Commun. Math. Phys. 247, 1–47 (2004) Digital Object Identifier (DOI) 10.1007/s00220-003-0996-0
Communications in
Mathematical Physics
A Two Dimensional Fermi Liquid. Part 1: Overview Joel Feldman1, , Horst Kn¨orrer2 , Eugene Trubowitz2 1 2
Department of Mathematics, University of British Columbia, Vancouver, B.C., Canada V6T 1Z2. E-mail:
[email protected] Mathematik, ETH-Zentrum, 8092 Z¨urich, Switzerland. E-mail:
[email protected];
[email protected]
Received: 21 September 2002 / Accepted: 12 August 2003 Published online: 6 April 2004 – © Springer-Verlag 2004
Abstract: In a series of ten papers (see the flow chart at the end of §I), of which this is the first, we prove that the temperature zero renormalized perturbation expansions of a class of interacting many–fermion models in two space dimensions have nonzero radius of convergence. The models have “asymmetric” Fermi surfaces and short range interactions. One consequence of the convergence of the perturbation expansions is the existence of a discontinuity in the particle number density at the Fermi surface. Here, we present a self contained formulation of our main results and give an overview of the methods used to prove them.
Contents I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Flow Chart for the 2–d Fermi Liquid Construction . . . . . . . . II. An Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Renormalization of the Fermi Surface and the Dispersion Relation 2. Multi Scale Analysis . . . . . . . . . . . . . . . . . . . . . . . . 3. Integrating Out a Scale . . . . . . . . . . . . . . . . . . . . . . . 4. Overlapping Loops . . . . . . . . . . . . . . . . . . . . . . . . . 5. Particle–Particle Bubbles . . . . . . . . . . . . . . . . . . . . . . 6. Particle–Hole Ladders . . . . . . . . . . . . . . . . . . . . . . . 7. Power Counting in Position Space . . . . . . . . . . . . . . . . . 8. Sectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9. Cancellation Between Diagrams . . . . . . . . . . . . . . . . . . 10. The Counterterm . . . . . . . . . . . . . . . . . . . . . . . . . .
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Research supported in part by the Natural Sciences and Engineering Research Council of Canada and the Forschungsinstitut f¨ur Mathematik, ETH Z¨urich.
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J. Feldman, H. Kn¨orrer, E. Trubowitz
III. Formal Renormalization Group Maps Appendix A. Model Computations . . . . Notation . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . .
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I. Introduction The concept of a Fermi liquid was introduced by L. D. Landau in [L1, L2, L3] and has become the generally accepted explanation for the unexpected success of the independent electron approximation. An elementary sketch of Landau’s well known physical arguments can be found in [AM, pp. 345–351]. More thorough and technical discussions are presented in [AGD, pp. 154–203] and [N]. Roughly speaking, at temperature zero, the single particle excitations of a noninteracting Fermi gas become (almost stable) “quasi–particles” in a Fermi liquid. The quasi–particle spectrum has the “same structure” as the noninteracting single particle excitation spectrum and the quasi–particle density function n(k) still has a jump at the “Fermi surface”. The quasi–particle interaction at temperature zero is encoded in Landau’s f –function f (kF , kF ). It is well known that there are a number of potential instabilities that can drive an interacting fermi gas away from the Fermi liquid state. See, for example, [MCD, §1.2,4.5]. One of the most celebrated is the BCS instability for the formation of Cooper pairs leading to superconductivity in 2 and 3 dimensions. This is a potential instability for any time reversal invariant system [KL, L]. Another important instability is the Luttinger instability. There are solvable models in one space dimension that exhibit qualitatively different behavior from that of a three dimensional Landau Fermi liquid. In particular, the quasi–particle density function n(k) is continuous across the “Fermi surface” but has infinite slope there. These systems are called Luttinger liquids. For a rigorous treatment of Luttinger liquids in one dimension, see [BG] and the references therein. Anderson [A1, A2] suggested that a two dimensional Fermi gas should exhibit behavior similar to a one dimensional Luttinger liquid. Theorem I.4, which is proven in this series of papers, rigorously shows that this is not the case for the class of models considered here. In particular, we show that the density function n(k) has a jump discontinuity across the Fermi surface (Theorem I.5). The existence of the Landau f-function and its basic regularity properties follow directly from Theorem I.7. The standard model for a gas of weakly interacting fermions in a d-dimensional crystal at low temperature is given in terms of • a single particle dispersion relation (shifted by the chemical potential) e(k) on Rd , • an ultraviolet cutoff U (k) on Rd , • an interaction V . Here k is the momentum variable dual to the position variable x ∈ Rd . The Fermi surface associated to the dispersion relation e(k) is by definition F = k ∈ Rd e(k) = 0 . The ultraviolet cutoff is a smooth function with compact support that fulfills 0 ≤ U (k) ≤ 1 for all k ∈ Rd . We assume that it is identically one on a neighbourhood of the Fermi surface1 . 1
In particular, we assume that F is compact.
Two Dimensional Fermi Liquid. 1: Overview
3
We use renormalization group techniques to show that, for d = 2 and under the assumptions on the dispersion relation e(k) specified in Hypotheses I.12below, such a system is a Fermi liquid whenever V is small enough (the precise statement is given in Theorem I.5 below). Renormalization is necessary since the Fermi surfaces for the noninteracting (that is V = 0) and interacting systems (V = 0) do not, in general, agree. We therefore select (V –dependent) counterterms δe(k), from the space in Definition I.1, below, in such a way that the Fermi surface of the model with dispersion relation e(k) − δe(k) and interaction V is equal to F . Definition I.1. The δe(k) on Rd that counterterms, E, consists of all functions space of d 1 are supported in k ∈ R U (k) = 1 and for which the L –norm of the Fourier transform is finite. That is d d x δe∧ (x) < ∞, where δe∧ (x) =
dd k (2π)d
e−ık·x δe(k) .
The temperature Green’s functions at temperature zero (also known as the Euclidean Green’s functions) for this model can be described in field theoretic terms using the anticommuting fields ψσ (x0 , x), ψ¯ σ (x0 , x), where x0 ∈ R is the temperature (or Euclidean time) argument and σ ∈ {↑, ↓} is the spin argument. For x = (x0 , x, σ ) we ¯ write ψ(x) = ψσ (x0 , x) and ψ(x) = ψ¯ σ (x0 , x). For a model with dispersion relation e(k) − δe(k) and interaction V = 0, the Green’s functions are the moments of the Grassmann Gaussian measure, dµC(δe) , whose covariance is the Fourier transform of C(k0 , k; δe) =
U (k) . ık0 − e(k) + δe(k)
Precisely, for x = (x0 , x, σ ), x = (x0 , x , σ ) ∈ R × Rd × {↑, ↓}, C(x, x ; δe) =
¯ ) dµC(δe) (ψ, ψ) ¯ = δσ,σ ψ(x)ψ(x
d d+1 k ı
− e C(k; δe), (2π )d+1
where < k, x − x >− = −k0 (x0 − x0 ) + k · (x − x ) for k = (k0 , k) ∈ R × Rd . To simplify notation we set C(k) = C(k; 0)
,
C(x, x ) = C(x, x ; 0).
The interaction between the fermions is determined by the effective potential ¯ 1 )ψ(x2 )ψ(x ¯ 3 )ψ(x4 ) dx1 dx2 dx3 dx4 . ¯ V(ψ, ψ) = V (x1 , x2 , x3 , x4 ) ψ(x (R×Rd ×{↑,↓})4
We assume that V is translation invariant and spin independent. For some results, we also assume that V obeys V (R0 x1 , R0 x2 , R0 x3 , R0 x4 ) = V (−x1 , −x2 , −x3 , −x4 )
(I.1)
V (−x2 , −x1 , −x4 , −x3 ) = V (x1 , x2 , x3 , x4 ),
(I.2)
and
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J. Feldman, H. Kn¨orrer, E. Trubowitz
where R0 (x0 , x, σ ) = (−x0 , x, σ ) and −(x0 , x, σ ) = (−x0 , −x, σ ). We call (I.1) “k0 –reversal reality” and (I.2)“bar/unbar exchange invariance”. Precise definitions and a discussion of the properties of these symmetries are given in Appendix B of [FKTo2]. In the case of a two–body interaction v(x0 , x), the interaction kernel is V ((x1,0 ,x1 ,σ1 ),··· ,(x4,0 ,x4 ,σ4 )) = − 21 δ(x1 ,x2 )δ(x3 ,x4 )δ(x1,0 −x3,0 )v(x1,0 −x3,0 ,x1 −x3 ),
(I.3)
ˇ 0 , k), of where δ((x0 ,x,σ ),(x0 ,x ,σ )) = δ(x0 −x0 )δ(x−x )δσ,σ . If the Fourier transform, v(k ˇ ˇ 0 , k) , then the interaction the two–body interaction v(x0 , x) obeys v(−k 0 , k) = v(k kernel V has all four symmetries mentioned above. In addition, V always conserves particle number. We briefly discuss the norms imposed on interaction kernels. For a function f (x1 , · · · , xn ) on (R × Rd × {↑, ↓})n we define its L1 –L∞ –norm as sup dxj |f (x1 , · · · , xn )|. |||f |||1,∞ = max 1≤j0 ≤n x ∈R×Rd ×{↑,↓} j0
j =1,··· ,n j =j0
A multiindex is an element δ = (δ0 , δ1 , · · · , δd ) ∈ N0 × Nd0 . The length of a multiindex δ = (δ0 , δ1 , · · · δd ) is |δ| = δ0 + δ1 + · · · + δd and its factorial is δ! = δ0 !δ1 ! · · · δd !. For two multiindices δ, δ we say that δ ≤ δ if δi ≤ δi for i = 0, 1, · · · , d. The spatial part of the multiindex δ = (δ0 , δ1 , · · · δd ) is δ = (δ1 , · · · δd ) ∈ Nd0 . It has length |δδ | = δ1 +· · ·+δd . For a multiindex δ and x = (x0 , x, σ ), x = (x0 , x , σ ) ∈ R×Rd ×{↑, ↓} set (x − x )δ = (x0 − x0 )δ0 (x1 − x1 )δ1 · · · (xd − xd )δd . We fix r0 , r ≥ 6 for the numbers of temporal and spatial momentum derivatives that we will control. The norm imposed on an interaction kernel will be δi,j 1 (x − x ) V (x , x , x , x ) . max j 1 2 3 4 δi,j ! i δi,j ∈N0 ×Nd0 for 1≤i<j ≤4 |δδ i,j |≤r, |δi,j ;0 |≤r0
When V is of the form (I.3), max
δi,j ∈N0 ×Nd0 for 1≤i<j ≤4 |δδ i,j |≤r, |δi,j ;0 |≤r0
1,∞
1≤i<j ≤4
δi,j (x − x ) V (x , x , x , x ) i j 1 2 3 4 δi,j ! 1
1≤i<j ≤4
= max δ ∈Nd0 |δδ |≤r
1 δ!
1,∞
δ x v(x) dx.
Formally, the generating function for the connected Green’s functions is ¯ ¯ ¯ ¯ δe) = log 1 eφJ ψ eV (ψ,ψ) e−K(ψ,ψ;δe) dµC (ψ, ψ), G(φ, φ; Z where the source term is φJ ψ =
¯ ¯ dx φ(x)ψ(x) + ψ(x)φ(x).
(I.4)
(I.5)
Two Dimensional Fermi Liquid. 1: Overview
5
The counterterm is implemented in ¯ δe) = 1 dτ d d xd d y δe∧ (x − y) ψ¯ σ (τ, x)ψσ (τ, y) K(ψ, ψ; 2 σ ∈{↑,↓}
¯ ¯ ¯ is the partition function. The fields φ, φ¯ are and Z = eV (ψ,ψ) e−K(ψ,ψ;δe) dµC (ψ, ψ) called source fields and the fields ψ, ψ¯ are called internal fields. The connected Green’s functions themselves are determined by n ∞ n 1 ¯ δe) = ¯ i )φ(yi ). G(φ, φ; φ(x dxi dyi G2n (x1 , y1 , · · · , xn , yn ; δe) (n!)2 n=1
i=1
i=1
Observe that for δe ∈ E, formally, ¯ δe) = log 1 G(φ, φ; Z
¯ ¯ eφJ ψ eV (ψ,ψ) dµC(δe) (ψ, ψ),
¯ ¯ See [FKTo2, Lemma C.2]. where Z = eV (ψ,ψ) dµC(δe) (ψ, ψ). We show in this paper that, for d = 2 and under the Hypotheses I.12 on e(k), there exists, for every sufficiently small interaction, a counterterm δe ∈ E such that connected Green’s functions G2n (·; δe) exist and have Fermi surface F . This statement needs to be made precise, because the functional integrals in the definition of G are not, a priori, well defined due to the singularities of the propagator. This problem is dealt with by a multiscale analysis. We introduce scales by slicing momentum space into shells around the Fermi surface. 1 We choose a “scale parameter” M > 1 and a function ν ∈ C0∞ ([ M , 2M]) that takes 2 values in [0, 1], is identically 1 on [ M , M] and obeys ∞
ν M 2j x = 1 j =0
for 0 < x < 1 (see also [FKTo2, §VIII]). The function ν may be constructed by
choosing a function ϕ ∈ C0∞ (−2, 2) that is identically one on [−1, 1] and setting ν(x) = ϕ(x/M) − ϕ(Mx) for x > 0 and zero otherwise. Definition I.2. i) For j ≥ 1, the j th scale function on R × Rd is defined as
ν (j ) (k) = ν M 2j (k02 + e(k)2 ) . By construction, ν (j ) is identically one on √ 2 1 k = (k0 , k) ∈ R × Rd M ≤ |ik0 − e(k)| ≤ M M1j . Mj The support of ν (j ) is called the j th shell. By construction, it is contained in √ k ∈ R × Rd √1 M1j ≤ |ik0 − e(k)| ≤ 2M M1j . M
The momentum k is said to be of scale j if k lies in the j th shell.
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J. Feldman, H. Kn¨orrer, E. Trubowitz
ii) For real j ≥ 1, set
ν (≥j ) (k) = ϕ M 2j −1 (k02 + e(k)2 ) .
By construction, ν (≥j ) is identically 1 on √ k ∈ R × Rd |ik0 − e(k)| ≤ M M1j . Observe that if j is an integer, then for |ik0 − e(k)| > 0, (i) ν (≥j ) (k) = ν (k). i≥j
The support of ν (≥j ) is called the j th neighbourhood of the Fermi surface. By construction, it is contained in √ k ∈ R × Rd |ik0 − e(k)| ≤ 2M 1j . The support of ϕ is contained in
M 2j −2 (k02
+ e(k)2 )
M
is called the
j th
extended neighbourhood. It
√ k ∈ R × Rd |ik0 − e(k)| ≤ 2M M1j .
iii) For real j¯ ≥ 1, the covariance with infrared cutoff at scale j¯ and counterterm δe is U (k) − ν (≥j¯) (k) , ık0 − e(k) + δe(k)[1 − ν (≥j¯) (k)] d d+1 k ı− IR(j ) C IR(j¯) (x, x ; δe) = δσ,σ e C ¯ (k; δe). (2π)d+1
C IR(j¯) (k0 , k; δe) =
Also write C IR(j¯) (k) = C IR(j¯) (k; 0)
,
C IR(j¯) (x, x ) = C IR(j¯) (x, x ; 0).
The Grassmann Gaussian measure dµC IR(j ) (δe) is characterized by ¯ ¯ ) dµC IR(j ) (δe) (ψ, ψ) ¯ = C IR(j¯) (x, x ; δe), ψ(x)ψ(x ¯ ¯ ¯ ψ(x ¯ ) dµC IR(j ) (δe) (ψ, ψ) ¯ = 0. ψ(x)ψ(x ) dµC IR(j ) (δe) (ψ, ψ) = ψ(x) ¯ ¯ Remark I.3. i) As the scale parameter M > 1, the shells near the Fermi curve have j¯ near +∞, and the neighbourhoods shrink as j¯ → ∞. Also, lim C IR(j¯) (k) = C(k) ¯
j →∞
pointwise. ii) Our ultraviolet cutoff U (k) cuts off spatial directions only and does not restrict k0 . The entire ultraviolet regime, that is, large k0 , will be treated in a first step. To do so, we pick an integer j0 ≥ 2 and integrate out a covariance containing the factor U (k) − ν (>j0 ) (k). See Theorem V.8.
Two Dimensional Fermi Liquid. 1: Overview
7
Even for the cutoff, and hence bounded, covariance C IR(j¯) , it is not a priori clear that the generating functional ¯ ¯ V, δe) = log 1 ¯ Gj (φ, φ; eφJ ψ eV (ψ,ψ) dµC IR(j ) (δe) (ψ, ψ), Z ¯ ¯ ¯ ¯ where Z = eλV (ψ,ψ) dµC IR(j ) (δe) (ψ, ψ), ¯ or the corresponding connected Green’s functions, G2n;j (x1 , y1 , · · · , xn , yn ), defined ¯ by n ∞ n 1 ¯ V, δe) = ¯ i )φ(yi ), dxi dyi G2n;j (x1 , y1 , · · · , xn , yn ) φ(x Gj (φ, φ; 2 (n!) ¯ ¯ i=1 i=1 n=1
make sense for a reasonable set of V ’s and δe’s. On the other hand, it is easy to see, using graphs, that each term in the formal Taylor expansion of the Grassmann function2 ¯ V, δe(V )) in powers of V is well–defined for a large class of V ’s and δe(V )’s. Gj (φ, φ; ¯ ¯ ¯ is ∞ The Taylor expansion of eφJ ψ eV (ψ,ψ) dµC IR(j ) (ψ, ψ) n=1 Gj ,n (V, . . . , V), where ¯ ¯ the nth term is the multilinear form 1 ¯ · · · Vn (ψ, ψ) ¯ dµC IR(j ) (ψ, ψ) ¯ Gj ,n (V1 , · · · , Vn ) = n! eφJ ψ V1 (ψ, ψ) ¯ ¯ restricted to the diagonal. Explicit evaluation of the Grassmann integral expresses Gj ,n ¯ as the sum of all graphs with vertices V1 , · · · , Vn and lines C IR(j¯) . The (formal) ¯ t1 V1 + · · · + tn Vn , 0) ¯ V, 0) Taylor coefficient ddt1 · · · ddtn Gj (φ, φ; of Gj (φ, φ; t1 =···=tn =0 ¯ ¯ is similar, but with only connected graphs. Choosing δe to be an appropriate function of V produces renormalized connected graphs3 . We prove here that,
under suitable ¯ V, δe(V ) hypotheses, for each j¯ , the renormalized formal Taylor series for Gj φ, φ; ¯ 4 converges to an analytic function of V with a radius of convergence that is independent of j¯ . We further show that the limit as j¯ → ∞ exists. Theorem I.4. Assume that d = 2 and that e(k) fulfills the Hypotheses I.12 below. There is an open ball, centered on the origin, in the Banach space of translation invariant and spin independent interaction kernels V with norm δi,j 1 max ||| δi,j ! (xi − xj ) V (x1 , x2 , x3 , x4 )|||1,∞ , δi,j ∈N0 ×Nd0 for 1≤i<j ≤4 |δδ i,j |≤r, |δi,j ;0 |≤r0
1≤i<j ≤4
and an analytic counterterm function V → δe(V ) ∈ E on the ball, that vanishes for V = 0, such that the following holds: For any real j¯ ≥ 1, the formal Taylor series ¯ ¯ = log 1 ¯ Gj (φ, φ) eφJ ψ eλV (ψ,ψ) dµC IR(j ) (δe(V )) (ψ, ψ) Z ¯ ¯ 2 We shall, in Def. VI.7, introduce a norm on the Grassmann algebra generated by φ and φ. ¯ All of our generating functionals will in fact have finite norm. 3 Under the hypotheses of Theorem I.4, below, when j is finite, both the propagator C IR(j ) and the ¯ ¯ vertex V are continuous in momentum space. The values of all connected graphs, whether renormalized or not, are well–defined. However renormalization is essential for the limit j¯ → ∞. 4 For an elementary discussion of analytic maps between Banach spaces see, for example, Appendix A of [PT].
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J. Feldman, H. Kn¨orrer, E. Trubowitz
converges to an analytic function on the ball. As j¯ → ∞, the Green’s functions G2n;j converge uniformly, in x1 , · · · , yn and V , to a translation invariant, spin independent,¯
2n particle number conserving function G2n on R × Rd × {↑, ↓} that is analytic in V . If, in addition, V is k0 –reversal real, as in (I.1), then δe(k; V ) is real for all k. The proof of Theorem I.4 follows the statement of Theorem VIII.5 in [FKTf2]. Theorem I.5. Under the hypotheses of Theorem I.4 and the assumption that V obeys the symmetries (I.1) and (I.2), the Fourier transform
ˇ 2 (k0 , k) = dx0 d d x eı(−k0 x0 +k·x) G2 (0, 0, ↑), (x0 , x, ↑) G
= dx0 d d x eı(−k0 x0 +k·x) G2 (0, 0, ↓), (x0 , x, ↓) of the two–point function exists and is continuous, except on the Fermi surface (precisely, except when k0 = 0 and e(k) = 0). Define dk0 ık0 τ ˇ G2 (k0 , k). n(k) = lim 2π e τ →0+
Then n(k) is continuous except on the Fermi surface F . If k¯ ∈ F , then lim n(k) and k→k¯ e(k)>0
lim n(k) exist and obey
k→k¯ e(k)<0
lim n(k) − lim n(k) > 21 .
k→k¯ e(k)<0
k→k¯ e(k)>0
The proof of Theorem I.5 follows Lemma XII.4 in [FKTf3]. Remark I.6. The quantity n(k) is known as the momentum distribution function. The jump discontinuity in n(k) at the Fermi surface exhibited in Theorem I.5 is generally viewed as the most basic characteristic of a Fermi liquid [MCD, §4.1]. The number 21 in the bound of I.5 has no special significance. It may be replaced by any number strictly smaller than one, provided the interaction is made sufficiently weak. Theorem I.7. Let ˇ 4;σ1 ,σ2 ,σ3 ,σ4 (k1 , k2 , k3 ) = G
×
G4 x1 , x2 , x3 , (0, 0, σ4 ) 3
e−(−1)
ı −
dx0, d d x
=1
be the Fourier transform of the four–point function and ˇA ˇ G 4;σ1 ,σ2 ,σ3 ,σ4 (k1 , k2 , k3 ) = G4;σ1 ,σ2 ,σ3 ,σ4 (k1 , k2 , k3 )
4 =1
1 ˇ 2 (k ) k =k −k +k G 4 1 2 3
Two Dimensional Fermi Liquid. 1: Overview
9
ˇ A is its amputation by the physical propagator. Under the hypotheses of Theorem I.5, G 4 continuous on
(k1 , k2 , k3 ) k1 = k2 , k2 = k3 ,
U (k1 ) = U (k2 ) = U (k3 ) = U (k1 − k2 + k3 ) = 1 .
ˇ A has a decomposition Furthermore, G 4 ˇA G 4;σ1 ,σ2 ,σ3 ,σ4 (k1 , k2 , k3 ) = Nσ1 ,σ2 ,σ3 ,σ4 (k1 , k2 , k3 )
2 k3 +k4 , , k − k + 21 Lσ1 ,σ2 ,σ3 ,σ4 k1 +k 2 1 2 2 k4 =k1 −k2 +k3
k3 +k2 k1 +k4 1 − 2 Lσ1 ,σ2 ,σ3 ,σ4 2 , 2 , k2 − k3 k4 =k1 −k2 +k3
with ◦ Nσ1 ,σ2 ,σ3 ,σ4 continuous ◦ Lσ1 ,σ2 ,σ3 ,σ4 (q1 , q2 , t) continuous except at t = 0 ◦ Lσ1 ,σ2 ,σ3 ,σ4 (q1 , q2 , (0, t)) having an extension to t = 0 which is continuous in (q1 , q2 , t) ◦ Lσ1 ,σ2 ,σ3 ,σ4 (q1 , q2 , (t0 , 0)) having an extension to t0 = 0 which is continuous in (q1 , q2 , t0 ). The proof of Theorem I.7 is at the end of §XV in [FKTf3]. Remark I.8. Lσ1 ,σ2 ,σ3 ,σ4 (q1 , q2 , t) consists of particle–hole ladder contributions of the form q1 + q1 −
t 2 t 2
q2 + q2 −
t 2 t 2
Remark I.9. Theorem I.4 defines G2n;j (V ) through its renormalized perturbation expan¯ an additional finite volume cutoff by replacsion. Alternatively, one could introduce d ing the space–time R × R with (R/LZ) × (Rd /LZd ). The associated Green’s functions, G2n;j ,L (V ), are trivially meromorphic functions of V that are analytic at V = 0. ¯ of these papers could be used to show that the domain of analyticity of The methods G2n;j ,L (V ) includes the ball of Theorem I.4 and that, as L → ∞, G2n;j ,L converges ¯ ¯ uniformly to G2n;j . We do not do so. ¯ We also do not deal with the question of independence of the renormalization prescription. Suppose that e(k), δe(k; V ) and e (k), δe (k; V ) satisfy the appropriate regularity conditions and that e(k) − δe(k; V ) = e (k) − δe (k; V ), for some specific V . Observe that G2n;j depends on e(k) and δe(k; V ) only through the combinations ¯ e(k) − δe(k; V )[1 − ν (≥j¯) (k)] and ν (≥j¯) (k) = ϕ M 2j −1 (k02 + e(k)2 ) . Hence, it is likely that the limiting Green’s functions constructed using e(k), δe(k; V ) coincide with those constructed using e (k), δe (k; V ), for the specific V , but we do not attempt to prove so here.
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J. Feldman, H. Kn¨orrer, E. Trubowitz
We now state the hypotheses on the dispersion relation e(k) used in Theorems I.4, I.5 and I.7. First, we assume that the dispersion relation e(k) is r + 6 times continuously differentiable. Furthermore, we assume that the Fermi curve F = k ∈ R2 e(k) = 0 is a strictly convex, smooth connected curve with curvature bounded away from zero. Since F is strictly convex, for each point k ∈ F there is a unique point a(k) ∈ F different from k such that the tangent lines to F at k and a(k) are parallel. a(k) is called the antipode of k. Choose an orientation for F . Definition I.10. i) Let k ∈ F , t the oriented unit tangent vector to F at k and n the inward pointing unit normal vector to F at k. Then there is a function ϕk (s), defined on a neighbourhood of 0 in R, such that s → k + st + ϕk (s)n is an oriented parametrization of F near k. ii) We say that F is strongly asymmetric if there is n0 ∈ N, with n0 ≤ r, such that for each k ∈ F there exists an n ≤ n0 with (n)
(n)
ϕk (0) = ϕa(k) (0). Remark I.11. i) By construction, ϕk (0) = ϕ˙k (0) = 0 and ϕ¨k (0) is the curvature of F at k. ii) If F is symmetric about a point p ∈ R2 , that is F = { 2p − k k ∈ F }, then ϕk = ϕa(k) for all k ∈ F . Symmetry of the Fermi curve about a point promotes the formation of Cooper pairs and the phase transition to a superconducting state. Theorem I.4 shows that – at temperature zero – this is the only instability in a broad class of short range many fermion models, at least when d = 2. Sufficiently high temperature also blocks the Cooper instability and leads to Fermi liquid behaviour, even for a round Fermi surface. This was shown in [DR1, DR2] using the criterion proposed in [S]. See also [PS]. When d = 1, fermionic many–body models exhibit Luttinger liquid rather than Fermi liquid behaviour. See Chapter 11 of [BG] and the references therein. We would expect that results like Theorems I.4 and I.5 also hold for d = 3. There has been some progress in this direction [MR, DMR]. iii) In [FKTa] we show that independent fermions in a suitably chosen periodic electromagnetic background field have a dispersion relation whose associated Fermi curve, for suitably chosen chemical potential, is smooth, strictly convex, strongly asymmetric and has nonzero curvature everywhere. Hypothesis I.12 (on the dispersion relation). We assume that e(k) is r + 6 times continuously differentiable with r ≥ 6, that the Fermi curve F is a strictly convex, smooth, strongly asymmetric, connected curve whose curvature is bounded away from zero and that ∇e(k) does not vanish on F . This paper is divided into three parts. This first part, which consists of Sects. I through III and Appendix A, contains an overview of the main ideas involved in the construction (§II) and the algebraic structure of the construction (§III). Part 2, [FKTf2], consists of Sect. IV through X and Appendix B and contains the proof of Theorem I.4 — using the results of the auxiliary papers [FKTl, FKTo1, FKTo2, FKTo3, FKTo4 and FKTr1,
Two Dimensional Fermi Liquid. 1: Overview
11
FKTr2]. The existence of a discontinuity in the particle number density at the Fermi surface, stated in Theorem I.5, is proven in the third part, [FKTf3]. Continuity properties of the amputated four point Green’s functions, stated in Theorem I.7, are also proven in [FKTf3]. Our proof uses a multiscale analysis and discrete renormalization group flow, in the framework of fermionic functional integrals, and also uses renormalization of the Fermi surface. In the j th step of the renormalization group flow we turn on interactions amongst particles whose momenta have distance from the Fermi surface about M1j . The output of the j th step is an effective interaction for particles whose momenta have distance from the Fermi surface less than M1j . In [FKTr1] we build a tool for controlling Gaussian fermionic functional integrals of exponentials of effective interactions in a quite abstract setting5 , using Gram bounds for determinants. In [FKTo3, FKTo4] this tool is combined with a technique developed specifically to handle problems created by the fact that the singularity of the original propagator lies on an object with nontrivial geometry, the Fermi surface. The technique is “sectorization of the Fermi surface”. It allows us to use the abstract results of [FKTr1] while retaining the ability to exploit conservation of momentum. In particular, conservation of momentum leads to improved power counting, due to “overlapping loops”, for the two point function and the non– ladder contributions to the four point function. This mechanism is formulated abstractly in [FKTr2] and implemented concretely in [FKTo3]. Generalized particle–particle and particle–hole ladder diagrams require special treatment. Particle–particle ladders have improved power counting due to the assumed asymmetry of the Fermi surface6 . This is proven in §XXII of [FKTo4]. The estimate on the sum of all contributions from particle–hole ladders exploits a sign cancellation between scales and is developed in [FKTl]. The treatment of the first renormalization group step, i.e. the ultraviolet regime, is a simple application of the expansion of [FKTr1] and appears in [FKTo2]. Control of the change of sectorization forced by the change of scales between renormalization group steps is achieved in [FKTo4]. As mentioned above, §II and III of this paper give an overview of the construction. The paper [FKTo1], where nonsingular propagators (as is the case for insulators) are considered, can also serve as an introduction to some of the techniques. The set of eleven papers is self–contained. [FKTr1+r2], [FKTl] and [FKTo4] are each reasonably independent of the other papers in the series. The interface between each of these and the other papers is through a relatively small number of definitions and theorems. The flow chart below provides a thumbnail sketch of the contents of each of the papers and gives the logical connections between them. Cumulative notation tables are provided at the end of each part. The construction described in these papers was outlined in [FKLT1, FKLT2, FKLT3].
5 We expect that the results of [FKTr1] may be usable beyond the present problem and so may be of more general interest. 6 The fact that Schr¨ odinger operators with periodic magnetic potentials usually have such asymmetric Fermi surfaces is proven in the auxiliary paper [FKTa].
12
J. Feldman, H. Kn¨orrer, E. Trubowitz
1. Flow Chart for the 2–d Fermi Liquid Construction
A Two Dimensional Fermi Liquid Part 1:
Overview
Statement of results. Description of problems involved and techniques used. FKTf1 Part 2:
Convergence of Perturbation Expansions in Fermionic Systems
Convergence
Construction of the model by a multiscale analysis with renormalization. FKTf2 Part 3: The
Fermi Surface
Control of the two–point function. The jump at the Fermi surface. FKTf3
Particle Hole Ladders Estimates on generalized particle–hole ladders. FKTl
Asymmetric Fermi Surfaces for Magnetic Schr¨odinger Operators Asymmetry condition on the Fermi surface is fulfilled for generic periodic magnetic Schr¨ odinger operators. FKTa
Single Scale Analysis of Many Fermion Systems Part 1:
Nonperturbative Bounds
Abstract bounds for Fermionic functional integrals. FKTr1
Insulators
Fermi systems with nonsingular propagators. FKTo1 Part 2: The
Part 1:
First Scale
Part 2:
Overlapping Loops
Isolating and exploiting overlapping loops in the expansion of Part 1. Isolating ladders. FKTr2
The ultraviolet end of the model and the first scale of the infrared end. FKTo2 Part 3:
Sectorized Norms
Sectors introduced. One renormalization group step. Exploitation of overlapping loops. FKTo3 Part 4:
Sector Counting
Passage from scale to scale for sectorized norms. Particle–particle ladders. FKTo4
II. An Overview In this section, we describe the main difficulties in the proof of Theorems I.4 and I.5 and outline our strategy to overcome them. For simplicity, we omit spins in this discussion. The notation in this section is close, but not always identical to that used in the rest of this paper. As the main theorems are for d = 2 space dimensions, we describe all constructions only for d = 2, even those that can be extended to other dimensions. 1. Renormalization of the Fermi Surface and the Dispersion Relation The Fermi surface of a fermionic many particle system is the locus in momentum space where the two ˇ 2 (0, k) has a discontinuity – if such a discontinuity occurs at all.7 For point function G a system of non-interacting fermions, the Fermi surface coincides with the zero–set of the dispersion relation. In a metal or a crystal, the dispersion relation is a datum derived 7
For example, in a superconductor there is no such discontinuity.
Two Dimensional Fermi Liquid. 1: Overview
13
from first principles; it is determined by the associated periodic Schr¨odinger operator. On the other hand, the Fermi surface of the system of interacting fermions is accessible to measurement. See, for example, [AM]. As already mentioned in §I, the Fermi surface of a system of interacting fermions is, in general, different from that of the system of non-interacting fermions with the same dispersion relation. This shift in the Fermi surface is responsible for the divergence of many coefficients in naive perturbation expansions. It is controlled by renormalizing the dispersion relation. Theorems I.4 and I.5 state that, given a function e(k) and an interaction V fulfilling all of the theorems’ hypotheses, there is a function δe(k), called the “counterterm”, such that the system with dispersion relation e(k)−δe(k) and interaction V has a Fermi surface and that Fermi surface is precisely F = {k e(k) = 0 }. We pointed out in the previous paragraph that the data derived from first principles are the dispersion relation and the interaction V . Therefore it is desirable to prove that every reasonable function e (k) is of the form e(k) − δe(k) as above. This could be done by proving the invertibility of the map e(k) → e(k) − δe(k) in an appropriate function space. To all orders in perturbation theory, this has been achieved in [FST4]. The bounds of this paper are not yet strong enough to prove the corresponding result non-perturbatively. Even for C ∞ functions e(k) and V , it is not known how smooth the counterterm δe(k) is. In this paper, we show that δe is C . In [FST3] it is shown that δe is C 2+ to all orders in perturbation theory. Later in this overview (Subsect. 10) we shall point out where this lack of smoothness in the counterterm creates difficulties for the construction. 2. Multi Scale Analysis We cannot treat the functional integral (I.4) defining the formal Green’s functions in one piece, because the propagator is singular. Similarly it is probably impossible to determine the counterterm δe(k) in one step. Therefore we introduce scales adjusted to the size of the propagator in momentum space and to the infrared cut off propagators C IR(j ) (k0 , k; δe) of Definition I.4, construct an appropriate counterterm for each scale j and take the limit j → ∞. The limit is controlled by comparing, for each j , the model with covariance C IR(j +1) (k0 , k; δe) to that with covariance C IR(j ) (k0 , k; δe). This comparison amounts to “integrating out scale j ”. We give an introduction to “integrating out a scale” in the next subsection. In §III, we describe, formally but in more detail, how the limit j → ∞ is taken. 3. Integrating Out a Scale The discussion of the previous subsection shows that the essential estimates in our construction concern the effect of integrating out one scale as above. To simplify the discussion, we consider the case in which the covariance C IR(j ) (k; δe) is replaced by C IR(j ) (k; 0). That is, for the moment, we ignore the effect of (j ) (k) the counterterm. Since C IR(j +1) (k; 0) = C IR(j ) (k; 0) + C (j ) (k) with C (j ) (k) = ıkν0 −e(k) , ¯ ¯ ¯ = log 1 ¯ eφJ (ψ+ζ )+V (ψ+ζ,ψ+ζ ) dµC IR(j ) (ζ, ζ¯ )dµC (j ) (ψ, ψ) Gj +1 (φ, φ) Z ¯ ¯ ¯ = log Z1 eφJ ψ+W (φ,φ,ψ,ψ) dµC (j ) (ψ, ψ), where ¯ ψ, ψ) ¯ = log 1 W(φ, φ, Zj
¯
¯
eφJ ζ +V (ψ+ζ,ψ+ζ ) dµC IR(j ) (ζ, ζ¯ )
14
J. Feldman, H. Kn¨orrer, E. Trubowitz
is the effective interaction at scale j . The partition functions Z, Z and Zj are chosen so that Gj +1 (0, 0) = W(0, 0, 0, 0) = 0. To iterate this construction, we need an effective interaction ¯ ¯ 1 ¯ ¯ W (φ, φ, ψ, ψ) = log Zj +1 eφJ ζ +V (ψ+ζ,ψ+ζ ) dµC IR(j +1) (ζ, ζ¯ ) at scale j + 1. So the problem is to estimate ¯ ¯ ¯ ¯ ψ, ψ) ¯ = log 1 W (φ, φ, eφJ ζ +W (φ,φ,ψ+ζ,ψ+ζ ) dµC (j ) (ζ, ζ¯ ) Z in terms of estimates on W. The main difficulties already occur when φ = φ¯ = 0, so we concentrate on this special case. Write ¯ W(0, 0, ψ, ψ) = dp1 ···dpn dq1 ···dqn w2n (p1 ,··· ,pn , q1 ,··· ,qn ) n≥0
δ (p1 +···+pn −q1 −···−qn )ψ¯ (p1 )···ψ¯ (pn ) ψ (q1 )···ψ (qn ) and ¯ = W (0, 0, ψ, ψ)
n≥0
dp1 ···dpn dq1 ···dqn w2n (p1 ,··· ,pn , q1 ,··· ,qn )
δ (p1 +···+pn −q1 −···−qn )ψ¯ (p1 )···ψ¯ (pn ) ψ (q1 )···ψ (qn ), ¯ ¯ where ψ(k) and ψ(k) are the Fourier transforms of ψ(x) and ψ(x) respectively8 . Then w2n can be written as a sum of values of connected directed graphs with vertices w2 , w4 , · · · and propagator C (j ) (k). See [FW, Chap. 3]. Naive power counting just uses that C (j ) (k)∞ = sup |C (j ) (k)| is of order M j
(II.1)
k
and volume of the j th shell is of order
1 M 2j
.
(II.2)
This is because the j th shell has width of order M1j in the k0 direction and in the k–direction transversal to F and has circumference, in the direction along F , of order one. If one assumes that w2n ∞ is of order M j (n−2) for all n
(II.3)
is again of order then every graph contributing to w2n
M j i (ni −2) M j i (2ni −2n)/2 M −2j [i (2ni −2n)/2−i 1+1] = M j (n−2) . The three factors come from the suprema of the vertex functions w2ni , the suprema of the i 2n2i −2n propagators and the volume of the domain of integration respectively. For n ≥ 3 the Condition (II.3) grows nicely in j . Four legged vertices (n = 2) are marginal — the estimate (II.3) alone would lead to divergences in powers of j when the sum 8
Precise Fourier transform conventions are formulated in §VI.
Two Dimensional Fermi Liquid. 1: Overview
15
over j is performed. We exploit “overlapping loops” and special estimates on ladders to derive better bounds on the four point functions. The counterterm is built up from contributions at each scale chosen so that (II.3) holds for n=1, too. Overlapping loops will be discussed in Subsect. 4, ladders in Subsect. 5 and 6, and the construction of the counterterm in Subsect. 10. It is well known that the sum of the norms of all graphs diverges. In this paper we use cancellations between different graphs to derive convergent bounds. There are several schemes to implement such cancellations, all variants of the basic scheme of [C]. In all of these schemes the cancellation is only seen when one writes the model in position space variables. This is a major technical difficulty, since the geometry of the Fermi surface and special effects like “overlapping loops” and ladder estimates are naturally formulated in momentum space. We use the cancellation scheme developed in [FMRT, FKTcf, FKTr1, FKTr2]. It is sufficiently close to the graphical picture to allow us to implement overlapping loops in collections of diagrams within which cancellations take place. 4. Overlapping Loops We first describe the effect of overlapping loops in an example. Consider the diagram p1
q1
k1 p2
k2
q2
contributing to w4 (p1 , p2 , q1 , q2 ). If w4 = 1, its value is (p1 , p2 , q1 , q2 ) = dk1 dk2 C (j ) (k1 +q1 −p1 ) C (j ) (k1 ) C (j ) (k1 −k2 ) C (j ) (q2 −k2 ). Naive power counting gives ∞ ≤ C (j ) 4∞ · (volume of the j th shell)2 = O(1). However, taking into account that C (j ) is supported on the j th shell, ∞ ≤ C (j ) 4∞ · (volume of T ), where
T = {(k1 , k2 ) k1 , k1 − k2 , q2 − k2 ∈ j th neighbourhood}.
(II.4)
Let S be the set of all k2 for which q2 − k2 lies in the j th neighbourhood and S be the set of all k2 for which |k2 | ≥ √ 1 j . Then T ⊂ T1 ∪ T2 , where M
T1 = {(k1 , k2 ) k1 , k1 − k2 ∈ j th neighbourhood, k2 ∈ S ∩ S }, T2 = {(k1 , k2 ) k1 ∈ j th neighbourhood, k2 ∈ S \ S }.
16
J. Feldman, H. Kn¨orrer, E. Trubowitz
For k2 ∈ S the volume of the set
{k1 k1 , k1 −k2 ∈ j th neighbourhood} = j th neighbourhood ∩ k2 +j th neighbourhood
k2 + j th neighbourhood
j th neighbourhood
is of order
1 M 2j
· √ 1 j , since this set has width of order M
1 in the k0 direction and in the Mj 1 order √ j in the direction along F . M
k direction transversal to F and width at most of Here we use that the Fermi surface F has curvature bounded away from zero. Therefore the volume of T1 is of order
1
1 1 · (volume of S) = M 5j/2 . · O M12j = O M 9j/2 M 5j/2 Similarly the volume of T2 is bounded by (volume of j th neighbourhood) · (volume of S \ S ) = O( M12j ) ·
1 M 5j/2
1 = O( M 9j/2 ).
Therefore, using (II.1), ∞ = O( √ 1 j ). The volume estimate on T derived above is M
not optimal. By [FST2, Theorem 1.1], the volume of T is O Mj j . Similar improvements are possible for all diagrams with “overlapping loops”, i.e. with two different simple loops which have at least one line in common. If w2 = 0, the only four legged diagrams without overlapping loops and tadpoles are particle–particle ladders
and particle–hole ladders
See [FST1, §2.4]. We shall Wick order the effective interactions with respect to the future covariance in order to exclude tadpoles. The condition that w2 = 0 is achieved by moving the two legged part of the effective interaction into the covariance. We have already mentioned that the effect of overlapping loops and special ladder estimates are used to get convergent bounds on the four point functions. The effect of overlapping loops has to be combined with the cancellation scheme between diagrams mentioned at the end of Subsect. 3 and discussed in Subsect. 9 below. Thus the cancellation scheme has to be sensitive enough to detect simultaneous overlapping loops in all mutually cancelling diagrams and to isolate ladder diagrams. Furthermore the geometric estimates on T above have to be exploitable in a position space setting. This is done using sectors. See Subsect. 8.
Two Dimensional Fermi Liquid. 1: Overview
17
5. Particle–Particle Bubbles The strong asymmetry condition of Definition I.10 is used to get improved power counting on particle–particle ladders. We describe the effect in the example of the particle–particle bubble
p1
q1 k
p2
q2
again assuming that w4 = 1. The value of this graph is dk C (j ) (t − k) C (j ) (k), where t = p1 + p2 = q1 + q2 is the transfer momentum. Naive power counting again gives that this value is O(1). On the other hand, taking the support condition of C (j ) into account, the value of this graph is bounded by C (j ) 2∞ · (volume of {k k and t − k lie in the j th neighbourhood}. By the strong asymmetry condition of Definition I.10, F and the shifted reflected Fermi surface t − F = {t − k k ∈ F } have tangency of order at most n0 . From this one deduces that
{k k, t − k ∈ j th neighbourhood} = j th neighbourhood ∩ t − (j th neighbourhood) has width of order M1j in the k0 direction and in the k direction transversal to F and 1 in the direction along F . Therefore its volume is of order width at most of order M j/n 0 t − (j th neighbourhood)
j th neighbourhood
1 1 M 2j M j/n0
1 and the value of the particle–particle bubble is of order M j/n . 0 As in the case of overlapping loops, this estimate is based on a volume estimate in momentum space. Again sectors will used to implement this in position space variables.
6. Particle–Hole Ladders Our estimate on particle–hole ladders is not based on a geometric argument as in the case of particle–particle ladders or overlapping loops, but
18
J. Feldman, H. Kn¨orrer, E. Trubowitz
on cancellations between scales. The limit as j → ∞ of the particle–hole bubble Bj (p1 , p2 , q1 , q2 ) p1 q1
p2
q2
with w4 = 1 and propagator i≤j C (i) has a discontinuity in the transfer momentum t = p1 − q1 at t = 0, but is continuous for t = 0 and smooth in a neighbourhood of the origin in {(t0 , t) t0 = 0}. See the introduction to [FKTl], and in particular Lemma I.1 there. The proof of this lemma is based on integration by parts and thus cancellation between scales. In the multi-scale analysis, we combine all the contributions of particle–hole ladders created at scales ≤ j , and give a uniform estimate on the result. A vertex of a particle– hole ladder at scale j may be a particle–hole ladder created at a previous scale i < j , as in the diagram: p1 q1
C (j )
C (j ) C (i)
p2
q2 C (i)
The iteration of these effects (and of the Wick ordering with respect to future covariances) leads to the concept of iterated particle–hole ladders of Definition VII.7. Uniform estimates on these iterated particle–hole ladders are stated in Theorem VII.8 and proven in [FKTl]. They have to be in position space, as they have to be combined with the other estimates derived from the “cancellation scheme” mentioned in Subsect. 3. This is technically difficult, because it amounts to taking Fourier transforms of quantities whose limit, as j → ∞, is discontinuous. 7. Power Counting in Position Space Proving power counting bounds on graphs is usually split into two steps. To illustrate the two steps, we consider the situation that we have two vertices ϕ1 and ϕ2 with n1 and n2 legs, respectively. For simplicity we ignore orientation of the lines. We assume we form a diagram by connecting ϕ1 and ϕ2 by 1 ≤ r ≤ min{n1 , n2 } lines. =
ϕ1
ϕ2
This can be done in two steps: First connect ϕ1 and ϕ2 by one line and call the result . =
ϕ1
ϕ2
Two Dimensional Fermi Liquid. 1: Overview
19
Secondly, pairwise contract 2r − 2 legs of to form r − 1 lines. ϕ1
ϕ2
The first estimate is to bound the norm of in terms of a constant times the product of the norms of ϕ1 and ϕ2 . We call such a constant a “contraction bound” c. If one uses · ∞ norms in momentum space, then C (j ) (k)∞ is a contraction bound. A tadpole bound is a number b with the following property. Let ϕ be any graph with at least two legs, and ϕ the graph obtained from ϕ by connecting two legs to form a line.
Then the norm of ϕ is bounded by b times the norm of ϕ. If one uses · ∞ norms in momentum space, then C (j ) (k)1 is a tadpole bound. Applying one contraction bound and r − 1 tadpole bounds, one sees that the norm of is bounded by c br−1 times the product of the norms of ϕ1 and ϕ2 . By (II.1) and (II.2), for the · ∞ norms in momentum space, the contraction bound is of order M j , while the tadpole bound is of order M1j . The power counting for the · ∞ norm in momentum space described in Subsect. 3 can be generalized to the abstract setting of contraction and tadpole bounds: If one assumes that the norm of w2n is of order
1 cbn−1
for all n
(II.5)
is again of order 1 . For example, if such a graph then every graph contributing to w2n cbn−1 has two vertices, w2n1 and w2n2 , then there are r = n1 + n2 − n connecting lines and the norm of is bounded by
cbn1 +n2 −n−1 norm of w2n1 norm of w2n2 = O cbn1 +n2 −n−1 cbn11 −1 cbn12 −1
1 = O cbn−1 .
A general graph may be bounded by building it up one vertex at a time. As pointed out in Subsect. 3, we need to use position space variables. A natural position space norm for translation invariant vertices ϕ(x1 , · · · , xn ), which mimics the · ∞ norm in momentum space, is |||ϕ|||1,∞ = max sup dxj |ϕ(x1 , · · · , xn )|. 1≤p≤n xp
j =1,··· ,n j =p
It is easily seen that the L1 norm, C (j ) (x)1 , of the Fourier transform of C (j ) (k) is a contraction bound for this norm and that the L∞ norm of C (j ) in position space, C (j ) (x)∞ , is a tadpole bound. Clearly, C (j ) (x)∞ ≤ C (j ) (k)1 , so that we again have a tadpole bound of order M1j . A naive computation, given in the next paragraph, gives a bound on C (j ) (x)1 that is of order M 2j . A more refined argument, sketched in the next but one paragraph, gives a (realistic – see Example A.1) bound of order M 3j/2 . In any event, M 3j/2 C (j ) (k)∞ and naive power counting in position space does not coincide with power counting in momentum space. Substituting c = O M 3j/2 and
20
J. Feldman, H. Kn¨orrer, E. Trubowitz
5 b = O M1j into (II.5) yields the requirement that |||w2n |||1,∞ be order M j (n− 2 ) . In particular the norm of the four point function would have to decrease like √ 1 j as j M increased. This is absurd, since the original interaction V is, at each scale, the dominant part of the four point function. Again, we use sectors to cope with this problem. We first sketch the standard calculation that gives the naive bound on C (j ) (x)1 . For a multi-index δ = (δ0 , δ1 , δ2 ) of non-negative integers write |δ| = δ0 + δ1 + δ2 and x δ = x0δ0 x1δ1 x2δ2 . Then, integrating by parts |δ| times,
x δ (j ) |C (x)| Mj |δ|
≤
|δ| 1 ∂ M j |δ| ∂k δ
C (j ) (k)1 = O( M1j ), |δ|
since the support of ∂∂k δ C (j ) (k) has volume of order M12j and ∂∂k δ C (j ) (k)∞ is of order M j (|δ|+1) . Therefore
2
2
2 (j ) 1 + Mx0j 1 + Mx1j 1 + Mx2j |C (x)| = O( M1j ). Dividing by
1+
ν=0,1,2
xν 2 Mj
and integrating over R3 gives the bound C (j ) (x)1 =
O(M 2j ). As it motivates the construction of sectors, we indicate how the more refined bound on C (j ) (x)1 is derived. Assume first that F has a straight segment of length l on the k2 axis9 and that e(k) = k1 in a neighbourhood of this segment. Choose a cutoff function χ(k2 ) that is identically one on most of the segment, zero outside the segment and for (j ) ∂n 1 (j ) (k). An argument similar to which | ∂k n χ (k2 )| is of order ln . Set Cs (k) = χ (k2 )C 2 that of the previous paragraph shows that
2
2 (j ) 1 + Mx0j 1 + Mx1j 1 + (lx2 )2 |Cs (x)| = O( Ml j ) (j )
and therefore that Cs (x)1 = O(M j ). The same argument also works for a realistic Fermi surface, if one cuts out a “sector” of length l ≤ √ 1 j as indicated in the figure M below.
The precise computation is in [FKTo3, Prop. XIII.1 and Lemma XIII.2]. If the sector is too long, the curvature of the Fermi surface causes deterioration of the bounds on the derivatives parallel to the Fermi surface. One can divide up the j th neighbourhood into O( 1l ) sectors and use a partition of unity by functions like χ , to see that C (j ) (x)1 ≤ (number of sectors) · O(M j ) = O( 1l M j ). If one chooses l =
√1 Mj
, one gets the bound C (j ) (x)1 = O(M 3j/2 ).
9 This of course contradicts our hypotheses that F is strictly convex. We will remove this assumption immediately.
Two Dimensional Fermi Liquid. 1: Overview
21
8. Sectors We cover the j th neighbourhood by slightly overlapping sectors of length l ≤ √ 1 j as indicated in the figure below. M
The set of sectors is called asectorization of scale j and length l. Furthermore we select a partition of unity {χs (k) s ∈ } subordinate to the sectorization, such that each χs has properties analogous to those of the function χ in the last subsection10 . Recall that we want to integrate out scale j and that ¯ ¯ ¯ = 1 W (0, 0, ψ, ψ) eW (0,0,ψ+ζ,ψ+ζ ) dµC (j ) (ζ, ζ¯ ). Z Decompose the kernel w2n of the part of W that is homogeneous of degree 2n in the fields w2n (p1 ,··· ,pn , q1 ,··· ,qn ) = ω2n ((p1 ,s1 ),··· ,(pn ,sn ), (q1 ,sn+1 ),··· ,(qn ,s2n )) s1 ,··· ,s2n ∈
for pi , qi in the j th neighbourhood. Here, ω2n is a function that vanishes unless pi lies in the sector si and qi lies in the sector sn+i . ω2n is called a –sectorized representative of w can then be written as a sum of values of w2n . A –sectorized representative ω2n 2n of connected directed graphs with vertices ω2 , ω4 , · · · and propagator C (j ) (k), where the momentum integrals in the graphs also include sector sums. The main norm we use for the Fourier transforms ωˆ 2n ((x1 ,s1 ),··· ,(x2n ,s2n )) of ω2n is |||ωˆ 2n |||1, = max max |||ωˆ 2n ((x1 ,s1 ),··· ,(x2n ,s2n ))|||1,∞ . (II.6) 1≤i0 ≤2n si0 ∈
si ∈ for i=i0
That is, we first fix one sector and then take the sum over all other sectors of the ||| · |||1,∞ norms of the functions ωˆ 2n ((·,s1 ),··· ,(·,s2n )). With respect to this norm, we have a contraction bound of order M j and a tadpole bound of order Ml j . We first indicate how the contraction bound is derived. Let ϕ1 ((x1 ,s1 ),··· ,(xn ,sn )) and ϕ2 ((x1 ,s1 ),··· ,(xm ,sm )) be vertices, and ((x1 ,s1 ),··· ,(xn−1 ,sn−1 ), (x2 ,s2 ),··· ,(xm ,sm )) (j ) ,s )) dxn dx1 ϕ1 ((x1 ,s1 ),··· ,(xn ,sn )) C (xn −x1 ) ϕ2 ((x1 ,s1 ),··· ,(xm = m sn ,s1 ∈
be the graph constructed by connecting ϕ1 and ϕ2 with one line. Write C (j ) (k) = (j ) (j ) (j ) Cs (k), where Cs (k) = χs (k) C (j ) (k). Let Cs (x) be the Fourier transform of
s∈ (j ) Cs (k). 10
Then
For precise definitions of sectors, sectorizations, and the partition of unity see Def. VI.2 through Def. VI.5.
22
J. Feldman, H. Kn¨orrer, E. Trubowitz
=
dxn dx1
sn ,s,s1 ∈
(j )
ϕ1 ((x1 ,s1 ),··· ,(xn ,sn )) Cs
,s )). (xn −x1 ) ϕ2 ((x1 ,s1 ),··· ,(xm m
(II.7)
By conservation of momentum, the integral in (II.7) vanishes if sn ∩ s ∩ s1 = ∅. For , s, by double convolution fixed sectors s1 , · · · , sn , s1 , · · · , sm (j ) dxn dx1 ϕ1 ((x1 ,s1 ),··· ,(xn ,sn )) Cs (xn −x1 ) ϕ2 ((x1 ,s1 ),··· ,(xm ,sm )) 1,∞
≤
(j ) |||ϕ1 ((·,s1 ),··· ,(·,sn ))|||1,∞ Cs (x)1 |||ϕ2 ((·,s1 ),··· ,(·,sm ))|||1,∞
(II.8)
We consider the contribution to ||| |||1, having the sector s1 fixed. By (II.7), (II.8) and conservation of momentum it is bounded by (j ) |||ϕ1 ((·,s1 ),··· ,(·,sn ))|||1,∞ Cs (x)1 |||ϕ2 ((·,s1 ),··· ,(·,sm ))|||1,∞ . (II.9) s2 ,··· ,sn−1 ∈ sn ,s,s ∈ 1 ∈ s2 ,··· ,sm sn ∩s∩s1 =∅
Observe that, for given sn , there are at most three sectors s and at most three sectors s1 with sn ∩ s = ∅, sn ∩ s1 = ∅. Therefore (II.9) is bounded by (j ) 9 max |||ϕ1 ((·,s1 ),··· ,(·,sn ))|||1,∞ Cs (x)1 |||ϕ2 (·,s1 ),··· ,·,sm ))|||1,∞ s,s1 ∈
s2 ,··· ,sn−1 ,sn ∈ ∈ s2 ,··· ,sm
(j )
≤ 9 |||ϕ1 |||1, |||ϕ2 |||1, max Cs (x)1 . s∈
Fixing si with 2 ≤ i ≤ n − 1 or si with 2 ≤ i ≤ m leads to the same bound, so that ||| |||1, ≤ 9 |||ϕ1 |||1, |||ϕ2 |||1, max Cs (x)1 . (j )
s∈
(II.10)
(j )
As at the end of the previous subsection, one sees that max Cs (x)1 = O(M j ) . This s∈
gives the contraction bound of order M j . To derive the tadpole bound, let ϕ((x1 ,s1 ),··· ,(xn ,sn )) be a vertex and ((x1 ,s1 ),··· ,(xn−2 ,sn−2 )) (j ) = dxn−1 dxn ϕ((x1 ,s1 ),··· ,(xn−2 ,sn−2 ),(xn−1 ,sn−1 ),(xn ,sn )) C (xn−1 −xn ) sn−1 ,sn ∈
be obtained by joining the last two legs of ϕ to form a tadpole. As above, by conservation of momentum, for each choice of sectors s1 , · · · , sn−2 , |||((·,s1 ),··· ,(·,sn−2 ))|||1,∞ (j ) ≤ dxn−1 dxn ϕ(·,s1 ),··· ,(·,sn−2 ),(xn−1 ,sn−1 ),(xn ,sn )) Cs (xn−1 −xn ) sn−1 ,s,sn ∈ sn−1 ∩s∩sn =∅
≤3
sn−1 ,sn ∈
(j )
|||ϕ((·,s1 ),··· ,(·,sn−2 ),(·,sn−1 ),(·,sn ))|||1,∞ max Cs (x)∞ s∈
1,∞
Two Dimensional Fermi Liquid. 1: Overview
23
as, for a given sector sn−1 , there are at most three sectors s for which s ∩ sn−1 = ∅. Consequently (j ) ||||||1, ≤ 3 |||ϕ|||1, max Cs (x)∞ . s∈
As in the previous subsection, = O Ml j and the tadpole bound of order l follows. Mj
Substituting c = O M j and b = O Ml j into (II.5) yields the requirement that (j ) Cs (x)∞
|||ωˆ 2n |||1, is of order
M j (n−2)
ln−1
for all n.
(II.11)
In contrast to the norm ||| · |||1,∞ of Subsect. 7, this norm is compatible with change of scale. In §VI, we choose, for each scale i, a sectorization i of length li , where li goes to zero as some power of M1 i 11 . In going from scale j to scale j + 1, we construct from of w a a j –sectorized representative ω2n j +1 –sectorized representative ω2n of w2n 2n that fulfills (II.11) with j replaced by j + 1 and l replaced by lj +1 . It is constructed using a partition of unity subordinate to j +1 . See Example A.3. To give an idea of the underlying mechanism, we show that the problem with the ||| · |||1,∞ norm of the four point function described in Subsect. 7 does not occur for the ||| · |||1,j norm. To do so, we need a = j –sectorized representative for the original interaction kernel V (p1 , p2 , q1 , q2 ) whose ||| · |||1, norm is of order 1l . A natural such representative is
v (p1 , s1 ), (p2 , s2 ), (q1 , s3 ), (q2 , s4 ) = χs1 (p1 ) χs2 (p2 ) χs3 (q1 ) χs4 (q2 ) ×V (p1 , p2 , q1 , q2 ).
By conservation of momentum, the Fourier transform vˆ (·, s1 ),(·, s2 ), (·, s3 ), (·, s4 ) vanishes if (s1 + s2 ) ∩ (s3 + s4 ) = ∅. Here, s1 + s2 = {p1 + p2 p1 ∈ s1 , p2 ∈ s2 } . s1 0 s1 +s2
s2
One sees, by the same method that yielded Cs (x)1 = O(M j ), that the Fourier transform of each χs (k) fulfills χˆ s (x)1 = O(1). Therefore, for each choice of sectors s1 , s2 , s3 , s4 ,
|||vˆ (·, s1 ), (·, s2 ), (·, s3 ), (·, s4 ) |||1,∞ = O |||Vˆ |||1,∞ . Therefore, |||v||| ˆ 1, is of the order max (s2 , s3 , s4 ) ∈ 3 (s1 + s2 ) ∩ (s3 + s4 ) = ∅ . s1 ∈
(II.12)
To estimate this number, fix a sector s1 . Observe that the map F × F → R2 , (k1 , k2 ) → k1 + k2
(II.13)
This is forced on us by the condition li ≤ √1 i which is imposed to ensure that the curvature of the M sector does not affect Fourier transform estimates. 11
24
J. Feldman, H. Kn¨orrer, E. Trubowitz
is locally invertible at every (k1 , k2 ) ∈ F × F for which the tangent vector to F at k1 is not parallel to the tangent vector to F at k2 . From this one concludes that for a general choice of s2 (such that s1 + s2 is not close to 2k or k + antipode(k) for all k ∈ F ), there are only there are O(1) choices of sectors s3 , s4 such that (s1 +s2 )∩(s3 +s4 ) = ∅ . Since O 1l sectors, the contribution to (II.12) from all general sectors s2 is O 1l . One can see
that the few other sectors s2 also only contribute O 1l . Thus, indeed |||v||| ˆ 1, = O 1l . The precise argument is given in [FKTo4, Prop. XIX.1]. The estimates on the contraction and tadpole bounds and about change of sectorization in this subsection all are specific to two space dimensions. In three space dimensions, similar arguments would give that: a contraction bound for C (j ) is of order M j , 2 a tadpole bound for C (j ) is of order Ml j ,
1 |||v||| ˆ 1, = O l3 , since the map (II.13) has one dimensional fibers in this case. Thus, for the case of three space dimensions, the power counting suggested by sectors is not compatible with change of sectorization. This is the reason why, in this paper, we restrict ourselves to two space dimensions12 . The advantage of sectors is that they allow exploitation of conservation of momentum, while working in position space. One example is the estimate on |||v||| ˆ 1, derived two paragraphs ago. Another important example is that the improvements due to overlapping loops and particle–particle bubbles described in Subsect. 4 and 5 can be implemented using sectors. To this end we use a variant of the norm (II.6), with three sectors held fixed, max max |||ωˆ 2n ((x1 ,s1 ),··· ,(x2n ,s2n ))|||1,∞ . (II.14) |||ωˆ 2n |||3, = 1≤i1
si ∈ for i=i1 ,i2 ,i3
Its power counting is better by one factor of l than the power counting of the ||| · |||1, norm. That is, we expect |||ωˆ 2n |||3, is of order
M j n−2 l
for all n,
(II.15)
and in particular that |||ωˆ 4 |||3, is of order one13 . This norm is particularly useful for the four point function. Since a given momentum can lie in at most two sectors, ω4 (p1 , p2 , q1 , q2 )∞ ≤ 23 |||ωˆ 4 |||3, . In Example A.2, we show how (II.11) and (II.15) can be used to get improved power counting for the ||| · |||1, norm of the diagram discussed in Subsect. 4. 9. Cancellation Between Diagrams In Subsect. 7 and 8, we described how the power counting bound on a diagram formed by connecting two vertices ϕ1 and ϕ2 by r ≥ 1 lines14 is obtained. First we connect ϕ1 and ϕ2 by one line and call the resulting graph 12
Progress in the use of sectorization in three space dimensions has been made in [MR, DMR]. The ˆ 1, = argument above that the sectorized representative v of the original interaction fulfills |||v||| O 1l also shows that |||v||| ˆ 3, = O(1). 13
14
Again, for simplicity, we ignore orientation of the lines.
Two Dimensional Fermi Liquid. 1: Overview
25
and apply a contraction bound. Then we form the remaining r − 1 lines, each time applying a tadpole bound. Iterating this procedure, adding vertex after vertex, leads to the power counting for arbitrary diagrams. Since we are dealing with fermions, we may assume in our discussion that all vertex functions are antisymmetric. Let n1 ≥ r and n2 ≥ r be the number of legs of ϕ1 and ϕ2 , respectively. Assume that max{n1 , n2 } > r. Denote by G the
set of all diagrams obtained from joining ϕ1 and ϕ2 by r lines. G has cardinality nr1 nr2 r! . If one bounds each individual diagram by power counting, and sums over all diagrams in G, this large number of diagrams leads to divergences (due to the factor r!). We first describe how one finds cancellations between diagrams of G, then describe a blocking of diagrams that allows one to find similar cancellations for arbitrary numbers of vertices, and then show how, using this blocking, one may simultaneously exploit both these cancellations and overlapping loops. The first step in constructing the diagrams of G is again to choose one leg of ϕ1 and one leg of ϕ2 , form a line between these two legs, call the resulting graph and estimate the norm of in terms of the norms of ϕ1 and ϕ2 using a contraction bound. The second step is to choose (r − 1) additional legs of ϕ1 and (r − 1) additional legs of ϕ2 . r −1 r −1 =
ϕ1
ϕ2
The third step is to form all possible connections between the (r − 1) legs of ϕ1 and the (r 1) legs of ϕ2 chosen in the second step. There are n1 n2 choices in the first step,
n1− −1 n2 −1 r−1 r−1 choices in the second step and (r − 1)! choices in the third step. Each diagram of G is obtained r times, since with the first step we distinguish one of the r lines. Observe that
n1 −1 n2 −1
n1 n2 1 r n1 n2 r−1 r r! r−1 (r − 1)! = r There are cancellations amongst the diagrams formed in the third step described above. For simplicity, assume that the (r − 1) legs of ϕ1 chosen in the second step are labeled by 2, · · · , r and are all incoming, and that the (r − 1) legs of ϕ1 are labeled by n2 − r + 2, · · · , n2 and are all outgoing. Then the sum of all diagrams formed in the third step is dx2 ···dxr dxn −r+2 ···dxn (x2 ,··· ,xr , ··· xn −r+2 ,···xn ) 2 2 2 2 × ψ¯ (x2 ) · · · ψ¯ (xr ) ψ (xn 2 −r+2 ) · · · ψ (xn 2 ) dµC (j ) . Its · 1,∞ norm is bounded by
1,∞
sup x2 ,··· ,xr xn −r+2 ,···xn 2 2
ψ¯ (x2 ) · · · ψ¯ (xr ) ψ (xn 2 −r+2 ) · · · ψ (xn 2 ) dµC (j ) .
The magnitude of the functional integral is ψ¯ (x2 ) · · · ψ¯ (xr ) ψ (xn 2 −r+2 ) · · · ψ (xn 2 ) dµC (j ) = det C (j ) (xµ −xn 2 −r+ν ) µ,ν=2,··· ,r .
26
J. Feldman, H. Kn¨orrer, E. Trubowitz
Observe that the µ–ν matrix entry ı− (j ) 2 C (j ) (xµ − xn 2 −r+ν ) = dk e C (k) ı− C (j ) (k) 2 √ = eı− |C (j ) (k)| , e (j ) |C
(k)|
L2
(we are deliberately ignoring some unimportant factors of 2π) is the L2 inner product of ı− C (j ) (k) 2 √ the vector vµ (k) = eı− |C (j ) (k)| and the vector vν (k) = e . |C (j ) (k)| The vectors vµ , vν all have L2 norm C (j ) (k)1 . Therefore, by Gram’s bound on determinants, ψ (x2 ) · · · · · · ψ¯ (xn 2 ) dµC (j ) ≤ C (j ) (k)r−1 1 . Thus the bound on the sum of all diagrams formed at the third step is ||| |||1,∞ C (j ) (k)r−1 ≤ c C (j ) (k)r−1 |||ϕ1 |||1,∞ |||ϕ2 |||1,∞ , 1 1 where c is a contraction bound. Consequently, the ||| · |||1,∞ norms of the sum of all graphs in G is bounded by
n1 −1 n2 −1 (j ) r−1 (j ) 1 |||ϕ1 |||1,∞ |||ϕ2 |||1,∞ r n1 n2 r−1 r−1 C (x)1 C (k)1 since, as seen in Subsect. 7, C (j ) (x)1 is a contraction bound. This is smaller than the bound (G) C (j ) (x)1 C (j ) (k)r−1 |||ϕ1 |||1,∞ |||ϕ2 |||1,∞ 1 1 . derived by summing the graphwise bounds, by a factor of (r−1)! In general, we say that b is an integral bound, if the following holds: Let ϕ be any antisymmetric vertex, 2r ≤ n, and S(x2r+1 , · · · , xn ) be the sum of the values of all diagrams obtained from ϕ by joining each of the legs labeled 1, · · · , r to one and only one of the legs labeled r + 1, · · · , 2r.
