Commun. Math. Phys. 270, 1–12 (2007) Digital Object Identifier (DOI) 10.1007/s00220-006-0139-5
Communications in
Mathematical Physics
Three Applications of Instanton Numbers Elizabeth Gasparim1 , Pedro Ontaneda2 1 Department of Mathematical Sciences, New Mexico State University, Las Cruces, NM 88003-8001, USA.
E-mail:
[email protected]
2 Department of Mathematical Sciences, SUNY at Binghamton, Binghamton, NY 13902-6000, USA.
E-mail:
[email protected] Received: 14 August 2005 / Accepted: 7 September 2006 Published online: 23 November 2006 – © Springer-Verlag 2006
Abstract: We use instanton numbers to: (i) stratify moduli of vector bundles, (ii) calculate relative homology of moduli spaces and (iii) distinguish curve singularities. 1. Introduction Instantons on a blow-up have two local numerical invariants, which we name height and width. Their sum gives the instanton charge. In this paper we present some ways in which this pair of invariants gives finer information than the charge alone. Firstly, we show that instanton numbers give the coarsest stratification of moduli of bundles on blow-ups for which the strata are separated. Secondly, we show that the relative homology H2 (Mk ( X ), Mk (X )) is nontrivial, where Mk denotes moduli of charge k instantons, and X is obtained from X by blowing up a point. This shows that, despite the fact that Mk (X ) and Mk ( X ) have the same dimension, there is a significant topological difference between them. Thirdly, we give examples of analytically distinct curve singularities, which are not distinguished by any of the classical invariants (δ P , Milnor number, Tjurina number, and multiplicity) but have distinct instanton numbers. This paper focuses on rank 2 instantons on blown-up surfaces. We are specially interested in the behavior of instantons near an exceptional divisor. We give an explicit con2 . We show that such struction of instantons on the blow-up of C2 at the origin, denoted C instantons are determined by the data : = ( j, p, t∞ ), formed by an integer j, a polynomial p, and a framing at infinity, that is, a holomorphic map t∞ : C2 − {0} → SL(2, C). The charge of takes values between j and j 2 depending on p. However, unlike instantons on S 4 , whose charge is given locally by a unique invariant, called the multiplicity, these instantons have two independent local holomorphic invariants. These invariants do not depend on the choice of framing, and can therefore be calculated directly from the algebraic data ( j, p). A Macaulay2 algorithm that calculates the instanton numbers out of this data is available in [8]. The authors acknowledge support from NSF and NSF/NMSU Advance.
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E. Gasparim, P. Ontaneda
The connection between holomorphic vector bundles and instantons is made through the Kobayashi–Hitchin correspondence. In Sect. 2, we use this correspondence to con2 . In Sect. 3, we use instanton numbers to stratify moduli of bundles struct instantons on C on the blown-up plane with a fixed splitting type over the exceptional divisor. In Sect. 4, we consider the moduli spaces Mk (X ) and Mk ( X ) of rank 2 instantons on a compact surface X and on the surface X = the blow up of X at a point, and prove that H2 (Mk ( X ), Mk (X )) = 0. In Sect. 5, we use instanton numbers as invariants of curve singularities. For curves, the trick is as follows. Given a plane curve p(x, y) = 0 with singularity at the origin, chose an integer j, and construct an instanton with data ( j, p). We then use the numerical invariants of the instanton as analytic invariants of the curve. 2 2. Instantons on C 2 is determined by a triple : = ( j, p, t∞ ), where j Every rank 2 instanton on C is an integer, p a polynomial and t∞ a trivialization at infinity. This characterization comes from putting together two results: on one side, the proof due to King [11] of the 2 and, on the other Kobayashi–Hitchin correspondence over the noncompact surface C 2 side, the description of rank two holomorphic bundles on C given in [7]. We review these two results. 2 Instantons on the blown-up plane are naturally identified with instantons on CP 2 framed at infinity; this is a simple consequence of the fact that CP is the conformal 2 . In his Ph.D. thesis, A. King [11] identifies the moduli space compactification of C 2 MI(C ; r, k) of instantons on the blown-up plane of rank r and charge k, with the mod2 2 uli space MI(CP , ∞ : r, k) of instantons on CP , framed at ∞, whose underlying vector bundle has rank r, and Chern classes c1 = 0 and c2 = k. On the other hand, consider the Hirzebruch surface 1 , as the canonical complex 2 by adding a line ∞ at infinity. Essentially by deficompactification obtained from C 2 ; r, k) of “stable” holomorphic bundles nition King identifies the moduli space MH(C 2 on C with rank r and c2 = k with the moduli space MH(1 , ∞ ; r, k) of holomorphic bundles on 1 with a trivialization along ∞ and whose underlying vector bundle has rank r, c1 = 0 and c2 = k. King then proves the Kobayashi–Hitchin correspondence in this case, namely that the map 2 ; r, k) → MH(C 2 ; r, k) MI(C given by taking the holomorphic part of an instanton connection is a bijection. Therefore, 2 is completely determined by a rank two holomorphic bundle a rank 2 instanton on C 2 on C with vanishing first Chern class, together with a trivialization at infinity. The instanton has charge k if and only if the corresponding holomorphic bundle extends to a bundle on 1 trivial on ∞ having c2 = k. We are led to study holomorphic rank two 2 are algebraic extensions of 2 . As shown in [6], holomorphic bundles on C bundles on C line bundles; moreover, by [4], if the first Chern class vanishes, then such bundles are trivial on the complement of the exceptional divisor. Note. Triviality outside the exceptional divisor is very useful and is intrinsically related 2 are algebraic, cf. [4]. It is of course not true to the fact that holomorphic bundles on C 2 minus the exceptional divisor in general that a holomorphic bundle defined only on C
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is trivial; we make essential use of the fact that our bundles/instantons are defined over 2 . the entire C Now that we have established the equivalence between instantons and bundles, we 2 . Because of the triviality at infinity, it give an explicit construction of instantons on C follows that we have also existence of instantons on any surface containing a P1 with self-intersection −1. 2 with vanishing first Chern class splits over A holomorphic rank 2 bundle E on C the exceptional divisor as O( j) ⊕ O(− j) for some nonnegative integer j, called the splitting type of the bundle, and, in this case, E is an algebraic extension 0 → O(− j) → E → O( j) → 0
(2.1)
(here by abuse of notation we write O(k) both for the line bundle OP1 (k) as well as 2 ). A bundle E fitting in an exact sequence (1) is determined by for its pull-back to C its extension class p ∈ E xt 1 (O(− j), O( j)). We fix, once and for all, the following coordinate charts: 2 = U ∪ V, C where U = {(z, u)} C2 {(ξ, v)} = V with
(ξ, v) = (z −1 , zu)
(2.2)
in U ∩ V. Then in these coordinates, the bundle E has a canonical transition matrix of the form j z p (2.3) 0 z− j from U to V, where p: =
2 j−2
j−1
pil z l u i
(2.4)
i=1 l=i− j+1
is a polynomial in z, z −1 and u ( [6] Thm. 2.1). Hence E is completely determined by the pair ( j, p). To have an instanton we need also a trivialization at infinity. By [4] Cor. 4.2, E is trivial outside the exceptional divisor. Therefore we may assign to E a trivialization at infinity t∞ ∈ SL(2, C2 − {0}) thus obtaining an instanton. As a consequence every 2 is determined by a triple rank–two instanton on C := ( j, p, t∞ ).
(2.5)
) determine the same instanton if and Generically, two triples ( j, p, t∞ ) and ( j , p , t∞
only if j = j, p = λp and t∞ = A t∞ , where λ = 0, and A ∈ C2 − {0}, SL(2, C) . To define the topological charge of we need to extend to a compact surface. This (local) charge is independent of the chosen compactification, and in fact only depends on an infinitesimal neighborhood of the exceptional divisor. For simplicity we take the com2 a line at pactification given by the Hirzebruch surface 1 obtained by adding to C infinity. An instanton on C2 corresponds to a bundle E on 1 trivial on ∞ , together
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with a trivialization over this line. Let π : 1 → Z be the map that contracts the −1 line. The charge of is by definition c() = c(E) : = c2 (E) − c2 ((π∗ E)∨∨ ).
(2.6)
The instanton is generic if and only if its charge equals its splitting type. Moreover, for every j > 1 there are nongeneric instantons ( j, p, t∞ ), with charge varying from j + 1 up to j 2 (see Theorem 3.6). 3. Local Moduli Spaces
) are equivalent if they represent the Two triples = ( j, p, t∞ ) and = ( j , p , t∞ 2 are isomorsame instanton; consequently their corresponding bundles E and E over C
phic, hence must have the same splitting type, i.e. j = j . Consider two triples ( j, p, t∞ )
), with the same j, and corresponding bundles (E, t ) and (E , t ) over and ( j, p , t∞ ∞ ∞ 1 . An isomorphism of framed bundles is a bundle isomorphism : E → E such
. Two framings t and t for the same underlying bundle E over that (t∞ ) = t∞ ∞ 1 ∞ differ by a holomorphic map : ∞ → SL(2, C) and, since ∞ is compact, must be constant. Hence, projecting (E, t∞ ) on the first coordinate we obtain a fibration of the space of framed bundles over 1 over the space of bundles over 1 which are trivial on the line at infinity, with fibre SL(2, C),
SL(2, C) ↓ { f ramed rank − 2 bundles over 1 } ↓ {rank − 2 bundles over 1 trivial on ∞ } .
(3.1)
We are thus led to study the base space of this fibration. We define M j to be the space 2 with vanishing first Chern class and with of rank two holomorphic bundles on the C splitting type j, modulo isomorphism, that is,
2 M j = E hol. bundle over C : ∼. (3.2) E| O( j) ⊕ O(− j) Fix the splitting type j and set J = ( j − 1)(2 j − 1). Then the polynomial p has J coefficients and we identify p with the J −tuple of complex numbers formed by its coefficients written in lexicographical order. We define in C J the equivalence relation p ∼ p if ( j, p) and ( j, p ) represent isomorphic bundles. This gives a set-theoretical identification (3.3) M j = CJ / ∼ . We give C J / ∼ the quotient topology and M j the topology induced by (3.3). M j is generically a complex projective space of dimension 2 j − 3 ( [6] Thm. 3.5), and is included in M j+1 by: Proposition 3.1. The following map defines a topological embedding j : M j → M j+1 . ( j, p) → ( j + 1, zu 2 p)
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The proof is in Sect. 5. The map j takes M j into the least generic strata of M j+1 . In fact, im equals the subset of M j+1 consisting of all bundles that split on the second formal neighborhood of the exceptional divisor. The complexity of the topology of M j increases with j. M2 is non-Hausdorff and by the embedding given in Proposition 3.1 this property persists in M j for j ≥ 2. Example 3.2. The description of M2 as a quotient C3 / ∼ as in (8) gives M2 P1 ∪ {A, B}, where points in the generic set P1 represent bundles that do not split on the first formal neighborhood. The point A corresponds to a bundle that splits on the first formal neighborhood but not on higher neighborhoods, and B corresponds to the split bundle (see [6]). The topological counterpart of this decomposition is understood by calculating the instanton charge. If E ∈ P1 , then c(E) = 2, whereas c(A) = 3 and c(B) = 4. The good stratification of M2 by topological invariants agrees with the expectation we might have based on our experience from the case of bundles over compact surfaces. It then appears natural to hope that topological charges stratify M j into Hausdorff components. This is however entirely false. For each j > 2 there are non-Hausdorff subspaces of M j where the topological charge remains constant. We now define the finer instanton numbers that stratify the spaces M j into Hausdorff components. 3.1. Instanton numbers. Consider a compact complex surface X together with the blow–up π : X → X of a point x ∈ X and denote by the exceptional divisor. By the Kobayashi–Hitchin correspondence instantons on X (resp.X ) correspond to stable bundles on X (resp.X ). be a holomorphic bundle over O Let E X satisfying det E X and E| O( j) ⊕ ∨∨ O(− j) with j ≥ 0. Set E = (π∗ E) . Friedman and Morgan ([3], p. 393) gave the following estimate − c2 (E) ≤ j 2 . j ≤ c2 ( E) (3.4) Sharpness of these bounds was shown in [5]. Since the n th infinitesimal neighborhood 2 of on X is isomorphic (as a scheme) to the n th infinitesimal neighborhood of in C 2 is we are able to use the explicit description for bundles on C , given in (2.3). Hence E determined on a neighborhood V () of the exceptional divisor by a pair ( j, p). Define a sheaf Q by the exact sequence, → E → Q → 0. 0 → π∗ E − c2 (E) = l(Q). An application of Then Q is supported at the point x and c2 (π∗ E) Grothendieck–Riemann–Roch gives − c2 (E) = l(Q) + l(R 1 π∗ E). c2 ( E) are local analytic invariants and depend only on the data ( j, p) Both l(Q) and l(R 1 π∗ E) is stable on defining E over V (). Suppose E is stable on X, then E X see [3]. Hence, corresponds to an instanton on E. This if E corresponds to an instanton on X, then E justifies the following terminology. over V () = ( j, p, t∞ ) Definition 3.3. A holomorphic bundle E X such that E| ∨∨ and (π∗ E) E is said to be obtained by holomorphic patching of to E. If E is given a frame at the point x, then E and uniquely determine E.
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and call them the Definition 3.4. We set w() : = l(Q) and h() : = h(R 1 π∗ E), height and the width of . The charge of is given by c() := w() + h(). Remark 3.5. The charge addition given by the patching of can be calculated by a Macaulay2 program [8]. The program has as input j and p and as outputs w() and h(). The following result shows that instanton numbers provide good stratifications for moduli of instantons on C2 . In fact, these numbers give the coarsest stratification of M j for which the strata are Hausdorff. In [1] it is shown that the stratification by Chern numbers is not fine enough to have this property. Theorem 3.6 ([1] Thm. 4.1). The numerical invariants w and h provide a decomposition M j = ∪Si , where each Si is homeomorphic to an open subset of a complex projective space of dimension at most 2 j − 3. For j > 0, the lower bounds for these invariants are (1, j − 1) and this pair of invariants takes place on the generic part of M j which is homeomorphic to CP 2 j−3 minus a closed subvariety having codimension at least 2. The upper bounds are ( j ( j + 1)/2, j ( j − 1)/2) and this pair occurs at the single point of M j that represents the split bundle. Note that the M j are labeled by splitting type, however, we also need the loci of fixed local charge i. Definition 3.7. The local moduli Ni of bundles with fixed local charge i is
2 : c1 (E) = 0, c(E) = cloc (E) = i ∼. Ni = E hol. bundle over C 2 Corollary 3.8. N0 is just a point, N1 is also just a point, and N2 CP 1 . For i ≥ 2, Ni has dimension 2i − 3. Proof. Just use Theorem 3.6 and Example 3.2.
4. Topology of Instanton Moduli Spaces Let X be a compact complex surface. By the Kobayashi–Hitchin correspondence (cf. [13]), we know that irreducible SU (2) instantons of charge k on X are in one-to-one correspondence with rank 2 stable holomorphic bundles on X with Chern classes c1 = 0 and c2 = k. Given a complex surface Y, let Mk (Y ) denote the moduli of irreducible instantons on Y with charge k, or equivalently, moduli of stable bundles on X having zero first Chern class and second Chern class k. Let π : X → X be the blow up of a point x ∈ X. The aim of this section is to show that there is a significant difference between the moduli spaces of instantons on X and X , despite the fact that their dimensions coincide. To this purpose we show that H2 (Mk ( X ), Mk (X )) = 0. To see Mk (X ) as a subspace of Mk ( X ), the polarizations on the two surfaces have to be chosen appropriately. If L is an ample divisor on X then for large N the divisor L = N L − is ample on X . We fix, once and for all, the polarizations L and L on X and X respectively. From now on, Mk (Y ) stands for moduli of rank two bundles on Y slope stable with respect to the fixed polarization. If E is L–stable on X, then π ∗ (E) is L– stable on X . Therefore, the pull back map induces an inclusion of moduli spaces Mk (X ) → Mk ( X ). We proceed to show that for all k ≥ 1 the relative homology H2 (Mk ( X ), Mk (X )) does not vanish. We use holomorphic patching as defined in 3.3, and to this end we introduce framings.
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Definition 4.1. Framed bundles. • Let π F : F → Z be a bundle over a surface Z that is trivial over Z 0 := Z − Y. Given two pairs f = ( f 1 , f 2 ) : Z 0 → π F−1 (Z 0 ) and g = (g1 , g2 ) : Z 0 → π F−1 (Z 0 ) of linearly independent sections of F| Z 0 , we say that f is equivalent to g if there exist a map holomorphic φ : Z 0 → S L(2, C) satisfying f = φg such that φ extends to a holomorphic map over the entire Z . A frame of F over Z 0 is an equivalence class of linearly independent sections over Z 0 . f on → • A framed bundle E X is a pair consisting of a bundle π E : E X together over N 0 := N () − . with a frame of E 2 is a pair consisting of a bundle πV : V → C 2 together • A framed bundle V f on C 2 with a frame of V over C − . • A framed bundle E on X is a pair consisting of a bundle E → X together with a frame of E over N (x) − x, where N (x) is a small disc neighborhood of x. We will always consider N (x) = π E(N ()). X ), Mk (X ), and Ni denote the framed versions of Mk ( X ), Mk (X ), Notation 4.2. Mk ( and Ni respectively. f
f
f
For any k, there is a stratification of the moduli space of framed bundles on X as f X) ≡ Mk (
k
f
f
Mk−i (X ) × Ni .
i=0
More details of this decomposition are given in [9]. We use the notation f
f
K i := Mk−i (X ) × Ni . Lemma 4.3. Removing the singular points of Mk ( X ) does not change homology up to dimension k. That is, if Sing denotes the singularity set of Mk (X ), then for q < k, X ) = Hq Mk ( X ) − Sing . Hq Mk ( X ) satisfying H 2 (End0 E) = 0 are smooth Proof. By Kuranishi theory, points E ∈ Mk ( points. Therefore, the singularity set of Mk ( X ) is contained in k = {E ∈ Mk ( X) : H 2 (End0 E) = 0}. Moreover, the moduli space is defined on a neighborhood of a singular point by dim H 2 (End √ 0 E) equations. In ([2] Thm. 5.8), Donaldson shows that dim H 2 (End0 E) ≤ a + b k + 3k. Therefore, an application of Kirwan’s result √ ([12] Cor. 6.4) gives Hq (Mk ( X )) = Hq (Mk ( X ) − Sing) for q < dim Mk ( X ) − 2(a + b k + X ) − 7k = 8k − 3 − 7k < k. 3k) < dim Mk ( f
A similar argument holds for Mk (X ). In what follows we work only with the smooth f f X ) and Mk (X ) which, by abuse of notation, we still denote by the same part of Mk ( symbols. Lemma 4.4. For q ≤ 2,
f Hq Mk ( X ) = Hq (K 0 ∪ K 1 ).
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E. Gasparim, P. Ontaneda
Proof. The subset of pull-back bundles K 0 = {π ∗ (E), E ∈ Mk (X )} is well known to f X ). For i ≥ 1, the subset be open and dense in Mk ( f f f X ) : c2 (π∗ E ∨∨ ) = k − i = Mk−i (X ) × Ni K i = E ∈ Mk ( f
k K . Then S has has real codimension at least 2i by [9], Lemma 6.3. We set S2 = ∪i=2 i 2 f codimension at least 4 in Mk ( X ). Consequently, using Lemma 4.3 for q < 3, we have isomorphisms f f Hq Mk ( X ) = Hq Mk ( X ) − S2 = Hq (K 0 ∪ K 1 ).
Lemma 4.5. The real codimension of K 1 in Mk ( X ) is exactly 2. f
f X ) it follows that the codimension of K 1 Proof. Since K 0 is open and dense in Mk ( f in Mk ( X ) equals the codimension of K 1 inside K 0 ∪ K 1 . By definition, any bundle = h( E) = 0. On the ∈ K 0 is trivial around the divisor, and therefore satisfies w( E) E ∈ K 1 satisfies w( F) = 1 (by Theorem 3.6). Consequently, other hand, any bundle F ∈ K 0 ∪ K 1 : w( F) = 1} is the zero locus of a single analytic (in fact algebraic) K1 = {F equation in K 0 ∪ K 1 ; hence K 1 has complex codimension one. f
Theorem 4.6. Let k ≥ 1 and suppose Mk (X ) is non-empty, then f f X ), Mk (X ) = 0. H2 Mk ( Proof. By Lemma 4.4 the map
f X) Hq (K 0 ∪ K 1 ) → Hq Mk (
(4.1)
is an isomorphism, for q = 0, 1, 2. f The map of pairs (K 0 ∪ K 1 , K 0 ) → (Mk ( X ), K 0 ) induces a map between the long exact sequences of these pairs. Using (4.1) and the five lemmas we conclude that the map H2 (K 0 ∪ K 1 , K 0 ) → H2 (Mk ( X ), K 0 ) f
is an isomorphism. Since K 1 is closed in K 0 ∪ K 1 we have that H2 (K 0 ∪ K 1 , K 0 ) = H2 (ν(K 1 ), ν(K 1 ) − K 1 ) = H2 (T ν(K 1 )) (by excision), where ν(K 1 ) is the normal bundle of K 1 in K 0 ∪ K 1 , and T ν(K 1 ) is the Thom space of this bundle. By Lemma 4.5, K 1 has codimension exactly 2, therefore the fiber of T ν(K 1 ) has dimension 2. Consequently (by the Thom isomorphism or duality theorems): H2 (T ν(K 1 )) = H0 (K 1 ) = r Z, where r is the number of components of K 1 , and it follows that H2 (K 0 ∪ K 1 , K 0 ) = r Z. If K 1 is connected H2 (K 0 ∪ K 1 , K 0 ) = Z. Now, the theorem follows from the simple f observation that K 0 is the set of pull-back bundles, which is isomorphic to Mk (X ).
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2 with any prescribed charge. Note that in Sect. 2 we constructed instantons on C However, existence of irreducible instantons on a compact surface follows from existence of the corresponding stable bundles; such bundles in general are only known to exist for large c2 . On a surface containing a −1 line, however, many nontrivial semista2 ble bundles can be constructed using our holomorphic patching 3.3, by patching any C instanton bundle to a trivial bundle on X. 5. Curve Singularities Here is how to use instanton numbers to distinguish curve singularities. Start with a curve p (x, y) = 0 on C2 . Choose your favorite integer j and construct an instanton 2 having data ( j, p). Calculate the height, width and charge of the instanton, use on C them as invariants of the curve. In other words, we are using the polynomial defining the plane curve as an extension class in Ext1 (O( j), O(− j)). This defines a bundle E( j, p) as in (2.3). We then regard the instanton numbers of this bundle as being associated to the curve. Note that to perform the computations we must choose a representative for the curve and coordinates for the bundle. Here we use the canonical choice of coordinates for 2 as in Sect. 2 and consider only either quasi-homogeneous curves, or else reducible C curves which are products of two quasi homogeneous curves. For these curves there is a preferred choice of representative. Whereas this is certainly restrictive, it is nevertheless true that interesting results appear, given that instanton numbers distinguish some of these singularities which are not distinguished by any of the classical invariants: the δ P invariant, the Milnor number, or the Tjurina number of the singularity (see Table II) and in addition the multiplicity (Table III). Taking into account that the blow-up map in our canonical coordinates is given by x → u and y → zu the bundle E( j, p) is represented by j p(u, zu) z E( j, p) : = . 0 z− j In this paper we give a few results to illustrate the behavior of the instanton numbers when applied to singularities. Explicit hand-made computations of these invariants for small values of j appear in [1] and [5]. A Macaulay2 algorithm is available to compute the invariants in the general case, see Remark 3.5. Theorem 5.1. Instanton numbers distinguish nodes (tacnodes) from cusps (higher order cusps). Proof. These singularities have quasi–homogeneous representatives of the form y n −x m , n < m, n even for nodes and tacnodes, and n odd for cusps and higher order cusps. We want to show that instanton numbers detect the parity of the smallest exponent. In fact, more is true, instanton numbers detect the multiplicity itself. Suppose n 1 < n 2 . We claim that if j > n 2 then w( j, p1 ) = w( j, p2 ). In fact, for n < m and large enough j the width takes the value w( j, y n − x m ) = n(n + 1)/2. Alternatively, by vector bundle reasons we have that w( j, p1 ) < w( j, p2 ). The second assertion is easier to show. The holomorphic bundle E( j, p1 ) restricts as a non-trivial extension on the n 1 th formal neighborhood ln 1 whereas E( j, p2 ) splits on ln 1 . These
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E. Gasparim, P. Ontaneda j =4
TABLE I polynomial
δP
μ
τ
w
h
x 5 y − y4 x 8 − x 5 y2 − x 3 y2 + y4
9 9
17 17
17 15
10 8
6 6 j =4
TABLE II polynomial
δP
μ
τ
w
h
x 2 − y7 x 3 − y4
3 3
6 6
6 6
3 6
5 6
TABLE III polynomial
mult.
δP
μ
τ
w
j =4 h
charge
x 3 − x 2 y + y3 x 3 − x 2 y2 + y3
3 3
3 3
4 4
4 4
4 5
3 3
7 8
bundles therefore belong to different strata of M j and by Theorem 3.6 must have distinct instanton numbers. We consider the following classical invariants: P /O P ), • δ P = dim( O • Milnor number μ = dim(O/ < J (P) >), • Tjurina number τ = dim(O/ < P, J (P) >). Note. The first table is motivated by Exercise 3.8 of Hartshorne [10], p. 395. However, in the statement of the problem, the first polynomial contains an incorrect exponent. It is written as “x 4 y − y 4 ” but it should be “x 5 y − y 4 .” Theorem 5.2. In some cases instanton numbers give finer information than the classical invariants. Proof. Table II gives 2 singularities that are obviously distinct, since they have different multiplicities, but are not distinguished by δ p , Milnor and Tjurina numbers. Table III shows that instanton numbers are the only invariants to distinguish the irreducible singularity x 3 − x 2 y + y 3 from the reducible singularity x 3 − x 2 y 2 + y 3 . Remark 5.3. The idea of using the polynomial defining a singularity as the extension class of a holomorphic bundle can be further generalized in several ways. For curves themselves, one can use other base spaces. For instance, constructing bundles on the total space of OP1 (−k) requires very little modifications, but give quite different results. One can also generalize to hypersurfaces in higher dimensions, using the equation of the hypersurface to define an extension of line bundles. 6. Embedding Theorem an Proof of Proposition 3.1. We want to show that ( j, p) → ( j + 1, zu 2 p) defines j z p embedding M j → M j+1 . We first show that the map is well defined. Suppose 0 z− j
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ab represent isomorphic bundles. Then there are coordinate changes cd αβ holomorphic in z, u and holomorphic in z −1 , zu such that γ δ −j j p
−p a b z α β z . = c d γ δ 0 zj 0 z− j
and
z j p
0 z− j
Therefore these two bundles are isomorphic exactly when the system of equations α β a + z − j p c z 2 j b + z j ( p d − ap) − pp c = (∗) γ δ z −2 j c d − z − j pc αβ ab holomorphic holomorphic in z, u which makes can be solved by a matrix γ δ cd in z −1 , zu. On the other imagesof these two bundles are given by transition matrices j+1 hand, the z z j+1 z u 2 p
z u2 p and − j−1 , which represent isomorphic bundles iff there are 0 z − j−1 0 z α¯ β¯ a¯ b¯ holomorphic in z, u and holomorphic in z −1 , zu coordinate changes γ¯ δ¯ c¯ d¯ satisfying the equality − j−1 j+1 z u 2 p
−z u 2 p α¯ β¯ a¯ b¯ z z . = γ¯ δ¯ 0 z j+1 0 z − j−1 c¯ d¯ That is, the images represent isomorphic bundles if the system α¯ β¯ a¯ + z − j u 2 p c¯ z 2 j+2 b¯ + z j+2 u 2 ( p d¯ − a¯ p) − z 2 u 4 pp c¯ = γ¯ δ¯ z −2 j−2 c¯ d¯ − z − j u 2 p c¯
(∗∗)
has a solution. ¯ c, ¯ and choose a¯ i = ai+2 , b¯i = bi+2 u 2 , Write x = xi u i for x ∈ {a, b, c, d, ¯ b, ¯ d} a, ab a¯ b¯ solves (*), one verifies that solves (**), c¯i = ci+2 u −2 , d¯i = di+2 . Then if cd c¯ d¯ which implies that the images represent isomorphic bundles and therefore j is well defined. To show that the map is injective just reverse the previous argument. Continuity is obvious. Now we observe also that the image j (M j ) is a saturated set in M j+1 (meaning that if y ∼ x and x ∈ j (M j ) then y ∈ j (M j )). In fact, if E ∈ j (M j ) then E splits in the 2nd formal neighborhood. Now if E ∼ E then E must also split in the 2nd formal neighborhood, therefore the polynomial corresponding to E is of the form u 2 p and hence j (z −1 p ) gives E . Note also that j (M j ) is a closed subset of M j+1 , given by the equations pil = 0 for i = 1, 2 and i − j + 1 ≤ l ≤ j − 1. Now the fact that j is a homeomorphism over its image follows from the following easy lemma. Lemma 6.1. Let X ⊂ Y be a closed subset and ∼ an equivalence relation in Y, such that X is ∼ saturated. Then the map I : X/∼ → Y/∼ induced by the inclusion is a homeomorphism over the image. Proof. Denote by π X : X → X/∼ and πY : Y → Y/∼ the projections. Let F be a closed subset of X/∼. Then π X−1 (F) is closed and saturated in X and therefore π X−1 (F) is also closed and saturated in Y. It follows that πY (π X−1 (F)) is closed in Y/∼.
12
E. Gasparim, P. Ontaneda
References 1. Ballico, E., Gasparim, E.: Numerical Invariants for Bundles on Blow-ups. Proc. Amer. Math. Soc. 130(1), 23–32 (2002) 2. Donaldson, S. K.: Polynomial invariants for smooth four-manifolds. Topology 29(3), 257–315 (1990) 3. Friedman, R., Morgan, J.: On the diffeomorphism types of certain algebraic surfaces II. J. Diff. Geom. 27, 371–398 (1988) 4. Gasparim, E.: Holomorphic Bundles on O(−k) are algebraic. Comm. Algebra 25(9), 3001–3009 (1997) 5. Gasparim, E.: Chern Classes of Bundles on Blown-up Surfaces. Comm. Algebra 28(10), 4919–4926 (2000) 6. Gasparim, E.: Rank Two Bundles on the Blow up of C2 . J. Alg. 199, 581–590 (1998) 7. Gasparim, E.: On the Topology of Holomorphic Bundles. Bol. Soc. Parana. Mat. 18(1–2), 113–119 (1998) 8. Gasparim, E., Swanson, I.: Computing instanton numbers of curve singularities. J. Sym. Comp. 40, 965–978 (2005) 9. Gasparim, E.: The Atiyah–Jones conjecture for rational surfaces. http://arxiv.org/list/ math.AG/0403138, 2004 10. Hartshorne, R.: Algebraic Geometry. Graduate Texts in Mathematics 56, Berlin-Heidelberg-New York: Springer Verlag, 1977 11. King, A.: Ph.D. Thesis, Oxford (1989) 12. Kirwan, F.: On spaces of maps from Riemann surfaces to Grassmanians and applications to the cohomology of vector bundles. Arch. Math. 24(2), 221–275 (1986) 13. Lübke, M., Teleman, A.: The Kobayashi–Hitchin correspondence. River Edge, NJ: World Scientific Publishing Co., Inc., (1997) Communicated by N.A. Nekrasov
Commun. Math. Phys. 270, 13–51 (2007) Digital Object Identifier (DOI) 10.1007/s00220-006-0141-y
Communications in
Mathematical Physics
Shuffle Relations for Regularised Integrals of Symbols Dominique Manchon, Sylvie Paycha Mathém˙atiques, Université Blaise Pascal (clermont II), 63177 Aubière cedex, France. E-mail:
[email protected];
[email protected] Received: 21 October 2005 / Accepted: 15 June 2006 Published online: 2 December 2006 – © Springer-Verlag 2006
Abstract: We prove r egshuffle relations which relate a product of regularised integrals of classical symbols σi dξi , i = 1, . . . , k to regularised nested iterated integrals: k i=1
r eg
σi dξi =
τ ∈k
r eg
dξ1
|ξ2 |≤|ξ1 |
dξ2 · · ·
|ξ L |≤|ξk−1 |
k dξk ⊗i=1 στ (i) ,
where k is the group of permutations over k elements. We show that these shuffle relations hold if all the symbols σi have vanishing residue; this is true of non-integer order symbols on which the regularised integrals have all the expected properties such as Stokes’ property [MMP]. In general the shuffle relations hold up to finite parts of corrective terms arising from a renormalisation on tensor products of classical symbols, a procedure adapted from renormalisation methods to compute Feynman diagrams familiar to physicists. We relate the shuffle relations for regularised integrals of symbols with shuffle relations for multiple zeta functions adapting the above constructions to the case of a symbol on the unit circle. Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Regularised Integrals of log-Polyhomogeneous Symbols . . . . . . . . 3.1 Cut-off integrals of log-polyhomogeneous symbols . . . . . . . . 3.2 Integrals of holomorphic families of log-polyhomogenous symbols 3.3 Regularised integrals of log-polyhomogeneous symbols . . . . . . 4. Regularised Integrals on Tensor Products of Classical Symbols . . . . 4.1 Tensor products of symbols . . . . . . . . . . . . . . . . . . . . . 4.2 A first extension of regularised integrals to tensor products . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
14 18 19 19 23 24 25 25 26
14
D. Manchon, S. Paycha
5. An Alternative Extension of Regularised Integrals to Tensor Products of Classical Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Rota-Baxter relations . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Iterated cut-off integrals of classical symbols . . . . . . . . . . . . . 5.3 Iterated integrals of holomorphic families of classical symbols . . . 5.4 Obstructions to shuffle relations for regularised integrals of general classical symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Feynman graphs and tensor products of symbols . . . . . . . . . . . 6. Relation to Multiple Zeta Functions . . . . . . . . . . . . . . . . . . . . 6.1 The symbol of invariant operators on the unit circle . . . . . . . . . 6.2 Discrete sums of symbols and the Euler-MacLaurin formula . . . . 6.3 Discrete Chen sums of symbols . . . . . . . . . . . . . . . . . . . . 6.4 Multiple zeta functions . . . . . . . . . . . . . . . . . . . . . . . .
. . . .
. . . .
28 29 31 33
. . . . . . .
. . . . . . .
35 39 41 41 44 47 48
1. Introduction Before describing the contents of the paper, let us give some general motivation. Starting ˜ f ) : N → C: from a function f : N → C, one can build functions P( f ) : N → C and P( ˜ f )(n) = P( f )(n) = f (m), P( f (m). n>m>0
n≥m>0
The operators P and P˜ obey Rota-Baxter relations and define Rota-Baxter type operators of weight −1 and 1 respectively: P( f ) P(g) = P ( f P(g)) + P (g P( f )) + P( f g) and ˜ f ) P(g) ˜ P( = P˜
˜ ˜ f ) − P( ˜ f g). f P(g) + P˜ g P(
When applied to f (n) = n −z 1 , g(n) = n −z 2 , these relations lead to the “second shuffle relations” for zeta functions [ENR]: ζ (z 1 ) ζ (z 2 ) = ζ (z 1 , z 2 ) + ζ (z 2 , z 1 ) + ζ (z 1 + z 2 ), 1 −z 2 where ζ (z) = n>0 n −z and ζ (z 1 , z 2 ) = n 1 >n 2 n −z 1 n 2 . Similarly, ζ (z 1 ) ζ (z 2 ) = ζ˜ (z 1 , z 2 ) + ζ˜ (z 2 , z 1 ) − ζ˜ (z 1 + z 2 ), 1 −z 2 where ζ˜ (z 1 , z 2 ) = n 1 ≥n 2 n −z 1 n2 . Correspondingly, starting from f ∈ L 1 (R, C), one can build P( f ) : R → C: P( f )(y) = f (x) d x. y≥x
Then the classical Rota-Baxter relation (of weight zero) P( f ) P(g) = P( f P(g)) + P(g P( f ))
Shuffle Relations for Regularised Integrals of Symbols
15
is an integration by parts in disguise. It leads to to shuffle relations for integrals: k i=1 R
fi =
τ ∈k
R
P P · · · P( f τ (k) ) f τ (k−1) · · · f τ (2) f τ (1) ∀ k ≥ 2
under adequate integrability assumptions on the functions f i . Zeta functions generalize to zeta functions associated to elliptic classical pseudodifferential operators on a closed manifold M defined by ζ A (z) = λ−z n λn ∈Spec(A),λn =0
modulo some extra under assumptions on the leading symbol of the operator A to ensure the existence of its complex power A−z . If σ A (z) denotes the symbol of this complex power then provided the order of A is positive, for Re(z) large enough, ζ A is actually an integral of the symbol on the cotangent bundle T ∗ M: ζ A (z) = dx tr x (σ A (z))(x, ξ ) d¯ξ M
Tx∗ M
dξ with d¯ξ := (2π function on )n , n being the dimension of M. It extends to a meromorphic the whole plane replacing the ordinary integral by a cut-off integral −T ∗ M . The main purpose of this paper is to establish shuffle relations for cut-off integrals of classical symbols σi ∈ C S αi (Ui ) (see notations in the Preliminaries): k
− σi = − P · · · P P(στ (k) ) στ (k−1) · · · στ (2) στ (1) ∀ k ≥ 2 i=1
τ ∈k
and other regularised integrals built from cut-off integrals. We give sufficient assumptions on the symbols for such shuffle relations to hold, conditions which we shall specify below, once we have introduced the necessary technical tools. It turns out that on the class of non-integer order classical symbols, on which these regularised integrals have the expected properties such as Stokes’ property, translation invariance...(see [MMP]), these shuffle relations hold. Otherwise a renormalisation procedure is needed to take care of obstructions to these shuffle relations. In order to make this statement precise, we first need to extend cut-off and other regularised integrals on classical symbols to cut-off and other regularised iterated integrals on tensor products of classical symbols; they are all continuous linear forms on spaces of symbols which naturally extend to continuous linear forms on the (closed) tensor product. The Wodzicki residue, which is also continuous on classical symbols of fixed order, extends in a similar way to a higher order residue density r esx,k at point k C S(U ) and the ˆ i=1 x = (x1 , . . . , xk ) ∈ U = U1 × · · · × Uk on the tensor product ⊗ i well-known relation expressing the ordinary residue density resx := resx,0 as a complex residue: 1 Resz=0 − σ (z)(x, ξ ) dξ = − resx (σ (0)) ∀σ ∈ C S(U ) ∗ α (0) Tx U k C S(U ). Here σ (z) is a holomorphic family of classical symbols with ˆ i=1 extends to ⊗ i order α(z) such that α (0) = 0.
16
D. Manchon, S. Paycha
k C S αi (U ) is meromorphic ˆ i=1 Indeed, the map z → −T ∗ U σ (z)(x, ξ ) dξ with σ ∈ ⊗ i x with poles of order no larger than k and we have (see Theorem 2) (−1)k k ˆ i=1 r esx,k (σ (0)) ∀σ ∈ ⊗ C S(Ui ), (1) Reskz=0 − σ (z)(x, ξ ) dξ = k
(0) α Tx∗ U i=1 i which is independent of the choice of regularisation R : σ → σ (z) which sends the symbol σ to a holomorphic family of symbols σ (z) such that σ (0) = σ . Another approach to regularised iterated integrals is to consider the operator σ → P(σ ), P(σ )(η) = σ (ξ ) dξ. |ξ |≤|η|
It maps σ ∈ C S(U ) to a symbol P(σ ) which is not anymore classical, since it raises the power of the logarithm entering the asymptotic expansion of the symbol by one. The fact that the algebra of classical symbols is not stable under the action of P justifies the introduction of log-polyhomogeneous symbols in this context (see e.g. [L] for an extensive study of log-polyhomogeneous symbols and operators). Indeed, the operator P satisfies a Rota-Baxter relation (of weight zero): P(σ ) P(τ ) = P(σ P(τ )) + P(τ P(σ )) and defines a Rota-Baxter operator on the algebra of logpolyhomogeneous symbols (see Proposition 3). In one dimension the Rota-Baxter relation is an integration by parts formula in disguise but for higher dimensions, this Rota Baxter formula does not merely reduce to an integration by parts formula. However, similarities are to be expected between the obstructions to shuffle relations for regularised integrals studied here and the obstructions to Stokes’ formula for regularised integrals of symbol valued forms studied in [MMP]. In both cases the obstructions disappear under a non-integrality assumption on the orders of the symbols involved. It is interesting to note that regularised integrals behave nicely specifically on symbols of non-integer order, namely when they obey Stokes’ property [MMP] and have good transformation properties [L, MMP]. Unlike in the previous approach, we now take a fixed open subset U ∈ Rn so that Ui = U, i = 1, . . . , k. From a tensor product σ = σ1 ⊗ · · · ⊗ σk of classical symbols σi ∈ C S(U ) and operators σ → Pk (σ ), (σ )(x; ξ1 , . . . , ξk ) := P (σ (x; ξ1 , . . . , ξk , ·)) (ξk ), for fixed x ∈ U , one builds a map (x, ξ ) → (P1 ◦ · · · ◦ Pk−1 (σ )) (x, ξ ) which is logpolyhomogeneous. The regularised cut-off iterated integral of σ can then be seen as an ordinary regularised cut-off integral (extended by M. Lesch [L] to logpolyhomogeneous symbols) on the logpolyhomogeneous symbol P1 ◦ · · · ◦ Pk−1 (σ ) in our case1 : − dξ1 P1 ◦ · · · ◦ Pk−1 (στ ) . − σ = Tx∗ U
∗ τ ∈k Tx U
1 Similar nested integrals arise in D.Kreimer’s work [K1] in relation to a change of scale in the renormalisation procedure. His rooted trees describing nested integrations can be adapted to our context, decorating trees with symbols σi . We thank D. Kreimer for pointing this reference out to us, which we read after this article was completed.
Shuffle Relations for Regularised Integrals of Symbols
17
When σ = ⊗ σi and the (left) partial sums α1 + α2 + · · · + α j , j = 1, . . . , k of the orders αi of the symbols σi ∈ C S(U ) are non-integer, the following shuffle relations hold (see Theorem 4): k −
∗ i=1 Tx U
σi =
−
∗ τ ∈k Tx U
dξ1 P1 ◦ · · · ◦ Pk−1 (στ ) ,
(2)
k σ where we have set στ := ⊗i=1 τ (i) . A holomorphic regularisation procedure R : σ → σ (z) on C S(U ) (with some continuity assumption) induces a regularisation procedure σ1 ⊗· · ·⊗σk → σ1 (z)⊗· · ·⊗σk (z) ˆ k C S(U ). Using results by Lesch [L] on cut-off integrals of holomorphic on ⊗ families of logpolyhomogeneous symbols we build meromorphic maps z → −T ∗ U σ (z) with poles x ˆ k C S(U ). of order at most k for any σ ∈ ⊗ When σ (z) has order Eq. (2) implies the following equality of meromorphic functions k −
∗ i=1 Tx U
σi (z) =
−
∗ τ ∈k Tx U
k dξ P1 ◦ · · · ◦ Pk−1 ⊗i=1 στ (i) (z) .
(3)
But in general, the constant term in the meromorphic expansion on the l.h.s does not R coincide with the product of the regularised integrals −T ∗ U σi := fpz=0 − σi (z), namely x in general fpz=0
k −
∗ i=1 Tx U
σi (z) =
k R − σi . ∗ i=1 Tx U
However, shuffle relations extend to these regularised integrals provided the symbols involved have vanishing Wodzicki residue (see Corollary 2): k R R − σi = − dξ1 P1 ◦ · · · ◦ Pk−1 (στ ) . ∗ i=1 Tx U
∗ τ ∈k Tx U
For general symbols, a renormalisation procedure borrowed from physicists keeps track of counterterms one needs to introduce in order to pick the “right” finite part thereby circumventing the problem that “taking finite parts” does not commute with “taking products” of meromorphic functions. The above constructions are adapted in Sect. 5 to invariant classical pseudodifferential operators acting on sections over the unit circle S 1 . Using the identification S 1 R/2π Z, one can relate the shuffle relations for integrals of the symbol of the modulus of the Dirac operator on the circle with “second shuffle relations” for multiple zeta-functions. The adaptation is not straightforward as the symbol is not a smooth function anymore; since it involves Dirac measures the integrals turn out to be discrete sums. The Euler-MacLaurin formula is the main tool which enables us to go from integrals of symbols to discrete sums of symbols. These shuffle relations for regularised integrals of symbols and their link with shuffle relations for zeta functions are a hint towards deeper algebraic structures underlying cut-off multiple integrals on one hand and renormalisation procedures in quantum field theory on the other hand (see Sect. 5.5).
18
D. Manchon, S. Paycha
It appears from the investigations carried out here, that iterated integrals of symbols seem to provide a stepping stone between Feynman type integrals in physics and the renormalisation procedures used to handle their divergences on one hand and multiple zeta functions and the regularised shuffle relations they obey, a line of thought we pursue further in [MP]. 2. Preliminaries For α ∈ R, k ∈ N, the set C S α,k (U ) of scalar valued logpolyhomogeneous symbols of order α on an open subset U of Rn can be equipped with a Fréchet structure. Such a symbol reads: σ =
N −1
ψ σα−m + σ(N ) ,
(4)
m=0
where ψ is a smooth function which vanishes at 0 and equals to one outside a compact, where σα−m (x, ξ ) = kp=0 σα−m, p (x, ξ ) log p |ξ | ∈ C ∞ (S ∗ U ) with σα−m, p (x, ξ ) positively homogeneous in ξ of order α − m and where σ(N ) ∈ C ∞ (S ∗ U ) is a symbol of order α − N . The following semi-norms labelled by multiindices γ , β and integers m ≥ 0, p ∈ {1, . . . , k}, N give rise to a Fréchet topology on C S α,k (U ): γ β
supx∈K ,ξ ∈Rn (1 + |ξ |)−α+|β| |∂x ∂ξ σ (x, ξ )|;
N −1 −α+N +|β| γ β supx∈K ,ξ ∈Rn |ξ | |∂x ∂ξ σ − ψ(ξ ) σα−m (x, ξ )|; m=0 γ β supx∈K ,|ξ |=1 |∂x ∂ξ σα−m, p (x, ξ )|,
where K ranges over compact sets in U . Remark 1. Note that the first set of norms corresponds to the ordinary symbol topology, the second set of norms controls the rest term σ(N ) whereas the last set of norms is the ordinary supremum norm on the homogeneous components of the symbol. Let us introduce some notations. The set C S −∞ (U ) := m∈R C S m (U ) corresponds to the algebra of smoothing symbols. The set C S Z,∗ (U ) := C S m,k (U ) m∈Z k∈N
of integer order log-polyhomogeneous symbols, which is equipped with an inductive limit topology of Fréchet spaces is strictly contained in the algebra generated by logpolyhomogeneous symbols of any order C S ∗,∗ (U ) := C S m,k (U ). m∈R k∈N
Following [KV] (see also [L]), we extend the continuity on symbols of fixed order to families of symbols with varying order as follows:
Shuffle Relations for Regularised Integrals of Symbols
19
Definition 1. Let k be a non-negative integer. A map b → σ (b) ∈ C S ∗,k (U ) of symbols parametrized by a topological space B is continuous if the following assumptions hold: 1. the order α(b) of σ (b) is continuous in b, 2. for any non-negative integer j, the homogeneous components σα(b)− j,l (b)(x, ξ ), 0 ≤ l ≤ k of the symbol σ (b)(x, ξ ) yield continuous maps b → σα(b)− j (b) := k l ∞ ∗ l=0 σα(b)− j,l log |ξ | into C (T U ), 3. for any sufficiently large integer N , the truncated kernel (N ) K (b)(x, y) := dξ eiξ ·(x−y) σ(N ) (b)(x, ξ ), Tx∗ U
where σ(N ) (b)(x, ξ ) := σ (b)(x, ξ ) −
N
ψ(ξ ) σα(b)− j (b)(x, ξ )
j=0
yields a continuous map b → σ(N ) (b) into some C K (N ) (U × U ) where lim N →∞ K (N ) = +∞. 3. Regularised Integrals of log-Polyhomogeneous Symbols We recall for completeness, well-known results on regularisation techniques of integrals of ordinary log-polyhomogeneous symbols which lead to trace functionals on the corresponding pseudodifferential operators. 3.1. Cut-off integrals of log-polyhomogeneous symbols. We start by recalling the construction of cut-off integrals of log-polyhomogeneous symbols [L] which generalizes results previously established by Guillemin and Wodzicki in the case of classical symbols. Lemma 1. Let U be an open subset of Rn and for any non-negative integer k, let σ ∈ C S ∗,k (U ) be a log-polyhomogeneous symbol, then for any x ∈ U , • B ∗ (0,R) σ (x, ξ )dξ has an asymptotic expansion in R → ∞ of the form: x
Bx∗ (0,R)
+
k
Pl (σα− j,l )(log R) R α− j+n
j=0,α− j+n=0 l=0
k resx,l (σ ) l=0
∞
σ (x, ξ )dξ ∼ R→∞ C x (σ ) +
l +1
logl+1 R,
(5)
where Pl (σα− j,l )(X ) is a polynomial of degree l with coefficients depending on σα− j,l and where C x (σ ) is the constant term corresponding to the finite part: C x (σ ) := σ(N ) (x, ξ ) dξ + ψ(ξ )σ (x, ξ ) dξ Tx∗ U
+
Bx∗ (0,1)
N
k
j=0,α− j+n=0 l=0
(−1)l+1 l! (α − j + n)l+1
which is independent of N ≥ α + n − 1.
Sx∗ U
σα− j,l (x, ξ )d S ξ
20
D. Manchon, S. Paycha
• For any fixed μ > 0, fp R→∞
Bx∗ (0,μ R)
σ (x, ξ )dξ = fp R→∞
Bx∗ (0,R)
σ (x, ξ ) dξ +
k logl+1 μ l=0
l +1
· resl,x (σ ).
Remark 2. If σ is a classical operator, setting k = 0 in the above formula yields
fp R→∞
Bx∗ (0,R)
σ (x, ξ )dξ :=
Tx∗ U
σ(N ) (x, ξ ) dξ +
N j=0
N
−
j=0,α− j+n=0
1 α− j +n
Bx∗ (0,1)
ψ(ξ ) σα− j (x, ξ ) dξ
Sx∗ U
σα− j (x, ω)dω.
Proof. Given a log-polyhomogeneous symbol σ ∈ C S α,∗ (U ), for any N ∈ N we write: σ (x, ξ ) =
N
ψ(ξ )σα− j (x, ξ ) + σ(N ) (x, ξ ) ∀(x, ξ ) ∈ T ∗ U,
(6)
j=0
where σ(N ) ∈ S α−N −1 (U ). • For some fixed N ∈ N chosen large enough such that α − N − 1 < −n, we write σ (x, ξ ) = Nj=0 ψσα− j (x, ξ ) + σ(N ) (x, ξ ) and split the integral accordingly: Bx∗ (0,R)
σ (x, ξ )dξ =
N j=0
ψ(ξ )σα− j (x, ξ )dξ +
Bx∗ (0,R)
Bx∗ (0,R)
σ(N ) (x, ξ )dξ.
Since α − N − 1 < −n, σ(N ) lies in L 1 (Tx∗ U ) and the integral B ∗ (0,R) σ(N ) (x, ξ )dξ x converges when R → ∞ to T ∗ U σ(N ) (x, ξ )dξ . On the other hand, for any j ≤ N , x
Bx∗ (0,R)
ψ(ξ )σα− j (x, ξ ) =
Bx∗ (0,1)
ψ(ξ )σα− j (x, ξ ) +
Dx∗ (1,R)
σα− j (x, ξ ), (7)
since ψ is constant equal to 1 outside the unit ball. Here Dx∗ (1, R) = Bx∗ (0, R) − Bx∗ (0, 1). The first integral on the r.h.s. converges and since σα− j (x, ξ ) =
k
σα− j,l (x, ξ ) logl |ξ |,
l=0
the second integral reads: Dx∗ (1,R)
σα− j (x, ξ )dξ =
k l=0
R 1
r α− j+n−1 logl r dr ·
Sx∗ U
σα− j,l (x, ω)dω.
Shuffle Relations for Regularised Integrals of Symbols
21
Hence the following asymptotic behaviours: k logl+1 R dξ σα− j (x, ξ ) ∼ R→∞ σα− j,l (x, ω)dω · l +1 Dx∗ (1,R) Sx∗ U l=0
=
k logl+1 R
l +1
l=0
resl,x (σ ) if α − j = −n
whereas:
l l! k (−1)i+1 (l−i)! logi R α− j+n σα− j ∼ R→∞ · R σα− j,l (x, ω)dω ∗U (α − j + n)i Dx∗ (1,R) S x l=0 i=0 α− j+n R l · σα− j,l (x, ω)dω + (−1) l! (α − j + n)l+1 Sx∗ U (−1)l+1 l! + · σα− j,l (x, ω)dω if α − j = −n. (α − j + n)l+1 Sx∗ U Putting together these asymptotic expansions yields the statement of the proposition with L N N (−1)l+1 l! C x (σ ) = σ(N ) + ψσa j + σa ,l . (a j + n)l+1 Sx∗ U j Tx∗ U Bx∗ (0,1) j=0,a j +n=0 l=0
j=0
• The μ-dependence follows from
logl+1 (μ R) = logl+1 R
1+
log μ log R
∼ R→∞ logl+1 R
l+1
l+1 k Cl+1
k=0
log μ log R
k .
k resl,x (σ ) l+1 The logarithmic terms l=0 l+1 log (μ R) therefore contribute to the finite k logl+1 μ part by l=0 l+1 · resl,x (σ ) as claimed in the lemma. Discardingthe divergences, we can therefore extract a finite part from the asymptotic expansion of B(0,R) σ (x, ξ )dξ and set for σ ∈ C S ∗,k (Rn ): Definition 2. Given a non-negative integer k, an open subset U ⊂ Rn and a point x ∈ U , for any σ ∈ C S α,k (U ), the cut-off integral − σ (x, ξ )dξ := fp R→∞ σ (x, ξ )dξ Tx∗ U
Bx∗ (0,R)
=
+
Tx∗ U
σ(N ) (x, ξ ) dξ +
N j=0
N
k
j=0,α− j+n=0 l=0
is independent of N > α + n − 1.
Bx∗ (0,1)
(−1)l+1 l! (α − j + n)l+1
ψ(ξ )σα− j (x, ξ ) dξ
Sx∗ U
σα− j,l (x, ξ ) d S ξ
(8)
22
D. Manchon, S. Paycha
It is independent of the parametrisation R provided the higher Wodzicki residue σ−n,l (x, ξ )d S ξ resx,l := Sx∗ U
vanishes for all integer 0 ≤ l ≤ k. This explicit description of the finite part leads to the following continuity result. Proposition 1. For any fixed α ∈ R and any non-negative integer k, and given an open subset U ∈ Rn , a point x ∈ U , the map C S α,k (U ) → C ∞ (U, C)
x → −
σ →
Tx∗ U
σ (x, ξ ) dξ
is continuous in the Fréchet topology of C S α,k (U ) and the natural topology of C ∞ (U, C). Remark 3. The assumption that α be fixed is essential here. Proof. From formula (8) and the fact that symbols are smooth functions on U × Rn , it follows that the cut-off is C ∞ (U, C)-valued. integral The maps σ → x → B ∗ (0,1) ψ(ξ )σα− j (x, ξ ) dξ and σ → x → S ∗ U σα− j,l x x (x, ξ ) d S ξ are clearly continuous as integrals over compact sets of continuous maps. On the other hand the map σ → x → T ∗ U σ(N ) (x, ξ ) dξ is continuous since σ → σ(N ) x
is continuous and σ(N ) (x, ξ ) ≤ C(1 + |ξ |)−N can be uniformly bounded by an L 1 function. As well as the higher order residue density function resx,k , one can define on C S ∗,k (U ) an extension of the ordinary residue density function resx as follows: resx (σ ) := (σ (x, ξ ))−n d S ξ, Sx∗ U
where d S ξ is the volume measure on the unit cotangent sphere Sx∗ U induced by the canonical volume measure on Tx∗ U . Even though it certainly does not induce a graded trace on the algebra of log-polyhomogeneous operators on a closed manifold as the higher order residue does [L], it is a useful tool for what follows since we have the following continuity result: Lemma 2. Given any non-negative integer k, and given any α ∈ R, the map: C S α,k (U ) → C ∞ (U, C) σ → (x → resx (σ )) is continuous for the Fréchet topology on C S α,k (U ).
Shuffle Relations for Regularised Integrals of Symbols
23
3.2. Integrals of holomorphic families of log-polyhomogenous symbols. Following [KV] (see also [L]), we define a holomorphic family of log-polyhomogeneous symbols in C S ∗,k (U ) in the same way as in Definition 1 replacing continuous by holomorphic. We quote from [PS] the following theorem which extends results of [L] relating the Wodzicki residue of holomorphic families of log-polyhomogeneous symbols with higher Wodzicki residues. For simplicity, we restrict ourselves to holomorphic families with order α(z) given by an affine function of z, a case which covers natural applications. Theorem 1. Let U be an open subset of Rn and let k be a non-negative integer. For any holomorphic family z → σ (z) ∈ C S α(z),k (U ) of symbols parametrised by a domain W ⊂ C such that z → α(z) = α (0) z+α(0) is an affine function with α (z) = α (0) = 0, then for any x ∈ U , there is a Laurent expansion in a neighborhood of any z 0 ∈ P, −
Tx∗ U
σ (z)(x, ξ )dξ = fpz=z 0 −
Tx∗ U
+
K
σ (z)(x, ξ )dξ +
k+1 r j (σ )(z 0 )(x) j=1
(z − z 0 ) j
s j (σ )(z 0 )(x) (z − z 0 ) j + o (z − z 0 ) K ,
j=1
where for 1 ≤ j ≤ k + 1, R j (σ )(z 0 )(x) is locally explicitly determined by a local expression (see [L] for the case α (0) = 1) r j (σ )(z 0 )(x) :=
k l= j−1
(−1)l+1 l! (l+1− j) res σ (z 0 ). x (l) (α (z 0 ))l+1 (l + 1 − j)!
(9)
Here σ(l) (z) is the local symbol given by the coefficient of logl |ξ | of σ , i.e. σ (z) =
k
σ(l) (z) logl |ξ |.
l=0
On the other hand, the finite part fpz=z 0 −T ∗ U σ (z)(x, ξ )dξ consists of a global piece x −Rn σ (z 0 )(x, ξ ) dξ and a local piece: σ (z 0 )(x, ξ ) dξ fpz=z 0 − σ (z)(x, ξ )dξ = − Tx∗ U
Tx ∗U
+
k
(−1)l+1 1 (l+1) res σ (z 0 ). (10) x (l) (α (z 0 ))l+1 l + 1 l=0
Finally, for 1 ≤ j ≤ K , S j (σ )(z 0 )(x) reads s j (σ )(z 0 ) := − σ ( j) (z 0 ) dξ Tx∗ U
+
k l=0
(−1)l+1 l! j! ( j+l+1) res σ (z ) . x 0 (l) (α (z 0 ))l+1 ( j + l + 1)!
(11)
24
D. Manchon, S. Paycha
As a consequence, the finite part fpz=z 0 −T ∗ U σ (z)(x, ξ )dξ is entirely determined by the x
derivative α (z 0 ) of the order and by the derivatives of the symbol σ (l) (z 0 ), l ≤ k + 1 via the cut-off integral and the Wodzicki residue density. 3.3. Regularised integrals of log-polyhomogeneous symbols. Let us briefly recall the notion of holomorphic regularisation taken from [KV] (see also [PS]).
Definition 3. A holomorphic regularisation procedure on C S ∗,k (U ) for any fixed nonnegative integer k is a map R : C S ∗,k (U ) → Hol C S ∗,k (U ) σ → σ (z),
∗,k where Hol C S (U ) is the algebra of holomorphic maps with values in C S ∗,k (U ), such that 1. σ (0) = σ , 2. σ (z) has holomorphic order α(z) (in particular, α(0) is equal to the order of σ ) such that α (0) = 0. We call a regularisation procedure R continuous whenever the map R : C S ∗,k (U ) → Hol C S ∗,k (U ) σ → (z → σ (z)) is continuous. Remark 4. It is easy to check [PS] that if z → σ (z) ∈ C S α(z),k (U ) then σ ( j) (z 0 ) ∈ C S α(z 0 ),k+ j (U ). Examples of holomorphic regularisations are the well known Riesz regularisation σ → σ (z)(x, ξ ) := σ (x, ξ ) · |ξ |−z and generalisations of the type σ → σ (z)(x, ξ ) := H (z) · σ (x, ξ ) · |ξ |−z , where H is a holomorphic function such that H (0) = 1. The latter include dimensional regularisation (see [P]). These regularisation procedures are clearly continuous. As a consequence of the results of the previous paragraph, given a holomorphic reg∗,k ∗,k ularisation procedure R : σ → σ (z) on C S (U ) and a symbol σ ∈ C S (U ), for every point x ∈ U , the map z → −T ∗ U σ (z)(x, ξ ) dξ is meromorphic with poles of x
order at most k + 1 at points in α −1 ([−n, +∞[ ∩ Z), where α is the order of σ (z) so that we can define the finite part when z → 0 as follows.
Definition 4. Given a holomorphic regularisation procedure R : σ → σ (z) on C S ∗,k (U ), a symbol σ ∈ C S ∗,k (U ) and any point x ∈ U , we define the regularised integral R σ (x, ξ ) dξ := fpz=0 − σ (z)(x, ξ ) dξ Tx∗ U
T ∗U
⎛ x := lim ⎝− z→0
Tx∗ U
⎞ k+1 1 j dξ σ (z)(x, ξ ) − Resz=0 − dξ σ (z)(x, ξ )⎠ . zj Tx∗ U
We have the following continuity result.
j=1
Shuffle Relations for Regularised Integrals of Symbols
25
Proposition 2. Given a continuous holomorphic regularisation procedure R : σ → σ (z) on C S ∗,k (U ), where k is a non-negative integer, for any fixed α ∈ R, there is a discrete set Pα ⊂ C such that the map C S α,k (U ) → C ∞ (U, Hol(C − Pα )) σ → − σ (x, ξ )(z) dξ Tx∗ U
is continuous on
C ∞ (U, Hol(C −
Pα )). Moreover the map
C S α,k (U ) → C ∞ (U, C) R σ → − σ (x, ξ ) dξ Tx∗ U
is continuous on C S α,k (U ). Remark 5. The assumption that α be constant is essential here. Proof. From Theorem 1 we know that the map z → −T ∗ U σ (z)(x, ·) is meromorphic x with simple poles in some discrete set Pα . From Proposition 1 we know that the map σ → −σ is continuous. Combining these two results gives the continuity of the map σ → z → −T ∗ U σ (x, ξ )(z) dξ , where the r.h.s. is understood as a holomorphic map x on C − Pα . We now prove the second part of the proposition. By Theorem 1 applied to z 0 = 0, it is sufficient to check that the maps σ → −T ∗ U σ (0)(x, ξ ) dξ and the maps σ → x
resx σ ( j) (0) are C ∞ (U, C) valued and continuous for any 1 ≤ j ≤ k + 1 for the Fréchet topology on log-polyhomogenous symbols and the Fréchet topology on smooth functions. From the continuity assumption on the regularisation R combined with Proposition 1 and Lemma 2 it follows that for a log-polyhomogeneous symbol τ , both (xj) → −T ∗ U τ (x, ξ ) dξ and x → resx (τ ) are smooth functions. Applying this to τ = σ (0) x (which is log-polyhomogeneous by the above remark) with 0 ≤ j ≤ k + 1 yields the result. 4. Regularised Integrals on Tensor Products of Classical Symbols 4.1. Tensor products of symbols. Let U1 , . . . , U L be open subsets of Rn . Since the spaces C S m i (Ui ) and C S m i ,ki (Ui ) are Fréchet spaces, we can form their closed tensor products, where the closed tensor product of two Fréchet spaces E and F is the Fréchet space ˆ built as the closure of E ⊗ F for the finest topology for which ⊗ : E × F → E ⊗ F E ⊗F is continuous. Definition 5. For any multiindices (m 1 , . . . , m L ) ∈ R L , (k1 , . . . , k L ) ∈ N L we set L ˆ i=1 C S m i (Ui ) C Sw(m 1 ,...,m L ) (U1 × · · · × U L ) := ⊗
and L ˆ i=1 C Sw(m 1 ,...,m L ) ,(k1 ,...,k L ) (U1 × · · · × U L ) := ⊗ C S m i ,ki (Ui ) .
The multiindex (m 1 , . . . , m L ) is called the multiple order of σ and m 1 + · · · + m L its total order.
26
D. Manchon, S. Paycha
There are at least two ways of continuously extending regularised integrals to tensor products of symbols. 4.2. A first extension of regularised integrals to tensor products. Definition 6. Let U = U1 × · · · × U L with x = (x1 , . . . , x L ), xi ∈ Ui , i = 1, . . . , L open subsets in Rn . Let (α1 , . . . , α L ) ∈ Cl and let (k1 , . . . , k L ) be a multiindex of non-negative integers. The continuous maps C S αi ,ki (Ui ) → C ∞ (Ui , C)
σi →
xi → −
Tx∗ Ui
σi (xi , ξi ) dξi , i = 1, . . . , L
i
induce a uniquely defined map: C Sw(α1 ,...,α L ),(k1 ,...,k L ) (U ) → C ∞ (U, C)
σ →
x →=
Tx∗ U
σ (x, ξ ) dξ1 · · · dξ L
ˆ k C S(Ui ) called the multiple regularised cut-off which gives rise to a linear map on ⊗ integral of σ (x, ·). k σ (x , ·) we have: Clearly, if σ (x, ·) = ⊗i=1 i i
=
Tx∗ U
σ (x, ξ ) dξ1 · · · dξ L =
L = i=1
Tx∗ Ui
σ (xi , ξi ) dξi .
i
The following extends holomorphic regularisations to tensor products of symbol spaces. Definition 7. Let U = U1 × · · · × U L be a product of open subsets of Rn . For a given multiindex (k1 , . . . , k L ) with ki non-negative integers, a regularisation procedure R on C Sw∗,(k1 ,...,k L ) (U ) is a map: R : C Sw∗,(k1 ,...,k L ) (U ) → Hol C Sw∗,(k1 ,...,k L ) (U ) σ → R(σ ) : z → σ (z) such that 1. σ (0) = σ , 2. σ (z) has holomorphic (multiple) order α(z) = (α1 (z), . . . , α L (z)) ∈ R L (in par ticular, α(0) is equal to the (multiple) order of σ ) such that Re αi (0) > 0 for all i ∈ {1, . . . , L}. ∗,(k ,...,k L ) Here Hol C Sw 1 (U ) is the algebra of holomorphic maps with values in C S ∗,k (U ). Clearly, regularisation procedures R1 , . . . , R L on C S ∗,k1 (U1 ), . . . , C S ∗,k L (U L ) induce L R on C S ∗,(k1 ,...,k L ) (U ), which we refer to as a ˆ i=1 a regularisation procedure R = ⊗ w i product regularisation procedure.
Shuffle Relations for Regularised Integrals of Symbols
27
Definition 8. Let U = U1 × · · · × U L with Ui , i = 1, . . . , L open subsets in Rn and let (k1 , . . . , k L ) be a multiindex of non-negative integers. Given a product regularisation procedure L k k ˆ i=1 R=⊗ Ri : σ = ⊗i=1 σi → σ (z) = ⊗i=1 σi (z) k C S(U ) of continuous regularisations R , i = 1, . . . , L, the continuous maps ˆ i=0 on ⊗ i i
C S αi (Ui ) → C ∞ (Ui , Hol(C − Pi ))
σi →
xi →
Tx∗ Ui
Ri (σi )(z)(xi , ξi ) dξi , i = 1, . . . , L
i
induce a uniquely defined map: k k ˆ i=0 C S αi (Ui ) → C ∞ (U, Hol(C − ∪i=1 Pi )) ⊗
σ →
x →=
Tx∗ U
R(σ )(z)(x, ξ ) dξ1 · · · dξ L .
Similarly the continuous maps C S αi (Ui ) → C ∞ (Ui , C)
σi →
xi → −
Ri
Tx∗ Ui
σi (xi , ξi ) dξi , i = 1, . . . , L
i
induce a uniquely defined map: k ˆ i=0 C S αi (Ui ) → C ∞ (U, C) ⊗
σ →
x →=
R
Tx∗ U
σ (x, ξ ) dξ ,
k C S(U ) called the multiple regularised integral ˆ i=0 which induces a linear map on ⊗ i associated with the product regularisation R.
The Wodzicki residue density resxi on C S(Ui ) similarly give rise by continuity to k C S(U ) in such a way that for any x = (x , . . . , x ) ∈ U × · · · × U : ˆ i=1 r esx,k on ⊗ i 1 L 1 L r esx,L (⊗σi (xi , ·)) =
L
resxi (σi (xi , ·)).
i=1
Theorem 2. Let U = U1 × · · · × U L with Ui , i = 1, . . . , L open subsets in Rn and let L C S(U ). Given a product regularisation procedure ˆ i=1 σ ∈⊗ i L L L ˆ i=1 R=⊗ Ri : ⊗i=1 σi → ⊗i=1 σi (z)
on C Sw (U ) of continuous regularisations Ri , i = 1, . . . , L such that Ri (σ )(z) has order αi (z), the map z →= R(σ )(z)(x, ξ ) dξ1 · · · dξ L Tx∗ U
28
D. Manchon, S. Paycha
is meromorphic with poles at most of order L and: L Resz=0 =
Tx∗ U
(−1) L r esx,L (σ ). R(σ )(z)(x, ξ ) dξ1 · · · dξ L = L
i=1 αi (0)
In particular, when αi (0) = α (0) is constant this yields L Resz=0
=
Tx∗ U
R(σ )(z)(x, ξ ) dξ1 · · · dξ L =
(−1) L (α (0)) L
r esx,L (σ ).
Proof. By a continuity argument, this follows from the fact that this same relation holds L σ : on products σ = ⊗i=1 i L Resz=0 =
L
Tx∗ U i=1
Ri (σi )(z)(xi , ξi ) dξ1 · · · dξ L =
L
Resz=0 −
i=1
=
Tx∗ Ui
Ri (σi )(z)(xi , ξi ) dξi
i
L −1 resxi (σi ) αi (0) i=1
(−1) L r esx,L (σ ). = L
i=1 αi (0) On the grounds of this theorem, taking finite parts we set: L R on C S (U ), for any σ ∈ ˆ i=1 Definition 9. Given a product regularisation R = ⊗ i w C Sw (U ) we call
R = σ (x, ξ ) := fpz=0 = Tx∗ U
Tx∗ U
σ (z)(x, ξ ) dξ
with R : σ → σ (z), the R-regularised iterated integral of σ . Remark 6. With these notations we have: R L Ri L = dξ ⊗i=1 σi (xi , ξi ) = = dξi σi (xi , ξi ). Tx∗ U i
i=1
Tx∗ Ui i
5. An Alternative Extension of Regularised Integrals to Tensor Products of Classical Symbols We now give an alternative extension of regularised integrals to tensor products of classical symbols which we then compare withthe one previously defined. For this purpose we consider a map similar to the map σ → |ξ |≤R σ (x, ξ )dξ underlying the construction of cut-off integrals. We will henceforth work under the assumption U1 = · · · = Uk = U an open subset of Rn .
Shuffle Relations for Regularised Integrals of Symbols
29
5.1. Rota-Baxter relations Proposition 3. 1. The map σ → P(σ ) defined by σ (x, ξ ) dξ P(σ )(x, η) := |ξ |≤|η|
maps C S ∗,k−1 (U ) to C S ∗,k (U ). Given σ ∈ C S ∗,k−1 (U ), P(σ ) = C+τ for some constant C and with τ ∈ C S α+n,k . In particular, when α ∈ R, it has order max(0, α + n). For any σ ∈ C S ∗,k−1 (U )
2.
P(σ )(x, η) −
resx,k−1 (σ ) logk |η| ∈ C S ∗,k−1 (U ) k
(12)
so that if σ has vanishing residue of order k − 1 then P(σ ) also lies in C S ∗,k−1 (U ). 3. P obeys the following Rota-Baxter relation [EGK]: P(σ ) P(τ ) = P(σ P(τ )) + P(τ P(σ )).
(13)
Proof. Replacing R by |η| in the asymptotic expansion (5) yields: P(σ )(x, η) ∼ C x (σ ) +
∞
k−1
Pl (σα− j,l )(log |η|) |η|α− j+n
j=0,α− j+n=0 l=0
+
k−1 l=0
resx,l (σ ) logl+1 |η|, l +1
(14)
where Pl (σα− j,l )(X ) is a polynomial of degree l with coefficients depending on σα− j,l and where C x (σ ) is the constant term corresponding to the finite part. P(σ ) is therefore the sum of a symbol of order zero (the constant C x (σ )) and a symbol τ of order α + n so that when α ∈ R, its order is max(0, α + n). Furthermore, it res (σ ) lies in C S α,k (U ) and the coefficient of logk |η| is x,k−1 . k The Rota-Baxter relation then follows from: P(σ )(η) P(τ )(η) = σ (ξ ) dξ τ (ξ ) dξ |ξ |≤|η| |ξ |≤|η| ˜ ˜ = σ (ξ ) dξ τ (ξ ) d ξ + τ (ξ ) dξ σ (ξ˜ ) d ξ˜ |ξ |≤|η|
|ξ˜ |≤|ξ |
|ξ |≤|η|
|ξ˜ |≤|ξ |
= P(σ P(τ ))(η) + P(τ P(σ ))(η). k+1 C S ∗,∗ (U ) be the space of k-chains built from C S ∗,∗ (U ). Using the ˆ i=1 Let Ck := ⊗ Rota-Baxter map we define a map
P• : C•+1 → C• by k+1 k ˆ i=1 ˆ i=1 C S ∗,∗ (U ) → ⊗ C S ∗,∗ (U ) Pk : ⊗ Pk (σ )(x, ξ1 , . . . , ξk ) := P (σ (·, ξ1 , . . . , ξk , ·)) (x, ξk ).
In particular we have: Pk (σ1 ⊗ · · · ⊗ σk+1 ) (x, ξ1 , . . . , ξk ) = σ1 (x, ξ1 ) · · · σk (x, ξk ) P(σk+1 )(x, ξk ).
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D. Manchon, S. Paycha
Theorem 3. Let U be an open subset of Rn . For any integer k > 1, k C S αi (U ) to C S ∗, k−1 (U ). ˆ i=1 1. the composition P1 ◦ · · · ◦ Pk−1 maps ⊗ For σi ∈ C S(U ),
P1 ◦ P2 ◦ · · · ◦ Pk−1 (σ1 ⊗ · · · ⊗ σk ) = P (· · · P(σk )σk−1 · · ·)σ2 ) σ1
(15)
is a finite sum of log-polyhomogeneous symbols of order given by the partial sum α1 + α2 + · · · + α j + ( j − 1)n with j = 1, . . . , k. In particular, when α1 , . . . , αk ∈ R, then P1 ◦ P2 ◦ · · · ◦ Pk−1 (σ ) has order given by o (P1 ◦ P2 ◦ · · · ◦ Pk−1 (σ )) = max (0, . . . , max(0, max(0, αk + n) + αk−1 + n), . . .) + α2 + n) + α1 . 2. Furthermore, k−1 P1 ◦ · · · ◦ Pk−1 (σ1 ⊗ · · · ⊗ σk )(x, ξ1 ) −
j=1 resx (σ j )
(k − 1)!
logk−1 |ξ1 | ∈ C S ∗,k−2 (U ). (16)
3. The following shuffle (or iterated Rota-Baxter) relations hold: k
P(σi ) =
τ ∈k
i=1
=
P ◦ P1 ◦ · · · ◦ Pk−1 (στ (1) ⊗ · · · ⊗ στ (k) )
P P · · · P(στ (k) )στ (k−1) · · ·)στ (2) στ (1) .
(17)
τ ∈k
Remark 7. For k = 2 Eq. (17) yields back Eq. (13). Proof. 1. By a continuity argument, it suffices to show that P1 ◦ P2 ◦ · · · ◦ Pk−1 (σ ) ∈ C S ∗,k−1 (U ) for any σ = σ1 ⊗ · · · ⊗ σk . This follows from the first point in Proposition 3 by induction on k. Indeed, applying it to k = 2, we first check that P1 (σ2 ) ∈ C S ∗,1 (U ); then assuming that the statement holds for k we can apply Proposition 3 to P2 ◦ P3 ◦ · · · ◦ Pk (σ2 ⊗ · · · ⊗ σk+1 ) ∈ C S ∗,k−1 (U ) from which we infer that P1 ◦ P2 ◦ · · · ◦ Pk (σ1 ⊗ σ2 ⊗ · · · ⊗ σk+1 ) = P (P2 ◦ P3 ◦ · · · ◦ Pk (σ1 ⊗ σ2 ⊗ · · · σk+1 )) ∈ C S ∗,k (U ). This formula combined with Proposition 3 also yields in a similar manner that P1 ◦ P2 ◦ P3 ◦ · · · ◦ Pk−1 (σ1 ⊗ · · · ⊗ σk ) is a finite sum of log-polyhomogeneous symbols of order α1 + · · · + α j + ( j − 1)n with j = 1, . . . , k. From there we easily derive the formula for degree of P1 ◦ P2 ◦ P3 ◦ · · · ◦ Pk−1 (σ1 ⊗ · · · ⊗ σk ) when the αi ’s are real. 2. Similarly, an induction using Eq. (12) implies Eq. (16). 3. Equation (17) follows from Eq. (13) in a similar manner.
Shuffle Relations for Regularised Integrals of Symbols
31
5.2. Iterated cut-off integrals of classical symbols. By the results of the previous parak C S(U ) to C S ∗,k−1 (U ), a space on which ˆ i=1 graph, the operator P1 ◦ · · · ◦ Pk−1 sends ⊗ we can apply cut-off regularisation described in Sect. 2. k C S(U ) and given a point ˆ i=1 Definition 10. Let U ⊂ Rn be an open subset. For σ ∈ ⊗ x ∈ U we set − σ (x, η)dη := − dξ P1 ◦ · · · ◦ Pk−1 (σ ◦ τ )(x, ξ ) (Tx∗ U )k
∗ τ ∈k Tx U
=
−
∗ τ ∈k Tx U
dξ1
|ξ2 |≤|ξ1 |
dξ2 · · ·
|ξk |≤|ξk−1 |
dξk σ (x, ξτ (1) , . . . , ξτ (k) ).
Lemma 3. Rn be an open subset. For σ1 , . . . , σk ∈ C S(U ) such that all the (left) partial sums of the orders α1 + α2 + · · · + α j , j = 1, . . . , k are non-integer valued, then k −
∗ i=1 Tx U
σi (x, ξi ) dξi = fp R→∞
k i=1 |ξi |≤R
σi (x, ξi ) dξi .
Proof. We need to show that k i=1
fp Ri →∞
|ξi ≤|Ri
σi (x, ξi ) dξi = fp R→∞
k i=1 |ξi |≤R
σi (x, ξi ) dξi .
For each i ∈ {1, . . . , k} we have the following asymptotic expansion (see Eq. (5)): |ξi |≤Ri
+
ki p=0
σi (x, ξi )dξi ∼ Ri →∞ C x (σi ) +
∞
ki
Pp (σαi −m, p )(log Ri )Riαi −m+n
m=0,αi −m+n=0 p=0
res p,x (σi ) log p+1 Ri . p+1
Multiplying these asymptotic expansions and setting Ri = R can give rise to new finite k k fp Ri →∞ |ξi |≤Ri σi (x, ξi ) dξi = i=1 C x (σi ). Indeed, when setparts other than i=1 = R = R, positive powers of R arising from the asymptotic expansion of ting R i j i σ (x, ξ ) dξ might compensate negative powers of R arising from the asympi i i j |ξi |≤Ri totic expansion of |ξ j |≤R j σi (x, ξ j ) dξ j thus leading to a new constant term. But since
such powers arise in the form R α1 +α2 +···+α j −m+ j n such a compensation can only happen if α1 + α2 + · · · + α j takes integer values. One therefore avoids such compensations assuming that none of all the (left) partial sums of the orders α1 + α2 + · · · + α j are non-integers.
We deduce from the definition and the above lemma that cut-off regularisation “commutes” with products of symbols in certain special cases: the cut-off iterated integral of a product of symbols coincides with the product of the cut-off integrals of the symbols provided these have orders whose (left) partial sums are non-integer valued.
32
D. Manchon, S. Paycha
Proposition 4. Let σi ∈ C S αi (U ), i = 1, . . . , k be such that all the (left) partial sums of the orders α1 + α2 + · · · + α j , j = 1, . . . , k are non-integer valued. Then −
k
(Tx∗ U )k i=1
k −
σi (x, ξi ) dξi =
σi (x, ξi ) dξi .
(18)
i=1 Tx U
Proof. From the above lemma it follows that k − σi (x, ξi ) dξi i=1 Tx U
= fp R→∞ = fp R→∞
k i=1 |ξi |≤R
τ ∈k
=−
σi (x, ξi ) dξi
k
|ξ1 |≤R
dξ1
|ξ2 |≤|ξ1 |
···
|ξk |≤|ξ L−1 |
dξk
k
στ (i) x, ξτ (i)
i=1
σi (x, ξi ) dξi .
(Tx∗ U )k i=1
Theorem 4. Let σi ∈ C S αi (U ), i = 1, . . . , k be such that all the (left) partial sums α1 + α2 + · · · + α j , j = 1, . . . , k are non-integer valued. Then the following shuffle relations hold: k − dξi σi ∗ i=1 Tx U
= =
−
P1 ◦ · · · ◦ Pk−1 στ (1) ⊗ · · · ⊗ στ (k) (ξ ) dξ
−
dξ1
∗ τ ∈k Tx U
∗ τ ∈k Tx U
dξ2 · · ·
|ξ2 |≤|ξ1 |
|ξk−1 |≤|ξk |
dξk−1 στ (k) (x, ξk ) · · · στ (1) (x, ξ1 ). (19)
Proof. Recall that P(σi )(xi , ηi ) = |ξ |≤|ηi | σi (x, ξ ) dξ. Applying Eq. (17) to ηi = R for i = 1, . . . k and then taking the finite part when R → ∞ yields the result: k −
∗ i=1 Tx U
σi =
k
fp R→∞
i=1
⎛
= fp R→∞ ⎝ − = τ ∈k
Tx∗ U
Bx (0,R)
σi
τ ∈k
Bx (0,R)
⎞ P1 ◦ · · · ◦ Pk−1 (στ (1) ⊗ · · · ⊗ στ (k) )⎠
P · · · P(στ (k) )στ (k−1) · · ·)στ (2) στ (1) dξ1 .
The above lemma then yields the result under the assumption that all partial orders are non-integer.
Shuffle Relations for Regularised Integrals of Symbols
33
5.3. Iterated integrals of holomorphic families of classical symbols. When the symbols have integer order, neither does the iterated cut-off integral of the tensor product of the symbols coincide with the product of their cut-off integrals (see Eq. (18)), nor do the shuffle relations (19) hold for cut-off integrals. However holomorphic perturbation of these symbols will have holomorphic orders, the (left) partial sums of which will be non-integer outside a discrete set and both Eq. (18) and the shuffle relations (19) hold for these perturbed symbols. Proposition 5. Let U be an open subset of R n . Let R : σ → σ (z) be a holomorphic regularisation procedure on C S ∗,∗ (U ) such that σ (z) has order α(z) = q z + α(0) with q = 0. For any σi ∈ C S ∗,ki (U ), i = 1, 2, with σi (z) of order αi (z) = q z + αi (0) 1. the map
z → −
Tx∗ U
P(σ2 (z))(ξ ) σ1 (z)(ξ ) dξ
is meromorphic with at most poles of order k1 + k2 + 2 in the discrete set P2 := q −1 (Z − α1 (0)) ∪ (2 q)−1 (Z − α1 (0) − α2 (0)) . 2. We have the following identity of meromorphic functions: − dξ1 σ1 (z) − dξ2 σ2 (z) Tx∗ U
=−
Tx∗ U
Tx∗ U
P (σ1 (z)) (ξ ) σ2 (z)(ξ ) dξ + −
Tx∗ U
P (σ2 (z)) (ξ ) σ1 (z)(ξ ) dξ. (20)
Proof. 1. We first observe that P(σ2 (z)) σ1 (z) is the sum of a symbol τ1 (z) ∈ C S α1 (z),k1 (U ) proportional to σ1 (z) and a symbol τ2 (z) σ1 (z) ∈ C S α1 (z)+α2 (z)+n,k1 +k2 +1 (U ) with τ2 (z) ∈ C S α2 (z)+n,k2 +1 (U ) (see Proposition 3). By Theorem 1 and using the linearity of the cut-off integral, we find that the cut-off integral − P(σ2 (z))(ξ ) σ1 (z)(ξ ) dξ = − τ1 (z)(x, ξ ) dξ + − τ2 (z)(ξ ) σ1 (z)(ξ ) dξ Tx∗ U
Tx∗ U
Tx∗ U
is meromorphic with poles of order at most k1 + k2 at points in P2 defined as in the proposition since α1 (z) = q z +α1 (0) and α1 (z)+α2 (z)+n = 2q z +α1 (0)+α2 (0)+n. 2. Equation (20) then follows from applying (19) to σi := σi (z) (with k = 2) outside the discrete set of poles. This generalises to the tensor product of k symbols. Theorem 5. Let U be an open subset of Rn and let R be a holomorphic regularisation procedure σ → σ (z) on C S(U ) such that σ (z) has order α(z) = q z + α(0) with q = 0. For any σi ∈ C S(U ) with σi (z) of order αi (z) = q z + αi (0), 1. the map z → −T ∗ U dξ P1 ◦ · · · ◦ Pk−1 (σ1 (z) ⊗ · · · ⊗ σk (z)) (x, ξ ) is meromorphic x with poles of order at most k in Pk :=
k
( j q)−1 Z − α1 (0) − α2 (0) − · · · − α j (0) .
j=1
34
D. Manchon, S. Paycha
2. The map z → −
(Tx∗ U )k
k ⊗i=1 σi (z) dξ
is meromorphic with poles of order at most k and we have the following equality of meromorphic functions: k −
∗ i=1 Tx U
=
dξi σi (z)
−
∗ τ ∈k Tx U
P1 ◦ · · · ◦ Pk στ (1) (z) ⊗ · · · ⊗ στ (k) (z) (x, ξ ) dξ,
(21)
where k denotes the group of permutations on k elements. Proof. Statements 1 and 2 in the theorem follow by induction on k from statements 1 and 2 of Proposition 5. Indeed, Proposition 5 with k1 = k2 = 0 yields the theorem for k = 1. Replacing σ2 in Proposition 5 by P2 ◦ · · · ◦ Pk (σ2 ⊗ · · · σk+1 ) ∈ C S ∗,k−1 (U ) (so that k2 = k − 1 here) then yields the induction step k → k + 1 since P1 ◦ P2 ◦ · · · ◦ Pk (σ1 (z) ⊗ σ2 (z) ⊗ · · · ⊗ σk+1 (z)) = P (P2 ◦ P3 ◦ · · · ◦ Pk (σ1 (z) ⊗ σ2 (z) ⊗ · · · σk+1 (z))) . Corollary 1. Under the same assumptions and using the same notations as in Theorem 5, we have the following equality of meromorphic maps: k k − ⊗i=1 σi (z) dξ = = ⊗i=1 σi (z) dξ (Tx∗ U )k
(Tx∗ U )k
k − =
∗ i=1 Tx U
σi (z)(x, ξi ) dξi .
(22)
The highest order pole is given by: Reskz=0 −
Tx∗ U
k −1 (−1)k k k r esx,k (⊗i=1 ⊗i=1 σi (z) dξ = k σi ) =
(0) resx (σi ).
α i=1 αi (0) i=1 i
Proof. As a consequence of the shuffle relations (21), we have the following equality of meromorphic functions:
k − − ⊗i=1 σi (z) dξ = P1 ◦ · · · ◦ Pk στ (1) (z) ⊗ · · · ⊗ στ (k) (z) (x, ξ ) dξ Tx∗ U
∗ τ ∈k Tx U
k − =
∗ i=1 Tx U
σi (z)(x, ξi ) dξi .
Shuffle Relations for Regularised Integrals of Symbols
35
On the other hand, by the results of Sect. 3 we have a further equality of meromorphic functions: k k ⊗i=1 σi (z) dξ = − ⊗i=1 σi (z) dξ, = (Tx∗ U )k
(Tx∗ U )k
which shows that the two regularised integrals − and = both coincide on tensor products of holomorphic symbols with the product of the regularised integral of each of the symbols. The Wodzicki residue formula then follows from Theorem 2.
5.4. Obstructions to shuffle relations for regularised integrals of general classical symbols. The finite part of a product of meromorphic functions with poles generally does not coincide with the product of the finite parts. As a result, when the symbols have non-vanishing residues, taking finite parts of the above shuffle relations on the level of meromorphic functions does not yield the expected shuffle equations for the corresponding finite parts. However, in that case a renormalisation procedure familiar to physicists provides the obstruction in terms of counterterms arising in the renormalisation. Let M(C) denote the algebra of meromorphic functions on C, and let Mk (C) denote the space of meromorphic functions on C with poles of order at most k at z = 0. k Clearly, if f 1 , . . . , f k ∈ M1 (C) then i=1 f i ∈ Mk (C). Let as before f.p.z=0 f = 1 lim z→0 [ f (z) − z resz=0 f (z)] denote the finite part at z = 0 of a function f ∈ M1 (C). Then, in general k
f.p.z=0 f i (z) = f.p.z=0
i=1
k
f i (z).
i=1
A renormalisation procedure taken from physics provides a recursive procedure to com k pute the obstruction to the equality; when the products i=1 f i (z) arise from applying dimensional regularisation to Feynman type functions in the language of Etingof [E], this comes down to applying the renormalisation procedure used by physicists for connected Feynman graphs to a concatenation of disjoint one loop diagrams. The underlying Hopf algebra ([K2, CK]) in the situation considered here is the symk metric algebra H := ⊕∞ C S(U )2 built on the vector space C S(U ). It is in particuk=0 lar commutative and cocommutative. Although very simple, this toy model is instructive. The (deconcatenation) coproduct on σ = σ1 · · · σk reads:
σ = σ ⊗ 1 + 1 ⊗ σ +
σj ⊗
J {1,...,k},J =φ j∈J
σi .
i ∈J /
A regularisation procedure R : σ → σ (z) induces a map φ : C S(U ) → M1 (C) defined by φ(σ )(z) = −
Tx∗ U
2 denotes the symmetrised tensor product.
σ (z)(x, ξ ) dξ.
36
D. Manchon, S. Paycha
Our previous constructions show it extends to an algebra morphism : H → M(C), σ = σ1 · · · σk → −
T ∗ U ×···×T ∗ U
σ1 (z)(x, ξ1 ) · · · σk (z)(x, ξk )dξ1 · · · dξk .
The Hopf structure on H provides a recursive procedure to get a Birkhoff decomposition of the corresponding loop (σ ) for any σ ∈ H, i.e. a factorisation of the form (σ ) = − (σ )−1 + (σ ), where + (σ ) is holomorphic at 0. Namely, with Sweedler’s notations x = x ⊗ 1 + 1 ⊗ x + x ⊗ x
, − (σ )(σ
) , − (σ ) := −T (σ ) + + (σ ) := (σ ) + − (σ ) + − (σ )(σ
), where T is the projection on the pole part. This corresponds to Bogolioubov’s prescription by which one first “prepares”3 the symbol σ . As our Hopf algebra is a symmetric algebra, the picture drastically simplifies in our situation: indeed H is generated as an algebra by the space C S(U ) of its primitive elements. As both − and + are algebra morphisms [CK] we get the following explicit expressions: − (σ1 · · · σk ) = (−1)k
k
T φ(σ j ) ,
(23)
j=1
+ (σ1 · · · σk ) =
k
(I − T ) φ(σ j ) .
(24)
j=1
By evaluating + at z = 0 we then see that the renormalised value of the quantity (σ1 · · · σk ) at z = 0 is given by the product of the finite parts of the φ(σ j )’s, j = 1, . . . , k. There is another way of describing this renormalisation procedure via a renormalisation operator R on the space of Laurent series (z 1 , . . . , z k ) → f (z 1 , . . . , z k ) in several variables. For this, instead of σ1 (z)(x, ξ1 ) · · · σk (z)(x, ξk ) dξ1 · · · dξk , (σ ) : z → − Tx∗ U ×···×Tx∗ U
let us consider the map (z 1 , . . . , z k ) → Symm−
Tx∗ U ×···×Tx∗ U
σ1 (z 1 )(x, ξ1 ) · · · σk (z k )(x, ξk ) dξ1 · · · dξk
which defines a (symmetric) Laurent series in (z 1 , . . . , z k ); setting z 1 = z 2 = · · · = z k = z gives back the meromorphic function (σ ). Given a nonempty subset J = 3 We borrow this expression and the notations that follow from [CM] but we refer the reader to Kreimer [K2], see also [CK] for the Hopf algebra that underlies this renormalisation procedure.
Shuffle Relations for Regularised Integrals of Symbols
37
{i 1 , . . . , i |J | } {1, . . . , k}, setting J¯ := {i |J |+1 , . . . , i k } to be its complement in {1, . . . , k}, from such a Laurent series f we build the map f J : (z; z i|J |+1 , . . . z ik ) → f (z 1 , . . . , z ik )|zi =z,∀i∈J .
When f = f 1 ⊗· · ·⊗ f k (with e.g. f i = φ(σi )) then f J (z; z i|J |+1 , · · · z ik ) = j∈J f j (z)· j∈J f j (z j ). Let us set
¯ f )(z) := f (z 1 , . . . , z k )|z =z,1≤i≤k + C f (z; −) (z) R( J i φ= J {1,...,k}
which, in the case f =
k ⊗i=1 fi
¯ k f i )(z) := R(⊗ i=1
considered above reads
k
f i (z) +
C(⊗ j∈J f j )(z)
φ= J {1,...,k}
i=1
f i (z).
i ∈J /
The counterterm C is defined inductively on the number k of variables by
¯ f) , C ( f ) := −T R( where T is the projection onto the pole part of the Laurent series in z. The renormalisation operator R is then defined by ¯ f ) + C( f ) R( f ) := R(
¯ f) = (1 − T ) R( k which for f = ⊗i=1 f i reads: k ¯ k f i ) + C(⊗k f i ) f i ) := R(⊗ R(⊗i=1 i=1 i=1 ⎛
k f i + (1 − T ) ⎝ = (1 − T )
C(⊗ j∈J f j )
J {1,...,k},J =φ
i=1
⎞ fi ⎠ .
i ∈J /
To illustrate this construction, let us take k = 2 and compute R( f ) with f a Laurent series in z 1 , z 2 in each variable z i . There are only two subsets J ⊂ {1, 2} to consider in the renormalisation procedure J1 = {1} and J2 = {2} and we set f i := f Ji so that
R( f ) = (1 − T )( f ) − (1 − T ) T ( f 1 ) + T ( f 2 ) . Writing
f (z 1 , z 2 ) =
j
ai j z 1i z 2 + o(sup(|z 1 |, |z 2 |)),
−I ≤i≤1;−J ≤ j≤1
where I , resp. J is the largest order of the poles at 0 of f 1 , resp. f 2 respectively, we get R( f )(z) = (1 − T )( ai j z i+ j + o(z)) −I ≤i≤1;−J ≤ j≤1
− (1 − T )(( =
j
ai j z i z 2 )|z2 =z + (
i>0
ai j z
i+ j
0≤i+ j
= a0 0 + o(1).
+ o(z) − (
ai j z 1i z j )|z1 =z )
j>0
i>0,i+ j≥0
ai j z i+ j +
j>0,i+ j≥0
ai j z i+ j )
38
D. Manchon, S. Paycha
In particular, for two meromorphic functions f 1 and f 2 with simple poles: fpz=0 (R( f 1 ⊗ f 2 )) (z) = R( f 1 f 2 )(0) = fpz=0 f 1 (z) fpz=0 f 2 (z). More generally, an induction procedure yields: Theorem 6. Let (z 1 , . . . , z k ) → f (z 1 , . . . , z k ) have a Laurent expansion in each of the variables z i . The map z → R( f )(z) is holomorphic at z = 0 and its value at z = 0 coincides with the constant term in the Laurent expansion in (z 1 , . . . , z k ). k In particular, when f = ⊗i=1 f i where the functions f i , i = 1, . . . , k are meromorphic at z = 0, then R( f )(0) coincides with the product of the finite parts of the f i ’s: fpz→0 (R( f 1 ⊗ · · · ⊗ f k )(z)) = R( f 1 ⊗ · · · ⊗ f k )(0) =
k
fpz=0 f i (z).
i=1
Proof. The operator R yields an algebra morphism on the algebra of Laurent series and takes values in meromorphic functions which are holomorphic at z = 0 [CK]. As f → R( f )(z) restricted to M(C) takes f to a holomorphic function at 0 with value R( f )(0) given by the finite part of f at z = 0, on a tensor product f 1 ⊗ · · · ⊗ f k → R( f 1 ⊗ · · · ⊗ f k )(0) picks up the product of the finite parts of the f i ’s at z = 0. By a closure argument, we conclude that the map z → R( f )(z) is holomorphic at z = 0 on the whole algebra of Laurent series and that its value at z = 0 coincides with the constant term in the Laurent expansion in (z 1 , . . . , z k ). The second assertion is straightforward. Remark 8. As a consequence, if instead of using one complex parameter z, we regularise each σi by σi → σi (z i ) using a different complex parameter z i we can avoid this renormalisation procedure: k k k fpz 1 ,...,z k →0 ⊗i=1 − σi (z i ) = fpz 1 ,...,z k →0 (⊗i=1 f i )(z 1 , . . . , z k ) = fpz=0 f i (z). i=1
Applying the above theorem to f i : z → −T ∗ Ui σi (z) we get the following description xi
of the obstructions to shuffle relations for general classical symbols: Corollary 2. Given a regularisation procedure R on C S(U ) for any i = 1, . . . , k, for any σi ∈ C S(U ), k R R − σi − − dξk P1 ◦ · · · ◦ Pk−1 (στ ) ∗ i=1 Tx U
= fpz=0 −
∗ τ ∈k Txk U
Tx∗ U ×···Tx∗ U
=
( f 1 (z) · · · f k (z) − R( f 1 ⊗ · · · ⊗ f k )(z))
i 1 +···+i k =0,(i 1 ,...,i k )=0
ai11 · · · aikk ,
where as before, στ (i) := στ (i) and where the ai ’s correspond to the coefficients in ai i the meromorphic expansion at z = 0 of the cut-off integrals −T ∗ U σi (z) = −1 z + a0 + x
a1i z + o(z). In particular, the shuffle relations therefore hold if all the σi ’s have vanishing residue.
Shuffle Relations for Regularised Integrals of Symbols
39
Proof. As in the proof of Corollary 1 we have k − σi (z) dξi fpz=0 ∗ i=1 Tx U
R
− dξ1 P · · · P(στ (k) )στ (k−1) · · ·)στ (2) (ξ1 )στ (1) (ξ1 ) = ∗ τ ∈k Tx U
=
R − dξ1
∗ τ ∈k Tx U
|ξ2 |≤|ξ1 |
dξ2 · · ·
|ξk |≤|ξk−1 |
dξk στ (1) (ξ1 ) · · · στ (k) (ξk ).
On the other hand, Theorem 6 applied to f i : z → −T ∗ U σi (z) yields
k
k − − σi (z) fpz=0 σi (z) − R ∗ i=1 Txi Ui
k − = fpz=0
∗ i=1 Tx U
=
∗ i=1 Tx U
k R − σi (z) σi (z) − ∗ i=1 Tx U
i 1 +···+i k =0,(i 1 ,...,i k )=0
ai11 · · · aikk
which in turn yields the result of the theorem.
5.5. Feynman graphs and tensor products of symbols. Propagators in quantum field theory, when considered in momentum space in the euclidean setup and in absence of infrared divergences, are classical symbols: for example the propagator of a massive scalar field with mass m is a classical symbol of order −2: σ (ξ ) =
1 . |ξ |2 + m 2
Let be any one-particle-irreducible Feynman diagram with I internal edges, E external edges and V vertices. The loop number of is defined by: L := I − V + 1. Each (internal or external) edge e comes with its propagator σe . The Feynman rules associate to each 1PI graph (up to a symmetry factor and up to powers of the coupling constants), together with external momenta ( p1 , . . . , p E ) ∈ Rn E , the following (often ill-defined) integral: I ( p1 , . . . , p E ) = σe ( pe ) σe (ξe ) dξ. e external edge
V e internal edge
Here V is the affine subspace of dimension Rn L inside the space Rn I of internal momenta defined by the vanishing of the sum of momenta at each vertex. The sum of all external momenta also vanishes: pe = 0. e external edge
40
D. Manchon, S. Paycha
I σ considered previously with injective affine We can combine tensor products σ = ⊗i=1 i ∼ n L n I maps B : R −→ V ⊂ R with L ≤ I , which encode the choice of L independent internal momenta for the integration. One can then build a class of functions
f (ξ1 . . . , ξ L ) = σ ◦ B(ξ1 , . . . , ξ L ) in the momenta ξ1 , . . . , ξ L which, for a rather large choice of propagators σi ’s are of Feynman type in the language of Etingof [E]. The integral I ( p1 , . . . , p E ) is given, up to the external momenta factor, by the integration on Rn L of the function σ ◦ B above. A regularisation procedure R on classical symbols as described in paragraph 3.2 gives rise to holomorphic families z → σi (z) from which we can consider the map I σ (z ). We address here the following open questions: (z 1 , . . . , z I ) → σz 1 ,...,z I = ⊗i=1 i i B being injective, it is reasonable to expect the map (z 1 , . . . , z I ) → − σz 1 ,...,z I ◦ B(ξ1 , . . . , ξ L ) dξ1 · · · dξ L to give rise to a Laurent expansion in the z i ’s, on the grounds of work by Speer [S]4 who proves this fact when σi (ξ ) = (|ξ |2 + m i2 )−1 ∀i ∈ {1, . . . , I } and σ (z) = σ 1+z . Alternatively, following a dimensional regularisation type procedure, one can build maps (z 1 , . . . , z L ) → σ ◦ B(ξ1 , . . . , ξ L ) |ξ1 |−z 1 · · · |ξ L |−z L dξ1 · · · dξ L , which again can be expected to give rise to Laurent expansions and hence to a meromorphic function at 0 when z 1 = · · · = z L = z. Etingof’s results on dimensional regularisation [E] imply this meromorphicity property when σi (ξ ) = (|ξ |2 + m i2 )−1 ∀i ∈ {1, . . . , I } on the grounds of a theorem by Bernstein but further investigations are needed to prove the first part of the statement on the existence of a Laurent expansion in several variables. Theorem 6 shows that transposing a renormalisation procedure “à la Connes and Kreimer” to the rather trivial Hopf algebra given by the symmetric tensor algebra of meromorphic functions (equipped with the symmetrised product and the deconcatenation coproduct) boils down to picking up the constant term in the Laurent expansion in (z 1 , . . . , z k ) in the tensor product f 1 (z 1 ) · · · f k (z k ), thus providing the “renormalised” value of the tensor product f 1 (z) · · · f k (z) at z = 0. The fact that the “renormalised value” at 0 can be reached by distinguishing the parameters z 1 , . . . , z k had already been proved by Speer [S] in the particular case we briefly described above in relation to his work. Implementing a renormalisation procedure on the symmetric algebra of meromorphic functions (where the R¯ operation is almost a triviality) as we do here corresponds in physics to implementing it on regularised Feynman integrals, a rather elementary procedure which of course does not entail the complexity and subtlety of the original renormalisation procedure which physicists implement on Feynman diagrams. The latter would correspond here to implementing a renormalisation procedure on the σ ◦ B, which is work in progress. The nested structure of the Feynman integral I ( p1 , . . . , p E ) is reminiscent to Chen’s iterated integrals; in particular, a formal inductive integration procedure as performed by physicists shows that each integration w.r.to pi can potentially bring in an extra logarithmic power, just as each nested integral does inside a Chen integral. This analogy has 4 We thank Dirk Kreimer for drawing our attention to this reference. Speer’s results are transposed here to the euclidean set up.
Shuffle Relations for Regularised Integrals of Symbols
41
been carefully made precise and investigated in [DK, K1, K3]. Although a commutative associative product analogous to the shuffle product can be built on 1PI graphs (see [K3] Sect. 2.3), shuffle relations seem to be lost in that context since the boundaries of the integrals are expressed in terms of decorated rooted trees ([DK] Sect. IV). 6. Relation to Multiple Zeta Functions We want to adapt the previous results to symbols of operators on the unit circle. But instead of using an atlas on S 1 and expressing the symbol of the operators in local charts (e.g. using stereographic projections), we view S 1 as the Lie group U (1) seen as the range of (R, +) under the group morphism: : R → S1 x → ei x which has kernel 2π Z π1 (S 1 ). This amounts to identifying S 1 with the quotient R/2π Z. In this picture, the additive group structure on R/2π Z is identified with the multiplicative group structure on S 1 : (x + y + 2π n) = (x + 2π k)(y + 2πl) ∀k, l, n ∈ Z, an important fact for what follows. 6.1. The symbol of invariant operators on the unit circle. We then identify S 1 with R/2π Z and note the group law additively. The kernel K (x, y) of an invariant operator P depends only on the difference x − y. It lifts to a 2π -periodic function K˜ on R. The Fourier transform of K˜ is a linear combination of Dirac masses at the integers, and can reasonably be taken as a symbol for the operator P. It defines then a S 1 -invariant distribution on the cotangent T ∗ S 1 . The trace of P, when it exists, will be given by the integral of the symbol on T ∗ S 1 . We will illustrate this principle on complex powers of the laplacian. The Laplacian = −∂t2 on S 1 has discrete spectrum {n 2 , n ∈ Z}. The operator := |Ker⊥ , where Ker⊥ 2 denotes √ the orthogonal space to the kernel, has spectrum {n ; n ∈ Z−{0}} and its square
root has spectrum {|n|, n ∈ Z − {0}} as a consequence of which its zeta function is given by: |n|−z ζ√ (z) := n∈Z−{0} ∞ −z
= 2
n
n=1
where ζ is the Riemann zeta function.
= 2ζ (z),
42
D. Manchon, S. Paycha
ζ√ (z) can also be seen as the canonical trace of the operator ζ√ (z) = TR =
√
T ∗ S1
−z
√ −z so that:
σz (x, ξ )d x dξ,
√ where σz is the symbol of (still to be defined). We use the Mellin transform to √ −z express in terms of the heat-kernel of on S 1 : ∞ √ −z z 1
= t 2 −1 e−t dt. ( 2z ) 0 We want to compute its symbol. √
−z
, where is the Laplacian on S 1 reads for ξ ∈ R: σz (x, ξ ) = |k|−z δk (ξ ).
Proposition 6. The symbol of
k∈Z−{0}
Proof. If Ht (x, y) = h t (x − y) denotes the heat-kernel of on S 1 we have for every f ∈ C ∞ (S 1 , R) ∩ Ker⊥ : ∞ √ −z z 1
f = t 2 −1 h t f dt. 2z 0 Taking Fourier transforms we get σz =
1 z
2
∞
t 2 −1 h t dt, z
0
t · fˆ. We therefore need to compute the Fourier transform of h t and since h t f = h hence an explicit expression for the heat-kernel of the Laplace operator on S 1 . The heat kernel of the corresponding Laplace operator on R at time t is given by K t (x, y) = kt (x − y) with: x2 1 kt (x) := √ e− 4t , 4π t
and when identifying S 1 with R/2π Z, the heat-kernel of the Laplacian on S 1 is given by Ht (x, y) = kt (x − y + 2π n). n∈Z
The fact that it is “translation invariant modulo 2π ” enables us to define the symbol using an ordinary Fourier transform. Setting Ht (x, y) = h t (x − y) we have: −t f = hˆ fˆ e−t f = h t ∗ f ⇒ e t
Shuffle Relations for Regularised Integrals of Symbols
43
so that the Fourier transform of h t can be intepreted as the symbol of e−t . We first derive h t using the Poisson summation formula:
f (x + n) =
n∈Z
e2iπ kx
k∈Z
+∞
−∞
f (y)e−2iπ ky dy.
Hence
x + n) (with k˜t (y) := kt (2π y)) k˜t ( 2π n∈Z +∞ = eikx kt (2π y)e−2iπ ky dy
h t (x) =
−∞
k∈Z
1 ikx +∞ e kt (y)e−iky dy 2π −∞ k∈Z +∞ y2 1 ikx = e e− 4t e−iky dy √ 2π 4π t k∈Z −∞ 1 2 = eikx e−tk , 2π
=
k∈Z
√ +∞ λy 2 1 2 since for any λ > 0 we have −∞ e−i yξ e− 2 dy = √π e− 2λ ξ . Considering any test λ function ϕ ∈ Cc∞ (R) and taking Fourier transforms we find:
< hˆ t , ϕ > = < h t , ϕˇ > +∞ 2 e−iky e−tk ϕ(y) ˇ dy = −∞
=
k∈Z
=
k∈Z
e−tk
2
+∞ −∞
e−iky ϕ(y) ˇ dy (by Fubini’s theorem)
ϕ(k)e−tk . 2
k∈Z
On the other hand the orthogonal projection p on Ker (i.e. the constant functions) is given by: p( f )(x) = f (y) dy. S1
Its kernel K p is then the constant function on S 1 × S 1 equal to 1. The associated function K˜ p is the constant function 1 on R, so the symbol of p is the Dirac mass at 0. From that
we deduce that the symbol of e−t is given by: k∈Z−{0}
e−tk δk . 2
44
D. Manchon, S. Paycha
Applying the Mellin transform we finally get: ∞ z 1 2 2 −1 σz (x, ξ ) = t e−tk δk (ξ ) dt z ( 2 ) 0 k∈Z−{0} −z |k| δk (ξ ). = k∈Z−{0}
6.2. Discrete sums of symbols and the Euler-MacLaurin formula. The symbol σz just described involves Dirac measures so that we cannot directly apply the results of Sects. 2, 3 and 4 derived for smooth symbols to define its truncated and regularised integrals. The presence of Dirac measures leads to discrete sums which we need to truncate and regularise all the same; we therefore focus in this paragraph on truncated and regularised discrete sums of symbols. As we shall see, the Euler-MacLaurin formula ([Ha] Chap. 13) builds a bridge between discrete sums on one hand and continuous integrals of symbols on the other hand. It enables to transpose the properties derived previously for regularised integrals and iterated nested integrals to regularised sums and iterated nested sums. Let us consider symbols (x, ξ ) → σ (x, ξ ) of log-polyhomogeneous symbols on R in the class C S ∗,k (see Sect. 1 and Subsect. 2.2) “with constant coefficients”, i.e. independent of the first variable x. They clearly define symbols on the quotient S 1 = R/2π Z which we also call σ . We drop the first variable x ∈ S 1 and consider σ as a function of a single variable ξ ∈ R (here identified with Tx∗ S 1 for any x ∈ S 1 ). Let us denote by C S ∗,k (R) the class of such symbols and C S ∗,∗ (R) the algebra generated by the union over l ∈ N of these sets. There is a discretised version P of the Rota-Baxter P(σ )(η) = |ξ |≤|η| σ (ξ ) dξ of Sect. 4: P(σ )(n) = σ (k) ∀σ ∈ C S ∗,∗ (R), (25) |k|≤|n|
which has properties similar to those of P as the following lemma shows. Lemma 4. For any σ ∈ C S ∗,k (R), there is a symbol P(σ ) ∈ C S ∗,k+1 (R) with same order max(0, α + 1) (where α is the order of σ ) as P(σ ), which interpolates P(σ ). More precisely, P(σ )(n) = P(σ )(n) ∀n ∈ N and for any σ ∈ C S ∗,k (R), P(σ ) − P(σ ) lies in C S ∗,k (R). Remark 9. Let σ1 and σ2 be two classical symbols of order α1 , α2 respectively. It follows from the above lemma and Proposition 3 that σ1 P(σ2 ) has order α1 + max(0, α2 + 1) so that if α1 < −1 and α2 ≤ −1 it lies in L 1 (R) ∩ C S ∗,1 (R). Proof. The results of Subsect. 2.1 and the Euler-MacLaurin formula are the essential ingredients. We set τ (t) := σ (t) + σ (−t), so that we have: P(σ )(m) =
m j=0
τ ( j).
Shuffle Relations for Regularised Integrals of Symbols
45
Let us first recall the Euler-MacLaurin formula (formula (13.6.3) in G.H. Hardy’s monograph [Ha], with adapted notations): Consider the Bernoulli numbers, defined by: et
Bj t = t j, −1 j! j
so that 1 1 1 B0 = 1, B1 = − , B2 = , B4 = − , . . . , 2 6 30 and B2 j+1 = 0 for j ≥ 1. Define for any n the function φn by the equation: t
e xt − 1 tn = , φ (x) n et − 1 n!
(26)
n≥1
and define ψn as the 1-periodic function equal to φn on the interval [0, 1[. We then have for N ∈ N: N N P(σ )(N ) − P(σ )(N ) = τ (m) − τ (t) dt 0
m=0
B2r 1 τ (N ) + τ (2r −1) (N ) + C j + T j,N 2 (2r )! j
=
(27)
r =1
with Cj =
B2r 1 τ (2r −1) (1) τ (t) dt + τ (1) − 2 (2r )! 0 r =1 +∞ 1 ψ2 j+2 (t)τ (2 j+2) (t) dt − (2 j + 2)! 1
and T j,N =
j
1
1 (2 j + 2)!
+∞
(28)
ψ2 j+2 (t)τ (2 j+2) (t) dt.
N
Setting B2r 1 τ (2r −1) (ξ ) + C j + T j,|ξ | P(σ )(ξ ) := P(σ )(ξ ) + τ (ξ ) + 2 (2r )! j
r =1
then yields a symbol P(σ ) in C S ∗,k+1 (R). Indeed, we know by Proposition 3 in Sect. 4 that P(σ ) lies in C S ∗,k+1 (R) and has order max(0, α + 1) where α is the order of σ . The other terms on the r.h.s lie in C S ∗,k (R) as a result of the fact that σ itself lies in C S ∗,k (R) and have order ≤ α. Indeed, since τ lies in C S ∗,k (R), τ (2 j+2) also lies in C S ∗,k (R) and the remainder term ξ → T j,|ξ | has decreasing order with j. In particular, we see that P(σ ) − P(σ ) lies in C S ∗,k (R) and has order ≤ max(0, α) (0 is due to the presence of the constant C j ) so that P(σ ) and P(σ ) have the same order.
46
D. Manchon, S. Paycha
Remark 10. Formula (27) applied for k and k + 1 respectively shows Ck+1 = Ck so that Ck stabilises at a constant C for large k. On the grounds of this result we set the following definition. Definition 11. For any σ ∈ C S ∗,k (R) the expression: N σ (k) := fp R→+∞ P(σ )(R) − σ := fp N →+∞ k∈Z
k=−N
defines the cut-off sum of σ on the integers. N Remark 11. Since P(σ ) has the same order as P(σ ), the sum k=−N σ (k) converges N when the corresponding integral −N σ (ξ ) dξ converges, namely when σ has order < −1 in which case we have: − σ (k) = σ (k). k∈Z
k∈Z
Let us now consider holomorphic perturbations of a symbol σ ∈ C S ∗,k (R) (these are closely related to the “gauged symbols” of [G2]). Proposition 7. Let z → σz be a holomorphic family of log-polyhomogeneous symbols on R of order α(z) = −qz + α(0) with q > 0 that lie in the class C S ∗,k (R). 1. The cut-off sum: R σz (k) − σz (k) := fp R→+∞ k∈Z
(29)
k=−R
is a meromorphic function of z which coincides with +∞ k=−∞ σz (k) on the half-plane Re α(0)+1− j Re α(0)+1 , with poles in , q ∈ N of order ≤ l + 1. Re z > q q 2. The difference: − σz (ξ ) dξ − − σz (k) R
k∈Z
is a holomorphic function of z. Proof. As can be seen from the expression of P(σ ), a holomorphic perturbation of σ in C S ∗,k (R) induces a holomorphic perturbation P(σ )(z) := P(σz ) of P(σ ) in C S ∗,k+1 (R) which reads: |ξ | 1 P(σ )(z)(ξ ) = σz (t) dt + τz (ξ ) 2 −|ξ | +
k B2r (2r −1) (−1)r −1 (ξ ) + Ck (z) + Tk,|ξ | (z), τ (2r )! z r =1
where the various terms are obtained by substituting σz to σ in the r.h.s. of (27). By |ξ | Theorem 1 the integral term −|ξ | σz shares all properties listed in Proposition 7. The
Shuffle Relations for Regularised Integrals of Symbols
47
term 21 τz (ξ ) and each term inside the sum yields a holomorphic family in the symbol class C S ∗,k (R). The remainder term ξ → Tk,|ξ | (z) yields a holomorphic family of symbols of decreasing order according to k. Finally, formula (28) shows that C(z) = Ck (z) is holomorphic in the half-plane: Hk := {z ∈ C, Re z >
Re α(0) − 2k − 1 }. q
As this holds for any k, the function C(z) is holomorphic in the whole complex plane, and Proposition 7 is proven. As a fundamental example, consider the holomorphic family: σz (ξ ) = χ (ξ )|ξ |−z , where χ is a cut-off function which vanishes around 0 and such that χ (ξ ) = 1 for |ξ | ≥ 1. One gets the expected relation between the cut-off sum of the symbol σz and the zeta function: Corollary 3. We have the following equality of meromorphic functions with simple poles at integer numbers: − σz (k) = 2 ζ (z). k∈Z
Proof. Since the cut-off sum coincides with the ordinary sum of the series when it converges absolutely, the equality holds for z in the half-plane {Re z > 1}. By Item 1 of Proposition 7 the cut-off sum is a meromorphic function of z, which therefore coincides with the well-known meromorphic continuation of 2ζ . Remark 12. A simple computation shows that the cut-off integral of σz reads: 2 + h(z), − σz (ξ ) dξ = z−1 where h is holomorphic. We then recover from Item 3 of Proposition 7 that ζ (z) − is holomorphic in the whole complex plane.
1 z−1
6.3. Discrete Chen sums of symbols. Similarly to the operator P, the operator P satisfies relations reminiscent of Rota-Baxter relations of weight −1: P(σ )(n) P(τ )(n) = P(σ P(τ ))(n) + P(τ P(σ ))(n) + P(σ τ )(n) ∀n ∈ N with an extra term P(σ τ ) that did not arise in the weight zero Rota-Baxter relations for integrals we considered previously. We want to build from P discrete Chen sums of symbols inductively in a similar manner to the way we built continuous Chen integrals of symbols from P. We first define from P the operators ˆ i=1 C S(R) → ⊗ ˆ i=1 Map(N, C), Pj : ⊗
P j (σ )(n 1 , . . . , n j ) := P σ (n 1 , . . . , n j , ·) (n j ). j+1
j
On the grounds of Lemma 4 we derive the following result.
48
D. Manchon, S. Paycha
ˆ i=1 C S(R), then Lemma 5. Let σ ∈ ⊗ j+1
1. P(σ ) j defined by
P(σ ) j (ξ1 , . . . , ξ j ) := P σ (ξ1 , . . . , ξ j , ·) (ξ j ) ˆ i=1 C S(R) ⊗ C S ∗,1 (R). lies in ⊗ k C S(R), then P ◦ · · · ◦ P ˆ i=1 2. Let σ = σ1 ⊗ · · · ⊗ σk ∈ ⊗ 1 k−1 (σ1 ⊗ · · · ⊗ σk ) defined inductively by P1 ◦ · · · ◦ Pk−1 (σ1 ⊗ · · · ⊗ σk ) := P P2 ◦ · · · ◦ Pk−1 (σ1 ⊗ · · · ⊗ σk ) j−1
lies in C S ∗,k−1 (R) and has the same order as P1 ◦ · · · ◦ Pk−1 (σ1 ⊗ · · · ⊗ σk ), given by max(0, . . . , max(0, max(0, αk + 1) + αk−1 + 1), . . .) + α2 + n) + α1 , where αi is the order of σi . Proof. The first assertion is a direct consequence of Lemma 4. The second assertion then follows from an induction procedure on j to check that Pk− j ◦ · · · ◦ Pk−1 = ˆ k C S(R) to ⊗ ˆ k− j−1 C S(R) ⊗ C S ∗, j (R). The computation of (σ1 ⊗ · · · ⊗ σk ) maps ⊗ the order also follows by induction using the fact that by Lemma 4, P(σ ) and P(σ ) have the same order derived in Theorem 3. We are now ready to define discrete Chen sums of symbols. Combining Lemma 5 with Lemma 4 shows that the cut-off sum of the symbol P1 ◦ · · · ◦ Pk−1 (σ1 ⊗ · · · ⊗ σk ) is well defined so that we can set the following definition. Definition 12. For σ1 , . . . , σk ∈ C S(R), we call Chen
− σ1 ⊗ · · · ⊗ σk := − P1 ◦ · · · ◦ Pk−1 (σ1 ⊗ · · · ⊗ σk )
the cut-off Chen sum of σ := σ1 ⊗ · · · ⊗ σk . Remark 13. Given the expression of the order of P1 ◦ · · · ◦ Pk−1 (σ1 ⊗ · · · ⊗ σk ) explicited in the above lemma, it converges whenever α1 < −1 and αi ≤ −1 for all i = 1 in which case we have that Chen
− σ1 ⊗ · · · ⊗ σk =
Chen
σ1 ⊗ · · · ⊗ σk
is an ordinary discrete Chen sum. 6.4. Multiple zeta functions. We now apply the above results to σi := σsi := χ (ξ ) |ξ |−si , where s1 , . . . , sk are real numbers and χ is a cut-off function which vanishes around 0 and such that χ (ξ ) = 1 for |ξ | ≥ 1. We want to generalise Corollary 3 to integrals k σ (s ) relating them to multiple zeta functions (investigated in of tensor products ⊗i=1 i i [H] and [Z], see also [C] or [Wa] for a review on the subject). Applying the results of the previous paragraph to the σi ’s of order −si leads to the following result which gives back a known domain of convergence for multiple zeta functions.
Shuffle Relations for Regularised Integrals of Symbols
49
Theorem 7. If s1 > 1 and si ≥ 1 for i = 2, . . . , k the discrete Chen sum · · · ⊗ σsk converges and is proportional to the multiple zeta function: Chen
σs1 ⊗ · · · ⊗ σsk = 2k ζ˜ (s1 , . . . , sk ) := 2k
Chen
σs1 ⊗
−s1 k n −s k · · · n1 .
1≤n k ≤n k−1 ≤···≤n 1
It extends to all si ∈ R by a cut-off Chen integral of the type defined above: Chen ζ˜ (s1 , . . . , sk ) := 2−k − σs1 ⊗ · · · ⊗ σsk ,
where we have used the same symbol for the extended mutiple ζ -function. Proof. It follows immediately from applying the results of the previous paragraph to σi = σsi of order −si . As a consequence we can also write: ∞ ζ˜ (s1 , . . . , sk ) = − P˜1 ◦ · · · ◦ P˜k−1 (σs1 ⊗ · · · ⊗ σsk )(n), n=1
where ˜ f )(m) := P(
f (m), ∀ f ∈ F(N, C)
1≤n≤m
and j+1 j ˆ i=1 ˆ i=1 P˜ j : ⊗ Map(N, C) → ⊗ Map(N, C)
˜ ˜ P j ( f )(n 1 , . . . , n j ) := P f (n 1 , . . . , n j , ·) (n j ).
If s1 > 1 and si ≥ 1 for i = 1 then clearly, we have ordinary sums: ζ˜ (s1 , . . . , sk ) =
∞
P˜1 ◦ · · · ◦ P˜k−1 (σs1 ⊗ · · · ⊗ σsk )(n).
n=1
Remark 14. • One can check that the same type of results holds with the usual multiple zeta functions −s1 k ζ (s1 , . . . , sk ) := n −s k · · · n1 1≤n k
instead of ζ˜ (s1 , . . . , sk ) provided the large inequalities between the |ξ j |’s and |n j |’s are replaced by strict ones. • The above results can be extended5 to complex numbers z i instead of real numbers si replacing s1 ≥ 1 and si > 1, i = 1 in the convergence assumptions by Re(z 1 ) ≥ 1 and Re(z i ) > 1, i = 1. 5 via an extra statement on Chen sums of holomorphic families which we omit here, but which can be established along the same lines as was the meromorphicity result on Chen integrals of holomorphic families.
50
D. Manchon, S. Paycha
The well known “second shuffle relations” for multiple zeta functions [ENR] come from the natural partition of the domain: Pk,l := {x1 > · · · > xk > 0} × {xk+1 > · · · > xk+l > 0} ⊂]0, +∞[k+l into:
Pk,l =
Pσ ,
σ ∈mix sh(k,l)
where mix sh(k, l) stands for the mixable shuffles, i.e. the surjective maps σ from {1, . . . k + l} onto {1, . . . m(σ )} (for some m(σ ) ≤ k + l) such that σ1 < · · · < σk and σk+1 < · · · < σk+l . The domain Pσ is defined by: Pσ = {(x1 , . . . , xk+l ) / xσr > xσr +1 if σr = σr +1 and xr = xr +1 if σr = σr +1 }. The second shuffle relations are: ζk (z 1 , . . . , z k ) ζl (z k+1 , . . . , z k+l ) =
ζm(σ ) (Z σ ),
(30)
σ ∈mix sh(k,l)
where Z σ is the m(σ )-uple defined by: Z (σ ) j =
zi .
i∈{1,...,k+l}, σ (i)= j
For k = l = 1 they read: ζ (z 1 )ζ (z 2 ) = ζ (z 1 , z 2 ) + ζ (z 2 , z 1 ) + ζ (z 1 + z 2 ). Using the identification −R σz (ξ )dξ = 2ζ (z) derived previously we can indeed compute: 4ζ (z 1 )ζ (z 2 ) =
2 − D(σzi ) i=1 R
D(σz 1 ) + − D(σz 1 ) D(σz 2 ) = − D(σz 2 ) R |ξ1 <|ξ2 | R |ξ2 |<|ξ1 | D(σz 1 ) ⊗ D(σz 2 ) +− |ξ1 |=|ξ2 |
= 4ζ (z 1 , z 2 ) + 4ζ (z 2 , z 1 ) + 4ζ (z 1 + z 2 ). The verification of the general formula (30) goes along the same lines. Acknowledgements. The first author would like to thank K. Ebrahimi-Fard for stimulating discussions around renormalisation and both authors are thankful to Li Guo for interesting discussions around the Rota-Baxter relation, which served as a motivation for parts of this work. We also wish to thank D. Kreimer for pointing out to us some valuable references. The second author is grateful to the Max Planck Institute where parts of this article were worked out and to D. Zagier with whom some informal discussions around zeta functions associated with operators took place as well as to M. Marcolli for stimulating discussions on renormalisation and G. Racinet for his patient explanations concerning multiple zeta functions. Last but not least, let us address our thanks to the referee for insisting on the need to clarify the relations of our constructions with known renormalisation procedures in physics, a line of thought we also pursue in [MP].
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References [C]
Cartier, P.: Fonctions polylogarithmes, nombres polyzeta et groupes pro-unipotents. Séminaire Bourbaki, n.885 (2001) [CM] Connes, A., Marcolli, M.: From Physics to Number theory via Noncommutative Geometry. http:// arxiv.org/list/hep-th/041114, 2004 [CK] Connes, A., Kreimer, D.: Hopf algebras, Renormalisation and Noncommuttive Geometry. Commun. Math. Phys. 199, 203–242 (1988) [DK] Delbourgo, R., Kreimer, D.: Using the Hopf algebra structure of QFT in calculations. Phys. Rev. D60, 105025 (1999) [E] Etingof, P.: Notes on dimensional regularization. In: Quantum Fields; a course for mathematicians. Providence, RI: AMS-IAS, 597–607, 1999 [EGK] Ebrahimi-Fard, K., Guo, L., Kreimer, D.: Integrable renormalization I: the ladder case. J. Math. Phys. 45, 3758–3769 (2004); Integrable renormalization II: the general case. Ann. Inst. H. Poincaré 6, 369–395 (2004) [EGGV] Ebrahimi-Fard, K., Gracia-Bondia, J., Guo, L., Varilly, J.: Combinatorics of renormalization as matrix calculus. Phys. Lett. B 363, 552–558 (2005) [ENR] Espie, M., Novelli, J-Ch., Racinet, G.: Formal computations about multiple zeta values. IRMA Lect. Math. Theor. Phys. 3, Berlin: De Cruyler, 2003 [FG] Friedlander, L., Guillemin, V.: Determinants of zeroth order operators. To appear in a special volume of “surveys in Differential Geometry” dedicated to S.S. Chern. Available at http:// arxiv.org/list/math.SP/0601743, 2006 [G] Guillemin, V.: Residue traces for certain algebras of Fourier integral operators. J. Funct. Anal. 115, 391–417 (1993); A new proof of Weyl’s formula on the asymptotic distribution of eigenvalues. Adv. Math. 55, 131–160 (1985) [G2] Guillemin, V.: Gauged lagrangian distributions. Adv. Math. 102(2), 184–201 (1993) [GSW] Guillemin, V., Sternberg, S., Weitsman, J.: The Ehrhart function for symbols. http:// arxiv.org/list/math.CO/0601714, 2006 [Ha] Hardy, G.H.: Divergent series. Oxford: Clarendon, 1949 [H] Hoffman, M.: Multiple harmonic series. Pacific J. Math. 152, 275–290 (1992); The algebra of multiple harmonic series. J. Algebra 194, 477–495 (1997); The Hopf algebra structure of multiple harmonic sums. Med. Phys. B (Proc. Suppl.) 135, 215–219(2004) [K1] Kreimer, D.: Chen’s iterated integral represents the operator product expansion. Adv. Theo. Math. Phys. 3, 627–690 (1999) [K2] Kreimer, D.: On the Hopf algebra of perturbative quantum field theory. Adv. Theo. Math. Phys. 2, 303–334 (1998) [K3] Kreimer, D.: New mathematical structures in renormalizable quantum field theory. Ann. Phys. 303, 179–202 (2003) [KV] Kontsevich, M., Vishik, S.: Geometry of determinants of elliptic operators. In: Func. Anal. on the Eve of the XXI century, Vol I, Progress in Mathematics 131, Basel: Birkhäuser 1994 pp.173–197; Determinants of elliptic pseudodifferential operators. Max Planck Preprint (1994) [L] Lesch, M.: On the non-commutative residue for pseudo-differential operators with log-polyhomogeneous symbols. Ann. Global Anal. Geom. 17, 151–187 (1998) [MMP] Maeda, Y., Manchon, D. Paycha, S.: Stokes’ formulae on classical symbol valued forms and applications. http://arxiv.org/list/math.DG/0510454, 2005 [MP] Manchon, D., Paycha, S.: Renormalised Chen integrals for symbols on Rn and renormalised polyzeta functions. http://arxiv.org/list/math.NT/0604562, 2006 [P] Paycha, S.: From heat-operators to anomalies; a walk through various regularization techniques in mathematics and physics. Emmy Nöther Lectures, Göttingen, 2003, available at http://www.math.uni-goettingen.de; Anomalies and regularisation techniques in mathematics and physics. Lecture Notes, Preprint, Colombia, 2003 available at http://www.lma.univ-bpclermont.fr/ paycha/publications/html [PS] Paycha, S., Scott, S.: A Laurent expansion for regularised integrals of holomorphic symbols. To appear in Geom. Funct. Anal. [S] Speer, E.: Analytic renormalization. J. Math. Phys. 9, 1404–1410 (1968) [Wa] Waldschmidt, M.: Valeurs zêta multiples. Une introduction. J. Theorie des Nombres de Bordeaux 12, 1–16 (2000) [W] Wodzicki, M.: Non-commutative residue. In: Lecture Notes in Math. 1283, Berlin Heidelberg New York: Springer Verlag, 1987; Spectral asymmetry and noncommutative residue (in Russian). Thesis, Steklov Institute (former) Soviet Academy of Sciences, Moscow, 1984 [Z] Zagier, D.: Values of zeta functionals and their applications. In: First European Congress of Mathematics, Vol. II, Basel Birkhaüser, 1994 pp.497–512; Multizeta values, manuscript Communicated by A. Connes
Commun. Math. Phys. 270, 53–67 (2007) Digital Object Identifier (DOI) 10.1007/s00220-006-0145-7
Communications in
Mathematical Physics
On the Geometry of Closed G2 -Structures Richard Cleyton1 , Stefan Ivanov2 1 Humboldt-Universität zu Berlin, Institut für Mathematik, Unter den Linden 6, D-10099 Berlin, Germany.
E-mail:
[email protected]
2 University of Sofia “St. Kl. Ohridski”, Faculty of Mathematics and Informatics. Blvd. James Bourchier 5,
1164 Sofia, Bulgaria. E-mail:
[email protected] Received: 9 December 2005 / Accepted: 29 June 2006 Published online: 5 December 2006 – © Springer-Verlag 2006
Abstract: We give an answer to a question posed in physics by Cvetiˇc et al. [9] and recently in mathematics by Bryant [3], namely we show that a compact 7-dimensional manifold equipped with a G 2 -structure with closed fundamental form is Einstein if and only if the Riemannian holonomy of the induced metric is contained in G 2 . This could be considered to be a G 2 analogue of the Goldberg conjecture in almost Kähler geometry and was indicated by Cvetiˇc et al. in [9]. The result was generalized by Bryant to closed G 2 -structures with too tightly pinched Ricci tensor. We extend it in another direction proving that a compact G 2 -manifold with closed fundamental form and divergence-free Weyl tensor is a G 2 -manifold with parallel fundamental form. We introduce a second symmetric Ricci-type tensor and show that Einstein conditions applied to the two Ricci tensors on a closed G 2 -structure again imply that the induced metric has holonomy group contained in G 2 . 1. Introduction A 7-dimensional Riemannian manifold is called a G 2 -manifold if its structure group reduces to the exceptional Lie group G 2 . The existence of a G 2 -structure is equivalent to the existence of a non-degenerate three-form on the manifold, sometimes called the fundamental form of the G 2 -manifold. From the purely topological point of view, a 7-dimensional paracompact manifold is a G 2 -manifold if and only if it is an oriented spin manifold [23]. The geometry of G 2 -structures has also attracted much attention from physicists. The central issue in physics is that connections with holonomy contained in G 2 plays a rôle in string theory [14, 9, 24, 15, 18]. The G 2 -connections admitting three-form torsion have been of particular interest. In [11], Fernández and Gray divide G 2 -manifolds into 16 classes according to how the covariant derivative of the fundamental three-form behaves with respect to its decomposition into G 2 -irreducible components (see also [7]). If the fundamental form is parallel
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with respect to the Levi-Civita connection then the Riemannian holonomy group is contained in G 2 , we will say that the G 2 -manifold or the G 2 -structure on the manifold is parallel. In this case the induced metric on the G 2 -manifold is Ricci-flat, a fact first observed by Bonan [1]. It was shown by Gray [17] (see also [2, 25]) that a G 2 -manifold is parallel precisely when the fundamental form is harmonic. The first examples of complete parallel G 2 -manifolds were constructed by Bryant and Salamon [5, 16]. Compact examples of parallel G 2 -manifolds were obtained first by Joyce [19–21] and recently by Kovalev [22]. Compact parallel G 2 -manifolds will be referred to as Joyce spaces. Examples of G 2 -manifolds in other Fernández-Gray classes may be found in [10, 6]. A central point in our argument is that the Riemannian scalar curvature of a G 2 -manifold may be expressed in terms of the fundamental form and its derivatives and furthermore the scalar curvature carries a definite sign for certain classes of G 2 -manifolds [13, 3]. In the present paper we are interested in the geometry of closed G 2 -structures i.e., G 2 -manifolds with closed fundamental form (sometimes these spaces are called almost G 2 -manifolds or calibrated G 2 -manifolds). In the sense of the Fernández-Gray classes, this is complementary to the physicists’ requirement of three-form torsion [12]. Compact examples of closed G 2 -manifolds were presented by Fernández [10]. Topological quantum field theory on closed G 2 -manifolds were discussed in [24]. Supersymmetric string solutions on closed G 2 -manifolds were investigated in [9] where the authors indicated the G 2 -analogue of the Goldberg conjecture in almost Kähler geometry. Bryant shows in [3] that if the scalar curvature of a closed G 2 -structure is non-negative then the G 2 -manifold is parallel. The question whether there are closed G 2 -structures which are Einstein but not Ricci-flat then naturally arises. We investigate this question in the compact and in the non-compact cases. In the first version of the present article [8] we answered negatively to the G 2 -version of the Goldberg conjecture, namely, we proved that there are no closed Einstein G 2 -structures (other than the parallel ones) on a compact 7-manifold. In [4] Bryant generalized this non-existence result for closed G 2 -structures on compact 7-manifold whose Ricci tensor is too tightly pinched. In the present article we obtain a non-existence result involving third derivatives of the fundamental form. Namely, we prove the following Main Theorem. A compact G 2 -manifold with closed fundamental form and harmonic Weyl tensor (divergence-free Weyl tensor) is a Joyce space. The second Bianchi identity leads to Corollary 1.1. A compact G 2 -manifold with closed fundamental form and harmonic curvature (divergence-free curvature tensor) is a Joyce space. Corollary 1.2. A compact Einstein G 2 -manifold with closed fundamental form is a Joyce space. The latter may be considered to be a G 2 analogue of the Goldberg conjecture in almost Kähler geometry (see e.g. [9]). The representation theory of G 2 gives rise to a second symmetric Ricci type tensor on G 2 -manifolds. Therefore one may consider two complementary Einstein equations. We find a connection between the two Ricci tensors and show in Theorem 5.7, with no compactness assumption, that if both Einstein conditions hold simultaneously on a G 2 -manifold with closed fundamental form then the fundamental form is parallel. Our main tool is the canonical connection of a G 2 -structure and its curvature. We will show that the Ricci tensor of the canonical connection is proportional to the Riemannian
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Ricci tensor. This leads to the corollary that a compact G 2 -manifold with closed fundamental form which is Einstein with respect to the canonical connection is a Joyce space. Our main technical tool is an integral formula which holds on any compact G 2 manifold with closed fundamental form. We derive the Main Theorem as a consequence of a more general result, Theorem 7.1, which shows that the vanishing of the 27 -part of the divergence of the Weyl tensor implies that a closed G 2 -structure is parallel on a compact 7-manifold. 2. General Properties of G 2 -Structures We recall some notions of G 2 geometry. Endow R7 with its standard orientation and inner product. Let e1 , . . . , e7 be an oriented orthonormal basis which we identify with the dual basis via the inner product. Write ei1 i2 ...i p for the monomial ei1 ∧ ei2 ∧ · · · ∧ ei p . We shall omit the -sign understanding summation on any pair of equal indices. Consider the three-form ω on R7 given by ω = e124 + e235 + e346 + e457 + e561 + e672 + e713 .
(2.1)
The subgroup of G L(7) fixing ω is the exceptional Lie group G 2 . It is a compact, connected, simply-connected, simple Lie subgroup of SO(7) of dimension 14 [2]. The Hodge star operator supplies the 4-form ∗ω given by ∗ω = −e3567 − e4671 − e5712 − e6123 − e7234 − e1345 − e2456 .
(2.2)
We let the expressions 1 ωi jk ei jk , 6 1 ∗ω = ωi jkl ei jkl 24 and ωi jkl . We then obtain the following set of formulae: ω=
define the symbols ωi jk ωi pq ω j pq ωi pq ω jkpq ωi j p ωklp ωi j pq ωklpq ωi j p ωklmp
= = = = =
6δi j , −4ωi jk , −ωi jkl + δik δ jl − δil δ jk , (2.3) −2ωi jkl + 4(δik δ jl − δil δ jk ), δik ω jlm − δ jk ωilm + δil ω jmk − δ jl ωimk + δim ω jkl − δ jm ωikl .
Definition 2.1. A G 2 -structure on a 7-manifold M is a reduction of the structure group of the tangent bundle to the exceptional group G 2 . Equivalently, there exists a nowhere vanishing differential three-form ω on M and local frames of the cotangent bundle with respect to which ω takes the form (2.1). The three-form ω is called the fundamental form of the G 2 -manifold M [1]. We will say that the pair (M, ω) is a G 2 -manifold with G 2 -structure (determined by) ω. Remark 2.2. Alternatively, a G 2 -structure can be described by the existence of a two-fold vector cross product P on the tangent spaces of M. The fundamental form of a G 2 -manifold determines a metric through gi j = 16 ωikl ω jkl . This is referred to as the metric induced by ω. We write ∇ g for the associated Levi-Civita connection, ||.||2 for the tensor norm with respect to g. In addition we will freely identify vectors and co-vectors via the induced metric g.
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Let (M, ω) be a G 2 -manifold. The action of G 2 on the tangent space induces an action of G 2 on k (M) splitting the exterior algebra into orthogonal subspaces, where lk corresponds to an l-dimensional G 2 -irreducible subspace of k : 1 (M) = 17 , 2 (M) = 27 ⊕ 214 , 3 (M) = 31 ⊕ 37 ⊕ 327 , where 27 = {α ∈ 2 (M)|∗(α ∧ ω) = −2α}, 214 = {α ∈ 2 (M)|∗(α ∧ ω) = α}, 31 = {t.ω| t ∈ R}, 37 = {∗(β ∧ ω)| β ∈ 1 (M)}, 327 = {γ ∈ 3 (M)| γ ∧ ω = 0, γ ∧ ∗ω = 0}. The Hodge star ∗ gives an isometry between lk and l7−k . d will denote the G 2 representation of highest weight (λ1 , λ2 ) More generally, V(λ 1 ,λ2 ) 1 7 ∼ is of dimension d. Note that V(0,0) = 31 ∼ = 41 is the trivial representation, 17 ∼ = V(1,0) 14 7 ∼ ∼ the standard representation of G 2 on R , and the adjoint representation is g2 = V(0,1) = 27 ∼ 3 ∼ 4 is isomorphic to the space of traceless sym214 . Also note that V(2,0) = 27 = 27 2 7 7 metric tensors S0 V on V(1,0) . 3. Ricci Tensors on G 2 -Manifold Let (M, ω) be a G 2 -manifold with fundamental form ω. Let g be the associated Riemannian metric; R X,Y = [∇ g X , ∇ g Y ] − ∇ g [X,Y ] is then the curvature tensor of the Levi-Civita connection ∇ g of the metric g. The Ricci tensor ρ is defined as usual as the contraction ρi j = Rsi js , where Rsi js are the components Rsi jk := g(R(es , ei )e j , ek ) of the curvature tensor with respect to an orthonormal basis e1 , . . . , e7 . Definition 3.1. On (M, ω) we may define a second symmetric tensor ρ by
:= Ri jkl ωi js ωklm . ρsm
(3.4)
We will call the ρ the -Ricci tensor of the G 2 -manifold. The two Ricci tensors have common trace in the following sense. Let s = tr g ρ = ρii be the scalar curvature and let the trace of ρ be denoted by s = tr g ρ = ρii . Proposition 3.2. On a G 2 -manifold we have s = −2s. Proof. Apply (2.3) to the definition of s and use skew-symmetry of ∗ω and the Bianchi identity to conclude that Ri jkl ωi jkl = 0.
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Definition 3.3. We shall use the term -Einstein for G 2 -manifold (M, ω) when the traceless part of the -Ricci tensor vanishes, i.e., when the equation ρ =
s
g 7
holds. We define associated Ricci three-forms by ρi jk := Ri jlm ωlmk + R jklm ωlmi + Rkilm ωlm j , ρi jk := ρis ωs jk + ρ js ωski + ρks ωsi j . In terms of the Ricci forms, the Weitzenböck formula for the fundamental form can be written as follows: Proposition 3.4. On any G 2 -manifold the following formula holds: dδω + δdω = ∇ g ∗ ∇ g ω + ρ + ρ .
(3.5)
4. Closed G 2 -Structures Let (M 7 , ω) be a G 2 -manifold with closed fundamental form. The two-form δω then takes values in 214 [3]. As a consequence we get Proposition 4.1. The following formulas are valid on a closed G 2 -structure: δωi j ωi jk = 0,
δωi p ω pjk + δω j p ω pki + δωkp ω pi j = 0.
(4.6)
It is well-known [17] that a G 2 -structure is parallel if and only if it is closed and co-closed, dω = δω = 0. The two-form δω thus may be interpreted as the deviation of ω from a parallel G 2 -structure. We are going to find explicit formulae for the covariant derivatives of the fundamental form of a closed G 2 -structure in terms of δω and its derivatives. Definition 4.2. The canonical connection ∇˜ of a closed G 2 -structure may be defined by the equation 1 g(∇˜ X Y, Z ) = g(∇ g X Y, Z ) − δω(X, ei )ω(ei , Y, Z ) 6 for vector fields X, Y, Z .
(4.7)
Using (4.6) it is easy to see that ∇˜ is a metric G 2 -connection, i.e., it satisfies ˜ = 0, ∇ω
˜ = 0. ∇g
The torsion T of ∇˜ is determined by 1 δω(Z , ei )ω(ei , X, Y ). 6 On a compact G 2 -manifold the canonical connection may be characterized as the unique G 2 -connection of minimal torsion with respect to the L 2 -norm on M. It may also be described by the fact that the difference ∇ g − ∇˜ takes values in 27 , the orthogonal complement of g2 ⊂ 2 with respect to the metric induced by g. From the properties of the canonical connection and δω one derives g(T (X, Y ), Z ) =
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Proposition 4.3. For a closed G 2 -structure the following relations hold: 1 δωi p ω pjkl , 2 1 = − δωi j ωklm − δωik ωlm j + δωil ωm jk − δωim ω jkl 2
∇ g i ω jkl = ∇ g i ω jklm
(4.8) (4.9)
and ∇ g ∗ ∇ g ω jkl =
1 1 δω 2 ω jkl − δωi p δωi j ω pkl + δωi p δωik ω pl j + δωi p δωil ω pjk . 4 4 (4.10)
Applying (4.7) and (4.8) we get that the curvature R˜ of the canonical connection ∇˜ is related to the curvature of the Levi-Civita connection by: 1 g 1 Ri jkl = R˜ i jkl + ∇ i δω j p − ∇ g j δωi p ω pkl + δωis δω j p ωspkl 6 9 1 δωik δω jl − δωil δω jk . (4.11) − 36 5. Curvature of Closed G 2 -Structures From here on (M 7 , ω) will be a G 2 -manifold with closed G 2 -structure. We have Proposition 5.1. The Ricci tensors of a closed G 2 -structure (M, ω) are given by 1 1 ρlm = − dδωs jm ωs jl + δωl j δωm j ; (5.12) 4 2 1
ρlm = dδωs jm ωs jl + δωl j δωm j − δω 2 δml . (5.13) 2 Proof. The Ricci identities for ω, ∗ω together with (4.8) and (4.9) lead to the following useful Lemma 5.2. If ω is a closed G 2 -structure on M 7 then 1 1 1 ρsr ωr kl + Rskir ωlir − Rslir ωkir = − dδωs j p + ∇ g s δω j p ω pjkl 2 2 4 1 + δω pj δωs j ωklp , 2
(5.14)
−Rsi jr ωr klm − Rsikr ω jrlm − Rsilr ω jkrm − Rsimr ω jklr 1 g ∇ i δωs j − ∇ g s δωi j ωklm − ∇ g i δωsk − ∇ g s δωik ωlm j = 2 1 g + ∇ i δωsl − ∇ g s δωil ωm jk − ∇ g i δωsm − ∇ g s δωim ω jkl 2 1 − δωi j δωsp − δωs j δωi p ω pklm − δωik δωsp − δωsk δωi p ω plm j 4 1 − δωil δωsp − δωsl δωi p ω pm jk − δωim δωsp − δωsm δωi p ω pjkl . (5.15) 4
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Using (4.7) we get 1 ∇ g k δωis ωism = ∇˜ k δωis ωism + δωkr δωrq ωsiq ωism = δωkr δωmr , 6
(5.16)
˜ since ∇δω ∈ 214 . If we multiply (5.14) by ωmkl and use the Bianchi identity as well as (5.16) we obtain (5.12). Multiplying (5.15) by ωml j , and again using the Bianchi identity (alternatively: multiply (4.11) by ωklm ), we get 1 δωi j δωsp − δωs j δωi p ω j pk . Rsilr ωklr = ∇ g s δωik − ∇ g i δωsk + 4
(5.17)
From (5.17) we get that
= Rsilr ωklr ωsim ρkm
1 = dδωsik ωsim − ∇ g k δωis ωsim + δωi j δωsp ω j pk ωsim . 2
(5.18)
The second term is calculated in (5.16). The last term is manipulated using (4.6) and (2.3): δωi j δωsp ω j pk ωsim = −δω pj ω jki − δωk j ω ji p δωsp ωsim = δωsp δω j p ωki j ωsim + δω jk ω ji p ωsim = δωsp δω j p −ωk jsm + δks δ jm − δkm δ js +δωsp δω jk −ω j psm + δ js δ pm − δ jm δ ps = − δω 2 δkm + 4δω jm δω jk ,
(5.19)
again, since δω ∈ 214 . Substituting (5.19) and (5.16) into (5.18) we obtain (5.13). The equality (5.16) leads to dδωs jm ωs jm = 3 δω 2 .
(5.20)
Taking the trace in (5.12) and using (5.20), we get the formula for the scalar curvature of a closed G 2 -structure discovered recently by Bryant in [3]. Corollary 5.3. The scalar curvature of a closed G 2 -structure is non-positive while the
-scalar curvature is non-negative. These functions are given by 1 s = − δω 2 , 4
s =
1 δω 2 . 2
(5.21)
In view of (5.21), the trace-free part of the Ricci tensor ρ 0 has the expression ρ0 = ρ +
1 δω 2 g. 28
(5.22)
Definition 5.4. The canonical connection gives us a third Ricci tensor which we denote by ρ: ˜ ρ˜i j = R˜ si js .
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Corollary 5.5. On a 7-manifold with closed G 2 -structure the Ricci tensor of the canonical connection is related to the Riemannian Ricci tensor through the following formula: ρ˜ =
2 ρ. 3
Proof. Taking the trace of (4.11) we get 1 1 dδω pji ω pjl + δωis δωls . 12 6 This equality and (5.12) completes the proof. ρil = ρ˜il −
Furthermore, we have Proposition 5.6. The Ricci three-forms of a closed G 2 -structure are given by 1 1 δωli δω pi ω pjk + δωki δω pi ω pl j ρ jlk = − δω 2 ω jlk − dδω jlk − 4 2 (5.23) + δω ji δω pi ω pkl , 1
δωi j δω pj ω psk + δωk j δω pj ω pis + δωs j δω pj ω pki . (5.24) ρsik = 2dδωsik + 4 Proof. Substitute (4.10) and (5.24) into the Weitzenböck formula (3.5) to get (5.23). The cyclic sum in the equality (5.17) gives (5.24). Theorem 5.7. Let (M 7 , ω) be a G 2 -manifold with closed fundamental form. If (M, ω) is Einstein and -Einstein then M is parallel. Proof. Let (M, ω) be a G 2 -manifold with closed G 2 -structure ω. Suppose that both the Einstein and -Einstein equations are satisfied. Proposition 5.1 in this case yields 1 3 δω 2 δi p , dδωi jk ωi jm = δω 2 δkm . 7 7 Taking into account (5.25) and the equalities (4.10), (5.23), we obtain δωi j δω pj =
ρi jk = 2dδωi jk −
3 δω 2 ωi jk , 28
1 δω 2 ωi jk , 7 3 ρi jk = − δω 2 ωi jk . 28 In view of (5.26), the Weitzenböck formula (3.5) gives ∇ g ∗ ∇ g ωi jk =
dδω =
(5.25)
1 δω 2 ω. 14
(5.26)
(5.27)
Bryant shows in [3] that on an Einstein manifold with closed G 2 -structure the 327 -part of dδω is given by the 327 -part of ∗(δω ∧ δω). Comparing with (5.27), we see that the 327 -part of ∗(δω ∧ δω) vanishes. We need the following algebraic Lemma 5.8. Let α be a two-form in 214 . Then the 327 -part of ∗(α ∧ α) vanishes if and only if α = 0.
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Proof. First note that if α is a two-form in 214 then α ⊗ α ∈ S 2 214 ⊂ S 2 (2 ). The space of symmetric tensors on 214 decomposes as follows: 77 27 1 S 2 214 = V(0,2) + V(0,2) + V(0,0) .
Recall that the map S 2 2 → 4 given by β ∨ γ → β ∧ γ is surjective and equivariant. By Schur’s Lemma we may conclude that α ∧ α ∈ 427 ⊕ 41 if α ∈ 214 . Now suppose (α ∧ α)4 = 0. We may then conclude that α ∧ α = c∗ω for some 27 constant c. However, as α is a two-form on an odd-dimensional space it is degenerate. Let X ∈ R7 be a non-zero vector such that X α = 0. Then cX ∗ω = X (α ∧ α) = 2(X α) ∧ α = 0. But the left-hand side vanishes only if c = 0. Now, Lemma 5.8 implies δω = 0, whence ∇ g ω = 0. Remark 5.9. Bryant observe [4] that the following identity holds. α ∧ α 2 = 6 α 4 , α ∈ 214 .
(5.28)
Clearly (5.28) implies the lemma. Note that the constants have been changed to fit our conventions. 6. An Integral Formula on Closed G 2 -Manifold Our main technical tool to handle the closed G 2 -structure on a compact manifold is the next Proposition 6.1. Let (M, ω, g) be a compact G 2 -manifold with closed fundamental form. Then the following integral formula holds: 1 28 7 7 δω 4 + ||ρ 0 ||2 − ρ 0pl δωbl δωbp − (δ R)bkl ω jkl δωbj d V = 0. 9 18 9 M 24 (6.29) Proof. The first Pontrjagin form p1 (∇) of a connection ∇ may be defined by p1 (∇) :=
1 Ri jab Rklab ei ∧ e j ∧ ek ∧ el . 16π 2
The first Pontrjagin class of T M which is the de Rham cohomology class whose representative element is the first Pontrjagin form of a some on M is affine connection ˜ is an exact independent of the connection on M. This implies that p1 (∇ g ) − p1 (∇) ˜ ∧ω 4-form. Since the fundamental form ω is closed, the wedge product p1 (∇ g )− p1 (∇) is exact. From Stokes’ theorem we obtain ˜ ∧ ω = 0. p1 (∇ g ) − p1 (∇) (6.30) M
However, we may also express the integrand in terms of the curvatures of ∇ g and ∇˜ using that 16π 2 p1 (∇ g ) ∧ ω = Ri jab Rklab ωi jkl = Rabi j Rklab ωi jkl ,
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and ˜ ∧ ω = R˜ i jab R˜ klab ωi jkl . 16π 2 p1 (∇) From this point on the proof is essentially a brute force calculation reducing the difference of these two expressions to the form (6.29). Since ∇˜ is a G 2 -connection we get R˜ abi j ωi jkl = 2 R˜ abkl .
(6.31)
Using (4.11), (2.3) and (6.31) we calculate 2 dδωabp − ∇ g p δωab ω pkl Rklab 3 5 8 − δωai δωbj ωi jkl Rklab + δωak δωbl Rklab . (6.32) 18 9
16π 2 p1 (∇ g ) ∧ ω = 2Rklab R˜ abkl −
Applying (4.11) to (6.32) and using (2.3) we obtain after some calculations that 16π 2 p1 (∇ g ) ∧ ω
2 1 = 2 R˜ klab R˜ abkl + dδωabp − ∇ g p δωab ω pkl − Rklab + R˜ klab 3 3 5 8 2 ˜ 1 +δωai δωbj ωi jkl − Rklab + Rklab + δωak δωbl Rklab − R˜ klab . (6.33) 18 9 9 9
˜ term of the integrand we first observe that (4.11) implies To calculate the p1 (∇) 1 R˜ i jkl = R˜ kli j + dδωklp − ∇ g p δωkl ω pi j 6 1 1 1 − dδωi j p − ∇ g p δωi j ω pkl + δωks δωlp ωspi j − δωis δω j p ωspkl . (6.34) 6 9 9 Taking into account (6.31), (6.34) and (2.3), we calculate ˜ ∧ ω = 2 R˜ klab R˜ abkl − 2 dδωabp − ∇ g p δωab ω pkl R˜ klab 16π 2 p1 (∇) 3 4 8 − δωai δωbj ωi jkl R˜ klab + δωak δωbl R˜ klab . 9 9 Subtracting (6.35) from (6.33) we get ˜ ∧ ω = A + B + C, 16π 2 p1 (∇ g ) − p1 (∇)
(6.35)
(6.36)
where A, B, C are given by 1 A = dδωabp − ∇ g p δωab ω pkl Rklab − Rklab − R˜ klab , 3 7 2 B = δωai δωbj ωi jkl Rklab − Rklab − R˜ klab , 18 3 1 C = δωak δωbl − Rklab + Rklab − R˜ klab . 9 We shall calculate each term in (6.36).
(6.37) (6.38) (6.39)
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First observe that (5.28) implies the useful identity 4δωai δωbi δωa j δωbj = ||δω||4 .
(6.40)
Taking into account (4.11), (5.17), (5.12), (5.16), (5.19), (5.20) and (6.40) we obtain after some calculation that
2 8 19 1
δω 4 + dδωabp − ∇ g p δωab − ||ρ 0 ||2 24.7 3 3 4 0 1 + ρsp δωst δω pt − dδωabp − ∇ g p δωab ωi j p δωai δωbj 3 9 1 − dδωabp − ∇ g p δωab ω pkl ωsmab δωks δωlm . 9
A=−
(6.41)
Let X s = Rakbl ωsak δωbl , Ys = δωbl δωli δωbp ωi ps . Using the first and second Bianchi identity as well as (5.17), we get Rklab δωak δωbl = = = = =
1 Rakbl δωak δωbl 2 1 − Rakbl ∇ g s ωsak δωbl 2 1 1 δ X + Rakbl ωsak ∇ g s δωbl 2 2
2 1 1 1 1
δ X + dδω 2 − ∇ g δω + ∇ g s δωbl δωli δωbp ωi ps 2 6 2 4
2 1 1 1 1
2 g δ X − δY + dδω − ∇ δω
2 4 6 2 1 1 − ||δω||4 + dδωsil δωbl δωbp ωi ps . (6.42) 8 4
Applying (5.12) to (6.42), we obtain Rklab δωak δωbl =
1 1 1 δ X − δY + dδω 2 2 4 6
2 1 1
g 0
δω 4 − ρsp − ∇ δω + δωst δω pt . 2 28
(6.43)
Remark 6.2. Notice that (6.43) is the Weitzenböck formula δdδω = ∇ g ∗ ∇ g δω −
1 δω 2 δω + ρ 0 (δω, .) + ρ 0 (., δω) + R(δω) 14
for the 2-form δω on a closed G 2 -structure. Using (4.11), (5.28) and (6.43) we get that
25 1 1
1
∇ g δω 2 δω 4 − dδω 2 + δ(2X − Y ) − 36 63.16 54 18 1 0 1 + ρsp δωst δω pt + dδωkls − ∇ g s δωkl ωsab δωak δωbl . 9 6
C=−
(6.44)
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Now we calculate B. Denote Z a = Rklab ω jkl δωbj . Using (4.8) and (5.17) we have the following sequence of equalities: 1 Rklab δωai ωi jkl δωbj = Rklab ∇ g a ω jkl δωbj 2 = −δ Z + (δ R)bkl ω jkl δωbj − Rklab ω jkl ∇ g a δωbj
2 1 1
= −δ Z + (δ R)bkl ω jkl δωbj − dδω 2 − ∇ g δω
6 2 1 + dδωabj − ∇ g j δωab ωsl j δωas δωbl . (6.45) 4 Applying (4.11), we get 2 Rklab − R˜ klab δωai δωbj ωi jkl 3 1 = − dδωkls − ∇ g s δωkl ωsab ωi jkl δωai δωbj 9 2 − δωks δωlm δωai δωbj ωsmab ωi jkl . 27 Denote Vk = ωmab δωlm δωbj δωai ωi jkl . Using (4.8), we obtain −
(6.46)
1 δωks ωsmab δωlm δωai δωbj ωi jkl = ∇ g k ωmab δωlm δωai δωbj ωi jkl 2 3 = −δV − dδωklm − ∇ g m δωkl ωmab ωi jkl δωai δωbj . 2 (6.47) Getting together (6.45), (6.46) and (6.47) we obtain
7 7 7
1
∇ g δω 2 dδω 2 − δ(21Z − 4V ) + (δ R)bkl ω jkl δωbj − 27 9 54 18 1 + dδωklm − ∇ g m δωkl ωmab ωi jkl δωai δωbj 9 7 dδωabj − ∇ g j δωab ωsl j δωas δωbl . + (6.48) 36 Collecting terms from (6.41), (6.44) and (6.48) we obtain the following expression: ˜ ∧ω = δ − 1 X + 1 Y − 7Z + 4 V 16π 2 p1 (∇ g ) − p1 (∇) 18 36 9 27 1 8 139 4 δω − dδω 2 − ||ρ 0 ||2 − 18.56 27 3 7 1 + (δ R)bkl ω jkl δωbj + dδωkls 9 4 13 − ∇ g s δωkl ωsab δωak δωbl + ρ 0pl δωbl δωbp . (6.49) 9 We express the last two terms in a more tractable form. Applying (4.6) we have dδωabp δωai δωbj ω pi j = −dδωabp δωai δω pj ωibj + δωi j ωbpj = −dδωapb δωai δωbj ωi pj − dδωabp δωai δωi j ωbpj , B=−
On the Geometry of Closed G2 -Structures
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whence dδωabp δωai δωbj ω pi j =
1 1 dδωabp ωbpj δωai δω ji = −2ρa j δωai δω ji + ||δω||4 , 2 4 (6.50)
where we used (6.40) and (5.12). The next equalities are a consequence of (5.12), (5.28) and (5.16), 1 ∇ g s δωkl ωsab δωak δωbl = δY + ||δω||4 − dδωsak − ∇ g a δωks ωsab δωkl δωbl 4 = δY + 4ρkb δωkl δωbl . (6.51) Substitute (6.50), (6.51) into (6.49), use (5.12) and integrate over the compact M to get 11 1 8 1 ||δω||4 − ||dδω||2 − ||ρ 0 ||2 − ρ 0pl δωbl δωbp − 18.28 27 3 18 M 7 (6.52) + (δ R)bkl ω jkl δωbj d V = 0. 9 We calculate from (5.23) that ||dδω||2 =
15 ||δω||4 + 12||ρ 0 ||2 − 12ρ 0pl δωbl δωbp . 28
(6.53)
Substitute (6.53) into (6) to obtain (6.29). Corollary 6.3. (Integral Weitzenböck formula.) On a compact G 2 -manifold with closed fundamental form we have
2 1 dδω 2 − ∇ g δω − 2ρsp δωst δω pt d V. Rakbl δωak δωbl d V = M M 3 Proof. The first Bianchi identity implies Rakbl δωak δωbl = 2Rklab δωak δωbl . Apply (5.22) to (6.43), multiply by two and integrate the obtained equality over the compact manifold to get the result. 7. Proof of the Main Theorem We consider the co-differential of the Weyl tensor, δW as an element of T ∗ M ⊗ 2 (T ∗ M). According to the splitting of the space of two forms, δW splits as follows: 2 ⊕ δW72 , δW = δW14 2 is a section of T ∗ M ⊗ 2 (T ∗ M) ∼ R7 ⊗ g while δW 2 is a section of where δW14 = 2 14 7 ∗ T M ⊗ 27 (T ∗ M). The Main Theorem is a consequence of the following general
Theorem 7.1. Let (M, ω, g) be a compact G 2 -manifold with closed fundamental form ω. 2 . Suppose that the 27 -part of the co-differential of the Weyl tensor vanishes, δW = δW14 Then (M, ω, g) is a Joyce space.
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Proof. On a 7-dimensional Riemannianmanifoldthe Weyl tensor is expressed in terms s of the normalized Ricci tensor h = − 15 ρ − 12 g as follows: g
Wi jkl = Ri jkl − h ik g jl + h jk gil − h jl gik + h il g jk . The second Bianchi identity implies ∇ g i ρ jk −∇ g j ρik = (δ R)ki j , dsi = 2∇ g j ρ ji , (δW )ki j = −4 ∇ g i h jk −∇ g j h ik . (7.54) The condition of the theorem reads 4 g 2 0 = (δW )ki j ωi jl = ∇ i ρ jk − ∇ g j ρik ωi jl − dsi ωikl . 5 15 Consequently, 4 (δ R)ki j ωi jl δωkl = ∇ g i ρ jk − ∇ g j ρik ωi jl δωkl = 0, 5
(7.55)
since δω ∈ 214 . To apply effectively our integral formula (6.29) we have to evaluate one more term. Denote K i = ωi jl ρ jk δωkl . Equation (7.55) together with (5.12) leads to the equality 1 δ K = − ρ jk δω jl δωkl − 2||ρ||2 2 which implies, by an integration over the compact M, that ρ jk δω jl δωkl d V = −4 ||ρ||2 d V. M
(7.56)
M
The equalities (7.55), (7.56), (5.22) and (6.29) yield 1 14 ||δω||4 + ||ρ 0 ||2 d V = 0. 3 M 24 Hence, Theorem 7.1 follows. Clearly, our Main Theorem follows from Theorem 7.1. Corollary 5.5 and the main theorem lead to Theorem 7.2. Any compact 7-manifold with closed G 2 -structure which is Einstein with respect to the canonical connection is a Joyce space. Acknowledgements. The authors wish to thank Andrew Swann for useful discussions and remarks. This research was supported by the Danish Natural Science Research Council, Grant 51-00-0306. The authors were members of the EDGE, Research Training Network HPRN-CT-2000-00101, supported by the European Human Potential Programme. The final part of this paper was done during the visit of S.I. at the Abdus Salam International Centre for Theoretical Physics, Trieste, Italy. S.I. thanks the Abdus Salam ICTP for providing support and an excellent research environment.
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References 1. Bonan, E.: Sur le variétés riemanniennes a groupe d’holonomie G 2 ou Spin(7). C. R. Acad. Sci. Paris 262, 127–129 (1966) 2. Bryant, R.: Metrics with exceptional holonomy. Ann. Math. 126, 525–576 (1987) 3. Bryant, R.: Some remarks on G 2 -structures. http://arxiv.org/list/math.DG/0305124, 2003 4. Bryant, R.: Some remarks on G 2 -structures. To appear in the 2005 Gkova Geometry/Topology Conference Proceedings, Cambridge, MA: International Press 5. Bryant, R., Salamon, S.: On the construction of some complete metrics with exceptional holonomy. Duke Math. J. 58, 829–850 (1989) 6. Cabrera, F., Monar, M., Swann, A.: Classification of G 2 -structures. J. London Math. Soc. 53, 407–416 (1996) 7. Chiossi, S., Salamon, S.: The intrinsic torsion of SU(3) and G 2 -structures. In: Differential Geometry, Valencia 2001, River Edge, NJ: World Sci. Publishing, 2002, pp. 115–133 8. Cleyton, R., Ivanov, S.: On the geometry of closed G 2 - structures. http://arxiv.org/list/math.DG/0306362, 2003 9. Cvetiˇc, M., Gibbons, G.W., Lu, H., Pope, C.N.: Almost Special Holonomy in Type IIA&M Theory. Nucl. Phys. B638, 186–206 (2002) 10. Fernández, M.: An Example of compact calibrated manifold associated with the exceptional Lie group G 2 . J. Diff. Geom. 26, 367–370 (1987) 11. Fernández, M., Gray, A.: Riemannian manifolds with structure group G 2 . Ann. Mat. Pura Appl. (4)32, 19–45 (1982) 12. Friedrich, Th., Ivanov S.: Parallel spinors and connections with skew-symmetric torsion in string theory. Asian J. Math. 6, 303–336 (2002) 13. Friedrich, Th., Ivanov S.: Killing spinor equations in dimension 7 and geometry of integrable G 2 manifolds. J. Geom. Phys. 48, 1–11 (2003) 14. Gauntlett, J., Kim, N., Martelli, D., Waldram, D.: Fivebranes wrapped on SLAG three-cycles and related geometry. JHEP 0111 018 (2001) 15. Gauntlett, J., Martelli, D., Waldram, D.: Superstrings with Intrinsic torsion. Phys. Rev. D69 086002 (2004) 16. Gibbons, G.W., Page, D.N., Pope, C.N.: Einstein metrics on S 3 , R3 , and R4 bundles. Commun. Math. Phys. 127, 529–553 (1990) 17. Gray, A.: Vector cross product on manifolds. Trans. Am. Math. Soc. 141, 463–504 (1969); Correction 148, 625 (1970) 18. Ivanov, P., Ivanov, S.: SU (3)-instantons and G 2 , Spin(7)-heterotic string solitons. Commun. Math. Phys. 259, 79–102 (2005) 19. Joyce, D.: Compact Riemannian 7-manifolds with holonomy G 2 . I. J. Diff. Geom. 43, 291–328 (1996) 20. Joyce, D.: Compact Riemannian 7-manifolds with holonomy G 2 . II. J. Diff. Geom. 43, 329–375 (1996) 21. Joyce, D.: Compact Riemannian manifolds with special holonomy. Oxford: Oxford University Press, 2000 22. Kovalev, A.: Twisted connected sums and special Riemannian holonomy. J. Reine Angew. Math. 565, 125–160 (2003) 23. Lawson, B., Michelsohn, M.-L.: Spin Geometry. Princeton, NJ: Princeton University Press, 1989 24. Leung, N.C.: TQFT for Calabi-Yau three folds and G 2 manifolds. Adv. Theor. Math. Phys. 6, 575–591 (2002) 25. Salamon, S.: Riemannian geometry and holonomy groups. London: Pitman Res. Notes Math. Ser., 201 1989 Communicated by G.W. Gibbons
Commun. Math. Phys. 270, 69–108 (2007) Digital Object Identifier (DOI) 10.1007/s00220-006-0147-5
Communications in
Mathematical Physics
Superselection Sectors and General Covariance. I Romeo Brunetti1 , Giuseppe Ruzzi2 1 II Institute für Theoretische Physik, Universität Hamburg, Luruper Chaussee 149, D-22761 Hamburg,
Germany. E-mail:
[email protected]
2 Dipartimento di Matematica, Università di Roma “Tor Vergata,” Via della Ricerca Scientifica, I-00133
Roma, Italy. E-mail:
[email protected] Received: 21 December 2005 / Accepted: 2 August 2006 Published online: 21 November 2006 – © Springer-Verlag 2006
Abstract: This paper is devoted to the analysis of charged superselection sectors in the framework of the locally covariant quantum field theories. We shall analyze sharply localizable charges, and use net-cohomology of J.E. Roberts as a main tool. We show that to any 4-dimensional globally hyperbolic spacetime a unique, up to equivalence, symmetric tensor C∗ -category with conjugates (in case of finite statistics) is attached; to any embedding between different spacetimes, the corresponding categories can be embedded, contravariantly, in such a way that all the charged quantum numbers of sectors are preserved. This entails that to any spacetime is associated a unique gauge group, up to isomorphisms, and that to any embedding between two spacetimes there corresponds a group morphism between the related gauge groups. This form of covariance between sectors also brings to light the issue whether local and global sectors are the same. We conjecture this holds that at least on simply connected spacetimes. It is argued that the possible failure might be related to the presence of topological charges. Our analysis seems to describe theories which have a well defined short-distance asymptotic behaviour.
Contents 1. 2. 3. 4. 5. 6. 7. A.
Introduction . . . . . . . . . . . . . . . . Spacetime Geometry . . . . . . . . . . . . Locally Covariant Quantum Field Theory . Homotopy of Posets and Net Cohomology Charged Superselection Sectors . . . . . . Local Completeness . . . . . . . . . . . . Conclusions and Outlook . . . . . . . . . Symmetric Tensor C∗ -Categories . . . . .
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1. Introduction The present paper represents a first step toward a new investigation of superselection sectors based on the general framework of locally covariant quantum field theory [13]. One of the milestones of quantum field theory is the investigation of the superselection rules. They are, in essence, constraints to the nature and scope of possible measurements, proving that in quantum physics the superposition principle does not hold unrestrictedly. Stimulated by the preliminary investigation about the existence of superselection rules in quantum field theory by Wick, Wightman and Wigner [58], Haag and Kastler [33] found that the correct interpretation resides in the inequivalent representations of abstract C∗ algebras describing local observables, i.e. equivalence classes of local measurements. Some years later, Doplicher, Haag and Roberts [21, 22] undertook an awesome analysis which culminated at the beginning of the nineties with a long series of remarkable papers (Doplicher and Roberts, e.g. see [24, 25]). The main result that has been proved is that besides any algebra of observables there is associated, canonically, a field algebra with normal Bose-Fermi commutation relations, on which a compact gauge group of the first kind acts. This group determines the superselection structure of the theory, in the sense that its irreducible representations are in one-to-one correspondence with the superselection sectors describing sharply localized charges. Well beyond the importance for the physical interpretation, they also obtained outstanding mathematical results, e.g. by founding an abstract theory of group duals extending that of Tannaka and Krein. Let us describe in more specific terms the basic ideas of superselection sectors in the algebraic approach of Haag and Kastler [33, 32]. One assumes that all the physical information is contained in the a priori correspondence between (open, bounded) regions of Minkowski spacetime and algebras of observables that can be measured inside it, i.e., we have an abstract net of local observables, namely the correspondence O → A (O) which associates, for instance, to every element O of the family Kdc of double cones of Minkowski spacetime the (unital) C∗ -algebra A (O). The C∗ -algebra generated by all local ones is termed the quasi-local algebra of observables and denoted by AKdc . A few basic principles can be adopted, among which are causality and Poincaré covariance. The claim is that this is all one needs for investigating (structural) properties of a physical theory. This claim is substantiated by many results, among which the theory of superselection sectors is the leading example [32], as far as the intrinsic perspective is concerned, although one should not forget to cite that also the time-honored perturbation theory can be fruitfully formulated in the algebraic framework [9, 26, 34]. An important point, where the physical interpretation enters heavily, is in the choice of the correct physical states for the theory at hand. As far as applications to elementary particle physics is concerned, one assumes the existence of a canonical state, that of the (pure) vacuum state ω0 . Doplicher, Haag and Roberts invoked the intuitive idea that any particle (at least in the case in which no massless excitations are present in the physical theory) arises as an elementary and localized excitation of the vacuum state. Hence, they put forward that, if π0 is the representation of the algebra AKdc associated with the vacuum state ω0 , any representation π which describes an elementary and localized excitation of the vacuum should satisfy π A (O⊥ ) ∼ = π0 A (O⊥ ), O ∈ Kdc ,
(1.1)
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where the symbol ∼ = means unitary equivalence, and ⊥ denotes the causal complement. This put the previous intuitive remark of particles as localized excitations of the vacuum on rigorous footing by declaring those representations to be “physically relevant” that are unitarily equivalent to the vacuum one in the causal complement of any double cone. Superselection sectors of the quasi-local algebra AKdc are then defined to be the unitary equivalence classes of those irreducible representations, satisfying the selection condition, and where the labels distinguishing them are called charged quantum numbers. One then considers the representations satisfying condition (1.1) as referring to “localizable charges.” The domain of validity of such intrinsic development is, for instance, that of the massive sector of hadronic theories. Borchers [7], in a previous attempt, based the selection of physical sectors on the requirement of positivity of the energy, since also in this case one can exclude states for which the matter density does not vanish at infinity. On the basis of his remarks, the analysis of Doplicher, Haag and Roberts was further refined by Buchholz and Fredenhagen [17], at least for the case of gauge theories without massless excitations, by requiring the selection condition on unbounded regions called spacelike cones. These are regions extending to spacelike infinity. Their criterion is essentially that shown before, where one changes the double cones with the new regions. The so selected representations bear charges termed “topological charges,” and the results are similar to those of Doplicher, Haag and Roberts. However, there are important theories for which the above requirements are not fulfilled, for instance in abelian gauge theories like QED. There the above localization is not pertinent since any gauge charge has attached, via Gauss’ Law, its own flux line or string which forces a poorer localization. QED itself is still awaiting a complete determination of its sectors, but see e.g. [16] and [42, 43]. In both localizable and topological cases, the analysis leads to the following remarkable results shedding light on deep characteristic aspects of elementary particle physics, namely; 1. Particle-Antiparticle symmetry. To each charge corresponds a unique conjugated charge which entails the particle-antiparticle symmetry; 2. Particle Statistics. Each charge has a permutation symmetry to which there corresponds uniquely a sign and an integer d, its statistical dimension; the corresponding particles, if any, then satisfy parastatistics of order d, the sign distinguishes between para-Bose and para-Fermi statistics. Usual Bose and Fermi statistics correspond to d = 1. 3. Composition of Charges. Two charges can be composed because they can be created in spacelike separated regions. The analysis developed so far has been fruitfully applied in other contexts, for instance in (chiral) conformal quantum field theory by Fredenhagen, Kawahigashi, Longo, Rehren, Schroer and various other collaborators [28, 36, 35], for quantum field theory on curved spacetime in a paper by Guido, Longo, Roberts and Verch [31], and initially for soliton physics and then towards more mathematical aims by Müger [39, 40]. In a more mathematical direction one should cite the work done by Baumgärtel and Lledó [2] and by Vasselli [52]. On the previous investigation of the superselection structure in the context of curved backgrounds Guido, Longo, Roberts and Verch [31] used as a major tool the results by Roberts on the connection between net cohomology and superselection sectors [44, 45].
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Net cohomology has made clearer that it is the causal and topological structure of spacetime that are the basic relevant features when studying superselection sectors. In particular, we now understand that the main point is how these two properties are encoded in the structure of the index set of the net (e.g. the set of double cones Kdc for Minkowski spacetime) as a poset (partially ordered set) ordered under inclusion. Let us then call K• the preferred choice of index set on the spacetime M, and call AK• the corresponding net of local observables. One proves, under some conditions, that representations satisfying the selection criterion (1.1) are, up to equivalence, in one-to-one correspondence with 1-cocycles of the poset with values in the (vacuum) representation of the net of local algebras, BK• : O → B(O) where B(O) ≡ π0 (A (O)) , for any O ∈ K• . They define a C∗ −category Zt1 (BK• ). In order to be a good starting point for the analysis of superselection structure, it should possess good features like being a tensor category, having a permutation symmetry and a conjugation (in the case of finite statistics). Recently, in [50], one of us completed the investigation of [31] in the case where the spacetime is globally hyperbolic, relying on a refined form of net cohomology where a more precise investigation of the properties of indices was made clear, much stimulated by some previous results in [46]. Among several things, it is discussed why the choice made in [31] of basing the net structure on the so-called “regular diamonds” index set Kr d is unfortunate. There are two main problems with such a choice; the first is that this index set is not directed, causing problems with the definition of the tensor product structure of the category of 1-cocycles; the second is that it has elements with a non connected causal complement, a topological feature that seems, at least, to forbid a straightforward application of Haag duality, if possible at all. In [50] a better choice was made, that of using as a basic index set that of “diamonds,” noted Kd , having only elements with connected causal complements. On this basis the analysis was easier and more powerful, and one arrives at a category of 1-cocycles Zt1 (BKd ) that indeed possesses all the previously stated properties. Our aims, as announced at the beginning, is to initiate a study of superselection sectors for the case of locally covariant quantum field theories [13], stepping forward from the previous developments [31, 50, 56]. This new framework deals with a non-trivial blend of two principles, locality and general covariance, in quantum physics. We refer to the review paper [10] for further discussion. It contains the Haag-Kastler setting as a subcase, it found deep application to perturbation theory on curved spacetime [34] and appears to be well suited to developing a new look at perturbative quantum gravity [11]. Especially, it gives a new perspective in quantum field theory, for instance see [55, 49]. Hence, it is an important point to determine and study the features of the superselection structure. The basic features of the approach will be recalled in Sect. 3, here we just notice that it is a categorical approach in which a quantum field theory is considered as a covariant functor from a category Loc whose objects are 4-dimensional globally hyperbolic spacetimes, with isometric embeddings as arrows, to a category of C∗ -algebras of observables, with injective ∗ -homomorphisms as arrows. Aiming at founding superselection theory for this more general setting, we are then faced with the following list of issues: (a) the good choice of the index sets for any globally hyperbolic spacetime and their stability under isometric embeddings, (b) a pertinent choice of a reference state (space), with a minimal set of assumptions for nets of observable algebras,
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(c) the choice of what kind of 1-cocycles one wants to investigate, i.e., regarding their localization properties and the construction of their C∗ −categories, with the sought for properties, and especially their behaviour under isometric embeddings. For the first problem, a “canonical” choice was made in [13]. Namely, a family of regions of each M with the properties of being relatively compact, causally convex, i.e., such that all causal curves starting and ending inside such a region would always remain inside it, and with non-empty causal complement. We denote it by Kh (M). One immediately faces a problem with such a family. Indeed, it shares with the regular diamonds the same annoying topological feature of having nonconnected causal complements. However, we learned in [50] how to overcome this by passing to the diamonds’ family Kd (M). Although for the first family stability under isometric embeddings is essentially obvious, our proof for the second relies on a nice recent result due to Bernal and Sánchez [6] on the extension of compact hypersurfaces. A choice of a reference state is more involved and delicate. Notice that, in the Minkowskian analysis of sectors á la Doplicher-Haag-Roberts, a choice is made in terms of a single pure reference state, typically the vacuum. In the locally covariant setting one can prove [13] that there is no choice of single states on each M, pure or not, which is covariant under local diffeomorphisms. However, the covariance works well [13], in examples, if instead one chooses folia of states for each manifold. This entails that it is appropriate [13] to consider a functorial description of state space that takes into account the covariance under local diffeomorphisms. It is then rather natural to choose, for our present purposes, a locally quasi-equivalent state space So (M) for any M, that satisfies some further technical conditions as Borchers property and such that there is at least one state for which the associated net of von Neumann algebras satisfies irreducibility and punctured Haag duality. Since no preferred choice of states is at hand, one of our main tasks would be therefore to exploit the relation between nets defined in terms of different states belonging to the reference state space. It is perhaps one of the virtues of our approach that such a relation can be fully discussed, and brings, under certain technical conditions, an isomorphism between the nets, which moreover behaves well under isometric embeddings. At the level of categories of 1-cocycles, we have been able to prove that they do not depend on the choice of the states inside the reference state space. As far as assumptions for the net are concerned, we have chosen mainly to require irreducibility and punctured Haag duality. This is really a minimal requirement since a form of Haag duality seems to be really necessary for exploiting properties of superselection sectors. Innocent as it looks, this choice will turn out to be crucial for our construction. However, the real crux of the matter comes with the last issue. Here, there are several possibilities, depending on the choice of the index set and the topological features of the spacetimes. We recall that according to the results in general relativity [59], Cauchy surfaces can have any topology as (closed) 3-dimensional topological spaces. For instance, they can be compact and/or non-simply connected, e.g. as in de Sitter space and for the RP3 geon (see, e.g., [29]), respectively. Hence, the problem lies in the large number of possibilities that we have for choosing the localization properties of the 1-cocycles, namely what kind of index set we would like to use, and accordingly, what features of the charges do we want to highlight. This is additionally complicated by the fact that some of the index sets have the ability to recover the topology of the spacetime while others do not [50]. It is in order to clarify another point, namely, although we are working within the locally covariant setting, this does not mean that the charges have to be necessarily
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localized in relatively compact regions. Local covariance only requires that the charges are local functionals of the metric, i.e., they depend only on local geometrical data, nothing forbids the charges testing the topological structure of spacetime. Something like the Buchholz and Fredenhagen analysis of topological charges can be envisaged. See, however, Remark 6.3. In this first paper, however, we restrict our attention to localizable charges, namely we fix as a main index set that of diamonds Kd (M) on any globally hyperbolic spacetime M, and we choose as 1-cocycles those having the property of being path-independent, in the precise sense of paths in the poset Kd (M). In this case the 1-cocycles provide trivial representations of the first fundamental group of the manifold [50], hence they do not probe the topology of the spacetime. This choice corresponds to that of localizable charges in the Doplicher, Haag and Roberts sense on Minkowski spacetime. In the forthcoming paper [14] we will discuss how different choices of the posets are related to each other, still in the localizable case. We are working on the more complicated analysis of “charges carrying topological structure” i.e., on path-dependent 1-cocycles. Even in the easier case of localizable charges there is a subtlety. Although diamonds form a well behaved family for several of the questions underlying the mathematical development of the structure of sectors, one needs the family Kh (M) for some crucial results, since it contains nonsimply connected regions. For instance, under the isomorphism between nets for different choices of states, a 1-cocycle of Kd (M) does not remain necessarily path-independent. The stability under net isomorphisms seems to hold only when the nets are extended to Kh (M) (see Sect. 5.1.2). Nonetheless, these results involve only investigating the category of 1-cocycles, not the tensor structure. Hence, one of the main points of the analysis is to prove that the categories of 1-cocycles associated to the different families are actually equivalent. Granted that, one can proceed to the analysis of the tensor structure of the categories. It is not clear to us, at the moment, whether the presence of both families underlies some further subtle point of more physical origin. A similar situation has been studied by Ciolli [18] for massless scalar fields in 1+1 dimensions. The main result of the paper is the following: Calling R the restriction operation of 1-cocycles defined on spacetime M to a subspacetime N ⊂ M, we have Main Technical Theorem. The restriction R lifts to a full and faithful covariant ∗ -functor between Z 1 (ω, Kd (M)) and Z 1 (ω, Kd (N )), for any choice of the state ω t t in the reference state space So . This result entails the good behaviour under isometric embeddings of spacetimes. Furthermore, it shows that the kind of theories we study behave well under the scaling limit. Indeed, the Main Technical Theorem seems to be the cohomological counterpart of the “equivalence between local and global intertwiners” that “good” theories possess in the scaling limit [20]. Furthermore, the main result of the paper is that, according to the expectations coming out the locally covariant approach, the suitably defined superselection structure is also functorial, namely, one can define a map S from the category of spacetimes Loc to the category Sym of symmetric tensor C∗ -categories, with full and faithful symmetric tensor ∗ -functors as arrows, for which there holds Main Theorem. The map S : Loc → Sym is a contravariant functor. Roughly speaking, the message coming out the theorem is that the physical content of superselection sectors based on different spacetimes can be faithfully transported
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provided the spacetimes can be embedded. Namely, all charged quantum numbers are preserved in this embedding. An important byproduct of the Main Theorem is that to any spacetime there is associated a unique gauge group, up to isomorphism, and that to any embedding between spacetimes there corresponds a group morphism between the respective gauge groups. Another consequence of the Main Theorem is that it makes possible a clear investigation of the possible relation between local and global superselection sectors of a spacetime. Because of the sharp localization, one expects an equivalence between local and global superselection sectors. However there is no general evidence of this equivalence, that we call local completeness of superselection sectors. Indeed, one can prove it only in models derived from free quantum fields. We point out that a possible violation of local completeness might be related to the nontrivial topology of spacetimes and, in particular, to the existence of path-dependent 1-cocycles. We now pass on to outline the content of the paper. In Sect. 2 we start by elaborating geometrical features which are crucial for the following parts. We discuss some background results on Lorentzian spacetime mainly for fixing our notation. Then we pass to the discussion of families of subsets for each globally hyperbolic spacetime in 4-dimensions and their stability properties under isometric embeddings. Some categorical notions are briefly outlined. In Sect. 3 we start by recalling the basic definitions of the locally covariant quantum field theory as a covariant functor. There we introduce some crucial definitions and assumptions. Moreover we prove some results on state spaces. Sect. 4 contains only a brief discussion of the basic definitions and properties of net cohomology. The heart of the paper is in Sect. 5. There we prove our Main Theorem and several related results. Eventually, these last results are applied, in Subsect. 5.3, to prove the generally covariant behaviour of the superselection structures. In Sect. 6 we point out the difficulties in proving the equivalence between local and global sectors. Conclusions and Outlook follow. An Appendix is provided where some categorical notions and some results about Doplicher-Roberts Reconstruction Theorem are briefly recalled. 2. Spacetime Geometry In the first part of this section we review some basics notions of Lorentzian geometry, introduce the category of spacetimes and define some families of sets of technical importance. We prove some results of independent interest.
2.1. Lorentzian spacetimes. We recall some basics on the causal structure of spacetimes and establish our notation. Standard references for this topic are [3, 41, 27, 57]. A spacetime M, in our framework, consists of a Hausdorff, paracompact, connected, without boundary, smooth, oriented 4-dimensional manifold M endowed with a smooth metric g with signature (−, +, +, +), and with a time-orientation, that is a smooth vector field v satisfying the equation g p (v p , v p ) < 0 for each p ∈ M. (Throughout this paper smooth means C ∞ ). A curve γ in M is a continuous, piecewise smooth, regular function γ : I → M, where I is a connected subset of R with nonempty interior. It is called timelike, lightlike, spacelike if respectively g(γ˙ , γ˙ ) < 0, = 0, > 0 all along γ , where γ˙ = dγ dt . Assume that γ is causal, i.e. a nonspacelike curve; we can classify it according to the time-orientation v as future-directed or past-directed if respectively g(γ˙ , v) < 0, > 0 all along γ . When γ is future-directed and there exists limt→sup I γ (t) (limt→inf I γ (t)), then it is said to
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have a future (past) endpoint. In the negative case, it is said to be future (past) endless; γ is said to be endless if none of them exist. Analogous definitions are assumed for past-directed causal curves. The chronological future I + (S), the causal future J + (S) and the future domain of dependence D + (S) of a subset S ⊂ M are defined as: I + (S) ≡ {x ∈ M| exists a future-directed timelike curve from S to x }; J + (S) ≡ S ∪ {x ∈ M| exists a future-directed causal curve from S to x }; D + (S) ≡ {x ∈ M| any past-directed endless causal curve through x meets S}. These definitions have duals in which “future” is replaced by “past” and the + by −. By this, we define I (S) ≡ I + (S) ∪ I − (S), J (S) ≡ J + (S) ∪ J − (S) and D(S) ≡ D + (S) ∪ D − (S). Remarks 2.1. It is worth recalling the following properties of causal sets1 . (a) Let S ⊆ M. Then I + (S) is an open set and I +(cl(S)) = I +(S); cl(J + (S)) = cl(I + (S)) and int (J + (S)) = I + (S); cl J + (cl(S)) = cl J + (S) . (b) Let S1 , S2 , S3 ⊆ M with S1 ⊆ J + (S2 ) and S2 ⊆ I + (S3 ). Then S1 ⊆ I + (S3 ). The causal disjointness relation is a symmetric binary relation ⊥ on the subsets of M defined as follows: S ⊥ V ⇐⇒ V ⊆ M \ J (S).
(2.1)
The causal complement of a set S is the open set S ⊥ defined as S ⊥ ≡ M \ cl(J (S)).
(2.2)
A set S is acausal if { p} ⊥ {q} for each pair p, q ∈ S. A set S is achronal if I + (S)∩S = ∅. A (acausal) Cauchy surface C of M is an achronal (acausal) set verifying D(C) = M. Any Cauchy surface is a closed, connected, Lipschitz hypersurface of M. A spacelike Cauchy surface is a smooth Cauchy surface whose tangent space is everywhere spacelike. Any spacelike Cauchy surface is acausal. Global hyperbolicity and the category Loc. A spacetime M is globally hyperbolic if it admits a smooth [4, 5] foliation by spacelike Cauchy surfaces, namely, there is a 3-dimensional smooth manifold and a diffeomorphism F : R × → M such that: for each t ∈ R the set Ct ≡ {F(t, y) | y ∈ } is a spacelike Cauchy surface of M; the curve t ∈ R → F(t, y) ∈ M is a future-directed (by convention) endless timelike curve for any y ∈ . We recall some causality properties of a globally hyperbolic spacetime. Remarks 2.2. Assume that M is a globally hyperbolic spacetime. Let K be a rela tively compact subset of M. Then, J + (cl(K )) is closed, and J + (cl(K )) = cl J + (K ) ; D + (cl(K )) is compact. If C is a spacelike Cauchy surface, then J (cl(K ))∩C is a compact subset of C. Two useful properties of causal sets are given in the next two lemmas. Lemma 2.3. Let M be a globally hyperbolic spacetime. If S is an open subset of M, then J + (S) = I + (S), J − (S) = I − (S) and J (S) = I (S). 1 cl(S) and int (S) denote respectively the closure and the interior of the set S.
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Proof. It is clear that I + (S) ⊆ J + (S). Now, let x ∈ J + (S). Take a point x0 in J − (x)∩S = ∅. As S is an open set there is y ∈ S such that x0 ∈ I + (y) (global hyperbolicity, in fact, implies that the collection I + (x) ∩ I − (y), as x, y vary in M, is a basis for the topology of M [41, 27]). By Remark 2.1.b, we have that x ∈ J + (x0 ), x0 ∈ I + (y) ⇒ x ∈ I + (y) ⇒ x ∈ I + (S). Hence J + (S) = I + (S). Analogously, J − (S) = I − (S); hence J (S) = I (S).
Lemma 2.4. Let K , S be two subsets of a globally hyperbolic spacetime M. Assume that K is relatively compact with K ⊥ = ∅, and that S is open. If cl(K ) ⊂ S, then K ⊥ ∩ S = ∅. Proof. Assume that K ⊥ ∩ S = ∅. This is equivalent to say that S ⊆ (M \ K ⊥ ) = cl(J (K )) = J (cl(K )) (see Remark 2.2). Since cl(K ) ⊂ S, we have that J (cl(K )) ⊆ J (S) ⊆ J (cl(K )). Hence J (S) = J (cl(K )) and this leads to a contradiction. Indeed, by Lemma 2.3 J (S) is an open set while J (cl(K )) is closed and different from M since K ⊥ = ∅. Therefore J (S) is an open and closed proper subset of M; this is not possible because M is connected. Let M and M1 be globally hyperbolic spacetimes with metrics g and g1 respectively. A smooth function ψ from M1 into M is called an isometric embedding if ψ : M1 → ψ(M1 ) ⊆ M is a diffeomorphism and ψ∗ g1 = gψ(M1 ) . The category Loc is the category whose objects are the 4-dimensional globally hyperbolic spacetimes; the arrows (M1 , M) are the isometric embeddings ψ : M1 −→ M preserving the orientation and the time-orientation of the embedded spacetime, and that satisfy the property ∀x, x1 ∈ ψ(M1 ), J + (x) ∩ J − (x1 ) is contained in ψ(M1 ). The composition law between two arrows ψ and φ, denoted by ψφ, is given by the usual composition between smooth functions; the identity arrow id M is the identity function of M. 2.2. Stable families of indices. Given M ∈ Loc, the aim is to introduce families of open subsets of M which will be used as index sets for nets of local algebras. We prove their stability under isometric embeddings. Globally hyperbolic regions. Let Kh (M) be the collection of subsets O ⊆ M satisfying the following properties: (i) O is an open, connected, relatively compact set, and O⊥ = ∅; (ii) if x1 , x2 ∈ O, then J + (x1 ) ∩ J − (x2 ) contained in O. It turns out by this definition that Kh (M) is a basis for the topology of M and that any element O of Kh (M) – with the metric gO and with the induced orientation and time orientation – is a globally hyperbolic spacetime, hence O ∈ Loc. We now show some straightforward geometrical results. Lemma 2.5. Let O ∈ Kh (M). The following assertions hold. (a) There are O1 , O2 ∈ Kh (M) such that cl(O1 ) ⊥ O and cl(O ∪ O1 ) ⊆ O2 . (b) For any O1 ∈ Kh (M), with cl(O1 ) ⊥ O, there is O2 ∈ Kh (M) such that cl(O ∪ O1 ) ⊂ O2 .
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Proof. (a) follows from (b). (b) Since cl(O1 ) ⊂ M \ J (cl(O)), Lemma 2.4 implies that O1⊥ ∩ O⊥ = M \ (J (cl(O)) ∪ J (cl(O1 ))) is open and nonempty. Pick x ∈ O1⊥ ∩ O⊥ and a spacelike Cauchy surface C that meets x. Define K ≡ J (cl(O)) ∩ C and K 1 ≡ J (cl(O1 )) ∩ C. Observe x ∈ K ∪ K 1 and that, in the relative topology of C, both K and K 1 are compact and connected because so are cl(O) and cl(O1 ) (for compactness see Remark 2.2). Let W , W1 be two connected and relatively compact open sets of C such that: K ⊂ W , K 1 ⊂ W1 and x ∈ cl(W ∪ W1 ). Now, if W ∩ W1 = ∅, then we define O2 ≡ D(W ∪ W1 ). O2 verifies the properties of the statement. In fact as W ∪ W1 is an open and connected subset of C, O2 is open and globally hyperbolic [41, Lemma 43]. Moreover, since W ∪ W1 is relatively compact, O2 is relatively compact. Finally, O2⊥ = ∅ because x ∈ cl(W ∪ W1 ) = ∅. Hence O2 ∈ Kh (M), and it is also clear that cl(O ∪ O1 ) ⊂ O2 . Conversely, assume that W ∩ W1 = ∅. Let γ : [0, 1] → C be a curve that does not meet x and such that γ (0) ∈ W and γ (1) ∈ W1 . We can find a family Ui , i = 1, . . . n of open, relatively compact and connected sets of C which cover the curve γ and such that, if G ≡ W ∪ W1 ∪ Un ∪ · · · ∪ U1 , then x ∈ cl(G). By setting O2 ≡ D(G), as before we have that O2 verifies the properties of the statement. (b) follows from (a): it is enough to observe that, since O⊥ = ∅ and since Kh (M) is a basis for the topology of M, there is O1 ∈ Kh (M), with cl(O1 ) ⊥ O. We now use the previous lemma to show that compactness of Cauchy surfaces is the only obstruction to the directedness of the family Kh (M), namely that for any O1 , O2 ∈ Kh (M) there is O ∈ Kh (M) such that O1 ∪ O2 ⊆ O. Lemma 2.6. If M ∈ Loc has noncompact Cauchy surfaces, then Kh (M) is directed under inclusion. Proof. Let O1 , O2 ∈ Kh (M) and let C be a spacelike Cauchy surface. Since the set K ≡ (J (cl(O1 )) ∩ C) ∪ (J (cl(O2 )) ∩ C) is compact, the set C \ K is open and nonempty. From now on the proof is identical to that of Lemma 2.5.b. Consider M1 , M ∈ Loc and let ψ ∈ (M1 , M). Define ψ(Kh (M1 )) ≡ {ψ(O) ⊆ M | O ∈ Kh (M1 )}, Kh (M)ψ(M1 ) ≡ {O ∈ Kh (M) | cl(O) ⊆ ψ(M1 )}.
(2.3)
Lemma 2.7. Given M1 , M ∈ Loc and ψ ∈ (M1 , M), then Kh (M)ψ(M1 ) = ψ(Kh (M1 )). Proof. It is clear that if O1 ∈ Kh (M1 ), then ψ(O1 ) ∈ Kh (M)ψ(M1 ) . Conversely, let O ∈ Kh (M)ψ(M1 ) . By Lemma 2.4, O⊥ ∩ ψ(M1 ) = ∅. Then the causal complement of ψ −1 (O) in M1 is nonempty, hence ψ −1 (O) ∈ Kh (M1 ). Finally, we want to stress that Kh (M) have elements which are nonsimply connected subsets of M and elements whose causal complement is nonconnected. This fact creates some problems in studying superselection sectors of a net indexed by Kh (M). However, as we shall see in Sect. 4, these problems will be overcome by basing the theory on a particular subfamily of Kh (M) whose elements do not present the above topological features. To this end, define Sub(Kh (M)) ≡ {K1 ⊆ Kh (M) | K1 is a basis for the topology of M}. Clearly Kh (M) ∈ Sub(Kh (M)). Further properties are discussed in Sect. 4.
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Diamonds regions. A relevant element of Sub(Kh (M)) is the set Kd (M) of diamonds of M [50]. An open subset O of M is called a diamond if there is a spacelike Cauchy surface C, a chart (U, φ) of C, and an open ball B of R3 such that O = D(φ −1 (B)), cl(B) ⊂ φ(U ) ⊂ R3 . We will say that O is based on C and call φ −1 (B) the base of O. Any diamond O is an open, relatively compact, connected and simply connected subset of M, and its causal complement O⊥ is connected. Furthermore, for any diamond O there exists a pair of diamonds O1 , O2 such that cl(O), cl(O1 ) ⊂ O2 , O ⊥ O1 .
(2.4)
The set of diamonds Kd (M) is a basis for the topology of M and Kd (M) ∈ Sub(Kh (M)). Let ψ(Kd (M1 )) ≡ {ψ(O) ⊆ M | O ∈ Kd (M1 )}, Kd (M)ψ(M1 ) ≡ {O ∈ Kd (M) | cl(O) ⊆ ψ(M1 )}.
(2.5)
Lemma 2.8. Given M1 , M ∈ Loc and ψ ∈ (M1 , M), then Kd (M)ψ(M1 ) = ψ(Kd (M1 )). Proof. We prove the inclusion (⊇). Let O be a diamond of M1 . This means that there exists a spacelike Cauchy surface C1 of M1 , a chart (U, φ1 ) such that O1 = D(φ1−1 (B)), where B is a ball of R3 such that cl(φ1−1 (B)) ⊆ U . Let B1 be a ball of R3 such that cl(B) ⊆ cl(B1 ) and cl(φ1−1 (B1 )) ⊆ U . Observe that ψφ1−1 (B1 ) is a relatively compact, spacelike acausal open set of M with boundaries and with a nonempty complement. By [6], there exists in M a spacelike Cauchy surface C such that cl(ψφ1−1 (B1 )) ⊆ C. Define V ≡ ψφ1−1 (B1 ) and φ ≡ φ1 ψ −1 . The pair (V, φ) is a chart of C and cl(φ −1 (B)) ⊂ V . Finally observe that by the properties of ψ we have that ψ(D(φ1−1 (B))=D(ψφ1−1 (B))= D(φ −1 (B)), namely Kd (M)ψ(M1 ) ⊇ ψ(Kd (M1 )). The proof of the reverse inclusion is very similar to the previous one and we omit it. There are many other elements of Sub(Kh (M)) which may be of a certain interest. For instance, one such is the family of regular diamonds used in [31] (see it for the exact definition, or [54]). It contains the family of diamonds as a subset, and has the same stability property as the others. We refrain from giving details here, since this family does not play a rôle hereafter. More discussion will be found in [14]. 3. Locally Covariant Quantum Field Theory Locally covariant quantum field theory is a categorical approach to the quantum theory of fields which incorporates the locality principle of classical field theory in a generally covariant manner [13, 10]. In order to introduce the axioms of the theory, we give a preliminary definition. Let us denote by Obs the category whose objects A are unital C∗ -algebras and whose arrows (A1 , A2 ) are the unit-preserving injective C∗ -morphisms from A1 into A2 . The composition law between the arrows α1 and α2 , denoted by α1 α2 , is given by the usual composition between C∗ -morphisms; the unit arrow idA of (A, A) is the identity morphism of A.
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A locally covariant quantum field theory is a covariant functor A from the category Loc (see Sect. 2) into the category Obs. We denote A (ψ) by αψ for any ψ ∈ (M1 , M). The functor A is said to be causal if, given ψi ∈ (Mi , M) for i = 1, 2, ψ1 (M1 ) ⊥ ψ2 (M2 ) ⇒ αψ1 (A (M1 )), αψ2 (A (M2 )) = 0, where ψ1 (M1 ) ⊥ ψ2 (M2 ) means that ψ1 (M1 ) and ψ2 (M2 ) are causally disjoint in M. From now on A will denote a causal locally covariant quantum field theory. In conclusion, let us see how a net of local algebras over M ∈ Loc can be recovered from a locally covariant quantum field theory A . To this end, recall that any O ∈ Kh (M), considered as a spacetime with the metric gO , belongs to Loc. The injection ι M,O of O into M is an element of (O, M) because of the definition of Kh (M). Then, using αι M,O ∈ (A (O), A (M)) to define A(O) ≡ αι M,O (A (O)), it turns out [13] that the correspondence AKh (M) : Kh (M) O −→ A(O) ⊂ A (M),
(3.1)
is a net of local algebras satisfying the Haag-Kastler axioms: O1 ⊆ O2 ⇒ A(O1 ) ⊆ A(O2 ), isotony, O1 ⊥ O2 ⇒ [A(O1 ), A(O2 )] = 0, causalit y. As for the local covariance of the theory, let M1 ∈ Loc with the metric g1 , and let ψ ∈ (M1 , M). Because of Lemma 2.7, ψ(O) ∈ Kh (M) for each O ∈ Kh (M1 ). Since ι−1 M,ψ(O ) ψ ι M1 ,O is an isometric embedding of the spacetime O onto the spacetime ψ(O) — the latter equipped with the metric gψ(O) — one has that αψ : A(O) ⊂ A (M1 ) → A(ψ(O)) ⊂ A (M),
(3.2)
is a C∗ -isomorphism. As ψ(Kh (M1 )) ∈ Sub(Kh (M)), denote by Aψ(Kh (M1 )) the net index by ψ(Kh (M1 )) obtained by restricting AKh (M) to ψ(Kh (M1 )). Then, relation (3.2) says that the mapping αψ : AKh (M1 ) → Aψ(Kh (M1 )) is a net-isomorphism. 3.1. States and representations of nets. Fix M ∈ Loc, and consider the net AKh (M) . A state ω of the algebra A (M), is defined to be a positive (ω(A∗ A) ≥ 0, A ∈ A (M)), and normalized (ω( 11) = 1) linear functional on it. For any state ω of the algebra A (M) we will denote by (πω , Hω , ω ) the corresponding GNS-construction; by ω∗ AKh (M) we will denote the net of von Neumann algebras defined as the correspondence ω∗ AKh (M) : Kh (M) O → Aω (O) ⊆ B(Hω ),
(3.3)
where Aω (O) ≡ πω (A(O)) for any O ∈ Kh (M), and B(Hω ) is the C∗ –algebra of linear bounded operators of Hω . Furthermore, we set Aω (M) ≡ πω (A (M)). Once ω∗ AKh (M) is given, to any element K1 ∈ Sub(Kh (M)) there corresponds a net of von Neumann algebras ω∗ AK1 obtained by restricting ω∗ AKh (M) to K1 . Properties of such a net of local algebras, which will be important for our purposes, are the following: • ω∗ AK1 is said to be irreducible whenever, given T ∈ B(H) such that T ∈ Aω (O) for any O ∈ K1 , then T = c · 11.
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• ω∗ AK1 satisfies the Borchers property if given O ∈ K1 for any O1 ∈ K1 with cl(O) ⊂ O1 any nonzero orthogonal projection E ∈ Aω (O) is equivalent to 11 in Aω (O1 ). • ω∗ AK1 is locally definite if C · 11 = ∩{Aω (O) | O ∈ K1 , x ∈ O}, for any point x of M. • ω∗ AK1 satisfies punctured Haag duality if given O ∈ K1 , with cl(O) ⊥ {x}, then Aω (O) = ∩{Aω (O1 ) | O1 ∈ K1 , O1 ⊥ O, cl(O1 ) ⊥ {x}}, for any point x of M. Some observations are in order. First, given K1 ∈ Sub(Kh (M)), the irreducibility of the net ω∗ AK1 is, in general, stronger than irreducibility of the representation πω of A (M). This is because the collection ∪O∈K1 Aω (O) needs not to be dense in Aω (M) (see also [13]). Moreover, it is clear that if ω∗ AK1 satisfies punctured Haag duality, then it satisfies Haag duality; given O ∈ K1 , then Aω (O) = ∩{Aω (O1 ) | O1 ∈ K1 , O1 ⊥ O}. If ω∗ AK1 satisfies punctured Haag duality and is irreducible, then it is locally definite [49]. From now on we will say that a state ω ∈ S (M) satisfies punctured Haag duality if ω∗ AKh (M) is irreducible and satisfies punctured Haag duality. We note the following straightforward results: Lemma 3.1. The following assertions hold: (a) If AK1 is irreducible, then AK2 is irreducible, for any K2 ∈ Sub(Kh (M)) such that K1 ⊆ K2 ; (b) If AK1 satisfies the Borchers property, then AK2 satisfies the Borchers property for any K2 ∈ Sub(Kh (M)) such that K2 ⊆ K1 ; (c) If there is K1 ∈ Sub(Kh (M)) such that AK1 is locally definite, then AK2 is locally definite for any K2 ∈ Sub(Kh (M)). Proof. (a) and (b) are obvious. (c) derives from the fact that any element of Sub(Kh ) is a basis for the topology of M. 3.2. State Space. We now turn to the notion of a state space of A [13]. A state space of a unital C∗ -algebra A is a family of states S (A) of A which is closed under finite convex combinations and operations ω(·) → ω(A∗ · A)/ω(A∗ A) for A ∈ A. We denote by Sts the category whose objects are the state spaces S (A) of unital C∗ -algebras A whose arrows are the positive maps γ ∗ : S (A) → S (A ), arising as dual maps of injective morphisms of C∗ -algebras γ : A → A, by γ ∗ ω(A) ≡ ω(γ (A)) for each A ∈ A . The composition law between two arrows, as the definition of the identity arrow of an object, are obvious. A state space for a locally covariant quantum field theory A is a contravariant functor S between Loc and Sts. Some of the properties, involving nets and states, can be generalized to state spaces. First we recall that a state space S is said to be locally quasi-equivalent if for any M ∈ Loc and for any pair ω, σ ∈ S (M) we have that F(πω A(O)) = F(πσ A(O)) for any O ∈ Kh (M), where F(πω A(O)) is the local folium, i.e. the collection of normal states of the algebras πω (A(O)). Definition 3.2. We say that a locally quasi-equivalent S satisfies the Borchers property (resp. local definiteness) if for any M ∈ Loc and for any ω ∈ S (M) the net ω∗ AKh (M) satisfies the Borchers property (resp. local definiteness).
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Lemma 3.3. Let S be a locally quasi-equivalent state space. Given M ∈ Loc, for any pair σ, ω ∈ S (M) the nets ω∗ AKh (M) and σ ∗ AKh (M) are isomorphic: There is a collection ρω,σ ≡ {ρO : Aω (O) −→ Aσ (O) | O ∈ Kh (M)} made of ∗ -isomorphisms of von Neumann algebras such that ρO Aω (O1 ) = ρO1 if O1 ⊆ O. Proof. Local quasi-equivalence means that for any O ∈ Kh (M) there exists a unique isomorphism ρO : Aω (O) −→ Aσ (O) such that ρO πω (A) = πσ (A) for any A ∈ A(O). The collection ρω,σ ≡ {ρO | O ∈ Kh (M)} is compatible with the net structure. Indeed, note that given O1 ⊆ O, then ρO πω (A) = πσ (A) = ρO1 πω (A) for any A ∈ Aω (O1 ). By uniqueness we have ρO Aω (O1 ) = ρO1 if O1 ⊆ O. As an easy consequence of this lemma we have Corollary 3.4. Let S be a locally quasi-equivalent state space. Assume that for any M ∈ Loc there is ω ∈ S (M) such that ω∗ AKh (M) satisfies the Borchers property (resp. local definiteness). Then S fulfills the Borchers property (resp. local definiteness). Now, consider ψ ∈ (M1 , M) and the associated injective C∗ -morphism αψ : A (M1 ) → A (M). By means of αψ , a state ω ∈ S (M) induces two different representations on A (M1 ). On the one hand, if (πω , Hω , ω ) is the GNS construction associated with ω, then πω αψ is a representation of A (M1 ). On the other hand, by local covariance ωαψ ∈ S (M1 ). If (πωαψ , Hωαψ , ωαψ ) is the GNS construction associated with ωαψ , then πωαψ is a representation of A (M1 ). In order to understand the relation between πω αψ and πωαψ , define V πω αψ (A) ω ≡ πωαψ (A) ωαψ ,
A ∈ A (M1 ).
(3.4)
Since ωαψ is cyclic, V is a partial isometry from Hω onto Hωαψ such that V V ∗ = 11Hωα and V ∗ V ∈ πω (αψ (A (M1 ))) . Let ψ
τψω (πω αψ (A)) ≡ πωαψ (A),
A ∈ A (M1 ).
(3.5)
Clearly τψω : πω αψ (A (M1 )) → πωαψ (A (M1 )) is a C∗ −morphism. However, it is also clear that τψω : ω∗ Aψ(Kh (M1 )) → (ωαψ )∗ AKh (M1 ) defines a net-morphism, because αψ is a net-isomorphism, see (3.2). Proposition 3.5. Assume that ω∗ AKh (M) satisfies the Borchers property. (a) πω αψ and πωαψ are locally quasi-equivalent representations of A (M1 ). (b) τψω is a net-isomorphism. Proof. (a) Since V ∗ V ∈ (πω αψ (A (M1 ))) , we have V ∗ V ∈ Aω (O) = πω (A(O)) , ∀O ∈ ψ(Kh (M1 )). We show that V ∗ V has central support equal to 11 on Aω (O) for O ∈ ψ(Kh (M1 )). This is equivalent to showing that for any orthogonal projection E of Aω (O), then E · V ∗ V = 0 if, and only if, E = 0. To this end, assume that E = 0. By Lemma 2.5.a, there is O1 ∈ ψ(Kh (M1 )) such that cl(O) ⊂ O1 . By the Borchers property there is an isometry W ∈ Aω (O1 ) such that W W ∗ = E. Then E · V ∗ V = 0 ⇐⇒ W W ∗ · V ∗ V = 0 ⇐⇒ V ∗ V · W ∗ = 0 ⇐⇒ V ∗ V = 0. This leads to a contradiction. Hence V ∗ V has central support 11 on Aω (O) . (b) The proof follows from (a) because by (3.4) τψω (πω αψ (A)) = V πω αψ (A)V ∗ for any A ∈ A (M1 ).
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We now study the relations between the net-isomorphisms introduced in this section. Lemma 3.6. Let S be a locally quasi-equivalent state space which satisfies the Borchers property, and let ψ ∈ (M1 , M). Then: (a) ρωαψ ,σ αψ τψω = τψσ ρω,σ , for any pair σ, ω ∈ S (M); ωαψ ω τψ ,
ω =τ (b) τψφ φ
for any φ ∈ (M2 , M1 ) and ω ∈ S (M).
Proof. (a) Let O ∈ Kh (M1 ) and A ∈ A(O). By the definition of ρω,σ (see Lemma 3.3) and by (3.5), we have that τψσ (πσ (αψ (A))) = πσ αψ (A) = ρωαψ ,σ αψ (πωαψ (A)) = ρωαψ ,σ αψ τψω (πω (αψ (A))) = ρωαψ ,σ αψ τψω ρσ,ω (πσ (αψ (A))).
(b) Observe that ωαψφ ∈ S (M2 ). By definition (3.5) we have that ω τψφ (πω αψφ (A)) = πωαψφ (A),
τψω (πω (αψ (A))) = πωαψ (A),
ωαψ
τφ
(πωαψ (αφ (A))) = πωαψφ (A),
A ∈ A (M2 ), A ∈ A (M1 ), A ∈ A (M2 ).
ωα
ωαψ
Using these relations we have that τφ ψ τψω (πω αψφ (A)) = τφ ω (π α (A)), for any A ∈ A (M ). πωαψφ (A) = τψφ ω ψφ 2
(πωαψ (αφ (A))) =
4. Homotopy of Posets and Net Cohomology The mathematical framework under which we will study superselection sectors is that of the net cohomology of posets. In the present section we give some notions and results referring to [45–47, 50]. Throughout this section, we fix M ∈ Loc, and denote by P a basis for the topology of M whose elements are connected open subsets of M, and have a nonempty causal complement. Furthermore, by the same symbol P, we denote the partially ordered set (poset) formed by the basis P ordered under inclusion ⊆. 4.1. Homotopy of posets. We give a brief outline of the definition of the set n (P) of singular n-simplices of the poset P. Details can be found in the cited references. A 0-simplex a of P is nothing but an element of P. Inductively, for n ≥ 1, an n-simplex d is an (n + 2)-tuple d = (|d|, ∂0n d, . . . , ∂nn d), where |d|, called the support of d, is an element of P, (∂0n d, . . . , ∂nn d), called boundaries of d, are (n−1)-simplices, satisfying the following properties: (i)|∂0n d|, . . . , |∂nn d| ⊆ |d|; (ii) the mapping ∂in : n (P) → n−1 (P) which associates the i-boundary ∂i d to an n-simplex d, enjoys the relation n ◦ ∂i+1 , i ≥ j. ∂in−1 ◦ ∂ nj = ∂ n−1 j
(4.1)
The simplicial set of P, denoted by ∗ (P), is the collection of all singular simplices. From now on, we will omit the superscripts from the symbol ∂in , and will denote 0-simplices by the letter a; 1-simplices by b and 2-simplices by c.
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Given a 1-simplex b, the reverse of b is the 1-simplex b defined as ∂0 b = ∂1 b, ∂1 b = ∂0 b and |b| = |b|. A 1-simplex b is said to be degenerate to a 0-simplex a whenever ∂0 b = a0 = ∂1 b and a = |b|. We will denote by b(a) the 1-simplex degenerate to a. 4.1.1. Pathwise and simple connectedness. Given a0 , a1 ∈ 0 (P), a path from a0 to a1 is a finite ordered sequence p = {bn , . . . , b1 } of 1-simplices enjoying the relations ∂1 b1 = a0 , ∂0 bi = ∂1 bi+1 with i ∈ {1, . . . , n − 1}, ∂0 bn = a1 . The starting point of p, written ∂1 p, is the 0-simplex a0 , while the end point of p, written n |b |. ∂0 p, is the 0-simplex a1 . The support | p| of the path p is the open set | p| ≡ ∪i=1 i We denote the set of paths from a0 to a1 by P(a0 , a1 ), and by P(a0 ) the set of closed paths whose end point is a0 . Finally, we note that the poset P is pathwise connected, i.e., P(a0 , a1 ) is never empty for any pair of 0–simplices a0 ,a1 . This is because M is connected and because the elements of P are connected subsets of M Consider a path p = {bn , . . . , b1 } ∈ P(a0 , a1 ). The reverse of p is a path p, the P(a1 , a0 ) defined by p ≡ {b1 , . . . , bn }. The composition of p with a path q = {bk , . . . b1 } of P(a1 , a2 ) is the path q ∗ p of P(a0 , a2 ), defined by q ∗ p ≡ {bk , . . . , b1 , bn , . . . , b1 }. The reverse and the composition of paths are homotopically stable. To be precise, the set of paths with the same end points is endowed with a homotopy equivalence relation, written ∼ (see [46, 50]). Compositions and reverses of homotopic paths lead to homotopic paths. This property and pathwise connectedness of P allow one to introduce the notion of the fundamental group of the poset P, denoted π1 (P). The poset P is simply connected whenever π1 (P) is trivial. It turns out that if P is directed, then P is simply connected. The following lemma will be useful for later purposes. Lemma 4.1. Let p = {bn , . . . , b1 } be a path, and let q = qn ∗ · · · ∗ q1 be a path such that |qi | ⊆ |bi |, ∂0 qi ⊆ ∂0 bi and ∂1 qi ⊆ ∂1 bi for i = 1, . . . , n. Then: ˜ bˆ such that p ∼ b˜ ∗ q ∗ b; ˆ (a) there exist a pair of 1-simplices b, (b) if p and q are closed paths, there is a 1-simplex b such that p ∼ b ∗ q ∗ b, where b is the reverse of b. Proof. (a) Since composition of paths is homotopically stable, we can assume, without loss of generality, p to be a 1-simplex b. So, let q be a path such that |q| ⊆ |b|, ∂0 q ⊆ ∂0 b, ˆ = |b|, ∂1 bˆ = ∂1 b, and ∂1 q ⊆ ∂1 b. Let bˆ and b˜ be 1-simplices defined, respectively, by |b| ˜ ˜ ˆ ˜ ˜ ∂0 b = ∂1 q and |b| = |b|, ∂1 b = ∂0 q, ∂0 b = ∂0 b. Note that b and b ∗ q ∗ bˆ have the same end points. These two paths are homotopic because the poset formed by the collection O ∈ P such that O ⊆ |b| is directed under inclusion, hence it is simply connected. (b) follows from (a). The link with the topology of M. Given a curve γ : [0, 1] → M, a path p = {bn , . . . , b1 } is an approximation of γ if there is a partition 0 = s0 < s1 < . . . < sn = 1 of the interval [0, 1] such that γ ([si−1 , si ]) ⊆ |bi |, γ (si−1 ) ∈ ∂1 bi , γ (si ) ∈ ∂0 bi , for i = 1, . . . n. We denote the set of approximations of γ by App(γ ), and observe App(γ ) = ∅ for any curve γ . The link between the fundamental group of P and the fundamental group of the spacetime M is given by the following theorem.
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Theorem 4.2 ([50]). Assume that the elements of P are simply connected. Then the fundamental group of P and that of M are isomorphic. In particular, if γ , β are curves with the same end points and p,q are paths with the same end points such that p ∈ App(γ ) and q ∈ App(β), then γ and β are homotopic if, and only if, p and q are homotopic. Since the set Kd (M) of diamonds of M verifies the hypotheses of the statement, we have that π1 (Kd (M)) π1 (M). The statement fails if P admits nonsimply connected elements. A counterexample is provided by Kh (M) which is simply connected irrespectively of the topology of M. To show this a preliminary result is necessary. Lemma 4.3. Let p be a closed path of Kd (M) with end point a. There exists a closed path q of Kd (M), with end point a, and an element O ∈ Kh (M) such that p is homotopic to q and |q| ⊆ O . Proof. Let γ be a closed curve with p ∈ App(γ ). Note that, if a is a diamond of the form D(G), with G contained in a spacelike Cauchy surface C, then we can assume that γ meets G in a point x. By [50, Lemma 3.1] γ is homotopic to a closed curve β lying on C, and with end point x. Let q = {bn , . . . b1 } be a closed path, with end point a, formed by diamonds bi = D(G i ), with G i ⊆ C for any i, and such that q ∈ App(β). Theorem 4.2 implies that p ∼ q. Now, define K ≡ ∪i G i and note that K ⊂ C is open, relatively compact and connected. Let O ≡ D(K ) and note that |q| ⊆ O. O is open, relatively compact, connected and, since K ⊂ C, by a standard argument (see, e.g. [41]) O is globally hyperbolic. It is also clear that we can shrink the sets G i in such a way that O⊥ = ∅. Hence O ∈ Kh (M). Proposition 4.4. The poset Kh (M) is simply connected for any M ∈ Loc. Proof. Let p be a closed path of Kh (M), with end point a0 , and let γ be a closed curve with p ∈ App(γ ). Since Kd (M) is a basis for the topology of M there is a closed path q in Kd (M), with end point, to say, a, such that: q ∈ App(γ ); q and p satisfy the properties of the statement of Lemma 4.1. By Lemma 4.1.b there is b ∈ 1 (Kh (M)), with ∂1 b = a0 , and ∂0 b = a, such that p ∼ b∗q ∗b, where b is the reverse of b. By Lemma 4.3 there is a closed path q1 in Kd (M), with end point a, and O ∈ Kh (M) such that q1 ∼ q and |q1 | ⊆ O. As the subposet of Kh (M) formed by O and {O1 ∈ Kh (M) | O1 ⊂ O} is directed under inclusion, it is simply connected, hence q1 ∼ b(a). Homotopic stability of compositions and reverses of paths and the relations p ∗b(∂1 p) ∼ p and p ∗ p ∼ b(∂1 p), which hold for any path p [50, Lemma 2.3], imply that p ∼ b∗q∗b ∼ b∗b(a)∗b ∼ b(a0 ). 4.2. Net cohomology. Consider a net AP : P O → A (O) ⊆ B(H) of von Neumann algebras over a Hilbert space H. A 1-cocycle z of the poset P, with values in AP , is a field z : 1 (P) b → z(b) ∈ B(H) of unitary operators of B(H) satisfying the properties: (i) z(∂0 c) · z(∂2 c) = z(∂1 c), (ii) z(b) ∈ A(|b|),
c ∈ 2 (P) ; b ∈ 1 (P).
Property (i) is called the 1-cocycle identity, while (ii) is called the locality condition for cocycles. Given a pair z, z 1 of 1-cocycles an intertwiner t between z, z 1 is a field
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t : 0 (P) a → ta ∈ B(H) satisfying the properties: (iii) t∂0 b · z(b) = z 1 (b) · t∂1 b , (iv) ta ∈ A(a),
b ∈ 1 (P) ; a ∈ 0 (P).
(iii) is the intertwining property while (iv) is the locality condition for intertwiners. We denote the set of intertwiners between z and z 1 by (z, z 1 ). The category of 1-cocycles Z 1 (P) is the category whose objects are 1-cocycles and whose arrows are the corresponding set of intertwiners. It turns out that this is a C∗ -category, details can be found in [50]. Two 1-cocycles z, z 1 are equivalent (or cohomologous) if there exists a unitary arrow t ∈ (z, z 1 ). A 1-cocycle z is trivial if it is equivalent to the identity cocycle ι defined as ι(b) = 11 for any 1-simplex b. Lemma 4.5. If AP is either irreducible or locally definite, then the identity cocycle ι is irreducible, that is (ι, ι) = C · 1. 1 Proof. If t ∈ (ι, ι), then t∂0 b = t∂1 b for any 1-simplex b ∈ 1 (P). This entails that t is a constant field, that is, ta = ta1 for any pair a, a1 ∈ 0 (P), because P is pathwise connected. Hence ta ∈ A(O) for any O ∈ P. This in turn entails that ta belongs both to ∩{A(O) | O ∈ Kh (M)} and to ∩{A(O) | O ∈ Kh (M) , x ∈ O} for any x ∈ M. Hence, the proof follows from the definition of irreducibility and local definiteness (see Sect. 3.1). From now on we assume that the identity 1-cocycle ι of Z 1 (P) is irreducible. We extend a 1-cocycle z ∈ Z 1 (P) from 1-simplices to paths by defining z( p) ≡ z(bn ) · · · z(b2 ) · z(b1 ),
p = {bn , . . . , b1 }.
A 1-cocycle z is path-independent if for any pair a0 , a1 ∈ 0 (P) we have z( p) = z(q) for any pair of paths p, q ∈ P(a0 , a1 ). It turns out that a 1-cocycle z is path-independent if, and only if, it is trivial in B(H), namely, if there exists a field V : 0 (P) a → Va ∈ B(H) of unitary operators such that z(b) = V∂0 b · V∂∗1 b for any b ∈ 1 (P). We call the full C∗ -subcategory Zt1 (P) of Z 1 (P), whose objects are path-independent 1-cocycles, the category of path-independent 1-cocycles. Connection between homotopy and net cohomology. We recall two important properties of 1-cocycles: First, any 1-cocycle z is invariant for homotopic paths, i.e. if p ∼ q, then z( p) = z(q); secondly, z( p) = z( p)∗ and z(b(a)) = 11. These properties, homotopic stability of the reverse and composition of paths, and Theorem 4.2 find their main application in the following Theorem 4.6 ([50]). The following assertions hold. (a) To any z ∈ Z 1 (P) there corresponds a unitary representation σz of the fundamental group π1 (P) of P; σz trivial if, and only if, z is trivial. (b) Assume that the elements of P are simply connected. Then, to any z ∈ Z 1 (P) there corresponds a unitary representation σz of the fundamental group π1 (M) of M; σz is trivial if, and only if, z is trivial.
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This theorem has two important consequences. First, if P is simply connected, then any 1-cocycle of P is path-independent. Lemma 4.4 says that the poset Kh (M) is simply connected independently of the topology of M. Then Z 1 (Kh (M)) = Zt1 (Kh (M)),
M ∈ Loc.
(4.2)
Secondly, note that the set Kd (M) of diamonds of M verifies the hypotheses of Theorem 4.6.b. This allows us to give a topological characterization of the set Z 1 (Kd (M)) of 1-cocycles of Kd (M). (1) The 1-cocycles of Kd (M) which are path-independent do not carry any information about the topology of the spacetime since they provide trivial representations of the fundamental group. On the other hand, this type of 1-cocycles has a direct physical interpretation; they represent sharply localized sectors of the net of local observables (see Sect. 5.2). The inspection of their charge structure and of their local covariance is the subject of the present paper, and it will be carried out throughout Sect. 5. (2) The 1-cocycles of Kd (M) which are path-dependent are of a topological nature since they provide nontrivial representations of the fundamental group of the spacetime M (clearly, if M is nonsimply connected). In our opinion, path-dependent 1-cocycles might be charged sectors induced by the nontrivial topology of the spacetime, a phenomenon predicted and studied in the literature [30, 1, 51]. However, until now, there is no interpretation of these 1-cocycles in terms of superselection sectors, namely representations of the net of local observables. We investigate in Sect. 6 a notion that possibly points towards the correct interpretation. 5. Charged Superselection Sectors By charged superselection sectors in the Minkowski space M4 , it is meant the unitary equivalence classes of the irreducible representations of a net of local observables which are local excitations of the vacuum representation. We can distinguish two types of charged sectors according to the regions of the spacetime used as an index set of the net of local observables. Charged sectors of Doplicher-Haag-Roberts type, when one considers double cones of M4 , and the charges of Buchholz-Fredenhagen type associated with a particular class of nonrelatively compact regions like spacelike cones. In both cases, sectors define a C∗ -category in which the charge structure manifests itself by the existence of a tensor product, a permutation symmetry, and a conjugation (DoplicherHaag-Roberts analysis [21, 22], and Buchholz-Fredenhagen analysis [17]). Furthermore, it is possible to reconstruct the (unobservable) fields and the gauge group underlying the theory [25]. Our purpose is to study superselection sectors of Doplicher-Haag-Roberts type in the framework of a locally covariant quantum field theory A . The first step is to introduce the notion of reference state space, which will play for the theory the same rôle played by the vacuum representation. To this aim, it is worth observing that the vacuum representation plays the rôle of a reference representation that singles out charged sectors, and that in both cited analysis it is enough to take as vacuum representation one satisfying the Borchers property and Haag duality2 [25]. 2 It is possible to make do with less than the Borchers property, [48]. On the contrary, up to now, Haag duality or a weaker form of it [45], seems to be an essential requirement on the vacuum representation for the theory of superselection sectors.
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Definition 5.1. A reference state space for A is a locally quasi-equivalent state space So satisfying the Borchers property, and such that for any M ∈ Loc there is at least one state ω ∈ So (M) satisfying punctured Haag duality. An example of a locally covariant quantum field theory with a state space verifying the properties of Definition 5.1 is provided by the Klein-Gordon scalar field and by the space of quasi-free states satisfying the microlocal spectrum condition [13, 49] (see also Sect. 6). We stress that we require the existence for any M ∈ Loc of at least one state ω ∈ So (M) satisfying punctured Haag duality. This property seems to be the right generalization of Haag duality, which apparently seems to deal well with the nontrivial topology of arbitrary globally hyperbolic spacetimes [50]. The reason why punctured Haag duality is assumed on the net indexed by Kd (M), is that Kh (M) contains elements which are not simply connected, and elements whose causal complement is not connected. So, punctured Haag duality (and also Haag duality) might not hold for a net indexed by Kh (M) (see [49]). Finally, recall that with our convention a state ω satisfying punctured Haag duality means that the net ω∗ AKd (M) is irreducible and satisfies punctured Haag duality. These two properties entail local definiteness (see Sect. 3.1). Then, by Corollary 3.4, So is locally definite. Definition 5.2. The charged superselection sectors of A , with respect to the reference state space So , are the unitary equivalence classes of the irreducible elements of the categories Zt1 (ω, Kd (M)) of path-independent 1-cocycles of Kd (M) with values in ω∗ AKd (M) , as ω varies in So (M) and as M varies in Loc. Our program for investigating superselection sectors is divided in two parts. Our first aim is to understand the charge structure of the categories Zt1 (ω, Kd (M)) on a fixed background spacetime M and how these categories are related as ω varies in So (M). Secondly, we will inspect the locally covariant structure of sectors. This means that we will study the connection of sectors associated with different isometrically embedded background spacetimes . Now, some observations concerning the definition of superselection sectors in a locally covariant quantum field theory are in order. (1) Our definition of superselection sectors in terms of net cohomology is equivalent to the usual one given in terms of representations of the net of local observables which are sharp excitations of a reference representation. In particular, we will show in Sect. 5.2, that for any spacetime M and for any σ ∈ So (M), to any 1-cocycle z ∈ Zt1 (σ, Kd (M)) there corresponds, up to equivalence, a unique representation π z of the net of local observables which is a sharp excitation of a representation πω associated with a state ω ∈ So (M) satisfying punctured Haag duality. (2) There are several reasons why we choose to study 1-cocycles of the poset Kd (M) instead of 1-cocycles of Kh (M). On the one hand, Kd (M) reflects the topological and causal properties of M better than Kh (M): The fundamental group of Kd (M) is the same as that of M and any diamond has a connected causal complement. These two properties have been one of the keys of the paper [50] where, in the Haag-Kastler framework, the charge structure of sharply localized sectors in a fixed background spacetime has been investigated. On the contrary, Kh (M) is simply connected irrespective of the topology of M (Proposition 4.4), and it has elements with a nonconnected causal complement. On the other hand, there is no loss of generality in studying path-independent 1-cocycles of Kd (M) instead of those of Kh (M), because the corresponding categories are equivalent (see [50, Thms. 2.12
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and 2.23]). We have to say, however, that the cited result provides an equivalence of C∗ −categories, but ignores the tensorial structure of the categories. This topic will be analyzed in [14] where we will provide a symmetric tensor equivalence between the categories associated with Kd (M) and Kh (M). (3) Since So satisfies the Borchers property, by a routine calculation (see [46]), it turns out that the category Zt1 (ω, Kd (M)) is closed under direct sums and subobjects for any ω ∈ So (M) and any M ∈ Loc. As observed above So is locally definite. Then, by Lemma 4.5, the identity cocycle of Zt1 (ω, Kd (M)) is irreducible for any ω ∈ So (M) and any M ∈ Loc. 5.1. Fixed background spacetime. In the present section we investigate the charge structure of superselection sectors in a fixed background spacetime M ∈ Loc. We will start by noticing that for a state ω ∈ So (M) satisfying punctured Haag duality, the corresponding category has a tensor product, a permutation symmetry, and the objects with a finite statistics have conjugates. Afterwards, we will show that all the constructions can be coherently extended to the categories Zt1 (σ, Kd (M)) for any σ ∈ So (M). We conclude this section by studying the behaviour of these categories under restriction to subregions of M. 5.1.1. A preferred reference state. As a starting point of our investigation we apply to our framework the results of the analysis [50] of sharply localized sectors on a fixed background spacetime M, carried out in the Haag-Kastler framework. Given M ∈ Loc, let ω ∈ So (M). Consider the category Zt1 (ω, Kd (M)), and for any tuple z, z 1 , z 2 , z 3 ∈ Zt1 (ω, Kd (M)) and t ∈ (z, z 1 ), s ∈ (z 2 , z 3 ), define (z ⊗ω z 1 )(b) ≡ z(b) · ad z( p) (z 1 (b)), b ∈ 1 (Kd (M)), (t ⊗ω s)a ≡ ta · ad z(q) (sa ), a ∈ 0 (Kd (M)),
(5.1)
where p is a path with ∂1 p ⊥ |b| and ∂0 p = ∂1 b; q is a path with ∂1 q ⊥ a and ∂0 p = a; ad z( p) denotes the adjoint action. Since the elements of Kd (M) have connected causal complements, path-independence of 1-cocycles implies that these definitions do not depend on the choice of the paths p and q. Now, given z, z 1 ∈ Zt1 (ω, Kd (M)), for any 0-simplex a define εω (z, z 1 )a ≡ z 1 (b)∗ · ad z 1 ( p) (z(b1 )∗ ) · z(b1 ) · ad z( p1 ) (z 1 (b)),
(5.2)
where b1 , b are 1-simplices such that ∂0 b1 ⊥ ∂0 b and ∂1 b1 = ∂1 b = a; p is a path from the causal complement of |b1 | to ∂1 b; p1 is a path from the causal complement of |b| to ∂0 b1 . This expression is independent of the choices of b1 , b, p1 , p. We will refer to the pair (⊗ω , εω ) as the tensor structure of Zt1 (ω, Kd (M)), although, up until now, we cannot affirm either that ⊗ω is a tensor product or that εω is a permutation symmetry. One can easily see, for instance, that z ⊗ω z 1 satisfies the 1-cocycle identity, but it is not clear if it satisfies the locality condition nor if it is path-independent. However, for a particular choice of ω we have the following result Theorem 5.3 ([50]). Let ω ∈ So (M) satisfy punctured Haag duality. Then relations (5.1) and (5.2) define, respectively, a tensor product ⊗ω and a permutation symmetry εω of Zt1 (ω, Kd (M)); the category has left inverses and a notion of statistics of objects; the objects with finite statistics have conjugates.
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Using this result, in the next two sections we will prove that (⊗σ , εσ ) define a tensor product and a permutation symmetry of Zt1 (σ, Kd (M)) for any σ ∈ So (M). Thus, the existence, for any M ∈ Loc, of at least one state ω ∈ So (M) satisfying punctured Haag duality is a cornerstone for our analysis. Remarks 5.4. It is worth observing that it is possible to choose the paths b, b1 , p, p1 involved in definition (5.2) in such a way that all their supports are contained in a unique diamond O. In particular it is possible to replace the paths p and p1 by two 1-simplices b˜ and b˜1 which have the same endpoints of p and p1 , respectively, and whose support is O. This is an easy consequence of the definition of diamonds and of property (2.4). The same holds true for the paths involved in definition (5.1). 5.1.2. Independence of the choice of states I. The aim of this section is to show that for any pair σ, ω ∈ So (M) the corresponding categories Zt1 (ω, Kd (M)) and Zt1 (σ, Kd (M)) are ∗ -isomorphic. Let ρω,σ : ω∗ AKh (M) −→ σ ∗ AKh (M) , be the net-isomorphism defined in Sect. 3.2 (see Lemma 3.3). We stress here that despite considering the categories associated with the set Kd (M), the fact that ρω,σ is a net-isomorphism of the nets indexed by Kh (M) is of a crucial importance when stating the claimed isomorphism (see proof of Lemma 5.5). Now, for any pair z, z 1 ∈ Zt1 (ω, Kd (M)) and t ∈ (z, z 1 ) define Fω,σ (z)(b) ≡ ρ|b| (z(b)), b ∈ 1 (Kd (M)), Fω,σ (t)a ≡ ρa (ta ), a ∈ 0 (Kd (M)).
(5.3)
Clearly, Fω,σ (z)(b) ∈ Aσ (|b|) and Fω,σ (t)a ∈ Aσ (a). For any c ∈ 2 (Kd (M)) we have that Fω,σ (z)(∂0 c) · Fω,σ (z)(∂2 c) = ρ|∂0 c| (z(∂0 c)) · ρ|∂2 c| (z(∂2 c)) = ρ|c| (z(∂0 c)) · ρ|c| (z(∂2 c)) = ρ|c| (z(∂0 c) · z(∂2 c)) = ρ|∂1 c| (z(∂1 c)) = Fω,σ (z)(∂1 c). Hence Fω,σ (z) is a 1-cocycle of Kd (M). Observe that if M is simply connected, then, by Theorem 4.6.b, Fω,σ (z) is a path-independent 1-cocycle. For the general case we have the following Lemma 5.5. Fω,σ (z) is a path-independent 1-cocycle of Zt1 (σ, Kd (M)), for any pair ω, σ ∈ So (M). Proof. Let p be a closed path of Kd (M) with end point a. By Lemma 4.3, there exists a closed path q of Kd (M), with end point a, and an element O ∈ Kh (M) such that p ∼ q and |q| ⊆ O. Assume q = {bn , . . . , b1 }. By homotopic invariance of 1-cocycles we have Fω,σ (z)( p) = Fω,σ (z)(q) = ρ|bn | (z(bn )) · · · ρ|b1 | (z(b1 )) = ρO (z(bn )) · · · ρO (z(b1 )) = ρO (z(bn ) · · · z(b1 )) = ρO (z(q)) = ρO ( 11) = 11, where the path-independence of z has been used.
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It is clear that Fω,σ (t) ∈ (Fω,σ (z), Fω,σ (z 1 )). Moreover, since ρω,σ is a net-isomorphism, Fω,σ is an isomorphism, namely Fσ,ω ◦ Fω,σ = idZ 1 (ω,Kd (M)) , Fω,σ ◦ Fσ,ω = idZ 1 (σ,Kd (M)) , t
t
where Fσ,ω is the functor associated with the net-isomorphism ρσ,ω . In conclusion we have the following Proposition 5.6. Fω,σ : Zt1 (ω, Kd (M)) → Zt1 (σ, Kd (M)) is an isomorphism of C∗ categories for any pair ω, σ ∈ So (M). We will refer to the functor Fω,σ as the flip functor. 5.1.3. Independence of the choice of states II. The aim of this section is to show that the superselection sectors of the category Zt1 (σ, Kd (M)), for any σ ∈ So (M), have a charge structure, and that all these categories carry the same physical information. In other words we want to show that for any σ ∈ So (M) the category Zt1 (σ, Kd (M)) has a tensor product, a permutation symmetry, and that the objects with finite statistics have conjugates; furthermore, that all these categories are symmetric tensor ∗ -isomorphic. To this end, we will first note that, thanks to the flip functor, the category associated with a state satisfying punctured Haag duality, induces a tensor structure on Zt1 (σ, Kd (M)) for any σ ∈ So (M). Secondly, we will show that the induced structure coincides with the ambient one, defined by (5.1) and (5.2). This, in turn, will entail that the flip functor is a symmetric tensor ∗ -isomorphism. Consider a state ω ∈ So (M) satisfying punctured Haag duality. Recall that by Theorem 5.3, ⊗ω and εω are, respectively, a tensor product and a permutation symmetry of Zt1 (ω, Kd (M)). Now, given σ ∈ So (M), for any z, z ∈ Zt1 (σ, Kd (M)) and for any pair of arrows t, s of the category Zt1 (σ, Kd (M)), define z ⊗ωσ z 1 ≡ Fω,σ Fσ,ω (z) ⊗ω Fσ,ω (z 1 ) , (5.4) t ⊗ωσ s ≡ Fω,σ Fσ,ω (t) ⊗ω Fσ,ω (s) , and εσω (z, z 1 ) ≡ Fω,σ (εω (Fσ,ω (z), Fσ,ω (z 1 ))).
(5.5)
∗ -isomorphism,
the above formulas define, respectively, a Since the flip functor is a tensor product and a permutation symmetry of Zt1 (σ, Kd (M)). We will refer to the pair (⊗ωσ , εσω ) as the tensor structure of Zt1 (σ, Kd (M)) induced by ω. Lemma 5.7. Let ω ∈ So (M) satisfy punctured Haag duality. Then, the following assertions hold for any σ ∈ So (M). (a) ⊗ωσ = ⊗σ and εσω = εσ ; (b) The pair (⊗σ , εσ ) define a tensor product and a permutation symmetry of the category Zt1 (σ, Kd (M)). Proof. (a) Given a 1-simplex b ∈ 1 (Kd (M)), by Remark 5.4 there is a 1-simplex b1 and a diamond O such that ∂0 b1 = ∂1 b, ∂0 b1 ⊥ |b|, and |b|, |b1 | ⊆ O. Now recalling definition (5.1), we have that Fσ,ω (z) ⊗ω Fσ,ω (z 1 )(b) = Fσ,ω (z)(b) · adFσ,ω (z)(b1 ) Fσ,ω (z 1 )(b)) = ρ|b| (z(b)) · adρ|b1 | (z(b1 )) (ρ|b| (z 1 (b)) = ρO (z(b)) · adρO (z(b1 )) ρO (z 1 (b)) = ρO z(b) · ad z(b1 ) (z 1 (b)) .
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Hence, since Fσ,ω (z) ⊗ω Fσ,ω (z 1 )(b) ∈ Aω (|b|), we have −1 (z ⊗ωσ z 1 )(b) = Fω,σ Fσ,ω (z) ⊗ω Fσ,ω (z 1 ) (b) = ρ|b| Fσ,ω (z) ⊗ω Fσ,ω (z 1 )(b) −1 −1 = ρO Fσ,ω (z) ⊗ω Fσ,ω (z 1 )(b) = ρO ρO z(b) · ad z(b1 ) (z 1 (b)) = z(b) · ad z(b1 ) (z 1 (b)) = (z ⊗σ z 1 )(b). The same argument leads to the identity t ⊗ωσ s = t ⊗σ s for any pair of arrows t, s of Zt1 (σ, Kd (M)). So we have ⊗ωσ = ⊗σ . We now apply a similar argument to the permutation symmetry εσω (see definition (5.2)). Consider a 0-simplex a. By Remark ˜ b˜1 which fulfill, with respect to a, the properties 5.4, there are four 1-simplices b, b1 , b, ˜ |b˜1 | are contained in a of definition (5.2), and such that all the supports |b|, |b1 |, |b|, diamond O. So εω (Fσ,ω (z), Fσ,ω (z 1 ))a = Fσ,ω (z 1 )(b1 )∗ · adFσ,ω (z 1 )(b˜1 ) (Fσ,ω (z)(b)∗ ) · · Fσ,ω (z)(b) · adFσ,ω (z)(b) ˜ (Fσ,ω (z 1 )(b1 ))
= ρ|b1 | (z 1 (b1 )∗ ) · adρ ˜
˜
|b1 | (z 1 (b1 ))
(ρ|b| (z(b)∗ ) ·
· ρ|b| (z(b)) · adρ ˜ (z(b)) ˜ (ρ|b1 | (z 1 (b1 ))) |b| = ρO z 1 (b1 )∗ · ad z 1 (b˜1 ) (z(b)∗ ) · ρO z(b) · ad z(b) ˜ (z 1 (b1 )) = ρO z 1 (b1 )∗ · ad z 1 (b˜1 ) (z(b)∗ ) · z(b) · ad z(b) ˜ (z 1 (b1 )) = ρO (εσ (z, z 1 )a ). Hence, since εω (Fσ,ω (z), Fσ,ω (z 1 ))a ∈ Aω (a), we have εσω (z, z 1 )a = Fω,σ (εω (Fσ,ω (z), Fσ,ω (z 1 )))a = ρa−1 (εω (Fσ,ω (z), Fσ,ω (z 1 ))a ) −1 −1 = ρO (εω (Fσ,ω (z), Fσ,ω (z 1 ))a ) = ρO (ρO (εσ (z, z 1 )a )) = εσ (z, z 1 )a ,
which completes the proof. (b) follows from (a).
The following relations are a consequence of the previous lemma and of the invertibility of the flip functor. Consider ω, σ ∈ So (M) such that ω satisfies punctured Haag duality. For any z, z 1 ∈ Zt1 (σ, Kd (M)) and for any pair t, s of arrows of Zt1 (σ, Kd (M)), we have Fσ,ω (z) ⊗ω Fσ,ω (z 1 ) = Fσ,ω (z ⊗σ z 1 ), Fσ,ω (t) ⊗ω Fσ,ω (s) = Fσ,ω (t ⊗σ s),
(5.6)
εω (Fσ,ω (z), Fσ,ω (z 1 )) = Fσ,ω (εσ (z, z 1 )).
(5.7)
and
In conclusion we have the following Theorem 5.8. For any ω ∈ So (M) the category Zt1 (ω, Kd (M)) equipped with ⊗ω and εω is a symmetric tensor C∗ -category with left inverses. Any object with finite statistics has conjugates. For any ω, σ ∈ So (M), the functor Fω,σ : Zt1 (ω, Kd (M)) → Zt1 (σ, Kd (M)) is a covariant symmetric tensor ∗ -isomorphism.
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Proof. Consider a pair of states ω, σ ∈ So (M). In order to prove that Fω,σ : Zt1 (ω, Kd (M)) → Zt1 (σ, Kd (M)) is a symmetric tensor ∗ -isomorphism, we consider a state ωo ∈ So (M) satisfying punctured Haag duality and use the fact that the tensor structures of Zt1 (ω, Kd (M)) and Zt1 (σ, Kd (M)) are equal to the tensor structure induced by ωo . Given z, z 1 ∈ Zt1 (ω, Kd (M)), we have Fω,σ (z ⊗ω z 1 ) = Fω,σ (z ⊗ωωo z 1 ) = Fω,σ ◦ Fωo ,ω Fω,ωo (z) ⊗ωo Fω,ωo (z 1 ) = Fωo ,σ Fω,ωo (z) ⊗ωo Fω,ωo (z 1 ) = Fωo ,σ Fσ,ωo ◦ Fω,σ (z) ⊗ωo Fσ,ωo ◦ Fω,σ (z 1 ) by (5.6) = Fωo ,σ ◦ Fσ,ωo Fω,σ (z) ⊗σ Fω,σ (z 1 ) = Fω,σ (z) ⊗σ Fω,σ (z 1 ). The same reasoning applied to arrows and to the permutation symmetry shows that the flip functor is a covariant symmetric tensor ∗ -isomorphism. As the category associated with a state satisfying punctured Haag duality has left inverses and the object with finite statistics has conjugates, the same holds for the category associated with any state of So (M). This is because the flip functor is a covariant symmetric tensor ∗ -isomorphism (see in Appendix). 5.1.4. Restriction to subregions. Let N ⊂ M be an open connected subset of M, such that for any pair x1 , x2 ∈ N then J + (x1 ) ∩ J − (x2 ) is contained in N . This property says that N is a globally hyperbolic spacetime. As N is isometrically embedded in M and as diamonds are stable under isometric embeddings (Lemma 2.8) we have Kd (M) N ≡ {O ∈ Kd (M) | O ⊂ N } = Kd (N ). Let AKd (N ) be the net of local algebras indexed by Kd (N ), obtained by restricting AKd (M) to Kd (N ). Let ω ∈ So (M), ω∗ AKd (N ) inherits from ω∗ AKd (M) the Borchers property and local definiteness, as follows from Lemma 3.1. However, it needs not be irreducible. Let Zt1 (ω, Kd (N )) be the category of path-independent 1-cocycles of Kd (N ) with values in ω∗ AKd (N ) . This is a C∗ -category closed under direct sums and subobjects, and, by local definiteness, the identity cocycle is irreducible (Lemma 4.5). Now, let ω ∈ So (M). For any z, z 1 ∈ Zt1 (ω, Kd (M)) and t ∈ (z, z 1 ) define R(z)(b) ≡ z(b), ∈ 1 (Kd (N )), R(t)(a) ≡ ta , a ∈ 0 (Kd (N )). R : Zt1 (ω, Kd (M)) → Zt1 (ω, Kd (N )) is a covariant ∗ -functor, called the restriction functor. Our aim is to show that R is full and faithful. Note that if σ ∈ So (M), then it is easily seen that the following diagram is commutative: Zt1 (ω, Kd (M)) R
Fω,σ
/ Z 1 (σ, Kd (M)) t
R
Fω,σ / Z 1 (σ, Kd (N )) Zt1 (ω, Kd (N )) t Therefore if we prove that R is full and faithful for a particular choice of ω then it would be full and faithful for any other element σ ∈ So (M).
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Theorem 5.9. R is a full and faithful ∗ −functor. Proof. In the first part of the proof we follow [46, Theorem 30.2]. As observed above it is enough to prove the assertion when ω satisfies punctured Haag duality. Given z, z 1 ∈ Zt1 (ω, Kd (M)), let t ∈ (R(z), R(z 1 )). This means that t∂0 b · z(b) = z 1 (b) · t∂1 b for any b ∈ 1 (Kd (N )). We want prove that there exists t ∈ (z, z 1 ) such that ta = ta whenever a ∈ 0 (Kd (N )). Fix a0 ∈ Kd (N ), define ta ≡ z 1 ( pa ) · ta0 · z( pa )∗ a ∈ 0 (Kd (M)), where pa is a path in Kd (M) from a0 to a. This definition does not depend on the chosen a0 ∈ 0 (Kd (N )) nor on the chosen path pa . Moreover t∂0 b · z(b) = z 1 (b) · t∂1 b for any b ∈ 1 (Kd (M)), and ta = ta for any a ∈ 0 (Kd (N )). So, what remains to be shown is that ta ∈ Aω (a) for any a ∈ 0 (Kd (M)). The proof is very similar to the proof of [50, Prop. 4.19]. Let x0 ∈ N ; we first prove that ta ∈ Aω (a) for any a ∈ 0 (Kd (M)) whose closure cl(a) is causally disjoint from {x0 }. Fix a 0-simplex a1 of Kd (M) to be such that cl(a1 ) ⊥ {x0 } and a1 ⊥ a. We now make use of the fact that the definition of t depends neither on the choice of a0 nor the choice of the path. Note that we can always find a0 ∈ 0 (Kd (N )) such that a0 ⊥ a1 and cl(a0 ) ⊥ {x0 }. Furthermore, since the causal complement of a1 is connected, there is a path pa which lies in the causal complement of a1 . Therefore: ta · A = z 1 ( pa ) · ta0 · z( pa )∗ · A = z 1 ( pa ) · ta0 · A · z( pa )∗ = A · ta , ta ∈ Aω (a1 ) for any a1 ⊥ a and cl(a1 ) ⊥ x. By punctured for any A ∈ Aω (a1 ). Hence Haag duality ta ∈ Aω (a). Thus, we have shown that ta ∈ Aω (a) for any 0-simplex a such that cl(a) ⊥ {x0 }. By [50, Prop. 4.19] the proof follows. Three comments about Theorem 5.9 are in order. (1) This is a key result for our aims. It will entail that the embedding of a sector into a different spacetime preserves the statistical properties (see Remark 5.16), this being the genesis of the local covariance of gauge groups (see Sect. 5.3.3). (2) Theorem 5.9 is nothing but the cohomological version of the “equivalence between local and global intertwiners,” a property that the superselection sectors which are preserved in the scaling limit fulfills [20] (see also [44]). We emphasize that in the present paper this equivalence arises as a natural consequence of punctured Haag duality. (3) It can be easily shown that the restriction functor R is a symmetric tensor functor. The proof is contained, implicitly, in the proof of Proposition 5.12.
5.2. Net cohomology and sharply localized sectors. The purpose of this section is to provide the interpretation of our definition of superselection sectors in terms of representations of the net of local observables which are a sharp excitation of a reference representation. A representation π on a Hilbert space Hπ of the net AKd (M) is a collection {πO }O∈Kd (M) of representations πO of the algebras A(O) on Hπ , which is compatible with the net structure, i.e. πO1 A(O) = πO if O ⊆ O1 . Given ω ∈ So (M) let πω be the GNS representation of the algebra A (M), on the Hilbert space Hω , which is
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associated with ω. A representation π = {πO }O∈Kd (M) is a sharp excitation of πω if there exists a family {VO }O∈Kd (M) of unitary operators from Hπ onto Hω such that VO1 πO (A) = πω (A)VO1 A ∈ A(O), O ⊥ O1 .
(5.8)
Let (π, π1 ) be the collection of linear bounded operators T : Hπ → Hπ1 such that T πO (A) = π1 O (A)T for A ∈ A(O) and for O ∈ Kd (M). We denote by Rep(ω) the category whose objects are those representations of AKd (M) which are a sharp excitation of πω and with arrows the corresponding intertwiner operators. Proposition 5.10. For any σ ∈ So (M), the category Zt1 (σ, Kd (M)) is equivalent to the category Rep(ω) for any ω ∈ So (M) satisfying punctured Haag duality. Proof. We first prove that Zt1 (ω, Kd (M)) is equivalent to the category Rep(ω) for a state ω as in the statement. We give only a sketch of the proof because it is very similar to the proof of [31, Lemma 3A.6]. Consider z ∈ Zt1 (ω, Kd (M)). Fix O1 ∈ Kd (M) and define z πO (A) ≡ z( pO1 ) · πω (A) · z( pO1 )∗ ,
A ∈ A(O),
where pO1 is a path with ∂0 pO1 = O1 and ∂1 pO1 ⊥ O. This definition is well posed as diamonds have connected causal complements, and it can be easily shown that π z is a representation of AKd (M) . For any O let qO be a path from O1 to O, and let VO ≡ z(qO ). One can easily check that VO πO2 (A) = πω (A)VO , for any A ∈ A(O2 ) with O2 ⊥ O. Hence π z ∈ Rep(ω). Conversely, given π ∈ Rep(ω), let VO , O ∈ Kd (M), be the collection of unitary operators associated with π by (5.8). Define z π (b) ≡ V∂0 b · V∂∗1 b , b ∈ 1 (Kd (M)). z π satisfies the 1-cocycle identity and is path-independent. Punctured Haag duality (hence, Haag duality) entails that z π fulfills the locality condition, hence z π ∈ Zt1 (ω, Kd (M)). Following the cited reference, one arrives at the categorical equivalence between Zt1 (ω, Kd (M)) and Rep(ω). Now the proof follows from Proposition 5.6. It is now clear that, in a fixed background spacetime M all the categories Zt1 (σ, Kd (M)) carry the same physical information for any choice of σ ∈ So (M). Indeed, they are associated with representations of the net of local observables which are a sharp excitation of the representation πω associated with a state ω ∈ So (M) satisfying punctured Haag duality.
5.3. Locally covariant structure of sectors. We show how the locally covariant structure of superselection sectors arises. We introduce the embedding functor which gives a first important information on the covariant structure of sectors. Such a structure is encoded in the superselection functor analyzed in the subsequent section. Finally, by applying the Doplicher-Roberts duality theory of compact groups to the superselection functor, we investigate the locally covariant properties of the associated gauge groups.
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5.3.1. The Embedding functor. Consider M1 , M ∈ Loc, with ψ ∈ (M1 , M), let αψ : A (M1 ) → A (M) be the C∗ -morphism associated with ψ. Given ω ∈ So (M), let τψω : ω∗ Aψ(Kd (M1 )) → (ωαψ )∗ AKd (M1 ) be the corresponding net-isomorphism introduced in Sect. 3.2. Definition 5.11. Given ψ ∈ (M1 , M) and ω ∈ So (M). We call embedding the map of categories Eψω : Zt1 (ω, Kd (M)) → Zt1 (ωαψ , Kd (M1 )) defined, for z, z 1 ∈ Zt1 (ω, Kd (M)) and t ∈ (z, z 1 ), as Eψω (z)(b) ≡ τψω (z(ψ(b))), b ∈ 1 (Kd (M1 )), Eψω (t)a ≡ τψω (tψ(a) ), a ∈ 0 (Kd (M1 )), where ψ(b) is the 1-simplex of Kd (M) defined by ∂0 ψ(b) ≡ ψ(∂0 b), ∂1 ψ(b) ≡ ψ(∂1 b) and |ψ(b)| ≡ ψ(|b|). We have the following Proposition 5.12. Given ψ ∈ (M1 , M) and ω ∈ So (M). The embedding Eψω : Zt1 (ω, Kd (M)) → Zt1 (ωαψ , Kd (M1 )) is a covariant symmetric tensor ∗ -functor which is full and faithful. Proof. That Eψω is a covariant ∗ -functor is obvious from the fact that τψω is a net-isomorphism. Given z, z 1 ∈ Zt1 (ω, Kd (M)) and t ∈ (Eψω (z), Eψω (z 1 )), define sa ≡ (τψω )−1 (tψ −1 (a) ), a ∈ 0 (ψ(Kd (M1 ))). s ∈ (R(z), R(z 1 )), where R is the restriction functor from Zt1 (ω, Kd (M)) into Zt1 (ω, Kd (ψ(M1 ))). Since R is full, there is t ∈ (z, z 1 ) such that (τψω )−1 (tψ −1 (a) ) = R(t )a = ta , a ∈ 0 (ψ(Kd (M1 ))), ) = Eψω (t )a for any a ∈ 0 (Kd (M1 )). This which is equivalent to ta = τψω (tψ(a) proves that Eψω is full. If Eψω were not-faithful, there would be t1 , t2 ∈ (z, z 1 ) such that Eψω (t1 ) = Eψω (t2 ). This is equivalent to t1ψ(a) = t2ψ(a) , with a ∈ 0 (Kd (M1 )), because τψω is a net-isomorphism. This, in turns, is equivalent to R(t1 )a = R(t2 )a , with a ∈ 0 (ψ(Kd (M1 ))), and this leads to a contradiction because R is faithful. What remains to be shown is that the embedding is a symmetric and tensor ∗ -functor. To this end let z, z 1 ∈ Zt1 (ω, Kd (M)) and b ∈ 1 (Kd (M1 )). Using definition (5.1) we have Eψω (z ⊗ω z 1 )(b) = τψω (z ⊗ω z 1 )(ψ(b)) = τψω z(ψ(b)) · ad z( p) z 1 (ψ(b)) = τψω (z(ψ(b))) · adτψω (z( p)) τψω (z 1 (ψ(b)))
= (Eψω (z) ⊗ωαψ Eψω (z 1 ))(b),
where the fact that τψω is a morphism of C∗ -algebras has been used (see Sect. 3.2). The same reasoning leads to Eψω (t ⊗ω s) = Eψω (t) ⊗ωαψ Eψω (s), for any pair t, s of arrows
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of Zt1 (ω, Kd (M)). Now, let z, z 1 ∈ Zt1 (ω, Kd (M)) and a ∈ 0 (Kd (M1 )). Using (5.2) we have ω ∗ Eψω (ε(z, z 1 ))a = τψω z 1 (b1 )∗ · ad z 1 (q) ˜ (z(b) ) · τψ z(b) · ad z( p) ˜ (z 1 (b1 )) , where b1 , b are 1-simplices in Kd (M) such that ∂1 b = ∂1 b1 = ψ(a) and ∂0 b ⊥ ∂0 b1 . Let us consider the first term of the product. We can take in Kd (M1 ) two 1-simplices b1 , b satisfying the relations ∂1 b1 = ∂1 b = a and ∂0 b1 ⊥ ∂0 b . So the 1-simplices ψ(b1 ), ψ(b ) of Kd (M) fulfill the hypotheses of the definition of a permutation symmetry. Furthermore, let q be a path in Kd (M1 ) such that ∂0 q = ∂1 b1 and ∂1 q ⊥ |b |. Then ψ(q ) has the same properties of q . Therefore ∗ ω ∗ ∗ τψω z 1 (b1 )∗ · ad z 1 (q) ˜ (z(b) ) = τψ z 1 (ψ(b1 )) · ad z 1 (ψ(q )) (z(ψ(b1 )) ) = τψω (z 1 (ψ(b1 )))∗ · adτψω (z 1 (ψ(q ))) τψω (z(ψ(b1 )))∗ ∗ = Eψω (z 1 )(b1 )∗ · adEψω (z 1 )(q ) Eψω (z)(b ) . Applying the same reasoning to the other term of the product we arrive at Eψω (ε(z, z 1 ))a = ε(Eψω (z), Eψω (z 1 ))a for any a ∈ 0 (Kd (M1 )). Lemma 5.13. Given ψ ∈ (M1 , M), let ω, σ ∈ So (M). Then Eψω ◦ Fσ,ω = Fσ αψ ,ωαψ ◦ Eψσ . Proof. Given z ∈ Zt1 (σ, Kd (M)) and b ∈ 1 (Kd (M1 )), by Lemma 3.6.a we have that Fσ αψ ,ωαψ ◦ Eψσ (z)(b) = ρσ αψ ,ωαψ (τψσ (z(ψ(b)))) = τψω (ρσ,ω (z(ψ(b))) = Eψω ◦ Fσ,ω (z)(b). 5.3.2. The Superselection Functor. We now are in a position to show the covariant structure of superselection sectors. Let Sym be the category whose objects are symmetric tensor C∗ -categories, and whose arrows are the full and faithful, symmetric tensor ∗ -functors. According to the philosophy of the locally covariant quantum field theories, we expect that the superselection sectors can exhibit a structure of functor from the category Loc into the category Sym. We know that the superselection sectors of any spacetime M ∈ Loc identify a family of categories within the same isomorphism class, any such category Zt1 (ω, Kd (M)) is labeled by an element ω ∈ So (M). Since there is no natural way to associate an element of this isomorphism class to the spacetime M, as M varies in Loc, we will be forced to make a choice. Given a locally covariant quantum field theory A and a reference state space So , let ω ≡ {ω M ∈ So (M) | M ∈ Loc}
(5.9)
be a choice of states. The superselection functor associated with the choice ω, is the categories map Sω : Loc → Sym defined as Sω (M) ≡ Zt1 (ω M , Kd (M)), M ∈ Loc, (5.10) Sω (ψ) ≡ Fω M αψ ,ω M1 ◦ Eψω M , ψ ∈ (M1 , M). Theorem 5.14. Sω : Loc −→ Sym is a contravariant functor.
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Proof. Let ψ ∈ (M1 , M). Since Sω (ψ) is defined as the composition of the flip and of the embedding functor, Proposition 5.12 and Theorem 5.8 imply that Sω (ψ) : Sω (M) → Sω (M1 ) is a full, faithful symmetric tensor ∗ −functor. Given φ ∈ (M2 , M1 ), by Lemma 5.13, we have ω M1
◦ Fω M αψ ,ω M1 ◦ Eψω M
Sω (φ) ◦ Sω (ψ) = Fω M1 αφ ,ω M2 ◦ Eφ
ω αψ
= Fω M1 αφ ,ω M2 ◦ Fω M αψφ ,ω M1 αφ ◦ Eφ M ωM = Fω M αψφ ,ω M2 ◦ Eψφ
◦ Eψω M
= Sω (ψφ). Finally, by the definitions of the flip and of the embedding functors we have Sω (id M ) = Fω M ,ω M ◦ EidωMM = idZ 1 (ω M ,Kd (M)) = idSω (M) . t
Proposition 5.15. If ω and σ is a pair of choice of states, then the functors Sω and Sσ are isomorphic. Proof. Define u ω,σ (M) = Fω M ,σ M ,
M ∈ Loc.
By Theorem 5.8, it follows that u ω,σ (M) : Sω (M) → Sσ (M) is a symmetric tensor isomorphism. Given ψ ∈ (M1 , M). By Lemma 5.13 we have that u ω,σ (M1 ) ◦ Sω (ψ)
= Fω M1 ,σ M1 ◦ Fω M αψ ,ω M1 ◦ Eψω M = Fω M αψ ,σ M1 ◦ Eψω M
= Fσ M αψ ,σ M1 ◦ Fω M αψ ,σ M αψ ◦ Eψω M = Fσ M αψ ,σ M1 ◦ Eψσ M ◦ Fω M ,σ M = Sσ (ψ) ◦ u ω,σ (M). Hence, u ω,σ : Sω → Sσ is a natural isomorphism.
Remarks 5.16. This theorem is the main result of this paper because it shows the covariance of charged superselection sectors: if ψ ∈ (M1 , M), then to any sector of M there correspond a unique sector of M1 with the same charged quantum numbers. To be precise, let z ∈ Sω (M) be an irreducible object with statistical parameter λ([z]) = χ ([z])·d([z]), where [z] denotes the equivalence class of z. Let z be the conjugate of z (see Appendix A). Then Sω (ψ)(z) is an irreducible object of Sω (M1 ) such that Sω (ψ)(z) = Sω (ψ)([z]). Furthermore z and Sω (ψ)(z) have the same statistics. Moreover, Sω (ψ)([z]) is the conjugate sector of Sω (ψ)(z). 5.3.3. The Gauge Weak Functor. The next natural step of the present investigation should be the application of Doplicher-Roberts reconstruction theorem [25] to the pair of functors (A , Sω ) in order to analyze the local covariance of fields and of the gauge groups underlying the theory. In the present section we will follow partially this investigation line by showing the covariant structure of gauge groups associated with the superselection sectors. Because of some technical problems, that will be exposed below, we will not deal with the reconstruction of fields. Conversely, gauge groups can be easily reconstructed by using the Doplicher-Roberts duality theorem (hereafter, DR-theorem)
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for compact groups [24] (note that in the reconstruction theorem [25] the gauge group of fields is the dual of the category of superselection sectors). Before beginning the analysis we need a preliminary observation on the DR-theorem whose functorial properties are described in some detail in Appendix A. The DR-theorem states that any tensor C∗ -category with a permutation symmetry, and conjugates, is the abstract dual of a compact group which is uniquely associated with the category only up to isomorphism. To any full and faithful, symmetric tensor ∗ -functor between categories there corresponds, in a contravariant fashion, a group morphism. Also in this case the correspondence between functors and group morphisms is not injective. These degrees of freedom will reflect in the weakening of the local covariance of the gauge groups (see below). Consider a choice of states ω. For any spacetime M, we choose a compact group G ω (M) among the isomorphism class of compact groups associated with the full subcategory of Sω (M) whose objects have conjugates. Furthermore, for any ψ ∈ (M1 , M), we choose a group morphism αω (ψ) : G ω (M1 ) → G ω (M), among the set of group morphisms associated with the functor Sω (ψ) : Sω (M) → Sω (M1 ). Let Grp be the category whose objects are compact groups and whose arrows are the corresponding set of group morphisms. Now, for any choice of states ω we denote by Gω the categories map Gω : Loc → Grp defined as Gω (M) ≡ G ω (M) M ∈ Loc, (5.11) Gω (ψ) ≡ αω (ψ) ψ ∈ (M1 , M). We have the following Theorem 5.17. The map Gω satisfies the following properties: (a) given M ∈ Loc, there exists gω (M) ∈ Gω (M) such that Gω (id M ) = ad gω (M) ; (b) given φ ∈ (M2 , M1 ) and ψ ∈ (M1 , M), there exists gω (φ, ψ) ∈ Gω (M2 ) such that Gω (φ) ◦ Gω (ψ) = ad gω (φ,ψ) ◦ Gω (φψ). Proof. The proof is an easy application of the DR-theorem to Theorem 5.14.
