⊗ V + V ⊗ C2 (V ). Thus if V is C2 -cofinite then so is V ⊗ V . We consider a map σ ∈ G L(V ⊗ V ) defined by σ (a ⊗ b) = b ⊗ a for a, b ∈ V . Then we see that σ is an automorphism of V ⊗ V of order 2. Now we set := (V ⊗ V )σ . V is a vertex operator subalgebra of V ⊗ V with vacuum vector 1 ⊗ 1 and Virasoro Then V vector ω ⊗ 1 + 1 ⊗ ω. We introduce some notations. For a, b ∈ V , we set . φ(a, b) = a ⊗ b + b ⊗ a, and η(a) = a ⊗ 1 + 1 ⊗ a ∈ V It is clear that φ(a, b) = φ(b, a) for a, b ∈ V and η(a) = φ(a, 1) = φ(1, a). Lemma 3.1. For any a, u, v ∈ V and n ∈ Z, η(a)(n) φ(u, v) = φ(a(n) u, v) + φ(u, a(n) v).
(3.1)
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Proof. We have η(a)(n) φ(u, v) = (a ⊗ 1 + 1 ⊗ a)(n) (u ⊗ v + v ⊗ u) = a(n) u ⊗ v + a(n) v ⊗ u + u ⊗ a(n) v + b ⊗ a(n) v = φ(a(n) u, v) + φ(u, a(n) v). This proves the lemma.
Lemma 3.2. For any a, b ∈ V, η(a)(−1) η(b) = η(a(−1) b) + φ(a, b). Proof. Substitute u = b, v = 1 and n = −1 in (3.1).
is spanned by Im φ, we have the following proposition. Since V is strongly generated by Im η. Proposition 3.3. V ). We denote by φ and η the compositions of η and φ with /C2 (V Now we consider V /C2 (V ) respectively. Since L −1 (V ) ⊂ C2 (V ), we the canonical projection · : V → V have φ(L −1 a, b) = −φ(a, L −1 b)
(3.2)
for any a, b ∈ V . In particular one sees that η(L −1 a) = 0. Therefore we have L −1 V ⊂ Ker η,
(3.3)
η(a(−n) 1) = 0
(3.4)
and
for a ∈ V and n ≥ 2. The following lemma is an important property of φ. Lemma 3.4. For any a, b, u ∈ V and n ≥ 2, φ(a(−n) u, b) = −φ(u, a(−n) b). Proof. By Lemma 3.1, ). φ(a(−n) u, b) + φ(u, a(−n) b) = η(a)(−n) φ(u, b) ∈ C2 (V Thus we have the lemma.
Now we show more complicated identities. Lemma 3.5. For a, b, u ∈ V and n ≥ 2, φ(a, b(−n) u) + φ(b, a(−n) u) ∞
= η(a(−i−2) b(−n+1+i) u) + η(b(−i−2) a(−n+1+i) u) . i=0
(3.5)
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Proof. We have φ(a, b)(−n) η(u) = (a ⊗ b + b ⊗ a)(−n) (u ⊗ 1 + 1 ⊗ u) (a(i) u ⊗ b(−n−1−i) 1 + a(i) 1 ⊗ b(−n−1−i) u = i∈Z
+ b(i) u ⊗ a(−n−1−i) 1 + b(i) 1 ⊗ a(−n−1−i) u) ∞ = (a(i) u ⊗ b(−n−1−i) 1 + b(i) u ⊗ a(−n−1−i) 1) i=−n ∞
(a(−i−1) 1 ⊗ b(−n+i) u + b(−i−1) 1 ⊗ a(−n+i) u)
+
=
i=0 ∞
(φ(a(−n+i) u, b(−1−i) 1) + φ(b(−n+i) u, a(−1−i) 1)).
i=0
By Lemma 3.4, we see that if i > 0, then φ(a(−n+i) u, b(−1−i) 1) + φ(b(−n+i) u, a(−1−i) 1) = −φ(b(−1−i) a(−n+i) u, 1) − φ(a(−1−i) b(−n+i) u, 1) ∞
=− η(a(−i−2) b(−n+1+i) u) + η(b(−i−2) a(−n+1+i) u) . i=0
), we have (3.5). Since φ(a, b)(−n) η(u) ∈ C2 (V
Therefore we have φ(a, b(−n) u) = −φ(b, a(−n) u)
mod Im η
(3.6)
for a, b, u ∈ V and n ≥ 2. Finally we show the following. Lemma 3.6. Let a, b, u ∈ V and n ∈ Z>0 . If n ≥ 3 then φ(a, b(−n) u) ∈ Im η. Proof. Since n − 1 ≥ 2, by using (3.5), one has 1 φ(a, (L −1 b)(−n+1) u) n−1 1 φ(L −1 b, a(−n+1) u) mod Im η =− n−1 1 η(b(−2) a(−n+1) u) ∈ Im η. = n−1
φ(a, b(−n) u) =
Suppose that V is of CFT type and take a subset S of V such that V = C1 ⊕ SC ⊕ C2 (V ). Then by Theorem 2.3, V is spanned by vectors of the form v = x 1 (−n 1 ) · · · x r (−nr ) 1, x j ∈ S, n 1 > · · · > nr ≥ 1.
(3.7)
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By (3.4), if r = 1 and n 1 ≥ 2, then η(v) = 0. Next we consider the case r ≥ 2. Since n 1 > · · · > nr −1 ≥ 2, for u ∈ V , we have φ(v, u) = (−1)r −1 φ(x r (−nr ) 1, x r −1 (−nr −1 ) · · · x 1 (−n 1 ) u). If nr ≥ 2, then φ(v, u) ∈ Im η by Lemma 3.4. On the other hand in the case nr = 1, we can write x r −1 (−nr −1 ) · · · x 1 (−n 1 ) u as a linear combination of vectors of the form (3.7), say w = y 1 (−m 1 ) · · · y r (−m s ) 1,
y j ∈ S, m 1 > · · · > m s ≥ 1.
Then if m 1 ≥ 3 then φ(x r , w) ∈ Im η by Lemma 3.6. Consequently, we find that for any vector v of the form (3.7) and u ∈ V, φ(v, u) is equivalent to a linear combination of vectors of the form φ(x, y), φ(x, y(−1) z) or φ(x, y(−2) z) for x, y, z ∈ S modulo Im η. Therefore we have ) = φ(x, y) x, y ∈ S /C2 (V V C + φ(x, y(−i) z) x, y, z ∈ S, i = 1, 2 C + Im η. (3.8) Consequently we have the following lemma. /C2 (V ) Lemma 3.7. If V is C2 -cofinite and of CFT type, then the quotient space of V by Im η is finite dimensional. Proof. If V is C2 -cofinite, then we may take S to be a finite set. Thus (3.8) proves the lemma.
Now the following corollary holds. Corollary 3.8. Let V be a C2 -cofinite vertex operator algebra of CFT type. If Im η is is C2 -cofinite. finite dimensional, then V 4. On the Finiteness of dim Im η In this section we consider the finiteness of dimension of the image Im η and show the following theorem: Theorem 4.1. Let V be a C2 -cofinite simple vertex operator algebra of CFT type, and suppose that V is strongly generated by a finite subset T ⊂ ∪∞ d=1 Vd . Then V is C 2 -co finite if and only if η(x(−n) y) x, y ∈ T, n ≥ 1 C is finite dimensional. In Theorem 4.1, the “only if” part is clear. Hence it suffices to prove that the following lemma by Corollary 3.8. Lemma 4.2. Let V and T be as in Theorem 4.1. Then Im η is finite dimensional if η(x(−n) y) x, y ∈ T, n ≥ 1 C is finite dimensional. We give a proof of Lemma 4.2 at the end of this section. We first recall the standard filtration of a vertex operator algebra (see [GN] or [M1]). Set
r 1 r i j Fk V := a(n 1 ) · · · a(nr ) 1 r ≥ 1, a ∈ V, homogeneous, |a | ≤ k j=1
C
for k ∈ Z≥0 . The following proposition is well known (see [Li] for its proof).
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Proposition 4.3. Let a 1 , . . . , a r ∈ V be homogeneous vectors, n 1 , . . . , nr ∈ Z, k, l ∈ Z≥0 and u ∈ Fl (V ). i (1) If |a | = k, then a 1 (n 1 ) . . . a r (nr ) u ≡ a i1 (n i1 ) . . . a ir (n ir ) u modulo the subspace Fk+l−1 (V ) for any permutation {i 1 , . . . , ir } of {1, . . . , r }. (2) If |a i | = k and n i ≥ 0 for some i, then a 1 (n 1 ) · · · a r (nr ) u ∈ Fk+l−1 (V ). i (3) If |a | = k, then (a 1 (n 1 ) . . . a r (nr ) 1)(−1) u ≡ a 1 (n 1 ) . . . a r (nr ) u modulo the subspace Fk+l−1 (V ). /C2 (V ). Then PropNow we consider a filtration η(Fk V ) for k ∈ Z≥0 of Im η ⊂ V osition 4.3 shows the following proposition. Proposition 4.4. Let a 1 , . . . , a r ∈ V be homogeneous, n 1 , . . . , nr ∈ Z, k, l ∈ Z≥0 and u ∈ Fl (V ). i (1) If |a | = k, then η(a 1 (n 1 ) . . . a r (nr ) u) ≡ η(a i1 (n i1 ) . . . a ir (n ir ) u) modulo η(F (V )) for any permutation {i 1 , . . . , ir } of {1, . . . , r }. k+l−1 (2) If |a i | = k and n i ≥ 0 for some i, then η(a 1 (n 1 ) . . . a r (nr ) u) ≡ 0 modulo η(F (V )). k+l−1 (3) If |a i | = k, then η((a 1 (n 1 ) . . . a r (nr ) 1)(−1) u) ≡ η(a 1 (n 1 ) . . . a r (nr ) u) modulo η(Fk+l−1 (V )). Suppose that V is strongly generated by a subset T consisting of homogeneous vectors. By definition, we see that V is spanned by vectors of the form 1 r x(−n . . . x(−n 1 with n i ≥ 1 and xi ∈ T. r) 1)
(4.1)
Moreover, each subspace Fk (V ) is spanned by vectors of the form (4.1) subject to r i i=1 |x | ≤ k. The following lemma plays an essential role in the proof of Lemma 4.2. Lemma 4.5. For any a, b, u ∈ V and m, n ≥ 2, η(a(−m) b(−n) 1) · η(u) = 2η(a(−m) b(−n) u) +
∞ −n η((b(i) a)(−m−n−i) u) i i=0
+
∞
αm,n:i η(a(−m−n−i) b(i) u) + αm,n;i η(b(−m−n−i) a(i) u) ,
i=0
where αm,n:i is a constant defined in (2.6). Proof. Set w=
∞
αm,n:i a(−m−n−i) b(i) u + αm,n;i b(−m−n−i) a(i) u
i=0
for simplicity. Then by (2.1), we have (a(−m) b(−n) 1)(−1) u = a(−m) b(−n) u + w. Hence η(a(−m) b(−n) 1) · η(u) = η((a(−m) b(−n) 1)(−1) u) + φ(a(−m) b(−n) 1, u) = η(a(−m) b(−n) u) + η(w) + φ(a(−m) b(−n) 1, u).
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By Lemma 3.4, we see that φ(a(−m) b(−n) 1, u) = φ(1, b(−n) a(−m) u) = η(b(−n) a(−m) u), since m, n ≥ 2. Now Commutativity formula (2.2) shows the lemma.
Lemma 4.5 implies that if m, n ≥ 2 then η(a(−m) b(−n) 1) · η(u) is in Im η for any a, b, u ∈ V . We also find that by Proposition 4.4, 1 η(a(−m) b(−n) 1) · η(u) (4.2) 2 modulo η(F|a|+|b|+k−1 (V )) for any m, n ≥ 2, homogeneous a, b ∈ V and u ∈ Fk (V ). Let a i ∈ V, m i ∈ Z>0 (i = 1, 2, . . . , r ). We introduce a notation η(a(−m) b(−n) u) ≡
(a 1 , . . . , a r )m 1 ,...,m r := (m 1 − 1)! . . . (m r − 1)!a 1 (−m 1 ) . . . a r (−m r ) 1. It is clear that L −1 (a 1 , . . . , a r )m 1 ,...,m r =
r
(a 1 , . . . , a r )m 1 ,...,m i +1,...,m r
i=1
holds. Thus (3.3) shows that r η (a 1 , . . . , a r )m 1 ,...,m i +1,...,m r = 0.
(4.3)
i=1
By using this identity for r = 2, we have the following lemma. Lemma 4.6. For a, b ∈ V, m, n ∈ Z>0 and −m < l < n,
(4.4) η (a, b)m,n = (−1)l η (a, b)m+l,n−l .
In particular, if m + n is odd, then η (a, a)m,n = 0.
Proof. By (4.3), we have η (a, b)m,n = −η (a, b)m+1,n−1 for m ≥ 1 and n ≥ 2. Induction on l > 0 proves (4.4). We can also show that (4.4) is valid if l ≤ 0. If m + n is odd and m + n = 2k + 1 (k ∈ Z>0 ), then
η (a, a)m,n = (−1)n−k−1 η (a, a)k,k+1 = (−1)n−k η (a, a)k+1,k . Since (a, a)k,k+1 ≡ (a, a)k+1,k
modulo L −1 V by Commutativity formula (2.2), we have η( (a, a)k+1,k ) = 0 and η (a, a)m,n = 0.
We shall start a proof of Lemma 4.2. For simplicity, we set Uk := η(Fk (V )) for any k ∈ Z≥0 . For s ≥ 1, we set
Bks;r = η (x 1 , . . . , x s )m 1 ,...,m r ,1s x i ∈ T, |x i | = k, m i ≥ 2 C , for k ∈ Z≥0 and r, s ∈ Z≥0 with r ≤ s, where (x 1 , . . . , x s )1s denotes i |x | = k for any x i ∈ T . We note (x 1 , . . . , x s )1,...,1 . We define Bks;r = 0 if that if s > k then Bks;r = 0 because T contains only homogeneous vectors of positive weight. We also find that each Bks,0 is finite dimensional for each k, s.
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Lemma 4.7. (Bks;r + Uk−1 )/Uk−1 is finite dimensional for any r, k ≥ 1. Proof. In the case s = 1, Bk1;1 = η( (x)m ) | x ∈ T, |x| = k, m ≥ 2 C . Since (x)m ∈ L −1 V for m ≥ 2, we have Bk1;1 = 0. Next we consider the case s ≥ 2. If r ≥ 2, then (4.2) shows that η( (x 1 , . . . , x s )m 1 ,...,m r ,1s−r ) 1 ≡ η( (x 1 , x 2 )m 1 ,m 2 ) · η( (x 3 , . . . , x s )m 3 ....,m r ,1s−r ) 2 modulo Uk−1 . It follows from Lemma 4.6 that η( (x 1 , x 2 )m 1 ,m 2 ) = (−1)m 2 η( (x 1 , x 2 )m 1 +m 2 −2,2 ). Therefore we have η( (x 1 , . . . , x s )m 1 ,··· ,m r ,1s−r ) ≡ (−1)m 2 η( (x 1 , . . . , x s )m 1 +m 2 −2,2,m 3 ,...,m r ,1s−r ) modulo Uk−1 . Proposition 4.4 (1) and the argument above show that η( (x 1 , . . . , x s )m 1 ,...,m r ,1s−r ) r
≡ (−1)
i=2 m i
η( (x 1 , . . . , x s )2+ri=1 (m i −2),2,...,2,1s−r )
modulo Uk−1 . Hence we have Bks;r + Uk−1 = η( (x 1 , . . . , x s )m,2,...,2,1s−r ) x i ∈ T, |x i | = k, m ≥ 2 + Uk−1 =
k
C
Bl2;2 . . . η( (x 3 , . . . , x s )2,...,2,1s−r ) x i ∈ T, |x i | = k − l + Uk−1 .
C
l=2
From the assumption of Lemma 4.2, Bl2;2 is finite dimensional for any l. We also find that i |x | = the subspace spanned by η( (x 3 , . . . , x s )2,...,2,1r −s ) with x i ∈ T satisfying k −l is also finite dimensional for each l. Hence (Bks;r +Uk−1 )/Uk−1 is finite dimensional if r ≥ 2. Finally we consider the case r = 1. For m ≥ 3, by (4.3) and Proposition 4.4 (1), we have η( (x 1 , . . . , x s )m,1s−1 ) = −
s
η( (x 1 , . . . , x s )m−1,1i−2 ,2,1s−i ) ∈ Bks;2 + Uk−1
i=2
for x i ∈ T with
Bks;1
|x i | = k. This shows that ⊂ η( (x 1 , . . . , x s )2,1,...,1 ) x i ∈ T + Bks;2 + Uk−1 . C
Hence (Bks;1 +Uk−1 )/Uk−1 is finite dimensional. This completes the proof of Lemma 4.7.
Lemma 4.8. Uk is finite dimensional for any k ≥ 0.
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Proof. By definition and Proposition 4.4 (3), for k ≥ 1, Uk =
k s
Bks;r + Uk−1 .
(4.5)
s=1 r =0
Since Bks,0 is finite dimensional, Lemma 4.7 proves the quotient space Uk /Uk−1 is of finite dimension. Thus U0 = C1 implies that Uk is finite dimensional for all k ∈ Z≥0 .
Now we suppose that V is C2 -cofinite and show that Uk = Uk−1 for sufficiently large k. Since V is C2 -cofinite, there exists p ∈ Z>0 such that (x 1 , . . . , x t )1t is in F |x i |−1 (V ) for any t ≥ p and x i ∈ T . Lemma 4.9. For any k ≥ s ≥ r ≥ 1, if s − r ≥ p then Bks,r ⊂ Uk−1 . Proof. If s − r ≥ p, then we have (x 1 , · · · , x s )m 1 ,...,m r ,1s−r ∈ F |x i |−1 (V ) for m i ∈ Z>0 . Thus Bks,r ⊂ Uk−1 .
We set κ := max{|x||x ∈ T } and ν := T . One notices that Bks,r = 0 if s ≤ [k/κ], where [x] denotes the maximal integer less than or equal to q for q ∈ Q. Therefore by 4.9, if k ≥ pκ then Uk =
k
s
Bks,r + Uk−1 .
(4.6)
s=[k/κ]+1 r =s− p+1
Lemma 4.10. If k ≥ κ(5ν + p), then Bks;r ⊂ Uk−1 for any s ≥ r ≥ 1. Proof. Since k/κ ≥ 5ν + p, by (4.6), we may assume that s ≥ 5ν + p + 1 and r ≥ 5ν + 1. For such k, s, r we show that the image of v := (x 1 , . . . , x s )m 1 ,...,m r ,1s−r by η is in Uk−1 for any x i ∈ T and m i ≥ 2. Since r ≥ 5ν + 1, there are 1 ≤ i 1 < . . . < i 6 ≤ r such that x i1 = . . . = x i6 . Thus by Proposition 4.4 (1), we may assume that x 1 = x 2 = x 3 = x 4 = x 5 = x 6 , say y. The proof of Lemma 4.7 implies that we may also assume that m 2 = . . . = m r = 2. Suppose that m 1 ≥ 3. Then we have 1 η( (y, y)m 1 ,2 ) . . . η( (x 3 , . . . , x s )2,...,2,1s−r ) 2 1 ≡ − η( (y, y)m 1 −1,3 ) . . . η( (x 3 , . . . , x s )2,...,2,1s−r ) 2 ≡ −η( (x 1 , . . . , x s )m 1 −1,3,2,...,2,1s−r )
η(v) ≡
≡ −η( (x 2 , x 3 , x 1 , x 4 , . . . , x s )3,2,m 1 −1,2,...,2,1s−r ) 1 ≡ − η( (y, y)3,2 ) . . . η( (y, y, x 5 , . . . , x s )m 1 −1,2,...,2,1s−r ) 2 =0 modulo Uk−1 . Hence η(v) ∈ Uk−1 .
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We consider the case m 1 = 2. In this case, v = y(−2) 6 w with w = (x 7 , . . . , x s )2,...,2,1s−r . Then Associativity formula shows that v ≡ (y(−1) y)(−3) y(−2) 4 w − 2y(−3) y(−1) y(−2) 4 w ≡ (y(−1) y)(−3) 2 y(−2) 2 w − 2(y(−1) y)(−3) y(−3) y(−1) y(−2) 2 w − 2y(−3) y(−1) y(−2) 4 w ≡ (y(−1) y)(−3) 3 w − 2(y(−1) y)(−3) 2 y(−3) y(−1) w −2(y(−1) y)(−3) y(−3) y(−1) y(−2) 2 w − 2y(−3) y(−1) y(−2) 4 w modulo Fk−1 (V ). The images of the last two terms by η are in Uk−1 because both of them are congruent to vectors of the form y(−3) y(−2) u for some suitable vectors u modulo Fk−1 (V ) and η(y(−3) y(−2) u) ≡ 21 η(y(−3) y(−2) 1) · η(u) = 0 modulo Uk−1 . As for the first and second terms, we see that they are congruent to vectors of the form (a, a, b, u)3,3,3,1 for some suitable vectors a, b, u modulo Fk−1 (V ). Then by using the same argument in the proof of Lemma 4.7, we have η( (a, a, b, u)3,3,3,1 ) ≡ −η( (a, a, b, u)3,4,2,1 ) 1 ≡ − η( (a, a)3,4 ) · η( (b, u)2,1 ) = 0. 2 Therefore we have η(v) ∈ Uk−1 . Consequently one has Bks,r ⊂ Uk−1 for k ≥ κ(5ν + p).
Finally we have the following lemma. Lemma 4.11. Uk = Uk−1 for k ≥ κ(5ν + p). Proof. By (4.6) and Lemma 4.10, we have Uk = Uk−1 for k ≥ κ(5ν + p).
Now we can prove Lemma 4.2 and Theorem 4.1. Proof of Lemma 4.2. By Lemma 4.11, Im η = Uk for sufficiently large k. Hence by Lemma 4.8 we have Lemma 4.2.
Proof of Theorem 4.1. It follows from Corollary 3.8 and Lemma 4.2.
5. C2 -Cofiniteness of L(c p,q , 0) /C2 (V ) concerning with the Virasoro vector In this section we find some identities in V and show that L(c p,q , 0) is C 2 -cofinite for any coprime p, q ≥ 2. In this section we call vectors of the form L m 1 . . . L m r 1 with m i < 0 a monomial type vector of length r . We set cm,n;i := αm,n:i + αn,m:i for m, n ≥ 1. Then by Lemma 2.1, (L −m L −n 1)(−1) L − p L −q 1 = L −m L −n L − p L −q 1 ∞ cm−1,n−1;i L −m−n+1−i L i−1 L − p L −q 1 (5.1) + i=0
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holds for m, n, p, q ≥ 2. By calculating L i−1 L − p L −q 1 for i ≥ 0 and using the fact that L i−1 1 = 0 for i ≥ 0, (5.1) implies (L −m L −n 1)(−1) L − p L −q 1 = L −m L −n L − p L −q 1 +
+
+
p−1
cm−1,n−1:i ( p + i − 1)L −m−n+1−i L −q L i− p−1 1
i=0 q−1
cm−1,n−1:i (q i=0 p+q−1
+ i − 1)L −m−n+1−i L − p L i−q−1 1
cm−1,n−1:i ( p + i − 1)(q + i − p − 1)L −m−n+1−i L i− p−q−1 1
i=0
q + 1 cV L −m−n− p L −q 1 + cm−1,n−1; p+1 3 2 p + 1 cV L −m−n−q L − p 1 + cm−1,n−1;q+1 3 2 q + 1 cV + cm−1,n−1; p+q+1 (2 p + q) L −m−n− p−q 1. 2 3
(5.2)
By applying η to the both sides in (5.2) we shall find some identities between /(C2 (V ) for sufficiently large integer m. To get η(L −m+2 L −2 ω) and η(L −m ω) in V them, we need more lemmas. First it is clear from (3.3) that η(L −m 1) = 0 for m ≥ 3.