2r + 1, · · · , n
ϕ
Then the norm of S is bounded by b2r times the norm of ϕ. The argument in the previous paragraph shows that C (j ) has an integral bound with respect to the ||| · |||1,∞ norm that is of order C (j ) (k)1 = O( √ 1 j ). Similarly one sees that C (j ) has an integral bound, M
l with respect to the ||| · |||1, norm, that is of order O . Thus, in both cases, the Mj integral bound is of the order of the square root of the tadpole bound found before. The discussion above shows how to get cancellations between diagrams that have only two vertices. The combinatorics for treating diagrams of arbitrary size was developed in [FMRT, FKTcf, FKTr1, FKTr2]. We sketch it here. As mentioned in Subsect. 4, we Wick order with respect to future covariances in order to avoid tadpoles. Thus, we write ¯ = :U (ψ, ψ): ¯ C (≥j ) , W(0, 0, ψ, ψ) ¯ = :U (ψ, ψ): ¯ C (≥j +1) , W (0, 0, ψ, ψ)
Two Dimensional Fermi Liquid. 1: Overview
where C (≥j ) =
27
C (i) . Then the kernels of U are sums of diagrams whose vertices
i≥j
are kernels of U and which have two kinds of lines. The first arises from integrating with respect to dµC (j ) and has propagator C (j ) . The second arises in Wick ordering W and has propagator C (≥j +1) . The subgraph of each diagram obtained by deleting the C (≥j +1) lines must be connected and tadpole–free. For clarity, we ignore the Wick lines. That is, we discuss a simplified situation in which W is Wick ordered with respect to C (j ) and W is not Wick ordered at all15 , so that we write ¯ = :W(ψ, ¯ C (j ) W(0, 0, ψ, ψ) ψ): ¯ = r W(ψ, ψ) dp1 ···dqn w ˜ 2n (p1 ,··· ,qn ) δ (p1 +···+pn −q1 −···−qn ) n≥0
× ψ¯ (p1 ) · · · ψ¯ (pn ) ψ (q1 ) · · · ψ (qn ). Then w2n =
1 connected diagrams without tadpoles with . ! 2n external legs and vertices from w˜ 2 , w˜ 4 , · · ·
All diagrams are labeled. A rooted diagram is a diagram with one distinguished vertex, called the root. Clearly, 1 1 connected, rooted, tadpole–free diagrams with w2n = . ! 2n external legs and vertices from w˜ 2 , w˜ 4 , · · ·
The distance between two vertices in a diagram is the minimal number of lines needed to form a path connecting these two vertices. In a rooted diagram, the r th ring is defined as the set of vertices of distance r from the root. Thus, the zeroth ring is the root itself, and the first ring consists of all vertices that are directly connected to the root by a line. Observe that the full subgraph formed by the union of the first r rings of a diagram G is again a connected diagram Gr . Each leg emanating from a vertex of the r th ring that is not part of a line in Gr (that is, each external leg of Gr ) is either an external leg for the whole diagram or is connected to a vertex of the (r + 1)st ring. Observe that, for each graph, there is an r0 such that the r th ring is empty for all r ≥ r0 .
For simplicity, we only discuss the case that n = 0, i.e. that there are no external legs. Given a rooted diagram without external legs, we have the following combinatorial data: • r = (vertices of the r th ring), 15
General Wick ordering is treated in [FKTr2].
28
J. Feldman, H. Kn¨orrer, E. Trubowitz
− γi,r 0 be the number of legs of the i th vertex in the r th ring • For i = 1, · · · , r , let γi,r + γi,r r − 1 r that are connected to a vertex in ring number . r + 1 − 0 +γ + legs and γ − ≥ By the definition of “ring”, the i th vertex in the r th ring has γi,r +γi,r i,r i,r − 0 ,γ+) 1 for all 1 ≤ r ≤ r0 . The sequences = ( 1 , 2 , · · · ) and γ = (γi,r , γi,r r=1,2,··· i,r i=1,··· r are called the combinatorial data of the rooted graph. We only exploit cancellations between connected rooted graphs with the same com γ and let G be the set of all diagrams binatorial data. So fix some combinatorial data , ˜ with these combinatorial data. Denote by Gr the sum of the values of all subgraphs Gr of graphs G ∈ G. View it as a single vertex. It has r i=1
+ γi,r =
r+1 i=1
− γi,r+1
˜ r is formed from G ˜ r−1 and how the norm of G ˜ r is external legs. We describe how G th ˜ bounded in terms of the norm of Gr−1 : Connect each of the r vertices of the r ring ˜ to G by one line and apply contraction bounds. Then apply an integral bound for the r r−1 − th ˜ (γ i=1 i,r − 1) remaining connections between Gr−1 and the r ring. + γ1,r
1 0 γ1,r
˜ r−1 G
0 γ2,r
2 + γ2,r 0 lines Then, using a variant of the integral bound, form all connections between the γ1,r 0 coming from the first vertex, the γ2,r lines coming from the second vertex, · · · and the γ 0r ,r lines coming from the last vertex, always avoiding tadpoles. Repeat this procedure for all r for which r = 0. The bounds obtained in this way are summable over all combinatorial data. Observe that ladders
− 0 =0 have very special combinatorial data: Each r is either zero, one or two, γi,r = 2, γi,r + and γi,r is either zero or two. In the example
root
Two Dimensional Fermi Liquid. 1: Overview
29
+ + + + + we have 1 = 2 = 2, 3 = 1, γ1,1 = γ2,1 = γ1,2 = 2, γ2,2 = γ1,3 = 0. Not all diagrams with such combinatorial data are ladders, but there are so few of them that the non–ladder diagrams with these combinatorial data can be bounded individually without generating divergences. In many cases, the combinatorial data of a diagram alone allow the detection of an overlapping loop. The two basic cases are: − 0 ≥ 3 then there is an overlap+ γi,r (i) If, for some r ≥ 1 and 1 ≤ i ≤ r , we have γi,r ping loop, as indicated in the following figures. Here vi denotes the i th vertex of the r th ring. − vi γi,r ≥3
˜ r−1 G
vi ˜ r−1 G
vi
vi
˜ r−1 G
or
vi
vi
˜ r−1 G
vi
vi
− 0 ≥1 γi,r = 2, γi,r
− 0 ≥2 γi,r = 1, γi,r
− 0 ≥ 2, γ + ≥ 1, γ + ≥ 1 then there is an + γ1,r (ii) If r = 2, r+1 = 1 and γ1,r 1,r 2,r overlapping loop as seen in the following figures.
r th ring (r + 1)st ring
˜ r−1 G
r th ring (r + 1)st ring
˜ r−1 G − ≥2 γ1,r
− 0 ≥1 = 1, γ1,r γ1,r
These overlapping loops can be used to generate improved estimates by the techniques mentioned at the end of Subsect. 8 without seriously affecting the cancellations described above. If one also takes external legs into account and if w2 = 0, then cases (i) and (ii) are enough to identify at least one overlapping loop in each four–legged diagram that does not have the combinatorial data of a ladder diagram. See §VII of [FKTr2]. 10. The Counterterm The counterterm δe is constructed so that the proper self energy ˇ 2,j (p) δ(p − q) be the Fourier transform of is bounded by 21 |ik0 − e(k)|. That is, let G the two point Green’s function G2,j (x, y) constructed in Theorem I.4, and define j (k) by (≥j ) ˇ 2,j (k) = U (k)−ν (k) . G ık0 −e(k)−j (k) Then |j (k)| ≤ 21 |ik0 − e(k)|. To achieve this, we specify, for each scale j , a space Kj of “allowed future counterterm contributions for scales after j ”. It consists of functions K(k) of the vector part of
30
J. Feldman, H. Kn¨orrer, E. Trubowitz
k = (k0 , k) only. These functions are required to be bounded by a small constant times lj +1 . The numerator reflects overlapping loop volume improvement. We construct a M j +1 map δej from Kj to the space E of counterterms such that, if one writes
ˇ 2,j k; δej (K) = G
U (k)−ν (≥j ) (k) ık0 −e(k)−j (k;K)
then j (0, k); K = K(k) in the j th neighbourhood. Think of δej (k; K) as being of the form δ e˜j (k; K) + K(k) with δ e˜j (k; K) implementing cancellations that have already been identified and K(k) reserved to implement as yet unidentified cancellations. Since
j (k; K) equals δej (k; K) plus higher order contributions, it is easy to solve j (0, k); K = K(k) for δ e˜j (k; K). The algebra of this (recursive) construction is presented in detail in §III. We prove that δe = lim δej (0) exists and has the required j →∞
properties. At each scale, the properties of the counterterms are used to obtain the bound for the w2 (p, q) of order M1j necessary for power counting. The choice indicated above guarantees that such a bound holds for momenta k = q − p for which the “temperature th part” k0 vanishes. To get this bound
1 for arbitrary k = (k0 , k) in the j neighbourhood, in particular when |k0 | = O M j , we show that the k0 derivative of this function is of order one. For this, we have to control derivatives (in momentum space) of all the data in the renormalization construction. Control of derivatives in momentum space is also needed to provide decay in position space. This is used, for example, in proving contraction bounds through L1 norms in position space. We pointed out in Subsect. 1 that derivatives of δej (0) might blow up as j → ∞. Therefore we have to pay special attention to the behaviour of derivatives. This is the reason why we introduce, in Definition V.2, the “norm domain” that gives a convenient notation for bounds on derivatives of functions. III. Formal Renormalization Group Maps To simplify notation involving the fields, we define, for ξ = (x0 , x, σ, a) = (x, a) ∈ R × Rd × {↑, ↓} × {0, 1}, the internal fields ψ(x) = ψσ (x0 , x) if a=0 ψ(ξ ) = ¯ . ψ(x) = ψ¯ σ (x0 , x) if a=1 Similarly, we define for an external variable η = (y0 , y, τ, b) = (y, b) ∈ R × Rd × {↑, ↓} × {0, 1}, the source fields φ(y) = φσ (y0 , y) if b=0 φ(η) = ¯ . φ(y) = φ¯ σ (y0 , y) if b=1 B = R × Rd × {↑, ↓} × {0, 1} is called the “base space” parameterizing the fields. An antisymmetric function S(ξ, ξ ) on B × B is called a covariance and determines a Grassmann Gaussian measure by ψ(ξ )ψ(ξ ) dµS (ψ) = S(ξ, ξ ).
Two Dimensional Fermi Liquid. 1: Overview
31
A function S(k) on momentum space, R × Rd , defines a function S(x, x ) on position 2
space R × Rd × {↑, ↓} by
d d+1 k ı− e S(k). (III.1) (2π)d+1
2 Any function S(x, x ) on position space, R × Rd × {↑, ↓} defines a unique antisymmetric function S(ξ, ξ ) on B × B by if a=0, a =1 S(x, x ) S((x,a),(x ,a )) = −S(x , x) if a=1, a =0 . (III.2) 0 if a=a
S(x, x ) = δσ,σ
We denote the associated Grassmann Gaussian measure again by dµS . With the notation introduced above the source term of (I.5) is φJ ψ = dξ dξ φ(ξ )J (ξ, ξ )ψ(ξ ) = ψJ φ, where the operator J has kernel
if a=1, a =0 1
J (x0 , x, σ, a), (x0 , x , σ , a ) = δ(x0 − x0 )δ(x − x )δσ,σ −1 if a=0, a =1. 0 otherwise (III.3)
Definition III.1 (Renormalization Group Maps). Let S be a covariance and W(φ, ψ) a Grassmann function for which Z = eW (0,ζ ) dµS (ζ ) = 0. We set S (W)(φ, ψ) = log Z1 eW (φ,ψ+ζ ) dµS (ζ ), 1 ˜ S (W)(φ, ψ) = log Z eφJ ζ eW (φ,ψ+ζ ) dµS (ζ ). ˜ S map Grassmann functions in the variables φ, ψ, to Grassmann functions in S and the same variables. They obey the semigroup property S1 +S2 = S1 ◦ S2
,
˜ S1 +S2 = ˜ S1 ◦ ˜ S2 .
(III.4)
By Lemma VII.3 of [FKTo2], they are related by ˜ S (W)(φ, ψ) = 1 φJ SJ φ + S (W)(φ, ψ + SJ φ) 2 where, for any covariance S, φSφ = dξ1 dξ2 φ(ξ1 ) S(ξ1 , ξ2 ) φ(ξ2 ). Clearly ˜ C IR(j ) (δe) (V)(φ, Gj (φ; δe) = 0) ¯ ¯ Observe that
with
ψ) = V(ψ). V(φ,
C IR(j¯) (k; 0) = C IR(1) (k; 0) + C (1) (k) + · · · + C (j −2) (k) + C (j −1) (k),
(III.5)
32
J. Feldman, H. Kn¨orrer, E. Trubowitz
where C (i) (k0 , k) =
ν (i) (k) ık0 −e(k) .
Therefore, by induction and the semigroup property, ˜ C (j −1) ◦ ˜ C (j −2) ◦ · · · ◦ ˜ C (1) ◦ ˜ C IR(1) (0) (V)(φ, 0). Gj (φ; 0) = ¯
(III.6)
Since C (i) (k) is supported on the i th shell only, (III.6) would provide a convenient framework for a multiscale analysis of the unrenormalized generating functional Gj (φ; 0), by ¯ recursively controlling the effective interactions ˜ C (i−1) ◦ · · · ◦ ˜ C (1) ◦ ˜ C IR(1) (0) (V)(φ, ψ) Gi (φ, ψ; 0) =
˜ = C (i−1) Gi−1 (0) (φ, ψ).
(III.7)
We now describe an analog of (III.6) for the case δe = 0. It incorporates a number of technical modifications needed to maintain control over the bounds. These modifications include the introduction of a scale–dependent Wick ordering and a scale–dependent contribution to the counterterm, periodic shifting of a portion of the interaction into the covariance and the periodic isolation of purely φ dependent terms. To this end, the effective interaction Gi (φ, ψ; 0) of (III.7) is replaced by a triple (W, G, u) with ◦ G(φ) being the purely φ dependent part of the effective interaction, ◦ W(φ, ψ) being the rest of the interaction, and ◦ u being the kernel of a quadratic Grassmann function that has been moved from the effective interaction into the covariance. To help clarify the algebraic structure of this more complicated setting, we outline the construction in the category of formal power series, without specifying the bounds that will ultimately be proven. To avoid formal power series in infinitely many variables, we introduce a coupling constant λ into the interaction ¯ 1 )ψ(x2 )ψ(x ¯ 3 )ψ(x4 ) dx1 dx2 dx3 dx4 V(ψ) = λ V (x1 , x2 , x3 , x4 ) ψ(x (R×Rd ×{↑,↓})4
and deal with Grassmann algebra valued formal power series in λ. Correspondingly, the final counterterm, δe(k) is made λ–dependent. Definition III.2. The space of formal (final) counterterms, E f orm , consists of the space ∞ of all formal power series δe(k, λ) = δen (k)λn in λ each of whose coefficients is n=1 supported in k ∈ Rd U (k) = 1 . As indicated above the final counterterm δe is built up from contributions at each scale. f orm
of formal (future) counterterms for scale j is the Definition III.3. The space Kj space of all formal power series in λ whose coefficients are antisymmetric, translation invariant functions of x, x . The coefficient of λ0 vanishes and the Fourier transform of each coefficient is supported on supp ν (≥j +1) (0, k) . Definition III.4. A formal interaction triple at scale j is a triple (W, G, u) that obeys the following conditions:
Two Dimensional Fermi Liquid. 1: Overview
33
◦ W(φ, ψ; K) is a formal power series in λ, whose coefficients are functions of K ∈ f orm Kj that take values in the Grassmann algebra generated by the fields φ(ξ ) and ψ(ξ ). Furthermore W(φ, 0; K) = 0 and W(φ, ψ; K)λ=0 = 0. The coefficients of W are translation invariant, spin independent and particle–number conserving. f orm ◦ G(φ; K) is a formal power series in λ whose coefficients are functions of K ∈ Kj that take values in the Grassmann algebra generated by the fields φ(ξ ). The constant term G(0; K) = 0. The coefficients of G are translation invariant, spin independent and particle–number conserving. ◦ u(ξ1 , ξ2 ; K) is a formal power series in λ whose coefficients are antisymmetric, spin independent, particle number conserving, translation invariant functions of ξ1 , ξ2 ∈ B f orm and K ∈ Kj . The Fourier transform16 u(k; ˇ K) of u obeys
ˇ uˇ (0, k); K = −K(k)
ˇ uˇ k; K λ=0 = −K(k).
The condition that W(φ, 0; K) = 0 ensures that G(φ; K) contains the full pure φ part of the effective interaction. As mentioned above, u is the kernel of a quadratic Grassmann function that has been moved from the effective interaction into the covariance. Precisely, Definition III.5. (i) Let u(ξ1 , ξ2 ) be a formal power series in λ whose coefficients are antisymmetric, spin independent, particle number conserving, translation invariant functions of ξ1 , ξ2 ∈ B. Then (j )
Cu (k) =
ν (j ) (k) . ik0 − e(k) − u(k) ˇ
(ii) Let u(ξ1 , ξ2 ; K) be a formal power series in λ whose coefficients are antisymmetric, spin independent, particle number conserving, translation invariant functions of f orm ξ1 , ξ2 ∈ B. Then, for K ∈ Kj , Cj (u; K)(k) =
ν (≥j ) (k) , (≥j +2) (k) ˇ ık0 −e(k)−u(k;K)− ˇ K(k)ν
Dj (u; K)(k) =
ν (≥j +1) (k) . (≥j +2) (k) ˇ ık0 −e(k)−u(k;K)− ˇ K(k)ν
For the formal interaction triple (W, G, u) at scale j , integrating out scale j involves the evaluation of an integral with respect to the Gaussian measure with covariance (j ) Cu(K) . The effective interaction W will be Wick ordered with respect to the covariance Cj (u(K); K). One important property of the Wick ordering covariances is that (j ) Dj (u; K) = Cj (u; K) − Cu(K) , so that :f (ψ + ζ ):Cj (u,K) dµC (j ) (ζ ) = :f (ψ):Dj (u,K) u(K)
for all Grassmann functions f (ψ), by Proposition A.2.i and Lemma A.4.ii of [FKTr1]. This property prevents the formation of Wick self–contractions 16 A systematic set of Fourier transform conventions will be given in Def. VI.1. In the present context u(k; ˇ K) is the Fourier transform of u ((0, 0, ↑, 1), (x0 , x, ↑, 0); K) as in Theorem I.5.
34
J. Feldman, H. Kn¨orrer, E. Trubowitz
and ensures that the effective interaction resulting from integrating out scale j is naturally Wick ordered with respect to the “output Wick ordering covariance” Dj (u, K). Also the Wick ordering covariances have been chosen so that, for k of scale at least (≥j +2) (k) = u(k; ˇ ˇ j + 3, u(k; ˇ K) + K(k)ν ˇ K) + K(k) vanishes for k0 = 0. This property ensures that the denominator still vanishes only on the Fermi surface. Definition III.6. Integrating out the fields of scale j is implemented by the map j , which maps a formal interaction triple (W, G, u) of scale j to the triple (W , G , u) determined by :W (φ,ψ+ζ ;K):ψ,Cj (u;K) 1 :W (φ, ψ; K):ψ,Dj (u;K) = log Z(φ) eφJ ζ e dµC (j ) (ζ ) u(K)
G (φ) = G(φ) + log where log Z(φ) =
eφJ ζ e
log
Z(φ) Z(0) ,
:W (φ,ψ+ζ ;K):ψ,Cj (u;K)
dµC (j ) (ζ ) dµDj (u;K) (ψ). u(K)
(W , G , u)
In fact is again a formal interaction triple of scale j . That W (φ, 0; K) = 0 follows by inserting the definitions into W (φ, 0; K) = :W (φ, ψ; K):ψ,Dj (u,K) dµDj (u,K) (ψ). To verify W (φ, ψ; K)λ=0 = 0, observe that for λ = 0, (j ) 1 φJ ζ :W (φ,ψ+ζ ;K):ψ,Cj (u;K) e e dµC (j ) (ζ ) = eφJ ζ dµC (j ) (ζ ) = e 2 φJ Cu(K) J φ u(K)
u(K)
is independent of ψ. To verify the various symmetries, apply Remark B.5 of [FKTo2]. Remark III.7. Define, for all 1 ≤ i ≤ j ≤ ∞, ν (≥i) (k)−ν (≥j ) (k) (≥j ) (k)] [i,j ) ik −e(k)− u(k)[1−ν ˇ 0 Cu (k) = ν (≥i) (k) ik0 −e(k)−u(k) ˇ
(≥i)
We also write Cu
[i,∞)
= Cu
if j < ∞ if j = ∞
.
.
Let (W , G , u) = j (W, G, u). Then, for any infrared cutoff j + 2 ≤ j¯ ≤ ∞, formally, ignoring the problems engendered by the infrared singularity, φJ ψ :W (φ,ψ;K):ψ,C (u;K) j e dµC [j,j ) (ψ) e u ¯ G(φ) + log :W (0,ψ;K): ψ,Cj (u;K) e dµC [j,j ) (ψ) u ¯ φJ ψ :W (φ,ψ;K):ψ,D (u;K) j e dµC [j +1,j ) (ψ) e u ¯ . = G (φ) + log :W (0,ψ;K): ψ,Dj (u;K) e dµC [j +1,j ) (ψ) u ¯
Two Dimensional Fermi Liquid. 1: Overview [j,j )
(j )
35
[j +1,j )
Proof. Since Cu ¯ = Cu + Cu ¯, f (φ, , ψ + ζ ) dµC (j ) (ζ ) dµC [j +1,j ) (ψ) f (φ, ψ) dµC [j,j ) (ψ) = u u ¯ u ¯ by Proposition I.21 of [FKTffi]. Hence, by Proposition A.2.ii of [FKTr1], :W (φ,ψ;K):ψ,Cj (u;K) log eφJ ψ e dµC [j,j ) (ψ) u ¯ :W (φ,ψ+ζ ;K):ψ,Cj (u;K) = log eφJ (ψ+ζ ) e dµC (j ) (ζ ) dµC [j +1,j ) (ψ) u u ¯ :W (φ,ψ;K):ψ,Dj (u;K) +log Z(φ) = log eφJ ψ e dµC [j +1,j ) (ψ) u ¯ (φ,ψ;K): : W ψ,Dj (u;K) = log eφJ ψ e dµC [j +1,j ) (ψ) + log Z(φ) u ¯ :W (φ,ψ;K):ψ,Dj (u;K) = log eφJ ψ e dµC [j +1,j ) (ψ) + log Z(0) + G (φ) − G(φ). u ¯ Subtracting the same equation with φ = 0 gives the desired result.
When we derive bounds on the map j , we get improvements on the two– and four– legged contributions to W (0, ψ) by exploiting overlapping loops. See Subsect. 4 of §II and the introduction to [FKTr2]. However to ensure the presence of and to detect sufficiently many overlapping loops, we need that the two–legged part of W(0, ψ) vanishes. See the end of Subsect. 9 of §II and Remark VI.7 of [FKTr2]. Therefore, we wish that the formal interaction triple (W, G, u) input to j be an element of (j,f orm)
Definition III.8 (Formal Input Data). The space Din of formal input data consists of the set of all formal interaction triples (W, G, u) at scale j , in the sense of Definition III.4, obeying (i) If the effective interaction W(K) = Wm,n with m,n≥0
Wm,n =
m i=1
dηi
n
dξ Wm,n (η1 ,··· ,ηm ,ξ1 ,··· ,ξn )φ(η1 ) · · · φ(ηm )ψ(ξ1 ) · · · ψ(ξn ),
=1
then W0,2 = 0. (<j ) (<j ) (ii) The coefficient of λ0 in G(φ; K) is 21 φJ C−K J φ. Here Cu =
U (k)−ν (≥j ) (k) . ik0 −e(k)−u(k) ˇ
(j,f orm)
, the output no longer satisfies condiWhen j is applied to an element of Din tion (i) of Definition III.8. Rather, the output lies in (j,f orm)
Definition III.9 (Formal Output Data). The space Dout of formal output data consists of the set of all formal interaction triples (W, G, u) at scale j , in the sense of (≤j ) Definition III.4, for which the coefficient of λ0 in G(φ; K) is 21 φJ C−K J φ. (j,f orm)
Lemma III.10. Let (W, G, u) ∈ Din
(j,f orm)
. Then j (W, G, u) ∈ Dout
.
36
J. Feldman, H. Kn¨orrer, E. Trubowitz
Proof. Set (W , G , u) = j (W, G, u). Then G (φ)λ=0 = G(φ)λ=0 + log Z(φ) Z(0) λ=0
(<j ) = 21 φJ C−K J φ + log Z(φ) Z(0) λ=0 .
Since log Z(φ)λ=0 = the result follows.
(j )
eζ J φ dµC (j ) (ζ ) dµDj (u;K) (ψ) = 21 J φC−K J φ
log
−K
(j,f orm)
Elements of the space Dout are not of the form desired for the application of j +1 , the map implementing the integration out of scale j + 1. In particular, the two– f orm point part of the effective interaction is nonzero and Kj is not the appropriate space of counterterms. Below, just before Proposition III.12, we construct maps (j,f orm)
Oj : Dout (j,f orm)
and, for each (W, G, u) ∈ Dout
(j +1,f orm)
→ Din
,
, f orm
f orm
renj,j +1 ( · , W, u) : Kj +1 → Kj
with the following property: If (W, G, u) ∈ Dout and (W , G , u ) = Oj (W, G, u) f orm and if K ∈ Kj +1 and K = renj,j +1 (K , W, u), then, for any infrared cutoff j + 1 ≤ j¯ ≤ ∞, formally, ignoring the problems engendered by the infrared singularity, (j,f orm)
:W (φ,ψ;K):
ψ,Dj (u;K) dµC [j +1,j ) (ψ) eφJ ψ e u(K) ¯ G(φ; K) + log :W (0,ψ;K): ψ,Dj (u;K) e dµC [j +1,j ) (ψ) u(K) ¯ φJ ψ :W (φ,ψ;K ):ψ,C (u ;K ) j +1 e dµC [j +1,j ) (ψ) e u (K )¯ . = G (φ; K ) + log :W (0,ψ;K ): ψ,Cj +1 (u ;K ) e dµC [j +1,j ) (ψ) u (K )¯
(III.8)
The map Oj moves the two–point part of W into the covariance, through u , and updates the Wick ordering covariance (for a precise statement, see (III.11) below). The map renj,j +1 introduces the contribution of the current scale into the counterterm. Using the maps j of Definition III.6 and the maps Oj and renj,j +1 we can describe the renormalization group flow. We start by choosing an arbitrary but fixed j0 ≥ 2 and integrate out all scales from 1 to j0 to arrive at the initial effective interaction triple (j ,f orm) (Wjout , Gjout , uj0 ) ∈ Dout0 with 0 0 ˜ = Wjout 0 ˜ Gjout = 0
(≤j )
˜ (V)(φ, ψ) −
(≤j )
(V)(φ, 0),
C−K 0 C−K 0
uj0 = −K.
(≤j )
C−K 0
(V)(φ, 0),
Two Dimensional Fermi Liquid. 1: Overview
37
The renormalization group flow is the concatenation of the maps Oj0 , j0 +1 , Oj0 +1 , j0 +2 , · · · , j , Oj , · · · applied to the initial datum. Set , Gjout , uj0 ) (Wjin , Gjin , uj ) = Oj −1 ◦ j −1 ◦ Oj −2 ◦ · · · ◦ j0 +1 ◦ Oj0 (Wjout 0 0 (j,f orm)
∈ Din (Wjout , Gjout , uj )
,
= j ◦ Oj −1 ◦ j −1 ◦ Oj −2 ◦ · · · ◦ j0 +1 ◦ Oj0 (Wjout , Gjout , uj0 ) 0 0 (j,f orm)
∈ Dout
(III.9)
. f orm
We recursively define maps reni,j : Kj
f orm
→ Ki
, j0 ≤ i ≤ j by
renj,j (K) = K,
reni,j (K) = reni,j −1 renj −1,j (K) f orm
We define for K ∈ Kj
for j > i.
, δej (K) = renj0 ,j (K)
and show that, for the generating function of the connected Green’s functions at scale j¯ of Theorem I.4, φJ ψ :W in (φ,ψ;K):ψ,C (u ;K) j j e j dµC [j,j ) (ψ) e ¯ uj (K) in ¯ Gj (φ, φ; δej (K)) = Gj (φ; K) + log :Wjin (0,ψ;K):ψ,Cj (uj ;K) ¯ e dµC [j,j ) (ψ) ¯ uj (K) φJ ψ :W out (φ,ψ;K):ψ,D (u ;K) j j e j dµC [j +1,j ) (ψ) e uj (K)¯ out = Gj (φ; K) + log out e:Wj (0,ψ;K) :ψ,Dj (uj ;K) dµC [j +1,j ) (ψ) uj (K)¯ (III.10) f orm
for K ∈ Kj and j0 ≤ j ≤ j¯ − 2. When j¯ = ∞, (III.10) holds with G∞ being the formal generating function of the connected Green’s functions (I.4). Equation (III.10) is proven by induction on j , combining Remark III.7 and (III.8). so that, ˜ (≤j0 ) (V), To start the induction, observe that G out (φ; K) + W out (φ, ψ; K) = j0
j0
C−K
by the semigroup property (III.4), ¯ K) = Gjout (φ; K) + log Gj (φ, φ; 0 ¯
=
Wjout (φ,ψ;K) 0
dµ
W out (0,ψ;K) e j0 dµ
Gjout (φ; K) + log 0
eφJ ψ e
eφJ ψ e
(j ,j )
0 (ψ) C−K ¯
(j0 ,j ) C−K (ψ) out :Wj (φ,ψ;K):ψ,Dj (uj ;K) 0 0 0
¯
dµ
(j ,j )
Cu 0 (K) ¯
(ψ)
j0
e
:Wjout (0,ψ;K) 0
:ψ,Dj0 (uj0 ;K) dµ
(j ,j )
Cu 0 (K) ¯
(ψ)
j0
with Dj0 = 017 . 17 Wick ordering with respect to the covariance zero is the same as not Wick ordering at all. D = 0 j0 is a convention that we introduce purely for notational consistency.
38
J. Feldman, H. Kn¨orrer, E. Trubowitz
In Theorem VIII.5, we shall prove bounds that show that the limits δe = lim δej (0) j →∞
¯ δe) = lim G out (φ; 0) = lim G in (φ; 0) exist. To prove Theorem I.4 we and G(φ, φ; j j
j →∞ j →∞ out ¯ show that lim Gj (φ, φ) − Gj (φ; 0) = 0. j →∞
We now describe the passage from output data to input data, that is the maps (j,f orm)
Oj : Dout
(j +1,f orm)
→ Din
f orm
f orm
renj,j +1 ( · , W, u) : Kj +1 → Kj
.
(j,f orm)
Let (W, G, u) ∈ Dout
and write W(0, ψ; K) = dξ1 · · · dξm W0,m (ξ1 , · · · , ξm ; K) ψ(ξ1 ) · · · ψ(ξm ) m≥2
with each W0,m (ξ1 , · · · , ξm ; K) antisymmetric under permutation of the ξi ’s. To perform the reWick ordering, observe that, if E is any covariance and ˜ :W(φ, ψ; K):ψ,Dj (u;K) = :W(φ, ψ; K):ψ,Dj (u;K)+E then, by Lemma A.2.i of [FKTr1], ˜ W(φ, ψ; K) =
W(φ, ψ + ψ ; K) dµE (ψ ).
We wish to choose the covariance E such that, after we reWick order :W(φ, ψ; ˜ ˜ K):ψ,Dj (u;K) to :W(φ, ψ; K):ψ,Dj (u;K)+E and move the quadratic part of W(φ, ψ; K) into the covariance, replacing u(K) by u (K ), then the Wick ordering covariance Dj (u; K) + E is exactly Cj +1 (u ; K ). When we replace u(K) by u (K ), we choose f orm f orm K ∈ Kj as a function of K ∈ Kj +1 in such a way that u (K ) fulfills the third condition of Definition III.4. This function will be denoted K(K ) = renj,j +1 (K , W, u) = K + δK(K ). The unknowns in the scheme outlined in the last paragraph are E and δK. They are determined implicitly by the requirements of the last paragraph. We choose to express E ˜ 0,2 . Once and δK in terms of one function q(ξ1 , ξ2 ; K ), with 21 q being the kernel of W q is determined, we set
ˇ δ K(k; K ; q) = qˇ (0, k); K ν (≥j +1) ((0, k)), K(K ; q) = K + δK(K ; q), ˇ K(K ; q)) + q(k; ˇ K )ν (≥j +1) (k), uˇ (k; K ; q) = u(k;
E(K ; q) = Cj +1 u ( · ; q); K − Dj (u; K(K ; q)). Set ˜ W(φ, ψ; K ; q) = and expand ˜ W(0, ψ; K ; q) =
(III.11)
W(φ, ψ + ψ ; K(K ; q)) dµE(K ;q) (ψ )
dξ1 · · · dξm W˜ 0,m (ξ1 , · · · , ξm ; K ; q) ψ(ξ1 ) · · · ψ(ξm ).
m≥0
˜ 0,2 is now an implicit equation. The requirement that 21 q be the kernel of W
Two Dimensional Fermi Liquid. 1: Overview
39
Lemma III.11. There is a unique formal power series q0 (ξ1 , ξ2 ; K ) in λ that solves the equation 1 2 q(K )
= W˜ 0,2 K ; q(K ) .
(III.12)
The coefficient of λ0 in q0 vanishes. Proof. Equation (III.12) is the form q = F (λ, q), with F being C ∞ in λ and q and with F (λ, 0) being of order at least λ. An easy formal power series argument yields the result. We define, for each K ∈ Kj +1 , f orm
˜ ˜ W(φ, ψ; K ) = W(φ, ψ; K ; q0 (K )),
˜ φ, ψ; K − W ˜ φ, 0; K − 1 dξ1 dξ2 q0 (ξ1 , ξ2 ; K )ψ(ξ1 )ψ(ξ2 ), W (φ, ψ; K ) = W 2 ˜ ˜ G (φ; K ) = G(φ; K(K )) + W(φ, 0; K ) − W(0, 0; K ), u (K ) = u (K ; q0 (K )), and Oj (W, G, u) = (W , G , u ). We also define renj,j +1 (K , W, u) = K(K ) = K(K ; q0 (K )) ∈ Kj
f orm
(j,f orm)
Proposition III.12. Let j ≥ j0 , (W, G, u) ∈ Dout Then
.
and (W , G , u ) = Oj (W, G, u).
a) (j +1,f orm)
(W , G , u ) ∈ Din
.
b) If K ∈ Kj +1 and K = renj,j +1 (K , W, u) then, formally, ignoring the problems engendered by the infrared singularity, f orm
:W (φ,ψ;K):
ψ,Dj (u;K) eφJ ψ e dµC [j +1,j ) (ψ) u(K) ¯ G(φ; K) + log :W (0,ψ;K): ψ,Dj (u;K) e dµC [j +1,j ) (ψ) u(K) ¯ φJ ψ :W (φ,ψ;K ):ψ,C (u ;K ) j +1 e dµC [j +1,j ) (ψ) e u (K )¯ = G (φ; K ) + log :W (0,ψ;K ): ψ,Cj +1 (u ;K ) e dµC [j +1,j ) (ψ) u (K )¯
if j + 1 < j¯ ≤ ∞. Proof. Let K ∈ Kj +1 and set K = renj,j +1 (K , W, u). f orm
40
J. Feldman, H. Kn¨orrer, E. Trubowitz
a) We first verify that (W , G , u ) is a formal interaction triple. The only condition of Definition III.4 that is not trivially satisfied is
uˇ (0, k); K = uˇ (0, k); K K ; q0 (K ) + qˇ0 ((0, k); K )ν (≥j +1) (0, k) ˇ ˇ = −K(k; K ; q0 (K )) + δ K(k; K ; q0 (K )) = −Kˇ (k). vanishes amounts to We now verify the conditions of Definition III.8. That W0,2 W˜ 0,2 = 21 q0 which is Lemma III.11. Condition (ii) of Definition III.8 is trivially fulfilled. b) Observe that [j +1,j )
[j +1,j )
Cu (K )¯ = Cu(K(K¯ ))+q
0 (K
)ν (≥j +1)
.
(III.13)
Set U(K ) = 21 dξ1 dξ2 q0 (ξ1 , ξ2 ; K ):ψ(ξ1 )ψ(ξ2 ):Cj +1(u ;K ) . By Lemma C.2 of [FKTo2] and (III.13), :W (φ,ψ;K):ψ,Dj (u;K) dµC [j +1,j ) (ψ) G(φ; K) + log eφJ ψ e u(K) ¯ ˜ (φ,ψ;K): : W ψ,Cj +1 (u ;K ) −U +U +(G (φ;K)−G (φ;K )) dµC [j +1,j )(ψ) = G (φ; K )+log eφJ ψ e u(K) ¯ :W (φ,ψ;K ):ψ,C +U j +1 (u ;K ) = G (φ; K ) + log eφJ ψ e dµC [j +1,j ) (ψ) + const u(K) ¯ (φ,ψ;K ): : W ψ,Cj +1 (u ;K ) = G (φ; K ) + log eφJ ψ e dµC [j +1,j ) (ψ) + const ¯ u(K)+q0 (K )ν (≥j +1) :W (φ,ψ;K ):ψ,C j +1 (u ;K ) dµ [j +1,j ) (ψ) + const . = G (φ; K ) + log eφJ ψ e Cu (K )¯ Subtracting the same equation with φ = 0 gives part b).
Appendix A. Model Computations This appendix provides a number of model computations that illustrate important features of the present construction. Various other model computations are given in the introductory sections of this paper and other papers in this series. Here is a table of model computations and their locations. Topic Overlapping loop volume improvement Particle–particle bubble volume improvement Particle–hole bubble sign cancellation Sectorization and conservation of momentum Power counting with sectorization Overlapping loops and sectors Sectorization and change of scale Cancellations at high orders of perturbation theory
Location §II, Subsect. 4 §II, Subsect. 5 [FKTl, Lemma I.1] Example A.1 §II, Subsect. 8 Example A.2 Example A.3 §II, Subsect. 9 [FKTr2, §X] [FKTo1, §V]
Two Dimensional Fermi Liquid. 1: Overview
41
Example A.1 (Sectorization and Conservation of Momentum). Sectors enable us to apply L1 norms to functions for which the L1 norm well approximates the L∞ norm of the Fourier transform. We provide a simple illustration of the use of sectorization as a tool for bounding 1,∞ norms of Green’s functions built from C (j ) . (j ) Recall that C (j ) (k) = ν (k) and that, on the support of ν (j ) , ik0 − e(k) ≈ 1j . ik0 −e(k)
M
To make this example as explicit as possible, we suppress k0 , choose d = 3, choose |k| (j ) (k) by ϕ M j (|k| − 1) , e(k) = |k|2 − 1, replace ik0 − e(k) by M j and replace ν where ϕ ∈ C ∞ ([−1, 1]) is real and even. So we define
j (j ) (j ) Mj dd k c (x, x ) = eik·(x−x ) c(j ) (k). c (k) = |k| ϕ M (|k| − 1) (2π)d We have rigged things so that we can compute c(j ) (x, x ) relatively explicitly using the
d d Fourier transform S d−1 dσ (k ) eirk ·(x−x ) = 2 2 −1 d2 ωd (r|x − x |)1− 2 J d −1 (r|x − 2
x |) of the unit sphere in Rd . Here , ωd and Jν are the Gamma function, the surface area of S d−1 and the Bessel function of order ν, respectively. The answer is c(j ) (x, x ) =
1 2π 2 |x−x |
| sin |x − x | ϕˆ |x−x . Mj
In particular ! (j ) ! !c (x, x )!
1,∞
=
3
d x
1
2π 2 |x|
|x| sin |x| ϕˆ M j =
2 π
For large j , | sin(r)| is much more rapidly varying than ϕˆ by its average value, π2 , introduces only a small error, ! (j ) !
1 ∞ 4 !c (x, x )! = π2 + O Mj dr 1,∞ 0
1 2j 4 + O Mj =M π2
0
dr r sin(r)ϕˆ Mr j .
∞
r Mj
and replacing | sin(r)|
r ϕˆ Mr j dr r ϕ(r) ˆ .
∞
(A.1)
0
! ! Observe that !c(j ) (x, x )!1,∞ is a factor of about M j larger than supk c(j ) (k) ∼ M j . Now introduce a sectorization of the Fermi surface k ∈ R3 |k| = 1 as in Subsect. 8 of §II (for details, see Def. VI.2) using approximately square sectors of side 1 . We may construct, also as in Subsect. 8 of §II (for details, see (XIII.2) of [FKTo3] M j/2 and Lemma XII.3
of [FKTo3]), a partition of unity, χs , s ∈ , of the support of ϕ M j (|k| − 1) such that
j (j ) (j ) (j ) Mj dd k cs (x, x ) = eik·(x−x ) cs (k) cs (k) = |k| ϕ M (|k| − 1) χs (k) (2π)d ! ! (j ) obeys !cs (x, x )!1,∞ ≤ const M j . The proof of this bound was sketched in Subsect. 7 of §II. For details, see Prop. XIII.1and Lemma XIII.2of [FKTo3]. Observe that, with sectorization, ! (j ) (j ) ! !cs (x, x )! ≤ const sup cs (k). 1,∞
k
42
J. Feldman, H. Kn¨orrer, E. Trubowitz
with ℵ < 21 , this would no longer be the case. Also 2 observe that, since || is of order M j/2 , 1 M ℵj
Had we chosen sectors of side
! (j ) ! (j ) ! !
2 !c (x, x )! !cs (x, x )! ≤ ≤ const M j/2 M j = const M 2j 1,∞ 1,∞ s∈
recovers (A.1), up to an unimportant constant, by a technique that extends to nonround Fermi surfaces, for which explicit computations of c(j ) (x, x ) are not available. Finally, consider G2 (x, x ) = dy c(j ) (x, y)c(j ) (y, x ). This would be a (first order) contribution to a model with!covariance c(j )!and an ultralocal ! ! ! quadratic interaction. If we attempt to bound G2 (x, x )!1,∞ just using !c(j ) (x, x )!1,∞ , we get ! ! !G2 (x, x )! G2 (x, 0) = dx dy c(j ) (x, y)c(j ) (y, 0) = dx 1,∞ ! !2 ≤ dx dy c(j ) (x, y)c(j ) (y, 0) = !c(j ) (x, x )!1,∞ ∼ const M 4j , which is a terrible answer: G2 (x, x ) is the Fourier transform of c(j ) (k)2 = M 2j 1 ϕ(M j (|k|−1))2 and |k|1 2 ϕ(M j (|k|−1))2 is very much like |k| ϕ(M j (|k|−1)), so the real |k|2 ! ! behaviour of !G2 (x, x )! is M 3j . We may recover this real behaviour using sectors, 1,∞
! ! !G2 (x, x )!
1,∞
dx dy c(j ) (x, y)c(j ) (y, 0) (j ) (j ) = dx dy cs (x, y)cs (y, 0)
=
≤
s,s ∈
s,s ∈
(j ) (j ) dx dy cs (x, y)cs (y, 0).
(j ) (j ) (j ) (j ) But dy cs (x, y)cs (y, 0) is the Fourier transform of cs (k)cs (k) = c(j ) (k)2 χs (k) χs (k), which vanishes identically unless the supports of χs and χs overlap. For each fixed s ∈ , there are at most 9 sectors s ∈ that overlap with s. Hence ! ! (j ) (j ) !G2 (x, x )! dx dy c ≤ 9|| max (x, y)c (y, 0) s s 1,∞ s,s ∈
s,s ∈
! (j ) ! ! ! !cs (x, x )! !c(j ) (x, x )! ≤ const ||M 2j ≤ 9|| max 1,∞ s 1,∞ s,s ∈ 3j
≤ const M as desired.
Two Dimensional Fermi Liquid. 1: Overview
43
Example A.2 (Overlapping Loops and Sectors). Let be the diagram of Subsect. 4 in §II with vertices w4 fulfilling the bounds
|||ωˆ 4 |||1, = O 1l , |||ωˆ 4 |||3, = O 1) ˆ 1, = of (II.11) and (II.15). The naive power counting bound (II.11) gives that |||||| 1 O l . We show that exploiting overlapping loop volume improvement leads to the ˆ 1, = O(1) and |||||| ˆ 3, = O(l). bounds |||||| Let (p1 , p2 , p3 , q1 , q2 , q3 ) be the six legged subgraph consisting of the left two vertices of . p1 q1 w4 q2
= p2
w4
p3 q3
As in (II.10),
j (j ) |||ˆ |||1, ≤ 9 |||ωˆ 4 |||21, max Cs (x)1 ≤ O Ml2 , s∈ |||ˆ |||3, ≤ 9 |||ωˆ 4 |||3, |||ωˆ 4 |||1, max Cs (x)1 ≤ O (j )
s∈
Mj l
.
Clearly, is obtained by joining to w4 with three lines. q1 q2 p1 p3 q2 w4 = p2 q3 As in Subsect. 8 of §II, by conservation of momentum, ˆ (x1 ,s1 ),(x2 ,s2 ),(x3 ,s3 ),(x4 ,s4 )) ( ˆ ((x1 ,s1 ),(x2 ,s2 ),(y1 ,σ1 ),(x3 ,s3 ),(y2 ,σ2 ),(y3 ,σ3 )) = dy1 dy2 dy3 dz1 dz2 dz3 σi ,σi ,σi ∈ σi ∩σi ∩σi =∅ for i=1,2,3
(j )
× Cσ
1
(y1 −z1 ) C
(j ) (j ) (z −y ) C (z3 −y3 ) σ2 2 2 σ3
ωˆ 4 ((z2 ,σ2 ),(z3 ,σ3 ),(z1 ,σ1 ),(x4 ,s4 )),
so that ! ! !( ˆ (·,s1 ),(·,s2 ),(·,s3 ),(·,s4 ))! 1,∞ ≤ ˆ ((·,s1 ),(·,s2 ),(·,σ1 ),(·,s3 ),(·,σ2 ),(·,σ3 ))1,∞ σi ,σi ,σi ∈ σi ∩σi ∩σi =∅ for i=1,2,3
(j )
× Cσ
1
(x)1 C
(j ) (j ) (x)∞ C (x)∞ σ2 σ3
ωˆ 4 ((·,σ2 ),(·,σ3 ),(·,σ1 ),(·,s4 ))1,∞ .
44
J. Feldman, H. Kn¨orrer, E. Trubowitz
For fixed sectors s1 , s2 , s3 , ! ! !( ˆ (·,s1 ),(·,s2 ),(·,s3 ),(·,s4 ))! s4
≤ 36
σ1 ,σ2 ,σ3 ∈
(j ) ˆ ((·,s1 ),(·,s2 ),(·,σ1 ),(·,s3 ),(·,σ2 ),(·,σ3 ))1,∞ max C (x)1 σ
2 (j ) C (x) × max ∞ σ σ ∈
1,∞
σ1 ∈
max
σ1 ,σ2 ,σ3 ∈ s 4
2 |||ωˆ 4 |||3, ≤ O M j Ml j
1
ωˆ 4 ((·,σ2 ),(·,σ3 ),(·,σ1 ),(·,s4 ))1,∞
ˆ ((·,s1 ),(·,s2 ),(·,σ1 ),(·,s3 ),(·,σ2 ),(·,σ3 ))1,∞ ,
σ1 ,σ2 ,σ3 ∈
since, for a given sector σi , there are three sectors σi and three sectors σi with σi ∩σi = ∅, ˆ 3, with fixed s1 , s2 , s3 is bounded by σi ∩ σi = ∅. We see that the contribution to ||||||
2 2 j O M j Ml j |||ωˆ 4 |||3, ||| |||3, = O M j Ml 2j 1 Ml = O(l) ˆ 1, with fixed s1 is bounded and, taking the sector sum over s2 , s3 , the contribution to |||||| by
2 2 j O M j Ml j |||ωˆ 4 |||3, ||| |||1, = O M j Ml 2j 1 Ml2 = O(1). The contributions with other sectors fixed are estimated in the same way. Example A.3 (Sectorization and Change of Scale). Each time the renormalization group flows to a new scale, the associated sector decomposition changes. Therefore, in the notation of §II, we have to choose new sectorized representatives for w2 , w4 , · · · and compare the norms of these new sectorized representatives with respect to the new sectorization to the norms of the old sectorized representatives with respect to the old sectorization. To isolate this problem, suppose that j > j , that and , respectively, are sectorizations of scale j and j , respectively, of length l and l , respectively and that l < l. Furthermore, let ω2n ((p1 ,s1 ), ··· , (pn ,sn ), (q1 ,sn+1 ), ··· , (qn ,s2n )) be a –sectorized representative of w2n . Using the partition of unity {χs }s ∈ subordinate so , one constructs the –sectorized representative )) = (qn ) ω2n ((p1 ,s1 ), ··· , (qn ,s2n )) ω2n ((p1 ,s1 ), ··· , (qn ,s2n χs1 (p1 ) · · · χs2n si ∈ si ∩si =∅ 1≤i≤2n
of (II.6) is defined in terms of a supremum over s ∈ of w2n . The norm ωˆ 2n 1, of a sum over Momi (s ) ) ∈ 2n si = s and there exist p ∈ s , q ∈ sn+ , 1≤ ≤n = (s1 , · · · , s2n such that p1 + · · · + pn = q1 + · · · + qn . It is natural to partition the sum over Momi (s ) into a sum over (s1 , · · · , s2n ) ∈ 2n followed by a sum over elements of Momi (s ) that obey s ∩ s = ∅ for all 1 ≤ ≤ 2n. Now, we will try to motivate that, “morally”, for any fixed s1 , · · · s2n ∈ , there are at
Two Dimensional Fermi Liquid. 1: Overview
45
2n−3 most const ll elements of Momi (s ) obeying s ∩ s = ∅ for all 1 ≤ ≤ 2n. We may assume that i = 1. Then s1 must be s . Denote by I the interval on the Fermi curve F that has length l + 2l and is centered on s ∩ F . If s ∈ intersects s , then s ∩ F is contained in I . Every sector in contains an interval of F of length 43 l that does not intersect any other sector in . (The specific number 43 comes from Def. VI.2. l of these “hard core” intervals can be contained in It is not important.) At most 43 l+2 l 4 l 2n−3 choices for si , i = 1, n, 2n. I . Thus there are at most 3 l + 3 is essentially uniquely determined by con Fix si , i = n, 2n. Once sn is chosen, s2n servation of momentum. But the desired bound demands more. It says, roughly speaking, are essentially uniquely determined. As p and q run over s and that both sn and s2n sn+ , respectively, for 1 ≤ ≤ n−1, the sum p1 +· · ·+pn−1 −q1 −· · ·−qn−1 runs over ) to be in Mom (s ), there a small set centered on some point k. In order for (s1 , · · · , s2n 1 ∩F with q −p very close to k. But q −p is a secant must exist p ∈ sn ∩F and q ∈ s2n joining two points of the Fermi curve F . We have assumed that F is strictly convex. q
q
k p
p
Consequently, for any given k = 0 in R2 there exist at most two pairs (p , q ) ∈ F 2 with are almost uniquely determined. q − p = k. So, if k is not near the origin, sn and s2n If k is close to zero, then p1 + · · · + pn−1 − q1 − · · · − qn−1 must be close to zero and the number of allowed si , i = n, 2n is reduced. Careful application of these types of arguments yields (Prop. XIX.4 of [FKTo4])
ωˆ ≤ const n l 2n−3 |||ωˆ 2n |||1, + 1 |||ωˆ 2n |||3, , 2n 1, l l (A.2) 2n−4 |||ωˆ 2n |||3, ≤ const n ll |||ωˆ 2n |||3, . For this reason, we usually estimate the combination |||ωˆ 2n |||1, + 1l |||ωˆ 2n |||3, . The analog of (II.11) and (II.15) for this combination is j (n−2)
|||ωˆ 2n |||1, + 1l |||ωˆ 2n |||3, is of order Mln−1 for all n. Thanks to (A.2), this bound is preserved, for n > 2, when the sector decomposition is refined,
n−1 ωˆ 2n 1, + l1 ωˆ 2n M −(j +1)(n−2) l 3,
n−1 l 2n−3 1 ≤ const n M −(j +1)(n−2) l ||| ω ˆ ||| + ||| ω ˆ ||| 2n 1, 2n 3, l l
−(n−2) l n−2 −j (n−2) n−1 n = const M M l |||ωˆ 2n |||1, + 1l |||ωˆ 2n |||3, l
= const n M −(1−ℵ)(n−2) M −j (n−2) ln−1 |||ωˆ 2n |||1, + 1l |||ωˆ 2n |||3, . We even have a small factor, M −(1−ℵ)(n−2) , available for eating up constants like const n .
46
J. Feldman, H. Kn¨orrer, E. Trubowitz
Notation Not’n
Description
Reference
E r0 r M ν (j ) (k) ν (≥j ) (k) n0 ||| · |||1, ||| · |||3, J S (W )(φ, ψ) ˜ C (W )(φ, ψ)
counterterm space number of k0 derivatives tracked number of k derivatives tracked scale parameter, M > 1 th scale function j (j ) i≥j ν (k) degree of asymmetry no derivatives, all but 1 sector summed no derivatives, all but 3 sectors summed particle/hole swap operator log Z1 eW (φ,ψ+ζ ) dµS (ζ ) log Z1 eφJ ζ eW (φ,ψ+ζ ) dµC (ζ )
Definition I.1 following (I.3) following (I.3) before Definition I.2 Definition I.2 Definition I.2 Definition I.10 (II.6) (II.14) (III.3) Definition III.1 Definition III.1
References [A1]
Anderson, P.W.: Luttinger-liquid behavior of the normal metallic state of the 2D Hubbard model. Phys. Rev. Lett. 64, 1839–1841 (1990) [A2] Anderson, P.W.: Singular forward scattering in the 2D Hubbard model and a renormalized Bethe Ansatz ground state. Phys. Rev. Lett. 65, 2306–2308 (1990) [AGD] Abrikosov, A.A., Gorkov, L.P., Dzyaloshinski, I.E.: Methods of Quantum Field Theory in Statistical Physics. New York: Dover Publications, 1963 [AM] Ashcroft, N.W., Mermin, N.D.: Solid State Physics, Philadelphia, PA: Saunders College, 1976 [BG] Benfatto, G., Gallavotti, G.: Renormalization Group. Physics Notes, Vol. 1, Princeton, NJ: Princeton University Press, 1995 [C] Caianello, E.R.: Number of Feynman Diagrams and Convergence. Nuovo Cimento 3, 223– 225 (1956) [DMR] Disertori, M., Magnen, J. Rivasseau, V.: Interacting Fermi liquid in three dimensions at finite temperature: Part I: Convergent Contributions. Ann. Henri Poincar´e 2, 733–806 (2001) [DR1] Disertori, M., Rivasseau, V.: Interacting Fermi liquid in two dimensions at finite temperature. Part I: Convergent Attributions. Commun. Math. Phys. 215, 251–290 (2000) [DR2] Disertori, M., Rivasseau, V.: Interacting Fermi liquid in two dimensions at finite temperature. Part II: Renormalization. Commun. Math. Phys. 215, 291–341 (2000) [FKLT1] Feldman, J., Kn¨orrer, H., Lehmann, D., Trubowitz, E.: Fermi Liquids in Two-Space Dimensions. In: Constructive Physics, V. Rivasseau, ed., Springer Lecture Notes in Physics 446, Berlin-Heidelberg-New York: Springer Verlag, 1995, pp. 267–300 [FKLT2] Feldman, J., Kn¨orrer, H., Lehmann, D., Trubowitz, E.: Are There Two Dimensional Fermi Liquids?. In: Proceedings of the XIth International Congress of Mathematical Physics, D. Iagolnitzer, ed., Cambridge, MA: International Press, 1995, pp. 440–444 [FKLT3] Feldman, J., Kn¨orrer, H., Lehmann, D., Trubowitz, E.: A Class of Fermi Liquids. In: Particles and Fields, G. Semenoff, L. Vinet, eds., CRM Series in Mathematical Physics, New York: Springer-Verlag, 1999, pp. 35–62 [FKTa] Feldman, J., Kn¨orrer, H., Trubowitz, E.: Asymmetric Fermi Surfaces for Magnetic Schr¨odinger Operators. Commun. Partial Differ.Eq. 25, 319–336 (2000) [FKTcf] Feldman, J., Kn¨orrer, H., Trubowitz, E.: A Nonperturbative Representation for Fermionic Correlation Functions. Commun. Math. Phys. 195, 465–493 (1998) [FKTf2] Feldman, J., Kn¨orrer, H., Trubowitz, E.: A Two Dimensional Fermi Liquid, Part 2: Convergence. Commun. Math. Phys. 247, 49–111 (2004) [FKTf3] Feldman, J., Kn¨orrer, H., Trubowitz, E.: A Two Dimensional Fermi Liquid, Part 3: The Fermi Surface. Commun. Math. Phys. 247, 113–177 (2004) [FKTffi] Feldman, J., Kn¨orrer, H., Trubowitz, E.: Fermionic Functional Integrals and the Renormalization Group, CRM Monograph Series 16, American Mathematical Society (2002) [FKTl] Feldman, J., Kn¨orrer, H., Trubowitz, E.: Particle–Hole Ladders. Commun. Math. Phys. 247, 179–194 (2004)
Two Dimensional Fermi Liquid. 1: Overview [FKTo1] [FKTo2] [FKTo3] [FKTo4] [FKTr1] [FKTr2] [FMRT] [FST1] [FST2] [FST3] [FST4] [FW] [KL] [L] [L1] [L2] [L3] [MR] [MCD] [N] [PS] [PT] [S]
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Feldman, J., Kn¨orrer, H., Trubowitz, E.: Single Scale Analysis of Many Fermion Systems, Part 1: Insulators. Rev. Math. Phys. 15, 949–993 (2003) Feldman, J., Kn¨orrer, H., Trubowitz, E.: Single Scale Analysis of Many Fermion Systems, Part 2: The First Scale. Rev. Math. Phys. 15, 995–1037 (2003) Feldman, J., Kn¨orrer, H., Trubowitz, E.: Single Scale Analysis of Many Fermion Systems, Part 3: Sectorized Norms. Rev. Math. Phys. 15, 1039–1120 (2003) Feldman, J., Kn¨orrer, H., Trubowitz, E.: Single Scale Analysis of Many Fermion Systems, Part 4: Sector Counting. Rev. Math. Phys. 15, 1121–1169 (2003) Feldman, J., Kn¨orrer, H., Trubowitz, E.: Convergence of Perturbation Expansions in Fermionic Models, Part 1: Nonperturbative Bounds. Commun. Math. Phys. 247, 195–242 (2004) Feldman, J., Kn¨orrer, H., Trubowitz, E.: Convergence of Perturbation Expansions in Fermionic Models, Part 2: Overlapping Loops. Commun. Math. Phys. 247, 243–319 (2004) Feldman, J., Magnen, J., Rivasseau, V., Trubowitz, E.: An Infinite Volume Expansion for Many Fermion Green’s Functions. Helv. Phys. Acta. 65, 679–721 (1992) Feldman, J., Salmhofer, M., Trubowitz, E.: Perturbation Theory around Non–Nested Fermi Surfaces, I. Keeping the Fermi Surface fixed. J. Stat. Phys. 84, 1209–1336 (1996) Feldman, J., Salmhofer, M., Trubowitz, E.: Regularity of the Moving Fermi Surface: RPA Contributions. Commun. Pure and Appl. Math. LI, 1133–1246 (1998) Feldman, J., Salmhofer, M., Trubowitz, E.: Regularity of Interacting Nonspherical Fermi Surfaces: The Full Self–Energy. Commun. Pure and Appl. Math. LII, 273–324 (1999) Feldman, J., Salmhofer, M., Trubowitz, E.: An inversion theorem in Fermi surface theory. Commun. Pure and Appl. Math. LIII, 1350–1384 (2000) Fetter, A.L., Walecka, J.D.: Quantum Theory of Many-Particle Systems. New York: McGrawHill, 1971 Kohn, W., Luttinger, J.M.: New Mechanism for Superconductivity. Phys. Rev. Lett. 15, 524– 526 (1965) Luttinger, J.M.: New Mechanism for Superconductivity. Phys. Rev. 150, 202–214 (1966) Landau, L.D.: The Theory of a Fermi Liquid. Sov. Phys. JETP 3, 920 (1956) Landau, L.D.: Oscillations in a Fermi Liquid. Sov. Phys. JETP 5, 101 (1957) Landau, L.D.: On the Theory of the Fermi Liquid. Sov. Phys. JETP 8, 70 (1959) Magnen, J., Rivasseau, V.: A Single Scale Infinite Volume Expansion for Three-Dimensional Many Fermion Green’s Functions. Math. Phys. Electronic Journal 1, 3 (1995) Metzner, W., Castellani, C., Di Castro, C.: Fermi Systems with Strong Forward Scattering. Adv. Phys. 47, 317–445 (1998) Nozi`eres, P.: Theory of Interacting Fermi Systems. New York: Benjamin, 1964 Pedra, W., Salmhofer, M.: Fermi Systems in Two Dimensions and Fermi Surface Flows. To appear in the Proceedings of the 14th International Congress of Mathematical Physics. Lisbon, 2003 P¨oschel, J., Trubowitz, E.: Inverse Spectral Theory. New York: Academic Press, 1987 Salmhofer, M.: Continuous renormalization for Fermions and Fermi liquid theory. Commun. Math. Phys. 194, 249–295 (1998)
Communicated by J.Z. Imbrie
Commun. Math. Phys. 247, 49–111 (2004) Digital Object Identifier (DOI) 10.1007/s00220-003-0997-z
Communications in
Mathematical Physics
A Two Dimensional Fermi Liquid. Part 2: Convergence Joel Feldman1, , Horst Kn¨orrer2 , Eugene Trubowitz2 1
Department of Mathematics, University of British Columbia, Vancouver, B.C., Canada V6T 1Z2. E-mail: [email protected] 2 Mathematik, ETH-Zentrum, 8092 Z¨ urich, Switzerland. E-mail: [email protected]; [email protected] Received: 21 September 2002 / Accepted: 12 August 2003 Published online: 6 April 2004 – © Springer-Verlag 2004
Abstract: Using results established in other papers in our series, we prove the existence of the infinite volume, temperature zero, thermodynamic Green’s functions of a two dimensional, weakly coupled fermion gas with an asymmetric Fermi curve and short range interactions. This is done by showing that our sequence of renormalization group maps converges.
Contents IV. V. VI. VII. VIII. IX.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . The First Scales . . . . . . . . . . . . . . . . . . . . . . . . . Sectors and Sectorized Norms . . . . . . . . . . . . . . . . . Ladders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Infrared Limit of Finite Scale Green’s Functions . . . . . . . . One Recursion Step . . . . . . . . . . . . . . . . . . . . . . . 1. Input and Output Data . . . . . . . . . . . . . . . . . . . . 2. Integrating Out a Scale . . . . . . . . . . . . . . . . . . . . 3. Sector Refinement, ReWick Ordering and Renormalization X. The Recursive Construction of the Green’s Functions . . . . . 1. Initialization at j = j0 . . . . . . . . . . . . . . . . . . . . 2. Recursive Step j → j + 1 . . . . . . . . . . . . . . . . . . Appendix B. Self–Consistent ReWick Ordering . . . . . . . . . . . Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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50 50 55 63 67 75 75 77 83 94 95 97 100 110 111
Research supported in part by the Natural Sciences and Engineering Research Council of Canada and the Forschunginstitut f¨ur Mathematik, ETH Z¨urich.
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J. Feldman, H. Kn¨orrer, E. Trubowitz
IV. Introduction This paper, together with [FKTf1] and [FKTf3] provides a construction of a two dimensional Fermi liquid at zero temperature. It contains Sects. IV through X and Appendix B. Sects. I through III and Appendix A are in [FKTf1] and Sects. XI through XV and Appendices C and D are in [FKTf3]. Cumulative notation tables are provided at the end of each part. The main goal of this part is the proof of convergence of the Green’s functions stated in Theorem I.4. In the proof of this theorem, which follows the state¯ with a generating functional G rg (φ, φ) ¯ ment of Theorem VIII.5, we compare Gj (φ, φ) j ¯ constructed by iterating a renormalization group map j times for some j¯ − 2 < j < j¯ . See also §III. To aid in the derivation of bounds on the renormalization group map, we fix a scale parameter M that is sufficiently big (depending on the dispersion relation e(k) and the ultraviolet cutoff U (k)). This M is used throughout the rest of this paper, with the exception of the proof of Theorem I.4 from Theorem VIII.5, where we also explain that fixing M gives no loss of generality. V. The First Scales In §III, we outlined the algebraic aspects of our strategy for proving Theorem I.4. To state and prove the convergence of δej (0) and G˜j (φ, 0), we clearly have to introduce norms for these and various related objects. There are at least two places where control over derivatives will be needed. The analog, Lemma IX.7, of the formal power series Lemma III.11 will involve an application of the implicit function theorem and will require control of derivatives with respect to K. Secondly, we need to control the size of uˇ (k0 , k); K in a neighbourhood of the Fermi surface when k0 = 0, using the fact that this quantity is small when k0 = 0. This is done using the k0 –derivatives. For this reason we shall also control momentum space derivatives, through position space decay, of quantities appearing in the strategy outlined in the last section. The notations in Definitions V.1 and V.2, below, are introduced to aid in keeping track of the effect of the chain rule and Leibniz’s rule on the estimates of derivatives (see [FKTo1, §II]). We use these notations to formulate the bounds of Theorem V.8, in which all scales up to some fixed index j0 are integrated out. These bounds would deteriorate badly with j0 and not permit the limit j0 → ∞ to be taken. Theorem V.8 (which will be reformulated in Theorem VI.12) provides the starting point for the renormalization group analysis of Chapter IX, in which scale dependent power counting is crucial. Definition V.1 (Decay operators). i) Recall that, for a multiindex δ, x = (x0 , x, σ ), x = (x0 , x , σ ) ∈ R × Rd × {↑, ↓}, (x − x )δ = (x0 − x0 )δ0 (x1 − x1 )δ1 · · · (xd − xd )δd . If ξ = (x, a), ξ = (x , a ) ∈ B, we define (ξ − ξ )δ = (x − x )δ . ii) Let n be a positive integer. For a function f (ξ1 , · · · , ξn ) on B n , a multiindex δ, and 1 ≤ i, j ≤ n; i = j set δ Di,j f (ξ1 , · · · , ξn ) = (ξi − ξj )δ f (ξ1 , · · · , ξn ).
A decay operator D on the set of functions on B n is an operator of the form (1)
(k)
D = Duδ 1 ,v1 · · · Duδ k ,vk
Two Dimensional Fermi Liquid. 2: Convergence
51
with multiindices δ (1) , · · · , δ (k) and 1 ≤ uj , vj ≤ n, uj = vj . The indices uj , vj are called variable indices. The total order of D is δ(D) = δ (1) + · · · + δ (k) . In a similar n way, we define the action n of a decay operator on the set of functions on R × Rd or on R × Rd × {↑, ↓} . Definition V.2. i) On R+ ∪ {∞} = x ∈ R x ≥ 0 ∪ {+∞}, addition and the total ordering ≤ are defined in the standard way. With the convention that 0 · ∞ = ∞, multiplication is also defined in the standard way. ii) Let d ≥ 0. The (d + 1)–dimensional norm domain Nd+1 is the set of all formal power series X= Xδ t0δ0 t1δ1 · · · tdδd δ∈N0 ×Nd0
in the variables t0 , t1 , · · · , td with coefficients Xδ ∈ R+ ∪ {∞}. To shorten notation, we set t δ = t0δ0 t1δ1 · · · tdδd . Addition and partial ordering on Nd+1 are defined componentwise. Multiplication is defined by (X · X )δ =
β+γ =δ
Xβ Xγ .
The max and min of two elements of Nd+1 are again defined componentwise. We identify R+ ∪{∞} with the set of all X ∈ Nd+1 having Xδ = 0 for all δ = 0 = (0, · · · , 0). If a > 0, X0 = ∞ and a − X0 > 0 then (a − X)−1 is defined as (a − X)−1 = For an element X = ∂ ∂tj
δ∈N0 ×Nd0
1 a−X0
∞ X−X0 n n=0
a−X0
.
Xδ t δ of Nd+1 and 0 ≤ j ≤ d the formal derivative
X is defined as ∂ ∂tj
X=
(δj + 1)Xδ+ j t δ ,
δ∈N0 ×Nd0
where j is the j th unit vector. iii) For j ≥ 0 we set cj =
M j |δ| t δ +
|δδ |≤r |δ0 |≤r0
and for X ∈ Nd+1 with X0 <
∞ t δ ∈ Nd+1 ,
|δδ |>r or |δ0 |>r0
1 Mj
,
ej (X) =
cj 1−M j X
.
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J. Feldman, H. Kn¨orrer, E. Trubowitz
Definition V.3. For a function f on B m ×B n we define the (scalar valued) L1 –L∞ –norm as max sup dξ |f (ξ1 , · · · , ξn )| ifm = 0 1≤j0 ≤n ξj ∈B j =1,··· ,n j 0 |||f |||1,∞ = j =j0 sup dξj |f (η1 , · · · , ηm ; ξ1 , · · · , ξn )| if m = 0 η1 ,··· ,ηm ∈B
j =1,··· ,n
and the (d + 1)–dimensional L1 –L∞ seminorm
f 1,∞ =
δ∈N0 ×Nd0
1 max |||D f |||1,∞ t δ ifm = 0 δ! D decay operator with δ(D)=δ
|||f |||1,∞
.
if m = 0
Here |||f |||1,∞ stands for the formal power series constant coefficient with |||f |||1,∞ and all other coefficients zero and dξ g(ξ ) = a∈{0,1} σ ∈{↑,↓} dx0 dx g (x0 , x, σ, a) . Given a function on momentum space, we apply the above norms using the Notation V.4. If χ (k) is a function on R × Rd , we define the Fourier transform χˆ by a d d+1 k e(−1) ı− χ (k) (2π) χ(ξ, ˆ ξ ) = δσ,σ δa,a d+1 for ξ = (x, a) = (x0 , x, σ, a), ξ = (x , a ) = (x0 , x , σ , a ) ∈ B. Remark V.5. i) Let V (x1 , x2 , x3 , x4 ) be an interaction kernel as in Theorem I.4 and define, by abuse of notation, the function V on B 4 by V ((x1 ,b1 ),(x2 ,b2 ),(x3 ,b3 ),(x4 ,b4 )) = δb1 ,1 δb2 ,0 δb3 ,1 δb4 ,0 V (x1 , x2 , x3 , x4 ). Then the hypothesis of Theorem I.4 is equivalent to V 1,∞ ≤ εc0 for some sufficiently small ε. ii) The constants cj will be used to describe the behaviour of momentum space derivatives of the covariance C (j ) . The quantities ej (X) are used in bounding the differentiability properties of various kernels depending on a counterterm whose norm is bounded by X. This allows us to take into account the fact that the characteristics, as regards both size and smoothness, of counterterms are very different from the characteristics of kernels built purely from C (j ) and various smooth functions. The characteristics of the counterterms are a consequence of their construction from var ious C (j ) ’s, including those with j j . As j increases, the contribution to the counterterm from C (j ) becomes smaller and smaller, and more and more concentrated near the Fermi surface, but less and less smooth. We also wish to use our norms to control the coupling constant dependence of various kernels. This is done using
Two Dimensional Fermi Liquid. 2: Convergence
53
Definition V.6. Fix 0 < υ < 41 . Set, for a coupling constant 0 < λ < 1, λ−(1−υ)(m+n−2)/2 if m + n ≥ 4 1 ρm;n (λ) = (1−υ) max{m+n−2,2}/2 = . λ if m + n = 2 λ−(1−υ) Remark V.7. The exponent of Definition V.6 is motivated by the following considerations. For this discussion, introduce a coupling constant λ and replace V(ψ) by λV(ψ). The exponent of the initial generating functional contains, aside from the counterterm, two vertices with ψ fields. One, λV(ψ), has four ψ fields and is proportional to the coupling constant λ. The other, ψφ, has one ψ field, one φ field and is independent of λ. Consider any connected graph G with m external φ legs, n external ψ legs, v ≥ 1 of the λV(ψ) vertices and m of the ψφ vertices. Since the φ field is always external, G must have precisely m ψφ vertices to have m external φ legs. The graph has 4v+2m−(m+n) 2 internal lines. To be connected, G must have at least v + m − 1 internal lines, so that 4v+2m−(m+n) 2
≥v+m−1
Thus G is proportional to λv with v ≥ max
⇒
m+n−2 2
v≥
m+n−2 . 2
,1 .