Theorem 5.17.b says that the map Gω is not, in general, a covariant functor because the defining properties of functors are verified in a weak sense, namely up to isomorphisms of the set of arrows. We will refer to the map Gω : Loc → Grp as the gauge weak functor 3 . Theorem 5.18. Given a pair ω, σ of choices of states, then for any M ∈ Loc there exists a group isomorphism αω,σ (M) : Gσ (M) → Gω (M), such that: if ψ ∈ (M1 , M), then there exists gω,σ (ψ) ∈ Gω (M) such that αω,σ (M) ◦ Gσ (ψ) = ad gω,σ (ψ) ◦ Gω (ψ) ◦ αω,σ (M1 ). 3 Note that the notion of a weak covariant (or contravariant) functor can be given in terms of a 2-category, see for instance [37].
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Proof. Consider the natural isomorphism u ω,σ : Sω → Sσ defined in the proof of Theorem 5.15. Recall that u ω,σ satisfies the following properties: for any M ∈ Loc we have that u ω,σ (M) : Sω (M) → Sσ (M), (∗) is a covariant symmetric tensor ∗ -isomorphism which satisfies, for any ψ ∈ (M1 , M) the following equation: u ω,σ (M) ◦ Sω (ψ) = Sσ (ψ) ◦ u ω,σ (M1 ). (∗∗) Consider the equation (∗). Since u ω,σ (M) is an isomorphism, by the DR-theorem there corresponds a group isomorphism αω,σ (M) : Gσ (M) → Gω (M). Consider now the equation (∗∗), and note that u ω,σ (M1 ) ◦ Sω (ψ) is a full faithful symmetric tensor functor from Sω (M) into Sσ (M1 ). Hence there is g ∈ Gω (M) such that Gω (ψ) ◦ αω,σ (M1 ) = ad g (αu ω,σ (M1 )◦Sω (ψ) ). In an analogous fashion αω,σ (M)◦Gσ (ψ)=ad g1 (αSσ (ψ)◦u ω,σ (M1 ) ) for some g1 ∈ Gω (M). Therefore by (∗∗) we have ad g−1 ◦ αω,σ (M) ◦ Gσ (ψ) = ad g−1 ◦ Gω (ψ) ◦ αω,σ (M1 ), 1
and the proof follows.
In conclusion, this theorem shows that the gauge weak functor does not depend on the choice of states ω. In particular the groups Gω (M) belong to the same isomorphism class for any possible choice ω. Therefore we call Gω (M), for some choice ω, the gauge group associated with M ∈ Loc. Remarks 5.19. We point out two problems connected to the study of the local covariance of fields. First, in the case that the set Kh (M) is nondirected, the Doplicher-Roberts reconstruction theorem does not apply straightforwardly. This happens for instance when either M is nonsimply connected or M has compact Cauchy surfaces. Secondly, consider M, M1 ∈ Loc such that there is ψ ∈ (M1 , M). Assume that Kh (M) and Kh (M1 ) are directed.4 In this case one can apply the reconstruction theorem and obtain the algebras of fields, say F (M) and F (M1 ). One expects that F (M1 ) is isomorphic to the subalgebra F (ψ(M1 )) of F (M). It is not clear how to show this because in the definition of F (M1 ) the category Sω (M1 ) intervenes while in that of F (ψ(M1 )) Sω (M) intervenes, and we do not know whether these categories are equivalent. 6. Local Completeness The local covariance of the theory makes possible the analysis of the relation between local and global superselection sectors. This section is devoted to a preliminary analysis of this topic. We will discuss how the possible nonequivalence between local and global superselection sectors might be related to the nontrivial topology of spacetimes and in particular to the existence of path-dependent 1-cocycles. Fix a spacetime M ∈ Loc and consider the set of diamonds Kd (M). Any diamond O is a globally hyperbolic spacetime, and the injection ι M,O : O → M provides an embedding of O into M. Given a choice ω of states, we focus on the superselection 4 For instance we can take as M the Minkowski space, and as M a diamond of the Minkowski space. One 1 can easily check that the set of diamonds of these two spaces are directed.
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sectors Sω (M) associated with M and to those associated with any diamond O, that is Sω (O). We know that Sω (ι M,O ) : Sω (M) → Sω (O)
(6.1)
∗ -functor
is a covariant symmetric tensor which is full and faithful. From the physical point of view, we expect that this functor is an equivalence. The sectors under investigation are sharply localized, hence they should not be affected by the nontrivial topology of the spacetime and there should be no difference between local and global behaviour. However, up until now, we have no argument to establish this equivalence. We explain the technical problem, to which we will refer as the extension problem. Given O ∈ Kd (M), let Kd (O) be the set of diamonds of O considered as a globally hyperbolic spacetime. Given ω ∈ So (M), let Zt1 (ω, Kd (O)) be the category of pathindependent 1-cocycles of Kd (O) with values in the net ω∗ AKd (O) . This category is a tensor C∗ -category with a permutation symmetry and its objects with finite statistics have conjugates. One can easily see that the functor (6.1) is an equivalence if, and only if, for any diamond O and for any 1-cocycle z O of Zt1 (ω, Kd (O)), having finite statistics, there exists z ∈ Zt1 (ω, Kd (M)) such that zKd (O) = z O .
(6.2)
Now, it is convenient to introduce a new terminology. Definition 6.1. Given M ∈ Loc, we will say that the superselection sectors of (A (M), So (M)) are locally complete whenever Sω (M) is equivalent to Sω (O) for any O ∈ Kd (M); conversely, we will say that they are locally incomplete. Examples of theories with locally complete sectors can be easily provided (see below), however we have no argument to prove that this holds true in general5 . The, possible, failure of local completeness seems to be related in a subtle way to the nontrivial topology of spacetimes: in particular to nonsimply connectedness, as we explain by means of the following reasonings: (1) The first example of a locally covariant quantum field theory F with a locally quasiequivalent state space Sμ has been provided in [13]. F is the mapping Loc M → F (M), where F (M) is the CCR algebra of the scalar Klein-Gordon field over M; Sμ is the mapping Loc M → Sμ (M), where Sμ (M) is the space of quasi-free states of F (M) satisfying the microlocal spectrum condition [12]. Pure states of Sμ (M) fulfill the split property, and, as shown in [49], punctured Haag duality. Thus, according to Definition 5.1, Sμ is a reference state space for F . Now, consider a pure state ω of Sμ (M). By means of the same reasoning used in [15] it turns out that any 1-cocycle of the category Zt1 (ω, Kd (M)) is a finite direct sum of the identity 1-cocycle ι. By Proposition 5.6, this holds for any σ ∈ Sμ (M). So for any choice ω and for any M the category Sω (M) is trivial. Hence the functor Sω (ψ) : Sω (M1 ) → Sω (M) is an equivalence for any ψ ∈ (M1 , M) and the theory is locally complete. This easy6 example says that the spacetime topology, in particular the nonsimply connectedness, does not represent an obstruction to local completeness. 5 The only attempts known by the authors to solve this problem in the Haag-Kastler framework are due to Roberts [44] and Longo (private communication). 6 Modifying this example one can construct examples of locally complete theories having a nontrivial superselection content. It is enough to enlarge the symplectic space associated with any M ∈ Loc in a such way that it is possible to introduce an action of a compact group.
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(2) The existence of a possible relation between topology and local incompleteness, can be seen by analyzing a situation easier than the extension problem. Consider a spacetime M, and let ω be a state of So (M). Assume that there is a family {z O } with z O ∈ Zt1 (ω, Kd (O)) for any diamond O, satisfying the following property: given b ∈ 1 (Kd (M)), then z O (b) = z O1 (b)
(6.3)
for any pair O, O1 of diamonds of M such that cl(|b|) ⊆ O ∩ O1 . Now define z(b) ≡ z O (b), b ∈ 1 (Kd (M)), where O is a diamond which contains cl(|b|). By (6.3), this definition does not depend on the chosen diamond O. Using property (2.4) it can be easily seen that z is a 1-cocycle of Kd (M) which extends the family {z O }. However, in the case that M is nonsimply connected, we may not exclude that z is path-dependent (see Sect. 4.2). Hence a failure of local completeness might be related to the existence of path-dependent 1-cocycles. (3) In order to strengthen the idea in (2), consider a nonsimply connected spacetime M, and let ω be a state of So (M). Assume that there exists a path-dependent 1-cocycle z of Kd (M) with values in ω∗ AKd (M) . For any O ∈ Kd (M) define z O (b) ≡ z(b), b ∈ 1 (Kd (O)). Since O is simply connected, z O is a path-independent 1-cocycle of Kd (O). Hence, the family {z O } satisfies the condition (6.1) and its extension to Kd (M) is a pathdependent 1-cocycle. One should be cautious at this point. The above does not imply the violation of local completeness because one should first ensure that z O has finite statistics. Unfortunately, at the moment we are unable to prove this. Remarks 6.2. According to the above discussion, the obstruction to local completeness of sectors seems to be the presence of topological 1-cocycles. So, we expect that, in a simply connected spacetime, the sectors are locally complete. Remarks 6.3. To avoid confusion, we want to stress that there is no relation between the path-dependent 1-cocycle of a poset and the topological charged sectors discovered by Buchholz and Fredenhagen in [17]. As observed the former are of a topological nature because they provide nontrivial representation of the fundamental group of the poset. The latter are called topological because of their localization properties. In this case the poset underlying the theory is that formed by the set of spacelike cones of the Minkowski space. One can easily see that the 1-cocycles associated with this type of charges are path-independent. Thus, they provide only trivial representations of the fundamental group of the poset. 7. Conclusions and Outlook The paper is concerned with the analysis of superselection sectors in the framework of a locally covariant quantum field theory A . The main purpose was the understanding of the covariance behaviour of those sectors which describe sharply localized charges, namely those type of sectors that on a fixed background spacetime are a sharp excitation of a reference representation, the vacuum in the Minkowski space, of the observable net.
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As the present paper is the first investigation on this topic, it is worth recalling in some detail the basic assumptions and results. The first, very useful result, is of a geometrical nature: (1) The set of diamonds Kd (M) of globally hyperbolic spacetimes M is stable under isometric embeddings (Lemma 2.8). This allows to express the locally covariant principle in terms of nets of local observables AKd (M) indexed by the set of diamonds Kd (M) of M. The set Kh (M) introduced in [13] does not fit the topological and causal properties of the spacetime M, compared to Kd (M). The first step needed to introduce the superselection sectors of the quantum field theory functor A has been the definition of a reference state space So for the theory. We required that So is locally quasi-equivalent, that it satisfies the Borchers property and punctured Haag duality (see Definition 5.1). In particular, we emphasize that the local quasi-equivalence is required not only on diamonds but also on the larger family Kh (M). This was an important assumption, as we are requiring that the elements So (M) behave locally in the same way also on non-simply connected regions. Given So we have defined the sectors as follows: The superselection sectors associated with a spacetime M are the families Zt1 (ω, Kd (M)), as ω varies in So (M), of path-independent 1-cocycles of the poset obtained by ordering under inclusion Kd (M), which take their values in the net ω∗ AKd (M) (see Definition 5.2). (2) For any spacetime M the sets Zt1 (σ, Kd (M)), as σ varies in So (M), carry the same physical information: 1-cocycles of Zt1 (σ, Kd (M)) are, up to equivalence, in bijective correspondence with representations of the observable net which are sharp excitations of the representation πω induced by a state ω in So (M) satisfying punctured Haag duality (see Proposition 5.10). (3) Sectors manifest a charge structure: Their quantum numbers have a composition law, a statistics and a charge conjugation symmetry. This structure is independent from the choice of the state ω in So (M). This is expressed by the fact that Zt1 (ω, Kd (M)) is a symmetric tensor C∗ −category whose objects with finite statistics have conjugates; all the categories Zt1 (ω, Kd (M)) belong to the same isomorphism class (see Theorem 5.8). (4) The charge structure is contravariant: If a spacetime M1 can be isometrically embedded in M, then to any sector of M there corresponds a unique sector of M1 with the same charged quantum numbers; the correspondence associating to any spacetime M the category Zt1 (ω M , Kd (M)), with ω M ∈ So (M), is a contravariant functor (see Theorem 5.14 and Remark 5.16) 7 . These results imply that the physical content of superselection sectors carried by each spacetime remains stable when the spacetimes can be coherently embedded. Notice that in this first paper we took a conservative point of view, namely, we work at a level that is as faithful as possible to the tradition in superselection theory. Indeed, one might have even started in a more abstract way by defining representations in the generally covariant sense from the beginning.This amounts to view them as natural transformations. However, at the moment it is not clear how to generalize some of the important technical features that we used. We now pass on to briefly outline the prospect for future work. 7 A similar functorial structure arises in the theory of subsystems [19].
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An important point that will be tackled in a forthcoming contribution [14] is how various indices are related to each other, in the sense of tensor categories. Here, one makes use of more machinery from algebraic topology, in the sense of injections and retractions, always tailored to the poset situation. New directions of research can be envisaged. For instance, an important step would be to clarify whether the reconstruction procedure of Doplicher and Roberts can be fully implemented, in general, by reconstructing the field nets, and proving them to be locally covariant. In case the theory is locally complete and the family index is directed, the reconstruction can be done, and it will appear in the third paper of the series. A most important direction would be the incorporation of the case when the 1-cocycles are not path-independent. Here the full topological structure of the spacetimes enters the game. This is strongly related also to the problem of proving, or disproving, local completeness in the sense of Sect. 6. It looks like a difficult task but worth exploring. We hope to come back to this point in the near future. We conclude with a couple of possible further directions of research. The first deals with the gauge groups. One may imagine that the gauge groups associated with any local region act as local gauge groups, thus opening a fresh look at this problem in the algebraic setting. Here, a more geometrical setting might prove helpful [47]. The second direction deals with the fact that the theories described in this paper seem to have a well-defined ultraviolet behaviour from the outset, a fact which is exemplified by the properties of the functor of restriction R (see Theorem 5.9). A closer connection with the work done by D’Antoni, Morsella and Verch [20], possibly within the framework outlined in [8], would be desirable. Acknowledgements. We are particularly thankful to J.E. Roberts and K. Fredenhagen for their kind remarks during this investigation. We thank as well E. Vasselli for his precious help with the intricacies of the Doplicher-Roberts reconstruction theorem. We are grateful to the DFG and to the European Network “Quantum Spaces – Noncommutative Geometry” for financial support.
A. Symmetric Tensor C∗-Categories Let C be a C∗ -category. We denote the objects of C by z, z 1 , z 2 , . . . and the set of arrows between z, z 1 by (z, z 1 ). The composition of arrows is indicated by “·” and the identity arrow of (z, z) by 1z . References for this appendix are [38, 24, 37]. Symmetric tensor C∗ -categories. A tensor C∗ -category C is a C∗ -category equipped with an associative bilinear functor ⊗ : C × C → C, called a tensor product, with a unit ι and commuting with the adjoint ∗. We will consider only tensor C∗ -categories, with irreducible unit ι, which are closed under direct sums and subobjects. A tensor C∗ -category C is said to be symmetric if it has a permutation symmetry ε, i.e., a mapping ε : C z 1 , z 2 −→ ε(z 1 , z 2 ) ∈ (z 1 ⊗ z 2 , z 2 ⊗ z 1 ) satisfying the relations: (i) (ii) (iii) (iv)
ε(z 3 , z 4 ) · t ⊗ s = s ⊗ t · ε(z 1 , z 2 ); ε(z 1 , z 2 )∗ = ε(z 2 , z 1 ); ε(z 1 , z 2 ⊗ z) = 1z 2 ⊗ ε(z 1 , z) · ε(z 1 , z 2 ) ⊗ 1z ; ε(z 1 , z 2 ) · ε(z 2 , z 1 ) = 1z 2 ⊗z 1 ,
where t ∈ (z 2 , z 4 ), s ∈ (z 1 , z 3 ). By ii) − iv), ε(z, ι) = ε(ι, z) = 1z for any z. Let C1 , C2 be symmetric tensor C∗ -categories with tensor products and permutation symmetries,
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respectively, (⊗1 , ε1 ), (⊗2 , ε2 ). A ∗ -functor F : C1 → C2 is said to be a symmetric tensor ∗ -functor, if for any pair of objects z 1 , z 2 of C1 and any pair t, s of arrows of C1 , we have F(z ⊗1 z 1 ) = F(z) ⊗2 F(z 1 ), F(t ⊗1 s) = F(t) ⊗2 (F(s)), F(ε1 (z, z 1 )) = ε2 (F(z), F(z 1 )). We say that C1 and C2 are equivalent (isomorphic) if there is a symmetric tensor ∗ -functor F : C1 → C2 which is an equivalence (isomorphism). A tensor natural transformation u : F → G between two symmetric tensor ∗ -functors G, F : C1 → C2 is a natural transformation such that u(z ⊗1 z 1 ) = u(z) ⊗2 u(z 1 ). It will be said to be a tensor natural isomorphism (tensor natural unitary) if it is a natural isomorphism (a natural unitary). Statistics and conjugation. Consider a symmetric tensor C∗ -category C with left inverses (see [24, 37] for the definition of left inverse). By means of left inverses one can select a full C∗ -subcategory Cf of C, called the category of objects with finite statistics. Cf is closed under direct sums, subobjects, tensor products, and equivalence. Any object of Cf is a direct sum of irreducible objects. The possible statistics of an irreducible object z are classified by a rational number λ(z), called the statistics parameter, which is an invariant of the equivalence class of z. It turns out that λ(z) is the product of two invariants: λ(z) = χ (z) · d(z)−1 where χ (z) ∈ {1, −1}, d(z) ∈ N. The statistical phase χ (z) distinguishes para-Bose (1) and para-Fermi (−1) statistics, while the statistical dimension d(z) gives the order of the parastatistics. Ordinary Bose and Fermi statistics correspond to d(z) = 1. An object z of C has conjugates if there exists an object z and a pair of arrows r ∈ (ι, z ⊗ z), r ∈ (ι, z ⊗ z) satisfying the conjugate equations r ∗ ⊗ 1z · 1z ⊗ r = 1z , r ∗ ⊗ 1z · 1z ⊗ r = 1z . Conjugation is a property stable under subobjects, direct sums, tensor products and, furthermore, it is stable under equivalence. It turns out that if z has conjugates, then z has finite statistics. Consider two symmetric tensor C∗ -categories C1 and C2 , and let F : C1 → C2 be a full and faithful symmetric tensor ∗ -functor. Let r ∈ (ι, z ⊗1 z) and r ∈ (ι, z ⊗1 z) solve the conjugate equations with respect to z and z. Then F(r ) ∈ (ι, F(z) ⊗2 F(z)) and F(r ) ∈ (ι, F(z) ⊗2 F(z)) solve the conjugate equations in C2 with respect to F(z) and F(z). Moreover, F(z) is irreducible if, and only if, z is irreducible, and F(z) has the same statistics as z. A symmetric tensor C∗ -category is said to have conjugates if any object of the category has conjugates. Doplicher-Roberts duality theorem. We recall some basic facts on the Doplicher-Roberts duality theorem of compact groups [24] (DR-theorem) focusing on its functorial properties. Denote by Sym the full subcategory of Sym (see Sect. 5.3.2) whose objects are symmetric tensor C∗ -categories having conjugates. The main technical result of the DRtheorem is the embedding theorem: Any C ∈ Sym admits an embedding into a category of finite dimensional Hilbert spaces. To be precise, an embedding of C is a pair (H, H), where H ∈ Sym is a category of finite dimensional Hilbert spaces, while H : C → H
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is a symmetric tensor ∗ −functor which is full and faithful. Given an embedding (H, H) of C, the set End⊗ (H ) ≡ {tensor natural unitaries u : H → H } equipped with the composition (u 1 ◦ u)(z) ≡ u 1 (z) · u(z) is, with a suitable topology, a compact group. It turns out that C is the abstract dual of End⊗ (H ). This group is uniquely associated with the category C only up to isomorphism, because the embedding of C is not uniquely determined. However for any other embedding (H , H ) of C there exists a tensor natural unitary equivalence w : H → H
(A.1)
which associates to any object z ∈ C a unitary operator w(z) from the Hilbert space H (z) onto the Hilbert space H (z). w preserves the tensor products w(z ⊗ z 1 ) = w(z) ⊗ w(z 1 ), z, z 1 ∈ C, ⊗ is the tensor product of H , and w(z 1 ) · H (t) = H (t) · w(z) for any z, z 1 ∈ C and t ∈ (z, z 1 ). It turns out that the mapping End⊗ (H ) u → w ◦ u ◦ w ∗ ∈ End⊗ (H ) is a group isomorphism, where (w ◦ u ◦ w ∗ )(z) ≡ w(z) · u(z) · w ∗ (z), z ∈ C. We now want to show the functorial property of the DR-theorem used in the proofs of Theorems 5.17 and 5.18. To this end, let C, C1 , C2 ∈ Sym and let (H, H), (H1 , H1 ) and (H2 , H2 ) be choices of embeddings of these three categories. Given F ∈ (C1 , C) and L ∈ (C2 , C1 ), observe that F ◦ L ∈ (C2 , C). Note that (H ◦ F, H) is an embedding of C1 while (H1 ◦ L , H1 ) and (H ◦ F ◦ L , H) are two embeddings of C2 . Let w F , w L and w F L be tensor natural unitary equivalences w F : H1 → H ◦ F, w L : H2 → H ◦ L and w F L : H2 → H ◦ F ◦ L (see (A.1)). Define α F (u)(z 1 ) ≡ w ∗F (z 1 ) · u(F(z 1 )) · w F (z 1 ), z 1 ∈ C1 , u ∈ End⊗ (H ); α L (u 1 )(z 2 ) ≡ w ∗L (z 2 ) · u 1 (L(z 2 )) · w L (z 2 ), z 2 ∈ C2 , u 1 ∈ End⊗ (H1 ); α F L (u)(z 2 ) ≡ w ∗F L (z 2 ) · u(F ◦ L(z 2 )) · w F L (z 2 ), z 2 ∈ C2 , u ∈ End⊗ (H ). It turns out that α F : End⊗ (H ) → End⊗ (H1 ), α L : End⊗ (H1 ) → End⊗ (H2 )α F L : End⊗ (H ) → End⊗ (H2 ) are group morphisms. Now, observe that α L ◦ α F is a group morphism from End⊗ (H ) into End⊗ (H2 ), and that (α L ◦ α F )(u)(z 2 ) = w ∗L (z 2 ) · α F (u)(L(z 2 )) · w L (z 2 ) = w ∗L (z 2 ) · w ∗F (L(z 2 )) · u(F ◦ L(z 2 )) · w F (L(z 2 )) · w L (z 2 ), where u ∈ End⊗ (H ) and z 2 ∈ C2 . Hence, if we define w F,L;F L (z 2 ) ≡ w ∗L (z 2 ) · w ∗F (L(z 2 )) · w F L (z 2 ), z 2 ∈ C2 , then it turns out that w F,L;F L is an element of End⊗ (H2 ) and that (α L ◦ α F )(u)(z 2 ) = adw F,L;F L (z 2 ) α F L (u)(z 2 ) , for any z 2 ∈ C2 and u ∈ End⊗ (H ).
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References 1. Ashtekar, A., Sen., A.: On the role of space-time topology in quantum phenomena: superselection of charge and emergence of nontrivial vacua. J. Math. Phys. 21, 526–533 (1980) 2. Baumgärtel, H., Lledó, F.: Duality of compact groups and Hilbert C∗ -systems for C∗ -algebras with nontrivial center. Internat. J. Math. 15, 759–812 (2004) 3. Beem, J.K., Ehrlich, P.E., Easley, K.L.: Global Lorentzian geometry. 2nd edition, New York: Marcel Dekker, Inc., 1996 4. Bernal, A.N., Sánchez, M.: On smooth Cauchy hypersurfaces and Geroch’s splitting theorem. Commun. Math. Phys. 243, 461–470 (2003) 5. Bernal, A.N., Sánchez, M.: Smoothness of time functions and the metric splitting of globally hyperbolic spacetimes. Commun. Math. Phys. 257, 43–50 (2005) 6. Bernal, A.N., Sánchez, M.: A note on the extendability of compact hypersurfaces to smooth Cauchy hypersurfaces. http://arxiv.org/list/gr-qc/0507018, 2005 7. Borchers, H.J.: Local rings and the connection of spin with statistics. Commun. Math. Phys. 1, 281–307 (1965) 8. Brunetti, R.: Locally covariant quantum field theories. “Rigorous Quantum Field Theory. A Festschrift for Jacques Bros.” Progress in Mathematics, vol. 251, Birkhauser (2006) 9. Brunetti, R., Fredenhagen, K.: Microlocal analysis and quantum field theory: Renormalization on physical backgrounds. Commun. Math. Phys. 208, 623–661 (2000) 10. Brunetti, R., Fredenhagen, K.: Algebraic Quantum Field Theory. In: Francoise, J.-P., Naber, G., Tsun, T.S. (eds.) Encyclopedia of Mathematical Physics. Elsevier (2006) 11. Brunetti, R., Fredenhagen, K.: Towards a background independent formulation of perturbative quantum gravity. To appear in the Proceedings of the Workshop on “Mathematical and Physical Aspects of Quantum Gravity”, held in Blaubeuren 28/7-01/08, 2005. http://arxiv.org/list/gr-qc/0603079 (2006) 12. Brunetti, R., Fredenhagen, K., Köhler, M.: The microlocal spectrum condition and Wick polynomials of free fields on curved spacetimes. Commun. Math. Phys. 180, 633–652 (1996) 13. Brunetti, R., Fredenhagen, K., Verch, R.: The generally covariant locality principle – A new paradigm for local quantum physics. Commun. Math. Phys. 237, 31–68 (2003) 14. Brunetti, R., Ruzzi, G.: Superselection sectors and general covariance II. In preparation 15. Buchholz, D., Doplicher, S., Longo, R., Roberts, J.E.: A new look at Goldstone’s theorem. Rev. Math. Phys. Special Issue, 49–83 (1992) 16. Buchholz, D.: The physical state space of quantum electrodynamics. Commun. Math. Phys. 85, 49–71 (1982) 17. Buchholz, D., Fredenhagen, K.: Locality and the structure of particle states. Commun. Math. Phys. 84, 1–54 (1982) 18. Ciolli, F.: Massless scalar free Field in 1+1 dimensions II: Net Cohomology and Completeness of Superselection Sectors. In preparation 19. Conti, R., Doplicher, S., Roberts, J.E.: Superselection theory for subsystems. Commun. Math. Phys. 218(2), 263–281 (2001) 20. D’Antoni, C., Morsella, G., Verch, R.: Scaling algebras for charged fields and short-distance analysis for localizable and topological charges. Ann. Henri Poincaré 5, 809–870 (2005) 21. Doplicher, S., Haag, R., Roberts, J.E.: Local observables and particle statistics I. Commun. Math. Phys. 23, 199–230 (1971) 22. Doplicher, S., Haag, R., Roberts, J.E.: Local observables and particle statistics II. Commun. Math Phys. 35, 49–85 (1974) 23. Doplicher, S., Longo, R.: Standard and split inclusions of von Neumann algebras. Invent. Math. 35, 493–536 (1984) 24. Doplicher, S., Roberts, J.E.: A new duality theory for compact groups. Invent. Math. 98(1), 157–218 (1989) 25. Doplicher, S., Roberts, J.E.: Why there is a field algebra with a compact gauge group describing the superselection sectors in particle physics. Commun. Math. Phys. 131(1), 51–107 (1990) 26. Dütsch, M., Fredenhagen, K.: Causal perturbation theory in terms of retarded products and, and a proof of the action Ward identity. Rev. Math. Phys. 16 (10), 1291–1348 (2004) 27. Ellis, G.F.R., Hawking, S.W.: The large scale structure of space-time. Cambridge: Cambridge University Press, 1973 28. Fredenhagen, K., Rehren, K.-H., Schroer, B.: Superselection sectors with braid group statistics and exchange algebras. II: Geometric aspects and conformal invariance. Rev. Math Phys. Special Issue, 113–157 (1992) 29. Friedman, J.L., Schleich, K., Witt, D.M.: Topological censorship. Phys. Rev. Lett. 71, 1486–1489 (1993) 30. Fuller, R.W., Wheeler, J.A.: Causality and multiply connected space-time. Phys. Rev. 128(2), 919–929 (1962)
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31. Guido, D., Longo, R., Roberts, J.E., Verch, R.: Charged sectors, spin and statistics in quantum field theory on curved spacetimes. Rev. Math. Phys. 13(2), 125–198 (2001) 32. Haag, R.: Local Quantum Physics. 2nd edition Springer Texts and Monographs in Physics, BerlinHeidelberg-New York: Springer, 1996 33. Haag, R., Kastler, D.: An algebraic approach to quantum field theory. J. Math. Phys. 5, 848–861 (1964) 34. Hollands, S., Wald, R.M.: Local Wick polynomials and time ordered products of quantum fields in curved spacetime. Commun. Math. Phys. 223, 289–326 (2001) 35. Kastler, D., Mebkhout, M., Rehren, K-H.: Introduction to the algebraic theory of superselection sectors. Space-time dimension = 2 – Strictly localizable morphisms. In: The algebraic theory of superselection sectors, D. Kastler (ed.) (Palermo 1989), River Edge, NJ: World Sci. Publishing, 1990, pp. 113–214 36. Kawahigashi, Y., Longo, R.: Classification of local conformal nets. Case c < 1. Ann. of Math. 160 (2), 493–522 (2004) 37. Longo, R., Roberts, J.E.: A theory of dimension. K-Theory 11(2), 103–159 (1997) 38. Mac Lane, S.: Categories for the working mathematician. New York Heidelberg-Berlin: Springer Verlag, 1971 39. Müger, M.: The superselection structure of massive quantum field theories in 1 + 1 dimensions. Rev. Math. Phys. 10, 1147–1170 (1998) 40. Müger, M.: Frobenius algebras in and Morita equivalence of tensor categories I. J. Pure Appl. Alg. 180, 81–157 (2003) 41. O’Neill, B.: Semi–Riemannian geometry. New York: Academic Press, 1983 42. Porrmann, M.: Particle weights and their disintegration I. Commun. Math. Phys. 248, 269–304 (2004) 43. Porrmann, M.: Particle weights and their disintegration II. Commun. Math. Phys. 248, 305–333 (2004) 44. Roberts, J.E.: Local cohomology and superselection structure. Commun. Math. Phys 51(2), 107–119 (1976) 45. Roberts, J.E.: Lectures on algebraic quantum field theory. In: The algebraic theory of superselection sectors, D. Kastler (ed.) (Palermo 1989), River Edge, NJ: World Sci. Publishing, 1990, pp 1–112 46. Roberts, J.E.: More lectures in algebraic quantum field theory. In: Noncommutative geometry edited by S. Doplicher, R. Longo, C.I.M.E. Lectures, Martina Franca, Italy, 2000. Berlin-Heidelberg-New York: Springer, 2003 47. Roberts, J.E., Ruzzi, G.: A cohomological description of connections and curvature over posets. http:// arxiv.org/list/math.AT/0604173, 2006 48. Ruzzi, G.: Essential properties of the vacuum sector for a theory of superselection sectors. Rev. Math. Phys. 15(10), 1255–1283 (2003) 49. Ruzzi, G.: Punctured Haag duality in locally covariant quantum field theories. Commun. Math. Phys. 256, 621–634 (2005) 50. Ruzzi, G.: Homotopy of posets, net-cohomology, and theory of superselection sectors in globally hyperbolic spacetimes. Rev. Math. Phys. 17(9), 1021–1070 (2005) 51. Sorkin, R.: The quantum electromagnetic field in multiply connected space. J. Phys. A 12, 403–421 (1979) 52. Vasselli, E.: Continuous Fields of C∗ -algebras arising from extensions of tensor C∗ -categories. J. Funct. Anal. 199, 123–153 (2003) 53. Verch, R.: Continuity of symplectically adjoint maps and the algebraic structure of Hadamard vacuum representations for quantum fields in curved spacetime. Rev. Math. Phys. 9(5), 635–674 (1997) 54. Verch, R.: Notes on regular diamonds., Preprint, available as ps-file at http://www/lqp.uni-goettingen.de/lqp/papers/ 55. Verch, R.: A spin-statistics theorem for quantum fields on curved spacetimes manifolds in a generally covariant framework. Commun. Math. Phys. 223, 261 (2001) 56. Verch, R.: Stability Properties of Quantum Fields on Curved Spacetimes. Habilitation Thesis, University of Göttingen, March 2003 57. Wald, R.M.: General Relativity. Chicago: University of Chicago Press, 1984 58. Wick, G.C., Wightman, A.S., Wigner, E.P.: The intrinsic parity of elementary particles. Phys. Rev. 88, 101–105 (1952) 59. Witt, D.M.: Vacuum space-times that admit no maximal slice. Phys. Rev. Lett. 57, 1386–1389 (1986) Communicated by Y. Kawahigashi
Commun. Math. Phys. 270, 109–139 (2007) Digital Object Identifier (DOI) 10.1007/s00220-006-0156-4
Communications in
Mathematical Physics
Exponential Mixing for the 3 D Stochastic Navier–Stokes Equations Cyril Odasso1,2 1 Ecole Normale Supérieure de Cachan, Antenne de Bretagne, Avenue Robert Schuman, Campus de Ker
Lann, 35170 Bruz France. E-mail:
[email protected]
2 IRMAR, UMR 6625 du CNRS, Campus de Beaulieu, 35042 Rennes Cedex, France
Received: 20 February 2006 / Accepted: 1 July 2006 Published online: 1 December 2006 – © Springer-Verlag 2006
Abstract: We study the Navier–Stokes equations in dimension 3 (NS3D) driven by a noise which is white in time. We establish that if the noise is at the same time sufficiently smooth and non-degenerate in space, then the weak solutions converge exponentially fast to equilibrium. We use a coupling method. The arguments used in dimension two do not apply since, as is well known, uniqueness is an open problem for NS3D. New ideas are introduced. Note however that many simplifications appear since we work with non degenerate noises. Introduction We are concerned with the stochastic Navier–Stokes equations on a three dimensional bounded domain (NS3D) with Dirichlet boundary conditions. These equations describe the time evolution of an incompressible fluid subjected to a determinist and a random exterior force and are given by ⎧ ⎪ d X + ν(−)X dt + (X, ∇)X dt + ∇ p dt = φ(X )dW + f dt, ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ (div X ) (t, ξ ) = 0, for ξ ∈ D, t > 0, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
X (t, ξ ) = 0,
for ξ ∈ ∂ D, t > 0,
X (0, ξ ) = x0 (ξ ),
for ξ ∈ D.
(0.1)
Here D is an open bounded domain of R3 with smooth boundary ∂ D or D = (0, 1)3 . We have denoted by X the velocity, by p the pressure and by ν the viscosity. The external force field acting on the fluid is the sum of a random force field of white noise type φ(X )dW and a determinist one f dt.
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In the deterministic case (φ = 0), there exists a global weak solution (in the PDE sense) of (0.1) when x0 is square integrable, but uniqueness of such a solution is not known. On another hand, there exists a unique local strong solution when x0 is smooth, but global existence is an open problem (see [36] for a survey on these questions). In the stochastic case, there exists a global weak solution of the martingale problem, but pathwise uniqueness or uniqueness in law remain open problems (see [10] for a survey on the stochastic case). The main result of the present article is to establish that, if φ is at the same time sufficiently smooth and non-degenerate, then the solutions converge exponentially fast to equilibrium. More precisely, given a solution, there exists a stationary solution (which might depends on the given solution), such that the total variation distance between the laws of the given solution and of the stationary solution converges to zero exponentially fast. Due to the lack of uniqueness, it is not straightforward to define a Markov evolution associated to (0.1). Some recent progress have been obtained in this direction. In [3, 7], under conditions on φ and f very similar to ours, it is shown that every solution of (0.1) limit of Galerkin approximations verify the weak Markov property. Uniqueness in law is not known but we think that this result is a step in this direction. Our result combined with this result implies that the transition semi-group constructed in [3] is exponentially mixing. Note also that recently, a Markov selection argument has allowed the construction of a Markov evolution in [13]. Our result does not directly apply since we only consider solutions which are limit of Galerkin approximations. However, suitable modifications of our proof might imply that under suitable assumptions on the noise, the Markov semi-group constructed in [13] is also exponentially mixing. Our proof relies on coupling arguments. These have been introduced recently in the context of stochastic partial differential equations by several authors (see [15, 21, 24–27, 30–32]). The aim was to prove exponential mixing for degenerate noise. It was previously observed that the degeneracy of the noise on some subspace could be compensated by dissipativity arguments [1, 8, 22]. More recently, highly degenerate noises have been considered in [17, 28]. In all these articles, global well posedness of the stochastic equation is strongly used in many places of the proof. As already mentioned, this is not the case for the three dimensional Navier–Stokes equations considered here. Thus substantial changes in the proof have to be introduced. However, we require that the noise is sufficiently non-degenerate and many difficulties of the above mentioned articles disappear. The main idea is that coupling of solutions can be achieved for initial data which are small in a sufficiently smooth norm. A coupling satisfying good properties is constructed thanks to the Bismut-Elworthy-Li formula. Another important ingredient in our proof is that any weak solution enters a small ball in the smooth norm and that the time of entering in this ball admits an exponential moment. We overcome the lack of uniqueness of solutions by working with Galerkin approximations. We prove exponential mixing for these with constants which are controlled uniformly. Taking the limit, we obtain our result for solutions which are the limit of Galerkin approximations 1. Preliminaries and Main Result 1.1. Weak solutions. Here L(K 1 ; K 2 ) (resp. L2 (K 1 ; K 2 )) denotes the space of bounded (resp Hilbert-Schmidt) linear operators from the Hilbert space K 1 to K 2 .
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We denote by |·| and (·, ·) the norm and the inner product of L 2 (D; R3 ) and by |·| p the norm of L p (D; R3 ). Recall now the definition of the Sobolev spaces H p (D; R3 ) for p ∈ N, ⎧ ⎨ H p (D; R3 ) = X ∈ L 2 (D; R3 ) ∂α X ∈ L 2 (D; R3 ) for |α| ≤ p , ⎩ |X |2 = |∂ X |2 . Hp
|α|≤ p
α
It is well known that (H p (D; R3 ), |·| H p ) is a Hilbert space. The Sobolev space H01 (D; R3 ) is the closure of the space of smooth functions on D with compact support by |·| H 1 . Setting X = |∇ X | , we obtain that · and |·| H 1 are two equivalent norms on H01 (D; R3 ) and that (H01 (D; R3 ), ·) is a Hilbert space. Let H and V be the closure of the space of smooth functions on D with compact support and free divergence for the norm |·| and ·, respectively. Let π be the orthogonal projection in L 2 (D; R3 ) onto the space H . We set A = π (−) , D(A) = V ∩ H 2 (D; R3 ), B(u, v) = π ((u, ∇)v) and B(u) = B(u, u). Let us recall the following useful identities ⎧ ⎨ (B(u, v), v) = 0, u, v ∈ V, ⎩ (B(u, v), w) = −(B(u, w), v), u, v, w ∈ V. As is classical, we get rid of the pressure and rewrite problem (0.1) in the form ⎧ ⎨ d X + ν AX dt + B(X )dt = φ(X )dW + f dt, ⎩ X (0) = x ,
(1.1)
0
where W is a cylindrical Wiener process on H and with a slight abuse of notation, we have denoted by the same symbols the projections of φ and f . It is well-known that (A, D(A)) is a self-adjoint operator with discrete spectrum. See [2, 34]. We consider (en )n an eigenbasis of H associated to the increasing sequence (μn )n of eigenvalues of (A, D(A)). It will be convenient to use the fractional power (As , D(As )) of the operator (A, D(A)) for s ∈ R, ⎧ ⎨ D(As ) = X = ∞ xn en ∞ μ2s |xn |2 < ∞ , n=1 n=1 n ⎩ As X = ∞ μs x e where X = ∞ x e . n=1
We set for any s ∈ R,
n n n
n=1 n n
s s X s = A 2 X , Hs = D(A 2 ).
It is obvious that (Hs , ·s ) is a Hilbert space, that (H0 , ·0 ) = (H, |·|) and that (H1 , ·1 ) = (V, ·). Moreover, recall that, thanks to the regularity theory of the Stokes operator, Hs is a closed subspace of H s (D, R3 ) and ·s is equivalent to the usual norm of H s (D; R3 ) when D is an open bounded domain of R3 with smooth boundary ∂ D. When D = (0, 1)3 , it remains true for s ≤ 2.
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Let us define
⎧ 2 + + + ⎪ X = L∞ ⎪ loc (R ; H ) ∩ L loc (R ; V ) ∩ C(R ; Hs ), ⎪ ⎨ W = C(R+ ; H−2 ), ⎪ ⎪ ⎪ ⎩ = X × W, ∗
where s is any fixed negative number. Remark that the definition of X is not depending on s < 0. Let X ∗ (resp. W∗ ) be the projector ∗ → X (resp. ∗ → W). The space ∗ is endowed with its borelian σ -algebra F ∗ and with Ft∗ t≥0 the filtration generated by (X ∗ , W∗ ). Recall that W is said to be a (Ft )t –cylindrical Wiener process on H if W is (Ft )t – adapted, if W (t + ·) − W (t) is independent of Ft for any t ≥ 0 and if W is a cylindrical Wiener process on H . Let E be a Polish space. We denote by P(E) the set of probability measure on E endowed with the borelian σ –algebra. Definition 1.1 (Weak solutions). A probability measure Pλ on ( ∗ , F ∗ ) is said to be a weak solution of (1.1) with initial law λ ∈ P(H ) if the three following properties hold. i) The law of X ∗ (0) under Pλ is λ. ii) The process W∗ is a (Ft∗ )t –cylindrical Wiener process on H under Pλ . iii) We have Pλ -almost surely (X ∗ (t), ψ) + ν
t
0 (X ∗ (s),
= (X ∗ (0), ψ) + t
t
0 (B(X ∗ (s)), ψ)ds t ( f, ψ) + 0 (ψ, φ(X ∗ (s))dW∗ (s)),
Aψ)ds +
(1.2)
for any t ∈ R+ and any ψ smooth mapping on D with compact support and divergence zero. When the initial value λ is not specified, x0 is the initial value of the weak solution Px0 (i.e. λ is equal to δx0 the Dirac mass at point x0 ). These solutions are weak in both probability and PDE sense. On the one hand, these are solutions in law. Existence of solutions in law does not imply that, given a Wiener process W and an initial condition x0 , there exist a solution X associated to W and x0 . On the other hand, these solutions live in H and it is not known if they live in H1 . This latter fact causes many problems when trying to apply the Ito Formula on F(X ∗ (t)) when F is a smooth mapping. Actually, we do not know if we are allowed to apply it. That is the reason why we do not consider any weak solution but only those which are limits in the distribution of solutions of Galerkin approximations of (1.1). More precisely, for any N ∈ N, we denote by PN the eigenprojector of A associated to the first N eigenvalues. Let ( , F, P) be a probability space and W be a cylindrical Wiener process on H for P. We consider the following approximation of (1.1): ⎧ ⎨ d X N + ν AX N dt + PN B(X N )dt = PN φ(X N )dW + PN f dt, (1.3) ⎩ X (0) = P x . N
N 0
In order to have existence of a weak solution, we use the following assumption. Hypothesis 1.2. The mapping φ is bounded Lipschitz H → L2 (H ; H1 ) and f ∈ H .
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We set B1 = sup |φ(x)|2L2 (H ;H1 ) + x∈H
| f |2 . νμ1
It is easily shown that, given x0 ∈ H , (1.3) has a unique solution X N = X N (·, x0 ). Proceeding as in [11], we can see that the laws (PxN0 ) N of (X N (·, x0 ), W ) are tight in a well chosen functional space. Then, for a subsequence (Nk )k , (X Nk , W ) converges in law to Px0 a weak solution of (1.1). Hence we have existence of the weak solutions of (1.1), but uniqueness remains an open problem. Remark 1.3. We only consider weak solutions constructed in that way. This allows to make some computations and to obtain many estimates. For instance, when trying to estimate the L 2 -norm of X ∗ (t) under a weak solution Px0 , we would like to apply the Ito Formula on |X ∗ |2 . This would give d |X ∗ |2 + 2ν X ∗ 2 dt = 2 (X ∗ , φ(X ∗ )dW∗ ) + 2( f, X ∗ )dt + |φ(X ∗ (t))|2L2 (H ;H ) dt. Integrating and taking the expectation, we would deduce that, if f = 0 and φ constant,
t 2 2 X ∗ (s) dt = |x0 |2 + t |φ|2L2 (H ;H ) . Ex0 |X ∗ (t)| + 2ν 0
Unfortunately, those computations are not allowed. However, analogous computations are valid if we replace Px0 by PxN0 , which yields
t 2 2 X N (s) dt = |PN x0 |2 + t |PN φ|2L2 (H ;H ) . E |X N (t)| + 2ν 0
Then, we take the limit and we infer from the Fatou Lemma and from the semi-continuity of |·|, · in Hs that
t X ∗ (s)2 dt ≤ |x0 |2 + t |φ|2L2 (H ;H ) , Ex0 |X ∗ (t)|2 + 2ν 0
provided f = 0 and φ constant and provided Px0 is the limit in distribution of solutions of (1.3). Let P and Y be a probability measure and a random variable on ( ∗ , F ∗ ), respectively. The distribution DP (Y ) denotes the law of Y under P . A weak solution Pμ with initial law μ is said to be stationary if, for any t ≥ 0, μ is equal to DPμ (X ∗ (t)). We define PtN ψ (x0 ) = E (ψ(X N (t, x0 ))) = ExN0 (ψ(X ∗ (t)) , where ExN0 is the expectation associated to PxN0 . It is easily shown that X N (·, x0 ) verifies the strong Markov property, which obviously implies that (PtN )t∈R+ is a Markov transition semi-group on PN H . Ito Formula on |X N (·, x0 )|2 gives d |X N |2 + 2ν X N 2 dt = 2 (X N , φ(X N )dW ) + 2(X N , f )dt + |PN φ(X N )|2 dt,
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which yields, by applying arithmetico-geometric inequality and Hypothesis 1.2, d |X N |2 + ν X N 2 dt ≤ 2 (X N , φ(X N )dW ) + cB1 dt. Integrating and taking the expectation, we obtain c B1 . E |X N (t)|2 ≤ e−νμ1 t |x0 |2 + νμ1
(1.4)
(1.5)
Hence, applying the Krylov-Bogoliubov Criterion (see [4]), we obtain that (PtN )t admits an invariant measure μ N and that every invariant measure has a moment of order two in H . Let X 0N be a random variable whose law is μ N and which is independent of W , then X N = X N (·, X 0N ) is a stationary solution of (1.3). Integrating (1.4), we obtain
t
E |X N (t)|2 + νE
X N (s)2 ds ≤ E |X N (0)|2 + cB1 t.
0
Since the law of X N (s) is μ N for any s ≥ 0 and since μ N admits a moment of order 2, it follows c x2 μ N (d x) ≤ B1 . (1.6) ν PN H Moreover the laws (PμNN ) N of (X N (·, X 0N ), W ) are tight in a well chosen functional
space. Then, for a subsequence (Nk )k , PμNNk k converges in law to Pμ a weak stationary solution of (1.1) with initial law μ (see [11] for details). We deduce from (1.6) that c x2 μ(d x) ≤ B1 , ν H which yields (see [12]) Pμ (X ∗ (t) ∈ H1 ) = 1 for any t ≥ 0.
(1.7)
We do not know if X ∗ (t) ∈ H1 for all t holds Pμ –almost surely. This would probably imply strong uniqueness μ–almost surely. Remark that it is not known in general if μ is an invariant measure because, due to the lack of uniqueness, it is not known if (1.1) defines a Markov evolution. We will see below that this is the case under suitable assumptions.
1.2. Exponential convergence to equilibrium. In the present article, the covariance operator φ of the noise is assumed to be at the same time sufficiently smooth and nondegenerate with bounded derivatives. More precisely, we use the following assumption. Hypothesis 1.4. There exist ε > 0 and a family (φn )n of continuous mappings H → R with continuous derivatives such that ⎧ ⎨ φ(x)dW = ∞ φn (x)en dWn where W = ∞ Wn en , n=1 n=0 ⎩ κ = ∞ sup 2 1+ε |φ (x)| μ < ∞. 0
n=1
x∈H
n
n
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Moreover there exists κ1 such that for any x, η ∈ H2 , ∞ φ (x) · η2 μ2 < κ1 η2 . n n 2 n=1
For any x ∈ H and N ∈ N, we have φn (x) > 0 and 2 κ2 = sup φ −1 (x) x∈H
L(H3 ;H )
< ∞,
(1.8)
where φ(x)−1 · h =
∞
φn (x)−1 h n en
for h =
n=1 s
For instance, φ = A− 2 fulfills Hypothesis 1.4 provided s ∈
∞
h n en .
n=0
5 2, 3
. We set
B0 = κ0 + κ1 + κ2 + | f |2 . Remark 1.5 (Additive noise). If the noise is additive, Hypothesis 1.4 simplifies. Indeed in this case, we do not need to assume that φ and A commute. This requires a different but simpler proof of Lemma 3.2 below. Remark 1.6 (Large viscosity). Another situation where we can get rid of the assumption that the noise is diagonal is when the viscosity ν is sufficiently large. The proof is simpler in that case. Remark 1.7. It is easily shown that Hypothesis 1.4 and f ∈ H imply Hypothesis 1.2. Therefore, solutions of (1.3) are well-defined and, for a subsequence, they converge to a weak solution of (1.1). The aim of the present article is to establish that, under Hypothesis 1.4 and under a condition of smallness of f ε , the law of X ∗ (t) under a weak solution Px0 converges exponentially fast to equilibrium provided Px0 is a limit in distribution of solutions of (1.3). Before stating our main result, let us recall some definitions. Let E be Polish space. The set of all probability measures on E is denoted by P(E). The set of all bounded measurable (resp. uniformly continuous) maps from E to R is denoted by Bb (E; R) (resp. U Cb (E; R)). The total variation μvar of a finite real measure λ on E is given by λvar = sup {|λ()| | ∈ B(E)} , where we denote by B(E) the set of the Borelian subsets of E. The main result of the present article is the following. Its proof is given in Sect. 4 after several preliminary results.
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Theorem 1.8. Assume that Hypothesis 1.4 holds. There exists δ 0 , C and γ > 0 only depending on φ, D, ε and ν such that, for any weak solution Pλ with initial law λ ∈ P(H ) which is a limit of solutions of (1.3), there exists a weak stationary solution Pμ with initial law μ such that
2 DP (X ∗ (t)) − μ ≤ Ce−γ t 1 + |x| λ(d x) , (1.9) λ var H
provided f 2ε Hs for s < 0.
≤
δ0
and where ·var is the total variation norm associated to the space
Moreover, for a given Pλ , μ is a unique and Pμ is a limit of solutions of (1.3).
It is well known that ·var is the dual norm of |·|∞ which means that for any finite measure λ on Hs for s < 0, λ = sup g(x) λ (d x) , var |g|∞ ≤1
Hs
where the supremum is taken over g ∈ U Cb (Hs ) which verifies |g|∞ ≤ 1. Hence (1.9) is equivalent to
2 Eλ (g(X ∗ (t))) − ≤ C |g|∞ 1 + |x| g(x) μ(d x) λ(d x) , (1.10) H
H
for any g ∈ U Cb (Hs ). Remark 1.9 (Topology associated to the total variation norm). Remark that if λ is a finite measure of Hs0 , then the value of the total variation norm of λ associated to the space Hs is not depending of the value of s ≤ s0 . Hence, since DPλ (X ∗ (t)) is a probability measure on H then (1.9) (resp. (1.10)) remains true when ·var is the total variation norm associated to the space H (resp. for any g ∈ Bb (H ; R)). Moreover, we see below that, under suitable assumptions, if λ is a probability measure on H2 , then DPλ (X ∗ (t)) is still a probability measure on H2 . It follows that (1.9) (resp (1.10)) remains true when ·var is associated to H2 (resp. for any g ∈ Bb (H2 ; R)). Our method is not influenced by the size of the viscosity ν. Then, for simplicity in the redaction, we now assume that ν = 1. 1.3. Markov evolution. Here, we take into account the results of [3, 7] and we rewrite Theorem 1.8. This section is not necessary in the understanding of the proof of Theorem 1.8. Let (Nk )k be an increasing sequence of integer. In [3, 7], it is established that it is possible to extract a subsequence (Nk )k of (Nk )k such that, for any x0 ∈ H2 , PxN0k converges in distribution to a weak solution Px0 of (1.1) provided the following assumption holds. Hypothesis 1.10. There exist ε, δ > 0 such that the mapping φ is bounded in L2 (H; H1+ε ). Moreover, for any x, ker φ(x) = {0} and there exits a bounded map φ −1 : H → L (H3−δ ; H ) such that for any x ∈ H , φ(x) · φ −1 (x) · h = h Moreover f ∈ V .
for any h ∈ H3−δ .
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The method to extract (Nk )k is based on the investigation of the properties of the Kolmogorov equation associated to (1.1) perturbed by a very irregular potential. It follows that (Px0 )x0 ∈H2 is a weak Markov family, which means that for any x0 ∈ H2 , Px0 (X ∗ (t) ∈ H2 ) = 1 for any t ≥ 0,
(1.11)
and that, for any t1 < · · · < tn , t > 0 and any ψ ∈ Bb (H2 ; R) Ex0 ( ψ(X ∗ (t + tn ))| X ∗ (t1 ), . . . , X ∗ (tn )) = Pt ψ(X ∗ (tn )),
(1.12)
where (Pt ψ) (x0 ) = Ex0 (ψ(X ∗ (t))) . Note that (1.11) was known only for a stationary solution (see [12]). Remark 1.11. Assume that Hypothesis 1.4 holds. If we strengthen (1.8) into 2 κ2 = sup φ −1 (x) < ∞, x∈H
L(H3−δ ;H )
for some δ > 0, then Hypothesis (1.10) holds. Hence, we immediately deduce the following corollary from Theorem 1.8. Corollary 1.12. Assume that Hypothesis 1.4 and 1.10 hold. Then there exits a unique invariant measure μ for (Pt )t∈R+ and C, γ > 0 such that for any λ ∈ P(H2 ),
∗ 2 P λ − μ ≤ Ce−γ t 1 + |x| λ(d x) , (1.13) t var H2
provided f 2ε ≤ δ 0 and where ·var is the total variation norm associated to the space H2 . Remark 1.13 (Uniqueness of the invariant measure μ). Assume that Hypothesis 1.10 holds. Let Px0 and P x0 be two weak solutions of (1.1) which are limits in distribution of solutions of (1.3). Then we build (Pt )t and (Pt )t as above associated to Px0 and P x0 , respectively. It follows that there exists μ and μ such that (1.13) and (1.10) hold for ((Pt )t , Px0 , μ) and ((Pt )t , P x0 , μ ). Although we have uniqueness of the invariant measures μ and μ associated to (Pt )t and (Pt )t , we do not know if μ and μ are equal. 1.4. Coupling methods. The proof of Theorem 1.8 is based on coupling arguments. We now recall some basic results about coupling. Moreover, in order to explain the coupling method in the case of non-degenerate noise, we briefly give the proof of exponential mixing for Eq. (1.3). Let (λ1 , λ2 ) be two distributions on a polish space (E, E) and let ( , F, P) be a probability space and let (Z 1 , Z 2 ) be two random variables ( , F) → (E, E). We say that (Z 1 , Z 2 ) is a coupling of (λ1 , λ2 ) if λi = D(Z i ) for i = 1, 2. We have denoted by D(Z i ) the law of the random variable Z i . The next result is fundamental in the coupling methods, the proof is given for instance in the Appendix of [30].
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Lemma 1.14. Let (λ1 , λ2 ) be two probability measures on (E, E). Then λ1 − λ2 var = min P(Z 1 = Z 2 ). The minimum is taken over all couplings (Z 1 , Z 2 ) of (λ1 , λ2 ). There exists a coupling which reaches the minimum value. It is called a maximal coupling. Let us first consider the case of the solutions of (1.3). Assume that Hypothesis 1.4 holds. Let N ∈ N and (x01 , x02 ) ∈ R2 . Combining arguments from [23, 27], it can be shown that there exists a decreasing function p N (·) > 0 such that ∗ ∗ N (1.14) δx 2 − P1N δx 1 ≤ 1 − p N x01 + x02 . P1 0
0
var
Applying 1.14, webuild a maximal coupling (Z 1 , Z 2 ) = (Z 1 (x01 , x02 ), Z 2 (x01 , x02 )) Lemma ∗
∗ of P1N δx 1 , P1N δx 2 . It follows 0
0
P (Z 1 = Z 2 ) ≥ p N x01 + x02 > 0.
(1.15)
) be a couple of independent cylindrical Wiener processes and δ > 0. We Let (W, W , respecdenote by X N (·, x0 ) and X N (·, x0 ) the solutions of (1.3) associated to W and W 1 2 tively. Now we build a couple of random variables (V1 , V2 ) = (V1 (x0 , x0 ), V2 (x01 , x02 )) on PN H as follows ⎧ ⎪ (X (·, x0 ), X N (·, x0 )) if x01 = x02 = x0 , ⎪ ⎪ ⎨ N (V1 , V2 ) = (Z 1 (x01 , x02 ), Z 2 (x01 , x02 )) if (x01 , x02 ) ∈ B H (0, δ)\{x01 = x02 }, (1.16) ⎪ ⎪ ⎪ ⎩ (X (·, x 1 ), 2 else, N 0 X N (·, x 0 )) where B H (0, δ) is the ball of H × H with radius δ.
∗ ∗ Then (V1 (x01 , x02 ), V2 (x01 , x02 )) is a coupling of ( P1N δx 1 , P1N δx 2 ). It can be 0
0
shown that it depends measurably on (x01 , x02 ). We then build a coupling (X 1 , X 2 ) of (D(X N (·, x01 )), D(X N (·, x02 ))) by induction on N. We first set X i (0) = x0i for i = 1, 2. Then, assuming that we have built (X 1 , X 2 ) on {0, 1, . . . , k}, we take (V1 , V2 ) as above independent of (X 1 , X 2 ) and set X i (k + 1) = Vi (X 1 (k), X 2 (k))
for i = 1, 2.
Taking into account (1.5), it is easily shown that the time of return of (X 1 , X 2 ) in B(0, 4(c/μ1 )B1 ) admits an exponential moment. We choose δ = 4(c/μ1 )B1 . It follows from (1.15), (1.16) that (X 1 (n), X 2 (n)) ∈ B(0, δ) implies that the probability of having (X 1 , X 2 ) coupled (i.e. equal) at time n + 1 is bounded below by p N (2δ) > 0. Finally, remark that if (X 1 , X 2 ) are coupled at time n + 1, then they remain coupled for any time after. Combining these three properties and using the fact that (X 1 (n), X 2 (n))n∈N is a discrete strong Markov process, it is easily shown that
2 2 1 2 −γ N n 1 + x01 + x02 , (1.17) P X (n) = X (n) ≤ C N e with γ N > 0.
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Recall that (X 1 , X 2 ) is a coupling of (D(X N (·, x01 )), D(X N (·, x02 ))) on N. It follows that (X 1 (n), X 2 (n)) is a coupling of ((PnN )∗ δx 1 , (PnN )∗ δx 2 ). Combining Lemma 1.14 0 0 and (1.17), we obtain, for n ∈ N,
2 2 ∗ ∗ 1 2 N N −γ N n 1 + P δ − P δ ≤ C e 2 1 x0 + x0 . n N n x x 0
0
var
N )∗ λ)⊗μ , where μ is an invariant Setting n = t and integrating (x02 , x01 ) over ((Pt−n N N measure, it follows that, for any λ ∈ P(PN H ),
∗ N |x|2 λ(d x) . λ − μ N ≤ C N e−γ N t 1 + (1.18) Pt var
PN H
This result is useless when considering Eq. (1.1) since the constants C N , γ N strongly depend on N . If one tries to apply directly the above arguments to the infinite dimensional equation (1.1), one faces several difficulties. First it is not known whether Px0 is Markov. We only know that, as explained in Sect. 1.3, a Markov transition semi-group can be constructed. This is a major difficulty since this property is implicitly used in numerous places above. Another strong problem is that the Girsanov transform is used in order to obtain (1.14). Contrary to the two dimensional case, no Foias-Prodi estimate is available for the three dimensional Navier–Stokes equations and the Girsanov transform should be done in the infinite dimensional equation. This seems impossible. We will show that we are able to prove an analogous result to (1.14) by a completely different argument. However, this will hold only for small initial data in H2 . Another problem will occur since it is not known whether solutions starting in H2 remain in H2 . We remedy the lack of Markov property by working only on Galerkin approximations and prove that (1.18) holds with constants uniform in N . As already mentioned, we prove that (1.14) is true for x01 , x02 in a small ball of H2 and uniformly in N . Then, following the above argument, it remains to prove that the time of return in this small ball admits an exponential moment. Note that the smallness assumption on f is used at this step. In the following sections, we prove Proposition 1.15. Assume that Hypothesis 1.4 holds. Then there exist δ 0 = δ 0 (B0 ,D,ε,ν), 0 C = C(φ, D, ε, ν) > 0 and γ = γ (φ, D, ε, ν) > 0 such that if f 2ε ≤ δ holds, then, N for any N ∈ N, there exists a unique invariant measure μ N for Pt t∈R+ . Moreover, for any λ ∈ P(PN H ),
∗ N |x|2 λ(d x) . λ − μ N ≤ Ce−γ t 1 + (1.19) Pt var
PN H
We now explain why this result implies Theorem 1.8. Let λ ∈ P(H ) and X λ be a random variable on H whose law is λ and which is independant of W . Since ·var is the dual norm of |·|∞ , then (1.19) implies that
2 E (g(X N (t, X λ ))) − |g| |x| ≤ C 1 + g(x) μ (d x) λ(d x) , (1.20) N ∞ PN H
H
for any g ∈ U Cb (Hs ) for s < 0. Assume that, for a subsequence (Nk )k , X N (t, X λ ) converges in distribution in Hs to the law X ∗ (t) under the weak solution Pλ of (1.1). Recall that the family (PμNN ) N is tight. Hence, for a subsequence (Nk )k of (Nk )k , Pμ Nk converges to Pμ a weak stationary solution of (1.1) with initial law μ. Taking the limit, (1.10) follows from (1.20), which yields Theorem 1.8.
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2. Coupling of Solutions Starting from Small Initial Data The aim of this section is to establish the following result. A result analogous to (1.15) but uniform in N . Proposition 2.1. Assume that Hypothesis 1.4 holds and that f ∈ H . Then there exist (T, δ) ∈ (0, 1)2 such that, for any N ∈ N, there exists a coupling (Z 1 (x01 , x02 ), Z 2 (x01 , x02 )) of ((PTN )∗ δx 1 , (PTN )∗ δx 2 ) which measurably depends on (x01 , x02 ) ∈ H2 and which veri0 0 fies 3 P Z 1 (x01 , x02 ) = Z 2 (x01 , x02 ) ≥ (2.1) 4 provided 2 2 1 (2.2) x0 ∨ x02 ≤ δ. 2
2
Assume that Hypothesis 1.4 holds and that f ∈ H . Let T ∈ (0, 1). Applying Lemma 1.14, we build (Z 1 (x01 , x02 ), Z 2 (x01 , x02 )) as the maximal coupling of (PT∗ δx 1 , PT∗ δx 2 ). 0 0 Measurable dependence follows from a slight extension of Lemma 1.17 (see [30], Remark A.1). In order to establish Proposition 2.1, it is sufficient to prove that there exists c(B0 , D) not depending on T ∈ (0, 1) and on N ∈ N such that ∗ ∗ √ N (2.3) PT δx 2 − PTN δx 1 ≤ c(B0 , D) T , 0
provided
var
0
2 2 1 x0 ∨ x02 ≤ B0 T 3 . 2
2 D))2
and δ = B0 T 3 . Then it suffices to choose T ≤ 1/(4c(B0 , Since ·var is the dual norm of |·|∞ , (2.3) is equivalent to √ E g(X N (T, x02 )) − g(X N (T, x01 )) ≤ 8 |g|∞ c(B0 , D) T
(2.4)
(2.5)
for any g ∈ U Cb (PN H ). It follows from the density of Cb1 (PN H ) ⊂ U Cb (PN H ) that, in order to establish Proposition 2.1, it is sufficient to prove that (2.5) holds for any N ∈ N, T ∈ (0, 1) and g ∈ Cb1 (PN H ) provided (2.4) holds. The proof of (2.5) under this condition is split into the next three subsections. 2.1. A priori estimate. For any process X , we define the H1 –energy of X at time t by t H1 2 X (s)22 ds. E X (t) = X (t) + 0
Now we establish the following result which will be useful in the proof of 2.5. Lemma 2.2. Assume that Hypothesis 1.4 holds and that f ∈ H . There exist K 0 = K 0 (D) and c = c(D) such that for any T ≤ 1 and any N ∈ N, we have
B0 √ H1 P sup E X N (·,x0 ) > K 0 ≤ c 1 + T, K0 (0,T ) provided x0 2 ≤ B0 T .
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Let X N = X N (·, x0 ). Ito Formula on X N 2 gives d X N 2 + 2 X N 22 dt = d MH1 + IH1 dt + PN φ(X N )2L2 (H ;H1 ) dt + I f dt, where ⎧ ⎨ I = −2 (AX N , B(X N )) , I f = 2 (AX N , f ) , H1 ⎩ M (t) = 2 t (AX (s), φ(X (s))dW (s)) . H1
N
0
(2.6)
N
Combining a Hölder inequality, an Agmon inequality and an arithmetico-geometric inequality gives 3 3 1 IH1 ≤ 2 X N 2 |X N |∞ X N ≤ c X N 22 X N 2 ≤ X N 22 + c X N 6 . (2.7) 4 Similarly, using Poincaré inequality and Hypothesis 1.4, 1 1 I f ≤ X N 22 + c | f |2 ≤ X N 22 + cB0 . (2.8) 4 4 We deduce from (2.6), (2.7), (2.8), Hypothesis 1.4 and Poincaré inequality that d X N 2 + X N 22 dt ≤ d MH1 + cB0 dt + c X N 2 X N 4 − 4K 02 dt, (2.9)
where K0 = Setting
μ1 . 8c
(2.10)
σH1 = inf t ∈ (0, T ) X N (t)2 > 2K 0 ,
we infer from x0 2 ≤ B0 T that for any t ∈ (0, σH1 ), 1 EH X N (t) ≤ cB0 T + MH1 (t).
(2.11)
We deduce from Hypothesis 1.4 and from Poincaré inequality that φ(x)∗ A is bounded in L(H1 ; H1 ) by cB0 . It follows that for any t ∈ (0, σH1 ), t t PN φ(X N (s))∗ AX N (s)2 dt ≤ cB0 X N (s)2 ds ≤ 2cK 0 B0 T. MH1 (t) = 4 0
0
Hence a Burkholder-Davis-Gundy inequality gives √ E sup MH1 ≤ cE MH1 (σH1 ) ≤ c K 0 B0 T ≤ c(K 0 + B0 ) T . (0,σH1 )
It follows from (2.11) and T ≤ 1 that E
1 sup E H XN
(0,σH1 )
√ ≤ c(B0 + K 0 ) T ,
which yields, by a Chebyshev inequality, P
sup
(0,σH1 )
1 EH XN
> K0
B0 √ T. ≤c 1+ K0
1 Now, since sup(0,σH ) E H X N ≤ K 0 implies σH1 = T , we deduce Lemma 2.2. 1
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2.2. Estimate of the derivative of X N . Let N ∈ N and (x0 , h) ∈ (H2 )2 . We are concerned with the following equation ⎧ ⎨ dη N + Aη N dt + PN B(X N , η N ) dt = PN (φ (X N ) · η N ) dW, ⎩ η (s, s, x ) · h = P h, N
0
(2.12)
N
where B(X N , η N ) = B(X N , η N ) + B(η N , X N ), X N = X N (·, x0 ) and η N (t) = η N (t, s, x0 ) · h for t ≥ s. Existence and uniqueness of the solutions of (2.12) are easily shown. Moreover if g ∈ Cb1 (PN H ), then, for any t ≥ 0, we have ∇ PtN g (x0 ), h = E (∇g(X N (t, x0 )), η N (t, 0, x0 ) · h) . (2.13) For any process X , we set σ (X ) = inf t ∈ (0, T )
t X (s)22 ds ≥ K 0 + 1 ,
(2.14)
0
where K 0 is defined in Lemma 2.2. We establish the following result: Lemma 2.3. Assume that Hypothesis 1.4 holds and that f ∈ H . Then there exists c = c(B0 , D) such that for any N ∈ N, T ≤ 1 and (x0 , h) ∈ (H2 )2 , σ (X N (·,x0 )) η N (t, 0, x0 ) · h23 dt ≤ c h22 . E 0
For better readability, we set η N (t) = η N (t, 0, x0 )·h and σ = σ (X N (·, x0 )). Ito Formula on η N (t)22 gives
2 d η N 22 + 2 η N 23 dt = d Mη N + Iη N dt + PN φ (X N ) · η N L
2 (U ;H2 )
dt, (2.15)
where ⎧
⎨ Mη (t) = 2 t A2 η N , (PN φ (X N ) · η N ) dW ds, N 0 3 1 ⎩ B(X N , η N ) . Iη N = −2 A 2 η N , A 2 It follows from Hölder inequalities, Sobolev Embedding and an arithmetico-geometric inequality Iη N ≤ c η N 3 η N 2 X N 2 ≤ η N 23 + c η N 22 X N 22 . Hence, we deduce from (2.15) and Hypothesis 1.4, d η N 22 + η N 23 dt ≤ d Mη N + c η N 22 X N 22 + B0 η N 22 dt. Integrating and taking the expectation, we obtain
σ E E(σ, 0) η N (σ )22 + E(σ, t) η N (t)23 dt ≤ h22 , 0
(2.16)
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123
where E(t, s) = e−B0 t−c
t s
X N (r )22 dr
.
Applying the definition of σ , we deduce σ η N (t)23 dt ≤ h22 exp (c(K 0 + 1) + B0 T ) , E
(2.17)
0
which yields Lemma 2.3. 2.3. Proof of (2.5). Let ψ ∈ C ∞ (R; [0, 1]) be such that ψ = 0 on (K 0 + 1, ∞)
and
ψ = 1 on (−∞, K 0 ).
For any process X , we set
ψX = ψ 0
Remark that
T
X (s)22 ds
.
E g(X N (T, x02 )) − g(X N (T, x01 )) ≤ I0 + |g|∞ (I1 + I2 ) ,
(2.18)
where ⎧ 1 ))ψ ⎨ I0 = E g(X N (T, x 2 ))ψ − g(X (T, x 2 1 N X N (·,x0 ) X N (·,x0 ) , 0 0 ⎩I = P T X N (s, x i )2 ds > K 0 . i 0 2 0 For any θ ∈ [1, 2], we set ⎧ ⎨ x θ = (2 − θ )x 1 + (θ − 1)x 2 , X θ = X N (·, x θ ), 0 0 0 0 ⎩ η (t) = η (t, 0, x θ ), σ = σ (X ). θ
N
0
θ
θ
Recall that σ was defined in (2.14). For better readability, the dependence on N has been omitted. Setting h = x02 − x01 , we have
2
I0 ≤ 1
|Jθ | dθ Jθ = ∇E g(X θ (T ))ψ X θ , h .
(2.19)
To bound Jθ , we apply a truncated Bismut-Elworthy-Li formula (see Appendix 4) Jθ =
1 J + 2Jθ,2 , T θ,1
(2.20)
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C. Odasso
where ⎧ σθ −1
= E g(X (T ))ψ ⎪ Jθ,1 ⎪ θ X θ 0 (φ (X θ (t)) · ηθ (t) · h, dW (t)) , ⎪ ⎨ σθ
t = E g(X (T ))ψ Jθ,2 1 − (t), A(η (t) · h)) dt , (AX θ θ θ Xθ 0 T ⎪ ⎪ ⎪ ⎩ ψ = ψ T X (s)2 ds . θ 2 X 0 It follows from Hölder inequality that ! ! σθ 2 J ≤ |g|∞ ψ X θ (t)2 dt E E θ,2 ∞ 0
σθ 0
ηθ (t) · h22 dt
and from Hypothesis 1.4 that ! J ≤ |g|∞ B0 E θ,1
σθ 0
ηθ (t) · h23 dt.
Hence for any T ≤ 1, |Jθ | ≤ c(B0 , D) |g|∞
1 T
! E
σθ 0
ηθ (t) · h23 dt.
(2.21)
Combining (2.21) and Lemma 2.3, we obtain |Jθ | ≤ c(B0 , D) |g|∞
h2 , T
which yields, by (2.4) and (2.19), I0 ≤ c(B0 , D) |g|∞
√
T.
Since B0 T 3 ≤ B0 T , we can apply Lemma 2.2 to control I1 + I2 in (2.18) if (2.4) holds. Hence (2.5) follows provided (2.4) holds, which yields Proposition 2.1. 3. Time of Return in a Small Ball of H2 Assume that Hypothesis 1.4 holds. Let N ∈ N and T, δ, Z 1 , Z 2 be as in Proposition 2.1. ) be a couple of independent cylindrical Wiener processes on H . We denote Let (W, W , respectively. by X N (·, x0 ) and X N (·, x0 ) the solutions of (1.3) associated to W and W We build a couple of random variables (V1 , V2 ) = (V1 (x01 , x02 ), V2 (x01 , x02 )) on PN H as follows ⎧ ⎪ (X (·, x0 ), X N (·, x0 )) if x01 = x02 = x0 , ⎪ ⎪ ⎨ N (V1 , V2 ) = (Z 1 (x01 , x02 ), Z 2 (x01 , x02 )) if (x01 , x02 ) ∈ BH2 (0, δ)\{x01 = x02 }, (3.1) ⎪ ⎪ ⎪ ⎩ (X (·, x 1 ), 2 N 0 X N (·, x 0 )) otherwise.
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125
We then build (X 1 , X 2 ) by induction on T N. Indeed, we first set X i (0) = x0i for i = 1, 2. Then, assuming that we have built (X 1 , X 2 ) on {0, T, 2T, . . . , nT }, we take (V1 , V2 ) as above independent of (X 1 , X 2 ) and we set X i ((n + 1)T ) = Vi (X 1 (nT ), X 2 (nT ))
for i = 1, 2.
It follows that (X 1 , X 2 ) is a discrete strong Markov process and a coupling of (D(X N (·, x01 )), D(X N (·, x02 ))) on T N. Moreover, if (X 1 , X 2 ) are coupled at time nT , then they remain coupled for any time after. We set 2 2 τ = inf t ∈ T N\{0} X 1 (t) ∨ X 2 (t) ≤ δ . (3.2) 2 2 The aim of this section is to establish the following result. Proposition 3.1. Assume that Hypothesis 1.4 holds. There exist δ 3 = δ 3 (B0 , D, ε, δ), α = α(φ, D, ε, δ) > 0 and K = K (φ, D, ε, δ) such that for any (x01 , x02 ) ∈ H × H and any N ∈ N,
E e
ατ
2 2 ≤ K 1 + x01 + x02 ,
provided f 2ε ≤ δ 3 . The result is based on the fact that, in the absence of noise and a forcing term, all solutions go to zero exponentially fast in H . A similar idea is used for the two-dimensional Navier–Stokes equations in [23]. The proof is based on the following four lemmas. The first one allows to control the probability that the contribution of the noise is small. Its proof strongly uses the assumption that the noise is diagonal in the eigenbasis of A. As already mentioned, in the additive case, the proof is easy and does not need this assumption. Lemma 3.2. Assume that Hypothesis 1.4 holds. For any t, M > 0, there exists p0 (t, M) = p0 (t, M, ε, (|φn |∞ )n , D) > 0 such that for any adapted process X , P
sup Z 22 (0,t)
≤M
≥ p0 (t, M),
where
t
Z (t) =
e−A(t−s) φ(X (s))dW (s).
0
It is proved in Sect. 3.1. Then, using this estimate and the smallness assumption on the forcing term, we estimate the moment of the first return time in a small ball in H .Let δ3 > 0. We set 2 ∗ 1 τ L 2 = τ ∧ inf t ∈ T N X (t) ∨ X 2 (t) ≥ δ3 .
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C. Odasso
Lemma 3.3. Assume that Hypothesis 1.4 holds. Then, for any δ3 > 0, there exist C3 (δ3 ), C3 (δ3 ) and γ3 (δ3 ) such that for any (x01 , x02 ) ∈ (H2 )2 ,
2 2
γτ 2 3 E e L ≤ C3 1 + x01 + x02 , provided | f | ≤ C3 . The proof is postponed to Sect. 3.2. Then, we need to get a finer estimate in order to control the time necessary to enter a ball in stronger topologies. To prove the next two lemmas, we use an argument similar to one used in the determinist theory (see [35], Chap. 7). Lemma 3.4. Assume that Hypothesis 1.4 holds. Then, for any δ4 , there exist p4 (δ4 ) > 0, C4 (δ4 ) > 0 and R4 (δ4 ) > 0 such that for any x0 verifying |x0 |2 ≤ R4 , we have for any T ≤ 1, P X N (T, x0 )2 ≤ δ4 ≥ p4 , provided | f | ≤ C4 . The proof is postponed to Sect. 3.3. Lemma 3.5. Assume that Hypothesis 1.4 holds. Then, for any δ5 , there exist p5 (δ5 ) > 0, C5 (δ5 ) > 0 and R5 (δ5 ) > 0 such that for any x0 verifying x0 2 ≤ R5 and for any T ≤ 1, P X N (T, x0 )22 ≤ δ5 ≥ p5 , provided f ε ≤ C5 . The proof is postponed to Sect. 3.4. Proof of Proposition 3.1. We set δ5 = δ, δ4 = R5 (δ5 ), δ3 = R4 (δ4 ),
p4 = p4 (δ4 ),
p5 = p5 (δ5 ),
and δ 3 = C3 (δ3 ) ∧ C4 (δ4 ) ∧ C5 (δ5 ). By the definition of τ L 2 , we have 2 2 1 X (τ L 2 ) ∨ X 2 (τ L 2 ) ≤ R4 (δ4 ). We distinguish three cases.
p 1 = ( p 4 p 5 )2 ,
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2 2 The first case is X 1 (τ L 2 )2 ∨ X 2 (τ L 2 )2 ≤ δ, which obviously yields
P
2 min max X i (τ L 2 + kT ) ≤ δ
k=0,...,2 i=1,2
2
≥ p1 . X 2 (τ L 2 ), X 2 (τ L 2 )
(3.3)
We now treat the case x0 = X 1 (τ L 2 ) = X 2 (τ L 2 ) with x0 22 > δ. Combining Lemma 3.4 and Lemma 3.5, we deduce from the weak Markov property of X N that P X N (2T, x0 )22 ≤ δ ≥ p5 p4 , provided |x0 |2 ≤ R4 . Recall that, in that case, X 1 (τ L 2 + 2T ) = X 2 (τ L 2 + 2T ). Hence, since the law of X 1 (τ L 2 + 2T ) conditioned by (X 1 (τ L 2 ), X 2 (τ L 2 )) is D(X N (2T, x0 )), it follows
2 ≥ p4 p5 ≥ p1 , P max X i (τ L 2 + 2T ) ≤ δ X 1 (τ L 2 ), X 2 (τ L 2 ) i=1,2
2
and then (3.3). 2 2 The last case is X 1 (τ L 2 ) = X 2 (τ L 2 ) and X 1 (τ L 2 )2 ∨ X 2 (τ L 2 )2 > δ. In that case, (X 1 (τ L 2 + T ), X 2 (τ L 2 + T )) conditioned by (X 1 (τ L 2 ), X 2 (τ L 2 )) are independent. Hence, since the law of X i (τ L 2 + T ) conditioned by (X 1 (τ L 2 ), X 2 (τ L 2 )) = (x01 , x02 ) is D(X N (T, x0i )), it follows from Lemma 3.4 that
2 i 1 2 P max X (τ L 2 + T ) ≤ δ4 X (τ L 2 ), X (τ L 2 ) ≥ p42 . i=1,2
1
2 Then, we distinguish the three cases ( X i (τ L 2 + T )1 )i=1,2 in the small ball of H2 , equal or different and we deduce from Lemma 3.5 by the same method
P
2 i 1 2 ≥ p52 , min max X (τ L 2 + kT ) ≤ δ X (τ L 2 + T ), X (τ L 2 + T )
k=1,2 i=1,2
2
provided 2 max X i (τ L 2 + T ) ≤ δ4 .
i=1,2
1
Combining the two previous inequalities, we deduce (3.3) for the latter case. We have thus proved that (3.3) is true almost surely. Integrating (3.3), we obtain
P
2 min max X i (τ L 2 + kT ) ≤ δ ≥ p1 .
k=0,...,2 i=1,2
Combining Lemma 3.3 and (3.4), we conclude.
2
(3.4)
128
C. Odasso
3.1. Probability of having a small noise. We now establish Lemma 3.2.We deduce from Hölder inequality and from n μ−2 n < ∞ that Hypothesis 1.4 implies the following fact: for any ε0 ∈ (0, ε), there exists α ∈ (0, 1), a family (φ¯ n )n of measurable maps H → R and a family (bi )i of positive numbers such that ⎧ ⎨ φ(x) · en = bn φ¯ n (x)en , (3.5) ⎩ sup φ¯ (x) ≤ 1, B ∗ = μ1+ε0 (b )2(1−α) < ∞. n
x∈H
n
n
n
For simplicity we restrict our attention to the case t = 1. The generalization is easy. Remark that Z (t) = bn Z n (t)en , n
where
t
Z n (t) =
e−μn (t−s) φ¯ n (X (s))dWn (s), where W =
0
Wn en .
n
√ μn Z n 2 and from (3.5) that √ 2 P sup Z 22 ≤ B ∗ M ≥ P sup μn Z n ≤ M μεn0 (bn )−2α , ∀ n .
It follows from Z 22 =
2 n bn μn
(0,1)
(0,1)
(3.6)
Setting Wn (t) =
√ μn Wn
t μn
,
we obtain (Wn )n a family of independent brownian motions. Moreover we have √ μn Z n (t) = Z n (μn t), where Z n (t) =
0
t
e−(t−s) ψn (s)dWn (s), ψn (s) = φ¯ n
Hence, it follows from (3.6) that P sup
(0,1)
Z 22
∗
≤B M
≥P
s X . μn
2 ε −2α sup Z n ≤ M μn0 (bn ) , ∀n .
(0,μn )
= W (i + ·) − W (i) on (0, 1). We set Let Wn,i n n
⎧ ⎨0 Mn,i (t) = ⎩ 1∧t 0
if t ≤ 0, (s) es ψn (i + s)dWn,i
if t ≥ 0.
(3.7)
Exponential Mixing for the 3D Stochastic Navier–Stokes Equations
129
Remark that Z n (t) =
∞
e−(t−i) Mn,i (t − i),
i=1
which yields for any q ∈ N, sup Z n 2 ≤
(0,q)
e e−1
max
sup Mn,i .
i=0,...,q−1 (0,1)
(3.8)
) Remark that (Wn,k n,k is a family of independent brownian motions on (0, 1). It follows that (Mn,k )n,k are martingales verifying Mn,k , Mn ,k = 0 if (n, k) = (n , k ). Hence, combining a theorem by Dambis, Dubins and Schwartz (Theorem 4.6, p. 174 of [20]) and a theorem by Knight (Theorem 4.13, p. 179 of [20]), we obtain a family (Bn,k )n,k of independent brownian motions verifying Mn,k (t) = Bn,k ( Mn,k (t)). (3.9)
Remark 3.6. In the two previous theorems (Theorem 4.6, p. 174 and Theorem 4.13, p. 179 of [20]), it is assumed that P (< M > (∞) < ∞) = 0. However, as explained in Problem 4.7 of [20], the proof is easily adapted to the case P (< M > (∞) < ∞) > 0. Remarking that for any t ∈ (0, 1),
Mn,k (t) =
t
|ψn (k + s)|2 ds ≤ 1,
0
we deduce from (3.8) and (3.9) that for any q ∈ N∗ ,
e max sup Bn,i . sup Z n 2 ≤ e − 1 i=0,...,q−1 (0,1) (0,q) Hence it follows from (3.7) that P sup
(0,1)
Z 22
≤M
2 ε0 −2α ≥ P sup Bn,i ≤ cM μn (bn ) , ∀ n, ∀ i ≤ μn + 1 , (0,1)
2 1 where c = e−1 e B∗ . We deduce from the independence of (Bn,k )n,k that μn +1 " 2 ε0 −2α P cM μn (bn ) , P sup Z 2 ≤ M ≥ (0,1)
where
n∈N∗
2 P(d0 ) = P sup B1,1 ≤ d0 . (0,1)
Recall there exists a family (c p ) p such that 2 p E sup B1,1 ≤ cp. (0,1)
(3.10)
130
C. Odasso
It follows from Chebyshev inequality and from 1 − x ≥ e−ex for any x ≤ e−1 that for 1
any d0 ≤ d p = e−1 c p p , −p
P(d0 ) ≥ 1 − c p d0
Applying (3.10), we obtain for any p > 0 ⎛ c p 2 P sup Z 2 ≤ M ≥ C p (M) exp ⎝− p M (0,1) where
−p
≥ e−ec p d0 .
n>N( p,M)
⎞ μn + 1 2αp ⎠ bn , ε p μn0
(3.11)
⎧ ⎨ N ( p, M) = sup n ∈ N\{0} M μεn0 (bn )−2α ≤ d p ,
ε0 ⎩ C (M) = −2α μn +1 . p n≤N( p,M) P cM μn (bn )
Choosing p sufficiently high, we deduce from H0 that μn + 1 2αp bn ≤ C p < ∞, ε0 p μ n n which yields, by (3.11), that for any M > 0 and for p sufficiently high, c p 2 P sup Z 2 ≤ M ≥ C p (M) exp − p . M (0,1)
(3.12)
Remark that for any p, ε0 we have N ( p, M) < ∞. Moreover, it is well-known that for any d0 > 0, P(d0 ) > 0, which yields C p (M) > 0 and then Lemma 3.2. 3.2. Proof of Lemma 3.3. For simplicity in the redaction, we restrict our attention to the case f = 0. The generalisation is easy. Recall (1.5) c E |X N (t)|2 ≤ e−μ1 t |x0 |2 + B0 . μ1 Since (X 1 , X 2 ) is a coupling of (D(X N (·, x01 )), D(X N (·, x01 ))) on T N, we obtain
2 2 c 2 2 E X 1 (nT ) + X 2 (nT ) ≤ e−μ1 nT x01 + x02 + 2 B0 . (3.13) μ1 Since (X 1 , X 2 ) is a strong Markov process, it can be deduced that there exist C6 and γ6 such that for any x0 ∈ H ,
2 2 γ τ (3.14) E e 6 L 2 ≤ C6 1 + x01 + x02 , where τ L 2
= inf t ∈ T N\{0}
1 2 2 2 X (t) + X (t) ≥ 4cB0 .
Exponential Mixing for the 3D Stochastic Navier–Stokes Equations
131
Taking into account (3.14), a standard argument gives that, in order to establish Lemma 3.3, it is sufficient to prove that there exist ( p7 , T7 ) such that P |X N (t, x0 )|2 ≤ δ3 ≥ p7 (δ3 , t) > 0, (3.15) provided N ∈ N, t ≥ T7 (δ3 ) and |x0 |2 ≤ 4cB0 . We set t e−A(t−s) φ(X N (s))dW (s), Y N = X N − PN Z , Z (t) = 0
M = sup Z 22 . (0,t)
Assume that there exist M7 (δ3 ) > 0 and T7 (δ3 ) such that M ≤ M7 (δ3 )
implies
δ3 , 4
|Y N (t)|2 ≤
(3.16)
provided t ≥ T7 (δ3 ) and |x0 |2 ≤ 4cB0 . Then (3.15) results from Lemma 3.2 with δ3 . M = min M7 (δ3 ), 4 We now prove (3.16). Remark that d Y N + AY N + PN B(Y N + PN Z ) = 0. dt Taking the scalar product of (3.17) with Y N , it follows that d |Y N |2 + 2 Y N 2 = −2(Y N , B(Y N + PN Z )). dt Recalling that (B(y, x), x) = 0, we obtain
(3.17)
(3.18)
−2(Y N , B(Y N + PN Z )) = −2(Y N , (Y N , ∇)PN Z ) − 2(Y N , B(PN Z )). We deduce from Hölder inequalities and Sobolev embedding that −(z, (x, ∇)y) ≤ c z x y . Hence it follows from (3.18) that d |Y N |2 + 2 Y N 2 ≤ c Z 2 Y N + c Z Y N 2 , dt which yields, by an arithmetico-geometric inequality, 1 3 d |Y N |2 + 2 Y N 2 ≤ cM 2 Y N 2 + cM 2 . dt
It follows that M ≤
1 c2
implies 3 d |Y N |2 + Y N 2 ≤ cM 2 dt
Integrating, we deduce from |x0 |2 ≤ 4cB0 that |Y N (t)| ≤ 4ce 2
−μ1 t
B0 + c
on (0, t).
3
M2 μ1
(3.19)
.
Choosing t sufficiently large and M sufficiently small we obtain (3.16) which yields (3.15) and then Lemma 3.3.
132
C. Odasso
Remark 3.7. In order to avoid a lengthy proof, we have not split the arguments in several cases as in the proof of Proposition 3.1. The reader can complete the details. 3.3. Proof of Lemma 3.4. We use the decomposition X N = Y N + PN Z defined in Sect. 3.2 and set M = sup Z 22 . (0,T )
Integrating (3.19), we obtain for M satisfying the same assumption M ≤ 1 T
T
Y N (t)2 dt ≤
0
1 , c2
3 1 |x0 |2 + cM 2 , T
which yields, by a Chebyshev inequality,
3 T 2 λ t ∈ (0, T ) Y N (t)2 ≤ |x0 |2 + 2cM 2 ≥ , T 2 where λ denotes the Lebesgue measure on (0, T ). Setting 3 2 τH1 = inf t ∈ (0, T ) Y N (t)2 ≤ |x0 |2 + 2cM 2 T
(3.20)
,
we deduce from (3.20) and the continuity of Y N that Y N (τH )2 ≤ 2 |x0 |2 + 2cM 23 . 1 T
(3.21)
Taking the scalar product of 2 AY and (3.17), we obtain d Y N 2 + 2 Y N 22 = −2(AY N , B(Y N + PN Z )). dt
(3.22)
It follows from Hölder inequalities, Sobolev Embeddings and Agmon inequality that 1
−2(Ay, B(x, z)) ≤ c y2 z 2 z22 x , 1
where B(x, y) = (x, ∇)y + (y, ∇)x. Hence, we obtain by applying arithmetico-geometric inequalities ⎧ 3 3 ⎪ ≤ c Y N 22 Y N 2 ≤ 41 Y N 22 + c Y N 6 , ⎪ ⎪ −2(AY N , B(Y N )) ⎨ ⎪ ⎪ ⎪ ⎩
−2(AY N , B(PN Z ))
1
3
≤ c Y N 2 Z 2 Z 22 ≤ 3 2
1 4
Y N 22 + c Z 42 ,
B(Y N , PN Z )) ≤ c Y N 2 Y N 2 Z ≤ c Z Y N 22 . −2(AY N , 1
B(Y N , PN Z ) + B(PN Z ), it follows from Remarking that B(Y N + PN Z ) = B(Y N ) + 1 (3.22) that M ≤ 4c implies d Y N 2 + Y N 22 ≤ c Y N 2 Y N 4 − 4K 02 + cM 2 , dt
(3.23)
Exponential Mixing for the 3D Stochastic Navier–Stokes Equations
133
where K 0 is defined in (2.10). Let us set σH1 = inf t ∈ (τH1 , T ) Y N (t)2 > 2K 0 , and remark that on (τH1 , σH1 ), we have d Y N 2 + Y N 22 ≤ cM 2 . dt
(3.24)
Integrating, we obtain that Y N (σH )2 + 1
σH1
τH1
2 Y N (t)22 dt ≤ Y N (τH1 ) + cM 2 .
(3.25)
Combining (3.21) and (3.25), we obtain that, for M and |x0 |2 sufficiently small, Y N (σH )2 ≤ δ4 ∧ K 0 , 1 4 which yields σH1 = T . It follows that X N (T )2 ≤ δ4 ,
(3.26)
provided M and |x0 |2 sufficiently small. It remains to use Lemma 3.2 to get Lemma 3.4. 3.4. Proof of Lemma 3.5. It follows from (3.24) that T Y N (t)22 dt ≤ x0 2 + cM 2 , 0
1 provided M ≤ 4c and x0 2 + cM 2 ≤ K 0 . Applying the same argument as in the previous subsection, it is easy to deduce that there exists a stopping time τH2 ∈ (0, T ) such that Y N (τH )2 ≤ 2 x0 2 + cM 2 , (3.27) 2 2 T
provided M and x0 are sufficiently small. Taking the scalar product of (3.17) and 2 A2 Y N , we obtain 3 1 d Y N 22 + 2 Y N 23 = −2 A 2 Y N , A 2 B(Y N + PN Z ) . dt
(3.28)
Applying Hölder inequality, Sobolev Embeddings H2 ⊂ L ∞ and H1 ⊂ L 4 and an arithmetico-geometric inequality, we obtain 3 1 1 −2 A 2 y, A 2 B(x, y) ≤ c y3 x2 y2 ≤ y23 + c x42 + y42 . 4 Hence we deduce from (3.28) and from B(Y N + PN Z ) = B(Y N ) + B(Y N , PN Z ) + B(PN Z ), d Y N 22 + Y N 23 ≤ c Y N 22 (Y N 22 − 2K 1 ) + c Z 42 , (3.29) dt
134
C. Odasso
where K 1 is defined as K 0 in (2.10) but with a different c. We set σH2 = inf t ∈ (τH2 , T ) Y N (t)22 > 2K 1 . Integrating (3.29), we obtain Y N (σH )2 + 2 2
σH2 τH2
2 Y N (t)23 dt ≤ Y N (τH2 )2 + cM 2 .
Taking into account (3.27) and choosing x0 2 and M 2 sufficiently small, we obtain Y N (σH )2 ≤ δ ∧ K 1 . 2 2 4 It follows that σH2 = T and that X N (T )2 ≤ δ,
(3.30)
provided M and x0 sufficiently small, which yields (2.2). 4. Proof of Theorem 1.8 As already explained, Theorem 1.8 follows from Proposition 1.15. We now prove Proposition 1.15. Let (x01 , x02 ) ∈ (H2 )2 . Let us recall that the process (X 1 , X 2 ) is defined at the beginning of Sect. 3. Let δ > 0, T ∈ (0, 1) be as in Proposition 2.1 and τ defined in (3.2), setting 2 2 τ1 = τ, τk+1 = inf t > τk X 1 (t) ∨ X 2 (t) ≤ δ , 2
2
it can be deduced from the strong Markov property of (X 1 , X 2 ) and from Proposition 3.1 that
2 2
, E eατk+1 ≤ K E eατk 1 + X 1 (τk ) + X 2 (τk ) which yields, by the Poincaré inequality, ⎧ ⎨ E (eατk+1 ) ≤ cK (1 + 2δ)E (eατk ) , ⎩ E (eατ1 ) ≤ K 1 + x 1 2 + x 2 2 . 0 0 It follows that there exists K > 0 such that
2 2
ατ k k E e ≤ K 1 + x01 + x02 . Hence, applying the Jensen inequality, we obtain that, for any θ ∈ (0, 1),
2 2
E eθατk ≤ K θk 1 + x01 + x02 .
(4.1)
Exponential Mixing for the 3D Stochastic Navier–Stokes Equations
135
We deduce from Proposition 2.1 and from (3.1) that 1 P X 1 (T ) = X 2 (T ) ≤ , 4 provided (x01 , x02 ) is in the ball of (H2 )2 with radius δ. Setting k0 = inf k ∈ N X 1 (τk + T ) = X 2 (τk + T ) , it follows that k0 < ∞ almost surely and that
n 1 P (k0 > n) ≤ . 4
(4.2)
Let θ ∈ (0, 1). We deduce from Schwartz inequality that ∞ ∞ θ θ
E e 2 ατk0 = E e 2 ατn 1k0 =n ≤ P (k0 ≥ n) E eθατn . n=1
n=1
Combining (4.1) and (4.2), we deduce ∞ 2 2 θ K θ n ατ E e 2 k0 ≤ 1 + x01 + x02 . 2 n=0
Hence, choosing θ ∈ (0, 1) sufficiently small, we obtain that there exists γ > 0 nondependent on N ∈ N such that
2 2
γτ k 0 E e (4.3) ≤ 4 1 + x01 + x02 . Recall that if (X 1 , X 2 ) are coupled at time t ∈ T N, then they remain coupled for any time after. Hence X 1 (t) = X 2 (t) for t > τk0 . It follows
2 2 1 2 −γ nT 1 + x01 + x02 . P X (nT ) = X (nT ) ≤ 4e N )∗ δ , (P N )∗ δ ), we deduce from Since (X 1 (nT ), X 2 (nT )) is a coupling of ((PnT x01 x02 nT Lemma 1.14
2 2 ∗ N ∗ N −γ nT 1 + x01 + x02 , (4.4) PnT δx 1 − PnT δx 2 ≤ 4e 0
0
var
for any n ∈ N and any (x01 , x02 ) ∈ (H2 )2 . Recall that the existence of an invariant measure μ N ∈ P(PN H ) is justified in Sect. 1.3. Let λ ∈ P(H ) and t ∈ R+ . We set n = Tt and C = 4eγ T . Integrating N (x01 , x02 ) over ((Pt−nT )∗ λ) ⊗ μ N in (4.4), we obtain
∗ N |x0 |2 λ(d x) , λ − μ N ≤ Ce−γ t 1 + Pt var
which establishes (1.19).
H
136
C. Odasso
Appendix A. Proof of (2.20) For simplicity in the redaction, we omit θ and N in our notations. Remark that J = (∇E (g(X (T ))ψ X ) , h) = J1 + 2J2 ,
(A.1)
where ⎧ ⎨ J1 = E ((∇g(X (T )), η(T, 0) · h) ψ X ) , ⎩ J = E g(X (T ))ψ T (AX (t), A(η(t, 0) · h)) ds . 2 X 0 According to [29], let us denote by Ds F the Malliavin derivative of F at time s. We have the following formula of the Malliavin derivative of the solution of a stochastic differential equation Ds X (t) = 1t≥s η(t, s) · φ(X (s)), which yields
t
Ds X (t) · m(s) ds = G(t) · m,
(A.2)
0
where
t
G(t) · m =
η(t, s) · φ(X (s)) · m(s) ds.
0
The uniqueness of the solutions gives η(t, 0) · h = η(t, s) · (η(s, 0) · h) for any 0 ≤ s ≤ t, which yields η(T, 0) · h =
1 T
T
η(t, s) · (η(s, 0) · h) ds.
0
Setting w(s) = φ −1 (X (s)) · η(s, 0) · h, we infer from (A.2) η(T, 0) · h =
1 1 G(T ) · w = T T
T
Ds X (T ) · w ds,
0
which yields 1 (∇g(X (T )), η(T, 0) · h) = T
T
(∇g(X (T )), Ds X (T ) · w) ds.
0
Remark that (Ds g(X (T )), w) = (∇g(X (T )), Ds X (T ) · w) .
(A.3)
Exponential Mixing for the 3D Stochastic Navier–Stokes Equations
137
It follows
which yields J1 =
1 E T
T
1 T
(∇g(X (T )), η(T, 0) · h) =
T
(Ds g(X (T )), w) ds,
0
ψ X (Ds g(X (T )), w) ds.
(A.4)
0
Recall that the Skohorod integral is the dual operator of the Malliavin derivative (see [29]). It follows
T 1 ψ X (w(t), dW (t)) . (A.5) J1 = E g(X (T )) T 0 Recall the formula of integration of a product
T
ψ X (w(t), dW (t)) = ψ X
0
T
T
(w(t), dW (t)) −
0
(Ds ψ X , w(s)) ds.
(A.6)
0
Remark that Ds ψ X = 2ψ X
T
ADs X (t) · (AX (t)) dt,
0
which yields, by (AX (t), ADs X (t) · w(s)) = (w(s), ADs X (t) · (AX (t))), 0
T
(Ds ψ X , w(s)) ds = 2ψ X
T
0
T
(AX (t), ADs X (t) · w(s)) dtds.
0
We deduce from (A.3) that 0
T
(Ds ψ X , w(s)) ds = 2ψ X
T
t (AX (t), Aη(t, 0) · h) dt.
(A.7)
0
Remark that ψ X = ψ X = 0
if
σ < T.
Hence combining (A.6) and (A.7), we obtain
T 0
ψ X (w(t), dW (t)) = ψ X 0
σ
(w(t), dW (t)) − 2ψ X
Thus, (2.20) follows from (A.1) and (A.5).
σ 0
t (AX (t), Aη(t, 0) · h) dt.
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References 1. Bricmont, J., Kupiainen, A., Lefevere, R.: Exponential mixing for the 2D Stochastic Navier–Stokes Dynamics. Commun. Math. Phys. 230(1), 87–132 (2002) 2. Constantin, P., Foias, C.: Navier–Stokes Equations. University of Chicago Press, Chicago, IL (1988) 3. Da Prato, G., Debussche, A.: Ergodicity for the 3D stochastic Navier–Stokes equations. J. Math. Pures Appl. (9) 82(8), 877–947 (2003) 4. Da Prato, G., Zabczyk, J.: Stochastic equations in infinite dimensions. Encyclopedia of Mathematics and its Applications. Cambridge: Cambridge University Press, 1992 5. Da Prato, G., Zabczyk, J.: Ergodicity for Infinite Dimensional Systems. London Mathematical Society Lecture Notes, 229,Cambridge: Cambridge University Press, 1996 6. Debussche, A., Odasso, C.: Ergodicity for the weakly damped stochastic Non-linear Schrödinger equations.. J.Evolution Eqs. 5(3), 317–356 (2005) 7. Debussche, A., Odasso, C.: Markov solutions for the 3D stochastic Navier–Stokes equations with state dependent noise. J. Evol. Eq. 6(2), 305–324 (2006) 8. E, W, Mattingly, J.C., Sinai, Y.G.: Gibbsian dynamics and ergodicity for the stochastically forced Navier–Stokes equation. Commun. Math. Phys. 224, 83–106 (2001) 9. Flandoli, F.: Irreducibilty of the 3-D stochastic Navier–Stokes equation. J. Funct. Anal. 149, 160– 177 (1997) 10. Flandoli, F.: An introduction to 3D stochastic Fluid Dynamics. CIME Lecture Notes, 2005 11. Flandoli, F., Gatarek, D.: Martingale and stationary solutions for stochastic Navier–Stokes equations. PTRF 102, 367–391 (1995) 12. Flandoli, F., Romito, M.: Partial regularity for the stochastic Navier–Stokes equations. Trans. Am. Math. Soc. 354(6), 2207–2241 (2002) 13. Flandoli, F., Romito, M.: Markov selections for the 3D stochastic Navier–Stokes equations. In preparation 14. Foias, C., Temam, R.: Gevrey class regularity for the solutions of the Navier–Stokes equations. J. Funct. Anal. 87(2), 359–369 (1989) 15. Hairer, M.: Exponential Mixing Properties of Stochastic PDEs Through Asymptotic Coupling. Prob. Theory Rel. Fields 124(3), 345–380 (2002) 16. Hairer, M., Mattingly, J.C.: Ergodic properties of highly degenerate 2D stochastic Navier–Stokes equations. C.R. Math. Acad. Sci. Paris 339(12), 879–882 (2004) 17. Hairer, M., Mattingly, J.: Ergodicity of the 2D Navier–Stokes equations with degenerate forcing. Preprint 18. Henshaw, W.D., Kreiss, H.O., Reyna, L.G.: Smallest scale estimate for the Navier–Stokes equations for incompressible fluids. Arch. Rat. Mech. Anal. 112(1), 21–44 (1990) 19. Huber, G., Alstrom, P.: Universal Decay of vortex density in two dimensions. Physica A 195, 448– 456 (1993) 20. Karatzas, I., Shreve, S.E.: Brownian motion and stochastic calculus. Second edition, Graduate Texts in Mathematics. 113, New York: Springer-Verlag, 1991 21. Kuksin, S.: On exponential convergence to a stationary mesure for nonlinear PDEs. The M. I. Viishik Moscow PDE seminar, Amer. Math. Soc. Trans. (2), 206,Providence, RI Amer. Math. Soc., 2002 22. Kuksin, S., Shirikyan, A.: Stochastic dissipative PDE’s and Gibbs measures, Commun. Math. Phys. 213, 291–330 (2000) 23. Kuksin, S., Shirikyan, A.: Ergodicity for the randomly forced 2D Navier–Stokes equations. Math. Phys. Anal. Geom. 4, 147–195 (2001) 24. Kuksin, S., Shirikyan, A.: A coupling approach to randomly forced randomly forced PDE’s I. Commun. Math. Phys. 221, 351–366 (2001) 25. Kuksin, S., Piatnitski, A., Shirikyan, A.: A coupling approach to randomly forced randomly forced PDE’s II. Commun. Math. Phys. 230(1), 81–85 (2002) 26. Kuksin, S., Shirikyan, A.: Coupling approach to white-forced nonlinear PDEs. J. Math. Pures Appl. 1, 567–602 (2002) 27. Mattingly, J.: Exponential convergence for the stochastically forced Navier–Stokes equations and other partially dissipative dynamics. Commun. Math. Phys. 230, 421–462 (2002) 28. Mattingly, J., Pardoux, E.: Ergodicity of the 2D Navier–Stokes Equations with Degenerate Stochastic Forcing. Preprint 2004 29. Nualart, D.: Malliavin Calculus and related topic. Probability and its Applications, 93BNewyork: Springer, 1995 30. Odasso, C.: Ergodicity for the stochastic Complex Ginzburg–Landau equations. to appear in Annales de l’institut Henri-Poincaré, Probabilités et Statistiques. 31. Odasso, C.: Exponential Mixing for Stochastic PDEs: The Non-Additive Case, preprint available on http://www.bretagne.ens-cachan.fr/math/people/cyril.odasso/ and http://arxiv.org/list/math.AP/0505502, 2005
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32. Shirikyan, A.: Exponential mixing for 2D Navier–Stokes equation pertubed by an unbounded noise. J. Math. Fluid Mech. 6(2), 169–193 (2004) 33. Shirikyan, A.: Analyticity of solutions of randomly perturbed two-dimensional Navier–Stokes equations. Usp. Mat. Nauk 57(4(346)), 151–166 (2002); translation in Russian Math. Surveys 57(4), 785–799 (2002) 34. Temam, R.: Navier–Stokes Equations. Theory and Numerical Analysis. Amsterdam-NewyorkOxford, North-Holland (1977) 35. Temam, R.: Navier–Stokes equations and nonlinear functional analysis. CBMS-NSF Regional Conference Series in Applied Mathematics, Philadelphia PA: SIAM , 1995 36. Temam, R.: Some developments on Navier–Stokes Equations in the second half of the 20th Century. In: Développement des Mathématiques au cours de la seconde moitié du XXème siécl, J.P. Pier, ed., Basel: Birkhauser Verlag, 2000 Communicated by G. Gallavotti
Commun. Math. Phys. 270, 141–161 (2007) Digital Object Identifier (DOI) 10.1007/s00220-006-0153-7
Communications in
Mathematical Physics
Lie Superalgebras and Irreducibility of A(11) –Modules at the Critical Level Dražen Adamovi´c Department of Mathematics, University of Zagreb, Bijeniˇcka 30, 10000 Zagreb, Croatia. E-mail:
[email protected] Received: 24 February 2006 / Accepted: 12 June 2006 Published online: 5 December 2006 – © Springer-Verlag 2006
Abstract: We introduce the infinite-dimensional Lie superalgebra A and construct a family of mappings from a certain category of A–modules to the category of A(1) 1 – modules at the critical level. Using this approach, we prove the irreducibility of a large (1) family of A1 –modules at the critical level parameterized by χ (z) ∈ C((z)). As a consequence, we present a new proof of irreducibility of certain Wakimoto modules. We also give natural realizations of irreducible quotients of relaxed Verma modules and calculate characters of these representations.
1. Introduction In the analysis of Fock space representations of infinite-dimensional Lie (super)algebras, one of the main problems is to prove irreducibility of these representations. Irreducible highest weight representations of affine Lie algebras at the critical level can be realized by using certain bosonic Fock representations, called the Wakimoto modules (cf. [FF1, FF2, FB, F, S, W]). Irreducibility of certain Wakimoto modules gave a very natural proof of the Kac-Kazhdan conjecture on characters of irreducible representations at the critical level (cf. [F, FF1, KK]). The category of representations at the critical level is much richer than the category O. So one can investigate the modules outside the category O and try to understand their structure. In particular one can investigate the relaxed Verma modules, their irreducible quotients and the corresponding characters. These representations have not been fully investigated and deserve more attention in the literature. However, such kinds of repre2 sentations appeared in the context of representation theory of the affine Lie algebra sl on non-critical levels (cf. [AM, FST]). In the present paper we shall demonstrate that the relaxed representations also appear naturally at the critical level. We will give a free Partially supported by the MZOS grant 0037125 of the Republic of Croatia
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field realization of the irreducible quotients of relaxed Verma modules at the critical 2 . level and calculate their characters in the case of the affine Lie algebra sl Outside the critical level, the representation theory of the affine Lie algebra A(1) 1 is related to the representation theory of the N = 2 superconformal algebra. Recall that the N = 2 superconformal algebra is the infinite-dimensional Lie superalgebra with basis L(n), H(n), G ± (r ), C, n ∈ Z, r ∈ 21 + Z and (anti)commutation relations given by C 3 (m − m)δm+n,0 , 12 1 C ± m − r G ± (m + r ), [H(m), H(n)] = mδm+n,0 , [L(m), G (r )] = 3 2 [L(m), H(n)] = −nH(n + m), [H(m), G ± (r )] = ±G ± (m + r ), C 1 r2 − δr +s,0 , {G + (r ), G − (s)} = 2L(r + s) + (r − s)H(r + s) + 3 4 [L(m), C] = [H(n), C] = [G ± (r ), C] = 0, {G + (r ), G + (s)} = {G − (r ), G − (s)} = 0 [L(m), L(n)] = (m − n)L(m + n) +
for all m, n ∈ Z, r, s ∈ 21 + Z. In [FST], the authors investigated the Kazama-Suzuki and anti-Kazama-Suzuki mappings which give rise to an equivalence of certain categories of representations 2 and an N = 2 superconformal algebra. In the context of vertex algebras these of sl mappings were considered in [A1]. But at the critical level the representation theory of 2 is very different from the representation theory outside the critical level. In particular, sl the associated vertex algebra N (−20 ) contains an infinite-dimensional center (cf. [F]). In the present paper we find an infinite-dimensional Lie superalgebra A with the 2 at the critical important property that its representation theory is related to that of sl level. This Lie superalgebra has generators S(n), T (n), G ± (r ), C, n ∈ Z, r ∈ 21 + Z, which satisfy the following relations: S(n), T (n), C are in the center of A, C {G (r ), G (s)} = 2S(r + s) + (r − s)T (r + s) + 3 {G + (r ), G + (s)} = {G − (r ), G − (s)} = 0 +
−
1 2 r − δr +s,0 , 4
for all n ∈ Z, r, s ∈ 21 + Z. The main difference between our algebra A and the N = 2 superconformal algebra is in the fact that A contains a large center and that it does not contain the Virasoro and Heisenberg subalgebra. Next we consider the vertex superalgebra V associated to a vacuum representation for A. This vertex superalgebra is introduced in Sect. 4 as a vertex subalgebra of F ⊗ M(0), where F is a Clifford vertex superalgebra and M(0) a commutative vertex algebra. Then following [A1] we show that there is a non-trivial vertex algebra homomorphism g : N (−20 ) → V ⊗ F−1 , where F−1 is a lattice vertex superalgebra associated to the lattice Zβ, β, β = −1. This result allows us to construct N (−20 )–modules from V–modules. Moreover, we prove that if U is an irreducible V–module satisfying a certain grading condition, then U ⊗ F−1 = ⊕s∈Z Ls (U ) is a completely reducible N (−20 )–module. Therefore every component Ls (U ) is an irre(1) ducible A1 –module at the critical level. It is important to notice that the irreducibility result is proved by using the theory of vertex algebras (cf. Lemma 6.1).
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143 (1)
In this way the problem of constructing irreducible A1 –modules is reduced to the construction of irreducible V–modules. But on irreducible V–modules, the action of the Lie superalgebra A can be expressed by the action of generators of infinite-dimensional Clifford algebras. By using this fact, in Sect. 5 we prove the irreducibility of a large family of V–modules. These modules are parameterized by χ (z) ∈ C((z)). In Sect. 6 we 2 modules construct mappings Ls which send irreducible V–modules to the irreducible sl at the critical level. As an application, in Sect. 7 we present a proof of irreducibility of a large family of the Wakimoto modules. In Sect. 8 we study the irreducible highest 2 –modules at the critical level. In particular, we study the simple vertex algeweight sl bra L(−20 ). In Sect. 9 we get realization of irreducible quotients of relaxed Verma modules. It turns out that these irreducible modules can be realized on a certain lattice type vertex algebra. 2. Vertex Algebra N(k0 ) We make the assumption that the reader is familiar with the axiomatic theory of vertex superalgebras and their representations (cf. [DL, FHL, FLM, LL, K2, Z]). In this section we recall some basic facts about vertex algebras associated to affine Lie algebras (cf. [FZ, Li1, MP]). Let g be a finite-dimensional simple Lie algebra over C and let (·, ·) be a nondegenerate symmetric bilinear form on g. Let g = n− + h + n+ be a triangular decomposition for g. The affine Lie algebra gˆ associated with g is defined as g ⊗ C[t, t −1 ] ⊕ CK , where K is the canonical central element [K1] and the Lie algebra structure is given by [x ⊗ t n , y ⊗ t m ] = [x, y] ⊗ t n+m + n(x, y)δn+m,0 K . We will write x(n) for x ⊗ t n . Let P = g ⊗ C[t] ⊕ CK be an upper parabolic subalgebra. For every k ∈ C, let Cvk be the 1–dimensional P–module such that the subalgebra g ⊗ C[t] acts trivially, and the central element K acts as multiplication by k ∈ C. Define the generalized Verma module N (k0 ) as N (k0 ) = U (ˆg) ⊗U (P) Cvk . Then N (k0 ) has a natural structure of a vertex algebra. The vacuum vector is 1 = 1⊗vk . The vertex algebra N (k0 ) has very rich representation theory. Let U be any g– module. Then U can be considered as a P–module. The induced gˆ –module N (k, U ) = U (ˆg) ⊗U (P) U is a module for the vertex algebra N (k0 ). Let N 1 (k0 ) be the maximal ideal in the vertex algebra N (k0 ). Then L(k0 ) = N (k0 ) is a simple vertex algebra. N 1 (k0 ) Let now g = sl2 (C) with generators e, f , h and relations [h, e] = 2e, [h, f ] = −2 f , [e, f ] = h. We fix the normalized Killing form (·, ·) on g such that (h, h) = 2. Let 0 , 1 ∈ (Ch ⊕CK )∗ be the fundamental weights for gˆ . For arbitrary λ ∈ (Ch ⊕CK )∗ , let L(λ) be the irreducible highest weight gˆ –module with highest weight λ. For s ∈ Z, we define H s = −s h2 . Then (H s , h) = −s. Define ∞ H s (n) H s (0) −n s (z) = z (−z) exp . −n n=1
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Applying the results obtained in [Li2] on N (k0 )–modules we get the following proposition. Proposition 2.1. Let s ∈ Z. For any N (k0 )–module (M, Y M (·, z)), s (πs (M), Y M (·, z)) := (M, Y M (s (z)·, z))
is a N (k0 )–module. πs (M) is an irreducible N (k0 )–module if and only if M is an irreducible N (k0 )–module. By definition we have: s (z)e(−1)1 = z −s e(−1)1, s (z) f (−1)1 = z s f (−1)1, s (z)h(−1)1 = h(−1)1 − skz −1 1. In other words, the corresponding automorphism πs of U (ˆg) satisfies the condition: πs (e(n)) = e(n − s), πs ( f (n)) = f (n + s), πs (h(n)) = h(n) − skδn,0 . In the case s = −1, one can see that π−1 (L((k − n)0 + n1 )) = L(n0 + (k − n)1 ), for every n ∈ Z≥0 . It is also important to notice the following important property: πs+t (M) ∼ = πs (πt (M)), (s, t ∈ Z). In particular, M∼ = π0 (M) ∼ = πs (π−s (M)). Lemma 2.1. Let x ∈ C. Assume that U is an irreducible N (k0 )–module which is generated by the vector vs (s ∈ Z) such that: e(n − s)vs = f (n + s + 1)vs = 0 (n ≥ 0), h(n)vs = δn,0 (x + ks)vs (n ≥ 0).
(2.1) (2.2)
Then U∼ = π−s (L((k − x)0 + x1 )). Proof. We consider the N (k0 )–module πs (U ). By construction, we have that πs (U ) is an irreducible highest weight gˆ –module with the highest weight (k − x)0 + x1 . Therefore, πs (U ) ∼ = L((k − x)0 + x1 ), which implies that U∼ = π−s (πs (U )) ∼ = π−s (L((k − x)0 + x1 )), and the lemma holds.
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3. Clifford Vertex Superalgebras The Clifford algebra C L is a complex associative algebra generated by
± (r ), r ∈
1 2
+ Z,
and relations { ± (r ), ∓ (s)} = δr +s,0 ; { ± (r ), ± (s)} = 0, where r, s ∈ 21 + Z. Let F be the irreducible C L–module generated by the cyclic vector 1 such that
± (r )1 = 0 for r > 0. A basis of F is given by
+ (−n 1 − 21 ) · · · + (−nr − 21 ) − (−k1 − 21 ) · · · − (−ks − 21 )1, where n i , ki ∈ Z≥0 , n 1 > n 2 > · · · > nr , k1 > k2 > · · · > ks . Define the following fields on F:
+ (n + 21 )z −n−1 , − (z) =
− (n + 21 )z −n−1 .
+ (z) = n∈Z
n∈Z
− (z)
+ (z)
The fields and generate on F the unique structure of a simple vertex superalgebra (cf. [Li1, K2, FB]). Define the following Virasoro vector in F: 1 + 3 − 1 ( (− 2 ) (− 2 ) + − (− 23 ) + (− 21 ))1. 2 ( f ) (n)z −n−2 Then the components of the field L ( f ) (z) = Y (ω( f ) , z) = n∈Z L defines on F a representation of the Virasoro algebra with central charge 1. Set J ( f ) (z) = Y ( + (− 21 ) − (− 21 )1, z) = J ( f ) (n)z −n−1 . ω( f ) =
n∈Z
Then we have [J ( f ) (n), ± (m + 21 )] = ± ± (m + n + 21 ). = Ker F − ( 1 ) be the subalgebra of the vertex superalgebra F generated by Let F 2 the fields −n + (n − 21 )z −n−1 and − (z) =
− (n + 21 )z −n−1 . ∂ + (z) = n∈Z
n∈Z
is a simple vertex superalgebra with basis Then F
+ (−n 1 − 21 ) · · · + (−nr − 21 ) − (−k1 − 21 ) · · · − (−ks − 21 )1, where n i , ki ∈ Z≥0 , n 1 > n 2 > · · · > nr ≥ 1, k1 > k2 > · · · > ks ≥ 0.
(3.3)
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Let F = Ker F − ( 21 ) ∩ Ker F + ( 21 ) be the subalgebra of the vertex superalgebra F generated by the fields ∂ + (z) =
n∈Z
−n + (n − 21 )z −n−1 and ∂ − (z) =
n∈Z
−n − (n − 21 )z −n−1 .
Then F is a simple vertex superalgebra with basis
+ (−n 1 − 23 ) · · · + (−nr − 23 ) − (−k1 − 23 ) · · · − (−ks − 23 )1,
(3.4)
where n i , ki ∈ Z≥0 , n 1 > n 2 > · · · > nr , k1 > k2 > · · · > ks . 4. The Vertex Superalgebra V In this section we shall define the vertex superalgebra V and study its representation theory. The vertex superalgebra V contains a large center. Moreover, the vertex superalgebra F is a simple quotient of V. Let M(0) = C[γ + (n), γ − (n) | n < 0] be the commutative vertex algebra generated by the fields γ ± (z) =
γ ± (n)z −n−1
n<0
± −n−1 ∈ C((z)). Let M(0, χ + , χ − ) denote the (cf. [F]). Let χ ± (z) = n∈Z χn z 1–dimensional irreducible M(0)–module with the property that every element γ ± (n) acts on M(0, χ + , χ − ) as multiplication by χn± ∈ C. Let now F be the vertex superalgebra generated by the fields ± (z) and γ ± (z). Therefore F = F ⊗ M(0). Denote by V the vertex subalgebra of the vertex superalgebra F generated by the following vectors τ ± = ( ± (− 23 ) + γ ± (−1) ± (− 21 ))1,
γ + (−1) − γ − (−1) j= 1, 2 2γ + (−1)γ − (−1) + γ + (−2) + γ − (−2) 1. ν= 4
(4.5) (4.6) (4.7)
Then the vertex superalgebra structure on V is generated by the following fields: G ± (z) = Y (τ ± , z) =
n∈Z
S(z) = Y (ν, z) =
G ± (n + 21 )z −n−2 ,
(4.8)
S(n)z −n−2 ,
(4.9)
T (n)z −n−1 .
(4.10)
n∈Z
T (z) = Y ( j, z) =
n∈Z
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Lie Superalgebras and Irreducibility of A1 –Modules
Denote by A the Lie superalgebra with basis S(n), T (n), G ± (r ), C, n ∈ Z, r ∈ and (anti)commutation relations given by
147 1 2
+Z
[S(m), S(n)] = [S(m), T (n)] = [S(m), G ± (r )] = 0, [T (m), T (n)] = [T (m), G ± (r )] = 0, [C, S(m)] = [C, T (n)] = [C, G ± (r )] = 0, {G + (r ), G − (s)} = 2S(r + s) + (r − s)T (r + s) + C3 (r 2 − 41 )δr +s,0 , {G + (r ), G + (s)} = {G − (r ), G − (s)} = 0
for all n ∈ Z, r, s ∈ 21 + Z. By using the commutator formulae for vertex superalgebras, we have that the components of fields (4.8)–(4.10) satisfy the (anti)commutation relation for the Lie superalgebra A such that the central element C acts as multiplication by c = −3. So the vertex superalgebra V is generated by the Lie superalgebra A. Thus we can study V–modules as modules for the Lie superalgebra A. The proof of the following proposition is standard. Proposition 4.1. We have: (1) V = U (A).1. (2) Assume that U is a V–module. Then U is an irreducible V–module if and only if U is an irreducible A–module. Let V com be the vertex subalgebra of V generated by the fields S(z) and T (z). V com is a commutative vertex algebra. The operator J f (0) acts semisimply on the vertex superalgebra V and defines the following Z–gradation: V=
V m , where
m∈Z
V
m
= {v ∈ V | J f (0)v = mv} = spanC {G + (−n 1 − 23 ) · · · G + (−nr − 23 )G − (−k1 − 23 ) · · · G − (−ks − 23 )w| w ∈ V com , n i , k j ∈ Z≥0 , r − s = m}. (4.11)
It is clear that V com ⊂ V 0 . Every U (A)-submodule of V becomes an ideal in the vertex superalgebra V. Let I com = U (A).V com be the ideal in V generated by V com . From the definition of vertex superalgebras V and F we get the following result. Proposition 4.2. The quotient vertex superalgebra V/I com is isomorphic to the simple vertex superalgebra F. Remark 4.1. In this paper we always assume that C acts as multiplication by c = −3. But one can also construct a vertex superalgebra, isomorphic to V, such that C acts as multiplication by any nonzero complex number. This vertex superalgebra can be constructed by using the theory of local fields (cf. [K2, Li1]).
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5. Irreducibility of Certain V–Modules In this section we shall consider a family of irreducible V–modules. For χ + , χ − ∈ C((z)) we set F(χ + , χ − ) := F ⊗ M(0, χ + , χ − ). Then F(χ + , χ − ) is a module for the vertex superalgebra V, and therefore for the Lie superalgebra A. Since M(0, χ + , χ − ) is one-dimensional, we have that as a vector space ( ± (−i − 21 ) | i ≥ 0). (5.12) F(χ + , χ − ) ∼ =F∼ = This actually shows that for every χ + , χ − ∈ C((z)) there exists a structure of an A–module on the vertex superalgebra F. In this section we shall use this identification. Proposition 5.1. Assume that λ ∈ C \ Z, p ∈ Z≥0 and that χ (z) =
∞
χ−n z n−1 ∈ C((z))
n=− p
satisfies the following conditions: χ p = 0, χ0 ∈ C \ Z if p = 0.
(5.13) (5.14)
Then F( λz , χ ) is an irreducible V–module. Proof. Since F( λz , χ ) is a V–module, it remains to prove that it is an irreducible module for the Lie superalgebra A. Recall (5.12). The A–module structure on F( λz , χ ) is uniquely determined by the following action of generators of A on F: G + (i − 21 ) = (λ − i) + (i − 21 ), G − (i − 21 ) = −i − (i − 21 ) +
∞
(5.15) χ−k − (k + i − 21 ).
(5.16)
k=− p
First we shall prove that the vacuum vector is a cyclic vector of the U (A)–action,i.e., U (A).1 = F.
(5.17)
Take an arbitrary basis element v = + (−n 1 − 21 ) · · · + (−nr − 21 ) − (−k1 − 21 ) · · · − (−ks − 21 )1 ∈ F,
(5.18)
where n i , ki ∈ Z≥0 , n 1 > n 2 > · · · > nr ≥ 0, k1 > k2 > · · · > ks ≥ 0. Let N ∈ Z≥0 be such that N ≥ k1 . By using (5.16) we get that G − ( p − N − 21 )· · ·G − ( p − 23 )G − ( p − 21 )1 = B − (−N − 21 ) · · · − (− 23 ) −(− 21 )1, where
B=
χ pN +1 χ0 (χ0 + 1) · · · (χ0 + N )
if p ≥ 1 . if p = 0
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So B = 0, and we have that
− (−N − 21 ) · · · − (− 23 ) − (− 21 )1 ∈ U (A).1. By using this fact and the action of elements G + (i − 21 ), i ∈ Z, we obtain that v ∈ U (A).1. In this way we proved (5.17). In order to prove irreducibility, it is enough to show that arbitrary basis element v of the form (5.18) is cyclic in F. This follows from (5.17) and the fact that G − (nr + p + 21 ) · · · G − (n 1 + p + 21 )G + (ks + 21 ) · · · G + (k1 + 21 )v = B 1, where the non-trivial constant B is given by B = (−1)r s (λ − k1 − 1) · · · (λ − ks − 1) · B and if p ≥ 1 χ rp . B = (χ0 − n 1 − 1) · · · (χ0 − nr − 1) if p = 0
This proposition has the following important consequence. Corollary 5.1. Assume that λ, μ ∈ C \ Z. Then F( λz , μz ) is an irreducible V–module. Now let χ (z) ∈ C((z)). Define : ⊗ M(0, 0, χ ). χ := F F χ is a submodule of the V–module F(0, χ ). Now we shall prove the It is clear that F following important irreducibility result: Proposition 5.2. Assume that p ∈ Z≥0 and that χ (z) =
∞
χ−n z n−1 ∈ C((z))
n=− p
satisfies the following conditions: χ p = 0, χ0 ∈ {1} ∪ (C \ Z) if p = 0.
(5.19) (5.20)
χ is an irreducible V–module. Then F χ is a V–module, it remains to prove that F χ is an irreducible module for Proof. Since F the Lie superalgebra A. The A–module structure on Fχ is uniquely determined by the following action of the Lie superalgebra A on F: G + (i − 21 ) = −i + (i − 21 ), G − (i − 21 ) = −i − (i − 21 ) +
(5.21) ∞
χ−k − (k + i − 21 ).
(5.22)
k=− p
and the same proof as that of By using this action, the basis description (3.3) of F, Proposition 5.1 we get the irreducibility result.
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Corollary 5.2. Assume that λ ∈ C\Z or λ = −1. Then F
λ −z
is an irreducible V–module.
Proposition 5.3. Assume that m, n ∈ Z≥0 . Then
F(− mz , − nz ) = Ker F(− m ,− n ) G − (m + 21 ) Ker F(− m ,− n ) G + (n + 21 ) z z z z + − 1 1 ∼ ( (−i − 2 ), (− j − 2 ) | i, j ∈ Z≥0 , i = m, j = n) = is an irreducible V–module. In particular, the V–module F(0, − nz ) is irreducible. Proof. First we notice that on F(− mz , − nz ) G + (i − 21 ) = −(i + m) + (i − 21 ), G − (i − 21 ) = −(i + n) − (i − 21 ). This implies that n ∼ + m ( (−i − 21 ), − (− j − 21 )|i, j ∈ Z≥0 , i = m, j = n) F(− , − ) = z z
(5.23)
(5.24)
is a V–submodule of F(− mz , − nz ). It remains to prove that F(− mz , − nz ) is an irreducible A–module. By using (5.23) we have that F(− mz , − nz ) has the structure of a module for the subalgebra m,n C L of the Clifford algebra C L generated by
+ (−i − 21 ), − (− j − 21 ), i, j ∈ Z, i = m, j = n.
Since ( + (−i − 21 ), − (− j − 21 ) | i, j ∈ Z≥0 , i = m, j = n) is an irreducible m,n C L–module, we have that F(− m , − n ) is an irreducible A–module.
z z 6. Vertex Algebras at the Critical Level In previous sections we investigated the properties of the vertex superalgebra V. This vertex superalgebra is generated by the Lie superalgebra A which is similar to the N = 2 superconformal algebra. This makes the vertex superalgebra V similar to the N = 2 vertex superalgebras investigated in [EG, FST] and [A1]. The main difference is that V does not contain the Virasoro and the Heisenberg subalgebra. On the other hand, V contains a large center. One important property of the N = 2 vertex superalgebras is 2 –vertex algebras. The most effective way for studytheir connection to the affine sl ing this connection is by using the Kazama-Suzuki and anti-Kazama-Suzuki mappings (cf. [KS, FST]). Motivated by the anti-Kazama-Suzuki mapping, we shall get a realiza2 –modules at the critical level. tion of the vertex algebras associated to the sl Let F−1 , be the lattice vertex superalgebra VZβ , associated to the lattice L = Zβ, where β, β = −1. As a vector space, F−1 is isomorphic to Mβ (1) ⊗ C[L], where Mβ (1) is a level one irreducible module for the Heisenberg algebra hˆ Z associated to the one dimensional abelian algebra h = L ⊗Z C and C[L] is the group algebra with a generator eβ . The generators of F−1 are eβ and e−β . Moreover, F−1 is a simple vertex superalgebra and a completely reducible Mβ (1)–module isomorphic to
m F−1 ∼ F−1 , = m∈Z
where
m F−1
is an irreducible Mβ (1)–module generated by emβ .
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We shall now consider the vertex superalgebra V ⊗ F−1 . Let Y be the vertex operator defining the vertex operator superalgebra structure on V ⊗ F−1 . For every v ∈ V ⊗ F−1 , let Y (v, z) = n∈Z vn z −n−1 . Define e = G + (− 23 )1 ⊗ eβ , (6.25) h = −2(1 ⊗ β(−1) − T (−1) ⊗ 1), (6.26) − −β 3 f = G (− 2 )1 ⊗ e . (6.27) For x ∈ spanC {e, f, h} set x(z) = Y (x, z) = n∈Z x(n)z −n−1 . Then the components of the fields e(z), f (z) and h(z) satisfy the commutation relations for the affine Lie 2 of level −2. In particular we have algebra sl β G + (i − 21 ) ⊗ en−i−1 , e(n) = i∈Z
h(n) = −2β(n) + 2T (n), −β G − (i − 21 ) ⊗ en−i−1 . f (n) = i∈Z
Denote by V the subalgebra of V ⊗ F−1 generated by e, f and h, i.e., V = spanC {u 1n 1 · · · u rnr (1 ⊗ 1)| u 1 , . . . , u r ∈ {e, f, h}, n 1 , . . . , nr ∈ Z, r ∈ Z≥0 }. As a U (ˆg)–module, V is a cyclic module generated by the vacuum vector 1 ⊗ 1. This implies that V is a certain quotient of the vertex algebra N (−20 ). So we have: Proposition 6.1. There exists a non-trivial homomorphism of vertex algebras g : N (−20 ) → V ⊗ F−1 , which is uniquely determined by (6.25)–(6.27). Define the vector H = + (− 21 ) − (− 21 )1 ⊗ 1 + 1 ⊗ β(−1) ∈ F ⊗ F−1 ⊂ F ⊗ F−1 .
(6.28)
Then the operator H (0) acts semisimply on vertex superalgebras F ⊗ F−1 and V ⊗ F−1 . Let W (s) = {v ∈ V ⊗ F−1 |H (0)v = sv}. Then we have the following decomposition:
W (s). V ⊗ F−1 ∼ = s∈Z
Let tˆ be the (commutative) Lie algebra generated by the components of the field T (z), and let MT (0) ∼ = C[T (−1), T (−2), . . . ] be the associated (commutative) vertex algebra. Let gˆ ext = gˆ ⊕ tˆ be the extension of the affine Lie algebra gˆ by the Lie algebra tˆ such that every element of tˆ is in the center of gˆ ext .
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Theorem 6.1. The vertex algebra W (0) is generated by e, f, h, j and W (0) = U (ˆgext ).(1 ⊗ 1) ∼ = V ⊗ MT (0). Proof. Let U be the vertex subalgebra of V ⊗ F−1 generated by the set {e, f, h, j}. The components of the fields e(z), f (z), h(z), T (z) span the Lie algebra gˆ ext . Since the field T (z) commutes with the action of gˆ we have that U = U (ˆgext ).(1 ⊗ 1) ∼ = V ⊗ MT (0). Since the operator H (0) acts trivially on the vertex algebra U, we conclude that U ⊂ W (0). We shall now prove that the vectors e, f, h, j generate W (0). First we notice that
m W (0) = V m ⊗ F−1 . (6.29) m∈Z
Since S(−2)1 ⊗ 1 = 41 (e(−1) f (−1) + f (−1)e(−1) + 21 h(−1)2 ).(1 ⊗ 1) − 21 T (−1)2 1 ⊗ 1, we have that S(−2)1 ⊗ 1 ∈ U. Therefore, V com ⊂ U.
(6.30)
We note that the following relations hold: G + (− 23 )1 ⊗ 1 = e(−2).(1 ⊗ e−β ), [e(m), (1 ⊗ e
−β
)n ] 3 G (− 2 )1 ⊗ 1 [ f (m), (1 ⊗ eβ )n ] −
= 0 for every m, n ∈ Z, β
= f (−2).(1 ⊗ e ),
= 0 for every m, n ∈ Z, 1 ⊗ β(−1) = − 21 (h(−1) − 2T (−1)).(1 ⊗ 1).
(6.31) (6.32) (6.33) (6.34) (6.35)
Relations (6.30) and (6.35) imply that 0 V com ⊗ F−1 = V com ⊗ Mβ (1) ⊂ U.
(6.36)
We shall now prove that V m ⊗ ⊂ U, where m ∈ Z. By using the description of V m from (4.11), we conclude that it is enough to show that elements of the form m F−1
v = G + (−n 1 − 23 ) · · · G + (−nr − 23 )G − (−k1 − 23 ) · · · G − (−ks − 23 )w ⊗ w (1) m w ∈ V com , w (1) ∈ F−1 , n i , k ∈ Z≥0 , r − s = m,
belong to U. By using (6.31) and (6.32) we get that (2) v= g˜i . G − (−k1 − 23 ) · · · G − (−ks − 23 )w ⊗ wi(2) i∈I (2)
(2)
−s for certain wi ∈ F−1 , g˜i ∈ U (Ce ⊗ C[t, t −1 ]) ⊂ U (ˆg) and a finite index set I . Similarly, applying (6.33) and (6.34) we get (3) (2) (3) G − (−k1 − 23 ) · · · G − (−ks − 23 )w ⊗ wi = g˜i, .(w ⊗ wi, ) (i ∈ I ) ∈Ii
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153
(3)
0 = M (1), g˜ −1 for certain wi, ∈ F−1 g), and a finite index β i, ∈ U (C f ⊗ C[t, t ]) ⊂ U (ˆ (3)
set Ii . Since w ⊗ wi, ∈ V com ⊗ Mβ (1), we obtain v ∈ U (ˆg).(V com ⊗ Mβ (1)) ⊂ U (ˆgext ).(1 ⊗ 1) = U. Therefore, m V m ⊗ F−1 ⊂ U for every m ∈ Z.
Now (6.29) implies that W (0) = U.
The next lemma follows form Corollary 4.2 of [DM]. This result will be our important tool in the irreducibility analysis. Lemma 6.1. Assume that M is an irreducible V ⊗ F−1 –module. Then for each 0 = w ∈ M, M is spanned as a V ⊗ F−1 –module by u n w, for u ∈ V ⊗ F−1 and n ∈ Z. Lemma 6.2. Assume that R is an irreducible W (0)–module. Then R is an irreducible V –module. In particular, R is an irreducible gˆ –module at the critical level. ∼ S ⊗ N , where S is an irreducible Proof. Since W (0) ∼ = V ⊗ MT (0), we have that R = V –module and N is an irreducible MT (0)–module. Since MT (0) is a commutative vertex algebra, we have that N is one dimensional and that every element T (n) acts on R as a scalar multiplication. Therefore R is irreducible as a V –module.
Theorem 6.2. Assume that U is a V–module such that U admits the following Z–gradation:
U= U j , V i .U j ⊂ U i+ j . (6.37) j∈Z
Then U ⊗ F−1 =
Ls (U ), where Ls (U ) :=
s∈Z
i∈Z
−s+i U i ⊗ F−1
is a W (0)–module. If U is irreducible, then for every s ∈ Z Ls (U ) is an irreducible V –module. Proof. Since V ⊗ F−1 = ⊕∈Z W (), relation (6.37) implies that W ().Ls (U ) ⊂ Ls+ (U ) for every , s ∈ Z.
(6.38)
This proves that Ls (U ) is a W (0)–module for every s ∈ Z. Assume now that U is irreducible. Then U ⊗ F−1 is an irreducible V ⊗ F−1 –module. Let 0 = v ∈ Ls (U ). Since U ⊗ F−1 is an irreducible V ⊗ F−1 –module, by Lemma 6.1 we get that U ⊗ F−1 = spanC {u n v | u ∈ V ⊗ F−1 , n ∈ Z}.
(6.39)
By using (6.38) and (6.39) we conclude that Ls (U ) = spanC {u n v | u ∈ W (0), n ∈ Z}.
(6.40)
So Ls (U ) is an irreducible W (0)–module. Now Lemma 6.2 gives that Ls (U ) is an irreducible V –module, and therefore an irreducible gˆ –module at the critical level.
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Corollary 6.1. Assume that U ⊂ F(χ + , χ − ) is an irreducible V–module. Then U ⊗ F−1 is a completely reducible V –module
U ⊗ F−1 = Ls (U ) s∈Z
and Ls (U ) = {v ∈ U ⊗ F−1 | H (0)v = sv} is an irreducible V –module. Moreover, Ls (U ) is an irreducible gˆ –module at the critical level. Proof. The operator J f (0) acts semisimply on U ⊂ F(χ + , χ − ) and it defines on U the following gradation:
U= U j , where U j = {u ∈ U | J f (0)u = ju}. j∈Z
Now Theorem 6.2 implies that Ls (U ) is an irreducible gˆ –module at the critical level.
2 at the Remark 6.1. The results obtained in this section show that representations of sl critical level are connected with representations of the vertex superalgebra V. One can also consider V as a certain limit (in the sense of [FB]) of vertex superalgebras associated to the N=2 superconformal algebra. Therefore we expect that the representation theory of general affine Lie (super)algebras at the critical level is connected with the theory of certain vertex superalgebras which are limits of W-algebras associated to general Kazama-Suzuki models. 7. Weyl Vertex Algebra and Irreducibility of the Wakimoto Modules In this section we will see that our gˆ –modules include the Wakimoto gˆ –modules at the critical level defined by using vertex algebra W associated to the Weyl algebra. As an application, we present a proof of irreducibility for a family of Wakimoto modules. ⊗ F−1 . The operator H (0) First we shall consider the simple vertex superalgebra F acts semismply on F ⊗ F−1 , and we have that Ker F⊗F −1 H (0) = L0 ( F) is a simple vertex algebra. We shall now identify this vertex algebra. Define: a := + (− 23 )1 ⊗ eβ , a ∗ := − − (− 21 )1 ⊗ e−β , and a(z) = Y (a, z) =
n∈Z
a(n)z −n−1 , a ∗ (z) = Y (a ∗ , z) =
(7.41)
a ∗ (n)z −n .
n∈Z
Then [a(n), a(m)] = [a ∗ (n), a ∗ (m)] = 0, [a(n), a ∗ (m)] = δn+m,0 . Therefore, the components of the fields a(z) and a ∗ (z) span the infinite dimensional ⊗ F−1 generated by a(z) and a ∗ (z) Weyl algebra. Let W be the vertex subalgebra of F (cf. [FMS, F, A2]). By using a similar proof to that of Theorem 6.1 we get.
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is generated by a and a ∗ . Thus we have: Proposition 7.1. The vertex algebra L0 ( F) ∼ L0 ( F). (7.42) W = For every χ ∈ C((z)), the vector space χ ⊗ F−1 ∼ ⊗ F−1 F =F χ ) ⊂ F ⊗ F−1 is a carries the structure of a gˆ –module at the critical level, and L0 ( F gˆ –submodule. By construction, we have that as a vector space L0 ( Fχ ) is isomorphic to ∼ L0 ( F) = W . Therefore for every χ ∈ C((z)) there exists a structure of a gˆ –module on the vertex algebra W . By using the definition of the gˆ –module structure on W and (7.41) one gets: e(z) = a(z), h(z) = −2 : a ∗ (z)a(z) : −χ (z), f (z) = − : a ∗ (z)2 a(z) : −2∂z a ∗ (z) − a ∗ (z)χ (z). Therefore the structure of a gˆ –module on the vertex algebra W coincides with the Wakimoto module W−χ (see [F] and references therein). Combining Proposition 5.2 and Corollary 6.1 we obtain the following irreducibility result. χ ) is isomorphic to the Theorem 7.1. (1) For every χ ∈ C((z)), the gˆ –module L0 ( F Wakimoto module W−χ . (2) Assume that χ (z) =
∞
χ−n z n−1 ∈ C((z)) ( p ∈ Z≥0 )
n=− p
χ ) satisfies the conditions (5.19) and (5.20) of Proposition 5.2. Then W−χ ∼ = L0 ( F is an irreducible gˆ –module at the critical level. 8. Construction of Irreducible Highest Weight Modules In this section we apply the results from previous sections and obtain a construction of all irreducible highest weight gˆ –modules at the critical level. It turns out that these modules are realized inside the Weyl vertex algebra W , and therefore they are submodules of certain Wakimoto modules. By using the methods developed in [A3], we shall identify modules obtained from irreducible highest weight modules by applying the automorphism πs . We will also show the vertex superalgebra F ⊗ F−1 is a completely reducible module for the simple vertex algebra L(−20 ). First we shall study non-generic highest weight representations. Theorem 8.1. (i) For every n ∈ Z≥0 the vector space F(0, − nz ) ⊗ F−1 carries a gˆ – module structure uniquely determined by β e(m) = − i + (i − 21 ) ⊗ em−i−1 , i∈Z
f (m) = −
i∈Z
−β
(i + n) − (i − 21 ) ⊗ em−i−1 ,
h(m) = −2β(m) + nδm,0 , K = −2,
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where m ∈ Z. Moreover, F(0, − nz ) ⊗ F−1 is a completely reducible gˆ –module and
F(0, − nz ) ⊗ F−1 ∼ πs (L(−(2 + n)0 + n1 )). = s∈Z
(ii) U (ˆg).(1 ⊗ 1) ∼ = L(n0 −(n + 2)1 ). = L(−(2 + n)0 +n1 ), U (ˆg).(1 ⊗ e−β ) ∼ Proof. By using Proposition 5.3 we have that for every n ∈ Z≥0 , F(0, − nz ) is an irreducible V–module. Then Corollary 6.1 gives that F(0, − nz ) ⊗ F−1 is a completely reducible gˆ –module isomorphic to ⊕s∈Z Ls (F(0, − nz )), where Ls (F(0, − nz )) is an irreducible gˆ – module. For every s ∈ Z, we set vs = 1 ⊗ e−sβ ∈ Ls (F(0, − nz )). Now using Lemma 2.1 one obtains that Ls (F(0, − nz )) = U (ˆg).vs = π−s (L(−(2 + n)0 + n1 )). This proves (i). The second assertion follows form (i) and from the fact that π−1 (L(−(2+ n)0 + n1 )) ∼
= L(n0 − (2 + n)1 ). When n ∈ Z>0 , then L(−(2 + n)0 + n1 ) is not a module for the simple vertex algebra L(−20 ). So Theorem 8.1 can be applied only in the framework of N (−20 )– modules. But when n = 0 we have that V–module F = F(0, 0) is a simple vertex superalgebra (see Proposition 4.2), and we have the following realization of L(−20 )– modules. Corollary 8.1. The simple vertex algebra L(−20 ) is a subalgebra of the vertex superalgebra F ⊗ F−1 , and we have the following decomposition of L(−20 )–modules:
F ⊗ F−1 ∼ πs (L(−20 )). = s∈Z
Proof. Since F ⊗ F−1 is a vertex superalgebra we have that Ker F⊗F−1 H (0) ∼ = L0 (F) ∼ = L(−20 ) is a vertex subalgebra of F ⊗ F−1 . Now the statement follows from Theorem 8.1.
By using Corollary 5.2 and the proof similar to that of Theorem 8.1 one obtains the following result. Theorem 8.2. For every λ ∈ {−1} ∪ C \ Z, the vector space F gˆ –module structure uniquely determined by : β e(m) = − i + (i − 21 ) ⊗ em−i−1 , i∈Z
f (m) = −
i∈Z
−β
(i + λ) − (i − 21 ) ⊗ em−i−1 ,
h(m) = −2β(m) + λδm,0 , K = −2,
λ −z
⊗ F−1 carries a
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where m ∈ Z. Moreover, F
λ −z
F
λ −z
157
⊗ F−1 is a completely reducible gˆ –module and
⊗ F−1 ∼ =
πs (L(−(2 + λ)0 + λ1 )).
s∈Z
9. Realization of Irreducible Modules on the Vertex Algebra (0) So far we studied irreducible gˆ –modules realized on the Weyl vertex algebra. In this section we shall see that there exists a family of irreducible gˆ –modules realized on a larger vector space. In order to construct new irreducible representations, we shall study the vertex algebra (0) = L0 (F) which contains the Weyl vertex algebra W as a subalgebra. We shall also identify irreducible gˆ –modules which are Z≥0 –graded but don’t belong to the category O. These modules are irreducible quotients of relaxed Verma modules studied in [FST]. First we recall that the boson-fermion correspondence gives that the fermionic vertex superalgebra F is isomorphic to the lattice vertex superalgebra VZ . Therefore, F ⊗ F−1 ∼ = VL = M(1) ⊗ C[L], where VL is the lattice vertex superalgebra associated to the lattice L = Zα + Zβ, α, α = −β, β = 1, α, β = 0. (As usual, M(1) is a level one irreducible module for the Heisenberg algebra hˆ Z associated to the abelian algebra h = L ⊗Z C and C[L] is the group algebra with generators eα and eβ .) The operator H from (6.28) coincides with α(−1) + β(−1). We conclude that the vertex algebra (0) = Ker F⊗F−1 (α + β)(0) = L0 (F) is isomorphic to the vertex algebra (0) ∼ = M(1) ⊗ C[Z(α + β)].
(9.43)
Remark 9.1. By using different methods, the vertex algebra (0) was also studied by E. Frenkel in [F] and by S. Berman, C. Dong and S. Tan in [BDT]. Now we shall apply the results from Sects. 5 and 6. The following result gives a construction of a large class of irreducible gˆ –modules on the vertex algebra (0). Combining Proposition 5.1 and Corollary 6.1 we obtain the following result: Theorem 9.1. Assume that λ ∈ C \ Z and that χ (z) =
∞
χ−n z n−1 ∈ C((z)) ( p ∈ Z≥0 )
n=− p
satisfies the conditions (5.13) and (5.14) of Proposition 5.1. Then on the vertex algebra (0) = L0 (F) the structure of an irreducible gˆ –module isomorphic to L0 (F( λz , χ )) exists.
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Define the following Virasoro vector in (0) ⊂ F ⊗ F−1 : ω = ω( f ) ⊗ 1 − 21 1 ⊗ β(−1)2 = 21 (α(−1)2 − β(−1)2 ).(1 ⊗ 1). Let L(z) = Y (ω, z) = n∈Z L(n)z −n−2 . Then the components of the field L(z) satisfies the commutation relations for the Virasoro algebra with central charge 2. Moreover, L(0) acts semisimply on F ⊗ F−1 with integer and half-integer eigenvalues, and it defines a Z≥0 –gradation on the vertex algebra (0):
(0) = (0)m , (0)m = {v ∈ (0) | L(0)v = mv}. (9.44) m∈Z≥0
Let ch (0) (q, z) = tr q L(0) z −2β(0) . By using relation (9.43) and the properties of the δ-function one can easily show the following result. Proposition 9.1. We have: ch (0) (q, z) = δ(z 2 ) = δ(z 2 )
∞
(1 − q n )−2
n=1 ∞
(1 − q n z 2 )−1 (1 − q n z −2 )−1 .
(9.45)
n=1
We are now interested in gˆ –modules from Theorem 9.1 such that the gˆ –action is compatible with the L(0)–gradation on the vertex algebra (0). But the action is compatible with the gradation if and only if χ (z) = μz for certain μ ∈ C. Therefore we should consider the irreducible V–module F( λz , μz ), where λ, μ ∈ C \ Z. The module F( λz , μz ) has the simple structure as an A–module. In fact, the action of the Lie superalgebra A is (up to a scalar factor) the same as the action of the Clifford algebra C L on F. When we apply Corollary 6.1, we get a family of irreducible gˆ –modules Ls (F( λz , μz )) at the critical level. Now we want to identify these irreducible gˆ –modules. λ μ For every s ∈ Z, we define a family of vectors w(s) j ∈ Ls (F( z , z )), by (s)
w0 := 1 ⊗ e−sβ , + + ( j−s)β 1 1 w (s) ( j ∈ Z>0 ), j := (− j + 2 ) · · · (− 2 )1 ⊗ e (s)
w− j := − (− j + 21 ) · · · − (− 21 )1 ⊗ e−( j+s)β ( j ∈ Z>0 ). By using a direct calculation, one can prove the following lemma. Lemma 9.1. Assume that s, j ∈ Z and n ∈ Z≥0 . Then we have (s) e(n − s).w (s) j = δn,0 (λ + j)w j+1 , (s)
(s)
h(n).w j = δn,0 (2 j − 2s + λ − μ)w j , (s)
(s)
f (n + s).w j = δn,0 (μ − j)w j−1 .
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Let us first consider the case s = 0. Define: E λ,μ := L0 (F( λz , μz )). By construction, E λ,μ ∼ = (0) as a vector space. Introduce the gradation operator L(0) on the gˆ –module E λ,μ by using the vertex algebra gradation (9.44) on (0). So let E λ,μ =
E λ,μ (m), E λ,μ (m) = {v ∈ E λ,μ | L(0)v = mv}.
m∈Z≥0
Since [L(0), x(n)] = −nx(n) for every x ∈ g, we have that gˆ –action on E λ,μ is compatible with the gradation. In other words, E λ,μ is a gˆ ⊕ CL(0)–module. Lemma 9.1 shows that the top level (0)
E λ,μ (0) = spanC {w j | j ∈ Z} is an irreducible U (g)–module which is neither highest nor lowest weight with respect to g. Next we consider the general case. By using Lemma 9.1 we have that πs (Ls (F( λz , μz ))) is an irreducible Z≥0 –graded gˆ –module whose top level is isomorphic to E λ,μ (0). This proves that πs Ls (F( λ , μ )) ∼ = E λ,μ . Therefore, z
z
Ls (F( λz , μz )) ∼ = π−s (E λ,μ ). In this way we have proved the following result. Theorem 9.2. For every λ, μ ∈ C \ Z, the vector space F( λz , μz ) ⊗ F−1 ∼ = F ⊗ F−1 carries a gˆ –module structure uniquely determined by: e(m) =
β (λ − i) + (i − 21 ) ⊗ em−i−1 , i∈Z
−β (μ − i) − (i − 21 ) ⊗ em−i−1 , f (m) = i∈Z
h(m) = −2β(m) + (λ − μ)δm,0 , K = −2, where m ∈ Z. Moreover, F( λz , μz ) ⊗ F−1 is a completely reducible gˆ –module and F( λz , μz ) ⊗ F−1 ∼ =
πs (E λ,μ ).
s∈Z
Although modules E λ,μ don’t belong to the category O, one can investigate their characters. The operators h(0) and L(0) act semisimply on E λ,μ with finite-dimensional common eigenspaces. Since we need the degree operator L(0) we shall consider E λ,μ as a module for the Lie algebra gˆ ⊕ CL(0).
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Corollary 9.1. For every λ, μ ∈ C \ Z, the vertex algebra (0) carries the structure of an irreducible Z≥0 –graded gˆ ⊕ CL(0)–module isomorphic to E λ,μ . We have the following character formulae: ch E λ,μ (q, z) = tr q L(0) z h(0) = z λ−μ δ(z 2 )
∞
(1 − q n z 2 )−1 (1 − q n z −2 )−1
n=1
= z λ−μ δ(z 2 )
∞
(1 − q n )−2 .
n=1
Proof. We already showed that E λ,μ is an irreducible Z≥0 –graded gˆ ⊕ CL(0)–module. By using (9.45) we get ch E λ,μ (q, z) = tr q L(0) z h(0) = z λ−μ ch(0) (q, z) = z λ−μ δ(z 2 )
∞
(1 − q n )−2 .
n=1
This proves the character formula.
Remark 9.2. Modules E λ,μ are irreducible quotients of certain relaxed Verma modules which are introduced and studied in [FST]. In our terminology, the relaxed Verma modules are N (−2, E λ,μ (0)) and they have the following character: ch N (−2,E λ,μ (0)) (q, z) = z λ−μ δ(z 2 )
∞
(1 − q n )−3 .
n=1
In Corollary 9.1 we calculate the characters of irreducible quotients of relaxed Verma modules by using our explicit realization. So our methods don’t use structure theory of relaxed Verma modules. Acknowledgements. We would like to thank the referees for their valuable comments.
References Adamovi´c, D.: Representations of the N = 2 superconformal vertex algebra. Internat. Math. Res. Notices 2, 61–79 (1999) [A2] Adamovi´c, D.: Representations of the vertex algebra W1+∞ with a negative integer central charge. Comm. Algebra 29(7), 3153–3166 (2001) (1) [A3] Adamovi´c, D.: A construction of admissible A1 –modules of level − 43 . J. Pure Appl. Algebra 196, 119–134 (2005) [AM] Adamovi´c, D., Milas, A.: Vertex operator algebras associated to the modular invariant representations (1) for A1 . Math. Res. Lett. 2, 563–575 (1995) [BDT] Berman, S., Dong, C., Tan, S.: Representations of a class of lattice type vertex algebras. J. Pure Appl. Algebra 176, 27–47 (2002) [DL] Dong, C., Lepowsky, J.: Generalized vertex algebras and relative vertex operators. Boston: Birkhäuser, (1993) [DM] Dong, C., Mason, G.: On quantum Galois theory. Duke Math. J. 86, 305–321 (1997) [EG] Eholzer, W., Gaberdiel, M. R.: Unitarity of rational N = 2 superconformal theories. Commun. Math. Phys. 186, 61–85 (1997) [FB] Frenkel, E., Ben-Zvi, D.: Vertex algebras and algebraic curves. Mathematical Surveys and Monographs, Vol. 88, Providence, RI: Amer. Math. Soc., 2001 [F] Frenkel, E.: Lectures on Wakimoto modules, opers and the center at the critical level. Adv. Math. 195, 297–404 (2005) [A1]
(1)
Lie Superalgebras and Irreducibility of A1 –Modules [FF1]
161
Feigin, B., Frenkel, E.: Representations of affine Kac–Moody algebras and bosonization. In: Physics and mathematics of strings, Singapore. World Scientific, 1990, pp. 271–316 [FF2] Feigin, B., Frenkel, E.: Affine kac-moody algebras and semi-infinite flag manifolds. Commun. Math. Phys. 128, 161–189 (1990) [FHL] Frenkel, I. B., Huang, Y.-Z., Lepowsky, J.: On axiomatic approaches to vertex operator algebras and modules. Mem. Amer. Math. Soc. 104, (1993) [FLM] Frenkel, I. B., Lepowsky, J., Meurman, A.: Vertex Operator Algebras and the Monster. Pure and Applied Math. Vol. 134, New York: Academic Press, 1988 [FMS] Friedan, D., Martinec, E., Shenker, S.: Conformal invariance, supersymmetry and string theory. Nucl. Phys. B 271, 93–165 (1986) [FST] Feigin, B. L., Semikhatov, A. M., Tipunin, I. Y.: Equivalence between chain categories of representations of affine sl(2) and N = 2 superconformal algebras. J. Math. Phys. 39, 3865–390 (1998) [FZ] Frenkel, I. B., Zhu, Y.: Vertex operator algebras associated to representations of affine and virasoro algebras. Duke Math. J. 66, 123–168 (1992) [K1] Kac, V.: Infinite dimensional Lie algebras. Third edition, Cambridge: Cambridge Univ. Press, 1990 [K2] Kac, V.: Vertex Algebras for Beginners. University Lecture Series. Volume 10, Second Edition, Providence, RI: Amer. Math. Soc., 1998 [KK] Kac, V., Kazhdan, D.: Structure of representations with highest weight of infinite dimensional lie algebras. Adv. Math. 34, 97–108 (1979) [KS] Kazama, Y., Suzuki, H.: New N = 2 superconformal field theories and superstring compactifications. Nucl. Phys. B 321, 232–268 (1989) [Li1] Li, H.: Local systems of vertex operators, vertex superalgebras and modules. J. Pure Appl. Algebra 109, 143–195 (1996) [Li2] Li, H.: The phyisical superselection principle in vertex operator algebra theory. J. Alg. 196, 436– 457 (1997) [LL] Lepowsky, J., Li, H.: Introduction to vertex operator algebras and their representations. Progress in Math. 227, Boston: Birkhäuser, 2004 ˜ [MP] Meurman, A., Primc, M.: Annihilating fields of standard modules of sl(2, C) and combinatorial identities. Mem. Amer. Math. Soc. 652, 1999 [S] Szczesny, M.: Wakimoto modules for twisted affine lie algebras. Math. Res. Lett. 9(4), 433– 448 (2002) (1) [W] Wakimoto, M.: Fock representations of affine lie algebra A1 . Commun. Math. Phys. 104, 605– 609 (1986) [Z] Zhu, Y.: Vertex operator algebras, elliptic functions and modular forms. Ph. D. thesis, Yale University, 1990; Modular invariance of characters of vertex operator algebras. J. Amer. Math. Soc. 9, 237–302 (1996) Communicated by Y. Kawahigashi
Commun. Math. Phys. 270, 163–196 (2007) Digital Object Identifier (DOI) 10.1007/s00220-006-0162-6
Communications in
Mathematical Physics
C2 /Zn Fractional Branes and Monodromy Robert L. Karp Department of Physics, Rutgers University, Piscataway, NJ 08854-8019, USA. E-mail:
[email protected] Received: 27 February 2006 / Accepted: 12 July 2006 Published online: 6 December 2006 – © Springer-Verlag 2006
Abstract: We construct geometric representatives for the C2 /Zn fractional branes in terms of branes wrapping certain exceptional cycles of the resolution. In the process we use large radius and conifold-type monodromies, and also check some of the orbifold quantum symmetries. We find the explicit Seiberg-duality which connects our fractional branes to the ones given by the McKay correspondence. We also comment on the Harvey-Moore BPS algebras. Contents 1. Introduction . . . . . . . . . . . . . . . . . . . 2. C2 /Z3 Geometries . . . . . . . . . . . . . . . . 2.1 The toric geometry of C2 /Z3 . . . . . . . . 2.2 The C2 /Z3 moduli space . . . . . . . . . . 3. C2 /Z3 Monodromies . . . . . . . . . . . . . . 3.1 Fourier-Mukai functors . . . . . . . . . . . 3.2 Monodromies in general . . . . . . . . . . 3.3 C2 /Z3 monodromies . . . . . . . . . . . . 3.4 The Z2 kernel squared: MZ2 ◦ MZ2 . . . . . 4. The C2 /Z3 Fractional Branes . . . . . . . . . . 4.1 Generating fractional branes . . . . . . . . 4.1.1 Computing MZ3 (i ∗ OC3 ). . . . . . . . . 4.1.2 Computing (MZ3 )2 (i ∗ OC3 ). . . . . . . 4.2 Consistency check . . . . . . . . . . . . . . 5. Generalizations . . . . . . . . . . . . . . . . . . 5.1 Connection with the McKay correspondence 5.1.1 The quiver. . . . . . . . . . . . . . . . 5.1.2 Seiberg duality. . . . . . . . . . . . . . 5.1.3 Quiver L E G O. . . . . . . . . . . . . .
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1. Introduction Understanding the physical properties of D-branes throughout the entire moduli space of a given Calabi-Yau compactification is an important and so far unsolved problem. Nevertheless much progress has been made in this direction. For N = 2 Type II compactifications, π -stability and derived categories seem to provide the most general framework so far, as it has been argued that topological B-branes are in one-to-one correspondence with the objects of the derived category of coherent sheaves on the Calabi-Yau variety [1]. For non-linear sigma-models on a large Calabi-Yau this has been carefully checked [2]. Since the Kahler deformations are exact in the topological B-model, one expects the derived category description of topological B-branes to be valid at any point of the moduli space. There is ample evidence for this by now. N = 2 Type II compactifications generically have a rich phase structure. The description of B-branes in the various phases is quite different, and the expected equivalence gives rise to interesting mathematical statements. The best known example of this sort is the celebrated McKay correspondence. This shifted the question from asking what the D-branes are at a given point in moduli space to asking which ones are stable, more precisely π -stable, and therefore physical. Determining the set of stable branes is cumbersome. The most workable method suggests to start at a point where one has a good understanding of stability, e.g., a large radius point, where π -stability reduces to μ-stability, and try to catalog what objects are lost and gained as the Kahler moduli are varied “adiabatically” [3]. Orbifolds provide a rich testing ground for these ideas. In this case D-branes can be described explicitly as boundary states in a solvable conformal field theory (CFT). But there is another description, originating from the world-volume theory of the D-brane probing the orbifold singularity, which is a quiver gauge theory [4]. In this language D-branes are objects in the derived category of representations of the quiver. The McKay correspondence gives an equivalence between this category and the derived category of coherent sheaves on the resolved space [5, 6]. As we said earlier, the McKay correspondence is a prototype of what happens in general: in different patches of the moduli space one has very different looking descriptions for the D-branes, which sit in inequivalent categories, but if one passes to the derived category then they become equivalent. Therefore it makes sense to talk about a geometric representation for a brane at any point in moduli space. Passing from an abelian category to the derived category is physically motivated by brane–anti-brane annihilation, thorough tachyon condensation [1, 7]. In the quiver representation language one can consider the simple representations, i.e., those that have no non-trivial subrepresentations. These correspond to fractional branes [8]. Their physical interpretation using π -stability is quite simple: at the orbifold point the space-filling D3-brane becomes marginally stable against decaying into the fractional branes. In fact this phenomenon is more general than orbifolds, and should apply to a D3-brane at any Calabi-Yau singularity. This has been understood in great detail for the conifold [9].
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Although the fractional branes are obvious in the quiver representation language, their geometric interpretation is quite unclear. The McKay correspondence tells us that there should be objects (bundles or perhaps complexes) on the resolved space whose Ext 1 -quiver is the one we started with. One of the central problems in this area is to find these objects. As a warm-up exercise one can try to determine the K-theory class of the fractional branes. So far, even this question has been answered only in a limited context, using mirror symmetry techniques [10] or the McKay correspondence.1 Our general goal is to get a deeper understanding of the geometry of fractional branes, going beyond K-theory. And we will do this without resorting to mirror symmetry or the McKay correspondence. Instead we use the quantum symmetry of an orbifold theory to generate the fractional branes as an orbit. In the present paper we investigate several collections of fractional branes for the C2 /Zn orbifolds, while the case of Cn /G will be investigated elsewhere [12]. Ultimately one would like to understand the world-volume theory of D-branes at an arbitrary point in the moduli space of a compact Calabi-Yau. Studying examples where one has at his disposal different methods hopefully will teach us the “mechanics” of the geometric approach. We hope to return to some phenomenologically more interesting examples, like [13] and [14], in the future. As a byproduct of the techniques developed in this paper, by the end, we will have performed some very strong consistency checks of the “D-brane derived category” picture. As we will see shortly, the functors implementing the monodromy transformations are not simple by any measure. The relations they satisfy would be very hard to guess without physical input. In the last part of the paper we use stacky methods to provide several collections of fractional branes. This relies on an extension of the McKay correspondence due to Kawamata. These branes are naturally associated to regions of the moduli space where we have no solvable CFT description, or a reliable supergravity approximation either. Fortunately, the algebro-geometric tools are powerful enough to deduce what we want. The organization of the paper is as follows. In Sect. 2 we use toric methods to investigate the geometry of the C2 /Z3 model together with its Kahler moduli space. Section 3 starts with a review of the Fourier-Mukai technology, and then it is applied to the various C2 /Z3 monodromies. Using the detailed structure of the moduli space we also prove the Z2 quantum symmetry at the Z2 point. To our knowledge, this is the first proof of this sort. In Sect. 4 we use the Z3 monodromy to produce a collection of fractional branes. In Sect. 5 this collection is compared to the one given by the McKay correspondence. This produces an interesting Seiberg duality, which allows us to extend our result to C2 /Zn in general. Then we turn to a collection of fractional branes on the partially resolved C2 /Zn orbifold, with the use of a generalization of the McKay correspondence by Kawamata. The partially resolved orbifold is singular, and it is not a global quotient either, hence it is particularly pleasing that we can handle it directly by geometric methods. We conclude with some thoughts on the algebra of BPS states and the superpotential. The appendixes contain some spectral sequences that are used endlessly throughout the paper.
1 For the ample physics literature on this subject see, e.g., [11] and references therein.
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Fig. 1. The toric fan for the resolution of the C2 /Z3 singularity
2. C2 /Z3 Geometries In this section we review some aspects of the C2 /Z3 geometric orbifold and the associated CFT. First we work out the relevant toric geometry of C2 /Z3 , then we turn our attention to the moduli space of complexified Kahler forms, and in particular its discriminant loci. We pay particular attention to the singularities in the moduli space. Most of the material in this section must be known to the experts, but we could not find suitable references for it. 2.1. The toric geometry of C2 /Z3 . The C2 /Z3 variety with the supersymmetric Z3 action (z 1 , z 2 ) → (ωz 1 , ω2 z 2 ),
ω3 = 1,
(1)
is toric, and a convenient representation for it is provided by the fan in Fig. 1. More precisely, the C2 /Z3 variety consists of only one cone, generated by the vertices v1 and v2 . In this figure we also included the divisors corresponding to the crepant resolution of the singularity. We denote the resolved space by X . The exceptional locus of the blow-up consists of the divisors corresponding to the rays v3 and v4 . Let us denote the divisors associated to vi by Di . As we will see shortly, in Eq. (4) below, D3 and D4 are both −2 curves. There are two linear equivalence relations among the divisors: 3D1 + D3 + 2D4 ∼ 0, −2D1 + D2 − D4 ∼ 0.
(2)
From the toric diagram in Fig. 1 one can immediately read out the intersection relations:2 D2 D3 = D1 D4 = 1,
D3 D4 = 1.
(3)
Using the linear equivalences (2) we obtain the expected results D32 = D42 = −2,
D2 D4 = D1 D3 = 0.
(4)
From the geometry it is also clear that the curves D3 and D4 are the generators of the Mori cone of effective curves. By Lemma 3.3.2 in [15] there is a bijection between 2 Since C2 /Z is non-compact, one restricts to the intersections of the compact cycles, in this case 3 D3 and D4 .
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Fig. 2. The phase structure of the C2 /Z3 model
the Mori cone generators and the generators of the lattice of relations of the point-set A = {v1 , . . . , v4 }. In particular, D3 and D4 yield the relations3 0 1 −2 1 Q= . (5) 1 0 1 −2 The Kahler cone is dual to the Mori cone, and in our case both are two dimensional. It is immediate from the intersection products in (4) that the ordered pair {D2 , D1 } is dual to the ordered pair {D3 , D4 }. The precise ordering will play an important role in what follows. Since we are in two complex dimensions, an irreducible divisor is a curve. This leads to potential confusion. To avoid it, we refer to the curves represented by the divisors Di as Ci , and these live in the second homology H2 (X, Z), while Di will refer to their Poincaré duals, which live in the second cohomology H2 (X, Z). In this notation we can rephrase the earlier result: {D2 , D1 } ∈ H2 (X, Z) is dual to {C3 , C4 } ∈ H2 (X, Z).
(6)
2.2. The C2 /Z3 moduli space. The point-set A = {v1 , . . . , v4 } admits four obvious triangulations. Therefore in the language of [16, 17] the gauged linear sigma model has four phases. The secondary fan has its rays given by the columns of the matrix (5), and is depicted in Fig. 2. The four phases are as follows: the completely resolved smooth phase, the two phases where one of the P1 ’s has been blown up to partially resolve the Z3 fixed point to a Z2 fixed point, and finally the Z3 orbifold phase. The Z2 phases corresponding to the cones C3 and C4 can be reached from the smooth phase C1 by blowing down the curves C3 resp. C4 . The discriminant locus of singular CFT’s can be computed using the Horn parametrization [18, 19]. We briefly review the general construction. Let us denote by Q = (Q ia ), i = 1, . . . , n, a = 1, . . . , k, the matrix of charges appearing in Cox’s holomorphic quotient construction of a toric variety [20]. The primary component of the discriminant, 3 In the gauged linear sigma model language the rows of this matrix represent the U(1) charges of the chiral superfields.
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0 , is a rational variety4 . Horn uniformization gives an explicit rational parametrization for 0 . Accordingly, we introduce k auxiliary variables, s1 , . . . , sk and form the linear combinations ξi =
k
Q ia sa ,
for all i = 1, . . . , n.
(7)
a=1
Let xa , a = 1, . . . , k be local coordinates on the moduli space of complex structures of the mirror. 0 then has the following parameterization: xa =
n
ξi
Q ia
,
for all a = 1, . . . , k.
(8)
i=1
In our context the matrix of charges in question is Q from Eq. (5). Let x3 and x4 be the local coordinates on the moduli space of complex structures of the mirror. One can use the mirror map so that x3 and x4 give coordinates on the Kahler moduli space of X as well. Applying the Horn uniformization equations (7) and (8) gives x3 =
s1 (s1 − 2s2 ) , (2s1 − s2 )2
x4 = −
s2 (2s1 − s2 ) . (s1 − 2s2 )2
(9)
4 Since in our case i=1 Q ia = 0, for both a = 1, 2 the above equations are homogeneous, and therefore x3 and x4 depend only on the ratio s1 /s2 . Eliminating s1 /s2 gives the sought after equation for 0 : 0 = 27x32 x42 − 18x3 x4 + 4x3 + 4x4 − 1.
(10)
In fact this is the only component of the discriminant. The discriminant curve itself is singular. It has a unique singular point at (x3 , x4 ) = (1/3, 1/3). To see the nature of the singularity we choose convenient coordinates around the singularity x3 =
1 1 + (y1 + y2 ), x4 = + (y1 − y2 ), 3 3
(11)
in terms of which the leading terms of 0 are5 3(y13 − y1 y22 ) + y22 .
(12)
Since the y1 y22 term is subleading compared to the other two, 0 has a cusp at (x3 , x4 ) = (1/3, 1/3).6 This will be important later on, in Sect. 3.4, since it allows one to have different monodromies around different parts of the primary component of the discriminant. The orbifold points in the moduli space are themselves singular points. This fact is related to the quantum symmetry of an orbifold theory. For either of the Z2 points, the homogenous coordinate ring construction of Cox [20] shows a C2 /Z2 singularity with weights (1, −1). Alternatively, using the “old” – group algebra C[σ∨ ] of the dual cone σ∨ – construction [22], one arrives at the affine scheme Spec C[y −1 , x 2 y, x] = Spec C[u, v, w]/(uv − w2 ). At the Z3 point the moduli space locally is of the form C2 /Z3 , with weights (1, 2). 4 Meaning that it is birational to a projective space. 5 We discarded an overall factor of 12. 6 A similar fact has been noted in [21].
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3. C2 /Z3 Monodromies We start this section with a brief review of Fourier-Mukai functors. Then we express the various monodromy actions on D-branes in terms of Fourier-Mukai equivalences. The remaining part of the section deals with rigorously proving an identity which is the analog in the Fourier-Mukai language of the Z2 quantum symmetry at a partially resolved point in the moduli space.
3.1. Fourier-Mukai functors. For the convenience of the reader we review some of the key notions concerning Fourier-Mukai functors, and at the same time specify the conventions used. We will make extensive use of this technology in the rest of the paper. Our notation follows [23]. Given two non-singular proper algebraic varieties, X 1 and X 2 , an object K ∈ D(X 1× X 2 ) determines a functor of triangulated categories K : D(X 1 ) → D(X 2 ) by the formula7 L K (A) := R p2∗ K ⊗ p1∗ (A) ,
(13)
where pi : X × X → X is projection to the i th factor: X 1 × XH2 HH p vv HH 2 v v HH v HH vv v $ v {
(14)
p1
X1
X 2.
The object K ∈ D(X 1 × X 2 ) is called the kernel of the Fourier-Mukai functor K . It is convenient to introduce the external tensor product of two objects A ∈ D(X 1 ) and B ∈ D(X 2 ) by the formula L
A B = p2∗ A ⊗ p1∗ B.
(15)
The importance of Fourier-Mukai functors when dealing with derived categories stems from the following theorem of Orlov:8 Theorem 3.1. Let X 1 and X 2 be smooth projective varieties. Suppose that F : D(X 1 ) → D(X 2 ) is an equivalence of triangulated categories. Then there exists an object K ∈ D(X 1 × X 2 ), unique up to isomorphism, such that the functors F and K are isomorphic. The first question to ask is how to compose Fourier-Mukai (FM) functors. Accordingly, let X 1 X 2 and X 3 be three non-singular varieties, while letting F ∈ D(X 1 × X 2 ) and G ∈ D(X 2 ×X 3 ) be two kernels. Let pi j : X 1 ×X 2 ×X 3 → X i ×X j be the projection map. A well-known fact is the following: 7 D(X ) denotes the bounded derived category of coherent sheaves on X . R p is the total right derived 2∗ L functor of p2∗ , i.e., it is an exact functor from D(X ) to D(X ). Similarly, ⊗ is the total left derived functor
of ⊗. In later sections these decorations will be subsumed. 8 Theorem 2.18 in [24]. The theorem has been generalized for smooth quotient stacks associated to normal projective varieties [25].
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Proposition 3.2. The composition of the functors F and G is given by the formula ∗ L ∗ G ◦ F H , where H = R p13∗ p23 (G) ⊗ p12 (F) .
(16)
Proposition 3.2 shows that composing two FM functors gives another FM functor, with a simple kernel. The composition of the kernels F and G ∈ D(X × X ) is therefore defined as ∗ L ∗ G F := R p13∗ p23 (G) ⊗ p12 (F) .
(17)
There is an identity element for the composition of kernels: δ∗ (O X ), where δ : X → X × X is the diagonal embedding. For brevity we will denote δ∗ (O X ) by O : O := δ∗ (O X ).
(18)
O = δ∗ (O X ) has the expected properties: O G = G O = G, for all G ∈ D(X × X ).
(19)
Finally, the functors 23 : D(X 1 × X 2 ) → D(X 1 × X 3 ), G23 ∈ D(X 2 × X 3 ), 23 (−) := G23 −, 12 : D(X 2 × X 3 ) → D(X 1 × X 3 ), G12 ∈ D(X 1 × X 2 ), 12 (−) := − G12 ,
(20)
are morphisms between triangulated categories, i.e., they preserve distinguished triangles. The composition of kernels is also associative G3 (G2 G1 ) ∼ = (G3 G2 ) G1 .
(21)
Now we have all the technical tools ready to study the monodromy actions of physical interest.
3.2. Monodromies in general. The moduli space of CFT’s contains the moduli space of Ricci-flat Kahler metrics. This, in turn, at least locally has a product structure, with the moduli space of Kahler forms being one of the factors. This is the moduli space of interest to us. In what follows we study the physics of D-branes as we move in the moduli space of complexified Kahler forms. This space is a priori non-compact, and its compactification consists of two different types of boundary divisors. First we have the large volume divisors. These correspond to certain cycles being given infinite volume. The second type of boundary divisors are the irreducible components of the discriminant. In this case the CFT becomes singular. Generically this happens because some D-brane (or several of them, even infinitely many) becomes massless at that point, and therefore the effective CFT description breaks down. For the quintic this breakdown happens at the well known conifold point [26]. The monodromy actions around the above divisors are well understood. We will need a more abstract version of this story, in terms of Fourier-Mukai functors acting on the derived category, which we now recall.9 9 For an extensive treatment of monodromies in terms of Fourier-Mukai functors see [27, 28].
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Large volume monodromies are shifts in the B field: “B → B + 1”. If the Kahler cone is higher dimensional, then we need to be more precise, and specify a two-form, or equivalently a divisor D. Then the monodromy becomes B → B + D. We will have more to say about the specific D’s soon. The simplest physical effect of this monodromy on a D-brane is to shift its charge, and this translates in the Chan-Paton language into tensoring with the line bundle O X (D). This observation readily extends to the derived category: Proposition 3.3. The large radius monodromy associated to the divisor D is L
L D (B) = B ⊗ O X (D),
for all B ∈ D(X ).
(22)
Furthermore, this is a Fourier-Mukai functor L , with kernel L = δ∗ O X (D),
(23)
where δ : X → X × X is again the diagonal embedding. Since we haven’t found a suitable reference for this statement, we are going to include its proof. This will also serve as a warm-up exercise in the techniques that we use later L
on. From now on for brevity we are going to suppress most decorations L, R and ⊗ from the derived functors during computations. Proof. All we need to show is that the Fourier-Mukai functor δ∗ O X (D) has the desired action. By its definition L δ∗ O X (D) (B) = p2∗ δ∗ O X (D) ⊗ p1∗ (B) .
(24)
Using the projection formula gives L δ∗ O X (D) (B) = p2∗ δ∗ O X (D) ⊗ δ ∗ p1∗ (B) .
(25)
But p2∗ δ∗ = ( p2 ◦ δ)∗ = 1X ∗ and δ ∗ p1∗ = ( p1 ◦ δ)∗ = 1X∗ , and this completes the proof. Now we turn our attention to the conifold-type monodromies. We will need the following conjecture from [27]:10 Conjecture 3.4. If we loop around a component of the discriminant locus associated with a single D-brane A (and its translates) becoming massless, then this results in a relabeling of D-branes given by the autoequivalence of the derived category D(X ), L B −→ Cone RHomD(X ) (A, B) ⊗ A −→ B . (26) This action is again of Fourier-Mukai type. Lemma 3.2 of [29] provides us with the following simple relation for any B ∈ D(X ): L ∼ (27) Cone(A∨ A→O) (B) = Cone RHomD(X ) (A, B) ⊗ A −→ B , 10 The conjecture goes back to Kontsevich, Morrison and Horja. We refer to [27] for more details.
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where for an object A ∈ D(X ) its dual by definition is A∨ = RHomD(X ) (A, O X ).
(28)
Since the functor Cone(A∨ A→O) will play a crucial role, we introduce a special notation for it: L TA := Cone(A∨ A→O) , TA (B) = Cone RHomD(X ) (A, B) ⊗ A −→ B . (29) The question of when is TA = Cone(A∨ A→O) an autoequivalence has a simple answer. For this we need the following definition: Definition 3.5. Let X be smooth projective Calabi-Yau variety of dimension n. An object E in D(X ) is called n-spherical if ExtrD(X ) (E, E) is equal to Hr (S n , C), that is C for r = 0, n and zero in all other degrees. One of the main results of [29] is the following theorem: Theorem 3.6 (Prop. 2.10 in [29]). If the object E ∈ Db (X ) is n-spherical then the functor TE is an autoequivalence. This brief review brings us to a point where we can apply this abstract machinery to study the C2 /Z3 monodromies, and eventually use them to construct the fractional branes. 3.3. C2 /Z3 monodromies. Now we have all the ingredients necessary for constructing the monodromy actions needed to generate the fractional branes. The toric fan for the moduli space of complexified Kahler forms was depicted in Fig. 2. The four maximal cones C1 , . . . , C4 correspond to the four distinguished phase points. The four edges correspond to curves in the moduli space. It is immediate to see these are all P1 ’s, at least topologically. For us the analytic structure will be important, and we need to be more careful here. The four curves connecting the different phase points are sketched in Fig. 3. The curves in question are weighted projective lines: L1 = L2 = P1 (1, 2), L3 = L4 = P1 (3, 2). Since the singularities are in codimension one, these spaces in fact are not singular, and they are all isomorphic to P1 . In terms of the coordinates (x3 , x4 ) introduced in Eq. (9) we have L1 : (x4 = 0) and L3 : (x3 = 0). The discriminant 0 intersects the four lines, and it is now clear from Eq. (10) that all intersections are transverse. We depicted this fact in Fig. 3 using short segments. When talking about monodromy there are two cases to be considered. One can loop around a divisor, i.e., real codimension two objects; or one can loop around a point inside a curve. Of course the two notions are not unrelated. Our interest will be the second type of monodromy: looping around a point inside a P1 . What we would like to write down is the monodromy inside L3 around the Z3 point. Since there is no direct approach to doing this, we follow an indirect way: both L1 and L3 are spheres, with three marked points, and we can compute the corresponding monodromies. Our approach is to go from the smooth point to the Z3 point by first “moving” inside L1 and then L3 . We start with L1 , which has three distinguished points: the smooth point, L1 ∩ 0 and the Z2 point. Monodromy around the smooth point inside L1 is a large radius monodromy, and (6) together with Prop. 3.3 tell us that it is precisely L D2 .
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Fig. 3. The moduli space of the C2 /Z3 model
L1 ∩ 0 is a conifold-type point. Following [26, 30] we know that it is the D-brane wrapping the shrinking cycle C3 that should go massless at this point. But the mass depends on the central charge, which in turn is only a function of the K-theory class. At the K-theory level this fact has been verified [31]. Conjecture 3.4 then tells us the monodromy: Ti∗ OC3 . Putting these two facts together we have the monodromy around the Z2 point inside L1 : MZ2 = Ti∗ OC3 ◦ L D2 . We can immediately evaluate the kernel of this Fourier-Mukai functor KZ2 = Cone (i ∗ OC3 )∨ i ∗ OC3 −→ O δ∗ O X (D 2) = Cone (i ∗ OC3 )∨ i ∗ OC3 (1) −→ δ∗ O X (D2 ) .
(30)
(31)
This expression will be useful when dealing with quantum symmetries. Now we can continue our march towards the Z3 point inside L3 . Once again there are three distinguished points: the Z3 point, L3 ∩ 0 and the Z2 point. By the same token as before, monodromy around L3 ∩ 0 is T j∗ OC4 . Of course we want the monodromy around the Z3 point, so we need the monodromy around this Z2 point as well, which a priori has nothing to do with the previous Z2 monodromy inside L1 . But the monodromy around the Z2 point is more subtle. Figure 3 in fact is quite misleading, since in reality the spheres L1 and L3 intersect transversely in 4-space. Moreover, the intersection point is an orbifold itself: C2 /Z2 . To see what happens, we need to work out the fundamental group of the complement of L1 and L3 .11 Since the intersection point is a C2 /Z2 orbifold, we surround it by the lens space L = S 3 /Z2 , instead of the usual sphere S 3 . L1 and L3 are both smooth curves and therefore intersect L in unknotted circles. This way we reduced the problem of computing π1 (C2 /Z2 − {L1 ∪ L3 }) to computing = π1 (L − {L1 ∪ L3 }). To evaluate this consider the covering map q : S 3 → L, with free Z2 action, induced by the Z2 action on C2 . The intersection of both L1 and L3 with L lift under q −1 to unknotted circles in 11 This situation is similar to the one analyzed in [21].
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S 3 . These circles are linked once and thus π1 (S 3 − {q −1 (L1 ) ∪ q −1 (L3 )}) = Z ⊕ Z.12 The generators are the loops around q −1 (L1 ) and q −1 (L3 ), we call them g1 and g2 . Since q is a normal cover we have a short exact sequence of abelian groups 0
/Z⊕Z
/
/ Z2
0/ .
(32)
We can easily show that = Z ⊕ Z as well, by choosing a convenient fundamental domain, and two generators for : l1 encircles L1 , while l2 goes from a basepoint to its antipodal. This second generator is a closed curve in L = S 3 /Z2 because of the quotienting, but it does not lift to S 3 . Nevertheless, 2l2 does lift to S 3 , and q −1 (2l2 ) = g1 + g2 . In terms of the two basis g1 , g2 and l1 , l2 we have the non-trivial map in (32):
1 −1 0 2
Z⊕Z
/ Z ⊕ Z.
(33)
Now we can continue our monodromy calculation. We claim that the loop around L1 ∩ L3 inside L1 is homotopic to the loop around L1 ∩ L3 inside L3 . This statement is not to be taken literally though. Neither L1 nor L3 are part of the moduli space, so we are not looping inside them. What we have are loops that are infinitesimally close to such loops, but lie outside L1 or L3 . This distinction is usually irrelevant, but for us the singularity brings it to the forefront. What we need to do is to deform the loop inside L1 around L1 ∩ L3 so that it doesn’t intersect L1 or L3 , and similarly for the loop inside L3 around L1 ∩ L3 . The reader can convince himself that the generic deformations are indeed both homotopic to l2 . Therefore the monodromy inside L3 around the Z3 point is given by MZ3 = T j∗ OC4 ◦ MZ2 = T j∗ OC4 ◦ Ti∗ OC3 ◦ L D2 . The associated Fourier-Mukai kernel is KZ3 = Cone ( j∗ OC4 )∨ j∗ OC4 −→ O KZ2 = Cone(( j∗ OC4 )∨ k∗ OC3 +C4 (D1 + D2 ) −→ −→ ((i ∗ OC3 )∨ i ∗ OC3 (1) → δ∗ O X (D2 ))).
(34)
(35)
3.4. The Z2 kernel squared: MZ2 ◦ MZ2 . It is an interesting question to ask how the Z2 quantum symmetry of the partially resolved Z2 orbifold is realized in the derived category setup. Accordingly, we would like to compute the action of (MZ2 )2 on a generic object. To get a feel for what to expect, we compute the Chern character of (MZ2 )2 acting on the trivial bundle O X . Inspecting the form of MZ2 in Eq. (30) we see that in order to compute ch((MZ2 )2 O X ) some general properties might be of use. Taking the Chern character of both sides in Eq. (29) one obtains [29, 21]: ch TA (B) = ch(B) − A, B ch(A), (36) where A, B is an Euler characteristic: A, B = (−1)i dim ExtiD(X ) (A, B).
(37)
i 12 An equivalent way of seeing this is to note that C2 − {L ∪ L } is homotopic to C∗ × C∗ , and that 1 3 S 1 × S 1 is a deformation retract of C∗ × C∗ .
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The Grothendieck-Riemann-Roch theorem gives a useful way to compute this: A, B = ch(A∨ ) ch(B) td(X ).
(38)
X
Using Eq. (36) and Eq. (38) one obtains ch(MZ2 O X )
= e D2 ,
ch((MZ2 )2 O X ) = e D1 .
(39)
This suggests that (MZ2 )2 acts like a large radius monodromy. A more involved computation shows that indeed on any object x, ch((MZ2 )2 x) = e D1 ch(x).
(40)
This result is a bit surprising, but a similar fact has been observed before [21]. As discussed in Sect. 3.3 the loop inside L1 and around L1 ∩ L3 after it is deformed becomes homotopic to l2 . (MZ2 )2 thus corresponds to 2l2 , which lifts to the S 3 , and encircles both L1 and L3 . Equation (6) tells us that monodromy around the divisor L1 is LO X (D1 ) . Assuming that monodromy around L3 is trivial, we have a perfect agreement with (40). This is also consistent with the general statement that monodromy at an orbifold point has to be associated with B-field components other than the blow-up mode(s) of the orbifold, in this case D2 . From now on we assume that monodromy around L3 is trivial. We will see that this leads to consistent results. Similarly we can ask about the Z3 quantum symmetry at the Z3 orbifold point. By the same argument, we could have a mismatch caused by monodromy around L3 and L4 . But as argued in the previous paragraph, there is no monodromy around L3 , and due to the symmetry of the problem there should be no monodromy around L4 either. We need to get (MZ3 )3 = 1 on the nose. And indeed for a general object x, one has ch((MZ3 )3 x) = ch(x).
(41)
We note that ch((MZ2 ) x) does not have a simple expression, and similarly for ch((MZ3 ) x) and ch((MZ3 )2 x). Using the Fourier-Mukai technology we can go beyond the K-theory analysis and evaluate (MZ2 )2 as it acts on the derived category. We can prove the following Proposition 3.7. In the notation of Prop. 3.3 we have an equivalence of functors: (MZ2 )2 ∼ = L D1 .
(42)
Proof. We have seen in the previous section, Eq. (31), that the Fourier-Mukai kernel associated to monodromy around the Z2 point C3 is given by KZ2 = Cone (i ∗ OC3 )∨ i ∗ OC3 (1) −→ δ∗ O X (D2 ) . (43) Let us make the following abbreviations: A = (i ∗ OC3 )∨ i ∗ OC3 (1),
B = δ∗ O X (D2 ), KZ2 = Cone (A −→ B) .
(44)
According to Proposition 3.2 the kernel of (MZ2 )2 is given by KZ2 KZ2 . As reviewed in Eq. (20), − KZ2 is an exact functor, and therefore KZ2 KZ2 = Cone (A → B) KZ2 = Cone (Cone (A A → A B) −→ Cone (B A → B B)) .
(45)
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R. L. Karp
Using the spectral sequences from Sect. A one can show that A A = ((i ∗ OC3 )∨ i ∗ OC3 (1))⊕2 , B A = (i ∗ OC3 )∨ (1) i ∗ OC3 (1),
A B = (i ∗ OC3 )∨ i ∗ OC3 (2), B B = δ∗ O X (2D2 ).
(46)
First we simplify
Cone (A A → A B) = (i ∗ OC3 )∨ Cone i ∗ OC3 (1)⊕2 → i ∗ OC3 (2) .
(47)
The Euler sequence for the tangent bundle of P1 , or the Koszul resolution of the complete intersection of two generic “lines” on P1 , gives the short exact sequence 0
/ P1 (−2) O
/ 1 (−1) ⊕2 O P
/ P1 O
0/ .
(48)
This translates into the statement that
Cone OP1 (1) ⊕2 −→ OP1 (2) = OP1 [1].
(49)
Therefore Cone (A A → A B) = (i ∗ OC3 )∨ i ∗ OC3 .
(50)
Using this and the results in Eq. (46) we have13 KZ2 KZ2 = (i ∗ OC3 )∨ i ∗ OC3 −→ (i ∗ OC3 (−1))∨ i ∗ OC3 (1) −→ δ∗ O X (2D2 ) . (51) This is reminiscent of Beilinson’s resolution of the diagonal in Pd × Pd [32]: 0 → OPd (−d) dPd (d) → · · · → OPd (−1) 1Pd (1) → OPd ×Pd → OPd → 0, (52) where iPd is the sheaf of holomorphic i-forms on Pd . Beilinson’s resolution for C3 = P1 is in fact very simple: 0
/ C3 (−1) C3 OC3 (−1) O
/ C3 C3 OC3 O
/ j3 ∗ OC3
0/ ,
(53)
where j3 : C3 → C3 ×C3 is the diagonal map. The notation C3 makes explicit where the exterior product is considered. But this is a short exact sequence on C3 ×C3 , while Eq. (51) is a statement in X ×X . Fortunately we can relate the two: Lemma 3.8. Let Z be a subvariety of the variety X, and let i : Z → X be the embedding. For two sheaves A and B, on Z one has that i ∗ A X i ∗ B = (i ×i)∗ (A Z B). 13 We underlined the 0th position in the complex.
(54)
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177
Proof of the lemma. First we specify some notation: pi denotes projection on the i th factor of X × X . Similarly si projects on the i th factor of Z × Z . By definition then i ∗ A X i ∗ B = p2∗ i ∗ A⊗ p1∗ i ∗ B. But p ∗j ◦ i ∗ = (i×i)∗ ◦ s ∗j , as follows from the following fiber square: Z×Z
si
/Z
pi
/X
i×i
X×X
(55) i
Thus p2∗ i ∗ A ⊗ p1∗ i ∗ B = (i ×i)∗ s2∗ A ⊗ p1∗ i ∗ B. By the projection formula this is (i × i)∗ (s2∗ A ⊗ (i ×i)∗ p1∗ i ∗ B). Now (i ×i)∗ p1∗ = ( p1 ◦ (i ×i))∗ . Next we can use the fact that the above diagram is commutative and rewrite further: ( p1 ◦ (i ×i))∗ = (i ◦ s1 )∗ = s1∗ i ∗ . Since A Z B = s2∗ A ⊗ s1∗ B this completes the proof of the lemma. Returning to Eq. (51), let us look at the first two terms in the complex: (i ∗ OC3 )∨ i ∗ OC3 −→ (i ∗ OC3 (−1))∨ i ∗ OC3 (1).
(56)
The dual of the short exact sequence 0
/ O X (−D)
/ OX
/ OD
/0
(57)
shows that O ∨D = O D (D)[−1].
(58)
Therefore (56) can be rewritten as {i ∗ OC3 (−2) i ∗ OC3 −→ i ∗ OC3 (−1) i ∗ OC3 (1)}[−1].
(59)
Using (54) this becomes (i ×i)∗ {OC3 (−2) OC3 −→ OC3 (−1) OC3 (1)}[−1],
(60)
where now the is on C3 rather than X . To bring the last expression to the Beilinson form we rewrite it as (i ×i)∗ OC3 (−1)OC3 (−1) −→ OC3 OC3 ⊗s2∗ OC3 (−1)⊗s1∗ OC3 (1) [−1], (61) where si : C3 × C3 → C3 is projection on the i th factor. Now using the Beilinson resolution from Eq. (53) this becomes (i ×i)∗ { j3 ∗ OC3 ⊗ s2∗ OC3 (−1) ⊗ s1∗ OC3 (1)}[−1].
(62)
Using the projection formula we have: (i ×i)∗ j3 ∗ {OC3 ⊗ j3 ∗ s2∗ OC3 (−1) ⊗ j3 ∗ s1∗ OC3 (1)}[−1].
(63)
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R. L. Karp
By construction si ◦ j3 = id. From the commutative diagram j3 i×i / C3 ×C3 / X×X C3 Ms MMM 7 p MMM ppp p p MMM p i MMM ppp δ * ppp & X
(64)
we also have that (i×i)◦ j3 = δ ◦ i. This simplifies (63) and it becomes simply δ∗ i ∗ OC3 [−1]. Using this (51) is nothing but KZ2 KZ2 = Cone δ∗ i ∗ OC3 [−1] −→ δ∗ O X (2D2 ) = δ∗ Cone i ∗ OC3 [−1] −→ O X (2D2 ) .
(65)
The Chern character of this is e D1 , and a bit more work shows that indeed KZ2 KZ2 = δ∗ O X (D1 ).
(66)
To see this note that Cone i O [−1] −→ O (2D ) = Cone i ∗ OC3 [−1] −→ ∗ C X 2 3 the O X (2D2 − D1 ) ⊗ O X (D1 ). The relations (2) imply that 2D2 − D1 = −D3 . Using exact triangle storming from (57) shows that Cone i ∗ OC3 [−1] −→ O X (−D3 ) = O X . Finally, the use of Prop. 3.3 then completes the proof of the theorem. Based on (41) and our experience with MZ2 , we conjecture that KZ3 KZ3 KZ3 = δ∗ O X , but judging by the above proof and the fact that KZ3 is much more complicated than KZ2 , we didn’t even attempt proving this. Nevertheless, we will have evidence for the conjecture in Sect. 4.2 4. The C2 /Z3 Fractional Branes In this section we use the Z3 monodromy action found in the previous section to generate a collection of fractional branes, and study some of their properties. As a starting point we need to know one of the fractional branes. We assume that the D5-brane wrapping the exceptional divisor C3 is one of the fractional branes. This is a natural assumption as long as we do not make any claims about the rest of the fractional branes. It is reasonable to expect that by various monodromy transformations any one of the fractional branes can be brought to this form. Instead of guessing the other two fractional branes, we look at the orbit of this D5-brane under the Z3 monodromy action. In the quiver language the fractional branes are the simple representations of the quiver, and are mapped into each other under the Z3 quantum symmetry. Therefore, the fractional branes will necessarily form a length three orbit of the Z3 monodromy, which is an incarnation of the Z3 quantum symmetry. By the same token we could have chosen C4 to be the fractional branes to start with. The C2 /Z3 geometry is completely symmetric with respect to C3 and C4 , and therefore we expect that whichever we start with, the other one will show up in the orbit. This is precisely what we are going to find.
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179
4.1. Generating fractional branes. We start by recalling Eq. (34), which gives us the form of the Z3 monodromy MZ3 : MZ3 = T j∗ OC4 ◦ Ti∗ OC3 ◦ L D2 .
(67)
By the assumption made above the 1st fractional brane is i ∗ OC3 , and the other two are MZ3 (i ∗ OC3 ) and (MZ3 )2 (i ∗ OC3 ). We start out by computing MZ3 (i ∗ OC3 ). 4.1.1. Computing MZ3 (i ∗ OC3 ). The first step is quite trivial: L D2
i ∗ OC3
/ i ∗ OC3 (1) .
(68)
We can act on this with TOC3 , and use the fact that Ri ∗ is a triangulated functor, to obtain i ∗ OC3 (1)
TOC
3
/ Cone i ∗ O ⊕2 −→ i ∗ OC (1) = i ∗ Cone O ⊕2 −→ OC (1) . 3 3 C3 C3 (69)
The intermediate steps above involved using the spectral sequence (120), but we suppress the details. Now we can use the Euler sequence (48) and simplify i ∗ OC3 (1)
TOC
3
/ i ∗ OC3 (−1)[1] .
(70)
The final leg involves using the spectral sequence (122) and results in i ∗ OC3 (−1)[1]
TOC
4
/ Cone j∗ OC −→ i ∗ OC (−1)[1] = k∗ OC +C [1]. 4 3 3 4 (71)
Here in the last equality we used the exact triangle i ∗ OC3 (−1)
/ k∗ OC3 +C4
/ j∗ OC4
/ i ∗ OC3 (−1)[1]
(72)
/0.
(73)
stemming from the short exact sequence:14 0
/ OC (−D)
/ OC+D
/ OD
Thus the 2nd fractional brane is the D5-brane that wraps both exceptional divisors, C3 and C4 . 14 The intuition behind this formula is easy to understand. The inclusion map of D into C + D allows us to restrict functions from C + D to D. This is the map OC+D → O D . The kernel of this map consists of those functions on C that vanish at the intersection point with D: OC (−D). For a rigorous proof see [33].
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R. L. Karp
4.1.2. Computing (MZ3 )2 (i ∗ OC3 ). To determine the 3rd fractional brane we apply the Z3 monodromy again. Now the starting point is the second fractional brane k∗ OC3 +C4 (D2 )[1] from the previous section: L D2
k∗ OC3 +C4 [1]
/ k∗ OC3 +C4 (D2 )[1].
(74)
For the next step we need RHom X (i ∗ OC3 , k∗ OC3 +C4 (D2 )[1]). We will determine these Ext groups in two different ways. The first method will use the cohomology long exact sequence associated to an exact triangle. The second method will use the spectral sequence derived in Appendix B. Although the first method is a priori more straightforward, we will see that once the general spectral sequence result is established, it is much more efficient. The fact that the two methods give the same result provides a consistency check for our calculations. We start with the long exact sequence associated to the exact triangle (72) once the covariant functor ExtiX (i ∗ OC3 , −) is applied to it: i ExtiX (i ∗ OC3 , j∗ OC4 ) −→ Exti+1 X (i ∗ OC3 , i ∗ OC3 ) −→ Ext X (i ∗ OC3 , k∗ OC3 +C4 (D2 )[1]). (75)
The spectral sequence (122) tells us that ExtiX (i ∗ OC3 , j∗ OC4 ) = δi,1 , while using the spectral sequence (120) gives us 15 C for i = 0, 2 ExtiX (i ∗ OC3 , i ∗ OC3 ) = (76) 0 otherwise. Using these two facts, the long exact sequence (75) tells us that ExtiX (i ∗ OC3 , k∗ OC3 +C4 (D2 )[1]) = δi,−1 . The same result can be obtained much quicker, if we apply the spectral sequence (127) to our case. What we need to compute is ExtiX (i ∗ OC3 , k∗ OC3 +C4 (D2 )[1]) = Exti+1 X (i ∗ OC3 (−1), k∗ OC3 +C4 ). By Serre duality this becomes Ext1−i X (k∗ OC3 +C4 , i ∗ OC3 (−1)). The spectral sequence (127) then reads O
H p (P1 , O(−2))
O0
C
0
0 /
(77)
q
q =
H p (P1 , O(−1)) /
p
p
and therefore RHom X (i ∗ OC3 , k∗ OC3 +C4 (D2 )[1]) = C[1], as we saw before. With the “algebra” out of the way, we can head back to monodromy and establish that k∗ OC3 +C4 (D2 )[1]
TOC
3
/ Cone i ∗ OC [1] → k∗ OC +C (D2 )[1] . 3 3 4
(78)
15 The same result for Exti (i O , i O ) follows if we use Prop. 3.15 of [29], which guarantees that X ∗ C3 ∗ C3 i ∗ OC3 is a spherical object in the sense of Definition 3.5. The same is true for j∗ OC4 .
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The RHS can be simplified using (72), and gives a simple answer: Cone i ∗ OC3 [1] → k∗ OC3 +C4 (D2 )[1] = j∗ OC4 [1].
(79)
Therefore the last step of the computation involves TOC4 ( j∗ OC4 )[1]. Here we can use a more general result Lemma 4.1. If A is an n-spherical object, then TA (A) = A[1 − n]. Proof. By the definition of TA (A) and the n-sphericity of A one has L L Cone RHomD(X ) (A, A) ⊗ A → A = Cone ( C → 0 → . . . → 0 → C) ⊗ A → A = Cone (A ⊕ A[−n] −→ A) ∼ = A[1−n].
(80)
For n = 2 and A = j∗ OC4 16 the lemma gives T j∗ OC4 ( j∗ OC4 ) = j∗ OC4 [−1], and thus j∗ OC4 [1]
TOC
/ j∗ OC4 .
4
(81)
This establishes j∗ OC4 as the 3rd fractional brane. 4.2. Consistency check. In order to test the Z3 quantum symmetry conjecture we verify the closure of the Z3 orbit i ∗ OC3 , k∗ OC3 +C4 (D2 )[1], j∗ OC4 under the Z3 monodromy. The first two steps are: j∗ OC4
L D2
/ j∗ OC4
TOC
3
/ Cone i ∗ OC [−1] −→ j∗ OC . 3 4
(82)
The next step is to compute the action of TOC4 on this. Since TOC4 is an exact functor, i.e. a functor between triangulated categories, we have that
TOC4Cone i ∗ OC3 [−1] −→ j∗ OC4 = Cone TOC4 (i ∗ OC3 )[−1] −→ TOC4 ( j∗ OC4 ) . (83) But
TOC4 (i ∗ OC3 ) = Cone j∗ OC4 [−1] −→ i ∗ OC3 = k∗ OC3 +C4 (D2 ),
(84)
and by Lemma 4.1 TOC4 ( j∗ OC4 ) = j∗ OC4 [−1].
(85)
Therefore
TOC4 Cone i ∗ OC3 [−1] −→ j∗ OC4 = Cone k∗ OC3 +C4 (D2 ) −→ j∗ OC4 [−1] = i ∗ OC3 .
(86)
In other words (MZ3 )3 (i ∗ OC3 ) = i ∗ OC3 . A similar computation shows that (MZ3 )3 ( j∗ OC4 ) = j∗ OC4 . These two facts provide evidence for the (MZ3 )3 = id conjecture. 16 Equation (76) guarantees that j O ∗ C4 is 2-spherical.
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R. L. Karp
5. Generalizations 5.1. Connection with the McKay correspondence. The original version of the McKay correspondence [5] relates the representations of a finite subgroup of SL(2, C) to the cohomology of the minimal resolution of the Kleinian singularity C2 / . Gonzalez-Sprinberg and Verdier [34] reinterpreted the McKay correspondence as a K-theory isomorphism, observing that the representation ring of is the same as the -equivariant K-theory of C2 . But even at this point a deeper understanding of the correspondence was lacking, all the results were based on a case by case analysis. A solid understanding of the McKay correspondence culminated with the work of Bridgeland, King and Reid [6], who showed that in dimensions two and three the McKay correspondence is an equivalence of two very different derived categories.17 In fact the seeds of the Bridgeland-King-Reid construction were already implicit in the work of Gonzalez-Sprinberg and Verdier [34], which inspired Kapranov and Vasserot [35] to prove the derived McKay correspondence for C2 / prior to the general BridgelandKing-Reid proof. As we will see, Kapranov and Vasserot implicitly provide a collection of fractional branes for arbitrary ⊂ SL(2, C), which is different from what we obtained by monodromy for Z3 . Let us review their construction. First of all, we have the covering map p : C2 −→ C2 / . Then we have the map p˜ : X −→ C2 / corresponding to the resolution of singularities. Using these two maps we can consider the fiber product Y of C2 and X over C2 / : X ×C2
(87)
p2
Y
p1
q2
/ C2 p
q1
( X
#
p˜
/ C2 /
On the same diagram we depicted the projection maps p1 and p2 of the product X ×C2 . Y is in fact an incidence subscheme in X ×C2 , and qi is the restriction of pi to Y . Let Coh (C2 ) be the category of -equivariant coherent sheaves on C2 , and let Coh(X ) be the category of coherent sheaves on X . Kapranov and Vasserot define two functors: : D(Coh (C2 )) −→ D(Coh(X )),
(F) = (Rq1∗ Lq2∗ F) ,
: D(Coh(X )) −→ D(Coh (C2 )),
(G) = R p2∗ RHom(OY , p1∗ G).
(88)
The main result of their paper shows that and are mutually inverse equivalences of categories. Moreover, they also determine the images under of some special objects in D(Coh (C2 )). To define what these objects are, let us recall that a finite-dimensional representation V of gives rise to two equivariant sheaves on C2 : 1. the skyscraper sheaf V ! , whose fiber at 0 is V and all the other fibers vanish, and 2. the locally free sheaf V˜ = V ⊗OC OC2 . 17 The Bridgeland-King-Reid proof formally generalizes to higher dimensions, but then one needs more information about the existence of crepant resolutions, and in particular about Nakamura’s -Hilbert scheme.
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There is a one-to-one correspondence between the representations of the McKay quiver and the category of -equivariant coherent sheaves on C2 (for a review of this see, e.g., [11]). In the language of quiver representations the fractional branes are the simple objects, i.e. with no sub-objects; the representations with all but one node assigned the trivial vector space, and all arrows are assigned the 0 morphisms. The non-trivial node is assigned the vector space C. Under this equivalence, the simple objects correspond to the skyscraper sheaves π ! , which are assigned to the irreducible representations π of . Therefore the fractional branes are the -equivariant sheaves π ! . Let Ci be the exceptional divisors of the crepant resolution of C2 / , for a subgroup of SL(2, C). A theorem in Sect. 2.3 of [35] asserts that the image of the collection {π ! : π irrep of } under the functor from (88) is O j C j and OCi (−1)[1].
(89)
Specializing this result to the C2 /Z3 case, and relabeling, gives the collection OC3 +C4 , OC3 (−1)[1], OC4 (−1)[1].
(90)
This collection is to be contrasted with the one obtained in Sect. 4: OC3 +C4 [1], OC3 , OC4 .
(91)
Next we show that both collections give what is expected of them, and also elucidate their connection. We will also investigate to what extent do these collections generalize for C2 /Zn with n > 3. 5.1.1. The quiver. Let us start with the collection (91) obtained by monodromy. Using the spectral sequence (127) it is immediate that ExtiX (k∗ OC3 +C4 [1], i ∗ OC3 ) = δi,1 . By Serre duality this also means that ExtiX (i ∗ OC3 , k∗ OC3 +C4 [1]) = δi,1 . A similar result holds for OC4 as well. In fact all the results are invariant under the interchange of C3 and C4 . Finally, the spectral sequence (122) shows that ExtiX ( j∗ OC4 , i ∗ OC3 ) = δi,1 . These results give us the well-known C2 /Z3 quiver, depicted in Fig. 4. Note that in writing down the quiver we used the result of Eq. (76) to draw the loops corresponding to the adjoint fields. This took care only of OC3 and OC4 . For ExtiX (k∗ OC3 +C4 [1], k∗ OC3 +C4 [1]) we can use the exact sequence (72) and apply the covariant functor HomD(X ) (k∗ OC3 +C4 [1], −) to it, and get the same answer: C for i = 0, 2 ExtiX (k∗ OC3 +C4 , k∗ OC3 +C4 ) = . (92) 0 for i = 0, 2 A similar computation shows that the collection (90) gives the same quiver. Therefore we have two potential sets of fractional branes. The next thing we need to check is whether their central charges add up to that of the D3-brane. The various Chern characters are easiest to compute from (73). In general, for C2 /Z n , we have:18 ch(i ∗ O C j ) = p +
n−1
C j , ch(i ∗ OCi (−1)) = Ci , ch(i ∗ OCi ) = Ci + p,
j=1 18 For brevity every embedding map is denoted by i.
(93)
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R. L. Karp
Fig. 4. The quivers for the two fractional C2 /Z3 collections
where p denotes the class of a point. Therefore ch(i ∗ O C j ) +
n−1
ch(i ∗ OCi (−1)[1]) = ch(O p ).
(94)
i=1
On the other hand ch(i ∗
O
C j [1]) +
n−1
ch(i ∗ OCi ) = (n − 2) ch(O p ).
(95)
i=1
This shows that both collections (90) and (91) give fractional branes for n = 3. Naturally, one would still need to check that the central charge, and hence the mass of these branes, is a third of the D0-brane central charge at the Z3 point in moduli space. This can be done very easily using local mirror symmetry and the expression for the central charges in terms of the periods given in [36]. The reason why this works is that the central charges are determined by the large volume asymptotics, which depend only on the Chern character of the brane. A simple computation shows that the collection (89) gives the C2 /Z n quiver for any n ≥ 2, not only for n = 3: • 5 • o
/
•
u o
/
i •
...
• o
/
) • o
/
•
It is also clear from (95) that the naive generalization of the monodromy collection (91) cannot be fractional. There is a simple explanation of why the n = 3 case is singled out. Intuitively there are two types of P1 ’s in the An−1 chain of the resolution: the two P1 ’s at the end of the chain are different from those in the middle in that they intersect only one other P1 , as opposed to two other P1 ’s that a middle P1 intersects. This makes a big difference in the spectral sequence calculation. For n = 3 these “middle” P1 ’s are absent, hence the simplification. We will return to this shortly.
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5.1.2. Seiberg duality. Now that we know that both (90) and (91) give fractional branes, the question of how they are related arises naturally. We turn to answering it in this section using Seiberg duality. Seiberg duality was originally formulated [37] as a low-energy equivalence between N = 1 supersymmetric gauge theories: an SU (Nc ) theory with N f fundamental flavors and no superpotential, and an SU (N f − Nc ) theory with N f fundamental magnetic flavors with a superpotential containing mesons. The duality says that both flow to the same point in the infrared. This was shown to arise as a consequence of an N = 2 duality [38]. From our point of view the Berenstein-Douglas [39] extension of Seiberg duality is the relevant one. The Berenstein-Douglas formulation of Seiberg duality has a natural stratification. In its simplest form it amounts to a base change for the branes. Since the new basis usually involves anti-branes in the language of the old basis, this change is most naturally done in the derived category of coherent sheaves, rather than sheaves alone. Therefore in this form Seiberg duality is an autoequivalence of the derived category of coherent sheaves, which by Orlov’s theorem (Theorem 3.1) is a Fourier-Mukai functor. The most general form of Seiberg duality arises when the t-structure of the derived category is changed. This is usually achieved by the use of tilting complexes [39]. What makes this possible is the underlying fact that there are different abelian categories with equivalent derived categories. Thus, in general, the difference between two collections of fractional branes can only be partially attributed to a choice of basepoint, since tiltings are more general than autoequivalences. The McKay collection, although not explicitly, but inherently is associated to the vicinity of the orbifold point. The collection obtained by monodromies explicitly involved the choice of a basepoint for the loops in the moduli space, and this basepoint was in the vicinity of the large volume point. Therefore it is reasonable to expect that the two collections differ only by a change in basepoint. Changing the basepoint amounts to conjugating the branes, and as we said earlier is an autoequivalence of the derived category. This gives a Parseval-type equality for the Ext-groups, which leads to the same quiver. This is in line with the fact that both collections gave the same C2 /Z3 quiver. Indeed, after some educated guesswork, one finds that the two collections (90) and (91) are related by monodromy around a point in moduli space where the brane wrapping both C3 and C4 once is becoming massless.19 In other words, by Conjecture 3.4 the two collections go into each other under the action of TOC3 +C4 . First, Eq. (92) shows that TOC3 +C4 (k∗ OC3 +C4 [1]) = TOC3 +C4 (k∗ OC3 +C4 )[1] = k∗ OC3 +C4 [1 − 1] = k∗ OC3 +C4 . (96) Second, by a similar computation: TOC3 +C4 (i ∗ OCi ) = i ∗ OCi (−1)[1], for i = 3, 4.
(97)
What happens to this relationship for n > 3? Since we know that the collection O C j and OCi (−1)[1],
(98)
is fractional for any n, and for n = 3 it gave under the Seiberg duality (TOC3 +C4 )−1 the n = 3 version of the collection O C j [1] and OCi , 19 The existence of such a point has been established in [31].
(99)
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R. L. Karp
one might ask what is the image of (98) under (TO C )−1 . We need to be more careful j
here, since the inverse functor (TO C )−1 is not of the form (29). It is most simply j presented in the form (Definition 2.7 of [29]) TA−1 (B) := Cone B −→ lin(RHomD(X ) (B, A), A) [−1]. (100) For a complex of vector spaces b◦ , and A ∈ D(X ), the q th term in the complex lin(b◦ , A) is given by p (Aq+ p )⊕ dim b . (101) lin q (b◦ , A) = p∈Z
Using this definition, it is easy to check that for an n-spherical object A, one has that −1 (O C j ) = TA−1 (A) = A[n − 1], as expected from Lemma 4.1. This shows that TO O
Cj
Cj
[1].20
Using (100) one obtains −1
TO (OCi (−1)[1]) = Cj
O j=i C j
for i = 1 or n − 1
OCi (−1)[1] otherwise
.
(102)
Thus the new collection is O C j [1], O j>1 C j , O j
OC
and a fractional collection is given by OC , OC (−1)[1]. Eyeballing the collection (89) we see that it contains only the OC (−1)[1] part of the collection (104), for every Ci . It is natural to assume that the decay of OC (−1)[1] goes through the channel 0
/ OC (−1)
/ OC
/ Op
0/ .
(105)
20 One can show by induction using the spectral sequence (127) that O C j is 2-spherical for any n. 21 For more on Seiberg duality for N = 2 theories we refer to [40].
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The physics of this decay in the C2 /Z2 case is very simple: in the large volume limit the D3-brane O p , which is space filling and point-like in the internal space, is stable; while at the orbifold point it is marginally unstable under decay into OC (−1) and OC [41, 9]. This gives an interesting physical interpretation of the C2 /Zn result. We have already seen that ch(i ∗ OC (−1)) = C, ch(i ∗ OC ) = C + p.
(106)
Therefore the D5-brane i ∗ OC has a D3-brane flux turned on, while the D5-brane i ∗ OC (−1) does not. Now let us look at the chain of n − 1 P1 ’s in the completely resolved space, and start shrinking say C1 . When the volume of C1 reaches 0, by the above C2 /Z2 arguments, the points on C1 become unstable, with decay products i ∗ OC1 and i ∗ OC1 (−1). Next we shrink C2 . This will destabilize the points of C2 , which decay into i ∗ OC2 and i ∗ OC2 (−1), provided that these were stable. Now C1 and C2 intersect in a point, and that point is linearly equivalent to any other point either on C1 or C2 . But all these points are destabilized, and hence we expect that so is any brane that carries the charge of such a point. This argument can be continued until all the Ci ’s have shrunk, at which point (106) would dictate that the i ∗ OCi ’s have decayed. On the other hand, the OC (−1)’s cannot entirely account for the decay of all the OC ’s if we use only the channel (105). This is another reason why O C j is needed, and then a repeated use of the octahedral axiom, (73) and (105) give a complete understanding of the decays. This argument therefore suggests that only the i ∗ OC (−1)’s are stable at the Zn point, and indeed these are the ones that show up in (89). In the above physical argument we used linear equivalence to conclude that any two points on Ci are equivalent. In fact we can deform them into each other. But this is not the case for the singular curve C j . This fact can be understood using some technology. First, for an integral scheme X there is an isomorphism between the divisor class group CaCl X , i.e., Weil divisors modulo linear equivalence, and the Picard group Pic X , i.e., Cartier divisors modulo isomorphisms (Prop. II.6.15 of [42]). On the other hand, if C˜ is the normalization of the curve C, then [42] provides a short exact sequence connecting ˜ Pic C and Pic C: 0
/
P∈C
O˜ ∗P /O∗P
/ Pic C
/ Pic C˜
/ 0.
(107)
The normalization of C = C j is the disjoint union of n − 1 lines, and therefore Pic C˜ = Z⊕n−1 . The Weil divisors on C = C j are points, and (107) shows that the different points living on the different components are not linearly equivalent. Furthermore, the divisors supported on the intersection points are also linearly inequivalent. Therefore the curve C = C j has a rich structure of inequivalent divisors, and it would be interesting to understand what the physical implications of this fact are, e.g., in connection with moduli stabilization. 5.2. The C2 /Zn quiver from partial resolutions. The partial resolutions of the C2 /Zn singularity form a partially ordered set. The simplest partial resolutions involve blowing up only one of the n − 1 exceptional divisors. This is particularly easy to do torically. We sketched the general situation in Fig. 5.
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R. L. Karp
The McKay correspondence gives an equivalence between quiver representations and sheaves on the resolved space, but it glosses over the partial resolutions. One can fill in the gap, by recasting it slightly into the language of stacks. First recall that there is an equivalence of categories between Cn /G quiver representations and coherent sheaves on the quotient stack [Cn /G]. Therefore the McKay correspondence reads as D([Cn /G]) ∼ = D(crepant resolution of Cn /G).
(108)
Kawamata generalized the above statement, and for G abelian he proved that [43]: D([Cn /G]) ∼ = D(partial crepant stacky resolution) ∼ = D(crepant resolution), (109) where in the middle one has to consider the partially resolved space as a stack. Therefore it makes sense to talk about fractional branes on the partially resolved space, and ask what they are. The strategy of this section is to use an appropriate set of objects on the exceptional divisor of the resolution to model the fractional branes. This strategy was successfully deployed in [44] as well. All the lattice points corresponding to the crepant partial resolutions lie on the line x + y = 1, and are of the form vk = (k, 1 − k), where 1 ≤ k ≤ n − 1. These points are equidistant, and therefore the star of the associated toric divisor Dk is given by the following 1-dimensional fan: ◦o v1
◦
◦
vk−1
• vk
◦
vk+1
◦
◦
◦/ vn
One immediately recognizes this fan as corresponding to the weighted projective line P1 (k, n − k). Without loss of generality we can assume that k < n − k. As a variety, or scheme, P1 (k, n − k) is isomorphic to P1 . The origin of this “smoothing-out” is identical to the one that underlies the isomorphism C/Zn ∼ = C. Algebraically this isomorphism is in fact a trivial statement: Spec C[x n ] ∼ = Spec C[x]. In order for the weighted projective line P1 (k, n − k) to be able to capture the fact that it provides a partial resolution for the C2 /Zn singularity, we have to retain more information than its scheme structure. In fact, the toric fan contains this data. We choose to retain this extra embedding information by using the language of stacks. We can consider the stack P1 (a, b) from two different points of view: as a toric stack [45], or as a quotient stack [46, 47]. We find it very convenient to work with the latter description.
Fig. 5. The toric fan for a partial resolution of the C2 /Zn singularity
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The stack P1 (k, n − k) has a full and strong exceptional collection of length n [46, 47]: O, O(1) . . . O(n − 1).
(110)
The mutation-theoretic dual of this exceptional collection was thoroughly investigated in [47].22 In particular, Proposition 2.5.11 of [47] states that the mutation-theoretic left dual of the collection O, . . . , O(n − 1) is given by the full exceptional sequence M(1−n) [1 − n], M(2−n) [2 − n], . . . , M(−1) [−1], M(0) .
(111)
In order to explain the previous expression we need to introduce some notation. Let 1 I ⊆ {1, 2} be a subset, and consider the weighted projective line P (w1 , w2 ). Then #I will denote the number of elements in I , while |w I | = i∈I wi . In this notation, for 0 ≤ l < n, the complex M(−l) is defined as a subcomplex of the Koszul complex K twisted by O(−l) [47], with j th term given by: j O(l − |w I |) ⊆ O(l − |w I |) = K j (−l). (112) M(−l) := #I =− j,|w I |=l
#I =− j
In other words M(l) has non-zero components only in non-positive degrees. For the stack P1 (k, n − k) the explicit expressions for the M(l) ’s are easy to write down: ⎧ O(l) f or 0 ≤ l < k ⎨ / O(l) f or k ≤ l < n − k . (113) O(l − k) M(−l) = ⎩ / O(l) f or l ≥ n − k O(l − k) ⊕ O(l − n + k) For brevity let us denote the stack P1 (k, n −k) by Yk . Similarly, the partially resolved quotient stack Blk [C2 /Zn ], with exceptional divisor Dk , is denoted by Xk . Let i : Yk = P1 (k, n − k) −→ Xk = Blk [C2 /Zn ] denote the embedding morphism of stacks. Proposition 5.1. For n and k relatively prime, the pushed-forward complexes i ∗ M(1−n) , i ∗ M(2−n) , . . . , i ∗ M(−1) , i ∗ M(0) provide a model for the C2 /Zn fractional branes. Proof. First we show that the Ext-quiver of the collection i ∗ M(1−n) , . . . , i ∗ M(−1) , i ∗ M(0) is the C2 /Zn quiver. To evaluate the Ext-groups we use the stacky version of the spectral sequence (120), as presented for example in [49], and adapted to our case i : Y = P1 (k, n − k) → X = Blk [C2 /Zn ].23 For E and F two objects in the bounded derived category of the stack Y = P1 (k, n − k) the spectral sequence reads p,q
E2
p
p+q
= Ext Y (E, F ⊗ q NY /X ) =⇒ Ext X (i ∗ E, i ∗ F) .
22 For the definition and properties of mutations see, e.g., [48]. 23 For simplicity we dropped the subscript k.
(114)
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R. L. Karp
Since NY /X = K Y has rank one,24 the spectral sequence degenerates at E 2 , and we have that Ext 1X (i ∗ E, i ∗ F) = Ext 1Y (E, F) ⊕ Ext 0Y (E, F ⊗ K Y ).
(115)
Serre duality then gives Ext 0Y (E, F ⊗ K Y ) = Ext 1Y (F, E)∨ , and therefore Ext 1X (i ∗ E, i ∗ F) = Ext 1Y (E, F) ⊕ Ext 1Y (F, E)∨ .
(116)
This shows that Ext1X is automatically symmetric, and the resulting quiver has bidirectional arrows. The Ext 1X (M(l), M( j)) groups are easily computed using Lemma 2.5.12 of [47]: dim Ext kX (M(l), M( j)) = #{J ⊆ {1, 2} | # J = k, |w J | = j − l}.
(117)
This observation reduces the problem of determining the Ext-quiver to a purely graph theoretic one, which is easy to solve. Let the n objects M(1−n) , M(2−n) , . . . , M(−1) , M(0) be the nodes of a graph. For every non-zero Ext 1Y (M(−i) , M(− j) ) we put a directed arrow. For the first k nodes, i.e., 1 ≤ l < k, there are two outgoing arrows (one to node l + k and another one to node l + n − k). Similarly, for k ≤ l ≤ n − k there is one incoming and one outgoing arrow. Finally, for l > n − k there are two incoming arrows. Therefore we have a graph with two arrows originating or ending at every node. Going from the collection M(1−n) , . . . , M(−1) , M(0) to the collection i ∗ M(1−n) , . . . , i ∗ M(−1) , i ∗ M(0) by virtue of Eq. (116) simply makes every arrow bidirectional. So the orientations of the arrows can be dropped, and they become paths. Now we have a graph such that every node is visited by two paths. After reordering this is precisely the C2 /Zn quiver, provided that we can show that the graph is connected. This is where the gcd(n, k) = 1 condition comes in, which is equivalent to the condition gcd(n − k, k) = 1. The Euclidean algorithm then guarantees the existence of two integers a and b such that a(n − k) + bk = 1. Since we have only links of length n − k and k the above equation shows that there is a path connecting any two neighboring nodes, where the path in question has a links of length n − k and b links of length n. This establishes that the Ext-quiver of the collection i ∗ M(1−n) , . . . , i ∗ M(−1) , i ∗ M(0) is connected, and hence it is the C2 /Zn quiver. The next step is to show that the i ∗ M(l) ’s indeed “add up” to the D3-brane. For this we 1−n need to compute i=0 ch i ∗ M(l) . We use the stacky version [50] of the Grothendieck-Riemann-Roch theorem (GRR): 1−n l=0
ch(i ∗ M(l) ) td(X ) =
1−n l=0
24 We used the fact that K is trivial. X
i ∗ (ch M(l) td Y) = i ∗ (
1−n l=0
ch M(l) td Y).
(118)
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The Chern characters ch M(l) are straightforward to compute from (113), in terms of the class of a point on Y = P1 (k, n − k): 1−n
ch M(l) =
l=0
=
k−1
el H +
n−k−1
l=0
l=k
n−1
k−1
l=0
el H −
l=0
el H − e(l−k)H +
n−1
el H − e(l−k)H − e(l−n+k)H
l=n−k
el H −
n−k−1
(119)
el H = k(n − k)H.
l=0
In the toric description of Y = P1 (k, n − k) [45] it is clear that k(n − k)H is the Chern character of a “non-singular” point on the stack Y = P1 (k, n − k).25 The GRR theorem 1−m ch i ∗ M(l) equals the Chern character of a non-singular point on then shows that i=0 X , which is what we wanted to prove. We believe that a similar result holds even without the technical condition gcd(n, k) = 1. More precisely, if gcd(n − k, k) = d and n − k = n 1 d and k = n 2 d, then the stack P1 (k, n − k) is a Zd quotient of P1 (n 1 , n 2 ). Let D be a d-torsion divisor on P1 (k, n − k), and H be the hyperplane-divisor. Then one would have to consider the mutation-theoretic dual of the exceptional collection O, . . . , O(nH), O(D), . . . , O(D + nH), . . . , O((d − 1)D), . . . , O((d − 1)D + nH). But proving this is beyond the scope of the present paper. Instead we just note that a similar problem was treated in [44]. 5.3. BPS algebras. Ten years ago, while computing threshold corrections in N = 2 heterotic compactifications, Harvey and Moore observed that these were closely related to product formulas in generalized Kac-Moody (GKM) algebras. Inspired by the heterotic/IIA duality they introduced the algebra of BPS states [51, 52] to explain the appearance of the GKM algebras. The product structure was defined in terms of an integral on the correspondence variety of certain moduli spaces of sheaves. At that time D-branes were thought of as sheaves with charge valued in K-theory [53, 7]. Currently we have a refinement of this picture, with D-branes as objects in D(X ), while the charge is determined by the natural map D(X ) → K(D(X )) = K(X ). Therefore one needs to revisit the way the algebra of BPS states is defined. An algebraic model for the algebra of BPS states could be provided by Ringel-Hall algebras. For an abelian category A in which all the Ext1 groups are finite, Ringel [54] defines an algebra R(A), which is the free abelian group on the isomorphism classes of A, endowed with a multiplication whose structure constants are suitably normalized Ext 1 ’s. Unfortunately it seems hard to extend this construction to a triangulated category, such as D(A). Mimicking the Hall algebra construction, with exact triangles replacing exact sequences, fails to give an associative multiplication [55]. This suggests that one should perhaps define the BPS algebra of D-branes in terms of abelian categories. This seems possible throughout the moduli space MK of complexified Kahler forms. Douglas [1] proposed that MK can be covered by open subsets Ui , and for each Ui there is an abelian category Ai , such that D(Ai ) ∼ = D(X ). This was proven recently in a different form for the local Calabi-Yau OP2 (−3) by Bridgeland [56]. 25 This guarantees that we need not deal with the subtleties of the inertia stack.
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R. L. Karp
Bridgeland proves that for every point in an open subset Stab0 (X ) ⊂ Stab(X ) of the space of π -stability conditions there exists a bounded t-structure, and that the heart of this t-structure, which is an abelian category, is invariant for an open subset of Stab0 (X ). Therefore the natural proposal is to take the algebra of BPS states to be the Ringel-Hall algebra of the π -stable objects. On the other hand, defining the BPS algebra of D-branes in terms of abelian categories seems aesthetically unsatisfactory. Another approach would be to consider all BPS D-branes, not only the stable ones. As we move in the Kahler moduli space MK , the collection of stable objects changes, and so does our proposed BPS algebra. The most extreme case of this is Seiberg-Witten theory [57]. For simplicity let us focus on the pure N = 2 case with gauge group SU(2). Geometric engineering tells us how to obtain this theory from string theory, and the BPS states of the field theory become a subset of those of the string theory. In the derived category context this map was worked out in [58]. The spectrum of stable objects is particularly simple: there is an almost circle-shaped region delimited by two lines of marginal stability. Outside of it there are infinitely many BPS states, but inside there are only two: the monopole and the dyon. Therefore the BPS algebra jumps from an infinite dimensional one to become finite dimensional. At this point very little is known about the algebras of BPS states. Ringel originally the associproved the following: let be a simply laced Dynkin diagram , with ated quiver, g the corresponding Lie algebra, and let A() be the abelian category of Then R(A()) is the positive part (nilpotent finite-dimensional representations of . subalgebra) of the enveloping Hopf algebra U(g ). Working over the finite field Fq , is the positive part of the quantum group Uq (g ). The same result was proven R(A()) in [35] using the derived category of the resolution of C2 /G , where G is the finite subgroup of SL(2, C) corresponding to . Ringel-Hall algebras were also investigated recently for elliptic curves [59]. It would be even more interesting if one could associate an algebra D R(X ) to D(X ) directly, without resorting to its abelian hearts. Moreover, one would like to see the might R(Ai ) heart-algebras as subalgebras in D R(X ). It was suggested that D R(A()) yield the whole quantum group [55]. D R(X ) would have interesting physical content, as it would integrate both perturbative and non-perturbative information, and would be inherently characteristic of the D-branes throughout the entire moduli space. Progress in this direction has recently been made by Toen [60], who defines an associative multiplication on the rational vector space generated by the isomorphism classes of a triangulated category T , where T is the perfect derived category per (T ) of a proper dg-category T . The resulting Q-algebra is the derived Hall algebra DH(T ). The multiplication involves all the Exti ’s, not only Ext 1 , and therefore is more natural. Toen proves that if A is the heart of a non-degenerate t-structure, then the RingelHall algebra R(A) is a subalgebra of DH(T ). Unfortunately for us, the relationship and the quantum group Uq (g ) remains to be investigated, and between DH(D(A())) an explicit example is yet to be worked out. Acknowledgements. It is a pleasure to thank Alastair Craw, Alberto Canonaco, Emanuel Diaconescu, Mike Douglas, Chris Herzog, Greg Moore, Tony Pantev, Ronen Plesser, Martin Rocek, Seiji Terashima and Eric Sharpe for useful conversations. I am especially indebted to Paul Aspinwall, from whom I learned most of the subject, for discussions at early stages of this project. I would also like to thank the 2005 Simons Workshop on Mathematics and Physics, and the 2005 Summer Institute in Algebraic Geometry at the University
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193
of Washington, for providing a stimulating environment where part of this work was done. The author was supported in part by the DOE grant DE-FG02-96ER40949.
A. Some Useful Spectral Sequences In the bulk of the paper we make extensive use of spectral sequences. This is a well known device in algebraic geometry, but so far had a limited appearance in the physics literature. The three spectral sequences we use have had extensive recent treatment [61]. Let us state them one by one, following the presentation of [61], and at the same time rephrasing their results in terms of Ext groups. The simplest case concerns a smooth subvariety S of a smooth variety X . Let i : S → X be the embedding, and N S/ X the normal bundle of S in X . Then for two vector bundles E and F, or more precisely locally free sheaves on S, we have the first spectral sequence: p,q
E2
p
p+q
= Ext S (E, F ⊗ q N S/ X ) =⇒ Ext X
(i ∗ E, i ∗ F) ,
(120)
where q denotes the q th exterior power. A more general case is when you are given two nested embeddings: j : T → S and i : S → X , a vector bundle F on T , and a vector bundle E on S. Then we have the spectral sequence: p,q
E2
p
p+q
= Ext T (E|T , F ⊗ q N S/ X |T ) =⇒ Ext X
(i ∗ E, j∗ F) .
(121)
The symbol |T means restriction to T . The final and most general case deals with two subvarieties T and S of X . Now the embeddings are i : S → X and j : T → X . Once again F is a vector bundle on T , and E is a vector bundle on S. The spectral sequence is: p,q
E2
= Ext S∩T (E| S∩T , F| S∩T ⊗ q−m N˜ ⊗ top N S∩T /T ) =⇒ Ext X p
p+q
(i ∗ E, j∗ F) , (122)
where N˜ = T X | S∩T /(T S| S∩T ⊕ T T | S∩T ) is a quotient of tangent bundles, while m = rk N S∩T /T . Although these spectral sequences were derived for sheaves, they extend to the derived category. It is also clear that (120) is a particular case of (121), which in turn is a particular case of (122). B. A Simple Spectral Sequence In this appendix we derive a spectral sequence that is used extensively in the paper. Our derivation follows ideas from the Appendix of [61]. Let X be a smooth algebraic variety. Consider two divisors C and D on X , and the embedding maps: i : C + D → X and j : C → X . Our task is to compute ExtiX (i ∗ OC+D , j∗ F) for a coherent sheaf F on C. It is worth pointing out that the divisor C + D is singular. We start with the short exact sequence of sheaves on X 0
/ O X (−C − D)
s
/ OX
/ i ∗ OC+D
/ 0.
(123)
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This is a locally free, and therefore projective resolution for i ∗ OC+D . Now we apply the left exact contravariant functor Hom X (−, j∗ F), 0
/ Hom X (i ∗ OC+D , j∗ F )
s
/ Hom X (O X , j∗ F )
/ Hom X (O X (−C−D), j∗ F ) . (124)
The Ext groups are given by the homology of this complex. Since supp( j∗ F) ⊂ C, and s vanishes on C + D, it follows that s = 0. Therefore Ext0 (i ∗ OC+D , j∗ F) = Hom X (O X , j∗ F) = j∗ F, Ext1 (i ∗ OC+D , j∗ F) = Hom X (O X (−C − D), j∗ F) = j∗ F(C 2 + C D).
(125)
At this point we can use the local to global spectral sequence on X , p,q
q
p+q
= H p (X, Ext X (A, B)) =⇒ Ext X (A, B),
E2
(126)
and the fact that H p (X, j∗ A) = H p (C, A) to conclude that the spectral sequence with p,q the following E 2 term: q
O
(127)
0
H p (C, F(C 2 +C D))
H p (C, F)
/ p
p+q
converges to Ext X (i ∗ OC+D , j∗ F). References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.
Douglas, M.R.: D-branes, categories and N = 1 supersymmetry. J. Math. Phys. 42, 2818–2843 (2001) Aspinwall, P.S., Lawrence, A.E.: Derived categories and zero-brane stability. JHEP 08, 004 (2001) Aspinwall, P.S., Douglas, M.R.: D-brane stability and monodromy. JHEP 05, 031 (2002) Douglas, M.R., Moore, G.W.: D-branes, Quivers, and ALE Instantons. http://arxiv.org/list/ hep-th/9603167, 1996 McKay, J.: Graphs, singularities, and finite groups. In: The Santa Cruz Conference on Finite Groups. Proc. Sympos. Pure Math. 37. Providence, R.I.: Amer. Math. Soc. 1980, pp 183–186 Bridgeland, T., King, A., Reid, M.: Mukai implies McKay: The McKay correspondence as an equivalence of derived categories. J. Amer. Math. Soc. 14(3), 535–554 (2001) Witten, E.: D-branes and K-theory. JHEP 12, 019 (1998) Diaconescu, D.-E., Douglas, M.R., Gomis, J.: Fractional branes and wrapped branes. JHEP 02, 013 (1998) Aspinwall, P.S.: A point’s point of view of stringy geometry. JHEP 01, 002 (2003) Diaconescu, D.-E., Gomis, J.: Fractional branes and boundary states in orbifold theories. JHEP 10, 001 (2000) Aspinwall, P.S.: D-branes on Calabi-Yau manifolds. In: Recent Trends in String Theory. River Edge, NJ: World Scientific, 2004, pp 1–152 Karp, R.L.: On the Cn /Zm fractional branes. http://arxiv.org/list/hep-th/0602165, 2006 Berenstein, D., Herzog, C.P., Ouyang, P., Pinansky, S.: Supersymmetry breaking from a Calabi-Yau singularity. JHEP 0509, 084 (2005)
C2 /Zn Fractional Branes and Monodromy
195
14. Verlinde, H., Wijnholt, M.: Building the standard model on a D3-brane. http://arxiv.org/list/hepth/0508089, 2005 15. Cox, D.A., Katz, S.: Mirror symmetry and algebraic geometry. Mathematical Surveys and Monographs 68, Providence, RI: Amer. Math. Soc. 1999 16. Witten, E.: Phases of N = 2 theories in two dimensions. Nucl. Phys. B 403, 159–222 (1993) 17. Aspinwall, P.S., Greene, B.R., Morrison, D.R.: Calabi-Yau moduli space, mirror manifolds and spacetime topology change in string theory. Nucl. Phys. B 416, 414–480 (1994) 18. Gelfand, I.M., Kapranov, M.M., Zelevinsky, A.V.: Discriminants, resultants, and multidimensional determinants. Boston, MA: Birkhäuser, 1994 19. Morrison, D.R., Plesser, M.R.: Summing the instantons: Quantum cohomology and mirror symmetry in toric varieties. Nucl. Phys. B 440, 279–354 (1995) 20. Cox, D.A.: The homogeneous coordinate ring of a toric variety. J. Alg. Geom. 4(1), 17–50 (1995) 21. Aspinwall, P.S.: Some navigation rules for D-brane monodromy. J. Math. Phys. 42, 5534–5552 (2001) 22. Fulton, W.: Introduction to toric varieties. Ann. Math. Studies 131, Princeton, NJ: Princeton University Press, 1993 23. Horja, R.P.: Derived category automorphisms from mirror symmetry. Duke Math. J. 127(1), 1–34 (2005) 24. Orlov, D.O.: Equivalences of derived categories and K 3 surfaces. J. Math. Sci. (New York) 84(5), 1361– 1381 (1997) 25. Kawamata, Y.: Equivalences of derived categories of sheaves on smooth stacks. Amer. J. Math. 126(5), 1057–1083 (2004) 26. Strominger, A.: Massless black holes and conifolds in string theory. Nucl. Phys. B 451, 96–108 (1995) 27. Aspinwall, P.S., Karp, R.L., Horja, R.P.: Massless D-branes on Calabi-Yau threefolds and monodromy. Commun. Math. Phys. 259, 45–69 (2005) 28. Distler, J., Jockers, H., Park, H.-j.: D-brane monodromies, derived categories and boundary linear sigma models. http://arxiv.org/list/hep-th/0206242, 2002 29. Seidel, P., Thomas, R.: Braid group actions on derived categories of coherent sheaves. Duke Math. J. 108(1), 37–108 (2001) 30. Greene, B.R., Kanter, Y.: Small volumes in compactified string theory. Nucl. Phys. B 497, 127–145 (1997) 31. De la Ossa, X., Florea, B., Skarke, H.: D-branes on noncompact Calabi-Yau manifolds: K -theory and monodromy. Nucl. Phys. B 644, 170–200 (2002) 32. Beilinson, A.A.: Coherent sheaves on Pn and problems in linear algebra. Funkt. Anal. i Pril. 12(3), 68– 69 (1978) 33. Barth, W., Peters, C., Van de Ven, A., Hulek, K.: Compact complex surfaces. 2nd Enlarged ed., Ergebnisse der Mathematik 4, Berlin: Springer-Verlag, 2004 34. Sprinberg, G.G., Verdier, J.-L.: Construction géométrique de la correspondance de McKay. Ann. Sci. École Norm. Sup. (4) 16(3), 409–449 (1983) 35. Kapranov, M., Vasserot, E.: Kleinian singularities, derived categories and Hall algebras. Math. Ann. 316(3), 565–576 (2000) 36. Skarke, H.: Non-perturbative gauge groups and local mirror symmetry.. JHEP 11, 013 (2001) 37. Seiberg, N.: Electric-magnetic duality in supersymmetric nonAbelian gauge theories. Nucl. Phys. B 435, 129–146 (1995) 38. Argyres, P.C., Plesser, M.R., Seiberg, N.: The Moduli Space of N=2 SUSY QCD and Duality in N=1 SUSY QCD. Nucl. Phys. B471, 159–194 (1996) 39. Berenstein, D., Douglas, M.R.: Seiberg duality for quiver gauge theories. http://arxiv.org/list/hepth/0207027, 2002 40. Robles-Llana, D., Rocek, M.: Quivers, quotients, and duality. http://arxiv.org/list/hep-th/0405230, 2004 41. Douglas, M.R., Fiol, B., Romelsberger, C.: The spectrum of BPS branes on a noncompact CalabiYau. JHEP 09, 057 (2005) 42. Hartshorne R: Algebraic geometry. GTM, No. 52, New York: Springer-Verlag, 1977 43. Kawamata, Y.: Log crepant birational maps and derived categories. J. Math. Sci. Univ. Tokyo 12(2), 211– 231 (2005) 44. Herzog, C.P., Karp, R.L.: Exceptional collections and D-branes probing toric singularities. JHEP 02, 061 (2006) 45. Borisov, L.A., Chen, L., Smith, G.G.: The orbifold Chow ring of toric Deligne-Mumford stacks. J. Amer. Math. Soc. 18(1), 193–215 (2005) 46. Auroux, D., Katzarkov. L., and Orlov, D.: Mirror symmetry for weighted projective planes and their noncommutative deformations. http://arxiv.org/list/math.AG/0404281, 2004 47. Canonaco, A.: The Beilinson complex and canonical rings of irregular surfaces. Mem. Amer. Math. Soc. 862, Providence, RI: Amer. Math. Soc., 2006 48. Rudakov, A.N., et al.,: Helices and vector bundles. London Math. Soc. Lecture Note Ser. 148, Cambridge: Cambridge Univ. Press, 1990 49. Katz, S., Pantev, T., Sharpe, E.: D-branes, orbifolds, and Ext groups. Nucl. Phys. B673, 263–300 (2003)
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50. Toen, B.: Théorèmes de Riemann-Roch pour les champs de Deligne-Mumford.. K -Theory 18(1), 33– 76 (1999) 51. Harvey, J.A., Moore, G.W.: Algebras, BPS States, and Strings. Nucl. Phys. B 463, 315–368 (1996) 52. Harvey, J.A., Moore, G.W.: On the algebras of BPS states. Commun. Math. Phys. 197, 489–519 (1998) 53. Minasian, R., Moore, G.W.: K-theory and Ramond-Ramond charge. JHEP 11, 002 (1997) 54. Ringel, C.M.: Hall algebras and quantum groups. Invent. Math. 101(3), 583–591 (1990) 55. Kapranov, M.: Heisenberg doubles and derived categories. J. Algebra 202(2), 712–744 (1998) 56. Bridgeland, T.: Stability conditions on a non-compact Calabi-Yau threefold. Commun. Math. Phys. 266, 715–733 (2006) 57. Seiberg, N., Witten, E.: Electric-magnetic duality, monopole condensation, and confinement in N = 2 supersymmetric Yang-Mills theory. Nucl. Phys. B 426, 19–52 (1994) 58. Aspinwall, P.S., Karp, R.L.: Solitons in Seiberg-Witten theory and D-branes in the derived category. JHEP 04, 049 (2003) 59. Burban, I., Schiffmann, O.: On the Hall algebra of an elliptic curve, I. http://arxiv.org/list/math.AG/ 0505148, 2005 60. Toen, B.: Derived Hall Algebras. http://arxiv.org/list/math.QA/0501343, 2005 61. Katz, S., Sharpe, E.: D-branes open string vertex operators, and Ext groups. Adv. Theor. Math. Phys. 6, 979–1030 (2003) Communicated by M.R. Douglas
Commun. Math. Phys. 270, 197–231 (2007) Digital Object Identifier (DOI) 10.1007/s00220-006-0125-y
Communications in
Mathematical Physics
Multidimensional Continued Fractions, Dynamical Renormalization and KAM Theory Kostya Khanin1 , João Lopes Dias2 , Jens Marklof3 1 Department of Mathematics, University of Toronto, 100 St George Street,
Toronto, Ontario M5S 3G3, Canada. E-mail:
[email protected]
2 Departamento de Matemática, ISEG, Universidade Técnica de Lisboa, Rua do Quelhas 6,
Lisboa, 1200-781, Portugal. E-mail:
[email protected]
3 School of Mathematics, University of Bristol, Bristol BS8 1TW, U.K. E-mail:
[email protected]
Received: 26 March 2006 / Accepted: 27 April 2006 Published online: 10 October 2006 – © Springer-Verlag 2006
Abstract: The disadvantage of ‘traditional’ multidimensional continued fraction algorithms is that it is not known whether they provide simultaneous rational approximations for generic vectors. Following ideas of Dani, Lagarias and Kleinbock-Margulis we describe a simple algorithm based on the dynamics of flows on the homogeneous space SL(d, Z)\ SL(d, R) (the space of lattices of covolume one) that indeed yields best possible approximations to any irrational vector. The algorithm is ideally suited for a number of dynamical applications that involve small divisor problems. As an example, we explicitly construct a renormalization scheme for the linearization of vector fields on tori of arbitrary dimension. 1. Introduction The aims of this paper are two-fold. The first objective is to introduce a new multidimensional continued fraction algorithm that is ideally suited for different dynamical applications. The algorithm can be used in order to effectively deal with small divisors whenever quasi-periodicity with several frequencies is an essential feature of a problem. Our second goal is to demonstrate the strength of the algorithm by developing a renormalization approach to KAM theory. The method, being conceptually very simple, is also very general, and allows us to consider a wide class of frequency vectors. For reasons of clarity we restrict our attention to vectors satisfying an explicit Diophantine condition (valid for a set of vectors of full Lebesgue measure); the extension to more general frequency vectors follows straightforwardly from the approach presented here, and will be detailed in a separate publication.
1.1. Continued fractions. The classical continued fraction algorithm produces, for every irrational α ∈ R, a sequence of rational numbers pn /qn that approximate α up to an error of order 1/qn2 . The first objective of this paper is to develop a multidimensional
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analogue that allows us to approximate any irrational α ∈ Rd−1 by rational vectors. The theory of multidimensional continued fractions has a long history going back to Jacobi and Perron (see [30] for extensive references). The “traditional” m.c.f. algorithms (Jacobi-Perron algorithm, etc.) have many beautiful ergodic properties. Unfortunately, the quality of approximations provided by them is very difficult to control. In fact, it is not even known whether those algorithms give simultaneous rational approximations for Lebesgue almost all α. In the case d ≥ 4 the only result in this direction is a recent computer-assisted proof of the almost everywhere strong convergence for the ordered Jacobi-Perron algorithm [11, 12]. However, even in this case an explicit description of the set of bad vectors seems difficult. For example, the existence of ‘noble’ vectors, that is vectors corresponding to a periodic continued-fraction expansion, for which approximations do not converge, is rather unsatisfactory. For these reasons, there has been only very limited success in the application of traditional multidimensional algorithms in renormalization schemes, in particular those involving typical frequency vectors. The algorithm we employ here does not suffer from such pathologies. Following Lagarias’ seminal ideas in [21], our approach is based on the dynamics of the geodesic1 flow on the homogeneous space \G with G = SL(d, R) and = SL(d, Z). Notice that \G may be identified with the space of lattices in Rd of covolume one or, equivalently, with the Teichmüller space of flat d-dimensional tori. Lagarias algorithm is, to the best of our knowledge, the first that provides a strongly convergent multidimensional continued fraction expansion for all vectors. The main advantage of our algorithm is that in addition it allows for effective hyperbolicity estimates which are crucial in dynamical renormalizations. The problem of multidimensional continued fractions may be formulated in the following way. Given a vector α ∈ Rd−1 , find a sequence of matrices T (n) ∈ GL(d, Z), n ∈ N, such that the ‘cocycle’ corresponding to the products P (n) = T (n) T (n−1) . . . T (1) exponentially contracts in the direction of the vector ω = ( α1 ) ∈ Rd and exponentially expands in all other directions. Thus, the cocycle should have one negative Lyapunov exponent and d − 1 positive. In this spirit, our algorithm comprises the following steps:2 (1) With every α ∈ Rd−1 associate the orbit {C(t) : t ≥ 0} ⊂ SL(d, R), where C(t) =
1d−1 0
α 1
1d−1 e−t 0
0
e(d−1)t
,
(1.1)
1d−1 denotes the (d − 1) × (d − 1) unit matrix. (2) Fix a Siegel set S.3 Given a sequence of times t1 < t2 < . . . → ∞, use classical reduction theory to find matrices P (n) ∈ SL(d, Z) that map the points C(tn ) to S. 1 The term ‘geodesic’ is slightly inaccurate when d > 2. The orbits of the flow on \G we are discussing here correspond in fact to geodesics on the unit cotangent bundle of the space \G/ SO(d) for a certain family of initial conditions. Only for d = 2 the cotangent bundle can be identified with \G. 2 The fact that we restrict our attention here to SL(d, Z) is not essential; the algorithm can readily be extended to allow for approximation by elements from GL(d, Z). 3 A Siegel set has the property that (a) it contains a fundamental domain F of SL(d, Z) in SL(d, R), and (b) it is contained in a finite number of translates PF , P ∈ SL(d, Z).
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(3) Define the n th continued fraction map by α (n−1) → α (n) = where α (0) = α and
T
(n)
=
(n)
(n) (n−1) α + t (n) T11 12 t (n) α (n−1) 21
(n)
T11 t (n) 21
(n)
+ t22
,
(1.2)
t 12 (n) t22
∈ SL(d, Z)
(1.3)
is the n th transfer matrix defined by P (n) = T (n) P (n−1) . Note that the choice of matrices P (n) is not unique, and thus different choices will lead to different algorithms. Uniqueness is guaranteed if one imposes the additional requirement that P (n) C(tn ) belongs to a fundamental domain F, as in Lagarias’ algorithm. However, in this case, due to a very complicated geometry of fundamental domains, it is difficult to control hyperbolic properties of the matrices T (n) . On the other hand, the use of Siegel sets allows us to relate these hyperbolic properties with the Diophantine properties of vector α. Another important difference with the Lagarias algorithm is connected with a choice of times tn . Lagarias requires a specific choice of tn that corresponds to the times the geodesic exits the fundamental domains. On the contrary, our algorithm selects the tn according to the diophantine properties of α: the better approximable by rationals α is, the faster the tn have to increase with n in order to ensure the hyperbolicity of the transfer matrices. Only in the special case of badly approximable α the times tn can be chosen to have bounded gaps tn − tn−1 . Notice that in dimension d = 2 Lagarias’ algorithm, and thus also our algorithm, do not reduce to the classical continued fraction dynamics, see [10] for details. However, a subsequence of exit times, called Hermite critical times, gives an accelerated version of the standard continued fractions [21]. Since the action of SL(d, R) on Rd−1 by fractional linear transformation defines a group action, we have (n) (n) P α + p12 α (n) = 11(n) , (1.4) p α + p (n) 21 22
where P
(n)
=T
(n)
T
(n−1)
···T
(1)
=
(n) P11 p(n) 21
p(n) 12 (n) p22
∈ SL(d, Z).
(1.5)
1.2. Renormalization. Dani [5] and Kleinbock-Margulis [17] observed that Diophantine properties of α translate to divergence properties of the corresponding orbit {C(t) : t ≥ 0} ⊂ \G in the cusps of \G. We exploit these results to show that, under mild Diophantine conditions on α (satisfied by a set of α of full Lebesgue measure, cf. Sect. 2.6), there is a sequence of times tn such that the transfer matrices T (n) are uniformly hyperbolic in a sense made precise in Sect. 2.8. This fact allows us to develop renormalization schemes for vector fields and Hamiltonian flows that had previously been constructed only in dimension one [23] or for very special choices of α [1, 18, 19, 22]. In this paper we focus on the case of vector fields, where, as in the traditional approaches (cf. [28]), the proofs are technically slightly simpler than in the case of Hamiltonian flows. Full details of the latter are given in [15].
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We say ω ∈ Rd is Diophantine if there are constants > 0, C > 0 such that k (d−1)(1+) |k · ω| > C,
(1.6)
for all k ∈ Zd − {0}. Note that we may assume without loss of generality that ω is of the form ω = ( α1 ) with α ∈ Rd−1 . Condition (1.6) then translates to a standard Diophantine condition on α, see Sect. (2.6) for details. Theorem 1.1. For any real analytic vector field v on Td , d ≥ 2, sufficiently close to a constant vector field with Diophantine vector ω ∈ Rd , there is b > 0, an analytic curve p : (−b, b) → Rd , s → ps , and an analytic conjugacy h isotopic to the identity between the flow generated by v + ps and the linear flow φt (x) = x + t (1 + s)ω on Td , t ≥ 0, for each |s| < b. Moreover, the maps v → h and v → p are analytic. Let us emphasize that the result holds for all analytic vector fields close to a constant one without any additional conditions, such as preservation of volume, etc. The main strategy of the proof of the above theorem is as follows. Consider a vector field X (x, y) = ω + f (x) + y, where x ∈ Td , and y ∈ Rd is an auxiliary parameter. The vector field f (x) is a sufficiently small analytic perturbation of a constant vector field. We may furthermore assume that ω = ( α1 ) for some Diophantine α ∈ Rd−1 ; this is achieved by a rescaling of time. The aim is to find a value of parameter y = yω such that the vector field X (x, yω ) is linearizable to a constant vector field identically equal to ω by means of an analytic transformation of the coordinates on Td . Renormalization is an iterative process, and we thus assume that after the (n − 1)th renormalization step the vector field is of the slightly more general form X n−1 (x, y) = ω(n−1) + f n−1 (x, y),
(1.7)
where ω(n−1) = ( α 1 ) and α (n−1) is given by the continued fraction algorithm, cf. (1.4). The Fourier modes of f n−1 are smaller than in the previous step, and decay expo+ nentially as k → +∞. We define a cone of resonant modes by a relation In−1 = {k ∈ d (n−1) th Z : |k · ω | ≤ σn−1 k }. The n step requires the following operations: (n−1)
+ . (1) Eliminate all Fourier modes outside of the resonant cone In−1 (2) Apply a linear operator corresponding to a coordinate transformation given by the −1 inverse transfer matrix T (n) . (n) (3) Rescale time to ensure that the frequency vector is of the form ω(n) = ( α1 ). −1
The conjugate action on the Fourier modes is given by k → T (n) k. It follows from + if σ the hyperbolicity of T (n) that this transformation contracts for k ∈ In−1 n−1 is small enough. This gives a significant improvement of the analyticity domain which results in the decrease of the estimates for the corresponding Fourier modes. As a result, all Fourier modes apart from the zero modes get smaller. To decrease the size of the latter, we choose a parameter y = yn in such a way that the corresponding zero modes vanish, and then consider a neighbourhood of y-values centred at yn . That is, the auxiliary parameter y is used to eliminate an instability in the direction of constant vector fields. To get enough control on the parameter dependence we perform an affine rescaling of this parameter on every renormalization step. One can then show that the corresponding sequence of parameter domains is nested and converges to a single point y = yω for which the initial vector field is indeed linearizable.
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In order for the scheme to be effective, the sequence of stopping times tn and the sizes of the resonant cones defined by the sequence of σn must be chosen properly. Large intervals δtn = tn − tn−1 improve hyperbolicity but, on the other hand, worsen −1 estimates for the norms T (n) , T (n) . Similarly, if σn is too small the elimination of non-resonant modes will give large contributions; on the other hand, for large values −1 of σn the multiplication by T (n+1) will not yield a contraction for k ∈ In+ . As we shall show below a right choice of sequences {(tn , σn )} can be made, depending on the Diophantine properties of the vector ω. The mechanism of convergence of renormalizations is well understood by now. The main framework was developed in [18] for special frequency vectors. Our proof of convergence of the renormalizations follows the same basic steps. However, there are several essential differences. The most important one is related to our choice of a sequence tn which we discussed above. In fact, the use of long intervals δtn = tn −tn−1 simplifies the estimates and makes the proof more transparent. Another important point is connected with the construction of the analytic conjugacy h. That is built as h = lim h n , where h n is the composition of P (n) with the coordinate transfomations used in all renormalization steps up to the n th one. The domain of analyticity for each h n is a complex strip around Rd with width ρn P (n) −1 , where ρn is the width of analyticity strip for the vector fields at the corresponding step. In d = 2, using standard continued fractions, the ratio ρn P (n) −1 is bounded from below ([24]). However, in the multidimensional case, since we do not have such a bound, we have to consider shrinking domains. As a result, the conjugacy h is only C 1 on Rd . To recover analyticity, we use the analytic dependence of the conjugacy on the initial vector field (Theorem 3.19). The above arguments are adapted in [15] to the case of Hamiltonian flows, which is technically slightly more challenging. Let B ⊂ Rd , d ≥ 2, be an open set containing the origin, and let H 0 be a real-analytic Hamiltonian function H 0 (x, y) = ω · y +
1 yQ y, (x, y) ∈ Td × B; 2
(1.8)
with ω ∈ Rd and a real symmetric d × d matrix Q. It is said to be non-degenerate if det Q = 0. Theorem 1.2. Suppose H 0 is non-degenerate and ω is Diophantine. If H is a real analytic Hamiltonian on Td × B sufficiently close to H 0 , then the Hamiltonian flow of H leaves invariant a Lagrangian d-dimensional torus where it is analytically conjugated to the linear flow φt (x) = x + tω on Td , t ≥ 0. The conjugacy depends analytically on H . 1.3. A brief review. The idea of renormalization was introduced to the theory of dynamical systems by Feigenbaum [6] in the late 1970’s. In the case of Hamiltonian systems with two degrees of freedom MacKay proposed in the early 1980’s a renormalization scheme for the construction of KAM invariant tori [25]. The scheme was realized for the construction of invariant curves for two-dimensional conservative maps of the cylinder. An important feature of MacKay’s approach is the analysis of both smooth KAM invariant curves and so-called critical curves corresponding to critical values of a parameter above which invariant curves no longer exist. From the point of view of renormalization theory the KAM curves correspond to a trivial linear fixed point for the renormalization transformations, while critical curves give rise to very
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complicated fixed points with nontrivial critical behavior. MacKay’s renormalization scheme was carried out only for a small class of Diophantine rotation numbers with periodic continued fraction expansion (such as the golden mean). Khanin and Sinai studied a different renormalization scheme for general Diophantine rotation numbers [16]. Both of the above early approaches were based on renormalization for maps or their generating functions. Essentially, the renormalization transformations are defined in the space of pairs of mappings which, being iterates of the same map, commute with each other. These commutativity conditions cause difficult technical problems, and led MacKay [26] to propose the development of alternative renormalization schemes acting directly on vector fields. The same idea was realized by Koch [18] who proves a KAM type result for analytic perturbations of linear Hamiltonians H 0 (x, y) = ω · y, for frequencies ω which are eigenvectors of hyperbolic matrices in SL(d, Z) with only one unstable direction. Notice that the set of such frequencies has zero Lebesgue measure and in the case d = 2 corresponds to vectors with a quadratic irrational slope. Further improvements and applications of Koch’s techniques appeared in [1, 7, 19, 22, 23], emphasizing the connection between KAM and renormalization theories. The results of this paper illustrate that such a programme can indeed be carried out in considerable generality. Another direction was followed in [20], presenting a computer-assisted proof of the existence of MacKay’s golden mean critical renormalization fixed point in the context of Hamiltonian vector fields with two degrees of freedom. Other renormalization ideas have appeared in the context of the stability of invariant tori for nearly integrable Hamiltonian systems inspired by quantum field theory and an analogy with KAM theory (see e.g. [3], and [8, 9] where a graph representation of the invariant tori in terms of Feynman diagrams is used). For the sake of transparency we have restricted our attention to Diophantine vectors ω, which form a set of full Lebesgue measure and are thus ‘typical.’ A more detailed analysis under weaker Diophantine conditions is, in principle, possible within the present framework. It is however a fundamental open problem to state a sharp (i.e. the weakest possible) Diophantine condition under which the above conjugacy can be established. The answer to this question is known only in the case d = 2, namely in the discrete-time situation for the Siegel problem [31] and for circle diffeormophisms [32], where the Diophantine condition is of Brjuno type. It would also be interesting to see whether the multidimensional continued fraction algorithm presented here will allow generalizations of other one-dimensional renormalization constructions. A concrete challenge is for instance the extension of the recent results on the reducibility of cocycles over irrational rotations by Ávila and Krikorian [2]. In the next section we introduce the multidimensional continued fraction algorithm, and include a discussion of its hyperbolicity properties required in the renormalization schemes. Section 3 provides a detailed account of one exemplary case, the renormalization of vector fields.
2. Multidimensional Continued Fractions and Flows on Homogeneous Spaces 2.1. Flows on homogeneous spaces. Let us set G = SL(d, R) and = SL(d, Z), and define the diagonal subgroup {E t : t ∈ R} in G, where E t = diag(er1 t , . . . , erd t )
(2.1)
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203
with constants satisfying the conditions r1 , . . . , rd−1 < 0 < rd ,
d
r j = 0.
(2.2)
j=1
The right action of E t on the homogeneous space \G generates the flow t : \G → \G,
M → M E t .
(2.3)
Since G is a simple Lie group with finite center, t is ergodic and mixing [27]. Let F ⊂ G be a fundamental domain of the left action of on G. Recall that, by definition of the fundamental domain of a free group action, PF = G, F ∩ PF = ∅ for all P ∈ − {1}, (2.4) P∈
and hence, for any given M ∈ G, there is a unique family of P(t) ∈ such that M(t) := P(t)M E t ∈ F
(2.5)
holds for all t ∈ R. 2.2. A convenient parametrization. Let us consider those M ∈ G which can be written as A 0 1 α M= , (2.6) β 0 1 γ where A ∈ Mat d−1 (R) (the space of real (d − 1) × (d − 1) matrices), α, β ∈ Rd−1 are column vectors, γ ∈ R with γ > 0. This yields a local parametrization of G for the set A 0 1 α d−1 G + := ∈ G : A ∈ Mat (R), α, β ∈ R , γ ∈ R d−1 >0 , β 0 1 γ (2.7) which is particularly convenient for our purposes. All other matrices are either of the above form with γ < 0 instead, or may be written as A 0 1 α M=S , (2.8) β 0 1 γ where S ∈ is a suitably chosen “signed permutation matrix”, i.e., every row and every column contains one and only one non-zero coefficient, which is either 1 or −1. In the following we will stay clear of the parameter singularity at γ = 0, and thus may assume without loss of generality S = 1. To work out the action of a general element T ∈ G in the above parametrization, consider T : M → M˜ := T M, (2.9)
where T =
T11 t 21
t 12 , t22
(2.10)
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1 M˜ = 0
A˜ 0 β ˜ γ˜ . A short calculation yields the fractional linear action M is as above and
α˜ 1
T11 α + t 12 , t α + t 21 22
α → α˜ = and γ → γ˜ =
(2.11)
(2.12)
t 21 α + t22 γ ,
(2.13)
˜ β˜ which will not be needed in the following. and more complicated expressions for A, 2.3. Multidimensional continued fractions. Let t0 = 0 < t1 < t2 < . . . → ∞ be sequence of times, with gaps δtn := tn − tn−1 (2.14) chosen large enough so that P(tn ) = P(tn−1 ), where P(t) is defined by (2.5). The sequence P (n) := P(tn ) of matrices in may be viewed as the continued fraction approximants of the vector α, which are the best possible for suitable choices of a fundamental domain F and times tn , see [21]. Let us furthermore put M (n) := M(tn ) with M(t) as in (2.5), and define α (n) , γ (n) by the decomposition (2.6), i.e., by (n) 0 A 1 α (n) (n) (2.15) M = β (n) γ (n) . 0 1 From M (n) = P (n) M E tn and (2.12), (2.13) we deduce α (n) = and
γ (n) =
(n)
p(n) α 21
(n) p21 α
where P
(n)
(n) + p22 (n)
+ p22
(n)
P11 p(n) 21
=
(n)
P11 α + p12
,
(2.16)
erd tn γ ,
(n) p12 (n) . p22
(2.17)
(2.18)
It is evident that if the components of ( α, 1) are linearly independent over Q, then γ = 0 implies γ (n) = 0 for all n ≥ 0. We shall later employ the transfer matrices T (n) defined by P (n) = T (n) P (n−1) . Here, (n) M = T (n) M (n−1) E δtn implies α (n) = and
γ (n) =
(n) (n−1) α + t (n) T11 12 t (n) α (n−1) 21
(n) (n−1) t 21 α
where T
(n)
=
(n)
+ t22
,
(n) erd δtn γ (n−1) , + t22
(n) T11 t (n) 21
t (n) 12 . (n) t22
(2.19)
(2.20)
(2.21)
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205
2.4. Siegel sets. In dimensions d>2 it is difficult to describe the geometry of a fundamental domain F. To overcome this problem, C. Siegel introduced simply connected sets Sd ⊂ G which have the property that they contain F and are contained in a finite number of translates PF, P ∈ . Consider the Iwasawa decomposition M = nak,
(2.22)
⎛ ⎞ ⎞ a1 . . . u 1d ⎜ .. ⎟ .. ⎟ .. ⎜ ⎟ . . . ⎟ ⎜ ⎟, ⎟, a = (2.23) ⎜ ⎟ ⎟ .. .. ⎝ . u d−1,d ⎠ . ⎠ ad 1 and k ∈ SO(d), with u i j , a j ∈ R, a j > 0, a1 · · · ad = 1. Then √ 3 a j+1 > 0 ( j = 1, . . . , d − 1), k ∈ SO(d) (2.24) Sd = nak : n ∈ F N , a j ≥ 2 where
⎛
1 u 12 ⎜ .. ⎜ . n=⎜ ⎜ ⎝
is an example of a Siegel set [29]; here F N denotes a compact fundamental region of ( ∩ N )\N , where N is the upper triangular group of elements of the form n as above. 2.5. Dani’s correspondence. We assume from now on that r1 , . . . , rd−1 = −1, rd = d − 1, i.e.,
(2.25) E t = diag e−t , . . . , e−t , e(d−1)t . Let us denote by | · | the maximum norm in Rd−1 . A vector α ∈ Rd−1 is called badly approximable or of bounded type, if one of the following equivalent conditions is satisfied: (i) There exists a constant C > 0 such that |kα + m|d−1 |k| > C
(2.26)
for all m ∈ Zd−1 , k ∈ Z − {0}. (ii) There exists a constant C > 0 such that |m|d−1 |m · α + k| > C
(2.27)
− {0}, k ∈ Z. for all m ∈ The statements (i) and (ii) are equivalent in view of Khintchine’s transference principle ([4] Chap. V). We recall Dani’s correspondence in the following proposition (cf. [5], Theorem 2.20). Zd−1
Proposition 2.1. The orbit { M E t : t ≥ 0}, with M as in (2.6), is bounded in \G if and only if the vector α is of bounded type. The reason why the parameters A, β, γ are irrelevant in the statement is that the family of matrices A 0 −t W (t) = E Et (2.28) β γ is bounded in G for all t ≥ 0. The boundedness of the orbit { M E t : t ≥ 0} implies of course that there is a compact set C ∈ G such that M(t) ∈ C for all t ≥ 0, with M(t) as in (2.5).