(5.3)
We next have the following lemma by applying Lemma 4.6 to a = b = ω. Lemma 5.1. For m, n ≥ 2,
m+n−4 η(L −m L −n 1) = (−1) η(L −m−n+2 ω). n−2 n
Moreover if m + n is odd then η(L −m L −n ) = 0. As for monomial type vectors of length 3, we have the following two lemmas. Lemma 5.2. If m, n, l ≥ 3, then η(L −m L −n L −l 1) = − f (m, n, l)η(L −m−n−l+2 ω), where
m +n +l −4 1 (m − n) (−1)l f (m, n, l) = 2 l −2 m +n +l −4 + (m − l) (−1)m+l n−2 m +n +l −4 m + (n − l) (−1) . m−2
(5.4)
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Proof. It follows from the commutation relations of the Virasoro algebra that L −l L −n L −m 1 = L −m L −n L −l 1 + (n − l)L −n−l L −m 1 + (m − l)L −n L −l−m 1 + (m − n)L −m−n L −l 1. Since m, n, l ≥ 3, we have η(L −l L −n L −m 1) = φ(L −l L −n L −m 1, 1) = −φ(1, L −m L −n L −l 1) = −η(L −m L −n L −l 1) by Lemma 3.4. It follows from Lemma 5.1 that m +n +l −4 η(L −m−n L −l ) = (−1)l η(L −m−n−l+2 w), l −2 m +l +n −4 η(L −n L −l−m ) = (−1)m+l η(L −m−n−l+2 ω), n−2 m +n +l −4 η(L −l−n L −m ) = (−1)m η(L −m−n−l+2 ω). m−2
(5.5) (5.6) (5.7)
Therefore, −η(L −m L −n L −l ) = η(L −m L −n L −l ) + 2 f (m, n, l)η(L −m−n−l+2 ω) by means of f (m, n, l) given in (5.4). This proves the lemma.
Next we consider the vector η(L −m L −n ω) for m, n ≥ 3. By Lemma 2.1, for m, n ≥ 2, one gets the following identity: (L −m L −n 1)(−1) ω = L −m L −n ω + cm−1,n−1:0 L −m−n+1 L −3 1 cV +2cm−1,n−1:1 L −m−n ω + cm−1,n−1;3 L −m−n−2 1. (5.8) 2 Hence by (5.3), (5.8) and Lemma 5.1 we have η((L −m L −n 1)(−1) ω) = η(L −m L −n ω) − dm−1,n−1 η(L −m−n ω)
(5.9)
for m, n ≥ 2, where dm,n = (m + n)cm,n:0 − 2cm,n:1 . We also note that η(L −m L −n 1) · η(ω) = η((L −m L −n 1)(−1) ω) + φ(L −m L −n 1, ω)
(5.10)
for m, n ≥ 2. Hence if m, n ≥ 3, then (5.9) and Lemma 3.4 give η(L −m L −n 1) · η(ω) = η(L −m L −n ω) + η(L −n L −m ω) − dm−1,n−1 η(L −m−n ω) = 2η(L −m L −n ω) + (m − n − dm−1,n−1 )η(L −m−n ω). Therefore we have 2η(L −m L −n ω) = η(L −m L −n 1) · η(ω) +(−m + n + dm−1,n−1 )η(L −m−n ω)
(5.11)
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for m, n ≥ 3. Finally by Lemma 5.1 and (5.11), we see that for m, n ≥ 3, m+n−4 (−1)n η(L −m−n+2 ω) · η(ω) 2η(L −m L −n ω) = n−2 +(−m + n + dm−1,n−1 )η(L −m−n ω).
(5.12)
On the other hand, it also follows from (5.10) that for m ≥ 3, η(L −m ω) · η(ω) = η((L −m ω)(−1) ω) + φ(L −m ω, ω). If m ≥ 3, then φ(L −m ω, ω) = −φ(ω, L −m ω) = −φ(L −m ω, ω), and this shows φ(L −m ω, ω) = 0. Thus by (5.9), we have η(L −m ω) · η(ω) = η(L −m L −2 ω) − dm−1,1 η(L −m−2 ω).
(5.13)
Therefore by (5.12) and (5.13), we have the following lemma. Lemma 5.3. For m, n ≥ 3, m+n−4 2η(L −m L −n ω) = (−1)n η(L −m−n+2 L −2 ω) + g(m, n)η(L −m−n ω), n−2 where
m+n−4 g(m, n) = − (−1)n dm+n−3,1 − m + n + dm−1,n−1 . n−2
Now we return to (5.2). For m, n ≥ 3 and p, q ≥ 2, we have η(L −m L −n 1) · η(L − p L −q 1) = η((L −m L −n 1)(−1) L − p L −q 1) + φ(L −m L −n 1, L − p L −q 1) = η((L −m L −n 1)(−1) L − p L −q 1) + η(L −n L −m L − p L −q 1) = η((L −m L −n 1)(−1) L − p L −q 1) + η(L −m L −n L − p L −q 1) + (m − n)η(L −m−n L − p L −q 1). By (5.2), the right hand side is a sum of 2η(L −m L −n L − p L −q 1) and η(w) such that w is a linear combination of monomial type vectors whose lengths are 2 or 3 and wights are s := m + n + p + q. Lemmas 5.1–5.3 show that η(w) is a linear combination of η(L −s+2 ω) and η(L −s+4 L −2 ω). Therefore we get an identity of the form 2η(L −m L −n L − p L −q 1) = η(L −m L −n 1) · η(L − p L −q 1) + αη(L −s+4 L −2 ω) + βη(L −s+2 ω)
(5.14)
for some scalars α, β which are able to be calculated explicitly by using Lemmas 5.1–5.3. Now we take m, n, p, q so that m ≥ 14 and even, n = 13 − 2k, p = 3 + 2k for 0 ≤ k ≤ 2 and q = 2. Then we see that s = m + 18 is an even integer greater than or equal to 32, and η(L −m L −n 1) · η(L − p L −q 1) is zero by Lemma 5.1. Thus we have constants αk , βk such that 2η(L −s+18 L −13+2k L −3−2k ω) = αk η(L −s+4 L −2 ω) + βk η(L −s+2 ω).
(5.15)
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On the other hand by calculating (5.14) again after changing (m, n, p, q) to (m, p, n, q), we find constants γk , δk ∈ C such that 2η(L −s+18 L −3−2k L −13+2k ω) = γk η(L −s+4 L −2 ω) + δk η(L −s+2 ω),
(5.16)
where we note that η(L −13+2k ω) = 0. We also have L −m L − p L −n L −q 1 = L −m L −n L − p L −q 1 + (n − p)L −m L −n− p L −q 1. Hence by (5.16) and by Lemmas 5.2 and 5.3, we have an another identity 2η(L −s+18 L −13+2k L −3−2k ω) = γk η(L −s+4 L −2 ω) + δk η(L −s+2 ω).
(5.17)
Finally we have an identity ξk η(L −s+4 L −2 ω) + ζk η(L −s+2 ω) = 0,
(5.18)
where ξk = ξk (s, cV ) = αk − γk and ζk = ζk (s, cV ) = βk − δk . In Appendix, we give the explicit forms of the coefficients ξk (s, c) and ζk (s, c) for k = 0, 1, 2, s ≥ 32 and c ∈ C. By means of the explicit forms of ξk and ζk for k = 0, 1, 2, we can show the following lemma. Lemma 5.4. For any even s ≥ 32 and c ∈ C, one of the determinants of matrices ξ1 (s, c) ζ1 (s, c) ξ2 (s, c) ζ2 (s, c) ξ0 (s, c) ζ0 (s, c) , and is nonzero. ξ1 (s, c) ζ1 (s, c) ξ2 (s, c) ζ2 (s, c) ξ0 (s, c) ζ0 (s, c) We give a proof of Lemma 5.4 in the Appendix. This lemma implies that η(L −s+4 L −2 ω) = η(L −s+2 ω) = 0 for any even integer s ≥ 32 and any central charge c = cV ∈ C. We recall η(L −n ω) = 0 if n is odd. Therefore, we have the following theorem. Theorem 5.5. Let V be a vertex operator algebra. Then η(L −n ω) = 0 for n ≥ 30. In particular, the subspace η(L −n ω) | n ≥ 2 C is finite dimensional. As an application of Theorems 4.1 and 5.5, we consider a 2-cyclic permutation orbifold models of the Virasoro vertex operator algebras. Let V = L(c p,q , 0) be the simple Virasoro vertex operator algebra of central charge c = c p,q for coprime integers p, q ≥ 2. It is well known that V is strongly generated by ω and C2 -cofinite. Therefore by Theorems 4.1 and 5.5, we have the following theorem. Theorem 5.6. Let p, q be coprime integers greater than or equal to 2. Then the 2-cyclic permutation orbifold model L(c p,q , 0) is C 2 -cofinite. Acknowledgements. The author would like to thank Professor Masahiko Miyamoto for insightful comments and helpful discussion. He is also thankful to Prof. Hiromich Yamada for his encouragement and finding some typos in the first version. He greatly appreciates the referees’ useful comments and suggestions.
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6. Appendix In this section we give the explicit forms of ξk (m, c) and ζk (m, c) for k = 0, 1, 2 and prove Lemma 5.4. We use Mathematica for the computations in this section. First we give data to compute ξ0 and ζ0 . We take (m, n, p, q) = (m, 13, 3, 2) with even integer m. Then we have 1 (m − 13)(m + 12)(−60354201600 − 25041744000m 14! −11025031680m 2 + 2218757736m 3 + 4290676052m 4 + 2061162870m 5 +561027415m 6 + 98527338m 7 + 11580231m 8 + 907530m 9 + 45565m 10 +1326m 11 + 17m 12 ), 1 (m − 13)(m + 12)(m + 13)(m + 14) β0 = 13! · 8 ×(−5748019200 − 2706163200m − 1031677920m 2 + 416682968m 3 +502648380m 4 + 206064690m 5 + 49210811m 6 + 7683234m 7 +814359m 8 + 58630m 9 + 2769m 10 + 78m 11 + m 12 ) 91(241m + 738)(m − 13) m + 14 c. + 2(m + 2)(m + 3) 16
α0 =
As well, we get 1 (m + 12)(784604620800 + 322088054400m + 146756039040m 2 14! −84093309768m 3 − 95650195420m 4 − 30695547818m 5 −491574005m 6 + 2565009941m 7 + 883762815m 8 + 155173941m 9 +16374865m 10 + 1047527m 11 + 37505m 12 + 577m 13 ), 1 (m + 11)(m + 12)(m + 14)(264931430400 + 137634854400m δ0 = − 2 · 16! +43159534560m 2 − 53918986488m 3 − 38531775476m 4 −7272782558m 5 + 1442243915m 6 + 1002076031m 7 +232782927m 8 + 31029951m 9 + 2612465m 10 + 140117m 11 +4489m 12 + 67m 13 ) 33(192721m + 2502378)(m − 3) m + 14 c. + 2(m + 2)(m + 13) 16
γ0 =
Finally we get m + 14 2284800 , (m + 1)(m + 11)(m + 13) 17 152320(m 3 + 30m 2 + 437m + 2628) m + 14 ζ0 = (m + 1)(m + 3)(m + 11)(m + 13) 17 ξ0 = −
(3168931m 3 + 41305158m 2 − 26765899m − 366373782) m + 14 − c. (m + 2)(m + 3)(m + 13) 16
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We can compute the following coefficients by the similar way. m + 14 17821440 ξ1 = − , (m + 5)(m + 7)(m + 13) 17 8 (13m 6 + 559m 5 + 10914m 4 + 113042m 3 + 541013m 2 ζ1 = 5 · 13! +743199m + 141660)(m − 2)m(m + 2)(m + 4)(m + 6)(m + 7)(m + 8) ×(m + 10)(m + 12)(m + 14) 15(1595885m 3 + 18046170m 2 − 32028341m − 411762282) m + 14 − c, (m + 2)(m + 5)(m + 11) 16 m + 14 18670080 , ξ2 = − (m + 3)(m + 9)(m + 13) 17 32 ζ2 = (143m 6 + 6435m 5 + 129602m 4 + 1390950m 3 + 7149347m 2 15! +12054375m + 5315868)(m − 2)m(m + 2)(m + 4)(m + 5)(m + 6) ×(m + 8)(m + 10)(m + 12)(m + 14) 156(152331m 3 + 1559062m 2 − 4454319m − 55321438) m + 14 c. − (m + 2)(m + 7)(m + 9) 16 Proof of Lemma 5.4. By using the explicit ξk (m, c) and ζk (m, c) for k = forms of ξk ζ k for k = 0, 1, 2 as pk (m) + qk (m)c 0, 1, 2, we can describe the determinants ξk+1 ζk+1 with polynomials pk (m) and qk (m) in m, where we identify ξ3 and ζ3 with ξ0 and ζ0 respectively. The system of equations pk (m) + qk (m)c = 0 for all k = 0, 1, 2 leads three equations pk (m)qk+1 (m) − pk+1 (m)qk (m) = 0 for k = 0, 1, 2. The polynomials pk (m)qk+1 (m) − pk+1 (m)qk (m) for k = 0, 1, 2 are given as products of nonzero constants, powers of factors (m + r ) with −2 ≤ r ≤ 18 and a polynomial f (m) given by f (m) = −5823421556567940 − 13295522326219116m − 7085484924471269m 2 −1746250016719384m 3 − 310878749441408m 4 − 41974581663344m 5 −4071611633914m 6 − 252490022696m 7 − 6600424292m 8 +133103900m 9 + 7930183m 10 . Since f (m + 39) is a polynomial in m whose coefficients are all positive, for m ≥ 39, f (m) = 0. We also see that f (m) = 0 for 32 ≤ m ≤ 38. Therefore if m ≥ 32, one of pk (m)qk+1 (m) − pk+1 (m)qk (m) for k = 0, 1, 2 is nonzero. This implies that one of determinants pk (m) + qk (m)c is nonzero for m ≥ 32 and arbitrary c ∈ C.
References [Ar] [Ban1] [Ban2] [Ban3]
Arakawa, T.: A Remark on the C2 -cofiniteness condition on vertex algebras. http://arxiv.org/ abs/1004.1492v2 [math.QA], 2010 Bantay, P.: Characters and modular properties of permutation orbifolds. Phys. Lett. B419, 175–178 (1998) Bantay, P.: Permutation orbifolds. Nucl. Phys. B633(3), 365–378 (2002) Bantay, P.: The kernel of the modular representation and the galois action in rcft. Commun. Math. Phys. 233, 423–438 (2003)
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Barron, K., Dong, C., Mason, G.: Twisted sectors for tensor product vertex operator algebras associated to permutation groups. Commun. Math. Phys. 227, 349–384 (2002) Borisov, L., Halpern, M.B., Schweigert, C.: Systematic approach to cyclic orbifolds. Int. J. Mod. Phys. A13, 125–168 (1998) Dijkgraaf, R., Moore, G., Verlinde, E., Verlinde, H.: Elliptic genera of symmetric products and second quantized strings. Commun. Math. Phys. 185, 197–209 (1997) Frenkel, I., Huang, Y.-Z., Lepowsky, J.: On axiomatic approaches to vertex operator algebras and modules. Mem. Amer. Math. Soc. 104(494), (1993) Fuchs, J., Klemm, A., Schmidt, M.G.: Orbifolds by cyclic permutations in gepner type superstrings in the corresponding calabi-yau manifolds. Ann, Phys. 214, 221–257 (1992) Gaberdiel, M., Neitzke, A.: Rationality, quasirationality and finite w-algebras. Commun. Math. Phys. 238(1-2), 305–331 (2003) Huang, Y.-Z.: Cofiniteness conditions, projective covers and the logarithmic tensor product theory. J. Pure Appl. Algebra 213(4), 458–475 (2009) Klemm, A., Schmidt, M.G.: Orbifolds by cyclic permutations of tensor product conformal field theories. Phys. Lett. B245, 53–58 (1990) Li, H.-S.: Vertex algebras and vertex poisson algebras. Commun. Contemp. Math. 6(1), 61–110 (2005) Li, H.-S., Lepowsky, J.: Introduction to vertex operator algebras and their representations, Prog. Math., Basel-Boston: Birkhäuser, 2004 Matsuo, A., Nagatomo, K.: Axioms for a vertex algebra and the locality of quantum fields. MSJ Memoirs 4, Tokyo: Mathematical Society of Japan, 1999 Miyamoto, M.: Modular invariance of vertex operator algebras satisfying c2 -cofiniteness. Duke Math. J. 122(1), 51–91 (2004) Miyamoto, M.: Flatness of tensor products and semi-rigidity for C2 -cofinite vertex operator algebras I. http://arxiv.org/abs/0906.1407v2 [math.QA], 2009 Miyamoto, M.: Flatness of tensor products and semi-rigidity for C2 -cofinite vertex operator algebras. II (Functional part), http://arxiv.org/abs/0909.3665v1 [math.QA], 2009 Miyamoto, M.: A Z3 -orbifold theory of lattice vertex operator algebra and Z3 -orbifold constructions, http://arxiv.org/abs/1003.0237vL [math.QA], 2010 Yamskulna, G.: C2 -cofiniteness of the vertex algebra VL+ when L is a non-degenerate even lattice, http://arxiv.org/abs/0903.2458v1 [math.QA], 2009 Zhu, Y.-C.: Modular invariance of characters of vertex operator algebras. J. Amer. Math. Soc. 9, 237–302 (1996)
Communicated by Y. Kawahigashi
Commun. Math. Phys. 303, 845–896 (2011) Digital Object Identifier (DOI) 10.1007/s00220-010-1136-2
Communications in
Mathematical Physics
Cyclotomic Integers, Fusion Categories, and Subfactors Frank Calegari1 , Scott Morrison2 , Noah Snyder3 1 Department of Mathematics, Northwestern University, Evanston, IL, 60208-2730, USA 2 Department of Mathematics, University of California, Berkeley, CA 94720-3840, USA 3 Department of Mathematics, Columbia University, New York, NY 10027, USA.
E-mail:
[email protected] Received: 3 May 2010 / Accepted: 31 May 2010 Published online: 26 September 2010 – © The Author(s) 2010. This article is published with open access at Springerlink.com
Abstract: Dimensions of objects in fusion categories are cyclotomic integers, hence number theoretic results have implications in the study of fusion categories and finite depth subfactors. We give two such applications. The first application is determining a complete list of numbers in the interval (2, 76/33) which can occur as the FrobeniusPerron dimension of an object in a fusion category. The smallest number on this list is realized in a new fusion category which is constructed in the Appendix written by V. Ostrik, while the others are all realized by known examples. The second application proves that in any family of graphs obtained by adding a 2-valent tree to a fixed graph, either only finitely many graphs are principal graphs of subfactors or the family consists of the An or Dn Dynkin diagrams. This result is effective, and we apply it to several families arising in the classification of subfactors of index less than 5.
1. Introduction Let C be a fusion category and f any ring map from the Grothendieck ring K (C) to C. If X is an object in C, then Etingof–Nikshych–Ostrik proved in [13] that f ([X ]) is a cyclotomic integer. This result allows for applications of algebraic number theory to fusion categories and subfactors. The first such application was given by Asaeda and Yasuda [1,3] who excluded a certain infinite family of graphs as possible principal graphs of subfactors. We prove two main results, one a classification of small Frobenius–Perron dimensions of objects in fusion categories, and the other a generalization of Asaeda–Yasuda’s result to arbitrary families of the same form. Theorem 1.0.1. Let X be an object in a fusion category whose Frobenius–Perron dimension satisfying 2 < FP(X ) ≤ 76/33 = 2.303030 . . . then FP(X ) is equal to one of the following algebraic integers:
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√
√ 7+ 3 = 2√ 5= 1 + 2 cos(2π/7) = √ 1+ 5 = √ 2 √ 1 + 13 = 2
2.188901059 . . . , 2.236067977 . . . , 2.246979603 . . . , 2 cos(π/20) + 2 cos(9π/20) = 2.288245611 . . . , 2.302775637 . . . .
Remark 1.0.2. Each of the numbers in Theorem 1.0.1 can be realized as the Frobenius–Perron dimension of an object in a fusion category. See §3.1 and Appendix A (written by Ostrik). Theorem 1.0.3. Let be a connected graph with || vertices. Fix a vertex v of , and let n denote the sequence of graphs obtained by adding a 2-valent tree of length n − || to at v (see Fig. 1). For any fixed , there exists an effective constant N such that for all n ≥ N , either: (1) n is the Dynkin diagram An or Dn . (2) n is not the principal graph of a subfactor.
Fig. 1. The family of graphs n
Remark 1.0.4. The main theorem of Asaeda–Yasuda [3] is the particular case where is the Dynkin diagram A7 and v is the central vertex. See Example 10.1.9 to see our results applied to this case and two others arising in the classification of subfactors of small index. Perhaps surprisingly, both Theorem 1.0.1 and Theorem 1.0.3 can be deduced purely from arithmetic considerations. The first main result follows immediately from the following theorem. Theorem 1.0.5. Let β ∈ Q(ζ ) be a real algebraic integer in some cyclotomic extension of the rationals. Let β denote the largest absolute value of all conjugates of β. If β ≤ 2 then β = 2 cos(π/n) for some integer n. If 2 < β < 76/33, then β is one of the five numbers occurring in Theorem 1.0.1. The second main result is a consequence of the following theorem, combined with the fact that the even part of a finite depth subfactor is a fusion category. Theorem 1.0.6. For any , there exists an effective constant N such that for all n ≥ N , either: (1) All the eigenvalues of the adjacency matrix Mn are of the form ζ + ζ −1 for some root of unity ζ , and the graphs n are the Dynkin diagrams An or Dn .