We set aside λυ max{m+n−2,2}/2 , which we bound by λυn/10 to achieve good inductive behaviour, i.e. to control various constants that arise in the course of the expansion. 1 Ultimately, we choose a maximum allowed coupling constant λ0 , rename υ/10 = α0 λ0
and consider |λ| < λ0 and α ≥ α0 . Then, our bound on the m φ–legged, n ψ–legged part of the effective interaction will be proportional to 1 (1−υ) max{m+n−2,2}/2 . α n λ0
We now further explain the phrase “good inductive behaviour” used in the last paragraph. Consider, more generally, a connected graph G with m external φ legs, n external ψ legs, m ˜ of the ψφ vertices and v ≥ 1 other vertices. Suppose that the i th other vertex m−(m+n) ˜ has mi φ–legs and ni ψ–legs. The number i (mi +ni )+2 of internal lines must 2 be at least v + m ˜ − 1 so i (mi +ni )+2m−(m+n) ˜ 2
≥v+m ˜ −1
⇒
v m +n −2 i
i
2
≥
m+n−2 . 2
i=1
As v ≥ 1,
v
max
mi +ni −2 2
, 1 ≥ max m+n−2 ,1 . 2
i=1
We thus have αn (1−υ) max{m+n−2,2}/2 λ0
≤
1
v
α ni −n i=1
α ni (1−υ) max{mi +ni −2,2}/2 λ0
.
The small factors α n1i −n are available for controlling various constants that arise in ˜ ≤ ni − n, the the course of the expansion. Observe that, as m = m ˜ + mi and m m−(m+n) ˜ number of internal lines of G, i (mi +ni )+2 is bounded by n i − n. 2
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J. Feldman, H. Kn¨orrer, E. Trubowitz
We choose an arbitrary but fixed scale, j0 ≥ 2, and integrate the first scales, between 1 and j0 , in one fell swoop. Theorem V.8. There are (M and j0 –dependent) constants µ, λ¯ and β0 such that, for all 1 λ < λ¯ and β0 ≤ β ≤ λυ/5 , the following holds: Let X ∈ Nd+1 with X0 < µ, δe ∈ E with δ e ˆ 1,∞ ≤ X and V(ψ) = dξ1 ···dξ4 V (ξ1 ,··· ,ξ4 ) ψ(ξ1 ) · · · ψ(ξ1 ) B4
with an antisymmetric function V fulfilling V 1,∞ ≤ λ e0 (X). Write ˜
(≤j )
C−δe0
(≤j ) V (ψ) (φ, ψ) = V(ψ) + 21 φJ C−δe0 J φ + dη1 ···dηm m,n≥0 m+n even
Bm+n
dξ1 ···dξn Wm,n (η1 ,··· ,ηm ,ξ1 ,··· ,ξn ; δe)
×φ (η1 ) · · · φ (ηm ) ψ (ξ1 ) · · · ψ(ξn ) with kernels Wm,n that are separately antisymmetric under permutations of their η and ξ arguments. Then β n ρm;n (λ) Wm,n (δe) 1,∞ ≤ const β 3 λυ e0 (X) m+n≥2 m+n even
and
m+n≥2 m+n even
β n ρm;n (λ) dds Wm,n (δe + sδe )s=0 1,∞ ≤ const β 3 λυ e0 (X) δ eˆ 1,∞ .
Furthermore, each Wm,n is jointly analytic1 in V and δe. If V fulfills the reality condition ˜ (≤j0 ) V (ψ) (φ, ψ) is k0 –reversal real, in the of (I.1) and δe(k) is real valued, then sense of Definition B.1.R of [FKTo2]2 .
C−δe
Proof. Apply Theorem VIII.6 of [FKTo2] with ρm,n = ρm;n (λ) and ε = const β 4 λυ . Observe that, by Remark VIII.7.iii of [FKTo2], the hypotheses on ρm;n are fulfilled. If λ¯ is chosen small enough, then the hypothesis ε < ε0 is also fulfilled.The reality statement is a consequence of Remark B.5 of [FKTo2]. In Theorem V.8, we integrated out the part of the field ψ with covariance C (≤j0 ) . To recover the full, infrared cutoff covariance C IR(j¯) of Theorem I.4, we must also integrate out the part of the field with covariance (i,j )
Cu
(k) =
ν (>i) (k) − ν (≥j ) (k) . ˇ − ν (≥j¯) (k)] ık0 − e(k) − u(k)[1
1 As in the discussion leading up to Theorem I.4, the W m,n ’s are initially defined as formal Taylor series in V . The conclusions of the theorem implicitly include the convergence of the formal Taylor series for V obeying V 1,∞ ≤ λ e0 (X) and δe obeying δ e ˆ 1,∞ ≤ X. 2 This definition is the obvious generalization of (I.1) to functions of arbitrarily many variables.
Two Dimensional Fermi Liquid. 2: Convergence
55
Lemma V.9. Let j¯ ≥ j0 +2 be an infrared cutoff. For |||V |||1,∞ and |||δe|||1,∞ sufficiently small ˜ (≤j0 ) V (ψ) = ˜ C IR(j ) (δe) V (ψ) . ˜ (j0 ,j ) C−δe ¯ C−δe ¯ Proof. Since |||V |||1,∞ and |||δe|||1,∞ are sufficiently small, both sides of the desired identity are well-defined. It is proven by applying the semi–group property (III.4) using (≤j )
(j ,j )
0 C IR(j¯) (δe) = C−δe0 + C−δe ¯.
VI. Sectors and Sectorized Norms From now on we consider only d = 2, so that the Fermi “surface” is a curve in R×R2 . We choose a projection πF from the first extended neighbourhood onto the Fermi surface. Convention. Generic constants that depend only on the dispersion relation e(k) and the ultraviolet cutoff U (k) will be denoted by “const ”. Generic constants that may also depend on the scale parameter M, but still not on the scale j , will be denoted “const ”. To systematically deal with Fourier transforms, we call Bˇ = R × Rd × {↑, ↓} × {0, 1} “momentum space”. For ξˇ = (k, σ , a ) = (k0 , k, σ , a ) ∈ Bˇ and ξ = (x, a) = (x0 , x, σ, a) ∈ B we define the inner product ξˇ , ξ = δσ ,σ δa ,a (−1)a k, x− = δσ ,σ δa ,a (−1)a − k0 x0 + k1 x1 + · · · + kd xd , “characters”
ˇ
a
ı ξ ,ξ E+ (ξˇ , ξ ) = δσ ,σ δa ,a e = δσ ,σ δa ,a eı(−1)
E− (ξˇ , ξ ) = δσ ,σ δa ,a e and integrals dξ · = a∈{0,1} σ ∈{↑,↓}
−ı ξˇ ,ξ
= δσ ,σ δa ,a e
R×Rd
dx0 d d x ·
−k0 x0 +k1 x1 +···+kd xd
,
−ı(−1)a −k0 x0 +k1 x1 +···+kd xd
d ξˇ · =
a∈{0,1} σ ∈{↑,↓}
R×Rd
,
dk0 d d k ·
For ξˇ = (k, σ, a), ξˇ = (k , σ , a ) ∈ Bˇ we set ξˇ + ξˇ = (−1)a k + (−1)a k ∈ R × Rd .
Definition VI.1 (Fourier transforms). Let f (η1 ,··· ,ηm ; ξ1 ,··· ,ξn ) be a translation invariant function on B m × B n . The total Fourier transform fˇ of f is defined by d+1 fˇ(ηˇ 1 ,··· δ(ηˇ 1 +···+ηˇ m +ξˇ1 +···+ξˇn ) m ; ξˇ1 ,··· ,ξˇn ) (2π) ,ηˇm n
= E+ (ηˇ i , ηi ) dηi E+ (ξˇj , ξj ) dξj f (η1 ,··· ,ηm ; ξ1 ,··· ,ξn ) i=1
j =1
56
J. Feldman, H. Kn¨orrer, E. Trubowitz
or, equivalently, by f (η1 ,··· ,ηm ; ξ1 ,··· ,ξn ) =
m
i=1
E− (ηˇ i ,ηi ) d ηˇ i (2π)d+1
n
j =1
E− (ξˇj ,ξj ) d ξˇj (2π)d+1
fˇ(ηˇ 1 ,··· ,ηˇ m ; ξˇ1 ,··· ,ξˇn )
× (2π)d+1 δ(ηˇ 1 +···+ηˇ m +ξˇ1 +···+ξˇn ). fˇ is defined on the set (ηˇ 1 ,··· ,ηˇ m ; ξˇ1 ,··· ,ξˇn ) ∈ Bˇ m × B n ηˇ 1 +···+ηˇ m +ξˇ1 +···+ξˇn =0 . If m = 0, n = 2 and f (ξ1 ,ξ2 ) conserves particle number and is spin independent and antisymmetric, we define fˇ(k) by fˇ((k,σ,1),(k,σ ,0)) = δσ,σ fˇ(k). We now introduce sectors. Definition VI.2 (Sectors and sectorizations). i) Let I be an interval on the Fermi surface F and j ≥ 2. Then s = k in the j th neighbourhood πF (k) ∈ I is called a sector of length |I | at scale j . Two different sectors s and s are called neighbours if s ∩ s = ∅. ii) If s is a sector at scale j , its extension is s˜ = k in the j th extended neighbourhood πF (k) ∈ s . iii) A sectorization of length l at scale j is a set of sectors of length l at scale j that obeys – the set of sectors covers the Fermi surface – each sector in has precisely two neighbours in , one to its left and one to its right 1 – if s, s ∈ are neighbours then 16 l ≤ |s ∩ s ∩ F | ≤ 18 l. Observe that there are at most 2 length(F )/l sectors in . s
s
Definition VI.3 (Sectorized representatives). Let be a sectorization at scale j , and let m, n ≥ 0. n i) The antisymmetrization of a function ϕ on B m × B × is Ant ϕ(η1 ,··· ,η n ,sn )) m ; (ξ1 ,s1 ),··· ,(ξ = m!1n! sgnπ sgnπ ϕ(ηπ(1) ,··· ,ηπ(m) ; (ξπ (1) ,sπ (1) ),··· ,(ξπ (n) ,sπ (n) )). π ∈Sm π ∈Sn
Two Dimensional Fermi Liquid. 2: Convergence
57
ii) Denote by Fm (n; ) the space of all translation invariant, complex valued functions ϕ(η1 ,··· ,ηm ; (ξ1 ,s1 ),··· ,(ξn ,sn )) n on B m × B × that are antisymmetric in their external (= η) variables and whose Fourier transform ϕ( ˇ ηˇ 1 ,··· ,ηˇ m ; (ξˇ1 ,s1 ),··· ,(ξˇn ,sn )) vanishes unless ki ∈ s˜i for all ˇ 1 ≤ i ≤ n. Here, ξi = (ki , σi , ai ). iii) Let f be a translation invariant, complex valued function on B m × B n that is antisymmetric in its first m variables. A –sectorized representative for f is a function ϕ ∈ Fm (n; ) obeying fˇ(ηˇ 1 ,··· ,ηˇ m ; ξˇ1 ,··· ,ξˇn ) =
si ∈ i=1,··· ,n
ϕ( ˇ ηˇ 1 ,··· ,ηˇ m ; (ξˇ1 ,s1 ),··· ,(ξˇn ,sn ))
for all ξˇi = (ki , σi , ai ) with ki in the j th neighbourhood. iv) Let u((ξ,s), (ξ ,s )) be a translation invariant, spin independent, particle number conserving function on (B × )2 . We define u(k) ˇ by ˇ = δσ,σ u(k)
u( ˇ (k,σ,1,s), (k,σ ,0,s )).
s,s ∈
We now fix a constant 21 < ℵ < 23 , and for each scale j ≥ 2, a sectorization j of length lj = M1ℵj . Also, we fix for each j ≥ 2, a system χs (k), s ∈ j of functions that take values in [0, 1] such that i) χs is supported in the extended sector s˜ and
χs (k) = 1
for k in the j th neighbourhood.
s∈
ii) χˆ s 1,∞ , ≤
const cj −1
with a constant const that does not depend M, j , or s. The existence of such a “partition of unity” is shown in Lemma XII.3 of [FKTo3]. They are used to construct sectorized representatives. n Definition VI.4. Let j, i ≥ 2. If i = j , define, for functions ϕ on B m × B × i and n f on Bˇ m × B × i , ϕj (η1 ,··· ,ηm ; (ξ1 ,s1 ),··· ,(ξn ,sn )) n
= dξ1 ···dξn ϕ(η1 ,··· ,ηm ; (ξ1 ,s1 ),··· ,(ξn ,sn )) χˆ s (ξ , ξ ), s1 ,··· ,sn ∈i
=1
fj (ηˇ 1 ,··· ,ηˇ m ; (ξ1 ,s1 ),··· ,(ξn ,sn )) n
= dξ1 ···dξn f (ηˇ 1 ,··· ,ηˇ m ; (ξ1 ,s1 ),··· ,(ξn ,sn )) χˆ s (ξ , ξ ). s1 ,··· ,sn ∈i
=1
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J. Feldman, H. Kn¨orrer, E. Trubowitz
If ϕ is antisymmetric under permutation of its η arguments, then ϕj ∈ Fm (n, j ). For i = j define ϕj = ϕ and fj = f . Similarly, define, for functions ϕ on B m × B n and f on Bˇ m × B n , ϕj (η1 ,··· ,ηm ; (ξ1 ,s1 ),··· ,(ξn ,sn )) =
dξ1 ···dξn ϕ(η1 ,··· ,ηm ; ξ1 ,··· ,ξn )
fj (ηˇ 1 ,··· ,ηˇ m ; (ξ1 ,s1 ),··· ,(ξn ,sn )) =
dξ1 ···dξn f (ηˇ 1 ,··· ,ηˇ m ; ξ1 ,··· ,ξn )
n
=1 n
χˆ s (ξ , ξ ),
=1
χˆ s (ξ , ξ ).
They are j –sectorized representatives for ϕ resp. f . Definition VI.5. Let j ≥ 2 be a scale. We consider fermionic fields φ(η), η ∈ B and ψ(ξ,s ), ξ ∈ B, s ∈ j . i) A j –sectorized Grassmann function is of the form w=
m
n
dξj wm,n (η1 ,··· ,ηm ; (ξ1 ,s1 ),··· ,(ξn ,sn )) m,n≥0 s1 ,··· ,sn ∈j i=1 j =1 × φ (η1 ) · · · φ (ηm ) ψ (ξ1 ,s1 ) · · · ψ (ξn ,sn ). dηi
(VI.1)
ii) Let W=
m,n≥0
m
i=1
dηi
n
j =1
dξj Wm,n (η1 ,··· ,ηm ; ξ1 ,··· ,ξn )φ(η1 ) · · · φ(ηm )ψ(ξ1 ) · · · ψ(ξn )
be a Grassmann function with each Wm,n a function on B m × B n that is separately antisymmetric in its external (= φ) variables and in its internal (= ψ) variables. A j –sectorized representative for W is a j –sectorized Grassmann function of the form (VI.1), where, for each m, n, wm,n is a j –sectorized representative for Wm,n that is also antisymmetric in the variables (ξ1 , s1 ), · · · , (ξn , sn ). Definition VI.6 (Norms for sectorized functions). Let j ≥ 2 and m, n ≥ 0. i) For a function ϕ on B m × (B × j )n and an integer p > 0 we define the seminorm |ϕ||p,j to be zero if m ≥ 1, p ≥ 2 or if m = 0, p > n. In the case m ≥ 1, p = 1 we set |ϕ||p,j =
si ∈j
ϕ(η1 ,··· ,ηm ; (ξ1 ,s1 ),··· ,(ξn ,sn )) 1,∞ .
In the case m = 0, p ≤ n we set |ϕ||p,j =
max
max
1≤i1 <···
ϕ((ξ1 ,s1 ),··· ,(ξn ,sn )) 1,∞ .
si ∈j for i=i1 ,··· ,ip
In both cases, the · 1,∞ norm applies to all the position space variables.
Two Dimensional Fermi Liquid. 2: Convergence
59
ii) We shall fix a λ0 > 0, sufficiently small depending on j0 and the scale parameter M. For ϕ ∈ Fm (n, j ) set |ϕ||1,j + 1 |ϕ||3,j + 12 |ϕ||5,j if m = 0 lj lj , |ϕ||j = ρm;n lj |ϕ|| if m = 0 2j 1, j M where ρm;n =
(j ) ρm;n
=
1 (1−υ) max{m+n−2,2}/2 λ0
1 4
if m = 0 lj
Mj
if m > 0
and υ was fixed in Definition V.6. Definition VI.7 (Norms for sectorized Grassmann functions). i) A j –sectorized Grassmann function w can be uniquely written in the form dη1 ···dηm dξ1 ···dξn wm,n (η1 ,··· ,ηm (ξ1 ,s1 ),··· ,(ξn ,sn )) w(φ, ψ) = m,n s1 ,··· ,sn ∈j
× φ(η1 ) · · · φ(ηm ) ψ((ξ1 ,s1 )) · · · ψ((ξn ,sn ) )
with wm,n antisymmetric separately in the η and in the ξ variables. Set, as in Definition XV.1 of [FKTo3], for α > 0 and X ∈ Nd+1 , l B n/2 2j α n Mj j |wm,n|j . Nj (w, α, X) = Mlj ej (X) m,n≥0
The constant B depends on M, but not j and was specified in Definition XV.1 of [FKTo3]. ii) A Grassmann function G(φ) can be uniquely written in the form G(φ) = dη1 ···dηm Gm (η1 ,··· ,ηm ) φ(η1 ) · · · φ(ηm ) m
with Gm antisymmetric. Set N (G) =
1 (1−υ) max{m−2,2}/2
m>0 λ0
|||Gm |||∞ ,
where |||Gm |||∞ = supη1 ,··· ,ηm Gm (η1 ,··· ,ηm ). Remark VI.8. i) The system ρ = (ρm;n ) of Definition VI.6.ii fulfill the inequalities (XV.1) of [FKTo3]. ii) If w(φ, ψ) is a j –sectorized Grassmann function, then 1 N w(φ, 0) ≤ √ Nj (w, α, X) 4 j lj M
for all α and X ∈ Nd+1 . iii) The j independent part of the coefficient of |wm,n|1,j in Nj (w, α, X) is, up to a factor of Bn/2 , equal to
αn
(1−υ) max{m+n−2,2}/2
λ0
. This choice was motivated in Remark V.7.
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J. Feldman, H. Kn¨orrer, E. Trubowitz
iv) If Nj (w, α, X)0 ≤ 1, then, up to |wm,n|1,j ≤
δ=0 ∞ t 1
n −1 lj2
δ,
M j ( 2 −2) n
M j n/2 1 √ 4 M j lj lj
if m = 0 if m = 0
(VI.2)
.
The case m = 0 was motivated in (II.11). Next consider the case that m + n = 4, m, n ≥ 1 and wm,n is the coefficient of φ m ψ n in V(φ + ψ). Then, allowing a full sector sum for each ψ leg, |||V |||1,∞ < ∞ implies that |wm,n|1,j = O l1n , which j
is a tighter bound than (VI.2). An argument similar to that in Subsect. 8 of §II may also be used to show that if wm,n is a graph with vertices obeying (VI.2), then wm,n obeys a bound of the same order as (VI.2). In Definition VI.6, we use the norm · 1,∞ of Definition V.3, to measure kernel sizes. When m = 0, i.e. there are no external legs, this norm takes a “supremum norm in momentum space”. When m = 0 and there are external legs, this norm sups over the positions of external arguments in position space, which corresponds to an L1 norm in momentum space. Additional integrals in momentum space tend to smooth out kernel singularities and reduce norm sizes. The resulting improved power counting is captured by the factor (M j lj )1/4 in the definition of ρm;n (part (ii) of Definition VI.6) and correspondingly in (VI.2). In [FKTf3] we make a more detailed analysis of the two– and four–point functions. There we apply the “supremum norm in momentum space” to external as well as internal arguments. The improved power counting discussed in the previous paragraph does not appear and we cannot include a factor like (M j lj )1/4 in the norms. See Definition XIII.15.ii. v) The quantity of main interest in the norm |ϕ||j of Definition VI.6 is |ϕ||1,j . As seen in Example A.2 of [FKTf1], we need | · |3,j norms to get improved bounds on |ϕ||1,j by exploiting overlapping loops. For four point functions ϕ, |ϕ||3,j is also useful because it mimics the supremum in momentum space. To get improved bounds on |ϕ||3,j by exploiting overlapping loops, we need to use the | · |5,j norm. We now define the space of functions from which the various counterterm kernels will be chosen. The Fourier transforms of counterterms are functions of the spatial momenta components k only. The norms introduced earlier in this section for k0 dependent functions have obvious generalizations to k0 independent functions. For example, 2 if K (x, s), (x , s ) is a translation invariant function on R2 × j , we define 1 K 1,j = max δ! i1 =1,2 si ∈j 1
×
d sı 1 ∈j δ∈N0
max
max sup
D decay operator i2 =1,2 with δ(D)=(0,δ)
xi2
dxı 2 DK (x1 , s1 ), (x2 , s2 ) t δ ,
where ı = 3−i. Other obvious generalizations of this nature to k0 independent functions are formulated precisley in Appendix E of [FKTo4]. Definition VI.9. Let Kj be the space of all translation invariant, sectorized functions 2 K (x, s), (x , s ) on R2 × j for which
Two Dimensional Fermi Liquid. 2: Convergence
61
lj +1 i) K 1,j < λ1−υ + δ=0 ∞ t δ , 0 M j +1 ˇ ii) the Fourier transform K(k) is supported on supp ν (≥j +1) (0, k) . The counterterm K is said to be real if, for each s, s ∈ j , the Fourier transform Kˇ (k, s), (k , s ) is real valued. Remark VI.10. If K ∈ Kj , then Kj −1 ∈ Kj −1 . To see this, observe that, by part (iii) of Proposition XIX.4 of [FKTo4], Kj −1 1,j −1 ≤
lj −1
l
l
j +1 cj −2 K 1,j < const jl−1 λ1−υ 0 M j +1 j lj + ∞ t δ < λ1−υ + ∞ tδ, 0 Mj
const l j δ=0
δ=0
if M is large enough. Remark VI.11. The final counterterm δe(k) will be constructed in Theorem VIII.5 using bounds proven in Lemma X.1. As in Definition III.5, Remark III.7 and Lemma V.9, we have the following covariances. Definition VI.12. (i) Let u((ξ,s), (ξ ,s )) be a translation invariant, spin independent, particle number conserving function on (B × )2 . Then ν (j ) (k) , ik0 − e(k) − u(k) ˇ ν (≥j ) (k) (≥j ) Cu (k) = , ik0 − e(k) − u(k) ˇ ν (≥i) (k) − ν (≥j ) (k) [i,j ) Cu (k) = . ık0 − e(k) − u(k)[1 ˇ − ν (≥j¯) (k)] (j )
Cu (k) =
(ii) Let u be a function from Kj to the space of antisymmetric, translation invariant, spin independent, particle number conserving functions on (B × j )2 . Then, for K ∈ Kj , Cj (u; K)(k) = Dj (u; K)(k) =
ν (≥j ) (k) , (≥j +2) (k) ˇ ık0 −e(k)−u(k;K)− ˇ K(k)ν ν (≥j +1) (k) . (≥j +2) (k) ˇ ık0 −e(k)−u(k;K)− ˇ K(k)ν
(iii) Let Cu (ξ, ξ ), Cu (ξ, ξ ), Cu (ξ, ξ ), Dj (u; K)(ξ, ξ ) and Cj (u; K)(ξ, ξ ) be their Fourier transforms as in (III.1) and (III.2). (j )
(≥j )
[i,j )
To start the recursive construction of the Green’s functions, we reformulate Theorem V.8 in terms of sectorized objects. Theorem VI.13. For K ∈ Kj0 , set u(K) = − Kext ∈ F0 (2, j0 ), j0
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J. Feldman, H. Kn¨orrer, E. Trubowitz
where Kext was defined in Definition E.3 of [FKTo4]3 . Then there exist constants λ¯ , α¯ > 1 0 such that for all 0 ≤ λ0 < λ¯ , α¯ < α < υ/10 and all λ0
V 1,∞ ≤ λ0 ej0 K 1,j0 ,
K ∈ Kj0 the Grassmann function
˜
(≤j ) Cu(K)0
(≤j ) V (ψ) (φ, ψ) − 21 φJ Cu(K)0 J φ
has a j0 –sectorized representative w(φ, ψ; K) = dη1 ···dηm dξ1 ···dξn wm,n (η1 ,··· ,ηm (ξ1 ,s1 ),··· ,(ξn ,sn ); K) m,n s1 ,··· ,sn ∈j0
× φ(η1 ) · · · φ(ηm ) ψ((ξ1 ,s1 )) · · · ψ((ξn ,sn ) )
with wm,n antisymmetric separately in the η and in the ξ variables, w0,0 = 0 and Nj0 w(K), α, K 1,j0 ≤ const α 4 λυ0 ej0 K 1,j0 , Nj0 dds w(K + sK )s=0 , α, K 1,j0 ≤ M j0 ej0 K 1,j0 K 1,j0 for all K . w is analytic in V and K. If V fulfills the reality condition of (I.1) and K is real, then w(φ, ψ; K) is k0 –reversal real, in the sense of Definition B.1.R of [FKTo2]. Proof. Write ˜
(≤j )
Cu(K)0
(≤j ) V (ψ) (φ, ψ) = V(ψ) + 21 φJ Cu(K)0 J φ +
m,n≥0 m+n even
dη1 ···dηm dξ1 ···dξn Bm+n
× Wm,n (η1 ···ηm ,ξ1 ,··· ,ξn )φ (η1 ) · · · φ (ηm ) ψ (ξ1 ) · · · ψ(ξn ) and set wm,n
Wm,n j0 = W0,4 + V
j0
if (m, n) = (0, 4) if (m, n) = (0, 4)
using the sectorization f of Definition VI.4. By Proposition XIX.15 of [FKTo4] l B n/2 2j0 α n Mj0j0 |wm,n|j0 Nj0 w(K), α, K 1,j0 = Mlj ej0 K 1,j0 0 m,n≥0 α4 ≤ const cj0 ej0 K 1,j0 V 1,∞ λ1−υ 0 + (const α)n ρm;n (λ0 ) Wm,n 1,∞ m,n≥0 (j )
0 since ρm;n ≤ const ρm;n (λ0 ). By hypothesis
α4 λ1−υ 0 3
V 1,∞ ≤ α 4 λυ0 ej0 K 1,j0
ˇ By Remark E.4.i of [FKTo4], under this definition, Kˇ ext ((k0 , k)) = K(k).
Two Dimensional Fermi Liquid. 2: Convergence
63
and by Theorem V.8, with δe = −u, ˇ X = const K 1,j0 and β = const α, m,n≥0
(const α)n ρm;n (λ0 ) Wm,n 1,∞ ≤ const β 3 λυ e0 (X) ≤ const α 3 λυ0 ej0 K 1,j0 .
Therefore, by Corollary A.5.ii of [FKTo1], 2 Nj0 w(K), α, K 1,j0 ≤ const α 4 λυ0 cj0 ej0 K 1,j0 ≤ const α 4 λυ0 ej0 K 1,j0 . The proof of the bound on Nj0 dds w(K + sK )s=0 , α, K 1,j0 is similar.
VII. Ladders In naive power counting for our model, four-legged vertices are neutral. So there is a danger that the size of four legged kernels after j steps of the renormalization group flow is of order j . We shall show that this logarithmic divergence does not occur. More (j ) precisely, let (W, u, G) ∈ Din be an input datum before integrating out the jth scale (see Definition IX.1) and let (W , G, u, p) = j (W, G, u, p) be the result of integrating out scale j . Assume that (W, G, u, p) is bounded – the precise hypothesis is given in Definition IX.1. We shall show that the norm of the four point part of W does not exceed the norm of the four point part of W by more than const with constants , const indeM j pendent of j . Most contributions to the four point part of W − W are controlled using “overlapping loops”, see [FKTr1]. The only exceptions are ladders. A ladder consists of a sequence of four legged kernel “rungs” connected by pairs of propagators.
For a formal definition, see §XIV of [FKTo3]. Taking creation and annihilation indices into account, such a ladder is either a “particle–particle ladder”
or a “particle–hole ladder”.
The strong asymmetry of the Fermi curve (see Definition I.10) allows us to bound particle–particle ladders of scale j by const . This estimate is stated precisely in PropM j osition VII.6 below and proven in Theorem XXII.7 of [FKTo4]. This is in contrast to the case of a symmetric Fermi curve, where the particle–particle ladders generate the Cooper instability (see [FW, Chapter 10], [FMRT, §4]). The main estimates on particle– hole ladders are stated in Theorem VII.8 below. They are proven in [FKTl] for arbitrary strictly convex Fermi curves.
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Before we formulate the estimates on particle–hole and particle–particle ladders we give a precise definition of ladders. To treat “undirected ladders”, particle–particle and particle–hole ladders with or without spin and with or without external momenta simultaneously, we first work over arbitrary measure spaces, like, for example, R × R2 or R × R2 × {↑, ↓} or B = R × R2 × {↑, ↓} × {0, 1}. See also §XIV of [FKTo3]. Definition VII.1. Let X be a measure space. i) A complex valued function on X × X is called a propagator over X. ii) A four legged kernel over X is a complex valued function on X2 × X2 . We sometimes consider it as a bubble propagator over X, graphically depicted by
or as a rung over X, graphically depicted by
iii) If A and B are propagators over X then the tensor product A ⊗ B(x1 , x2 , x3 , x4 ) = A(x1 , x3 )B(x2 , x4 ) is a bubble propagator over X. We set C(A, B) = A ⊗ A + A ⊗ B + B ⊗ A. iv) Let F be a four legged kernel over X. The antisymmetrization of F is the four legged kernel 1 Ant F (x1 , x2 , x3 , x4 ) = 4! sign(π ) F (xπ(1) , xπ(2) , xπ(3) , xπ(4) ). π∈S4
F is called antisymmetric if F = Ant F . We will consider ladders with rungs taking values in the measure space B × , where is a sectorization. As propagators, we will use the unsectorized propagators (j ) A = Cu(K) and B = Dj (u(K); K) of Definition III.6. Definition VII.2. Let X be a measure space and let S be a finite set4 . It is endowed with the counting measure. Then X × S is also a measure space. i) Let P be a propagator over X, f a four legged kernel over X × S and F a function on (X × S)2 × X2 . We define (f • P )((x1 ,s1 ),(x2 ,s2 );x3 ,x4 ) = dx1 dx2 f ((x1 ,s1 ),(x2 ,s2 ),(x1 ,s1 ),(x2 ,s2 )) P (x1 ,x2 ;x3 ,x4 ), s1 ,s2 ∈S
(F • f )((x1 ,s1 ),··· ,(x4 ,s4 )) = dx1 dx2 F ((x1 ,s1 ),(x2 ,s2 );x1 ,x2 ) f ((x1 ,s1 ),(x2 ,s2 ),(x3 ,s3 ),(x4 ,s4 )) s1 ,s2 ∈S
whenever the integrals are well–defined. Observe that (f • P ) is a function on (X × S)2 × X2 and F • f is a four legged kernel over X × S. 4
In practice, S will be a set of sectors and X will be B or R × R2 × {↑, ↓} or R × R2 .
Two Dimensional Fermi Liquid. 2: Convergence
65
ii) Let ≥ 1 , r1 , · · · , r+1 rungs over X × S and P1 , · · · , P bubble propagators over X. The ladder with rungs r1 , · · · , r+1 and bubble propagators P1 , · · · , P is defined to be r1 • P1 • r2 • P2 • · · · • r • P • r+1 . If r is a rung over X × S and A, B are propagators over X, we define L (r; A, B) as the ladder with + 1 rungs r and bubble propagators C(A, B). When we integrate out scale j in our model, the contributions to the four legged kernel that are not controlled by “overlapping loops” are antisymmetrizations of ladders that are defined over B × , where is a sectorization. Such ladders decompose into particle–particle ladders and particle–hole ladders that are defined over smaller spaces that do not have creation/annihilation components. Definition VII.3. Set B = (x0 , x, σ ) x0 ∈ R, x ∈ R2 , σ ∈ {↑, ↓} . If is a sectorization and z = (x, σ, b, s) ∈ B ×, we define its undirected part u(z) ∈ B × and its creation/annihilation index b(z) ∈ {0, 1} by u(z) = (x, σ, s) and b(z) = b, respectively. If z = (x, σ, s) ∈ B × and b ∈ {0, 1}, we define ιb (z ) = (x, σ, b, s) ∈ B ×. Definition VII.4. i) Let f be a four legged kernel over B × . When f is a rung, its particle–particle reduction is the four legged kernel over B × given by f pp (z1 ,z2 ,z3 ,z4 ) = f (ι0 (z1 ),ι0 (z2 ),ι1 (z3 ),ι1 (z4 )) = and its particle–hole reduction is f ph (z1 ,z2 ,z3 ,z4 ) = f (ι0 (z1 ),ι1 (z2 ),ι1 (z3 ),ι0 (z4 )) =
1
3 f
2
4
1
3 f
2 When f is a bubble propagator, the corresponding reductions are
4
f (z1 ,z2 ,z3 ,z4 ) = f (ι1 (z1 ),ι1 (z2 ),ι0 (z3 ),ι0 (z4 )), f (z1 ,z2 ,z3 ,z4 ) = f (ι1 (z1 ),ι0 (z2 ),ι0 (z3 ),ι1 (z4 )).
pp ph
ii) Let f be a four legged kernel over B × . Its particle–particle value is the four legged kernel over B × given by Vpp (f )(z1 ,z2 ,z3 ,z4 ) = δb(z1 ),0 δb(z2 ),0 δb(z3 ),1 δb(z4 ),1 f (u(z1 ),u(z2 ),u(z3 ),u(z4 )) +δb(z1 ),1 δb(z2 ),1 δb(z3 ),0 δb(z4 ),0 f (u(z3 ),u(z4 ),u(z1 ),u(z2 )) and its particle–hole value is Vph (f )(z1 ,z2 ,z3 ,z4 ) = δb(z1 ),0 δb(z2 ),1 δb(z3 ),1 δb(z4 ),0 f (u(z1 ),u(z2 ),u(z3 ),u(z4 )) +δb(z1 ),1 δb(z2 ),0 δb(z3 ),0 δb(z4 ),1 f (u(z2 ),u(z1 ),u(z4 ),u(z3 )) −δb(z1 ),1 δb(z2 ),0 δb(z3 ),1 δb(z4 ),0 f (u(z2 ),u(z1 ),u(z3 ),u(z4 )) −δb(z1 ),0 δb(z2 ),1 δb(z3 ),0 δb(z4 ),1 f (u(z1 ),u(z2 ),u(z4 ),u(z3 )). The decomposition of ladders over B into particle–particle and particle–hole ladders is given by the following lemma, whose proof is trivial.
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Lemma VII.5. i) Let f1 , · · · , f+1 be particle number preserving four legged kernels over B × that are separately antisymmetric in their first two and their last two arguments. Let P1 , · · · , P be particle number preserving bubble propagators over B that satisfy Pi (ξ1 , ξ2 , ξ3 , ξ4 ) = Pi (ξ2 , ξ1 , ξ4 , ξ3 ) for i = 1, · · · , . Then pp pp pp f1 • P1 • · · · • P • f+1 = f1 • ppP1 • · · · • ppP • f+1 , ph ph ph = 2 f1 • phP1 • · · · • phP • f+1 . f1 • P1 • · · · • P • f+1 ii) Let f be an antisymmetric, particle number preserving, four legged kernel over B×. Then f = Vpp (f pp ) + Vph (f ph ). We now state the ladder estimates used in the rest of the paper. τ2 Proposition VII.6. Let 0 < < 2M j , where τ2 is the constant of Lemma XIII.6 of [FKTo3]. Let u((ξ,s), (ξ ,s )) ∈ F0 (2, j ) be an antisymmetric, spin independent, particle number conserving function whose Fourier transforms obey |u(k)| ˇ ≤ 21 |ık0 − e(k)| and such that |u||1,j ≤ cj . Furthermore let f ∈ F0 (4, j ). Then for all ≥ 1, L f ; Cu(j ) , Cu(≥j +1) ≤ const cj |f |+1 3,j , 3,j pp 1/n0 (j ) (≥j +1) Vpp L f ; Cu , Cu ≤ const lj cj |f |+1 3,j , 3, j
if the Fermi curve F is strongly asymmetric in the sense of Definition I.10. Here, n0 is the constant of Definition I.10. Proposition VII.6 is a special case of Proposition XIV.9. See Remark XIV.10. The first inequality of Proposition VII.6 is not good enough for the control of the (j ) (≤j ) four point function, since replacing Cu by Cu would give an additional factor of j n . The second inequality of Proposition VII.6 gives estimates for particle–particle ladders at each individual scale j that are good enough to be summable over j . Particle–hole ladders do not, at least in general, obey such estimates. If they did, they would be continuous in momentum space, even after all cutoffs are removed. Therefore, we bound the sum of all particle–hole ladders of scales up to j together, making use of cancellations between neighbouring scales. Building up such sums of ladders leads to “compound particle–hole ladders”. Definition VII.7. Let F = F (i) i = 2, 3, · · · be a family of antisymmetric functions (2) (3) in F0 (4, i ). Let p = p , p , · · · be a sequence of antisymmetric, spin independent, particle number conserving functions p (i) ((ξ,s), (ξ ,s )) ∈ F0 (2, i ). We define, recursively on 0 ≤ j < ∞, the iterated particle hole (or wrong way) ladders up to scale j , denoted by L(j ) (p, F ) , as L(0) (p, F ) = 0 L(j +1) (p, F ) = L(j ) (p, F )j + 2
∞
(−1) (12)+1 L
=1
where uj =
j −1 i=2
(i)
pj and wj =
j
(i) i=2 Fj
(j )
(≥j +1) ph
wj ; Cuj , Cuj
+ 18 Ant Vph L(j ) (p, F )
j
,
. The re-
sectorization L(j ) (p, F )j is defined by the natural analog of Definition VI.4. For the details, see Definition XIX.6 of [FKTo4].
Two Dimensional Fermi Liquid. 2: Convergence
67
Observe that L(j ) (p, F ) is a four legged kernel over B × j −1 and depends only on the components F (2) , · · · , F (j −1) of F and p (2) , · · · , p(j −2) of p . Also observe that w0 , L(1) (p, F ), w1 and L(2) (p, F ) all vanish. When we apply Definition VII.7, F (i) will be the volume improved part of the contribution to the four–point function generated by integrating out scale i. Furthermore, p(i) will be, roughly speaking, the contribution to the renormalized two–point function at K = 0 that is moved into the covariance at scale i. In particular, F (2) through F (j0 ) and p (2) through p (j0 −1) will be zero. The main estimate on iterated particle hole ladders is Theorem VII.8. For every ε > 0there are constants ρ0 , const such that the following holds. Let F = F (2) , F (3) , · · · be a sequence of antisymmetric, spin independent, particle number conserving functions F (i) ∈ F0 (4, i ) and p = p (2) , p(3) , · · · be a sequence of antisymmetric, spin independent, particle number conserving functions p(i) ∈ F0 (2, i ). Assume that there is ρ ≤ ρ0 such that for i ≥ 2 |F (i)|3,i ≤ Then for all j ≥ 2
ρ c M εi i
|p (i)|1,i ≤ ρMlii ci
pˇ (i) (0, k) = 0.
(j ) Vph L (p, F )j 3, ≤ const ρ 2 cj . j
Theorem VII.8 is a special case of Theorem XIV.12 in part 3. See Remark XIV.13. Both Theorems are consequences of the estimates on iterated particle hole ladders derived in [FKTl]. VIII. Infrared Limit of Finite Scale Green’s Functions The nonperturbative construction of the infrared limit will be similar to the formal construction outlined in §III. We first adapt the notion of a formal interaction triple (W, G, u) at scale j of Definition III.4 to the needs of the nonperturbative construction. The function u modifying the covariance at scale j is built up of contributions created at scales up to j − 1. To bound u, we keep track of all of these individual contributions. They are encoded in the additional datum p of Definition VIII.1 (Interaction Quadruple). An interaction quadruple at scale j is a quadruple (W, G, u, p) that obeys the following conditions. ◦ W is a map from the space Kj of counterterms to the space of even, translation invariant, spin independent, particle number conserving Grassmann functions in φ and ψ, that obeys W(φ, 0, K) = 0. ◦ G is a map from Kj to the space of even, translation invariant, spin independent, particle Grassmann functions in φ, that obeys G(0, K) = 0. number conserving ◦ p = p(2) , · · · , p (j −1) where each p (i) ((ξ,s), (ξ ,s )) is an antisymmetric, spin independent, particle number conserving function in F0 (2, i ) that obeys (i) li p ≤ λ1−υ c. (VIII.1) 0 1, Mi i i
The Fourier transform pˇ (i) (k) of p (i) is supported in the i th neighbourhood and vanishes at k0 = 0.
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◦ u is a map from Kj to the space of antisymmetric, spin independent, particle number conserving functions in F0 (2, j ). The function u(K) has a decomposition u(K) =
j −1 i=2
(i) pj + δu(K) − Kext
j
(VIII.2)
with δu((ξ,s), (ξ ,s ); K) an antisymmetric function in F0 (2, j −1 ) that vanishes at k0 = 0 and when K = 0 and obeys d δu(K + sK ) ≤ λ1−υ ej K 1,j K 1,j 0 ds s=0 1,j −1 (1,0,0) d 1−υ j −3 D ej K 1,j K 1,j 1,2 ds δu(K + sK ) s=0 1,j −1 ≤ λ0 M + ∞ tδ (VIII.3) δ0 =r0
for all K ∈ Kj and all K . The interaction quadruple (W, G, u, p) is said to be real, if W(φ, ψ, K), G(φ, K), u(K) and p(1) , · · · , p (j −1) are k0 –reversal real, in the sense of Definition B.1.R of [FKTo2], for all real K ∈ Kj . In particular pˇ (i) (−k0 , k) = pˇ (i) (k0 , k). Remark VIII.2. i) We remind the reader that was defined in Definition VI.9. The extension The space Kj of counterterms Kext (ξ, s), (ξ , s ) of K (x, s), (x , s ) was defined in Definition E.1 of [FKTo4]. The sector length lj was fixed after Definition VI.3, the resectorization pj was defined in Definition VI.4, the space F0 (2, j ) was defined in Definition VI.3.ii and the seminorm | · |1,j was defined in Definition VI.6.i. δ was defined in Definition V.1.ii and the elements c and The decay operator Di,j j ej (X) of the norm domain were defined in Definition V.2.iii. ii) Observe that p is independent of K, so that, in (VIII.2), the only K dependence of u is through δu(K) − Kext . iii) The representation (VIII.2) of u implies that u(k; ˇ K) =
j −1
ˇ pˇ (i) (k) + δ u(k; ˇ K) − K(k)
i=2
for k in the j th neighbourhood. Bounds on u(K) will be provided in Lemma VIII.7. The relation between counterterms at different scales is formalized in Definition VIII.3. A projective system of counterterms consists of analytic maps reni,j : Kj +1 −→ Ki+1 δej : Kj +1 −→ E
for j0 ≤ i ≤ j, for j0 ≤ j,
such that renj,j is the identity map of Kj +1 , reni,i ◦ reni ,j = reni,j for j0 ≤ i ≤ i ≤ j, δei ◦ reni,j = δej for j0 ≤ i ≤ j,
Two Dimensional Fermi Liquid. 2: Convergence
69
and sup |||δ eˆj (K)|||1,∞ ≤ λ1−υ 0 ,
K∈Kj
1−υ 1 reni,j (0) − reni,j (0) ≤ λ + ∞t δ , j 0 1, 2 i
δ=0
1 |||δ eˆj (0) − δ eˆj (0)|||1,∞ ≤ λ1−υ 0 2j
for all j0 ≤ i ≤ j ≤ j . A projective system is said to be real if reni,j (K) is real and δej (k; K) is real–valued for all i, j and all real K. Remark VIII.4. For any projective system of counterterms, the sequence δej (K)K=0 of infrared cutoff counterterms converges in the topology of E. We shall prove, in §X, the following bounds on the analogs of the formal interaction triple (Wjout , Gjout , uj ) of (III.9). Theorem VIII.5. Assume that d = 2, that e(k) fulfills the Hypotheses I.12 and that the scale parameter M has been chosen big enough. Then there exist constants α, ¯ λ¯ > 0 1 such that for all 0 ≤ λ0 < λ¯ , α¯ < α < υ/10 the following holds. For each translation λ0
invariant, spin independent interaction kernel V obeying V 1,∞ ≤ λ0 c0 there exist ◦ a projective system of counterterms reni,j , δej (2) (3) ◦ a family p = p , p · · · of functions p(i) ∈ F0 (2, i ) ◦ a family F = F (2) , F (3) · · · of antisymmetric kernels F (i) ((ξ1 ,s1 ), · · · , (ξ4 ,s4 )) ∈ F0 (4, i ) such that
(i) F ≤ 3, i
1/n λ1−υ 0 l 0 ci . α7 i
All of this data depends analytically on V . Also, for each scale j ≥ j0 there exist Wj, rg rg Gj and uj , depending analytically on V and K, such that Wj , Gj , uj , (p(2) ,··· ,p(j −1) ) is an interaction quadruple at scale j . Furthermore (R1) Wj (K) has a j –sectorized representative, w(φ, ψ; K) = dη1 ···dηm dξ1 ···dξn wm,n (η1 ,··· ,ηm (ξ1 ,s1 ),··· ,(ξn ,sn ); K) m,n s1 ,··· ,sn ∈j
× φ(η1 ) · · · φ(ηm ) ψ((ξ1 ,s1 )) · · · ψ((ξn ,sn ) )
with wm,n antisymmetric separately in the η and in the (ξ, s) variables, wm,0 = 0 for all m ≥ 0 and λ1−υ l w0,2 (K) ≤ α0 7 Mjj ej K 1,j , 1,j Nj w(K), α, K 1,j ≤ ej K 1,j , Nj dds w(K + sK )s=0 , α, K 1,j ≤ M j ej K 1,j K 1,j for all K ∈ Kj and all K . w depends analytically on V and K.
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(R2) The function w0,4 (K) has a decomposition w0,4 (K) = δF (j +1) (K) +
j i=2
(i) Fj + 18 Ant Vph L(j +1) (p, F )
with an antisymmetric kernel δF (j +1) ((ξ1 ,s1 ), · · · , (ξ4 ,s4 ); K) ∈ F0 (4, j ) such that (j +1) δF (K)3, ≤ j
λ1−υ 0 α4
l1/n0 j +1 α4
+
1 M j K 1,j B2
ej ( K 1,j ) for all K ∈ Kj .
The particle–hole projector Vph is defined in Definition VII.4. (R3) For each K ∈ Kj , j rg 1 √ N Gj (K) − 21 φJ C (≤j ) J φ ≤ 4 4 i=2
li M i
for all K ∈ Kj .
rg
Let the part of Gj (K) that is homogeneous of degree two be rg Gj,2 (K)
=
rg
dη1 dη2 Gj,2 (η1 , η2 , K) φ(η1 )φ(η2 ).
Then N
d rg rg ) − G (K + sK ) s=0 ≤ M j K 1,j ds Gj (K + sK d rg j,2 G (K + sK ) ≤ M j K 1, ds
s=0 ∞
j,2
j
for all K ∈ Kj and all K . (R4) For K ∈ Kj +2 and K = renj,j +1 (K), rg rg 4 N Gj +1 (K) − 21 φJ C (j +1) J φ − Gj (K ) ≤ √ 4
lj M j
.
(R5) For infrared cutoffs j¯ ≥ j + 2, the generating function of the connected Green’s functions at scale j¯ of Theorem I.4, considered as a formal Taylor series in V , fulfills ¯ δej (K)) = G rg (φ; K) + log Gj (φ, φ; j ¯
:W (φ,ψ;K):
ψ,Dj (uj ;K) dµC [j +1,j ) (ψ) eφJ ψ e j uj (K)¯ :W (0,ψ;K) j e :ψ,Dj (uj ;K) dµC [j +1,j ) (ψ) uj (K)¯
for K ∈ Kj . If, in addition, V satisfies the reality condition of (I.1) then ◦ the projective system reni,j , δej is real ◦ each F (i) is k0 –reversal real, of [FKTo2] in thergsense of Definition B.1.R ◦ each interaction quadruple Wj , Gj , uj , (p(2) ,··· ,p(j −1) ) is real ◦ for real K, the j –sectorized representative w(φ, ψ; K) of Wj (K) is k0 –reversal real.
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Proof of Theorem I.4 from Theorem VIII.5. Observe that Gj (φ; δe) depends on M and ¯ j¯ only through the combination Mj¯ . Hence, for constructing lim Gj (φ; δe), we may, j →+∞ ¯ j ∈R ¯ ¯ without loss of generality, use any M > 1 we wish. Choose M, α and λ0 fulfilling the hypotheses of Theorem VIII.5. By Remark V.5, the conditions on the interaction kernel in Theorems I.4 and VIII.5 agree. By Remark VIII.4, δe = lim δej (0) j →∞
exists. If V is k0 –reversal real, as in (I.1), then δe(k) is real for all k. By Definition VIII.3, |||δ e||| ˆ 1,∞ ≤ λ1−υ 0
1 |||δ eˆj (0) − δ e||| ˆ 1,∞ ≤ λ1−υ 0 2j
for all j ≥ j0 . By Weierstrass’ Theorem, δe is analytic in V . rg We now show that the sequence Gj (0) − 21 φJ C (≤j ) J φ is a Cauchy sequence. Let l j ≥ j0 and K = renj,j +1 (0). By Definition VIII.3, K ≤ λ1−υ jj + ∞t δ . 1,j
0
M
δ=0
Hence, by (R3), (R4) and the Definition VI.7.ii of the norm N (G), rg rg N Gj +1 (0) − 21 φJ C (≤j +1) J φ − Gj (0) + 21 φJ C (≤j ) J φ rg rg rg rg ≤ N Gj +1 (0) − 21 φJ C (j +1) J φ − Gj (K ) + N Gj (K ) − Gj,2 (K ) rg rg rg rg −Gj (0) + Gj,2 (0) + N Gj,2 (K ) − Gj,2 (0) 4 1 ≤ √ + M j K 1,j + 1−υ M j K 1,j 4 j λ0
lj M
4 ≤ √ 4
lj M j
+ 2lj .
Let G(φ) =
∞ n=1
1 n!
n
dξi Gn (ξ1 , · · · , ξn )
i=1
n
φ(ξi )
i=1
rg
be the limit of the Gj (0)’s. It is analytic in V . We now show that the generating functionals Gj (φ) of the connected Green’s functions at scale j¯ also converge to G(φ). Let j¯ ≥ j0 ¯+ 3 and let j = j (j¯ ) be the integer with j + 1 < j¯ ≤ j + 2. By Definition VIII.3, K = lim renj,j (0) j →∞
exists in Kj and obeys δe = δej (K). Observe that, by (R5), rg
Gj (φ; δe) − Gj (φ; 0) ¯ rg rg rg = Gj (φ; δej (K)) − Gj (φ; K) + Gj (φ; K) − Gj (φ; 0) ¯ :W (φ,ψ;K):ψ,Dj (uj ;K) eφJ ψ e j dµS (ψ) rg rg = log :W (0,ψ;K) + Gj (φ; K) − Gj (φ; 0) j e :ψ,Dj (uj ;K) dµS (ψ) ˜ S :Wj (K):ψ,Dj (uj ;K) (φ, 0) + G rg (φ; K) − G rg (φ; 0), (VIII.4) = j j
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where S= =
ν (≥j +1) (k) − ν (≥j¯) (k) ık0 − e(k) − uˇ j (k; K)[1 − ν (≥j¯) (k)] ν (≥j +1) (k) − ν (≥j¯) (k) . ık0 − e(k) − uˇ j (k; K)[ν (≥j ) (k) − ν (≥j¯) (k)]
As in the previous paragraph
rg rg N Gj (K) − Gj (0) ≤ 2lj .
(VIII.5)
In Lemma VIII.7, below, we prove that the hypotheses of Proposition XV.10 of [FKTo3], (≥j +2) (k) and ˇ ˇ = uˇ j (k; K) + K(k)ν with u(k) ˇ = uˇ j (k; K)[ν (≥j ) (k) − ν (≥j¯) (k)], v(k) X = K 1,j are satisfied. Consequently, √ 4 ˜ S :Wj (K):ψ,Dj (uj ;K) (φ, 0) − 1 φJ SJ φ ≤ 10 lj M j N 2 As N
1
2 φJ SJ φ
= ≤ ≤
1 1 |||J SJ |||∞ ≤ 1−υ S(k) L1 λ1−υ λ0 0 (≥j +1) 1 − ν (≥j¯) 1−υ Vol support ν λ0 const , j λ1−υ 0 M
sup |S(k)| k
we have that
10 ˜ S :Wj (K):ψ,Dj (uj ;K) (φ, 0) ≤ √ N 4
lj M j
+
const j λ1−υ 0 M
.
(VIII.6)
Combining (VIII.4), (VIII.5) and (VIII.6), rg lim N Gj (φ; δe) − Gj (j ) (φ; 0) = 0 j →∞ ¯ ¯ ¯ so that lim Gj = G j →∞ ¯ ¯ in the N ( · ) norm. Consequently, for each n, Gn;j converges uniformly to Gn . ¯ Definition VIII.6. If u((ξ,s), (ξ ,s )) is an antisymmetric, translation invariant, spin independent, particle number conserving function on (B × )2 and µ(k) is a function on R × R2 , set dζ u((ξ,s), (ζ,s ))µ(ζ, ˆ ξ ), (u ∗ µ)( ˆ (ξ,s), (ξ ,s )) = B (µˆ ∗ u)((ξ,s), (ξ ,s )) = dζ u((ζ,s), (ξ ,s ))µ(ζ, ˆ ξ ), B
where µˆ was defined in Notation V.4. ˇ With this definition (u ∗ µ) ˆ ˇ(k) = (µˆ ∗ u)ˇ(k) = u(k)µ(k). Lemma VIII.7. Let W, G, u, p be an interaction quadruple at scale j . Then
Two Dimensional Fermi Liquid. 2: Convergence
73
i) (≥j +2) ≤ λ1−υ ık0 − e(k) ≤ 1 ık0 − e(k), u(k; ˇ (k) ˇ K) + K(k)ν 0 2 3 d ˇ K + sK )s=0 + Kˇ (k)ν (≥j +2) (k) ≤ 4M j + 2 K 1,j ık0 − e(k) ds u(k; for all k in the j th neighbourhood, all K ∈ Kj and all K . ii) 1−υ λ |u(K)||1,j ≤ const M0j −1 + K 1,j ej ( K 1,j ), d u(K + sK ) ≤ const ej ( K 1,j ) K 1,j . ds s=0 1, j
iii) Let j¯ ∈ (j, j + 2]. Then 1−υ λ0 u(K) ∗ ν (≥j ) − ν (≥j¯) ≤ const + K 1, j −1 j ej ( K 1,j ). 1, M j
Proof.
i) By Remark VIII.2.iii, (≥j +2) u(k; ˇ (k) ˇ K) + K(k)ν j −1 (i) ˇ = pˇ (k) + δ u(k; ˇ K) − K(k) 1 − ν (≥j +2) (k) ≤
i=2 j −1
pˇ (i) (k) + δ u(k; ˇ 1 − ν (≥j +2) (k) ˇ K) + K(k)
i=2
and
d ˇ K + sK )s=0 + Kˇ (k)ν (≥j +2) (k) ≤ dds δ u(k; ˇ K + sK )s=0 ds u(k; +Kˇ (k) 1 − ν (≥j +2) (k)
for all k in the j th neighbourhood. Because pˇ (i) vanishes at k0 = 0, Lemma XII.12 of [FKTo3] and (VIII.1) imply that (i) li (i) pˇ (k) ≤ 2|k0 | ∂ | p | ≤ 2|k0 | λ1−υ Mi. (VIII.7) 1, 0 i ∂t(1,0,0) Mi t=0
Similarly, by Lemma XII.12 of [FKTo3], (VIII.3) and Definition VI.9, (1,0,0) δ u(k; ˇ K) ≤ 2|k0 | |D1,2 δu(K)||1,j −1 t=0 1 lj +1 1−υ j −3 ≤ 2|k0 | λ0 M λ1−υ j +1 0 M l j +1 1−υ 1 − M j λ0 M j +1 lj (VIII.8) ≤ 4|k0 | λ2−2υ 0 d (1,0,0) d ˇ K + sK )s=0 ≤ 2|k0 | D1,2 ds δu(K + sK )s=01, ds δ u(k; j −1 t=0 1 1−υ j −3 ≤ 2|k0 | λ0 M K 1,j 1−υ lj +1 j 1 − M λ0 M j +1 j −3 M K 1,j ≤ 4|k0 | λ1−υ 0
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and, by Lemma XII.12 of [FKTo3], Definition VI.9 and Definition VIII.1.i of [FKTo2], K(k) ˇ 1 − ν (≥j +2) (k) ≤ 2 K 1,j 1 − ν (≥j +2) (k) t=0 lj +1 ≤ 2λ1−υ 1 − ν (≥j +2) (k) 0 M j +1 lj +1 |ık0 − e(k)| ≤ 2λ1−υ 0 M j +1 √1/M 1 M j +1 √ ≤ 2 Mλ1−υ l |ık − e(k)| j +1 0 0 1 1−υ ≤ 10 λ0 lj |ık0 − e(k)| (VIII.9) (≥j +2) Kˇ (k) 1 − ν (≥j +2) (k) ≤ 2 K 1, (k) j t=0 1 − ν |ık0 − e(k)| ≤ 2 K 1,j √ 1/M M j1+1 3
≤ 2M j + 2 K 1,j |ık0 − e(k)|
ˇ 1 − ν (≥j +2) (k) , if M is large enough. In the last inequality of the bound on K(k) l
we used that jl+1 = M1ℵ with ℵ > 21 . Combining (VIII.7), (VIII.8) and (VIII.9), j (≥j +2) u(k; ˇ ˇ K) + K(k)ν (k) j −1 1−υ 1 λ1−υ ≤ 2li + 4λ1−υ l + l j j 0 0 |ık0 − e(k)| ≤ λ0 |ık0 − e(k)| 10 i=2 d ˇ K + sK )s=0 + Kˇ (k)ν (≥j +2) (k) ds u(k; 3 −3 + 2M 2 M j K 1,j |ık0 − e(k)| ≤ 4λ1−υ 0 M 3 ≤ 4M j + 2 K 1,j ık0 − e(k). ii) By Corollary XIX.13, Remark XIX.5 and Proposition E.10.i of [FKTo4], |u(K)||1,j j −1 (i) = pj + δu(K) − Kext j i=2
j −1
1,j
li 1−υ K K ≤ const cj −1 λ1−υ M + λ e ( K ) + 1, j j j 0 0 1,j 1,j M 1−υ i=2 λ0 ≤ const M j −1 cj + λ1−υ 0 ej ( K 1,j )cj K 1,j + cj K 1,j 1−υ λ K ≤ const M0j −1 + 1 + λ1−υ e ( K 1,j ) 0 1,j j d u(K + sK ) ds s=0 1,j ≤ const cj −1 λ1−υ 0 ej ( K 1,j ) K 1,j + K 1,j ≤ const ej ( K 1,j ) K 1,j .
iii) By Lemma XIII.7 of [FKTo3] with µ(t) = ϕ(t/M) − ϕ(M 2(j¯−j ) t/M), where ϕ is the function used in Definition I.2, and = M j we have
Two Dimensional Fermi Liquid. 2: Convergence
75
u(K) ∗ ν (≥j ) − ν (≥j¯) ≤ const cj |u(K)||1,j 1,j 1−υ λ ≤ const cj M0j −1 + K 1,j ej ( K 1,j ) 1−υ λ ≤ const M0j −1 + K 1,j ej ( K 1,j ). IX. One Recursion Step The data of Theorem VIII.5 are constructed recursively. In this section, we implement one recursion step, analogous to the map j ◦ Oj of §III. 1. Input and Output Data. We now impose the actual conditions on the input and output data, analogous to Definitions III.8 and III.9. Definition IX.1 (Input Data). The input data just before integrating out the j th scale (j ) is the set Din of interaction quadruples, in the sense of Definition VIII.1, (W, G, u, p) that fulfill (I1) W(K) has a j –sectorized representative w(φ, ψ; K) = dη1 ···dηm dξ1 ···dξn wm,n (η1 ,··· ,ηm (ξ1 ,s1 ),··· ,(ξn ,sn ); K) m,n s1 ,··· ,sn ∈j
× φ(η1 ) · · · φ(ηm ) ψ((ξ1 ,s1 )) · · · ψ((ξn ,sn ) )
with wm,n antisymmetric separately in the η and in the ξ variables, w0,2 = 0, wm,0 = 0 for all m ≥ 0 and Nj w(K), 64α, K 1,j ≤ ej K 1,j , d Nj ds w(K + sK )s=0 , 64α, K 1,j ≤ M j ej K 1,j K 1,j , for all K ∈ Kj and all K . (I2) There is a family F of antisymmetric kernels F (i) ((ξ1 ,s1 ), · · · , (ξ4 ,s4 )) ∈ F0 (4, i ),
2≤i ≤j −1
(independent of K) and an antisymmetric kernel δF (j ) ((ξ1 ,s1 ), · · · , (ξ4 ,s4 ); K) ∈ F0 (4, j ) such that w0,4 (K) = δF (j ) (K) +
j −1 i=2
(i) Fj + 18 Ant Vph L(j ) (p, F )
j
,
where the particle–hole value Vph was defined in Definition VII.4. Furthermore, for all 2 ≤ i ≤ j − 1, (i) F ≤ 3, i
1/n λ1−υ 0 l 0 ci α7 i
and (j ) δF (K) ≤ 3, j
λ1−υ 0 α4
l1/n0 j
α3
+
1 M j K 1,j B2
ej ( K 1,j )
for all K ∈ Kj .
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(I3) For each K ∈ Kj , j −1 1 √ N G(K) − 21 φJ C (<j ) J φ ≤ 4 4 i=2
li M i
.
Let G2 (K) = dη1 dη2 G2 (η1 , η2 , K) φ(η1 )φ(η2 ) be the part of G(K) that is homogeneous of degree two. Then N dds G(K + sK ) − G2 (K + sK ) s=0 ≤ 21 M j K 1,j d G2 (K + sK ) ≤ 1 M j K 1, j ds 2 s=0 ∞ for all K ∈ Kj and all K . The input data is said to be real if ◦ the interaction quadruple (W, G, u, p) is real ◦ for real K, the j –sectorized representative w(φ, ψ; K) of W(K) is k0 –reversal real, in the sense of Definition B.1.R of [FKTo2] and ◦ each F (i) is k0 –reversal real. Definition IX.2 (Output Data). The output data just after integrating out the j th scale (j ) is the set Dout of interaction quadruples (W, G, u, p) that fulfill (O1) W(K) has a j –sectorized representative dη1 ···dηm dξ1 ···dξn wm,n (η1 ,··· ,ηm (ξ1 ,s1 ),··· ,(ξn ,sn ); K) w(φ, ψ; K) = m,n s1 ,··· ,sn ∈j
× φ(η1 ) · · · φ(ηm ) ψ((ξ1 ,s1 )) · · · ψ((ξn ,sn ) )
with wm,n antisymmetric separately in the η and in the ξ variables, wm,0 = 0 for all m ≥ 0 and λ1−υ l w0,2 (K) ≤ α0 7 Mjj ej K 1,j , 1,j Nj w(K), α, K 1,j ≤ ej K 1,j , Nj dds w(K + sK )s=0 , α, K 1,j , ≤ M j ej K 1,j K 1,j for all K ∈ Kj and all K . (O2) There is a family F of antisymmetric kernels F (i) ((ξ1 ,s1 ), · · · , (ξ4 ,s4 )) ∈ F0 (4, i ),
2≤i≤j
(independent of K) and an antisymmetric kernel δF (j +1) ((ξ1 ,s1 ), · · · , (ξ4 ,s4 ); K) ∈ F0 (4, j ) such that w0,4 (K) = δF (j +1) (K) +
j i=2
Furthermore,
(i) F ≤ 3, i
and (j +1) δF (K)3, ≤ j
λ1−υ 0 α4
1/n λ1−υ 0 l 0 ci α7 i
l1/n0 j +1 α4
(i) Fj + 18 Ant Vph L(j +1) (p, F ) .
+
for all 2 ≤ i ≤ j
1 M j K 1,j B2
ej ( K 1,j ) for all K ∈ Kj
Two Dimensional Fermi Liquid. 2: Convergence
77
(O3) For each K ∈ Kj , j −1 1 √ N G(K) − 21 φJ C (≤j ) J φ ≤ 4 4 i=2
li M i
2 + √ 4
lj M j
for all K ∈ Kj
Let G2 (K) = dη1 dη2 G2 (η1 , η2 , K) φ(η1 )φ(η2 ) be the part of G(K) that is homogeneous of degree two. Then N dds G(K + sK ) − G2 (K + sK ) s=0 ≤ M j K 1,j d G2 (K + sK ) ≤ M j K 1, j ds s=0 ∞ for all K ∈ Kj and all K . The output data is said to be real if ◦ the interaction quadruple (W, G, u, p) is real, ◦ for real K, the j –sectorized representative w(φ, ψ; K) of W(K) is k0 –reversal real, ◦ each F (i) is k0 –reversal real. Remark IX.3. Conditions (O1) and (O2) coincide with Conditions (R1) and (R2) of Theorem VIII.5, while Condition (O3) implies Condition (R3). (j )
(j )
2. Integrating Out a Scale. In this subsection we implement the map j : Din → Dout , analogous to that of Definition III.6, that integrates out fields of scale j . We use the covariances (j )
Cu (k) = (≥j )
Cu
(k) =
Dj (u; K)(k) = Cj (u; K)(k) =
ν (j ) (k) , ık0 −e(k)−u(k;K) ˇ ν (≥j ) (k) , ık0 −e(k)−u(k;K) ˇ (≥j +1) ν (k) , (≥j +2) (k) ˇ ık0 −e(k)−u(k;K)− ˇ K(k)ν (j ) Cu (k) + Dj (u; K)(k)
from Definition III.5 and Remark III.7. Definition IX.4. Integrating out the fields of scale j is implemented by the map j , (j ) which maps an interaction quadruple (W, G, u, p) ∈ Din to the quadruple (W , G , u, p) determined by :W (φ,ψ+ζ ;K):ψ,Cj (u;K) 1 :W (φ, ψ; K):ψ,Dj (u;K) = log Z(φ) eφJ ζ e dµC (j ) (ζ ), u(K)
G (φ) = G(φ) + log Z(φ) Z(0) , where log Z(φ) =
log
eφJ ζ e
:W (φ,ψ+ζ ;K):ψ,Cj (u;K)
dµC (j ) (ζ ) dµDj (u;K) (ψ). u(K)
(j )
(j )
Theorem IX.5. If (W, G, u, p) ∈ Din then j (W, G, u, p) ∈ Dout .
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Proof. Let (W , G , u, p) = j (W, G, u, p) . As in Definition III.6, one easily verifies that (W , G , u, p) is an interaction quadruple of scale j . Define W by . W (φ, ψ; K) . ˜ (j ) .. W(φ, ψ; K) .. . ψ,Dj (u;K) = . ψ,C (u;K) Cu j . W (φ,ψ+ζ ;K) . . . ψ,Cj (u;K) 1 = log Z(0) eφJ ζ e dµC (j ) (ζ ). u
Then W (φ, ψ; K) = W (φ, ψ; K) − W (φ, 0; K), G (φ) = G(φ) + W (φ, 0; K).
(IX.1)
We now apply Theorems XV.3 and XV.7 of [FKTo3] to bound W . By (I1) and parts (i) and (ii) of Lemma VIII.7, the hypotheses of Theorem XV.3 of [FKTo3], with µ = const , λ1−υ = 0j −1 and X = K , are fulfilled. Therefore W has a sectorized representative 1,j
M
w obeying
Nj (w − 21 φJ Cu J φ − w, α, K 1,j ) (j )
≤
Nj (w,64α, K 1,j ) const α 1− const Nj (w,64α, K 1, ) , α j
|w0,2 |1,j
Nj (w,64α, K 1,j ) const lj , α 8 ρ0;2 M j 1− const α Nj (w,64α, K 1,j ) ∞ w −w0,4 − 1 (−1) (12)+1 L (w0,4 ; Cu(j ) ,Dj (u;K)) 0,4 4 3,j =1 2 (w,64α, K ) N 1, j j ≤ αconst lj 1− const N (w,64α, K . 10 ρ ) 2
≤
0;4
α
1,j
j
(IX.2)
The hypotheses of Theorem XV.7 of [FKTo3], with, in addition, Y = const K 1, and j ε = const M j K 1, , are fulfilled. Hence j
t=0
(j ) Nj dds w (K + sK ) − 21 φJ Cu(K+sK ) J φ s=0 , α, K 1,j ≤ Nj dds w(K + sK )s=0 , α, K 1,j d + const α Nj ds w(K + sK ) s=0 , 16α, K 1,j Nj (w,64α, K 1, )
j + const α 1− const N (w,64α, K 1, ) j α2 j × Nj dds w(K + sK )s=0 , 16α, K 1,j + M j Y Nj (w,64α, K 1,j )2 j Y + ε . M + const const 2 α 1− N (w,63α, K ) 1,j
j
α2
By (I1) and Corollary A.5.ii of [FKTo1], Nj (w,64α, K 1,j ) 1− const α Nj (w,64α, K 1,j )
and
Nj (w,64α, K 1,j )2 1− const α Nj (w,64α, K 1,j )
≤
ej ( K 1,j ) 1− const α ej ( K 1,j )
≤ const ej ( K 1,j )
≤ const ej ( K 1,j )2 ≤ const ej ( K 1,j ).