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2.6. Diophantine conditions. A vector α ∈ Rd−1 is called Diophantine, if there exist constants > 0, C > 0 such that |m|(d−1)(1+) |m · α + k| > C
(2.29)
for all m ∈ Zd−1 − {0}, k ∈ Z. It is well known that Diophantine vectors form a set of full Lebesgue measure [4]. Let us show that (2.29) implies the inequality k (d−1)(1+) |k · ω| > C,
(2.30)
for all k ∈ Zd − {0}, where ω = ( α1 ), cf. (1.6). With k = ( m k ), (2.29) yields |k · ω| > C|m|−(d−1)(1+) ≥ C|k|−(d−1)(1+) ≥ C k −(d−1)(1+)
(2.31)
for all m ∈ Zd−1 − {0}, k ∈ Z. In the case when m = 0, we have k = 0 (since k = 0) and thus (2.30) holds trivially.4 Note also that (2.30) evidently implies (2.29), however with different choices for C in both inequalities. Following [17] we define the following function on G: δ(M) =
inf
kM .
k∈Zd −{0}
(2.32)
It is easily checked that δ(M) is invariant under left action of , and may thus be viewed as a function on \G. In terms of the Iwasawa parametrization (2.22) and the Siegel set Sd defined in (2.24) we have the following estimate. Lemma 2.2. For M = nak ∈ Sd as in (2.22), (2.24), there are constants 0 < C1 ≤ C2 such that for all 0 < ad ≤ 1, C1 ad ≤ δ(M) ≤ C2 ad .
(2.33)
Proof. Since x |x| x for all x ∈ Rd , we may prove the statement of the lemma for the function δ˜d (M) = inf kM (2.34) k∈Zd −{0}
instead.5 Due to the rotational invariance of the Euclidean distance we may assume that k ∈ SO(d) is the identity. Proof by induction. The statement trivially holds for d = 1. Therefore let us assume the assertion is true for dimension d − 1. The j th coefficient of the vector kM is
kM
⎛
j
= ⎝k j +
j−1
⎞ ki u i j ⎠ a j .
(2.35)
i=1
4 Note that every admissible constant in (2.29) needs to satisfy C < 1/2; to see this, choose m = (1, 0, . . . , 0), and k ∈ Z such that |α1 + k| ≤ 1/2. 5 In the following, A B means ‘there is a constant C > 0 such that A ≤ C B’. If A B A we will also use the notation A B.
Multidimensional Renormalization
207
Since a1 → ∞ when ad → 0, this implies that when taking the infimum in (2.32) we must take k1 = 0 for all sufficiently small ad . Thus we now need to estimate ⎛ ⎞ j−1 ⎝ ⎠ max k j + ki u i j a j inf k˜ ∈Zd−1 −{0} 2≤ j≤d i=1 ⎛ ⎞ j−1 −1/(d−1) ⎝ ⎠ = a1 (2.36) inf max k j + ki u i j a˜ j , k˜ ∈Zd−1 −{0} 2≤ j≤d i=1 1/(d−1) where k˜ = (k2 , . . . , kd ), a˜ j = a1 a j so that a˜ 2 · · · a˜ d = 1. Now ⎛ ⎞ j−1 ˜ δ˜d−1 ( M), ˜ max ⎝k j + ki u i j ⎠ a˜ j =: δd−1 ( M) inf k˜ ∈Zd−1 −{0} 2≤ j≤d i=1
where M˜ = n˜ a˜ with
⎛
1 u 23 ⎜ .. ⎜ . n˜ = ⎜ ⎜ ⎝
⎞ . . . u 2d .. ⎟ .. . . ⎟ ⎟, ⎟ .. . u d−1,d ⎠ 1
⎛ a˜ 2 ⎜ .. ⎜ . a˜ = ⎜ ⎜ .. ⎝ .
(2.37)
⎞ ⎟ ⎟ ⎟. ⎟ ⎠
(2.38)
a˜ d
It is easily checked that M˜ ∈ Sd−1 , so by the induction hypothesis, for suitable constants 0 < C1,d−1 ≤ C2,d−1 , we have ˜ ≤ C2,d−1 a˜ d , C1,d−1 a˜ d ≤ δ˜d−1 ( M) 1/(d−1)
provided a˜ d = a1 have
(2.39) 1/(d−1)
ad ≤ 1. So for ad sufficiently small and a1 C1,d−1 ad ≤ δ˜d (M) ≤ C2,d−1 ad .
ad ≤ 1, we (2.40)
In the remaining case a˜ d > 1, all a˜ j are bounded from above and below by positive ˜ is bounded from above and below by positive constants. constants, and hence δ˜d−1 ( M) −1/(d−1) < ad , and, in view of our choice of the Siegel Furthermore a˜ d > 1 implies a1 −1 d−1 set, a1 = a2 · · · ad ad . So −1/(d−1)
ad a1
< ad
and the required bound follows from (2.36) also for the case a˜ d > 1.
(2.41)
Lemma 2.3. Choose M as in (2.6) and suppose α satisfies condition (2.29). Then there exists a constant C > 0 such that for all t ≥ 0, δ(M E t ) > C e−θt ,
(2.42)
(d − 1) . d + (d − 1)
(2.43)
where θ=
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K. Khanin, J. Lopes Dias, J. Marklof
Proof. Let us put k = ( m, k) with m ∈ Zd−1 and k ∈ Z. Then
−t δ(M E t ) = inf me , ( mα + k)e(d−1)t W (t) (m,k)∈Zd −{0}
−t inf me , ( mα + k)e(d−1)t , (m,k)∈Zd −{0}
(2.44)
since W (t), as defined in (2.28), is bounded in G for all t ≥ 0. Furthermore for t sufficiently large inf
(m,k)∈Zd −{0}
|( me−t , ( mα+k)e(d−1)t )| =
inf
m∈Zd−1−{0}, k∈Z
|( me−t , ( mα+k)e(d−1)t )| (2.45)
which, in view of the Diophantine condition (2.29), is bounded from below by ≥
inf
m∈Zd−1−{0}
|( me−t , C|m|−(d−1)(1+) e(d−1)t )| = e−θt
inf
m∈Zd−1 −{0}
|( x, C|x|−(d−1)(1+) )|, (2.46)
where x = e(θ−1)t m. We conclude the proof by noting that inf
m∈Zd−1 −{0}
|( x, C|x|−(d−1)(1+) )| ≥
inf
y∈Rd−1 −{0}
|( y, C| y|−(d−1)(1+) )| > 0. (2.47)
The fact that = 0 implies θ = 0 is consistent with Dani’s correspondence. On the other hand, θ < 1 for any < ∞. 2.7. Norm estimates. Let · denote the usual matrix norm M x . x =0 x
M := sup
(2.48)
Proposition 2.4. Choose M = M (0) as in (2.6), and suppose α satisfies condition (2.29). Then there are constants c1 , c2 , c3 , c4 , c5 , c6 > 0 such that for all n ∈ N ∪ {0}, M (n) ≤ c1 exp[(d − 1)θ tn ], M
(n) −1
(2.49)
≤ c2 exp(θ tn ),
(2.50)
P (n) ≤ c3 exp[(d θ + 1 − θ )tn ], P
(n) −1
T T
(n)
(n) −1
(2.51)
≤ c4 exp[(d − 1 + θ )tn ],
(2.52)
≤ c5 exp[(1 − θ )δtn + d θ tn ],
(2.53)
≤ c6 exp[(d − 1)(1 − θ )δtn + d θ tn ].
(2.54)
Proof. For any M ∈ Sd as in (2.22) we have, for all 0 < ad ≤ 1, −(d−1)
M a1 = (a2 · · · ad )−1 ad and
M −1 ad−1 .
,
(2.55) (2.56)
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209
Combine this with Lemmas 2.2 and 2.3 to obtain the bounds −1 −1
−1 −1 M (n) C2 δ M (n) = C2 δ M (0) E tn < C2 C exp(θ tn )
(2.57)
and M (n) C2d−1 δ(M (n) )−(d−1) = C2d−1 δ(M (0) E tn )−(d−1) < C2d−1 C
−(d−1)
exp[(d − 1)θ tn ].
(2.58)
The remaining estimates follow immediately from (2.49), (2.50) and the equations P (n) = M (n) E −tn M (0)
−1
T (n) = M (n) E −δtn M (n−1)
,
−1
.
(2.59)
Proposition 2.5. Choose M = M (0) as in (2.6), and suppose α satisfies condition (2.29). Then there is a constant c7 > 0 such that for all n ∈ N ∪ {0}, 2 d − (d − 1) tn ≤ |γ (n) | ≤ c1 exp[(d − 1)θ tn ] (2.60) c7 exp − θ 1−θ with c1 as in (2.49). Proof. The upper bound for |γ (n) | follows from (2.49), since γ (n) = (M (n) )dd and hence |γ (n) | ≤ M (n) . From (2.17) and the Diophantine condition (2.29) we have (n)
(n)
(n)
|γ (n) | = γ exp[(d − 1)tn ]| p21 α + p22 )| > C exp[(d − 1)tn ]| p21 |−(d−1)(1+) . (2.61) Since 1+ = and
d −1+θ (d − 1)(1 − θ )
(n)
| p21 | ≤ P (n)
the proposition follows from the estimate (2.51).
(2.62)
(2.63)
2.8. Hyperbolicity of the transfer matrices. Let d (n) = 0} ω(n) ⊥ = {ξ ∈ R : ξ · ω
be the orthogonal complement of the vector α (n) (n) ω = ∈ Rd . 1 (n−1)
Lemma 2.6. For all ξ ∈ ω⊥
(n) −1
T
(2.64)
(2.65)
, n ∈ N,
−1 ξ = exp(−δtn ) M (n−1) M (n) ξ .
(2.66)
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K. Khanin, J. Lopes Dias, J. Marklof
Proof. This follows directly from the relation A(n−1) ξ δtn (n−1) = exp(−δtn ) M (n−1) ξ , M ξ = exp(−δtn ) E 0
(2.67)
where ξ ∈ Rd−1 comprises the first d − 1 components of ξ . Proposition 2.7. Choose M = M (0) as in (2.6), and suppose α satisfies condition (2.29). (n−1) Then there is a constant > 0 such that for all ξ ∈ ω⊥ , n ∈ N, T (n) with
−1
ξ ≤
1
exp(−ϕn ) ξ 2
(2.68)
ϕn = (1 − θ )δtn − d θ tn−1 .
(2.69)
Proof. From Lemma 2.6, T (n)
−1
ξ ≤ exp(−δtn ) M (n−1) M (n)
and the proposition follows from the bounds (2.49), (2.50).
−1
ξ ,
(2.70)
Given any positive sequence ϕ0 , ϕ1 , . . ., the values tn that solve Eq. (2.69) with t0 = 0 are tn =
n 1 (1 + β)n− j ϕ j , 1−θ
(2.71)
dθ . 1−θ
(2.72)
j=1
where β=
E.g., for constant ϕn = ϕ > 0, we have nϕ tn = ϕ n dθ [(1 + β) − 1]
(θ = 0) (0 < θ < 1).
(2.73)
2.9. The resonance cone. As we shall see, a crucial step in our renormalization scheme is to eliminate all far-from-resonance modes in the Fourier series, i.e., all modes labeled by integer vectors outside the cone K (n) = {ξ ∈ Rd : |ξ · ω(n) | ≤ σn ξ }
(2.74)
for a given σn > 0. Lemma 2.8. Choose M = M (0) as in (2.6), and suppose α satisfies condition (2.29). Then −1 T (n) ξ
sup ≤ + c6 σn−1 ed δtn exp − (1 − θ )δtn + d θ tn−1 , (2.75) ξ 2 ξ ∈K (n−1) −{0} for all n ∈ N.
Multidimensional Renormalization
211
Proof. We write ξ = ξ 1 + ξ 2 , where ξ1 =
ξ · ω(n−1) (n−1) ω , ω(n−1) 2
(n−1)
ξ 2 ∈ ω⊥
.
(2.76)
Firstly, |ξ · ω(n−1) | −1 ≤ σn−1 T (n) ξ , ω(n−1) (2.77) since ξ ∈ K (n−1) and ω(n−1) = ( α (n−1) , 1) ≥ 1. Hence in view of (2.54), T (n)
−1
ξ 1 ≤ T (n)
T (n)
−1
−1
ξ 1 = T (n)
−1
ξ 1 ≤ c6 σn−1 exp[(d − 1)(1 − θ )δtn + d θ tn ] ξ .
(2.78)
Secondly, from Proposition 2.7 we infer T (n) This proves (2.75).
−1
ξ 2 ≤
1
exp − (1 − θ )δtn + d θ tn−1 ξ . 2
(2.79)
Remark 2.9. Note that if the tn are chosen as in (2.73), and σn−1 ≤ then T (n) for all ξ ∈
K (n−1) ,
1 −1 c exp(−d δtn ), 2 6
−1
ξ ≤ exp(−ϕ) ξ
(2.80)
(2.81)
n ∈ N and ϕ > 0.
3. Renormalization of Vector Fields 3.1. Definitions. The transformation of a vector field X on a manifold M by a diffeomorphism ψ : M → M is given by the pull-back of X under ψ: ψ ∗ X = (Dψ)−1 X ◦ ψ. As the tangent bundle of the d-torus is trivial, T Td Td × Rd , we identify the set of vector fields on Td with the set of functions from Td to Rd , that can be regarded as maps of Rd by lifting to the universal cover. We will make use of the analyticity to extend to the complex domain, so we will deal with complex analytic functions. We will also be considering an extra variable related to a parameter. Remark 3.1. We will be using maps between Banach spaces over C with a notion of analyticity stated as follows (cf. e.g. [14]): a map F defined on a domain is analytic if it is locally bounded and Gâteux differentiable. If it is analytic on a domain, it is continuous and Fréchet differentiable. Moreover, we have a convergence theorem which is going to be used later on. Let {Fk } be a sequence of functions analytic and uniformly locally bounded on a domain D. If limk→+∞ Fk = F on D, then F is analytic on D.
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K. Khanin, J. Lopes Dias, J. Marklof
Let ρ, a, b > 0, r = (a, b) and consider the domain Dρ × Br , where Dρ = {x ∈ Cd : Im x < ρ/2π } for the norm u = i |u i | on Cd , and d−1 d Br = y = (y1 , . . . , yd ) ∈ C : |yi | < a and |yd | < b . (3.1) i=1
Take complex analytic functions f : Dρ × Br → Cd that are Zd -periodic on the first coordinate and on the form of the Fourier series f (x, y) = f k ( y)e2π i k· x . (3.2) k∈Zd
Its coefficients are analytic functions f k : Br → Cd with a continuous extension to the closure Br , endowed with the sup-norm: f k r = sup f k ( y) . y∈Br
The Banach spaces Aρ,r and Aρ,r are the subspaces of such functions such that the respective norms f ρ,r = f k r eρ k , k∈Zd
f ρ,r
=
(1 + 2π k ) f k r eρ k
k∈Zd
are finite. Also, write the constant Fourier mode of f ∈ Aρ,r through the projection E f ( y) = f (x, y)d x = f 0 ( y) (3.3) Td
into the projected space denoted by EAr . The norm of its derivative D f 0 is given by the operator norm D f 0 r = sup g r =1 D f 0 g r . Some of the properties of the above spaces are of easy verification. For instance, given any f, g ∈ Aρ,r we have: • f (x, y) ≤ f ρ,r ≤ f ρ,r where (x, y) ∈ Dρ × Br , • f ρ−δ,r ≤ f ρ,r with δ < ρ. In order to setup notations write, according to Sect. 2, ω(0) = ω ∈ Rd − {0}, λ0 = 1 and, for n ∈ N, ⎛ ⎞ 0 −1 . ω(n) = γ (n) M (n) ⎝ .. ⎠ = λn P (n) ω = ηn T (n) ω(n−1) , (3.4) 0 1
where λn =
γ γ (n)
e(d−1)tn and ηn =
λn . λn−1
(3.5)
In the following, we will be interested in equilibria-free vector fields with a “twist” along the parameter direction. By rescaling this direction we will find the right parameter
Multidimensional Renormalization
213
which guarantees the conjugacy to a linear flow. For a fixed n ∈ N ∪ {0}, we will be studying vector fields of the form X (x, y) = X n0 ( y) + f (x, y), where f ∈ Aρ,r and X n0 ( y) = ω(n) + γ (n)
(x, y) ∈ Dρ × Br , −1
M (n) y.
(3.6)
(3.7)
(We drop the second coordinate of the vector field because it will always be equal to zero – there is no dynamics along the parameter direction.) The linear transformation on y deforms the set Br along the directions of the columns of M (n) (see (2.15)). In particular, its d th column corresponds to ω(n) . For the space of the above vector fields we use the same notation Aρ,r and the same norm · ρ,r without ambiguity. 3.2. Resonance modes. Given σn > 0 we define the far from resonance Fourier modes f k as in (3.2) with respect to ω(n) to be the ones whose indices k are in the cone In− = {k ∈ Zd : |k · ω(n) | > σn k }.
(3.8)
Similarly, the resonant modes correspond to the cone In+ = Zd − In− .
(3.9)
It is also useful to define the projections I+n and I− n on Aρ,r and Aρ,r by restricting + − the Fourier modes to In and In , respectively. The identity operator is I = I+n + I− n. Moreover, take
An =
T (n+1) k k∈In+ −{0}
−1
sup
k
.
(3.10)
A useful property of the above cones is included in the lemma below. Lemma 3.2. If k ∈ In− and y ∈ Brn with rn = (an , bn ), an ≤ σn
1 − bn 2
then
|γ (n) | M (n) −1 and bn <
1 , 2
σ n k . k · X n0 ( y) > 2
(3.11)
(3.12)
Proof. For every y ∈ Brn and k ∈ In− , |k · (ω(n) + γ (n)
−1
M (n) y)| = |(1 + yd )k · ω(n) + γ (n)
−1
k · M (n) (y1 , . . . , yd−1 , 0)|
> (1 − bn )σn k − an |γ (n) |−1 M (n) k . Our choice of an yields (3.12).
(3.13)
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3.3. Basis change, time rescaling and reparametrization. The fundamental step of the renormalization is a transformation of the domain of definition of our vector fields. This is done by a linear change of basis (coming essentially from the multidimensional continued fraction expansion of ω – see Sect. 2), a linear rescaling of time because the orbits take longer to cross the new torus, and a change of variables for the parameter y in order to deal with the zero mode of the perturbation. Let ρn−1 , an−1 , bn−1 > 0, rn−1 = (an−1 , bn−1 ) and consider a vector field 0 ( y) + f (x, y), X (x, y) = X n−1
(x, y) ∈ Dρn−1 × Brn−1 ,
(3.14)
with f ∈ Aρn−1 ,rn−1 . We are interested in the following coordinate and time linear changes: x → T (n)
−1
x,
t → ηn t.
(3.15)
Notice that negative time rescalings are possible, meaning that we are inverting the direction of time. In addition to (3.15) we will use a transformation on y, a map y → n (X )( y) depending on X in a way to be defined later. Therefore, consider the transformation
−1 L n (x, y) = T (n) x, n (X )( y) , (x, y) ∈ C2d , (3.16) that determines a vector field in the new coordinates as the image of the map X → Ln (X ) = ηn L ∗n X. That is, for (x, y) ∈ L −1 n Dρn−1 × Brn−1 , Ln (X )(x, y) = ηn T (n) [ω(n−1) + γ (n−1)
−1
M (n−1) n (X )( y) + f 0 ◦ n (X )( y)]
+ ηn T (n) ( f − f 0 ) ◦ L n (x, y).
(3.17)
In order to eliminate the k = 0 mode of the perturbation of X in the new coordinates −1 and to normalise the linear term in y to γ (n) M (n) y, using the definitions of T (n) and ηn we choose −1
−1 e−dδtn y1 , . . . , e−dδtn yd−1 , yd , (3.18) n (X ) : y → Id +γ (n−1) M (n−1) f 0 if possible. Hence,
where
n ( f − f 0 )(x, y), Ln (X )(x, y) = X n0 (x, y) + L
(3.19)
n : f → ηn T (n) f ◦ L n . L
(3.20)
Denote by μ the set of X ∈ Aρn−1 ,rn−1 such that f 0 rn−1 < μ. Lemma 3.3. Let rn = (an , bn ) and μn−1 > 0 such that
−1 an ≤ edδtn an−1 − 1 + |γ (n−1) | M (n−1) μn−1 ,
−1 bn ≤ bn−1 − 1 + |γ (n−1) | M (n−1) μn−1 .
(3.21)
Multidimensional Renormalization
215
There exist an analytic map n : μn−1 → Diff(Brn , Cd ) such that, for each X ∈ μn−1 , n (X ) is given by (3.18) and n (X )(Brn ) ⊂ Brn−1 .
(3.22)
In case f 0 is real-analytic, n (X )|Rd is also real-valued. Proof. For X ∈ Aρn−1 ,rn−1 with f 0 rn−1 < μn−1 and δn−1 = (δn−1 , δn−1 ) with δn−1 = μn−1 |γ (n−1) | M (n−1)
−1
,
(3.23)
we have by the Cauchy estimate D f 0 rn−1 −δn−1 ≤
f 0 rn−1 1 . < −1 (n−1) δn−1 |γ | M (n−1)
−1
So, F = Id +γ (n−1) M (n−1) f 0 is a diffeomorphism on Brn−1 −δn−1 . Now, if R1 < an−1 − δn−1 − μn−1 , R2 < bn−1 − δn−1 − μn−1 and R = (R1 , R2 ), we have B R ⊂ F(Brn−1 −δn−1 ) and F −1 (B R ) ⊂ Brn−1 −δn−1 . Therefore, n (X ) as given by (3.18) is a diffeomorphism on Brn by choosing R = (e−dδtn an , bn ), and thus we get (3.22). In addition, X → n (X ) is analytic from its dependence on f 0 . When restricted to a real domain for a real-analytic f 0 , n (X ) is also real-analytic. Let the translation R z on C2d be defined for z ∈ Cd and given by R z : (x, y) → (x + z, y).
(3.24)
Notice that we have the following “commutative” relation: L ∗n R ∗z = RT∗ (n) z L ∗n ,
z ∈ Cd .
(3.25)
This also follows from the fact that n is unchanged by the introduction of the translation R z .
3.4. Analyticity improvement. Lemma 3.4. If δ > 0 and ρn ≤
ρn−1 − δ, An−1
(3.26)
n as a map from (I+ − E)Aρn−1 ,rn−1 ∩ μn−1 into (I − E)A is continuous then L n−1 ρn ,rn and compact with n ≤ |ηn | T (n) 1 + 2π . (3.27) L δ Remark 3.5. This result means that every vector field in I+n−1 Aρn−1 ,rn−1 ∩ μn−1 , i.e. a function on Dρn−1 × Brn−1 into Cd , has an analytic extension to T (n)
−1
Dρn × Brn−1 .
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Proof. Let f ∈ (I+n−1 − E)Aρn−1 ,rn−1 ∩ μn−1 . Then, n
(n) −1 k −1 1 + 2π T (n) k f k ◦ Mn rn e(ρn −δ+δ) T .
f ◦ L n ρ ,rn ≤
+ −{0} k∈In−1
By using the relation ξ e−δ ξ ≤ δ −1 with ξ ≥ 0, (3.10) and (3.22), we get f k rn−1 e An−1 (ρn +δ) k f ◦ L n ρ ,rn ≤ (1 + 2π/δ) n
(3.28)
+ −{0} In−1
≤ (1 + 2π/δ) f ρn−1 ,rn−1 .
(3.29)
n f ≤ |ηn | T (n) f ◦ L n . Finally, L ρn ,rn ρn ,rn The above for Dρn × Brn is also valid for Dζ × Brn , ζ > ρn but satisfying a similar n = I ◦ J , where J : (I+ − E)Aρn−1 ,rn−1 → A inequality to (3.26). Therefore, L ζ,rn n−1 n , and the inclusion map I : A → A is compact. is bounded as L ζ,rn ρ ,rn n
For 0 <
ρn
≤
ρn ,
consider the inclusion In : Aρ ,rn → Aρ ,rn n
(3.30)
n
by restricting X ∈ Aρ ,rn to the smaller domain Dρn × Brn . When restricted to nonn constant modes, its norm can be estimated as follows: Lemma 3.6. If φn ≥ 1 and then
0 < ρn ≤ ρn − log(φn ),
(3.31)
In (I − E) ≤ φn−1 .
(3.32)
Proof. For f ∈ (I − E)Aρ ,rn , we have n
In ( f ) ρ ,rn ≤ n
k =0
(1 + 2π k ) f k rn eρn k φn− k ≤ φn−1 f ρ ,rn . n
(3.33)
3.5. Elimination of far from resonance modes. The theorem below (to be proven in Appendix A) states the existence of a nonlinear change of coordinates U , isotopic to the identity, that cancels the In− modes of any X as in (3.6) with sufficiently small f . We are eliminating only the far from resonance modes, this way avoiding the complications usually related to small divisors. We remark that the “parameter” direction y is not affected by this change of coordinates. For given ρn , rn , ε, ν > 0, denote by Vε the open ball in Aρn +ν,rn centred at X n0 with radius ε. Theorem 3.7. Let rn be as in (3.11), σn < ω(n) and ν σn σn εn = min , . 42 4π 72 ω(n)
(3.34)
Multidimensional Renormalization
217
For all X ∈ Vεn there exists an isotopy Ut : Dρn × Brn → Dρn +ν × Brn , (x, y) → (x + u t (x, y), y),
(3.35)
of analytic diffeomorphisms with u t in Aρn ,rn , t ∈ [0, 1], satisfying ∗ − I− n Ut X = (1 − t) In X,
U0 = Id .
(3.36)
This defines the maps Ut : Vεn → Aρn ,rn , X → Id +u t ,
(3.37)
and Ut : Vεn → I+ Aρn ,rn ⊕ (1 − t)I− n Aρn +ν,rn ,
X → Ut∗ X,
(3.38)
which are analytic, and satisfy the inequalities 42t − I f ρn ,rn , σn n ≤(3 − t) f ρn +ν,rn .
Ut (X ) − Id ρn ,rn ≤ Ut (X ) −
X n0 ρn ,rn
(3.39)
If X is real-analytic, then Ut (X )(R2d ) ⊂ R2d . Remark 3.8. Further on we will be using the above result for t = 1. So that all far from resonance modes are eliminated. Recall the definition of the translation R z in (3.24). Lemma 3.9. In the conditions of Theorem 3.7, if x ∈ Rd and X ∈ Vεn , then Ut (X ◦ R x ) = R x−1 ◦ Ut (X ) ◦ R x
(3.40)
on Dρn ,rn . Proof. Notice that R x (Dρn × Brn ) = Dρn × Brn . If Ut = Ut (X ) is a solution of the homotopy equation (3.36) on Dρn × Brn , then U˜ t = R x−1 ◦ Ut (X ) ◦ R x solves the same − ˜ ˜ ˜ equation for X˜ = X ◦ R x , i.e. I− n X ◦ Ut = (1 − t)In X , on Dρn × Brn . 3.6. Trivial limit of renormalization. Let a sequence of “widths” 0 < σn < 1 of the resonance cones In+ be given. The n th step renormalization operator is thus Rn = Un ◦ In ◦ Ln ◦ Rn−1 and R0 = U0 ,
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K. Khanin, J. Lopes Dias, J. Marklof
where Un is the full elimination of the modes in In− as in Theorem 3.7 (for t = 1). Notice that Rn (X 0 + v) = X n0 , for every v ∈ Cd . From the previous sections the map Rn on its domain is analytic. Also, in case a vector field X is real-analytic, the same is true for Rn (X ). Fix the constants ν and δ as in Theorem 3.7 and Lemma 3.4, respectively, and choose 0 < λ < 1. Take ⎧ ⎫ σn+1 |γ (n+1) | ⎬ σn |γ (n) | ⎨ n 2 − dδt (n) (n+1) λ σn M e n+1 M n = min εn , $n , λn (3.41) −1 2 −1 (i) (n) (n) ⎩ T 1 + |γ | M ⎭ i=1
by assuming that the sequence of times tn guarantees that n > 0. Now, write Bn =
n (
Ai ,
(3.42)
i=0
with Ai given by (3.10). By recalling the inequalities (3.26) and (3.31) we choose, for a given ρ0 > 0, ) * n−1 n−1 1 ρn = Bi log (φi+1 ) − (δ + ν) Bi , ρ0 − (3.43) Bn−1 i=0
where
i=0
n−1 ,1 ≥ 1 φn = max 2|ηn | T (n) (1 + 2π δ −1 ) n
(3.44)
is to be used in Lemma 3.6. Define the following function for every ω ∈ Rd associated to the choice of σn : B(ω) =
+∞
Bi log (φi+1 ) + (δ + ν)
i=0
+∞
Bi .
(3.45)
i=0
The convergence of the renormalization scheme now follows directly from our construction. Theorem 3.10. Suppose that
B(ω) < +∞
(3.46)
and ρ > B(ω) + ν. There is K , b > 0 and rn = (an , bn ) with an > 0 and bn > b > 0, such that if X is in a sufficiently small open ball around X 0 in Aρ,r0 , then (i) X is in the domain of Rn and Rn (X ) − Rn (X 0 ) ρn ,rn ≤ K n X − X 0 ρ,r0 , n ∈ N ∪ {0},
(3.47)
(ii) for each |s| < b there exists in Brn−1 ⊂ Cd the limits pns (X ) = lim n (Rn−1 (X )) . . . m (Rm−1 (X ))(0, . . . , 0, s) m→+∞
and
lim pns (X ) − (0, . . . , 0, s) = 0,
n→+∞
(3.48)
(3.49)
Multidimensional Renormalization
219
(iii) the map X → pn (X ) is analytic and takes any real-analytic X into an analytic curve s → pns (X ) in Rd . Proof. Let ξ > 0 and ρ0 = ρ − ν − ξ > 0 such that ρ0 > B(ω). Hence, by (3.43), we −1 for all n ∈ N. have R > 0 satisfying ρn > R Bn−1 Denote by c the radius of an open ball in Aρ,r0 centred at X 0 and containing X . If c ≤ ε0 we can use Theorem 3.7 to obtain R0 (X ) ∈ I+0 Aρ0 ,r0 with r0 = (a0 , b0 ) satisfying (3.11) and R0 (X ) − R0 (X 0 ) ρ0 ,r0 ≤ 2 X − X 0 ρ+ξ,r0 ≤ 2ξ −1 X − X 0 ρ,r0 . Let K = 2(ξ 0 )−1 and assume that c ≤ K −1 min{b0 (1 − λ), 21 − b0 }. So, (3.47) holds for n = 0. Now, with n ∈ N we choose the following rn : an = σn
1 − b0 2
n−1 |γ (n) | = b − cK λi , and b n 0 M (n)
(3.50)
i=0
so that 1/2 > bn > b = b0 −cK (1−λ)−1 . The inequalities in (3.11) follow immediately. Moreover, (3.21) is also satisfied with μn−1 = cK n−1 because
1 −1 − b0 1 + |γ (n−1) | M (n−1) n−1 an−1 − e−dδtn an ≥ 2
−1 ≥ 1 + |γ (n−1) | M (n−1) cK n−1 , (3.51) bn−1 − bn = cK λn−1
−1 ≥ 1 + |γ (n−1) | M (n−1) cK n−1 . Suppose that X n−1 = Rn−1 (X ) ∈ I+n−1 Aρn−1 ,rn−1 and 0 X n−1 − X n−1 ρn−1 ,rn−1 ≤ K n−1 X − X 0 ρ,r .
Since (3.21) holds, Lemmas 3.3 and 3.4 are valid and, together with (3.19) and Lemma 3.6, can be used to estimate In ◦ Ln (X n−1 ): In ◦ Ln (X n−1 ) − X n0 ρ ,rn ≤ |ηn | T (n) (1 + 2π δ −1 )φn−1 K n−1 X − X 0 ρ,r0 n
=
1 K n X − X 0 ρ,r0 . 2
(3.52)
This vector field is inside the domain of Un as (3.11) and 21 c K n < εn are satisfied. Thus (3.47) follows from (3.39). (n) Denote by f 0 the constant mode of the perturbation term of X n . By Lemma 3.3, n (X n−1 ) : Brn → Brn−1 is given by y → (Id +gn ) diag(e−dδtn , . . . , e−dδtn , 1) y, where
−1 (n−1) −1 − Id gn = Id +γ (n−1) M (n−1) f 0
(3.53)
220
K. Khanin, J. Lopes Dias, J. Marklof
is defined on Brn with rn = (e−dδtn an , bn ). So, for z ∈ Brn there is ξ ∈ Brn such that −1 (n−1) f0 (0)] − z −1 −1 (n−1) (n−1) (n−1) = −[I +γ (n−1) M (n−1) D f 0 (ξ )]−1 γ (n−1) M (n−1) [D f 0 (ξ ) z+ f 0 (0)]
gn (z) = [I + γ (n−1) M (n−1)
−1
D f 0(n−1) (ξ )]−1 [z − γ (n−1) M (n−1)
(3.54) and gn rn ≤
|γ (n−1) | M (n−1) 1 − |γ (n−1) | M
(n−1) −1
−1
(n−1) D f 0 rn
rn D f 0(n−1) rn + f 0(n−1) rn . (3.55)
The choice of rn means that
(n−1) | σ |γ n−1 min{an−1 − e−dδtn an , bn−1 − bn } min , λn−1 . M (n−1)
(3.56)
By using (3.41) and the Cauchy estimate, D f 0(n−1) rn ≤
(n−1)
rn−1 f0 λn−1 . (3.57) −1 min{an−1 − e−dδtn an , bn−1 − bn } |γ (n−1) | M (n−1)
Thus,
gn rn λn−1 .
(3.58)
Writing ys = (0, . . . , 0, s), by induction we have m n (X n−1) . . . m (X m−1)( ys ) = ys + diag(e−d(ti−1 −tn−1 ) , . . . , e−d(ti−1 −tn−1 ) ,1)gi (ξ i), i=n
(3.59) for some ξ k ∈ Brk . Therefore, from (3.58), there exists pns (X ) ∈ Cd unless X is real which clearly gives pns (X ) ∈ Rd . In addition, pns (X ) −
ys ≤
+∞ i=n
gi ri
λn−1 . 1−λ
(3.60)
The maps X → pns (X ) are analytic since the convergence is uniform. Lemma 3.3 gives us the nested sequence n (X n−1 )(Brn ) ⊂ Brn−1 . So, as ys ∈ ∩i∈N Bri , it follows that pns (X ) ∈ Brn−1 . Remark 3.11. The above can be generalised for a small analyticity radius ρ by consider = U N L N . . . U1 L1 U0 (X ), ing a sufficiently large N and applying the above theorem to X where X is close enough to X 0 . We recover the large strip case since ρ N is of the order of B N−1−1 . It remains to check that ρ N > B(ω(N ) ) + ν. This follows from the fact that B(ω(N ) ) = B N−1−1 [B(ω) − B N (ω)], where B N (ω) is the sum of the first N terms of B(ω) so that B N (ω) → B(ω) as N → +∞. Lemma 3.12. If ω = α1 in Rd is diophantine, i.e. α satisfies (2.29) with exponent (related to θ by (2.43) and to β by (2.72)), then (3.46) is verified.
Multidimensional Renormalization
221
Proof. Let us set δtn = ξ tn−1 , σn = exp(−cδtn ), n ≥ 1, where positive constants ξ, c will be chosen later. Obviously, tn = (1+ξ )tn−1 = [(1+ξ )/ξ ]δtn and δtn = (1+ξ )δtn−1 . We shall assume that c < d(1 + ξ ), (3.61) so that σn−1 exp(dδtn ) = exp(−cδtn−1 + dδtn ) = exp[(d − c/(1 + ξ ))δtn ] is much larger than given by Proposition 2.7. Hence, using (2.75) we have dθ c An−1 exp − (3.62) + d − (1 − θ ) + δtn . 1+ξ ξ −1
We next estimate ω(n) and εn . It follows from (3.4) that ω(n) M (n) |γ (n) |. Thus, using (2.49), (2.60) we have θ 1+ξ θ (3.63) d 2 tn = exp d2 δtn . ω(n) exp 1−θ 1−θ ξ Since ω(n) ≥ 1 one gets from (3.34) that εn ∼ σn2 / ω(n) which together with (3.63) implies θ 21+ξ d δtn εn exp(−2cδtn ). (3.64) exp −2c − 1−θ ξ Here X ∼ Y means that there exist two positive constants C1 , C2 > 0 such that C1 Y < X < C2 Y . Using again (2.49), (2.60) we get σn |γ (n) | θ 21+ξ (3.65) d δtn . exp −c − 1−θ ξ M (n) Also, since M (n+1) ≥ γ (n+1) , σn+1 |γ (n+1) | exp[−(c + d)(1 + ξ )δtn ]. M (n+1) exp(dδtn+1 )
(3.66)
We shall assume that c and ξ are chosen in such a way that −c − so that
θ 1+ξ d2 > −(c + d)(1 + ξ ), 1−θ ξ
σn |γ (n) | σn+1 |γ (n+1) | σn |γ (n) | − . M (n) M (n+1) exp(dtn+1 ) M (n)
(3.67)
(3.68)
Inequality (3.67) is equivalent to the following condition: c>
θ 1+ξ 2 1+ξ d. d − 1 − θ ξ2 ξ
(3.69)
Finally, we want An to be small and, hence, require the exponent in (3.62) to be negative dθ c + d − (1 − θ ) + < 0. (3.70) − 1+ξ ξ
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Suppose that conditions (3.61), (3.69), (3.70) are satisfied. It follows immediately from the estimates above and (3.5), (2.49), (2.50), (2.54), (2.60) that | log n |, | log n−1 |, log T (n) , | log |ηn || δtn . At the same time Bn =
n (
Ai C n exp(−αtn+1 ),
(3.71)
(3.72)
i=0
where α=
c dθ − d + (1 − θ ) − > 0. 1+ξ ξ
(3.73)
Since Bn decays exponentially with tn and log φn grows at most linearly the series (3.46) converges. To finish the proof it is enough to show that conditions (3.61), (3.69), (3.70) can be satisfied. Indeed, since 0 < θ < 1 we can choose ξ so large that 1−θ −dθ/ξ > 0 and θ 1+ξ 2 1+ξ d < 0. (3.74) d − 1 − θ ξ2 ξ It is easy to see that all three inequalities (3.61), (3.69), (3.70) are satisfied if (1 + ξ )(d − β) < c < (1 + ξ )d, where β = 1 − θ − dθ/ξ > 0. 3.7. Analytic conjugacy to linear flow. As a consequence of Theorem 3.10, we obtain an analytic conjugacy between a vector field and the linear flow, thus proving Theorem 1.1. In the following we always assume the conditions of Sect. 3.6. Let r = r0 and = {X ∈ Aρ,r : X − X 0 ρ,r < c} (3.75) inside the domain of Rn for all n ∈ N ∪ {0}. By taking X ∈ , we denote X n = Rn (X ) ∈ I+n Aρn ,rn so that X n = λn (U0 ◦ L 1 ◦ U1 · · · L n ◦ Un )∗ (X ),
(3.76)
where Uk = Uk (Ik Lk (X k−1 )) is given by Theorem 3.7 for t = 1 at the k th step and L k is the linear rescaling as in (3.16) for X k−1 . Denote by Vn the coordinate change
−1 Vn : (x, y) → P (n) x, 1 (X 0 ) . . . n (X n−1 )( y) (3.77) −1 ◦ Vn and and set V0 = Id. Thus, L n = Vn−1
∗ ∗
−1 X n = λn (Vn ◦ Un )∗ Vn−1 ◦ Un−1 ◦ Vn−1 · · · V1 ◦ U1 ◦ V1−1 U0∗ (X ). (3.78)
In particular, the y-coordinate is only transformed by the second component of Vn . Notice that if X n = X n0 for some n ∈ N, y = 1 (X 0 ) . . . n (X n−1 )(0, . . . , 0, s) ∈ Cd , with |s| < b, corresponds to the parameter for which X is conjugated to (1 + s) ω(n) . The parameter value for the general case X n − X n0 → 0 as n → +∞ is p s (X ) = p1s (X ).
Multidimensional Renormalization
223
Lemma 3.13. There is an open ball B about X 0 in such that we can find a sequence Rn > 0 satisfying R−1 = ρ, 1/2
Rn + 2π 42K n X − X 0 ρ,r ≤ Rn−1 ≤ and
lim Rn−1 n
1/2
n→+∞
ρn−1 , X ∈ B, P (n−1)
= 0.
(3.79)
(3.80)
$n 1/2 Proof. Let ρ∗ = min ρn . It is enough to check that n λn ρ∗ i=1 T (i) −1 with 1/2 0 < λ < 1 and taking Rn = cλ−n n for some positive constant c. This immediately implies (3.80) and (3.79) by considering a small enough upper bound for X − X 0 ρ,r . Let Diff per (Dζ , Cd ), ζ > 0, be the Banach space of Zd -periodic diffeomorphisms g : Dζ → Cd with finite norm g ζ = k g k eζ k , where g k ∈ Cd are the coefficients of the Fourier representation. It is simple to check that g ◦ P (n) Rn ≤ g ρn . Denote by u n the analytic function u n : → Diff per (Dρn , Cd ) s X → Un (In Ln (X n−1 ))(·, pn+1 (X )).
(3.81)
s (X ) ∈ B , D ×{ y = p s (X )} is inside the domain D As pn+1 rn ρn ρn ,rn of Un (In Ln (X n−1 )) n+1 given in Theorem 3.7. Now, for each X , define the isotopic to the identity diffeomorphism
Wn (X ) = P (n)
−1
◦ u n (X ) ◦ P (n) ,
(3.82)
−1
on P (n) Dρn . If X is real-analytic, then Wn (X )(Rd ) ⊂ Rd , since this property holds for u n (X ). We also have Wn (X 0 ) = Id. Lemma 3.14. For all n ∈ N ∪ {0}, Wn : B → Diff per (D Rn , Cd ) is analytic satisfying Wn (X ) : D Rn → D Rn−1 and 1/2
Wn (X ) − Id Rn ≤ 42K n X − X 0 ρ,r , X ∈ B.
(3.83)
Proof. For any X ∈ , in view of (3.39) we get −1
Wn (X ) − Id Rn = P (n) ◦ [u n (X ) − Id] ◦ P (n) Rn 42 (n) −1 ≤ P In Ln (X n−1 ) − X n0 ρn ,rn . σn We can bound the above by (3.83). Now, for x ∈ D Rn and X ∈ B ⊂ , Im Wn (X )(x) ≤ Im(Wn (X )(x) − x) + Im x < Wn (X ) − Id Rn + Rn /2π ≤ Rn−1 /2π. So we have Wn (X ) : D Rn → D Rn−1 and Wn (X ) ∈ Diff per (D Rn , Cd ). From the properties of Un , Wn is analytic as a map from B into Diff per (D Rn , Cd ).
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Consider the analytic map Hn : B → Diff per (D Rn , Cd ) defined by the coordinate transformation Hn (X ) : D Rn → Dρ0 as Hn (X ) = W0 (X ) ◦ · · · ◦ Wn (X ).
(3.84)
Lemma 3.15. There exists c > 0 such that for X ∈ B and n ∈ N, 1/2
Hn (X ) − Hn−1 (X ) Rn ≤ cn X − X 0 ρ,r .
(3.85)
Proof. For each k = 0, . . . , n − 1, consider the transformations G k (z, X ) = (Wk (X ) − Id) ◦ (Id +G k+1 (z, X )) + G k+1 (z, X ), G n (z, X ) = z(Wn (X ) − Id), with (z, X ) ∈ {z ∈ C : |z| < 1 + dn } × B, where we have c > 0 such that dn =
c 1/2
n X − X 0 ρ,r
− 1 > 0.
If the image of D Rn under Id +G k+1 (z, X ) is inside the domain of Wk (X ), or simply G k+1 (z, X ) Rn ≤ (Rk − Rn )/2π, then G k is well-defined as an analytic map into Diff per (D Rn , Cd ), and G k (z, X ) Rn ≤ Wk (X ) − Id Rk + G k+1 (z, X ) Rn . An inductive scheme shows that G n (z, X ) Rn ≤(Rn−1 − Rn )/2π, G k (z, X ) Rn ≤
n−1
Wi (X ) − Id Ri + |z| Wn (X ) − Id Rn
i=k
≤(Rk−1 − Rn )/2π. By Cauchy’s formula Hn (X ) − Hn−1 (X ) Rn = G 0 (1, X ) − G 0 (0, X ) Rn + + + + , + 1 G 0 (z, X ) + + + dz + =+ + 2πi z(z − 1) + + + |z|=1+dn /2
, Rn
and Hn (X ) − Hn−1 (X ) Rn ≤
2 dn
sup
|z|=1+dn /2 1/2
G 0 (z, X ) Rn
n X − X 0 ρ,r .
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225
Consider C 1per (Rd , Cd ) to be the Banach space of the Zd -periodic C 1 functions between Rd and Cd with norm f C 1 = max max D k f (x) .
(3.86)
k≤1 x ∈Rd
Lemma 3.16. There exists C > 0, an open ball B ⊂ B about X 0 and an analytic map H : B → Diff per (Rd , Cd ) such that for X ∈ B , H (X ) = limn→+∞ Hn (X ) and H (X ) − Id C 1 ≤ C X − X 0 ρ,r .
(3.87)
If X ∈ B is real-analytic, then H (X ) ∈ Diff per (Rd , Rd ). Proof. As the domains D Rn are shrinking, we consider the restrictions of Wn (X ) and Hn (X ) to Rd , and estimate their C 1 norms from the respective norms in Diff per (D Rn , Cd ). More precisely, for any X ∈ B, making use of Lemma 3.15, Hn (X ) − Hn−1 (X ) C 1 ≤ max sup D k [Hn (X )(x) − Hn−1 (X )(x)] k≤1 x ∈D R /2 n
≤
4 Hn (X ) − Hn−1 (X ) Rn , Rn
(3.88)
which goes to zero by (3.80). Notice that here we have used Cauchy’s estimate D 1 g ζ ≤ (2π/eδ) g ζ +δ with ζ, δ > 0. Therefore, the existence of the limit Hn (X ) → H (X ) is shown as n → +∞, in the Banach space C 1per (Rd , Cd ). Moreover, H (X ) − Id C 1 X − X 0 ρ,r . The convergence of Hn is uniform in B so H is analytic. As the space of close to identity diffeomorphisms is closed for the C 1 norm, H (X ) is a diffeomorphism for any X sufficiently close to X 0 , i.e. X ∈ B . The fact that, for real-analytic X , H (X ) takes real values for real arguments, follows from the same property of each Wn (X ). To simplify notation, write π y X = X (·, y). Lemma 3.17. For every real-analytic X ∈ B and |s| < b, [H (X )]∗ (π ps (X ) X ) = (1 + s) ω on Rd . Proof. For each n ∈ N the definition of Hn (X ) and (3.78) imply that ∗
−1 Hn (X )∗ (π ps (X ) X ) = λ−1 (X n ). n π p s (X ) Vn
(3.89)
The r.h.s. can be written as −1
−1
∗
(n) −1 0 [ω(n) +γ (n) M (n) n (X n−1 )−1· · · 1 (X 0 )−1 p s (X )]+λ−1 λ−1 n P n π p s (X ) Vn (X n−X n ) (n) = (1 + s) ω + λ−1 n P
−1 (n) −1
γ
∗
s −1 M (n) pn+1 (X ) − s ω + λ−1 (X n − X n0 ). n π p s (X ) Vn (3.90)
Its terms can be estimated, for x ∈ Rd , by ∗
−1 (n) λ−1 (X n − X n0 )(x) ≤ |λ−1 n π p s (X ) Vn n | P
−1
1/2
X n − X n0 ρn ,rn n ,
(3.91)
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K. Khanin, J. Lopes Dias, J. Marklof
and 1 M (0) −1 s s P (n) M (n) [ pn+1 (X ) − (0, . . . , 0, s)] = E tn [ pn+1 (X ) − (0, . . . , 0, s)] (n) |λn γ | |γ |e(d−1)tn s pn+1 (X ) − (0, . . . , 0, s) , (3.92)
which is controlled by (3.49). Consequently, the limit of (3.89) as n → +∞ is (1 + s) ω. Using the convergence of Hn we complete the proof. Lemma 3.18. If X ∈ B and x ∈ Rd , then H (X ◦ R x ) = R x−1 ◦ H (X ) ◦ R x ,
(3.93)
where R x : z → z + x is a translation on Cd . Proof. For each n ∈ N, (3.40) and (3.25) yield that Un (X ◦ R x ) = Un (X ) ◦ R x and Ln (X ◦ R x ) = Ln (X ) ◦ RT (n) x . This implies immediately that Rn (X ◦ R x ) = Rn (X ) ◦ R P (n) x . Next, from a simple adaptation of (3.40) and the formula R P (n) z = P (n) R z P (n) z ∈ Cd , we get
(3.94) −1
for
−1
Wn (X ◦ R x ) = P (n) ◦ Uan (Ln Rn−1 (X ◦ R x )) ◦ P (n) = R x−1 ◦ Wn (X ) ◦ R x .
(3.95)
Thus, Hn (X ◦ R x ) = R x−1 ◦ Hn (X ) ◦ R x . The convergence of Hn implies (3.93).
Theorem 3.19. If v ∈ Vect ω (Td ) is sufficiently close to ω, then there exists an analytic curve p : (−b, b) → Rd , s → ps , and h ∈ Diff ω (Td ) homotopic to the identity such that h ∗ (v + ps ) = (1 + s)ω. (3.96) The maps v → p and v → h are analytic. Proof. The lift v to Rd of v is assumed to have an analytic extension in Dρ . Consider the real-analytic vector field Y (x, y) = v(x) + y in Aρ,r . Suppose that v is close enough to ω such that Y ∈ B and Y ◦ R z ∈ B for some η > 0 and z ∈ Dη . Then, the parameter ps = p s (Y ) ∈ Rd and the C 1 -diffeomorphism h = H (Y ) mod 1 verify (3.96). We now want to extend h analytically to a complex neighbourhood of its domain. Take h(z) = z + H (Y ◦ R z )(0), z ∈ Dη . The maps z → Y ◦ R z and X → H (X ) are analytic and C 1per (Rd , Cd ) g → g(0) is bounded. As h involves their composition, it is analytic on the domain Dη and Zd -periodic. From (3.93), for any x ∈ Rd , we have h(x) mod 1 = (x + R x−1 ◦ H (Y ) ◦ R x (0)) mod 1 = (x + H (Y )(x) − x) mod 1 = h(x).
(3.97)
The extension of h is a complex analytic diffeomorphism, thus h is a real-analytic diffeomorphism.
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227
Appendix A. Homotopy Method for Vector Fields In this section we prove Theorem 3.7 using the homotopy method (cf. [22]). As n is fixed, we will drop it from our notations. In addition we write ρ = ρn and ρ = ρn + ν. We will be using the symbol D x for the derivative with respect to x. Firstly, we include a technical lemma that will be used in the following. Lemma A.1. Let f ∈ Aρ,r . If U = Id +(u, 0), where u : Dρ × Br → D(ρ−ρ )/2 is in Aρ ,r and u ρ ,r < (ρ − ρ )/4π, then • f ◦ U ρ ,r ≤ f (ρ+ρ )/2,r , • D x f ◦ U ≤ f (ρ+ρ )/2,r , • f ◦ U − f ρ ,r ≤ f (ρ+ρ )/2,r u ρ ,r , • D x f ◦ U − D x f ≤
4π ρ−ρ f ρ,r
u ρ ,r .
The proof of these inequalities is straightforward and thus will be omitted. Now, assume that δ = 42ε/σ < 1/2. For vector fields in the form X = ω + π2 + f , where π2 : (x, y) → y is seen as a function in Aρ,r , consider f to be in the open ball in Aρ,r centred at the origin with radius ε. The coordinate transformation U is written as U = Id +(u, 0), with u in . B = u ∈ I− Aρ ,r : u : Dρ × Br → Dρ , u ρ ,r < δ . Notice that we have I− U ∗ (X ) = I− (DU )−1 (ω + π2 + f ◦ U, 0) = (I− (I + D x u)−1 (ω + π2 + f ◦ U ), 0). From now on the parameter r is omitted whenever there is no ambiguity. Define the operator F : B → I− Aρ , F(u) = I− (I + D x u)−1 (ω + π2 + f ◦ U ).
(A.1)
F(u) takes real values for real arguments whenever u has that property. It is easy to see that the derivative of F at u is the linear map from I− Aρ to I− Aρ : D F(u) h = I− (I + D x u)−1 [D x f ◦ U h −D x h (I + D x u)−1 (ω + π2 + f ◦ U )].
(A.2)
We want to find a solution of F(u t ) = (1 − t)F(u 0 ),
(A.3)
with 0 ≤ t ≤ 1 and “initial” condition u 0 = 0. Differentiating the above equation with respect to t, we get du t D F(u t ) = −F(0). (A.4) dt Proposition A.2. If u ∈ B, then D F(u)−1 is a bounded linear operator from I− Aρ to I− Aρ and D F(u)−1 < δ/ε.
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From the above proposition (to be proved in Sect. A.1) we integrate (A.4) with respect to t, obtaining the integral equation: t ut = − D F(u s )−1 F(0) ds. (A.5) 0
In order to check that u t ∈ B for any 0 ≤ t ≤ 1, we estimate its norm: u t ρ ≤ t sup D F(v)−1 F(0) ρ v∈B
≤ t sup D F(v)−1 I− f ρ < tδ f ρ /ε, v∈B
so, u t ρ < δ. Therefore, the solution of (A.3) exists in B and is given by (A.5). Moreover, if X is real-analytic, then u t takes real values for real arguments. It is now easy to see that (−D(Ut − Id))n X 0 + I+ Ut∗ f + (1 − t)I− f. Ut∗ X − X 0 = I+ n≥2
So, using Lemma A.1, 1 2 ( ω u t ρ + f ρ ) + (1 − t) f ρ 1 − u t ρ 1 2 δ ω f ρ /ε2 + 1 f ρ + (1 − t) f ρ < 1−δ 2 δ ω 1 + 1 + 1 − t f ρ . < 1−δ ε
Ut∗ X − X 0 ρ ≤
Moreover, Ut∗ X − X 0 − I+ f − (1 − t)I− f ρ = O( f 2ρ ), hence the derivative of X → Ut∗ X at X 0 is I − tI− . A.1. Proof of Proposition A.2 Lemma A.3. If f ρ < ε < σ/6, then D F(0)−1 : I− Aρ → I− Aρ is continuous and D F(0)−1 <
3 . σ − 6 f ρ
Proof. From (A.2) one has D F(0) h = I−( f − Dω ) h = − I − I− f Dω−1 Dω h, where f h = D f h − Dh f and Dω h = D x h (ω +π2 ). Thus, the inverse of this operator, if it exists, is given by
−1 D F(0)−1 = −Dω−1 I − I− f Dω−1 .
Multidimensional Renormalization
229
The inverse of Dω is the linear map from I− Aρ to I− Aρ : g k ( y) Dω−1 g(x, y) = e2π i k· x , 0 ( y) 2π ik · X − k∈I
and is well-defined since Lemma 3.2 implies that |k · X 0 ( y)| > σ k /2, with k ∈ I − and y ∈ Br . So, + + 1 + 2π k + g k ( y) + ρ k −1 +e + sup + Dω g ρ = 2π k · X 0 ( y) + k∈I −
y∈Br
1 + 2π k 3 g k r eρ k ≤ g ρ . < π σ k σ − k∈I
Hence, D −1 ω
< 3/σ . It is possible to bound from above the norm of f : I− Aρ → Aρ
by f ≤ 2 f ρ . Therefore, I− f Dω−1 < σ6 f ρ < 1, and +
−1 + + + σ +< + I − I− f D −1 ω + σ − 6 f . + ρ
The statement of the lemma is now immediate.
Lemma A.4. Given u ∈ B, the linear operator D F(u) − D F(0) mapping I− Aρ into I− Aρ , is bounded and ) u ρ 4 − 2 u ρ 4π D F(u) − D F(0) < + f ρ 1 − u ρ ρ − ρ 1 − u ρ * 2 − u ρ + ω + π2 . 1 − u ρ Proof. The formula (A.2) gives [D F(u) − D F(0)] h = I− (I + D x u)−1 [D x f ◦ U h − (I + D x u)D x f h −D x h (I + D x u)−1 (ω + π2 + f ) ◦ U +(I + D x u)D x h (ω + π2 + f )] = I− (I + D x u)−1 {A + B + C}, where A = [D x f ◦ U − D x f − D x u D x f ] h, B = D x u D x h (ω + π2 + f ), C = −D x h (I + D x u)−1 [ f ◦ U − f − D x u (ω + π2 + f )] . Using Lemma A.1, 4π A ρ ≤ f u + f u ρ ρ ρ ρ h ρ , ρ − ρ B ρ ≤ ω + π2 + f ρ u ρ h ρ , 1 f (ρ+ρ )/2 u ρ + u ρ ω + π2 r + f ρ h ρ . C ρ ≤ 1 − u ρ
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To conclude the proof of Proposition A.2, notice that
−1 D F(u)−1 ≤ D F(0)−1 −1 − D F(u) − D F(0) −1 σ δ 4π 4 − 2δ 2−δ − 2ε − ω + π < + ε + 2 r 3 1−δ ρ − ρ 1−δ 1−δ δ < . ε The last inequality is true if −1 δ2 σ 2δ 4π 4 − 2δ ε<δ − ω + π2 r 1 + 2δ + + 3 (1 − δ)2 1 − δ ρ − ρ 1−δ with a positive numerator N and denominator D in the r.h.s. This is so for our choices of ε and δ < 21 , by observing that π2 r = sup y ≤ a |γ (n) |−1 M (n) + b ω < y∈Br
1 ω , 2
(A.6)
thus ω + π2 r < 23 ω and 2δ σ ω + π2 r < 12δ ω < . 2 (1 − δ) 6 So, N > δσ/6, D < 7, and finally ε ≤
σ2 42 ω
<
σ 42
< N /D.
Acknowledgements. We would like to express our gratitude to D. Kleinbock, H. Koch, R. S. MacKay, G. Margulis, Ya. Sinai and J.-C. Yoccoz for useful discussions and comments. JLD was supported by Fundação para a Ciência e a Tecnologia, and JM by an EPSRC Advanced Research Fellowship. We would also like to thank the Isaac Newton Institute, CAMGSD/IST and Cemapre/ISEG (through FCT’s Program POCTI/FEDER) for travel support.
References 1. Abad, J.J., Koch, H.: Renormalization and periodic orbits for Hamiltonian flows. Commun. Math. Phys. 212, 371–394 (2000) 2. Ávila, A., Krikorian, R.: Reducibility or non-uniform hyperbolicity for quasiperiodic Schrödinger cocycles. Ann. of Math., to appear 3. Bricmont, J., Gaw¸edzki, K., Kupiainen, A.: KAM theorem and quantum field theory. Commun. Math. Phys. 201, 699–727 (1999) 4. Cassels, J.W.S.: An Introduction to Diophantine Approximation. Cambridge: Cambridge University Press, 1957 5. Dani, S.G.: Divergent trajectories of flows on homogeneous spaces and Diophantine approximation. J. Reine Angew. Math. 359, 55–89 (1985) 6. Feigenbaum, M.J.: Quantitative universality for a class of non-linear transformations. J. Stat. Phys. 19, 25–52 (1978) 7. Gaidashev, D.G.: Renormalization of isoenergetically degenerate Hamiltonian flows and associated bifurcations of invariant tori. Discrete Contin. Dyn. Syst. 13, 63–102 (2005) 8. Gallavotti, G.: Twistless KAM tori. Commun. Math. Phys. 164, 145–156 (1994) 9. Gentile, G., Mastropietro, V.: Methods for the analysis of the Lindstedt series for KAM tori and renormalizability in classical mechanics. A review with some applications. Rev. Math. Phys. 8, 393–444 (1996) 10. Grabiner, D.J., Lagarias, J.C.: Cutting sequences for geodesic flow on the modular surface and continued fractions. Monatsh. Math. 133, 295–339 (2001)
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11. Hardcastle, D.M.: The three-dimensional Gauss algorithm is strongly convergent almost everywhere. Experiment. Math. 11, 131–141 (2002) 12. Hardcastle, D.M., Khanin, K.: The d-dimensional Gauss transformation: strong convergence and Lyapunov exponents. Experiment. Math. 11, 119–129 (2002) 13. Herman, M.R.: Inégalités “a priori” pour des tores lagrangiens invariants par des difféomorphismes symplectiques. Inst. Hautes Études Sci. Publ. Math. 70, 47–101 (1989) 14. Hille, E., Phillips, R. S.: Functional analysis and semi-groups, Volume 31. AMS Colloquium Publications, Rev. ed. of 1957, Providence, RI:Amer. Math. soc., 1974 15. Khanin, K., Lopes Dias, J., Marklof, J.: Renormalization of multidimensional Hamiltonian flows. Nonlinearity (2006) (to appear) Preprint, 2005, available at http://pascal.iseg.utl.pt/∼jldias/KLM.pdf 16. Khanin, K., Ya. Sinai.: The renormalization group method and Kolmogorov-Arnold-Moser theory. In: Sagdeev, R.Z. ed., Nonlinear phenomena in plasma physics and hydrodynamics, Moscow: Mir, 1986 pp. 93–118 17. Kleinbock, D.Y., Margulis, G.A.: Flows on homogeneous spaces and Diophantine approximation on manifolds. Ann. of Math (2), 148, 339–360 (1998) 18. Koch, H.: A renormalization group for Hamiltonians, with applications to KAM tori. Erg. Theor. Dyn. Syst. 19, 475–521 (1999) 19. Koch, H.: On the renormalization of Hamiltonian flows, and critical invariant tori. Discrete Contin. Dyn. Syst. 8(3), 633–646 (2002) 20. Koch, H.: A renormalization group fixed point associated with the breakup of golden invariant tori. Discrete Contin. Dyn. Syst. 11, 881–909 (2004) 21. Lagarias, J.C.: Geodesic multidimensional continued fractions. Proc. London Math. Soc. 69, 464–488 (1994) 22. Lopes Dias, J.: Renormalization of flows on the multidimensional torus close to a K T frequency vector. Nonlinearity 15, 647–664 (2002) 23. Lopes Dias, J.: Renormalization scheme for vector fields on T2 with a diophantine frequency. Nonlinearity 15, 665–679 (2002) 24. Lopes Dias, J.: Brjuno condition and renormalization for Poincaré flows. Discrete Contin. Dyn. Syst. 15, 641–656 (2006) 25. MacKay, R.S.: Renormalisation in area-preserving maps. River Edge, NJ: World Scientific Publishing Co. Inc., 1993 26. MacKay, R.S.: Three topics in Hamiltonian dynamics. In: Aizawa, Y., Saito, S., Shiraiwa, K. (eds.) Dynamical Systems and Chaos, Volume 2. Singapore: World Scientific, 1995 27. Moore, C.C.: Ergodicity of flows on homogeneous spaces. Am. J. Math. 88, 154–178 (1966) 28. Moser, J.: Stable and random motions in dynamical systems. Annals Math. Studies, Princeton, NJ: Princeton Univ. Press, 1973 29. Raghunathan, M.S.: Discrete subgroups of Lie groups. Berlin-Heidelberg-New York Springer-Verlag, 1972 30. Schweiger, F.: Multidimensional continued fractions. Oxford Science Publications. Oxford: Oxford University Press, 2000 31. Yoccoz, J.-C.: Petits diviseurs en dimension 1 (Small divisors in dimension one). Astérisque 231, 1995 32. Yoccoz, J.-C.: Analytic linearization of circle diffeomorphisms. In: Dynamical systems and small divisors, Eds. J. Marmi, C. Yoccoz, Lecture Notes in Mathematics, 1784, Berlin-Heidelberg-New York: Springer-Verlag, 2002 Communicated by G. Gallavotti
Commun. Math. Phys. 270, 233–265 (2007) Digital Object Identifier (DOI) 10.1007/s00220-006-0126-x
Communications in
Mathematical Physics
Nonequilibrium Statistical Mechanics and Entropy Production in a Classical Infinite System of Rotators David Ruelle1,2 1 Mathematics Dept., Rutgers University, 110 Frelinghuysen Road, Piscataway, NJ 08854-8019, USA 2 IHES, 91440 Bures sur Yvette, France. E-mail:
[email protected]
Received: 31 March 2006 / Accepted: 4 April 2006 Published online: 7 October 2006 – © Springer-Verlag 2006
Abstract: We analyze the dynamics of a simple but nontrivial classical Hamiltonian system of infinitely many coupled rotators. We assume that this infinite system is driven out of thermal equilibrium either because energy is injected by an external force (Case I), or because heat flows between two thermostats at different temperatures (Case II). We discuss several possible definitions of the entropy production associated with a finite or infinite region, or with a partition of the system into a finite number of pieces. We show that these definitions satisfy the expected bounds in terms of thermostat temperatures and energy flow. 0. Introduction In the present paper, we study certain classical Hamiltonian systems consisting of an infinite number of coupled degrees of freedom (rotators or “little wheels”). For a system in the class considered, the time evolution ( f t ) is well defined, and given by the limit (in some sense) of the Hamiltonian time evolution for finite subsystems. [Note that other infinite systems, like gases of interacting particles, would be much more difficult to control.] A probability measure on the phase space of the infinite system is called a state, and it has a well-defined time evolution. We introduce a family of initial states called -states (they are Gibbs states of some sort). Some of these -states describe a situation where parts of our infinite system (thermostats) are at given temperatures. For a -state , the time-evolved state f t gives a finite Gibbs entropy S t (X ) to each finite subsystem X of the infinite system L. If X is infinite (but has finite interaction with the rest of the system) the difference S t (X ) = limY →∞ (S t (X ∩ Y ) − S 0 (X ∩ Y )) still makes sense. The bulk of the paper is dedicated to a discussion of the (nontrivial) dynamics of our infinite system of rotators. Understanding the dynamics of the system is a necessary prerequisite to analyzing its nonequilibrium statistical mechanics. We shall in fact examine a specific nonequilibrium problem: is it possible to define a local rate of entropy
234
D. Ruelle
production (associated with a finite region X ) in a nontrivial manner? This possibility has been suggested by Denis Evans and coworkers [16]. We examine their proposal and some alternatives, but obtain only partial results. Because of the obvious physical interest of the problem, we now give some details. By time-averaging f t or dS t (X )/dt (over a suitable sequence of intervals [0, T ] → ∞) we may define a nonequilibrium steady state 1 ρ = lim T →∞ T
T dt f t 0
and an average rate of entropy growth σ (X ) = lim
T →∞
1 S T (X ) T
(we do not know that σ (X ) is uniquely determined by ρ and X ). We ask if an entropy production rate e(X ) can be meaningfully associated with a finite set X ⊂ L. For definiteness we shall think of two physical situations. In Case I there is a finite set X 0 such that an external force acts on X 0 , and the initial state restricted to L\X 0 corresponds to thermal equilibrium at temperature β −1 . In Case II we have L = X 0 L 1 L 2 , where X 0 is finite, L 1 and L 2 are infinite and restricted to L i corresponds to thermal equilibrium at temperature βi−1 (with β1−1 < β2−1 ). There is a thermodynamic formula for the global rate of entropy production: (Case I), e = β × energy flux to thermostat e = (β1 − β2 ) × energy flux to thermostat 1 (Case II). [Note that in our Case II, the energy flux to thermostat 2 is minus the energy flux to thermostat 1. Case I resembles Case II, with thermostat 2 replaced by the external force, and ascribed an infinite temperature (β2 = 0).] The question is how to define a local rate of entropy production e(X ) ≥ 0 such that sup X finite e(X ) = e . The original proposal by Evans and coworkers1 is to take, for X finite, e(X ) = −σ (X ). This is shown to be the average rate of volume contraction in the phase space [X ] of the subsystem X due to the fluctuating forces to which it is subjected by the complementary subsystem L\X . Another idea is to replace the entropy S(X ) by the conditional entropy given formally ˇ ) = S(L) − S(L\X ). The corresponding rate of entropy production is by S(X e(X ˇ ) = σ (L\X ). 1 Actually, the ideas presented in [16] are formulated for Case I, and for the large volume limit of a system thermostatted at the boundary rather than for an actually infinite system. While the two idealizations are technically quite different, they are expected to give the same results in cases of physical interest. A proposal similar to that of [16], but with a continuous system limit instead of a large volume limit occurs in an unpublished note of G. Gallavotti [“Large deviations, fluctuation theorem and Onsager-Machlup theory in nonequilibrium statistical mechanics”].
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235
We shall make the important physical assumption that the expectation value of the energy for each finite system X has a bound independent of time2 . It follows that e(X ˇ ) is finite, and one has 0 ≤ e(X ) ≤ e(X ˇ ). Instead of using a finite set X one may base a definition of entropy production rate on a finite partition A = (X 0 , X 1 , . . . , X n ) of L, with finite boundary (this will be made precise later). We define e(A) =
n j=0
σ (X j ),
e(A) ˇ =
σ (X j ).
j:X j infinite
In particular, in Case II, for X finite ⊃ X 0 , we have e(X ˇ ) = e((X, ˇ L\X )) ≤ e((X, ˇ L 1 \X, L 2 \X )), and the right-hand side e((X, L 1 \X, L 2 \X )) seems a rather natural definition of entropy production rate. We shall later study further properties of the entropy production rates defined above, but we note here that they are all bounded by the thermodynamic expression e . The problem is to prove that they depend effectively on X or A, and are not identically equal to 0 or e . We now recall some earlier work to put the problem of defining a local entropy production rate in perspective. In studies of quantum spin systems [15, 11], using the infinite system dynamics proved to exist by D.W. Robinson, the global entropy production (for Case II) was defined by the thermodynamic relation e = (β1 − β2 ) × energy flux to thermostat 1, but the quantities e(X ), e(X ˇ ) were not introduced because they would automatically vanish. This is because, for quantum spin systems we have | Sˇ t (X )| < S t (X ) (see [3] Proposition 6.2.28(b)); for classical rotators by contrast, the entropy is not bounded below. The statistical mechanics of classical systems outside of equilibrium can be studied in models with nongradient forces and a “deterministic thermostat” [7, 10]. Such a nonhamiltonian system corresponds in effect to a rather general time evolution ( f t ) defined by a vector field X on a finite dimensional manifold M. In general, no absolutely continuous invariant measure (i.e., “phase space volume” m) on M is preserved by the time evolution, but one may assume that there is a natural (singular) measure ρ describing a nonequilibrium steady state. One can argue that the average phase space volume contraction ρ(d x)(−divm X )(x) is the rate of entropy production by the system. This identification (for which see Andrei [1]) has been used in particular by Evans, Cohen, and Morriss [6], and by Gallavotti and Cohen [9] in the study of fluctuations of the entropy production. See also the work of Posch and Hoover [13], Gallavotti [8]. Note now that if we introduce a nongradient force ξ(q) in the Hamiltonian equations of motion, the volume dp dq is preserved, but energy conservation is lost 2 Note that in Case I, if the system has dimension ≤ 2, the external force may cause an infinite accumulation of energy in a finite region. Our assumption that the nonequilibrium steady state ρ gives a finite expectation to the energy of finite subsystems is thus invalid, and so is our analysis.
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and this is why a thermostat is needed. In the case of a deterministic thermostat, the phase space contraction is caused by the thermostat (as one can check in the example of the isokinetic thermostat corresponding to an added “force” −α( p, q) p, where α( p, q) = p · ξ(q)/ p · p). In the lab however the thermostat is of a different nature: it is typically a large system (reservoir) with which the small system of interest can exchange heat, and it is not clear at first how to define entropy production. In particular, a nonequilibrium steady state for the infinite system L may well have absolutely continuous projection on the phase space of the small system X [4, 5, 2], which contradicts e(X ) > 0 but may allow e(X ˇ ) > 0. To indicate the difficulty of the problems considered here, and in particular of proving e(X ˇ ) > 0, consider Case II in dimension ≤ 2. There (as indicated by the macroscopic continuous limit), f t presumably tends to an equilibrium state ρ and the entropy production e(X ˇ ) vanishes for all X . 0.1. Discussion and conjectures. The thermodynamic formula for the global rate of entropy production e uses explicitly the thermostat temperature(s). Can one give a formula for e not involving temperature(s) explicitly? Yes, in the case of isokinetic thermostat(s): e is the rate of phase space volume contraction. The answer is less clear for the infinite Hamiltonian systems discussed in the present article, but one may conjecture that e is the sup over X or A of one or the other of the quantities e(X ), e(X ˇ ), e(A), e(A) ˇ introduced earlier, these quantities are candidates to describe a local rate of entropy production associated with a region X or a partition A. The proposed locality, originally formulated for e(X ), is the great interest of the paper [16], even though the existence of a nontrivial local entropy production is not yet clarified. Note that X is any finite set and need not be in any particular position with respect to the thermostat(s). The partition A may be used to separate a “small system” from one or more thermostats. It is known that the quantities e(X ), e(X ˇ ), e(A), e(A) ˇ are in the interval [0, e ], but not clear if they can take values strictly between 0 and e . In particular there are results [4, 5, 2] to the effect that the nonequilibrium steady state in L may have absolutely continuous projection to a finite region X , implying that S t (X ) is bounded, and that e(X ) = 0. It is not known that e(X ˇ ) = 0, but we do know that 0 ≤ e(X ) ≤ e(X ˇ ) ≤ e , and that e(X ˇ ) is a subadditive function of X , while e(X ) is superadditive (see Sect. 5.2). Among several possible conjectures, I would find most interesting the possibility that e(X ˇ ) is generally nonzero, with sup X e(X ˇ ) = e . What I would find most plausible, is that e(X ˇ ) = 0 in Case I when X ⊃ X 0 , and that e(X, ˇ L 1 \X, L 2 \X ) = 0 in Case II when X ⊃ X 0 . 1. Description of the Model Our system will be an infinite collection of rotators labelled by x ∈ L, each with Hamiltonian Hx ( px , qx ) = px2 /2 + Vx (qx ), where px ∈ R, qx ∈ T. [This is for simplicity; it would probably be easy to replace the rotators by more complicated systems.] We let be a set of unordered pairs {x, y} of points in L, i.e., is a graph with vertex set L, and we define a formal Hamiltonian for the infinite system of little wheels: Hx ( px , qx ) + W{x,y} (qx , q y ). x∈L
{x,y}∈
The functions Vx , W{x,y} are assumed to be smooth.
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For X ⊂ L, let X = {{x, y} ∈ : x, y ∈ X } and, when X is finite, write HX ( p X , q X ) =
Hx ( px , qx ) +
W{x,y} (qx , q y ),
{x,y}∈ X
x∈X
where p X = ( px )x∈X ∈ R X , q X = (qx )x∈X ∈ T X . We shall also make use of a constant external force3 F ∈ R X 0 acting on a finite set X 0 . For finite X , a time evolution ( f Xt ) on R X × T X is defined by d dt
pX qX
=
FX − ∂q X H X ( p X , q X ) , pX
where the term FX is the component of F in R X , and is present only in Case I. We have thus f Xt ( p X (0), q X (0)) = ( p X (t), q X (t)). We shall suppose that is connected and, for x, y ∈ L, define d(x, y) = min{k : ∃x0 , . . . , xk ∈ L with x0 = x, xk = y and {x j−1 , x j } ∈ for j = 1, . . . k}. We write then Bxk = {y : d(x, y) ≤ k}. Assumption 1.1 (Finite dimensionality). There is a polynomial P(k) such that for all x ∈ L and k ≥ 0, |Bxk | ≤ P(k). [We may take P(k) = 1 + ak b for some a, b > 0; is thus assumed to have order ≤ a, and “dimension” ≤ b.] Note that by compactness of T and the assumed smoothness of Vx , W{x,y} , every “force” term (i.e., each component Fx of F for x ∈ X 0 , each ∂qx Vx , each ∂qx W{x,y} , ∂q y W{x,y} ) is bounded, and has bounded derivatives with respect to its arguments qz . Assumption 1.2 (Uniform boundedness). The force terms ∂qx Vx , ∂qx W{x,y} , ∂q y W{x,y} and their qz -derivatives (up to any finite order) are bounded uniformly in x, y ∈ L. Lemma 1.3 (Uniform boundedness of forces). The forces Fx − ∂qx H X ( p X , q X ) or −∂qx H X ( p X , q X ) and their qz -derivatives have modulus bounded respectively by constants K , K . [We shall also denote by K¯ a constant ≥ 2K , P(1)K ,1.] This follows from Assumptions 1.1 and 1.2 [only the bounded order of is used from Assumption 1.1]. 3 F is taken constant for simplicity. More generally one could consider the case of a smooth function F(q X 0 , φ t α) of q X 0 and φ t α with values in R X 0 , where (φ t ) is a smooth dynamical system on a compact manifold A, and α is distributed according to some prescribed (φ t )-ergodic measure on A.
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2. Time Evolution of Infinite Systems For X ⊂ L, we shall from now on write [X ] = (R × T) X . We note the following facts which follow from Lemma 1.3. (i) For X finite and ξ ∈ [X ], if f Xt ξ = ( px (t), qx (t))x∈X we have the estimate | px (t) − px (0)| ≤ K |t| when x ∈ X. (ii) For X˜ finite and ξ˜ ∈ [ X˜ ] let also f t˜ ξ˜ = ( p˜ x (t), q˜ x (t))x∈ X˜ . Then, if px (0) = X p˜ x (0), qx (0) = q˜ x (0) for some x ∈ X ∩ X˜ we have | px (t) − p˜ x (t)| ≤ 2K |t|, |qx (t) − q˜ x (t)| ≤ K |t|2 , and since |qx (t) − q˜ x (t)| ≤ 1 ≤ K¯ , we also have |qx (t) − q˜ x (t)| ≤ [K |t|2 . K¯ ] 2 ≤ K¯ |t|, 1
so that max(| px (t) − p˜ x (t)|, |qx (t) − q˜ x (t)|) ≤ K¯ |t|. (iii) Let k ≥ 0 and X ⊃ Bxk , X˜ ⊃ Bxk . Then, with the notation of (ii), if (∀y ∈ Bxk )
p y (0) = p˜ y (0) and q y (0) = q˜ y (0)
we have max(| px (t) − p˜ x (t)|, |qx (t) − q˜ x (t)|) ≤
( K¯ |t|)k+1 , (k + 1)!
indeed, by the equation of motion and induction on k we have |
d t ( K¯ |t|)k ( f X ξ − f Xt ξ˜ )| ≤ K¯ , dt k!
and the desired result follows by integration. Lemma 2.1 (A priori estimates) . For finite X, X˜ ⊂ L , let ξ ∈ [X ], ξ˜ ∈ [ X˜ ], and f Xt ξ = (ξx (t))x∈X = ( px (t), qx (t)), f t˜ ξ˜ = (ξ˜x (t))x∈ X˜ = ( p˜ x (t), q˜ x (t)). With this X notation, (a) | px (t) − px (0)| ≤ K |t|, hence | px (t)| ≤ | px (0)| + K |t|, (b) if k > 0, and Bxk−1 ⊂ X ∪ X˜ , and ξ y (0) = ξ˜ y (0) for all y ∈ Bxk−1 , then |ξx (t) − ξ˜x (t)| = max[| px (t) − p˜ x (t)|, |qx (t) − q˜ x (t)|] ≤
( K¯ |t|)k . k!
This follows from (i) and (iii) above [(b) is a rather rough estimate, but sufficient for our purposes].
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Proposition 2.2 (Time evolution). Let ( px (0), qx (0))x∈L ∈ [L] be given. For finite X ⊂ L, write ( pxX (t), qxX (t))x∈X = f Xt ( pxX (0), qxX (0))x∈X . Then for each x ∈ L the limits lim
d(x,L\X )→∞
pxX (t) = px (t),
lim
d(x,L\X )→∞
qxX (t) = qx (t)
exist, and ( px (t), qx (t))x∈L is the unique solution of the infinite system evolution equation with initial condition ( px (0), qx (0))x∈L . We write ( px (t), qx (t))x∈L = f t ( px (0), qx (0))x∈L . The existence of the limit follows from Lemma 2.1. Writing the infinite system evolution equation is left to the reader, as well as checking that ( px (t), qx (t))x∈L is the unique solution. Remark 2.3. The limits in Proposition 2.2 are faster than exp(−k d(x, L\X )) for any k > 0, independently of ( px (0), qx (0))x∈L , and uniformly for t in any compact interval [−T, T ]. Existence and uniqueness theorems are known in more difficult situations; see for instance [12]. The proof of Proposition 2.2 does not use the finite dimensionality of , only its finite order. Notation 2.4. In principle we use the notation ( pxX (t), qxX (t))x∈X for the finite system time evolution ( f Xt ), and ( px (t), qx (t))x∈L for the infinite system evolution ( f t ), but it will often be convenient to drop the superscript X . ˙ by addition of a point It is useful to compactify the momentum space R to a circle R ˙ × T) X for X ⊂ L. The phase space of at infinity for each x ∈ L, and write [ X˙ ] = (R ˙ × T) L = [ L]. ˙ We shall use the product our infinite system is then [L] = (R × T) L ⊂ (R L ˙ ˙ ˙ is compact and [L] topologies on [L] = (R × T) and [ L] = (R × T) L ; therefore [ L] ˙ has the topology it inherits as subset of [ L]. If U ⊂ L we denote by πU the projection ˙ = [U˙ ] × [ L\U ˙ ] → [U˙ ]. πU : [ L] Proposition 2.5 (Continuity of f t ). The map (ξ, t) → f t ξ is continuous [L]×R → [L] and, for each t, f t : [L] → [L] is a homeomorphism. To prove the continuity of (ξ, t) → f t ξ , it suffices to prove the continuity of (ξ, t) → ( px (t), qx (t)) for each x ∈ L, and this results from the uniformity of the limits in Proposition 2.2 (see Remark 2.3). By uniqueness of f t , the map f −t is the inverse of f t and, since f −t : [L] → [L] is continuous, f t is a homeomorphism. Proposition 2.6 (Smoothness of f t ). Let X ⊂ Y, X finite and Y finite or = L. For ξ ∈ [X ], η ∈ [Y \X ], write f Yt (ξ, η) = ( px (t), qx (t))x∈Y . Then, for fixed η and each x ∈ Y, the map (ξ, t) → ( px (t), qx (t)) is smooth [X ] × R → R × T. This results from the bounds on the derivatives (uniform in Y ) obtained in Proposition 2.7 below. Proposition 2.7 (Estimate of derivatives). Let Y ⊂ L , Y finite or = L , and f Yt (ηx (0))x∈Y = (ηx (t))x∈Y = ( px (t), qx (t)).
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Write (i, j)
rx
(t; x1 , . . . , x j ) =
∂ ∂i ∂ ··· ηx (t), ∂t i ∂ηx1 (0) ∂ηx j (0)
where x1 , . . . , x j need not be all distinct; then (i, j)
|r x
(t; x1 , . . . , x j )| ≤ Pi j ((| p y (0)| + K |t|)),
where Pi j (( p y )) is a polynomial of degree ≤ i + 1 (the degree is 1 if (i, j) = (0, 0), ≤ i otherwise) in the p y with y ∈ Bxi . Let σ = σ (x, x1 , . . . , x j ) denote the smallest number of edges of a connected subgraph of having x, x1 , . . . , x j among its vertices. Then the coefficients of Pi j are positive and ¯
≤ Mi j e j K |t|
(L j |t|)τ τ!
with suitable L j , Mi j > 0, for all τ such that 0 ≤ τ ≤ [σ − i]+ , where we have written [σ − i]+ = max(0, σ − i). The proof is given in Appendix A.1. Proposition 2.8 (Estimate of differences). We use the notation of Proposition 2.7. Let (i, j) (i, j) ˜ For finite X ⊂ L (ηx (0)), (η˜ x (0)) ∈ [Y ] and define r˜x as r x with η replaced by η. we assume η y (0) = η˜ y (0) when y ∈ / X, and write (i, j)
r x
(i, j)
(t; x1 , . . . , x j ; X ) = r x
(i, j)
(t; x1 , . . . , x j ; X ) − r˜x
(t; x1 , . . . , x j ; X ).
If d(x, X ) > i, we have (i, j)
|r x
(t; x1 , . . . , x j ; X )| ≤ Q((| p y (0)| + K |t|)),
where Q(( p y )) is a polynomial of degree ≤ i + 1 (the degree is 1 if (i, j) = (0, 0), ≤ i otherwise) in the p y with y ∈ Bxi . Let σ = σ (x, x1 , . . . , x j ; X ) denote the smallest number of edges of a subgraph of (not necessarily connected) connecting each point x, x1 , . . . , x j to some point of X . Then the coefficients of Qi j are positive and ¯
≤ Mi j e j K |t|
(L j |t|)τ τ!
for all τ such that 0 ≤ τ ≤ σ − i The proof is given in Appendix A.2. Remark 2.9. Propositions 2.7, 2.8 will be used in the proof of Theorem 4.5 below. In view of these applications the following facts should be noted. (a) The condition d(x, X ) > i in Proposition 2.8 is not a serious limitation because, for the finitely many values of x such that d(x, X ) ≤ i, one can estimate r (i, j) by Proposition 2.7 applied to r (i, j) and r˜ (i, j) .
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(b) Write σ = σ (y, y1 , . . . , y j ) and let y be fixed, then y1 ,...,y j
1 < ∞. (σ − i)!
Indeed, we have |yk − y| ≤ σ , hence σ −i ≥
j |yk − y| − i , j k=1
so that 1/(σ − i)! decreases faster than exponentially with respect to r1 = r |y1 − y|, . . . , r j = |y j − y|, while |B yr1 | · · · |B y j | is polynomially bounded. 3. Time Evolution for Probability Measures ˙ × T) L carried by [L] = (R × T) L ˙ = (R Consider any probability measure on [ L] (i.e., gives zero measure to the points at infinity). We can find constants κnx > 0 such that, if we write Bn = {( px , qx )x∈L : | px | ≤ κnx for all x ∈ L}, we have (Bn ) > 1−1/n. We may thus write limn→∞ ||−n || = 0, where the measure n has support in the compact set Bn ⊂ [L], and (t, ξ ) → f t ξ is continuous on R × Bn . We define then f t = lim f t n n→∞
(norm limit).
Notice also that f t n has support in the compact set Bn defined like Bn with κnx replaced
= κ + K |t| (see Lemma 2.1(a)). Therefore f t is again carried by [L]. by κnx nx ˙ is Proposition 3.1 (Continuity of time evolution). If the probability measure on [ L] t carried by [L], then the probability measure f is well defined, carried by [L], and ˙ with the vague topology. t → f t is continuous R → measures on [ L] ˙ → R we have ( f t n )(A) = n (A ◦ f t ), where For any continuous function A : [ L] A ◦ f t restricted to Bn depends continuously on t with respect to the uniform norm on C(Bn → R). Therefore ( f t n )(A) is a continuous function of t, and so is its uniform limit t → f t (A). This shows that t → f t is continuous with respect to the w ∗ ˙ concluding the proof. (=vague) topology of measures on [ L], Let X be a probability measure on [ X˙ ] for finite X ⊂ L. We write X → ∞ when, for every finite U ⊂ L, eventually X ⊃ U . Suppose that for every finite U and A ∈ C([U˙ ] → R) the limit lim X (A ◦ πU X )
X →∞
˙ ] × [U˙ ] → [U˙ ]. This limit is then of exists, where πU X is the projection [ X˙ ] = [ X \U ˙ which we the form (A ◦ πU ), where is a uniquely defined probability measure on [ L] call the thermodynamic limit of the X : = θ lim X →∞ X .
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This means that πU = w ∗ lim X →∞ πU X X or (modulo the identifications A → A ◦ πU X , A → A ◦ πU ) is the limit of the X on ∪U finite C([U˙ ] → R) ◦ πU ˙ → R). In particular, if is any probability measure on [ L], ˙ which is dense in C([ L] we have = θ lim X →∞ π X . We shall later also consider thermodynamic limits associated with a sequence X n → ∞, writing = θ limn→∞ X n if πU = w ∗ limn→∞ πU X n X n for all finite U ⊂ L. Proposition 3.2 (Time evolution of thermodynamic limits). Suppose that θ lim X →∞ X = , where X , are probability measures carried by [X ], [L] respectively. Then θ lim X →∞ f Xt X = f t uniformly for t ∈ [−T, T ]. We have to prove that, for every finite U ⊂ L, and A ∈ C([U˙ ] → R), lim ( f Xt X )(A ◦ πU X ) = ( f t )(A ◦ πU ).
X →∞
We may (and shall) assume that |A| ≤ 1. Given > 0, we know that ||A ◦ πU ◦ f t − A ◦ πU X ◦ f Xt ◦ π X || < 2 for sufficiently large X, say X ⊃ V for suitable V ⊃ U , for all t ∈ [−T, T ]. Under these conditions we have thus ||A ◦ πU ◦ f t − A ◦ πU V ◦ f Vt ◦ πV || < /2 and ||A ◦ πU X ◦ f Xt ◦ π X − A ◦ πU V ◦ f Vt ◦ πV || < which we shall use below in the form ||A ◦ πU X ◦ f Xt − A ◦ πU V ◦ f Vt ◦ πV X || < . Take now a function ∈ C([V ] → R) with compact support and || ≤ 1, such that || − ( ◦ πV )|| < .
Using the notation a ∼ b to mean |a − b| < , we have
( f t )(A ◦ πU ) ∼ ( f t (( ◦ πV )))(A ◦ πU ) = (( ◦ πV )(A ◦ πU ◦ f t ))
∼ (( ◦ πV )(A ◦ πU V ◦ f Vt ◦ πV )) = (((A ◦ πU V ◦ f Vt )) ◦ πV ) = (t ◦ πV ),
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where the function t = (A ◦ πU V ◦ f Vt ) : [V ] → R is continuous with compact support, hence extends to a continuous function on [V˙ ]. By assumption we have |(t ◦ πV ) − X (t ◦ πV X )| < for sufficiently large X , uniformly with respect to t ∈ [−T, T ] (this is because t → t is continuous with respect to the uniform norm of C([V˙ ] → R)). We may thus take W ⊃ V such that, if X ⊃ W and t ∈ [−T, T ],
(t ◦ πV ) ∼ X (t ◦ πV X ) = X (((A ◦ πU V ◦ f Vt )) ◦ πV X )
∼ X (( ◦ πV X )(A ◦ πU X ◦ f Xt )) = ( f Xt (( ◦ πV X ) X ))(A ◦ πU X ). We have thus |( f t )(A ◦ πU ) − ( f Xt (( ◦ πV X ) X ))(A ◦ πU X )| < 4 when X ⊃ W , t ∈ [−T, T ]. We may now let → 1, obtaining |( f t )(A ◦ πU ) − ( f Xt X )(A ◦ πU X )| ≤ 4 as announced. Proposition 3.2 also holds for the thermodynamic limit associated with a sequence Xn → ∞ 4. -States and Their Time Evolution We introduce now a special set of probability measures. Definition 4.1 (-states). We say that the probability measure carried by [L] is a state if there exist constants β˜x > 0 (for x ∈ L), smooth functions V˜x : T → R (for x ∈ L) and W˜ {x,y} : T × T → R (for {x, y} ∈ ) such that the β˜x , β˜x−1 , V˜x , W˜ {x,y} , ∂qx W˜ {x,y} , ∂q y W˜ {x,y} are bounded uniformly in x, y ∈ L , and the following holds: For every finite X ⊂ L , the conditional measure X (dξ |η) of on [X ] given η ∈ [L\X ] is of the form ⎡ ⎤ 1 ∗ X (dξ |η) = const. exp ⎣− β˜x px2 + V˜x (qx ) − W˜ {x,y} (qx , q y )⎦ dξ, 2 x∈X
{x,y}
∗
where extends over those {x, y} ∈ such that x ∈ X, and we have written ξ = ( px , qx )x∈X , η = ( px , qx )x∈L\X . [The -states are Gibbs states4 for a certain interaction given by the β˜x , V˜x , W˜ {x,y} .] If is a -state we may, for finite U ⊂ L, write (πU )(dξ ) = U (ξ )dξ , where U is smooth on [U ]. Note that U (ξ ) has a ( p, q)-factorization: it is the product of a smooth function of the qx for x ∈ U , and of a Gaussian β˜x /2π exp(−β˜x px2 /2) for each x ∈ U . 4 See [14] for a discussion of Gibbs states in the simpler case of spin systems. We shall not make use of the theory of Gibbs states in the present paper.
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We shall now take X finite and Y = X ∪ X˜ 1 ∪ . . . ∪ X˜ k¯ , where X˜ k = {y ∈ L\X : d(y, X ) = k} ¯ the -state (k¯ is thus the “size” of Y ). If ξ ∈ [X ], and ηk ∈ [ X˜ k ] for k = 1, . . . , k, property of then gives Y (ξ, η1 , . . . , ηk¯ ) = Ck¯ .0 (ξ |η1 ).1 (η1 |η2 ) · · · k−1 ¯ (ηk−1 ¯ , ηk¯ ).k¯ (ηk¯ ), / ): where Ck¯ is a normalization constant and (putting W˜ {x,y} = 0 if {x, y} ∈ ⎡ 0 (ξ |η1 ) = exp ⎣− (β˜x px2 /2 + V˜x (qx )) − W˜ {x,y} (qx , q y ) x∈X
−
⎤
W˜ {x,y} (qx , q y )⎦,
x∈X y∈ X˜ 1
⎡ k (ηk |ηk+1 ) = exp ⎣−
− k¯ (ηk¯ ) =
(β˜x px2 /2 + V˜x (qx )) −
x∈ X˜ k
x,y∈X
W˜ {x,y} (qx , q y )
x,y∈ X˜ k
⎤
W˜ {x,y} (qx , q y )⎦ when k > 0, and
x∈ X˜ k y∈ X˜ k+1
˜ ¯ ]. ν(dηk+1 ¯ ) k¯ (ηk¯ |ηk+1 ¯ ) for some probability measure ν on [ X k+1
Using the fact that the Jacobian of fUt is 1 ( fUt preserves dξ ) we have fUt ((πU )(dξ )) = fUt (U (ξ )dξ ) = U ( fU−t ξ )dξ. Thus, by Proposition 3.2, if X ⊂ Y as above, (π X f t )(dξ ) = w ∗ limY →∞ π X (Y ( f Y−t (ξ, ηY ))dξ dηY ). We may write f Y−t (ξ, ηY ) = ( f Y−t0 (ξ, ηY ), f Y−t1 (ξ, ηY ), . . . , f Y−tk¯ (ξ, ηY )) with f Y−t0 (ξ, ηY ) ∈ [X ],
¯ f Y−tk (ξ, ηY ) ∈ [ X˜ k ] for k = 1, . . . , k.
If ξ, ξ˜ ∈ [X ], ηY ∈ [Y \X ], the quotient Y ( f Y−t (ξ, ηY )) Y ( f −t (ξ˜ , ηY )) Y
is thus a product of quotients k ( f Y−tk (ξ, ηY )| f Y−t(k+1) (ξ, ηY ))
k ( f Y−tk (ξ˜ , ηY )| f Y−t(k+1) (ξ˜ , ηY ))
for k = 0, . . . , k¯ − 1
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and k¯ ( f Y−tk¯ (ξ, ηY ))
k¯ ( f Y−tk¯ (ξ˜ , ηY ))
,
where the arguments f Y−tk (ξ, ηY ), f Y−t(k+1) (ξ, ηY ) and their derivatives have dependence on ξ that decreases faster than exponentially with respect to k (Propositions 2.7 and 2.8). Let us define tY k (ξ, ηY ) by (−β˜x px (0)2 /2) k ( f Y−tk (ξ, ηY )| f Y−t(k+1) (ξ, ηY )) = tY k (ξ, ηY ). exp x∈ X˜ k
for k = 0, . . . , k¯ − 1, where X˜ k is replaced by X for k = 0, and k¯ ( f −t (ξ, ηY )) = t ¯ (ξ, ηY ). exp (−β˜x px (0)2 /2). Y k¯
Yk
x∈ X˜ k¯
We shall also use tk (ξ, η) defined by −t (ξ, η)) = tk (ξ, η). exp k ( f k−t (ξ, η)| f k+1
(−β˜x px (0)2 /2),
x∈ X˜ k
where f −t (ξ, η) = ( f 0−t (ξ, η), . . . , f k−t (ξ, η), . . .) with f 0−t (ξ, η) ∈ [X ], and f k−t (ξ, η) ∈ [ X˜ k ] for k ≥ 1. From our definitions it follows that ⎡ ⎤ k¯
tY k (ξ, ηY ) Y ( f Y−t (ξ, ηY )) exp x∈X (−β˜x px (0)2 /2) ⎣ ⎦ = · . Y ( f −t (ξ˜ , ηY )) t (ξ˜ , ηY ) exp x∈X (−β˜x p˜ x (0)2 /2) Y
k=0
Yk
Lemma 4.2 (Basic uniform estimates). In the above formula, we have, uniformly in t ∈ [−T, T ] and the size k¯ of Y, the estimates (a)
| log tY 0 (ξ, ηY )| < const.(1 + sup | px (0)|), x∈X
(b)
t log Y k (ξ, ηY ) < polyn.(k) (1 + sup | px (0)|) k! tY k (ξ˜ , ηY ) x∈ X˜ k
if k ≥ 1.
These estimates remain true when tY k is replaced by tk . We note that, by Lemma 2.1(a), | px (t)2 − px (0)2 | ≤ K |t|(2| px (0)| + K |t|). From this, and the definitions, the first inequality of the lemma follows. The second inequality is obtained by using also the finite dimensionality Assumption 1.1 and Lemma 2.1(b). Define now the regions Ru , Rv× ⊂ [L] = [X ] × [L\X ] such that Ru = {(ξ, η) : | px | ≤ u if x ∈ X },
Rv× = {(ξ, η) : | px | ≤ kv if x ∈ X˜ k for k ≥ 1}.
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Lemma 4.3 (Existence of limit in Ru ∩ Rv× ). In Ru ∩ Rv× , the expression Y ( f Y−t (ξ, ηY )) (−β˜x px (0)2 /2)]−1 · [tY 0 (ξ, ηY ) exp −t ˜ Y ( f (ξ˜ , ηY )) d ξ Y [X ] x∈X ⎤ ⎡ ⎡ ⎤−1 k¯
tY k (ξ˜ , ηY ) t ⎦ (ξ˜ , ηY ) exp =⎣ d ξ˜ ⎣ (−β˜x p˜ x (0)2 /2)⎦ tY k (ξ, ηY ) Y 0 [X ]
k=1
x∈X
¯ has upper and lower bounds exp(±const.(1 + v)) uniformly in u, t ∈ [−T, T ], and k, and tends when k¯ → ∞, uniformly for (ξ, η) ∈ Ru ∩ Rv× , to
∞ −1
t (ξ˜ , η) k t 2 d ξ˜ (−β˜x p˜ x (0) /2) . (ξ˜ , η) exp tk (ξ, η) 0 [X ] k=1
x∈X
The limit is continuous. Let (ξ, η) ∈ Ru ∩ Rv× , and assume k¯ to be large. The quotients tY k (ξ˜ , ηY ) tY k (ξ, ηY )
(k ≥ 1)
¯ for small k, and (using Lemma 4.2) very close to are nearly independent of Y (i.e., of k) 1 for large k, so that ∞ k¯
tY k (ξ˜ , ηY ) tk (ξ˜ , η) = ¯ tY k (ξ, ηY ) tk (ξ, η) k→∞
lim
k=1
k=1
uniformly, and we have bounds exp(±const.(1 + v)) by Lemma 4.2(b). Note now that tY 0 (ξ˜ , ηY ) tends to t0 (ξ˜ , η) uniformly for (ξ˜ , η) ∈ Ru ∩ Rv× , and we can extend the integral over ξ˜ from | px | < u to [X ] because the Gaussian exp
(−β˜x p˜ x (0)2 /2)
x∈X
beats the exponential growth of tY 0 (ξ˜ , ηY ) given by Lemma 4.2(a). Bounds of the form exp(±const.(1 + v)) hold again after integration. Lemma 4.4 (Large v Gaussian estimate). For large v, (πY \X f Yt πY )(dηY ) has mass < exp(−const.v 2 ) outside of πY \X Rv× , uniformly in the size k¯ of Y . The ( p, q)-factorization of Y (ξ, ηY ) shows that the mass outside of Rv× is bounded, ¯ by a Gaussian < exp(−const.v 2 ) for large v. But the time evolution f t uniformly in k, Y changes | px | (additively) by at most K |t|, so that the Gaussian estimate remains valid.
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Theorem 4.5 (Smooth density of evolved states). Let be a -state. For finite X, and Y of size k¯ as above, we write t 2 ¯ ˜ Y X (ξ ) exp (−βx px (0) /2) = dηY Y ( f Y−t (ξ, ηY )). x∈X
There is a smooth function ¯tX (ξ ) of ξ and t such that
(π X f t )(dξ ) = ¯tX (ξ ) exp
(−β˜x px (0)2 /2) dξ,
x∈X
and we have, uniformly for | px | < u (x ∈ X ) and |t| ≤ T, ¯tX (ξ ) = lim ¯tY X (ξ ). ¯ k→∞
The limit also holds for the derivatives with respect to ξ, t. The ¯tX (ξ ), ¯tY X (ξ ) have upper and lower bounds exp(±const.(1 + u)), and the absolute values of their derivatives have ¯ bounds polyn.(u). exp(const.(1 + u)) uniformly in t ∈ [−T, T ] and k. We start with the remark that
Y ( f Y−t (ξ, ηY )) dξ d ξ˜ Y ( f −t (ξ˜ , ηY ))
[X ]
Y
is the conditional measure of f Yt πY on [X ] given ηY ∈ [Y \X ]. Integrating this conditional measure with respect to (πY \X f Yt πY )(dηY ) yields π X f Yt πY . Thus ¯tY X (ξ ) =
(πY \X f Yt πY )(dηY )
Y ( f Y−t (ξ, ηY )) (β˜x px (0)2 /2). · exp d ξ˜ Y ( f −t (ξ˜ , ηY ))
[X ]
Y
x∈X
The integrand in the right-hand side is the product of a factor controlled by Lemma 4.3, and a factor tY 0 (ξ, ηY ) which has upper and lower bounds exp(±const.(1 + u)) uniformly in k¯ (by Lemma 4.2(a)) and tends to t0 (ξ, η) when k¯ → ∞, uniformly for (ξ, η) ∈ Ru ∩ Rv× . Using also Lemma 4.4 and the fact that πY \X f Yt πY has the w ∗ limit π L\X f t when k¯ → ∞ (Proposition 3.2) we find that t ¯ lim Y X (ξ ) = (π L\X f t )(dη) t0 (ξ, η) ¯ k→∞
×
[X ]
d ξ˜
∞ t ˜
k (ξ , η) tk (ξ, η)
k=1
t0 (ξ˜ , η) exp
−1 (−β˜x p˜ x (0)2 /2)
x∈X
uniformly when | px (0)| ≤ u for x ∈ X , with uniform upper and lower bounds exp(± const. (1 + u)). We call the limit ¯tX (ξ ). Since ¯tY X (ξ ) exp
x∈X
(−β˜x px (0)2 /2) dξ = (π X f Yt πY )(dξ )
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has the w ∗ limit (π X f t )(dξ ), it follows that this limit has a density (−β˜x px (0)2 /2) ¯tX (ξ ) exp x∈X
as asserted. Using the notation f t (ξˆ , η) = ( f 0t (ξˆ , , η), f ˆt (η)), ξ
we may write ¯tX (ξ )
=
( f )(d ξˆ dη)
∞
t (ξ˜ , η) k d ξ˜ t (ξ, η) [X ] k=1 k −1
t
× l0t (ξ˜ , η) exp
x∈X
⎡
(−β˜x p˜ x (0)2 /2)
(d ξˆ dη) ⎣
=
[X ]
× l0t (ξ˜ , f ˆt (η)) exp ξ
⎡ d ξ˜ ⎣
∞ t (ξ˜ , f t (η))
k ξˆ k=1
x∈X
l0t (ξ, η) ⎤
⎦ tk (ξ, f ˆt (η)) ξ ⎤−1
(−β˜x p˜ x (0)2 /2)⎦
l0t (ξ, f ˆt (η)) ξ
and remember that this is the limit of a similar expression for ¯tX Y (ξ ). We want to show that ¯tX (ξ ) has derivatives (of all orders) with respect to ξ, t by showing that the deriv¯ Note that in atives of ¯tX Y (ξ ), for ξ, t in a compact set, are bounded with respect to k. estimating the integrals of polynomials in p, the px (0) integral always has a Gaussian factor exp(−βx px (0)2 /2) (remember the ( p, q) factorization of ). Therefore we only have to worry about bounding the coefficients of the polynomials. Inspection of the above expression shows that computing a first order derivative essentially involves multiplying the integrand by the logarithmic derivatives of l0t (ξ, f ˆt (η)), or tk (ξ˜ , f ˆt (η))/tk (ξ, f ˆt (η)) and summing over k. In view of the very ξ ξ ξ explicit form of the logarithm of the k , we just have to estimate the derivatives of f −t (ξ, f ˆt (η)) with respect to ξ and t. ξ
As far as t0 (ξ, f ˆt (η)) is concerned, we have to consider the derivatives of ξ
f 0−t (ξ, f ˆt (η)) (or f 1−t (ξ, f ˆt (η)), which is similar). The ξ -derivative is of the form (0,1)
ξ
ξ
r x (−t; x1 ) with x, x1 ∈ X , giving a bounded contribution by Proposition 2.7. The t-derivative is of the form r x(1,0) (−t) + r x(0,1) (−t; x1 )r x(1,0) (t) 1 x1
with x ∈ X , and we may take x1 ∈ X˜ k1 . By Proposition 2.7, |r (0,1) (±t)| is bounded by a polynomial in p with bounded coefficients, and r x(0,1) (−t; x1 ) ≤ const./k1 !, again giving a bounded contribution because k1 | X˜ k1 |/k1 ! is bounded.
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We turn now to tk (ξ˜ , f ˆt (η))/tk (ξ, f ˆt (η)), i.e., we have to consider the derivatives of
ξ ξ (0,1) −t (or f k+1 ). The ξ -derivative is of the form r x (−t; x1 ) with x ∈ (0,1) that |r x (−t; x1 )| ≤ const./d(x, x1 )! = const./k!, which makes a
f k−t (ξ, f ˆt (η)) ξ
X˜ k , x1 ∈ X , so bounded contribution because
| X˜ k |/k! is bounded. The t-derivative is of the form r x(1,0) (−t; X ) + r x(0,1) (−t; x1 ; X )r x(1,0) (t), 1 k
x1
where x ∈ X˜ k and we may take x1 ∈ X˜ k1 . By Proposition 2.8 we have |r x(1,0) (−t; X )| ≤ polyn.( p)/k!, |r x(0,1) (−t; x1 ; X )| ≤ const./σ ! ≤ const./ max(k, k1 )!, and since k | X˜ k |/k!, k,k1 | X˜ k |.| X˜ k1 |/ max(k, k1 )! are bounded, we also have bounded contributions for the t-derivative. We consider now higher order derivatives with respect to ξ, t. The computation of such a derivative gives terms where the integrand is multiplied by a product of logarithmic derivatives of the type discussed above; the contribution is again seen to be bounded. There are also terms containing derivatives of the logarithmic derivatives, and these are expressed in terms of higher order derivatives of f −t (ξ, f ˆt (η)) with ξ (0, j) (−t; x1 , . . . , x j )| ≤ x1 · · · ∂ξx j is estimated by |r x contribution. The derivative ∂ i /∂t i is a sum of terms
respect to ξ, t. The derivative ∂ j /∂ξ
const./d(x, X )! giving a bounded r x(i1 ,i2 ) (−t; y1 , . . . , yi2 ; X ) (multiplied by derivatives ∂ k f ˆt /∂t k = r (k,0) ), where i 1 + ξ i 2 = i; these terms can be estimated by Proposition 2.8, and give a bounded contribution. The general mixed derivative ∂ i+ j /∂t i ∂ξx1 · · · ∂ξx j , with j ≥ 1, is a sum of (i , j+i )
terms r x 1 2 (−t; x1 , . . . , x j , y1 , . . . , yi2 ) with x1 , . . . , x j ∈ X (multiplied by derivatives of the form r (k,0) ) which can be estimated by Proposition 2.7, and give a bounded contribution. Remark 4.6 (Uniform bounds). The proof of Theorem 4.5 gives estimates of ¯tY X (ξ ) and its derivatives with respect to t and ξ , which are uniform with respect to the size k¯ of Y . They are also uniform with respect to the -states with conditional measures corresponding to a fixed choice of β˜x , V˜x , W˜ {x,y} , and remain uniform if some of the W˜ {x,y} are replaced by 0. For Y finite and η = ( px , qx )x∈Y define ⎡ ⎤ 1 β˜x px2 + V˜x (qx ) − W˜ {x,y} (qx , q y )⎦, ˜Y (η) = Z Y−1 exp ⎣− 2 x∈Y
x,y∈Y
where Z Y−1 is a normalization factor, and write (−β˜x px (0)2 /2) = ¯˜tY X (ξ ) exp x∈X
[Y \X ]
dηY ˜Y ( f Y−t (ξ, ηY )).
Then the above remarks show that the uniform estimates on ¯tY X (ξ ) and its derivatives given by Theorem 4.5 can be taken to hold also for ¯˜t . YX
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5. Entropy Given a -state , and X ⊂ Y finite, we write π X f Yt πY = tY X (ξ ) dξ, π X f t = tX (ξ ) dξ, where (t, ξ ) → tY X (ξ ), tX (ξ ) are smooth on R × [X ]. In Theorem 4.5, we used the notation tY X (ξ ) = ¯tY X (ξ ) exp (−β˜x px (0)2 /2), tX (ξ ) = ¯tX (ξ ) exp (−β˜x px (0)2 /2) x∈X
x∈X
and saw that ¯tY X (ξ ) tends to ¯tX (ξ ) together with its derivatives, uniformly on compacts of R × [X ], when Y → ∞. We can now define a (Gibbs) entropy SYt (X ) or S t (X ) by SYt (X ) = −
[X ]
tY X (ξ ) log tY X (ξ ) dξ, S t (X ) = −
[X ]
tX (ξ ) log tX (ξ ) dξ.
These are convergent integrals in view of the uniform bounds given in Theorem 4.5. Furthermore SYt (X ) → S t (X ), uniformly for |t| ≤ T , when Y → ∞. We may assume that Y ⊃ X˜ 1 = {y ∈ L : dist(y, X ) = 1}. If ξ ∈ [X ], η ∈ [Y \X ] or [L\X ], let η1 ∈ X˜ 1 be obtained from η by restricting the index set to X˜ 1 . Then the equations of motion for ξ, η show that we may write dξ = X (ξ, η1 ), dt
dη = Y(ξ, η), dt
where X does not depend on Y . Writing ˆtY = Y ◦ f Y−t , we have tY X (ξ )
=
[Y \X ]
dη ˆtY (ξ, η)
and the “continuity equation” d ˆt (ξ, η) + ∇ξ · (ˆtY (ξ, η)X (ξ, η1 )) + ∇η · (ˆtY (ξ, η)Y(ξ, η)) = 0 dt Y so that d t (ξ ) = − dt Y X
[Y \X ]
dη ∇ξ · (ˆtY (ξ, η)X (ξ, η1 ))
dη ˆtY (ξ, η)X (ξ, η1 ), d 1 t log Y X (ξ ) = − t ∇ξ · dη ˆtY (ξ, η)X (ξ, η1 ), dt X Y (ξ ) [Y \X ] d t t t [ (ξ ) log Y X (ξ )] = −(log Y X (ξ ) + 1)∇ξ · dη ˆtY (ξ, η)X (ξ, η1 ). dt Y X [Y \X ] = −∇ξ ·
[Y \X ]
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Using the estimates of Theorem 4.5 we find that t → SYt (X ) is a smooth function of t, with d t S (X ) = dt Y
[X ]
dξ log tY X (ξ ) ∇ξ ·
=−
[X ]
[Y \X ]
dξ (∇ξ tY X (ξ )) · tY X (ξ )
dη ˆtY (ξ, η)X (ξ, η1 )
[Y \X ]
dη ˆtY (ξ, η)X (ξ, η1 ).
Write now X + = X ∪ X˜ 1 = {y ∈ L : d(y, X ) ≤ 1}. The probability measure tY X + (ξ, η1 ) dξ dη1 conditioned on ξ ∈ [X ] is denoted by tY X + (η1 |ξ ) dη1 where tY X + (ξ, η1 ) . tY X (ξ )
tY X + (η1 |ξ ) = Theorem 4.5 gives uniform estimates for
tY X + (η1 |ξ ), ∇ξ tY X + (η1 |ξ ) so that d t S (X ) = − dt Y =
dξ ∇ξ tY X (ξ ) · [X ] dξ tY X (ξ )∇ξ ·
[ X˜ 1 ]
[ X˜ 1 ]
[X ]
dη1 tY X + (η1 |ξ )X (ξ, η1 )
dη1 tY X + (η1 |ξ )X (ξ, η1 ).
It follows also that, when Y → ∞, d SYt (X )/dt tends to
[X ]
dξ
tX (ξ )∇ξ
·
[ X˜ 1 ]
dη1 tX + (η1 |ξ )X (ξ, η1 )
uniformly for |t| ≤ T , and the limit is d S t (X )/dt. Proposition 5.1 (Time derivative of S(X ), X finite). When Y → ∞, the derivative d SYt (X )/dt tends, uniformly for |t| ≤ T, to d t S (X ) = dt
[X ]
dξ tX (ξ )∇ξ ·
[ X˜ 1 ]
dη1 tX + (η1 |ξ )X (ξ, η1 )
which is a smooth function of t. The proof, as given above, is essentially a corollary of Theorem 4.5.
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D. Ruelle
Suppose now that X˜ 1 = {y ∈ L : d(x, y) = 1} is finite, but X is not necessarily finite. We still have, for Y finite, d t S (X ∩ Y ) = dt Y
[X ∩Y ]
dξ
tY,X ∩Y (ξ )∇ξ
·
[ X˜ 1 ]
dη1 tY,X +∩Y (η1 |ξ )X (ξ, η1 ),
and this can be bounded independently of Y . Proposition 5.2 (Time derivative of S t (X )). If X ⊂ L , and X is not necessarily finite, but X˜ 1 = {y ∈ L : d(X, y) = 1} is finite, we may define S t (X ) = lim (S t (X ∩ Y ) − S 0 (X ∩ Y )) Y →∞
and we have d SYt (X ) = dt
(π X f )(dξ )∇ξ · t
[X ]
[ X˜ 1 ]
dη1 tX + (η1 |ξ )X (ξ, η1 )
which is a smooth function of t. This follows from the usual estimates. Note that ∇ξ is a derivative with respect to a finite number of variables corresponding to nonzero components of X (ξ, η1 ). ˇ We shall now study a conditional, or “external” entropy S defined for X finite by SˇYt (X ) = SYt (Y ) − SYt (Y \X ). Using the notation ˆtY = Y ◦ f Y−t , Y,Y \X (η) =
[X ]
dξ ˆtY (ξ, η), tY X (ξ |η) =
ˆtY (ξ, η) Y,Y \X (η)
as above, we find ˆt (ξ, η) ˇS t (X ) = − dξ dη ˆt (ξ, η) log Y Y Y Y,Y \X (η) = Y,Y \X (η) dη[− dξ tY X (ξ |η) log tY X (ξ |η)] = Y,Y \X (η) dη SYt (X |η), where we have written SYt (X |η) = −
dξ tY X (ξ |η) log tY X (ξ |η).
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Let also
∞ −1
t (ξ˜ , η) k tX (ξ |η) = d ξ˜ (−β˜x p˜ x (0)2 /2) t (ξ˜ , η) exp tk (ξ, η) 0 [X ] k=1 x∈X t 2 × 0 (ξ, η) exp (−βx px (0) /2) x∈X
S (X |η) = − t
dξ
tX (ξ |η) log tX (ξ |η),
then Lemma 4.2(a) and Lemma 4.3 imply that SYt (X |η) → S t (X |η) when Y → ∞, uniformly for η ∈ π L\X Rv× and t ∈ [−T, T ], and with uniform bounds |SYt (X |η)| ≤ const.(1 + v). Therefore (using Lemma 4.4 and Proposition 3.2) we see that when Y → ∞ we have SˇYt (X ) → Sˇ t (X ) uniformly for t ∈ [−T, T ], where Sˇ t (X ) = (π L\X f t )(dη) S t (X |η). Note that Sˇ t (X ) is obtained by taking the mean entropy S t (X |η) associated with tX (ξ |η) and averaging over η, while S t (X ) is the entropy associated with the average X (ξ ) of tX (ξ |η) over η. In particular, concavity gives Sˇ t (X ) ≤ S t (X ). Since d SYt (Y )/dt = 0, we have d ˇt d SY (X ) = − SYt (Y \X ) dt dt dη tY,Y \X (η)∇η · dξ tY X (ξ |η)Y(ξ, η) =− [Y \X ] [X ] =− dη tY,Y \X (η) ∂η y · dξ tY X (ξ |η)Y y (ξ, η), [Y \X ]
y∈ X˜ 1
[X ]
where X˜ 1 = {y ∈ Y : d(y, X ) = 1} and Y y is the y-component of Y. We may now let Y → ∞, finding: ˇ When Y → ∞, the derivative Proposition 5.3 (Time derivative of the entropy S). t ˇ d SY (X )/dt tends, uniformly for t ∈ [−T, T ], to d ˇt t ∂η y · dξ tX (ξ |η)Y y (ξ, η) S (X ) = − (π L\X f )(dη) dt [X ] y∈ X˜ 1
which is a smooth function of t. This is again a corollary of Theorem 4.5.
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Note that d ˇt d S (X ) = − S t (L\X ). dt dt Assumption 5.4 (Bounded energy). For every finite X ⊂ L the kinetic energy is bounded independently of t: dξ tX (ξ ) p 2X /2 ≤ const.(X ) [it would be equivalent to assume a bound on the total energy H X ]. We have the general inequality √ S t (X ) ≤ dξ tX (ξ ) p 2X /2 + |X | log 2π . [this follows from the “variational principle for the free energy”, and can be proved by using the concavity of the log:
e− p X /2 ≤ log tX (ξ ) 2
dξ tX (ξ ) log
dξ e− p X /2 = |X | log 2
√
2π .
Therefore the bounded energy assumption gives a bound on the entropy: S t (X ) ≤ const.(X ). Similarly, we find
S (X |η) ≤ t
hence Sˇ t (X ) ≤
dξ tX (ξ |η) p 2X /2 + |X | log
dξ tX (ξ ) p 2X /2 + |X | log
√
√ 2π ,
2π ≤ const.(X ).
In particular we have Sˇ t (X ) ≤ S t (X ) ≤ const.(X ). Definitions 5.5 (Large volume limit). We may take a sequence (Tn ) tending to +∞ such Tn 1 ˙ dt f t has a limit ρ in the vague topology of measures on [ L]: that Tn 0 Tn 1 dt f t → ρ tn → ∞, Tn 0 We call the probability measure ρ a nonequilibrium steady state (NESS). In view of Assumption 5.4, ρ is carried by [L]. Furthermore ρ is invariant under ( f t ). We can also (by going to a subsequence) assume that Tn S Tn (X ) 1 d = dt S t (X ) → σ (X ) Tn Tn 0 dt
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when X˜ 1 = {y ∈ L : d(X, y) = 1} is finite (X need not be finite). Note that σ (X ) might not be determined by ρ and X . For notational simplicity we shall write T → ∞ instead of Tn , n → ∞. 5.1. Interpretation (Entropy production [16]). As mentioned in the Introduction, Denis Evans and coworkers [16] have proposed to identify the mean entropy production rate in a finite region X to e(X ) = −σ (X ) = − lim
T →∞
S T (X ) − S 0 (X ) . T
According to Proposition 5.1 this is the mean rate of volume contaction in [X ], and e(X ) corresponds to the accepted definition of entropy production in the presence of a deterministic thermostat. A related choice is e(X ˇ ) = σ (L\X ) = − lim
T →∞
Sˇ T (X ) − Sˇ 0 (X ) . T
This is the mean rate of volume expansion in [L\X ], and corresponds to the rate of entropy growth due to X , as seen by the “external world” L\X . Since Sˇ t (X ) ≤ S t (X ) ≤ const.(X ), we have 0 ≤ e(X ) ≤ e(X ˇ ). We may also define mean entropy production rates associated with a finite partition A = (X 0 , X 1 , . . . , xn ) of L provided X 0 , X 1 , . . . , X n have finite “boundaries” {y ∈ L : d(X i , y) = 1}. We write e(A) =
n j=0
σ (X j ), e(A) ˇ =
σ (X j ).
j:X j infinite
In particular, in Case II, for X finite ⊃ X 0 , we have e(X ˇ ) = e((X, ˇ L\X )) ≤ e((X, ˇ L 1 \X, L 2 \X )). We proceed now with some general inequalities satisfied by σ , e, and e. ˇ 5.2. Basic inequalities. We have e(∅) = e(∅) ˇ = 0 by definition, and remember that 0 ≤ e(X ) ≤ e(X ˇ ). The strong subadditivity of the entropy implies that, if U , V have finite boundaries, σ (U ∪ V ) − σ (U ) − σ (V ) + σ (U ∩ V ) ≤ 0 (we have used the fact that S 0 ((U ∪ V ) ∩ Y ) − S 0 (U ∩ Y ) − S 0 (V ∩ Y ) + S 0 (U ∩ V ∩ Y ) is bounded independently of Y ). This implies the strong superadditivity of e, and subadditivity of e. ˇ In particular e(U ∪ V ) ≥ e(U ) + e(V ) if U ∩ V = ∅ and e(U ˇ ∪ V ) ≤ e(U ˇ ) + e(V ˇ ).
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If U ⊂ V we also have5 e(U ) ≤ e(V ), e(U ˇ ) ≤ e(V ˇ ) i.e., e(X ), e(X ˇ ) are increasing functions of X . We can extend the definition of e(X ), e(X ˇ ) to infinite X : e(X ) =
sup
finite U ⊂X
e(U ), e(X ˇ )=
sup
finite U ⊂X
e(U ˇ ).
In the situations of interest for us e(L) ˇ will be finite and we may call this quantity the total entropy production rate. Note that the entropy production rate e(X ˇ ) is not an additive function of X , but that its subadditivity amounts to some kind of locality. Note also that if e(X ) > 0 we must have S T (X ) → −∞, in particular the TX cannot remain bounded when T → ∞, contrary to some evidence [4, 5]. But there is no obvious objection to having an entropy production rate e(X ˇ ) > 0. 6. Thermodynamic Bound on Entropy Production We shall show that in Case I (an external force and a thermostat at temperature β −1 ) we have e(X ˇ ) ≤ β × energy flux to thermostat, where the right-hand side is the thermodynamic rate of entropy production. A more general result is given below (see Proposition 6.3). In Case I we have introduced a finite set X 0 on which external forces act. As initial state we shall use the thermodynamic limit of a sequence: = θ limY →∞ ˜Y (η) dη, where6 ˜Y (η) = Z Y−1 exp[− H˜ X 0 − β HY \X 0 ]. In this formula,
1 2 ˜ ˜ βx px + Vx + W˜ {x,y} , 2 x∈X 0 x,y∈X 0 1 px2 + Vx + = W{x,y} , 2
H˜ X 0 = HY \X
x∈Y \X
x,y∈Y \X
and Z Y−1 is a normalization factor. In this section it will be convenient to use X 0 instead of X in the definition of Y , so that Y = X 0 ∪ X˜ 1 ∪ · · · ∪ X˜ k¯ . We take X of the form X 0 ∪ X˜ 1 ∪ · · · ∪ X˜ k (this is no serious restriction) and choose a subsequence k¯ → ∞ such that the π X Y (˜Y (η) dη) converge vaguely (we use here the thermodynamic limit for a sequence as explained in Sect. 3). 5 Note that, for U ⊂ V , we have S T (Y \U ) ≤ S T (Y \V ) + S T (V \U ), hence Sˇ T (V ) ≤ Sˇ T (U ) + S T (V \U ), Y Y Y Y Y Y
hence e(V ˇ ) ≥ e(U ˇ ) + e(V \U ). 6 More generally we could allow a term ˜ x∈X 0 ,y ∈X / 0 W{x,y} of interaction between X 0 and Y \X 0 .
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The state is a -state corresponding to the choice β˜x = β, V˜x = βVx for x ∈ / X 0, / X 0 . We define and W˜ {x,y} = βW{x,y} for x, y ∈ ˜tY X (ξ ) = ˜Y ( f Y−t (ξ, ηY )) dηY [Y \X ]
and S˜Yt (X ) = −
[X ]
˜tY X (ξ ) log ˜tY X (ξ ) dξ, Sˇ˜Yt (X ) = S˜Yt (Y ) − S˜Yt (Y \X ).
Writing ˜tY X (ξ |η) = ˜tY Y (ξ, η)/˜Y,Y \X (η), S˜Yt (X |η) = − dξ ˜tY X (ξ |η) log ˜tY X (ξ |η), we find Sˇ˜Yt (X ) =
˜tY,Y \X (η)dη S˜Yt (X |η).
ˇ˜ Lemma 6.1 (Thermodynamic limit for S). Sˇ˜Yt (X ) → Sˇ t (X ) together with the t-derivatives, uniformly for t ∈ [−T, T ], when k¯ → ∞. We have shown (in the proof of Proposition 5.2) how SˇYt (X ) → Sˇ t (X ). We proceed in the same way here, using the uniform estimates of Theorem 4.5 which hold again ˜ as explained in Remark 4.6. when is replaced by , Since f Yt is volume preserving in [Y ] we have S˜Yt (Y ) = S˜Y0 (Y ), therefore Sˇ˜Y0 (X ) − Sˇ˜Yt (X ) = S˜Yt (Y \X ) − S˜Y0 (Y \X ). We fix now X , with X 0 ⊂ X ⊂ Y as indicated above. Note that, by the ( p, q) factorization, S˜Y0 (Y \X 0 ) = S˜Y \X 0 (Y \X 0 ) is the sum of a momentum term S˜ 0 p (integral over p, trivial) and a configuration term S˜ 0q (integral over q). The configuration part q q ˜Y,Y \X (qY \X ) of ˜Y,Y \X (qY \X ) differs from ˜Y,Y \X 0 (qY ) = ˜Y \X 0 (q X \X 0 (qY \X ) by a factor bounded independently of Y (because there is a finite number of bounded terms V˜x q q and W˜ {x,y} with x ∈ X \X 0 ). Therefore | log ˜Y,Y \X (qY \X ) − log ˜Y \X 0 (qY )| is bounded independently of Y , hence | S˜Y0 (Y \X 0 ) − S˜Y0 (Y \X )| ≤ C0 with C0 independent of Y . Define now a function ∗ on [Y \X 0 ] = [X \X 0 ] × [Y \X ] to be the product of Z −1 exp(−βp 2X \X 0 /2)
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D. Ruelle
on [X \X 0 ] (with Z −1 a normalization factor) and ˜tY,Y \X on [Y \X ]. Then there is a constant C1 such that ∗
S =−
[Y \X 0 ]
dη ∗ (η) log ∗ (η) = S˜Yt (Y \X ) + C1
and the “variational principle for the free energy” gives S − S˜Y0 (Y \X 0 ) ≤ ∗
[Y \X 0 ]
dη (∗ (η) − ˜Y \X 0 (η)) β HY \X 0 (η)
so that S˜Yt (Y \X ) − S˜Y0 (Y \X ) ≤ C0 − C1 + S ∗ − S˜Y0 (Y \X ) dη (∗ (η) − ˜Y \X 0 (η)) β HY \X 0 (η). ≤ C0 − C1 + [Y \X 0 ]
There are also constants C2 , C3 such that
[Y \X 0 ] [Y \X 0 ]
dη ∗ (η)β HY \X 0 (η) ≤
dη ˜Y \X 0 (η)) β HY \X 0 (η) ≥
[Y \X ]
[Y \X ]
dη ˜tY,Y \X (η)β HY \X (η) + C2 , dη ˜0Y,Y \X (η)β HY \X (η) + C3 .
Therefore, with a constant C = C0 − C1 + C2 − C3 independent of Y and t, we have Sˇ˜Y0 (X ) − Sˇ˜Yt (X ) ≤ C +
[Y \X ]
= C +β = C +β = C +β
[Y ] [Y ]
[Y ]
dη[˜tY,Y \X (η) − ˜0Y,Y \X (η)]β HY \X (η) dη [˜tY (η) − ˜0Y (η)]HY \X (η) dη ˜0Y (η)[HY \X ( f Yt η) − HY \X (η)] dη ˜0Y (η)
t
dτ 0
d HY \X ( f Yτ η). dτ
The equations of motion yield d HY \X ( f Yτ η) = (π X +,Y f Yτ η), dτ where X + = {x ∈ L : d(x, X ) ≤ 1} and ( pY , qY ) is given by =−
x∈X y ∈X /
py
∂ W{x,y} (qx , q y ). ∂q y
Nonequilibrium Statistical Mechanics and Entropy Production in Rotators
Therefore [Y ]
dη ˜0Y (η)
t 0
dτ (π X +,Y f Yτ η) =
t
dτ 0
t
=
dτ
Sˇ˜Y0 (X ) − Sˇ˜Yt (X ) ≤ C + β We may now let Y → ∞, obtaining Sˇ 0 (X ) − Sˇ t (X ) ≤ C + β
t
dτ 0
dτ 0
dξ ˜τY X + (ξ )(ξ )
dξ ˜τY X + (ξ )(ξ ).
[X +]
t
dη ˜τY (η)(π X +,Y η)
[X +]
0
so that
[Y ]
259
[X +]
dξ τX + (ξ )(ξ ).
Proposition 6.2 (Bound on entropy production, Case I). In Case I the mean rate of entropy production of the finite set X is ≤ β×energy flux out of X 0 : 0 ≤ e(X ˇ ) ≤ β (π X 0 + ρ)(dξ )0 (ξ ), where 0 is the function computed for X = X 0 . It suffices to consider the case of large X , so we assume X ⊃ X 0 . Taking in the previous inequality the large time limit described in Assumption 5.4 we obtain 0 ≤ e(X ˇ ) ≤ β (π X + ρ)(dξ ) (x), where the right-hand side is independent of X , and we may thus take X = X 0 . We now give without proof a general bound on σ (X ), which can be obtained using the same ideas as for Proposition 6.2. Proposition 6.3 (Bound on σ (X )). Let X be infinite, with finite “boundary” X˜ 1 = {y ∈ L : d(X, y) = 1}. We let the initial state be the thermodynamic limit of a sequence: = θ limY →∞ ˜Y (η) dη, where ˜ ˜Y (η) = Z Y−1 e− HY , 1 β˜x px2 + V˜x + H˜ Y = W˜ {x,y} . 2 x∈Y
x,y∈Y
We assume that β˜x = β, V˜x = βVx , W˜ {x,y} = βW{x,y} when x, y ∈ X, i.e., X is a thermostat at temperature β −1 . Then σ (X ) ≤ β × energy flux to X .
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Note that if the energy flows out of X , then σ (X ) < 0, and in particular σ (X ) does not vanish. Applications of Proposition 6.3, in particular to Case II, are left to the reader. A. Appendices A.1 (Proof of Proposition 2.7). By uniformity of the bounds it suffices to consider the situation when Y is finite. The case (i, j) = (0, 0) follows from Lemma 2.1(a), with τ = σ = 0. Differentiating the time evolution equation for ηx (t) = ( px (t), qx (t)), we find d (0,1) rx = x y (qx , q y )r y(0,1) , dt 1 y∈Bx
where x y depends smoothly on the q’s, is independent of the p’s, and there is a uniform bound |x y | ≤ K¯ . y (0,1)
Therefore, if r (t) = supx,x1 ∈Y |r x
(t; x1 )|, we have d r (t) ≤ K¯ r (t) dt ¯
with r (0) = 1, so that |r x(0,1) (t; x1 )| ≤ r (t) ≤ e K |t| . With the notation k = d(x, x1 ) we claim that ¯
|r x(0,1) (t; x1 )| ≤ {e K |t| }k , n u where we have defined {eu }k = eu − k−1 n=0 u /n! ≤ e inf ≤k u /! For k = 0 the claim has been proved above. For k > 0 we have by induction, d (0,1) r ¯ K¯ |t| dt x (t; x1 ) ≤ K {e }k−1 , (0,1)
and our claim follows by integration, using r x (0, x1 ) = 0. Remember that σ (x, x1 , . . . , x j ) is the smallest length of a subgraph of connecting x, x1 , . . . , x j . We claim that for j ≥ 1 there are constants L j > 0 and M j > 0 such that (0, j)
|r x
¯
(t; x1 , . . . , x j )| ≤ M j e j K |t| inf
0≤τ ≤σ
(L j |t|)τ , τ!
(1)
where σ = σ (x, x1 , . . . , x j ). We have already proved (1) for j = 1, with L 1 = K¯ , M1 = 1. For j > 0 we have d (0, j) (0, j) r x (t; x1 , . . . , x j ) = x y r y (t; x1 , . . . , x j ) + r est. dt 1 y∈Bx
(2)
Nonequilibrium Statistical Mechanics and Entropy Production in Rotators
261 (0, j )
The r est is a sum, over y ∈ Bx1 and X , of products of factors r z n n (t; X n ) where z n is x or y, X = (X n ) is a partition of (x1 , . . . , x j ) into |X | > 1 subsequences of length jn , and each product has a coefficient which is a smooth function of qx , q y . Thus
(L jn |t|)τn (L j |t|)τ ¯ j K |t| ≤ Cje , jn K¯ |t| |r est| ≤ C j max exp τn ! τ ! X n n where L j = maxX n L jn and τ must be of the form n τn , i.e., 0 ≤ τ ≤ n σ (z n , X n ), where each z n is either x or y, and therefore σ (z n , X n ) + 1 ≥ σ (x, x1 , . . . , x j ) = σ, n
so that all values of τ between 0 and [σ − 1]+ are allowed. We have thus |r est| ≤ C j e
j K¯ |t|
inf
0≤τ ≤[σ −1]+
(L j |t|)τ τ !
.
We shall prove (1) by induction on j, assuming now j > 1. First let us write (0, j)
sup |r x x
¯
(t; x1 , . . . , x j )| = r (t : x1 , . . . , x j ) = e K |t| s(t)
and let σ = σ (x1 , . . . , x j ) = min σ (x, x1 , . . . , x j ). x
Then, for 0 ≤ τ ≤ [σ − 1]+ and t ≥ 0,
τ d ¯ (L j t) r (t; x1 , . . . , x j ) ≤ K¯ r (t; x1 , . . . , x j ) + C j e j K t dt τ !
or
τ d ¯ (L j t) s(t) ≤ C j e( j−1) K t . dt τ !
Thus
(C j t)(L j t)τ ds d ( j−1) K¯ t ≤ e for 0 ≤ τ ≤ σ − 1, dt dt (τ + 1)!
and also (taking τ = 0) Cj d ds ¯ ≤ e( j−1) K t . dt dt ( j − 1) K¯ We shall take L j = L j + C j and M j = 1 + C j /L j + C j [( j − 1) K¯ ]−1 . In particular, we have s(t) ≤ M j e
( j−1) K¯ t ¯
(L j t)τ
+1
(τ + 1)!
s(t) ≤ M j e( j−1) K t ,
for 0 ≤ τ ≤ σ − 1,
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D. Ruelle
hence ¯
s(t) ≤ M j e( j−1) K t
(L j t)τ
inf
τ!
0≤τ ≤σ
and (0, j)
|r x
¯
(t : x1 , . . . , x j )| ≤ M j e j K |t| inf
(L j |t|)τ
0≤τ ≤σ
τ!
.
To prove (1), it suffices to show that if σ (x, x1 , . . . , x j ) ≥ σ + k then (0, j)
|r x
¯
(t; x1 , . . . , x j )| ≤ M j e j K |t|
inf
0≤τ ≤σ +k
(L j |t|)τ τ!
and we have just shown this for k = 0. We proceed now by induction on k for k > 0. For y ∈ Bx1 we have σ (y, x1 , . . . , x j ) ≥ σ (x, x1 , . . . , x j ) − 1 ≥ σ + k − 1. Therefore, our induction asssumption and estimate of |r est| give
d (0, j) (L j |t|)τ r x (t; x1 , . . . , x j ) ≤ K¯ M j e j K¯ |t| inf dt τ ! 0≤τ ≤σ +k−1 + Cje
j K¯ |t|
inf
0≤τ ≤σ +k−1
(L j |t|)τ τ !
.
Since L j = L j + C j ≥ C j + K¯ , and 1 ≤ M j , we may write for t ≥ 0,
(L j t)τ d (0, j) ¯ |r x (t; x1 , . . . , x j )| ≤ M j e j K |t| , inf L j dt τ ! 0≤τ ≤σ +k−1 hence (0, j)
|r x
¯
(t; x1 , . . . , x j )| ≤ M j e j K |t|
(L j t)τ τ!
if 1 ≤ τ ≤ σ + k, but the above inequality also holds by the induction assumption when τ = 0, and this completes the proof of (1). We discuss now the case i > 0. We have an explicit expression for r (1, j) = dr (0, j) /dt, given by the evolution equation for ( p, q) if j = 0, by (2) if j ≥ 1. We may differentiate repeatedly with respect to t, replacing the derivatives in the right-hand side by using either the evolution equation for ( p, q) or (2). We express thus r (i, j) as a polynomial (0,) in the p y and the r y with ≤ j, with coefficients that are smooth functions of the (0,) that occur satisfy d(x, y) ≤ i. Furthermore, q y . The indices y of the p y , q y , and r y
Nonequilibrium Statistical Mechanics and Entropy Production in Rotators
263
(0, )
(0,m )
in any given term of the polynomial, the factors r y1 1 (X 1 ), . . . , r ym (with (X 1 , . . . , X m ) forming a partition of (x1 , . . . , x j )) satisfy
(X m ) that occur
σ (x, y1 , . . . , ym ) ≤ i, σ (x, y1 , . . . , ym ) + σ (y1 , X 1 ) + · · · + σ (ym , X m ) ≥ σ (x, x1 , . . . , x j ) = σ so that σ (y1 , X 1 ) + · · · + σ (ym , X m ) ≥ σ − i. Therefore, using (1), we see that r (i, j) is a polynomial of degree ≤ i in the p y such that y ∈ Bxi , with coefficients bounded in absolute value by ¯
const.e j K |t|
inf
0≤τ ≤σ −i
(L j |t|)τ τ!
if i ≤ σ , otherwise by ¯
const.e j K |t| , concluding the proof of the proposition.
A.2 (Proof of Proposition 2.8). Note that the conditions on the coefficients of Q are of the same form as those on the coefficients of P in Proposition 2.7, but σ has a new definition (and the L j , Mi j may have to be chosen larger than for Proposition 2.7). Note also that Lemma 2.1(b) gives, for d(x, X ) > 0, |r x(0,0) (t; X )| ≤
( K¯ |t|)τ τ!
if 1 ≤ τ ≤ d(x, X ).
Using also Lemma 2.1(a), this proves the proposition in the case (i, j) = (0, 0) since here σ = d(x, X ). We shall later use the fact that for the q-component we have actually, for d(x, X ) ≥ 0, |qx (t; X )| = |qx (t) − q˜ x (t)| ≤
( K¯ |t|)τ τ!
if 0 ≤ τ ≤ d(x, X )
[this is because qx (t), q˜ x (t) ∈ T, so that |qx (t) − q˜ x (t)| ≤ 1]. (0, j) In the study of r x for j > 0 we do not impose the condition d(x, X ) > 0. If j > 0, we claim that there are constants L j > 0 and M j > 0 such that (0, j)
|r x
¯
(t; x1 , . . . , x j ; X )| ≤ M j e j K |t| inf
0≤τ ≤σ
(L j |t|)τ , τ!
(3)
where σ = σ (x, x1 , . . . , x j ; X ). This will be proved by induction on j, using (1) and the equation d (0, j) (0, j) x y (qx , q y )r y (t; x1 , . . . , x j ; X ) + r est. r x (t; x1 , . . . , x j ; X ) = dt 1 y∈Bx
The r est here is a finite sum of products, each of which has exactly one factor with a in front of it. The factors are: a coefficient depending smoothly on qx , q y , and factors
264
D. Ruelle
(0, j )
r z n n , where z n is x or y and the X n form a partition X = (X n ) of (x1 , . . . , x j ) into |X | subsequences of length jn . (0,1) In particular, for j = 1, the r est is y x y .r y . Using the remark above on (0,1)
|qx |, and the earlier bound on |r x |r est| ≤ const.
| one finds
k+ ( K¯ |t|)k K¯ |t| ( K¯ |t|) ¯ (2 K¯ |t|) .e ≤ const.e K |t| , k! ! (k + )!
where k is allowed values in [0, d(y, X )] or [0, d(x, X )] and is allowed values in [0, d(y, x1 )], so that k + is allowed all values in [0, [σ − 1]+ ]. (0, j) For general j > 0, using induction on j, and the bounds on |qx |, |r x | shows that the products appearing in the r est have, in absolute value, bounds of the form
(L jn |t|)n ( K¯ |t|)k ¯ . exp const. jn K |t| . k! n ! n n
k+ n ¯ (L j |t|)τ n L jn )|t|] j K¯ |t| [( K + j K¯ |t| ≤ const.e , ≤ const.e (k + n )! τ ! where L j = K¯ + maxX n L jn , and we must now discuss the range of τ = k + n . Remember that there is a in front of one of the factors of the product we are considering. If the is in front of the coefficient depending smoothly on qx , q y , this corresponds to k ∈ [0, d(z, X )] with z = x or y, while n ∈ [0, σ (z n , X n )] with z n = x or y. Since d(x, y) = 1, we have d(z, X ) + σ (z , X ) + 1 ≥ σ (x, x1 , . . . , x j ; X ) = σ ; therefore τ = k + n is allowed all values such that 0 ≤ τ ≤ [σ − 1]+ . (0, j) If the is in front of one of the r z n , say for n = a, the corresponding a is (0, j ) ∈ [0, σ (z a , X a ; X )] by the induction assumption, the other r z n n are ∈ [0, σ (z n , X n )], and we have k = 0. Note that σ (z a , X a ; X ) + σ (z n , X n ) + 1 ≥ σ (x, x1 , . . . , x j ; X ) = σ, n =a
therefore τ = n is again allowed all values such that 0 ≤ τ ≤ [σ − 1]+ . In conclusion we have |r est| ≤ C j e
j K¯ |t|
inf
0≤τ ≤[σ −1]+
(L j |t|)τ τ !
.
To start the proof of (3) we write (0, j)
sup |x x
¯
(t; x1 , . . . , x j ; X )| = r (t; x1 , . . . , x j ; X ) = e K |t| s(t)
and let σ = σ (x1 , . . . , x j ; X ) = min x σ (x, x1 , . . . , x j ; X ), Then for τ ∈ [0, [σ −1]+ ] and t ≥ 0 we obtain, as in the proof of Proposition 2.7,
τ d ¯ (L j t) s(t) ≤ C j e( j−1) K t , dt τ !
Nonequilibrium Statistical Mechanics and Entropy Production in Rotators
265
hence (0, j)
|r x
¯
(t; x1 , . . . , x j ; X )| ≤ M j e j K |t| inf
0≤τ ≤σ
(L j |t|)τ , τ!
and the proof of (3) continues as the proof of (1). The case i > 0 (taking now d(x, X ) > i) is treated as in the proof of Proposition 2.7. Acknowledgements. During the lengthy elaboration of the present paper, I have benefitted from useful correspondence with D. Evans, and many discussions with J.-P. Eckmann, G. Gallavotti, J.L. Lebowitz, H. Moriya, H. Posch, and L.-S. Young.
References 1. Andrey, L.: The rate of entropy change in non-Hamiltonian systems. Phys. Letters 11A, 45–46 (1985) 2. Bonetto, F., Kupiainen, A., Lebowitz, J.L.: Absolute continuity of projected SRB measures of coupled Arnold cat map lattices. Ergod. Th. Dynam. Syst. 25, 59–88 (2005) 3. Bratteli, O., Robinson, D.W.: Operator Algebras and Quantum Statistical Mechanics 2, Second edition. Berlin: Springer, 1997 4. Eckmann, J.-P., Pillet, C.-A., Rey-Bellet, L.: Nonequilibrium statistical mechanics of anharmaonic chains coupled to two baths at different temperatures. Commun. Math. Phys. 201, 657–697 (1999) 5. Eckmann, J.-P., Pillet, C.-A., Rey-Bellet, L.: Entropy production in non-linear, thermally driven Hamiltonian systems. J. Stat. Phys. 95, 305–331 (1999) 6. Evans, D.J., Cohen, E.G.D., Morriss, G.P.: Probability of second law violations in shearing steady states. Phys. Rev. Letters 71, 2401–2404 (1993) 7. Evans, D.J., Morriss, G.P.: Statistical Mechanics of Nonequilibrium Fluids. New York: Academic Press, 1990 8. Gallavotti, G.: Nonequilibrium statistical mechanics (stationary): overview. In: Encyclopedia of Mathematical Physics J.-P. Françoise, G.L. Naber, T.-S. Tsun, eds., Vol 3, Elsevier, 2006 pp. 530–539 9. Gallavotti, G., Cohen, E.G.D.: Dynamical ensembles in stationary states. J. Stat. Phys. 80, 931–970 (1995) 10. Hoover, W.G.: Molecular Dynamics. Lecture Notes in Physics 258. Heidelberg: Springer, 1986 11. Jakši´c, V., Pillet, C.-A.: On entropy production in quantum statistical mechanics. Commun Math. Phys. 217, 285–293 (2001) 12. Lanford, O.E., Lebowitz, J.L., Lieb, E.H.: Time evolution of infinite anharmonic systems. J. Stat. Phys. 16, 453–461 (1977) 13. Posh, H.A., Hoover, W.G.: Large-system phase-space dimensionality loss in stationary heat flows. Physica D 187, 281–293 (2004) 14. Ruelle, D.: Thermodynamic Formalism. Reading MA: Addison-Wesley, 1978 15. Ruelle, D.: Entropy production in quantum spin systems. Commun. Math. Phys. 224, 3–16 (2001) 16. Williams, S.R., Searles, D.J., Evans, D.J.: Independence of the transient fluctuation theorem to thermostatting details. Phys. Rev. E 70, 066113–1-6 (2004) Communicated by G. Gallavotti
Commun. Math. Phys. 270, 267–293 (2007) Digital Object Identifier (DOI) 10.1007/s00220-006-0127-9
Communications in
Mathematical Physics
Nuclearity and Thermal States in Conformal Field Theory Detlev Buchholz1 , Claudio D’Antoni2, , Roberto Longo2, 1 Institut für Theoretische Physik, Universität Göttingen, 37077 Göttingen, Germany.
E-mail:
[email protected]
2 Dipartimento di Matematica, Università di Roma “Tor Vergata”, Via della Ricerca Scientifica, 1-I-00133
Roma, Italy. E-mail:
[email protected];
[email protected] Received: 3 April 2006 / Accepted: 20 April 2006 Published online: 4 November 2006 – © Springer-Verlag 2006
Dedicated to László Zsidó on the occasion of his sixtieth birthday Abstract: We introduce a new type of spectral density condition, that we call L 2 nuclearity. One formulation concerns lowest weight unitary representations of S L(2, R) and turns out to be equivalent to the existence of characters. A second formulation concerns inclusions of local observable von Neumann algebras in Quantum Field Theory. We show the two formulations to agree in chiral Conformal QFT and, starting from the trace class condition Tr(e−β L 0 ) < ∞ for the conformal Hamiltonian L 0 , we infer and naturally estimate the Buchholz-Wichmann nuclearity condition and the (distal) split property. As a corollary, if L 0 is log-elliptic, the Buchholz-Junglas set up is realized and so there exists a β-KMS state for the translation dynamics on the net of C∗ -algebras for every inverse temperature β > 0. We include further discussions on higher dimensional spacetimes. In particular, we verify that L 2 -nuclearity is satisfied for the scalar, massless Klein-Gordon field. 1. Introduction As is known the general principles of Quantum Field Theory (locality, Poincaré covariance, positivity of the energy) are compatible with physically unrealistic models. A simple example is obtained by the infinite tensor product of copies of a fixed model: although the basic structure is preserved under this infinite tensoring, the degrees of freedom may arbitrarily increase. In order to select physically acceptable theories, two kinds of conditions have been introduced in particular. The first are compactness/nuclearity conditions. The HaagSwieca compactness condition requires the set of vectors localized in a given bounded spacetime region with a common energy bound to be a (norm) pre-compact subset of the Hilbert space. Supported by MIUR, GNAMPA-INDAM and EU network “Quantum Spaces—Non Commutative Geometry” HPRN-CT-2002-00280
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A quantitative, efficient refinement of the above compactness is the BuchholzWichmann nuclearity condition [6]: for given double cone O of the Minkowski spacetime and β > 0, the map O (β) : A(O) → H,
X → e−β H X
has to be a nuclear operator. Here H is the state Hilbert space, H is the energy operator and A(O) is the von Neumann algebra of observables localized within O. Possibly conditions on the nuclear norm ||O (β)||1 as O increases or β → 0 may be added. There are several different versions of the nuclearity condition that reflect structural properties of the considered theory. The second type condition is the split property. This property is intrinsically encoded in the net A of local observable algebras. It has several equivalent formulations; we shall state here that A is defined to be split if the natural map ˜ → X X ∈ A(O) · A(O) ˜ X ⊗ X ∈ A(O) A(O) (from the algebraic tensor product) extends to an isomorphism between the von Neumann ˜ and A(O) ∨ A(O) ˜ (statistical independence). Here O ⊂ O˜ algebras A(O) ⊗ A(O) 1 ˜ are double cones and O O. Several structure properties follow from the split assumption, for example one can derive the local current algebra (in integrated form) [16]. The original motivation to introduce the nuclearity was indeed to derive the split property [6]. The nuclearity condition implies the split property, is directly analyzable and has a number of important consequences in itself, as the construction of thermal equilibrium states. This paper primarily concerns conformal quantum field theory, say chiral conformal QFT on the circle. In this context the energy operator is usually replaced with the conformal Hamiltonian L 0 and so there is a another natural nuclearity condition, namely the trace class condition for the semigroup generated by L 0 : Tr(e−β L 0 ) < ∞, β > 0 , namely the characters have to be defined or, from the physical viewpoint, Gibbs states for L 0 must exist at inverse temperature β > 0. It is known that the trace class condition for all β > 0 implies the split property, although the proof is rather indirect [13]. It is natural to study the relations between the trace class condition and the more physical Buchholz-Wichmann nuclearity property; in particular one would derive this latter property from the trace class condition which is a standard assumption in most approaches to conformal QFT. Note the different character of the two conditions: the trace class condition only refers to the associated unitary representation of the Möbius group while Buchholz-Wichmann nuclearity is expressed in terms of local observables too. We shall indeed prove that Buchholz-Wichmann nuclearity follows from the trace class condition, with a rather precise description of this dependence. One most interesting ingredient of our analysis is the introduction of a new nuclearity condition that we call L 2 -nuclearity condition. The key point is that this condition can be phrased in two different ways: one only refers to the unitary representation U of the Möbius group, the other to the net of local algebras A, so we will have a bridge to relate the trace class condition to the Buchholz-Wichmann nuclearity. To be explicit denote by 1 O O ˜ means that the closure of O is contained in the interior of O. ˜
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1 K I ≡ − 2π log I the generator of the dilation one-parameter unitary group associated with the interval I , namely Eq. (2.2) below holds. Then the L 2 -nuclearity condition requires that the operator2 1/4 I
−1/4
TI˜U,I ≡ ˜ I
be a trace class operator on H. Clearly this is a condition on the representation U . Now the operator I is the Tomita-Takesaki modular operator associated with the pair 1/4 −1/4 A(I ) and the vacuum vector, so the L 2 -nuclearity condition || ˜ I ||1 < ∞ can I be expressed in term of the net A. It is then straightforward to infer modular nuclearity 2 from L -nuclearity. Modular nuclearity [7] is a sort of “local version” of BuchholzWichmann nuclearity where the energy operator is replaced with a modular operator (generator of a “local dynamics”). As shown in [8] modular nuclearity and BuchholzWichmann nuclearity are closely related conditions, indeed they are equivalent with the assumption of certain growth conditions. Our chain of implications becomes closed by showing that L 2 -nuclearity is equivalent to the trace class condition. Indeed we shall have the equality ||TI˜U,I ||1 = Tr(q L 0 ) ,
(1.1)
where I˜ is a symmetric interval of R and I = q I˜ is a dilated interval, 0 < q < 1. As we shall see that ||T ˜U ||1 < ∞ immediately implies that A(I ) ⊂ A( I˜) is split, we shall I ,I have in this way a precise dependence from the trace class condition at a fixed β > 0 to the split property at a certain distance (distal split property). This paper is organized as follows. We begin by recalling a formula long pointed out by Schroer and Wiesbrock (see [26]), that we prove here. This formula concerns an unbounded similarity between the semigroup generated by the conformal Hamiltonian and a dilation one-parameter unitary group. We then discuss certain operator identities that are quite powerful for our analysis and have their own interest. The first TI˜,I TI˜∗,I = q 2L 0 , directly relates the conformal Hamiltonian and the T operators. The second one s
s
e−2s L 0 = e− tanh( 2 )H e− sinh(s)H e− tanh( 2 )H relates the conformal Hamiltonian with the translation Hamiltonian H and its ray inversion conjugate H . This shows that T operators are naturally related both to the translation semigroup (see also Proposition 3.2) and to the rotation semigroup and are thus a natural link between these semigroups. We now come to one most interesting consequence of our analysis, the existence of temperature states for the translation dynamics, starting from natural requirements for the conformal Hamiltonian that partly motivated our paper. Quantum Field Theory is a scheme to analyze finitely many quantum particles and one primarily considers the homogeneous fundamental state, the vacuum state; at this stage there is no thermodynamical consideration as the vacuum is a state of zero temperature. In recent years however the study of finite-temperature states has acquired a 2 We often denote by the same symbol a linear operator and its closure.
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definite interest because of various reasons, in particular the desire to study extreme situations where gravitational effects make matter so dense that the system becomes thermodynamical (e.g. black holes). A general construction of thermal states in Quantum Field Theory has been made by Buchholz and Junglas [11]. One of our aims is to derive their assumptions in the context of conformal QFT starting from natural assumptions in this context. As noted, with our formulas we can easily discuss the relations among various forms of nuclearity. As a corollary we see that if L 0 is log-elliptic in the sense that (cf. [21]) log Tr(e−s L 0 ) ∼ const.
1 , sα
s → 0+ ,
for some α > 0, then the Buchholz-Junglas nuclearity assumptions are satisfied and so there exists a β-KMS state with respect to translations on the net of C∗ -algebras for every inverse temperature β > 0 (see also [23] for general considerations). Note that log-ellipticity naturally holds with α = 1 in rational conformal field theory (modular nets) [21]. We end our paper with a discussion concerning QFT on higher dimensional spacetimes. In particular we will use our results for chiral conformal QFT to verify the L 2 nuclearity condition for the neutral, massless, free field on the Minkowski spacetime. 2. An Operator Identity Associated with SL(2, R) The upper and the right semicircle will be denoted respectively by I∩ and I⊃ . We shall often pass from the “circle picture” to the “real line picture”, namely we identify S 1 \{−1} with R by the stereographic map. Then I∩ (0, ∞) and I⊃ (−1, 1). We shall denote by G the universal cover of the Möbius group ( PSL(2, R)), that acts on S 1 as usual. With I an interval of S 1 , we denote by I the one-parameter subgroup of PSL(2, R) of “dilations” associated with I ; namely R+ (s) : x ∈ R → es x ∈ R , and I is then defined by conjugation by any g ∈ PSL(2, R) such that g I∩ = I . I has a unique lift to a one-parameter subgroup of G, that we will still denote by I . Given a unitary, positive energy representation U of G on a Hilbert space H we shall denote by L 0 ≡ L U 0 the conformal Hamiltonian, namely the infinitesimal generator of the rotation subgroup. Thus L 0 is a positive selfadjoint operator on H. We shall denote by K I the selfadjoint generator on H of U ( I (·)), namely eis K I = U ( I (s)) .
(2.1)
1 We shall set K I ≡ − 2π log I , thus I ≡ I,U is the unique positive, non-singular, selfadjoint operator on H such that
is I = U ( I (−2π s)) .
(2.2)
Given the positive energy, unitary representation U of G on H we choose an antiunitary involution J = J I∩ such that J U (g)J = U (rgr −1 ),
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where r is any pre-image in G of the Möbius transformation z → z¯ ; then one defines J I for any interval I by requiring that Jg I ≡ U (g)J I U (g)∗ . Following [5] we define the 1/2 unbounded anti-linear involution S I ≡ J I I and set H (I ) ≡ {ξ ∈ H : S I ξ = ξ }. H (I ) is a real, standard Hilbert subspace of H and one has (see [5]): Proposition 2.1. I is the modular operator associated with H (I ). Clearly the family of real Hilbert spaces H (I ) is covariant for the representation U . A non-trivial fact is that if U is a representation of S L(2, R) (not of its cover), then {H (I )} is isotone, namely I ⊂ I˜ ⇒ H (I ) ⊂ H ( I˜) . Given an inclusion of real, standard Hilbert subspaces H ⊂ H˜ of H we shall consider the linear operator on H, TH˜ ,H (λ) ≡ λH˜ −λ H , 0 ≤ λ ≤ 1/2 . For each λ, TH˜ ,H (λ) is densely defined, bounded and ||TH˜ ,H (λ)|| ≤ 1. Denoting the closure by the same symbol TH˜ ,H (λ), the map λ → TH˜ ,H (λ) is holomorphic in the strip 0 < λ < 1/2 and continuous on the closure. These facts can be proved by the same arguments as in [7]. Given the unitary representation U of G as above, we then set TI˜,I (λ) ≡ TH ( I˜),H (I ) (λ) so we have: Corollary 2.2. If U is a positive energy, unitary representation of S L(2, R) on H, and I ⊂ I˜ are intervals of S 1 , then the associated operator TI˜,I (λ) is bounded with ||TI˜,I (λ)|| ≤ 1 and the map λ → TI˜,I (λ) is holomorphic in the strip 0 < λ < 1/2 and continuous on its closure. We shall see that Cor. 2.2 holds also for all positive energy, unitary representations of G, but the proof is non-trivial as H (I ) is not included in H ( I˜) in this general case. The case λ = 1/4 is of particular relevance and we set TI˜,I ≡ TI˜,I (1/4) . We now prove a formula pointed out by Schroer and Wiesbrock [26]. We give here below a proof in the case U is a representation of S L(2, R), a case that covers most needs in this paper. The proof will be continued in Appendix 9.3 to treat the general case of representations of the cover G. Theorem 2.3 cf. [26]. Let U be a positive energy unitary representation of G. For every s ≥ 0, the following identity holds: −1/4
1 −is 2 1 1/4
= e−2π s L 0 ,
(2.3)
where 1 ≡ I∩ , 2 ≡ I⊃ and L 0 are associated with U . 1/4 −1/4 −1/4 More precisely the domain of 1 −is is a core for 1 and the closure of 2 1 1/4 −1/4 −2π s L 0 . is equal to e 1 −is 1 2
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Proof. We split the proof in parts; a part may rely on a (temporary) independent extra assumption. Part 1. We denote by k1 , k2 and l0 the elements in the Lie algebra sl(2, C) corresponding to K 1 ≡ K I1 , K 2 ≡ K I2 and L 0 . Then [k1 , k2 ] = −il0 , [k1 , l0 ] = k2 . Denoting by Ad(g) the adjoint action of g ∈ G on sl(2, C) we then have Ad(e−2πitk1 )(k2 ) =
∞ n t n=0
n!
δkn1 (k2 ) = cosh(2π t)k2 − sinh(2π t)l0 ,
where δk1 ≡ 2π [k1 , · ], therefore for all s, t ∈ R we have the identity in G e−2πitk1 e2πisk2 e2πitk1 = e2πis cosh(2π t)k2 −sinh(2π t)l0 that of course gives the operator identity e−2πit K 1 e2πis K 2 e2πit K 1 = e2πis
cosh(2π t)K 2 −sinh(2π t)L 0
.
(2.4)
Consider the right-hand side of Eq. (2.4) that we denote by Ws (t). Given r > 1/2, by Lemma 2.4 there exist s0 > 0, a dense set D ⊂ H of joint analytic vectors for K 1 , K 2 , L 0 such that, for any fixed s ∈ R with |s| ≤ s0 and η ∈ D, the vector-valued function t → Ws (t)η ≡ e2πis cosh(2π t)K 2 −sinh(2π t)L 0 η has a bounded analytic continuation in the ball Br ≡ {z ∈ C : |z| < r }. −it Consider now the left hand side of Eq. (2.4). By definition it is equal to it1 −is 2 1 . Part 2. In this part we assume that U is a representation of S L(2, R) (rather than of its cover G). Now the map −it t ∈ R → it1 −is (2.5) 2 1 has a uniformly bounded, strongly operator continuous, analytic extension in the strip −λ −is S(0, −1/2) ≡ {z ∈ C : −1/2 < z < 0}. Indeed λ1 −is 2 1 = TI1 ,I1,s (λ)2 , where I1,s = I2 (2π s)I1 ⊂ I1 , so the analyticity of (2.5) follows from Lemma 2.2 in the S L(2, R)-case. Taking matrix elements −it ∗ (η, it1 −is 2 1 ξ ) = (Ws (t) η, ξ ) = (W−s (t)η, ξ )
with η ∈ D and ξ an entire vector for 1 , both functions defined by the left and right side of the above equation have an analytic extension in S(0, −1/2) ∩ Br . Taking the value at t = −i/4 we have −1/4
(η, 1 −is 2 1 1/4
ξ ) = (e−2π s L 0 η, ξ ) = (η, e−2π s L 0 ξ ),
−1/4
is equal to e−2π s L 0 if s is real and |s| ≤ s0 , hence hence the closure of 1 −is 2 1 for all s ∈ R by the group property. This ends the proof in the S L(2, R)-case. 1/4
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The proof is continued in Appendix 9.3. We now recall the following result by Nelson used in Th. 2.3. Lemma 2.4. [24]. Let U be a unitary representation of a Lie group G on a Hilbert space H and X 1 , X 2 , . . . , X n a basis for the associated Lie algebra generators. There exist a neighborhood U of the origin in Cn and a dense set of vectors D ⊂ H of smooth vectors for U such that ∞ ||(u 1 X 1 + u 2 X 2 + · · · + u n X n )k η||
k!
k=0
<∞
for all (u 1 , u 2 , . . . u n ) ∈ U and η ∈ D. 3. Trace Class Property and L 2 -Nuclearity for Representations of SL(2, R) 3.1. First formula. Given intervals I I˜ we consider the inner distance between I˜ and I to be the number ( I˜, I ) defined as follows. First assume I˜ = I∩ and I symmetric with respect to the vertical axis, namely I has boundary points {z, −¯z } with z ∈ I∩ ∩ I⊃ . Then z = I⊃ (s)1 for a unique s > 0 and we put ( I˜, I ) ≡ s. For a general inclusion I I˜ there exists a unique g ∈ G such that g I˜ = I∩ and g I = −g I as above and we put ( I˜, I ) ≡ (g I˜, g I ). Of course ( I˜, I ) can be any positive real number. A simple expression is given in the real line picture if I, I˜ ⊂ R are symmetric intervals: then ( I˜, I ) = log(d I˜ /d I ), where d I is the usual length of I . (See Appendix 9.1 for more.) 1 Tr(|A|), If A ∈ B(H), √ the nuclear norm ||A||1 of A is the L norm, namely ||A||∗1 ≡1/2 where |A| ≡ A∗ A; the Hilbert-Schmidt norm is given by ||A||2 = Tr(A A) . We shall consider the property that ||T ˜U ||1 < ∞, that we call L 2 -nuclearity (with I ,I respect to I I˜). Note that ||T U ||1 depends only on ( I˜, I ), namely ||T U ||1 does not I˜,I
I˜,I
change if we replace I I˜ by h I h I˜ with h ∈ G.
Proposition 3.1. In every positive energy unitary representation U , we have TI˜U,I = e−s L 0 2
is/2π
,
where I˜ = I∩ , I I˜ is symmetric w.r.t. the vertical axis, s = ( I˜, I ) and 2 is as above. Therefore ||TI˜U,I ||1 = ||e−s L 0 ||1 (3.1) for any inclusion I I˜ such that s = ( I˜, I ). Proof. Denote by {z, −¯z } the boundary points of I . By multiplying both sides of formula (2.3) by is 2 on the right, we get the equality 1/4 −is −1/4 is e−2π s L 0 is 2 1 2 2 =1 1/4 −1/4 =1 U ( I2 (2π s))1 U ( I2 (−2π s)) 1/4
−1/4
=1 I1,2π s , where I1,s ≡ I2 (s)I1 ; that is to say T ˜U = e−s L 0 2
is/2π
is/2π
then 2
I ,I
is unitary we are done.
, where s = ( I˜, I ). Since
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As a consequence we have a key equation for the T operator TI˜,I TI˜∗,I = e−2s L 0 , s ≡ ( I˜, I )
(3.2)
with I˜ the upper semicircle and I symmetric w.r.t. the vertical axis. Of course by Möbius covariance we have a general formulation of the above proposition and the above equation for arbitrary inclusions I I˜. In particular formula (3.1) holds true for any inclusion of intervals I I˜ with s = ( I˜, I ). 3.2. Second formula. Denote by τ the one parameter group of translations on R and τ the translations associated with (−∞, 0), namely the conjugate group of τ by the ray inversion map x → −1/x. Both τa and τ−a maps R+ into itself for positive a and a . Set I ≡ R+ and Ia ,a ≡ τ−a τa I with a, a > 0, so that Ia ,a I . Let U be a positive energy unitary representation of G and denote by H ≡ H I and H ≡ H I = H I the positive generators of the one-parameter unitary subgroups corresponding to τ and τ . We have: Proposition 3.2. U TI,I
a ,a
= e−a
H I
e−a HI e−ia HI eia
H I
.
Proof. Indeed U TI,I
1/4
a ,a
−1/4
≡ I I = =
a ,a
1/4 −1/4 I e−ia HI eia HI I e−ia HI eia HI 1/4 −1/4 1/4 −1/4 I e−ia HI I I eia HI I e−ia HI eia HI −a H I −a H I −ia H I ia H I
=e
e
e
(3.3)
,
e
where we have used the Borchers commutation relation ia H I −is is I = ei(e I e
−2π s )a H
I
and the analogous one with H instead of H . If a > 0, the above equation holds true for all complex s with −1/2 ≥ s ≥ 0 and we have applied it with s = −i/4 [2]. As a consequence we have another key equation for the T operator: ∗ TI,Ia ,a TI,I = e−a HI e−2a a ,a
H I
e−a HI .
(3.4)
Note also that by Prop. 3.1 we also have −1
−1
√
||TI,Ia ,a ||1 = Tr(e−(I,Ia ,a )L 0 ) = Tr(e−2 sinh ( (I,Ia ,a ))L 0 ) = Tr(e−2 sinh ( aa )L 0 ), √ where (I, Ia ,a ) = a a is the second inner distance (Appendix 9.1), thus = sinh( 2 ) by Prop. 9.1. 3 We now have some of our basic formulas. 3 As 2L = H + H one could more directly use the Golden-Thompson inequality Tr(e−a H e−2a H e−a H ) 0 ≥ Tr(e−2a H −2a H ) ) (see [25]), yet this only gives an inequality.
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Theorem 3.3. In any positive energy unitary representation of G we have
s
s
e−2s L 0 = e− tanh( 2 )H e− sinh(s)H e− tanh( 2 )H ,
(3.5)
therefore s
e−2s L 0 ≤ e−2 tanh( 2 )H
(3.6)
for all s > 0. Proof. Consider an inclusion of intervals I I˜ with I˜ = I∩ , I symmetric with respect is/2π to the vertical axis and ( I˜, I ) = s. By Prop. 3.1 we have TI˜,I = e−s L 0 2 thus TI˜,I TI˜∗,I = e−2s L 0 . On the other hand by Prop. 3.2 we have TI˜,I TI˜∗,I = e−a H e−2a
H
e−a H ,
where a > 0 and a > 0 satisfy τ−a τa I˜ = I . By Eq. (9.4) we have a = sinh(s)/2 and a = tanh(s/2), so we have formula (3.5). s Equation (3.5) immediately entails e−2s L 0 ≤ e−2 tanh( 2 )H . Note that the inequality s
e−s L 0 ≤ e− tanh( 2 )H
(3.7)
follows from (3.6) because the square root is an operator-monotone function.4 Note also that the equation
s
s
e−2s L 0 = e− tanh( 2 )H e− sinh(s)H e− tanh( 2 )H
follows by (3.5) by applying a conjugation by a π -rotation on both sides. Remark. We may formally analytically continue the parameter s in formula (3.5) to the imaginary axis and get the equality
s
s
e−2is L 0 = ei tan( 2 )H ei sin(s)H ei tan( 2 )H ,
(3.8)
in particular
e−iπ L 0 = ei H ei H ei H .
(3.9)
Indeed Eq. (3.8) holds true as one can check that it corresponds to an identity in the group S L(2, R). One can then use (3.8) to give an alternative derivation of the identity (3.6). This shows that Eq. (3.6) holds for all unitary representations of G, without assuming positive energy, although the involved operators become unbounded in the general case. 4 The inequality (3.7) does not follow from L ≥ 1 H because the exponential is not operator monotone. 0 2
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3.3. More general embeddings. With U a positive energy representation of G as above, we shall need to estimate the nuclear norm of the more general embedding operators U U TI,I (λ) = TI,I (λ) ≡ λI −λ I0 , 0 < λ < 1/2, 0 0
associated with an inclusion of intervals I0 I . Clearly U U = TI,I (1/4). TI,I 0 0 τ I as above, Proposition 3.4. For an inclusion of intervals It ⊂ I with It ≡ It,t = τ−t t thus (I, It ) = t, we have TI,It (λ) = e−i cos(2π λ)t HI e− sin(2π λ)t HI e− sin(2π λ)t HI ei cos(2π λ)t HI e−it HI eit HI .
Proof. U TI,I (λ) ≡ λI −λ It t
−it H I it H I = λI e−it HI eit HI −λ e I e λ −it H −λ λ it H −λ −it H it H Ie I I I e I I e = I e I
= e−i(e
−2πiλ )t H I
ei(e
2πiλ )t H
I
(3.10)
e−it HI eit HI
= e−i(cos(2π λ)−i sin(2π λ))t HI ei(cos(2π λ)+i sin(2π λ))t HI e−it HI eit HI
= e−i cos(2π λ)t HI e− sin(2π λ)t HI e− sin(2π λ)t HI ei cos(2π λ)t HI e−it HI eit HI . Corollary 3.5. TI,It (λ)1 = TI,Isin(2π λ)t 1 . Proof. Immediate because by Proposition 3.4 the operator TI,Isin(2π λ)t is obtained by left and right multiplication of TI,It (λ) by unitary operators. 4. A Further Operator Inequality We shall need a version of an inequality proved in [8], see Appendix 9.2. Proposition 4.1. Let U be a positive energy, unitary representation of G. We have ||e− tan(2π λ)d I H −λ I || ≤ 1 ,
0 < λ < 1/4 ,
(4.1)
where I is an interval of R with usual length d I . Proof. Assume first that U is a representation of PSL(2, R). We may consider U as a representation of PSL(2, R) × PSL(2, R) which is trivial on the second component. As PSL(2, R) × PSL(2, R) is the symmetry group of a two-dimensional Möbius covariant net on the 2-dimensional Minkowski spacetime, following the comments in Appendix 9.2 we get the bound (4.1). If U is a representation of a finite n-cover of PSL(2, R), we consider the n-time tensor product representation U ⊗ · · · ⊗ U , which is a representation of PSL(2, R). Inequality (4.1) follows also in this case. Finally, if U is any representation of G, the result follows by a continuity argument by using Lemma 9.2.
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Note that Inequality (4.1) gives −2 tan(2π λ)d I H −λ −λ I ≤ 1 I e
namely e−2 tan(2π λ)d I H ≤ 2λ I . In particular, if I = I2 is the interval (−1, 1) of the real line, we have e−4 tan(2π λ)H ≤ 2λ 2 . By conjugating both members of the inequality with the modular conjugation J I2 (ray inversion map) we get e−4 tan(2π λ)H ≤ −2λ 2 , namely 4 tan(2π λ)H e−4 tan(2π λ)H ≤ 2λ 2 ≤e
(4.2)
and, by rescaling with the dilation unitaries we obtain the inequality e−2 tan(2π λ)d I H ≤ 2λ I ≤e
2 tan(2π λ) d1 H I
.
(4.3)
In particular, evaluating at λ = 1/8, we get e−2d I H ≤ I
1/4
2
≤ e dI
H
.
(4.4)
5. Modular Nuclearity and L 2 -Nuclearity 5.1. Basic abstract setting. We now introduce the concept of L 2 -nuclearity in an abstract setting of inclusions of von Neumann algebras with a distinguished cyclic and separating vector. This will immediately provide the notion of L 2 -nuclearity in conformal QFT by considering an inclusion of local observable von Neumann algebras and the vacuum vector. Let M be a von Neumann algebra on a Hilbert space H and cyclic and separating unit vector . We set L ∞ (M) = M,
L 2 (M) = H,
L 1 (M) = M∗ .
Then we have the embeddings x→(x,J · ) / L 1 (M) L ∞ (M) M ;; ∞,1 B ;; ;; M M ;; ∞,2 2,1 ; x→1/4 x ;;; ξ →(ξ,J · ) ;; ;
L 2 (M)
All embeddings are bounded with norm one. Let now N ⊂ M be an inclusion of von Neumann algebras with cyclic and separating unit vector .
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Given p, q = 1, 2, ∞, p ≥ q, we shall say that L p,q -nuclearity holds for N ⊂ M, 5 with respect to , if M p,q | N is a nuclear operator. ∞,2 L nuclearity has played an important rôle and has been named modular nuclearity [7]. M = M M , we have As ∞,1 2,1 ∞,2 M M M M | N ||1 ≤ ||2,1 || · ||∞,2 | N ||1 ≤ ||∞,2 | N ||1 , ||∞,1
where ||·||1 denotes the nuclear norm. Thus modular nuclearity implies L ∞,1 nuclearity. Indeed the following result from [7, 8] holds: M | || ≤ Proposition 5.1 [7, 8]. Modular nuclearity implies L ∞,1 nuclearity and ||∞,1 N 1 M ||∞,2 | N ||1 . M | is nuclear of type s, then also M | is nuclear of type s. Conversely if ∞,2 N ∞,1 N
Of course the first part of the above statement is immediate by the above diagram: M | = N · M | and || N || ≤ 1. indeed ∞,1 N 2,1 2,1 ∞,2 N M | is nuclear (thus if modular Proposition 5.2 [7, 8]. If N or M is a factor and ∞,1 N nuclearity holds) then N ⊂ M is a split inclusion. M | nuclear means that there exist Proof We recall the short proof. By definition ∞,1 N sequences of elements ϕk ∈ N ∗ and ψk ∈ M ∗ ( L 1 (M)) such that k ||ϕk || ||ψk || < ∞ and ϕk (n)ψk (m ) , n ∈ N , m ∈ M , ω(nm ) = k M | is normal the ϕ can be chosen normal (take the normal where ω ≡ ( · , ). As ∞,1 N k part). Thus the state ω on N M extends to N ⊗ M and this gives the split property.
The above proposition also holds in the non-factor case with N and M generating algebraically a tensor product (which is automatic in the factor case) if M is properly infinite. Consider now the commutative diagram M | ∞,1 N
L ∞ (N ) −−−−−−−−−−−−→ L 1 (M) ⏐ ⏐ M N ⏐ ∞,2 ⏐ 2,1 1/4
−1/4
TM,N ≡ M N
L 2 (N ) −−−−−−−−−−−−−→ L 2 (M) 1/4
−1/4
Recall that the operator M N is densely defined with norm one; its closure TM,N here above is the canonical embedding of L 2 (N ) into L 2 (M). We shall now consider the condition that TM,N be a nuclear operator that we call the L 2 -nuclearity condition. 5 Recall that a linear operator A : X → Y between Banach spaces X, Y is called nuclear if there exist ∗ sequences of elements f k ∈ X and yk ∈ Y such that k || f k || ||yk || < ∞ and Ax = k f k (x)yk . The infimum ||A||1 of k || f k || ||yk || over all possible choices of { f k } and {yk } as above is the nuclear norm of A.
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M | || ≤ ||T Proposition 5.3. L 2 -nuclearity implies modular nuclearity and ||∞,2 N 1 M,N ||1 . N N M | =T Proof. Immediate because ∞,2 N M,N · ∞,2 and ||∞,2 || ≤ 1.
Remark. As shown in [7], the split property for an inclusion of von Neumann algebras M | is a compact operator, q = 1 as in Sect. 5.1 implies modular compactness, i.e. ∞,q N or 2, (independently of the choice of the vector ). It is natural to wonder whether the split property implies the compactness of the operators TM,N . We shall see in Sect. 8.1 that this is not the case by computing the operator TM,N for inclusions of local von Neumann algebras associated to wedge regions on Minkowski spacetime. 5.2. Varying the exponent. We shall consider the condition TM,N (λ)1 < ∞ with TM,N (λ) ≡ λM −λ N for general exponents 0 < λ < 1/2. Note that TM,N (λ) ≤ 1 by a standard interpolation argument [7]. Consider the map λM : M → H, λM : x ∈ M → λM x ∈ H, M = M . We have || M || ≤ 1 if 0 ≤ λ ≤ 1/2. Since thus 1/4 λ ∞,2
λM | N = TM,N (λ) · λN , we have ||λM | N ||1 ≤ ||TM,N (λ)||1 . 6. Conformal Nets and Nuclearity Conditions We now consider a Möbius covariant net A. Namely A is a map I → A(I ) S1
from the set of proper intervals of to von Neumann algebras on a fixed Hilbert space H. We assume: Isotony, i.e. I ⊂ I˜ ⇒ A(I ) ⊂ A( I˜); Möbius covariance, i.e. there exists a unitary representation U of G on H such that U (g)A(I )U (g)∗ = A(g I ); Positive energy, i.e. the conformal Hamiltonian L 0 , the generator of the one-parameter rotation subgroup, is positive; Vacuum vector, i.e. there is a U -invariant unit
vector ; Irreducibility, i.e. I A(I ) = B(H) and I A(I ) = C. We refer e.g. to [4, 13] for what we need here. We do not assume locality nor diffeomorphism covariance. It follows that is cyclic and separating for each fixed von Neumann algebra A(I ), so the modular operator I is defined. Moreover itI = Z (t)U ( I (−2π t)). Here Z is a one-parameter group of internal symmetries (Z (t)A(I )Z (−t) = A(I ) and Z (t) = ). If the net A is local, then Z is trivial. Note that the operator TI˜,I (λ) ≡ λ˜ −λ I here I
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below is equal to e−8λπ K I˜ e8λπ K I also in the non-local case as Z does not depend on the interval and thus the corresponding factors cancel out. Consider the following nuclearity conditions for A: Trace class condition: Tr(e−s L 0 ) < ∞, s > 0; L 2 -nuclearity: ||TI˜,I (λ)||1 < ∞, ∀I I˜, 0 < λ < 1/2; Modular nuclearity: I˜,I (λ) : x ∈ A(I ) → λ˜ x ∈ H is nuclear ∀I I˜, 0 < λ < I 1/2; −s H x ∈ H is nuclear, I Buchholz-Wichmann nuclearity: BW I (s) : x ∈ A(I ) → e interval of R, s > 0 (H the generator of translations); Conformal nuclearity: I (s) : x ∈ A(I ) → e−s L 0 x ∈ H is nuclear, I interval of S 1 , s > 0. We shall show the following chain of implications: Trace class condition L 2 − nuclearity ⇓ Modular nuclearity ⇓ Buchholz-Wichmann nuclearity ⇓ Conformal nuclearity Where all the conditions can be understood for a specific value of the parameter, that will be determined, or for all values in the parameter range. Remark. The implication “BW-nuclearity ⇒ Modular nuclearity” also holds true if one assumes “nuclearity of higher order”, e.g. type s [8]. The argument goes through the A( I˜)
following chain of implications: “BW-nuclearity of type s ⇒ ∞,1 |A(I ) is nuclear of
A( I˜) type s ⇒ ∞,2 |A(I ) is nuclear of type s, with I I˜ ”.
We have already discussed the implications “Trace class condition ⇔ L 2 -nuclearity ⇒ Modular nuclearity”. 6.1. Modular nuclearity ⇒ BW-nuclearity. Equation (4.1) gives ||e− tan(2π λ)d I H −λ I || ≤ 1 for all 0 < λ < 1/4, so the following holds: Proposition 6.1. Let I0 I be a an inclusion of intervals of R. We have ||BW tan(2π λ)d I ||1 ≤ || I,I0 (λ)||1 , I0 where d I is the length of I , 0 < λ < 1/4. Proof. With X ∈ A(I0 ) we have λ −s H BW I X , X = e−s H −λ I0 (s)X = e I
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thus
BW tan(2π λ)d I = e− tan(2π λ)d I H −λ · I,I0 (λ), I0 I tan(2π λ)d I ||1 ≤ || I,I0 (λ)||1 as desired. and so ||BW I0
6.2. Quantitative and asymptotic estimates. At this point we have the following chain of inequalities: ||BW tan(2π λ)d I ||1 ≤ || I,I0 (λ)||1 I0 ≤ ||TI,I0 (λ)||1 = ||TI,I1 ||1
(I, I1 ) = sin(2π λ) (I, I0 )
= Tr(e−s L 0 ), s = (I, I1 ). −1 −1 Note that s = 2 sinh (I, I1 ) = 2 sinh sin(2π λ) (I, I0 ) by Prop. 9.1. As λ → 0+ one has tan(2π λ) ∼ 2π λ and s ∼ 4π λ (I, I0 ) so we have the asymptotic inequality −(2 (I,I0 )/d I )a L 0 ), a → 0+ . (6.1) ||BW I0 (a)||1 ≤ Tr(e Here below we have our last estimate that will give a relation to conformal nuclearity. 6.3. BW-nuclearity ⇒ Conformal nuclearity. By Eq. (3.6) there exists a bounded operator B with norm ||B|| ≤ 1 such that s
e−s L 0 = Be− tanh( 2 )H , therefore
I (s) = BBW I (tanh(s/2))
(6.2)
and so we have Proposition 6.2. || I (s)||1 ≤ ||BW I (tanh(s/2))||1 . One more consequence of Eq. (6.2) is the following. Corollary 6.3. If the split property holds for A then “conformal compactness” holds, namely I (s) is a compact operator for all s > 0 and all intervals of S 1 . Proof. The split property implies “modular compactness” ( I˜,I (λ) is compact) [7] and “modular compactness” implies “BW-compactness”(BW I (s) is compact) by Eq. (9.7) [8]. The corollary thus follows by Eq. (6.2). 6.4. Deriving the split property. The implication “trace class condition ⇒ modular nuclearity” has the following corollary (cf. [13]), namely a simple and direct derivation of the split property. Corollary 6.4. Let A satisfy the trace class condition at a fixed s0 > 0, i.e. Tr(e−s0 L 0 ) < ∞. Then the distal split property holds, more precisely A(I ) ⊂ A( I˜) is a split inclusion if I˜ ⊃ I and ( I˜, I ) > s0 . Proof. By Eq. (3.1) we have L 2 -nuclearity holds if ( I˜, I ) > s0 . Thus modular nuclearity holds for the inclusion I ⊂ I˜ by Prop. 5.1. The split property then holds for A(I ) ⊂ A( I˜) by Prop. 5.2.
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7. Constructing KMS States Let A0 be an irreducible net of von Neumann algebras on R, which is translation covariant with positive energy and vacuum vector (with the Reeh-Schlieder cyclic and separating properties). We denote by the same symbol A0 the quasi-local C∗ -algebra, i.e. the norm closure of ∪ I A(I ) as I varies in the bounded intervals of R. Let A ⊂ A0 be the C∗ -algebras of elements with norm continuous orbit, namely A = {X ∈ A0 : lim ||τt (X ) − X || = 0}, t→0
where as above τ is the translation automorphism one-parameter group. Consider the BW-nuclearity condition for A0 with the asymptotic bound || IBW (β)||1 ≤ ecr
m β −n
, β → 0+ ,
(7.1)
where c, m and n are positive constant and r is the length of I . Let now A be a Möbius covariant net on S 1 and denote by A0 its restriction to R S 1 \ {−1}. Theorem 7.1 [11, Thm. 3.4]. If BW-nuclearity holds for A with the asymptotic bound (7.1), then for every β > 0 there exists a state ϕβ on A which is β-KMS with respect to τ . With A a conformal net as above, fix an interval I0 of the real line. Let I ≡ λ−1 I0 , where λ > 0. We now show that, because of dilation covariance, the behavior of the BW nuclearity index as I increases at fixed inverse temperature β is the same as I is fixed and β → 0. Lemma 7.2. || IBW (s)||1 = || IBW (λs)||1 . 0 Proof. We have e−sλH A(I0 ) = e−sλH A(λI ) = e−sλH D(λ)A(I ) = D(λ) D(λ)∗ e−sλH D(λ) A(I ) = D(λ)e−s H A(I ), where D(et ) = U ( R+ (t)), and this readily implies the statement. Theorem 7.3. Let A be a Möbius covariant net on S 1 . If the trace class condition holds for A with the asymptotic bound 1
Tr(e−s L 0 ) ≤ econst. s α , s → 0+ for some α > 0, then the bound (7.1) holds with m = n = α. As a consequence for every β > 0 there exists a translation β-KMS states on A. Proof. Immediate by the Buchholz-Junglas Theorem 7.1, the estimate (6.1) and the above lemma. In case the net A is diffeomorphism covariant, a direct construction of translation KMS states has been pointed out to us by Mihály Weiner (work in progress).
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8. A Look at Nets in Higher Dimension Let A be a net of von Neumann algebras on the Minkowski spacetime. We shall need only isotony, positive energy and vacuum vector with Reeh-Schlieder property for wedge regions (neither locality or Bisognano-Wichmann property will be used). We shall say that L 2 -nuclearity holds for A if the inclusion of von Neumann algebras ˜ satisfies L 2 -nuclearity with respect to the vacuum vector for double A(O) ⊂ A(O) ˜ cones O O, namely ||TO˜ ,O ||1 < ∞, 1/4
−1/4
where TO˜ ,O ≡ ˜ O and O is the modular operator associated with (A(O), ). O We shall consider the operator TO˜ ,O also in the case of different regions (wedges). 8.1. On wedge inclusions on Minkowski spacetime. Let then A be a net on the Minkowski spacetime as above. We consider the wedge W ≡ {x : x1 > |x0 |} and a sub-wedge Wa ≡ {x : x1 > |x0 | + a} with a > 0. Analogously as in formula (3.3) have: Proposition 8.1. TW,Wa = e−a P0 e−ia P1 , where P0 , P1 are the generator of time translations (energy operator) and space translation in the x1 -direction (momentum operator). Proof. Clearly A(Wa ) = eia P1 A(W )e−ia P1 . Therefore −1/4
Wa
−1/4 −ia P1
= eia P1 W
e
a
−1/4 i a2 (P0 −P1 ) −i a2 (P0 +P1 )
a
= ei 2 (P0 +P1 ) e−i 2 (P0 −P1 ) W
e
e
.
Since the light-like translations in the x0 ±x1 direction maps W into itself for positive/negative translations and the corresponding unitary groups have positive generators P0 ± P1 , we can proceed similarly as in (3.3): 1/4
−1/4
W Wa
a
−1/4
a
a
a
=W ei 2 (P0 +P1 ) e−i 2 (P0 −P1 ) W ei 2 (P0 −P1 ) e−i 2 (P0 +P1 ) 1/4 a −1/4 1/4 −i a2 (P0 −P1 ) −1/4 i a2 (P0 −P1 ) −i a2 (P0 +P1 ) W e e = W ei 2 (P0 +P1 ) W W e 1/4
a
a
a
a
=e− 2 (P0 +P1 ) e− 2 (P0 −P1 ) ei 2 (P0 −P1 ) e−i 2 (P0 +P1 ) =e−a P0 e−ia P1 , where we have used Borchers commutation relations [2]. Remark. Suppose now A to be the net generated by a free scalar field of mass m > 0 on the two dimensional Minkowski spacetime. It is known that in this case the inclusion A(Wa ) ⊂ A(W ) is split if a > 0, see [15]. As the spectrum of P0 contains a continuous part, e−a P0 is definitely not compact. Assuming the split property, the local von Neumann algebras are isomorphic to the unique Connes-Haagerup injective factor of type I I I1 . Moreover all split inclusions N ⊂ M with N and M the injective I I I1 -factor are isomorphic. Therefore we have shown that there exists joint cyclic and separating vectors ξ1 and ξ2 for this inclusion such that L 2 -nuclearity holds w.r.t. ξ1 but even L 2 -compactness fails to hold w.r.t. ξ2 . Note also that, if we consider the associated one-particle real Hilbert space structure, that is a unitary, massive representation of the two-dimensional Poincaré group, then as above TW,Wa is not compact; moreover, as ||TW,Wa || = ||e−a P0 || = e−am < 1 we have examples where the uniform norm ||TW,Wa || is arbitrarily small and the corresponding real Hilbert space inclusion is not split.
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8.2. L 2 -nuclearity for the scalar, massless, free field. With O a double cone in the Minkowski spacetime Rd+1 , 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 = x0 , x" : x = 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 translation-dilation covariant net on R. A0 is local if d is odd and twisted local if d is even. Moreover A0 extends to a Möbius covariant net on S 1 (d odd) or on the double cover of S 1 (d even). For simplicity here below we treat the case of d odd; the case d even may be dealt with analogously. The one particle Hilbert space K is (a complex span of) the completion of S(Rd+1 ) equipped with the scalar product ( f, g) =
1 (2π )d
dd p ˜¯ f (−|p|, −p)g(|p|, ˜ p) . 2|p|
Consider a distribution H on Rd+1 of the form H (x) =
∂ k1 ∂ x1k1
···
∂ kd ∂ xdkd
δ(x)h(x0 ),
(8.1)
where δ(x) is the Dirac function concentrated on the time-axis and h is a real smooth function in S(R). Call k ≡ k1 + · · · kd the order of H . The anti-Fourier transform H˜ is a homogeneous degree k polynomial in p1 , . . . pd ˜ p0 ). Denote by Tk the linear span of distributions as in (8.1) with order k. So times h( Tk ⊂ K and the linear span T of the Tk ’s is dense in K (see [20]). Lemma 8.2. Given an interval I ⊂ R, A(O I ) is generated by the Weyl unitaries W (H ), as H varies in k Tk with supp(H ) ⊂ I (i.e. supp(h) ⊂ I ). Proof. Denote by B(I ) the von Neumann algebra generated by the Weyl unitaries W (H ), as H varies in k Tk with supp(H ) ⊂ I . Clearly B is a translation-dilation covariant net on R with positive energy and B(I ) ⊂ A0 (I ). By the above density of T in K, the net B is cyclic on the vacuum vector of A. By the Reeh-Schlieder theorem B(I ) is cyclic on the vacuum. By the geometric expression of the vacuum modular group of A(O I ) [20], B(I ) is globally invariant under the modular group of A(O I ). By the Tomita-Takesaki theory B(I ) = A(O I ). Lemma 8.3. If d is odd, then A0 =
∞
Nd (k)A(k) ,
k=0
where A(k) is the Möbius covariant net on S 1 associated with the k th -derivative of the U (1)-current algebra and Nd (k) is a multiplicity factor (see below).
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Proof. Let U be the irreducible, positive energy representation of the Poincaré group with mass m = 0 and zero helicity. Then A is associated with the second quantization of U as in [5]. Then U extends to an irreducible unitary representation of the d + 1-dimensional conformal group Cd ≡ S O0 (2, d + 1) on the same Hilbert space, that we still denote by U . The subgroup of Cd generated by time-translations, dilations and ray inversion is isomorphic to the Möbius group (= P S L(2, R)). Denote by U0 the restriction of U to this copy of P S L(2, R). Then A0 is associated to the second quantization of U0 . Indeed the real Hilbert subspace (in the one-particle Hilbert space) associated with O I by the representation U of Cd is defined only in terms of U0 (K I generates a one-parameter unitary subgroup of U0 ). Clearly U0 =
∞
Nd (k)U (k) ,
k=1
where U (k) is the positive energy irreducible representation of P S L(2, R) with lowest weight k. As the second quantization net associated with U (k) is A(k) , we get the thesis. We now determine the multiplicity factor Nd (k) in Lemma 8.3. As in the case d = 1 we have A0 = A(0) ⊗ A(0) , cf. [20], we may assume d > 1. Now if H1 , H2 ∈ Tk then ∞ 1 2k+d−2 ˜¯ ˜ p0 (H1 , H2 ) = P¯1 (p)P2 (p)||p|=1 dσd−1 , h 1 (− p0 )h 2 ( p0 )d p0 (2π )d 0 S d−1 (8.2) where P1 and P2 are the homogeneous degree k polynomials in p ≡ p1 , . . . pd " appearing in the Fourier anti-transform of H1 and H2 and σd−1 is the volume element of the unit d − 1-dimensional sphere in Rd . Remark (Spherical harmonics, cf. [22]). Let m d (k) be the number of monomials p1k1 · · · pdkd with order k = k1 + · · · + kd . Note that we have m d (k) =
k
m d−1 (h)
(8.3)
h=0
so m 1 (k) = 1, m 2 (k) = k + 1, m 3 (k) = (k + 1)(k + 2)/2, …, and we have m d (k) ∼
1 k d−1 , (d − 1)!
k →∞.
Denote by Pk the functions on S d−1 that are restrictions of homogeneous degree k polynomials in p1 , . . . pd . Clearly Pk ⊂ L 2 (S d−1 , dσd−1 ). Moreover Pk−2 ⊂ Pk . The natural unitary representation of S O(d) on L 2 (S d−1 , dσd−1 ) leaves Pk globally invariant and the corresponding decomposition into irreducible subspace is L 2 (S d−1 , dσd−1 ) =
∞ k=0
(Pk # Pk−2 ).
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It follows that the irreducible S O(d)-subspaces of L 2 (S d−1 , dσd−1 ) have dimension dim(Pk # Pk−2 ) = m d (k) − m d (k − 2) = m d−1 (k − 1) + m d−1 (k) .
(8.4)
The elements of Pk # Pk−2 are the harmonic spherical functions of degree k, namely a polynomial P ∈ Pk is orthogonal to Pk−2 iff P, as a function on Rd , is annihilated by the Laplace operator. Now Tk is an invariant subspace for the representation U of Cd . The restriction U0 of U to P S L(2, R) obviously commutes with the restriction of U to S O(d). Therefore the restriction of U to Tk # Tk−2 is the tensor product of an irreducible representation of S O(d) and an irreducible, positive energy representation Vk of P S L(2, R). We now show that the lowest weight of Vk is equal to k + (d − 1)/2. Indeed by Eq. (8.2) the Hilbert space Kk of Vk is the completion of S(R) equipped with scalar product ∞ (h 1 , h 2 )k ≡ p02k+d−2 h˜¯ 1 (− p0 )h˜ 2 ( p0 )d p0 0 d−3 d−3 d−3 d−3 ∞ k+ 2 k+ 2 k+ 2 k+ 2 ˜ = p0 h¯ (− p0 )h˜ ( p0 )d p0 = (h ,h )1 , 0
1
2
1
2
where h (k) is the k-derivative of h. Translation and dilation unitaries have a natural expression on Kk . Clearly K1 is the one particle Hilbert space of the U (1)-current algebra. In the x0 configuration space the ray inversion unitary corresponds to the geometric action x0 → −1/x0 (h(x0 ) goes to h(−1/x0 ) multiplied by some power of x0 depending on k). This suffices to conclude that Vk is the irreducible unitary representation of S L(2, R) with lowest weight k + (d − 1)/2. By formula (8.4) we thus have Nd (k) = 0 if k < (d − 1)/2 and, as k → ∞, Nd (k+
(d−1) 2 ) = dim(Pk # Pk−2 ) = m d−1 (k−1)+m d−1 (k) ∼ k d−2 , k → ∞. 2 (d−2)!
Lemma 8.4. We have 2 sd where L 0 is the conformal Hamiltonian of A0 log Tr(e−s L 0 ) ∼
s → 0+ ,
(k)
Proof. With L (k) the generator of rotations in the representation U (k) we have Tr(e−s L 0 ) e−sk + = 1−e −s , therefore as s → 0 we have: (d−1) ∞ ∞ e− 2 s d−2 −sk 1 2 −sk N (k)e ∼ k e d 1 − e−s (d − 2)! (1 − e−s ) k=1 k=0 ∞ ∞ 1 1 2 2 d−2 −st ∼ t e dt = t d−2 e−t dt (d − 2)! (1 − e−s ) 0 (d − 2)!(1 − e−s ) s d−1 0 1 (d − 1) 1 2 2 2 = ∼ d, = −s d−1 −s d−1 (1 − e ) s (d − 2)! (1 − e ) s s
Tr(e−s L 0 ) = U
where is the Euler Gamma-function. U As log Tr(e−s L 0 ) ∼ Tr(e−s L 0 ) as s → 0+ (see [21, Appendix]) we have completed our proof.
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Remark. We give the explicit expression for the trace in the above estimate in the case d = 3. In this case N3 (k) = 2k − 1, therefore ∞ ∞ ∞ 1 2 1 −sk −sk (2k − 1)e = ke − e−sk 1 − e−s 1 − e−s 1 − e−s k=1 k=1 k=1 1 d 2 e−s (1 + e−s ) cosh(s/2) e−s =− = = − 1 − e−s ds 1 − e−s (1 − e−s )2 (1 − e−s )3 4 sinh3 (s/2)
Tr(e−s L 0 ) = U
that goes as 2s −3 when s → 0+ . Corollary 8.5. Let A be the net of von Neumann algebras on Rd+1 associated with the free massless scalar field. Then L 2 -nuclearity holds for A, indeed ||TOr ,O1 ||1 ∼ e
const.
1 (log r )d
as r → 1+ , where Or is a double cone in Rd , centered at the origin, of radius r > 1. By our general results (see Sect. 6) we find in particular the nuclearity estimates of Buchholz-Jacobi [11]. Note that we have proved the following. Proposition 8.6. Let U be the unitary irreducible representation of the conformal group S O(2, d + 1) (d odd) whose restriction to the d + 1-dimensional Poincaré group is the positive energy, massless, zero helicity representation. Then the irreducible decomposition of the restriction of U to the P S L(2, R) subgroup of S O(2, d + 1) generated by time-translation, dilations and ray inversion is given by U | P S L(2,R) =
∞ m d−1 (k − 1) + m d−1 (k) U (k+1) , k=0
where U (k) is the k-lowest weight representation of P S L(2, R) and m d (k) is given by (8.4). 9. Appendix 9.1. Inner distance. Our estimates are based on the introduction of a certain inner distance for inclusions of intervals of S 1 . Such concepts are discussed and clarified here below. With I I˜ the inner distance ( I˜, I ) introduced in Sect. 3.2 can be more intrinsically defined as follows. Let w1 , w2 be the boundary points of I˜ and z 1 , z 2 be the boundary points of I in the counterclockwise order, thus w1 ≺ z 1 ≺ z 2 ≺ w2 (in the counterclockwise order). Let z be a point between z 1 and z 2 . The reflection r I˜ associated with I˜ maps z in a point z ∈ I˜ and let I0 be the interval (z , z). Choose t such that z 1 ≡ I0 (t)z 1 and z 2 ≡ I0 (t)z 2 are conjugate under the reflection r I0 . Then ( I˜, I ) is the unique s > 0 such that I0 (s)w1 = z 1 . It is easily seen that the inner distance satisfies the following properties: • Positivity: ( I˜, I ) > 0 if I I˜ and all positive values are attained.
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• Monotonicity: If I1 I2 I3 we have (I3 , I1 ) > (I3 , I2 ) and (I3 , I1 ) > (I2 , I1 ). • Möbius invariance: ( I˜, I ) = (g I˜, g I ) for all g ∈ G. • Super-additivity: (I3 , I1 ) ≥ (I3 , I2 ) + (I2 , I1 ) if I1 I2 I3 . (See below.) In Sect. 3.2 we have considered a second inner distance defined as follows. First I ) = t if in the real line picture I˜ = R+ and It = τ −t τt I˜, (t > 0). is the conjugate of τ by the ray inversion Here τt is the translation by t and τ−t t
( I˜,
: x → τ−t
x . 1 + tx
More generally we put for any a, a > 0, √ ˜ ( I˜, I ) = aa if I = τ−a τa I . , the above is a Möbius Since conjugating τt and τt by a dilation by λ gives τλ−1 t and τλt ˜ ˜ invariant definition for I ⊂ I when I is the positive half-line. In general, with I1 I2 intervals of S 1 , let g ∈ G be such that g I˜ = I∩ ( R+ ); then we put
( I˜, I ) ≡ (g I˜, g I ) . By the above comments, is well-defined. It easily seen that also is positive, monotone and Möbius covariant. The two distances are related as follows. Proposition 9.1. = sinh(/2). Proof. Given t > 0 and an inclusion of intervals I ⊂ I˜ with ( I˜, I ) = t we want to calculate s ≡ ( I˜, I ). We may assume that I˜ = R+ and I = Iλ−1 t,λt ≡ τλ −1 t τλt I˜. Then −1 λ t 1 ˜ . τ (λt, ∞) = , I = τ I = τ−λ −1 t λt −λ−1 t 1 + t 2 λt We may further choose λ so that I symmetric under ray inversion, namely λ−1 = √ 1 + t 2 . Thus √ t 1 + t2 I = √ , . (9.1) t 1 + t2 Now in the real line picture the right semicircle I⊃ corresponds to the interval (−1, 1), thus s is determined by t . (9.2) (−1,1) (s)0 = √ 1 + t2 Now (−1,1) (s) : x →
x + 1 − e−s (x − 1) x + 1 + e−s (x − 1)
(see [20]) thus Eq. (9.2) gives 1 − e−s sinh(s/2) t = = tanh(s/2) = √ −s 2 1 + e 1+t 1 + sinh2 (s/2) and this implies t = sinh(s/2).
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√ 2 Note that (−1,1) (s)(0, ∞) = Iλ−1 t,λt ≡ τ−λ −1 t τλt with λ = 1/ 1 + t , namely (−1,1) (s)(0, ∞) = τ
√ τ √ (0, ∞) −t 1+t 2 t/ 1+t 2
.
(9.3)
In terms of the inner distance s the interval I in (9.1) is given by I = tanh(s/2), coth(s/2) = τ− sinh(s/2) cosh(s/2) τtanh(s/2) (0, ∞) = τ− sinh(s)/2 τtanh(s/2) (0, ∞).
(9.4)
9.1.1. Super-additivity of the inner distance. The super-additivity of the inner distance can certainly be shown directly. Moreover notice that, setting Ir ≡ (−r, r ), we have (I R , Ir ) = log
R , r
R>r >0,
and so in this case we have the additive property (Ir1 , Ir3 ) = (Ir1 , Ir2 ) + (Ir2 , Ir3 ), r1 > r2 > r3 > 0 . Yet, it is instructive to give a functional analytic proof of the super-additivity. For each k ∈ N there exists exactly one irreducible, positive energy, unitary representation (up to unitary equivalence) of G with lowest weight k, that we denote by U (k) with corre(k) (k) sponding conformal Hamiltonian L 0 . Then each eigenvalue of L 0 has multiplicity one. Now ∞ (k) (k) tk, I˜,I ≡ ||TI˜U,I ||1 = Tr e−s L 0 = e−sn = n=k
s = ( I˜, I ). We thus have t I˜,I ≡ lim
k→∞
k
e−sk , 1 − e−s
˜
tk, I˜,I = e−( I ,I ) .
(9.5)
Then is super-additive, namely t = e− is sub-multiplicative: t I3 ,I1 ≤ t I3 ,I2 t I2 ,I1 (k)
(k)
(k)
with I3 I2 I1 . This follows at once by Eq. (9.5) because TIU3 ,I1 = TIU2 ,I1 TIU3 ,I2 and the nuclear norm is sub-multiplicative. 9.2. Comments on an inequality in [8]. Let A be a Poincaré covariant net local von Neumann algebra as in Sect. 8. The following inequality is proved in [8]: ||e− tan(2π λ)dO H −λ O || ≤ 2 ,
0 < λ < 1/4 .
(9.6)
Here H is the time-translation generator, O is a double cone of Rd+1 whose axis is contained in the time axis, O is the vacuum modular operator associated with A(O), and dO is the time extension of O. We comment here more that the left-hand side is bounded by 1, namely ||e− tan(2π λ)dO H −λ O || ≤ 1 ,
0 < λ < 1/4 .
(9.7)
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Indeed we may consider the n-tensor product net A ⊗ · · · ⊗ A and apply Eq. (9.6) to it. Then n ||e− tan(2π λ)dO H −λ O || ≤ 2
that gives (9.6) as n is arbitrary. If one reads (9.7) in the one-particle Hilbert space of a free field, one finds an inequality that only refers to representations of the Poincaré group. In other words, if U is a positive energy, unitary representation of the Poincaré group, then Eq. (9.7) holds true, where O is the modular operator of the real Hilbert space associated with O (intersection of all real Hilbert subspaces associated with wedges that contain O, see [5]). Note that letting λ → 0+ in the inequality in the remark following [8, Lemma 3.6] we get the operator inequality K O ≤ dO · H, where K O =
1 − 2π
log O .
9.3. Proof of Theorem 2.3 in the general case. We end now the proof of Theorem 2.3 for general representations of the universal cover G of S L(2, R). This is necessary for further applications on nets on the cover of S 1 . We begin to restate a result contained in [18]. Lemma 9.2 [18, Prop. 2.2]. Let U = U (1) be the irreducible unitary representation of G with lowest weight 1 on a Hilbert space H and denote by K 1 , H and L 0 the associated generators of the I∩ -dilations, translations and rotations. There exists an irreducible unitary representation U (α) of G with lowest weight α on H, for all α ≥ 1 such that K 1(α) = K 1 , H (α) = H, (α)
L 0 = L 0 − λH −1 ,
λ = α(α − 1)/2 .
The above operator identities hold on a common core for all K 1(α) , H (α) and L (α) 0 , the (α) corresponding generators for U , jointly for all α ≥ 1. We recall that the construction in [18, Prop. 2.2] was set up with U (α) extending the Schroedinger representation for K 1 − log H on H = L 2 (R). It is implicit in that proof that S(R) is a joint core as stated in the above lemma. Now we have 2L 0 = H + H , therefore H (α) = H − 2λH −1 . With K 2 ≡ K I⊃ we have 2K 2 = H − H , therefore K 2(α) = K 2 + λH −1 .
(9.8)
End of proof of Theorem 2.3. Note first that in the S L(2, R) case we have (e.g. by Cor. 2.2) −i z ||i1z −is 2 1 || ≤ 1, z ∈ S(0, −1/2), s ∈ R
and therefore, by what is already proved, ||e2πis cosh(2π z)K 2 −sinh(2π z)L 0 || ≤ 1, for all z ∈ S(0, −1/2) and s ∈ R.
(9.9)
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Part 3. In this part we assume that Eq. (9.9) holds true. Let η ∈ H and choose a sequence of vectors ηn ∈ D norm converging to η. Then Ws (z)ηn → Ws (z)η uniformly in z showing that Ws (·) is holomorphic in S(0, −1/2) ∩ Br . Because of the identity (2.4) we infer that the map −it t → it1 −is 2 1
has an analytic continuation in S(0, −1/2) ∩ Br . According to Thm. 2.3 of Araki and Zsido [1] the value of this analytic continuation at t = −1/4 is the closure of 1/4 −1/4 1 −is . This must coincide with Ws (t)t=−i/4 , namely Eq. (2.3) holds true. 2 1 We have therefore proved the theorem by assuming Eq. (9.9). Part 4. In this part we assume that U (2π ), the 2π -rotation in the representation U is a scalar and U (2π ) = ei2π ϑ with ϑ rational. Thus U is a representation of a finite cover of S L(2, R). Then there exists n ∈ N such that the n-tensor product U˜ ≡ U ⊗ U ⊗ · · · ⊗ U is a representation of S L(2, R). By Part 1, Eq. (9.9) holds in the representation U˜ , namely ||Ws (z) ⊗ · · · ⊗ Ws (z)|| ≤ 1. Therefore ||Ws (z)|| ≤ 1, namely Eq. (9.9) holds in the representation U . By Part 3 we conclude that Eq. (2.3) is true in the representation U . Part 5. In this part we assume that U = U (α0 ) is the irreducible representation of G with lowest weight α0 irrational. Therefore U (2π ) is a scalar. We assume first that α0 > 1 so we may be in the setting of Lemma 9.2 and Eq. (9.8) holds too. Let η ∈ D be a unit vector and consider the function 2πis cosh(2π z)K 2α −sinh(2π z)L α0 η F(s, z, λ) ≡ e −1 −1 = e2πis cosh(2π z)K 2 −sinh(2π z)L 0 +λ (cosh(2π z)+sinh(2π z))H η −1 −1 = e2πis cosh(2π z)K 2 −sinh(2π z)L 0 +λ (exp(2π z))H η, λ = α(α − 1). Note that F is defined |s| ≤ s0 , z ∈ S(0, 1/2) and α the given α0 . By Part 4, F is also defined if α ≥ 1 is rational (and all s ∈ R and z ∈ S(0, 1/2)) and in this case ||F(s, z, λ)|| ≤ 1. As the map 1 × R+ → sλ−1 e2π z s, z, λ ∈ [−s0 , s0 ] × S 0, 2 is open (by the holomorphic open mapping theorem), it follows that if λ is sufficiently close to λ0 = α0 (α0 − 1), there exists s and z such that F(s , z , λ) = F(s, z, λ0 ). Taking α ∈ Q we conclude that ||F(s, z, λ0 )|| ≤ 1. Therefore ||Ws (z)|| ≤ 1. By Part 3 we have proved the lemma for U (α0 ) with α0 > 1. Concerning the case 0 < α0 < 1, by considering U (α0 ) ⊗ U (2−α0 ) we conclude as in Part 4 that ||Ws (z)|| ≤ 1 for U (α0 ) , thus the theorem holds in this case by Part 3. Part 6. We may now finish our proof. Indeed if U is any unitary, positive energy representation of G, then U has a direct integral decomposition ⊕ U= N (α)U (α) dα R+
for some multiplicity function N . As the statement holds for U (α) it then holds for U .
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As a corollary, setting t = −i/2 in Theorem 2.3 we see that a formula in [17] concerning a class of unitary representations of S L(2, R) also holds for all unitary, positive energy representations of G: Corollary 9.3. If U is a unitary, positive energy representation of G and s > 0, the associated operators as in Th. 2.3 satisfy −1/2
1 −is 2 1 1/2
−1/2
for ξ in a core for 1
ξ = is 2ξ
(9.10)
.
10. Concluding Comments There are two natural problems left open in this paper. The first is whether L 2 -nuclearity holds for double cone inclusion associated with the free, scalar, massive field. An answer to this question would clarify the meaning of the L 2 -nuclearity condition, in particular whether it is tight up with conformal invariance and/or spacetime with a space compactification. The second problem is whether BW-nuclearity in chiral conformal QFT implies the trace class condition. In other words one would infer from BW-nuclearity condition conformal nuclearity with a uniformity in interval length. There is a natural continuation of our work concerning a discussion of the nuclearity condition for chiral conformal nets in non-vacuum sectors, but this goes beyond our aim in this article. Acknowledgements. We thank H. Bostelmann for a helpful comment on the material in Section 8.2.
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. Borchers, H.J.: The CPT theorem in two-dimensional theories of local observables. Commun. Math Phys. 143, 315–332 (1992) 3. Borchers, H.J., Buchholz, D.: Global properties of vacuum states in de Sitter space. Ann. Inst. H. Poincaré Phys. Théor. 70, 23–40 (1999) 4. Brunetti, R., Guido, D., Longo, R.: Modular structure and duality in conformal quantum field theory. Commun. Math. Phys. 156, 201–219 (1993) 5. Brunetti, R., Guido, D., Longo, R.: Modular localization and Wigner particles. Rev. Math. Phys. 14, 759–785 (2002) 6. Buchholz, D., Wichmann, E.: Causal independence and the energy-level density of states in local quantum field theory. Commun. Math. Phys. 106, 321 (1986) 7. Buchholz, D., D’Antoni, C., Longo, R.: Nuclear maps and modular structures. I. General properties. J. Funct. Anal. 88, 233–250 (1990) 8. Buchholz, D., D’Antoni, C., Longo, R.: Nuclear maps and modular structures II: application to quantum field theory. Commun. Math. Phys. 129, 115–138 (1990) 9. Buchholz, D., Lechner, G.: Modular nuclearity and localization. Ann. H. Poincaré 5, 1065–1080 (2004) 10. Buchholz, D., Jacobi, P.: On the nuclearity condition for massless fields. Lett. Math. Phys. 121, 255–270 (1989) 11. Buchholz, D., Junglas, P.: On the existence of equilibrium states in local quantum field theory. Commun. Math. Phys. 13, 313–323 (1987) 12. Buchholz, D., Yngvason, J.: Generalized nuclearity conditions and the split property in quantum field theory. Lett. Math. Phys. 23, 159–167 (1991) 13. D’Antoni, C., Longo, R., Radulescu, F.: Conformal nets, maximal temperature and models from free probability. J. Op. Theory 45, 195–208 (2001)
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14. D’Antoni, C., Doplicher, S., Fredenhagen, K., Longo, R.: Convergence of local charges and continuity properties of W∗ -inclusions. Commun. Math. Phys. 110, 325–348 (1987) 15. D’Antoni, C., Fredenhagen, K.: Charges in space-like cones. Commun. Math. Phys. 94, 537–544 (1984) 16. Doplicher, S., Longo, R.: Standard and split inclusions of von Neumann algebras. Invent. Math. 75, 493–536 (1984) 17. Guido, D., Longo, R.: An algebraic spin and statistics theorem. Commun. Math. Phys. 172, 517–533 (1995) 18. Guido, D., Longo, R., Wiesbrock, H.-W.: Extensions of conformal nets and superselection structures. Commun. Math. Phys. 192, 217–244 (1998) 19. Haag, R.: Local Quantum Physics. Berlin: Springer-Verlag, 1996 20. Hislop, P.D., Longo, R.: Modular structure of the local algebras associated with the free massless scalar field theory. Commun. Math. Phys. 84, 71–85 (1982) 21. Kawahigashi, Y., Longo, R.: Noncommutative spectral invariants and black hole entropy. Commun. Math. Phys. 257, 193–225 (2005) 22. Kirillov, A.A.: Elements of the Theory of Representations. New York: Springer-Verlag, 1976; Sugiura, M.: Unitary Representations and Harmonic Analysis. An Introduction. Amsterdam, Tokyo: North-Holland, 1990 23. Longo, R.: Notes for a quantum index theorem. Commun. Math. Phys. 222, 45–96 (2001) 24. Nelson, E.: Analytic vectors. Ann. Math. 70, 572–615 (1959); Goodman, R.: Analytic and entire vectors for representations of Lie groups. Trans. Amer. Mat. Soc. 143, 55–76 (1969) 25. Ruskai, M.B.: Inequalities for traces on von Neumann algebras. Commun. Math. Phys. 26, 280–289 (1972) 26. Schroer, B.: Two-dimensional models as testing ground for principles and concepts of local quantum physics. Ann. Phys. 321, 435–479 (2006) 27. Stratila, S., Zsido, L.: Lectures on von Neumann Algebras. Tunbridge Wells, UK: Abacus Press, 1979 28. Wiesbrock, H.-W.: Half-sided modular inclusions of von Neumann algebras. Commun. Math. Phys. 157, 83–92 (1993) Communicated by Y. Kawahigashi
Commun. Math. Phys. 270, 295–333 (2007) Digital Object Identifier (DOI) 10.1007/s00220-006-0148-4
Communications in
Mathematical Physics
Spectral Curves and the Mass of Hyperbolic Monopoles Paul Norbury1 , Nuno M. Romão2 1 Department of Mathematics and Statistics, University of Melbourne, 3010, Melbourne, Australia.
E-mail:
[email protected]
2 School of Mathematical Sciences, University of Adelaide, 5005, Adelaide, Australia.
E-mail:
[email protected] Received: 3 January 2006 / Accepted: 26 July 2006 Published online: 13 December 2006 – © Springer-Verlag 2006
Abstract: The moduli spaces of hyperbolic monopoles are naturally fibred by the monopole mass, and this leads to a nontrivial mass dependence of the holomorphic data (spectral curves, rational maps, holomorphic spheres) associated to hyperbolic multimonopoles. In this paper, we obtain an explicit description of this dependence for general hyperbolic monopoles of magnetic charge two. In addition, we show how to compute the monopole mass of higher charge spectral curves with tetrahedral and octahedral symmetries. Spectral curves of euclidean monopoles are recovered from our results via an infinite-mass limit. 1. Introduction Magnetic monopoles are paradigmatic examples of topological solitons in gauged field theories. Once a metric on R3 has been fixed, a monopole is defined as a pair (d A , ) satisfying the Bogomol’ny˘ı equation B A = ∗d A
(1)
modulo gauge equivalence. Here, B A is the curvature of the SU(2)-connection d A on End E, where E → R3 is the trivial vector bundle associated to the defining representation, and (the Higgs field) is a section of End E with constant and nonzero norm at infinity, lim ||(x)|| = m > 0.
|x|→∞
In the radial compactification of R3 , the boundary condition (2) provides a map 2 2 | S∞ 2 : S → S
(2)
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from the 2-sphere at infinity to the 2-sphere of radius m centred at the origin of su(2) ∼ = R3 , whose degree k is interpreted as the magnetic charge of the monopole. The Bogomol’ny˘ı equation (1) turns out to be quite tractable when the metric has constant curvature, for then twistor methods can be used to characterise the solutions in terms of objects from complex geometry (holomorphic bundles, spectral curves, rational maps). There is an extensive literature focused on the case where the metric is euclidean. Then the space of all monopoles with a given topology carries a natural L 2 metric which is hyperkähler, and its geodesic flow can be interpreted physically as slow motion in the Yang–Mills–Higgs model in R1,3 [22, 5, 23]. The algebraic topology of these moduli spaces captures the spectrum of bound states at the quantum level, and it has been shown to be consistent with certain S-duality conjectures in quantum field theory [28, 27]. Monopoles in hyperbolic space, where the metric in polar coordinates is taken to be r 2 2 2 ds 2 = dr 2 + R 4 sinh2 dθ , + sin θ dφ R2 were first studied by Atiyah [3], and they were soon perceived as being quite distinct from euclidean monopoles. They have a remarkably rich structure, which somewhat degenerates when the euclidean limit R → ∞ is taken. One interesting fact about hyperbolic monopoles is that they are completely determined by the value of the connection at infinity [26, 24], which reduces to a U(1)-connection on a 2-sphere; this can be regarded as a classical manifestation of the AdS/CFT correspondence [21]. The connection at infinity also plays a rôle in a characterisation of hyperbolic monopoles through holomorphic spheres in projective spaces [24], which is not available in the euclidean case. A fundamental feature of hyperbolic monopoles that makes them different from their euclidean counterparts is that they come with different values of the mass m ∈ ]0, ∞[ defined by Eq. (2). This parameter is related to the radius of curvature R of hyperbolic space: a rescaling of the radius by R → λR maps monopoles of mass m to monopoles of mass m/λ. (In the euclidean case, the radius of curvature is infinite and this means that euclidean monopoles of different masses can be identified.) Fixing the radius of hyperbolic space, the mass is a physical parameter and naturally appears as one of the moduli for the solutions of the Bogomol’ny˘ı equations. The objects obtained in the limit of mass zero (sometimes called “nullarons”1 ) are somewhat special and simpler to study; their twistor data have been pointed out to be directly related to complex curves arising in a construction of solutions to the Yang–Baxter equation for the chiral Potts model [6, 4]. By the same scaling argument, hyperbolic monopoles with infinite mass can be identified with euclidean monopoles [18]. One of the most basic questions one can ask about the moduli space of hyperbolic monopoles with a given topology is how this space is foliated by the mass. The answer to this question is trivial for monopoles of magnetic charge one. For charge two, a partial answer has been given in reference [24], where a distribution tangent to the leaves of this foliation was constructed. In this paper, we will give a much better characterisation of the mass of a two-monopole. We focus on the description of hyperbolic monopoles in terms of spectral curves, and parametrise spectral curves of 2-monopoles explicitly in terms of the mass parameter. In this way, we obtain a complete characterisation of all the moduli of two-monopoles in hyperbolic space. Our methods will apply to calculate the monopole mass in a continuous family of spectral curves of any charge, but are most effective in situations analogous to the charge two case, where we can take advantage of the fact that each spectral curve is elliptic. In higher charge, one can try to obtain spectral 1 This term is due to Michael Murray.
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curves that are Galois covers of an elliptic curve by imposing platonic symmetries on the monopole fields, although such symmetric curves will not always correspond automatically to smooth solutions of the Bogomol’ny˘ı equation (1). We shall undertake a systematic study of this class of curves, and use their symmetry to reduce the computation of the mass dependence, once again, to a calculation on elliptic curves. Thus we shall provide the first explicit examples of spectral curves of hyperbolic monopoles of arbitrary mass m > 0 and charges two, three and four beyond the (rather degenerate) axially symmetric case treated in reference [24]. We also investigate some limiting cases of our constructions; in particular, we will show how one can study the infinite-mass limit to recover spectral curves of euclidean monopoles that have been featured in the existing literature [17, 14, 16]. 2. Spectral Curves of Hyperbolic Monopoles The setup for twistor theory of hyperbolic space H 3 is the correspondence μ
H3
C
ν Z
(3)
that we now explain. We consider the model of H 3 as the upper half-plane x3 > 0 in R3 , which we parametrise using cartesian coordinates (x1 , x2 , x3 ). It is useful to com2 at infinity) obtained as pactify H 3 by adding a boundary ∂ H 3 ∼ = P1 (the 2-sphere S∞ a one-point compactification of the 2-plane of equation x3 = 0. We use z = x1 + i x2 as a stereographic coordinate on this Riemann sphere (z = zz01 in terms of homogeneous coordinates for this P1 , and z = ∞ denotes the point at infinity). The geodesics on H 3 are the half-circles in R3 lying on planes perpendicular to and centred at points of x3 = 0, together with the half-lines perpendicular to this plane, and are uniquely determined by a pair of points of intersection with ∂ H 3 . So the space of all oriented geodesics (the twistor space of H 3 ) is the complex surface Z = P1 × P1 − P1¯ , where P1¯ := {(w, z) ∈ P1 × P1 : wˆ = z} is the antidiagonal; (w, z) denotes the oriented geodesic starting at the antipodal point wˆ := −1/w¯ of w and ending at z. The correspondence space C in (3) is the subset of H 3 × Z defined by the incidence relation (x1 , x2 , x3 ) lies on (w, z),
(4)
while the maps μ and ν are the natural projections. Thus μ ◦ ν −1 (w, z) is the geodesic of H 3 corresponding to the oriented geodesic (w, z), while ν ◦ μ−1 (x1 , x2 , x3 ) is called the star at (x1 , x2 , x3 ) and is the set of all oriented geodesics through this point of H 3 . There are two natural maps ε± : Z → P1 , ε− (w, z) = w, ˆ giving the endpoints of an oriented geodesic.
ε+ (w, z) = z
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Lemma 2.1. The star at (x1 , x2 , x3 ) is the projective line in Z given by the equation (x1 − i x2 )wz − (x12 + x22 + x32 )w + z − (x1 + i x2 ) = 0.
(5)
Proof. We observe that if (w, z) is in the star at (0, 0, x3 ), then z and w must have the same argument, and Pythagoras’ theorem gives 2 wˆ + z 2 = x 2 + wˆ − z ⇔ z = x32 w, (6) 3 2 2 which is the equation for the star at (0, 0, x3 ). This star is related to the star at (x1 , x2 , x3 ) by a translation which maps an oriented geodesic (w , z ) satisfying (6) to an oriented geodesic (w, z) through (x1 , x2 , x3 ), where z = z − (x1 + i x2 ) and w . w = 1 + (x1 − i x2 )w Substituting this in Eq. (6), we obtain (5). It is clear that (6) defines a projective line, and so the same must be true for the general star (5). Note that the statements (4) and (5) are equivalent. A star (5) is invariant under the map σ : (w, z) → (ˆz , w) ˆ
(7)
reversing the orientation of oriented geodesics. This map is antiholomorphic and squares to the identity, so it is a reality structure for the complex surface Z ; it has no fixed points. A set invariant under σ is said to be real. Twistor theory consists of interpreting analytic objects in physical space H 3 in terms of algebraic objects in twistor space Z , and vice-versa. In our context, the correspondence will relate hyperbolic monopoles (d A , ) on H 3 to their spectral curves, which are complex curves ⊂ Z satisfying certain conditions. We now review briefly how they arise and refer the reader to references [25] and [24] for further details. The twistor correspondence for the Bogomol’ny˘ı equation (1) makes use of the family of operators H : Z −→ End H 0 (μ ◦ ν −1 (w, z), E) (w,z)∈Z
introduced by Hitchin (for euclidean monopoles) in [12], and defined by H(w, z) := (d A − i) |μ◦ν −1 (w,z) . The Bogomol’ny˘ı equation (1) turns out to be the integrability condition for the holomorphic structure (μ ◦ ν −1 )∗ ∂¯ A on the bundle ker H → Z induced by the connection d A . Thus, for a solution (d A , ) of (1), ker H is a holomorphic complex bundle of rank two, and it has two distinguished line subbundles L ± with fibres L± = s ∈ ker H(w, z) : lim s(x) = 0 . (w,z) x→ε± (w,z)
The spectral curve ⊂ Z of the monopole (d A , ) is defined as the support of the cokernel of a morphism of coherent sheaves on Z given by the composition O(L − ) −→ O(ker H) −→ O(ker H/L + ),
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which is the same as saying
:= (w, z) ∈ Z : L +(w,z) = L − (w,z) .
Holomorphic line bundles on Z can be constructed by purely algebraic means. They are obtained as tensor products of bundles pulled back from each factor of P1 × P1 , ∗ OP1 (k1 ) ⊗ ε+∗ OP1 (k2 ), O Z (k1 , k2 ) := εˆ −
k1 , k2 ∈ Z,
and complex powers L λ (λ ∈ C) of the topologically trivial line bundle L := O Z (1, −1). The bundles
Lλ
do not extend to P1 × P1 ⊃ Z unless λ ∈ Z. We shall write L λ (k1 , k2 ) := L λ ⊗ O Z (k1 , k2 ).
(8)
These line bundles can be constructed as follows. Introduce U1 := {(w, z) ∈ Z : w = ∞, z = ∞}, U2 := {(w, z) ∈ Z : w = 0, z = 0}.;
(9)
since (0, ∞), (∞, 0) ∈ P1¯ , these contractible sets provide an open cover of the twistor space Z . It can be taken as a trivialising cover for the line bundle (8), with transition function g12 : U1 ∩ U2 → C∗ given by g12 (w, z) = w k1 +λ z k2 −λ .
(10)
The connection between the holomorphic theory of the Bogomol’ny˘ı equation above and algebraic geometry is provided by the identifications L + = L m (0, −k) and L − = L −m (−k, 0) for a hyperbolic monopole of charge k and mass m. This result is obtained from an analysis of the asymptotics of the fields (d A , ) near the boundary ∂ H 3 [25]. Spectral curves of hyperbolic monopoles satisfy the following three conditions: 1. ⊂ Z is a real curve in the linear system |O Z (k, k)|; 2. L 2m+k ∼ = O ; 3. H 0 ( , L s (k − 2, 0)) = 0 for all 0 < s < 2m + 2. In the broader context of integrable systems, these conditions lead to an ODE of Lax type. It is believed that they are also sufficient to guarantee the existence of a hyperbolic monopole to which is associated. Although sufficiency has not yet been proven, for the purposes of this paper we will use the terms hyperbolic monopole and spectral curve to refer to complex curves in Z satisfying the conditions above. Using the coordinates w, z for Z , one can write a polynomial equation for as ψ(w, z) = (−w)k q(w), ˆ q(z), where ·, · is the standard inner product in Ck+1 and q : C → Ck+1 is defined in terms of k + 1 vectors v j ∈ Ck+1 by k
k j q(z) = z vj; j j=0
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in fact, q induces a holomorphic map P1 → Pk that also characterises the monopole up to the action of PU(k + 1) on the target [24]. The Conditions 1–3 above yield a model for the moduli space of hyperbolic k-monopoles Mk in terms of spectral curves. A version of this was used in reference [24] to give a description of M2 . The group of (direct) isometries of H 3 can be identified with PSL2 C, and it acts on Mk . There is a moment map associated to this action, which can be used to define a centre of a monopole: spectral curves in the zero-set of the moment map correspond to centred monopoles and satisfy [24] k
(2 j − k)||v j || = 0,
j=0
2
k−1
( j + 1)(k − j)v j , v j+1 = 0. j=0
3. The Mass of Hyperbolic 2-Monopoles Our aim in this section is to obtain an explicit equation for the spectral curve ⊂ Z of a hyperbolic 2-monopole in terms of its mass. We shall first deal with the case where is smooth, and then show that the result can be extended by continuity to arbitrary spectral curves.
3.1. A model for the spectral curve. We begin with a lemma giving a standard form for smooth spectral curves of hyperbolic 2-monopoles. Lemma 3.1. If the spectral curve of a 2-monopole is smooth, its PSL2 C-orbit contains a curve ⊂ Z of the form ψ(w, z) = w2 z 2 +
u2 − 4 u 2 − 2uv + 4 wz − (w 2 + z 2 ) + 1 = 0. 2(u − v) 4(u − v)
(11)
Here, u, v ∈ ]2, ∞[ depend only on the monopole mass and on an internal modulus, and they satisfy u 2 − 2uv + 4 > 0.
(12)
Proof. The spectral curve of a 2-monopole is the vanishing locus of a polynomial of the form √ ψ(w, z) = v0 , v2 w 2 z 2 − 2v1 , v2 (w 2 z + wz 2 ) + ||v0 ||2 (w 2 + z 2 ) √ −2||v1 ||2 wz + 2v2 , v1 (w + z) + v2 , v0 , (13) where v j ∈ C3 . The intersection of with the diagonal P1 ⊂ P1 × P1 , which we parametrise using the coordinate z, is then given by the zeroes of the quartic polynomial ψ(z, z). From (13) we find ψ(ˆz , zˆ ) = z¯ −4 ψ(z, z), which implies that these zeroes occur in antipodal pairs. Moreover, these points must be distinct for to be smooth. So an element of Stab(0,0,1) PSL2 C ∼ = SO(3) to √ we can use 1 rotate them to the positions ± λ and ± √ on the real axis of P1 . The value of λ ∈ ]0, 1[ λ
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is uniquely determined by this process, once the mass of the monopole has been fixed. Thus we have shown that a rotation can be applied to have
1 2 ψ(z, z) = z 4 − λ + z + 1; (14) λ in this step, we have used the freedom of multiplying ψ by an overall factor in C∗ . The most general form for ψ(w, z) consistent with (14) is 1 1 ψ(w, z) = w 2 z 2 + 2 wz − (λ + + 2 )(w 2 + z 2 ) + 1 = 0. 2 λ Invariance under the reality map only constrains 2 to be a real number. However, given this, the positivity condition on spectral curves can easily be seen to be equivalent to 2 > 0. To complete the proof, we need only to observe that (by an easy continuity argument) the map (u, v) → (λ + λ1 , 2 ) defined by ∞[ (u, v) ∈ ]2, ∞[2 : u 2 − 2uv + 4 > 0 −→ ]2, ∞[×]0, (15) uv−4 u 2 −2uv+4 (u, v) −→ u−v , 2(u−v) is bijective, and that u>v
(16)
is always satisfied in the domain of (15).
Our task is to calculate the dependence of u and v on the mass m and an extra real parameter. As a preliminary step, we need to obtain a good description of the family of elliptic curves ⊂ P1 × P1 given by (11). We introduce the map π : P1 × P1 −→ P2 , ([w0 : w1 ] , [z 0 : z 1 ]) −→ [w0 z 0 : w0 z 1 + w1 z 0 : w1 z 1 ] ,
(17)
which can be regarded as a projection onto the space of orbits of the automorphism σ+ : (w, z) → (z, w).
(18)
This map has order two, commutes with the reality structure σ defined in (7), and can be written as σ+ = σ ◦ σ− = σ− ◦ σ, where σ− : (w, z) → (w, ˆ zˆ ); σ− is induced on P1 × P1 by the parity transformation on hyperbolic space x1 x2 1 , , P(0,0,1) : (x1 , x2 , x3 ) → − 2 , x1 + x22 x12 + x22 x3
(19)
(20)
which can be described as reflection across (0, 0, 1). Since P(0,0,1) reverses the orientation of H 3 , σ− is antiholomorphic, and it is a reality structure on P1 × P1 or Z alternative
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to (7). Thus there is a Vierergruppe of diffeomorphisms of twistor space associated with (0, 0, 1) (and likewise with any other point of H 3 ), V(0,0,1) := {id, σ, σ− , σ+ } ∼ = Z2 ⊕ Z2 , which is naturally Z2 -graded by (anti)holomorphicity. That σ+ should arise as a symmetry of spectral curves (13) of centred 2-monopoles is unsurprising, as we expect their fields to be symmetric under the parity transformation (20). The fixed points of σ+ form the diagonal P1 of P1 × P1 (i.e. the star at (0, 0, 1)), and 2 σ+ = id. Thus π is two-to-one on P1 × P1 − P1 and one-to-one on P1 . In homogeneous coordinates Z 0 , Z 1 and Z 2 for P2 , π(P1 ) is the conic given by C(Z 0 , Z 1 , Z 2 ) := Z 12 − 4Z 0 Z 2 = 0.
(21)
The image of under π is given by the equation ˜ 1 , ζ2 ) := ζ22 − ψ(ζ
u2 − 4 2 ζ + uζ2 + 1 = 0 4(u − v) 1
in affine coordinates ζ1 = ZZ 01 = w + z and ζ2 = ZZ 02 = wz. This is also a plane conic, so we can find a rational parametrisation f : P1 → π( ) for it. In fact, if we write (22) f (t) = 1 − αt 2 : 2αβt : t 2 − α , we will have f (P1 ) = π( ) if and only if we choose α and β such that the conditions u=α+
1 α
(23)
and u − v = αβ 2
(24)
are satisfied; α and β are then real. Notice that Eq. (23) determining α indeed gives two real solutions of the form α, α1 since u > 2, and because of the invariance under α → α1 we are free to assume that α > 1, which we shall do from now on; that β is real then follows from (16). We now realise as a double cover of the P1 where f is defined, branching on the set of four points f −1 (π( ) ∩ π(P1 )). One way to do this is to regard as the Riemann surface of the global function of t extending the germ t → F(t) (25) at t = 0, where F(t) := C ◦ f (t) = 4α(t 4 − vt 2 + 1).
(26)
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√ (In (25) and henceforth, we use · to denote the principal branch of the square root.) √ The branch points of → P1 are the zeroes of F, and they are of the form ± κ, ± √1κ with κ+
1 = v. κ
(27)
Again, v > 2 ensures that κ is real and positive, and by invariance of (27) under κ → we may assume from now on that 0 < κ < 1.
1 κ
(28)
The elliptic curve given by (11) can now be realised as a two-sheeted cover of the projective line parametrised by t as follows (cf. Fig. 1). Introduce branch cuts on the √ segments [− √1κ , − κ] and [κ, √1κ ], and label the two sheets as (i) and (ii), where sheet (i) is the one on which the germ (25) (which takes a positive value at t = 0) is defined. Clearly, on sheet (i)/(ii), the analytic continuation of (25) then takes negative/positive √ √ values on ] − ∞, − √1κ [ and ] √1κ , +∞[, and positive/negative values on ] − κ, κ[, respectively. To complete the correspondence between our descriptions of as the Riemann surface of the germ (25) and as a hypersurface in P1 × P1 , we just need to make the identification of the two descriptions at a single point. We choose this point p1 to be √ an element of the fibre of → P1 over t = − α. In the description of by analytic continuation, we specify p1 by saying that it lies on sheet (i). In the description ⊂ P1 × P1 , we will need to√specify the coordinates (w, z) of p1 . Notice that from (22) we have that when t = ∓ α wz =
√ (∓ α)2 − α = 0, √ 1 − α(∓ α)2
so either w = 0 orz = 0; we specify p1 by saying that it has z = 0. Then we find from (11) that w = ±2 uu−v 2 −4 ; we must set u−v ,0 , p1 := 2 u2 − 4
Fig. 1. Hyperbolic 2-monopole: contours of integration in a realisation of
(29)
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since (17) implies that the point
u−v p2 := −2 ,0 u2 − 4
(30)
√ projects to t = + α under (17). For further reference, we note that the other points on √ the fibres over t = ∓ α have w = 0 and z = ±2 uu−v 2 −4 , and we label them as
u−v u−v q1 := 0, 2 , q2 := 0, −2 . u2 − 4 u2 − 4
(31)
By construction, p1 is on sheet (i), so q1 must be on sheet (ii) as it belongs to the same fibre. Moreover, one can write √ Z 1 (t) ± F(t) , w(t) = 2Z 0 (t) where the signs ± distinguish √ between the two sheets; this √ equation implies that, on a given sheet, if w = 0 at t = ∓ α, then w = 0 at t = ± α, and the same is true for z. So we conclude that p2 is on sheet (ii), while q2 is on sheet (i). We want to discuss the conditions characterising a spectral curve from the point of view of the function theory on , and for that we shall need to fix generators for the spaces of 1-forms and 1-cycles on the elliptic curve. A global holomorphic 1-form on is given by the standard formula ω=
dt √ ± F(t)
(32)
which follows from taking Poincaré residues of (11) [11]; again, the signs label sheets, and we make the convention of taking the top sign on sheet (i) near t = 0. To describe a standard basis (a, b) for H1 ( ; Z) ∼ = Z2 we draw representatives for a and b in Fig. 1. The convention is that paths on sheet (i) are drawn with continuous lines, whereas paths on sheet (ii) are drawn with dotted lines. 3.2. Reciprocity on . To make the connection between the algebraic-geometric Condition 2 and complex analysis on , we shall use the reciprocity law for differentials of first and third kinds on a compact Riemann surface (cf. [11], pp. 229–230). This technique is in the same spirit of previous work on euclidean monopoles, where a different reciprocity law has been applied to deduce constraints on the coefficients of polynomials defining spectral curves [17, 8, 15]. We start by rephrasing L 2m+k ∼ = O in terms of the existence of a global holomorphic trivialisation for the line bundle L 2(m+1) on , or equivalently a nowhere-vanishing holomorphic section ξ ∈ H 0 ( , L 2(m+1) ). Such a section is locally represented by nowhere-vanishing holomorphic functions ξ j defined on the basic open sets U j ∩ ( j = 1, 2) (cf. (9)), and which are patched together using the appropriate transition function (10): 2(m+1) w ξ1 (w, z) = ξ2 (w −1 , z −1 ). (33) z
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To begin with, this patching condition should hold on U1 ∩ U2 ∩ , which consists of with eight points deleted, namely the points of the form (w, 0), (w, ∞), (0, z) and (∞, z). There are two of each, since the polynomial ψ(w, z) defining has bidegree (2, 2); for example, the points of the form (0, z) are p1 and p2 given by Eqs. (29) and (30). Taking logarithmic differentials in both sides of (33), we obtain the relation
dw dz ω1 = 2(m + 1) − + ω2 , (34) w z where ω j := d log ξ j =
dξ j ξj
j = 1, 2.
(35)
The definition (35) implies the following integrality property on periods: ω j , ω j ∈ 2πi Z. a
We set 1 1 := 2πi
(36)
b
ω1 , b
1 2 := − 2πi
ω1 .
(37)
a
Notice that we can replace 2(m + 1) by s and go through the same argument to conclude that Condition 3 can be rephrased as: (1 , 2 ) must be primitive in Z2 .
(38)
Each ω j is a holomorphic 1-form on U j ∩ (since ξ j are holomorphic and never vanish there), but they extend to global meromorphic 1-forms on as a consequence of (34). To see this, notice that ω2 is holomorphic on a small neighbourhood of − U 1 = { p1 , p2 , q 1 , q 2 }
dz in , and then (34) implies that ω1 must have the same principal part as 2(m+1) dw − w z on this set. Since w is a good coordinate on around p1 and p2 , while z is a good coordinate around q1 and q2 , we can conclude that ω1 has simple poles at each of these points, and we find Res p j (ω1 ) = −2(m + 1)
and
Resq j (ω1 ) = 2(m + 1),
j = 1, 2.
(39)
A similar analysis can be done for ω2 . In classical terminology, a holomorphic 1-form (such as ω) is called a differential of the first kind, while a meromorphic 1-form with only single poles (such as ω1 ) is called a differential of the third kind. A standard result of complex analysis on compact Riemann surfaces is the reciprocity law [11] q
ω a a ω1 = 2πi Res (ω ) ω. (40) q 1 ω b b ω1 p0 q
Here, p0 ∈ is any basepoint, the integration paths on the right-hand side can be deformed to avoid the paths representing the homology basis, and the sum is over the set
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of poles of ω1 . Equation (40) relates two elements of the covering space H 1 ( , O ) ∼ = TO Jac( ) to the jacobian variety of . On a complex curve of higher genus g, the left-hand-side of the reciprocity relation would have a sum over conjugate pairs of a standard (symplectic) basis of H1 ( ; Z), and there would be g equations. Note that by Liouville’s theorem
Resq (ω1 ) = 0, q
which explains why the right-hand side of (40) is in fact independent of p0 . In our context, using (37) and (39), Eq. (40) can be written as q1 q2 1 ω + 2 ω = 2(m + 1) ω+ ω . a
b
p1
(41)
p2
∼ H 1 ( , O ) are realised in differWe note in passing that elements of H 0 ( , K )∗ = ent ways in the two sides of this equation, namely as a pairing (via ) with a 1-cycle 1 a + 2 b ∈ H1 ( ; Z) in the left-hand side, and as a multiple of an Abel sum associated to the divisor q1 + q2 − p1 − p2 ∈ Div0 ( ) representing the class O (1, −1) in the right-hand side; in fact, if we interpret the integrals in terms of the Abel–Jacobi map, and use congruence on the period lattice of , we can read (41) as a rather transparent statement of L 2(m+1) | being trivial. Of course, there is a freedom of changing the subscripts of the p j and q j in this expression, but we find it convenient to choose paths of integration as in (41). More precisely, we choose paths γ j ( j = 1, 2) on starting at p j and ending at q j as illustrated in Fig. 1, but defined only as a class in the √ relative homology of ( , { p j , q j }). Notice that the locations of the projections t = ± α of the poles of ω1 with respect to the branch points are as shown, since u>v⇔α+
1 1 1 >κ+ ⇒α> . α κ κ
To understand what constraint (41) imposes on the coefficients of , we have to compute the integrals in this equation. Under our conventions, it is clear that all of them are real except for the period a ω, which is pure imaginary. Therefore 1 = 0, and (38) imposes 2 = ±1. So we only need to evaluate ω = −2I1 , b
γ1
and
ω = 2(I1 − I2 ), γ2
ω = −2I2 ,
where I1 :=
√
κ
√ − κ
dt , √ F(t)
I2 :=
√ √1 κ
α
√
dt . F(t)
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In terms of standard notation for elliptic integrals of the first kind ϕ dθ F(ϕ, κ) := , 0 1 − κ 2 sin2 θ π ,κ , K (κ) := F 2 we calculate
I1 =
and 1 I2 = 2
κ α
κ K (κ) α
1 K (κ) − F arcsin √ , κ ακ
.
Substituting in Eq. (41), we find that 1 = −1 and
1 K (κ) . F arcsin √ , κ = 2(m + 1) ακ
(42)
3.3. Mass parametrisation. Equation (42) can be solved for α to give α=
κ sn2
1 K (κ) 2(m+1) , κ
(43)
in terms of Jacobi’s sine amplitude function. (In the following, we will often drop the modulus argument κ in jacobian elliptic functions.) Using (27), (43) and standard algebra of jacobian functions [7], we find
u 2 − 2uv + 4 2 K (κ) K (κ) = cs ds (44) 2(u − v) κ m+1 m+1 and u2 − 4 1 = ns2 4(u − v) κ
K (κ) . m+1
(45)
So we have shown that Eq. (11) defines a 2-monopole of mass m if and only if (44) and (45) hold. Moreover, we shall see in Sect. 4.1 below that we can extend our results by continuity to the κ → 0 limit, where the spectral curve becomes reducible and singular. In other words, we have the following result. Proposition 3.2. A spectral curve of a 2-monopole of mass m centred at (0, 0, 1) ∈ H 3 can be rotated to the form κ sn2 ρ (w 2 z 2 + 1) + 2 cn ρ dn ρ wz − (w 2 + z 2 ) = 0,
(46)
where 0 ≤ κ < 1 and ρ=
K (κ) . m+1
(47)
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When m is a rational number, the coefficients of (46) turn out to be algebraic over Q(κ). For example, if m = 1 we can use the bisection formulae for jacobian elliptic functions [7] to write the spectral curve in the form √ κ(w 2 z 2 + 1) + 2κ 1 + κ wz − (1 + κ )(w 2 + z 2 ) = 0, √ where κ := 1 − κ 2 . We shall find later that a similar phenomenon occurs in examples of spectral curves of higher charge. 4. Limiting Cases of 2-Monopoles In this section, we examine and interpret the limits of our result in Proposition 3.2 as κ → 0, κ → 1, m → 0 and m → ∞. 4.1. Axial symmetry. When we let κ → 0, sn ρ → sin ρ, cn ρ → cos ρ, dn ρ → 1, whereas (47) implies ρ→
π . 2(m + 1)
Thus (46) becomes
π w − 2 cos wz + z 2 = 0. 2(m + 1) 2
(48)
This is the spectral curve of an axially symmetric 2-monopole at (0, 0, 1), as computed in reference [24]. This curve is reducible, since the polynomial in (48) factorises as iπ iπ w − e 2(m+1) z w − e− 2(m+1) z . The reduced components are two projective lines in P1 × P1 , intersecting at the points (0, 0) and (∞, ∞) in the diagonal, which are related by the real structure σ . Notice that the lines iπ
w = e± 2(m+1) z iπ
are real and can be thought of as stars at the complex conjugate points (0, 0, e± 4(m+1) ) (cf. Eq. (6) for a star at (0, 0, x3 ) ∈ H 3 ). This situation is analogous to axially symmetric monopoles in euclidean space [14]. We conclude that we can extend the range of κ from 0 < κ < 1 to 0 ≤ κ < 1, as in Proposition 3.2.
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4.2. Large separation. To study the limit κ → 1, we use that [7] sn ρ → tanh ρ, cn ρ → sech ρ, dn ρ → sech ρ, together with ρ → +∞ from Eq. (47). We find that the spectral curve (46) then degenerates to (w 2 − 1)(z 2 − 1) = 0.
(49)
This reduces to four real lines in P1 × P1 . The two lines w = ∓1 and z = ±1 together can be regarded as a limiting star at the point (±1, 0, 0) ∈ ∂ H 3 , for either choice of signs; these two lines intersect at the point (∓1, ±1) ∈ P1¯ , respectively. We interpret (49) as the spectral curve of two 1-monopoles that are infinitely separated at each of the ends of the geodesic μ ◦ ν −1 (1, 1) = μ ◦ ν −1 (−1, −1) of H 3 . The analysis of the limiting examples κ → 0 and κ → 1 suggests that κ can roughly be thought of as a parameter of the separation of two single monopole cores in configurations of 2-monopoles centred at (0, 0, 1). As κ varies from 1 to 0, the cores approach symmetrically along the geodesic given by x2 = 0, x12 + x32 = 1 from infinite distance to coincidence at (0, 0, 1) (the axially symmetric configuration). Strictly speaking, the description of a 2-monopole configuration in terms of two superposed 1-monopoles is only appropriate in the asymptotic limit of large separation, and becomes worse and worse as κ decreases. This is why the 1-monopole cores in the axially symmetric configuration are found to be located at points of the complexification of H 3 rather than at the centre (0, 0, 1) itself. 4.3. 2-Nullarons. If we let m → 0, then sn ρ → sn K (κ) = 1, cn ρ → cn K (κ) = 0, √ dn ρ → dn K (κ) = 1 − κ 2 , and the spectral curve (46) becomes κ(w 2 z 2 + 1) − (w 2 + z 2 ) = 0.
(50)
This result can be obtained by more direct means. In fact, (50) is the standard Z2 -symmetric curve encoding a solution of the Potts model [4], and can be computed as = R(ˆz ) R(w) with
(51)
√ R(z) =
1 − κ2 . z2 − κ
More generally, Eq. (51) produces all the spectral curves of massless monopoles (nullarons) of charge k from rational maps R : P1 → P1 of degree k.
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4.4. Euclidean 2-monopoles. In the limit m → ∞, which is equivalent to ρ → 0, the curve (46) tends to two copies of the diagonal P1 ⊂ P1 ×P1 , with equation (w−z)2 = 0. However, at the next asymptotic order, we can recover the spectral curve of a euclidean monopole embedded in T P1 . To explain this, we must first make more precise what is meant by the euclidean limit of a hyperbolic monopole. As discussed in the introduction, rescaling the hyperbolic metric to have larger and larger curvature radius R is equivalent to rescaling the fields (equivalently, the spectral curves), and in particular their mass, while keeping the metric constant; the infinite radius limit R → ∞ can then be thought of as an infinite mass limit m → ∞, to be interpreted as a euclidean monopole. Implicit in this rescaling is the choice of a point in H 3 . We choose to rescale around the centre (0, 0, 1) ∈ H 3 , and this forces us to consider only centred hyperbolic monopoles, hence with limit centred euclidean monopoles. More generally, rescaling around a different point in H 3 would lead to limit spectral curves concentrated around the corresponding star in P1 × P1 . Recall that a point (w, z) ∈ P1 × P1 is interpreted as a geodesic in H 3 running from wˆ to z, both points on the sphere at infinity. From the perspective of the centre (0, 0, 1), the geodesic is viewed as the vector pointing to its closest point from (0, 0, 1) and the (orthogonal) tangent direction at that point. Now take a sequence of geodesics in H 3 in the spectral curves converging to a geodesic through (0, 0, 1), or equivalently a sequence of points (wm , z m ) ∈ P1 × P1 for m → ∞ with limit (w∞ , z ∞ ) = (z ∞ , z ∞ ) on the diagonal. For each value of m, rescale the radius of curvature of the metric on H 3 by m. Then a simple geometric argument (involving similarity of infinitesimal triangles on the 2-plane containing the sequence of geodesics) shows that, in the limit, one obtains the euclidean geodesic (52) (wm , z m ) → (η, ζ ) = lim m(z m − wm ), z ∞ , m→∞
where we are using the standard coordinates on T P1 introduced by Hitchin [12]. Given the ρ → 0 asymptotics sn2 (ρ, κ) = ρ 2 + O(ρ 3 ) and cn(ρ, κ) dn(ρ, κ) = 1 −
1 + κ2 2
ρ 2 + O(ρ 3 ),
we can write η2 = lim m 2 (wm − z m )2 m→∞
(wm − z m )2 ρ→0 ρ2 2 z 2 + 1) (2 cn ρ dn ρ − 2)wm z m + κ sn2 ρ (wm m = K (κ)2 lim ρ→0 ρ2 2 2 = K (κ)2 lim −(1 + κ 2 )wm z m + κ(wm z m + 1) = K (κ)2 lim
m→∞
= K (κ) (−(1 − κ 2 )ζ 2 + κ(ζ 4 − 1)), 2
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where we have used (52) in the first and last steps. Hence we obtain the curve in |OT P1 (4)| η2 − K (κ)2 (ζ 2 − κ)(κζ 2 − 1) = 0
(53)
as the euclidean limit of (46). To compare this with the spectral curve of a generic euclidean 2-monopole [17], as given in reference [23] (say), η2 −
K (k)2 2 4 k (ζ + 1) − 2(2 − k 2 )ζ 2 = 0, 4
(54)
where 0 < k < 0, one can start by relating κ and k by computing the coordinate ζ at 1 the four intersection points of (53) and (54) with the zero √ section η = 0 of T P (i.e. the branch points of → P1 ). In (53) we find ζ = ± κ, ± √1κ , whereas for (54) the solutions have the same form but are parametrised differently, namely √
2 2 2 2 κ . κ = 2 −1− −1 −1 ⇒ k = k k2 1+κ Using a descending Landen’s transformation [9], we can now write √ 2 κ = (1 + κ)K (κ) K (k) = K 1+κ and conclude that (53) and (54) simply give two different parametrisations of the same curve. An alternative way of checking that our limit curve is correct is to start from our parametrisation of the branch points and use the characterisation of spectral curves in Sect. 3 of [15] to deduce (53). 5. Platonic Monopoles in Hyperbolic Space For charge k > 2, the spectral curves are of higher genus (k − 1)2 > 1 and depend on 4k − 3 internal moduli, so it becomes a difficult task to compute any one of them, let alone obtain a complete picture of the whole moduli space. A technique that has been fruitful is to impose invariance of the monopoles under certain isometries [14]; in some cases, this cuts the number of moduli down to a manageable number, while important features of the moduli space are preserved. For instance, such symmetry constraints define totally geodesic subsets of the moduli space, whose geodesic flow is simply a restriction of the ambient geodesic flow; moreover, one expects these subsets to carry significant information about the topology of the whole moduli space. In the euclidean case, beautiful examples of scattering of symmetric multi-monopoles have been found in this way [14, 23], as well as new insight into the structure of the monopole fields themselves [29]. The construction of spectral curves of symmetric monopoles on euclidean space has largely relied on the study of Nahm’s equations. For hyperbolic monopoles of general mass this route cannot be taken, as we do not yet have enough understanding of the generalisation of Nahm’s equations appropriate for the problem. In the following sections, we shall obtain results about symmetric hyperbolic monopoles by attacking the problem directly via the geometry of spectral curves. Although our methods apply more generally, we will restrict ourselves to monopoles with the rotational symmetry of a platonic solid, and would like to construct associated spectral curves with arbitrary mass.
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The platonic solids we consider are the tetrahedron, the octahedron and the icosahedron, with (special) symmetry groups isomorphic to A4 , S4 and A5 , respectively. Instead of the octahedron and the icosahedron, we could have taken the dual solids (cube and dodecahedron), which have the same group of symmetries. We start by writing down Ansätze for real (k, k) curves in Z with the symmetry of a given platonic solid. Again, we may restrict ourselves to centred curves without losing generality. The following elementary observation will be useful: Lemma 5.1. Let ⊂ Z be a centred (k, k) curve. If (w, z) ∈ is fixed by a rotation preserving (0, 0, 1), then w = z. Proof. The set of points of H 3 fixed by a (nontrivial) rotation in Stab(0,0,1) PSL2 C ∼ = SO(3) is a geodesic through (0, 0, 1). Lemma 2.1 then implies that the spectral lines fixed by this rotation must lie on the diagonal P1 ⊂ P1 × P1 . ∼ SO(3) has been fixed; a Suppose that a platonic group G ⊂ Stab(0,0,1) PSL2 C = pictorial way of doing this is to map the corresponding solid onto the star at (0, 0, 1) (which we can think of as a two-sphere centred at (0, 0, 1) and identify with P1 ), using central projection. In the following, we shall be interested in G-symmetric (k, k) curves for which the space of orbits /G is an elliptic curve. Note that, if is smooth, so too will be E. Proposition 5.2. Let ⊂ P1 ×P1 be a smooth (k, k) curve invariant under G = A4 , S4 or A5 symmetry with quotient /G = E an elliptic curve. Then k = 3 or 4 in the G = A4 case, and k = 4 and 6 in the S4 , respectively A5 cases. Remark. In the course of the proof we will see that, together with the diagonal P1 ⊂ P1 × P1 , the curves satisfying the conditions of the proposition are the smallest degree smooth (k, k) curves in P1 × P1 with tetrahedral, octahedral or icosahedral symmetry. Proof. The quotient /G parametrises the orbits of G on ⊂ Z . Generic orbits of G on P1 have |G| points and thus the same is true of generic orbits of G on Z . There are three exceptional orbits of G on P1 , and hence Z , given by orbits of vertices, (midpoints of) edges and (midpoints of) faces under the identification of P1 with the symmetric polyhedron. Thus generic G-orbits on a G-symmetric (k, k) curve have |G| points, and there are at most three exceptional orbits. In the tetrahedral case, there are three orbits consisting of 4, 6 and 4 points — the orbits of vertices, edges and faces — on the (k, k) curve . For the octahedral and icosahedral cases, the exceptional orbits correspond to 6, 12 and 8 vertices, edges and faces, and 12, 30 and 20 vertices, edges and faces. We say that any point in an exceptional orbit is an exceptional point. The Euler characteristic of (given by 2k(2 − k)) minus its exceptional points is divisible by |G| since it admits a free G-action: |G| | 2k(2 − k) − #{exceptional points}.
(55)
Any point of an exceptional orbit of G is fixed by some element of G and hence by Lemma 5.1 it lies in the diagonal P1 ⊂ Z . Since we assume to be smooth, it cannot contain the component P1 (except in the non-reduced case = kP1 ), so · P1 = (k, k) · (1, 1) = 2k = #{exceptional points} + |G|,
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where the right-hand side consists of the exceptional orbits and of ≥ 0 orbits of size |G|. Thus |G| | 2k(1 − k)
(56)
and we immediately deduce that k = 0 or 1 mod 3, 4 or 5 in the respective tetrahedral, octahedral and icosahedral cases. Since the quotient /G has genus one, the Euler characteristic gives 2k(2 − k) − #{exceptional points} = −#{exceptional orbits} ≥ −3. |G|
(57)
In the tetrahedral case, |G| = 12, and if k ≥ 5, then the left-hand side of (57) is less than −3 which contradicts the inequality, so for 1 < k < 5, k = 3 and 4 give solutions of (56). The exceptional orbits consist of 6 points in the k = 3 case, and 4 plus 4 points in the k = 4 case to give a solution of (57). In the octahedral case, |G| = 24, so (57) implies that k < 8. Solutions of (56) must be 0 or 1 mod 4, hence k = 4 or 5 and only 4 gives a solution of (56), with exceptional orbit of 8 points. In the icosahedral case, |G| = 60, so (57) implies that k < 11, and together with the mod 5 condition we need only check k = 5, 6 or 10. Both k = 6 and k = 10 give solutions of (57), however only k = 6 with exceptional orbit of 12 points allows a solution of (57) and the proposition is proven. It is useful to describe each of the exceptional G-orbits on P1 as the zeroes of a binary form — a homogeneous polynomial in the two homogeneous coordinates for P1 . One then obtains three forms K v , K e , K f ∈ C[ζ0 , ζ1 ]hom describing the positions of the vertices, (midpoints of) edges and (midpoints of) faces of the corresponding polyhedron. G acts on P1 and this action can be transferred to the vector spaces of binary forms of each degree. By construction, K e , K f and K v are projectively invariant under G, and for each of these forms the scalar factors under elements of G give an abelian character of the platonic group. Since G is finite, suitable products of these three forms must be strictly invariant. In his famous book [20] of 1884 on Galois theory, Felix Klein described the ring of G-invariant forms for the three platonic groups; in particular, the unique monic elements of minimal positive degree can be given in each case as follows, in an obvious orientation of the polyhedra: • For G = A4 , K e (ζ0 , ζ1 ) = ζ0 ζ1 (ζ14 − ζ04 ); • For G = S4 , K f (ζ0 , ζ1 ) = ζ18 + 14ζ04 ζ14 + ζ08 ; • For G = A5 , K v (ζ0 , ζ1 ) = ζ0 ζ1 (ζ110 + 11ζ05 ζ15 − ζ010 ); in this case, all projectively invariant forms are strictly invariant, a consequence of simplicity. Given the result of Lemma 5.1, candidates for smooth G-invariant (k, k) curves of lowest k can now be written as (w − z)k + α K˜ (w, z) = 0,
(58)
where K˜ (w, z) ∈ C[w, z] can be polarised to a G-invariant element of (C[w0 , w1 ] ⊗ C[z 0 , z 1 ])hom that projects to the G-invariant forms K e, f,v above under the dual ι∗ of the inclusion ι : P1 → P1 × P1 . The value of k corresponds to half of the degree of Klein’s invariant form on P1 (i.e., k = 3, 4 or 6), since ι∗ projects ( p, q)-forms onto
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( p + q)-forms. Thus for each G, this procedure yields the symmetric curves of lowest k in Proposition 5.2. One way to find the appropriate function K˜ on P1 × P1 from Klein’s invariant forms on P1 is to start with a polynomial Ansatz incorporating invariance under σ+ in (18) and impose G-invariance, as illustrated in the following example. A reason why we should give σ+ the chance of being an extra symmetry of K˜ is that the parity transformation P(0,0,1) in (20) leaves the zero-set of the particular Klein form associated with each platonic group invariant; notice that this is not true in general for the other two forms associated with a given platonic solid. Example 5.3. Let G = A4 . In terms of the inhomogeneous coordinate ζ := ζ1 /ζ0 , Klein’s form K v (ζ0 , ζ1 ) corresponds to the polynomial ζ (ζ 4 − 1). The projection ι∗ is described by its action on generators ι∗ (w) = ι∗ (z) = ζ , thus we write 2 4 c1 (wz)2 + c2 (wz) w+z + (1 − c1 − c2 ) w+z −1 . K˜ v (w, z) = w+z 2 2 2 In the orientation chosen, A4 is generated by the rotations
w−i z−i 1 1 , , (w, z) → , . (w, z) → (−w, −z), (w, z) → w z w+i z+i
(59)
Imposing invariance under these, one finds c1 = 1 and c2 = 0. Thus we write the Ansatz (58) as (w − z)3 + iα (w + z)((wz)2 − 1) = 0.
(60)
An easy check shows that this curve is real with respect to (7) if and only if α ∈ R. √ The argument we used in Sect. 4.3 shows that m = 0 corresponds to α = 3, given that the rational map of degree 3 with the A4 -symmetry that we are using is [23] √ 1 − i 3z 2 R(z) = √ . i 3z − z 3 Moreover, m → ∞ should correspond to α = 0 (as the limit spectral curve will be SO(3)-symmetric, thus three copies of the diagonal P1 ). Thus in fact we can take √ 0<α< 3 and it is easy to show that (60) is smooth and irreducible for all these values of α. Using the same technique, we can find the corresponding polynomial K˜ f for G = S4 and write (58) as (w − z)4 + α (w 4 z 4 + 6w 2 z 2 + 4wz(w 2 + z 2 ) + 1) = 0.
(61)
This curve lies in the 2-dimensional family of A4 -symmetric degree (4, 4) curves (w−z)4 +α (w 4 z 4 +6w 2 z 2 +4wz(w 2 +z 2 ) + 1)+iβ(w−z)(w+z)((wz)2 −1) = 0 (62) which are real provided α, β ∈ R. It is easy to check that (62) are all invariant under the rotations (59), and if β = 0 also under the extra order four rotation (w, z) → (iw, i z).
(63)
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In fact, the curves (62) are still invariant under (w, z) → (i z, iw),
(64)
which together with (59) generates S4 , and then /S4 is also an elliptic curve, double-covered by /A4 ; however, since (64) is not a rotation we do not call this S4 an octahedral symmetry group. Note that the curves (60) in the k = 3 case are also invariant under this hidden S4 symmetry, but in this case /S4 is a rational curve. As for k = 3, the limit m → ∞ of (60) should give α = 0, and the value of α for m = 0 can be calculated from the S4 -invariant rational map [23] √ z 4 + 2 3i z 2 + 1 R(z) = √ z 4 − 2 3i z 2 + 1 to be α = 1, thus we take 0 < α < 1; again, is smooth and irreducible for all these values of α. For G = A5 we have one parameter α ∈ R in (w − z)6 + α (9(w 6 + z 6 ) + 9wz(w 4 + z 4 ) + 10(wz)2 (w 2 + z 2 ) + 10(wz)3 +3(w + z)((wz)5 − 1)) = 0.
(65)
This final example fails the vanishing cohomology Condition 3 (cf. Propositions 8.1 and 8.2). As in the euclidean case, it is necessary to multiply the degree 6 polynomial by (w − z) to obtain a reducible degree (7, 7) curve, so that the holomorphic sections on the degree (6, 6) component have to satisfy enough further conditions to be forced to vanish. We will not treat this reducible spectral curve here. 6. Tetrahedral and Octahedral Symmetry In the rest of the paper, we shall relate the symmetric (k, k) curves above to spectral curves of hyperbolic monopoles with a given mass. Our main aim is to obtain the mass associated with these curves, and the basic strategy will consist of transferring calculations on to the quotients by their platonic symmetry groups, π : → /G =: E.
(66)
In this way, the complex analysis needed to relate the monopole mass m to the parameter α in the Ansätze will have the same flavour as the k = 2 discussion above. However, unlike the 2-monopole case, we will now have to deal with a nontrivial Condition 3, which we shall approach using some classical algebraic geometry. It will turn out that Condition 3 will now also play a rôle in the mass calculation itself. Our results can be summarised as follows: Theorem 6.1. There is a unique PSL2 C-orbit of tetrahedrally symmetric 3-monopoles with mass m > 0; a representative is the centred monopole with spectral curve (60), where α and m satisfy the relation
2 1 1 1 ℘ , (67) − 2 1 2 = 2m + 3 12 α involving the Weierstraß ℘-function of the elliptic curve with invariants (83).
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Theorem 6.2. There is a unique PSL2 C-orbit of octahedrally symmetric 4-monopoles with representative (61), whose mass is determined by the relation
3 1 −4α 4 + 10α 3 − 115α 2 + 60α − 3 ℘ = , , (68) 1 2 m + 2 54α 2 (α + 1)2 where the Weierstraß ℘-function has invariants (93) and (94). In both cases, we actually construct a curve together with a linear flow on its jacobian variety that avoids the theta-divisor; this will be discussed in Sects. 7 and 8. It would be interesting to know if this flow arises from a Lax system as in [1, 10, 12]. We express the relation between α and m for each Ansatz in terms of the 1-parameter family of elliptic curves E ≡ E α ; thus the half-periods 1 and 2 in Eqs. (67) and (68), which we shall specify carefully below, are functions of α. To obtain this relation, our starting point is a reciprocity argument similar to the one in Sect. 3.2. We let ξ j denote trivialising sections of L 2m+k | over the open sets U j ∩ ( j = 1, 2) defined by (9), and use them to obtain differentials of the third kind ω j = d log ξ j on , related through
dw dz + ω2 . ω1 = (2m + k) − w z We want to apply the reciprocity law generalising (40) to ω1 and to π ∗ ω, where ω is a global holomorphic 1-form on E; obviously, π ∗ ω is a differential of the first kind on g which is invariant under the G-action. Fix a canonical basis {a , b }=1 for H1 ( ; Z) ∼ = Z2g , where g = (k − 1)2 , and define integers m , n by 1 1 m := ω1 , n := − ω1 . 2πi b 2πi a dz We denote by p j , q j the poles of dw w − z ; they can be found explicitly using the equation for . Our conventions are such that the residues of ω1 at these poles are given by
Res p j (ω1 ) = −(2m + k) and Resq j (ω1 ) = 2m + k
j = 1, . . . , 2(k − 1).
Using the explicit description in Sect. 10, we shall see that in both cases the points p j and q j lie on two separate G-orbits, and we set p := π( p j ),
q := π(q j ).
The reciprocity law can now be written as g 2(k−1)
qj a π ∗ ω −n ∗ = (2m + k) π ∗ ω, π ω m b pj =1
(69)
j=1
where the paths√of integration in the right-hand side avoid the 1-homology basis. Let I = ]0, 3[ if k = 3, respectively I = ]0, 1[ if k = 4, denote the range of α. Equation (69) determines α ∈ I as a function of m, once the integers m , n are known. In fact, these integers are constant along the isotopy of curves = α given by (60) or (61): Lemma 6.3. The integers m , n defined by Eq. (69) are independent of α ∈ I .