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(2) The largest eigenvalue λ of the adjacency matrix Mn is greater than 2, and the field Q(λ2 ) is not abelian. Although Theorem 1.0.6 is, in principle, effective, it is difficult to apply in practice. We also give a logically weaker but more effective version of Theorem 1.0.6 which is sufficient to prove Theorem 1.0.3 and is practical for many examples. We briefly summarize the main ideas in the proofs of these arithmetic theorems. A key idea of Cassels [7] is to study elements with small normalized trace M (β) = 1 deg β Tr(β · β) ∈ Q rather than work directly with bounds on β . A key principle, made rigorous by Loxton [25], says that if β is a cyclotomic integer and M (β) is small, then β can be written as a sum of a small number of roots of unity. This principle was first applied by Cassels to √study cyclotomic integers of small norm [7]. In fact, Theorem 1.0.5 (at least for β ≤ 5) is a consequence of the main theorem of Cassels with finitely many exceptions. A careful study of Cassels’ analysis shows that any exceptions must lie in the field Q(ζ N ) with N = 4692838820715366441120 = 25 · 33 · 5 · 7 · 11 · 13 · 17 · 19 · 23 · 29 · 31 · 37 · 41 · 47 · 53. Given that the problem of finding small vectors inside a lattice (say, of algebraic integers) is NP-complete, this is not immediately useful. We improve on Cassels argument in three main ways. First, we show that β < 76/33 implies that M (β) < 23/6 (which improves substantially on the obvious bound of (76/33)2 ). Second, we systematically exploit the condition that β is real (an assumption that Cassels did not make). In particular, we adapt techniques of A. J. Jones [20] and Conway and A. J. Jones [8] for classifying small sums of three roots of unity to understand real sums of five roots of unity. Finally, we engage in a detailed case-by-case analysis to complete the argument and remove all exceptions. We now sketch the ideas of the proof of Theorem 1.0.6. Let λn be the FrobeniusPerron eigenvalue of the graph n . The average n1 μ |μ2 − 2|2 over all eigenvalues μ of the adjacency matrix of n can be shown to converge to 2 as n increases without bound. Since all Galois conjugates of λn are eigenvalues of the adjacency matrix, this suggests that M (λ2n − 2) should also be small. By the Cassels-Loxton principle, if λ2n is cyclotomic, one would expect that λ2n − 2 should be a sum of a small number of roots of unity. Explicitly, we deduce for all n greater than some explicit bound (depending on ) that either λ2n is not cyclotomic or λ2n − 2 is the sum of at most two roots of unity. The latter case can only occur if |λn | ≤ 2, in which case the characteristic polynomial of n is a Chebyshev polynomial, and n is necessarily an extended Dynkin diagram. In order to make this argument rigorous, one needs to understand the relationship between all eigenvalues and the subset of eigenvalues conjugate to λn . We do this in two different ways. First, we use the result of Etingof-Nikshych-Ostrik to show that all non-repeating eigenvalues are cyclotomic integers. In light of this result, we need only control the repeated eigenvalues and the eigenvalues of the form ζ + ζ −1 for roots of unity ζ . This can be done using techniques of Gross-Hironaka-McMullen [15]. To finish the argument, we use a much easier version of Theorem 1.0.5 to get a contradiction. For the second proof, we use some height inequalities to show that the degree of λn grows linearly in n. Again this is enough to get a bound on M (λ2n − 2), as well as bounds on M (P(λ2n )) for other polynomials in λ2n . The desired contradiction then follows from Loxton’s result applied to a particular polynomial in λ2n .
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Remark 1.0.7. The methods used in our proof of Theorem 1.0.1 can certainly be extended further than 76/33, at the cost of a certain amount of combinatorial explosion. √ there do exist limit points of the set of possible β , including at √ However, 2 2 = lim 2 2 cos(π/n) and 3 = lim 1 + 2 cos(2π/n). The best general “sparse−→
−→
ness” result we have is Theorem 9.1.1 which states that the set of values of M (β) for β a cyclotomic integer is a closed subset of Q. Theorem 1.0.1 √ is similar in spirit to Haagerup’s classification of all subfactors of index less than 3 + 3 = 4.73205... [16]. In fact, a version of Theorem 1.0.1 follows from Haagerup’s classification, for example, “if X is an object in a unitary tensor cate√
gory with duals then the dimension of X does not lie in the interval (2, 5+ 2 13 ) = (2, 2.074313 . . .).” Our result is weaker in that we assume finiteness, but stronger in that it does not assume unitarity and applies to larger dimensions. In the other direction, one might wonder if purely arithmetic considerations have implications for finite depth subfactors of small index larger than 4. Indeed, using only arithmetic we can prove the following result. Theorem 1.0.8. Suppose that 4 < α < 4 + 10/33 = 4.303030 √ . . . is the index of a finite 5+ 13 depth subfactor. Then either α = 3 + 2 cos(2π/7), or α = 2 .
1.1. Detailed summary. The proof of Theorem 1.0.5 proceeds in several steps. We first prove the theorem for those β which can be written as the sum of at most 5 roots of unity (Theorem 4.2.10). This argument requires some preliminary analysis of vanishing sums of roots of unity, which we undertake in §4. Having done this, we prove Theorem 5.0.13, which shows that any exception to Theorem 1.0.5 lies in Q(ζ N ) with N = 420. A useful technical tool is provided by Lemma 5.1.1, which allows us to reduce our search to β satisfying M (β) < 23/6 rather than M (β) < 5 as in Cassels. In Lemma 7.0.8 and Corollary 7.0.10, we prove Theorem 1.0.5 for β ∈ Q(ζ84 ). In §8 we make the final step of showing that any counterexample β ∈ Q(ζ420 ) must actually lie in Q(ζ84 ). There is a certain amount of combinatorial explosion in this section which we control as much as possible with various tricks. Although our paper is written to be independent, it would probably be useful to the reader to consult A. J. Jones [20] when reading §4.2, and Cassels [7] when reading §§5.2–8. In §9, we prove an easier version of Theorem 1.0.5 which will be used to prove the effective version of Theorem 1.0.3. In this section, we also prove that the values of M (β) for β a cyclotomic integer are a closed subset of Q. We then prove an effective version of Theorem 1.0.3 in §10 and give applications to several families which appear in the classification of small index subfactors. In §11, we prove Theorem 1.0.6 which is logically stronger but less effective than the result in the previous section. A reader mainly interested in the applications to subfactors may wish to skip directly to §10 & §11.
2. Definitions and Preliminaries If N is an integer, let ζ N denote exp(2πi/N ). Having fixed this choice for all N , there is no ambiguity when writing expressions such as ζ12 + ζ20 —a priori, such an expression is not even well defined up to conjugation.
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Suppose that Q(β) is an abelian extension. By the Kronecker–Weber theorem, β is contained inside some minimal cyclotomic field Q(ζ N ). (N is the conductor of Q(β).) If β ∈ Q(ζ N ) is an algebraic integer, we shall consider several invariants attached to β: Definition 2.0.1. Fora cyclotomic integer β, we denote by N (β) the size of the smallest set S such that β = S ξi for ξi a root of unity. Definition 2.0.2. If β is any algebraic integer, we let β denote the maximum of the absolute values of all the conjugates of β, and let M (β) denote the average value of the real numbers |σβ|2 , where σβ runs over all conjugates of β. Remark 2.0.3. If β ∈ K , where K is Galois and G = Gal(K /Q), then M (β) is well behaved whenever complex conjugation is central in G, since then |σβ|2 = σ |β|2 , and [K : Q]M (β) = Tr(|β 2 |). This is the case, for example, whenever K is totally real or abelian. In particular, in these cases, M (β) ∈ Q. There are inequalities N (β) ≥ β , which follows from the triangle inequality, and β 2 ≥ M (β) ≥ |N K /Q (β)|1/[K :Q] , which is ≥ 1 if β is non-zero. Note that α + β = α + β in general. Example 2.0.4. Suppose that β is a totally real algebraic integer and that β ≤ 2. If α + α −1 = β, then all the conjugates of α have absolute value 1. A theorem of Kronecker [24] implies that α is a root of unity, and then an easy computation shows that β = 2 cos(π/n) for some integer n. This example shows that the values β are discrete in [0, θ ] for any θ < 2. On the other hand, it follows from Theorem 1 of [31] that the values of β for totally real algebraic integers β are dense in [2, ∞). Thus, the discreteness implicit in Theorem 1.0.5 reflects a special property of cyclotomic integers. It also follows from Theorem 1 of [31] that the values M (β) (for totally real β) are dense in [2, ∞). On the other hand, a classical theorem of Siegel [29] says that M (β) ≥ 3/2 for any totally real algebraic integer β of √ 1+ 5 degree ≥ 2, the minimum value occurring for β = 2 , and, furthermore, the values of M (β) are discrete in [0, θ ] for any θ < λ = 1.733610 . . . In the cyclotomic case, we once more see a limit point of M (β) at 2 followed by a region beyond 2 where M (β) is discrete (Theorem 9.0.1). Moreover, the closure of M (β) on [0, ∞) is, in fact, a closed subset of Q (Theorem 9.1.1). 3. Background on Fusion Categories and Subfactors In this section, we rapidly review some notions about fusion categories and subfactors, and collect a few remarks and examples. Although the applications of our main results are to fusion categories and subfactors, their proofs are purely arithmetic and can be read independently from this section. A fusion category C over a field k is an abelian, k-linear, semisimple, rigid, monoidal category with finitely many isomorphism classes of simple objects. In this paper, all fusion categories are over the complex numbers. A subfactor is an inclusion A < B of von Neumann algebras with trivial centers. We will only consider subfactors in this paper which are irreducible (B is an irreducible A-B bimodule) and type I I1 (there exists a unique normalized trace). A subfactor is called
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finite depth if only finitely many isomorphism classes of simple bimodules appear as summands of tensor powers of A B A . In particular, to every finite depth subfactor there is an associated fusion category C, called the principal even part which is the full subcategory of the category of A-A bimodules whose objects are summands of tensor powers of A B A . The principal graph of a subfactor is a bipartite graph whose even vertices are the simple A-A bimodules which occur as summands of tensor powers of A B A , whose odd vertices are the simple A-B bimodules which occur as summands of tensor powers of A B A tensored with A B B , and where X and Y are connected by dim(X ⊗ A B B , Y ) edges. Remark 3.0.5. Usually included in the data of a principal graph is the choice of a fixed leaf which corresponds to the monoidal unit A A A . All the techniques in our paper which eliminate a graph as a possible principal graph eliminate the graph for any choice of leaf. Nonetheless, techniques in other papers often depend on the choice of fixed leaf. In particular, the families in Haagerup’s list of potential principal graphs of small index [16] have modularity restrictions on the length of the degree 2 tree which depend on the choice of leaf. Strictly speaking, our main result when applied to = A7 with v the middle vertex is stronger than the result in [3] where they only check noncyclotomicity after assuming Haagerup’s modularity conditions. Nonetheless, we will often elide this issue, and when we say a paper eliminated a family of potential principal graphs we will mean that they eliminated the principal graphs in that family which had not already been eliminated by Haagerup. A dimension function on a fusion category C is a ring map f : K (C) → C, where K (C) is the Grothendieck group thought of as a ring with the product induced by the tensor product. We often abuse notation by applying f directly to objects in C. There exists a unique dimension function FP called the Frobenius–Perron dimension which assigns a positive real number to each simple object [13, §8]. The Frobenius–Perron dimension of X ∈ C is given by the unique largest eigenvalue of left multiplication by [X ] in K (C) ⊗ C. The Frobenius–Perron dimension of A B A in C is the index of A < B which is denoted [B : A]. The index of A < B is the square of the largest eigenvalue of the adjacency matrix of the principal graph. For the applications in our paper, we need the following strong arithmetic condition on dimensions. Theorem 3.0.6 [13, Corollary 8.53]. If C is a fusion category, X is an object in C, and f is a dimension function, then Q( f (X )) is abelian. We will also want a version of this result that more easily applies to principal graphs: Lemma 3.0.7. If is the principal graph of a finite depth subfactor A < B and λ is an eigenvalue of M() of multiplicity one, then Q(λ2 ) is abelian. Proof. Let C be the fusion category which is the principal even part of the subfactor. Let X be the object A B A inside C. From the definition of the principal graph it follows that λ2 is a multiplicity 1 eigenvalue for left multiplication by [X ] in the base extended Grothendieck group K (C) ⊗ C. Decompose K (C) ⊗ C as a product of matrix algebras End(Vi ). An element of End(Vi ) can be thought of as acting by left multiplication on itself or as acting on Vi . The eigenvalues of the former action are exactly the eigenvalues of the latter action but each repeated dim Vi times. In particular, if x is an element of a multi-matrix algebra then any multiplicity one eigenvalue of x acting on the algebra
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by left multiplication must be a component of x in one of the 1-dimensional matrix summands. In particular, we see that there is a map of rings f : K (C) ⊗ C → C such that λ2 = f (X ). Our result now follows immediately from Theorem 3.0.6.
The following well-known arithmetic arguments proving two versions of the V. Jones index restriction [21] are baby examples of the main idea of this paper: Lemma 3.0.8. If X is an object in a fusion category with FP(X ) ≤ 2, then FP(X ) = 2 cos(π/n) for some integer n. Lemma 3.0.9. If A < B is a finite depth subfactor with index [B : A] ≤ 4, then [B : A] = 4 cos(π/n)2 = 2 + 2 cos(2π/n). Proofs. In light of Theorem 3.0.6, Lemma 3.0.8 follows directly from Example 2.0.4. In light of Lemma 3.0.7, Lemma 3.0.9 follows either from applying Example 2.0.4 to λ, where λ2 = [B : A], or to λ2 − 2.
Remark 3.0.10. This is weaker than the V. Jones index restriction since we are making a finite depth assumption. Indeed, all our results in this paper about subfactors and monoidal categories depend crucially on finiteness assumptions.
3.1. Realizing the possible dimensions. As mentioned in the Introduction, each of the numbers in Theorem 1.0.5 can in fact be realized as the dimension of an object in a fusion category. Nonetheless, we do not necessarily expect that every number of the form x for x a real cyclotomic integer can be realized as a dimension of an object in a fusion category.√We quickly summarize how each of these numbers can be realized. √ The dimension ( 3 + 7)/2 occurs in a fusion category constructed by Ostrik in the Appendix based on an unpublished construction via a conformal inclusion √ (due to Xu [33]) of a subfactor originally constructed by Izumi [18]. The dimension 5 can be achieved by a Tambara–Yamigami category associated to Z/5Z [32]. The dimension th 1 + 2 cos(2π/7) occurs as a dimension √ √ of an object in quantum SU(2) at a 14 root of unity. The dimension (1 + 5)/ 2 occurs in the Deligne tensor product of quantum √ SU(2) at a 10th root of unity and quantum SU(2) at an 8th root of unity. Finally, (1 + 13)/2 occurs as the dimension of an object in the dual even part of the Haagerup subfactor [2]. 3.2. Deduction of Theorem 1.0.8 from Lemma 3.0.7. Suppose that α is the index of a finite depth subfactor and 4 < α < 4 + 10/33 = 4.303030 . . . Then α is a cyclotomic integer by Lemma 3.0.7, and α = λ2 for a totally real algebraic integer λ which is the Perron–Frobenius eigenvalue of the principal graph. Thus −2 ≤ (σ λ)2 − 2 ≤ 76/33 for every conjugate σ λ of λ. In particular, if β = α − 2, then 2 < β < 76/33, and by Theorem 1.0.5, we deduce that β is one of the five numbers occurring in Theorem 1.0.1. On the other hand, for three of these five numbers β has a conjugate smaller than −2, and hence the √ corresponding field Q(λ) is not totally real. Thus, either α = 3 + 2 cos(2π/7) or α = 5+ 2 13 .
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4. The Case when β is a Sum of at most 5 Roots of Unity The goal of this section is to prove Theorem 1.0.5 in the case that N (β) ≤ 5 (see Theorem 4.2.6). The outline of this argument is that we first use the Conway–A. J. Jones classification of small vanishing sums of roots of unity in order to show that, outside a few exceptional cases, any real sum of five roots of unity is of the obvious form (with pairs of complex conjugate terms). We then make a more in depth analysis of small sums of the form ζ Na + ζ N−a + ζ Nb + ζ N−b . 4.1. Vanishing Sums. Consider a vanishing sum: ξi = 0, S
where the ξi are roots of unity. Such a sum is called primitive if no proper subsum vanishes. We say that such a sum has |S| terms. We may normalize any such sum up to a finite ambiguity by insisting that one of the summands be 1. Theorem 4.1.1 (Conway–A. J. Jones [8]). For every |S|, there are only finitely many primitive normalized vanishing sums i∈S ξi = 0. The Conway and A. J. Jones result is more precise, in that they give explicit bounds on the conductor of the cyclotomic field generated by the ξi in a vanishing sum with a fixed number of terms. For our purposes, it will be useful to have a more explicit description of the primitive normalized vanishing sums for small |S|. The following result is a small extension of Theorem 6 of [8] which can be found in Table 1 of [28]. Theorem 4.1.2 (Conway–A. J. Jones, Poonen–Rubinstein). The primitive vanishing sums with |S| even and |S| ≤ 10 are as follows: • |S| = 2: 1 + (−1) = 0. • |S| = 6: ζ6 + ζ65 + ζ5 + ζ52 + ζ53 + ζ54 = 0. • |S| = 8: ζ6 + ζ65 + ζ7 + ζ72 + ζ73 + ζ74 + ζ75 + ζ76 = 0. 4 10 11 17 23 24 ζ6 + ζ65 + ζ30 + ζ30 + ζ30 + ζ30 + ζ30 + ζ30 = 0. 2 12 13 19 20 ζ6 + ζ65 + ζ30 + ζ30 + ζ30 + ζ30 + ζ30 + ζ30 = 0.
• |S| = 10: 3 7 9 + ζ10 + ζ10 = 0. ζ7 + ζ72 + ζ73 + ζ74 + ζ75 + ζ76 + ζ10 + ζ10 10 13 25 31 37 1 + ζ3 + ζ7 + ζ72 + ζ21 + ζ21 + ζ42 + ζ42 + ζ42 + ζ42 = 0. 10 16 19 31 37 1 + ζ3 + ζ7 + ζ73 + ζ21 + ζ21 + ζ42 + ζ42 + ζ42 + ζ42 = 0. 10 19 19 25 37 1 + ζ3 + ζ7 + ζ74 + ζ21 + ζ21 + ζ42 + ζ42 + ζ42 + ζ42 = 0. 10 19 25 31 1 + ζ3 + ζ7 + ζ75 + ζ21 + ζ21 + ζ42 + ζ42 + ζ42 + ζ42 = 0. 13 19 13 25 37 + ζ21 + ζ42 + ζ42 + ζ42 + ζ42 = 0. 1 + ζ3 + ζ72 + ζ74 + ζ21
In particular, there does not exist any vanishing sum with |S| = 4.
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Note that any vanishing sum of roots of unity with |S| terms decomposes as a sum of primitive vanishing sums each with |Si | terms, where |S| = |Si | is a partition of |S|. We are interested in cyclotomic integers β that are totally real. Lemma 4.1.3. Suppose that N (β) ≤ 5, and that β = 0 is real. Then there exists integers a, b, and a root of unity ζ such that, up to sign, one of the following holds: N (β) = 1, and β = 1. N (β) = 2 and β = ζ a + ζ −a . N (β) = 3, and β = ζ a + ζ −a + 1. N (β) = 4, and β = ζ a + ζ −a + ζ b + ζ −b . N (β) = 5, and β = ζ a + ζ −a + ζ b + ζ −b + 1. 17 . N (β) = 3, and β is Galois conjugate to ζ12 + ζ20 + ζ20 N (β) = 4, and β is Galois conjugate to one of −9 −7 3 + ζ 15 , (a) ζ84 + ζ84 + ζ84 84 −9 −7 3 + ζ 27 , (b) ζ84 + ζ84 + ζ84 84 17 . (c) 1 + ζ12 + ζ20 + ζ20 (8) N (β) = 5, and β is Galois conjugate to one of 17 + ζ a + ζ −a for some root of unity ζ , (a) ζ12 + ζ20 + ζ20 −9 −7 3 + ζ 15 , (b) 1 + ζ84 + ζ84 + ζ84 84 −9 −7 3 + ζ 27 , (c) 1 + ζ84 + ζ84 + ζ84 84 −9 −7 3 + ζ 13 + ζ84 + ζ84 + ζ84 (d) ζ84 84 −9 −7 15 + ζ 25 + ζ 73 . (e) ζ84 + ζ84 + ζ84 84 84 Proof. Let I denote a set of size N (β) such that β = I ξi . Note that −1 is a root of unity. If β is real, then we have a vanishing sum β −β = ξi + −ξi−1 = 0
(1) (2) (3) (4) (5) (6) (7)
I
I
with 2N (β) ≤ 10 terms. This sum can be decomposed into primitive sums whose number of terms sum to 2N (β). Write such a primitive vanishing sum as ξi + −ξi−1 = 0, A
B
where A and B are disjoint subsets of I . Suppose that |A| + |B| is odd. Since the sum is invariant under complex conjugation, we may assume that |A| > |B|. It follows that β= ξi = ξi + ξi = ξi + ξi−1 , I
I \A
A
I \A
B
and hence N (β) ≤ |I | − |A| + |B| < |I |, a contradiction. Thus, every such vanishing subsum must have an even number of terms. Suppose that there is a vanishing subsum with 2 terms. Then we have the following options: (1) If ξi + ξ j = 0, then β = I −{i, j} ξi , and hence N (β) ≤ |I | − 2, a contradiction. (2) If ξi − ξi−1 = 0, then ξi = ±1. Let γ = β − ξi . Then γ real and satisfies N (γ ) = N (β) − 1. −1 (3) If ξi − ξ −1 j = 0, let γ = β − ξi − ξi . Then γ is real and N (γ ) = N (β) − 2.
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In all these cases, the result follows by induction on N (β). So we may assume that there are no vanishing subsums with 2 terms. Since there exists no primitive vanishing sum with 4 terms, and since 10 < 6 + 6, it follows that I ξi + I −ξi−1 is itself primitive. Suppose that 2|I | = 6 and our sum is proportional to a primitive vanishing sum with 6 terms. Hence our sum is proportional to ζ6 + ζ65 + ζ5 + ζ52 + ζ53 + ζ54 . By construction, there exists a decomposition of the sum I ξi + I −ξi−1 into pairs with product −1. Rescaling, we have a decomposition of the sum ζ6 + ζ65 + ζ5 + ζ52 + ζ53 + ζ54 into pairs with constant product. Since the product of all of these numbers is 1, this constant product must be a third root of unity. But at least one pair consists only of fifth roots of unity, so the product of this pair is a fifth root of unity. Hence the product of each pair must be 1. It follows that the constant of proportionality is ζ4±1 . Hence, up to sign and Galois 17 . conjugation, β = ζ4 ζ6 + ζ4 ζ5 + ζ4 ζ52 , which is Galois conjugate to η := ζ12 + ζ20 + ζ20 8 6 4 2 The minimal polynomial of this number is x − 8x + 14x − 7x + 1, and its largest Galois conjugate is 2.40487 . . .. We note in passing that √ √ 4 + 5 + 15 + 6 5 2 η = 2 cos(π/30) + 2 cos(13π/30), and η = . 2 Suppose that 2|I | = 8 and our sum is proportional to a primitive vanishing sum with 8 terms. First suppose that the vanishing sum is proportional to ζ6 +ζ65 +ζ7 +ζ72 +ζ73 +ζ74 + ζ75 + ζ76 . Again, we look for a way of decomposing this sum into four pairs with a fixed constant product. Since the product of all the terms is 1, the constant must be a fourth root of unity. However, at least one pair has product which is a seventh root of unity. Hence, the constant product must be 1. Hence, β must be a sum of four elements, each consisting of one term from each of the pairs (ζ6 , ζ6−1 ), (ζ7 , ζ7−1 ), (ζ72 , ζ7−2 ), (ζ73 , ζ7−3 ) all scaled by a fixed primitive 4th root of unity. This leads to sixteen possibilities, which fall under two Galois orbits. One Galois orbit consists of the twelve Galois conjugates −9 −7 3 + ζ 15 , which has minimal polynomial + ζ84 + ζ84 of ζ84 84 x 12 − 15x 10 + 64x 8 − 113x 6 + 85x 4 − 22x 2 + 1, and largest root β = 3.056668 . . . . The other orbit consists of the four conjugates of −9 −7 3 + ζ 27 , which has minimal polynomial x 4 − 5x 2 + 1. We have ζ84 + ζ84 + ζ84 84 √ √ 3+ 7 −9 −7 3 27 = 2.188901 . . . . ζ84 + ζ84 + ζ84 + ζ84 = 2 Now suppose that the vanishing sum is proportional to a sum of the form ζ5a1 + ζ5a2 − ζ5a3 (ζ3 + ζ32 ) − ζ5a4 (ζ3 + ζ32 ) − ζ5a5 (ζ3 + ζ32 ), where the ai are some permutation of {0, 1, 2, 3, 4}. This includes the last two vanishing sums with 8 terms. Here the product of all the terms is ζ5a3 +a4 +a5 . Rescaling the sum by a fifth root of unity, we may assume that the product of all the terms is 1. So the product of each pair must be a fourth root of unity. Furthermore, at least one pair consists of two 30th roots of unity, hence the product of each pair must be a 30th root of unity. Hence the product of each pair must be ±1. Since the fifth roots of unity must then pair with each other the product must be 1. However, at most one other pair multiplies to 1 (if ai = 0 for i = 3, 4, 5). Hence there are no β that yield this vanishing sum.