(IX.3)
Two Dimensional Fermi Liquid. 2: Convergence
79
Therefore, by (IX.2) and (I1), recalling that w vanishes when ψ = 0, Nj (w − 21 φJ Cu J φ, α, K 1,j ) ≤ Nj (w, α, K 1,j ) + const α ej ( K 1,j ) 1 ≤ 64 Nj (w, 64α, K 1,j ) + const α ej ( K 1,j ) ≤ ej ( K 1,j ) (IX.4) (j )
and |w0,2 |1,j ≤ const
λ1−υ lj 0 α8 M j
ej ( K 1,j ) ≤
λ1−υ lj 0 α7 M j
(IX.5)
ej ( K 1,j )
and, by (IX.3) and (I1), d (j ) (K + sK ) − 21 φJ Cu(K+sK ) J φ s=0 , α, K 1,j ds w ej ( K 1,j ) M j K 1,j ≤ Nj dds w(K + sK )s=0 , α, K 1,j + const α const 1 d ≤ 64 Nj ds w(K + sK ) s=0 , 64α, K 1,j + α ej ( K 1,j ) M j K 1,j ≤ M j ej ( K 1,j ) K 1,j (IX.6)
Nj
By (IX.1), w (φ, ψ; K) = w (φ, ψ; K) − w (φ, 0; K) is a j –sectorized representative for W . (O1) now follows from (IX.5), (IX.4) and (IX.6). In preparation for the verification of (O2), set δF1 (K) = w0,4 (K) − w0,4 (K) ∞ (j ) (−1) (12)+1 Ant L w0,4 (K); Cu(K) , Dj (u(K); K) . − 41 =1
By (IX.2),
δF (K) ≤ 1 3, j
λ1−υ 0 l e ( K 1,j ). α9 j j
In particular δF (0) ≤ 1 3, j
λ1−υ 0 l c . α9 j j
(IX.7)
Define δF2 (K) = w0,4 (K) − w0,4 (0) ∞ (j ) (≥j +1) pp δF3 = 41 (−1) (12)+1 Ant Vpp L w0,4 (0); Cu(0) , Cu(0) =1
and
δF
(j +1)
(K) = δF1 (0) + δF2 (K) + δF3 .
(≥j +1)
Observe that Dj (u(0); 0) = Cu(0)
(j ) (≥j +1) L w0,4 (0); Cu(0) , Cu(0)
and, by Lemma VII.5.ii,
=
(j ) (≥j +1) pp Vpp L w0,4 (0); Cu(0) , Cu(0)
(j ) (≥j +1) ph +Vph L w0,4 (0); Cu(0) , Cu(0) .
80
J. Feldman, H. Kn¨orrer, E. Trubowitz
Therefore (j +1)
(K) = w0,4 (0) + δF (K) w0,4 ∞ (j ) (≥j +1) ph 1 +1 +4 (−1) (12) Ant Vph L w0,4 (0); Cu(0) , Cu(0) . (IX.8) =1
We now estimate δF (j +1) (K)3, . By (IX.6), j
δF 2 3,
j
≤ ≤
1
d ds w0,4 (sK
+ s K)
s =0 0 λ1−υ j 0 e ( K 1,j ) M K 1,j . α 4 B2 j
By Proposition VII.6, for ≥ 1, (j ) (≥j +1) pp Vpp L w0,4 (0); Cu(0) , Cu(0)
3,j
ds
3,j
(IX.9)
1/n ≤ const lj 0 cj |w0,4 (0)||+1 3,j .
The hypotheses of this proposition are fulfilled by parts (i) and (ii) of Lemma VIII.7. Observe that, by (I1), 4 lj B 2 1 1 M 2j 1−υ lj | w0,4 (K)||3,j ≤ ej ( K 1,j ) lj ej ( K 1,j )(64α) M j λ0
so that |w0,4 (K)||3,j ≤
λ1−υ 0 e ( K 1,j ). (64α)4 B2 j
In particular, |w0,4 (0)||3,j ≤
λ1−υ 0 c . (64α)4 B2 j
Therefore, by Corollary A.5.ii of [FKTo1], ∞ 1/n δF const lj 0 cj |w0,4 (0)||+1 3 3, ≤ 3,j j
≤
=1 ∞ =1
1/n0
const lj
≤ const ≤ const
λ1−υ 2 0 α4
cj
λ1−υ 0
1/n0
lj
α4
c3j
1/n λ2−2υ 0 l 0 cj . α8 j
cj
+1 1
1 − const
λ1−υ 1/n 0 l 0 c2j α4 j
(IX.10)
Hence, by (IX.7), (IX.9) and (IX.10), (j +1) 1/n0 λ1−υ l const λ1−υ 0 δF (K)3, ≤ α0 4 αj5 + B12 M j K 1,j + l ej ( K 1,j ) 4 j α j 1−υ l1/n0 λ j +1 ≤ α0 4 + B12 M j K 1,j ej ( K 1,j ). (IX.11) α4 To verify (O2), set F (i) = F (i) for all 0 ≤ i ≤ j − 1, F (j ) = δF (j ) (0). By (I2), w0,4 (0) =
j i=2
(i) F j + 18 Ant Vph L(j ) (p, F )
j
.
Two Dimensional Fermi Liquid. 2: Convergence
81
Therefore, by (IX.8) and the Definition VII.7 of iterated particle–hole ladders w0,4 (K) =
j i=2
+ 41 =
j
F j + δF (i)
∞ =1
(j +1)
(K) + 18 Ant Vph L(j ) (p, F )
j
(j ) (≥j +1) ph (−1) (12)+1 Ant Vph L w0,4 (0); Cu(0) , Cu(0)
F j + δF (i)
(j +1)
(K) + 18 Ant Vph L(j +1) (p, F ) .
i=2
The estimate on δF (j +1) (K)3, required for (O2) was verified in (IX.11). By (I2), j
(j ) F = δF (j ) (0)3, ≤ 3, j
j
1/n λ1−υ 0 l 0 cj α7 j
λ1−υ 1/n and F (i)3, = F (i)3, ≤ α0 7 li 0 ci for 2 ≤ i ≤ j − 1. i i This leaves only the verification of (O3). By (IX.1), G (φ; K) − 21 φJ C (≤j ) J φ = G(φ; K) − 21 φJ C (<j ) J φ (j ) + w (φ, 0; K) − 21 φJ Cu(K) J φ (j ) − 21 φJ C (j ) − φCu(K) J φ. Now
u(k;K) ˇ
(j )
C (j ) (k) − Cu(K) (k) = −ν (j ) (k) [ık
ˇ 0 −e(k)][ık0 −e(k)−u(k;K)]
.
By Lemma VIII.7.i, (≥j +2) u(k; ˇ ˇ K) + K(k)ν (k) ≤ λ1−υ |ık0 − e(k)|. 0 On the support of ν (j ) , ν (≥j +2) (k) vanishes and u(k; ˇ K) ≤ λ1−υ |ık0 − e(k)| so that 0 (j ) C (k) − C (j ) (k) ≤ u(K)
2λ1−υ (j ) 0 |ık0 −e(k)| ν (k).
√
On the support of ν (j ) , |ık0 − e(k)| ≤ M2M j , so that (j ) 2λ1−υ 1−υ (j ) d3k 0 C − C (j ) ≤ ν (k) ≤ const λ0 u(K) ∞ (2π)3 |ık0 −e(k)| ≤
const
√ M 1−υ λ Mj 0
and (j ) N φJ (C (j ) − Cu(K) )J φ =
1 λ1−υ 0
(j ) J C − C (j ) J ≤ u(K) ∞ √ M Mj
≤
const
≤
1/2 √ 4 lj M j
= const
√ M
√1
M j (3+ℵ)/4 4
|ıx−y|≤
1 λ1−υ 0
lj M j
√
2M Mj
dx dy
1 |ıx−y|
(j ) C − C (j ) u(K) ∞
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J. Feldman, H. Kn¨orrer, E. Trubowitz
if j ≥ 1 and M is big enough. Therefore, by (IX.4) and Remark VI.8.ii, N G (φ; K) − 21 φJ C (j ) J φ − G(φ; K) (j ) (j ) ≤ N w (φ, 0; K) − 21 φJ Cu(K) J φ + 21 N φJ (C (j ) − Cu(K) )J φ (j ) 1/2 ≤ Nj w (K) − 21 φJ Cu(K) J φ , α , K 1,j 0 + √ 4 j 2 ≤ √ 4
lj M
lj M
(IX.12)
. j
Consequently, by (I3), N G (φ; K) − 21 φJ C (≤j ) J φ ≤ N G(φ; K) − 21 φJ C (<j ) J φ + N G (φ; K) − 21 φJ C (j ) J φ − G(φ; K) ≤4
j −1
1 √ 4
i=2
li M i
2 + √ 4
lj M j
.
By (IX.1), (IX.6) and Remark VI.8.ii, (j ) N dds G (φ; K+sK ) − G(φ; K+sK ) − 21 φJ Cu(K+sK ) J φ s=0 (j ) = N dds w (φ, 0; K+sK ) − 21 φJ Cu(K+sK ) J φ s=0 1 d ) − 1 φJ C (j ) ≤ √ N ( K+sK J φ , α , K w j 1, j 4 ds 2 u(K+sK ) s=0 lj M j 0 j 1 ≤ √ M e ( K 1,j ) K 1,j j 4 j t=0 lj M
≤
1 4
M j K 1,j .
(IX.13)
It follows directly from this bound and (I3) that N dds G(K + sK ) − G2 (K + sK ) s=0 . ≤ M j K 1,j . To prove the last inequality of (O3), observe that, by parts (i) and (ii) of Lemma VIII.7 and Lemma XII.12 of [FKTo3], )| d u(k;K+sK d (j ) s=0 (j ) ds ˇ (k) ds Cu(K+sK ) s=0 = [ık 2 ν −e(k)−u(k;K)] ˇ 0
≤
const
K 1,j
|ık0 −e(k)|2
Hence d (j ) C ds u(K+sK ) s=0 ∞ ≤ ≤
const
K 1,j
const
K 1,j
d 3k
1 √1 M Mj
= const K 1,j ln ≤ 41 M j K 1,j Combining (I3), (IX.37) and (IX.14),
ν (j ) (k).
ν (j ) (k) |ık0 −e(k)|2 √ ≤r≤ 2M
√ 2M
1 Mj
dr
1 r
(IX.14)
Two Dimensional Fermi Liquid. 2: Convergence
83
d G (K + sK ) ds 2 s=0 ∞ ≤ dds G2 (K + sK )s=0 ∞ d ) − G(φ; K+sK ) − 1 φJ C (j ) G +λ1−υ N (φ; K+sK J φ 0 ds 2 u(K+sK ) s=0 (j ) + 1 d C 2
ds
u(K+sK ) s=0 ∞
≤ M j K 1,j .
Remark IX.6. Let (W, G, u, p) ∈ Din and (W , G , u, p) = j (W, G, u, p). (j )
(i) The data F (2) , · · · , F (j −1) of (O2) coincides with the data F (2) , · · · , F (j −1) of (I2). 2 (ii) By (IX.12), N G (φ; K) − 21 φJ C (j ) J φ − G(φ; K) ≤ √ . 4 j lj M
(iii) By Remark XV.11 of [FKTo3], the sectorized representative w of W and the function G (φ) may be obtained from the sectorized representative w of W and the function G(φ) by (j ) . w (φ, ψ; K) . 1 . . ψ,Dj (u;K) = 2 φJ Cu J φ + C (j ) (:w:ψ,Cj (u;K)j ) j
u,j
(j )
× (φ, ψ + Cu J φ), w (φ, ψ; K) = w (φ, ψ; K) − w (φ, 0; K), G (φ) = G(φ) + w (φ, 0; K). (j )
The covariances Cu,j , Cj (u; K)j and Dj (u; K)j are defined as follows. Let (j )
Dj (u; K)(k), as specified just before DefiC(k) be one of Cu (k), Cj (u; K)(k), nition IX.4 and let c (·, s), (·, s ) be the Fourier transform of χs (k) C(k) χs (k) as in (III.1) and (III.2). Then C ((ξ,s),(ξ ,s )) = c((ξ,t),(ξ ,t )). t∩s=∅ t ∩s =∅
(iv) If the input data are analytic functions of K, then, by Remark XV.11 of [FKTo3], the output data are analytic functions of the input data and K. (v) If the input data is real, then, by Remark B.5 of [FKTo2], the output data is real.
3. Sector Refinement, ReWick Ordering and Renormalization. We now implement an analog with estimates of the formal power series map (j )
(j +1)
Oj : Dout → Din (j )
that reWick orders. For each (W, G, u, p) ∈ Dout we choose a sectorized representative w(φ, ψ; K) = dη1 ···dηm dξ1 ···dξn wm,n (η1 ,··· ,ηm (ξ1 ,s1 ),··· ,(ξn ,sn ); K) m,n s1 ,··· ,sn ∈j
× φ(η1 ) · · · φ(ηm ) ψ((ξ1 ,s1 )) · · · ψ((ξn ,sn ) )
for W satisfying (O1) and (O2).
84
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We first solve the re-Wick ordering equation, that – in the context of formal power series – was discussed in Lemma III.11. As in (III.11), we set, for any function q : Kj +1 → F0 (2, j ), δK (x, s), (x , s ); K ; q = dx0 q(K ) ∗ νˆ (≥j +1) (x0 , x, s), (0, x , s ) K(K ; q) = K j + δK(K ; q)
u (K ; q) = u(K(K ; q))j +1 + q(K )j +1 ∗ νˆ (≥j +1) E(K ; q) = Cj +1 u ( · ; q); K − Dj (u; K(K ; q)).
(IX.15)
ˇ K ; q) χs (k) in the sense of Let e (·, s), (·, s ) be the Fourier transform of χs (k) E(k; Definition IX.3 of [FKTo2]. As in Proposition XII.8 of [FKTo3], e defines a covariance on the vector space Vj , generated by the fields ψ(ξ, s), by e (ξ, t), (ξ , t ) . Ej ψ(ξ,s ), ψ(ξ ,s ); K ; q = t∩s=∅ t ∩s =∅
Set
w(φ, ˜ ψ; K ; q) =
w(φ, ψ + ψ ; K(K ; q)) dµEj (K ;q) (ψ )
and expand w(0, ˜ ψ; K ; q) =
n≥0 s1 ,··· ,sn ∈j
dξ1 ···dξn w ˜ 0,n ((ξ1 ,s1 ),··· ,(ξn ,sn ); K
(IX.16)
; q)
× ψ((ξ1 ,s1 )) · · · ψ((ξn ,sn ) ). Lemma IX.7. For each K ∈ Kj +1 there exists an antisymmetric, spin independent, particle number conserving function q0 (K ) ∈ F0 (2, j ) that solves the equation 1 2 q(K )
= w˜ 0,2 (K ; q(K ))
and fulfills |q0 (K )||1,j ≤ d q0 (K + sK ) ≤ ds s=0 1, j
λ1−υ lj 0 e ( K 1,j +1 ), α6 M j j λ1−υ 0 α ej ( K 1,j +1 ) K 1,j +1 .
If w and u are analytic in K, then q0 is jointly analytic in K , w and u. Proof. The proof is an application the implicit function theorem and is given following Lemma B.3 in Appendix B. Define, for K ∈ Kj +1 ,
δK(K ) = δK K ; q0 (K ) , renj,j +1 (K , W, u) = K(K ) = K(K ; q0 ),
(IX.17)
If w and u are analytic in K, then δK and renj,j +1 are analytic in K , w and u. If the output data is real and K is real, then renj,j +1 (K , W, u) is real.
Two Dimensional Fermi Liquid. 2: Convergence
85
Lemma IX.8. There is a constant const , independent of M and j , such that if K ∈ Kj +1 , then (i) δK(K ) 1,j ≤ d δK(K + sK ) ≤ ds s=0 1, j
λ1−υ lj 0 e ( K 1,j +1 ), α6 M j j 1−υ λ0 α ej ( K 1,j +1 ) K 1,j +1 ,
(ii) K(K ) 1,j ≤ d K(K + sK ) ≤ ds s=0 1, j
const l const
cj −1 K 1,j +1 + j +1 lj
λ1−υ lj 0 e ( K 1,j +1 , ) α6 M j j
M ℵ ej ( K 1,j +1 ) K 1,j +1 ,
(iii) ej ( K(K ) 1,j ) ≤ const ej +1 ( K 1,j +1 , ) Proof. The proof is given following Lemma B.3 in Appendix B.
Proposition IX.9. Let (W, G, u, p) ∈ Dout and K ∈ Kj +1 . Then (j )
renj,j +1 (K , W, u) ∈ Kj Proof. As renj,j +1 (K , W, u) = K(K ; q0 ), the proposition follows from Lemma B.1.i and Lemma IX.7. We define, for each K ∈ Kj +1 , ˜ ψ; K ; q0 (K )) w(φ, ˜ ψ; K ) = w(φ, w (φ, ψ; K ) = w˜ φ, ψ; K − w˜ φ, 0; K −
1 2
dξ1 dξ2
s1 ,s1 ∈j
× q0 ((ξ1 ,s1 ),(ξ2 ,s2 ); K )ψ((ξ1 ,s1 ))ψ((ξ2 ,s2 )),
(IX.18)
and let W be the Grassmann function having sectorized representative w . Also set ˜ 0; K ) − w(0, ˜ 0; K ), G (φ; K ) = G(φ; K(K )) + w(φ, u (K ) = u (K ; q0 ), (i) p = p (i) for all 2 ≤ i ≤ j − 1, (j ) p = δu(δK(0))j ∗ νˆ (≥j ) + q0 (0) ∗ νˆ (≥j +1) − δK(0)ext ∗ νˆ (≥j ) .
(IX.19)
ˇ K(K )) ν˜ (≥j +1) (k) + qˇ0 (k; K ) ν (≥j +1) (k), uˇ (k; K ) = u(k; (j ) pˇ (k) = δ u(k; ˇ δK(0)) ν (≥j ) (k) ˇ +qˇ0 (k; 0) ν (≥j +1) (k) − δ K(k; 0) ν (≥j ) (k).
(IX.20)
j
Observe that
We now define
= (W , G , u , p ). Oj (W, G, u, p)
86
J. Feldman, H. Kn¨orrer, E. Trubowitz (j +1)
(j )
Theorem IX.10. Let (W, G, u, p) ∈ Dout . Then Oj (W, G, u, p) ∈ Din
.
Let w be the sectorized representative Proof. Set (W , G , u , p ) = Oj (W, G, u, p). of W and j −1 (i) u(K) = pj + δu(K) − Kext j
i=2
be the decomposition of u specified in (VIII.2). Let w, ˜ w be as in (IX.18), q0 be the function of Lemma IX.7 and let δK(K ), K(K ) be as in (IX.17). Verification that (W , G , u , p ) is an interaction quadruple of scale j + 1 We first check the properties of p . As p (i) = p (i) , for all 2 ≤ i ≤ j − 1, we need only ˇ (j ) (k) is supported on the j th neighbourhood and, since discuss p (j ) . By (IX.20), p(≥j +1) ˇ δ K(k; 0) = qˇ0 (0, k); 0 ν ((0, k)), vanishes at k0 = 0. By Lemma XIII.7 of [FKTo3], Remark XIX.5 of [FKTo4], Lemma E.5 of [FKTo4], (VIII.3), Lemma IX.7, Lemma IX.8.i and Corollary A.5.i of [FKTo1], (j ) p ≤ δu(δK(0))j ∗ νˆ (≥j ) (k)1, + q0 (0) ∗ νˆ (≥j +1)1, 1,j j j + δK(0)ext ∗ νˆ (≥j )1, j j ≤ const cj δu(δK(0)) 1, + q0 (0)1, + δK(0) 1,j j −1 j λ1−υ l 1−υ ≤ const λ0 ej ( δK(0) 1,j ) δK(0) 1,j + α0 6 Mjj cj 1−υ λ1−υ l λ0 lj ≤ const α0 6 Mjj cj λ1−υ e c + 1 j j 6 j 0 α M ≤
λ1−υ lj 0 c . α5 M j j
(IX.21)
Thus, (VIII.1) holds for p . Next, we construct the decomposition (VIII.2) for u . Set L=K δu (K ) = δu(K(L))j + q0 (L) ∗ νˆ (≥j +1) − δK(L)ext ∗ νˆ (≥j ) . L=0
By construction,
δu
∈ F0 (2, j ) and
δu (0)
= 0. Since δ uˇ (0, k); K vanishes,
L=K ˇ δ uˇ (0, k); K = qˇ0 (0, k); L ν (≥j +1) (0, k) − δ K(k; =0 L) ν (≥j ) (0, k) L=0
by (IX.15) and (IX.17). Observe that, by Remark XIX.3.iii of [FKTo4], L=K δu (K )j +1 = δu(K(L))j +1 + q0 (L)j +1 ∗ νˆ (≥j +1) − δK(L)ext , j +1 L=0 (j ) (≥j +1) pj +1 = δu(K(L))j +1 + q0 (L)j +1 ∗ νˆ − δK(L)ext . j +1
Since
K(K )
=
K j
u (K ) = u(K(K ))j +1 + q0 (K )j +1 ∗ νˆ (≥j +1) j −1 (i) = pj +1 + δu(K(K )) − K(K )ext i=2
L=0
+ δK(K ),
j +1
+ q0 (K )j +1 ∗ νˆ (≥j +1)
Two Dimensional Fermi Liquid. 2: Convergence
=
j −1 i=2
=
87
(i) pj +1 + δu(K(L))j +1 + q0 (L)j +1 ∗ νˆ (≥j +1) − [δK(L)ext ]j +1
−[Kext ]j +1 j −1 (i) (j ) p j +1 + p j +1 i=2
L=K
+ δu (K ) − Kext
j +1
and u (K ) has the desired form. (1,0,0) d δu (K + sK )s=0 1, , for K , K ∈ Kj +1 . By definition We next bound D1,2 ds j
(1,0,0) d D 1,2 ds δu (K + sK ) s=0 1,j (1,0,0) d ≤ D1,2 ds δu(K j + sK j + δK(K + sK ))j s=0 1, j (1,0,0) d +D1,2 ds q0 (K + sK )s=0 ∗ νˆ (≥j +1)1, (1,0,0) d j +D1,2 ds δK(K + sK )ext ∗ νˆ (≥j ) s=0 1, . j
(IX.22)
By Remark XII.11 of [FKTo3] and Lemma E.5 of [FKTo4], Lemma IX.8.i and Remark XV.2.iv of [FKTo3], the third term is bounded by (1,0,0) d (≥j ) D 1,2 ds δK(K + sK )ext ∗ νˆ s=0 1,j ≤ ∂∂t0 dds δK(K + sK )ext ∗ νˆ (≥j ) s=0 1, j ∂ d ≤ const ∂t0 cj ds δK(K + sK ) s=0 1, j 1−υ λ0 ∂ ≤ const ∂t0 cj α ej ( K 1,j +1 ) K 1,j +1 λ1−υ ≤ const 0α K 1,j +1 ∂∂t0 ej ( K 1,j +1 ) λ1−υ ≤ const 0α M j ej ( K 1,j +1 ) K 1,j +1 + ∞ tδ ≤
M2 const α
+
δ0 =r0 1−υ (j +1)−3 ej +1 ( K 1,j +1 ) K 1,j +1 λ0 M δ
∞t .
(IX.23)
δ0 =r0
Similarly, by Remark XII.11 and Lemma XIII.7 of [FKTo3], and Lemma IX.7, the second term is bounded by (1,0,0) d (≥j +1) D 1,2 ds q0 (K + sK ) s=0 ∗ νˆ 1,j ≤ ∂∂t0 dds q0 (K + sK )s=0 ∗ νˆ (≥j +1) 1, j ∂ d ≤ const ∂t0 cj ds q0 (K + sK ) s=0 1, j 1−υ λ0 ∂ ≤ const ∂t0 cj α ej ( K 1,j +1 ) K 1,j +1 2 (j +1)−3 ≤ const Mα λ1−υ ej +1 ( K 1,j +1 ) K 1,j +1 0 M + ∞ tδ. δ0 =r0
(IX.24)
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J. Feldman, H. Kn¨orrer, E. Trubowitz
By the chain rule,
d ds δu(Kj + sKj + δK(K + sK )) s=0 d = ds δu(Kj + δK(K ) + sKj ) s=0 + dds δu(K j + δK(K ) + s dδK(Kdt+xK ) x=0 )s=0 = dds δu(K(K ) + sK j )s=0 + dds δu(K(K ) + s ddx δK(K + xK )x=0 )s=0 .
(IX.25)
We bound these two contributions to the first term of the right hand side of (IX.22) separately. By Remark XIX.5 of [FKTo4], (VIII.3), with K replaced by K(K ), Proposition E.10.ii of [FKTo4] and Lemma IX.8.iii, (1,0,0) d D 1,2 ds δu(K(K ) + sKj )j s=0 1,j (1,0,0) d ≤ const cj −1D1,2 ds δu(K(K ) + sK j )s=0 1, j −1 j −3 ≤ const λ1−υ M e ) + ∞ tδ K(K K j 1, 1, j j j 0 ≤ ≤
δ0 =r0 lj 1−υ j −3 const λ0 M ej +1 K 1,j +1 lj +1 cj −1 K 1,j +1
1 lj const M l
+
δ0 =r0
j +1
+
δ0 =r0
∞ tδ
(j +1)−3 λ1−υ ej +1 K 1,j +1 K 1,j +1 0 M
∞ tδ.
(IX.26)
Similarly, by Remark XIX.5 of [FKTo4], (VIII.3), parts (i) and (iii) of Lemma IX.8 and Corollary A.5.ii of [FKTo1], (1,0,0) d d D 1,2 ds δu(K(K ) + s dx δK(K + xK ) x=0 )j s=0 1,j (1,0,0) d ≤ const cj −1D1,2 ds δu(K(K ) + s ddx δK(K + xK )x=0 )s=0 1, j −1 d j −3 ≤ const λ1−υ M e ) δK(K + xK ) + ∞ tδ K(K j 1,j 0 dx x=0 1, j
≤ ≤
1−υ j −3 const λ0 M ej +1
λ1−υ const 0α
K 1,j +1
λ1−υ 0 α
ej (
K 1,j +1
δ0 =r0
) K 1,j +1 +
δ0 =r0
∞ tδ
(j +1)−3 λ1−υ ej +1 K 1,j +1 K 1,j +1 + ∞ t δ . (IX.27) 0 M δ0 =r0
Substituting (IX.25) into (IX.22) and applying the bounds (IX.23), (IX.24), (IX.26) and (IX.27), we have (1,0,0) d D 1,2 ds δu (K + sK ) s=0 1,j 2 λ1−υ (j +1)−3 1 lj 0 ≤ const 0 Mα + M λ1−υ + ej +1 K 1,j +1 K 1,j +1 0 M α lj +1 + ∞ tδ δ0 =r0
with an M–independent constant const 0 . If M is sufficiently large 1 lj const 0 M l
j +1
1 = const 0 M 1−ℵ ≤ 21 .
Two Dimensional Fermi Liquid. 2: Convergence
89
If α is sufficiently large and λ0 is sufficiently small, depending on M, 2 λ1−υ M const 0 α + 0α ≤ 21 and (1,0,0) d 1−υ (j +1)−3 D ej +1 ( K 1,j +1 ) K 1,j +1 1,2 ds δu (K + sK ) s=0 1,j ≤ λ0 M + ∞ tδ. δ0 =r0
The remaining bound required in (VIII.3), namely, d δu (K + sK ) ≤ λ1−υ ej +1 K 1,j +1 K 1,j +1 0 ds s=0 1, j
is now a consequence of Corollary XIX.12 of [FKTo4]. The remaining requirements of Definition VIII.1 are easily verified. Preparation for the verification of (I1), (I2) and (I3) Clearly w (K ) = w (K ) j +1
(IX.28)
is a j +1 –sectorized representative of W . Write dη1 ···dηm dξ1 ···dξn w ˜ m,n (η1 ,··· ,ηm (ξ1 ,s1 ),··· ,(ξn ,sn ); K ) w(φ, ˜ ψ; K ) = m,n s1 ,··· ,sn ∈j
× φ(η1 ) · · · φ(ηm) ψ((ξ1 ,s1 )) · · · ψ((ξn ,sn ) ), w (φ, ψ; K ) = dη1 ···dηm dξ1 ···dξn wm,n (η1 ,··· ,ηm (ξ1 ,s1 ),··· ,(ξn ,sn ); K )
m,n s1 ,··· ,sn ∈j
× φ(η1 ) · · · φ(ηm) ψ((ξ1 ,s1 )) · · · ψ((ξn ,sn ) ), w (φ, ψ; K ) = dη1 ···dηm dξ1 ···dξn wm,n (η1 ,··· ,ηm (ξ1 ,s1 ),··· ,(ξn ,sn ); K )
m,n s1 ,··· ,sn ∈j
× φ(η1 ) · · · φ(ηm ) ψ((ξ1 ,s1 )) · · · ψ((ξn ,sn ) ).
= 0. Consequently, By Lemma IX.7, w˜ 0,2 (K ) = 21 q0 (K ) and hence, by (IX.18), w0,2 by (IX.28), w0,2 = 0. By (IX.16) and Proposition A.2 of [FKTr1],
w(φ, ˜ ψ; K ) = :w(φ, ψ; K(K )):ψ,−Ej (K ;q0 ) .
(IX.29)
lj B By Lemma B.1.iii, λ1−υ l j M j is an integral bound for Ej . Hence by Corollary 0 II.32.ii of [FKTr1], with f (ψ) = w(φ, ψ; K(K )) and f (ψ) = :w(φ, ψ; K(K )):ψ,−Ej , Nj w(K ˜ ) − w K(K ) , α2 , X ≤ for all X ∈ Nd+1 . In particular ˜ ), α2 , X ≤ Nj w(K
8λ1−υ 0 l α2 j
3 2
Nj (w(K(K )), α, X)
Nj w(K(K )), α, X .
(IX.30)
(IX.31)
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To get a similar bound on d ˜ ds w(K
+ sK )s=0 , observe that, by (IX.29) =: dds w(K(K +sK )) .. ψ,−E (K ;q ) s=0 j 0 + dds :w(K(K )):ψ,−Ej (K +sK ;q0 ) s=0 .
d ˜ ds w(K
+ sK )s=0
(IX.32)
By Corollary II.32 of [FKTr1], (O1), Lemma IX.8, parts (ii) and (iii), and Corollary A.5.ii of [FKTo1], the first term is bounded by Nj .. dds w(K(K +sK )) .. ψ,−E (K ;q ) s=0 , α2 , K 1,j +1 j 0 d ≤ Nj ds w(K(K +sK )) s=0 , α , K 1,j +1 ≤ ej ( K 1,j +1 )Nj dds w(K(K +sK ))s=0 , α , K(K ) 1,j ≤ M j ej ( K 1, )ej ( K(K ) 1, ) d K(K + xK ) j +1
j +ℵ
dx
j
1,j x=0
≤ const M ej +1 ( ) K 1,j +1 ≤ const M j +ℵ ej +1 ( K 1,j +1 ) K 1,j +1 . K 1,j +1
3
(IX.33)
In preparation for bounding the second term of (IX.32), observe that, by Remark II.30 of [FKTr1] and Lemma B.1, d d ds Ej (K +sK ; q0 ) s=0 = ds Ej (K +sK ;q0 (K )) s=0 d + dds Ej (K ;q0 (K )+s dx q0 (K +xK )|x=0 ) s=0 has integral bound
lj K 1,j +1 t=0 + const ≤ const lj K 1,j +1 t=0
const
since, by Lemma IX.7, d q0 (K + xK ) ≤ dx x=0 1, j
λ1−υ 0 α
lj
λ1−υ 0 α K 1,j +1 t=0
ej ( K 1,j +1 ) K 1,j +1 .
Consequently, by Corollary II.32.iii of [FKTr1], (O1), Lemma IX.8, parts (ii) and (iii), and Corollary A.5.ii of [FKTo1], Nj dds .. w(K(K )) .. ψ,−E (K +sK ;q ) s=0 , α2 , K 1,j +1 j 0 const ≤ (α−1)2 Nj w(K(K )) , α , K 1,j +1 M j K 1,j +1 j const ≤ (α−1) 2 ej ( K 1,j +1 ) Nj w(K(K )) , α , K(K ) 1,j M K 1,j +1 ≤ ≤
const e ( K 1,j +1 )ej ( K(K ) 1,j ) M j K 1,j +1 (α−1)2 j const e ( K 1,j +1 ) M j K 1,j +1 . (α−1)2 j +1
Combining (IX.32), (IX.33) and (IX.34), Nj dds w(K ˜ + sK )s=0 ,
α 2
,
K 1,j +1
(IX.34)
≤ const M j +ℵ ej +1 ( K 1,j +1 ) K 1,j +1
(IX.35)
Two Dimensional Fermi Liquid. 2: Convergence
91
Verification of (I3) By the definition, (IX.19), of G N G (φ; K ) − G(φ; K(K )) = N w(φ, ˜ 0; K ) − w(0, ˜ 0; K ) 1 ≤ √ ˜ 0; K ) , α2 , 0 0 Nj w(φ, 4 j lj M 3/2 ≤ √ Nj w(K(K )) , α , 0 0 4 j lj M 3/2 ≤ √ Nj w(K(K )) , α , K(K ) 1,j 0 4 j lj M
2 ≤ √ 4
(IX.36)
lj M j
by Remark VI.8.ii, (IX.31) and (O1). Therefore, by (O3), N G (φ; K ) − 21 φJ C (<j +1) J φ ≤ N G(φ; K(K )) − 21 φJ C (≤j ) J φ + N G (φ; K ) − G(φ; K(K )) j
≤4
1 √ 4
li M i
i=0
.
From (IX.19), N
G (φ; K +sK ) − G2 (φ; K +sK ) s=0 ≤ N dds G(φ; K(K +sK )) − G2 (φ; K(K +sK )) s=0 + N dds w(K ˜ + sK ) s=0 ψ=0 ≤ N dds G φ; K(K )+sL − G2 φ; K(K )+sL s=0 1 d α +√ N w(K ˜ + sK ) , , K , 1, j 4 j +1 ds j s=0 2 d ds
lj M
0
where L = ddx K(K +xK )|x=0 obeys L 1,j ≤ const M ℵ ej ( K 1,j +1 ) K 1,j +1 , by Lemma IX.8.ii. Hence, by (O3) and (IX.35) N
d ds
G (φ; K +sK ) − G2 (φ; K +sK ) s=0
≤ M j L 1,j + const M j +ℵ ej +1 ( K 1,j +1 ) K 1,j +1 ≤ const M j +ℵ ej ( K 1,j +1 ) K 1,j +1 ≤ 21 M j +1 K 1,j +1 + ∞ tδ.
(IX.37)
δ=0
Similarly, d G (φ; K + sK ) ds 2 s=0 ∞ ≤ d G2 (φ; K(K +sK )) ds j
≤M ≤
1 2
+ λ1−υ 0 N ∞
s=0 1−υ √ 1 L 1,j + const λ0 4 l Mj
M
j +1
K 1,j +1 +
δ=0
j
∞ tδ.
d ˜ ds w(K
+ sK ) s=0 ψ=0
M j +ℵ ej +1 ( K 1,j +1 ) K 1,j +1
92
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Verification of (I2) Observe that by (IX.30), (O1) and Lemma IX.8.iii 2 M 2j α 4 lj B 1 w˜ 0,4 (K ) − w0,4 K(K ) 1−υ j lj
16
≤ ≤
M
8λ1−υ 0 α2 8λ1−υ 0 α2
λ0
lj
lj Nj w(K(K )), α, K(K ) 1,j
3,j
lj ej ( K(K ) 1,j )
λ1−υ 0 l e ( K 1,j +1 ) α 2 j j +1
≤ const
so that, by Remark XIX.5 of [FKTo4] and Corollary A.5.ii of [FKTo1] λ2−2υ ≤ const 0α 6 lj ej +1 ( K 1,j +1 ). w˜ 0,4 (K ) − w0,4 K(K )
(IX.38)
j +1 3,j +1
Set F (i) = F (i) for all 2 ≤ i ≤ j and δF
(j +1)
(K ) = δF (j +1) (K(K ))j +1 + w˜ 0,4 (K ) − w0,4 (K(K ))
j +1
.
F ) depends only on p (2) , · · · , p (j −1) and F (2) , · · · , F (j ) . By DefinitionVII.7, L(j +1) (p, (j +1) In particular L (p , F ) = L(j +1) (p, F ). Therefore w0,4 (K ) = w˜ 0,4 (K )j +1
= δF (j +1) (K ) +
j i=2
and
(i) F ≤ 3, i
(i) Fj +1 + 18 Ant Vph L(j +1) (p , F )
1/n λ1−υ 0 l 0 ci α7 i
j +1
for all 2 ≤ i ≤ j.
Furthermore, by Remark XIX.5 of [FKTo4], (IX.38), (O2), Lemma IX.8 and Corollary A.5.ii of [FKTo1] (j +1) δF (K )3, j +1 (j +1) λ2−2υ ≤ const cj δF (K(K ))3, + const 0α 6 lj ej +1 ( K 1,j +1 ) j 1/n0 λ1−υ lj +1 j 1 ej ( K(K ) 1,j ) + M K(K ) ≤ const α0 4 1, 4 2 j α B
≤ ≤
+const 1−υ λ0 α4
λ1−υ 0 α4
λ2−2υ 0 l e ( K 1,j +1 ) α 6 j j +1
λ1−υ lj + const B12 lj +1 M j K 1,j +1 + const α0 2 lj ej +1 ( K 1,j +1 ) j +1 1 1 ej +1 ( K 1,j +1 ), + const M 1−ℵ M K (IX.39) 1, 2 j +1 B 1/n
const
l1/n0 j +1 α3
lj +10 α4
1/n
lj +10 2α ,
since const λ1−υ 0 lj ≤
by the hypothesis α <
requirement of Definition V.6 that 0 < υ < (j +1) δF (K )3,
j +1
≤
λ1−υ 0 α4
l1/n0 j +1 α3
+
1 4.
1 υ/10 λ0
of Theorem VIII.5 and the
In particular
1 M j +1 K 1,j +1 B2
ej +1 ( K 1,j +1 ).
Two Dimensional Fermi Liquid. 2: Convergence
93
Verification of (I1) Set ω˜ m,n =
dη1 ···dηm dξ1 ···dξn
s1 ,··· ,sn ∈j
w˜ m,n (η1 ,··· ,ηm (ξ1 ,s1 ),··· ,(ξn ,sn ))
× φ(η1 ) · · · φ( ηm ) ψ((ξ1 ,s1 )) · · · ψ((ξn ,sn ) ), = dη1 ···dηm dξ1 ···dξn wm,n (η1 ,··· ,ηm (ξ1 ,s1 ),··· ,(ξn ,sn ))
ωm,n
s1 ,··· ,sn ∈j +1
×φ(η1 ) · · · φ(ηm ) ψ((ξ1 ,s1 )) · · · ψ((ξn ,sn ) ).
Then w = w =
m,n≥1 m,n≥1
ω˜ m,n + ωm,n +
n≥4 n≥4
ω˜ 0,n , ω0,n .
By (O2), (IX.39), Remark XIX.5 of [FKTo4] and Theorem VII.8 (with ρ = λ1−υ 0 , ℵ ε=n ) 0
w (K ) 0,4 3,
j +1
j F (i) ≤ δF (j +1) (K )3, + j +1 3,j +1 j +1 i=2 (j +1) 1 + 8 Ant Vph L (p , F ) 3,
≤
≤
λ1−υ 0 α4
j +1
j +1
1/n li 0 α3
+
1
j +1
1 M j +1 K 1,j +1 B2
ej +1 ( K 1,j +1 ) i=2 +const λ2−2υ cj +1 0 λ1−υ j +1 1 1 1 0 const 4 + M K 1, 3 2 1−ℵ j +1 ej +1 ( K 1,j +1 ). α α M B const
M 1−ℵ
Consequently, by Proposition XIX.1 of [FKTo4], w (K ) 0,4 1,
j +1
λ1−υ
≤ const α 4 0l
j +1
1 α3
+
1 M 1−ℵ
1 M j +1 K 1,j +1 B2
ej +1 ( K 1,j +1 ).
Therefore, by Corollary A.5 of [FKTo1] , (K ), 64α, K 1,j +1 ) Nj +1 (ω0,4
l +1 = 224 α 4 B2 j1−υ ej +1 ( K 1,j +1 ) w0,4 (K )1, + lj1+1 w0,4 (K )3, j +1 j +1 2 λ0 j +1 B 1 ≤ const α 3 + M 1−ℵ M K 1,j +1 ej +1 ( K 1,j +1 ) ≤ 21 ej +1 ( K 1,j +1 ).
By (IX.31), (O1) and Lemma IX.8.iii Nj (w (K ), α2 , 0) ≤ 23 Nj (w(K(K )), α, 0) ≤ 23 Nj (w(K(K )), α, K(K ) 1,j ) ≤ 23 ej ( K(K ) 1,j ) ≤ const ej +1 ( K 1,j +1 )
(IX.40)
94
J. Feldman, H. Kn¨orrer, E. Trubowitz
so that, by Corollary XIX.8 of [FKTo4] and Corollary A.5 of [FKTo1], Nj +1 (w (K ) − ω0,4 (K ), 64α, K 1,j +1 ) 1 ≤ M (1−ℵ)/8 ej +1 ( K 1,j +1 ) Nj w (K ) − ω˜ 0,4 (K ), α2 , K 1,j +1 1 ej +1 ( K 1,j +1 )2 Nj w (K ), α2 , 0 ≤ M (1−ℵ)/8 1 ej +1 ( K 1,j +1 )3 ≤ const M (1−ℵ)/8 1 ≤ 2 ej +1 ( K 1,j +1 ). (IX.41)
Combining (IX.40) and (IX.41), we get Nj +1 (w (K ), 64α, K 1,j +1 ) ≤ ej +1 ( K 1,j +1 ). By (IX.18) and (IX.35), Nj
d ds w (K
+ sK )s=0 ,
α 2
,
K 1,j +1
≤ const M j +ℵ ej +1 ( K 1,j +1 ) K 1,j +1 .
Therefore by Corollary XIX.8 of [FKTo4] and Corollary A.5 of [FKTo1],
+ sK )s=0 , 64α, K 1,j +1 ≤ const ej +1 ( K 1,j +1 ) Nj dds w (K + sK )s=0 , α2 , K 1,j +1
Nj +1
d ds w (K
≤ const M j +ℵ ej +1 ( K 1,j +1 )2 K 1,j +1 ≤ M j +1 ej +1 ( K 1,j +1 ) K 1,j +1 .
∈ Remark IX.11. Let (W, G, u, p) ∈ Dout and (W , G , u , p ) = Oj (W, G, u, p) (j +1) Din . (j )
(i) The data F (2) , · · · , F (j ) of (I2) coincides with the data F (2) , · · · , F (j ) of (O2). Also, by (IX.19), p (i) = p (i) for all 2 ≤ i ≤ j − 1. (ii) By (IX.36), 2 N G (φ; K ) − G φ; renj,j +1 (K , W, u) ≤ √ 4
lj M j
.
(iii) If the output data are analytic functions of K, then, by Lemma IX.7, the input data are analytic functions of the output data and K. (iv) If the output data is real, then the input data is real.
X. The Recursive Construction of the Green’s Functions In this section, we construct the data of Theorem VIII.5, recursively in j .
Two Dimensional Fermi Liquid. 2: Convergence
95
1. Initialization at j = j0 . We set ˇ for K ∈ Kj0 , ◦ δej0 (K)(k) = K(k) (2) (3) ◦ p = p = · · · = p(j0 −1) = 0, ◦ F (2) = · · · = F (j0 ) = 0 , rg
and define Wj0 , Gj0 and uj0 as follows. uj0 (K) = − Kext , ˜ Wj0 (K) = rg ˜ Gj0 (K) =
j0
(≤j ) Cu 0(K) j0 (≤j0 ) j0 (K)
Cu
˜ (V (ψ))(φ, ψ) −
(≤j0 ) j0 (K)
Cu
(V (ψ))(φ, 0),
(V (ψ))(φ, 0).
rg Clearly, Wj0 , Gj0 , uj0 , (p(2) ,··· ,p(j0 −1) ) is an interaction quadruple at scale j0 . Next, we (j )
˜ ψ; K) be the j0 –sectorized representative for verify that it is in Dout0 . Let w(φ, ˜ (≤j0 ) V (ψ) (φ, ψ)− 1 φJ C (≤j0 ) J φ = Wj0 (K)(φ, ψ)+G rg (K)(φ)− 1 φJ C (≤j0 ) J φ j0 uj (K) uj (K) 2 2 Cu(K)
0
0
chosen in Theorem VI.12 and set w(φ, ψ; K) = w(φ, ˜ ψ; K) − w(φ, ˜ 0; K) dη1 ···dηm dξ1 ···dξn wm,n (η1 ,··· ,ηm (ξ1 ,s1 ),··· ,(ξn ,sn ); K) = m,n s1 ,··· ,sn ∈j0
× φ(η1 ) · · · φ(ηm ) ψ((ξ1 ,s1 )) · · · ψ((ξn ,sn ) ).
By Theorem VI.12, ˜ α, K 1,j0 ≤ const α 4 λυ0 ej0 K 1,j0 Nj0 w(K), α, K 1,j0 ≤ Nj0 w(K), and Nj0
d
ds w(K
+ sK )s=0 , α, K 1,j0 ≤ Nj0 dds w(K ˜ + sK )s=0 , α, K 1,j0 ≤ M j0 ej0 K 1,j0 K 1,j0 .
In particular,
Nj0 w(K), α, K 1,j0 ≤ ej0 K 1,j0 w0,2 (K) ≤ M j0 Bα1 2 ρ Nj0 w(K), α, K 1,j0 1,j0 0;2 ≤ const α 2 λ0 ej0 K 1,j0 λ1−υ l ≤ α0 7 Mjj00 ej0 K 1,j0
so that (O1) is satisfied. Similarly, w0,4 (K) ≤ B2 α 4 ρ1 3, j0
Nj0 w(K), α, K 1,j0 λ1−υ ≤ α0 9 ej0 K 1,j0 . 0;4 (λ0 )
Setting δF (j0 +1) (K) = w0,4 (K) , (O2) is satisfied. Observe that rg (≤j0 ) Gj0 (K) − 21 φJ C (≤j0 ) J φ = w(φ, ˜ 0; K) − 21 φJ C (≤j0 ) − Cuj (K) J φ. 0
96
J. Feldman, H. Kn¨orrer, E. Trubowitz
Now (≤j0 ) (k) 0 (K)
C (≤j0 ) (k) − Cuj
=
ˇ K(k)[U (k)−ν (>j0 ) (k)] . ˇ [ık0 −e(k)][ık0 −e(k)+K(k)]
j0 +1 ˇ On the support of U (k)−ν (>j0 ) (k), |ık0 −e(k)| ≥ √ 1 j0 and |K(k)| ≤ const λ1−υ 0 M j0 +1 MM so that (≤j ) (≤j0 ) (≤j0 ) 1 N φJ C (≤j0 ) − Cuj (K) J C 0 − Cuj (K) J φ = 1−υ J ∞ λ0 0 0 l d 3 k U (k)−ν (>j0 ) (k) ≤ const Mjj00+1 +1 (2π)3 |ık0 −e(k)|2 lj0 +1 1 ≤ const M j0 +1 |ıx−y|≥ √ 1 dx dy |ıx−y| 2
l
M M j0 |y|≤const
≤ ≤
lj
+1
const j0 +1 M 0 1 4
ln
√ M M j0
lj 0 M j 0
if M is big enough. Therefore, by Remark VI.8.ii and Theorem VI.12, rg 1 Nj0 w(K), ˜ α, K 1,j0 0 N Gj0 (K) − 21 φJ C (≤j0 ) J φ ≤ 4 j lj 0 M
0
(≤j0 ) +N φJ C (≤j0 ) − Cuj (K) Jφ ≤ Similarly, N
0
2 . 4 l M j0 j0
d rg ds Gj0 (K + sK ) − Gj0 ,2 (K + sK ) s=0 = N dds w(φ, ˜ 0; K + sK )s=0 1 d ≤ N w(K ˜ + sK ) , α, K j 1, 0 j ds s=0 0 0 4 j lj 0 M 0 j0
≤ M K 1,j0 (≤j0 ) Kˇ (k)[U (k)−ν (>j0 ) (k)] and, since dds Cuj (K+sK , ) (k) s=0 = − 2 ˇ [ık0 −e(k)+K(k)] 0 d Gj ,2 (K + sK ) 0 ds s=0 ∞ d (≤j0 ) ≤ ds w˜ 2,0 (K + sK ))s=0 ∞ + 21 J dds Cuj (K+sK ) s=0 J ∞ 0 d d 3 k U (k)−ν (>j0 ) (k) ≤ λ1−υ ˜ 0; K + sK )s=0 + const sup Kˇ (k) (2π) 3 0 N ds w(φ, |ık0 −e(k)|2 k j0 1 ˇ ≤ λ1−υ dx dy |ıx−y| 2 0 M K 1,j0 + const sup K (k) |ıx−y|≥ √ 1 k
j0 ≤ λ1−υ 0 M K 1,j0 + const K 1,j0 j0 ≤ M K 1,j0
M M j0
√ |y|≤const ln M M j0
if M is big enough, for all K ∈ Kj0 and all K . This completes the verification that (O3) is satisfied. As pointed out in Remark IX.3, conditions (O1–3) imply conditions (R1–3) of Theorem VIII.5. For j = j0 , (R4) is vacuous. Condition (R5) was verified formally as (III.10). The analyticity and reality conditions of Theorem VIII.5 follow from Theorem VI.12.
Two Dimensional Fermi Liquid. 2: Convergence
97
2. Recursive Step j → j + 1. Fix j ≥ j0 and assume that ◦ maps δej , reni,j , j0 ≤ i ≤ j ≤ j , ◦ p(2) , · · · , p(j −1) , ◦ F (2) , · · · , F (j ) , rg and output data Wj , Gj , uj , (p(2) ,··· ,p(j −1) ) at scale j have been constructed and fulfill the conclusions of Theorem VIII.5. rg Define Wj +1 , Gj +1 , uj +1 and p (j ) by
rg rg Wj +1 , Gj +1 , uj +1 , (p(2) ,··· ,p(j ) ) = j +1 ◦ Oj Wj , Gj , uj , (p(2) ,··· ,p(j −1) ) .
By Theorems IX.10 and IX.5, the left hand side is an output datum of scale j + 1 and, by Remark IX.3, satisfy conditions (R1–3). By Remarks IX.11.i and IX.6.i, the F (j +1) of (O2) may be appended to F (2) , · · · , F (j ) so that (R2) is satisfied. The analyticity and reality conditions of Theorem VIII.5, follow from Remarks IX.6.iv,v and IX.11.iii,iv. Define renj,j +1 to be the map renj,j +1 ( · , Wj , uj ) of (IX.17). By Remarks IX.11.ii and IX.6.ii, (R4) is satisfied. Define, for j0 ≤ i ≤ j , reni,j +1 = reni,j ◦ renj,j +1 and
ˇ δej +1 (K)(k) = renj0 ,j +1 (K) (k).
Then the algebraic conditions of Definition VIII.3 are fulfilled. The analyticity and reality of renj,j +1 was observed following (IX.17). That the estimates are also fulfilled is proven in Lemma X.1.
i) For all K ∈ Kj +1 , li reni,j +1 (K) ≤ λ1−υ + ∞ tδ, 0 1, Mi i
δ=0
|||δ eˆj +1 (K)|||1,∞ ≤
λ1−υ 0 .
ii) There is a universal constant const such that, for all j0 ≤ i ≤ j +1 and K, K ∈ Kj +1 , d i reni,j +1 (K + sK ) ≤ const j +1−i ljl+1 K 1,j +1 + ∞ tδ. ds s=0 1, i
δ=0
iii) For all j0 ≤ j ≤ j + 1, reni,j +1 (0) − reni,j (0) ≤ λ1−υ 0 1, i
|||δej +1 (0) − δej (0)|||1,∞ ≤ λ1−υ 0 Proof.
1 2j
+
δ=0
1 . 2j
i) We prove, by induction on j − i that reni,j +1 (K) =
j +1
P,j +1 (K)i
=i
with
∞ tδ,
P,j +1 (K) ∈ K
Pj +1,j +1 (K) = K
(X.1)
98
J. Feldman, H. Kn¨orrer, E. Trubowitz
and P,j +1 (K) 1, ≤
λ1−υ l 0 e ( K 1,j +1 ) α5 M j
(X.2)
for < j + 1. This will then imply reni,j +1 (K) 1,i
j +1 P,j +1 (K) ≤ i
=i
≤ const
1,i
j +1 li l ci−1 P,j +1 (K) 1, by Proposition E.10.ii of [FKTo4]
=i
i ≤ const ljl+1 ci−1 K 1,
j +1
li ci−1 K
≤ const lj +1
1,j +1
+ const
j
λ1−υ li 0 c e ( K 1,j +1 ) α 5 M i−1 j
=i
+ const
j
λ1−υ li 0 e ( K 1,j +1 ) α5 M j
=i
by Example A.3 of [FKTo1] λ1−υ i ≤ const ljl+1 ci−1 K 1, + const α0 5 Mli i ej ( K 1,j +1 ) j +1 λ1−υ i ci−1 K 1, + α0 4 Mli i ej ( K 1,j +1 ). ≤ const ljl+1
(X.3)
j +1
In particular, setting t = 0, reni,j +1 (K) ≤ 1,i t=0
l
const l i j +1
l
j +2 λ1−υ + 0 M j +2
li ≤ λ1−υ +2 0 M j +2 li ≤ λ1−υ 0 Mi
λ1−υ li 0 α4 M i
λ1−υ li 0 α4 M i
1 l
j +2 1 − M j λ1−υ 0 M j +2
(X.4)
by Definitions VI.9 and V.2.iii, if M is big enough. Substituting i = j0 and using ϕ ( · , s), ( · , s ) ≤ const ϕ 1, 1,∞
s,s ∈i
li
i
2 which applies to any translation invariant sectorized function on R2 × i , also gives the desired bound on δej +1 . Now suppose that (X.1) and (X.2) hold for reni+1,j +1 (K). Then, defining δK (i+1) (K ) = reni,i+1 (K ) − K i , reni,j +1 (K) = reni,i+1 reni+1,j +1 (K) = reni+1,j +1 (K) + δK (i+1) reni+1,j +1 (K) i
=
j +1
P,j +1 (K)i + δK (i+1) reni+1,j +1 (K)
=i+1
=
j +1 =i
P,j +1 (K)i
Two Dimensional Fermi Liquid. 2: Convergence
99
Pi,j +1 (K) = δK (i+1) reni+1,j +1 (K) .
if we choose By Lemma IX.8.i,
δK (i+1) (K ) 1,i ≤
λ1−υ li 0 e ( K 1,i+1 ). α6 M i i
By the inductive hypothesis, (X.3) applies when i is replaced by i + 1, so Pi,j +1 (K) 1,i
≤
λ1−υ 0 α6 λ1−υ 0 α6 λ1−υ 0 α6
≤
λ1−υ 0 α6
≤ ≤
li
e ( reni+1,j +1 (K) 1,i+1 ) Mi i
λ1−υ li+1 l li e const l i+1 ci K 1,j +1 + 0 4 e ( K 1,j +1 ) i i α M M i+1 j j +1 ci li Mi λ1−υ li+1 1 − const lj +1 M i ci K 1,j +1 − α0 4 li+1 M ej ( K 1,j +1 ) li ci . 1−υ M i 1−const M j cj K 1, j +1 −λ0 ej ( K 1,j +1 )
By Lemma A.4.ii and Corollary A.5.ii of [FKTo1], Pi,j +1 (K) 1,i ≤
const
≤
const
λ1−υ li ci 1 0 α 6 M i 1−const M j cj K 1,j +1 1−λ1−υ 0 ej ( K 1,j +1 ) λ1−υ cj li 0 α 6 M i 1−constM j cj K 1,j +1
ej ( K 1,j +1 ).
Corollary A.5.i of [FKTo1], with µ = const and X = M j K 1,j +1 , yields cj 1−constM j cj K 1,j +1
≤ const ej ( K 1,j +1 )
which, by Corollary A.5.ii of [FKTo1], implies Pi,j +1 (K) 1,i ≤
λ1−υ li 0 e ( K 1,j +1 ) α5 M i j
as desired. ii) We use induction on j − i. Introduce the local notation |||K|||j +1 = K 1, j +1
t=0
As reni,j +1 is the identity map for i = j + 1, the case i = j + 1 is trivial. As d d ds reni,j +1 (K + sK ) s=0 = ds reni,i+1 reni+1,j +1 (K) +s dds reni+1,j +1 (K + s K )s =0 s=0
we have, by Lemma IX.8.ii, d reni,j +1 (K + sK ) ds
s=0 1,i
≤
ℵ const M ei reni+1,j +1 (K) 1,i+1 d × ds reni+1,j +1 (K + s K )s =0 1, i+1
100
J. Feldman, H. Kn¨orrer, E. Trubowitz
Setting t = 0, d reni,j +1 (K + sK ) ds s=0 i ≤ const M ℵ 1−M i |||ren 1 (K)||| dds reni+1,j +1 (K + sK )s=0 i+1 i+1,j +1 i+1 1 d reni+1,j +1 (K + sK ) ≤ const M ℵ by (X.4) l ≤
ds i+1 1−M i λ1−υ 0 M i+1 ℵ d const M ds reni+1,j +1 (K + sK ) s=0 i+1 .
s=0 i+1
By induction d reni,j +1 (K + sK ) ≤ const M ℵ j +1−i |||K |||j +1 ds s=0 i as desired. iii)
reni,j +1 (0) − reni,j (0) = reni,j renj ,j +1 (0) − reni,j (0). Hence, by part (ii), l reni,j +1 (0) − reni,j (0) ≤ const j −j0 lj0 renj ,j +1 (0) 1,j + ∞ tδ. 1, i
j
By (X.4) reni,j +1 (0) − reni,j (0) ≤ 1, i
≤
const
δ=0
j −j0 lj0
lj
λ1−υ 0
const j λ1−υ 0 M
≤ λ1−υ 0
1 2j
+
+
δ=0
lj Mj
δ=0 δ
+
∞t
δ=0 δ
∞ tδ
∞t ,
and, setting i = j0 ,
renj ,j +1 (0) − renj ,j (0) |||δej +1 (0) − δej (0)|||1,∞ ≤ const 0 0 lj 0 1,j0 t=0 1 ≤ const const j −j0 λ1−υ j 0 ≤ λ1−υ 0
1 . 2j
M
This completes the proof of Theorem VIII.5. Appendix B. Self–Consistent ReWick Ordering In this appendix, we prove Lemma IX.7 and parts (i) and (ii) of Lemma IX.8. We view any fixed Q ∈ F0 (2, j ) as the constant function K → Q on Kj +1 . In this sense, the definitions of (IX.15) apply. For example, δK (x, s), (x , s ); Q = dx0 Q ∗ νˆ (≥j +1) (x0 , x, s), (0, x , s ) . Lemma B.1. Assume that K ∈ Kj +1 and Q ∈ F0 (2, j ) obeys |Q|1,j ≤ Then
λ1−υ lj 0 α M j ej ( K 1,j +1 ).
Two Dimensional Fermi Liquid. 2: Convergence
101
i) K(K ; Q) ∈ Kj . ii) There are constants const , independent of j but possibly depending on M, and const , independent of M and j , such that ej ( K(K ;Q) 1,j ) ≤ const ej ( K 1,j +1 ), ej ( K(K ;Q) 1,j ) ≤ const ej +1 ( K 1,j +1 ). iii)
lj B λ1−υ l j M j is an integral bound for Ej (K ; Q), 0 ∂ cj |Q |1,j l is an integral bound for dds Ej (K ; • const Mjj M j |Q|1,j + ∂ t0 t=0 . In particular, if d ∈ Nd+1 is independent of t0 and |Q|1,j ≤ dcj , Q + s Q ) s=0 ! then const lj d0 is an integral bound for dds Ej (K ; Q + s Q )s=0 . • const lj K 1,j +1 t=0 is an integral bound for dds Ej (K + sK ; Q)s=0 .
•
Proof.
i) Observe that δ Kˇ (k; Q) = Qˇ (0, k) ν (≥j +1) ((0, k)).
By Proposition E.10.ii of [FKTo4] and Lemma XIII.7 of [FKTo3], K j + δK(Q) 1,j ≤ K j 1,j + Q ∗ νˆ (≥j +1)1, ≤
const l
≤
const
j c K + const cj +1Q1, j −1 1,j +1 j +1
lj
j
M ℵ cj −1 K 1,j +1 + const cj +1 × ( K 1,j +1 )
j +2 + const ≤ const M ℵ λ1−υ 0 M j +2
l
l
j +1 ≤ λ1−υ + 0 M j +1
δ=0
λ1−υ lj 0 α Mj
∞ tδ
λ1−υ lj 0 α M j ej
+
δ=0
∞ tδ (B.1)
if M is large enough and α is large enough, depending on M. By definition, supp Kˇ ⊂ supp ν (≥j +2) (0, k) ⊂ supp ν (≥j +1) (0, k), and by construction ˇ ; Q) fulfills the required support property. ˇ ⊂ supp ν (≥j +1) (0, k), so K(K supp δK ii) By (B.1) ej ( K(K ;Q) 1,j ) = ≤ ≤ ≤
cj 1 − M j K j + δK(Q) 1,j
1 − Mj
const
cj
M ℵ cj −1 K 1,j +1 + const cj +1 cj
1 − M j +1 cj K 1,j +1 − const cj
λ1−υ 0 α
λ1−υ lj 0 α M j ej ( K 1,j +1 )
cj +1 ej ( K 1,j +1 )
1 − M j +1 cj K 1,j +1 − λ1−υ ej ( K 1,j +1 ) 0
102
J. Feldman, H. Kn¨orrer, E. Trubowitz
if α is large enough, since cj +1 ej ( K 1,j +1 ) ≤ M r+r0 cj ej ( K 1,j +1 ) ≤ const ej ( K 1,j +1 ).
(B.2)
By Lemma A.4.ii and Corollary A.5.ii of [FKTo1], ej ( K(K ;Q) 1,j ) ≤ const ≤
const
cj 1 1−M j +1 cj K 1,j +1 1−λ1−υ ej ( K 1,j +1 ) 0 cj
1−M j +1 cj K 1,j +1
ej ( K 1,j +1 ).
Corollary A.5.i of [FKTo1], with µ = M and X = M j K 1,j +1 , yields cj
1−M j +1 cj K 1,j +1
≤ const ej ( K 1,j +1 )
which, by Corollary A.5.ii of [FKTo1], implies the first bound. On the other hand, Corollary A.5.i of [FKTo1], with µ = 1 and X = M j +1 K 1,j +1 , yields cj
1−M j +1 cj K 1,j +1
≤
cj +1
1−M j +1 cj +1 K 1,j +1
≤ const ej +1 ( K 1,j +1 )
which, by Corollary A.5.ii of [FKTo1], implies the second bound. iii) Set V (K ; Q) = u (K ; Q) + Kext ∗ νˆ (≥j +3) , v(K ; Q) = u(K(K ; Q)) + K(K ; Q)ext ∗ νˆ (≥j +2) .
Then E(K ; Q) = Cj +1 u ( · ; Q); K − Dj u; K(K ; Q) =
ν (≥j +1) (k)
ık0 −e(k)−uˇ (k;K ;Q)−Kˇ (k)ν (≥j +3) (k) ν (≥j +1) (k)
− =
;Q))−K(k;K ;Q)ν (≥j +2) (k) ˇ ık0 −e(k)−u(k;K(K ˇ ν (≥j +1) (k) ν (≥j +1) (k) − ık −e(k)−v(k;K ;Q) . ˇ 0 ık0 −e(k)−Vˇ (k;K ;Q)
(B.3)
Also Vˇ (k; K ; Q) − v(k; ˇ K ; Q) ; Q)) + Kˇ j + δ Kˇ (Q) ν (≥j +2) ˇ = uˇ (K ; Q) + Kˇ ν (≥j +3) − u(K(K = u(K(K ˇ ; Q)) + Qˇ ν (≥j +1) + Kˇ ν (≥j +3) − u(K(K ˇ ; Q)) + Kˇ j + δ Kˇ (Q) ν (≥j +2) = −Kˇ (k)ν (j+2) (k) + Qˇ (k) ν (≥j +1) (k) −Qˇ (0, k) ν (≥j +1) ((0, k))ν (≥j +2) (k). (B.4) For the last equality, we used that Kˇ (k) = Kˇ j (k), by Definitions E.7 of [FKTo4] and XII.4 of [FKTo3], since Kˇ (k) vanishes outside the support of ν (≥j +2) ((0, k)). By Definition VI.9, Lemma XII.12 of [FKTo3] and Definition VIII.1 of [FKTo2],
Two Dimensional Fermi Liquid. 2: Convergence
103
lj +2 |ık0 −e(k)| Kˇ (k)ν (j +2) (k) ≤ 2λ1−υ ljj+2 ν (j +2) (k) ≤ 2λ1−υ 1 0 0 M +2 M j +2 √1 M M j +2 √ = 2 Mλ1−υ 0 lj +2 |ık0 − e(k)| 1 1−υ ≤ 10 λ0 lj |ık0 − e(k)|. Similarly, using Lemma XIII.7 of [FKTo3], Qˇ (k) ν (≥j +1) (k) − Qˇ (0, k) ν (≥j +1) ((0, k))ν (j +2) (k) ≤ 2|k0 | ∂∂t0 Q ∗ νˆ (≥j +1)1, j t=0 ≤ const |k0 | ∂∂t0 cj +1Q1, j t=0
(B.5)
(B.6)
and 0 −e(k)| Qˇ (k) 1 − ν (≥j +2) (k) ν (≥j +1) (k) ≤ 2Q ν (j +1) (k) ≤ 2Q1, |ık 1 1,j j √1 M M j +1 3 = 2M j + 2 Q1, |ık0 − e(k)|. (B.7) j
Combining (B.4)–(B.7) |Vˇ (k;K ; Q) − v(k; ˇ K ; Q)| ≤
≤
1 1−υ 10 λ0 lj +2 1 1−υ 4 λ0 lj |ık0
+ const
3 + 2M j + 2 Q1,
∂ cj +1 | Q||1,j ∂ t0
j
− e(k)|
t=0
|ık0 − e(k)| (B.8)
if α is large enough. Lemma VIII.7.i implies ˇ |v(k; ˇ K ; Q)| = u(k; ˇ K(K ; Q)) + K(k; K ; Q)ν (≥j +2) (k) ≤ λ1−υ 0 |ık0 − e(k)|
(B.9)
as well as d v(k; K ; Q + s Q )|s=0 = dds u(k; ˇ K j + δK(Q) + sδK(Q ))|s=0 ds ˇ +δ Kˇ (k; Q )ν (≥j +2) (k) 3
≤ 4M j + 2 δK(Q ) 1,j |ık0 − e(k)| ≤
3
const
M j + 2 |Q|1,j |ık0 − e(k)|
(B.10)
and d v(k; K + sK ; Q)|s=0 = dds u(k; ˇ K j + δK(Q) + sK j )|s=0 ds ˇ +Kˇ (k)ν (≥j +2) (k) 3
≤ 4M j + 2 K j 1,j |ık0 − e(k)| lj
≤
const l
≤
const M
j +1
3
M j + 2 cj −1 K 1,j +1 |ık0 − e(k)|
j + 23 +ℵ
cj −1 K 1,j +1 |ık0 − e(k)|. (B.11)
From (B.4) d ds
ˇ K ; Q + s Q ) Vˇ (k; K ; Q + s Q ) − v(k; = Qˇ (k) ν (≥j +1) (k) − Qˇ (0, k) ν (≥j +1) ((0, k))ν (≥j +2) (k)
104
J. Feldman, H. Kn¨orrer, E. Trubowitz
so that
d ˇ ˇ K ; Q + s Q ) ds V (k; K ; Q + s Q ) − v(k; ∂ cj +1 | Q | 1,j 3 + 2M j + 2 Q1, |ık0 − e(k)| ≤ const ∂ t0 j
t=0
(B.12)
by (B.6) and (B.7). Similarly, d ˇ ˇ K + sK ; Q) = −Kˇ (k)ν (≥j +2) (k) ds V (k; K + sK ; Q) − v(k; so that
d ˇ ˇ K + sK ; Q) ds V (k; K + sK ; Q) − v(k; ≤ 2K 1, ν (j +2) (k) j +1 −e(k)| 0 ≤ 2K 1, |ık 1 j +1 √1 M M j +2 5 = 2M j + 2 K 1, |ık0 − e(k)| j +1
(B.13)
as in (B.7). Using (B.8) and (B.9) ν (≥j +1) (k) ν (≥j +1) (k) E(K ; Q) = − ık0 −e(k)−v(k;K ˇ (k;K ;Q) ˇ ;Q) ık −e(k)− V 0 ;Q) (≥j +1) Vˇ (k;K ;Q)−v(k;K ˇ (k) = ν ˇ [ık −e(k)−V (k;K ;Q)] [ık −e(k)−v(k;K ˇ ;Q)] 0
≤
0
λ1−υ 0 lj |ık0 −e(k)| .
The integral bound for Ej (K ; Q) now follows from Proposition XII.16 of [FKTo3]. Similarly, d E (K ; Q + s Q ) ds j s=0 d ;Q+sQ )| d Vˇ (k;K ;Q+sQ )|s=0 ˇ s=0 (≥j +1) ds v(k;K = ds − (k) ;Q)]2 ν 2 ˇ [ık −e(k)−v(k;K ˇ [ık0 −e(k)−V (k;K ;Q)]
≤ const
0
∂ cj | Q | 1, j M j Q 1, + ∂ t0 j
| |
|ık0 −e(k)|
t=0
and the first integral bound for dds Ej (K ; Q + s Q )s=0 also follows from Proposition XII.16 of [FKTo3]. If if d ∈ Nd+1 is independent of t0 and |Q|1,j ≤ dcj , then
M j |Q|1,j +
∂ cj | Q | 1,j ∂ t0
t=0
≤ const M j d0
and the second integral bound for dds Ej (K ; Q + s Q )s=0 follows from the first. Finally, using (B.11) and (B.13), d E(K + sK ; Q) ds s=0 d +sK ;Q)| d Vˇ (k;K +sK ;Q)|s=0 ˇ s=0 (≥j +1) ds v(k;K = ds − (k) ;Q)]2 ν 2 ˇ [ık −e(k)− v(k;K ˇ [ık0 −e(k)−V (k;K ;Q)] 0 ≤ const
M j K 1,j +1 |ık0 −e(k)|
Two Dimensional Fermi Liquid. 2: Convergence
and the integral bound for XII.16 of [FKTo3].