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Proof. Suppose that α and m are fixed. For each element ωi of a basis of global holomorphic 1-forms on , a reciprocity relation like (69) can be written, where π ∗ ω is replaced by ωi . Taking real and imaginary parts of these g equations, one obtains 2g real equations in the 2g real unknowns m , n . If the basis {ωi }i is dual to the 1-cycles a , then we see that the equations corresponding to the imaginary parts decouple the variables n , and we can solve for them as the matrix of coefficients is nonsingular by the second Riemann bilinear relations [11]. Substitution in the equations corresponding to the real parts give immediately the m , as the matrix of coefficients is then the identity by construction. We conclude that we can always solve for m , n , and the solutions are given as linear fractional functions of the periods of and the real and imaginary parts p of the integrals q j j ωi . We now argue that the periods of are continuous functions of α. Let {ϕα }α∈I be an isotopy of Z such that ϕα ( 1/2 ) = α for all α ∈ I , say. Fix representatives of a basis of 1-cycles {c } on 1/2 , and transfer them to each α using ϕα . Moreover, equip each curve with a basis of global holomorphic 1-forms ωiα obtained by adjunction (i.e. taking Poincaré residues of a fixed set of g holomorphic 2-forms on a neighbourhood of the family of curves in Z ); this is possible as each α is smooth. Then one finds ϕα∗ c
ωiα =
c
ϕα∗ ωiα
and the right-hand side is an integral of a continuous function of α, which is itself continuous in α. Clearly, the same type of argument works for the paths connecting poles dz of dw w to poles of z , and the lemma follows. To transfer the calculation to E, we define c := π∗
g j=1 (m j a j
+ n j b j ) =: 1 a + 2 b,
(70)
where {a, b} is a standard basis of H1 (E; Z), and obtain
ω = 2(2m + k) c
q
ω
(71)
p
after using π to change variables and clearing a factor of deg π = |G|. As a consequence of Lemma 6.3, the components of c will not change in an isotopy of bases of H1 (E; Z) defined for α ∈ A. Note that, although the paths connecting q j to p j on had to be chosen to have zero intersection with the elements of H1 ( ; Z) → H1 ( , { p j , q j }; Z), this does not necessarily hold as we project to E. However, there is a simple criterion to test whether a path γ from q to p on E is the image of a suitable path on , namely, the components of the 1-cycle c ∈ H1 (E; Z) must be independent of m for solutions (α, m) of (71); once at least two such solutions are obtained, γ can be found systematically by expanding in a basis of H1 (E, { p, q}; Z) ∼ = Z3 . In Sect. 9, we shall describe a general procedure that allows one to obtain data (α, m) for half-integer values of m. From these, we will be able to calculate both a suitable path γ and the 1-cycle c from an explicit construction of E in Sect. 10. Finally, we will use uniformisation on E to evaluate the mass constraint (71) in terms of elliptic functions.
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7. Canonical Embedding The canonical embedding of a genus g curve is a map → Pg−1 defined by z → [ω1 (z) : . . . : ωg (z)] with respect to a local trivialisation of the canonical line bundle K . This map is an embedding except when the curve is hyperelliptic, in which case it maps two-to-one onto a rational curve in Pg−1 . It gives a useful geometric version of Riemann–Roch which we will use. By adjunction, the canonical embedding of a curve embedded in a surface X can be induced by an embedding X → Pg−1 . We will use this to give an explicit description of the canonical embedding of a curve in the quadric Q := P1 × P1 . (This can be done similarly for P2 . An immediate consequence is that no smooth hyperelliptic curve with g > 1 embeds in P2 or P1 × P1 .) Lemma 7.1. For k > 2, the canonical embedding of a smooth (k, k) curve ⊂ P1 × P1 is the composition → P1 × P1 → Pk(k−2) (w, z) → 1 : w : ... : w k−2 : z : wz : ... : w k−2 z : ... : z k−2 : wz k−2 : ... : w k−2 z k−2 . Proof. As mentioned above we will use the adjunction formula K = O(k −2, k −2)| . Sections of K can be identified with sections that extend to all of Q = P1 ×P1 since the relation introduced by the equation of is in higher degree than k − 2, or equivalently the outer two cohomology groups vanish in the following exact sequence: H 0 (Q, O(−2,−2)) → H 0 (Q, O(k −2, k −2)) → H 0 ( , O(k −2, k −2)) → H 1 (Q, O(−2, −2)). The polynomials 1, w, . . . , w k−2 , z, wz, . . . , w k−2 z, . . . , z k−2 , wz k−2 , . . . , w k−2 z k−2 give a basis of the space of sections of O(k − 2, k − 2), where we are using U1 in (9) as a local trivialising set. Lemma 7.2. For any smooth (3, 3) curve ⊂ P1 × P1 , a nontrivial linear equivalence relation p1 + p2 + p3 ∼ q 1 + q 2 + q 3 is equivalent to p1 + p2 + p3 ∼ O (1, 0) or O (0, 1). Proof. This is a well-known fact. Clearly any two fibres of the projection of P1 × P1 to the first factor are linearly equivalent, and the same for projection to the second factor. It ends up that these are the only linear equivalences between triples of points on a smooth degree (3, 3) curve in P1 × P1 . An equivalent statement is that a g31 on must be O (1, 0) or O (0, 1). (Recall that a gdr is any linear system of degree d and dimension r + 1, see for example [2].) We include the proof as a warm-up for similar results. We use the canonical embedding φ : → P3
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to prove this. The images φ( p1 ), φ( p2 ) and φ( p3 ) lie inside a P2 ⊂ P3 . When there exists the nontrivial relation p1 + p2 + p3 ∼ q1 + q2 + q3 , the geometric version of Riemann–Roch says that φ( p1 ), φ( p2 ) and φ( p3 ) lie inside a P1 ⊂ P3 . Suppose two of the points lie on a (1, 0) or (0, 1) curve, say p1 and p2 lie on a (1, 0) curve. Then their images determine a line in P3 , [1 : w : c : wc] for some constant c. The image of p3 , given by [1 : w : z : wz], lies on the curve only if z = c, i.e. p3 lies on the same (1, 0) curve, proving the lemma. If no two points lie on a (1, 0) or (0, 1) curve, then the three points lie on a smooth (1, 1) curve. Any smooth (1, 1) curve is equivalent to the diagonal P1 ⊂ P1 × P1 which has images [1 : w : w : w2 ] inside P3 , and any three different points of this form are linearly independent due to the 1, w, w2 terms. This contradicts the fact that the points p1 , p2 and p3 span P1 . Essentially the same argument is used when some of the pi coincide, so the result follows. Recall that a divisor D is special if H 1 (D) = 0, or equivalently if it has more sections than a generic divisor of the same degree. Lemma 7.3. A positive divisor D of degree at most 7 on a (4, 4) curve ⊂ Q is special precisely when one of the following holds: (a) D contains four points on a (1, 0) or (0, 1) curve; (b) D contains six points on a (1, 1) curve. Proof. The proof of this requires a systematic analysis of many separate cases. To avoid this we will instead refer to an exercise from [2], p. 199, stating that any collection of at most seven points in P3 that fails to impose independent conditions on quadrics contains one of the following: (i) four collinear points; (ii) six points on a conic; (iii) seven coplanar points. Lemma 7.1 shows that the canonical embedding of factors through the quadric Q. More is true. It also factors through P3 , so we have → Q → P3 → P8 with rightmost map [z 0 : z 1 : z 2 : z 3 ] → [z 02 : z 0 z 1 : z 0 z 2 : z 0 z 3 : z 12 : z 1 z 3 : z 22 : z 2 z 3 : z 32 ] induced from the degree two Veronese map, where the monomial z 1 z 2 is missing since it is equal to z 0 z 3 on Q. Thus a collection of points on representing a positive divisor D gives a collection of points in P3 . By the geometric version of Riemann–Roch, D is special when the images of the points in P8 are dependent, so one of (i), (ii), or (iii) occurs. The intersection of a line L ⊂ P3 and Q either consists of two points (counted with multiplicity) or L ⊂ Q. If (i) occurs then four points from Q lie on a line L ⊂ P3 and hence L ⊂ Q. The only such lines are (1, 0) or (0, 1) curves so case (a) holds. The intersection of a conic C ⊂ P3 and Q either consists of four points or C ⊂ Q. If (ii) occurs, then six points from Q lie on a conic C ⊂ P3 and hence C ⊂ Q. Conics in Q are (1, 1) curves so (b) holds. The intersection of a plane and the quadric in P3 is a conic in P2 and a (1, 1) curve in Q, so if (iii) occurs then seven points lie on a (1, 1) curve and again (b) holds. Lemma 7.4. For any smooth (4, 4) curve ⊂ P1 × P1 , a nontrivial linear equivalence relation p1 + p2 + p3 + p4 ∼ q 1 + q 2 + q 3 + q 4 implies p1 + p2 + p3 + p4 ∼ O (1, 0) or O (0, 1).
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Proof. This is immediate from Lemma 7.3. Lemma 7.5. For any smooth (4, 4) curve ⊂ P1 ×P1 , the existence of two independent linear equivalence relations p1 + · · · + p8 ∼ q 1 + · · · + q 8 ∼ r 1 + · · · + r 8 implies that there exists p9 , . . . , p12 such that p1 + · · · + p8 + p9 + · · · + p12 ∼ O (2, 1) or O (1, 2). Proof. By independent relations we mean that the points satisfy one of the three equivalent conditions: h 0 ( p1 + . . . + p8 ) > 2; or p1 + . . . + p8 defines a g82 ; or the images of p1 , …, p8 in P8 live in a P5 . Since the images of p1 , …, p8 in P8 satisfy two relations, any subset of seven points satisfies a relation and we can apply Lemma 7.3 to all eight such subsets. (I) If seven points contain six points on a smooth (1, 1) curve, then no four points of the eight points are contained on a line, so all subsets of seven points must contain six points in a conic. Since a conic is determined by three points, the eight conics must coincide. Thus the eight points consist of seven points on a smooth conic and an eighth general point. In particular, the eight points lie inside a (2, 1) or (1, 2) curve as claimed. (II) If four of the eight points are contained on a line, say a (1, 0) curve, then since five points cannot lie on a line — is a smooth (4, 4) curve — when a subset of seven points does not include these four points, it must include either four points on another line, or three points on a (0, 1) curve so that it has 3 + 3 points on a reducible (1, 1) curve. In both these cases, seven points lie on two lines, and the eighth point is general, so in particular the eight points lie inside a (2, 1) or (1, 2) curve as claimed. (III) If neither (I) nor (II), i.e. no four points lie on a line and no six points lie on a smooth (1, 1) curve then case (b) of Lemma 7.3 must apply to all eight subsets, with a reducible (1, 1) curve, i.e. 3 + 3 points must lie on a (1, 0) and (0, 1) curve. A subset of seven points may not contain the three points on the (1, 0) curve, so another (1, 0) curve must contain three points, and this takes at least two extra points from the eight points, i.e. 3 + 2 + 3 points distributed on a (1, 0), (1, 0) and (0, 1) curve. Similarly we also need another (0, 1) curve with three points and this requires at least one more point, so nine points are required. Thus, Case (III) is empty and the lemma is proven. 8. Crossing the Theta-Divisor 8.1. k = 3 tetrahedral symmetry. Consider the smooth genus four curve ⊂ P1 × P1 with Eq. (60) for α ∈ R+ . The bundle L s (1, 0)| is a degree 3 bundle, or equivalently a degree 3 divisor on . Recall that the divisor D of L s (1, 0)| meets the theta-divisor if one of the following equivalent properties holds: 1. 2. 3. 4.
D ∼ p1 + p2 + p3 , i.e. D is linearly equivalent to a positive divisor; there exists a meromorphic function f on such that ( f ) + D ≥ 0; there exists a nontrivial holomorphic section of the line bundle L s (1, 0)| ; H 0 ( , L s (1, 0)) = 0.
Here we prove that, when 0 < s < 2m + 2, the divisor of L s (1, 0)| does not meet the theta-divisor. Proposition 8.1. H 0 ( , L s (1, 0)) = 0 for 0 < s < 2m + 2.
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Proof. The geometry implies that L is invariant under the A4 -action, since L comes from the standard U(1) monopole which is symmetric under SO(3). In other words, O(1, −1) is invariant under the SO(3)-action on P1 × P1 so L := O(1, −1)| is invariant under A4 . In particular, L is the pull-back of a degree zero line bundle Lˆ on E = /A4 , and L s = π ∗ Lˆ s for s ∈ C. Since E is an elliptic curve, as a divisor Lˆ s ∼ p − p , and all such divisors are represented as s varies over the complex numbers. Thus, since Lˆ s pulls back to L s , we can represent the divisor of any L s by L s ∼ Orb p − Orb p , where Orb p is the A4 -orbit in that lies over p ∈ E. So questions involving L s are questions about orbits of A4 . We will show that the difference of two orbits of A4 in plus the divisor class O (1, 0) lies in the theta-divisor, Orb p − Orb p + O (1, 0) ∼ q1 + q2 + q3 ,
(72)
in the following trivial cases: Orb p − Orb p ∼ 0 or Orb p − Orb p ∼ L −1 = O (−1, 1). The action of A4 on the left-hand side of (72) preserves the two orbits and also preserves the linear equivalence class O (1, 0) since the action, given in (59), preserves the two P1 factors of P1 × P1 . Thus, for any g ∈ A4 , q1 + q2 + q3 ∼ gq1 + gq2 + gq3 .
(73)
The collection {q1 , q2 , q3 } cannot be invariant under A4 , since the orbits of A4 have size 12 and 6. Thus we can choose a g ∈ A4 that does not preserve {q1 , q2 , q3 }, so the linear equivalence relation (73) is a nontrivial relation between degree 3 positive divisors. By Lemma 7.2, this implies one of the two cases q1 + q2 + q3 ∼ O (1, 0) or q1 + q2 + q3 ∼ O (0, 1). If q1 + q2 + q3 ∈ O (1, 0), then (72) reduces to Orb p − Orb p ∼ 0, ∼ in other words L s is trivial, so s = 0 (or a multiple of 2m + 3 since L 2m+3 = O .) If q1 + q2 + q3 ∈ O (0, 1), then 0 ∼ Orb p − Orb p + O (1, 0) − (q1 + q2 + q3 ) ∼ L s+1 , so s = −1 (plus a multiple of 2m + 3.) In particular, when 0 < s < 2m + 2, L s (1, 0)| does not meet the theta-divisor.
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8.2. k = 4 octahedral symmetry. Orbits of S4 on consist of 24 points, except for the one exceptional orbit of eight points given by the (1, 1) divisor P1 ∩ . Proposition 8.2. H 0 ( , L s (2, 0)) = 0 for 0 < s < 2m + 2. Proof. As before, we reduce questions involving L s to questions about orbits of S4 using L s ∼ Orb p − Orb p for p, p ∈ E = /S4 . The difference of two orbits of S4 in plus a O (2, 0) divisor lies in the theta-divisor, Orb p − Orb p + O (2, 0) ∼ q1 + q2 + · · · + q8 ,
(74)
in the following trivial cases: Orb p − Orb p ∼ L , = 0, −1 or − 2.
(75)
The action of S4 on the left-hand side of (74) preserves the two orbits and also preserves the linear equivalence class O (2, 0). Thus, for any g ∈ S4 , q1 + q2 + · · · + q8 ∼ gq1 + gq2 + · · · + gq8 .
(76)
If the collection {q1 , q2 , . . . , q8 } is invariant under S4 then it consists of the exceptional orbit and q1 + q2 + · · · + q8 ∼ O (1, 1), which yields = −1 in (75). If the collection {q1 , q2 , . . . , q8 } is not invariant under S4 , then we can choose a g ∈ S4 that does not preserve {q1 , q2 , . . . , q8 }, so the linear equivalence relation (76) is a nontrivial relation between degree 8 positive divisors, or equivalently dim H 0 ( , L s (2, 0)) ≥ 2. We may assume that there is no section in H 0 ( , L s (2, 0)) that is invariant under S4 , or more generally generates a 1-dimensional representation of S4 , since the zero set of such a section would have to be the exceptional orbit of S4 and we get the previous case of = −1. Thus, if H 0 ( , L s (2, 0)) = C2 it must be an irreducible representation of S4 . The standard 2-dimensional representation is given by the action on H 0 ( , O(1, 0)) and thus the tensor product H 0 ( , L s (3, 0)) is the 4-dimensional representation with a 1-dimensional irreducible component generated by a section ξ = wχ ∈ H 0 ( , L s (3, 0)) for χ ∈ H 0 ( , L s (2, 0)). The zero set of ξ is invariant under S4 and contains a (1, 0) divisor. But this is impossible simply because any orbit of S4 , or collection of exceptional orbits, contains at most three points in a (1, 0) curve. Thus dim H 0 ( , L s (2, 0)) > 2, so by Lemma 7.5 there exist points q9 , q10 , q11 , q12 such that q1 + q2 + · · · + q12 ∼ O (2, 1) or O (1, 2) and (74) becomes Orb p − Orb p + O (2, 0) + q9 + q10 + q11 + q12 ∼ O (2, 1) or O (1, 2).
(77)
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Since Orb p − Orb p = L s and any multiple of O (1, −1) is a power of L , we can adjust this expression and choose two new orbits over p˜ and p˜ so that Orb p˜ − Orb p˜ + O (1, 0) ∼ q9 + q10 + q11 + q12 ,
(78)
which is a reduction of the original problem. Again we use g ∈ S4 to get a nontrivial linear equivalence q9 + q10 + q11 + q12 ∼ gq9 + gq10 + gq11 + gq12 and apply Lemma 7.4 to obtain q9 + q10 + q11 + q12 = O (1, 0) or O (0, 1). Put this back into (77) to get q1 + q2 + · · · + q8 = O (2, 0), O (1, 1), or O (0, 2) and hence Orb p − Orb p ∼ L , = 0, −1 or − 2. In particular, we have proved that when 0 < s < 2m + 2, L s (2, 0)| does not meet the theta-divisor. Remark. Hitchin [13] proves a result analogous to Proposition 8.2 in the euclidean case using a slightly different method. His treatment of the tetrahedral case is essentially the same as ours, relying on knowledge of g31 divisors on a genus four curve. Whereas we extend this approach through an analysis of g41 and g82 divisors on a genus nine curve, Hitchin analyses the representation theory of S4 more thoroughly and produces a beautiful application of the McKay correspondence. We need a small amount of representation theory to exclude a two-dimensional representation of S4 , mainly because the space of g81 divisors is too large to manage. The family of tetrahedrally symmetric (4, 4) curves (62) cannot be treated using the representation theory approach, while the method here does generalise. 9. Half-Integer Mass We have shown that triviality of L 2m+k on determines a relation between α and m. The vanishing of H 0 ( , L 2m (k − 2, 0)) also has a direct consequence on the parameter α of the symmetric spectral curve. Since H 0 ( , L 2m (k − 2, 0)) = H 0 ( , O(−2, k)), a section can be described by ξ1 (w, z) = w −2 z k ξ2 (w −1 , z −1 ) + ψ(w, z)χ (w −1 , w, z −1 , z), where ξ1 , ξ2 give the section over the two coordinate patches U1 and U2 as in (9), ˇ w −2 z k = g12 (w, z) is the transition function of O(−2, k), i.e. its Cech cohomology class, and we work in the Laurent polynomial ring C[z, z −1 , w, w −1 ] modulo ψ(w, z), the defining polynomial of . A section exists if we can find χ (w−1 , w, z −1 , z) so that ψ(w, z)χ cancels all of the negative powers of w and z in w −2 z k ξ2 (w −1 , z −1 ). So ψ(w, z) acts as a type of Toeplitz operator on χ . In fact χ = χ (w−1 , w) and only the coefficients of w −1 need to be taken
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care of. This is not hard to prove, and is most efficiently expressed in terms of an exact ˇ ˇ sequence in Cech cohomology in which we see that χ is also a Cech cocycle, ψ
0 → H 0 ( , O(−2, k)) → H 1 (Q, O(−2 − k, 0)) → H 1 (Q, O(−2, k)); so χ ∈ H 1 (Q, O(−2 − k, 0)) and multiplication by ψ(w, z) yields a linear map ψ Ck+1 ∼ = Ck+1 = H 1 (Q, O(−2 − k, 0)) → H 1 (Q, O(−2, k)) ∼
(79)
with kernel corresponding to sections of H 0 ( , O(−2, k)). Since H 0 ( , O(−2, k)) vanishes, the kernel is trivial and the determinant of (79) is nonzero. In the tetrahedral case (79) is represented by the matrix ⎛ ⎞ 1 0 −iα 0 3 0 iα ⎟ ⎜0 =⎝ , det = (3 − α 2 )2 = 0 iα 0 3 0⎠ 0 −iα 0 1 and in the octahedral case by ⎛ 1 0 0 0 0 0 ⎜ 0 4 − 4α ⎜ 0 6 + 6α 0 = ⎜0 ⎝0 0 0 4 − 4α α 0 0 0
⎞ α 0⎟ ⎟ 0 ⎟ , det 0⎠ 1
= 96(1 + α)2 (1 − α)3 = 0.
√ The nonvanishing determinant restricts 0 < α < 3 in the tetrahedral case and 0 < α < 1 in the octahedral case, in agreement with the discussion in Sect. 5. More significant information is obtained from ψ1
0 → H 0 ( , O(−3, k + 1)) → H 1 (Q, O(−3 − k, 1)) → H 1 (Q, O(−3, k + 1)), where ψ1 is again a Toeplitz type operator induced from multiplication by ψ. Trivial kernel, and hence nonzero determinant of 1 , occurs when m > 21 , since H 0 ( , L 2m−1 (k − 2, 0)) = H 0 ( , O(−3, k + 1)) vanishes when 0 < 2m − 1 < 2m + 2. In the tetrahedral case, det and hence α =
√1 3
for m >
1 1 2.
= 4(1 − 3α 2 )2 (3 − α 2 )2
But α tends to zero as m → ∞, and α begins at
when m = 0, so by continuity, α =
√1 3
√
3
for some value of m. In fact,
1 1 α=√ ⇔ m= , 2 3 since H 0 ( , L 2m−1 (k − 2, 0)) = H 0 ( , O(k − 2, 0)) has nontrivial sections, and thus 1 has nontrivial kernel, so det 1 = 0. The preceding calculation is sufficient for our purposes to get the relation (67) once we have found a suitable path γ on E from q to p, but we need at least one more
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Table 1. The parameter α for some half-integer values of m m 0 1 2
G = A4 k=3 √ 3
G = S4 k=4
√1 3 √
1 3 1 7
2− 3 √ 23 − 4 33 0
1 3 2
∞
1 √ 7−4 3 0
value of (α, m) to be able to find γ , as explained in Sect. 6. However, similar calculations on H 0 ( , L 2m−2 (k − 2, 0)) = H 0 ( , O(−4, k + 2)) yield a map 2 with 2 2 2 2 2 2 det 2 = 4(α 2 + 5)2 (1 − √ 3α ) (α − 4α + 1) (α + 4α + 1) , and this enables one to conclude that α = 2 − 3 when m = 1. This technique applies to yield an algebraic value of α for any half-integer mass m; Table 1 summarises these results for m ≤ 23 . In the octahedral case, det
1
= 16(1 + 5α)2 (α + 5)3 (3α − 1)3 (α − 1)4
and using the argument above we deduce that α=
1 1 ⇔m= . 3 2
(80)
The next values for α, found by computing the determinants up to m = 23 , are displayed in Table 1. 10. The General Mass Constraints We start by constructing the quotients π in (66) explicitly, and in particular a realisation of the elliptic curves E, by means of invariant theory. To do this, we use the procedure illustrated in Example 5.3 to find rational invariants v, ˇ x, ˇ yˇ ∈ C( ) A4 given in the 1 1 coordinates of P × P by P3 (w + z)((wz)2 − 1) := , P1 3 (w − z)3 w 4 z 4 + w 4 + z 4 + 12w 2 z 2 + 1 P4 , xˇ = 4 := P1 (w − z)4 (wz)6 − ((wz)2 + 1)((w + z)4 + 4wz(w + z)2 + (wz)2 ) + 1 P6 . yˇ = 6 := P1 (w − z)6 vˇ =
Here, P denote forms of degree (, ) that are projectively invariant under (59). They satisfy a relation (in degree 12) which can be written in terms of the invariants above as ˇ + 9vˇ 2 ) − 3 = 0. 4xˇ 3 − 11xˇ 2 − 4 yˇ 2 − 14vˇ 2 − 27vˇ 4 + 2 x(5
(81)
Out of these rational functions, we produce maps into P2 whose restriction to each has A4 -orbits as fibres. The images are determined by a single relation among the invariants, which can be used to describe the elliptic curves /A4 . This procedure is easily adapted
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to the octahedral case, as we shall explain below. In practice, it is convenient to choose the coordinates of the embeddings to obtain an image in standard Weierstraß form. 10.1. k = 3 tetrahedral symmetry. Using Eq. (60) to eliminate vˇ = αi , we obtain from (81) the relation on ,
14 18 27 2 3 2 4 yˇ = 4xˇ − 11xˇ + 10 − 2 xˇ − 3 + 2 − 4 . α α α So we redefine the invariants as x := xˇ − plane cubic
11 12 ,
y := 2 yˇ , in order to obtain the standard
y 2 = 4x 3 − g2 x − g3 =: F(x),
(82)
where g2 =
18 1 + , 12 α 2
g3 = −
5 1 27 + + . 216 2α 2 α 4
(83)
The picture is that π := [1 : x : y] maps A4 -orbits in (given by Eq. (60)) into P2 , and the image E is realised as the elliptic curve given by (82). From this equation, we read off that x ◦ u −1 = ℘ is a Weierstraß function of E and y ◦ u −1 = ±℘ its derivative up to sign, where u −1 : C → E is the well-defined inverse to a “uniformisation map”. We use e1 , e2 , e3 to denote the zeroes of F, F(x) = 4(x − e1 )(x − e2 )(x − e3 ) with e1 + e2 + e3 = 0. From (83), we compute the j-invariant of E to be j (α) =
g2 3 α 2 (α 2 + 216)3 = . g2 3 − 27g3 2 26 33 (α 2 − 27)3
One way of realising E is as a degree 3 branched cover [1 : x ◦ π −1 ] : E → P1 , with branch points [1 : e1 ], [1 : e2 ] and [1 : e3 ] and suitable branch cuts. Clearly, two zeroes of F must be nonreal and complex conjugates, while the other (say e2 ) is real and positive since g3 = 4e1 e2 e3 > 0. We take Im (e1 ) > 0. Notice that Re e1 = Re e3 = − 21 e2 and Im e3 = −Im e1 . We must define branch cuts, the sheet decomposition, a holomorphic 1-form ω and a basis of 1-homology for E before we can compute the integrals in the relation (71). We consider the standard holomorphic 1-form on E given by ω = ±√
dx F(x)
(84)
up to sign. The conventions that we shall follow are illustrated in Fig. 2. One branch cut runs from e2 to ∞ downwards while the other one joins e1 to e3 avoiding the negative dz real axis; these separate E into two sheets (i) and (ii). The poles of dw w − z are easily seen to be √ √ p1 = (ρ¯ α, 0), p2 = (−ρ¯ α, 0), √ √ q1 = (0, ρ α), q2 = (0, −ρ α),
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Fig. 2. Tetrahedrally symmetric 3-monopole: contours of integration on E = /A4
where ρ := eπi/4 . Their images on E ⊂ P2 can be computed as $
% 1 2i 1 1 p := π( p1 ) = 1 : = π( p2 ), − 2 :− 1+ 2 12 α α α
% $ 1 1 2i 1 = π(q2 ), 1+ 2 q := π(q1 ) = 1 : − 2 :+ 12 α α α and they lie on the√same fibre of the double cover [1 : x] : E → P1 , as we claimed 1 − α12 < 0, the common projection of both p and q onto before. Since α < 3 implies 12 the x-plane lies on the negative real axis. As before, paths are drawn with continuous or dotted lines according to whether they lie on sheets (i) or (ii), and we have defined a, b and the path from p to q (avoiding a and b) following this notation; in particular, q lies on sheet (i) and p on sheet (ii). We choose the signs in (84) so that the coefficient at p has positive imaginary part. Equation (71) relates the periods +∞ dx 2 := ω = 2 , (85) √ F(x) a e2 e1 dx (86) 2 := ω = −2i √ −F(x) b e2 to the integral
q p
ω = 2i
−∞ 1 1 12 − α 2
√
dx −F(x)
(87)
as 1 + 2 = 2i(2m + 3)
−∞ 1 1 12 − α 2
√
dx . −F(x)
(88)
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Notice that is real, while the integral (87) is pure imaginary, and that one has e1 e1 dx dx 2 Re = −i +i √ √ −F(x) −F(x) e e 2e3 2e2 dx dx +i = −i √ ¯ −F(x) e1 e2 − F(x) e3 dx =i √ −F(x) e1 = . Taking real parts in both sides of Eq. (88), one then obtains the relation 2 = −21 .
(89)
These integrals are most conveniently dealt with using uniformisation. The standard uniformisation point is the set of 2-poles of ℘, thus of 2-poles of x: x = ∞ ⇔ w = z; an easy check gives that all the six points of ∩ P1 are poles, and they are mapped to [0 : 0 : 1] by π , the branch point at infinity of [1 : x ◦ π −1 ] : E → P1 . One has ℘ ( ) = e2 and ℘ ( ) = e3 , so to conform with standard practice we should use the half-periods 2 = , 1 + 2 ⇒ 1 = 2 − . = 2
(90)
Now we can write formally ℘ (0) −∞ dx d℘ = i √
1 1 1 −F(x) − 12 ) ℘ ℘◦℘ −1 ( 12 12 − α 2 α 0 = du 1 − ℘ −1 ( 12
= ∓℘ −1
1 ) α2
1 1 − 2 , 12 α
(91)
where ℘ −1 is determined only up to sign (as ℘ has order 2 in a fundamental region) and up a point in the lattice = 2 1 Z ⊕ 2 2 Z inside the u-plane. Equation (88) then gives, making use of (89),
1 1 ( − 2 )1 mod . ≡ ∓℘ −1 − 2 2(2m + 3) 12 α The ambiguity introduced in (91) is removed once we apply ℘ to both sides of this equation:
1 1 1 1 ℘ = − 2. 2(2m + 3) 12 α
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The only information missing is the integer 1 determining c = 1 (1, −2) ∈ H1 (E; Z), but this can be obtained directly from (88) and (89) using the data (α, m) we calculated in Sect. 9. We always find 1 = 4 by evaluating the integrals numerically for all the positive values of m in Table 1. This confirms that we are working with a path γ on E that is the image of a path with zero intersection with the 1-cycles of . We end up with Eq. (67), and this completes the proof of Theorem 6.1. 10.2. k = 4 octahedral symmetry. When (62) is used to eliminate xˇ = 1 − (1 + iβ v)/α, ˇ one obtains the relation on
2
12 β 2iβ 2β 2 18 3 vˇ 2 + − 1 − 9 v ˇ + 4 − −4 yˇ 2 − 27vˇ 4 + 2 α α α α α2
6 4 1 2iβ vˇ + 2 1 − = 0. (92) + 2 1− α α α α In principle, one could follow the same procedure as in the k = 3 case to obtain the relation among α, β and m, but a calculation for general values of the parameters is out of hand as one now must deal with the roots of a quartic polynomial, and these then give rise to a less symmetric form of the Weierstraß equation (82). In the following, we shall only consider the case of octahedral symmetry β = 0, working with /S4 rather than /A4 , which considerably simplifies the calculations as one should expect. The extension from tetrahedral to octahedral symmetry can be realised by adding the rotation (63), which changes the tetrahedral invariants as vˇ → −v, ˇ xˇ → x, ˇ
yˇ → − yˇ .
Therefore, x, ˇ vˇ yˇ , and vˇ 2 are octahedral invariants, and a cubic relation among them on can be obtained simply by multiplying both sides of (92) by vˇ 2 . This relation is brought to Weierstraß form (82) if one uses the coordinates y :=
√
2i vˇ yˇ ,
x :=
2 3 2 1 vˇ + − ; 2 3α 27
the elliptic invariants are now 16 5 16 4 − + 2− 3 253 27α 3α 3α
(93)
32 2 64 41 4 − + − + . 19683 729α 9α 2 81α 3 9α 4
(94)
g2 = and g3 =
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These yield the j-invariant j (α) =
(432α 3 − 4048α 2 + 11385α − 9108)3 . 22 33 113 233 (α − 4)2 (α − 3)3
Again, we realise E := /S4 as the image of the map π := [1 : x : y] to P2 . It is easy to see that F(x) has again a real positive zero e2 and two distinct complex conjugate zeroes e1 , e3 ; we choose the convention Im e1 > 0 as before. dz The differential dw w − z on has four poles with positive residue and four poles with negative residue, and their images on E ⊂ P2 are, respectively, % $ 7 √ 1+α 2 : 2i 2 , p= 1:− − 27 6α α % $ √ 7 2 1+α : − 2i 2 . q= 1:− − 27 6α α We should integrate on along four paths, from π −1 ( p) to π −1 (q) points, which do not cross a basis of 1-cycles of . Contrary to the k = 3 case, it will turn out that the image of such paths will have nontrivial intersection with a standard basis of E. We follow the same conventions as above for the covering [1 : x] : E → P1 and the periods (85)–(86), but not for the path γ , which we define as having #γ , b = −1 as illustrated in Fig. 3. We obtain the relation for ω and c as in (84) and (70), q ω = 8(m + 2) ω, c
p
and using our conventions we find
1 + 2 = 4i(m + 2)
2 7 − 27 − 6α
e2
dx . √ −F(x)
Again, (89) holds, and we are led to the relation
−4α 4 + 10α 3 − 115α 2 + 60α − 3 1 1 = ℘ , 2(m + 2) 54α 2 (α + 1)2
Fig. 3. Octahedrally symmetric 4-monopole: contours of integration on E = /S4
(95)
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where we made use of the duplication formula for ℘. Given (80), we find from (95), 1 = 6, and the same result is obtained for the other (α, m) data, which confirms that the right path γ has been chosen. This completes the proof of Theorem 6.2. 11. Rational and Infinite Mass This final section illustrates how one may deal with the general mass constraints (67) and (68) in two important concrete cases, providing nontrivial checks to our calculations. 11.1. Solutions for rational mass. We shall now explain briefly how to solve the mass conditions (67) and (68) explicitly to calculate α for rational values of the monopole mass. We shall be making use of the fact that, for m ∈ Q, the left-hand side of either of these conditions is a division value of the ℘-function, i.e., a complex number of the form
2k1 1 + 2k2 2 , ℘ n where n ∈ N and k1 , k2 ∈ Z are such that (k1 , k2 ) = (0, 0) mod n. Classical techniques in the theory of elliptic functions [19] can be used to show that division values correspond exactly to the roots of the so-called special division equation Dn (℘ (u)) = 0, 2 2 where Dn (℘) is a polynomial of degree n 2−1 or n 2−4 defined by ψ2+1 (u) ψ2+2 (u) or D2+2 (℘ (u)) := − 2 + 1 ( + 1)℘ (u) according to whether n is odd or even; here, ℘ (u) ℘
(u) · · · ℘ (n−1) (u)
℘
(u) · · · ℘ (n) (u) (n − 1)n−1 ℘ (u) ψn (u) := .. .. .. .. &n−1 2 . . . . j=1 j! ℘ (n−1) (u) ℘ (n) (u) · · · ℘ (2n−3) (u) D2+1 (℘ (u)) :=
is easily seen to satisfy ψn (−u) = (−1)n+1 ψn (u), so indeed Dn (℘) ∈ Q[g2 , g3 ][℘]. To calculate α for m ∈ Q, one replaces ℘ (u) by the right-hand side of (67) or (68) in the special division equation for the corresponding elliptic curve E = /G, thus obtaining a polynomial equation for α; by inspection, the minimal polynomial for α is recovered as one of the factors, and hence α itself. This argument shows that, for m ∈ Q, Eqs. (67) and (68) determine α as an algebraic number. The results of Table 1 in Sect. 9 are easily checked using this procedure; in particular, we can compute limm→0 α and write the limit “nullaron” curves in the form (51), from which we can recover the corresponding rational maps. For m ≥ 2, the algorithm in Sect. 9 becomes impractical, whereas the special division equation can still be used to generate minimal polynomials for the parameter α. In addition, one can calculate in principle the value of α for any m ∈ Q. For example, in the k = 3, G = A4 case, for m = 13 , we obtain the minimal polynomial α 10 − 39α 8 + 506α 6 + 866α 4 − 715α 2 − 11, which yields α
0.791875 as the unique positive real root larger than α|m= 1 = 2
√1 . 3
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11.2. Euclidean limit. As explained in Sect. 4.4, centred spectral curves degenerate to k copies of the diagonal P1 when the limit m → ∞ is taken, but a rescaling of the metric yields curves in T P1 which can be interpreted as spectral curves of euclidean monopoles. This limit process allows one to derive the spectral curves of platonic monopoles in euclidean space from our results for hyperbolic monopoles; it will also provide a nontrivial check on our calculations. We focus on the case k = 3, G = A4 for brevity. Given (52), we expect the rescaled parameter α˜ := m 3 α to tend to a finite limit as m → ∞ (and α → 0), and our aim is to calculate this limit using the relation (67). The natural way to proceed is to rescale the A4 -invariants on in such a way that the quotient elliptic curve E = /A4 ⊂ P2 has a sensible limit; thus we use x˜ := x/m 4 and y˜ := y/m 6 to embed E in P2 . When m → ∞, one obtains the Weierstraß equation for E, 27 ˜ x), =: 4(x˜ − e˜1 )(x˜ − e˜2 )(x˜ − e˜3 ) =: F( ˜ α˜ 2
y˜ 2 = 4x˜ 3 −
where we use conventions for the invariants (and periods) consistent with Sect. 10.1. This is the same elliptic curve as in the calculations in Sect. 9 of [14], since the j-invariant is zero in both cases. Clearly, the periods transform as ˜ i = m 2 i under our rescaling. 1 2 Using the asymptotics ℘ (ρ) = ρ 2 + O(ρ ) as ρ → 0, one obtains from (67) the relation m6 m6 1 1 − + O( ) = m6 12 α˜ 2 ˜ 12
m→∞
=⇒
˜ 1 = i α˜
(96)
for the limit elliptic curve. Now we evaluate, in the limit, +∞ d x˜ d x˜ − e˜2 e˜2 ˜ x) ˜ x) − F( ˜ F( ˜ √ √ 2/3 1 1 1/3 2πi/3 2/3 1/3 2 e π a˜ ! 3 π a˜ ! 6 2 =− − . √ 3! 56 9 3! 53
˜ 1 = −2i
e˜1
(97)
Equating the right-hand sides of (96) and (97), we find 9 ! 13 α˜ = 6 3 ; 2 π thus the limit curve in T P1 η3 + 2i αζ ˜ (ζ 4 − 1) = 0 is precisely the spectral curve of a euclidean 3-monopole with tetrahedral symmetry as found in [14, 15]. Acknowledgements. The authors are grateful to Michael Murray for discussions and support. The second author’s work is financed by the Australian Research Council, and he would like to thank the Department of Mathematics and Statistics of the University of Melbourne for hospitality.
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References 1. Adler, M., van Moerbeke, P.: Linearization of Hamiltonian systems, Jacobi varieties and representation theory. Adv. in Math. 38, 318–379 (1980) 2. Arbarello, E., Cornalba, M., Griffiths, P., Harris, J.: Geometry of Algebraic Curves. Volume I. Grundlehren der Mathematischen Wissenschaften 267, Berlin Heidelberg New York: Springer-Verlag, 1985 3. Atiyah, M.F.: Magnetic monopoles in hyperbolic spaces. In: Vector Bundles in Algebraic Geometry (Bombay Colloquium 1984), Tata Institute, Oxford: Oxford University Press, 1987, pp. 1–33 4. Atiyah, M.F.: Magnetic monopoles and the Yang–Baxter equations. Int. J. Mod. Phys. A 6, 2761–2774 (1991) 5. Atiyah, M.F., Hitchin, N.J.: The Geometry and Dynamics of Magnetic Monopoles. Princeton, NJ: Princeton University Press, 1988 6. Atiyah, M.F., Murray, M.K.: Monopoles and Yang–Baxter equations. In: Further Advances in Twistor Theory, Vol. II. (eds.) Mason, L.J., Hughston, L.P., Kobak, P.Z., Essex: Longman, pp. 13–14 (1995) 7. Byrd, P.F., Friedman, M.D.: Handbook of Elliptic Integrals for Engineers and Physicists. BerlinHeidelberg-New York: Springer-Verlag, 1954 8. Ercolani, E., Sinha, A.: Monopoles and Baker functions. Commun. Math. Phys. 125, 385–416 (1989) 9. Fricke, R.: Elliptische Funktionen. In: Encyklopädie der mathematischen Wissenschaften II 2, Burkhardt, H., Wirtinger, W., Fricke, R., Hilb, E. (eds.) Leipzig: Teubner, 1913, pp. 177–348 10. Griffiths, P.: Linearizing flows and a cohomological interpretation of Lax equations. Amer. J. Math. 107, 1445–1484 (1985) 11. Griffiths, P., Harris, J.: Principles of Algebraic Geometry. New York: Wiley, 1978 12. Hitchin, N.J.: Monopoles and geodesics. Commun. Math. Phys. 83, 579–602 (1982) 13. Hitchin, N.J.: Magnetic monopoles with Platonic symmetry. In: Moduli of Vector Bundles (Sanda and Kyoto, 1994), ed. Maruyama, M.. Lecture Notes in Pure and Applied Mathematics 179, BerlinHeidelberg-New York: Springer-Verlag, 1996, pp. 55–63 14. Hitchin, N.J., Manton, N.S., Murray, M.K.: Symmetric monopoles. Nonlinearity 8, 661–692 (1995) 15. Houghton, C.J., Manton, N.S., Romão, N.M.: On the constraints defining BPS monopoles. Commun. Math. Phys. 212, 219–243, (2000) 16. Houghton, C.J., Sutcliffe, P.M.: Tetrahedral and cubic monopoles. Commun. Math. Phys. 180, 343–361 (1996) 17. Hurtubise, J.: SU (2) monopoles of charge 2. Commun. Math. Phys. 92, 195–202 (1983) 18. Jarvis, S., Norbury, P.: Zero and infinite curvature limits of hyperbolic monopoles. Bull. London Math. Soc. 29, 737–744 (1997) 19. Kiepert, L.: Wirkliche Ausführung der ganzzahligen Multiplication der elliptischen Functionen. J. reine angew. Math. 76, 21–33 (1872) 20. Klein, F.: Vorlesungen über das Ikosaeder und die Auflösung der Gleichung vom fünften Grade. Leipzig: Teubner, 1884 21. Maldacena, J.: The large N limit of superconformal field theories and supergravity. Adv. Theor. Math. Phys. 2, 231–252, (1998) 22. Manton, N.S.: A remark on the scattering of BPS monopoles. Phys. Lett. B 110, 54–56 (1982) 23. Manton, N.S., Sutcliffe, P.M.: Topological Solitons. Cambridge: Cambridge University Press, 2004 24. Murray, M.K., Norbury, P., Singer, M.A.: Hyperbolic monopoles and holomorphic spheres. Ann. Global Anal. Geom. 23, 101–128, (2003) 25. Murray, M., Singer, M.: Spectral curves of non-integral hyperbolic monopoles. Nonlinearity 9, 973–997 (1996) 26. Norbury, P.: Boundary algebras of hyperbolic monopoles. J. Geom. Phys. 51, 13–33 (2004) 27. Segal, G.B., Selby, A.: The cohomology of the space of magnetic monopoles. Commun. Math. Phys. 177, 775–787 (1996) 28. Sen, A.: Dyon-monopole bound states, self-dual harmonic forms on the multi-monopole moduli space, and S L(2, Z)-invariance of string theory. Phys. Lett. B 329, 217–221, (1994) 29. Sutcliffe, P.M.: Monopole zeros. Phys. Lett. B 376, 103–110 (1996) Communicated by G.W. Gibbons
Commun. Math. Phys. 270, 335–358 (2007) Digital Object Identifier (DOI) 10.1007/s00220-006-0152-8
Communications in
Mathematical Physics
Random Graph Asymptotics on High-Dimensional Tori Markus Heydenreich, Remco van der Hofstad Department of Mathematics and Computer Science, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands. E-mail:
[email protected];
[email protected] Received: 9 January 2006 / Accepted: 9 August 2006 Published online: 13 December 2006 – © Springer-Verlag 2006
Abstract: We investigate the scaling of the largest critical percolation cluster on a large d-dimensional torus, for nearest-neighbor percolation in sufficiently high dimensions, or when d > 6 for sufficiently spread-out percolation. We use a relatively simple coupling argument to show that this largest critical cluster is, with high probability, bounded above by a large constant times V 2/3 and below by a small constant times V 2/3 (log V )−4/3 , where V is the volume of the torus. We also give a simple criterion in terms of the subcritical percolation two-point function on Zd under which the lower bound can be improved to small constant times V 2/3 , i.e. we prove random graph asymptotics for the largest critical cluster on the high-dimensional torus. This establishes a conjecture by [1], apart from logarithmic corrections. We discuss implications of these results on the dependence on boundary conditions for high-dimensional percolation. Our method is crucially based on the results in [11, 12], where the V 2/3 scaling was proved subject to the assumption that a suitably defined critical window contains the percolation threshold on Zd . We also strongly rely on mean-field results for percolation on Zd proved in [17–20].
1. Introduction 1.1. The model. We consider Bernoulli bond percolation on the graph G, where G is either the hypercubic lattice Zd , or the finite torus Tr,d = {−r/2, . . . , r/2 − 1}d . For G = Zd , we consider two sets of bonds. In the nearest-neighbor model, two vertices x and y are linked by a bond whenever |x − y| = 1, whereas in the spread-out model, they are linked whenever 0 < x − y ≤ L. Here, and throughout the paper, we write · for the supremum norm, and | · | for the Euclidean norm on Zd . The integer parameter L is typically chosen large. We let each bond independently be occupied with probability p, or vacant otherwise. The resulting product measure is denoted by PZ, p , and the corresponding expectation EZ, p . We write {0 ←→ x} for the event that there
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exists a path of occupied bonds from the origin 0 to the lattice site x, and define τZ, p (x) := PZ, p (0 ←→ x)
(1.1)
to be the two-point function. We further write CZ (x) := {y ∈ Zd | x ←→ y} for the cluster or connected component of x, |CZ (x)| for the number of vertices in CZ (x) and χZ ( p) := x∈Zd τZ, p (x) = EZ, p |CZ (0)| for the expected cluster size. The degree of the graph, which we denote by , is thus = 2d in the nearest-neighbor case and = (2L + 1)d − 1 in the spread-out case. It is well-known that bond percolation on Zd in dimension d ≥ 2 obeys a phase transition, i.e. there exists a critical threshold pc (Zd ) ∈ (0, 1) such that pc (Zd ) = inf{ p : PZ, p (|CZ (0)| = ∞) > 0}. Furthermore, by the results in [3, 34], pc (Zd ) can be expressed as pc (Zd ) = sup{ p : χZ ( p) < ∞}. For G = Tr,d , we also consider two related settings: 1. The nearest-neighbor torus: an edge joins vertices that differ by 1 (modulo r ) in exactly one component. For d fixed and r large, this is a periodic approximation to Zd . Here = 2d for r ≥ 3. We study the limit in which r → ∞ with d > 6 fixed, but large. 2. The spread-out torus: an edge joins vertices x = (x1 , . . . , xd ) and y = (y1 , . . . , yd ) if 0 < maxi=1,...,d |xi − yi |r ≤ L (with | · |r the metric on Zr ). We study the limit r → ∞, with d > 6 fixed and L large (depending on d) and fixed. This gives a periodic approximation to range-L percolation on Zd . Here = (2L + 1)d − 1 provided that r ≥ 2L + 1, which we will always assume. We consider bond percolation on these tori with bond occupation probability p and write PT, p and ET, p for the product measure and corresponding expectation, respectively. We use the notation τT, p (·), χT ( p) and CT (·) analogously to the corresponding Zd -quantities. In this paper, we will investigate the size of the maximal cluster on Tr,d , i.e., |Cmax | := max |CT (x)|, x∈Tr,d
(1.2)
at the critical percolation threshold pc (Zd ). An alternative definition for the critical percolation threshold on the torus, denoted by pc (Tr,d ), was given in [11, (1.7)] as the solution to χT ( pc (Tr,d )) = λV 1/3 ,
(1.3)
where λ is a sufficiently small constant, and V = |Tr,d | = r d denotes the volume of the torus. The definition of pc (Tr,d ) in (1.3) is an internal definition only, due to the fact that [12] deals with rather general tori, for which an external definition (such as pc (Zd )) does not always exist. On the other hand, the internal definition in (1.3) assumes a priori mean-field behavior, and is therefore unsuitable outside this setting. On the high-dimensional torus Tr,d , we therefore have two sensible critical values, the externally defined pc (Zd ), and the internally defined pc (Tr,d ) in (1.3). One of the goals of this paper is to investigate how close these two critical values are. The most prominent example of percolation on a finite graph is the random graph, which is obtained by applying percolation to the complete graph. This has been first studied by Erd˝os and Rényi in 1960 [15]. They showed that, when p is scaled as (1 + ε)V −1 , there is a phase transition at ε = 0. For ε < 0, the size of the largest cluster is proportional to log V , whereas for ε > 0, it is proportional to V . For ε = 0, the size of the
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largest cluster divided by V 2/3 weakly converges to some (non-trivial) limiting random variable, while the expected cluster size is, as in (1.3), proportional to V 1/3 . This follows from results by Aldous [5]. See also [10] for results up to 1984, and [26, 27, 33] for references to subsequent work. We will refer to the V 2/3 -scaling as random graph asymptotics. In this paper, we study the size of the largest cluster on the torus for p = pc (Zd ). It has been shown by Borgs, Chayes, van der Hofstad, Slade and Spencer [11, 12] that, if pc (Zd ) = pc (Tr,d ) + O(V −1/3 ),
(1.4)
then, with probability at least 1 − O(ω−1 ), |Cmax | is in between ω−1 V 2/3 and ωV 2/3 as V → ∞, for ω ≥ 1 sufficiently large. Here, we write f (x) = O(g(x)) for functions f, g ≥ 0 and x converging to some limit, if there exists a constant C > 0 such that f (x) ≤ Cg(x) in the limit, and f (x) = o(g(x)) if g(x) = O( f (x)). Furthermore, we write f = (g) if f = O(g) and g = O( f ). Aizenman [1] conjectured that this random graph asymptotics holds for the maximal critical cluster in dimension d > 6, as we explain in more detail below. Also in [12] it was conjectured that (1.4) holds. By means of a coupling argument, we prove that a slightly weaker statement than (1.4) (with a logarithmic correction in the lower bound, see (1.6) below) indeed holds for d sufficiently large in the nearest-neighbor model, or d > 6 and L sufficiently large in the spread-out model. Furthermore, we give a criterion which we believe to hold, and which implies (1.4) without logarithmic corrections. Note that all our results assume that d is large in the nearest-neighbor model or d > 6 and L large in the spread-out model. That is, we require the torus to be in some sense high-dimensional. We do believe that the results hold for all d > 6 and L ≥ 1, however, the proof relies on various lace expansion results, which require that the degree is large. On the other hand, we do not expect these asymptotics to be true for d ≤ 6. Aizenman [1] studied a similar question, but now for percolation on a box of width r under bulk boundary conditions, where clusters are defined to be the intersection of the box {−r/2, . . . , r/2 − 1}d with clusters in the infinite lattice (and thus clusters need not be connected within the box). Aizenman assumed that the probability, at criticality, that x is connected to the origin is bounded above and below by constants times x−(d−2) . This assumption was established in [19] for the spread-out model for d > 6 and sufficiently large, but finite L ≥ 1, and in [18] for the nearest-neighbor model above 19 dimensions. Aizenman showed that, under this condition on the two-point function, the size of the largest connected component under bulk boundary conditions is, with high probability, bounded from above by a constant times r 4 log r , and bounded from below by εr r 4 for any sequence εr → 0 as r → ∞. Furthermore, he conjectures that the r 4 -scaling for the size of the largest cluster holds for dimension d > 6 also under free boundary conditions (where no connections outside the box are allowed), but changes to V 2/3 = r 2d/3 r 4 under periodic boundary conditions. This indicates the importance of boundary conditions at criticality in high dimensions. We will further elaborate on the role of boundary conditions in Sect. 6.
1.2. Results. Our first result gives asymptotic bounds on the size of the largest cluster. Theorem 1.1. Fix d > 6 and L sufficiently large in the spread-out case, or d sufficiently large for nearest-neighbor percolation. Then there exist constants b1 , b2 , C > 0, such
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that for all ω1 ≥ C and ω2 ≥ 1, PT, pc (Zd ) ω1−1 V 2/3 (log V )−4/3 ≤ |Cmax | ≤ ω2 V 2/3 ≥1−
b1 3/2 ω1 (log V )2
−
b2 ω2
as r → ∞.
(1.5)
The constant b1 can be chosen as 288 · 1203 , and b2 equal to b6 in [11, Theorem 1.3]. To prove Theorem 1.1, we will use a coupling argument relating χT ( p) and χZ ( p) to show that there exists a constant ≥ 0 such that, when r → ∞, pc (Tr,d ) −
−1/3 (log V )2/3 ≤ pc (Zd ) ≤ pc (Tr,d ) + V −1/3 . V
(1.6)
Relying on results in [11], (1.6) implies (1.5). Inequality (1.5) implies that |Cmax |V −2/3 is a tight random variable, but it does not rule out that |Cmax |V −2/3 → 0 as V → ∞. Our method is crucially based on the results in [11, 12], but we also rely on mean-field results for percolation on Zd by Hara [17, 18], Hara and Slade [20], and Hara, van der Hofstad and Slade [19]. Each of these papers relies on the lace expansion, but the lace expansion will not be used in this paper. The lecture notes by Slade [38] and Hara and Slade [21] provide a general introduction to the lace expansion and its role in proving mean-field critical behavior for percolation and related models. Note that in [11, 12], pc (Tr,d ) was defined as in (1.3). The results in [11, 12], however, do not establish rigorously that the exponent 1/3 in (1.3) is the only correct choice. Indeed, [11, 12] suggest that a smaller exponent would also do, since the supercritical results proved there are not sufficiently sharp. Theorem 1.1 shows that, at least in terms of the power of V , the scaling of |Cmax | at pc (Zd ) and at pc (Tr,d ) is identical, thus establishing that on Zd the choice (1.3) is appropriate. Unfortunately, the lower bound in Theorem 1.1 does not quite meet the upper bound. Under a condition on the percolation two-point function, we can prove the matching lower bound. To state this result, we introduce the quantity
χ Z ( p, r ) := sup y
τZ, p (z),
(1.7)
r
z ∼y, z≥ r2
r
where we write that x ∼ y when x (mod r ) = y (mod r ). We will call x and y r r-equivalent when x ∼ y. Theorem 1.2. Under the assumptions in Theorem 1.1, suppose that there exists a K > 0 such that, for p = pc (Zd ) − K −1 V −1/3 and some C K > 0, the bound χ Z ( p, r ) ≤ C K V −2/3 holds. Then, for all ω ≥ 1 and b equal to b6 in [11, Theorem 1.3], 1 2/3 b 2/3 ≥1− . PT, pc (Zd ) V ≤ |Cmax | ≤ ωV ω ω
(1.8)
(1.9)
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Inequality (1.9) implies that |Cmax |V −2/3 is tight, and that each possible weak limit along any subsequence is non-zero. The result in (1.9) combined with the results in [11] would indicate that the scaling of |Cmax | at pc (Tr,d ) and at pc (Zd ) agree, thus showing that there is no significant difference between the internally and externally defined critical values. Analogously to Theorem 1.1, we will show that when (1.8) holds, there exists a constant ≥ 0 such that, when r → ∞, −1/3 V ≤ pc (Zd ) ≤ pc (Tr,d ) + V −1/3 , (1.10) and deduce (1.9) using [11]. We strongly believe that (1.8) holds. Indeed, (1.8) follows when the two-point function is sufficiently smooth. For example, the condition pc (Tr,d ) −
max r
z≥ 2 , x∈Tr,d
τZ, p (z) ≤C τZ, p (z + x)
(1.11)
for some positive constant C, where we consider Tr,d = {−r/2, . . . , r/2 − 1}d as a subset of Zd , implies that C r (1.12) τZ, p (z + x) for z ≥ . τZ, p (z) ≤ V 2 x∈Tr,d
Note that, for every y ∈ Zd , we have that all functions f :
Zd
χ Z ( p, r ) = sup y
≤
→ R. Consequently, τZ, p (z)
r
z ∼y
x∈Tr,d
f (z + x) =
x∈Zd
f (x) for
r
z ∼y, z≥ r2
C sup V y
z ∼y, z≥ r2 x∈Tr,d r
τZ, p (z + x) ≤
C C τZ, p (x) = χZ ( p). V V d
(1.13)
x∈Z
Thus, for p = pc (Zd ) − K −1 V −1/3 , we obtain by the fact that γ = 1 (see [4 and 20], or Theorem 3.3 below) that χ Z ( p, r ) is bounded from above by a constant multiple of V −2/3 . 1.3. Related results. In this section we discuss the relation between Theorems 1.1 and 1.2 and the literature. Hara and Slade [22, 23] study the geometry of large critical clusters on the rescaled lattice. Under the conditions of Theorem 1.1, they show that critical clusters with size of order n on the lattice rescaled by n −1/4 converge to integrated super-Brownian excursion as n → ∞. Together with the results in [20], and using [3, 4], these papers prove that various critical exponents for percolation exist and take on mean-field values. Borgs, Chayes, Kesten and Spencer [13, 14] consider the largest cluster in a finite box of width r under free boundary conditions, i.e., clusters are connected only within the box. They show that, for p = pc (Zd ), the largest critical cluster scales like V δ/(1+δ) , where the critical exponent δ is defined by
(1.14) P pc (Zd ) |CZ (0)| ≥ n ≈ n −1/δ as n → ∞,
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under some conditions related to the so-called scaling and hyperscaling postulates. The hyperscaling postulates are proven in dimension d = 2, and are widely believed to hold up to the upper critical dimension 6. For the mean-field value δ = 2, proved in [22, 23], we would obtain the V 2/3 asymptotics. However, in [13, 14], it was assumed that crossing probabilities of a cube of dimensions (r, 3r, . . . , 3r ) remain uniformly bounded away from 1 as r → ∞. In high dimensions, Aizenman [1] proves that any cube {0, . . . , r }d has crossings with high probability, so that the results in [13, 14] do not apply. Also, in high dimensions, the hyperscaling relations are not valid. More specifically, one hyperscaling relation is that δ−1 . (1.15) δ+1 Under the conditions of Theorem 1.1, due to [6, 18–20], we have that η = 0, δ = 2, so that this hyperscaling relation fails for d > 6. Theorems 1.1 and 1.2 study the scaling of the largest critical percolation cluster on the high-dimensional torus. These results indicate that the scaling limit of the largest critical cluster should be described by |Cmax |V −2/3 . We conjecture that, at p = pc (Zd ), the random variables |Cmax |V −2/3 converge as r → ∞ to some (non-trivial) limiting distribution. It would be of interest to investigate whether, if the rescaled largest cluster converges, the limit law is identical to the limit of |Cmax |r −2/3 for the largest cluster of the random graph on r vertices, as identified by Aldous [5]. The convergence of |Cmax |V −2/3 would describe part of the incipient infinite cluster (IIC) for percolation on the torus, as described by Aizenman [1]. Aizenman’s IIC is closely related to the scaling limit of percolation on large cubes, see [1, Sect. 5]. Mind also the warning at the bottom of [1, p. 553]. Another approach to the incipient infinite cluster is described by Kesten [30]. Indeed, Kesten investigates the local configuration close to the origin in Z2 , conditioned on the critical cluster of the origin to be infinite. Since, at criticality, the cluster of the origin is infinite with probability 0, an appropriate limit needs to be taken. Kesten offers two alternatives: 2−η =d
(i) To condition the origin to be connected to infinity at p > pc (Z2 ) and take the limit p pc (Z2 ). (ii) To condition the critical cluster of the origin to be connected to the boundary of the box {−r, . . . , 0, . . . , r }2 and take the limit r → ∞. Kesten proves that both limits exist and are equal. This limit is Kesten’s incipient infinite cluster. Kesten was motivated to describe this IIC in order to study random walk on large critical clusters [31], for which physicists have performed simulations showing subdiffusive behavior. Járai [28, 29] extended these results, and proved that several other natural conditioning and limiting schemes give the same limit. In one of these constructions, Járai takes a uniform point in the largest critical cluster on a box {0, . . . , r − 1}d , shifts it to the origin and takes the limit r → ∞. In [25], the proof of existence of the IIC was extended to high-dimensional percolation, under the assumptions of Theorems 1.1 and 1.2. The proof in [25] follows the proof in [24], where the IIC was constructed for spread-out oriented percolation above 4 spatial dimensions. We conjecture that, as r → ∞, the law of local configurations around a uniform point in Cmax at criticality converges to the IIC as constructed in [25]. This result would give a natural link between the scaling limit of critical percolation on a large box in [1] and Kesten’s notion of the IIC in [30].
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We recall that, following the conjecture in [1], the size of the largest connected component |Cmax | on the cube {0, . . . , r −1}d under free boundary conditions scales like r 4 (as under bulk boundary conditions), in contrast to the V 2/3 -scaling under periodic boundary conditions. Such qualitatively different behavior between free and periodic boundary conditions has also been observed when studying loop-erased random walks and uniform spanning trees on a finite box in high dimensions, as we will explain now. Choose two uniform points x and y from the d-dimensional box of side length r , with d > 4. We are interested in the graph distance between these two points on a uniform spanning tree. Pemantle [35] showed that this graph distance has the same distribution as the length of a loop-erased random walk starting in x and stopped when reaching y. Loop-erased random walk above 4 dimensions converges to Brownian motion (cf. [32, Sect. 7.7]), that is, it scales diffusively. This suggests that, under free boundary conditions, the graph distance between x and y scales like r 2 . On the other hand, the combined results of Benjamini and Kozma [7] and Peres and Revelle [36] show that the distance between x and y on a uniformly chosen spanning tree on the torus Tr,d is of the order V 1/2 = r d/2 > r 2 for d > 4. Schweinsberg [37] identifies the logarithmic correction for the scaling on the 4-dimensional torus. 1.4. Organization. This paper is organized as follows. In Sect. 2, we state a coupling result that is crucial for all our subsequent bounds. In Sect. 3, we collect the main results from previous work that are used in our arguments. In Sect. 4, we prove the upper bound in Theorem 1.1. In Sect. 5, we prove the complementary lower bounds in Theorems 1.1 and 1.2. Finally, in Sect. 6, we discuss how the growth of the maximal cluster depends on the precise boundary conditions. 2. A Coupling Result for Clusters on the Torus and Zd In this section, we prove that the cluster size for percolation on the torus is stochastically smaller than the one on Zd by a coupling argument. We fix p, and omit the subscript p from the notation. We use subscripts Z and T to denote objects on Zd and Tr,d , respectively. The goal of this section is to give a coupling of the Tr,d -cluster and the Zd -cluster of the origin. This will be achieved by constructing these two clusters simultaneously from a percolation configuration on Zd , as we explain in more detail now. The basic idea is that, on any graph, it is well-known that the law of a cluster C(0) can be described by subsequently exploring the bonds one can reach from 0. We will first describe this exploration of a cluster in some detail, before giving the coupling, which is described by a more elaborate way of exploring the percolation clusters on the torus and on Zd simultaneously from a percolation configuration on Zd . The exploration process is defined in terms of colors of the bonds. Initially, all bonds are uncolored, which means that they have not yet been explored. During the exploration process we will color the bonds black if they are found to be occupied, and white if they are found to be vacant. Furthermore, we distinguish between active and inactive vertices. Initially, only the origin 0 is active, and all other vertices are inactive. We now explore the bonds in the graph according to the following scheme. We order the vertices in an arbitrary way. Let v be the smallest active vertex. Now we explore (and color) all uncolored bonds that have an endpoint in v, i.e., we make the bond black with probability p and white with probability 1 − p, independently of all other bonds. In case
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we have assigned the black color, we set the vertex at the other end of the bond active (unless it was already active and none of its neighboring edges are now uncolored, in which case we make it inactive). In particular, the active vertices are those vertices that are part of a black bond, as well as an uncolored bond. Finally, after all bonds starting at v have been explored, we set v inactive. We repeat doing so until there are no more active vertices. In the latter case, the exploration process is completed, i.e., there are no more black bonds that share a common endpoint with an uncolored bond. The cluster C(0) is equal to the set of vertices that are part of the black bonds. When the graph is finite, then this procedure always stops. When the graph is infinite, then the exploration process continues forever precisely when |C(0)| = ∞. This completes the exploration of a single cluster on a general graph. The exploration of a single cluster will be extended to explore the cluster on the torus CT (0) and the cluster on the infinite lattice CZ (0) simultaneously from a percolation configuration on Zd . For CZ (0), the result of the exploration will be identical to the exploration of a single cluster described above. The related cluster CT (0) is a subset of all vertices that are r -equivalent to vertices part of a black bond. The main result in this section is Proposition 2.1, whose proof gives the details of this simultaneous construction of the two clusters. T To state the result, we need some notation. For x, y ∈ Tr,d , we write x ←→ y when x is connected to y in the percolation configuration on the torus, while, for x, y ∈ Zd , we Z write x ←→ y when x is connected to y in the percolation configuration on Zd . Also, we call two distinct bonds {x1 , y1 } and {x2 , y2 } r -equivalent if there exists an element z ∈ Zd such that {x1 , y1 } = {x2 + r z, y2 + r z}. We sometimes abbreviate r -equivalent to equivalent. For a directed bond b = (x, y), we write b = x and b = y, and for two bonds r b1 = (x1 , y1 ) and b2 = (x2 , y2 ), we write b1 ∼ b2 when (x1 , y1 ) = (x2 + r z, y2 + r z) for some z ∈ Zd . For A and B increasing events, we denote by A ◦ B the event that A and B occur on disjoint sets of bonds, see [9]. Proposition 2.1 (The coupling). Consider nearest-neighbor percolation for r ≥ 3 or spread-out percolation for r ≥ 2L + 1, in any dimension. There exists a probability law PZ,T on the joint space of Zd - and Tr,d -percolation such that, for all events E, PZ,T (CT (0) ∈ E) = PT (CT (0) ∈ E),
PZ,T (CZ (0) ∈ E) = PZ (CZ (0) ∈ E), (2.1)
and PZ,T -almost surely, for all x ∈ Tr,d , T
{0 ←→ x} ⊆
Z
{0 ←→ y}.
(2.2)
r
y∈Zd : y ∼x r
In particular, |CT (0)| ≤ |CZ (0)|. Moreover, for x ∼ y, and PZ,T -almost surely, Z
T
{0 ←→ y} ∩ {0 ←→ x}c (2.3) Z Z Z Z ⊆ (0←→z) ◦ (z←→b1 ) ◦ (z←→b2 ) ◦ (b2 is Z − occ.) ◦ (b2←→y). r d b1 =b2 : b1 ∼b2 z∈Z
Equation (2.2) will be used to conclude that the expected cluster size on Tr,d is bounded from above by the one on Zd . In order to prove our main results, we use (2.3) to prove a related lower bound on the expected cluster size on Tr,d in terms of the one on Zd .
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See Sects. 4 and 5 for details. The inequality on the cluster sizes of Tr,d - and Zd -percolation also follows from the coupling used by Benjamini and Schramm [8, Theorem 1]. However, (2.3) does not follow immediately from their work. Proof of Proposition 2.1. The exploration of a single cluster, as described above, will be generalized to construct CT (0) and CZ (0) simultaneously from a percolation configuration on Zd . The difference between percolation on the torus and on Zd can be summarised by saying that, on the torus, r -equivalent bonds have the same occupation status, while on Zd , equivalent and distinct bonds have an independent occupation status. For the exploration of the torus CT (0), we have to make sure that we explore equivalent bonds at most once. We therefore introduce a third color, gray, indicating that the bond itself has not been explored yet, but one of its equivalent bonds has. Therefore, at each step of the exploration process, we have 4 different types of bonds on Zd : • • • •
uncolored bonds, which have not been explored yet; black bonds, which have been explored and found to be occupied; white bonds, which have been explored and found to be vacant; gray bonds, of which an equivalent bond has been explored.
As in the exploration of a single cluster, we number the vertices of Zd in an arbitrary way, and start with all bonds uncolored and only the origin active. Then we repeat choosing the smallest active bond, and explore all uncolored bonds containing it. However, after exploring a bond (and coloring it black or white), we color all bonds that are r -equivalent to it gray. Again, this exploration is completed when there are no more active vertices. This is equivalent to the fact that there are no more black (and therefore occupied) bonds sharing a common endpoint with an uncolored bond. The exploration process of CT (0) must be completed at some point, since the number of bonds within CT (0) is finite and vertices turn active only if a bond containing it is explored and is found to be occupied. We call the result of this exploration process the T-exploration. The cluster CT (0) consists of all vertices in Tr,d that are contained in a bond that is r -equivalent to a black bond. However, we have embedded the cluster CT (0) into Zd , which will be useful when we also wish to describe the related cluster CZ (0). For CZ (0), the exploration of the cluster is similar, but there are no gray bonds. We start with the final configuration of the T-exploration, and set all vertices that are a common endpoint of a black bond and a gray bond active. Then we make all gray bonds uncolored again. From this setting we apply the coloring scheme that colors the uncolored bonds that contain an active vertex. The coloring scheme is the one for the exploration cluster on Zd , where no gray bonds are created. Again, we perform this exploration until there are no more active vertices, i.e., no more black bonds attached to uncolored bonds. The result is called the Z-exploration. In particular, the black bonds in the T-exploration are a subset of the black bonds in the Z-exploration, which proves T Z 1 r {0 ←→ y}, and hence |CT (0)| ≤ |CZ (0)|. that {0 ←→ x} ⊆ y ∼x Z
We now show (2.3). When {0 ←→ y} occurs, then picture all Z-occupied paths from r T 0 to y in mind. Since x ∼ y and {0 ←→ x}c occurs, each of these paths Z-connecting 0 and y should contain a bond which is T-vacant. Fix such a (self-avoiding) path 1 To obtain a coupling for the full percolation configurations on T d r,d and Z , we can finally let all bonds that have not been explored be independently occupied with probability p, both in Tr,d and in Zd , indepen-
dently of each other. However, we do not rely on the coupling of the percolation configuration, but only on the coupling of the clusters CT (0) and CZ (0).
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Fig. 1. Illustration of the right-hand side in (2.3). The bond b1 has been explored first and is found to be Z-vacant and T-vacant. The bond b2 is r -equivalent and thus has been explored in the Z-exploration only, it is T-vacant (determined by b1 during the T-exploration), but Z-occupied
ω : 0 ←→ y that is Z-occupied, and denote by b2 the first bond that is T-vacant, but Z Z Z-occupied, so that (0 ←→ b2 ) ◦ (b2 is Z − occ.) ◦ (b2 ←→ y). Due to our coupling, this implies that there exists a previously explored bond b1 that is r -equivalent to b2 , which is (T- and Z-) vacant. This, in turn, implies that there exists a vertex z that is Z visited by ω such that (z ←→ b1 ) without using any of the bonds in ω. Therefore, the Z
Z
Z
Z
event (0 ←→ z) ◦ (z ←→ b1 ) ◦ (z ←→ b2 ) ◦ (b2 is Z − occ.) ◦ (b2 ←→ y) occurs (see Fig. 1). 3. Previous Results In this section we cite results of previous works that will be used in our analysis later on. We make essential use of results by Borgs, Chayes, van der Hofstad, Slade and Spencer [11, 12] for percolation on Tr,d , which we cite in the following two theorems. Theorem 3.1 (Subcritical phase). Under the conditions in Theorem 1.1, for λ sufficiently small and any q ≥ 0, −1 −1 λ−1 V −1/3 + q ≤ χT pc (Tr,d ) − −1 q ≤ λ−1 V −1/3 + q/2 . (3.1) Also, for p = pc (Tr,d ) − −1 q and ω ≥ 1, −1 36 χT3 ( p) χ 2 ( p) ≥ 1+ . PT, p |Cmax | ≥ T 3600 ω ωV Instead of the upper bound in (3.1), we will mainly use the cruder bound 2 χT pc (Tr,d ) − −1 q ≤ . q
(3.2)
(3.3)
Theorem 3.2 (Scaling window). Assume the conditions in Theorem 1.1. Let λ > 0 and < ∞. Then there is a finite positive constant b (depending on λ and ) such that, for p = pc (Tr,d ) + −1 with | | ≤ V −1/3 and ω ≥ 1, b PT, p ω−1 V 2/3 ≤ |Cmax | ≤ ωV 2/3 ≥ 1 − . (3.4) ω
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Theorems 3.1 and 3.2 have been proven in [11, Theorems 1.2 and 1.3] subject to the triangle condition. Using the lace expansion, the triangle condition was established in [12, Prop. 1.2 and Theorem 1.3] for d > 6 and sufficient spread-out or d sufficiently large for nearest-neighbor percolation. Furthermore, we will use two properties of Zd -percolation in high dimensions formulated in the following two theorems. Theorem 3.3 (Expected cluster size). Under the conditions in Theorem 1.1, there exists a positive constant Cχ , such that 1 Cχ
≤ χZ ( p) ≤
pc (Zd ) − p pc (Zd ) − p
as p pc (Zd ).
(3.5)
In other words, the critical exponent γ exists and takes on the mean-field value 1. This theorem is proven in [20] and [4]. According to [21], d ≥ 19 is sufficient for the nearest-neighbor model. We also need (sub)critical bounds on the decay of the connectivity function. These are expressed in the following theorem. Theorem 3.4 (Bounds on the two-point function). Under the conditions in Theorem 1.1, there exist constants cτ , Cτ , cξ , Cξ > 0 such that, Cτ cτ ≤ τZ, pc (Zd ) (x) ≤ . d−2 (|x| + 1) (|x| + 1)d−2
(3.6)
In other words, the critical exponent η exists and takes the value 0. Furthermore, for any p < pc (Zd ), τZ, p (x) ≤ e
x − ξ( p)
,
(3.7)
where the correlation length ξ( p) is defined by ξ( p)−1 = − lim
n→∞
1 log PZ, p (0, . . . , 0) ←→ (n, 0, . . . , 0) , n
(3.8)
and satisfies cξ ( pc (Zd ) − p)−1/2 ≤ ξ( p) ≤ Cξ ( pc (Zd ) − p)−1/2 .
(3.9)
The power law bound (3.6) is due to Hara [18] for the nearest-neighbor case, and to Hara, van der Hofstad and Slade [19] for the spread-out case. For the exponential bound (3.7), see e.g. Grimmett [16, Prop. 6.47]. Hara [17] proves the bound (3.9). 4. The Upper Bound on the Maximal Critical Cluster The following corollary establishes the upper bound on pc (Zd ) and the upper bound on |Cmax | in Theorem 1.1. For the proof, we first use Proposition 2.1 to obtain that χT ( p) ≤ χZ ( p). Then we use (3.5) to turn this into relations between pc (Tr,d ) and
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pc (Zd ). Finally, using Theorem 3.2 we obtain a bound on |Cmax |. We now present the details of the proof. Corollary 4.1. Under the conditions of Theorem 1.1 there exists a constant ≥ 0 such that, when r → ∞, pc (Zd ) ≤ pc (Tr,d ) +
−1/3 V .
(4.1)
Consequently, for b as in Theorem 3.2 and all ω ≥ 1, b PT, pc (Zd ) |Cmax | ≤ ωV 2/3 ≥ 1 − . ω
(4.2)
Proof. By Proposition 2.1, χT ( pc (Tr,d )) ≤ χZ ( pc (Tr,d )).
(4.3)
When pc (Tr,d ) ≥ pc (Zd ), then (4.1) holds with = 0, so we will next assume that pc (Tr,d ) < pc (Zd ). Using (1.3), (4.3) and (3.5), we obtain that λV 1/3 ≤
Cχ
, pc (Zd ) − pc (Tr,d )
(4.4)
Cχ −1/3 V , λ
(4.5)
so that pc (Zd ) ≤ pc (Tr,d ) +
which is (4.1) with = λ−1 Cχ . The bound (4.2) follows from the fact that, with p = pc (Tr,d ) + −1 V −1/3 ≥ pc (Zd ) by (4.1), b PT, pc (Zd ) |Cmax | ≤ ωV 2/3 ≥ PT, p |Cmax | ≤ ωV 2/3 ≥ 1 − ω
(4.6)
for some constant b > 0 depending on λ and , and all ω ≥ 1. We have used Theorem 3.2 in the last bound. 5. The Lower Bound on the Maximal Critical Cluster In this section, we will bound χT ( p) − χZ ( p) from below. First we use the results and framework of Sect. 2 to prove such a lower bound in terms of χ Z ( p, r ). Subsequently, assuming (1.8), we use Theorem 3.2, Corollary 4.1 and the bounds (3.3) and (3.4) to prove Theorem 1.2. Some more work is required if we do not assume (1.8). We first deduce a bound on χ Z ( p, r ) using the bounds in Theorem 3.4. Then we use this bound together with Lemma 5.1 to show Theorem 1.1 with the same ingredients as for the proof of Theorem 1.2.
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5.1. A lower bound on χT ( p) in terms of χZ ( p) Lemma 5.1. For all p ∈ [0, 1] and r ≥ 3 in the nearest-neighbor model or r ≥ 2L + 1 in the spread-out model,
χT ( p) ≥ χZ ( p) 1 − χZ ( p) χ Z ( p, r ) − p 2 χZ ( p)2 χ Z ( p, r ) . (5.1) Proof. The bound (5.1) will be achieved by comparing the two-point functions on the torus and on Zd . We will write P = PZ,T and omit the percolation parameter p from the notation. Using (2.2), we write ⎞ ⎛ ⎞ ⎛ Z Z T ⎟ ⎜ ⎟ ⎜ τT (x) = P ⎝ {0 ←→ y}⎠ − P ⎝ {0 ←→ y} ∩ {0 ←→ x}c ⎠ . (5.2) r
r
y∈Zd :y ∼x
y∈Zd :y ∼x
We further bound, using inclusion-exclusion, ⎛ ⎞ Z 1 Z ⎜ ⎟ P⎝ {0 ←→ y}⎠ ≥ P 0←→y − 2 r r y∈Zd :y ∼x
y∈Zd :y ∼x
Z P 0←→y1 , y2 , r
y1 = y2 ∈Zd :y1 ,y2 ∼x
(5.3) so that
τT (x) ≥
τZ (y) − r
y∈Zd :y ∼x
1 2
⎛
⎜ −P⎝
Z P 0 ←→ y1 , y2 r
y1 = y2 ∈Zd :y1 ,y2 ∼x
⎞
Z T ⎟ {0 ←→ y} ∩ {0 ←→ x}c ⎠ .
(5.4)
r
y∈Zd :y ∼x
Summation over x ∈ Tr,d and using that
x∈Tr,d
r
y∈Zd :y ∼x
=
y∈Zd
yields that
χT ( p) ≥ χZ ( p) − χT,1 ( p) − χT,2 ( p),
(5.5)
where χT,1 ( p) =
1 2
Z P 0 ←→ y1 , y2 ,
(5.6)
r y1 = y2 ∈Zd :y1 ∼y2
⎞
⎛
χT,2 ( p) =
x∈Tr,d
⎜ P⎝
Z T ⎟ {0 ←→ y} ∩ {0 ←→ x}c ⎠ .
(5.7)
r y∈Zd :y ∼x
r
Here we use that the sum over x and over y1 and y2 such that y1 , y2 ∼ x is the same as r the sum over y1 and y2 such that y1 ∼ y2 . We are left to bound χT,1 ( p) and χT,2 ( p). We start by bounding χT,1 ( p). Using the tree-graph inequality (see [4]),
Z τZ (z) τZ (x − z) τZ (y − z), (5.8) P 0 ←→ x, y ≤ z∈Zd
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(for which the proof easily follows using the BK-inequality [9]), we obtain 1 χT,1 ( p) ≤ τZ (z) τZ (y1 − z) τZ (y2 − z) 2 z r y1 = y2 :y1 ∼y2
1 = χZ ( p) 2
τZ (y1 ) τZ (y2 ).
(5.9)
r y1 = y2 :y1 ∼y2
Here, and in the remainder of the proof, all sums over vertices will be over Zd unless written explicitly otherwise. r Since y1 = y2 and y1 ∼ y2 , we must have that y1 ≥ r2 or y2 ≥ r2 . By symmetry, these give the same contributions, so that, τZ (y1 ) τZ (y2 ) ≤ χZ ( p)2 χ Z ( p, r ), (5.10) χT,1 ( p) ≤ χZ ( p) y1
r
y2 :y1 ∼y2 ,y2 ≥ r2
where we recall (1.7). We are left to prove that χT,2 ( p) ≤ p 2 χZ3 ( p) χ Z ( p, r ). We use (2.3) and note that the right-hand side of (2.3) does not depend on x. Since r = y∈Zd , this brings us to x∈Tr,d y∈Zd :y ∼x ⎛ ⎜ Z Z Z χT,2 ( p) ≤ P⎝ (0 ←→ z) ◦ (z ←→ b1 ) ◦ (z ←→ b2 ) y∈Zd
r d b1 =b2 :b1 ∼b2 z∈Z
⎞
Z ⎟ ◦ (b2 is Z − occ.) ◦ (b2 ←→ y)⎠ .
(5.11)
Therefore, by the BK-inequality [9], Z Z Z χT,2 ( p) ≤ P (0 ←→ z) ◦ (z ←→ b1 ) ◦ (z ←→ b2 ) y,z
r
b1 =b2 :b1 ∼b2 Z
≤ p
y,z
◦(b2 is Z − occ.) ◦ (b2 ←→ y)
τZ (z) τZ (b1 − z) τZ (b2 − z) τZ (y − b2 ).
(5.12)
r
b1 =b2 :b1 ∼b2
We can perform the sums over z, y to obtain, with bi = bi − z, χT,2 ( p) ≤ p χZ ( p)2 τZ (b1 ) τZ (b2 ) = p 2 χZ ( p)2 b1 =b2 :
r b1 ∼b2
τZ (u) τZ (v), r
u =v : u ∼v
(5.13) where the factor 2 arises from the number of choices for b1 and b2 for fixed b1 and b2 . Therefore, by (5.10), we arrive at the bound Z ( p, r ). χT,2 ( p) ≤ p 2 χZ3 ( p) χ The bounds (5.10) and (5.14) complete the proof of (5.1).
(5.14)
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5.2. Proof of the main results Proof of Theorem 1.2. We assume (1.8) and take p = pc (Zd ) − K 1 −1 V −1/3 for K 1 sufficiently large. Choose V sufficiently large to ensure that p > 0. When K 1 ≥ K , the bound (1.8) still holds. We obtain from Lemma 5.1 together with Theorem 3.3 and (1.8) that χT ( p) ≥ K 1−1 V 1/3 1 − Cχ C K K 1−1 V −1/3 − p 2 Cχ2 C K K 1−2 ≥ c K 1 V 1/3 , (5.15) where c K 1 is chosen appropriately. Let K 1 be so large that p < pc (Tr,d ), which can be done by (4.1). Then, by (3.3), 2 ≥ χT ( p) ≥ c K 1 V 1/3 , ( pc (Tr,d ) − pc (Zd ) + K 1 V −1/3 ) so that
pc (Zd ) ≥ pc (Tr,d ) + K 1 −
2 cK1
V −1/3 ,
(5.16)
(5.17)
−1
which is (1.10) with = 2 c K 1 − K 1 ∨ 0. This, together with (4.1), permits using Theorem 3.2. By doing so, we obtain that, for p equal to the right-hand side of (5.17), PT, pc (Zd ) ω−1 V 2/3 ≤ |Cmax | ≤ ωV 2/3 ≥ PT, p ω−1 V 2/3 ≤ |Cmax | ≤ ωV 2/3 ≥ 1−
b . ω
(5.18)
This completes the proof of Theorem 1.2. Unfortunately, we cannot quite prove (1.8) so we will give cruder upper bounds on χ Z ( p, r ). This is the content of the following lemma: Lemma 5.2. Under the conditions in Theorem 1.1, choose K sufficiently large, and let R = K (log V )( pc (Zd ) − p)−1/2 . Then for all p ≤ pc (Zd ) − K −1 V −1/3 , χ Z ( p, r ) ≤
Cχ˜ R 2 V
(5.19)
for some constant Cχ˜ > 0. Note that it is here where the power of log V comes into play. Proof. For p ≤ pc (Zd ) − K −1 V −1/3 , we bound χ Z ( p, r ) = sup τZ, p (z) ≤ sup τZ, p (z) + sup y
r
z ∼y,z≥ r2
y
y
r
z ∼y,z≥R
τZ, p (z).
r
z ∼y, r2 ≤z≤R
(5.20) We start with the second contribution, for which we use (3.6). Since z ≥ r2 , we have that Cτ Cτ C1 ≤ (5.21) τZ, p (z) ≤ τZ, pc (Zd ) (z) ≤ d−2 (|z| + 1) V (|z + x| + 1)d−2 x∈Tr,d
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for constants Cτ , C1 > 0, where C1 depends on the dimension d only. Therefore, Cτ C1 sup τZ, p (z) ≤ sup V y (|z + x| + 1)d−2 y r r x∈Tr,d z ∼y:z≤R
z ∼y: r2 ≤z≤R
≤
C1 V
z:z≤2R
Cτ C2 R 2 , ≤ (|z| + 1)d−2 V
(5.22)
where the positive constant C2 depends on d and L only. For the sum due to z ≥ R, we use (3.7) and (3.9) to see that sup τZ, p (z) ≤ sup exp −Cξ−1 z ( pc (Zd ) − p)1/2 . (5.23) y
y
r
z ∼y, z≥R
r
z ∼y, z≥R
Since z ≥ d1 (|z 1 | + · · · + |z d |) for all z = (z 1 , . . . , z d ) ∈ Zd , (5.23) can be further bounded from above by exp −Cξ−1 z ( pc (Zd ) − p)1/2 z:z≥R
⎛
≤⎝
⎞d 1/2 ⎠ exp −(dCξ )−1 |z 1 | pc (Zd ) − p
z 1 :|z 1 |≥R
≤ C3 pc (Z ) − p d
−d/2
1/2 R d , pc (Z ) − p exp − Cξ
(5.24)
for some constant C3 > 0. Since R = K (log V )( pc (Zd ) − p)−1/2 , the exponential term
−d/2 ≤ can be bounded by V −K /Cξ . Furthermore, by our choice of p, pc (Zd ) − p
−d/2 d/6 −1 K V . Choose K so large that K /Cξ − d/6 > 1. Then the upper bound is of the order o(V −1 ). This, together with (5.22), proves the claim. We next use Lemma 5.2 to prove the lower bound in (1.6): Lemma 5.3. Under the conditions in Theorem 1.1, there exists a constant ≥ 0 such that (5.25) pc (Zd ) ≥ pc (Tr,d ) − V −1/3 (log V )2/3 . Proof. By Lemma 5.2, for all p ≤ pc (Zd ) − K −1 V −1/3 , χ˜ Z ( p, r ) ≤
Cχ˜ K 2 (log V )2
. V pc (Zd ) − p
(5.26)
With Theorem 3.3, this can be further bounded as χ˜ Z ( p, r ) ≤ Cχ˜ K 2
(log V )2 χZ ( p). V
Then, by Lemma 5.1 and χZ ( p) ≥ 1, χT ( p) ≥ χZ ( p) 1 − (1 + p2 )χ˜ ( p, r ) χZ ( p)2
(5.27)
(5.28)
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if and r are sufficiently large. Combining (5.27) and (5.28) yields 2 2 2 (log V ) 3 χZ ( p) . χT ( p) ≥ χZ ( p) 1 − (1 + p ) Cχ˜ K V
(5.29)
Let pˆ := pc (Zd ) −
Cˆ −1/3 (log V )2/3 V
(5.30)
ˆ we take V large to ensure for some (sufficiently large) constant Cˆ > 0. Depending on C, that pˆ > 0. Then, by Theorem 3.3, 3 Cχ3 Cχ 3 χZ ( p) ˆ ≤ V (log V )−2 . (5.31)
3 = 3 d Cˆ pc (Z ) − pˆ Substituting (5.31) into (5.29) for p = p, ˆ using pˆ ≤ 1 and the lower bound in (3.5) give (1 + 2 ) Cχ˜ K 2 Cχ3 1 V 1/3 χT ( p) ˆ ≥ 1− . (5.32) Cˆ 3 Cˆ (log V )2/3 We make the Cˆ in (5.30) so large that (1 + 2 ) Cχ˜ K 2 Cχ3 1 cˆ := 1 − > 0, Cˆ 3 Cˆ
(5.33)
so that (5.32) simplifies to χT ( p) ˆ ≥ cˆ V 1/3 (log V )−2/3 . The quantity
q := pc (Tr,d ) − pˆ = Cˆ V −1/3 (log V )2/3 − pc (Zd ) − pc (Tr,d ) ,
(5.34)
(5.35)
is positive if V is large enough, by (4.1). Hence Theorem 3.1 is applicable, and (3.3) yields
χT ( p) ˆ = χT pc (Tr,d ) − q −1 ≤
2
. pc (Tr,d ) − pˆ
Merging (5.34) and (5.36), we arrive at 1 2 pc (Tr,d ) − pc (Zd ) ≤ − Cˆ V −1/3 (log V )2/3 , cˆ
(5.36)
(5.37)
ˆ ∨ 0. which is (5.25) with = (2cˆ−1 − C) Corollary 5.4. Under the conditions in Theorem 1.1, there exists a constant C > 0 such that, for all ω1 ≥ C, −1 1203/2 · 288 −1 −4/3 2/3 ≥ 1 + 3/2 PT, pc (Zd ) |Cmax | ≥ ω1 (log V ) V . (5.38) ω1 (log V )2
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√ Proof. Take V so large that λ−1 ≤ ω1 120−1 (log V )2/3 , and let √ pˆ = pc (Tr,d ) − ω1 120−1 −1 V −1/3 (log V )2/3 .
(5.39)
Then, by (3.1), 2 1 χT2 ( p) ˆ ≥ ≥ 3600 ω1−1 (log V )−4/3 V 2/3 . √ λ−1 V −1/3 + ω1 120−1 V −1/3 (log V )2/3 (5.40) This enables the bound 2 pˆ χ T PT, pc (Zd ) |Cmax | ≥ ω1−1 (log V )−4/3 V 2/3 ≥ PT, pc (Zd ) |Cmax | ≥ . (5.41) 3600 By Lemma 5.3, pˆ ≤ pc (Zd ) for ω1 ≥ C, and C > 0 large enough. Thus, we use (3.2) and (3.3) to bound (5.41) further from below by −1 −1 χT2 pˆ 36 χT3 pˆ 36 · 23 · 1203 ≥ 1 + 3/2 . (5.42) PT, pˆ |Cmax | ≥ ≥ 1+ 3600 V ω (log V )2 1
Combining our results from Sects. 4 and 5, we finally prove Theorem 1.1: Proof of Theorem 1.1. By Corollaries 4.1 and 5.4, for ω1 ≥ C for some sufficiently large C and ω2 ≥ 1, ⎛ ⎞ 1203 ·288 3/2 ⎜ ω (log V )2 ⎟ b2 PT, pc (Zd ) ω1−1 (log V )−4/3 V 2/3 ≤ |Cmax | ≤ ω2 V 2/3 ≥ 1− ⎝ 1 ⎠− , 1203 ·288 ω2 1+ 3/2 2 ω1 (log V )
(5.43) where the term in brackets on the right-hand side vanishes for V → ∞. Then b1 in (1.5) can be taken as 1203 · 288. This proves Theorem 1.1. 5.3. Discussion of (1.8). At the end of Sect. 1.2 we argued why we believe that (1.8) holds. Another approach to (1.8) is to split the sum over z in (1.7). Note that, for p = pc (Zd ) − K 1 −1 V −1/3 , the sum due to z ≤ K 1 V 1/6 can be bounded, for any K 1 > 0, as τZ (z) ≤ C sup (z + 1)−(d−2) ≤ C K 12 V −2/3 . sup y
r
z ∼y, r2 ≤z≤K 1 V 1/6
y
r
z ∼y, r2 ≤z≤K 1 V 1/6
(5.44) Therefore, we are left to give a bound on the contribution from z ≥ K 1 V 1/6 . The restriction z ≥ K 1 V 1/6 is equivalent to z ≥ C K ,K 1 ξ( p), where ξ( p) denotes the correlation length. Indeed, ξ( p) is comparable in size to ( pc (Zd ) − p)1/2 as proven by Hara [17] (see also (3.9)), and the constant C K ,K 1 can be made arbitrarily large by taking K 1 large. This contribution could be bounded by investigating τZ (z) for z ≥ C K ,K 1 ξ( p).
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6. The Role of Boundary Conditions In this section, we discuss the impact of boundary conditions on the geometry of the largest critical cluster. For the d-dimensional box {−r/2, . . . , r/2 − 1}d , we write Br,d if we consider it with free boundary conditions, and we write Tr,d if it is equipped with periodic boundary conditions. We fix p = pc (Zd ) and further omit this subscript. Furthermore, we write C for a positive constant, whose value may change from line to line. With Michael Aizenman [2], we have discussed the role of boundary conditions for critical percolation above the upper critical dimension. We will summarize the consequences of Theorems 1.1 and 1.2 in this discussion now. Assume that the conditions in Theorem 1.1 are satisfied. Let X 1 , X 2 , X 3 and X 4 be 4 uniformly chosen vertices in Br,d . Then, Aizenman notices that, with bulk boundary conditions, PZ (X 1 ←→ X 3 | X 1 ←→ X 2 , X 3 ←→ X 4 ) → 0,
(6.1)
as the width of the torus r tends to infinity. Indeed, PZ (X 1 ←→ X 3 | X 1 ←→ X 2 , X 3 ←→ X 4 ) =
PZ (X 1 ←→ X 2 , X 3 , X 4 ) . PZ (X 1 ←→ X 2 , X 3 ←→ X 4 ) (6.2)
We can compute PZ (X 1 ←→ X 2 , X 3 , X 4 ) = V −4
EZ [|CZ (x, r )|3 ],
(6.3)
x∈Br,d Z
where CZ (x, r ) is the set of vertices y ∈ Br,d for which x ←→ y, and the right-hand side will be bounded from above by the following lemma. Lemma 6.1. Under the conditions of Theorem 1.1, for p = pc (Zd ) and r ≥ 3∨(2L +1), EZ [|CZ (x, r )|3 ] ≤ Cr 10 ,
for all x ∈ Br,d .
(6.4)
The proof will be given at the end of this section. On the other hand, Z Z PZ (X 1 ←→ X 2 , X 3 ←→ X 4 ) = V −4 x,y,u,v∈Br,d PZ (x ←→ u, y ←→ v). By the FKG-inequality, for all x, y, u, v ∈ Br,d , Z
Z
Z
Z
PZ (x ←→ u, y ←→ v) ≥ PZ (x ←→ u) PZ (y ←→ v),
(6.5)
so that ⎛ PZ (X 1 ←→ X 2 , X 3 ←→ X 4 ) ≥ V −4 ⎝
⎞2 Z
PZ (x ←→ u)⎠ .
(6.6)
x,u∈Br,d
For fixed x, by (3.6), u∈Br,d
Z
PZ (x ←→ u) ≥
u∈Br,d
cτ ≥ Cr 2 . (|x − u| + 1)d−2
(6.7)
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Summing over x gives an extra factor V . We obtain that PZ (X 1 ←→ X 2 , X 3 ←→ X 4 ) ≥ C V −2 r 4 .
(6.8)
Therefore, when d > 6, PZ (X 1 ←→ X 2 , X 3 , X 4 ) PZ (X 1 ←→ X 2 , X 3 ←→ X 4 ) V −3r 10 r6 → 0. (6.9) ≤ = −2 4 CV r CV
PZ (X 1 ←→ X 3 | X 1 ←→ X 2 , X 3 ←→ X 4 ) =
All this changes when we consider the torus with periodic boundary conditions and we assume that
χT ( pc (Zd )) = V 1/3 . (6.10) Note that (6.10) is a consequence of (1.10), which follows from (1.8), and Theorem 3.1. In this case, PT (X 1 ←→ X 2 , X 3 , X 4 ) ≥ PT (X 1 , X 2 , X 3 , X 4 ∈ Cmax ).
(6.11)
Thus, for ω ≥ 1 sufficiently large, 1 PT (X 1 ←→ X 2 , X 3 , X 4 ) ≥ PT X 1 , X 2 , X 3 , X 4 ∈ Cmax , |Cmax | ≥ V 2/3 ω 1 2/3 1 −4 −4/3 −4 −4/3 ≥ ω V ≥ω V PT |Cmax | ≥ V . (6.12) ω 2 On the other hand, we have that
PT (X 1 ←→ X 2 , X 3 ←→ X 4 ) ≤ PT (X 1 ←→ X 2 ) ◦ (X 3 ←→ X 4 ) + PT (X 1 ←→ X 2 , X 3 , X 4 ),
(6.13)
and, by the BK-inequality,
PT (X 1 ←→ X 2 ) ◦ (X 3 ←→ X 4 ) ≤ PT (X 1 ←→ X 2 ) PT (X 3 ←→ X 4 ) = V −2 χT ( pc (Zd ))2 ≤ C V −4/3 .
(6.14)
Therefore, assuming (6.10), we obtain lim sup PT (X 1 ←→ X 3 | X 1 ←→ X 2 , X 3 ←→ X 4 ) > 0. V →∞
(6.15)
The difference between (6.9) and (6.15) was conjectured by Aizenman [2]. The obvious conclusion is that boundary conditions play a crucial role for high-dimensional percolation on finite cubes. We do not know that (6.10) holds, so now we will investigate the changes using the results in Theorem 1.1 in the above discussion. Thus, we will use that, with high probability, V 2/3 (log V )−4/3 ≤ |Cmax | ≤ ωV 2/3 ,
(6.16)
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and 1 1/3 V (log V )−2/3 ≤ χT ( pc (Zd )) ≤ ωV 1/3 . ω
(6.17)
We will see that the conclusion weakens. Indeed, by (6.9), PZ (X 1 ←→ X 3 | X 1 ←→ X 2 , X 3 ←→ X 4 ) ≤ Cr 6 V −1 = Cr 6−d → 0, (6.18) where the convergence is as an inverse power of r , while by an argument as in (6.12), now using (6.16), 16
PT (X 1 ←→ X 3 | X 1 ←→ X 2 , X 3 ←→ X 4 ) ≥ C(log V )− 3 ,
(6.19)
which only converges to zero as a power of log r . Therefore, the main conclusion that boundary conditions play an essential role is preserved. We have argued that the largest critical cluster with bulk boundary conditions is much smaller than the one with periodic boundary conditions. We will now argue that, under the condition of Theorem 1.1, critical percolation clusters on the periodic torus Tr,d are similar to percolation clusters on a finite box with bulk boundary conditions, where the box has width V 1/6 = r d/6 r . Here we rely on the coupling in Proposition 2.1. In particular, when the origin 0 is T-connected to a uniformly chosen point X , then, with high probability, there is no Z-connection at distance o(V 1/6 ) from 0 to a point that is r -equivalent to X . This will be illustrated by the following calculation. Assume for the moment that χT ( pc (Zd )) = (V 1/3 ). This follows from our assumption (1.8). Choose the vertex X uniformly from the torus Tr,d . Then, for any ε > 0,
r Z T PZ,T ∃y ∈ Zd : y ∼ X, |y| ≤ εV 1/6 , 0 ←→ y | 0 ←→ X
r Z PZ,T ∃y ∈ Zd : y ∼ X, |y| ≤ εV 1/6 , 0 ←→ y ≤ .
T PZ,T 0 ←→ X For the denominator, we rewrite
T PZ,T 0 ←→ X = V −1 χT ( pc (Zd )) ≥ C V −2/3 ,
(6.20)
(6.21)
whereas the numerator in (6.20) is bounded from above by r 1 1 Z PZ,T y ∼ X, |y| ≤ εV 1/6 , 0 ←→ y ≤ ≤ Cε2 V −2/3 . d−2 V + 1) (|y| 1/6 d y∈Z
y:|y|≤εV
(6.22) Thus,
r Z T PZ,T ∃y ∈ Zd : y ∼ X, |y| ≤ εV 1/6 , 0 ←→ y | 0 ←→ X ≤ Cε2 .
(6.23)
We have seen that, under bulk boundary conditions, |Cmax | is of the order (r 4 ), whereas under periodic boundary conditions, it is of the order (V 2/3 ). Thus, the maximal critical percolation cluster on a high dimensional torus is (R 4 ) with R = V 1/6 r , so that Cmax is 4-dimensional, but now in a box of width R. This suggests that percolation
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on a box Br,d with periodic boundary conditions is similar to percolation on the larger box B R,d under bulk boundary conditions, with R = V 1/6 r . Without assuming (1.8), the lower bound on the denominator is only C V −2/3 (log V )2/3 , thus we obtain the weaker bound r Z T (6.24) PZ ∃y ∈ Zd : y ∼ X, |y| ≤ εV 1/6 (log V )−1/3 , 0 ←→ y | 0 ←→ X ≤ ε2 . The conclusion that occupied paths are long is preserved. We conclude this section with the proof of Lemma 6.1. Proof of Lemma 6.1. For all x ∈ Br,d , Z Z Z EZ [|CZ (x, r )|3 ] = PZ x ←→ s ←→ t ←→ u ≤3
s,t,u∈Br,d
Z Z Z Z Z PZ (x ←→ v) ◦ (v ←→ s) ◦ (v ←→ w) ◦ (w ←→ t) ◦ (w ←→ u) .
s,t,u∈Br,d v,w∈Zd
(6.25) Using the BK-inequality, this can be further bounded from above by ⎛ ⎞2 3 ⎝ sup τZ (s − v)⎠ τZ∗3 (x − u), v∈Zd s∈Br,d
(6.26)
u∈Br,d
τ ∗3
where Z denotes the threefold convolution of τZ . We begin to bound the expression in parenthesis. Fix v ∈ Zd . If the distance between v and the box Br,d is larger than r , then τZ (s − v) ≤ Cr −(d−2) for all s ∈ Br,d , by (3.6). Hence, in this case, τZ (s − v) ≤ Cr 2 . (6.27) s∈Br,d
Otherwise, s − v ≤ 2r for all s ∈ Br,d , and therefore, by (5.22), τZ (s − v) ≤ τZ (z) ≤ Cr 2 . s∈Br,d
(6.28)
z∈B2r,d
Using [19, Prop. 1.7 (i)] for d > 6, we see that, for all z ∈ Zd , the upper bound in (3.6) implies that τZ∗2 (z) ≤
C , (|z| + 1)d−4
(6.29)
which in turn implies, when d > 6, so that (d − 2) + (d − 4) > d, τZ∗3 (z) = (τZ ∗ τZ∗2 )(z) ≤
C . (|z| + 1)d−6
Thus we obtain, for x ∈ Br,d , τZ∗3 (x − u) ≤ τZ∗3 (z) ≤ u∈Br,d
z∈B2r,d
z∈B2r,d
C ≤ Cr 6 . (|z| + 1)d−6
(6.30)
(6.31)
The combination of the bounds (6.25)–(6.31) yields the desired upper bound Cr 10 .
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Acknowledgement. This work was supported in part by the Netherlands Organisation for Scientific Research (NWO). The research of RvdH was performed in part while visiting Microsoft Research in the summer of 2004. We thank Christian Borgs, Jennifer Chayes and Gordon Slade for valuable discussions at the start of this project, and Michael Aizenman for useful discussions on the relations between scaling limits and IIC’s, as well as the role of boundary conditions for high-dimensional percolation as described in Sect. 6. We thank Jeffrey Steif for pointing our attention to [8], and the referee for numerous suggestions to improve the presentation.
References 1. Aizenman, M.: On the number of incipient spanning clusters. Nucl. Phys. B [FS] 485, 551–582 (1997) 2. Aizenman, M.: Private communication (2005) 3. Aizenman, M., Barsky, D.J.: Sharpness of the phase transition in percolation models. Commun. Math. Phys. 108, 489–526 (1987) 4. Aizenman, M., Newman, C.M.: Tree graph inequalities and critical behavior in percolation models. J. Stat. Phys. 36, 107–143 (1984) 5. Aldous, D.J.: Brownian excursions, critical random graphs and the multiplicative coalescent. Ann. Probab. 25, 812–854 (1997) 6. Barsky, D.J., Aizenman, M.: Percolation critical exponents under the triangle condition. Ann. Probab. 19, 1520–1536 (1991) 7. Benjamini, I., Kozma, G.: Loop-erased random walk on a torus in dimension 4 and above. Commun. Math. Phys. 259, 257–286 (2005) 8. Benjamini, I., Schramm, O.: Percolation beyond Zd , many questions and a few answers. Elect. Comm. in Probab. 1, 71–82 (1996) 9. van den Berg, J., Kesten, H.: Inequalities with applications to percolation and reliability. J. Appl. Probab. 2, 556–569 (1985) 10. Bollobás, B.: Random Graphs. London Academic Press (1985) 11. Borgs, C., Chayes, J.T., van der Hofstad, R., Slade, G., Spencer, J.: Random subgraphs of finite graphs: I. The scaling window under the triangle condition. Random Struct. Alg. 27, 331–357 (2005) 12. Borgs, C., Chayes, J.T., van der Hofstad, R., Slade, G., Spencer, J.: Random subgraphs of finite graphs: II. The lace expansion and the triangle condition. Ann. Probab. 33, 1886–1944 (2005) 13. Borgs, C., Chayes, J.T., Kesten, H., Spencer, J.: Uniform boundedness of critical crossing probabilities implies hyperscaling. Random Struct. Alg. 15, 368–413 (1999) 14. Borgs, C., Chayes, J.T., Kesten, H., Spencer, J.: The birth of the infinite cluster: finite-size scaling in percolation. Commun. Math. Phys. 224, 153–204 (2001) 15. Erdös, P., Rényi, A.: On the evolution of random graphs. Magyar Tud. Akad. Mat. Kutató Int. Közl. 5, 17–61 (1960) 16. Grimmett, G.: Percolation. 2nd edition Berlin, Springer (1999) 17. Hara, T.: Mean-field critical behaviour for correlation length for percolation in high dimensions. Prob. Th. Rel. Fields 86, 337–385 (1990) 18. Hara, T.: Decay of correlations in nearest-neighbour self-avoiding walk, percolation, lattice trees and animals. http://arxiv.org/abs/math-ph/0504021 (2005) 19. Hara, T., van der Hofstad, R., Slade, G.: Critical two-point functions and the lace expansion for spread-out high-dimensional percolation and related models. Ann. Probab. 31, 349–408 (2003) 20. Hara, T., Slade, G.: Mean-field critical behaviour for percolation in high dimensions. Commun. Math. Phys. 128, 333–391 (1990) 21. Hara, T., Slade, G.: Mean-field behaviour and the lace expansion. In: Probability and Phase Transition, Grimmett, G. ed., Dordrecht: Kluwer (1994) 22. Hara, T., Slade, G.: The scaling limit of the incipient infinite cluster in high-dimensional percolation. I. Critical exponents. J. Stat. Phys. 99, 1075–1168 (2000) 23. Hara, T., Slade, G.: The scaling limit of the incipient infinite cluster in high-dimensional percolation. II. Integrated super-Brownian excursion. Commun. Math. Phys. 128, 333–391 (2000) 24. van der Hofstad, R., den Hollander, F., Slade, G.: Construction of the incipient infinite cluster for spreadout oriented percolation above 4 + 1 dimensions. Commun. Math. Phys. 231, 435–461 (2002) 25. van der Hofstad, R., Járai, A.A.: The incipient infinite cluster for high-dimensional unoriented percolation. J. Stat. Phys. 114, 625–663 (2004) 26. Janson, S., Knuth, D.E., Łuczak, T., Pittel, B.: The birth of the giant component. Random Struct. Alg 4, 71–84 (1993) 27. Janson, S., Łuczak, T., Rucinski, A.: Random Graphs. New York: Wiley (2000) 28. Járai, A.A.: Incipient infinite clusters in 2D. Ann. Probab. 31, 444–485 (2003) 29. Járai, A.A.: Invasion percolation and the incipient infinite cluster in 2D. Commun. Math. Phys. 236, 311–334 (2003)
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30. Kesten, H.: The incipient infinite cluster in two-dimensional percolation. Prob. Th. Rel. Fields. 73, 369– 394 (1986) 31. Kesten, H.: Subdiffusive behavior of random walk on a random cluster. Ann. Inst. Henri Poincaré 22, 425–487 (1986) 32. Lawler, G.F.: Intersections of Random Walks. Boston: Birkhäuser (1991) 33. Łuczak, T., Pittel, B., Wierman, J.C.: The structure of a random graph at the point of the phase transition. Trans. Amer. Math. Soc. 341, 721–748 (1994) 34. Menshikov, M.V.: Coincidence of critical points in percolation problems. So. Math. Dokl. 33, 856–859 (1986) 35. Pemantle, R.: Choosing a spanning tree for the integer lattice uniformly. Ann. Probab. 19, 1559–1574 (1991) 36. Peres, Y., Revelle, D.: Scaling limits of the uniform spanning tree and loop-erased random walk on finite graphs. http://arxiv.org/abs/math.PR/0410430 (to appear in Ann. Probab) (2005) 37. Schweinsberg, J.: The loop-erased random walk and the uniform spanning tree on the four-dimensional discrete torus. http://arxiv.org/abs/math.PR/0602515 (2006) 38. Slade, G.: The Lace Expansion and its Applications. Springer Lecture Notes in Mathematics, Ecole d’Eté Probabilitès Saint-Flour, Vol. 1879, Berlin-Heidelberg-New York: Springer (2006) Communicated by M. Aizenman
Commun. Math. Phys. 270, 359–371 (2007) Digital Object Identifier (DOI) 10.1007/s00220-006-0150-x
Communications in
Mathematical Physics
Efficient Quantum Algorithms for Simulating Sparse Hamiltonians Dominic W. Berry1,2 , Graeme Ahokas2,3 , Richard Cleve2,3,4,5 , Barry C. Sanders2,6 1 2 3 4 5 6
Department of Physics, The University of Queensland, Queensland 4072, Australia Institute for Quantum Information Science, University of Calgary, Alberta T2N 1N4, Canada Department of Computer Science, University of Calgary, Alberta T2N 1N4, Canada School of Computer Science, University of Waterloo, Ontario N2L 3G1, Canada Institute for Quantum Computing, University of Waterloo, Ontario N2L 3G1, Canada Centre for Quantum Computer Technology, Macquarie University, Sydney, New South Wales 2109, Australia
Received: 18 January 2006 / Accepted: 24 July 2006 Published online: 14 December 2006 – © Springer-Verlag 2006
Abstract: We present an efficient quantum algorithm for simulating the evolution of a quantum state for a sparse Hamiltonian H over a given time t in terms of a procedure for computing the matrix entries of H . In particular, when H acts on n qubits, has at most a constant number of nonzero entries in each row/column, and H is bounded by a constant, we may select any positive integer k such that the simulation requires O((log∗ n)t 1+1/2k ) accesses to matrix entries of H . We also show that the temporal scaling cannot be significantly improved beyond this, because sublinear time scaling is not possible. 1. Introduction There are three main applications of quantum computer algorithms: the hidden subgroup problem, with Shor’s factorization algorithm one important example [1], search problems [2], and simulation1 of quantum systems [4, 5]. Lloyd’s method for simulating quantum systems [5] assumes a tensor product structure of smaller subsystems. Aharonov and Ta-Shma (ATS) [6] consider the alternative case where there is no evident tensor product structure to the Hamiltonian, but it is sparse and there is an efficient method of calculating the nonzero entries in a given column of the Hamiltonian. Such representations of Hamiltonians can arise as encodings of computational problems, such as simulations of quantum walks [7–11]. A quantum simulator approximates the unitary evolution e−i H t . Given that the Hamiltonian may be expressed as a sum of simple Hamiltonians H = j H j , the evolution is approximated by a sequence of unitary operators of the form e−i H j t/r . The cost of the simulation is quantified by the number of operators in this sequence, Nexp . For the 1 The term “simulation” is used here to mean simulation of the evolution of a state under a Hamiltonian. This is distinct from another use of the term concerning finding the ground state of a Hamiltonian, for which evidence exists of intractability (QMA-hard) [3].
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problem considered by ATS, the decomposition of H is not known, but H is given by a “black-box” (also known as an “oracle”) which can be queried. The cost is then quantified by the number of calls to this black-box, Nbb . Our goal is to make the quantum simulation as efficient as possible by optimizing the sequence of unitary steps e−i H j t/r to minimize Nexp if the decomposition of H is known, or to minimize Nbb otherwise. To reduce Nexp , we apply the higher-order integrators of Suzuki [12, 13] to reduce the temporal scaling from t 3/2 [6] or t 2 [5] to the slightly superlinear scaling t 1+1/2k , where k is the order of the integrator and may be an arbitrarily large integer. We determine an upper bound on the number of exponentials required to approximate the evolution with a given accuracy. This enables us to estimate the optimal value of k, and therefore the k-independent scaling in t. We then prove that, in the black-box setting, this scaling is close to optimal, because it is not possible to perform simulations sublinear in t. We also provide a superior method for decomposing the Hamiltonian into a sum for the problem considered by ATS, which dramatically reduces the scaling of n 2 [14] or n 9 [6] to log∗ n for n qubits. This method is similar to “deterministic coin tossing” [15], as well as Linial’s graph coloring method [16]. 2. Problems and Results We commence with a statement of the problems that we consider in this paper and follow with the solutions that will be proven. Problem 1. The Hamiltonian is of the form H = mj=1 H j . The problem is to simulate
the evolution e−i H t by a sequence of exponentials e−i H j t such that the maximum error in the final state, as quantified by the trace distance, does not exceed . Specifically we wish to determine an upper bound on the number of exponentials, Nexp , required in this sequence.
For this problem, the H j should be of a form that permits e−i H j t to be accurately and efficiently simulated for arbitrary evolution time t . It is therefore reasonable to quantify the complexity of the calculation by the number of exponentials required. This problem includes the physically important case of simulating tensor product systems considered by Lloyd [5], for which each H j can be considered to be an interaction Hamiltonian. It also may be applied to the case where there is a procedure for calculating the nonzero elements in the columns [6]. In that case, each H j is a 1-sparse Hamiltonian. The decomposition must be calculated, which requires additional steps in the algorithm. Our general result for Problem 1 is the following theorem. Theorem 1. When the permissible error is bounded by , Nexp is bounded by Nexp ≤ m52k (mτ )1+1/2k / 1/2k ,
(1)
for ≤ 1 ≤ 2m5k−1 τ , where τ = H t, and k is an arbitrary positive integer. By taking k to be sufficiently large, it is possible to obtain scaling that is arbitrarily close to linear in τ . However, for a given value of τ , taking k to be too large will increase Nexp . To estimate the optimum value of k to take, we express Eq. (1) as Nexp ≤ m 2 τ e2k ln 5+ln(mτ/)/2k .
Efficient Quantum Algorithms for Simulating Sparse Hamiltonians
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The right-hand side has a minimum for k = round
1 log5 (mτ/) + 1 . 2
Here we have added 1 and rounded because k must take integer values. Adopting this value of k provides the upper bound Nexp ≤ 2m 2 τ e2
√
ln 5 ln(mτ/)
,
(2)
for ≤ 1 ≤ mτ/25. Equation (2) is an expression for Nexp that is independent of k. The scaling in Eq. (2) is close to linear for large mτ . We show that this scaling is effectively optimal, because it is not possible to perform general simulations sublinear in τ (see Sect. 4). This result applies in the “black-box” setting, so it does not rule out the possibility that individual Hamiltonians have structure that allows them to be simulated more efficiently. The second problem which we consider is that of sparse Hamiltonians. Problem 2. The Hamiltonian H has no more than d nonzero entries in each column, and there exists a black-box function f that gives these entries. The dimension of the space that H acts upon does not exceed 2n . If the nonzero elements in column x are y1 , . . . , yd , where d ≤ d, then f (x, i) = (yi , Hx,yi ) for i ≤ d , and f (x, i) = (x, 0) for i > d . The problem is to simulate the evolution e−i H t such that the maximum error in the final state, as quantified by the trace distance, does not exceed . We wish to determine the scaling of the number of calls to f , Nbb , required for this simulation. For each x, the order in which the corresponding yi are given can be arbitrary. The function f is an arbitrary black-box function, but we assume that there is a corresponding unitary U f such that U f |x, i|0 = |φx,i |yi , Hx,yi , and we may perform calls to both U f and U †f . Here |φx,i represents any additional states which are produced in the reversible calculation of f . ATS approached the problem by decomposing the Hamiltonian into a sum of H j . We apply a similar approach to obtain the following theorem. Theorem 2. The number of black-box calls for given k is Nbb ∈ O (log∗ n)d 2 52k (d 2 τ )1+1/2k / 1/2k
(3)
(r )
with log∗ n ≡ min{r | log2 n < 2} (the (r ) indicating the iterated logarithm). The log∗ n scaling is a dramatic improvement over the n 9 scaling implicit in the method of ATS, as well as the n 2 scaling of Childs [14].
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D. W. Berry, G. Ahokas, R. Cleve, B. C. Sanders
3. Higher Order Integrators To prove Theorem 1, we apply the method of higher-order integrators. Following Suzuki [12, 13], we define S2 (λ) =
m j=1
e H j λ/2
1
e H j λ/2 ,
j =m
and the recursion relation S2k (λ) = [S2k−2 ( pk λ)]2 S2k−2 ((1 − 4 pk )λ)[S2k−2 ( pk λ)]2 with pk = (4 − 41/(2k−1) )−1 for k > 1. Suzuki then proves that [12] ⎛ ⎞ m H j λ⎠ − S2k (λ) ∈ O(|λ|2k+1 ) exp ⎝
(4)
j=1
for |λ| → 0. The parameter λ corresponds to −it for Hamiltonian evolution. We first assess the higher-order integrator method in terms of all quantities t, m, k, and H . Our result is Lemma 1. Using integrators of order k and dividing the time into r intervals, we have the bound ⎛ ⎞ m H j ⎠ − [S2k (−it/r )]r ≤ 2(2m5k−1 τ )2k+1 /r 2k , (5) exp ⎝−it j=1
for (2m5k−1 τ )2k+1 /r 2k ≤ 1.
(6)
Proof. Consider a Taylor expansion of both terms in the left-hand side (LHS) of Eq. (4). Those terms containing λ to powers less than 2k + 1 must cancel because the correction term is O(|λ|2k+1 ), and terms with λl for l ≥ 2k + 1 must contain a product of l of the H j terms; thus ⎛ ⎞ Ll l m ∞ exp ⎝ H j λ⎠ = S2k (λ) + λl C lp H j pq . j=1
l=2k+1
p=1
q=1
The constants C lp and the number of terms L l depend on m and k. In order to bound C lp and L l , first consider the Taylor expansion of the exponential in the LHS of Eq. (4). Because the operators H j are in general noncommuting, expanding (H1 + · · · + Hm )l yields m l terms. Therefore the Taylor expansion contains m l terms with λl . These terms have multiplying factors of 1/l! because this is the multiplying factor given by the Taylor expansion of the exponential. To place a bound on the number of terms in the Taylor expansion of S2k (λ), note that S2k (λ) consists of a product of 2(m − 1)5k−1 + 1
Efficient Quantum Algorithms for Simulating Sparse Hamiltonians
363
exponentials. The expansion for S2k (t) may be obtained by expanding each of the exponentials individually. To place a bound on the contribution to the error from terms containing λl , we can replace each of the terms in this expansion with the upper bounds on their norms. Thus the bounds may be obtained from the expansion of (1 + |λ| + |λ|2 /2! + . . .)2(m−1)5
k−1 +1
,
(7)
where ≡ H . Each H j satisfies H j ≤ H [6], so ≥ max H j . This equation is just the expansion of exp{|λ|[2(m − 1)5k−1 + 1]}, so the bound for each term is {|λ|[2(m − 1)5k−1 + 1]}l . l!
(8)
Using standard inequalities we obtain Ll l ∞ ∞ |λ|l l m + [2(m − 1)5k−1 + 1]l λl C lp H j pq ≤ l! p=1 ∞
l=2k+1
≤2
l=2k+1
q=1
l=2k+1
|λ|l [2m5k−1 ]l ≤ (1/3)|2m5k−1 λ|2k+1 exp |2m5k−1 λ|. l!
Therefore we obtain the inequality ⎛ ⎞ m H j ⎠ − S2k (λ) ≤ |2m5k−1 λ|2k+1 , exp ⎝λ j=1
provided |2m5k−1 λ| ≤ 1. Substituting λ = −it/r , where r is an integer, and taking the power of r , gives the error bound ⎛ ⎞ m H j ⎠ − [S2k (−it/r )]r ≤ [1 + (2m5k−1 t/r )2k+1 ]r − 1, (9) exp ⎝−it j=1
for 2m5k−1 t/r ≤ 1. This may alternatively be expressed as in Lemma 1.
By placing limits on the norm of the difference in the unitaries, we limit the trace distance of the output states. This is because U1 − U2 ≥ U1 |ψ − U2 |ψ 1 ≥ Tr U1 |ψψ|U1† − U2 |ψψ|U2† 2 = D U1 |ψψ|U1† , U2 |ψψ|U2† ,
with D the trace distance. We now use this to prove Theorem 1. Proof (of Theorem 1). Let us take r = 21/2k (2m5k−1 τ )1+1/2k / 1/2k .
(10)
Given the restriction ≤ 1, it is easily seen that Eq. (6) holds. In addition, the right-hand side of Eq. (5) does not exceed , so the error can not exceed .
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D. W. Berry, G. Ahokas, R. Cleve, B. C. Sanders
Because the number of exponentials in S2k (λ) does not exceed 2m5k−1 , we have Nexp ≤ 2m5k−1r . If we take r as in Eq. (10), then we find that Nexp ≤ m52k (mτ )1+1/2k / 1/2k .
(11)
Here the multiplying factor has been changed to take into account the ceiling function. Hence the order scaling is as in Eq. (1). This result may be used for any case where the Hamiltonian is a sum of terms that may be simulated efficiently. It may therefore be applied to the case of tensor product systems, where the individual H j are interaction Hamiltonians. It can be also used for cases of the type of Problem 2, where the Hamiltonian is sparse. In this case we have the additional task of decomposing the Hamiltonian into a sum. 4. Linear Limit on Simulation Time We have shown that the simulation of any Hamiltonian may be performed arbitrarily close to linearly in the scaled time τ . We now show that the scaling cannot be sublinear in τ , provided the number of qubits can grow at least logarithmically with respect to τ . The result is Theorem 3. For all positive integers N there exists a row-computable 2-sparse Hamiltonian H such that simulating the evolution of H for scaled time τ = π N /2 within precision 1/4 requires at least τ/2π queries to H . Here a row-computable Hamiltonian means one where there is a method for efficiently calculating the nonzero elements in each row. Proof. The idea is to construct a 2-sparse Hamiltonian such that the simulation of this Hamiltonian determines the parity of N bits. It has been shown that the parity of N bits requires N /2 queries to compute within error 1/4 [19, 20]; therefore the Hamiltonian can not be simulated any more efficiently. First consider a Hamiltonian H acting on orthogonal basis states |0, . . . , |N , for which the nonzero matrix entries are j + 1|H | j = j|H | j + 1 = (N − j)( j + 1)/2. This Hamiltonian is equivalent to a Jx operator for a spin N /2 system, with the | j being Jz eigenstates. It is therefore clear that e−iπ H |0 = |N and H = N /2. Now we construct an augmented version of the above Hamiltonian, that corresponds to a graph with two disjoint lines with weights as above, where the lines “cross over” at the positions where bits X 1 , . . . , X N are 1. We add an ancilla qubit so the Hamiltonian H acts on basis states |0, 0, . . . , |0, N , |1, 0, . . . , |1, N . The nonzero matrix entries of H are k , j + 1|H |k, j = k, j|H |k , j + 1 = (N − j)( j + 1)/2 for values of k and k such that k ⊕ k = X j+1 (where ⊕ is XOR). Thus, if X j+1 is zero, then there is a nonzero matrix element between |0, j and |0, j + 1, as well as between |1, j and |1, j + 1. If X j+1 is equal to 1, then the nonzero matrix elements are between |0, j and |1, j + 1, as well as |1, j and |0, j + 1. We may determine a sequence of bits k0 , . . . , k N such that k j ⊕ k j+1 = X j+1 . The Hamiltonian
Efficient Quantum Algorithms for Simulating Sparse Hamiltonians
Xj =
365
0
1
1
0
1
0
0
1
0,0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
1,0
1,1
1,2
1,3
1,4
1,5
1,6
1,7
1,8
Fig. 1. Graph representing the example Hamiltonian in the proof of Theorem 3. States are represented by ellipses, and nonzero elements of the Hamiltonian are indicated by lines. The sequence of states |k j , j with k0 = 0 is indicated by the solid line
acting on the set of states |k j , j will then be equivalent to the original Hamiltonian acting on the states | j. It is therefore clear that e−iπ H |k0 , 0 = |k N , N . The graph corresponding to a Hamiltonian of this type is shown in Fig. 1. The system separates into two distinct sets of states which are not connected. If the system starts in one of the states on the path indicated by the solid line, it can not evolve under the Hamiltonian to a state on the dotted line. From the definition of the k j , if k0 = 0, then k j is the parity of bits X 1 to X j , and in particular k N gives the parity of all N bits. Thus if we start with the initial state |0, 0 and simulate the evolution e−iπ H , we obtain the state |k N , N , where k N is the parity of the N bits X 1 , . . . , X N . Thus measuring the state of the ancilla qubit will give the parity. Let us denote the final state obtained by the simulation by |ψ, and the reduced density operator for the ancilla by ρanc . If the error probability is no less than 1/4, then D(ρanc , |k N k N |) ≥ 1/4, which implies that D(|ψψ|, |k N , N k N , N |) ≥ 1/4. Hence, if there are fewer than N /2 queries to the X j , the error in the simulation as quantified by the trace distance must be at least 1/4. Each query to a column of H requires no more than two queries to the X j (for column j we require a query to X j and X j+1 ). Thus, if there are fewer than N /4 queries to H , then there are fewer than N /2 queries to the X j . In addition, the scaled time for the simulation is τ = H t = π N /2. Thus the simulation of H requires at least N /4 = τ/2π queries to obtain trace distance error less than 1/4. The form of this result differs slightly from that in the previous section, in that the cost is specified in terms of the number of queries to the Hamiltonian, rather than the number of exponentials. It is straightforward to show the following result for the number of exponentials. Corollary 1. There is no general integrator for Hamiltonians of the form H = H1 + H2 such that (trace distance) error < 1/4 may be achieved with the number of exponentials Nexp < τ/2π . By general integrator we mean an integrator that depends only on τ , and not the Hamiltonian.
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Proof. We take H as in the preceding proof. This Hamiltonian may be expressed in the form H = H1 + H2 by taking H1 to be the Hamiltonian with k , j + 1|H1 |k, j nonzero only for even j, and H2 to be the Hamiltonian with k , j + 1|H2 |k, j nonzero only for odd j. Each query to the Hk requires only one query to the X j . For H1 (H2 ), determining the nonzero element in column j requires determining which of j and j + 1 is odd (even), and performing a query to the corresponding X j or X j+1 . Both H1 and H2 are 1-sparse, and therefore may be efficiently simulated with only two queries [17, 18]. If Nexp < τ/2π , the total number of queries to the X j is less than τ/π . Taking t = π and H = N /2, the number of queries is less than N /2. However, from Refs. [19, 20] the error rate can be no less than 1/4. Hence error rate < 1/4 cannot be achieved with Nexp < τ/2π . 5. Efficient Decomposition of Hamiltonian Next we consider the problem of simulating general sparse Hamiltonians, as in Problem 2. Given that the dimension of the space does not exceed 2n , we may represent the state of the system on n qubits, and x and y may be n-bit integers. The real and imaginary parts of the matrix elements will be represented by n bit integers (for a total of 2n bits for each matrix element), where n must be chosen large enough to achieve the desired accuracy. In order to simulate the Hamiltonian, we decompose it into the form H = mj=1 H j , where each H j is 1-sparse (i.e., has at most one nonzero entry in each row/column). If H j is 1-sparse then it is possible to directly simulate exp(−i H j t) with just two black-box queries to H j [17, 18]. Since the value of m directly impacts the total cost of simulating H , it is desirable to make m as small as possible. The size of the sum may be limited as in the following lemma. Lemma 2. There exists a decomposition H = mj=1 H j , where each H j is 1-sparse, such that m = 6d 2 and each query to any H j can be simulated by making O(log∗ n) queries to H . Proof. From the black-box function for H , we wish to determine black-box functions for each H j that give the nonzero row number, y, and matrix element, corresponding to each column x. This black-box for H j is represented by the function g(x, j), with output (y, (H j )x,y ). If there is no nonzero element in column x, the output is (x, 0). Intuitively, it is helpful to consider the graph G H associated with H whose vertex set is {0, 1}n . Each vertex corresponds to a row or column number, and there is an edge between vertex x and y if the matrix element Hx,y is nonzero. As H is Hermitian we take the graph to be undirected. We wish to determine an “edge-coloring” of G H , which is a labeling of the edges such that incident edges have different colors. Each edge color, j, then corresponds to a different Hamiltonian H j in the decomposition of H . The basic idea is as in the following labeling scheme, where the labels are indexed from the set {1, . . . , d}2 . We take f y to be the y-component of f ; then f y (x, i) gives the i th neighbor of vertex x in the graph. Let (x, y) be an edge of G H such that y = f y (x, i) and x = f y (y, j). Thus edge (x, y) is labeled with the ordered pair (i, j) for x ≤ y, or ( j, i) for x ≥ y. This labeling is not quite an edge-coloring; for w < x < y it is possible for edges (w, x) and (x, y) to both have the label (i, j). That will be the case if y and w are the i th and j th neighbors of x, respectively, and x is the i th neighbor of w and the j th neighbor of y. To ensure that the labels are unique, we add the additional parameter ν, so the label is (i, j, ν).
Efficient Quantum Algorithms for Simulating Sparse Hamiltonians
367 (0)
We assign ν via a method similar to “deterministic coin tossing” [15]. We set x0 = x, then determine a sequence of vertices (0)
(0)
(0)
x0 < x1 < x2 < · · · (0) (0) (0) such that xl+1 = f y (xl(0) , i) and f y (xl+1 , j) = xl(0) . That is, the edges (xl(0) , xl+1 ) are labeled (i, j, ν), with the same values of i and j for each edge. We need to choose values of ν for the edges such that the same value is never repeated in this chain. A typical chain may have only two elements; however, there exist Hamiltonians such that long chains may be formed. In the case that the chain is long, we do not determine it any further than x z(0) . Here z n is the number of times we must iterate l → 2log2 l n +1 (starting at 2n ) to obtain 6 or less. This quantity is of order log∗ n, and for any realistic problem size z n itself will be no more than 6 2 . (1) Now we determine a second sequence of values xl . This sequence is taken to have (0) (0) the same length as the first sequence. For each xl and xl+1 , we determine the first bit (0) position where these two numbers differ, and record the value of this bit for xl , followed (1) by the binary representation of this position, as xl . The bit positions are numbered from (0) zero; that is, the first bit is numbered 00 . . . 0. If xl is at the end of the sequence, we (1) (0) simply take xl to be the first bit of xl , followed by the binary representation of 0. (0) There are 2n different possible values for each of the xl , and 2n different possible (1) values for each of the xl . (0) (1) (1) From the definition, each xl is unique. Also xl must differ from xl+1 . This is (0) (0) (0) because, even if the positions of the first bit where xl differs from xl+1 and xl+1 differs (0) (0) from xl+2 are identical, the value of this bit for xl will of course be different from the (0) (1) (1) value for xl+1 . As the xl contain both the position and the value of the bit, xl must (1) . differ from xl+1 (0) (1) is at the end of the sequence. Then xl+1 contains the There is a subtlety when xl+1 (0) , and the position of the first bit which differs is taken to be 1. In that case, first bit of xl+1 (0) (1) at the first bit (so the bit positions recorded in xl(1) and xl+1 are if xl(0) differs from xl+1 (1) (1) identical), then the bit values which are recorded in xl and xl+1 must be different. Thus (1) . it is still not possible for xl(1) to be equal to xl+1 We repeat this process until we determine the sequence of values xl(z n ) . We determine ( p+1) ( p) ( p) the xl from the xl in exactly the same way as above. At each step, xl differs ( p) from xl+1 for exactly the same reasons as for p = 1. As we go from p to p + 1, the ( p) number of possible values for the xl is reduced via the mapping k → 2log2 k. Due (z ) to our choice of z n , there are six possible values for x0 n . (0) Now if w < x with x = f y (w, i) and w = f y (x, j), then we may set w0 = w and perform the calculation in exactly the same way as for x in order to determine w0(z n ) . If ( p) ( p) the chain of xl(0) ends before z n , then the xl will be the same as the wl+1 . In particular x0(z n ) will be equal to w1(z n ) , so it is clear that w0(z n ) will differ from x0(z n ) . 2 For z > 6 we require n > 101037 ; clearly an unrealistic problem size. n
368
D. W. Berry, G. Ahokas, R. Cleve, B. C. Sanders (0)
(0)
On the other hand, if there is a full chain of x0 up to x z n +1 , then the chain for w will
(1) end at wz(0) , which is equivalent to x z(0) n . Then wz n +1 will be calculated in a different n +1 (1)
(1)
( p)
(1)
way to x z n , and may differ. However, wz n will be equal to x z n −1 . At step p, wz n − p+1 ( p)
(z )
(z n )
will be equal to x z n − p . In particular, at the last step, w1 n will be equal to x0 (z ) w0 n
. Thus
(z ) x0 n .
we find that again differs from (z ) As x0 n has this useful property, we assign the edge (x, y) the color (i, j, ν), where ν = x0(z n ) . Due to the properties of the above scheme, if the edge (w, x) has the same values of i and j as (x, y), it must have a different value of ν. Therefore, via this scheme, adjacent edges must have different colors. Now we describe how to calculate the black-box function g using this approach. We replace j with (i, j, ν) to reflect the labeling scheme, so the individual Hamiltonians are H(i, j,ν) . The black-box function we wish to calculate is g(x, i, j, ν). We also define the function ϒ(x, i, j) to be equal to the index ν as calculated in the above way. There are three main cases where we give a nontrivial output: 1. 2. 3.
f y (x, i) = x, i = j and ν = 0, f y (x, i) > x, f y ( f y (x, i), j) = x and ϒ(x, i, j) = ν, f y (x, j) < x, f y ( f y (x, j), i) = x and ϒ( f y (x, j), i, j) = ν.
In Cases 1 and 2 we return g(x, i, j, ν) = f (x, i), and for Case 3 we return g(x, i, j, ν) = f (x, j); in all other cases we return g(x, i, j, ν) = (x, 0). Case 1 corresponds to diagonal elements of the Hamiltonian. We only return a nonzero result for ν = 0, in order to prevent this element being repeated in different Hamiltonians H(i, j,ν) . Case 2 corresponds to there existing a y > x such that y is the i th neighbor of x and x is the j th neighbor of y. Similarly Case 3 corresponds to there existing a w < x such that w is the j th neighbor of x and x is the i th neighbor of w. The uniqueness of the labeling ensures that Cases 2 and 3 are mutually exclusive. As there are d possible values for i and j, and ν may take six values, there are 6d 2 colors. Thus we may take m = 6d 2 . In determining ν, we need a maximum of 2(z n + 2) queries to the black-box; this is of order log∗ n. ( p)
To illustrate the method for determining ν, an example is given in Table 1 for {xl } ( p) and Table 2 for {wl }. Figure 2 shows a portion of the graph corresponding to the values in Tables 1 and 2. In the Tables n = 18, so there are 218 possible values in the first column. Then there are 36 possible values in the second column, 12 in the third, 8 in the (0) fourth and 6 in the fifth. Thus z n is equal to 4 in this case, and the sequence of xl is (0) determined up to x5 . ( p)
Table 1. Example values of xl under our scheme for calculating ν. The value of ν obtained is in the upper right, and is shown in bold. For this example n = 18 and z n = 4. The values in italics are those that may differ ( p) from wl+1 (there are no corresponding values for the bottom row) l\ p 0 1 2 3 4 5
0 001011100110011010 010110101010011011 011011101110101101 101011101011110100 101011101011110101 111000010110011010
1 000001 000010 000000 010001 000001 100000
2 0100 1100 0001 1001 0000 1000
3 000 100 000 100 000 100
4 000 100 000 100 000 100
Efficient Quantum Algorithms for Simulating Sparse Hamiltonians
369
( p)
Table 2. Example values of wl under our scheme for calculating ν. The value of ν obtained is in the upper right, and is shown in bold. For this example n = 18 and z n = 4. The values in italics are those which may ( p) differ from xl−1 l\ p 0 1 2 3 4 5
0 000010010110111001 001011100110011010 010110101010011011 011011101110101101 101011101011110100 101011101011110101 (1,3,ν)
(1,3,ν)
1 000010 000001 000010 000000 010001 100000 (1,3,ν)
2 1100 0100 1100 0001 0000 1000 (1,3,ν)
(1,3,ν) 178932
4 100 000 001 100 000 100
(1,3,ν)
9657
47514
92827
w
x
y
x(0) 0
x(0) 1
(0) 2
(0) 3
(0) 4
w1(0)
w2(0)
w3(0)
w4(0)
w5(0)
w0(0)
113581
3 100 000 100 111 000 100
178933
230810 (0) 5
Fig. 2. A portion of the graph for the example given in Tables 1 and 2. The vertices w, x, y, etc each have i = 1 and j = 3 for the edge labels, so it is necessary for the ν to differ to ensure that adjoining edges have (0) (0) distinct labels. The xl and wl which the vertices correspond to are also given. The numbers in the first columns of Tables 1 and 2 are the binary representations of the vertex numbers given here ( p)
In Table 1 the values of xl are given, and the elements in the first column are values (0) (1) (0) (0) of xl . As an example of calculation of xl , note that x0 differs from x1 in the second (0) (1) bit position. The second bit for x0 is 0, so this is the first bit for x0 . We subtract 1 (1) from the bit position to obtain 1, and take the remaining bits of x0 to be the binary (0) representation of 1. For the case of x5 , this is the end of the chain, so we simply take (1) (0) x5 to be the first bit of x5 , which is 1, and the binary representation of 0. ( p) In Table 2 the values of wl are given, where these are calculated from a w < x such that x = f y (w, i) and w = f y (x, j). The example given illustrates the case where (0) (0) (0) (0) the sequence of wl (with wl = xl−1 ) ends before the sequence of xl . In this case, (z )
(z )
we find that the differences propagate towards the top, but we still have x0 n = w1 n . (4) Thus different values of ν are obtained, as expected. For x we obtain ν = x0 = 000, and for w we obtain ν = w0(4) = 100. We can use Lemma 2 to prove Theorem 2. Proof (of Theorem 2). Overall the number of Hamiltonians H(i, j,ν) in the decomposition is m = 6d 2 . To calculate g(x, i, j, ν), it is necessary to call the black-box 2(z n + 2) times. To simulate evolution under the Hamiltonian H(i, j,ν) , we require g to be implemented by a unitary operator Ug satisfying Ug |x, i, j, ν|0 = |x, i, j, ν|y, (H(i, j,ν) )x,y . As discussed above, the function f may be represented by a unitary U f ; using this unitary it is straightforward to obtain a unitary U˜ g such that U˜ g |x, i, j, ν|0 = |φx,i, j,ν |y, (H(i, j,ν) )x,y .
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We may obtain the unitary Ug in the usual way by applying U˜ g , copying the output, then applying U˜ g† [21]. Using the method of Ref. [6], the Hamiltonian H(i, j,ν) may be simulated using a call to Ug and a call to Ug† . As z n is of order log∗ n, the number of black-box calls to f for the simulation of each H(i, j,ν) is O(log∗ n). Using these values, along with Eq. (1), we obtain the number of black-box queries as in Eq. (3). Another issue is the number of auxiliary operations, which is the number of operations that are required due to the overhead in calculating ϒ(x, i, j). It is necessary to perform bit comparisons between a maximum of z n + 2 numbers in the first step, and each has n bits. This requires O(n log∗ n) operations. In the next step the number of bits is O(log2 n) bits, which does not change the scaling. Hence the number of auxiliary operations is O n(log∗ n)2 d 2 52k (d 2 τ )1+1/2k / 1/2k . This scaling is superior to the scaling n 10 in Ref. [6]. Next we consider the error introduced by calculating the matrix elements to finite precision. Given that the matrix elements are represented by 2n bit integers, the error cannot exceed H /2n . The error in calculating exp(−i H(i, j,ν) t) will not exceed τ/2n [6], so the error in the integrator due to the finite precision does not exceed 4m5k τ/2n . This error can then be kept below /2 by choosing n > 5 + log2 (τ d 2 5k /). The total error may be kept below by choosing the integrator such that the integration error does not exceed /2. 6. Conclusions We have presented a scheme for simulating sparse Hamiltonians that improves upon earlier methods in two main ways. First, we have examined the use of higher order integrators to reduce the scaling to be close to linear in H t. Second, we have significantly improved the algorithm for the decomposition of the Hamiltonian, so the scaling of the number of black-box calls is close to log∗ n, rather than polynomial in n. In addition we have shown that the scaling cannot be sublinear in H t (for reasonable values of n). Acknowledgement. This project has been supported by the Australian Research Council, Alberta’s Informatics Circle of Research Excellence, the Canadian Institute for Advanced Research, the Canadian Network of Centres of Excellence for the Mathematics of Information Technology and Complex Systems (MITACS), and the U.S. Army Research Office. R.C. thanks Andrew Childs for helpful discussions.
References 1. Shor, P. W.: Algorithms for quantum computation: Discrete logarithms and factoring. In: Proc. 35th Symp. on Foundations of Computer Science, Los Alamitos, CA:IEEE, 1994, pp. 124–134 2. Grover, L.: Quantum mechanics helps in searching for a needle in a haystack. Phys. Rev. Lett. 79, 325–328 (1997) 3. Kempe, J., Kitaev, A., Regev, O.: The complexity of the local Hamiltonian problem. SIAM J. Computing 35, 1070–1097 (2006)
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4. Feynman, R.P.: Simulating physics with computers. Int. J. Theoret. Phys. 21, 467–488 (1982) 5. Lloyd, S.: Universal quantum simulators. Science 273, 1073–1078 (1996) 6. Aharonov, D., Ta-Shma, A.: Adiabatic quantum state generation and statistical zero knowledge. In: Proc. 35th Annual ACM Symp. on Theory of Computing, New York:ACM, 2003, pp. 20–29 7. Childs, A., Farhi, E., Gutmann, S.: An example of the difference between quantum and classical random walks. J. Quant. Inf. Proc. 1, 35–43 (2002) 8. Shenvi, N., Kempe, J., Whaley, K.B.: Quantum random-walk search algorithm. Phys. Rev. A 67, 052307 (2003) 9. Childs, A., Goldstone, J.: Spatial search by quantum walk. Phys. Rev. A 70, 022314 (2004) 10. Ambainis, A.: Quantum walk algorithm for element distinctness. In: Proc. 45th Symp. on Foundations of Computer Science, Los Alamitos, CA: IEEE, 2004, pp. 22–31 11. Ambainis, A., Kempe, J., Rivosh, A.: Coins make quantum walks faster. In: Proc. 16th ACM-SIAM SODA, Philadelphia, PA:SIAM, 2005, pp. 1099–1108 12. Suzuki, M.: Fractal decomposition of exponential operators with applications to many-body theories and Monte Carlo simulations. Phys. Lett. A 146, 319–323 (1990) 13. Suzuki, M.: General theory of fractal path integrals with applications to many-body theories and statistical physics. J. Math. Phys. 32, 400–407 (1991) 14. Childs, A.M.: Quantum information processing in continuous time. Ph.D. Thesis, Massachusetts Institute of Technology, 2004 15. Cole, R., Vishkin, U.: Deterministic coin tossing with applications to optimal parallel list ranking. Inform. and Control 70, 32–53 (1986) 16. Linial, N.: Locality in distributed graph algorithms. SIAM J. Comput. 21, 193–201 (1992) 17. Childs, A.M., Cleve, R., Deotto, E., Farhi, E., Guttman, S., Spielman, D.A.: Exponential algorithmic speedup by quantum walk. In: Proc. 35th Annual ACM Symp. on Theory of Computing, New York: ACM, 2003, pp. 59–68 18. Ahokas, G.: Improved algorithms for approximate quantum Fourier transforms and sparse Hamiltonian simulations. M.Sc. Thesis, University of Calgary, 2004 19. Beals, R., Buhrman, H., Cleve, R., Mosca, M., de Wolf, R.: Quantum lower bounds by polynomials. J. ACM 48, 778–797 (2001) 20. Farhi, E., Goldstone, J., Gutmann, S., Sipser, M.: Limit on the speed of quantum computation in determining parity. Phys. Rev. Lett. 81, 5442–5444 (1998) 21. Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge: Cambridge University Press, 2000 Communicated by M.B. Ruskai
Commun. Math. Phys. 270, 373–443 (2007) Digital Object Identifier (DOI) 10.1007/s00220-006-0154-6
Communications in
Mathematical Physics
Structure of the Space of Ground States in Systems with Non-Amenable Symmetries M. Niedermaier1, , E. Seiler2 1 Laboratoire de Mathematiques et Physique Theorique, CNRS/UMR 6083, Université de Tours,
Parc de Grandmont, 37200 Tours, France. E-mail:
[email protected]
2 Max-Planck-Institut für Physik (Werner-Heisenberg-Institut), Föhringer Ring 6, 80805 Munich, Germany
Received: 24 January 2006 / Accepted: 13 September 2006 Published online: 14 December 2006 – © Springer-Verlag 2006
Abstract: We investigate classical spin systems in d ≥ 1 dimensions whose transfer operator commutes with the action of a nonamenable unitary representation of a symmetry group, here SO(1, N ); these systems may alternatively be interpreted as systems of interacting quantum mechanical particles moving on hyperbolic spaces. In sharp contrast to the analogous situation with a compact symmetry group the following results are found and proven: (i) Spontaneous symmetry breaking already takes place for finite spatial volume/finitely many particles and even in dimensions d = 1, 2. The tuning of a coupling/temperature parameter cannot prevent the symmetry breaking. (ii) The systems have infinitely many non-invariant and non-normalizable generalized ground states. (iii) The linear space spanned by these ground states carries a distinguished unitary representation of SO(1, N ), the limit of the spherical principal series. (iv) The properties (i)–(iii) hold universally, irrespective of the details of the interaction. Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Group Decomposition of the State Space . . . . . . . . . . . . . . . . . . . 2.1 Generalized spin systems . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 The spectral problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Orbit decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 The reduction of ρ(G) on L 2 (Mr ) . . . . . . . . . . . . . . . . . . . . 3. The Spectral Problem on the Invariant Fibers . . . . . . . . . . . . . . . . . 3.1 Basic consequences of the ρ(G) invariance . . . . . . . . . . . . . . . 3.2 Fiber decomposition of invariant selfadjoint operators . . . . . . . . . . 3.3 Relating the spectral problems of A and Aσ : Aσ compact . . . . . . . . 3.4 Relating the spectral problems of A and Aσ : Aσ not necessarily compact Membre du CNRS
374 379 379 380 383 387 395 395 396 400 406
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3.5 Absence of the discrete series . . . . . . . . . . . . . . . . . 4. The Structure of the Ground State Sector . . . . . . . . . . . . . 4.1 Gσ (T) is empty for all but one principal series representation 4.2 The role of strict positivity . . . . . . . . . . . . . . . . . . 4.3 Gσ (T) for Tσ compact . . . . . . . . . . . . . . . . . . . . 5. Conclusions and Outlook . . . . . . . . . . . . . . . . . . . . . Appendix A: Harmonic Analysis on Noncompact Lie Groups . . . . . Appendix B: The Restricted Dual of SO(1,N) . . . . . . . . . . . . . Appendix C : The Amenable Case ISO(N) . . . . . . . . . . . . . . .
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1. Introduction Spontaneous symmetry breaking is typically discussed for compact internal or for abelian translational symmetries, see e.g. [45, 48, 37]. Both share the property of being amenable [42] and their spontaneous breakdown is a specific dynamical property of the interaction. Here we consider systems with a nonamenable symmetry, by which we mean that the dynamics is invariant under a nonamenable unitary representation [4] of a locally compact group (which then by necessity is also nonamenable). One goal of this note is to show, roughly, that whenever the dynamics of a system of classical statistical mechanics is invariant under a nonamenable symmetry, this symmetry is always spontaneously broken, irrespective of the details of the interaction. Neither does the long or short ranged nature of the interaction matter, nor can the tuning of a (temperature) parameter prevent the symmetry breaking. Spontaneous symmetry breaking even occurs in one and two dimensions, where for compact symmetries this is ruled out by the Mermin-Wagner theorem. The phenomenon is not limited to a semiclassical regime and occurs already for systems with finitely many degrees of freedom. The systems will be defined on a finite lattice of arbitrary dimension and connectivity. The dynamical variables are ‘spins’ attached to the vertices of the lattice, taking values in some noncompact Riemannian symmetric space Q = G/K , where G is the noncompact symmetry group and K a maximally compact subgroup. The dynamics is specified by a transfer operator acting on the square integrable functions on the configuration manifold Q . Such a system can alternatively be interpreted as a quantum mechanical system of finitely many particles living on Q; we only have to interpret the transfer matrix as exp(−H ) and thereby define the Hamiltonian H ; the inevitable spontaneous symmetry breaking appears then as degeneracy of the generalized ground states of this system. Conversely, given a quantum mechanical system with a Hamiltonian H , we can re-interpret the system as one of classical statistical mechanics with exp(−H ) as the transfer matrix. To fix ideas one may take a Hamiltonian of the conventional form H =−
ν ν 1 i + Vik , 2 i=1
(1.1)
i,k=1
where i is the Laplace-Beltrami operator on Q for the ith particle and the Vik are some potentials describing the interaction of particle i and k, depending only on the geodesic distances of the particles. Typically one would also require that the set = {1, 2 . . . ν} has the structure of a lattice and that the interaction links neighboring sites only. The Hamiltonians (1.1) are however only one class of examples, many others are covered. Indeed apart from some technical conditions on the transfer matrix it is mostly the invariance that matters.
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To analyze these systems, it is necessary to perform something analogous to the wellknown separation of the center of mass motion from the relative motion in Euclidean space. This turns out to be considerably more involved in our setting. It is convenient not to define an actual center of mass of the ν spins or particles, respectively, but rather take simply one of them as parameterizing the global position of the configuration. The universality of the resulting symmetry breaking is then related to the fact that the global motion of these systems never allows for a symmetric (proper or generalized) ground state. Normally spontaneous symmetry breaking can only happen in the thermodynamic limit. We stress again that here the situation is different: spontaneous symmetry breaking takes place already on a finite spatial lattice. The systems exhibit a remarkable universality in the structure of their generalized ground states. Namely, there are always infinitely many non-normalizable and non-invariant ground states which transform irreducibly under a preferred representation of the group – the same for a large class of transfer operators! All generalized ground states can be generated by forming linear combinations of factorized wave functions, where one factor describes the global and the other one the relative motion; of course the second factor will be sensitive to the interaction as far as the relative motion is concerned; what is universal is the transformation law under global symmetry transformations of the first factor. In quantum one-particle systems described by an exactly soluble Schrödinger equation an infinite degeneracy in the ground state energy has been found earlier: first of all in the well known problem of the Landau levels in the Euclidean plane, closer to our situation explicitly in [8] for the supersymmetric SO(1,2) invariant quantum mechanics and implicitly in [31] (p. 172), [61] and in [7] for the lowest Landau level. The interplay between spontaneous symmetry breaking, nonamenability and properties of the transfer operator was understood in [38], initially for the hyperbolic spin chain. The thermodynamic limit can usually only be taken on the level of correlation functions, so that the ‘fate’ of the ground state orbit cannot directly be traced. By means of an Osterwalder-Schrader reconstruction one can in principle recover a Hilbert space description, however inevitably one with an exotic structure; cf. [38]. Our main example for Q will be the hyperboloids H N := SO0 (1, N )/SO(N ), N ≥ 2, in part because of the importance of the Lorentz and de Sitter groups in physics, and in part because already for the groups SO0 (1, N ) the harmonic analysis exhibits all of the characteristic complications. Most of the constructions however generalize to a large class of noncompact coset spaces and are actually easier to understand in a general setting. We thus specialize to Q = H N only when needed. Let us now make things a little more explicit: the configuration space M is the direct product of ν := || copies of the space Q. A hypercubical lattice ⊂ Zd of arbitrary dimension d is a prime example, however neither the dimension nor the connectivity of the lattice is essential. The pure states of the system are described by elements of L 2 (M), i.e. functions ψ : M → C, square integrable with respect to the invariant measure dγ . The left diagonal G action d on M induces a unitary representation M of G on L 2 (M) via ( M (g)ψ)(m) = ψ(g −1 m). Since L 2 (M) [L 2 (Q)]⊗ ν it can be identified with the ν-fold inner tensor product of the left quasiregular representation
1 . On this L 2 space we consider bounded selfadjoint operators A commuting with the group action, i.e. M ◦ A = A ◦ M . Such operators A can only have essential spectrum. Specifically we consider so-called transfer operators defining the dynamics of the system. A precise definition is given in Definition 2.1 below. Here it may suffice to say that a transfer operator is a bounded selfadjoint operator on L 2 which is positive as well
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as positivity improving; as usual the latter property is realized by taking for T an integral operator with strictly positive kernel, T (m, m ) > 0 for all m, m ∈ M. We are interested in an invariant dynamics, so we assume M ◦ T = T ◦ M . Important examples are T = exp(−H ), with H as in (1.1), but Hamiltonians with more complicated ‘time derivative’ terms and non-pair potentials would also be allowed. The latter is welcome because such complicated Hamiltonians naturally arise as the result of blocking transformations. It is easy to see that an invariant transfer operator T cannot be a compact operator, furthermore it cannot even have normalizable ground states, i.e. solutions of Tψ = T ψ, with ψ ∈ L 2 . (In fact T must either lie in the continuous spectrum of T or be a limit of eigenvalues with infinite multiplicity). Instead our setting is such that T has a continuous extension to an operator from L p to L p for 1 ≤ p ≤ ∞ and we identify conditions under which T has generalized ground states. These we take as almost everywhere defined functions (not just distributions) which are eigenfunctions with spectral value T . The set of these generalized ground states forms a linear space which we call the ground state sector G(T) of T. It is important that T is defined as a selfadjoint operator on a Hilbert space, here L 2 (M), so that the spectral theorem can be applied to provide a resolution of the identity. The notion of a generalized ground state is then unambiguous, although there is some freedom in the choice of the topological vector space in which the ground state wave functions live. The separation of global and relative configurations is achieved by writing M = Q × N , where N collects the ‘relative degrees of freedom’; the construction is done in such a way that M = (G × N )/d(K ), where d(K ) is the right diagonal action of K on G × N . The details of this construction will be given in Sect. 2. The global part of the 2 configurations can now ⊕ be Fourier transformed: the L functions on G have a Plancherelˇ type decomposition dν(σ ) Lσ ⊗ Lσˇ , where the fiber spaces Lσ carry the unitary (infinite dimensional) ν-almost always irreducible representations πσ , and ν is carried by r of G. We show in Sect. 3 that one can associate to an the so-called restricted dual G invariant selfadjoint operator A on L 2 (M) a ν-measurable field of bounded selfadjoint r , via operators Aσ on L 2 (N ) ⊗ Lσ , σ ∈ G ˇ σˇ ) , L 2 (M) = dν(σ ) L2σ (M) . (1.2) A = dν(σ )(1I ⊗ A In the second formula we indicated that the state space L 2 (M) decomposes into fibers ˇ σˇ . which carry the representation πσ and which are preserved under the action of 1I ⊗ A 2 (N ); however its ˇ Each of the fiber spaces is isometric to L2σ (M) ∼ L ⊗ L ⊗ L = σ σˇ elements will be realized as functions on M. This is done by constructing for v ∈ Lσ an (antilinear) map τvσ : L 2 (N ) ⊗ Lσ → L2σ (M), f → τvσ ( f ). Roughly, the image function arises by reinterpreting the matrix element ( f (n), πσ (g)v)σ as a (generically not square integrable) function on M. The map is designed such that it is an isometry onto its image and has the following intertwining properties: τvσ ( f )(g −1 m) = τπσ (g)v,σ ( f )(m) , [Aτvσ ( f )](m) = τvσ (Aσ f )(m) .
(1.3a) (1.3b)
According to the first equation the G-action on the argument of the function just rotates the reference vector v ∈ Lσ with the representation πσ . In the second relation we anticipated that the action of A can be extended to the (in general non-L 2 ) functions τvσ ( f ). In view of (1.3) it is plausible that the spectral problems of A and Aσ are related as follows: suppose first that the eigenvalue equations Aσ χ = λχ and A = λ are well-defined,
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with λ ∈ Spec(A) and (generically non-L 2 ) eigenfunctions χ , , and second that the map τvσ admits an extension to the generalized eigenfunctions χ of Aσ . Then by (1.3b) the image function τvσ (χ ) should be an eigenfunction of A enjoying the equivariance property (1.3a). This construction principle can be implemented for a large class of invariant selfadjoint operators A specified in Sect. 3.3 and 3.4. In overview the result is that a set of r with generalized eigenfunctions λσ exists for almost all λ ∈ Spec(A) and σ ∈ G the following properties: (i) they are almost everywhere defined functions (not distributions) λ,σ : M → C. (ii) they are σ -equivariant, i.e. they lie in the image of the maps τei σ , where {ei } is an orthonormal basis of Lσ and where the domain is the linear hull of a complete set of eigenfunctions of Aσ with spectral value λ ∈ SpecAσ . (iii) λσ (gm) → 0, as g leaves compact subsets of G. (iv) As λ runs through Spec(A) and r , the eigenfunctions λσ are complete, in the sense that any smooth σ runs through G function can be expanded in terms of the λσ and that a Parseval relation holds on (L 1 ∩ L 2 )(M). The transfer operators T considered are special cases of such invariant selfadjoint operators, which in addition are positive and positivity improving. The spectral value relevant for the ground states of a transfer operator T is λ = T . The important “almost all λ ∈ Spec(T)” clause in the above completeness result prevents one from getting all the generalized ground states simply by specialization. However, whenever for some r a complete set of eigenfunctions of Tσ with spectral value T can be found, σ ∈G their images under τei σ will produce Gσ (T), the space of σ -equivariant generalized ground states. r could occur as a “representation carried by the Offhand of course every σ ∈ G ground state sector” in Gσ (T). Remarkably this is not the case: under fairly broad conditions only one representation occurs and always the same! For definiteness we formulate the following results for G = SO0 (1, N ), K = SO(N ), M = SO(N −1); many aspects however are valid for any noncompact linear reductive Lie group. Theorem 1.1. Let T be a transfer operator on L 2 (M, dγ ) commuting with the unitary representation induced by a proper action d of G. In terms of the fiber decomposition (1.2) one has: (a) Gσ (T) is empty for all but the principal series representations whenever one of the following holds: (i) Gσ (T) contains a strictly positive function. (ii) Gσ (T) contains a K -singlet. (iii) Tσ is compact. (b) Gσ (T) is non-empty for at most one of the principal series representations – the limit of the spherical (M-singlet) principal series. Combined (a) and (b) imply that if there are generalized ground states which transform equivariantly according to some unitary irreducible representation πσ , this representation must be – under any of the conditions (i)–(iii) and possibly others – the limit of the spherical principal series, for which we write π00 . It remains to establish the existence of such generalized ground states. The known construction principles for generalized eigenfunctions (the classic ones [17, 34], as well as the one described above) are not sufficient to assure the existence of generalized ground states (neither as functions nor as distributions) – so any of the fiber spaces in Theorem 1.1 could be empty, including G00 (T). In a follow-up paper [39] we describe a construction principle which ensures the existence of generalized ground states in various situations; a preview is given in the conclusions. There is a simple case that is, however, important for applications, in which the existence of generalized ground
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states is immediate: the case that all the fiber operators Tσ are compact. The eigenspaces with eigenvalue Tσ can in principle be constructed via the well-known projector s-limt→∞ (Tσ / Tσ )t (where the limit exists in the strong operator sense). One then has the following concrete variant of Theorem 1.1: Theorem 1.2. Let T be as in Theorem 1.1 and assume in addition that the fiber operators r are compact. Then all fiber spaces except G00 (T) are empty. Further T00 is Tσ , σ ∈ G itself a transfer operator which has a unique ground state in G00 (T) ⊂ L 2 (N ) ⊗ Lσ =00 . In the realization of Lσ =00 as L 2 (S N −1 ) this ground state can be represented by a unique a.e. positive function ψ0 . G(T) is the linear span of functions of the form ψ0 (n, p) (q, n) = d S( p) , (1.4) N −1 N −1 S (q0 − q · p) 2 where q = gq ↑ = (q0 , q). Here M was identified with Q × N in a way that replaces the original diagonal left G action by g → gq. For comparison we mention here the corresponding results for amenable (compact or abelian) Lie groups. When T is invariant under the action of a compact Lie group, the very same setting entails that the ground states have three concordant properties: normalizability, uniqueness, and invariance. That is, there exists a normalizable, nondegenerate ground state, which is a group singlet. This ground state can be obtained by acting with a projector P on an arbitrary L 2 function, where P is obtained as the strong operator limit of the iterated (compact or trace class) transfer operator: P = s − limt→∞ (T/ T )t . When T does not have normalizable ground states this limit does not exist; the corresponding weak limit will be the null vector. A construction of generalized ground states based on a similar but more subtle fixed point principle will be discussed in [39]; see the conclusions for a preview. When T is invariant under a noncompact amenable Lie group, for instance in the Euclidean case Q = ISO(N )/SO(N ), our construction resembles the well-known procedure of separating the center-of-mass motion (see for instance [44]). It yields a ground state sector carrying the trivial representation of ISO(N ) and for which the centerof-mass wave function is unique (the properties of the ‘internal’ ground state sector replacing G(T00 ) again depend on the details of the interaction). Of the three concordant properties above only the normalizability is lost. In Appendix C we specialize our constructions to this degenerate situation, to contrast it with the non-amenable case. Both amenable cases have in common that the representation carried by the ground state sector is uniquely determined and always the same, namely the trivial one. The above results show which aspects of this picture generalize to the case of non-amenable symmetries (namely, the uniqueness of the representation and its universality) and which do not (viz, the uniqueness of the ground state). In view of Theorems 1.1 and 1.2 the limit of the spherical principal series appears to be the natural generalization of the singlet in the noncompact setting. The article is organized as follows. In Sect. 2 we describe the setting in more detail and explain why the Gel’fand-Maurin theory is insufficient to account for the results we are aiming at. The fiber decomposition of L 2 (M) is introduced. Section 3 provides the adapted fiber decomposition of a wider class of selfadjoint operators A and relates the spectral problem of A to that of the fiber operators Aσ . These results are then applied in Sect. 4 to the ground state sector of transfer operators, giving a proof of Theorem 1.1 and ramifications of it. Appendices A and B provide the necessary background on
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the harmonic analysis of non-compact Lie groups. In Appendix C we collect counterparts of some of the results in the simple case of a flat symmetric space, for the sake of contradistinction. For further orientation we refer to the table of contents. 2. Group Decomposition of the State Space Here we introduce the class of systems considered and prepare an orbit decomposition of the configuration manifold. The wave functions (square integrable functions on this manifold) are then subjected to a Plancherel decomposition with respect to their global position, which is parameterized by an element of G. Eventually this gives rise to a decomposition of the state space L 2 (M) into fibers labeled by irreducible unitary representations of the original group action.
2.1. Generalized spin systems. We consider generalized spin systems of the following type: the dynamical variables take values in an indecomposable Riemannian symmetric space Q := G/K , with G a noncompact Lie group and K a maximal compact subgroup. The Lie groups will be taken to be linear reductive, meaning that G is a closed subgroup of GL(N , R) or GL(N , C) which is stable under conjugate transpose. K then is the isotropy group of the point q ↑ = eK ∈ Q. We further assume that there is an involution ι such that K consists of its fixed points, and that gι(g)−1 has unit determinant. Then G and K form a symmetric pair. Examples are SO0 (1, N )/SO(N ), SL(N , R)/SO(N ), U( p, q)/U( p) × U(q). In Appendix C we will also consider the degenerate case R N = ISO(N )/SO(N ). The configuration manifold M is the direct product of ν := || ≥ 2 copies of this space. For much of the following only has to have the structure of a point set; it is assumed though that ν → ∞ captures the physical intuition of a thermodynamic limit. A hypercubical lattice ⊂ Zd of arbitrary dimension d is a prime example, however neither the dimension nor the structure of the lattice is essential. Ordering the points in some way, we write m = (q1 , . . . , qν ) for the points in M. Further we denote by γ Q and dγ Q the invariant metric and the measure on Q. Equipped with the product metric γ (m) := i γ Q (qi ) and the product measure dγ (m) := i dγ Q (qi ) the configuration space M is a simply connected Riemannian manifold, and in fact a reducible Riemannian symmetric space. Further M carries an action d : G ×M → M of the group G, via d(g)(m) = (gq1 , . . . , gqν ), where q → gq is the left (transitive) action of G on Q. Clearly d(g) is an isometry and d(g)(m) = m for all m implies that g is the identity in G, that is, the action of G is effective. Since d(G) := {d(g), g ∈ G} is a subgroup of the full isometry group which is closed in the compact-open topology, (M, γ ) also is a proper Riemannian G-manifold in the sense of [35], Sect. 5. In fact, the main reason for considering product manifolds of the above type is that they have a well defined orbit decompositon M = Q ×N , N /d(K ) = M/d(G), to be described later. With certain refinements this generalizes to all proper Riemannian G-manifolds, see [35]. The pure states of the system are described by elements of L 2 (M), i.e. functions ψ : M → C, square integrable with respect to dγ . The proper G action d on M induces a unitary representation M of G on L 2 (M) via ( M (g)ψ)(m) = ψ(d(g −1 )(m)). Since L 2 (M) [L 2 (Q)]⊗ ν it can be identified with the ν-fold inner tensor product of the ν left quasi-regular representation 1 of G on L 2 (Q); we write M ⊗ 1 . As outlined before, we call a bounded selfadjoint T on L 2 (M) a transfer operator if it is positive as well as positivity improving. Positivity of T means (ψ, Tψ) ≥ 0,
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which is equivalent to the spectrum being nonnegative. Typical transfer operators are also positivity improving because they arise as integral operators with a positive kernel; the formalization as a positivity improving map turned out to be useful, see [44], p. 201 ff. For convenience we recall the definitions: a function M m → ψ(m) ∈ C has some property almost everywhere (a.e.) if it holds for all m ∈ M\I with γ (I ) = 0. A nonzero function is called positive if ψ ≥ 0 a.e. and strictly positive if ψ > 0 a.e. Then T is called positivity preserving if (Tψ)(m) ≥ 0 a.e. and positivity improving if (Tψ)(m) > 0 a.e. for any positive ψ. Equivalently T is positivity improving iff (φ, Tψ) > 0 for all positive φ, ψ ∈ L 2 ; see [44] p. 202. Our notion of transfer operators requires an additional condition: Definition 2.1. A transfer operator T is a positive integral operator, given by (Tψ)(m) = dγ (m ) T (m, m )ψ(m ) ,
(2.1)
where the kernel T : M × M → R+ is symmetric, continuous and strictly positive, i.e. T (m, m ) > 0 a.e. and satisfies sup dγ (m ) T (m, m ) < ∞ . (2.2) m
The second condition is sufficient (but by no means necessary) to ensure that T defines a bounded operator from L p to L p for 1 ≤ p ≤ ∞; see [30], p.173 ff. The operator norm T L p →L p = sup φ p =1 Tφ p is bounded by the integral in (2.2) and coincides with it for p = 1, ∞. Positivity of the kernel entails that T is positivity improving. Positivity of the operator (that is, of its spectrum) does not follow from this. However if it is not satisfied we can switch to T2 and the associated integral kernel, where positivity is manifest. Without much loss of generality we assume therefore the kernel to be such that T is positive. As a bounded symmetric operator on L 2 the integral operator defined by T (m, m ) has a unique selfadjoint extension which we denote by the same symbol T. The kernel of Tt will be denoted by T (m, m ; t) for t ∈ N. In this situation T and all its powers are transfer operators in the sense of the previous definition. An invariant dynamics is specified by a G-invariant transfer operator, i.e. one which commutes with M on L 2 .
M (g) ◦ T = T ◦ M (g) , ∀ g ∈ G .
(2.3)
It is easy to see that T then cannot have normalizable ground states (see Proposition 3.1 below). In this situation one will naturally search for generalized eigenstates of T, which in our case will be simply solutions of T = T with an almost everywhere defined function (not just a distribution) on M. The set of generalized ground states forms a linear space which we call the ground state sector G(T) of T. 2.2. The spectral problem. Since T commutes with the G action M , one expects that the transfer operator and a set of operators Z whose diagonal action characterizes an irreducible representation can be diagonalized simultaneously. In a sense this is correct and the generalized eigenfunctions with spectral value T in fact belong to a special irreducible representation of G, see Theorem 1.1. The purpose of this interlude is to explain why the general results available in the literature on such (simultaneous) spectral decompositions are insufficient to produce the generalized ground states sought for.
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Let A be a bounded selfadjoint operator on the separable Hilbert space L 2 (M). Let ⊂ L 2 (M) ⊂ be a Gel’fand triple [17, 6] (rigged Hilbert space) for A. An element ∈ is called an eigendistribution or generalized eigenstate of A with spectral value λ ∈ Spec(A) if (φ, (A − λ)) = 0, for all φ ∈ . The set of generalized eigenstates for some λ ∈ Spec(A) forms a linear subspace of which is called the generalized eigenspace Eλ (A) for the spectral value λ ∈ Spec(A). If L 2 (M) carries the unitary representation M of a connected (noncompact) Lie group G there is a natural action of G on the distributions ∈ , viz ( M (g) ◦ , φ) := (, M (g)−1 ◦ φ) for all φ ∈ and g ∈ G. Naturally is called invariant if M (g) ◦ = for all g ∈ G. If A commutes with M , M (g) ◦ A = A ◦ M (g) for all g ∈ G, one expects that the generalized eigenspaces can be decomposed into components irreducible with respect to M . Under mild extra conditions this is indeed the case. The nuclear spectral theorem (Gel’fand-Maurin theorem, [17, 34]) guarantees the existence of direct integral decompositions of the form L 2 (M) = dμ(λ, σ ) Eλ,σ (A) . (2.4) Spec(A)×G
is the dual of G and μ(λ, σ ) is a measure on Spec(A)× G defining the decompoHere G with a set of parameters sition. To simplify the notation we identified elements πσ of G σ uniquely specifying an equivalence class of unitary irreducible representations. The precise version of the nuclear spectral theorem can be found in [17, 34, 6]. The fiber spaces Eλ,σ (A) contain the generalized eigenfunctions of A in transforming irreducibly under G. The nuclear spaces are much smaller than L 1 , the dual spaces therefore much larger than L ∞ , and the generalized eigenfunctions supplied by the Gel’fand type constructions may be genuine distributions. The fact that A commutes with the elliptic Nelson operator of G (built from the Casimirs of G and K ) entails [34] that the (averaged) eigendistributions M (g) ◦ are smooth functions in g, but little can be said about their distributional type. A result by Berezanskii (described and proven in [34]) specifies sufficient conditions under which the dual space of a triple ⊂ L 2 ⊂ consists of almost everywhere defined functions (not distributions). We shall make use of this result for the ‘relative motion’ alluded to in the introduction. Irrespective of the distributional type of the generalized eigenfunctions the decomposition (2.4) has however two important drawbacks: – generalized eigenfunctions are assured to exist only for μ-almost all spectral values λ ∈ Spec(A), and the measure μ is usually not known explicitly. λ ⊂ G – whenever generalized eigenfunctions exist for a given λ ∈ Spec(A), let G denote a set that carries the restricted measure in (2.4). Then the spaces Eλ,σ (A) are λ . assured to be irreducible only for μ-almost all σ ∈ G The first of these ‘almost all’ caveats presents a major obstruction if one wants to apply the general framework to a specific spectral value, like T , the ground state value, which is our main concern here. The following example illustrates the problem. Let T be the integral operator on L 2 (R+ ) defined by the kernel T (x, y) = e−|x−y| . It can be seen to be a transfer operator in the above sense with spectrum Spec(T) = [0, 2]. For all spectral values different from λ = 2 there exist generalized L ∞ eigenstates, yet the operator does not have a generalized ground state. The point to observe is that for all ψ ∈ L 2 (R+ ) the image function (Tψ)(x) is twice differentiable with
(Tψ) = (Tψ) − 2ψ .
(2.5)
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All must be linear combinations of e±iωx with ω = √ solutions of Tψ = λψ therefore 2 2/λ(ω) − 1, i.e. λ(ω) = 2/(1 + ω ). One finds 1 [sin ωx + ω cos ωx] = cos(ωx − b) , ω ≥ 0 , b = arccot ω , ψω (x) = √ 1 + ω2 (2.6) where the normalization has been chosen such that ψω ∞ = 1. The explicit construction shows that ψω ∈ L ∞ , although the space of test functions is slightly smaller than L 1 in that twice differentiable L 1 functions φ have to satisfy φ(0) = ∂x φ(0) (which can be seen by averaging Eq. (2.5) with a test function). The fact that the generalized eigenfunctions (2.6) also satisfy ψω (0) = ∂x ψω (0) ensures their completeness; it is easy to verify that ∞ π dω ψω (x)ψω (y) = δ(x − y) , x, y ≥ 0 , 2 0 ∞ d x ψω1 (x)ψω2 (x) = 0 , ω1 = ω2 . (2.7) 0
Using Eq. (2.7) we find the following spectral resolution of the integral kernel T : n 2 2 ∞ dω ψω (x)ψω (y) . (2.8) T (x, y; n) = π 0 1 + ω2 For example 1 T (x, y; 2) = e−|x−y| (1 + |x − y|) − e−(x+y) . 2
(2.9)
Despite these nice properties no generalized ground state exists. This is because candidates for it must be contained in the set (2.6); however the relevant limit (λ → 2 i.e. ω → 0) vanishes pointwise, while the closest maximum of ψω , lying at b(ω)/ω ∼ π/(2ω), moves out to ∞. The upshot is that the ‘almost all’ caveat in the general theorems is crucial for their validity and renders them at the same time useless for the construction e.g. of the ground state sector. Even transfer operators with a complete system of regular (here: L ∞ ) generalized eigenfunctions may fail to have a ground state. This explains why in the constructive Theorem 5.1 certain subsidiary conditions must be present; we do expect however that the ones given can still be weakened. In Sect. 3 we will analyze the spectral problem for invariant selfadjoint operators A as defined in Definition 3.1. Under mild subsidiary conditions ((C) in Sect. 3.3 and (C1), (C2) in Sect. 3.4) a complete set of eigenfunctions λσ in Eλσ (A) can be found. Some of the properties of the λσ have been anticipated in the introduction. In this context it is worth emphazising two points. First, in contrast to the familiar situation with normalizable eigenfunctions the existence of a ∈ such that A = λ does in itself not imply λ ∈ Spec(A). A simple counterexample is the hyperbolic spin chain discussed in detail in [38]: there the constant functions are eigenfunctions of the transfer operator T, but the corresponding eigenvalue lies above the spectrum of T. When solving the spectral problem A = λ with a nonnormalizable ∈ , the information that λ is a point in the spectrum therefore has to be supplied independently. A second point worth repeating is that all known construction principles for generalized eigenfunctions
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(including the one presented in Sect. 3) are guaranteed to work only for almost all points in the spectrum. For a prescribed λ ∈ Spec(A) additional considerations are necessary to show that sufficiently many eigenfunctions exist. This applies in particular to the σ -equivariant eigenspaces Eλσ (A) of an invariant selfadjoint operator A and to the ground state fibers Gσ (T) := E T ,σ (T) of a transfer operator. Whenever the fiber spaces Eλ,σ (A) in (2.4) are nonempty for a fixed λ ∈ Spec(A) one can match the decomposition in (2.4) with the purely group theoretical one. Since
M is a unitary representation on general grounds it can be decomposed into irreducible such that components [9]. That is, there exists a measure μ M on G L 2 (M) =
⊕ G
dμ M (σ ) L2σ (M) ,
(2.10)
in the support of the where the fibers L2σ (M) are irreducible for μ M -almost all σ ∈ G measure. On the other hand from (2.4) one can define generalized eigenspaces Eλ (A) by Eλ (A) = dμ(λ, σ ) Eλ,σ (A) , (2.11) λ G
for which there is a non-empty eigenspace Eλ,σ (A). λ denotes the part of G where G Thus Spec(A) dμ(λ, σ ) = dμ M (σ ). There are no obvious strategies to determine λ ⊂ G of a given spectral value λ (for which generalized the representation content G eigenfunctions exist). For a transfer operator T we identify its ground state sector G(T) with E T (T), and similarly for the equivariant fibers Gσ (T) = E T ,σ (T). We shall thus T for its apply the decomposition (2.11) also for the ground state sector and write G representation content. One of the main goals later on will be to show that under mod T consists of a single point (a single representation) only, erate extra assumptions G which is always the same for all transfer operators considered.
2.3. Orbit decomposition. A simple but crucial fact about the configuration manifolds M = Q × · · · × Q is that they have a well defined orbit decomposition which eventually carries over to the states and the operators acting on them. The idea of the decomposition is to single out one of the variables in m = (q1 , . . . , qν ), say q1 , to parameterize the location on the orbits and to define coordinate functions n i transversal to it to describe the relative location of the points such that they change only by elements of K as one moves along an orbit. To this end we fix some q ↑ ∈ Q = eK with isotropy group K , i.e. kq ↑ = q ↑ for all k ∈ K . Based on it we wish to define a section gs : Q → G such that q = gs (q)q ↑ for all q ∈ Q. Clearly this condition defines gs only up to right multiplication by some ks = ks (g, q) ∈ K , gs (gq) = ggs (q)ks (g, q) .
(2.12)
For gs to be well-defined the element k has to be uniquely determined for given g and q. It is easy to see that this is the case whenever G admits an Iwasawa decomposition, which is the case for all connected simple noncompact Lie groups, in particular the ones considered. Consistency requires the cocycle condition ks (g1 g2 , q) = ks (g2 , q)ks (g1 , g2 q), in particular ks (e, q) = e for all q ∈ Q and e ∈ G the identity. If we normalize gs such that gs (q ↑ ) = e it follows that ks (gs (q), q ↑ ) = e = ks (gs (q)−1 , q) and ks (g, q) =
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ks (ggs (q), q ↑ ). For elements k ∈ K of the subgroup one has ks (k, q ↑ ) = k −1 . Generally the Iwasawa decomposition entails that the cocycle ks ( · , q) : G → K is surjective for all q ∈ Q. Our main example for the symmetric space Q will be H N = SO0 (1, N )/SO(N ), the N -dimensional hyperboloid with the Riemannian metric γH N induced by the indefinite metric q · q = (q 0 )2 − (q 1 )2 − · · · − (q N )2 in the imbedding linear space. Explicitly H N = {q ∈ R1,N | q · q = (q 0 )2 − (q 1 )2 − · · · − (q N )2 = 1, q 0 > 0}. The invariant measure on H N is dγH N (q) = d N +1 qδ(q 2 − 1)θ (q 0 ) and will be denoted by dq for short. The ν-fold product M = HνN equipped with the product metric then is a simply connected Riemannian manifold (and, in fact, a reducible symmetric space). ν We denote the product measure i=1 dqi on M by dγM . For G = SO0 (1, N ) and K = SO(N ) the section gs (q) is just the familar expression for the pure boost mapping q ↑ = (1, 0, . . . , 0) to q = (q0 , q1 , . . . , q N −1 ) =: (q0 , q). Explicitly ⎛ ⎞ q0 q T ⎠. (2.13) gs (q) = ⎝ q 1I + q01+1 q q T It satisfies gs (q) = e iff q = q ↑ . The cocycle ks (g, q) = gs (q)−1 g −1 gs (gq) based on this section is known as ‘Wigner rotation’ [59] and satisfies ks (k, q) = k −1 for all q ∈ Q, k ∈ K ,
(2.14)
that is, not only for q = q ↑ . Using the section gs we now define the following diffeomorphism on M = Q ν : ϑ(q1 , . . . , qν ) = (q1 , gs (q1 )−1 q2 , . . . , gs (q1 )−1 qν ) =: (q1 , n 2 , . . . , n ν ) , ϑ −1 (q1 , n 2 , . . . , n ν ) = (q1 , gs (q1 )n 2 , . . . , gs (q1 )n ν ) .
(2.15)
This diffeomorhism is measure preserving due to the invariance of the measures dqi : let f ∈ L 1 (dγM ) and dn := dγN (n) := i=1 dγ Q (n i ). Then dqi f (q1 , gs (q1 )−1 q2 , . . . , gs (q1 )−1 qν ) ( f ◦ ϑ)(q1 , n)dγM (m) = dq1 =
dq1
i=1
dqi f (q1 , . . . , qν ) =
dγM (m) f (m) .
(2.16)
i=1
So the measure dγM can also be factorized as dγM (m) = dq1 dn ,
(2.17)
The product Q × N equipped with the product metric γ Q × γN and the measure dq dn is a Riemannian manifold Ms which by construction is isometric to M, and with the isometry given by the above ϑ: ϑ : (M, γ ) −→ (Ms , γs ) := (Q × N , γ Q × γN ) .
(2.18)
The manifold Ms also has the structure of a G space which it inherits from M. Recall that the transversal coordinate functions n i = n i (q1 , qi ) ∈ Q are defined by n i :=
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gs (q1 )−1 qi , for i = 2, . . . , ν. On them M acts by M (g)(n i ) = gs (g −1 q1 )−1 g −1 qi = ks (g −1 , q1 )−1 n i . The original diagonal action d(g)m = (gq1 , . . . , gqν ) becomes a twisted action ds (depending on a choice of section) in the coordinates (q, n), i.e. ϑ ◦d = ds ◦ ϑ, with ds (g −1 )(q, n) = (g −1 q, ks (g −1 , q)−1 n) .
(2.19)
As ds (gs (q))(q0 , n) = (q, n) and ds (gs (q)−1 )(q, n) = (q0 , n) it acts transitively on the first variable. However as one moves along Q the action ds co-rotates the n variables in a q-dependent way. For a generic symmetric space this happens even on the subgroup K ; for H N and the Wigner rotation one has ds (k −1 )(q, n) = (k −1 q, k −1 n), though. In the terminology of [60], Sect. 4, Ms := Q ×ks N is the skew product G-space induced from the K -space N . Indeed, N equipped with the diagonal action of K : dN (k)n = (k −1 n 2 , . . . , k −1 n ν ) is a K -space, and by construction (M, d) and (Ms , ds ) are isometric as G spaces. Since the cocycle is surjective for fixed q we gained a less redundant description of the space of orbits: space of orbits: M/d(G) = Ms /ds (G) = N /dN (K ) ,
(2.20)
where, importantly, K is compact. On the other hand, the twisted action (2.19) is cumbersome when one tries to decompose the unitary representation based on ds into irreducible components. But the left twisted action ds can be traded for an untwisted right action r by the following construction (based on a remark in [60], p.75). Consider Mr := (G × N )/d(K ),
(2.21)
that is, the space of equivalence classes (g, n) ∼ (k −1 g, k −1 n), k ∈ K in G × N . In order not to clutter the notation we also write (g, n) for the equivalence class generated by a point in G × N . Maps and functions on G × N that are constant on d(K ) orbits then lift unambiguously to maps and functions on Mr . On G × N and Mr we define a right G-action r by r (g )(g, n) = (gg , n) ,
(2.22)
which is just the standard right action of G on itself leaving the n variables untouched. The action (2.22) is constant on the equivalence classes because the right r (G) action and the left d(K ) action on G ×N commute, r (g)d(k) = d(k)r (g). In fact Mr equipped with the right G action is isomorphic to the original manifold M = Q ν with the diagonal left action M (G) (and thereby also to the skew product G space Ms = Q ×ks N ). The isomorphism is given by first considering the following map φ˜ : M → G × N : φ˜ : M −→ G × N , ˜ 1 , . . . , qν ) = (gs (q1 )−1 , gs (q1 )−1 q2 , . . . , gs (q1 )−1 qν ) . φ(q
(2.23)
This map is injective, but not surjective. Because gs is a global section of G/K its range intersects each d(K ) orbit exactly once, so that it determines uniquely a diffeomorphism φ : M −→ Mr .
(2.24)
We define an inverse of φ˜ −1 : G × N −→ M (initially only defined on the range of ˜ by φ) φ˜ −1 (g, n) = d(g −1 )(q ↑ , n) .
(2.25)
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We can immediately interpret this as a map from all of G × N to M which is constant on the equivalence classes under d(K ) φ˜ −1 (kg, kn) = d(kg)(q ↑ , kn) = (g −1 q ↑ , g −1 n) = φ˜ −1 (g, n) ,
(2.26)
and hence lifts to a map φ −1 : Mr −→ M. By direct computation one verifies φ˜ ◦ φ˜ −1 = id ,
(2.27)
φ˜ only maps orbits of d(K ) into themselves. But this is enough to see are really inverse to each other. According to (2.22) the map φ intertwines the left action d(G) with the right action r (G): whereas φ˜ −1 ◦ that φ and φ −1
φ◦d =r ◦φ .
(2.28)
In addition the map φ is measure preserving; this can be seen similarly as the measure preserving property of ϑ: consider a L 1 function f on M and let fr = f ◦ φ −1 . Then, using the G-invariance of the measure dn and the fact that the invariant measure dq is the push-forward of Haar measure dg under the canonical projection G −→ G/K , one sees that ν dγ Q (qi ) f (q1 , . . . , qν ) = dg dn fr (g, n) . (2.29) i=1
For completeness we also note explicitly the isometry χ = φ ◦ ϑ −1 between the skew product Q space Ms = Q ×ks N and the G-manifold Mr with the diagonal right action: χ : Q ×ks N → Mr , χ −1 : Mr → Q ×ks N ,
χ (q, n) = (gs (q)−1 , n) , χ −1 (g, n) = (g −1 q ↑ , ks (g −1 , q ↑ )−1 n) .
(2.30)
In summary we have two equivalent descriptions of the original G-manifold (M, d), namely (Ms , ds ) and (Mr , r ). The second one is more convenient for the reduction problem because the space of orbits M/d(G) = Mr /r (G) is now described by equivalence classes with respect to the usual right action of G on itself. This structure of course carries over to the function spaces and the unitary representations on them induced by the G-actions. We have L 2 (M) ψ → ( M (g)ψ)(m) = ψ(d(g −1 )(m)) , (2.31a) 2 −1 L (Ms ) ψs → ( s (g)ψs )(q, n) = ψs (ds (g )(q, n) , (2.31b) 2 L (Mr ) ψr → (ρ(g0 )ψr )(g, n) = ψr (r (g0 )(g, n)), r (K )ψr = ψr , (2.31c) where r (k)ψr (g, n) = ψr (k −1 g, k −1 n). As Hilbert spaces of course all three L 2 spaces are isometric to L 2 (Q × N ) and the three representations M , s , and ρ are likewise unitarily equivalent. Explicitly : L 2 (Mr ) −→ L 2 (M) , (ψr )(m) := ψr (φ(m)) , ρ = −1 ◦ M ◦ ,
(2.32)
and similarly for L 2 (Ms ). By (2.31c) L 2 (Mr ) can also be identified with the subspace invariant under r (K ) of L 2 (G × N ). By (2.29) the map is indeed an isometry. As (−1 ◦ M (g0 ) ◦ ψr )(g, n) = ψr (φ ◦ d(g0−1 ) ◦ φ −1 (g, n)) the unitary equivalence of the representations follows from (2.28). We summarize our results in
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Proposition 2.1. There is a diffeomorphism φ from the configuration manifold M = Q ν to Mr = (G × Q ν−1 )/d(K ) such that the diagonal left action d(G) on M gets mapped into the right action r (G) on the first factor of Mr . φ is measure preserving for the natural measures on M and Mr and therefore induces a natural isomorphism of the spaces L 2 (M) and L 2 (Mr ). 2.4. The reduction of ρ(G) on L 2 (Mr ). With these preparations at hand we can now address the reduction problem of M (G) in the variant where it acts as ρ(G) on L 2 (Mr ). As noted above, the latter is the r (K ) invariant subspace of L 2 (G×N ), which carries the commuting unitary representations ρ(G) and r (K ). Moreover ρ(G) for fixed n ∈ N is just the right regular representation of G mapping ψr (g, n) into ρ(g0 )ψr (g, n) = r , is ψr (gg0 , n). Its decomposition into unitary irreducible representations πσ , σ ∈ G thus given by the Plancherel decomposition (A.1). The precise form used and the notar , tions are summarized in Appendix A. In particular dν is the Plancherel measure on G the restricted dual of G, and g → πσ (g) denotes the irreducible representation assor . It acts on a separable Hilbert space Lσ with inner product ciated with some σ ∈ G (·, ·)σ . Applying the expansion (A.2) to the G-part of a function in ψ : G × N → C gives (σ, n)] , dν(σ ) Tr[πσ (g) ψ ψ(g, n) = r G (2.33) −1 (σ, n) = dg πσ (g )ψ(g, n) . ψ G
For functions ψ that are r (K ) invariant this will lead to the desired decomposition of L 2 (Mr ) into irreducible components. We consider here first the decomposition of the larger space L 2 (G × N ). Provided suitable conditions are imposed on the function (σ, n) for fixed n are trace class or ψ (which we describe shortly) the transforms ψ Hilbert-Schmidt operators on Lσ . Further they satisfy (σ, n)πσ (g −1 ) [ρ(g0 ) (g1 )ψ] (σ, n) = πσ (g0 )ψ 1 (σ, n) , = (πσ × πσˇ )(g0 , g1 )ψ
(2.34)
using (A.6) and the notation πˇ σ = πσˇ in the last equation. This states that the map intertwines the outer tensor product ρ × of the left and the right regular ψ → ψ representation of G with πσ × πσˇ . Throughout we shall adopt the following conventions for compact operators A, B on some separable Hilbert space H with orthonormal basis ei , i ∈ N, and its dual space Hˇ with dual basis eˇi , i ∈ N: A= ei Ai j eˇ j , [A† ]i j = A∗ji . (2.35) ij
Compact operators that are even trace class arise for example as Fourier transforms of functions ψ ∈ D, where D is the space of functions ψ(g, n) that are smooth with compact support in g ∈ G and square integrable in n ∈ N : for such ψ the Fourier transform (σ, n) is a trace class operator on a separable Hilbert space Lσ , for all ψ ∈ D, and ψ r , n ∈ N . Moreover the Fourier expansion (2.34) then is valid pointwise almost all σ ∈ G in g. If ψ is in L 2 (G) ∩ L 1 (G) as a function of g and square integrable in n the Fourier
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r , n ∈ N . (σ ) are still Hilbert-Schmidt operators for almost all σ ∈ G coefficients ψ We identify the trace class operators with a subspace of Lσ ⊗ Lˇ σˇ , which in turn can be identified with the space of Hilbert-Schmidt operators on Lσ ; see Appendix A3. This (σ, · ) : N → Lσ ⊗ Lˇ σˇ . The Parseval identity, means the coefficients are functions ψ (σ, n)† ψ(σ, n )] , dg φ(g, n)∗ ψ(g, n ) = dν(σ ) Tr[φ (2.36) G
r G
is valid for all functions in L 2 (Mr ) which for fixed n lie in L 1 (G) ∩ L 2 (G). It implies (σ, n)] is integrable with respect to dν(σ )dn; hence it is an (σ, n)† ψ that the trace Tr[ψ r . integrable function on N for almost all σ ∈ G r with the structure of a Hilbert space This suggests to equip the fibers at fixed σ ∈ G 2 which we denote by L σ σˇ (N ). We shall also need various subspaces of L 2σ σˇ (N ) and for convenient reference we collect them in the following definition: Definition 2.2. The Hilbert spaces L 2σ σˇ (N ) := F : N → Lσ ⊗ Lˇ σˇ (F, F)σ σˇ < ∞ ∼ = Lσ ⊗ Lˇ σˇ ⊗ L 2 (N ) , (2.37a) L 2σ σˇ (N )0 := F : N→Lσ ⊗ Lˇ σˇ (F, F)σ σˇ < ∞, F(k −1 n)πσ (k)† = F(n) , (2.37b) where (F1 , F2 )σ σˇ := dn Tr[F1† (n)F2 (n)] are called the fiber spaces of L 2 (G × N ) be the unitary dual of K , κ ∈ K , and Vκˇ ⊂ Lσˇ the and L 2 (Mr ), respectively. Let K subspaces in Eq. (2.45) below. Then L 2σ κˇ (N ) := { f : N → Lσ ⊗ Vκˇ | ( f, f )σ κˇ < ∞} ∼ = Lσ ⊗ Vκˇ ⊗ L 2 (N ) , (2.38a) L 2σ κˇ (N )0 := { f : N → Lσ ⊗ Vκˇ | f (kn) = f (n)rκ (k)† , ( f, f )σ κˇ <∞} , (2.38b) where ( f 1 , f 2 )σ κˇ := dnTr Vκˇ [ f 1 (n)† f 2 (n)] are called the κ-channels of L 2σ σˇ (N ) and L 2σ σˇ (N )0 , respectively. The adjoints of the singlet channels κ = 0 lead to spaces L 2σ (N) := f : N→Lσ dn( f (n), f (n))σ < ∞ ∼ = Lσ ⊗ L 2 (N ) , (2.39a) L 2σ (N )0 := { f ∈ L 2σ (N ) | f (kn) = πσ (k) f (n)} .
(2.39b)
With the definition (2.37a) the Fourier transformation (2.34) becomes an isometry ⊕ ∈ D ψ −→ ψ dν(σ ) L 2σ σˇ (N ) , , ψ )σ σˇ , dν(σ ) (ψ dgdn ψ(g, n)∗ ψ(g, n) =
(2.40)
r G
which extends uniquely to an isometry between Hilbert spaces. Since the trace class operators form an ideal in the algebra of all bounded linear operators on Lσ for all ψ ∈ D the (σ, n)] is defined pointwise for all (g, n) ∈ G ×N and it is a continuous trace Tr[πσ (g) ψ (σ, n)πσ (g1 )† is function in g. For the same reason [ρ(g0 ) (g1 )ψ] (σ, n) = πσ (g0 )ψ
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(σ, n) is. As a consequence L 2 (N ) carries a a trace class operator for all g0 , g1 if ψ σ σˇ unitary representation πσ σˇ of G × G, πσ σˇ (g0 , g1 )F(n) := πσ (g0 )F(n)πσ (g1 )† , πσ σˇ (g0 , g1 )F1 , πσ σˇ (g0 , g1 )F2 σ σˇ = F1 , F2 σ σˇ .
(2.41)
It coincides with πσ × πσˇ , the outer tensor product of the two representations (which is irreducible [15], Thm 7.20]) whenever both of the factors are. The isometry (2.41) therefore also provides the decomposition of ρ × , the outer tensor product of the right and the left regular representation of G into a direct integral of (ν-almost everywhere) irreducible representations, ρ × = dν(σ ) πσ × πσˇ . (2.42) With these preparations at hand we can turn to the decomposition of L 2 (Mr ), which we naturally identified with the subspace of L 2 (G × N ) consisting of left K invariant functions. Clearly the left K invariance of the function ψ translates into the following condition on the Fourier coefficients: ! (σ, k −1 n)πσ (k)† = ψ (σ, n) . [ r (k)ψ] (σ, n) = ψ
(2.43)
We also introduce the corresponding K -singlet subspace of L 2σ σˇ (N ) as in (2.37b). Since πσ† ‘acting from the right’ is unitarily equivalent to πσˇ , in representation theoretical terms (2.43) means !
N × πσˇ | K = id,
(2.44)
where N (k)F(n) := F(k −1 n). The condition (!) can be understood as the projection onto the subspace of left K singlets in a decomposition of L 2 (G) ⊗ L 2 (N ) which we prepare now. First recall that the restriction of πσˇ to the subgroup K decomposes as follows: m κˇ rκˇ , Lσˇ = m κˇ Vκˇ . (2.45) πσˇ | K = σˇ K
σˇ κ∈ ˇ K
σˇ ⊂ K for which the irreducible representations rκˇ on the finite dimenHere the subset K sional vector space Vκˇ occurs with nonzero multiplicity m κˇ is called the K content of πσˇ ; see Appendix A. Often it is convenient to use a basis of Lσˇ obtained by concatenation of the bases eκs ˇ , s = 0, . . . , m κ dim Vκˇ − 1, of m κˇ Vκˇ . (Here of course for fixed κˇ the basis vectors eκs ˇ , s = 0, . . . , dim Vκˇ −1, eκs ˇ , s = dim Vκˇ , . . . , dim 2Vκˇ −1, etc are likewise orthogonal.) We shall call eκs ˇ , s = 0, . . . , m κˇ dim Vκˇ − 1, κˇ ∈ K σˇ ,
(2.46)
the K -adapted basis of Lσˇ . For operators F ∈ L 2σ σˇ (N ) the components with respect to an orthonormal basis {ei , i ∈ N}, and its dual {eˇi = (ei , · )σ , i ∈ N} are Fi j := eˇi (Fe j ), so that F = i j ei Fi j eˇ j . In the K -adapted basis these become Fκs,κ s = eκs ˇ (Feκ s ). In view of (2.44) one has to decompose the representation ρ × | K × N of G × K in order to decompose L 2 (G × N ) ∼ = L 2 (G) ⊗ L 2 (N ). Since the group that is represented is really only G × K , the second tensor product should be read as an inner one; however
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for our analysis it is convenient first to regard it as an outer one too: combining (2.42) with (2.45) gives for the first factor ⊕ ρ × | K = dν(σ ) πσ × m κˇ rκˇ . (2.47) σˇ κ∈ K
For the second factor we write m Nκ rκ ,
N =
L 2 (N ) =
κ
κ
m Nκ L 2κ (N ) ,
(2.48)
where L 2κ (N ) is the subspace of functions transforming irreducibly according to f (k −1 n) = rκ (k) f (n) and m Nκ are some multiplicities. Combining (2.47) and (2.48) results in L 2 (G) ⊗ L 2 (N ) = dν(σ ) Lσ ⊗ m κˇ Vκˇ ⊗ m Nκ L 2κ (N ) . (2.49) κ,κ
The Fourier coefficients transform according to (σ, n) = πσ (g)ψ (σ, n)(rκˇ × rκ )(k, k ) . (ρ × | K × N )(g, k, k )ψ
(2.50)
It remains to implement the condition (!) in (2.43), (2.44). Since ψ(k −1 g, k −1 n) = ( | K × N )(k, k)ψ(g, n) this amounts to considering now the inner tensor product
| K ⊗ N and projecting onto the singlet sector. The reduction of | K ⊗ N produces in a first step a direct double sum over κ, κ of terms of the form rκˇ ⊗rκ with multiplicities m κ m Nκ . In the next step we use that rκˇ ⊗ rκ contains the singlet if and only if κ = κ . The latter readily follows from the general result of [4, 5] (described in Appendix A4) on the singlet content of a tensor product of two unitary representations. As a consequence the direct double sum in (2.49) reduces to a single sum and one can check that the condi(σ, n)(rκˇ × rκ )(k, k) = ψ (σ, k −1 n)rκ (k −1 ) = ψ (σ, n). tion (!) comes out correctly: ψ −1 Viewed as functions of n alone the Fourier coefficients obey f (k n) = f (n)rκ (k) or f (kn) = f (n)rκ (k)† . Therefore, in order to go from L 2 (G × N ) to L 2 (Mr ) it is useful to consider the subspaces L 2σ κˇ (N ) of L 2σ σˇ (N ) and L 2σ σˇ (N )0 defined in (2.38). Here we interpret the elements of Lσ ⊗ Vκˇ as linear maps from Vκ ⊂ Lσ to Lσ ; the trace in the inner product (2.37) for generic Hilbert-Schmidt operators reduces to a trace on Vκ for operators with values in Lσ ⊗ Vκˇ . In components ∗ f 1 (n)iκs f 2 (n)iκs , f (n) = ei f (n)iκs eˇκs , (2.51) Tr Vκ [ f 1 (n)† f 2 (n)] = i,s
i,s
where ei , i ∈ N, is a basis on Lσ and eˇκs , s = 0, . . . , m κ dim Vκ −1, is a basis of linear forms on m κ Vκ . The space L 2σ κˇ (N ) carries a unitary representation πσ κˇ of G × K , πσ κˇ (g, k) f (n) := πσ (g) f (n)rκ (k)† , Tr Vκ [[πσ κˇ (g, k) f 1 (n)]† πσ κˇ (g, k) f 2 (n)] = Tr Vκ [ f 1 (n)† f 2 (n)] .
(2.52)
Note that the product πσ (g) f (n) still transforms nontrivially under the left diagonal action of K according to πσ (g) f (n) → πσ (kg) f (kn) = πσ (k)[πσ (g) f (n)]rκ (k −1 ). However, introducing Pκ as the orthogonal projection from L 2 (G) onto the subspace
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transforming according to κ, traces of the form Tr Vκ [Pκ πσ (g) f (n)] are invariant under the left diagonal action of K . Finally we arrive at the desired decomposition of L 2 (Mr ) into a direct integral of irreducible spaces L 2 (Mr ) = dν(σ ) m Mκˇ L 2σ κˇ (N )0 , (2.53) r G
σ κ∈ K
for some multiplicities m Mκˇ . The representation ρ itself has been decomposed into a direct integral of primary representations (that is, [15], p. 206, ones which are direct sums of identical copies of some irreducible representation). Here ρ acts as the right regular representation of G on the first argument of the functions L 2 (Mr ), i.e. ρ(g0 )ψ(g, n) = ψ(gg0 , n). The left K invariance of the functions is broken up into the different κ ‘channels’ on the Fourier coefficients; generic left K invariant functions are built from simple ones. Consistency with the initial decomposition (2.41) fixes the multiplicities m Mκ = m κ ,
(2.54)
where m κ are the multiplicities occurring in the decomposition of πσ | K , see (2.45). Indeed, disregarding the breakup into the irreducible κ channels the fiber spaces have the following isometric descriptions: L2σ (Mr ) = m Mκˇ L 2σ κˇ (N )0 ∼ = L 2σ σˇ (N )0 ⊂ L 2σ σˇ (N ) σ κ∈ K
∼ = Lσ ⊗
m Mκˇ Vκˇ ⊗ L 2 (N ) ,
(2.55)
σ κ∈ K
where we used (2.38) in the first isometry and inferred (2.54) from the required match with L 2σ σˇ (N ) ∼ = Lσ ⊗ Lˇ σˇ ⊗ L 2 (N ). The significance of the breakup into the irreducible κchannels can be seen more clearly by identifying the pre-images of the functions in dν(σ ) m Mκˇ L 2σ κˇ (N )0 with on L 2 (G × N ). Here we take L 2 (N ) to consist of respect to the isometry ψ → ψ σ κˇ σ . Roughly, the pre-image consists of functions invariant the zero vector only, if κ ∈ / K σ . Specifically we set under left convolution with the character χκˇ of κˇ ∈ K (E κ ∗ ψ)(g, n) := dκ dk χκ (k −1 )ψ(kg, n) , (2.56) L 2κ (Mr ) := ψκ ∈ L 2 (Mr ) E κ ∗ ψκ = δκκ ψκ . The properties of the functions in this subspace are summarized in the following lemma. Lemma 2.1. (a) For ψκˇ ∈ L 2κˇ (Mr ), the Fourier coefficients obey 0 for κ = κ , ψκˇ Pκˇ = 2 ψκˇ ∈ L σ κˇ (N )0 for κ = κ .
(2.57)
In terms of components with respect to the K -adapted basis (2.46) this is equivalent to κˇ (σ, n)κ1 s1 ,κ2 s2 = f κˇ (n)κ1 s1 ,s2 δκκ2 , with f κˇ ∈ L 2 (N )0 . ψ σ κˇ
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(b) For ψκˇ ∈ D ∩ L 2κˇ (Mr ) (i.e. g → ψκˇ (g, n) is smooth in g with compact support) we have κˇ (σ, n)] . ψκˇ (g, n) = dν(σ ) Tr Vκˇ [Pκˇ πσ (g) ψ (2.58) r G
(c) For φκˇ , ψκˇ ∈ L 2κˇ (Mr ), a Parseval identity holds κˇ (σ, n)] . κˇ (σ, n)† ψ dν(σ ) dnTr[φ dgdnφκˇ (g, n)∗ ψκˇ (g, n) = r G
(2.59)
Proof. (a) We compute (E κ ∗ ψ)(g, n) by inserting the Fourier decomposition (2.34) for ψ ∈ L 2 (Mr ). This gives (σ, n) dκ dk χκ (k −1 )πσ (k) πσ (g) . (E κ ∗ ψ)(g, n) = dν(σ ) Tr ψ =
r G
r G
K
(σ, n)Pκ ] . dν(σ ) Tr Vκ [πσ (g)ψ
(2.60)
In the second step we identified the K integral as the projector (A.14) onto the m κ Vκ κˇ , or in components with respect to the κˇ Pκˇ = δκκ ψ subspace of Lσ . This shows ψ κˇ (σ, n)κ1 s1 ,κ2 s2 = f κˇ (n)κ1 s1 ,s2 δκκ2 , with f κˇ ∈ L 2 (N )0 . The equiK -adapted basis ψ σ κˇ (σ, kn)Pκˇ = variance property in the definition of L 2σ κˇ (N )0 follows from (2.43) and ψ † † (σ, n)πσ (k) Pκˇ = ψ (σ, n)Pκˇ rκˇ (k) . The fact that f κˇ is square integrable in the norm ψ ( , )σ κˇ follows from the Plancherel identity (2.58). (b) Equation (2.58) follows from (2.60) and the definition of L 2κˇ (Mr )0 . Note that the trace is constant on the equivalence classes (g, n) ∼ (kg, kn), although the product κˇ (σ, n) itself transforms nontrivially under (g, n) → (kg, kn). πσ (g)ψ (c) This follows from (a) and (2.36). We add some comments on Lemma 2.1. First, in view of (2.53) the result can be κˇ in (2.58) provides a partial isometry summarized by stating that the map ψκˇ → ψ ⊕ 2 dν(σ )m Mκˇ L 2σ κˇ (N )0 . (2.61) L κˇ (Mr ) −→ r G
Next we discuss two special cases, where in the decomposition (2.58) only “class 1” representations occur (i.e. representations which contain a vector invariant under the action of the subgroup K ). The first case is that of ρ(K ) singlets in L 2 (Mr ). In the decomposition (2.34) functions obeying ψ(gk, n) = ψ(g, n) are characterized by Fou(σ, n) = ψ (σ, n), where P0 is the projector onto the singlet rier coefficients with P0 ψ sector (‘κ = 0’). In this case Eq. (2.34) reduces to (σ, n) πσ (g K )], ψ(g K , n) = dν(σ ) P0 [ψ G /K (2.62) −1 −1 dγG/K (g K ) ψ(g K , n) P0 πσ (K g ) . P0 ψ (σ, n) = G/K
Since the functions g → P0 πσ (g) are left K -invariant by definition only ‘class 1’ representations (with respect to K ) occur in (2.62). Specifically, consistency with the
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harmonic analysis on Q = G/K requires that only those (class 1) irreducible represenr ⊂G tions occur which enter the harmonic analysis on G/K . We have written G/K for this subset; as described in Appendix A.7 it contains the spherical principal series representations only. (σ, n) πσ (g K )] For Fourier coefficients obeying in addition (2.43) the projection P0 [ψ is invariant under (g, n) → (kg, kn). In this case the functions in (2.62) define a subspace of L 2 (Mr ). Note however that it is not ρ(G) irreducible; the latter requires Fourier . To proceed recall (σ, n)Pκ = P0 ψ (σ, n) for some κ ∈ K coefficients satisfying P0 ψ that the section gs : Q → G provides an injective imbedding of Q in G. We set E σ,κˇ (q) := Pκˇ πσ (gs (q))P0 ,
E σ,κˇ (q)† = P0 πσ (gs (q)−1 )Pκˇ ,
ψκˇ (q, n) := ψκˇ (gs (q)−1 , n) ,
(2.63)
in terms of which the expansion (2.58) takes the form κˇ (σ, n)] , dν(σ ) Tr[E σ,κˇ (q)ψ ψκˇ (q, n) = κˇ (σ, n) = ψ
G /K
(2.64)
dγ Q (q) ψκˇ (q, n)E σ,κˇ (q) . †
Q
Another subspace of L 2 (Mr ) in whose decomposition only class 1 irreducible representations appear is the singlet (κ = 0) sector of (2.53). On account of Lemma 2.1 this sector arises from functions ψ0 ∈ L 0 (Mr ), that is, functions obeying ψ0 (g, k −1 n) = ψ0 (g, n) = ψ0 (kg, n). The Fourier decomposition takes the form 0 (σ, n)] , ψ0 (g, n) = dν(σ ) P0 [πσ (g)ψ G /K (2.65) −1 0 (σ, n) = ψ 0 (σ, n)P0 = dg ψ0 (g, n)πσ (g )P0 . ψ We can now summarize the results on the group theoretical decomposition. Since the action of ρ(G) on L 2 (Mr ) corresponds to the original M action of G on L 2 (M) (not to be confused with the left regular representation of G on itself denoted by ) we the have found the desired group theoretical decomposition (2.10): only a subset of G, restricted dual G r appears, and each σ ∈ G r occurs with infinite multiplicity dim (Lσ ). r and the copies are counted with the The measure dμ M is the Plancherel measure on G . We continue to use the realization L 2 (Mr ), where the group counting measure on K acts via the right regular representation on the first argument of the functions ψ = ψr . For convenient reference we collect the results in: Proposition 2.2. (a) The Hilbert space L 2 (Mr ) decomposes under the action of the unitary representation ρ(G) according to dν(σ ) m κˇ L 2σ κˇ (N )0 =: dν(σ ) L2σ (Mr ) , (2.66) L 2 (Mr ) = r G
σ κ∈ K
r G
with L 2σ,κˇ (N )0 defined in (2.38), and m κ the multiplicities in (2.45). (b) Disregarding the breakup into irreducible κ channels the fiber spaces have the following descriptions m κˇ L 2σ κˇ (N )0 ∼ L2σ (Mr ) = = Lσ ⊗ Lˇ σˇ ⊗ L 2 (N ) ,(2.67) = L 2σ σˇ (N )0 ⊂ L 2σ σˇ (N ) ∼ σ κ∈ K
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with L 2σ σˇ (N ) defined in (2.37). (c) On the subspace of L 2 (Mr ) containing a ρ(K ) singlet the decomposition has support only on the spherical principal series representations. The content of Proposition 2.2 can be illustrated by matching (2.66) against a direct decomposition of L 2 (M) L 2 (Q)⊗ν into irreducibles. From (A.26) one has dω1 dω1 ··· Lω1 ⊗ · · · ⊗ Lων , (2.68) L 2 (Q)⊗ν 2 |c(ω1 )| |c(ω1 )|2 ⊂ Rdim A carries the spherical principal series represenation πω,0 . where Lω , ω ∈ Q The decomposition problem essentially amounts to decomposing arbitrary tensor products Lω1 ⊗ . . . ⊗ Lων of spherical principal series representations. This can be done inductively by first decomposing Lω1 ⊗ Lω2 = dμ(ω1 , ω2 |σ )Lσ , then Lσ ⊗ Lω3 for in the support of the measure dμ(ω1 , ω2 |σ ), and so on. Depending on what any σ ∈ G this strategy requires knowledge of is known about the support of dμ(ω1 , ω2 |σ ) in G large portions of the complete branching rules. Since the complete branching rules are known only for a few noncompact groups (like SL(2, R), see [36]) this would be tedious, to say the least. Information about the support of the measure dμ(ω1 , ω2 |σ ), seems to be available only in a few cases, like SO0 (1, N ) from [11]. According to Theorem 10.5 given there, a tensor product of two spherical principal series unitary irreducible representations (UIR) of SO0 (1, N ) decomposes into principal series UIR only, for N ≥ 4 even. (For N odd this holds trivially and for N = 2 it is manifestly not true, see [36]). For more than two tensor copies (or for N = 2), however, the result is insufficient to exclude the occurrence of UIR other than principal series representations. For ν = 3 one would need to know how the tensor product of a spherical and a generic (non-spherical) principal series decomposes. This might include discrete series representations and possibly others. So, for the direct decomposition of ν ≥ 4 fold tensor products of the spherical principal series, one needs to know essentially the complete branching rules. As a spin-off of Proposition 2.2 we have Corollary 2.3. (a) Let Lω1 ⊗ · · · ⊗ Lων be an arbitrary tensor product of spherical principal series representations. Then for almost all ω1 , . . . , ων with respect to the measure in (2.68) its decomposition does not contain the singlet (as the only finite dimensional UIR). (b) Let Q(ω1 , . . . , ων |σ )(q1 , . . . , qν ) be an intertwiner from Lω1 ⊗ · · · ⊗ Lων to Then, for almost all ω1 , . . . , ων , Lσ , σ ∈ G. dk Q(ω1 , . . . , ων |σ )(kq1 , . . . , kqν ) = 0 , (2.69) K
unless σ is again a spherical principal series representation. Part (a) generalizes a result by Fulling [16] for tensor products of spherical principal representations to all reductive linear Lie groups. Part (b) provides a nontrivial ‘sumrule’ whenever explicit expressions for the interwiners are available. The ‘almost all’ caveat is needed because the Harish-Chandra c-function defining the measure in (2.68) (viz. a ∗ , where a is the Lie is a meromorphic function over the complexification of Q C algebra of the subgroup A in the Iwasawa decomposition.) See e.g. [24], Chap., II.3. At the position of a pole the decomposition (2.68) yields no information about the fiber spaces, but otherwise all spherical principal series representations occur [24], Chap. VI.3. The location of the poles of c can be analyzed from the explicit expressions [24], which could be used to specify the exceptional sets.
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3. The Spectral Problem on the Invariant Fibers We proceed to study the consequences of the orbit decomposition for selfadjoint operators commuting with the group action. It is natural to consider the same class of integral operators as in Definition 2.1, but drop the positivity requirements. Definition 3.1. A standard invariant selfadjoint operator A on L 2 (M) is an integral operator with a symmetric continuous kernel A : M × M → C, which obeys (3.1) below and satisfies A(d(g)m, d(g)m ) = A(m, m ) for all g ∈ G, m, m ∈ M. Its image −1 ◦ A ◦ under the isometry in (2.32) is also denoted by A and is an integral operator on L 2 (Mr ) with kernel A : G × N × N → C, subject to (3.5) below and (3.1). Here sup dm |A(m, m | = sup dn dg |A(g, n, n )| =: K A < ∞ . (3.1) m
n
As in Definition 2.1 the condition (3.1) entails that A is well-defined as a bounded selfadjoint operator from L p to L p for all 1 ≤ p ≤ ∞. The structure of the kernels A and A of course implies that the operators A commute with M (G) and ρ(G), respectively. 3.1. Basic consequences of the ρ(G) invariance. As a first result we have: Proposition 3.1. (a) Every standard invariant selfadjoint operator A on L 2 (M) has purely essential spectrum, Spec (A) = ess-Spec (A). (b) A transfer operator commuting with M (G) cannot have normalizable ground states; hence T ∈ c-Spec(T), or T is a limit point of eigenvalues of infinite multiplicity. defining the Proof. (a) Recall from Eqs. (2.10) and (2.11) that the measure μ M on G decomposition of the unitary representation
is related to that defining the generalized M eigenspaces of A by Spec(A) dμ(λ, σ ) = dμ M (σ ). By Proposition 2.2 the measure μ M (σ ) can be identified with the Plancherel measure. For the noncompact Lie groups considered the support of the Plancherel measure contains only infinite dimensional representations (see Appendix A). Therefore, whenever a generalized eigenspace Eλ,σ (A) occurs in (2.4) its spectral value λ has infinite multiplicity. This excludes that λ lies in the discrete spectrum of the operator; hence λ ∈ ess-Spec(A). Note that it is not excluded that A has point spectrum of infinite multiplicity, i.e. infinite multiplets of normalizable eigenfunctions for some spectral value λ. (b) Let now T be a transfer operator on L 2 (M) commuting with M (G). Assume it has a normalizable ground state, i.e. a solution of Tψ = T ψ, with ψ ∈ L 2 (M). By a well known result, based on the Perron-Frobenius theorem (see for instance [19, 44]), this ground state would be unique and therefore invariant under the action of M (G), i.e. an M singlet. This entails that (ψ, ψ) diverges as the infinite volume of the group is ‘overcounted’. Thus T cannot be a eigenvalue. By part (a) it can also not be a limit point of the discrete spectrum (since there is none), leaving only the possibilities: T ∈ c-Spec(T) or T a limit point of eigenvalues of infinite multiplicity. To proceed, let us momentarily denote the selfadjoint operator A in the realization acting on L 2 (Mr ) by Ar := −1 ◦A◦, with as in (2.32). We write Ar (g, n, g , n ; t) for the corresponding kernel. Writing out (−1 ◦ A ◦ ψr )(g, n) one finds Ar (g, n, g , n ) = A(φ −1 (g, n), φ −1 (g , n )) = A(g g −1 q ↑ , g g −1 n, q ↑ , n ) . (3.2)
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In the last step we used the invariance of A, i.e. M (g) ◦ A = A ◦ M (g) for all g ∈ G. Since Art = −1 ◦ At ◦ for t ∈ N, also all iterated kernels will be related by (3.2). For the reasons explained in Sect. 2.3 the kernel Ar is most useful. From the last expression in (3.2) one sees explicitly that Ar only depends on the right invariant combination g g −1 , consistent with ρ(g) ◦ Ar = Ar ◦ ρ(g) for the operator. Further Ar is manifestly constant on the left equivalence classes (kg, kn) ∼ (g, n) and (k g , k n ) ∼ (g , n ) with (k, k ) ∈ K × K . We set A(gg
−1
, n, n ; t) := Ar (g, n, g , n ; t) ,
(3.3)
and note A(g, n, n ) = A(g −1 , n , n) , A(kg, kn, n ; t) = A(g, n, n ; t) = A(gk −1 , n, kn ; t) , k ∈ K .
(3.4)
We may thus interpret Ar either as an integral operator acting on L 2 (Mr ) or on L 2 (G × N ); in the latter interpretation it automatically projects onto the r (K ) invariant subspace (with r defined in (2.31)). This means that the nonzero spectrum of A lies automatically in that subspace. For convenience, we will therefore work with this interpretation of Ar . To simplify the notation we drop the subscript r in the following and write A for Ar etc. Due to the properties (3.3), (3.5) A respects the fiber decomposition (2.66) in the following sense: r , c its complement and let H and Hc Lemma 3.1. Let be a measurable subset of G be the corresponding subspaces of L 2 (G) (which are orthogonal complements of each other): ⊕ H= dν(σ ) Lσ ⊗ Lˇ σˇ ⊗ L 2 (N ) , (3.5) ⊕ c 2 ˇ H = dν(σ ) Lσ ⊗ Lσˇ ⊗ L (N ) . c
Then AH ⊂ H and AHc ⊂ Hc . Proof. Let ψ ∈ H, ψc ∈ Hc and consider s(g) := (ψc , Aρ(g)ψ) = (ρ(g)−1 ψc , Aψ) .
(3.6)
The first expression shows that the Fourier transform s of s is supported in . On the other hand, the second expression shows that the Fourier transform is supported in c . This is possible only if s vanishes identically. Putting g = e, the lemma follows. 3.2. Fiber decomposition of invariant selfadjoint operators. In view of Lemma 3.1 the operator A should also map the fiber spaces L2σ (Mr ) in (2.67) onto itself. To give precise meaning to this statement we have to construct fiber operators A(σ ) from A. In a first step one applies the expansion (2.34) to the kernel A(g, n, n ), n , n)πσ (g) , A(g, n, n ) = dν(σ )Tr A(σ, r G (3.7) −1 n , n) := dg A(g, n, n )πσ (g ) . A(σ, G
Structure of the Space of Ground States in Systems with Non-Amenable Symmetries
The swapped order in the (n, n ) arguments was chosen in order to have k n , k −1 n)κ1 s1 ,κ2 s2 = n , n)κ s,κ s rκ2 (k)s s , A(σ, rκ1 (k )s1 s A(σ, 1 2 2
397
(3.8)
ss
where the indices κ, s, etc. refer to the K -adapted basis (2.46). Note further the hermiticity and diagonal K invariance imply n , n) = A(σ, kn , kn) = Tr Vκ A(σ, n , n) . n, n )† , Tr Vκ A(σ, A(σ,
(3.9)
n, n ) is Since by (3.1) for a.e. n, n the kernel A is in L 1 (G), its Fourier transform A(σ, a compact operator on Lσ for a.e. n, n , see [15], Theorem 7.6. Moreover (3.1) implies n, n ): the following bound on A(σ, n, n ) ≤ K A , sup dn A(σ, (3.10) n
where . denotes the operator norm on Lσ . If we momentarily introduce the spaces p L σ (N ) := ψ : N → Lσ (3.11) dn (ψ(n), ψ(n))σp/2 < ∞ , n, n ) defines a bounded linear operator from L 1σ (N ) to L 1σ (N ) it follows first that A(σ, ∞ as well as from L σ (N ) to L ∞ σ (N ); by complex interpolation it follows then essentially n, n ) defines a bounded linear operator from L σp (N ) to as in [30], p. 173 ff that A(σ, √ p L σ (N ) for all p ∈ [1, ∞], where the p = ∞ norm is given by supn (ψ(n), ψ(n))σ . Furthermore the projection property of A onto the r (K ) invariant subspace translates into an equivariance property analogous to (2.43): n , k −1 n)πσ (k −1 ) = A(σ, n , n) = πσ (k)A(σ, k −1 n , n) , for k ∈ K , (3.12) A(σ, using (2.34), (3.5), (3.8). In the following we wish to convert the action of A on L 2 (Mr ) ⊕ n) = A(σ )ψ (σ, n) into an action A(σ ) on the fibers dν(σ ) L 2σ (Mr ) such that Aψ(σ, holds. To this end we note −1 (3.13) (Aψ)(g, n) = dg dn A(gg , n, n )ψ(g , n ) . From the definitions (2.34) one readily computes n , n) . n) = dn ψ (σ, n ) A(σ, Aψ(σ,
(3.14)
acts from the right on ψ ; it acts only on the factor Lˇ σˇ ⊗ L 2 (N ). If one views Note that A ψ as a matrix, the ‘column’ index is unaffected. We can now define the fiber operator A(σ ) by this action from the right, Eq. (3.14), on the elements of L 2σ σˇ (N ) introduced in (2.37): n); (σ, n) := Aψ(σ, A(σ )ψ the fiber operators A(σ ) then form a measurable field of operators such that ⊕ A= dν(σ ) A(σ ). r G
(3.15)
(3.16)
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The spectrum of A(σ ) has infinite multiplicity because of the spectator role of the first tensor factor: we have ˇ σˇ , A(σ ) = 1I ⊗ A
(3.17)
ˇ σˇ maps the space of bounded linear ˇ σˇ acts on ‘from the right’. This means that A where A 2 forms on Lσ with values in L (N ) into itself, or equivalently, it is a map from Lˇ σˇ ⊗L 2 (N ) ˇ σˇ is the (partial) dual of an operator Aσ mapping Lσ ⊗ L 2 (N ) into into itself. Thus A itself. We will see that in many cases the spectrum of Aσ has only finite multiplicity. As will become clear below, however, both A(σ ) and Aσ in general mix the different κ channels in the decomposition (2.53). Since A is a bounded operator, it follows from general considerations (see [15], (7.30)) that A(σ ) : L 2σ σˇ (N ) → L 2σ σˇ (N ) is a bounded operator for almost all σ (see also Proposition 3.4 below). Similarly Aσ is a bounded operator L 2σ (N ) → L 2σ (N ) for almost all σ , with L 2σ (N ) as in (2.39a) of Definition 2.2. On this space the left action
r (K ) corresponds to f (n) → πσ (k) f (k −1 ); the singlet subspace of L 2σ (N ) under this action is just L 2σ (N )0 as defined in (2.39b). r nonexceptional. Whenever A(σ ) and hence Aσ are bounded operators we call σ ∈ G This is the case for all σ in the discrete part G d of the restricted dual (see Proposition r \G d . Whenever σ does not occur as 3.4 below) and for almost all σ in the remainder G an integration variable we will assume it to be non-exceptional. Let us now describe the operator Aσ explicitly. In terms of the operator valued kernel its action is given by A, n , n)† f (n ) = dn A(σ, n, n ) f (n ) . (Aσ f )(n) = dn A(σ, (3.18) The equivariance property carries over to Aσ : (Aσ f )(kn) = πσ (k)(Aσ f )(n) ,
(3.19)
or, in the K -adapted basis (Aσ f )(kn)κs = s rκ (k)ss (Aσ f )(n)κs . This is true whether or not f satisfies the corresponding equivariance property; it is a reflection of the fact that A projects onto the r (K ) invariant subspace of L 2 (G × N ). Proposition 3.4. Let A be a standard invariant selfadjoint operator on L 2 (Mr ) in the sense of Definition 3.1. Then: ⊕ ˇ σˇ ), respecting the fibers (a) The operator A has a decomposition A = dν(σ )(1I ⊗ A in Proposition 2.2. Further Aσ is well-defined as a bounded linear and selfadjoint d and almost all σ ∈ G r \G d (called non-excepoperator on L 2σ (N ) for all σ ∈ G tional). The associated norms satisfy ess sup Aσ = A . σ
(3.20)
r | Spec(A(σ )) ∩ (λ − , (b) λ ∈ ess-Spec(A) = Spec(A) iff for all > 0 the set {σ ∈ G ˇ λ+) = ∅} has positive measure, where A(σ ) = 1I⊗ Aσ . Further λ is an eigenvalue r | λ is an eigenvalue of A(σ )} has positive measure. of A iff the set {σ ∈ G
Structure of the Space of Ground States in Systems with Non-Amenable Symmetries
399
Proof. (a) The decomposition has been just explained; the boundedness a.e. as well as (3.20) follow from the general theory of direct integral decompositions of operators (see [15] or [44], Theorem XIII.83). For the sake of completeness we sketch the argument: For all ψ ∈ L 2 (Mr ) one has from (2.34) and (3.15), (σ, n)πσ (g)] . (Aψ)(g, n) = dν(σ ) Tr [A(σ )ψ (3.21) r G
n) by the Parseval identity (2.36) The norm of Aψ is related to that of Aψ(σ, (σ ), A(σ )ψ (σ ))σ σˇ , dν(σ ) (A(σ )ψ (Aψ, Aψ) = r G
(3.22)
with ( , )σ σˇ defined in (2.37). Since A is a bounded operator the left-hand side of (3.22) is finite, hence the integrand on the right hand side is finite ν-almost everywhere. The bound (3.20) follows by choosing sequences of ψ such that their Fourier transforms (σ, n) are becoming concentrated at a particular value σ (see [15], Prop. 7.33). Further ψ A(σ ) is symmetric and has a unique selfadjoint extension for a.e. σ , which we denote ˇ σˇ , the same holds for the fiber operators Aσ . by the same symbol. Since A(σ ) = 1I ⊗ A (b) The statements about the spectrum in general as well as the eigenvalues follow as in Theorem XIII.85 of [44]. From our analysis we haveobtained the following: The Hilbert space L 2 (Mr ) is ⊕ resolved into the direct integral dν(σ )Lσ ⊗ Lˇ σˇ ⊗ L 2 (N ). Likewise A can be resolved ⊕ ⊕ ˇ σˇ . The fiber operators Aσ are (ν into a direct integral A(σ )dν(σ ) = 1ILσ ⊗ A a.e.) selfadjoint and therefore have a spectral resolution, which leads to a direct integral decomposition of the spaces L 2σ (N ) as well as of Aσ and A(σ ): L 2σ (N )0
=
Aσ =
⊕
⊕
dμσ (λ)L2λσ (N ) ,
(3.23)
dμσ (λ)λ 1IL2 (N ) , λσ
as well as L2σ (Mr ) = A(σ ) =
⊕
⊕
dμσ (λ) Lσ ⊗ L2λσˇ (N ) ,
(3.24)
dμσ (λ)λ 1ILσ ⊗L2 (N ) . λσˇ
Combining these decompositions, we have L 2 (Mr ) =
⊕
A=
dν(σ )dμσ (λ) Lσ ⊗ L2λσˇ (N ) ,
⊕
dν(σ )dμσ (λ)λ 1ILσ ⊗L2 (N ) . λσˇ
So we have identified the measure dμ(λ, σ ) in (2.4) as dν(σ ) dμσ (λ).
(3.25)
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3.3. Relating the spectral problems of A and Aσ : Aσ compact. Our goal is to analyze concretely how the spectral resolution of the fiber operators Aσ relates to the spectral problem of A, including the construction of generalized eigenvectors. At first we consider an important special case in which the kernel of A is ‘almost’ square integrable implying that the fiber operators Aσ are compact. Explicitly we require: (C) dgdndn |A(g, n, n )|2<∞, or equivalently dq dm|A(q ↑ , q, m)|2 < ∞. Q ν−1
A(m, m )
Here is the kernel of the original operator acting on L 2 (M) while A is the kernel of its image under the isometry in Proposition 2.1. Of course since A(m, m ) is invariant it can never be square integrable in the proper sense, see Proposition 3.1. However (C) implies that the fiber operators are Hilbert-Schmidt for almost all σ . Lemma 3.2. Let A be a standard invariant selfadjoint operator as in Definition 3.1 whose kernel in addition satifies Condition (C). Then the fiber operators Aσ and A(σ ) d and for almost all σ ∈ G c . These σ ∈ G r are called are Hilbert-Schmidt for all σ ∈ G non-exceptional. Proof. The Parseval identity (2.36) applied to A(g, n, n ) gives n, n )] . n, n )† A(σ, dν(σ ) Tr[A(σ, dg|A(g, n, n )|2 = r G
(3.26)
The integral over n, n of the left-hand side is finite by assumption; that of the right-hand side can be expressed in terms of the Hilbert-Schmidt norm of the integral operator A(σ ) and gives dν(σ ) A(σ ) 22 . Thus A(σ ) must have finite Hilbert-Schmidt norm r , as asserted. The equivalence to the second condition in (C) can for almost all σ ∈ G be seen from (3.2). In detail dgdndn |A(g, n, n )|2 = dgdndn |A(gq ↑ , gn, q0 , n )|2 = dγ Q (q1 )dndn |A(q1 , gs (q1 )n, q ↑ , n )|2 = dq dm|A(q ↑ , q, m)|2 . (3.27) Q ν−1
In the second equality we used the fact that the section gs : Q → G provides a one-to-one correspondence between points in Q and right K orbits in G. By definition (q1 , gs (q1 )n) = (q1 , q2 , . . . qν ) = m and the dγ Q (q1 )dn = dγ (m) integral just defines the iteration of the modulus of the kernel. The spectral problem for the fiber operators Aσ and A(σ ) is now trivial: these operators have discrete spectrum except for 0, which is an accumulation point of eigenvalues of finite multiplicity. All eigenvectors lie in the respective L 2 spaces, i.e. in L 2σ (N ) for ˇ σˇ a dense set of eigenvecAσ and in L 2σ σˇ (N ) for A(σ ). Since A(σ ) is of the form 1I ⊗ A ˇ tors F : N → Lσ ⊗ Lσˇ exists such that F(n) is trace class (not just Hilbert-Schmidt as in the definition of L 2σ σˇ (N ). The goal in the following is to ‘lift’ these normalizable eigenvectors of the fiber operators to non-normalizable σ -equivariant eigenfunctions of A. To this end the action of A has to be extended to functions outside of L 2 (Mr ). Our Definition 3.1 ensures that A can be extended naturally as an operator from L ∞ (Mr ) to itself, bounded in the sup norm. In order to relate the eigenvectors of the fiber operators to the eigenfunctions of A (which will no longer be square integrable), we need a generalization of the Fourier transformation. In preparation we introduce two pairs of Banach spaces as follows:
Structure of the Space of Ground States in Systems with Non-Amenable Symmetries
Definition 3.2. L 1,2 (G × N ) := L 1 (G) ⊗ L 2 (N ) ,
φ 1,2 :=
L ∞,2 (G × N ) := L ∞ (G) ⊗ L 2 (N ) , t ∞,2 :=
dn
dg|φ(g, n)|
401
2 1/2
,
2 1/2 dn sup |t (g, n)| . g
r Further, for σ ∈ G
L 2 (N , C(Lσ )) := C : N → C(Lσ ) dn C(n) 2 < ∞ ,
(3.28)
where C(Lσ ) is the space of compact operators on Lσ and C(n) is the operator norm. Similarly 2 L (N , J1 (Lσ )) := F : N → J1 (Lσ ) dn F(n) 21 < ∞ , (3.29) where J1 (Lσ ) is the space of trace class operators on Lσ and F(n) 1 = Tr[|F(n)|] is the trace norm of F(n). Denoting by the topological duals one has L ∞,2 (N × G) = L 1,2 (G × N ) , L 2 (N , J1 (Lσ )) = L 2 (N , C(Lσ )) .
(3.30a) (3.30b)
Of course the spaces are not reflexive, the double duals are much larger than the original spaces. The relations (3.30) follow from the well-known facts L 1 (G) = L ∞ (G) and C(Lσ ) = J1 (Lσ ), see e.g. [43], Theorem VI.26, for the latter. Concretely one has |(t, φ)| ≤ t ∞,2 φ 1,2 , t ∈ L ∞,2 , φ ∈ L 1,2 . Thus (t, · ) defines a linear bounded functional on L 1,2 ; moreover every such functional is of that form. Similarly, 1/2 1/2 † 2 (F, C)σ σˇ = dnTr[F(n) C(n)] ≤ dn C(n) 2 dn F(n) 1 , (3.31) so that (F, · )σ σˇ defines a linear bounded functional on L 2 (N , C(Lσ )); moreover every such functional is of this form. The two pairs of spaces (3.30a,b) are related by the generalization of the Fourier transform we are aiming for. To see this we introduce Definition 3.3. A function t = tσ ∈ L ∞,2 (G × N ) is called σ -equivariant for some r , if there exists a σ ∈ L 2 (N , C(Lσ )) such that σ ∈G (σ )) for all φ ∈ L 1,2 (G × N ) , (tσ , φ) = σ (φ
(3.32)
(σ, n) is the Fourier transform (2.34) of φ (which is C(Lσ )-valued for φ( · , n) ∈ where φ 1 L (G)). Such tσ intertwine the right G action ρ on L ∞ (G) with that of πσ , i.e. ) , (ρ(g −1 )tσ , φ) = σ (πσ (g)φ
(3.33)
for all g ∈ G and φ ∈ L 1,2 (G × N ). The σ -equivariant subspace of L ∞,2 (G × N ) is denoted by Lσ∞,2 (G × N ).
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Proposition 3.5. Every σ -equivariant tσ ∈ L ∞,2 (G × N ) is of the form tσF (g, n) = Tr[F(n)πσ (g)] ,
(3.34)
for a unique F ∈ L 2 (N , J1 (Lσ )). Equivalently the map tσ : L 2 (N , J1 (Lσ )) → Lσ∞,2 (G × N ), F → tσF , is a bounded injection. The intertwining relation becomes π (g )F tσ σ 0 (g, n) = tσF (gg0 , n). A substitute for the Parseval relation is (σ ))σ σˇ , (tσF , φ) = (F, φ
(3.35)
for all φ ∈ L 1,2 (G × N ). Proof. We begin by showing that the map F(n) → tσF ( · , n) is injective for almost all n ∈ N. Here we can appeal to a theorem due to Glimm [18, 15] that implies that for a type I group (which we have here) the closure in operator norm of the algebra generated by πσ (g), g ∈ G consists of all compact operators on Lσ . Then if tσF (φ) = 0 for all φ ∈ D(G) also Tr[Fπσ (g)] = 0 for all g ∈ G and by the above mentioned theorem Tr[F A] = 0 for all compact A, which implies F = 0. A slightly more down to earth argument goes as follows: Assume again that Tr Fπσ (φ) = 0 for all φ ∈ D(G). Replacing φ by E κ ∗ φ ∗ E κ we see that Tr[Pκ F Pκ πσ (φ)] = 0 for all φ ∈ D(G) and κ, κ . Note that Ran Pκ = m κ Vκ and Ran Pκ = m κ Vκ have finite dimension. By a theorem of Kadison [28], the norm closure of the algebra generated by πσ (g), g ∈ G, contains operators that map any finite set of linearly independent vectors into any other given finite set of vectors of the same cardinality. This means that for any linear map L κ ,κ from m κ Vκ to m κ Vκ there is an element A in that norm closure such that L κ ,κ = Pκ A Pκ . Hence Tr[F L κ ,κ ] = 0 for any such linear map, which implies F = 0. To complete the proof, let σ ∈ L 2 (N , C(Lσ )) be the element associated with tσ via (3.32). On account of (3.30b) there exist some F ∈ L 2 (N , J1 (Lσ )) such that σ (C) = (σ, n) (F † , C)σ σˇ for all C ∈ L 2 (N , C(Lσ )). This holds in particular for C(σ, n) = φ (σ )) = (tσF , φ), with for φ ∈ L 1 (G) ⊗ L 2 (N ). A simple computation then gives σ (φ tσF as in (3.34). The uniqueness of F follows the above injectivity result. For the norm of tσF one has tσF ∞,2 ≤ ( dn F(n) 21 )1/2 . We add some remarks. Proposition 3.5 allows to give the space Lσ∞,2 (G × N ) the structure of a pre-Hilbert space by defining a scalar product (tσF , tσF ) := dnTr[F(n)† F (n)] ; (3.36) and of course by completion this gives rise to a Hilbert space, which can be identified with L 2 (N , J2 (Lσ )), where J2 (Lσ ) is the space of Hilbert-Schmidt operators on Lσ . Comparing (3.34) with (2.34) one sees that F can be viewed as the unique Fourier r ). The equivaritransform of t = tσ , but in a fixed σ fiber (it is a ‘δ function’ in G ance property (3.32) characterizes the fiber. Equivalently the σ -equivariant subspace of L ∞,2 (G × N ) can be identified with L 2 (N , J1 (Lσ )) by the bounded injection tσ . The map tσ can also be turned into an isometry by transporting the norm to its image. r , of G × G, Since L 2 (N , J1 (Lσ )) carries the representation πσ × πσˇ , σ ∈ G where (πσ × πσˇ )(g0 , g1 )F(n) = πσ (g0 )F(n)πσ (g1−1 ), the intertwining property in
Structure of the Space of Ground States in Systems with Non-Amenable Symmetries
403
Proposition 3.5 in principle generalizes to (πσ ×πσˇ )(g0 , g1 ) tσF (g, n) = tσF (g1−1 gg0 , n). However since we eventually are interested in functions on Mr = (G × N )/d(K ), only left actions with g1 = k ∈ K are useful. Whenever F(n) satisfies the equivariance condition (2.43) the function tσF (g, n) projects to one on Mr , i.e. tσF (kg, kn) = tσF (g, n) for k ∈ K . We denote the left K invariant subspace of L ∞,2 (G × N ) by L ∞,2 (Mr ) and the subspace of L 2 (N , J1 (Lσ )) whose elements satisfy (2.43) by L 2 (N , J1 (Lσ ))0 . Proposition 3.5 evidently remains valid as an injective identification of L 2 (N , J1 (Lσ ))0 with the σ -equivariant subspace of L ∞,2 (Mr ), for which we write Lσ∞,2 (Mr ). To proceed we now show that in addition to intertwining the group actions, the map tσ also intertwines the action of A with that of A(σ ). Indeed, using only the definitions (3.8), (3.13), (3.34) one computes n , n) πσ (g) [AtσF ](g, n) = dn Tr F(n ) A(σ, = Tr (A(σ )F)(n)πσ (g) = tσA(σ )F (g, n) .
(3.37)
If the Fourier transform were well-defined on these functions the relation (3.37) would amount to AtσF = A(σ )F, just as in (3.14), (3.15). The point of (3.37) is that it remains valid for A acting on Lσ∞,2 (G × N ) and A(σ ) acting on L 2σ σˇ (N ). acts from Since n → F(n) takes values in the trace class operators and the kernel A the right one can expand F(n) into rank 1 operators of the form F(n) =
∞
vi ⊗ fˇi (n) ,
(3.38)
i=1
with vi ∈ Lσ and f i : N → Lσ in L 2σ (N ). Mostly it is therefore the restriction of tσ to rank 1 operators F(n) = v ⊗ fˇ(n) that is needed and it is convenient to introduce a separate notation for it. We define τvσ : L 2σ (N ) → Lσ∞,2 (G × N ) , τvσ ( f )(g, n) :=
ˇ tσv⊗ f (g, n)
f → τvσ ( f ) , (3.39) = Tr v ⊗ fˇ(n) πσ (g) = ( f (n), πσ (g)v)σ .
Note that the map is antilinear in f but linear in fˇ, and that tσF (g, n) = i τvi σ ( f i )(g, n), for F of the form (3.38). Whenever f is K equivariant the image function is left K invariant, i.e. (3.39) also defines a map τvσ : L 2σ (N )0 → Lσ∞,2 (Mr ). It is straightforward to see that the map is bounded in the norm · ∞,2 . It is thus a bounded injection and can be made into an isometry onto its image by transferring the norm to the image. The representation πσ acts on rank 1 valued operators merely by rotating the reference vector: v ⊗ fˇ(n) → (πσ (g0 )v) ⊗ fˇ(n), so that τvσ ( f )(gg0 , n) = τπσ (g0 )v,σ ( f )(g, n). Further τvσ interwines the action of A and Aσ : either from (3.37) or directly from (3.13), (3.18) one finds (Aτvσ ( f ))(g, n) = (Aσ f σ (n), πσ (g)v)σ = (τvσ (Aσ f ))(g, n) .
(3.40)
This intertwining relation now allows to reduce the spectral problem for ⊕A to the simpler ˇ σˇ ). dν(σ )(1I⊗A spectral problems of the fiber operators Aσ in the decomposition A =
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We write Eλ,σ (A) = σ ∈ Lσ∞,2 (Mr ) σ is eigenfunction of
A with spectral value λ ∈ Spec(A) , (3.41a)
Eλ (Aσ ) = χσ ∈ L 2σ (N )0 χσ is eigenvector of
Aσ with eigenvalueλ ∈ Spec(Aσ ) ,
(3.41b)
ˇ σˇ ) ⊂ L 2 (N ) and Eλ (A(σ )) ⊂ L 2 (N , J1 (Lσ ))0 . and similarly for Eλ (A σˇ The space Eλ,σ (A) inherits the pre-Hilbert space structure from Lσ∞,2 (Mr ) and we denote its completion by Eλ,σ (A), whereas Eλ (Aσ ) is already a Hilbert space. The relation of these spectral problems is described by Proposition 3.6. Let A be a standard invariant selfadjoint operator on L 2 (Mr ) in the sense of Definition 3.1 and assume that its kernel satisfies in addition Condition (C). r Then the fiber operators A(σ ), Aσ are Hilbert-Schmidt for all non-exceptional σ ∈ G by Lemma 3.2. Further for these σ ∈ G r (a) For each v ∈ Lσ the map τvσ : Eλ (Aσ ) → Eλ,σ (A) is a bounded injection. d , has an eigenvector (normalizable eigenfunction) for some λ, the (b) If Aσ , σ ∈ G associated generalized eigenfunction in (a) of A is normalizable (i.e. λ is an eigenvalue of A of infinite multiplicity). (c) A complete orthonormal set of eigenfunctions of A(σ ) in Eλ (A(σ )) is given by ei ⊗ χˇ λσ r ; (3.42) a complete orthonormal set of eigenfunctions A in Eλ,σ (A) is given by
τei σ (χλσ r )(g, n) ,
(3.43)
where χλσ r runs through an orthonormal basis of Eλ (Aσ ) and ei runs through an orthonormal basis of Lσ . The eigenfunctions (3.43) vanish pointwise for g → ∞ (i.e. as g leaves any compact subset of G). Eλ,σ (A) and Eλ (A(σ )) are isometric as Hilbert spaces. In particular the spectrum of A(σ ) is contained in the spectrum of r . A for almost all σ ∈ G (d) An orthonormal basis of L 2σ (N )0 is given by
χλσ r ,
(3.44)
where χλσ r runs through an orthonormal basis of Eλ (Aσ ), and λ runs through the eigenvalues of Aσ . An orthonormal basis of L 2 (N , J2 (Lσ ))0 is given by (3.45) ei ⊗ χˇ λσ r , where in addition ei runs through an orthonormal basis of Lσ .
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(e) The set of all eigenfunctions λσ Eλ,σ (A) is complete in the sense that for each φ ∈ (L 2 ∩ L 1 )(Mr ) the following Parseval relation holds: (σ, n)ei 2 , (φ, φ) = dν(σ )dμσ (λ) (3.46) dn χλσ r (n), φ σ i,r
where the measure dμσ (λ) introduced in (3.24) here is just the counting measure of the eigenvalues λ of Aσ . Proof. (a) Since Aσ is selfadjoint, we can assume that λ is real. Clearly, if χ ∈ Eλ (Aσ ), i.e. Aσ χ = λχ , then Aτvσ (χ ) = τvσ (Aσ χ ) = λτvσ (χ ), i.e. τvσ (χ ) ∈ Eλ,σ (A). By Proposition 3.5 the map Eλ (Aσ ) χ → τvσ (χ ) ∈ Eλ,σ (A) is injective. (b) Recall from [9], Theorem 14.3.3, that the coefficients of a square integrable d , satisfy, using the K -adapted basis (2.46), representation πσ , σ ∈ G (3.47) dg πσ (g)∗κ1 s1 ,κ2 s2 πσ (g)κ3 s3 ,κ4 s4 = dσ−1 δκ1 κ3 δs1 s3 δκ2 κ4 δs2 s4 , where dσ is the formal degree of πσ . Let χ ∈ L 2σ (N ) be the normalizable eigenfunction d , and consider the associated generalized eigenfunction τvσ (χ ) of Aσ for some σ ∈ G of A. Using (3.47) one can verify by direct computation that (3.48) dσ dgdn |τvσ (χ )(g, n)|2 = (v, v)σ (χ , χ )σ , holds. It follows that τvσ (χ ) is actually normalizable, i.e. an element of L 2 (Mr ) ∩ Lσ∞,2 (Mr ). For the spectral values this implies: if λ ∈ d-Spec(Aσ ) then it is an eigenvalue of infinite multiplicity of A. (c) Equation (3.42) follows directly from the completeness of the eigenfunctions of Aσ and the relation between the operators A(σ ) and Aσ . The second statement follows from the fact that all the functions in Eλ,σ (A) are σ -equivariant and hence by Proposition 3.5 of the form σ (g, n) = Tr[F(n)πσ (g))]. The eigenfunction condition translates into the condition that F is an eigenvector of A(σ ) with eigenvalue λ; a complete set of such eigenvectors is given by (3.42), as we have just seen. By construction the map F → tσF preserves the scalar product, which implies the orthonormality of the set (3.43). So tσF provides an isometry between Eλ,σ (A) and Eλ (A(σ )). The vanishing at infinity in G of the eigenfunctions (3.42) follows from the Howe-Moore theorem, see [60], Theorem 2.2.20. (d) This follows from the spectral theorem applied to Aσ and A(σ ) together with part (c). (σ, n) is a.e. a Hilbert-Schmidt (e) Since φ ∈ L 2 ∩ L 1 , its partial Fourier transform φ operator in Lσ . The statement then follows from the completeness of the fiber eigenvectors (part (a)) and the Plancherel theorem of Appendix A1. We repeat a cautioning remark made in Sect. 2.2: the fact that λ0 ∈ Spec(Aσ0 ) for some σ0 and hence Eλ0 ,σ0 (A) is nontrivial does in itself not imply that λ0 ∈ Spec(A). In fact there are explicit counterexamples. By Proposition 3.4b one needs a set of nonzero Plancherel measure, such that for each neighborhood U of λ0 we have σ ∈ Spec(Aσ )∩ U = ∅. In the case of compact fiber operators Aσ we have however obtained the following relation between the spectral problems of A and Aσ : for every spectral value |λ| ≤ A
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of A there is an eigenvalue of Aσ . The corresponding normalizable eigenvector χ of Aσ generates a G orbit of Lσ∞,2 eigenfunctions of A via τvσ (χ ), and all σ -equivariant eigenfunctions of A arise in that way by taking linear combinations in χ and the reference vectors v ∈ Lσ . In addition there may be ‘fake’ solutions of A = λ, where λ = Spec(A). The value |λ| = A of course always belongs to the spectrum of A. 3.4. Relating the spectral problems of A and Aσ : Aσ not necessarily compact. The discussion becomes a little more complicated in the case of not necessarily compact fiber operators. These naturally arise if the original kernel is not ‘almost’ square integrable as in Condition (C). In this section we require instead the weaker conditions (C1) supm dm |A(m, m )|2 < ∞ . (C2) There is an invariant multiplication operator M given by a d(G) invariant function in L 2 (M) also denoted by M, such that M −1 AM defines a bounded (invariant) operator on L 2 (M). For a positive kernel A(m, m ) > 0 the condition (C1) amounts to supm A(m, m; 2) < ∞, which is weaker than the integrability condition in Lemma 3.1 (at least as far as decay properties of A are concerned). The multiplication operator acts by (Mψ)(m) = M(m)ψ(m), for ψ ∈ L 2 (M). Invariance M(d(g)m) = M(m), for all g ∈ G, means that M maps into multiplication by a function M˜ on N , given by ˜ M(n) := M(q ↑ , n)
(3.49)
under the isometry −1 : L 2 (M) → L 2 (Mr ) in (2.32). We drop the tilde and continue to write M ∈ L 2 (N ) for this function and note that the kernel of M −1 AM, viewed as a bounded operator on L 2 (Mr ) is given by M(n)−1 A(gg −1 , n, n )M(n ). The counterpart of Lemma 3.2 is now given by d Lemma 3.3. Under the conditions (C1), (C2) the fiber operators Aσ have for all σ ∈ G c the properties and for almost all σ ∈ G n, n ) 2 =: C A < ∞, where . 2 denotes the Hilbert-Schmidt (C1’) supn dn A(σ, 2 norm for operators on Lσ . (C2’) There is a function M ∈ L 2 (N ) such that M −1 Aσ M defines a bounded operator on L 2σ (N ), where M maps each L 2σ (N ) into itself via (Mφ)(n) := M(n)φ(n). Proof. (C1’) follows from (C1) by applying (3.26). (C2’) is obvious. The goal in the following will be to first gain control over the eigenfunctions Aσ (a step which was trivial in Sect. 3.3) and then to lift them to πσ equivariant generalized eigenfunctions of A. For the first step we will make use of the Gel’fand-Maurin theory of generalized eigenvectors and eigenspaces, in particular of a version due to Berezanskii as described (and proven) in Maurin [34]. Berezanskii’s construction, applied to the present situation, r a triplet requires to set up for almost each σ ∈ G σ (N ) ⊂ L 2σ (N ) ⊂ σ (N ) ,
(3.50)
in which σ = σ (N ) is the domain of an unbounded, densely defined closed operator Bσ whose inverse is Hilbert-Schmidt and σ = σ (N ) is the topological dual of σ . Choosing
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Bσ := M −1 A−1 σ ,
(3.51)
we compute −1 † −1 n, n ) 22 ≤ M 22 C A , (3.52) Tr L 2σ (N ) (Bσ ) (Bσ ≤ dndn |M(n)M(n )| A(σ, n, n ) 2 where in the last step we used property (C1)’ and the fact that therefore A(σ, 2 2 is the kernel of a bounded operator on L σ (N ) with norm ≤ C A , see [30], p.176. This shows that Bσ−1 is indeed Hilbert-Schmidt. Property (C2)’ now guarantees that Aσ maps σ into itself. Indeed from Aσ M L 2σ (N ) = M L 2σ (N ) one sees that M L 2σ (N ) coincides both with the range of Bσ−1 = Aσ M and with the domain of Bσ . σ is a pre-Hilbert space with the scalar product (φ, ψ) B := (Bσ φ, Bσ ψ)σ + (φ, ψ)σ ;
(3.53) √ we denote its completion with respect to the norm φσ B := (φ, φ) B by σ ⊂ L 2σ (N ). The dual of Aσ acting on σ will also be denoted by Aσ . Berezanskii’s theorem guarantees that Aσ has a complete set of eigendistributions in σ . We now proceed to show that in the situation at hand these are in fact almost everywhere defined functions from N with values in Lσ (which are of course not square integrable). More precisely we have r the Proposition 3.7. Under Conditions (C1) and (C2) above, for almost all σ ∈ G fiber operator Aσ has a complete set of generalized eigenfunctions χλσ r , where r = 1, . . . g(λ) with g(λ) ∈ N ∪ {∞} denotes the multiplicity of the spectral value λ. The generalized eigenfunctions satisfy for almost all λ ∈ Spec(Aσ ): (a) Aσ χλσ = λ χλσ , (b) Mχλσ ∈ L 2σ (N ). (c) For φ ∈ M L 2σ (N )0 we have the Parseval (completeness) relation g(λ) 2 dn (φ(n), φ(n))σ = dn dμσ (λ) χλσ r (n), φ(n) σ . (3.54) r =1
We denote the space of these eigenfunctions by Eλ (Aσ ). Proof. We first show that (the dual of) Aσ maps σ into almost everywhere defined functions from N with values in Lσ . More precisely, for any χ ∈ σ , M † Aσ χ = (Bσ−1 )† χ ∈ L 2σ (N ) .
(3.55)
Since σ is a pre-Hilbert space, by the Riesz representation theorem each element χ of σ can be represented by an element in σ . This means that there is a ψχ ∈ σ ⊂ L 2σ (N ) such that for all φ ∈ σ , |(χ , φ)σ | = |(ψχ , φ) B | ≤ ψχ B φ B . Bσ−1 φ,
Replacing φ by
Bσ−1 2 ) φ 2σ , gives
and using
Bσ−1 φ 2B
= (φ, φ)σ +
(Bσ−1 φ,
(3.56) Bσ−1 φ)σ
|(χ , Bσ−1 φ)σ | ≤ ψχ B Bσ−1 φ B ≤ const ψχ B φ σ . Bσ−1 φ)σ
≤ (1 + (3.57)
This shows that (χ , defines a bounded linear functional on the dense domain σ ⊂ L 2σ (N ) and hence is given by an element of L 2σ (N ), as claimed in (3.55). Now (a) and (c) follow from Berezanskii’s theorem and (b) follows from (a) and (3.55).
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Consistency requires that the dual Aσ of Aσ (with a separate notation only at this point) is such that MAσ M −1 is a bounded operator on L 2σ (N ), which is however just the relation dual to (C2)’. Let us add that it is not possible to apply this method directly to obtain the σ -equivariant generalized eigenfunctions of A itself. As long as A and M are invariant the product AM will never be Hilbert-Schmidt, as the infinite group volume is once more overcounted. One would have to replace the invariant multiplication operator M by an operator of multiplication by a (necessarily noninvariant) function in L 2 (Mr ), which would make the strategy at least cumbersome. We rather proceed as in Sect. 3.3. In order to avoid having to introduce further notation we continue to use the spaces in Definition 3.2 but indicate the modified notion of square integrability by a mnemonic prefactor M or M −1 . For example F ∈ M −1 L 2 (N , J1 (Lσ )) if t∈M
−1 ∞,2
L
dn|M(n)|2 F(n) 21 < ∞ ,
(G × N )
if
dn|M(n)|2 sup |t (g, n)|2 < ∞ .
(3.58)
g
The σ -equivariant subspace of M −1 L ∞,2 (G × N ) is denoted by M −1 Lσ∞,2 (G × N ), etc. We define tσF (g, n) := Tr[F(n)πσ (g)] as in (3.34). With these modified spaces Proposition 3.5 carries over, and states that every element of M −1 Lσ∞,2 (G × N ), is of the form tσF (g, n) with a unique F ∈ M −1 L 2 (N , J1 (Lσ )). The intertwining property A(σ )F carries over from (3.37). Since F(n) is for the G actions is manifest and AtσF = tσ trace class we can expand it as in (3.38), F(n) =
vi ⊗ χˇ i (n) ,
vi ∈ Lσ , χi ∈ M −1 L 2σ (N ) ,
(3.59)
i
and define τvσ : M −1 L 2σ (N ) −→ M −1 Lσ∞,2 (Mr ) , χ → τvσ (χ ) ,
τvσ (χ )(g, n) = (χ (n), πσ (g)v)σ ,
(3.60)
where again the spaces and their norms are defined in the obvious way. In parallel to Eq. (3.41) we write Eλ,σ (A) = σ ∈ M −1 Lσ∞,2 (Mr ) σ is eigenfunction of
A with spectral value λ ∈ Spec(A) ,
Eλ (Aσ ) = χσ ∈ M −1 L 2σ (N )0 χσ is eigenfunction of
(3.61a)
Aσ with spectral value λ ∈ Spec(Aσ ) , (3.61b)
ˇ σˇ ) ⊂ M −1 L 2 (N )0 and Eλ (A(σ )) ⊂ M −1 L 2 (N , J1 (Lσ ))0 . and similarly for Eλ (A σˇ
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We obtain the following counterpart of Proposition 3.6: r non-exceptional as in Proposition Proposition 3.8. Let A and A(σ ), Aσ with σ ∈ G 3.4, and assume that A satisfies Conditions (C1), (C2). Then (a) τvσ : Eλ (Aσ ) → Eλ,σ (A) is a bounded injection. (b) A complete set of generalized eigenfunctions of A(σ ) in Eλ (A(σ ) is given by ei ⊗ χˇ λσ r ; (3.62) a complete set of generalized eigenfunctions of A in Eλ,σ (A) is given by τei σ (χλσ r )(g, n) ,
(3.63)
where χλσ r runs through the complete set of Propostion 3.7 and ei runs through an orthonormal basis of Lσ . The eigenfunctions (3.63) vanish at infinity in G. Eλ,σ (A) and Eλ (A(σ )) are homeomorphic as Banach spaces, provided one uses on the image the norm transported from the preimage. In particular the spectrum of r . A(σ ) is contained in the spectrum of A for almost all σ ∈ G (c) A complete set of of generalized eigenfunctions, spanning M −1 L 2σ (N )0 is given by (3.64) χλσ r , where χλσ r runs through the complete set above and λ runs through the eigenvalues of Aσ . A complete set of generalized eigenfunctions spanning M −1 L 2 (N , J2 (Lσ ))0 is given by ei ⊗ χˇ λσ r , (3.65) where in addition ei runs through an orthonormal basis of Lσ . (d) The set of all eigenfunctions λσ Eλ,σ (A) is complete in the sense that for each φ ∈ (L 2 ∩ L 1 )(Mr ) the following Parseval relation holds: (σ, n)ei 2 , (φ, φ) = dν(σ )dμσ (λ) (3.66) dn χλσ r (n), φ σ i,r
with the measure dμσ (λ) introduced in (3.24). Proof. The proof parallels that of Proposition 3.6. The only change is that the completeness of the set of eigenvectors has to be replaced by the completeness relation of Proposition 3.7 (c). Symbolically we can summarize the content of Propositions 3.6b and 3.8b in the isometry Eλ,σ (A) Eλ (A(σ )) .
(3.67)
That is, under Conditions (C) or (C1), (C2) the space of eigenfunctions of the fiber operator A(σ ) is naturally isometric to the space of σ -equivariant eigenfunctions of A. When A = T is a transfer operator and the spectral value is λ = T , the generalized eigenspace E T ,σ (T) = Gσ (T) is the σ -equivariant piece of ground state sector of T.
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3.5. Absence of the discrete series. A priori the spectral value of λ of the operator A is of course unrelated to the type σ of (irreducible) representation with respect to which a generalized eigenfunction transforms. The goal of Sect. 4 will be to show that for a r transfer operator T and the spectral value λ = T , the possible representations σ ∈ G are severely constrained. A first flavor of the correlation between the spectral value of A and the type σ of irreducible representations allowed can be obtained from the ˇ σˇ one of its fiber operCorollary 3.1. Let A be as in Sect. 3.3 or 3.4 and A(σ ) = 1I ⊗ A ators. Under any one of the following conditions the eigenspaces Eλ,σ (A) Eλ (A(σ )) d , the discrete series of G, are empty: for σ ∈ G (a) Eλ (A(σ )) contains a left K singlet, i.e. a generalized eigenfunction F obeying F(kn) = F(n), for all k ∈ K . (b) Eλ,σ (A) contains a right K singlet, i.e. a generalized eigenfunction obeying (gk, n) = (g, n), for all k ∈ K . (c) A = T is a transfer operator, λ = T and it is an eigenvalue of T(σ ). Proof. (a), (b) For definiteness let us consider the setting in Sect. 3.3, for the one in Sect. 3.4 only notational changes are required. Via the isometry F → tσF the assumptions in (a) and (b) amount to Tr[F(kn)πσ (g)] = Tr[F(n)πσ (g)] and Tr[F(n)πσ (gk)] = Tr[F(n)πσ (g)], respectively, for all k ∈ K , g ∈ G and almost all n ∈ N . In terms of P0 , the projector onto the κ = 0 singlet subspace in Lσ , this becomes Tr[F(n)πσ (g)] = Tr[F(n)P0 πσ (g)] and Tr[F(n)πσ (g)] = Tr[F(n)πσ (g)P0 ], respectively. However these can be nonvanishing functions only if the projections P0 πσ (g) or πσ (g)P0 are nontrivial, which requires πσ to be of K -type 1. All K -type 1 representations are however non-discrete (in fact spherical principal series) representations. d be given and suppose that the point (c) The proof is by contradiction. Let σ ∈ G spectrum of Tσ is nonempty and contains T . The spectral radius equals T and the normalizable eigenfunction χσ of Tσ has eigenvalue T . By Prop. 3.6(b) the associated eigenfunction τvσ (χσ ) of T is normalizable. This contradicts Prop. 3.1(b). Part (c) of the corollary is useful because in many cases all the fiber operators Tσ are compact (see Sect. 3.3). Then the nonzero spectrum of Tσ is discrete for all nonexceptional σ . Parts (a), (b) are useful because often one has independent reasons to expect that a ground state should exist, which is at least a singlet under a maximal compact subgroup K of G. In all cases the corollary entails that all discrete series representations are ruled out as candidates for the representation carried by the ground state sector. The remaining part of the restricted dual corresponds to noncompact Cartan subalgebras (see Appendix A.2) and the associated irreducible representations are cuspidial principal r of generic linear reductive Lie series. To avoid having to discuss the restricted dual G groups, we focus from now on on the case G = SO0 (1, N ). Its representation theory shows all the typical complications (existence of discrete series, in particular) and one can reasonably expect that the subsequent results will generalize to other Lie groups. 4. The Structure of the Ground State Sector For definiteness we consider from now on the case G = SO0 (1, N ). The restricted dual r = G d ∪ G p , of a set G d describing discrete UIR (unitary then is a disjoint union G p describing principal series UIR. A UIR irreducible series representations) and a set G d is parameterized for a tuple of integers ±s ∈ N, ξ ∈ M s ⊂ M, a UIR σ p ∈ G p σd ∈ G
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Here M is labeled by a real parameter ω ≥ 0 and again by a tuple of integers ξ ∈ M. is the dual of the subgroup SO(N − 2). We refer to Appendices A and B for a brief survey of the relevant aspects of the representation theory of linear reductive Lie groups in general, and of SO0 (1, N ) in particular. Recall from Sect. 3 that the space of σ -equivariant ground states Gσ (T) can be isometrically identified with the eigenspace E T (T(σ )) of the fiber operator T(σ ) = 1I ⊗ Tˇ σˇ . r for The goal in the following is to successively rule out more and more UIRs σ ∈ G which the fiber spaces in Gσ (T) can be nonempty. This will lead to Theorems 1.1 and 1.2, as announced in the introduction. 4.1. Gσ (T) is empty for all but one principal series representation. We begin by studying in more detail the restricted transfer operators Tωξ := Tσ , where we now write the label σ as (ω, ξ ) corresponding to a member of the principal series (see Appendix B). the subset of G r corresponding to the principal series. We also denote by M Here the realization of the model space Lσ as L 2ξ (K ) and the realization of the infinite sums over κ in terms of k-integrations is useful. See Appendix A.8. We first show that in this realization the function space L 2σ (N )0 in (2.39), for which we now write L 2ξ (N ), takes the following form: f ∈ L 2 (N × K , Vξ ) , f (k0 n, k) = f (n, kk0 ) , k0 ∈ K , f (n, mk) = rξ (m) f (n, k) , m ∈ M .
(4.1)
For fixed n ∈ N the second equation is just the defining relation for L 2ξ (K ), see (A.43). The first relation is the realization of the equivariance condition under r (K ) in the compact model of the principal representation. Indeed the right action f (n, k) → f (n, k)πωξ (g) can be read off from Eq. (A.56b) and is given by f (n, k)πωξ (g) = (δν)−1 (a(kg −1 )) f (n, g −1 [k]) .
(4.2)
For g = k0 ∈ K the right-hand side of (4.2) reduces to f (n, kk0−1 ) so that the equivariance condition assumes the form given in (4.1). As a check one can verify the left K invariance f (k0 n, k)πωξ (k0 g) = f (n, k)πωξ (g) , k0 ∈ K ,
(4.3)
using a(kg −1 k0−1 ) = a(kg −1 ) and k(kg −1 k0−1 ) = k(kg −1 )k0−1 via (A.37). The fiber operator Tωξ will now act on the space of functions (4.1) as an integral operator with kernel Tωξ (n, n ; k, k ) := dg T (g, n , n)[πωξ (g)](k , k) G
dp T (k
= P
−1
pk, n , n) (δ −1 ν)( p) rξ (m( p)) = Tωξ (kn, k n ; e, e) , (4.4)
using (A.54). Explicitly ν = νω and δ are given by (B.9). From (3.5) and (A.54) one verifies the symmetries Tωξ (k0 n, n ; kk0−1 , k ) = Tωξ (n, n ; k, k ) = Tωξ (n, k0 n ; k, k k0−1 ) ,
(4.5)
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which ensure that Tωξ acts consistently on the function space (4.1). For fixed arguments the kernel (4.4) is a linear map on Vξ whose matrix elements obey (4.6) v, Tωξ (n, n ; k, k ) v Vξ ≤ T0,0 (n, n ; k, k ) sup | v, rξ (m)v Vξ | , m∈M
v, v
∈ Vξ . The inequality is strict unless ω = 0 and ξ = 0. Here we wrote for all ξ = 0 for the trivial representation of M and ω = 0 refers to the ω → 0 limit of the principal series. The bound (4.6) is a manifest consequence of (4.4) and the unitarity of ν : A → U (1). With these preparations we can show the crucial Proposition 4.1. (a) The operators Tωξ : L 2ξ (N ) → L 2ξ (N ) are bounded for all (not Their norms Tωξ are continuous functions of ω and obey almost all) ω ≥ 0, ξ ∈ M. and ω ≥ 0 ,
Tωξ ≤ T00 for all ξ ∈ M
(4.7)
where the inequality is strict unless ω = 0 and ξ = 0. (b) T00 is a transfer operator in the sense of Definition 2.1 uniquely associated with T. Proof. (a) For the boundedness we show that the kernel of Tωξ obeys the stronger condition (4.8) sup sup dn dk v, Tωξ (n, n , k, k )v Vξ < ∞ .
v = v =1 n,k
Clearly the expression under the first sup is bounded by sup dgdn T (g, n , n) sup dk v, πω,ξ (g0 )(k, k )v Vξ . n
(4.9)
k,g0
The first factor is bounded by the constant K T in (3.1). To estimate the second factor, let φl be a sequence of positive normalized functions on G approximating a delta distribution centered around g0 ∈ G. From (A.54), v, πωξ (φl )(k, k )v V ≤ dp δ( p)−1 φl (k −1 pk ) sup v, rξ (m)v V . (4.10) ξ ξ m
P
The second factor is a finite constant and it suffices to consider the dk integral of the first factor. Recall from Appendix A that P = N AM, dp = d(na)dm = δ(a)2 dndadm. Parameterizing k as k = k (s )m , with m ∈ M, k (s ) ∈ K /M (see Appendix B) one has dk d(na)dm δ(a)−1 φn (k −1 namk ) = dm dg δ(a(g))φn (k −1 gm ) ≤ 1 , K
P
M
G
(4.11) using dg = dndadmd S(s ) from (B.5), (B.6), and the fact that δ(a(g)) ≤ 1. This gives (4.8). From here it follows (see Definition 2.1) that Tωξ defines a bounded operator from p p L ξ (N ) to L ξ (N ), for all 1 ≤ p ≤ ∞. For p = 2 this is the assertion; for p = 2 it gives an alternative (though less general) proof of the extension discussed after Eq. (3.11). To see the continuity of the norms, recall
Tωξ = sup |( f, Tωξ f )| , f
f ∈ L 2ξ (N ) , ( f, f ) = 1 .