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Suppose that 2|I | = 10 and our sum is proportional to a primitive vanishing sum with 10 terms. First suppose our vanishing sum is proportional to the first 10-term vanishing sum: ζ7 + ζ72 + ζ73 + ζ74 + ζ75 + ζ76 − (ζ5 + ζ52 + ζ53 + ζ54 ). The product of all of the terms in this sum is 1, hence the product of each pair must be a 5th root of unity. However, at least one pair consists only of 7th roots of unity, hence the product of each pair must be 1. Hence, β = ζ4±1 (ζ7±1 + ζ7±2 + ζ7±3 − ζ5±1 − ζ5±2 ). Up to Galois conjugation there are two numbers of this form. Their minimal polynomials are x 24 − 36x 22 + 506x 20 − 3713x 18 + 15825x 16 − 40916x 14 + 64917x 12 − 62642x 10 + 35684x 8 − 11253x 6 + 1717x 4 − 90x 2 + 1 and x 8 − 12x 6 + 34x 4 − 23x 2 + 1. The largest roots of these are 3.7294849 . . . and 2.861717 . . . respectively. Now suppose our vanishing sum is proportional to a sum of the form ζ7a1 + ζ7a2 + ζ7a3 + ζ7a4 − ζ7a5 (ζ3 + ζ32 ) − ζ7a6 (ζ3 + ζ32 ) − ζ7a7 (ζ3 + ζ32 ), where the ai are a permutation of the numbers {0, . . . , 6}. This form includes the remaining 5 vanishing sums. After possibly rescaling by a 7th root of unity, the product of all the terms is 1, and hence the product of each pair is a 5th root of unity. Since the only fifth root of unity that appears as a product of two terms is 1, the product of each pair must be 1. Hence, without loss of generality the pairs must be {ζ7a1 , ζ7−a1 }, {ζ7a3 , ζ7−a3 }, {−ζ3 , −ζ32 }, {−ζ7a6 ζ3 , −ζ7−a6 ζ32 }, {−ζ7−a6 ζ3 , −ζ7a6 ζ32 }. Thus β is Galois conjugate to something of the form ζ4 (ζ7 + ζ7x − ζ3±1 − ζ7 ζ3 − If the last sign is positive then β can be rewritten, using ζ3 + ζ32 = −1, as a sum of 4 terms. Hence the last sign is negative. Now, if the first sign is positive we can also rewrite β as a sum of four roots of unity. Namely, we see that y
y (ζ7 ζ32 )±1 ).
−y
−y
ζ4 (ζ7 + ζ7x − ζ3 − ζ7 ζ3 − ζ7 ζ3 ) = −ζ4 ζ3 (−ζ3−1 ζ7 − ζ3−1 ζ7x + 1 + ζ7 + ζ7 ) y
y
= −ζ4 ζ3 (−ζ7a − ζ7b + ζ3 ζ7 + ζ3 ζ7x ), where a, b, x, ±y are a permutation of 2, . . . , 6. The relation that we used is ζ7a1 + ζ7a2 + ζ7a3 + ζ7a4 + ζ7a5 − (ζ3 − ζ3−1 )ζ7a6 − (ζ3 − ζ3−1 )ζ7a7 , where the ai are a premutation of 0, . . . , 6. Hence β is Galois conjugate to y
−y
ζ4 (ζ7 + ζ7x − ζ32 − ζ7 ζ3 − ζ7 ζ3 ), where x and y are each one of {2, 3, 4, 5} such that x is not congruent to ±y modulo 7. There are two different Galois orbits of that form. The roots of x 12 − 16x 10 + 60x 8 − 78x 6 + 44x 4 − 11x 2 + 1, the largest of which is approximately 3.354753 . . . ; and the roots of x 12 − 22x 10 + 85x 8 − 113x 6 + 64x 4 − 15x 2 + 1, the largest of which is approximately 4.183308 . . . . These correspond to Galois conjugates of the roots occurring in (8d) and (8e) in the statement of the theorem. Note the curious identities (of sums of real numbers): −9 −7 −9 −7 3 13 3 15 13 15 (ζ84 + ζ84 + ζ84 + ζ84 + ζ84 ) = (ζ84 + ζ84 + ζ84 + ζ84 ) + (ζ84 + ζ84 − ζ84 ), −9 −7 −9 −7 15 25 73 3 15 25 73 3 + ζ84 + ζ84 + ζ84 + ζ84 ) = (ζ84 + ζ84 + ζ84 + ζ84 ) + (ζ84 + ζ84 − ζ84 ). (ζ84 13 − ζ 15 and ζ 25 + ζ 73 − ζ 3 are equal to Here the “exotic” real numbers ζ84 + ζ84 84 84 84 84 2 cos(13π/84) and 2 cos(25π/84) respectively, and so can actually be written as the sum of two roots of unity.
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4.2. Estimates. In this section, we analyze in more detail sums of the form β = ζ Na + ζ N−a + ζ Nb + ζ N−b . We wish to find all such sums which have β < 4 cos(2π/7). Our argument in this section closely follows the paper of A. J. Jones [20], who studies expressions of the form β = 1 + ζ Na + ζ Nb with β small. In outline, this argument uses the geometry of numbers, as follows. The Galois conjugates of ζ Na + ζ N−a + ζ Nb + ζ N−b are all of the form ζ Nak + ζ N−ak + ζ Nbk + ζ N−bk for (k, N ) = 1. Using Minkowski’s theorem, we can find a k such that all four roots of unity are all “close” to one, and thus the expression above is large. However, it is not immediately apparent that one can choose such a k co-prime to N . Instead, using certain estimates involving the Jacobsthal function, we show that either there exists a suitable k co-prime to N or the integers (a, b) satisfy a linear relation ax − by = 0 mod N with (x, y) one of a small explicit finite set of integers ((2, 2), (3, 3), or (2, 4)). In the latter case, we may study β directly. A much simpler 1-dimensional argument, also using estimates on the Jacobsthal function, gives 17 + ζ a + ζ −a such that β < 4 cos(2π/7). a description of all β = ζ12 + ζ20 + ζ20 N N Definition 4.2.1. The Jacobsthal function j (N ) is defined to be the smallest m with the following property: In every arithmetic progression with at least one integer co-prime to N , every m consecutive terms contains an element co-prime to N . Lemma 4.2.2. Suppose that M|N has one fewer distinct prime factors than N . Then there is an inequality j (M)2 ≤ N /11 for N > 210 and N = 330, 390. Proof. A result of Kanold [22] shows that j (N ) ≤ 2ω(N ) , where ω(N ) is the number of distinct primes dividing N . Note that j (N ) only depends on the square-free part of N . Suppose that N has at least d ≥ 7 prime factors. Then j (M) ≤ 2d−1 , whereas N /11 ≥ (11)−1
d
pn ≥ 2 · 3 · 5 · 7 · 13 · 4d−6 =
n=1
1365 d−1 ·4 ≥ j (M)2 . 512
For smaller d, we note the following bounds on j (M), noting that M has one less distinct prime divisor than N . These bounds were computed by Jacobsthal [19]: if d if d if d if d
= 2, = 3, = 4, = 5,
then then then then
j (M) ≤ 2, j (M) ≤ 4, j (M) ≤ 6 and j (M) ≤ 10.
Thus, if N has 5 prime factors, we are done if N ≥ 1100, if N has 4 prime factors, we are done if N ≥ 396, and if N has less than three prime factors, we are done if N ≥ 176. Yet if N has 5 prime factors, then N ≥ 2310, and if N has 4 prime factors, then N ≥ 396 unless N = 210, 330, or 390.
Lemma 4.2.3. j (M) ≤ 2M/5 − 1 for all M except M ∈ {1, 2, 3, 4, 5, 6, 7, 10, 12}. Proof. As in Lemma 4.2.2 we use the result of Kanold to see that this theorem is true for all M which is divisible by 3 or more primes. If M is a product of two primes, then j (M) ≤ 4, so the inequality follows so long as M > 12. If M is prime then j (M) ≤ 2, so the inequality follows so long as M > 7.
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Remark 4.2.4. The known asymptotic bounds for j (N ) are much better, see, for example, Iwaniec [17]. Now we apply these bounds on Jacobsthal functions to finding small sums of roots of unity. 17 + ζ a + ζ −a , then β = 2 cos(2π/60), or β = Lemma 4.2.5. If β = ζ12 + ζ20 + ζ20 N N 17 , or β ≥ 4 cos(2π/7). ζ12 + ζ20 + ζ20 17 . Write N = AM, where A = (N , 60). We see that β is Proof. Let η = ζ12 + ζ20 + ζ20 −b conjugate to η + ζ Nb + ζ N for any (b, N ) = 1 such that b ≡ a mod A. If there exists such a b satisfying b/N ∈ [−1/5, 1/5], then
β ≥ η + 2 cos(2π/5) = 3.022901 . . . , which is certainly greater than 4 cos(2π/7). To guarantee the existence of such a b, we need to ensure that at least one term of the arithmetic progression of integers congruent to a mod A in the range [−N /5, N /5] is co-prime to M (it is automatically co-prime to A). The length of this arithmetic progression is at least 2N /5A − 1 = 2M/5 − 1. Such a b always exists provided that j (M) ≤ 2M/5 − 1, where j (M) denotes the Jacobsthal function. By Lemma 4.2.3, this inequality holds for all M except M ∈ {1, 2, 3, 4, 5, 6, 7, 10, 12}. This leaves a finite number of possible N and β to consider, which we can explicitly compute. In particular, we look at N ∈ {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 24, 25, 28, 30, 35, 36, 40, 42, 45, 48, 50, 60, 70, 72, 75, 80, 84, 90, 100, 105, 120, 140, 144, 150, 180, 200, 210, 240, 300, 360, 420, 600, 720}. Indeed, in this range, the smallest largest conjugate of β is 2 cos(2π/60) (with, e.g., N = 60, a = 17), the second smallest is η = 2.40487 . . . (with, e.g., N = 4, a = 1), and the next smallest is 19 = 2.71559 . . . > 4 cos(2π/7). η + 2 cos 2π 60
Theorem 4.2.6. Suppose that N > 230 (we will need a slightly higher bound than the 210 of the Lemma 4.2.2), and N = 330, or 390. Let β be a number of the form ζ Na + ζ N−a + ζ Nb + ζ N−b , where a and b are relatively prime, then either: (1) β is the sum of at most√two roots √ of√unity, and thus β ≤ 2, (2) β is conjugate to (1 + 5)/ 2 or 6, (3) β has a positive conjugate whose absolute value is bigger than 4 cos(2π/7), in particular, β ≥ 4 cos(2π/7). Before proving this theorem we prove several lemmas. Let us fix once and for all the constant K = 2/49. Lemma 4.2.7. Let x, y ∈ R. Suppose that x 2 +y 2 < K . Then 2 cos(2π x)+2 cos(2π y) > 4 cos(2π/7).
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Proof. The minimum value of 2 cos(2π x) + 2 cos(2π y) occurs when x = y = 1/7.
Definition 4.2.8. Denote by a,b,N ⊂ Z2 the set of integer vectors x such that x.(a, −b) ≡ 0 mod N . The determinant of the lattice a,b,N is N . We may describe it explicitly as follows. Let u = (b, a), and fix a vector v such that v.(a, −b) = 1. Then Z2 is generated by u and v, and a,b,N is generated by u and N v. Up to scalar, there is a canonical map φ : a,b,N → Z/N Z obtained by reduction modulo N . We say that a vector λ ∈ a,b,N is co-prime to N if and only if the image φ(λ) of λ in Z/N Z lands in (Z/N Z)× . Denote by Q the quadratic form Q(x, y) = x 2 + y 2 on Z2 restricted to a,b,N ; it has discriminant −4N 2 . Lemma 4.2.9. If λ is co-prime to N , and Q(λ) ≤ K · N 2 , then β = ζ Na + ζ N−a + ζ Nb + ζ N−b has β ≥ 4 cos(2π/7). Proof. We may write (r, s) = λ = k(b, a) mod N , for some k co-prime to N . Replacing ζ by ζ k is thus an automorphism of Q(ζ ), which has the effect of replacing β by ζ ka + ζ −ka + ζ kb + ζ −kb = 2 cos(2πr/N ) + 2 cos(2π s/N ) ≤ 4 cos(2π/7). Hence, from Lemma 4.2.7, we deduce the result.
Proof of Theorem 4.2.6. It suffices to assume that β < 4 cos(2π/7) and derive a contradiction. Note that the Galois conjugates of β can be obtained by replacing ζ N by ζ Nk for some integer k such that (k, N ) = 1. Hence the Galois conjugates are exactly the numbers of the form ζ Na + ζ N−a + ζ Nb + ζ N−b for (a , b ) ∈ a,b,N which is relatively prime to N . By reduction theory for quadratic forms, there exists a basis μ, ν of a,b,N for which Q(x · μ + y · ν) = Ax 2 + Bx y + C y 2 ,
|B| ≤ A ≤ C, := B 2 − 4 AC = −4N 2 .
Now A2 ≤ AC ≤ AC + 13 (AC − B 2 ) = − 43 = 3N 2 ≤ K 2 · N 4 , providing that N > 43. Hence Q(μ) < K · N 2 , and thus, by Lemma 4.2.9, μ is not co-prime to N . Since φ : a,b,N → Z/N Z is surjective, there exists an integer k such that kμ + ν is co-prime to N . By assumption, N has a prime factor q that divides μ. The terms in this sequence must all be automatically co-prime to q. Let M be N divided by the highest power of q dividing N . In order to find something of the form kμ + ν is co-prime to N , it suffices to find one that is co-prime to M. By definition of the Jacobsthal function, it follows that we may take a
− j (M) B j (M) B k∈ + , + 2 2A 2 2A such that kμ + ν is co-prime to N , and hence Q(kμ + ν) > K · N 2 . Yet Q(kμ+ν) = Ak 2 + Bk +C = A(k − B/2 A)2 + (4 AC − B 2 )/4 A ≤ j (M)2 A/4 + N 2 /A, and thus j (M)2 A2 /4 + N 2 ≥ K N 2 A.
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Since this inequality holds for A = 0, and since j (M)2 > 0, we see that the inequality holds exactly on the complement of some (possibly empty) interval. Using the assumption that √ N ≥ 230 and Lemma 4.2.2, we see that the inequality does not hold for A = 3N . Namely, 3 3 3 j (M)2 N 2 + N 2 ≤ N + N 2 < K N 2 A. 4 44 Similarly, using that N ≥ 28, we also see that the inequality does not hold for A = 25. Namely,
25 N2 N j (M)2 + N 2 /25 ≤ + 4 44 25
A < K · N 2 · A.
√ Hence the inequality does not hold for any A in the interval [25, 3N ]. Since A is positive and A2 ≤ 3N 2 it follows that A ≤ 24, and hence Q(μ) ≤ 24. Write μ = (x, y). Then x 2 + y 2 ≤ 24, and ax − by ≡ 0 mod N . Recall that μ is not co-prime to N , and thus x must not be co-prime to y. It follows that (x, y), up to sign and ordering, is one of the pairs (2, 2), (3, 3) or (2, 4). We consider each of these in turn. (1) (x, y) = (2, 2). It follows that (a, b) = (a, a) or (a, a + N /2). In the first case, the maximum absolute value of any conjugate of β is of the form 4 cos(π/M) for some M. In the second case, ζ a = −ζ b , so β = 0. (2) (x, y) = (3, 3), either (a, b) = (a, a), or, after making an appropriate permutation, (a, b) = (a, a + N /3). In this case, with ω3 = 1, β = ζ a + ζ −a + ζ a ω + ζ −a ω−1 = −ω−1 ζ a − ωζ a is a sum of two roots of unity. (3) (x, y) = (2, 4). The only new possibility is (a, b) = (a, a + N /4). Letting i 4 = 1, we find that β = ζ a (1 + i) + ζ −a (1 − i) =
√ a √ 2(ζ ζ8 + ζ −a ζ8−1 ) = 2(ζ + ζ −1 ),
√ and hence β = 2 2 cos(π/M) for some M. The only numbers of this kind between and 4 cos(π/7) occur for M = 5 and 6, for which we obtain the values √ 2√ √ (1 + 5)/ 2 and 6. This completes the proof of the theorem.
Theorem 4.2.10. Let β be totally real, and suppose that N (β) ≤ 5. Then either (1) β is a sum of at most two roots of unity. (2) β ≥ 4 cos(2π/7). (3) A conjugate of β is one of the following numbers, listed in increasing order:
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√
√ 3+ 7 2√ 5 1 + 2 cos(2π/7) = 2 cos(π/7) + 2 cos(3π/7) √ 1+ 5 = 2 cos(π/20) + 2 cos(9π/20) √ 2 1 + 2 cos(4π/13) + 2 cos(6π/13) 1 + 2 cos(2π/11) + 2 cos(6π/11) 2 cos(π/30) + 2 cos(13π/30) √ 1+ 2 √ 6 = 2 cos(π/12) + 2 cos(5π/12) 2 cos(11π/42) + 2 cos(13π/42)
= 2.18890105931673 . . . , = 2.23606797749978 . . . , = 2.24697960371746 . . . , = 2.28824561127073 . . . , = 2.37720285397295 . . . , = 2.39787738911579 . . . , = 2.40486717237206 . . . , = 2.41421356237309 . . . , = 2.44948974278317 . . . , = 2.48698559166908 . . . .
Proof. We split into cases using the classification of Lemma 4.1.3. If N (β) = 3 and β = 1+ζ a +ζ −a , then the largest conjugate of β is 1+2 cos(2π/N ). For N less than 7 we could rewrite this as a sum √ of fewer than three terms. If N = 7, then β = 1+2 cos(2π/7). If N = 8, then β = 1 + 2. If N ≥ 9, then β > 4 cos(2π/7). If N (β) = 4, and β = ζ a + ζ −a + ζ b + ζ −b , then the previous theorem applies if N > 230 and N = 330 or 390. If N (β) = 5, and β = 1 + ζ a + ζ −a + ζ b + ζ −b , then the previous theorem applies to β − 1 if N > 230 and N = 330 or 390. If β − 1 has a positive conjugate whose absolute value is larger than 4 cos(2π/7), it follows that β > 1 + 4 cos(2π/7) 17 + ζ a + ζ −a , then we apply Lemma 4.2.5. If N (β) = 5 and β = ζ12 + ζ20 + ζ20 N N Hence we need only consider finitely many remaining numbers. First, we may have that N ≤ 230 or N = 330 or N = 390. Second, we may be in one of the finitely many exceptional cases in Lemma 4.1.3. In the former case, we compute directly that the largest conjugates all have absolute value at least 4 cos(2π/7), except for exceptions √ the√ listed above. For the latter case, only one of the exception numbers, ( 3 + 7)/2, has β small enough.
Remark 4.2.11. It is a consequence of this computation and Theorem 1.0.5 that the smallest largest conjugate of a real cyclotomic integer which is not a sum of 5 or fewer roots of unity is √
1 + 13 = − ζ 2 + ζ −2 + ζ 6 + ζ −6 + ζ 8 + ζ −8 = 2.30277 . . . , 2 where ζ is a 13th root of unity. We shall use the following result, which follows directly from Theorem 4.2.10. Corollary 4.2.12. Let β be integer such that √ √ 3 ≤√N (β) ≤ 5. Then √a real √ cyclotomic either β is conjugate to 21 ( 3 + 7), 5, 1 + 2 cos(2π/7), (1 + 5)/ 2, or β ≥ 76/33. 5. The Normalized Trace The goal of the next two sections is to prove that
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Theorem 5.0.13. If β is a cyclotomic √ integer such that β is real, β < 76/33, and N (β) ≥ 3, then either β = (1 + 13)/2, or β ∈ Q(ζ N ), where N = 4 · 3 · 5 · 7 = 420. So suppose that β is real, that β ∈ Q(ζ N ) with N minimal, that N (β) ≥ 3, and β < 76/33. First we prove a lemma which allows us to reduce to studying β with M (β) < 23/6. Second, we show that if p k | N with k > 1 then p k = 4, this argument uses techniques developed by Cassels [7]. In the√ next section, we will show that if p > 7 then either p N or p = 13 and β = (1 + 13)/2. Again this argument will use techniques generalizing those of Cassels. 5.1. Relationship between β and M (β). The following lemma allows us to reduce to considering β with M (β) small. Lemma 5.1.1. Let β ∈ Q be a totally real algebraic integer, and suppose that β < 76/33 = 2.303030 . . . Then either β 2 = 4 or 5, or M (β) < 23/6 = 3.833333 . . . . √
Proof. Let κ = 23/6, and let α = 1+ 2 13 . Since √ √ 23 1 7 + 13 7 − 13 7 + , M (α) = = < 2 2 2 2 6 √ we may assume that β is not a conjugate of α. Similarly M ( 3) = 3, and so we may assume that β 2 = 3. The inequality (x) = 120(κ − x) − 36 log |x − 4| + 160 log |x − 5| + 9 log |x − 3|
+ 2 log |x 2 − 7x + 9| > 0 for x ∈ [0, (76/33)2 ] = [0, 5.303948 . . .] is an easy calculus exercise. (Note that the roots of the polynomial x 2 − 7x + 9 are the conjugates of α 2 .) The critical points are the roots of −40200 + 68381x − 44376x 2 + 13814x 3 − 2071x 4 + 120x 5 . The absolute minimum value in this range occurs at approximately x = 3.320758 . . . , where obtains its minimum of roughly 0.394415 . . . (Fig. 2). Let S = {xi } be a finite set of real numbers in [0, (76/33)2 ] whose average is greater than κ = 23/6. Then the average of κ − xi is less than zero, and hence 0 > 120 (κ − xi ) ≥ 36 log |xi − 4| + 160 log |xi − 5| + 9 log |xi − 3| +2 log |xi2 − 7xi + 9|. Suppose that S consists of the squares of the conjugates of β ∈ K = Q(β). Since β < 76/33, it follows that all the xi lie in [0, (76/33)2 ]. Since we are assuming that β 2 = 3, 4, 5, nor a conjugate of α 2 (which is a root of x 2 − 7x + 9), it follows that the norms of β 2 − 3, β 2 − 4, and β 2 − 5, as well as β 4 − 7β 2 + 9, are non-zero algebraic integers. Hence the absolute value of their norms are at least one. Taking logarithms, we deduce that every sum occurring on the right-hand side of the inequality above is non-negative, which is a contradiction, and the lemma is established.