105
d ds Ej (K + sK ; Q) s=0
also follows from Proposition
Recall that w˜ 0,2 (K ; Q) ∈ F0 (2, j ) was defined, following (IX.15), to be the coefficient of ψ((ξ1 ,s1 ))ψ((ξ2 ,s2 ) ) in .. w(K(K ; Q)) .. −E (K ;Q) . j
Lemma B.2. Assume that K ∈ Kj +1 , d ∈ Nd+1 is independent of t0 and λ1−υ l Q ≤ 0α Mjj ej K 1,j +1 , 1, j Q ≤ dcj . 1, j
Then |w˜ 0,2 (K ; Q)||1,j ≤ d w˜ 0,2 (K ; Q + s Q ) ≤ ds s=0 1,j d w˜ 0,2 (K + sK ; Q) ≤ ds s=0 1, j
λ1−υ lj 0 e K 1,j +1 , α 6.5 M j j λ1−υ 0 α d ej K 1,j +1 , λ1−υ 0 e ( K 1,j +1 ) K 1,j +1 . α 1.5 j
Bl
Proof. We use the notation of §XII of [FKTo3]. Set α˜ = (1−υ)/2 , b = M jj , X = 2λ0 K(K ; Q) 1,j and c = const 1 M j ej X , where const 1 is the constant of Lemma XV.5 " of [FKTo3]. Let, for a sectorized Grassmann function v = n vn with vn ∈ C⊗ n V , α
N (v; α) ˜ = Observe that, if V = v = n V0,n , then
m,n Vm,n
1 c b2
n
α˜ n bn |vn|1, .
with Vm,n ∈ Am ⊗
N (v; 2α) ˜ ≤
const 1 Bλ1−υ 0
"n
V and V0,2 = 0, and if
Nj V , α, X .
Set, using the notation of Definition XII.6 of [FKTo3], W (K ; Q) = w(K(K ; Q))φ=0, W2 (K ; Q) = Gr w(K(K ; Q))0,2 , W4 (K ; Q) = W (K ; Q) − W2 (K ; Q). Then ej (X)||w˜ 0,2 (K ; Q)||1,j =
N Gr(w˜ 0,2 (K ; Q)); α˜ + const 1α˜ 2 M j N Gr(w˜ 0,2 (K ; Q)) − W2 (K ; Q); α˜
1 const 1 α˜ 2 M j
≤ ej (X)||w(K(K ; Q))0,2|1,j
≤ ej (X)||w(K(K ; Q))0,2|1,j + const 1α˜ 2 M j N .. W4 (K ; Q) .. −E 1
≤ ej (X)||w(K(K ; Q))0,2|1,j
1
− W4 (K ; Q); α˜ λ1−υ l + const α˜04 M jj N W4 (K ; Q); 2α˜ j (K
;Q)
by Corollary II.32 of [FKTr1] and Lemma B.1.iii. By the observation above
106
J. Feldman, H. Kn¨orrer, E. Trubowitz λ1−υ 0 lj α˜ 4 M j
lj (j ) 1 N W4 (K ; Q); 2α˜ ≤ const α˜ 4 M φ, α, X j Nj w(K(K ; Q)) + 2 φC 2(1−υ) λ l ≤ const 0α 4 M j j Nj w(K(K ; Q)) + 21 φC (j ) φ, α, X .
Hence, by (O1), |w˜ 0,2 (K ; Q)||1,j ≤ ≤ ≤
2(1−υ)
λ1−υ λ l 0 lj e (X)2 + const 0α 4 M j j ej (X) α7 M j j λ1−υ l const α07 M jj ej ( K 1,j +1 ) λ1−υ 0 lj e ( K 1,j +1 ) α 6.5 M j j
by Corollary A.5.ii of [FKTo1]and Lemma B.1.ii. We now prove the bound on dds w˜ 0,2 (K ; Q + s Q )s=01, . This time we use α = j " and, for any sectorized Grassmann function v = n vn with vn ∈ C ⊗ n V , n n α b |vn|1, . N (v; α ) = b12 c n
The other notation is as in the first part of this proof. This time, if V = " Vm,n ∈ Am ⊗ n V (V0,2 need not vanish), and if v = n V0,n , then const N (v; 2α ) ≤ B 1 λ1−υ Nj V , α, X . 0 Hence
m,n Vm,n
α 2
with
ej (X) dds w˜ 0,2 (K ; Q + s Q )s=01, j d 1 = ˜ 0,2 (K ; Q + s Q ))s=0; α 2 j N ds Gr(w const1 α M 1 ≤ N dds .. W (K ; Q + s Q ) .. −E (K ;Q+sQ ) s=0; α j const1 α 2 M j 1 ≤ N .. dds W (K ; Q + s Q )s=0 .. −E (K ;Q) ; α j const1 α 2 M j . d . 1 + 2 j N ds . W (K ; Q) . −E (K ;Q+sQ ) s=0; α const 1 α M j d 1 ≤ N W (K ; Q + s Q ) ; 2α 2 ds s=0 const1 α M j j 1 1 const d0BM N W (K ; Q); 2α + const 1 α 2 M j (α −1)2 λ1−υ ≤ const α 20M j Nj dds W (K ; Q + s Q )s=0, α, X λ1−υ d +const 0 α 4 0 Nj W (K ; Q), α, X .
In the second last inequality, we used Corollary II.32.i,iii of [FKTr1] and Lemma B.1.iii. Since s=0 d d ds W (K ; Q + s Q ) s=0 = ds w Kj + δK(Q + s Q ) φ=0 = dds w K j + δK(Q) + sδK(Q ) s=0 . φ=0
(O1) implies that Nj dds W (K ; Q + s Q )s=0, α, X ≤ M j ej (X) δK(Q ) 1,j ≤
const M
j
cj +1 ej (X) Q 1,j .
Two Dimensional Fermi Liquid. 2: Convergence
107
(O1) also implies that Nj W (K ; Q), α, X = Nj w K(K ; Q) φ=0, α, X ≤ ej (X). Hence
ej (X) dds w˜ 0,2 (K ; Q + s Q )s=01, j 1−υ λ0 j ≤ const α 2 M j M cj +1 ej (X) dcj + ≤ const
λ1−υ 0 dej ( K 1,j +1 ) α2
≤
λ1−υ 0 d0 ej (X) α4
λ1−υ 0 α dej ( K 1,j +1 )
by Lemma B.1.ii, (B.2) and Corollary A.5.ii of [FKTo1]. Finally, we prove the bound on dds w˜ 0,2 (K + sK ; Q)s=01, . We have j
ej (X) dds w˜ 0,2 (K + sK ; Q)s=01, j d 1 = ˜ 0,2 (K + sK ; Q))s=0; α 2 j N ds Gr(w const1 α M 1 ≤ N dds .. W (K + sK ; Q) .. −E (K +sK ;Q) s=0; α j const 1 α 2 M j . .d 1 ≤ 2 j N . ds W (K + sK ; Q) s=0 . −E (K ;Q) ; α const 1 α M j . d . 1 ; α + N ; Q ) W (K . . ds −Ej (K +sK ;Q) s=0 const 1 α 2 M j d 1 ≤ 2 j N ds W (K + sK ; Q) s=0; 2α const 1 α M M j K 1,j +1 1 1 + N W (K ; Q); 2α 2 j (α −1)2 const B const 1 α M λ1−υ ≤ const α 20M j Nj dds W (K + sK ; Q)s=0, α, X λ1−υ +const α0 4 K 1,j +1 Nj W (K ; Q), α, X . In the second last inequality, we used Corollary II.32.i,iii of [FKTr1] and Lemma B.1.iii. Since s=0 d d . ds W (K + sK ; Q) s=0 = ds w Kj + δK(Q) + sKj φ=0
(O1) implies that Nj dds W (K + sK ; Q)s=0, α, X ≤ M j ej (X) K j 1,j ≤
const M
j +ℵ
cj −1 ej (X) K 1,j +1
and, as we have already observed, Nj w K(K ; Q) φ=0, α, X ≤ ej (X). Hence
1−υ λ ej (X) dds w˜ 0,2 (K + sK ; Q)s=01, ≤ const α0 2 cj −1 + j
by Lemma B.1.ii.
K 1,j +1 ej (X)
λ1−υ 0 e ( K 1,j +1 ) K 1,j +1 α2 j 1−υ λ0 e ( K 1,j +1 ) K 1,j +1 α 1.5 j
≤ const ≤
λ1−υ 0 α4
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J. Feldman, H. Kn¨orrer, E. Trubowitz
We now solve q(K ) = 2w˜ 0,2 (q(K ); K ) by a standard contraction mapping argument. Define q (0) = 0, q (1) = 2w˜ 0,2 (0; K ), (n+1) q = 2w˜ 0,2 (q (n) ; K ),
n ≥ 1.
We use the shorthand notation ej = ej ( K 1,j +1 ). Lemma B.3. Let K ∈ Kj +1 . Then 1−υ n−1 1−υ (n) λ λ l q − q (n−1) 2 α06.5 Mjj ej ≤ κ 0α 1, j
Proof. The proof is by induction on n. By Lemma B.2 (1) λ1−υ l q ≤ 2 α06.5 Mjj ej 1, j
and the conclusion of the lemma is true for n = 1. If the lemma is satisfied for some n, then, by Lemma B.2 with 1−υ n−1 1−υ λ λ l d = κ 0α 2 α06.5 Mjj 1−M j K1 1,j +1
we have
(n+1) q − q (n)1, = 2 w˜ 0,2 (q (n) ; K ) − w˜ 0,2 (q (n−1) ; K ) 1, j
j
≤ ≤ ≤
λ1−υ 2 0α dej 1−υ n−1 1−υ λ1−υ λ λ l 2 α06.5 Mjj e2j 2 0α κ 0α 1−υ n 1−υ λ λ l κ 0α 2 α06.5 Mjj ej .
Proof of Lemma IX.7. Fix any K ∈ Kj +1 . By Corollary A.5.ii there is a constant κ such that e2j ≤ κ2 ej . If α is small enough, Lemma B.3 implies that every λ1−υ l
2 α06.5 Mjj (n) λ1−υ l q ≤ e ≤ 4 α06.5 Mjj ej 1−υ j 1,j λ 1 − κ 0α and that the sequence {q (n) }n≥1 converges to a q0 (K ) also obeying λ1−υ l q0 (K ) ≤ 4 α06.5 Mjj ej ≤ 1, j
d ds
λ1−υ lj 0 e . α6 M j j
(B.14)
Fix any K and denote Q0 = q0 (K ) and Q = dds q0 (K + sK )s=0 . Applying to q0 (K + sK ) = 2w˜ 0,2 (q0 (K + sK ); K + sK ) yields s=0 d Q = ds q0 (K + sK ) s=0 = 2 dds w˜ 0,2 (q0 (K + sK ); K )s=0 + 2 dds w˜ 0,2 (q0 (K ); K + sK )s=0 = 2 dds w˜ 0,2 (Q0 + s Q ; K )s=0 + 2 dds w˜ 0,2 (Q0 ; K + sK )s=0 .
Two Dimensional Fermi Liquid. 2: Convergence
109
As, for fixed j , w˜ 0,2 (K ; Q) is analytic in Q and K and as ej K 1,j +1 has only finitely many finite coefficients, there is some finite β such that |Q|1,j ≤ βej K 1,j +1 . Choose a β that is within K 1,j +1
d = β 1−M j K
1,j +1
λ1−υ 0 2α 1.5
of the infimum of all β’s that work. By Lemma B.2, with
,
λ1−υ
K 1,
λ1−υ
|Q|1,j ≤ 2 0α β 1−M j K j +1 ej + 2 α01.5 ej ( K 1,j +1 ) K 1,j +1 1,j +1 1−υ λ0 λ1−υ 0 ≤ κ α β + 2 α 1.5 ej K 1,j +1 λ1−υ ≤ 41 β + 2 α01.5 ej K 1,j +1 if α is large enough. Thus |Q|1,j ≤ β ej K 1,j +1 with β =
1 4β
λ1−υ
β ≥ 4 α01.5 , then
λ1−υ
λ1−υ
+ 2 α01.5 . If
λ1−υ
β − β = 2 α01.5 − 43 β ≤ − α01.5 which violates the requirement that β that is within work. Hence λ1−υ
|Q|1,j ≤ 4 α01.5 ej K 1,j +1 ≤
λ1−υ 0 2α 1.5
of the infimum of all β’s that
λ1−υ 0 α ej K 1,j +1 .
(B.15)
Proof of Lemma IX.8. δK(K )
1,j
and
(i) By (B.14), (B.15) and Lemma XIII.7 of [FKTo3], ≤ const cj +1|q0 (K )||1,j ≤ const
d δK(K + sK ) ≤ ds s=0 1,
λ1−υ lj 0 e α 6.5 M j j
d const cj +1 ds q0 (K
j
≤
λ1−υ lj 0 e α6 M j j
+ sK )s=01,
j
λ1−υ l const α01.5 Mjj ej K 1,j +1 λ1−υ lj 0 α M j ej K 1,j +1 .
≤ ≤
(ii) By Proposition E.10.ii of [FKTo4] and part (i), K(K ) 1,j ≤
const l
j +1
≤
const l
cj −1 K 1,j +1 + j +1
lj
cj −1 K 1,j +1 + δK(K ) 1,j
lj
λ1−υ lj 0 e α6 M j j
and d K(K + sK ) = dds K j + sK j + δK(K + sK ) s=0 1, ds s=0 1,j j ≤ K 1, + d δK(K + sK ) j
≤
const
(iii) is contained in Lemma B.1.ii.
ds
s=0 1,j
λ1−υ 0 α ej ( K 1,j +1 ) K 1,j +1 ℵ const M ej ( K 1,j +1 ) K 1,j +1 .
+ ≤
j
M ℵ cj −1 K 1,j +1
110
J. Feldman, H. Kn¨orrer, E. Trubowitz
Notation
Norms Norm ||| · |||1,∞ · 1,∞ ||| · |||∞ | · |p, ||| · |||1, ||| · |||3, · 1,
Characteristics no derivatives, external positions, acts on functions derivatives, external positions, acts on functions no derivatives, external positions, acts on functions derivatives, external positions, all but p sectors summed no derivatives, all but 1 sector summed no derivatives, all but 3 sectors summed 2 like | · |1, , but for functions on R2 × 1 1 |ϕ||1,j + l |ϕ||3,j + 2 |ϕ||5,j if m = 0 j lj ρm;n lj |ϕ|| if m = 0 1,j M 2j lj B n/2 M 2j e (X) n |wm,n |j j m,n≥0 α l Mj j 1 |||G ||| Definition VI.7 m ∞ m>0 (1−υ) max{m−2,2}/2
|ϕ||j Nj (w, α, X) N(G )
Reference Definition V.3 Definition V.3 Definition VI.7 Definition VI.6 (II.6) (II.14) [Def’n E.3, FKTo4] Definition VI.6 Definition VI.7
λ0
Spaces Not’n E Kj B Bˇ
Description counterterm space space of future counterterms for scale j R × Rd × {↑, ↓} × {0, 1} viewed as position space R × Rd × {↑, ↓} × {0, 1} viewed as momentum space R × Rd × {↑, ↓} viewed as position space n functions on Bm × B × , internal momenta in sectors
Reference Definition I.1 Definition VI.9 before Def VII.1 beginning of §VI Definition VII.3 Definition VI.3.ii
Din
(j,form)
formal input data for scale j
Definition III.8
Dout
(j,form)
formal output data for scale j
Definition III.9
Din
(j )
input data for scale j
Definition IX.1
B Fm (n; )
(j ) Dout
output data for scale j Definition IX.2
Other Notation Not’n r0 r M const const ν (j ) (k)
ν (≥j ) (k) n0 J S (W )(φ, ψ)
Description number of k0 derivatives tracked number of k derivatives tracked scale parameter, M > 1 generic constant, independent of scale generic constant, independent of scale and M j th scale function (j ) i≥j ν (k) degree of asymmetry particle/hole swap operator log Z1 eW (φ,ψ+ζ ) dµS (ζ )
Reference following (I.3) following (I.3) before Definition I.2
Definition I.2 Definition I.2 Definition I.10 (III.3) Definition III.1
Two Dimensional Fermi Liquid. 2: Convergence Not’n ˜ C (W )(φ, ψ) ℵ λ0 υ ρm;n (λ) ρm;n
Description log Z1 eφJ ζ eW (φ,ψ+ζ ) dµC (ζ ) 1 <ℵ< 2 2 3 maximum allowed “coupling constant” 0 < υ < 41 , power of λ0 eaten by bounds λ−(1−υ) max{m+n−2,2}/2 ρm;n (λ0 ) 1 if m = 0 ; 4 lj M j if m > 0 1 M ℵj
lj
=
j cj
the sectorization at scale j of length lj j –independent constant j |δ| t δ + = ∞ t δ ∈ Nd+1 |δδ |≤r M |δδ |>r
ej (X)
=
B
fext ∗ • µˆ
= length of sectors of scale j
|δ |≤r
cj0 0 1−M j X
or |δ0 |>r0
extends f (x, x ) to fext (x0 , x, σ, a), (x0 , x , σ , a ) convolution ladder convolution Fourier transform
111 Reference Definition III.1 following Definition VI.3 Theorem VIII.5 Definition V.6 Definition V.6 Definition VI.6.ii following Definition VI.3 following Definition VI.3 Definition VI.7 Definition V.2 Definition V.2 [Definition E.1, FKTo4] Definition VIII.6 Definition VII.2, Notation V.4
References [FKTf1] Feldman, J., Kn¨orrer, H., Trubowitz, E.: A Two Dimensional Fermi Liquid, Part 1: Overview. Commun. Math. Phys. 247, 1–47 (2004) [FKTf3] Feldman, J., Kn¨orrer, H., Trubowitz, E.: A Two Dimensional Fermi Liquid, Part 3: The Fermi Surface. Commun. Math. Phys. 247, 113–177 (2004) [FKTl] Feldman, J., Kn¨orrer, H., Trubowitz, E.: Particle–Hole Ladders. Commun. Math. Phys. 247, 179–194 (2004) [FKTo1] Feldman, J., Kn¨orrer, H., Trubowitz, E.: Single Scale Analysis of Many Fermion Systems, Part 1: Insulators. Rev. Math. Phys. 15, 949–993 (2003) [FKTo2] Feldman, J., Kn¨orrer, H., Trubowitz, E.: Single Scale Analysis of Many Fermion Systems, Part 2: The First Scale. Rev. Math. Phys. 15, 995–1037 (2003) [FKTo3] Feldman, J., Kn¨orrer, H., Trubowitz, E.: Single Scale Analysis of Many Fermion Systems, Part 3: Sectorized Norms. Rev. Math. Phys. 15, 1039–1120 (2003) [FKTo4] Feldman, J., Kn¨orrer, H., Trubowitz, E.: Single Scale Analysis of Many Fermion Systems, Part 4: Sector Counting. Rev. Math. Phys. 15, 1121–1169 (2003) [FKTr1] Feldman, J., Kn¨orrer, H., Trubowitz, E.: Convergence of Perturbation Expansions in Fermionic Models, Part 1: Nonperturbative Bounds. Commun. Math. Phys. 247, 195–242 (2004) [FMRT] Feldman, J., Magnen, J., Rivasseau, V., Trubowitz, E.: Fermionic Many-Body Models. In: Mathematical Quantum Theory I: Field Theory and Many-Body Theory, J. Feldman, R. Froese, L. Rosen, (eds.), CRM Proceedings & Lecture Notes 7, Providence, RI: Am. Math. Soc., 1994, pp. 29–56 [FW] Fetter, A.L., Walecka, J.D.: Quantum Theory of Many-Particle Systems. New York: McGrawHill, 1971 Communicated by J.Z. Imbrie
Commun. Math. Phys. 247, 113–177 (2004) Digital Object Identifier (DOI) 10.1007/s00220-003-0998-y
Communications in
Mathematical Physics
A Two Dimensional Fermi Liquid. Part 3: The Fermi Surface Joel Feldman1, , Horst Knörrer2 , Eugene Trubowitz2 1 2
Department of Mathematics, University of British Columbia, Vancouver, B.C., Canada V6T 1Z2. E-mail: [email protected] Mathematik, ETH-Zentrum, 8092 Zürich, Switzerland. E-mail: [email protected]; [email protected]
Received: 21 September 2002 / Accepted: 12 August 2003 Published online: 6 April 2004 – © Springer-Verlag 2004
Abstract: We show that the particle number density derived from the thermodynamic Green’s function at temperature zero constructed in the second part of this series has a jump across the Fermi curve, a basic property of a Fermi liquid. We further show that the two particle thermodynamic Green’s function at temperature zero has the regularity behavior expected in a Fermi liquid. Contents XI. XII. XIII. XIV. XV.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . Momentum Green’s Functions . . . . . . . . . . . . . . . . . Momentum Space Norms . . . . . . . . . . . . . . . . . . . Ladders with External Momenta . . . . . . . . . . . . . . . . Recursion Step for Momentum Green’s Functions . . . . . . . 1. More Input and Output Data . . . . . . . . . . . . . . . . 2. Integrating Out a Scale . . . . . . . . . . . . . . . . . . . 3. Sector Refinement, ReWick Ordering and Renormalization Appendix C. Hölder Continuity of Limits . . . . . . . . . . . . . . Appendix D. Another Description of Particle–Hole Ladders . . . . . Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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113 114 128 134 143 144 146 157 169 170 175 177
XI. Introduction This paper, together with [FKTf1] and [FKTf2] provides a construction of a two dimensional Fermi liquid at temperature zero. This paper contains Sects. XI through XV and Research supported in part by the Natural Sciences and Engineering Research Council of Canada and the Forschunginstitut für Mathematik, ETH Zürich.
114
J. Feldman, H. Knörrer, E. Trubowitz
Appendices C and D. Sections I through III and Appendix A are in [FKTf1] and Sects. IV through X and Appendix B are in [FKTf2]. Notation tables are provided at the end of the paper. The main goal of this part is the proof of the existence of a Fermi surface, stated in Theorem I.5. The proof of this theorem follows Lemma XII.4. We assume for the rest of this paper that the interaction V satisfies the reality condition (I.1) and is bar/unbar exchange invariant in the sense of (I.2). The latter is not essential1 . It is used only for notational convenience at intermediate stages of the proof.
XII. Momentum Green’s Functions Recall that the momentum distribution function n(k) of Theorem I.5 is expressed in ˇ 2 (k) of the two point Green’s function G2 (x, y). In terms of the Fourier transform G rg Theorem VIII.5, we defined a generating functional Gj (K). Following the statement rg of Theorem VIII.5, we constructed G2 (x, y) as the limit of functions 2Gj,2 ((x,1),(y,0)), rg rg where dη1 dη2 Gj,2 (η1 , η2 ) φ(η1 )φ(η2 ) = 2 dxdy Gj,2 ((x,1),(y,0)) φ(x,1)φ(y,0) is rg the part of Gj (K)K=0 that is homogeneous of degree two. In §XV, we shall prove the ˇ rg (k), of these functions. following decomposition of the Fourier transforms, 2G j,2
Theorem XII.1. Let ℵ < ℵ < 23 . If the constants α, ¯ λ¯ > 0 of Theorem VIII.5 are big, ˇ rg (k) respectively small, enough, the data of Theorem VIII.5 may be chosen such that 2G j,2 has the decomposition
ˇ (k) = C 2G j,2 uj (0) (k) + rg
(≤j )
1 [ik0 −e(k)]2
j j
q (i,) (k).
i=2 =i
Here uj (0) is the sectorized function uj (ξ1 , s1 ), (ξ2 , s2 ); K K=0 and q (i,) (k), ≥ i ≥ 2 is a family of functions with q (i,) (k) vanishing when k is in the (i + 2)nd neighbourhood and and when k is not in the support of U (k) and obeying, for each multiindex δ = (δ0 , δ ) with |δ| ≤ 2, l sup Dδ q (i,) (k) ≤ 2λ1−2υ M ℵ (−i) M δ0 i M |δδ |. 0 M
(XII.1)
k
Furthermore q (i,) (−k0 , k) = q (i,) (k0 , k). For the rest of this section, we deduce consequences of Theorem XII.1. In Lemma ˇ rg (k) and in Lemma XII.3 we show XII.2, we describe regularity properties of lim G j,2 j →∞
that Fourier transforms commute with the limit j → ∞. From this we derive properties of the proper self–energy and use them to prove that there is a jump in the momentum distribution function n(k), thus showing that Theorem XII.1 implies Theorem I.5. 1
See footnote (2) in §XV.
Two Dimensional Fermi Liquid. 3: Fermi Surface
115
Lemma XII.2. i) The sequence of functions uˇ j (k; 0) converges uniformly to a function P (k) that vanishes at k0 = 0 and obeys P (k) ≤ λ1−2υ min{k0 , 1 , 0 ∇P (k) ≤ λ1−2υ , 0 ∇P (k) − ∇P (k ) ≤ λ1−2υ |k − k | 21 , 0 P (k) − uˇ j (k; 0) ≤ λ1−2υ lj min{k0 , 1 . 0
j j ii) The sequence of functions Qj (k) = i=2 =i q (i,) (k) converges uniformly to a function Q(k) that vanishes unless k is in the support of U (k) and obeys 3 Q(k) ≤ λ1−3υ min{ik0 − e(k) 2 , 1 , 0 1 ∂Q (k) ≤ λ1−3υ min{ik0 − e(k) 2 , 1 , 0 ∂k0 Q(k) − Q(k ), ∂Q (k) − ∂Q (k ) ≤ λ1−3υ |k − k | 21 , 0 ∂k0 ∂k0 Q(k) − Qj (k) ≤ λ1−3υ lj min{ik0 − e(k), 1 . 0 iii) P (−k0 , k) = P (k0 , k) and Q(−k0 , k) = Q(k0 , k). Proof.
i) By Lemma XII.12 of [FKTo3] and (VIII.1), li sup Dδ pˇ (i) (k) ≤ 4λ1−υ M i|δ| 0 Mi
(XII.2)
k
for all |δ| ≤ 2. Consequently, the sequence of functions uˇ j (k; 0) = converges uniformly to P (k) =
∞
j −1
pˇ (i) (k)
i=2
pˇ (i) (k) and, by Lemma C.1, with α = ℵ and
i=2
β = 1 − ℵ,
≤ λ1−2υ , |P (k)|, ∇P (k) ≤ const λ1−υ 0 0 ∇P (k) − ∇P (k ) ≤ const λ1−υ |k − k |ℵ ≤ λ1−2υ |k − k | 21 , 0 0 if λ0 is small enough. Since each pˇ (i) (k) vanishes at k = 0, the same is true for |k0 |. Furthermore, P (k) and |P (k)| ≤ λ1−2υ 0 j −1 ∞ P (k) − pˇ (i) (k) ≤ pˇ (i) (k) i=2
≤
i=j ∞ i=j
4λ1−υ 0 li min{ k0 |, 1
≤ const λ1−υ 0 lj min{ k0 |, 1 ≤ λ1−2υ lj min{k0 |, 1 . 0
116
J. Feldman, H. Knörrer, E. Trubowitz
l ii) As q (i,) (k) ≤ 2λ1−2υ M ℵ (−i) and 0 M ∞ ∞ ∞ l ℵ (−i) −ℵ i M = M M i=2 =i
−(1+ℵ−ℵ )i 1 M 1−M −(1+ℵ−ℵ )
< ∞,
i=2
the sequence of functions Qj (k) =
q (i,) (k) converges uniformly to
2≤i≤≤j
Q(k) =
q (i,) (k).
2≤i≤
li By Lemma C.1, with j = − i, C0 = λ1−2υ , C1 = λ1−2υ li , α = 1 − ℵ + ℵ 0 0 Mi and β = ℵ − ℵ, ∞ ∞ q (i,) (k) − q (i,) (k ) ≤ const λ1−2υ li |k − k |1−ℵ +ℵ , 0 =i
=i
and by Lemma C.1, with j = −i, C0 = λ1−2υ li , C1 = λ1−2υ li M i , α = 1−ℵ +ℵ 0 0 and β = ℵ − ℵ, ∞ =i
∂q (i,) ∂k0 (k) −
∞ =i
∂q (i,) ∂k0 (k )
≤ const λ1−2υ li M (1−ℵ +ℵ)i |k − k |1−ℵ +ℵ . 0
ℵ Pick any 21 < ℵ < ℵ and set γ = 1−ℵ +ℵ . Note that, since 1 − ℵ > 0, 0 < γ < 1. th Taking the γ power of this bound and multiplying by the (1 − γ )th power of the bound (i,) ∞ ∞ ∞ ∂q ∂q (i,) ∂q (i,) (k) − (k ) ≤ 2 sup λ1−2υ li (1−ℵ1+ℵ)(−i) ∂k0 (k) ≤ 4 0 ∂k0 ∂k0 =i
k
=i
M
=i
≤ const λ1−2υ li 0 gives ∞ =i
∂q (i,) ∂k0 (k) −
∞ =i
∂q (i,) ∂k0 (k )
≤ const λ1−2υ li M ℵ i |k − k |ℵ . 0
Hence Q(k) − Q(k ) ≤ const λ1−2υ |k − k |1−ℵ +ℵ ≤ λ1−3υ |k − k | 56 , 0 0 ∂Q 1 ∂Q 1−2υ 1−3υ ℵ (k) − |k − k | ≤ λ0 |k − k | 2 , (XII.3) ∂k0 ∂k0 (k ) ≤ const λ0 if λ0 is small enough. By hypothesis, every q (i,) (k) vanishes on the (i + 2)nd neighbourhood. Hence, for k in the support of ν (m) (k), q (i,) (k) vanishes when i + 2 ≤ m and ∞ Q(k) ≤
∞
i=m−2 =i
∞ q (i,) (k) ≤ 2λ1−2υ 0
lm ≤ const λ1−2υ 0 Mm
∞ l M ℵ (−i) M
i=m−2 =i
Two Dimensional Fermi Liquid. 3: Fermi Surface
117
1+ℵ ≤ const λ1−2υ min ik0 − e(k) ,1 0 3 1−3υ min ik0 − e(k) 2 , 1 , ≤ λ0 ∞ (i,) ∞ ∂Q (k) ≤ ∂q (k) ∂k0 ∂k0 ≤ ≤ ≤
i=m−2 =i ∞ ∞ 2λ1−2υ li M −(1−ℵ +ℵ)(−i) 0 i=m−2 =i ℵ const λ1−2υ min ik0 − e(k) , 1 0 1 min ik0 − e(k) 2 , 1 . λ1−3υ 0
≤ const λ1−2υ lm 0
(XII.4)
In general Q(k) − Qj (k) =
j ∞ i=2 =j +1
q (i,) (k) +
∞ ∞
q (i,) (k).
i=j +1 =i
So, for k in the support of ν (m) (k), ∞ l Q(k) − Qj (k) ≤ 2λ1−2υ M ℵ (−i) 0 M m−2≤i≤j =j +1 ∞ ∞
+
i=max{m−2,j +1} =i
≤ const λ1−2υ 0 ≤ ≤
l 2λ1−2υ M ℵ (−i) 0 M
m−2≤i≤j
lj M ℵ (j −i) Mj
∞
+
i=max{m−2,j +1}
lmax{m,j } l } M ℵ (max{m,j }−m) + Mmax{m,j max{m,j } M max{m,j }
lj (ℵ −1)(max{m,j }−m) const λ1−2υ +1 0 Mm M
const λ1−2υ 0
li Mi
l
j ≤ const λ1−2υ 0 Mm ≤ λ1−3υ lj min ik0 − e(k), 1 .
0
iii) That uˇ j (−k0 , k; 0) = uˇ j (k0 , k; 0) and q (i,) (−k0 , k) = q (i,) (k0 , k) are consequences of Theorems VIII.5 and XII.1, respectively. Lemma XII.3.
0 d d k −i(−k x +k·x) U (k) Q(k) 0 0 G2 (0, 0, ↑), (x0 , x, ↑) = dk e + 2π (2π)d ik0 −e(k)−P (k) [ik0 −e(k)]2 (with the value of G2 (0, 0, ↑), (x0 , x, ↑) at x0 = 0 defined through the limit x0 → 0+) U (k) Q(k) and the Fourier transform of G2 (0, 0, ↑), (x0 , x, ↑) is ik0 −e(k)−P (k) + [ik0 −e(k)]2 , which is continuous at all points (k0 , k) for which ik0 − e(k) = 0. Proof. By the definitions of Qj and u(k) ˇ (Definition VI.3.iv) rg dk0 d d k −i(−k0 x0 +k·x) 2Gj,2 (0, 0, ↑, 1), (x0 , x, ↑, 0) = 2π (2π)d e
Qj (k) ν (≤j ) (k) . × ik −e(k)− + uˇ (k;0) [ik −e(k)]2 0
j
0
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J. Feldman, H. Knörrer, E. Trubowitz
Hence
U (k) Q(k) e−i(−k0 x0 +k·x) ik0 −e(k)−P + 2 (k) [ik0 −e(k)] rg −2Gj,2 (0, 0, ↑, 1), (x0 , x, ↑, 0) d −i(−k0 x0 +k·x) 0 d k = dk 2π (2π)d e ν (≤j ) (k)[P (k)−uˆ (k;0)] U (k)−ν (≤j ) (k) × ik + [ik −e(k)−uˆ (k;0)][ik j−e(k)−P (k)] + 0 −e(k)−P (k) dk0 d d k 2π (2π )d
0
j
0
Q(k)−Qj (k) [ik0 −e(k)]2
.
Let U˜ (k) be the characteristic function of the support of U (k). By Lemma XII.2, all three of l min{|ik0 −e(k)|,1} ˜ Q(k)−Qj (k) U (k) [ik −e(k)]2 ≤ j |ik −e(k)| 2 0 0 U (k)−ν (≤j ) (k) ν (>j ) (k) ν (>j ) (k) ik0 −e(k)−P (k) = ik0 −e(k)−P (k) ≤ 2 |ik0 −e(k)| ν (≤j ) (k)[P (k)−uˆ (k;0)] l min{|k0 |,1} [ik −e(k)−uˆ (k;0)][ik j−e(k)−P (k)] ≤ 4 j|ik −e(k)| 2 U (k) 0
j
0
0
rg converge to zero in L1 (Rd+1 ) as j → ∞. Consequently, 2Gj,2 (0, 0, ↑, 1), (x0 , x, ↑,
0 d d k −i(−k x +k·x) U (k) Q(k) 0 0 0) converges uniformly to dk as e + 2 d 2π (2π) ik0 −e(k)−P (k) [ik0 −e(k)] j tends to infinity. In the proof of Theorem I.4, we defined G2 ((0,0,↑),(x0 ,x,↑)) as the rg pointwise limit of the 2Gj,2 ((0,0,↑,1),(x0 ,x,↑,0))’s. Hence
U (k) Q(k) dk0 d d k −i(−k0 x0 +k·x) . e + G2 (0, 0, ↑), (x0 , x, ↑) = 2 d 2π (2π) ik0 −e(k)−P (k) [ik −e(k)] 0
Write
G2 (0, 0, ↑), (x0 , x, ↑) = a(x0 , x) + b(x0 , x) + c(x0 , x)
with
dk0 d d k 2π (2π)d
e−i(−k0 x0 +k·x)
U (k) i(1−w(k))k0 −e(k) ,
dk0 d d k 2π (2π)d
e−i(−k0 x0 +k·x)
U (k)[P (k)−iw(k)k0 ] [i(1−w(k))k0 −e(k)][ik0 −e(k)−P (k)] ,
c(x0 , x) =
dk0 d d k 2π (2π)d
e−i(−k0 x0 +k·x)
Q(k) , [ik0 −e(k)]2
w(k) =
1 ∂P i ∂k0 (0, k).
a(x0 , x) = b(x0 , x) =
By repeated use of Lemma XII.2, 3 Q(k) min{|ik0 −e(k)| 2 ,1} ˜ U (k) ∈ L2 (Rd+1 ), [ik −e(k)]2 ≤ λ1−3υ 0 |ik0 −e(k)|2 0 U (k)[P (k)−iw(k)k0 ] 1−2υ U (k) min{|k0 |3/2 ,|k0 |} ∈ L2 (Rd+1 ), [i(1−w(k))k0 −e(k)][ik0 −e(k)−P (k)] ≤ 8λ0 |ik −e(k)|2 0
so the Fourier transform of b(x0 , x) + c(x0 , x) exists and equals U (k)[P (k)−iw(k)k0 ] [i(1−w(k))k0 −e(k)][ik0 −e(k)−P (k)]
+
Q(k) [ik0 −e(k)]2
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Observe that a(x0 , x) = where
1 1−w(k)
dd k (2π)d
e−ik·x U (k)ex0 e(k)/(1−w(k)) χ (k, x0 ),
if e(k) < 0 and x0 ≥ 0 1 χ (k, x0 ) = −1 if e(k) > 0 and x0 < 0 . 0 otherwise
1 As 1−w(k) U (k)ex0 e(k)/(1−w(k)) is in L2 (Rd ), the spatial Fourier transform d d x eik·x 1 a(x0 , x) exists and equals 1−w(k) U (k)ex0 e(k)/(1−w(k)) χ (k, x0 ). A direct computation 1 dx0 e−ik0 x0 U (k)ex0 e(k)/(1−w(k)) now shows that the temporal Fourier transform 1−w(k) U (k) χ(k, x0 ) exists and equals i(1−w(k))k −e(k) . Thus the Fourier transform of G2 (0, 0, ↑ 0 U (k) Q(k) ), (x0 , x, ↑) exists and equals ik0 −e(k)−P (k) + [ik0 −e(k)]2 , which, by Lemma XII.2, is continuous except when ik0 − e(k) = 0. Lemma XII.4. Let S(k0 , k) be a function that obeys • S(k0 , k) and
∂S ∂k0 (k0 , k)
are continuous in (k0 , k) and there are C, ε > 0 such that ∂S (k0 , k) − ∂S (0, k) ≤ C|k0 |ε . ∂k0 ∂k0
∂S • S(0, k) and 1i ∂k (0, k). are real. ∂S 0 • S(k0 , k), ∂k0 (k0 , k) ≤ 21 and S(0, k) ≤ 21 |e(k)|.
Define
N (k, τ ) =
dk0 eik0 τ 2π ik0 −e(k)−S(k0 ,k)
N (k) = lim N (k, τ ). τ →0+
Then N(k) is continuous except on the Fermi surface F . If k¯ ∈ F , then lim N (k) and k→k¯ e(k)>0
lim N (k) exist and obey
k→k¯ e(k)<0
lim N (k) − lim N (k) = 1 −
k→k¯ e(k)<0
k→k¯ e(k)>0
1 ∂S ¯ −1 . i ∂k0 (0, k)
Proof. Define A(k) = 1 −
1 ∂S i ∂k0 (0, k),
E(k) = e(k) + S(0, k), R(k0 , k) = S(k0 , k) − S(0, k) −
∂S ∂k0 (0, k) k0 ,
and observe that ∂S ik0 − e(k) − S(k0 , k) = ik0 − ∂k (0, k) k0 − e(k) − S(0, k) − R(k0 , k) 0 = iA(k)k0 − E(k) − R(k0 , k).
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Hence, for any η > 0, N(k, τ ) = I1 (k, τ ) + I2 (k, τ ) + I3 (k, τ ) − I3 (k, τ ) + I4 (k, τ ) with I1 (k, τ ) = I2 (k, τ ) = I3 (k, τ ) = I3 (k, τ ) =
|k0 |<η
R
I4 (k, τ ) =
|k0 |<η
dk0 eik0 τ 2π iA(k)k0 −E(k) , dk0 2π
eik0 τ iA(k)k0 −E(k)−R(k0 ,k)
−
eik0 τ iA(k)k0 −E(k)
,
dk0 eik0 τ 2π ik0 −e(k) ,
|k0 |<η |k0 |≥η
dk0 eik0 τ 2π ik0 −e(k) , dk0 2π
eik0 τ ik0 −e(k)−S(k0 ,k)
−
eik0 τ ik0 −e(k)
.
We shall later fix some small η. Control of I1 . Let > 0. On the set D = (τ, k) τ ∈ R, |e(k)| > , the integrand eik0 τ iA(k)k0 −E(k)
is continuous in (τ, k), for each fixed k0 , and is uniformly bounded by 2 . Hence, by the Lebesgue dominated convergence theorem, I1 (τ, k) is continuous on D and obeys dk0 1 lim I1 (τ, k) = 2π iA(k)k0 −E(k) τ →0 |k0 |<η dk0 iA(k)k0 +E(k) =− 2π A(k)2 k 2 +E(k)2 0 |k0 |<η E(k) dk0 =− 2π A(k)2 k 2 +E(k)2 0 |k0 |<η dk E(k) = − sgnA(k) 2π0 21 A(k) k0 +1 |k0 |<η E(k) e(k) A(k) −1 = − sgn . η E(k) πA(k) tan
Since A(k) = lim tan−1 η E(k)
e(k)→0 e(k) =0
π 2,
we have ¯ lim lim I1 (τ, k) − lim lim I1 (τ, k) = 1/A(k).
k→k¯ e(k)<0
τ →0
k→k¯ e(k)>0
τ →0
(XII.5)
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Control of I2 . Since |iA(k)k0 − E(k)| ≥ 21 |k0 | and, for some k˜0 between 0 and k0 , R(k0 , k) = k0 ∂S (k˜0 , k) −
∂S ∂k0 (0, k)
∂k0
≤ C|k0 |1+ε
and iA(k)k0 − E(k) − R(k0 , k) ≥ A(k)k0 − Im R(k0 , k) = A(k)k0 − Im [R(k0 , k) − R(0, k)] ∂R ˜ = k0 A(k) − Im ∂k (k0 , k) 0 ∂S = k0 A(k) − Im ∂k (k˜0 , k) 0 ∂S − ∂k (0, k) ≥ 41 |k0 |. 0 If η is small enough, the integrand
eik0 τ iA(k)k0 −E(k)−R(k0 ,k)
−
eik0 τ iA(k)k0 −E(k)
=
eik0 τ R(k0 ,k) [iA(k)k0 −E(k)][iA(k)k0 −E(k)−R(k0 ,k)]
is uniformly bounded in magnitude, on the domain of integration, by the integrable function |k 8C 1−ε . Since the integrand is, for each fixed k0 = 0, continuous in (τ, k), 0| the Lebesgue dominated convergence theorem implies that I2 (τ, k) is continuous (τ, k) and, in particular, lim lim I2 (τ, k) − lim lim I2 (τ, k) = 0.
k→k¯ e(k)<0
τ →0
k→k¯ e(k)>0
τ →0
(XII.6)
Control of I3 and I3 . By residues, for all τ > 0 and e(k) = 0, 0 I3 (τ, k) = e(k)τ e
if e(k) > 0 . if e(k) < 0
Hence I3 (τ, k) is continuous in (τ, k) for all k with e(k) = 0, and obeys lim lim I3 (τ, k) − lim lim I3 (τ, k) = 1.
k→k¯ e(k)<0
τ →0+
k→k¯ e(k)>0
τ →0+
(XII.7)
As in the “Control of I1 ”, but with A(k) = 1 and E(k) = e(k), I3 (τ, k) is continuous in (τ, k) for all k with e(k) = 0, and obeys lim lim I3 (τ, k) − lim lim I3 (τ, k) = 1.
k→k¯ e(k)<0
τ →0
k→k¯ e(k)>0
τ →0
(XII.8)
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J. Feldman, H. Knörrer, E. Trubowitz
Control of I4 . Since S(k0 , k) ≤ 21 , |ik0 − e(k)| ≥ |k0 | and, for some k˜0 between 0 and k0 , ik0 − e(k) − S(k0 , k) ≥ k0 − Im S(k0 , k) = k0 − Im [S(k0 , k) − S(0, k)] = k0 1 − Im ∂S (k˜0 , k) ≥ 1 |k0 |, 2
∂k0
the integrand
eik0 τ ik0 −e(k)−S(k0 ,k)
−
eik0 τ ik0 −e(k)
=
eik0 τ S(k0 ,k) [ik0 −e(k)][ik0 −e(k)−S(k0 ,k)]
is uniformly bounded in magnitude, on the domain of integration, by the integrable function k12 . Since the integrand is, for each fixed k0 , continuous in (τ, k), the Lebes0
gue dominated convergence theorem implies that I4 (τ, k) is continuous (τ, k) and, in particular, lim lim I4 (τ, k) − lim lim I4 (τ, k) = 0.
k→k¯ e(k)<0
τ →0
k→k¯ e(k)>0
(XII.9)
τ →0
Combining (XII.5–XII.9) gives the desired jump.
Proof of Theorem I.5 from Theorem XII.1. Define U (k) dk0 ik0 τ N1 (k, τ ) = lim N1 (k, τ ), 2π e ik0 −e(k)−P (k) N1 (k) = τ →0+ Q(k) dk0 ik0 τ N2 (k, τ ) = N2 (k) = lim N2 (k, τ ). 2π e [ik −e(k)]2 τ →0+
0
Then, by Lemma XII.3, n(k) = N1 (k) + N2 (k). By Lemma XII.2.ii, the integrand eik0 τ [ik Q(k) 2 is continuous in τ and k, except pos0 −e(k)] sibly when ik0 − e(k) = 0, and is uniformly bounded by
3 ik0 τ Q(k) 1−3υ min{|ik0 −e(k)| 2 ,1} 1−3υ 1 √1 ≤ λ ∈ L1 (R). ≤ λ min , e 2 2 2 0 0 [ik −e(k)] |ik −e(k)| |k | k 0
0
0
0
Hence, by the Lebesgue dominated convergence theorem, N2 (k, τ ) is continuous. Hence, so is N2 (k). In Lemma XII.2, parts (i) and (iii), we showed that P (k) satisfies the conditions imposed on the function S(k) in Lemma XII.4. So, by Lemma XII.4, with S(k) replaced by P (k), N1 (k) is continuous except on F and if k¯ ∈ F , then lim N1 (k) and k→k¯ e(k)>0
lim N1 (k) exist and obey
k→k¯ e(k)<0
lim N1 (k) − lim N1 (k) = 1 −
k→k¯ e(k)<0
k→k¯ e(k)>0
1 ∂P ¯ −1 . i ∂k0 (0, k)
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123
Since N2 (k) is continuous, n(k) is continuous except on F and if k¯ ∈ F , then lim n(k) k→k¯ e(k)>0
and lim n(k) exist and obey k→k¯ e(k)<0
lim n(k) − lim n(k) = 1 −
k→k¯ e(k)<0
k→k¯ e(k)>0
1 ∂P ¯ −1 . i ∂k0 (0, k)
The remaining conclusions of Theorem I.5 were proven in Lemma XII.3.
If k is such that U (k) = 1, then U (k) ik0 −e(k)−P (k)
+
Q(k) [ik0 −e(k)]2
=
1 ik0 −e(k)−(k)
with (k) =
(k) P (k) + Q(k) − Q(k) ik0P−e(k)
1+
Q(k) ik0 −e(k)
−
P (k) Q(k) ik0 −e(k) ik0 −e(k)
.
(k) is usually called the proper self–energy. For the sake of completeness, we summarize the properties of (k) proven in this paper and its relationship to the function P (k) used above. < ℵ < ℵ < 23 . Then 3 (k) − P (k) ≤ λ1−4υ min{ik0 − e(k) 2 , 1 , 0 (k), ∂ (k) ≤ λ1−4υ , 0 ∂k0 ∂ 1 1−2υ ∂ (k) − (k ), (k) − |k − k |ℵ ≤ λ1−4υ |k − k | 2 . 0 ∂k0 ∂k0 (k ) ≤ const λ0
Lemma XII.5. Let
1 2
Proof. Writing E(k) = ik0 − e(k) and differentiating (k) =
E(k)2 P (k)+E(k)2 Q(k)−E(k)P (k)Q(k) E(k)2 +E(k)Q(k)−P (k)Q(k)
with respect to k0 yields ∂ ∂k0 (k)
=
∂Q ∂Q ∂P ∂P + 2iEQ + E 2 ∂k − iP Q − E ∂k Q − EP ∂k 2iEP + E 2 ∂k 0 0 0 0
E(k)2 + E(k)Q(k) − P (k)Q(k) −
∂Q [E 2 P + E 2 Q − EP Q][2iE + iQ + E ∂k − 0
∂Q ∂P ∂k0 Q − P ∂k0 ] . [E(k)2 + E(k)Q(k) − P (k)Q(k)]2
Implementing the cancellation (of terms of the form E n P ) 2iEP E 2 +EQ−P Q
−
[E 2 P +E 2 Q−EQP ]2iE [E 2 +EQ−P Q]2
= − 2iE
3 Q+2iEP 2 Q−4iE 2 P Q
[E 2 +EQ−P Q]2
we have ∂ ∂k0 (k)
−
=
∂Q ∂Q ∂P ∂P + 2iEQ + E 2 ∂k − iP Q − E ∂k Q − EP ∂k E 2 ∂k 0 0 0 0
E(k)2 + E(k)Q(k) − P (k)Q(k)
∂Q ∂Q ∂P − ∂k Q−P ∂k ]+2iE 3 Q+2iEP 2 Q−4iE 2 P Q [E 2 P +E 2 Q−EP Q][iQ+E ∂k 0 0 0
[E(k)2 + E(k)Q(k) − P (k)Q(k)]2
.
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∂Q ∂P Observe that, if we weight E, P and Q with degree one and we weight ∂k and ∂k with 0 0 degree zero, then both the numerator and denominator of the first term are homogeneous of degree two and both the numerator and denominator of the second term are homogeneous of degree four. This sort of homogeneity information can be used in place of the ∂ explicit formula for ∂k (k). 0 Define, for m = 0, 1, 2,
˜ (m) (k) = Q
(ik0 )m Q(k) [ik0 −e(k)]m+1
=
˜ (m) (k) = Q 0
(ik0 )m ∂Q [ik0 −e(k)]m ∂k0
=
P (k) ik0
=
P˜ (k) =
with q˜ (j,,m) (k) = (j
+ 2)nd
(ik0 )m q (j,) (k) [ik0 −e(k)]m+1
2≤j ≤
(ik0 )m q (j,) (k) [ik0 −e(k)]m+1
=
∂q (j,) (ik0 )m [ik0 −e(k)]m ∂k0
=
2≤j ≤ ∞ pˇ (j ) (k) ik0 j =2
(j,,m)
and q˜0
q˜ (j,,m) (k),
2≤j ≤
=
(j,,m)
q˜0
2≤j ≤ ∞ (j )
p˜
(k),
(k)
j =2
(k) =
∂q (j,) (ik0 )m [ik0 −e(k)]m ∂k0
vanishing on the
neighbourhood and obeying
(j,,m) l sup Dδ q˜ (j,,m) (k), sup Dδ q˜0 (k) ≤ const λ1−2υ M ℵ (−j ) M j M j δ0 M |δδ | 0 M k
k
≤ const λ1−2υ lj M −(1+ℵ−ℵ )(−j ) M |δ| 0 (XII.10) for |δ| ≤ 1 and p˜ (j ) (k) =
pˇ (j ) (k) ik0
= −i
1 0
∂ pˇ (j ) ∂ k0 (tk0 , k) dt
obeying
j |δ| sup Dδ p˜ (j ) (k) ≤ const λ1−υ 0 lj M k
(i,)
for all |δ| ≤ 1. The right-hand side of (XII.10) coincides with the bounds on ∂q∂k0 that were used in the derivation of the second bounds of (XII.3) and (XII.4). Hence, for m = 0, 1, 2, ˜ (m) (k)|, |Q ˜ (k)| ≤ const λ1−2υ min{|ik0 − e(k)|ℵ , 1} |Q 0 0 (m)
|P˜ (k)| ≤ λ1−3υ 0 (XII.11)
and (m) (m) Q ˜ (k) − Q ˜ (m) (k ), Q ˜ (k) − Q ˜ (m) (k ), P˜ (k) − P˜ (k ) ≤ const λ1−2υ |k − k |ℵ. 0 0 0 (XII.12)
Two Dimensional Fermi Liquid. 3: Fermi Surface
125
The bound on P˜ is a direct application of Lemma C.1 with α = ℵ and β = 1 − ℵ. In terms of these new functions, (k) = (k) − P (k) = ∂ ∂k0 (k)
=
˜ (0) (k) P (k) + Q(k) − P (k)Q ˜ (0) (k) − P˜ (k)Q ˜ (1) (k) 1+Q
˜ (0) (k) + P (k)P˜ (k)Q ˜ (1) (k) Q(k) − 2P (k)Q ˜ (0) (k) − P˜ (k)Q ˜ (1) (k) 1+Q ∂P ∂k0
−
˜ (0) + Q ˜ (0) − i P˜ Q ˜ (1) − + 2i Q 0
∂P ˜ (0) ∂k0 Q ˜ (0) − P˜ Q ˜ (1) 1+Q
˜ (1) − P˜ Q 0
˜ (1) − ˜ (1) + Q P˜ [i Q 0
∂P ˜ (1) ˜ (2) ] − P˜ Q 0 ∂k0 Q ˜ (0) − P˜ Q ˜ (1) ]2 [1 + Q
˜ (0) − P˜ Q ˜ (1) ][i Q ˜ (0) + Q ˜ (0) − ∂P Q ˜ (1) ] ˜ (0) − P˜ Q [Q 0 0 ∂k0 ˜ (0) + 2i P˜ 2 Q ˜ (2) − 4i P˜ Q ˜ (1) +2i Q − . ˜ (0) − P˜ Q ˜ (1) ]2 [1 + Q (XII.13) ∂ ∂P ˜ (0) , Q ˜ (1) , (k) are rational functions in the variables P , P˜ , ∂k , Q, Q Both (k) and ∂k 0 0 ˜ (0) , Q ˜ (1) and Q ˜ (2) . As all of these variables are bounded in magnitude by λ1−3υ , ˜ (2) , Q Q 0 0 0 0 the numerators contain no constant terms and the denominators are bounded away from zero, (k), ∂ (k) ≤ const λ1−3υ ≤ λ1−4υ . 0 0 ∂k0
Each term in the numerator of (k) − P (k) contains either a factor of Q(k) or a factor ˜ (m) (k) so that, using the bound on Q(k) in (XII.4), of P (k)Q (k) − P (k) ≤ const λ1−2υ min{|ik0 − e(k)|1+ℵ , 1} ≤ λ1−4υ min{|ik0 − e(k)| 23 , 1}. 0 0 Applying a Taylor expansion with degree one remainder and with expansion point x0 = ˜ (2) (k) and evaluation point x = P (k ), y = P˜ (k ), · · · , P (k), y0 = P˜ (k), · · · , z0 = Q 0 ˜ (2) (k ), z=Q 0 (k) − (k )
˜ (2) (k) − Q ˜ (2) (k )| ≤ const max |P (k) − P (k )|, |P˜ (k) − P˜ (k )|, · · · , |Q 0 0
|k − k |ℵ ≤ const λ1−2υ 0 ∂ (k) − ∂ (k ) ∂k0 ∂k0
˜ (2) (k) − Q ˜ (2) (k )| ≤ const max |P (k) − P (k )|, |P˜ (k) − P˜ (k )|, · · · , |Q 0 0
|k − k |ℵ . ≤ const λ1−2υ 0
Remark XII.6. We have proven, in Lemma XII.5, only very limited regularity properties of the proper self–energy (k). It is proven in [FST3], that, to all finite orders of perturbation theory, the proper self–energy is C 2−ε for every ε > 0. While we prove
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J. Feldman, H. Knörrer, E. Trubowitz
the convergence of perturbation theory in the course of proving Theorems I.4 and I.5, the convergence proof does not give sufficient control over derivatives to allow us to rigorously conclude that is C 2−ε . The amputation used in the input and output data of §XV multiplies all external legs by A(k) = ik0 − e(k). The following lemma bounds the functions used to change to the amputation in which one multiplies by ik0 − e(k) − (k), the inverse of the physical two–point function, instead. Lemma XII.7. (k) . Then i) Let A1 (k) = ν (≤i) (k) ik0 −e(k)−P ik0 −e(k)
A1 (k)˜ ≤ const ci . ii) Let A2 (k) =
ik0 −e(k)−(k) ik0 −e(k)−P (k) .
Then
A2 (k) ≤ 2
A2 (k) − A2 (k ) ≤ λ1−3υ |k − k | 21 . 0
Proof. Since pˇ (j ) (k) is supported on the j th neighbourhood, (k) A1 (k) = ν (≤i) (k) − ν (≤i) (k) ik0P−e(k)
= ν (≤i) (k) − Since cj ≤ 1 + const
Mj c Mi i
ν (≤i) (k) ik0 −e(k)
i+1 j =2
pˇ (j ) (k).
for all j ≤ i + 1,
i+1 1−υ lj (j ) ˜ i+1 pˇ (k) ≤ 2λ0 M j cj j =2
j =2
1+ ≤ const λ1−υ 0
1 c Mi i
.
(≤i) (k) ˜ ≤ const M i ci , so that Also ν (≤i) (k)˜≤ const ci and ikν 0 −e(k) δ sup ∂∂k δ ν (≤i) (k) ≤ const M i|δ| k
and δ sup ∂∂k δ k
δ i+1 ∂ (j ) ν (≤i) (k) sup p ˇ (k) ik0 −e(k) ∂k δ k j =2
i(|δ |−1) ≤ const M i(|δ|+1) λ1−υ 0 M
i|δ+δ | ≤ const λ1−υ 0 M
provided δ = 0. This handles all contributions to A1 (k)˜ except those for which no derivatives act on P (k). For those contributions, we write P (k) = ik0 P˜ (k) and use that |P˜ (k)| ≤ λ1−3υ so that 0 δ (≤i) (≤i) (k) δ ∂ ν (k) sup P (k) ∂∂k δ ikν 0 −e(k) sup M i|δ| . ≤ λ1−3υ k0 ∂k δ ik0 −e(k) ≤ const λ1−3υ 0 0 k
k
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127
Now we move on to A2 (k) = 1 −
(k)−P (k) ik0 −e(k)−P (k) .
In the notation of the proof of Lemma XII.5, (k)−P (k) ik0 −e(k)−P (k)
A4 (k) =
= = =
where K˜ 0 (k) =
ik0 E(k)
and A3 (k) = ˜ (m) and P˜ (XII.11) and (XII.12) on Q
˜ (0) (k)+P (k)P˜ (k)Q ˜ (1) (k) Q(k)−2P (k)Q ˜ (0) (k)−P˜ (k)Q ˜ (1) (k)][E(k)−P (k)] [1+Q ˜ (1) (k)+P˜ (k)2 Q ˜ (0) (k)−2P˜ (k)Q ˜ (2) (k) Q ˜ (0) (k)−P˜ (k)Q ˜ (1) (k)][1−K˜ 0 (k)P˜ (k)] [1+Q A3 (k) , 1−K˜ 0 (k)P˜ (k)
˜ (1) (k)+P˜ (k)2 Q ˜ (0) (k)−2P˜ (k)Q ˜ (2) (k) Q . ˜ (0) (k)−P˜ (k)Q ˜ (1) (k) 1+Q
By the bounds
|A3 (k)| ≤ const λ1−2υ min{|ik0 − e(k)|ℵ , 1} 0 A3 (k) − A3 (k ) ≤ const λ1−2υ |k − k |ℵ . 0
(XII.14)
To bound A3 (k)−A3 (k ), we used a Taylor expansion argument as in Lemma XII.5. M 1 1 Let be such that M1 ≤ |k − k | ≤ M . If either |E(k)| ≤ M or |E(k )| ≤ M , then const both |E(k)|, |E(k )| ≤ M and, by (XII.11) and (XII.14), 1 |A4 (k)|, |A4 (k )| ≤ const λ1−2υ ≤ const λ1−2υ |k − k |ℵ . 0 0 M ℵ
If both |E(k)| ≥
1 M
and |E(k )| ≥
1 , M
then
max{|E(k)|, |E(k )|} ≤ const min{|E(k)|, |E(k )|} and K˜ 0 (k) − K˜ 0 (k ) ≤ min 2, const
|k−k | min{|E(k)|,|E(k )|}
≤ const
|k−k | max{|E(k)|,|E(k )|}
Hence, by (XII.14), (XII.11) and (XII.12) A4 (k) − A4 (k ) K˜ 0 (k)P˜ (k)−K˜ 0 (k )P˜ (k ) 3 (k ) = A3 (k)−A + A (k) 3 1−K˜ 0 (k )P˜ (k ) [1−K˜ 0 (k)P˜ (k)][1−K˜ 0 (k )P˜ (k )] ≤ const λ1−2υ |k − k |ℵ + const λ1−2υ |E(k)|ℵ K˜ 0 (k) − K˜ 0 (k )P˜ (k) 0 0 +const λ1−2υ K˜ 0 (k )P˜ (k) − P˜ (k ) ≤
0 1−2υ const λ0 |k
− k |ℵ + const λ1−2υ |k − k |ℵ . 0
The desired bounds on A2 follow.
ℵ
.
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XIII. Momentum Space Norms Theorem XII.1 is concerned with the Fourier transforms of the two point functions rg rg Gj,2 (η1 , η2 ). In the proof of Theorem VIII.5, Gj,2 is built up recursively from contriˇ rg , we butions to effective interactions with two, one or no external fields. To control G j,2 control the partial Fourier transforms, with respect to the external variables, of these contributions. In this chapter, we provide the notation to do this. In particular, we specify norms for functions that depend on “external” momentum and “internal” sectorized position variables. These norms coincide with those of §VI when there are no external legs. When there are external legs, the new norms sup over the momenta of external as well as internal arguments and also allow momentum space derivatives to act on external arguments. We do this because, in this part, we are interested in the momentum space regularity properties of the two and four point functions. This contrasts with the norms of §VI, which were used to prove the existence of all n–point functions but which provided very limited momentum space regularity. Definition XIII.1 (Partial Fourier transforms). Let f (η1 ,··· ,ηm ; ξ1 ,··· ,ξn ) be a translation invariant function on B m ×B n . If n ≥ 1, the partial Fourier transform f ∼ is defined, using the notation of Definition VI.1, by m ∼ E+ (ηˇ i , ηi ) dηi f (η1 ,··· ,ηm ; ξ1 ,··· ,ξn ) f (ηˇ 1 ,··· ,ηˇ m ; ξ1 ,··· ,ξn ) = i=1
or, equivalently, by f (η1 ,··· ,ηm ; ξ1 ,··· ,ξn ) =
m i=1
d ηˇ i ∼ E− (ηˇ i , ηi ) (2π) d+1 f (ηˇ 1 ,··· ,ηˇ m ; ξ1 ,··· ,ξn ).
If n = 0, we set f ∼ = fˇ. ˇ Definition XIII.2. If φ(η) is a Grassmann field, we set for, ηˇ = (k, σ, a) ∈ B, a ˇ η) φ( ˇ = dη E− (η, ˇ η)φ(η) = dx0 d d x e−(−1) ı− φ(x0 , x, σ, a). Remark XIII.3. A translation invariant sectorized Grassmann function W can be uniquely written in the form W(φ, ψ) = dη1 ···dηm dξ1 ···dξn Wm,n (η1 ,··· ,ηm ; ξ1 ,··· ,ξn ) φ(η1 ) · · · φ(ηm ) m,n
× ψ(ξ1 ) · · · ψ(ξn )
with Wm,n antisymmetric separately in the η and in the ξ variables. Then, under the Fourier transform Definitions XIII.1 and XIII.2, W(φ, ψ) m = m
+
i=1
m,n n≥1
d ηˇ i (2π )d+1 m
i=1
∼ ˇ ηˇ 1 ) · · · φ( ˇ ηˇ m ) Wm,0 (ηˇ 1 ,··· ,ηˇ m ) (2π)d+1 δ(ηˇ 1 +···+ηˇ m ) φ(
d ηˇ i (2π)d+1
n i=1
dξi
∼ ˇ ηˇ 1 ) · · · φ( ˇ ηˇ m )ψ(ξ1 ) · · · ψ(ξn ). Wm,n (ηˇ 1 ,··· ,ηˇ m ; ξ1 ,··· ,ξn ) φ(
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Definition XIII.4. A function f (ηˇ 1 ,··· ,ηˇ m ; ξ1 ,··· ,ξn ) on Bˇ m × B n , with n ≥ 1, is said to be translation invariant if f (ηˇ 1 ,··· ,ηˇ m ; ξ1 +t,··· ,ξn +t ) = eı ηˇ 1 +···+ηˇ m ,t − f (ηˇ 1 ,··· ,ηˇ m ; ξ1 ,··· ,ξn ) for all t ∈ R × Rd . A distribution f (ηˇ 1 ,··· ,ηˇ m ) on Bˇ m is said to be translation invariant if it is supported on ηˇ 1 + · · · + ηˇ m = 0. Recall, from just before Definition VI.1, that if ˇ then ηˇ + ηˇ = (−1)a k + (−1)a k ∈ R × Rd . ηˇ = (k, σ, a), ηˇ = (k , σ , a ) ∈ B, The partial Fourier transform of a translation invariant function on B m × B n is a translation invariant function on Bˇ m × B n . Definition XIII.5 (Differential–decay operators). Let m, n ≥ 0. If n ≥ 1, let f be a function on Bˇ m × B n . If n = 0, let f be a function on Bˇm =
(ηˇ 1 , · · · , ηˇ m ) ∈ Bˇ m ηˇ 1 + · · · + ηˇ m = 0 .
i) For 1 ≤ j ≤ m and a multiindex δ set Dδj f ((p1 ,τ1 ,b1 ),··· ,(pm ,τm ,bm ); ξ1 ,··· ,ξn ) d δ δ δ 0 = ı(−1)bj 0 − ı(−1)bj ∂ δ0
∂pj,0
=1
× f ((p1 ,τ1 ,b1 ),··· ,(pm ,τm ,bm ); ξ1 ,··· ,ξn ).
∂ δ1 δ1 ∂pj,1
δd δd ∂pj,d
··· ∂
ii) Let 1 ≤ i = j ≤ m + n and δ a multiindex. Set Dδi;j f Dδi;j f Dδi;j f Dδi;j f
= (Di − Dj )δ f = (Di − ξj −m )δ f = (ξi − Dj −m )δ f δ = (ξi−m − ξj −m )δ f = Di−m,j −m f
if if if if
1 ≤ i < j ≤ m, 1 ≤ i ≤ m, m + 1 ≤ j ≤ m + n, m + 1 ≤ i ≤ m + n, 1 ≤ j ≤ m, m + 1 ≤ i < j ≤ m + n.
The decay operator Di,j was defined in Definition V.1. iii) A differential–decay operator (dd–operator) of type (m, n), with m + n ≥ 2, is an operator D of the form (1)
(r)
D = Dδi1 ;j1 · · · Dδir ;jr with 1 ≤ i = j ≤ m + n for all 1 ≤ ≤ r. A dd–operator of type (1, 0) is an oper(1) (r) ator of the form D = Dδ1 · · · Dδ1 . The total order of D is δ(D) = δ (1) + · · · + δ (r) . Remark XIII.6. i) For a translation invariant function ϕ on B m × B n , Di;j (ϕ ∼ ) = (Di,j ϕ)∼ . In particular, Leibniz’s rule also applies for differential–decay operators.
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ii) Let f be a translation invariant function on Bˇ m × B. Then, for ξ = (x0 , x, σ, a) ∈ B, f (ηˇ 1 , · · · , ηˇ m ; ξ ) = eı ηˇ 1 +···+ηˇ m ,(x0 ,x)− f (ηˇ 1 , · · · , ηˇ m ; (0, σ, a)). Consequently, for 1 ≤ i ≤ m and a multiindex δ Dδi;m+1 f (ηˇ 1 ,··· ,ηˇ m ;ξ ) = eı ηˇ 1 +···+ηˇ m ,(x0 ,x)− Dδi f (ηˇ 1 , · · · , ηˇ m ; (0, σ, a)). Definition XIII.7. For a function f on Bˇm , set f ˜=
δ∈N0 ×N20
1 δ!
max
D dd−operator with δ(D)=δ
sup ηˇ 1 ,··· ,ηˇ m ∈Bˇ
Df (ηˇ 1 ,··· ,ηˇ m ) t δ .
Let f be a function on Bˇ m × B n with n ≥ 1. Set f ˜=
δ∈N0 ×N20
1 δ!
max
D dd−operator with δ(D)=δ
sup ηˇ 1 ,··· ,ηˇ m ∈Bˇ
D f (ηˇ 1 ,··· ,ηˇ m ;ξ1 ,··· ,ξn ) tδ. 1,∞
The norm ||| · |||1,∞ of Definition V.3 refers to the variables ξ1 ,··· ,ξn . That is, D f (ηˇ 1 ,··· ,ηˇ m ;ξ1 ,··· ,ξn ) = max 1,∞
sup
1≤j0 ≤n ξj ∈B 0
dξj |D f (ηˇ 1 ,··· ,ηˇ m ;ξ1 ,··· ,ξn )|.
j =1,··· ,n j =j0
Remark XIII.8. In the case m = 0 the norm · 1,∞ of Definition V.3 and the norm · ˜of Definition XIII.7 agree. Definition XIII.9. We amputate a Grassmann function by applying the Fourier transˆ in the sense of Notation V.4, of A(k) = ik0 − e(k) to its external arguments. form A, Precisely, if W(φ, ψ) is a Grassmann function, then ˆ ψ), W a (φ, ψ) = W(Aφ, where ˆ (Aφ)(ξ )=
ˆ ξ )φ(ξ ). dξ A(ξ,
Remark XIII.10. If C(ξ, ξ ) is the covariance associated to C(k) in the sense of (III.1) and (III.2) and J is the particle hole swap operator of (III.3), then, by parts (i) and (ii) of Lemma IX.5 of [FKTo2], ˆ , ξ ) = E(ξ, ˆ dηdη C(ξ, η)J (η, η )A(η ξ ), where Eˆ is the Fourier transform of ik0 − e(k) C(k) in the sense of Notation V.4. Integrating out the first few scales is controlled much as in Theorem V.8.