(4.12)
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We may assume that the supremum is reached on a sequence ( fl )l∈N of normalized functions in L 2ξ (N ), in which case it suffices to show that liml→∞ ( fl , Tωξ fl ) is continuous in ω. Each matrix element ( fl , Tωξ fl ) is manifestly a continuous function of ω. To show that this remains true for the limit l → ∞ we establish a uniform bound: using (4.1), (4.4) one has for the matrix elements −1 ( fl , Tωξ fl ) = dndn dkdk dp T (k pk, n , n)(δ −1 ν)( p) P
× fl (n, k), rξ (m( p)) fl (n , k ) Vξ ,
(4.13)
where ω enters only through the phase ν( p) = νω ( p), see (B.9). The modulus of these matrix elements is easily seen to have the l and ω independent upper bound −1 sup dp T (k pk, n , n) δ( p)−1 < ∞ . (4.14) n,k,n ,k
P
By dominated convergence one can take the l → ∞ limit inside the integrals, after which continuity in ω is manifest. Finally, to verify (4.7) we return to (4.12). From (4.6) the modulus of the matrix elements can be bounded by ( f, Tωξ f ) ≤ dkdndk dn f (n, k), Tωξ (n, n ; k, k ) f (n , k ) V ξ ≤
dkdndk dn T0,0 (n, n ; k, k ) f (n, k) Vξ f (n , k ) Vξ ,
(4.15)
where the second inequality is strict unless ω = 0 and ξ = 0. Here both norms are pointwise nonnegative functions in L 20 (N ). In fact one gets a norm preserving map from L 2ξ (N ) to a subset of L 20 (N ) by mapping f (n, k) to f 0 (n, k) := f (n, k) Vξ . In particular f has unit norm in L 2ξ (N ) if and only if f 0 has unit norm in L 20 (N ). On the other hand to get the norm of T00 the supremum of ( f 0 , T00 f 0 ) over all normalized f 0 ∈ L 20 (N ) has to be taken. These are complex valued functions but by the integral representation (4.4) T00 (n, n , k, k ) ≥ 0 and the supremum is reached on the subset of (all) nonnegative functions in L 20 (N ). Among those are the ones in the range of the map L 2ξ (N ) → L 20 (N ), f → f Vξ , described above. So one has for (ξ, ω) = (0, 0),
Tωξ ≤
sup f ∈L 2ξ (N )
|( f, Tωξ f )| < sup(| f |, T00 | f |) ≤
= T00 .
f
sup
( f 0 , T00 f 0 )
f 0 ∈L 20 (N )
(4.16)
(b) From part (a) we know that T00 is a bounded linear integral operator with spectrum contained in [0, T ]. The integral representation (4.4) shows that its kernel is strictly positive almost everywhere. Thus T00 is also positivity improving and by (4.8) it is a transfer operator in the sense of Definition 2.1. As a corollary we obtain part (b) of Theorem 1.1 announced in the introduction.
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Corollary 4.1. Gσ (T) is non-empty for at most one principal series representation – the limit of the spherical principal representation, σ = (ω = 0, ξ = 0). Further G00 (T) can be isometrically identified with the ground state sector G(T00 ) of the transfer operator T00 . We add some comments on the discrete series. Recall from (3.20),
T = ess sup Tσ , r σ ∈G
(4.17)
s ⊂ M. and that a discrete series UIR is labeled by a tuple of integers ±s ∈ N, ξ ∈ M Remark 4.1. We expect that a preferred class of discrete series representations exists such that a bound similar to (4.7) holds. If there are discrete series UIR for which
T±sξ = T , a natural conjecture is that they have the minimal possible M weight for a given s ∈ N, i.e. ξ = ξs := (|s|, . . . , |s|). This requires that the norms T±sξ of the operators T±sξ obey s ,
T±sξ ≤ T±s,ξs for all ξ ∈ M
(4.18)
where the inequality is strict unless ξ = ξs := (|s|, . . . , |s|). A verification would require a concrete model for the discrete series (like that of the principal series described in Appendix A.8) producing a counterpart of Eq. (4.4). We leave this for a future investigation. Here we proceed by identifying simple conditions under which discrete series representations are ruled out as candidates for the representation carried by Gσ (T). A set of such conditions has already been identified in Corollary 3.1. We proceed by studying the role of positivity. 4.2. The role of strict positivity. One property which one expects from a ground state is “not to have nodes”. This means the ground state wave function can be chosen strictly positive a.e. In this section we show that, somewhat surprisingly, the positivity requirement uniquely determines the representation carried by σ -equivariant eigenfunctions. As before we denote by π00 the limiting representation of the spherical principal series πω0 , ω ≥ 0, of G = SO0 (1, N ), and by A00 the corresponding fiber operator in the decomposition of A (see Sect. 3). Further we set P = { ∈ L ∞,2 (Mr ) or ∈ M −1 L ∞,2 (Mr ) | (g, n) > 0 a.e.} ,
(4.19)
for the settings in Sect. 3.3 or 3.4, respectively. Proposition 4.2. Eλ,σ (A) ∩ P = {0} unless πσ = π00 . Further the intersection is isometric to E A (A00 ) and is spanned by a.e. strictly positive functions. Proof. Recall from Sect. 3 that the generalized eigenspaces Eλ,σ (A) invariant under πσ are generated by ‘factorized’ functions of the form i τvi σ (χi )(g, n). Here χ (n)∗κs (πσ (g)v)κs , (4.20) τvσ (χ )(g, n) = (χ (n), πσ (g)v)σ = κs
with v ∈ Lσ and χ ∈ Eλ (Aσ ) a generalized eigenfunction of Aσ . By assumption at least one such function is strictly positive a.e. on Mr , for which we write (g, n) =
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i (χi (n), πσ (g)vi )σ . Replacing g with kg, k ∈ K , and averaging over K must still produce an a.e. strictly positive function. On the other hand, evaluated in the K adapted basis the average reads dk (kg, n) = χi (n)∗00 P0 πσ (g)vi > 0 . (4.21) K
i
Importantly the K average cannot vanish, so at least one of the projections P0 πσ (g)vi must be nonzero. This means Lσ must contain a K -invariant vector, namely P0 πσ (g)vi for some i, and πσ is by definition of K -type 1. But for the UIR of SO0 (1, N ) this is the case only for πω,ξ =0 , see Appendix B. Discrete series representations and non-spherical principal series representations are ruled out. Thus, for Eλ,σ (A) ∩ P to be non-empty the corresponding UIR πσ must be a member of the spherical principal series, πσ = πω0 , for some ω ≥ 0. It remains to show that only the ω → 0 limit can occur. To this end we replace g in (4.21) by gk and average over K . Using the fact that πω,0 (g)00,00 evaluates to the Legendre function E ω,00 (q0 ), q = g −1 q ↑ = (q0 , q), in (B.23) one gets the condition dkdk (kgk , n) = χi (n)∗00 (vi )00 E ω,00 (q0 ) > 0 , (4.22) K ×K
i
for almost all n ∈ N and q0 > 1. However, E ω,00 is an oscillating function of its argument unless ω = 0. This means (4.22) cannot be satisfied for ω = 0. We remark that the first part of the argument extends to all Lie groups of the type considered: Eλ,σ (A)∩P = {0} unless πσ is of K -type 1. The second part of the argument shows that πσ must have a K -spherical function p(g) = πσ (g)00,00 which is real and positive on all of G, i.e. on K \G/K . An interesting (apparently open) question is which nonamenable Lie groups have a UIR with a pointwise positive K -spherical function (not just one of positive type). As a corollary we now obtain part (a) of Theorem 1.1 as anticipated in the introduction. Corollary 4.2. Gσ (T) is empty for all but the principal series representations whenever one of the following holds: (i) Gσ (T) contains a strictly positive function. (ii) Gσ (T) contains a K -singlet. (iii) Tσ is compact. Proof. (i) is a special case of Proposition 4.2. (ii) and (iii) follow from Corollary 3.1. Combined parts (a) and (b) of Theorem 1.1 show that under broad conditions the representation σ carried by the equivariant fibers Gσ (T) of the ground state sector is uniquely determined and always the same for every invariant transfer operator, namely the limit of the spherical principal series. The existence of generalized ground states in G00 (T) G(T00 ) is however not guaranteed from the outset. A simple but important special case where it is, is when all the fiber operators Tσ are compact. 4.3. Gσ (T) for Tσ compact. Here we present the proof of Theorem 1.2, as anticipated in the introduction. We reformulate the theorem in a more precise way as follows: Theorem 1.2’ Let T be an invariant transfer operator in the sense of Definition 2.1 and 3.1, and assume that its kernel satisfies in addition Condition (C) in Sect. 3.3. Then all its
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ground state fibers Gσ (T) but G00 (T) E T (T00 ) are empty. Further T00 is a transfer operator whose ground state sector G(T00 ) = E T (T00 ) is nonempty and is spanned by a unique normalizable and a.e. strictly positive function ψ0 ∈ L 2 (N × S N −1 ) L 2 (N ) ⊗ L00 . Explicitly G(T) is the linear hull of functions ψ0 (n, p) (g, n) = d S( p) , (4.23) N −1 S N −1 (q0 − q · p) 2 where q = gq ↑ = (q0 , q). Proof. By Theorem 1.1 we know that at most G(T00 ) is nonempty. All functions in G(T00 ) are thus M-singlets and take values in C = Vξ =0 . By Lemma 3.2 and Proposition 4.1 T00 is a Hilbert-Schmidt operator and a transfer operator in the sense of Definition 2.1. Its eigenspace E T (T00 ) coincides with its ground state sector G(T00 ). By the generalized Perron-Frobenius theorem [19, 44] it follows that the ground state sector of T00 is one-dimensional (non-empty, in particular) and is spanned by a unique a.e. strictly positive function ψ0 (n, k) > 0 on N × K , with ψ0 (k0 n, k) = ψ0 (n, kk0 ), namely the ground state of T00 . Note that via the realization (4.1) the result is applied to scalar valued functions on N × K rather than to L00 -valued functions on N . Since ξ = 0 the function also obeys ψ0 (n, mk) = ψ0 (n, k), for all m ∈ M, and thus can be viewed as a function on N × S N −1 , with S N −1 = M\K . This is related to the fact that the K -content of the ξ =0 = {(0, . . . , 0, ) , ≥ 0}. principal series representations πω,ξ with ξ = 0 equals K These are the SO(N ) irreducible representations which can be realized on the space of symmetric traceless tensors, or equivalently on the harmonic symmetric polynomials of degree on the sphere S N −1 . In the original model, where ψ0 : N → L00 , we relabel the components ψ0 (n)κs in the K -adapted basis such that ψ0 (n)κs = ψ0 (n) m , where ψ0 (n) m are the components of ψ0 (n, p) in a basis of spherical harmonics Y m ( p) on S N −1 . The matrix element τv,00 (ψ0 )(g, n) = (ψ0 (n), π00 (g)v0 ), then evaluates to a double sum over labels of the spherical harmonics. This holds for any reference vector v ∈ L00 . Specifically we choose now for v the unique K -invariant vector v0 ∈ L00 , i.e. the vector satisfying π00 (k)v0 = v0 for all k ∈ K . In the K -adapted basis of L00 this vector has components (v0 )κs = (v0 )00 δκ=0 δs=0 . The matrix element then evaluates to τv0 ,00 (ψ0 )(g, n) = (ψ0 (n), π00 (g)v0 ) = ψ0 (n) m π00 (g) m,00 (v0 )00 . (4.24)
m
The matrix elements π00 (g) m,00 are right K invariant and reduce to functions on Q = G/K , with the identification q = gq ↑ . In the case at hand these are the Legendre functions E ω=0, m described in (B.24). Using the integral representationin (B.24) the sum N −1 : can be replaced by an integral over S
m ψ0 (n) m E 0, m (q) = d S( p) 0, p (q)ψ0 (n, p), with ψ0 (n, p) = m ψ0 (n) m Y m ( p). Inserting (B.18) and omitting the overall constant (v0 )00 gives the announced expression (4.23). It remains to show that the linear hull of these functions is dense in G(T). We proceed in two steps. First we show that the choice of the K -invariant vector v0 was inessential. This is because on account of the irreducibilityof L00 every vector v ∈ L00 lies in the closure of linear combinations of the form i ci π00 (gi )v0 . Thus τv,00 (g, n) with v ∈ L00 generic lies in the linear hull of the functions (4.24). In a second step we appeal to Proposition 3.6, according to which linear combinations of the form (3.43) (but now with a unique χλ= T ,σ =00,r =1 = ψ0 ) are dense in Eλ= T ,σ =(00) (T) = G(T).
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Remark 4.2. Theorem 1.2’ is a generalization of the result announced in Sect. 3.3 of [12]; as a special case it contains the omitted proof of (a slightly corrected variant of) Theorem 2 in [12]. Explicitly: Let T be the transfer operator of the hyperbolic nonlinear sigma-model (see Eqs. (2.3), (2.4) of [12]). Then T2 satisfies the premise of Theorem 1.2. Proof. One only has to check that Condition (C) in Sect. 3.3 is satisfied. This follows from the Lemma in Sect. 3.3 of [12]. 5. Conclusions and Outlook The set of ground states of a quantum mechanical many particle system – equivalently, a statistical mechanics system on a finite lattice – whose dynamics is invariant under a nonamenable unitary representation of a locally compact group has been shown here to have a remarkable structure: despite the invariance of the dynamics there are always infinitely many noninvariant and nonnormalizable ground states. This spontaneous symmetry breaking cannot be prevented by tuning a coupling or temperature parameter, and it takes place even in one and two dimensions and even for finite systems. The phenomenon is also not limited to a semiclassical regime. The linear space spanned by these ground states carries a distinguished unitary irreducible representation of the group, which for SO0 (1, N ) has been identified as the limit of the spherical principal series. These properties hold for a large class of transfer operators or Hamiltonians. Compactness of the transfer operators restricted to the group invariant fibers is a sufficient but by no means necessary condition. The same is true for the very existence of generalized ground states. As explained in Sect. 2.1 the well-known theory of generalized eigenfunctions due to Gel’fand [17] and their application to group representations due to Maurin [34] is not sufficient to assure the existence of generalized ground states (neither as functions nor as distributions). However, we can guarantee their existence as functions in various situations considerably broader than the above one. These involve by necessity a subsidiary condition on the transfer operator. One can take [39] (s1) T00 is a compact operator. (s2) the integral kernel of T has an extremizing configuration m ∗ ∈ M, in the sense that T (m, m ; t) ≤ T (m ∗ , m ∗ , t) for all m, m ∈ M, and possibly others. In these cases one has the following construction principle: Theorem 5.1. Let T be as in Theorem 1.1 and such that the subsidiary condition (s1) or (s2) is satisfied. Set t (v) := Tt v/(v, Tt v) for v ∈ L 2 (M) and t ∈ N. Then there exist (explicitly known) vectors v∗ ∈ L 2 (M) and a sequence of integers (t j ) j∈N such that the limit ∞ := w ∗ − lim t j (v∗ ) , j→∞
(5.1)
exists and is an element of G(T). The point of Theorem 5.1 is that even a weak limit has the tendency to either diverge or to vanish. It is only by a carefully tuned ‘renormalization’ that a finite and nonzero limit can be obtained which is a generalized ground state of T. The choice of the sequence t j is presumably inessential, but the choice of specific vectors v∗ ∈ L 2 (M) is crucial. Typically these ‘seed’ vectors form a set of measure zero in L 2 (M) and their identification is a major part of the theorem.
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The results for these ‘finite’ systems can be generalized in a number of ways. The choice of M as a product manifold M = Q ×ν is natural from the viewpoint of a quantum mechanical or statistical mechanics interpretation. One could also replace M with the phase space of a dynamical system with constraints generating the Lie algebra of a nonamenable group, in which case M could be any proper riemannian G manifold (in the sense of [35]). We expect that the results of Sect. 2 and 3 remain valid in this more general setting. The main reason for specializing to G = SO0 (1, N ) in Sect. 4 was that these groups have split rank 1, so their restricted dual is relatively simple. Nevertheless we expect that counterparts of Theorem 1.1 exist for basically every noncompact semi-simple Lie group with a finite center: only a tiny subset of unitary irreducible representations are potential candidates for the representation carried by the ground state sector of a system with a G invariant dynamics. Another extension is to supersymmetric multiparticle systems with nonamenable symmetries (see [8] for one-particle systems). The transfer operators associated with graded homogeneous spaces are relevant for the description of disordered systems [13, 14, 63]. They are not symmetric with respect to a positive definite scalar product, but have a unique ground state in a certain sense. Restricted to a bosonic subspace transfer operators related to the ones considered here arise. This leads to the expectation that the lowest lying excitations of the original system would have a structure analogous to that of the ground state space found here. The analysis of the thermodynamic limit is an open problem. In the operator/Hilbert space formulation the thermodynamic limit can presumably at best be taken for specific lattices (hierarchical, Cayley trees, etc.). For the hypercubic lattices directly relevant to lattice quantum field theories one needs to analyze the limit in terms of correlation functions, see [49]. Expectations are however that the structure of the ground state sector, if it changes, will ‘fragment’ even more [38, 12]. In fact the existence of non-invariant expectation values in the thermodynamic limit is a more-or-less direct consequence of the non-amenability [12]. This should lead to a general no-go theorem (inverse, nonconstructive Coleman theorem). We believe that this circle of ideas will be relevant in a variety of contexts, ranging from localization phenomena in solid state physics and quantum Kaluza-Klein theories to AdS duality and the ground states of quantum gravity. We hope to return to these applications elsewhere. Acknowledgements. We wish to thank M. Zirnbauer for calling our attention to the transfer operators on graded homogeneous spaces. E. S. would like to thank H. Saller and S. Ruijsenaars for useful discussions.
Appendix A: Harmonic Analysis on Noncompact Lie Groups In this appendix we summarize the results on the representation theory and the harmonic analysis of non-compact Lie groups needed in the main text. In Appendix B we present more explicit results for SO(1, N ). 1. Basic setting. The harmonic analysis on a noncompact Lie group is governed by Plancherel-type decompositions L 2 (G) =
⊕ r G
dν(σ ) Lσ (G) ⊗ Lˇ σˇ (G) ,
ρ× =
⊕ r G
dν(σ ) πσ ⊗ πσˇ , (A.1)
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which has support only on a proper subset where dν is the Plancherel measure on G r ⊂ G, the restricted dual of the group G. The πσ are unitary continuous represenG r . In the following we tations and Lσ := Lσ (G) is irreducible for ν almost all σ ∈ G present the precise version of (A.1) used in the main text and set the notation. We use the variant of (A.1) valid for any linear connected reductive Lie group G (linear meaning that G is a closed subgroup of GL(N , R) or GL(N , C), reductive meaning that it is stable under conjugate transpose). Examples are SO(1, N ), SO(N , C), SL(N , R), SU(m, n), Sp(2N , R). The Plancherel measure dν(σ ) is then a combination of a counting measure for discrete variables and a Lebesgue measure for continuous variables (see [25] and references therein); for groups G having no compact Cartan subgroup the discrete part is absent (by a theorem of Harish-Chandra, see e.g. [53]). This holds in particular for the complex connected simple Lie groups [33, 55]. The spaces Lσ are Hilbert spaces equipped with an inner product ( , )σ depending on σ . As a consequence of the multiplicity theorems decribed below one can identify each Lσ with a subspace of L 2 (K ), where K is a maximal compact subgroup of G. The πσ are distributions over a space of test functions D on G which take values in the trace class operators on Lσ . For the class of Lie groups considered the distributions arise by integrationwith respect to a L ∞ (G) function πσ (g). Thus, for each φ ∈ D the integral πσ (φ) := dgφ(g)πσ (g) (where dg is Haar measure on G) is a trace class operator whose matrix elements in some orthonormal basis ei in Lσ we denote by dgφ(g)πσ (g)i j . A typical choice for D is Cc∞ (G), the smooth compactly supported functions on G. For ψ ∈ L 2 (G) ∩ L 1 (G) the operator πσ (ψ) is Hilbert-Schmidt, for ψ ∈ L 1 (G) it is still compact [18, 15]. The C-valued functions g → πσ (g)i j are called the coefficients of πσ . They are continuous functions but in general do not have sufficient decay to be (square) integrable on G. This weak continuity is equivalent ([15], p. 68) to the continuity of the Lσ -valued functions. The Plancherel theorem states that φ ∈ D can be expanded in terms of eigenfunctions as (σ )i j πσ (g)i j , φ φ(g) = dν(σ ) r G
(σ )i j = φ
G
i, j
dg φ(g) πσ (g −1 )i j .
The Plancherel-Parseval identity (σ )i∗j ψ (σ )i j , φ dg φ(g)∗ ψ(g) = dν(σ ) r G
G
(A.2)
(A.3)
ij
(σ ). expresses the unitarity of the map φ → φ The irreducible representations on which the Plancherel measure has support turn out to come in families parameterized by conjugacy classes of Cartan subgroups of G. The Cartan subgroups are of the form H = T × R, where T is compact and R Rd for has discrete parameters coming from T and continuous parameters some d. Its dual H The labels σ should be thought of as elements of H and the Plancherel coming from R. measure takes the form dν(σ ) = dσ ν(H : σ ) , (A.4) r G
H
H
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where the sum extends over the (finite number of) conjugacy classes of Cartan subgroups of G and the functions ν(H : σ ) are known ([25] and references therein). A Cartan subgroup here is defined as a maximal abelian subalgebra all of whose elements r are diagonalizable as matrices over the complex numbers. The representations σ ∈ G associated with a compact Cartan subgroup are called discrete series representations, all others are called (on the basis of a theorem) cuspidial principal series. Among them are the principal series proper associated with a unique conjugacy class to be described below. For example SL(2, R) has two conjugacy classes of Cartan subgroups, H = SO(2) and H = {diag(a, a −1 ) | a ∈ R\{0}}. In contrast, SL(n, R) with n > 2 has [n/2]+1 conjugacy classes of Cartan subgroups, none of which is compact. The case G = SO0 (1, N ) will be discussed in more detail in Appendix B. 2. Cartan subgroups. The structure of the Cartan subgroups H entering (A.4) is best described in terms of their Lie algebra h. For the Lie groups considered there exists an involution ι such that the set of its fixed points generates a maximal compact subgroup K of G. (More precisely (G ι )0 ⊂ K ⊂ G ι , where G ι is the fixed point set and (G ι )0 is its identity component.) The involution ι of G induces a decomposition of the Lie algebra g = k ⊕ q, where k and q are even and odd under (the differential of) ι, respectively. The Lie algebras h can be assumed to be invariant under ι. They are of the form h = k0 + a0 , where k0 ⊂ k, a0 ⊂ a. Here a is the Lie algebra of the subgroup A in the Iwasawa decomposition G = N AK . To k0 , a0 one can associate a subgroup P0 = N0 A0 M0 of G called a “cuspidial parabolic subgroup”. Here N0 is a nilpotent subgroup of N , A0 is a subgroup of A, and M0 is such that A0 M0 is the centralizer of A0 in G. There is a systematic technique, called parabolic induction, which allows one to construct unitary representations of G from those of P0 . Almost all of them are irreducible and provide the above “cuspidial principal series” representations. For our purposes the most important one is the principal series proper, which is associated with the (up to conjugacy) unique Cartan subgroup H for which h ∩ q = a, i.e. for which A0 is all of A. The associated subgroup P = N AM is called minimal parabolic and the construction of the associated principal series representations will be detailed later. The other extreme case is when A0 = {e} consists of the identity only, in which case P0 is all of G. The associated representations are precisely the above discrete series representations. By a theorem of Harish-Chandra G has discrete series representations if and only it has a compact Cartan subgroup. Equivalently ([53], p. 282) rank K = rank M + dim A ,
(A.5)
where rank K is the dimension of the maximal torus of K . For example all SO0 ( p, q) groups with pq even have discrete series, as do Sp(n, R), SU( p, q), and SL(2, R). Groups which do not have discrete series are SL(n, R), n > 2, and SO0 ( p, q) with pq odd. Several constructions of (all) discrete series representations are known, see [46]. In particular there is an elegant construction based on the kernel of the Dirac operator on homogeneous spinor bundles, see [41, 1]. r is The content of the theorem mentioned after (A.4) is that the restricted dual G exhausted by the above cuspidial principal series representations and the discrete series representations. 3. Tensor product conventions. Let H be a separable Hilbert space with inner product ( , ), linear in the right and anti-linear in the left argument. The conjugate Hilbert space Hˇ is the Hilbert space with underlying additive group identical to that of H but with scalar ˇ and inner product (v, multiplication defined by (λ, v) ˇ → λ∗ v, ˇ for λ ∈ C, vˇ ∈ H, ˇ w) ˇ := (w, v). The Hilbert space Hˇ can canonically be identified with the space of linear forms
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L(H, C) on H. Indeed by Riesz theorem L(H, C) (v, ·) → vˇ ∈ Hˇ is a linear isomorphism of complex vector spaces. The map H → Hˇ is anti-linear; it associates to each v ∈ H a linear form as usual, vˇ = (v, · ), v(u) ˇ = (v, u). The orthonormality relations of a basis ei , i ∈ N, then amount to eˇ j (ei ) = δi j . If vi = eˇi (v) are the components of a vector v = i vi ei , then the associated linear form vˇ has the complex conjugate components, vˇ = i vi∗ eˇi . Suppose that H carries in addition a representation π of some Lie group G. Then π(g)v has components (π(g)v)i = j π(g)i j v j , where π(g)i j := eˇi (π(g)e j ) are the matrix elements of π(g). Note that in these conventions π(g) acts by ‘matrix multiplication’ on the components v j = eˇ j (v). We define the conjugate representa−1 )u)). Since ((π tion πˇ on Hˇ by (πˇ (g)v)(u) ˇ := v(π(g ˇ ˇ (g)v) ˇ ∨ , u) = (π(g) ˇ v)(u) ˇ = −1 )u) = (v, π(g −1 )u) = (π(g)v, u), this amounts to [π ∨ = π(g)v, or v(π(g ˇ ˇ (g)v] ˇ ∗ ∗ πˇ (g)vˇ = [π(g)v]∨ . In particular π(g) ˇ vˇ has components (πˇ (g)v) ˇ i = j π(g)i j v j , if v j are the components of v ∈ H. For separable Hilbert spaces H1 and H2 with orthonormal bases ei , i ∈ N, and f j , j ∈ N, repectively, we define the tensor product H2 ⊗ Hˇ 1 as the Hilbert space spanned by f i eˇ j =: f i ⊗ eˇi and completed with respect to the inner product ( f i ⊗ eˇ j , f k ⊗ eˇl )2 = δik δ jl . The tensor product H2 ⊗ Hˇ 1 can be canonically identified with the space J2 (H1 , H2 ) of Hilbert-Schmidtoperators F : H1 → H2 , equipped with the inner product (F, F )2 = Tr[F † F ] = i j F ji∗ F ji , where F = i j f i Fi j eˇ j are the components Hˇ 1 → J2 (H1 , H2 ) is simply given by the exten of F. Theisometry H2 ⊗ sion of ( i vi f i ) ⊗ ( j w j e j )∨ → i j vi w ∗j f i eˇ j . (This isometry was in fact already used before in the identification f i eˇ j = f i ⊗ eˇi .) Suppose now that H1 , H2 carry unitary representations π1 , π2 , respectively. Then H2 ⊗ Hˇ 1 carries a unitary representation π2 × πˇ 1 of G × G, the outer tensor product of π2 and πˇ 1 . It is given by (π2 × πˇ 1 )(g2 , g1 )(v2 ⊗ vˇ1 ) := π2 (g2 )v2 ⊗ πˇ 1 (g1 )vˇ1 . In the realization as Hilbert-Schmidt operators this means (π2 × πˇ 1 )(g2 , g1 )F = π2 (g2 )Fπ1 (g1 )† = π2 (g2 )Fπ1 (g1 )−1 .
(A.6)
The outer tensor product of two unitary representations is irreducible if and only if both constituents are. The diagonal representation (π2 ⊗ πˇ 1 )(g) := (π2 × πˇ 1 )(g, g) of G is called the inner tensor product of π2 and πˇ 1 ; of course it is in general not irreducible if π2 and πˇ 1 are. 4. Relation to amenability. The support of the Plancherel measure in (A.1) also reflects the amenability or nonamenability of the underlying group (see [42] for the definition of amenable topological groups). For a continuous unitary representation π of a locally weakly contained in π . Here π1 compact group G the support of π is the set σ ∈ G is said to be weakly contained in π2 if every function of positive type can be approximated, uniformly on compact subsets of G, by finite sums of functions of positive type associated with π2 . Here functions of positive type can be identified with the diagonal r , the reduced dual matrix elements g → (v, π(g)v) of a representation. By definition G of G, is the support of the (left or right) regular representation of G; see [9], Definitions 18.1.7 and 18.3.1. An amenable locally compact group G is characterized by the r = G ([9], Prop. 18.3.6). In fact, whenever G r weakly contains a single property that G r = G follows ([9], Prop. 18.3.6 finite dimensional continuous unitary representation G and Addendum 18.9.5). For the group itself one has: a connected semisimple Lie group with a finite center is amenable if and only if it is compact (see [42] or [60], Prop. 4.1.8). All the noncompact (linear reductive) Lie groups considered here are therefore nonamer is a proper subset of G and that G r cannot contain any finite nable. It follows that G
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dimensional continuous unitary representation. In contrast the Euclidean group ISO(N ) considered in Appendix C is amenable. Sometimes also the concept of an amenable representation is useful, which relates to (A.6). Specifically one is interested in situations where π1 ⊗ π2 contains the unit representation of G, i.e. the singlet. A simple case is when π is finite dimensional; then π ⊗ πˇ always contains the unit representation (as F = 1I by (A.6) is clearly an invariant tensor). More generally one has ([5], Prop. A.1.11 and Corollary A.1.12): Let π1 , π2 be unitary representations of the Lie group G on H1 , H2 . Then the following are equivalent: (i) π1 ⊗ π2 contains the unit representation. (ii) There exists a finite dimensional representation π which is a subrepresentation of both π1 and π2 . Further, if π1 is irreducible condition (ii) can be replaced by: (ii)’ π1 is finite dimensional and is contained in πˇ 2 . The unitary representation π is called amenable if for π1 = π , π2 = πˇ any one of the conditions (i)–(iii) is satisfied. This in turn can be shown to be equivalent to [4]: a unitary representation π of a Lie group G on a Hilbert space H is amenable if there exists a positive linear functional ω over B(H) (the C ∗ -algebra of bounded linear operators on H) such that ω(π(g)T π(g)−1 ) = ω(T ), for all g ∈ G and all T ∈ B(H). Further is amenable [4]. For a simple a locally compact group is amenable iff every π ∈ G noncompact Lie group (connected with finite center and rank > 1) the only amenable representations are those containing the trivial one. 5. Characters. The coefficient functions g → πσ (g)i j form a unitary irreducible representation, viz πσ (g1 g2 )i j = πσ (g1 )ik πσ (g2 )k j , k
πσ (e)i j = δi j ,
πσ (g −1 )i j = [πσ (g) ji ]∗ .
(A.7)
For a compact Lie group the coefficients also obey orthogonality and completeness relations essentially summarizing the content of the Plancherel (or Peter-Weyl) expansion. In the case of a noncompact Lie group these have no direct counterpart in that double sums over products of the matrix coefficients or traces diverge. Instead character functions and the associated spectral projectors provide the appropriate substitute for orthogonality and completeness relations. (σ ) is a trace class operator for all Characters are defined as follows [55]. Since φ φ ∈ D the trace σ (φ) := φ (σ )ii = dg φ(g)πσ (g −1 )ii , (A.8) i
i
G
is well-defined and independent of the choice of orthonormal basis on Lσ . Thus σ : D → C is a distribution over D for every unitary irreducible representation πσ . It characterizes such a representation in that σ1 = σ2 holds if and only if the representations πσ1 and πσ2 are unitarily equivalent. A representation πσ is called tempered if the distribution σ extends continuously to S(G), the Schwartz space of functions φ ∈ C ∞ (G) such that φ and all its derivatives are square integrable on G. The sum in (A.8) can of course not be pulled inside the integral: as all eigenvalues of πσ (g) have modulus one the sum over πσ (g)ii diverges. However for the class of Lie groups considered a regularity theorem ensures the existence of a locally integrable function σ such that (A.9) σ (φ) = dg φ(g) σ (g −1 ) for all φ ∈ D . G
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The function σ is constant on conjugacy classes in the sense that σ (gg g −1 ) = σ (g ) for all g ∈ G and g ∈ G . Here G is a dense open subset of G characterized by the fact that each of its elements lies in precisely one Cartan subgroup H of G, see e.g. [55]. The character function σ is also an eigenfunction of Z (G), the abelian algebra of all bi-invariant differential operators on G. In terms of them the Plancherel expansion (A.2) can be rewritten as φ(g) = dν(σ ) ( σ ∗ φ)(g) , φ ∈ D . (A.10) r G
Here
( σ ∗ φ)(g) :=
−1
dg σ (gg )φ(g ) G −1 = dg φ(g ) πσ (g)i j πσ (g ) ji . i
G
(A.11)
j
Due to the properties of σ the above tempered irreducible representations come in families parameterized by conjugacy classes of Cartan subgroups of G. The bi-invariant function (g, g ) → σ (gg −1 ) in (A.11) can be viewed as a ‘regularized’ version of the formal double sum that would arise by pulling the sum over i inside the integral. There are two natural ways to achieve such a regularization. One is r , which gives rise to spectral projectors: by performing averages over Borel sets in G −1 −1 E I (gg ) := dν(σ ) σ (gg ) , (A.12a) I (A.12b) dg E I (g1 g −1 )E J (gg2−1 ) = E I ∩J (g1 g2−1 ) , −1 −1 EG )= dν(σ ) σ (gg ) = δ(g, g ) . (A.12c) r (gg r G
6. K-finite functions. A literal way to take the sum over i in (A.11) inside the integral is by restricting the class of functions φ to the K -finite ones. A function f ∈ Cc∞ (G) is called left (resp. right) K -finite ([57] p. 236) if the set { f (kg), k ∈ K } (resp. { f (gk), k ∈ K }) lies in a finite dimensional subspace of C(G), the continuous functions on G. It is called , be the unitary irreducible (bi)-K-finite if both holds. Let K k → rκ (k), κ ∈ K representation of K with highest weight κ on a vector space Vκ , dκ := dim Vκ . Their characters k → χκ (k) := Tr[rκ (k)] obey χκ ∗ χκ = δκ,κ dκ−1 χκ (where we took κ to label a unitary equivalence class and ∗ denotes the convolution product with respect to then E I := κ∈I dκ χκ is the spectral projector; in K ). For any finite subset I ⊂ K particular E I ∗ E J = E I ∩J . The Fourier expansion takes the form f = κ∈ K dκ f ∗ χκ and converges in the L 2 (K ) norm. One can view the characters as functions on G with support on K only and convolute functions in Cc∞ (G) with the projectors E I (with the convention ( f ∗ h)(g) = dg f (gg −1 )h(g ) = dg f (g −1 )h(g g)). Then ([57], p.237) left K -finite iff E I ∗ f = f , f ∈ Cc∞ (G) is (A.13) right K -finite iff f ∗ E I = f , . The function f is bi-K -finite if E I ∗ f ∗ E I = f holds. for some finite I ⊂ K
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A representation π of G on a Hilbert space H is called K -finite ([57], p.232) if its restriction to the compact subgroup K is unitary and decomposes into a unitary sum , each occurring with finite multiplicity m κ . That is, of irreducibles (rκ , Vκ ), κ ∈ K ! H = κ∈ K m κ Vκ as a representation of K . Let Pκ : H → m κ Vκ be the orthogonal projection. Note that an alternative characterization of a K -finite representation π is that . A vector v ∈ H is called K -finite Pκ is an operator of finite rank on H for all κ ∈ K if π(k)v, k ∈ K , generates a finite dimensional subspace of H ([29], p. 25). Evidently . To study the relation between K this is the case iff PI v = v for some finite I ⊂ K finite functions and K -finite vectors the following explicit realization of the projectors is useful: dk χκ (k −1 )π(k)v , v ∈ H . (A.14) Pκ v = dκ K
Consistency requires that π(k) ◦ Pκ = Pκ ◦ π(k), k ∈ K , which indeed is a property of the right-hand side of (A.14) using that the character χκ is constant on K conjugacy classes. To verify (A.14) first note that the matrix elements of π(k), k ∈ K , are }, where eκ,m+dκ = eκ,m , blockdiagonal in the basis {eκ,s , s = 0, . . . , m κ dκ −1, κ ∈ K m = 0, . . . , dκ −1, is an orthonormal basis of Vκ . Explicitly (eκ s , π(k)eκs ) = δκ ,κ rκ (k)s s . (A.15) Any v ∈ H can by assumption be expanded as v = κ,s (eκs , v) eκs and π(k) acts on m κ Vκ as the blockdiagonal matrix rκ (k). Thus, to verify (A.14) one only has to show that (eκ m , Pκ v) = δκ ,κ (eκm , v). Taking the trace over m 1 = m 2 in 1 dk rκ (k −1 )m 1 m 2 rκ (k)m 3 m 4 = δκκ δm 1 m 4 δm 2 m 3 , (A.16) d κ K (see e.g. [3], p. 170) this readily follows. The interest of these constructions lies in fact that any K -finite representation is a direct orthogonal sum of unitary irreducible representations ([57], Lemma 8.6.22). The irreducible representions are of course also K finite and for them explicit bounds on the multiplicities are available. For the purposes here two results are relevant: Let G be a semi-simple linear connected real Lie group and K a maximal compact subgroup. Then and the irreducible representation κ ∈ K occurs in πσ | K at πσ is K -finite for any σ ∈ G most with multiplicity dκ (see [9], p.331). A theorem by Harish-Chandra ([58], p.319) states that essentially the same is true for any connected semi-simple Lie group with finite center (with a technically slightly different notion of irreducibility). ! On account of these multiplicity bounds one can identify Lσ as a vector space with κ m κ Vκ and since the multiplicities m κ ≤ dκ do not exceed those in the decomposition of L 2 (K ) (which equal dκ ) one can identify each Lσ with a subspace of L 2 (K ). In certain cases the upper bound on the multiplicities is even sharper. In the case G = SO0 (1, N ), K = SO(N ) can occur at most once in πσ | K , by a result due to Dixmier we focus on, each κ ∈ K [10]. Another case when this happens is for G = SL(2, C) and K = SU(2), see [58], p.317, where also the general conditions for m κ ≤ 1 are discussed. Let now π be a K -finite representation and write π(ψ) = dg ψ(g)π(g) for ψ ∈ ∞ Cc (G). Using Eq. (A.14) one readily verifies the following relations: π(E κ ∗ ψ) = π(ψ)Pκ ,
π(ψ ∗ E κ ) = Pκ π(ψ) ,
(A.17)
and the two-sided projections. The same holds for the Fourier coefficients similarly for and dg ψ(g)π(g −1 ) = (π ∗ ψ)(e). For an irreducible representation πσ , σ ∈ G,
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ψ ∈ L 1 (G) ∩ L 2 (G), the relations (A.17) then imply the following decomposition of (σ ) : Lσ → Lσ : the Hilbert-Schmidt operator ψ (σ ) = (σ )κ2 s2 ,κ1 s1 eˇκ1 s1 ψ eκ2 s2 ψ κ2 ,s2 ,κ1 s1
=
κ2 ,κ1
(σ )Pκ1 = Pκ2 ψ
(E κ1∗ψ ∗E κ2 ) (σ ) .
(A.18)
κ2 ,κ1
For left K -finite functions the sum over κ1 is finite, for right K -finite functions the one over κ2 is, and for bi-K finite functions both sums are finite. Here eˇκs is the basis dual (σ )κ2 s2 ,κ1 s1 = eˇκ2 s2 (ψ (σ )eκ1 s1 ) = to eκs , s = 0, . . . , m κ dim Vκ − 1. In particular ψ (eκ2 s2 , ψ (σ )eκ1 s1 )σ . 7. Harmonic analysis on G/K . If instead of Fourier analyzing functions in L 2 (G) one is only interested in L 2 (G/K ) functions, where Q G/K is a symmetric space of noncompact type, the harmonic analysis simplifies considerably. In group theoretical terms it amounts to the decomposition of the quasiregular representation 1 of G on L 2 (Q). We resume the specifications and notations of Sect. A.1; in particular K is a maximal compact subgroup of G, so that Q is an indecomposable Riemannian symmetric space. The key simplification is that for the harmonic analysis on G/K only a subset of the principal series representations proper is needed (see [23] and [24], Sect. IV.7). Recall that the principal series representations proper are those induced by the minimal parabolic subgroup P = N AM, where G = N AK is the Iwasawa decomposition of G and M is the centralizer of A in K . The inducing construction will be described below. The upshot is that the principal series representations πν,ξ are labeled by a character ν : A → U (1) of M. The of A and by a unitary (finite dimensional) irreducible representation ξ ∈ M principal series representations associated with the singlet ξ = 0 of M are called the spherical principal series representations (or minimal or class 1 principal series; see e.g. [58], Vol.1, p.462). These representations πν,0 are thus labeled by elements of A for this subset and only, which can be identified with a subset of Rdim A . We write Q label the characters ν = νω and the representations πω,0 := πνω ,0 by points in ω ∈ Q. The abstract definition of G/K as the subset of G r needed for the harmonic analysis ⊂ Rdim A . Moreover the Plancherel Q on G/K thus turns into the bijection G/K measure dν(σ )| restricted to the spherical principal series is absolutely continuous G /K
with respect to the Lebesgue measure dω on Rdim A , dω = . dν(σ ) G /K |c(ω)|2
(A.19)
Here c(ω) is the Harish-Chandra c-function, for which an explicit formula in terms of the structure of G/K is known. Concerning the K content of the πω,0 representations, it is known that all of them contain the K -singlet with multiplicity 1 ([23], p. 414). As a consequence the representations πω,0 can be set into one-to-one correspondence to K -spherical functions. A continuous function p on G is called K -spherical ([24], p. 357) if it satisfies dk p(g1 kg2 ) = p(g1 ) p(g2 ) . (A.20) This implies that p is K -bi-invariant and normalized p(e) = 1.
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One can explicate this structure by writing out the coefficients of πω,0 in the K adapted basis of Sect. 2.4 using (A.15). An equivalent characterization of the matrix elements in (A.15) is then πω,0 (k1 gk2 )κs,κ s = rκ (k1 )ss1 πω,0 (g)κs1 ,κ s2 rκ (k2 )s2 s . (A.21) s1 ,s2
In particular one sees: πω,0 (g)00,κs is left K -invariant, πω,0 (g)κs,00 is right K -invariant, and πω,0 (g)00,00 is K -spherical. The right K-invariant functions g → [πω,0 (g)]κs,00 are sufficient for the harmonic analysis on the right coset space G/K . To see this observe that for a left K -invariant function (φ(gk) = φ(g), for all k ∈ K ) the decomposition (A.2) specializes to (σ )00,κs δκ ,0 δs ,0 , φ (σ )κs,κ s = dγG (g K ) φ(g K ) dγ K (k) πσ (k −1 g −1 )κs,κ s = φ φ(g K ) =
G/K
K
dν(σ )
G /K
(σ )00,κs πσ (g K )κs,00 . φ
(A.22)
κ,s
r for which the matrix ele for the subset of representations in G Here we wrote G/K ments φ (σ )κs,00 are nonzero. According to the above results it consists of spherical ⊂ Rdim A to a Q principal series representations only, and there is a bijection G/K dim A subset of R . Combined with (A.19) this allows for a very explicit description of the harmonic analysis on G/K . Via the Iwasawa decomposition the section gs (q) provides a one-to-one correspondence between points q ∈ Q and right K -orbits (recall that gs (q) equals na viewed as a function of q). We define E ω,κs (q) := [πω,0 (gs (q))]0 0,κs .
(A.23)
The functions E ω,κs (q) are equivariant with respect to πω,0 , i.e. E ω,κ s (q) πω,0 (g)κ s ,κs . E ω,κs (g −1 q) =
(A.24)
κ s
The spectral synthesis formulas then assume the form dω (ω)κs E ω,κs (q) , ψ ψ(q) = |c(ω)|2 Q κs (ω)κs = dγ Q (q) ψ(q)E ω,κs (q) . ψ
(A.25)
Q
Finally we mention that in the fiber decomposition ⊕ dω L 2 (G/K ) L ,
ω 1 |c(ω)|2
⊕
dω πω,0 , |c(ω)|2
(A.26)
all fiber spaces Lω are isometric to L 2 (K /M). This can be understood from the fact that generic (nonspherical) principal series representations can be modeled on L 2 (K ) (see below); so for the M singlets this gives model spaces isometric to L 2 (K /M). 8. Principal series. As is clear from the preceding discussion the principal series of unitary irreducible representations is at the core of the harmonic analysis for noncompact
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Lie groups. For complex (noncompact semisimple connected) Lie groups (with a finite center) it suffices in fact for the harmonic analysis. Since we will need a number of results that occur in its construction we present here a concise but in principle selfcontained summary thereof. For definiteness we consider again the class of linear reductive Lie groups, though everything goes through also for arbitary non-compact semi-simple Lie groups with a finite center. The principal series arise as special cases of so-called multiplier representations defined as follows. Consider a diffentiable manifold M carrying a differentiable G action M m → g.m ∈ M. For a finite dimensional vector space with inner product , V let L 2 (M, V ) be the Hilbert space of functions f : M → V square integrable with respect to ( f 1 , f 2 ) := dγ (m) f 1 (m), f 2 (m) V , f 1 (m), f 2 (m) V :=
f 1 (m)∗s f 2 (m)s .
(A.27)
s
Let further G × M (g, m) → χ (g, m), χ (g, m) : V → V be a cocycle satisfying χ (g1 g2 , m) = χ (g1 , m)χ (g2 , g1 .m) , χ (e, m) = 1I . Set
" [π(g) f ](m) := χ (g, m) f (g.m)
dγg (m) , dγ
(A.28)
(A.29)
where dγg = d(γ ◦ g) is the translated measure and dγg /dγ is the Radon-Nikodym derivative. The latter ensures both the representation property and the unitarity with respect to the inner product (A.27), π(g1 )π(g2 ) = π(g1 g2 ) , (π(g1 ) f 1 , π(g2 ) f 2 ) = ( f 1 , f 2 ) .
(A.30)
One now applies this construction principle to the group manifold K , where K is the maximal compact subgroup of G. Via the Iwasawa decomposition G = N AK = B K it carries a G-action induced by the right translations. Explicitly let g = n(g)a(g)k(g) = b(g)k(g) be the unique Iwasawa decomposition of some g ∈ G. Then B\G K , k0 g = b(k0 g)k(k0 g) , k0 ∈ K ,
(A.31)
that is, the coset space B\G of left equivalence classes bg ∼ g, b ∈ B, is isometrically identified with K by picking the representative k(k0 g) =: g[k0 ] in the Iwasawa decomposition. For each g ∈ G the map k0 → g[k0 ] defines a diffeomorphism on K , which is the G-action on K needed to define the multiplier representation (A.29). We write d(g[k]) for the translated bi-invariant Haar measure dk in K and d(g[k])/dk for the corresponding Radon-Nikodym derivative. According to the general construction # d(g[k]) [π(g) f ](k) = χ (g, k) f (g[k]) , (A.32) dk defines a unitary representation on L 2 (K , V ) for any cocycle χ (g, k) : V → V . It remains to compute the Radon-Nikodym derivative. It comes out to be (b(kg))−1 ,
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where b → (b) is the (right) modular function of the (non-unimodular) subgroup B = AN . That is, dk f (k) = dk f (g[k])(b(kg))−1 , (A.33) K
K
for all f ∈ Cc (K ). The proof is simple and instructive, so we present it here, also in order to set the conventions (see e.g. [29], p. 44f and [55], p. 83f), with the opposite conventions). Let db be the left invariant Haar measure on B and dg, dk the bi-invariant Haar measures on G, K , respectively. Then dg = dbdk
for functions of g = bk ,
(A.34)
where the order bk (rather than kb) is important. Indeed, by definition the map B × K → G, (b, k) → bk is an isomorphism. There exists therefore an analytical function J : B × K → R+ such that dg = J (b, k)dbdk. Since dg = d(b0 gk0 ) for all b0 ∈ B, k0 ∈ K , it follows that J (b0 b, kk0 ) = J (b, k). Thus J must be a constant, which by a change of normalization can be set to unity. Note that if functions of kb were considered the counterpart of (A.34) would read dg = dr bdk, with dr b the right Haar measure on B. Our convention for the modular function is that of [15], p. 46ff, d(bb0 ) = (b0 ) db ,
(A.35)
(b−1 )db.
so that dr b = With these preparations at hand the verification of (A.33) is straightforward. Let E ∈ Cb (B) be a function such that B dbE(b) = 1. Given an arbitrary f ∈ Cc (K ) define a function F ∈ Cc (G) by g = bk → E(b) f (k). Then dk f (k) = dg F(g) = dg F(gg0 ) = dkdbF(bkg0 ) , (A.36) G
G
for all g0 ∈ G. By (A.31) kg0 decomposes as kg0 = b(kg0 )g0 [k], so that F(bkg0 ) = E(bb(kg0 )) f (g0 [k]). Inserting into (A.36) and shifting the integration variable b → bb(kg0 )−1 gives (A.33). To proceed we describe the action of G on itself in terms of the Iwasawa decomposition G = N AK . Converting the relations in [55], p. 84 into the present conventions one has k(g1 g2 ) = k(k(g1 )g2 ) = g2 [k(g1 )] , a(g1 g2 ) = a(g1 )a(k(g1 )g2 ) , n(g1 g2 ) = n(g1 )[a(g1 )n(k(g1 )g2 )a(g1 )
(A.37) −1
].
As a check, note n(g1 g2 )a(g1 g2 )k(g1 g2 ) = n(g1 )a(g1 )(nak)(k(g1 )g2 ) = g1 g2 . In particular it follows that a(g, k) := a(kg) is an A-valued cocycle, a(g1 g2 , k) = a(g1 , k)a(g2 , g1 [k]) , a(e, k) = e .
(A.38)
We evaluate it on two types of continuous group homomorphisms. The first, δ : A → R+ , is obtained from d(ana −1 ) =: δ(a)2 dn ,
(an) = δ(a)2 ,
(A.39)
where g = nak is the Iwasawa decomposition. The first relation in (A.39) defines δ, the second readily follows from db = dadn for functions of b = an (see [55], p. 80,
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[29], p. 39). The second group homomorphism ν : A → U (1) assigns to each a ∈ A a complex phase ν(a) ∈ U (1). We define a representation of G on L 2 (K , V ) by [πν (g) f ](k) := δ(a(g, k))−1 ν(a(g, k)) f (g[k]) .
(A.40)
Since it is obtained by specialization of (A.32) it is unitary. The following property of πν will allow one to restrict it to subspaces of L 2 (K , V ) such that the resulting representations are equivalent to the principal series representations sought after. Namely: (i) The restriction of πν to K is the right regular representation of K , i.e. πν (k0 ) f (k) = f (kk0 ). (ii) πν commutes with the left regular representation of M,
(m) ◦ πν (g) = πν (g) ◦ (m), for all m ∈ M .
(A.41)
Here M is the centralizer of A in K , that is the subgroup M ⊂ K whose elements commute with all elements of A. Property (i) follows from a(k0 , k) = e and k0 [k] = kk0 . Equation (A.40) is equivalent to a(mg) = a(g) , k(mg) = mk(g) , for all m ∈ M .
(A.42)
To verify (A.41) it suffices to note that n(mg) = mn(g)m −1 since M normalizes N . Thus mg = n(mg)a(mg)k(mg) = mn(g)a(mg)m −1 k(mg), which gives (A.41). be an irreducible representation of M on Vξ and Let now m → rξ (m), ξ ∈ M, consider L 2ξ (K ) = { f ∈ L 2 (K , Vξ ) | f (mk) = rξ (m) f (k) , m ∈ M} .
(A.43)
Equation (A.40) implies that for f ∈ L 2ξ (K ) one has [πν (g) f ](mk) =rξ (m)[πν (g) f ](k). Thus πν : L 2ξ (K ) → L 2ξ (K ) and the restriction , (A.44) πν,ξ (g) := πν (g) 2 L ξ (K )
is well defined. The subgroup P := N AM is a (minimal) parabolic subgroup of G and the representations (A.44) are the P-principal series representations of G in the so-called ‘compact model’. Explicitly [πν,ξ (g) f ](k) = δ(a(kg))−1 ν(a(kg)) f (g[k]) , g = nak .
(A.45)
The standard definition of the P-principal series is by ‘parabolic induction’. For completeness we briefly recap this construction and verify that it is equivalent to (A.45). First note that in the above notation χν,ξ (nam) = ν(a)rξ (m) is a unitary representation of P = N AM on Vξ . Indeed, rearranging the Iwasawa components of p1 , p2 ∈ P one finds p1 p2 = n( p1 )n(m( p1 )a( p1 ) p2 )a( p1 )a( p2 )m( p1 )m( p2 ), using an(g)a −1 = n(ag) and mn(g)m −1 = n(mg). The P-principal series is then defined as the representation of G induced by χν,ξ , Ind GP χν,ξ . This means one considers the linear space of functions F : G → Vξ such that F( pg) = ( p)−1/2 χν,ξ ( p)F(g) , dk F1 (k), F2 (k) Vξ , (F1 , F2 ) := K
(A.46)
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which upon completion with respect to the norm given forms a Hilbert space Hχν,ξ . Here is the modular function of P = N AM, which (since M is compact) coincides with the modular function of B = N A and hence is given by (nam) = δ(a)2 . Explicitly the covariance equation in (A.46) thus reads F(namg) = δ(a)−1 ν(a)rξ (m) F(g) .
(A.47)
The induced representation then is defined as the restriction of the right regular representation to Hχν,ξ , [Ind GP (g0 )F](g) := F(gg0 ) ,
F ∈ Hχν,ξ .
(A.48)
The unitarity follows from (A.33) applied to f (k) = F1 (g0 [k]), F2 (g0 [k]) Vξ . The equivalence to the ‘compact model’ (A.44), (A.45) comes about as follows. Via F(nak) = δ(a)−1 ν(a) f (k) ,
(A.49)
one can set up a correspondence between functions F ∈ Cc (G) and f ∈ Cc (K ). Moreover F ∈ Hχν,ξ if and only if f ∈ L 2ν,ξ (K ). To see this one rewrites the argument namg in (A.47) as namg = nn(amg)aa(g) mk(g); in this form the equivariance properties are directly mapped onto each other. The square integrability in both function spaces is the same, as F| K = f in (A.49). Finally the correspondence (A.49) maps the representations onto each other: We use the relation kg0 = (na)(kg0 )g0 [k] to rewrite gg0 = nakg0 as gg0 = nn(akg0 )aa(kg0 )g0 [k] . [Ind GP (g0 )F](g)
(A.50)
= F(gg0 ) = [πν,ξ (g0 ) f ](k), for g = From here one easily verifies nak. Taken together, the restriction map F → F| K intertwines the unitary representations Ind GP χν,ξ on Hχν,ξ and πν,ξ on L 2ξ (K ). In summary, the principal series representations of a semisimple Lie group are pa and by an element ξ ∈ M, rameterised by a unitary character ν : A → U (1), i.e. ν ∈ A, where M is the centralizer of A in K and G = N AK is the Iwasawa decomposition. From the viewpoint of the Plancherel measure (A.4) they account for the conjugacy classes of Cartan subgroups of the form H = T × R, where the noncompact part R (see App.A.1.2), while the compact part T comes from the Cartan is isomorphic to A × HM , σ = (ν, ξ ). The construction does subgroup HM of M. Schematically H = A not ensure irreducibility in itself; typically however principal series representations are irreducible. By Kostant’s theorem ([58], Thm. 5.5.2.3) this is the case whenever G is a semisimple connected Lie group with finite center and P is a minimal parabolic subgroup. In particular this holds for all SO0 (1, N ), N ≤ 3, and for the two-fold simply connected covering Spin(1, N ) of SO0 (1, N ), with N odd. In rare cases a principal series representation may fail to be irreducible; however it then decomposes into a direct sum of irreducible representations, the number of which cannot exceed the order of the Weyl group (see [58], Vol.1, Corr. 5.5.2.2, p. 461). The action of πν,ξ on L 2ξ (K ) and its matrix elements can be described fairly explicitly. To this end set πν (φ) := G dgφ(g)πν (g), for φ ∈ Cc (G). Then [πν (φ) f ](k0 ) = dk πν (φ)(k0 , k) f (k) , (A.51) −1 −1 πν (φ)(k0 , k) = d(na) φ(k0 nak)(δ ν)(a) ,
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that is, πν (φ) acts as an integral operator with the given kernel. If the function φ has support only on K the kernel reduces to πν (φ)(k0 , k) = φ(k0−1 k). Both statements follow directly from (A.45) taking into account that g[k0 ] = k(k0 g) and dg = d(na)dk for functions of g = nak. Since πν commutes with the left regular representation of M, see Eq. (A.40), the kernel obeys πν (φ)(mk0 , mk) = πν (φ)(k0 , k) , m ∈ M .
(A.52)
Using the fact that for any m ∈ M the map n → m −1 nm is a diffeomorphism of N onto itself with unit Jacobian, dn = d(m −1 nm), the invariance (A.52) is also readily verified directly. Both the trace Tr[πν (φ)] = K dkπν (φ)(k, k) and the Hilbert-Schmidt norm πν (φ) 22 = Tr[πν (φ)∗ πν (φ)] are finite for φ ∈ Cc∞ (G). This remains true upon restriction to L 2ξ (K ), where πνξ (φ) basically gives the Fourier coefficients entering the Plancherel decomposition. In detail let [Pξ f ](k) :=
M
dm rξ (m −1 ) f (mk) , [Pξ f ](mk) = rξ (m)[Pξ f ](k) , (A.53)
be the projector from L 2 (K , Vξ ) to L 2ξ (K ). Then πνξ (φ) := πν (φ)Pξ = Pξ πν (φ) = (ν, ξ )† , acts as an integral operator with matrix valued kernel φ πνξ (φ)(k0 , k) =
M
= P
dm πν (φ)(k0 , mk)rξ (m)
dpφ(k0−1 pk)(δ −1 ν)( p)rξ (m( p)) ,
(A.54)
where P = N AM, dp = d(na)dm and m(n a m ) = m . Note that the (M) invariance (A.52) of the kernel πν (φ)(k0 , k) has turned into a covariance πνξ (φ)(mk0 , mk) = rξ (m) πνξ (φ)(k0 , k) rξ (m)−1 .
(A.55)
As noted before, for φ ∈ Cc∞ (G) the kernel (A.54) defines a trace class operator on Both the left and the right action of πνξ (g) on πν,ξ (φ) produces again a trace class operator whose kernel is readily worked out. One finds L 2ξ (K ).
kernel of πν,ξ (g)πνξ (φ) = πν,ξ ( (g)φ)(k, k ) = (δ −1 ν)(a(kg)) πν,ξ (φ)(g[k], k ) , kernel of πν,ξ (φ)πνξ (g) = πν,ξ (ρ(g
−1
(A.56a)
)φ)(k, k )
= (δν)−1 (a(k g −1 )) πν,ξ (φ)(k, g −1 [k ])) . Note that (A.56b) is the formal adjoint of (A.56a), as it should.
(A.56b)
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Appendix B: The Restricted Dual of SO(1, N) Here we explicate some of the general results in Appendix A for the case SO0 (1, N ). These groups have split rank 1, i.e. the subgroup A in the Iwasasa decomposition is one-dimensional. As a consequence the restricted dual is exhausted by the principal series proper and the discrete series; cuspidial principal series are absent. Further all unitary irreducible representations are multiplicity free, which gives rise to very explicit descriptions of their SO(N ) content. Finally one can get explicit expressions for the coefficients of the spherical principal series in terms of simple special functions (generalized Legendre functions). 1. Group decompositions and orbits. The groups SO0 (1, N ) are generalizations of the Lorentz group (N = 3) and the de-Sitter group (N = 4). They are connected and locally compact; the two-fold simply connected covering group of SO0 (1, N ) is Spin(1, N ). The natural action of SO0 (1, N ) on R1,N decomposes it into 6 types of orbits: the origin {x = 0}, the two-sheeted hyperboloids {x · x = r, ±x 0 > 0}, the one-sheeted hyperboloids (de-Sitter spaces) {x · x = −r } (with r > 0 in both cases), and finally the cones {x · x = 0, ±x 0 > 0}. Both for the detailed description of these orbits and for the representation theory the Iwasawa decomposition G = N AK is instrumental. For G = SO0 (1, N ) it takes the following form: K SO(N ) is the isotropy group of q ↑ = (1, 0, . . . , 0). A is the one-dimensional subgroup generated by a(θ ), θ ∈ R, and N is the N −1 dimensional subgroup generated by n(t), t = (t1 , . . . , t N −1 )T ∈ R N −1 , where ⎞ ⎛ ⎞ ⎛ 2 chθ shθ 1 + 21 t 2 − t2 tT ⎟ ⎜ ⎟ ⎜ shθ chθ 2 n(t) = ⎝ t 2 a(θ ) = ⎝ ⎠, 1− t t T ⎠ (B.1) 2
1I N −1
t
2
−t
1I N −1
with t 2 := t12 + · · · + t N2 −1 . Observe that n(t)n(t ) = n(t + t ) and a(θ )a(θ ) = a(θ + θ ), so that N R N −1 and A R. Each element g ∈ G admits a unique decomposition g = nak with n ∈ N , a ∈ A, and k ∈ K . Since g −1 = k −1 a −1 n −1 the same holds for a decomposition with the subgroups oppositely ordered, G = K AN . Let η = diag(1, −1, . . . , −1) be the bilinear form on R1,N . As before let M denote the centralizer of A in K . Clearly M SO(N − 1), with SO(N − 1) acting on the lower (N −1) × (N −1) block of the matrices. One has a(θ )n(t)a(θ )−1 = n(eθ t) ,
mn(t)m −1 = n(mt) , m ∈ M .
(B.2)
This shows that as subgroups AN = N A and that N M is the semidirect product of N R N −1 with M. Explicit parameterizations of the various G-orbits in R1,N can be obtained from the Iwasawa decomposition by letting it act on a reference vector of the orbit. For the upper sheet H N = {q · q = 1 , q0 > 0} of the two-sheeted hyperboloid we take q ↑ = (1, 0, . . . , 0) as the reference vector. The action of G via the N AK decomposition then gives the ‘horispherical’ coordinates on H N . Indeed, n(t)a(θ )q ↑ parameterizes a unique point q = (q0 , . . . , q N ) in H N , 1 1 q0 = chθ + t 2 e−θ , q1 = shθ + t 2 e−θ , qi = e−θ ti−1 , i = 2, . . . , N , (B.3) 2 2 and (θ, t1 , . . . , t N −1 ) are its horospherical coordinates. One can invert the transformation, in which case the product n(t)a(θ ) viewed as a function of q gives back the section
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gs (q) of Eq. (2.13). The isometry H N SO(1, N )/SO(N ) is likewise manifest. Similar descriptions – not needed here – exist for the cone {x · x = 0, x 0 > 0} and the one-sheeted hyperboloid {q · q = −1}. Finally we note the relevant invariant measures. Let dk denote the normalized Haar measure on K = SO(N ). Set da = dθ for a = a(θ ) ∈ A which gives Haar measure on A. Then d(na) = e−θ(N −1) dθ dt1 . . . dt N −1 ,
(B.4)
is the left invariant measure on N A. The left invariance d(n(t0 )a(θ0 )na) = d(na) is easily checked from a(θ0 )n(t) = n(eθ0 t)a(θ0 ). The Haar measure on G in the N AK Iwasawa decomposition is then given by dg = d(na)dk ,
g ∈ N AK .
(B.5)
If elements k ∈ K are decomposed according to k = k(s )m, m ∈ M, s ∈ S N −1 K /M, the normalized measures on K , M and S N −1 are related by dk = d S(s )dm .
(B.6)
2. Dual and restricted dual of SO0 (1, N ). For G = SO0 (1, N ) and Spin(1, N ) the dual space (as a topological space) is known completely [2]. The lists in [40] and [26] are not quite complete and do not discuss the square integrability of the representations. r we briefly sketch these results To illustrate the relation to the restricted dual G here. All unitary irreducible representations (UIR) come from those of the Lie algebra. For general N the relevant UIR of the Lie algebra so(1, N ) have been classified by Ottoson [40] and Schwarz [47]. Apart from the singlet π0 there are three main types of UIR. For the harmonic analysis on L 2 (SO0 (1, N )) only two of them are needed, the principal π(princ) and the discrete series π(disc). In addition there are complementary series π(comp) (subdivided in [40, 47] into supplementary series and exceptional series). Each UIR is labeled by r := rank SO0 (1, N ) = [(N + 1)/2] real parameters, (ξ1 , . . . , ξr −1 , s). The ξi are nonnegative and are either all integers or all halfintegers and are ordered ξ1 ≤ ξ2 ≤ · · · ≤ ξr −1 . The remaining parameter s can be real or complex. Depending on the value of s the parameters ξ are constrained by further conditions, which together with the value of s specify to which series a given UIR belongs. For the principal series one has −1) , π(princ) : s = iω , ω ≥ 0 , ξ ∈ SO(N
(B.7)
where the explicit description of the weights ξ is given in Eq. (B.11) below. The general discussion of the principal series from Appendix A can be straightforwardly specialized to the case of SO0 (1, N ). The A homomorphisms ν =: νω and δ are given by νω :A → U (1) , δ:A → R+ ,
νω (a(θ )) = eiωθ , ω ≥ 0 , δ(a(θ )) = e
−θ
N −1 2
(B.8)
.
For N = 2r even there are also discrete series representations which enter the Plancherel decomposition. One has [27, 32] π ± (disc) :
±s ≤ ξ1 ≤ · · · ≤ ξr −1 , ±s ∈ N, ξi ∈ N ,
(B.9)
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where N are the positive integers and the π ± series are labeled by the sign of s. The (−s, ξ ) representation is the mirror image of the (s, ξ ) representation in the following sense. Define a spatial reflection by θ (q0 , q1 , . . . , q N ) = (q0 , −q1 , . . . , q N ), which induces an outer automorphism g → θgθ −1 of SO0 (1, N ) (in the defining matrix representation). Then the mirror image πθ(s,ξ ) is defined by πθ(s,ξ ) (g) = πs,ξ (θgθ −1 ); the result mentioned is that πθ(s,ξ ) is unitary equivalent to π−s,ξ . An explicit formula for the value of the quadratic Casimir on any of these UIR (in terms of their parameters) is known [27, 11]. As described in Appendix A.1 the Cartan subgroups are relevant for the harmonic analysis on G. In the case of G = SO0 (1, N ), there is a single conjugacy class of Cartan subgroups when N is odd and two when N is even [57], p. 188,212. In the notation of the previous section this arises because dim A = 1 in the Iwasawa decomposition G = K AN . In this case the number of conjugacy classes of G is 2rankK −rank M . (Recall that rank SO(N ) = [N /2], rank SO(1,N) = [(N+1)/2]). Explicitly, if HM is the (up to conjugacy unique) Cartan subalgebra of M one can adjoin either the generator of A or the generator of HK /HM (where HK is the up to conjugacy unique Cartan subalgebra of K ) in order to obtain an abelian subgroup of G. For N even both subgroups obtained in this way have the same dimension N /2 = rankSO(N ) + 1. For N odd HM and HK have the same dimension N2 − 1 (a 2 × 2 block was needed to add a new Cartan generator). The compact abelian subalgebra HM = HK is thus not maximal and only the noncompact Cartan subalgebra exists. According to the general discussion the Plancherel decomposition takes the form L 2 (SO0 (1, N )) = L 2 (SO0 (1, N ))disc ⊕
L 2 (SO0 (1, N ))disc =
0 ξ ∈M
⎧ ⎨
{0}
⎩!
2 σ ∈disc L σ (SO0 (1, N ))
∞
dμ(ω, ξ )Lω,ξ ,
N odd , N even ,
(B.10)
where the sum in the discrete part ranges over the set in (B.9). An explicit formula for dμ(ω, ξ ) is known [27]. 3. K-content of principal and discrete series. Restricted to the subgroup K = SO(N ) the irreducible representations πσ of SO0 (1, N ) decompose into a direct sum of irreducible representations rκ of K , each of which occurs with multiplicity at most one [10]. The σ ⊂ K which occurs with nonzero (and hence unit) multiplicity is called the subset of K K content of the representation. Here we describe it explicitly for the principal and the discrete series. Recall from (B.7) that a principal series representation is labeled by a real parameter ω ≥ 0 and by a highest weight ξ of M = SO(N −1). Explicitly the latter means N even : ξ = ξ1 , . . . , ξ N −2 , 0 ≤ ξ1 ≤ · · · ≤ ξ N −2 , 2 2 N odd : ξ = ξ1 , . . . , ξ N −1 , |ξ1 | ≤ ξ2 ≤ · · · ≤ ξ N −1 . 2
(B.11)
2
Here we used that UIR of the orthogonal groups are labeled by highest weights which are ordered sets (m 1 , . . . , m r ), where r = rank SO(N ) equals N /2 and (N −1)/2 for N even and odd, respectively. The m i are either all integer or all halfinteger and are subject
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to the constraints SO(2r ) : |m 1 | ≤ m 2 ≤ · · · ≤ m r , SO(2r + 1) :
0 ≤ m 1 ≤ · · · ≤ mr .
(B.12)
The singlet corresponds to κ = (0, . . . , 0); the symmetric traceless tensor representions have κ = (0, . . . , 0, m r ), m r ∈ N. The mirror image θ κ of an UIR κ is defined as follows. Let θ ∈ O(N ) be a reflection, i.e. θ 2 = 1I, det θ = −1. Since θ kθ −1 ∈ SO(N ) for all k, we can define a representation of SO(N ) by rθκ (k) := rκ (θ kθ −1 ) called the mirror image of κ. It is again an irreducible highest weight representation and unitarily equivalent to rκ , where SO(2r ) : κ = (−m 1 , m 2 , . . . , m r ) , SO(2r + 1) : κ = (m 1 , m 2 , . . . , m r ) = κ .
(B.13)
The K = SO(N ) content πω,ξ can now be described explicitly πωξ = rκ , K
(B.14)
ξ κ∈ K
ξ ⊂ K where each UIR of K occurs with multiplicity precisely one and the subsets K are characterized by [40, 47] N even : κ = (m 1 , . . . , m N ) , 2 N odd : κ = (m 1 , . . . , m N −1 ) , 2
|m 1 | ≤ ξ1 ≤ m 2 ≤ · · · ≤ ξ N −2 ≤ m N , 2 2 (B.15) |ξ1 | ≤ m 1 ≤ ξ2 ≤ · · · ≤ ξ N −1 ≤ m N −1 . 2
2
These are precisely the same conditions under which the UIR ξ of M = SO(N − 1) occurs (with unit multiplicity) in the restriction of rκ | M (see e.g. [3]). The result (B.11) thus exemplifies the general reciprocity rule mentioned in part A: if πωξ | K contains (with unit multiplicity) then rκ | M contains ξ ∈ M (with unit multiplicity). κ∈K Since the last label m r , r = rank SO(N ), can be made arbitrarily large the subsets K ξ are always infinite. However with the exception of ξ = 0 (the M-singlet) they never contain κ = 0 (the K singlet). The K content of πω,ξ =0 (the spherical principal series) is given by ξ =0 = {(0, . . . , 0, m r ) , K
m r ≥ 0} .
(B.16)
As can be seen from (B.17) below the discrete series representations never contain a K singlet. Irreducible representations containing a vector invariant under a compact subgroup are often called “class 1”. One sees that among the representations of the restricted dual of SO0 (1, N ) the only class 1 representations are those of the spherical principal series πω,0 , and for them the class 1 property with respect to M and with respect to K is equivalent. The K content of the discrete series (B.9) comes out as follows: π ± (disc) :
±s ≤ m 1 ≤ ξ1 ≤ m 2 ≤ · · · ≤ ξr −1 ≤ m r ,
(B.17)
either by direct investigation of the Harish-Chandra characters [32], or by specialization of Blattner’s formula [22]. One sees from (B.13) that a discrete series representation together with its mirror image θ κ: m 1 is strictly positive for never contains some κ ∈ K
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π + ∩ K π − = ∅. In the π + series and strictly negative for the π − series. Equivalently K particular the π ± series never contain the K singlet. 4. Harmonic analysis on H N . Here we explicate the reduction of the harmonic analysis on L 2 (SO0 (1, N )) to that on L 2 (H N ), where H N = SO0 (1, N )/SO(N ), N ≥ 2, is the upper part of the two-sheeted hyperboloid in R N +1 . The result will be the decomposition (A.26) of the quasiregular representation 1 of SO0 (1, N ) on H N . Since K /M S N −1 all fiber spaces will be isometric to L 2 (S N −1 ). The matrix elements (A.23) come out to be certain Legendre functions which are also generalized eigenfunctions of the LaplaceBeltrami operator on H N . Let −H N be minus the Laplace-Beltrami operator on H N . Its spectrum is absolutely continuous and is given by the interval 41 (N−1)2 +ω2 , ω > 0. There are several complete orthogonal systems of improper eigenfunctions. From a group theoretical viewpoint the most convenient systems are the ‘principal plane waves’ ω, p (q) (see [56] and the references therein) labeled by ω > 0 and a ‘momentum’ vector p ∈ S N −1 . Parameterizing ) q = (ξ, ξ 2 − 1 s), they read ) 1 ω, p (q) := [ξ − ξ 2 − 1 s · p]− 2 (N −1)−iω . (B.18) The completeness and orthogonality relations take the form dγ Q (q) ω, p (q)∗ ω , p (q) = d(ω)−1 δ(ω − ω )δ( p, p ) ,
∞
dω d(ω) 0
d S( p) ω, p (q)∗ ω, p (q ) = δ(q, q ) ,
S N −1
(B.19)
where δ(q, q ) and δ( p, p ) are the normalized delta distributions with respect to the invariant measures dγ Q (q) and d S( p) on H N and S N −1 , respectively. In terms of the coordinates (ξ, s) the former reads ∞ dγ Q (q) = dξ(ξ 2 − 1) N /2−1 d S(s ) . (B.20) 1
S N −1
The spectral weight is determined by the Harish-Chandra c-function for SO0 (1, N ) and is given by 2 1 N 2−1 + iω d(ω) = (B.21) . (2π ) N (iω) The main virtue of these functions is their simple transformation law under SO0 (1, N ), see e.g. Appendix A of [12]. It characterizes the spherical principal unitary series πω,0 , ω ≥ 0, of SO0 (1, N ), where πω,0 and its complex conjugate are unitary equivalent (see e.g. [56], Sects. 9.2.1 and 9.2.7). The orthogonality and completeness relations (B.19) amount to the decomposition (A.26) of the quasi-regular representation 1 on L 2 (H N ). Spectral projectors E I commuting with 1 are defined in terms of their kernels E I (q · q ), I ⊂ R+ by E I (q · q ) := dω d(ω) d S( p) ω, p (q)∗ ω, p (q ) , N −1 I S (B.22) d(q )E I (q · q )E J (q · q ) = E I ∩J (q · q ) .
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Combined with the completeness relation in (B.19) this shows that the spectrum of −H N is absolutely continuous. A complete orthogonal set of real eigenfunctions of −H N is obtained by taking the d S( p) average of the product of ω, p (q) with some spherical harmonics on the p-sphere. This amounts to a decomposition in terms of SO↑(N ) UIR, where the ‘radial’ parts of the resulting eigenfunctions are given by Legendre functions. Using the normalization and the integral representation from ([20] p. 1000) one has in particular
1−N /2
1
S N −1
d S( p) ω, p (q) = (2π ) N /2 (ξ 2 − 1) 4 (2−N ) P−1/2+iω (ξ ) .
(B.23)
As a check on the normalizations one can take the ξ → 1+ limit in (B.23). The limit on the rhs is regular and gives 2π N /2 / (N /2), which equals the area of S N −1 as required by the limit of the lhs. Denoting the set of real scalar spherical harmonics by Y m (k),
∈ N0 , m = 0, . . . , d −1, with d = (2 + N − 2)(κ + N − 3)!/( !(N − 2)!) we set E ω, m (q) :=
d S( p) Y m ( p)ω, p (q)
(B.24a) 1−N /2−
1
= n (ω) Y m (s ) (ξ 2 − 1) 4 (2−N ) P−1/2+iω (ξ ) ,
(B.24b)
N /2
with n 0 (ω) = (2π ) , n (ω) ⎞1/2 ⎛
−1
2 ⎠ , ≥ 1. ω2 + N 2−1 + j = (2π ) N /2 ⎝ j=0
The expression (B.24b) is manifestly real, the equivalence to (B.24a) can be seen as follows: from (B.19), (B.20), and the orthogonality and completeness of the spherical harmonics one readily verifies that both (B.24a) and (B.24b) satisfy
d(q) E ω, m (q)∗ E ω , m (q) = d(ω)−1 δ(ω − ω )δ , δm,m ,
∞
dω d(ω) 0
(B.25) E ω, m (q)∗ E ω, m (q ) = δ(q, q ) .
,m
Further both (B.24a) and (B.24b) transform irreducibly with respect to the real d dimensional matrix representation of SO↑(N ) carried by the spherical harmonics. Hence they must coincide. A drawback of the functions (B.24) is that the k integration spoils the simple transformation law of the ω, p under SO(1, N ). The transformation law can now be inferred from the addition theorem
,m
1
1−N /2
E ω, m (q)E ω, m (q ) = (2π ) N /2 [(q ·q )2 − 1] 4 (2−N ) P−1/2+iω (q ·q ) .
(B.26)
For example for q = gq ↑ this describes the transformation of the SO↑(N ) singlet E ω,0,0 (q) under g ∈ SO(1, N ).
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Appendix C : The Amenable Case ISO (N) Here we outline counterparts of our main results for the coset space ISO(N )/SO(N ) R N . This coset can be viewed as the flat space limit of the hyperboloid SO0 (1, N )/SO(N ). The underlying Euclidean group ISO(N ) is noncompact but amenable; in accordance with the general picture [38] the generalized spin systems turn out to have a unique non-normalizable ground state. The group ISO(N ) is the semi-direct product of the amenable groups SO(N ) and R N , and hence is itself amenable, [9], Lemma 18.3.7. As the defining representation one can take a subgroup of N +1 × N +1 matrices acting on vectors (x, 1)T in R N +1 k a x kx + a g(k, a) = , g(k, a) = , k ∈ SO(N ) , a, x ∈ R N . (C.1) 0 1 1 1 The composition law is g(k1 , a1 )g(k2 , a2 ) = g(k1 k2 , a1 +k1 a2 ), which gives g(k, a)−1 = g(k −1 , −k −1 a) and g(k, a) = g(e, a)g(k, 0) = g(k, 0)g(e, k −1 a). Haar measure on ISO(N ) is dkda, where dk is the normalized Haar measure on SO(N ) and da Lebesgue measure on R N . We shall need a section gs : R N → ISO(N ) such that x = gs (x)x ↑ , where x ↑ = (0, 1)T is fixed by the rotation subgroup. An obvious choice is e x , ks (g(k, a), x) = k −1 , (C.2) gs (x) = 0 1 for which the cocycle ks in (2.12) is independent of x and a. The configuration manifold is M = Q ν = R N ν , the state space is L 2 (M) = 2 L (R N ν ), and ISO(N ) acts on it via the ν-fold inner product of the left quasiregular representation 1 . Explicitly [ M (k, a)ψ](x) = ψ(k −1 (x − a)) , x = (x1 , . . . , xν ) ,
(C.3)
where we wrote M (k, a) for M (g(k, a)). A transfer operator T in the sense of Definition 2.1 is described by a kernel T : R N ν × R N ν → R+ , which is symmetric, continuous, pointwise strictly positive, and subject to the condition (2.2). The invariance
M ◦ T = T ◦ M translates into T (k(x + a), k(y + a)) = T (x, y), for all k ∈ SO(N ) and a ∈ R N . To exploit this symmetry we proceed as in Sect. 2 and switch to a model of L 2 (M) where M acts via right multiplication on a single group-valued argument. Recall that Mr = (G × N )/d(K ) and that the isometry to M is constructed from the map φ˜ : M → ISO(N ) × N , N R N (ν−1) , ˜ 1 , . . . , xν ) = (gs (x1 )−1 , x2 − x1 , . . . , xν − x1 ) , φ(x ˜ −1 x1 , . . . , g −1 xν ) = (gs (x1 )−1 g, x2 − x1 , . . . , xν − x1 ) . φ(g
(C.4)
As expected the group acts on the image points by right multiplication on the first argument. Since g(k0 , 0)gs (x1 )−1 g(k, a) = g(k0 k, −k0 (x1 − a)) the left SO(N ) invariant functions on ISO(N ) × N are characterized by ψr (gs (e, −k −1 (x1 − a)), k −1 n) = ψr (g(k, −x1 + a), n) .
(C.5)
Effectively the functions ψr thus project onto functions on Q × N , however at the price of a more complicated group action. In fact ψs (x, n) := ψr (g(e, −x), n) defines an element of L 2 (Ms ) with the group action [ s (k, a)ψs ](x, n) = ψs (k −1 (x − a), k −1 n).
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The unitary irreducible representations (UIR) entering the decomposition of the regular representation of ISO(N ) on L 2 (ISO(N )) can be described as follows (Gross and Kunze, [21]). For 0 = ν ∈ R N , let K ν be the isotropy group of ν in K = SO(N ). Then each K ν is conjugate to M := SO↑ (N − 1), defined as the subgroup leaving be an irreducible repree1 = (1, 0, . . . , 0)T ∈ R N invariant. Let m → rξ (m), ξ ∈ M, sentation of M on Vξ and consider (as in Eq. (A.43)) L 2ξ (K ) = { f ∈ L 2 (K , Vξ ) , f (mk) = rξ (m) f (k) , m ∈ M} , ( f 1 , f 2 ) := dk f 1 (k), f 2 (k) Vξ .
(C.6)
K
On L 2ξ (K ) define a unitary representation by [πν,ξ (k0 , a0 ) f ](k) = eiν·ka0 f (kk0 ) .
(C.7)
This is well-defined because πν,ξ commutes with the left regular representation of M. In particular [πν,ξ (k0 , a0 ) f ](mk) = rξ (m)[πν,ξ (k0 , a0 ) f ](k). Moreover [21]: – πν,ξ is irreducible for all 0 = ν ∈ R N and ξ ∈ M. – Every infinite dimensional unitary representation is equivalent to some such πν,ξ . the representations πν,ξ and πν ,ξ are equiv– Given 0 = ν, ν ∈ R N and ξ, ξ ∈ M, alent if and only if first ν and ν belong to the same SO(N ) orbit and second ξ and ξ are equivalent under the identification of K ν with K ν . These representations constitute the principal series of ISO(N ), the subset with ξ = 0 (M-singlets) is called the spherical principal series. We now fix a representative from each equivalence class as follows. If ν · ν = ν · ν = ω2 , ω ∈ R+ , then by abuse of notation we denote the N -tuple (0, . . . , 0, ω)T ∈ R N also by ω. In this case we write K ω = SO↑ (N −1) for K ν and πω,ξ for πν,ξ . The K -content of these representations is the same as that of the corresponding principal series representations of SO0 (1, N ), as their restrictions to K coincide. This gives πω,ξ = r , (C.8) K
ξ
∈ K
ξ ⊂ K as in (B.15). with K The finite dimensional representations of ISO(N), obtained from the irreducible representations of SO(N) by representing the abelian normal subgroup of translations trivially are not in the support of the regular representation. So the Plancherel decomposition takes the form ∞ 2 L (ISO(N )) = dωd(ω) dim Vξ Lω,ξ ⊗ Lˇ ω,ξˇ , 0
ξ ∈M
∞
ρ × =
dωd(ω) 0
ξ ∈M
dim Vξ πω,ξ ⊗ πˇ ω,ξˇ
(C.9)
with d(ω) =
ω N −1 2π N /2 (2π ) N (N /2)
(C.10)
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(which can be viewed as the squared inverse of the Harish-Chandra c function for ISO(N ); see [52] for N = 2). The formulae for the harmonic analysis and synthesis will be given below. In relation to (C.9) the singlet representation deserves special consideration. Since ISO(N ) is amenable the singlet must weakly be contained in (equivalently: lie in the support of) the regular representation (see e.g. [9], Prop.18.3.6, and Definitions 18.3.1, 18.1.7). By definition this support is the restricted dual of the locally compact group under consideration, here ISO(N ). As a matter of fact [9], 18.8.4, it also coincides with the carrier of the Plancherel measure. The upshot is that the limit limω→0 πω,ξ =0 is weakly contained in the decomposition (C.9) and coincides with the singlet representation of ISO(N ). We shall write π00 for it. To every φ ∈ L 1 (ISO(N )) one can asign a compact operator as its Fourier transform (ν, ξ )† = dkda φ(k, a) πν,ξ (k, a) = πν,ξ (φ) , φ (C.11) and for φ ∈ (L 1 ∩ L 2 )(ISO(N )) the image is a Hilbert-Schmidt operator for almost all (ω, ξ ) with respect to the Plancherel measure. Since the latter is for fixed ξ absolutely continuous with respect to Lebesgue measure on R+ , the Hilbert-Schmidt property will hold for almost all ω > 0 with respect to the Lebesgue measure. Indeed, πω,ξ (φ) can be realized explicitly as an integral operator on L 2ξ (K ). Repeating the steps in Appendix A.8 one finds [πν,ξ (φ) f ](k) = dk πν,ξ (φ)(k, k ) f (k ) , (C.12) πν,ξ (φ)(k, k ) =
M×R N
dmda φ(k −1 mk , a)eiν·ka rξ (m) .
The formula for the Fourier synthesis reads ∞ (ω, ξ )πω,ξ (k, a)] . dω d(ω) dim Vξ Tr[φ φ(k, a) = 0
(C.13)
ξ ∈M
Formally this can be verified by evaluating the trace in terms of a kernel of the form (C.13) and freely exchanging the order of integrations. For a proof see [21]. With these preparations at hand we can proceed with the group theoretical decomposition of the Hilbert space L 2 (Mr ) and of standard invariant selfadjoint operators A acting on it. The constructions of Sect. 2 and 3 carry over with minor modifications; mainly to fix the notations we run through the main steps. Proposition 2.2 remains valid with the Plancherel measure from (C.9) substituted and with the K -content from (C.8). We use pairs σ = (ω, ξ ) and σˇ = (ω, ξˇ ) to label the representations and their conjugates. Proposition 3.4 likewise carries over and provides the decomposition of the operators. We write ∞ L 2 (Mr ) = dω d(ω) dim Vξ L2ωξ (Mr ) , 0
A=
ξ ∈M
∞
dω d(ω) 0
ξ ∈M
(C.14) ˇ ˇ) , dim Vξ (1I ⊗ A ωξ
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for the respective decompositions. The fiber spaces L2ωξ (Mr ) are isometric to Lωξ ⊗ ˇ ˇ acts for almost all ω > 0 as a bounded linear and selfadjoint Lˇ ωξˇ ⊗ L 2 (N ) and 1I ⊗ A ωξ operator on these fiber spaces. The spectral problems of A and Aωξ can now be related as in Sect. 3.3 and 3.4. We maintain the definitions of the generalized eigenspaces Eλ,ωξ (A) and Eλ (Aωξ ), in Eqs. (3.41) and (3.61), respectively. Then the map τv,ωξ : Eλ (Aωξ ) → Eλ,ωξ (A) , f → τv,ωξ ( f ) , dk f (n, k )∗ [πωξ (k, a)v](k ), v ∈ L 2ξ (K ) , (C.15) τv,ωξ ( f )(k, a, n) = K
again provides an isometry onto its image. In the second line we used the realization of Lω,ξ as L 2ξ (K ) in (C.7). The intertwining properties τv,ωξ ( f )(gg0 , n) = τπωξ (g0 )v,ωξ ( f )(g, n) and Aτv,ωξ ( f ) = τv,ωξ (Aωξ f ) remain valid. Then Propositions 3.6 and 3.8 carry over. Major modifications however occur in the structure of the ground state sector of a transfer operator T. Its fiber operators Tωξ can be realized as integral operators on L 2ξ (K ) with the following kernel: dmda T (k −1 mk , a, n , n) eiω e1 ·ka rξ (m) . (C.16) Tωξ (n, n ; k, k ) = M×R N
For these integral operators a counterpart of Proposition 4.1 holds: the operators Tωξ : Their norms are L 2ξ (N ) → L 2ξ (N ) are bounded for all (not almost all) ω ≥ 0, ξ ∈ M. continuous functions of ω and obey ω ≥ 0,
Tωξ ≤ T00 for all ξ ∈ M,
(C.17)
where the inequality is strict unless ξ = 0 and ω = 0. Further T00 is a transfer operator in the sense of Definition 2.1. This result entails that all the generalized eigenspaces E T (Tωξ ), ω > 0, ξ ∈ M, must be empty. The remaining E T (T00 ) coincides with the ground state sector G(T00 ) of T00 . Provided the map (C.15) is defined also for ω = 0, ξ = 0 it assigns to every generalized ground state f ∈ G(T00 ) a generalized ground state τv,00 ( f ) of T. Moreover by forming linear combinations i τvi ,00 ( f i ) one can generate a dense set in G(T). Since the singlet representation π00 is one-dimensional, while all the πωξ , ω > 0, are infinite dimensional, the family of maps τv,ω0 is of course not continuous for ω → 0. However from (C.16) one can verify directly that every eigenfunction of T00 , viewed as a function on G/K × N /dN (K ) constant in the first argument, is also a generalized eigenfunction of T. Indeed the counterpart of (C.15) for the singlet representation is sim ply τ00 ( f )(e, n) := dk f (n, k)∗ , as the function v is constant and can be omitted. Then [Tτ00 ( f )](e, n) = τ00 (T00 f )(e, n), where the relevant kernel dkda T (k, a, n, n ) is symmetric in n and n . It follows that linear combinations i ci τ00 ( f i ), ci ∈ C, generate a dense subspace of G(T). In particular all generalized ground states of T are functions on G/K ×N /dN (K ) constant in the first argument. This means τ00 : G(T00 ) → G(T) is an isometry. Whenever T is an eigenvalue of T00 both G(T00 ) and G(T) are one-dimensional. Viewed as an element of G(T) however the wave function is not normalizable as the infinite volume of ISO(N ) is overcounted. In summary, we arrive at Theorem C.1. Let T be a transfer operator on L 2 (R N ν ) commuting with the unitary representation (C.3) of ISO(N ). Let Gωξ (T) denote the space of generalized ground states
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−1), and whose elements transform equivariantly according to πωξ , ω ≥ 0, ξ ∈ SO(N let 1I ⊗ Tˇ ωξˇ be the component of T in the fiber πωξ . Then: (a) Gωξ (T) is empty unless ω = 0, ξ = 0, where π00 is the singlet representation. (b) G(T) can isometrically be identified with G(T00 ) and is generated by rotationally invariant functions of x2 − x1 , . . . , xν − x1 . Whenever T is an eigenvalue of T00 the transfer operator T has a unique ground state, which is up to a phase an a.e. strictly positive function of the above type. References 1. Atiyah, M., Schmid, W.: A geometric construction of the discrete series for semisimple Lie groups. Invent. Math. 42, 1–62 (1977) 2. Baldoni-Silva, M.W., Barbasch, D.: The unitary spectrum of real rank one groups. Invent. Math. 72, 27 (1983) 3. Barut, A., R¸aczka, R.: Theory of group representations and applications. Singapore: World Scientific, 1986 4. Bekka, B.: Amenable unitary representations of locally compact groups. Invent. Math. 100, 383 (1990) 5. Bekka, B., de la Harpe, P., Valette, A.: Kazhdan groups. Forthcoming book, see http://www.mmas.univmetz.fr/ bekka 6. Bohm, A., Gadella, M.: Dirac Kets, Gamow Vectors and Gel’fand Triplets. The Rigged Hilbert Space Formulation of Quantum Mechanics. Berlin, etc.: Springer-Verlag, 1989 7. Comtet, A.: On the Landau levels on the hyperbolic plane. Ann. Phys. 173, 185 (1987) 8. Davies, A., MacFarlane, A., van Holten, J.: The quantum-mechanical supersymmetric O(3) and O(2,1) sigma-models. Nucl. Phys. B 216, 493 (1983) 9. Dixmier, J.: C ∗ -Algebras. Amsterdam, etc.: North Holland, 1977 10. Dixmier, J.: Sur les représentations de certains groupes orthogonaux. Compt. Rend. 250, 3263 (1960) 11. Dobrev, V., Mack, G., Petkova, V., Petrova, S., Todorov, I.: Harmonic analysis on the n-dimensional Lorentz group and its applications to conformal quantum field theory. Lecture Notes in Physics, Vol. 63, Berlin, etc.: Springer, 1977 12. Duncan, A., Niedermaier, M., Seiler, E.: Vacuum orbit and spontaneous symmetry breaking in hyperbolic sigma-models. Nucl. Phys. B720, 235–288 (2005). (Erratum) Nucl. Phys. B758, 330–331 (2006) 13. Efetov, K.B.: Supersymmetry and theory of disordered metals. Adv. Phys. 32, 53 (1983) 14. Efetov, K.B.: Supersymmetry in Disorder and Chaos. Cambridge: Cambridge University Press, 1997 15. Folland, G.: A course in abstract harmonic analysis. Boca Raton, etc.: CRC Press 1995 16. Fulling, S.: Absence of trivial subrepresentations from tensor products of unitary representations of pseudo-orthogonal groups. J. Math. Phys. 15, 1567 (1974) 17. Gel’fand, I.M., Vilenkin, N.Y.: Generalized Functions. Volume IV, New York: Academic Press, 1964 18. Glimm, J.: Type I C ∗ -Algebras. Ann. Math. 73, 572 (1961) 19. Glimm, J., Jaffe, A.: Quantum Physics. Berlin, etc.: Springer-Verlag, 1987 20. Gradshteyn, I., Ryzhik, I.: Table of integrals and products. New York-London: Academic Press, 1980 21. Gross, K., Kunze, R.: Fourier decompositions of certain representations. In: Symmetric Spaces, W. Boothby, G. Weiss (eds), New York: Marcel Decker, 1972, pp. 119–139 22. Hecht, H., Schmid, W.: A proof of Blattner’s conjecture. Inv. Math. 31, 129 (1975) 23. Helgason, S.: Differential geometry and symmetric spaces. New York: Academic Press, 1962 24. Helgason, S.: Geometric analysis on symmetric spaces. Providence, RI: Amer. Math. Soc., 1991 25. Herb, R., Wolf, J.: The Plancherel theorem for general real semi-simple Lie groups. Compos. Math. 57, 271 (1986) 26. Hirai, T.: On irreducible representations of the Lorentz group of n-th order. Proc. Japan Acad. 38, 258 (1962) 27. Hirai, T.: The Plancherel formula for the Lorentz group of n-th order. Proc. Japan Acad. 42, 323 (1965) 28. Kadison, R.V.: Irreducible Operator Algebras. Proc. Nat. Acad. Sci. U.S.A. 43, 273 (1957) 29. Lang, S.: SL(2, R). Reading, MA, etc.: Addison-Wesley, 1975 30. Lax, P.: Functional Analysis. New York, etc.: Wiley-Interscience, 2002 31. Lott, J.: Renormalization group flow for general sigma-models. Commun. Math. Phys. 107, 165 (1986) 32. Mack, G.: On Blattner’s formula for the discrete series of SO0 (2n, 1). J. Funct. Anal. 23, 311 (1976) 33. Mackey, G.: Induced Representations. New York, etc.: W.A. Benjamin, 1968 34. Maurin, K.: General Eigenfunction Expansions and Unitary Representations of Topological Groups. Warsaw: Polish Scientific Publishers, 1968
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35. Michor, P.: Isometric actions of Lie groups and invariants. Lecture notes, Univ. Vienna, 1997, available at www.mat.univie.ac.at/ michor/tgbook.pdf 36. Mukunda, N., Radhakrishnan, B.: Clebsch-Gordon problem and coefficients of the 3-dimensional Lorentz group in a continuous basis, I-IV. J. Math. Phys. 15, 1320, 1332, 1643, 1656 (1974) 37. Narnhofer, H., Thirring, W.: Spontaneously broken symmetries. Ann. Inst. Henri Poincaré 70, 1 (1999) 38. Niedermaier, M., Seiler, E.: Nonamenability and spontaneous symmetry breaking – the hyperbolic spin chain. Ann. Henri Poincaré 6, 1025–1090 (2005) 39. Niedermaier, M., Seiler, E.: Generalized ground states as a limit of the iterated transfer operator. Unpublished notes, 2005 40. Ottoson, U.: A classification of the unitary irreducible representations of SO0 (1, N ). Commun. Math. Phys. 8, 228 (1968) 41. Parthasarathy, R.: Dirac operator and discrete series. Ann. Math. 96, 1–30 (1972) 42. Paterson, A.: Amenability. Providence, RI: Amer. Math. Soc., 1988 43. Reed, M., Simon, B.: Methods of Modern Mathematical Physics. Volume 1, revised edition, New YorkLondon: Academic Press, 1980 44. Reed, M., Simon, B.: Methods of Modern Mathematical Physics. Volume 4, New York-London: Academic Press, 1978 45. Ruelle, D.: Statistical mechanics. New York: W.A. Benjamin, 1968 46. Schmid, W.: Discrete Series. Proc. Symposia in Pure Mathematics 61, 83–113 (1997) 47. Schwarz, F.: Unitary irreducible representations of SO0 (1, N ). J. Math. Phys. 12, 131 (1971) 48. Sewell, G.L.: Quantum Mechanics and its Emergent Macrophysics. Princeton, N.J.: Princeton University Press, 2002 49. Spencer, T., Zirnbauer, M.: Spontaneous symmetry breaking of a hyperbolic sigma-model in three dimensions. Commun. Math. Phys. 252, 167 (2004) 50. Strichartz, R.: Harmonic analysis on hyperboloids. J. Funct. Anal. 12, 341 (1973) 51. Strocchi, F.: Symmetry Breaking. Berlin, etc.: Springer, 2005 52. Sugiura, M.: Unitary representations and harmonic analysis. Tokyo: Kodansha Scientific Books, 1975 53. Taylor, M.: Noncommutative harmonic analysis. Providence, RI: Amer. Math. Soc., 1986 54. van den Ban, E., Flensted-Jensen, M., Schichtkrull, H.: Basic harmonic analysis on pseudo-riemannian symmetric spaces. In: Noncompact Lie groups and some of their applications, Tanner, E. Wilson, R. (eds), Dordrecht: Kluwer, 1994, pp. 61–101 55. Varadarajan, V.S.: An Introduction to Harmonic Analysis on Semisimple Lie Groups. Cambridge: Cambridge University Press, 1989 56. Vilenkin, N., Klimyk, A.: Representations of Lie groups and special functions. Volume 2, Dordrecht: Kluwer, 1993 57. Wallach, N.: Harmonic analysis on homogeneous spaces. New York: Marcel Dekker, 1973 58. Warner, G.: Harmonic Analysis on semisimple Lie groups. Berlin, etc.: Springer-Verlag (1972) 59. Weinberg, S.: The Quantum Theory of Fields. Volume I, Cambridge: Cambridge University Press, (1995) 60. Zimmer, R.: Ergodic theory and semi-simple groups. Boston, etc.: Birkhäuser, 1984 61. Zirnbauer, M.: Migdal-Kadanoff fixed point in the graded nonlinear sigma-model for disordered single particle systems without time reversal symmetry. Phys. Rev. Lett. 60, 1450 (1988) 62. Zirnbauer, M.: Fourier analysis on a hyperbolic supermanifold with constant curvature. Commun. Math. Phys. 141, 503 (1991) 63. Zirnbauer, M.: Riemannian symmetric spaces and their origin in random matrix theory. J. Math. Phys. 37, 4986 (1996) Communicated by J.Z. Imbrie
Commun. Math. Phys. 270, 445–480 (2007) Digital Object Identifier (DOI) 10.1007/s00220-006-0155-5
Communications in
Mathematical Physics
Semi-Classical Properties of Geometric Quantization with Metaplectic Correction L. Charles Université Pierre et Marie Curie-Paris 6, UMR 7586 Institut de Mathématiques de Jussieu, Paris, F-75005, France. E-mail:
[email protected] Received: 16 February 2006 / Accepted: 14 June 2006 Published online: 14 December 2006 – © Springer-Verlag 2006
Abstract: The geometric quantization of a symplectic manifold endowed with a prequantum bundle and a metaplectic structure is defined by means of an integrable complex structure. We prove that its semi-classical limit does not depend on the choice of the complex structure. We show this in two ways. First, by introducing unitary identifications between the quantum spaces associated to the various complex polarizations and second, by defining an asymptotically flat connection in the bundle of quantum spaces over the space of complex structures. Furthermore Berezin-Toeplitz operators are intertwined by these identifications and have principal and subprincipal symbols defined independently of the complex structure. The relation with the Schrödinger equation and the group of prequantum bundle automorphisms is considered as well.
1. Introduction Geometric Quantization of Kostant [14] and Souriau [17] is a procedure which associates a quantum space to a symplectic manifold endowed with a prequantum bundle and a polarization. Since its introduction, there has been some attempt to find natural identifications between quantum spaces associated to different polarizations (cf. [4, 16]). In the case of symplectic compact manifolds with complex polarizations, Ginzburg and Montgomery observed in [10] that a natural identification does not exist for a broad class of manifolds. Recently Foth and Uribe [9] obtained semi-classical results in the same direction. We prove that there exists a natural semi-classical identification when the definition of the quantum spaces is altered with the metaplectic correction. This result is a consequence of our study undertaken in [8] of the symbolic calculus of Toeplitz operators and Lagrangian sections that we extend in this paper to Fourier integral operators. Before we state our results, let us discuss quantization without metaplectic correction.
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1.1. Ordinary quantization. Let (M, ω) be a symplectic compact manifold with a prequantization bundle L → M, i.e. a Hermitian line bundle with a connection of curvature 1 i ω. Denote by Jint the space of integrable complex structure of M compatible with ω and positive. To any j ∈ Jint is associated a sequence of quantum spaces Qk ( j) := { j-holomorphic sections of L k },
k = 1, 2, . . . .
Here the holomorphic structure of L is the one compatible with the connection. The semi-classical limit corresponds to k → ∞. When k is sufficiently large, the Kodaira vanishing theorem and Riemann-Roch-Hirzebruch theorem imply that the dimension of Qk ( j) is given by a Riemann-Roch number, which only depends on the symplectic structure of M and k. Assume that we can choose such an integer k independently of the complex structure j. Then for any ja , jb ∈ Jint we can identify Qk ( ja ) with Qk ( jb ) by means of a unitary map Uk ( ja , jb ) : Qk ( ja ) → Qk ( jb ). These identifications are mutually compatible if they satisfy: • (functoriality) Uk ( jb , jc ) ◦ Uk ( ja , jb ) = Uk ( ja , jc ), for any ja , jb , jc ∈ Jint . Moreover if these maps are canonical in the sense that they only depend on the complex, symplectic and prequantum structures, they should satisfy: • (naturality) for any prequantization bundle automorphism of L k and complex structures ja , jb ∈ Jint , the diagram Qk ( ja ) ⏐ ⏐ ∗
Uk ( ja , jb )
−−−−−→
Qk ( jb ) ⏐ ⏐ ∗
Uk (∗ ja ,∗ jb )
Qk (∗ ja ) −−−−−−−−−→ Qk (∗ jb ) commutes. Here the vertical maps are pull-back by , sending a j-holomorphic section into a section holomorphic with respect to ∗ j := φ ∗ j, where φ is the symplectomorphism of M covered by . Sometimes one only requires an identification between the projectivised quantum spaces. It is important to observe that if there exists such a collection {Uk ( ja , jb ), ( ja , jb ) ∈ 2 } which is both functorial and natural, then for any complex structure j ∈ J , Jint int the quantum space Qk ( j) becomes a representation of the group G of prequantization bundle automorphism of L k . Indeed let us set Vk () := Uk (∗ j, j) ◦ ∗ : Qk ( j) → Qk ( j),
∈ G.
Then for any prequantization bundle automorphisms 1 and 2 , we have Vk (1 ) ◦ Vk (2 ) =Uk (∗1 j, j) ◦ ∗1 ◦ Uk (∗2 j, j) ◦ ∗2 =Uk (∗1 j, j) ◦ Uk (∗1 ∗2 j, ∗1 j) ◦ ∗1 ◦ ∗2 by naturality, =Uk (∗1 ∗2 j, j) ◦ (2 ◦ 1 )∗ by functoriality, =Vk (2 ◦ 1 ).
(1)
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Considering the associated infinitesimal representation, Ginzburg and Montgomery proved in [10] that the existence of such a representation contradicts “no go” theorems in many cases. Indeed one can view the Lie algebra of G as C ∞ (M, R), the Lie bracket being the Poisson bracket. Then assuming that the maps Uk ( ja , jb ) depend smoothly on ja and jb , we obtain a Lie algebra representation C ∞ (M, R) → End(Qk ( j)). By [10], since M is compact and Qk ( j) is finite dimensional, the associated projective representation is trivial. But for a broad class of manifolds M, G contains a finite dimensional subgroup which preserves a complex structure j and whose induced representation on Qk ( j) is not projectively trivial. The same arguments contradict also the existence of an identification between the projectivised quantum spaces. In spite of this result, there is a natural identification of a particular interest which has been introduced for the quantization of the moduli spaces of flat connections (cf. [12] and [1]). To define it consider the quantum spaces Qk ( j) as the fibers of a bundle Qk → Jint . Then introducing a functorial and natural family (Uk ( ja , jb )) which depends smoothly on ja and jb amounts to endowing this bundle with a flat G-invariant connection. Now consider Qk as a subbundle of Pk := C ∞ (M, L k ) × Jint → Jint . Since Pk is trivial, it has a natural flat connection and Qk is equipped with the projected connection. Because of the previous result, the curvature Rk of Qk can not vanish in general. On the other hand by the theory of Boutet de Monvel and Guillemin [5], the Toeplitz operators provide an asymptotic representation of the Poisson algebra C ∞ (M) as operators on End(Qk ( j)) when k → ∞. So it is possible that the curvature Rk is asymptotically flat (cf. end of Sect. 8.4 for a quantitative argument). Foth and Uribe compute the asymptotics of Rk in [9] and prove the following: for any j ∈ Jint and tangent vectors η, μ ∈ T j Jint , there exists a function f (η, μ) ∈ C ∞ (M) such that Rk (η, μ) = k ( j) f (η, μ) + O(k −1 ) : Qk ( j) → Qk ( j), where k ( j) is the orthogonal projector of C ∞ (M, L k ) onto Qm k ( j). Furthermore, they give a simple formula for the multiplicator f (η, μ), which shows that it does not vanish for a generic choice of (η, μ). Consequently the curvature is not asymptotically flat. Neither is it asymptotically projectively flat. 1.2. Main results. Let us turn to geometric quantization with metaplectic correction. The metaplectic structures were introduced by Kostant in [15] as metaplectic principal bundles lifting the symplectic frame bundle (cf. also [11] and [2]). Here we use the half-form bundle approach (cf. [18]) more convenient for our purpose. Given a complex structure j ∈ Jint , a half-form bundle (δ, ϕ) of (M, j) is a line bundle δ → M together with an isomorphism of line bundles ∗ ϕ : δ 2 → n,0 j T M
covering the identity of M. (M, j) admits a half-form bundle if and only if the second Stiefel-Whitney class of M vanishes. From now on, we assume this condition is satisfied and we set k Qm k ( j, δ, ϕ) := { j-holomorphic sections of δ ⊗ L }, k = 1, 2, . . . ,
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where the superscript “m” stands for metaplectic. Here the holomorphic structure of δ is such that ϕ is an isomorphism of holomorphic bundles. There is an obvious notion of isomorphism between two half-form bundles associated to the same complex structure and these isomorphisms give rise to isomorphisms between the associated quantum spaces. Our aim is to extend this to the whole collection D of triples ( j, δ, ϕ), where j ranges through Jint . In Sect. 5.1, we define a collection M of morphisms, which makes (D, M) a category such that every morphism is an isomorphism. Important facts are that the automorphism group of any a ∈ D is Z2 and the isomorphism classes are in one to one correspondence with the elements of H 1 (M, Z2 ). Furthermore isomorphism classes correspond to equivalence classes of metaplectic structures of M. Theorem 1.1. There exists a family ((Ukm ( ))k ; ∈ M) such that for any morphism
: a → b, the sequence (Ukm ( ))k consists of operators m Ukm ( ) : Qm k (a) → Qk (b)
that are unitary if k is sufficiently large. Furthermore, for any composable morphisms
and ∈ M, we have Ukm ( ) ◦ Ukm ( ) = Ukm ( ◦ ) + O(k −1 ), where the estimate O(k −1 ) is for the uniform norm of operators. One of the original motivations to introduce the metaplectic correction was to define some natural pairings between the quantum spaces associated to different polarizations, which are called now Blattner-Kostant-Sternberg pairings. Our construction of the operators Uk ( ) is rather different. These are Fourier integral operators with a prescribed principal symbol and the functoriality property is a consequence of the symbolic calculus. We interpret this theorem as a semi-classical functoriality of quantization with halfform bundle. Moreover, the family ((Ukm ( ))k ; ∈ M) is natural with respect to a suitable action of the group G of prequantization bundle automorphisms of L on (D, M). We can therefore adapt the previous construction (1) and we obtain for any a ∈ D an asymptotic representation on Qm k (a) of a central extension by Z2 of the identity component of G. This is in some sense a generalization of the standard metaplectic representation. We will also prove that the operators Ukm ( ) can be defined as parallel transport in an appropriate bundle. Let us consider a smooth family ((δ j , ϕ j ), j ∈ Jint ) of isomorphic half-form bundles. Let Qm k → Jint be the quantum space bundle, whose fiber over j is the space of j-holomorphic sections of L k ⊗ δ j . Theorem 1.2. For any positive integer k, the bundle (Qm k → Jint ) has a canonical m m connection ∇ Qk . The sequence (∇ Qk , k ∈ N∗ ) satisfies • for any j ∈ Jint and tangent vectors η, μ ∈ T j Jint , the uniform norm of the curvature m R Qk (η, μ) is O(k −1 ), • the parallel transport in Qm k along a curve γ with endpoints ja and jb is equal to Ukm ( ) modulo O(k −1 ). Here Ukm ( ) is the sequence of Theorem 1.1 and is the half-form bundle morphism ( ja , δ ja , ϕ ja ) → ( jb , δ jb , ϕ jb ) obtained by extending continuously the identity of ( ja , δ ja , ϕ ja ) in morphisms ( ja , δ ja , ϕ ja ) → (γ (t), δγ (t) , ϕγ (t) ).
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The connection ∇ Qk is induced by a connection on the prequantum space bundle. However the latter bundle is not trivial contrary to the case without metaplectic correction. The paper is organized as follows. Section 2 is devoted to preliminary material. Section 3 contains our results about symbolic calculus for Fourier integral operators. These results are reformulated in Sect. 5.3 with the half-form bundle formalism. In Sect. 5, we deduce Theorem 1.1 and related facts on the representation of the prequantization bundle automorphisms and the Schrödinger equation. The study of the quantum space bundle and its connection is in Sects. 7. Section 8 is devoted to the action of the prequantization bundle automorphism group on the quantum space bundle. m
2. Preliminaries Let (E, ω) be a symplectic real vector space of dimension 2n. Let J (E, ω) be the space of complex structures j of E compatible with ω and positive. Given j ∈ J (M, ω), we ∗ denote by n,0 j E the line of complex linear forms of E of type (n, 0) for the complex structure j. ∗ Definition 2.1. Given ja and jb ∈ J (E, ω), let ja , jb be the linear map from n,0 ja E
∗ to n,0 jb E such that
¯ ∀ α, β ∈ n,0 E ∗ .
ja , jb (α) ∧ β¯ = α ∧ β, ja Let us give some elementary properties of these maps. First, ja , jb is well-defined and invertible because the sesquilinear pairing n,0 ∗ ∗ ¯ a /ω∧n n,0 jb E × ja E → C, (αb , αa ) → αb ∧ α
is non-degenerate, ja and jb being positive. Whenever ja = jb , ja , jb is the identity. ∗ With the usual scalar product on n,0 j E defined by means of ω and j, the adjoint of ∗
ja , jb is jb , ja . This is easily checked using that the scalar product of α, β ∈ n,0 j E is given by
¯ n. i n(2−n) α ∧ β/ω Last definition that we need is the following. Definition 2.2. Given ja , jb , jc in J (E, ω), let ζ ( ja , jb , jc ) be the complex number such that
ja , jc = ζ ( ja , jb , jc ) jb , jc ◦ ja , jb . As we will see in the next section, the symbols of Fourier integral operators behave in part as square roots of the ja , jb . This will appear first via the continuous square root 1 1 ζ 2 of the complex function ζ : J 3 (E, ω) → C∗ determined by ζ 2 ( j, j, j) = 1, for any 1 j ∈ J (M, ω). ζ 2 is well-defined and analytic because J 3 (E, ω) is contractible and ζ is an analytic function (cf. (5)). It follows from the associativity of the composition that ζ is a cocycle ζ ( jb , jc , jd ).ζ −1 ( ja , jc , jd ).ζ ( ja , jb , jd ).ζ −1 ( ja , jb , jc ) = 1.
(2)
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Furthermore j, j = Id implies ζ ( ja , jb , jb ) = ζ ( ja , ja , jb ) = 1.
(3)
1
The function ζ 2 satisfies the same equations. To prepare further proofs, we compute the function ζ in the following parametrization of J (E, ω). Let us choose a fixed complex structure j0 ∈ J (E, ω). Then for any j ∈ J (E, ω), the space of linear forms of type (1, 0) with respect to j, viewed as a subspace of 0,1 ∗ ∗ E ∗ ⊗ C = 1,0 j0 E ⊕ j0 E ,
is the graph of a complex linear map 0,1 ∗ ∗ μ : 1,0 j0 E → j0 E .
(4)
The condition that j is compatible with ω is that ∀ X, Y ∈ E 0,1 j0 ,
ω(μt X, Y ) + ω(X, μt Y ) = 0,
1,0 where μt : E 0,1 j0 → E j0 is the transposed of μ. And the positivity of j translates into the positivity of the Hermitian map: 1,0 Id − μt μ¯ t : E 1,0 j0 → E j0 .
This defines a one-to-one correspondence between J (M, ω) and an open set of a sub0,1 ∗ ∗ space of Hom( 1,0 j0 E , j0 E ). For any j, let us identify the (n, 0)-forms with respect to j with the (n, 0)-forms with respect to j0 by the map n,0 ∗ ∗ n,0 j0 E → j E ,
α 1 ∧ . . . ∧ α n → (α 1 + μ(α 1 )) ∧ . . . ∧ (α n + μ(α n )).
Then straightforward computations prove the following lemma. ∗ Lemma 2.3. With the previous identifications, ja , jb regarded as a map from n,0 j0 E to itself is the multiplication by Id μ¯ a Id μ¯ a . det −1 , det μa Id μb Id 0,1 ∗ ∗ where the matrices represent maps from 1,0 j0 E ⊕ j0 E to itself. Consequently,
det
ζ ( ja , jb , jc ) = det
Id μ¯ a μb Id Id μ¯ a μc Id
. det
. det
Id μ¯ b μc Id Id μ¯ b μb Id
.
(5)
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3. Fourier Integral Operator Let (M, ω) be a symplectic compact connected manifold with a prequantization bundle (L , ∇), i.e. L is a Hermitian line bundle and ∇ a connection of curvature 1i ω. The quantizations of (M, ω) we will consider depend on two additional datas: a complex structure j of M compatible with ω and positive, and a holomorphic Hermitian line bundle K over the complex manifold (M, j). Let us denote by K the collection of such pairs ( j, K ). To any a = ( ja , K a ) ∈ K, we associate the sequence of Hilbert spaces Hk (a) := {holomorphic sections of L k ⊗ K a },
k = 1, 2, . . . ,
where the holomorphic structure of L is the one compatible with the connection ∇ such that L → M is holomorphic with respect to ja . The scalar product is defined by means of the Hermitian structure of L k ⊗ K a and the Liouville measure of M. For any a, b ∈ K, let us introduce the space F(a, b) of Fourier integral operators from H(a) to H(b). Their definition is a slight generalization of the one in [7] because of the fiber bundles K a and K b . Consider a sequence (Sk ) such that for every k, Sk is an operator Hk (a) → Hk (b). The scalar product of Hk (a) gives us an isomorphism Hom(Hk (a), Hk (b)) Hk (b) ⊗ Hk (a). The latter space can be regarded as the space of holomorphic sections of (L k ⊗ K b ) ( L¯ k ⊗ K a ) → M 2 , where M 2 is endowed with the complex structure ( jb , − ja ). The section Sk (x, y) associated in this way to Sk is its Schwartz kernel. We say that (Sk ) is a Fourier integral operator of F(a, b) if k n Sk (x, y) = E k (x, y) f (x, y, k) + O(k −∞ ), (6) 2π where • E is a section of L L¯ → M 2 such that E(x, y) < 1 if x = y, E(x, x) = u ⊗ u, ¯ ∀u ∈ L x such that u = 1, •
and ∂¯ E ≡ 0 modulo a section vanishing to any order along the diagonal. f (., k) is a sequence of sections of K b K¯ a → M 2 which admits an asymptotic expansion in the C ∞ topology of the form f (., k) = f 0 + k −1 f 1 + k −2 f 2 + . . . whose coefficients satisfy ∂¯ f i ≡ 0 modulo a section vanishing to any order along the diagonal.
Let us define the principal symbol of (Sk ) to be the map x → f 0 (x, x). Using the Hermitian structure of K a , we regard it as a section of Hom(K a , K b ) → M. The principal symbol map σ : F(a, b) → C ∞ (M, Hom(K a , K b )) satisfies the expected property.
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Theorem 3.1. The following sequence is exact σ
0 → F(a, b) ∩ O(k −1 ) → F(a, b) − → C ∞ (M, Hom(K a , K b )) → 0, where the O(k −1 ) is for the uniform norm of operators. The composition of these operators is also as expected, with some complications regarding the product of the symbols. Given three complex structures ja , jb and jc of 1 M, we denote by ζ 2 ( ja , jb , jc ) the function of C ∞ (M) whose values at x is the complex 1 number ζ 2 ( ja (x), jb (x), jc (x)) defined in Sect. 2 with E = Tx M. Theorem 3.2. Let a, b and c belong to K. If T ∈ F(a, b) and S ∈ F(b, c), then S ◦ T is a Fourier integral operator of F(a, c). Furthermore, 1
σ (S ◦ T ) = ζ 2 ( ja , jb , jc ) σ (S) ◦ σ (T ). The two previous theorems were essentially proved in Chapter 4.1 of [7] except the formula for the composition of the symbols, which will be proved in Chapter 4. Since the composition of operators is associative, the same holds for the symbol. Observe that this can be directly checked with the cocycle relation (2). F(a, a) is the space T (a) of Toeplitz operators of H(a). Equivalently, a Toeplitz operator is any sequence (Tk : Hk (a) → Hk (a)) of operator of the form Tk = k g(., k) + Rk , where k is the orthogonal projector of L 2 (M, L k ⊗ K a ) onto Hk (a), g(., k) is a sequence of C ∞ (M) with an asymptotic expansion g0 + k −1 g1 + . . . in the C ∞ topology and Rk is O(k −∞ ). As a result the principal symbol σ (Tk ) is the function g0 . Let us define the normalized symbol of (Tk ) to be the formal series g(., ) +
g(., ), 2
where g(., ) = g0 + g1 + . . . and is the holomorphic Laplacian. We are actually only interested in the two first terms of this series, which are the principal symbol and the subprincipal symbol g1 + 21 g0 . As a consequence of the works of Boutet de Monvel and Guillemin, the product of the normalized symbol is a star-product ([5]). Theorem 3.3. Let S be a Fourier integral operator of F(a, b) and Ta ∈ T (a), Tb ∈ T (b) be two Toeplitz operators with the same principal symbol f . Then Tb ◦ S − S ◦ Ta = k1 R with R ∈ F(a, b). Furthermore the principal symbol of R is Hom(K a ,K b ) σ (S), σ (R) = f 1,b − f 1,a + 21 α ja , jb , X f σ (S) + 1i ∇ X f where • f 1,a , f 1,b are the subprincipal symbols of Ta and Tb respectively, • X f is the Hamiltonian vector field of f ,
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• α ja , jb is the one-form of M such that ∇ ja , jb = 1i α ja , jb ⊗ ja , jb , where ja , jb is the section of n,0 ∗ ∗ Hom( n,0 ja T M, jb T M) → M
whose value at x is the endomorphism ja (x), jb (x) defined in 2.1 with E = Tx M. The n,0 ∗ ∗ connection ∇ is induced by the Chern connections of n,0 ja T M and jb T M. By Theorem 3.2, Tb S − STa is Fourier integral operator of F(a, b). Since 1
1
ζ 2 ( ja , ja , jb ) = ζ 2 ( ja , jb , jb ) = 1, its principal symbol vanishes and consequently R ∈ F(a, b). So the proof of the theorem consists in computing the principal symbol of R. This is postponed to Chapter 6. Let us deduce some interesting consequences. Applying the theorem with two Toeplitz operators S and T of T (a), we recover that the principal symbol of ki [S, T ] is the Poisson bracket of the principal symbols of S and T . Actually, we can also compute the subprincipal symbol of ki [S, T ] with the previous theorem. First, the operators of F(a, b) may be used to identify H(a) with H(b) in a semiclassical sense. More precisely, we consider the space U F(a, b) consisting of Fourier integral operators of F(a, b) which are unitary in the sense that Sk Sk∗ = IdHk (a) and Sk∗ Sk = IdHk (b) when k is sufficiently large. By some standard argument that we briefly recall now, U F(a, b) is not empty if and only if K a and K b are isomorphic as line bundles. First it follows directly from the definition of a Fourier integral operator that the adjoint of an operator S ∈ F(a, b) belongs to F(b, a) and its principal symbol is the adjoint of the principal symbol of S. So if S ∈ U F(a, b), Theorem 3.2 implies that the principal symbol of S is a line bundle isomorphism K a → K b . Conversely if K a and K b are isomorphic, there exists an elliptic R ∈ F(a, b), meaning that its principal symbol doesn’t vanish anywhere. Then R ∗ R is an elliptic Toeplitz operator by Theorem 3.2. 1 1 So (R ∗ R)− 2 is a Toeplitz operator (cf. as instance [6]). Finally R(R ∗ R)− 2 belongs to U F(a, b). Now if S ∈ U F(a, b) and Ta is a Toeplitz operator of H(a), then by Theorem 3.2 Tb = STa S ∗ is a Toeplitz operator of H(b) with the same principal symbol as Ta . Applying Theorem 3.3, we compute its subprincipal symbol in terms of the principal and subprincipal symbols of Ta : f 1,b = f 1,a + α S − 21 α ja , jb , X f , where α S is such that ∇ Hom(K a ,K b ) σ (S) = 1i α S ⊗ σ (S). A consequence of this formula is the following result.
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Theorem 3.4. The composition law ∗a of the normalized symbols of the Toeplitz operators of H(a) satisfies: f ∗a g = f g + 2i { f, g} + O(2 ) and i ( f
∗a g − g ∗a f ) = { f, g} − ω ja − 21 ω K a , X f ∧ X g + O(2 ),
∗ where 1i ω ja and 1i ω K a are the Chern curvatures of n,0 ja T M and K a respectively.
Indeed, if K a and K b are isomorphic and ∗a satisfies the result, the same holds for ∗b because of (7) and the relations ω jb = dα ja , jb + ω ja , ω K b = dα S + ω K a . Furthermore we can explicitly compute ∗a , in the case where M is Cn with H(a) the Bargmann space, and the result is satisfied. Of course, this is not sufficient to conclude. But it appears in the proofs of the previous theorems that all the results about the symbolic calculus are completely local and we can really deduce in this way Theorem 3.4. 4. Proof of Theorem 3.2 The proof relies on the complex stationary phase lemma. We only sketch the first part, because the details appeared in [7], with some typos however. The Schwartz kernel of an operator S ∈ F(a, b) is by definition of the form k n k E a,b (x, y) f (x, y, k) + O(k −∞ ). 2π Let us write on a neighborhood of the diagonal ¯
∇ L L E a,b = 1i αa,b ⊗ E a,b . The following lemma is proved in [7]. Lemma 4.1. The one-form αa,b vanishes along the diagonal of M 2 . Furthermore, for every vector field X 1 , X 2 , Y1 , Y2 of M, L(X 1 ,X 2 ) αa,b , (Y1 , Y2 )(x, x) = ω(q¯a,b (X 1 − X 2 ), Y1 )(x) + ω(qb,a (X 1 − X 2 ), Y2 )(x), where q¯a,b (x) and qb,a (x) are respectively the projections onto (Tx M)0,1 jb with kernel 1,0 0,1 (Tx M)1,0 ja and onto (Tx M) ja with kernel (Tx M) jb .
Consider now S ∈ F(a, b) and S ∈ F(b, c). The Schwartz kernel of S S is k 2n
k k E b,c (x, y).E a,b (y, z) f (x, y, k). f (y, z, k) μ M (y) + O(k −∞ ) 2π M with μ M the Liouville form of M. Since |E b,c (x, y).E a,b (y, z)| < 1 outside the diagonal of M 3 , this integral is O(k −∞ ) outside the diagonal of M 2 , and to estimate it on a neighborhood of (x, z) = (x, x) it suffices to integrate on a neighborhood of x. We evaluate the result by applying the stationary phase lemma. Let us write E b,c (x, y).E a,b (y, z) = ei(x,y,z) t (x) ⊗ t¯(z)
(8)
with t a unitary local section of L → M. We deduce from the previous lemma the following facts:
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• d y vanishes along the diagonal 3 of M 3 . • If Y1 and Y2 are two tangent vectors of M at x, d y2 (Y1 , Y2 )(x, x, x) = ω(qc,b Y1 , Y2 )(x) − ω(q¯a,b Y1 , Y2 )(x).
(9)
In particular d y2 is non-degenerate along 3 . • The kernel of the tangent map to d y at (x, x, x) is
1,0 T(x,x,x) 3 ⊗ C ⊕ (Tx M)0,1 jc × (0) × (0) ⊕ (0) × (0) × (Tx M) ja .
These ensure that we can apply the stationary phase lemma (cf. [13] or the appendix of [8]). Thus the Schwartz kernel of S S is of the form k n F k (x, z)g(x, z, k) + O(k −∞ ), 2π where g(., k) is a sequence of sections of K c K¯ a which admits an asymptotic expansion in negative power of k and r (x,z)
F(x, z) = ei
t (x) ⊗ t¯(z)
with r (x, z) ≡ (x, y, z)
(10)
modulo a linear combination with C ∞ coefficient of the functions ∂ y i (x, y, z), i = 1, . . . , 2n. Let us check that the section F satisfies the assumptions following Eq. (6). Since ∂ y i vanishes along the diagonal, it follows from (8) that r (x, x) = 0. Furthermore, we have Lemma 4.2. Consider M 2 as a complex manifold with complex structure ( jc , − ja ). Then ∂¯ F ≡ 0 modulo a section vanishing to any order along the diagonal. Proof. Introduce complex coordinates x 1 , . . . , x n on M for jc . Let us write ∇t = 1i t ⊗ a j d x j − a¯ j d x¯ j . Derivating Eq. (8) and using that ∇(∂x¯ i ,0) E b,c vanishes to any order along the diagonal of M 2 , we get ∂x¯ i (x, y, z) + a¯ i (x) ≡ 0
(11)
modulo I3 (∞), i.e. modulo a function vanishing to any order along the diagonal of M 3 . Thus ∂x¯ i ∂ y j (x, y, z) ≡ 0
mod I3 (∞).
In the same way, if z 1 , . . . , z n are complex coordinates for ja , we show that ∂z i ∂ y j (x, y, z) ≡ 0
mod I3 (∞).
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Then we deduce from (10) and (11) that for any multi-index α and β, the function β(1)
. . . ∂xα(n) ∂xα(1) ¯ n ∂z 1 ¯1
β(n)
∂x¯ i r (x, z) + a¯ i (x)
. . . ∂z n
vanishes along the diagonal 2 of M 2 . This implies that ¯
L L ∇(∂ F ≡0 i ,0) x¯
modulo a section vanishing to any order along 2 . We treat in the same way the covariant derivatives of F with respect to the vector fields (0, ∂z i ). Then since the kernel of S S is a holomorphic section of (L k ⊗ K b ) ( L¯ k K a ) the coefficients of the asymptotic expansion of g(., k) satisfy ∂¯ gl ≡ 0 modulo a section vanishing at any order along the diagonal. So we proved that S S is a Fourier integral operator of F(a, c). The final step is to compute its symbol. By the stationary phase lemma, we have g(x, x, k) = f (x, x, k). f (x, x, k)
δ(x) 1 2
det [−i∂ y j ∂ y k (x, x, x)] j,k
+ O(k −1 ),
where μ M (y) = δ(y).|dy 1 . . . dy 2n |. We deduce from (9) that −id y2 (Y1 , Y2 )(x, x, x) = −ω(iqc,b Y1 − i q¯a,b Y1 , Y2 )(x) = −ω( jb qc,b Y1 + jb q¯a,b Y1 , Y2 )(x) = gb (qc,b Y1 + q¯a,b Y1 , Y2 )(x), where gb is the metric ω(X, jb Y ). Since the Liouville form μ M is the Riemannian volume for gb , it comes that δ(x) 1 2
det [−i∂ y j ∂ y k (x, x, x)] j,k
1
= det − 2 [qc,b + q¯a,b ](x).
Thus to obtain the formula in Theorem 3.2, we have to show that 1
1
det− 2 [qc,b + q¯a,b ] = ζ 2 ( ja , jb , jc ). To see this, let us choose jb as the reference complex structure and let us associate 0,1 ∗ ∗ to ja and jc the bundle maps μa and μc from 1,0 jb T M to jb T M as in (4). On one hand, we have by (5) (since μb = 0) Id μ¯ a . ζ ( ja , jb , jc ) = det −1 μc Id
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On the other hand, T j0,1 M is the graph of −μat : T j0,1 M → T j1,0 M. It follows that q¯a,b a b b is the map 0 0 : T j1,0 M ⊕ T j0,1 M → T j1,0 M ⊕ T j0,1 M. b b b b μ¯ at Id Similarly, qc,b is the map Id μtc : T j1,0 M ⊕ T j0,1 M → T j1,0 M ⊕ T j0,1 M. b b b b 0 0 The result follows. 5. Half-Form Bundle and Quantization 5.1. Preliminaries on half-form bundle. Let j be an almost-complex structure of M. Recall that a half-form bundle of (M, j) is a complex line bundle δ → M together with a line bundle isomorphism ∗ ϕ : δ 2 → n,0 j T M
which covers the identity of M. Two half-form bundles (δa , ϕa ) and (δb , ϕb ) are isomorphic if there exists a line bundle isomorphism : δa → δb covering the identity and such that ϕb ◦ 2 = ϕa . In the case where there exists a half-form bundle, there are #H 1 (M, Z2 ) isomorphism classes of half-form bundles. The existence and the choice up to isomorphism of a half-form bundle over a symplectic manifold (M, ω) is in some sense independent of the almost complex structure, providing it is compatible with ω and positive. To see this we extend the previous notion of half-form bundle isomorphisms to the collection D consisting of the triples ( j, δ, ϕ), where j is an almost-complex structure of M compatible with ω and positive, and (δ, ϕ) is a half-form bundle for (M, j). Let us define a half-form bundle morphism ( ja , δa , ϕa ) → ( jb , δb , ϕb ) to be an isomorphism of line bundles : δa → δb such that ϕb ◦ 2 = ja , jb ◦ ϕa .
(12)
n,0 ∗ ∗ Here ja , jb is the morphism n,0 ja T M → jb T M defined over x ∈ M as in Definition 2.1 with E = Tx M and the complex structures ja (x) and jb (x). The composition of a morphism : ( ja , δa , ϕa ) → ( jb , δb , ϕb ) with a morphism
: ( jb , δb , ϕb ) → ( jc , δc , ϕc ) is defined as 1
◦m := ζ 2 ( ja , jb , jc ) ◦ , where the product ◦ on the right-hand side is the usual composition of maps and the 1 function ζ 2 ( ja , jb , jc ) is defined as in Sect. 2. Observe that ◦m is the product of symbol appearing in Theorem 3.2.
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It is easily checked that D with this collection of morphisms is a groupoid. The associativity of ◦m follows from the cocycle condition (2). Equations (3) imply that the identity 1a of δa is the unit of ( ja , δa , ϕa ), i.e. 1a ◦m = ,
◦m 1a = ,
if and are any morphisms ( jb , δb , ϕb ) → ( ja , δa , ϕa ) and ( ja , δa , ϕa ) → ( jb , δb ,ϕb ) respectively. Moreover, for any ( j, δ, ϕ) ∈ D, define the Hermitian structure of δ in such a way that ϕ becomes an isomorphism of Hermitian bundles. Then since
∗ja , jb = jb , ja , the adjoint ∗ of any morphism : ( ja , δa , ϕa ) → ( jb , δb , ϕb ) is a morphism ( jb , δb , ϕb ) → ( ja , δa , ϕa ) satisfying
∗ ◦m = 1a ,
◦m ∗ = 1b .
(13)
∗.
So is invertible, with inverse If a and b in D are isomorphic, there exist exactly two morphisms a → b. Observe also that given an almost complex structure j, each isomorphism class of D has a representative whose almost complex structure is j. So the existence of a half-form bundle doesn’t depend on the almost complex structure. And there are # H 1 (M, Z2 ) isomorphism classes in D if it is not empty. 5.2. Quantization. Let us consider now the collection Dint consisting of triples ( j, δ, ϕ) ∈ D with an integrable complex structure j. Given a ∈ Dint , let us denote by Qm k (a) the k Hilbert space of holomorphic sections of L ⊗ δa . With our previous notations Qm k (a) := Hk ( ja , δa ). ∗ Here the holomorphic and Hermitian structures of δa are such that ϕa : δa2 → n,0 ja T M is an isomorphism of holomorphic Hermitian bundle. If a and b belong to Dint , any half-form bundle morphism : δa → δb is the symbol of a unitary Fourier integral operator of F(( ja , δa ), ( jb , δb )), m Ukm ( ) : Qm k (a) → Qk (b), k = 1, 2, . . . .
Indeed if S is a Fourier integral operator with symbol , it follows from (13) and Theorem 3.2 that S ∗ S is a Toeplitz operator with symbol 1. Hence S(S ∗ S)−1/2 is a unitary Fourier integral operator with symbol . Contrary to the notations, U m ( ) = (Ukm ( ))k is not uniquely determined by . It is unique modulo multiplication by a unitary Toeplitz operator of symbol 1. So strictly speaking, U m ( ) is an equivalence class of Fourier integral operators. To avoid any confusion we will say that two such operators are equal modulo O(). Theorem 5.1. U m is functorial, that is if is the composition of the morphisms of half-form bundle and , then U m ( ) = U m ( ) ◦ U m ( ) modulo O(). Furthermore if is a half-form bundle morphism a → b, the map sending the Toeplitz operator T : Qm (a) → Qm (a) into (U m ( ))∗ T U m ( ) : Qm (b) → Qm (b) preserves the normalized symbols modulo O(2 ).
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The first part is an immediate consequence of Theorem 3.2 because the composition of half-form bundle morphisms is the same as the composition of symbols. The second part follows from Theorem 3.3, or more directly from formula (7). The group G of connection-preserving Hermitian automorphisms of L acts on the quantum spaces as follows. First an automorphism of G covers a symplectomorphism φ of M. Then acts on Dint by sending a = ( j, δ, ϕ) into ∗ a := (φ ∗ j, φ ∗ δ, φ ∗ ϕ), where φ ∗ ϕ is defined in such a way that the diagram φ∗
∗ ∗ −−−→ n,0 n,0 j Tφ(x) M − φ ∗ j Tx M ⏐ ∗ ⏐ ϕ⏐ ⏐φ ϕ 2 δφ(x)
(φ ∗ )2
−−−−→
(φ ∗ δ)2x
commutes. Finally the operator C ∞ (M, L k ⊗ δa ) → C ∞ (M, L k ⊗ φ ∗ δa ),
s → ((k )−1 ⊗ φ ∗ ) ◦ s ◦ φ
m ∗ ∗ restricts to a unitary operator Qm k (a) → Qk ( a) that we denote by . Let us consider now a ∈ D, fixed until the end of this section. If belongs to the identity component Go of G, then a and ∗ a are isomorphic half-form bundles. In this case, we associate to any morphism : ∗ a → a the sequence of operators m Vkm ( , ) := Ukm ( ) ◦ ∗ : Qm k (a) → Qk (a).
As the operators U m ( ), V m ( , ) is uniquely defined up to multiplication by a unitary Toeplitz operator with symbol 1. Denote by G0m the set of pairs ( , ), where ∈ Go and is a half-form bundle morphism ∗ a → a. Theorem 5.2. For any half-form bundle a ∈ Dint , G0m endowed with the product ( 1 , 1 ).( 2 , 2 ) := ( 2 ◦m (∗2 1 ), 1 ◦ 2 ) is a central extension of G0 by Z2 . Furthermore V m is a right-representation of G0m up to O() in the sense that V m ( 2 , 2 ) ◦ V m ( 1 , 1 ) ≡ V m (( 1 , 1 ).( 2 , 2 ))
mod O().
In the definition of the product of G0m , we used the following action of G on the halfform bundle morphisms. If is prequantization bundle automorphism of L covering the symplectomorphism φ and is a morphism a → b, then ∗ is the morphism ∗ a → ∗ b defined in such a way that the diagram δa ⏐ ⏐ φ∗
−−−−→ δb ⏐ ⏐ ∗ φ ∗
φ ∗ δa −−−−→ φ ∗ δb commutes. One deduces easily from the relations ∗2 (∗1 ) = (1 ◦ 2 )∗ ,
∗ ( 1 ◦m 2 ) = (∗ 1 ) ◦m (∗ 2 )
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that G0m is a group. Furthermore, one has U m (∗ ) = ∗ ◦ U m ( ) ◦ (∗ )−1
mod O()
which implies the last part of the theorem: V m ( 2 , 2 ) ◦ V m ( 1 , 1 ) =U m ( 2 ) ◦ ∗2 ◦ U m ( 1 ) ◦ ∗1 =U m ( 2 ) ◦ ∗2 ◦ U m ( 1 ) ◦ (∗2 )−1 ◦ (1 ◦ 2 )∗ =U m ( 2 ) ◦ U m (∗2 1 ) ◦ (1 ◦ 2 )∗ mod O() =U m ( 2 ◦m ∗2 1 ) ◦ (1 ◦ 2 )∗ mod O() =V m (( 1 , 1 ).( 2 , 2 )) by Theorem 5.1. It is well-known that the Lie algebra of G0 is C ∞ (M, R), the Lie bracket being the Poisson bracket (cf. (16) for an explicit formula for the exponential map). Let us associate to any f ∈ C ∞ (M) a Toeplitz operator Qm ( f ) of Qm (a) whose normalized symbol is f modulo O(2 ). By Theorem 3.4, we obtain a Lie algebra representation up to O() in the sense that
m m m −1 1 i kQk ( f ), kQk (g) = kQk ({ f, g}) + O(k ). By exponentiating we recover the representation of Theorem 5.2. Theorem 5.3. For any f ∈ C ∞ (M, R), we have m exp it.kQm k ( f ) = Vk ( t , t )
mod O(),
where t = exp(t f ) and ( t ) is the continuous family of half-form bundle morphisms
t : a → ∗t a such that 0 is the identity of δa . This last result will be proved in Sect. 8 (cf. the remark after Corollary 8.3). 5.3. Reformulation of the results of Chapter 3. Assume that (M, ω) admits a unique half-form bundle up to isomorphism. If this is not the case we can still apply what follows by restricting to an open contractible set of M. Let us return to the quantum spaces H(a) defined from a complex structure ja and a Hermitian holomorphic line bundle K a → M. As in [8], we introduce a half-form bundle (δa , ϕa ) and a holomorphic Hermitian line bundle L 1,a such that K a = δa ⊗ L 1,a . For another pair ( jb , K b ), introduce in the same way (δb , ϕb ) and L 1,b . Then rewriting the formulas of Chapter 3 with these data, we get more transparent results: • The formula for the commutators in Theorem 3.4 becomes i ( f
∗a g − g ∗a f ) = { f, g} − ω1,a , X f ∧ X g + O(2 ),
where 1i ω1,a is the curvature of L 1,a .
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• Denote by Ma,b the set of half-form bundle morphisms ( ja , δa , ϕa ) → ( jb , δb , ϕb ), then Hom(K a , K b ) = Ma,b ×Z2 Hom(L 1,a , L 1,b ), where we divided by Z2 to identify ( , 1 ) with (− , − 1 ). The composition of symbols in Theorem 3.3 is then the product of the composition of half-form bundle morphisms with the usual composition. • The symbol of a unitary operator S ∈ U F(a, b) is of the form [ , 1 ] with ∈ Ma,b and 1 a unitary isomorphism L 1,a → L 1,b . Furthermore the equivalence of the starproducts ∗a and ∗b induced by S is up to second order f a → f a + α1 , X fa + O(2 ), where α1 is such that ∇ Hom(L 1,a ,L 1,b ) 1 = 1i α1 ⊗ 1 . This point of view will also be useful to prove Theorem 3.3 in the following section. 6. Proof of Theorem 3.3 To prove the theorem, we consider the kernels of the Fourier integral operators as Lagrangian sections and interpret Tb S − STa as the result of the action of a Toeplitz operator on a Lagrangian section. The computation of the symbol is then a corollary of Theorem 3.4 in [8]. Let us regard M 2 as a symplectic manifold with symplectic form ω M 2 = πl∗ ω − πr∗ ω, where πl and πr are the projections onto the first and second factor respectively. Then L L¯ is a prequantization bundle of M 2 with curvature 1i ω M 2 and the diagonal map : M → M 2 is a Lagrangian embedding. Furthermore ( jb , − ja ) is a complex structure of M 2 compatible with ω M 2 and positive. Denote by H(b, −a) the associated Hilbert space H(b, −a) = holomorphic sections of (L k L¯ k ) ⊗ (K b K¯ a ) → M 2 . Then the Fourier integral operators of F(a, b) are defined in such a way that their kernel is a Lagrangian section of H(b, −a) associated to the diagonal. Let S ∈ F(a, b) with kernel S(.). Let Ta and Tb be Toeplitz operators of H(a) and H(b) with normalized symbols f a (., ) and f b (., ) respectively. Then it is easily checked that the kernel of Tb S − STa is T S(.) where T is a Toeplitz operator of H(b, −a) with normalized symbol g(x, y, ) = f b (x, ) − f a (y, ). Assume now that Ta and Tb have the same principal symbol. Then the principal symbol of T vanishes along the diagonal and consequently the principal symbol of T S(.) vanishes. By applying Theorem 3.4 of [8], we obtain the principal symbol of k −1 T S(.) which corresponds to the principal symbol of k −1 (Tb S − STa ). Let us use the half-form bundles as in Chapter 5.3. The symbol of S as a Fourier integral operator of F(a, b) is a class [ , 1 ], where is a half-form bundle morphism
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δa → δb and 1 a bundle morphism L 1,a → L 1,b . We have to show that the symbol of k −1 (Tb S − STa ) is [ , 1 ] with Hom(L 1,a ,L 1,b )
1 = ( f 1,b − f 1,a ) 1 + 1i ∇ X
1 ,
(14)
where X is the Hamiltonian vector field of f and f 1,a , f 1,b are the subprincipal symbols of Ta and Tb respectively. To the morphisms and 1 correspond two sections ˜ ∈ C ∞ (M, δb ⊗ δ a ),
˜ 1 ∈ C ∞ (M, L 1,b ⊗ L 1,a ).
˜ ⊗
˜ 1 . The restriction to the The principal symbol of the Lagrangian section S(.) is
diagonal of the Hamiltonian vector field of the principal symbol of T is ∗ X . Then it follows from Theorem 3.4 of [8] that the principal symbol of k −1 T.S(.) is L 1,b ⊗L 1,a ˜ 1 ˜ ˜ ˜ ˜ ⊗
˜ 1 + 1 (D
1 . ( f 1,b − f 1,a )
X ) ⊗ 1 + i ⊗ ∇ X i
˜ It remains to explain how the section D X of δb ⊗ δ a is defined and to prove that it vanishes. This will imply (14). Consider the isomorphism n,0 ∗ 2n,0 ∗ ∗ 2 ∗ ∗ ξ : n,0 jb T M ja T M → jb ,− ja T M , β α → πl β ∧ πr α
(δb δ¯a , ξ ◦ (ϕb ϕ¯a )) is a half-form bundle of M 2 for the complex structure ( jb , − ja ). Then δb ⊗ δ¯a = ∗ (δb δ¯a ) is a square root of 2n T ∗ M ⊗ C through the map ϕδ : δb2 ⊗ δ¯a2 → 2n M ⊗ C, u b ⊗ u a → ∗ (ξ(ϕb (u b ) ϕ¯a (u a ))) ˜ and D X is defined in such a way that 1 ˜ ⊗ D ˜ ⊗2 ˜ ϕδ (
X ) = 2 L X .ϕδ ( ).
˜ Then D X = 0 follows from the following lemma and Liouville theorem. ˜ ⊗2 ) = i n(n−2) ωn /n!. Lemma 6.1. ϕδ (
˜ ja , jb the section of n,0 M ⊗ n,0 M → M associated to ja , jb . Proof. Denote by
jb ja Since is a half-form bundle morphism, we have ˜ ⊗2 ) =
˜ ja , jb . (ϕa ⊗ ϕ¯b )(
˜ Introduce a unitary section α of n,0 ¯ Consequently ja M. We have ja , jb = ja , jb (α) ⊗ α. ˜ ⊗2 ) =∗ (πl∗ ja , jb (α) ∧ πr∗ α)) ϕδ (
¯ = ja , jb (α) ∧ α¯ =α ∧ α¯ by definition of ja , jb =i n(n−2) ωn /n! because α is unitary.
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7. Geometric Interpretation Consider as previously a symplectic manifold (M, ω) with a prequantization bundle (L , ∇). The space J of almost complex structures of M compatible with ω and positive may be regarded as the space of sections of a fiber bundle over M, which turns it into an infinite dimensional manifold. Let us fix an isomorphism class D of half-form bundles and choose for any j ∈ J a half-form bundle of (M, j) which represents D and depends “smoothly” on j. One way to do that is first to choose ( j0 , δ0 , ϕ0 ) representing D and then to set δ j := δ0 ,
ϕ j := j0 , j ◦ ϕ0 ,
∀j ∈ J.
Let Pkm
→ J be the bundle of prequantum spaces, whose fiber at j is the space of smooth sections of L k ⊗ δ j . Let us consider now a submanifold Jint of J which contains only integrable complex structures. Assume that the family of Hilbert spaces k Qm k, j := {holomorphic sections of L ⊗ δ j },
j ∈ Jint
m defines a smooth subbundle Qm k → Jint of Pk → Jint , when k is sufficiently large. This assumption is satisfied as soon as the dimension of Qm k, j is constant when j runs over Jint . This follows from Fredholm theory because Qm is the kernel of the holomork, j phic Laplacian, an elliptic second order differential operator whose coefficients depend smoothly on the complex structure. Furthermore as noticed by Foth and Uribe [9], for any complex structure j0 , there exists an integer N such that the dimension of Qm k, j is constant when j describes a C 2 neighborhood of j0 and k is larger than N . The C 2 topology is involved here to control the curvature term in the Bochner-Kodaira identity and deduce a uniform vanishing theorem. Then the dimension of Qm k, j is given by the Riemann-Roch theorem. Before we continue, let us note that Pkm and Qm k depend only on the isomorphism class D, providing we regard them as the orbifold bundles Pkm /Z2 and Qm k /Z2 , where Z2 acts trivially on the base J and by ±Id on the fibers. Indeed, let us consider another smooth family (δ˜ j , ϕ˜ j ) j∈J , obtained as above by choosing a half-form bundle ( j˜0 , δ˜0 , ϕ˜0 ) representing D and denote by P˜ km the associated bundle of prequantum spaces. Then there exist exactly two continuous families
j : (δ j , ϕ j ) → (δ˜ j , ϕ˜ j ); j ∈ J
of half-form bundle morphisms. These families induce isomorphisms Pkm → P˜ km and ˜m Qm k → Qk , which are unique up to the Z2 -action. All the constructions which follow only depend on D in this sense. First we define a connection on Pkm . Given a tangent vector μ of J at j0 , let us introduce a curve jt of J tangent to μ at t = 0 and consider the continuous family ( t ) of half-form bundle morphism (δ j0 , ϕ j0 ) → (δ jt , ϕ jt ) such that 0 is the identity of δ j0 . Then we define the covariant derivative of a section of Pkm with respect to μ to be d Pm ∇μ k ( j0 ) :=
−1 .( jt ), dt t=0 t m . The connection on Qm is then where the derivative is in the t-independent space Pk, j0 k defined as ∇ Qk := k ◦ ∇ Pk , m
m
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where k is the section of End(Pkm ) which at j is the orthogonal projector onto Qm k, j . Theorem 7.1. For any k, the connection ∇ Qk is compatible with the Hermitian structure. Furthermore, m
• For any j ∈ Jint and η, μ ∈ T j Jint , the sequence of curvature m R Qk (η, μ) : Qm k, j → Qk, j , k = 1, 2, . . . m
is a Toeplitz operator whose principal symbol vanishes. • For any curve γ of Jint with endpoints ja and jb , the sequence of parallel transport γ in Qm k is a unitary Fourier integral operator m Qm k, ja → Qk, jb , k = 1, 2, . . .
of F(( ja , δ ja ), ( jb , δ jb )). Its principal symbol is the half-form bundle morphism δ ja → δ jb obtained by extending continuously the identity of δ ja in half-form bundle morphisms δ ja → δγ (t) . The proof is postponed to Sect. 10. Let us compute the curvature of Pkm . Given an almost complex-structure j0 ∈ J , we can represent any j ∈ J as a section 0,1 ∗ ∗ μ ∈ C ∞ (M, Hom( 1,0 j0 T M, j0 T M)) ∗ such that the graph of μ(x) is 1,0 j Tx M for any x ∈ M. In this way, we identify the tangent space to J at j0 with 0,1 ∗ ∗ t t T j0 J μ ∈ C ∞ (M, Hom( 1,0 j0 T M, j0 T M)); ω(μ (.), .) + ω(., μ (.)) = 0
and J becomes a neighborhood of the zero section of T j0 J . Theorem 7.2. The connection ∇ Pk is compatible with the Hermitian structure. Its curvature at η, μ ∈ T j J is given by m
R Pk (η, μ) = m
1 tr(η.μ¯ − μ.η) ¯ , 2
m ∈ Pk, j.
It is interesting to compare the previous theorems with the results of Foth and Uribe [9]. The curvature of Qm k is the sum of two terms which cancel each other at first order. The first term is the curvature of Pkm and the second one is a commutator (cf. Lemma 8.6). In the case considered by Foth and Uribe, the prequantum spaces are defined without half-form and consequently don’t depend on the complex structure. Then the bundle Jint ×C ∞ (M, L k ) is endowed with the trivial connection, and composing with the Szegö projector, we obtain a connection on the quantum space bundle. Its curvature equals a commutator (cf. Lemma 2.1 of [9]), which is essentially the same as in our situation, and isn’t canceled by the curvature of the prequantum bundle, flat in this case. Proof of Theorem 7.2. Let j0 be a fixed almost-complex structure and let us identify J with an open convex set O of T j0 J as previously. Let us compute the connection in the trivialization m Pkm O × Pk, j0
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induced by the continuous family of half-form bundle isomorphisms (δ j0 , ϕ j0 ) → (δ j , ϕ j ) extending the identity of δ j0 . Let μ(t) be a curve of O covered by a section (t). By Lemma 2.3, the continuous curve of half-form bundle morphisms t : δμ(0) → δμ(t) is in the previous trivialization the multiplication by the continuous square root of t → det
Id μ(0) ¯ Id μ(0) ¯ . det −1 μ(0) Id μ(t) Id
equal to 1 at t = 0. Then we have d −1 .(t) dt t=0 t −1 1 ˙ ˙ μ(0) ¯ Id − μ(0)μ(0) ¯ (0) + (0). = − tr μ(0) 2
Pm
k ∇μ(0) ˙ (0) =
Thus we have ∇ Pk = d + α with m
−1 1 , α, μ. ˙ = − tr μ˙ μ¯ Id − μμ¯ 2
m ∀ μ˙ ∈ Tμ O, ∈ Pk, j0 .
Finally it is easy to compute the curvature at the origin of O, where α vanishes which leads to the formula of the theorem. To check the compatibility of the connection with the scalar product, observe that our trivialization doesn’t preserve the scalar product. Actually since ∗j0 , j ◦ j0 , j = ζ ( j0 , j, j0 ), we have (, )Pk,m j =
1 M
1 2
ζ ( j0 , j, j0 )(x)
(x), (x) L k ⊗δ μ M (x). j0
Then using again that the connection form α vanishes at the origin and that ζ ( j0 , j, j0 ) = m , 1 + O(| j − j0 |2 ) we deduce that for every section , of O × Pk, j0 d(, ) = (∇ Pk , ) + (, ∇ Pk ) m
at the origin.
m
Remark 7.3. It is immediate to deduce the first part of Theorem 7.1. Since is selfadjoint, if and are section of Qm k → Jint , then 2(∇ Qk , ) + (, ∇ Qk ) = (∇ Pk , ) + (, ∇ Pk ) m
m
m
m
= (∇ Pk , ) + (, ∇ Pk ) = d(, ), m
which proves that ∇ Qk is Hermitian. m
m
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8. Action of the Prequantization Bundle Automorphisms Adapting the constructions of Sect. 5, we define an action of the identity component G0 of the group of prequantization bundle automorphisms of (L , ∇) on Pkm and Qm k . For any equivariant vector bundle equipped with an invariant connection, one defines a moment (cf. Definition 7.5 in [3]). This notion makes sense in our infinite dimensional setting. In the first part of this section, we prove the moment of a function f in the Poisson Lie algebra C ∞ (M) is a Toeplitz operator. From this we compute the solution of the Schrödinger equation in terms of parallel transport. This last result was obtained in [9] in the case without metaplectic correction. This enables us to deduce that the quantum propagator is a Fourier integral operator from the fact that parallel transport is such an operator. Next we compute the commutator of Toeplitz operators in terms of the curvature of the quantum space bundle. As a corollary we obtain that in general the curvature is not O(k −2 ). Finally we explain how the same ideas apply in the case without metaplectic correction. 8.1. The infinitesimal action of Go on the bundles P m and Qm . Let us start with the definition of the action of G0 on Pkm . An automorphism ∈ Go covering the symplectomorphism φ acts on the base J by sending j into φ ∗ j. Let us lift this action to Pkm . Given j ∈ J , there are exactly two bundle maps φ ∗ : δ j → δφ ∗ j covering φ such that the diagram φ∗
∗ ∗ n,0 −−−→ n,0 j T M − φ∗ j T M ⏐ ϕφ ∗ j ϕj⏐ ⏐ ⏐
δ 2j
(φ ∗ )2
−−−−→
δφ2 ∗ j
commutes. Then the pull-back by k ⊗ (φ ∗ )−1 : L k ⊗ δ j → L k ⊗ δφ ∗ j m → Pm ∗ is a linear map Pk, j k,∗ j . Choosing the bundle map δ j → δ j in such a way m that it depends continuously on j, we obtain the action of on Pk . Since this action is only defined up to multiplication by −1, we obtain merely a Go -action on the orbibundle (Pkm , Z2 ). Given a function f ∈ C ∞ (M), let us define the operator
Opk ( f ) := f +
1 Lk m m (∇ ⊗ Id + Id ⊗ D X ) : Pk, j → Pk, j , ik X
j ∈ J,
where X is the Hamiltonian vector field of f and D X is the first order differential operator of C ∞ (M, δ j ) such that p j L X ϕ j (β 2 ) = 2ϕ j β ⊗ (D X β) ,
∀β ∈ C ∞ (M, δ j )
(15)
n−1,1 ∗ ∗ with p j the projection of n T ∗ M ⊗ C onto n,0 T M ⊕...⊕ j T M with kernel j ∗ 0,n j T M.
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Recall that the Lie algebra of Go may be viewed as C ∞ (M, R). Given f ∈ C ∞ (M, R), t := exp(t f ) is the automorphism of L which covers the Hamiltonian flow φt of f at time t and is given by t (ξ ) = eit f (x) Tt .ξ,
if ξ ∈ L x ,
(16)
where Tt is the parallel transport from L x to L φt (x) along the Hamiltonian flow. Theorem 8.1. Let f ∈ C ∞ (M, R) and denote by Ut : Pkm → Pkm the action of exp(t f ) on P m . Let j0 ∈ J and j : R → J be the curve jt = φt∗ j0 . For m the section s of j ∗ P m defined by s = U .s satisfies any s0 ∈ Pk, t t 0 j0 k 1 j ∗ Pkm s = Opk ( f ) s. ∇ ik ∂t The action of the symplectomorphism group on J preserves the subspace of integrable almost complex structures. Assume that j0 ∈ Jint and that the submanifold Jint is invariant under the action of exp(t f ). Then the operator Ut restricts to an operator m ∗ m Qm k, j0 → Qk, jt and the section s of j Qk defined as above satisfies 1 j ∗ Qmk ∇ s = Qm k ( f ) s, ik ∂t where Qm k ( f ) is defined by m m Qm k ( f ) := k, j Opk ( f ) : Qk, j → Qk, j .
We proved in [8] (cf. Theorem 1.5) the sequence (Qm k ( f ))k is a Toeplitz operator whose normalized symbol is f + O(2 ). Proof. First we may assume that s0 = α⊗β with α ∈ C ∞ (M, L k ) and β ∈ C ∞ (M, δ j0 ). Furthermore since Ut ◦ Us = Ut+s , it is sufficient to prove the result at t = 0. Let us write αt = ∗t α,
βt = φt∗ β ∈ C ∞ (M, δ jt ).
Then Ut .(α ⊗ β) = αt ⊗ βt and consequently j ∗ Pkm
∇∂t
s = α˙ t ⊗ βt + αt ⊗ β˙t ,
where the derivative α˙ t is in the t-independent vector space C ∞ (M, L k ) and d β˙0 := t−1 ∗t β dt t=0
(17)
with t : δ j0 → δ jt the continuous family of half-form bundle morphisms such that 0 is the identity of δ j0 . It is a classical result that 1 k α˙ t = ik f + ∇ XL f αt . ik
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So it remains to prove that β˙0 = D X β. Denote by D˜ the map sending β ∈ δ j0 into β˙0 defined in (17). D˜ is a first order differential operator. We have to prove that ˜ p j0 L X ϕ j0 (β 2 ) = 2ϕ j0 (β ⊗ Dβ). We have d φ ∗ ϕ j (β 2 ) dt t=0 t 0 d = p j0 ϕ jt ((φt∗ β)2 ) dt t=0 d = p j0 ϕ jt ((φt∗ β)2 ). dt t=0
p j0 L X ϕ j0 (β 2 ) = p j0
Now it follows from the definition of j0 , jt that p j0 γ = −1 j0 , jt γ ,
∗ ∀ γ ∈ n,0 jt T M.
Since t is a half-form bundle morphism, we obtain d ϕ j ((φ ∗ β)2 ) 2 p j0 L X ϕ j0 (β 2 ) = −1 dt t=0 j0 , jt t t d = ϕ j0 (( t−1 φt∗ β)2 ) dt t=0 d = ϕ j0 (( t−1 φt∗ β)2 ) dt t=0 ˜ = 2ϕ j0 (β ⊗ Dβ) which was to be proved.
8.2. Schrödinger equation. As a corollary of Theorem 8.1, we obtain the relation between the parallel transport in Pkm , the action of G0 and the Schrödinger equation with Hamiltonian Opk ( f ). Pm
m m Corollary 8.2. Let Tt k : Pk, j0 → Pk, jt be the parallel transport along the curve t → jt . Then the family of operators
Pt := (Tt
Pkm −1
)
m m ◦ Ut : Pk, j0 → Pk, j0
satisfies 1 d Pt s0 = Opk ( f ) Pt s0 ik dt m . for any s0 ∈ Pk, j0
Semi-Classical Properties of Geometric Quantization
Proof. Since Tt
Pkm
469
is parallel transport, 2
1 Pm 1 d j ∗P m Pt s0 = (Tt k )−1 ∇∂t k Ut s0 ik dt ik = (Tt
Pkm −1
)
Opk ( f )Ut s0
by Theorem 8.1. Furthermore for any function g ∈ C ∞ (M), we have Ut Opk (g) = Opk (∗t g)Ut . So Ut and Opk ( f ) commutes, because f is preserved by its Hamiltonian flow. Consequently 1 d Pt s0 = Pt Opk ( f )s0 . ik dt To conclude, we prove that Pt and Opk ( f ) commute. We have 2
(18)
d Pt Opk ( f )(Pt )−1 = P˙t Opk ( f )(Pt )−1 − Pt Opk ( f )(Pt )−1 P˙t (Pt )−1 dt = ik Pt Op2k ( f )(Pt )−1 − Pt Opk ( f )(Pt )−1 Pt Opk ( f )(Pt )−1 =0
because of (18).
Let us assume again that j0 ∈ Jint and that Jint is preserved by the action of exp(t f ). Then arguing as in the previous proof, we can deduce the similar result for the bundle Qm k . Qm
m m Corollary 8.3. Let Tt k : Qm k, j0 → Qk, jt be the parallel transport in Qk along the curve t → jt . Then we have
1 d Qmk Qm k Pt s0 = Qm s0 , k ( f )Pt ik dt Qm k
where Pt
is the operator (Tt
Qm k −1 )
∀s0 ∈ Qm k, j0 ,
m ◦ Ut : Qm k, j0 → Qk, j0 .
Recall that Qm ( f ) is a Toeplitz operator whose normalized symbol is f + O(2 ). Then Theorem 5.3 follows from the fact that the parallel transport in Qm is a unitary Fourier integral operator (cf. Theorem 7.1). 8.3. Commutators and curvature. In the next theorem, we compute the commutator of m m Opk ( f ) and Opk (g) (resp. Qm k ( f ) and Qk (g)) in terms of the curvature of Pk → J (resp. Qm → J ). k Theorem 8.4. Let f and g be two functions of C ∞ (M, R). Then ik[Opk ( f ), Opk (g)] = Opk ({ f, g}) + (ik)−1 R Pk (η, μ), m
where η and μ are the vector fields of J corresponding to the infinitesimal action of f and g on J . Furthermore, m m −1 Qk (η, μ) ik[Qm k ( f ), Qk (g)] = Qk ({ f, g}) + (ik) R m
when η and μ are tangent to Jint .
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L. Charles
Since Qm ( f ) is a Toeplitz operator with normalized symbol f + O(2 ), it follows from Theorem 3.4 that m m −2 ik[Qm k ( f ), Qk (g)] = Qk ({ f, g}) + O(k ).
(19)
This is consistent with the second equation of the previous theorem and the fact that m m R Qk (η, μ) is O(k −1 ). Moreover, one can prove in this way that R Qk can’t be O(k −2 ) except for particular submanifolds Jint . Indeed given a complex structure j, there is a star-product ∗ j such that for any functions f and g, m m −∞ ), Qm k ( f ) j ◦ Qk (g) j ≡ Qk (h(., k)) j + O(k −1 where h(., k) lhas an asymptotic expansion h 0 + k h 1 + .. whose coefficients satisfy f ∗ j g = h l . One can prove that ∗ j is a Vey star-product, i.e. the bidifferential operators defining ∗ j have the same principal symbol as the bidifferential operators defining the Moyal-Weyl star-product. Hence i−1 f ∗ j g − g ∗ j f = { f, g} + 2 A( f, g) + O(3 ),
where A is a non-vanishing bidifferential operator. So if η and μ are the infinitesimal actions of f and g, −2 R Qk (η, μ) j = ik −1 Qm k (A( f, g)) j + O(k ), m
−1 and Qm k (A( f, g)) j is not O(k ) as soon as A( f, g) doesn’t vanish. The first equation of the theorem can be deduced from the expression of the curvature in Theorem 7.2 as follows. First recall that
k k Lk ik f + ik1 ∇ XL , g + ik1 ∇YL = { f, g} + ik1 ∇[X,Y ],
where X and Y are the Hamiltonian vector fields of f and g. Then we compute the bracket of the operators D X , DY entering in the definition of Opk ( f ) and Opk (g) (cf. (15)) in terms of the infinitesimal actions η and μ on J of X and Y respectively.
¯ − ημ). ¯ Lemma 8.5. We have D X , DY = D[X,Y ] + 21 tr(μη 0,1 Proof. Given any one-form β and complex structure j, denote by p 1,0 j β and p j β the component of β of type (1, 0) and (0, 1) for j. The vector field η is given at j by 0,1 ∗ ∗ η j ∈ C ∞ (M, Hom( 1,0 j T M, j T M)),
η j (α) = p 0,1 j L X α.
Consequently, if α ∈ 1,0 j M, 1,0 1,0 0,1 1,0 0,1 [ p 1,0 j L X , p j LY ]α = p j L X (LY − p j LY )α − p j LY (L X − p j L X )α
= ( p 1,0 ¯ j η j )α. j L[X,Y ] − η¯ j μ j + μ n−1,1 ∗ ∗ So if p j is the projection from n T ∗ M ⊗ C onto n,0 T M⊕ j T M with kernel j ∗ . . . ⊕ 0,n j T M, we have for any (n, 0)-form α,
[ p j L X , p j LY ]α = ( p j L[X,Y ] + tr(μ¯ j η j − η¯ j μ j ))α, which implies the result.
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471
Consequently, ik[Opk ( f ), Opk (g)] = Opk ({ f, g}) +
1 ¯ 2ik tr(μη
− ημ), ¯
and we deduce the first equation of Theorem 8.4 from Theorem 7.2. To prove the second equation, we start with the following relation between the curvatures of Pkm and Qm k . Lemma 8.6. For every vector field η and μ of Jint , we have
End(Pkm ) m m End(Pkm ) k , ∇μ k + k R Pk (η, μ), R Qk (η, μ) = k ∇η where ∇ End(Pk ) k is the commutator [∇ Pk , k ]. m
m
Proof. We have Qm
Qm
Pm
Pm
∇η k ∇μ k =k ∇η k k ∇μ k k End(Pkm ) Pkm Pm Pm =k ∇η k ∇ μ k + k ∇ η k ∇ μ k k End(Pkm ) End(Pkm ) End(Pkm ) Pm =k ∇η k ∇ μ k + k ∇η k k ∇ μ k Pm
Pm
+ k ∇ η k ∇ μ k k . m Since 2k = k , we have k ∇ End(Pk ) k k = 0. So the second term of the sum vanishes. Using this it is easy to compute the curvature of Qm k , Qm
Qm
Qm
k R Qk (η, μ) = [∇η k , ∇μ k ] − ∇[η,μ] m
End(Pk ) End(Pkm ) Pm Pm Pkm = k ∇η k , ∇μ k + k [∇η k , ∇μ k ] − k ∇[η,μ]
End(Pkm ) m End(Pkm ) = k ∇η k , ∇μ k + k R Pk (η, μ), m
which proves the result.
On the other hand we can compute the commutator of k with Opk ( f ) in terms of the covariant derivative of k . Lemma 8.7. Let f ∈ C ∞ (M, R) and η be the infinitesimal action of f on J , then m
1 End(Pk ) k = Opk ( f ), k . ik ∇η Proof. This follows from Theorem 8.1 by derivating the relation k, jt Ut = Ut k, j0 . Applying twice this last lemma, we obtain m 1 End(Pk ) k ) ik ∇μ m End(Pkm ) 1 End(Pk ) k )(Opk (g) + ik1 ∇μ k ). ik ∇η
k Opk ( f )k Opk (g)k =k Opk ( f )k (Opk (g) + =k (Opk ( f ) + Hence
k Opk ( f )k , k Opk (g)k
= k Opk ( f ) +
End(Pkm )
1 ik ∇η
k , Opk (g) +
End(Pkm )
1 ik ∇μ
k .
(20)
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L. Charles
Similarly, we have k Opk ( f )k Opk (g)k =(Opk ( f ) − =(Opk ( f ) − So
Opk ( f ) −
m 1 End(Pk ) k )k Opk (g)k ik ∇η m End(Pkm ) 1 End(Pk ) k )(Opk (g) − ik1 ∇μ k )k . ik ∇η
k Opk ( f )k , k Opk (g)k = m 1 End(Pk ) k , ik ∇η
Opk (g) −
m 1 End(Pk ) k k . ik ∇μ
(21)
Now Eqs. (20) and (21) imply
k Opk ( f )k , k Opk (g)k =
End(Pkm ) End(Pkm ) k Opk ( f ), Opk (g) k + k ik1 ∇η k , ik1 ∇μ k k , and we deduce the second equation of Theorem 8.4 from the first one and Lemma 8.6. 8.4. An analog result in finite dimension. It is interesting to note that the expression for the curvature in Theorem 7.2 is a direct consequence at least formally of Theorem 8.1 on the infinitesimal action. Consider a finite dimensional vector bundle E → X endowed with a connection ∇. Assume a Lie group G acts on E preserving the connection. Given η in the Lie algebra g of G, denote by η X the vector field corresponding to the infinitesimal action on the base X and by Lη the infinitesimal action on C ∞ (X, E). Then Lη − ∇η X acts by exterior multiplication by a section M(η) ∈ C ∞ (M, End(E)) called the moment of η. Proposition 8.8. For any vectors η, μ ∈ g, we have [M(η), M(μ)] = M([η, μ]) + R E (η X , μ X ), where R E is curvature of ∇. Proof. Since the connection in invariant, we have [Lη , ∇] = 0. Replacing Lη with ∇η X + M(η), we obtain that ∇ End(E) M(η) = R E (η X , .).
(22)
Since η → ∇η X + M(η) is a Lie algebra representation, we have for any η, μ ∈ g, [∇η X + M(η), ∇μ X + M(μ)] =∇[η,μ] X + M([η, μ]) =∇[η X ,μ X ] + M([η, μ]). Assuming that ∇ is G-invariant, we can compute by the lemma the commutators [∇η X , M(μ)] = R E (μ X , η X ),
[M(η), ∇μ X ] = −R E (η X , μ X ) = R E (μ X , η X ).
Then using that [∇η X , ∇μ X ] = ∇[η X ,μ X ] + R E (η X , μ X ) we obtain the proposition.
Semi-Classical Properties of Geometric Quantization
473
If we apply this equation in our infinite dimensional setting with E the bundle of quantum spaces or prequantum spaces and G the group of prequantization bundle automorphism, we obtain Theorem 8.4. It is also interesting to consider the situation of the introduction without half-form bundle (cf. Sect. 1.1). Let Pk := C ∞ (M, L k ) × Jint be the prequantum space bundle and Qk be the subbundle of quantum spaces. As explained in the introduction the Group G of prequantization bundle automorphisms acts on these bundles. Moreover these bundles are endowed with invariant connection. Then one proves that the moment of f ∈ C ∞ (M) on Qk is the operator (ik)Qk ( f ), where Qk ( f ) is the Toeplitz operator 1 k Qk ( f ) := k f + ∇ XL ik with X the Hamiltonian vector field of f . Consequently, one has ik[Qk ( f ), Qk (g)] = Qk ({ f, g}) + (ik)−1 R Qk (η, μ), where η and μ are the infinitesimal actions of f and g respectively on Jint . Then we recover the main point of the argument of Ginzburg and Montgomery: if the curvature vanishes, the map f → (ik)Qk ( f ) is a Lie algebra representation. Furthermore, the result of Foth and Uribe gives the first correction terms in the computation of the commutator of two Toeplitz operators. 9. Preliminaries for the Proof of Theorem 7.1 m to Given two complex structures ja , jb , we introduce a class of operators from Pk, ja m Pk, jb extending the class of Fourier integral operator we considered previously. First by m , the Schwartz kernels of these operators can be regarded using the scalar product of Pk, ja as C ∞ sections of the bundle k L ⊗ δ jb L¯ k ⊗ δ¯ ja → M 2 .
Let N be a non-negative integer. We say that (Tk )k∈N∗ is an operator of A N ( ja , jb ) if its Schwartz kernel is of the form k n E k (x, y) f (x, y, k) + O(k −∞ ), Tk (x, y) = 2π where • E is a section of L L¯ → M 2 such that E(x, y) < 1 if x = y, E(x, x) = u ⊗ u, ¯ ∀u ∈ L x such that u = 1, and ∂¯ E ≡ 0 modulo a section vanishing to any order along .
474
•
L. Charles
f (., k) is a sequence of sections of δ jb δ¯ ja → V which has an asymptotic expansion in the C ∞ topology, f (., k) = k N f −N + k N −1 f −N +1 + . . . , where for all 0 l N , f −l vanishes to order 2l along the diagonal of M 2 . 1
As a result, the Schwartz kernel of Tk is uniformly O(k n+N ). It is O(k n+N − 2 ) if and only if fl vanishes at order 2l + 1 along the diagonal whenever 2l + 1 0. This follows from the fact that ln E < 0 outside the diagonal and its Hessian along the diagonal is non-degenerate in the transverse directions (cf. Lemma 1 in [6]). We define the symbol of (Tk ) as −N [ f −N ]2N + −N +1 [ f −N +1 ]2(N −1) + . . . . + −1 [ f −1 ]2 + [ f 0 ], where [ f −N +l ]2(N −l) is the equivalence class of f N −l modulo the functions vanishing at order 2(N − l) + 1 along the diagonal. So with the usual identification, the space of symbol is the space of sections of j ,j S Na b := δ jb ⊗ δ¯ ja ⊗ −N Sym2N C ⊕ −N +1 Sym2(N −1) C ⊕ . . . ⊕ Sym0 C , where C is the conormal bundle of the diagonal of M 2 . Theorem 9.1. The composition of S ∈ A N ( jb , jc ) with S ∈ A N ( ja , jb ) is an operator of A N +N ( ja , jc ). Furthermore there exists a bilinear bundle map j , j , jc
L Na,Nb
j , jc
: S Nb
j , jb
× S Na
j ,j
→ S Na+Nc
j ,j ,j
such that the principal symbol of SS is L Na,Nb c (σ, σ ) if σ and σ are the symbols of S and S respectively. Proof. The proof is essentially the same as the one of Theorem 3.2 about the composition of Fourier integral operators. The computation of the symbol follows from the version of the stationary phase lemma stated in the appendix of [8]. By extending the Fourier integral operators of F(( ja , δ ja ), ( jb , δ jb )) to operators m → P m in such a way that they satisfy Pk, ja k, jb k, ja Tk k, jb = Tk , k = 1, 2, . . . , F(( ja , δ ja ), ( jb , δ jb )) becomes a subspace of A0 ( ja , jb ). Both definitions of principal j ,j symbols are the same if we identify the sections of S0a b = δ jb ⊗ δ¯ ja with the fiber bundle morphisms δ ja → δ jb by using the scalar product of δ ja . Theorem 9.2. If T is an operator of A N ( ja , jb ) with symbol σ , then k, ja T k, jb is a Fourier integral operator of F(( ja , δ ja ), ( jb , δ jb )). Furthermore the symbol of j ,j k, ja T k, jb is L Na b (σ ), where j , jb
L Na is a fiber-bundle morphism.
j , jb
: S Na
j , jb
→ S0a
Semi-Classical Properties of Geometric Quantization
475
Proof. Again the proof relies on the methods of Sect. 4. To show that k, ja Sk, jb is a Fourier integral operator, we argue as in the following of Lemma 4.2. The other part is an application of the stationary phase lemma in the appendix of [8]. Finally let us describe explicitly the symbol product for the composition of operators of A N ( ja , ja ). To do this it is convenient to introduce complex coordinates (U, z i ) for ja and write the symbol in the following way: σ (, Z¯ , Z , x) =
N
−l σl ( Z¯ , Z )(x),
x ∈U
l=0
with σl ( Z¯ , Z )(x) =
|α|+|β|=2l
1 α(1) β(1) β(n) α(n) ∇(∂ 1 ,0) . . . ∇(∂ n ,0) ∇(0,∂ 1 ) . . . ∇(0,∂ n ) fl (x, x) Z¯ α Z β z¯ z z¯ z α!β!
and ∇ a covariant derivation of δ ja δ¯ ja . Theorem 9.3. With the previous notations, the map of Theorem 9.1 is ¯ L a,a,a N ,N (σ, σ )(, Z , Z , x) =
N +N l=0
l l σ (, Z¯ −Y¯ , Y, x).σ (, Y¯ , Z − Y, x) , Y¯ =Y =0 l!
where is the operator :=
G i, j (x)∂Y i ∂Y¯ j
i, j
with (G i, j ) the inverse matrix of (G j,i ) whose coefficients are such that ω = i G i, j dz i ∧ d z¯ j . There isn’t any difficulty to extend these results to the case where the complex structures depend smoothly on a parameter. We end these preliminaries with the variations of the section E as a function of the complex structure. Let x ∈ M and be a germ at x of a Lagrangian submanifold of M. Let us fix a unitary section s of L → . Let jt be a curve in Jint . Then consider a smooth family E t of sections of L → M such that E t = s along and ∂¯ jt E t ≡ 0 modulo a section vanishing to any order along . Let us write d Et = ft Et dt on a neighborhood of x. Proposition 9.4. The function f 0 and its first derivatives vanish over . Furthermore, if Z and Z are holomorphic vector fields for the complex structure j0 , then 1 Z¯ . Z¯ . f 0 = ω( Z¯ , μ( Z¯ )) i 0,1 ∗ ∗ along , with μ ∈ C ∞ (M, Hom( 1,0 j0 T M, j0 T M)) the tangent vector to jt at j0 .
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L. Charles
Since is Lagrangian, T j0,1 M and T ⊗ C are transverse. So the result gives the t Hessian of f along . Proof. Since E t (y) is constant for all y ∈ , f t vanishes along . Let us write ∇ E t = 1i E t ⊗ αt . We can prove that αt vanishes along and Z¯ .α0 , W = ω( Z¯ , W )
(23)
d if Z is a holomorphic vector field for j0 (cf. Lemma 4.2 in [8]). Since dt and ∇ commute, 1 we have d f t = i α˙ t . So d f t vanishes along because the same holds for αt . 0,1 ∗ ∗ Now let us associate to jt ∈ J the section μt of Hom( 1,0 j0 T M, j0 T M) as we
did in (4). Then T j0,1 M is the graph of t
−μtt : T j0,1 M → T j1,0 M. 0 0 So if Z is a section of T j1,0 M, Z¯ − μtt ( Z¯ ) is a section of T j0,1 M and t 0 αt , Z¯ − μtt ( Z¯ ) vanishes to any order along . Thus the same holds for the derivative α˙ t , Z¯ − μtt ( Z¯ ) − αt , μ˙ tt ( Z¯ ). In particular, we have along , Z¯ .α˙ 0 , Z¯ = Z¯ .α0 , μ˙ t0 ( Z¯ ). Here we used that μ0 = 0. Finally the results follow from d f t = 1i α˙ t and (23).
10. Proof of Theorem 7.1 Let us start with the computation of the curvature. By Lemma 8.6,
End(Pkm ) m m End(Pkm ) R Qk (η, μ) = k ∇η k , ∇μ k + k R Pk (η, μ). By Theorem 7.2, k R Pk (η, μ) is at j a Toeplitz operator of Qmj with principal symbol m
− μ.η)( ¯ j). Then it follows from the following proposition that R Qk (η, μ)( j) is a Toeplitz operator with vanishing principal symbol. m
1 ¯ 2 tr(η.μ
Proposition 10.1. For any tangent vector η, μ ∈ T j Jint , the operator
End(Pkm ) End(Pkm ) k ∇η k , ∇μ k ¯ j). is a Toeplitz operator of Qmj with principal symbol − 21 tr(η.μ¯ − μ.η)(
Semi-Classical Properties of Geometric Quantization
477
End(P m )
k Proof. First we prove that ∇η k is an operator of A2 ( j, j) and compute its symbol. Let jt be a curve of Jint whose tangent vector at 0 is η. Let t : δ j0 → δ jt be the continuous family of half-form bundle morphisms such that 0 is the identity of δ j0 . We have at j0 , d End(Pkm ) m k . = t−1 k, jt t , ∈ Pk, ∇η j0 . dt t=0 Recall that k, jt is an operator of A0 ( jt , jt ) with symbol 1. Thus its kernel is of the form k n E tk (x, y) f t (x, y, k) + O(k −∞ ), 2π where f t (., k) is a sequence of sections of δ jt δ¯ jt equal to 1 + O(k −1 ) on the diagonal.
We obtain the kernel of t−1 k, jt t by replacing f t with
t−1 (x) f t (x, y, k) t (y)
which again is equal to 1 + O(k −1 ) on the diagonal. Derivating with respect to t, we End(Pkm )
deduce that ∇η
k is an operator of A2 ( j, j) with symbol
−1 [g], d where g is the function of M 2 such that dt Et = g E0 . t=0
[g]. Let (z i ) be a complex coordinate system for j0 such that ω = Let jus compute j i dz ∧ d z¯ at x. Denote by U (t) the symmetric matrix such that the family i = 1, . . . , n dz i + j Ui j (t)d z¯ j , ∗ ˙ is a base of 1,0 jt Tx M. So the derivative U of U (t) at t = 0 is the matrix of η( j0 ). By End(Pkm )
Proposition 9.4, the symbol of ∇η
k at x is 1 ˙ ¯ i ¯ j ˙¯ i j (Ui j Z Z + Ui j Z Z ), −1 [g]( Z¯ , Z , x) = − 2 where we used the notations of Theorem 9.3. Then it follows from Theorems 9.1 and 9.2 that
End(Pkm ) End(Pkm ) k ∇η k , ∇μ k is a Toeplitz operator and we compute its symbol by applying Theorem 9.3. First the
End(Pkm ) End(Pkm ) k , ∇μ k is at x symbol of ∇η −2 ˙ ˙¯ Ui j Vkl − V˙i j U˙¯ kl Z¯ i Z¯ j Z k Z l , U˙ ik V˙¯ jk − V˙ik U˙¯ jk Z¯ i Z j − 4
End(Pkm ) End(Pkm ) where V˙ is associated to μ as U˙ to η. Then the symbol of k ∇η k , ∇μ k is at x, −1
1 − tr (U˙ V˙¯ − V˙ U˙¯ ). 2 Since U˙ and V˙ are the matrices of η( j0 ) and μ( j0 ), we obtain the result.
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Let us prove now the last part of Theorem 7.1. Consider a curve j : [0, 1] → Jint . Denote by t : δ j0 → δ jt the continuous family of half-form bundle morphisms such that 0 is the identity. Proposition 10.2. Consider a smooth family of Fourier integral operator Pt ∈ F(( j0 , δ j0 ), ( jt , δ jt )); t ∈ [0, 1] with symbol (σt t )t . Then the operator j ∗ Qmk j ∗ Qm m k ◦ P t : Qm )(t) with (t) = Pt 0 ∇∂t k, j0 → Qk, jt , 0 → (∇∂t is a Fourier integral operator of F(( j0 , δ j0 ), ( jt , δ jt )) and its symbol is σ˙ t t . Proof. The Schwartz kernel of Pt is of the form k n E tk (x, y) f t (x, y, k) 2π j ∗ Pkm
with f t (x, x, k) = σt (x) t (x) + O(k −1 ). So the kernel of (∇∂t
◦ P)t is
k n d k n d −1 E tk (x, y) f t (x, y, k) + ( t,s E tk (x, y) (x). f s (x, y, k)) , s=t 2π dt 2π ds where ( t,s : δ jt → δ js )s is the continuous family of half-form bundle morphisms such that t,t is the identity of δ jt . By Proposition 9.4, d E t = gt E t , dt j ∗ Pkm
where gt and its first derivatives vanish along the diagonal. So (∇∂t A2 ( j0 , jt ). Since
◦P
t
belongs to
jt , js ◦ j0 , jt = ζ ( j0 , jt , js ) j0 , js we have −1
t,s ◦ 0,s = j ∗ Pkm
Thus the symbol of (∇∂t
−1
◦P
d [gt ] + ds
is
1 1 2
ζ ( j0 , jt , js )
t
1 1
ζ 2 ( j0 , jt , js )
s=t
0,t .
σt + σ˙ t 0,t .
(24)
Then it follows from Theorem 9.2 that j ∗ Qm j ∗ Qm (∇∂t k ◦ P t = k, jt ◦ (∇∂t k ◦ P t belongs to F(( j0 , δ j0 ), ( jt , δ jt )). Furthermore its symbol is of the form (at σt + σ˙ t ) t with at a C ∞ function. To end the proof, it suffices to show that if σt = 1 for every t, j ∗ Qm then the symbol of (∇∂t k ◦ P t vanishes.
Semi-Classical Properties of Geometric Quantization
479
1
Let us check it at t = 0. Since ζ 2 ( j0 , j0 , js ) = 1 for every s and 0,0 is the identity, formula (24) simplifies into −1 [g0 ]σ0 + σ˙ 0 which is equal to −1 [g0 ] because σt = 1. Introduce complex coordinates (z i ) for the complex structure j0 such that ω = idz i ∧ d z¯ i at x. Let μ be the tangent vector of jt at t = 0. Let U be the matrix at x of μ in the bases dz i , d z¯ i . Then by Proposition 9.4, 1 Ui j Z¯ i Z¯ j . [g0 ] = − 2 Qm
Finally an application of Theorem 9.3 proves that the symbol of ∇μt k ◦ Pt vanishes at t = 0. Let us compute now the symbol at any t. Since σt = 1, the operator Pt is invertible with an inverse in F(( jt , δ jt ), ( j0 , δ j0 )). So the operator m Pt,s = Ps ◦ Pt−1 : Qm k, jt → Qk, js
belongs to F(( jt , δ jt ), ( js , δ js )) and by Theorem 5.2, its symbol is t,s . It follows from the previous computation that j ∗ Qmk ◦ Pt,. s ∇∂s belongs to F(( jt , δ jt ), ( js , δ js )) and its symbol vanishes at t = s. Consequently, the symbol of j ∗ Qm j ∗ Qmk ◦ P t = ∇∂s k ◦ Pt,. s=t ◦ Pt ∇∂t vanishes.
Then it is easy to construct by successive approximations a smooth family of operators Pt : Qm ( j0 ) → Qm ( jt ) in F(( j0 , δ j0 ), ( jt , δ jt )) such that P0 is the identity of Qm k, j0 and that the total symbol of j ∗ Qm k
(∇∂t
◦ P)t vanishes. Consequently, j ∗ Qmk ∇∂t ◦ P t = O(k −∞ ),
where the big O is for the uniform norm of operators and is uniform with respect to t. m If Tt : Qm k, j0 → Qk, jt is the parallel transport along jt , then
t j ∗ Qm Tt = Pt − Tt T−s ∇∂s k ◦ P s ds. 0
By the first part of Theorem 7.1, Tt is unitary. Consequently Tt = Pt + O(k −∞ ). Then using that k, jt (Tt − Pt )k, j0 = Tt − Pt and k, j ∈ F(( j, δ j ), ( j, δ j )), we show that the Schwartz kernel of Tt − Pt is uniformly O(k −∞ ) with its successive covariant derivatives. This proves Theorem 7.1.
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References 1. Axelrod, S., Della Pietra, S., Witten, E.: Geometric quantization of Chern-Simons gauge theory. J. Differential Geom. 33(3), 787–902 (1991) 2. Bates, S., Weinstein, A.: Lectures on the geometry of quantization. Volume 8 of Berkeley Mathematics Lecture Notes. Providence, RI: Amer. Math. Soc. (1997) 3. Berline, N., Getzler, E., Vergne, M.: Heat kernels and Dirac operators. Volume 298 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Berlin: SpringerVerlag (1992) 4. Blattner, R.J.: Quantization and representation theory. In: Harmonic analysis on homogeneous spaces (Proc. Sympos. Pure Math., Vol. XXVI, Williams Coll., Williamstown, Mass., 1972), Providence, RI: Amer. Math. Soc., pp. 147–165 (1973) 5. Boutet de Monvel, L., Guillemin, V.: The spectral theory of Toeplitz operators. Volume 99 of Annals of Mathematics Studies. Princeton NJ: Princeton University Press (1981) 6. Charles, L.: Berezin-Toeplitz operators, a semi-classical approach. Commun. Math. Phys. 239(1–2), 1– 28 (2003) 7. Charles, L.: Quasimodes and Bohr-Sommerfeld conditions for the Toeplitz operators. Comm Partial Differ. Eqs. 28(9–10), 1527–1566 (2003) 8. Charles, L.: Symbolic calculus for Toeplitz operators with half-forms. J. Sympl. Geom. 4(2), 171–198 (2006) 9. Foth, T., Uribe, A.: Remarks on the naturality of quantization. http://arxiv.org/abs/math/0410016, 2004 10. Ginzburg, VL., Montgomery, R.: Geometric quantization and no-go theorems. In: Poisson geometry (Warsaw, 1998), Vol 51 of Banach Center Publ., Warsaw Polish Acad. Sci., 2000, pp. 69–77 11. Guillemin, V., Sternberg, S.: Geometric asymptotics. Mathematical Surveys, No. 14, Providence, RI: Amer. Math. Soc., 1977 12. Hitchin, N.J.: Flat connections and geometric quantization. Commun Math. Phys. 131(2), 347–380 (1990) 13. Hörmander, L.: The analysis of linear partial differential operators. I. Volume 256 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Berlin: SpringerVerlag, second edition, 1990 14. Kostant, B.: Quantization and unitary representations. I. Prequantization. In: Lectures in modern analysis and applications, III, Lecture Notes in Math., Vol. 170, Berlin: Springer, 1970, pp. 87–208 15. Kostant, B.: Symplectic spinors. In: Symposia Mathematica, Vol. XIV (Convegno di Geometria Simplettica e Fisica Matematica, INDAM, Rome, 1973), London: Academic Press 1974, pp 139–152 16. Rawnsley, J.H.: On the pairing of polarizations. Commun Math. Phys. 58(1), 1–8 (1978) 17. Souriau, J.-M.: Structure des systèmes dynamiques. Maîtrises de mathématiques. Paris: Dunod (1970) 18. Woodhouse, N.M.J.: Geometric quantization. 2nd edition, Oxford Mathematical Monographs, New York: The Clarendon Press/Oxford University Press (1992) Communicated by B. Simon
Commun. Math. Phys. 270, 481–517 (2007) Digital Object Identifier (DOI) 10.1007/s00220-006-0159-1
Communications in
Mathematical Physics
Large n Limit of Gaussian Random Matrices with External Source, Part III: Double Scaling Limit Pavel M. Bleher1, , Arno B. J. Kuijlaars2, 1 Department of Mathematical Sciences, Indiana University-Purdue University Indianapolis,
402 N. Blackford St., Indianapolis, IN 46202, USA. E-mail:
[email protected]
2 Department of Mathematics, Katholieke Universiteit Leuven, Celestijnenlaan 200 B, B-3001 Leuven,
Belgium. E-mail:
[email protected] Received: 28 February 2006 / Accepted: 5 July 2006 Published online: 9 December 2006 – © Springer-Verlag 2006
Abstract: We consider the double scaling limit in the random matrix ensemble with an external source 1 −nTr( 1 M 2 −AM) 2 e dM Zn defined on n × n Hermitian matrices, where A is a diagonal matrix with two eigenvalues ±a of equal multiplicities. The value a = 1 is critical since the eigenvalues of M accumulate as n → ∞ on two intervals for a > 1 and on one interval for 0 < a < 1. These two cases were treated in Parts I and II, where we showed that the local eigenvalue correlations have the universal limiting behavior known from unitary random matrix ensembles. For the critical case a = 1 new limiting behavior occurs which is described in terms of Pearcey integrals, as shown by Brézin and Hikami, and Tracy and Widom. We establish this result by applying the Deift/Zhou steepest descent method to a 3×3-matrix valued Riemann-Hilbert problem which involves the construction of a local parametrix out of Pearcey integrals. We resolve the main technical issue of matching the local Pearcey parametrix with a global outside parametrix by modifying an underlying Riemann surface. 1. Introduction and Statement of Results 1.1. The random matrix model. This is the third and final part of a sequence of papers on the Gaussian random matrix ensemble with external source 1 −nTr( 1 M 2 −AM) 2 e d M, (1.1) Zn The first author was supported in part by the National Science Foundation (NSF) Grant DMS-0354962.
The second author was supported by FWO-Flanders project G.0455.04, by K.U. Leuven research grant
OT/04/24, by INTAS Research Network 03-51-6637, by a grant from the Ministry of Education and Science of Spain, project code MTM2005-08648-C02-01, and by the European Science Foundation Program MISGAM.
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ξ1
ξ1
ξ1
ξ2
ξ2
ξ2
ξ3
ξ3
ξ3
Fig. 1. The structure of the Riemann surface for Eq. (1.2) for the values a > 1 (left), a = 1 (middle) and a < 1 (right). In all cases the eigenvalues of M accumulate on the interval(s) of the first sheet with a density given by (1.3)
defined on n × n Hermitian matrices, where A is a diagonal matrix with two eigenvalues ±a (with a > 0) of equal multiplicities (so that, n is even). This matrix ensemble was introduced by Brézin and Hikami [9, 10] as a simple model for a phase transition that is expected to exhibit universality properties. The phase transition can be seen from the behavior of the eigenvalues of M in the large n limit, since for a > 1, the eigenvalues accumulate on two intervals, while for 0 < a < 1, the eigenvalues accumulate on one interval. The limiting mean density of eigenvalues follows from earlier work of Pastur [22]. It is based on an analysis of the equation (Pastur equation) ξ 3 − zξ 2 + (1 − a 2 )ξ + a 2 z = 0,
(1.2)
which yields an algebraic function ξ(z) defined on a three-sheeted Riemann surface. The restrictions of ξ(z) to the three sheets are denoted by ξ j (z), j = 1, 2, 3. There are four real branch points if a > 1 which determine two real intervals. The two intervals come together for a = 1, and for 0 < a < 1, there are two real branch points, and two purely imaginary branch points. Figure 1 depicts the structure of the Riemann surface ξ(z) for a > 1, a = 1, and a < 1. In all cases we have that the limiting mean eigenvalue density ρ(x) = ρ(x; a) of the matrix M from (1.1) is given by ρ(x; a) =
1 Im ξ1+ (x), π
x ∈ R,
(1.3)
where ξ1+ (x) denotes the limiting value of ξ1 (z) as z → x with Im z > 0. For a = 1 the limiting mean eigenvalue density vanishes at x = 0 and ρ(x; a) ∼ |x|1/3 as x → 0. We note that this behavior at the closing (or opening) of a gap is markedly different from the behavior that occurs in the usual unitary random matrix ensembles Z n−1 e−nTrV (M) d M, where a closing of the gap in the spectrum typically leads to a limiting mean eigenvalue density ρ that satisfies ρ(x) ∼ (x − x ∗ )2 as x → x ∗ if the gap closes at x = x ∗ . In that case the local eigenvalue correlations can be described in terms of ψ-functions associated with the Painlevé II equation, see [5, 11]. The phase transition for the model (1.1) is different, and it cannot be realized in a unitary random matrix ensemble. 1.2. Non-intersecting Brownian motion. The nature of the phase transition at a = 1 may also be seen from an equivalent model of non-intersecting Brownian paths, see
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1.5
1
0.5
0
–0.5
–1
–1.5
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Fig. 2. Non-intersecting Brownian paths that start at one point and end at two points. At any intermediate time the positions of the paths are distributed as the eigenvalues of a Gaussian random matrix ensemble with external source. As their number increases the paths fill out a region whose boundary has a cusp
Fig. 2. Consider n independent one-dimensional Brownian motions that are conditioned to start at time t = 0 at the origin, end at time t = 1 at ±1, where half of the paths ends at +1 and the other half at −1, and that are conditioned not to intersect at intermediate times t ∈ (0, 1). As explained in [2], at any intermediate time t, the positions of the n Brownian motions, have the same distribution as the eigenvalues of a Gaussian random matrix ensemble with external source (up to trivial scaling). Now, as n → ∞ and under appropriate scaling of the variance of the Brownian motions, the paths fill out a region in the t − x-plane. Then for small time the paths are in one group, which at a certain critical time tcr splits into two groups, where one group ends at x = +1 and the other group at x = −1. The situations t < tcr , t = tcr , and t > tcr correspond to a < 1, a = 1, and a > 1, respectively, in the Gaussian random matrix model with external source. The boundary curve has a cusp singularity at the critical time as shown in Fig. 2.
1.3. Correlation kernel. Brézin and Hikami [9, 10] showed, see also [28], that the eigenvalues of the random matrix ensemble (1.1) are distributed according to a determinantal point process. There is a kernel K n (x, y; a) so that the eigenvalues x1 , . . . , xn have the joint probability density pn (x1 , . . . , xn ) =
1 det(K n (x j , xk ; a)) j,k=1,...,n n!
and so that for each m ≤ n, the m-point correlation function n! Rm (x1 , . . . , xm ) = · · · pn (x1 , . . . , xn )d xm+1 · · · d xn (n − m)! n−m times
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takes determinantal form as well: Rm (x1 , . . . , xm ) = det(K n (x j , xk ; a)) j,k=1,...,m . In [6] we pointed out that the kernel can be built out of multiple Hermite polynomials [3, 8, 26] in much the same way that the correlation kernel for unitary random matrix ensembles (without external source) is related to orthogonal polynomials. The Christoffel-Darboux formula for multiple orthogonal polynomials [6, 14] allows one to express the kernel in terms of the Riemann-Hilbert problem for multiple Hermite polynomials (see [27] and below). Applying the Deift/Zhou steepest descent analysis [15, 18] to the Riemann-Hilbert problem in the non-critical case, we were able to show that the kernel has the usual scaling limits from random matrix theory. That is, we obtain the sine kernel sin π(x − y) π(x − y)
(1.4)
Ai(x)Ai (y) − Ai (x)Ai(y) x−y
(1.5)
K bulk (x, y) = in the bulk, and the Airy kernel K edge (x, y) =
at the edge of the spectrum, as scaling limits of K n (x, y; a) if a > 1 [7] or 0 < a < 1 [2].
1.4. Double scaling limit. In this paper we consider the double scaling limit at the critical parameter a = 1 of the Gaussian random matrix ensemble with external source, or equivalently, of the non-intersecting Brownian motion model at the critical time t = tcr . As is usual in a critical case, there is a family of limiting kernels that arise when a changes with n and a → 1 as n → ∞ in a critical way. These kernels are constructed out of Pearcey integrals and therefore they are called Pearcey kernels. The Pearcey kernels were first described by Brézin and Hikami [9, 10]. A detailed proof of the following result was recently given by Tracy and Widom [25]. Theorem 1.1. We have for every fixed b ∈ R, x 1 y b lim 3/4 K n = K cusp (x, y; b), , ; 1 + √ n→∞ n n 3/4 n 3/4 2 n
(1.6)
where K cusp is the Pearcey kernel K cusp (x, y; b) =
p(x)q (y) − p (x)q (y) + p (x)q(y) − bp(x)q(y) x−y
(1.7)
with 1 p(x) = 2π
∞ −∞
e
− 14 s 4 − b2 s 2 +isx
ds
and
1 q(y) = 2π
1 4 b 2 + 2 t +it y
e4t
dt. (1.8)
Large n Limit of Gaussian Random Matrices with External Source. Part III
Im z
485
Σ
Re z
Fig. 3. The contour that appears in the definition of q(y)
The contour consists of the four rays arg z = ±π/4, ±3π/4, with the orientation shown in Fig. 3. The functions (1.8) are called Pearcey integrals [23]. They are solutions of the third order differential equations p (x) = x p(x) + bp (x) and q (y) = −yq(y) + bq (y), respectively. Away from the critical point x = 0, the usual scaling limits (1.4) and (1.5) from random matrix theory continue to hold in the case a = 1 (also in the double scaling regime). This can be proved for example as in [7, 25], and we will not consider this any further here. Theorem 1.1 implies that local eigenvalue statistics of eigenvalues near 0 are expressed in terms of the Pearcey kernel. For example we have the following corollary of Theorem 1.1. Corollary 1.2. The probability that a matrix of the ensemble (1.1) with a = 1+bn −1/2 /2 has no eigenvalues in the interval [cn −3/4 , dn −3/4 ] converges, as n → ∞, to the Fredholm determinant of the integral operator with kernel K cusp (x, y; b) acting on L 2 (c, d). Similar expressions hold for the probability to have one, two, three, …, eigenvalues in an O(n −3/4 ) neighborhood of x = 0. Tracy and Widom [25] and Adler and van Moerbeke [1] gave differential equations for the gap probabilities associated with the Pearcey kernel and with the more general Pearcey process which arises from considering the non-intersecting Brownian motion model at several times near the critical time. See also [20] where the Pearcey process appears in a combinatorial model on random partitions.
1.5. Steepest descent method for RH problems. Brézin and Hikami and also Tracy and Widom used a double integral representation for the kernel in order to establish Theorem 1.1. In this paper we use the Deift/Zhou steepest descent method for the Riemann-Hilbert problem for multiple Hermite polynomials. This method is less direct than the steepest descent method for integrals. However, an approach based on the Riemann-Hilbert problem may be applicable to more general situations, where an integral representation is not available. This is the case, for example, for the general
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(non-Gaussian) unitary random matrix ensemble with external source 1 −nTr(V (M)−AM) e dM Zn
(1.9)
with a general potential V . The Riemann-Hilbert problem is formulated in Sect. 2. The asymptotic analysis of the Riemann-Hilbert problem presents a new feature that we feel is of importance in its own right. We will not use the Pastur equation (1.2) which defines the ξ -functions and the Riemann surface that corresponds to it, but instead we use a modified equation to define the ξ -functions. We discuss this in Sect. 3. The modification may be thought of in potential theoretic terms and we briefly discuss this in Sect. 3 as well. The anti-derivatives of the modified ξ -functions are introduced in Sect. 4 and they play an important role in the steepest descent analysis of the Riemann-Hilbert problem in the rest of the paper. The main issue is the construction in Sect. 8 of the local parametrix around 0 with the aid of Pearcey integrals. The modification of the ξ -functions is used here to be able to match the local Pearcey parametrix with the outside parametrix. Even so it turns out that we cannot achieve the matching condition on a fixed circle around the origin, but only on circles with radii n −1/4 that decrease as n increases. However, the circles are big enough to capture the behavior (1.6) which takes place at a distance to the origin of order n −3/4 . The precise estimates that lead to the proof of Theorem 1.1 are given in the final Sects. 9 and 10. 2. Riemann-Hilbert Problem As shown in our paper [6], the correlation kernel is expressed in terms of the solution to the following 3 × 3 matrix valued Riemann-Hilbert (RH) problem. Find Y : C \ R → C3×3 such that • Y is analytic on C \ R, • for x ∈ R, we have
⎛
1 2 −ax)
1 e−n( 2 x Y+ (x) = Y− (x) ⎝0 1 0 0
1 2 +ax)
e−n( 2 x 0 1
⎞ ⎠,
(2.1)
where Y+ (x) (Y− (x)) denotes the limit of Y (z) as z → x from the upper (lower) half-plane, • as z → ∞, we have ⎞ ⎛ z n 0 0 1 ⎝ 0 z −n/2 0 ⎠ , (2.2) Y (z) = I + O z 0 0 z −n/2 where I denotes the 3 × 3 identity matrix. The RH problem has a unique solution, given explicitly in terms of the multiple Hermite polynomials. The correlation kernel of the Gaussian random matrix model with external source is equal to ⎛ ⎞ 1 − 14 n(x 2 +y 2 ) e 0 enay e−nay Y+−1 (y)Y+ (x) ⎝0⎠ . K n (x, y; a) = (2.3) 2πi(x − y) 0
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In what follows we are going to apply the Deift/Zhou steepest descent method for RH problems to the above RH problem for Y . It consists of a sequence of explicit transformations Y → T → S → R which leads to a RH problem for R in which all jumps are close to the identity matrix and which is normalized at infinity. Then R is close to the identity matrix, and analyzing the effect of the transformations on the kernel (2.3) we will be able to prove Theorem 1.1. 3. Modification of the ξ -Functions 3.1. Modified Pastur equation. The analysis in [2, 7] for the cases a > 1 and 0 < a < 1 was based on Eq. (1.2) and it would be natural to use (1.2) also in the case a = 1. Indeed, that is what we tried to do, and we found that it works for a ≡ 1, but in the double scaling b regime a = 1 + 2√ with b = 0, it led to problems that we were unable to resolve in a n satisfactory way. A crucial feature of our present approach is a modification of Eq. (1.2) when a is close to 1, but different from 1. At x = 0 we wish to have a double branch point for all values of a so that the structure of the Riemann surface is as in the middle figure of Fig. 1 for all a. For c > 0, we consider the Riemann surface for the equation w3 , (3.1) w 2 − c2 where w is a new auxiliary variable. The Riemann surface has branch points at z ∗ = √ 3 3 ∗ 2 c, −z and a double branch point at 0. There are three inverse functions wk , k = 1, 2, 3, that behave as z → ∞ as 1 c2 w1 (z) = z − +O 3 , z z 1 c2 w2 (z) = c + (3.2) +O 2 , 2z z 1 c2 w3 (z) = −c + +O 2 , 2z z z=
and which are defined and analytic on C \ [−z ∗ , z ∗ ], C \ [0, z ∗ ] and C \ [−z ∗ , 0], respectively. Then we define the modified ξ -functions p , for k = 1, 2, 3, (3.3) ξk = w k + wk which we also consider on their respective Riemann sheets. In what follows we take √ a + a2 + 8 and p = c2 − 1. (3.4) c= 4 Note that a = 1 corresponds to c = 1 and p = 0. In that case the functions coincide with the solutions of Eq. (1.2) that we used in our earlier works. From (3.1), (3.3), and (3.4) we obtain the modified Pastur equation ξ 3 − zξ 2 + (1 − a 2 )ξ + a 2 z + where c is given by (3.4).
(c2 − 1)3 = 0, c2 z
(3.5)
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Lemma 3.1. Let a > 0 and take c and p as in (3.4). Then at infinity we have 1 1 ξ1 (z) = z − + O 3 , z z 1 1 +O 2 , ξ2 (z) = a + 2z z 1 1 +O 2 . ξ3 (z) = −a + 2z z
(3.6)
Proof. This follows from direct calculations using (3.2), (3.3) and the fact that 2c − 1c = a. Alternatively, one could also use (3.5). The new ξ -functions have the same asymptotic behavior (3.6) as z → ∞ (up to order 1/z 2 ) as the solutions of (1.2). This is important for the first two transformations of the Riemann-Hilbert problem. The situation at z = 0 is different. The fact that we can control the behavior at z = 0 as well is the reason for the introduction of the modified ξ -functions. 3.2. Behavior at z = 0. We start with the behavior of the functions wk . Lemma 3.2. There exist analytic functions f 1 and g1 defined in a neighborhood U1 of z = 0 so that for z ∈ U1 and k = 1, 2, 3, ⎧ z 2k 1/3 k 5/3 ⎪ for Im z > 0, ⎨ −ω z f 1 (z) − ω z g1 (z) + 3 (3.7) wk (z) = ⎪ ⎩ −ωk z 1/3 f 1 (z) − ω2k z 5/3 g1 (z) + z for Im z < 0. 3 In addition, we have f 1 (0) = c2/3 , and f 1 (z) and g1 (z) are real for real z ∈ U1 . Proof. Putting z = x 3 and w = x y in (3.1) we obtain y 3 = x 2 y 2 − c2 ,
(3.8)
which has a solution y = y(x) that is analytic in a neighborhood U1 of 0 and satisfies y(0) = −c2/3 and y (0) = 0. Then we can write y(x) = − f 1 (x 3 )−x 4 g1 (x 3 )+x 2 h 1 (x 3 ) with f 1 , g1 and h 1 analytic in U1 and f 1 (0) = c2/3 . Putting this back in (3.8) we find after straightforward calculations that (with z = x 3 ) f 1 (z)g1 (z) =
1 , 9
f 1 (z)3 − c2 +
2 2 z + g1 (z)3 z 4 = 0, 27
(3.9)
and h 1 (z) = 13 . Going back to z and w variables, we see that there is a solution w = w(z) to (3.1) with z w(z) = −z 1/3 f 1 (z) − z 5/3 g1 (z) + , 3
for z ∈ C \ (−∞, 0],
where we take the principal branches of the fractional powers. This solution is real for z real and positive, and so it coincides with the solution w3 (z). This proves (3.7) for k = 3. The expressions (3.7) for k = 1, 2 follow by analytic continuation. Since y(x) is real for real x, we also find that f 1 (z) and g1 (z) are real if z is real.
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From (3.9) it is easy to give explicit expressions for f 1 and g1 . However we will not use this in the future. From Lemma 3.2 and (3.3) we get the following behavior for the functions ξk near z = 0. Lemma 3.3. There exist analytic functions f 2 and g2 defined in a neighborhood U2 of z = 0 so that for z ∈ U2 and k = 1, 2, 3, ⎧ z 2k 1/3 k −1/3 ⎪ g2 (z) + for Im z > 0, ⎨ −ω z f 2 (z) − ω z 3 (3.10) ξk (z) = ⎪ ⎩ −ωk z 1/3 f 2 (z) − ω2k z −1/3 g2 (z) + z for Im z < 0. 3 In addition, we have 1 f 2 (0) = c2/3 + c−4/3 (c2 − 1), 3
g2 (0) = c−2/3 (c2 − 1),
(3.11)
and f 2 (z) and g2 (z) are real for real z ∈ U2 . Proof. This follows from (3.3) and the previous lemma. Indeed from (3.7) where f 1 and g1 satisfy the relations (3.9), we can deduce 1 1 1 = −z 1/3 2 ( f 1 (z) + 3z 2 g1 (z)2 ) − z −1/3 2 (3 f 1 (z)2 + z 2 g1 (z)). w3 (z) 3c 3c Then (3.10) follows from (3.7) and (3.3) if we take f 2 (z) = f 1 (z) +
c2 − 1 ( f 1 (z) + 3z 2 g1 (z)2 ) 3c2
(3.12)
c2 − 1 (3 f 1 (z)2 + z 2 g1 (z)). 3c2
(3.13)
and g2 (z) = z 2 g1 (z) +
Because of Lemma 3.2 this also implies (3.11) and the fact that f 2 (z) and g2 (z) are real for real z ∈ U2 . 3.3. Potential theoretic interpretation. As an aside we want to mention that the modified ξ -functions may be thought of in terms of a modified equilibrium problem for logarithmic potentials. For a > 1, it was noted in [7], that the limiting mean eigenvalue density ρ(x) = ρ(x; a) may be characterized as follows. We minimize 1 1 dμ1 (x)dμ1 (y) + dμ2 (x)dμ2 (y) E(μ1 , μ2 ) = log log |x − y| |x − y| 1 2 1 dμ1 (x)dμ2 (y) + x − ax dμ1 (x) (3.14) + log |x − y| 2 1 2 + x + ax dμ2 (x) 2 among all non-negative measures μ1 , μ2 on R with dμ1 = dμ2 = 21 . There is a unique minimizer [24], and for a > 1, we have that supp(μ1 ) ⊂ [0, ∞), supp(μ2 ) ⊂
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(−∞, 0], and ρ is the density of μ1 + μ2 . For a < 1, the minimizing measures for (3.14) do not have disjoint supports, and in fact these minimizers are not related to our random matrix ensemble (1.1) at all. The modification we are alluding to is to minimize (3.14) among signed measures − ± + μ1 = μ+1 − μ− 1 , μ2 = μ2 − μ2 , where μ j are non-negative measures, that satisfy dμ1 = dμ2 = 21 and in addition (1) supp(μ1 ) ⊂ [0, ∞), supp(μ2 ) ⊂ (−∞, 0], and, − (2) there is a δ > 0, such that supp(μ− 1 ) ⊂ [0, δ) and supp(μ2 ) ⊂ (−δ, 0]. The condition (1) plays a role for a < 1, since it prevents the supports of μ1 and μ2 to overlap. For a > 1, condition (2) plays a role, since it allows the measures to become negative near 0. Now let μ1 , μ2 be the minimizers for this modified equilibrium problem, and let ρ˜ be the density of μ1 + μ2 . Then it can be shown that the density of μ1 + μ2 is equal to π1 Im ξ1+ (x), where ξ1 is the modified ξ1 -function introduced in this section. We will not use this potential-theoretic connection in the analysis that follows in this paper, but we anticipate that it might be important for the general unitary random matrix ensemble with external source (1.9). We finally note that a modified equilibrium problem was also used in [11, 12] in order to analyse the double scaling limit in unitary random matrix ensembles (without external source), so one might speculate that such an approach might be characteristic for double scaling limits in random matrix ensembles. 4. The λ-Functions 4.1. Definition and first properties. The main role is played by the λ-functions which are anti-derivatives of the ξ -functions. They are defined here as z λk (z) = ξk (s)ds, (4.1) 0+
where the path of integration starts at 0 on the upper side of the cut and is fully contained (except for the initial point) in C \ (−∞, z ∗ ] for k = 1, 2, and in C \ (−∞, 0] for k = 3. Then λ1 and λ2 are defined and analytic on C \ (−∞, z ∗ ], and λ3 is defined and analytic on C \ (−∞, 0]. As follows from (3.6) and (4.1), the λ-functions behave at infinity as 1 λ1 (z) = z 2 − log z + 1 + O(1/z), 2 1 λ2 (z) = az + log z + 2 + O(1/z), (4.2) 2 1 λ3 (z) = −az + log z + 3 + O(1/z), 2 for certain constants k , k = 1, 2, 3, where log z is taken as the principal value, that is, with a cut along the negative real axis. From contour integration based on (3.6) where we use the residue of ξ2 at infinity, we find λ1− (0) = πi and λ2− (0) = −πi. Then we get the following jump properties of the λ-functions on the cuts (−∞, 0] and (−∞, z ∗ ]: λ1+ = λ2− + πi,
λ2+ = λ1− − πi, λ3+ = λ3− ,
on [0, z ∗ ],
λ1+ = λ3− ,
λ2+ = λ2− + πi, λ3+ = λ1− − πi,
on [−z ∗ , 0],
λ1+ = λ1− − 2πi, λ2+ = λ2− + πi, λ3+ = λ3− + πi,
on
(−∞, −z ∗ ].
(4.3)
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Re(λ1) = Re(λ2) Re(λ1) = Re(λ3) Re(λ2) = Re(λ3)
1.5 1 0.5 0 –0.5 –1 –1.5 –2
–4
–3
–2
–1
0
1
2
3
4
Fig. 4. Curves where Re λ j = Re λk for the value a = 2.0
4.2. Behavior near z = 0. Near the origin the λ-functions behave as follows. Lemma 4.1. There exist analytic functions f 3 and g3 in a neighborhood U3 of z = 0 so that ⎧ 3 2k 4/3 1 k 2/3 z2 ⎪ ⎪ for Im z > 0, ⎨ − ω z f 3 (z) − ω z g3 (z) + 4 2 6 (4.4) λk (z) = 2 ⎪ 3 1 z ⎪ ⎩ λ (0) − ωk z 4/3 f (z) − ω2k z 2/3 g (z) + for Im z < 0. k− 3 3 4 2 6 In addition, we have 1 f 3 (0) = f 2 (0) = c2/3 + c−4/3 (c2 − 1), g3 (0) = 3g2 (0) = 3c−2/3 (c2 − 1),(4.5) 3 and f 3 (z) and g3 (z) are real for real z ∈ U3 . Proof. The relations (4.4) follow by integrating (3.10). Note that λ1− (0) = πi, λ2− (0) = −πi, and λ3− (0) = 0. The other statements of the lemma also follow directly from Lemma 3.3. 4.3. Critical trajectories. Curves where Re λ j = Re λk for some j = k are shown in Figs. 4, 5, and 6, for the cases a > 1, a = 1, and a < 1, respectively. These are critical curves that play a crucial role in the asymptotic analysis. The curves are critical trajectories of the quadratic differentials (ξ j (z) − ξk (z))2 dz 2 (and their analytic continuations beyond the branch cuts in case a < 1). The solid curves in Figs. 4–6 are the critical trajectories of the quadratic differential (ξ1 (z) − ξ2 (z))2 dz 2 . The quadratic differential has a simple zero at z = z ∗ . Three trajectories are emanating from z = z ∗ at equal angles, one of these being the real interval (0, z ∗ ). For a > 1, the quadratic differential has a double zero at z = x0 for some x0 ∈ (0, z ∗ ). Four trajectories are emanating from the double zero at equal angles as can be seen in Fig. 4.
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Re(λ1) = Re(λ2) Re(λ1) = Re(λ3) Re(λ2) = Re(λ3)
1.5 1 0.5 0 –0.5 –1 –1.5 –2 –4
–3
–2
–1
0
1
2
3
4
Fig. 5. Curves where Re λ j = Re λk for the value a = 1.0
The dashed curves are the critical trajectories of the quadratic differential (ξ1 (z) − ξ3 (z))2 dz 2 . Because of symmetry, these are the mirror images of the trajectories of the quadratic differential (ξ1 (z) − ξ2 (z))2 dz 2 with respect to the imaginary axis. For a > 1, the solid curve that passes vertically through x0 and its dashed mirror image with respect to the imaginary axis meet in two points ±i y0 on the imaginary axis. Together they enclose a neighborhood of the origin. The dashed-dotted curves are the critical trajectories of (ξ2 (z) − ξ3 (z))2 dz 2 . For a < 1, the quadratric differential has two double zeros at z = ±i y0 for some y0 > 0. Four trajectories are emanating from these double zeros at equal angles as shown in Fig. 6. Besides the imaginary axis there are curves passing horizontally through ±i y0 , and these curves meet each other at two points ±x0 on the real axis and they enclose a neighborhood of the origin. Beyond these two points the quadratic differentials have analytic continuations, but the formula changes since either ξ2 or ξ3 reaches its branch cut and changes into ξ1 . Consequently, the dashed-dotted curves in Fig. 6 continue beyond ±x0 as either solid or dashed curves. The relative orderings of the real parts Re λ1 , Re λ2 and Re λ3 changes if we cross one of the critical trajectories, but it remains constant in the regions bounded by the critical trajectories. For each of the unbounded regions we can determine the ordering from the behavior at infinity (4.2). For example, we have in the right-most region Re λ1 > Re λ2 > Re λ3 , and if we cross the solid curve where Re λ1 = Re λ2 , the ordering becomes Re λ2 > Re λ1 > Re λ3 , and so on. In the cases a < 1 and a > 1 the trajectories enclose a bounded neighborhood of the origin. There is no such neighborhood in case a = 1. The neighborhood is small if a is close to 1. In this neighborhood the relative ordering of the real parts is different. So we can easily verify the following. Lemma 4.2. Except for z in the exceptional bounded neighborhood of the origin, we have that Re λ2 (z) > max(Re λ1 (z), Re λ3 (z))
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2
Re(λ1) = Re(λ2) Re(λ1) = Re(λ3) Re(λ2) = Re(λ3)
1.5 1 0.5 0 –0.5 –1 –1.5 –2 –4
–3
–2
–1
0
1
2
3
4
Fig. 6. Curves where Re λ j = Re λk for the value a = 0.5
in the region in the right-half plane, bounded by the solid and dashed-dotted curves, and Re λ3 (z) > max(Re λ1 (z), Re λ2 (z)) in the region in the left-half plane bounded by the dashed and dashed-dotted curves. The exceptional neighborhood will not cause a problem to us, since it turns out to shrink fast enough if a = 1 + (b/2)n −1/2 and n → ∞. For n large enough, the exceptional neighborhood is well within the disk around the origin of radius n −1/4 where we are going to construct a special parametrix with Pearcey integrals. Then the different ordering of the real parts of the λk will not play a role.
5. First Two Transformations of the RH Problem The first and second transformation of the RH problem are the same as in our earlier paper [7], except that we use the λ-functions that were introduced in the last section via the modified ξ -functions.
5.1. First transformation Y → T . Using the functions λk and the constants k , k = 1, 2, 3, we define 1 2 T (z) = diag e−n 1 , e−n 2 , e−n 3 Y (z)diag en(λ1 (z)− 2 z ) , en(λ2 (z)−az) , en(λ3 (z)+az) . (5.1)
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Then by (2.1) and (5.1) and the jump properties (4.3) we have T+ (x) = T− (x) jT (x) for x ∈ R, where ⎞ ⎛ n(λ −λ ) 1 en(λ3 −λ1− ) e 1 2+ ⎠, jT = ⎝ x ∈ (0, z ∗ ), (5.2) 0 en(λ1 −λ2 )− 0 0 0 1 ⎞ ⎛ n(λ −λ ) n(λ −λ ) 1 e 1 3 + e 2+ 1− ⎠, 0 1 0 jT = ⎝ x ∈ (−z ∗ , 0), (5.3) 0 0 en(λ1 −λ3 )− ⎞ ⎛ 1 en(λ2+ −λ1− ) en(λ3+ −λ1− ) ⎠ , x ∈ (−∞, −z ∗ ) ∪ (z ∗ , ∞). j T = ⎝0 (5.4) 1 0 0 0 1 The function T (z) solves the following RH problem: • T is analytic on C \ R, • for x ∈ R, we have T+ (x) = T− (x) jT (x), where jT is given by (5.2)–(5.4), • as z → ∞, T (z) = I + O
1 . z
(5.5)
(5.6)
The asymptotic property (5.6) follows from (2.2), (5.1), and the behavior (4.2) of the λ-functions at infinity. 5.2. Second transformation T → S. The second transformation of the RH problem consists of opening of lenses around the intervals [0, z ∗ ] and [−z ∗ , 0]. The lenses are as shown in Fig. 7. We define (see also Sect. 5 in [7]) ⎛ ⎞ 1 0 0 S = T ⎝−en(λ1 −λ2 ) 1 −en(λ3 −λ2 ) ⎠ in the upper right lens region, (5.7) 0 0 1 ⎛ ⎞ 1 0 0 S = T ⎝en(λ1 −λ2 ) 1 −en(λ3 −λ2 ) ⎠ in the lower right lens region, (5.8) 0 0 1 ⎛ ⎞ 1 0 0 0 1 0⎠ in the upper left lens region, S=T⎝ (5.9) n(λ −λ ) n(λ −λ ) 1 3 2 3 −e −e 1 ⎛ ⎞ 1 0 0 0 1 0⎠ S=T⎝ in the lower left lens region, (5.10) n(λ −λ ) n(λ −λ ) 1 3 2 3 e −e 1 and S = T outside the lenses. It leads to a matrix valued function S which is defined and analytic in C \ S , where S consists of the real line and the upper and lower lips of the lenses. On S we have
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Re(λ1) = Re(λ2) Re(λ1) = Re(λ3) Re(λ2) = Re(λ3)
1.5 1 0.5 0 –0.5 –1 –1.5 –2
–4
–3
–2
–1
0
1
2
3
4
Fig. 7. Opening of lenses around the intervals [0, z ∗ ] and [−z ∗ , 0] for the value a = 2.0. The upper and lower
lips of the lenses together with the real axis form the contour S which is shown in bold. Also shown are the critical trajectories as in Fig. 4
S+ = S− jS where jS is defined as follows (the orientation on S is taken from left to right): ⎛ ⎞ 0 10 jS = ⎝−1 0 0⎠ , on (0, z ∗ ), (5.11) 0 01 ⎛ ⎞ 0 01 jS = ⎝ 0 1 0⎠ , on (−z ∗ , 0), (5.12) −1 0 0 ⎞ ⎛ 1 en(λ2+ −λ1− ) en(λ3+ −λ1− ) ⎠ , on (−∞, −z ∗ ) ∪ (z ∗ , ∞), jS = ⎝0 (5.13) 1 0 0 0 1 ⎛ ⎞ 1 0 0 jS = ⎝en(λ1 −λ2 ) 1 en(λ3 −λ2 ) ⎠ , on the upper lip of the right lens, (5.14) 0 0 1 ⎞ ⎛ 1 0 0 0 1 0⎠ , on the upper lip of the left lens, jS = ⎝ (5.15) n(λ −λ ) n(λ −λ ) 1 3 2 3 e 1 e ⎛ ⎞ 1 0 0 0 1 0⎠ , on the lower lip of the left lens, jS = ⎝ (5.16) n(λ −λ ) n(λ −λ ) 1 3 2 3 e −e 1 ⎛ ⎞ 1 0 0 jS = ⎝en(λ1 −λ2 ) 1 −en(λ3 −λ2 ) ⎠ , on the lower lip of the right lens. (5.17) 0 0 1 Thus S solves the following RH problem: • S is analytic on C \ S ,
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• for z ∈ S , we have S+ (z) = S− (z)jS(z), where jS is given by (5.11)–(5.17), • as z → ∞, we have S(z) = I + O 1z . Now the ordering of the real parts of the λk in various regions in the complex plane (see Lemma 4.2) shows that the jump matrices in (5.13)–(5.17) are all close to the identity matrix if n is large, except in a neighborhood of the origin. To be precise, if a < 1 then Re λ3 > Re λ2 near the origin in the right half-plane, which means that the entries ±en(λ3 −λ2 ) in the jump matrices in (5.14) and (5.17) are not small near the origin but instead grow exponentially if n gets large. Similarly the entries ±en(λ2 −λ3 ) in the jump matrices in (5.15) and (5.16) also grow exponentially near the origin. On the other hand, if a > 1, then Re λ1 is bigger than the other two in the exceptional neighborhood of the origin, so that the other non-zero off-diagonal entries in the jump matrices in (5.14)– (5.17) grow exponentially in a neighborhood of the origin. For a = 1 there are no such exceptions and all jump matrices in (5.13)–(5.17) are close to the identity matrix if n is large. When we wrote that certain entries grow exponentially as n gets large, it was understood that the value of a = 1 remained fixed. However, eventually we are going to take a = 1 + O(n −1/2 ) as n → ∞. Then it will turn out that the possible growth of certain entries in the jump matrices is confined to a small enough region near the origin, which shrinks sufficiently fast as n → ∞, so that we can still ignore the jumps (5.13)–(5.17) in the next step. 6. Model RH Problem We consider the following auxiliary model RH problem: find M : C\[−z ∗ , z ∗ ] → C3×3 such that • M is analytic on C \ [−z ∗ , z ∗ ], • for x ∈ (−z ∗ , z ∗ ) we have M+ (x) = M− (x) j M (x), where ⎛ ⎞ 0 10 for x ∈ (0, z ∗ ), j M (x) = ⎝−1 0 0⎠ , 0 01 and
⎞ 0 01 j M (x) = ⎝ 0 1 0⎠ , −1 0 0
(6.1)
⎛
• as z → ∞, M(z) = I + O
for x ∈ (−z ∗ , 0),
(6.2)
1 . z
(6.3)
This RH problem has a solution, see [7, Sect. 6], that can be explicitly given in terms of the mapping functions wk , k = 1, 2, 3, from (3.1) and (3.2). The solution takes the form ⎛ ⎞ M1 (w1 (z)) M1 (w2 (z)) M1 (w3 (z)) M(z) = ⎝ M2 (w1 (z)) M2 (w2 (z)) M2 (w3 (z))⎠ , (6.4) M3 (w1 (z)) M3 (w2 (z)) M3 (w3 (z))
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where M1 , M2 , M3 are the three scalar valued functions w+c w 2 − c2 −i , M2 (w) = √ , √ √ 2 w w 2 − 3c2 w w 2 − 3c2 w−c −i M3 (w) = √ . √ 2 w w 2 − 3c2
M1 (w) =
(6.5)
Note that by (3.2) we have that wk (z) is of order z 1/3 as z → 0. By (6.4) and (6.5) this implies that M(z) = O(z −1/3 )
as z → 0.
(6.6)
The RH problem for M easily gives that det M(z) ≡ 1. Thus M −1 (z) exists for z ∈ C \ [−z ∗ , z ∗ ] and from (6.6) it follows that M −1 (z) = O(z −2/3 ). However, the special form of the solution (6.4)–(6.5) shows that all cofactors of M are actually O(z −1/3 ) as z → 0. Thus M −1 (z) = O(z −1/3 )
as z → 0.
(6.7)
This may also be understood from the fact that M −1 = M t , since together with M it is easy to see that also M −t is a solution of the RH problem (6.1)–(6.3). The model solution M will be used to construct a parametrix for S outside of small neighborhoods of the edge points and the origin. Namely, we consider disks of fixed radius r around the edge points and a shrinking disk D(0, n −1/4 ) of radius n −1/4 around the origin. At the edge points and at the origin M is not analytic (it is not even bounded) and in the disks around the edge points and the origin the parametrix is constructed differently. 7. Parametrix at Edge Points The construction of a parametrix P at the edge points ±z ∗ can be done with Airy functions in a by now standard way, see [6, 15–17]. We omit details. We only note that √ 3 3 ∗ ±z = ± 2 c depends on c and therefore on a. As a → 1 we have c → 1 and so √
√
±z ∗ → ± 3 2 3 . We construct the Airy parametrices in fixed neighborhoods D(± 3 2 3 , r ) √
of ± 3 2 3 so that √
• P is analytic on D(± 3 2 3 , r ) \ S , √
• for z ∈ D(± 3 2 3 , r ) ∩ S , we have P+ (z) = P− (z) jS (z),
(7.1)
where jS is given by (5.11)–(5.17), • as n → ∞, P(z) = M(z) I + O n −1
√ 3 3 uniformly for z ± = r. 2
(7.2)
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8. Parametrix at the Origin The main issue is the construction of a parametrix at the origin and this is where the Pearcey integrals come in. For a sufficiently close to 1, we want to define Q in a neighborhood D(0, r ) of the origin such that • Q is analytic on D(0, r ) \ S , • for z ∈ D(0, r ) ∩ S , we have Q + (z) = Q − (z) jS (z),
(8.1)
where jS is given by (5.11)–(5.17), • as n → ∞, and with a = 1 + O(n −1/2 ), we have Q(z) = M(z) I + O n −1/2
uniformly for |z| = n −1/4 .
(8.2)
The parametrix Q will be constructed with the aid of Pearcey integrals. To motivate the construction, we note that the jump matrices for S can be factored as jS = e−n − jSo en + ,
(8.3)
where = diag(λ1 , λ2 , λ3 ) and ⎛
jSo
jSo
jSo
jSo
jSo
jSo
⎞ 0 10 = ⎝−1 0 0⎠ 0 01 ⎛ ⎞ 0 01 = ⎝ 0 1 0⎠ −1 0 0 ⎛ ⎞ 100 = ⎝1 1 1⎠ 001 ⎛ ⎞ 100 = ⎝0 1 0⎠ 111 ⎛ ⎞ 1 0 0 = ⎝0 1 0⎠ 1 −1 1 ⎛ ⎞ 10 0 = ⎝1 1 −1⎠ 00 1
on (0, z ∗ ),
(8.4)
on (−z ∗ , 0),
(8.5)
on the upper lip of the right lens,
(8.6)
on the upper lip of the left lens,
(8.7)
on the lower lip of the left lens,
(8.8)
on the lower lip of the right lens.
(8.9)
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We show in the next subsection that the Pearcey integrals satisfy a RH problem with exactly the above jump matrices except that these jumps are situated on six rays emanating from the origin.
8.1. The Pearcey parametrix. Let b ∈ R be fixed. The Pearcey differential equation p (ζ ) = ζ p(ζ ) + bp (ζ ) admits solutions of the form p j (ζ ) =
1 4 b 2 − 2 s +isζ
j
e− 4 s
ds
(8.10)
for j = 0, 1, 2, 3, 4, 5, where 0 = (−∞, ∞), 2 = (i∞, 0] ∪ [0, −∞), 4 = (−i∞, 0] ∪ [0, ∞),
1 = (i∞, 0] ∪ [0, ∞), 3 = (−i∞, 0] ∪ [0, −∞), 5 = (−i∞, i∞),
(8.11)
or any other contours that are homotopic to them as for example given in Fig. 8. The formulas (8.11) also determine the orientation of the contours j . Define = (ζ ; b) in six sectors by ⎛
⎞ − p2 p1 p5 = ⎝− p2 p1 p5 ⎠ for 0 < arg ζ < π/4, − p2 p1 p5 ⎛ ⎞ p0 p1 p4 = ⎝ p0 p1 p4 ⎠ for π/4 < arg ζ < 3π/4, p0 p1 p4 ⎛ ⎞ − p3 − p5 p4 = ⎝ − p3 − p5 p4 ⎠ for 3π/4 < arg ζ < π, − p3 − p5 p4
(8.12)
(8.13)
(8.14)
Γ1
Γ2 Γ5 Γ0
Γ3
Γ4
Fig. 8. The contours j , j = 0, 1, . . . , 5, equivalent to those in (8.11), that are used in the definition of the Pearcey integrals p j
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⎛
p4 = ⎝ p4 p4 ⎛ p0 = ⎝ p0 p0 ⎛ p1 = ⎝ p1 p1
⎞ − p5 p3 − p5 p3 ⎠ − p5 p3 ⎞ p2 p3 p2 p3 ⎠ p2 p3 ⎞ p2 p5 p2 p5 ⎠ p2 p5
for − π < arg ζ < −3π/4,
(8.15)
for − 3π/4 < arg ζ < −π/4,
(8.16)
for − π/4 < arg ζ < 0.
(8.17)
Then has jumps on the six rays. We choose an orientation on these rays so that the rays in the right half-plane are oriented from 0 to ∞, and the rays in the left half-plane are oriented from ∞ to 0. Then the integral representations (8.10)–(8.11) easily imply that + = − j , where ⎛ ⎞ ⎛ ⎞ 100 0 10 j = ⎝1 1 1⎠ on arg ζ = π/4, j = ⎝−1 0 0⎠ on arg ζ = 0, (8.18) 001 0 01 ⎛ ⎞ ⎛ ⎞ 10 0 100 j = ⎝1 1 −1⎠ on arg ζ = −π/4, j = ⎝0 1 0⎠ on arg ζ = 3π/4, (8.19) 00 1 111 ⎛ ⎞ ⎛ ⎞ 0 01 1 0 0 j = ⎝ 0 1 0⎠ on arg ζ = −π, j = ⎝0 1 0⎠ on arg ζ = −3π/4. (8.20) −1 0 0 1 −1 1 So these are indeed the jump matrices of (8.4)–(8.9). 8.2. Asymptotics of Pearcey integrals. A classical steepest descent analysis of the integral representations gives the following result for the asymptotic behavior of (ζ ; b) as ζ → ∞. As always we use the principal branches of the fractional powers, that is, with a branch cut along the negative axis. Lemma 8.1. For every fixed b ∈ C, we have as ζ → ∞, ⎛ −1/3 ⎞⎛ ⎞ 0 0 −ω ω2 1 2π b2 ⎝ζ 0 1 0 ⎠ ⎝ −1 1 1⎠ ie 8 (ζ ; b) = 3 0 0 ζ 1/3 −ω2 ω 1 ⎛ θ (ζ ;b) ⎞ 0 0 e 1 × I + O ζ −2/3 ⎝ 0 eθ2 (ζ ;b) (8.21) 0 ⎠ θ (ζ 3 0 0 e ;b) for Im ζ > 0, and
⎛ −1/3 ⎞⎛ 2 ⎞ 0 0 ω ω 1 2π b2 ⎝ζ 0 1 0 ⎠ ⎝ 1 1 1⎠ ie 8 (ζ ; b) = 3 0 0 ζ 1/3 ω ω2 1 ⎛ θ (ζ ;b) ⎞ 0 0 e 2 × I + O ζ −2/3 ⎝ 0 eθ1 (ζ ;b) 0 ⎠ 0 0 eθ3 (ζ ;b)
(8.22)
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for Im ζ < 0, where ω = e2πi/3 and 3 2k 4/3 b k 2/3 ω ζ + ω ζ , k = 1, 2, 3. (8.23) 4 2 The O-terms in (8.21) and (8.22) are uniform for b in a bounded subset of the complex plane. θk (ζ ; b) =
Proof. We give an outline of the proof, cf. also the calculations in [19]. Let θ (s; ζ, b) = − 41 s 4 − b2 s 2 + iζ s. The saddle point equation for (8.10) is ∂θ = −s 3 − bs + iζ = 0. ∂s For b = 0 there are three solutions sko = −iωk ζ 1/3 , k = 1, 2, 3, and as ζ → ∞, while b remains bounded, the three saddles sk = sk (ζ ; b) are close to sko , and in fact b sk (ζ ; b) = −iωk ζ 1/3 − iω2k ζ −1/3 + O(ζ −5/3 ) 3 The value at the saddles is
as ζ → ∞.
3 k 4/3 b 2k 2/3 b2 ω ζ + O(ζ −2/3 ) + ω ζ + as ζ → ∞. 4 2 6 Then, if Ck is the steepest descent path through sk , we obtain from classical steepest descent arguments 2π − 14 s 4 − b2 s 2 +iζ s eθ(sk (ζ ;b);ζ,b) (1 + O(ζ −2/3 )) e ds = ± 2θ ∂ Ck − ∂s 2 (sk (ζ, b); ζ, b) 2π 2k −1/3 3 ωk ζ 4/3 + b ω2k ζ 2/3 + b2 2 6 (1 + O(ζ −2/3 )). iω ζ =± e4 3 θ (sk (ζ ; b); ζ, b) =
The choice of ± sign depends on the orientation of the steepest descent path. Now take any of the six sectors that appear in the definition (8.12)–(8.17) of and take some p j that appears in the definition of in that sector. The contour j in the definition (8.10) of p j can be deformed to the steepest descent contour through one of the saddles, or to the union of two or three such steepest descent contours. However, in the latter case, it turns out that there is always a unique dominant saddle for p j in that particular sector. Thus for some k and some choice of ± sign, we have 2π 2k −1/3 3 ωk ζ 4/3 + b ω2k ζ 2/3 + b2 2 6 (1 + O(ζ −2/3 )) iω ζ p j (ζ ) = ± e4 (8.24) 3 as ζ → ∞ in the chosen sector. Similarly, 2π 3 ωk ζ 4/3 + b ω2k ζ 2/3 + b2 2 6 (1 + O(ζ −2/3 )), ie 4 p j (ζ ) = ± 3 2π k 1/3 3 ωk ζ 4/3 + b ω2k ζ 2/3 + b2 2 6 (1 + O(ζ −2/3 )). iω ζ e 4 p j (ζ ) = ± 3
(8.25) (8.26)
A further analysis reveals which value of k and what sign is associated with p j in the particular sector. We will not go through this analysis here, but the result is given by (8.21)) and (8.22). This completes the proof of the lemma.
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Note that in the above lemma we only state the leading term in a full asymptotic expansion, which is enough for the purposes of this paper. We also stay away from situations where saddles coalesce. For more asymptotic results on Pearcey integrals in various regimes, see [4, 19, 21] and the references cited therein. 8.3. Definition of Q. We are going to define the local parametrix Q in the form Q(z) = E(z)(n 3/4 ζ (z); n 1/2 b(z))en (z) e−nz
2 /6
,
= diag(λ1 , λ2 , λ3 ),
(8.27)
where E is an analytic prefactor, z → ζ (z) is a conformal map from a neighborhood of 0 in the z-plane to a neighborhood of 0 in the ζ -plane, and z → b(z) is analytic. We choose ζ (z) and b(z) so that the exponential factors in the asymptotic behavior of 2 (n 3/4 ζ (z); n 1/2 b(z)) are cancelled when we multiply them by en (z) e−nz /6 . We use the functions f 3 and g3 from Lemma 4.1 in the following definition. These functions depend on a, and to emphasize the a-dependence we write f 3 (z; a) and g3 (z; a). The functions ζ (z) and b(z) also depend on a. Definition. For z in a sufficiently small neighborhood of 0, we define ζ (z) = ζ (z; a) = z [ f 3 (z; a)]3/4 ,
(8.28)
and b(z) = b(z; a) =
g3 (z; a) . f 3 (z; a)1/2
(8.29)
In (8.28) and (8.29) the branch of the fractional powers is chosen which is real and positive for real values of z near 0. Lemma 8.2. (a) There is an r > 0 and a δ > 0 so that for each a ∈ (1 − δ, 1 + δ) we have that z → ζ (z; a) is a conformal map on the disk D(0, r ) and z → b(z; a) is analytic on D(0, r ). (b) In addition we have b(z; a) = O(a − 1) + O(z 2 ) as a → 1 and z → 0.
(8.30)
Proof. Following the constructions of f j and g j for j = 1, 2, 3 in Lemmas 3.2, 3.3, and 4.1 and their proofs, we easily see that f 3 (z; a) = f 3 (z; 1) + O(a −1), g3 (z; a) = g3 (z; 1) + O(a −1), as a → 1,
(8.31)
uniformly for z in a neighborhood of 0, and f 3 (z; 1) = 1 + O(z 2 ), g3 (z; 1) = O(z 2 )
as z → 0.
(8.32)
Both parts of the lemma follow from (8.31) and (8.32), and the definitions (8.28) and (8.29). From now on we assume that |a − 1| < δ, where δ > 0 is as in part (a) of Lemma 8.2, so that z → ζ (z; a) is a conformal map. Near 0 we choose the precise form of the lenses so that the lips of the lenses are mapped by z → ζ (z; a) to the rays arg ζ = ±π/4 and arg ζ = ±3π/4. Then from the fact that the jump matrices (8.18)–(8.20) of agree with those in (8.4)–(8.9), it follows that the jump condition (8.1) for Q is satisfied. This holds for any choice of analytic prefactor E that is used in (8.27) to define Q. We are going to define E so that the matching condition (8.2) is satisfied as well.
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8.4. Matching condition. To obtain the matching condition (8.2) we first note that the definitions (8.28) and (8.29) give us (we drop the a-dependence in the notation) ζ (z)4/3 = z 4/3 f 3 (z),
b(z)ζ (z)2/3 = g3 (z).
Hence by (4.4) and (8.23) we have for Im z > 0 with |z| < r , θk (ζ (z); b(z)) + λk (z) − z 2 /6 = 0,
k = 1, 2, 3,
(8.33)
while for Im z < 0 with |z| < r , θ2 (ζ (z); b(z)) + λ1 (z) − z 2 /6 = λ1− (0) = πi, θ1 (ζ (z); b(z)) + λ2 (z) − z 2 /6 = λ2− (0) = −πi,
(8.34)
θ3 (ζ (z); b(z)) + λ3 (z) − z /6 = λ3− (0) = 0. 2
Assume a = 1 + O(n −1/2 ). Then it follows from (8.30) that n 1/2 |b(z; a)| ≤ C
for |z| ≤ 2n −1/4
(8.35)
for every n large enough, with a value C that is independent of n. As a consequence we can use the expansions (8.21), (8.22) as n → ∞, because of Lemma 8.1. We find from (8.21) and (8.22) and the relations (8.33) and (8.34) between θk and λk that the exponential factors in the asymptotic behavior (as n → ∞) of (n 3/4 ζ (z); n 1/2 b(z))en e−nz
2 /6
cancel if we take z so that 0.9n −1/4 ≤ |z| ≤ 1.1n −1/4 . So we have proved the following. Lemma 8.3. Let a = 1 + O(n −1/2 ). Then we have as n → ∞, uniformly for z so that 0.9n −1/4 ≤ |z| ≤ 1.1n −1/4 , that Q(z) = E(z)(n 3/4 ζ (z); n 1/2 b(z))en (z) e−nz /6 ⎛ −1/4 ⎞ n 0 0 2π nb(z)2 /8 ie = E(z) ⎝ 0 1 1 ⎠ K (ζ (z))(I + O(n −1/3 )), 3 0 0 n 1/4 2
where
⎞⎛ ⎞ ⎧ ⎛ −1/3 2 1 ζ 0 0 −ω ω ⎪ ⎪ ⎪ ⎝ 0 1 0 ⎠ ⎝ −1 1 1⎠ ⎪ ⎪ ⎪ ⎪ ⎨ 0 0 ζ 1/3 −ω2 ω 1 K (ζ ) = ⎛ ⎞ ⎞⎛ 2 ⎪ ⎪ ζ −1/3 0 0 ω ω 1 ⎪ ⎪⎝ ⎪ ⎪ 0 1 0 ⎠ ⎝ 1 1 1⎠ ⎪ ⎩ 0 0 ζ 1/3 ω ω2 1
(8.36)
for Im ζ > 0, (8.37) for Im ζ < 0.
Proof. This follows from the asymptotic behavior (8.21) and (8.22), since we have shown in the above that the exponential factors in (8.21) and (8.22) are cancelled when 2 we multiply them by en (z) e−nz /6 . As for the O-term, we note that n 3/4 ζ (z) = O(n 1/2 ) if |z| = cn −1/4 with 0.9 ≤ c ≤ 1.1, so that the O(ζ −2/3 ) term in (8.21)–(8.22) leads to the O(n −1/3 ) term in (8.36).
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In order to achieve the matching (8.2) of Q(z) with M(z) we now define the prefactor E by ⎛ 1/4 ⎞ 0 0 n 3 −nb(z)2 /8 E(z) = − ie M(z)K (ζ (z))−1 ⎝ 0 1 0 ⎠ . 2π 0 0 n −1/4
(8.38)
Then the matching condition (8.2) follows from (8.36) and (8.38). It only remains to check that E is analytic in a full neighborhood of the origin. This follows since M and K satisfy the same jump relations on the real line. Indeed we have from the expressions (8.37) for K , for real ζ with ζ > 0, ⎛ 2 ⎞−1 ⎛ ⎞ ⎛ ⎞ ω ω 1 −ω ω2 1 0 10 K − (ζ )−1 K + (ζ ) = ⎝ 1 1 1⎠ ⎝ −1 1 1⎠ = ⎝−1 0 0⎠ , 0 01 ω ω2 1 −ω2 ω 1 while for real ζ < 0 we have to take into account that ζ 1/3 and ζ −1/3 have different ±-boundary values, so that for ζ < 0, ⎛ 2 ω K − (ζ )−1 K + (ζ ) = ⎝ 1 ω ⎛ 2 ω =⎝1 ω
⎞⎛ ⎞−1 ⎛ 1/3 −1/3 ⎞ ω 1 0 0 ζ− ζ+ −ω ω2 1 ⎠ ⎝ −1 1 1⎠ 1 1⎠ ⎝ 0 1 0 −1/3 1/3 ω2 1 −ω2 ω 1 0 0 ζ− ζ+ ⎞⎛ ⎞−1 ⎛ ⎞ ⎛ ⎞ −ω ω2 1 ω 1 0 01 ω2 0 0 1 1⎠ ⎝ 0 1 0 ⎠ ⎝ −1 1 1⎠ = ⎝ 0 1 0⎠ . −1 0 0 0 0ω ω2 1 −ω2 ω 1
These are indeed equal to the jumps satisfied by M; see (6.1) and (6.2). Since ζ (z) is a conformal map on D(0, r ) that is real and positive for z ∈ (0, r ), and real and negative for z ∈ (−r, 0), we find that M(z)K (ζ (z))−1 is analytic across both (0, r ) and (−r, 0). Thus E(z) is analytic in D(0, r ) \ {0}. The isolated singularity at 0 is removable, since the entries in M(z) and K (ζ (z))−1 have at most z −1/3 -type singularity at the origin, and they cannot combine to form a pole. The conclusion is that E is analytic. This completes the construction of the local parametrix Q at the origin.
9. Final Transformation We now fix b ∈ R and let a = 1 +
b √ . 2 n
Now we define
⎧ √ ⎪ S(z)M(z)−1 , for z ∈ C\ S outside the disks D(0, n −1/4 ) and D(± 3 2 3 , r ), ⎪ ⎪ ⎪ ⎨ √ R(z) = S(z)P(z)−1 , for z ∈ D(± 3 3 , r ) \ S , 2 ⎪ ⎪ ⎪ ⎪ ⎩ S(z)Q(z)−1 , for z ∈ D(0, n −1/4 ) \ S . (9.1) Then R is analytic inside the disks and also across the real interval between the disks. Thus R is analytic outside the contour R shown in Fig. 9.
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• 0
505
•* z
Fig. 9. The contour R . The matrix-valued function R is analytic on C \ R . The disk around 0 has radius n −1/4 and is shrinking as n → ∞. The disks are oriented counterclockwise and the remaining parts of R are oriented from left to right
Lemma 9.1. We have R+ = R− j R where j R (z) = I + O(n
−1
)
√ 3 3 uniformly for z ∓ = r, 2
j R (z) = I + O(n −1/6 ) uniformly for |z| = n −1/4 , and there exists c > 0 so that −cn 2/3 j R (z) = I + O e1+|z|2
(9.2) (9.3)
uniformly for z on the remaining parts of R . (9.4) √
Proof. The behavior (9.2) of the jump matrix on the circles around the endpoints ± 3 2 3 is a result of the construction of the Airy parametrix. It follows as in [16, 17]. The jump matrix for |z| = n −1/4 is by (9.1) and (8.2) (we use positive orientation) j R = M Q−1 = M(I + O(n −1/3 ))M −1 = M + M O(n −1/3 )M −1 (z). Since M(z) = O(z −1/3 ) and M −1 (z) = O(z −1/3 ) as z → 0 by (6.6) and (6.7), we obtain (9.3). The jump matrix j R (z) on √ the remaining part of R is I + O(e−cn ) if z ∈ R stays at a fixed distance of 0 and ± 3 2 3 . But now the disk around 0 is shrinking as n increases, and so we have to be more careful here. We note that the jump matrix is j R (z) = M(z) jS (z)M −1 (z) and we want to know its behavior as n → ∞ for z on the lips of the lenses near 0 and |z| ≥ n −1/4 . The jump matrices jS in (5.14)–(5.17) contain off-diagonal entries ±en(λk −λ j ) . For a = 1 these entries are decaying on the contours and so we have for some positive constant c1 > 0, Re ((λ j − λk )(z; 1)) ≥ c1 |z|4/3 for z on the lips of the lenses near 0. Since λ j (z; a) = λ j (z; 1) + z 2/3 O(a − 1) as a → 1, we then get that Re ((λ j − λk )(z; a)) ≥ c1 z −4/3 − c2 |z|2/3 |a − 1|. Then if a − 1 = (b/2)n −1/2 and |z| ≥ n −1/4 we easily get that Re (λ j − λk )(z; a)) ≥ c3 n −1/3 for some positive constant c3 > 0. Then it follows from (5.14)–(5.17) that jS (z) = I + O(e−c3 n
2/3
).
This leads to (9.4) since M(z) = O(z −1/3 ) and M −1 (z) = O(z −1/3 ) as z → 0, see (6.6) and (6.7).
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To summarize, we find that R solves the following RH problem: • R is analytic on C \ R , • for z ∈ R , we have R+ = R− j R , where j R satisfies (9.2)–(9.4), • as z → ∞, we have R(z) = I + O(1/z). The RH problem for R is posed on a contour that is varying with n. This is a slight complication. However we still can guarantee the following behavior of R as n → ∞. Proposition 9.2. As n → ∞ we have that R(z) = I + O
n −1/6 1 + |z|
(9.5)
uniformly for z ∈ C \ R . Since the proof of Proposition 9.2 is somewhat technical due to the fact that the contours are varying with n, we give it in Appendix A. 10. Proof of Theorem 1.1 Now we are ready for the proof of Theorem 1.1. We fix b ∈ R and take b a =1+ √ . 2 n 10.1. The effect of the transformations Y → T → S → R. We are going to follow the effect of the transformations on the correlation kernel K n (x, y; a) for real values of x and y close to 0. We start from (2.3) which gives K n (x, y; a) in terms of the solution of the RH problem for Y . The transformation (5.1) then implies that ⎞ ⎛ 1 2 2 e−nλ1+ (x) e 4 n(x −y ) nλ2+ (y) nλ3+ (y) −1 ⎠ . (10.1) K n (x, y; a) = T+ (y)T+ (x) ⎝ 0e e 0 2πi(x − y) 0 According to the transformation T → S given in (5.7)–(5.10) we now have to distinguish between x and y being positive or negative. We will do the calculations explicitly for x > 0 and y < 0. The other cases are treated in the same way. So we assume that x > 0 and y < 0, and both of them are close to 0. The formulas (5.7) and (5.9) applied to (10.1) then give ⎛ −nλ (x) ⎞ 1 2 2 e 1+ e 4 n(x −y ) nλ1+ (y) −1 nλ (y) 3+ ⎝ K n (x, y; a) = S+ (y)S+ (x) e−nλ2+ (x) ⎠ . (10.2) −e 0e 2πi(x − y) 0 Next we note that for z close to 0, inside the disk or radius n −1/4 , we have by (9.1), n (z) −nz S(z) = R(z)Q(z) = R(z) Q(z)e e
2 /6
,
where = Q(z)e−n (z) enz 2 /6 = E(z)(n 3/4 ζ (z; a); n 1/2 b(z; a)), Q(z)
(10.3)
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see (8.36). Thus if 0 < x < n −1/4 and −n −1/4 < y < 0, we have ⎛ ⎞ ⎛ −nλ (x) ⎞ 1 e 1+ ⎝1⎠ e−nx 2 /6 S+ (x) ⎝e−nλ2+ (x) ⎠ = R(x) Q(x) 0 0 and
nλ (y)
−1 2 (y)R −1 (y). −e 1+ 0 enλ3+ (y) S+−1 (y) = eny /6 −1 0 1 Q Inserting these two relations into (10.2) we find that ⎛ ⎞ 1 2 2 1 −1 e 12 n(x −y ) + (y)R −1 (y)R(x) Q + (x) ⎝1⎠ . (10.4) −1 0 1 Q K n (x, y; a) = 2πi(x − y) 0 To obtain the scaling limit (1.6) of K n we need the following lemma. Lemma 10.1. Let an = 1 + (b/2)n −1/2 . (a) Let xn = xn −3/4 , where x ∈ R is fixed. Then lim n 3/4 ζ (xn ; an ) = x
(10.5)
lim n 1/2 b(xn ; an ) = b.
(10.6)
n→∞
and n→∞
(b) Let also yn = yn −3/4 where y ∈ R is fixed. Then lim E −1 (yn )R −1 (yn )R(xn )E(xn ) = I
n→∞
(10.7)
Proof. (a) Since ζ (z; a) = z [ f 3 (z; a)]3/4 by (8.28) and f 3 (z; a) → 1 if z → 0 and a → 1 by (8.31) and (8.32), we get that the limit (10.5) immediately follows. For (10.6) we need to go back to the definitions in Lemmas 3.2, 3.3, and 4.1 of f 1 and g j , for j = 1, 2, 3. From 3.2 and its proof it follows that f 1 (z; a) = f 1 (0; a) + O(z 2 ) and g1 (z; a) = g1 (0) + O(z 2 ) as z → 0, and the O-terms are uniform with respect to a in a neighborhood of 1. Then by (3.13) we have g2 (z; a) = g2 (0; a) + O(z 2 )
as z → 0
(10.8)
uniformly for a in a neighborhood of 1. Since we have, cf. Lemmas 3.3 and 4.1, z 1 2/3 z g3 (z; a) = s −1/3 g2 (s; a)ds, 2 0 we get from (10.8) that g3 (z; a) = g3 (0; a) + O(z 2 )
as z → 0
(10.9)
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−2/3 (c2 −1), again uniformly for √ a in a neighborhood of 1. By (4.5) we have g3 (0; a) = 3c 2 where c = (a + a 2 + 8)/4 = (a − 1)/3 + O((a − 1) ) as a → 1. Thus g3 (0; a) = 2(a − 1) + O((a − 1)2 ) as a → 1, and it follows from (10.9) and the definitions of an and xn that
n 1/2 g3 (xn ; an ) = n 1/2 g3 (0; an ) + O(n −1 ) = 2(an − 1) + O(n −1/2 ) = b + O(n −1/2 ) as n → ∞. Then (10.6) follows because of the definition (8.29) and the fact that f 3 (xn ; an ) → 1 as n → ∞. (b) Since M(z)K (ζ (z))−1 is analytic in a neighborhood of the origin and xn − yn = O(n −3/4 ), we have K (ζ (yn ))M(yn )−1 M(xn )K (ζ (xn ))−1 = I + O(n −3/4 ) as n → ∞. Hence by (8.38), ⎛ −1/4 ⎞ 0 0 n n ;an ⎝ 0 1 0 ⎠ E −1 (yn )E(xn ) = en(b(yn ;an 0 0 n 1/4 ⎛ 1/4 ⎞ 0 0 n ×(I + O(n −3/4 )) ⎝ 0 1 0 ⎠ 0 0 n −1/4 )2 −b(x
= en(b(yn ;an )
2 −b(x
)2 )/8
n ;an )
2 )/8
(I + O(n −1/4 )).
(10.10)
Note that both nb(yn ; an )2 and nb(xn ; an )2 tend to b2 as n → ∞ because of (10.6). Thus we also get from (8.38) that E(xn ) = O(n 1/4 ),
and
E −1 (yn ) = O(n 1/4 ).
(10.11)
Next, we get from (9.5) and Cauchy’s theorem that for z = O(n −3/4 ) we have 1 R(s) d R(z) = ds = O(n −1/6 ) as n → ∞. dz 2πi |s|=n −1/4 s − z Then by the mean-value theorem, R(xn ) − R(yn ) = O((xn − yn )n −1/6 ) = O(n −11/12 ) so that R −1 (yn )R(xn ) = I + R −1 (yn )(R(xn ) − R(yn )) = I + O(n −11/12 ).
(10.12)
Combining (10.6), (10.10), (10.11) and (10.12) we obtain (10.7). Now we can compute the double scaling limit of K n . Indeed, it follows from (10.3), (10.4), and Lemma 10.1 that x 1 y b = K cusp (x, y; b), Kn , ;1 + √ (10.13) lim n→∞ n 3/4 n 3/4 n 3/4 2 n
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where K cusp (x, y; b) =
1 −1 0 1 2πi(x − y) ⎛ ⎞ 1 ⎝ ⎠ if x > 0 and y < 0. (10.14) ×−1 + (y; b)+ (x; b) 1 0
Similar calculations show that the limit (10.13) exists for all x and y, and
1 −1 1 0 2πi(x − y) ⎛ ⎞ 1 ⎝ ⎠ if x > 0 and y > 0, (10.15) ×−1 + (y; b)+ (x; b) 1 0
1 −1 1 0 K cusp (x, y; b) = 2πi(x − y) ⎛ ⎞ 1 ⎝ ⎠ 0 ×−1 (y; b) (x; b) if x < 0 and y > 0, (10.16) + + 1
1 −1 0 1 K cusp (x, y; b) = 2πi(x − y) ⎛ ⎞ 1 ⎝ ⎠ ×−1 if x < 0 and y < 0. (10.17) + (y; b)+ (x; b) 0 1 K cusp (x, y; b) =
10.2. Different formula for K cusp . To complete the proof of Theorem 1.1 we show that the formulas (10.14)–(10.17) for K cusp can be rewritten in the form (1.7) given in the theorem. This involves the Pearcey integrals p(x) and q(y) of (1.8). Define ⎛ ⎞ p0 p1 p4 = ⎝ p0 p1 p4 ⎠ . (10.18) p0 p1 p4 (ζ ) agrees with (ζ ) in the sector π/4 < arg ζ < 3π/4, Then by (8.13) we have that in the full complex ζ -plane, and in particular on the real axis. but (10.18) defines Using the jump relation + = − j for arg ζ = π/4 and arg ζ = 3π/4, see (8.18) and (8.19), we find that ⎛ ⎞ 1 0 0 (x; b) ⎝−1 1 −1⎠ + (x; b) = if x > 0 0 0 1 and
⎛
⎞ 1 0 0 (x; b) ⎝ 0 1 0⎠ + (x; b) = −1 −1 1
if x < 0.
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Inserting this into (10.14)–(10.17) we find that all four cases lead to
⎛ ⎞ 1
1 −1 (y; b) (x; b) ⎝0⎠ 011 K cusp (x, y; b) = 2πi(x − y) 0
(10.19)
which is the same expression for all x, y ∈ R. −1 . The inverse of is built out of solutions of Our next task is to compute q (z) = −zq(z) + bq (z).
(10.20)
It is easy to see that for any solution q of (10.20) and any solution p of the Pearcey equation p (z) = zp(z) + bp (z)
have pq − p q + p q − bpq = 0 so that
(10.21)
[ p, q] := pq − p q + p q − bpq = const.
−1 has the form q − bq −q q for some particular It follows that each row of is given by (10.18), we have solution of (10.20). More precisely, since ⎛ ⎞ q1 − bq1 −q1 q1 −1 = ⎝q2 − bq2 −q2 q2 ⎠ , (10.22) q3 − bq3 −q3 q3 where [ p0 , q1 ] = 1, [ p1 , q1 ] = 0, [ p4 , q1 ] = 0, [ p0 , q2 ] = 0, [ p1 , q2 ] = 1, [ p4 , q2 ] = 0, [ p0 , q3 ] = 0, [ p1 , q3 ] = 0, [ p4 , q3 ] = 0.
(10.23)
Then if q0 = q2 + q3 we have [ p0 , q0 ] = 0, [ p1 , q0 ] = 1, [ p4 , q0 ] = 1,
(10.24)
and from (10.18), (10.19), and (10.22) it follows that K cusp (x, y; b) =
p0 (x)q0 (y) − p0 (x)q0 (y)+ p0 (x)q0 (y) − bp0 (x)q0 (y) .(10.25) 2πi(x − y)
Recall that (10.21) has solutions with integral representations 1 4 b 2 p(z) = e− 4 s − 2 s +isz ds,
(10.26)
where is a contour in the complex plane that starts and ends at infinity at one of the angles 0, ±π/2, or π . Similarly, there are solutions of (10.20) with integral representation 1 4 b 2 1 q(z) = e 4 t + 2 t +it z dt, (10.27) 2πi where is a contour in the complex plane that starts and ends at infinity at one of the angles ±π/4 or ±3π/4.
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Lemma 10.2. Let p and q be given by (10.26) and (10.27) such that ∩ = ∅. Then [ p, q] = 0. If ∩ = {z 0 } and and intersect transversally at z 0 , and if the contours are oriented so that meets in z 0 on the −-side of , then [ p, q] = 1. Proof. We write [ p, q] = pq − p q + p q − bpq as a double integral, and for convenience we take z = 0. So from (10.26) and (10.27), 1 4 b 2 1 4 b 2 1 [ p, q] = (−t 2 − st − s 2 − b)e 4 t + 2 t − 4 s − 2 s dsdt 2πi 3 t + bt − s 3 − bs 1 t 4 + b t 2 − 1 s 4 − b s 2 1 2 dsdt. e4 2 4 = 2πi s−t If ∩ = ∅ then we can write this as 1 1 t 4+ b t 2 ∂ − 1 s4− b s2 1 2 e 4 dsdt [ p, q] = e4 2 2πi s − t ∂s 1 − 1 s4− b s2 ∂ 1 t 4+ b t 2 1 2 e 4 e 4 2 dtds + 2πi s − t ∂t and we can apply integration by parts to both inner integrals. The integrated terms vanish because of the choice of contours and the result is 1 4 b 2 1 4 b 2 ∂ 1 1 dsdt e4t +2t −4s −2s [ p, q] = − 2πi ∂s s − t 1 4 b 2 1 4 b 2 ∂ 1 1 dtds = 0. − e4t +2t −4s −2s 2πi ∂t s − t If ∩ = ∅ then we cannot make the splitting of integrals as above, and we have to proceed differently. If and intersect at z 0 as in the statement of the second part of the lemma, then we can deform contours so that and intersect in 0, and that for some δ > 0, contains the real interval [−δ, δ] oriented from left to right, and contains the vertical interval [−iδ, iδ] oriented from bottom to top. Let ε ∈ (0, δ) and write ε = \ (−ε, ε). Then it follows as above that 1 4 b 2 1 4 b 2 1 [ p, q] = lim (−t 2 + st − s 2 − b)e 4 t + 2 t − 4 s − 2 s dsdt ε→0 2πi ε 1 1 1 t 4+ b t 2 ∂ − 1 s4− b s2 2 e4 2 e 4 = lim dsdt ε→0 2πi ε s − t ∂s 1 1 − 1 s4− b s2 ∂ 1 t 4+ b t 2 2 + e 4 e 4 2 dtds . 2πi ε s − t ∂t If we now do an integration by parts, integrated terms at ±ε appear from the second double integral. The other terms vanish and the result is 1 4 b 2 1 4 b 2 1 1 1 [ p, q] = lim − e− 4 s − 2 s e 4 ε + 2 ε ds. ε→0 2πi s + ε s−ε
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Now we deform so that instead of the vertical segment [−iδ, iδ] it contains the semicircle |s| = δ, Re s > 0. Then we pick up a residue contribution from s = ε which is equal to 1. The remaining integral vanishes in the limit ε → 0, so that we find [ p, q] = 1, as claimed by the lemma. −1 explicitly. We claim that for j = 1, 2, 3, Lemma 10.2 allows us to compute 1 4 b 2 1 q j (z) = e 4 t + 2 t +it z dt, (10.28) 2πi j where 1 is a contour in the left half-plane from e−3πi/4 ∞ to e3πi/4 ∞, 2 is a contour in the upper half-plane from eπi/4 ∞ to e3π/4 ∞, and 3 is a contour in the lower halfplane from e−3πi/4 ∞ to e−πi/4 ∞. Indeed, with these contours j , and taking note of the definition and orientation of 0 , 1 , and 4 in (8.11), we easily get from Lemma 10.2 that the relations (10.23) hold. Thus for q0 = q2 + q3 we find that q0 = −iq, where q is defined as in (1.8). Since p0 = 2π p, it is then easy to check that the formula (10.25) for the kernel is equivalent to the formula (1.7) in the statement of the theorem. This completes the proof of Theorem 1.1. Appendix A. Proof of Proposition 9.2 Let R be the contour depicted on Fig. 9, with orientation from the left to the right and in the positive direction on the circles. As usual, we will assume that the minus side of the contour is on the right. By a simple arc on R we will mean a connected, relatively open, with respect to R , subset 0R ⊂ R , which does not contain any triple point of R , a point where three curves meet. By L 2 ( R ) we will mean, as usual, the space of measurable functions with f 2 =
1 | f | |dz| 2
R
2
< ∞.
(A.1)
We have the following general proposition. Proposition A.1. Suppose that a 3 × 3 matrix-valued function v(z), z ∈ R , belongs to L 2 ( R ) and it is Lipschitz on some simple arc 0R ⊂ R . Suppose also that on 0R , v(z) solves the equation v(s) j R0 (s) 1 ds, z ∈ 0R , (A.2) v(z) = I − 2πi R z − − s where z − means the value of the limit of the integral from the minus side, and j R = I + j R0 . Then v(s) j R0 (s) 1 ds, z ∈ C \ R , R(z) = I − (A.3) 2πi R z − s satisfies on 0R the jump condition, R+ (z) = R− (z) j R (z),
z ∈ 0R .
(A.4)
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Proof. From (A.2), (A.3), R− (z) = v(z), z ∈ R .
(A.5)
By the jump property of the Cauchy transform, R+ (z) − R− (z) = v(z) j R0 (z) = R− (z) j R0 (z),
(A.6)
hence R+ (z) = R− (z) j R (z). Proposition A.1 is proved. We will solve Eq. (A.2) by the series, v(z) = v0 (z) + v1 (z) + v2 (z) + . . . ,
(A.7)
where v0 (z) = I ;
v j (z) = −
1 2πi
R
v j−1 (s) j R0 (s) ds, z ∈ R , z− − s
j ≥ 1. (A.8)
We will inductively estimate v j (z). We begin with some general definitions and results. Introduce the operators v(s) 1 C± v(z) = − ds, z ∈ , (A.9) 2πi z ± − s where is a contour on the complex plane. We assume that v is Lipschitz and L 2 integrable if is unbounded. We have that C+ − C− = Id and C+ + C− = C = −
1 v.p. πi
(A.10)
v(s) ds, z ∈ . z−s
(A.11)
Suppose that the contour is given by the parametric equations, = {x = t, y = ϕ(t), −∞ < t < ∞},
(A.12)
where ϕ is uniformly Lipschitz, so that there exists M ≥ 0 such that |ϕ(x) − ϕ(y)| ≤ M|x − y|.
(A.13)
Then as shown in [13], there exists an absolute constant K 0 such that C f 2 ≤ K 0 (1 + M)10 f 2 ,
(A.14)
where f 2 =
1 | f | |dz| 2
2
.
(A.15)
This implies similar estimates for C± . If 0 ⊂ then C0 = PC P,
P f = χ0 f,
(A.16)
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hence C0 2 ≤ C 2 .
(A.17)
Therefore, estimate (A.14) holds, with the same constant, for any contour = {x = t, y = ϕ(t), a < t < b}.