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Fig. 2. The function (x) + 1, on a log scale. The four visible peaks, and one that is not apparent on the graph, near x = 5.30278, are actually asymptotes
Remark 5.1.2. The constants (120, 36, 160, 9, 2) chosen in this proof are somewhat arbitrary and mysterious, and fine tuning would certainly lead to an improved result. However, to increase 76/33 substantially one would need to allow M (β) to increase, which would increase the combinatorial difficulty of our later arguments. It follows that in order to prove Theorem 5.0.13, we may assume that M (β) < 23/6. We shall also frequently use the following lemmas: Lemma 5.1.3. (Cassels’ Lemma 2 [7]) If N (α) ≥ 2, then M (α) ≥ 3/2. Lemma 5.1.4. (Cassels’ Lemma 3 [7]) If N (α) ≥ 3, then M (α) ≥ 2. 5.2. The case when p 2 |N . Suppose that β ∈ Q(ζ N ), and suppose that N is minimal with respect to this property. We start with what Cassels calls the second case, that is, the case when N admits a prime p such that p 2 |N . Explicitly, assume that p m N for an integer m ≥ 2. Let N = p m−1 M, so pM. Let ζ be a p mth root of unity. We may write ζ i αi , β= S
where αi ∈ Q(ζ M ). Here S denotes any set of p m−1 integers that are distinct modulo p m−1 . After having chosen an S, the αi are determined uniquely by β. Since β is real, it is invariant under complex conjugation. It follows that ζ i αi = ζ −i αi . S
S
If p is odd, let S denote the set {− ( p 2 −1) , − ( p 2 −3) , . . . , −1, 0, 1, 2, . . . ( p 2 −1) }. If p = 2, let S = {−(2m−2 − 1), . . . , −1, 0, 1, 2, . . . , 2m−2 }. From the uniqueness of this expansion we deduce, if p is odd, that αi = α−i for all i ∈ S. If p = 2, we deduce m−2 m−2 that αi = α−i if i < 2m−2 , and that ζ 2 α2m−2 = ζ 2 α2m−2 . m−1
m−1
m−1
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Lemma 5.2.1. There is an equality M (β) =
863
M (αi ).
Proof. Our proof is essentially that of Cassels (who proves it under extra hypotheses that are not required for the proof of this particular statement). We reproduce the proof here. The conjugates of ζ over Q(ζ M ) are ζ · ζ pn for n = 0 to p m−1 − 1. Let M (θ ) denote the average of the conjugates of |θ |2 over Q(ζ M ). Then p m−1 M (β) =
n
=
i
n
=
ζ i αi ζ pni
ζ − j α j ζ − pn j
j
ζ
i− j
ζ
i, j
ζ i− j αi α j
pn(i− j)
αi α j
ζ pn(i− j) .
n
i, j
Now i ≡ j mod p m−1 if and only if i = j, and thus ζ p(i− j) = 1 if and only if i = j. For all other pairs (i, j), the final sum is a power sum of a non-trivial root of unity over a complete set of congruence classes, and is thus 0. Hence, as in Cassels, we find that M (β) =
|αi |2 ,
and the result follows upon taking the sum over the conjugates of Q(ζ M ) over Q.
Let X denote the number of αi which are non-zero. In order to prove Theorem 5.0.13 in this case, we must show that if p 2 | N , then p = 2 and 4N . 5.3. The case when X = 1. If p is odd, then β = α = α. In this case we find that β ∈ Q(ζ M ), contradicting the minimality assumption on N . If p = 2, then either β = α = α, or β = ζ2
m−2
α2m−2 = ζ 2
m−2
α2m−2 .
By minimality, we deduce that 2m−2 = 1, and hence 2m = 4. (The number in fact, of this form.)
√
√ 3+ 7 2
is,
5.4. The case when X = 2. If p is odd, we deduce that β = ζ α + ζ −1 α. If N (α) ≤ 2, then we are done, by Corollary 4.2.12. If N (α) > 2, then by Lemma 3 of Cassels, M (α) ≥ 2, and M (β) ≥ 4. If p = 2, the same argument applies, except in this case it could be that β = α0 + ζ 2
m−2
α2m−2 .
Once more, since N is minimal with respect to β, it must be the case that 2m−2 = 1 and 2m = 4.
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5.5. The case when X = 3. If p is odd, then, for some primitive p mth root of unity ζ , we have β = ζ α + γ + ζ −1 α. If α is a root of unity, then, by Corollary 4.2.12, we may assume that N (γ ) ≥ 3 and hence (by Lemma 3 of Cassels) that M (γ ) ≥ 2, and thus M (β) ≥ 1 + 1 + 2 = 4. If N (α) = 2, then by Lemma 2 of Cassels, M (α) ≥ 3/2, and hence M (β) ≥ 3/2 + 3/2 + 1 = 4. If p = 2, there is at least one i such that αi = 0 and i = 0, 2m−2 . It follows that β = ζ α + γ + ζ −1 α for some γ such that γ = γ , and the proof proceeds as above. 5.6. The case when X ≥ 4 It is immediate that M (β) ≥ 4. . 6. The Case when p Exactly Divides N We now consider what Cassels calls the first case, where p || N . So suppose that β is real, that β ∈ Q(ζ N ) with N minimal, that N (β) ≥ 3, and β <√76/33. We will show in this section that if p | N then p ≤ 7 or p = 13 and β = (1 + 13)/2. (In particular, we may assume that p is odd.) This will complete the proof of Theorem 5.0.13. Write N = pM once again, and let ζ be a primitive p th root of unity. The conjugates of ζ are now ζ · ζ k for any k except k ≡ −1 mod p. We write ζ i αi , β= S
where αi ∈ Q(ζ M ) and S denotes {−( p − 1)/2, . . . , 0, 1, . . . , ( p − 1)/2}. This expansion is no longer unique; there is ambiguity given by a fixed constant for each element. Since β is real, it is invariant under complex conjugation. It follows that there exists a fixed λ ∈ Q(ζ M ) such that αi = α−i + λ.
√ The element λ itself must satisfy λ = −λ, or equivalently, that λ · −1 is real. Let X denote the number of terms occurring in S such that αi = 0.
√ Lemma 6.0.1. If λ = 0, then X ≥ ( p + 1)/2. If λ is a root of unity, then λ = ± −1.
Proof. If λ = 0, then since α−i − αi = λ, at least one of {αi , α−i } must be non-zero. Since there are ( p + 1)/2 such pairs not containing any √common element, the result follows. The second claim follows from the fact that λ · −1 is real.
6.1. The case when X = 1. We deduce that β = α = α, contradicting the minimality of N . 6.2. The case when X = 2. If p ≥ 7, by Lemma 6.0.1, we may assume that λ = 0, and hence β = ζ α + ζ −1 α. If α is a root of unity, √ then N (β) ≤ 2. Hence, we may assume (replacing α by a conjugate) that |α| ≥ 2. Note that we may choose ζ to be primitive, since N was chosen
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to be minimal with respect to β. Write ζ α = |α|e2πiθ . The conjugates of ζ are ζ · ζ k , where k is any integer such that k ≡ −1 mod p. We replace ζ by a conjugate to make θ as close to 0 or 1/2 as possible. By Dirichlet’s box principle, with no constraint on k we could insist that θ ≤ 1/2 p, or, if we liked, that θ − 1/2 ≥ 1/2 p. Given our single constraint, we may at least find a conjugate of ζ such that θ satisfies one of these inequalities. In either case, we deduce that √ |β| > 2|α| cos(π/7) ≥ 2 2 cos(π/7) = 2.548324 . . . > 2.303030 . . . = 76/33. 6.3. The case when X = 3. Suppose that X = 3, and suppose that p ≥ 11. By Lemma 6.0.1, we may assume that λ = 0. We may therefore assume that β = ζ α + γ + ζ −1 α, where γ = γ . After conjugating, we may assume that |αγ | ≥ 1. After possibly negating β, we may assume that γ is positive. Write ζ α = |α|e2πiθ . Now we must insist that θ is small rather than θ − 1/2, and thus may only deduce that θ ≤ 1/ p. It follows that 1 2|α| cos(2π/11) ≥2· = 2.594229 . . . > 76/33. β ≥ 2|α| cos(2π/11) + |α| |α| 6.4. An interlude. We recall some facts that will be used heavily in the sequel. There is always a formula: ( p − 1)M (β) = ( p − X ) M (αi ) + M (αi − α j ), (1) (This is Eq. 3.9 of Cassels, his argument is similar to that in Lemma 5.2.1.) We often use this equation in the following way. Suppose that the X non-zero terms break up into sets of size X j consisting of equal terms. Then, since M (αi − α j ) ≥ 1 if αi = α j , we deduce that 1 ( p − 1)M (β) ≥ ( p − X ) M (αi ) + X j (X − X j ) 2 1 = (p − X) M (αi ) + X2 − (2) X 2j . 2 We also note the following lemma, whose proof is obvious. Lemma 6.4.1. Suppose that at least Y of the αi are equal to α. Then we may — after subtracting α from each αi — assume that X ≤ p − Y . Finally, we note the following. p−1 2 and λ = 0. 1 < 4 ( p + 3) then at
Lemma 6.4.2. Suppose that p ≥ 13. Then we may assume that X ≤
Proof. The Corollary to Lemma 1 of Cassels states that if M (β) least p+1 2 of the αi are equal to each other. By Lemma 6.4.1 it follows that we can assume that X ≤ p−1 2 . Hence, we need only compute that ( p + 3) ≥ 4 > 23/6 > M (β). 4
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6.5. The case when X = 4, and p ≥ 11. By Lemma 6.0.1, we may write β = ζ α + ζ −1 α + ζ i γ + ζ −i γ . If α and γ are roots of unity, then we are done by Corollary 4.2.12. Thus, we may assume that N (α) ≥ 2, and hence that M (α) = M (α) ≥ 3/2. If γ is not equal to α or α, then {α, α} are certainly both distinct from {γ , γ }. Hence evaluating M on the corresponding differences is at least one. Using Eq. 1, we deduce that ( p − 1)M (β) ≥ ( p − 4)(3/2 · 2 + 2) + 4, and hence, if p ≥ 11, that M (β) ≥ 3.9 > 23/6. This contradicts Lemma 5.1.1. Suppose that γ = α. If α is not real, then α and γ are distinct from α and γ , and hence ( p − 1)M (β) ≥ ( p − 4)(3/2 · 4) + 4, from which we deduce a contradiction as above. If γ = α is real, then
β = α ζ + ζ −1 + ζ i + ζ −i . Since α and ζ + ζ −1 + ζ i + ζ −i lie in disjoint Galois extensions, the maximal conjugate of β is the product of the maximal conjugate of α and the maximal conjugate of the second factor. Since p > 5, the latter factor cannot be written √ as a√sum of a smaller number of roots of unity, and hence its maximum is at least ( 3 + 7)/2, by Corollary 4.2.12. Yet, since M (α) ≥ 3/2, at least one conjugate of α has absolute value √ ≥ 2, and hence √ √ 14 + 6 = 3.095573 . . . > 76/33. β ≥ 2 6.6. The case when X = 5, and p ≥ 11. Once more by Lemma 6.0.1, we may write that β = ζ α + ζ i γ + δ + ζ −i γ + ζ −1 α. If α, δ, and γ are roots of unity, then we are done by Corollary 4.2.12. We break up our argument into various subcases. 6.6.1. X = 5 and M (α) = M (γ ) = 1, M (δ) ≥ 3/2 If α = γ are both real, then, after replacing β by −β if necessary, they are both one, and
β = δ + ζ + ζ −1 + ζ i + ζ −i . We deduce that ( p − 1)M (β) ≥ ( p − 5)(3/2 + 4) + 4. This implies that M (β) ≥ 4 if p ≥ 13. By computation, if p = 11, there exist two conjugates of the right-hand side, one positive and one negative, both of which have absolute value at least 2 cos(2π/11) + 6 cos(3π/11) = 1.397877 . . . .
Cyclotomic Integers, Fusion Categories, and Subfactors
On the other hand, there exists a conjugate of δ with absolute value at least hence there exists a conjugate of β with absolute value at least √
867
√ 2, and
2 + 2 cos(2π/11) + 6 cos(3π/11) = 2.812090 . . . > 2.303030 . . . = 76/33.
Thus we may assume that either α is real and γ is not, or that they are both not real. Thus δ is distinct from the four terms {α, α, γ , γ } and either {α, α} has no intersection with {γ , γ } or {α, γ } has no intersection with {α, γ }. In either case, we deduce that ( p − 1)M (β) ≥ ( p − 5)(3/2 + 4) + 8, which implies that M (β) ≥ 4.1 > 23/6. 6.6.2. X = 5 and M (α) ≥ 3/2. We break this case up into further subcases. (1) M (γ ) = M (δ) = 1: Clearly the terms involving α are distinct from the other terms, and hence ( p − 1)M (β) ≥ 6( p − 5) + 6, and thus M (β) ≥ 4.2 > 23/6. (2) M (δ) ≥ 3/2, and M (γ ) = 1: In this case, ( p − 1)M (β) ≥ ( p − 5)(3/2 · 3 + 2) + 6, which implies that M (β) ≥ 4.5 > 23/6. (3) M (γ ) ≥ 3/2, M (δ) = 1: In this case, ( p − 1)M (β) ≥ ( p − 5)(3/2 · 4 + 1) + 4, and thus M (β) ≥ 4.6 > 23/6. (4) M (αi ) ≥ 3/2 for all i: In this case, ( p − 1)M (β) ≥ ( p − 5)(3/2 · 5), and hence M (β) ≥ 4.5 > 23/6. 6.7. The case when X = 6, p ≥ 11, and λ = 0. If X = 6, then Lemma 5.1.1 no longer applies when p = 11. We consider this possibility at the end of this subsection. Thus, we assume that β = αi ζ i + α j ζ j + αk ζ k + αi ζ −i + α j ζ − j + αk ζ −k . Again, we break up into subcases.
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6.7.1. X = 6, all the αi are roots of unity. If all the αi are the same, they must be (after changing the sign of β if necessary) equal to 1. We compute in this case that ( p − 1)M (β) = ( p − 6)6. If p = 11, 13, then M (β) ≥ 4.125 > 23/6. Otherwise, we may write β = 2 cos(2πi/ p) + 2 cos(2π j/ p) + 2 cos(2π k/ p). Note that (i, p) = ( j, p) = (k, p) = 1. Without loss of generality, we may assume that i = 1. The smallest value of β for p = 11 or p = 13 of this kind may easily be computed to be
1+
√ 2
−2(cos(4π/11) + cos(8π/11) + cos(12π/11)) = 2.397877 . . . , 13
= −2(cos(4π/13) + cos(12π/13) + cos(16π/13)) = 2.302775 . . . ,
the former of which is larger than 76/33, the latter which is on our list. The second smallest number for p = 13 is 3.148114 . . . > 76/33. Suppose that one of the αi is not real. Then αi is certainly distinct from αi , and either α j = α j or α j and α j are both distinct from αi and αi , and similarly with k. It follows that there are at least 9 pairs of numbers which are distinct, the minimum occurring when αi = α j = αk or when α j = αk = ±1. In either case, we find that ( p − 1)M (β) ≥ ( p − 6)6 + 9, and hence M (β) ≥ 3.9 > 23/6. Finally, suppose that all the αi are real, but that they are not all equal. Then, up to sign, β = 2 cos(2πi/ p) + 2 cos(2π j/ p) − 2 cos(2π k/ p). In this case, we compute that ( p − 1)M (β) ≥ ( p − 6)6 + 8, which is larger than 23/6 if p = 11. If p = 11, we enumerate the possibilities directly, and find that the smallest value of β is 2 cos(2π/11) − 2 cos(8π/11) − 2 cos(16π/11) = 3.276858 . . . > 76/33. 6.7.2. X = 6, and M (αi ) ≥ 3/2. If M (α j ) ≥ 3/2 also then ( p − 1)M (β) ≥ ( p − 6)8, and hence M (β) ≥ 4. Thus we may assume that M (α j ) = M (αk ) = 1. In this case, there are clearly at least 8 distinct pairs, and thus ( p − 1)M (β) ≥ ( p − 6)7 + 8, and hence M (β) ≥ 4.3 > 23/6.
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6.8. The case when X≥ 7, and p ≥ 11. Note that we make no assumptions on λ in this case. Write β = S αi ζ i . From Eq. 2, we deduce that ( p − 1)M (β) ≥ X ( p − X ) +
1 2 2 X − Xj . 2
If p ≥ 13, then by Lemma 6.4.2, we may assume that X ≤ ( p − 1)/2. In particular, this implies that p ≥ 17. In this case, the inequality ( p − 1)M (β) ≥ X ( p − X ) already implies that M (β) ≥ 4.375 > 23/6. Hence we may reduce to the case when p = 11. By Lemma 6.4.1, we may assume that X j ≤ 11 − X . We consider the various possibilities: 2 (1) Suppose that X = 7. Then X j ≤ 4, and hence X j ≤ 25, and 10M (β) ≥ 7(11 − 7) +
1 (49 − 25) = 40, 2
and M (β) ≥ 4 > 23/6. 2 X j ≤ 22, and (2) Suppose that X = 8. Then X j ≤ 3, and hence 10M (β) ≥ 8(11 − 8) +
1 (64 − 22) = 45, 2
and M (β) ≥ 4.5 > 23/6. 2 X j ≤ 17, and (3) Suppose that X = 9. Then X j ≤ 2, and hence 10M (β) ≥ 9(11 − 9) +
1 (81 − 17) = 50, 2
and M (β) ≥ 5 > 23/6. 2 X j ≤ 10, and (4) Suppose that X = 10. Then X j ≤ 1, and hence 10M (β) ≥ 10(11 − 10) +
1 (100 − 10) = 55, 2
and M (β) ≥ 5.5 > 23/6. 6.9. The case when X = 6, p = 11, and λ = 0. Write β = S αi ζ i . Since λ = 0, it must be the case that either αi or α−i is non-zero. Moreover, by cardinality reasons, at least one of these must be zero, and hence λ = αi − α−i = αi . Thus, in this case, it must be the case that β =α+λ ζi, T
where T is a subset√of S of cardinality 5 such that T ∪ {−T } ∪ {0} = S. Moreover, α − α = λ, and λ · −1 is real. If λ is not a root of unity, then 10M (β) ≥ (11 − 6)(3/2 · 5 + 1),
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and hence M (β) ≥ 4.25 ≥√23/6. Hence λ is a root of unity, √ which must be equal (after changing the sign of β) to −1. Clearly α is not equal to −1. Hence M (β) ≥ (11 − 6)(5 + M (α)) + 5 = 30 + 5M (α). It follows that M (α) < 8/5 < 2, and thus α is the sum of at most two roots of unity. If α is a root of unity, then α = α −1 , and hence α − α −1 = λ =
√ −1.
5 . In this case we may check every possibility for β (the This implies that α = ζ12 or ζ12 set of possible T has cardinality 25 since it requires a choice of one of {i, −i} for each non-zero i mod 11), and the smallest such (largest conjugate) is:
−1 −3 −4 −5 2 ζ12 + ζ4 ζ11 = 2.524337 . . . > 2.303030 . . . = 76/33. + ζ11 + ζ11 + ζ11 + ζ11
Suppose √ that N (α) = 2. Then either M (α) = 3/2 and α is a root of unity √ times (1 + 5)/2, or M (α) ≥ 5/3 > 8/5. Hence we may now assume that α = (1 + 5)/2 · ξ for a root of unity ξ . We now obtain the equation √ √ 1+ 5 (ξ − ξ −1 ) = −1. 2 9 . Again, we check the possibilities for From this equation we deduce that ξ = ζ20 or ζ20 β, the smallest being: √
1+ 5 −1 −3 −4 −5 2 = 3.197154 . . . > 76/33. + ζ11 + ζ11 + ζ11 + ζ11 ζ20 + ζ4 ζ11 2
This completes the proof of Theorem 5.0.13. 7. An Analysis of the Field Q(ζ84 ) In order to progress further, we require some more precise analysis of certain elements α in the field Q(ζ84 ) with M (α) small. Lemma 7.0.1. Suppose that α ∈ Q(ζ7 ) satisfies M (α) ≤ 4. Then, up to sign and rescaling by a 7th root of unity, either: (1) (2) (3) (4) (5) (6) (7)
α α α α α α α
= 0 or α = 1, and M (α) = 0 or 1. = 1 + ζ7i with i = 0, and M (α) = 5/3. = 1 − ζ7i with i = 0, and M (α) = 7/3. j = 1 + ζ7i + ζ7 with (i, j) distinct and non-zero, and M (α) = 2. j = 1 + ζ7i − ζ7 with (i, j) distinct and non-zero, and M (α) = 10/3. = 2 and M (α) = 4. j = ζ7i + ζ7 + ζ7k − 1 with (i, j, k) distinct and non-zero, and M (α) = 4.
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Proof. Write α = ai ζ7i , where ai ∈ Z. We may assume that all the ai are nonnegative, and that at least one ai is equal to 0. Suppose that Ai of the ai are equal to i. Then (i − j)2 Ai A j . 6M (α) = (ai − a j )2 = 2 A A , we deduce Suppose that M (α) ≤ 4. From the inequality 48 ≥ 12M (α) ≥ n n 0 that An = 0 if n ≥ 7. It is easy to enumerate the partitions of 7 = Ai satisfying the inequality 24 ≥ (i − j)2 Ai A j . We write A as (A0 , A1 , . . .), showing only up until the last nonzero value, and find a strict inequality for
A ∈ {(7), (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1), (2, 4, 1), (1, 5, 1), (1, 4, 2)} (giving cases (1), (1), (2), (4), (4), (2), (1), (5), (3) and (5) of the statement, respectively) and equality for A ∈ {(6, 0, 1), (3, 3, 1), (1, 3, 3), (1, 0, 6)} (giving cases (6),(7),(7) and (6) of the statement, respectively). The result follows.
Corollary 7.0.2. Suppose that α ∈ Q(ζ7 ) satisfies N (α) ≥ 4, then M (α) ≥ 4. Lemma 7.0.3. Suppose that α ∈ Q(ζ21 ) satisfies M (α) < 17/6. Then, up to sign and a 21st root of unity, either: (1) (2) (3) (4) (5)
α is a sum of at most three roots of unity. α lies in the field Q(ζ7 ). j α = ζ7i + ζ7 + ζ7k − ζ3 , where (i, j, k) are distinct and non-zero, and M (α) = 5/2. j α = 1+ζ7i −(ζ7 +ζ7k )ζ3 , where (i, j, k) are distinct and non-zero, and M (α) = 8/3. j j α = ζ7i + ζ7 + (ζ7 + ζ7k )ζ3 , where (i, j, k) are distinct, and M (α) = 8/3.
Proof. We may write α = γ + δζ3 , where M (α) =
1 (M (γ ) + M (δ) + M (γ − δ)). 2
We may assume that γ = δ, since otherwise α = −ζ32 γ is, up to a root of unity, in Q(ζ7 ), giving case (2). In general, we note that α = (γ − δ) − δζ32 = (δ − γ )ζ3 − γ ζ32 , Hence, after re-ordering if necessary, we may assume that N (γ − δ) ≥ N (γ ) ≥ N (δ). Assume that M (α) ≤ 17/6. If N (δ) ≥ 3, then M (γ − δ), M (γ ), and M (δ) are all ≥ 2, and thus M (α) ≥ 3, a contradiction. We consider various other cases. (i) N (δ) = 1 and N (γ ) ≤ 2: In this case, N (α) ≤ 3, giving case (1). (ii) N (δ) = 1 and N (γ ) = 3: If N (γ − δ) ≥ 4, then M (α) ≥ (1 + 2 + 10/3)/2 ≥ 19/6. Thus N (γ − δ) = 3. In particular, (δ − γ ) + (γ ) − (δ) = 0 is a vanishing sum of length 3 + 1 + 3. The only primitive vanishing sums in Q(ζ7 ) have length 7 or 2. Thus, the expression above must be a multiple of the vanishing sum 1 + ζ7 + ζ72 + ζ73 + ζ74 + ζ75 + ζ76 + ζ77 = 0.