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Theorem XIII.11. There are (M and j0 –dependent) constants λ¯ , µ and β0 such that, 1 for all λ0 < λ¯ and β0 ≤ β ≤ υ/5 , the following holds: Let X ∈ Nd+1 with X0 < µ, δe ∈ E with δ e ˆ 1,∞ ≤ X and V(ψ) =
λ0
B4
dξ1 ···dξ4 V (ξ1 ,··· ,ξ4 ) ψ(ξ1 ) · · · ψ(ξ1 )
with an antisymmetric function V fulfilling V 1,∞ ≤ λ0 e0 (X). Write ˜
(≤j ) C−δe0
(≤j ) V (ψ) (φ, ψ) = V(ψ) + 21 φJ C−δe0 J φ +
m,n≥0 m+n even
Bm+n
dη1 ···dηm dξ1 ···dξn
× Wm,n (η1 ···ηm ,ξ1 ,··· ,ξn ; δe) φ (η1 ) · · · φ (ηm ) ψ (ξ1 ) · · · ψ(ξn ) with kernels Wm,n that are separately antisymmetric under permutations of their η and ξ arguments. Then a∼ β m+n ρ˜m;n Wm,n (δe)˜≤ β 4 λυ0 e0 (X) m+n≥2 m+n even
and m+n≥2 m+n even
a∼ β m+n ρ˜m;n dds Wm,n (δe + sδe )s=0 ˜≤ β 4 λυ0 e0 (X) δ eˆ 1,∞ ,
where ρ˜m;n =
mυ/7 λ0 (1−υ) max{m+n−2,2}/2 λ0
and υ was fixed in Definition V.6. υ/5
Proof. Apply Theorem X.12 of [FKTo2] with ρm,n = ρ˜m;n , ε = const β 4 λυ0 ≤ const λ0 υ/7 mυ/7 and ε = λ0 . Observe that, ρ˜m;n is λ0 times the ρm;n of Remark VIII.7.iii of [FKTo2]. So the hypotheses of Theorem X.12 concerning ρm;n are fulfilled. Choosing λ¯ small enough ensures that the hypotheses ε, ε ≤ ε0 of Theorem X.12 are satisfied. In the course of exhibiting control over amputated Green’s functions in momentum space, we must deal with kernels having external arguments in momentum space and internal arguments sectorized and in position space. Definition XIII.12 (Momentum Space Norms). Let p be a natural number and a sectorization. (i) For a function f on Bˇm we define ˜ ˜ = f if p = m − 1, m = 2, 4. . |f |p, 0 otherwise
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(ii) For a translation invariant function f on Bˇ m × (B × )n with n ≥ 1, we set ˜ |f |p, = 0 when p > m + n or p < m, and ˜ = |f |p,
sup
1 δ!
δ∈N0 ×N20 1≤i1 <···
×
max
D dd−operator with δ(D)=δ
Df (ηˇ 1 ,··· ,ηˇ m ;(ξ1 ,s1 ),··· ,(ξn ,sn )) tδ 1,∞
when m ≤ p ≤ m + n . The norm ||| · |||1,∞ of Definition V.3 refers to the variables ξ1 ,··· ,ξn . Remark XIII.13. In the case m = 0 the norm | · |p, of Definition VI.6 and the norm ˜ of Definition XIII.12 agree. | · |p, Definition XIII.14. Let m, n ≥ 0 and a sectorization. For n ≥ 1, denote by Fˇ m (n; ) the space of all translation invariant, complex valued functions f (ηˇ 1 ,··· ,ηˇ m ; (ξ1 ,s1 ),··· ,(ξn ,sn )) n on Bˇ m × B × whose Fourier transform fˇ(ηˇ 1 ,··· ,ηˇ m ; (ξˇ1 ,s1 ),··· ,(ξˇn ,sn )) vanishes unless ki ∈ s˜i for all 1 ≤ j ≤ n. Here, ξˇi = (ki , σi , ai ). Also, let Fˇ m (0; ) be the space of all momentum conserving, complex valued functions f (ηˇ 1 ,··· ,ηˇ m ) on Bˇ m . ˜ We now provide the analogue of Definition VI.6.ii for the | |–norms. Let j ≥ 2 and let j be the sectorization of scale j and length lj = M1ℵj fixed just before Definition VI.4. Definition XIII.15. i) For f ∈ Fˇ m (n; j ) set ˜ + |f |2, ˜ + 1 |f |3, ˜ |f |1, j j j lj 1 1 1 ˜ ˜ ˜ + | f | + | f | + |f |6, if m = 0 , 4,j 5,j j |f |j˜ = ρ˜m;n lj l2j l2j |f |1, ˜ + 12 |f |5, ˜ ˜ + 1 |f |3, if m = 0 j j j lj l j
˜ ,ρ˜ = ρ˜m;n|f |p, ˜ . |f |p, j j ii) An even sectorized Grassmann function w can be uniquely written in the form w(φ, ψ) = dη1 ···dηm dξ1 ···dξn wm,n (η1 ,··· ,ηm (ξ1 ,s1 ),··· ,(ξn ,sn )) m,n
× φ(η1 ) · · · φ(ηm ) ψ((ξ1 ,s1 )) · · · ψ((ξn ,sn ) ) with wm,n antisymmetric separately in the η and in the ξ variables. Set, for α > 0 and X ∈ Nd+1 , Nj∼ (w, α, X) = Mlj ej (X) 2j
m,n≥0
α m+n
lj B (m+n)/2 Mj
∼ ˜ |j , |wm,n
∼ is the partial Fourier transform of w where wm,n m,n of Definition XIII.1 and B = 4 max{4B3 , B4 } with B3 , B4 being the constants of Proposition XVI.8 of [FKTo3].
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Remark XIII.16. In particular, for the “pure φ” part of w, ∼ ∼ ˜ α2 B Nj (w(φ, 0), α, X) = ej (X) 1−9υ/7 |1,j + M j |w2,0 λ0
∼ ˜ 1−11υ/7 | w4,0 | 3,j λ α 4 B2
.
0
The sectorized version of Theorem XIII.11 is Theorem XIII.17. For K ∈ Kj0 , set u(K) = − Kext ∈ F0 (2, j0 ), j0
where Kext was defined in Definition E.3 of [FKTo4]. Then there exist constants α, ¯ λ¯ > 0 1 ¯ such that for all 0 ≤ λ0 < λ, α¯ < α < υ/10 and all λ0
V 1,∞ ≤ λ0 ej0 K1,j0
K ∈ Kj0
˜ the j0 –sectorized representative w(φ, ψ; K) of
(≤j )
Cu(K)0
(≤j ) V (ψ) (φ, ψ) − 21 φJ Cu(K)0 J φ
constructed in Theorem VI.12 may be chosen to obey Nj∼0 w a (K), α, K1,j0 ≤ const α 4 λυ0 ej0 K1,j0 , Nj∼0 dds w a (K + sK )s=0 , α, K1,j0 ≤ const α 4 λυ0 ej0 K1,j0 K 1,j0 for all K . Furthermore w a (φ, ψ, K)−w a (0, ψ, K) vanishes unless νˆ (≤j0 ) φ is nonzero. Proof. Write ˜
(≤j )
Cu(K)0
V (ψ) (φ, ψ) = V(ψ) +
(≤j0 ) 1 2 φJ Cu(K) J φ
+
m,n≥0 m+n even
Bm+n
dη1 ···dηm dξ1 ···dξn
× Wm,n (η1 ,··· ,ηm ,ξ1 ,··· ,ξn )φ (η1 ) · · · φ (ηm ) ψ (ξ1 ) · · · ψ(ξn ) and set, as in Theorem VI.12, wm,n
Wm,n j0 = W0,4 + V
j0
if(m, n) = (0, 4) if(m, n) = (0, 4)
using the sectorization f of Definition XIX.14 of [FKTo4], By Proposition XIX.15 of [FKTo4], l B (m+n)/2 a∼ 2j0 Nj∼0 w a (K), α, K1,j0 = Mlj ej0 K1,j0 α m+n Mj0j0 |wm,n|j˜0 0 m,n≥0 α 4 ≤ const cj0 ej0 K1,j0 V 1,∞ λ1−υ 0
a∼ ˜ . + (const α)m+n ρ˜m;n Wm,n m,n≥0
By hypothesis α4 λ1−υ 0
V 1,∞ ≤ α 4 λυ0 ej0 K1,j0 ,
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J. Feldman, H. Knörrer, E. Trubowitz
and by Theorem XIII.11, with δe = −u, ˇ X = const K1,j0 and β = const α, a∼ ˜ ≤ β 4 λυ0 e0 (X) ≤ const α 4 λυ0 ej0 K1,j0 . (const α)m+n ρ˜m;n Wm,n m,n≥0
Therefore, by Corollary A.5.ii of [FKTo1], 2 Nj∼0 w a (K), α, K1,j0 ≤ const α 4 λυ0 cj0 ej0 K1,j0 ≤ const α 4 λυ0 ej0 K1,j0 . The proof of the bound on Nj∼0 dds w a (K + sK )s=0 , α, K1,j0 is similar. By Lemma VII.3 of [FKTo2], ˜ (≤j0 ) V (ψ) (φ, ψ) − 1 φJ C (≤j0 ) J φ = (≤j0 ) V (ψ) (φ, ψ + C (≤j0 ) J φ). u(K) u(K) 2 Cu(K)
Since
(≤j )
Cu(K)0
Cu(K)
(≤j )
V (ψ) (φ, ψ) is independent of φ and since Cu(K)0 J φ vanishes unless
νˆ (≤j0 ) φ is nonzero, w a (φ, ψ, K) − w a (0, ψ, K) vanishes unless νˆ (≤j0 ) φ is nonzero. XIV. Ladders with External Momenta In the proof of Theorem VIII.5, ladders and iterated particle hole ladders needed special attention. Due to the “external improvement” of Lemma XII.19 of [FKTo3], we needed to consider only ladders all of whose “ends” correspond to ψ fields and are integrated out at a later scale. This is not the case here. We consider ladders some of whose “ends” correspond to ψ fields and have sectorized position space arguments (ξ, s) ∈ B × j , and some of whose ends correspond to φ fields and have momentum space arguments ˇ To do this, we extend the definitions and estimates of ladders and iterated particle ηˇ ∈ B. hole ladders from §VII. The following definition extends Definition VII.3. Definition XIV.1. Let be a sectorization. i) Define Bˇ = (k0 , k, σ ) k0 ∈ R, k ∈ R2 , σ ∈ {↑, ↓} and the disjoint unions Y = Bˇ ∪· (B × ), X = Bˇ ∪· (B × ).
ii) Let z ∈ X . Then we define its undirected part u(z) ∈ Y and its creation/annihilation index b(z) ∈ {0, 1} by (k, σ ) if z = (k, σ, b) ∈ Bˇ u(z) = (x, σ, s) if z = (x, σ, b, s) ∈ B × , b if z = (k, σ, b) ∈ Bˇ b(z) = b if z = (x, σ, b, s) ∈ B × .
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135
iii) Let z ∈ Y and b ∈ {0, 1}. Then we define ιb (z ) as the unique point z ∈ X with u(z) = z and b(z) = b. With this notation, Definition VII.4 and Lemma VII.5.ii about particle–particle and particle–hole reductions and values carry over almost verbatim. Definition XIV.2. i) Let f be a four legged kernel over X . When f is a rung, its particle–particle reduction is the four legged kernel over Y given by f pp (z1 ,z2 ,z3 ,z4 ) = f (ι0 (z1 ),ι0 (z2 ),ι1 (z3 ),ι1 (z4 )) =
1
3 f
2
4
and its particle–hole reduction is f ph (z1 ,z2 ,z3 ,z4 ) = f (ι0 (z1 ),ι1 (z2 ),ι1 (z3 ),ι0 (z4 )) =
1
3 .
f
2 When f is a bubble propagator, the corresponding reductions are
4
f (z1 ,z2 ,z3 ,z4 ) = f (ι1 (z1 ),ι1 (z2 ),ι0 (z3 ),ι0 (z4 )), f (z1 ,z2 ,z3 ,z4 ) = f (ι1 (z1 ),ι0 (z2 ),ι0 (z3 ),ι1 (z4 )).
pp ph
ii) Let f be a four legged kernel over Y . Its particle–particle value is the four legged kernel over X given by Vpp (f )(z1 ,z2 ,z3 ,z4 ) = δb(z1 ),0 δb(z2 ),0 δb(z3 ),1 δb(z4 ),1 f (u(z1 ),u(z2 ),u(z3 ),u(z4 ))
+δb(z1 ),1 δb(z2 ),1 δb(z3 ),0 δb(z4 ),0 f (u(z3 ),u(z4 ),u(z1 ),u(z2 )),
and its particle–hole value is Vph (f )(z1 ,z2 ,z3 ,z4 ) = δb(z1 ),0 δb(z2 ),1 δb(z3 ),1 δb(z4 ),0 f (u(z1 ),u(z2 ),u(z3 ),u(z4 ))
+δb(z1 ),1 δb(z2 ),0 δb(z3 ),0 δb(z4 ),1 f (u(z2 ),u(z1 ),u(z4 ),u(z3 )) −δb(z1 ),1 δb(z2 ),0 δb(z3 ),1 δb(z4 ),0 f (u(z2 ),u(z1 ),u(z3 ),u(z4 )) −δb(z1 ),0 δb(z2 ),1 δb(z3 ),0 δb(z4 ),1 f (u(z1 ),u(z2 ),u(z4 ),u(z3 )).
Lemma XIV.3. Let f be an antisymmetric, particle number preserving, four legged kernel over X . Then f = Vpp (f pp ) + Vph (f ph ). In §XIII we developed norms for functions on Bˇ m × (B × )n , since we usually write Grassmann monomials in a way such that all φ fields stand before all ψ fields. However the “ends” of ladders have a natural ordering, and φ and ψ fields may be arbitrarily distributed among them. Therefore we extend some of the notation of §XIII to this situation and repeat the detailed Definition of §XVI of [FKTo3]. Definition XIV.4. Set X0 = Bˇ and X1 = B × . Let i = (i1 , · · · , in ) ∈ {0, 1}n . i) The inclusions of Xij , j = 1, · · · , n, in X induce an inclusion of Xi1 × · · · × Xin in Xn . We identify Xi1 × · · · × Xin with its image in Xn .
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ii) Set m(i) = n − (i1 + · · · + in ). Clearly, m(i) is the number of copies of Bˇ in Xi1 × · · · × Xin . iii) If f is a function on Xi1 × · · · × Xin , then Ord f is the function on Bˇ m(i) × (B × )n−m(i) obtained from f by shifting all of the Bˇ arguments before all of the B × arguments, while preserving the relative order of the Bˇ arguments and the relative order of the B × arguments and multiplying by the sign of the permutation that implements the reordering of the arguments. That is, Ord f (x1 , · · · , xn ) = sgnπf (xπ(1) , · · · , xπ(n) ), where the permutation π ∈ Sn is determined by π(j ) < π(j ) if ij < ij or ij = ij j < j . Remark XIV.5. Using the identification of Definition XIV.4.i, Xn = · Xi1 × · · · × Xin , i1 ,··· ,in ∈{0,1}
where, on the right-hand side we have a disjoint union. If f is a function on Xn and i = (i1 , · · · , in ) ∈ {0, 1}n , we denote by f the restriction of f to Xi1 × · · · × Xin . i Definition XIV.6. i) We denote by Fˇ n; the set of functions on Xn with the property that for each i = (i1 , · · · , in ) ∈ {0, 1}n with m(i) < n, Ord f i ∈ Fˇ m(i) (n − m(i); ), and as such there is a function g on Bˇn such that f (0,··· ,0) (ηˇ 1 , · · · , ηˇ n ) = (2π)3 δ(ηˇ 1 + · · · + ηˇ n ) g(ηˇ 1 , · · · , ηˇ n ). ii) For a function f ∈ Fˇ n; and a natural number p we set Ord f ˜ ˜ = |g|p, ˜ + |f |p, i
p,
,
i∈{0,1}n m(i) =0
˜ ρ˜ + ˜ ρ˜ = |g|p,, |f |p,,
Ord f ˜ , i p,,ρ˜
i∈{0,1}n m(i) =0
where g is the function on Bˇn such that f (0,··· ,0) (ηˇ 1 , · · · , ηˇ n ) = (2π)3 δ(ηˇ 1 + · · · + ηˇ n ) g(ηˇ 1 , · · · , ηˇ n ). We extend Definition VII.2 of ladders to the case that the rungs are all defined over X = Bˇ ∪· (B × ) and the bubble propagators are defined over B. Definition XIV.7. Let be a sectorization. i) Let P be a bubble propagator over B and r a rung over X . We set (r • P )(y1 ,y2 ;x3 ,x4 ) = dx1 dx2 r(y1 ,y2 ,(x1 ,s1 ),(x2 ,s2 )) P (x1 ,x2 ;x3 ,x4 ). s1 ,s2 ∈ B×B
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(r • P ) is a function on X2 × B 2 . For a general function F on X2 × B 2 , define the rung (F • r) over X by (F • r)(y1 ,y2 ,y3 ,y4 ) = dx1 dx2 F (y1 ,y2 ;x1 ,x2 ) r((x1 ,s1 ),(x2 ,s2 ),y3 ,y4 ) s1 ,s2 ∈ B×B
if at least one of the arguments y1 , · · · , y4 lies in B× ⊂ X , and for ηˇ 1 , ηˇ 2 , ηˇ 3 , ηˇ 4 ∈ Bˇ ⊂ X , (F • r)(ηˇ 1 ,ηˇ 2 ,ηˇ 3 ,ηˇ 4 ) (2π)3 δ(ηˇ 1 +ηˇ 2 +ηˇ 3 +ηˇ 4 ) = dx1 dx2 F (ηˇ 1 ,ηˇ 2 ;x1 ,x2 ) r((x1 ,s1 ),(x2 ,s2 ),ηˇ 3 ,ηˇ 4 ). s1 ,s2 ∈ B×B
ii) Let ≥ 1 , r1 , · · · , r+1 rungs over X and P1 , · · · , P bubble propagators over B. The ladder with rungs r1 , · · · , r+1 and bubble propagators P1 , · · · , P is defined to be r1 • P1 • r2 • P2 • · · · • r • P • r+1 . If r is a rung over X and A, B are propagators over B, we define L (r; A, B) as the ladder with + 1 rungs r and bubble propagators C(A, B). iii) Definitions (i) and (ii) apply verbatim to the situation when creation/annihilation indices are ignored, that is, when B and X are replaced by B and Y , respectively. Remark XIV.8. The ladder r1 • P 1 • r2 • P2 • · ·· • r • P • r+1 depends only on r1 X2 ×(B×)2 , r2 (B×)4 , · · ·, r (B×)4 and r+1 (B×)2 ×X2 . If r1 , · · · , r+1 are sup
ported on (B × )4 ⊂ X4 then Definitions VII.2.ii and XIV.7.ii agree. Also Lemma VII.5.i carries over verbatim. The analog of Proposition VII.6 is
τ2 Proposition XIV.9. Let 0 < < 2M j , where τ2 is the constant of Lemma XIII.6 of [FKTo3]. Let u((ξ,s), (ξ ,s )) ∈ F0 (2, j ) be an antisymmetric, spin independent, particle number conserving function whose Fourier transforms obeys |u(k)| ˇ ≤ 21 |ık0 − e(k)| and such that |u||1,j ≤ cj . Furthermore let f ∈ Fˇ 4,j . Then for all ≥ 1,
L f ; Cu(j ) , Cu(≥j +1) ˜ ˜ +1 , ≤ const cj |f |3, 3,j j 1/n0 Vpp L f ; Cu(j ) , Cu(≥j +1) pp ˜ ˜ +1 , ≤ const lj cj |f |3, 3, j j
if the Fermi curve F is strongly asymmetric in the sense of Definition I.10. Here, n0 is the constant of Definition I.10. Proof. The first inequality is a direct consequence of Remark D.8 of [FKTo3] with X = 2M j cj and v = u = v = u followed by Corollary A.5.i of [FKTo1] with (j ) (≥j +1) , then, by Lemma VII.5.i and Remark X = τ2 , µ = 1. If we set C = C Cu , Cu XIV.8, (j ) (≥j +1) pp L f ; Cu , Cu = f pp • pp C • · · · • pp C • f pp . Thus the second inequality follows from Theorem XXII.7 of [FKTo4].
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Remark XIV.10. By Remarks XIII.13 and XIV.8, Proposition VII.6 is a special case of Proposition XIV.9. Definition VII.7 and Theorem VII.8 carry over almost verbatim to the present situation. Definition XIV.11. Let F = F (i) i = 2, 3, · · · be a family of antisymmetric functions in Fˇ 4,i . Let p = p(2) , p(3) , · · · be a sequence of antisymmetric, spin independent, particle number conserving functions p(i) ((ξ,s), (ξ ,s )) ∈ F0 (2, i ). We define, recursively on 0 ≤ j < ∞, the iterated particle hole (or wrong way) ladders up to scale j , denoted by L(j ) (p, F ) , as F ) = 0 L(0) (p, L(j +1) (p, F ) = L(j ) (p, F )j + 2
∞
(−1) (12)+1 L
(j )
(≥j +1) ph
wj ; Cuj , Cuj
=1
where uj =
j −1 i=2
(i)
pj and wj =
j
(i) i=2 Fj
+ 18 Ant Vph L(j ) (p, F )
j
.
Theorem XIV.12. For every ε > 0 there are constants ρ0 , const such that the following holds. Let F = F (2) , F (3) , · · · be a sequence of antisymmetric, spin indepen dent, particle number conserving functions F (i) ∈ Fˇ 4,i and p = p (2) , p(3) , · · · be a sequence of antisymmetric, spin independent, particle number conserving functions p(i) ∈ F0 (2, i ). Assume that there is ρ ≤ ρ0 such that for i ≥ 2, ˜ ≤ |F (i)|3, i
ρ c M εi i
|p (i)|1,i ≤ ρMlii ci
pˇ (i) (0, k) = 0.
Then for all j ≥ 2, (j ) ˜ Vph L (p, F )j 3, ≤ const ρ 2 cj . j
Theorem XIV.12 follows from Theorem D.2 below. Theorem D.2, in turn, is proven in [FKTl]. Remark XIV.13. With the same argument as in Remark XIV.10, Theorem VII.8 is a special case of Theorem XIV.12 . Theorem XIV.12 is not general enough for controlling the effect of a renormalization ˜ norms. Consider an iterated particle–hole ladder L(j ) (p, group step on the | · |3, f ). j Integrating out subsequent scales can result in a propagator and a φ field being hooked (j ) to the ψ legs of the ladder. See the ψ + Cu,j J φ in Remark IX.6.iii. The resulting object is no longer an iterated particle–hole ladder. In §IX, this was harmless because “external improvement” with respect to the | · |3,j norm (see Lemma XII.19 of [FKTo3]) led ˜ , we to bounds that were summable over scales. With the more sensitive norm | · |3, j have to control the “shear” from ψ to φ fields in particle–hole ladders. Under a shear ˆ transformation, a Grassmann function W(φ, ψ) is mapped to W(φ, ψ + Bφ). Definition XIV.14. Let be a sectorization, B(k) a function on R × R2 and f ∈ Fˇ n; .
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i) The shear of f with respect to B is the element shear(f, B) ∈ Fˇ n; defined by
dξν E+ (ηˇ ν , ξν )B(kν ) shear(f, B)i (y1 , · · · , yn ) = 1≤ν≤n j∈{0,1}n jp ≥ip , 1≤p≤n iν =0, jν =1
sν ∈ B
× f j(z1 , · · · , zn ) zν =(ξν ,sν ) if zν =yν
iν =0, jν =1 , if iν =jν ,
where, for iν = 0, yν = ηˇ ν = (kν , σν , aν ) ∈ Bˇ and E+ was defined before Definition VI.1. ii) We use Gr(φ, ψ; f ) to denote the Grassmann function with kernel f . That is, n φ(yp ) ifip = 0 Gr(φ, ψ; f ) = dy1 · · · dyn fˆi (y1 , · · · , yn ) , ψ(yp ) ifip = 1 i∈{0,1}n
p=1
where factors in the product are in the order specified by the index p, fˆ is the Fourier transform of f with respect to its φ arguments and yν runs over B when ν = 0 and over B × when ν = 1. The definition of shear has been chosen so that ˆ f) Gr(φ, ψ; shear(f, B)) = Gr(φ, ψ + Bφ; (XIV.1) ˆ ˆ ξ )φ(ξ ), for all where, with some abuse of notation, we set (Bφ)(ξ, s) = B dξ B(ξ, s ∈ , retaining the Bˆ defined in Notation V.4. Corollary XIV.15 (to Theorem XIV.12). For every ε > 0 and cB > 0 there are constants ρ0 , const such that the following holds. Let v = v (2) , v (3) , · · · be a sequence of antisymmetric, independent, particle number conserving functions v (i) ∈ Fˇ 4,i (2) spin (3) and p = p , p , · · · be a sequence of antisymmetric, spin independent, particle number conserving functions p(i) ∈ F0 (2, i ). Assume that there is ρ ≤ ρ0 such that for i ≥ 2, ˜ ≤ |v (i)|3, i
ρ c, M εi i
|p (i)|1,i ≤ ρMlii ci ,
pˇ (i) (0, k) = 0.
Let B(k) be a function obeying B(k)˜≤ cB cj , set f (i) = shear v (i) , ν (≥i) B ∈ Fˇ 4,i (2) (3) and let f = f , f , · · · . i) For all j ≥ 2, (j ) ˜ Vph L (p, f )j 3, ≤ const ρ 2 cj . j
(i) ii) Let B (k) obey B (k)˜ ≤ c cB cj , set fs = shear v (i) , ν (≥i) (B + sB ) ∈ Fˇ 4,i (2) (3) and let fs = fs , fs , · · · . For all j ≥ 2 and all c > 0, d ˜ Vph L(j ) (p, fs )j s=0 3, ≤ const c ρ 2 cj . ds j
In the proof, which follows Remark XIV.20, we will use auxiliary external fields, named φ . We now extend the notation of Definitions XIV.4 and XIV.6 to include them.
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ˇ X1 = B × and Definition XIV.16. Let be a sectorization. Set X−1 = X0 = B, X = X−1 ∪· X0 ∪· X1 . Let i = (i1 , · · · , in ) ∈ {−1, 0, 1}n . i) The inclusions of Xij , j = 1, · · · , n, in X induce an inclusion of Xi1 × · · · × Xin in X n . We identify Xi1 × ·· · × Xin with its image in X n . ii) Set m (i) = # 1 ≤ j ≤ n ij = −1 and m(i) = # 1 ≤ j ≤ n ij = 0 . iii) If f is a function on Xi1 ×· · ·×Xin , then Ord f is the function on Bˇ m (i) × Bˇ m(i) ×(B×
)n−m (i)−m(i) obtained from f by permuting the arguments so that all X−1 arguments appear before all X0 arguments and all X0 arguments appear before all X1 arguments, while preserving the relative order of the Xj arguments, j = −1, 0, 1, and multiplying by the sign of the permutation that implements the reordering of the arguments. That is, Ord f (x1 , · · · , xn ) = sgn σ f (xσ (1) , · · · , xσ (n) ), where the permutation σ ∈ Sn is determined by σ (j ) < σ (j ) if ij < ij or ij = ij , j < j . iv) Using the identification of part (i), n X = · Xi1 × · · · × Xin , i1 ,··· ,in ∈{−1,0,1}
where, on the right-hand side we have a disjoint union. If f is a function on X n and i = (i1 , · · · , in ) ∈ {−1, 0, 1}n , we denote by f the restriction of f to Xi1 ×· · ·×Xin . i Definition XIV.17. Let be a sectorization and m , m, n ≥ 0. i) For n ≥ 1, denote by Fˇ m ,m (n; ) the space of all translation invariant, complex n m valued functions f (ηˇ 1 ,··· ,ηˇ m +m ; (ξ1 ,s1 ),··· ,(ξn ,sn )) on Xm whose −1 × X0 × B × ˇ ˇ ˇ Fourier transform f (ηˇ 1 ,··· ,ηˇ m ; (ξ1 ,s1 ),··· ,(ξn ,sn )) vanishes unless ki ∈ s˜i for all 1 ≤ j ≤ n. Here, ξˇi = (ki , σi , ai ). Also, let Fˇ m ,m (0; ) be the space of all momentum m conserving, complex valued functions f (ηˇ 1 ,··· ,ηˇ m +m ) on Xm −1 × X0 . ii) We denote by Fˇ n; the set of functions on X n with the property that for each i = (i1 , · · · , in ) ∈ {−1, 0, 1}n with m (i) + m(i) < n, Ord f i ∈ Fˇ m (i),m(i) (n − m (i) − m(i); ), and such that for each i = (i1 , · · · , in ) ∈ {−1, 0, 1}n with m (i) + m(i) = n there is a function gm ,m ∈ Fˇ m ,m (0; ) such that Ord f i (ηˇ 1 , · · · , ηˇ n ) = (2π)3 δ(ηˇ 1 + · · · + ηˇ n ) gm ,m (ηˇ 1 , · · · , ηˇ n ). iii) There is a natural identification : Fˇ m ,m (n; ) → Fˇ m +m (n; ) obtained by iden m +m m ˇ m ˇ m tifying Xm = Bˇ m +m . Similarly, if i = (i1 , · · · , in ) ∈ −1 ×X0 = B × B with X0 n {−1, 0, 1} and f is a function on Xi1 × · · · × Xin then the function (f ) on Xπ(i1 ) × · · · × Xπ(in ) , where π(−1) = π(0) = 0 and π(1) = 1, is obtained by → Fˇ n; by identifying X−1 with X0 . We extend the map to : Fˇ n; (f )i =
j∈{−1,0,1}n π(j)=i
f j
for all i ∈ {0, 1}.
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iv) For a function f ∈ Fˇ m ,m (n; ) and a natural number p we set ˜ = |(f )|p, ˜ . |f |p, For a function f ∈ Fˇ n; and a natural number p we set
˜ = |f |p,
˜ f . i p,
i∈{−1,0,1}n → Fˇ n, by Lemma XIV.18. For κ ∈ Cn , define Sκ : Fˇ n,
n 1−i κj j f i Sκ f i =
for all i ∈ {−1, 0, 1}n .
j =1
, Then, for all f ∈ Fˇ n,
˜ ≤ 3n ˜ ≤ |f |p, |(f )|p,
sup κ ∈Cn |κj |≤1, 1≤j ≤n
˜ . |(Sκ f )|p,
Proof. The first bound is just the triangle inequality, ˜ = |(f )|p,
(f ) ˜ . ≤ i p, i∈{0,1}n
i∈{0,1}n
˜ f ˜ . = |f |p, j p,
j∈{−1,0,1}n π(j)=i
For the second bound, we just use the Cauchy integral formula n n 1−i dκj d j 1 S f i = f = κ 1−i (1−ij )! 2πı j j =1
κ =0
dκj
j =1 |κj |=1
1 2−ij κj
Sκ f
to prove that, for each i ∈ {−1, 0, 1}n , ˜ ˜ f = f i p, ≤ i p, and then sum over i ∈ {−1, 0, 1}n .
sup κ ∈Cn |κj |≤1, 1≤j ≤n
˜ |(Sκ f )|p,
Let B(k) be a kernel which is the product of two other kernels B1 (k) and B2 (k). Then, for any Grassmann function W(φ, ψ), ˆ = W(φ, ψ + Bˆ 1 φ ) ˆ . W(φ, ψ + Bφ) φ =B φ 2
Thus the shear transformation with respect to B may be written as the composition of another shear–like transformation with respect to B1 and a “scaling transformation” with respect to B2 . To make this precise, we have Definition XIV.19. Let be a sectorization and B(k) a function on R × R2 .
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i) If f ∈ Fˇ n; , then the element shear (f, B) ∈ Fˇ n; is defined by
shear (f, B)i (y1 , · · · , yn ) = dξν E+ (ηˇ ν , ξν )B(kν ) 1≤ν≤n iν =−1
sν ∈ B
× f |i| (z1 , · · · , zn ) zν =(ξν ,sν ) if
iν =−1 , if iν =−1
zν =yν
where |i| = (|i1 |, · · · , |in |) and yν = ηˇ ν = (kν , σν , aν ) ∈ Bˇ when iν = −1. , then the element sct (f, B) ∈ Fˇ ii) If f ∈ Fˇ n; n; is defined by
sct (f, B)i (y1 , · · · , yn ) = B(kν ) f i (y1 , · · · , yn ), 1≤ν≤n iν =−1
where yν = (kν , σν , aν ) ∈ Bˇ if iν = −1. iii) If f ∈ Fˇ n; , then the element sct(f, B) ∈ Fˇ n; is defined by
sct(f, B)i (y1 , · · · , yn ) = B(kν ) f i (y1 , · · · , yn ), 1≤ν≤n iν =0
where yν = (kν , σν , aν ) ∈ Bˇ if iν = 0. Remark XIV.20.
i) If B1 (k) and B2 (k) are functions on R × R2 and f ∈ Fˇ n; , then shear(f, B1 B2 ) = sct shear (f, B1 ), B2 .
and B(k) be a function obeying B(k)˜≤ cB ci . Then, ii) Let f ∈ Fˇ n;i , f ∈ Fˇ n; i by repeated application of Eq. (XVII.3) of [FKTo3], with j replaced by i, X = 0, XB = 1, there is a constant const , depending on n, such that
˜ |shear (f, B)|p, ≤ i ˜ |sct(f, B)|p,i ≤ ˜ |sct (f , B)|p, ≤ i Proof of Corollary XIV.15.
const const const
˜ , max{1, cB }n ci |f |p, i n ˜ , max{1, cB } ci |f |p, i ˜ . max{1, cB }n ci |f |p, i
i) Let v˜ (i) = shear (v (i) , ν (≥i) ). Then, by Remark XIV.20.i, f (i) = sct (v˜ (i) , B)
F ) to F (i) ∈ Fˇ n; in the obvious way (just and, extending the definition of L(j ) (p, ˇ ˇ ˇ replace B by B ∪· B in Definitions XIV.7 and XIV.11),
˜ B)) Vph L(j ) (p, (sct (v, f )j = Vph L(j ) p, j (j ) = sct Vph L p, v˜ ) , B . j
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143
Hence, by Lemma XIV.18 and Remark XIV.20.ii, ˜ (j ) ˜ Vph L (p, f )j 3, ≤ sct Vph L(j ) p, v˜ ) , B j
j
3,j
˜ v˜ ) 3, ≤ const cj Vph L(j ) p, j j (j ) ˜ ≤ const cj sup v˜ ) Sκ Vph L p, j
3,j
κ ∈C4 |κν |≤1, 1≤ν≤4
.
Observe that v˜ ) = Vph L(j ) p, (Sκ v˜ ) Sκ Vph L(j ) p, j
By Definition I.2.ii, ν (≥i) (k)˜≤ XIV.20.ii, sup κ ∈C4 |κν |≤1, 1≤ν≤4
Sκ v˜ (i) ˜ ≤ 3, i
j
const ci .
.
Hence, by Lemma XIV.18 and Remark
sup κ ∈C4 |κν |≤1, 1≤ν≤4
Sκ v˜ (i) ˜ 3,
i
˜ ˜ ≤ v˜ (i)3, ≤ const ci v (i)3, ≤ const i
≤
const
i
ρ 2 c M εi i
ρ c. M εi i
So Theorem XIV.12 gives (j ) ˜ Vph L (p, f )j 3, ≤ j
const cj
sup κ ∈C4 |κν |≤1, 1≤ν≤4
˜ ˜ (Sκ v) Vph L(j ) p,
j 3,j
≤ const ρ 2 cj . ii) Part (i), with cB replaced by 2cB and B replaced by B + sB , implies that, for all s ∈ C obeying |s| ≤ c1 , (j ) ˜ Vph L (p, fs )j 3, ≤ const ρ 2 cj . j
fs )j is a polynomial in s, the desired bound now follows from Since Vph L(j ) (p, the Cauchy integral formula (j ) d 1 V ( p, f ) = ds s12 Vph L(j ) (p, fs )j . L s j s=0 2πi ds ph |s|= c1
XV. Recursion Step for Momentum Green’s Functions ˜ This section provides the analog of §IX for the | |–norms. Recall, from §XI, that we are assuming that the interaction V satisfies the reality condition (I.1) and is bar/unbar exchange invariant in the sense of (I.2).
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1. More Input and Output Data. We now supplement the conditions on the input and output data of Definitions IX.1 and IX.2 in order get more detailed information on the behaviour of the two– and four–point functions. Recall from Theorem XII.1 that 1 2 2 < ℵ < ℵ < 3 . We generalize the notation cj and ej (X) of Definition V.2 to ci,j =
M iδ0 M j |δδ | t δ +
|δδ |≤r |δ0 |≤r0
ei,j (X) =
∞ t δ ∈ Nd+1 ,
|δδ |>r or |δ0 |>r0
ci,j 1−M j X
(XV.1)
,
so that we can track different degrees of smoothness in temporal and spatial directions. (j ) Definition XV.1 (More Input Data). Let D˜ in be the set of interaction quadruples, (W, G, u, p), that fulfill Definitions VIII.1, IX.1 and the following. Let w(φ, ψ; K) be the j –sectorized representative of W(K) specified in (I1) and w a (φ, ψ; K) its amputation in the sense of Definition XIII.9.
(I ∼ 1) The Grassmann function w a (φ, ψ; K) − wa (0, ψ; K) vanishes unless νˆ (<j ) φ is nonzero and Nj∼ w a (K), 64α, K1,j ≤ ej K1,j , Nj∼ dds w a (K + sK )s=0 , 64α, K1,j ≤ M j ej K1,j K 1,j (I ∼ 2)
for all K ∈ Kj and all K . There is a family v of antisymmetric kernels v (i) ∈ Fˇ 4,i ,
2≤i ≤j −1
(independent of K) and an antisymmetric kernel δf (j ) (K) ∈ Fˇ 4,j obeying 1/n (i) ˜ li 0 v ≤ c for all 2 ≤ i ≤ j − 1, 3,i ,ρ˜ α7 i 1/n0 (j ) ˜ l δf (0) ≤ jα 7 cj , 3,j ,ρ˜ d δf (j ) (K + sK ) ˜ ≤ 64α14 B2 ej (K1,j )M j K 1,j ds s=0 3, ,ρ˜ j
for all K ∈ Kj , from which the quartic parts of w a and G a are built as follows. [i,j ) Let f (i) = shear v (i) , Cu(0) (k)A(k) ∈ Fˇ 4,i , 2 ≤ i ≤ j −1, where shear(f, B) was defined in Definition XIV.14. Then the kernel of the quartic part of wa + G a is δf (j ) (K) +
j −1 i=2
(i) fj + 18 Ant Vph L(j ) (p, f)
j
,
where the particle–hole value Vph was defined in Definition XIV.2. The kernel v (i) vanishes unless all of its φ momenta are in the support of ν (
Two Dimensional Fermi Liquid. 3: Fermi Surface
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(I ∼ 3) Let dη1 dη2 G2 (η1 , η2 ; K) φ(η1 )φ(η2 ) be the part of G(K) that is homogeneous ˇ 2 (k; K) has the decomposition of degree two. Then 2G ˇ 2 (k; K) = C 2G u(0) (k) + (<j )
1 (ik0 −e(k))2
j −1
δq (i) (k; K) +
j
q (i,) (k)
=i
i=2
with q (i,) (k), i ≤ ≤ j and δq (i) (k; K) vanishing in the (i + 2)nd neighbourhood and when k is not in the support of U (k), δq (i) (k; 0) = 0 for 2 ≤ i ≤ j − 1 and (i,) ˜ l ℵ (−i) q (k) ≤ λ1−2υ ci, , M 0 M d ˜ δq (i) (K + sK ) ≤ M ℵ (j −i) e 1 1 K K . ds
i+ 2 ,j + 2
s=0
1,j
1,j
a (η, (ξ, s); K) be the kernel of the part of w a (K) that is of degree 1 in φ Let w1,1 and degree 1 in ψ. Then a∼ ˜ ,ρ˜ ≤ |w1,1 (K)|1, j
1 lj e (K1,j ). α6 M j j
(I ∼ 4) W, G, the j –sectorized representative w(φ, ψ; K) of W(K) and all of the F (i) ’s and v (i) ’s are bar/unbar invariant in the sense of Definition B.1.B of [FKTo2]. If K is real, then W, G, the j –sectorized representative w(φ, ψ; K) of W(K) and all of the q (i,) ’s are k0 –reversal real in the sense of Definition B.1.R of [FKTo2]. Remark XV.2. By Remark B.3.ii of [FKTo2], for any interaction quadruple (W, G, u, p), u and every p (i) is bar/unbar invariant. Definition XV.3 (More Output Data). Let D˜ out be the set of interaction quadruples (W, G, u, p) that fulfill Definitions VIII.1, IX.2 and the following. Let w(φ, ψ; K) be the j –sectorized representative of W(K) specified in (O1) and w a (φ, ψ; K) its amputation in the sense of Definition XIII.9. (j )
(O ∼ 1) The Grassmann function w a (φ, ψ; K) − w a (0, ψ; K) vanishes unless νˆ (≤j ) φ is nonzero and Nj∼ w a (K), α, K1,j ≤ ej K1,j , Nj∼ dds w a (K + sK )s=0 , α, K1,j ≤ M j ej K1,j K 1,j (O ∼ 2)
for all K ∈ Kj and all K . There is a family v of antisymmetric kernels v (i) ∈ Fˇ 4,i ,
2≤i≤j
(independent of K) and an antisymmetric kernel δf (j +1) (K) ∈ Fˇ 4,j obeying 1/n (i) ˜ l 0 v ≤ iα 7 ci 3, ,ρ˜ i
for all 2 ≤ i ≤ j ,
1/n (j +1) ˜ l 0 δf (0)3, ,ρ˜ ≤ jα+1 8 cj , j d δf (j +1) (K + sK ) ˜ ≤ α 41B2 ej (K1,j )M j K 1,j ds s=0 3, ,ρ˜ j
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for all K ∈ Kj , from which the quartic parts of w a and G a are built as follows. [i,j ] Let f (i) = shear v (i) , Cu(0) (k)A(k) ∈ Fˇ 4,i , 2 ≤ i ≤ j . Then the kernel of the quartic part of wa + G a is δf (j +1) (K) +
j i=2
(O ∼ 3)
(i) fj + 18 Ant Vph L(j +1) (p, f) .
The kernel v (i) vanishes unless all of its φ momenta are in the support of ν (
1 (ik0 −e(k))2
j
δq (i) (k; K) +
i=2
j
q (i,) (k)
=i
with q (i,) (k), i ≤ ≤ j and δq (i) (k; K) vanishing in the (i + 2)nd neighbourhood and when k is not in the support of U (k), δq (i) (k; 0) = 0 for 2 ≤ i ≤ j and (i,) ˜ l ℵ (−i) q (k) ≤ λ1−2υ ci, , M 0 M d ˜ δq (i) (K + sK ) ≤ M ℵ (j −i) e 1 1 K1, K 1, for i < j , j j i+ 2 ,j + 2 ds s=0 d ˜≤ M ℵ −ℵ e 1 K1, K 1, . δq (j ) (K + sK ) j j j+ ds s=0 2
a (η, (ξ, s); K) w1,1
be the kernel of the part of wa (K) that is of degree 1 in φ Let and degree 1 in ψ. Then a∼ 1 lj w (K) ˜ j +1 1,j +1 ,ρ˜ ≤ α 7 M j ej (K1,j ), 1,1 d a∼ υ/8 w (K+sK ) ˜ ≤ α1 + λ0 α 21B ej (K1,j ) K 1,j . j +1 s=0 1, ds 1,1 ,ρ˜ j +1
(O ∼ 4) W, G, the j –sectorized representative w(φ, ψ; K) of W(K) and all of the F (i) ’s and v (i) ’s are bar/unbar invariant. If K is real, then W, G, the j –sectorized representative w(φ, ψ; K) of W(K) and all of the q (i,) ’s are k0 –reversal real in the sense of Definition B.1.R of [FKTo2]. 2. Integrating Out a Scale. (j ) (j ) ∈ D˜ out . Theorem XV.4. If (W, G, u, p) ∈ D˜ in , then j (W, G, u, p)
The rest of this subsection is devoted to the proof of this theorem. Let w= ωm,n (K) m,n
with ωm,n (φ, ψ; K) =
s1 ,··· ,sn ∈j
dη1 ···dηm dξ1 ···dξn
wm,n (η1 ,··· ,ηm (ξ1 ,s1 ),··· ,(ξn ,sn ); K)
× φ(η1 ) · · · φ(ηm ) ψ((ξ1 ,s1 )) · · · ψ((ξn ,sn ) )
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be the j –sectorized representative of W specified in (I1). Let (W , G , u, p) = j (W, G, u, p) and choose the sectorized representative w of W as in Remark IX.6.iii. Define z by . a . z(φ, ψ; K) . a . . . ψ,Dj (u;K) = C (j ) . w − ω1,1 . ψ,Cj (u;K) j
u,j
j
ˆ + δBφ), × (φ, ψ + Bφ
(XV.2)
where (j ) B(k) = ik0 − e(k) Cu(0) (k), (j ) (j ) δB(k; K) = ik0 − e(k) Cu(K) (k) 1 + wˇ 1,1 (k; K) − Cu(0) (k) , a (k; K) is the Fourier transform, as in the last part of Definition VI.1, of the and wˇ 1,1 a (η, (ξ, s); K). As in Proposition XII.8 of [FKTo3], with function (η, ξ ) → s∈j w1,1 some abuse of notation, ˆ ˆ (Bφ)(ξ, s) = dξ B(ξ, ξ )φ(ξ )
for all s ∈ j . Recall that if, for each s, s ∈ j , c ( · , s), ( · , s ) is the Fourier transform, as in (III.1) and (III.2), of χs (k)C(k)χs (k), then Cj (ξ, s), (ξ , s ) = t,t ∈j c (ξ, t), (ξ , t ) . The integration and initial/final Wick ordering covariances s∩t =∅ s ∩t =∅
(j )
(j )
Cu,j , Cj (u; K)j , Dj (u; K)j are constructed in this way from the Cu (k), Cj (u; K)(k), Dj (u; K)(k) specified in Definition VI.11. Lemma XV.5. a w (φ, ψ; K) = z(φ, ψ; K) + ω1,1 (φ, ψ; K) − z(φ, 0; K), a
a (j ) ˆ ˆ ˇ 1,1 )ˆAφ. G (φ) = G a (φ) + z(φ, 0; K), − 21 (J (1 + wˇ 1,1 )ˆAφ)C u(K) J (1 + w (j )
Proof. In this proof, set C = Cu,j and Cj = Cj (u; K)j . Define, for each s ∈ j , Bs (k) to be the Fourier transform, as in the last part of Definition VI.1, of the function (η, ξ ) → w1,1 (η, (ξ, s); K). Observe that
Bs (k) = wˇ 1,1 (k; K).
s∈j
By Lemma B.62 of [FKTo2], ω1,1 (φ, ψ) =
dηdξ ψ(ξ, s)(J Bˆ s )(ξ, η)φ(η).
(XV.3)
s∈j 2 This is the only step in the proof that substantially uses bar/unbar invariance. Without it, the argument ˆ + δBφ of (XV.2) would have a more general, but still controllable form. Bφ
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Consequently, by Lemma C.1 of [FKTo2], C :w:ψ,Cj (φ, ψ) = log Z1
e
:w(φ,ψ+ζ ;K):ψ,Cj
dµC (ζ ) :(w−ω
= log Z1
1,1 eω1,1 (φ,ψ)+ω1,1 (φ,ζ ) e ˆ = ω1,1 (φ, ψ) + log Z1 es ζ (s)J Bs φ
:(w−ω
)(φ,ψ+ζ ;K):ψ,Cj
dµC (ζ )
)(φ,ψ+ζ ;K):
1,1 ψ,Cj ×e dµC (ζ ) 1 (J Bˆ s φ)C(s, s )(J Bˆ s φ) = ω1,1 (φ, ψ) − 2 s,s ∈j ˆ s φ+ζ ;K):ψ,C :(w−ω 1,1 )(φ,ψ+s C( · ,s)J B j dµ (ζ ) + log Z1 e C = ω1,1 (φ, ψ) − 21 (J Bˆ s φ)C(s, s )(J Bˆ s φ)
s,s ∈j
+C :w − ω1,1 :ψ,Cj (φ, ψ + s C( · , s)J Bˆ s φ) (j )
= ω1,1 (φ, ψ) − 21 (J wˇ 1,1 φ)Cu (J wˇ 1,1 φ) (j ) +C :w − ω1,1 :ψ,Cj (φ, ψ + Cu J wˇ 1,1 φ). Therefore the Grassmann function w of Remark IX.6.iii is determined by (j ) (j ) . w (φ, ψ; K) . 1 . . ψ,Dj (u;K) = 2 φJ Cu J φ + C (:w:ψ,Cj )(φ, ψ + Cu J φ) j
(j )
(j )
= − 21 (J φ)Cu J φ + ω1,1 (φ, Cu J φ) (j ) − 21 (J wˇ 1,1 φ)Cu (J wˇ 1,1 φ) + ω1,1 (φ, ψ) (j ) +C :w − ω1,1 :ψ,Cj (φ, ψ + Cu J (1 + wˇ 1,1 )ˆφ). (j )
By (XV.3), the antisymmetry of Cu and repeated use of Lemma IX.5 of [FKTo2], (j )
ω1,1 (φ, Cu J φ) =
(j ) (j ) (Cu J φ)J Bˆ s φ = (Cu J φ)J wˇ 1,1 φ s (j )
(j )
= −(J φ)Cu (J wˇ 1,1 φ) = −(J wˇ 1,1 φ)Cu (J φ), so that (j ) . w (φ, ψ; K) . ˆ ˆ 1 . ψ,Dj (u;K) = − 2 (J (1 + wˇ 1,1 ) φ)Cu J (1 + wˇ 1,1 ) φ + ω1,1 (φ, ψ) . j (j ) +C :w − ω1,1 :ψ,Cj (φ, ψ + [Cu (k)(1 + wˇ 1,1 )]ˆφ).
Amputating and applying parts (i) and (ii) of Lemma IX.5 of [FKTo2], a (j ) a ˆ ˆ + ω1,1 w (φ, ψ; K) = − 21 (J (1 + wˇ 1,1 )ˆAφ)C ˇ 1,1 )ˆAφ (φ, ψ) + z(φ, ψ; K). u J (1 + w
The lemma now follows by Remark IX.6.iii.
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Lemma XV.6. (j ) (j ) ˆ ˆ (j ) ˆ ˆ = (J Aφ)C ˆ ˇ 1,1 )ˆAφ (J (1 + wˇ 1,1 )ˆAφ)C u(K) J (1 + w u(0) (J Aφ) + φJ E φ
with
(j ) (j ) (j ) E (j ) (k; K) = −A(k)2 Cu(K) (k)wˇ 1,1 (k; K) 2 + wˇ 1,1 (k; K) + Cu(K) (k) − Cu(0) (k) . Proof. Set B (k; K) = 1 + wˇ 1,1 (k; K). By repeated use of Lemma IX.5 of [FKTo2] and the facts that J 2 = −1l and J t = −J , 2 ˆ ˆ ˆ (J Bˆ Aφ)C u(K) (J B Aφ) = −(J B AJ φ)Cu(K) (J B Aφ) = (B A J φ)Cu(K) (J B Aφ) (j )
t
(j )
(j )
(j )
(j )
= (J φ)B ACu(K) J B Aφ
= −φJ B ACu(K) J B Aφ.
Similarly, (j ) ˆ ˆ ˆ (j ) ˆ (J Aφ)C u(0) (J Aφ) = −φJ ACu(0) J Aφ.
Subtracting and observing that (j ) (j ) (j ) −A(k)2 Cu(K) (k)B (k)2 − Cu(0) (k) = −A(k)2 Cu(K) (k) B (k)2 − 1 (j ) (j ) +Cu(K) (k) − Cu(0) (k) = E (j ) (k; K) yields ˆ ˆ ˆ ˆ ˆ ˆ (j ) (J Bˆ Aφ)C u(K) (J B Aφ) − (J Aφ)Cu(0) (J Aφ) = φJ E φ. (j )
(j )
Lemma XV.7. i) ˜ (j ) j C u(K) (k) ≤ const M ej K1,j , ik0 − e(k) C (j ) (k)˜≤ const ej K1, , j u(K) ˜ d (j ) 2j C ds u(K+sK ) (k) s=0 ≤ const M ej K1,j K 1,j , ˜ d j (ik0 − e(k))C (j ) ds u(K+sK ) (k) s=0 ≤ const M ej K1,j K 1,j , ˜ d (ik0 − e(k))2 C (j ) ds u(K+sK ) (k) s=0 ≤ const ej + 1 K1,j K 1,j . 2
ii) B ˜≤ const cj , δB(K)˜≤ const ej K1,j M j K1,j + d δB(K + sK ) ˜≤ const M j ej K1, K 1, . j j ds s=0
λ1−2υ lj 0 α6
,
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J. Feldman, H. Knörrer, E. Trubowitz
iii) 1−8υ/7
λ l E (j ) (k; 0)˜≤ const 0 α 6 Mjj cj , d (j ) E (K + sK ) ˜≤ const e 1 K1, K 1, . j j j+ ds s=0 2
Proof.
i) Observe that ν (j ) (k) ˇ 0 −e(k)−u(k;K)
(j )
(ik0 − e(k))Cu(K) (k) = (ik0 − e(k)) ik
=
ν (j ) (k) 1−
u(k;K) ˇ ik0 −e(k)
and d ds (ik0
(j )
− e(k))2 Cu(K+sK ) (k)
d = (ik0 − e(k))2 ds ik
= (ik0 − e(k))2 [ik
ν (j ) (k)
ˇ 0 −e(k)−u(k;K+sK ν (j ) (k)
ˇ 0 −e(k)−u(k;K+sK
=
)
)]2
d ˇ K ds u(k;
+ sK )
ˇ K + sK ) ν (j ) (k) dds u(k; . ) 2 ˇ 1 − u(k;K+sK ik0 −e(k)
Next use the fact that, on the support of ν (j ) ,
show that, if f (k) vanishes except on the support of (j ) ˜ ν (k) ≤
f (k) ik0 −e(k)
˜ ≤
≤ |ik0 − e(k)| ≤
√ 1 M Mj
√
2M Mj
to
ν (≤j ) (k),
const cj + 1 , 2
M j + 2 cj + 1 f (k).˜ 1
const
2
(XV.4)
Also use Lemma VIII.7 and Lemma XII.12 of [FKTo3] to show that 1−υ
˜ u(K) ≤ 2||u(K)||1, ≤ const λ0j −1 + K1, ej (K1, ) ˇ j j j M d u(K + sK )s=0 ds ˇ
≤ const M1j ej (K1,j ), ˜ ≤ 2 d u(K + sK ) ds
≤ ˇ Since iku(k;K) ≤ −e(k) 0
1 2
s=0 1,j const ej (K1,j ) K 1,j .
(XV.5)
on the support of ν (j ) , we have
(ik0 − e(k))C (j ) (k)˜≤ u(K) d ˜ (ik0 − e(k))2 C (j ) ds u(K+sK ) (k) s=0 ≤
K1,j , const ej + 1 K1,j K 1,j . const ej + 1 2
2
The remaining bounds follow from these, the second bound of (XV.4), Corollary A.5 of [FKTo1] and the observation that ej + 1 (K1,j ) ≤ const ej (K1,j ) for some 2 M, r0 and r dependent constant.
Two Dimensional Fermi Liquid. 3: Fermi Surface
151
ii) By (I ∼ 3), Remark XIII.6 and (I ∼ 1), ik0 − e(k) wˇ 1,1 (k; K)˜≤ 2||w a∼ (K)|1, ˜ 1,1 j 1−8υ/7
= 2λ0
1−8υ/7
λ0
≤2 ik0 − e(k) d wˇ 1,1 (k; K + sK ) ˜≤ ds s=0
α6
1−8υ/7
λ0 Bα 2
a∼ ˜ ,ρ˜ |w1,1 (K)|1, j
lj e (K1,j ), Mj j
(XV.6)
ej (K1,j ) K 1,j .
All three bounds follow by using Leibniz and Corollary A.5.ii of [FKTo1] to combine (XV.6) with the bounds of part (i). (j ) iii) For the first bound, just apply A(k)Cu(0) (k)˜ ≤ const cj from part (i), the first bound of (XV.6) and wˇ 1,1 (k; K)˜≤ const
1−8υ/7
λ0
α6
lj ej (K1,j ),
(XV.7)
which follows from (XV.6), (XV.4) and the fact that wˇ 1,1 (k; K) vanishes except on the support of ν (>j ) . Now for the second bound. By part (i), d ˜ A(k)2 C (j ) ds u(K+sK ) (k) s=0 ≤ const ej + 1 K1,j K 1,j . 2
By part (i), (XV.6) and (XV.7), ˜ A(k)wˇ 1,1 (k;K ) 2+ wˇ 1,1 (k;K ) d A(k)C (j ) ds u(K+sK ) (k) s=0 1− 8 υ
≤ ≤
λ 7 const 0α 6 lj ej (K1,j )K 1,j , λ1−2υ 0 l e (K1,j ) K 1,j . α6 j j A(k)C (j ) (k) 1 + wˇ 1,1 (k;K ) d A(k)wˇ 1,1 (k;K+sK ) ˜ u(K) ds s=0 1− 8 υ
≤ const ≤
λ1−2υ 0 α2
λ0 7 α2
ej (K1,j ) K 1,j
ej (K1,j ) K 1,j .
Proof of Theorem XV.4. We have already proven in Theorem IX.5 that (W , G , u, p) ∈ (j ) Dout . We shall use ωm,n , zm,n , · · · to denote the part of w, z, · · · that is of degree m in φ and degree n in ψ and wm,n , zm,n , · · · to denote the corresponding kernels. We shall also use ωn , zn , · · · to denote the part of w, z, · · · that is of degree n in φ and ψ combined and wn , zn , · · · to denote the corresponding kernels. Recall that . . z(φ, ψ; K) . a . ˆ + δBφ), ˆ . . ψ,D (u;K) = C (j ) . w a − ω1,1 . ψ,C (u;K) (φ, ψ + Bφ j
j
j
u,j
j
and define z by . . z (φ, ψ; K) . a . . . ψ,Dj (u;K) = C (j ) . w a − ω1,1 . ψ,Cj (u;K) (φ, ψ) j
u,j
j
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so that ˆ + δB(K)φ; K). z(φ, ψ; K) = z (φ, ψ + Bφ
(XV.8)
Preparation for the verification of (O ∼ 1), (O ∼ 2) and (O ∼ 3). We now apply Theorems XVII.3 and XVII.6 of [FKTo3] to bound z. By (I ∼ 1), parts (i) and (ii) of Lemma VIII.7 a , and Lemma XV.7.ii, the hypotheses of Theorem XVII.3 of [FKTo3], with w = w a −ω1,1 B replaced by B + δB, w = z, w = z , µ = X = K 1, , are fulfilled. Therefore,
const ,
=
λ1−υ 0 , M j −1
υ/7
γ = λ0
and
j
a , α, K1,j ) ≤ const Nj∼ (z − w a + ω1,1
˜ ,ρ˜ , |z1,1|1, ˜ ,ρ˜ ≤ |z2,0|1, j j
1 α
υ/7
+ λ0
Nj∼ (wa ,64α,K1,j ) const ∼ 1− α Nj (wa ,64α,K1,j )
Nj∼ (wa ,64α,K1,j )2 const lj ∼ a α 8 M j 1− const α Nj (w ,64α,K1,j )
(XV.9)
and z − wa − 4 4 ≤
1 4
∞ =1
a (j ) ˜ (−1) (12)+1 Ant L (w4 ; Cu ,Dj ) 3,j ,ρ˜
Nj∼ (wa ,64α,K1,j )2 const l . j const 10 α 1− α Nj∼ (wa ,64α,K1,j )
(XV.10)
The hypotheses of Theorem XVII.6 of [FKTo3], with, in addition, Y = const K 1, , j Z = const M j K and ε = const M j K 1, , are fulfilled. Hence 1,j
j
t=0
a (K + sK ) − w a (K + sK ) s=0 , α, K1,j Nj∼ dds z(K + sK ) + ω1,1
N ∼ (wa ,64α, K1, ) υ/7 ≤ const λ0 + α12 1− constj N ∼ (wa ,64α, Kj ) 1,j α2 j × Nj∼ dds w a (K + sK )s=0 , 16α, K1,j N ∼ (wa ,64α, K1, )
+ const 1− constj N ∼ (wa ,64α, Kj ) 1,j j α2
υ/7 ∼ a 1 × α 2 Nj (w , 64α, K1,j ) + λ0 M j K 1,j .
(XV.11)
By (I ∼ 1) and Corollary A.5.ii of [FKTo1], Nj∼ (wa ,64α, K1,j ) const ∼ 1− α Nj (wa ,64α, K1,j )
≤
ej (K1,j ) 1− const α ej (K1,j )
≤ const ej (K1,j )
and Nj∼ (wa ,64α, K1,j )2 const ∼ 1− α Nj (wa ,64α, K1,j )
≤ const ej (K1,j )2 ≤ const ej (K1,j ).
Therefore, by (XV.9) and (I ∼ 1), recalling that w a vanishes when ψ = 0, υ/7 a Nj∼ (z + ω1,1 , α, K1,j ) ≤ Nj∼ (w a , α, K1,j ) + const α1 + λ0 ej (K1,j ) 1 ≤ 64 Nj∼ (w a , 64α, K1,j ) + ≤ ej (K1,j )
1 2 ej (K1,j )
(XV.12)
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153
and ˜ ,ρ˜ , |z1,1|1, ˜ ,ρ˜ ≤ |z2,0|1, j j and z − wa − 4 4
1 4
∞ =1
const lj α8 M j
ej (K1,j ) ≤
1 lj α7 M j
a (j ) ˜ (−1) (12)+1 Ant L (w4 ; Cu ,Dj ) 3,j ,ρ˜
≤
ej (K1,j ) (XV.13)
const l e (K1,j ). α 10 j j
(XV.14) Furthermore, by (XV.11), (I ∼ 1) and Corollary A.5.ii of [FKTo1], a Nj∼ dds z(K + sK ) + ω1,1 (K + sK ) − w a (K + sK ) s=0 , α, K1,j υ/7 ≤ const α12 + λ0 ej (K1,j ) M j K 1,j . (XV.15) and
a Nj∼ dds z(K + sK ) + ω1,1 (K + sK ) s=0 , α, K1,j ≤ Nj∼ dds w a (K + sK )s=0 , α, K1,j υ/7 +const α12 + λ0 ej (K1,j ) M j K 1,j 1 ≤ 64 Nj∼ dds w a (K + sK )s=0 , 64α, K1,j + 21 ej (K1,j ) M j K 1,j ≤ M j ej (K1,j ) K 1,j .
(XV.16)
Verification of (O ∼ 1). Observe that B(k) and δB(k; K) vanish unless k is in the support of ν (j ) . Also, by (I ∼ 1), w a (φ, ψ; K) − wa (0, ψ; K) vanishes unless νˆ (<j ) φ is nonzero. Consequently, z(φ, ψ; K) − z(0, ψ; K), and hence w a (φ, ψ; K) − w a (0, ψ; K), vanishes unless νˆ (≤j ) φ is nonzero. The bounds of (O ∼ 1) follow from Lemma XV.5, (XV.12) and (XV.16). Verification of (O ∼ 2). By Lemma XV.5, ω4a (φ, ψ; K) + G4a (φ; K) = z4 (φ, ψ; K) + G4a (φ; K) = z4 (φ, ψ; 0) + G4a (φ; 0) + δf0 (φ, ψ; K), where δf0 (φ, ψ; K) = z4 (φ, ψ; K) + G4a (φ; K) − z4 (φ, ψ; 0) − G4a (φ; 0). As the corresponding kernel δf0 (K) obeys
a d d δf (K + sK ) = (K + sK ) − w (K + sK ) + z 4 0 4 ds ds
(j ) d (K ds δf
(XV.15) and (I ∼ 2) give d δf (K + sK ) ˜ ds 0 s=0 3,
+
j ,ρ˜
≤
const α4
υ/7
α 2 + λ0
+ sK ),
1 64α 4 B2 × ej (K1,j ) M j K 1,j .
By (XV.8), ˆ + δB(0)φ; ˆ z4 (φ, ψ; 0) + G4a (φ; 0) = z4 (φ, ψ + Bφ 0) + G4a (φ; 0) ˆ 0) + G4a (φ; 0) + δf1 (φ, ψ), = z4 (φ, ψ + Bφ;
(XV.17)
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J. Feldman, H. Knörrer, E. Trubowitz
where ˆ + δB(0)φ; ˆ 0). ˆ 0) − z4 (φ, ψ + Bφ; δf1 (φ, ψ) = z4 (φ, ψ + Bφ a (0), By Lemma XV.7.ii and Theorem XVII.6 of [FKTo3], with w = w a (0) − ω1,1 (j )
Cκ = Cu(0) , Dκ = Dj (u; 0), independent of κ, Bκ = B + (κ0 + κ)δB(0), where υ/7
0 ≤ κ0 ≤ 1, γ = λ0 , µ = Z=
lj λ1−2υ 0 α6
const ,
=
λ1−υ 0 , M j −1
cB = const , X = Y = = 0 and
, the corresponding kernel δf1 is bounded by ˜ δf 1 3,
j ,ρ˜
1−2υ
υ/7 λ0
≤ const λ0
lj
α6
1 c . α 4 B2 j
(XV.18)
Next, let δf2 = shear z4 (0) − w4a (0) −
1 4
∞
(−1) (12)+1 Ant L
=1
(j ) w4a (0); Cu(0) , Dj (u(0); 0) , B .
By (XV.14), Lemma XV.7.ii and Lemma XVII.5 of [FKTo3], with cB = const , X = 0, XB = 1, const 1 δf (XV.19) 2 3, ,ρ˜ ≤ α 10 lj cj ≤ α 9 lj cj . j
Define δf3 = shear
! 1 4
∞
(−1) (12)+1 Ant Vpp
=1
(j ) (≥j +1) pp L w4a (0); Cu(0) , Cu(0) ,B
and δf
(j +1)
(K) = δf0 (K)+δf1 +δf2 +δf3
δf
(j +1)
"
(j +1) (φ, ψ; K) = Gr φ, ψ; δf (K) ,
where Gr(φ, ψ, f ) is the Grassmann function associated to the kernel f as in Definition XIV.14.ii. By Proposition VII.6, for ≥ 1, 1/n (j ) (≥j +1) pp ≤ const lj 0 cj |w4a (0)||+1 Vpp L w4a (0); Cu(0) , Cu(0) 3,j 3,j
so that (j ) (≥j +1) pp Vpp L w4a (0); Cu(0) , Cu(0)
3,j ,ρ˜
a 1/n0 +1 ≤ const λ1−υ 0 lj cj | w4 (0)||3,j ,ρ˜ .
The hypotheses of this proposition are fulfilled by parts (i) and (ii) of Lemma VIII.7. Observe that, by (I ∼ 1), 4 lj B 2 1 M 2j lj ej (K1,j )(64α) M j lj | w0,4 (K)||3,j ,ρ˜ ≤ ej (K1,j ) so that |w0,4 (K)||3,j ,ρ˜ ≤
1 e (K1,j ). (64α)4 B2 j
In particular, |w4a (0)||3,j ,ρ˜ ≤
1 c . (64α)4 B2 j
Two Dimensional Fermi Liquid. 3: Fermi Surface
155
Therefore, by Lemma XVII.5 of [FKTo3] and Corollary A.5.ii of [FKTo1], δf 3 3,
≤
j ,ρ˜
≤
∞ =1 ∞ =1
1/n0
cj
1/n0
cj
const λ1−υ 0 lj const λ1−υ 0 lj
≤ const
1/n λ1−υ 0 l 0 c3j α8 j
≤ const
1/n λ1−υ 0 l 0 cj . α8 j
|w4a (0)||+1 3,j ,ρ˜
+1 1 c α4 j 1
1 − const
λ1−υ 1/n 0 l 0 c2j α4 j
(XV.20)
Hence, by (XV.18), (XV.19) and (XV.20), (j +1) δf (0)3,
j ,ρ˜
≤
λ1−2υ 0 l α 10 j
and by (XV.17), d δf (j +1) (K + sK ) ˜ ds s=0 3,
lj α9
+
j ,ρ˜
≤
(≥j +1)
Observe that Dj (u(0); 0) = Cu(0)
+
1/n const λ1−υ 0 lj 0 α8
1/n
cj ≤
lj +10 c α8 j
(XV.21)
1 e (K1,j ) M j K 1,j . α 4 B2 j
(XV.22)
and, by Lemma XIV.3,
(j ) (≥j +1) (j ) (≥j +1) pp = Vpp L w4a (0); Cu(0) , Cu(0) L w4a (0); Cu(0) , Cu(0) (j ) (≥j +1) ph +Vph L w4a (0); Cu(0) , Cu(0) ˆ 0) is so that the kernel of z4 (φ, ψ + Bφ; shear w4a (0) +
1 4
∞
(−1) (12)+1 Ant Vph
=1
(j ) (≥j +1) ph L w4a (0); Cu(0) , Cu(0) ,B
+δf2 + δf3 .
(XV.23)
To verify (O ∼ 2), set v (i) = v (i) for all 2 ≤ i ≤ j − 1, v (j ) = δf (j ) (0) and define f (i) , 2 ≤ i ≤ j , by (i) [i,j ] shear f (i) , B if 2 ≤ i ≤ j − 1 (i) (j ) f = shear v , Cu(0) (k)A(k) = , shear δf (0), B if i = j Observe that v (j ) vanishes unless all of its φ momenta are in the support of ν (<j ) , since the same is true for δf (j ) (0) by (I ∼ 1) and (I ∼ 2). Since B(k) is supported on the j th (i) (i) neighbourhood f j = shear fj , B for all 2 ≤ i ≤ j − 1. Consequently, observing that the pure ψ parts of f (i) and f (i) coincide and that only the pure ψ parts contribute to internal ladder vertices, shear Ant Vph L(j ) (p, f) , B = Ant Vph L(j ) (p, f ) . j
j
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j −1 (i) By (I ∼ 2), the kernel of ω4a (φ, ψ; 0) + G4a (φ; 0) is δf (j ) (0) + i=2 fj + 1 (j ) f) ˆ 0) + G a (φ; 0) is so that the kernel of ω4a (φ, ψ + Bφ; 4 8 Ant Vph L (p, j (j ) j (i) 1 f ) . Therefore, by the Definition VII.7 of iteri=2 f j + 8 Ant Vph L (p, j
ated particle–hole ladders, ω4a (φ, ψ; K) + G4a (φ; K) ∞ ˆ w4a (0) + 1 (−1) (12)+1 Ant = Gr φ, ψ + Bφ; 4 =1 (j +1) (j ) (≥j +1) ph + G4a (φ; 0) + δf (φ, ψ; K) × Vph L w4a (0); Cu(0) , Cu(0) j (i) (j +1) = Gr φ, ψ; f j + δf (K) + 18 Ant Vph L(j ) (p, f ) i=2
ˆ 1 +Gr φ, ψ + Bφ; 4 = Gr φ, ψ;
j
∞
(−1) (12)+1 Ant Vph
=1 (j +1)
f j + δf (i)
j
(j ) (≥j +1) ph L w4a (0); Cu(0) , Cu(0)
(K) + 18 Ant Vph L(j +1) (p, f ) .
i=2
The estimates on δf (j +1) (K)3, ,ρ˜ required for (O ∼ 2) were proven in (XV.21) and j (XV.22). By (I ∼ 2), (j ) v 3,
j ,ρ˜
= δf (j ) (0)3,
j ,ρ˜
1/n
≤
lj 0 c α7 j
1/n l 0 and v (i)3, ,ρ˜ = v (i)3, ,ρ˜ ≤ iα 7 ci for 2 ≤ i ≤ j − 1. i
i
Verification of (O ∼ 3). Let ˇ 2 (k; K) = C 2G u(0) (k) + (<j )
1 (ik0 −e(k))2
j −1
δq (i) (k; K) +
j
q (i,) (k)
=i
i=2
be the decomposition of (I ∼ 3) and set q (i,) (k) = q (i,) (k), q
(j,j )
2 ≤ i ≤ j − 1, i ≤ ≤ j,
(k) = 2ˇz2,0 (k; 0) − E (j ) (k; 0),
δq (i) (k; K) = δq (i) (k; K),
2 ≤ i ≤ j − 1,
δq (j ) (k; K) = 2ˇz2,0 (k; K) − 2ˇz2,0 (k; 0) − E (j ) (k; K) + E (j ) (k; 0). By Lemmas XV.5 and XV.6, both conclusions of Lemma IX.5.i of [FKTo2] and the fact that the transpose of J is −J , ˇ 2 (k; K) = C 2G u(0) (k) + (≤j )
1 (ik0 −e(k))2
j i=2
δq (i) (k; K) +
j =i
q (i,) (k) .