(A.18)
Furthermore, it holds, with the same constant, for any complex linear transformation of contour (A.18). Let us denote by G M the set of all contours which can be obtained by a complex linear transformation from a contour (A.18), where ϕ satisfies (A.13) and is differentiable. Observe that any interval of a straight line belongs to G0 , and any circular arc of angular measure less than or equal to π2 belongs to G1 . Suppose now that = 1 ∪ . . . ∪ m is a piecewise contour such that (1) j belongs to G M , j = 1, . . . , m; (2) the closed contours, j and k , j = k, can intersect only at their end-points; (3) if j and k intersect then the angle between them at the intersection point is positive,
( j , k ) > ε > 0;
(A.19)
(4) if j and k , j = k, are two infinite contour then they “well diverge” at infinity, so that there exists a constant c > 0 such that |γ j (s) − γk (t)| ≥ c(|s| + |t|),
(A.20)
where γ j , γk are the parametric equations of the contours j , k , induced by parametrization (A.12). Theorem A.2. If is a piecewise contour which satisfies Conditions (1)–(4), then C is bounded in L 2 , and C 2 is estimated from above by a constant which depends only on the Lipschitz constants M j of the contours j , j = 1, . . . , m, and on the constants ε and c of conditions (A.19), (A.20). Proof. We have to prove that for some K 1 > 0, |(C f, g)| ≤ K 1 f 2 g2 .
(A.21)
To that end, it is sufficient to prove that for some K 2 > 0, |(C (χ j f ), χk g)| ≤ K 2 f 2 g2 ,
1 ≤ j, k ≤ m.
(A.22)
For j = k, it follows from estimate (A.13) applied to a linear transformation of j . For j = k it follows from (A.19), (A.20), and the estimate, 1 ∞ ∞ | f (s)g(t)|dsdt ≤ f 2 g2 . (A.23) π 0 s+t 0 Theorem A.2 is proved.
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When applied to the contour R , Theorem A.2 gives that there exists a constant K , independent of n, such that C R 2 ≤ K .
(A.24)
By (A.10), (A.11) this implies that ± ≤ K0 = C R 2
K +1 . 2
(A.25)
From (A.8) we have that − ( j R0 v j−1 ). v j = C R
j ≥ 1.
(A.26)
j R0 v j−1 2 ≤ j R0 C v j−1 2
(A.27)
Since
and 1
j R0 C = sup | j R0 (z)| ≤ K 1 n − 6 ,
(A.28)
z∈ R
we obtain the recursive estimate, 1
v j 2 ≤ K n − 6 v j−1 2 ,
K = K0 K1.
(A.29)
For v1 we have that − ( j R0 )2 ≤ K 0 j R0 2 ≤ K 3 n − 6 − 8 . v1 2 = C R 1
1
(A.30)
Thus, 1
1
v j 2 ≤ K 3 (K n − 6 ) j n − 8 .
(A.31)
This implies the convergence of series (A.7) in L 2 , for large n. Let us discuss analytic properties of the functions v j . Denote R =
16
lR ,
(A.32)
l=1
the partition of the contour R (see Fig. 9) into 16 simple arcs. Fix any ε > 0. Let z 0 be any point on lR such that the distance from z 0 to the end-points of lR is bigger than 1
εn = εn − 4 .
(A.33)
The function j R0 (z) can be analytically continued from lR to the εn -neighborhood of the point z 0 , D(z 0 , εn ) = {z : dist(z, z 0 ) < εn }.
(A.34)
This implies that 1 v1 (z) = − 2πi
R
j R0 (s) ds z− − s
(A.35)
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can be also analytically continued from lR to D(z 0 , εn ), because we can deform the contour of integration, R . Then, inductively, we can analytically continue v j (z) from lR to D(z 0 , εn ), by deforming the contour of integration in (A.8). Observe that on the deformed contour we have the L 2 -estimate, (A.31), hence by the Cauchy-Schwarz inequality we obtain that 1
1
1
|v j (z)| ≤ K 4 ε−1 n 4 (K n − 6 )( j−1) n − 8 ,
z ∈ D(z 0 , εn /2).
(A.36)
This proves the convergence of series (A.7) in the neighborhood D(z 0 , εn /2) to an analytic v(z). Thus, v(z) is analytic on R outside of the triple points. Observe that the function R(z) defined by formula (A.3), denote it for a moment ˜ ˜ R(z), coincides with R(z) defined by (9.1). Indeed, both R(z) and R(z) solve the same RH problem, and if z 0 is any triple point of R , then in some neighborhood of z 0 , ˜ | R(z)| ≤ C|z − z 0 |− 2 , 1
|R(z)| ≤ C, (A.37) ˜ for some C > 0. For R(z) it follows from (A.3) by the Cauchy-Schwarz inequality, and for R(z) it is obvious from (9.1). If we consider now −1 ˜ , (A.38) X (z) = R(z)R(z) then X (z) has no jumps on R and in a neighborhood of the triple points it satisfies the estimate 1
|X (z)| ≤ C|z − z 0 |− 2 .
(A.39)
Therefore, the triple points are removable singularities and X (z) is analytic on C. Also, ˜ X (∞) = I , hence X (z) = I everywhere on C, and R(z) = R(z). Now we can estimate R(z). From (A.3), ∞ v j (s) j R0 (s) 1 ds. (A.40) R(z) = I + R j (z), R j (z) = − 2πi R z−s j=0
Suppose that 1
dist(z, R ) > 0.1n − 4 . Then, 1 |R0 (z)| = 2π
(A.41)
1 j R0 (s) K 0 n − 6 ds ≤ , R z − s 1 + |z|
(A.42)
and by (A.31), 1
1
1
1
1
K 1 (K n − 6 ) j n − 24 K 1 n 4 (K n − 6 ) j+1 n − 8 |R j (z)| ≤ = , 1 + |z| 1 + |z| By summing all these inequalities from j = 0 to ∞, we obtain that ! 1 1 n− 6 R(z) = I + O , dist(z, R ) > 0.1n − 4 . 1 + |z| 1
j ≥ 1.
(A.43)
(A.44)
In fact, the restriction dist(z, R ) > 0.1n − 4 is not essential, because we can deform the contour R . This completes the proof of Proposition 9.2.
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References 1. Adler, M., van Moerbeke, P.: PDE’s for the Gaussian ensemble with external source and the Pearcey distribution. http://arxiv.org/list/math.PR/0509047, 2005 2. Aptekarev, A.I., Bleher, P.M., Kuijlaars, A.B.J.: Large n limit of Gaussian random matrices with external source, Part II. Commun. Math. Phys. 259, 367–389 (2005) 3. Aptekarev, A.I., Branquinho, A., Van Assche, W.: Multiple orthogonal polynomials for classical weights. Trans. Amer. Math. Soc. 355, 3887–3914 (2003) 4. Berry, M.V., Howls, C.J.: Hyperasymptotics for integrals with saddles. Proc. Roy. Soc. London Ser. A 434, 657–675 (1991) 5. Bleher, P., Its, A.: Double scaling limit in the random matrix model. The Riemann-Hilbert approach. Commun. Pure Appl. Math. 56, 433–516 (2003) 6. Bleher, P.M., Kuijlaars, A.B.J.: Random matrices with external source and multiple orthogonal polynomials. Int. Math. Research Notices 2004, no. 3, 109–129 (2004) 7. Bleher, P.M., Kuijlaars, A.B.J.: Large n limit of Gaussian random matrices with external source, Part I. Commun. Math. Phys. 252, 43–76 (2004) 8. Bleher, P.M., Kuijlaars, A.B.J.: Integral representations for multiple Hermite and multiple Laguerre polynomials. Ann. Inst. Fourier 55, 2001–2004 (2005) 9. Brézin, E., Hikami, S.: Universal singularity at the closure of a gap in a random matrix theory. Phys. Rev. E 57, 4140–4149 (1998) 10. Brézin, E., Hikami, S.: Level spacing of random matrices in an external source. Phys. Rev. E 58, 7176– 7185 (1998) 11. Claeys, T., Kuijlaars, A.B.J.: Universality of the double scaling limit in random matrix models. Commun. Pure Appl. Math. 59, 1573–1603 (2006) 12. Claeys, T., Kuijlaars, A.B.J., Vanlessen, M.: Multi-critical unitary random matrix ensembles and the general Painlevé II equation. http://arxiv.org/list/math-ph/0508062, 2005, to appear in Ann. Math. 13. Coifman, R.R., McIntosh, A., Meyer, Y.: L’intégrale de Cauchy définit un opérateur borné sur L 2 pour les courbes Lipschitziennes. Ann. Math. 116, 361–387 (1982) 14. Daems, E., Kuijlaars, A.B.J.: A Christoffel-Darboux formula for multiple orthogonal polynomials. J. Approx. Theory 130, 190–202 (2004) 15. Deift, P.: Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert approach, Courant Lecture Notes in Mathematics, Vol. 3, Providence R.I.: Amer. Math. Soc. (1999) 16. Deift, P., Kriecherbauer, T., McLaughlin, K.T-R, Venakides, S., Zhou, X.: Uniform asymptotics of polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory. Commun. Pure Appl. Math. 52, 1335–1425 (1999) 17. Deift, P., Kriecherbauer, T., McLaughlin, K.T-R, Venakides, S., Zhou, X.: Strong asymptotics of orthogonal polynomials with respect to exponential weights. Commun. Pure Appl. Math. 52, 1491–1552 (1999) 18. Deift, P., Zhou, X.: A steepest descent method for oscillatory Riemann-Hilbert problems. Asymptotics for the MKdV equation. Ann. Math. 137, 295–368 (1993) 19. Miyamoto, T.: On an Airy function of two variables. Nonlinear Anal. 54, 755–772 (2003) 20. Okounkov, A., Reshetikhin, N.: Random skew plane partitions and the Pearcey process. http:// arxiv.org/list/math.CO/0503508, 2005 21. Paris, R.B., Kaminski, D.: Asymptotics and Mellin-Barnes integrals. Cambridge: Cambridge University Press (2001) 22. Pastur, L.: The spectrum of random matrices (Russian). Teoret. Mat. Fiz. 10, 102–112 (1972) 23. Pearcey, T.: The structure of an electromagnetic field in the neighborhood of a cusp of a caustic. Philos. Mag. 37, 311–317 (1946) 24. Saff, E.B., Totik, V.: Logarithmic Potentials with External Field. Berlin-Hedielberg-NewYork: SpringerVerlag (1997) 25. Tracy, C., Widom, H.: The Pearcey process. Commun. Math. Phys. 263, 381–400 (2006) 26. Van Assche, W., Coussement, E.: Some classical multiple orthogonal polynomials. J. Comput. Appl. Math. 127, 317–347 (2001) 27. Van Assche, W., Geronimo, J., Kuijlaars, A.B.J.: Riemann-Hilbert problems for multiple orthogonal polynomials. In: Special Functions 2000, J. Bustoz et al., eds., Dordrecht: Kluwer, 2001, pp. 23–59 28. Zinn-Justin, P.: Random Hermitian matrices in an external field. Nucl. Phys. B 497, 725–732 (1997) Communicated by H. Spohn
Commun. Math. Phys. 270, 519–544 (2006) Digital Object Identifier (DOI) 10.1007/s00220-006-0161-7
Communications in
Mathematical Physics
Hausdorff Dimension for Randomly Perturbed Self Affine Attractors Thomas Jordan1 , Mark Pollicott1 , Károly Simon2 1 Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK.
E-mail:
[email protected],
[email protected]
2 Institute of Mathematics, Technical University of Budapest, H-1529 B.O. box 91, Budapest, Hungary.
E-mail:
[email protected] Received: 28 February 2006 / Accepted: 2 August 2006 Published online: 15 December 2006 – © Springer-Verlag 2006
Abstract: In this paper we shall consider a self-affine iterated function system in Rd , d ≥ 2, where we allow a small random translation at each application of the contractions. We compute the dimension of a typical attractor of the resulting random iterated function system, complementing a famous deterministic result of Falconer, which necessarily requires restrictions on the norms of the contraction. However, our result has the advantage that we do not need to impose any additional assumptions on the norms. This is of benefit in practical applications, where such perturbations would correspond to the effect of random noise. We also give analogous results for the dimension of ergodic measures (in terms of their Lyapunov dimension). Finally, we apply our method to a problem originating in the theory of fractal image compression.
1. Introduction In this article we consider families of self-affine Iterated Function Systems (IFS) defined on Rd . More precisely, we consider contractions m F := { f i (x) = Ai · x + ti }i=1 ,
(1)
for x ∈ Rd , where the Ai are d × d non-singular matrices satisfying 0 < Ai < < 1, ∀1 ≤ i ≤ m,
(2)
and the vectors ti are in Rd . The following definition is standard. Research of Jordan and Pollicott was supported by the EPSRC and the research of Simon was supported by an EU-Marie Curie grant and the OTKA Foundation #T42496.
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Definition 1.1 (The attractor of F ). Let B be a ball in Rd centered at the origin with radius larger than max1≤i≤m ti /(1 − ). Then the attractor of F is defined by: :=
∞
f i0 ...in−1 (B),
(3)
n=1 i 0 ...i n−1
where we denote f i0 ...in−1 = f i0 ◦ · · · ◦ f in−1 . It is easy to see that this definition does not depend on the choice of B and that is the unique non-empty compact set for which =
m
f i ().
i=1
Since we use the translations as parameters it is sometimes convenient to explicitly show the dependence by writing F t and f it , instead of F and f i , respectively. We let dimH t and dimB t denote the Hausdorff dimension and Box dimension of the attractor, respectively. (We refer the reader to [4] for the definitions.) The following result is due to Falconer [3] and Solomyak [17]. Theorem 1.2 (Falconer, Solomyak). If Ai <
1 , for all 1 ≤ i ≤ m, 2
(4)
then for Lebm·d -almost all vectors t := (t1 , . . . , tm ) ∈ Rm·d the dimension of the attracm f t (t ) is tor t = ∪i=1 i dimH t = dimB t = min {d, s(A1 , . . . , Am )} ,
(5)
where s(A1 , . . . , Am ) is the singularity dimension (see [3, Prop. 4.1] and Definition 1.3 below.) Theorem 1.2 was originally proved by Falconer in 1988 under the stronger hypothesis that Ai < 13 , for all 1 ≤ i ≤ m. Ten years later Solomyak weakened the hypotheses to their present form. Previously, Edgar [1] had already observed that the bound in (4) is optimal. Moreover, it was shown by Simon and Solomyak in [16] that the bound 21 in (4) cannot be improved even in the special case that all the maps f i are similarities. Finally, the “almost all” hypothesis is necessary, in light of the construction by Bedford and McMullen of sporadic examples where the equality (5) fails even when the norms are smaller than 21 [12]. If the matrices Ai are orthogonal then the singularity dimension is simply the usual similarity dimension, while in the general case it is defined in terms of the singular values of repeated products of the matrices A1 , . . . , Am . More precisely, let T be a non-singular linear mapping from Rd to Rd . The singular values α1 ≥ α2 ≥ · · · ≥ αd of T are the positive square roots of the eigenvalues of the positive definite symmetric matrix T ∗ T . The singular value function φ s (T ) is defined for s > 0 by s−(k−1) , if k − 1 < s ≤ k ≤ d; α1 · · · αk−1 αk φ s (T ) := s/d if s ≥ d. (α1 · · · αd ) , We can now present the definition of the singularity dimension.
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Definition 1.3 (Singularity dimension). For a contracting self-affine IFS defined by (1) the singularity dimension is ⎧ ⎫ ∞ ⎨ ⎬ φ t (Ai0 · · · Ain ) < ∞ . (6) s(A1 , . . . , An ) := inf t > 0 : ⎩ ⎭ n=0 i 0 ...i n
The following conjecture is widely believed to hold. Conjecture 1.4. For a typical (in an appropriate sense) self-affine IFS the Hausdorff dimension of the limit set is equal to min {d, s(A1 , . . . , Am )}. In this general direction, we will prove a statement for a random perturbation of a given attractor . Of particular importance is the fact that our theorem will make no assumptions regarding the matrix norms. More precisely, we assume that with each application of the functions from the given IFS we make a random additive error Y . We assume that these errors have distribution η, where η is an absolutely continuous distribution with bounded density supported on a disk D which is centered at the origin and can be chosen to be arbitrarily small. We saw in (3) that is defined by all possible compositions f i0 ···in−1 . For each such function we assume that the perturbations are independent. More precisely, let T be the m-adic tree with m n nodes on the n th level. Each of these n th level nodes corresponds to a word in := (i 0 , . . . , i n−1 ) ∈ {1, . . . , m}n . To obtain a random perturbation of the attractor we consider the random perturbations of the n th level node maps f in : Rd → Rd of the form y
f in in := ( f i0 + yi0 ) ◦ ( f i1 + yi0 i1 ) ◦ · · · ◦ ( f in−1 + yi0 ...in−1 ), where in = (i 0 , . . . , i n−1 ) ∈ T , and the elements of
· · × D yin := yi0 , yi0 i1 , . . . , yi0 ...in−1 ∈ D × · n
are i.i.d. with distribution η. The sequence of all random errors is given by y := yin i ∈T ∈ D T . n
(7)
Given y ∈ D T we denote the associated attractor by y . More precisely, := y
∞
y
f in in (B),
n=0 in
where B is a sufficiently large ball in Rd , centered at the origin. Let y : → Rd be the natural projection from the symbolic space = {1, . . . , m}Z
+
to the attractor y given by
y
y (i) := lim f in in (0), n→∞
(8)
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where i ∈ and in denotes the truncation to the first n terms. More explicitly, n y (i) := lim ti0 + Ai0 · · · Aik−1 · tik n→∞
k=1
+ yi0 +
n
(9)
Ai0 · · · Aik−1 · yi0 ...ik .
k=1
On D T we define the infinite product measure by P := η × · · · × η × · · · .
(10)
The following theorem gives a rigorous formulation of the intuitive notion that random attractors should have the expected dimension. Theorem 1.5 (Main Theorem). Consider contracting self-affine IFS of the form (1). For P-almost all y ∈ D T we have: (1) If s(A1 , . . . , Am ) ≤ d then dimH y = s(A1 , . . . , Am ); (2) If s(A1 , . . . , Am ) > d then Lebd (y ) > 0. There is a closely related result by Peres, Simon and Solomyak in [14] for self-similar IFS with random multiplicative errors. Moreover, there the authors consider IFS which contract on average and where the n th level maps are assumed to have the same error. Our method of proof of this theorem involves an investigation of the dimension of certain measures defined on the attractor (Theorem 1.7). These are the images, under the natural projection, of ergodic, shift invariant Borel measures on . For the rest of the paper when we refer to an ergodic measure on , or a compact subset of , we shall always assume it is both Borel and shift invariant. We use this method since s(A1 , . . . , Am ) can be expressed in terms of the Lyapunov exponents of an ergodic measure introduced by Käenmäki in [7]. More precisely, let ν be an ergodic measure on and let A : → M, where M denotes the set of d × d matrices with real entries, be defined by A(i) := Ai0 . Then for the stationary process given by the measure ν and ∞ Pn (A, i) := Ai∗n−1 · · · Ai∗0 n=1
(11)
we denote the Lyapunov exponents [9, Theorem 5.7] by λ1 (ν) ≥ λ2 (ν) ≥ · · · ≥ λd (ν).
(12)
Indeed, the main reason for introducing the transposed matrices Ai∗ is to reverse the order of the product of such matrices, and so be able to directly apply known results on Lyapunov exponents. This leads naturally to the following definition. Definition 1.6 (Definition of the Lyapunov dimension D(ν)). (i): If
k := k(ν) = max {i : 0 < h ν + λ1 (ν) + · · · + λi (ν)} ≤ d,
(13)
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then we define D(ν) := k +
h ν + λ1 (ν) + · · · + λk (n) ; −λk+1 (ν)
(ii): If h ν + λ1 (ν) + · · · + λd (ν) > 0 then we define D(ν) := d ·
hν , −(λ1 (ν) + · · · + λd (ν))
where h ν is the entropy of the measure ν. Note that in both cases above the Lyapunov dimension is simply the generalization of the similarity dimension (which can be higher than the dimension of the space) if the system is self-similar. The following theorem gives a characterization of properties of the image of an ergodic measure in terms of the Lyapunov dimension. Theorem 1.7. Consider a contracting self-affine IFS of the form (1) and an ergodic measure ν on . For P-almost all y ∈ D T the following hold: (a):
y
dimH ∗ (ν) = min {d, D(ν)} . (b): If D(ν) > d then
y
∗ (ν) Lebd .
We refer the reader to [5] for the definition of the Hausdorff dimension of a measure. Theorem 1.5 is an immediate consequence of Theorem 1.7 and the following proposition. Let E( ) denote the ergodic probability measures on . Proposition 1.8. There exists an ergodic probability measure μ on such that s(A1 , . . . , Am ) = D(μ) = sup D(ν). ν∈E ( )
The measure μ in Proposition 1.8 is the same as that constructed by Käenmäki in [7]. We briefly explain the scheme of the proofs of Theorem 1.5 and Theorem 1.7. The key point is the introduction of a new self-affine transversality condition (26) for certain families of self-affine IFS, which was motivated by Solomyak’s general projection scheme in [17]. We show that if this condition holds then for a typical parameter value the Hausdorff dimension of the attractor is the minimum of d and the singularity dimension (Theorem 4.2). Furthermore, the Hausdorff dimension of the image ν = y (μ) of an ergodic measure μ on is the Lyapunov dimension if the Lyapunov dimension D(ν) is smaller than d. Alternatively, if D(ν) is larger than d then the image measure is absolute continuous (Theorem 4.3). As another application of this method we can solve a randomized version of a long standing open problem in fractal image compression [6, 8] (see Sect. 6). The techniques we use in this paper also apply to the deterministic case, giving the following analogue of Theorem 1.2. We define a natural projection π t : → Rd by n t π (i) := lim ti0 + Ai0 · · · Aik−1 · tik . n→∞
k=1
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Theorem 1.9. Consider a contracting self-affine IFS of the form (1) and an ergodic measure ν on . As in Theorem 1.2 we assume that Ai <
1 , for all 1 ≤ i ≤ m. 2
Then for Lebmd a.e. t ∈ Rmd we have: (a): dimH π∗t (ν) = min {D(ν), d}; (b): If D(ν) > d then π∗t (ν) Lebd ; (c): If s(A1 , . . . , Am ) > d then Lebd (t ) > 0, where t is the attractor. That is t = π t ( ). In Sect. 2 we describe the main properties of the singular value function and singularity dimension. In Sect. 3 we introduce a variant on thermodynamic formalism, which is important in relating singularity dimension and Lyapunov dimension. The self-affine transversality condition is introduced in Sect. 4 and used to get lower bounds on the dimensions, and to verify absolute continuity, in a general setting. Section 5 contains proofs of Theorems 1.2–1.7. Finally, in Sect. 6 we apply our method to a problem in fractal image compression. 2. Main Properties of the Singular Value Function Here we summarize those properties of the singular value function and singularity dimension which will be used in our proofs. One can easily see that for every s, h > 0 we have α1h ≥
φ s+h (T ) ≥ αdh . φ s (T )
(14)
The singular value function is submultiplicative [3]: For each s ≥ 0, φ s (T1 T2 ) ≤ φ s (T1 )φ s (T2 ). We denote by ∗ the set of finite words from the alphabet {1, . . . , m}. Fix a contractive IFS F, of the form (1). We choose 0 < aF < bF < 1 such that for each i = 1, . . . , m, 1 > bF ≥ α1 (Ai ) ≥ · · · ≥ αd (Ai ) ≥ aF > 0. We then obtain
|i|
|i|
bF ≥ α1 (Ai ) ≥ · · · ≥ αd (Ai ) ≥ aF ,
(15)
where |i| denotes the length of i. Furthermore, it follows from the definition that for all i ∈ ∗ and s ≥ 0 we have s|i| s|i| (16) bF ≥ φ s (Ai ) ≥ aF .
For an arbitrary i ∈ ∗ with |i| ≥ n, or i ∈ , we define
ψns (i) := log φ s (Ai0 · · · Ain−1 ). It follows from (16) that ns log bF ≥ ψns (i) ≥ ns log aF .
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Using
ψn+k (i) ≤ ψn (i) + ψk (σ n i), it follows from the sub-additive ergodic theorem that for every ergodic measure ν on and for ν almost all i ∈ we have that 1 1 lim ψns (i) = lim ψns (i)dν(i). (17) n→∞ n n→∞ n
Finally, we need the following simple result to show that the singular values are the same if we reverse the order of the product of the matrices. Lemma 2.1. For every i ∈ ∗ and for every 1 ≤ k ≤ d we have αk (Ai ) = αk (Pn (A, i)), where Pn (A, i) was defined in (11). 2.1. A Corollary of a proof of Oseledeˇc’s Theorem. Using the proof [9, p. 43–47] of Oseledeˇc’s Theorem we can deduce the following result. Lemma 2.2. Consider a contractive IFS F defined by (1). Let ν be an arbitrary ergodic measure on . We define Pn (A, i) and λi (ν) for i = 1, . . . , d as in (11) and (12). Then for ν-almost all i ∈ and 1 ≤ k ≤ d we have lim
n→∞
1 log αk (Pn (A, i)) = λk (ν), n
(18)
where αk (Pn (A, i)) is the k th largest singular value of Pn (A, i). This is an immediate consequence of the proof [9, Theorem 5.7]. 2.2. The singularity dimension and finite measures. Consider the d × d non-singular matrices A1 , . . . , Am satisfying (2) and ⊂ a σ -invariant compact set. For = the singularity dimension was defined by Falconer [3, p. 344]. For the convenience of the reader we summarize the most important facts about the singularity dimension s (A1 , . . . , Am ) for ⊂ compact σ -invariant analogous to those in [3, p. 344.]. For every > 0 and s > 0 we define a set function Ns on the subsets of by Ns (A) := inf φ s (Aωk ) : ωk ∈ ∗ , A ⊂ ∪k ωk and |ωk | ≥ . k
Then it follows from [10, Theorem 4.2] that the measure N s (A) := lim Ns (A) = sup Ns (A) →∞
is a Borel regular measure. It follows from (14) and (15) that if t < s and N t () < ∞ then N s () = 0. This implies that we can define s (A1 , . . . , Am ) by (19) s (A1 , . . . , Am ) = inf s : N s () = 0 = sup s : N s () = ∞ . It follows from [3, p. 344] that if = then s (A1 , . . . , Am ) is equal to s(A1 , . . . , Am ) defined in (9). The following lemma allows us to find a finite measure whose properties are closely related to the singularity dimension.
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Lemma 2.3. If N s () = ∞ for some s then there exists a finite measure μ supported on and a constant c0 such that μ(ω) ≤ c0 · φ s (Aω ) for every ω ∈ ∗ , where μ(ω) represents the measure of the corresponding cylinder set. Proof. Fix an s such that N s () = ∞. It follows from [15, Theorem 54] that there exists a compact set ⊂ such that 0 < N s ( ) < ∞. Let i ∧ j denote the common part of i and j. We define a metric ρ on as follows: s φ (Ai∧j ), if i = j; ρ(i, j) := 0, if i = j. We can see that ρ is a metric since for every ω ∈ ∗ we have φ s (Ai∧j ) ≤ max φ s (Ai∧ω ), φ s (Aω∧j ) . Then it follows from [15, Theorem 53] that the measure N s is the 1-dimensional Hausdorff measure on the compact set . Now the assertion of the lemma immediately follows from [10, Theorem 8.17]. 3. Thermodynamical Formalism Following [7], we define the energy of an ergodic measure ν by 1 E ν (s) := lim ψns (i)dν(i). n→∞ n The following lemma relates the energy to the Lyapunov exponents.
(20)
Lemma 3.1. Assume the same hypotheses as in Lemma 2.2. For k < s ≤ k + 1, we can write λ (ν) + · · · + λk (ν) + [s − k] λk+1 (ν), if s < d; E ν (s) = s 1 (21) if s ≥ d. d [λ1 (ν) + · · · + λd (ν)] , From this we obtain that
h ν + E ν (D(ν)) = 0.
(22)
Proof. It follows from (17) that for ν-almost all i we have E ν (s) = follows from Lemma 2.2 that for ν-almost all i we have right-hand side of (21).
lim 1 ψ s (i). n→∞ n n
lim 1 ψ s (i) n→∞ n n
It
is equal to the
We can define the pressure function by 1 log φ s (Ai ). n→∞ n
P(s) := lim
|i|=n
In particular, we see that P(0) = log m > 0. Comparing (14) and (15) we see that for h>0 s+h s 1 φ (Ai ) − n1 · log φ (Ai ) n · log |i|=k
|i|=k
≥ log aF h and from this we see that the function P(s) is strictly decreasing, and has a unique zero which we denote by t0 > 0. The following result is due to Käenmäki [7]. log bF ≥
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Theorem 3.2 (Käenmäki). For any ergodic measure ν we have P(s) ≥ h ν + E ν (s). Moreover, (a): There exists an ergodic measure μ on such that 0 = P(t0 ) = h μ + E μ (t0 ),
(23)
(b): s(A1 , . . . , Am ) = t0 . We are now in a position to prove Proposition 1.8, assuming Theorem 3.2. Proof of Proposition 1.8. It follows from (21) that the function s → E μ (s) is strictly decreasing. Therefore it follows from (22) and (23) that t0 = D(μ). Using part (b) of Theorem 3.2 the result follows. 4. Generalized Projection Scheme for Self Affine IFS In [17, p. 542], Solomyak presented a generalized projection scheme applicable to selfconformal IFS. In this section we construct an analogous scheme suitable for self-affine IFS. In the next section we give the details of the proofs. Assume that we are given a contractive IFS F of the form (1). For every distinct i, j ∈ we write αk (i ∧ j) for the k th largest singular value of Ai∧j if 1 ≤ k ≤ d and α0 (i ∧ j) := ∞, αd+1 (i ∧ j) := 0. Let J := [α+1 (i ∧ j), α (i ∧ j)]. We define a function Z i∧j : [0, ∞) → [0, 1] (see Fig. 1) by Z i∧j (ρ) :=
d d min {ρ, αk (i ∧ j)} ρ = · 11 J + 11 J0 . αk (i ∧ j) α1 (i ∧ j) · · · α (i ∧ j)
(24)
=1
k=1
Note that if all the maps of F are similarities then the similarity ratio of f i∧j is λi∧j and Z i∧j (ρ) = min 1,
ρd λdi∧j
.
The motivation for the next theorem is that we want to consider a one-parameter family of self-affine IFS defined by random perturbations of F. The parameter space is a compact metric space U on which we are given a probability measure M. We want to give a dimension estimate for M almost all parameters u ∈ U . Definition 4.1. Consider a contractive IFS F of the form (1). Let U be a compact metric space (parameter space) with a finite Borel measure M. Assume we are also given a con+ tinuous map : U × → Rd (the natural projection), where ⊂ = {1, . . . , m}Z is compact and σ -invariant. For any u ∈ U, i ∈ we write u (i) := (u, i). We make the following two definitions. Self-affine Hölder condition. There exists a constant K > 0 such that for every u ∈ U and i ∈ and for every n ∈ N we can find an isometry G = G(i, n) : Rd → Rd with u (i|n) ⊂ K · G ([0, α1 (i|n)] × · · · × [0, αd (i|n)]) , (25) where i|n denotes the truncation of i ∈ to a word of length n (i.e., we scale the image of the isometry by the constant K ).
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Fig. 1. The function ρ → Z i∧j (ρ) for αk := αk (i ∧ j)
This condition is used only for upper bounds on the Hausdorff dimension. Self-affine transversality condition. There is a constant C > 0 (independent of i, j) such that for all i, j ∈ , i = j we have M u ∈ U : u (i) − u (j) < ρ < C · Z i∧j (ρ). (26) With this definition, we can now state the following theorem. Theorem 4.2. Consider a contractive IFS of the form (1) and a compact metric space U with a finite Borel measure M. Let and be as in Definition 4.1. If the self-affine Hölder condition and self-affine transversality condition hold then for M-almost all u ∈ U we have: (a): dimH (u ()) = min {d, s (A1 , . . . , Am )} ; and (b): If s (A1 , . . . , Am ) > d then Lebd (u ()) > 0. Note that the self-affine Hölder condition (25) in the self-similar case reduces to the formulae in [17, p. 542, formulae (10)]. Similarly the self-affine transversality condition (26) in the self-similar case reduces to the usual transversality condition (see e.g. [17, p. 542, formulae (11)]). We also have the following measure theoretic result. Theorem 4.3. Consider a contractive IFS of the form (1) and a compact metric space U with a finite Borel measure M. Let and be as in Definition 4.1. Let ν be an ergodic measure on . If the self-affine Hölder condition and self-affine transversality
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condition hold then for M-almost all u ∈ U we have: (a): dimH u∗ (ν) = min {d, D(ν)} ; and (b): If h ν > −(λ1 + · · · + λd ) then u∗ (ν) Lebd . To get the lower bound in Theorems 4.2 and 4.3 we need the following proposition. Proposition 4.4. Consider a contractive IFS of the form (1) and a compact metric space U with a finite Borel measure M. Let and be as in Definition 4.1. Assume that the self-affine transversality condition holds. Consider a Radon measure μ on and a positive number s such that there exists c1 > 0 which satisfies that for all finite words ω ∈ ∗, μ([ω]) ≤ c1 φ s (Aω ). (27) Then for M-almost all u ∈ U we have:
(a): If s ≤ d then dimH u∗ μ ≥ s.; and (b): If s > d then u∗ (μ) Lebd . In turn, to prove part (a) of Proposition 4.4 we need the following lemma. Lemma 4.5. Consider a contractive IFS of the form (1) and a compact metric space U with a finite Borel measure M. Let and be as in Definition 4.1. Assume that the self-affine transversality condition holds. Then for every t ∈ N, 0 < t < d there exists a constant c2 = c2 (t) (independent of i, j) such that for all i, j ∈ , i = j we have u
(i) − u (j) −t dM(u) < c2 · φ t (Ai∧j ) −1 . (28) u∈U
Proof. Fix t > 0 and i, j ∈ . Observe that we can write
u (i) − u (j) −t dM(u)
u∈U
∞ =t
M u ∈ U : u (i) − u (j) < ρ ρ −t−1 dρ.
ρ=0
Thus using (28) it is enough to prove that there exists a constant such that if we write ∞ t Ii∧j
:=
Z i∧j (ρ)ρ −t−1 dρ
(29)
ρ=0
then
t ≤ const · α1 (i ∧ j) · · · αk−1 (i ∧ j) · αkt−(k−1) (i ∧ j) Ii∧j
where k − 1 < t ≤ k.
−1
,
(30)
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T. Jordan, M. Pollicott, K. Simon
Henceforth, for notational simplicity, we shall write Z (ρ), αk and I in place of Z i∧j (ρ), t . We can write the integral in (29) as a sum of the integrals over the αk (i ∧ j) and Ii∧j intervals [0, αk ] and [αk , ∞]. More precisely, if we write αk I1 :=
Z (ρ)ρ
−t−1
∞ dρ and I2 :=
Z (ρ)ρ −t−1 dρ,
ρ=αk
ρ=0
then we have t Ii∧j = I1 + I2 .
(31)
First we prove I1 is bounded by the expression on the right-hand side of (30) and then we show the same holds for I2 . The proof then follows. Next observe that 0 < ρ < αi implies Thus
αk I1 < ρ=0
ρi ρ i−1 < . α1 · · · αi α1 · · · αi−1
ρk 1 α1 · · · αk−1 αkt−(k−1) ρ −t−1 dρ = α1 · · · αk k−t
which verifies the required inequality with the constant I2 =
k−1
1 k−t .
−1
,
(32)
We now bound
Z (ρ)ρ −t−1 dρ.
(33)
=0 J
Observe that
J
Z (ρ)ρ −t−1 dρ =
1 −t − α−t , (α1 · · · α )−1 α+1 t −
where for = 0 we set α0−t = 0. Using that 1 1 +1−t −t > (α1 · · · α+1 )−1 α+1 (α1 · · · α )−1 α+1 t − ( + 1) t − one can see that the number which we add in the sum (33) for each is smaller than that which we subtract for + 1. Therefore ! "−1 1 · α1 · · · αk−1 αkt−(k−1) I2 < . t − (k − 1) This completes the proof of the required inequality for I2 . It follows from this (31) and 1 1 (32) that (30) holds for I with the constant t−(k−1) + k−t . We now use Lemma 4.5 to prove Proposition 4.4. This proposition is important in the proofs of all of the main results in this paper.
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The proof of Proposition 4.4. We first prove part (a). Assume that s < d. It follows from the potential theoretic characterization of the Hausdorff dimension (see e.g. [4, Theorem 4.13]) that for every measure ν on Rd we have dimH ν ≥ sup α : |x − y|−α dν(x)dν(y) < ∞ . (34) Fix t < s. It suffices to prove that u t (i) − u (j) −t dμ(i)dμ(j)dM(u) < ∞, I := u∈U(i,j)∈×
since this, together with (34), implies that dimH (u∗ μ) ≥ t for M-almost all u ∈ U . Using (27) and Lemma 4.5 we obtain ∞ t
−1 I < c2 φ (Aω ) dμ(i)dμ(j) t
k=0 |ω|=ki∧j=ω
≤ c2 c1
∞
−1 μ([ω])φ s (Aω ) φ t (Aω ) .
(35)
k=0 |ω|=k
It follows from (14) and (15) that
−1 k·(s−t) ≤ bF . φ s (Aω ) · φ t (Aω ) Using this and (35) we obtain that I t ≤ c1 c2
∞
k·(s−t)
bF
k=0
|ω|=k
μ([ω]) < ∞.
=1
Since t can be chosen arbitrarily close to s this completes the proof of part (a). We next prove part (b). It follows from the definition (24) of Z i∧j (ρ) that for distinct i, j ∈ and ρ > 0 we have the bound Z i∧j (ρ) ≤
ρd . φ d (Ai∧j )
Thus it follows from the self-affine transversality condition (26) that for all distinct i, j ∈ and ρ > 0, M u ∈ U : u (i) − u (j) < ρ < C ·
ρd . φ d (Ai∧j )
(36)
Furthermore, from (16) we obtain that for ω ∈ ∗ , μ([ω]) φ s (Aω ) |ω|(s−d) ≤ c ≤ bF · . 1 φ d (Aω ) φ d (Aω )
(37)
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To show absolute continuity of u∗ μ for M almost all u we will follow a standard approach (introduced by Peres and Solomyak in [13]). In particular, it suffices to show that u μ(B(x, r )) u I := lim inf ∗ d∗ μ(x)dM(u) < ∞. r →0 2r d We apply Fatou’s Lemma, lift to the shift space, and apply Fubini’s Theorem. Finally, using inequality (36) we have that 1 I ≤ lim inf d 11{(i,j):|u (i)−u (j)|≤r } dμ(i)dμ(j)dM(u) r →0 2r 1 ≤ lim inf d M{u : |u (i) − u (j)| ≤ r }dμ(i)dμ(j) r →0 2r 1 dμ(i)dμ(j). ≤C φ d (Ai∧j ) To complete the proof we split the integral up into an infinite summation and use (37) to get the bound I ≤C
∞
μ[ω]2 (φd (A[ω] ))−1
k=0 |ω|=k ∞
≤ c1 C + C
k(s−d)
bF
k=0
μ(ω) < ∞.
|ω|=k
This suffices to show the result (cf. [13]). 5. The Proof of the Theorems In this section we will prove Theorems 4.2 and 4.3, and then deduce from these Theorems 1.5 and 1.7. The proof of Theorem 4.2. We first prove part (a). The upper bound follows immediately from the definition of the singular value function. To get the lower bound we fix an arbitrary s < min {d, s (A1 , . . . , Am )}. In particular, N s () = ∞ by (19). It then follows from Lemma 2.3 that we can find a measure μ which is supported on satisfying (27), i.e., μ([ω]) ≤ c1 φ s (Aω ). In particular, we now can apply Proposition 4.4 to deduce that for M almost all u we have dimH u () ≥ s. This completes the proof of part (a) of the theorem. Next we prove part (b). Since we are assuming that d < s (A1 , . . . , Am ) we can choose d < s < s (A1 , . . . , Am ). Using definition (19) of s (A1 , . . . , Am ) it follows from Lemma 2.3 that there exists a measure μ supported by and a constant c0 > 0 such that for every ω ∈ ∗ we have μ([ω]) ≤ c0 · φ s (Aω ). Thus we can apply part (b) of Proposition
4.4 to deduce that for M a.e. u ∈ U we have u∗ (μ) Lebd and thus Lebd u∗ () > 0.
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Proof of Theorem 4.3. We first prove part (b). It is apparent from the definitions of Z i∧j (ρ) and φ d that Z i∧j (ρ) ≤
ρd φ d (Ai∧j )
for distinct i, j (cf. Fig. 1). Since we are assuming the self-affine transversality condition (26), there exists C > 0 such that M{u ∈ U : |u (i) − u (j)| ≤ ρ} ≤ C
ρd . φ d (Ai∧j )
Recall that by the Shannon-McMillan-Breiman Theorem [18, p. 93] we have that for ν almost all i ∈ , 1 lim log ν([i 0 , . . . , i n−1 ]) = −h(ν) (38) n→∞ n and that we have already shown that for ν almost all i ∈ lim
n→∞
1 log αk (Ai|n ) = λk (ν), n
which implies that for ν almost all i ∈ , # lim n α1 (Ai|n ) · · · αd (Ai|n ) = eλ1 (ν)+···+λd (ν) . n→∞
(39)
To show that νu := u∗ ν is absolutely continuous it suffices to show that for an arbitrary ε > 0 we can find a measure νε which is the restriction of ν to a set with measure greater than 1 − ε and for which u∗ νε is absolutely continuous. For any ε > 0 we have that by Egorov’s Theorem there exists a set X ε ⊂ such that ν(X ε ) > 1 − ε and the convergence in both (38) and (39) is uniform. We let νε be ν restricted to this set. Choose δ > 0 such that h ν − δ > −(λ1 + · · · + λd ) + δ
(40)
(where we write λi for λi (ν) for convenience). By the uniform convergence in (38) and (39) we can choose N ∈ N such that for n ≥ N and i ∈ X ε , ν([i|n]) ≤ en(−h ν +δ) and φ d (Ai|n ) ≥ en(λ1 +···+λd −δ) . It is now enough to check the absolute continuity of the measure u∗ νε . To see this we observe that by (40) there exists d < s and a constant c1 such that for every ω ∈ ∗ we have νε ([ω]) < c1 φ s (Aω ). Using Proposition 4.4 we obtain that u∗ νε Lebd . Since this holds for all ε > 0 this completes the proof of part (b). We next turn to the proof of part (a). We can assume that D(ν) < d. Let k := k(ν) be defined as in (13). In particular, k < D(ν) ≤ k + 1 and we can choose k < s < D(ν). Let ε > 0 satisfy D(ν) − s =
2ε . −λk+1
Then it follows from Lemma 3.1 that E ν (s) > −h ν + 2ε.
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Using Egorov’s Theorem it follows from (20), the definition of E ν (s), (17) and the Shannon-McMillan-Breiman Theorem that for every δ > 0 there exists a set Hδ ⊂ with ν(Hδ ) > 1 − δ and there exists an N such that for all n ≥ N and for all i ∈ Hδ we have ν(i|n) ≤ exp(n(−h ν + ε)) < exp(n E ν (s)) < φ s (Ai|n ). In this way we see there is a c > 0 such that for all i ∈ Hδ we have ν(i|n) < cφ s (Ai|n ). Let νδ := ν| Hδ . Then by Proposition 4.4 we obtain that for M a.e. u ∈ U we have dimH (u∗ ν) ≥ dimH (u∗ νδ ) ≥ s. Since s < D(ν) was arbitrary this gives the lower estimate in part (a) of the theorem. To get the upper bound in part (a) assume that D(ν) < d. We fix an arbitrary u ∈ U and then we can prove the upper estimate for := u (). Since u is fixed, we write instead of u to simplify the notation. Given any γ > D(ν), it is enough to prove that dimH (∗ ν) ≤ γ . We choose ε > 0 to satisfy γ >
(k + 1)ε h ν + λ1 + · · · λk +k+ . −λk+1 + ε −λk+1 + ε
(41)
By the self-affine Hölder condition (25) there is a K > 0 such that for every i ∈ and n ∈ N the set (i|n) can be covered by a rectangular box Bi|n with sides K · α1 (i|n), . . . , K · αd (i|n). Without loss of generality, let us assume that K = 1. Let k := k(ν) be as in (13). For every i ∈ and n ∈ N we fix a subdivision of the box Bi|n into N (i|n) :=
α1 (i|n) · · · αk (i|n) k (i|n) αk+1
boxes of sides (i|n) := αk+1 (i|n), . . . , αk+1 (i|n), αk+2 (i|n), . . . , αd (i|n). k+1
Let Pn (i) denote the box which contains (i). We denote by Q n (i) the set in which corresponds to Pn (i). That is Q n (i) := {j ∈ : j ∈ [i|n] and (j) ∈ Pn (i)} , where [i|n] denotes the n th level cylinder. In general Q n (i) −1 (Pn (i)) for is not 1 − 1. Furthermore, let ν(i|n) Anε := i ∈ : ν(Q n (i)) ≥ ε · . N (i|n) Observe that for every n we have ν
Anε
c
≤ ε.
(42)
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In particular, ν
Anε
c
=
|ω|=n
c ν([ω]) ≤ ε. ≤ ν [ω] ∩ Anε N (ω) · ε · N (ω) |ω|=n
Using (42) we see that the set Aε := lim sup Anε = n→∞
satisfies
∞ ∞
Anε
k=1 n=k
ν (Aε ) > 1 − ε.
It follows from Egorov’s Theorem that there exists a G ε ⊂ Aε such that ν(G ε ) > 1 − 2ε and we have that 1 1 log ν(i|n) → −h ν and log α (i|n) → λ (ν), n n where the convergence is uniform for all 1 ≤ ≤ d and all i ∈ G ε . Thus, for every i ∈ G ε we can find arbitrary large n simultaneously satisfying the following four conditions: (i): ν (Q n (i)) > ε Nν(i|n) (i|n) ; n(−h −ε) ν (ii): e < ν(i|n) < en(−h ν +ε) ; (iii): en(λ −ε) < α (i|n) < en(λ +ε) for every 1 ≤ ≤ d; ε < ε. (iv): log αlog k+1 (i|n) Fix i ∈ G ε and choose n satisfying (i)–(iv) above. We want to apply Frostman’s Lemma to the measure ∗ ν, and to this end we need to estimate Rn (i) :=
log ∗ ν [B ((i), αk+1 (i|n))] . log αk+1 (i|n)
In particular, log ε log ν(i|n) − log N (i|n) log ν(Q n (i)) ≤ + log αk+1 (i|n) log αk+1 (i|n) log αk+1 (i|n) log N (i|n) − log ν(i|n) ≤ε+ − log αk+1 (i|n) n (λ1 + · · · λk + kε − kλk+1 + kε) + n(h ν + ε) ≤ε+ −n(λk+1 − ε) (k + 1)ε h ν + λ1 + · · · + λk +k+ . = −λk+1 + ε −λk+1 + ε
Rn (i) ≤
For ε satisfying (41) we can find n arbitrarily large such that Rn (i) < γ . From this we deduce that for every i ∈ G ε we can bound lim inf n→∞
log ∗ ν [B ((i), αk+1 (i|n))] < γ. log αk+1 (i|n)
(43)
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By construction, G ε1 ⊃ G ε2 whenever ε1 < ε2 and thus we obtain that (43) holds for ν-a.e. i ∈ . Frostman’s Lemma ([2], Theorem 3.3.14) now implies that, dimH ν ≤ γ . This completes the proof of the upper estimate since γ > D(ν) was chosen arbitrarily. In order to apply Theorems 4.2 and 4.3 to deduce Theorems 1.5 and 1.7, we need to show that the self-affine transversality condition holds. This is the purpose of the next lemma. Lemma 5.1. The self-affine transversality condition (26) holds for y , y ∈ D T , i.e., there exists a constant C > 0 such that for all distinct i, j ∈ we have P y ∈ D T : y (i) − y (j) < ρ < C · Z i∧j (ρ).
(44)
Proof. Fix y ∈ D T and consider the natural projection y : → y defined in (8). It follows from (9) that for every i, j ∈ with |i ∧ j| = n there exists a random variable qn (i, j, y) which is independent of yi0 ...in such that % $ (45) |y (i) − y (j)| = Ai∧j yi0 ...in + qn (i, j, y) . Let us fix all of the terms in y except yi0 ...in ∈ D. Using the product structure of P and the fact that the measure η is absolutely continuous with respect to Lebd with bounded density we see that in order to verify (44) it is enough to prove that Lebd yi0 ...in ∈ D : y (i) − y (j) < ρ < C · Z i∧j (ρ). Let us write Boxρ := [−ρ, ρ]d and let r be the radius of the ball D. Let ϕ be a rotation of Rd which sends the coordinate axes to the mutually orthogonal singular vectors of A−1 i∧j and let Boxi∧j := ϕ([−r, r ]d ). It is clear that for all distinct i, j we have D ⊂ Boxi∧j . Then using (45) we get that
Lebd yi0 ...in ∈ D : y (i) − y (j) < ρ ≤ Lebd A−1 i∧j Boxρ ∩ Boxi∧j . Finally, it follows from elementary geometry that
Lebd A−1 i∧j Boxρ ∩ Boxi∧j ≤ C · Z i∧j (ρ), which completes the proof of the lemma. We are now in a position to complete the proofs of Theorems 1.5 and 1.7. Proof of Theorem 1.5. Theorem 1.5 is an immediate consequence of Lemma 5.1 and Theorem 4.2. Proof of Theorem 1.7. Theorem 1.7 is an immediate consequence of Lemma 5.1 and Theorem 4.3. In order to deduce Theorem 1.9, we need the following lemma. This is an adaptation of related results proved by Falconer [3] and Solomyak [17].
Randomly Perturbed Self-Affine Sets
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Lemma 5.2. Consider a contractive self-affine IFS F of the form (1). Let U be an arbitrary ball in Rmd and let M be the measure Lebmd restricted to U . Given u = (u 1 , . . . , u m ) ∈ U we define u (i) := lim ( f i0 + u i0 ) ◦ · · · ◦ ( f in−1 + u in−1 )(0). n→∞
Then the self-affine transversality condition (26) holds. Proof. Let T be a d × d matrix with singular values 0 < αd ≤ · · · ≤ α1 < 1. We can bound d ρ Leb{x ∈ B(0, δ) : |T x| ≤ ρ} ≤ C0 min ,δ (46) αi i=1
for some constant C0 > 0. This is easily seen by observing that T −1 B(0, ρ) is contained in a box with sides of length 2r αi aligned in the direction of the singular vector associated with αi . Fix distinct i, j ∈ and let L := Lebmd {u ≤ δ : |u (i) − u (i)| ≤ ρ}. Let k = |i ∧ j| and then we can write |u (i) − u (j)| = Ai∧j (u ik+1 − u jk+1 + ( f ik+1 u ik+2 + f ik+1 f ik+2 u ik+3 + · · · ) − ( f jk+1 u jk+2 + f jk+1 f jk+2 u jk+3 + · · · )) = Ai∧j (u ik+1 − u jk+1 + E(u)). It follows from [17, p. 540] that provided Ai < 21 for all 1 ≤ i ≤ m then E < 1. Thus as in [3] the linear transformation from Rmd → Rmd defined by (u 1 , u 2 , . . . , u m ) → (y, u 2 , . . . , u m ) , where y = u ik+1 − u jk+1 + E(u) will be invertible. Thus there exists a constant C1 > 0 such that L ≤ C1 Lebd {(y, u 2 , . . . , u d ) ∈ B : f i∧j (y) ≤ ρ}, where B is the product of the interval [−(2 + m)δ, (2 + m)δ] in the y direction and the d − 1 dimensional δ-ball. It now follows from (46) that there exists C2 = C2 (δ) and C3 = C3 (δ) such that L ≤ C2
d i=1
ρ min , δ ≤ C3 Z i∧j (ρ). αi
Finally, we can use Lemma 5.2 to prove Theorem 1.9. Proof of Theorem 1.9. Theorem 1.9 immediately follows from Theorem 4.3, Theorem 4.2 and Lemma 5.2.
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6. An Application in Fractal Image Compression In this section we will compute the almost sure Hausdorff dimension of certain randomly perturbed graph directed IFS which appear in the theory of fractal image compression. The following graph directed Iterated Function System (GIFS) arose naturally in the field of fractal image compression. Given a large natural number K we partition the unit square Q into M := 2 K × 2 K subsquares of size 2−K which we denote by R1 , . . . , R M . This is the family of range squares. We define the family of domain squares as the set of all N := (2 K − 1) × (2 K − 1) squares D1 , . . . , D N which are the union of four range squares having a common vertex. We associate a domain square to every range square. More precisely, we are given a map ϕ : {1, . . . , M} → {1, . . . , N } .
(47)
This defines a directed graph = (V, E) as follows. The vertices are V = {1, . . . , M}. There is a directed edge (k, ) ∈ E from k ∈ V to ∈ V if R is one of the four squares of Dϕ(k) . In this way is a directed graph with exactly four edges going out of every vertex. We also assume that for each 1 ≤ k ≤ M and = ϕ(k) we have associated a self-affine map Sk : (D × [0, 1]) → (Rk × [0, 1]) of the form Sk (x, z) := (Tk (x), f k (z)) , where Tk : D → Rk is onto, f k : [0, 1] → [0, 1], and they satisfy &1 ' 0 2 DTk (x) = and f k (z) := λk · z + tk , 0 21
(48)
where 0 < λk < 1. See Fig. 2. We define the graph directed sets 1 , . . . , M to be the unique family of compact sets satisfying i = Si ( j ), (49) (i, j)∈E
Fig. 2. The function Sk
Randomly Perturbed Self-Affine Sets
539
and the attractor of the GIFS is :=
M
i .
(50)
i=1
Note that if we write Se := Si whenever (i, j) = e ∈ E then (49) can be written as i =
m
Se ( j ).
j=1 e=(i, j)∈E M In particular, {Si }i=1 is a graph directed IFS in the sense of [11]. The attractor can also be written as
=
∞
Si0 ...in−1 (B),
n=1 i 0 ...i n−1
where B := [0, 1]3 and the union is over admissible words i 0 , . . . , i n−1 . We suppose that at each application of the functions we make a random translational error in the vertical direction. More precisely, for every admissible in := (i 0 , . . . , i n−1 ) we consider
y (51) Sinin := Si0 + y i1 ◦ · · · ◦ Sin−1 + y i1 ...in−1 , where y i0 ...in−1 = (0, 0, yi0 ...in−1 ) ∈ R3 and the numbers yi0 ...in−1 are chosen independently in every step, from an arbitrary small interval I centered at the origin using an absolutely continuous distribution η with bounded density. Gathering the random errors in an infinite vector as in (7) we can write,
y := y1 , . . . , ym , y11 , . . . , ym,m , . . . ∈ I × I × · · · =: I∞ . As in (10) we define the product measure on I∞ by P := η × η × · · · .
(52)
It is our aim to compute the dimension of the attractor, y :=
∞
y
Sinin (B),
n=0 in
where B = [0, 1]2 × [−u, u] ⊂ R3 , where u is sufficiently large, and the union is over admissible words in , for P a.e perturbation y. Theorem 6.1. Let E be a d × d matrix such that E i j = λi if (i, j) ∈ E and E i j = 0 otherwise.We denote by (E) the spectral radius of the matrix E. Assume that the matrix E is irreducible. Then for P a.e. y ∈ I∞ we have log (E) . (53) dimH y = max 2, 1 + log 2 We remark that a related result for box dimension was obtained in [8] for almost all deterministic attractors of a similar type. The main difference between the GIFS considered here and the ones considered in [8] is that in [8] the authors considered deterministic GIFS with a restriction on the family of domain squares. This restriction allowed the use of a technique which cannot be used in this more general setting.
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Proof. Let := i ∈ : E ik ik+1 > 0 and let n be the set of n cylinders of . The upper estimate. This is independent of the vertical translations. Let n be the n th approximation of the Cantor set consisting of 4n boxes Bi0 ...in−1 , for (i 0 . . . i n−1 ) ∈ n . Each of these has a face parallel to the (x, y)-plane which is a square of side 21n and a vertical side of length 2uλi0 ...in−1 . Thus, for all (i 0 . . . i n−1 ) ∈ n the box Bi0 ...in−1 can be covered by
2uλi 0 ...i n−1 1/2n
cubes of size 1/2n , where λi0 ,...,in−1 = λi0 · · · λin−1 . Therefore, ⎡
s H1/2 n ()
≤
2u 2n(s−1)
⎤
⎢ ⎥ ⎢ ⎥ ⎢ s n⎥ λi0 ...in−1 + 1/2 ⎥ ⎢ ⎢ ⎥ i 0 ...i n−1 ⎣i0 ...in−1 ⎦ E n 1
(54)
2n
where the summations are taken over all (i 0 . . . i n−1 ) ∈ n . This implies that whenever (E) ≤ 2 we have dimH () = 2. If (E) > 2 it also follows immediately from (54) that dimH () ≤ 1 + loglog(E) 2 . The lower estimate. In the case that (E) ≤ 2 we have already seen that dim H () = 2. Therefore, for the rest of the proof we may assume that (E) > 2. If we could verify the self-affine transversality condition (26) then we would be able to apply Theorem 4.2 which would imply the required lower estimate. Unfortunately there is no way to check condition (26) for the GIFS under consideration. However, we will introduce another GIFS which is the same in the vertical direction and which is shrunk by a small amount in the direction of both the x and y axes. In this way there will be a gap in between any two shrunken range squares in the plane which will allows us to verify the self-affine transversality condition. Thus the Hausdorff dimension of its attractor gives a lower bound for the Hausdorff dimension of the original attractor. Let us fix an arbitrary 0 < r < 1/2 K . We use the same directed graph as above and let ϕ be the same as in (47). For every 1 ≤ k ≤ M let Rkr be a square having the same center and sides parallel to the sides of Rk , but with sides of length 21K − r . Similarly, for 1 ≤ ≤ N we define Dr as the square having the same center as D and sides of length 2 K1−1 − r parallel to the sides of D . For every k and = ϕ(k) then we define the surjective affine map Skr : Dr → Rkr of the form Skr (x, z) := (Tkr (x), f k (z)), where f k is defined as in (48) and Tkr : Dr → Rk is surjective. Furthermore, & DTkr (x) =
' β 0 , 0 β
where β = β(r ) =
1 · 2
1 M 1 M
− −
2 r 1 r
.
Randomly Perturbed Self-Affine Sets
541
Then as in (49) and in (50) there exists a unique family of compact nonempty sets r1 , . . . , rm and r satisfying ri
=
Srj ()
and := r
(i, j)∈E
M
rj .
j=1
(r ) Let us write s for the singularity dimension of this system (see (19)). To proceed we need the following result.
Claim 1. If (E) > 2 and 0 < r <
1 2K
is sufficiently small such that
1 < (E), β then (r )
s = 1 +
(55)
log (E) . − log β
Proof of the Claim. For (i 0 , . . . , i n−1 ) ∈ n we have ⎡
DSir0 ,...,in−1
⎤ 0 βn 0 0 ⎦. ≡ ⎣ 0 βn 0 0 λi0 ...in−1
Since the dimension of the attractor is larger than 2 it suffices to consider s > 2 for which we have that 2n φ s (DSir0 ,...,in−1 ) = max λi0 ...in−1 β n(s−1) , λis−2 . β 0 ...i n−1 Using that on the one hand i 0 ...i n−1
and on the other hand
λis−2 β 2n < 0 ...i n−1
i 0 ...i n−1
β 2n = (2β)2n < 1
i 0 ...i n−1
λi0 ...in−1 = E n 1 we obtain that
β n(s−1) E n 1 ≤
i 0 ...i n−1
φ s (DSir0 ,...,in−1 ) ≤ β n(s−1) E n 1 + 1,
which immediately implies that n i 0 ...i n−1
φ s (DSir0 ,...,in−1 )
Using (6) this completes the proof of the claim.
(r )
< ∞, if s > s ; (r ) = ∞, if s < s .
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To complete the proof of the lower estimate in (53) the only we need to do is to thing r verify the self-affine transversality condition 26 for the GIFS S j . It would then follow from Theorem 4.2 that for P a.e. y ∈ I∞ we have (r )
dimH ry = s .
(56) (r )
Since this holds for all sufficiently small r > 0 and since s = limr →0 s this would complete the proof of the lower estimate. Now we prove that the self-affine transversality condition holds for the GIFS Srj . Claim 2. For sufficiently small r > 0 there exists a constant, C > 0, such that for all distinct i, j ∈ we have P y ∈ I∞ : |ry (i) − ry (j)| < ρ < C · Z i∧j (ρ), where ry : → ry is the natural projection defined by: r,yin
ry (i) := lim Sin n→∞
where
r,yin
Sin
(0, 0, 0),
" !
:= Sir0 + yi0 ◦ · · · ◦ Sirn−1 + yi0 ...in−1 .
(57)
Proof. We fix 0 < r < 1/2 K which is sufficiently small such that β = β(r ) satisfies (55). Henceforth we will not explicitly show the dependence of variables on r . For simplicity we write ⎡ n ⎤ 0 0 β 0 ⎦. Ai0 ,...,in−1 := DSir0 ,...,in−1 ≡ ⎣ 0 β n 0 0 λi0 ...in−1 We can write the projection y as . (i) + / y (i), y (i) = where . (i) := ti0 +
∞
Ai0 ...ik−1 tik ,
k=1
is the deterministic part and / y (i) := y i + 0
∞
Ai0 ...ik−1 y i0 ...ik ,
k=1
. (i) in the (x, y) is the random part. Furthermore, let p12 (i) ∈ R2 be the component of plane and let p3 (i) be the third component. Given distinct i, j ∈ set ω := i ∧ j. Let k := |ω| and set γk (i, y) := λik yi0 ...ik+1 + λik ik+1 yi0 ...ik+2 + · · · .
Randomly Perturbed Self-Affine Sets
543
Finally we denote qk (i, j, y) := −y j0 ... jk + γk (i, y) − γk (j, y) + p3 (σ k i) − p3 (σ k j), which is independent of yi0 ...ik . With this notation we can write & ' p12 (σ k i) − p12 (σ k j) y (i) − y (j) = Aω · yi0 ...ik + qk (i, j, y) & k
' β p12 (σ k i) − p12 (σ k j) . = λω yi0 ...ik + λω qk (i, j, y) For all and for all (n 1 . . . n ) ∈ we fix yn 1 ...n , with the exception of yi0 ...ik , which is allowed to vary. For convenience, we denote our free variable yi0 ...ik by y ∈ I . Recall that the measure η is absolutely continuous with bounded density. Thus it follows from (52) that it is sufficient to check that there exists a constant c (independent of i and j) such that for every ρ > 0 we have & k
' β p12 (σ k i) − p12 (σ k j) ∈ Boxρ < c · Z ω (ρ), Leb y ∈ I : (58) λω yi0 ...ik + λω qk (i, j, y) where Boxρ := [−ρ, ρ]3 . Since we have assumed that |ω| = k the vectors p12 (σ k (i)) and p12 (σ k (j)) are in the same domain square, which we denote by Dr . However, these two vectors are in two different range squares Rur , Rvr , say, contained in Dr . In particular, at least one of the two components of the vector p12 (σ k i) − p12 (σ k j) has absolute value greater than 2r . Thus, whatever value y = yi0 ...ik takes we have & k
' β p12 (σ k i) − p12 (σ k j) ∈ Boxρ . if ρ < 2rβ k then λω yi0 ...ik + λω qk (i, j, y) Hence, in this case (58) holds. Next we consider the case when ρ > β k . If we additionally assume ρ > λω then Z (ρ) = 1 and (58) holds with c = 1. Therefore we may assume that β k < ρ < λω
holds. Then, by definition, Z ω(ρ) = λρω . On the other hand, since qk (i, j, y) is independent of y = yi0 ...ik the Lebesgue measure of those y for which the absolute value of the third component of the vector & k
' β p12 (σ k i) − p12 (σ k j) λω yi0 ...ik + λω qk (i, j, y)
is smaller than ρ is equal to λ2ρω . Hence, (58) holds with constant c = 2. Finally, we assume that 2rβ k < ρ < β k . By an obvious case analysis one can see that for every ω, 1 Z ω (β k ) < =: c1 . k Z ω (2rβ ) (2r )3 Using that the function ρ → Z ω (ρ) is monotone increasing we obtain that for all such ρ, & k
' β p12 (σ k i) − p12 (σ k j) ∈ Boxρ < 2c1 · Z ω (ρ). Leb y ∈ I : λω yi0 ...ik + λω qk (i, j, y) Hence, (58) holds with constant c = 2c1 . This completes the proof of the claim.
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We can now complete the proof of Theorem 6.1. Using Theorem 4.2 we obtain that for P a.e y ∈ I∞ (56) holds. A simple computation shows that for every r > 0 and for every y we have dimH y ≥ dimH ry . This completes the proof of the theorem. References 1. Edgar, G.A.: Fractal dimension of self-similar sets: some examples. Supplemento ai Rendiconti del Circolo Matematico di Palermo, Serie II. 28, 341–358 (1992) 2. Edgar, G.A.: Integral, Probability and Fractal measures. Berlin-Heidelberg-NewYork: Springer, 1998 3. Falconer, K.: The Hausdorff dimension of self-affine fractals. Math. Proc. Camb. Phil. Soc. 103, 339– 350 (1988) 4. Falconer, K.J.: Fractal Geometry. NewYork: Wiley, 2003 5. Falconer, K.J.: Techniques in Fractal Geometry. NewYork: Wiley, 1997 6. Fisher, Y., Dudbridge, F., Bielefeld, B.: On the Dimension of fractally encoded images. In: Fractal Image Encoding and Analysis, Proceedings of the NATO ASI (Trondheim, 1995). Berlin-Heidelberg-NewYork: Springer, 1998, pp. 89–94 7. Käenmäki, A.: On natural invariant measures on generalised iterated function systems. Ann. Acad. Sci. Fenn. Math. 29(2), 419–458 (2004) 8. Keane, M., Simon, K., Solomyak, B.: The dimension of graph directed attractors with overlaps on the line, with an application to a problem in fractal image recognition. Fund. Math. 180(3), 279–292 (2003) 9. Krengel, U.: Ergodic Theorems. Berlin: Walter de Gruyter, 1985 10. Mattila, P.: Geometry of Sets and Measures in Euclidean Spaces. Cambridge: Cambridge University Press, 1995 11. Mauldin, R.D., Williams, S.C.: Hausdorff dimension in graph directed constructions. Trans. Amer. Math. Soc. 309, 811–829 (1998) 12. McMullen, C.: The Hausdorff dimension of general Sierpi´nski carpets. Nagoya Math. J. 96, 1–9 (1984) 13. Peres, Y., Solomyak, B.: Absolute continuity of Bernoulli convolutions, a simple proof. Math. Res. Lett. 3(2), 231–239 (1996) 14. Peres, Y., Simon, K., Solomyak, B.: Absolute continuity for random iterated function systems with overlaps. http://arxiv.org/list/math.ds/0502200, 2005 15. Rogers, C.A.: Hausdorff Measures. Cambridge: Cambridge University Press, 1970 16. Simon, K., Solomyak, B.: On the dimension of self-similar sets. Fractals 10, 59–65 (2002) 17. Solomyak, B.: Measure and dimension for some fractal families. Math. Proc. Camb. Phil. Soc. 124(3), 531–546 (1998) 18. Walters, P.: An introduction to Ergodic Theory. Berlin-Heidelberg-NewYork: Springer, 1982 Communicated by J.L. Lebowitz
Commun. Math. Phys. 270, 545–572 (2007) Digital Object Identifier (DOI) 10.1007/s00220-006-0166-2
Communications in
Mathematical Physics
Quantum Field Theory on Curved Backgrounds. I. The Euclidean Functional Integral Arthur Jaffe, Gordon Ritter Harvard University, 17 Oxford St., Cambridge, MA 02138, USA. E-mail:
[email protected];
[email protected] Received: 31 March 2006 / Accepted: 11 September 2006 Published online: 21 December 2006 – © Springer-Verlag 2006
Abstract: We give a mathematical construction of Euclidean quantum field theory on certain curved backgrounds. We focus on generalizing Osterwalder Schrader quantization, as these methods have proved useful to establish estimates for interacting fields on flat space-times. In this picture, a static Killing vector generates translations in Euclidean time, and the role of physical positivity is played by positivity under reflection of Euclidean time. We discuss the quantization of flows which correspond to classical space-time symmetries, and give a general set of conditions which imply that broad classes of operators in the classical picture give rise to well-defined operators on the quantum-field Hilbert space. In particular, Killing fields on spatial sections give rise to unitary groups on the quantum-field Hilbert space, and corresponding denselydefined self-adjoint generators. We construct the Schrödinger representation using a method which involves localizing certain integrals over the full manifold to integrals over a codimension-one submanifold. This method is called sharp-time localization, and implies reflection positivity. Introduction This article presents a construction of a Euclidean quantum field theory on time-independent, curved backgrounds. Earlier work on field theories on curved space-time (Kay [33], Dimock [14], Bros et al. [7]) uses real-time/Lorentzian signature and algebraic techniques reminiscent of P(ϕ)2 theory from the Hamiltonian point of view [22]. In contrast, the present treatment uses the Euclidean functional integral [23] and Osterwalder-Schrader quantization [38, 39]. Experience with constructive field theory on Rd shows that the Euclidean functional integral provides a powerful tool, so it is interesting also to develop Euclidean functional integral methods for manifolds. Euclidean methods are known to be useful in the study of black holes, and a standard strategy for studying black hole (BH) thermodynamics is to analytically continue time in the BH metric [10]. The present paper implies a mathematical construction of
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scalar fields on any static, Euclidean black hole background. The applicability of the Osterwalder-Schrader quantization procedure to curved space depends on unitarity of the time translation group and the time reflection map which we prove (Theorem 2.5). The Osterwalder-Schrader construction has universal applicability; it contains the Euclidean functional integral associated with scalar boson fields, a generalization of the Berezin integral for fermions, and a further generalization for gauge fields [2]. It also appears valid for fields on Riemann surfaces [28], conformal field theory [17], and may be applicable to string theory. The present paper extends this construction to models on curved backgrounds. Our paper has many relations with other work. Wald [42] studied metrics with Euclidean signature, although he treated the functional integral from a physical rather than a mathematical point of view. Brunetti et al [8] developed the algebraic approach (Haag-Kastler theory) for curved space-times and generalized the work of Dimock [13]. They describe covariant functors between the category of globally hyperbolic spacetimes with isometric embeddings, and the category of ∗-algebras with unital injective ∗-monomorphisms. The examples studied in this paper—scalar quantum field theories on static spacetimes—have physical relevance. A first approximation to a full quantum theory (involving the gravitational field as well as scalar fields) arises from treating the sources of the gravitational field classically and independently of the dynamics of the quantized scalar fields [6]. The weakness of gravitational interactions, compared with elementary particle interactions of the standard model, leads one to believe that this approximation is reasonable. It exhibits nontrivial physical effects which are not present for the scalar field on a flat spacetime, such as the Hawking effect [25] or the Fulling-Unruh effect [41]. Density perturbations in the cosmic microwave background (CMB) are calculated using scalar field theory on certain curved backgrounds [35]. Further, Witten [45] used quantum field theory on Euclidean anti-de Sitter space in the context of the AdS/CFT correspondence [24, 36]. Some of the methods discussed here in Sect. 2 have been developed for the flat case in lecture courses; see [27]. Notation and conventions. We use notation, wherever possible, compatible with standard references on relativity [44] and quantum field theory [23]. We use Latin indices a, b = 0 . . . d − 1 for spacetime indices, reserving Greek indices μ, ν = 1 . . . d − 1 for spatial directions. We include in our definition of ‘Riemannian manifold’ that the underlying topological space must be paracompact (every open cover has a locally finite open refinement) and connected. The notation L 2 (M) is used when M is a C ∞ Riemannian manifold, and implicitly refers to the Riemannian volume measure on M, which we sometimes denote by dvol. Also U(H) denotes the group of unitary operators on H. Let G = I (M) = Iso(M) denote the isometry group, while K is its Lie algebra, the global Killing fields. For ψ a smooth map between manifolds, we use ψ ∗ to denote the pullback operator (ψ ∗ f )( p) = f (ψ( p)). The notation Δ = Δ M means the Laplace operator for the Riemannian metric on M. 1. Reflection Positivity 1.1. Analytic continuation. The Euclidean approach to quantum field theory on a curved background has advantages since elliptic operators are easier to deal with than hyperbolic operators. To obtain physically meaningful results one must perform the analytic
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continuation back to real time. In general, Lorentzian spacetimes of interest may not be sections of 4-dimensional complex manifolds which also have Riemannian sections, and even if they are, the Riemannian section need not be unique. Thus, the general picture of extracting physics from the Euclidean approach is a difficult one where further investigation is needed. Fortunately, for the class of spacetimes treated in the present paper (static spacetimes), the embedding within a complex 4-manifold with a Euclidean section is guaranteed, and in such a way that Einstein’s equation is preserved [11]. 1.2. Time reflection. Reflection in Euclidean time plays a fundamental role in Euclidean quantum field theory, as shown by Osterwalder and Schrader [38, 39]. Definition 1.1 (Time reflection). Let M be a Riemannian manifold. A time reflection θ : M → M is an isometric involution which fixes pointwise a smooth codimension-one hypersurface Σ. This means that θ ∈ Iso(M), θ 2 = 1 and θ (x) = x for all x ∈ Σ. We now discuss time reflection for static manifolds, which is the example that we will study in this paper. Example 1.1 (Static manifolds). Suppose there exists a globally defined, static Killing field ξ . Fix a hypersurface Σ ⊂ M to which ξ is orthogonal. Define a global function t : M → R by setting t = 0 on Σ, and otherwise define t ( p) to be the unique number t such that φt (x) = p for some x ∈ Σ, where {φt } is the one-parameter group of isometries determined by ξ . Finally, define θ to map a point p ∈ M to the corresponding point on the same ξ -trajectory but with t (θ ( p)) = −t ( p). This defines a decomposition M = Ω− ∪ Σ ∪ Ω+ , θ Ω± = Ω∓ , θ Σ = Σ.
(1.1)
In past work [28], we have considered time-reflection maps which fall outside the bounds of Example 1.1 ([28] applies to compact Riemann surfaces, which cannot support Killing fields), but we will not do so here. The time-reflection map given by a hypersurface-orthogonal Killing field is not unique, but depends on a choice of the initial hypersurface, which we fix. The initial hypersurface will be used to define time-zero fields. Reflection of the Euclidean time coordinate t → −t analytically continues to Hermitian conjugation of e−it H . 1.3. Fundamental assumptions. Let C = (−Δ + m 2 )−1 be the resolvent of the Laplacian, also called the free covariance, where m 2 > 0. Then C is a bounded self-adjoint operator on L 2 (M). For each s ∈ R, the Sobolev space Hs (M) is a real Hilbert space, which can be defined as completion of Cc∞ (M) in the norm f 2s = f, C −s f .
(1.2)
We work with test functions in H−1 (M). This is a convenient choice for several reasons: the norm (1.2) with s = −1 is related in a simple way to the free covariance, and further, Dimock [15] has given an elegant proof of reflection positivity for Sobolev test functions. Another motivation is as follows. Suppose we wish to prove that ϕ(h) is a bounded perturbation of the free Hamiltonian H0 for a scalar field on Rd . The first-order perturbation is ˆ 1 |h( p)|2 d p, (1.3) − Ω1 , H0 Ω1 = − 2 ω( p)2
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where we used Ω1 = −H0−1 ϕ(h)Ω. Existence of (1.3) is equivalent to h ∈ H−1 (Rd ), so this is a natural condition for test functions. Therefore we choose H−1 (M) for the generalization to curved manifolds. The Sobolev spaces give rise to a natural rigging, or Gelfand triple, and various associated Gaussian measures [18, 40]. The inclusion Hs+k → Hs for k > 0 is Hilbert-Schmidt, so the spaces H∞ ≡
Hs (M) ⊂ H−1 (M) ⊂
s
Hs (M) ≡ H−∞
s
form a Gelfand triple, and H∞ is a nuclear space. There is a unique Gaussian measure μ defined on the dual H−∞ with covariance C. This means that 1 S( f ) ≡ eiΦ( f ) dμ(Φ) = e− 2 f,C f , f ∈ H∞ . H−∞
Define E := L 2 (H−∞ , μ). The space E is unitarily equivalent to Euclidean Fock space over H−1 (M) (see for example [40, Theorem I.11]). The algebra generated by monomials of the form Φ( f1 ) . . . Φ( f n ) is dense in E. This is a special case of a general construction discussed in the reference. Definition 1.2 (Standard domain). For an open set Ω ⊆ M, the standard domain in E corresponding to Ω is: E Ω = span{eiΦ( f ) : f ∈ H−1 (M), supp( f ) ⊂ Ω}. Let EΩ denote the closure in E of E Ω . Definition 1.2 refers to subspaces of E generated by functions supported in an open set. This includes empty products, so 1 ∈ EΩ for any Ω. Of particular importance for Euclidean field theory is the positive-time subspace E+ := EΩ+ , where the notation Ω+ refers to the decomposition (1.1). A linear operator on E which maps E+ → E+ is said to be positive-time invariant. 1.4. Operator induced by a diffeomorphism. We will consider the effect which diffeomorphisms of the underlying spacetime manifold have on the Hilbert space operators which arise in the quantization of a classical field theory. For f ∈ C ∞ (M) and ψ : M → M a diffeomorphism, define f ψ ≡ ψ∗ f = (ψ −1 )∗ f = f ◦ ψ −1 .
(1.4)
The reason for using ψ −1 here is so that Definition 1.3 gives a group representation.
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Definition 1.3 (Induced operator). Let ψ be a diffeomorphism, and A(Φ) = Φ( f 1 ) . . . Φ ( f n ) ∈ E a monomial. Define Γ (ψ) : A : ≡ : Φ( f 1 ψ ) · · · Φ( f n ψ ) : .
(1.5)
This extends linearly to a dense domain in E. We refer to Γ (ψ) as the operator induced by the diffeomorphism ψ. Note that if ψ is an isometry, then (1.5) is equivalent to the definition Γ (ψ)A ≡ Φ( f 1 ψ ) . . . Φ( f n ψ ) without Wick ordering, as follows from (1.8) below. The induced operators Γ (ψ) are not necessarily bounded on E. In fact, for a general diffeomorphism ψ, the operator ψ ∗ may fail to be bounded on L 2 (M) or H−1 (M). If the Jacobian |dψ| satisfies uniform upper and lower bounds, i.e. (∃ c1 , c2 > 0) c1 < sup |dψx | < c2 ,
(1.6)
x∈M
then (ψ −1 )∗ is bounded on L 2 (M), but Γ (ψ) may still be unbounded on E, because the operator norm of Γ (ψ) on the degree-n subspace of E may fail to have a limit as n → ∞. In this situation, Γ (ψ) is to be regarded as a densely-defined unbounded operator whose domain includes all finite particle vectors. If (ψ −1 )∗ is a contraction on H−1 (M), then Γ (ψ) is a contraction on E (in particular, bounded). A special case of this is ψ ∈ Iso(M), which implies that Γ (ψ) is unitary and Γ (ψ) E = 1. Lemma 1.1 (Naturalness property). Let ψ : M → M be a diffeomorphism, and consider the pullback ψ ∗ acting on L 2 (M), with its Hermitian adjoint (ψ ∗ )† . Then det(dψ) = 1 ⇔ (ψ ∗ )† = (ψ −1 )∗ ⇔ ψ is volume-preserving.
(1.7)
Furthermore, ψ ∈ Iso(M) ⇔ Γ (ψ) ∈ U(E) ⇔ [ψ ∗ , Δ] = 0 ⇔ [ψ ∗ , C] = 0.
(1.8)
The last part of (1.8) follows from [32, Theorem III.6.5], while the rest of the statements in (1.7) and (1.8) are standard calculations. It follows that Γ restricts to a unitary representation of G = Iso(M) on E. For an open set Ω ⊂ M, define Iso(M, Ω) = {ψ ∈ Iso(M) : ψ(Ω) ⊂ Ω}, and similarly Diff(M, Ω). These are not subgroups of Diff(M) but they are semigroups under composition. If ψ ∈ Diff(M, Ω) we say ψ preserves Ω. Lemma 1.2 (Presheaf property). Let ψ : U → V be a diffeomorphism, where U, V are open sets in M. Let EU , E V be the corresponding standard domains (cf. Definition 1.2). Then Γ (ψ)EU = E V . In particular, if ψ : M → M preserves Ω ⊂ M, then Γ (ψ) preserves the corresponding subspace EΩ ⊂ E.
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For maps ψ : U → V which are subset inclusions U ⊆ V , Lemma 1.2 asserts that the association U → EU is a presheaf. It also follows from Lemma 1.2 that the mappings U → EU and ψ → Γ (ψ) define a covariant functor from the category of open subsets of M with invertible, smooth maps between them into the category of Hilbert spaces and densely defined operators. Lemma 1.2 implies that if ψ(Ω+ ) ⊂ Ω+ then Γ (ψ) is positive-time invariant. This is necessary but not sufficient for Γ (ψ) to have a quantization. A sufficient condition is that Γ (ψ) and ΘΓ (ψ)† Θ both preserve E+ , where Θ = Γ (θ ), as shown by Theorem 2.1. 1.5. Continuity results Lemma 1.3 (Sobolev continuity). For the free covariance C = (−Δ + m 2 )−1 , { f 1 , . . . , f n } −→ A(Φ) := Φ( f 1 ) . . . Φ( f n ) ∈ E is a continuous map from (H−1 )n → E, where we take the product of the Sobolev topologies on (H−1 )n . Proof. Since Φ is linear, it is sufficient to show that A(Φ) E is bounded by const. i f i −1 . As a consequence of the Gaussian property of the measure dμC , one needs only bound the linear case. But 1/2 (1.9) Φ( f ) E = (Φ( f )Φ( f )) dμC = f −1 . Theorem 1.1 (Strong continuity). Let {ψn } be a sequence of orientation-preserving isometries which converge to ψ in the compact-open topology. Then Γ (ψn ) → Γ (ψ) in the strong operator topology on B(E). The proof of Theorem 1.1 follows standard arguments in analysis. Let us give a sense of how it is to be used. If all the elements of a certain one-parameter group of isometries ψt are such that Γ (ψt ) have bounded quantizations, then t → Γˆ (ψt ) defines a oneparameter group of operators on H (the quantum-field Hilbert space). In this situation, Theorem 1.1 justifies the application of Stone’s theorem. This picture is to be developed in Sect. 2.
1.6. Reflection positivity Definition 1.4. With θ as in Definition 1.1, let Θ = Γ (θ ) be the induced reflection on E. A measure μ on H−∞ is said to be reflection positive if Θ(F) F dμ ≥ 0 for all F ∈ E+ . (1.10) A bounded operator T on L 2 (M) is said to be reflection positive if supp f ⊆ Ω+ ⇒ f, θ T f L 2 (M) ≥ 0.
(1.11)
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Reflection positivity for the measure μ is equivalent to the following inequality for operators on E = L 2 (dμ): 0 ≤ Π+ ΘΠ+ , where Π+ : E → E+ is the canonical projection. A Gaussian measure with mean zero and covariance C is reflection positive iff C is reflection positive in the operator sense, Eq. (1.11). An equivalent condition is that for any finite sequence { f i } of real functions supported in Ω+ , the matrix Mi j = exp f i , θC f j has no negative eigenvalues. For Riemannian manifolds which possess an isometric involution whose fixed-point set has codimension one, there is a simple potential-theoretic proof of reflection positivity [12]. The relation between reflection positivity and operator monotonicity under change of boundary conditions for the Laplacian was discovered in [21]. A different proof of reflection positivity on curved spaces was given by Dimock [15], based on Nelson’s proof using the Markov property [37]. We give a third proof later in this paper based on our sharp-time localization theorem. The result is summarized as follows. Theorem 1.2 (Reflection positivity). Let M be a Riemannian manifold with a time reflection. Then the covariance C = (−Δ + m 2 )−1 and its associated Gaussian measure are reflection positive.
2. Osterwalder-Schrader Quantization and the Feynman-Kac Formula The Osterwalder-Schrader construction is a standard feature of quantum field theory. It begins with a “classical” Euclidean Hilbert space E and leads to the construction of a Hilbert space H = Π E+ , which is the projection Π of the Euclidean space E+ . It also yields a quantization map T → Tˆ from a classical operator T on E to a quantized operator Tˆ acting on H. In this section we review this construction, dwelling on the quantization of bounded operators T on E that may yield a bounded or an unbounded quantization Tˆ , as well as the quantization of an unbounded operator T on E. We give a variation of the previously unpublished treatment in [27], adapted to curved space-time. 2.1. The Hilbert space. Define a bilinear form (A, B) on E+ by (A, B) = Θ A, B E for A, B ∈ E+ .
(2.1)
Using self-adjointness of Θ on E, one can show that this form is sesquilinear,
(B, A) =
Θ B A dμ =
∗ B Θ A dμ = (A, B) .
(2.2)
If θ is not an isometry, then Θ is non-unitary in which case Osterwalder-Schrader quantization is not possible. Therefore, it is essential that θ ∈ Iso(M). The form (2.1) is degenerate, and has an infinite-dimensional kernel which we denote N . Therefore (2.1) determines a nondegenerate inner product , H on E+ /N , making the latter a pre-Hilbert space.
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Definition 2.1 (Hilbert space). The (Osterwalder-Schrader) physical Hilbert space H is the completion of E+ /N , with inner product , H . Let Π : E+ → H denote the natural quotient map, a contraction mapping from A ∈ E+ to Aˆ := Π A. There is an exact sequence: 0
/N
incl.
/ E+
Π
//H
/0 .
2.2. Quantization of operators. Assume that T is a densely defined, closable operator on E with domain D ⊂ E. Define T + := Θ T ∗ Θ, and assume there exists a subdomain D0 ⊂ D ∩ E+ on which T + is defined and for which both T : D0 → E+ , and T + : D0 → E+ .
(2.3)
Theorem 2.1 (Condition for quantization). Assume that Dˆ 0 := Π (D0 ) is dense in H. Condition (2.3) ensures that T has a quantization Tˆ with domain Dˆ 0 . Furthermore Tˆ ∗ is defined, Tˆ has a closure, and on Dˆ 0 , we have: + . Tˆ ∗ = T
(2.4)
Proof. First, we check that Tˆ is well-defined. Suppose A ∈ N ∩ D0 . Let B ∈ E+ range over a set of vectors in the domain of Θ T ∗ Θ such that the image of this set under Π is dense in H. Then
ˆ H = T ∗ Θ B, A E = Θ B, T A E = B, ˆ T 0 = Θ T ∗ Θ B ˆ , A A H . Thus T A ∈ N , and hence T is well-defined on D0 /D0 ∩ N . To check (2.4) is a routine calculation. The main content of Theorem 2.1 can be expressed as a commutative diagram. For bounded transformations, Theorem 2.1 simply means that if T : E+ → E+ and the dotted arrow in the following diagram is well-defined, then so are the two solid arrows: 0
/N
0
/N
incl.
/ E+
incl.
/ E+
Π
Θ T ∗Θ
T
Π
/H
/0
Tˆ
/H
/0
Lemma 2.1 (Contraction property). Let T be a bounded transformation on E such that T and Θ T ∗ Θ each preserve E+ . Then Tˆ is a bounded transformation on H and Tˆ H ≤ T E . Proof. This proceeds by the multiple reflection method [23].
(2.5)
We now discuss some examples of operators satisfying the hypotheses of Theorem 2.1. Theorem 2.2 (Self-adjointness). Let U be unitary on E, and U (E+ ) ⊂ E+ . If U −1 Θ = ΘU then U admits a quantization Uˆ and Uˆ is self-adjoint. (Do not assume U −1 preserves E+ .)
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Proof. The operator ΘU ∗ Θ = Θ 2 U = U preserves E+ , so Theorem 2.1 ⇒ U has a quantization Uˆ . Self-adjointness of Uˆ follows from Eq. (2.4). Theorem 2.3 (Unitarity). Let U be unitary on E, and U ±1 (E+ ) ⊂ E+ . If [U, Θ] = 0 then U admits a quantization Uˆ and Uˆ is unitary. Proof. The operator ΘU ∗ Θ = U ∗ = U −1 preserves E+ by assumption, so U has a quantization. Also, Θ(U −1 )∗ Θ = U preserves E+ , so U −1 also has a quantization. Obviously, the quantization of U −1 is the inverse of Uˆ . Equation (2.4) implies that the adjoint of Uˆ is the quantization of ΘU ∗ Θ = U ∗ = U −1 . Examples of operators satisfying the conditions of Theorems 2.2 and 2.3 arise from isometries on M with special properties. We now discuss two classes of isometries, which give rise to self-adjoint and unitary operators as above. Example 2.1 (Reflected Isometries). An element ψ ∈ Iso(M) is said to be a reflected isometry if ψ −1 ◦ θ = θ ◦ ψ .
(2.6)
If additionally ψ(Ω+ ) ⊆ Ω+ then Theorem 2.2 implies that Γˆ (ψ) : H → H exists and is self-adjoint. If ψ satisfies (2.6) then so does ψ −1 ; hence if ψ −1 (Ω+ ) ⊆ Ω+ , then Γ (ψ −1 ) has a quantization and Γˆ (ψ −1 ) is the inverse of Γˆ (ψ). Example 2.2 (Reflection-Invariant Isometries). A reflection-invariant isometry is an element ψ ∈ Iso(M) that commutes with time-reflection, ψθ = θ ψ. It follows that [Γ (ψ), Θ] = 0. If ψ and ψ −1 both preserve Ω+ then Γ (ψ ±1 )E+ ⊂ E+ , and Theorem 2.3 implies that Γˆ (ψ) : H → H is unitary. The set of reflection-invariant isometries form a subgroup of the full isometry group. 2.3. Quantization domains. Quantization domains are subsets of Ω+ which give rise to dense domains in H after quantization. This is important for the analysis of unbounded operators on H. For example, an isometry which satisfies (2.6) may only map a proper subset O ⊂ Ω+ into Ω+ , and in this case Γ (ψ) is only defined on a non-dense subdomain of E+ . If O is a quantization domain, then Π EO may still be dense in H, and can serve as a domain of definition for Γˆ (ψ). Definition 2.2. A quantization domain is a subspace Ω ⊂ Ω+ with the property that Π (EΩ ) is dense in H. Example 2.3. Perhaps the simplest quantization domain is a half-space at times greater than T > 0,
O+,T = x ∈ Rd : x0 > T . (2.7) Let D+,T = E O+,T = Γ (ψT )E + where ψT (x, t) = (x, t + T ); then Π (D+,T ) is dense in H, as follows from Theorem 2.4. Theorem 2.4 generalizes (2.7) to curved spacetimes, and also allows one to replace the simple half-space O+,T with a more general connected subset of Ω+ .
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Theorem 2.4 (Construction of quantization domains). Suppose that O := ψ(Ω+ ) ⊂ Ω+ . If [Γ (ψ), Θ] = 0 or Γ (ψ)Θ = ΘΓ (ψ −1 ) (i.e. ψ is reflection-invariant or reflected) then O is a quantization domain. Proof. By Lemma 1.2, we have E O = Γ (ψ)E + .
(2.8)
Let Cˆ ∈ H be orthogonal to every vector Aˆ ∈ Π (EO ). Choose B ∈ E+ and let A := Γ (ψ)B ∈ EO . Then ˆ Π (Γ (ψ)B) H = ΘC, Γ (ψ)B E . ˆ A ˆ H = C, 0 = C, Since Γ (ψ)−1 is unitary on E, apply it to the inner product to yield
Γ (ψ −1 )ΘC, B E = 0 (∀ B ∈ E+ ). Therefore Γ (ψ −1 )ΘC is orthogonal (in E) to the entire subspace E+ . First, suppose that [Γ (ψ −1 ), Θ] = 0. Then we infer ˆ B ˆ H (∀ Bˆ ∈ Π (E + )), 0 = ΘΓ (ψ −1 )C, B E = Γˆ (ψ −1 )C, i.e. Cˆ ∈ ker Γˆ (ψ −1 ). Therefore, (Π (EO ))⊥ = ker Γˆ (ψ −1 ).
(2.9)
Since [Γ (ψ −1 ), Θ] = 0, Theorem 2.3 implies that Γˆ (ψ) is unitary, hence the kernel of Γˆ (ψ −1 ) is trivial and Π (EO ) is dense in H. We have thus completed the proof in this case. Now, assume that Γ (ψ)Θ = ΘΓ (ψ −1 ). Example 2.1 implies that Γˆ (ψ) exists and is self-adjoint on H, and moreover (by the same argument used above), (Π (EO ))⊥ = ker Γˆ (ψ) . If ψ = ψt , where {ψs } is a one-parameter group of isometries, and if Γˆ (ψt ) is a strongly continuous semigroup, then by Stone’s theorem, Γˆ (ψt ) = e−t K for K self-adjoint. Since e−t K clearly has zero kernel, the proof is also complete in the second case. Corollary 2.1. The set O+,T is a quantization domain. The problem of characterizing all quantization domains appears to be open.
2.4. Construction of the Hamiltonian and ground state. Theorem 2.5 (Time-translation semigroup). Let ξ = ∂/∂t be the time-translation Killing field on the static spacetime M. Let the associated one-parameter group of isometries be denoted φt : M → M. For t ≥ 0, U (t) = Γ (φt ) has a quantization, which we denote R(t). Further, R(t) is a well-defined one-parameter family of self-adjoint operators on H satisfying the semigroup law.
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Proof. Lemma 1.1 implies that U (t) is unitary on E, and it is clearly a one-parameter group. Also, φt ◦ θ = θ ◦ φ−t and U (t)E+ ⊂ E+ for t ≥ 0, so this is a reflected isometry; see Example 2.1. Theorem 2.2 implies R(t) = Uˆ (t) is a self-adjoint transformation on H for t ≥ 0, which satisfies the group law R(t)R(s) = R(t + s) for t, s ≥ 0 wherever it is defined.
Theorem 2.6 (Hamiltonian and ground state). R(t) is a strongly continuous contracˆ There exists a densely tion semigroup, which leaves invariant the vector Ω0 := 1. defined, positive, self-adjoint operator H such that R(t) = exp(−t H ), and H Ω0 = 0. Thus Ω0 is a quantum-mechanical ground state. Proof. It is immediate that R(t)Ω0 = Ω0 . The contraction property R(t) ≤ I follows from the multiple reflection method, as explained in [23]. The remaining statements are consequences of Stone’s theorem. The operator H is the quantum mechanical generator (in the Euclidean picture) of translations in the direction ξ . When ξ = ∂/∂t, then H is called the Hamiltonian. It is immediate from the definition that Ω0 is also invariant under the quantizations of any spacetime symmetries. 2.5. Feynman-Kac theorem. ˆ Bˆ ∈ H, and let H be the Hamiltonian conTheorem 2.7 (Feynman-Kac). Let A, structed in Theorem 2.6. Each matrix element of the heat kernel e−t H is given by a Euclidean functional integral, ˆ H = Θ A U (t)B dμ(Φ). ˆ e−t H B (2.10)
A, The right-hand side of (2.10) is the Euclidean path integral [16] of quantum field theT ory. Mark Kac’ method [30, 31] for calculating the distribution of the integral 0 v(X t )dt, where v is a function defined on the state space of a Markov process X , gives a rigorous version of Feynman’s work, valid at imaginary time. In the present setup, (2.10) requires no proof, since the functional integral on the right-hand side is how we defined the matrix element on the left-hand side. However, some work is required (even for flat spacetime, M = Rd ) to see that the Hilbert space and Hamiltonian given by this procedure take the usual form arising in physics. This is true, and was carried out for Rd by Osterwalder and Schrader [38] and summarized in [23, Ch. 6]. Since H is positive and self-adjoint, the heat kernels can be analytically continued t → it. We therefore define the Schrödinger group acting on H to be the unitary group R(it) = e−it H .
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Given a time-zero field operator, action of the Schrödinger group then defines the corresponding real-time field. For flat spacetimes in d ≤ 3 it is known [23] that Theorem 2.7 has a generalization to non-Gaussian integrals, i.e. interacting quantum field theories: t ˆ e−t HV B ˆ H = Θ A, exp −
A, dt d x V (Φ(x, t )) Bt E 0 V = Θ A e−S0,t Bt dμ(Φ). (2.11) Construction of the non-Gaussian measure (2.11) in finite volume can presumably be completed by a straightforward extension of present methods, while the infinite-volume limit seems to require a cluster expansion. Work is in progress to address these issues for curved spacetimes. 2.6. Quantization of subgroups of the isometry group. Physics dictates that after quantization, a spacetime symmetry with p parameters should correspond to a unitary representation of a p-dimensional Lie group acting on H. The group of spacetime symmetries for Euclidean quantum field theory should be related to the group for the real-time theory by analytic continuation; this was shown for flat spacetime by Klein and Landau [34]. For curved spacetimes, no such construction is known, and due to the intrinsic interest of such a construction, we give further details, and show that the methods already discussed in this paper suffice to give a unitary representation of the purely spatial symmetries on H. Example 2.2 introduced reflection-invariant isometries. We now discuss an important subclass of these, the purely spatial isometries, which are guaranteed to have well-defined quantizations. We continue to assume we have a static manifold M with notation as in Example 1.1. There is a natural subgroup G space of G = Iso(M) consisting of isometries which map each spatial section into itself. We term these purely spatial isometries. The classic constructions [29] of finite-volume interactions in two dimensions work on a cylinder M = S 1 × R, in which case G space is the subgroup of Iso(S 1 × R) corresponding to rotations around the central axis. Since G space ⊂ G as a Lie subgroup, gsp = Lie(G space ) is a subalgebra of K, the Lie algebra of global Killing fields. Consider the restriction of the unitary representation Γ to the subgroup G space . By a standard construction, the derivative DΓ is a unitary Lie algebra representation of gsp on E, for which E+ is an invariant subspace. The latter property is crucial; if E+ is not an invariant subspace for an operator, then that operator does not have a quantization. As with many aspects of Osterwalder-Schrader quantization, a commutative diagram is helpful: G space Lie
Γ
(2.12)
Lie
gsp
/ U(E)
DΓ
/ u(E)
Note that U(E) is an infinite-dimensional Lie group. Further, there are delicate analytic questions involving the domains of the symmetric operators in u(E). In the present paper we investigate only the algebraic structure.
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By Theorem 2.3, each one-parameter unitary group U (t) on E+ coming from a oneparameter subgroup of G space has a well-defined quantization Uˆ (t) which is a unitary group on H. The methods of Sect. 1.5 establish strong continuity for these unitary groups, so their generators are densely-defined self-adjoint operators as guaranteed by Stone’s theorem. Suppose that [X, Y ] = Z for three elements X, Y, Z ∈ gsp . Let Xˆ : H → H be the quantization of DΓ (X ), and similarly for Y and Z . Our assumptions guarantee that [DΓ (X ), DΓ (Y )] = DΓ (Z ) is null-invariant, therefore we have [ Xˆ , Yˆ ] = Zˆ ,
(2.13)
valid on the domain of vectors in H where the expressions are defined. One-parameter subgroups coming from G space always admit unitary representations on H, but for other subgroups of G, the analogous theory is much more subtle. Since any element of K is a vector field acting on functions as a differential operator, it is local (does not change supports) and hence positive-time invariant, so quantization applied directly to infinitesimal generators may be possible. There, one runs into delicate domain issues. A discussion of the domains of some self-adjoint operators obtained by this procedure was given in Sect. 2.3, and some variant of this could possibly be used to treat the domains of the quantized generators. When applied to isometry groups, Osterwalder-Schrader quantization of operators involves the procedure of taking the derivative of a representation, applied to the infinite-dimensional group U(E). Thus, it is not surprising that it is functorial, adding to its intrinsic mathematical interest. These connections are likely to lead to an interesting new direction in representation theory, especially for noncompact groups. 3. Variation of the Metric 3.1. Metric dependence of matrix elements in quantum field theory. We wish to obtain rigorous analytic control over how quantum field theory on a curved background depends upon the metric. Definition 3.1 (Stable family). Let Mλ denote the Riemannian manifold diffeomorphic to R × S, endowed with the product metric dsλ2 := dt 2 + G μν (λ)d x μ d x ν ,
(3.1)
where G(λ) is a metric on S, and G μν (λ) depends smoothly on λ ∈ R. We refer to a family {Mλ }λ∈R satisfying these properties as a stable family. We denote the full metric (3.1) as g(λ) or gλ . For a stable family, it is clearly possible to choose Ω± , Σ in a way that is independent of λ. Let t denote the coordinate which is defined so that t|Σ = 0 and ξ = ∂/∂t. Then the data (Ω− , Σ, Ω+ , ξ, t) is constant in λ. However, the Hilbert spaces L 2 (Mλ ), the covariance C(λ) = Cλ := (−Δg(λ) + m 2
−1
),
and the test function space H−1 (Mλ ) all depend upon λ, as does the Gaussian measure described in Sect. 1.3. These dependences create many subtleties in the quantization
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procedure. In particular, the usual theory of smooth or analytic families of bounded operators does not apply to the family of operators λ → C(λ), because if λ = λ then C(λ) and C(λ ) act on different Hilbert spaces. It is clearly of interest to have some framework in which we can make sense out of the statement “λ → C(λ) is smooth.” More generally, we would like a framework to analyze the λ-dependence of the Osterwalder-Schrader quantization. Our approach to this set of problems is based on the observation that, for a stable family, there exist test functions f : M → R which are elements of H−1 (Mλ ) for all λ. For example, H−1 (Mλ ). (3.2) Cc∞ (M) ⊂ H−1 := λ∈R
Such test functions can be used to give meaning to formally ill-defined expressions such as ∂Cλ /∂λ. To give meaning to the naive expression ∂Cλ 1 f := lim (Cλ f − Cλ+ f ) , →0 ∂λ
(3.3)
we must specify the topology in which the limit is to be taken. Suppose that f ∈ Cc∞ as before. A natural choice is the topology of L 2 (Mλ ), but some justification is necessary in the noncompact case. Clearly Cλ f ∈ L 2 (Mλ ), but it is not clear that Cλ+ f also determines an element of L 2 (Mλ ). After all, the covariance operators are nonlocal, and Cλ+ f generally does not have compact support (unless of course M itself is compact). In order that the limit (3.3) can be taken in the topology of L 2 (Mλ ), it is necessary and sufficient that ∃ 1 > 0 such that Cλ+ f ∈ L 2 (M √ λ ) for all < 1 . In other words, the limit (3.3) makes sense iff F() ≡ M |Cλ+ f |2 |gλ | d x < ∞ for all < 1 . Since obviously F(0) < ∞, it suffices to show F() is continuous at = 0. If we write the expressions in terms of coordinate charts and assume f > 0, then we can translate the problem into one of classical analysis. Indeed, ⎛ ⎞2 ⎜ ⎟ F() = d x |gλ (x)| ⎝ dy |gλ+ (y)| Cλ+ (x, y) f (y)⎠ . (3.4) M
supp f
Thus the condition for differentiability of F() at = 0 becomes one of “differentiating under the integral,” which can be treated by standard methods. The overall conclusion: if F() is continuous at = 0, then (3.3) makes sense. Anticipating what is to come, this condition implies that (3.7) also makes sense. We now return to the study of the full quantum theory on Mλ . Define Eλ := L 2 (dμλ ), where dμλ is the unique Gaussian probability measure associated to C(λ) by Minlos’ theorem.1 If f ∈ H−1 , then A f,λ = : e−iΦ( f ) : C(λ)
(3.5)
1 As before, E iΦ( f ) | f ∈ H (M ), supp( f ) ⊂ Ω , with completion + +,λ = span e λ −1
E+,λ = E +,λ . Also define E λ to be the (incomplete) linear span of eiΦ( f ) for f ∈ H−1 (Mλ ).
Quantum Field Theory on Curved Backgrounds. I
defines a canonical element of Eλ for each λ. Then
A f,λ , A g,λ E ,λ = exp f, Cλ g L 2 (Mλ ) .
559
(3.6)
Lemma 3.1 (Smoothness of covariance). Assume that {Mλ }λ∈R is a stable family. Then
f, C(λ)g L 2 (Mλ ) is a smooth function of λ, for any f, g ∈ Cc∞ (M). √ Proof. The integral f, C(λ)g L 2 (Mλ ) = M f Cλ g |gλ | d x is localized over the support of f , which is compact. The dominated convergence theorem shows that we can interchange ∂/∂λ with the integral. It follows immediately that the matrix element (3.6) on E of the canonical elements A f,λ and A g,λ is a smooth function of the parameter λ. When we change λ, the measure dμλ follows a path in the space of all Gaussian measures. This change in the measure can be controlled through operator estimates on the covariance. Using formula 9.1.33 from [23, p. 208] we have: d 1 (3.7) A dμλ = (ΔdC/dλ A) dμλ . dλ 2 In particular, if C(λ) is smooth then so is A dφC(λ) . Here we must interpret dC/dλ as in the discussion following (3.3). The null space Nλ of OS quantization also depends on the metric, as we discuss presently. When it is necessary to distinguish the time direction, we denote local coordinates by x = (x, t). The subspace of Nλ corresponding to monomials in the field is canonically isomorphic to the space of test functions f such that2 f (x, −t) (Cλ f ) (x, t) |gλ (x)| d x = 0. (3.8) M
All of the quantities in the integrand (3.8) which depend on λ do so smoothly. Assuming the applicability of dominated convergence arguments similar to those used above, it should be possible to show that Nλ varies continuously in the Hilbert Grassmannian, but we do not address this here. For each λ, the Osterwalder-Schrader theory gives unambiguously a quantization H(λ) ≡ E+,λ /Nλ . Theorem 3.1 (Smoothness of matrix elements in H). Assume that {Mλ }λ∈R is a stable family. Define the canonical element A f,λ as in (3.5). Then λ → Aˆ f,λ , Rλ (t) Aˆ g,λ H(λ) is smooth. ˆ Rλ (t) B ˆ H(λ) = exp θ f, (Cλ h) ◦ φ −1 , where φλ,t is the time t Proof. Calculate A, λ,t map of the Killing field ∂/∂t on the spacetime Mλ . Since f has compact support, the −1 . dominated convergence theorem applies to the integral θ f, (Cλ h) ◦ φλ,t One class of examples which merits further consideration is the class formulated on M = Rd+1 with ds 2 = dt 2 + g(λ)i j d x i d x j , i, j = 1 . . . d. Assume that G(λ)i j depends analytically on λ ∈ C, and to order zero it is the flat metric δi j . Theorem 3.1 implies that the matrix elements of H have a well-defined series expansion about λ = 0, and we know that precisely at λ = 0 they take their usual flat-space values. 2 For integrals such as this one, we can factorize the Laplacian as in Sect. 4.
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3.2. Stably symmetric variations. It is of interest to extend the considerations of the previous section to the quantizations of symmetry generators. For this we continue to consider variations of an ultrastatic metric, as in Eq. (3.1). One important aspect of the quantization that is generally not λ-invariant is the symmetry structure of the Riemannian manifold. We assume M = R × M , where M is a Riemannian manifold with metric gμν (λ). In this section we study a special case in which the perturbation does not break the symmetry. Let Kλ denote the algebra of global Killing fields on (M , g(λ)). In certain very special cases we may have the following. Definition 3.2 (Stable symmetry). The family of metrics λ → g(λ) is said to be stably symmetric over the subinterval I ⊂ R if for each λ ∈ I , there exists a basis {ξi (λ) : 1 ≤ i ≤ n} of Kλ , and the family of bases can be chosen in such a way that λ → ξi (λ) is smooth ∀ i. Equivalently, the condition of stable symmetry is that Kλ = K F(Mλ ) gives a rank n vector bundle over R (or some subinterval thereof) and we have chosen a complete set {ξi : i = 1 . . . n} of smooth sections. Example 3.1. (Curvature variation) The most general constant-curvature hyperbolic metric on H has arc length ds =
c |dz| (z)
(3.9)
and curvature −c−2 . Consider the spacetime R × H(c) where H(c) is the upper halfplane with metric (3.9). Variation of the curvature parameter c satisfies the assumptions of Definition 3.2. Example 3.2. (ADM mass, charge, etc.) Many spacetimes considered in physics seem to have the property of stable symmetry under variation of parameters, at least for certain ranges of those parameters. For the Euclidean continuation of the Reissner-Nordström black hole, where λ plays the role of either mass m or charge e, one may observe that the assumptions of Definition 3.2 hold. However, the Euclidean RN metrics are not ultrastatic as was assumed above. Therefore, it would be interesting to extend the analysis of this section to static metrics of the form F(λ, x)dt 2 + G(λ, x)d x 2 , where x is a d − 1 dimensional coordinate. For each i, λ, the Killing field ξi (λ) gives rise to a one-parameter group of isometries on M, which we denote by φi,λ,x ∈ Iso(M), where x ∈ R is the flow parameter. These flows act on the spatial section of M for each fixed time; they are purely spatial isometries in the sense considered above. Therefore, the map Ti (λ, x) = Γ (φi,λ,x ) : E −→ E
(3.10)
is positive-time invariant, null-invariant, and has a unitary quantization Tˆi (λ, x) : H −→ H .
(3.11)
None of the following constructions depend on i, so for the moment we fix i and suppress it in the notation. Since each T (λ, x) depends on a Killing field ξ , the first step is to determine how the Killing fields vary as a function of the metric. Since the Killing fields are solutions to a first-order partial differential equation, one possible method
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561
of attack could proceed by exploiting known regularity properties of solutions to that equation. If one were to pursue that, some simplification may be possible due to the fact that a Killing field is completely determined by its first-order data at a point. We obtain a more direct proof. The T operators depend on the Killing field through its associated one-parameter flow. For each fixed λ, the construction gives a one-parameter subgroup (in particular, a curve) in G space . If we vary λ ∈ [a, b], we have a free homotopy between two paths in G space . Each cross-section of this homotopy, such as λ → φλ,x ( p) with the pair (x, p) held fixed, describes a continuous path in a particular spatial section of M. Theorem 3.2. Assume stable symmetry and define T (λ, x) as in (3.10). Then for each x (held fixed), the map λ −→ Tˆ (λ, x) ∈ U(H) is a strongly continuous operator-valued function of λ. Proof. First, we claim that λ → φλ,x is continuous in the compact-open topology. The latter follows from standard regularity theorems for solutions of ODEs, since we have assumed λ → ξ(λ) is smooth, and φλ,x ( p) is the solution curve of the differential operator ξ(λ) p . Theorem 1.1 implies that Γ (φλ,x ) ∈ U(E) is strongly continuous with respect to λ. By Theorem 2.1, the embedding of bounded operators on E into B(H) is norm-continuous. Composing these continuous maps gives the desired result. 4. Sharp-time Localization The goal of this section is to establish an analog of [23, Theorem 6.2.6] for quantization in curved space, and to show that the Hilbert space of Euclidean quantum field theory may be expressed in terms of data local to the zero-time slice. This is known as sharp-time localization. We first define the type of spacetime to which our results apply. Definition 4.1. A quantizable static spacetime is a complete, connected Riemannian manifold M with a globally defined (smooth) Killing field ξ which is orthogonal to a codimension one hypersurface Σ ⊂ M, such that the orbits of ξ are complete and each orbit intersects Σ exactly once. Under the assumptions for a quantizable static spacetime, but with Lorentz signature, Ishibashi and Wald [26] have shown that the Klein-Gordon equation gives sensible classical dynamics, for sufficiently nice initial data. These assumptions guarantee that we are in the situation of Definition 1.1. The main difficulty in establishing sharp-time localization comes when trying to prove the analog of formula (6.2.16) of [23] in the curved space case, which would imply that the restriction to E0 of the quantization map is surjective. The proof given in [23] relies on the formula (6.2.15) from Prop. 6.2.5, and it is the latter formula that we must generalize. 4.1. Localization on flat spacetime. The Euclidean propagator on Rd is given explicitly by the momentum representation 1 1 C(x; y) = C(x − y) = e−i p·(x−y) dp , d 2 p + m2 (2π ) Rd
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A. Jaffe, G. Ritter
for x, p ∈ Rd . Let f = f (x) denote a function on Rd−1 , and define f t (x, t ) = f (x)δ(t − t ) . Theorem 4.1 (Flat-space localization). Let M = Rd with the standard Euclidean metric. Then 1 (t−s)μ e g 2 d−1 ,
f t , Cgs L 2 (Rd ) = f, L (R ) 2μ 1/2
2 . where μ is the operator with momentum-space kernel μ( p) = p + m 2 4.2. Splitting the Laplacian on static spacetimes. Consider a quantizable static spacetime M, defined in Definition 4.1. Use Latin indices a, b, etc. to run from 0 to d − 1 and Greek indices μ, ν = 1 . . . d − 1. Denote the spatial coordinates by x = (x 1 , . . . , x d−1 ) = (x μ ) , and set t = x 0 . Write g in manifestly static form, F 0 1/F gab = , with inverse g ab = 0 G μν 0
0 , G μν
(4.1)
where F and G depend only on x, and not on t = x 0 . It is then clear that G := det(gab ) = F G, where G = det(G μν ) .
(4.2)
−1 It follows that g 0ν = g μ0 = 0, and g 00 = F −1 = g00 , does not depend upon time. Using the formula, Δf = G −1/2 ∂a G 1/2 g ab ∂b f , the Laplacian on M may be seen to be 1 Δ M = ∂t2 + Q, where (4.3) F √ 1 Q := √ ∂μ ( G G μν ∂ν ). (4.4) G
The operator Q is related to the Laplacian ΔΣ for the induced metric on Σ. Applying the product rule to (4.3) yields 1 ∂α (ln F) G αβ ∂β + ΔΣ . (4.5) 2 Note that a formula generalizing (4.5) to “warped products” appears in Bertola et.al. [5]. In order that the operator μ = (−Q + m 2 )1/2 exists for all m 2 > 0, we require that −Q is a positive, self-adjoint operator on an appropriately-defined Hilbert space. The correct Hilbert space is √ KΣ := L 2 (Σ, G d x) . (4.6) √ Here G d x denotes the Borel measure on Σ which has the indicated form in each local coordinate system, and G = F G as in Eq. (4.2). Spectral theory of the operator −Q considered on KΣ is mathematically equivalent to that of the “wave operator” A defined by Wald [42, 43] and Wald and Ishibashi [26]. In those references, the Klein-Gordon equation has the form (∂t2 + A)φ = 0. The relation between Wald’s notation and ours is that Q = −(1/F)A − m 2 , and Wald’s function V is our F 1/2 . As pointed out by Wald, we have the following, Q=
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Theorem 4.2 (Q is symmetric and negative). Let (M, gab ) be a quantizable static spacetime. Then −Q is a symmetric, positive operator on the domain Cc∞ (Σ) ⊂ KΣ . Proof. It is easy to see that Q is symmetric on Cc∞ (Σ) with the metric of KΣ ; it remains to show −Q ≥ 0 on the same domain. Using (4.4), the associated quadratic form is √ √ 1 f √ ∂μ ( G G μν ∂ν f ) G d x
f, (−Q) f KΣ = − G √ = ∇ f 2G G d x ≥ 0, where we used integration by parts to go from the first line to the second.
4.3. Hyperbolic space. It is instructive to calculate Q in the explicit example of H d , often called Euclidean AdS in the physics literature because its analytic continuation is the Anti-de Sitter spacetime. The metric is ds 2 = r −2
d−1
d xi2 , r = xd−1 .
i=0
The hyperbolic Laplacian in d dimensions is (see for instance [4]): ΔH d = (2 − d)r
∂ + r 2 ΔR d . ∂r
(4.7)
Any vector field ∂/∂ xi where i = d − 1 is a static Killing field. We have set up the coordinates so that it is convenient to define t = x0 as before, and we can quantize in the t direction. Comparing (4.4) with (4.7), we find that F = r −2 and ∂2 ∂ ∂ + r2 + ΔH d−1 , = −r 2 ∂r ∂r ∂ xi d−1
Q = (2 − d)r
(4.8)
i=1
which matches (4.5) perfectly. We return to this example spacetime in Appendix A, where we calculate its Green function, and discuss the analytic continuation.
4.4. Curved space localization. To generalize Theorem 4.1 to curved space, choose static coordinates x, t near the time-zero slice Σ. If f = f (x) is a function on the slice Σ, we define f t (x, t ) = f (x)δ(t − t ), which is a distribution on the patch of M covered by this coordinate chart. For the moment, we assume that this coordinate patch is the region of interest. By Eq. (4.4), we infer that the integral kernel C(x, y) of the operator C = (−Δ+m 2 )−1 is time-translation invariant, so that we may write C(x, y) = C(x, y, x0 − y0 ).
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In order to apply spectral theory to Q, we choose a self-adjoint extension of the symmetric operator constructed by Theorem 4.2. For definiteness, we may choose the Friedrichs extension, but any ambiguity inherent in the choice of a self-adjoint extension will not enter into the following analysis. We denote the self-adjoint extension also by Q, which is an unbounded operator on KΣ . The following is a generalization of Theorem 4.1 to curved space. Theorem 4.3 (Localization of sharp-time integrals). Let M be a quantizable static spacetime (Definition 4.1). Then:
f t , Cgs M = f,
F
1/2 e
−|t−s|ω
2ω
F
1/2
g KΣ
,
(4.9)
√ √ where μ = (−Q + m 2 )1/2 and ω = ( Fμ2 F)1/2 . Hence C is reflection positive on L 2 (M). Proof. Because M was assumed to be a quantizable static spacetime, F = ξ, ξ Σ ≥ 0. Moreover, if F( p) = 0 then ξ p = 0, for any p ∈ Σ. A non-trivial Killing field cannot vanish on an open set, so the zero-set of F has measure zero in Σ. From this we infer that multiplication by the function F −1 defines a (possibly-unbounded) but densely-defined self-adjoint multiplication operator on KΣ . For simplicity of notation, assume f is real-valued. Perform a partial Fourier transform with respect to the time variable: √ 1 ei E(t−s)
f t , Cgs M = f (x) g (x) G d x . (4.10) d E −1 2 2 2π F E − Q+m 1/2
, where the square root is defined through the spectral Define μ := −Q + m 2 calculus on KΣ . As a consequence of Theorem 4.2, μ and ω are positive, self-adjoint operators on KΣ . The integrand of (4.10) contains the operator: i E(t−s) ei E(t−s) ei E(t−s) 1/2 e
= = F F 1/2 . F −1 E 2 + μ2 E 2 + ω2 F −1/2 E 2 + F 1/2 μ2 F 1/2 F −1/2
We next establish that ω is invertible. Since μ2 > I , where > 0, we have ω2 =
√
√ Fμ2 F > F
and therefore, √ √ 1 . ω−2 < ( Fμ2 F)−1 < F Since 1/F is a densely defined operator on KΣ , it follows that ω2 (hence ω) is invertible. For λ > 0,
π e−|τ |λ ei Eτ . dE = 2 2 E +λ λ
(4.11)
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565
Decompose the operator ω according to its spectral resolution, with ω = λ d Pλ and I = d Pλ the corresponding resolution of the identity, and apply (4.11) in this decomposition to conclude π e−|t−s|ω 1/2 ei E(t−s) d E = F 1/2 (4.12) F . −1 2 2 F E +μ ω Inserting (4.12) into (4.10) gives −|t−s|ω ! √ e (F 1/2 g) (x) G d x F 1/2 f (x)
f t , Cgs M = 2ω Σ
e−|t−s|ω 1/2 = f, F 1/2 F g , 2ω KΣ also demonstrating reflection positivity.
(4.13)
The operator ω2 may be calculated explicitly if the metric is known, and is generally not much more complicated than Q. For example, using the conventions of Sect. 4.3, one may calculate ω2 for H d : ω2 = −
d−1
∂i2 + d r −1 ∂r + (m 2 − d)r −2 .
i=1
For H 2 , the eigenvalue problem ω2 f = λ f becomes a second-order ODE which is equivalent to Bessel’s equation. The two linearly independent solutions are √ √ r 3/2 J 1 √4m 2 +1 (r λ) and r 3/2 Y 1 √4m 2 +1 (r λ). 2
2
The spectrum of ω2 on H 2 is then [0, +∞). Given a function f on Σ, we obtain a distribution f t supported at time t as follows: f t (x, t ) = f (x)δ(t − t ). It may appear that this is not well-defined because it depends on a coordinate. However, given a static Killing vector, the global time coordinate is fixed up to an overall shift by a constant, which we have determined by the choice of an orthogonal hypersurface where t = 0. Thus a pair ( p, t), where p ∈ Σ and t ∈ R uniquely specify a point in M. Theorem 4.4 (Localization of H). Let M be a quantizable static spacetime. Then the vectors exp(iΦ( f 0 )) lie in E+ , and quantization maps the span of these vectors isometrically onto H. Proof. Since E+ is the closure of the set E + of vectors exp(iΦ( f )) with supp( f ) ⊂ Ω+ , it follows that any sequence in E+ which converges in the topology of E has its limit in E+ . The L 2 norm in E, 2 1 iΦ( f ) − eiΦ(g) dμC (Φ) = 2(1 − e− 2 f −g −1 ), e is controlled in terms of the norm −1 on Sobolev space, which is the space of test functions. This will give us the first part of the theorem.
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A. Jaffe, G. Ritter
If t > 0, then there exists a sequence of smooth test functions {gn } with compact, positive-time support such that lim gn = f t
n→∞
in the Sobolev topology, hence exp(iΦ( f t )) ∈ E+ . Define the time-t subspace Et ⊂ E+ to be the subspace generated by vectors of the form exp(iΦ( f t )). By taking the t → 0 limit, we see that exp(iΦ( f 0 )) ∈ E+ and the first part is proved. It is straightforward to see that the quantization map Π (A) ≡ Aˆ is isometric when restricted to vectors of the form exp(iΦ( f 0 )), since the time-reflection θ acts trivially on these vectors. It remains to see that the restriction to such vectors is onto H. Then we wish to prove (E0 ) ˆ ⊃ Et ˆ . (4.14) t>0
" First, let us see why (4.14), if true, finishes the proof. We must show that t>0 Et is dense in E+ . Of course, E+ is spanned by polynomials in classical fields of the form √ Φ( f ) = Φ(x, t) f (x, t) G d xdt . Write the t integral as a Riemann sum: Φ( f ) = lim
N →∞
where
N
(δt)i Φ ( f i )ti ,
i=1
Φ ( f i )ti =
(4.15)
√ Φ(x, ti ) f i (x) G d x,
(4.16)
and where f i (x) = f (x, ti ). Equation (4.15) represents Φ( f ) as a limit of linear combinations of elements Φ( f ti ) ∈ Eti . A similar argument applies " to polynomials A(Φ) of classical fields, and to L 2 limits of such polynomials. Thus t>0 Et is dense in E+ . Then (4.14) implies (E0 ) ˆ is also dense in E+ . Equation (4.14) is proved by means of the following identity: ˆ : exp(i Φ( f t )) : ˆ H = A, ˆ : exp(i Φ( f t 0 )) : ˆ H ,
A,
(4.17)
f t := (F −1/2 e−tω F 1/2 ) f,
(4.18)
where where f is a function on Σ, and hence so is f t . Thus f t 0 ( p, t ) = δ(t )(F −1/2 e−tω F 1/2 f )( p) To prove (4.17), we first suppose A = :
eiΦ(gs )
for
p∈Σ.
: where g ∈ TΣ and s > 0. Then
ˆ : exp(iΦ( f t )) : ˆ H = : eiΦ(θgs ) : , : eiΦ( ft ) : E
A, = exp θgs , C f t M # e−(t+s)ω 1/2 $ F f K , = exp g, F 1/2 Σ 2ω where we have used localization (Theorem 4.3) in the last line.
(4.19)
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567
Computing the right side of (4.17) gives # iΦ(θg ) $ # $ t t s : , : eiΦ( f 0 ) : :e E = exp θgs , C( f 0 ) M # e−sω 1/2 t $ = exp g, F 1/2 F f K Σ 2ω −(t+s)ω # $ e F 1/2 f K = (4.19). = exp g, F 1/2 Σ 2ω We conclude that Eqs. (4.17)–(4.18) hold"true for A = : eiΦ(gs ) : . We then infer the validity of (4.17) for all A in the span of t>0 Et by linear combinations and limits. Equation (4.17) says that for every vector v in a set that is dense in H, there exists v ∈ (E0 ) ˆ such that L(v) = L(v ) for any linear functional L on H. If v = v then we could find some linear functional to separate them, so they are equal. Therefore (E0 ) ˆ is a dense set, completing the proof of Theorem 4.4. Theorem 4.4 implies that the physical Hilbert space is isometrically isomorphic to E0 , and to an L 2 space of the Gaussian measure with covariance which can be found by the t, s → 0 limit of (4.19), to be:
∗ 1 1/2 F , H = L 2 Nd−1 , dφC , where C = F 1/2 2ω
(4.20)
and Nd−1 denotes the nuclear space over the (d − 1)-dimensional slice. Compare (4.20) with [23], Eq. (6.3.1). By assumption, 0 lies in the resolvent set of ω, implying that C is a bounded, self-adjoint operator on KΣ . 4.5. The ϕ bound. Here we prove that an estimate known in constructive field theory as the Glimm-Jaffe ϕ bound (see [20]) is also true for curved spacetimes. Theorem 4.5 (ϕ bound). Let T > 0. There exists a constant M such that ˆ e−(H0 +ϕ(h))T Aˆ ≤ exp(T h 2G M) A ˆ 2, A, H
(4.21)
H
where h G = h, Gh 1/2 and G is the resolvent of Q at −m 2 . Proof. Apply the Schwartz inequality (for the inner product on H) n times, to obtain 1/2 ˆ e−2T (H0 +ϕ(h)) Aˆ ˆ e−(H0 +ϕ(h))T Aˆ ≤ A ˆ H A, A, H H 2−n −(n−1) n T (H +ϕ(h)) 2−2 −2 0 ˆ e ˆ ˆ A, ≤ A . A H H
Apply the Feynman-Kac formula to the very last expression, to obtain
ˆ e−(H0 +ϕ(h))T Aˆ A,
H
2−n 2n T 2−2−(n−1) − 0 Φ(h,t)dt n ˆ Θ A, e ≤ A H U (2 T )A . E
We cannot take the n → ∞ limit at this point, because the object depends on A. It suffices to establish the desired result for A in a dense subspace, so take A ∈ L 4 ∩ E+ .
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We now use the Schwartz inequality on E as well as the fact that Θ is unitary on E, to obtain 2−(n+1) 2n T ˆ e−(H0 +ϕ(h))T Aˆ ˆ 2−2−(n−1) A 2−n A, e2 0 Φ(h,t)dt A A, ≤ A . E H H
E
Now Hölder’s inequality with exponents
1 4
+
ˆ e−(H0 +ϕ(h))T A ˆ H
A, ˆ 2−2 ≤ A H
−(n−1)
−n
1 4
−n
+
1 2
A 2E A 2L 4
e4
= 1 implies
2n T 0
Φ(h,t)dt
2−n−2 dμ0
.
(4.22)
Up to this point, the argument applies to a general measure dμ on path space. Now assume that the measure is Gaussian. The function f = 4h(x)χ[0,2n T ] (t) has the desir 2n T able property that Φ( f ) = 4 0 Φ(h, t)dt, so the Gaussian integral in (4.22) equals S(i f ) = e f,C f /2 . Therefore, −n−2 ˆ e−(H0 +ϕ(h))T Aˆ ≤ A ˆ 2−2−(n−1) A 2−n A 2−n A, S(i f )2 . (4.23) E H L4 H
For H1 and H2 self-adjoint operators with 0 ≤ H1 ≤ H2 , we have (H2 + a)−1 ≤ (H1 + a)−1 for any a > 0. By Theorem 4.2, −Q ≥ 0, so take H1 = −Q, and H2 = −(1/F)∂t2 − Q. We conclude3 C = (−Δ + m 2 )−1 ≤ (−Q + m 2 )−1 ≡ G. Since ker(G) = {0}, G determines a norm h G = h, Gh 1/2 . Then S(i f ) ≤ e8 h,Gh 2
nT
= e2
n+3 T h 2 G
.
Raising this to the power 2−n−2 , and taking the n → ∞ limit we see that the factors −n −n A 2E A 2L 4 approach 1, and thus (4.23) becomes:
ˆ e−(H0 +ϕ(h))T Aˆ A,
H
ˆ 2. ≤ e2T h G A H 2
This establishes (4.21), completing the proof of Theorem 4.5.
4.6. Fock representation for time-zero fields. To obtain a Fock representation of the time-zero fields we mimic the construction of [23, §6.3] with the covariance (4.20). To simplify the constructions in this section, we assume the form ds 2 = dt 2 + G μν d x μ d x ν and F = 1. Then Q = ΔΣ , the Laplacian on the time-zero slice, and μ = (−ΔΣ + m 2 )1/2 . The set of functions h ∈ L 2 (Σ) such that μ p h ∈ L 2 (Σ) is precisely the Sobolev space H p (Σ), which is also the set of h such that C− p h ∈ L 2 . Sobolev spaces satisfy the reverse inclusion relation p ≥ q ⇒ Hq ⊆ H p . Also Cq f ∈ H p ⇔ f ∈ Hq− p . 3 Compare this with the analogous estimate valid in Rd , C ≤ (−∇ 2 + m 2 )−1 , which may be proved by a x Fourier transform of the resolvent kernel.
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569
This allows us to determine the natural space of test functions for the definition of the Fock representations: ! ! 1 φ C−1/2 f + iπ C1/2 f , 2 ! ! 1 ∗ a ( f ) = φ C−1/2 f − iπ C1/2 f . 2 a( f ) =
In particular, if the natural domain of φ is H−1 as discussed following Eq. (1.3), then f must lie in the space where C−1/2 f ∈ H−1 , i.e. f ∈ H1/2 . 5. Conclusions and Outlook We have successfully generalized Osterwalder-Schrader quantization and several basic results of constructive field theory to the setting of static spacetimes. Dimock [14] constructed an interacting P(ϕ)2 model with variable coefficients, with interaction density ρ(t, x) : ϕ(x)4 :, and points out that a Riemannian (ϕ 4 )2 theory may be reduced to a Euclidean (ϕ 4 )2 theory with variable coefficients. However, the main constructions of [14] apply to the Lorentzian case and for curved spacetimes no analytic continuation between them is known. Establishing the analytic continuation is clearly a priority. Also, there are certain advantages to a perspective which remembers the spacetime structure; for example, in this picture the procedure for quantizing spacetime symmetries is more apparent. In the present paper we have not treated the case of a non-linear field, though all of the groundwork is in place. Such construction would necessarily involve a generalization of the Feynman-Kac integral (2.11) to curved space, and would have far-reaching implications, and one would like to establish properties of the particle spectrum for such a theory. The treatment of symmetry in this paper is only preliminary. We have isolated two classes of isometries, the reflected and reflection-invariant isometries, which have welldefined quantizations. We believe that this construction can be extended to yield a unitary representation of the isometry group, and work on this is in progress. This, together with suitable extensions of Sect. 2.6 could have implications for the representation theory of Lie groups, as is already the case for the geometric quantization of classical Hamiltonian systems. The treatment of variation of the metric in Sect. 3 is also preliminary; it does not cover the full class of static spacetimes. Geroch [19] gave a rigorous definition of the limit of a family of spacetimes, which formalizes the sense in which the Reissner-Nordström black hole becomes the Schwarzschild black hole in the limit of vanishing charge. It would be interesting to combine the present framework with Geroch’s work to study rigorously the properties of the quantum theory under a limit of spacetimes. Another direction is to isolate specific spacetimes suggested by physics which have high symmetry or other special properties, and then to extend the methods of constructive field theory to obtain mathematically rigorous proofs of such properties. Several studies along these lines exist [7, 26], but there is much more to be done. We hope that the Euclidean functional integral methods developed here may facilitate further progress. Rigorous analysis of thermal properties such as Hawking radiation should be possible. Given that new mathematical methods are available which pertain to Euclidean quantum field theory in AdS, a complete, rigorous understanding of the holographically dual
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theory on the boundary of Ad S suggested by Maldacena [1, 24, 36, 45] may be within reach of present methods. Constructive field theory on flat spacetimes has been developed over four decades and comprises thousands of published journal articles. Every statement in each of those articles is either: (i) an artifact of the zero curvature and high symmetry of Rd or Td or (ii) generalizable to curved spaces with less symmetry. The present paper shows that the Osterwalder-Schrader construction and many of its consequences are in class (ii). For each construction in class (ii), investigation is likely to yield non-trivial connections between geometry, analysis, and physics. Acknowledgements. We would like to thank Jonathan Weitsman and Joachim Krieger for interesting discussions, and Jon Dimock for his earlier work [14, 15] which sparked our interest in these models.
A. Euclidean Anti-de Sitter and its Analytic Continuation The Green’s function G on a general curved manifold is the inverse of the corresponding positive transformation, so it satisfies (Δ − μ2 )G = −g −1/2 δ ,
(A.1)
where G( p, q) is a function of two spacetime points. By convention Δ acts on G in the first variable, and δ denotes the Dirac distribution of the geodesic distance d = d( p, q). Translation invariance implies that G only depends on p and q through d( p, q). We note that solutions of the homogeneous equation (Δ − μ2 )φ = 0 may be recovered from the Green’s function. Conversely, we may deduce the Green’s function by solving the homogeneous equation for d > 0 and enforcing the singularity at d = 0. Equation (A.1) for the Green’s function takes a simple form in geodesic polar coordinates on H n with r = d = geodesic distance; the Green’s function has no dependence on the angular variables and the radial equation yields ! (A.2) ∂r2 + (n − 1) coth(r )∂r − μ2 G(r ) = −δ(r ) . We find it convenient to write the homogeneous equation in terms of the coordinate u = cosh(r ). When u = 1, (A.2) becomes (Δ − μ2 )G(u) = −(1 − u 2 )G (u) + nuG (u) − μ2 G(u) = 0 .
(A.3)
For n = 2 and μ2 = ν(ν +1), Eq. (A.3) is equivalent to Legendre’s differential equation: (1 − u 2 )Q ν (u) − 2u Q ν (u) + ν(ν + 1)Q ν (u) = 0 .
(A.4)
Note that (A.4) has two independent solutions for each ν, called Legendre’s P and Q functions, but the Q function is selected because it has the correct singularity at r = 0. Thus 1 1 1 1/2 2 2 Q ν (cosh r ), where ν = − + μ + G 2 (r ; μ ) = . (A.5) 2π 2 4 The case μ2 = 0 is particularly simple; there the Legendre function becomes elementary: r! 1 1 ln tanh = Q 0 (cosh r ) . G 2 (r ; 0) = − (A.6) 2π 2 2π
Quantum Field Theory on Curved Backgrounds. I
For n = 3, one has
√ ±r μ2 +1 e 1 . G 3 (r ; μ2 ) = 4π sinh(r )
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(A.7)
Finally, we note that the analytic continuation of (A.5) gives the Wightman function on Ad S2 . The real-time theory on Anti-de Sitter, including its Wightman functions, were discussed by Bros et al. [7]. In particular, our Eq. (A.5) analytically continues to their Eq. (6.8). Given a complete set of modes, one may also calculate the Feynman propagator by using the relation i G F (x, x ) = 0 | T {φ(x)φ(x )} | 0 and performing the mode sum explicitly as in [9]; the answer may be seen to be related to the above by analytic continuation. Here, T denotes an Ad S-invariant time-ordering operator. A good general reference is the classic paper [3]. References 1. Aharony, O., Gubser, S.S., Maldacena, J., Ooguri, H., Oz, Y.: Large N field theories, string theory and gravity. Phys. Rep. 323(3–4), 183–386 (2000) 2. Ashtekar, A., Lewandowski, J., Marolf, D., Mourão, J., Thiemann, T.: SU(N ) quantum Yang-Mills theory in two dimensions: a complete solution. J. Math. Phys. 38(11), 5453–5482 (1997) 3. Avis, S.J., Isham, C.J., Storey, D.: Quantum field theory in anti-de Sitter space-time. Phys. Rev. D (3) 18(10), 3565–3576 (1978) 4. Beardon, A.F. The Geometry of Discrete Groups. Volume 91 of Graduate Texts in Mathematics. New York: Springer-Verlag, (1995) (Corrected reprint of the 1983 original) 5. Bertola, M., Bros, J., Gorini, V., Moschella, U., Schaeffer, R.: Decomposing quantum fields on branes. Nucl. Phys. B 581(1–2), 575–603 (2000) 6. Birrell, N.D., Davies, P.C.W.: Quantum Fields in Curved space. Volume 7 of Cambridge Monographs on Mathematical Physics. Cambridge: Cambridge University Press (1982) 7. Bros, J., Epstein, H., Moschella, U.: Towards a general theory of quantized fields on the anti-de Sitter space-time. Commun. Math. Phys. 231(3), 481–528 (2002) 8. Brunetti, R., Fredenhagen, K., Verch, R.: The generally covariant locality principle—a new paradigm for local quantum field theory. Commun. Math. Phys. 237(1–2), 31–68 (2003) 9. Burgess, C.P., Lütken, C.A.: Propagators and effective potentials in anti-de Sitter space. Phys. Lett. B 153(3), 137–141 (1985) 10. Carlip, S., Teitelboim, C.: Aspects of black hole quantum mechanics and thermodynamics in 2 + 1 dimensions. Phys. Rev. D (3) 51( 2), 622–631 (1995) 11. Chru´sciel, P.T.: On analyticity of static vacuum metrics at non-degenerate horizons. Acta Phys. Polon. B 36(1), 17–26 (2005) 12. De Angelis, G.F., de Falco, D., Di Genova, G.: Random fields on Riemannian manifolds: a constructive approach. Commun. Math. Phys. 103(2), 297–303 (1986) 13. Dimock, J.: Algebras of local observables on a manifold. Commun. Math. Phys. 77(3), 219–228 (1980) 14. Dimock, J.: P(ϕ)2 models with variable coefficients. Ann. Phys. 154(2), 283–307 (1984) 15. Dimock, J.: Markov quantum fields on a manifold. Rev. Math. Phys. 16(2), 243–255 (2004) 16. Feynman, R.P.: Space-time approach to non-relativistic quantum mechanics. Rev. Mod. Phys. 20, 367– 387 (1948) 17. Gaw¸edzki, K.: Lectures on conformal field theory. In: Quantum Fields and Strings: a Course for Mathematicians, Vol. 1, 2 (Princeton, NJ, 1996/1997) Providence, RI: Amer. Math. Soc., (1999) pp. 727–805 18. Gel’fand, I.M., Vilenkin, N.Ya.: Generalized Functions. Volume 4, Applications of harmonic analysis, Translated from the Russian by A. Feinstein, New York: Academic Press [Harcourt Brace Jovanovich Publishers], 1964 [1977] 19. Geroch, R.: Limits of spacetimes. Commun. Math. Phys. 13, 180–193 (1969) 20. Glimm, J., Jaffe, A.: The λφ24 quantum field theory without cutoffs. IV. Perturbations of the Hamiltonian. J. Math. Phys. 13, 1568–1584 (1972) 21. Glimm, J., Jaffe, A.: A note on reflection positivity. Lett. Math. Phys. 3(5), 377–378 (1979) 22. Glimm, J., Jaffe, A. Quantum Field Theory and Statistical Mechanics. Boston, MA: Birkhäuser Boston Inc., 1985
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23. Glimm, J., Jaffe, A.: Quantum Physics: a Functional Integral Point of View. New York: Springer-Verlag, Second edition, 1987 24. Gubser, S.S., Klebanov, I.R., Polyakov, A.M.: Gauge theory correlators from non-critical string theory. Phys. Lett. B 428(1–2), 105–114 (1998) 25. Hawking, S.W.: Particle creation by black holes. Commun. Math. Phys. 43(3), 199–220 (1975) 26. Ishibashi, A., Wald, R.M.: Dynamics in non-globally-hyperbolic static spacetimes. II. General analysis of prescriptions for dynamics. Class. Quantum Grav. 20(16), 3815–3826 (2003) 27. Jaffe, A.: Introduction to Quantum Field Theory, 2005. Lecture notes from Harvard Physics 289r, available online at http://www.arthurjaffe.com/Assets/pdf/IntroQFT.pdf 28. Jaffe, A., Klimek, S., Lesniewski, A.: Representations of the Heisenberg algebra on a Riemann surface. Commun. Math. Phys. 126(2), 421–431 (1989) 29. Jaffe, A., Lesniewski, A., Weitsman, J.: The two-dimensional, N = 2 Wess-Zumino model on a cylinder. Commun. Math. Phys. 114(1), 147–165 (1988) 30. Kac, M.: On distributions of certain Wiener functionals. Trans. Amer. Math. Soc. 65, 1–13 (1949) 31. Kac, M.: On Some Connections Between Probability Theory and Differential and Integral Equations. In: Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, 1950. Berkeley and Los Angeles: University of California Press, 1951, pp. 189–215 32. Kato, T.: Perturbation theory for linear operators. Classics in Mathematics. Berlin: Springer-Verlag, 1995, reprint of the 1980 edition 33. Kay, B.S.: Linear spin-zero quantum fields in external gravitational and scalar fields. I. A one particle structure for the stationary case. Commun. Math. Phys. 62(1), 55–70 (1978) 34. Klein, A., Landau, L.J.: From the Euclidean group to the Poincaré group via Osterwalder-Schrader positivity. Commun. Math. Phys. 87(4), 469–484 (1982/83) 35. Lyth, D.H., Riotto, A.: Particle physics models of inflation and the cosmological density perturbation. Phys. Rep. 314(1–2), 146 (1999) 36. Maldacena, J.: The large N limit of superconformal field theories and supergravity. Adv. Theor. Math. Phys. 2(2), 231–252 (1998) 37. Nelson, E.: Construction of quantum fields from Markoff fields. J. Funct. Anal. 12, 97–112 (1973) 38. Osterwalder, K., Schrader, R.: Axioms for Euclidean Green’s functions. Commun. Math. Phys. 31, 83– 112 (1973) 39. Osterwalder, K., Schrader, R.: Axioms for Euclidean Green’s functions. II. Commun. Math. Phys. 42, 281–305, (1975) (with an appendix by Stephen Summers) 40. Simon, B.: The P(φ)2 Euclidean (quantum) Field Theory. Princeton, NJ: Princeton University Press, 1974 41. Unruh, W.G.: Notes on black hole evaporation. Phys. Rev. D14, 870 (1976) 42. Wald, R.M.: On the Euclidean approach to quantum field theory in curved spacetime. Commun. Math. Phys. 70(3), 221–242 (1979) 43. Wald, R.M.: Dynamics in nonglobally hyperbolic, static space-times. J. Math. Phys. 21(12), 2802– 2805 (1980) 44. Wald, R.M.: General Relativity. Chicago, IL: University of Chicago Press, 1984 45. Witten, E.: Anti de Sitter space and holography. Adv. Theor. Math. Phys. 2(2), 253–291 (1998) Communicated by J.Z. Imbrie
Commun. Math. Phys. 270, 573 (2007) Digital Object Identifier (DOI) 10.1007/s00220-006-0138-6
Communications in
Mathematical Physics
Publisher’s Erratum
Nonlinear Maxwell Theory and Electrons in Two Dimensions Artur Sowa 120 Monticello Avenue, Durham, NC 27707, USA. E-mail:
[email protected] Published online: 4 November 2006 – © Springer-Verlag 2006 Commun. Math. Phys. 226, 559–566 (2002)
The original online version of this article unfortunately contains a mistake. The name of the author, Artur Sowa, is missing on the journal’s website.
The online version of the original article can be found at http://dx.doi.org/10.1007/s002200200619
Commun. Math. Phys. 270, 575–585 (2007) Digital Object Identifier (DOI) 10.1007/s00220-006-0157-3
Communications in
Mathematical Physics
Nonzero Kronecker Coefficients and What They Tell us about Spectra Matthias Christandl1 , Aram W. Harrow2 , Graeme Mitchison1 1 Centre for Quantum Computation, DAMTP, University of Cambridge, Wilberforce Road,
Cambridge, CB3 0WA, UK. E-mail:
[email protected];
[email protected]
2 Department of Computer Science, University of Bristol, Bristol, BS8 1UB, UK. E-mail:
[email protected]
Received: 20 February 2006 / Accepted: 27 June 2006 Published online: 9 January 2007 – © Springer-Verlag 2006
Abstract: A triple of spectra (r A , r B , r AB ) is said to be admissible if there is a density operator ρ AB with (Spec ρ A , Spec ρ B , Spec ρ AB ) = (r A , r B , r AB ). How can we characterise such triples? It turns out that the admissible spectral triples correspond to Young diagrams (μ, ν, λ) with nonzero Kronecker coefficient gμνλ [5, 14]. This means that the irreducible representation of the symmetric group Vλ is contained in the tensor product of Vμ and Vν . Here, we show that such triples form a finitely generated semigroup, thereby resolving a conjecture of Klyachko [14]. As a consequence we are able to obtain stronger results than in [5] and give a complete information-theoretic proof of the correspondence between triples of spectra and representations. Finally, we show that spectral triples form a convex polytope. 1. Introduction A curious connection between representation theory and the spectra of operators was discovered recently. Suppose we are given a bipartite density operator ρ AB , and suppose this has spectrum r AB = Spec (ρ AB ). Let r A be the spectrum of the marginal operator ρ A = Tr B ρ AB , and r B that of the other marginal operator ρ B . Then clearly there are restrictions on the possible spectral triples (r A , r B , r AB ) as ρ AB ranges over all density operators. For instance, if ρ AB is pure, so r AB = (1, 0, . . . ), then r A = r B . How does one characterise the set of possible spectral triples? One way to do this is via representation theory [5, 14]: there is a correspondence between triples of spectra and irreducible representations of the symmetric group Vμ , Vν and Vλ , where Vλ ⊂ Vμ ⊗ Vν .
(1)
Two rather different methods were used to prove this. In [14] a body of powerful techniques from invariant theory [11, 15, 2] were harnessed (see also [16, 7]). In [5], the
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approach came from the direction of quantum information theory, and a key ingredient was a theorem relating spectra and Young diagrams due to Alicki, Rudnicki and Sadowski [1] and Keyl and Werner [12]. This theorem can be given a short and elegant proof [10] (see also [5]) that has interesting parallels with classical information theory. To those with an information theory background, therefore, the approach taken in [5] has some advantages of accessibility. It is shown there that for every density operator ρ AB there is a sequence of triples (μ( j) , ν ( j) , λ( j) ) satisfying relation (1) that converges to the spectra: lim (μ¯ ( j) , ν¯ ( j) , λ¯ ( j) ) = (r A , r B , r AB ),
j→∞
where the bar denotes normalisation. Klyachko [14] proves this as well as a converse that says that to every (μ, ν, λ) with Vλ ⊂ Vμ ⊗ Vν there is a density operator with ¯ spectra (μ, ¯ ν, ¯ λ). One aim of this paper is to show that informational methods can be used to prove Klyachko’s converse. On our way to this result we prove his conjecture [14, Conjecture 7.1.4] that triples (μ, ν, λ) with Vλ ⊂ Vμ ⊗ Vν form a semigroup. We also prove that the semigroup is finitely generated. Together with our previous results on the correspondence with spectral triples this will imply that the set of admissible spectral triples is a convex polytope.
2. Background Let us consider in more detail the relation between irreducible representations and spectra. The irreducible representations of both unitary and symmetric groups are labelled by Young diagrams. If λ denotes a Young diagram, its row lengths are λ1 ≥ λ2 ≥ . . . ≥ λd d and its size is |λ| := i=1 λi . We denote the corresponding irreducible representations of U(d) (or GL(d)) with highest weight λ by Uλd and those of the symmetric group Sk by Vλ . Schur-Weyl duality states that (Cd )⊗k decomposes as a direct sum of irreducible representations: (Cd )⊗k ∼ =
λ∈Par(k,d)
Uλd ⊗ Vλ ,
(2)
where Par(k, d) indicates the set of partitions of k into ≤ d parts; i.e. the Young diagrams with no more than d rows and a total of k boxes. Consider a density operator ρ on Cd . We can take k copies of it and measure the label λ on (Cd )⊗k . The estimation theorem [1, 12] states that, as k increases, the spectrum r of ρ is increasingly well approximated by the normalised row lengths of λ, i.e. by the distribution λ¯ = λ/|λ|. Formally: Theorem 2.1 (Estimation Theorem). Let Pλ be the projection onto Uλd ⊗ Vλ . Then ¯ ) , Tr Pλ ρ ⊗k ≤ (k + 1)d(d−1)/2 exp −k D(λ||r where D( p||q) =
i
pi log( pi /qi ) is the Kullback-Leibler distance.
(3)
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Let us now return to the case of bipartite states, and consider the Clebsch-Gordan series for the symmetric group: Vμ ⊗ Vν ∼ gμνλ Vλ , = λ∈Par(k,k)
where the multiplicities gμνλ are known as the Kronecker coefficients. Since Vλ ∼ = Vλ , the Kronecker coefficients can also be defined in terms of the Sk -invariant subspace of Vμ ⊗ Vν ⊗ Vλ , i.e. gμνλ = dim(Vμ ⊗ Vν ⊗ Vλ ) Sk .
(4)
There is also a way of viewing the Kronecker coefficients in terms of the irreducible representations of the general linear group. It is arrived at by equating the Schur-Weyl decompositions of (Cm ⊗ Cn )⊗k and of (Cmn )⊗k (see [5]) and reads Uλmn↓GL(m)×GL(n) ∼ gμνλ Uμm ⊗ Uνn . = μ∈Par(k,m) ν∈Par(k,n)
This interpretation of the Kronecker coefficients can equivalently be stated in terms of invariants as gμνλ = dim(Uμm ⊗ Uνn ⊗ Uλmn )GL(m)×GL(n) ,
(5)
where GL(m)×GL(n) acts on Uμm ⊗Uνn and simultaneously on Uλmn , the representation dual to Uλmn , according to the inclusion GL(m)×GL(n) → GL(m)⊗GL(n) ⊂ GL(mn). In [5] Theorem 2.1 was applied to give the following: Theorem 2.2. For every density operator ρ AB , there is a sequence (μ( j) , ν ( j) , λ( j) ) of partitions, labeled by natural numbers j, such that gμ( j) ,ν ( j) ,λ( j) = 0 for all j and lim μ¯ ( j) = Spec ρ A ,
(6)
lim ν¯ ( j) = Spec ρ B ,
(7)
lim λ¯ ( j) = Spec ρ AB .
(8)
j→∞
j→∞ j→∞
Klyachko [14] derived a very similar theorem: Theorem 2.3. For a density operator ρ AB with the rational spectral triple (r A , r B , r AB ) there is an integer m > 0 such that gmr A ,mr B ,mr AB = 0. He also proved a converse, that is given in the following section as Theorem 3.2. We now give a resumé of our new results.
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3. Summary of the Results Let KRON denote the set of triples (μ, ν, λ) with nonzero Kronecker coefficients. Our first result is Theorem 3.1. KRON is a finitely generated semigroup with respect to row-wise addition, i.e. gμνλ = 0 and gμ ν λ = 0 implies gμ+μ ,ν+ν ,λ+λ = 0. This was conjectured in Klyachko’s paper [14, Conjecture 7.1.4 and below]. It implies stability of the Kronecker coefficients: i.e. if gμνλ = 0 then g N μN ν N λ = 0, for integers N > 0. This was announced by Kirillov [13, Theorem 2.11] but without proof. A simple corollary of stability is that non-vanishing Kronecker coefficients obey entropic relations (as explained in [5]). More importantly, it plays a key role in our information-theoretic proof of the following theorem. Theorem 3.2. Let μ, ν and λ be diagrams with k boxes and at most m, n and mn rows, respectively. If gμνλ = 0, then there exists a density operator ρ AB on H A ⊗ H B = Cm ⊗ Cn with spectra Spec ρ A = μ, ¯
(9)
Spec ρ = ν¯ , ¯ Spec ρ AB = λ.
(10)
B
(11)
We also give a compact version of the proof of Theorem 2.2, which was presented in [5]. In this way we obtain a simple proof of the full correspondence between Kronecker coefficients and admissible spectral triples. Furthermore, it will allow us to draw the following corollary. Corollary 3.3. Theorem 2.2 and Theorem 2.3 are equivalent. Using the correspondences to spectral triples, the fact that KRON is a finitely generated semigroup can be given the following geometrical interpretation. Theorem 3.4. Spect, the set of admissible spectral triples, is a convex polytope. 4. The Set of Nonzero Kronecker Coefficients is a Finitely Generated Semigroup In order to prove Theorem 3.1, we introduce a representation of GL(n) which we call the Cartan product ring Q m := Uμm . k≥0 μ∈Par(k,m)
Define Q n similarly and also Q mn :=
k≥0 λ∈Par(k,mn)
Uλmn ,
where Uλmn is the representation dual to Uλmn . We assume here that μm , νn and λmn are non-negative because we are ultimately interested in combining irreducible representations of Sk , which are only defined for nonnegative λ. However, all of our results can be
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easily generalized for dominant weights μ, ν, λ without the non-negativity restrictions. To establish Q n as a graded ring, we introduce the Cartan product [8] that maps Uμ ⊗Uν to Uμ+ν by projecting onto the unique Uμ+ν -isotypic subspace of Uμ ⊗ Uν . We denote the Cartan product by ◦, so that for |u μ ∈ Uμ , |u ν ∈ Uν , |u μ ◦ |u ν is defined to be the projection of |u μ ⊗ |u ν onto the Uμ+ν -isotypic subspace of Uμ ⊗ Uν . Clearly Q n is graded under the action of ◦, GL(n) preserves this grading and GL(n) acts properly on products, i.e. g(|α ◦ |β) = (g|α) ◦ (g|β). The proof of Theorem 3.1 now rests on the following lemma: Lemma 4.1. (a) Q n has no zero divisors. That is, if |α, |β ∈ Q n are nonzero, then |α ◦ |β = 0. (b) Q m ⊗ Q n ⊗ (Q mn ) has no zero divisors. n Here we have defined (Q n ) = λ (Uλ ) with the corresponding Cartan product n ) and we have extended the Cartan product to tensor products (Uμn ) ◦ (Uνn ) → (Uμ+ν in the natural way. Proof. Although only statement (b) of the lemma is used in the proof of the theorem, for ease of exposition we will prove part (a) and then sketch how similar arguments can establish (b). Our proof follows the treatment of the Borel-Weil theorem in [3, p. 115], with notational changes (e.g. we write α|g|v for what would be called α(gv) there). Let |vλ be a highest weight vector for Uλ . For any |α ∈ Uλ , note that α|g|vλ is (i) a polynomial in the matrix elements of g, (ii) identically zero only if |α = 0 (due to the irreducibility of Uλ ). Now define the set X α := {g ∈ GL(n)|α|g|vλ = 0}. The above two claims mean that X α is a proper closed subset of GL(n) in the Zariski topology whenever |α = 0. Similarly, if |β ∈ Uλ and |vλ is a highest weight vector for Uλ then X β := {g ∈ GL(n)|β|g|vλ = 0} is a proper Zariski-closed subset of GL(n) if and only if |β = 0. The fact that |vλ and |vλ are highest weight vectors means that |vλ ⊗ |vλ = |vλ+λ ∈ Uλ+λ and thus α|g|vλ β|g|vλ = (α| ⊗ β|)g(|vλ ⊗ |vλ ) = (α| ◦ β|)g|vλ+λ .
(12)
We are free to replace α| ⊗ β| with α| ◦ β| in the last step because we are taking the inner product with a vector that lies entirely in Uλ+λ . Now suppose |α ◦ |β = 0. Then for all g at least one of the terms on the LHS of Eq. (12) vanishes, and thus GL(n) = X α ∪ X β . Since GL(n) is irreducible, it cannot be the union of two proper closed subsets, and we conclude |α and |β cannot both be nonzero. The proof of (b) is almost identical, but consider instead |α ∈ Uμ ⊗ Uν ⊗ Uλ , |β ∈ Uμ ⊗ Uν ⊗ Uλ and the group GL(m) × GL(n) × GL(mn) (which is still irreducible). Note that we could relax the restriction that λn ≥ 0 by multiplying the inner products of the form α|g|vλ by a high enough power of det g (guaranteed to be nonzero for g ∈ GL(n)) to obtain a polynomial in the matrix elements of g. Proof of Theorem 3.1. Given any ring R with an action of G on it, let R G denote the ring of G-invariants in R. Now recall Eq. (5): gμνλ = dim(Uμ ⊗ Uν ⊗ Uλ )GL(m)×GL(n) .
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M. Christandl, A. W. Harrow, G. Mitchison
If gμνλ = 0 and gμ ν λ = 0 then according to this equation there exist nonzero vectors |u μνλ ∈ (Uμ ⊗ Uν ⊗ Uλ )GL(m)×GL(n) and |u μ ν λ ∈ (Uμ ⊗ Uν ⊗ Uλ )GL(m)×GL(n) . Define |u μ+μ ,ν+ν ,λ+λ = |u μ,ν,λ ◦ |u μ ,ν ,λ . Then |u μ+μ ,ν+ν ,λ+λ = 0 by part (b) of Lemma 4.1 and |u μ+μ ,ν+ν ,λ+λ ∈ Uμ+μ ⊗ Uν+ν ⊗ Uλ+λ is GL(m) × GL(n)-invariant, since this property is preserved by the Cartan product. Thus (Uμ+μ ⊗ Uν+ν ⊗ )GL(m)×GL(n) = 0 and we conclude that g Uλ+λ μ+μ ,ν+ν ,λ+λ = 0. We continue to show that KRON is finitely generated. Note that Q m , Q n and Q mn are each generated by a basis for the respective fundamental representations1 and therefore are finitely generated. This implies that also R = Q m ⊗ Q n ⊗ Q mn is finitely generated. We now consider the ring of invariants R G of R under the simultaneous action of the group G = GL(m) × GL(n) on Q m ⊗ Q n and as a subgroup of GL(mn) on Q mn . This action is algebraic, i.e. every element of R is contained in a finite-dimensional representation of G. Thus we can apply a generalisation of Hilbert’s theorem [18, Theorem 3.6] to conclude that R G is finitely generated. Since RG ∼ (Uμm ⊗ Uνn ⊗ Uλmn )GL(m)×GL(n) = μ,ν,λ
and gμνλ = dim(Uμm ⊗ Uνn ⊗ Uλn )GL(m)×GL(n) we see that KRON is finitely generated, too. A similar proof was given in [6] which proved that under the diagonal action of a connected reductive group G, the triples (μ, ν, λ) with dim(Uμ ⊗ Uν ⊗ Uλ )G = 0 form a finitely generated semigroup. 5. The Correspondence of Nonzero Kronecker Coefficients to Spectra Proof of Theorem 2.2. Rather than working with the mixed state ρ AB we will consider a purification |ψ ABC of ρ AB , which has Spec ρ C = Spec ρ AB . Let r A := Spec ρ A , r B := Spec ρ B and r C := Spec ρ C . PμA denotes the projector onto the Young subspace Uμm ⊗ Vμ in system A, and PνB , PλC are the corresponding projectors onto Young subspaces in B and C, respectively. As a consequence of Theorem 2.1 (see [5, Corollary 2]), for given > 0 one can find a k0 such that the following inequalities hold simultaneously for all k ≥ k0 : Tr PX (ρ A )⊗k ≥ 1 − , PX := PμA , (13) μ:μ−r ¯ A 1 ≤ B ⊗k
Tr PY (ρ )
≥ 1 − ,
PY :=
ν:¯ν −r B C ⊗k
Tr PZ (ρ )
≥ 1 − ,
PZ :=
PνB ,
(14)
PλC .
(15)
1 ≤
λ:λ¯ −r C 1 ≤
For 0 < < 13 , the estimates (13)-(15) can be combined to yield Tr [(PX ⊗ PY ⊗ PZ ) (|ψψ| ABC )⊗k ] ≥ 1 − 3 > 0.
(16)
1 The fundamental representations of GL(d) are those with Young diagrams of the form (1 , 0d− ) for
∈ {1, . . . , d}.
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581
Since (|ψ ABC )⊗k is evidently invariant under permutation of its k subsystems, it takes the form |αμνλ , (|ψ ABC )⊗k = μνλ
where |αμνλ ∈ Uμm ⊗ Uνn ⊗ Uλmn ⊗ (Vμ ⊗ Vν ⊗ Vλ ) Sk . Equation (16) then implies that there must be at least one triple (μ, ν, λ) with μ¯ − r A 1 ≤ , ¯ν − r B 1 ≤ , λ¯ − r C 1 ≤ and |αμνλ = 0. Thus (Vμ ⊗ Vν ⊗ Vλ ) Sk = 0 implies that gμνλ = 0. It remains to pick a sequence of decreasing j with corresponding triples (μ( j) , ν ( j) , λ( j) ). It has been observed in different contexts that the speed √ of convergence in Theorem 2.1 and consequently in Theorem 2.2 is proportional to 1/ k [1, 9]. We will now prove Corollary 3.3, the equivalence of Theorem 2.2 and 2.3. Proof of Corollary 3.3. We start by showing how Theorem 2.3 follows from Theorem 2.2. Let (r A , r B , r AB ) ∈ Spect, the set of admissible spectral triples. According to Theorem 2.2, there is a sequence of elements in KRON, whose normalised values converge to (r A , r B , r AB ). By Theorem 3.1, the set KRON is a finitely generated semigroup. With a finite set of generators (μ(i) , ν (i) , λ(i) ) of KRON we can therefore express (r A , r B , r AB ) in the form (r A , r B , r AB ) =
xi (μ¯ (i) , ν¯ (i) , λ¯ (i) ),
(17)
i
for a set of nonnegative numbers xi which sum to one. Since the union of the t + 1-vertex simplices equals the whole polytope, every point in it can be taken to be the sum of just t + 1 normalised generators (μ¯ (i) , ν¯ (i) , λ¯ (i) ) (cf. Carathéodory’s theorem). From the set of m + n + mn equations in the variablesxi in Eq. (17), choose a set of t linearly independent ones, add the (t + 1)th constraint i xi = 1 and write the set of equations as AB } and M x = r, i.e. r = (r1 , . . . , rt , 1) for r j ∈ {r1A , . . . , rmA , r1B , . . . rnB , r1AB , . . . , rmn x = (x1 , . . . , xt+1 ). If (r A , r B , r AB ) is rational, the xi will be rational as well, since M is rational. This shows that (r A , r B , r AB ) = i nni (μ¯ (i) , ν¯ (i) , λ¯ (i) ), where we set xi = nni for n i , n ∈ N. Multiplication by |μ|n results in |μ|n(r A , r B , r AB ) =
n i (μ(i) , ν (i) , λ(i) ).
i
Since the right-hand side of this equation is certainly an element of KRON this shows that for rational (r A , r B , r AB ) there is a number m := |μ|n such that gmr A ,mr B ,mr AB = 0. It remains to show that Theorem 2.2 follows from Theorem 2.3. Suppose (r A , r B , r AB ) is a spectral triple corresponding to some ρ AB . Then we can construct a series of rational triples (r A( j) , r B( j) , r AB( j) ) that approaches (r A , r B , r AB ) and by Theorem 2.3, there exists a series (μ( j) , ν ( j) , λ( j) ) such that (μ¯ ( j) , ν¯ ( j) , λ¯ ( j) ) approaches (r A , r B , r AB ) and gμ( j) ,ν ( j) ,λ( j) = 0 for all j.
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Note that there are two ways in which Klyachko’s Theorem 2.3 does not quite give the full strength of Theorem 2.2. First, it does not guarantee the speed of convergence. Second, it does not imply that, in the case of rational triples (r A , r B , r AB ), there is an increasing sequence of values of k for which gkr A ,kr B ,kr AB = 0; this follows from Theorem 2.2 and can be thought of as a sort of stability obtained without appeal to Theorem 3.1. We now turn our attention to Theorem 3.2, beginning with an asymptotic result. Lemma 5.1. Let μ, ν and λ be diagrams with k boxes and at most m, n and mn rows, respectively. If gμνλ = 0, then there exists a density operator ρ AB on Cm ⊗ Cn with spectra satisfying ¯ 1 ≤ δ, Spec ρ A − μ
(18)
Spec ρ − ν¯ 1 ≤ δ, Spec ρ AB − λ¯ 1 ≤ δ,
(19)
B
for δ = O(mn (log k)/k).
Here if p, q are probability distributions then p − q1 :=
(20) x
| p(x) − q(x)|.
Proof. It will suffice to construct a pure state |ϕ ABC ∈ Cm ⊗Cn ⊗Cmn with Spec ϕ A − μ ¯ 1 ≤ δ, Spec ϕ B − ν ¯ 1 ≤ δ and Spec ϕ C − λ¯ 1 ≤ δ, since Spec ϕ AB = Spec ϕ C . Since gμνλ = 0, there exists a unit vector |ψ ∈ (Vμ ⊗Vν ⊗Vλ ) Sk . Writing = m 2 n 2 , |ψ ∈ ((Cm ⊗ Cn ⊗ Cmn )⊗k ) Sk = ((C )⊗k ) Sk ∼ = Uσ , where Uσ denotes the symmetric representation of GL( ), and the superscript emphasizes which GL(·) we are using. Denote the projector onto Uσ ⊂ (C )⊗k by Pσ . Note
that Tr Pσ = dim Uσ = k+ −1
−1 ≤ k . Fix a vector |φ0 ∈ C and let dU denote a Haar measure for the unitary group U ( ) with normalisation dU = 1. Then by Schur’s lemma Pσ = dim Uσ
dg (g|φ0 φ0 |g † )⊗k .
g∈U ( )
Thus 1 = Tr Pσ |ψψ|
dg Tr |ψψ|(g|φ0 φ0 |g † )⊗k = dim Uσ g∈U ( )
≤ dim Uσ max Tr |ψψ|(g|φ0 φ0 |g † )⊗k . g∈U ( )
Let g ∈ U ( ) be the unitary operator achieving the above maximisation, and define |ϕ := g|φ0 . Then |ψ| |ϕ⊗k |2 ≥
1 ≥ k − . dim Uσ
Nonzero Kronecker Coefficients and Spectra
583
Let Pμm , Pνn and Pλmn denote the projectors onto Uμm ⊗Vμ ⊂ (Cm )⊗k , Uνn ⊗Vν ⊂ (Cn )⊗k and Uλmn ⊗Vλ ⊂ (Cmn )⊗k , respectively. Then by construction Pμm ⊗ Pνn ⊗ Pλmn |ψ = |ψ, so |ψψ| ≤ Pμm ⊗ Pνn ⊗ Pλmn and Tr Pμm ⊗ Pνn ⊗ Pλmn |ϕϕ|⊗k ≥ Tr (|ψψ|)|ϕϕ|⊗k = |ψ| |ϕ⊗k |2 ≥
1 ≥ k − . dim Uσ
Focussing for now on the A subsystem, we have Tr Pμm (ϕ A )⊗k ≥ k − .
(21)
On the other hand, Theorem 2.1 (Spectrum Estimation) states that Tr Pμm (ϕ A )⊗k ≤ (k + 1)m(m−1)/2 exp(−k D(μSpec ¯ ϕ A )).
(22)
Combining Eqs. (21) and (22), we find that D(μSpec ¯ ϕ A) ≤
1 2 m(m
− 1) log(k + 1) + log k k
and for k > 1, we can apply Pinsker’s inequality [17] (which states that p − q21 /2 ≤ D( pq) for any probability distributions p, q) to bound μ¯ − Spec ϕ A 1 ≤ 3mn (log k)/k. This proves Eq. (18). Equations (19) and (20) follow by repeating this argument (starting with Eq. (21)) for Pνn and Pλmn . Theorem 3.2 now follows by observing that, if gμ,ν,λ = 0, then g jμ, jν, jλ = 0 for any integer j ≥ 1, because of the semigroup property (Theorem 3.1). The above lemma then gives us a sequence of density operators ρ(AB ¯ ν, ¯ λ¯ ), j) whose spectra converge to (μ, AB m n and, by compactness of the set of density matrices on C ⊗ C , the sequence ρ( j) has a limit ρ AB satisfying Eqs. (9), (10) and (11). 6. Convexity Let us now gather together some implications of the theorems. Let Kron denote the normalised triples (μ, ¯ ν, ¯ λ¯ ), where (μ, ν, λ) ∈ KRON. From Theorem 2.2 we know that any admissible spectral triple, i.e. any point in Spect, can be approximated by a sequence in Kron and therefore lies in Kron, the closure of Kron; thus Spect ⊆ Kron. From Theorem 3.2 we know that Kron ⊆ Spect, and hence, since Spect is closed, Kron ⊆ Spect. Thus we have Spect = Kron. Note that Kron consists of rational points (normalised row lengths of diagrams) and there are certainly operators with irrational spectra. So Kron, unlike its closure, is a proper subset of Spect.
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M. Christandl, A. W. Harrow, G. Mitchison
Theorem 3.1 allows us to say more about Kron, and hence Spect. The semigroup ¯ (μ¯ , ν¯ , λ¯ ) ∈ Kron, then property of KRON (Theorem 3.1) implies that if (μ, ¯ ν, ¯ λ), ( p μ¯ + (1 − p)μ¯ , p ν¯ + (1 − p)¯ν , p λ¯ + (1 − p)λ¯ ) ∈ Kron, for every p with 0 ≤ p ≤ 1. Thus Kron is convex. Furthermore, Theorem 3.1 implies that there is a finite set of generators (μ(i) , ν (i) , λ(i) ) of KRON, so any (μ, ¯ ν, ¯ λ¯ ) ∈ Kron can be written (μ, ¯ ν, ¯ λ¯ ) = xi (μ¯ (i) , ν¯ (i) , λ¯ (i) ). i
Thus Kron is a convex polytope. We enshrine this in Theorem 3.4. An alternative proof for Theorem 3.4 that makes use of Kirwan’s convexity theorem for moment maps can be found in [4, Chap. 2.3.6]. Acknowledgements. The inspiration for this paper came from a discussion by one of us (MC) with Allen Knutson, who essentially sketched the arguments of Theorems 3.1. He gave us further helpful advice at several points, always worded in a lively and indeed unforgettable way. We also had valuable advice from Graeme Segal. Finally, we thank Koenraad Audenaert for useful pointers to the literature, Alexander Klyachko for many stimulating discussions, and an anonymous referee for suggestions that simplified the proof of Theorem 3.1. This project was supported by the EU under projects PROSECCO (IST-2001-39227), RESQ (IST-200137559) and the Integrated Project FET/QIPC SCALA. MC acknowledges the support of a DAAD Doktorandenstipendium, the U.K. Engineering and Physical Sciences Research Council and a Nevile Research Fellowship, which he holds at Magdalene College Cambridge. AWH thanks the Centre for Quantum Computation for hospitality while completing this work and acknowledges partial support from the ARO and ARDA under ARO contract DAAD19-01-1-06.
References 1. Alicki, R., Rudnicki, S., Sadowski, S.: Symmetry properties of product states for the system of N n-level atoms. J. Math. Phys. 29(5), 1158–1162 (1988) 2. Berenstein, A., Sjamaar, R.: Coadjoint orbits, moment polytopes and the Hilbert-Mumford criterion. J. Amer. Math. Soc. 13(2), 433–466 (2000) 3. Carter, R., Segal, G., MacDonald, I.: Lectures on Lie Groups and Lie Algebras. Volume 32 of London Mathematical Society Student Texts. 1st edition, Cambridge Univ. Press, (1995) 4. Christandl, M.: The Structure of Bipartite Quantum States: Insights from Group Theory and Cryptography. PhD thesis, University of Cambridge, February 2006. Available at http://arxiv.org/abs/quantph/0604183, 2001 5. Christandl, M., Mitchison, G.: The spectra of density operators and the Kronecker coefficients of the symmetric group. Commun. Math. Phys. 261(3), 789–797 (2005) 6. Elashvili, A.G.: Invariant algebras. Advances in Soviet Math. 8, 57–64 (1992) 7. Franz, M.: Moment polytopes of projective G-varieties and tensor products of symmetric group representations. J. Lie Theory 12, 539–549 (2002) 8. Fulton, W., Harris, J.: Representation Theory: A First Course. New York: Springer, (1991) 9. Harrow, A.: Applications of coherent classical communication and the Schur transform to quantum information theory. Thesis, Doctor of philosophy in physics, Massachusetts Institute of Technology, September 2005, available at http://arxiv.org/abs/quant-ph/0512255, 2005 10. Hayashi, M., Matsumoto, K.: Quantum universal variable-length source coding. Phys. Rev. A 66(2), 022311 (2002) 11. Heckman, G.J.: Projections of orbits and asymptotic behaviour of multiplicities for compact connected Lie groups. Invent. Math. 67, 333–356 (1982) 12. Keyl, M., Werner, R.F.: Estimating the spectrum of a density operator. Phys. Rev. A 64(5), 052311 (2001) 13. Kirillov, A.N.: An invitation to the generalized saturation conjecture. http://arxiv.org/abs/ math.CO/0404353, 2004 14. Klyachko, A.: Quantum marginal problem and representations of the symmetric group http:// arxiv.org/list/quant-ph/0409113, 2004
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15. Mumford, D., Fogarty, J., Kirwan, F.: Geometric invariant theory, Volume 34 of Ergebnisse der Mathematic und ihrer Grenzgebiete (2). Third edition, Berlin: Springer-Verlag, (1994) 16. Ness, L.: A stratification of the null cone via the moment map. Amer. J. Math. 106, 1281–1329 (1984) 17. Pinsker, M.S.: Information and Information Stability of Random Variables and Processes. San Francisco: Holden-Day, 1964 18. Popov, V.L., Vinberg, E.B.: Algebraic Geometry IV. Volume 55 of Encyclopaedia of Mathematical Sciences, Chapter II. Invariant Theory, Berlin-Heidelberg-New York: Springer-Verlag 1994, pp. 123–278 Communicated by M.B. Ruskai
Commun. Math. Phys. 270, 587–634 (2007) Digital Object Identifier (DOI) 10.1007/s00220-006-0163-5
Communications in
Mathematical Physics
On Genus Two Riemann Surfaces Formed from Sewn Tori Geoffrey Mason1, , Michael P. Tuite2, 1 Department of Mathematics, University of California Santa Cruz, CA 95064, U.S.A.
E-mail:
[email protected]
2 Department of Mathematical Physics, National University of Ireland, Galway, Ireland.
E-mail:
[email protected] Received: 4 March 2006 / Accepted: 14 June 2006 Published online: 9 January 2007 – © Springer-Verlag 2006
Abstract: We describe the period matrix and other data on a higher genus Riemann surface in terms of data coming from lower genus surfaces via an explicit sewing procedure. We consider in detail the construction of a genus two Riemann surface by either sewing two punctured tori together or by sewing a twice-punctured torus to itself. In each case the genus two period matrix is explicitly described as a holomorphic map from a suitable domain (parameterized by genus one moduli and sewing parameters) to the Siegel upper half plane H2 . Equivariance of these maps under certain subgroups of Sp(4, Z) is shown. The invertibility of both maps in a particular domain of H2 is also shown.
1. Introduction This paper is the second of a series intended to develop a mathematically rigorous theory of chiral partition and n-point functions on Riemann surfaces at all genera, based on the theory of vertex operator algebras. The purpose of the paper is to provide a rigorous and explicit description of the period matrix and other data on a higher genus Riemann surface in terms of data coming from lower genus surfaces via an explicit sewing procedure. In particular, we consider and compare in some detail the construction of a genus two Riemann surface in two separate ways: either by sewing two tori together or by sewing a torus to itself. Although our primary motivation is to lay the foundations for the explicit construction of the partition and n-point functions for a vertex operator algebra on higher genus Riemann surfaces [MT2], we envisage that the results herein may also be of wider interest. Support provided by the National Science Foundation DMS-0245225, and the Committee on Research at the University of California, Santa Cruz. Supported by The Millenium Fund, National University of Ireland, Galway.
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G. Mason, M. P. Tuite
As is well-known (cf. [H] for a systematic development), the axioms for a vertex (operator) algebra V amount to an algebraicization of aspects of the theory of gluing spheres, i.e., compact Riemann surfaces of genus zero. A quite complete theory of (bosonic) n-point functions at genus one was developed by Zhu in his well-known paper [Z]. For particularly well-behaved vertex operator algebras (we have in mind rational vertex operator algebras in the sense of [DLM]), Zhu essentially showed that the npoint functions, defined as particular graded traces over V , are elliptic and have certain S L(2, Z) modular-invariance properties1 with respect to the torus modular parameter τ . A complete description of bosonic Heisenberg and lattice VOA n -point functions is given in [MT1]. Apart from its intrinsic interest, modular-invariance is an important feature of conformal field theory and its application to string theory e.g., [P, GSW]. Mathematically, it has been valuable in recent developments concerning the structure theory of vertex operator algebras [DM1, DM2], and it may well play an important role in geometric applications such as elliptic cohomology and elliptic genus. These are all good reasons to anticipate that a theory of n -point functions at higher genus will be valuable; another is the consistency of string theory at higher loops (genera). Our ultimate goal, then, is to emulate Zhu’s theory by first defining and then understanding the automorphic properties of n-point functions on a higher genus Riemann surface where the modular variable τ is replaced by the period matrix Ω (g) , a point in the Siegel upper half-space Hg at genus g. When one tries to implement the vision outlined above at genus g ≥ 2, difficulties immediately arise which have no analog at lower genera [T]. In Zhu’s theory, there is a clear relation between the variable τ and the relevant vertex operators: it is not the definition of n-point functions, but rather the elucidation of their properties, that is difficult. At higher genus, the very definition of n-point function is less straightforward and raises interesting issues. Many (but not all) of these are already present at genus 2, and it is this case that we mainly deal with in the present paper. A very general approach to conformal field theory on higher genus Riemann surfaces has been discussed in the physics literature [MS, So, DP, VV, P]. In particular, there has been much progress in recent years in understanding genus two superstring theory [DPI, DPVI, DGP]. The basic idea we follow is that by cutting a Riemann surface along various cycles it can be reduced to (thrice) punctured spheres, and conversely one can construct Riemann surfaces by sewing punctured spheres. It is interesting to note that Zhu’s g = 1 theory is not formulated by sewing punctured spheres per se, but rather by implementing a conformal map of the complex plane onto a cylinder. Tracing over V has the effect of sewing the ends of the cylinder to obtain a torus. This idea does not generalize to the case when g = 2, and we must hew more closely to the sewing approach of conformal field theory. Roughly speaking, what we do is sew tori together in order to obtain a compact Riemann surface S of genus 2 and which is endowed with certain genus 1 data encoded by V via the Zhu theory [T]. There are two essentially different ways to obtain S (which, for simplicity, we take to have no punctures in this work) from genus 1 data: either by sewing a pair of once-punctured complex tori, or by sewing a twice punctured torus to itself (attaching a handle). These two sewing schemes will give rise to seemingly different theories and different definitions of n-point functions. These issues are discussed in detail in the sequel [MT2]. We now give a more technical introduction to the contents of the present paper. Notwithstanding our earlier discussion of the role of vertex operators, they do not appear explicitly in the present work! We are concerned here exclusively with setting up 1 The general conjecture that the partition function is a modular function if V is rational remains open.
On Genus Two Riemann Surfaces Formed from Sewn Tori
589
foundations so that the ideas we have been discussing are rigorous and computationally effective. Section 2 records the many modular and elliptic-type functions that we will need. In the paper [Y], which is very important for us, Yamada developed a general approach to computing the period matrix of a Riemann surface S obtained by sewing Riemann surfaces S1 , S2 (which may coincide) of smaller genus. In Sects. 3 and 4 we develop the theory in the case that S1 and S2 are distinct. We refer to this as the -formalism, itself being a complex number which is a part of the data according to which the sewing is performed. (See Fig. 1 below.) We begin in Sect. 3 with some of the details of Yamada’s general theory, and make some explicit computations. In particular, we introduce infinite matrices Aa for a = 1, 2, whose entries are certain weighted moments of the normalized differential of the second kind on Sa . These matrices determine another infinite matrix X whose entries are weighted moments of the normalized differential of the second kind on S (Proposition 1), and this in turn determines the period matrix Ω (g) of S (Theorem 3). In particular, the infinite matrix I − A1 A2
(1)
plays an important role (I is the infinite unit matrix). The entries of this matrix depend on data coming from Sa , and in particular they are power series in . We show (Theorem 2) that (1) has a well-defined determinant det(I − A1 A2 ) which is holomorphic for small enough . Section 3 ends with some additional results concerning the holomorphy of det(I − A1 A2 ) and Ω (g) in various domains. In Sect. 4 we study in more detail the case in which the Sa have genus 1, so that they have a modulus τa ∈ H1 . The triple (τ1 , τ2 , ) determines a genus 2 surface as long as the three parameters in question satisfy a certain elementary inequality. This defines a manifold D (τ1 , τ2 , ) ⊆ H1 × H1 × C consisting of all such admissible triples. Associating to this data the genus two period matrix Ω (2) = Ω (2) (τ1 , τ2 , ) of S defines a map F : D (τ1 , τ2 , ) → H2 (τ1 , τ2 , ) → Ω (2) (τ1 , τ2 , )
(2)
which is important for everything that follows. When we introduce partition functions in the -formalism at g = 2 in the sequel to the present paper [MT2], they will be functions on D , not H2 . The map F interpolates between the two domains. We obtain (Theorem 4) an explicit expression for the genus 2 period matrix (2) (2) Ω11 Ω12 Ω (2) (τ1 , τ2 , ) = (3) (2) (2) Ω12 Ω22 determined by an admissible triple. Each Ωi(2) j turns out to be essentially a power series 2 in with coefficients which are quasimodular forms, i.e., certain polynomials in the Eisenstein series E 2 (τi ), E 4 (τi ), E 6 (τi ) for i = 1, 2. Moreover F is an analytic map, and we show (Theorem 5) that it is equivariant with respect to the action of a group G∼ = (S L(2, Z) × S L(2, Z)) Z2 (the wreathed product of S L(2, Z) and Z2 ). G embeds into Sp(4, Z) in a standard way, and this defines the action of G on H2 . The action on D is explained in Subsect. 4.4. These calculations are facilitated by an alternate description (Proposition 4 ) of the entries of (3) in terms of combinatorial gadgets that 2 Notation for modular and elliptic-type functions is covered in Sect. 2.
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we call chequered necklaces. They are certain kinds of graphs with nodes labelled by positive integers and edges labelled by quasimodular forms, and they play a critically important role in the sequel to the present paper. We show (Proposition 5) that about any degeneration point p where = 0 (i.e., the two tori S1 , S2 touch at a point), there is a G-invariant neighborhood of p throughout which F is invertible. Sections 5 and 6 are devoted to development of the corresponding formalism in the case that S is obtained by self-sewing (i.e., attaching a handle to) a surface S1 of one lower genus. We refer to this as the ρ-formalism. Although we are able to achieve results that parallel the development of the -formalism outlined in the previous paragraph, it is fair to say that of the two, the ρ-formalism is the more complicated. In Sect. 5 we first discuss the results of Yamada (loc. cit.) in a general ρ -formalism, and calculate weighted moments as before. This leads us to introduce the analog of (1), namely I − R,
(4)
where R is an infinite matrix whose entries are 2 × 2 block matrices determined by weighted moments of the normalized differential of the second kind on S1 . As before, the entries of R are holomorphic in ρ, and we show (Theorem 7) that det(I − R) is defined and holomorphic in a certain ρ-domain. The matrix R then determines the period matrix on S (Theorem 8). We also discuss several sewing scenarios for self-sewing a sphere, including one (Proposition 9) where the Catalan series, familiar from combinatorics [St], plays an unexpected role. In Sect. 6 we investigate in detail the self-sewing of a twice-punctured torus with modulus τ ∈ H1 to form a genus two Riemann surface. As before sewing determines a map F ρ : Dρ (τ, w, ρ) → H2 (τ, w, ρ) → Ω (2) (τ, w, ρ),
(5)
where now Dρ (τ, w, ρ) ⊆ H1 ×C×C determines the admissible sewing parameters (w describes the relative position of the punctures). Again we obtain (Theorem 9) explicit formulas for the entries of the matrix (2) (2) Ω11 Ω12 (2) Ω (τ, w, ρ) = (6) (2) (2) Ω12 Ω22 and show that F ρ is holomorphic. Roughly speaking, the entries of R and the Ωi(2) j in this case are power series in ρ with coefficients which are quasimodular and elliptic functions in the variables τ, w. We provide a combinatorial description of (6) in terms of a notion of chequered necklace suitably modified compared to the case. A com(2) plicating factor is that Ω22 involves a logarithm of (the inverse square of) the prime form on S. Because of this, it is necessary to pass to a covering space Dˆ ρ of Dρ before equivariance properties can be considered. This is carried-out in Sect. 6.3, where we construct a diagram Fρ
Dρ −→ H2
Fˆ ρ Dˆ ρ
On Genus Two Riemann Surfaces Formed from Sewn Tori
591
We show (Theorem 11) that the map Fˆ ρ is equivariant with respect to a group L described as a semi-direct product of S L(2, Z) and the (nonabelian) Heisenberg group H∼ = Z1+2 (a 2-step nilpotent group with center Z). Again L acts on H2 via an embedding into Sp(4, Z) and on Dˆ ρ in a manner prescribed in Theorem 10. In Subsect. 6.4 we obtain the expected local invertibility of F ρ about a point of degeneration, which is a bit more subtle than degeneration in the -formalism. One of the reasons for establishing the local invertibility results is that once obtained, we have a way of comparing the two sewing domains D and Dρ , at least in some regions, by looking at (F ρ )−1 ◦ F .
(7)
In the final Sect. 7, (7) is briefly discussed, where we observe that it is equivariant with respect to a common subgroup of G and L isomorphic to S L(2, Z). The Appendix concludes with the explicit formulas for Ω (2) to O( 9 ) in the -formalism and to O(ρ 5 ) in the ρ -formalism. 2. Some Elliptic Functions We briefly discuss a number of modular and elliptic-type functions that we will need. The notation we introduce will be in force throughout the paper. The Weierstrass elliptic function with periods3 σ, ς ∈ C∗ is defined by 1 1 1 ℘ (z, σ, ς ) = 2 + . (8) − z (z − mσ − nς )2 (mσ + nς )2 m,n∈Z,(m,n) =(0,0)
Choosing ς = 2πi and σ = 2πiτ (τ will always lie in the complex upper half-plane H), we define P2 (τ, z) = ℘ (z, 2πiτ, 2πi) + E 2 (τ ) ∞ 1 = 2+ (k − 1)E k (τ )z k−2 . z
(9)
k=2
Here, E k (τ ) is equal to 0 for k odd, and for k even is the Eisenstein series [Se] E k (τ ) = −
2 Bk + σk−1 (n)q n . k! (k − 1)! n≥1
Here and below, we take q = exp(2πiτ ); σk−1 (n) = Bernoulli number defined by
d|n
d k−1 , and Bk is the k th
tk t t − 1 + = Bk et − 1 2 k! k≥2
=
1 4 1 6 1 2 t − t + t + O(t 8 ). 12 720 30240
3 The period basis is more usually denoted by ω , ω . 1 2
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P2 can be alternatively expressed as P2 (τ, z) =
nq n qz + (q n + qz−n ), (qz − 1)2 1 − qn z
(10)
n≥1
where qz = exp(z). If k ≥ 4 then E k (τ ) is a holomorphic modular form of weight k on S L(2, Z). That is, it satisfies E k (γ τ ) = (cτ + d)k E k (τ ) for all γ =
ab cd
∈ S L(2, Z), where we use the standard notation γτ =
aτ + b . cτ + d
(11)
On the other hand, E 2 (τ ) has an exceptional transformation law E 2 (γ τ ) = (cτ + d)2 E 2 (τ ) −
c(cτ + d) . 2πi
(12)
The first three Eisenstein series E 2 (τ ), E 4 (τ ), E 6 (τ ) are algebraically independent and generate a weighted polynomial algebra Q = C[E 2 (τ ), E 4 (τ ), E 6 (τ )] which, following [KZ], we call the algebra of quasimodular forms. We define P1 (τ, z) by P1 (τ, z) =
1 − E k (τ )z k−1 , z
(13)
k≥2
d P1 and P1 + z E 2 is the classical Weierstrass zeta function. P1 is where P2 = − dz quasi-periodic with
P1 (τ, z + 2πi) = P1 (τ, z), P1 (τ, z + 2πiτ ) = P1 (τ, z) − 1.
(14)
We also define P0 (τ, z), up to a choice of the logarithmic branch, by P0 (τ, z) = − log(z) +
1 k≥2
k
E k (τ )z k ,
(15)
d P0 . We define the elliptic prime form K (τ, z) by [Mu] where P1 = − dz
K (τ, z) = exp(−P0 (τ, z)),
(16)
2
d 2 so that P2 = dz 2 log K . (exp(z E 2 /2)K (τ, z) is the classical Weierstrass sigma function.) K (τ, z) is quasi-periodic with
K (τ, z + 2πi) = −K (τ, z), K (τ, z + 2πiτ ) = −qz−1 q −1/2 K (τ, z).
(17)
On Genus Two Riemann Surfaces Formed from Sewn Tori
593
K (τ, z) is an odd function of z and can be expressed as K (τ, z) = − where θ1 (τ, z) =
iθ1 (τ, z) = z + O(z 3 ), η(τ )3
+ 1/2)2 + (n + 1/2)(z + iπ )) and 1
(1 − q n ) η(τ ) = q 24
(18)
n∈Z exp(πiτ (n
(19)
n≥1
is the Dedekind eta function. Define elliptic functions Pk (τ, z) for k ≥ 3 from the analytic expansion P1 (τ, z − w) = Pk (τ, z)w k−1 ,
(20)
k≥1
where Pk (τ, z) =
(−1)k−1 d k−1 1 P1 (τ, z) = k + E k + O(z). k−1 (k − 1)! dz z
(21)
Finally, it is convenient to define for k, l = 1, 2, . . ., (k + l − 1)! E k+l (τ ), (k − 1)!(l − 1)! (k + l − 1)! Pk+l (τ, z). D(k, l, z) = D(k, l, τ, z) = (−1)k+1 (k − 1)!(l − 1)! C(k, l) = C(k, l, τ ) = (−1)k+1
(22) (23)
Note that C(k, l) = C(l, k) and D(k, l, z) = (−1)k+l D(l, k, z). These naturally arise in the analytic expansions (in appropriate domains) 1 P2 (τ, z − w) = + C(k, l)z l−1 w k−1 , (24) 2 (z − w) k,l≥1
and for k ≥ 1, Pk+1 (τ, z) = Pk+1 (τ, z − w) =
1 z k+1
+
1 C(k, l)z l−1 , k
(25)
l≥1
1 D(k, l, w)z l−1 . k
(26)
l≥1
3. The Formalism for Sewing Together Two Riemann Surfaces In this section we review a general construction due to Yamada [Y] for “sewing” together two Riemann surfaces of genus g1 and g2 to form a surface of genus g1 + g2 . The principle aim is to describe various structures such as the genus g1 + g2 period matrix in terms of data coming from the genus g1 and genus g2 surfaces. The basic method described below follows that of Yamada. However, a significant number of changes have been made in order to express the final formulas more neatly. We also discuss the holomorphic properties of the period matrix and of a certain infinite dimensional determinant. In the next section, this general formalism will be applied to the construction of a genus two Riemann surface.
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3.1. The Bilinear Form ω(g) and the Period Matrix Ω (g) . Consider a compact Riemann surface S of genus g with canonical homology basis a1, . . . ag , b1 , . . . bg . In general (g) there exists g holomorphic 1-forms νi , i = 1, . . . g which we may normalize by [FK1, Sp] (g) ν j = 2πiδi j . (27) ai
These forms can be neatly encapsulated in a unique singular bilinear two form ω(g) , known as the normalized differential of the second kind. It is defined by the following properties [Sp, Mu, Y]: 1 ω(g) (x, y) = + regular terms d xd y (28) (x − y)2 for any local coordinates x, y, with normalization ω(g) (x, ·) = 0
(29)
ai
for i = 1, . . . , g. Using the Riemann bilinear relations, one finds that (g) νi (x) = ω(g) (x, ·),
(30)
bi (g)
with νi normalized as in (27). The genus g period matrix Ω (g) is then defined by 1 (g) (g) Ωi j = νj (31) 2πi bi
for i, j = 1, . . . , g. It is useful to also introduce the normalized differential of the third kind [Mu, Y] (g) ω p2 − p1 (x)
p2 =
ω(g) (x, ·),
(32)
p1
for which
ai
(g)
ω p2 − p1 = 0. For a local coordinate x near pa for a = 1, 2 we have (−1)a (g) + regular terms d x. ω p2 − p1 (x) = x − pa (g)
Both ω(g) (x, y) and ω p2 − p1 (x) can be expressed in terms of the prime form K (g) (x, y) (d x)−1/2 (dy)−1/2 , a holomorphic form of weight (− 21 , − 21 ) with [Mu] ω(g) (x, y) = ∂x ∂ y log K (g) (x, y)d xd y, (g)
ω p2 − p1 (x) = ∂x log
K (g) (x, K (g) (x,
p2 ) d x. p1 )
(33) (34)
We also note that K (g) (x, y) = −K (g) (y, x) and that K (g) (x, y) = x − y + O((x − y)3 ).
On Genus Two Riemann Surfaces Formed from Sewn Tori
595
Example 1. For the genus one Riemann torus with periods 2πi and 2πiτ along the a and b cycles, the holomorphic 1-form satisfying (27) in the usual parameterization is ν1(1) = dz . The normalized differential of the second kind is determined by P2 (τ, z) via ω(1) (x, y) = P2 (τ, x − y)d xd y.
(35)
(1)
In this case, (29) and (30) follow from (14), and Ω11 = τ . The normalized differential of the third kind is ω(1) p2 − p1 (x) = (P1 (τ, x − p2 ) − P1 (τ, x − p1 ))d x and the prime form (1) is K (x, y) = K (τ, x − y) of (16). It is well-known that Ω (g) is a complex symmetric matrix with positive-definite imaginary part, i.e., Ω (g) ∈ Hg , the genus g Siegel complex upper half-space. The intersection form is a natural non-degenerate symplectic bilinear form on the first homology group H1 (S, Z) ∼ = Z2g , satisfying (ai , a j ) = (bi , b j ) = 0, (ai , b j ) = δi j , i, j = 1, . . . , g. The genus g symplectic group4 is Sp(2g, Z) = {γ =
A B C D
∈ S L(2g, Z)|
AB T = B A T , C D T = D T C, AD T − BC T = Ig }. It acts on Hg via γ .Ω (g) = (AΩ (g) + B)(CΩ (g) + D)−1 ,
(36)
and naturally on H1 (S, Z), where it preserves . 3.2. The Formalism for Sewing Two Riemann Surfaces. We now discuss a general method described by Yamada [Y] for calculating the bilinear form (28) and hence the period matrix on the surface formed by sewing together two Riemann surfaces. Consider two Riemann surfaces Sa of genus ga for a = 1, 2. Choose a local coordinate z a on Sa in the neighborhood of a point pa , and consider the closed disk |z a | ≤ ra for ra > 0 sufficiently small. (Note that the choice ra = 1 is made in ref. [Y]). Introduce a complex sewing parameter where || ≤ r1r2 , and excise the disk {z a , |z a | ≤ ||/ra¯ } ⊂ Sa to form a punctured surface Sˆa = Sa \{z a , |z a | ≤ ||/ra¯ }. Here and below, we use the convention 1 = 2, 2 = 1.
4 Here and elsewhere, the transpose of a matrix or vector is denoted by T.
(37)
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G. Mason, M. P. Tuite
Fig. 1. Sewing Two Riemann Surfaces
Define the annulus Aa = {z a , ||/ra¯ ≤ |z a | ≤ ra } ⊂ Sˆa , and identify A1 and A2 as a single region A = A1 A2 via the sewing relation z 1 z 2 = .
(38)
ˆ 1 \A1 } ∪ {Sˆ2 \A2 } ∪ A of genus In this way we obtain a compact Riemann surface {S g1 + g2 . The sewing relation (38) can be considered to be a parameterization of a cylinder connecting the two punctured Riemann surfaces. Noting the notational differences with ref. [Y], the genus g1 + g2 normalized differential of the second kind ω(g1 +g2 ) of ( 28) enjoys the following properties: Theorem 1 (Ref. [Y], Theorem 1, Theorem 4). (a) ω(g1 +g2 ) is holomorphic in for || < r1r2 ; (b) lim→0 ω(g1 +g2 ) (x, y) = ω(ga ) (x, y)δab for x ∈ Sˆa , y ∈ Sˆb , a, b = 1, 2. Regarded as a power series in , the coefficients of ω(g1 +g2 ) can be calculated from and ω(g2 ) as follows. Let Ca (z a ) ⊂ Aa denote a simple, closed, anti-clockwise oriented contour parameterized by z a , surrounding the puncture at z a = 0. Note that C1 (z 1 ) may be deformed to −C2 (z 2 ) via (38). Then one finds [Y]: ω(g1 )
Lemma 1 (op.cit., Lemma 4). ω
(g1 +g2 )
(x, y) = ω
(ga )
1 (x, y)δab + 2πi
(ω
(g1 +g2 )
za (y, z a )
ω(ga ) (x, ·)) (39)
Ca (z a )
for x ∈ Sˆa , y ∈ Sˆb and a, b = 1, 2 . Define weighted moments for ω(g1 +g2 ) for k, l = 1, 2, . . . by X ab (k, l) = X ab (k, l, ) (k+l)/2 1 = √ kl (2πi)2
u −k v −l ω(g1 +g2 ) (u, v).
(40)
Ca (u) Cb (v)
√ The (k+l)/2 / kl factor is introduced for later convenience. Note that X ab (k, l) = X ba (l, k)
(41)
and that −(k+l)/2 X ab (k, l, ) is holomorphic in for || < r1r2 from Theorem 1. We define X ab = (X ab (k, l)) to be the infinite matrix indexed by k, l.
On Genus Two Riemann Surfaces Formed from Sewn Tori
597
Next define a set of holomorphic 1-forms on Sˆa by k/2 aa (k, x) = aa (k, x, ) = z a−k ω(ga ) (x, z a ), √ 2πi k
(42)
Ca (z a )
and define aa (x) = (aa (k, x)) to be the infinite row vector indexed by k. Note from (28) that for x, y ∈ Sˆa with x = 0 we have ⎡ ⎤ ⎢ 1 ⎥ ω(ga ) (x, y) = z a−k ω(ga ) (x, z a )⎦ y k−1 dy, ⎣ 2πi k≥1
=
√
Ca (z a )
k
−k/2
aa (k, x)y k−1 dy.
(43)
k≥1
Using Lemma 1 we have: Lemma 2. ω(g1 +g2 ) (x, y) is given by (g ) ω a (x, y) + aa (x)X a¯ a¯ aaT (y) x, y ∈ Sˆa , ω(g1 +g2 ) (x, y) = T x ∈ Sˆa , y ∈ Sˆa¯ . aa (x)(−I + X aa ¯ )aa¯ (y)
(44)
Proof. From (43) it follows that za
ω(ga ) (x, ·) =
0
−k/2 √ aa (k, x)z ak . k k≥1
(45)
Let x, y ∈ Sˆ1 . Using (38), (39) and (45) we find that ω(g1 +g2 ) (x, y) − ω(g1 ) (x, y) is given by ⎛ ⎞ −k/2 k ⎜ ⎟ z 2−k ω(g1 +g2 ) (y, z 2 )⎠ (46) √ a1 (k, x) ⎝− 2πi k k≥1
C2 (z 2 )
(k+l)/2 1 = a1 (k, x)a1 (l, y) √ (2πi)2 kl k,l≥1
z 2−k w2−l ω(g1 +g2 ) (z 2 , w2 ),
C2 (w2 ) C2 (z 2 )
giving (44) for x, y ∈ Sˆ1 . For x ∈ Sˆ1 , y ∈ Sˆ2 we find that ω(g1 +g2 ) (x, y) is given by ⎛ ⎞ k −k/2 ⎜ ⎟ z 2−k ω(g1 +g2 ) (y, z 2 )⎠ √ a1 (k, x) ⎝− 2πi k k≥1
=−
k≥1
(47)
C2 (z 2 )
a1 (k, x)a2 (k, y)
(k+l)/2 1 + a1 (k, x)a2 (l, y) √ (2πi)2 kl k,l≥1
z 1−l z 2−k ω(g1 +g2 ) (z 1 , z 2 ),
C1 (z 1 ) C2 (z 2 )
giving (44). A similar analysis follows for x, y ∈ Sˆ1 and x ∈ Sˆ2 , y ∈ Sˆ1 .
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G. Mason, M. P. Tuite
We next compute the explicit form of the moment matrix X ab in terms of the moments of ω(ga ) , which we denote by (k+l)/2 √ (2πi)2 kl
Aa (k, l) = Aa (k, l, ) =
=
k/2
√ 2πi k
x −k y −l ω(ga ) (x, y)
Ca (x) Ca (y)
x −k aa (l, x).
(48)
Ca (x)
Note from (28) that for x, y ∈ Sˆa we have d xd y ω(ga ) (x, y) − (x − y)2 ⎫ ⎧ ⎪ ⎬ ⎨ 1 ⎪ −k −l (ga ) = u v ω (u, v) x k−1 y l−1 d xd y, ⎪ ⎪ (2πi)2 ⎭ ⎩ k,l≥1 Ca (u) Ca (v) √ −(k+l)/2 = kl Aa (k, l, )x k−1 y l−1 d xd y.
(49)
k,l≥1
Proposition 1. The matrices X ab are given in terms of Aa by X aa = Aa (I − Aa¯ Aa )−1 , X a a¯ = I − (I − Aa Aa¯ )−1 .
(50) (51)
Here, (I − Aa Aa¯ )−1 =
(Aa Aa¯ )n
(52)
n≥0
and is convergent as a power series in for || < r1r2 . Proof. Compute X 11 from (46) to find (k+l)/2 1 X 11 (k, l) = √ kl (2πi)2 −
m≥1
[
x −k y −l ω(g1 ) (x, y)
C1 (x) C1 (y)
k/2
1 √ 2πi k
(m+l)/2
1 √ 2 (2πi) ml
C1 (x)
x −k a1 (m, x).
y −l z 2−m ω(g1 +g2 ) (y, z 2 )],
C1 (y) C2 (z 2 )
and similarly for X 22 . Thus using (48) and recalling (41) we have X aa = Aa (I − X aa ¯ ).
(53)
On Genus Two Riemann Surfaces Formed from Sewn Tori
599
We may find X 12 from (47) as follows: k/2 1 X 12 (k, l) = − √ 2πi k ⎛
m≥1
C1 (x)
(m+l)/2
⎜ ⎝ √
a1 (m, x)x −k
1 ml (2πi)2
⎞
⎟ y −l z 2−m ω(2) (y, z 2 )⎠ ,
C2 (y) C2 (z 2 )
and similarly for X 21 , i.e., X a a¯ = −Aa X a¯ a¯ . Define infinite block matrices A1 0 X 11 X 12 , A= , X= 0 A2 X 21 X 22
(54) Q=
0 −A1 −A2 0
(55)
so that (53, 54) can be combined as X = A + Q X,
(56)
X = (I − Q)−1 A.
(57)
so that
Here (I − Q)−1 = n≥0 Q n which we now show converges for || < r1r2 . Consider X = A + AX + Q 2 X . Then since Q 2 = diag(A1 A2 , A2 A1 ) we obtain iterative relations X aa = Aa (I + Aa¯ X aa ), I − X a a¯ = I + Aa Aa¯ (I − X a a¯ ).
(58) (59)
Now −(k+l)/2 X ab (k, l) is holomorphic in for || < r1r2 by Theorem 1. Therefore, X ab (k, l) has a series expansion in 1/2 convergent for || < r1r2 . Then (58) implies N (Aa Aa¯ )n Aa )(k, l) + O( (k+l)/2+2N +1 ) where the coefficient of each X aa (k, l) = ( n=0 power of consists of a finite sum of finite products of A1 and A2 . Hence X aa =
∞
Aa (Aa¯ Aa )n = Aa (I − Aa¯ Aa )−1 ,
n=0
converges for || < r1r2 . A similar argument holds for X a a¯ where one finds X a a¯ =
∞
(Aa Aa¯ )n = I − (I − Aa Aa¯ )−1
n=1
converges for || < r1r2 . Finally (I − Q)−1 =
Qn =
n≥0
is therefore also convergent for || < r1r2 .
n≥0
Q 2n (I + Q)
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G. Mason, M. P. Tuite
The invertibility of the infinite matrix I − A1 A2 for || < r1r2 is crucial in the sewing formalism. We now define an infinite determinant of I − A1 A2 which we show is a holomorphic function of for || < r1r2 . This determinant plays a dominant role in the sequel to this work [MT2]. Firstly, since A1 (k, m)A2 (m, l) = O( m+(k+l)/2 ) we may define a (2N − 3) × (2N − 3) matrix TN (k, l) = A1 (k, m)A2 (m, l), (60) 1≤m≤N −(k+l)/2
for 1 ≤ k, l ≤ 2N − 3. TN is a truncated approximation for A1 A2 to O( N ), ⎞ ⎛ TN 0 · · · ⎟ ⎜ A1 A2 = ⎝ 0 0 · · · ⎠ + O( N +1 ). .. .. . . . . . We may then define as formal power series in to O( N ) the expressions5 det(I − A1 A2 ) = det(I N − TN ) + O( N +1 ), (61) N +1 T r log(I − A1 A2 ) = T r log(I N − TN ) + O( ), (62) N /2 1 where T r log(I N − TN ) = − n=1 n T r (TNn ) + O( N +1 ). Comparing the finite matrix contributions of (61) and (62) we have Lemma 3. As formal power series in , log det(I − A1 A2 ) = T r log(I − A1 A2 ).
(63)
We now show that T r log(I − A1 A2 ) is holomorphic in for || < r1r2 so that: Theorem 2. det(I − A1 A2 ) is non-vanishing and holomorphic in for || < r1r2 . Proof. Let ω(g1 +g2 ) = f (z 1 , z 2 , )dz 1 dz 2 for |z a | ≤ ra . Then f (z 1 , z 2 , ) is holomorphic in for || ≤ r for r < r1r2 from Theorem 1. Apply Cauchy’s inequality to the coefficient functions for f (z 1 , z 2 , ) = n≥0 f n (z 1 , z 2 ) n to find M , rn for M = sup|za |≤ra ,||≤r | f (z 1 , z 2 , )| . Consider 1 I= ω(g1 +g2 ) (z 1 , z 2 ) log(1 − ), 2 (2πi) z1 z2 | f n (z 1 , z 2 )| ≤
(64)
(65)
Cr1 (z 1 ) Cr2 (z 2 )
for Cra (z a ) the contour with |z a | = ra . Then using (64) we find ||n |I| ≤ | f n (z 1 , z 2 ) log(1 − )dz 1 dz 2 | 2 (2π ) z1 z2 n≥0
≤
n≥0
Cr1 (z 1 ) Cr2 (z 2 )
M.
|| |.r1r2 , .| log 1 − rn r1 r2
||n
5 For the sake of notational simplicity we denote both the usual finite dimensional and the defined infinite dimensional determinants by det.
On Genus Two Riemann Surfaces Formed from Sewn Tori
601
i.e., I is absolutely convergent and thus holomorphic in for || ≤ r < r1r2 . Since |z 1 z 2 | = r1r2 we may alternatively expand in /z 1 z 2 to obtain k 1 ω(g1 +g2 ) (z 1 , z 2 )z 1−k z 2−k I=− k (2πi)2 k≥1
where T r X 12
Cr1 (z 1 ) Cr2 (z 2 )
= −T r X 12 , = k≥1 X 12 (k, k) for X 12 of (51). But (59) implies T r X 12 = − T r ((A1 A2 )n ), n≥1
is absolutely convergent for || < r1r2 . Hence we find 1 T r ((A1 A2 )n ), T r log(I − A1 A2 ) = − n n≥1
is also absolutely convergent for || < r1r2 . Thus from Lemma 3, det(I − A1 A2 ) is non-vanishing and holomorphic for || < r1r2 . These determinant properties can also be expressed in terms of the block matrix Q using6 Lemma 4. det(I ± Q) = det(I − A1 A2 ). Proof. Let Q N be the truncated approximation for Q to O( N ). Then one finds det(I + Q N ) = det(I − Q N ). But det(I + Q N ) det(I − Q N ) = det(I − Q 2N ) = det(I − TN )2 for TN of (60) and the result follows. The sewn genus g1 + g2 Riemann surface naturally inherits a basis of cycles labelled {as1 , bs1 |s1 = 1, . . . , g1 } and {as2 , bs2 |s2 = g1 + 1, . . . , g1 + g2 } from the genus g1 and genus g2 surfaces respectively. Integrating ω(g1 +g2 ) along the b cycles then gives us the holomorphic 1-forms and period matrix. For a = 1, 2 we define (ga ) (66) αsa (k) = aa (k), bsa (g )
(g )
and the infinite vector αsa a = (αsa a (k)). Then we find using (30) and (31) together with Lemma 2 and Proposition 1 that [Y]: Theorem 3 (op. cit. Theorem 4.). Ω (g1 +g2 ) is holomorphic in for || < r1r2 , and is given by (g +g2 )
= 2πiΩs1 t11 + αs1 1 (A2 (I − A1 A2 )−1 )αt1 1 ,
(g +g ) 2πiΩs2 t12 2 (g +g ) 2πiΩs1 s12 2
(g ) (g ) (g )T 2πiΩs2 t22 + αs2 2 (A1 (I − A1 A2 )−1 )αt2 2 , (g ) (g )T −αs1 1 (I − A2 A1 )−1 αs2 2 (g ) (g )T −αs2 2 (I − A1 A2 )−1 αs1 1 ,
2πiΩs1 t11
(g )
= = =
(g )
with s1 , t1 ∈ {1, . . . , g1 } and s2 , t2 ∈ {g1 + 1, . . . , g1 + g2 }. 6 See the previous footnote.
(g )T
(67) (68)
(69)
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Example 2. Let S1 be a genus g surface and S2 the Riemann sphere C ∪ {∞} with bilinear form ω(0) (x, y) =
d xd y , x, y ∈ S2 . (x − y)2
(70)
√ Choose p2 = 0 with z 2 ∈ C as the local coordinate on S2 . Then a2 (k, x) = k k/2 x −k−1 d x from ( 42) and hence A2 = 0 from (48). Thus X 22 = A1 and X 11 = X 12 = X 21 = 0. Then one can check that the RHS of (44) reproduces ω(g) directly for a = b = 1 whereas, using (38 ), it follows for a = b = 2 from (49) and for a = b from (43). Let us now consider the holomorphic properties of Ω (g1 +g2 ) . From Theorem 1, ω(g1 +g2 ) (x, y) is holomorphic in for || < r1r2 and therefore Ω (g1 +g2 ) is also. We now show that if ω(ga ) is holomorphic with respect to a complex parameter (such as one of the modular parameters of the Riemann surface Sa ) then ω(g1 +g2 ) and therefore Ω (g1 +g2 ) is also holomorphic with respect to that parameter. To this end we firstly prove the following elementary lemma: Lemma 5. Let f (x, function holomorphic in x for |x| < R with expan y) be a complex m sion f (x, y) = m≥0 cm (y)x . Suppose that each cm (y) is holomorphic and that f (x, y) is continuous in y for |y| < S. Then f (x, y) is also holomorphic in y for |y| < S. Proof. Define the compact region R = {(x, y) : |x| ≤ R− , |y| ≤ S− } for R− = R − δ1 and S− = S − δ2 for δ1 , δ2 > 0. f is continuous in the compact region R and hence | f (x, y)| ≤ M ≡ supR | f |. Apply Cauchy’s inequality to f as a holomorphic function of x for |x| ≤ R− to find |cm (y)| ≤
sup|x|≤R− | f (x, y)| m R−
≤
M m. R−
But cm (y) is holomorphic for |y| ≤ S− with expansion cm (y) = applying Cauchy’s inequality again gives |cmn | ≤
sup|y|≤S− |cm (y)| n S−
≤
n≥0 cmn y
n
so that
M
m Sn . R− −
Thus f (x, y) = m≥0 x m y n is absolutely convergent for |x| < R− , |y| < n≥0 cmn m S− . Hence cn (x) = m≥0 cmn x converges for |x| < R− and f is holomorphic in y with convergent expansion f (x, y) = n≥0 cn (x)y n for |y| < S− . We may then choose δ1 , δ2 sufficiently small to show that the result follows for all |x| < R, |y| < S. Proposition 2. Suppose that ω(ga ) is a holomorphic function of a complex parameter μ for |μ| < S. Then for || < r1r2 , ω(g1 +g2 ) is also holomorphic in μ for |μ| < S. Proof. Suppose that ω(g1 ) is holomorphic in μ wlog. Then a1 (k) and A1 (k, l) are holomorphic (and continuous) in μ for |μ| < S. We now show that X ab is holomorphic in μ for |μ| < S using Lemma 5. Using continuity of A1 and (58) of Proposition 1 we find that for |μ + δ| < S, (I − A2 A1 ) lim (X 11 (μ + δ) − X 11 (μ)) = 0. δ→0
On Genus Two Riemann Surfaces Formed from Sewn Tori
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But (I − A2 A1 ) is invertible for || < r1r2 from Proposition 1 and so X 11 (μ) is continuous for |μ| < S. A similar result holds for X 12 , X 21 and X 22 . From Theorem 1, −(k+l)/2 X ab (k, l) is holomorphic in for || < r1r2 . Furthermore, as explained in Proposition 1, the expansion coefficients consist of a finite sum of finite products of A1 and A2 terms and thus they are holomorphic in μ for |μ| < S. We may therefore apply Lemma 5 to −(k+l)/2 X ab (k, l) which is continuous in μ for |μ| < S and holomorphic in for || < R = r1r2 with expansion coefficients holomorphic in μ for |μ| < S. Thus −(k+l)/2 X ab (k, l) and therefore X ab (k, l) is holomorphic in μ for |μ| < S. Finally consider ω(g1 +g2 ) as given in (44) of Lemma 2. Using arguments similar to those above we see that ω(g1 +g2 ) is continuous in μ for |μ| < S and holomorphic in for || < r1r2 with expansion coefficients holomorphic in μ for |μ| < S. Thus ω(g1 +g2 ) is holomorphic in μ for |μ| < S. Corollary 1. Given the previous conditions then Ω (g1 +g2 ) is holomorphic in μ for |μ| < S where || < r1r2 . In conclusion, we similarly find by applying Proposition 2 to (65) that Proposition 3. Suppose that ω(ga ) is holomorphic in μ for |μ| < S. Then det(I − A1 A2 ) is non-vanishing and holomorphic in μ for |μ| < S and || < r1r2 . 4. Sewing Two Tori to Form a Genus Two Riemann Surface We now specialize to the case of two tori sewn together to form a genus two Riemann surface. We first consider an elementary description of a disk on a torus compatible with S L(2, Z) modular-invariance. We then apply the formalism in order to sew two punctured tori with modular parameters τ1 and τ2 together to form a genus two Riemann surface with period matrix Ω (2) (τ1 , τ2 , ) ∈ H2 , where Ω (2) is holomorphic in (τ1 , τ2 , ) ∈ D for a suitably defined domain D . We provide an alternative description of Ω (2) in terms of the sum of weights of particular “necklace” graphs. We then describe the equivariance properties of this holomorphic mapping from D to H2 with respect to a certain subgroup G ⊆ Sp(4, Z), and prove that it is invertible in a certain G-invariant domain. 4.1. A Closed Disk on a Torus. A complex torus S (that is, a compact Riemann surface of genus 1), can be represented as a quotient C/, where is a lattice in C. Moreover, two such tori Sa , a = 1, 2 are isomorphic if, and only if, the lattices are homothetic , that is, there is a ξ ∈ C∗ such that 2 = ξ 1 . A framing of S = C/ is a choice of basis (σ, ς ) such that the modulus τ = σ/ς satisfies τ ∈ H1 . We say that the basis (σ, ς ) is positively oriented in this case. A pair of framed tori (C/a , σa , ςa ), a = 1, 2 are isomorphic if, and only if, there is a ξ as above such that (σ2 , ς2 ) = ξ(σ1 , ς1 ). The modulus τ depends only on the isomorphism class of the framed torus, and there is a bijection {isomorphism classes of framed tori} → H1 , (C/, σ, ς ) → σ/ς.
(71)
S L(2, Z) is the group of automorphisms of which preserves oriented bases. It acts on isomorphism classes of framed tori via ab : (C/, σ, ς ) → (C/, aσ + bς, cσ + dς ), cd
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and on H1 via fractional linear transformations aτ + b ab : τ → . cd cτ + d With respect to these two actions, the bijection (71) is S L(2, Z)-equivariant. In the following it is convenient to identify S with the standard fundamental region for determined by the basis, and with appropriate identifications of boundary. To describe a well-defined disk in S, define the minimal length of as D() = min |λ|.
(72)
D(ξ ) = |ξ |D(). (ξ = 0).
(73)
0 =λ∈
It obviously satisfies
We may now describe a closed disk on S. The proof follows from the triangle inequality. Lemma 6. For p ∈ S, the points z ∈ S satisfying |z − p| ≤ k D() define a closed disk centered at p provided k < 21 . Let S be a complex torus of modulus τ . Among all homothetic lattices for which we define S ∼ = C/, it will be convenient to work with the lattice τ which has basis ab ∈ S L(2, Z) we have (2πiτ, 2πi). Note that for γ = cd D(γ τ ) =
1 D(τ ). |cτ + d|
(74)
4.2. The Genus Two Period Matrix in the Formalism. We now apply the -formalism to a pair of tori Sa = C/a with local co-ordinates z a , where a has oriented basis (σa , ςa ) and τa = σa /ςa ∈ H1 for a = 1, 2. We shall sometimes refer to S1 and S2 as the left and right torus respectively. After Lemma 6 we may consider the annuli Aa centred at the origin of Sa described above, with outer radius ra < 21 D(a ). Following the prescription of Subsect. 3.2, we sew the two tori by identifying the annuli A1 and A2 via the relation z 1 z 2 = as in (38), where || ≤ r1r2 < 41 D(1 )D(2 ). As discussed in Subsect. 4.1 we take (σa , ςa ) = (2πiτa , 2πi) and qa = exp(2πiτa ) for a = 1, 2. Define the domain7 D = {(τ1 , τ2 , ) ∈ H1 ×H1 ×C | || <
1 D(τ1 )D(τ2 )}. 4
(75)
We now explicitly determine the period matrix8 . 7 The superscript merely denotes that we are working in the -formalism, and should not be interpreted as a variable of any kind. 8 The genus two period matrix is also described in [T] without proof and in a different notation.
On Genus Two Riemann Surfaces Formed from Sewn Tori
605
Theorem 4. Sewing determines a holomorphic map F : D → H2 , (τ1 , τ2 , ) → Ω (2) (τ1 , τ2 , ).
(76)
Moreover Ω (2) = Ω (2) (τ1 , τ2 , ) is given by (2)
2πiΩ11 = 2πiτ1 + (A2 (I − A1 A2 )−1 )(1, 1), (2) 2πiΩ22 (2) 2πiΩ12
= 2πiτ2 + (A1 (I − A2 A1 ) = −(I − A1 A2 )
−1
−1
)(1, 1),
(1, 1).
(77) (78) (79)
Notation here is as follows: Aa (τa , ) is the infinite matrix with (k, l)-entry, Aa (k, l, τa , ) =
(k+l)/2 C(k, l, τa ); √ kl
(80)
(1, 1) refers to the (1, 1)-entry of a matrix with C(k, l, τ ) of (22). Proof. The bilinear two form ω(1) is given by (35). Using (20), the basis of 1-forms (42) with periods (2πiτa , 2πi) is then given by k/2 d x z −k P2 (τa , x − z)dz, √ 2πi k Ca (z) √ k/2 = k Pk+1 (τa , x)d x.
aa (k, x) =
(81)
Now (80) follows from (48), (25) and (22). Note that (14) implies that αa (k) of ( 66) is αa (k) = 1/2 δk,1 .
(82)
We therefore find Ω (2) to be given by (77)–(79) for || < r1r2 < 41 D(τ1 )D(τ2 ). By Theorem 3, Ω (2) is holomorphic in for || < r1r2 < 41 D(τ1 )D(τ2 ). The left torus bilinear form ω(1) (x, y, τ1 ) is holomorphic in some neighborhood |τ1 − τ10 | < S of any point τ10 ∈ H1 . Hence we may apply Corollary 1 with μ = τ1 − τ10 for |μ| < S to conclude that Ω (2) is holomorphic in τ1 . Similarly Ω (2) is holomorphic in τ2 , and by Hartog’s theorem (e.g., [Gu]) Ω (2) is holomorphic on D . The infinite matrices Aa (τa , ) will play a crucial rôle in the further analysis of the formalism. Dropping the subscript, they are symmetric and have the form: √ 2 ⎞ E 2 (τ ) 3 E 4 (τ ) 0 0 ··· √ ⎜ −3 2 E 4 (τ ) 0 −5 2 3 E 6 (τ ) · · · ⎟ ⎜√ 0 ⎟ 3 ⎜ 3 2 E 4 (τ ) ⎟ 0 10 E (τ ) 0 · · · 6 A(τ, ) = ⎜ ⎟. √ 3 ⎜ 4 0 −5 2 E 6 (τ ) 0 −35 E 8 (τ ) · · · ⎟ ⎝ ⎠ .. .. .. .. .. . . . . . ⎛
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4.3. Chequered Necklace Expansion for Ω (2) . It is useful to introduce an interpretation for the expressions for Ω (2) found above in terms of the sum of weights of certain graphs. Let us introduce the set of chequered necklaces N . By definition, these are connected graphs with n ≥ 2 nodes, (n −2) of which have valency 2 and two of which have valency 1 (these latter are the end nodes), together with an orientation, say from left to right, on the edges. Moreover vertices are labelled by positive integers and edges are labelled alternatively by 1 or 2 as one moves along the graph, e.g., k1
k2
1
k3
2
k4
1
2
k5
1
k6
• −→ • −→ • −→ • −→ • −→ •
We also define the degenerate necklace N0 to be a single node with no edges. Define a weight function ω : N −→ C[E 2 (τa ), E 4 (τa ), E 6 (τa ), | a = 1, 2], k
a
l
as follows: if a chequered necklace N has edges E labelled as • −→ • then we define ω(E) = Aa (k, l, τa , ),
ω(N ) = ω(E),
(83)
where Aa (k, l, τa , ) is given by (80) and the product is taken over all edges E of N . We further define ω(N0 ) = 1. Among all chequered necklaces there is a distinguished set for which both end nodes are labelled by 1. There are four types of such chequered necklaces, which may be further distinguished by the labels of the two edges at the extreme left and right. We use the notation (37) for a = 1, 2 , and say that the chequered necklace 1
a
i
j
b
1
• −→ • . . . • −→ • is of type ab . We then set Nab = {isomorphism classes of chequered necklaces of type ab}, ωab = ω(N ), N ∈Nab
where ωab is considered as an element in C[E 2 (τa ), E 4 (τa ), E 6 (τa ), | a = 1, 2]. It is clear that we may use this formalism to represent matrix expressions like those appearing earlier. Then we have Lemma 7. For a = 1, 2 we have −1 ωa a¯ = ωaa ¯ = (I − Aa Aa¯ ) (1, 1), ωaa = (Aa¯ (I − Aa Aa¯ )−1 )(1, 1).
Thus we may conclude from Theorem 4 that Proposition 4. For a = 1, 2 we have (2) Ωaa = τa + (2)
Ωa a¯ = −
ωaa , 2πi
ωa a¯ . 2πi
On Genus Two Riemann Surfaces Formed from Sewn Tori
607
4.4. Equivariance of F . In Theorem 4 we established the existence of the analytic map F . Here we establish the equivariance of this map with respect to a certain subgroup G of Sp(4, Z). We will employ the graphical representation for Ω = Ω (2) (τ1 , τ2 , ) in terms of chequered necklaces discussed in the last subsection. As an abstract group, G is isomorphic to (S L(2, Z) × S L(2, Z)) Z2 , i.e., the direct product of two copies of S L(2, Z) which are interchanged upon conjugation by an involution. There is a natural injection G → Sp(4, Z) in which the two S L(2, Z) subgroups are mapped to ⎧⎡ ⎧⎡ ⎤⎫ ⎤⎫ a 0 b1 0 ⎪ 1 0 0 0 ⎪ ⎪ ⎪ ⎨ 1 ⎨ ⎬ ⎬ ⎢ 0 1 0 0⎥ ⎢ 0 a 0 b2 ⎥ 1 = ⎣ , (84) , 2 = ⎣ 2 ⎦ ⎦ ⎪ ⎪ ⎩ c1 0 d1 0 ⎪ ⎩ 0 0 1 0 ⎪ ⎭ ⎭ 0 0 0 1 0 c2 0 d2 and the involution is mapped to
⎡
0 ⎢1 β=⎣ 0 0
1 0 0 0
0 0 0 1
⎤ 0 0⎥ . 1⎦ 0
(85)
In this way we obtain a natural action of G on H2 . The action on D is described in the next lemma. Lemma 8. G has a left action on D as follows: ), c1 τ1 + d1 ), γ2 .(τ1 , τ2 , ) = (τ1 , γ2 τ2 , c2 τ2 + d2 β.(τ1 , τ2 , ) = (τ2 , τ1 , ), γ1 .(τ1 , τ2 , ) = (γ1 τ1 , τ2 ,
(86) (87) (88)
for (γ1 , γ2 ) ∈ S L(2, Z) × S L(2, Z). Proof. It is straightforward (and quite standard) to see that (86) - (88) formally define a (left) action of G on H1 × H1 × C. What we must show is that this action preserves the domain D . For β this is obvious, and for elements (γ1 , γ2 ) it follows from (74). We now establish the following result: Theorem 5. F is equivariant with respect to the action of G, i.e., there is a commutative diagram for γ ∈ G, F
D → H2 γ ↓ ↓γ F
D → H2
Ω11 Ω12 . Of course, each Ω12 Ω22 Ωi j is a function of (τ 1 , τ2 , ). The action of G on H2 is given in (36), and in particular Ω22 Ω12 . Therefore from (88) we have β : Ω → Ω12 Ω11 Ω22 Ω12 F (β(τ1 , τ2 , )) = F (τ2 , τ1 , ) = = β(F (τ1 , τ2 , )). Ω12 Ω11
Proof. Fix (τ1 , τ2 , ) ∈ D , with Ω = F (τ1 , τ2 , ) =
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So the theorem is true in case γ = β. To complete the proof of the theorem, it suffices to consider the case when γ = γ1 lies in the ‘left’ modular group acting on τ1 . From (36), a Ω +b Ω 1
γ1 : Ω →
11
1
c1 Ω11 +d1 Ω12 c1 Ω11 +d1
12
c1 Ω11 +d1 c Ω2 Ω22 − c1 Ω1 1112 +d1
,
(89)
and we are obliged to show that the matrix in display (89) coincides with F (γ1 (τ1 , τ2 , )) = F (γ1 τ1 , τ2 , c1 τ1+d1 ). In other words, we must establish the following identities: a1 Ω11 + b1 = Ω11 (γ1 τ1 , τ2 , ), c1 Ω11 + d1 c1 τ1 + d1 Ω12 = Ω12 (γ1 τ1 , τ2 , ), c1 Ω11 + d1 c1 τ1 + d1 2 c1 Ω11 Ω22 − = Ω22 (γ1 τ1 , τ2 , ). c1 Ω11 + d1 c1 τ1 + d1
(90) (91) (92)
Aa (k, l, τa , ) of (80) is a modular form of weight k + l for k + l > 2, whereas Aa (1, 1, τa , ) = E 2 (τa ) enjoys an exceptional transformation law thanks to (12). Using Lemma 8 we then find that )) = (c1 τ1 + d1 )(k+l)/2 (A1 (τ1 , ) + κδk1 δl1 ), c1 τ1 + d1 )) = (c1 τ1 + d1 )−(k+l)/2 A2 (τ2 , ), A2 (k, l, τ2 , c1 τ1 + d1
A1 (k, l, γ1 τ1 ,
(93) (94)
where κ=−
c1 . 2πi c1 τ1 + d1
(95)
It follows from Proposition 4 both that 1 − κω11 = and
c1 Ω11 + d1 , c1 τ1 + d1
(96)
Ω11 γ1 τ1 , τ2 , c1 τ1 + d1 1 a1 τ1 + b1 + . ω11 γ1 τ1 , τ2 , = c1 τ1 + d1 2πi c1 τ1 + d1
(97)
Consider a necklace N ∈ N11 of weight ω(N ) and let S11 (N ) denote the set of all 1
1
1
“broken” graphs formed from N by deleting any n edges of type • −→ • for all n ≥ 0. Every such graph consists of n + 1 connected graphs N1 , . . . Nn+1 of type 11. From (93) and (94) it therefore follows that ω(N )(γ1 τ1 , τ2 ,
1 )= κn c1 τ1 + d1 c1 τ1 + d1 n≥0
N1 ,...Nn+1
ω(N1 ) . . . ω(Nn+1 ).
On Genus Two Riemann Surfaces Formed from Sewn Tori
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Summing over all N we then find ω11 (γ1 τ1 , τ2 ,
1 n+1 )= κ n ω11 c1 τ1 + d1 (c1 τ1 + d1 ) n≥0
ω11 1 = (c1 τ1 + d1 ) 1 − κω11 ω11 = , c1 Ω11 + d1 where for the last equality we used (96). Now (97) yields Ω11 γ1 τ1 , τ2 ,
c1 τ1 + d1
1 Ω11 − τ1 = a1 τ1 + b1 + c1 τ + d1 c1 Ω11 + d1 a1 Ω11 + b1 = , c1 Ω11 + d1
which is the desired (90). Similarly from Proposition 4 we have Ω12
γ1 τ1 , τ2 , c1 τ1 + d1
1 =− . ω12 γ1 τ1 , τ2 , (c1 τ1 + d1 ) 2πi c1 τ1 + d1
Breaking necklaces of type 12 results in products over necklaces of type 11 together with one necklace of type 12. Hence by a similar argument to that above we find ω12 γ1 τ1 , τ2 ,
c1 τ1 + d1
ω12 1 − κω11 (c1 τ1 + d1 )ω12 = , c1 Ω11 + d1
=
so that Ω12 (γ1 τ1 , τ2 , cτ1+d1 ) is as in (91). Finally, 1 . ω22 γ1 τ1 , τ2 , Ω22 (γ1 τ1 , τ2 , ) = τ2 + cτ1 + d1 c1 τ1 + d1 2πi cτ1 + d1 Breaking necklaces of type 22 results in products over necklaces of type 11 together with one necklace of type 12 and another of type 21. Hence by a similar argument to that above we find 1 ω22 (γ1 τ1 , τ2 , )= (c1 τ1 + d1 ) 2πi cτ1 + d1 2πi = Ω22 − τ2 −
2 κω12 ω22 + 1 − κω11
2 c1 Ω12 , c1 Ω11 + d1
leading to (92). This completes the proof of the theorem.
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4.5. Local Invertibility of F about the Two Tori Degeneration Point = 0. Let D0 be the subset of D for which = 0. From Theorem 4 it is clear that the restriction of F to D0 induces the natural identification ∼
F : D0 → H1 × H1 ⊆ H2 , τ1 0 . (τ1 , τ2 , 0) → 0 τ2
(98)
D0 corresponds to the set of points where the genus 2 Riemann surface degenerates into a (2) (2) pair of genus 1 surfaces with Ω (2) = diag(Ω11 , Ω22 ). We will consider the invertibility of the map F in a neighborhood of a point in D0 . First we prepare a lemma. Recall (e.g., [FK2]) that a group H of homeomorphisms of a space X is said to act discontinuously on X if each point x ∈ X has a precisely invariant open neighborhood under the action of H in the following sense (loc. cit.): the stabilizer Stab(x) of x in H is finite, and there is an open neighborhood N of x such that h N ∩ N = φ if h ∈ / Stab(x) and h N = N if h ∈ Stab(x). Lemma 9. Suppose that H acts discontinuously on a pair of spaces X, Y , and that F : X → Y is a continuous H -equivariant map. Then the following hold: (a) If x ∈ X and F(x) = y then there are precisely invariant open neighborhoods (under the action of H ) U ⊆ X and V ⊆ Y of x and y respectively with F(U ) ⊆ V ; (b) Suppose further that Stab(x) = Stab(y) and " that the restriction of F to U is 1 − 1. Then F is 1 − 1 on the H -invariant domain h∈H hU . Proof. For part (a), let V be a precisely invariant open neighborhood of y in Y , U a precisely invariant neighborhood of x in X , and set U = F −1 (V ) ∩ U . Because F is H -equivariant then Stab(x) ⊆ Stab(y), and from this it follows that U is also precisely invariant under the action of H . Now (a) follows. As for (b), suppose the contrary so that there exist u 1 , u 2 ∈ U and h 1 , h 2 ∈ H such that h 1 u 1 = h 2 u 2 and F(h 1 u 1 ) = F(h 2 u 2 ). Thanks to the equivariance of F it is no loss to assume that h 2 = 1, so that h 1 u 1 = u 2 and F(h 1 u 1 ) = F(u 2 ). From the last equality we see that h 1 V ∩ V = φ, so that h 1 V = V and h 1 ∈ Stab(y). Therefore, h 1 ∈ Stab(x) by hypothesis, and therefore h 1 U = U . But then h 1 u 1 and u 2 ∈ U are distinct points of U on which F takes the same value. This contradicts the assumption that F is 1 − 1 on U , and completes the proof of the lemma. We now have Proposition 5. Let x ∈ D0 . Then there exists a G−invariant neighborhood Nx ⊆ D of x throughout which F is invertible. Proof. Let x = (τ1 , τ2 , 0). From Theorem 4, the Jacobian of F at x satisfies # # #1 0 0 # # # # # # ∂(Ω11 , Ω22 , Ω12 ) # # # # # # ∂(τ , τ , ) # = # 0 1 0 # = −1. 1 2 # 0 0 −1 # x By the inverse function theorem, there exists an open neighborhood of x in D throughout which F is invertible. Set F (x) = y. It follows immediately from (98) that the stabilizers (in G) of x and y are equal.
On Genus Two Riemann Surfaces Formed from Sewn Tori
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Next, it is well-known that the action (36) of Sp(2g, Z) on Hg is discontinuous. In particular, the action of G on H2 is discontinuous: furthermore from the case g = 1 it is easy to see that the action of G on H1 × H1 × C (and hence also on D ) is also discontinuous. Choose precisely invariant neighborhoods (under the action of G) U, V of x, respectively y such that the conditions of part (a) of Lemma 9 hold. It is clear from the non-vanishing of the Jacobian that we may also assume that F is 1 − 1 on U . Thus, we have achieved the hypotheses of part (b) of Lemma 9. That result tells us that the open neighborhood $ Nx = γU γ ∈G
has the desired properties.
We conclude this section with the explicit form of Ω = Ω (2) (τ1 , τ2 , ) to order 3 . We have from ( 77) to (79) that 2πiΩ11 = 2πiτ1 + 2 E 2 (τ2 ) + O( 4 ), 2πiΩ22 = 2πiτ2 + 2 E 2 (τ1 ) + O( 4 ), 2πiΩ12 = −(1 + 2 E 2 (τ1 )E 2 (τ2 ) + O( 4 )).
(99) (100) (101)
It is straightforward to check the equivariance properties described in Theorem 5 to the given order. In Appendix A more detailed expansions are provided. We may invert this 3 that relationship using Proposition 5 to find to order Ω12 2 4 τ1 = Ω11 − 2πiΩ12 E 2 (Ω22 ) + O(Ω12 ),
τ2 = =
2 4 Ω22 − 2πiΩ12 E 2 (Ω11 ) + O(Ω12 ), 2 2 −2πiΩ12 (1 − (2πi) Ω12 E 2 (Ω11 )E 2 (Ω22 ) +
(102) (103) 4 O(Ω12 )).
(104)
5. The ρ Formalism for Self-Sewing a Riemann Surface 5.1. The General ρ Formalism. In this section we review the general Yamada construction [Y] for sewing a Riemann surface of genus g to itself to form a surface of genus g + 1. We consider examples of sewing a Riemann sphere to itself in some detail where the Catalan series arise in a surprising way. In the next section, this general formalism will be applied to the construction of a genus two surface where the Catalan series again plays an important role. Consider a Riemann surface S of genus g and let z 1 , z 2 be local coordinates on S in the neighborhood of two separated points p1 and p2 . Consider two disks |z a | ≤ ra for ra > 0 sufficiently small and a = 1, 2. Note that r1 , r2 must be sufficiently small to also ensure that the disks do not intersect. Introduce a complex parameter ρ where |ρ| ≤ r1r2 and excise the disks {z a , |z a | < |ρ|ra−1 ¯ }⊂S for a = 1, 2 to form a twice-punctured surface $ Sˆ = S\ {z a , |z a | < |ρ|ra−1 ¯ }. a=1,2
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G. Mason, M. P. Tuite
Here we again use the convention (37). We define annular regions Aa ⊂ Sˆ with Aa = {z a , |ρ|ra−1 ¯ ≤ |z a | ≤ ra } and identify them as a single region A = A1 A2 via the sewing relation z 1 z 2 = ρ,
(105)
ˆ to form a compact Riemann surface S\{A 1 ∪A2 }∪A of genus g +1. The sewing relation (105) can be considered to be a parameterization of a cylinder connecting the punctured Riemann surface to itself. Using the Yamada formalism [Y], and noting the notational differences, the genus g + 1 normalized differential of the second kind ω(g+1) of (28) obeys Theorem 6 (Ref. [Y], Theorem 1, Theorem 4). (a) ω(g+1) is holomorphic in ρ for |ρ| < r1r2 . ˆ (b) limρ→0 ω(g+1) (x, y) = ω(g) (x, y) for x, y ∈ S. Regarded as a power series in ρ, the coefficients of the analytic expansion of ω(g+1) in ρ can be calculated from ω(g) . Let Ca (z a ) ⊂ Aa denote a closed anti-clockwise oriented ˆ Note that C1 (z 1 ) contour parameterized by z a surrounding the puncture at z a = 0 on S. may be deformed to −C2 (z 2 ). Then similarly to Lemma 1 we find [Y] Lemma 10. ω
(g+1)
(x, y) = ω
(g)
z 1 (g+1) (x, y) + (ω (y, z) ω(g) (x, ·)), 2πi
(106)
a=1,2C (z) a
ˆ for x, y ∈ S. For a, b = 1, 2 and k, l = 1, 2, . . . we define weighted moments Yab ¯ (k, l) =
ρ (k+l)/2 1 √ kl (2πi)2
u −k v −l ω(g+1) (u, v).
(107)
Ca (u) Cb (v)
Note that Yab (k, l) = Yb¯ a¯ (l, k). We also define Y = (Yab (k, l)) to be the infinite matrix ˆ indexed by the pairs a, k and b, l. We define a set of holomorphic 1-forms on S, ρ k/2 aa (k, x) = √ 2πi k
z a−k ω(g) (x, z a ),
(108)
Ca (z a )
and define a(x) = (aa (k, x)) and a(x) ¯ = (aa¯ (k, x)) to be the infinite row vectors indexed by a, k . In a similar way to Lemma 2 we then have Lemma 11. ω(g+1) (x, y) for x, y ∈ Sˆ is given by ω(g+1) (x, y) = ω(g) (x, y) − a(x)(I − Y )a(y) ¯ T.
(109)
On Genus Two Riemann Surfaces Formed from Sewn Tori
613
We next compute the explicit form of Y in terms of the following weighted moments of ω(g) : ρ (k+l)/2 1 Rab x −k y −l ω(g) (x, y) ¯ (k, l) = − √ kl (2πi)2 ρ k/2
=−√
1 2πi k
Ca (x) Cb (y)
x −k ab (l, x),
(110)
Ca (x)
where Rab (k, l) = Rb¯a¯ (l, k) and the extra minus sign is introduced for later convenience. We may consider R as an infinite block matrix (similar to Q of (55)) B A , (111) R = (Rab (k, l)) = − A BT with A(k, l) = A(k, l, ρ) =
B(k, l) = B(k, l, ρ) =
ρ (k+l)/2 √ (2πi)2 kl
√
(2πi)2 kl
x −k y −l ω(g) (x, y),
C1 (x) C1 (y)
ρ (k+l)/2
x −k y −l ω(g) (x, y).
(112)
C2 (x) C1 (y)
Similarly to Proposition 1 we find: Proposition 6. Yab (k, l) is given in terms of R by I − Y = (I − R)−1 . Here (I − R)−1 =
(113)
Rn
n≥0
and is convergent in ρ for |ρ| < r1r2 . Likewise, similarly to Theorem 2 we may define det(I − R) and find: Theorem 7. det(I − R) is non-vanishing and holomorphic in ρ for |ρ| < r1r2 . We can define a standard basis of cycles {a1 , b1 , . . . ag+1 , bg+1 } on the sewn genus g + 1 surface as follows, where the set {a1 , b1 , . . . ag , bg } is the original basis. Then ag+1 is defined as the contour C2 on Sˆ whereas bg+1 is defined to be a path chosen in Sˆ from z 1 = z 0 to z 2 = ρ/z 0 which points are identified on the sewn surface. Integrating (109) ˆ along a br cycle on S for r = 1, . . . g gives g holomorphic 1-forms for x ∈ S, (g+1)
νr
(g)
(x) = νr (x) − a(x)(I − R)−1 α¯ rT ,
where α¯ r = (αr,a¯ (k)) with
(114)
αr,a (k) =
aa (x, k). br
(115)
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G. Mason, M. P. Tuite
We then find from (30), (31), (114) and (115) that for r, s = 1, . . . g, (g+1)
2πiΩr s
(g)
= 2πiΩr s − αr (I − R)−1 α¯ sT . (g+1)
The remaining normalized holomorphic one form νg+1 can be expressed in terms (g)
of the normalized differential of the third kind ω p2 − p1 of (32) with weighted moments dz a −k ρ k/2 1 (g) βa (k) = √ (ω p2 − p1 + (−1)1+a )z . (116) za a k 2πi Ca (z a )
Then by Cauchy’s theorem we find that [Y] (g+1)
Lemma 12 (op. cit, Corollary 5). The normalized holomorphic one form νg+1 is given by (g+1) νg+1 (x)
=
(g) ω p2 − p1 (x) +
1 2πi
ω
(g+1)
a=1,2 C (z) a
z dz a (g) (x, z) (ω p2 − p1 + (−1)1+a ). (117) za
Hence integrating (117) over a br cycle and using ( 109) we find for r = 1, . . . , g that p2 (g+1) (g) 2πiΩrg+1 = νr − αr (I − R)−1 β¯ T . p1 (g+1)
Finally Ωg+1g+1 is described in [Y]: (g+1)
Lemma 13 (op. cite. Lemma 5). Ωg+1g+1 is given by (g+1)
2πiΩg+1g+1 = log(
+
ρ )+ z 02
a=1,2
z 2−1 (z 0 )
(g)
ω p2 − p1 z 1−1 (z 0 )
1 2πi
(g+1) νg+1 (z)
Ca
z
(g)
(ω p2 − p1 + (−1)1+a
z a−1 (z 0 )
dz a ), za
where the logarithmic branch is determined by the choice of the cycle bg+1 as a path in Sˆ from z 1 = z 0 to z 2 = ρ/z 0 . (g+1)
Substituting νg+1 from (117) one eventually obtains [Y] (g+1) 2πiΩg+1g+1 = log ρ + C0 − β(I − R)−1 β¯ T ,
where
⎡ ⎢ C0 = lim ⎢ u→0 ⎣
z 2−1 (u)
z 1−1 (u)
⎤
⎥ (g) ω p2 − p1 − 2 log u ⎥ ⎦.
On Genus Two Riemann Surfaces Formed from Sewn Tori
615
However from (34) we may express C0 in terms of the prime form C0 = lim log u→0
K (g) (z 2−1 (u), p2 )K (g) (z 1−1 (u), p1 )
u 2 K (g) (z 2−1 (u), p1 )K (g) (z 1−1 (u), p2 )
= − log(−z 1 ( p1 )z 2 ( p2 )K (g) ( p2 , p1 )2 ), d −1 z a (u)|u=0 = 1/z a ( pa ) and using K (g) ( p2 , p1 ) = −K (g) ( p1 , p2 ). We therewhere du fore find altogether that
Theorem 8. The genus g + 1 period matrix for |ρ| < r1r2 is given by (g+1)
2πiΩr s
(g+1) 2πiΩrg+1
(g)
= 2πiΩr s − αr (I − R)−1 α¯ sT , r, s = 1, . . . , g, p2 =
(g)
νr
− β(I − R)−1 α¯ rT , r = 1, . . . , g,
(118)
(119)
p1 (g+1) 2πiΩg+1g+1
−ρ = log z 1 ( p1 )z 2 ( p2 )K (g) ( p2 , p1 )2
− β(I − R)−1 β¯ T ,
(120)
where Ω (g+1) is holomorphic in ρ for 0 < |ρ| < r1r2 and the logarithmic branch is determined by the choice of the cycle bg+1 . We finally obtain the following holomorphic properties for ω(g+1) , Ω (g+1) and det(I − R). The proof follows a similar argument to that for Propositions 2 and 3. Proposition 7. Suppose that ω(g) is a holomorphic function of a complex parameter μ for |μ| < S. Then det(I − R) is non-vanishing and both ω(g+1) and det(I − R) are holomorphic in μ for |μ| < S with |ρ| < r1r2 whereas Ω (g+1) is holomorphic in μ for |μ| < S with 0 < |ρ| < r1r2 . 5.2. Self-Sewing a Sphere to form a Torus. It is instructive to consider two separate examples of sewing a Riemann sphere to itself to form a torus. The first is mainly illustrative whereas the second is related to some later genus two considerations wherein the Catalan numbers arise in an interesting and surprising way. In both cases ω(1) is given by (35) with an appropriately identified modular parameter τ . 5.2.1. Simplest Case. Let S0 = C∪{∞} be the Riemann sphere with bilinear form (70). Choose local coordinates z 1 = z in the neighborhood of the origin and z 2 = 1/z for z in the neighborhood of the point at infinity. Identify the annular regions |q|ra−1 ¯ ≤ |z a | ≤ ra for a complex parameter q obeying |q| ≤ r1r2 via the sewing relation z = qz .
(121)
Note that the annular regions do not intersect on the sphere provided r1r2 < 1 so that |q| < 1. We then find [Y]
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G. Mason, M. P. Tuite
Proposition 8. q = exp(2πiτ ), where τ is the torus modular parameter. Proof. The 1-forms (108) are √ k/2 −k−1 kq x d x, √ k/2 k−1 a2 (k, x) = − kq x d x, a1 (k, x) =
so that A(k, l) = 0 and B(k, l) = q k δk,l in (111) giving I − R = diag(1 − q, 1 − q, . . . , 1 − q k , 1 − q k , . . .). Hence Lemma 11 gives for x, y ∈ Sˆ0 , ⎧ ⎨
kq k xy + ω(1) (x, y) = ⎩ (x − y)2 1 − qk k≥1
% (⎫ x k & y 'k ⎬ d xd y . + ⎭ xy y x
Under the conformal map z → log z we then verify ω(1) (u, v) = P2 (τ, u − v)dudv with u = log x and v = log y using (10), where q = exp(2πiτ ). The sewing relation (121) is then just the standard torus periodicity relation log z = log z + 2πiτ . Alternatively, we may apply (120) of Theorem 8 using z 2 = 1/z − 1/ p2 and then (1) consider p2 → ∞. Then K (0) ( p2 , 0) = p2 with ω p2 −0 (x) = ( x−1p2 − x1 )d x so that βa (k) = 0 and z 1 (0)z 2 ( p2 )K (0) ( p2 , 0)2 = −1 independent of p2 . This implies that (1) 2πiτ = 2πiΩ11 = log q again. Remark 1. The modular transformation τ → τ +1 is generated by a continuous variation in the sewing parameter exp(iθ )q for 0 ≤ θ ≤ 2π . This corresponds to a Dehn twist (1) b1 → a1 +b1 in the b1 cycle chosen in Lemma 13 and Theorem 8 so that 2πiΩ11 = log q is evaluated on the next logarithmic branch. ) Remark 2. ω(1) (x, y) and det(1 − R) = k≥1 (1 − q k )2 are clearly holomorphic for |q| < 1 as expected from Theorems 6 and 7. 5.2.2. General Self-Sewing of a Sphere and the Catalan Series. For z ∈ S0 choose local coordinates z 1 = z in the neighborhood of the origin and z 2 = z − w for z# in the neigh# borhood of w ∈ S0 . Identify the annuli |ρ|r2−1 ≤ |z| ≤ r1 and |ρ|r1−1 ≤ #z − w # ≤ r2 for |ρ| ≤ r1r2 via the sewing relation z(z − w) = ρ.
(122)
The two annular regions do not intersect provided |w| > r1 +r2 ≥ r1 +|ρ|r1−1 ≥ 2|ρ|1/2 . The lower bound occurs for r1 = r2 = |ρ|1/2 and is realized when the two annuli become degenerate (infinitesimally thin) and touch at the point z 1 = −z 2 = w/2 with w 2 = −4ρ. Thus defining χ =−
ρ , w2
On Genus Two Riemann Surfaces Formed from Sewn Tori
617
then χ = 41 is the degenerate point. We define the Catalan series9 to be the series f (χ ) convergent for |χ | < 41 satisfying χ= Thus
f . (1 + f )2
√
1 2n 1 − 4χ −1= χn 2χ n n+1 n≥1 & ' 2 3 4 = χ + 2χ + 5χ + 14χ + O χ 5 .
f (χ ) =
1−
(123)
(124)
* 2n + are the Catalan numbers which occur in a remarkably wide range The coefficients n1 n+1 of combinatorial settings e.g., [St]. Proposition 9. For the sewing described by (122), the torus modular parameter is q = f (χ ), the Catalan series. Proof. Define the Möbius transformation z → γ .z =
w z− f ( ), 1+ f z−1
(125)
where f = f (χ ). Then with z = γ .Z and z = γ .Z the sewing relation (122) becomes, on using (123), Z = f Z . Thus we recover the earlier sewing relation of (121) with modular parameter q = f (χ ). This result can be verified from (120) of Theorem 8 as follows. With ω(0) of (70) the basis of 1-forms (108) is given by √ (0) a1 (k, x) = kρ k/2 x −k−1 d x, √ (0) a2 (k, x) = kρ k/2 (x − w)−k−1 d x, (126) with R
(0)
B (0) A(0) =− A(0) B (0)T
A(0) (k, l) = 0,
B (0) (k, l) =
, (−χ )(k+l)/2 (−1)k+1 (k + l − 1)! , √ (k − 1)!(l − 1)! kl
1 1 − )d x, x −w x K (0) (w, 0) = w, (−χ )k/2 β (0) (k) = √ [−1, (−1)k ], k (0)
ωw−0 (x) = (
9 The Catalan series is more usually defined to be 1 + f (χ ).
(127)
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G. Mason, M. P. Tuite
where the 0 superscript indicates the genus of the sphere. After some calculation, we find that τ is given by (1)
2πiτ = 2πiΩ11 = log χ + 2
1 k≥1
where S1,k (χ ) = 1 and Sn,k (χ ) =
χ kn−1 +...+k1
kn−1 ,...k1 ≥1
k2 + k1 − 1 , ... k1
k
χk
Sn,k (χ ),
n≥1
k + kn−1 − 1 kn−1 + kn−2 − 1 kn−1 kn−2 (128)
for n > 1. We will show below that Sn,k (χ ) = (1 + f (χ ))k ,
(129)
n≥1
which implies
1 k k≥1 k χ
Sn,k (χ ) = − log(1 − χ (1 + f )) = log(1 + f ) from (124).
n≥1
Therefore 2πiτ = log χ + 2 log(1 + f ) = log f* so that+q = f as claimed. It remains to prove (129). Since k1 ≥1 χ k1 k2 +kk11 −1 = (1 − χ )−k2 − 1 we find for n > 1 that kn−1 k + kn−1 − 1 ... χ Sn,k (χ ) = kn−1 kn−1 ≥1 χ k2 k3 + k2 − 1 − Sn−1,k (χ ). 1−χ k2 k2 ≥1
Repeating this process leads to N
% Sn,k (χ ) =
n=1
where [ 1−
1
χ 1−χ /···
1 1−
χ 1−χ /···
( k , N
] N denotes the N th term in the continued fraction expansion of F =
1/(1 − χ F) whose solution from (124) is F = 1 + f .
Remark 3. The modular transformation τ → τ +1 is generated by a continuous variation in the sewing parameter exp(iθ )ρ for 0 ≤ θ ≤ 2π . Using Lemma 11 and comparing to ω(1) of (35) results in novel expressions for Eisenstein series E n (q) for q = f (χ ). Thus, for example, one finds Proposition 10. E 2 (q = f (χ )) = −
2χ 1 + (1 + B (0) )−1 (1, 1), 12 1 − 4χ
(130)
where (1, 1) refers to the (k, l) = (1, 1) element of the infinite matrix (1 + B (0) )−1 .
On Genus Two Riemann Surfaces Formed from Sewn Tori
619
Proof. From (109) and (126) we have ω(1) (x, y) =
d xd y − a (0) (x)(I − R (0) )−1 (a¯ (0) )T (y). (x − y)2
1 But ω(1) (x, y) = ω(1) (u, v) = P2 (u − v, τ )dudv with τ = 2πi log f (χ ) from Propu v osition 9 with x = γ .e and y = γ .e using (125). Then, on substituting for u, v into ω(1) (x, y) one eventually finds using ( 127) that
ω(1) (x, y) =
(k+l)/2 √ eu−v dudv χ (0) −1 − (1 + B ) (k, l) kl − (eu−v − 1)2 1 − 4χ k,l≥1 % 1 − f k+1 1 − f l+1 .(eu − 1)k−1 (ev − 1)l−1 eu − f 1 − f ev ( 1 − f k+1 1 − f l+1 u+v k+l +(−1) e dudv, 1 − f eu ev − f
√ using 1 − f = (1 + f ) 1 − 4χ . Expanding in u, v we then find that ω(1) (x, y) = [
2χ 1 1 + (1 + B (0) )−1 (1, 1) + O(u, v)]dudv, (131) − (u − v)2 12 1 − 4χ
from which the result follows on comparison with (9).
6. Self-Sewing a Torus to Form a Genus Two Riemann Surface 6.1. The Genus Two Period Matrix in the ρ Formalism. We now apply the ρ-formalism to sew a twice punctured torus with modulus τ and punctures separated by w to form a genus two Riemann surface with period matrix Ω (2) (τ, w, ρ). We will see that Ω (2) is holomorphic for (τ, w, ρ) in an appropriate domain Dρ . We again provide a description of Ω (2) in terms of a sum of weights of necklaces. There is a holomorphic mapping F ρ : Dρ → H2 , and we describe its equivariance properties with respect to a certain (2) group. The logarithmic contribution log(−ρ/K 2 ) to Ω22 in (120) gives rise to a subtle analytic structure which we discuss in some detail. Finally, we prove that F ρ is invertible in a certain domain. Consider a framed torus (cf. Subsect. 4.1) S = C/, where ⊆ C is a lattice with positively oriented basis (σ, ς ) and modulus τ = σ/ς ∈ H1 . Define annuli Aa , a = 1, 2, centered at z = 0 and z = w of S with local coordinates z 1 = z and z 2 = z − w respectively. Take the outer radius of Aa to be ra < 21 D() and the inner radius to be |ρ|/ra¯ , with |ρ| ≤ r1r2 < 14 D()2 (cf. Lemma 6). Identifying the annuli according to the sewing relation (105) z 1 z 2 = ρ gives rise to a compact Riemann surface of genus 2. As in the remarks following Lemma 6, we now take = τ with basis (2πiτ, 2πi) and with w in the fundamental parallelogram for τ with sides (2πiτ, 2πi). As with the sphere example above, the two annuli must not intersect. This requires the inequalities |w − λ| > r1 + r2 ≥ 2|ρ|1/2 to hold for λ ∈ τ . Thus we find 2|ρ|1/2 < |w| < D(τ ) − 2|ρ|1/2 .
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G. Mason, M. P. Tuite
1 Notice that this implies |ρ| < 16 D(τ )2 , which refines the inequality satisfied by ρ discussed above. As a result of this discussion, we see that the relevant domain in the ρ -formalism is the following:10
Dρ = {(τ, w, ρ) ∈ H1 × C × C | |w − λ| > 2|ρ|1/2 > 0, λ ∈ τ }.
(132)
We may apply Theorem 8 to determine Ω (2) (τ, w, ρ). We find: Theorem 9. Sewing determines a holomorphic map F ρ : Dρ → H2 , (τ, w, ρ) → Ω (2) (τ, w, ρ).
(133)
Proposition 11. Ω (2) = Ω (2) (τ, w, ρ) is given by (2)
2πiΩ11 = 2πiτ − ρσ ((I − R)−1 (1, 1)), (2) 2πiΩ12 (2) 2πiΩ22
= w − ρ 1/2 σ ((β(I − R)−1 (1)), ρ = log(− ) − β(I − R)−1 β¯ T , K (τ, w)2
(134) (135) (136)
where the branch of the log function in (136) is determined by the choice of the cycle b2 . Here, R = R(τ, w, ρ) = (Rab (k, l)) is an infinite matrix with indices k, l = 1, 2, 3, . . . and a, b = 1, 2; β = β(τ, w, ρ) = (βa (k)) is an infinite row vector; (1, 1) and (1) are the (1, 1)- and (1)- (block) entries of a matrix; σ (M) denotes sum over the entries of a finite matrix; and ρ (k+l)/2 D(k, l, τ, w) C(k, l, τ ) R(k, l) = − √ , (137) C(k, l, τ ) D(l, k, τ, w) kl ρ k/2 β(k) = √ (Pk (τ, w) − E k (τ ))[−1, (−1)k ], (138) k with notation as in Sect. 2. Proof. Since ω(1) (x, y) = P2 (x − y)d xd y from ( 20) we find that the set of 1-forms (108) with periods (2πiτ, 2πi) is given by √ a1 (k, x) = a1 (k, x, τ, ρ) = kρ k/2 Pk+1 (τ, x)d x, a2 (k, x) = a2 (k, x, τ, ρ) = a1 (k, x − w). The matrices A(k, l), B(k, l) in (112) are given directly from the expansions (25) and (26) which are convergent on Dρ resulting in (137). α1,a (k) of (115) is independent of a = 1, 2 with α1,a (k) = aa (k, ·) = ρ 1/2 δk,1 . b1 (2)
Hence 2πiΩ11 is as stated from (118) of Theorem 8 for (τ, w, ρ) ∈ Dρ . From Exam(1) (x) = (P1 (τ, x − w) − P1 (τ, x))d x and the prime form is ple 1 we know that ωw−0 10 The footnote relating to (75) concerning notation applies here too.
On Genus Two Riemann Surfaces Formed from Sewn Tori
621 (1)
K (1) (x, y) = K (τ, x − y). We obtain the given moments (138) of ωw−0 (x) from ( 20). (2)
Hence since ν (1) (x) = d x, we find 2πiΩ12 is as given from (119) of Theorem 8 for (τ, w, ρ) ∈ Dρ . Finally applying (120) with K (1) (w, 0) = K (τ, w) we obtain (136) for (τ, w, ρ) ∈ Dρ . (2) Ωi j (τ, w, ρ) is holomorphic in ρ for 0 < |ρ| < r1r2 from Theorem 8. Proposition 7 (2)
then states that Ωi j (τ, w, ρ) is also holomorphic in τ ∈ H1 . We also need to show that −1 converges for |ρ| < r r Ωi(2) 1 2 j (τ, w, ρ) is holomorphic in w. Since Y = (I − R) (2)
then, following an argument similar to that in Proposition 2, we find that Ωi j (τ, w, ρ) is continuous in w for (τ, w, ρ) ∈ Dρ . The Weierstrass functions Pk (τ, w) for k ≥ 1 (2) are holomorphic in w. Hence the ρ expansion coefficients Ωi j (τ, w, ρ) are holomorphic functions in w since they consist of finite sums and products of these Weierstrass (2) functions. Hence, by Lemma 5, Ωi j (τ, w, ρ) is holomorphic in w. Finally, by Hartog’s (2)
Theorem, Ωi j is holomorphic on Dρ .
6.2. Necklace Expansion for Ω (2) . We introduce a graphical interpretation for the ρ period matrix formulas analogous to that described earlier for the -expansion. Consider the set of necklaces N = {N }: they are connected graphs with n ≥ 2 nodes, n − 2 of which have valency 2 and two of which have valency 1, together with an orientation, say from left to right. Furthermore, each vertex carries two labels k, a with k a positive integer and a = 1 or 2. A typical necklace in the ρ-formalism looks as follows: k1 ,a1
k2 ,a2
k3 ,a3
k4 ,a4
• −→ • −→ • −→ •
We define the degenerate necklace N0 to be a single node with no edges. Next we define a weight function ω : N −→ C[P2 (τ, w), P3 (τ, w), E 2 (τ ), E 4 (τ ), E 6 (τ ), ρ 1/2 ]. k,a
l,b
If N ∈ N has edges E labelled as • −→ • then we define ω(E) = Rab (k, l, τ, w, ρ),
ω(N ) = ω(E), with Rab (k, l) as in (137) and where the product is taken over all edges of N . We further define ω(N0 ) = 1. The necklaces with prescribed end nodes labelled (k, a; l, b) look as follows: k,a
k1 ,a1
k2 ,a2
l,b
• −→ • . . . • −→ •
(type (k, a; l, b)).
We set Nk,a;l,b = {isomorphism classes of necklaces of type (k, a; l, b)}. As in Lemma 7 we obtain
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Lemma 14. We have for k, l ≥ 1,
−1 (k, l) = (I − R)ab
ω(N ).
N ∈Nk,a;l,b
Finally it is convenient to define ω11 =
ω(N ),
a,b=1,2 N ∈N1,a;1,b
ωβ1 =
ω1β¯ =
β¯b (k)
ω(N ),
N ∈N1,a;k,b
a,b=1,2 k≥1
ωβ β¯ =
ω(N ),
N ∈Nk,a;1,b
a,b=1,2 k≥1
βa (k)
βa (k)β¯b (l)
ω(N ).
(139)
N ∈Nk,a;l,b
a,b=1,2 k,l≥1
Note that Rab (k, l) = Rb¯a¯ (l, k), so that ωβ1 = ω1β¯ . Then from Theorem 9 we have Proposition 12. (2)
2πiΩ11 = 2πiτ − ρω11 , (2)
2πiΩ12 = w − ρ 1/2 ωβ1 , ρ (2) ) − ωβ β¯ . 2πiΩ22 = log(− K (τ, w)2 6.3. Equivariance of F ρ . In Subsect. 4.4 we defined a subgroup G ⊂ Sp(4, Z) which preserves the domain D , and proved the equivariance of F under the action of G. In this section we wish to establish analogous equivariance properties in the ρ-formalism. With this in mind, one might expect that the map F ρ occurring in Theorem 9 is the (2) correct analog of F . However, because of the logarithmic branch structure of Ω22 , it is necessary to lift F ρ to a single-valued function Fˆ ρ on a certain covering space Dˆ ρ for Dρ before the correct analogs can be established. 6.3.1. Some Heisenberg and Jacobi-type groups. In this subsection we consider some groups relevant to our enterprise, and start with certain subgroups of Sp(4, Z). For (a, b, c) ∈ Z3 set ⎛
1 ⎜a μ(a, b, c) = ⎝ 0 0
0 1 0 0
0 b 1 0
⎞ b c ⎟ , −a ⎠ 1
with A = μ(1, 0, 0), B = μ(0, 1, 0), C = μ(0, 0, 1).
(140)
On Genus Two Riemann Surfaces Formed from Sewn Tori
623
The matrices (140) form a subgroup Hˆ ⊆ Sp(4, Z) which is a 2-step nilpotent group with center isomorphic to Z and generated by C, and central quotient isomorphic to Z2 . Note that we have the presentation Hˆ = A, B, C | [A, B]C −2 = [A, B, C] = 1 .
(141)
The ‘left’ modular group 1 (84) is also a subgroup of Sp(4, Z), and indeed it normalizes Hˆ according to the conjugation formula γ −1 μ(u, v, w)γ = μ((u, v)γ , w), γ ∈ 1 . Here it was convenient to abuse notation, taking ⎛ ⎞ a0b0 ab ⎜0 1 0 0⎟ . γ =⎝ ∈ 1 and (u, v)γ = (u, v) c 0 d 0⎠ cd 0001
(142)
(143)
In this way we get a subgroup L = Hˆ 1 ⊆ Sp(4, Z) which is a split extension of S L(2, Z) by Hˆ . Note that Z (L) = C , and that the central quotient J = L/Z (L) ∼ = Z2 S L(2, Z) is the Jacobi group which figures in the transformation laws of Jacobi forms ([EZ]). Let H = A, B be the subgroup of Hˆ generated by A and B. It follows from (141) that H has the presentation H = A, B | [A, B] = C , [A, C ] = [B, C ] = 1 , where we have set C = C 2 . We call H the Heisenberg group, though H and Hˆ (which are not isomorphic) are often confused in this regard. We see from (142) that | Hˆ : H | = 2 and that 1 normalizes H . Thus L 0 = H 1 is a subgroup of L of index 2. Lemma 15. L acts on Dρ as follows: μ(a, b, c).(τ, w, ρ) = (τ, w + 2πiaτ + 2πib, ρ), aτ + b w ρ γ .(τ, w, ρ) = . , , cτ + d cτ + d (cτ + d)2
(144) (145)
The kernel of the action is Z (L), so that the effective action is that of J = L/Z (L). Proof. Let us first work with the larger domain whereby we allow the triple (τ, w, ρ) to lie in H1 × C × C. Then it is easy to see that the first equality defines an action of Hˆ with kernel C , and that the second equality defines a faithful action of S L(2, Z). Next we show that these two actions jointly define an action of the group L . To this end it is useful to rewrite (145) more functorially in terms of the cocycle j (γ , τ ) = cτ +d, which satisfies j (γ1 γ2 , τ ) = j (γ1 , γ2 τ ) j (γ2 , τ ), γ1 , γ2 ∈ 1 . Thus
γ .(τ, w, ρ) = γ τ,
w ρ , j (γ , τ ) j (γ , τ )2
,
(146)
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G. Mason, M. P. Tuite
and we have to show that γ −1 μ(x, y, z)γ .(τ, w, ρ) = μ((x, y)γ , z).(τ, w, ρ).
(147)
The right-hand-side of (147) is equal to (τ, w + 2πi((ax + cy)τ + bx + dy), ρ). The left-hand-side is equal to w ρ −1 , γ μ(x, y, z). γ τ, j (γ , τ ) j (γ , τ )2 ρ w −1 + 2πi(xγ τ + y), = γ . γ τ, j (γ , τ ) j (γ , τ )2 ρ w + 2πi(x(aτ + b) + y(cτ + d)) −1 , = γ . γ τ, j (γ , τ ) j (γ , τ )2 ρ w + 2πi(x(aτ + b) + y(cτ + d)) , = τ, j (γ , τ ) j (γ −1 , γ τ ) ( j (γ , τ ) j (γ −1 , γ τ ))2 = (τ, w + 2πi(x(aτ + b) + y(cτ + d)), ρ) , where we used (146) to get the last equality. This confirms (147). It remains to show that the action of L preserves Dρ , and for this it is enough to prove it for a set of generators. Bearing in mind the definition of Dρ (132), the result is clear for μ(x, y, z). To prove it for γ ∈ 1 , we must show that if (τ, w, ρ) ∈ Dρ then # # # #1/2 # # w # # ρ # # # # (148) # j (γ , τ ) − λ# > 2 # ( j (γ , τ ))2 # > 0 for all λ ∈ γ τ . But λ = j (γ1,τ ) λ for some λ ∈ τ , whence (148) reduces to | j (γ , τ )|−1 |w−λ | > 2| j (γ , τ )|−1 |ρ|1/2 > 0. This follows from the fact that (τ, w, ρ) ∈ Dρ , and the proof of the lemma is complete. 6.3.2. Some covering spaces. One sees that projection onto the first coordinate pr1 : Dρ → H1 , (τ, w, ρ) → τ, is locally trivial with contractible base H1 . From the long exact sequence associated to a fibration we obtain an exact sequence 0 = π2 (H1 ) → π1 (F) → π1 (Dρ ) → π1 (H1 ) = 0, where F is the fiber. Thus, we have π1 (Dρ ) ∼ = π1 (F). From Lemma 6.5, there is a free action of Z2 = Hˆ /Z (L) on each fiber pr1−1 (τ ). Furthermore, from the definition of Dρ we see that π1 (Dρ /Z2 ) ∼ = π1 (C/τ \ {0}) × π1 (C \ {0}) ∼ = H × Z. Here, H is the Heisenberg group of the previous subsection. We need to describe this identification carefully. Consider the usual realization of C/τ as the fundamental parallelogram for τ with identification of sides, and let α, β be the cycles along the sides with periods 2πiτ, 2πi respectively. Define δ to be a
On Genus Two Riemann Surfaces Formed from Sewn Tori
625
closed clockwise contour about an interior point of the parallelogram with local coordinate w = 0. Then there is an isomorphism of groups ∼ =
π1 (C/τ \ {0}) → H, α → A, β → B, δ → C . Similarly, let η denote a closed anti-clockwise contour about ρ = 0 in the complex plane. Then π1 (C \ {0}) = η . Let D˜ ρ be a universal covering space of Dρ with covering projection p1 : D˜ ρ → Dρ . There is a free action of the fundamental group H × Z on D˜ ρ , and we define Dˆ ρ = D˜ ρ /η−2 δ . Thus we have a sequence of covering projections p3 p4 p2 D˜ ρ −→ Dˆ ρ −→ Dρ −→ Dρ /Z2 ,
(149)
Dρ
lifts (modulo the fundamental group) to where p1 = p4 ◦ p3 . The action of 1 on an action on the universal cover. That is, there is a group G acting on D˜ ρ , where G fits into a short exact sequence 1 → H × Z → G → 1 → 1. We have Z (G) = Z (H ) × Z, in particular η−2 δ ∈ Z (G). It follows that G acts on Dˆ ρ , and there is a sequence of surjective group maps G → G/η−2 δ → L → 1
(150)
in which the four groups act on the corresponding spaces in (149 ). (2)
6.3.3. Lifting the logarithm l(x). From (136), the logarithmic contribution to Ω22 is ρ , x = (τ, w, ρ) ∈ Dρ . (151) l(x) = log − K (τ, w)2 The remaining parts ω11 , ωβ1 and ωβ β¯ of Ω (2) are single-valued on Dρ since they are expressible in terms of the Weierstrass functions and Eisenstein series. Now K (τ, w)2 = −θ1 (τ, w)2 /η(τ )6 is a Jacobi form of weight −2 and index 1 [EZ], so that exp l(x) = −ρ is single-valued. The way it transforms under the Jacobi group J can be read-off K (τ,w)2 of Lemma 15. We find that exp l((a, b).x) = exp(2πa 2 iτ + 2aw) exp l(x), (a, b) ∈ Z2 , 1 c1 w 2 exp l(x), γ1 ∈ 1 , exp l(γ1 .x) = exp − 2πi c1 τ + d1
(152) (153)
where (a, b) is the image of μ(a, b, c) in J . For a given choice of the branch l(x), we therefore find that l((a, b).x) = l(x) + 2πa 2 iτ + 2aw + 2πi N (a, b), (a, b) ∈ Z2 , for some N (a, b) ∈ Z.