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F. Calegari, S. Morrison, N. Snyder j
After scaling, we may assume that δ = −1, and thus γ = ζ7i + ζ7 + ζ7k for some triple (i, j, k) that are all distinct and non-zero. Since δ − γ is sum of 3 distinct 7th roots of unity in this case, we deduce that M (γ ) = M (δ − γ ) = 2, and hence M (α) = 5/2. We are thus in case (3). (iii) N (δ) = 1 and N (γ ) ≥ 4: It follows immediately that M (α) ≥ (1 + 10/3 + 10/3)/2 = 23/6, a contradiction. (iv) N (δ) = 2 and N (γ ) = 2: If N (δ −γ ) ≥ 4, then M (α) ≥ (5/3+5/3+10/3) = 20/6. If N (δ − γ ) = 3, we obtain a vanishing sum (δ − γ ) + (γ ) − (δ) = 0 j
of length 7, and hence γ = ζ7i + ζ7 and δ = −(ζ7k + ζ7l ), where (i, j, k, l) are all distinct. In this case, M (γ ) = M (δ) = 5/3, and γ − δ is minus a sum of three distinct 7th roots of unity, and so M (γ − δ) = 2. It follows that M (α) = 8/3 and we are in case (4). If N (δ − γ ) = 2, then the above sum is a vanishing sum of length 6. It follows that it is composed of vanishing subsums of length j j 2, from which it easily follows that δ = ζ7 + ζ7k and γ = ζ7i + ζ7 . In this case, M (δ) = M (γ ) = 5/3, and M (δ − γ ) = 2, and thus M (α) = 8/3, giving case (5). (v) N (δ) = 2 and N (γ ) ≥ 3: It follows immediately that M (α) ≥ (5/3+2+2)/2 = 17/6, a contradiction.
Corollary 7.0.4. Suppose that α ∈ Q(ζ21 ) satisfies M (α) < 9/4 and N (α) ≥ 3, then j α = 1 + ζ7i + ζ7 where (i, j) are distinct and non-zero and M (α) = 2. Lemma 7.0.5. Suppose that α ∈ Q(ζ21 ) satisfies M (α) < 23/6, then N (α) ≤ 5. Proof. As before we may write α = γ + δζ3 and we may assume that N (γ − δ) ≥ N (γ ) ≥ N (δ). If N (δ) ≤ 2, then we are done unless N (γ − δ) ≥ N (γ ) ≥ 4. In this case, we deduce from Corollary 7.0.2 that M (γ − δ) ≥ 4 and M (γ ) ≥ 4, from which it follows directly that M (α) ≥ (4 + 4 + 1)/2 > 23/6. Suppose that N (δ) ≥ 3. If N (δ − γ ) ≥ 4, then M (α) ≥ (2 + 2 + 4)/2 = 4 > 23/6. Thus, we may assume that N (δ) = N (γ ) = N (δ − γ ) = 3. Let us consider the resulting vanishing sum (δ − γ ) + (γ ) − (δ) = 0. It has length 9 = 7 + 2. After scaling α by a root of unity, we may assume that this sum is (having re-arranged the order of the roots of unity): (1 + ζ7 + ζ72 + · · · + ζ76 ) + (1 − 1) = 0. At least one of the three terms must be contained within the first sum. Furthermore, the (1 − 1) sum cannot be contained within a single term. Hence, we obtain the following two possibilities (up to symmetry): j
γ = 1 + ζ7i + ζ7 , δ = 1 − ζ7k − ζ7l , δ − γ = 1 + ζ7m + ζ7n , γ = 2 + ζ7i ,
j
δ = 1 − ζ7 − ζ7k , δ − γ = ζ7l + ζ7m + ζ7n ,
where (i, j, k, l, m, n) are distinct and non-zero. In the first case, we notice that since 1 + ζ3 = −ζ32 , in fact N (α) ≤ 5. In the second case, we compute that M (α) = (13/3 + 10/3 + 2)/2 = 29/6 > 23/6.
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Lemma 7.0.6. Suppose that α ∈ Q(ζ21 ) satisfies N (α) = 2, then M (α) ≥ 2, or M (α) = 5/3. Proof. Again we write α = γ + δζ3 . If either γ or δ is zero, then up to a root of unity α ∈ Q(ζ7 ) and we can apply Lemma 7.0.1. If neither γ nor δ is zero, then they must both be roots of unity, hence, M (α) = (2 + M (γ − δ))/2. Notice that γ − δ is not a root of unity, because there are no vanishing sums (γ − δ) + (δ) − (γ ) = 0 of length 3 in Q(ζ7 ). Since α is not a root of unity, γ = δ, and hence M (α) = (2 + M (γ − δ))/2 ≥ 2.
Lemma 7.0.7. Suppose that α ∈ Q(ζ84 ), that M (α) < 9/4, and that N (α) ≥ 3, then i (1 + ζ j + ζ k ). α = ζ84 7 7 Proof. Write α = γ + ζ4 δ. Since N (α) ≥ 3 it follows that one of γ or δ is not a root of unity. If γ and δ are both nonzero, then M (β) ≥ 1 + 3/2 > 9/4, hence γ or δ is zero, and up to a root of unity α ∈ Q(ζ21 ). The result then follows from Corollary 7.0.4.
Lemma 7.0.8. The elements α ∈ Q(ζ84 ) such that M (α) < 17/6 are, up to roots of unity, either a sum of at most 3 roots of unity, or are, up to a root of unity, one of the exceptional forms in Q(ζ21 ), specifically: j
(1) α = ζ7i + ζ7 + ζ7k − ζ3 , where (i, j, k) are distinct and non-zero, and M (α) = 5/2. j (2) α = 1+ζ7i −(ζ7 +ζ7k )ζ3 , where (i, j, k) are distinct and non-zero, and M (α) = 8/3. j j (3) α = ζ7i +ζ7 +(ζ7 +ζ7k )ζ3 , where (i, j, k) are distinct and non-zero, and M (α) = 8/3. Moreover, if N (α) = 2, then either M (α) ≥ 2 or M (α) = 5/3. Proof. If α = γ + δζ4 with γ , δ ∈ Q(ζ21 ), then M (α) = M (γ ) + M (δ). If γ = 0 or δ = 0 the problem reduces immediately to Lemma 7.0.3. So we may assume that γ = 0 and δ = 0. By symmetry, we may assume that M (γ ) ≥ M (δ) ≥ 1. It follows that M (γ ) < 11/6 < 2, and hence N (γ ) ≤ 2. If N (δ) = N (γ ) = 2, then M (α) ≥ 10/3. If N (α) = 2, then either γ and δ are non-zero, in which case M (α) = 2, or we may assume that α ∈ Q(ζ21c ), and apply Lemma 7.0.3.
Lemma 7.0.9. Suppose that α ∈ Q(ζ84 ). Then either M (α) ≥ 23/6, or N (α) ≤ 5. Proof. Assume that M (α) < 23/6. Write α = γ + δζ4 . If γ and δ are both non-zero, then we may assume that 17/6 > M (γ ) ≥ M (δ) ≥ 1. Suppose that N (δ) ≥ 2. Then M (δ) ≥ 5/3, and hence M (γ ) ≤ 13/6 < 5/2, from which we deduce from Lemma 7.0.8 that N (γ ) ≤ 3, and hence N (α) ≤ 5. Suppose that N (δ) = 1. Since M (γ ) ≤ 17/6, we see that N (γ ) ≤ 4 and N (α) ≤ 5. Thus we may assume that one of γ or δ is zero, and hence, up to a root of unity, α ∈ Q(ζ21 ). The result follows by Lemma 7.0.5.
Corollary 7.0.10. Suppose that√ β ∈ Q(ζ √ 84 ) is real. Then either β ≥ 76/33, N (β) ≤ 2, or β is either a conjugate of 21 ( 3 + 7) or 1 + 2 cos(2π/7). Proof. The result is an immediate consequence of Lemma 7.0.9, combined with Corollary 4.2.12 and Lemma 5.1.1.
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8. Final Reductions In this section, we complete the proof of Theorem 1.0.5 by proving the following. Theorem 8.0.1. If β is a real cyclotomic integer such √ that β √∈ Q(ζ √420 ), N (β) ≥ 3, and β < 76/33, then either β ∈ Q(ζ84 ), or β = 5 or (1 + 5)/ 2. The technique used in this section is to apply the style of arguments from Cassels “first case” which we used in Sect. 6 applied to the prime 5. The arguments are much more detailed than those in Sect. 6 and we exploit our understanding of small numbers in Q(ζ84 ). As in Sect. 6 we will use ζ to denote an arbitrary p th root of unity, and in this section p = 5. Recall that, on the other hand, ζ5 denotes the particular 5th root of unity e2πi/5 . Note that if N (β) ≤ 5, the result follows from Corollary 4.2.12. We consider various cases in turn. 8.1. The case when X = 1 and p = 5. The same proof in §6 holds verbatim. 8.2. The case when X = 2 and p = 5. Since p = 5, we √ may assume by Lemma 6.0.1 that λ = 0, and hence β = ζ α + ζ −1 α. Suppose that α ≥ 3. Then, as in §6, we deduce that √ β ≥ 2 α cos(π/5) ≥ 2 3 cos(π/5) = 2.802517 . . . > 2.303030 . . . = 76/33. It follows immediately from Lemma 6 of Cassels [7] that if α < N (α) ≤ 2, or α is a root of unity times one of √ 1 1 + −7 , 2
√ 3, then either
√ 1 √ −3 + 5 . 2
If N (α) ≤ 2, then N (β) ≤ 4 and we are done. Suppose that, up to a root of unity, α is one of the two exceptional cases. Since α√∈ Q(ζ84 ), only the first possibility may occur. Writing α as a root of unity times (1 + −7)/2 and enumerating all possibilities, the smallest possible element thus obtained is √ √ √ 7 + √−1 √ √ 1 7 − −1 −2 2 · ζ5 + · ζ5 = 13 + 3 5 + 14(5 + 5) 2 2 2 = 2.728243 . . . > 76/33. 8.3. The case when X = 3, p = 5, and λ = 0. We have that β = ζ α + γ + ζ −1 α. From Eq. 2, we deduce that 4M (β) = 2M (α) + 2M (α) + 2M (γ ) + M (α − α) + M (α − γ ) + M (α − γ ) = 4M (α) + 2M (γ ) + 2M (α − γ ) + M (α − α). We consider various subcases.
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8.3.1. X = 3, p = 5, λ = 0, and α = γ . We deduce that α is real, and hence β = α(ζ + 1 + ζ −1 ). It follows that β = α · 1 + ζ + ζ −1 = 2 cos(π/5) α > 76/33 if α ≥ 2. Thus α = 2 cos(π/n) √ for some √ n|84, and we quickly determine that the only β in the range [2, 76/33] is ( 5 + 1)/ 2. 8.3.2. X = 3, p = 5, λ = 0, α = γ , N (γ ) ≤ 2, N (α) ≥ 3, and α is not real. Since α is not real, M (α − α) ≥ 1. Since N (α) ≥ 3, if N (γ ) = 1 then N (α − γ ) ≥ 2, whereas if N (α − γ ) = 1 then N (γ ) ≥ 2. Thus 5 4M (β) ≥ 4M (α) + 2 + 1 + 1, 3 and hence M (α) < 9/4. It follows from Lemma 7.0.7 (and the fact that α ∈ Q(ζ84 )) that i (1 + ζ j + ζ k ). Moreover, we may assume that either γ = 1 or γ = ζ l + ζ −l for α = ζ84 7 7 84 84 i (1 + ζ j + ζ k ) (without the assumption some l. Enumerating all possibilities with α = ζ84 7 7 that α is not real), we find that the smallest largest conjugate is: 2 cos(π/5)(1 + 2 cos(2π/7)) − 1 = 2.635689 . . . > 76/33. 8.3.3. X = 3, p = 5, λ = 0, α = γ , N (γ ) ≤ 2, N (α) ≥ 3, and α is real. Suppose that N (γ ) and N (α − γ ) are both at least two. It follows from Lemma 7.0.7 that i (1 + ζ j + ζ k ), which was considered above. Thus, we may assume that at least α = ζ84 7 7 one of N (γ ) or N (α − γ ) equal to one. We show that N (α) ≤ 4. If N (γ ) = 1, and N (α) ≥ 5, then N (α − γ ) ≥ 4, and thus M (α) and M (α − γ ) are ≥ 8/3 by Lemma 7.0.8. Yet then M (β) ≥ 8/3 + (8/3 + 1)/2 = 9/2 > 23/6. Conversely, if N (α − γ ) = 1, then by assumption, N (γ ) ≤ 2, and so N (α) ≤ 3. It follows by Lemma 4.1.3 that we assume that α is one of the following forms, up to sign: (1) (2) (3) (4)
i + ζ −i , 1 + ζ84 84 i + ζ −i + ζ j + ζ − j , ζ84 84 84 84 −9 −7 3 + ζ 15 , + ζ84 + ζ84 ζ84 84 −9 −7 3 + ζ 27 , + ζ84 + ζ84 ζ84 84
k + ζ −k . (Here we are using the fact that α ∈ Q(ζ ) whereas we may assume that γ = ζ84 84 84 to eliminate some of the other exceptional possibilities in Lemma 4.1.3.) In √cases 3 and 4 every β has a conjugate of absolute value at least 3. In the first two cases, 5 occurs as a (degenerate) possibility for β. The second smallest largest conjugate is also degenerate, and occurs with α = 2 and γ = 1, where β = 2+2 cos(2π/5) = 2.618033 . . . > 76/33.
8.3.4. X = 3, p = 5, λ = 0, α = γ , N (γ ) ≤ 2, and N (α) ≤ 2. We may let i + ζ j and γ = ζ k + ζ −k . The smallest such largest conjugate (besides a α = ζ84 84 84 √84 degenerate 5) is 4 cos(π/5) cos(3π/7) + 2 cos(π/7) = 2.522030 . . . > 2.303030 . . . = 76/33.
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8.3.5. X = 3, p =√5, λ √ = 0, and N (γ ) ≥ 3. By Corollary 7.0.10, we may assume that either γ = 21 ( 3 + 7), 1 + 2 cos(2π/7), or γ = γ ≥ 76/33. In the latter case, −1 we choose √ a√conjugate of ζ such that ζ α + ζ α > 0, and then β > γ > 76/33. Since 1 M ( 2 ( 3 + 7)) = 5/2 and M (1 + 2 cos(2π/7)) = 2, we may deduce that M (γ ) ≥ 2. Thus 4M (β) ≤ 4M (α) + 4 + 2M (α − γ ) + M (α − α). The case γ = α has already been considered. Thus M (α − γ ) ≥ 1, and hence, since M (β) < 23/6, we deduce that M (α) < 7/3. By Lemma √ 7.0.8, √ it follows that N (α) ≤ 3. Enumerating over all α with N (α) ≤ 3 and γ = 21 ( 3 + 7) or 1 + 2 cos(2π/7), all the smallest conjugates (with α = 0) are at least 3, except for 1 + 2 cos(2π/7) + 2 cos(2π/5) = 2.865013 . . . > 76/33. 8.4. The case when X = 3, p = 5, and λ = 0. It follows, choosing ζ appropriately, that β = α + λ(ζ + ζ 2 ), where, as usual, α − α = λ. We do a brute force computation for all α with N (α) ≤ 3. i + ζ j + ζ k , where i is a divisor of Note that if N (α) = 3, we may assume that α = ζ84 84 84 84. The smallest resulting largest conjugate that arises is −7 7 7 ζ84 + (ζ84 − ζ84 )(ζ 3 + ζ 4 ) = 2 cos(π/30) + 2 cos(13π/30) = 2.404867 . . . ≥ 2.303030 . . . = 76/33.
We note that 4M (β) = (5 − 3)(M (α) + 2M (λ)) + 2M (α − λ). Since α − λ = α, we may write this as M (β) = M (α) + M (λ). Since λ = 0, it follows that M (β) < 17/6. We deduce by Lemma 7.0.8 that either N (α) ≤ 3, or α is one of three specific forms given in that lemma, that is, we may assume that α is, up to a root of unity, one of the following: j
n (ζ i + ζ + ζ k − ζ ), where (i, j, k) are distinct and non-zero modulo 7. (1) α = ζ84 3 7 7 7 j n i k (2) α = ζ84 (1 + ζ7 − (ζ7 + ζ7 )ζ3 ), where (i, j, k) are distinct and non-zero modulo 7. n (ζ i + ζ j + (ζ j + ζ k )ζ ), where (i, j, k) are distinct modulo 7. (3) α = ζ84 7 7 7 7 3
We compute in all cases that the smallest α + (α − α)(ζ + ζ 2 ) which occur are all ≥ 3.5, or α real and λ = 0.
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8.5. The case when X = 4, p = 5, and λ = 0. Since X = 4, by Lemma 6.4.1, we may assume that all the αi are distinct. We are assuming that λ = 0. Then α − α = λ. Write β = α + α1 ζ + α2 ζ 2 + α3 ζ 3 . Then α1 = λ, α2 − α3 = λ. Hence β = α + (α − α)ζ + (γ + α − α)ζ 2 + γ ζ 3 . There is some symmetry in this expression. If we let γ = θ + α − α, then γ + α − α = θ. This sends the pair (α − γ , γ + α − α) → (α − θ , θ ). It follows that the two terms γ and θ can be interchanged in various arguments. We compute that 4M (β) = M (α) + M (α − α) + M (γ + α − α) + M (γ ) + M (α) +M (α − γ ) + M (α − γ ) + M (γ ) + M (γ + α − α) + M (α − α) = 2M (α) + 2M (γ ) + 2M (α − α) + 2M (α − γ ) + 2M (γ + α − α). If α = γ then not every term is distinct, which is a contradiction, and hence all the five terms in the sum above are non-zero. Lemma 8.5.1. At least one of N (α) and N (γ ) is ≥ 3. Proof. We compute all numbers such that N (α) ≤ 2 or N (γ ) ≤ 2. We carry out the i + ζ j and γ = ζ k + ζ l . Then we may calculation as follows. Suppose that α = ζ84 84 84 84 assume that l ≥ k, and that either: (1) (2) (3) (4) (5) (6) (7) (8)
i i i i i i i i
= 1, = 3 and 3| j, = 4 and 2| j, = 7 and 7| j, = 12 and 6| j, = 21 and 21| j, = 28 and 14| j. = 84 and 42| j.
We remark that this computation also covers the cases where N (α) = 1 or N (γ ) = √1, k = ζ k−14 + ζ k+14 . The smallest largest conjugate which occurs is since ζ84 5, 84 84 which occurs in case 7, and the second smallest largest conjugate is 2 cos(π/30) + 2 cos(13π/30), in case 4. Thus we have shown that at least one of N (α) or N (γ ) is ≥ 3. By symmetry, the same argument also proves that at least one of N (α) or N (θ ) is ≥ 3.
Lemma 8.5.2. Either at least three of the terms M (α), M (γ ), M (α − α), M (α − γ ) and M (γ + α − α) above are roots of unity, or at least two terms are roots of unity and at least two other terms are the sum of at most two roots of unity.
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Proof. If there is at most one root of unity, then, by Lemma 7.0.8, M (β) ≥ 1/2(5/3 · 4 + 1) = 23/6. If there are only two roots of unity, and only one other term which can be expressed as the sum of exactly two roots of unity, then M (β) ≥ 1/2(2 · 2 + 5/3 + 1 + 1) = 23/6.
We now consider possible pairs of terms which are roots of unity. (1) α and γ : The result follows from Lemma 8.5.1. √ 21 , the former is (2) α and α − α: The latter is, up to a sign that we fix, −1 = ζ84 7 therefore, up to conjugation, ζ84 . By Lemma 8.5.2, either one of the other terms is a root of unity, or at least two terms are the sum of at most two roots of unity. If γ is a root of unity, we reduce immediately to case 1. If θ = γ + α − α is a root of unity, we also reduce to case 1, by symmetry. If α − γ is a root of unity, then N (γ ) ≤ 2. On the other hand, if at least two terms are the sum of at most two roots of unity, then either N (θ ) or N (γ ) is ≤ 2, and by symmetry, we may assume that N (γ ) ≤ 2, and we are done by Lemma 8.5.1. (3) α and α − γ : We deduce immediately that N (γ ) ≤ 2, and hence, we are done by Lemma 8.5.1. (4) α and θ = γ + α − α: This reduces to case 1 by symmetry. √ 21 . By (5) γ and α − α: The latter, after changing the sign of β, is −1 = ζ84 Lemma 8.5.2, either one of the other terms is a root of unity, or at least two terms are the sum of at most two roots of unity. Note that θ = γ + α − α is equal 21 . Suppose there is another root of unity. We consider various subcases: to γ + ζ84 21 = 0 we (a) θ is a root of unity: From the three term vanishing sum θ − γ − ζ84 49 77 deduce that γ = ζ84 or ζ84 . After conjugating we may assume it is the first. Then 21 35 2 49 3 β = α + ζ84 ζ + ζ84 ζ + ζ84 ζ .
Now 49 N (β) = 3/2 + M (α)/2 + M (α − ζ84 )/2. 49 ) ≤ 23/10. Since 23/10 < 5/2, it Either M (α) ≤ 23/10 or M (α − ζ84 49 ) ≤ 3. follows from Lemma 7.0.8 that either N (α) ≤ 3 or N (α − ζ84 Enumerating over all α with N (α) = 3, we find that the smallest value of the expression above is 42 49 21 3 35 49 4 |(1+ζ84 +ζ84 )+ζ84 ζ5 +ζ84 ζ5 +ζ84 ζ5 | = 1+4 cos2 (π/15) = 2.1970641 . . . ,
however, the β occurring here is not real, since we did not impose the condi21 . The second smallest value that tion (in our computation) that α − α = ζ84 occurs is 35 42 21 4 35 3 49 2 |(1+ζ84 +ζ84 )+ζ84 ζ5 +ζ84 ζ5 +ζ84 ζ5 | = 1 + 4 cos2 (π/30) = 2.226273 . . .