Two Dimensional Fermi Liquid. 3: Fermi Surface
157
By (XV.13), Lemma XII.12 of [FKTo3], Lemma XV.7.iii and (XV.16), 1−9υ/7 1−8υ/7 (j,j ) ˜ λ λ l l lj q (k) ≤ 4 0 α 7 Mjj cj + const 0 α 6 Mjj cj ≤ λ1−2υ c , 0 Mj j
1−9υ/7 d ˜≤ const λ02 δq (j ) (K+sK ) ej K1,j + const ej + 1 K1,j K 1,j ds s=0 α 2 ≤ M ℵ −ℵ ej + 1 K1,j K 1,j , 2
if M is large enough. a (φ, ψ) vanishes unless νˆ (<j ) φ is nonzero. Hence, by conservation of Recall that ω1,1 a momentum, w1,1 vanishes and, by (XV.13) and Proposition XIX.4.ii of [FKTo4], j +1
a∼ w
˜
1,1 j +1 1,j +1 ,ρ˜
∼ = z1,1
˜
j +1 1,j +1 ,ρ˜ const lj e (K1,j ) α8 M j j 1 lj e (K1,j ) α7 M j j
≤ ≤
∼ ˜ ≤ const cj z1,1 1,
j ,ρ˜
and, by (XV.15), d w a∼ (K+sK ) ˜ j +1 s=0 1, ds 1,1
j +1 ,ρ˜
˜ ∼ = dds z1,1 (K+sK )j +1 s=01, ,ρ˜ ˜j +1 ∼ ≤ const cj dds z1,1 (K+sK )s=01, j ,ρ˜ υ/7 const 1 ≤ α 2 B α 2 + λ0 ej (K1,j ) K 1,j υ/8 ≤ α1 + λ0 α 21B ej (K1,j ) K 1,j .
Verification of (O ∼ 4). Apply Remark B.5 of [FKTo2].
3. Sector Refinement, ReWick Ordering and Renormalization. (j ) (j +1) ∈ D˜ in . Theorem XV.8. If (W, G, u, p) ∈ D˜ out then Oj (W, G, u, p)
The rest of this subsection is devoted to the proof of this theorem. Let (W , G , u , p ) = Oj (W, G, u, p). Lemma XV.9. (≤j ) λ1−υ l ˜ (≤j ) A(k)2 Cu (0) (k) − Cu(0) (k) ≤ const α0 5 Mjj+1 +1 cj,j +1 , (≤j ) λ1−υ ˜ (≤j ) A(k) Cu (0) (k) − Cu(0) (k) ≤ const α0 5 lj cj +1 , [i,j ] λ1−υ ˜ [i,j ] A(k) Cu (0) (k) − Cu(0) (k) ≤ const α0 5 lj cj +1 for all 2 ≤ i ≤ j , ˜ ) A(k)C (≤j ) (k)˜, A(k)C (≤j u(0) u (0) (k) ≤ const cj +1 , ˜ ] A(k)C [i,j ] (k)˜, A(k)C [i,j for all 2 ≤ i ≤ j . u(0) u (0) (k) ≤ const cj +1
158
J. Feldman, H. Knörrer, E. Trubowitz
Proof. Recall from Definition VIII.1 that uˇ (k; 0) − u(k; ˇ 0) = pˇ (j ) (k). Hence ˇ 0)]ν (≤j ) (k) [ik0 − e(k)]2 [uˇ (k; 0) − u(k; [ik0 − e(k) − u(k; ˇ 0)][ik0 − e(k) − uˇ (k; 0)] (j ) pˇ (k)ν (≤j ) (k) = . ˇ 1 − iku(k;0) 1 − ikuˇ0 (k;0) −e(k) 0 −e(k)
(≤j ) (≤j ) A(k)2 Cu (0) (k) − Cu(0) (k) =
By (XV.4), (XV.5) and (IX.21), (≤j ) ˜ ν (k) ≤
const cj + 1 , 2
u(0) ˇ ˜, uˇ (0)˜≤ const M1j cj , 1−υ (j ) ˜ pˇ (k) ≤ 2 λ0 5 ljj cj . M α ˇ u (k;0) 1 1 (≤j ) , we have, using the Since iku(k;0) ≤ and 2 ik0 −e(k) ≤ 2 on the support of ν 0 −e(k) second bound of (XV.4) to control the denominator, ˜ pˇ (j ) (k)ν (≤j ) (k) ≤ const ˇ uˇ (k;0) 1 − 1 − iku(k;0) ik0 −e(k) 0 −e(k)
λ1−υ lj 0 c 1 α5 M j j + 2
≤ const
λ1−υ lj +1 0 c . α 5 M j +1 j,j +1
The proof of the second and third bounds are virtually identical, using [i,j ] ˜ ν (k) ≤ const c
j + 21 ,
and with one additional use of the second bound of (XV.4). The proof of the final two bounds uses ν I (k) ˇ 0 −e(k)−u(k;0)
I (k) = (ik0 − e(k)) ik (ik0 − e(k))Cu(0)
=
ν I (k) 1−
u(k;0) ˇ ik0 −e(k)
with I = [i, j ] and I = (≥ j ), and (XV.5) and the corresponding properties with u(0) replaced by u (0). Proof of Theorem XV.8. Let w=
m,n s1 ,··· ,sn ∈j
dη1 ···dηm dξ1 ···dξn
wm,n (η1 ,··· ,ηm (ξ1 ,s1 ),··· ,(ξn ,sn ); K)
× φ(η1 ) · · · φ(ηm ) ψ((ξ1 ,s1 )) · · · ψ((ξn ,sn ) ) be the j –sectorized representative of W specified in (O1). Choose the j +1 –sectorized representative w of W as in (IX.28), where w , the j –sectorized representative, was defined in (IX.18) and w˜ was defined in (IX.16). Recall that, by (IX.19), ˜ 0; K ) − w(0, ˜ 0; K ). G (φ; K ) = G(φ; K(K )) + w(φ,
(XV.24)
Two Dimensional Fermi Liquid. 3: Fermi Surface
159
Preparation for the verification of (I ∼ 1), (I ∼ 2) and (I ∼ 3). Recall from (IX.29) that w(φ, ˜ ψ; K ) = :w(φ, ψ; K(K )):ψ,−Ej (K ;q0 ) , where Ej (K ; q) was defined just before (IX.16). We proved in Lemma B.1.iii that # λ1−υ lj lj |E(K ; q0 )| ≤ |ik00−e(k)| . Hence, by Proposition XVI.8.ii of [FKTo3], 2λ1−υ 0 lj B3 M j ≤ # # lj B ˜ λ1−υ 0 lj M j is an integral bound for Ej for the configuration | · | p,j ,ρ˜ of seminorms. Hence by Corollary II.32.ii of [FKTr1], 8λ1−υ Nj∼ w˜ a (K ) − w a K(K ) , α2 , X ≤ α02 lj Nj∼ w a (K(K )), α, X (XV.25) for all X ∈ Nd+1 . In particular Nj∼ w˜ a (K ), α2 , X ≤ As in (IX.35), Nj∼ dds w˜ a (K + sK )s=0 − ≤
const (α−1)2
3 2
Nj∼ w a (K(K )), α, X .
a d α ds w (K(K +sK )) s=0 , 2
(XV.26)
K 1,j +1
,
M j ej +1 (K 1,j +1 )K 1,j +1
and Nj∼
≤
(XV.27)
d ˜ a (K + sK )s=0 , α2 , K 1,j +1 ds w j +ℵ ej +1 (K 1,j +1 )K 1,j +1 . const M
(XV.28)
Verification of (I ∼ 3). Let w˜ 2,0 be kernel of the part of w˜ that is of degree two in φ and degree zero in ψ. By (XV.24) and (O ∼ 3), ˇ 2 (k; K(K )) + 2(w˜ 2,0 )ˇ(k; K ) ˇ 2 (k; K ) = 2G 2G =
(≤j ) 1 Cu(0) (k) + (ik −e(k)) 2 0
j
δq (i) (k; K(K )) +
q (i,) (k)
=i
i=2 ˇ
j
+2(w˜ 2,0 ) (k; K ) (<j +1)
= Cu (0)
(k) +
1 (ik0 −e(k))2
j i=2
δq (i) (k; K ) +
j +1
q (i,) (k)
=i
with q (i,) (k) = q (i,) (k) q δq q
(i,j +1) (i)
2 ≤ i ≤ j, i ≤ ≤ j,
(k) = δq (k; K(0))
2 ≤ i ≤ j − 1,
(i)
(k; K ) = δq (k; K(K )) − δq (k; K(0)) 2 ≤ i ≤ j − 1, (≤j ) (≤j ) a ˇ (k) = δq (j ) (k; K(0))+2(w˜ 2,0 ) (k; 0)+[ik0 − e(k)]2 Cu(0) (k) − Cu (0) (k) , (i)
(i)
(j,j +1)
a ˇ a ˇ δq (j ) (k; K ) = δq (j ) (k; K(K )) − δq (j ) (k; K(0)) + 2(w˜ 2,0 ) (k; K ) − 2(w˜ 2,0 ) (k; 0).
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J. Feldman, H. Knörrer, E. Trubowitz
By (O ∼ 3), Lemma IX.8.i and Remark A.6 of [FKTo1], (i) δq (k; K(0))˜ = δq (i) (k; δK(0))˜ ≤M
ℵ (j −i)
ei+ 1 ,j + 1 δK(0)1,j δK(0)1,j 2
≤ const
2
M ℵ −ℵ M ℵ (j −i)
≤ M ℵ (j +1−i)
1√
M ℵ −ℵ
if i < j if i = j
λ1−υ lj 0 c 1 1 α 6 M j i+ 2 ,j + 2
λ1−υ lj +1 0 c α 5 M j +1 i,j +1
if α is large enough. This implies the desired bound on q (i,j +1) for i ≤ j −1. By Lemma a = 0), (O ∼ 1) and Lemma IX.8.iii, XII.12 of [FKTo3], (XV.25) (recall that w2,0 a ˇ (w˜ ) (k; 0) ˜≤ 2 |(w˜ a )ˇ(k; 0)|1, ˜ ≤ 2,0 2,0 j ≤
λ2−3υ lj 0 c . α 4 M j j,j +1
1−9υ/7
const λ0 α2 M j
So, by Lemma XV.9, ℵ (j,j +1) ˜ λ1−2υ M q ≤ λ1−υ ljj+1 c + 2 0α 4 M ℵ + 0 M +1 j,j +1 α 5 By (XV.28), d ˜ (w˜ a )ˇ(K + sK ) ≤ 2,0 ds s=0 ≤
8λ1−υ 0 l e (δK(0)1,j ) α2 j j
const α5
≤
λ1−υ lj +1 0 c . α 4 M j +1 j,j +1
1−9υ/7
const λ0 M j +ℵ ej +1 (K 1,j +1 )K 1,j +1 α2 M j λ1−2υ 0 e 1 (K 1,j +1 )K 1,j +1 (XV.29) α 2 j + 2 ,j +1
and, by (O ∼ 3) and parts (ii) and (iii) of Lemma IX.8, ˜ d ˜ (i) d ) + s d K(K +xK ) δq (i) (K(K +sK )) = K( δq K ds ds dx s=0 x=0 s=0 ≤ M ℵ (j −i) ei+ 1 ,j + 1 K(K )1,j 2 2 1 if i < j d √ × dx K(K +xK )x=0 ℵ −ℵ 1,j M if i = j ≤ const M ℵ (j −i) ei+ 1 ,j + 3 (K 1,j +1 ) M ℵ e0,j (K 1,j +1 )K 1,j +1 M ℵ −ℵ 2
≤
1 2
2
M ℵ (j +1−i) ei+ 1 ,j + 3 (K 1,j +1 ) K 1,j +1 2
2
if M is large enough. This and, when i = j , (XV.29) give the desired bound on dds δq (i) (K +sK ). a (K ) = w a (K ) As w1,1 ˜ 1,1 j +1 , we have, by Proposition XIX.4.ii of [FKTo4], (XV.25), ∼ ∼ (O 1), (O 3) and Lemma IX.8.iii, a∼ ˜ a∼ ˜ a∼ w (K ) K(K ) 1, ≤ const cj w˜ 1,1 (K ) − w1,1 1,1 1,j +1 ,ρ˜ j ,ρ˜ a∼ ˜ +w K(K ) 1,1
≤
j +1 1,j +1 ,ρ˜
8λ1−υ 1 0 const cj l N ∼ (w a (K(K )), α, K(K )1,j ) M j B(α/2)2 α 2 j j
Two Dimensional Fermi Liquid. 3: Fermi Surface
161
˜ a∼ +w1,1 (K(K ))j +1 1,
j +1 ,ρ˜
≤
const cj
≤ const ≤
λ1−υ 0 α4
lj Mj
lj l e (K(K )1,j ) + α17 Mjj ej (K(K )1,j ) Mj j
λ1−υ 0 α4
+
1 α7
ej +1 (K 1,j +1 )
1 lj +1 e (K 1,j +1 ). α 6 M j +1 j +1
(XV.30)
The derivative is bounded similarly, using (XV.27), (O ∼ 3), and parts (ii) and (iii) of Lemma IX.8, d a∼ w (K + sK ) ˜ ds 1,1 s=0 1,j +1 ,ρ˜ d a∼ ˜ a∼ K(K + sK ) s=01, ≤ const cj ds w˜ 1,1 (K + sK ) − w1,1 j ,ρ˜ d a∼ ˜ + w K(K + sK ) ≤
≤
ds 1,1 j +1 s=0 1,j +1 ,ρ˜ const e (K 1,j +1 )K 1,j +1 (α−1)2 α 2 B j +1 υ/8 + α1 + λ0 α 21B ej (K(K )1,j ) ddx K(K + xK )x=0 1, j 1
α2 B
const (α−1)2
+ const M ℵ
1
α
υ/8
+ λ0
ej +1 (K 1,j +1 )K 1,j +1 . (XV.31)
Verification of (I ∼ 2). Observe that by (XV.25), (O ∼ 1) and Lemma IX.8.iii, 2 ˜ M 2j α 4 lj B 1 a∼ ˜ 4 (K ) − w4a∼ K(K ) 3, lj 16 M j lj w ,ρ˜ ≤ ≤
8λ1−υ 0 l α2 j 1−υ 8λ0 l α2 j
≤ const
j
∼
Nj w a (K(K )), α, K(K )1,j
ej (K(K )1,j )
λ1−υ 0 l e (K 1,j +1 ) α 2 j j +1
so that, by Proposition XIX.4.ii of [FKTo4] and Corollary A.5.ii of [FKTo1], ˜ λ1−υ a∼ ≤ const α0 6 lj ej +1 (K 1,j +1 ). (XV.32) w˜ 4 (K ) − w4a∼ K(K ) j +1 3,j +1 ,ρ˜
Set, for all 2 ≤ i ≤ j , v (i) = v (i) and define [i,j ] (i) (i) [i,j +1) [i,j ] f = shear v , Cu (0) (k)A(k) = shear f (i) , Cu (0) (k) − Cu(0) (k) A(k) . Also set δf
(j +1)
(K ) = δf (j +1) (K(K ))j +1 + +
j (i) (i) fj +1 − fj +1
i=2 a a w˜ 4 (K ) − w4 (K(K )) j +1
+ 18 Ant Vph L(j +1) (p , f) j +1 (j +1) 1 (p , f ) . − 8 Ant Vph L j +1
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These definitions have been chosen so that w4a (K ) + Ga ˜ 4a (K )j +1 + Ga4 (K(K )) 4 (K ) = w
= δf (j +1) (K ) +
j i=2
(i) fj +1 + 18 Ant Vph L(j +1) (p , f )
j +1
.
Furthermore the required bounds 1/n (i) ˜ li 0 v ≤ c 3, ,ρ˜ α7 i
for all 2 ≤ i ≤ j
i
˜ are trivially satisfied, so it remains only to bound δf (j +1) (K )3, Since uˇ (k; 0) − u(k; ˇ 0) bourhood and
=
.
j +1 ,ρ˜ [i,j ] [i,j ] pˇ j (k), Cu (0) (k) − Cu(0) (k) is supported in the j th
neigh-
(i) [i,j ] (i) [i,j ] f j = shear fj , Cu (0) (k) − Cu(0) (k) A(k) .
Hence by Lemma XVII.5 of [FKTo3], with X = 0 and XB = const XV.9, followed by Proposition XIX.4.ii of [FKTo4], (i) (i) ˜ f j − fj 3,
j ,ρ˜
≤ const ≤ const
λ1−υ 0 l α5 j
and Lemma
(i) λ1−υ 0 l c f ˜ α 5 j j j 3,j ,ρ˜ ˜ λ1−υ 0 l c f (i)3, . α5 j j i ,ρ˜
(XV.33)
By Lemma XVII.5 of [FKTo3], with j replaced by i, X = 0, XB = const cj +1 and Lemma XV.9, followed by (O ∼ 2), 1/n ˜ (i) ˜ l 0 f ≤ const cj +1v (i)3, ≤ const iα 7 cj +1 3,i ,ρ˜ , ρ ˜ i 1/n (i) ˜ (i) ˜ li 0 f ≤ const c ≤ const c v j +1 3, ,ρ˜ 3, ,ρ˜ α 7 j +1 i
(XV.34)
i
so that, by Proposition XIX.4.ii of [FKTo4] and Corollary A.5 of [FKTo1], j (i) (i) ˜ fj +1 − fj +1 3,
j +1 ,ρ˜
i=2
≤ const
j i=2
1/n λ1−υ 0 l l 0 cj +1 α 12 j i
≤
λ1−υ 0 l c . α 11 j j +1
(XV.35) 1−9υ/7
, Also by Lemma XV.9 and Corollary XIV.15.ii, with j replaced by j + 1, ρ = λ0 (≤j ) (≤j ) (≤j ) ℵ ε = n0 , B = [Cu (0) (k) − Cu(0) (k)]A(k), B = Cu(0) (k)A(k) + sB , for 0 ≤ s ≤ 1, cB = const and c =
λ1−υ 0 l α5 j
(j +1) ˜ Ant Vph L(j +1) (p , f) (p , f )j +1 3, j +1 − Ant Vph L
j +1 ,ρ˜
≤ const
2−18υ/7 λ0 α5
lj cj +1 .
(XV.36)
f) depends only on p(2) , · · · , p(j −1) and f (2) , · · · , f (j ) . By DefinitionVII.7, L(j +1) (p, (j +1) In particular L (p , f) = L(j +1) (p, f).
Two Dimensional Fermi Liquid. 3: Fermi Surface
163
˜ To bound δf (j +1) (0)3, , we first use Proposition XIX.4.ii of [FKTo4], (XV.32), j +1 ,ρ˜ (XV.35) and (XV.36) to get the first line, then (O ∼ 2) to get the second line and Lemma IX.8 and Corollary A.5.ii of [FKTo1] to get the third line, (j +1) ˜ δf (0)3,
j +1 ,ρ˜
≤ const cj ≤ const
l1/n0 j +1 α8
l1/n0 j +1 α8
+
+
˜ ≤ const cj δf (j +1) (K(0))3,
j ,ρ˜
1 B2 α 4
λ1−υ 0 l α5 j
+ const
λ1−υ 0 l c α 5 j j +1
λ1−υ M j K(0)1,j ej (K(0)1,j ) + const α0 5 lj cj +1
cj +1
1/n
≤
lj +10 c , α 7 j +1
(XV.37) l
1/n0
j +1 (j +1) , we first use Proposition since const λ1−υ 0 lj ≤ 2α 2 . To bound the derivative of δf XIX.4.ii of [FKTo4] to get the first line, then (O ∼ 2) and (XV.27) to get the second line and Lemma IX.8 and Corollary A.5.ii of [FKTo1] to get the third line,
d δf (j +1) (K + sK ) ˜ ds s=0 3,j +1 ,ρ˜ d ˜ (j +1) ≤ const cj ds δf (K(K + sK ))s=03, j ,ρ˜ d a∼ ˜ +const cj ds w˜ 4 (K + sK )s=0 − dds w4a∼ (K(K +sK ))s=03, j ,ρ˜ jd 1 ≤ const α 4 B2 ej (K(K )1,j )M dx K(K + xK ) x=0 1, j
const + (α−1) 2 α 4 B2
≤ ≤
M ej +1 ( j
K 1,j +1
)K 1,j +1
1 1 const const 1−ℵ + e (K 1,j +1 )M j +1 K 1,j +1 (α−1)2 j +1 M α 4 B2 1 e (K 1,j +1 )M j +1 K 1,j +1 64α 4 B2 j +1
(XV.38)
if M and α are large enough. Verification of (I ∼ 1). Let ω˜ n and ωn be the parts of w˜ and w , respectively, that are of degree n in φ and ψ combined. By (XV.37), (XV.38), (XV.34), Proposition XIX.4.ii 1−9υ/7 (≤j ) of [FKTo4] and Corollary XIV.15.i (with ρ = λ0 , ε = nℵ , B = Cu (0) (k)A(k), 0 cB = const ) w a∼ (K ) ˜ 4 3,
j +1 ,ρ˜
˜ ≤ δf (j +1) (K )3,
j +1 ,ρ˜
+
j f (i) ˜ j +1 3,
j +1 ,ρ˜
i=2
˜ + 18 Ant Vph L(j +1) (p , f ) j +1 3,j +1 ,ρ˜ const const j +1 K ≤ α 41B2 M 1,j +1 ej +1 (K 1,j +1 ) 1−ℵ + (α−1)2 M +const ≤
1 α 4 B2
j +1 i=2
const α3
1/n
li 0 c α 7 j +1
+
const M 1−ℵ
+ const λ1−2υ cj +1 0
+
const (α−1)2
ej +1 (K 1,j +1 ).
164
J. Feldman, H. Knörrer, E. Trubowitz
Consequently, by Proposition XIX.1 of [FKTo4], a∼ ˜ 1 const const w (K ) ≤ +M 2 1−ℵ + 4 4 1, ,ρ˜ α3 α B l
const (α−1)2
j +1
j +1
ej +1 (K 1,j +1 ).
Therefore, by Corollary A.5 of [FKTo1],
Nj∼+1 (ω4a (K ), 64α, K 1,j +1 )
˜ ˜ ≤ 224 α 4 B2 lj +1 ej +1 (K 1,j +1 ) w4a (K )1, + w4a∼ (K )2, , ρ ˜ j +1 j +1 ,ρ˜ ˜ ˜ + lj1+1 w4a∼ (K )3, + lj1+1 w4a∼ (K )4, j +1 ,ρ˜ j +1 ,ρ˜ ˜ ˜ 25 4 2 a∼ ≤ 2 α B lj +1 ej +1 (K 1,j +1 ) w4 (K )1, ,ρ˜ + lj1+1 w4a∼ (K )3, , ρ ˜ j +1 j +1
const const const ≤ α 3 + M 1−ℵ + (α−1)2 ej +1 (K 1,j +1 ) ≤ 13 ej +1 (K 1,j +1 )
(XV.39)
˜ ˜ if M and α are large enough. We have used that | · |2, ≤ | · |1, and j +1 ,ρ˜ j +1 ,ρ˜ a a ˜ ˜ | · |4, ≤ | · |3, . Similarly, since ω2,0 (K ) = ω0,2 (K ) = 0, (XV.30) implies j +1 ,ρ˜ j +1 ,ρ˜ a∼ ˜ Nj∼+1 (ω2a (K ), 64α, K 1,j +1 ) ≤ 212 α 2 BM j +1 ej +1 (K 1,j +1 )w1,1 (K )1,
j +1 ,ρ˜
≤ ≤
const l e (K 1,j +1 )2 α 4 j +1 j +1 1 3 ej +1 (K 1,j +1 ).
(XV.40)
By (IX.18), (XV.26), (O ∼ 1) and Lemma IX.8.iii,
Nj∼ (w a (K ), α2 , 0) ≤ ≤ ≤ ≤
a 3 ∼ 2 Nj (w (K(K )), α, 0) a 3 ∼ 2 Nj (w (K(K )), α, K(K )1,j ) 3 2 ej (K(K )1,j ) const ej +1 (K 1,j +1 )
so that, by Corollary XIX.9 of [FKTo4] and Corollary A.5 of [FKTo1],
Nj∼+1 (w a (K ) − ω4a (K ) − ω2a (K ), 64α, K 1,j +1 ) 1 ≤ M (1−ℵ)/8 ej +1 (K 1,j +1 ) Nj∼ w a (K ) − ω4a (K ) − ω2a (K ), α2 , K 1,j +1 1 ≤ M (1−ℵ)/8 ej +1 (K 1,j +1 )2 Nj∼ w a (K ), α2 , 0 ≤ const
1 e (K 1,j +1 )3 M (1−ℵ)/8 j +1
≤ 13 ej +1 (K 1,j +1 ).
(XV.41)
Combining (XV.39), (XV.40) and (XV.41), we get Nj∼+1 (w (K ), 64α, K 1,j +1 ) ≤ ej +1 (K 1,j +1 ). By (IX.18) and (XV.28), Nj∼ dds w a (K + sK )s=0 ,
α 2
,
K 1,j +1
≤ const M j +ℵ ej +1 (K 1,j +1 )K 1,j +1 .
Two Dimensional Fermi Liquid. 3: Fermi Surface
165
Therefore by Corollary XIX.9 of [FKTo4] and Corollary A.5 of [FKTo1], Nj∼+1 dds w a (K + sK )s=0 − dds ω2a (K + sK )s=0 , 64α, K 1,j +1 ≤ const ej +1 (K 1,j +1 ) Nj∼ dds w (K + sK )s=0 , α2 , K 1,j +1 ≤ const M j +ℵ ej +1 (K 1,j +1 )2 K 1,j +1 ≤ 21 M j +1 ej +1 (K 1,j +1 )K 1,j +1 . By (XV.31), Nj∼+1 dds ω2a (K + sK )s=0 , 64α, K 1,j +1
υ/8 ℵ 1 const ≤ 642 M j +1 (α−1) ej +1 (K 1,j +1 )2 K 1,j +1 + λ 2 + const M 0 α ≤ 21 M j +1 ej +1 (K 1,j +1 )K 1,j +1 so Nj∼+1
a d ds w (K
+ sK )s=0 , 64α, K 1,j +1 ≤ M j +1 ej +1 (K 1,j +1 )K 1,j +1
as desired. That w a (φ, ψ; K) − w a (0, ψ; K) vanishes unless νˆ (<j +1) φ is nonzero is inherited from the same property of wa . Verification of (I ∼ 4). Bar/unbar invariance and k0 –reversal reality are inherited from (j ) the corresponding quantities in D˜ out by Remarks B.2 and B.5 of [FKTo2]. Proof of Theorem XII.1. Initialization at j = j0 . As at the beginning of §X, let ˜ (≤j0 ) V (ψ) (φ, ψ) − 1 φ w(φ, ˜ ψ; K) be the j0 –sectorized representative for 2 Cu(K)
(≤j0 ) Jφ J Cuj (K) 0
chosen in Theorems VI.12 and XIII.17 and set w(φ, ψ; K) = w(φ, ˜ ψ; K) − w(φ, ˜ 0; K).
Set ◦ ◦ ◦ ◦
v (2) = · · · = v (j0 ) = 0, f (2) = · · · = f (j0 ) = 0, q (i,) = 0 for all 2 ≤ i ≤ ≤ j0 δq (2) = · · · = δq (j0 −1) = 0,
except i = = j0 ,
and define δf (j0 +1) (K) = the kernel of the part of w(φ, ˜ ψ; K) that is quartic in (φ, ψ) , a q (j0 ,j0 ) = 2w˜ 2,0 (k; 0),
(≤j0 ) 2 (≤j0 ) a a δq (j0 ) (K) = 2w˜ 2,0 (k; K) − 2w˜ 2,0 (k; 0) + Cuj (K) (k) − Cuj (0) (k) ik0 − e(k) 0
=
0
(>j0 ) (k) a a ˇ K(k). 2w˜ 2,0 (k; K) − 2w˜ 2,0 (k; 0) − U ˇ(k)−ν 1+K(k)/(ik0 −e(k))
166
J. Feldman, H. Knörrer, E. Trubowitz
By Theorem XIII.17, with w a replaced by w˜ a , Nj∼0 w˜ a (K), α, K1,j0 ≤ const α 4 λυ0 ej0 K1,j0 , Nj∼0 dds w˜ a (K + sK )s=0 , α, K1,j0 ≤ const α 4 λυ0 ej0 K1,j0 . K 1,j0 . If λ0 is sufficiently small, depending on M, and α <
1 υ/10 λ0
these bounds and Lemma
XV.7.i (with C (≤j0 ) replacing C (j ) ) imply the bounds on
◦ wa (K) and dds w a (K + sK ) imposed by (O ∼ 1), ˜ ˜ ◦ δf (j +1) (0)3, and dds δf (j +1) (K + sK )s=03, imposed by (O ∼ 2), , ρ ˜ j j ,ρ˜ ◦ q (j0 ,j0 ) (k)˜and dds δq (j0 ) (K + sK )s=0 ˜imposed by (O ∼ 3), ◦ wa∼ (K) ˜ and d w a∼ (K+sK ) ˜ imposed by (O ∼ 3). 1,1
j0 +1
1,j0 +1 ,ρ˜
ds
j0 +1
1,1
s=0 1,j0 +1 ,ρ˜
The support properties required by (O ∼ 1) and (O ∼ 3) follow from the conclusion in Theorem XIII.17 that w˜ a (φ, ψ, K) − w˜ a (0, ψ, K) vanishes unless νˆ (≤j0 ) φ is nonzero. Finally (O ∼ 4) follows from Remark B.5 of [FKTo2] and the requirement, stated in the Introduction, §XI, to this part, that V satisfy the reality condition (I.1) and be bar/unbar exchange invariant. Recursive step j − 1 → j . In the proof of Theorem VIII.5 in §X, we constructed G rg rg (j ) so that Wj , Gj , uj , (p(2) ,··· ,p(j −1) ) ∈ Dout . By Theorems XV.4 and XV.8, we have, in rg (j ) addition, that Wj , Gj , uj , (p(2) ,··· ,p(j −1) ) ∈ D˜ out . Let ˇ (k) = C 2G j,2 u(0) (k) + rg
(≤j )
1 [ik0 −e(k)]2
j j
q (i,) (k)
i=2 =i
be the decomposition of (O ∼ 3), but with K = 0. By (O ∼ 3), l M ℵ (−i) M δ0 i M |δδ | . sup Dδ q (i,) (k) ≤ δ!λ1−2υ 0 M k
By (O ∼ 4), q (i,) (−k0 , k) = q (i,) (k0 , k) for all of the required i, .
Proof of Theorem I.7. By (O ∼ 2), the kernel of the quartic part of Gj (0), amputated by ik0 − e(k) rather than ˇ 1 = ik0 − e(k) − (k), is rg
G2 (k)
j (i) fj + 18 Ant Vph L(j +1) (p, fj ) , Pφ δf (j +1) (0) + j
i=2
j −1 (i) [i,j ] (i) where the i th component of fj is fj = shear v (i) , Cuj (k)A(k) , uj = i=2 pj and Pφ is projection onto the pure φ part. Recall from Definition VII.7 that the ladders L(n) (p, f ) were defined inductively by L(0) (p, f ) = 0, L(n+1) (p, f ) = L(n) (p, f )n + 2
∞ =1
(−1) (12)+1 L
wn ; Cu(n) , Cu(≥n+1) n n
ph
, (XV.42)
Two Dimensional Fermi Liquid. 3: Fermi Surface
where wn =
n
(i) 1 i=2 fn + 8 Ant
167
Vph L(n) (p, f )
n
. By the construction of Theorem
VIII.5, the quartic part of G, again amputated by ik0 −e(k) rather than ik0 −e(k)−(k), is ∞ Pφ f (i) + lim 81 Ant Vph L(j +1) (p, f ) i=2
j →∞
(≥i) i with f (i) = shear v (i) , CP (k)A(k) and P (k) = ∞ i=2 pˇ (k) as in Lemma XII.2. The sum makes sense even though different terms have different sectorization scales because φ arguments are not involved in sectorization. We shall prove convergence shortly. The quartic part of G, correctly amputated by ik0 − e(k) − (k), is ∞ Pφ F (i) + lim 81 Ant Vph L(j +1) (p, F ) , i=2
where
j →∞
F (i) = sct f (i) , ik0 −e(k)−(k) ik0 −e(k) ik0 −e(k)−P (k) ik0 −e(k) = sct shear v (i) , ν (≥i) ik0 −e(k)−P A (k) , 2 (k) ik0 −e(k)
(i) ik0 −e(k)−P (k) (≥i) = sct shear sct v , ik0 −e(k) , A2 (k) , ν
= sct shear sct v (i) , A1 (k) , ν (≥i) , A2 (k)
since, by (O ∼ 2), v (i) vanishes unless its φ momenta are in the support of ν (
j →∞
We have proven in Lemma XII.7.ii that A2 (k) is C 1/2 . So it suffices to prove that ∞ ˜ Pφ v˜ (i) and lim 81 Ant Vph L(j +1) (p, v) have the continuity properties specified i=2
j →∞
in the statement of the theorem. By (O ∼ 2), Lemma XII.7.i and Remark XIV.20.ii, (i) ˜ 1−11υ/7 1/n0 v˜ ≤ const λ0 li ci . 3, i
(XV.43)
Define, for f ∈ Fˇ 4, , |||f ||| ˜ = sup |g((k1 ,σ1 ,1),(k2 ,σ2 ,0),(k3 ,σ3 ,1),(k4 ,σ4 ,0))| σi ∈ {↑, ↓}, k1 + k3 = k2 + k4 , where g is the function on Bˇ4 such that f (0,··· ,0) (ηˇ 1 , · · · , ηˇ n ) = (2π)3 δ(ηˇ 1 + · · · + ηˇ n ) g(ηˇ 1 , · · · , ηˇ n ).
168
J. Feldman, H. Knörrer, E. Trubowitz
By Lemma XII.12 of [FKTo3], |||v˜ (i) ||| ˜ ≤ const λ0
1−11υ/7 1/n0 li
and, if D is dd–operator of type (4, 0) with |δ(D)| = 1, in the sense of Definition XIII.5, |||Dv˜ (i) ||| ˜ ≤ const λ0
1−11υ/7 1/n0 i li M .
By Lemma C.1, with α =
ℵ n0
and β = 1 −
ℵ n0 ,
∞
Pφ v˜ (i) is C ℵ/n0 .
i=2
The remaining estimates use cancellations between scales. For this reason, we want j −1 (i) to replace the j –dependent functions uˇ j = i=2 pˇ (k) in the recursive definition (XV.42) of iterated particle hole ladders by one j –independent function P (k) = ∞ (i) i=2 pˇ (k). As in Definition D.1, we define, recursively on 0 ≤ n < ∞, the compound particle hole ladder up to scale n as (0) LP (f ) = 0, (n+1)
LP
(f ) = LP (f )n + 2 (n)
where wP ,n =
∞
(−1) (12)+1 L
=1
(i) (n) 1 f + Ant V ( f ) L ph i=2 n P 8
n
n
w˜ j =
j
j
i=2 1 4
∞
(XV.44)
,
. For j ≥ 1 set
(i) ˜ v˜j + 18 Ant Vph L(j ) (p, v)
v˜ (j +1) = v˜ (j +1) −
(≥n+1) ph
(n)
wP ,n ; CP , CP
,
(−1) (12)+1 Ant
=1
+1) ph × Vph L (w˜ j ; CP(j ) ,CP(≥j +1) )ph −L (w˜ j ; Cu(jj) ,Cu(≥j ) j
j +1
.
By (D.3), with the replacements F (i) → v˜ (i) , F (i) → v˜ (i) , wj → w˜ j , vj → uj and v → P, j
w˜ j =
i=2
(j ) (i) v˜j + 18 Ant Vph LP (v˜ )
j
for all j ≥ 2. Set δL(n) = 2
∞
(−1) (12)+1
=1
(n) (≥n+1) ph L w˜ n ; Cu(n) , Cu(≥n+1) . − L w˜ n ; CP , CP n n
Subtracting (XV.44) from (XV.42) ˜ − L v) L(n+1) (p, P
(n+1)
˜ − L (v˜ ) + δL(n) . (v˜ ) = L(n) (p, v) n n P (n)
˜ = Lv (v˜ ) = 0, v) Since L(1) (p, (1)
˜ − L v) L(n+1) (p, P
(n+1)
(v˜ ) =
n i=1
(i)
δLn .
Two Dimensional Fermi Liquid. 3: Fermi Surface
169
˜ ρ = const λ1−11υ/7 By (XV.43), the hypotheses of Theorem XIV.12 with F = v, 0 and ε = nℵ0 are satisfied. Hence, by (D.10), (i) ˜ 3−33υ/7 Vph δL ≤ const λ0 li ci . 3, i
By Lemma XII.12 of [FKTo3], (i) ˜ Vph δL ≤ const λ3−33υ/7 li 0
DVph δL(i) ˜ ≤ const λ3−33υ/7 li M i
and
0
for all dd–operators, D, with |δ(D)| = 1. By Lemma C.1, with α = ℵ and β = 1 − ℵ, ∞ ∞ is C ℵ/n0 . We choose Pφ Vph δL(i) is C ℵ . Thus Pφ v˜ (i) + 18 Ant Vph δL(i) i=2 i=2 Nσ1 ,σ2 ,σ3 ,σ4 to be sct · , A2 (k) applied to this sum and Lσ1 ,σ2 ,σ3 ,σ4 (q1 , q2 , t) to be (n+1) (v˜ ) and evaluated at k1 = q1 + t , sct · , A2 (k) applied to lim Pφ Vph L P
n→∞
2
k2 = q1 − 2t , k3 = q2 + 2t and k4 = q2 + 2t . By [FKTl, Theorem I.22] the limit exists pointwise for all t = 0 and is continuous there. The same theorem provides the existence of continuous extensions of Lσ1 ,σ2 ,σ3 ,σ4 (q1 , q2 , (0, t)) and Lσ1 ,σ2 ,σ3 ,σ4 (q1 , q2 , (t0 , 0)) to t = 0.
Appendix C. Hölder Continuity of Limits Lemma C.1. Let α, β, C0 , C1 > 0 and M > 1. There is a constant C such that if f (t) =
∞
fj (t)
j =0
with
sup fj (t) ≤ t
sup fj (t)) ≤ C1 M βj ,
C0 M αj
t
then β α f (t) − f (t ) ≤ C C α+β C α+β |t − t |α/(α+β) . 0 1
Proof. Since
0 fj (t) − fj (t ) ≤ 2 min C1 M βj |t − t |, Cαj M
we have ∞ f (t) − f (t ) ≤ fj (t) − fj (t ) j =0
≤
2
C1 M βj |t − t | + 2
0≤j <∞ C 1 ≥ 1 |t−t | M (α+β)j C0 β
≤ 2C1 MMβ −1
C1
C0 |t
0≤j <∞ C 1 ≤ 1 |t−t | M (α+β)j C0
−β/(α+β) − t | |t − t |
C0 M αj
170
J. Feldman, H. Knörrer, E. Trubowitz 1 +2C0 1−M −α β
C1
C0 |t
α/(α+β) − t |
α
= C C0α+β C1α+β |t − t |α/(α+β) with C = 2
Mβ M β −1
Mα M α −1
+
.
Appendix D. Another Description of Particle–Hole Ladders The estimates on iterated particle hole ladders in [FKTl] use cancellations between scales. For this reason, we want to replace the j –dependent functions uj in the recursive Definition XIV.11 of iterated particle hole ladders by one j –independent function. Definition D.1. Let F = F (2) , F (3) , · · · be a sequence of antisymmetric, spin independent, particle number conserving functions F (i) ∈ Fˇ 4,i and v(k) a function on R × R2 such that |v(k)| ≤ 21 |ık0 − e(k)|. We define, recursively on 0 ≤ j < ∞, the compound (j ) particle hole (or wrong way) ladders up to scale j , denoted by L(j ) = Lv (F ) , as L(0) = 0, L(j +1) = Lj + 2 (j )
where w =
j
(i) i=2 Fj
∞
(−1) (12)+1 L
=1
+ 18 Ant Vph L(j )
j
w; Cv(j ) , Cv(≥j +1)
ph
,
.
In [FKTl, Remark I.21] we prove Theorem D.2. For every ε > 0 there are constants ρ˜0 , const such that the following holds. Let F = F (2) , F (3) , · · · be a sequence of antisymmetric, spin indepen dent, particle number conserving functions F (i) ∈ Fˇ 4,i and p = p (2) , p(3) , · · · be a sequence of antisymmetric, spin independent, particle number conserving functions p(i) ∈ F0 (2, i ). Assume that there is ρ ≤ ρ˜0 such that for i ≥ 2, ˜ ≤ |F (i)|3, i Set v(k) =
∞
i=2 pˇ
ρ c M εi i
(i) (k) .
|p (i)|1,i ≤ ρMlii ci
pˇ (i) (0, k) = 0.
Then for all j ≥ 1,
(j +1) ˜ Vph L (F ) 3, ≤ const ρ 2 cj . v j
Proof that Theorem XIV.12 follows from Theorem D.2. Let F = F (2) , F (3) , · · · and (2) (3) p = p , p , · · · be as in the hypotheses of Theorem XIV.12. Because pˇ (i) vanishes at k0 = 0, Lemma XII.12 of [FKTo3] implies that (i) ρ li (i) i pˇ (k) ≤ 2|k0 | ∂ ≤ 2|ık0 − e(k)| M i M = 2ρ li |ık0 − e(k)|. ∂t(1,0,0) | p | 1,i t=0
(D.1)
Two Dimensional Fermi Liquid. 3: Fermi Surface
Set wj =
j
(i) i=2 Fj
171
+ 18 Ant Vph L(j ) (p, F
j
and vj (k) =
j −1 i=2
pˇ (i) (k). By
definition F ) = L(j ) (p, F )j + 2 L(j +1) (p,
∞
(−1) (12)+1
=1
ph +1) L wj ; Cv(jj ) , Cv(≥j . (D.2) j
For j ≥ 1 set F (j +1) = F (j +1) −
1 4
∞
(−1) (12)+1 Ant
=1
+1) ph × Vph L (wj ; Cv(j ) ,Cv(≥j +1) )ph − L (wj ; Cv(jj ) ,Cv(≥j ) j
j +1
.
By induction ones sees that for j ≥ 2, wj =
j i=2
(i) ) Fj + 18 Ant Vph L(j v (F )
j
(D.3)
.
Since pˇ (i) (k) is supported on the i th extended neighbourhood (defined in Defini(j ) (j ) tion I.2) and ν (j ) is supported on the j th shell, Cv = Cv +pˇ (j ) +pˇ (j +1) . By Proposition j XIX.4.iii of [FKTo4], (j ) lj ρl p + p (j +1) ≤ const |p (j )|1,j + lj +1 cj p (j +1)1, ≤ const M jj cj +1 . j 1, j +1
j
It follows from (D.1) that |v(k) − vj (k)| ≤ const ρ lj |ık0 − e(k)|. By Corollary XIX.13 of [FKTo4], j −1 (i) pj
1,j
i=2
j +1 (i) , pj i=2
1,j
≤ const
We apply Proposition D.7.ii of [FKTo3] with u = j −1 i=2
j +1 i=2
ρ c . M j j +1
(i)
pj , v =
∞ i=2
pj , u = v = (i)
(i)
pj , ε = const ρ lj , X = τ2 cj +1 (where τ2 was defined in Lemma XIII.6 of
[FKTo3]), f = wj and get, that for all ≥ 1, ph +1) ph ˜ Vph L wj ; Cv(j ) , Cv(≥j +1) − L wj ; Cv(jj ) , Cv(≥j j
≤ ρ lj const cj +1
3,j
˜ +1 . |wj |3, j
(D.4)
Proposition XIX.4.ii of [FKTo4] implies that also ph +1) ph Ant Vph L wj ; Cv(j ) , Cv(≥j +1) − L wj ; Cv(jj ) , Cv(≥j j
≤ ρ lj const cj +1
˜
j +1 3,j +1
˜ +1 . |wj |3, j
(D.5)
172
J. Feldman, H. Knörrer, E. Trubowitz
We now prove by induction that, if ρ0 is small enough, then for all j ≥ 2, (j ) ˜ F ≤ 2 ρ max lj , M1εj cj , 3,j (j ) ˜ F − F (j )3, ≤ const ρ 3 lj −1 cj .
(D.6)
j
The induction beginning is trivial since F (2) = F (2) . For the induction step, assume that (D.6) holds for j ≤ j . Then, using Proposition XIX.4.ii of [FKTo4], j (i) F ˜ i=2
j
3,j
≤
const
ρ
j
max li ,
i=2
1 M εi
cj ≤
const
ρ cj .
(j ) Since Lv (F ) only depends on F (2) , · · · , F (j −1) , Theorem D.2 applies whenever 1 ρ ≤ 2 ρ˜0 , and
(j ) ˜ Vph L (F ) v 3,
≤ const0 ρ 2 cj −1
j −1
(D.7)
with a j –independent constant const 0 . Therefore, by (D.3) ˜ ≤ const (1 + const 0 ρ) ρ cj . |wj |3, j
(D.8)
Hence, by the Definition of F (j +1) and (D.5), ˜ (j +1) F − F (j +1)3,
j +1
≤ const ρ 2 lj cj
∞
const ρ cj cj +1
=1
≤ const ρ lj cj +1 . Therefore (j +1) ˜ F 3,
j +1
˜ ≤ F (j +1)3,
j +1
3
+const ρ 3 lj +1 cj +1 ≤ 2 ρ max lj +1 ,
1 M ε(j +1)
cj +1
and (D.6) is proven for all j . Set δL(j ) = 2
∞
(−1) (12)+1
=1
ph +1) (j ) (≥j +1) ; C , C . L wj ; Cv(jj ) , Cv(≥j − L w j v v j
Subtracting Definition D.1 from (D.2), +1) ) (j ) F ) − L(j (F ) = L(j ) (p, F )j − L(j L(j +1) (p, v v (F )j + δL .
Since L(1) (p, F ) = Lv (F ) = 0, (1)
+1) L(j +1) (p, F ) − L(j (F ) = v
j i=1
(i)
δLj .
(D.9)
Two Dimensional Fermi Liquid. 3: Fermi Surface
173
By (D.4) and (D.8), ∞ (j ) ˜ Vph δL ˜ +1 ≤ ρ lj const cj +1 |wj |3, 3, j j
=1
≤
∞ const
ρ 2 cj lj const ρ cj cj +1
=1
≤ const ρ 3 lj cj .
(D.10)
Hence by Proposition XIX.4.ii of [FKTo4]. (j ) ) ˜ Vph L (p, F ) − L(j v (F ) 3,
j −1
≤
j −1 (i) Vph δL
j −1
i=1
≤
j −1
const cj −2 Vph
˜
3,j −1
˜ δL(i) 3,
i
i=1
≤
j −1
const ρ 3 li ci cj −2
i=1
≤ const ρ 3 cj −1 . As pointed out in (D.7) , (D.6) implies that (j ) ˜ Vph L (F ) v 3,
j −1
≤ const0 ρ 2 cj −1
for all j ≥ 2 and Theorem XIV.12 follows by yet another application of Proposition XIX.4.ii of [FKTo4]. The inductive Definition XIV.11 of iterated particle–hole ladders is suited to direct application in the renormalization group analysis of §IX and §XV. Its relation to Definition D.1 of compound particle hole ladders has been exhibited above. In [FKTl] we use a more conceptual description of particle–hole ladders than that of Definition D.1. Corollary D.7 below and Remark I.21 of [FKTl] show that the description of particle– hole ladders in Definition D.1 agrees with that of Definition I.19 in [FKTl]. Therefore, Theorem D.2 follows from Theorem I.20 in [FKTl]. A compound particle–hole ladder of scale j may have a rung which is a particle–hole ladder of a scale i < j , and this rung may be “perpendicular” to the direction of the big ladder.
To make this concept precise, we define
174
J. Feldman, H. Knörrer, E. Trubowitz
Definition D.3. Let X be a measure space and F a four legged kernel on X. The associated flipped kernel is F f (x1 ,x2 ,x3 ,x4 ) = −F (x1 ,x3 ,x2 ,x4 ). The kernel F is called inversion symmetric if F (x4 ,x3 ,x2 ,x1 ) = F (x1 ,x2 ,x3 ,x4 ).
Lemma D.4. Let L be an inversion symmetric four legged kernel over Y . Then
ph = 13 (L + Lf ). Ant Vph (L)
The proof is left to the reader. Example D.5. (j ) (≥j +1) ph (j ) (≥j +1) over B and C Cu , Cu i) The propagators C Cu , Cu over B are inversion symmetric. ii) If f is an antisymmetric kernel over X then its particle hole reduction f ph is inversion symmetric. iii) If f1 , f2 are inversion symmetric four-legged kernels over Y and P is an inversion symmetric bubble propagator over B then (f1 • P • f2 )(z4 ,z3 ,z2 ,z1 ) = (f2 • P • f1 )(z1 ,z2 ,z3 ,z4 ).
iv) If f is an inversion symmetric four-legged kernel over Y , P an inversion symmetric bubble propagator over B and ≥ 1, then (f • P ) • f is also inversion symmetric. v) One proves by induction on j that the compound particle hole ladders L(j ) (p, F ) are inversion symmetric. Proposition D.6. Let be a sectorization. Let f be a particle number conserving, antisymmetric four legged kernel over X , L an inversion symmetric four legged kernel over Y and P a particle number conserving, inversion symmetric bubble propagator over B that obeys P (ξ1 , ξ2 , ξ3 , ξ4 ) = P (ξ2 , ξ1 , ξ4 , ξ3 ). Set w = f + 18 Ant Vph (L) . Then for ≥ 1, (12)+1
[w • P ] • w
ph
=
1 2
24 f
ph
+ L + Lf • phP • 24 f ph + L + Lf .
Proof. By Lemma VII.5.i and Lemma D.4,
ph [w • P ] • w ph ph = 21 (24)+1 f + 18 Ant Vph L • phP • · · · phP • f + 18 Ant Vph L = 21 24f ph + 3(Ant Vph L)ph • phP • · · · phP • 24f ph + 3(Ant Vph L)ph = 21 24f ph + L + Lf • phP • · · · phP • 24f ph + L + Lf .
(12)+1
Two Dimensional Fermi Liquid. 3: Fermi Surface
175
Corollary D.7. Let F = F (2) , F (3) , · · · be a sequence of antisymmetric, spin independent, particle number conserving functions F (i) ∈ Fˇ 4,i and v(k) a function on (j ) R × R2 such that |v(k)| ≤ 21 |ık0 − e(k)|. As in Definition D.1, let L(j ) = Lv (F ) be the compound particle hole ladder up to scale j . Then for j ≥ 0, L(j +1) = Lj + (j )
where F =
j
∞
(−1)
=1
(i) ph i=2 Fj
(j )
(j ) f
24F + Lj + Lj
(j ) (j ) f • C • · · · C • 24F + Lj + Lj ,
(j ) (≥j +1) and C = ph C Cv , Cv ).
Proof. Since L(j ) and C are inversion symmetric by Example D.5 and P = (j ) (≥j +1) C C v , Cv obeys P (ξ1 , ξ2 , ξ3 , ξ4 ) = P (ξ2 , ξ1 , ξ4 , ξ3 ), the corollary follows from Definition D.1 and Proposition D.6.
Notation Norms Norm
Characteristics
Reference
||| · |||1,∞
no derivatives, external positions, acts on functions
Definition V.3
· 1,∞
derivatives, external positions, acts on functions
Definition V.3
· ˜
derivatives, external momenta, acts on functions
Definition XIII.7
||| · |||∞
no derivatives, external positions, acts on functions
Definition VI.7
| · |p, ||| · |||1,
derivatives, external positions, all but p sectors summed
Definition VI.6
no derivatives, all but 1 sector summed
(II.6)
||| · |||3,
no derivatives, all but 3 sectors summed
(II.14)
˜ | · |p,
derivatives, external momenta, all but (p−# external momenta) sectors summed 2 like | · |1, , but for functions on R2 × 1 1 |ϕ||1,j + l |ϕ||3,j + 2 |ϕ||5,j if m = 0 j lj ρm;n lj |ϕ|| if m = 0 1,j M 2j lj B n/2 M 2j e (X) n |wm,n |j j m,n≥0 α lj Mj 1 m>0 (1−υ) max{m−2,2}/2 |||Gm |||∞ λ 06 1 ˜ ifm = 0 p=1 l[(p−1)/2] |f |p,j j ρ˜m;n ˜ ˜ ˜ |f |1, if m = 0 + l1 |f |3, + 12 |f |5, j j j
Definition XIII.12
˜ ,ρ˜ |f |p, j
˜ ρ˜m;n |f |p, j
Definition XIII.15
Nj∼ (w, α, X)
M 2j e (X) j m,n≥0 lj
· 1, |ϕ||j Nj (w, α, X) N(G ) |f |j˜
j
[Def’n E.3, [FKTo4]] Definition VI.6 Definition VI.7 Definition VI.7
Definition XIII.15
lj
l B (m+n)/2 ∼ α m+n j j |wm,n |j˜ M
Definition XIV.6 Definition XIII.15
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J. Feldman, H. Knörrer, E. Trubowitz
Spaces Not’n E Kj B Bˇ Bˇm
Y Fm (n; ) Fˇ m (n; ) Fˇ n; Fˇ m ,m (n; ) Fˇ
Description counterterm space space of future counterterms for scale j R × Rd × {↑, ↓} × {0, 1} viewed as position space R × Rd × {↑, ↓} × {0, 1} viewed as momentum space (ηˇ 1 , · · · , ηˇ m ) ∈ Bˇ m ηˇ 1 + · · · + ηˇ m = 0 = X0 ∪· X1 = Bˇ ∪· (B × ) = X−1 ∪· X0 ∪· X1 = Bˇ ∪· Bˇ ∪· (B × ) R × Rd × {↑, ↓} viewed as position space R × Rd × {↑, ↓} viewed as momentum space Bˇ ∪ · (B × ) n functions on Bm × B × , internal momenta in sectors n functions on Bˇ m × B × , internal momenta in sectors n functions on X that reorder to Fˇ m (n − m; )’s n m fns on Xm −1 × X0 × B × , internal momenta in sectors functions on X n that reorder to Fˇ m,m (n − m − m ; )’s
Reference Definition I.1 Definition VI.9 before Def VII.1 beginning of §VI Definition XIII.5 Definition XIV.1 Definition XIV.16 Definition VII.3 Definition XIV.1 Definition XIV.1 Definition VI.3.ii Definition XIII.14 Definition XIV.6 Definition XIV.17 Definition XIV.17
Din
formal input data for scale j
Definition III.8
Dout
formal output data for scale j
Definition III.9
Din
input data for scale j
Definition IX.1
output data for scale j
Definition IX.2
more input data for scale j
Definition XV.1
more output data for scale j
Definition XV.3
X X B Bˇ
n; (j,form ) (j,form ) (j )
(j ) Dout (j ) ˜ Din (j ) D˜ out
Other Notation Not’n r0 r M const const ν (j ) (k)
ν (≥j ) (k) n0 J S (W )(φ, ψ) ˜ C (W )(φ, ψ) ℵ ℵ λ0 υ ρm;n (λ) ρm;n ρ˜m;n lj j B
Description number of k0 derivatives tracked number of k derivatives tracked scale parameter, M > 1 generic constant, independent of scale generic constant, independent of scale and M j th scale function (j ) i≥j ν (k) degree of asymmetry particle/hole swap operator log Z1 eW (φ,ψ+ζ ) dµS (ζ ) log Z1 eφJ ζ eW (φ,ψ+ζ ) dµC (ζ ) 1 2 2 <ℵ< 3 ℵ < ℵ < 23 maximum allowed “coupling constant” 0 < υ < 41 , power of λ0 eaten by bounds λ−(1−υ) max{m+n−2,2}/2# ρm;n (λ0 ) 1 if m = 0 ; 4 lj M j if m > 0 mυ/7
ρm;n (λ0 )λ0 = 1ℵj = length of sectors of scale j M the sectorization at scale j of length lj j –independent constant
Reference following (I.3) following (I.3) before Definition I.2
Definition I.2 Definition I.2 Definition I.10 (III.3) Definition III.1 Definition III.1 following Definition VI.3 Theorem XII.1 Theorem VIII.5 Definition V.6 Definition V.6 Definition VI.6.ii Theorem XIII.11 following Definition VI.3 following Definition VI.3 Definitions VI.7, XIII.15
Two Dimensional Fermi Liquid. 3: Fermi Surface Not’n
Description j |δ| t δ + = ∞ t δ ∈ Nd+1 |δδ |≤r M |δδ |>r |δ0 |≤r0 or |δ0 |>r0 iδ j |δδ | t δ + = ∞ t δ ∈ Nd+1 |δδ |≤r M 0 M |δδ |>r
cj ci,j
|δ |≤r
ej (X)
=
ei,j (X)
=
cj0 0 1−M j X ci,j 1−M j X
or |δ0 |>r0
fext ∗ •
extends f (x, x ) to fext (x0 , x, σ, a), (x0 , x , σ , a ) convolution ladder convolution
shear( · , B) shear ( · , B) sct ( · , B) µˆ
maps kernel of W (φ, ψ) to kernel of W (φ, ψ + Bφ) maps kernel of W (φ, ψ) to kernel of W (φ, ψ + Bφ ) maps kernel of W (φ , φ, ψ) to kernel of W (Bφ , φ, ψ) maps kernel of W (φ , φ, ψ) to kernel of W (φ, φ, ψ) Fourier transform
177 Reference Definition V.2 (XV.1) Definition V.2 (XV.1) [Definition E.1, [FKTo4]] Definition VIII.6 Definition VII.2, Definition XIV.7 Definition XIV.14 Definition XIV.19 Definition XIV.19 Definition XIV.17 Notation V.4
References [FKTf1] [FKTf2] [FKTo1] [FKTo2] [FKTo3] [FKTo4] [FKTr1] [FST3]
Feldman, J., Knörrer, H., Trubowitz, E.: A Two Dimensional Fermi Liquid, Part 1: Overview. Commun. Math. Phys. 247, 1–47 (2004) Feldman, J., Knörrer, H., Trubowitz, E.: A Two Dimensional Fermi Liquid, Part 2: Convergence. Commun. Math. Phys. 247, 49–111 (2004) Feldman, J., Knörrer, H., Trubowitz, E.: Single Scale Analysis of Many Fermion Systems, Part 1: Insulators. Rev. Math. Phys. 15, 949–993 (2003) Feldman, J., Knörrer, H., Trubowitz, E.: Single Scale Analysis of Many Fermion Systems, Part 2: The First Scale. Rev. Math. Phys. 15, 995–1037 (2003) Feldman, J., Knörrer, H., Trubowitz, E.: Single Scale Analysis of Many Fermion Systems, Part 3: Sectorized Norms. Rev. Math. Phys. 15, 1039–1120 (2003) Feldman, J., Knörrer, H., Trubowitz, E.: Single Scale Analysis of Many Fermion Systems, Part 4: Sector Counting. Rev. Math. Phys. 15, 1121–1169 (2003) Feldman, J., Knörrer, H., Trubowitz, E.: Convergence of Perturbation Expansions in Fermionic Models, Part 1: Nonperturbative Bounds. Commun. Math. Phys. 247, 195–242 (2004) Feldman, J., Salmhofer, M., Trubowitz, E.: Regularity of Interacting Nonspherical Fermi Surfaces: The Full Self–Energy. Commun. Pure Appl. Math. LII, 273–324 (1999)
Communicated by J.Z. Imbrie
Commun. Math. Phys. 247, 179–194 (2004) Digital Object Identifier (DOI) 10.1007/s00220-004-1038-2
Communications in
Mathematical Physics
Particle–Hole Ladders Joel Feldman1, , Horst Kn¨orrer2 , Eugene Trubowitz2 1 2
Department of Mathematics, University of British Columbia, Vancouver, B.C., Canada V6T 1Z2. E-mail: [email protected] Mathematik, ETH-Zentrum, 8092 Z¨urich, Switzerland. E-mail: [email protected]; [email protected]
Received: 21 September 2002 / Accepted: 12 August 2003 Published online: 6 April 2004 – © Springer-Verlag 2004
Abstract: A self contained analysis demonstrates that the sum of all particle-hole ladder contributions for a two dimensional, weakly coupled fermion gas with a strictly convex Fermi curve at temperature zero is bounded. This is used in our construction of two dimensional Fermi liquids. This summary article contains the statements of the main results. The proofs are contained in the full, electronic, article. Electronic Supplementary Material: Supplementary material is available in the online version of this article at http:///dx.doi.org/10.1007/s00220-004-1038-2. Contents I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Ladders in Momentum Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Scales and Sectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Particle–Hole Ladders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. The Propagators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Resectorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7. Compound Particle–Hole Ladders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
180 181 184 187 188 191 191 192 194
Research supported in part by the Natural Sciences and Engineering Research Council of Canada and the Forschungsinstitut f¨ur Mathematik, ETH Z¨urich.
180
J. Feldman, H. Kn¨orrer, E. Trubowitz
Contents of the Electronic Supplement I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1. Ladders in Momentum Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2. Scales and Sectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3. Particle–Hole Ladders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 4. Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 5. The Propagators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 6. Resectorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 7. Compound Particle–Hole Ladders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 II. Reduction to Bubble Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1. Combinatorial Structure of Compound Ladders . . . . . . . . . . . . . . . . . . . . . . . . . 16 2. Spin Independence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3. Scaled Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 4. Bubble and Double Bubble Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 5. The Infrared Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 III. Bubbles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 1. The Infrared Limit – Nonzero Transfer Momentum . . . . . . . . . . . . . . . . . . . . . . 69 2. The Infrared Limit – Reduction to Factorized Cutoffs . . . . . . . . . . . . . . . . . . . . 71 IV. Double Bubbles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 Appendix A. Bounds on Propagators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 Appendix B. Bound on the Generalized Model Bubble . . . . . . . . . . . . . . . . . . . . . . . . 90 Appendix C. Sector Counting with Specified Transfer Momentum . . . . . . . . . . . . . . 103 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
I. Introduction This article is one of a series, starting with [FKTf1], that provides a construction of a class of two dimensional Fermi liquids. The concept of a Fermi liquid was introduced by L. D. Landau in [L1, L2, L3] and has become the generally accepted explanation for the success of the independent electron approximation. The phenomenological implications of Fermi liquid theory are derived from the structure of the single particle density nk and Landau’s quasiparticle interaction and forward scattering amplitude. The single particle density is constructed as a relatively straightforward limit of the one particle Green’s function. The quasiparticle interaction and forward scattering amplitude, by contrast, are defined through two different limits of the transfer momentum flowing through the particle/hole channel of the two particle Green’s function. This subtlety arises because the two particle Green’s function is bounded but not continuous at transfer momentum zero. In [FKTr2] we showed that the leading contributions to the two particle Green’s function are the, so-called, ladders. In this paper we extract sufficiently detailed information about particle/hole ladders to demonstrate the existence of the limits defining the quasiparticle interaction and forward scattering amplitude. In fact, in our construction of the full models, we are forced to this level of detail to formulate hypotheses on the sequence of effective interactions (see [FKTf2, Def. IX.1 and IX.2]) that enable us to make an inductive construction. In other words, we would not be able to construct any of the Green’s functions without the present fine analysis of the particle/hole channel.
Particle–Hole Ladders
181
The control of the particle hole channel is in some ways the most subtle part of the argument in the construction of a Fermi liquid in that it results from a cancellation involving essentially all scales. Roughly speaking, it is like picking up the singularity of a Fourier series from its partial trigonometric sums. Philip Anderson [A1, A2] suggested that, because of an instability arising from the particle/hole channel, two dimensional Fermi gases should exhibit behavior similar to a one dimensional Luttinger liquid, with the single particle density nk having a vertical tangent at the Fermi surface rather than a jump discontinuity. In this series of papers, we show that this is not the case for the class of models considered here. This summary article contains the definitions and statements of main results of the full article, which is available at http:///dx.doi.org/10.1007/s00220-004-1038-2. We use the same numbering here as in the full article. Formally, the amputated four–point Green’s function, G4 ((p1 ,σ1 ),(p2 ,σ2 ),(p3 ,σ3 ),(p4 ,σ4 )) with incoming particles of momenta p1 , p4 ∈ R × Rd and spins σ1 , σ4 ∈ {↑, ↓} and outgoing particles of momenta p2 , p3 and spins σ2 , σ3 , can be written as a sum of values of Feynman diagrams with four external legs. The propagator of these diagrams 1 is C(k) = ık0 −e(k) , where k = (k0 , k) ∈ R × Rd and the dispersion relation e(k) (into which the chemical potential has been absorbed) characterizes the independent fermion approximation. The interaction of the model determines the diagram vertices, V ((k1 ,σ1 ),(k2 ,σ2 ),(k3 ,σ3 ),(k4 ,σ4 )), k1 + k4 = k2 + k3 . Here, the incoming momenta are k1 , k4 and the outgoing momenta are k2 , k3 . 1
3 V
2
4
1. Ladders in Momentum Space. The most important contributions to this four–point function are ladders. The contribution of the particle–hole ladder with + 1 rungs
(p1 , σ1 )
(p3 , σ3 ) k1
V
V
V
k
V
(p2 , σ2 )
(p4 , σ4 )
is τi,1 ,τi,2 ∈{↑,↓} i=1,··· ,
d d+1 k1 (2π )d+1
···
d d+1 k (2π)d+1
V ((p1 ,σ1 ),(p2 ,σ2 ),(p1 +k1 ,τ1,1 ),(p2 +k1 ,τ1,2 ))C(p1 +k1 )C(p2 +k1 ) V ((p1 +k1 ,τ1,1 ),(p2 +k1 ,τ1,2 ),···) · · · V (··· ,(p1 +k ,τ,1 ),(p2 +k ,τ,2 )) C(p1 +k )C(p2 +k )V ((p1 +k ,τ,1 ),(p2 +k ,τ,2 ),(p3 ,σ3 ),(p4 ,σ4 )).
The contribution of the particle–particle ladder with + 1 rungs (p1 , σ1 ) (p4 , σ4 )
(p2 , σ2 ) (p3 , σ3 )
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is τi,1 ,τi,2 ∈{↑,↓} i=1,··· ,
d d+1 ki (2π)d+1
···
d d+1 k (2π)d+1
V ((p1 ,σ1 ),(p1 +k1 ,τ1,1 ),(p4 −k1 ,τ1,2 ),(p4 ,σ4 ))C(p1 +k1 )C(p4 −k1 ) V ((p1 +k1 ,τ1,1 ),··· ,(p4 −k1 ,τ1,2 )) · · · V (··· ,(p1 +k ,τ,1 ),(p4 −k ,τ,2 ),···) C(p1 +k )C(p4 −k )V ((p1 +k ,τ,1 ),(p2 ,σ2 ),(p3 ,σ3 ),(p4 −k ,τ,2 )).