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G. Mason, M. P. Tuite
˜ x) Let l( ˜ be a lifting of l(x) to a single-valued function on D˜ ρ . K (τ, w)2 is holomorphic for (τ, w) ∈ H1 ×C with a zero of order two for each w ∈ τ (see (18)). Let x˜ ∈ D˜ ρ and p1 (x) ˜ = x = (τ, w, ρ). Using (152) we find: ˜ x) ˜ x) l(α. ˜ = l( ˜ + 2πiτ + 2w + 2πi Nα , ˜ x) ˜ x) l(β. ˜ = l( ˜ + 2πi Nβ , ˜ x) ˜ x) l(η. ˜ = l( ˜ + 2πi, ˜ x) ˜ x) l(δ. ˜ = l( ˜ + 4πi, for some Nα , Nβ ∈ Z. In particular, note that by composing these transformations we confirm the relation [α, β] = δ. We may define new generators α = αη−Nα , β = βη−Nβ , which satisfy the same relations and for which Nα = Nβ = 0. Relabelling, we then obtain Lemma 16. With previous notation, we have ˜ a β b γ c δ d .x) ˜ x) l(α ˜ = l( ˜ + 2πia 2 τ + 2aw + 2πi(c + 2(ab + d)),
(154)
for a, b, c, d ∈ Z. In particular, ˜ x), ˜ −2 δ.x) ˜ = l( ˜ l(η so that l˜ pushes down to a single-valued function lˆ on Dˆ ρ . From (153) we find for γ ∈ 1 that l(γ1 .x) = l(x) −
1 c1 w 2 + 2πi N (γ1 ), . 2πi c1 τ + d1
(155)
for some N (γ1 ) ∈ Z. It is easy to see that (155) is consistent with respect to the composition of γ1 , γ2 ∈ 1 with a trivial cocycle condition N (γ1 γ2 ) = N (γ1 ) + N (γ2 ). Thus the extension (150) splits, in particular G contains the subgroup L = Hˆ 1 and G = L × Z. Since L ∩ η−2 δ = 1 there is an injection L −→ G/η−2 δ , and through this map L acts on Dˆ ρ . We can now read-off from (154) and (155) that lˆ transforms as follows: Theorem 10. The action of L on Dˆ ρ satisfies ˆ ˆ x) l(μ(a, b, c).x) ˆ = l( ˆ + 2πia 2 τ + 2aw + 2πi(ab + c), μ(a, b, c) ∈ Hˆ , (156) c1 w 2 1 ˆ 1 .x) ˆ x) l(γ . ˆ = l( ˆ − , γ1 ∈ 1 , 2πi c1 τ + d1 where xˆ ∈ Dˆ ρ , p4 (x) ˆ = (τ, w, ρ).
(157)
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627
6.3.4. Equivariance of Fˆ ρ and F ρ . After the results of the previous subsection we know that F ρ lifts to a single-valued holomorphic function Fˆ ρ on Dˆ ρ : Fˆ ρ : Dˆ ρ → H2 , xˆ → Ωˆ (2) (x). ˆ
(158)
By Proposition 12 we have (2) ˆ = 2πiτ − ρω11 (x), 2πi Ωˆ 11 (x) (2)
2πi Ωˆ 12 (x) ˆ = w − ρ 1/2 ωβ1 (x), (2) ˆ x) 2πi Ωˆ 22 (x) ˆ = l( ˆ − ωβ β¯ (x),
ˆ where (τ, w, ρ) = x = p4 (x). Theorem 11. Fˆ ρ is equivariant with respect to the action of L. Thus, there is a commutative diagram for γ ∈ L, ˆρ
F Dˆ ρ → H2 γ ↓ ↓γ ˆρ
F Dˆ ρ → H2
Proof. It suffices to consider the separate actions of Hˆ and 1 , and we first consider that of Hˆ . From ( 36), elements of Hˆ act as follows: Ωˆ 12 + a Ωˆ 11 + b Ωˆ 11 , . (159) μ(a, b, c) : Ωˆ → ˆ Ω12 + a Ωˆ 11 + b, Ωˆ 22 + a 2 Ωˆ 11 + 2a Ωˆ 12 + ab + c We must show that the matrix in (159) coincides with Fˆ ρ (μ(a, b, c).x). ˆ Set x = (τ, w, ρ). Using (144), the periodicity of Pk (τ, w) in w for k > 1, and the quasiperiodicity of P1 (τ, w) (14), we find that R(k, l) and β(k) satisfy R(k, l)((a, b).x) = R(k, l)(x), β(k)((a, b).x) = β(k)(x) + aρ 1/2 δk,1 . Thus ω11 , ωβ1 and ωβ β¯ satisfy ω11 ((a, b).x) = ω11 (x), ωβ1 ((a, b).x) = ωβ1 (x) + aρ 1/2 ω11 (x), ωβ β¯ ((a, b).x) = ωβ β¯ (x) + a 2 ρω11 (x) + 2aρ 1/2 ωβ1 (x). We therefore find ˆ =τ− Ωˆ 11 (μ(a, b, c).x)
ρ ω11 ((a, b).x) = Ωˆ 11 (x). ˆ 2πi
Similarly, we have 1 (w + 2πiaτ + 2πib − ρ 1/2 ωβ1 ((a, b).x)) 2πi = Ωˆ 12 (x) ˆ + a Ωˆ 11 (x) ˆ + b.
Ωˆ 12 (μ(a, b, c).x) ˆ =
(160) (161)
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Now application of (156) yields 1 ˆ (l(μ(a, b, c).x) ˆ − ωβ β¯ ((a, b).x)) 2πi 1 ˆ (l(x) ˆ + 2πia 2 τ + 2aw + 2πi(ab + c) = 2πi −ωβ β¯ (x) − a 2 ρω11 (x) − 2aρ 1/2 ωβ1 (x))
Ωˆ 22 (μ(a, b, c).x) ˆ =
= Ωˆ 22 + a 2 Ωˆ 11 + 2a Ωˆ 12 + ab + c. This establishes equivariance of Fˆ ρ with respect to Hˆ . As in the -formalism, the exceptional transformation law (12) for E 2 plays a critical rôle in establishing 1 -equivariance of Fˆ ρ . Consider the action (89) of 1 on H2 . Since E 2 appears only in R(1, 1) and β(1), (145) implies that R(k, l)(γ1 .x) = R(k, l)(x) + κδk,1 δl,1 , w β(k)(γ1 .x) = β(k)(x) − κ 1/2 δk,1 , ρ ρ c1 κ= . c1 τ + d1 2πi
(162) (163) (164)
We then have Ωˆ 11 (γ1 .x) ˆ =
ρ 1 1 (a1 τ + b1 − ω11 (γ1 .x)). c1 τ + d1 2πi c1 τ + d1
Similarly to Theorem 5 in the formalism, (162) implies that the transformation under γ1 of the weight ω(N ) for N ∈ N1,a;1,b is the sum of the weights of the product over all 1,a1
1,a2
possible necklaces in N1,a;1,b formed from N by deleting the edges of type • −→ • and multiplying by a κ factor for each such deletion. From Proposition 12 and (164) we obtain 1 − κω11 = (c1 Ωˆ 11 + d1 )/(c1 τ + d1 ). Then we find, much as before, that ω11 (γ1 .x) =
n+1 κ n ω11 (x)
n≥0
=
(c1 τ + d1 )ω11 (x) . c1 Ωˆ 11 + d1
(165)
Then Ωˆ 11 (γ1 .x) ˆ is as given in (89). We next have 1 1 (w − ρ 1/2 ωβ1 (γ1 .x)). c1 τ + d1 2πi Equations (162) and (163) imply that k βa (k)ω(N ) for N ∈ Nk,a;1,b transforms under γ1 as a sum over the product with κ factors of weights of necklaces in N1,a;1,b and at Ωˆ 12 (γ1 .x) ˆ =
On Genus Two Riemann Surfaces Formed from Sewn Tori
629
most one necklace in Nk,a;1,b with a βa (k) factor. Then one finds ρ 1/2 ωβ1 (x) − κwω11 (x) 1 − κω11 (x) Ωˆ 12 = w − 2πi(c1 τ + d1 ). . c1 Ωˆ 11 + d1
ρ 1/2 ωβ1 (γ1 .x) =
This implies Ωˆ 12 (γ1 .x) ˆ is as given in (89). Finally, using (157) we have ˆ = Ωˆ 22 (γ1 .x)
1 ˆ w2 (l(x) ˆ +κ − ωβ β¯ (γ1 .x)). 2πi ρ
Using a similar argument, (162) and (163) imply that 2 (x) − 2κ w ω (x) + κ 2 w ω (x) κωβ1 ρ 11 ρ 1/2 β1 2
ωβ β¯ (γ1 .x) = ωβ β¯ (x) +
1 − κω11 2 c1 Ωˆ 12 + 2πi = ωβ β¯ (x) − κ . ρ c1 Ωˆ 11 + d1 w2
Hence Ωˆ 22 (γ1 .x) ˆ is as given in (89) and hence 1 acts equivariantly. This completes the proof of the theorem. Remark 4. Much the same as in Remark 1, Ω22 → Ω22 + 1 is generated by C corresponding to a Dehn twist in the connecting cylinder. We may also choose a branch of l(x) and consider the equivariance of F ρ on Dρ under the action of the subgroup 1 . Corollary 2. For any choice of branch for l(x), F ρ is equivariant with respect to the action of 1 . Thus, there is a commutative diagram for γ ∈ 1 , Fρ
Dρ → H2 γ ↓ ↓γ Fρ Dρ →
H2
6.4. Local Invertibility About the Two Tori Degeneration Limiting Point. We now consider the invertibility of the F ρ map in the neighborhood of a degeneration point. In the ρ-formalism this degeneration is more subtle than that of the -formalism discussed in Subsect. 4.5. Define the 1 -invariant parameter χ =−
ρ . w2
(166)
We will show that ρ, w → 0 for fixed χ is the 2 -torus degeneration limit. From (132) we have |w| > 2|ρ|1/2 (for λ = 0) so that |χ | < 41 . (Similarly to Subsect. 5.2.2, χ = 14
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G. Mason, M. P. Tuite
is a singular point where two degenerate annuli touch at z 1 = −z 2 = w/2.) To describe this limit more precisely, we introduce the domain 1 }. (167) 4 Thanks to Theorem 9 and Corollary 2, there is a 1 -equivariant holomorphic map Dχ = {(τ, w, χ ) ∈ H1 × C × C | (τ, w, −w 2 χ ) ∈ Dρ , 0 < |χ | <
F χ : Dχ → H2 , (τ, w, χ ) → Ω (2) (τ, w, −w 2 χ ).
(168)
Let 1 } 4 denote the space of limit points where ρ, w → 0 for fixed χ = 0. Then χ
D0 = {(τ, 0, χ ) ∈ H1 × C × C|0 < |χ | < χ
Proposition 13. For (τ, w, χ ) ∈ Dχ ∪ D0 we have (2) = 2πiτ + w 2 (1 − 4χ )G(χ ) + O(w 4 ), 2πiΩ11 , (2) 2πiΩ12 = w 1 − 4χ (1 + w 2 (1 − 4χ )E 2 (τ )G(χ ) + O(w 4 )), (2)
2πiΩ22 = log f (χ ) + w 2 (1 − 4χ )E 2 (τ ) + O(w 4 ),
(169)
where G(χ ) =
1 + E 2 (q = f (χ )), 12
and f (χ ) is the Catalan series (124). Proof. Note that Pn (τ, w) = Then (137) and (138) imply
1 2 4 wn (1 + w E 2 (τ )(δn,2 − δn,1 ) + O(w )) from (9) and (13).
R(k, l) = R (0) (k, l) + w 2 χ E 2 (τ )δk,1 δl,1 + O(w 4 ), log −
β(k) = β (0) (k)(1 − w 2 E 2 (τ )δk,1 ) + O(w 4 ), ρ ) = log χ + w 2 E 2 (τ ) + O(w 4 ), K (τ, w)2
using (127). For w = 0 these expressions are exactly those found in Proposition 9 for a torus formed from a sphere by sewing an annulus centered at z = 0 to another centered at z = w. Expanding ( 134) with ρ = −χ w2 to order w 2 implies (2)
2πiΩ11 = 2πiτ + w 2 χ σ ((I − R (0) )−1 (1, 1)) + O(w 4 ). But (130) implies σ ((I − R (0) )−1 (1, 1)) = 2(I + B (0) )−1 (1, 1) = (2)
(1 − 4χ ) G(χ ), χ
(2)
leading to the stated result for Ω11 . Similarly, for Ω12 we find (2)
2πiΩ12 = w[1 − (−χ )1/2 σ (β (0) (1 − R (0) )−1 (1))], [1 + w 2 E 2 (τ )χ σ ((1 − R (0) )−1 (1, 1)) + O(w 4 )].
(170)
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631
After some algebra and using (123), (128) and (129 ) one finds 1 − (−χ )1/2 σ (β (0) (1 − R (0) )−1 (1)) = 1 − 2χ
Sn,1 (χ )
n≥1
= 1 − 2χ (1 + f (χ )) , = 1 − 4χ . (2)
(2)
The stated result for Ω12 then follows on using (170 ) again. Finally, for Ω22 we find as above, using Proposition 9, that (2)
2πiΩ22 = log χ − β (0) (1 − R (0) )−1 β¯ (0)T +E 2 (τ )w 2 [1 − (−χ )1/2 σ (β (0) (1 − R (0) )−1 (1))]2 + O(w 4 ) = log f (χ ) + w 2 (1 − 4χ )E 2 (τ ) + O(w 4 ). χ
The restriction of F χ to D0 induces the natural identification χ ∼
F χ : D0 → H1 × H1 ⊆ H2 τ 0 , (τ, 0, χ ) → 1 log f (χ ) 0 2πi (2)
(171)
(2)
i.e., Ω (2) = diag(Ω11 , Ω22 ). The invertibility of the 1 -equivariant map F χ in a χ neighborhood of a point in D0 then follows: χ
χ
Proposition 14. Let x ∈ D0 . Then there exists a 1 −invariant neighborhood Nx ⊆ Dχ of x throughout which F χ is invertible. Proof. The proof is very similar to Proposition 5. Let x = (τ, 0, χ ). The Jacobian of the F χ map at x is from (169) # #1 # # # # ∂(Ω11 , Ω12 , Ω22 ) # # = #0 # # # # ∂(τ, w, χ ) #0 x =
0 1 √ 1 − 4χ 2πi 0
# # # # # f (χ ) #
0 0
1 2πi f (χ )
1
= 0, 4π 2 χ
√ using f (χ ) = f (χ )/(χ 1 − 4χ ). By the inverse function theorem, there exists an χ open neighborhood of x ∈ D0 throughout which F χ is invertible. The result then follows by choosing precisely invariant neighborhoods (under the action of 1 ) U, V of x, respectively y = F χ (x) such that the conditions of part (a) of Lemma 9 hold. The open neighborhood Nxχ =
$ γ ∈1
has the required properties.
γU
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7. Mapping Between the and ρ Parameterizations We have described in the previous sections two separate parameterizations for the genus two period matrix Ω (2) based on either sewing two punctured tori in the -formalism or sewing a twice-punctured torus to itself in the ρ-formalism. In this final section we show that there is a 1-1 mapping between suitable 1 −invariant domains in both parameterizations. Theorem 12. There exists a 1-1 holomorphic mapping between 1 −invariant open domains I χ ⊂ Dχ and I ⊂ D , where I χ and I are open neighborhoods of a 2-torus degeneration point. Proof. From Proposition 5 there exists a G-invariant (and thus 1 −invariant) open domain N ⊂ D such that the holomorphic map F : N → F (N ) is invertible with F (N ) an open neighborhood of a given 2-torus degeneration point Ω (2) = (2) (2) diag(Ω11 , Ω22 ). Similarly, from Proposition 14 there exists a 1 -invariant open domain χ χ N ⊂ D such that the holomorphic map F χ : N χ → F χ (N χ ) is invertible with (2) (2) F χ (N χ ) an open neighborhood of diag(Ω11 , Ω22 ). Define a 1 -invariant open neigh(2) (2) borhood of diag(Ω11 , Ω22 ) by I Ω = F (N )∩ F χ (N χ ). Hence, defining 1 -invariant open domains I χ = (F χ )−1 (I Ω ) and I = (F )−1 (I Ω ), we find (F )−1 ◦ F χ : I χ → I is holomorphic and 1-1. We conclude by displaying the explicit form of the 1-1 mapping to order w3 , obtained by combining the expansions of (102)-(104) and Proposition 13: 1 2 1 w (1 − 4χ ) + O(w 4 ), τ1 (τ, w, χ ) = τ + 2πi 12 1 log( f (χ )) + O(w 4 ), τ2 (τ, w, χ ) = 2πi, (τ, w, χ ) = −w 1 − 4χ (1 + w2 E 2 (τ )(1 − 4χ ) + O(w4 )). (172) It is then straightforward to check that these relations are invariant under the action of 1 to the given order using (12), (86) and (145). Acknowledgement. The authors wish to thank Harold Widom and Alexander Zuevsky for useful discussions.
8. Appendix In this appendix we supply the explicit form of the genus two period matrix Ω (2) of Theorem 4 in the -formalism to O( 9 ) and of Theorem 9 in the ρ-formalism to O(ρ 5 ). (2) (2) 2πiΩ11 (τ1 , τ2 , ) = 2πiΩ22 (τ2 , τ1 , )
= 2πiτ1 + F2 2 + E 2 F2 2 4 + (E 2 2 F2 3 + 6 E 4 F2 F4 ) 6 + (E 2 3 F2 4 + 12 E 2 E 4 F2 2 F4 + 10 E 6 F2 F6 + 30 E 6 F4 2 ) 8 + O( 10 ), (2)
2πiΩ12 (τ1 , τ2 , ) = −[1 + E 2 F2 2 + (E 2 2 F2 2 + 3 E 4 F4 ) 4 + (E 2 3 F2 3 + 9 E 2 E 4 F2 F4 + 5 E 6 F6 ) 6 + (E 2 4 F2 4 + 15 E 2 2 E 4 F2 2 F4 + 5 E 2 E 6 F2 F6 + 30 E 2 E 6 F4 2 + 30 E 4 2 F2 F6 + 9 E 4 2 F4 2 + 7 E 8 F8 ) 8 ] + O( 11 ), where for brevity’s sake we have defined E k = E k (τ1 ) and Fk = E k (τ2 ).
On Genus Two Riemann Surfaces Formed from Sewn Tori
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Similarly, in the ρ-formalism we find that Ω (2) (τ, w, ρ) to O(ρ 4 ) is as follows: (2)
2πiΩ11 = 2πiτ − 2 ρ + 2 ( P2 + E 2 ) ρ 2 − 2 (P2 + E 2 )2 ρ 3 +2( (P2 + E 2 )3 + 2 P3 2 )ρ 4 + O(ρ 5 ), (2)
2πiΩ12 = w + 2 P1 ρ − 2 P1 (P2 + E 2 ) ρ 2
(2)
2πiΩ22
+2[ P1 (P2 + E 2 )2 + P3 (P2 − E 2 )]ρ 3 − 2[ P3 (P4 + E 4 ) + P1 (P2 + E 2 )3 + 2 P1 P3 2 + P3 (P2 2 − E 2 2 )]ρ 4 + O(ρ 5 ), ρ ) − 2 P1 2 ρ + [2 P1 2 (P2 + E 2 ) + (P2 − E 2 )2 ]ρ 2 = log(− 2 K (w, τ ) −[2 P1 2 (P2 + E 2 )2 + 2/3 P3 2 + 4 P1 P3 (P2 − E 2 )]ρ 3 +[1/2 P4 2 + 1/2 E 4 2 + 3 (P4 − E 4 ) (P2 − E 2 )2 + 2 P1 2 (P2 + E 2 )3 −E 4 P4 + 4 P3 P1 (P1 P3 + E 4 + P4 + P2 2 − E 2 2 )]ρ 4 + O(ρ 5 ),
where E k = E k (τ ) and Pk = Pk (w, τ ). References [DGP] [DLM] [DM1] [DM2] [DP] [DPI] [DPVI] [EZ] [FK1] [FK2] [GSW] [Gu] [H] [KZ] [MS] [MT1] [MT2] [Mu] [P] [Se] [St]
D’Hoker, E., Gutperle, M., Phong, D.H.: Two-loop superstrings and s-duality. Nucl. Phys. B 722, 81–118 (2005) Dong, C., Li, H., Mason, G.: Regularity of rational vertex operator algebras. Adv. Math. 132(1), 148– 166 (1997) Dong, C., Mason, G.: Rational vertex operator algebras and the effective central charge. Int. Math. Res. Not. No. 56, 2989–3008 (2004) Dong, C., Mason, G.: Holomorphic vertex operator algebras of small central charge. Pac. J. Math. 213(2), 253–266 (2004) D’Hoker, E., Phong, D.H.: The geometry of string perturbation theory. Rev. Mod. Phys. 60, 917– 1065 (1988) D’Hoker, E., Phong, D.H.: Two-loop superstrings. i: main formulas. Phys. Lett. B 529, 241– 255 (2002) D’Hoker, E., Phong, D.H.: Two-loop superstrings. vi: non-renormalization theorems and the 4-point function. Nucl. Phys. B 715, 3–90 (2005) Eichler, M, Zagier, D: The Theory of Jacobi Forms. Birkhäuser, Boston (1985) Farkas, HM, Kra, I: Riemann Surfaces. Springer-Verlag, New York (1980) Farkas, H.M., Kra, I.: Riemann Surfaces. New York: Springer-Verlag 1980 Farkas, H.M., Kra, I.: Theta Constants, Riemann Surfaces and the Modular Group. Graduate Studies in Mathematics, Providence, RI: Amer. Math. Soc. 2001 Green, M., Schwartz, J., Witten, E.: Superstring Theory Vol. 1. Cambridge: Cambridge University Press, 1987 Gunning, R.C.: Introduction to Holomorphic Functions of Several Variables Vol 1. Belmont, CA: Wadsworth & Brooks/Cole (1990) Huang, Y: Two-Dimensional Conformal Geometry and Vertex Operator Algebras. Birkhäuser, Boston (1997) Kaneko, M., Zagier, D.: A generalized Jacobi theta function and quasimodular forms. In: The Moduli Space of Curves (Texel Island, 1994), Progr. in Math. 129, Boston: Birkhauser, 1995 Moore, G., Seiberg, N.: Classical and quantum conformal field theory. Commun. Math. Phys. 123, 177–254 (1989) Mason, G., Tuite, M.P.: Torus chiral n-point functions for free boson and lattice vertex operator algebras. Commun. Math. Phys. 235, 47–68 (2003) Mason, G., Tuite, M.P.: The genus two partition function for free bosonic and lattice vertex operator algebras. To appear Mumford, D.: Tata Lectures on Theta I and II. Boston: Birkhäuser, 1983 Polchinski, J.: String Theory, Volume I. Cambridge: Cambridge University Press, 1998 Serre, J-P.: A Course in Arithmetic. Berlin: Springer-Verlag, 1978 Stanley, R.P.: Enumerative Combinatorics, Volume 2, Cambridge: Cambridge University Press, 1999
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Sonoda, H.: Sewing conformal field theories I. Nucl. Phys. B311, 401–416 (1988); Sewing conformal field theories II. Nucl. Phys. B311, 417–432 (1988) Springer, G.: Introduction to Riemann Surfaces. Reading, MA:, Addison-Wesley, (1957) Tuite, M.P: Genus two meromorphic conformal field theory, CRM Proceedings and Lecture Notes 30, Providence, RI: Amet. Math. Soc., 2001, pp. 231–251 Verlinde, E.P., Verlinde, H.L.: Chiral bosonization, determinants and the string partition function. Nucl. Phys. B 288, 357–396 (1987) Yamada, A.: Precise variational formulas for abelian differentials. Kodai Math. J. 3, 114–143 (1980) Zhu, Y.: Modular invariance of characters of vertex operator algebras. J. Amer. Math. Soc. 9, 237– 302 (1996)
Communicated by Y. Kawahigashi
Commun. Math. Phys. 270, 635–689 (2007) Digital Object Identifier (DOI) 10.1007/s00220-006-0164-4
Communications in
Mathematical Physics
Dynamics near Unstable, Interfacial Fluids Yan Guo1, , Chris Hallstrom2 , Daniel Spirn3, 1 Division of Applied Mathematics, Brown University, Providence, RI 02912, USA.
E-mail:
[email protected]
2 Department of Mathematics, University of Portland, Portland, OR 97203, USA.
E-mail:
[email protected]
3 School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA.
E-mail:
[email protected] Received: 16 March 2006 / Accepted: 13 July 2006 Published online: 9 January 2007 – © Springer-Verlag 2006
Abstract: We study three examples of unstable interfacial fluid motions: vortex sheets with surface tension, Hele-Shaw flows with surface tension, and vortex patches. In all three cases, the nonlinear dynamics of a large class of smooth perturbations is proven to be characterized by the corresponding fastest linear growing mode(s) up to the time scale of log 1δ , where δ is the magnitude of the initial perturbation. In all three cases, the analysis is based on an unified analytical framework that includes precise bounds on the growth of the linearized operator, given by an explicit solution formula, as well as a special sharp nonlinear energy growth estimate. Our main contribution is establishing this nonlinear energy growth estimate for each interface problem in certain high energy norms. 1. Abstract Framework In general, the passage from linear instability to nonlinear instability is delicate for conservative partial differential equations. Stability depends in a fundamental way on the norms, and the possible presence of continuous spectra, as well as severe nonlinearities with possible unbounded derivatives, both pose key difficulties in proving nonlinear instability in natural energy norms, such as · L 2 . One important difficulty to overcome is estimating derivatives that arise in the nonlinearity, and often one relies on a delicate interplay between the L 2 estimate and estimates in higher Sobolev norms. Strauss and the first author [15] developed a bootstrap method to treat such a difficulty. Grenier [13], on the other hand, introduced a second method that entails the construction of high order approximations to the nonlinear problem, well beyond the linear growing modes. The main idea of the Guo-Strauss approach is to show that on the time scale of the instability, the stronger Sobolev norm ||| · ||| of the solution is actually bounded by the growth of its weaker norm ·. Our implementation of this bootstrap framework is described as follows. Y. G. was supported in part by NSF grants DMS-0305161, INT-9815432 and a Salomon award of Brown University. D. S. was supported in part by NSF grant DMS-0510121.
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Lemma 1.1 (Bootstrap Instability) . Assume that L is a linear operator on a Banach space X with norm ·, and et L generates a strongly continuous semigroup on X such that et L (X,X ) ≤ C L eλt
(1.1)
for some C L and λ > 0. Assume a nonlinear operator N (y) on X and another norm ||| · |||, and constant C N , such that N (y) ≤ C N |||y|||2 for all y ∈ X and |||y||| < ∞. Assume for any solution y(t) to the equation y = L y + N (y)
(1.2)
with |||y(t)|||2 ≤ σ , there exists Cσ > 0 such that for any ε > 0, there exists Cε > 0 such that the following sharp energy estimate holds: 3 d |||y(t)||| ≤ ε|||y(t)||| + Cσ |||y(t)||| 2 + Cε y(t). dt
(1.3)
Consider a family of initial data y δ (0) = δy0 with y0 = 1 and |||y0 ||| < ∞ and let θ0 be a sufficiently small (fixed) number. Then there exists some constant C > 0 such that if 0 ≤ t ≤ Tδ ≡ we have
θ0 1 log λ δ
y(t) − δe Lt y0 ≤ C[|||y0 |||2 + 1]{δeλt }2 . e Lt y0 ≥ Cp eλt
In particular, if there exists a constant Cp such that time y(T δ ) ≥ τ0 > 0,
(1.4) then at the escape (1.5)
where τ0 depends explicitly on CL ,CN ,Cσ ,Cp ,λ,y0 ,σ and is independent of δ, see (1.9). In our applications of this lemma, y(t) represents the perturbation away from the trivial steady state y(t) ≡ 0, and λ can be chosen as the largest real part of eigenvalues of L. Moreover, for the general initial perturbation profile y0 which contains at least one such fastest growing mode (which we define as a generic perturbation), δe Lt y0 ≥ C p δeλt for all 0 ≤ t ≤ T δ , where the constant C p depends on the generic perturbation y0 . In the setting of this lemma, ||| · ||| is a stronger norm than · , and the assumption N (y) ≤ C N |||y|||2 means that the nonlinearity is not bounded in the weaker norm · (for instance, if there are higher order derivatives present). This is a key analytical difficulty in many instability problems. The assumptions in Lemma 1.1 can be relaxed in the following ways. (a) The small constant ε only needs to be less than, say, λ4 where λ is typically the largest positive real part of the set of discrete eigenvalues for L. (b) The second term of (1.3) can be replaced by a function h(|||y|||) such that lims→0 h(s) s = 0.
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Lemma 1.1 provides a general framework to reduce the study of nonlinear instability to the study of the linear instability of e Lt , see other applications [1, 4, 10, 12–15, 18, 20, 28]. Our explicit linear solutions (see formulas (2.21), (3.5), and (4.7)) contain no continuous spectra, and in all three applications e Lt is determined by a finite number of (at most two) fastest linear growing modes. We take advantage of the one-dimensional periodic structure of these problems to obtain a complete and precise linear analysis for e Lt . The most delicate part of applying this framework, on the other hand, is to observe and prove the energy estimate (1.3) for all three applications, which states that the higher Sobolev type of norm ||| · ||| does not create a faster growth rate than the weaker counterpart · , over the time scale of 0 ≤ t ≤ T δ . In other words, roughly speaking, the stronger norm ||| · ||| is controlled reversely by the weaker norm · . This is the key to close the instability argument, see the proof of Lemma 1.1 at the end of this section. Using this framework, we study three examples of unstable interfacial fluid motion: an unstable two dimensional vortex sheet with surface tension, an unstable HeleShaw flow and an unstable Kirchhoff elliptical vortex patch. In all three cases we prove nonlinear instability by showing that the nonlinear dynamics of any general perturbation is dominated by the fastest linear growing mode(s) up to the time scale of log 1δ . Here δ is the magnitude of the initial perturbation, see Theorem 2.4, Theorem 3.3, and Theorem 4.3 for details. Since the dynamics of these fluid flows are essentially simple, at least up to the time of instability, we can think of these results as an early description of the pattern formation of the PDE and a helpful predictor of the behavior of numerical simulations. In general such a sharp dynamical description may not hold. For instance Hwang and Guo [18] showed that the early dynamics of unstable Rayleigh-Taylor perturbations may not be described by the dominant eigenmode of the linearized problem. Therefore, Lemma 1.1 provides an important tool to distinguish those PDE’s whose early dynamics are dominated by linear behavior. The main contribution of this paper is the establishment of (1.3) for the vortex sheet, Hele-Shaw, and vortex patch problems for their respective higher Sobolev norms. These estimates are found in Proposition 2.2, Proposition 3.1, and Proposition 4.1. We first observed the possibility of this crucial estimate (1.3) holding by examining the associated estimate at the linear level (see (2.60), (3.11), and (4.13)), and we feel such behavior at the linear level may distinguish those PDE’s that satisfy the assumptions of Lemma 1.1. In Sect. 2 we study vortex sheets with surface tension. This system is a type of nonlinear wave equation [19], as can be seen from the linear analysis (see (2.18)). Our analysis is based on the approach developed in the recent work of Ambrose [2], who established the local-in-time well-posedness for the same system. Our main improvement is to further identify and prove a “quadratic” feature of the many nonlinear terms. In order to do this we revisit many estimates of [2] and carefully extract additional powers of the perturbation of the many singular integral estimates in Sobolev norms (see Subsect. 2.2). Section 3 examines Hele-Shaw flows with surface tension. This analysis is the simplest of the three problems, as the Hele-Shaw system is of parabolic type (see the linear equation (3.3)). As in the vortex sheet system, the Hele-Shaw equations are defined in terms of a Birkhoff-Rott operator and therefore we are able to make use of our new estimates from Subsect. 2.2. Moreover, due to the parabolic nature of the problem, we can establish a sharp nonlinear stability criterion: if the density stratification R < 0 (i.e. heavier fluid on the top), then it is nonlinearly unstable; if the density stratification R > 0 (i.e. heavier fluid on the bottom), then it is nonlinear stable. In particular, such stability leads to small data global existence for the Hele-Shaw flow.
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Finally, Sect. 4 concerns vortex patches, in which we revisit the nonlinear instability of rotating elliptical vortex patches. The governing equation is a nonlinear transport equation for the boundary of the patch, similar to the Contour Dynamics equation, see Majda–Bertozzi for a description [22]. In [14], we proved nonlinear instability by applying Grenier’s framework [13] of higher order linear approximations. Here, we reexamine the dynamics of unstable elliptical vortex patch perturbations by carefully extracting the linear growing terms in order to establish (1.3). Our new proof is more intuitive and reveals many deep structures of the problem. Finally, we conclude this section with a proof of the bootstrap lemma. Proof of Lemma 1.1. Since λ, σ , and Cσ are fixed positive constants, we define σ = min{σ, 4Cλσ } . We denote another constant θ0 = θ0 (C N , C L , λ, Cσ , σ, C p ) to be defined below in (1.8). Denote Tδ ≡
θ0 1 log , λ δ 1
T ∗ ≡ sup{|||y(t)||| 2 < σ }, t
T
∗∗
≡ sup{y(t) ≤ 2δeλt y0 }. t
For t ≤ min{T δ , T ∗ , T ∗∗ }, definitions of T ∗ , T ∗∗ that
we deduce from (1.3) with ε =
λ 4,
Cε = Cλ , and the
3 d λ |||y(t)||| ≤ |||y(t)||| + Cσ |||y(t)||| 2 + Cλ y(t) dt 4 λ ≤ |||y(t)||| + 2Cλ δeλt y0 2 λ ≤ |||y(t)||| + 2Cλ δeλt . 2 It follows from the Gronwall lemma that t λ λ t 2 |||y(t)||| ≤ δe |||y0 ||| + 2Cλ δ e 2 (t−τ ) eλτ dτ
0
4Cλ λt δe ≤ δe |||y0 ||| + λ 0 ||| + 1]δeλt , ≤ C[|||y λ 2t
= where C
(1.6) y
max{ 4Cλ λ , 1}.
Applying the Duhamel principle to = L y + N (y) yields t Lt L(t−τ ) y(t) − δe y0 = e N (y(τ ))dτ 0 t ≤ CL eλ(t−τ ) N (y(τ )) dτ 0 t ≤ CL CN eλ(t−τ ) |||y(τ )|||2 dτ 0 t 2 2 2 ≤ 2C C L C N [|||y0 ||| + 1]δ eλ(t−τ ) e2λτ dτ 0
≤ C[|||y0 |||2 + 1]δ 2 e2λt ,
(1.7)
Dynamics near Unstable, Interfacial Fluids
where C =
2CL CN 2C λ
639
. Finally, in order to conclude the lemma, it suffices to show that min{T δ , T ∗ , T ∗∗ } = T δ ,
by fixing θ0 a small enough. Set θ0 = min
Cp σ 1 . , , 0 ||| + 1] 2C[|||y0 |||2 + 1] 2C[|||y0 |||2 + 1] C[|||y
(1.8)
On the one hand, if T ∗ < T δ is the smallest, then we have from (1.6) for 0 ≤ t ≤ T ∗ , [|||y0 ||| + 1] δeλT ∗ |||y(T ∗ )||| ≤ C [|||y0 ||| + 1] δeλT δ
∗∗ ∗∗ δeλT ≤ 1 + C[|||y0 |||2 + 1]δeλT
δ ∗∗ ≤ 1 + C[|||y0 |||2 + 1]δeλT δeλT
∗∗ ∗∗ = 1 + C[|||y0 |||2 + 1]θ0 δeλT < 2δeλT , which is also a contradiction to the definition of T ∗∗ . To prove (1.5) we shall see that C p δeλt should dominate the nonlinear second order correction C[|||y0 |||2 + 1]{δeλt }2 for δeλt ≤ θ0 . In particular, at the escape time t = T δ , the dominant linear term is bounded from below by δ LT δ y0 ≥ C p δeλT = C p θ0 , δe whereas the nonlinear correction (1.4) satisfies
δ 2 C |||y0 |||2 + 1 δeλT = C |||y0 |||2 + 1 θ02 . Then (1.7) yields
y(T δ ) ≥ τ0 > 0,
where τ0 =
1 C p θ0 2
which depends on C L , C N , Cσ , C p , λ, y0 , σ and is independent of δ.
(1.9)
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2. Vortex Sheets The first case that we consider is that of vortex sheets in two space dimensions, describing the motion of the interface of two incompressible, inviscid, and irrotational fluids. We take into account the effect of surface tension, which plays a crucial role in the analysis of the PDEs. Without surface tension, the initial value problem is ill-posed, at least in some Sobolev spaces. Recently, local in time well-posedness in high Sobolev spaces was established. We are interested in the stability of the flat vortex sheet, which is a steady state. The well known Rayleigh-Taylor instability describes the situation where the fluid on the top is heavier, in which case the flat vortex sheet is (linearly) unstable; while the Kelvin-Helmholtz instability occurs if the strength of the vortex sheet is relatively large with respect to the surface tension. In two dimensions, the interface between two incompressible, inviscid, and irrotational fluids is described by a vortex sheet, a curve of localized vorticity which we denote by the parameterized curve z(α, t) = x(α, t) + i y(α, t) in the complex plane. The evolution of this curve is described in terms of the velocity of the interface, W, which is given by the Birkhoff-Rott integral 1 W= 2πi
∞ −∞
γ (α ) + γ0 dα , z(α) − z(α )
(2.1)
where γ + γ0 (γ0 being a steady state) is the vortex sheet strength. Here we have used the following Notation. We denote z(α, t), z 0 (α, t) to be parameterized curves in the complex plane and use boldface to denote complex numbers, for example W, t, n. Finally, we denote ¯ ab the complex inner product a, b = ab+¯ 2 . If the effects of surface tension are neglected, then the perturbation γ does not depend on time and in this case the problem is ill-posed. By including surface tension, the problem becomes well-posed and in this case, the evolution of γ (α, t) must also be considered. We will express the interface velocity in terms of a normal part U = W, n and a tangential part T, where ∂α z i∂α z t= , n=− . (2.2) |∂α z| |∂α z| Instead of using Cartesian coordinates for the curve, we will follow Hou-LowengrubShelley [16] and introduce θ , the tangent angle between the curve and the horizontal, as well as the arclength s. This parametrization of the curve simplifies the curvature boundary condition. Since we are considering periodic curves that satisfy the condition z(α + 2π, t) = z(α, t) + 2π , clearly there is a steady state z(α) ≡ α, γ ≡ 0. It is natural to use the length of the curve over one period l(t) + 2π = 0
2π
∂α s dα
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instead of s. The motion of the interface is then described by the evolution equations for θ and l plus an equation for the vortex sheet strength γ0 + γ . The first two equations come directly from taking time derivatives: ∂t θ = dl = dt
1
Uα +
∂α s 2π 0
T θa , ∂α s
(2.3)
∂αt s dα = −
2π 0
∂α θU dα.
(2.4)
We will make use of geometric identities that allow us to express the quantity U in terms of W. Finally, there is freedom to choose T , and following [2] we set T =
α
0
θα U dα +
α dl . 2π dt
The evolution equation for γ comes from the Laplace-Young condition on the pressure jump across the interface, along with Bernoulli’s equation for potential flows on opposite sides of the interface, see [5, 16, 2]. We allow for the possibility that the fluids on either side of the interface have different densities, and so introduce the Atwood number, A, defined by ρ1 − ρ2 A= , ρ1 + ρ2 where ρ1 and ρ2 are the densities below and above the interface respectively. We also introduce the Weber number W which is inversely proportional to the surface tension. The three evolution equations describing the motion of the interface are 2π 2π
∂α W, n + (T − W, t)∂α θ, (2.5) 2π + l 2π + l 1 dl 2π 2π ∂αα θ + − ∂α W, t (γ + γ0 ) ∂t γ = W (2π + l) 2π + l 2π dt 2π 2π + l π 2 (γ + γ0 )
∂t W, t + + ∂α γ (T − W, t) ∂α γ − 2 A 2π + l 2π (2π + l)2 2(2π + l)Ag − (T − W, t) ∂α W, t − sin(θ ), (2.6) 2π 2π dl =− ∂α θU dα. (2.7) dt 0 ∂t θ =
Again, we refer the reader to prior work on vortex sheet problems for further details regarding the derivation of these equations [2, 5, 16]. Since the Birkhoff-Rott operator (2.1) looks formally like a Hilbert transform, H( f ) =
1 π
∞ −∞
f (α ) dα , α − α
much progress has been achieved in the analysis of (2.5)–(2.7) by “forcing” the operator into a Hilbert transform plus perturbation. We consider curves such that z(α) − α are
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2π -periodic, and the corresponding expressions for the Birkhoff-Rott operator and for the Hilbert transform simplify: b+π z(α) − z(α ) 1 W= (γ (α ) + γ0 ) cot (2.8) dα , 4πi b−π 2 2π 1 1 α − α H(γ ) = P.V. dα , γ (α ) cot (2.9) 2 4π 2 0 which lets us write each expression as an integral over one period. Formally we believe (2.8) and (2.9) should be similar (at least in the tangential direction of W), and we spend a good deal of energy in Subsect. 2.2 showing this. Note that because of periodicity, b can be any real number. To push W into a Hilbert transform we define, as in [2], the operator
2π 1 α − α 1 z(α)−z(α ) − cot dα (2.10) K[z]( f ) = f (α ) cot 4πi 0 2 ∂α z(α ) 2 and the commutator
[H, f ] (g) = H( f g) − gH( f ),
(2.11)
which allows us to rewrite (2.8) as
We also define
1 1 1 H (γ ) + H, (γ ) + K[z](γ + γ0 ). W= 2i∂α z 2i ∂α z
(2.12)
1 1 H, J [z]( f ) = ( f ) , t + K[z] ( f ) , t 2i ∂α z
(2.13)
from which we find
W, t = J [z](γ ).
Letting = 2π/(2π + l), Eqs. (2.5)–(2.7) can be written in the following perturbation form: 1 2 H(∂α γ ) + (T − W, t) ∂α θ + R1 , n, 2 2A ∂αα θ 2 J [z] ∂t γ = + H (γ0 + γ )2 ∂α θ 1+ W 2 ∂t θ =
−{A2 (γ0 + γ ) + (T − W, t)}∂α γ 2 Ag sin(θ ) + F, − 2π dl =− ∂α θU dα, dt 0 where
(γ0 + γ )∂α2 z ∂α γ − R1 = ∂α zK[z] ∂α z (∂α z)2
∂α z 1 (γ0 + γ )∂α2 z + ∂ H, , γ − α 2i (∂α z)2 ∂α z
(2.14)
(2.15)
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and 1 d (γ0 + γ ) − (γ0 + γ ) R1 , t dt 2A
R2 , t + (T − W, t) 2 A R1 , t − A 1 H, (γ0 + γ ) (H (∂α γ )) − 2 H, (γ0 + γ ) ((γ0 + γ ) ∂α θ ) + 2 2 2 + AH ((γ0 + γ ) R1 , n) + A [H, (T − W, t)] (∂α θ ) ,
F =−
(2.16)
and ∂t z(α) − ∂t z(α ) (γ0 + γ ) (γ0 + γ ) ∂α − ∂α ∂t z R2 = K[z] ∂α z(α ) ∂α z(α ) ∂α z
1 1 1 t 1 (γ0 + γ ) ∂α + H, − [H, ∂t z] (γ0 + γ ) 2i ∂α z ∂α z 2i ∂α z
1 1 − H, (2.17) (∂t θ (γ0 + γ )) . 2 ∂α z We point out that the quantities R1 , R2 , F, (T − W, t), and J [z] are all formally second order in the perturbation, as is shown in Subsect. 2.3. Remark. Local well-posedness for (2.14)–(2.17) with g = 0 was established in the work of Iguchi and Ambrose using different methods [19, 2]. Here, we follow the energy approach of Ambrose [2]. The linearized vortex sheet satisfies the equations 1 H(∂α γ ), 2 γ2 1 A ∂t γ = ∂αα θ + 0 H(∂α θ ) − γ0 ∂α γ − 2 Agθ, W 2 2 dl = 0. dt ∂t θ =
(2.18)
α = |k|, By letting θ (t, x) = θk (t)eikx , γ (t, x) = γk (t)eikx , and using the fact that H∂ we get the system of ODEs, |k| γk , 2 k 2 γ02 |k| −i Akγ0 γk γ˙k = − + − 2 Ag θk − , W 2 2 θ˙k =
which has eigenvalues |k|1/2 k Aγ0 i ± = − λ± k 4 2
−4 Ag + (1 − A2 /4)γ02 |k| −
2|k|2 . W
(2.19)
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From this explicit formula for the eigenvalues, we deduce that there is a growing mode if and only if there exists an integer k such that −4 Ag + (1 − A2 /4)γ02 |k| −
2|k|2 > 0. W
(2.20)
Note that the number of possible such k is finite. We denote the maximal growth rate by λ (i.e., λ is the largest positive value of Re λ± k ) and the corresponding (finitely many) i wave number(s) by kmax . We also introduce χ > 0 to be the gap between λ and the next largest real part of λ± k . We will refer to this quantity later on. We can now express the solutions to the linearized vortex sheet equations in terms of the explicit Fourier series ⎞ ⎛ ⎛ ⎞ ⎛ ⎞ |k| t)θ (0) + tγ (0) (1 − λ k k k θ0 (0) θ (t) 2 2 ⎟ ⎜ ⎝γ (t)⎠ = ⎝−t Agθ0 (0) + γ0 (0)⎠ + eλk t ⎝− 2λk tθk (0) + (1 + λk t)γk (0)⎠ |k| l(t) l(0) λ+k =λ− 0 k ⎛ ⎞ 2λ+k θk (0)−kγk (0) λ− t+ikα 2λ− θ (0)−kγ (0) k λ+k t+ikα k k k e − e − − + + 2λk −2λk 2λk −2λk ⎜ − ⎟ ⎜ 2λk λ+k θk (0)−λ−k kγk (0) λ− t+ikα 2λ−k λ+k θk (0)−λ+k kγk (0) λ+ t+ikα ⎟ + k k ⎝ ⎠ e − e − − + + λ+k =λ− k
kλk −kλk
kλk −kλk
0
⎛
⎞ θ (0) ≡ e Lt ⎝γ (0)⎠ , l(0)
(2.21)
where λ± k are defined by (2.19). This defines the linear evolution operator L. Notice that " " 2 3/2 2 3/2 ± − + λk → ±i |k| , and λk − λk = 2i |k| W W as k tends to ∞. We easily conclude from (2.21) that # $ θ (t) H 1/2 + γ (t) L 2 ≤ Ceλt θ (0) H 1/2 + γ (0) L 2 .
(2.22)
2.1. Unstable vortex sheet dynamics. We first define the high energy norm as in [2]: k 1 %% k %%2 W Ek = E0 + H ∂αk γ ∂αk−1 γ dα %∂α θ % dα + 2 4 j=1 % %2 2 W 2 % % + (γ + γ0 )2 %∂αj−1 γ % dα 8 with E0 = l 2 +
1 2
(2.23)
(θ 2 + γ 2 )dα.
We note that E k , under the PsK conditions defined below, is an equivalent norm to E k l 2 + θ 2H k + γ 2
1
H k− 2
.
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We introduce a function space PsK which prevents our contour from having selfintersections and allows us to establish necessary lower and upper bounds. Let w = inf
η1 =η2
|w(η1 ) − w(η2 )| . |η1 − η2 |
(2.24)
If w % ≥ c %then w will have no self-intersections, and we also note that w ≥ c implies %∂η w % ≥ c. We now define a function space PsK . Definition 2.1. We say w(α) ∈ PsK if w(α) ∈ H s for some s ≥ 3 and ∂η w(α) 1 ≤ K
(2.25) 1 w(α) ≥ . (2.26) K This definition is used in [14], but similar types of function spaces are used in [22, 31, 2], among other places. Elements of PsK have no self-intersections and lack cusps. C
Proposition 2.2. Fix k ≥ 3 and let θ,& γ , l satisfy (2.14). If E k ≤ σ sufficiently small α (defined within the proof) and z(α) = 0 (2π + l)eiθ dα ∈ PsK , then for any ε > 0 there exists Cε such that d Ek 3/2 (2.27) ≤ εE k + Cσ E k + Cε E 0 . dt d Remark. In [2] an energy growth estimate of the form dt E k ≤ CeC E k was established; however, this is not strong enough to use Lemma 1.1, since CeC E k C + C E k + o(E k ), whereas we need an expansion of the form εE k + o(E k ). As a consed quence of our analysis, we achieve an improved large data estimate of the form dt Ek ≤ 12k−11 E k , see (2.59). C K (1 + E k )
The proof of Proposition 2.2 is found in Subsect. 2.4.1. We now define the generic profile for the vortex sheet problem. Definition 2.3. We define a smooth generic profile for the initial perturbation as ∞ ∞ ikα ikα ˜ = ˜ γ˜ , l) (θ, γ˜k e , l˜ (2.28) θ˜k e , k=0
k=0
such that θ˜kmax or γ˜kmax nonzero for some i. Notice that l˜ is a constant. i i From Proposition 2.2 and Lemma 1.1, we now establish the following Theorem 2.4. Given constants γ0 , A and g such that (2.20) is valid. Suppose θ δ (t, α), γ δ (t, α) and l δ (t) is a solution to the vortex sheet equations (2.14) with initial value ˜ (θ δ (0, α), γ δ (0, α), l δ (0)) = δ(θ˜ , γ˜ , l), ˜ is a generic profile defined in (2.28). Then, where (θ˜ , γ˜ , l) (θ δ (t), γ δ (t), l δ (t)) − δe Lt (θ˜k j , γ˜k j , 0) max
max
j
≤ C e−χ t + 1 + θ˜ 2H k + γ˜ 2
1
H k− 2
˜ 2 δeλt δeλt , + |l|
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for any δ ≤ δ0 sufficiently small, and 0 ≤ t ≤ T δ , where χ > 0 is the gap between λ and the next largest real part of λk . Here (θ, γ , l) = θ 1 + γ L 2 + l L 2 . H2
˜ is given explicitly in (2.21). Clearly, for Remark. We recall that e Lt (θ˜k j , γ˜k j , l) max max Lt ˜ ≥ ceλt for some positive constant c. θ˜ j , γ˜ j not both zero, e (θ˜ j , γ˜ j , l) kmax
kmax
kmax
kmax
The proof of Theorem 2.4 follows from Proposition 2.2 and Lemma 1.1 with the norms ||| · |||2 = E k and · = θ H 1/2 + γ L 2 + l L 2 . We use the explicit form ˜ the maximal growing modes of the Fourier series (2.21) to separate from δet L (θ˜ , γ˜ , l) Lt ˜ δe (θk j , γ˜k j , 0). max
max
2.2. Basic estimates. In this subsection, we estimate the error of using the Hilbert transform in lieu of the Birkhoff-Rott integral. These bounds will be used both for the vortex sheet and Hele-Shaw sections. Recall the Hilbert transform H f on a 2π periodic domain is defined as 2π 1 α − α Hf = f (α )dα . cot P.V. 2π 2 0 We will ignore the P.V. symbol in the text of the article. Lemma 2.5. The Hilbert transform has the following basic properties: f (ξ ) = −i sgn(ξ ) ' H f (ξ ) where ' f is the Fourier transform of f . H f L 2 = f L 2 . & H & (H f ) = − f as long as f dα = 0. f H( f )dα = 0, the Hilbert transform is antisymmetric. & 2 1/2 f + f H f α dα = f 1. H2
Before we establish our improvement bounds on K[z]( f ) and [H, f ](g) over the results in [2, 31], we have a few technical estimates on finite differences. Lemma 2.6. Let
1 z(α) − z(α ) Q 1 (z)(α, α1 ) = = ∂α z(τ α + (1 − τ )α )dτ, α − α 0 1 z(α) − z(α ) − ∂α z(α )(α − α ) Q 2 (z)(α, α1 ) = = (1 − τ )∂α2 z(τ α + (1 − τ )α )ds. (α − α )2 0
Then for any k ≥ 0, and 1 ≤ p ≤ ∞, we have the following bounds: 2π 2π
p |∂αk Q 1 | p + |∂αk Q 1 | p dαdα ≤ C∂α zW k, p ,
0
0
2π
0
0
2π
|∂αk Q 2 | p + |∂αk Q 2 | p dαdα ≤ C∂α2 z2W k, p ,
(2.29)
Q 1 (z)W k,∞ ≤ Cz H k+2 , 1
Q 1 (z)W k,∞ ≤ Cz H k+s |α − α |− 2s for s > 0.
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Proof. We deduce the first two inequalities from 1 k τ k ∂αk+1 z(τ α + (1 − τ )α )ds, ∂α Q 1 = 0
∂αk Q 2 =
1
0
(1 − τ )τ k ∂αk+2 z(τ α + (1 − τ )α )ds,
and splitting the interval of integration, using the change of variable α → y = τ α + (1 − τ )α for 0 ≤ τ ≤ 1/2, and α → y for 1/2 ≤ τ ≤ 1. Then integrate the p th power over both α and α variables. To prove the third estimate 1 d k Q 1 (z) = τ k ∂αk+1 z(τ α + (1 − τ )α )dτ ≤ C∂αk+1 z∞ ≤ Cz H k+2 . dα k 0 On the other hand, via an integration by parts in τ, we also have 1 τ k ∂αk+1 z(τ α + (1 − τ )α )dτ 0
1 1 d τ k {∂αk z(τ α + (1 − τ )α )}dτ α−α 0 dτ 1 1 1 k k τ =1 = {τ ∂ z(τ α + (1 − τ )α )}| − kτ k−1 ∂αk z(τ α + (1 − τ )α )dτ α τ =0 α − α α − α 0 1 ∂ k z(α) 1 = α − kτ k−1 ∂αk z(τ α + (1 − τ )α )dτ α − α α − α 0 1 ∂ k z(α) − ∂αk z(α ) 1 = α − kτ k−1 {∂αk z(τ α + (1 − τ )α ) − ∂αk z(α )}dτ. α − α α − α 0 =
Now we apply the Morrey estimate in one dimension to get |∂αk z(α) − ∂αk z(α )| + |∂αk z(τ α + (1 − τ )α ) − ∂αk z(α )| 1
≤ Cz H k+s |α − α |1− 2s . This completes the lemma.
Proposition 2.7. Let g ∈ H s and f ∈ H r , then for any s ≥ 0 , [H, g] ( f ) H s+r ≤ C ∂α g H s f H r ,
(2.30)
where r = {−1, 0}. We have for s ≥ 3 , [H, g] ( f ) H s ≤ C ∂α g H s−1 f H s−2 .
(2.31)
On the other hand we have
where i =
{− 21 , 0}.
[H, g] ( f ) H s+i ≤ C ∂α g H s−1+i f H s−2 ,
(2.32)
Furthermore we have s g∂ f H(∂ s f ) 2 ≤ C g H 3 f 2 s−1 . α α H L
(2.33)
In each of the results, C is a constant independent of f and g.
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Proof. To prove (2.30) we have α − α dα L 2 f ] (g) L 2 = f (α ) g(α) − g(α ) cot 2 1 g(α) − g(α ) k−1 α − α = f (α ) ∂ 4π α α − α α − α × cot dα L 2 . 2 , Using the integral Young’s inequality with smooth ζ (α, α ) = α − α cot α−α 2 ∂αk
∂αk−1 [H,
1 4πi
∂αk−1 [H,
f ] (g) L 2
g(α) − g(α ) ζ (α, α ) L 2 ≤ α,α α − α ≤ C f L 2 ∂α g H k−1 . C f L 2 ∂αk−1
If f ∈ H −1 then let f = ∂α ψ and integrate by parts once to get the same bound. To establish (2.31) and (2.32) we prove [H, g]( f ) H s+i ≤ C∂α g H s−1+i f H s−2 for s ≥ 2 and i = 0 or −1/2. We first prove the case i = 0 with the following estimate: [H, g]( f ) H s ≤ C∂α g H s−1 f H s−2 , [H, g]( f ) H s−1 ≤ C∂α g H s−2 f H s−2
(2.34)
for s ≥ 2. The proof of this is straightforward. In fact, we have l k−l α − α dα ∂αk [H, g]( f ) = ∂α g(α) ∂αl f (α ) cot 2 k l α − α dα . − ∂αk−l g(α )∂αl f (α ) cot 2 k In the case l ≤ k − 2, k − l ≥ 1, we can estimate the first terms by ∂α g H k−l−1 f H k−2 . By the usual Sobolev imbedding, the second term can be bounded by ∂αk−l g∂αl f L 2 ≤ C∂α g H k−1 f H k−2 . On the other hand, when l ≥ k − 2, we combine the two terms to get k−l ∂ g(α) − ∂αk−l g(α ) α − α (α − α dα ) cot ∂αl f (α ) α α − α 2 α − α dα . = ∂αk−2 f (α )∂α2−k+l Q 1 (∂αk−l g)(α − α ) cot 2 By (2.29), this is clearly bounded by f H k−2 × ∂α g H 2 for k large. This concludes the proof of (2.34).
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We now use standard interpolation to deduce the case for i = −1/2. Fix f and denote the operator acting on g by M(g) ≡ [H, g] ( f ). We can choose for any h ∈ L 2 such that h L 2 = 1, consider ' F(z) ≡ M(g),
' h , (1 + |ζ |s−1+z )
' denotes the Fourier transform, and z is complex number. Clearly, f is analytic where M ' on the stripe 0 ≤ Re z ≤ 1. Notice that for ' h 1 = (1+|ζ |hs−1+z ) , for z real, h 1 H −s+1−z ≤ 1. We thus have for z = 0 and z = 1, ' h ' F(z) = M(g), = M(g), h 1 ≤ C f H s−2 (1 + |ζ |s−1+z ) by (2.34). Therefore, Hadamard’s three line theorem implies that |F(z)| ≤ C f H s−2 for all 0 ≤ Re z ≤ 1, in particular, for z = 1/2. This conclude the proof since h 1 is an arbitrary normalized test function in H −s+1−1/2 . To prove (2.33), k k g∂α f H(∂α f )dα = − H(g∂αk f )∂αk f dα = − gH(∂αk f )∂αk f dα − ∂αk f [H, g] (∂αk f ) so g∂αk
f H(∂αk
1 f )dα = − [H, g] (∂αk f )∂αk f dα 2 1 = ∂α ([H, g] (∂αk f ))∂αk−1 f dα. 2
Therefore, by (2.30) with s = 2, | ∂α ([H, g] (∂αk f ))∂αk−1 f dα| ≤ ∂αk−1 f L 2 ∂α [H, g] (∂αk f ) L 2 ≤ ∂αk−1 f L 2 ∂αk f H −1 g H 3 . We now bound the operator K[z]( f ) defined in (2.10) in various Sobolev norms.
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Proposition 2.8. Let z(α) ∈ PsK then K[z]( f ) L 2 ≤ C K ∂α2 z L 2 f L 2 , # $2 ∂α K[z]( f ) L 2 ≤ C K 1 + ∂α z H 1 ∂α2 z H 1 f H 1 , # $2 ∂αk K[z]( f ) L 2 ≤ C K 1 + ∂α z H 2 ∂α2 z H k−1 f H 1 , for k ≥ 2. Proof. We first consider the L 2 estimate. We denote G(s) = cot(s) −
1 s
which is analytic in s, and we split K as 1 1 1 − f (α )dα , K1 [z]( f ) ≡ 2πi z(α) ∂α z(α )(α − α ) − z(α ) z(α) − z(α ) 1 α−α 1 K2 [z]( f ) ≡ G − G f (α )dα . 2πi 2 ∂α z(α ) 2
(2.35)
To estimate K1 , first note 1 Q2 1 − =− , z(α) − z(α ) ∂α z(α )(α − α ) ∂α z(α )Q 1 and since z(α) ∈ PsK , we have
1 1 + sup |Q 1 | α,α |∂α z(α )| Therefore, apply Lemma 2.6 to get Q2 K1 [z]( f ) L 2 ≤ ∂ z(α )Q 2 α 1 L
≤ CK .
· f L 2
α,α
≤ C K Q 2 L 2 f L 2 ≤ C K ∂α2 z α,α
L2
f L 2 .
To estimate K2 , we further split it as three parts: ∂α z(α )(α − α ) ∂α z(α )(α − α ) z(α) − z(α ) −G +G G 2 2 2 ∂α z(α ) − 1 α − α α − α + G . (2.36) −G 2 ∂α z(α ) 2 & & Since {∂α z − 1} = ∂α {z − α} = 0 by our periodic boundary condition, for the last term of (2.36) we have %2 % % % ∂α z(α ) − 1 α − α % 2 2 % f (α )dα G % dα ≤ C K ∂α z − 1 L 2 f L 2 % ∂α z(α ) 2 ≤ C K ∂α2 z2L 2 f 2L 2 ,
(2.37)
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from the Poincaré inequality. For the second term of (2.36) α − α ∂α z(α )(α − α ) −G G 2 2 1 (∂α z(α ) − 1)(α − α ) (∂α z(α ) − 1)(α − α ) α−α +s ds. G = 2 2 2 0
(2.38)
Therefore, we estimate the second term by ∂α z(α )(α − α ) α − α G −G L 2 α,α 2 2 ≤ C K ∂α z − 1 L 2 ≤ C K ∂α2 z L 2 . To estimate the first term of (2.36) we again use the mean value theorem to get ∂α z(α )(α − α ) z(α) − z(α ) −G G 2 2 (α − α )2 Q 2 1 ∂α z(α )(α − α ) G = 2 2 0 z(α) − z(α ) ∂α z(α )(α − α ) − ds. (2.39) +s 2 2 Hence, by Lemma 2.6, z(α) − z(α ) ∂α z(α )(α − α ) G −G L 2 ≤ C K Q 2 L 2 ≤ C K ∂α2 z L 2 , α,α α,α 2 2 which completes the estimate of the bound on K[z] in L 2 . We now consider the second and third part of the estimate on K. Again, we split K as in (2.35) and take k derivatives in α. For K1 , we take one α derivative and use the clever trick of Ambrose [2] to get
1 1 1 ∂αk K1 [z]( f ) = − dα f (α )∂αk 2πi z(α) − z(α ) ∂α z(α )(α − a )
1 −∂α z(α) 1 + dα = f (α )∂αk−1 2πi z(α) − z(α ) ∂α z(α )(α − a )2
∂α z(α) 1 1 f (α ) k−1 ∂ − dα = ∂ α 2πi ∂α z(α ) α z(α) − z(α ) (α − a )
f (α ) ∂α z(α) 1 1 k−1 ∂ − dα =− ∂α 2πi ∂α z(α ) α z(α) − z(α ) (α − a )
f (α ) 1 k−1 Q 2 (α , α) ∂ dα . =− ∂α 2πi ∂α z(α ) α Q 1 (α , α) Therefore, ∂αk K1 [z]( f ) L 2
f (α ) ≤ C ∂α ∂ z(α ) α
L
k−1 Q 2 (α , α) ∂ α Q (α , α) 2 1
L 2α,α
.
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By (5.4) the first factor on the right-hand side is bounded by: f (α ) ≤ C f L ∞ ∂ 2 z L 2 + C K f α L 2 ∂α α ∂α z(α ) L 2 ≤ C 1 + ∂α2 z L 2 f H 1 , and the second factor is bounded by ⎡ ⎤ 1 1 k−1 k−1 ⎦ C ⎣ Q ∞ ∂α Q 2 L 2 + Q 2 L ∞ ∂α Q 2 1 L 1 L α,α α,α , - . / 1 1 2 2 k−2 k−1 ≤ C K ∂α z k−1+∂α z ∞ 2 + k Q 1 L ∞ ∂α Q 1 2 Q1 ∞ Q1 ∞ H L L α,α L , / L ≤ C K ∂α2 z k−1 + ∂α2 z ∞ ∂αk−1 Q 1 2 ≤ C K ∂α2 z
H
H k−1
+ ∂α2 z
L
H1
L α,α
∂α z H k−1 .
We thus conclude the proposition for k ≥ 1, for the K1 part. For the second part K2 , we take ∂αk of (2.36). For the third part in (2.36), we have %2 % % k % α − α ∂α z(α ) − 1 % %∂ G f (α )dα % α % dα ∂α z(α ) 2 %2 % % ∂α z(α ) − 1 k % α − α % % ∂α G f (α )dα % dα = % ∂α z(α ) 2 ≤ C K ∂α z − 12L 2 f 2L 2 ≤ C K ∂α2 z2L 2 f 2L 2 . For the middle part in (2.36), we have by the Leibniz rule for derivatives, k ∂ G ∂α z(α )(α − α ) − G α − α 2 α 2 2 L α,α 1 k (∂α z(α )−1)(α−α ) (∂α z(α )−1)(α−α ) α−α × +s ds = ∂ α G 2 2 2 2 0 L
α,α
≤ C K {∂α z − 1 L 2 + ∂α z − 1 L ∞ ) ≤ C K ∂α z − 1 H 1 ≤ C K ∂α2 z 1 . H
Lastly, for the first part in (2.36), we first notice that # $ ∂αk {(α − a )2 Q 2 } = ∂αk−1 ∂α z(α) − ∂α z(α ) 1 ∂α2 z(α + s(α − α ))ds = ∂αk−1 (α − α ) 0
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so that as in the proof of the Lemma 2.6,
k ∂α (α − a )2 Q 2
L 2α,α
≤ C ∂α2 z
H k−1
.
By (2.39), we use both Leibniz rule and the Moser inequality: k ∂ G z(α) − z(α ) − G ∂α z(α )(α − α ) α 2 2 2 L α,α 1
2 ∂α z(α )(α − α ) ≤ C ∂αk α − α Q 2 2 G L α,α 0 2 z(α) − z(α ) ∂α z(α )(α − α ) +s − ds ∞ 2 2 L 1 ∂ z(α )(α − α) α k ∂ +C (α − α )2 Q 2 ∞ G α L 2 0 z(α) − z(α ) ∂α z(α )(α − α ) − ds +s 2 2 2 L α,α ⎫ ⎧ ⎬ ⎨ γ −1 ≤ C K ∂α2 z k−1 + Q 2 L ∞ ∂α z L ∞ [1 + ∂α z H k ] ⎭ ⎩ H γ ≤k
≤ C K ∂α2 z k−1 + ∂α2 z H 1 [1 + ∂α z H k ] , H
where we have used Lemma 2.6. In summary, we have
k ∂α K2 [z]( f ) 2 ≤ C K ∂α2 z k−1 + ∂α2 z 1 1 + ∂α z H k f L 2 , L
H
H
which concludes our proof. 2.3. Second order estimates. We now prove that R1 , R2 , (T − W, t), F are of high order in the perturbation. We also include an estimate on ∂t γ . For each estimate we derive large energy bounds, except for the bounds on ∂t γ (see (2.59)). Lemma 2.9. Recall the definition of R1 given by (2.15). For k ≥ 2, R1 H k ≤ C K (1 + E 2 )k+3 E k . Additional estimates for k = {0, 1} are included in the proof. Proof. We start with some simple bounds on which our argument relies. 1. Since ∂α z = −1 eiθ then 2π + l iθ l iθ e − 1 = eiθ − 1 + e , ∂α z − 1 = 2π 2π % % %%& θ % % and since %eiθ − 1% = % 0 ei x d x % ≤ |θ |, then ∂α z − 1 L 2 ≤ C l + θ L 2 .
(2.40)
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From (5.2) with k ≥ 1, k iθ ∂α e 2 ≤ C L
i x {γ } γ −1 k θ e max ◦ θ θ 2 ∂ ∞ α L ∞ 1≤γ ≤k−1 L L k ≤ C 1 + θ k−1 L ∞ ∂α θ 2 , L
so for k ≥ 2,
k ∂α ∂α z
L2
k−1 k ≤ C 1 + θ H 1 ∂α θ
L2
.
(2.41) (2.42)
We now have {γ } k 1 γ −1 k ∂ ≤ C max x −1 ◦ ∂α z ∂α z L ∞ ∂α ∂α z 2 α ∂ z 2 1≤γ ≤k L ∞ α L L −k−1 k−1 k ≤ C ∂α z−2 L ∞ + ∂α z L ∞ ∂α z L ∞ ∂α ∂α z L 2 ≤ C K ∂αk ∂α z 2 , L
where C K depends on K . Finally, if k ≥ 1, then since z ∈ PsK and k ≥ 1, k−1 ∂α z − 1 H k ≤ ∂α2 z k−1 ≤ C K 1 + θ H 1 θ H k , H 1 k−1 θ H k , ∂ z − 1 k ≤ ∂αa z H k−1 ≤ C K 1 + θ H 1 α H and ∂α z k ≤ C K + ∂ 2 z k−1 . H
α
(2.43)
H
2. We consider the L 2 norm of the first part of R1 in (2.15) then by Step 1 and Proposition 2.8, ∂α γ (γ0 + γ ) ∂α2 z R1 L 2 ≤ ∂ − zK[z] 2 α ∂α z (∂α z)2 L ∂ γ (γ + γ ) ∂ 2 z 0 2 α α − ≤ C K ∂α z 2 2 2 L ∂α z (∂α z) L 2 ≤ C K ∂α z 2 ∂α γ L 2 + γ + γ0 L ∞ ∂α2 z 2 . L
L
On the other hand applying the product estimate (5.4) repeatedly, we first estimate ∂α γ (γ0 + γ ) ∂α2 z ≤ ∂α γ H 1 + C K ∂α γ L 2 − ∂α2 z 2 1 ∂ z L (∂α z)2 α H 2 +C K γ + γ0 L ∞ ∂α z 1 H +C K γ0 + γ L ∞ ∂α z L ∞ ∂α2 z 2 . L
Therefore, the H 1 norm of the first part of R1 is 2 ∂α zK[z] ∂α γ − (γ0 + γ ) ∂α z 1 2 ∂α z (∂α z) H ∂ γ (γ + γ ) ∂ 2 z 2 α 0 2 α − ≤ C K 1 + ∂α z H 1 ∂α z 1 H ∂ z (∂ z)2 ≤ C K (1 + ∂α z H 1 )
4
α ∂α z2H 1 (∂α γ H 1
α
H1 + γ0 + γ L ∞ ∂α2 z H 1 ).
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We again use the product rule to estimate the last term as 2 ∂α zK[z] ∂α γ − (γ0 + γ ) ∂α z k 2 ∂α z (∂α z) H ∂α γ (γ0 + γ ) ∂α2 z 2 2 − ≤ C K 1 + ∂α z H 2 ∂α z k−1 1 ∂ z H (∂α z)2 α H 4 2 2 ∂ ∂ γ ≤ C K 1 + α z H 2 ∂α z k−1 αγ H1 + 0 + γ L ∞ ∂α z H
H1
.
3. Next we consider the second part of R1 in (2.15). First note that k 1 ∂ α (∂ z)2 α
so that
L2
∂α 1 (∂ z)2 α
≤ C K ∂αk−1 ∂α2 z
Hk
≤ C K ∂α2 z
L2
,
(2.44)
Hk
for any k ≥ 0. Therefore, by (2.30), the L 2 norm of the second part is bounded by
∂α z 1 (γ0 + γ ) ∂α2 z ∂ H, γ − α 2 2i (∂α z)2 ∂α z L 2 z 1 + γ ∂ ) (γ 0 α ∂α γ − ≤ CK 2 H, (∂ z)2 ∂α z α L 2 1 + γ ∂ z ) (γ 0 α . ≤ CK 2 ∂α (∂ z)2 2 ∂α γ − ∂ z α α L L Again by using product rule (5.4), the last factor is bounded by
∂α γ L 2 + γ + γ0 L ∞ ∂α2 z
L
. 2
For the H 1 estimates, we have from (2.30),
∂α z 1 (γ0 + γ ) ∂α2 z ∂α γ − 2i H, (∂ z)2 1 ∂α z α H
1 (γ0 + γ ) ∂α2 z H, ∂ ≤ C K 1 + ∂α2 z 1 γ − α 1 H (∂α z)2 ∂α z H 2z 1 + γ ∂ ) (γ 0 2 α ≤ C K 1 + ∂α z 1 1. ∂α (∂ z)2 1 ∂α γ − H ∂α z α H H By the product rule (5.4), the last factor above is bounded by 2 ∂α z
H
2 ∂ γ 1 + γ0 + γ L ∞ ∂α z α H 1
H1
.
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Finally, by (2.30) again,
k ∂α z 1 (γ0 + γ ) ∂α2 z ∂ ∂α γ − k α 2i H, (∂ z)2 ∂α z α H 2 1 (γ0 + γ ) ∂α z ∂α γ − ≤ CK k H, (∂ z)2 ∂α z α H
1 + γ ) ∂α2 z (γ 0 2 + C ∂α z k−1 H, ∂α γ − 1 H (∂α z)2 ∂α z H 2 1 + γ ∂ z ) (γ 0 α ≤ CK k−2 ∂α (∂ z)2 k−1 ∂α γ − ∂ z α α H H 1 + γ ) ∂α2 z (γ 0 2 ∂ ∂ + C K ∂α z k−1 γ − α (∂ z)2 1 α H ∂ z α
α
H
H1
.
We apply (5.3) to majorize the last factor by C K 1 + ∂α z H 2 ∂α2 z H k−1 ∂α γ H k−2 + γ0 + γ L ∞ ∂α2 z H k−2 . 4. We apply estimates (2.43) to the results of the second and third steps to conclude our lemma. In the course of the proof we have also established 1
R1 L 2 ≤ C K (1 + E 1 ) 2 E 1 , 7 2
R1 H 1 ≤ C K (1 + E 1 ) E 2 ,
(2.45) (2.46)
which will be used below. Lemma 2.10. For k ≥ 2, T − W, t H k ≤ C K (1 + E 2 )k+3 E k .
(2.47)
An estimate for k = 1 also holds and is included in the proof. Proof. Since W = 2i1 H γ∂0α+γz + K[z](γ0 + γ ) then by Proposition 2.8, W L 2
∂α z − 1 γ + K[z](γ0 + γ ) L 2 ≤C +C ∂α z L 2 ∂α z L 2 1
1
1
≤ C K E 02 + C K E 02 + C K E 12 γ0 + γ L 2 1
1
≤ C K (1 + E 0 ) 2 E 12 , where we used H(1) = 0. Next, by Proposition 2.8, 1 + ∂α γ + ∂α K[z](γ0 + γ ) L 2 ∂α W L 2 ≤ C − 1 1 ∂ z ∂α z L 2 α H 1 1 3 ≤ C K E 12 + C K γ H 1 +C K γ L ∞ ∂α2 z 2 + C K (1 + E 1 ) 2 E 22 γ0 + γ H 1 L
1 2
≤ C K (1 + E 2 ) E 2 . 2
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On the other hand for k ≥ 2 since ∂αk ∂α1 z k ∂α W
L2
L2
≤ ∂α1 z − 1
H k+1
then by (2.42)
k γ0 + γ k + ∂ ≤ K[z](γ + γ ) ∂ 2 0 α α L ∂α z L 2 1 k k 1 ≤C ∂α γ 2 + γ0 + γ L ∞ ∂α L ∂α z L ∞ ∂α z L 2 +C K (1 + E 2 ) ≤ C K (1 + E 2 )
k+3 2
k+4 2
1
E k2 γ0 + γ H 1 1
2 E k+1 .
We choose the exponent k+4 2 to satisfy the estimate for all k ≥ 0. We use the special structure of T − W, t to proceed. In particular, ∂α (T − W, t) = Tα − Wα , t − W, tα
lt − −π H (γ ∂α θ ) + R1 , t = W, n∂α θ + 2π − [ W, t t, tα + W, n n, tα ] lt + π H (γ ∂α θ ) − R1 , t, = 2π
(2.48)
where we used t, tα = 0 and n, tα = ∂α θ . Since % % 2π % dl % % %≤ ∂α θU dα, % dt % 0 then by (2.12), (2.7), (2.43), Propositions 2.7 and 2.8, % % % dl % % % ≤ C K E1. % dt %
(2.49)
First T − W, t L 2 ≤ C ∂α θ L ∞ W L 2 + C W L 2 1
1
1
1
1
≤ C K E 22 (1 + E 0 ) 2 E 12 + (1 + E 0 ) 2 E 12 1
≤ C K (1 + E 1 ) E 22 . Next, 1
∂α (T − W, t) L 2 ≤ C K (1 + E 0 ) 2 E 1 + γ ∂α θ L 2 + R1 , t L 2 1
≤ C K (1 + E 0 ) 2 E 1 + γ 1
1
H2 1
θ H 1 + R1 L 2 1
1
≤ C K (1 + E 0 ) 2 E 1 + C K E 12 E 12 + C K (1 + E 1 ) 2 E 1 1
≤ C K (1 + E 1 ) 2 E 1 . For higher Sobolev bounds, since t = eiθ then (2.41) implies 1 k−1 k ∂α t 2 ≤ C (1 + E 1 ) 2 E k2 . L
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We have from (2.48) that k ∂α (T − W, t)
L2
≤ C ∂αk−1 (γ ∂α θ ) 2 + ∂αk−1 R1 , t 2 L L k−1 ≤ C ∂α γ 2 ∂α θ L ∞ + C γ L ∞ ∂αk θ 2 L L k−1 k−1 +C ∂α R1 2 + C ∂α t 2 R1 L ∞ L
L
≤ C γ H k θ H 2 + C γ 1 θ H k H2 k−1 θ H k R1 H 1 +C R1 H k + C 1 + θ H 1 ≤ C K (1 + E 2 )k+3 E k .
(2.50)
We have also established the H 1 bound T − W, t H 1 ≤ C K (1 + E 1 ) E 2 .
(2.51)
Lemma 2.11. Let k ≥ 3 and (θ, γ , l) be a solution to (2.14), then R2 F
15
1
H k− 2
1
H k− 2
≤ C K (1 + E 3 )k+ 2 E k , ≤ C K (1 + E 3 )
k+ 15 2
Ek .
(2.52) (2.53)
Proof. 1. In order to estimate R2 we first bound ∂t θ and ∂t z. From (2.14), Lemma 2.9, and Lemma 2.10 we have ∂t θ H 1 ≤ C K γ H 2 + C K (T − W, t) ∂α θ + R1 , n H 1 1 ≤ C K E 32 + (1 + E 1 )4 E 2 , and for k ≥ 1, ∂t θ
H
k− 21
≤
2 ∂α γ k− 1 + (T − W, t) ∂α θ + R1 , n H k H 2 2 1
2 ≤ C K E k+1 + C K (1 + E 2 )k+3 E k+1
1/2
≤ C K (1 + E 2 )k+3 E k+1 . iθ
Since ∂t ∂α z = ∂t e =
t +i∂t θ iθ e
∂t ∂α z H 1
, then
% % % dl % % % ≤ C K % % + ∂t θ H 1 eiθ 1 H dt 1 1 ≤ C K (1 + E 0 ) 2 E 1 + (1 + E 1 )4 E 3 (1 + E 1 ) 2 9
1/2
≤ C K (1 + E 1 ) 2 E 3 ,
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where we have used (2.41). Finally, from (5.4) and (2.41) dl iθ 1 iθ ∂t ∂α z k− 1 ≤ e + ∂t θ e 1 1 H 2 dt H k− 2 H k− 2 % % % dl % ≤ %% %% eiθ k + C K ∂t θ L ∞ eiθ k− 1 + ∂t θ k− 1 eiθ ∞ H 2 H H 2 L dt 1/2
≤ C K (1 + E 2 )k+3 E k+1 .
(2.54)
2. We are now ready to estimate R2 term by term. The first term in R2 is bounded by K[z] ∂t z(α) − ∂t z(α ) ∂α γ0 + γ (α ) k− 1 ∂α z(α ) ∂α z(α ) H 2 γ + γ 1 0 ≤C k− 1 ∂t zK[z] ∂ z ∂α ∂α z α H 2 ∂ γ z + γ t 0 +C (2.55) k− 1 . K[z] ∂ z ∂α ∂α z α H 2 To estimate the first term in (2.55), we apply (2.43) and (5.4) and get 1 γ0 + γ K[z] ∂ C ∂t z L ∞ α k− 1 ∂α z ∂α z H 2 1 γ + γ 0 K[z] ∂α + C ∂t z k− 1 ∞ 2 H ∂α z ∂α z L 1 1 1 11 k+3 γ0 + γ ∂ ≤ C K (1 + E 3 ) 2 E 32 (1 + E 2 ) 2 E k2 ∂ z α ∂α z H 1 α 1 1 3 γ0 + γ ∂ +C K (1 + E 2 )k+2 E k (1 + E 1 ) 2 E 22 ∂ z α ∂ z 1 α
≤ C K (1 + E 3 )
k+7
α
H
3 2
Ek ,
where k ≥ 3, and via the product rule (5.4), 1 1 γ0 + γ ∂ ≤ C ∂ z α ∂ z 1 γ0 +γ H 21 ∂α z H 1 α α H 3 2
∂α 1 ∂ z α
1 +∂α γ H 1 ∂ z 1 α H1 H
1 2
≤ C K (1 + E 1 ) E 3 . On the other hand, the second term of (2.55) is estimated by Lemma 2.8, (2.43), and (2.54): 1 ∂ z k+1 γ0 + γ t C K (1 + E 1 ) 2 E k2 ∂ ∂ z α ∂α z H 1 α 1 1 k+1 γ0 + γ ∂ ≤ C K (1 + E 1 ) 2 E k2 ∂t z H 1 ∂ z α ∂α z H 1 α ≤ C K (1 + E 1 )
k+1 2
≤ C K (1 + E 3 )
k+15 2
1
11
1
3
1
E k2 (1 + E 3 ) 2 E 32 (1 + E 1 ) 2 E 32 3
E k2 .
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This implies the first term of R2 is bounded by C K (1 + E 3 )k+7 E k . For the second term in R2 we have (γ0 + γ ) K[z] ∂α ∂t z (γ0 + γ ) k− 1 ≤ K[z] ∂α ∂t z ∂ z k ∂α z α H 2 H 1 k+1 γ0 + γ 2 , ≤ C K (1 + E 1 ) 2 E k ∂t ∂α z ∂α z H 1 where ∂t ∂α z γ0 + γ ∂ z α
H1
1 − 1 ≤ C ∂t ∂α z H 1 1 + γ H 1 1 + 1 ∂α z H 13
1
≤ C K (1 + E 3 ) 2 E 32 , k+14
so the second term is bounded by C K (1 + E 3 ) 2 E k . The bound on the third term of R2 follows from (2.30): [H, ∂t z] 1 ∂α (γ0 + γ ) k− 1 ∂α z ∂α z 2 H 1 1 ≤ C K ∂t ∂α z k− 3 ∂α (γ0 + γ ) 1 H 2 ∂α z ∂α z H 3
1
≤ C K (1 + E 2 )k+2 E k (1 + E 1 ) 2 E 32 , 7
3
which is bounded by C K (1 + E 2 )k+ 2 E k2 . The fourth term in R2 satisfies 1 1 t 1 |l | γ0 + γ H 1 ∂ H, + γ ≤ C ) (γ 0 K t α k− 1 2i ∂α z ∂α z H k− 23 H 2 by the commutator estimate (2.30) and since ∂α ∂α1 z k− 3 ≤ ∂α ∂α1 z k−1 , we deduce k+1
H
H
2
that this term is bounded by C K (1 + E 3 ) 2 E k . Finally, the last term in R2 is bounded by
1 − H, 1 (∂t θ (γ0 + γ )) ≤ C K ∂α 1 ∂t θ (γ0 + γ ) H 1 2 ∂α z ∂α z H k−1 Hk 1 ∂ ≤ CK α ∂ z k−1 ∂t θ H 1 γ0 + γ H 1 α H ≤ C K (1 + E 3 )
k+12 2
Ek .
Combining the five terms yields (2.52). 4. We now estimate F term by term. The first term in F is bounded by % % 1 d % dl % − % % 1 + γ ≤ C K (1 + E 1 ) E k . + γ ≤ C ) (γ 1 0 K dt % dt % k− 1 H k− 2 H 2
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For the second term in F we use (5.4), (2.2) and Lemma 2.9, (γ0 + γ ) R1 , t k− 1 2 H ≤ C K 1 + γ L ∞ t L ∞ R1 k− 1 + C K 1 + γ L ∞ t k− 1 R1 L ∞ 2 H H 2 + C K 1 + γ k− 1 t L ∞ R1 L ∞ H 2 ≤ C K 1 + γ 1 R1 H k + C K 1 + γ 1 t H k R1 H 1 H2 H2 + C K 1 + γ k− 1 R1 H 1 , H
2
7
which is bounded by C K (1 + E 3 )k+ 2 E k thanks to Lemma 2.9 and (2.41). To estimate the third term in F, we notice from the product rule (5.4), 2A − R2 , t t t R R ∞ ∞ ≤ C + 1 k K 2 2 L L H k− 1 H k− 2 H 2 15
≤ C K (1 + E 3 )k+ 2 E k . The fourth term in F is bounded by (T − W, t) 2 A R1 , t
≤ C T − W, t L ∞ R1
1
H k− 2
1
H k− 2
t L ∞
+ C T − W, t L ∞ R1 L ∞ t + C T − W, t
1
H k− 2
1
H k− 2
R1 L ∞ t L ∞
≤ C K T − W, t H 1 R1 H k + C K T − W, t H 1 R1 H 1 t H k + C K T − W, t H k R1 H 1 , 15
which is further bounded by C K (1 + E 3 )k+ 2 E k thanks to the product rule (5.4), (2.40), (2.10), and (2.41). For the fifth and sixth terms in F we note H, γ0 ( f ) = 0; therefore, by (2.32), 1 2 − H, (γ0 + γ ) ((γ0 + γ ) ∂α θ ) 2 k− 1 ≤ C K H, γ ((γ0 + γ ) ∂α θ ) H k− 21 H 2 ≤ C K γ k− 1 (γ0 + γ ) ∂α θ H k−2 H
2 1 2
≤ C K (1 + E 2 ) E k , and
A 22 H, (γ0 + γ ) (H (∂α γ ))
1 H k− 2
≤ C K H, γ (H (∂α γ )) ≤ C K γ
1
H k− 2
1
H k− 2
∂α γ H k−2 ≤ C K E k .
The seventh term in F is bounded by (2.32): AH ((γ0 + γ ) { R1 , n})
1
H k− 2
≤ C K ∂α γ
3
H k− 2
R1 , n H k−2
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and from (5.4). Then by (2.40) and (2.2), AH ((γ0 + γ ) { R1 , n}) k− 1 H 2 γ R ≤ CK 1 1 H k−2 + R1 H 1 n H k−2 H k− 2 1 7 k−3 k+1 2 2 2 ≤ C K γ k− 1 (1 + E 2 ) E k−2 + (1 + E 1 ) E 2 (1 + E 1 ) E k−2 H
2
3
≤ C K (1 + E 2 )k+1 E k2 , and by (2.47), (2.32), the last term in F is bounded by: A [H, (T − W, t)] (∂α θ )
1
H k− 2
≤ C T − W, t
1
∂α θ H k−2
1 2
and CJ E k
H k− 2 3 2
≤ C K (1 + E 2 )k+2 E k . Summing over the eight terms yields bound (2.53). 1
Lemma 2.12. Suppose (θ, γ , l) is the solution to (2.14) with E k2 ≤ where CJ is defined below. Then for all k ≥ 3, 1/2
∂t γ H k−2 ≤ C K E k and
J [z](∂t γ ) H k ≤ C K E k .
1/2
≤ 21 ,
(2.56) (2.57)
Proof. Recall (2.13), the definition of J [z]. Using the commutator estimates (2.32) and Proposition 2.8, as well as the notation for the tangential part of the interface velocity, (2.2), we have
1 f H k + K[z]( f ) H k J [z] f H k ≤ Ct H k H, ∂α z 1 H k−1 f H 1 ≤ C K ∂α ∂α z 1 − 1 H k f H k−2 + C K (1 + ∂α z − 1 H k ) ∂α z ≤ CJ 1 + ∂α z2H 2 ∂α2 z H k−1 f H k−2 , (2.58) 1
since E k2 ≤ 21 . Hence from (2.43), 1/2
J [z]∂t γ H k ≤ CJ E k ∂t γ H k−2 . We now estimate ∂t γ H k−2 . Take k − 2 derivatives for the second equation in (2.14), 2 2A k I+ J [z] ∂t γ = ∂α θ + ∂αk−2 H (γ0 + γ )2 ∂α θ ∂αk−2 W 2
− ∂αk−2 A2 (γ0 + γ ) + (T − W, t) ∂α γ −
2 Ag k−2 ∂ sin(θ ) + ∂αk−2 F. α
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From (2.58), we have 13
1/2
1/2
∂t γ H k−2 − CJ E k ∂t γ H k−2 ≤ C K (1 + E 3 )k+ 2 E k . 1/2
Setting CJ E k
≤ 21 , then 13
1/2
∂t γ H k−2 ≤ C K (1 + E 3 )k+ 2 E k ≤ C K E k , since E k ≤ 21 . Finally, we obtain 1/2
J [z](∂t γ ) H k ≤ CJ E k ∂t γ H k−2 ≤ C K E k , which concludes our lemma. Remark. If we ignore the smallness assumption in Lemma 2.12, we can find an esti−1 (H k ,H k ) ≤ C K (1 + E k )10k+2 , which follows from mate of the form I + 2A J [z] Lemma 2.7, Lemma 2.8, and the L(L 2 , L 2 ) bound of [5]. This estimate yields large data bounds on ∂t γ and J [z](∂t γ ). Using Lemma 2.9, Lemma 2.10, Lemma 2.11, along with the proof of Proposition 2.2 yields an improved a priori estimate d E k ≤ C K (1 + E k )γ E k dt
(2.59)
with γ = 12k − 11 for k ≥ 3. 2.4. Energy inequality 2.4.1. Linear energy inequality. To illustrate the main ingredient of our energy estimate, we first consider the linear equations (2.18). To establish an energy inequality we differentiate the first energy term and use the linear evolution equation for ∂t θ = 21 H(∂α γ ) to get d 1 %% k %%2 1 ∂αk+1 θ ∂αk H(γ ) dα. %∂α θ % dα = − dt 2 2 Using the evolution equation for ∂t γ in (2.18), we can solve for W1 ∂αk+1 θ and get W γ02 1 k k−1 H(∂α θ ) + W Aγ0 ∂α γ + 2 AW gθ =− W ∂t γ − H(∂α γ )∂α 2 2 W γ02 W =− H(∂αk γ )∂t ∂αk−1 γ dα + H(∂αk γ )H(∂αk θ ) dα 2 4 W Aγ0 k k + H(∂α γ )∂α γ dα + W Ag H(∂αk−1 γ )∂αk θ dα. 2 The first of these integrals is equal to the negative of the time-derivative of the second energy integral W d − H(∂αk γ )∂αk−1 γ dα. 4 dt
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The third of these last integrals vanishes due to the antisymmetry of the Hilbert transform H. We again replace ∂αk+1 θ with the linear equation for ∂t γ . This lets us rewrite the second integral as W γ02 4
∂αk γ ∂αk θ dα
=−
W γ02 4
∂αk−1 γ ∂αk+1 θ dα W γ02 W γ02 k−1 k−1 =− H(∂α θ ) + W Aγ0 ∂α γ + 2 AW gθ dα W ∂t γ − ∂α γ ∂α 4 2 % % W 2 γ02 d % k−1 %2 =− %∂α γ % dα 8 dt 2 AW 2 γ02 g W γ04 k−1 k + ∂α γ H(∂α θ )dα − ∂αk−1 γ ∂αk−1 θ. 8 2
Let the zeroth order energy be E 0L
1 = 2
|θ |2 + |γ |2 + l 2 dα,
and define the k th order energy by (k ≥ 1) E kL
1 = 2
% % % % % k− 21 % W 2 γ02 %% k−1 %%2 W % k %2 % % %∂α θ % dα + %∂α γ % dα; %∂α γ % dα + 8 4
then collecting terms above, we obtain d E kL = −W Ag H(∂αk−1 γ )∂αk θ dα dt W 2 γ04 AW 2 γ02 g + ∂αk−1 γ H(∂αk θ )dα − ∂αk−1 γ ∂αk−1 θ dα. 8 2 Since the Hilbert transform H maps from L 2 to L 2 , the right-hand side is clearly bounded by Cγ H k−1 θ H k . Since E k contains k − 21 derivatives of γ , by (5.1) we have for any ε > 0, 1/2
γ H k−1 ≤ εE k
1/2
+ Cε E 0 .
We therefore deduce d E kL ≤ εE kL + Cε E 0 , dt where Cε depends on A,g, k, and γ0 . The last inequality follows from recursion.
(2.60)
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2.4.2. Nonlinear energy inequality. We expect to establish an energy inequality similar to (2.60) for the full nonlinear system (2.14). In particular, we use our improved bounds on R1 , R2 , T − W, t, F, and J [z] to establish Proposition 2.2. Proof of Proposition 2.2. Recall that our k th energy is E k = E 0 +
6k j=1
E˙ j , where
E˙ k = E k,1 + E k,2 + E k,3 % %2 1 %% k %%2 W 2 W 2 % % k k−1 = H ∂α γ ∂α γ + (γ0 + γ )2 %∂αk−1 γ % %∂α θ % + 2 4 8 and
E0 =
l 2 + γ 2 + θ 2 dα.
We now turn to study the nonlinear energy inequality. For a solution (θ, γ , l) we obtain 2 W 2 %% k−1 %%2 d ˙ 1 d E k,2 + E k,3 + Ek = %∂α γ % ∂t (γ + γ0 )2 dα dt dt 8 4 W 2 − ∂αk−1 γ (γ + γ0 )2 ∂αk−1 H (γ + γ0 )2 ∂α θ dα 8 AW 3 g − ∂αk−1 γ ∂αk−1 sin θ dα − W Ag H(∂αk−1 γ )∂αk θ dα 2 2 W2 − ∂αk−1 γ (γ + γ0 )2 ∂αk−1 ({A2 (γ0 + γ ) + (T − W, t)}∂α γ ) dα 4 W − H ∂αk γ ∂αk−1 ({A2 (γ0 + γ ) + (T − W, t)}∂α γ ) dα 2 3 W − ∂αk−1 γ ∂αk (γ + γ0 )2 ∂α θ − (γ + γ0 )2 ∂αk+1 θ dα 4 k k + ∂α θ ∂α [(T − W, t) ∂α θ ] dα + ∂αk θ ∂αk { R1 , n} dα W 2 W 2 k−1 2 k−1 − ∂α γ (γ + γ0 ) ∂α F dα − H ∂αk−1 γ ∂αk F dα 4 2 2 2 W − ∂αk−1 γ (γ + γ0 )2 ∂αk−1 (J [z]∂t γ ) dα 4 W − H(∂αk−1 γ )∂αk (J [z]∂t γ ) dα 2 = I + I I + I I I + I V + V + V I + V I I + V I I I + I X + X + X I + X I I. We claim, as in the linear energy flow, that each term I − X I I is either of the form 3
C E j , for j < k or C E k2 . In either case we can satisfy (2.27). We consider each of the 1
terms I − X I I individually. Since E k2 ≤
1 2
1
for t ≤ T1 then from (2.49) and E k2 we have
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3 3 % 1 d % dl 2 2 % % dt = C K | dt | ≤ C E k and |I | ≤ C 1 E k . To bound I I we use (2.56) so |I I | ≤ C 2 E k . From (5.4) we bound I I I by 3 1 + γ 1 θ H k + γ H k−1 ∂α θ L ∞ C γ H k−1 1 + γ 1
H2
H2
3 2
≤ C γ H k−1 θ H k + C E k . 3
Moreover, by (5.1) the first term is bounded by εE k +Cε E 0 . Thus |I I I | ≤ εE k +C3 E k2 + Cε3 E 0 . For the two terms of I V we use the (5.2) and (5.1) to easily get |I V | ≤ εE k + Cε4 E 0 . The bound on V is more subtle. We split ∂αk−1 ({A2 (γ0 + γ ) + (T − W, t)}∂α γ ) into {A2 (γ0 + γ ) + (T − W, t)}∂αk γ k−2 j ∂ k−1− j {A2 (γ0 + γ ) + (T − W, t)}∂αj+1 γ . + k−1 α
(2.61)
j=0
We first consider the second line of (2.61). From (2.10) there is an L 2 bound C(1 + 1
3
E k2 ) γ k−1 , and using (5.1) yields εE k + C51 E k2 + Cε51 E 0 . For the first line of (2.61) we integrate by parts and get 2 W 2 − ∂αk−1 γ (γ + γ0 )2 {A2 (γ0 + γ ) + (T − W, t)}∂αk γ 4 2 W 2 = ∂α ((γ + γ0 )2 {A2 (γ0 + γ ) + (T − W, t)})|∂αk−1 γ |2 8 ≤ εE k + Cε52 E 0 . 3/2
We thus conclude that |V | ≤ 2εE k +C5 E k +Cε5 E 0 , with C5 = C51 and Cε5 = Cε51 +Cε52 . The argument for V I is similar to V . We first split the integral: AW − H ∂αk γ (∂αk γ ){A2 (γ0 + γ ) + (T − W, t)} 2 k−2 AW j − (∂αj+1 γ )∂αk− j {A2 (γ0 + γ ) + (T − W, t)}. H(∂αk γ ) k−1 2 j=0
For the first term we use (2.33) and (5.1): AW − H ∂αk γ (∂αk γ ){A2 (γ0 + γ ) + (T − W, t)} 2 ≤ C(1 + γ H 3 + T − W, t H 3 ) γ 2H k−1 ≤ εE k + Cε61 E 0 . For the second term of the splitting we need to integrate by parts again.Before we & & & ˆ ˆ ˆ ˆ proceed, we note by Plancherel, H(∂ f ) f gdα = |ξ | f f gdξ = |ξ | f f gˆ dξ .
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%& % %& % & 1 1 1 1 % 1 % Since |ξ | 2 ≤ |ξ − η| 2 + |η| 2 , then % H(∂α f ) f gdα % ≤ |ξ | 2 % fˆ(ξ )% |ξ − η| 2 | % % & 1 % 1 %& fˆ(ξ − η)||g(η)|dηdξ ˆ + |ξ | 2 % fˆ(ξ )% | fˆ(ξ − η)| |η| 2 |g(η)|dηdξ ˆ ≤ f 2 1 H2 %& % g L 1 + f 1 f L 1 g 1 . Therefore, % H(∂α f ) f gdα % ≤ f 2 1 g H 1 , which H2
H2
H2
we use in the third step below: AW − H(∂αk γ )(∂αj+1 γ )∂αk− j {A2 (γ0 + γ ) + (T − W, t)} 2 AW = H(∂αk−1 γ )∂α (∂αj+1 γ )∂αk− j {A2 (γ0 + γ ) + (T − W, t)} 2 AW = H(∂αk−1 γ )(∂αk γ )∂α2 {A2 (γ0 + γ ) + (T − W, t)} 2 AW + (∂αj+1 γ )∂αk− j {A2 (γ0 + γ ) + (T − W, t)} H(∂αk−1 γ ) 2 k− j≥2
≤ C(γ H 3 + T − W, t H 3 )E k + C γ H k−1 γ H k−1 ≤ εE k + Cε62 E 0 for j + 1 ≤ k − 1. Thus |V I | ≤ 2εE k + Cε6 E 0 , where Cε6 = Cε61 + Cε62 . For the V I I term, we use (5.4) to simplify it as ∂αk−1 γ
k−1 j j=0
k
∂αj+1 θ ∂αk− j (γ + γ0 )2
=
∂αk−1 γ ∂α θ ∂αk (γ + γ0 )2 +
∂αk−1 γ
k−1 j j+1 k− j ∂ θ ∂α (γ + γ0 )2 k α j=1
=2
(γ
+ γ0 )∂α θ ∂αk−1 γ ∂αk γ
+
∂αk−1 γ ∂α θ
k−1 j j k− j ∂ γ∂ γ k α α j=1
+
∂αk−1 γ
k−1 j=1
j j+1 k− j ∂ θ ∂α (γ + γ0 )2 . k α
The first term is bounded by an integration by parts and (5.1), and the second and third terms are controlled again by (5.1). We find |V I I | ≤ εE k + Cε7 E 0 . For the V I I I term, ∂αk θ ∂αk ((T − W, t) ∂α θ ) dα = (T − W, t) ∂αk θ ∂αk+1 θ dα + ∂αk θ ∂αk ((T − W, t) ∂α θ ) − (T − W, t) ∂αk+1 θ dα. Upon integration by parts, the first term is bounded by 2 3/2 C ∂α (T − W, t) L ∞ ∂αk θ 2 ≤ C (1 + E 1 ) E k . L
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The second term is bounded by C ∂αk θ 2 ∂α (T − W, t) L ∞ ∂αk θ 2 + ∂αk (T − W, t) 2 ∂α θ L ∞ L L L 1 1 1 1 1 k+8 ≤ C E k2 (1 + E 1 ) E 22 E k2 + (1 + E 3 ) 2 E k2 E 12 ≤ C (1 + E 3 )
k+8 2
3/2
Ek .
3
Thus |V I I I | ≤ C8 E k2 . The term I X is bounded using (2.40) by
1 1 3 k−1 7 k+3 2 2 2 2 C E k (1 + E 2 ) E k + (1 + E 1 ) E k (1 + E 1 ) E 2 ≤ C E k2 , 3
thus |I X | ≤ C9 E k2 . For the X term, (2.53), we have the bound C 1 + γ
2 H
1 2
γ H k−1 F H k−1 ≤ C γ H k−1 E k 3
which lets us conclude |X | ≤ C10 E k2 . A similar calculation holds for the term X I . We note from (2.53), W − H ∂αk−1 γ ∂αk F ≤ C K γ k− 1 F k− 1 H 2 H 2 2 1
≤ C K E k2 (1 + E 3 )
k+20 2
k+20 2
3 2
≤ C K (1 + E 3 )
Ek
Ek ,
3
hence |X I | ≤ C11 E k2 . To estimate the last term X I I we use the bound (2.57) to control J [z](∂t γ ). The first term is bounded by C(1 + γ 1 )2 γ H k−1 J [z](∂t γ ) H k−1 ≤ H2
3 2
3
C E k . The second term has a similar bound, so |X I I | ≤ C12 E k2 . Summing over all terms, 3 ε 2 |I | + · · · + |X I I | ≤ 6εE k + Ck E k + Ck E 0 k
k
and redefine 6ε → ε. This yields Proposition 2.2. 3. Hele-Shaw Flow In this section, we consider Hele-Shaw flow in a vertical cell infinite in both directions with two fluids of differing but uniform viscosities and densities. As with the vortex sheet problem, we can describe the motion of the interface separating the fluids. Local well-posedness of Hele-Shaw flows with surface tension was established in [11] while local well-posedness for Hele-Shaw without surface tension was studied in [3] and [25]. The motion of the interface is driven by a pressure balance across the interface. As before, the interface velocity is given by The Birkhoff-Rott integral, but now the vortex
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sheet strength is given explicitly. For details, we refer the reader to the work of Hou, Lowengrub, and Shelly [16] (see also [25, 3]). The equations of motion are 1 γ (α ) dα , W= 2πi z(α) − z(α ) 1 γ = −2 A∂α s (W, t) + ∂α κ − R∂α y, W where W is the Weber number, R = g (ρ1 − ρ2 ) is the density stratification, and A is the Atwood number. As from before we employ the small-scale decomposition ∂α z = ∂α seiθ . If we reparametrize the tangential velocity on the interface T to be T (t, α) = ∂α θU, then the local arclength along the interface, ∂α s, is constant at each time t. We then get the system ∂t θ = Wα , n + (T − W, t) ∂α θ, 1 2 ∂ θ − R sin θ − 2 A W, t. γ = W α We are concerned with the stability of the trivial steady state θ ≡ 0 and γ ≡ 0. In particular, we can solve for γ via the second equation ∂α γ =
1 3 ∂ θ − R∂α sin(θ ) − 2 A∂α W, t, W α
so that
1 2 ∂ θ − R sin(θ ). W α Hence, we can write the Hele-Shaw equation for θ as (I + 2 AJ [z]) γ =
∂t θ = where
1 R H(∂α3 θ ) − ∂α H (sin(θ )) + F, 2W 2
(3.1)
(3.2)
F = (T − W, t) ∂α θ − AH (∂α W, t) + R1 , n.
We include Eq. (3.1) since (3.2) is not independent of γ . Equation (3.1) follows by (2.12):
1 1 1 H (γ ) + H, (γ ) + K[z](γ + γ0 ), W= 2i∂α z 2i ∂α z so
1 1 H, γ + K[z](γ + γ0 ) .
W, t = J [z](γ + γ0 ) = t, 2i ∂α z The linearized Hele-Shaw equations are 1 1 2 H(∂α γ ), γ = ∂ θ − Rθ. 2 W α Note that γ is in fact a dependent variable in the Hele-Shaw setting, and it can be recovered once θ has been determined. Combining these two equations yields 1 3 1 ∂α θ − R∂α θ . ∂t θ = H (3.3) 2 W ∂t θ =
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By letting θ (t, x) = θk (t)eikx , we get |k| dθk (t) =− dt 2 therefore, we set
In fact, we assume θ (0) =
6
|k| λk = − 2 k θk (0)e
θ (t) =
ikα
k2 + R θk (t), W
|k|2 +R . W
(3.4)
then the solution is given by
θk (0)eλk t+ikα ≡ e Lt θ (0),
(3.5)
k
where λk are given by (3.4). Clearly there are growing modes if there are integers k such that k 2 + RW < 0.
(3.6)
Therefore, to induce an instability we require the stratification to be negative (R < 0). If this occurs we set 7√ 8 −RW (3.7) kmax = to be the dominant growing eigenmode. Furthermore, there are only a finite number of growing modes possible. Denoting the maximal growth rate by λ, we have θ (t) L 2 ≤ Ceλt θ (0) L 2 .
(3.8)
We can recover the linear vortex sheet explicitly: 2 |k| + R eλk t+ikα . γ =− θk (0) W k
3.1. Unstable Hele-Shaw dynamics is straightforward.
The energy inequality for the Hele-Shaw equation
Proposition 3.1. We have the following sharp energy estimate: for E k ≤ σ sufficiently small and z ∈ PsK , then for any ε > 0, d θ 2H k ≤ ε θ 2H k + Cσ θ 3H k + Cε θ 2L 2 . dt
(3.9)
We now define the class of admissible perturbations, following the linear analysis above. Definition 3.2. We define a smooth generic profile for the initial perturbation as θ˜ =
∞ k=0
is nonzero for some i. such that θ˜kmax i
θ˜k eikα
(3.10)
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Then from Proposition 3.1 and Lemma 1.1, we have Theorem 3.3. Suppose R < 0 and −RW ≥ 1. Further, let θ δ (t, α) be a solution to the Hele-Shaw equations (3.2) with initial value θ δ (0, α) = δ θ˜ , where θ˜ is a generic profile defined in (3.10). Then,
θ δ (t) − δe Lt θ˜k j L 2 ≤ C e−χ t + (1 + θ˜ 2H k )δeλt δeλt , max
j
for any δ ≤ δ0 sufficiently small, and 0 ≤ t ≤ T δ , where χ > 0 is the gap between λ and the next largest real part of λk . On the other hand we have the following stability result Theorem 3.4. Suppose R > 0. Suppose θ δ (t, α) is a solution to the Hele-Shaw equations (3.2) with initial value θ δ (0, α) = δ θ˜ with δ θ˜ k ≤ C R , where C R is a small constant independent of the initial perturbaH tion. Then, the solution exists for all time and θ δ (t) H k ≤ Cθ˜ H k for any δ ≤ δ0 sufficiently small. The proof for Theorem 3.3 follows from Lemma 1.1 with ||| · ||| = · H k and · = · L 2 . We also use (3.5) to split the term δe Lt y0 into the fastest growing modes and the rest to conclude the proof. 3.2. Energy inequality 3.2.1. Linear energy inequality. To illustrate the main idea, we first recall that the linearized equation for θ satisfies ∂t θ =
1 3 R ∂α H(θ ) − H(∂α θ ). 2W 2
We then compute d 1 %% k %%2 %∂α θ % dα = ∂αk θ ∂αk ∂t θ dα dt 2 1 R = ∂αk θ1 H ∂αk+3 θ dα − ∂αk θ1 H(∂αk+1 θ )dα 2W 2 1 R =− ∂αk+1 θ (∂α H) ∂αk+1 θ dα − ∂αk θ H(∂αk+1 θ )dα, 2W 2 and therefore
d 1 θ (t)2H k + θ (t)2H k+3/2 = −R θ (t)2H k+1/2 . dt W
(3.11)
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We find standard parabolic estimates for the system. In fact if R > 0 then the system is completely diffusive and there are no growing modes. On the other hand if R < 0 then there are growing modes, but we have d 1 θ (t)2H k + θ (t)2H k+3/2 = |R| θ (t)2H k+1/2 dt W 1 R2 W θ (t)2H k+3/2 + C(k) θ (t)2L 2 ≤ 2W 2 or 1 R2 W d θ (t)2H k + θ (t)2H k+3/2 ≤ C(k) θ 2L 2 . dt 2W 2 3.2.2. Nonlinear energy inequality. We now turn to the full nonlinear problem. Proof of Proposition 3.1. From Eq. (3.2), we obtain % % R 1 % k %2 ∂αk+1 θ H ∂αk+2 θ − ∂αk θ ∂αk+1 (H[θ ])) %∂α θ % + 2W 2 R k k+1 =− ∂α θ ∂α H[sin θ − θ ] + ∂αk θ ∂αk F1 . 2
d 1 dt 2
(3.12)
We now estimate the cubic terms on the right-hand side. Clearly, the first term on the right-hand side can be written as |R| θ H k sin θ − θ H k+1 ≤ Cθ H k θ 2H k+1 . 2 On the other hand, the second term is bounded by θ H k F H k . By (2.40), (2.50), (2.58), together with product rule, we deduce (T − W, t) ∂α θ H k + A ∂α W, t H k + R1 , n H k ≤ C θ H k + γ k− 1 θ H k+1 + C θ H k + γ H
But from (3.1) and (2.58), γ
H
2
1
H k− 2
≤ Cθ
hand side of (3.12) is bounded by Cθ H k × small (Cθ H k ≤
1 2W ),
3
H k+ 2 θ 2
2 k− 21
.
. We therefore conclude the right3
H k+ 2
. Hence, for θ H k sufficiently
we have
d 1 θ (t)2H k + θ (t)2H k+3/2 ≤ −R θ (t)2H k+1/2 , dt 2W
(3.13)
which finishes the proof of Proposition 3.1. Proof of Theorem 3.4. Choose constant C R so that from the proof of Proposition 3.1, 1 Cθ H k ≤ 2W for all δθ˜ H k ≤ C R . Hence C R ≤ 2C1W , where C arises from the energy bounds. This is sufficient to satisfy (3.13) and hence stability for R > 0.
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4. Elliptical Vortex Patches The Kirchhoff elliptical vortex patch is an L 1 ∩ L ∞ weak solution to the twodimensional incompressible Euler equations, where the initial vorticity is a step function on the interior of an ellipse. Vortex patch solutions are globally well-posed, [8, 7]. It is well known that for an ellipse with major and minor axis a and b, the vortex patch rotates with a constant rate, = ab 2 . (a+b) The description of the elliptical patch patches is facilitated by the use of elliptical coordinates defined by x = c cosh(ξ ) cos(α), y = c sinh(ξ ) sin(α),
√ where c = a 2 − b2 . In complex notation we set z = x + i y = c cosh(ξ + iα). The unperturbed ellipse can then be expressed as z 0 (α) = c cosh(ξ0 + iα) = c cosh(ξ0 ) cos α + ic sinh(ξ0 ) sin α, where cosh(ξ0 ) = a/c and sinh(ξ0 ) = b/c. This implies c2 ab −2ξ0 1 ab = = sinh(2ξ0 )e−2ξ0 . = e 2 2(a + b)2 2 4
(4.1)
The boundary of a vortex patch is advected by the fluid velocity induced by the vorticity distribution and can be described in terms of an integro-differential equation referred to here as the Contour Dynamics Equation (see Constantin-Titi and Majda-Bertozzi for a more detailed description [9, 22]). For our perturbed elliptical patch, we will use an equation which we refer to as the Love equation that describes the evolution of radial perturbations ξ(α, t) about the base (unperturbed) ellipse (see [14] for a derivation). The Love equation is 2π % % 1 J ξt = −∂α ψr − i∂α z, (4.2) log %z(α) − z(α )% ∂α z(α )dα , 2π 0 where z(α) = c cosh(ξ(α, t) + ξ0 + iα) = c cosh(ξ + ξ0 ) cos α + ic sinh(ξ + ξ0 ) sin α, c2 (cosh(2ξ + 2ξ0 ) − cos(2α)) , 2 ψr = (cosh(2ξ + 2ξ0 ) + cos(2α)) . 2 J=
(4.3)
If we introduce the quantity q(α, t) = J0 (α)ξ(α, t), where J0 = terms of q, is
c2 2
(4.4)
(cosh(2ξ0 ) − cos(2α)) then the linearized Love equation, written in
∂t q = −∂α q −
1 ∂α 2π
% % log %z 0 (α) − z 0 (α )% q(α )dα ≡ Lq.
(4.5)
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A dispersion relationship for (4.5) can be computed. We will outline the results here and refer the reader to [14] for details. By setting q = eλk t eikα we obtain λ2k
ω0 = 2
,
γ −1 γ +1
2k
−
2 / 2kγ −1 , (γ + 1)2
(4.6)
from which we can conclude that if γ < 3 then the elliptical patch is neutrally stable, whereas if γ > 3 there exist linear growing modes. This is precisely the stability criterion shown by Love in his seminal paper on the Kirchhoff ellipse [21]. More recently, Tang and Wan translated the linear stability result into nonlinear orbital stability when γ < 3 [26, 29] while the authors have treated the nonlinear orbital instability of the Love equation for γ > 3 [14]. Given an initial perturbation q(α, 0) =
qk eikα ,
k
we can write down an explicit solution as in Subsect. 1.2 of [14]. Defining ω0 μ+k = 2 μ− k
ω0 = 2
, ,
γ −1 γ +1 γ −1 γ +1
k
+
k −
2kγ
(γ + 1)2 2kγ (γ + 1)2
/ −1 , / −1 ,
% % % the eigenvalues can be expressed by λ2k = %μ+k μ− k and the solution to (4.5) is given explicitly by q(α, t) =
k≥0,μ− k >0
+ μ + μ− μ+ − μ− qk cosh(λk t) − qk − q−k sinh(λk t) eikα 2λk 2λk
+ μ − μ− μ+ + μ− + q−k cosh(λk t) − qk − qk sinh(λk t) e−ikα 2λk 2λk + μ+ − μ− μ + μ− qk cos(λk t) − + qk − q−k sin(λk t) eikα 2λk 2λk − k≥0,μk <0
+ μ − μ− μ+ + μ− + q−k cos(λk t) − qk − q−k sin(λk t) e−ikα 2λk 2λk ≡ e Lt q(0).
(4.7)
This defines the linear evolution operator e Lt . If the quantity given in (4.6) is positive, i then there exist growing modes and we set kmax to be the eigenmode(s) associated to λ, the dominant eigenvalue. From the analysis of [14] we have q(t) L 2 ≤ Ceλt q(0) L 2 which bounds the growth of the perturbation in terms of λ.
(4.8)
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4.1. Unstable patch dynamics. In order to apply Lemma 1.1 to the vortex patch problem, we need to establish the following sharp energy estimate. Proposition 4.1. For q H k ≤ σ sufficiently small, and z ∈ PK for any ε > 0, d q2H k ≤ ε q2H k + Cσ q3H k + Cε q2L 2 . dt
(4.9)
We also define a class of perturbation that ensures the excitation of a dominant linear growing mode. Definition 4.2. A smooth generic profile for the initial perturbation is defined by q˜ = q˜k eikα (4.10) k
such that q˜kmax is nonzero for some i. i We then have the following main result for the vortex patch problem. Theorem 4.3. Given γ > 3, suppose q δ = J0 ξ δ is a solution to the Love equation (4.2) with initial value q δ (0, α) = δ q, ˜ where q˜ is a generic profile defined in (4.10). Then, δ $ # Lt q (t) − δe q ˜ ≤ C e−χ t + (1 + q ˜ H k )δeλt δeλt , j kmax 2 j L
for any δ ≤ δ0 sufficiently small, and 0 ≤ t ≤ T δ , where χ > 0 is the gap between λ and the next largest real part of λk . The proof of Theorem 4.3 follows immediately from Lemma 1.1 with ||| · ||| = · H k and · = · L 2 . We also use the explicit solution given by (4.7) to isolate the fastest growing modes in δe Lt y0 . 4.2 Energy inequality 4.2.1 Linear energy inequality. Recall that the linearized Love equation is % % 1 ∂t q = −∂α q − ∂α log %z 0 (α) − z 0 (α )% q(α )dα ≡ (L 1 + L 2 )q ≡ Lq, 2π which we use to define the linear operator L and its components L 1 and L 2 . Using the primitive of the Hilbert transform (2.9), we find that % % % α − α %% 1 1 ∂α (4.11) log %%sin % f (α )dα = 2 H f. 2π 2 This allows us to rewrite the linear Love equation (4.5) as % % % % % z 0 (α) − z 0 (α ) % 1 1 % q(α )dα . ∂t q = −∂α q − H(q) − ∂α log %% % α−α 2 2π % sin % 2
(4.12)
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From the form of (4.12) it follows that 1 1 k 2 k k+1 ∂t ∂α q 2 = − ∂α q∂α qdα − ∂αk qH ∂αk q dα L 2 2 % % % % ) % % z (α) − z (α 0 0 k k % %% q(α )dα dα. − ∂α q(α)∂α log % % sin α−α % 2 The first term is a perfect derivative and integrates to zero; the second term integrates to zero by the antisymmetry of the Hilbert transform. Therefore, we have the bound 2 2 ∂t ∂αk q 2 ≤ ε ∂αk q 2 + Cε q2L 2 L
L
(4.13)
from (5.1). We note that this bound can be improved, but the form of inequality (4.13) is instructive for the full nonlinear case. 4.2.2. Nonlinear energy inequality. We now turn to the proof of Proposition 4.1 which is an analogue to (4.13) for the full Love equation given by (4.9). The primary difficulty is that (4.2) is not in a suitable perturbation form, and we spend a good deal of effort extracting such a perturbation form. Our main goal is to show that the linearized Love equation (4.12) is the main part of the full nonlinear equation in the appropriate sense. We begin by establishing some useful identities. Recall that for the base ellipse E 0 we have the following definitions z 0 = c cosh(ξ0 + iα), c2 J0 = (cosh(2ξ0 ) − cos(2α)) = c2 |sinh(ξ0 + iα)|2 , 2 ab −2ξ0 e ψr0 = (cosh(2ξ0 ) + cos(2α)) . 4 We then have the following: 2π % % 1 0 % % 0 = −∂α ψr − i∂α z 0 (α), log z 0 (α) − z 0 (α ) ∂α z 0 (α )dα , 2π 0
(4.14)
which follows from the fact that an unperturbed Kirchhoff ellipse remains steady in the rotating frame. In the following lemma we prove (4.14) directly in order to establish some useful intermediate identities. Lemma 4.4. For z 0 defined above we have the following explicit formulas: 2π % % 1 −2ξ0 sin(2α), = ab i∂α z 0 (α), log %z 0 (α) − z 0 (α )% ∂α z 0 (α )dα 2 e 2π 0 (4.15) 9 < ; : i∂α z 0 (α), z 0 (α) − z 0 (α ) 1 2 i∂α z 0 , ∂α z 0 (α )dα = c4 sinh(2ξ0 ) sin(2α). 2 2π |z 0 (α) − z 0 (α )| (4.16)
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Proof. To evaluate the integral that appears in (4.15) we first write %% % c %% % % % |z 0 (α) − z 0 (α )| = ξ %1 − e−iα+iα % %1 − e2ξ0 +iα+iα % 2e 0 % % & % % and define Im (α + iβ) = π1 log %1 − eα+iβ+iα % eimα dα which can be computed (see [14, 26]): 1 −|mα|−imβ e Im (α + iβ) = − . |m| We then have % % 1 % % log %1 − eα+iβ+iα % cos(α )dα = 21 I1 (α + iβ) + I−1 (α + iβ) = −e−|α| cos(β), π % % 1 % % log %1 − eα+iβ+iα % sin(α )dα = 2i1 I1 (α + iβ) − I−1 (α + iβ) = −e−|α| sin(β), π and therefore % cos(α ) % 1 dα log %z 0 (α) − z 0 (α )% sin(α ) π % % 1 1 c cos(α ) % −iα+iα % cos(α ) dα dα − e = + log ξ log % %1 sin(α ) π 2e 0 sin(α ) π % % 1 % cos(α ) % dα + log %1 − e2ξ0 +iα+iα % sin(α ) π 2 cosh(ξ0 ) cos(α) . = −e−ξ0 2 sinh(ξ0 ) sin(α) Since ∂α z 0 (α) = ci sinh(ξ0 + iα) = −c cosh(ξ0 ) sin(α) + ic sinh(ξ0 ) cos(α) then % % 1 sin(α) % −ξ0 % . (4.17) sinh(2ξ0 ) log z 0 (α) − z 0 (α ) ∂α z 0 (α )dα = ce − cos(α) π Finally, we calculate the inner product % % 1 % % i∂α z 0 , log z 0 (α) − z 0 (α ) ∂α z 0 (α )dα 2π c −ξ0 −c sinh(ξ0 ) cos(α) sin(α) · e = sinh(2ξ0 ) −c cosh(ξ0 ) sin(α) − cos(α) 2 c2 −ξ0 e sin(2α) sinh(2ξ0 ) (cosh(ξ0 ) − sinh(ξ0 )) 4 c2 = e−2ξ0 sinh(2ξ0 ) sin(2α), 4
=
and since sinh(2ξ0 ) = 2 sinh(ξ0 ) cosh(ξ0 ) = 2 ab then c2 % % ab −2ξ0 1 % % i∂α z 0 , e sin(2α) log z 0 (α) − z 0 (α ) ∂α z 0 (α )dα = 2π 2 which proves (4.15).
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To show (4.16) we begin by simplifying several terms. First we note that : ; i∂α z 0 (α), ∂α z 0 (α ) = c2 sinh(ξ0 ) cosh(ξ0 ) cos(α) sin(α ) − c2 sinh(ξ0 ) cosh(ξ0 ) sin(α) cos(α ) c2 sinh(2ξ0 ) sin(α − α). = 2 Next, the numerator of the integral can be rewritten : ; i∂α z 0 (α), z 0 (α) − z 0 (α )
= −c2 sinh(ξ0 ) cosh(ξ0 ) cos2 (α) + sin2 (α) − cos(α) cos(α ) − sin(α) sin(α ) =−
c2 sinh(2ξ0 ) 1 − cos(α − α) . 2
Therefore, we have 9
1 i∂α z 0 , 2π
i∂α z 0 (α), z 0 (α) − z 0 (α )
<
∂α z 0 (α )dα |z 0 (α) − z 0 (α )|2 sin(α − α) 1 − cos(α − α) 1 c4 2 dα = − sinh (2ξ0 ) 4 2π |z 0 (α) − z 0 (α )|2
(4.18)
which we will compute via calculus of residues. In particular, if we set w = eiα −iα then we have 1 2 sin(α − α) 1 − cos(α − α) = − (1 − w)2 − (1 − w) . 4i To simplify the denominator we can write % %% %2 % % %z 0 (α) − z 0 (α )%2 = c e−ξ0 %%1 − eiα −iα %% %%1 − e2ξ0 +iα+iα %% 2 % %2 c2 −2ξ0 % % |1 − w|2 %1 − we2ξ0 +2iα % = e 4 1 1 c2 −2ξ0 1 − we2ξ0 +2iα 1 − e2ξ0 −2iα . = e (1 − w) 1 − 4 w w Now if γ is the contour defined by w then 2π sin(α − α) 1 − cos(α − α) c4 1 2 − sinh (2ξ0 ) dα 4 2π 0 |z 0 (α) − z 0 (α )|2 c2 1 = − sinh2 (2ξ0 ) 4 2π
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2 − 4i1 (1 − w)2 − (1 − w) dw 1 −2ξ0 1 1 2ξ +2iα 2ξ −2iα iw 1 − we 0 (1 − w) 1 − w 1 − we 0 γ 4e 2 wdw c = sinh2 (2ξ0 )e2ξ0 2ξ +2iα 0 8π w − e2ξ0 −2iα γ 1 − we dw c2 2 2ξ0 sinh (2ξ0 )e − 2ξ +2iα 0 8π w − e2ξ0 −2iα γ w 1 − we wdw e2ξ0 c2 sinh2 (2ξ0 ) 2ξ +2iα =− −2ξ0 −2iα w − e2ξ0 −2iα 8π e 0 w − e γ dw e2ξ0 c2 . + sinh2 (2ξ0 ) 2ξ +2iα −2ξ −2iα 0 0 8π e w − e2ξ0 −2iα γ w w−e
Returning to the integral in (4.18), we use the residue formula an analytic function f (z) inside the contour γ and obtain
&
f (z) γ z−a dz
= 2πi f (a) for
sin(α − α) 1 − cos(α − α) c4 1 sinh2 (2ξ0 ) dα 4 2π |z 0 (α) − z 0 (α )|2 4 , c e−2ξ0 −2iα 2 −2iα sinh (2ξ0 )e = (2πi) − −2ξ −2iα 8π − e2ξ0 −2iα e 0 ./ 1 1 + −4iα + −2ξ −2iα −2ξ −2iα e e 0 e 0 − e2ξ0 −2iα , / e−2ξ0 e−2ξ0 c2 2 −2iα 2iα e e + −2ξ = (i) sinh (2ξ0 ) − −2ξ 4 e 0 − e2ξ0 e 0 − e2ξ0
−
=
c2 −2ξ0 e sinh(2ξ0 ) sin(2α). 4
This finishes the proof of (4.16). As mentioned above, one of the difficulties of analyzing the Love equation is that the equation is not a perturbation expression. The integral operator in (4.2) has a nontrivial zeroth-order piece, as can be seen from (4.14). To facilitate the proof of Proposition 4.1, then, we subtract off the zeroth order terms and separate into linear and higher-order terms: J ∂t q = {−∂α ψr + ∂α ψr0 } J0 % % 1 − i∂α z, log %z(α) − z(α )% ∂α z(α )dα 2π % % 1 + i∂α z 0 , log %z 0 (α) − z 0 (α )% ∂α z 0 (α )dα 2π ≡ Lq + Rq,
(4.19)
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where the linear part Lq can be separated into four terms: Lq = {−∂α ψr + ∂α ψr0 } % % 1 − i∂α (z − z 0 ), log %z 0 (α) − z 0 (α )% ∂α z 0 (α )dα 2π % % % z(α) − z(α ) % 1 % ∂α z 0 (α )dα − i∂α z 0 , log %% 2π z 0 (α) − z 0 (α ) % % % 1 − i∂α z 0 , log %z 0 (α) − z 0 (α )% ∂α {z − z 0 }(α )dα 2π ≡ L1 q + L2 q + L3 q + L4 q
(4.20)
and the remainder Rq separates into three terms: % % % z(α) − z(α ) % 1 % % ∂α z(α )dα Rq = i∂α {z − z 0 }, log % 2π z 0 (α) − z 0 (α ) % % % % z(α) − z(α ) % 1 % % ∂α {z − z 0 }(α )dα + i∂α z 0 , log % 2π z 0 (α) − z 0 (α ) % % % 1 % % + i∂α {z − z 0 }, log z 0 (α) − z 0 (α ) ∂α {z − z 0 }(α )dα 2π ≡ R1 q + R2 q + R3 q. (4.21) We shall show that the remainder terms of R are higher order than the terms of L. Proof of Proposition 4.1. Taking k derivatives of the nonlinear Love equation (4.19) yields k j
J0 k− j ∂α {Lq + Rq}∂αk q k J j=1 J0 − J k ∂α {Lq + Rq}∂αk qdα + 2 ∂αk {Lq + Rq}∂αk qdα. (4.22) +2 J
j k− j For the first term of (4.22) we have the basic estimate ∂α JJ0 ∂α q L 2 ≤ Cq2H 3 along with the estimates on R and L below. The estimate on the second term of (4.22) follows from both the fact that the term J0 J−J makes the expression higher order and the estimates on R and L below. Therefore, we concentrate our energy on the estimate of last term of (4.22). Observe that d dt
|∂αk q|2 = 2
∂αj
R + L = R + (L − L) + L = R + (L1 + L2 + L3 − L 1 ) + (L4 − L 2 ) + L .
(4.23)
From the argument for (4.13) we have 2 ∂αk q∂αk Lqdα ≤ ε ∂αk q 2 + Cε2 q L 2 . L
It remains then to establish estimates for the nonlinear term and the linear comparison terms.
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For the nonlinear term, we claim that if q H k ≤ σ , then there exists C = C(σ ) such that % % % % % ∂ k Rq∂ k q % ≤ C q3 k . (4.24) α α % % H To prove this, we split R into R1 + R2 + R3 as shown in (4.21). By the basic estimate in Theorem 3 of the authors’ previous paper [14] (with an integration by parts if all the derivatives ∂αk hit on i∂α {z − z 0 }), % % % % % % % ∂ k R3 q∂ k q % ≤ Cz − z 0 k log %z 0 (α)−z 0 (α )% ∂α {z −z 0 }(α )dα q k H H α α % % Hk
≤C
q3H k
,
since z − z 0 H k ≤ C q H k . To control R1 and R2 , we use a direct consequence of the following estimate established in [14]: Let z 0 ∈ C ∞ , s ≥ 3, and z 0 , z and w ∈ PsK , then 2π |z(α) − z(α )| ∂α w(α )dα log (4.25) s ≤ C K z − z 0 H s w H s . |z 0 (α) − z 0 (α)| 0 H The proof of (4.25) follows exactly as in Lemma 3 in [14] with w instead of z a . Notice that only the H s norm (not the H s+1 norm) of w is needed; this is due to the fact D M 0 f = ∂α f − ∂α M 0 f , where M0 =
(∂α z 0 (α), z 0 (α) − z 0 (α )) (∂α z 0 (α ), z 0 (α) − z 0 (α ))
is smooth and therefore in Eq. (6.33) of [14] {μ(α, α )∂α w(α )} ∂α D s−1 M0 involves at most s derivatives of w. Here μ(α, α ) is a smooth cutoff function. This establishes (4.25). Based on (4.25) (and an integration by parts in the case where all the derivatives ∂αk hit on i∂α {z − z 0 }) we majorize the first and second terms as % % % % % ∂ k {R1 + R2 }q∂ k q dα % α α % % % % % z(α) − z(α ) % % % ≤ Cz − z 0 H k log % ∂α z 0 (α )dα q H k z 0 (α) − z 0 (α ) % Hk % % % z(α) − z(α ) % % % ∂α {z − z 0 }(α )dα q H k +Cz 0 H k log % % z 0 (α) − z 0 (α ) Hk ≤ Cq3H k . This completes the estimate of the nonlinear term and we now turn to the more delicate case of the linear comparison estimate.
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Because of the linear energy estimate (4.13), it suffices to show that for 0 ≤ k ≤ 3, and q H 3 ≤ σ, there exists Cσ > 0, for any ε > 0, there exists Cε > 0 such that ∂αk {Lq − Lq}∂αk q dα ≤ Cσ q3H 3 + εq2H 3 + Cε q2L 2 , (4.26) where L is defined by (4.12) and L is defined by (4.20). To establish this, we split L as shown in (4.20) and compare each Li with parts of L separately. The details are given in Lemmas 4.5 and 4.6 below. With the estimates for the nonlinear term and the linear comparison term established, we have 3 d q H k ≤ ε q H k + C q H2 k + Cε q L 2 dt which finishes the proof of Proposition 4.1. Lemma 4.5. For q H k < σ , there exists Cσ > 0 such that ∂αk {L4 q − L 2 q}∂αk q dα ≤ Cσ q3H k + εq2H k + Cε q2L 2 .
(4.27)
Proof. We begin by showing that there exists a smooth function h(s) with lims→0 h(s) s = 0 such that the term L4 q in (4.20) can be written % % : ; 1 − log %z 0 (α) − z 0 (α )% ∂α i∂α z 0 , {z − z 0 }(α ) dα 2π % % 1 ∂α log %z 0 (α) − z 0 (α )% q(α )dα = 2π % % 1 + ∂α log %z 0 (α) − z 0 (α )% h(q(α ))dα . (4.28) 2π Integrating by parts, % % : ; 1 − log %z 0 (α) − z 0 (α )% ∂α i∂α z 0 , {z − z 0 }(α ) dα 2π ; 1 (∂α z 0 (α ), z 0 (α ) − z 0 (α)) : = i∂α z 0 , {z − z 0 }(α ) dα . 2 2π |z 0 (α) − z 0 (α )| From the definition of z 0 , we have : ; ∂α z 0 (α ) − ∂α z 0 (α), z 0 (α ) − z 0 (α) = 0 so that inside the integral ; : ∂α z 0 (α ), z 0 (α ) − z 0 (α) |z 0 (α) − z 0 (α )|2
: =−
∂α z 0 (α), z 0 (α ) − z 0 (α) |z 0 (α) − z 0 (α )|2
; .
Moreover, : ; i∂α z 0 , {z − z 0 }(α ) = −c2 (sinh(ξ0 + iα), cosh(ξ0 + iα)(cosh ξ − 1) + sinh(ξ0 + iα) sinh ξ ) 2 c sinh(2ξ0 ) (cosh ξ − 1) − c2 J0 (sinh ξ − ξ ) , = −J0 ξ − 2
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which is of the form −q − h(q) where lims→0 h(s) s = 0, and h is a smooth function of q. This establishes (4.28). We note that the first term is exactly the linear integral operator given in the linear Love equation. Now we establish the quadratic nature of the function h. Observe that cosh(ξ ) − 1 = ξ 2
n=0
ξ 2n 2(n + 1)!
and set g(ξ ) =
n=0
ξ 2n ≤ cosh(ξ ). (2(n + 1))!
Note that g(ξ ) is also an analytic function, and a simple calculation shows that |g (γ ) (ξ )| ≤ cosh(ξ ) for all ξ . Therefore, by the generalized Möser inequality cosh(ξ ) − 1 H k ≤ C ξ L ∞ ξ g(ξ ) H k + ξ g(ξ ) L ∞ ξ H k ≤ C g(ξ ) H k ξ 2H k .
We now control the g(ξ ) term. First g(ξ ) L 2 ≤ Cg (ξ L ∞ ) ≤ C cosh ξ H k . Next (γ ) k k−1 k ∂α (g(ξ )) 2 ≤ C max g ◦ ξ ∞ ξ L ∞ ∂α ξ 2 1≤γ ≤k L L L γ −1 k ≤ C cosh(ξ ) L ∞ + cosh(ξ ) L ∞ ξ L ∞ ∂α ξ 2 L k−1 k ≤ C cosh (ξ L ∞ ) 1 + ξ L ∞ ∂α ξ 2 L k−1 k ≤ C cosh ξ H k 1 + ξ H k ∂α ξ 2 . L
This implies, g(ξ ) H k ≤ C cosh ξ H k 1 + ξ kH k , so
cosh(ξ ) − 1 H k ≤ C cosh ξ H k 1 + ξ kH k ξ 2H k .
(4.29)
A similar argument shows that sinh(ξ ) − ξ H k ≤ C cosh ξ H k 1 + ξ kH k ξ 2H k .
(4.30)
These estimates can then be combined with the previous bounds on the linear integral operator to finish the lemma. We now show that, through a delicate computation, L1 + L2 + L3 in (4.20) gives rise to the transport term L 1 q = −∂α q in the linear Love equation. Lemma 4.6. For k ≥ 1, the L1 + L2 + L3 satisfy ∂α (L1 + L2 + L3 ) q − ∂α L 1 q H k−1 ≤ ε q H k + q2H k + Cε q L 2 .
(4.31)
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Proof. The proof entails extracting the linear portion of the L j ’s from the singular integral operators. 1. We begin with L1 : − ∂α ψr + ∂α ψr0 ab ab −2ξ0 = − e−2ξ0 (sinh(2ξ + 2ξ0 )∂α ξ − sin(2α)) − sin(2α) e 2 2 c2 = − sinh(2ξ0 )e−2ξ0 sinh(2ξ + 2ξ0 )∂α ξ 4 2 = − c sinh(2ξ0 )∂α ξ 2 2 c sinh(2ξ0 ) (cosh(2ξ ) − 1) ∂α ξ + c2 cosh(2ξ0 ) sinh(2ξ )∂α ξ , (4.32) − 2 2 where we have used (4.1). 2. We now study L2 . From (4.17) we have 2π % % 1 i∂α {z − z 0 }, log %z 0 (α) − z 0 (α )% ∂α z 0 (α )dα 2π 0 2 c = − e−ξ0 sinh(2ξ0 )∂α ξ sin2 α cosh(ξ + ξ0 ) + cos2 α sinh(ξ + ξ0 ) 2 c2 −ξ0 + e sinh(2ξ0 ) sin(α) cos(α) [cosh(ξ + ξ0 ) − sinh(ξ + ξ0 ) 2 − cosh(ξ0 ) + sinh(ξ0 )] c2 = − e−ξ0 sinh(2ξ0 )∂α ξ eξ +ξ0 + cos(2α)e−ξ −ξ0 4 c2 −2ξ0 + e sinh(2ξ0 ) sin(2α) e−ξ − 1 . 4 Separating the linear terms and higher-order terms in ξ : −{ c2 e2ξ0 + c2 cos(2α)}∂α ξ + c2 sin(2α)ξ 2 2 2 + − c2 e2ξ0 eξ − 1 ∂α ξ − c2 cos(2α) e−ξ − 1 ∂α ξ 2 2 + c2 sin(2α) e−ξ − 1 + ξ . 2
(4.33)
Combining with the linear term in (4.32), L1 + L2 = −∂α q −
ξ ∂α J0 + h(q), 2
(4.34)
where the last term is of higher order. In particular we estimate h(q) in the same way we estimated the higher order terms in the proof of Lemma 4.5. Upon taking another derivative, the term −∂α 2 ξ ∂α J0 shall be canceled by a piece of ∂α L3 .
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% % & ) % % 3. We now turn to L3 . In [14] we prove an H k bound on log % z 0z(α)−z(α (α)−z 0 (α ) % f (α )dα , which is nearly strong enough for our purposes; however, the constant in front needs to evaluate L3 , since we would need be arbitrarily small, q H k . It is %difficult to simply % % z(α)−z(α ) % to extract the linear term from log % z 0 (α)−z 0 (α ) %. Instead we consider ∂α L3 . In particular % % % z(α) − z(α ) % 1 2 % % ∂α z 0 (α )dα ∂α L3 = i∂α z 0 , log % 2π z 0 (α) − z 0 (α ) % % % % z(α) − z(α ) % 1 % % ∂α z 0 (α )dα ∂α log % + i∂α z 0 , 2π z 0 (α) − z 0 (α ) % = L3,1 + L3,2 . The first term, L3,1 , is compact by (4.25), so L3,1 q H k−1 ≤ Cq H k−1 ≤ εq H k + Cε q L 2 , which follows from Sobolev interpolation (5.1). Therefore we pay particular attention to L3,2 : % % % z(α) − z(α ) % % ∂α z 0 (α )dα % ∂α log % z 0 (α) − z 0 (α ) % ; : ∂α z(α) − ∂α z 0 (α), z 0 (α) − z 0 (α ) = ∂α z 0 (α )dα |z 0 (α) − z 0 (α )|2 ; : ∂α z 0 (α), z 0 (α) − z 0 (α ) − ∂α z 0 (α )dα |z 0 (α) − z 0 (α )|2 ; : ∂α z(α), z(α) − z 0 (α) − z(α ) + z 0 (α ) + ∂α z 0 (α )dα |z 0 (α) − z 0 (α )|2 ; : ;/ ,: ∂α z(α), z(α) − z(α ) ∂α z(α), z(α) − z(α ) + − ∂α z 0 (α )dα |z(α) − z(α )|2 |z 0 (α) − z 0 (α )|2 = I1 + I2 + I3 + I4 .
(4.35)
There is a delicate extraction of the linear piece from both I1 and I2 . In particular ; : sinh(ξ + ξ0 + iα), z 0 (α) − z 0 (α ) ∂α z 0 (α )dα I 1 = ∂α ξ |z 0 (α) − z 0 (α )|2 ; : i sinh(ξ + ξ0 + iα), z 0 (α) − z 0 (α ) + ∂α z 0 (α )dα |z 0 (α) − z 0 (α )|2 ; : −i∂α z 0 (α), z 0 (α) − z 0 (α ) = ∂α ξ ∂α z 0 (α )dα |z 0 (α) − z 0 (α )|2 ; : sinh(ξ + ξ0 + iα) − sinh(ξ0 + iα), z 0 (α) − z 0 (α ) +∂α ξ ∂α z 0 (α )dα |z 0 (α) − z 0 (α )|2 ; : ∂α z 0 (α), z 0 (α) − z 0 (α ) + ∂α z 0 (α )dα 2 |z 0 (α) − z 0 (α )| ; : i sinh(ξ + ξ0 + iα) − i sinh(ξ0 + iα), z 0 (α) − z 0 (α ) + ∂α z 0 (α )dα , |z 0 (α) − z 0 (α )|2
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so that : I 1 + I 2 = ∂α ξ +∂α ξ
−i∂α z 0 (α), z 0 (α) − z 0 (α )
;
∂α z 0 (α )dα |z 0 (α) − z 0 (α )|2 ; : sinh(ξ + ξ0 + iα) − sinh(ξ0 + iα), z 0 (α) − z 0 (α )
|z 0 (α) − z 0 (α )|2 % % + (cosh(ξ ) − 1) ∂α log %z 0 (α) − z 0 (α )% ∂α z 0 (α )dα : ; − i z 0 (α), z 0 (α ) + sinh(ξ ) ∂α z 0 (α )dα |z 0 (α) − z 0 (α )|2
∂α z 0 (α )dα
1 2 3 4 = I1+2 + I1+2 + I1+2 + I1+2 . 1 is non-trivial, I2 is second order, and both I3 and I4 are compact. We claim that I1+2 1+2 1+2 1+2 In fact from (4.16) and (4.1):
c2 1 1 i∂α z 0 , I1+2 = − sinh(2ξ0 ) sin(2α)∂α ξ = − ∂α J0 ∂α ξ, 2π 4 2
(4.36)
which exactly cancels with the extra linear term in ∂α L 1 in (4.34)! Furthermore, by (5.1) i∂α z 0 , 1 Ij ≤ C q H k−1 ≤ ε q H k + Cε q L 2 (4.37) 1+2 2π H k−1 for 2 ≤ j ≤ 4 with C independent of ξ . Combining (4.34), (4.36), and (4.37) yields ∂α (L1 + L2 + L3 ) − ∂α2 q k−1 H
≤ ε q H k + C q2H k + Cε q L 2 + i∂α z 0 , I3 + I4 H k−1 . 4. The proof of Lemma 4.6 is then completed by the following estimates. For any ε > 0 there is a constant Cε > 0 such that sup |∂αk−1 i∂α z 0 (α), I3 | ≤ (ε + Cz − z 0 H k )z − z 0 H k + Cε z − z 0 L 2 , α
sup |∂αk−1 i∂α z 0 (α), I4 | α
≤ (ε + Cz − z 0 H k )z − z 0 H k
(4.38) + Cε z − z 0 L 2 . (4.39)
The proof the estimates of (4.38) and (4.39) are similar, so we pay particular attention to the more difficult (4.39) estimate. To show this, we rewrite i∂α z 0 (α), I4 as ; : ;/ ,: ; ∂α z(α), z(α) − z(α ) ∂α z(α), z(α) − z(α ) : − i∂α z 0 (α), ∂α z 0 (α ) dα 2 2 |z(α) − z(α )| |z 0 (α) − z 0 (α )| ; :
i∂α z 0 (α), ∂α z 0 (α ) 1 1 = ∂α z(α), Q 1 (z) − dα, |Q 1 (z 0 )|2 α − α |Q 1 (z)|2
Dynamics near Unstable, Interfacial Fluids
where Q 1 (z) = that
z(α)−z(α ) α−α .
is C ∞ . Hence we have
∂α z(α), Q 1 (z)
687
: ; Since i∂α z 0 (α), ∂α z 0 (α ) = 0 when α = α , we deduce ; : i∂α z 0 (α), ∂α z 0 (α ) Sα,α ≡ α − α
1 Sα,α dα − |Q 1 (z 0 )|2 |Q 1 (z)|2 / , Q 1 (z)Q 1 (z) − Q 1 (z 0 )Q 1 (z 0 ) = ∂α z(α), Q 1 (z) Sα,α dα |Q 1 (z)|2 |Q 1 (z 0 )|2
∂α z(α), Q 1 (z) Sα,α {Q 1 (z − z 0 )Q 1 (z) − Q 1 (z 0 )Q 1 (z − z 0 )}dα. = |Q 1 (z)|2 |Q 1 (z 0 )|2 1
We can extract the linear term in z − z 0 in the integrand as Sα,α ∂α z(α), Q 1 (z) {Q 1 (z − z 0 )Q 1 (z) − Q 1 (z 0 )Q 1 (z − z 0 )} |Q 1 (z)|2 |Q 1 (z 0 )|2 Sα,α ∂α z 0 (α), Q 1 (z 0 ) = {Q 1 (z − z 0 )Q 1 (z 0 ) − Q 1 (z 0 )Q 1 (z − z 0 )} |Q 1 (z 0 )|2 |Q 1 (z 0 )|2 Sα,α ∂α z 0 (α), Q 1 (z 0 ) + Q 1 (z − z 0 )Q 1 (z − z 0 ) |Q 1 (z 0 )|2 |Q 1 (z 0 )|2 {Q 1 (z − z 0 )Q 1 (z) − Q 1 (z 0 )Q 1 (z − z 0 )}2 |Q 1 (z 0 )|4 |Q 1 (z)|2 Sα,α ∂α z 0 (α), Q 1 (z − z 0 ) + {Q 1 (z − z 0 )Q 1 (z) − Q 1 (z 0 )Q 1 (z − z 0 )} |Q 1 (z)|2 |Q 1 (z 0 )|2 Sα,α ∂α {z − z 0 }(α), Q 1 (z) + {Q 1 (z − z 0 )Q 1 (z) − Q 1 (z 0 )Q 1 (z − z 0 )}, |Q 1 (z)|2 |Q 1 (z 0 )|2 +Sα,α ∂α z 0 (α), Q 1 (z 0 )
where the last four terms above are quadratic in z − z 0 . Notice that |Q 1 (z)| is bounded away from zero. We now take k − 1 order derivatives and use the product rule. For the first linear term in z − z 0 , by (2.29), for s < 1, % %
% % k−1 $ Sα,α ∂α z 0 (α), Q 1 (z 0 ) # % %∂ Q (z − z )Q (z ) − Q (z )Q (z − z ) dα 1 0 1 0 1 0 1 0 % % 2 2 |Q 1 (z 0 )| |Q 1 (z 0 )| ≤ Cz − z 0 H k−1+s |α − α |−1/2s dα ≤ Cεz − z 0 H k + Cε z − z 0 L 2 , by a compact Sobolev interpolation of H k and L 2 . For the remaining quadratic terms, if every factor has at most k − 2 order derivatives, then by (2.29) we can bound their derivatives by Cz − z 0 2H k for z − z 0 2H k ≤ σ. If one factor takes the full k − 1 derivatives, we apply the second estimate with s = 1 to majorize by Cz − z 0 2H k |α − α |−1/2 dα ≤ Cz − z 0 2H k .
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This establishes (4.39). Finally, we write
i∂α z 0 , I3 =
∂α z(α), Q 1 (z) |Q 1 (z 0 )|2
−
∂α z(α), Q 1 (z 0 ) |Q 1 (z 0 )|2
Sα,α dα .
Following the method for the estimate of I4 above yields (4.38) and completes the proof of Lemma 4.6. 5. Appendix We recall some classical Sobolev estimates that are used throughout the exposition. Lemma 5.1 (Interpolation) . Fix ε > 0 and 0 < r < s, then f 2H r ≤ ε f 2H s + Cε f 2L 2 , where Cε =
r s
1− rs ε
rs −1
(5.1)
.
Proof This follows from the Hölder inequality:
(1 + |k|2 )r/2 | fˆ|2 ≤
| fˆ|2 2γ
γ
≤ f L 2 f taking γ = 1 −
r s
2(1−γ ) r
H 1−γ
r
(1 + |k|2 ) 2(1−γ ) | fˆ|2
1−γ
,
and applying Young’s inequality.
The following are useful estimates for compositions and products. Lemma 5.2 (Möser Inequalities). Let u ∈ L ∞ such that ∂ k u ∈ L r (R) with 1 ≤ r ≤ ∞. If g ∈ C r (R), then ∂ k (g ◦ u) ∈ L r (R) and there is a constant C that depends on r and α such that k γ −1 (5.2) ∂ (g ◦ u) r ≤ C max g (γ ) ◦ u ∞ u L ∞ ∂ k u r , 1≤γ ≤k L L L k (5.3) ∂ (uv) 2 ≤ C u L ∞ ∂αk v 2 + ∂αk u 2 v L ∞ . L
L
L
Proof See [22] for example. Since we need to handle the H k norm of products for k non-integer, we include the following Lemma 5.3 (Generalized Möser Inequality) . Let k ∈ R+ . Let u ∈ L ∞ be such that ∂ k u ∈ L r (R) with 1 ≤ r ≤ ∞. There is a constant C that depends on r and α such that uv H k ≤ C u L ∞ v H k + u H k v L ∞ . (5.4) Proof See [27], Chap. 2.
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References 1. Almeida, L., Bethuel, F., Guo, Y.: A remark on the instability of symmetric vortices with large coupling constant. Comm. Pure Appl. Math. 50, 1295–1300 (1997) 2. Ambrose, D.: Well-Posedness of Vortex Sheets with Surface Tension. SIAM J. Math. Anal. 35, 221– 244 (2003) 3. Ambrose, D.: Well-posedness of two-phase Hele-Shaw flow without surface tension. Euro. J. Appl. Math. 15, 597–607 (2004) 4. Bardos, C., Guo, Y., Strauss, W.: Stable and unstable ideal plane flows. Dedicated to the memory of Jacques-Louis Lions. Chinese Ann. Math. Ser. B 23, 149–164 (2002) 5. Baker, G., Meiron, D., Orszag, S.: Generalized vortex methods for free-surface flow problems. J. Fluid Mech. 123, 477–501 (1982) 6. Beale, J.T., Hou, T., Lowengrub, J.: Growth rates for the linearized motion of fluid interfaces away from equilibrium. Comm. Pure Appl. Math. 46, 1269–1301 (1993) 7. Bertozzi, A., Constantin, P.: Global regularity for vortex patches. Commun. Math. Phys. 152, 19– 28 (1993) 8. Chemin, J.-Y.: Persistance de structures geometriques dans les fluides incompressibles bidimensionals. Ann. Ec. Norm. Super. 26, 1–16 (1993) 9. Constantin, P., Titi, E.: On the evolution of nearly circular vortex patches. Commun. Math. Phys. 119, 177– 198 (1988) 10. Cordier, S., Grenier, E., Guo, Y.: Two-stream instabilities in plasmas. Cathleen Morawetz: a great mathematician. Methods Appl. Anal. 7, 391–405 (2000) 11. Escher, J., Simonett, G.: Classical solutions for Hele-Shaw models with surface tension. Adv. Differ. Eqs. 2, 619–642 (1997) 12. Friedlander, S., Strauss, W., Vishik, M.: Nonlinear instability in an ideal fluid. Ann. Inst. H. Poincaré Anal. Non Linéaire 14, 187–209 (1997) 13. Grenier, E.: On the nonlinear instability of Euler and Prandtl equations. Comm. Pure Appl. Math. 53, 1067–1091 (2000) 14. Guo, Y., Hallstrom, C., Spirn, D.: Dynamics near an unstable Kirchhoff ellipse. Commun. Math. Phys. 245, 297–354 (2004) 15. Guo, Y., Strauss, W.: Instability of periodic BGK equilibria. Comm. Pure Appl. Math. 48, 861–894 (1995) 16. Hou, T., Lowengrub, J., Shelley, M.: Removing the stiffness from interfacial flows with surface tension. J. Comp. Phys. 114, 312–338 (1994) 17. Hou, T., Lowengrub, J., Shelley, M.: The long-time motion of vortex sheets with surface tension. Phys. Fluids 9, 1933–1954 (1997) 18. Hwang, H.J., Guo, Y.: On the dynamical Rayleigh-Taylor instability. Arch. Rat. Mech. Anal. 167, 235– 253 (2003) 19. Iguchi, T.: Well-posedness of the initial value problem for capillary-gravity waves. Funkcial. Ekvac. 44, 219–241 (2001) 20. Lin, Z.: Nonlinear instability of ideal plane flows. Inter. Math. Res. Not. 41, 2147–2178 (2004) 21. Love, A.E.: On the stability of certain vortex motion. Proc. Soc. Lond. 23, 18–42 (1893) 22. Majda, A., Bertozzi, A.: Vorticity and incompressible flow. Cambridge Texts in Applied Mathematics 27. Cambridge: Cambridge University Press (2002) 23. Ogawa, M., Tani, A.: Free boundary problem for an incompressible ideal fluid with surface tension. Math. Models Methods Appl. Sci. 12, 1725–1740 (2002) 24. Saffman, P.: Vortex dynamics. Cambridge Monographs on Mechanics and Applied Mathematics. New York: Cambridge University Press (1992) 25. Siegal, M., Caflisch, R., Howison, S.: Global existence, singular solutions, and ill-posedness for the Muskat problem. Comm. Pure Appl. Math. 57, 1374–1411 (2004) 26. Tang, Y.: Nonlinear stability of vortex patches. Trans. AMS, 304, 617–638 (1987) 27. Taylor, M.:Tools for PDE. In: Pseudodifferential Operators, Paradifferential Operators, and Layer Potentials. Providence, RI: Amer. Math. Soc., (1991) 28. Vishik, M., Friedlander, S.: Nonlinear instability in two dimensional ideal fluids: the case of a dominant eigenvalue. Commun. Math. Phys. 243, 261–273 (2003) 29. Wan, Y.H.: The stability of rotating vortex patches. Commun. Math. Phys. 107, 1–20 (1986) 30. Wan, Y.H., Pulvirenti, M.: Nonlinear stability of circular vortex patches. Commun. Math. Phys. 99, 435– 450 (1985) 31. Wu, S.: Mathematical analysis of vortex sheets. Comm. Pure Appl. Math. 59, 1065–1206 (2006) Communicated by P. Constantin
Commun. Math. Phys. 270, 691–708 (2007) Digital Object Identifier (DOI) 10.1007/s00220-006-0167-1
Communications in
Mathematical Physics
Vanishing Shear Viscosity Limit in the Magnetohydrodynamic Equations Jishan Fan1,2 , Song Jiang3 , Gen Nakamura4 1 College of Information Sciences and Technology, Nanjing Forestry University, Nanjing 210037, P.R. China.
E-mail:
[email protected]
2 Department of Mathematics, Suzhou University, Suzhou 215006, P.R. China 3 Institute of Applied Physics and Computational Mathematics, P. O. Box 8009, Beijing 100088, P.R. China.
E-mail:
[email protected]
4 Department of Mathematics, Hokkaido University, Sapporo, 060-0810, Japan.
E-mail:
[email protected] Received: 20 March 2006 / Accepted: 20 June 2006 Published online: 9 January 2007 – © Springer-Verlag 2006
Abstract: We study an initial boundary value problem for the equations of plane magnetohydrodynamic compressible flows, and prove that as the shear viscosity goes to zero, global weak solutions converge to a solution of the original equations with zero shear viscosity. As a by-product, this paper improves the related results obtained by Frid and Shelukhin for the case when the magnetic effect is neglected. 1. Introduction In this paper we study the problem of vanishing shear viscosity limit in one-dimensional magnetohydrodynamics. Magnetohydrodynamics (MHD) is concerned with the study of the interaction between magnetic fields and fluid conductors of electricity. The application of magnetohydrodynamics finds a very wide range of physical objects, from liquid metals to cosmic plasmas. The governing equations of plane magnetohydrodynamic compressible flows have the following form (see, e.g., [1, 18, 19, 24, 23]): ρt + (ρu)x 1 2 (ρu)t + ρu + p + |b|2 x 2 (ρw)t + (ρuw − b)x bt + (ub − w)x (ρe)t + (ρeu)x − (kex )x
= 0,
(1.1)
= (λu x )x ,
(1.2)
= (μwx )x , = (νbx )x , = λu 2x + μ|wx |2 + ν|bx |2 − pu x ,
(1.3) (1.4) (1.5)
where ρ denotes the density of the flow, u ∈ R the longitudinal velocity, w ∈ R2 the transverse velocity, and b ∈ R2 the transverse magnetic field, θ the temperature, Supported by NSFC (Grant No. 10301014, 10225105) and the National Basic Research Program (Grant No. 2005CB321700) of China.
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p = p(ρ, θ ) the pressure, and e = e(ρ, θ ) the internal energy; λ and μ are the bulk and the shear viscosity coefficients, respectively, ν is the magnetic diffusivity acting as a magnetic diffusion coefficient of the magnetic field, k = k(ρ, θ ) is the heat conductivity. In the present paper we focus on an initial boundary value problem for (1.1)–(1.5) in the domain Q T := (0, T ) × (0, 1) for the magnetohydrodynamic flow of a perfect gas with the following equations of state: p = Rρθ, e = cV θ, where R is the gas constant and cV is the heat capacity of the gas at constant volume, and for simplicity we put cV ≡ 1. As initial and boundary conditions, we consider (u, w, b, θx )|∂ = 0, (ρ, u, w, b, θ )|t=0 = (ρ0 , u 0 , w0 , b0 , θ0 ),
(1.6) (1.7)
where := (0, 1). The conditions (1.6) mean that the boundary is non-slip, impermeable and thermally insulated. In the past years, the well-posedness of the problem (1.1)–(1.7) and the large-time behavior of solutions have been investigated by several authors. The existence and uniqueness of local smooth solutions were proved first in [33], while the existence of global smooth solutions with small smooth initial data was shown in [32, 17]. In [32, 22], the exponential stability of solutions as t → ∞ was obtained. We mention that in the simpler case w ≡ 0 which corresponds to the longitudinal motion only, the existence and uniqueness of global strong solutions are studied in [16, 31]. Recently, under the technical condition that k(ρ, θ ) satisfies C −1 (1 + θ q ) ≤ k(ρ, θ ) ≡ k(θ ) ≤ C(1 + θ q )
(1.8)
for some q ≥ 2, Chen and Wang in [2] proved the existence, uniqueness and the Lipschitz continuous dependence of global strong solutions to (1.1)–(1.7) with large initial data satisfying 0 < inf ρ0 ≤ ρ0 (x) ≤ sup ρ0 < ∞, ρ0 , u 0 , w0 , b0 , θ0 ∈ H 1 (), θ0 (x) ≥ 0. A similar result is obtained in [3] for real gases and the existence of globally defined weak solutions is also studied under the same technical condition as (1.8), while the uniqueness of the weak solutions with Lebesgue initial data is given by the authors in [5] under the technical condition upon k(ρ): k(ρ, θ ) ≡ k(ρ) ≥ C/ρ,
(1.9)
which is also required in [16, 31]. The uniqueness and continuous dependence of weak solutions for the Cauchy problem have been proved recently by Hoff and Tsyganov in [13]. When b ≡ 0, the system (1.1)–(1.5) reduces to the Navier-Stokes equations for a (plane) compressible heat-conducting fluid (cf., e.g., [27]). In this case, the limit process as the shear viscosity μ → 0 for global strong solutions has been studied by Frid and Shelukhin [27, 9, 10]. Their proof is based on the a priori uniform-in-μ estimates derived in [28, 25] and the key estimates in [27, 10] are C −1 ≤ ρ,
ρ L ∞ (0,T ;W 1,1 ()) ≤ C, ρt L 1 ((0,T )×) ≤ C
(1.10)
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under the assumption on the initial density ρ0−1 ∈ L ∞ (), ρ0 ∈ W 1,1 ().
(1.11)
Thus, with the help of (1.10), they can show that as μ → 0, ρ → ρ strongly in L p ((0, T ) × )
for any p ≥ 1.
(1.12)
Here and in what follows we use f¯ to denote the weak limit as μ → 0 of f which depends on μ. Based on (1.12), Frid and Shelukhin then use the technique of Lagrangian coordinates and the nice property of the effective viscous flux, which has been discovered by several authors for the compressible Navier-Stokes equations (see, e.g., [26, 21, 12]), to complete their proof. However, they could not succeed to show pu x = p u x , u 2x = u 2x .
(1.13)
Instead, they prove a weaker identity (λ + 2μ)u 2x − Rρθ u x = λu 2x − Rρ θ u x ,
(1.14)
which is sufficient to get their zero shear viscosity limit. Notice that by (1.10), (1.13) implies (1.14). Very recently, by utilizing techniques from the study of the compressible Navier-Stokes equations and the weak convergence method, Fan and Jiang are able to show the zero shear viscosity limit for weak solutions to the multidimensional isentropic compressible Navier-Stokes equations with cylindric symmetry [4]. The aim of the present paper is to prove the convergence, as μ → 0, of weak solutions to the problem (1.1)–(1.7) with the initial data satisfying (the weaker conditions than in (1.11)) ρ0−1 , ρ0 ∈ L ∞ (), u 0 , w0 , b0 ∈ L 2 (), θ0 ∈ L 1 (), inf θ0 > 0
(1.15)
under the technical condition (1.8) but for some q ≥ 1 or (1.9). Moreover, the relations (1.13) are obtained. For the vanishing shear viscosity limit for the weak solutions here, compared with the strong solutions in the case b ≡ 0 dealt with in [27, 10] and the weak solutions in the case of b ≡ 0 and isentropy investigated in [4], the main difficulties lie in lack of (good) a priori estimates due to the additional magnetic effect, and in the derivation of the strong convergence of the density ρ due to lack of uniform a priori estimates on derivatives of ρ and in the treatment of the non-isentropic and/or magnetic effects, including the quadratic terms on the right-hand side of (1.5), in the limit process. To overcome such difficulties, we first utilize the technical condition (1.8) or (1.9) to derive the uniform in μ estimates adapted from [28, 25], and then we use the techniques from the study of the global existence of weak solutions to the multidimensional compressible Navier-Stokes equations (see, e.g., [21, 7, 14]), and exploit the feature of Eq. (1.2) to obtain ρ → ρ strongly in L 1 ((0, T ) × ), as μ → 0. With the help of the above strong convergence, we use a further compactness argument and Lemma 5.1 in [21] and Simon’s compactness principle [30] to complete our proof. We should point out here that if one neglects the magnetic effect, our result extends those in [27, 10] to weak solutions. Moreover, as a by-product of our proof, we obtain the existence of globally defined weak solutions to (1.1)–(1.7). Now we are ready to state the main result of this paper.
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Theorem 1.1. Assume that the initial data satisfy (1.15) and the heat conductivity k satisfies (1.8) for some q ≥ 1 or (1.9). Then the problem (1.1)–(1.7) has at least one weak solution (ρ, u, w, b, θ ), such that ρ → ρ strongly in L p (Q T ) and weak- ∗ in L ∞ (Q T ), ∀ 1 ≤ p < ∞, (u, b) → (u, b) strongly in L 2 (0, T ; H01 ()), w → w strongly in L 2 (Q T ), and μw2x = 0, θ → θ strongly in L r (Q T ),∀ r ∈ [1, 3), and θx → θ x weakly in L q (Q T ),∀ q < 3/2. Moreover, the limit functions ρ, u, w, b, θ solve (1.1)–(1.7) with μ = 0 in the sense of distributions. Remark 1.1. One can prove the existence, the uniqueness and Lipschitz continuous dependence on initial data of solutions (ρ, u, w, b, θ ) to the limit problem in the function class ρ, ρ −1 ∈ L ∞ (), u ∈ V2 (Q T ), b ∈ V2 (Q T ), θ ∈ V2 (Q T ), uu x ∈ L 2 (Q T ), b · bx ∈ L 2 (Q T ), w ∈ L ∞ (0, T ; L 2 ()),
(1.16)
under the technical condition bt ∈ L 2 (Q T ) when initial data satisfy ρ0−1 , ρ0 ∈ L ∞ (), u 0 ∈ L 4 (), b0 ∈ L 4 (), θ0 ∈ L 2 (), w0 ∈ L 2 () and the heat conductivity k satisfies (1.9), where V2 (Q T ) := L ∞ (0, T ; L 2 ()) ∩ L 2 (0, T ; H01 ()). The regularity (1.16) can be shown easily, while the uniqueness and the Lipschitz continuous dependence on initial data can be obtained by the same arguments as used in [5], and hence we omit the details here. This paper is organized as follows: Section 2 is devoted to the derivation of the uniform estimates which will be used in Sect. 3 to complete the proof of Theorem 1.1. In Sect. 4, we give a short but important remark on Frid and Shelukhin’s paper [10]. As the end of this section, we introduce some notation: Let I ⊂ R be an interval and B a Banach space. C(I, B − w) is the space of all functions which are in L ∞ (I, B) and continuous in t with values in B endowed with the weak topology. We will use the abbreviations: L q (0, T ; W m, p ) ≡ L q (0, T ; W m, p ()),
L p ≡ L p ().
The same letter C will denote various positive constants which do not depend on μ. f will denote the weak limit as μ → 0 of f which depends on μ. 2. Uniform A Priori Estimates This section is devoted to the derivation of a priori estimates of (ρ, u, b, w, θ ) which are independent of μ. To begin with, we notice that the total mass and energy are conserved. In fact, by rewriting (1.1)–(1.5) one has 1 E t + u E + p + |b|2 − w · b = (λuu x + μwwx + νb · bx + kθx )x , (2.1) x 2 k(ρ, θ ) λu 2x + μ|wx |2 + ν|bx |2 k(ρ, θ )θx2 θx + (ρs)t + (ρus)x − = , (2.2) θ θ θ2 x
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where E and s are the total energy and the entropy, respectively, 1 1 E := ρ e + (u 2 + w 2 ) + |b|2 , s := cV ln θ − (γ − 1)cV ln ρ. 2 2 Integrating (1.1), (2.1) and (2.2) over (0, t) × , we get Lemma 2.1. ρ(x, t)d x = ρ0 (x)d x =: m > 0,
(ρ ln ρ + ρ| ln θ |)(x, t)d x +
T
0
E(x, t)d x =
2 λu x + μ|wx |2 + ν|bx θ
2 |
E(x, 0)d x, +
kθx2 d xdt ≤ C, θ2
where C is a positive constant independent of μ. The following two lemmas give us pointwise uniform upper and lower bounds of the density. Lemma 2.2.
ρ(x, t) ≤ C
for any x ∈ , t ≥ 0.
(2.3)
Proof. It follows from (1.1) and (1.2) that (ρu)t = px , where p := Setting t x ϕ := p (x, τ )dτ + ρ0 (ξ )u 0 (ξ )dξ, 0
λu x −ρu 2 − p− 21 |b|2 .
0
we find that
ϕx = ρu, ϕt = p , ϕx |∂ = 0, ϕ|t=0 =
x
ρ0 u 0 dξ.
(2.4)
0
By virtue of Lemma 2.1 and the Cauchy-Schwarz inequality,
ϕx L ∞ (0,T ;L 1 ) ≤ C, ϕd x
≤ C,
which gives
ϕ L ∞ ((0,T )×) ≤ C.
(2.5)
Now, denoting F := exp(ϕ/λ) and using (2.4), we have after a straightforward calculation that 1 1 Dt (ρ F) := ∂t (ρ F) + u∂x (ρ F) = − p + |b|2 ρ F ≤ 0, λ 2 which together with (2.5) implies (2.3) immediately.
Lemma 2.3. Assume that either the condition (1.8) for some q ≥ 1 or (1.9) holds. Then, 1/C ≤ ρ(x, t), T (u 2x + μ|wx |2 + |bx |2 )d xdt ≤ C 0
with C being a positive constant independent of μ.
(2.6) (2.7)
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Proof. Firstly, we note that there exists an xˆ ∈ (0, 1) such that √ x √ √ ρ θd x |θx | 1 = θ − ( θ)x dy ≤ √ dy 2 θ xˆ ρd x 1/2 1/2 θ θ2 1 dy k x2 dy . ≤ 2 θ k
(2.8)
It is easy to see by the assumption on k(ρ, θ ) and Lemma 2.1 that θ dy ≤ C. k Inserting the above estimate into (2.8) and applying Lemma 2.1, we infer that T θ (·, t) L ∞ dt ≤ C.
(2.9)
0
On the other hand, it follows from Lemma 2.1 and the Cauchy-Schwarz inequality that x b · bx dy ≤ 2 |b| |bx |d x |b|2 = 2 0 √ bx ≤ 2 θ L ∞ b L 2 √ θ 2 L b x 1/2 ≤ Cθ L ∞ √θ 2 , L
which, by using Lemma 2.1 and (2.9), gives T b2L ∞ dt ≤ C.
(2.10)
0
Now, a calculation gives 1 1 p + |b|2 /2 1 |b|2 R = = + θ, Dt ρF ρF λ ρ F 2λ λF from which, (2.9) and (2.10), it follows that T T 1 R 1 2 dt exp 1 + θ ≤ b dt ∞ L ρF ∞ λF ∞ ρ F ∞ 2λ 0 0 0 L 0 L L ≤ C. Recalling the definition of F and (2.5), one obtains (2.6) immediately. As a consequence of Lemma 2.1, (2.6) and (2.9), we have θ L 2 (Q T ) ≤ C,
(2.11)
which, by integrating (1.5) over Q T and making use of the boundary conditions (1.6), yields (2.7).
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697
We can exploit the boundedness of the derivative of the temperature obtained so far to get higher integrability of the temperature as follows: Lemma 2.4. Assume that the heat conductivity k satisfies (1.8) for some q ≥ 1. Then, θ L 4−α (Q T ) + θx L 2−α/2 (Q T ) ≤ C k(θ )θx L β (Q T ) ≤ C
for any α ∈ (0, 1), for any 1 ≤ β < 1 + 1/(q + 2).
(2.12) (2.13)
Proof. Multiplying (1.5) by θ −α for α ∈ (0, 1), integrating by parts and applying (1.6), we get T k(θ )θx2 d xdt ≤ C, 0 < α < 1, (2.14) 1+α 0 θ which combined with (1.8) implies T θx2 d xdt ≤ C. α θ 0 Setting v := θ 1−α/2 , the above inequality is equivalent to T vx2 d xdt ≤ C. On the other hand, (2.11) implies
(2.15)
0
T
v 2 d xdt ≤ C.
(2.16)
θ L ∞ (0,T ;L 1 ) ≤ C,
(2.17)
v L ∞ (0,T ;L 1/(1−α/2) ) ≤ C.
(2.18)
0
From Lemma 2.1 and (2.6), we get
whence
By applying the Gagliardo-Nirenberg inequality (2−α)/(4−α)
2/(4−α)
v L (8−2α)/(2−α) ≤ Cv L 1/(1−α/2) v H 1
,
and making use of (2.15)–(2.18), one obtains θ L 4−α (Q T ) ≤ C. Interpolating the above estimate and (2.15), we obtain (2.12). To show (2.13), letting 1 ≤ r < q + 2, we take 0 < α < min{q + 2 − r, 1} in (2.14). Then, in view of the Sobolev imbedding theorem, the assumption (1.8), (2.14) and (2.17), we get T T T θ rL ∞ dt ≤ C θ r d xdt + C θ r −1 |θx |d xdt 0
0
0 T
2r −1+α k(θ )θ 2 θ ≤C + 1+αx d xdt θ rL−1 ∞ dt + C k(θ ) θ 0 0 T T 2r −2+α−q ≤ C +C θ rL−1 θ L ∞ dt, ∞ dt + C T
0
0
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which, by recalling the Young inequality and the fact that 2r − 2 + α − q < r , gives
T
0
θ rL ∞ dt ≤ C
for any 1 ≤ r < q + 2.
(2.19)
2(q+3)−2β(q+2) 1 , 1} in (2.14). q+2 , we take 0 < α < min{ β βq+(1+α)β − 1 < q + 2. Hence, we employ (2.14), (2.17), (1.8), 2−β
Now, letting 1 ≤ β < 1 +
Then, such α satisfies the Young inequality and (2.19) to arrive at
T 0
T β (1+α)β k(θ )θx2 d xdt + C |k(θ )| 2−β θ 2−β d xdt 1+α θ 0 0 T βq+(1+α)β ≤ C +C θ 2−β d xdt
|k(θ )θx |β d xdt ≤ C
T
0
≤ C +C ≤ C, which proves (2.13).
0
T
βq+(1+α)β
θ L ∞2−β
−1
dt
Lemma 2.5. Assume that the heat conductivity k satisfies (1.9). Then, θx L α (Q T ) ≤ C, ∀ α < 3/2, θ L 2α (Q T ) ≤ C, ∀ α < 3/2.
(2.20) (2.21)
Proof. Equation (2.20) is proved in [27] by applying the techniques in [25], and hence we omit its proof here. The estimate (2.21) follows from (2.17), (2.20) and the following Gagliardo-Nirenberg inequality: 1/2 1/2 (2.22) θ L 2α ≤ C θ L 1 θx L α + θ L 1 .
Remark 2.1. In [27, 10] the estimate (2.21) is not obtained, and thus the authors failed to prove (1.13). We shall see later that (2.21) is very important to get (1.13). Now, one can apply Lemmas 2.1–2.5, the Poincaré inequality, and the Sobolev imbedding theorem, and (2.10), (2.19) and (2.11) to the parabolic equations (1.2)–(1.5) to obtain a priori bounds on the time derivative of (ρ, u, b, w, θ ): Lemma 2.6. We have ρt L ∞ (0,T ;H −1 ) ≤ C, (ρu)t L 2 (0,T ;H −1 ) ≤ C, (ρw)t L 2 (0,T ;H −1 ) ≤ C, bt L 2 (0,T ;H −1 ) ≤ C, (ρθ )t L 1 (0,T ;W −1,β ) ≤ C for any 1 ≤ β < 1 + 1/(q + 2), where q is the same as in Lemma 2.4. We shall apply the following lemma on integration by parts, the proof of which can be found in [11]:
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Lemma 2.7 ([11]). Let H be a Hilbert space and V → H be dense in H . If u, v ∈ L p (0, T ; V ) with p ∈ (1, ∞), and u t , vt ∈ L q (0, T ; V ∗ ), 1/ p + 1/q = 1. Then, u, v ∈ C([0, T ], H ) and t
(u(t), v(t)) − (u(s), v(s)) = u (τ ), v(τ ) + u(τ ), v (τ ) dτ, s
where ·, · and (·, ·) denote the duality between V and V ∗ and the inner product in H , respectively. From this lemma, we can immediately deduce the following two elementary lemmas which will be used in our convergence proof. Lemma 2.8. Assume that u ∈ V2 (Q T ), u t ∈ L 2 (0, T ; H −1 ()), f ∈ L 2 (Q T ), u 0 ∈ L 2 (), and u solves u t + f x = 0, u|t=0 = u 0 in the sense of distributions, where V2 (Q T ) is the same as in Remark 1.1. Then, t 1 1 2 u dx − f u x d xds = u 2 d x, a.e. t ∈ (0, T ). 2 2 0 0 Lemma 2.9. Assume that u ∈ L ∞ (0, T ; L 2 ()), u t ∈ L 2 (Q T ), f ∈ L 2 (Q T ), u 0 ∈ L 2 (), and u solves u t + f = 0, u|t=0 = u 0 in the sense of distributions. Then, t 1 1 2 u dx + f ud xds = u 20 d x, a.e. t ∈ (0, T ). 2 2 0 3. Proof of Theorem 1.1 In this section we pass to the limit for (ρ, u, b, w, θ ) as μ → 0 in (1.1)–(1.7). First, it is easy to see by the uniform a priori estimates established in the last section and the compactness principle of Lions-Aubin and Simon [30] (also cf. [20, Lemma C.1] and the proof of (3.4) below) that one can extract a subsequence of (ρ, u, b, w, θ ), still denoted by (ρ, u, b, w, θ ) for simplicity, such that as μ → 0, ⎧ ρ ρ weakly-∗ in L ∞ (Q T ) and weakly in L p (Q T ), ∀ p ≥ 1, ⎪ ⎪ ⎪ ⎪ ρ → ρ in C([0, T ], L p − w) and hence in C([0, T ], H −1 ), ∀ p ≥ 1; ρ ≥ C, ⎪ ⎪ ⎪ ⎪ u u weakly-∗ in L ∞ (0, T ; L 2 ) and weakly in L 2 (0, T ; H 1 ), ⎪ ⎪ ⎨ b b weakly-∗ in L ∞ (0, T ; L 2 ) and strongly in L s (Q ), 0∀ s < 6, T (3.1) b → b in C([0, T ], L 2 − w) and hence in C([0, T ], H −1 ), ⎪ ⎪ ⎪ ⎪ b b weakly in L 2 (0, T ; H 1 ), w w weakly-∗ in L ∞ (0, T ; L 2 ), ⎪ ⎪ 0 ⎪ ⎪ θ θ weakly in L 4−α (Q T ) and θx θ x weakly in L 2−α/2 (Q T), ∀ α ∈ (0, 1). ⎪ ⎪ ⎩ θ θ weakly in L 2q (Q T ) and θx θ x weakly in L q (Q T ), ∀ q < 3/2.
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Therefore, using (3.1)1 , (3.1)3 , Lemma 2.6, and Lemma 5.1 in [21], we find that ρu ρ u weak-∗ in L ∞ (0, T ; L 2 ) and weakly in L s (Q T ), ∀ s < 6, ρu → ρ u in C([0, T ], L 2 − w) and hence in C([0, T ], H −1 ), which together with (3.1)3 results in ρu 2 ρ u 2 weakly in L 2 (Q T ), whence (recalling ρu 2 ρu 2 weakly), ρu 2 = ρ u 2 .
(3.2)
On the other hand, by virtue of Lemmas 2.1–2.3 and the Sobolev imbedding theorem, T 2 0 u(t) L ∞ dt ≤ C. Therefore, T u 2 (t)2H 1 dt ≤ C, uniformly in μ, 0
which, together with (3.1)2 , implies ρu 2 ρ u 2 , that is, ρu 2 = ρ u 2 .
(3.3)
From (3.2) and (3.3) it follows that u 2 = u 2 , which yields u → u strongly in L 2 (Q T ). Therefore, if we apply the interpolation theory and the following estimate obtained by means of Lemmas 2.1–2.3, T T u6L 6 dt ≤ u2L ∞ (0,T ;L 2 ) u4L ∞ dt 0
T
≤C
0
≤ C,
0
0
1
|u∂x u|dy
2
dt
0 T
≤C
u2L 2 ∂x u2L 2 dt uniformly in μ,
we conclude
(3.4) u → u strongly in L s (Q T ), ∀ s < 6. Similarly, using (3.1)1 , (3.1)7 , (3.1)8 , Lemma 2.6, and Lemma 5.1 in [21], we infer that ρθ ρ θ weakly in L s (Q T ), ∀ s < 3, whence
2
ρθ 2 ρ θ weakly in L p (Q T ), ∀ p < 3/2. Equation (3.5), by taking into account Lemma 2.4, gives then
(3.5)
2
ρθ 2 = ρ θ . Thus, from Lemma 2.4 and the same arguments as for (3.4) it follows that θ → θ strongly in L s (Q T ), ∀ s < 3.
(3.6)
Now we are in a position to prove the following key lemma on the strong convergence of the density.
Vanishing Shear Viscosity Limit in MHD
Lemma 3.1.
701
ρ → ρ strongly in L p (Q T ) for any 1 ≤ p < ∞.
(3.7)
Proof. Multiplying (1.2) by φ ∈ D() and integrating over (0, x), then multiplying the resulting equation by ρψ(t), ψ(t) ∈ D(0, T ), and integrating over (0, T ) × , we obtain T 1 2 ψ(t)dt ρ Rρθ − λu x + |b| φd x 2 0 x T 1 2 2 ρu + Rρθ + |b| − λu x φx dξ d x ψ(t)dt ρ = 2 0 0 T x + ψt dt ρ ρuφdξ d x. (3.8)
0
0
On the other hand, with the help of (3.1)–(3.6), we have by taking μ → 0 in (1.1) and (1.2) that ρ t + (ρ u)x = 0, 1 (ρ u)t + ρ u 2 + Rρ θ + |b|2 = (λu x )x x 2
(3.9) in D (Q T ).
(3.10)
Using the system (3.9), (3.10), and arguing similarly to the derivation of (3.8), we get 1 2 ψ(t)dt ρ Rρ θ − λu x + |b| φd x 2 0 T x 1 = ψ(t)dt ρ ρ u 2 + Rρ θ + |b|2 − λu x φx dξ d x 2 0 0 T x + ψt dt ρ ρ uφdξ d x.
T
0
(3.11)
0
Now, making use of (3.1)–(3.6) and taking to the limit as μ → 0 in (3.8), and then comparing the resulting identity with (3.11), we conclude that T T ψdt ρ[Rρθ − λu x ]φd x = ψdt ρ[Rρθ − λu x ]φd x, 0
0
which yields Rρ 2 θ − λρu x = Rρ 2 θ − λρ u x . Consequently, ρu x − ρ u x =
R 2 R (ρ θ − ρ 2 θ ) = (ρ 2 − ρ 2 )θ ≥ 0. λ λ
(3.12)
Applying the idea of the renormalized solutions to (1.1) introduced by DiPerna and Lions [21], we can show that (ρ log ρ)t + (u ρ log ρ)x + ρu x = 0, (ρ log ρ)t + (u ρ log ρ)x + ρ u x = 0
(3.13) (3.14)
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J. Fan, S. Jiang, G. Nakamura
in the sense of distributions. In fact, since ρ is uniformly bounded in L ∞ (Q T ) and u ∈ L 2 (0, T ; H01 ), we get from Proposition 4.2 in [8] that ρ is a renormalized solution of (1.1), i.e., ρ satisfies ∂t η(ρ) + [η(ρ)u]x + [η (ρ)ρ − η(ρ)]u x = 0
in D (Q T )
(3.15)
for any η ∈ C 1 (R), η (z) = 0 for z large enough. It is not difficult to verify that one can take η(z) = z log z in (3.15) by an approximate argument and the uniform a priori estimates established for ρ and u. Thus, we have (ρ log ρ)t + (uρ log ρ)x + ρu x = 0. Letting μ → 0 in the above equation and making use of (3.1)–(3.6), we obtain (3.13) immediately. Similarly, the limit functions ρ, u are still a renormalized solution to (3.9). That is, (3.15) with (ρ, u) replaced by (ρ, u) is still valid, and by an approximation one can take η(z) = z log z, and hence, Eq. (3.14) holds. Subtraction of (3.14) from (3.13) leads to (ρ log ρ − ρ log ρ)t + [u(ρ log ρ − ρ log ρ)]x + (ρu x − ρ u x ) = 0.
(3.16)
Consider a sequence of functions φm ∈ C0∞ (), such that 0 ≤ φm ≤ 1, φm (x) = 1 for all x such that dist(x, ∂) ≥ m −1 , |∂x φm (x)| ≤ 2m and dist(x, ∂)|∂x φm (x)| ≤ 2 for all x ∈ , φm (x) → 1 as m → ∞ for all x ∈ . Now, testing (3.16) by φm (x) and then sending m → ∞, employing (3.12), the Hardy inequality and (3.1)2 , we obtain (ρ log ρ − ρ log ρ)(x, t)d x ≤ (ρ log ρ − ρ log ρ)(x, 0)d x = 0,
which concludes ρ log ρ = ρ log ρ since ρ log ρ ≥ ρ log ρ a.e. on Q T . This identity, together with, for example, Theorem 2.11 in [8], yields ρ → ρ a.e. on Q T , which combined with the Egorov theorem proves the lemma. Remark 3.1. In fact, if we employ the same arguments as in [21, Step 5, pp. 23–24] and the interpolation theory, we can show that ρ→ρ
strongly in L ∞ (0, T ; L p ) for any 1 ≤ p < ∞.
By virtue of Lemma 3.1, it is easy to prove that ⎧ ρw ρ w weakly in L r (Q T ) (∀ r < 2) and weak-∗ in L ∞ (0, T ; L 2 ), ⎪ ⎪ ⎪ ⎪ ⎨ ρw ρ w in C([0, T ], L 2 − w) and hence in C([0, T ], H −1 ), ρuw ρ u w and weakly in L r0 (Q T ) for some r0 > 1, ⎪ ⎪ ρθ u ρ θ u weakly in L r1 (Q T ) for some r1 > 1, ⎪ ⎪ ⎩ ub → u b strongly in L 2 (Q T ),
(3.17)
and k(ρ, θ )θx = k(ρ, θ )θ x .
(3.18)
Vanishing Shear Viscosity Limit in MHD
703
Thus, by taking to the limit as μ → 0 in (1.1)–(1.5) and using (3.1), (3.17) and (3.18), the limit functions ρ, u, w, b, θ satisfy the following equations in the sense of distributions: ρ t + (ρ u)x 1 (ρ u)t + ρ u 2 + p + |b|2 x 2 (ρ w)t + (ρ u w − b)x bt + (u b − w)x
= 0,
(3.19)
= (λu x )x ( p = Rρ θ),
(3.20)
= 0, = (νbx )x ,
(3.21) (3.22)
(ρ θ )t + (ρ θ u)x − (k(ρ, θ )θ x )x = λu 2x + μ|wx |2 + ν|bx |2 − pu x ,
(3.23)
where μ|wx |2 , the weak limit of μ|wx |2 as μ → 0 in the space of signed Radon measures on Q T , is a nonnegative Radon measure on Q T . Next, we show the strong convergence of u x and w by applying the Lagrangian coordinates. To this end, we rewrite (1.2) in Lagrangian coordinates and applying Lemma 2.8 to the resulting equation (without the convection term), and then transform the resulting identity again to that in Eulerian coordinates, that is, 1 2
t
(ρu )(x, t)d x + λ u 2x d xds 0 t 1 1 t 2 2 = ρ0 u 0 d x + |b| u x d xds + Rρθ u x d xds. 2 2 0 0 2
(3.24)
In the same way, we have by transforming Eq. (3.20) to that in Lagrangian coordinates that t 1 ρ u2d x + λ u 2x d xds 2 0 t 1 1 t = ρ0 u 20 d x + |b|2 u x d xds + Rρ θ u x d xds. (3.25) 2 2 0 0 Thus, we take to the limit in (3.24) and apply (3.1)3 , (3.1)4 , (3.6) and (3.7) to the righthand side of (3.24), and then compare the resulting identity with (3.25), to deduce that 1 2
ρ u dx + λ 2
t 0
u 2x d xds
t 1 2 ρu d x + λ = lim u 2x d xds μ→0 0 2 t 1 2 ρu d x + λ u 2x d xds = lim sup μ→0 2 0 t 1 2 ρu d x + λ lim sup u 2x d xds, ≥ lim inf μ→0 2 μ→0 0 (3.26)
where for (3.26)2 the fact that the limit in (3.26)1 exists is also used. Since by convexity (see [8, Corollary 2.2]), 1 1 lim inf ρu 2 d x ≥ ρ u 2 d x, μ→0 2 2
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J. Fan, S. Jiang, G. Nakamura
we conclude from (3.26) that t lim sup μ→0
0
u 2x d xds
≤
t
0
u 2x d xds,
(3.27)
which, together with the convexity and the lower semicontinuity of weak convergence, results in T T u 2x d xds = u 2x d xds, lim μ→0 0
0
and consequently, u → u strongly in L 2 (0, T ; H01 ).
(3.28)
Now, transforming Eq. (1.3) into that in Lagrangian coordinates and applying Lemma 2.8 to the resulting equation, then changing the resulting identity to that in Eulerian coordinates, we obtain t t 1 1 ρ|w|2 d x + μ |wx |2 d xds − bx · wd xds = ρ0 |w0 |2 d x. (3.29) 2 2 0 0 Again applying Lemma 2.8 to (1.4) directly, we find that t t t 1 1 2 2 2 |b| d x+ν |bx | d xds+ bx ·w d xds = |b0 | d x+ ub · bx d xds. 2 2 0 0 0 (3.30) Summing up (3.29) and (3.30), one arrives at t t 1 ρ|w|2 d x + μ |wx |2 d xds + ν |bx |2 d xds 2 0 0 t 1 1 1 2 2 = ρ0 |w0 | d x + |b0 | d x + ub · bx d xds − |b|2 d x. (3.31) 2 2 2 0 Rewriting (3.21) in Lagrangian coordinates and applying Lemma 2.9, we argue in the same manner as in the derivation of (3.24) to infer t 1 1 2 ρ |w| d x − bx · w d xds = ρ0 |w0 |2 d x. (3.32) 2 2 0 Applying Lemma 2.8 to (3.22) directly, we get t t 1 2 2 |b| d x + ν |bx | d xds + bx · wd xds 2 0 0 t 1 = |b0 |2 d x + u b · bx d xds. 2 0 Adding the above identity to (3.32), one gets t 1 ρ |w|2 d x + ν |bx |2 d xds 2 0 t 1 1 1 2 = ρ0 |w0 | d x + |b0 |2 d x + u b · bx d xds − |b|2 d x. (3.33) 2 2 2 0
Vanishing Shear Viscosity Limit in MHD
705
Now, letting μ → 0 in (3.31), applying (3.1)4 and (3.7), and comparing the resulting identity with (3.33), we deduce t t 1 ρ|w|2 d x + μ |wx |2 d xds + ν |bx |2 d xds + |b|2 d x μ→0 2 2 0 0 t 1 1 ρ |w|2 d x + ν |bx |2 d xds + |b|2 d x. (3.34) = 2 2 0 lim
1
Since
lim inf μ→0
ρ|w|2 d x ≥
ρ w2 d x, lim inf μ→0
|b|2 d x ≥
|b|2 d x, a.e. t, (3.35)
the identity (3.34) gives (cf. the proof of (3.27))
T
lim sup μ→0
0
T
|bx | d x ≤ 2
0
|bx |2 d x,
which, recalling (3.1)4 and the proof of (3.28), implies b → b strongly in L 2 (0, T ; H01 ).
(3.36)
Substitution of (3.35) and (3.36) into (3.34) leads to μ|wx |2 = 0. Moreover, if we integrate (3.29) with respect to t, take to the limit and make use of (3.36), and compare then the resulting equation with (3.32), we easily obtain
T
lim
μ→0 0
T
ρ|w| d xds = 2
0
ρ|w|2 d xds.
(3.37)
On the other hand, note that |w − w|2 ≤ Cρ|w − w|2 = ρ|w|2 + ρ|w|2 − 2ρw · w. Integrating the above inequality over Q T , taking then to the limit as μ → 0, and using (3.37), (3.1)1 and (3.17)1 , we conclude that w → w strongly in L 2 (Q T ). Thus, the convergence part of Theorem 1.1 is proved. To prove the existence of global weak solutions to (1.1)–(1.7), one first mollifies the initial data, such that a global strong solution exists by [2], then one can take to the limit, and argue in the same manner as that used in the proof of the convergence part to show that the limit functions are indeed a weak solution of (1.1)–(1.7). The proof of Theorem 1.1 is complete.
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4. A Remark on the Cylindric Symmetry Case In this section we briefly discuss the following problem in the domain Q T := (0, T ) × with := {x : 0 < a < x < b} which corresponds to the multidimensional Navier–Stokes equations for compressible heat-conducting fluids in the case of cylindric symmetry: ρu ρt + (ρu)x + = 0, x v2 u + R(ρθ )x − (λ + 2μ) u x + = 0, ρ u t + uu x − x x x uv v − μ vx + = 0, ρ vt + uvx + x x x wx = 0, ρ(wt + uwx ) − μ wx x + x θx u cV ρ(θt + uθx ) − k θx x + + p ux + − Q = 0, x x
(4.1) (4.2) (4.3) (4.4) (4.5)
together with initial and boundary conditions
where
(u, v, w, θx )|∂ = 0, (ρ, u, v, w, θ )|t=0 = (ρ0 , u 0 , v0 , w0 , θ0 ),
(4.6) (4.7)
u 2 u 2 v 2 , Q := λ u x + + μ vx − + wx2 + 2u 2x + 2 x x x
(4.8)
x denotes radial variable, the velocity vector V := (u, v, w) is given by the radial, angular and axial velocities, respectively, and γ , λ, μ, cV and k are the physical positive constants. The zero shear viscosity limit for (4.1)–(4.6) has been studied by Frid and Shelukhin in [10]. More precisely, assuming that initial data satisfy (ρ0−1 , θ0−1 ) L ∞ < ∞, ρ0 > 0, θ0 > 0
¯ ρ0 , θ0 ∈ H 1 (), u 0 , v0 , w0 ∈ H01 (), on ,
they first prove in [27] that the problem (4.1)–(4.6) possesses a unique strong solution. Moreover, the following uniform in μ estimates hold: C −1 ≤ ρ ≤ C, ρx L ∞ (0,T ;L 1 ) ≤ C, ρt L 1 (Q T ) ≤ C, T (u 2x + μvx2 + μwx2 )d xdt ≤ C, u L ∞ (0,T ;L 2 ) ≤ C, 0
(4.9) (4.10)
uu x L 3/2 (Q T + u L 6 (Q T ) + u L 4 (0,T ;L ∞ ) ≤ C, (v, w) L ∞ (Q T ) ≤ C, (4.11) θ ≥ C > 0, θ L 2 (Q T ) ≤ C, θx L q (Q T ) ≤ C, ∀ q < 3/2.
(4.12)
Based on the estimates (4.9)–(4.12), Frid and Shelukhin [10] prove that the strong solution (ρ, u, v, w, θ ) converges to (ρ, u, v, w, θ ) which is a solution to (4.1)–(4.7) with μ = 0 in the sense of distributions. Using the techniques in Sect. 3, we can prove under weaker conditions on initial data that
Vanishing Shear Viscosity Limit in MHD
707
Theorem 4.1. Assume that inf ρ0 > 0, ρ0 ∈ L ∞ (), u 0 , v0 , w0 ∈ L 2 (), θ0 ∈ L 1 (), inf θ0 > 0.
(4.13)
Then, the following uniform in μ estimates hold: C −1 ≤ ρ ≤ C, ρt L ∞ (0,T ;H −1 ) ≤ C, u L ∞ (0,T ;L 2 ()) + u L 2 (0,T ;H 1 ) ≤ C, 0 √ (ρu)t L 2 (0,T ;H −1 ) ≤ C, v L ∞ (0,T ;L 2 ) + μv L 2 (0,T ;H 1 ) ≤ C, 0 √ (ρv)t L 2 (0,T ;H −1 ) ≤ C, w L ∞ (0,T ;L 2 ) + μw L 2 (0,T ;H 1 ) ≤ C, 0
(ρw)t L 2 (0,T ;H −1 ) ≤ C, θ L ∞ (0,T ;L 1 ) + θ L 2q (Q T ) + θx L q (Q T ) ≤ C, ∀ q < 3/2, (ρθ )t L 1 (0,T ;H −1 ) ≤ C. Moreover, as μ → 0, a weak solution (ρ, u, v, w, θ ) to (4.1)–(4.7) converges to (ρ, u, v, w, θ ) which is a solution to (4.1)–(4.7) with μ = 0 in the sense of distributions. Remark 4.1. Applying the method in [15, 6], one can prove the existence of a unique solution (ρ, u, v, w, θ ) to (4.1)–(4.7) with μ > 0 or μ = 0 under the conditions: C −1 ≤ ρ0 ≤ C, u 0 , v0 , w0 ∈ L 4 (), θ0 ∈ L 2 (), inf θ0 > 0. Remark 4.2. By applying the techniques of Sect. 3, a similar convergence result could be obtained for a free-boundary problem for the equations of compressible fluids investigated in [29]. References 1. Cabannes, H.: Theoretical Magnetofluiddynamics. New York-London: Academic Press, 1970 2. Chen, G.Q., Wang, D.: Existence and continuous dependence of large solutions for the magnetohydrodynamics equations. Z. Angew. Math. Phys. 54, 608–632 (2003) 3. Chen, G.Q., Wang, D.: Global solutions of nonlinear magnetohydrodynamics with large initial data. J. Diff. Eqs. 182, 344–376 (2002) 4. Fan, J., Jiang, S.: Zero shear viscosity limit for the Navier–Stokes equations of compressible isentropic fluids with cylindric symmetry. Rend. Sem. Mat. Univ. Pol. Torino (in press) 5. Fan, J., Jiang, S., Nakamura, G.: Stability of weak solutions to equations of magnetohydrodynamics with Lebesgue initial data. Preprint, 2005 6. Fan, J.: Stability of spherically symmetric weak solutions to the Navier–Stokes equations of compressible heat conductive fluids in bounded annular domains. Submitted, 2005 7. Feireisl, E.: On compactness of solutions to the compressible isentropic Navier-Stokes equations when the density is not square integrable. Comment. Math. Univ. Carlolinae 42, 83–98 (2001) 8. Feireisl, E.: Dynamics of Viscous Compressible Fluids. Oxford: Oxford Univ. Press, 2004 9. Frid, H., Shelukhin, V.V.: Boundary layers for the Navier–Stokes equations of compressible fluids. Comm. Math. Phys. 208, 309–330 (1999) 10. Frid, H., Shelukhin, V.V.: Vanishing shear viscosity in the equations of compressible fluids for the flows with the cylinder symmetry. SIAM J. Math. Anal. 31, 1144–1156 (2000) 11. Gajewski, H., Gröger, K., Zacharias, K.: Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen, Berlin: Akademil Verlag, 1974 12. Hoff, D.: Global solutions of the Navier–Stokes equations for multidimensional compressible flow with discontinuous initial data. J. Diff. Eqns. 120, 215–254 (1995) 13. Hoff, D., Tsyganov, E.: Uniqueness and continuous dependence of weak solutions in compressible magnetohydrodynamics. Z. Angew. Math. Phys. 56, 791–804 (2005) 14. Jiang, S., Zhang, P.: On spherically symmetric solutions of the compressible isentropic Navier–Stokes equations. Commun. Math. Phys. 215, 559–581 (2001)
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15. Jiang, S., Zlotnik, A.: Global well-posedness of the Cauchy problem for the equations of a one-dimensional viscous heat-conducting gas with Lebesgue initial data. Proc. Royal Soc. Edinburgh 134A, 939–960 (2004) 16. Kazhikhov, A.V., Smagulov, Sh.S.: Well-posedness and approximation methods for a model of magnetogasdynamics. Izv. Akad. Nauk Kazakh. SSR Ser. Fiz.-Mat. 5, 17–19 (1986) 17. Kawashima, S., Okada, M.: Smooth global solutions for the one-dimensional equations in magnetohydrodynamics. Proc Japan Acad. Ser. A, Math. Sci. 58, 384–387 (1982) 18. Landau, L.D., Lifshitz, E.M., Pitaevskii, L.P.: Electrodynamics of Continuous Media, 2nd ed., London: Butterworth-Heinemann, 1999 19. Li, T.-T., Qin, T.: Physics and Partial Differential Equations, 2nd. Volume 1, Beijing: Higher Education Press, 2005 (in Chinese) 20. Lions, P.-L.: Mathematical Topics in Fluid Dynamics. Volume 1, Incompressible Models, Oxford: Oxford Science Publication, 1996 21. Lions, P.-L.: Mathematical Topics in Fluid Dynamics. Volume 2, Compressible Models, Oxford: Oxford Science Publication, 1998 22. Liu, T.-P., Zeng, Y.: Long-time behavior of solutions for general quasilinear hyperbolic-parabolic systems of conservation laws. Mem. Amer. Math. Soc. 599, Providence, RI: Amer. Math. Soc. 1997 23. Moreau, R.: Magnetohydrodynamics. Dordrecht: Klumer Academic Publishers, 1990 24. Polovin, R.V., Demutskii, V.P.: Fundamentals of Magnetohydrodynamics. New York: Consultants. Bureau, 1990 25. Rakotoson, J.M.: A compactness lemma for quasilinear problem: application to parabolic equations. J. Funct. Anal. 106, 358–374 (1992) 26. Serre, D.: Solutions faibles globales des équations de Navier-Stokes pour un fluide compressible. C. R. Acad. Sci. Paris. 303, 639–642 (1986) 27. Shelukhin, V.V.: The limit of zero shear viscosity for compressible fluids. Arch. Rat. Mech. Appl. 143, 357–374 (1998) 28. Shelukhin, V.V.: A shear flow problem for the compressible Navier-Stokes equations. Int. J. Non-Linear Mech. 33, 247–257 (1998) 29. Shelukhin, V.V.: Vanishing shear viscosity in a free-boundary problem for the equations of compressible fluids. J. Diff. Eqs. 167, 73–86 (2000) 30. Simon, J.: Compact sets and the space L p (0, T ; B). Ann. Mat. Pura Appl. 146, 65–96 (1987) 31. Smagulov, Sh.S., Durmagambetov, A.A., Iskenderova, D.A.: The Cauchy problem for the equations of magnetogasdynamics. Diff. Eqs. 29, 278–288 (1993) 32. Ströhmer, G.: About compressible viscous fluid flow in a bounded region. Pacific J. Math. 143, 359–375 (1990) 33. Vol’pert, A.I., Hudjaev, S.I.: On the Cauchy problem for composite systems of nonlinear differential equations. Math. USSR-Sb. 16, 517–544 (1972) Communicated by P. Constantin
Commun. Math. Phys. 270, 709–725 (2007) Digital Object Identifier (DOI) 10.1007/s00220-006-0168-0
Communications in
Mathematical Physics
Poisson Quasi-Nijenhuis Manifolds Mathieu Stiénon1, , Ping Xu2, 1 Departement Mathematik, E.T.H. Zürich, Rämistrasse 101, 8092 Zürich, Switzerland.
E-mail:
[email protected]
2 Department of Mathematics, Pennsylvania State University, University Park, PA 16802, USA.
E-mail:
[email protected] Received: 1 April 2006 / Accepted: 18 September 2006 Published online: 9 January 2007 – © Springer-Verlag 2006
Abstract: We introduce the notion of Poisson quasi-Nijenhuis manifolds generalizing Poisson-Nijenhuis manifolds of Magri-Morosi. We also investigate the integration problem of Poisson quasi-Nijenhuis manifolds. In particular, we prove that, under some topological assumption, Poisson (quasi)-Nijenhuis manifolds are in one-one correspondence with symplectic (quasi)-Nijenhuis groupoids. As an application, we study generalized complex structures in terms of Poisson quasi-Nijenhuis manifolds. We prove that a generalized complex manifold corresponds to a special class of Poisson quasi-Nijenhuis structures. As a consequence, we show that a generalized complex structure integrates to a symplectic quasi-Nijenhuis groupoid, recovering a theorem of Crainic. Contents 1. 2. 3. 4. 5. 6. 7.
Introduction . . . . . . . . . . . . . . Preliminaries . . . . . . . . . . . . . . Poisson Quasi-Nijenhuis Manifolds . . Universal Lifting Theorem . . . . . . . Symplectic Nijenhuis Groupoids . . . Symplectic Quasi-Nijenhuis Groupoids Generalized Complex Structures . . . .
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1. Introduction Poisson Nijenhuis structures were introduced by Magri and Morosi [16, 18] in their study of bi-Hamiltonian systems, and intensively studied by many authors [12, 21]. Recall that a Poisson Nijenhuis manifold consists of a triple (M, π, N ), where M is a Francqui fellow of the Belgian American Educational Foundation. Research supported by NSF grant DMS03-06665 and NSA grant 03G-142.
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manifold endowed with a Poisson bivector field π , and a (1, 1)-tensor N whose Nijenhuis torsion vanishes, i.e. [N X, N Y ] − N ([N X, Y ] + [X, N Y ] − N [X, Y ]) = 0,
∀X, Y ∈ X(M),
together with some compatibility condition between π and N . Poisson Nijenhuis structures are very important in the study of integrable systems since they produce bi-Hamiltonian systems [16, 12]. As observed by Kosmann-Schwarzbach [11], given a Poisson Nijenhuis manifold (M, π, N ), ((T ∗ M)π , (T M) N ) constitutes a Lie bialgebroid, where (T ∗ M)π is equipped with the standard cotangent Lie algebroid structure induced by the Poisson tensor π while (T M) N is the deformed Lie algebroid on T M induced by the Nijenhuis endomorphism N . Indeed it is proved in [11] that the Lie bialgebroid condition on ((T ∗ M)π , (T M) N ) is equivalent to the triple (M, π, N ) being Poisson Nijenhuis. The main goal of the present paper is to introduce the notion of Poisson quasiNijenhuis structures. By definition, a Poisson quasi-Nijenhuis manifold is a quadruple (M, π, N , φ), where M is a manifold endowed with a Poisson bivector field π , a (1, 1)-tensor N and a closed 3-form φ such that π and N are compatible (in the usual Poisson-Nijenhuis sense) and [N X, N Y ] − N ([N X, Y ] + [X, N Y ] − N [X, Y ]) = π (i X ∧Y φ),
∀X, Y ∈ X(M).
Recall that Lie bialgebroids are pairs of transverse Dirac structures in a Courant algebroid [13]. When one of the two maximal isotropic direct summands fails to be Courant involutive, this becomes a quasi-Lie bialgebroid [20, 19]. Alternatively, a quasiLie bialgebroid is equivalent to the following data: a Lie algebroid A together with a degree 1 derivation δ of the associated Gerstenhaber algebra (∧• A), ∧, [·, ·] such that δ 2 = [φ, ·] and δφ = 0 for some φ ∈ (∧3 A) [9]. We prove Theorem A. Given (M, π, N , φ), the following are equivalent • (M, π, N , φ) is a Poisson quasi-Nijenhuis manifold; • (T ∗ M)π , (T M) N , φ is a quasi-Lie bialgebroid. It is well known that the global object corresponding to a Poisson manifold is a symplectic groupoid [2, 22]. It is natural to ask what is the global object integrating a Poisson Nijenhuis manifold. We prove Theorem B. The base manifold of a symplectic Nijenhuis groupoid is a Poisson Nijenhuis manifold. Moreover, there is a one-one correspondence between t-connected ) and integrable and t-simply connected symplectic Nijenhuis groupoids ( ⇒ M, ω, N Poisson Nijenhuis manifolds (M, π, N ). By a symplectic Nijenhuis groupoid, we mean a symplectic groupoid ( ⇒ M, ω) : T → T such that (, ) is a equipped with a multiplicative (1, 1)-tensor N ω, N symplectic Nijenhuis structure. The main idea of the proof of Theorem B can be outlined as follows. One proves that Poisson Nijenhuis structures on a manifold M are in one-one correspondence with Lie bialgebroids ((T ∗ M)π , δ) satisfying the condition that [δ, d] = 0, where d is the de Rham differential on M. The latter are the infinitesimal of symplectic Nijenhuis groupoids, as can be shown using the universal lifting theorem [9]. The same method can be used to prove an analogous result for Poisson quasiNijenhuis manifolds.
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Theorem C. The base manifold of a symplectic quasi-Nijenhuis groupoid is a Poisson quasi-Nijenhuis manifold. Moreover there is a one-one correspondence between t-con, nected and t-simply connected symplectic quasi-Nijenhuis groupoids ⇒ M, ω, N ∗ ∗ t φ − s φ and integrable Poisson quasi-Nijenhuis manifolds (M, π, N , φ). A symplectic quasi-Nijenhuis groupoid is a symplectic groupoid ( ⇒ M, ω) : T → T and a closed 3-form equipped with a multiplicative (1, 1)-tensor N , t ∗ φ − s ∗ φ is a symplectic quasi-Nijenhuis structure. φ ∈ 3 (M) such that , ω, N As an application, we study generalized complex structures in terms of Poisson quasiNijenhuis structures. The notion of generalized complex structures was introduced by Hitchin [8] and studied by Gualtieri [7] motivated by the study of mirror symmetry. It comprises both symplectic and complex structures as extreme cases. We show that on a generalized complex manifold (M, J ), where N π J= σ −N ∗ with N 2 + π σ = − id, the building units π , N and σ of J do exactly determine a Poisson quasi-Nijenhuis structure. Indeed, the endomorphism N can be used to define a derivation d N of the Gerstenhaber algebra associated to the Lie algebroid (T ∗ M)π . We prove Theorem D. The following are equivalent • J is a generalized complex structure; • (M, π, N , dσ ) is a Poisson quasi-Nijenhuis structure such that J
(T M) N ⊕ (T ∗ M)π − → T M ⊕ T ∗M is a Courant algebroid isomorphism. A similar result (in a different form) was already proved by Crainic using a direct argument [4]. Since a generalized complex structure corresponds to a quasi-Nijenhuis manifold according to Theorem D, as a consequence, we prove Theorem E. Let J be a generalized complex structure as given by Eq. (18), and ( ⇒ ∗ M, ω) a t-connected and t-simply connected symplectic groupoid integrating (T ∗ M)π . Then there is a multiplicative (1, 1)-tensor N on such that ⇒ M, ω, N , t dσ − s ∗ dσ is a symplectic quasi-Nijenhuis groupoid. This result, in a disguised form, was already proved by Crainic [4] using a different method. Notations. We denote the bracket on the sections of a Courant algebroid by ·, ·, except for the standard Courant bracket on T M ⊕ T ∗ M, which is denoted by ·, ·. The Lie bracket of vector fields and its extension to polyvector fields (i.e. the Schouten bracket) are denoted by [·, ·]. Any bundle map B : T ∗ M → T M induces a bracket on the space of 1-forms (see Eq. (8)). It is denoted by [·, ·] B as well as its extension to the space of differential forms of all degrees. Finally, if ·, · is a bracket on the space of sections of a vector bundle E of which J is a bundle endomorphism, then its deformation by J is denoted by ·, · J (see Eq. (19)).
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2. Preliminaries Definition 2.1 ([13]). A Courant algebroid is a triple consisting of • a vector bundle E → M equipped with a non-degenerate symmetric bilinear form ·, · , • a skew-symmetric bracket ·, · on (E), and ρ • a smooth bundle map E − → M called the anchor, which induces a natural differential operator D : C ∞ (M) → (E) defined by D f, A = 21 ρ(A) f for all f ∈ C ∞ (M) and A ∈ (E). These structures must be compatible in the following sense: ∀A, B, C ∈ (E) and ∀ f, g ∈ C ∞ (M), • ρ(A, B) = [ρ(A), ρ(B)], • A, B, C + B, C, A + C, A, B = 13 D A, B, C + B, C, A + C, A, B , • A, f B = f A, B + ρ(A) f B − A, B D f , • ρ ◦D = 0, i.e. D f, Dg = 0, • ρ(A)B, C = A, B + DA, B , C + B, A, C + DA, C . Note that a Courant algebroid is not a Lie algebroid as the Jacobi identity is not satisfied. Example 2.2 ([3]). The generalized tangent bundle T M ⊕ T ∗ M of a manifold M is a Courant algebroid, where the anchor is the projection onto the first component and the pairing and bracket are given, respectively, by X + ξ, Y + η = 21 ξ(Y ) + η(X ) , (1) 1 X + ξ, Y + η = [X, Y ] + L X η − LY ξ + 2 d ξ(Y ) − η(X ) , (2) ∀X, Y ∈ X(M), ∀ξ, η ∈ 1 (M). Definition 2.3. A Dirac structure is a smooth subbundle L of a Courant algebroid E, which is maximal isotropic with respect to ·, · and whose space of sections (L) is closed under ·, ·. It is thus naturally a Lie algebroid. It is well-known [23] that a Lie algebroid (A, [·, ·] A , ρ A ) gives rise to a Gerstenhaber algebra ((∧• A), ∧, [·, ·] A ), and a degree 1 derivation δ A of the graded commutative algebra ((∧• A∗ ), ∧) such that (δ A )2 = 0. Here δ A is given by (δ A α)(X 0 , X 1 , . . . , X n ) =
n
(−1)i (ρ A X i )α(X 0 , . . . , Xi , . . . , Xn)
i=0
+
(−1)i+j α([X i , X j ] A , X 0 ,. . . , X i ,. . . , X j ,. . . ,X n). (3) i<j
A Lie bialgebroid [15, 14] is a pair of Lie algebroid structures on A and its dual A∗ such that δ A∗ is a derivation of the Gerstenhaber algebra ((∧• A), ∧, [·, ·] A ) or, equivalently, such that δ A is a derivation of the Gerstenhaber algebra ((∧• A∗ ), ∧, [·, ·] A∗ ). Since the bracket [·, ·] A∗ can be recovered from the derivation δ A∗ , one is led to the following alternative definition.
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Definition 2.4. A Lie bialgebroid is a pair (A, δ) consisting of a Lie algebroid (A, [·, ·] A , ρ A ) and a degree 1 derivation δ of the Gerstenhaber algebra ((∧• A), ∧, [·, ·] A ) such that δ 2 = 0. More generally, we can speak about quasi-Lie bialgebroids [20, 9]. Definition 2.5 ([9]). A quasi-Lie bialgebroid is a triple (A, δ, φ) consisting of a Lie algebroid A, a degree 1 derivation δ of the Gerstenhaber algebra ((∧• A), ∧, [·, ·] A ) and an element φ ∈ (∧3 A) such that δ 2 = [φ, ·] A and δφ = 0. The link between Courant, Lie bi- and quasi-Lie bialgebroids is given by the following Theorem 2.6 ([13, 20, 19]). (i) There is a 1-1 correspondence between Lie bialgebroids and pairs of transversal Dirac structures in a Courant algebroid. (ii) There is a 1-1 correspondence between quasi Lie bialgebroids and Dirac structures with transversal isotropic complements in a Courant algebroid. Proof. The proof of (i) can be found in [13], and (ii) was proved in [20, 19]. Below we give an explicit formula describing such a correspondence, which will be needed later. Let (A, δ, φ) be a quasi Lie bialgebroid. Let ρ A∗ : A∗ → T M be the bundle map given by ρ A∗ (ξ )( f ) = ξ(δ f ),
∀ξ ∈ A∗ , ∀ f ∈ C ∞ (M).
Introduce a bracket on (A∗ ) by [ξ, η] A∗ (X ) = (ρ A∗ ξ )(ηX ) − (ρ A∗ η)(ξ X ) − (δ X )(ξ, η). (A∗ , ρ
∗ Note that A∗ , [·, ·] A∗ ) is in general not a Lie algebroid. Let E = A ⊕ A and ρ : E → T M be the bundle map
ρ(ξ + X ) = ρ A∗ (ξ ) + ρ A (X ). Define a non-degenerate symmetric pairing on E by ξ + X, η + Y = 21 ξ(Y ) + η(X ) , and a bracket ·, · on (E) by X, Y = [X, Y ] A , ξ, η = [ξ, η] A∗ + φ(ξ, η, ·), X, ξ = i X δ A∗ ξ + 21 δ A∗ (ξ X ) − i ξ δ A X + 21 δ A (ξ X ) ,
(4)
for all X, Y ∈ (A) and ξ, η ∈ (A∗ ). Here δ A∗ : (∧• A∗ ) → (∧•+1 A∗ ) is the derivation given by Eq. (3). Then (E, ·, · , ·, ·, ρ) is a Courant algebroid. Conversely, assume that (E, ·, · , ·, ·, ρ) is a Courant algebroid, and A is a Dirac structure with an isotropic complement B. The duality pairing A ⊗ B → R : X ⊗ ξ → 2ξ, X identifies B with A∗ . Let φ be the element in (∧3 A) defined by φ(ξ, η, ζ ) = 2ξ, η, ζ ,
∀ξ, η, ζ ∈ (B),
(5)
ρ B = ρ| B be the restriction of ρ to B and [·, ·] B be the bracket on (B) such that (6) ξ, η − [ξ, η] B ∈ (A), ∀ξ, η ∈ (B). • • ∗ •+1 •+1 ∗ ∼ ∼ Define a derivation δ : (∧ A)(= (∧ B )) → (∧ A)(= (∧ B ) as in Eq. (3). The triple (A, δ, φ) becomes a quasi-Lie bialgebroid.
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3. Poisson Quasi-Nijenhuis Manifolds Let M be a smooth manifold, π a Poisson bivector field, and N : T M → T M a (1, 1)-tensor. Definition 3.1 ([11]). The bivector field π and the tensor N are said to be compatible [12] if N ◦π = π ◦ N T where
and
CπN = 0,
(7)
CπN (α, β) := [α, β] N π − [N T α, β]π + [α, N T β]π − N T [α, β]π
and
[α, β] B := L Bα (β) − L Bβ (α) − d β(Bα)
(8)
for all α, β ∈ 1 (M) and any skew-symmetric bundle map B : T ∗ M → T M. The (1, 1)-tensor N is said to have zero Nijenhuis torsion if [N X, N Y ] − N [N X, Y ] + [X, N Y ] − N [X, Y ] = 0, ∀X, Y ∈ X(M). In [17], Magri and Morosi defined Poisson Nijenhuis manifolds as triples (M, π, N ) such that π and N are compatible and the Nijenhuis torsion of N vanishes. This definition is motivated by the following Fact 3.2 ([12, 21]). Assume that π ∈ X2 (M) is a Poisson tensor and N : T M → T M a (1, 1)-tensor on M. The tensor π N defined by π N (α, β) := β(N π α),
∀α, β ∈ 1 (M)
is skew-symmetric if, and only if, N ◦π = π ◦ N T . In this case, we have (i) [π, π N ] = 0 if CπN = 0; (ii) [π N , π N ] = 0 if the Nijenhuis torsion of N vanishes. Moreover the converse is true when π is non-degenerate. Hence, any Poisson Nijenhuis manifold (M, π, N ) is endowed with a bi-Hamiltonian structure (π, π N ), i.e. [π, π ] = 0,
[π, π N ] = 0,
[π N , π N ] = 0.
Similarly, one can define Poisson quasi-Nijenhuis manifolds. Let i N be the degree 0 derivation of ( • (M), ∧) defined by (i N α)(X 1 , . . . , X p ) =
p
α(X 1 , . . . , N X i , . . . , X p ),
∀α ∈ p (M).
i=1
Definition 3.3. A Poisson quasi-Nijenhuis manifold is a quadruple (M, π, N , φ), where π ∈ X2 (M) is a Poisson bivector field, N : T M → T M is a (1, 1)-tensor compatible with π , and φ is a closed 3-form on M such that [N X, N Y ] − N [N X, Y ] + [X, N Y ] − N [X, Y ] = π (i X ∧Y φ), ∀X, Y ∈ X(M) and i N φ is closed.
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It is well known that, on a Poisson manifold (M, π ), the bracket on 1 (M) associated to the bundle map π through Eq. (8) makes T ∗ M into a Lie algebroid with anchor π : T ∗ M → T M. The usual cotangent bundle will be denoted by (T ∗ M)π when equipped with this Lie algebroid structure. More precisely, we have the following Fact 3.4 ([2]). Let π be a bivector field on M. Then [π, π ] = 0 if, and only if, (T ∗ M)π is a Lie algebroid. On the other hand, defining a bracket [·, ·] N on X(M) by [X, Y ] N = [N X, Y ] + [X, N Y ] − N [X, Y ],
∀X, Y ∈ X(M)
as in [11], and considering N : T M → T M as an anchor map, we obtain a degree 1 derivation d N of ( • (M), ∧) inspired by Eq. (3): (d N α)(X 0 , X 1 , . . . , X n ) =
n (−1)i (N X i )α(X 0 , . . . , Xi , . . . , Xn) i=0
+
(−1)i+j α([X i , X j ] N , X 0 , . . . , Xi , . . . , X j , . . . , X n).
i< j
(9) Moreover, as proved in [11], we have the following identity d N = [i N , d] = i N ◦d − d ◦i N .
(10)
The following proposition extends a result of Kosmann-Schwarzbach [11, Prop. 3.2]. Proposition3.5. The quadruple (M, π, N , φ) is a Poisson quasi-Nijenhuis manifold if, and only if, (T ∗ M)π , d N , φ is a quasi Lie bialgebroid and φ is a closed 3-form. This is an immediate consequence of Fact 3.4 and the following two lemmas. Lemma 3.6 ([11, Proposition 3.2]). Assume that π ∈ X2 (M) is a Poisson tensor and N : T M → T M a (1, 1)-tensor on M. The differential d N is a derivation of the graded Lie algebra ( • (M), [·, ·]π ) if, and only if, π and N are compatible. Lemma 3.7. Let (M, π ) be a Poisson manifold and N : T M → T M a (1, 1)-tensor compatible with π . Then d N2 = [φ, ·]π if, and only if, [N X, N Y ] − N ([N X, Y ] + [X, N Y ] − N [X, Y ]) = π (i X ∧Y φ),
∀X, Y ∈ X(M)
and π # ◦(dφ) = 0, where (dφ) : ∧3 T M → T ∗ M is the bundle map defined by (dφ) (u, v, w) = i u∧v∧w dφ, ∀u, v, w ∈ T M. Proof. It follows from an easy computation that 2 d N f − [φ, f ]π (X, Y ) = (d f ) [N X, N Y ] − N ([N X, Y ] + [X, N Y ] − N [X, Y ]) − π (i X ∧Y φ) for all f ∈ C ∞ (M). Moreover, since d ◦d N + d N ◦d = 0, one has d N2 (d f ) − [φ, d f ]π = d(d N2 f ) − d[φ, f ]π − [dφ, f ]π = d(d N2 f − [φ, f ]π ) + [dφ, f ]π .
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Hence, d N2 − [φ, ·]π vanishes on 0- and exact 1-forms if, and only if, [N X, N Y ] − N ([N X, Y ] + [X, N Y ] − N [X, Y ]) = π (i X ∧Y φ),
∀X, Y ∈ X(M)
and [dφ, f ]π = 0, ∀ f ∈ C ∞ (M). The latter is easily seen to be equivalent to π # ◦(dφ) = 0. And in this case, since both d N2 and [φ, ·]π are derivations with re spect to ∧, we get d N2 = [φ, ·]π . As an immediate consequence, we obtain the following result of Kosmann-Schwarzbach [11]. Corollary 3.8. The triple (M, π, N ) is a Poisson Nijenhuis manifold if, and only if, ((T ∗ M)π , d N ) is a Lie bialgebroid. We now turn our attention to the particular case where the Poisson bivector field π is non-degenerate. Together with Lemma 3.6, the following two lemmas give another proof of the equivalence between the relation [π, π N ] = 0 and the compatibility condition (7) when π is non-degenerate (see Fact 3.2). Lemma 3.9. Assume that π ∈ X2 (M) is a Poisson tensor and N : T M → T M a (1, 1)-tensor on M. Then π N is a bivector field such that [π, π N ] = 0 if, and only if, all the squares in the following diagram commute. 0
/ C ∞ (M)
0
/ C ∞ (M)
dN
/ 1 (M)
dN
π
id
[π N ,·]
/ X1 (M)
dN
/ 2 (M) π
[π N
/ X2 (M) ,·]
/ 3 (M)
dN
/ ···
π
[π N
/ X3 (M) ,·] [π
(11) N ,·]
/ ···
Proof. We have π N T = N π (i.e. π N is a bivector field) if, and only if, ∀ f ∈ C ∞ (M), π N T d f = N π d f
⇔ π i N d f = πN d f ⇔ π d N f = [π N , f ].
(12)
And [π N , π ] = 0 is equivalent to [π N , π ] (d f ) = 0 ⇔ [[π N , π ], f ] = 0 ⇔ [[π N , f ], π ] + [π N , [π, f ]] = 0
⇔ [π N d f, π ] + [π N , π d f ] = 0 ⇔ [π, π N T d f ] = [π N , π d f ] ⇔ [π, π (i N d f )] = [π N , π d f ] ⇔ π d(i N d f ) = [π N , π d f ] ⇔ π d N (d f ) = [π N , π d f ]
(13)
for all f ∈ C ∞ (M). Since both π ◦d N and [π N , π (·)] are derivations of ( • (M), ∧), the equivalence follows from Eqs. (12)–(13).
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Lemma 3.10. Assume that π ∈ X2 (M) is a non-degenerate Poisson tensor, and N : T M → T M is a (1, 1)-tensor on M. If π N is a bivector field and Diagram (11) commutes, then d N is a derivation of [·, ·]π . Proof. Since π is Poisson, we have π [α, β]π = [π α, π β],
∀α, β ∈ • (M).
Then, the Jacobi identity for the Schouten bracket gives [π N , π [α, β]π ] = [[π N , π α], π β] + [π α, [π N , π β]], which can be rewritten as π d N [α, β]π = π [d N α, β]π + [α, d N β]π since π ◦d N = [π N , π (·)]. The conclusion follows from the invertibility of π .
The previous lemmas are used to prove the following Proposition 3.11.
(i) Let (M, π, N , φ) be a Poisson quasi-Nijenhuis manifold. Then, [π, π N ] = 0,
(14)
[π N , π N ] = 2π (φ).
(15)
and X2 (M)
(ii) Conversely, assume that π ∈ is a non-degenerate Poisson bivector field, N : T M → T M is a (1, 1)-tensor and φ is a closed 3-form. If Eqs. (14)–(15) are satisfied, then (M, π, N , φ) is a Poisson quasi-Nijenhuis manifold. Proof. (i) Fact 3.2 implies Eq. (14). By Proposition 3.5, (T ∗ M)π , d N , φ is a quasiLie bialgebroid. It is simple to see that its induced bivector field on M as in Proposition 4.8 of [9] is π N . From Proposition 4.8 of [9], it follows that [π N , π N ] = 2π (φ). (ii) Since [π, π N ] = 0, Lemma 3.9 implies that π ◦d N = [π N , π (·)] and Lemma 3.10 implies that d N is a derivation of [·, ·]π . Hence π and N are compatible by Lemma 3.6. Since π is non-degenerate, we may apply (π )−1 to Eq. (15). Then, making use of Lemma 3.9, we get back to d N2 = [φ, ·]π . Equation (15) and the graded Jacobi identity yield [π N , π (φ)] = 0. Applying (π )−1 , we get d N φ = 0. Corollary 3.12. Let ω be a symplectic 2-form and φ a closed 3-form on M. Then (M, ω, N , φ) is a symplectic quasi-Nijenhuis manifold if and only if [ω N , ω N ] = 2φ
dω N = 0,
and
• (M)
where [·, ·] stands for the Schouten bracket on (T ∗ M)π , and ω N is the 2-form on M defined by ω N (X, Y ) = ω(N X, Y ),
induced from the Lie algebroid
∀X, Y ∈ X(M).
Proof. It is well known that when π is non-degenerate, π is an isomorphism of differential Gerstenhaber algebras from ( • (M), d, [·, ·]) to (X• (M), [π, ·], [·, ·]) [23, 10]. The conclusion thus follows immediately from Proposition 3.11 since π ω N = π N . Remark 3.13. Poisson Nijenhuis structures arise naturally in the study of integrable systems. It would be interesting to find applications of Poisson quasi-Nijenhuis structures in integrable systems as well.
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4. Universal Lifting Theorem In this section, we recall the universal lifting theorem and its basic ingredients, as it plays a crucial role in the following sections. For details, see [9]. Let ⇒ M be a Lie groupoid, A → M its Lie algebroid and ∈ Xk () a k-vector field on . Define F ∈ C ∞ (T ∗ × .(k) . . × T ∗ ) by F (μ1 , . . . , μk ) = (μ1 , . . . , μk ). Definition 4.1. A k-vector field ∈ Xk () is multiplicative if, and only if, F is a (k)
. . × M A∗ . 1-cocycle with respect to the groupoid T ∗ × · · · × T ∗ ⇒ A∗ × M .(k) Remark 4.2. It is simple to see that a bivector field is multiplicative if, and only if, the ¯ graph of the multiplication ⊂ × × is coisotropic with respect to ⊕ ⊕ , ¯ denotes the opposite bivector field to . where − → ← − − → ← − Example 4.3. If P ∈ (∧k A), then P − P is multiplicative, where P and P denote, respectively, the right and left invariant k-vector fields on corresponding to P. By Xkmult () we denote the space of all multiplicative k-vector fields on . And Xmult () = k Xkmult (). Proposition 4.4 ([9]). The vector space Xmult () is closed under the Schouten bracket, and therefore is a graded Lie algebra. It is simple to show that for any given ∈ Xkmult () and any X ∈ (∧i A), the ← − ←−− (k + i − 1)-vector field [ X , ] is always left invariant. Define δ X ∈ (∧(k+i−1) A) by ← − ←−− δ X = [ X , ]. Thus one obtains a linear operator δ : (∧i A) → (∧(k+i−1) A). Here we use the ← − following convention: (∧0 A) ∼ = C ∞ (M) and for any f ∈ C ∞ (M), f = β ∗ f . One easily checks that the following identities are satisfied: δ (P ∧ Q) = (δ P) ∧ Q + (−1) p(k−1) P ∧ δ Q, δ [P, Q] = [δ P, Q] + (−1)( p−1)(k−1) [P, δ Q], for all P ∈ (∧ p A) and Q ∈ (∧q A). This leads to the following definition of k-differentials. Recall that for any Lie algebroid A → M, ((∧• A), ∧, [·, ·]) is a Gerstenhaber algebra [23]. Definition 4.5. A k-differential on a Lie algebroid A is a degree (k − 1) derivation of the Gerstenhaber algebra ((∧• A), ∧, [·, ·]); i.e. a linear operator δ : (∧• A) → (∧•+(k−1) A) satisfying δ(P ∧ Q) = (δ P) ∧ Q + (−1) p(k−1) P ∧ δ Q, δ[P, Q] = [δ P, Q] + (−1)( p−1)(k−1) [P, δ Q], for all P∈(∧ p A) and Q∈(∧q A). The set of k-differentials on A is denoted by Ak (A).
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The space of all multi-differentials A(A) = when endowed with the graded commutator: [δ1 , δ2 ] = δ1 ◦δ2 − (−1)(k−1)(l−1) δ2 ◦δ1 ,
k
Ak (A) becomes a graded Lie algebra
where δ1 ∈ Ak (A) and δ2 ∈ Al (A).
Below is a list of basic examples. Examples 4.6. (i) When A is a Lie algebra g, then k-differentials are in one-one correspondence with Lie algebra 1-cocycles δ : g → ∧k g with respect to the adjoint action. (ii) The 0-differentials correspond to sections φ ∈ (A∗ ) such that d A φ = 0, i.e. Lie algebroid 1-cocycles with trivial coefficients. (iii) The 1-differentials correspond to the infinitesimals of Lie algebroid automorphisms. (iv) If P ∈ (∧k A), then ad P = [P, ·] is clearly a k-differential, which is called the coboundary k-differential associated to P. (v) A Lie bialgebroid can be seen as a Lie algebroid together with a 2-differential of square zero. The converse is also true. From the previous discussion, we know that there exists a linear map X•mult () → A• (A) : → δ , which is a Lie algebra homomorphism since the graded Jacobi identity satisfied by the Schouten bracket implies that [δ , δ ] = δ[, ] .
(16)
Moreover, one has the following Universal Lifting Theorem ([9]). Assume that ⇒ M is a target-connected and target-simply connected Lie groupoid with Lie algebroid A. Then X•mult () → A• (A) : → δ is an isomorphism of graded Lie algebras.
5. Symplectic Nijenhuis Groupoids Definition 5.1. A symplectic Nijenhuis groupoid is a symplectic groupoid ( ⇒ M, ω) : T → T such that (, ) is a equipped with a multiplicative (1, 1)-tensor N ω, N symplectic Nijenhuis structure. The main result of this section is the following Theorem 5.2. (i) The unit space of a symplectic Nijenhuis groupoid is a Poisson Nijenhuis manifold. (ii) Every integrable Poisson Nijenhuis manifold is the unit space of a unique targetconnected, target-simply connected symplectic Nijenhuis groupoid.
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Here, by an integrable Poisson Nijenhuis manifold, we mean the corresponding Poisson structure is integrable, i.e. it admits an associated symplectic groupoid. See [5, 6] for the solution of the integrability problem for Poisson manifolds and, more generally, Lie algebroids. Recall that a Poisson Nijenhuis manifold (M, π, N ) gives rise to a Lie bialgebroid ((T ∗ M)π , d N ) according to Corollary 3.8. The following lemma gives a useful characterization of those Lie bialgebroids arising from Poisson Nijenhuis structures. Lemma 5.3. Let (M, π ) be a Poisson manifold. A Lie bialgebroid ((T ∗ M)π , δ) is induced by a Poisson Nijenhuis structure if and only if [δ, d] = 0, where d stands for the de Rham differential. Proof. If (M, π, N ) is a Poisson Nijenhuis manifold, then d N = i N ◦d − d ◦i N . Thus [d N , d] = d N ◦d + d ◦d N = (i N ◦d − d ◦i N )◦d + d ◦(i N ◦d − d ◦i N ) = 0. Conversely, given a Lie bialgebroid ((T ∗ M)π , δ) such that [δ, d] = 0, one obtains a Lie algebroid structure on T M. Let N : T M → T M be its anchor map. Thus δ = d N : C ∞ (M) → 1 (M). Since [δ, d] = 0, we have ∀ f ∈ C ∞ (M), δ(d f ) = −dδ f = −dd N f = d N (d f ). It thus follows that δ = d N on any differential forms since both δ and d N are derivations and they agree on 0- and exact 1-forms. According to Corollary 3.8, it follows that (M, π, N ) is a Poisson Nijenhuis manifold. Proof of Theorem 5.2. (i) From symplectic Nijenhuis groupoids to Poisson ) is a symplectic Nijenhuis grouNijenhuis manifolds. Assume that (, ω, N poid. Let π be the bivector field on which is the inverse of ω and let π N ∈ X2 () ◦ π . be the bivector field defined by π N = N π on the base manifold • Since [ π, π ] = 0, the induced bivector field π = t∗ of the symplectic groupoid ⇒ M is Poisson [22]. The Lie algebroid of → M is isomorphic to (T ∗ M)π [2]. And the multiplicative bivector field π corresponds to a 2-differential on (T ∗ M)π , which is the de Rham differential d. That is, ((T ∗ M)π , d) is the Lie bialgebroid corresponding to the symplectic groupoid (, ω). • As pointed out in Fact 3.2, π N is a Poisson tensor on [16, 12, 21]. Moreover, is a multiplicative (1, 1)-tensor π N is a multiplicative bivector field since N and π is a multiplicative bivector field. In other words, (, π N ) is a Pois• •+1 (M) be the 2-differential on son groupoid [14]. Let δ π N : (M) → (T ∗ M)π induced by the multiplicative Poisson bivector field π N on . Since [ π N , π N ] = 0, the universal lifting theorem implies that 2 0 = δ[ π N , π N ] = [δ π N , δ π N ] = δ π N ◦δ π N + δ π N ◦δ π N = 2δ π. N
Thus, ((T ∗ M)π , δ π N ) is a Lie bialgebroid. • Likewise, it is standard that [ π N , π ] = 0. Thus the universal lifting theorem implies that [δ , d] = 0. According to Lemma 5.3, δ π N π N = d N for some Nijenhuis tensor N on M and (M, π, N ) is a Poisson Nijenhuis manifold. (ii) From Poisson Nijenhuis manifolds to symplectic Nijenhuis groupoids. Given a Poisson Nijenhuis manifold (M, π, N ), then ((T ∗ M)π , d N ) is a Lie bialgebroid by Corollary 3.8. Assume that (T ∗ M)π is integrable (see [5, 6] for the integrability condition) and ( ⇒ M, ω) is a target-connected and target
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simply-connected symplectic groupoid of M. Since d N2 = 0 and [d N , d] = 0, the universal lifting theorem implies that d N corresponds to a multiplicative Poisson bivector field π N on such that [ π N , π ] = 0, where π is the Poisson tensor = is a on inverse to ω. Let N π N ◦ ω : T → T . Then it is clear that N multiplicative (1, 1)-tensor, and (, ω, N ) is a symplectic Nijenhuis groupoid. Since these two constructions are inverse to each other, the theorem is proved.
6. Symplectic Quasi-Nijenhuis Groupoids The goal of this section is to generalize Theorem 5.2 to the quasi-setting. More precisely, we will give an integration theorem for Poisson quasi-Nijenhuis manifolds. Definition 6.1. A symplectic quasi-Nijenhuis groupoid is a symplectic groupoid ( ⇒ : T → T and a closed 3-form M, ω) equipped with a multiplicative (1, 1)-tensor N 3 ∗ ∗ φ ∈ (M) such that , ω, N , t φ − s φ is a symplectic quasi-Nijenhuis structure. The following result is a generalization of Theorem 5.2. Theorem 6.2. (i) The unit space of a symplectic quasi-Nijenhuis groupoid is a Poisson quasi-Nijenhuis manifold. (ii) Every integrable Poisson quasi-Nijenhuis manifold (M, π, N , φ) is the unit space of a unique target-connected and target-simply connected symplectic quasi-Ni, t ∗ φ − s ∗ φ . jenhuis groupoid ⇒ M, ω, N Proof. The proof is similar to that of Theorem 5.2, so we will merely sketch it. Assume that (M, π, N , φ) is an integrable Poisson quasi-Nijenhuis manifold. Let ⇒ M be a target-connected and target-simply connected groupoid integrating the Lie algebroid (T ∗ M)π . By Proposition 3.5, (T ∗ M)π , d N , φ is a quasi-Lie bialgebroid, which integrates to a quasi-Poisson groupoid by the universal lifting theorem. Let π N ∈ X() be the bivector field on corresponding to d N . Then we have 1 π N , π N ] 2 [
− → ← − = φ − φ.
On the other hand, we know that ⇒ M is a symplectic groupoid, whose correspond ing Lie bialgebroid is (T ∗ M)π , d . The symplectic form on is denoted by ω. Let π ∈ X2 () be its corresponding Poisson tensor. Since [d N , d] = 0, we have [ π N , π] = 0 = ω : T → T . Then it is clear according to the universal lifting theorem. Let N π N ◦ − → ← − that N is a multiplicative Since π (t ∗ φ−s ∗ φ), from Proposition (1, 1)-tensor. φ−φ = ∗ ∗ 3.11, it follows that , ω, N , t φ − s φ is a symplectic quasi-Nijenhuis groupoid. The other direction can be proved by going backwards. Remark 6.3. Note that ω ( π N ) is a multiplicative 2-form on ⇒ M. It would be interesting to see what is the corresponding Dirac structure on M and how the integration result in [1] can be applied to this situation.
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7. Generalized Complex Structures This section is devoted to the investigation of the relationship between generalized complex structures and Poisson quasi-Nijenhuis structures. Let us first recall the definition of generalized complex structures [8, 7]. Definition 7.1. A generalized complex structure on a manifold M is a bundle map J : T M ⊕ T ∗M → T M ⊕ T ∗M satisfying the algebraic properties J 2 = −I
J v, J w = v, w
and
(17)
and the integrability condition J v, J w − v, w − J (J v, w + v, J w) = 0 ∀v, w ∈ (T M ⊕T ∗ M). Here ·, · and ·, · are the pairing and bracket on the standard Courant algebroid T M ⊕ T ∗ M as in Example 2.2. The first two algebraic conditions (17) imply that J must be of the form N π J= , σ −N ∗
(18)
where π ∈ X2 (M) is a bivector field, σ ∈ 2 (M) is a 2-form and N : T M → T M is a (1, 1)-tensor. Here σ : T M → T ∗ M is the map given by (σ X )(Y ) = σ (X, Y ), ∀X, Y ∈ X(M). On the other hand, a Courant algebroid can be deformed using a bundle map J . More precisely, let (E, ·, · , ·, ·, ρ) be a Courant algebroid over M and let E M
J
id
/E /M
be a vector bundle automorphism of E → M. Consider • the inner product
A, B J = J A, J B ,
• the bracket A, B J = J A, B + A, J B − J A, B, • and the bundle map
(19)
ρJ = ρ◦ J
induced by J . A natural question is Question 7.2. When is the quadruple (E, ·, · J , ·, · J , ρ J ) still a Courant algebroid? The next proposition gives a trivial sufficient condition.
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Proposition 7.3. The quadruple (E, ·, · J , ·, · J , ρ J ) is a Courant algebroid if J A, J B + J 2 A, B − J J A, B + A, J B = 0, ∀A, B ∈ (E). Moreover, in this case, J is a Courant algebroid isomorphism from (E, ·, · J , ·, · J , ρ J ) to (E, ·, · , ·, ·, ρ). We now give an answer to Question 7.2 in the special case of the standard Courant algebroid T M ⊕ T ∗ M, where J satisfies Eqs. (17), and is given by Eq. (18) . Lemma 7.4. Assume that J : T M ⊕T ∗ M → T M ⊕T ∗ M is given by Eq. (18). Let ·, · J be the deformed bracket on X(M) ⊕ 1 (M) as in Eq. (19). Then, for all X, Y ∈ X(M) and ξ, η ∈ 1 (M), we have ξ, η J = [ξ, η]π , X, Y J = [X, Y ] N + (dσ )(X, Y, ·), X, ξ J = [X, π ξ ] − π (L X ξ − 21 d(ξ X )) + L N X ξ − L X (N T ξ ) + N T (L X ξ − 21 d(ξ X )) .
(20) (21) (22)
Proof. This follows from a straightforward computation using Eqs. (2) and (19), and is left for the reader. Proposition 7.5. Let J : T M ⊕ T ∗ M → T M ⊕ T ∗ M be a bundle map which satisfies Eqs. (17), and is given by Eq. (18). Then (T M ⊕ T ∗ M, ·, · J , ·, · J , ρ J ) is a Courant algebroid if, and only if, (M, π, N , dσ ) is a Poisson quasi-Nijenhuis manifold. And in this case, (T M ⊕ T ∗ M, ·, · J , ·, · J , ρ J ) is naturally identified with the double of the quasi-Lie bialgebroid (T ∗ M)π , d N , dσ . Proof. Assume that (T M ⊕ T ∗ M, ·, · J , ·, · J , ρ J ) is a Courant algebroid. It is clear that A := T ∗ M and B := T M are transversal, maximal isotropic subbundles. By Eq. (20), A = T ∗ M is a Dirac structure with the induced bracket [·, ·]π . Thus, according to Theorem 2.6, we obtain a quasi-Lie bialgebroid. The construction of the corresponding derivation δ of ( • (M), ∧, [·, ·]π ) and the twisting 3-form φ was outlined in the proof of Theorem 2.6. In the present situation, we have ρ B (X ) = ρ J (X ) = ρ(J X ) = ρ(N X + σ X ) = N X,
∀X ∈ T M
and, combining Eqs. (21) and (6), [X, Y ] B = [X, Y ] N ,
∀X, Y ∈ X(M).
Therefore, comparing Eqs. (3) and (9), we conclude that δ = d N . And, combining Eqs. (5) and (21), we get φ(X, Y, Z ) = 2X, Y J , Z J = 2J X, Y J , J Z = 2X, Y J , Z = 2[X, Y ] N + dσ (X, Y, ·), Z = dσ (X, Y, Z ), ∀X, Y, Z ∈ X(M). ∗ Hence (T M)π , d N , dσ is a quasi-Lie bialgebroid or, equivalently according to Proposition 3.5, (M, π, N , dσ ) is a Poisson quasi-Nijenhuis manifold. Conversely, assume that (M, π, N , dσ ) is a Poisson quasi-Nijenhuis manifold. By Proposition 3.5, ((T ∗ M)π , d N , dσ ) is a quasi-Lie bialgebroid. Its double E is a Courant algebroid. We will show that E is indeed isomorphic to (T M ⊕T ∗ M, ·, · J , ·, · J , ρ J ).
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First, it is simple to check that their anchors and non-degenerate symmetric pairings coincide. It remains to check that their brackets coincide. According to Eq. (4), the bracket ·, · on (E) is given by ξ, η = [ξ, η]π , X, Y = [X, Y ] N + (dσ )(X, Y, ·), X, ξ = i X δT M ξ + 21 δT M (ξ X ) − i ξ δT ∗ M X + 21 δT ∗ M (ξ X )
(23) (24) (25)
for all X, Y ∈ X(M) and ξ, η ∈ 1 (M). In our case, we have δT ∗ M = [π, ·]
δT M = d N .
and
It follows from a straightforward verification that the right hand sides of Eqs. (20)–(22) and (23)–(25) coincide. Therefore, (T M ⊕ T ∗ M, ·, · J , ·, · J , ρ J ) is indeed a Courant algebroid. We are now ready to state the main result of this section. Theorem 7.6. Assume that J : T M ⊕ T ∗ M → T M ⊕ T ∗ M as given by Eq. (18) satisfies Eqs. (17). Then the following are equivalent • J is a generalized complex structure; • (M, π, N , dσ ) is a Poisson quasi-Nijenhuis manifold such that J
(T M) N ⊕ (T ∗ M)π − → T M ⊕ T ∗M is a Courant algebroid isomorphism. Here (T M) N ⊕ (T ∗ M)π denotes the Courant algebroid corresponding to the quasi-Lie bialgebroid ((T ∗ M)π , d N , dσ ). Proof. By Proposition 7.3, J is a generalized complex structure if, and only if, (T M ⊕ J
T ∗ M, ·, · J , ·, · J , ρ J ) is a Courant algebroid and (T M ⊕ T ∗ M, ·, · J , , J , ρ J ) − → (T M ⊕ T ∗ M, ·, · , ·, ·, ρ) is a Courant algebroid isomorphism. The result follows immediately from Proposition 7.5. Since any generalized complex structure naturally gives rise to a Poisson quasiNijenhuis manifold, as an immediate consequence of Theorem 6.2, we have the following Theorem 7.7. Let J be a generalized complex structure as given by Eq. (18), and ( ⇒ M, ω) a target-connected and target-simply connected symplectic groupoid on such that integrating (T ∗ M)π . Then there is a multiplicative (1, 1)-tensor N , t ∗ dσ − s ∗ dσ ) is a symplectic quasi-Nijenhuis groupoid. ( ⇒ M, ω, N Remark 7.8. Note that Theorem 3.3–3.4 in [4] essentially imply our Theorem 7.7. Our proof is conceptual, while Crainic used a direct argument. It would be interesting to see how Theorem 3.4 (ii) in [4] can be proved conceptually. Acknowledgement. We would like to thank several institutions for their hospitality while work on this project was being done: Erwin Schroedinger International Institute for Mathematical Physics (Stiénon and Xu), and Université Pierre et Marie Curie (Xu). Stiénon is grateful to the Émile Francqui and Belgian American Educational Foundations for supporting his stay at the Pennsylvania State University where this work was completed in 2005. We would also like to thank Marius Crainic and Jim Stasheff for many useful discussions, and the referee for helpful suggestions to improve the presentation of the paper.
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References 1. Bursztyn, H., Crainic, M., Weinstein, A., Zhu, C.: Integration of twisted Dirac brackets. Duke Math. J. 123(3), 549–607 (2004) 2. Coste, A., Dazord, P., Weinstein, A.: Groupoïdes symplectiques. Publications du Département dé Mathématiques. Nouvelle Série. A, Vol. 2, Publ–Dép Math Nouvelle Sér A, 87, Lyon: Univ Claude-Bernard, pp. i-ii, 162 (1987) 3. Courant, T.J.: Dirac manifolds. Trans. Amer. Math. Soc. 319(2), 631–661 (1990) 4. Crainic, M.: Generalized complex structures and Lie brackets. http://arxiv.org/list/math.DG/0412097, 2004 5. Crainic, M., Fernandes, R.L.: Integrability of Lie brackets. Ann. of Math. (2) 157(2), 575–620 (2003) 6. Crainic, M., Fernandes, R.L.: Integrability of Poisson brackets. J Differ Geom. 66(1), 71–137 (2004) 7. Gualtieri, M.: Generalized complex geometry. http://arxiv.org/list/math.DG/0401221, 2004 8. Hitchin, N.: Generalized Calabi-Yau manifolds. Q. J. Math. 54(3), 281–308 (2003) 9. Iglesias, D., Laurent-Gengoux, C., Xu, P.: Universal lifting theorem and quasi-Poisson groupoids. http://arxiv.org/list/math.DG/0507396, 2005 10. Kosmann-Schwarzbach, Y.: Exact Gerstenhaber algebras and Lie bialgebroids. Acta Appl. Math. 41(1-3), 153–165 (1995) 11. Kosmann-Schwarzbach, Y.: The Lie bialgebroid of a Poisson-Nijenhuis manifold. Lett. Math. Phys. 38(4), 421–428 (1996) 12. Kosmann-Schwarzbach, Y., Magri, F.: Poisson-Nijenhuis structures. Ann. Inst. H. Poincaré Phys. Théor. 53(1), 35–81 (1990) 13. Liu, Z.-J., Weinstein, A., Xu, P.: Manin triples for Lie bialgebroids. J. Differ. Geom. 45(3), 547–574 (1997) 14. Mackenzie, K.C.H., Xu, P.: Lie bialgebroids and Poisson groupoids. Duke Math. J. 73(2), 415– 452 (1994) 15. Mackenzie, K.C.H., Xu, P.: Integration of Lie bialgebroids. Topology 39(3), 445–467 (2000) 16. Magri, F., Morosi, C.: On the reduction theory of the Nijenhuis operators and its applications to Gel’ prime fand-Diki˘ı equations. Proceedings of the IUTAM-ISIMM symposium on modern developments in analytical mechanics, Vol. II (Torino, 1982) Atti. Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur. 117, pp. 599–626 (1983) 17. Magri, F., Morosi, C.: Old and new results on recursion operators: an algebraic approach to KP equation. Topics in soliton theory and exactly solvable nonlinear equations (Oberwolfach, 1986), Singapore: World Sci. Publishing, pp. 78–96 (1987) 18. Magri, F., Morosi, C., Ragnisco, O.: Reduction techniques for infinite-dimensional Hamiltonian systems: some ideas and applications. Commun. Math. Phys. 99(1), 115–140 (1985) 19. Roytenberg, D.: Courant algebroids, derived brackets and even symplectic supermanifolds. http:// arxiv.org/list/math.DG/9910078, 1999 20. Roytenberg, D.: Quasi-Lie bialgebroids and twisted Poisson manifolds. Lett. Math. Phys. 61(2), 123– 137 (2002) 21. Vaisman, I.: Complementary 2-forms of Poisson structures. Compositio Math. 101(1), 55–75 (1996) 22. Weinstein, A.: Symplectic groupoids and Poisson manifolds. Bull. Amer. Math. Soc. (N.S.) 16(1), 101– 104 (1987) 23. Xu, P.: Gerstenhaber algebras and BV-algebras in Poisson geometry. Commun. Math. Phys. 200(3), 545– 560 (1999) Communicated by Y. Kawahigashi
Commun. Math. Phys. 270, 727–758 (2007) Digital Object Identifier (DOI) 10.1007/s00220-006-0169-z
Communications in
Mathematical Physics
Dispersing Billiards with Cusps: Slow Decay of Correlations N. Chernov1 , R. Markarian2 1 Department of Mathematics, University of Alabama at Birmingham, Birmingham, AL 35294, USA.
E-mail:
[email protected]
2 Instituto de Matemática y Estadística “Prof. Ing. Rafael Laguardia”, Facultad de Ingeniería, Universidad de
la República, C.C. 30, Montevideo, Uruguay Received: 4 April 2006 / Accepted: 29 June 2006 Published online: 15 December 2006 – © Springer-Verlag 2006
Abstract: Dispersing billiards introduced by Sinai are uniformly hyperbolic and have strong statistical properties (exponential decay of correlations and various limit theorems). However, if the billiard table has cusps (corner points with zero interior angles), then its hyperbolicity is nonuniform and statistical properties deteriorate. Until now only heuristic and experimental results existed predicting the decay of correlations as O(1/n). We present a first rigorous analysis of correlations for dispersing billiards with cusps.
1. Introduction A billiard is a mechanical system in which a point particle moves in a compact container D and bounces off its boundary ∂D; in this paper we only consider planar billiards, where D ⊂ R2 or D ⊂ Tor 2 . The billiard dynamics preserves a uniform measure on its phase space, and the corresponding collision map (generated by the collisions of the particle with ∂D, see below) preserves a natural (and often unique) absolutely continuous measure on its own phase space, see definitions in Sect. 2. The dynamical properties of a billiard are determined by the shape of the boundary ∂D, and it may vary greatly from completely regular (integrable) to strongly chaotic. The first class of chaotic billiards was introduced by Ya. Sinai in 1970 [Si70]; he proved that if the boundary ∂D of a domain D ⊂ Tor 2 is smooth and strictly convex inward (with nowhere vanishing curvature), then the billiard map and flow are hyperbolic (moreover, uniformly hyperbolic), ergodic, mixing and K-mixing. He called such systems dispersing billiards, now they are often called Sinai billiards. In 1974, Gallavotti and Ornstein [GO74] proved that Sinai’s billiards were Bernoulli systems. Sinai’s billiards have strong statistical properties – exponential decay of correlations for the collision map [Y98], central limit theorem and weak invariance principle (for both map and flow, see [BS81, BSC91]), as well as strong invariance principle for the map [C06b].
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Fig. 1. Billiard table with three cusps
All these results have been extended to dispersing billiards with piecewise smooth boundary, where corner points exist, provided the interior angles made by the boundary at corner points are all positive [BSC91, C99]. On the contrary, dispersing billiards with corner points with zero internal angles (‘cusps’) are much harder to investigate; the main reason is a weak (non-uniform) hyperbolicity of the collision map. Indeed, whenever the moving particle gets deep into a cusp, it experiences a large number of rapid collisions that do not contribute much to the expansion or contraction of tangent vectors. Only in 1995, Reháˇcek proved that dispersing billiards with cusps were ergodic [R95], which implied K-mixing by a general argument, see [Si70] and also [CM06, Chap. 6], and Bernoulli property [CH96, OW98]. Statistical properties of dispersing billiards with cusps appear to be similar to those of expanding interval maps with indifferent fixed points (see, for example, [CGS92, CG93, Y99]). Just like a trajectory in an interval may be trapped in a vicinity of an indifferent fixed point, the billiard particle may be trapped in a cusp. Such phenomena result in an intermittent character of the dynamics (switching between regularity and chaos) and they are notoriously hard to analyze. In 1983, Machta [Mac83] investigated the rate of the decay of correlations for one particular billiard table made by three identical circular arcs tangent to each other at their points of contact (Fig. 1). He argued that correlation function Cn ( f, g), see definitions in the next section, should decay as O(1/n), which was much slower than the exponential decay than expected (and now established) for dispersing billiards without cusps. Machta’s arguments were almost entirely heuristic (he approximated the motion of the billiard particle in a cusp by a carefully constructed system of differential equations), but his analysis clearly demonstrated that the dynamics in a cusp were pretty complicated. Machta supported his conjecture by numerical experiments (see also [MR86]). The (rather unexpected) complexity of the dynamics in a cusp held back the mathematical studies of such billiards for quite a while. Only now we are able to prove Machta’s conjecture (in a slightly weaker form): Theorem 1.1. For dispersing billiards with cusps, the correlations Cn ( f, g) for the collision map and Hölder continuous observables f, g are bounded by |Cn ( f, g)| ≤ C(ln n)2 /n, where C > 0 is a constant.
Dispersing Billiards with Cusps: Slow Decay of Correlations
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Fig. 2. Orientation of r and ϕ
In most of our paper we deal with Machta’s three-arc table shown on Fig. 1. This allows us to present the arguments in a fairly tractable and geometrically transparent manner. In Sect. 6 we describe changes necessary for proving the theorem in the general case. Remark. Bounds on correlations similar to ours (with a logarithmic factor) have been established for Bunimovich’s stadium and other billiard models with polynomial decay of correlations [Mar04, CZ05a]. It is believed that the logarithmic factor is just an artefact of the method used, and a sharp bound on correlations is expected to be C/n. A work is currently underway to improve the argument and eliminate the logarithmic factor. Remark. In the studies of hyperbolic maps, if correlations decay as O(1/n), as in our case, the central limit theorem (CLT) usually fails. However, there are non-classical versions of the CLT that sometimes hold [BG06]. 2. Generalities Here we provide necessary facts from the theory of chaotic billiards. For a more detailed presentation of these and related facts see [BSC90, BSC91, C06a], as well as our recent book [CM06]. A planar billiard is a dynamical system where a point (particle) moves freely at unit speed in a domain D ⊂ R2 and reflects off its boundary ∂D by the rule “the angle of incidence equals the angle of reflection”. It is commonly assumed that ∂D is a finite union of C 3 curves (arcs). The phase space of this system is a three dimensional manifold = D × S 1 . The motion of the particle generates a Hamiltonian flow on preserving a Liouville measure, which is a product of uniform measures on D and S 1 . Let M = ∂D × [−π/2, π/2] be the standard cross-section of the billiard dynamics, we call M the collision space. Canonical coordinates on M are r and ϕ, where r is the arc length parameter on ∂D and ϕ ∈ [−π/2, π/2] is the angle between the postcollisional velocity vector v and the inward normal vector n to ∂D; the orientation of r and ϕ is shown on Fig. 2. The first return map F : M → M is called the collision map or the billiard map, it preserves smooth measure dμ = cos ϕ dr dϕ on M. Let f, g ∈ L 2μ (M) be two functions. Correlations are defined by ( f ◦ F n ) g dμ − f dμ g dμ. (2.1) Cn ( f, g) = M
M
M
It is well known that F : M → M is mixing if and only if lim Cn ( f, g) = 0
n→∞
∀ f, g ∈ L 2μ (M).
(2.2)
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The rate of mixing of F is characterized by the speed of convergence in (2.2) for smooth enough functions f and g. We will always assume that f and g are Hölder continuous or piecewise Hölder continuous with singularities that coincide with those of the map F k for some k. For example, the free path between successive reflections is one such function. We say that correlations decay exponentially if |Cn ( f, g)| < const · e−cn for some c > 0 and polynomially if |Cn ( f, g)| < const · n −a for some a > 0. Here the constant factor depends on f and g, but the exponent c (or a) only depends on the map and on the Hölder exponent of the functions f and g. Systems with strong (uniform) hyperbolicity are usually characterized by exponential decay of correlations; systems with weak (nonuniform) hyperbolicity usually have slow (polynomial) mixing rates. A general strategy for estimating the correlation function Cn ( f, g) for systems with weak hyperbolicity was developed in [CZ05a], it is based on recent Young’s results [Y98, Y99] and [Mar04]. That scheme is particularly convenient for billiards. First, one needs to ‘localize’ spots in the phase space where expansion (contraction) of tangent vectors slows down. Let M0 denote the union of all such spots and ˆ = M \ M0 . One needs to verify that the return map Fˆ : M ˆ →M ˆ (that avoids all M the ‘bad’ spots) is strongly (uniformly) hyperbolic. It preserves the measure μˆ obtained ˆ For any x ∈ M ˆ we call by conditioning μ on M. ˆ R(x) = min{n ≥ 1 : F n (x) ∈ M} the return time. For dispersing billiards with cusps, hyperbolicity deteriorates only as the moving particle gets deep down a cusp, where it experiences a large number of rapid collisions. We fix K 0 1 and call any sequence of successive collisions of length > K 0 in a cusp a corner series. We thus define M0 to be the set of all collision points during those corner series. Next the strategy developed in [CZ05a] consists of two steps; they are fully described in [CZ05a] (as well as applied to several classes of billiards with slow mixing rates), so we will not bring up unnecessary details here. ˆ →M ˆ has exponential decay At the first step one proves that the return map Fˆ : M of correlations. At the second step one obtains the following tail bound on the return time function: ˆ : R(x) > n) ≤ const · n −a μ(x ˆ ∈M (2.3) ˆ into the sets for some a > 1 and large n ≥ 1. This usually requires dividing M ˆ n ) for large n. E n = {x : R(x) = n + 1} and estimating the measure μ(E Lastly one uses the following theorem proven in [CZ05a, Sect. 3]: ˆ →M ˆ has exponential decay of correlations. Theorem 2.1. Suppose the map Fˆ : M If the tail bound (2.3) holds for the return time R(x), then correlations are bounded by |Cn ( f, g)| ≤ const (ln n)a n a−1 . 3. Corner Series Here we study the geometry of corner series. We examine a billiard trajectory entering a cusp and experiencing a large number of reflections there before getting out. To simplify our analysis we consider here a cusp made by two circular arcs of unit radius with a common tangent line.
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Fig. 3. The bottom of a corner series (here m = N¯ )
Let N be the number of reflections in the corner series and (rn , ϕn ), 1 ≤ n ≤ N , denote all the points of reflection in the cusp. We will also work with more convenient coordinates: γn = π/2 − |ϕn | and αn = |rn − r¯ |, where r¯ stands for the r coordinate of the vertex of the cusp (hence αn is the length of the arc of ∂D between the vertex and the n th collision point). Observe that γn are non-negative (in fact γn > 0 for 2 ≤ n ≤ N − 1); αn are all positive; αn are all small; γn are initially small, then slowly grow to about π/2 (for n ≈ N /2, as we prove below), and then again decrease and get small for n ≈ N . We first show that our trajectory comes closest to the vertex nearly in the middle of the corner series. Let α N¯ : = min αn . n
Lemma 3.1. We have | N¯ − N /2| ≤ 2. Proof. Consider two sequences of points (α N¯ + j , γ N¯ + j ) and (α N¯ − j , γ N¯ − j ) for j = 1, 2, . . .. Both sequences are going up, away from the corner, see Fig. 3. Without loss of generality, suppose α N¯ +1 ≥ α N¯ −1 . Then it is clear from Fig. 3 that γ N¯ +1 ≤ γ N¯ −1 ≤ γ N¯ . It is then an elementary geometric fact that α N¯ ≤ α N¯ −1 ≤ α N¯ +1 ≤ α N¯ −2 ≤ α N¯ +2 ≤ · · · and γ N¯ ≥ γ N¯ −1 ≥ γ N¯ +1 ≥ γ N¯ −2 ≥ γ N¯ +2 ≥ · · ·
(3.1)
(note that this is only true if the two circles making the corner are equal). Therefore, the number of collisions in the corner series occurring before N¯ and after N¯ differ by no more than one, i.e. | N¯ − N /2| ≤ 2.
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The two halves of the corner series, one before N¯ and the other after N¯ have very similar structure and properties. It will be enough to study in detail the first half of the series, 0 ≤ n ≤ N¯ . We further subdivide the corner series into three segments. We fix a small γ¯ ∈ (0, π/2) whose exact value is not important, say γ¯ = 10−10 . Now let N1 = max{n < N¯ : γn ≤ γ¯ }, denote N2 = N¯ and put N3 = min{n > N¯ : γn ≤ γ¯ }. Note that 0 < N1 < N2 < N3 < N . In what follows we use N2 instead of N¯ . We call the segment [1, N1 ] the entering period in the corner series, the segment [N1 +1, N3 −1] the turning period in it, and the segment [N3 , N ] its exiting period. It follows from (3.1) that |N1 − N3 | ≤ 2. Convention. We use the following notation: A B means that C −1 < A/B < C for some constant C = C(D) > 0. Also, A = O(B) means that |A|/B < C for some constant C = C(D) > 0. Proposition 3.2. We have N1 N2 − N1 N3 − N2 N − N3 N , hence all the three segments in the corner series have length of order N . Also, α1 N −2/3 and Also, and
αn n −1/3 N −2/3 γ1 = O(N −2/3 )
α N2 N −1 ,
and
∀n = 2, . . . , N1 ,
(3.3)
γ2 N −2/3 ,
(3.4)
∀n = 2, . . . , N1 .
(3.5)
and
γn nαn n 2/3 N −2/3
(3.2)
Proof. We consider the first half of the series, 1 ≤ n ≤ N2 . The following equations are simple geometric facts: γn+1 = γn + (αn + αn+1 ) (3.6) and sin αn+1 = sin αn − Due to (3.6) we have
2 − cos αn − cos αn+1 . tan(γn + αn )
γ2 = γ1 + α1 + α2 ≥ 2α2
(3.7) (3.8)
and γ1 + α1 + 2α2 + · · · + 2αn−2 + αn−1 = γn ≤ π/2, hence for all n ≤ N2 .
α1 + · · · + αn ≤ π/2
(3.9)
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Fig. 4. The first collision in a corner series.
At the very first collision, we have γ1 < γ1 , where γ1 denotes the angle made by the line passing through the first collision point and tangent to the other arc (Fig. 4). If we denote by α0 the coordinate of the point of tangency, then Eqs. (3.6)–(3.7) take form γ1 = α0 + α1 ,
sin α1 = sin α0 −
This easily gives α1 /α0 → 1 +
2 − cos α0 − cos α1 . tan α0
√
2, as N → ∞, hence √ γ1 < γ1 = (2 + 2 + o(1)) α1 ≤ 4α1 .
(3.10)
We introduce new variables: un =
αn+1 αn
and
wn =
γn , αn
hence αn = α1 u 1 u 2 · · · u n−1 .
(3.11)
It is important to find the asymptotics for wn . Equation (3.6) yields wn+1 = 1 +
wn + 1 . un
(3.12)
Since u n ≤ 1, we have wn+1 ≥ wn + 2. Since w2 ≥ 2 by (3.8), we obtain a lower bound for wn : wn ≥ 2n − 2. (3.13) To get an upper bound for wn , we first use (3.7) and obtain αn+1 > αn −
αn3 6
−
= αn −
2 /2) 2−(1−αn2 /2)−(1−αn+1 γn +αn
αn3 6
−
2 αn2 +αn+1 2(γn +αn ) .
This is equivalent to un > 1 −
1 + u 2n α2 1 αn2 − >1− n − . 6 2(1 + wn ) 6 1 + wn
(3.14)
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Combining this with (3.12) gives wn+1 < 1 +
wn + 1 1−
αn2 6
−
1 wn +1
.
Note that 1/(1 − x) < 1 + x + 2x 2 for small positive x (in fact, for all 0 < x < 1/2; and we indeed have αn2 /6 + 1/(wn + 1) < 1/2 since αn are all small and wn ≥ 2). Using this fact and making simple calculation yields wn+1 < wn + 3 + αn2 wn +
4 + αn4 (wn + 1). wn + 1
(3.15)
The lower bound (3.13) now implies wn < 3n + 2 ln n + 2
n−1
αi2 wi + C
i=0
for some absolute constant C (we note that w1 ≤ 4 due to (3.10)). The bound (3.9) implies n n n π π2 , (3.16) αi2 wi = αi γi ≤ αi ≤ 2 4 i=1
i=1
i=1
hence wn < 3n + 2 ln n + C.