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which is also not real. The third smallest value that occurs is 2.574706 . . . > 49 ) = 3, the smallest value thus obtained 2.303030 · · · = 76/33. If N (α − ζ84 is 49 28 56 21 3 35 49 4 + 1 + ζ84 + ζ84 ) + ζ84 ζ5 + ζ84 ζ5 + ζ84 ζ5 | = 1 + 4 cos2 (π/15), |(ζ84 the second smallest value is, as above, 1 + 4 cos2 (π/30), and the third smallest value is (once more) 2.574706 · · · > 2.303030 · · · = 76/33. (b) α is a root of unity: Since α and γ are roots of unity, we are reduced to case 1. (c) α − γ is a root of unity: If γ and α − γ are roots of unity, then N (α) ≤ 2, and we are done by Lemma 8.5.1. Hence we may assume that all other terms are not roots of unity, and hence there are at least two terms which are sums of at most 2 roots of unity. We consider various possibilities: (a) Suppose that N (α) = 2. Then we are done by Lemma 8.5.1. (b) We may assume that γ − α and θ are both at most the sum of two roots of i and α = ζ i + ζ j + ζ k with i ≤ j ≤ k. After conjugatunity. Write γ = ζ84 84 84 84 ing, we may assume that i divides 84. Enumerating all the possibilities, we find that the smallest number of this form is 2 cos(π/30) + 2 cos(13π/30) = 2.404867 . . . > 2.303030 . . . = 76/33. (6) γ and α − γ : Since N (α) ≤ 2 and N (γ ) = 1, we are done by Lemma 8.5.1. (7) γ and θ := γ + (α − α): If N (α − α) = 1 then we are back in case 5. If N (α) = 1 we are back in case 1. If N (α − γ ) = 1 we are back in case 6. Thus, by Lemma 8.5.2 it follows that at least one of N (α) or N (α − γ ) is equal to 2. In the first case, we are done by Lemma 8.5.1. In the second case, we may let i with i|84 and α = ζ i + ζ j + ζ k , and we are reduced to the computation γ = ζ84 84 84 84 in the final section of part 5. (8) α − α and α − γ : If either N (α) = 1 or N (γ ) = 1, then the other is the sum of at most two roots of unity, and we are done by Lemma 8.5.1. If θ is a root of unity, then by symmetry we can reduce to case 5. Thus, by Lemma 8.5.2, we may assume that at least two of γ , α and θ are the sums of at most two roots of unity. By Lemma 8.5.1, we are done unless N (γ ) = N (θ ) = 2, and N (α) ≥ 3. Since α − γ is a root of unity, it must be the case √ that N (α) = 3. Since α − α is a purely imaginary root of unity, it must be ± −1. Changing the sign of β if 21 . It follows that neccessary, we may assume that α − α = ζ84 7 7 ) = 0, ) − (α − ζ84 (α − ζ84 7 is real. Since 2 ≤ M (α − ζ 7 ) ≤ 4, and α lies in Q(ζ ), it and hence α − ζ84 84 84 7 is of the form: follows that α − ζ84 i + ζ −i , (a) ζ84 84 i + ζ −i + 1, (b) ζ84 84 i + ζ −i − 1, (c) ζ84 84 i + ζ −i + ζ j + ζ − j , (d) ζ84 84 84 84 −9 −7 3 + ζ 15 or ζ −9 + ζ −7 + ζ 3 + ζ 27 . + ζ84 + ζ84 (e) Galois conjugate to ζ84 84 84 84 84 84 j In all five cases, we let γ = ζ84 and enumerate all possibilities. The smallest largest conjugate is a relatively gargantuan 2.989043 . . . .
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(9) α − α and θ : By symmetry, we are reduced to case 5. (10) α − γ and θ : By symmetry, we are reduced to case 6. 8.6. The case when X = 4, p = 5, and λ = 0. We have β = ζ α + ζ −1 α + ζ 2 γ + ζ −2 γ . Note that every term is distinct. We have 4M (β) = 2M (α)+2M (γ )+2M (α − γ ) + 2M (α − γ )+M (α − α) + M (γ − γ ). Lemma 8.6.1. At least one of N (α) or N (γ ) is at least 3. Proof. We compute all numbers such that N (α) ≤ 2 or N (γ ) ≤ 2. We carry out the i + ζ j and γ = ζ k + ζ l . Then we may calculation as follows. Suppose that α = ζ84 84 84 84 assume that l ≥ k, and that either: (1) (2) (3) (4) (5) (6) (7) (8)
i i i i i i i i
= 1, = 3 and 3| j, = 4 and 2| j, = 7 and 7| j, = 12 and 6| j, = 21 and 21| j, = 28 and 14| j. = 84 and 42| j.
We remark that this computation also covers the cases where N (α) = 1 or√N (γ ) = 1, k = ζ k−14 + ζ k+14 . We find that the smallest largest conjugates are 5, which is since ζ84 84 84 on our list, and 2 cos(π/30) + 2 cos(13π/30) ≥ 76/33.
We note there is a symmetry between (α, γ ) and (α, γ ). Without loss of generality, we assume that N (α) ≥ N (γ ), and that N (α) ≥ 3. Lemma 8.6.2. At least one of the following holds: (1) At least two of {γ , α − γ , α − γ } are roots of unity. (2) Both α − α and γ − γ are roots of unity, and every element in {γ , α − γ , α − γ } is a sum of at most two roots of unity. Proof. Note that N (α) ≥ 3, and so M (α) ≥ 2. Suppose that α − α and γ − γ are not both roots of unity, and at most one of {γ , α − γ , α − γ } is a root of unity. then M (β) ≥ (1 + 5/3 + 5/3 + 2)/2 + (1 + 5/3)/4 = 23/6. Conversely, if α − α and γ − γ are both roots of unity, at most one of {γ , α − γ , α − γ } is a root of unity, and at most two of {γ , α − γ , α − γ } are the sum of 2 roots of unity, then M (β) ≥ (1 + 5/3 + 2 + 2)/2 + (1 + 1)/4 = 23/6.
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8.6.3. X = 4, p = 5, λ = 0, and two of {γ , α − γ , α − γ } are roots of unity. If γ is a root of unity, then so is γ . Since at least one of α − γ and α − γ is also a root of unity, we deduce that N (α) ≤ 2, and we are done by Lemma 8.6.1. Thus we may assume that N (α − γ ) = N (α − γ ) = 1. Recall that by Lemma 6.4.1, we may assume that α and i and α − γ = ζ j . We deduce γ are distinct from their conjugates. Write α − γ = ζ84 84 −j that α − γ = ζ84 . Thus j
i α − α = (α − γ ) − (α − γ ) = ζ84 − ζ84
and −j
i γ − γ = (α − γ ) − (α − γ ) = ζ84 − ζ84 j
i − ζ is purely imaginary, it follows that are purely imaginary. Since ζ84 84 −j
−i i ζ84 − ζ84 + ζ84 − ζ84 = 0. j
This is a vanishing sum of length four, so it must be comprised of two subsums of i = ζ j then α − α = 0, which is a contradiction. If ζ i = ζ − j , then length 2. If ζ84 84 84 84 i = −ζ −i and ζ j = −ζ − j . It follows γ − γ = 0, which is also a contradiction. Thus ζ84 84 84 84 √ √ i = ± −1 and ζ j = ± −1. Yet, for each of these possibilities, it is the case that ζ84 84 i is equal to ζ j or ζ − j , and hence either α = α or γ = γ , a contradiction. that ζ84 84 84 8.6.4. X = 4, p = 5, λ = 0, at most one of {γ , α −γ , α −γ } is a root of unity. It follows from Lemma 8.6.2 that either N (γ ) + N (α − γ ) ≤ 3 or N (γ ) + N (α − γ ) ≤ 3. i , and α = ζ i + ζ j + ζ k and enumerate, or α = If N (γ ) = 1, then we let γ = ζ84 84 84 84 j −i k and enumerate. If N (γ ) = 2, we let γ = ζ i +ζ j , and α = ζ i +ζ j +ζ k , ζ84 +ζ84 +ζ84 84 84 84 84 84 −j −i k . Enumerating over all such possibilities, we find that the smallest or ζ84 + ζ84 + ζ84 √ largest conjugates that arise are 5 and 2 cos(π/30) + 2 cos(13π/30). 8.7. The case when X = 5 and p = 5. In this case, by Lemma 6.4.1, we can reduce to the case that X < 5. This completes the proof of Theorem 1.0.5 9. M (β) is Discrete in an Interval Beyond 2 We have seen that the values of β for real cyclotomic integers are discrete in [0, 76/33] away from a limit point (from below) at 2. In this section, we show (now for all cyclotomic integers) that M (β) is discrete in [0, 9/4], away from a limit point (from both sides) at 2. This is an easy consequence of the following theorem. Theorem 9.0.1. Let β be a cyclotomic integer, and suppose that M (β) < 9/4. Then, up to a root of unity, either: (1) (2) (3) (4)
β β β β
= 0 or β = 1. is a sum of two roots of unity. j = 1 + ζ7i + ζ7 , where (i, j) are distinct and non-zero. j = ζ3±1 − (ζ5i + ζ5 ), where (i, j) are distinct and non-zero.
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Proof. Our proof follows the same lines as the arguments in Sects. 5.2–8, although it is much easier. Assume that M (β) < 9/4. Suppose that β ∈ Q(ζ N ), where N is the conductor of Q(β), and suppose that β is minimal, that is,no root of unity times β lies in a field of smaller conductor. Let p m N , and write β = αi ζ i , where ζ is a p mth root of unity and the αi ∈ Q(ζ M ), for N = pM. If p 2 |N , then β = M (αi ). If this sum consists of at least three non-zero terms, then M (β) ≥ 3. If this sum consists of two nonzero terms, and at least one of the αi is not a root of unity, then M (β) ≥ 1 + 3/2 > 9/4. Hence, either β is the sum of two roots of unity, or there is only one non-zero term, contradicting minimality. Thus we may suppose that N is squarefree. Suppose that p|N for p > 7. Since M (β) = 9/4 <
11 + 1 , 4
by Lemma 1 of [7] we deduce that one can write β as a sum of X ≤ ( p − 1)/2 non-zero terms. Suppose that X ≥ 3. It follows from Eq. 2 that ( p − 1)M (β) ≥ X ( p − X ) ≥ 3( p − 3), from which it follows that M (β) ≥ 12/5 > 9/4. Thus we may assume that X = 2, and β = α + ζ γ . If α and γ are roots of unity, then β is a sum of two roots of unity. If at least one of α or γ is not a root of unity, then ( p − 1)M (β) ≥ ( p − 2)(1 + 3/2), and hence M (β) ≥ 9/4, a contradiction. Thus, we may that N divides 105. assume Now let us consider β ∈ Q(ζ105 ). Write β = αi ζ i , and suppose there are X non-zero terms. We consider the various possible values of X , as in §8. (1) If X = 1, then β ∈ Q(ζ21 ). Hence the result follows from Corollary 7.0.4. (2) If X = 2, then β = α + γ ζ , and 4M (β) = 3M (α) + 3M (γ ) + M (α − γ ). If α and γ are roots of unity, then β is a sum of two roots of unity. If α = γ is not a root of unity, then M (β) ≥ 9/4. If α and γ are distinct, and at least one is not a root of unity, then 4M (β) ≥ 3(1 + 5/3) + 1, and it follows easily that M (β) ≥ 9/4. (3) If X = 3, β = αi ζ i , then we may assume that not all the αi are the same, since otherwise we may subtract ζ i α from β and assume that X = 2. Thus, at least two of the αi − α j are non-zero, and hence 4M (β) ≥ 2 M (αi ) + 2. If at least one of the αi is not a root of unity, then M (β) ≥ 7/3 > 9/4. Thus, we may assume that all the αi are roots of unity. Moreover, at least two of the αi must coincide, since otherwise 4M (β) ≥ 6 + 3 and thus M (β) ≥ 9/4. We may therefore assume, after multiplying by a root of unity, that β = α + ζi + ζ j,
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where (i, j) are distinct and non-zero, and α is a root of unity. Since 4M (β) = 6 + 2M (α − 1), we find that M (β) ≥ 9/4 unless α −1 is also a root of unity. If α −1 and α are both roots of unity then α = −ζ3±1 . Hence, up to a root of unity, β = ζ3±1 − (ζ i + ζ j ). (4) If X = 4, then we may assume that all the αi are distinct. Then 4M (β) ≥ M (αi − α j ) ≥ 10, and M (β) ≥ 5/2 > 9/4. (5) If X = 5, we may subtract a multiple of 1 + ζ + ζ 2 + ζ 3 + ζ 4 = 0 to reduce to a previous case.
Remark 9.0.2. The exceptional values (with M (α) = 2) occurring in Theorem 9.0.1 were already noticed by Cassels [7, Lemma 3]. The discreteness of M (β) away from 2 follows from the fact that, given an n th root of unity ζ , we have μ(n) M (1 + ζ ) = 2 1 + , ϕ(n) where μ(n) is the Möbius μ-function and ϕ(n) is Euler’s totient function — as n increases this converges to 2. We deduce the following: Corollary 9.0.3. Let β be a real cyclotomic integer, and suppose that M (β) < 9/4. Then, up to sign, either: (1) β is conjugate to 2 cos(2π/n) for some integer n. (2) β is conjugate to 1 + 2 cos(2π/7). 17 = 2 cos(π/30) + 2 cos(13π/30). (3) β is conjugate to η := ζ12 + ζ20 + ζ20 Proof. We use the fact (Lemma 4.1.3) that if β is totally real and N (β) ≤ 3, then, up to sign, β = 0, 1, η, ζ i + ζ −i , or 1 + ζ i + ζ −i (the sign is unnecessary in the first, third, or fourth cases).
9.1. A general sparseness result on the set of values of M (β) for β a cyclotomic integer. Theorem 9.1.1. Let L ⊂ R denote the closure of the set of real numbers of the form M (β) for cyclotomic integers β. Then L is a closed subset of Q. Proof. If U ⊂ R is a set, let U n for any positive integer n denote the set of sums of at most n elements of U . If U is closed, then so is U n . Let L (d) ⊂ L denote L ∩ [0, d]. Since L (1) = {0, 1}, it suffices to show that there exists an integer m (depending on d) such that L (d + 1/2) ⊂ L (d)m ∪ Q, since then the result follows by induction. Let γ denote a point in L (d + 1/2). There exists a sequence βk of cyclotomic integers with M (βk ) = γk such that lim γk = γ . We → note the following theorem of Loxton [25, §6.1, p. 81]:
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Theorem 9.1.2 (Loxton). There exists a continuous increasing unbounded function g(t) such that M (β) ≥ g(N (β)). In particular, any bound on M (β) yields an upper bound on N (β). Since M (βk ) = γk converges to γ ≤ d + 1/2, it follows that γk is bounded above by d + 1 for sufficiently large k. Without loss of generality, we may assume this bound holds for all k. It follows from Loxton’s theorem that the βk can be written as the sum of at most m = m(d) roots of unity for some m. Let Nk denote the conductor of βk . Recall that M (α) · [Q(α) : Q] ∈ Z. If the Nk are bounded, then the fields Q(βk ) are of bounded degree, and hence the M (βk ) = γk have bounded denominators, and M (β) ∈ Q. Hence, we may assume that the conductors Nk grow without bound. Let pkn k denote the largest prime power divisor of Nk . For each k, we may write βk = αi ζ i , where the sum runs over a set of cardinality m (allowing some of the αi to be zero). Assuming that βk is minimal (which we may do without changing the value of M (βk )) we may assume that there are at least two non-zero αi . We consider two cases: (1) Suppose that n k > 1 for infinitely many k. For such k, we have M (βk ) =
M (αi ).
Since at least two of the αi are non-zero, M (αi ) ≤ γk − 1 < d. Thus M (αi ) ∈ L (d), and M (βk ) ∈ L (d)m . Since the latter is closed, we deduce that M (β) ∈ L (d)m . (2) Suppose that n k = 1 for infinitely many k. We deduce that ( pk − 1)M (βk ) = ( pk − m)
M (αi ) +
M (αi − α j ).
Since at least two of the αi are non-zero, we deduce that pk − 1 · γk − 1 < d, M (αi ) ≤ pk − m the last inequality holding for sufficiently large k (equivalently, pk ). Thus M (αi ) ≤ d for sufficiently large k. From the AM-GM inequality, we deduce that m M (αi ) + 2 M (α j ) ≤ 4d . M (αi − α j ) ≤ 2 2 As k increases, therefore, the contribution of this term to M (βk ) converges to zero, and hence M (β) = lim M (βi ) = lim M (αi ), →
→
and thus γ = M (β) lies in the closure of L (d)m . Since L (d) is closed, γ ∈ L (d)m .
Remark 9.1.3. Since closed subsets of Q are very far from being dense, we see that this result is in stark contrast to the analogous set constructed out of M (β) for totally real integers β, which is dense in [2, ∞).
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10. Galois Groups of Graphs Let be a connected graph with || vertices. Fix a vertex v of , and let n denote the sequence of graphs obtained by adding a 2-valent tree of length n − || to at v. Let Mn denote the adjacency matrix of n , and let Pn (x) denote the characteristic polynomial of Mn . By construction, n has n vertices, and thus the degree of Pn (x) is n. The main result of this section is the following: Theorem 10.0.1. For any , there exists an effective constant N such that for all n ≥ N , either: (1) All the eigenvalues of Mn are of the form ζ + ζ −1 for some root of unity ζ , and the graphs n are the Dynkin diagrams An or Dn . (2) There exists at least one eigenvalue λ of Mn of multiplicity one such that Q(λ2 ) is not abelian. Remark 10.0.2. We shall also prove a stronger version of this result which only looks at the largest eigenvalue (Theorem 11.0.1). We include this result because, although Theorem 11.0.1 is also (in principle) effective, the bound on n arising in Theorem 10.0.1 is easily computed, and all our intended applications satisfy the conditions of Theorem 10.0.1. Corollary 10.0.3. For any , there exists an effective constant N such that for all n ≥ N , either: (1) n is the Dynkin diagram An or Dn . (2) n is not the principal graph of a subfactor. Proof. This is an immediate consequence of Theorem 10.0.1 and Lemma 3.0.7.
10.1. Adjacency matrices. We begin by recalling some basic facts about the eigenvalues of Mn . Lemma 10.1.1. Let x = t + t −1 , and write Pn (x) = Fn (t) ∈ Z[t, t −1 ]. (1) The matrix Mn is symmetric and the roots of Pn (x) are all real. (2) The polynomials Pn satisfy the recurrence: Pn (x) = x Pn−1 (x) − Pn−2 (x). (3) There is a fixed Laurent polynomial A(t) ∈ Z[t, t −1 ] such that: 1 = t n · A(t) − t −n · A(t −1 ). Fn (t) t − t We are particularly interested in the roots of Pn (x) of absolute value larger than 2, or, equivalently, the real roots of Fn (t) of absolute value larger than 1. The following facts will be useful to note. Lemma 10.1.2. Denote the roots of Pn (x) by λi for i = 1 to n. (1) If the roots of Pn−1 (x) are μi for i = 1 to n − 1, then, with the natural ordering of the roots, λ1 ≤ μ1 ≤ λ2 ≤ μ2 · · · ≤ μn−1 ≤ λn .
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(2) The number of roots of Pn (x) of absolute value larger than 2 are bounded. (3) The largest real root of Pn (x) is bounded. (4) For sufficiently large n, the real roots of Pn (x) of absolute value larger than 2 are bounded uniformly away from 2. Proof. The first claim is the interlacing theorem; see ([14], Theorem 9.1.1). By Descartes’ rule of signs, the polynomial Fn (t) has a bounded number of real roots, which implies the second claim. The largest real root of Fn (t) converges to the largest real root ρ∞ of A(t) (compare Lemma 12 of [26]) and hence the largest real root of Pn (x) −1 . The final claim follows immediately from the first converges to λ∞ = ρ∞ + ρ∞ two.
We use the letter λ to refer to a root of Pn (x), and the letter ρ to refer to the corresponding roots of Fn (t), where λ = ρ + ρ −1 . Lemma 10.1.3. There exists a polynomial B(t) such that for n larger than some effectively computable constant, every repeated root of Fn (t) on the unit circle is a root of B(t). Proof. The polynomial A(t) is monic. In particular, if A(t) has a root on the unit circle, then A(t) has a factor B(t) which is a reciprocal polynomial. It follows that we can write
1 t n · Fn (t) t − = B(t) t 2n · C(t) − C(t −1 ) , t where A(t) = B(t)C(t) and C(t) has no roots on the unit circle. Suppose that Fn (t) has a repeated root ρ on the unit circle. Then either ρ is a root of B(t), or it is a root of t 2n C(t) − C(t −1 ). Yet the absolute value of the derivative of this expression is, by the triangle inequality, greater than 2n|C(t)| − |C (t)| − |C (t −1 )|. Since C(t) has no roots on the unit circle, for all n larger than some effectively computable constant this expression is positive.
Lemma 10.1.4. For all sufficiently large n, there exists a constant K () such that (λ2 − 2)2 = 2n + K (). Proof. Clearly (λ2 − 2)2 = ρ 4 + 2 + ρ −4 . Since there isa pair of inverse roots of Fn (t) corresponding to every root λ of Pn (x), it follows that (λ2 − 2)2 = 2n + ρ 4 . The sum of the 4th powers of the roots of Fn (t) depends only on the first four coefficients of Fn (t), which is clearly independent of n, when n is sufficiently large compared to deg(A).
Recall that η := 2 cos(π/30) + 2 cos(13π/30) has degree 8 over Q. 8 Lemma 10.1.5. The polynomials 1,2,4 (x 2 − 3 − 2 cos(2π k/7)) and i=1 (x 2 − 2 − σi η) divide Pn (x) a uniformly bounded and effectively computable number of times. Proof. Since the polynomials in question have at least one real root larger than 2, the number of factors of Pn (x) of this form is clearly at most the number of real roots of Pn (x) of size larger than 2.
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Let us now complete the proof of Theorem 10.0.1. By Lemma 10.1.3, we deduce that for n sufficiently large, there is a uniformly bounded (with multiplicity) number of roots which have multiplicity ≥ 2. Moreover, if n is not An or Dn , then the number roots of the form ζ + ζ −1 is also uniformly and effectively bounded, by the main theorem of [15]. Finally, the number of roots λ such that λ2 − 2 = 1 + 2 cos(2π/7) or η is also uniformly bounded. Let R denote the set of roots in any of these three categories. Clearly, we have (λ2 − 2)2 ≤ 2n + K (). λ∈R /
On the other hand, by assumption, each λ2 − 2 with λ ∈ / R is a cyclotomic integer. If λ2 − 2 = ζ + ζ −1 , then λ = ζ 1/2 − ζ −1/2 lies in R. If λ2 − 2 = 1 + 2 cos(2π/7) or λ2 − 2 = η, then λ also lies in R. Thus, by Corollary 9.0.3, M (λ2 − 2) ≥ 9/4 for all λ∈ / R. Hence 9(n − |R|) . (λ2 − 2)2 ≥ 2n + K () ≥ 4 λ∈R /
Combining these two inequalities, we obtain a contradiction whenever n ≥ 4K ()+9|R|, as long as n is big enough for the conclusions of Lemma 10.1.3 and 10.1.4 to hold. Remark 10.1.6. In practice, one can improve the bound on n by noting that the cyclotomic and repeated factors (that one knows explicitly) to the sum 2 factors contribute 2 − 2)2 . (λ − 2)2 , thus enabling one to obtain a smaller bound on ∈R (λ / Remark 10.1.7. Suppose that A(t) has exactly one root of absolute value larger than 1. Then the polynomials Pn (x) have a unique root larger than 2, and Pn (x) factors as a Salem polynomial times a product of cyclotomic polynomials. (A Salem polynomial is an irreducible polynomial with a unique root of absolute value larger than 1.) Similarly, if is bipartite, and A(t) has a pair of roots (equal up to sign) of absolute value larger than 1, then Pn (x) factors into cyclotomic polynomials and a factor S(x 2 ), where S(x) is a Salem polynomial — in particular, in these cases, Pn (x) will never have repeating roots that are not cyclotomic. Remark 10.1.8. In practice, the limiting factor in applying this argument is the bound coming from Gross-Hironaka-McMullen [15] for roots of the form ζ N + ζ N−1 . The argument in [15] proceeds in two steps. First, there is a uniform bound on N . Second, for each fixed N the Pn which have such a root are precisely those in certain classes modulo be A divided by all its cyclotomic factors, let ( A) be the number of nonzero N . Let A The argument in [15] shows that if ζ N + ζ −1 is a root of Pn (x) for some coefficients of A. N n such that ζ N is not a root of An (t), then N divides m p≤2( A) p for some integer m ≤ 4 deg A (this is not the exact statement of [15, Thm 2.1], but the proof is the same). It seems in the cases that we have looked at that there is a much stronger bound on N , and proving an improved bound would substantially increase the effectiveness of our technique. Example 10.1.9. We compute three applications of Theorem 10.0.1. Consider the graphs i,n for i = 1, 2, 3, where the graphs i are given below in Fig. 3. The graphs 1,n and 2,n are the two infinite families which arise in the classification of Haagerup [16]. It was shown by Bisch [6] (using a fusion ring argument) that none of the 2,n are the principal graph of a subfactor. The corresponding result for 1,n and n > 10 was
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proved by Asaeda–Yasuda [3] using number theoretic methods. The family 3,n is one of several families arising in ongoing work of V. Jones, Morrison, √ Peters, Penneys, and Snyder, extending the classification of Haagerup beyond 3 + 3. We compute that K (1 ) = 2,
K (2 ) = 4,
K (3 ) = 8,
where Lemma 10.1.4 applies for n ≥ 8, n ≥ 7, and n ≥ 11 respectively. Similarly, we find that the cyclotomic factors of Pn (x) depend (for n ≥ 11) only on n mod 24, n mod 12, and n mod 24 for i = 1, 2, 3, and have degree at most 9, 6, and 8 respectively. The polynomials A(t) are given as follows: A1 (t) = (t 2 + 1)(t 4 + 1)(t 6 − t 4 − t 2 − 1)t −11 , A2 (t) = (t 2 − t + 1)(t 2 + t + 1)(t 6 − 2t 4 − 1)t −9 , A3 (t) = (t 2 − t + 1)(t 2 + t + 1)(t 10 − 2t 8 − t 6 − t 4 − 1)t −13 .