(3.17)
We will denote by C absolute constants (possibly different in different equations) whose exact values are not important. Now we have an upper bound for wn , and the overall asymptotic is wn n. In particular, as a result of (3.13) and (3.17) and the obvious γ N2 ≈ π/2 we have π π < αmin < . (3.18) 6N2 + 4 ln N2 + 2C 4N2 Next we focus on the entering period, i.e. on 1 ≤ n ≤ N1 . As long as γn ≤ γ¯ we have ¯ n + αn )3 , tan(γn + αn ) < γn + αn + c(γ where c¯ > 0 is a constant determined by γ¯ . Now Eq. (3.7) yields αn+1 −
3 2 /2 − α 4 /24 αn+1 α 2 /2 − αn4 /24 + αn+1 n+1 < αn − n . 6 γn + αn + c(γ ¯ n + αn )3
3 /6 > α 2 Note that αn+1 − αn+1 n+1 (1 − αn /6), hence 2 − α 4 /6 αn2 + αn+1 αn2 n 1 − αn+1 < αn − 2(γn + αn ) + 2c(γ ¯ n + αn )3 6
or, equivalently, un < 1 −
1 + u 2n − αn2 /6 2(wn + 1) + 8c(w ¯ n + 1)γn2
1−
αn2 . 6
(3.19)
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We now substitute (3.19) into (3.12) and after simple calculation arrive at ¯ n2 − wn αn2 . wn+1 > 2 + wn + (1 + u 2n )/2 − 8cγ Then we use the estimate (3.14) of u n and, with some more simple calculation, obtain ¯ n2 − 2wn αn2 wn+1 > 3 + wn − (wn + 1)−1 − (wn + 1)−2 − 8cγ > 3 + wn − (2n)−1 − (2n)−2 − 8cγ ¯ n2 − 2wn αn2 ,
(3.20)
where we also used (3.13). We now combine (3.12), (3.20) and (3.17): wn+1 − 1 wn + 1 ¯ n2 − 2wn αn2 1 − (2n)−1 − (2n)−2 − 8cγ > 1+ wn + 1 1 − (2n)−1 − (2n)−2 − 8cγ ¯ n2 − 2wn αn2 > 1+ − 2αn2 3n + 2 ln n + C 1 1 8cγ ¯ n2 − 2− − 2αn2 . > 1+ 3n + 2 ln n + C n n
u −1 n =
2
Observe that 1 + x > e x−x for small x, hence, with some simple calculation, we obtain 2 1 8cγ ¯ n2 4c¯2 γn2 64c¯2 γn4 2 (3.21) − − 2α u −1 > exp − − − n n 3n + 2 ln n + C n2 n n2 n2 or, lastly,
n
n n n n ¯ i2 100c¯2 γi4 1 2 8cγ 2 − −2 exp − αi − . 3i + 2 ln i + C i2 i i2 i=1 i=1 i=1 i=1 i=1 i=1 (3.22) Note that by (3.17) and (3.16),
n
u i−1 >
n γ2 i
i=1
i
<8
n γ2 i
i=1
wi
<8
n
αi γi < 2π 2 ,
i=1
and similarly n γ4 i=1
i i2
< 64
n γi4 i=1
wi2
< 64
n i=1
αi2 γi2 < 16π 2
n
αi2 < 8π 3 .
i=1
By a simple calculation n i=1
1 1 = ln n + n , 3i + 2 ln i + C 3
where |n |
1 ln n − C , u i−1 > exp 3 i=1
(3.23)
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where C > 0 is a constant. Combining this with (3.11) gives αn < Cn −1/3 α1
(3.24)
for all 1 ≤ n ≤ N1 . We now estimate αn from below in a similar way. By (3.20) wn > 3n − ln n − 2c¯
n
γi2 − C
i=1
for some constant C > 0, where (3.16) was used. For brevity, denote n =
n
γi2 .
i=1
We now use (3.14) and the obvious fact (1 − x)−1 < 1 + 2x for small positive x and arrive at α2 2 ln n 1 2C 4c ¯ n un > 1 − − n − − 2− . (3.25) 2 3n 6 9n 9n 9n 2 2 Using another obvious fact, 1 − x > e−x−x (for small x), we obtain n 4c ¯ i 1 . (3.26) u 1 · · · u n > exp − ln n − C − 3 9i 2 i=1 n We now show that i=1 i /i 2 is bounded. Indeed, n i i=1
i2
=
i γ2 n j
i2
i=1 j=1
=
n γ2 n j j=1 i= j
i2
<2
n γ2 j j=1
j
.
Since γ j = w j α j , using (3.17) gives n i i=1
i2
n
γjαj
j=1
for some constant C > 0. The last expression is bounded by (3.9). Now combining (3.11) with (3.26) gives αn > Cn −1/3 α1 for all 1 ≤ n ≤ N1 , with some C > 0. Along with (3.24) this gives αn n −1/3 α1 for all n ≤ N1 . Since −3/2 γ N1 ≈ γ¯ = const, the bounds (3.13) and (3.17) give N1 α1 . We now consider the turning period, where N1 ≤ n ≤ N2 , then the angle γn grows from γ¯ to about π/2. First, note that αn = γn /wn > γ¯ /(3n + ln n + C). By (3.6) we have N2
(γn − γn−1 ) ≥
n=N1
C , C
N2 n=N1
C N2 ≥ C ln 3n + ln n + C N1
for some constants > 0. Therefore, N1 < N2 < C N1 for some C > 0. We then −3/2 −1/3 N2 N , and for α N2 = minn αn we have α N2 N2 α1 N −1/3 α1 . obtain α1 The proposition is proved.
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For our future use we record some estimates obtained in the proof for the entering period of the corner series, i.e. for 1 ≤ n ≤ N1 . Due to (3.15) and (3.20) we have wn+1 − wn = 3 + O(n −1 + γn2 )
(3.27)
(we note that wn αn2 n 1/3 N −4/3 = O(n −1 ), so that the term wn αn2 is absorbed by others). Observe that n −1 γn2 for small n but n −1 γn2 for n ≈ N1 , so we have to keep both parts of the O(·) term in (3.27). Equation (3.27) immediately implies wn = 3n + O(ln n + n ). This estimate combined with (3.21) and (3.25) gives 2 γn ln n n 1 +O . un = 1 − +O +O 2 3n n n n2
(3.28)
(3.29)
Next, the following sums were proven to be uniformly bounded (by constants independent of n < N1 and N ): n γ2 i
i=1
i
= O(1) and
n i i=1
i2
= O(1).
(3.30)
We will also need asymptotic formulas for the intercollision times during a corner series. Denote by tn the time of the n th collision, 1 ≤ n ≤ N , and by τn = tn+1 − tn the time between successive collisions. It is a simple geometric fact that τn =
2 − cos αn − cos αn+1 sin(γn + αn )
(3.31)
for all 1 ≤ n < N2 (when the trajectory is going down the corner). Expanding into Taylor series and using (3.3) and (3.28)–(3.29) gives τn =
1 + u 2n + O(γn2 ) αn 2wn 1 + wn−1 + O(αn2 wn2 )
αn 2 + O(1/n) + O(αn2 ) wn 2 + O(1/n) + O(γn2 ) = αn wn−1 1 + O(1/n) + O(γn2 ) =
n −4/3 N −2/3 . This gives us another important relation αn wn−1 1 + O(1/n) + O(n 2 αn2 ) τn = sin γn αn wn 1 + O(γn2 ) 1 = 2 1 + O(1/n) + O(γn2 ) wn ln n γn2 n 1 + + = 2 +O 9n n3 n2 n3
(3.32)
(3.33)
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We need to estimate the ratio of neighboring τn ’s by using (3.31): 2 − cos αn+1 − cos αn+2 sin(γn + αn ) τn+1 = : Fn × Fn . = × τn 2 − cos αn − cos αn+1 sin(γn+1 + αn+1 ) The first fraction behaves as Fn =
2 (1 + u 2 ) + O(α 4 ) αn+1 n+1 n+1 αn2 (1 + u 2n ) + O(αn4 )
2 ) u 2n (1 + u 2n+1 ) + O(αn+1 1 + u 2n + O(αn2 ) ln n γn2 n 2 2 +O + + + α = 1− n , 3n n2 n n2
=
where we used (3.29) three times. Note that O(αn2 ) = O(γn2 /n 2 ), hence the last term is actually absorbed by the others. Next sin(γn+1 + αn+1 ) − sin(γn + αn ) sin(γn+1 + αn+1 ) [αn u n (wn+1 + 1) − αn (wn + 1)] cos θ , = 1− αn u n (wn+1 + 1) + O(γn3 )
Fn = 1 −
where θ ∈ (γn + αn , γn+1 + αn+1 ) by the mean value theorem, hence u n (wn+1 + 1) − (wn + 1) 1 + O(γn2 ) . Fn = 1 − u n (wn+1 + 1) According to (3.27), (3.28) and (3.29) the numerator behaves as u n (wn+1 + 1) − (wn + 1) = (u n − 1)wn + 4u n − 1 + O(n −1 + γn2 ) n ln n 2 + γn + , = 2+O n n hence Fn
ln n γn2 n 2 +O + 2 . =1− + 3n n2 n n
Again, O(nαn2 ) = O(γn2 /n), hence the last term is actually absorbed by the others. Combining our estimates for Fn and Fn gives ln n γn2 n τn+1 4 (3.34) +O + 2 =1− + τn 3n n2 n n for all n = 1, . . . , N1 . During the turning period, where N1 ≤ n ≤ N3 , we have αn 1/N by (3.2) and (3.3). Since γn ≥ γ¯ > 0, we easily obtain τn αn2 N −2 . Thus the time spent by the trajectory during each period in the corner series has the same order of magnitude: N 1 −1 n=1
τn 1/N
and
N3 n=N1
τn 1/N .
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Remark 3.3. Due to the time reversibility of billiard dynamics, all the asymptotic formulas obtained for the entering period remain valid for the exiting period. In particular, (3.35) and γ N = O N −2/3 . α N N −2/3 During the exiting period we will also use the ‘countdown’ index m = N + 1 − n, so that m = 1, . . . , N − N3 ; then in all our asymptotic formulas we can simply replace n by m. For example, αm m −1/3 N −2/3
and
γm mαm m 2/3 N −2/3
(3.36)
for all m = 2, . . . , N − N3 , etc. 4. Expansion of Unstable Curves In this section we estimate the rate of expansion of unstable vectors during corner series. First we recall general facts about unstable tangent vectors in dispersing billiards [BSC90, BSC91, C06a, CM06]. Let x = (r, ϕ) ∈ M. A tangent vector d x = (dr, dϕ) ∈ Tx M can be represented by an infinitesimal curve γ = γ (s) ⊂ M, where s ∈ (−ε, ε) is a parameter, such that d γ (0) = x and ds γ (0) = d x. The trajectories of the points y ∈ γ , after leaving M, make a one-parameter family (a bundle) of directed lines in ∂D. The curvature of the orthogonal cross-section of that bundle at x plays an important role; we denote it by B + = B + (x). Similarly, the past trajectories of the points y ∈ γ (before arriving at M) make a bundle of directed lines in ∂D whose curvature right before the collision with ∂D at x is denoted by B − . We have B+ = B− +
2K , cos ϕ
(4.1)
where K = K(r ) denotes the curvature of the boundary ∂D at the point r . For dispersing billiards K is positive and bounded away from zero and infinity. The tangent vector d x is said to be unstable if B − > 0 (hence B + > 0 as well). The slope of the vector d x is dϕ/dr = B − cos ϕ + K = B + cos ϕ − K, thus dϕ/dr > 0 for unstable vectors. At the next collision point x1 = F(x) ∈ M, the image vector d x1 = Dx F(d x) is characterized by the (precollisional) curvature B1− satisfying 1 B+ B1− = = , (4.2) 1 1 + τ B+ τ+ + B where τ is the time between collisions at the points x and F(x) (it is also the distance between the corresponding collision points, because the moving particle travels at unit speed). Note that B + > 0 implies B1− > 0, thus the image of an unstable vector will always be an unstable vector. We measure tangent vectors d x ∈ Tx M in the Euclidean norm 1/2 . d x = (dr )2 + (dϕ)2
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For unstable vectors, it is more convenient to use the p-norm defined by d x p = cos ϕ dr. The p-norm corresponds to the size of the orthogonal cross-section of the associated bundle of trajectories (it is the same before and after collision). In the p-norm, the expansion of unstable tangent vectors is given by Dx F(d x) p = 1 + τ B+ . d x p
(4.3)
Note that this ratio is > 1, i.e. unstable vectors expand monotonically in the p-norm (this is not necessarily true in the Euclidean norm, see [CM06, Chap. 4]). We return to our corner series. Again, for simplicity we analyze the three-arc billiard ˆ table shown on Fig. 1 and we assume that the arcs have unit radius. Let x = (r, ϕ) ∈ M N is going down a cusp (say, A) and comes back be a point whose trajectory {F i (x)}i=1 ˆ after N reflections. In that case F(x) = F N +1 (x), i.e. the return function takes value R(x) = N + 1. We denote by ˆ : R(x) = N + 1} E N = {x ∈ M the set of points whose trajectories go down a cusp for a corner series of exactly N collisions. We denote by xn = (rn , ϕn ) = F n (x) the images of the point x during the corner series, 1 ≤ n ≤ N , which corresponds to our notation in the previous section. Obviously, x has to start near the point D (opposite to the cusp A, see Fig. 1) and ˆ F(x) = F N +1 (x) has to land back near D again. At the point x = (r, ϕ), we have ϕ ≈ 0 and 0 < c < B − < C for some constants c, C > 0. Thus cos ϕ 1 and 0 < dϕ/dr 1, hence the Euclidean norm and the p-norm are uniformly equivalent on ˆ unstable vectors at our points x ∈ E N and F(x), i.e. right before and right after long corner series. Given an unstable vector d x ∈ Tx M, we denote by d xn = (drn , dϕn ) = Dx F n (d x) its images. We are interested in the total expansion factor of d x, ˆ x) Dx F N +1 (d x) Dx F(d = d x d x during the corner series. Proposition 4.1. For every x ∈ E N the total expansion factor for unstable vectors in the course of the corner series of N collisions has lower bound Dx F N +1 (d x) ≥ C N 5/3 , d x
(4.4)
where C > 0 is a constant. Its precise asymptotic is N −2/3 N −2/3 Dx F N +1 (d x) 5/3 1+ 1+ . N d x cos ϕ1 cos ϕ N
(4.5)
Dispersing Billiards with Cusps: Slow Decay of Correlations
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Proof. Since the Euclidean norm and the p-norm are uniformly equivalent at the points x ∈ E N and F N +1 (x), we can safely replace · with · p ; then we can use the formula (4.3) at every collision. Let tn denote the time of collision at xn and τn = tn+1 − tn the intercollision time (note that τn is the distance between the points of the n th and (n + 1)st collisions, as the speed of the moving particle equals one). Then the expansion factor for the vector d x under Dx F n is n−1
Dx F n (d x) p = (1 + τi Bi+ ), (4.6) d x p i=0
where for
Bn+
we have a recursive formula, due to (4.1)–(4.2): + Bn+1 =
2 Bn+ + sin γn+1 1 + τn Bn+
(4.7)
(remember that K = 1 and γn = π/2 − |ϕn |, so cos ϕn = sin γn ). Before we proceed, let us make an important remark. Recall that (3.4) and (3.35) only guarantee that γ1 = O(N −2/3 ) and γ N = O(N −2/3 ); in fact both γ1 and γ N may be arbitrarily close to zero. Thus the expansion of unstable vectors at the very first and the very last collision of the corner series may be arbitrarily strong. On the other hand, (4.7) shows that Bn+1 is a monotonically increasing function of both Bn+ and 1/ sin γn+1 . Thus if we increase γ1 and γ N , the total expansion factor Dx F N +1 (d x)/d x will only decrease. So we can assume that γ1 N −2/3
and
γ N N −2/3
(4.8)
and obtain an (asymptotical) lower bound on the total expansion factor. We will actually assume (4.8) and prove that Dx F N +1 (d x) p N 5/3 . d x p
(4.9)
This will give us, in particular, (4.4) for all x ∈ E N . For n = 0 we have B0+ 1 and τ0 1, hence 1 + τ0 B0+ 1, so the term i = 0 in (4.6) does not affect the asymptotics. For n ≥ 1, we put λn = τn Bn+ , then (4.6) takes form n−1
Dx F n (d x) p = (1 + λi ) (4.10) d x p i=0
and (4.7) takes form λn+1 =
2τn+1 τn+1 λn + . sin γn+1 τn 1 + λn
Lemma 4.2. For all x ∈ E N satisfying (4.8) we have λn 1/n λn 1/n 1/N λn 1/(N − n)
for 1 ≤ n ≤ N1 (entering period), for N1 ≤ n ≤ N3 (turning period), for N3 ≤ n < N (exiting period).
(4.11)
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Proof. During the entering period, we have λn a b λn+1 > 2 + 1 − n n 1 + λn for some a, b > 0 due to (3.33) and (3.34). Assuming that λn > c/n we get c/n a b λn+1 > 2 + 1 − n n 1 + c/n c + (a − bc + ac/n)/n . = n+c If c > 0 is small enough, the expression in parentheses is positive and we obtain λn+1 > c/(n + c) > c/(n + 1), thus completing the induction. Similarly, λn A B λn+1 < 2 + 1 − n n 1 + λn for some A, B > 0 due to (3.33) and (3.34). Assuming that λn < C/n we get C/n A B λn+1 < 2 + 1 − n n 1 + C/n C + (A − BC + AC/n)/n . = n+C If C > 0 is large enough, the expression in parentheses is negative (for large n), and we obtain λn+1 < C/(n + C) < C/(n + 1), thus completing the induction. Next we consider the turning period of the corner series. We just proved that λ N1 1/N1 1/N , and we noted in the previous section that τ N1 1/N 2 , hence B +N1 = λ N1 /τ N1 N . Then we can use the recursive formula + Bn+1 =
2 Bn+ + , sin γn+1 1 + τn Bn+
see (4.7), to estimate Bn+ for n ≥ N1 . Since 2<
2 2 =: G < ∞ < sin γn+1 sin γ¯
+ for all n ∈ (N1 , N3 ), we have Bn+1 ≤ G + Bn+ , thus Bn+ ≤ B N1 + N G N for N1 ≤ n ≤ N3 . To get a lower bound on Bn+ recall that τn ≤ D/N 2 for some D > 0. Assuming that Bn > d N for some small d > 0 we obtain + Bn+1 ≥2+
dN ≥ dN 1 + (D/N 2 )(d N )
for large N . Therefore, Bn+ N and λn = τn Bn+ 1/N for N1 ≤ n ≤ N3 . During the exiting period, we use the ‘countdown’ index m = N + 1 − n (see Remark 3.3), so that (4.11) takes form λm−1 =
2τm−1 τm−1 λm + . sin γm−1 τm 1 + λm
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Also, all the asymptotic formulas obtained in the previous section for the entering period remain valid for the exiting period if one replaces n by m; in particular, a 2τm−1 A b B τm−1 < < 2 and 1 + <1+ < 2 m sin γm−1 m m τm m for some constants 0 < a < A < ∞ and 0 < b < B < ∞ and all m ≥ 3. Next we use the ‘backward’ induction on m, going down from m = N3 to m = 1. Assuming that λm > c/m we get λm−1 > =
c/m a b + 1 + m2 m 1 + c/m c + [a + bc − c − c2 + (ac − a − bc − ac/m)/m]/(m + c) . m−1
If c > 0 is small enough, the expression in the brackets is positive (for large m), and we obtain λm−1 > c/(m − 1), thus completing the induction. Assuming that λm < C/m we get λm−1 < =
C/m A B + 1 + m2 m 1 + C/m C + [A + BC − C − C 2 + (AC − A − BC − AC/m)/m]/(m + C) . m−1
If C > 0 is large enough, the expression in the brackets is negative (for large m), and we obtain λm−1 < C/(m − 1), thus completing the induction. The lemma is proved.
Remark 4.3. Observe that during the exiting period λm 1/m and τm m −4/3 N −2/3 , + = λ /τ m 1/3 N 2/3 for all m = 2, . . . , N − N . The case m = 1 cf. (3.32), hence Bm m m 3 (i.e. n = N ) is not included in our estimates, because it is the last collision in the corner series, so that τ N 1, which affects λ N . However, for points x ∈ E N satisfying (4.8) we can still use (4.7), which gives us B +N B +N −1 N 2/3 . N −1 2 Lemma 4.2 implies that n=1 λn = O(1), i.e. this sum is bounded uniformly in N . Hence for any 1 ≤ N < N ≤ N we have −1 N
n=N
(1 + λn ) = exp
−1 N
ln(1 + λn ) exp
n=N
−1 N
λn .
(4.12)
n=N
N3 In particular, during the turning period, we have n=N λ 1, hence the expansion 1 −1 n is insignificant (it is uniformly bounded in N ). Next we estimate the expansion during the entering period. Lemma 4.4. For all x ∈ E N satisfying (4.8) we have
N1
n=1 (1 + λn )
N 1/3 .
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Proof. In view of (4.12), this is equivalent to O(1). It is enough to show that λn =
N1
n=1 λn
=
1 3
ln N1 + N , where N =
N1 1 + χn , where χn = O(1). 3n
(4.13)
n=1
The recursive formula (4.11) can be rewritten as λn 2 4 + bn λn+1 = 2 + an + 1 − , 9n 3n 1 + λn where, due to (3.33),
ln n γn2 n an = O + 2 + 3 n3 n n and, due to (3.34), bn = O
ln n γn2 n + 2 + n2 n n
(4.14)
.
Note that |an | ≤ c/n 2 and |bn | ≤ c/n for some small c > 0; in fact c > 0 can be made arbitrarily small by choosing sufficiently small γ¯ > 0. To verify (4.13) it is convenient to change the variable as 1 + Zn λn = . (4.15) 3n We substitute (4.15) into (4.14) and obtain by direct calculation 2
1 1 Zn 1 1 − Zn +O , +O Z n+1 = Rn + Z n 1 − + bn + O n n2 3n n2 n2 where Rn = 3nan + bn + O(1/n 2 ) ln n γn2 n + 2 . =O + n2 n n
(4.16)
Observe that Z n > −1 because λn > 0. It is clear that Z n gets closer to zero as n grows, but we need more precise asymptotics. If we fix a small δ > 0, then for large enough n we have δ . |Z n+1 | ≤ |Rn | + |Z n | 1 − n Without affecting the asymptotic behavior of Z n ’s we can assume that the above bound is valid for all n. Using it recurrently we obtain |Z n | ≤ |Rn | +
n−1 k=1
≤ const
n
|Rk |
n−1
1−
i=k
|Rk | e−
n i=k
k=1
≤ const
n k=1
|Rk |(k/n)δ .
δ i +1
δ/(i+1)
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Now we are ready to verify (4.13): N1
|χn | ≤ const
n=1
N1
|Z n |/n
n=1
≤ const
N1 n
|Rk |k δ /n 1+δ
n=1 k=1
≤ const
N1
|Rk |
k=1
≤ const
N1
N1
k δ /n 1+δ
n=k
|Rk |.
k=1
Due to (4.16), the last sum is bounded uniformly in N . The lemma is proved.
It remains to estimate the expansion during the exiting period: N −1 Lemma 4.5. For all x ∈ E N satisfying (4.8) we have n=N (1 + λn ) N 2/3 . 3 Proof. Our argument follows the lines of the previous proof and we again use the count N −N3 down index m = N − n + 1. In view of (4.12), the lemma is equivalent to m=2 λm = 2 3 ln(N − N3 ) + N , where N = O(1). It is enough to show that λm =
2 + χm , where 3m
N −N3
χm = O(1)
The recursive formula (4.11) now takes the form λm 2 4 λm−1 = + bm + am + 1 + , 9m 2 3m 1 + λm where, due to (3.33),
am = O
and, due to (3.34),
bm = O
(4.17)
m=2
ln m γm2 m + 2+ 3 m3 m m
ln m γm2 m + 2 + m2 m m
(4.18)
.
Note that |am | ≤ c/m 2 and |bm | ≤ c/m for some small c > 0 (and c can be made arbitrarily small by choosing sufficiently small γ¯ ). To verify (4.17) it is convenient to change the variable as 1 + Zm . (4.19) λm = 2 3m We substitute (4.19) into (4.18) and obtain by direct calculation 2
1 1 Zm 1 1 Z m−1 = Rm + Z m 1 − + bm + O − Z + O , + O m 2 2 m m 3m m m2
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N. Chernov, R. Markarian
where Rm = 3mam + bm + O(1/m 2 ) ln m γm2 m =O + + 2 . m2 m m
(4.20)
If we fix a small δ > 0, then for large enough m we have δ . |Z m−1 | ≤ |Rm | + |Z m | 1 − m Without affecting the asymptotic behavior of Z m ’s we can assume that the above bound is valid for all m ≥ 3. Using it recurrently we obtain |Z m | ≤
N −N3 k=m
≤ const
k
|Rk |
1−
i=m N −N3
|Rk | e−
δ i
k i=m
δ/i
k=m
≤ const
N −N3
|Rk |(m/k)δ .
k=m
Now we are ready to verify (4.17): N −N3
|χm | ≤ const
m=2
N −N3
|Z m |/m
m=2
≤ const
N −N3 N −N3 m=2
≤ const
N −N3
k=m
|Rk |
k=2
≤ const
N
|Rk |m δ−1 /k δ
k
m δ−1 /k δ
m=2
|Rk |.
k=2
Due to (4.20), the last sum is bounded uniformly in N . The lemma is proved.
After the last collision, the particle leaves the cusp and flies back to the vicinity of the point D ∈ ∂D. According to Remark 4.3, B +N N 2/3 and τ N 1, hence unstable vectors are additionally expanded by 1 + τ N B +N N 2/3 . Thus the total expansion factor for unstable vectors d x ∈ Tx M is Dx F N +1 (d x) p N 1/3 × N 2/3 × N 2/3 = N 5/3 d x p for all x ∈ E N satisfying (4.8).
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747
For points x ∈ E N where γ1 fails to satisfy (4.8), the expansion between the first and second collisions is Dx1 F(d x1 ) p N −2/3 = 1 + τ1 B1+ 1 + , d x1 p cos ϕ1 which accounts for the first extra factor in (4.5). For points x ∈ E N where γ N fails to ˆ (near D) is satisfy (4.8), the expansion between the last collision at x N and return to M Dx N F(d x N ) p = 1 + τ N B +N B +N d x N p = 2/ cos ϕ N + B − N [cos ϕ N ]−1 + N 2/3 N −2/3 2/3 1+ , N cos ϕ N which accounts for the second extra factor in (4.5). This completes the proof of Proposition 4.1.
5. Cell Structure Here we use the results of the previous two sections to analyze the sets E N , which consist of points whose trajectories go down a cusp and experience there a corner series of exactly N collisions. We will use standard facts of the theory of dispersing billiards [BSC90, BSC91, C99, CM06]. For example, the domains E N are bounded by singularity curves of the map Fˆ (which are singularity curves for the maps F i , i = 1, . . . , N , with N = R(x)); the latter are smooth compact curves whose slope in the r ϕ coordinates is negative and bounded away from zero and infinity, i.e. −∞ < C1 ≤ dϕ/dr ≤ C2 < 0 ˆ N ) are domains bounded by singufor some constants C1 , C2 . The images FN = F(E −1 ˆ larity curves of the map F , which are smooth compact curves with positive slope. Moreover, due to the time-reversibility of the billiard dynamics, we have a handy symmetry: a point (r, ϕ) belongs in E N if and only if (r, −ϕ) ∈ FN , hence FN is obtained by reflecting E N across the line ϕ = 0. More generally, a point (r, ϕ) is a singularity point for the map Fˆ if and only if (r, −ϕ) is a singularity point for its inverse Fˆ −1 . For simplicity, we again consider the three-arc billiard table shown on Fig. 1. There ˆ from which trajectories depart into cusps: their footpoints are three identical spots in M must be near D, E, or F (opposite to the cusps A, B, and C, respectively), and the velocities of such trajectories must be nearly orthogonal to ∂D. Consider one such spot, near the point x D = (r D , 0), where r D denotes the r -coordinate of D. A simple geometric inspection shows that x D itself belongs to a singularity curve, call it S0 (see the thick black line on Fig. 5, going from ‘northwest’ to ‘southeast’); it is made by trajectories whose very first collision in the cusp √ is grazing. One can easily check that the slope of the curve S0 at x D is dϕ/dr = −(3 + 3)/2. The domain E N (more precisely, its part near x D ) is a union of two ‘bent’ strips (colored grey on Fig. 5), we denote them by E N and E N . Each strip is bounded by an ‘outer’
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Fig. 5. The domain E N near the point x D
Fig. 6. Extremal trajectories in E N
curve, call it S N −1 , and an ‘inner’ curve, call it S N (as well as two short segments of S0 ). The curves S N and S N −1 bounding E N = E N ∪ E N terminate on S0 . The domains E N , N > K 0 , make a ‘nested’ structure and shrink to x D as N → ∞ (they are schematically shown by concentric ovals on Fig. 5). The curves S N separating the domain E N from E N +1 are made by trajectories whose last collision in the corner series is grazing. Why do we have two parts (two strips) of the domain E N , one above S0 and the other below S0 ? It is because the first collision of a corner series of length N may occur on either of the two arcs making the cusp (left or right), and each strip contains points coming down onto one of these arcs (the strip E N above S0 hits the left arc first, the strip E N below S0 hits the right arc first).
To determine the dimensions of the strips E N and E N observe that their extreme points (lying on the curve S0 and located farthest from the central point x D ) are made by trajectories whose very first collision in the cusp is grazing, see the solid lines on Fig. 6. Since the point of the first collision in the cusp is the distance N −2/3 from the vertex A, according to (3.2), we conclude that the trajectory originates at the distance N −2/3 from the point D. Thus the diameter of E N and E N (i.e. the ‘length’ of these strips) is N −2/3 .
Dispersing Billiards with Cusps: Slow Decay of Correlations
749
Fig. 7. An unstable curve W N ⊂ E N and its image Fˆ (W ) ⊂ FN
The middle parts of E N and E N (closest to the point x D ) are made by trajectories starting out at angles |ϕ| N −4/3 , see the dashed lines on Fig. 6, thus dist(E N , x D ) N −4/3 . This suggests that the width of the strips making E N is N −7/3 , but we will deduce this estimate from the results of the previous section. ˆ N ) is congruent to E N Due to the aforementioned symmetry, the image FN = F(E itself, in particular it consists of two strips, FN and FN of length N −2/3 , see Fig. 7. ˆ ) = F (this is the case when N is Without loss of generality we suppose that F(E N N is an unstable curve stretching ˆ ) = F ). If W ⊂ E even, otherwise we have F(E N N N ˆ ) will stretch ‘from top across E N (from S N to S N −1 ), see Fig. 7, then its image F(W −2/3 ˆ )| N . Due to Proposition 4.1, we to bottom’ of FN , so its length will be |F(W obtain |W | N −2/3 /N 5/3 = N −7/3 , which is exactly the width of the strip E N . It is now clear that μ(E ˆ N ) μ(E N ) N −2/3 × N −7/3 = N −3 , thus ˆ : R(x) > N ) N −2 , μ(x ˆ ∈M hence (2.3) holds with a = 2. This completes the proof of Theorem 1.1, except we have not yet verified all the conditions of Theorem 2.1: it remains to prove the following: ˆ →M ˆ has exponential decay of correlations. Proposition 5.1. The map Fˆ : M Proof. According to [CZ05a], it is enough to verify a set of standard conditions. These include several conditions of technical nature (distortion bounds, absolute continuity, curvature bounds for singularity lines, etc.), which for dispersing billiards without cusps have been verified in other papers [BSC91, C99] and in our book [CM06], and their verification for billiards with cusps only require minor changes. We only deal with the main condition on the expansion of unstable curves here. ˆ These include the points where Fˆ Let S denote the singularity set for the map F. is discontinuous as well as the preimages of the boundaries of homogeneity strips, see ˆ denote by Wi , i ≥ 1, the connected components below. For any unstable curve W ⊂ M of W \ S. For every i let i be the minimal factor of expansion of Wi under Fˆ (due to the distortion bounds, this factor does not vary much over Wi ). Then the expansion condition to be verified is sup i−1 < 1, (5.1) lim inf δ→0
W : |W |<δ i
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N. Chernov, R. Markarian
where the supremum is taken over unstable curves W of length |W | < δ. Let S1,d denote the set where the map Fˆ is discontinuous. In the vicinity of xd , the set S1,d is the union of the curve S0 and all the curves S N , N ≥ K 0 . ˆ is increasing in the r ϕ coordinates, hence it can only Any unstable curve W ⊂ M intersect any given discontinuity curve Si once. But it may intersect infinitely many (or all!) of them, hence W \ S1,d may have countably many connected components. Each component lies in one strip of E N for some N ≥ K 0 , and we denote them by W N = W ∩ E N and W N = W ∩ E N . Consider an arbitrary component W N , N ≥ K 0 . It must be further subdivided into finitely or countably many ‘homogeneous’ subcomponents in the following way. For every i = 1, . . . , N , if the image F i (W N ) crosses the boundary of a homogeneity strip (defined below) at a point y ∈ F i (W N ), then the curve W N must be subdivided at the point F −i (y). Homogeneity strips were introduced in [BSC91] for a better control over distortions, see also [CM06, Chap. 5]. We fix a large constant k0 1 and for each k ≥ k0 define two strips H±k ⊂ M by Hk = (r, ϕ) : π/2 − k −2 < ϕ < π/2 − (k + 1)−2 and H−k = (r, ϕ) : − π/2 + (k + 1)−2 < ϕ < −π/2 + k −2 . Now M is divided into homogeneity strips Hk bounded by the lines S±k = (r, ϕ) : ± ϕ = π/2 − k −2 for |k| ≥ k0 ; these are countably many horizontal lines on the r ϕ coordinate plane accumulating near the natural boundary |ϕ| = π/2, see Fig. 8. Consider the domains G i = F i (E N ) for i = 1, . . . , N . A direct geometric inspection shows that the very first one, G 1 , is a strip adjacent to the boundary ϕ = −π/2, see Fig. 8; its length in the ‘negative’ (northwest–southeast) direction is N −2/3 , and its width in the ‘positive’ (northeast–southwest) direction is N −5/3 (this follows from our estimates on the size of E N and our analysis of expansion of unstable curves in Sect. 4). −2/3 , hence k N 1/3 . It crosses infinitely many lines S−k , k ≥ k N , where k −2 N N N i Further images G , i ≥ 2, move away from the boundary |ϕ| = π/2, see Fig. 8, thus they can only cross 2k N N 1/3 lines S±k , k ≤ k N . When i approaches N , this picture is repeated in the reverse order: the domains G i , N /2 ≤ i ≤ N − 1, intersect finitely many lines S±k , k ≤ k N , and the last domain G N intersects countably many lines S−k , k ≥ k N . Since any unstable curve has a positive slope, dϕ/dr > 0, it may only intersect each line Sk once. Thus every line Sk can only induce one point in the curve W N where the latter must be subdivided. Most important are the intersections of S−k , k ≥ k N , with the very first image F(W N ) and the very last image F N (W N ). They induce a partition of W N into countably many subcomponents that we denote by W N ,k,m = W N ∩ F −1 (H−k ) ∩ F −N (H−m ),
(5.2)
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751
Fig. 8. The images G i = F i (E N ) and the lines S±k . The vertical line corresponds to r¯ = r A , the r -coordinate of the vertex A
k, m ≥ k N . Observe that cos ϕ k −2 in the strips H±k . Thus, according to (4.5), the map Fˆ expands the subcomponent W N ,k,m by a factor of N ,k,m N 5/3 (1 + k 2 N −2/3 )(1 + m 2 N −2/3 ). We now estimate from above the following sum: −5/3 −1 (1 + k 2 N −2/3 )−1 (1 + m 2 N −2/3 )−1 , N ,k,m N (k,m)∈Z N
(5.3)
(k,m)∈Z N
where Z N is the set of pairs (k, m) for which the intersection (5.2) is not empty. Observe that if the curve W N is traversed from one end to the other, then both indices k and m change monotonically. In the case treated here (which is shown on Figs. 7 and 8) both indices increase or both decrease depending on the direction in which the curve W N is traversed. Thus, if we join each pair of neighboring points of the set Z N ⊂ R2 by a unit segment, we will get a monotonically increasing polygonal line in the quadrant {k ≥ k N , m ≥ k N } starting at (k N , k N ). For every n ≥ 2k N there is at most one pair (k, m) ∈ Z N such that k + m = n. For a fixed value of k + m = n, by a simple application of Cauchy-Schwarz inequality we get (1 + k 2 N −2/3 )(1 + m 2 N −2/3 ) ≥ N −2/3 n 2 , thus (k,m)∈Z N
−5/3 −1 N ,k,m ≤ const · N
N −1
∞
N 2/3 n −2
n=2k N ∞ 2k N
dx x2
N −1 N −1/3 = N −4/3 .
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N. Chernov, R. Markarian
In other cases (say, for W N = W ∩ E N ) it might happen that, as the curve W N is traversed from one end to the other, then the indices k and m change in the opposite way: k increases and m decreases (or vice versa). Then, if we join each pair of neighboring points of the set Z N ⊂ R2 by a unit segment, we will get a monotonically decreasing polygonal line in the quadrant {k ≥ k N , m ≥ k N }. For every n ∈ Z there will be at most one pair (k, m) ∈ Z N such that k − m = n. For a fixed value of k − m = n, we obviously have (1 + k 2 N −2/3 )(1 + m 2 N −2/3 ) ≥ (1 + k 2N N −2/3 )(1 + (k N + |n|)2 N −2/3 ), thus (k,m)∈Z N
−1 N ,k,m
∞ 1 const · N −5/3 ≤ 1 + k 2N N −2/3 n=k 1 + n 2 N −2/3 N ∞ d x N −5/3 −2/3 x2 kN 1 + N
N −5/3 N 1/3 = N −4/3 , which is the same upper bound as in the previous case. Next, we need to add intersections of the lines S±k , k ≤ k N , with the intermediate images F i (W N ), 2 ≤ i ≤ N − 1. These contribute at most 2k N additional points of intersection, i.e. at most 2k N additional subcomponents in W N . The minimal expansion factor of the map Fˆ along the curve W N is N −5/3 , thus additional 2k N subcomponents will contribute the amount ≤ const · k N N −5/3 N −4/3 , which is of the same order of magnitude as the sum (5.3). Thus for every component W N = W ∩ E N of the original unstable curve W the sum of the reciprocals of the minimal expansion factors over all its subcomponents is bounded above by const · N −4/3 . It remains to sum up over N ≥ K 0 : const
∞
−1/3
N −4/3 ≤ const · K 0
< 1,
N =K 0
which is true if K 0 is chosen large enough. This proves (5.1) for unstable curves going through long corner series. For all the other unstable curves the dynamics is not different from that in ‘regular’ dispersing billiards (without cusps), where (5.1) has been verified in [Y98, C99], see also [CM06, Chap. 5]. Proposition 5.1 is proved.
This completes the proof of Theorem 1.1 for the special three-arc table shown on Fig. 1.
6. General Case In the previous sections we restricted our analysis to the three-arc billiard table with cusps introduced by Machta [Mac83] and shown on Fig. 1. This made our calculations relatively simple and geometrically transparent. Here we outline changes necessary for proving Theorem 1.1 in the general case. Let a cusp be made by two boundary components 1 , 2 ⊂ ∂D. Choose the coordinate system as shown on Fig. 9, then the equations of 1 and 2 are, respectively,
Dispersing Billiards with Cusps: Slow Decay of Correlations
753
Fig. 9. A cusp made by two curves, 1 and 2
y = f 1 (x) and y = − f 2 (x), where f i are convex C 3 functions, f i (x) > 0 for x > 0, and f i (0) = f i (0) = 0 for i = 1, 2. We will use the Taylor polynomial for the functions f i and their derivatives: f i (x) = 21 ai x 2 + O(x 3 ),
f i (x) = ai x + O(x 2 ),
f i (x) = ai + O(x),
where ai = f i (0). Since the curvature of the boundary of dispersing billiards must not vanish, we have ai > 0√for i = 1, 2. For the particular three-arc table analyzed earlier, f 1 (x) = f 2 (x) = 1 − 1 − x 2 . Consider a billiard trajectory entering the cusp and making a long series of N reflections there. We denote reflection points by (xn , yn ), where yn = f 1 (xn ) or yn = − f 2 (xn ) depending on which side of the cusp the reflection occurs. As in Sect. 3, we use ϕn and γn = π/2 − |ϕn | for the angle of reflection, but do not use αn any more (its role will be played by xn ). Generally, we will use the same symbols as in Sects. 3–4 (to make our presentations here and there comparable), but some symbols will now have a slightly different meaning. As in Sect. 3, we denote by N2 the deepest collision (closets to the vertex of the cusp). Clearly, the collisions occur alternatively from the two sides of the cusp, they go down the cusp monotonically, and then return back up monotonically as well: x1 > x2 > · · · > x N2 ≤ x N2 +1 < x N2 +2 < · · · < x N (possibly, two deepest collisions have equal x-coordinates). Lemma 3.1 partially extends to the general case. Namely, let xm = x N2 be the deepest collision, and assume without loss of generality that xm+1 ≥ xm−1 . Then xm+i ≥ xm−i
and
γm+i ≤ γm−i
for all i = 1, 2, . . ., as long as both collisions remain in the corner series. This implies that |N2 − N /2| = O(1). As in Sect. 3, we fix a small γ¯ > 0 and introduce N1 and N3 accordingly; this divides the corner series into three periods: entering, turning, and exiting ones. Our first task is to extend Proposition 3.2 to the general case. Consider two successive reflections at points (xn , yn ) and (xn+1 , yn+1 ) with angles γn and γn+1 . Without loss of generality, let yn = − f 2 (xn ), hence yn+1 = f 1 (xn+1 ). A direct geometric inspection shows that γn+1 = γn + tan−1 f 2 (xn ) + tan−1 f 1 (xn+1 ) (6.1)
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and xn+1 = xn −
f 2 (xn ) + f 1 (xn+1 ) , tan γn + tan−1 f 2 (xn )
(6.2)
as long as the trajectory goes down the cusp, i.e. n < N2 . Equations (6.1)–(6.2) are analogues of the simpler relations (3.6)–(3.7) used in Sect. 3. All the arguments of that section will carry over to the general case by way of Taylor expansion of all the functions involved in (6.1)–(6.2). We only outline main steps, the reader should have no trouble filling missing details. First, (6.1) gives γn+1 = γn + a2 xn + a1 xn+1 + O(xn2 ), and adding these up for 1 ≤ n < N2 gives x1 + · · · + x N2 = O(1) (remember that a1 , a2 > 0), which is an analogue of (3.9). Next we introduce variables: xn+1 γn and wn = . un = xn xn Due to (6.1), we obtain an analogue of (3.12): wn+1 = a1 +
wn + a2 + O(xn ) , un
(6.3)
¯ + O(1), where a¯ = (a1 + a2 )/2. and since u n ≤ 1 it follows that wn ≥ 2an Using (6.2) and the obvious tan x > x gives un > 1 −
a¯ [1 + O(xn )] . wn + a 2
Combining (6.3) with (6.4) gives
wn+1 < a1 + (wn + a2 + O(xn )) 1 +
a¯ [1 + O(xn )] + O(wn−2 ) wn + a 2
(6.4)
= wn + 3a¯ + O(xn ) + O(wn−1 ), ¯ + O(ln n). So we obtain wn n, hence γn nαn , in particular therefore wn ≤ 3an α N2 1/N2 . A more precise asymptotical formula follows from (6.2): un = 1 −
a2 + a1 u 2n + O(xn + γn2 ) , 2(wn + a2 )
(6.5)
and combining (6.3) with (6.5) gives a2 + a1 u 2n + O(xn + γn2 + n −1 ) 2 = wn + 3a¯ + O(xn + γn2 + n −1 ),
wn+1 = wn + a1 + a2 +
where we used the established fact u n = 1 − O(n −1 ). Therefore ¯ + O(ln n + n ), wn = 3an
(6.6)
Dispersing Billiards with Cusps: Slow Decay of Correlations
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where n = γ12 + · · · + γn2 , as in Sect. 3. It is easy to verify the relations (3.30). Next, (6.3) implies x wn+1 − a1 n 1+O u −1 n = wn + a 2 n 1 xn γn2 1 = 1+ +O + + 2 . 3n + O(ln n + n ) n n n Multiplying over n gives −1 1/3 x1 /xn = u −1 , 1 · · · u n−1 n −2/3
hence γn x1 n 2/3 . In particular, x1 N1 and x N1 1/N1 . The analysis of the turning period is easily done as in Sect. 3 and gives N1 N and x N2 1/N . Note that xn = O(1/n), hence the xn ’s can be absorbed by 1/n in the previous formulas, and then we get exactly the same formulas (3.27), (3.28), and (3.29) as in Sect. 3, except we now have an extra factor of a¯ in (3.27) and (3.28). Next, the intercollision time (=distance) is τn =
f 2 (xn ) + f 1 (xn+1 ) sin γn + tan−1 f 2 (xn )
xn a2 + a1 u 2n + O(xn ) 2wn 1 + a2 wn−1 + O(γn2 ) = xn wn−1 a¯ + O(n −1 + γn2 )
=
xn /n n −4/3 N −2/3 . It follows that τn a¯ = 2 1 + O(n −1 + γn2 ) sin γn wn ln n γn2 n 1 +O + 2 + 3 . = 9an ¯ 2 n3 n n Next we estimate
sin γn + tan−1 f 2 (xn ) τn+1 f 1 (xn+1 ) + f 2 (xn+2 ) = : Fn × Fn . × = τn f 2 (xn ) + f 1 (xn+1 ) sin γn+1 + tan−1 f 1 (xn+1 )
First, FN =
2 + a x 2 + O(x 3 ) a1 xn+1 2 n+2 n 2 + O(x 3 ) a2 xn2 + a1 xn+1 n
u 2n (a1 + a2 u 2n+1 ) + O(xn ) a2 + a1 u 2n + O(xn ) ln n γ 2 2+ n n +O + , = 1− + + x n 3n n2 n n2
=
(6.7)
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where = (a2 − a1 )/a. ¯ Using the same argument as in Sect. 3 we get sin γn+1 + tan−1 f 1 (xn+1 ) − sin γn + tan−1 f 2 (xn ) FN = 1 − sin γn+1 + tan−1 f 1 (xn+1 ) γn+1 − γn + a1 xn+1 − a2 xn + O(xn2 ) 1 + O(γn2 ) = 1− xn u n (wn+1 + a1 ) + O(xn2 + γn3 ) wn (u n − 1) + (3a¯ + a1 )u n − a2 + O(n −1 + γn2 ) 1 + O(γn2 ) = 1− u n (wn+1 + a1 ) + O(xn ) ln n γn2 n 2− . +O + + = 1− 3n n2 n n2 Therefore, τn+1 ln n γn2 n 4 +O + 2 + xn , =1− + τn 3n n2 n n (note that cancels out!). This is almost identical to (3.34); the extra term xn will not cause trouble as xn = O(1). In summary, all the main formulas here are similar to those in Sect. 3, with a notable exception: an extra factor of a¯ in the expressions for wn and in (6.7). The extension of the results of Sect. 4 to the general case is pretty straightforward, the only serious change involves the recursive formula (4.7) which now takes form + Bn+1 =
2Kn+1 Bn+ + , sin γn+1 1 + τn Bn+
where Kn+1 is the curvature of the boundary ∂D at the point (xn+1 , yn+1 ): f 1 (xn+1 ) Kn+1 = 3/2 = a1 + O(xn+1 ). 1 + [ f 1 (xn+1 )]2 The recurrence formula (4.11) changes accordingly: λn+1 =
2τn+1 Kn+1 τn+1 λn + . sin γn+1 τn 1 + λn
(6.8)
We observe two new elements here, as compared to (4.11) of Sect. 4: there is an extra factor of a¯ in the denominator, due to (6.7), and an extra factor a1 + O(xn ) in the numerator due to the curvature. Of course, when the collision occurs at the other side of the cusp, the curvature will be a2 + O(xn ). As the trajectory collides alternatively at both sides, the extra factors a1 and a2 alternate in the numerator. Due to the additive character of (6.8), the combined effect of the extra factors a1 and a2 in the numerator will be exactly opposite to that of the extra factor of a¯ = (a1 + a2 )/2 in the denominator, so in the end all the new factors will cancel out. This proves Proposition 4.1 in the general case. Lastly we extend the results of Sect. 5 to the general case. Our main task is to describe the structure of the cells E N . Let P denote the vertex of a cusp and L the common tangent line to the two boundary components making the cusp. Let Q(P) denote the other point of intersection of L with ∂D (opposite to P). For example, on Fig. 1 we have D = Q(A).
Dispersing Billiards with Cusps: Slow Decay of Correlations
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In generic billiard tables, L intersects ∂D at Q transversally, then, just as in Sect. 5, points x ∈ E N whose trajectories enter the cusp have to start near Q and their images ˆ F(x) = F N +1 (x) have to land back near Q again. Of course the ϕ-coordinate of x ∈ E N and F N +1 (x) ∈ FN need not be close to zero, so the cells E N may lie far away from their images FN . But all the estimates of Sect. 5 obviously remain valid. In the exceptional case, where the line L is tangent to ∂D at the point Q, the analysis requires modification. The boundary ∂D may be smooth at Q (thus L makes a ‘grazing collision’ at the point Q), or Q itself may be a corner point or even another cusp (!). For example, imagine a diamond-looking table made by four identical circular arcs tangent to each other at their endpoints – there are two pairs of cusps opposite to each other. In these exceptional cases one can analyze the cell structure directly, but this may be fairly complicated. A useful trick, however, may reduce the analysis to the generic case, in which the line L intersects ∂D transversally. One simply adds to ∂D a short ‘transparent’ line segment positioned inside D so that it cuts L transversally (or even orthogonally) between the points P and Q. We note that adding transparent walls to billiards is a standard trick [SC87]. Now the billiard trajectories going into the cusp for long corner series must first cross that newly added segment. They do not change their velocities (the segment is transparent, after all), but we register the point of intersection as an extra collision point. Therefore all the cells E N will appear on that extra segment added to the boundary. Their parameters will be obviously the same as described in Sect. 5. This allows us to prove Theorem 1.1 in the exceptional cases. We conclude with an open problem. We always assumed that the curvature of the boundary ∂D did not vanish, in particular we had a1 , a2 > 0 in this section. It is interesting to let the curvature vanish at the vertex of the cusp, so that a1 = 0 or a2 = 0, or both. Would this affect the rate of the decay of correlations? It seems that if a1 = 0 but a2 > 0 (or vice versa), then the rate will not change. But in the case a1 = a2 = 0 the cusp becomes very degenerate and may trap billiard trajectories for much longer than ‘regular’ cusps treated here. This may slow down the decay of correlations even further. A similar phenomenon was recently discovered in [CZ05b]. Acknowledgement. We thank our colleagues P. Balint, D. Dolgopyat, G. Galperin, and H.-K. Zhang for many useful discussions, and Alvaro Pardo for assisting us with computer simulations. N. C. was partially supported by NSF grant DMS-0354775. R. M. was partially supported by the Proyecto PDT 29/219 (CONICYT, Uruguay).
References [BG06]
Bálint, P., Gouëzel, S.: Limit theorems in the stadium billiard. Commun. Math. Phys. 263, 461– 512 (2006) [BS81] Bunimovich, L.A., Sinai, Ya.G.: Statistical properties of lorentz gas with periodic configuration of scatterers. Commun. Math. Phys. 78, 479–497 (1981) [BSC90] Bunimovich, L.A., Sinai, Ya.G., Chernov, N.I.: Markov partitions for two-dimensional billiards. Russ. Math. Surv. 45, 105–152 (1990) [BSC91] Bunimovich, L.A., Sinai, Ya.G., Chernov, N.I.: Statistical properties of two-dimensional hyperbolic billiards. Russ. Math. Surv. 46, 47–106 (1991) [C99] Chernov, N.: Decay of correlations and dispersing billiards. J. Stat. Phys. 94, 513–556 (1999) [C06a] Chernov, N.: Regularity of local manifolds in dispersing billiards. Math. Phys. Electr. J. 1, 54 (2006) [C06b] Chernov, N.: Advanced statistical properties of dispersing billiards. J. Stat. Phys. 122, 1061– 1094 (2006)
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[CH96] [CD06] [CZ05a] [CZ05b] [CM06] [CGS92] [CG93] [GO74] [Mac83] [MR86] [Mar04] [OW98] [R95] [Si70] [SC87] [Y98] [Y99]
N. Chernov, R. Markarian
Chernov, N., Haskell, C.: Nonuniformly hyperbolic k-systems are bernoulli. Ergod. Th. Dynam. Syst. 16, 19–44 (1996) Chernov, N., Dolgopyat, D.: Brownian Motion – I. To appear in Memoirs AMS; available at http://www.math.uab.edu/chernov/papers/brown15.pdf, (2006) Chernov, N., Zhang, H.-K.: Billiards with polynomial mixing rates. Nonlinearity 18, 1527– 1553 (2005) Chernov, N., Zhang, H.-K.: A family of chaotic billiards with variable mixing rates. Stochastics & Dynamics 5, 535–553 (2005) Chernov, N., Markarian, R.: Chaotic Billiards. Mathematical Survey and Monographs, Providence, RI: Amer. Marh. Sic. 2006 Collet, P., Galves, A., Schmitt, B.: Unpredictability of the ocurrence time of a long laminar period in a model of temporal intermittency. Ann. Inst. H. Poincaré. Phys. Théor. 57, 319–331 (1992) Collet, P., Galves, A.: Statistics of close visits to the indifferent fixed point of an interval map . J. Stat. Phys 72, 459–478 (1993) Gallavotti, G., Ornstein, D.S.: Billiards and bernoulli schemes. Commun. Math. Phys. 38, 83– 101 (1974) Machta, J.: Power law decay of correlations in a billiard problem. J. Stat. Phys. 32, 555–564 (1983) Machta, J., Reinhold, B.: Decay of correlations in the regular lorentz gas. J. Stat. Phys. 42, 949– 959 (1986) Markarian, R.: Billiards with polynomial decay of correlations. Ergod. Th. Dynam. Syst. 24, 177– 197 (2004) Ornstein, D., Weiss, B.: On the bernoulli nature of systems with some hyperbolic structure. Ergod. Th. Dynam. Syst. 18, 441–456 (1998) Reháˇcek, J.: On the ergodicity of dispersing billiards. Rand. Comput. Dynam. 3, 35–55 (1995) Sinai, Ya.G.: Dynamical systems with elastic reflections. ergodic properties of dispersing billiards. Russ. Math. Surv. 25, 137–189 (1970) Sinai, Ya.G., Chernov, N.I.: Ergodic properties of some systems of 2-dimensional discs and 3dimensional spheres. Russ. Math. Surv. 42, 181–207 (1987) Young, L.-S.: Statistical properties of dynamical systems with some hyperbolicity. Ann. Math. 147, 585–650 (1998) Young, L.-S.: Recurrence times and rates of mixing. Israel J. Math. 110, 153–188 (1999)
Communicated by G. Gallavotti
Commun. Math. Phys. 270, 759–775 (2007) Digital Object Identifier (DOI) 10.1007/s00220-006-0170-6
Communications in
Mathematical Physics
Universal Inequalities for Eigenvalues of the Buckling Problem on Spherical Domains Qiaoling Wang , Changyu Xia Departamento de Matemática, Universidade de Brasília, Brasília-DF, 70910-900, Brazil. E-mail:
[email protected],
[email protected] Received: 6 April 2006 / Accepted: 11 September 2006 Published online: 9 January 2007 – © Springer-Verlag 2006
Abstract: In this paper we study the eigenvalues of the buckling problem on domains in a unit sphere. We obtain universal bounds on the (k + 1)th eigenvalue in terms of the first k eigenvalues independent of the domains. 1. Introduction Let be a connected bounded domain with smooth boundary in an n(≥ 2)-dimensional Euclidean space IR n and let ν be the outward unit normal vector field of ∂. Denote by the Laplacian operator on IR n . Consider the following well-known eigenvalue problems : u = −λu
in ,
u = 0, on ∂, ∂u = 0, on ∂, 2 u = ηu in , u = ∂ν ∂u 2 u = − u in , u = = 0, on ∂. ∂ν
(1.1) (1.2) (1.3)
They are called the fixed membrane problem; the clamped plate problem and the bucking problem, respectively. Let 0 < λ1 < λ 2 ≤ λ 3 ≤ · · · , 0 < η1 ≤ η2 ≤ λ 3 ≤ · · · , 0 < 1 ≤ 2 ≤ 3 ≤ · · · denote the successive eigenvalues for (1.1), (1.2) and (1.3), respectively. Here each eigenvalue is repeated according to its multiplicity. In 1955 and 1956, Payne, Pólya and Partially supported by FEMAT.
Partially supported by CNPq, Pronex and Proex.
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Weinberger [PPW1, PPW2] proved that λ2 ≤ 3 for ⊂ IR 2 λ1 and conjectured that
λ2 λ2 ≤ λ1 λ1 disk
with equality if and only if is a disk. For n ≥ 2, the analogous statements are λ2 4 ≤1+ for ⊂ IR n , λ1 n and the PPW conjecture
λ2 λ2 ≤ , λ1 λ1 n−ball
with equality if and only if is an n-ball. This important PPW conjecture was solved by Ashbaugh and Benguria in their excellent papers [AB1, AB2, AB3]. In [PPW2], Payne, Pólya and Weinberger also proved the bound 2 λi , k = 1, 2, . . . , k k
λk+1 − λk ≤
(1.4)
i=1
for ⊂ IR 2 . This result easily extends to ⊂ IR n as λk+1 − λk ≤
k 4 λi , k = 1, 2, . . . . kn
(1.5)
i=1
Two main advances in extending (1.5) were made by Hile-Protter in [HP] and Yang [Y], respectively. Namely, in 1980, Hile and Protter proved k i=1
λi kn , for k = 1, 2, . . . . ≥ λk+1 − λi 4
(1.6)
In 1991, Yang proved the following much stonger inequality: k i=1
4 λi ≤ 0, for k = 1, 2, . . . . (λk+1 − λi ) λk+1 − 1 + n
(1.7)
By elementary calculations, one can show that Yang’s inequality (1.7) is sharper than the inequality (1.6) of Hile-Protter and that (1.6) is sharper than the inequality (1.5) of Payne-Pólya-Weinberger. For the eigenvalues of the clamped plate problem (1.2), Payne, Pólya established the following inequality (cf. [PPW2]): 8(n + 2) 1 ηi . n2 k k
ηk+1 − ηk ≤
i=1
(1.8)
Inequalities for Eigenvalues of the Buckling Problem on Spherical Domains
As a generalization of (1.8), Hile-Yeh obtained [HY] −1/2 k 1/2 k ηi n 2 k 3/2 ≥ ηi . ηk+1 − ηi 8(n + 2) i=1
761
(1.9)
i=1
Hook [H], Chen-Qian [CQ] proved, independently, the following inequality k k 1/2 η1/2 n2k 2 i ηi . ≤ 8(n + 2) ηk+1 − ηi i=1
(1.10)
i=1
In a survey paper on recent developments on eigenvalue problems, Ashbaugh [A] asked if one can obtain universal inequality for the eigenvalues of the clamped plate problem (1.2) which is similar to Yang’s universal inequality (1.7) for the eigenvalues of the fixed membrane problem (1.1). This problem has been solved by Cheng and Yang in [CY2]. Namely, they proved 1 ηk+1 − ηi ≤ k k
i=1
8(n + 2) n2
1/2
1 (ηi (ηk+1 − ηi ))1/2 . k k
(1.11)
i=1
For the buckling problem (1.3), Payne, Pólya and Weinberger [PPW] proposed the following Problem 1. Can one obtain a universal inequality for the eigenvalues of the buckling problem (1.3), which is similar to the universal inequalities for the eigenvalues of the fixed membrane problem (1.1) or of the clamped plate problem (1.2) ? Ashbaugh [A] mentioned this problem again. With respect to the above problem, Payne, Pólya and Weinberger proved 2 / 1 < 3 for ⊂ IR 2 . For ⊂ IR n this reads 2 / 1 < 1 + 4/n. Subsequently Hile and Yeh [HY] reconsidered this problem obtaining the improved bound 2 n 2 + 8n + 20 ≤ 1 (n + 2)2
for ⊂ IR n .
Ashbaugh [A] proved: n
i+1 ≤ (n + 4) 1 .
(1.12)
i=1
Recently, Cheng and Yang introduced a new method to construct trial functions for the buckling problem (1.3) and obtained the following universal bounds [CY4]: k i=1
4(n + 2) ( k+1 − i ) i . n2 k
( k+1 − i )2 ≤
i=1
(1.13)
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On the other hand, the inequalities on eigenvalues of the fixed membrane problem (1.1) on bounded domain in IR n obtained by Payne-Pólya-Weinberger, Hile-Protter, Yang have also been extended to some Riemannian manifolds other than the Euclidean space (cf. [CY1, CY3, H1, HM1, HM2, HS, Leu, Li, YY]). To the authors’ knowledge, little is known about the universal bounds on eigenvalues of the biharmonic operator for Riemannian manifolds other than the Euclidean space which are similar to the PaynePólya-Weinberger’s inequality (1.1), or the Hile-Protter’s inequality (1.6), or the Yang’s inequality (1.7). It is therefore natural to consider the following Problem 2. Let M be an n-dimensional complete Riemannian manifold and let be a bounded connected domain in M with smooth boundary ∂. Consider the eigenvalue problems: ∂u 2 u = ηu in , u = = 0, on ∂, (1.14) ∂ν ∂u 2 u = − u in , u = = 0, on ∂. (1.15) ∂ν For what kind of M, does there exist a universal bound on the (k + 1)th eigenvalue in terms of the first k eigenvalues of (1.14) or of (1.15)? In this paper we study the eigenvalues of the problem (1.15) when is a spherical domain and obtain the following Theorem 1. Let i be the i th eigenvalue of the following eigenvalue problem: ∂u 2 u = − u in , u|∂ = = 0, (1.16) ∂ν ∂
where is a connected domain in a unit sphere S n (1)(n ≥ 2) with smooth boundary ∂ and ν is the unit outward normal vector field of ∂. Then for any δ > 0, we have 2
k ( k+1 − i )2 i=1
≤
k i=1
+
δ 2 ( i − (n − 2)) ( k+1 − i )2 δ i + 4(δ i + n − 2)
k 1 (n − 2)2 . ( k+1 − i ) i + δ 4
(1.17)
i=1
From Theorem 1, we can obtain a more explicit inequality which is weaker than (1.17): Corollary 1. Under the same assumptions as in Theorem 1, we have k 1 9 1 (n − 2)2 1 k + k + i + k+1 − 2 4 k 4 4 i=1 ⎧ 2 k ⎨1 9 1 (n − 2)2 1 ≤ λk + k + i + ⎩4 4 k 4 4 −
i=1
5 k + 4
1 2 (n − 2)2 i + k 4 k
i=1
1/2 k 1 1 k + i . 4 k i=1
Inequalities for Eigenvalues of the Buckling Problem on Spherical Domains
763
2. Proofs of the Results Proof of Theorem 1 . First we recall Cheng-Yang’s method [CY4] of constructing trial functions. Denote by , the canonical metric on IR n+1 as well as that induced on S n (1). Let and ∇ be the Laplacian and the gradient operator of S n (1), respectively. Let x1 , x2 , . . . , xn+1 be the standard coordinate functions of the Euclidean space IR n+1 ; then n+1 n n+1 2 xα = 1 . S (1) = (x1 , . . . , xn+1 ) ∈ IR ; α=1
It is well known that xα = −nxα , α = 1, · · · , n + 1.
(2.1)
For functions f and g on , the Dirichlet inner product ( f, g) D of f and g is given by
( f, g) D = ∇ f, ∇g.
The Dirichlet norm of a function f is defined by || f || D = {( f, f ) D }1/2 =
1/2 |∇ f |2
.
Let u i be the i th orthonormal eigenfunction of the problem (1.16) corresponding to the eigenvalue i , i = 1, 2, . . . , that is, u i satisfies ∂u i = 0, 2 u i = − i u i in , u i |∂ = ∂ν ∂
(2.2) (u i , u j ) D = ∇u i , ∇u j = δi j , ∀ i, j.
Let ∇ 2 be the Hessian operator on S n (1). For a function f on , the squared norm of ∇ 2 f is defined as n 2 2 2 ∇ ∇ 2 f (ei , e j ) , f =
(2.3)
i, j=1
where e1 , . . . , en are orthonormal vector fields locally defined on . Define H22 () by H22 () = { f : f, |∇ f |, ∇ 2 f ∈ L 2 ()}. Then H22 () is a Hilbert space with respect to the norm || · ||2 : || f ||2 =
2 1/2 | f |2 + |∇ f |2 + ∇ 2 f .
(2.4)
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Q. Wang , C. Xia
2 () of H 2 () defined by Consider the subspace H2,D 2 ∂ f 2 2 H2,D () = f ∈ H2 () : f |∂ = =0 . ∂ν ∂ 2 () with The biharmonic operator 2 defines a self-adjoint operator acting on H2,D discrete eigenvalues 0 < 1 ≤ · · · ≤ k ≤ · · · for the buckling problem (1.16) and ∞ defined in (2.2) form a complete orthonormal basis for the the eigenfunctions {u i }i=1 2 2 () satisfies (φ, u ) = 0, ∀ j = 1, 2, . . . , k, then Hilbert space H2,D (). If φ ∈ H2,D j D the Rayleigh-Ritz inequality tells us that
k+1 ||φ||2D ≤ φ2 φ. (2.5)
For vector-valued functions F = ( f 1 , f 2 , . . . , f n+1 ), G = (g1 , g2 , . . . , gn+1 ) : → IR n+1 , we define an inner product (F, G) by
(F, G) ≡
F, G =
n+1 α=1
f α gα .
The norm of F is given by ||F|| = (F, F)
1/2
=
n+1 α=1
1/2 .
f α2
Let H21 () be the Hilbert space of vector-valued functions given by H12 () = F = ( f 1 , . . . , f n+1 ) : → IR n+1 ; f α , |∇ f α | ∈ L 2 (), for α = 1, . . . , n + 1 with norm ||F||1 = ||F|| + 2
n+1 α=1
1/2 |∇ f α |
2
.
Observe that a vector field on can be regarded as a vector-valued function from to 2 () ⊂ H2 () be a subspace of H2 () spanned by the vector-valued IR n+1 . Let H1,D 1 1 ∞ , which form a complete orthonormal basis of H2 (). For any functions {∇u i }i=1 1,D 2 (), we have ∇ f ∈ H2 () and for any X ∈ H2 (), there exists a f ∈ H2,D 1,D 1,D 2 () such that X = ∇ f . function f ∈ H2,D For any α = 1, . . . , n + 1 and each i ∈ {1, . . . , k}, we can decompose the vector-valued functions xα ∇u i as xα ∇u i = ∇h αi + Wαi ,
(2.6)
2 (), ∇h is the projection of x ∇u in H2 () and W ⊥ H2 (). where h αi ∈ H2,D αi α i αi 1,D 1,D Thus we have
2 Wαi |∂ = 0, and (Wαi , ∇u) = Wαi , ∇u = 0, for any u ∈ H2,D (). (2.7)
Inequalities for Eigenvalues of the Buckling Problem on Spherical Domains
765
2 () in L 2 () and of C 1 () in L 2 (), we conclude that By the denseness of H2,D
(Wαi , ∇h) = 0, ∀ h ∈ C 1 () ∩ L 2 (), which implies from the divergence theorem that
h divWαi = 0, ∀ h ∈ C 1 () ∩ L 2 (),
(2.8)
(2.9)
where for a vector field Z on , div Z denotes the divergence of Z . Consequently, we have div Wαi = 0.
(2.10)
Consider the functions φαi : → IR, given by φαi = h αi −
k
bαi j u j ,
(2.11)
j=1
where
bαi j =
xα ∇u i ∇u j = bα ji .
(2.12)
∂φαi = 0, ∂ν ∂
(2.13)
We have φαi |∂ =
(φαi , u j ) D =
∇φαi , ∇u j = 0, ∀ j = 1, . . . , k.
It therefore follows from the Rayleigh-Ritz inequality for k+1 that
|∇φαi |2 ≤ φαi 2 φαi , ∀α = 1, . . . , n + 1, i = 1, . . . , k. k+1
(2.14)
(2.15)
D
Since div Wαi = 0, we have from (2.2), (2.6) and (2.11) that φαi = h αi −
k
bαi j u j
j=1
= div(xα ∇u i ) −
k
bαi j u j
j=1
= ∇xα , ∇u i + xα u i −
k
bαi j u j
(2.16)
j bαi j u j .
(2.17)
j=1
and 2 φαi = (∇xα , ∇u i + xα u i ) +
k j=1
766
Q. Wang , C. Xia
Observe that
φαi u j = −
∇φαi , ∇u j = 0.
We have
φαi 2 φαi = φαi (∇xα , ∇u i + xα u i )
= h αi (∇xα , ∇u i + xα u i )
−
k j=1
bαi j
u j (∇xα , ∇u i + xα u i )
=
h αi (∇xα , ∇u i + xα u i ) − (∇xα , ∇u i + xα u i ) − 2
bαi j
j=1
=
k
k j=1
u j h αi
bαi j
2 u j h αi
∇xα , ∇u i 2 + 2xα ∇xα , ∇u i u i + xα2 (u i )2 =
+
k j=1
=
bαi j j
u j h αi
k ∇xα , ∇u i 2 + u i div(xα2 ∇u i ) − j bαi j ∇u j ∇h αi
j=1
k 2 ∇xα , ∇u i 2 + u i div(xα2 ∇u i ) − = j bαi j.
(2.18)
j=1
It is easy to see that ||xα ∇u i ||2 = ||∇h αi ||2 + ||Wαi ||2 , ||∇h αi ||2 = ||∇φαi ||2 +
k
2 bαi j.
(2.19)
j=1
Hence
u i div(xα2 ∇u i )
= − xα2 ∇u i , ∇(u i )
= u i div(xα2 ∇(u i ))
= u i ∇xα2 , ∇(u i ) + xα2 2 u i
(2.20)
Inequalities for Eigenvalues of the Buckling Problem on Spherical Domains
= =
=
=
767
u i ∇xα2 , ∇(u i ) − i u i ∇xα2 , ∇(u i ) + i u i ∇xα2 , ∇(u i ) + i
xα2 u i u i
∇(xα2 u i ), ∇u i
u i ∇xα2 , ∇u i + i ⎛
xα2 |∇u i |2
∇xα2 , u i ∇(u i ) + i u i ∇u i + i ⎝||∇φαi ||2 + ||Wαi ||2 +
k
⎞ 2 ⎠ bαi j .
j=1
Combining (2.15), (2.18) and (2.20), we get ( k+1 − i )||∇φαi ||2 ≤ pαi + ||∇xα , ∇u i ||2 + i ||Wαi ||2 +
k 2 ( i − j )bαi j, j=1
(2.21) where
pαi =
∇xα2 ,
u i ∇(u i ) + i u i ∇u i and || f || = 2
| f |2 .
(2.22)
Before we can finish the proof of Theorem 1, we will need the following lemma: Lemma. Set Z αi
n−2 xα ∇u i , cαi j = = ∇∇xα , ∇u i − 2
∇u j , Z αi ;
(2.23)
then cαi j = −cα ji .
(2.24)
Proof of Lemma. For any smooth function f on , we have from the Bochner formula that 1 |∇ f |2 = |∇ 2 f |2 + ∇ f, ∇( f ) + Ric(∇ f, ∇ f ) 2 (2.25) = |∇ 2 f |2 + ∇ f, ∇( f ) + (n − 1)|∇ f |2 , where Ric is the Ricci tensor of S n (1). Thus for any smooth function g on , it holds 1 |∇g|2 = |∇ 2 g|2 + ∇g, ∇(g) + (n − 1)|∇g|2 2
(2.26)
and 1 |∇( f + g)|2 = |∇ 2 ( f + g)|2 + ∇( f + g), ∇(( f + g)) + (n − 1)|∇( f + g)|2 . 2 (2.27) Subtracting the sum of (2.25) and (2.26) from (2.27), we get ∇ f, ∇g = 2∇ 2 f, ∇ 2 g + ∇ f, ∇(g) + ∇g, ∇( f ) + 2(n − 1)∇ f, ∇g, (2.28)
768
Q. Wang , C. Xia
where ∇ 2 f, ∇ 2 g =
n
∇ 2 f (es , et )∇ 2 g(es , et ),
s,t=1
being e1 , . . . , en orthonormal vector fields locally defined on . We infer from (2.28) by taking f = xα and g = u i that ∇xα , ∇u i = 2∇ 2 xα , ∇ 2 u i + ∇xα , ∇(u i ) + ∇u i , ∇(xα ) +2(n − 1)∇xα , ∇u i . (2.29) It is well-known that ∇ 2 xα = −xα , .
(2.30)
∇xα , ∇u i = −2xα u i + ∇xα , ∇(u i ) + (n − 2)∇xα , ∇u i .
(2.31)
Therefore,
From (2.23) and (2.29), we have
cαi j = = = = =
= = = =
n−2 ∇∇xα , ∇u i − xα ∇u i , ∇u j 2
n−2 bαi j − u j ∇xα , ∇u i − 2
n−2 bαi j − u j (−2xα u i + ∇xα , ∇(u i ) + (n − 2)∇xα , ∇u i ) − 2
n−2 bαi j 2 xα u j u i + u i div(u j ∇xα ) − (n − 2) u j ∇xα , ∇u i − 2
(2 − n) xα u j u i + u i ∇u j , ∇xα
n−2 −(n − 2) bαi j u j ∇xα , ∇u i − 2
n−2 bαi j (2 − n) u j div(xα ∇u i ) − ∇u i , ∇∇u j , ∇xα − 2
n−2 (n − 2) xα ∇u i , ∇u j − ∇u i , ∇∇u j , ∇xα − bαi j 2
n−2 ∇∇xα , ∇u j − xα ∇u j , ∇u i − 2 (2.32) −cα ji .
Thus (2.24) holds. This completes the proof of lemma.
Inequalities for Eigenvalues of the Buckling Problem on Spherical Domains
769
Let us continue on the proof of Theorem 1. It follows from (2.6), (2.8), (2.11), (2.14) and (2.23) that
xα ∇u i , Z αi = −2 ∇h αi + Wαi , Z αi −2
= −2 ∇h αi , ∇ Z αi + (n − 2) Wαi , xα ∇u i
k bαi j ∇u j , Z αi = −2 ∇φαi +
+(n − 2)
= −2
Wαi , xα ∇u i
∇φαi , Z αi − 2
=
j=1
bαi j cαi j + (n − 2)||Wαi ||2
j=1
(−2) ∇φαi , Z αi −
−2
k
k
cαi j ∇u j
j=1 k
bαi j cαi j + (n − 2)||Wαi ||2 .
(2.33)
j=1
We also have
xα ∇u i , Z αi = −2 xα ∇u i , ∇∇xα , ∇u i + (n − 2) −2 xα2 |∇u i |2
=2 div(xα ∇u i )∇xα , ∇u i + (n − 2) xα2 |∇u i |2
∇xα2 , u i ∇u i = 2 ∇xα , ∇u i 2 +
2 2 xα |∇u i | . (2.34) +(n − 2)
Combining (2.33) and (2.34), we get
k k bαi j cαi j = (−2) ∇φαi , Z αi − cαi j ∇u j + (n − 2)||Wαi ||2 , rαi + 2 j=1
j=1
(2.35) where rαi = 2||∇xα , ∇u i || + 2
∇xα2 , u i ∇u i + (n − 2)||xα ∇u i ||2 .
(2.36)
Multiplying (2.35) by ( k+1 − i )2 , one obtains from the Schwarz inequality that ⎛ ⎞ k ( k+1 − i )2 ⎝rαi + 2 bαi j cαi j ⎠ j=1
770
Q. Wang , C. Xia
⎧ ⎛ ⎫ ⎞ k ⎨ ⎬ ⎝(−2)∇φαi , Z αi − = ( k+1 − i )2 cαi j ∇u j ⎠ + (n − 2)||Wαi ||2 ⎩ ⎭ j=1 ⎛ 2 ⎞
k ⎟ k+1 − i ⎜ 3 2 Z ≤ δ( − ) |∇φ | + − c ∇u ⎝ k+1 i αi αi αi j j ⎠ δ j=1
+(n − 2)( k+1 − i ) ||Wαi || 2
2
⎞ ⎛ k − k+1 i ⎝ 2 ⎠ = δ( k+1 − i )3 ||∇φαi ||2 + cαi ||Z αi ||2 − j δ j=1
+(n − 2)( k+1 − i ) ||Wαi || , 2
2
(2.37)
where δ is a positive constant. Substituting (2.21) into (2.37), we get ⎛ ⎞ k ( k+1 − i )2 ⎝rαi + 2 bαi j cαi j ⎠ ⎛
j=1
≤ δ( k+1 − i )2 ⎝ pαi
⎞ k 2 ⎠ + ||∇xα , ∇u i ||2 + i ||Wαi ||2 + ( i − j )bαi j
⎛
+
⎞
j=1
k+1 − i ⎝ 2 ⎠ cαi + (n − 2)( k+1 − i )2 ||Wαi ||2 . ||Z αi ||2 − j δ k
(2.38)
j=1
Since bαi j = bα ji , cαi j = −cα ji , it follows from (2.38) by taking the sum for i from 1 to k that k
( k+1 − i )2 rαi − 2
i=1
k
( k+1 − i )( i − j )bαi j cαi j
i, j=1
⎛ ⎞ k k 2⎝ 2 2 2 ⎠ ≤δ ( k+1 − i ) ( i − j )bαi j pαi + ||∇xα , ∇u i || + i ||Wαi || + i=1
⎫ ⎧ k k ⎬ ⎨ 1 2 + ( k+1 − i )||Z αi ||2 − ( k+1 − i )cαi j ⎭ δ⎩ i=1
+(n − 2)
j=1
i, j=1
k
( k+1 − i )2 ||Wαi ||2
i=1
=
( k+1 − i )2 δpαi + (δ i + n − 2)||Wαi ||2 + δ||∇xα , ∇u i ||2
k i=1
Inequalities for Eigenvalues of the Buckling Problem on Spherical Domains
771
k k 1 2 ( k+1 − i )||Z αi ||2 − δ ( k+1 − i )( i − j )2 bαi j δ
+
i=1
−
i, j=1
k 1 2 ( k+1 − i )cαi j. δ
(2.39)
i, j=1
Therefore, we have from −2
k
( k+1 − i )( i − j )bαi j cαi j ≥ −δ
i, j=1
k
2 ( k+1 − i )( i − j )2 bαi j
i, j=1
−
k 1 2 ( k+1 − i )cαi j δ i, j=1
that k k ( k+1 − i )2 rαi ≤ ( k+1 − i )2 i=1
i=1
× δpαi + (δ i + n − 2)||Wαi ||2 + δ||∇xα , ∇u i ||2 1 ( k+1 − i )||Z αi ||2 . δ k
+
(2.40)
i=1
Let us compute
∇xα , ∇u i ∇u i , ∇xα ∇xα , ∇u i 2 =
=− xα div(∇xα , ∇u i ∇u i )
= − xα ∇u i , ∇∇xα , ∇u i − xα ∇xα , ∇u i u i
= − ∇h αi + Wαi , ∇∇xα , ∇u i − xα ∇xα , ∇u i u i
1 = − ∇h αi , ∇∇xα , ∇u i − ∇xα2 , ∇u i u i 2 δ i + n − 2 δ 2 ||∇h αi || + ||∇∇xα , ∇u i ||2 ≤ δ 4(δ i + n − 2)
1 − ∇xα2 , ∇u i u i . (2.41) 2 Introducing (2.41) into (2.40) and noticing (2.19), we get k k ( k+1 − i )2 rαi ≤ ( k+1 − i )2 i=1
i=1
772
Q. Wang , C. Xia
δ2 2 × δpαi + (δ i + n − 2)||xα ∇u i || + ||∇∇xα , ∇u i || 4(δ i + n − 2)
k k 1 1 2 2 − ( k+1 − i ) ∇xα , ∇u i u i + ( k+1 − i )||Z αi ||2 . (2.42) 2 δ 2
i=1
i=1
Summing over α, we have k k n+1 n+1 ( k+1 − i )2 rαi ≤ ( k+1 − i )2 α=1
i=1
i=1
× δpαi + (δ i + n − 2)||xα ∇u i ||2 +
α=1 δ2
4(δ i + n − 2) n+1
k 1 − ( k+1 − i )2 xα2 , ∇u i u i ∇ 2 α=1
i=1
+
k 1
δ
||∇∇xα , ∇u i ||
2
n+1
( k+1 − i )
||Z αi ||2 .
(2.43)
α=1
i=1
From (2.22), (2.36) and n+1 α=1
xα2 = 1,
we have n+1
pαi = 0,
α=1
n+1
rαi = 2
α=1
n+1
||∇xα , ∇u i ||2 + (n − 2)
α=1
n+1
||xα ∇u i ||2
α=1
= 2||∇u i ||2 + (n − 2)||∇u i ||2 = n,
(2.44)
where we have used the fact that n+1 α=1
∇xα , ∇u i 2 =
n+1
(∇u i (xα ))2 = |∇u i |2 .
α=1
Since n+1 α=1
∇xα , ∇u i ∇xα , ∇(u i ) = ∇u i , ∇(u i ),
(2.45)
Inequalities for Eigenvalues of the Buckling Problem on Spherical Domains
773
it follows from (2.31) that n+1
||∇∇xα , ∇u i ||2 = −
α=1
=−
n+1 α=1 n+1 α=1
∇xα , ∇u i ∇xα , ∇u i ∇xα , ∇u i (−2xα u i + ∇xα , ∇(u i )
+(n − 2)∇xα , ∇u i )
= − ∇u i , ∇(u i ) − (n − 2)||∇u i ||2
= ||u i ||2 − (n − 2) = i − (n − 2)
(2.46)
and n+1 α=1
2
n−2 ||Z αi || = ∇∇xα , ∇u i − 2 xα ∇u i 2
n+1 2 ||∇∇xα , ∇u i || − (n − 2) ∇∇xα , ∇u i , xα ∇u i = α=1
+
(n
− 2)2 4
||xα ∇u i ||2
(n − 2)2 (n − 2)2 = i + . 4 4 Consequently, we obtain by introducing (2.44)–(2.47) into (2.43) that = i − (n − 2) + (n − 2) +
(2.47)
k ( k+1 − i )2 n i=1
k δ 2 ( i − (n − 2)) 2 ≤ ( k+1 − i ) δ i + n − 2 + 4(δ i + n − 2) i=1
k 1 (n − 2)2 + , ( k+1 − i ) i + δ 4
(2.48)
i=1
that is 2
k ( k+1 − i )2 i=1
≤
k δ 2 ( i − (n − 2)) ( k+1 − i )2 δ i + 4(δ i + n − 2) i=1
+
k 1 (n − 2)2 . ( k+1 − i ) i + δ 4 i=1
This completes the proof of Theorem 1.
(2.49)
774
Q. Wang , C. Xia
Proof of Corollary 1. For any δ > 0, it follows from (1.17) that k k 1 2 2 2 ( k+1 − i ) ≤ δ ( k+1 − i ) i + 4 i=1
i=1
k 1 (n − 2)2 + . ( k+1 − i ) i + δ 4
(2.50)
i=1
Taking ⎧# 2 ⎫1/2 k ⎨ i=1 ⎬ ( k+1 − i ) i + (n−2) 4 δ= $ % #k 1 2 ⎩ ⎭ i=1 ( k+1 − i ) i + 4 in (2.50), we get k 1/2 k 1 2 2 ( k+1 − i ) ≤ ( k+1 − i ) i + 4 i=1 i=1 k 1/2 (n − 2)2 × ( k+1 − i ) i + . 4 i=1
It then follows from i ≤ k , i = 1, . . . , k, that k k 1 (n − 2)2 2 . ( k+1 − i ) ≤ k + ( k+1 − i ) i + 4 4 i=1
(2.51)
i=1
This is a quadratic inequality of k+1 . By elementary calculations, we obtain k 1 9 1 (n − 2)2 1 k + k + i + k+1 − 2 4 k 4 4 i=1 ⎧ 2 k ⎨1 9 1 (n − 2)2 1 ≤ λk + k + i + ⎩4 4 k 4 4 −
i=1
5 k + 4
1 k
k i=1
i2
(n − 2)2 + 4
1/2 k 1 1 k + i . 4 k i=1
This completes the proof of Corollary 1. References [A] [AB1]
Ashbaugh, M.S.: Isoperimetric and universal inequalities for eigenvalues. In: Spectral theory and geometry (Edinburgh, 1998), E. B. Davies, Yu Safalov (eds.), London Math. Soc. Lecture Notes, Vol. 273 Cambridge: Cambridge Univ. Press, 1999, pp. 95–139 Ashbaugh, M.S., Benguria, R.D.: Proof of the Payne-Pólya-Weinberger conjecture. Bull. Amer. Math. Soc. 25, 19–29 (1991)
Inequalities for Eigenvalues of the Buckling Problem on Spherical Domains
[AB2]
775
Ashbaugh, M.S., Benguria, R.D.: A sharp bound for the ratio of the first two eigenvalues of Dirichlet Laplacian and extensions. Ann. of Math. 135, 601–628 (1992) [AB3] Ashbaugh, M.S., Benguria, R.D.: A second proof of the Payne-Pólya-weinberger conjecture. Commun. Math. Phys. 147, 181–190 (1992) [CY1] Cheng, Q.M., Yang, H.C.: Estimates on eigenvalues of Laplacian. Math. Ann. 331, 445–460 (2005) [CY2] Cheng, Q.M., Yang, H.C.: Inequalities for eigenvalues of a clamped plate problem. Trans. Amer. Math. Soc. 358, 2625–2635 (2006) [CY3] Cheng, Q.M., Yang, H.C.: Inequalities for eigenvalues of Laplacian on domains and compact complex hypersurfaces in complex projective spaces. J. Math. Soc. Japan, 58, 545–561 (2006) [CY4] Cheng, Q.M., Yang, H.C.: Universal bounds for eigenvalues of a buckling problem. Commun. Math. Phsy. 262, 663–675 (2006) [CQ] Chen, Z.C., Qian, C.L.: Estimates for discrete spectrum of Laplacian operator with any order. J. China Univ. Sci. Tech. 20, 259–266 (1990) [H1] Harrell, E.M.: Some geometric bounds on eigenvalue gaps. Commun. Part. Differ. Eq. 18, 179– 198 (1993) [HM1] Harrell, E.M., Michel, P.L.: Commutator bounds for eigenvalues, with applications to spectral geometry. Commun. Part. Diff. Eq. 19, 2037–2055 (1994) [HM2] Harrell, E.M., Michel, P.L.: Commutator bounds for eigenvalues of some differential operators. In: Lecture Notes in Pure and Applied Mathematics, Vol. 168, G. Ferreyra, G.R. Goldstein, F. Neubrander (eds.) New York: Marcel Dekker 1995, pp. 235–244 [HS] Harrell, E.M., Stubbe, J.: On trace inequalities and the universal eigenvalue estimates for some partial differential operators. Trans. Am. Math. Soc. 349, 1797–1809 (1997) [HP] Hile, G.N., Protter, M.H.: Inequalities for eigenvalues of the Laplacian. Indiana Univ. Math. J. 29, 523–538 (1980) [HY] Hile, G.N., Yeh, R.Z.: Inequalities for eigenvalues of the biharmonic operator. Pacific J. Math. 112, 115–133 (1984) [H] Hook, S.M.: Domain independent upper bounds for eigenvalues of elliptic operator. Trans. Amer. Math. Soc. 318, 615–642 (1990) [Leu] Leung, P.F.: On the consecutive eigenvalues of the Laplacian of a compact minimal submanifold in a sphere. J. Austral. Math. Soc. (Series A) 50, 409–416 (1991) [Li] Li, P.: Eigenvalue estimates on homogeneous manifolds. Comment. Math. Helv. 55, 347–363 (1980) [PPW1] Payne, L.E., Pólya, G., Weiberger, H.F.: Sur le quotient de deux fréquences propres cosécutives. Comptes Rendus Acad. Sci. Paris 241, 917–919 (1955) [PPW2] Payne, L.E., Pólya, G., Weiberger, H.F.: On the ratio of consecutive eigenvalues. J. Math. and Phys. 35, 289–298 (1956) [Y] Yang, H.C.: An estimate of the difference between cosecutive eigenvalues. Preprint IC/91/60 of ICTP, Trieste, 1991 [YY] Yang, P.C., Yau, S.T.: Eigenvalues of the Laplacian of compact Riemann surfaces and minimal submanifolds. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 7, 55–63 (1980) Communicated by B. Simon
Commun. Math. Phys. 270, 777–788 (2007) Digital Object Identifier (DOI) 10.1007/s00220-006-0171-5
Communications in
Mathematical Physics
Remarks about the Inviscid Limit of the Navier–Stokes System Nader Masmoudi Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY 10012-1185, USA. E-mail:
[email protected] Received: 25 April 2006 / Accepted: 14 September 2006 Published online: 19 December 2006 – © Springer-Verlag 2007
Abstract: In this paper we prove two results about the inviscid limit of the NavierStokes system. The first one concerns the convergence in H s of a sequence of solutions to the Navier-Stokes system when the viscosity goes to zero and the initial data is in H s . The second result deals with the best rate of convergence for vortex patch initial data in 2 and 3 dimensions. We present here a simple proof which also works in the 3D case. The 3D case is new. 1. The Inviscid Limit The Navier-Stokes system is the basic mathematical model for viscous incompressible flows. In a bounded domain, it reads ⎧ ⎨ ∂t u + u.∇u − νu + ∇ p = 0, div(u) = 0, (1) ⎩ u = 0 on ∂, where u is the velocity, p is the pressure and ν is the kinematic viscosity. We can define a typical length scale L and a typical velocity U . The dimensionless parameter Re = U L/ν is very important to compare the properties of different flows. When Re is very large (ν very small), we expect that the Navier-Stokes system (N Sν ) behaves like the Euler system ⎧ ⎨ ∂t u + u.∇u + ∇ p = 0, div(u) = 0, (2) ⎩ u.n = 0 on ∂. The zero-viscosity limit of the incompressible Navier-Stokes equation in a bounded domain, with Dirichlet boundary conditions, is one of the most challenging open problems in Fluid Mechanics (see [20, 21] and the references therein). This is due to the formation of a boundary layer which appears because we can not impose a Dirichlet boundary
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condition for the Euler equation. This boundary layer satisfies formally the Prandtl equations, which seem to be ill-posed in general. In this paper we only deal with the inviscid limit in the whole space. All the results presented here can easily be extended to the periodic case. 2. Convergence in H s The inviscid limit in the whole space case was performed by several authors; we can refer for instance to Swann [22] and Kato [17, 18] (see also Constantin [10]). Here we would like to improve slightly the convergence stated in the previous works by proving the convergence in the H s space as long as the solution of the Euler system exists. Indeed, in most of the previous results only convergence in H s for s < s was proved. We also point out that in [18], Kato proved the convergence in H s for a short time, by using a general theory about quasi-linear equations. So, we do not claim that Theorem 2.1 is really new. Take the Navier-Stokes system in the whole space Rd : ∂t u n + div(u n ⊗ u n ) − νn u n = −∇ p n in Rd , div(u n ) = 0 in Rd , n u (t = 0) = u n0 with div(u n0 ) = 0,
(3) (4) (5)
where νn goes to 0 when n goes to infinity. Theorem 2.1. Let s > d/2 + 1, and u n0 ∈ H s (Rd ) such that u n0 goes to u 0 in H s (Rd ) when n goes to infinity. Let T ∗ be the time of existence and u ∈ Cloc ([0, T ∗ ); H s ) be the solution of the Euler system, ∂t u + div(u ⊗ u) = −∇ p in Rd , div(u) = 0 in Rd , u(t = 0) = u 0 with div(u 0 ) = 0.
(6) (7) (8)
Then, for all 0 < T0 < T ∗ , there exists ν0 > 0 such that for all νn ≤ ν0 , the NavierStokes system (3 – 5) has a unique solution u n ∈ C([0, T0 ]; H s (Rd )). Moreover, u n − u L ∞ (0,T0 ;H s ) → 0, n → ∞, (u n − u)(t) H s−2 ≤ C(νn t + u n0 − u 0 H s−2 ),
(u n − u)(t) H s ≤ C((νn t)(s−s )/2 + u n0 − u 0 H s )
(9) (10) (11)
for all 0 ≤ t ≤ T0 , s − 2 ≤ s ≤ s − 1 and C depends only on u and T0 . Remark 2.2. 1) The only relatively new part in Theorem 2.1 is the convergence in H s stated in (9) which holds for all T0 < T ∗ . 2) Interpolating between (10) and the uniform bound for wn in C([0, T ]; H s (Rd )), we deduce that u n converges to u in H s for any s < s and for s − 2 < s < s, we have (u n − u)(t) H s ≤ C(νn t + u n0 − u 0 H s−2 ) for all 0 ≤ t ≤ T0 .
s−s 2
(12)
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Proof. The proof of this theorem is based on a standard Grönwall inequality (see also [22, 17, 10]). Let us start by proving (10). First, we see that we can solve the Navier-Stokes system and Euler system in C([0, T ]; H s (Rd )) on some time interval independent of νn ≤ ν0 with bounds which are independent of n. This is because there is no boundary. Moreover, w n = u n − u satisfies ∂t w n + u n ∇w n + w n .∇u − νn w n + νn u + ∇( p n − p) = 0.
(13)
Then, we can write an energy estimate in H s−2 for w n = u n − u, namely ∂t w n 2H s−2 + νn ∇w n 2H s−2 ≤ C(u H s + w n H s )w n H s−2 + νn u H s−2 w n H s−2 ,
(14)
and by the Grönwall lemma, we can deduce that (10) holds for T0 = T . It was proved in [10] that the convergence holds as long as we can solve the Euler system and hence we can take any T0 such that T0 < T ∗ (see [10]). Notice that in [10], the regularity required is s − 2 > d/2 + 1. However, this is not necessary modulo the regularization argument which is used to prove the convergence in H s . To prove (11), we write an energy estimate in H s , s − 2 ≤ s ≤ s − 1, ∂t w n 2H s + νn ∇w n 2H s ≤ C(u H s + w n H s )w n 2H s + νn ∇u H s−1 ∇w n H 2s −(s−1) .
(15)
Then using an interpolation inequality and Holder inequality, we deduce that
2−(s−s ) Hs
−1 ∇wn H 2s −s+1 ≤ Cw n s−s ∇w n s H
≤
2− 2 1 ∇w n 2H s + C 2 w n s s−s . H C
Hence, 2 2− s−s
∂t w n 2H s ≤ Cw n 2H s + Cνn w n
Hs
(16)
and (11) follows. Getting the convergence in H s requires a regularization of the initial data. For all δ > 0, we take u δ0 such that u δ0 H s ≤ Cu 0 H s , u δ0 H s+1 ≤ Cδ , u δ0 H s+2 ≤ δC2 and for some s such that d/2 < s < s − 1, we have u δ0 − u 0 H s ≤ Cδ s−s . Such a u δ0 can be easily constructed by taking u δ0 = F −1 (1{|ξ |≤1/δ} Fu 0 ). Let v δ be the solution of the Euler system (6,7,8) with the initial data v δ (t = 0) = u δ0 . It is easy to see that v δ exists on some time interval [0, T ], T < T ∗ which only depends on u 0 H s and such that for 0 ≤ t ≤ T , we have v δ (t) H s ≤ C and v δ (t) H s+2 ≤ δC2 uniformly in δ. Indeed, the energy estimates at the level H s and H s+2 read ∂t v δ 2H s ≤ Cv δ 3H s , ∂t v δ 2H s+2
≤ Cv
δ
H s v δ 2H s+2 ,
(17) (18)
from which the uniform estimates follow. Then, setting wδ = v δ − u, we have ∂t w δ + w δ .∇v δ + u.∇w δ = −∇( p δ − p).
(19)
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Taking the energy estimate in H s yields ∂t w δ 2H s ≤ C(u H s + v δ H s )w δ 2H s + Cv δ H s+1 w δ H s w δ L ∞ . Then, we notice that on the time interval [0, T ], we have v δ H s+1 ≤ taking the energy estimate at the regularity s , we get
C δ.
(20)
Moreover,
∂t w δ 2H s ≤ C(u H s + v δ H s +1 )w δ 2H s ,
(21)
and since s + 1 < s, we get easily that wδ L ∞ (0,T ;H s ) ≤ Cδ s−s and by Sobolev
embedding, we have w δ L ∞ (0,T ;L ∞ ) ≤ Cw δ L ∞ (0,T ;H s ) ≤ Cδ s−s . Hence, (20) gives (22) ∂t w δ H s ≤ C(u H s + v δ H s ))w δ H s + Cδ s−s −1 . Hence w δ goes to zero in L ∞ (0, T ; H s ), namely v δ goes to v in L ∞ (0, T ; H s ) when δ goes to zero and we have
v δ − u L ∞ (0,T ;H s ) ≤ C(u δ0 − u 0 H s + δ s−s −1 T ).
(23)
Writing an energy estimate for w n,δ = u n − v δ , we get (here we drop the n and δ) ∂t w2H s + νn ∇w2H s ≤ C(w L ∞ v δ H s+1 w H s + (v δ H s + u n H s )w2H s ) + νn v δ H s+2 w H s .
(24)
Hence, we get ∂t w H s ≤ Cu n − u L ∞ v δ H s+1 + Cv δ − u L ∞ v δ H s+1 + νn v δ H s+2 + C(v δ H s + u n H s )w H s .
(25)
Since u n converges to u in H s−1 , we deduce that u n − u L ∞ (0,T ;L ∞ ) ≤ Cu n − u L ∞ (H s−1 ) ≤ C((νn T )1/2 + u n0 − u 0 H s−1 ). (26) Taking δ = δn such that δn , deduce that ∂t w n,δ H s ≤ C(
u n0 −u 0 H s−1 δn
and
νn δn2
go to zero when n goes to infinity, we
(νT )1/2 + u n0 − u 0 H s−1 ν + δ s−s −1 + 2 + w n,δ H s ). δ δ
(27)
Hence, by the Grönwall lemma, we deduce that wn,δ goes to zero in L ∞ (0, T ; H s ) and that u n goes to u in L ∞ (0, T ; H s ). Moreover, (νT )1/2 + u n0 − u 0 H s−1 ν −1 n s−s +δ u − u L ∞ (0,T ;H s ) ≤ C T + 2 δ δ
+ C(u n0 − u 0 H s + u δ0 − u 0 H s + δ s−s −1 T ).
(28)
We notice here that the rate of convergence gets better if we have a better approximation of u 0 by u δ0 . This will be studied in the next subsection. Since we have proved the convergence in H s till the time T , we can iterate the previous argument. Indeed, taking T as a new initial time and noticing that u n (T ) goes to u(T )
Remarks about the Inviscid Limit of the Navier–Stokes System
781
in H s , we see that we can iterate the previous argument on some time interval [T, T + T1 ] where T1 = T1 (u(T ) H s ) only depends on u(T ) H s and T1 ≥ C/u(T ) H s . Then, we can construct a sequence of times Tk , k ≥ 1 by this procedure. Now, it is clear that T + T1 + · · · + Tk goes to T ∗ when k goes to infinity. Indeed, the time Tk+1 goes to zero only if u(T + T1 + · · · + Tk ) H s goes to infinity, which means that T + T1 + · · · + Tk goes to T ∗ . This iteration argument allows us to get the convergence on any time interval [0, T0 ], T0 < T ∗ .
Remark 2.3. 1) We notice that the time T ∗ is related to the existence time for the Euler system (6). If d = 2 it is known [24, 23] that the Euler system (6) has a global solution and hence one can take any time T0 < ∞ in the above theorem. Moreover, since the 2D Navier-Stokes system has a global solution for all ν, we do not need the ν0 in the statement of the theorem. 2) The idea of using a regularization of the initial data was also used by Beirão da Veiga [2, 3] to prove a similar result in the compressible-incompressible limit. It is also used to prove the continuity of the solution with respect to the initial data in hyperbolic equations (see for instance Bona and Smith [5]). In the inviscid limit, this idea was used by Constantin and Wu [12] to prove some estimates on the rate of convergence of the vorticity. 2.1. Rate of convergence in H s . Take β such that 1 < β ≤ 2 and d/2 < s − β and for 0 ≤ δ < ∞, T > 0 we define u δ0 = F −1 (1{|ξ |≤1/δ} Fu 0 ), εT (δ) = u δ0 −u 0 H s +T δ β−1 , f T (δ) = δεT (δ) and gT (δ) = δ 2 εT (δ). We can see easily that for T > 0, f T and gT are increasing on [0, ∞). We denote by f T−1 and gT−1 their inverse. From the proof of Theorem 2.1, we can deduce the following corollary Corollary 2.4. Under the same hypotheses of Theorem 2.1, we have the following rate of convergence: (u n −u)(t) H s ≤ C
νt (gt−1 (νt))2
+C
t ((νt)β/2 + u n0 − u 0 H s−β )
f t−1 (t ((νt)β/2 + u n0 − u 0 H s−β ))
+Cu n0 −u 0 H s
for all 0 ≤ t ≤ T0 and C depends only on u and T0 .
(29)
Proof. Going back to the proof of Theorem 2.1, we see that (26) can be replaced by (u n − u)(t) L ∞ ≤ Cu n − u(t) H s−β ≤ C((νn t)β/2 + u n0 − u 0 H s−β ). Hence (28) can be replaced by
(u n − u)(t) H s ≤ Ct
(νt)β/2 + u n0 − u 0 H s−β ν + 2 δ δ
+ C(εt (δ) + u n0 − u 0 H s ).
(30)
(31)
Taking the optimum in δ and applying Lemma 2.5, we deduce easily that (29) holds.
Lemma 2.5. For a, b, t > 0, we have b a b a + 2 + εt (δ) ≤ 2 −1 + −1 . δ>0 δ δ f t (a) (gt (b))2 inf
The proof of this lemma is simple and is left for the reader.
(32)
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If we assume that u 0 is more regular, we can give a more precise rate. Corollary 2.6. We take the same hypotheses as in Theorem 2.1 and assume in addition that u 0 ∈ H s+α for some 0 < α ≤ 2. If 1 ≤ α ≤ 2, we have (u n − u)(t) H s ≤ C((νt)α/2 + u n0 − u 0 H s )
(33)
for all 0 ≤ t ≤ T0 and C depends only on u and T0 . If 0 < α < 1, then for all β such that 1 ≤ α + β ≤ 2 and s − β > d/2, we have (u n − u)(t) H s ≤ C(tu n0 − u 0 H s−β )α + C((νt)α/2 + u n0 − u 0 H s )
(34)
for all 0 ≤ t ≤ T0 and C depends only on u, β and T0 . Proof. First, notice that from the extra regularity of u 0 , we deduce that v δ H s+1 ≤ C(1 + δ α−1 ), v δ H s+2 ≤ Cδ α−2 and v δ − u H s ≤ Cδ α . If 1 ≤ α ≤ 2, then (24) yields ∂t w H s ≤ Cw H s + Cνn δ α−2 . (u n − u)(t) H s ≤ C νtδ α−2 + δ α + u n0 − u 0 H s . √ Taking the optimum in δ, namely δ = νt, we deduce that (33) holds. If 0 < α < 1, then arguing as in (15), we have
(35)
Hence
(36)
∂t w n 2H s−β + νn ∇w n 2H s−β ≤ C(u H s+α + w n H s )w n 2H s−β + νn ∇u H s+α−1 ∇w n H s+1−2β−α . (37) Then using an interpolation inequality and Holder inequality, we deduce that β+α−1
2−β−α
∇w n H s+1−2β−α ≤ Cw n H s−β ∇w n H s−β
≤
2 2− β+α 1 ∇w n 2H s−β + C 2 w n H s−β . C
Hence, we deduce that (u n − u)(t) H s−β ≤ C((νn t)(β+α)/2 + u n0 − u 0 H s−β ).
(38)
In the proof of Theorem 2.1, we see that (26) can be replaced by (u n − u)(t) L ∞ ≤ Cu n − u(t) H s−β ≤ C((νn t)(β+α)/2 + u n0 − u 0 H s−β ).
(39)
Moreover, v α − u L ∞ ≤ Cδ β+α . Hence (28) can be replaced by (u n − u)(t) H s ≤ Ct ((νt)(β+α)/2 + u n0 − u 0 H s−β )δ α−1 + νδ α−2 + C(δ α + tδ β+2α−1 + u n0 − u 0 H s ).
(40)
Taking the optimum in δ, we deduce that α (u n − u)(t) H s ≤ C t ((νt)(β+α)/2 + u n0 − u 0 H s−β ) + C(νt)α/2 + Cu n0 − u 0 H s . Hence (34) holds.
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783
3. Vortex Patches In this section d = 2 or 3. However, all the results can be easily extended to any dimension d ≥ 2 by using the notion of vortex patches in higher dimension (see [13]). For 2D vortex patches, namely the case where curl(u 0 ) is the characteristic function of a C 1+α domain α > 0, it was proved in [7] (see also [4]) that the Euler system (6,7,8) has a unique solution u such that the characteristic function of curl(u) remains a C 1+α domain ∞ (R; Li p). A similar result holds for 3D vortex patches, and that the velocity u is in L loc ∞ (0, T ∗ ; Li p) (see [15]). but only on a bounded interval, namely u ∈ L loc For vortex patches, Theorem 2.1 does not apply. Indeed, the velocity is not in H s for any s > d/2 + 1. For 2D vortex patches, it was proved in [11, 12] that the convergence to the Euler system still holds and that 1
u n − u L ∞ (0,T ;L 2 ) ≤ C(νn T ) 2 .
(42)
In [12], the authors also prove some estimate in L p spaces for the difference between the 1
vorticities, in particular they prove for p ≥ 2 that curl(u n − u) L ∞ (0,T ;L p ) ≤ Cνn4 p for some short time T and ε > 0. Also, in [1], a better rate of convergence is given for 2D vortex patches, namely 3
u n − u L ∞ (0,T ;L 2 ) ≤ C(νn T ) 4
−ε
(43)
which is optimal. Here we would like to extend the result of Abid and Danchin [1] to the 3D case and also give a slight improvement of their 2D result by allowing u n0 − u 0 to be just in L 2 . Moreover, the proof we present is much simpler. Let us recall the definition of a vortex patch Definition 3.1. Take 0 < r < 1. A vector field u is called a C r vortex patch if the following decomposition holds: curl(u) = χ P ωi + χ P c ωe ,
(44)
where P ⊂ Rd is an open set of class C 1+r and ωi , ωe ∈ C r (Rd )∩L 1 (Rd ) are compactly supported. Here χ P denotes the characteristic function of P. Notice that since curl(u) is divergencefree, we have ωi .n = ωe .n on ∂ P. This condition is always satisfied if d = 2. First, we recall that Gamblin and Saint-Raymond [15] proved the existence of a local ∞ (0, T ∗ ; Li p) to the vortex patch problem in 3D (see also [14 and 16]). solution u ∈ L loc Moreover, u remains a C r vortex patch. ∞ (0, T ∗ ; B ˙ α ), where α = min(r, 1/2) (see the Appendix). Hence, curl(u) ∈ L loc 2,∞ Theorem 3.2. Here d = 2 or 3. We assume that u n0 − u 0 goes to 0 in L 2 (Rd ) when n goes to infinity. We also assume that u 0 is a C r vortex patch. Then, if T ∗ is the time of existence and u ∈ Cloc ([0, T ∗ ); Li p) is the solution of the Euler system with initial data u 0 , then for all 0 < T < T ∗ , there exists ν0 such that for all νn ≤ ν0 and for all sequence of weak (Leray) solutions to the Navier-Stokes system (3 – 5), we have for 0 < t < T , (u n − u)(t) L 2 ≤ C((tνn )
1+α 2
+ u n0 − u 0 L 2 ),
where α = min(1/2, r ) and where C depends only on u and T .
(45)
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N. Masmoudi
1 ∞ Remark 3.3. In the 2D case, knowing that curl(u 0 ) ∈ L ∩ L does not imply that 2 2 u ∈ L and in general u 0 is not in L unless curlu = 0. In particular in the classical 2D vortex patch problem [19], namely the case curl(u) is the characteristic function of a C r +1 domain, u 0 is not in L 2 . However, in the 3D case, the fact that curlu 0 ∈ L 1 ∩ L ∞ implies that u 0 ∈ L 2 from the Biot-Savart formula.
Proof. Let us denote w n = u n − u, hence ∂t w n + w n .∇u + u.∇w n − νn w n − νn u = −∇( p n − p).
(46)
product with we get (at least formally) for 0 < t < T ,
t
t 1 n 1 w (t)2L 2 + νn ∇w n 2L 2 ≤ w n (0)2L 2 + C∇u L ∞ w n 2L 2 2 2 0 0
t
νn ∇u.∇w n . −
Taking the
wn ,
L2
(47)
0
In the 2D case, this computation is fully justified. We only point that in the 2D case, u and u n are not in general in L 2 but their difference is in L 2 . To prove (47) rigorously in the 3D case, we just add the energy inequality (48) and energy equality (49),
t 1 n 1 u (t)2L 2 + νn ∇u n 2L 2 ≤ u n (0)2L 2 , (48) 2 2 0 1 1 u(t)2L 2 = u(0)2L 2 , (49) 2 2 and subtract the weak formulation
t
n n u u(t) − u u(0) + (50) u n u.∇u + uu n .∇u n + νn ∇u n .∇u = 0. 0 −α α , the divergence-free property of u Besides, using the duality between B˙ 2,1 and B˙ 2,∞ and Lemma 4.1 (see the Appendix), we have
| ∇u.∇w n | ≤ C∇u B˙ α ∇w n B˙ −α (51) 2,∞
2,1
≤ Ccurlu B˙ α w n B˙ 1−α
(52)
≤ Ccurlu B˙ α w n αL 2 ∇w n 1−α . L2
(53)
2,∞
2,1
2,∞
By the Holder inequality, we have 2
2α
n 1+α νn curlu B˙ α w n αL 2 ∇w n 1−α ≤ Cνn curlu B1+α ˙ α w L 2 + L2 2,∞
2,∞
Hence, we get from (47), w
n
(t)2L 2 ≤ w n (0)2L 2 +C
0
t
νn ∇w n 2L 2 . (54) 2
2
2α
n 1+α ∇u L ∞ w n 2L 2 +Cνn curlu B1+α ˙ α w L 2 ds. (55) 2,∞
And by the Grönwall lemma, we deduce that 2
2
n 1+α wn (t) L1+α 2 ≤ Cw (0) L 2 + Cνn t
and (45) follows.
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785
From the proof, we can see that the only information we used about u is that u ∈ ∞ (0, T ∗ ; Li p) and curlu ∈ L ∞ (0, T ∗ ; B ˙ α ). Moreover, it is easy to see that if L loc 2,∞ loc α ∞ u ∈ L loc ([0, T ∗ ); Li p) then the B˙ 2,∞ , 0 < α < 1 regularity of curlu is propagated by α , then curlu ∈ L ∞ ([0, T ∗ ); B ˙ α ) (see [8]). Hence, the flow, namely if curlu 0 ∈ B˙ 2,∞ 2,∞ loc we have the following theorem Theorem 3.4. We assume that u n0 − u 0 goes to 0 in L 2 (Rd ) when n goes to infinity and α , 0 < α < 1. We also assume that the Euler system with the initial that curlu 0 ∈ B˙ 2,∞ ∞ ([0, T ∗ ); Li p). Then for all 0 < T < T ∗ , there data u 0 has a unique solution u ∈ L loc exists ν0 such that for all νn ≤ ν0 and for all sequences of weak (Leray) solutions to the Navier-Stokes system (3 – 5), we have for 0 < t < T , (u n − u)(t) L 2 ≤ C((tνn )
1+α 2
+ u n0 − u 0 L 2 ),
(57)
where C depends only on u and T . This theorem is an improvement of Theorem 1.1 of [1] since we only assume that the solution of the Euler system is Lipschitz. We would like to give two applications of this theorem which yield a better convergence rate than the simple application of Theorem 3.2. Consider a vector field u 0 which is a C r vortex patch with 0 < r < 1/2 and assume 1/2 in addition that curlu 0 ∈ B˙ 2,∞ , then Theorem 3.4 allows us to prove that 3
(u n − u)(t) L 2 ≤ C((tνn ) 4 + u n0 − u 0 L 2 ),
(58)
which is better than the rate we get from (45). There are several situations where u 0 is a C r vortex patch with 0 < r < 1/2 and 1/2 curlu 0 ∈ B˙ 2,∞ . For instance this is the case if curl(u 0 ) = χ P ωi0 + χ P c ωe0 is such that P ⊂ Rd is an open set of class C 1+r and ωi0 , ωe0 ∈ C 1/2 (Rd ) ∩ L 1 (Rd ). We notice here that for t > 0 we only know that u is a C r vortex patch, namely curl(u) = χ P (t) ωi (t) + ∞ (0, T ∗ ; C r (Rd ) ∩ L 1 (Rd )). χ P(t)c ωe (t) with P(t) of class C 1+r and ωi (t), ωe (t) ∈ L loc 1/2 yields that Hence, curlu ∈ L ∞ ([0, T ∗ ); B˙ r ). However, propagating the initial B˙ loc
2,∞
2,∞
∞ ([0, T ∗ ); B ˙ 1/2 ) and gives the better rate (58). curlu ∈ L loc 2,∞ In particular Theorem 3.4 applies to the classical 2D vortex patch, namely the case curlu 0 = χ P and P is of class C 1+r , r > 0 in which case (58) holds even if r < 1/2.
Remark 3.5. In the 2D case, one can lower the regularity of the initial data. Indeed Yudovich [24] proved that if ω0 = curl(u 0 ) ∈ L ∞ ∩ L p for some 1 < p < ∞ then the Euler system (6) has a unique global solution (see also [8]). It was proved in [9] that the solution to the Navier-Stokes system converges in L ∞ ((0, T ); L 2 ) to the solution of the Euler system if we only assume that ω0 = curl(u 0 ) ∈ L ∞ ∩ L p . More precisely, Chemin [9] proves that 1
u n − u L ∞ (0,T ;L 2 ) ≤ Ccurl(u 0 ) L ∞ ∩L 2 (νn T ) 2 ex p(−Ccurl(u 0 ) L ∞ ∩L 2 T ) .
(59)
Notice that here, the rate of convergence deteriorates with time. This does not happen if we also know that u is in L ∞ (0, T ; Li p) as was proved by Constantin and Wu [11]. Acknowledgements. The author is partially supported by an NSF grant DMS 0403983. He would like to thank Raphael Danchin for many remarks about an earlier version of the paper. He also would like to thank the referee for suggesting the study of the rate of convergence in Subsect. 2.1.
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4. Appendix We define C to be the ring of center 0, of small radius 1/2 and great radius 2. There exist two nonnegative radial functions χ and ϕ belonging respectively to D(B(0, 1)) and to D(C) so that χ (ξ ) + ϕ(2−q ξ ) = 1, (60) q≥0
| p − q| ≥ 2 ⇒ Supp ϕ(2−q ·) ∩ Supp ϕ(2−p ·) = ∅.
(61)
For instance, one can take χ ∈ D(B(0, 1)) such that χ ≡ 1 on B(0, 1/2) and take ϕ(ξ ) = χ (2ξ ) − χ (ξ ). Then, we are able to define the Littlewood-Paley decomposition. Let us denote by F the Fourier transform on Rd . Let h, h, q , Sq (q ∈ Z) be defined as follows: h = F −1 ϕ and h = F −1 χ , q u = F −1 (ϕ(2−q ξ )Fu) = 2qd Sq u = F
−1
(χ (2
−q
ξ )Fu) = 2
qd
h(2q y)u(x − y)dy, h(2q y)u(x − y)dy.
Then, we define the non-homogeneous and homogeneous Besov norms 1/r r r sq r s = S0 u 2 + 2 u , u B2,r q 2 L L u B˙ s =
2,r
q∈Z
q≥0
2r sq q urL 2
1/r
for s ∈ R and 1 ≤ r ≤ ∞. If r = ∞, then the summation over q is replaced by the L ∞ norm. Lemma 4.1. For 0 < α < 1, we have . w B˙ 1−α ≤ CwαL 2 ∇w1−α L2
(62)
2,1
Proof. This inequality can be easily deduced from real interpolation. We give here a direct proof. Actually, we will prove a stronger estimate, namely . w B˙ 1−α ≤ CwαB˙ 0 ∇w1−α B˙ 0 2,1
2,∞
2,∞
Indeed, we have q w L 2
≤ Cw B˙ 0 , 2,∞
q w L 2 ≤ C2−q w B˙ 1 . 2,∞
We take N such that 2 N w B˙ 0
2,∞
≤ w B˙ 1
2,∞
≤ 2 N +1 w B˙ 0 . 2,∞
(63)
Remarks about the Inviscid Limit of the Navier–Stokes System
787
Hence ∞
2(1−α)q q w L 2 ≤ C
q=−∞
2(1−α)q w B˙ 0
2,∞
+C
q≤N
≤ C2
2−αq w B˙ 1
2,∞
q≥N
(1−α)N
w B˙ 0
2,∞
+ C2
−α N
(64)
w B˙ 1 , 2,∞
from which (63) follows.
In the next two lemmas, we prove that if u 0 is a C r vortex patch then curl(u) ∈ ∞ (0, T ∗ ; B α ), where α = min(r, 1/2). L loc 2,∞ 1/2 Lemma 4.2. If P is a bounded open set of Rd of class C 1+r then χ P ∈ B˙ 2,∞ .
The proof is based on interpolation. Indeed, since P is C 1+r , it is Liptchiz and hence χ P ∈ L ∞ ∩ BV . Then q χ P L ∞ ≤ Cχ P L ∞ , q χ P L 1 ≤ C2−q χ P BV . Interpolating between L 1 and L ∞ , we deduce that q χ P L 2 ≤ C2−q/2 χ P BV χ P L ∞ , 1/2 and hence, χ P ∈ B˙ 2,∞ .
(65)
α , where α = min(r, 1/2). Lemma 4.3. If u is a C r vortex patch then curl(u) ∈ B˙ 2,∞
The proof uses the para-product decomposition of Bony ([6]) uv = Tu v + Tv u + R(u, v), where Tu v =
Sq−1 uq v and R(u, v) =
q uq v.
|q−q |≤1
q∈Z
We decompose χ P ωi = Tωi χ P + R(χ P , ωi ) + Tχ P ωi
(66)
and notice that since χ P and ωi are both in L ∞ , we get Tωi χ P + R(χ P , ωi ) B˙ α
≤ Cωi L ∞ χ P B˙ α ,
Tχ P ωi B˙ α
≤ Cχ P L ∞ ωi B˙ α .
2,∞ 2,∞
2,∞
2,∞
α . Hence Since, ωi is in C r and is compactly supported, we deduce that ωi ∈ B˙ 2,∞ α . χ P ωi ∈ B˙ 2,∞ α . The same proof holds for χ P c ωe and hence, curl(u) ∈ B˙ 2,∞
788
N. Masmoudi
References 1. Abidi, H., Danchin, R.: Optimal bounds for the inviscid limit of Navier-Stokes equations. Asymptot. Anal. 38(1), 35–46 (2004) 2. Beirão da Veiga, H.: On the singular limit for slightly compressible fluids. Calc. Var. Part. Differ. Eqs. 2(2), 205–218 (1994) 3. Beirão da Veiga, H.: Singular limits in compressible fluid dynamics. Arch. Rat. Mech. Anal. 128(4), 313–327 (1994) 4. Bertozzi, A.L., Constantin, P.: Global regularity for vortex patches. Commun. Math. Phys. 152(1), 19–28 (1993) 5. Bona, J.L., Smith, R.: The initial-value problem for the Korteweg-de Vries equation. Philos. Trans. Roy. Soc. London Ser. A 278(1287), 555–601 (1975) 6. Bony, J.-M.: Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires. Ann. Sci. École Norm. Sup. (4) 14(2), 209–246 (1981) 7. Chemin, J.-Y.: Persistance de structures géométriques dans les fluides incompressibles bidimensionnels. Ann. Sci. École Norm. Sup. (4) 26(4), 517–542 (1993) 8. Chemin, J.-Y.: Fluides parfaits incompressibles. Astérisque 230, 177 (1995) 9. Chemin, J.-Y.: A remark on the inviscid limit for two-dimensional incompressible fluids. Comm. Part. Differ. Eqs. 21(11-12), 1771–1779 (1996) 10. Constantin, P.: Note on loss of regularity for solutions of the 3-D incompressible Euler and related equations. Commun. Math. Phys. 104(2), 311–326 (1986) 11. Constantin, P., Wu, J.: Inviscid limit for vortex patches. Nonlinearity 8(5), 735–742 (1995) 12. Constantin, P., Wu, J.: The inviscid limit for non-smooth vorticity. Indiana Univ. Math. J. 45(1), 67–81 (1996) 13. Danchin, R.: Persistance de structures géométriques et limite non visqueuse pour les fluides incompressibles en dimension quelconque. Bull. Soc. Math. France 127(2), 179–227 (1999) 14. Dutrifoy, A.: On 3-D vortex patches in bounded domains. Comm. Part. Differ. Eqs. 28(7-8), 1237– 1263 (2003) 15. Gamblin, P., Saint Raymond, X.: On three-dimensional vortex patches. Bull. Soc. Math. France 123(3), 375–424 (1995) 16. Huang, C.: Singular integral system approach to regularity of 3D vortex patches. Indiana Univ. Math. J. 50(1), 509–552 (2001) 17. Kato, T.: Nonstationary flows of viscous and ideal fluids in R 3 . J. Funct. Anal. 9, 296–305 (1972) 18. Kato, T.: Quasi-linear equations of evolution, with applications to partial differential equations. In: Spectral Theory and Differential Equations (Proc. Sympos., Dundee, 1974; dedicated to Konrad Jörgens), Lecture Notes in Math., Vol. 448. Berlin: Springer, 1975, pp. 25–70 19. Majda, A.J., Bertozzi, A.L.: Vorticity and incompressible flow. Volume 27 of Cambridge Texts in Applied Mathematics. Cambridge: Cambridge University Press, 2002 20. Masmoudi, N.: The Euler limit of the Navier-Stokes equations, and rotating fluids with boundary. Arch. Rat. Mech. Anal. 142(4), 375–394 (1998) 21. Masmoudi, N.: Examples of singular limits in hydrodynamics. In: Handbook of Differential Equations, Vol. 3, pp. 195–275. North-Holland, Amsterdam (2007) 22. Swann, H.S.G.: The convergence with vanishing viscosity of nonstationary Navier–Stokes flow to ideal flow in R 3 . Trans. Amer. Math. Soc. 157, 373–397 (1971) 23. Wolibner, W.: Un theorème sur l’existence du mouvement plan d’un fluide parfait, homogène, incompressible, pendant un temps infiniment long. Math. Z. 37(1), 698–726 (1933) 24. Yudovich, V.I.: Non-stationary flows of an ideal incompressible fluid. u Z. Vy Cisl. Mat. i Mat. Fiz. 3, 1032–1066 (1963) Communicated by P. Constantin
Commun. Math. Phys. 270, 789–811 (2007) Digital Object Identifier (DOI) 10.1007/s00220-006-0183-1
Communications in
Mathematical Physics
Regularity of Coupled Two-Dimensional Nonlinear Fokker-Planck and Navier-Stokes Systems P. Constantin1 , C. Fefferman2 , E. S. Titi3,4 , A. Zarnescu1 1 Department of Mathematics, The University of Chicago, Chicago, IL 60637, USA.
E-mail:
[email protected]
2 Department of Mathematics, Princeton University, Princeton, NJ 08544, USA 3 Department of Mathematics and Department of Mechanics and Aerospace Engineering, University
of California, Irvine, CA 92697, USA
4 Department of Computer Science and Applied Mathematics, Weizmann Institute of Science,
Rehovot 76100, Israel Received: 27 April 2006 / Accepted: 1 September 2006 Published online: 9 January 2007 – © Springer-Verlag 2006
Abstract: We consider systems of particles coupled with fluids. The particles are described by the evolution of their density, and the fluid is described by the Navier-Stokes equations. The particles add stress to the fluid and the fluid carries and deforms the particles. Because the particles perform rapid random motion, we assume that the density of particles is carried by a time average of the fluid velocity. The resulting coupled system is shown to have smooth solutions at all values of parameters, in two spatial dimensions.
1. Introduction We discuss global regularity of solutions of systems of equations describing fluids with particle suspensions. The particles are parameterized by independent microscopic variables m that belong to a compact, connected, smooth Riemannian manifold M of dimension d. Derivatives with respect to the microscopic variables are designated by the subscript g. The particles are included in a fluid in Rn , n = 2 obeying the forced Navier-Stokes equations. The forces exerted by the particles on the fluid are expressed through the divergence of an added stress tensor. The added stress tensor τ p (x, t) is obtained after averaging out the microscopic variable and the Navier-Stokes equation is macroscopic. The microscopic inclusions at time t and macroscopic physical location x ∈ R2 are described by the density f (x, m, t)dm, where dm is the Riemannian volume element in M. The density is nonnegative, f ≥ 0 and ρ(x, t) =
f (x, m, t)dm ≤ 1 M
holds for every x, t ≥ 0.
(1)
790
P. Constantin, C. Fefferman, E. S. Titi, A. Zarnescu
The added stress tensor is given by an expansion τ p (x, t) =
∞
τ (k) (x, t),
(2)
k=1
where (1)
τi j (x, t) =
(1)
γi j (m) f (x, m, t)dm,
(3)
M (2)
τi j (x, t) =
(2)
γi j (m 1 , m 2 ) f (x, m 1 , t) f (x, m 2 , t)dm 1 dm 2 ,
(4)
M M
and, in general, (k)
τi j (x, t) = (k) γi j (m 1 , . . . , m k ) f (x, m 1 , t) f (x, m 2 , t) · · · f (x, m k , t)dm.
(5)
M×···×M
Expansions of this kind for the added stress tensor τ p are encountered in the polymer literature ([6]). In ([4]) it was proved that only two structure coefficients in the expan(1) (2) sion, γi j , γi j are needed in order to have energetically balanced equations, provided certain constitutive relations are imposed. The energy balance confers stability to certain time-independent solutions of the equations. In this work we are interested only in general existence results, and do not need to use special constitutive relations. We will (k) only use the fact that the coefficients γi j are smooth, time independent, x independent, f independent. When infinitely many coefficients are present, we will use a finiteness condition assuming that the series ∞
(k)
k 3 γi j H ρk (M×···×M)
(6)
k=1
converges for a sequence ρk > k+4d+6 . 2 From (1, 2, 5, 6 ) it follows that |τ p (x, t)| ≤ cρ(x, t)
(7)
holds with a constant that depends only on the coefficients γi(k) j . The spatial gradients of τ p are of particular importance for regularity. The fact that the constitutive coefficients (k) γi j are smooth functions of the microscopic variables allows us to relate the size of the spatial gradients of τ p to a rather coarse average on M: differentiating (5) with respect to x it follows from (6) that |∇x τ p (x, t)| ≤ cN (x, t)
(8) (k)
holds with a constant c that depends only on the smooth coefficients γi j . Here N (x, t) = R∇x f L 2 (M)
Regularity of Coupled 2D Nonlinear Fokker-Planck and Navier-Stokes Systems
791
and − s R = −g + I 2 with s > d2 + 1. The inequalities (7) and (8) are the only information concerning the relationship between τ p and f that we need for regularity results in this work. We can use the detailed form (2) and the finiteness condition (6) to deduce them, but we could just as well require them instead of (2). The particles are carried by the fluid, agitated by thermal noise and interact among themselves in a mean-field fashion, through potentials that depend linearly and nonlocally on the particle density distribution f ([15]). We assume that the fluid does not vary much in time during a characteristic relaxation time of the particles. Mathematically, this means that the particles are carried by a short time average of the fluid velocity. This assumption allows us to prove global existence of smooth solutions and to bound a priori the size of the physical space gradients of the stresses. The mathematical study of complex fluids is in a developing stage. Most results are for models that are macroscopic closures, that is, in which τ p has its own macroscopic evolution, coupled with the fluid: the microscopic variables do not appear at all. Existence theory for viscoelastic Oldroyd models is presented in ([13]); see also ([18]) for related issues. There are few other regularity results concerning complex fluids, including some that retain microscopic variables. Among them are ([7, 10, 11, 14, 17]). For Smoluchowski equations coupled with fluids, the case in which u is given by a time independent linear Stokes equation in n = 3, M = S2 with τ p given by a relation (3) was studied in ([16]) for the case of a linear Fokker-Planck equation and in ([4]) for general nonlinear Fokker-Planck equations. The proofs in the present work are based on a few key facts. The first one is that gradients of τ p are bounded by N , and N is controlled linearly by the advecting velocity, taking advantage of the a priori boundedness of ρ in L 1 ∩ L ∞ . This idea was used in ([4]) to prove regularity for the system of particles coupled with the Stokes system for the fluid. The second significant fact concerns the Navier-Stokes system driven by the divergence of bounded stresses. We are interested in the size of the time integral of the supremum of the norm of the gradient of velocity. This is an important nondimensional magnitude that controls the amplification of gradients of passively advected scalars. We prove a logarithmic bound for this amplification factor. The strategy of proof uses a natural idea introduced in ([3]): time integration is performed first in each wave-number shell, to take advantage of the rapid smoothing of small scales due to viscosity. 2. Estimates for 2-D Navier-Stokes Equations Consider a Navier-Stokes system in R2 : ∂u + u · ∇u − u + ∇ p = ∇ · σ, ∂t ∇ · u = 0, u|t=0 = u 0 ,
(9)
where σ (x, t) is a symmetric two-by-two matrix that, in this section, will be considered to be a given function. We will be interested in estimates when σ is integrable and bounded by constants that are known a priori and are of order one. The physical space
792
P. Constantin, C. Fefferman, E. S. Titi, A. Zarnescu
gradients of σ are possibly large. The aim of the bounds is to find the effect that these gradients have on the stretching amplification term t ∇u(t) L ∞ dt. 0
We take smooth, divergence-free initial velocities u(0) ∈ L 2 (R2 ) ∩ W 1+k,r (R2 ) with k ∈ R, k > 0, localized vorticity ω = ∇ ⊥ · u, ω(0) ∈ L 2 (R2 ) ∩ W k,r (R2 ) with r > 2. We recall the energy estimate T sup u(t)2L 2 t≤T
T ∇u(t)2L 2 dt
+ 0
≤
σ (t)2L 2 dt + u(0)2L 2
(10)
0
and the fact that in two dimensions the vorticity obeys ∂t ω + u · ∇x ω − ω = ∇x⊥ · divx σ.
(11)
Lemma 1. Let r ≥ 2. There exists a constant cr such that T sup ∇u(t)2L r t≤T
≤ cr
∇ · σ (t)2L r dt + cr ω(0)2L r
(12)
0
holds. For r = 2 we have additionally, T
T u(t) dt ≤ c2
∇ · σ 2L 2 (d x) + ω(0)2L 2 .
2
0
(13)
0
Proof. We multiply (11) by ωr −1 , r ≥ 2 and integrate by parts to obtain d ωrL r + ωr −2 |∇x ω|2 d x ≤ (r − 1) |∇x σ ||∇x ω|ωr −2 . r dt We use a Hölder inequality with exponents r, 2, r2r −2 , and then with exponents 2, 2: d ωrL r + ωr −2 |∇x ω|2 d x r dt 1/2 r −2 ≤ (r − 1)∇x · σ L r ω L 2r d x ωr −2 |∇x ω|2 d x ≤
1 (r − 1)2 ∇x · σ 2L r ωrL−2 ( r + 2 2
ωr −2 |∇x ω|2 d x)
(14)
Regularity of Coupled 2D Nonlinear Fokker-Planck and Navier-Stokes Systems
793
which implies that T sup
t∈[0,T ]
ω(t)2L r
≤ C(r − 1)
∇x · σ (t)2L r dt + ω(0)2L r
2
(15)
0
and thus (12) follows. In order to obtain (13) we integrate (14) in time at r = 2. We need a logarithmic inequality for u L ∞ . Such inequalities were first introduced in ([1]). We will write log∗ (λ) = log(2 + λ) for λ > 0. Note that log∗ (0) > 0 and log∗ (λμ) ≤ log∗ (λ) + log∗ (μ) holds for λ ≥ 0, μ ≥ 0. We check our inequality ⎧ ⎫ r ⎬ ⎨ r −2 u(t) 2 r ω(t) L L u(t) L ∞ ≤ Cr ω(t) L 2 1 + log∗ (16) ⎩ ω(t) L 2 ω(t) L 2 ⎭ directly from the Biot-Savart law: u(x, t) =
1 2π
dz z ⊥ ω(x − z, t) , |z|
R2
z . We pick two numbers 0 < l ≤ L, take a smooth radial function a(s) , where z = |z| 0 ≤ a(s) ≤ 1, that equals identically one for s ≤ 1 and identically zero for s ≥ 2, and write dz 1 u (l) (x, t) = z ⊥ ω(x − z, t) , 2π |z| |z|≤l
u (L) (l) (x, t)
1 = 2π
and 1 u (L) (x, t) = 2π
|z| dz ⊥ , z ω(x − z, t)a L |z|
|z|≥l
dz |z| ⊥ . z ω(x − z, t) 1 − a L |z|
|z|≥l
Clearly (L)
u = u (l) + u (l) + u (L) holds pointwise. It is also clear that |u (l) (x, t)| ≤ ω(t) L r l and that (L) |u (l) (x, t)|
r −2 r
≤ ω(t) L 2 log∗
2L l
.
We integrate by parts in the term u (L) , using ω = ∇ ⊥ · u and deduce |u (L) (x, t)| ≤ C
1 u(t) L 2 . L
794
P. Constantin, C. Fefferman, E. S. Titi, A. Zarnescu
We choose l=
ω(t) L 2 ω(t) L r
r r −2
and L=
u(t) L 2 ω(t) L 2
if, with this choice, it turns out that l < L. If not, then we still take L as above, but we take l = L. The inequality (16) follows. Note that (10) implies that T sup u(t)2L 2 + 0 ∇u(t)2L 2 dt ≤ σ 2L 2 (0,T ;L 2 ) + u 0 2L 2 , (17) t≤T
and note that (12) implies that sup ω(t) L r ≤ cr (∇ · σ L 2 (0,T ;L r ) + ω(0) L r ).
(18)
t≤T
Finally, using (13) and (17) we can write T
u(t)2H 2 dt ≤ c(1 + T ) σ 2L 2 (0,T ;H 1 ) + u(0)2H 1 .
(19)
0
Now we integrate the square of (16) in time, taking the supremum in time of the logarithmic part using (18), and bounding the time integral of the square of the gradients using (17). Lemma 2. For r > 2 there exists a constant cr such that T
u(t)2L ∞ dt ≤ cr K 0 log∗ ( r ) + log∗ ( 2 ) + log∗ K 0
(20)
0
holds with K 0 = σ 2L 2 (0,T ;L 2 ) + u 0 2L 2 ,
(21)
r = ∇ · σ L 2 (0,T ;L r ) + ω(0) L r
(22)
and 2 defined like r with r replaced by 2. Theorem 1. For r > 2 there exists a constant cr such that ∇x ∇x u L 2 (0,T ;(L r )) ≤ 1 k T ω0 W k,r + cr K 0 r log∗ ( r ) + log∗ ( 2 ) + log∗ (K 0 ) cr k holds with K 0 , r defined above in (21), (22).
(23)
Regularity of Coupled 2D Nonlinear Fokker-Planck and Navier-Stokes Systems
795
Proof. We represent ∇x ω(t) = T1 (∇x⊥ · σ ) − T2 (uω) + et (∇x ω(0)),
(24)
where T1 and T2 are operators of the form t h(t) → T h =
e(t−s) Hh(s)ds
0
with H = H(D) homogeneous of degree zero. Such operators are bounded in L p (dt; L q (d x)) for 1 < p, q < ∞ by the maximal regularity of the heat equation combined with the boundedness of the operators H in L q spaces (see, for example ([12])). The inequality follows then from the bound T
u(t)ω(t)2L r dt ≤ cr K 0 r2 log∗ ( r ) + log∗ ( 2 ) + log∗ (K 0 )
(25)
0
which, in turn, follows from (18) and (20). The requirement that ω(0) ∈ W k,r is a sufficient condition for et ∇x ω(0) ∈ L 2 (0, T ; L r ). p T We mention also an a priori bound for 0 u L ∞ dt for p < 2. This is obtained as follows: we write ω(t) = et ω0 + t + e(t−s) ∇ ⊥ · divx σ (s) + ∂2 ∂1 (u 22 − u 21 )(s) + (∂22 − ∂12 )u 1 u 2 (s) ds.
(26)
0
This follows from a well-known identity u · ∇ω = −∂2 ∂1 (u 22 − u 21 ) − (∂22 − ∂12 )u 1 u 2 . An easy calculation verifies this after one writes u 1 = −∂2 ψ, u 2 = ∂1 ψ, ω = ψ. From (26) we get, for any 1 < p, q < ∞, ω L p (0,t;L q ) ≤ C pq (t) + C pq σ L p (0,t;L q ) + C pq u2L 2 p (0,t;L 2q ) .
(27)
We know however from (17) that u L ∞ (0,t;L 2 ) ≤ K 0 with K 0 defined in (21) and u L 2 (0,t;L r ) ≤ Cr a priori, for any r < ∞, with K 0 , Cr independent of t ∈ [0, T ]. (These constants may depend on T because the norm of σ in L 2 may depend on T .) Then, by interpolation u L p (0,t;L q ) ≤ C pq
(28)
796
P. Constantin, C. Fefferman, E. S. Titi, A. Zarnescu
holds for q ≥ 2 and p <
2q q−2 .
In view of (27) we get that ω L p (0,t;L q ) ≤ C pq
holds for q ≥ 2 and p < theorem, we obtain that
q q−1 .
(29)
Then, taking q > 2 and using a Sobolev embedding T
p
u L ∞ dt ≤ C p
(30)
0
holds a priori, for any p < 2. T For the bound for 0 ∇u(t) L ∞ dt we will use the Littlewood-Paley decomposition. Let D( ) denote the set of C ∞ functions compactly supported in . Let C be the annulus centered at 0, and with radii 1/2 and 2. There exist two nonnegative, radial functions χ and ϕ, belonging respectively to D(B(0, 1)) and to D(C) so that χ (ξ ) + j≥0 ϕ(2− j ξ ) = 1, and | j − k| ≥ 2 ⇒ sup(ϕ(2− j )) ∩ sup(ϕ(2−k )) = ∅. ˜ j , S j ( j ∈ N) be defined We denote by F the Fourier transform on R2 and let h, h, by h = F −1 ϕ and h˜ = F −1 χ , j u = F −1 (ϕ(2− j ξ )Fu) = 22 j Sju = F
−1
(χ (2
−j
ξ )Fu) = 2
2j
h(2 j y)u(x − y)dy, ˜ j y)u(x − y)dy. h(2
Then u = S0 u +
j (u),
j≥0
where u ∈ S , the space of tempered distributions, and the equality holds in the sense of distributions. The well-known Bernstein inequalities (see, for instance [2]) express the fact that j is localized around the frequency 2 j . Proposition 1 (Bernstein Inequalities). Let a, b ∈ [1, ∞], j ≥ 0. There exist constants c, independent of a, b, j such that the following hold: j et u L a ≤ ce−t2 j f L ∞ ≤ c f L ∞ ,
2( j−1)
j u L a ,
(31) (32)
and, in two space dimensions:
S j f L ∞ ≤ c f L 2 + j∇ f L 2
(33)
Regularity of Coupled 2D Nonlinear Fokker-Planck and Navier-Stokes Systems
797
and j ∂ α u L a ≤ c2 j|α|+2 j (1/b−1/a) j u L b ,
(34)
where |α| is the length of the multiindex α. The Littlewood-Paley decomposition is best suited for Besov spaces B sp,q defined by requiring the sequence 2s j j (u) L p to belong to q and by requiring S0 (u) to be in L p . L 2 based Sobolev space norms can be computed in terms of the Littlewood-Paley decomposition: 22s j j u2L 2 , u2H s ∼ S0 (u)2L 2 + j≥0
where ∼ means equivalence of norms. However, the norm we are interested in is the L 1 (0, T ; W 1,∞ ) norm. The C s norms can be computed as uC s ∼ S0 (u)C s + sup 2 js j (u) L ∞ j≥0
but only if s is not an integer. In order to obtain L ∞ bounds for the gradient we will have to resort to the inequality: j (∇u) L ∞ . ∇u L ∞ ≤ S0 (∇u) L ∞ + j≥0 0 This inequality reflects the embedding B∞,1 ⊂ L ∞ , which is a strict inclusion. The 0 ) is that we can commute time integration advantage of using this sum (the norm in B∞,1 and summation, while time integration and supremum do not commute in general.
Theorem 2. Let u be a solution of the 2D Navier-Stokes system (9), with divergence-free initial data u 0 ∈ W 1,2 (R2 ) ∩ W 1,r (R2 ). Let T > 0 and let the forces ∇ · σ obey σ ∈ L 1 (0, T ; L ∞ (R2 )) ∩ L 2 (0, T ; L 2 (R2 )) and ∇ · σ ∈ L 1 (0, T ; L r (R2 )) ∩ L 2 (0, T ; L 2 (R2 )) with r > 2. There exists a constant c depending on r such that, for every > 0, T ∇u L ∞ dt ≤ c 0
√
T u(0) H 1 + cK 2(1) T
(1) + cK ∞ log∗
Br(1)
+
⎞ (1 + T ) σ 2L 2 (0,T ;H 1 ) + u(0)2H 1 ⎠ (35) + c(1 + T )u2L 2 (0,T ;H 1 ) log∗ ⎝ ⎛
holds, where ( p)
Kr
= σ L p (0,T ;L r )
(36)
798
P. Constantin, C. Fefferman, E. S. Titi, A. Zarnescu
and ( p)
= ∇ · σ L p (0,T ;L r ) .
Br
(37)
Consequently, in view of (17) and (35) above T
∇u(t) L ∞ dt ≤ c(1 + T )2 K log∗ Br(1) + σ 2L 2 (0,T ;H 1 ) + u(0)2H 1
(38)
0 (1)
(1)
with K = K 0 + K ∞ + K 2 + u(0) H 1 depending on norms of σ and the initial velocity, but not on gradients of σ , and only the argument of the logarithm depending on norms of the gradients of σ . Remark. The bound is in fact for the stronger norm of u in the inhomogeneous space 1 ). L 1 (0, T ; B∞,1 Proof. We start with the Duhamel formula for the gradient of solutions of (9), t ∇u = e ∇u 0 + t
e(t−s) H(D)(u(s) ⊗ u(s))ds
0
t −
e(t−s) H(D)(σ (s))ds.
(39)
0
The homogeneous operator H(D) is given by H(D)(u ⊗ u)i j = R j (δil + Ri Rl )Rk (u l u k )
(40)
1
with R j = ∂ j (−)− 2 Riesz transforms. The strategy is based on an idea of Chemin and Masmoudi ([3]) to take the time integral first, for each wave number shell. They did not use information about derivatives of σ , and therefore obtained only bounds for T supq≥1 0 q ∇u L ∞ ds. We will use the gradients of σ to bound the high frequencies and will sum in q in order to estimate ∇u L 1 ((0,t);L ∞ ) . Also, we use somewhat different estimates than them for the individual shell contributions, but like them, we take advantage of a time integration at each shell. We treat separately the contributions coming from σ and those coming from u ⊗ u: ∇u = F + U + et ∇u 0 , where t F(t) = −
e(t−s) H(D)(σ (s))ds,
(41)
0
and t U (t) = 0
e(t−s) H(D)((u ⊗ u)(s))ds.
(42)
Regularity of Coupled 2D Nonlinear Fokker-Planck and Navier-Stokes Systems
799
Clearly T
(1)
S0 F(t) L ∞ dt ≤ cK 2 T
(43)
0
is true using for instance S0 (F(t)) L ∞ (d x) ≤ (I − )S0 (F(t)) L 2 (d x) . We take q ≥ 0 and apply q : t q F(t) L ∞ ≤
e−(t−s)2
2(q−1)
22q σ (s) L ∞ ds.
(44)
0
Integrating on [0, T ] and changing order of integration we obtain T
T q F(t) L ∞ dt ≤
0
T σ (s) L ∞ (
e−(t−s)2
2(q−1)
22q dt)ds
s
0
T σ (s) L ∞ ds.
≤c
(45)
0
We bound the same quantity differently, with large q in mind, and use ∇σ ∈ L r with r > 2: t 2(q−1) (t−s) q(1+2/r ) 2 ∇ · σ L r ds. (46) q F(t) L ∞ ≤ c e−2 0
We integrate on [0, T ] and change the order of integration as above, to obtain T
T q F(t)
L∞
dt ≤
0
T ∇ · σ (s) ( Lr
e−2
2(q−1) (t−s)
2q(1+2/r ) dt)ds
s
0
≤
T
c
∇ · σ (s) L r ds.
2q(1−2/r )
(47)
0
Using (43), (45) to estimate the small wave numbers in F and (47) to estimate the high ones, we obtain T F(t) L ∞ dt ≤ 0
T
⎡ ⎣S0 F(t) L ∞ +
q F(t) L ∞ +
0≤q≤M
0
≤
(1) cK 2 T
T σ (s) L ∞ ds +
+ cM 0
⎤ q F(t) L ∞ ⎦ dt
q>M
c
T ∇ · σ (s) L r ds.
2 M(1/2−1/r ) 0
(48)
800
P. Constantin, C. Fefferman, E. S. Titi, A. Zarnescu
Then, choosing M,
T c 0 ∇ · σ (s) L r ds M = cr log∗ ,
(49)
we obtain
T F(t) L ∞ dt ≤
(1) cK 2 T
(1) + cK ∞ log∗
0
with
(1) Kr
(1)
defined in (36) and Br
(1)
Br
+
(50)
defined in (37).
Remark. We do not need to integrate the term F in time, if σ is bounded: we can obtain ( p) a pointwise logarithmic bound for F(t) in terms of Br with p > r2r −2 . Indeed, from (44) we have q F(t) L ∞ ≤ cσ L ∞ (dtd x)
(51)
and from (46) we obtain q F(t) L ∞ ≤ c2
−2 1 −2q( r2r − p)
∇ · σ L p (0,T ;L r ) .
(52)
Summing (51) from q = 0 to q = M, summing (52) from q = M to infinity and choosing M appropriately, we obtain ( p) Br (∞) sup F(t) L ∞ ≤ C K 0 T + cK ∞ log∗ (53) (∞) t≤T K∞ with (∞) = σ L ∞ (0,T ;L ∞ ) . K∞
(54)
This bound can be used to reprove the global existence of the Stokes system coupled with nonlinear Fokker Planck equations. We split the nonlinear term U (t) = S2 (U )(t) + V (t)
(55)
S2 (U ) = S0 (U ) + 1 (U ) + 2 (U ),
(56)
with
and V (t) =
q (U ).
(57)
q≥3
Clearly T S2 (U ) L ∞ dt ≤ cT u2L 2 (0,T ;H 1 ) . 0
(58)
Regularity of Coupled 2D Nonlinear Fokker-Planck and Navier-Stokes Systems
801
For the nonlinear term V we use Bony’s decomposition (see for instance [2]) into commensurate and incommensurate frequencies: V (t) = C(t) + I (t),
(59)
with
C(t) =
⎛
t
e(t−s) H(D)q ⎝
⎞ p (u(s)) ⊗ p (u(s))⎠ ds
(60)
| p− p |≤2
q≥3 0
and I (t) =
⎛
t
e(t−s) H(D)q ⎝
⎞ p (u) ⊗ p (u)⎠ ds.
(61)
| p− p |≥3
q≥3 0
In the decomposition above we made the convention that the indices p, p run from −1 to infinity and when p = −1 instead of p we have S0 , and of course, the same thing for p . We start by treating the term C(t). Because the range of q is q ≥ 3 it follows that the range of p, p is p ≥ 1, p ≥ 1. Then we estimate inside the integral q (( p u(s)) ⊗ ( p (u(s))) L ∞ ≤ c22q q (( p u(s)) ⊗ ( p (u(s))) L 1 ≤ c2(2q−2 p) ∇ p u(s) L 2 ∇ p u(s) L 2 . Clearly at least one of p, p , say p, satisfies p ≥ q − 2. Now, because | p − p | ≤ 2, it follows that p = p + j, j ∈ [−2, 2] and therefore, changing the order of summation in (60) ∞ 2 t ( ( C(t) L ∞ ≤ c p ∇u(s) L 2 p+ j ∇u(s) L 2 j=−2 p=1 0 p+2 ( −22(q−1) (t−s) 2q 2(q− p) × e 2 2 ds. q=3
Integrating in t and changing the order of integration we obtain T C(t) L ∞ dt ≤ 0 2 ∞
T
c
p (∇u(s)) L 2 p+ j (∇u(s)) L 2
j=−2 p=1 0 2 T ∞
≤c
j=−2 0
p=1
p+2
22(q− p) ds
q=3
p (∇u(s)) L 2 p+ j (∇u(s)) L 2 ds.
802
P. Constantin, C. Fefferman, E. S. Titi, A. Zarnescu
We have obtained thus T C(t) L ∞ dt ≤ cu2L 2 (0,T ;H 1 ) .
(62)
0
We turn now to the term I (t) of (61). This term is made up of two sums,
I1 (t) =
( t
I (t) = I 1 (t) + I2 (t), ( (t−s) e H(D)q p (u(s)) ⊗ p (u(s)) ds,
q≥3 0
I2 (t) =
( t
e(t−s) H(D)q
q≥3 0
p≥ p +3
( p ≥ p+3
(63)
p (u(s)) ⊗ p (u(s)) ds.
We will treat I1 because the treatment of I2 is the same, mutatis mutandis. Because p ≥ p + 3 it follows that, in order to have a nonzero contribution at q, the index p must belong to [q − 2, q + 2], i.e., p = q + j with j ∈ [−2, 2]. Then we can write I1 (t) =
2
t
e(t−s) H(D)q (Jq (s))ds
(64)
j=−2 q≥3 0
with Jq (s) = q+ j (u(s)) ⊗ Sq+ j−3 (u(s)).
(65)
q )(Jq (s)) L ∞ ≤ cSq+ j−3 (u(s)) L*∞ q+ j (u(s)) L ∞ √ ≤ c u(s) L 2 + M + 2∇u(s) L 2 q+ j (u(s)) L ∞ ,
(66)
For q ≤ M we estimate
where we used (33): Sq+ j (u(s)) L ∞ ≤ c u(s) L 2 + q + j∇u(s) L 2 . Using Bernstein’s inequality q+ j u(s) L ∞ ≤ cq+ j ∇u(s) L 2 we obtain √ q (Jq (s)) L ∞ ≤ c u(s) L 2 + M + 2∇u(s) L 2 q+ j ∇(u(s)) L 2 .
(67)
(68)
For q ≥ M we estimate q (Jq (s)) L ∞ ≤ c u(s) L 2 + u(s) L 2 2−q q+ j (u(s)) L 2 . We write T I1 (t) L ∞ dt ≤ A + B 0
(69)
Regularity of Coupled 2D Nonlinear Fokker-Planck and Navier-Stokes Systems
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with 2 M
T
A=c
⎛ T ⎞ 2(q−1) q (Jq (s)) L ∞ ⎝ 22q e−(t−s)2 dt ⎠ ds
j=−2 q=3 0
s
and 2 ∞ T
B=c
⎞ ⎛ T 2(q−1) q (Jq (s)) L ∞ ⎝ 22q e−(t−s)2 dt ⎠ ds.
j=−2 q=M 0
s
We use (68) for A:
A≤c
T
u(s) L 2 +
√
⎛ ⎞ M M + 2∇u(s) L 2 ⎝ q+ j ∇u(s) L 2 ⎠ ds q=3
0
T ≤c
u(s) L 2 +
√
M + 2∇u(s) L 2
√
M∇u(s) L 2 ds.
0
We have therefore A ≤ cMu2L 2 (0,T ;H 1 ) .
(70)
We use (69) for B: T B≤c
−q u(s) L 2 + u(s) L 2 2 q+ j (u(s)) L 2 ds q≥M
0
and therefore B ≤ c2
−M
T u(s)2H 2 ds
= c2
−M
T
0
u(t)2H 2 dt. 0
In view of (19) B ≤ c2−M (1 + T ) σ 2L 2 (0,T ;H 1 ) + u(0)2H 1 . For any > 0, we choose M = log∗
) ⎧ ⎨ c(1 + T ) σ 2 2 L
⎩
(0,T ;H 1 )
*⎫ + u(0)2H 1 ⎬ ⎭
(71)
804
P. Constantin, C. Fefferman, E. S. Titi, A. Zarnescu
and we obtain from (70) and (71) T I (t) L ∞ dt ≤ 0
≤ cu2L 2 (0,T ;H 1 ) log∗
c(1+T ) σ 2 2
+u(0)2 1 L (0,T ;H 1 ) H
(72) .
The sum of the inequalities (50), (58), (62), (72) and a straightforward estimate for the linear term carrying the initial data give the inequality (35) of the theorem. 3. Coupled Nonlinear Fokker-Planck and Navier-Stokes Systems in 2D We consider now the coupling between fluid and particles. The evolution of the density f is governed by a nonlinear Fokker-Planck equation ∂t f + v¯ · ∇x f + divg (G f ) =
1 g f. τ
(73)
The coefficient τ > 0 is the time scale associated with the particles. The microscopic variables m are non-dimensional. The tensor G is made of two parts, G = ∇g U + W.
(74)
The (0, 1) tensor field W is obtained from the macroscopic gradient of velocity in a linear smooth fashion, given locally as ⎛ ⎞ n ∂ v ¯ i W (x, m, t) = (Wα (x, m, t))α=1,...,d = ⎝ cαi j (m) (x, t)⎠ . (75) ∂x j i, j=1
α=1,...,d
ij
The smooth coefficients cα (m) do not depend on the solution, time or x and, like the (k) coefficients γi j , they are a constitutive part of the model. The potential U is given by U (x, m, t) =
b (K f ) (x, m, t), τ
(76)
where b is a nondimensional measure of the intensity of the inter-particles interaction. The nonlocal microscopic interaction potential (77) (K f ) (x, m, t) = K (m, q) f (x, q, t)dq M
is given by an integral operator with kernel K (m, q) which is a smooth, time independent, x independent, symmetric function K : M × M → R ([15]). The Navier-Stokes equations are ∂t v + v · ∇x v + ∇x p = νx v + ∇x · τ p
(78)
with ∇x · v = 0. The added stresses are given by the relations (2), with (5, 6). The added stresses are proportional to kT (where k is Boltzmann’s constant and T is temperature)
Regularity of Coupled 2D Nonlinear Fokker-Planck and Navier-Stokes Systems
805
and have units of energy per unit mass. The density of the fluid is normalized to one. The particles are advected by 1 v(x, ¯ t) = τ
t v(x, s)ds. (t−τ )+
We rescale the Navier-Stokes equations using the length scale λ = λ2 ν −1
(79)
τ δ,
ντ δ
and the time
= where δ is the Deborah number, the ratio between the relaxation time scale scale of the particles and the macroscopic (observation) advective time. We set x tδ λ , v(x, t) = δ u τ λ τ and we arrive at (9) with σ =
τ τp. δν
(80)
The Fokker-Planck equation becomes (∂t + u¯ · ∇x ) f + divg (G f ) =
1 g f δ
(81)
with 1 u(x, ¯ t) = δ and (74) with
t u(x, s)ds,
(82)
(t−δ)+
⎛
W (x, m, t) = (Wα (x, m, t))α=1,...,d = ⎝
n
i, j=1
⎞ ∂ u ¯ i cαi j (m) (x, t)⎠ ∂x j
(83) α=1,...,d.
and b (84) (K f ) (x, m, t). δ The forces applied by the particles are obtained after f is integrated along with (k) smooth coefficients γi j on M in order to produce σ . Therefore, only very weak regularity of f with respect to the microscopic variables m is sufficient to control σ . In order to take advantage of this, we consider the L 2 (M) selfadjoint pseudodifferential operator − s R = −g + I 2 (85) U (x, m, t) =
with s >
d 2
+ 1. We will use the following properties of R: [R, ∇x ] = 0, R∇g : L 1 (M) → L 2 (M) is bounded,
(86) (87)
R∇g : L 2 (M) → L ∞ (M) is bounded,
(88)
[∇g c, R
−1
] : H (M) → L (M) is bounded, s
2
(89)
806
P. Constantin, C. Fefferman, E. S. Titi, A. Zarnescu
for any smooth function c : M → R, and R : L 2 (M) → H s (M) is bounded.
(90)
We differentiate (81) with respect to x, apply R, multiply by R∇x f and integrate on M. Let us denote by N (x, t)2 = |R∇x f (x, m, t)|2 dm (91) M
the square of the L 2 norm of R∇x f on M. The following lemma was proved in ([4]): Lemma 3. Let u(x, ¯ t) be a smooth, divergence-free function and let f solve (81). There exists an absolute constant c > 0 (depending only on dimensions of space, the coeffiij cients cα and M, but not on u, ¯ f , δ) so that 1 ¯ + )N + c|∇x ∇x u| ¯ (∂t + u¯ · ∇x ) N ≤ c(|∇x u| δ
(92)
holds pointwise in (x, t). The proof is given below in the Appendix for completeness. It works independently of the dimension n of the variables x. The equation obeyed by ρ = M f dm is (∂t + u¯ · ∇x )ρ = 0.
(93)
We will take initial densities that obey 0 ≤ ρ(x, 0) ≤ 1. Therefore 0 ≤ ρ(x, t) ≤ 1
(94)
continues to be true and, in view of (7) and the fact that u¯ is divergence free, it follows from (93) that σ (t) L r ≤ cr
(95)
holds if we assume (as we do) that ρ(x, 0) ∈ L 1 (R2 ). The inequalities (7) and (8) use only the smoothness of the coefficients γ , the relations (2, 5), the condition (6) and (1), and therefore they hold throughout the evolution. We know thus that σ is bounded in T ( p) L ∞ and that K r (in (36)) are bounded a priori. (We also know that 0 u(t) L ∞ dt is bounded a priori (see (30).) This implies that the support of ρ, if initially compact, would expand only a finite amount in finite time. We do not use this for the proof, but obviously this is a physically important a priori quantitative information.) Theorem 3. Consider the coupled Fokker-Planck and Navier-Stokes system (73), (78) with arbitrary parameters ν, τ, b > 0. Assume that the initial velocity v(0) is divergence-free and smooth, v(0) ∈ W 1+k,r (R2 ) ∩ L 2 (R2 ) with k ∈ R, k > 0 and r > 2. Assume that the initial distribution of particles f (x, m, 0) is non-negative, smooth, in the sense that N (x, 0) = ∇x f (x, ·, 0) H −s (M) ∈ L r (R2 ) ∩ L 2 (R2 )
Regularity of Coupled 2D Nonlinear Fokker-Planck and Navier-Stokes Systems
807
for some s ∈ R, and localized, in the sense that f (x, m, 0)dm ρ(x, 0) = M
obeys 0 ≤ ρ(x, 0) ≤ 1 and ρ(·, 0) ∈ L 1 (R2 ). Then the solution of the system (73, 78) exists for all time and is smooth. In particular, the norms of v ∈ L ∞ (dt; W 1,r (R2 )) ∩ L 2 (dt; W 2,r (R2 )), f ∈ L ∞ (dt; W 1,r (d x; H −s (M))) can be bounded a priori in terms of the initial data, for arbitrary large finite intervals of time [0, T ]. Remark. The theorem implies that the important stretching magnitude T ∇v L ∞ dt 0
is bounded a priori in terms of the initial data. Proof. Let T > 0 be given. The sort time existence of solutions and the uniqueness of the solutions can be obtained following classical methods of proof. Let n(t) = ∇x σ 2L r + σ 2H 1 , t B(t) =
(96)
t ∇x σ 2L r dt +
0
σ 2H 1 dt,
(97)
0
g(t) = ∇ u(t) ¯ L∞ , γ (t) = sup g(s),
(98) (99)
0≤s≤t
and 2 2 ¯ ¯ . G(t) = ∇x ∇x u(t) L r + u(t) H2
(100)
dn ≤ cg(t)n(t) + cG(t). dt
(101)
Using (92) we have
Integrating on (0, t) t n(t) ≤ n(0) + cγ (t)B(t) + c
G(s)ds.
(102)
0
In view of (35) we have cγ (t) ≤ C0 1 + log∗ (B(t))
(103)
808
P. Constantin, C. Fefferman, E. S. Titi, A. Zarnescu
with C0 a constant depending on T and the initial data. In view of (19) and (23) we know t
G(s)ds ≤ C1 1 + B(t) log∗ B(t)
(104)
0
with C1 a constant. Note that n(t) =
d B(t). dt
(105)
Thus, from (102), (103) and (104) we deduce d B(t) ≤ C2 1 + B(t) log∗ B(t) dt
(106)
holds with C2 a constant that depends on T and the values ω(0)W k,r , ω(0) L 2 , u(0) L 2 and n(0). This produces a pointwise-in-time a priori finite bound for B(t) on the interval [0, T ], and retracing our steps, via (104) and (102), on n(t). Once the forces in the two-dimensional Navier-Stokes equations are known to be thus bounded, it follows (from (35)–but also much easier, from energy estimates) that the solution of the Navier-Stokes equation is smooth as stated. 4. Appendix: Proof of Lemma 3 The evolution equation of N is 1 (∂t + u¯ · ∇x ) N 2 = −D + I + I I + I I I + I V 2
(107)
with D=
+ + +∇g R∇x f +2 dm,
M
∂ u¯ j I =− ∂ xk II = −
2 α=1
III = −
M
∂f R ∂x j
∂ u¯ i (∇x ) ∂x j
(108) ∂f R ∂ xk
dm,
(109)
(Rdivg (cαi j f ))(∇x R f )dm,
(110)
M
2 ∂ u¯ i (Rdivg (cαi j ∇x f ))(R∇x f )dm, ∂x j
α=1
(111)
M
and IV = −
b δ
M
Rdivg (∇x f ∇g (K f ) )R∇x f dm.
(112)
Regularity of Coupled 2D Nonlinear Fokker-Planck and Navier-Stokes Systems
809
Now we start estimating these terms. We will use repeatedly (1) and (94). In view of the fact that D ≥ 0, we may discard this term. Clearly ¯ 2. |I | ≤ c|∇x u|N
(113)
In order to bound I I we use (87) to bound R∇g (cαi j f ) L 2 (M) ≤ c f L 1 (M) = c, so that we have ¯ N. |I I | ≤ c |∇x ∇x u|
(114)
In order to bound I I I we need to use the commutator carefully. We start by writing Rdivg (cαi j ∇x f ) = Rdivg (cαi j R −1 R∇x f ) = * ) divg (cαi j R∇x f ) + Rdivg cαi j , R −1 R∇x f. The second term obeys ,) * , , Rdivg cαi j , R −1 R∇x
, , f,
L 2 (M)
≤ cN
because, in view of (89) and (90) one has that * ) Rdivg cαi j , R −1 : L 2 (M) → L 2 (M) is bounded. The first term needs to be integrated against R∇x f and integration by parts gives 1 ij (divg (cα R∇x f ))R∇x f dm = (divg cαi j )|R∇x f |2 dm. 2 M
M
We obtain thus |I I I | ≤ c |∇x u| ¯ N 2. The term I V is split in two terms, I V = A + B: b Rdivg ( (∇x f )∇g (K f ) )R∇x f dm A=− δ
(115)
(116)
M
and B=−
b δ
Rdivg ( f ∇g (K∇x f ) )R∇x f dm.
(117)
M
The (0, 1) tensor (x, m, t) = (∇g K f )(x, m, t) is smooth in m for fixed x, t and (x, ·, t)W s,∞ (M) ≤ cs
810
P. Constantin, C. Fefferman, E. S. Titi, A. Zarnescu
holds for any s, with cs depending only on the kernel K . We write the term A,
b = − 2δ
A = − bδ Rdivg ({(∇x f )})R∇x f dm M = bδ R −1 (R∇x f ){ · ∇g R 2 ∇x f ))dm M . divg {} |R∇x f |2 dm + bδ (R∇x f ) R −1 , ∇g R(R∇x f )dm.
M
M
In view of (89), (90), the operator )
* R −1 , ∇g R : L 2 (M) → L 2 (M)
is bounded with norm bounded by an a priori constant. It follows that |A| ≤
cb 2 N (x, t) δ
holds. The term B is easier to bound, because (K∇x f )(x, m, t) =
R −1 K (m, n)R∇x f (x, n, t)dn
M
and thus (∇g K∇x f )(x, ·, t) L ∞ (M) ≤ cN (x, t). Using (87) it follows that |B| ≤
cb 2 N (x, t) δ
|I V | ≤
cb 2 N (x, t). δ
and consequently (118)
Putting together the inequalities (113), (114), (115) and (118) we finished the proof of the lemma. Acknowledgement. The work of P.C. is partially supported by NSF-DMS grant 0504213. The work of C.F. is partially supported by NSF-DMS grant 0245242. The work of E.S.T. is supported in part by the NSF grant no.DMS-0504619, the BSF grant no. 2004271, and by the MAOF Fellowship of the Israeli Council of Higher Education. We thank the anonymous referee for carefully reading the paper and suggesting improvements.
Regularity of Coupled 2D Nonlinear Fokker-Planck and Navier-Stokes Systems
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References 1. Brézis, H., Gallouet, T.: Nonlinear Schrödinger evolution equations. Nonlinear Anal. 4(4), 677–681 (1980) 2. Chemin, J.Y.: Perfect Incompressible Fluids. Oxford Lecture Series in Mathematics and its Applications. 14, New York: Clarendon Press Oxford University Press, 1998 3. Chemin, J.Y., Masmoudi, N.: About lifespan of regular solutions of equations related to viscoelastic fluids. SIAM J. Math. Anal. 33(1), 84–112 (2001) 4. Constantin, P.: Nonlinear Fokker-Planck Navier-Stokes systems. Commun. Math. Sciences 3(4), 531– 544 (2005) 5. Constantin, P., Foias, C.: Navier-Stokes Equations. Chicago, IL: U. Chicago Press, 1988 6. Doi, M., Edwards, S.F.: The Theory of Polymer Dynamics. Oxford: Oxford University Press, 1988 7. E, W., Li, T.J., Zhang, P-W.: Well-posedness for the dumbell model of polymeric fluids. Commun. Math. Phys. 248, 409–427 (2004) 8. Helgason, S.: Differential Geometry, Lie Groups and Symmetric spaces. London Academic Press, 1978 9. Hörmander, L.: The Analysis of Linear Partial Differential Operators. Vol. 3, Berlin-Heidelberg- New York-Tokyo: Springer-Verlag 1985 10. Jourdain, B., Lelievre, T., Le Bris, C.: Numerical analysis of micro-macro simulations of polymeric flows: a simple case. Math. Models in Appl. Science 12, 1205–1243 (2002) 11. Jourdain, B., Lelievre, T., Le Bris, C.: Esistence of solutions for a micro-macro model of polymeric fluid: the FENE model. J. Funct. Anal. 209, 162–193 (2004) 12. Lemarié-Rieusset, P.G.: Recent Developments in the Navier-Stokes Problem. Chapmann and Hall/CRC Research Notes in Mathematics 431, Boca Raton, FL: CRC, 2002 13. Lin, F-H., Liu, C., Zhang, P.: On hydrodynamics of viscoelastic fluids. CPAM 58, 1437–1471 (2005) 14. Li, T., Zhang, H., Zhang, P-W.: Local existence for the dumbell model of polymeric fluids. Commun. PDE 29, 903–923 (2004) 15. Onsager, L.: The effects of shape on the interaction of colloidal particles. Ann. N.Y. Acad. Sci 51, 627– 659 (1949) 16. Otto, F., Tzavaras, A.: Continuity of velocity gradients in suspensions of rod-like molecules. SFB Preprint 147 (2004), available at http://www.math.Umd.edu/Ntzavaras/reprints/existence.pdf 17. Renardy, M.: An existence theorem for model equations resulting from kinetic theories of polymer solutions. SIAM J. Math. Analysis 23, 313–327 (1991) 18. Sideris, T., Thomasses, B.: Global existence for 3D incompressible isotropic elastodynamics via the incompressible limit. CPAM 58, 750–788 (2005) Communicated by A. Kupiainen
Commun. Math. Phys. 270, 813–833 (2007) Digital Object Identifier (DOI) 10.1007/s00220-006-0176-0
Communications in
Mathematical Physics
High Energy Limits of Laplace-Type and Dirac-Type EigenFunctions and Frame Flows Dmitry Jakobson1, , Alexander Strohmaier2 1 Department of Mathematics and Statistics, McGill University, 805 Sherbrooke St. West, Montréal QC H3A
2K6, Canada. E-mail:
[email protected]
2 Mathematisches Institut, Universität Bonn, Beringstrasse 1, D-53115 Bonn, Germany.
E-mail:
[email protected] Received: 3 May 2006 / Accepted: 27 July 2006 Published online: 9 January 2007 – © Springer-Verlag 2007
Abstract: We relate high-energy limits of Laplace-type and Dirac-type operators to frame flows on the corresponding manifolds, and show that the ergodicity of frame flows implies quantum ergodicity in an appropriate sense for those operators. Observables for the corresponding quantum systems are matrix-valued pseudodifferential operators and therefore the system remains non-commutative in the high-energy limit. We discuss to what extent the space of stationary high-energy states behaves classically. 1. Introduction and Main Results If X is an oriented closed Riemannian manifold and the Laplace operator on X , then a complete orthonormal sequence of eigenfunctions φ j ∈ L 2 (X ) with eigenvalues λ j ∞ is known to converge in the mean to the Liouville measure, in the sense that 1 lim φ j , Aφ j = σ A (ξ )d L(ξ ), N →∞ N j≤N
T1∗ X
for any zero order pseudodifferential operator A, where integration is with respect to the normalized Liouville measure on the unit cotangent bundle T1∗ X , and σ A is the principal symbol of A. In particular, A might be a smooth function on X and the above implies that the sequence 1 |φ j (x)|2 N j≤N
converges to the normalized Riemannian measure in the weak topology of measures. In case the geodesic flow on T1 X is ergodic it is known that the following stronger result holds: The first author was supported by NSERC, FQRNT and Dawson fellowship.
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lim
N →∞
1 |φ j , Aφ j − σ A (ξ )d L(ξ )| = 0. N j≤N
T1∗ X
This property is commonly referred to as quantum ergodicity and it is equivalent to the existence of a density-one-subsequence φ j such that
σ A (ξ )d L(ξ ), lim φ j , Aφ j = j→∞
T1∗ X
for any zero order pseudodifferential operator A (see [Shn74, Shn93, CV85, Zel87]). We show in this paper that the high energy behavior of the Dirac operator D acting on spinors on a closed spin manifold X is determined by the frame flow in the same manner, as the geodesic flow determines the high energy limit of the Laplace operator. If Fk X is the bundle of oriented orthonormal k-frames in T ∗ X , then projection to the first vector makes Fk X → T1∗ X into a fiber bundle. In particular for k = n this is the full frame bundle and F X = Fn X is a principal fiber bundle over T1∗ X with structure group S O(n − 1). Transporting covectors parallel along geodesics extends the Hamiltonian flow on T1∗ X to a flow on Fk X . This is the so-called k-frame flow. In case k = n we will refer to it simply as the frame flow. Of course ergodicity of the k-frame flow for any k implies ergodicity of the geodesic flow, whereas the conclusion in the other direction is not always true (cf. Sect. 2). Still there are many examples investigated in the literature when the frame flow is ergodic. Our first main result is that quantum ergodicity holds for eigensections of the Dirac operator in case the frame flow is ergodic. Theorem 1.1. Let X be a closed Riemannian spin manifold of dimension n ≥ 3 with Dirac operator D acting on sections of the spinor bundle. Suppose that φ j ∈ L 2 (X ; S) is an orthonormal sequence of eigensections of D with eigenvalues λk ∞ such that the φk span1 the positive energy subspace of D. Then, if the frame flow on F X is ergodic, we have 1 1 lim |φ j , Aφ j − [ n ] Tr((1 + γ (ξ )) σ A (ξ ))d L(ξ )| = 0, N →∞ N 22 ∗ j≤N T1 X
for all A ∈ DO0cl (X, S). Here γ (ξ ) denotes the operator of Clifford multiplication with ξ . In particular there is a density one subsequence φ j such that 1
Tr((1 + γ (ξ )) σ A (ξ ))d L(ξ ). φ j , Aφ j → [ n ] 22 ∗ T1 X
A similar statement holds for the negative energy subspace. Another result is that the (2 min( p, n − p))-frame flow determines the high energy p behavior of the Laplace-Beltrami operator p acting on the space C ∞ (X, C X ) of complex-valued p-forms. Note that the Hodge decomposition implies that there are three p−1 p+1 invariant subspaces for p , namely the closures of dC ∞ (X, C X ), δC ∞ (X, C X ) and the finite dimensional space of harmonic forms. The latter subspace plays no role 1 In the sense that the linear hull is dense.
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for the high energy behavior. The eigenspaces of the first subspace consist of exterior derivatives of p − 1 eigenforms; their high energy behavior is therefore determined by the high energy behavior of p−1 . In particular the high energy behavior of 1 restricted to dC ∞ (X ) is controlled by the geodesic flow. We therefore look at the second subspace only. Note that any coclosed form which is a nonzero eigenvalue to p is coexact. Hence, to investigate the high energy behavior of p we have to look at the system pφ j = λ j φ j , δφ j = 0. In case p = 1 such systems appear in physics if one investigates electromagnetic fields (Maxwell’s equations) or the Proca equation for spin 1 particles. The restriction to the coclosed forms corresponds to a gauge condition which restricts to the transversal subspaces. Our result is that this system is quantum ergodic, if the 2 min( p, n − p)-frame flow is ergodic. Theorem 1.2. Let X be an oriented closed Riemannian manifold of dimension n ≥ 3 and let 0 < p < n. Suppose that φk is an orthonormal sequence of eigen- p-forms satisfying p φk = λ k φk , δφk = 0, such that the φk span ker(δ) and with λk ∞. Suppose that p = n−1 2 . Then, if the (2 min( p, n − p))-frame flow is ergodic, the system is quantum ergodic in the sense that 1 |φk , Aφk − ωt (σ A )| = 0, lim N →∞ N k≤N
for all A ∈ DO0cl (X ; C X ). In particular there is a density one subsequence φk such that p
lim φk , Aφk = ωt (σ A ), for all A ∈ DO0cl (X ; C X ). p
k→∞
Here ωt is a state on the C ∗ -algebra of continuous End( C X )-valued functions on T1∗ X which is defined by n − 1 −1 Tr i(ξ )i ∗ (ξ )a(ξ ) d L(ξ ), ωt (a) := p p
T1∗ X
where i(ξ ) is the operator of interior multiplication with ξ , and the adjoint i ∗ (ξ ) is the operator of exterior multiplication with ξ . Note that the system k φ j = λ j φ j , dφ j = 0 is equivalent to our system with p = n − k via the Hodge star operator. n−1 p+1 δ∗ leaves The restriction p = n−1 2 is necessary since if p = 2 the operator i the space Rg(δ) invariant and commutes with p (∗ is the Hodge star operator). In this case our result is
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Theorem 1.3. Let X be an oriented closed Riemannian manifold of odd dimension n ≥ 3. Let p = n−1 2 and suppose that φk is an orthonormal sequence of eigen- p-forms satisfying p φk = λ k φk , δφk = 0, p+1 i δ ∗ φk = ± λ k φk , −1/2
such that the φk span Ran(δ ± i p+1 p δ ∗ δ) and with λk ∞. Then, if the (n − 1)frame flow is ergodic, the system is quantum ergodic in the sense that 1 |φk , Aφk − ω± (σ A )| = 0, lim N →∞ N k≤N
for all A ∈ DO0cl (X ; C X ). In particular there is a density one subsequence φk such that p
lim φk , Aφk = ω± (σ A ), for all A ∈ DO0cl (X ; C X ). p
k→∞
Here the states ω± are defined by ( p!)2 Tr 1 ± i p i(ξ )∗ i(ξ )i ∗ (ξ )a(ξ ) d L(ξ ). ω± (a) := (2 p)! T1∗ X
As the example of Kähler manifolds (see Sect. 2) shows the above theorems do not hold if we assume ergodicity of the geodesic flow only. Our analysis is based on a version of Egorov’s theorem for matrix valued operators. A second order differential operator P acting on sections of a vector bundle E is said to be of Laplace type if σ P (ξ ) = g(ξ, ξ ) id E , i.e. if in local coordinates it is of the form P = − i,k g ik ∂i ∂k + lower order terms. Examples are the LaplaceBeltrami operator p acting on p-forms or the square D 2 of the Dirac operator on a Riemannian spin manifold. For such operators the first order term (the subprincipal symbol) defines a connection ∇ E on the bundle E. We will prove an Egorov theorem for matrix-valued pseudodifferential operators acting on sections of E. More precisely, for A ∈ DO0cl (X, E), a zero order classical pseudodifferential operator, the principal symbol σ A is an element in C ∞ (T1∗ X, End(π ∗ (E))), where π ∗ (E) is the pull-back of the bundle E → X under the projection π : T1∗ X → X . Note that the connection ∇ E determines a connection ∇ on End(π ∗ (E)). Parallel transport along the Hamiltonian flow of σ P then determines a flow βt acting on C ∞ (T1∗ X, End(π ∗ (E))). Our version of Egorov’s theorem specialized to Laplace type operators reads as follows. Theorem 1.4. If A ∈ DO0cl (X, E) and if P is a positive second order differential 1/2 1/2 operator of Laplace type then for all t ∈ R the operators At := e+ i t P Ae− i t P are 0 again in DOcl (X, E) and σ At = βt (σ A ). We actually prove a more general version of this theorem which applies to flows generated by first order pseudodifferential operators with real scalar principal symbols. Note that unlike in the scalar case the first order terms are needed to determine the flow. We show that for pseudodifferential operators with real scalar principal part the subprincipal symbol is invariantly defined as a partial connection along the Hamiltonian vector field, thus allowing us to define all flows without referring to local coordinate systems.
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1.1. Discussion. Dirac equation on R3 (and, more generally, on Rd ) has been studied from the semiclassical point of view in the papers [BoK98, BoK99, Bo101, BoG04, BoG04.2] of Bolte, Glaser and Keppeler. The authors would like to thank J. Bolte for bringing to their attention this problem on manifolds. Unlike in the works of Bolte, Glaser and Keppeler we investigate the high energy limit rather than the semiclassical limit. Therefore, all nontrivial dynamical effects are due to the nontrivial curvature of the spin-connection. This is conceptually different from the stated previous results where quantum ergodicity is due to a spin precession in an external magnetic field. External fields are not seen in the high energy limit and therefore the strict analog of the result of Shnirelman, Colin de Verdière and Zelditch [Shn74, Shn93, CV85, Zel87] can not be expected to hold in Rn or on manifolds with integrable geodesic flow. Apart from working on manifolds our methods also differ from those previously employed as we take the absolute value of the Dirac operator instead of the Dirac operator itself as the generator of the dynamics. This has the advantage of allowing for the full algebra of matrix valued functions as the observable algebra rather than a subalgebra. One can also justify this from a physical point of view. Namely, in a fully quantized theory the generator of the time evolution on the 1-particle Hilbert spaces is the absolute value of the Dirac operator. Furthermore, on the electron 1-particle subspace these two operators coincide. Our Theorem 1.3 for n = 3 and k = 1 deals with the electromagnetic field on a 3-dimensional compact manifold. The statement of Theorem 1.3 means that quantum ergodicity holds for circular polarized photons if the 2-frame flow is ergodic. We would also like to mention that the Egorov theorem as we state it is related to a work of Dencker ([D82]), who proved a propagation of the singularity theorem for systems of real principal type. It follows from his work that the polarization set of solutions to the Dirac equation is invariant under a certain flow similar to ours. The Egorov theorem in a form of Theorem 1.4 together with a similar conclusion as in Corollary 4.2 is already contained in a work of Bunke and Olbrich ([BuO104]). It is already noted there that the Laplace-Beltrami operator on p-forms fails to be Quantum ergodic in the direct sense because of the Hodge decomposition. We also refer the reader to [EW96, GMMP97] and references therein, and [San99] for discussion of semiclassical limits for matrix-valued operators, and relations to parallel transport. Since the high energy limit of the Quantum system associated to Laplace type operators on vector bundles is non-commutative the apropriate language to investigate questions of ergodicity is the language of C ∗ -dynamical systems and states (see Appendix A). This was already advertized by S. Zelditch in [Zel96] and it is shown there that for a large class of abstract C ∗ -dynamical systems classical ergodicity implies quantum ergodicity. The assumptions under which the theorems are stated in [Zel96] (G-abelianness or classical abelianness) are in general not satisfied in the examples we study. The method of the proof can be adapted to our situation, however. In our work we identify the classical flows corresponding to the Dirac operator and the Hodge Laplacian as frame flows, which allows us to use the results obtained by Brin, Arnold, Pesin, Gromov, Karcher, Burns and Pollicott to exhibit many examples of manifolds where quantum ergodicity holds for the Dirac operator, see Corollary 2.1. The connection to their work has not been made before in the literature on quantum ergodicity. Finally, our results on quantum ergodicity for p-eigenforms for Hodge Laplacian, and the role played by 2 min( p, n − p)-frame flow seem to be completely new. It is a hope of the authors that their results will stimulate further studies of relationship between ergodic theory of partially hyperbolic dynamical systems, and high energy behavior of eigenfunctions of matrix-valued operators.
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2. Ergodic Frame Flows: Known Examples The k-frame flow t , k ≥ 2 is defined as follows: let (v1 , . . . , vk ) be an ordered orthonormal set of k unit vectors in T p X . Then t v1 = G t v1 , where G t is the geodesic flow. t v j , 2 ≤ j ≤ k translates v j by the parallel translation at distance t along the geodesic determined by v1 . Here we summarize the cases when the frame flow is known to be ergodic. A k-frame flow is a SO(k − 1)-extension of the geodesic flow; on an n-dimensional manifold, k-frame flow is a factor of the n-frame flow for 2 ≤ k < n, so ergodicity of the latter implies ergodicity of the former. Frame flow preserves orientation, so in dimension 2 its ergodicity (restricted to positively-oriented frames, say) is equivalent to the ergodicity of the geodesic flow. Frame flows were considered by Arnold in [Arn61]. In negative curvature, they were studied by Brin, together with Gromov, Karcher and Pesin, in a series of papers [BrP74, Br75, Br76, BrG80, Br82, BrK84]. Recently, a lot of progress was made in understanding ergodic behavior of general partially hyperbolic systems, including frame flows. In the current paper, the authors are primarily interested in the ergodicity of the flow; the most recent paper dealing with that question appears to be [BuP03] by Burns and Pollicott, where the authors establish ergodicity under certain pinched curvature assumptions in “exceptional” dimensions 7 and 8, see below. In the sequel, we shall assume that M is negatively curved with sectional curvatures satisfying −K 22 ≤ K ≤ −K 12 . The frame flow is known to be ergodic and have the K property 1) if M has constant curvature [Br76, BrP74]; 2) for an open and dense set of negatively curved metrics (in the C 3 topology) [Br75]; 3) if n is odd, but not equal to 7 [BrG80]; or if n = 7 and K 1 /K 2 > 0.99023... [BuP03]; 4) if n is even, but not equal to 8, and K 1 /K 2 > 0.93, [BrK84]; or if n = 8 and K 1 /K 2 > 0.99023... [BuP03]. By Theorem 4.4 and results of Sect. 5, we have the following Corollary 2.1. Quantum ergodicity for Dirac operator and for Hodge Laplacian (conclusions of Theorems 1.1, 1.2 and 1.3) hold in each of the cases (1)–(4). The frame flow is not ergodic on negatively-curved Kähler manifolds, since the almost complex structure J is preserved. This is the only known example in negative curvature when the geodesic flow is ergodic, but the frame flow is not. In fact, given an orthonormal k-frame (v1 , . . . , vk ), the functions (vi , J v j ), 1 ≤ i, j ≤ k are first integrals of the frame flow, and in some cases it is possible to describe the ergodic components, [Br82, BrP74, BrG80]. Note that the conclusion of Theorem 1.2 is false in the Kähler case, because the decomposition into ( p, q)-forms is a decomposition into invariant subspaces of the Laplace-Beltrami operator. The Kähler case is interesting in its own right and will be discussed in a forthcoming paper. The frame flow is conjectured to be ergodic whenever the curvature satisfies −1 < K < −1/4, cf. [Br82]. That conjecture is still open. 3. Microlocal Analysis for Operators on Vector Bundles 3.1. The subprincipal symbol. Let X be a closed manifold. Suppose that n/2 P ∈ DOm X ). cl (X,
High Energy Limits and Frame Flows
819
Then the principal symbol σ P is well defined as a function on the cotangent bundle T˙ ∗ X = T ∗ X \0 which is smooth and positively homogeneous of degree m. The subprincipal symbol sub(P) is defined in local coordinates by sub(P) := pm−1 −
1 ∂ 2 pm , 2i ∂ x j ∂ξ j
(1)
j
where the functions pm are the terms homogeneous of degree m in the asymptotic expansion of the full symbol of P. Surprisingly, the subprincipal symbol turns out to be well defined as a function on T˙ ∗ X (see [DH72], ch 5.2). However, the situation changes if we n/2 X ⊗ E) for some vector bundle E. In this case the principal have P ∈ DOm cl (X,
symbol σ P is in C ∞ (T˙ ∗ X, End(π ∗ (E))), whereas the subprincipal symbol has a more complicated transformation law under a change of a bundle chart. More explicitly we have with φ(x, ξ ) = (x, ξ ) for some section w ∈ C ∞ (X ; E) using local coordinates and a local trivialization (see [DH72], Eq. 5.2.2) e− i φ P(e i φ w) = p(x, φx )w −
+
1 ∂2 p w 2i ∂ x j ∂ξ j j
j ( j)
where pm =
( j)
∂w 1 ∂ pm (x, φx ) + w i ∂x j 2 i ∂x j
1 ( j) pm (x, φx )
mod S m−2 , (2)
∂ pm ∂ξ j .
Suppose now that pm = σ P is scalar and real, i.e. σ P (x, ξ ) = h(x, ξ )id E x for some h ∈ C ∞ (T˙ ∗ X, R). The Hamiltonian vector field associated with the principal symbol σ P of P is a vector field on T˙ ∗ X and defined in local coordinates by ∂σ P ∂ ∂σ P ∂ . (3) HP = − ∂ξ j ∂ x j ∂ x j ∂ξ j j
( j) Now pm (x, φx ) has a nice interpretation in terms of H P . Namely, if (x, ξ ) ∈ T˙ ∗ X , ( j) then pm (x, ξ ) is the push forward of H P (x, ξ ) under the projection π : T˙ ∗ X → X expressed in local coordinates, i.e.
j
p ( j) (x, ξ )
∂ = π∗ (H P (x, ξ )). ∂x j
(4)
Therefore, in case E is trivial the last sum in (2) is exactly the Lie Derivative − i Lv w of half densities along the vector field v = π∗ (H P (x, φx )), which is defined without reference to the local coordinate system and depends on the function φ only. Hence, if we fix a local trivialization of E and change coordinates on the base manifold only, σ P and sub(P) ( j) is the transform as functions in C ∞ (T˙ ∗ X, End(π ∗ (E))). Note that j pm (x, φx ) 1i ∂∂w xj only term in (2) which depends on the derivative of w. Hence, under a change of bundle charts by the local function A ∈ C ∞ (X, GL(E)) we get the transformation law sub(P) → A−1 sub(P)A + A−1 p ( j) ∂ j A 1 = A−1 sub(P)A + A−1 H P A, i
(5)
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D. Jakobson, A. Strohmaier
whereas σ P transforms as a function in C ∞ (T˙ ∗ X, End(π ∗ (E))). If the principal symbol is scalar and real (5) is the transformation law of a partial connection2 along the vector field H P . Hence, by ∇ H P := H P + i sub(P) a covariant derivative is defined. We have therefore proved the following proposition. n/2 X ⊗ E) Proposition 3.1. Let E be a vector bundle and suppose that P ∈ DOm cl (X,
has real scalar principal symbol. Then the subprincipal symbol of sub(P) defined locally by (1) is invariantly defined as a partial connection on π ∗ E along the Hamiltonian vector field H P .
Let X be an oriented closed n-dimensional Riemannian manifold, let E → X be a hermitian vector bundle and suppose that P : C ∞ (X ; E) → C ∞ (X ; E) is a formally selfadjoint second order differential operator of Laplace type, i.e. σ P (ξ ) = g(ξ, ξ ) · 1. Then there exists a hermitian connection (see e.g. [BGV92] ch. 2.1) ∇ : C ∞ (X ; E) → C ∞ (X ; E ⊗ T ∗ X ), and a potential V ∈ C ∞ (X ; End(E)) such that P = ∇ ∗ ∇ + V. The connection and the potential are uniquely determined by these properties. The operator P is essentially selfadjoint on C ∞ (X ; E) and in case it is positive we may define the square root P 1/2 by functional calculus. By Seeley ([See67]) we know that P 1/2 is a classical pseudodifferential operator of order 1 and its principal symbol is given by σ P 1/2 (ξ ) = ||ξ ||g ·1. We use the metric to identify the bundle n/2 X with the trivial bundle and in this way we understand P and P 1/2 as operators acting on C ∞ (X ; E ⊗ n/2 X ). Since P is of Laplace type the Hamiltonian vector field H P 1/2 when restricted to the unit tangent bundle T1∗ X coincides with the geodesic spray. We will therefore write Hg for H P 1/2 in order to emphasize the dependence from the metric. Now ∇ determines uniquely a hermitian partial connection ∇˜ Hg along Hg on π ∗ E which satisfies
∇˜ Hg π ∗ ( f ) (x, ξ ) = ∇π∗ (Hg (x,ξ )) f for all f ∈ C ∞ (X, E), (6) where π ∗ ( f ) ∈ C ∞ (T ∗ X ; π ∗ (E)) is the pull back of a section f ∈ C ∞ (X ; E). If we fix a local trivialization of E we have i ik (∇˜ Hg f )(x, ξ ) − (Hg f )(x, ξ ) = g A i ξk , (7) ||ξ || i,k
where ∇i = ∂i + i Ai . The geometric meaning of this partial connection is as follows. The vector field Hg generates the geodesic flow. Hence, a partial connection along Hg allows to transport vectors along geodesics. The partial connection (6) is chosen in such a way that v ∈ π ∗ (E)(x,ξ ) = E x gets transported along the geodesic with the original connection ∇ on E. We have Proposition 3.2. For P = ∇ ∗ ∇ + V as above the partial connection determined by the subprincipal symbol sub(P 1/2 ) of P 1/2 coincides with ∇˜ Hg . 2 A partial connection along a vector field v can be defined by its covariant derivative which is a map ∇v : C ∞ (X ; E) → C ∞ (X ; E) satisfying ∇v ( f g) = v( f )g + f ∇v g for all f ∈ C ∞ (X ), g ∈ C ∞ (X ; E). Hence, parallel transport is defined along v only. Moreover, where v vanishes this is a bundle homomorphism.
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Proof. We calculate everything in local coordinates where |g| = 1 to keep the formulas as simple as possible. In such local coordinates we have ∇i = ∂i + i Ai . Then one easily calculates sub(P)(ξ ) = 2 i g ik Ai ξk . i,k
Since the principal symbol of P is scalar the formula as proved in [DG75], sub(P 1/2 )(ξ ) =
1 − 21 σ sub(P), 2 P
continues to hold and we obtain sub(P 1/2 )(ξ ) = i ||ξ ||−1
(8)
g ik Ai ξk .
i,k
This coincides with the claimed formula.
3.2. Egorov’s theorem. Suppose that A ∈ DO1cl (X ; E ⊗ n/2 ) has real scalar principal part, let H A be the associated Hamiltonian vector field on T˙ ∗ X and let ∇ H A = H A + i sub(A) be the partial connection on π ∗ E defined by the subprincipal symbol. Then this determines a geometric flow αt on the vector bundle π ∗ E such that the flow lines (x(t), ξ(t), v(t)) consist of the orbits (x(t), ξ(t)) of the Hamiltonian flow and v(t) expressed in coordinates of a local bundle chart satisfies dv(t) = i sub(A)v(t). dt
(9)
Note that αt lifts the Hamiltonian flow h t on T˙ ∗ X and makes π ∗ E an R-equivariant vector bundle. The induced flow αt∗ on C ∞ (T˙ ∗ X, π ∗ E) satisfies d ∗ α f = ∇ H A f, dt t
(10)
which shows that the flow is defined independent of a choice of local coordinates. Hence, there is also an action Ad(αt ) of R on π ∗ End(E) which extends the Hamiltonian flow and is compatible with αt . This defines a flow on C ∞ (T˙ ∗ X, π ∗ End(E)) which we denote by Ad(αt )∗ . Clearly, if f ∈ C ∞ (T˙ ∗ X, π ∗ End(E)), d Ad(αt )∗ f = [∇ H A , f ]. dt
(11)
[∇ H A , f ] = H A f + i [sub(A), f ].
(12)
In local coordinates one has
Egorov’s theorem now reads as follows. Proposition 3.3. Let A ∈ DO1cl (X ; E ⊗ n/2 ) and suppose that the principal symbol of A is of real scalar type, i.e. σ A (ξ ) = h(ξ ) · 1, where h ∈ C ∞ (T˙ ∗ M, R). Then, if B is n/2 ), also B := e+ i t A Be− i t A is in DOm (X ; E). Moreover, in DOm t cl (X ; E ⊗
cl σ Bt = Ad(αt )∗ (σ B ).
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Proof. As usual we have d Bt = i [A, B]. dt
(13)
The right-hand side is a pseudodifferential operator in DOncl (X ; E ⊗ n/2 ) and its principal symbol is given by σ i [A,B] = {σ A , σ B } + i [sub(A), B].
(14)
This follows immediately from the formulas for the asymptotic expansion of products of pseudodifferential operators. By (11) and (12) Eq. (13) is on the level of principal symbols the flow equation for Ad(αt )∗ . This equation can be solved on the symbol level order by order and one can construct a classical symbol for Bt in the usual way (as for example carried out in [Tay81], Ch VIII, §1). The only difference to the scalar case is the second term in (14) which causes the Hamiltonian flow to be replaced by Ad(αt )∗ . Suppose now that X is an oriented Riemannian manifold and let A = P 1/2 , where P = ∇ ∗ ∇ + V is a positive Laplace type operator. We may use the metric to identify
1/2 X with the trivial bundle. Moreover, the Hamiltonian vector field H A restricted to the unit cotangent bundle T1∗ X coincides with the geodesic spray. In this case it is convenient to identify positively homogeneous functions on T˙ ∗ X with smooth functions on n/2 ) T1∗ X by restriction. Hence, the principal symbol of an operator in DOm cl (X ; E ⊗
∗ ∞ ∗ is in C (T1 X, π End(E)). In this case the Egorov theorem says that the flow αt transports vectors in π ∗ E parallel with respect to the connection ∇ along the geodesic flow on T1∗ X . 4. The Dirac Operator and the Frame Flow In this section (X, g) is a compact oriented Riemannian manifold of dimension n ≥ 3. A spin structure (see e.g. [LM89, Fri]) on X is an Spin(n)-principal bundle P over X together with a smooth covering η from P onto the bundle F X of oriented orthonormal frames, such that the following diagram is commutative: P × Spin(n) −−−−→ ⏐ ⏐η×λ
P −−−−→ ⏐ ⏐η
X
(15)
F X × SO(n) −−−−→ F X −−−−→ X Here λ denotes the covering map Spin(n) → SO(n). The (complexified) Clifford algebra n n n Clc (Rn ) is isomorphic to Mat(2[ 2 ] , C) if n is even and to Mat(2[ 2 ] , C) ⊕ Mat(2[ 2 ] , C) if n is odd. The Clifford modules n are then defined by the action of this matrix algebra [n]
on C2 2 , where in case n is odd we project onto the first summand. Hence, n is an irreducible module for the Clifford algebra Clc (Rn ). Since Spin(n) ⊂ Clc (Rn ) the Clifford modules n are also modules for the group Spin(n). The corresponding representation ρ : Spin(n) → Aut(n ) is called the spinor representation of Spin(n). This representation is irreducible if n is odd. It is the direct sum of two irreducible components if n is even. The spinor bundle S associated with a Spin structure is the associated bundle P ×ρ n . The Levi-Civita connection on F X lifts naturally to a connection on P and this defines a connection ∇ S : C ∞ (X ; S) → C ∞ (X ; S ⊗ T ∗ X ) on S, the Levi-Civita
High Energy Limits and Frame Flows
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connection on the spinor bundle. The Dirac operator D : C ∞ (X ; S) → C ∞ (X ; S) is defined by − i γ ◦ ∇ S , where γ denotes the action of covector fields on sections of the spinor bundle by Clifford multiplication. D is of Dirac type and essentially selfadjoint on C ∞ (X ; S). The operator F = sign(D) is in DO0cl (X ; S) and its principal symbol σ F (ξ ) is given by Clifford multiplication by |ξ1| ξ . If n is even, then Clifford multiplin(n+1)
cation with the volume form times i 2 defines an involution on L 2 (X ; S) which anti-commutes with D and with F, but which commutes with |D|. The Lichnerowicz formula allows us to express the square of the Dirac operator by the spinor Laplacian D 2 = ∇ S∗ ∇ S +
1 R, 4
(16)
where R is the scalar curvature. By Prop. 3.2 the connection determined by the subprincipal symbol sub(|D|) of |D| transports spinors along geodesics with the spinor connection ∇ S . The corresponding flow on the bundle π ∗ S → T1∗ X will be denoted by αt . The induced flow on the bundle π ∗ End(S) → T1∗ X will be denoted as before by Ad(αt ). This induces a 1-parameter group βt = Ad(αt )∗ of ∗-automorphisms of the algebra C ∞ (T1∗ X, π ∗ End(S)). It extends continuously to the C ∗ -algebra A = C(T1∗ X, π ∗ End(S)). The Egorov theorem of the previous section therefore reads as follows. Proposition 4.1. Let D be the Dirac operator on a compact spin manifold X and let A ∈ DO0cl (X ; S). Then with At := e+ i t|D| Ae− i t|D| we have At ∈ DO0cl (X ; S) for all t ∈ R and σ At = βt (σ A ). And as a consequence we get Corollary 4.2. Let X be a closed Riemannian spin manifold of dimension n ≥ 3 and let D be the Dirac operator. Let φk be a sequence of eigensections to D 2 with D 2 φk = λk φk , φk , φ j = δk j , such that |λk | ∞ and such that the sequence of states ωi (A) := φi , Aφi on the C ∗ -algebra DO0cl (X ; S) converges in the weak-∗-topology. Then there is a βt -invariant state ω∞ on C(T1∗ X, π ∗ End(S)) such that lim ωn (A) = ω∞ (σ A ).
n→∞
Proof. Since the states ωi are invariant under the flow induced by conjugation with e i |D|t so is the limit state. The limit state ω vanishes on all operators of order −1 since their product with |D| is bounded. Since the norm closure of those operators are the compact operators K we have ω(K) = {0}. Hence, the state projects to a state on the quotient DO0cl (X ; S)/K. The symbol map is known to extend by continuity to a map
DO0cl (X ; S) → C(T1∗ X, π ∗ End(S)) with kernel K. Hence, the quotient is via the symbol map isomorphic to C(T1∗ X, π ∗ End(S)). Moreover, by the above the symbol map is equivariant with respect to the two flows.
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D. Jakobson, A. Strohmaier
There are two natural invariant states ω± on C(T1∗ X, π ∗ End(S)) given by 2 ω± (a) := Tr(P± (ξ )a(ξ )P± (ξ ))dμ L (ξ ), rk(S)
(17)
T1∗ X
where μ L is the normalized Liouville measure and P± := 21 (1 ± σ F ). The tracial state
ω(a) =
1 rk(S)
ω=
1 (ω+ + ω− ), 2
Tr(a(ξ ))dμ L (ξ ) T1∗ X
is therefore not ergodic (see Appendix A for the notion of ergodicity in this context). states with Theorem 4.3. Suppose the frame flow is ergodic. Then ω+ and ω− are ergodic respect to βt . If moreover n is odd the systems C(T1∗ X, π ∗ End(S)), ω± are βt -abelian. Proof. Note that since the Levi-Civita connection is compatible with the Clifford multiplication both P+ and P− are easily seen to be invariant under βt . To prove ergodicity we have to show (see Appendix A) that all βt -invariant elements in P± L ∞ (T1∗ X, π ∗ End(S)) P± are of the form c P± with c ∈ C. To show βt -abelianness we have to show that an invariant element in L 2 (T1∗ X, π ∗ End(S)) is of the form c1 P+ + c2 P− with c1 , c2 ∈ C. We therefore analyze how invariant elements in L 2 (T1∗ X, π ∗ End(S)) look like. Step 1. Denote by CX 1 , . . . , X n the space of noncommutative polynomials in the variables X 1 , . . . , X n . Now we define a continuous map T : L 2 (F X ) ⊗ CX 1 , . . . , X n → L 2 (T1∗ X, π ∗ End(S)), T ( f ⊗ p)(ξ ) := f (ξ, v) p(ξ, v)dμ(v),
(18) (19)
Fξ X
where the integration is over the invariant measure on the fiber Fξ X . The action of covectors on spinors here is by Clifford multiplication. The pullback h ∗t of the frame flow on F X defines a flow on L 2 (F X ) ⊗ CX 1 , . . . , X n by acting on the first tensor factor. Since the connection is compatible with Clifford multiplication and the Clifford action the map T intertwines the pullback h ∗t of the frame flow and the flow βt , i.e. T ◦ (h ∗t ⊗ 1) = βt ◦ T.
(20)
Note that the Clifford action on the spinor bundle is irreducible. This implies that T is onto. Step 2. Let ∈ L 2 (T1∗ X, π ∗ End(S)) be an invariant vector. By the above we can choose an element f ∈ L 2 (F X ) ⊗ CX 1 , . . . , X n such that T ( f ) = . Since by assumption T the frame flow is ergodic we have lim T →∞ T1 0 h ∗t ( f )dt = F X f (x)d x = m, where m is a constant polynomial. Since is invariant and T is equivariant we have almost everywhere
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825
⎛ ⎜ (ξ ) = T (m)(ξ ) = ⎝
⎞ ⎟ m(ξ, v)dμ(v)⎠ .
Fξ X
Since the measure on Fξ X is invariant under the action of S O(n − 1) the endomorphism Fξ X m(ξ, v)dμ(v) commutes with the action of Spin(n − 1). Clearly, the projections P+ (ξ ) and P− (ξ ) commute with the Spin(n − 1) action, and it is easy to see (for example by calculating the dimensions) that the representations of Spin(n − 1) on the ranges of P+ (ξ ) and P− (ξ ) are irreducible. Moreover, these two representations are equivalent iff n is even. In this case the algebra of invariant matrices in End(Sξ ) is generated by P+ (ξ ), P− (ξ ), . This shows that an invariant element of the form P± A P± is proportional to P± which proves ergodicity. If n is odd the two irreducible representations of Spin(n − 1) on the ranges of P+ (ξ ) and P− (ξ ) are inequivalent. Hence, any invariant element in A ∈ End(Sξ ) is of the form c1 P+ + c2 P− . Remark. That C(T1∗ X, π ∗ End(S)), ω± is not βt -abelian in even dimensions has a simple interpretation. It is that the space of invariant states on C(T1∗ X, π ∗ End(S)) is not a simplex, i.e. the decomposition of an invariant state into ergodic states is not unique. For example the tracial state ω has the decomposition ω = 21 (ω+ +ω− ), but it also has the decomposition ω = 21 (ω1 + ω2 ), where ω1 (a) = ω((1 + )a) and ω2 (a) = ω((1 − )a). The states ω1 and ω2 can also be shown to be ergodic if the frame flow is ergodic. Theorem 4.4. Let X be a compact Riemannian spin manifold with spinor bundle S and Dirac operator D : C ∞ (X ; S) → C ∞ (X ; S). Let φk be an orthonormal sequence of eigensections of D with eigenvalues λk ∞ such that φk spans3 L 2+ (X ; S) = 1+F 2 2 L (X ; S). If the frame flow of X is ergodic, then Quantum Ergodicity holds, i.e. lim
N →∞
N 1 |φk , Aφk − ω+ (σ A )| = 0, for all A ∈ DO0cl (X ; S). N k=1
Moreover, there is a density one subsequence φk such that lim φk , Aφk = ω+ (σ (A)), for all A ∈ DO0cl (X ; S).
k→∞
An analogous statement holds for eigensections with λk −∞ with ω+ replaced by ω− . 1 Proof. We denote ω = 21 (ω+ +ω− ) = rk(S) T1∗ X Tr(σ A (ξ ))dμ(ξ ) as the ordinary tracial state on C ∞ (T1∗ X, π ∗ End(S)). The heat trace asymptotics (cf. [GrSee95]) 2 Tr(A · e−D t ) = C(n) Tr(σ A (ξ ))dμ(ξ ) · t −n/2 + O(t −n/2+1/2 ) (21) T1∗ X
which one easily gets from the classical calculus of pseudodifferential operators together with Karamata’s Tauberian theorem implies that lim
N →∞
N 1 1+ F 1+ F φk , A φk = ω+ (σ A ), N 2 2 k=1
3 In the sense that the hull of the vectors is dense.
A ∈ DO0cl (X ; S).
(22)
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D. Jakobson, A. Strohmaier
The proof is now an analog to the proof of Shnirelman, Zelditch and Colin de Verdière ([Shn74, Shn93, CV85, Zel87]). To prove Quantum ergodicity it is obviously enough to show that for any selfadjoint A ∈ DO0cl (X ; S) with ω+ (σ A ) = 0 we have N 1 lim |φk , Aφk | = 0. N →∞ N
(23)
k=1
1+F Clearly, |φk , Aφk | = |φk , 1+F 2 A 2 φk | and therefore, we assume without loss of 1+F 1+F generality that A = 2 A 2 . Hence, σ A = P+ σ A P+ . Now let
1 A T := T
T
e+ i t|D| Ae− i t|D| dt
0
and since ω+ is ergodic we have ω+ (|σ AT |2 ) → 0 (see Lemma A.1) as T → ∞. This implies that also ω+ (|σ AT |) → 0 as T → ∞. On the other hand we have |φk , Aφk | = |φk , A T φk | ≤ φk , |A T |φk .
(24)
Then formula (22) applied to |A T | together with the fact that ω+ (σ|AT | ) < for T large enough allows us to conclude that N 1 lim sup |φk , Aφk | < , N →∞ N
(25)
k=1
for all > 0 from which the assertion follows. The existence of a density one sequence is based on a diagonalization argument which is well known (see e.g. [Zel96]). 5. The Hodge-Laplace Let X be a compact oriented Riemannian manifold, let ∗ X := ∗ T ∗ X be the exterior algebra bundle, and let ∗C X be its complexification. We denote by d the exterior derivative, δ the coderivative (i.e. the formal adjoint of d), and by ∗ the Hodge star operator. Then d + δ is a Dirac type operator acting on sections of ∗C X . Its square is the Hodge Laplace operator = dδ + δd.
(26)
Note that leaves the subspace of p-forms invariant. In the following we will denote the restriction of to p-forms by p . The ordinary Laplace operator acting on functions is therefore equal to 0 . If ∇ p is the Levi-Civita covariant derivative of p-forms the Weitzenböck formula states that p = ∇ ∗p ∇ p + H p ,
(27)
where H p is a section of End( p T ∗ X ) which can be expressed in terms of the curvature of the connection. For example H1 is equal to the Ricci curvature. We conclude 1/2 that the partial connection determined by the subprincipal symbol of p transports pmultivectors along the Hamiltonian vector field parallel with respect to the Levi-Civita
High Energy Limits and Frame Flows
827
connection. The corresponding flow on the C ∗ -algebra C(T1∗ X, π ∗ End( C X )) will be p denoted by βt . There is a natural invariant tracial state ωtr on C(T1∗ X, π ∗ End( C X )) given by p
−1 n a → ωtr (a) = Tr(a(ξ ))d L(ξ ). p
(28)
T1∗ X
As in the Dirac case this state is not ergodic for 0 < p < n. Let P ∈ C(T1∗ X, π ∗ End( C X )) p
defined by P(ξ )v := i(ξ )ξ ∧ v,
(29)
where i(ξ ) denotes the operator of interior multiplication with ξ . Then P is an orthogonal p projection in C(T1∗ X, π ∗ End( C X )) which is invariant under βt , and hence n−p p ωt + ωl , n n n ωl (a) = ωtr ((1 − P) · a), p n ωtr (P · a), ωt (a) = n−p ωtr =
(30) (31) (32)
is a decomposition into invariant states. The non-ergodicity of this state can be seen as the classical counterpart of the Hodge decomposition C ∞ (X ; C X ) = dC ∞ (X ; C p
p−1
X ) ⊕ δC ∞ (X ; C X ) ⊕ ker( p ), p+1
(33)
p
which induces a decomposition of L 2 (X ; C X ) into invariant subspaces of p . Theorem 5.1. Suppose 0 < p < n, n ≥ 3 and let k p = 2 min( p, n − p). Suppose that state with respect the k p -frame flow is ergodic and that p = n−1 2 . Then ωt is an ergodic p p to βt on C(T1∗ X, π ∗ C X ). If moreover p = n2 then the system C(T1∗ X, π ∗ C X ), ωt is βt -abelian. Proof. The proof is similar to the proof in the Dirac case. We investigate the set of p p invariant vectors in P L 2 (T1∗ X, π ∗ End( C X ))P and in L 2 (T1∗ X, π ∗ End( C X ))P. Step 1. Let N = k p . Denote by P ⊂ CX 1 , . . . , X N , Y1 , . . . , Y N the ring of polynomials in the noncommutative variables X 1 , . . . , X N and Y1 , . . . , Y N such that in each summand the same number of X and Y occur. Now define a map T : L 2 (FN X ) ⊗ P → L 2 (T1∗ X, π ∗ End( C X )), f (ξ, v) p(v)dμ(v), ˆ T ( f ⊗ p)(ξ ) = p
FN ,ξ X
828
D. Jakobson, A. Strohmaier
where integration is over the fibre FN ,ξ X of the bundle FN X over the point ξ ∈ T1∗ X . The endomorphism p(v) ˆ is defined by replacing all X i by exterior multiplication with vi and all Yi by interior multiplication with vi . Since the number of X and Y is the p ∗ same in each summand the operators leave the space C Tπ(ξ ) X invariant. Since exterior and interior multiplication are compatible with the Levi-Civita connection, the map T intertwines the pullback of N -frame flow and the flow βt , i.e. T ◦ (h ∗t ⊗ 1) = βt ◦ T. ∗ X Moreover, by an elementary exercise in linear algebra any endomorphism of C Tπ(ξ ) can be represented by a linear combination of elements of the form p(v ˆ 1 , . . . , v N ), where v1 , . . . , v N is a frame and p has degree at most 2N . Therefore, the map T is surjective. p
Step 2. Now let be an invariant element in L 2 (T1∗ X, π ∗ End( C X )). Then we may find an f ∈ L 2 (FN X ) ⊗ P such that = T ( f ). By the same argument as in the proof of Theorem 4.3 it follows that = T (m), where m is some constant polynomial p such that m(ξ ˆ ) commutes with the action of S O(n − 1) on each fiber of π ∗ ( C X ) . p Note that P(ξ ) and (1 − P(ξ )) project onto invariant subspaces. The fiber of π ∗ ( C X ) p ∗ ∗ ∗ at the point (x, ξ ) ∈ T1 X is given by C Tx X . On the other hand Tx X = Rξ ⊕ V , where V is the orthogonal complement of ξ in Tx∗ X . Hence, we have the decomposition p−1
p Tx∗ X = R⊗ p−1 V ⊕ p V . It is now easy to see that 1− P projects onto C⊗ C V , p whereas P projects onto C V . The representation of S O(n − 1) on p Cn−1 is irren ducible (see e.g. [FuHa91], Lecture 18) since we assumed p = n−1 2 . If p = 2 then all other components which occur in the decomposition into irreducible representations are inequivalent to this representation as one can see by calculating the dimensions (the exceptional case n = 3, p = 2 by other methods). Hence, in this case the algebra of p invariant elements in End( C Tx∗ X ) is generated by P(ξ ) and 1 − P(ξ ). Therefore, any invariant element is of the form c1 P + c2 (1 − P). This shows R-abelianness and ergodicity if p = n2 . Ergodicity for p = n2 follows from the fact that any invariant element p in P(ξ )End( C Tx∗ X )P(ξ ) is proportional to P(ξ ) which is a simple consequence of p the irreducibility of the S O(n − 1) action on C V . p
The orthoprojection onto the closure of δC ∞ (X ; p−1 T ∗ X ) is given by the zero order pseudodifferential operator ( p |ker( p )⊥ )−1 δd. The principal symbol of this operator is easily seen to coincide with P. Because of the Hodge decomposition quantum ergodicity for the operator p with 0 < p < n can never hold in the strict sense. An eigenform φ of p with nonzero eigenvalue can be decomposed uniquely as φ = dφ− + δφ+ ,
(34)
where φ− is an eigenform of p−1 and φ+ is an eigenform of p+1 . Hence, part of the spectrum of p comes from part of the spectrum of p−1 . The other part can be obtained by solving the system p φ = λφ, δφ = 0. We will show that in certain situations this system is quantum ergodic.
(35)
High Energy Limits and Frame Flows
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Theorem 5.2. Assume 0 < p < n, n ≥ 3 and suppose that φk is an orthonormal sequence of eigen- p-forms satisfying p φk = λ k φk , δφk = 0, such that the φk span ker(δ) and λk ∞. Suppose that p = n−1 2 . Then, if the 2 min( p, n − p)-frame flow is ergodic, the system is quantum ergodic in the sense that 1 lim |φk , Aφk − ωt (σ A )| = 0, N →∞ N k≤N
for all A ∈ DO0cl (X ; C X ). In particular there is a density one subsequence φk such that p
lim φk , Aφk = ωt (σ A ), for all A ∈ DO0cl (X ; C X ). p
k→∞
Proof. The proof is along the same lines as the proof of Theorem 4.4. Suppose that A ∈ p DO0cl (X ; C X ) with ωt (A) = 0 and A∗ = A. Let F be the operator ( p |ker( p )⊥ )−1 δd, then we have σ F = P. Since Fφk = φk we conclude that φk , Aφk = φk , F AFφk . Hence, we may assume without loss of generality that A = F AF and hence, σ A = Pσ A P. Again we have the heat asymptotics Tr(Ae− p t ) ∼ C( p, n)ωtr (σ A )t −n/2 ,
(36)
and from Lemma A.1 ωt (|σ AT |2 ) → 0 as T → ∞. Together these statements with the Karamata’s Tauberian theorem imply Quantum ergodicity in the stated form exactly in the same way as in the proof of Theorem 4.4. In the above proof it was necessary to exclude the case p = n−1 2 because in this p n−1 case the representation of S O(n − 1) on C is not irreducible but splits into a direct sum of two irreducible representations. One reason for this is the existence of an involution defined by the Hodge star operator which commutes with the S O(n − 1) action. This actually causes the state ωt to be non-ergodic in case p = n−1 2 . To see this let P± ∈ C(T1∗ X, π ∗ End( p X )) defined by P± (ξ )v := p−1
1 1 ± i p i(ξ )∗ i(ξ )ξ ∧ v, 2
p+1
(37)
where ∗ : C X → C X is the Hodge star operator. Then P± are orthogonal projecp tions C(T1∗ X, π ∗ End( C X )) which commute in addition with P. Therefore, we have P = P+ + P− and 1 (ω+ + ω− ), 2 ω+ (a) = 2ωt (P+ a), ω+ (a) = 2ωt (P− a). ωt =
Again these states are invariant. The same proof as for Theorem 5.1 now gives
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D. Jakobson, A. Strohmaier
Theorem 5.3. Suppose that n > 1 is odd. Let p = n−1 2 . If the (n − 1)-frame flow is p ergodic then ω± are ergodic states with respect to βt on C(T1∗ X, π ∗ C X ) and the p systems C(T1∗ X, π ∗ C X ), ω± are βt -abelian. It is amusing that also in case p = n−1 2 the non-ergodicity of the state ωt is related to the existence of a “quantum symmetry”, i.e. of a pseudodifferential operator which commutes with p and leaves the kernel of Rg(δ) invariant. Namely, for p = n−1 2 the −1/2 p+1 operator i p δ∗ is a selfadjoint involution on Rg(δ) whose principal symbol is precisely i p i(ξ )∗. Hence, P± are the principal symbols of the projections to the ±1 eigenspaces of this involution. The proof of Theorem 5.2 gives Theorem 5.4. Suppose that n is odd and 2 p = n − 1. Suppose that φk is an orthonormal sequence of eigen- p-forms satisfying p φk = λ k φk , δφk = 0, i p+1 δ ∗ φk = ± λk φk , −1/2
such that the φk span Ran(δ ± i p+1 p δ ∗ δ) and with λk ∞. Then, if the (n − 1)frame flow is ergodic, the system is quantum ergodic in the sense that lim
N →∞
1 |φk , Aφk − ω± (σ A (ξ ))| = 0, N k≤N
for all A ∈ DO0cl (X ; C X ). In particular there is a density one subsequence φk such that p
lim φk , Aφk = ω± (σ (A)), for all A ∈ DO0cl (X ; C X ). p
k→∞
Appendix A. Ergodicity of States for Noncommutative Classical Systems Let A be a unital C ∗ -algebra. A state ω over A is a positive linear functional ω : A → C with ω(1) = 1. The set of states E A of A is a convex weakly-∗-compact subset of A∗ and its extreme points are the pure states PA . Each state ω gives via the GNSconstruction rise to a representation πω : A → L(Hω ) with a cyclic vector ω such that ω(a) = ω , πω (a)ω . Up to equivalence the triple (πω , Hω , ω ) is uniquely determined by its properties and we refer to it as the GNS triple. In the following let αt be a strongly continuous group of ∗-automorphisms of A. The αt set of invariant states E A is again a weakly-∗-compact subset of A∗ . The extreme points αt in E A are called ergodic states. Hence, an invariant state ω is ergodic if it cannot be written as a convex linear combination of two other invariant states. If ω is an invariant state the group αt can be uniquely implemented by a strongly continuous unitary group U (t) on the GNS-Hilbert space Hω such that U (t)ω = ω and U (t)∗ πω (a)U (t) = πω (αt (a)). Let E ω be the orthogonal projection onto the space of αt -invariant vectors in Hω . Then the pair (A, ω) is called R-abelian if all operators in E ω πω (A)E ω commute pairwise. If we look at the following conditions:
High Energy Limits and Frame Flows
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(1) E ω has rank one, αt (2) ω is ergodic for αt , i.e. ω ∈ E(E A ), (3) {πω (A) ∪ U (t)} is irreducible on Hω , then it is known that (1) ⇒ (2) ⇔ (3). If, moreover, either (A, ω) is R-abelian, or ω is separating for πω (A)
, then all three conditions are equivalent (see [BR79], Prop. 4.3.7, Th. 4.3.17, Th 4.3.20). Now let E → X be a hermitian complex vector bundle over a compact Hausdorff space X . Then A = C(X ; End(E)) is a unital C ∗ -algebra. Suppose that μ is some finite Borel measure with μ(X ) = 1 and let P ∈ C(X ; End(E)) be a non-trivial orthogonal projection onto a subbundle, i.e. P has constant rank k. Then ω(a) := 1 Tr dμ(x) is a state over A. The GNS triple can be calculated (P(x)a(x)P(x)) k X explicitly and is given by Hω = L 2 (X ; End(E))P with scalar product 1 ∗ a, b = k X Tr(a (x)b(x))dμ(x) and ω = P. The action of A on Hω is by multiplication from the left. The von Neumann closure of πω (A) is given by πω (A)
= L ∞ (X ; End(E)). The commutant πω (A) of πω (A) can be identified with the opposite algebra of P L ∞ (X ; End(E))P which acts on L 2 (X ; End(E))P by right multiplication. Note that ω is separating for πω (A)
iff P = Id. Now any continuous geometric flow on E determines a continuous geometric flow on End(E). If the hermitian structure is preserved by the flow, this gives rise to a strongly continuous 1-parameter group αt on A. If P and μ are invariant under the flow, then ω is an invariant state. If all invariant vectors in L 2 (X ; End(E))P are of the form c P with c ∈ C, then by the above ω is ergodic. In this case E ω πω (A)E ω is just a multiplication by a number and the system is R-abelian. If a state ω is majorized by ω then there exists a positive element A ∈ P L ∞ (X ; End(E))P such that such that ω (a) = ω(a A). Hence, if P is up to a constant the only invariant element in P L ∞ (X ; End(E))P, then ω is ergodic. Now denote by ω the restriction of the state ω to the subalgebra B = PC(X ; End(E))P. If E denotes the subbundle onto which P projects then clearly B = C(X ; End(E )) and ω becomes the tracial state on B. Ergodicity of the state is equivalent to the condition that all invariant elements in P L ∞ (X ; End(E))P = L ∞ (X ; End(E )) are multiples of P. Therefore, ergodicity of ω is equivalent to the ergodicity of ω. Since ω is separating for πω
(B) ergodicity of ω is equivalent to E ω
having rank one. As a consequence we get Lemma A.1. If A ∈ P L ∞ (X, E)P such that ω(A) = 0, then ergodicity of ω implies that lim ω(|A T |2 ) = 0,
T →∞
(38)
where 1 AT = T
T αt (A)dt. 0
Proof. Since A = P A P we also have A2T = P A2T P. By the above E ω has rank one and its range is spanned by P. Now by the von Neumann ergodic theorem A T converges to E ω A in P L 2 (X, E)P. But since ω(A) = 0 we have E ω A = 0 and therefore A T converges to 0 in P L 2 (X, E)P. By definition this means that ω(|A T |2 ) converges to 0.
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Acknowledgements. The authors thank J. Bolte for bringing the problem to their attention and for many useful discussions. The authors would like to thank M. Brin, D. Dolgopyat, P. Gerard, N. Kamran, M. Lesch, W. Müller, M. Pollicott, I. Polterovich, P. Sarnak, A. Shnirelman, R. Schubert, J. Stix, J. Toth and S. Zelditch for useful discussions. The authors would also like to thank the anonymous referee whose comments helped improve the presentation of the results. The first author would like to thank the Department of Theoretical Physics at the University of Ulm for their hospitality during his visit. This paper was started while the first author visited Max Planck Institute for Mathematics in Bonn, and completed while he was visiting IHES; their hospitality is greatly appreciated. The second author would like to thank McGill University and CRM Analysis Laboratory for the hospitality during his Montréal visit.
References [Arn61] [BGV92] [BoK98] [BoK99] [Bo101] [BoG04] [BoG04.2] [BR79] [Br75] [Br76] [Br82] [BrG80] [BrK84] [BrP74] [BuO104] [BuP03] [CV85] [D82] [DG75] [DH72] [EW96] [Fri] [FuHa91]
Arnold, V.I.: Some remarks on flows of line elements and frames. Dokl. Akad. Nauk SSSR 138, 255–257 (1961) Berline, N., Getzler, E., Vergne, M.: Heat kernels and Dirac operators, Volume 298 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Berlin: Springer-Verlag, 1992 Bolte, J., Keppeler, S.: Semiclassical time evolution and trace formula for relativistic spin-1/2 particles. Phys. Rev. Lett. 81, no. 10, 1987–1991 (1998) Bolte, J., Keppeler, S.: A semiclassical approach to the Dirac equation. Ann. Physics 274, no. 1, 125–162 (1999) Bolte, J.: Semiclassical expectation values for relativistic particles with spin 1/2. In: Invited papers dedicated to Martin C. Gutzwiller, Part III. Found. Phys. 31, no. 2, 423–444 (2001) Bolte, J., Glaser, R.: Zitterbewegung and semiclassical observables for the Dirac equation. J. Phys. A 37, no. 24, 6359–6373 (2004) Bolte, J., Glaser, R.: A semiclassical Egorov theorem and quantum ergodicity for matrix valued operators. Commun. Math. Phys. 247, no. 2, 391–419 (2004) Bratteli, O., Robinson, D.W.: Operator algebras and quantum statistical mechanics. Vol. 1. C ∗ - and W ∗ -algebras, algebras, symmetry groups, decomposition of states, Texts and Monographs in Physics, New York: Springer-Verlag, 1979 Brin, M.: Topological transitivity of one class of dynamical systems and flows of frames on manifolds of negative curvature. Funct. Anal. Appl. 9, 8–16 (1975) Brin, M.: The topology of group extensions of anosov systems. Math. Notes 18, 858– 864 (1976) Brin, M.: Ergodic theory of frame flows. In: Ergodic Theory and Dynamical Systems II, Proc. Spec. Year, Maryland 1979-80, Progr. Math. 21, Boston: Birkhäuser, 1982, pp. 163–183 Brin, M., Gromov, M.: On the ergodicity of frame flows. Inv. Math. 60, 1–7 (1980) Brin, M., Karcher, H.: Frame flows on manifolds with pinched negative curvature. Comp. Math. 52, 275–297 (1984) Brin, M.I., Pesin, Ja.B.: Partially hyperbolic dynamical systems. Izv. Akad. Nauk SSSR Ser. Mat. 38, 170–212 (1974) Bunke, U., Olbrich, M.: Quantum ergodicity for vector bundles. Acta Applicandae Mathematicae. 90, 19–41 (2006) Burns, K., Pollicott, M.: Stable ergodicity and frame flows. Geom. Dedicata 98, 189– 210 (2003) Colin de Verdière, M. Y.: Ergodicité et fonctions propres du laplacien. Commun. Math. Phys. 102, 497–502 (1985) Dencker, N.: On the propagation of polarization sets for systems of real principal type. J. Funct. Anal. 46, 351–372 (1982) Duistermaat, J.J., Guillemin, V.W.: The spectrum of positive elliptic operators and periodic bicharacteristics. Invent. Math. 29(1), 39–79 (1975) Duistermaat, J.J., Hörmander, L.: Fourier integral operators ii. Acta Mathematica 128, 183– 269 (1972) Emmrich, G., Weinstein, A.: Geometry of the transport equation in multicomponent WKB approximations. Commun. Math. Phys. 176, no. 3, 701–711 (1996) Friedrich, T.: Dirac operators in Riemannian geometry, Volume 25 of Graduate Studies in Mathematics. Providence, RI: Amer. Math. Soc. 2000 Translated from the 1997 German original by Andreas Nestke. Fulton, W., Harris, J.: Representation theory, Volume 129 of Graduate Texts in Mathematics. New York: Springer-Verlag, 1991
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[GrSee95] [GMMP97] [LM89] [San99] [See67] [Shn74] [Shn93] [Tay81] [Zel87] [Zel96]
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Grubb, G., Seeley, R.: Weakly parametric pseudodifferential operators and Atiyah-PatodiSinger boundary problems. Invent. Math. 121, 481–529 (1995) Gérard, P., Markowich, P., Mauser, N., Poupaud, F.: Homogenization limits and Wigner transforms. Comm. Pure Appl. Math. 50, no. 4, 323–379 (1997). Erratum: Comm. Pure Appl. Math. 53, no. 2, 280–281 (2000) Lawson, H. B., Jr. Michelsohn, M.: Spin geometry. Volume 38 of Princeton Mathematical Series. Princeton, NJ: Princeton University Press, 1989 Sandoval, M.R.: Wave-trace asymptotics for operators of Dirac type. Comm. PDE 24, no. 9–10, 1903–1944 (1999) Seeley, R.T.: Complex powers of an elliptic operator. In: Singular Integrals (Proc. Sympos. Pure Math., Chicago, Ill., 1966), Providence, R.I.: Amer. Math. Soc., 1967, pp 288–307 Shnirelman, A.I.: Ergodic properties of eigenfunctions. (Russian). Uspehi Mat. Nauk 29, 181– 182 (1974) Shnirelman, A.I.: On the asymptotic properties of eigenfunctions in the regions of chaotic motion. In: Lazutkin, V., KAM theory and semiclassical approximations to eigenfunctions. Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 24, Berlin: Springer-Verlag, 1993 Taylor, M.E.: Pseudodifferential operators, Volume 34 of Princeton Mathematical Series. Princeton, N.J.: Princeton University Press, 1981 Zelditch, S.: Uniform distribution of eigenfunctions on compact hyperbolic surfaces. Duke Math. J. 55, 919–941 (1987) Zelditch, S.: Quantum ergodicity of C ∗ dynamical systems. Commun. Math. Phys. 177, 507– 528 (1996)
Communicated by B. Simon
Commun. Math. Phys. 270, 835–847 (2007) Digital Object Identifier (DOI) 10.1007/s00220-006-0184-0
Communications in
Mathematical Physics
Mirror Extensions of Local Nets Feng Xu Department of Mathematics, University of California at Riverside, Riverside, CA 92521, USA. E-mail:
[email protected] Received: 19 June 2006 / Accepted: 19 June 2006 Published online: 9 January 2007 – © Springer-Verlag 2006
Abstract: In this paper we prove a general theorem on the extensions of local nets which was inspired by recent examples of exotic extensions for Virasoro nets with central charge less than one and earlier work on cosets and conformal inclusions. When applying the theorem to conformal inclusions and diagonal inclusions, we obtain infinite series of new examples of completely rational nets. 1. Introduction Cosets, orbifolds and simple current extensions are efficient ways of constructing conformal field theories (CFT), and in fact they are so efficient that many believe that all rational CFT have been constructed (cf. p. 356 of [18]).1 This also motivates the following statement of E. Witten from p. 356 of [27], that “it is possible to conjecture that arbitrary rational conformal field theories in two dimensions can be derived from Chern-Simon theories in three dimensions” . In this paper we use the method of operator algebras, especially subfactor theory pioneered by Vaughan F.R. Jones (cf. [7]), to produce infinitely many new rational CFT which do not seem to come from cosets, orbifolds, simple current extensions or any combinations of them, and they do not come from CS theories in the usual way. Our results show that one has to modify the above statements . To describe our main result, let A be a completely rational net (cf. Definition 2.1). An irreducible extension of A is a net B such that A ⊂ B is an irreducible subnet (cf. Definition 2.2). The main result (cf. Th. 3.8) in this paper can be briefly described as follows: Let A ⊂ B be a normal subnet (cf. Definition 3.5 ) and A˜ the coset (cf. Definition 3.1). Note that A˜ depends on both A and B. Suppose that A ⊂ C is an irreducible extension. Then under certain conditions as described in Th. 3.8, there is an irreducible extension C˜ of A˜ which is called the “mirror” of A ⊂ C. Supported in part by NSF. 1 Also see p. 256 of [20] for a more recent appearance of similar statement.
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That A˜ ⊂ C˜ is called the “mirror” of A ⊂ C is explained after Th. 3.8. Roughly speaking the reason is that the link invariants labeled by the spectrum of A ⊂ C are the invariants labeled by the spectrum of A˜ ⊂ C˜ corresponding to the mirror image of the link. The motivations for this work come from the papers [9, 16, 23, 8, 28] and [29]. In [9] two exceptional extensions2 of Virasoro net with central charge c < 1 were constructed based on [16] and known A− D− E classifications of modular invariants, and in Sect. 4.2 of [28] it was observed that two series of conformal inclusions were closely related based on the level-rank duality as formulated in [29]. We will see that both cases are examples of mirror extensions in Sect. 4.3. We also note that in [23] certain two dimensional local extensions were constructed based on [16]. We will see that the results of [16] and [29] play an important role in the proof of Th. 3.8. The questions at the end of [8] also lead us to develop the general method in this paper. Perhaps the simplest example of a mirror extension is the mirror of conformal inclusion SU (2)10 ⊂ Spin(5). This is an extension C˜ of SU (10)2 , whose spectrum is L(20 ) + L(3 + 7 ) (cf. Definition 2.2 and Subsect. 4.3 ) based on level-rank duality. We note that the net C˜ is not the net associated to any affine Kac-Moody algebras. To the best of our knowledge, this and most of the infinite series of exotic extensions in Subsect. 4.3 have not appeared before, and hence we obtain a large class of new completely rational net. Based on the close relation of nets and vertex operator algebras (VOA) (cf. [30]), this also leads to a conjecture about the existence of a large class of new rational vertex operator algebras (VOA) (cf. the end of Subsect. 4.3). For an example, in the case of the simplest example above, the conjecture implies the existence of a rational VOA containing the affine VOA based on SU (10)2 with spectrum L(20 ) + L(3 + 7 ). We note that this is not an example of simple current extensions since the index of the √ inclusion (cf. Definition 2.2) is not an integer (it is 3 + 3 by Sect. 4.1 of [31]). The rest of this paper is organized as follows: In Sect. 2 we recall the basic notions about sectors, nets and examples to set up notations. In Sect. 3 we prove our main result on mirror extensions Th. 3.8. Using conformal inclusions, diagonal embeddings, and applying Th. 3.8 we obtain a large class of new completely rational nets in Sect. 4. 2. Preliminaries 2.1. Preliminaries on sectors. Given an infinite factor M, the sectors of M are given by Sect(M) = End(M)/Inn(M), namely Sect(M) is the quotient of the semigroup of the endomorphisms of M modulo the equivalence relation: ρ, ρ ∈ End(M), ρ ∼ ρ iff there is a unitary u ∈ M such that ρ (x) = uρ(x)u ∗ for all x ∈ M. Sect(M) is a ∗ -semiring (there are an addition, a product and an involution ρ → ρ) ¯ equivalent to the Connes correspondences (bimodules) on M up to unitary equivalence. If ρ is an element of End(M) we shall denote by [ρ] its class in Sect(M). We define Hom(ρ, ρ ) between the objects ρ, ρ ∈ End(M) by Hom(ρ, ρ ) ≡ {a ∈ M : aρ(x) = ρ (x)a ∀x ∈ M}.
2 One of the extensions is identified as a coset in [13].
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We use λ, μ to denote the dimension of Hom(λ, μ); it can be ∞, but it is finite if λ, μ have finite index. See [7] for the definition of the index for type I I1 case which initiated the subject and [19] for the definition of index in general. Also see §2.3 [11] for expositions. λ, μ depends only on [λ] and [μ]. Moreover we have if ν has finite index, then νλ, μ = λ, νμ , ¯
λν, μ = λ, μ¯ν which follows from Frobenius duality. μ is a subsector of λ if there is an isometry v ∈ M such that μ(x) = v ∗ λ(x)v, ∀x ∈ M. We will also use the following notation: if μ is a subsector of λ, we will write as μ ≺ λ or λ μ. A sector is said to be irreducible if it has only one subsector. 2.2. Local nets. By an interval of the circle we mean an open connected non-empty subset I of S 1 such that the interior of its complement I is not empty. We denote by I the family of all intervals of S 1 . A net A of von Neumann algebras on S 1 is a map I ∈ I → A(I ) ⊂ B(H) from I to von Neumann algebras on a fixed separable Hilbert space H that satisfies: A. Isotony. If I1 ⊂ I2 belong to I, then A(I1 ) ⊂ A(I2 ). If E ⊂ S 1 is any region, we shall put A(E) ≡ E⊃I ∈I A(I ) with A(E) = C if E has empty interior (the symbol ∨ denotes the von Neumann algebra generated). The net A is called local if it satisfies: B. Locality. If I1 , I2 ∈ I and I1 ∩ I2 = ∅ then [A(I1 ), A(I2 )] = {0}, where brackets denote the commutator. The net A is called Möbius covariant if in addition it satisfies the following properties C,D,E,F: C. Möbius covariance. There exists a non-trivial strongly continuous unitary representation U of the Möbius group Möb (isomorphic to P SU (1, 1)) on H such that U (g)A(I )U (g)∗ = A(g I ), g ∈ Möb, I ∈ I. D. Positivity of the energy. The generator of the one-parameter rotation subgroup of U (conformal Hamiltonian), denoted by L 0 in the following, is positive. E. Existence of the vacuum. There exists a unit U -invariant vector ∈ H (vacuum vector), and is cyclic for the von Neumann algebra I ∈I A(I ). By the Reeh-Schlieder theorem is cyclic and separating for every fixed A(I ). The modular objects associated with (A(I ), ) have a geometric meaning itI = U ( I (2π t)),
J I = U (r I ) .
Here I is a canonical one-parameter subgroup of Möb and U (r I ) is an antiunitary acting geometrically on A as an reflection r I on S 1 . This implies Haag duality: A(I ) = A(I ), where
I
is the interior of
S1
I.
I ∈I,
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F. Irreducibility. I ∈I A(I ) = B(H). Indeed A is irreducible iff is the unique U -invariant vector (up to scalar multiples). Also A is irreducible iff the local von Neumann algebras A(I ) are factors. In this case they are either C or III1 -factors with separable preduals in Connes classification of type III factors. By a conformal net (or diffeomorphism covariant net) A we shall mean a Möbius covariant net such that the following holds: G. Conformal covariance. There exists a projective unitary representation U of Diff(S 1 ) on H extending the unitary representation of Möb such that for all I ∈ I we have U (ϕ)A(I )U (ϕ)∗ = A(ϕ.I ), ϕ ∈ Diff(S 1 ), U (ϕ)xU (ϕ)∗ = x, x ∈ A(I ), ϕ ∈ Diff(I ), where Diff(S 1 ) denotes the group of smooth, positively oriented diffeomorphism of S 1 and Diff(I ) the subgroup of diffeomorphisms g such that ϕ(z) = z for all z ∈ I . A (DHR) representation π of A on a Hilbert space H is a map I ∈ I → π I that associates to each I a normal representation of A(I ) on B(H) such that π I˜ A(I ) = π I ,
I ⊂ I˜,
I, I˜ ⊂ I .
π is said to be Möbius (resp. diffeomorphism) covariant if there is a projective unitary representation Uπ of Möb (resp. Diff(S 1 )) on H such that πg I (U (g)xU (g)∗ ) = Uπ (g)π I (x)Uπ (g)∗ for all I ∈ I, x ∈ A(I ) and g ∈ Möb (resp. g ∈ Diff(S 1 )). By definition the irreducible conformal net is in fact an irreducible representation of itself and we will call this representation the vacuum representation. Let G be a simply connected compact Lie group. By Th. 3.2 of [3], the vacuum positive energy representation of the loop group LG (cf. [21]) at level k gives rise to an irreducible conformal net denoted by AG k . By Th. 3.3 of [3], every irreducible positive energy representation of the loop group LG at level k gives rise to an irreducible covariant representation of AG k . Given an interval I and a representation π of A, there is an endomorphism of A localized in I equivalent to π ; namely ρ is a representation of A on the vacuum Hilbert space H, unitarily equivalent to π , such that ρ I = id A(I ). We now define the statistics. Given the endomorphism ρ of A localized in I ∈ I, choose an equivalent endomorphism ρ0 localized in an interval I0 ∈ I with I¯0 ∩ I¯ = ∅ and let u be a local intertwiner in Hom(ρ, ρ0 ), namely u ∈ Hom(ρ I˜ , ρ0, I˜ ) with I0 following clockwise I , inside I˜, which is an interval containing both I and I0 . The statistics operator (ρ, ρ) := u ∗ ρ(u) = u ∗ ρ I˜ (u) belongs to Hom(ρ 2˜ , ρ 2˜ ). We I I will call (ρ, ρ) the positive or right braiding and ˜ (ρ, ρ) := (ρ, ρ)∗ the negative or left braiding. Next we recall some definitions from [10] . Recall that I denotes the set of intervals of S 1 . Let I1 , I2 ∈ I. We say that I1 , I2 are disjoint if I¯1 ∩ I¯2 = ∅, where I¯ is the closure of I in S 1 . When I1 , I2 are disjoint, I1 ∪ I2 is called a 1-disconnected interval in [30]. Denote by I2 the set of unions of disjoint 2 elements in I. Let A be an irreducible Möbius covariant net. For E = I1 ∪ I2 ∈ I2 , let I3 ∪ I4 be the interior of the complement of I1 ∪ I2 in S 1 , where I3 , I4 are disjoint intervals. Let ˆ := (A(I3 ) ∨ A(I4 )) . A(E) := A(I1 ) ∨ A(I2 ), A(E)
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ˆ Note that A(E) ⊂ A(E). Recall that a net A is split if A(I1 ) ∨ A(I2 ) is naturally isomorphic to the tensor product of von Neumann algebras A(I1 ) ⊗ A(I2 ) for any disjoint intervals I1 , I2 ∈ I. A is strongly additive if A(I1 ) ∨ A(I2 ) = A(I ), where I1 ∪ I2 is obtained by removing an interior point from I . Definition 2.1 [10, 17]. A conformal net A is said to be completely rational if A is ˆ split, and the index [A(E) : A(E)] is finite for some E ∈ I2 . The value of the index ˆ [A(E) : A(E)] (it is independent of E by Prop. 5 of [10]) is denoted by μA and is called the μ-index of A. Note that, by recent results in [17], every irreducible, split, local conformal net with finite μ-index is automatically strongly additive. Hence we have modified the definition in [10] by dropping the strong additivity requirement in the above definition. Also note that if A is completely rational, then A has only finitely many irreducible covariant representations by [10]. Let B be a Möbius net. By a Möbius subnet (cf. [15]) we shall mean a map I ∈ I → A(I ) ⊂ B(I ) that associates to each interval I ∈ I a von Neumann subalgebra A(I ) of B(I ), which is isotonic A(I1 ) ⊂ A(I2 ), I1 ⊂ I2 , and Möbius covariant with respect to the representation U , namely U (g)A(I )U (g)∗ = A(g.I ) for all g ∈ Möb and I ∈ I. Note that by Lemma 13 of [15] for each I ∈ I there exists a conditional expectation E I : B(I ) → A(I ) such that E preserves the vector state given by the vacuum of A. Definition 2.2. Let A be a Möbius covariant net. A Möbius covariant net B on a Hilbert space H is an extension of A if there is a DHR representation π of A on H such that π(A) ⊂ B is a Möbius subnet. The extension is irreducible if π(A(I )) ∩ B(I ) = C for some (and hence all) interval I , and is of finite index if π(A(I )) ⊂ B(I ) has finite index for some (and hence all) interval I . The index will be called the index of the inclusion π(A) ⊂ B. If π as as representation of A decomposes as [π ] = λ m λ [λ], where m λ are non-negative integers and λ are irreducible DHR representations of A, we say that [π ] = λ m λ [λ] is the spectrum of the extension. For simplicity we will write π(A) ⊂ B simply as A ⊂ B. Lemma 2.3. If A is completely rational, and a Möbius covariant net B is an irreducible extension of A. Then A ⊂ B has finite index and B is completely rational. Proof. A ⊂ B has finite index follows from Prop. 2.3 of [9], and it follows by Prop. 24 of [10] that B is completely rational. The following is essentially Th. 4.9 of [16] (cf. Sect. 2.4 of [9]) which is also used in Sect. 4.2 of [9]:
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Proposition 2.4. Let A be a Möbius covariant net, ρ a DHR representation of A localized on a fixed I0 with finite statistics, which contains id with multiplicity one, i.e., there is (unique up to a phase) isometry w ∈ Hom(id, ρ). Then there is a Möbius covariant net B which is an irreducible extension of A if and only if there is an isometry w1 ∈ Hom(ρ, ρ 2 ) which solves the following equations: w1∗ w = w1∗ ρ(w) ∈ R+ , w1 w1 = ρ(w1 )w1 , (ρ, ρ)w1 = w1 .
(1) (2) (3)
Proof. As in the proof of Th. 4.9 in [16], we just have to check the “if” part, and the only additional thing we need to check is that B is Möbius covariant. Since ρ is DHR with finite statistics, it follows that ρ is Möbius covariant by [5], and it follows by the formula in Cor. 19 of [15] that B is Möbius covariant. 2.3. Induction. Let B be a Möbius covariant net and A a subnet. We assume that A is strongly additive and A ⊂ B has finite index. Fix an interval I0 ∈ I and canonical endomorphism (cf. [16]) γ associated with A(I0 ) ⊂ B(I0 ). Given a DHR endomorphism ρ of B localized in I0 , the α-induction αρ of ρ is the endomorphism of B(I0 ) given by αρ ≡ γ −1 · Adε(ρ, λ) · ρ · γ , where ε denotes the right braiding (cf. Cor. 3.2 of [2]). In [31] a slightly different endomorphism was introduced and the relation between the two was given in Sect. 2.1 of [29]. Note that Hom(αλ , αμ ) =: {x ∈ B(I0 )|xαλ (y) = αμ (y)x, ∀y ∈ B(I0 )} and Hom(λ, μ) =: {x ∈ A(I0 )|xλ(y) = μ(y)x, ∀y ∈ A(I0 )}. We have the following lemma which will be used in the proof of Prop. 3.7. The proof is also implicitly contained in [31]. Lemma 2.5. Assume that λ, μ have finite index and λ, μ = αλ , αμ . Then Hom(αλ , αμ ) = Hom(λ, μ) ⊂ A(I0 ). Proof. By Lemma 3.6 of [2] Hom(λ, μ) ⊂ Hom(αλ , αμ ), and by assumption
λ, μ = αλ , αμ < ∞, hence Hom(αλ , αμ ) = Hom(λ, μ) ⊂ A(I0 ). 3. Mirror Extensions 3.1. Coset construction. Let B be a completely rational net and A ⊂ B be a subnet which is also completely rational. ˜ ) := A(I ) ∩ B(I ), ∀I ∈ I. Definition 3.1. Define a subnet A˜ ⊂ B by A(I
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We note that since A is completely rational, it is strongly additive and so we have ˜ ) = (∨ J ∈I A(J )) ∩ B(I ), ∀I ∈ I. The following lemma then follows directly from A(I the definition: ˜ ) is an irreducible Möbius Lemma 3.2. The restriction of A˜ on the Hilbert space ∨ I A(I covariant net. The net A˜ as in Lemma 3.2 will be called the coset of A ⊂ B. See [30] for a class of cosets from Loop groups. The following definition generalizes the definition in §3 of [30]: ˜ ) ∨ A(I ) ⊂ B(I ) has finite Definition 3.3. A ⊂ B is called cofinite if the inclusion A(I index for some interval I . Proposition 3.4. Let B be completely rational, and let A ⊂ B be a Möbius subnet which is also completely rational. Then A ⊂ B is cofinite if and only if A˜ is completely rational. ˜ ) ∨ A(I ) is naturally isomorphic to A(I ˜ ) ⊗ A(I ), and by Proof. Note that by [24] A(I ˜ the geometric nature of the modular group A(I ) is a factor. It follows that the inclusion ˜ ) ∨ A(I ) ⊂ B(I ) is irreducible. The “if” part now follows from Prop. 2.3 of [9], A(I and the “only if” part follows from Prop. 24 of [10]. Let B be completely rational, and let A ⊂ B be a Möbius subnet which is also completely rational. Assume that A ⊂ B is cofinite. We will use σi , σ j , . . . (resp. λ, μ . . .) to label irreducible DHR representations of B (resp. A) localized on a fixed interval I0 . Since A˜ is completely rational by Prop. 3.4, A˜ ⊗ A is completely rational, and so every irreducible DHR representation σi of B, when restricting to A˜ ⊗ A, decomposes as a direct sum of representations of A˜ ⊗ A of the form (i, λ) ⊗ λ by Lemma 27 of [10]. Here (i, λ) is a DHR representation of A˜ which may not be irreducible and we use the tensor notation (i, λ) ⊗ λ to represent a DHR representation of A˜ ⊗ A which is localized on I0 and defined by ˜ 0 ) ⊗ A(I0 ). (i, λ) ⊗ λ(x1 ⊗ x2 ) = (i, λ)(x1 ) ⊗ λ(x2 ), ∀x1 ⊗ x2 ∈ A(I We will also identify A˜ and A as subnets of A˜ ⊗ A in the natural way. We note that when no confusion arises, we will use 1 to denote the vacuum representation of a net. ˜ ) ∩ B(I ) = A(I ) for some I. Definition 3.5. A Möbius subnet A ⊂ B is normal if A(I The following is implied by Lemma 3.4 of [22] (also cf. p. 797 of [32]): Lemma 3.6. Let B be completely rational, and let A ⊂ B be a Möbius subnet which is also completely rational. Assume that A ⊂ B is cofinite. Then the following conditions are equivalent: (1) A ⊂ B is normal; (2) (1, 1) is the vacuum representation of A˜ and (1, λ) contains (1, 1) if and only if λ = 1. The following proposition will play a key role in the proof of Th. 3.8:
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Proposition 3.7. Let B be completely rational, and let A ⊂ B be a Möbius subnet which is also completely rational. Assume that A ⊂ B is cofinite and normal. Then: ˜ (1) Let γ be the restriction of the vacuum representation of B to A ⊗ A. Then [γ ] = λ∈exp [(1, λ) ⊗ λ], where each (1, λ) is irreducible and the set exp is determined by γ ; (2) Let λ ∈ exp be as in (1), then [α(1,λ)⊗1 ] = [α1⊗λ¯ ], and [λ] → [α1⊗λ ] is a ring isomorphism where the α-induction is with respect to A˜ ⊗ A ⊂ B as in Subsect. 2.3; Moreover the set exp is closed under fusion; (3) Let [ρ] = λ∈exp m λ [λ], where m λ = m λ¯ ≥ 0, ∀λ, and [(1, ρ)] = λ∈exp m λ [(1, λ)]. Then there exists an unitary element Tρ ∈ Hom(α(1,ρ)⊗1 , α1⊗ρ ) such that ((1, ρ), (1, ρ))Tρ∗ α1⊗ρ (Tρ∗ ) = Tρ∗ α1⊗ρ (Tρ∗ )˜ (ρ, ρ); (4) Let ρ, (1, ρ) be as in (3). Then Hom(ρ n , ρ m ) = Hom(α1⊗ρ n , α1⊗ρ m ), Hom((1, ρ)n , (1, ρ)m ) = Hom(α(1,ρ)n ⊗1 , α(1,ρ)m ⊗1 ), ∀n, m ∈ N; (5) Let ρ, (1, ρ) be as in (3), let L be an oriented framed link in three sphere with ˜ n components and let L˜ be the mirror image of L (cf. [25] ). Then L(ρ, . . . , ρ) = L((1, ρ), . . . , (1, ρ)) = L(ρ, . . . , ρ), where L(i 1 , . . . i k ) is defined as before Lemma 1.7.4 of [29]. Proof. (1), (2) follow from Th. 3.6 of [22] (also cf. Prop. 4.3 of [30]). As for (3), note that [ρ] ¯ = [ρ], and by (2) there exists an unitary element Tρ : α(1,ρ)⊗1 → α1⊗ρ . The equation in (3) follows from (3) of Prop. 2.3.1 in [29]. (4) follows from (2) and Lemma 2.5. (5) follows from Th. B in [29]: we note that even though Th. B of [29] is stated for cosets coming from Loop groups, the proof of Th. B applies verbatim to our case. Theorem 3.8. Let B be completely rational, and let A ⊂ B be a Möbius subnet which is also completely rational. Assume that A ⊂ B is cofinite and normal, and let exp be as in (1) of Prop. 3.7. Assume that A ⊂ C is an irreducible Möbius extension of A with spectrum [ρ] = λ∈exp m λ [λ], m λ ≥ 0. Then there is an irreducible Möbius extension C˜ of A˜ with spectrum [(1, ρ)] = λ∈exp m λ [(1, λ)]. Moreover C˜ is completely rational. Proof. Since A ⊂ C is an irreducible Möbius extension of A with spectrum [ρ] = 2 λ∈exp m λ [λ], m λ ≥ 0, by Prop. 2.4 there exist w ∈ Hom(id, ρ), w1 ∈ Hom(ρ, ρ ) which verifies Eqs. (1),(2) and (3). Note that [ρ] = [ρ] ¯ by [16]. Let Tρ be the unitary as given by (3) of Prop. 3.7 and define w˜ := Tρ∗ w, w˜ 1 := Tρ∗ α1⊗ρ (Tρ∗ )w1 Tρ . Note that by definitions 2 w˜ ∈ Hom(1, α(1,ρ)⊗1 ) = Hom(1, (1, ρ)), w˜ 1 ∈ Hom(α(1,ρ)⊗1 , α(1,ρ)⊗1 )
= Hom((1, ρ), (1, ρ)2 ), where we have also used (4) of Prop. 3.7. To prove the theorem, by Prop. 3.7 and Lemma 2.3 it is enough to check Eqs. (1), (2) and (3) with ρ replaced by (1, ρ) in Prop. 2.4. First let us check Eq. (1): w˜ 1∗ w˜ = Tρ∗ w1∗ α1⊗ρ (Tρ )Tρ Tρ∗ w = Tρ∗ w1∗ α1⊗ρ (Tρ )w = Tρ∗ w1∗ wTρ = w1∗ w,
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where we have used w ∈ Hom(1, ρ) = Hom(1, α1⊗ρ ) and w1∗ w ∈ R+ . similarly w˜ 1∗ (1, ρ)(w) ˜ = w1∗ ρ(w) ∈ R+ . Next we have w˜ 1 w˜ 1 = Tρ∗ α1⊗ρ (Tρ∗ )w1 Tρ Tρ∗ α1⊗ρ (Tρ∗ )w1 Tρ = Tρ∗ α1⊗ρ (Tρ∗ )w1 α1⊗ρ (Tρ∗ )w1 Tρ 2 = Tρ∗ α1⊗ρ (Tρ∗ )α1⊗ρ (Tρ∗ )w12 Tρ 2 = Tρ∗ α1⊗ρ (Tρ∗ )α1⊗ρ (Tρ∗ )α1⊗ρ (w1 )w1 Tρ , 2 ) = Hom(ρ, ρ 2 ), w 2 = ρ(w )w and where we have used w1 ∈ Hom(α1⊗ρ , α1⊗ρ 1 1 1 α1⊗ρ (w1 ) = ρ(w1 ) by definition. On the other hand
(1, ρ)(w˜ 1 )w˜ 1 = α(1,ρ)⊗1 (Tρ∗ α1⊗ρ (Tρ∗ )w1 Tρ )Tρ∗ α1⊗ρ (Tρ∗ )w1 Tρ
= α(1,ρ)⊗1 (Tρ∗ α1⊗ρ (Tρ∗ ))α(1,ρ)⊗1 (w1 )α(1,ρ)⊗1 (Tρ )Tρ∗ α1⊗ρ (Tρ∗ )w1 Tρ = α(1,ρ)⊗1 (Tρ∗ α1⊗ρ (Tρ∗ ))α(1,ρ)⊗1 (w1 )Tρ∗ w1 Tρ = α(1,ρ)⊗1 (Tρ∗ α1⊗ρ (Tρ∗ ))Tρ∗ α1⊗ρ (w1 )w1 Tρ 2 = Tρ∗ α1⊗ρ (Tρ∗ )α1⊗ρ (Tρ∗ )α1⊗ρ (w1 )w1 Tρ .
This proves Eq. (2). By (3) of Prop. 3.7, we have ((1, ρ), (1, ρ))w˜ 1 = ((1, ρ), (1, ρ))Tρ∗ α1⊗ρ (Tρ∗ )w1 Tρ = Tρ∗ α1⊗ρ (Tρ∗ )˜ (ρ, ρ)w1 Tρ and since (ρ, ρ)w1 = w1 , (ρ, ˜ ρ) = (ρ, ρ)∗ , we have proved ((1, ρ), (1, ρ))w˜ 1 = w˜ 1 which is Eq. (3). Remark 3.9. Due to (5) of Prop. 3.7, the extension A˜ ⊂ C˜ as given in Th. 3.8 will be called the mirror or the conjugate of A ⊂ C. Remark 3.10. The same idea in the proof of Th. 3.8 can also be used to obtain possibly non-local extension A˜ ⊂ C˜ when B is not necessarily local, and we plan to discuss applications elsewhere. 4. Applications 4.1. Extensions from conformal inclusions. Let G = SU (N ). We denote LG as the group of smooth maps f : S 1 → G under pointwise multiplication. The diffeomorphism group of the circle DiffS 1 is naturally a subgroup of Aut(LG) with the action given by reparametrization. In particular the group of rotations RotS 1 U (1) acts on LG. We will be interested in the projective unitary representation π : LG → U (H ) that are both irreducible and have positive energy. This means that π should extend to LG Rot S 1 so that H = ⊕n≥0 H (n), where the H (n) are the eigenspace for the action of RotS 1 , i.e., rθ ξ = expinθ for θ ∈ H (n) and dim H (n) < ∞ with H (0) = 0. It follows from [21] that for fixed level K which is a positive integer, there is only a finite number of such irreducible representations indexed by the finite set K P++ = λ ∈ P | λ = λi i , λi ≥ 0 , λi ≤ K , i=1,··· ,N −1
i=1,··· ,n−1
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where P is the weight lattice of SU (N ) and i are the fundamental weights. We will K to use 0 to denote the trivial representation of SU (N ). We will use L(λ), λ ∈ P++ label the irreducible representations of the net A SU (N ) K . Let G ⊂ H be inclusions of compact simple Lie groups. LG ⊂ L H is called a conformal inclusion if the level 1 projective positive energy representations of L H decompose as a finite number of irreducible projective representations of LG. LG ⊂ L H is called a maximal conformal inclusion if there is no proper subgroup G of H containing G such that LG ⊂ LG is also a conformal inclusion. A list of maximal conformal inclusions can be found in [6]. Let H 0 be the vacuum representation of L H , i.e., the representation of L H associated with the trivial representation of H . Then H 0 decomposes as a direct sum of irreducible projective representation of LG at level K . K is called the Dynkin index of the conformal inclusion. We shall write the conformal inclusion as G K ⊂ H1 . Note that it follows from the definition that A H1 is an extension of AG K . We shall limit our consideration to the following conformal inclusions so we can use the results of [30, 31] which is based on [26], though most of the arguments apply to other cases as well under certain finite index assumptions: SU (2)10 ⊂ S O(5)1 , SU (2)28 ⊂ (G 2 )1 , SU (3)5 ⊂ SU (6)1 , SU (3)9 ⊂ (E 6 )1 , SU (3)21 ⊂ (E 7 )1 ; (A8 )1 ⊂ (E 8 )1 ; and four infinite series: N (N − 1) SU (N ) N −2 ⊂ SU , N ≥ 4; (4) 2 1 N (N + 1) SU (N ) N +2 ⊂ SU ; (5) 2 1 SU (N ) M
SU (N ) N ⊂ Spin(N 2 − 1)1 , N ≥ 2; × SU (M) N ⊂ SU (M N )1 .
(6) (7)
Note that except for Eq. (7), the above cover all the maximal conformal inclusions of the form SU (N ) ⊂ H with H being a simple group. 4.2. Two series of normal inclusions. Lemma 4.1. The subnets A SU (N ) M ⊂ A SU (N M)1 are normal and cofinite. The set exp N +M which belongs to the root lattice of SU (N ). as in (1) Prop. 3.7 is the elements of P++ Proof. By Lemma 3.3 of [30] the coset of A SU (N ) M ⊂ A SU (N M)1 can be identified with A SU (M) N , and exchanging M and N we conclude that A SU (N ) M ⊂ A SU (N M)1 is normal, and it is cofinite by the remark after lemma 3.3 in [30]. The statement about exp follows from the branching rules in [1]. Lemma 4.2. (1) If A ⊂ B and B ⊂ C are normal subnets, then A ⊂ C is also normal. (2) Let A SU (N ) K ⊂ A SU (N ) K 1 ⊗ A SU (N ) K 2 ⊗ · · · ⊗ A SU (N ) Kl be subnets corresponding to the diagonal embedding of SU (N ) in SU (N ) × · · · × SU (N ) (l tensor factors), where K = K 1 + · · · + K l . Then A SU (N ) K ⊂ A SU (N ) K 1 ⊗ A SU (N ) K 2 ⊗ · · · ⊗ A SU (N ) Kl K is normal and cofinite. Moreover, the set exp as in (1) Prop. 3.7 is the elements of P++ which belong to the root lattice of SU (N ).
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Proof. Ad (1): For a fixed interval I , we have A(I ) ∩ C(I ) ⊃ A(I ) ∩ B(I ) ∨ B(I ) ∩ C(I ), hence (A(I ) ∩ C(I )) ∩ C(I ) ⊂ (A(I ) ∩ B(I )) ∩ (B(I ) ∩ C(I )) ∩ C(I ) = (A(I ) ∩ B(I )) ∩ B(I ) = A(I ) by the normality of A ⊂ B and B ⊂ C. It follows that (A(I ) ∩ C(I )) ∩ C(I ) = A(I ) and A ⊂ C is normal. Ad(2): By Cor. 3.4 of [30] the subnet is cofinite, so we just have to show the normality. We first show (2) for the case l = 2. By Lemma 3.6 it is sufficient to show that a representation (1, λ) contains the vacuum representation (1, 1) if and only if λ = 1, and this follows from the remark on p. 38 of [30] before equation (**). The statement about exp follows by p. 194 of [12]. The general l case follows by induction from the case l = 2 and (1). 4.3. Mirror extensions corresponding to conformal inclusions. We will apply Th. 3.8 to the cases when A ⊂ B is one of the subnets in Lemma 4.1 and Lemma 4.2, and the extension A ⊂ C corresponds to a conformal inclusion. Let us start with the normal subnet A SU (2)10 ⊂ A SU (20)1 as in Lemma 4.1 and extension A SU (2)10 ⊂ A Spin(5)1 . Note that the spectrum of A SU (2)10 ⊂ A Spin(5)1 is L(0) + L(3), where L(0) is the vacuum representation and L(3) is the representation of SU (2)10 corresponding to the spin 3 representation of SU (2). By Th. 3.8 we conclude that there is a mirror extension A SU (10)2 ⊂ C˜ whose spectrum is L(20 ) + L(3 + 7 ): here 3 + 7 is the representation of SU (10)2 corresponding to “(1, 3)” in the notation of Th. 3.8, and is determined uniquely by the branching rules for A SU (2)10 ⊂ A SU (20)1 . If we choose instead A SU (2)10 ⊂ A SU (2) K 1 ⊗ A SU (2) K 2 ⊗ · · · ⊗ A SU (2) Kl , K 1 + · · · + K l = 10 as the normal subnet in the previous paragraph and apply Th. 3.8, we obtain mirror extensions of the coset A˜ SU (2)10 whose spectrum is determined by Th. 3.8. When l = 2, K 1 = 1, K 2 = 9, the mirror extension is the extension labeled by (A10 , E 6 ) in [9]. Similarly we can obtain mirror extensions corresponding to SU (2)28 , SU (3)5 , SU (3)9 and SU (3)21 cases. As in the case of SU (2)10 , we obtain a finite number of mirror extensions. We note that in the case of SU (2)28 and the normal extension is A SU (2)28 ⊂ A SU (2)27 ⊗ A SU (2)1 , the mirror extension is the extension labeled by (A28 , E 8 ) in [9]. Next we construct infinite series of mirror extensions by using Eqs. (4), (5), (6). We note that if we choose the normal subnet as in Lemma 4.1, by Th. 3.8 and Sect. 4.2 of [28], we conclude that the extensions corresponding to SU (N ) N +2 ⊂ SU (N ) N (N +1) SU (N + 2 2) N ⊂ SU (N ) (N +2)(N +1) are mirrors of each other, while the extension corresponding to 2 SU (N ) N ⊂ Spin(N ) N 2 −1 is the mirror of itself. To obtain new extensions, we choose the normal subnets as in Lemma 4.2. We note that the spectrum of the extensions corresponding to the conformal inclusions of (4), (5) and (6) are given by [14] and Sect. 4.2 of [12]. Applying Th. 3.8, we obtain three infinite series of mirror extensions of the diagonal cosets SU (N ) N +2 ⊂ SU (N ) K 1 × SU (N ) K 2 × · · · × SU (N ) K l , K 1 + · · · K l = N + 2, SU (N ) N −2 ⊂ SU (N ) K 1 × SU (N ) K 2 × · · · × SU (N ) K m , K 1 + · · · K m = N − 2, SU (N ) N ⊂ SU (N ) M1 × SU (N ) M2 × · · · × SU (N ) M p , M1 + · · · M p = N , corresponding to the conformal inclusions of (4), (5) and (6) respectively. The spectrum of these extensions are determined as in Th. 3.8.
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Note that in all the new mirror extensions A˜ ⊂ C˜ constructed in this subsection, A˜ corresponds to a Vertex Operator Algebra (VOA) (cf. [4]) denoted by A˜ voa , and in fact A˜ voa correspond to either affine Kac-Moody algebras or cosets. We will use the same notation as in Th. 3.8 to label the representations of A˜ voa . Based on the close relations between nets and VOAs as implied by Sect. 2.2 of [30], we conjecture that each C˜ corresponds to a completely rational VOA (cf.[33]) denoted by C˜voa which contains A˜ voa , such that the branching rules of C˜voa when restricted to A˜ voa is given by the spectrum as in Th. 3.8. Acknowledgement. The author would like to thank Professor Vaughan F. R. Jones for valuable discussions and motivation for the problem.
References 1. Altschüler, D., Bauer, M., Itzykson, C.: The branching rules of conformal embeddings. Commun. Math. Phys. 132, 349–364 (1990) 2. Böckenhauer, J., Evans, D.: Modular Invariants, Graphs and α-Induction for Nets of Subfactors I. Commun. Math. Phys. 197, 361–386 (1998) 3. Fröhlich, J., Gabbiani, F.: Operator algebras and Conformal field theory. Commun. Math. Phys. 155, 569– 640 (1993) 4. Frenkel, I., Zhu, Y.: Vertex operator algebras associated to representations of affine and Virasoro algebras. Duke Math. J. 66(1) 123–168 (1992) 5. Guido, D., Longo, R.: The conformal spin and statistics theorem. Commun. Math. Phys. 181, 11–35 (1996) 6. Goddard, P., Nahm, W., Olive, D.: Symmetric spaces, Sugawara’s energy momentum tensor in two dimensions and free fermions. Phys. Lett. B 160(1-3) 111–116 (1985) 7. Jones, V.F.R.: Index for subfactors. Invent. Math. 72, 1–25 (1983) 8. Jones, V.F.R., Xu, F.: Intersections of finite families of finite index subfators. Int. J. Math. 15(7), 717–733 (2004) 9. Kawahigashi, Y., Longo, R.: Classification of local conformal nets. Case c < 1. Ann. Math. (2) 160(2) 493–522 (2004) 10. Kawahigashi, Y., Longo, R., Müger, M.: Multi-interval subfactors and modularity of representations in conformal field theory. Commun. Math. Phys. 219, 631–669 (2001) 11. Kac, V., Longo, R., Xu, F.: Solitons in affine and permutation orbifolds. Commun. Math. Phys. 253(3), 726–764 (2005) 12. Kac, V.G., Wakimoto, M.: Modular and conformal invariance constraints in representation theory of affine algebras. Adv. in Math. 70, 156–234 (1988) 13. Koester, S.: Local Nature of Coset Models. Rev. Math. Phys. 16, 353–382 (2004) 14. Levstein, F., Liberati, J.I.: Branching rules for conformal embeddings. Commun. Math. Phys. 173, 1–16 (1995) 15. Longo, R.: Conformal subnets and intermediate subfactors. Commun. Math. Phys. 237(1-2), 7–30 (2003) 16. Longo, R., Rehren, K.-H.: Nets of subfactors. Rev. Math. Phys. 7, 567–597 (1995) 17. Longo, R., Xu, F.: Topological sectors and a dichotomy in conformal field theory. Commun. Math. Phys. 251(2) 321–364 (2004) 18. Moore, G., Seiberg, N.: Lectures on RCFT. In: Physics Geometry and Topology, edited by H. C. Lee, Nato ASI Series, Vol. 238, London: Plenum, 1990. 19. Pimsner, M., Popa, S.: Entropy and index for subfactors. Ann. Scient. Ec. Norm. Sup. 19, 57–106 (1986) 20. Polchinski, J.: String theory, Vol. II, Cambridge: Cambridge University Press, 1998 21. Pressley, A., Segal, G.: Loop Groups’ Oxford: Oxford University Press, 1986. 22. Rehren, K.-H.: Chiral observables and modular invariants. Commun. Math. Phys. 208, 689–712 (2000) 23. Rehren, K.-H.: Canonical tensor product subfactors. Commun. Math. Phys. 211, 395–408 (2000) 24. Str˘atil˘a, S.: Modular theory in operator algebras. Tunbridge wells-kenti: Abucus Press, 1981 25. Turaev, V.G.: Quantum invariants of knots and 3-manifolds. Berlin, New York: Walter de Gruyter, 1994 26. Wassermann, A.: Operator algebras and Conformal field theories III. Invent. Math. 133, 467–538 (1998) 27. Witten, E.: The search for higher symmetry in string theory. In: Physics and mathematics of strings. Philos. Trans. Roy. Soc. London Ser. A 329(1605), 349–357 (1989) 28. Xu, F.: Applications of braided endomorphisms from conformal inclusions. Internat. Math. Res. Notices 5–23 (1998)
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29. Xu, F.: 3-manifold invariants from cosets. J. Knot Theory and its Ramifications 14(1), 21–90 (2005) 30. Xu, F.: Algebraic coset conformal field theories. Commun. Math. Phys. 211, 1–43 (2000) 31. Xu, F.: New braided endomorphisms from conformal inclusions. Commun. Math. Phys. 192, 347–403 (1998) 32. Xu, F.: Algebraic coset conformal field theories II. Publ. RIMS, Kyoto Univ. 35, 795–824 (1999) 33. Zhu, Y.: Modular invariance of characters of vertex operator algebras. J. Amer. Math. Soc. 9(1), 237–302 (1996) Communicated by Y. Kawahigashi