Fig. 3. The graphs i
In each case, we deduce that the only repeated factors of Fn (t) on the unit circle can occur at roots of unity. In all cases, the graphs i,n are bipartite, and, moreover, the polynomials Ai (t) have a unique pair of roots of absolute value larger than 1. It follows that Pn (x) can be written as the product of cyclotomic factors and a factor S(x 2 ), where S(x) is a Salem polynomial. From this we can directly eliminate the possible occurrence of a root λ of Pn (x) of the form λ2 − 2 = 1 + 2 cos(2π/7) or λ2 − 2 = η whenever the degree of S(x) is greater than 7, or when n ≥ 16. It follows that n,i does not correspond to a subfactor whenever n ≥ N , where N (1 ) = 9 · R(1 ) + 4 · K (1 ) = 9 · 9 + 4 · 2 = 89, N (2 ) = 9 · R(2 ) + 4 · K (2 ) = 9 · 6 + 4 · 4 = 70, N (3 ) = 9 · R(3 ) + 4 · K (3 ) = 9 · 8 + 4 · 8 = 104. We may explicitly enumerate the polynomials for smaller n, and our results are as follows:
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Corollary 10.1.10. The graphs i,n are not the principal graphs of subfactors for all (i, n) with the possible exception of the pairs (i, n) = (1, 7), (1, 8), (1, 10), (1, 14), (2, 6), (2, 7), (2, 8), (2, 9), (2, 11) and (3, 8). In these cases, we observe the following possibilities: √ (1) 1,7 = A7 , and = λ2 = (2 cos(π/8))2 = 2 + 2. (2) 1,8 = E 7 , the extended Dynkin diagram of E 7 , and = λ2 = 4. √ (3) 1,10 corresponds to the Haagerup subfactor [2], and = λ2 = 5+ 2 13 . (4) 1,14 corresponds to the extended Haagerup subfactor [5], and = λ2 = 3 + ζ + ζ −1 + ζ 3 + ζ −3 + ζ 4 + ζ −4 , with ζ 13 = 1. (5) (6) (7) (8) (9) (10)
2,6 = A5 , the extended Dynkin√ diagram of A5 , and = λ2 = 4. = 2,7 , and = λ2 = 3 + √2. = 2,8 , and = λ2 = (5 + √17)/2. = 2,9 , and = λ2 = (7 + 5)/2. = 2,11 , and = λ2 = 2 − ζ 4 − ζ −4 − ζ 6 − ζ −6 for ζ 13 = 1. = 3,8 = 2,8 .
In each of the cases 2,7 , 2,8 = 3,8 , 2,9 , and 2,11 , we may rule out the existence of a corresponding subfactor for each choice of fixed leaf by computing the global dimension and checking that, for some Galois automorphism σ , the ratio σ ()/ is not an algebraic integer [27]. 11. An Extension of Theorem 10.0.1 In this section, we prove the following extension of Theorem 10.0.1. Theorem 11.0.1. For sufficiently large n, either: (1) All the eigenvalues of Mn are of the form ζ + ζ −1 for some root of unity ζ , and the graphs n are the Dynkin diagrams An or Dn . (2) The largest eigenvalue λ of Mn is greater than 2, and the field Q(λ2 ) is not abelian. Remark 11.0.2. The proof of this theorem was found before the proof of Theorem 10.0.1. In our intended applications, all the conditions of Theorem 10.0.1 are met, however, this generalization may still be of interest. Definition 11.0.3. Let m (x) be the polynomial such that if x = t + t −1 , then m (x) = t m + t −m . Remark 11.0.4. The polynomials m (x) are the Chebyshev polynomials, appropriately scaled so that all their roots are contained in the interval [−2, 2]. If m is even, then m (x) is a polynomial in x 2 .
11.1. Heights and algebraic integers. The goal of this section is to show that the fields Q(ρ) for any real root ρ > 1 of Fn (t) have degree asymptotically bounded below by a linear function in n.
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Recall that the Weil height of an algebraic number γ = α/β such that K = Q(γ ) is defined to be 1 h(γ ) := log max{|α|v , |β|v }. [K : Q] v If λ∞ ≤ 2 then every root of Pn (x) has absolute value at most 2, and thus every root ρ of Fn (t) has absolute value 1. Yet then h(ρ) = 0 for all roots ρ of Fn (t). A theorem of Kronecker says that h(γ ) > 0 unless γ is zero or a root of unity. Hence, in this case, we are in the first case of Theorem 11.0.1. The following lemma is well known, and is a consequence of the triangle inequality. Lemma 11.1.1. If φ : P1 → P1 is a homomorphism of finite degree, then h(φ(P)) ≥ deg(φ) · h(P) + C(φ), for some constant C(φ) depending only on φ. Using this, we may deduce the following: Lemma 11.1.2. There exists an explicit constant c depending only on such that for sufficiently large n, and for every root ρ of Fn (t) there is an inequality: h(ρ) ≤
c . n
Proof. Consider the rational map φ : P1 → P1 defined by sending t to φ(ρ) = ρ 2n , we deduce that
A(t −1 ) A(t) .
Since
2n · h(ρ) = h(ρ 2n ) = h(φ(ρ)) ≤ deg(φ) · h(ρ) + C(φ). The lemma follows, taking c = C(φ) and n ≥ deg(φ).
Lemma 11.1.3. There exists a constant a such that if ρ is a root of Fn (t), then either ρ is a root of unity or [Q(ρ) : Q] ≥ a · n for sufficiently large n. Proof. For sufficiently large n, the real roots of absolute value larger than 1 of Fn (t) are bounded away from 1, by Lemma 10.1.2 (4). If ρ is a root of Fn (t) that is not a root of unity, then it has at least one conjugate of absolute value larger than 1, by Kronecker’s theorem. It follows from the definition of height that for sufficiently large n, [Q(ρ) : Q] · h(ρ) ≥ d for some absolute constant d. In light of the previous lemma, this suffices to prove the result with a = d/c.
Note that if λ = ρ + ρ −1 , then [Q(ρ) : Q(λ)] ≤ 2, and so the same result (with a different d) applies to [Q(λ) : Q]. Lemma 11.1.4. Fix an integer m. For sufficiently large n, if λ is a root of Pn (x), then 1 2m (σ λ) ≤ 5, [Q(λ) : Q] where the sum runs over all conjugates of λ.
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Proof. If |x| ≤ 2 then 2m (x) ≤ 4. If λ = ρ + ρ −1 and ρ is a root of unity the result is obvious. Thus we may assume (after conjugation if necessary) that ρ > 1. Suppose that λ has R conjugates of absolute value larger than 2. Each of these roots is bounded by λ∞ , and the number of such roots is also uniformly bounded, by Lemma 10.1.2. Note that 1 2 (λ∞ ) − 4 2m (σ λ) ≤ 4 + R · m . [Q(λ) : Q] [Q(λ) : Q] Since [Q(λ) : Q] becomes arbitrarily large by Lemma 11.1.3, the right-hand side is bounded by 5 for sufficiently large n.
The following result is an immediate consequence of Loxton’s theorem (Theorem 9.1.2) quoted previously: Corollary 11.1.5. If β is a cyclotomic integer such that M (β) ≤ 5, then N (β) is bounded by some absolute constant, which we denote by C. 11.2. Proof of Theorem 11.0.1. If λ∞ ≤ 2 then the first claim follows from [30, Theorem 2]. We may assume that λ∞ > 2. By Lemma 10.1.2 (4), we may assume that for all n, Pn (x) has no roots in the interval (2, α) for some α > 2. Choose an even integer m such that m (α) > C, where C is to be chosen later. By Lemma 11.1.4, we deduce that if n is sufficiently large, then for any root λ of Pn (x), 1 M (m (λ)) = 2m (σ λ) ≤ 5. [Q(λ) : Q] We assume that Q(λ2 ) is abelian for some λ > 2 and derive a contradiction. Since m is even, β = m (λ) ∈ Q(λ2 ), and hence β is cyclotomic. Moreover, M (β) ≤ 5. Choosing C to be as in the above corollary, we deduce that N (β) ≤ C. Since λ > 2, however, λ ≥ α and hence β > C. Yet the sum of C roots of unity has absolute value at most C, by the triangle inequality. This completes the proof of Theorem 11.0.1. Acknowledgements. We would like to thank MathOverflow where this collaboration began (see “Number theoretic spectral properties of random graphs” http://mathoverflow.net/questions/5994/). We would also like to thank Feng Xu for helpful conversations, and Victor Ostrik for writing the Appendix. Frank Calegari was supported by NSF Career Grant DMS-0846285, NSF Grant DMS-0701048, and a Sloan Foundation Fellowship, Scott Morrison was at the Miller Institute for Basic Research at UC Berkeley, and Noah Snyder was supported by an NSF Postdoctoral Fellowship. Open Access This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
Appendix A. A Pseudo-Unitary Fusion Category with an Object of Dimension √ √ 3+ 7 . by Victor Ostrik 2 A.1. The goal of this Appendix is to construct a fusion category V over C with an object √ √ √ √ √ V such that FP(V) = 3+2 7 (notice that since 3+2 7 < 1 + 2, the object V is automatically simple). We do not attempt to classify all fusion categories generated by such an object. The category we construct is pseudo-unitary (i.e. it is endowed with a spherical structure and FP(X ) = dim(X ) for any object X ); moreover all the categories considered in this Appendix are pseudo-unitary as well.
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A.2. Preliminaries. In this section we collect necessary definitions and results. We refer the reader to [11,13] for a general theory of fusion and braided fusion categories. Let C be a pre-modular fusion category, see e.g. [11, Def. 2.29]. Following [23] we will consider commutative associative unital algebras A ∈ C satisfying the following assumptions: (i) dim Hom(1, A) = 1; (ii) the pairing A ⊗ A → 1 defined as a composition of the multiplication A ⊗ A → A and a non-zero morphism A → 1 is non-degenerate and dim(A) = 0; (iii) the balance isomorphism θ A = id A . In [23] the algebras A satisfying these conditions were called “rigid C−algebras with θ A = id A ”; to abbreviate we will call such algebras “C−algebras” here. Given a pre-modular fusion category C and a C−algebra A ∈ C, one considers the category C A of right A−modules. The category C A has a natural structure of spherical fusion category, see [23, Theorem 3.3, Remark 1.19]. It contains a full fusion subcategory C 0A of dyslectic modules, see [23, Def. 1.8]. The category C 0A has a natural structure of pre-modular category. If C is pseudo-unitary the same is true for C A and C 0A . For a braided fusion category C let C op denote the opposite category (C op = C as a fusion category and the braiding in C op is the inverse of the braiding in C). Let Z(A) denote the Drinfeld center of a fusion category A. Theorem A.2.1 (cf. [23, Theorem 4.5], [10, Remark 4.3], [13, Theorem 2.15]). Assume that the category C is modular. We have dim C 0 dim(A) and dim C A category C 0A is modular;
(i) dim C A =
=
dim C ; dim(A)2
(ii) the (iii) there is a braided equivalence Z(C A ) = C (C 0A )op .
Recall (see e.g. [11, §2.12]) that a braided fusion category E is called Tannakian if it is braided equivalent to the representation category Rep(G) of a finite group G. Let E be a Tannakian subcategory of a braided fusion category C. Recall ([11, §5.4.1]) that in this situation one defines a fiber category EC E Vec. Theorem A.2.2 ([12, Theorem 1.3]). Let C be a modular category with Tannakian subcategory E = Rep(G). Assume that EC E Vec Z(A) for a fusion category A. Then C Z(B), where B = g∈G Bg is a faithfully G−graded fusion category with trivial component B1 equivalent to A.
A.3. Affine Lie algebras and conformal embeddings. Let g be a finite dimensional simple Lie algebra and let gˆ be the corresponding affine Lie algebra, see e.g. [4, §7.1]. For k ∈ Z>0 let C(g, k) denote the category of integrable highest weight gˆ −modules of level k (this category is denoted by Okint in loc. cit.). It is well known that the category C(g, k) has a natural structure of pseudo-unitary modular tensor category, see e.g. [4, Theorem 7.0.1]. The unit object of the category C(g, k) is the vacuum gˆ −module of level k. Let g ⊂ g be an embedding of simple (or, more generally, semisimple) Lie algebras. It defines an embedding gˆ ⊂ gˆ . This embedding does not preserve the level; we will write (ˆg)k ⊂ (ˆg )k if the pullback of a gˆ −module of level k under this embedding is a gˆ −module of level k (it is clear that k is uniquely determined by k ). Recall (see e.g. [9]) that a conformal embedding (ˆg)k ⊂ (ˆg )k is an embedding as above such that the
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pullback of any module from C(g , k ) is a finite direct sum of modules from C(g, k). Let (ˆg)k ⊂ (ˆg )k be a conformal embedding. Then the pullback of the vacuum gˆ −module of level k is an object A of C(g, k) which has a natural structure of C(g, k)−algebra, see [23, Theorem 5.2]. Moreover, there is a natural equivalence C(g, k)0A C(g , k ), see loc. cit. Example A.3.1. The following is a toy version of our main construction. There exists a conformal embedding (ˆs l2 )4 ⊂ (ˆs l3 )1 , see e.g. [9]. Let A0 ∈ C(sl2 , 4) be the corresponding C(sl2 , 4)−algebra. √ (cf. [4, §3.3]) that the category C(sl2 , 4) has 5 simple √ Recall objects of dimensions 1, 3, 2, 3, 1; in particular dim C(sl2 , 4) = 12. The category C(sl3 , 1) is pointed with underlying group Z/3Z; in particular dim C(sl3 , 1) = 3. We deduce from Theorem A.2.1 (i) that dim(A0 ) = 2 and dim C(sl √ 2 , 4) A0 = 6. Notice that the category C(sl2 , 4) A0 contains an object of dimension 3 since its center does (see Theorem A.2.1 (iii)); this object is automatically simple. It follows that the category C(sl2 , 4) A0 has precisely 4 simple objects: 3 from the subcategory C(sl2 , 4)0A0 √ and one more of dimension 3. Furthermore this implies that the category C(sl2 , 4) A0 is a Tambara-Yamagami category associated to Z/3Z [32]. In particular, C(sl2 , 4) A0 is Z/2Z−graded with trivial component C(sl2 , 4)0A0 = C(sl3 , 1). We now show that this example is an illustration of Theorem A.2.2. Since dim(A0 ) = 2, we see that A0 is a direct sum of two invertible objects. It follows that the subcategory E of C(sl2 , 4) generated by the invertible objects is Tannakian and is equivalent to Rep(Z/2Z) (see also [23, Theorem 6.5]). It follows from the definitions that in this case EC (sl2 ,4) E Vec = C(sl2 , 4)0A0 = C(sl3 , 1), see e.g. [11, Prop. 4.56 (i)]. Notice that E = E 1 can be considered as a subcategory of C(sl2 , 4) C(sl3 , 1)op . Clearly we have EC (sl2 ,4)C (sl3 ,1)op E Vec = (EC (sl2 ,4) E Vec) C(sl3 , 1)op = C(sl3 , 1) C(sl3 , 1)op . Since C(sl3 , 1) C(sl3 , 1)op = Z(C(sl3 , 1)) (see e.g. [11, Prop. 3.7]), Theorem A.2.2 says that C(sl2 , 4) C(sl3 , 1)op = Z(B), where B is a Z/2Z−graded category with trivial component C(sl3 , 1). This is indeed so since by Theorem A.2.1 (iii), Z(C(sl2 , 4) A0 ) = C(sl2 , 4) (C(sl2 , 4)0A0 )op = C(sl2 , 4) C(sl3 , 1)op . A.4. Izumi-Xu category IX . We will consider here another example for the formalism from §A.3. Let gG 2 and g E 6 be the simple Lie algebras of type G 2 and E 6 . There exists a conformal embedding (ˆgG 2 )3 ⊂ (ˆg E 6 )1 , see e.g. [9]. Let A1 ∈ C(gG 2 , 3) be the corresponding C(gG 2 , 3)−algebra. Proposition A.4.1. The category C(gG 2 , 3) A1 has precisely 4 simple objects 1, g, g2 and X. The subcategory generated by 1, g, g2 is pointed with underlying group Z/3Z. The remaining fusion rules are g ⊗ X = g2 ⊗ X = X ⊗ g = X ⊗ g2 = X; X ⊗ X = 1 ⊕ g ⊕ g2 ⊕ 3X. Proof. The category C(g E 6 , 1) is pointed with underlying group Z/3Z. Hence the category C(gG 2 , 3) A1 contains a pointed subcategory with underlying group Z/3Z, namely C(gG 2 , 3)0A1 C(g E 6 , 1). We will denote the simple objects of this subcategory by 1 (the unit object), g and g2 .
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Using [4, Theorem 7.0.2, Theorem 3.3.20] one computes dim C(gG 2 , 3) =
147 π 5π π 2π 3π 2 (64 sin( 21 ) sin( 4π 21 ) sin( 21 ) sin( 7 ) sin( 7 ) sin( 7 )) √ 2
=3
21
7+
2
. √
Since dim C(gG 2 , 3)0A1 = 3, we deduce from Theorem A.2.1 (i) that dim(A1 ) = 7+ 2 21 √ and dim C(gG 2 , 3) A1 = 21+32 21 . The sum of squares i di2 of the dimensions of simple √
objects of the category C(gG 2 , 3) A1 not lying in C(gG 2 , 3)0A1 is 15+32 21 . Notice that every α = di2 is a totally positive algebraic integer satisfying α = α. The proof of the following result is left to the reader: Lemma A.4.2. There are precisely three decompositions of positive algebraic integers α satisfying α = α, namely (1) (2) (3)
√ 15+3 21 2√ 15+3 21 2√ 15+3 21 2
= = =
√ 15+3 21 2
into a sum of totally
√ 15+3 21 ; 2 √ √ 5+ 21 + (5 + 21); 2 √ √ √ 5+ 21 + 5+ 2 21 + 5+ 2 21 . 2
Notice that in cases (2) and (3) the abelian subgroup Z ⊕ i Zdi ⊂ C is not closed under multiplication. Hence the only possibility is the decomposition (1); thus the category C(gG 2 , 3) A1 has precisely one simple object X that is not in C(gG 2 , 3)0A1 ; moreover √ √ dim(X) = 15+32 21 = 3+ 2 21 . The result follows.
A fusion category with fusion rules as in Proposition A.4.1 was constructed by Izumi in [18]. The construction presented here is due to Feng Xu [33] (note that it is not clear whether the two constructions produce equivalent categories). Thus we call the category C(gG 2 , 3) A1 the Izumi–Xu category and denote it by IX . Remark A.4.3. Both categories C(sl3 , 1) and C(g E 6 , 1) are pointed with underlying group Z/3Z. One observes (using [4, Theorem 3.3.20]) that these categories are opposite to each other. In particular, Theorem A.2.1 (iii) implies that Z(IX ) C(gG 2 , 3) C(g E 6 , 1)op C(gG 2 , 3) C(sl3 , 1). A.5. Main result. Theorem A.5.1. There exists a pseudo-unitary fusion category V such that (i) Z(V) C(gG 2 , 3) C(sl2 , 4); (ii) V = V0 ⊕ V1 is Z/2Z−graded with trivial component V0 equivalent to the IzumiXu category IX ; √ √ (iii) V1 contains three simple objects of dimensions 3+2 7 and a simple object of √ dimension 3.
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Proof. We recall that the category C(sl2 , 4) contains a Tannakian subcategory E Rep(Z/2Z) such that EC (sl2 ,4) E Vec C(sl3 , 1), see Example A.3.1. Now we con Vec sider E = 1 E as a subcategory of Z := C(gG 2 , 3) C(sl2 , 4). Clearly, EZ E C(gG 2 , 3) C(sl3 , 1). Thus Theorem A.2.2 and Remark A.4.3 imply that Z Z(V), where V is Z/2Z−graded fusion category with trivial component IX . Thus (i) and (ii) are proved. √ To prove (iii) we observe that the category Z contains an √ object of dimension 3; hence the category V contains an object M of dimension 3. The object M is automatically simple and is contained in V1 . Obviously, M ⊗ M = 1 ⊕ g ⊕ g2 . Hence M M∗ and Hom(M, X ⊗ M) = Hom(M ⊗ M∗ , X) = 0. Furthermore, Hom(X ⊗ M, X ⊗ M) = Hom(M, X∗ ⊗ X ⊗ M) = C3 . Thus, X ⊗ M ∈ V1 is a direct sum of three distinct simple objects V1 , V2 , V3 , none of which is isomorphic to M. Since √ 21+3 21 , we get that dim V1 = dim V0 = 2 √ 15 + 3 21 2 2 2 . dim(V1 ) + dim(V2 ) + dim(V3 ) = 2 Using Lemma A.4.2, we see that √ √ √ 5 + 21 3+ 7 dim(V1 ) = dim(V2 ) = dim(V3 ) = = . 2 2 Thus the theorem is proved.
A.6. Fusion rules of the category V. In this section we compute the fusion rules of the category V following a suggestion of Noah Snyder. First, at least one of the objects V1 , V2 , V3 is self dual; we assume that V1 is self dual and use notation V := V1 . The dimension count shows that V ⊗ V V2 ⊗ V2∗ V3 ⊗ V3∗ 1 ⊕ X. It follows that g⊗V V and g2 ⊗V V; thus we can (and will) assume that V2 = g⊗V and V3 = g2 ⊗ V. We claim that V ⊗ g g ⊗ V. Assume for the sake of contradiction that V ⊗ g g ⊗ V. It follows that the Grothendieck ring K (V) is commutative (since it is generated by the classes [g] and [V]). Thus [13, Lemma 8.49] implies that the map K (Z(V)) ⊗ Q → K (V) ⊗ Q is surjective. But this is impossible since any object of Z(V) = C(gG 2 , 3) C(sl2 , 4) is self dual and (g)∗ = g2 g. It follows that V ⊗ g g2 ⊗ V. The remaining fusion rules are easy to determine from the known information. We have Proposition A.6.1. The simple objects of the category V are 1, g, g2 , X, M, V, gV := g ⊗ V, g2 V := g2 ⊗ V. The fusion rules are uniquely determined by Proposition A.4.1 and V ⊗ g = g2 V, X ⊗ M = M ⊗ X = V ⊕ gV ⊕ g2 V, X ⊗ V = V ⊗ X = M ⊕ V ⊕ gV ⊕ g2 V, M ⊗ M = 1 ⊕ g ⊕ g2 , M ⊗ V = V ⊗ M = X, V ⊗ V = 1 ⊕ X.
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