Ladders with two rungs are called bubbles. The values of the bubbles with dispersion rela 2 − µ and interaction V ((p1 ,σ1 ),(p2 ,σ2 ),(p3 ,σ3 ),(p4 ,σ4 )) = λ δσ1 ,σ2 δσ3 ,σ4 − tion e(k) = |k| 2m δσ1 ,σ3 δσ2 ,σ4 are well–known for d = 2, 3 [FHN]. The particle–particle bubble has a logarithmic singularity [FKST, Prop. II.1b] at transfer momentum p1 + p4 = 0 which is responsible for the formation of Cooper pairs and the onset of superconductivity. This singularity persists in models having dispersion relations that are symmetric about the origin, i.e. e(k) = e(−k). On the other hand, if e(k) is strongly asymmetric in the sense of Definition I.10 of [FKTf1] then the particle–particle bubble remains continuous and, in particular, bounded [FKLT1, p. 297]. 2 For the particle–hole bubble with d = 2 and e(k) = |k| 2m − µ, d3k C(k + p1 ) C(k + p2 ) (2π )3 R3 1 d3k 1 = (2π)3 i(k0 +t0 /2)−e(k+t/2) i(k0 −t0 /2)−e(k−t/2) 3 R m m 2 2 2 2 2 2 − 2π + 2π|t|2 Re |t| (|t| −4kF )−4m t0 −4ımt0 |t| if t0 , |t| = 0 or |t| ≥ 2kF m = − 2π if t0 = 0 and 0 < |t| ≤ 2kF , 0 if t0 = 0 and t = 0 √ where t = p1 − p2 is the transfer momentum, kF = 2mµ is the radius of the Fermi √ surface and is the square root with nonnegative real part and cut along the nega∞ tive for example, [FHN (2.22) or FKST, Prop. II.1a]. This is C on
real axis.2See, t ∈ R × R t0 = 0 or |t| > 2kF , is H¨older continuous of degree 1 in a neighbourhood of any t with t0 = 0, 0 < |t| < 2kF , and is H¨older continuous of degree 1 2 in a neighbourhood of any t with t0 = 0, |t| = 2kF , but cannot be continuously extended to t = 0. However its restriction to t0 = 0 does have a C ∞ extension at the point t = 0. The discontinuity at t = 0 persists for general, even strongly asymmetric, e(k). For this reason, bounds on particle–hole ladders in position space are not straightforward. That the restriction of the particle–hole bubble to t0 = 0 does have a C ∞ extension for a large class of smooth dispersion relations may be seen by the following argument, which was shown to us by Manfred Salmhofer [S]. A generalization of this argument is used in Prop. III.27 of the full article. 1 Lemma I.1. Choose a “scale parameter” M > 1 and a function ν ∈ C0∞ M , 2M
2
1 2 that takes values in [0, 1], is identically 1 on M , M , is monotone on M , M and [M, 2M], and obeys ∞ ν M 2j x = 1 j =0
(I.1)
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j [0,j ] for 0 < x < 1. Set ν0 (k0 ) = =0 ν(M 2 k02 ) and let u(k, t) be a bounded C ∞ function with compact support in k and bounded derivatives. Let e(k) be a C ∞ function that obeys lim e(k) = +∞. Assume that the gradient of e(k) does not vanish on the |k|→∞
Fermi surface F = k ∈ Rd e(k) = 0 . Then
[0,j ]
B(t) = lim
dk
j →∞
ν0 (k0 )u(k, t) [ik0 − e(k)][ik0 − e(k + t)]
is C ∞ for t in a neighbourhood of 0. Proof. Write Bj (t) =
=
[0,j ]
ν0 (k0 )u(k,t) [ik0 −e(k)][ik0 −e(k+t)]
dk
[0,j ]
ν0 (k0 )u(k,t) 1 1 e(k)−e(k+t) ik0 −e(k) − ik0 −e(k+t) 1 [0,j ] ν0 (k0 )u(k,t) 1 ds dds ik0 −E(k,t,s) dk e(k)−e(k+t) 0 1 [0,j ] ν (k0 )u(k,t) ds [ik0 −E(k,t,s)] dk 2, 0 0
dk
=
= where
E(k, t, s) = se(k) + (1 − s)e(k + t). Make, for each fixed s and k0 , the change of variables from k to E and d − 1 variables θ on F . Denote by J (E, t, θ, s) the Jacobian of this change of variables and set f (k0 , E, θ, t, s) = u (k0 , k(E, θ, t, s)), t J (E, θ, t, s). Because u has compact support in k, f vanishes unless |E| ≤ E, for some finite E. Thus
1
Bj (t) =
ds
E
dθ
dk0
−E
0
Set Bj (t) Since
=
1
ds
E
dθ
dk0
−E
0
[0,j ] α ν0 (k0 )f (k0 ,E,θ,t,s) − ∂t [ik −E]2
[0,j ]
[0,j ]
ν0
0
ν0
dE
[0,j ]
dE
ν0
(k0 )f (k0 ,E,θ,t,s) . [ik0 −E]2
(k0 )f (k0 ,0,θ,t,s) . [ik0 −E]2
(k0 )f (k0 ,0,θ,t,s) [ik0 −E]2
is integrable on R × [−E, E], lim Bj (t) − j →∞
Bj (t)
≤ const α k 2|E| +E 2
exists and is
0
C∞
by the Lebesgue
dominated convergence theorem. So it suffices to consider Bj (t) = −2E
1
ds 0
dθ
[0,j ]
dk0
ν0
(k0 )f (k0 ,0,θ,t,s) . k02 +E 2
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Since
α ν0[0,j ] (k0 )f (k0 ,0,θ,t,s) ∂t ≤ constα 2 k +E 2 0
1 k02 +E 2
is integrable on R, lim Bj (t) exists and is C ∞ by the Lebesgue dominated convergence theorem.
j →∞
2. Scales and Sectors. In this paper, we derive position space bounds for generalized particle–hole ladders in two space dimensions as they arise in a multiscale analysis. The main result is Theorem I.20, which is used in [FKTf2], under the name Theorem D.2, to help construct a Fermi liquid. We assume that the dispersion relare +3 tion e(k)
is C2 for some re ≥ 6, that its gradient does not vanish on the Fermi curve F = k ∈ R e(k) = 0 and that the Fermi curve is nonempty, connected, compact and strictly convex (meaning that its curvature does not vanish anywhere). We also fix the number r0 ≥ 6 of derivatives in k0 that we wish to control. We introduce scales as in [FKTf1, Def. I.2] and [FKTo2, §VIII]: Definition I.2.
i) For j ≥ 1, the j th scale function on R × R2 is defined as ν (j ) (k) = ν M 2j (k02 + e(k)2 ) ,
where ν is the function of (I.1). It may be constructed by choosing a function ϕ ∈ C0∞ (−2, 2) that is identically one on [−1, 1] and setting ν(x) = ϕ(x/M)−ϕ(Mx) for x > 0 and zero otherwise. By construction, ν (j ) is identically one on √ 2 1
k = (k0 , k) ∈ R × R2 M ≤ |ik0 − e(k)| ≤ M M1j . Mj The support of ν (j ) is called the j th shell. By construction, it is contained in √
k ∈ R × R2 √1 M1j ≤ |ik0 − e(k)| ≤ 2M M1j M
The momentum k is said to be of scale j if k lies in the j th shell. ii) For j ≥ 1, set (i) ν (≥j ) (k) = ν (k) i≥j
for −e(k)| > 0 and ν (≥j ) (k) = 1 for |ik0 −e(k)| = 0. Equivalently, ν (≥j ) (k) = |ik2j0−1 ϕ M (k02 + e(k)2 ) . By construction, ν (≥j ) is identically 1 on √
k ∈ R × R2 |ik0 − e(k)| ≤ M M1j . The support of ν (≥j ) is called the j th neighbourhood of the Fermi surface. By construction, it is contained in √
k ∈ R × R2 |ik0 − e(k)| ≤ 2M M1j . The support of ϕ M 2j −2 (k02 + e(k)2 ) is called the j th extended neighbourhood. It is contained in √
k ∈ R × Rd |ik0 − e(k)| ≤ 2M M1j .
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To estimate functions in position space and still make use of conservation of momentum, we use sectorization. See [FKTf1, Example A.1]. The following definition is also made in [FKTf2, §VI] and [FKTo3, §XII]. Definition I.3 (Sectors and sectorizations). i) Let I be an interval on the Fermi surface F and j ≥ 1. Then
s = k in the j th neighbourhood πF (k) ∈ I is called a sector of length |I | at scale j . Here k → πF (k) is a projection on the Fermi surface. Two different sectors s and s are called neighbours if s ∩ s = ∅. ii) A sectorization of length l at scale j is a set of sectors of length l at scale j that obeys – the set of sectors covers the Fermi surface – each sector in has precisely two neighbours in , one to its left and one to its right 1 – if s, s ∈ are neighbours then 16 l ≤ |s ∩ s ∩ F | ≤ 18 l. Observe that there are at most 2 length(F )/l sectors in . In the renormalization group map of [FKTf1] and [FKTo3], we integrate over fields whose arguments (x, σ, s) lie in B × , where B = (R × R2 ) × {↑, ↓} is the set of all “(positions, spins)”. On the other hand, we are interested in the dependence of the two and four–point functions on external momenta. To distinguish between the set of all positions and the set of all momenta, we denote by M = R × R2 , the set of all possible momenta. The set of all possible positions shall still be denoted R×R2 . Thus the external variables (k, σ ) lie in Bˇ = M × {↑, ↓}. In total, legs of four–legged kernels may lie in the disjoint union Y = Bˇ ∪· (B × ) for some sectorization . The four–legged kernels over Y that we consider here arise in [FKTf2, §VII] as particle–hole reductions (as in Definition VII.4 of [FKTf2]) of four–legged kernels on X = Bˇ ∪· (B × ), where Bˇ = Bˇ × {0, 1} and B = B × {0, 1} and {0, 1} is the set of creation/annihilation indices. Particle–hole reduction sets the creation/annihilation index to zero for legs number one and four and to one for legs number two and three. To simplify the notation in this paper, we shall eliminate the spin variables so that the legs lie in Y = M ∪· (R × R2 ) × . Sometimes a four–legged kernel will have different sectorizations , on its two left hand legs and on its two right-hand legs. Therefore, we introduce the space (4)
Y , = Y2 × Y2 . (4)
Since Y is the disjoint union of M and (R×R2 )× , the space Y , is the disjoint union (4) Y , = · Yi1 , × Yi2 , × Yi3 , × Yi4 , , (I.2) i1 ,i2 ,i3 ,i4 ∈{0,1} (4)
where Y0, = M and Y1, = (R × R2 ) × . If f is a function on Y , , we denote by f (i ,··· ,i ) its restriction to Yi1 , × Yi2 , × Yi3 , × Yi4 , under the identification 1 4 (I.2).
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Definition I.4 (Translation invariance). Let and be sectorizations. i) Let y ∈ Y and t ∈ R × R2 . We set k if y = k ∈ M Tt y = . (x + t, s) if y = (x, s) ∈ R × R2 × ii) Let i1 , · · · , i4 ∈ {0, 1}. A function f on Yi1 , × Yi2 , × Yi3 , × Yi4 , is called translation invariant, if for all t ∈ R × R2 , bµ f (Tt y1 , · · · , Tt y4 ) = eı(−1) − f (y1 , · · · , y4 ), 1≤µ≤4 iµ =0
where
0 if µ = 1, 4 bµ = 1 if µ = 2, 3
(I.3)
and < k, x >− = −k0 x0 + k1 x1 + k2 x2 . This choice of bµ reflects our image of f as a particle–hole kernel, with first and fourth, resp. second and third, arguments being creation, resp. annihilation, arguments. (4) iii) A function f on Y , is translation invariant if f (i ,··· ,i ) is translation invariant 1 4 for all i1 , · · · , i4 ∈ {0, 1}. 4 A function f on Y is translation invariant if f (( · ,σ1 ),( · ,σ2 ),( · ,σ3 ),( · ,σ4 )) is translation invariant for all σ1 , · · · , σ4 ∈ {↑, ↓}. Definition I.5 (Fourier transform). Let , be sectorizations. Set Y2, = M × . i) Let i1 , · · · , i4 ∈ {0, 1, 2} and 1 ≤ µ ≤ 4 such that iµ = 1. The Fourier transform of a function f on Yi1 , × Yi2 , × Yi3 , × Yi4 , with respect to the µth variable is the function on Yi1 , × Yi2 , × Yi3 , × Yi4 , with iν if ν = µ
iν = 2 if ν = µ defined by (µ f )(y1 ,··· ,yµ−1 ,(k,s),yµ+1 ,··· ,y4 ) =
bµ −
eı(−1)
f (y1 ,··· ,yµ−1 ,(x,s),yµ+1 ,··· ,y4 ) d 3 x.
ii) Let i1 , · · · , i4 ∈ {0, 1} with iµ = 1 for at least one 1 ≤ µ ≤ 4. The total Fourier transform fˇ of a translation invariant function f on Yi1 , ×Yi2 , ×Yi3 , ×Yi4 ,
is defined by fˇ(y1 , y2 , y3 , y4 ) (2π)3 δ(k1 − k2 − k3 + k4 ) = µ f (y1 , y2 , y3 , y4 ), 1≤µ≤4 iµ =1
where yµ = kµ when iµ = 0 and yµ = (kµ , sµ ) when iµ = 1. fˇ is defined on the set of all (y1 , y2 , y3 , y4 ) ∈ Y2i1 , × Y2i2 , × Y2i3 , × Y2i4 , for which k1 − k2 = k3 − k4 .
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Definition I.6 (Sectorized functions). Let and be sectorizations. i) Let i1 , · · · , i4 ∈ {0, 1}. A translation invariant function f on Yi1 , ×Yi2 , ×Yi3 , × Yi4 , is sectorized if, for each 1 ≤ µ ≤ 4 with iµ = 1, the total Fourier transform fˇ(y1 ,··· ,yµ−1 ,(k,s),yµ+1 ,··· ,y4 ) vanishes unless k is in the j th extended neighbourhood and πF (k) ∈ s. (4) ii) A translation invariant function f on Y , is sectorized if f (i ,··· ,i ) is sectorized 1 4 for all i1 , · · · , i4 ∈ {0, 1}. 4 A translation invariant function f on Y is sectorized if f (( · ,σ1 ),( · ,σ2 ),( · ,σ3 ),( · ,σ4 )) is sectorized for all σ1 , · · · , σ4 ∈ {↑, ↓}. Remark I.7. If f is a function in the space Fˇ 4, of Definition XIV.6 of [FKTf2] (or Definition XVI.7.iii of [FKTo3]), then its particle–hole reduction is a sectorized function 4 on Y . 3. Particle–Hole Ladders. Definition I.8. i) A (spin independent) propagator is a translation invariant function 2 on R × R2 . If A(x, x ) is a propagator, then its transpose is At (x, x ) = A(x , x). ii) A (spin independent) bubble propagator is a translation invariant function on R × 4 R2 . If A and B are propagators, we define the bubble propagator A ⊗ B(x1 , x2 , x3 , x4 ) = A(x1 , x3 )B(x2 , x4 ). We set C(A, B) = = =
(A + B) ⊗ (A + B)t − B ⊗ B t A ⊗ At + A ⊗ B t + B ⊗ A t A A B + +
A B A iii) Let , ,
be sectorizations, P be a bubble propagator and F be a function on 2 Yi1 ,
× Yi2 ,
× (R × R2 ) . If K is a function on Y × Y × Y1, × Y1, , we set dx1 dx2 K(y1 ,y2 ,(x1 ,s1 ),(x2 ,s2 )) P (x1 ,x2 ;x3 ,x4 ). (K • P )(y1 ,y2 ;x3 ,x4 ) = s1 ,s2 ∈
If K is a function on Y1, × Y1, × Yi3 , × Yi4 , , we set, when i1 , i2 , i3 , i4 are not all 0, dx1 dx2 F (y1 ,y2 ;x1 ,x2 )K((x1 ,s1 ),(x2 ,s2 ),y3 ,y4 ) (F • K)(y1 ,y2 ,y3 ,y4 ) = s1 ,s2 ∈
and when i1 , i2 , i3 , i4 = 0, (F • K)(k1 ,k2 ,k3 ,k4 )(2π)3 δ(k1 −k2 −k3 +k4 ) = dx1 dx2 F (k1 ,k2 ;x1 ,x2 )K((x1 ,s1 ),(x2 ,s2 ),k3 ,k4 ). s1 ,s2 ∈
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Observe that K • P is a function on Y2 × (R × R2 )2 and F • K is a function on (4) Y , . If K is a function on (Y )4 and F is a function on (Y )2 × (B )2 we set (K • P )(( · ,σ1 ),( · ,σ2 ),( · ,σ3 ),( · ,σ4 )) = K (( · ,σ1 ),( · ,σ2 ),( · ,σ3 ),( · ,σ4 )) • P and (F • K )(( · ,σ1 ),( · ,σ2 ),( · ,σ3 ),( · ,σ4 )) F (( · ,σ1 ),( · ,σ2 ),( · ,τ1 ),( · ,τ2 )) • K (( · ,τ1 ),( · ,τ2 ),( · ,σ3 ),( · ,σ4 )). = τ1 ,τ2 ∈{↑,↓}
iv) Let ≥ 1 . Let, for 1 ≤ i ≤ + 1, (i) , (i) be sectorizations and Ki a function (4) on Y (i) (i) . Furthermore, let P1 , · · · , P be bubble propagators. The ladder with
, rungs K1 , · · · , K+1 and bubble propagators P1 , · · · , P is defined to be K1 • P1 • K2 • P2 • · · · • K • P • K+1 . 4
If is a sectorization and K1 , · · · , K+1 are functions on Y , the ladder with
rungs K1 , · · · , K+1 and bubble propagators P1 , · · · , P is defined to be
K1 • P1 • K2 • P2 • · · · • K • P • K+1 .
Remark I.9. We typically use C(A, B) with A being the part, ν (j ) (k)C(k), of the propagator, C(k), having momentum in the j th shell and B being the part, ν (≥j +1) (k)C(k), of the propagator having momentum in the (j + 1)st neighbourhood. The bubble propagator C(A, B) always contains at least one “hard line” A and may or may not contain one “soft line” B. The latter are created by Wick ordering. See [FKTf1, §II, Subsect. 9]. 4 Remark I.10. If F1 , F2 are functions on X and A, B are propagators over B in the sense of Definition VII.1.i of [FKTf2], then the particle–hole reduction of F1 •C(A, B)• F2 (with the C(A, B) of Definition VII.1.i of [FKTf2]) is equal to ph ph −F1 • C A(( · 1),( · 0)), B(( · 1),( · 0)) • F2 (with the C of Definition I.8) since B((x,σ,0),(x ,σ ,1)) = −B(( · 1),( · 0))t ((x,σ ),(x ,σ )). 4. Norms. In the momentum space variables, we take suprema of the function and its derivatives. In the position space variables, we will apply the L1 –L∞ norm of Definition I.11, below, to the function and to the function multiplied by various coordinate differences. n Definition I.11. Let f be a function on R × R2 . Its L1 –L∞ norm is |||f |||1,∞ = max sup dxj |f (x1 , · · · , xn )|. 1≤j0 ≤n x ∈R×R2 j0
j =1,··· ,n j =j0
Multiple derivatives are labeled by a multiindex δ = (δ0 , δ1 , δ2 ) ∈ N0 × N20 . For such a multiindex, we set |δ| = δ0 +δ1 +δ2 , δ! = δ0 ! δ1 ! δ2 ! and x δ = x0δ0 x1δ1 x2δ2 for x ∈ R×R2 .
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2 Definition I.12. Let be a sectorization and A a function on (R × R2 ) × . For a multiindex δ ∈ N0 × N20 , we define (x − y)δ A (x, s1 ), (y, s2 ) . |A|δ1, = max max 1,∞ i=1,2 si ∈
s3−i ∈
Variables for four–point functions may be momenta or position/sector pairs. Therefore we introduce differential–decay operators that differentiate momentum space variables and multiply position space variables by coordinate differences. We again use the identification (4) · Yi1 , × Yi2 , × Yi3 , × Yi4 ,
Y , = i1 ,i2 ,i3 ,i4 ∈{0,1}
of (I.2). Definition I.13 (Differential–decay operators). Let and be sectorizations, δ = (δ0 , δ1 , δ2 ) ∈ N0 × N20 a multiindex and µ, µ ∈ {1, 2, 3, 4} with µ = µ . i) Let i1 , · · · , i4 ∈ {0, 1} and f be a function on Yi1 , × Yi2 , × Yi3 , × Yi4 , . If iµ = 0, multiplication by the δth power of the position variable dual to kµ (see Definition I.5) is implemented by Dδµ f (· · · , kµ , · · · ) = (−1)δ1 +δ2 (−1)bµ |δ| ı |δ|
∂ δ1 ∂ δ2 ∂ δ0 δ0 δ1 δ2 f (· · · ∂kµ,0 ∂kµ,1 ∂kµ,2
, kµ , · · · ).
In general, set δ Dµ − Dδµ f Dδ − x δ f δ µ Dµ;µ f = δµ δ xµ − Dµ f δ xµ − xµδ f
if if if if
iµ iµ iµ iµ
= iµ = 0 = 0, iµ = 1 . = 1, iµ = 0 = iµ = 1
Here, when iµ = 1, the µth argument of f is (xµ , sµ ). (4) ii) If f is a function on Y , , then Dδµ;µ f (i ,··· ,i ) = Dδµ;µ f (i ,··· ,i ) for all 1 4 1 4 i1 , · · · , i4 ∈ {0, 1}. Definition I.14. Let , be sectorizations. i) Let i1 , · · · , i4 ∈ {0, 1} and f be a function on Yi1 , × Yi2 , × Yi3 , × Yi4 , . For multiindices δl , δc , δr ∈ N0 × N20 , we define l δc (δl ,δc ,δr ) r |f | , = max max
sup max Dδ1;2 Dµ;µ Dδ3;4 f 1,∞ .
sν ∈ ν=1,2 with iν =1
sν ∈ ν=3,4 with iν =1
µ=1,2 kν ∈M ν=1,2,3,4 µ =3,4 with iν =0
Here, the ν th argument of f is kν when iν = 0 and (xν , sν ) when iν = 1. The δr l c ||| · |||1,∞ of Definition I.11 is applied to all spatial arguments of Dδ1;2 Dδµ;µ
D3;4 f . (4)
ii) If f is a function on Y , , we define (δl ,δc ,δr ) = |f | ,
i1 ,i2 ,i3 ,i4 ∈{0,1}
f
(δl ,δc ,δr ) .
(i1 ,··· ,i4 ) ,
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In this definition, the system (δl , δc , δr ) of multiindices indicates, roughly speaking, that one takes δl derivatives with respect to the momentum flowing between the two left legs, δr derivatives with respect to the momentum flowing between the two right legs and δc derivatives with respect to momenta flowing from the left-hand side to the right-hand side. In [FKTf1, FKTf2, FKTf3] and [FKTo1, FKTo2, FKTo3, FKTo4], we combine the norms of all derivatives of a function in a formalδ power series. We denote by N3 the set of all formal power series X = Xδ t in the variables t = (t0 , t1 , t2 ) with δ∈N0 ×N20
coefficients Xδ ∈ R+ ∪{∞}. See Definition V.2 of [FKTf2] or Definition II.4 of [FKTo1]. A quantity in N3 characteristic of the power counting for derivatives in scale j is cj = M j |δ| t δ + ∞ tδ. (I.4) δ1 +δ2 ≤re |δ0 |≤r0
δ1 +δ2 >re or |δ0 |>r0
Definition I.15. Let be a sectorization. 2 i) For a function A on (R × R2 ) × , we define δ δ 1 |A|1, = δ! |A|1, t . δ∈N0 ×N20 (4)
ii) For a function f on Y4 = Y , , we define 1 max |f | = δ! δ∈N0 ×N20
δl +δc +δr =δ
(δ ,δ ,δr )
l c |f | ,
tδ.
4 iii) For a function f on Y , we define |f | = |f (( · ,σ1 ),( · ,σ2 ),( · ,σ3 ),( · ,σ4 ))| . σ1 ,··· ,σ4 ∈{↑,↓}
The following lemma, whose proof follows immediately from the various definitions and Lemma D.2.ii of [FKTo3], compares these norms with the norms of Definition VI.6 of [FKTf2]. Lemma I.16. Let be a sectorization.
4 (i) Let f be a sectorized, translation invariant function on Y and Vph (f ) its particle–hole value as in Definition VII.4 of [FKTf2]. Let | · |∼ 3, be the norm of Definition XIII.12 of [FKTf3] (or Definition XVI.4 of [FKTo3]). Then there is a constant const , that depends only on r0 and r, such that |Vph (f )|∼ ∞ tδ. 3, ≤ const |f | + δ1 +δ2 >r or δ0 >r0
(ii) Let g be a function in the space Fˇ 4, of Definition XIV.6 of [FKTf2] (or Definition XVI.7.iii of [FKTo3]) and g ph its particle–hole reduction as in Definition VII.4 of [FKTf2]. Then there is a universal const such that |g ph | ≤ const |g|∼ 3, .
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5. The Propagators. The propagators we use in the multiscale analysis of [FKTf1, FKTf2, FKTf3] are of the form Cv(j ) (k) =
ν (j ) (k) ik0 −e(k)−v(k)
Cv(≥j ) (k) =
ν (≥j ) (k) ik0 −e(k)−v(k)
with functions v(k) satisfying |v(k)| ≤ 21 |ık0 − e(k)|. Their Fourier transforms are (j ) d 3 k ı− (j ) e Cv (k) Cv (x, y) = (2π)3 d 3 k ı− (≥j ) Cv(≥j ) (x, y) = e Cv (k), (2π)3 d 3 k −ı− (j ) Cv(j ) (y) = e Cv (k) (2π)3 d 3 k −ı− (≥j ) Cv(≥j ) (y) = e Cv (k). (2π)3 The function v(k) will be the sum of Fourier transforms of sectorized, translation invari 2 ant functions p (x, s), (x, s ) on R × R2 × for various sectorizations . The Fourier transform of such a function is defined as d 3 x eı− p((0,s),(x,s )). p(k) ˇ = s,s ∈
6. Resectorization. We now fix 21 < ℵ < 23 and set lj = M1ℵj . Furthermore, we select, j ≥ 1, a sectorization j of length lj at scale j and a partition of unity
for each χs s ∈ j of the j th neighbourhood which fulfills Lemma XII.3 of [FKTo3] with
= j . The Fourier transform of χs is d3k χˆ s (x) = e−ı− χs (k) (2π) 3. Definition I.17 (Resectorization). Let j, j , jl , jl , jr , jr ≥ 1. 2 i) Let p be a sectorized, translation invariant function on R × R2 × j . Then, for j = j , the j –resectorization of p is dx1 dx2 χˆ s1 (x1 −x1 ) p((x1 ,s1 ),(x2 ,s2 )) χˆ s2 (x2 −x2 ). p j ( (x1 ,s1 ),(x2 ,s2 )) = s1 ,s2 ∈ j
2 It is a sectorized, translation invariant function on R × R2 × j . If j = j , we set p j = p. ii) Let i1 , · · · , i4 ∈ {0, 1} and f be a function on Yi1 , jl × Yi2 , jl × Yi3 , jr × Yi4 , jr that is sectorized and translation invariant. Then the (jl , jr )–resectorization of f is the sectorized, translation invariant function on Yi1 , j ×Yi2 , jl ×Yi3 , jr ×Yi4 , jr
l defined by
dxµ χˆ sµ ((−1)bµ (xµ −xµ )) f ( y1 ,y2 ,y3 ,y4 ), f j , j ( y1 ,y2 ,y3 ,y4 ) = l
r
∈
∈ sµ sµ jl jr µ∈{1,2}∩S µ∈{3,4}∩S
µ∈S
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{1, 2, 3, 4}
{1, 2} S = µ iµ = 1 ∩ {3, 4} ∅
where
if if if if
jl
jl
jl
jl
= jl , = jl , = jl , = jl ,
jr
jr
jr
jr
= jr = jr = jr = jr
and yµ = yµ for µ ∈ / S and, for µ ∈ S, yµ = (xµ , sµ ).
yµ = (xµ , sµ )
(4) iii) If f is a sectorized, translation invariant function on Y j , j , then f j , j (i ,··· ,i ) 1 4 r r l l = f (i ,··· ,i ) , for all i1 , · · · , i4 ∈ {0, 1}. If jl = jr = j , we set f j = 1
4
f j , j .
jl
jr
4 iv) If f is a sectorized, translation invariant function on Y j , then f j (( · ,σ1 ),( · ,σ2 ),( · ,σ3 ),( · ,σ4 )) = f (( · ,σ1 ),( · ,σ2 ),( · ,σ3 ),( · ,σ4 ))
j
for all σ1 , · · · , σ4 ∈ {↑, ↓}. (4)
Remark I.18. Let K and H be sectorized translation invariant functions on Y i , j and l
(4)
Y i
r , jr
l
respectively. Let P be a bubble propagator. If the Fourier transform
4
dxµ
µ=1
4
e−ı(−1)
bµ µ µ −
P (x1 , x2 , x3 , x4 )
µ=1
of P is supported on the max{jl , ir }th neighbourhood, then K •P •H = K i , j • P • H i , j .
i , j
l
r
l
l
r
r
7. Compound Particle–Hole Ladders. Define, for any set Z and any function K on Z 4 , the flipped function K f (z1 , z2 , z3 , z4 ) = −K(z1 , z3 , z2 , z4 ). (I.5) (2) (3) Definition I.19. Let F = F , F , · · · be a sequence of sectorized, translation 4 invariant functions F (i) on Y i and v(k) a function on M such that |v(k)| ≤ 21 |ık0 − e(k)|. We define, recursively on 0 ≤ j < ∞, the compound particle–hole (or wrong (j ) way) ladders up to scale j , denoted by L(j ) = Lv (F ) , as L(0) = 0, L(j +1) = L j + (j )
where F =
=1
(j )
(j ) f
F + L j + L j
(j ) (j ) f • C (j ) • · · · C (j ) • F + L j + L j ,
(i) (j ) = C C (j ) , C (≥j +1) . v v i=2 F j and the th term has bubble propagators C L(1) = L(2) = 0.
j
Observe that
∞
Particle–Hole Ladders
193
Theorem I.20. For every ε > 0 there are constants ρ0 , const 1 such that the following holds. Let F = F (2) , F (3) , · · · be a sequence of sectorized, translation invariant spin 4 independent 2 functions F (i) on Y i and p = p(2) , p(3) , · · · be a sequence of sec 2 torized, translation invariant functions p(i) on R × R2 × i . Assume that there is ρ ≤ ρ0 such that for i ≥ 2, |F (i) | i ≤ Set v(k) =
∞
i=2 pˇ
ρ c M εi i
(i) (k) .
|p (i) |1, i ≤ ρMlii ci
pˇ (i) (0, k) = 0.
Then for all j ≥ 1, +1) |L(j (F )| j ≤ const ρ 2 cj . v
Remark I.21. Theorem I.20 and Theorem D.2 of [FKTf3] are equivalent. If one replaces the functions F (i) of Theorem D.2 of [FKTf3] by 24 times their particle–hole reductions, then, by Corollary D.7 of [FKTf3] and Remark I.10, the concepts of compound ladders of Definition I.19 and Definition D.1 of [FKTf3] coincide. Hence Theorem I.20 and Theorem D.2 of [FKTf3] are equivalent by Lemma I.16. Theorem I.20 is proven in the full article following Corollary II.24. The core of the proof consists of bounds on two types of ladder fragments, that look like G1 G(i1 )
H
H G2
and are called particle–hole bubbles and double bubbles, and a combinatorial result, Corollary II.12, that enables one to express general ladders in terms of these fragments. The most subtle part of the bound, Theorem II.19, on particle–hole bubbles is a generalization of Lemma I.1. The bound, Theorem II.20, on double bubbles also exploits “volume improvement due to overlapping loops”. A simple introduction to this phenomenon is provided at the beginning of §IV of the full article. Ladders with external momenta have an infrared limit that behaves much like the model bubble of Lemma I.1. Theorem I.22. Under the hypotheses of Theorem I.20, the limit ) L(q, q , t, σ1 , · · · σ4 ) = lim L(j v (F ) i j →∞
1 ,i2 ,i3 ,i4 =0
((q+ 2t ,σ1 ),(q− 2t ,σ2 ),(q + 2t ,σ3 ),(q − 2t ,σ4 ))
exists for transfer momentum t = 0 and is continuous in (q, q , t) for t = 0. The restrictions to t = 0 and to t0 = 0, namely, L(q, q , (t0 , 0), σ1 , · · · σ4 ) and L(q, q , (0, t), σ1 , · · · σ4 ), have continuous extensions to t = 0. This theorem is proven in the full article following Lemma II.29. 1 Throughout this paper we use “const” to denote unimportant constants that depend only on the dispersion relation e(k) and the scale parameter M. In particular, they do not depend on the scale j . 2 “Spin independence” is formally defined in Definition II.6 of the full article.
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References [A1] [A2] [FHN] [FKLT1] [FKLT2] [FKST] [FKTf1] [FKTf2] [FKTf3] [FKTo1] [FKTo2] [FKTo3] [FKTo4] [FKTr1] [FKTr2] [FMRT] [L1] [L2] [L3] [N] [S]
Anderson, P.W.: “Luttinger-liquid” behavior of the normal metallic state of the 2D Hubbard model. Phys. Rev. Lett. 64, 1839–1841 (1990) Anderson, P.W.: Singular forward scattering in the 2D Hubbard model and a renormalized Bethe Ansatz ground state. Phys. Rev. Lett. 65, 2306–2308 (1990) Fukuyama, H., Hasegawa, Y., Narikiyo, O.: Two–Dimensional Hubbard Model at Low Electron Density. J. Phys. Soc. Jap. 60, 2013–2030 (1991) Feldman, J., Kn¨orrer, H., Lehmann, D., Trubowitz, E.: Fermi Liquids in Two Space Dimensions. In: Constructive Physics, V. Rivasseau, (ed.), 446, Springer LNP, Berlin-HeidelbergNew York: Springer, 1995, pp. 267–299 Feldman, J., Kn¨orrer, H., Lehmann, D., Trubowitz, E.: A Class of Fermi Liquids. In: Particles and Fields ’94, G. Semenoff, L. Vinet, (eds.), Berlin-Heidelberg-New York: Springer, 1999, pp. 35–62 Feldman, J., Kn¨orrer, H., Sinclair, R., Trubowitz, E.: Superconductivity in a Repulsive Model. Helv. Phys. Acta 70, 154–191 (1997) Feldman, J., Kn¨orrer, H., Trubowitz, E.: A Two Dimensional Fermi Liquid, Part 1: Overview. Commun. Math. Phys. 247, 1–47 (2004) Feldman, J., Kn¨orrer, H., Trubowitz, E.: A Two Dimensional Fermi Liquid, Part 2: Convergence. Commun. Math. Phys. 247, 49–111 (2004) Feldman, J., Kn¨orrer, H., Trubowitz, E.: A Two Dimensional Fermi Liquid, Part 3: The Fermi Surface. Commun. Math. Phys. 247, 113–177 (2004) Feldman, J., Kn¨orrer, H., Trubowitz, E.: Single Scale Analysis of Many Fermion Systems, Part 1: Insulators. Rev. Math. Phys. 15, 949–993 (2003) Feldman, J., Kn¨orrer, H., Trubowitz, E.: Single Scale Analysis of Many Fermion Systems, Part 2: The First Scale. Rev. Math. Phys. 15, 995–1037 (2003) Feldman, J., Kn¨orrer, H., Trubowitz, E.: Single Scale Analysis of Many Fermion Systems, Part 3: Sectorized Norms. Rev. Math. Phys. 15, 1039–1120 (2003) Feldman, J., Kn¨orrer, H., Trubowitz, E.: Single Scale Analysis of Many Fermion Systems, Part 4: Sector Counting. Rev. Math. Phys. 15, 1121–1169 (2003) Feldman, J., Kn¨orrer, H., Trubowitz, E.: Convergence of Perturbation Expansions in Fermionic Models, Part 1: Nonperturbative Bounds. Commun. Math. Phys. 247, 195–242 (2004) Feldman, J., Kn¨orrer, H., Trubowitz, E.: Convergence of Perturbation Expansions in Fermionic Models, Part 2: Overlapping Loops. Commun. Math. Phys. 247, 243–319 (2004) Feldman, J., Magnen, J., Rivasseau, V., Trubowitz, E.: Two Dimensional Many Fermion Systems as Vector Models. Europhys. Lett. 24, 521–526 (1993) Landau, L.D.: The Theory of a Fermi Liquid. Sov. Phys. JETP 3, 920 (1956) Landau, L.D.: Oscillations in a Fermi Liquid. Sov. Phys. JETP 5, 101 (1957) Landau, L.D.: On the Theory of the Fermi Liquid. Sov. Phys. JETP 8, 70 (1959) Nozi`eres, P.: Theory of Interacting Fermi Systems. New York: Benjamin, 1964 Salmhofer, M.: Private communication
Communicated by J.Z. Imbrie
Commun. Math. Phys. 247, 195–242 (2004) Digital Object Identifier (DOI) 10.1007/s00220-004-1039-1
Communications in
Mathematical Physics
Convergence of Perturbation Expansions in Fermionic Models. Part 1: Nonperturbative Bounds Joel Feldman1, , Horst Kn¨orrer2 , Eugene Trubowitz2 1
Department of Mathematics, University of British Columbia, Vancouver, B.C., Canada V6T 1Z2. E-mail: [email protected] 2 Mathematik, ETH-Zentrum, 8092 Z¨ urich, Switzerland. E-mail: [email protected]; [email protected] Received: 21 September 2002 / Accepted: 12 August 2003 Published online: 6 April 2004 – © Springer-Verlag 2004
Abstract: An estimate on the operator norm of an abstract fermionic renormalization group map is derived. This abstract estimate is applied in another paper to construct the thermodynamic Green’s functions of a two dimensional, weakly coupled fermion gas with an asymmetric Fermi curve. The estimate derived here is strong enough to control everything but the sum of all quartic contributions to the Green’s functions. Contents I. Introduction . . . . . . . . . . . . . . . . . . . . . . II. The Renormalization Group Map . . . . . . . . . . . II.1 Superalgebras . . . . . . . . . . . . . . . . . . II.2 Grassmann Gaussian Integrals . . . . . . . . . II.3 Tensor Algebra and Grassmann Algebras . . . . II.4 Seminorms . . . . . . . . . . . . . . . . . . . II.5 An Estimate of the Renormalization Group Map II.6 Contraction and Integral Bounds . . . . . . . . III. The Schwinger Functional . . . . . . . . . . . . . . III.1 Description of the Schwinger Functional . . . . III.2 Norm Estimates on the Schwinger Functional . III.3 Proof of Theorem II.28 . . . . . . . . . . . . . IV. More Estimates on the Renormalization Group Map . Appendix A. Wick–Ordering . . . . . . . . . . . . . . . Appendix B. Gram Bounds . . . . . . . . . . . . . . . . Notation . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . .
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Research supported in part by the Natural Sciences and Engineering Research Council of Canada and the Forschungsinstitut f¨ur Mathematik, ETH Z¨urich.
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I. Introduction In a Grassmann algebra A with generators ψi , the renormalization group map with respect to a covariance C is the map that associates to each element W (ψ) of the Grassmann algebra the element C (W )(ψ) = log
1 Z(W )
e
W (ψ+ξ )
dµC (ξ ),
where
Z(W ) =
eW (ξ ) dµC (ξ )
whenever Z(W ) = 0. Here, ξi is a second set of variables that anticommute amongst themselves and with the ψj ’s. · · · dµC (ξ ) is the Grassmann Gaussian integral with respect to these variables (see §II). The Schwinger functional with interaction W is the map that associates to a Grassmann function f (ξ ) the complex number S(f ) =
1 Z(W )
f (ξ ) eW (ξ ) dµC (ξ ).
Observe that, if Z(W ) = 0, we may always arrange that Z(W ) = 1 by adding a constant to W . In this case
1
C (W )(ψ) = 0
d dt C (tW ) dt
= 0
1
W (ψ + ξ ) etW (ψ+ξ ) dµC (ξ ) dt. etW (ψ+ξ ) dµC (ξ )
The t–integrand of the right-hand side is the Schwinger functional of W (ψ + ξ ) with interaction tW (ψ + ξ ), where the Schwinger functional is considered in the Grassmann algebra with generators ξi and coefficients in A. We exploit this observation and the representation of the Schwinger functional in [FKT1] to develop non–perturbative bounds for the renormalization group map. By “non–perturbative” we mean that we bound the sum of the perturbation expansion, not only individual terms in the expansion. In a perturbative analysis, one decomposes W = n Wn , where Wn is homogeneous of degree n in ψ. Then C (W ) is the sum of the values of all connected Feynman diagrams with vertices Wn and propagator C. In most applications, the kernels Wn are translation invariant. To bound the value of a Feynman diagram, one usually (see [FT]) selects a tree in the diagram (to exploit the connectedness of the diagram) and bounds the lines of the tree differently from the other lines.1 The non–perturbative analysis we present here is close to the diagrammatic analysis (see the introduction to [FKT1]), but it allows implementation of the Pauli exclusion principle for lines not on the tree. The norms we use in applications to many fermion systems are quite complicated. Therefore, here, we axiomatize their relevant properties: We assume that there is a system of norms · on the homogeneous subspaces of the Grassmann algebra. Furthermore, we assume that there is a “contraction bound” c for C, such that for any two homogeneous elements f, f in the Grassmann algebra, the norm of the diagram 1 Typically, the lines of the tree contribute a factor of the L norm of the propagator in position space 1 to the bound on the diagram, while the other lines contribute a factor of the L∞ norm of the propagator to the bound.
Convergence of Perturbation Expansions in Fermionic Models. Part 1
f
C
197
f
that is obtained by joining f and f by one line is bounded by c f f and we assume that there is an “integral bound” b that controls the effect of integrating out some of the fields attached to a single vertex. See Def. II.25. The contraction bound is analogous to the bound on the tree contribution to a Feynman diagram. The integral bound incorporates the Pauli exclusion principle and is often derived from Gram’s estimate for determinants. See Appendix B. It replaces the bound on the non–tree contribution to a Feynman diagram. When applying a renormalization group transformation, there are often other fields present, that do play no role in the renormalization group transformation. We suppress these fields by allowing a (super)algebra as coefficient ring for the Grassmann algebra on which the renormalization group map is analyzed. See Def. II.1. Also, in the analysis of many fermion systems, we have to control various derivatives (in momentum space) of the effective interactions involved. To get a coherent notation for derivative norms, we allow the norms to take values in a formal power series ring, where the powers code the degree of derivatives. See Def. II.14. In §II we introduce the concepts discussed above and formulate the main estimate on the renormalization group map in these terms (Theorem II.28). Section III discusses the connection with the Schwinger functional and gives the proof of Theorem II.282 . In §IV, further estimates of the renormalization group map are discussed, in particular on its derivative with respect to the interaction W and the covariance C. Part 2 of the paper (§VI–IX) discusses the phenomenon of “overlapping” loops, that is responsible for “improvements over natural power counting” and consequently is important for many fermion systems; see [S] and the introduction to Part 2 of this paper. A notation table is provided at the end of the paper.
II. The Renormalization Group Map II.1. Superalgebras. Definition II.1. (i) A superalgebra is an associative C-algebra A with unit 1, together with a decomposition A = A+ ⊕ A− such that 1 ∈ A+ and A+ · A+ ⊂ A+ , A+ · A− ⊂ A− ,
A− · A− ⊂ A+ , A− · A+ ⊂ A− ,
and ab = ba, if a ∈ A+ or b ∈ A+ , ab = −ba, if a, b ∈ A− . The elements of A+ are called even, the elements of A− odd. 2
For other approaches to controlling fermionic renormalization, see [DR] and [SW].
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(ii) A graded superalgebra is∞an associative C-algebra A with unit, together with a decomposition A = m=0 Am such that A0 = C, Am · An ⊂ Am+n for all m, n ≥ 0, and such that the decomposition A = A+ ⊕ A− with
A+ =
A− =
Am
m even
Am
m odd
gives A the structure of a superalgebra. (iii) Let A be a graded superalgebra, f = fm ∈ A with fm ∈ Am . Set Z(f ) = f0 ∈ m
A0 = C. Clearly, if f0 = 0, f Z (f )
=1+
fm f0 .
m≥1
In this case, set log Zf(f ) =
∞ (−1)n−1 n
f Z (f )
n −1 .
n=1
(iv) Let A and B be superalgebras. On the tensor product A⊗B we define multiplication by
a ⊗ (b+ + b− ) (a+ + a− ) ⊗ b = a(a+ + a− ) ⊗ (b+ + b− )b − 2 aa− ⊗ b− b = aa+ ⊗ b+ b + aa+ ⊗ b− b +aa− ⊗ b+ b − aa− ⊗ b− b for a ∈ A, b ∈ B , a± ∈ A± , b± ∈ B± . This multiplication defines an algebra structure on A ⊗ B. Setting (A ⊗ B)+ = (A+ ⊗ B+ ) ⊕ (A− ⊗ B− ), (A ⊗ B)− = (A+ ⊗ B− ) ⊕ (A− ⊗ B+ ) we get a superalgebra. If A and B are graded superal gebras then the decomposition A ⊗ B = ∞ m=0 (A ⊗ B)m with (A ⊗ B)m =
Am1 ⊗ Bm2
m1 +m2 =m
gives A ⊗ B the structure of a graded superalgebra. Example V = m II.2. Let V be a complex vector space. The Grassmann algebra V over V is a graded superalgebra. If A is any superalgebra, the Grassmann m≥0
algebra over V with coefficients in A is the superalgebra A
V =A⊗
V,
where the tensor product is taken as in Def. II.1.iv. If A is a graded superalgebra, so is V . A In fact almost all graded superalgebras A that will be used in this paper will be subalgebras of Grassmann algebras. The one exception is the “enlarged” algebra of Sect. VII.
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II.2. Grassmann Gaussian Integrals. Let A be a superalgebra, V be a complex vector space and C an antisymmetric bilinear form (covariance) on V . Then C determines an A–linear map from V to A that is called the Grassmann Gaussian A integral dµC on A V . Choose a set {ξi } of generators for V . We write elements of A V as Grassmann functions f (ξ ). A Grassmann function f (ξ ) is called even if it is an even element of the algebra A V . The Grassmann Gaussian integral of f (ξ ) is denoted f (ξ ) dµC (ξ ). Then
ξi1 ξi2 · · · ξin dµC (ξ ) = Pf Cik i 1≤k,≤n , where Cij = C(ξi , ξj ) and the Pfaffian of an n × n matrix M is denoted PfM. Observe that, for any even Grassmann function f (ξ ), the Grassmann Gaussian integral f (ξ ) dµC (ξ ) is an even element of the coefficient algebra A. Let U be another vector space. Using the canonical isomorphism A
(U ⊕ V ) =
r A
r,r
U
r A
V ∼ =
A
U ⊗
A
V ∼ =
AU
V,
the Grassmann Gaussian integral defines a map · dµC (ξ ) from A (U ⊕ V ) to A U . If {ζi } is any set of vectors in U then e ξi ζi dµC (ξ ) = e−1/2 ζi Cij ζj . Definition II.3. Choose a second copy V of V and denote the element of V corresponding to the element ξi of V by ψi . If dim V < ∞, the renormalization group map C is defined by
C (W )(ψ) = log Z1 eW (ψ+ξ ) dµC (ξ ) where Z = Z eW (ξ ) dµC (ξ ) for all W ∈
AV
for which Z
eW (ξ ) dµC (ξ ) = 0.
Remark II.4. i) If dim V < ∞, then C (W ) is a rational function of W . ii) By construction Z C (W ) = 0 for all W . iii) In Def. II.27, below, we extend the definition of C to normed vector spaces. In this paper we state estimates on the renormalization group map for Wick ordered interactions W . Recall that Wick ordering with respect to a covariance C,
is the A–linear map on
f (ξ ) → :f (ξ ):ξ,C AV
characterized by
:e ξi ζi :ξ,C = e1/2 ζi Cij ζj e ξi ζi . If the context admits, we delete the Wick ordering covariance C or the variable ξ (or both) from the symbol : · :ξ,C for Wick ordering. Also recall the integration by parts formula δ ξi g(ξ ) dµC (ξ ) = Cij δξj g(ξ ) dµC (ξ ) j
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J. Feldman, H. Kn¨orrer, E. Trubowitz
or more generally δ Cin j :ξi1 · · · ξin−1 : δξ g(ξ ) dµC (ξ ) :ξi1 · · · ξin : g(ξ ) dµC (ξ ) = j j
=
(−1)n−1 n
Ci j
. δ ξ ···ξ . . δξi i1 in .
δ δξj
g(ξ ) dµC (ξ ).
i,j
II.3. Tensor Algebra and Grassmann Algebras. Again, let V be a complex vector space with a set {ξi } of generators and A a superalgebra. We denote by V ⊗n the n–fold tensor product (over the complex numbers) of V with itself. The symmetric group Sn of all permutations of {1, · · · , n} acts on V ⊗n (from the right) in such a way that (v1 ⊗ · · · ⊗ vn )π = vπ(1) ⊗ · · · ⊗ vπ(n) for all v1 , · · · , vn ∈ V and π ∈ Sn . The nth exterior power n V of V can be identified ⊗n with the set of all antisymmetric elements in V . We have the canonical projection n 1 Antn : V ⊗n −→ V , f −→ n! sgn(π ) f π . n
π∈Sn
By A–linearity, Ant n induces a map from A ⊗ V ⊗n to A V . The image of v1 ⊗ · · · ⊗ vn under Ant n is denoted by v1 · · · vn . More generally, if n = n1 + · · · + nr with nonnegative integers n1 , · · · , nr we have the partial antisymmetrization nr n1 Antn1 ,··· ,nr : A ⊗ V ⊗n −→ V ⊗A · · · ⊗A V A A 1 f −→ n1 !···nr ! sgn(π ) f π (II.1) π∈Sn1 ×···×Snr
Here, Sn1 × · · · × Snr is viewed as a subgroup of Sn , and we view nA1 V ⊗A · · · ⊗A nr ⊗n . If the context allows, we delete the subscript A in this A V as a subspace of A ⊗ V tensor product. Elements of the r–fold tensor product A V ⊗· · ·⊗ A V are written as Grassmann functions f (ξ (1) , · · · , ξ (r) ), with ξ () the variable for the th copy of A V . Definition II.5. Let C be an antisymmetric bilinear form on V and let 1 ≤ i, j ≤ n. The contraction of the i th variable to the j th variable is the A–linear map ConC : A ⊗ V ⊗n −→ A ⊗ V ⊗(n−2) i→j
characterized by ConC (v1 ⊗· · ·⊗vn ) = ij C(vi , vj ) v1 ⊗· · · vi−1 ⊗vi+1 ⊗· · ·⊗vj −1 ⊗vj +1 ⊗· · ·⊗vn i→j
for all v1 , · · · , vn ∈ V . Here
j −i+1 if j > i (−1)
ij = 0 if j = i . (−1)i−j if j < i
Convergence of Perturbation Expansions in Fermionic Models. Part 1
201
Remark II.6. Let C1 , C2 be antisymmetric bilinear forms on V , λ1 , λ2 ∈ C, 1 ≤ i, j ≤ n and f ∈ A ⊗ V ⊗n . Then Con
i→j (λ C +λ C ) 1 1 2 2
f = λ1 ConC1 f + λ2 ConC2 f. i→j
i→j
Remark II.7. Assume that A = C and that {ξi }, i ∈ I is a basis of V . Then every element f of V ⊗ n can be uniquely written in the form f =
ϕ(i1 , · · · , in ) ξi1 ⊗ · · · ⊗ ξin
i1 ,··· ,in ∈I
with a function ϕ on I n . Then, for 1 ≤ µ < ν ≤ n, ConC f = µ→ν
j1 ,··· ,jn−2 ∈I
ϕ (j1 , · · · , jn−2 ) ξj1 ⊗ · · · ⊗ ξjn−2
with ϕ (j1 , · · · , jn−2 ) = (−1)ν−µ+1
i,j ∈I
Cij ϕ(j1 , · · · , jµ−1 , i, jµ · · · , jν−2 , j, jν−1 · · · , jn−2 ).
Lemma II.8. Let r ≥ 2, n = n1 + · · · + nr , 1 ≤ k = ≤ r and n1 + · · · + nk−1 + 1 ≤ µ, µ ≤ n1 + · · · + nk , n1 + · · · + n−1 + 1 ≤ ν, ν ≤ n1 + · · · + n . Let C be a covariance (antisymmetric bilinear form) on V and f ∈
n1 A
V ⊗· · ·⊗
nr A
V.
i) ConC f is partially antisymmetric, precisely µ→ν
ConC f ∈
n1
µ→ν
A
V ⊗ ··· ⊗
nk −1 A
V ⊗ ··· ⊗
n −1 A
V ⊗ ··· ⊗
nr A
V.
ii) ConC µ →ν
f = ConC f. µ→ν
Proof. We give the proof in the case r = 2, k = 1. The general case is analogous. ii) Clearly, it suffices to show that ConC f = ConC f for all f ∈ nA1 V ⊗ nA2 V . µ→ν
Let n = n1 + n2 and f = ConC f = µ→ν
ConC f =
1→n1 +1
i1 ,··· ,in ∈I
1→n1 +1
ϕ(i1 , · · · , in ) ξi1 ⊗ · · · ⊗ ξin ,
j1 ,··· ,jn−2 ∈I
j1 ,··· ,jn−2 ∈I
ϕ (j1 , · · · , jn−2 ) ξj1 ⊗ · · · ⊗ ξjn−2 , ϕ (j1 , · · · , jn−2 ) ξj1 ⊗ · · · ⊗ ξjn−2 .
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As f ∈ nA1 V ⊗ nA2 V , ϕ is antisymmetric under permutations of its first n1 arguments and under permutations of its last n2 arguments. Consequently, ϕ (j1 , · · · , jn−2 ) = (−1)ν−µ+1
i,j ∈I
Cij ϕ(j1 , · · · , jµ−1 , i, jµ · · · , jν−2 , j, jν−1 · · · , jn−2 )
= (−1)ν−µ+1 (−1)µ−1 (−1)ν−n1 −1 = (−1)n1 +1
i,j ∈I
i,j ∈I
Cij ϕ(i, j1 , · · · , jn1 −1 , j, jn1 , · · · , jn−2 )
Cij ϕ(i, j1 , · · · , jn1 −1 , j, jn1 , · · · , jn−2 )
= ϕ (j1 , · · · , jn−2 ). i) By part ii, we may assume that µ = 1 and ν = n1 + 1. By linearity, we may assume that f = v1 · · · vn1 ⊗ w1 · · · wn2 with v1 , · · · , vn1 , w1 , · · · , wn2 ∈ V . Using v1 · · · vn1 =
1 n1
n1
(−1)i−1 vi ⊗ v1 · · · vi−1 vi+1 · · · vn1
i=1
and its analog for w1 · · · wn2 , we have f =
1 n1 n2
n1 n2
(−1)i−1 (−1)j −1 vi ⊗ v1 · · · vi−1 vi+1 · · · vn1
i=1 j =1
⊗wj ⊗ w1 · · · wj −1 wj +1 · · · wn2 . Since ConC vi 1→n1 +1
⊗ v1 · · · vi−1 vi+1 · · · vn1 ⊗ wj ⊗ w1 · · · wj −1 wj +1 · · · wn2
= (−1)n1 +1 C(vi , wj )v1 · · · vi−1 vi+1 · · · vn1 ⊗ w1 · · · wj −1 wj +1 · · · wn2 , we have ConC f = µ→ν
1 n1 n2
n1 n2
(−1)n1 +j −i+1 C(vi , wj )v1 · · · vi−1 vi+1 · · · vn1
i=1 j =1
⊗ w1 · · · wj −1 wj +1 · · · wn2 . This shows that ConC f ∈ nA1 −1 V ⊗ nA2 −1 V .
(II.2)
µ→ν
Definition II.9. Let C be a covariance on V , r ≥ 1 and 1 ≤ k = ≤ r. i) Let n1 , · · · , nr ≥ 0. If nk , n ≥ 1 and f (ξ (1) , · · · , ξ (r) ) ∈ nA1 V ⊗ · · · ⊗ nAr V , the contraction of the ξ (k) –fields to the ξ () –fields is defined as ConC f ξ (k) →ξ ()
= n ConC f, µ→ν
where n1 + · · · + nk−1 + 1 ≤ µ ≤ n1 + · · · + nk and n1 + · · · + n−1 + 1 ≤ ν ≤ n1 + · · · + n . By Lemma II.8, this definition is independent of the choice of µ, ν. If nk = 0 or n = 0, we set ConC = 0. Observe that ConC maps n1 A
V ⊗ ··· ⊗
nr A
V to
n1 A
ξ (k) →ξ ()
V ⊗ ···
nk −1 A
V ⊗ ···
n −1 A
ξ (k) →ξ ()
V ⊗
nr A
V.
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ii) The maps ConC induce an A linear map from the r–fold tensor product ξ (k) →ξ ()
A
V ⊗ ··· ⊗
A
n1
V =
n1 ,··· ,nr ≥0
A
V ⊗ ··· ⊗
nr A
V
to itself, which is also denoted by ConC . ξ (k) →ξ ()
basis of V . Then, for every Grassmann function Lemma II.10. Assume that {ξi } is a f (ξ (1) , · · · , ξ (r) ) in nA1 V ⊗ · · · ⊗ nAr V , δ δ ConC (f ) = − n1k (k) Cij () f . ξ (k) →ξ ()
δξi
i,j
δξj
Proof. We give the proof in the case r = 2, k = 1, = 2. The general case is similar. By linearity we may assume that (1)
(1)
(2)
(2)
f = ξi1 · · · ξin ⊗ ξj1 · · · ξjn . 1
2
The claim then follows directly from (II.2).
To simplify r = 2, we write f (ξ, ξ ) instead of f (ξ (1) , ξ (2) ) for n1notation nwhen 2 elements of A V ⊗ A V . Similarly, in the case r = 3, we write f (ξ, ξ , ξ ) for f (ξ (1) , ξ (2) , ξ (3) ). Example II.11.
ConC ξk ξ = Ck , ξ →ξ
ConC ξk ξ ξm = Ck ξm − Ckm ξ , ξ →ξ
ConC ξj ξk ξ = 21 ξj Ck − ξk Cj . ξ →ξ
Remark II.12. Since taking partial derivatives commutes with Wick ordering, for any f (ξ, ξ ) ∈ A (V ⊕ V ) ConC :f (ξ, ξ ):ξ = :ConC f (ξ, ξ ) :ξ . ξ →ξ
ξ →ξ
The main reason for introducing the contraction operator is the following “integration by parts formula”: Lemma II.13. Let f (ξ, ξ , ξ ) be a Grassmann function of degree at least one in ξ . Set f˜(ξ, ξ , ξ ) = ConC f (ξ, ξ , ξ ), ξ →ξ
g(ξ, ˜ ξ ) = :f˜(ξ, ξ, ξ ):ξ , g(ξ, ξ ) = :f (ξ, ξ, ξ ):ξ .
Then
g(ξ, ˜ ξ ) dµC (ξ ) =
g(ξ, ξ ) dµC (ξ ).
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That is, :f (ξ, ξ, ξ ):ξ
ξ =ξ
. . Con f (ξ, ξ , ξ )
dµC (ξ ) = . . C ξ =ξ ξ ξ →ξ
ξ =ξ
dµC (ξ ).
Proof. It suffices to prove the statement in the case that f (ξ, ξ , ξ ) = ξi1 · · · ξin−r ξin−r+1 · · · ξin :ξjm ξjm−1 · · · ξj1 :ξ with 1 ≤ r ≤ n, m ≥ 1. Then
f˜(ξ, ξ , ξ ) = − 1r ξi1 · · · ξin−r
(−1)(n+m+)+(k−1) Cik j
k=n−r+1,···n =1,···m
× ξin−r+1 · · · ξik−1 ξik+1 · · · ξin :ξjm · · · ξj+1 ξj−1 · · · ξj1 : and g(ξ, ˜ ξ ) dµC (ξ ) =
1 r
(−1)n+m+k+ Cik j
k=n−r+1,···n =1,···m
× (:ξi1 · · · ξik−1 ξik+1 · · · ξin :)(:ξjm · · · ξj+1 ξj−1 · · · ξj1 :) dµC (ξ ). If m = n, both g(ξ, ˜ ξ ) dµC (ξ ) and g(ξ, ξ ) dµC (ξ ) are zero by A.1. In the case m = n, again by Lemma A.1 (−1)k+ Cik j det Cik j k =k g(ξ, ˜ ξ ) dµC (ξ ) = 1r k=n−r+1,···n =1,···m
=
1 r
k=n−r+1,···n
det Cik j
=
= det Cik j = (:ξi1 · · · ξir ξir+1 · · · ξin :) (:ξjm ξjm−1 · · · ξj1 :) dµC (ξ ) = g(ξ, ξ ) dµC (ξ ). II.4. Seminorms. We will formulate an estimate on the renormalization group map in terms of norms for which the contraction maps and Grassmann Gaussian integrals can be controlled. In this subsection we assume that A is a graded superalgebra. In practice we shall use families of norms on A V that encode information concerning various derivatives of the coefficient functions. To unify such families in a way that incorporates Leibniz’s rule we give Definition II.14. i) On R+ ∪ {∞} = x ∈ R x ≥ 0 ∪ {+∞}, addition and the total ordering ≤ are defined in the standard way. With the convention that 0 · ∞ = ∞, multiplication is also defined in the standard way.
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ii) Let d ≥ 0. The d–dimensional norm domain Nd is the set of all formal power series Xδ t1δ1 · · · tdδd X= δ=(δ1 ,··· ,δd )∈Nd0
in the variables t1 , · · · , td with coefficients Xδ ∈ R+ ∪ {∞}. Addition and partial ordering on Nd are defined componentwise: If δ1 X= Xδ t1δ1 · · · tdδd , X = Xδ t1 · · · tdδd , δ∈Nd0
δ∈Nd0
then X + X =
(Xδ + Xδ ) t1δ1 · · · tdδd , δ
X≤X
⇐⇒ Xδ ≤ Xδ for all δ ∈ Nd0 .
Multiplication is defined by (X · X )δ =
β+γ =δ
Xβ Xγ .
We identify R+ ∪ {∞} with the set of all X ∈ Nd with Xδ = 0 for all δ = 0 = (0, · · · , 0). If a > 0, X0 = ∞ and a − X0 > 0 then (a − X)−1 is defined as (a − X)−1 =
1 a−X0
∞ X−X0 n n=0
a−X0
.
Definition II.15. Let E be a complex vector space. A (d–dimensional) seminorm on E is a map · : E → Nd such that e1 + e2 ≤ e1 + e2 ,
λ e = |λ| e
for all e, e1 , e2 ∈ E and λ ∈ C. Example II.16. Let be the space of all functions f : Rd → C. Define sup ∂ δ f (x) t1δ1 · · · tdδd , f = δ∈Nd0
x∈Rd
where supx∈Rd ∂ δ f (x) = ∞ if ∂ δ f (x) is not everywhere defined. Remark II.17. i) By convention, N0 = R+ ∪ {∞}. ii) If E is a complex vector space and · is a 0–dimensional seminorm on E that obeys e < ∞ for all e ∈ E, then · is a seminorm on E in the conventional sense. Definition II.18. Let m, n ≥ 0. A seminorm · on the space Am ⊗ V ⊗n is called symmetric, if for all f ∈ Am ⊗ V ⊗n and all permutations π ∈ Sn , f π = f and f = 0 if m = n = 0.
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Remark II.19. Assume, as in Remark II.7, that A = C and that {ξi }, i ∈ I is a basis of V . Every element f of V ⊗ n can be uniquely written in the form f = ϕ(i1 , · · · , in ) ξi1 ⊗ · · · ⊗ ξin i1 ,··· ,in ∈I
with a function ϕ on I n . Therefore, a symmetric family of seminorms corresponds to a family of seminorms · on the spaces of functions ϕ on I n such that ϕ(i1 , · · · , in ) = ϕ(iπ(1) , · · · , iπ(n) )
for all π ∈ Sn .
Example II.20. Let A = C and V be a finite dimensional vector space with basis ξ1 , · · · , ξD . For a function ϕ on {1, · · · , D}n define the L1 –L∞ –norm, ϕ1,∞ = max
D
sup
1≤k≤n 1≤ik ≤D i1 ,··· ,ik−1 ,ik+1 ,··· ,,in =1
|ϕ(i1 , · · · , in )|.
This defines a family of symmetric (0–dimensional) seminorms on the spaces V ⊗n . Definition II.21. By Am [n1 , · · · , nr ] we denote the image of Am ⊗ V ⊗(n1 +···+nr ) under the partial antisymmetrization map Antn1 ,··· ,nr defined in (II.1). It is the C–linear subspace of n1 nr V (1) ⊗ · · · ⊗ V (r) A[n1 , · · · , nr ] = A
A
generated the form am p1 (ξ (1) ) · · · pr (ξ (r) ) with am ∈ Am and pν (ξ (ν) ) ∈ nν (ν) by elements of(1) V . Elements f (ξ , · · · , ξ (r) ) of Am [n1 , · · · , nr ] are called homogeneous, and nν is their degree of homogeneity in the variable ξν . Observe that Am [n1 , · · · , nr ] is a subspace of Am ⊗ V ⊗(n1 +···+nr ) . Lemma II.22. Let · be a family of symmetric seminorms on the spaces Am ⊗ V ⊗n . Then for all f (ξ (1) , · · · , ξ (r) ) ∈ Am [n1 , · · · , nr ], (i) for all permutations π ∈ Sr , f (ξ (1) , · · · , ξ (r) ) = f (ξ (π(1)) , · · · , ξ (π(r)) ). (ii) f (ξ (1) , · · · , ξ (r−2) , ξ (r−1) , ξ (r−1) ) ≤ f (ξ (1) , · · · , ξ (r−2) , ξ (r−1) , ξ (r) ). Here, f (ξ (1) , · · · , ξ (r−2) , ξ (r−1) , ξ (r−1) ) is an element of Am [n1 , · · · , nr−2 , nr−1 + nr ]. (iii) If ∈ C with | | = 1 and f (ξ (1) , · · · , ξ (r−1) , ξ (r) + ξ (r+1) ) =
nr
fk (ξ (1) , · · · , ξ (r−1) , ξ (r) , ξ (r+1) )
k=0
with fk ∈ Am [n1 , · · · , nr−1 , nr − k, k] then fk ≤ nkr f . (iv) f = 0 if f ∈ A0 [0, 0, · · · , 0].
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Proof. Parts (i) and (iv) are trivial. To prove part (ii) set f (ξ (1) , · · · , ξ (r−2) , ξ (r−1) ) = f (ξ (1) , · · · , ξ (r−2) , ξ (r−1) , ξ (r−1) ) = Antn1 ,··· ,nr−2 ,nr−1 +nr f. Then, by Def. II.18 f ≤
1 n1 !···nr−2 ! (nr−1 +nr )!
f π ≤ f .
π∈Sn1 ×···×Snr−2 ×Snr−1 +nr
We now prove part (iii). For any subset I of {n1 +· · ·+nr−1 +1, · · · , n1 +· · ·+nr } let πI−1 be the permutation that brings the sequence I, {n1 +· · ·+nr−1 +1, · · · , n1 +· · ·+nr }\I into standard order. Then
nr −k sgn(πI ) f πI . fk = I ⊂{n1 +···+nr−1 +1,··· ,n1 +···+nr } |I |=k
Again, by Def. II.18, fk ≤
f =
nr k
f .
|I |=k
Definition II.23. Let · be a family d–dimensional seminorms and let of symmetric f (ξ (1) , · · · , ξ (r) ) ∈ A V ⊗ · · · ⊗ A V ∼ = A (V ⊕ · · · ⊕ V ). Write fm;n1 ,··· ,nr f = m,n1 ,··· ,nr ≥0
with fm;n1 ,··· ,nr ∈ Am [n1 , · · · , nr ]. For any real α ≥ 1, b > 0 and c ∈ Nd set α |n| b|n| fm;n1 ,··· ,nr . N (f ; α) = b12 c m,n1 ,··· ,nr ≥0
We omit the dependence on b and c from the symbol N (f ; α). If the context allows, we will also delete the reference to α. In the applications we have in mind (see [FKTo1, Theorem V.2], [FKTo2, Theorem VIII.6], [FKTo3, Lemma XV.5], [FKTf1, Subsect. 7 of §II]), c is a bound for various weighted L1 –norms of the propagator C (in position space), while b2 is a bound for its L∞ –norm. α1 is a (possibly) fractional power of the coupling constant. Remark II.24. Let f (ξ (1) , · · · , ξ (r) ) ∈ A V ⊗ · · · ⊗ A V ∼ = A (V ⊕ · · · ⊕ V ). Then (i) for 1 ≤ s ≤ r, N f (ξ (1) + ξ (r+1) , · · · , ξ (s) + ξ (r+s) , ξ (s+1) , · · · , ξ (r) ); α ≤ N f (ξ (1) , · · · , ξ (r) ); 2α , (ii) for all a ≥ 1,
N (f ; α) ≤ N (f ; aα).
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Proof.
(i) Since f (ξ (1) + ξ (r+1) , · · · , ξ (s) + ξ (r+s) , · · · , ξ (r) ) = f (ξ (1) + ξ (r+1) , · · · , ξ (r) + ξ (2r) ) (r+1) ξ
=···=ξ (2r) =0
,
it suffices to prove the claim in the case s = r. Write fm;n1 ,··· ,nr f = m,n1 ,··· ,nr ≥0
with fm;n1 ,··· ,nr ∈ Am [n1 , · · · , nr ]. By part (iii) of Lemma II.22, fm;n1 ,··· ,nr (ξ (1) + ξ (r+1) , · · · , ξ (r) + ξ (2r) ) = fm; n1 −k1 ,··· ,nr −kr ,k1 ,··· ,kr (ξ (1) , · · · , ξ (r) , ξ (r+1) , · · · , ξ (2r) ) ki =0,··· ,ni for i=1,··· ,r
with fm; n1 −k1 ,··· ,nr −kr ,k1 ,··· ,kr ∈ Am [n1 − k1 , · · · , nr − kr , k1 , · · · , kr ] and fm; n
1 −k1 ,··· ,nr −kr ,k1 ,··· ,kr
r ≤ fm;n ,··· ,n ni . r 1 ki i=1
Consequently N f (ξ (1) + ξ (r+1) , · · · , ξ (r) + ξ (2r) ) = b12 c α |n| b|n| fm; n1 −k1 ,··· ,nr −kr ,k1 ,··· ,kr m,n1 ,··· ,nr ≥0
≤
1 b2
c
ki =0,··· ,ni for i=1,··· ,r
ki =0,··· ,ni for i=1,··· ,r
m; n1 ,···nr
=
1 b2
c
r ni i=1
ki
α |n| b|n| 2n1 +···+nr fm;n1 ,··· ,nr
m; n1 ,···nr (1)
= N f (ξ (ii) is trivial
α |n| b|n| fm;n1 ,··· ,nr
, · · · , ξ (r) ); 2α .
II.5. An Estimate of the Renormalization Group Map. Let A be a graded superalgebra, · be a family of symmetric d–dimensional seminorms on the spaces Am ⊗ V ⊗n and C be a covariance on V . Definition II.25. (i) We say that c ∈ Nd , with c0 = 0, ∞, is a contraction bound for C with respect to these seminorms if for all f ∈ Am ⊗ V ⊗n , f ∈ Am ⊗ V ⊗n and all 1 ≤ i ≤ n, 1 ≤ j ≤ n , ConC (f ⊗ f ) ≤ c f f . i→n+j
Observe that ConC (f ⊗ f ) ∈ Am+m ⊗ V ⊗(n+n −2) . i→n+j
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(ii) For n ≤ n define the partial antisymmetrization Antn = Antn ,1,1,··· ,1 . It is characterized by Antn (v1 ⊗ · · · ⊗ vn ⊗ w1 ⊗ · · · ⊗ wn−n ) n = v1 · · · vn ⊗ w1 ⊗ · · · ⊗ wn−n ∈ V ⊗A V ⊗(n−n ) A
for all v1 , · · · , vn , w1 · · · , wn−n ∈ V . We say that the real number b ≥ 0 is a (Grassmann) integral bound for C with respect to the family of seminorms if for every f ∈ Am ⊗ V ⊗n and every n ≤ n, Antn (f )dµC ≤ (b/2)n f . n
V ⊗A V ⊗(n−n ) to A⊗V ⊗(n−n ) . Example II.26. In Example II.20, a contraction bound for a covariance C = Cij 1≤i,j ≤D with respect to the L1 –L∞ –norm is given by Here, the Grassmann Gaussian integral maps
c = C1,∞ = max
1≤i≤D
A
Cij . 1≤j ≤D
Let b > 0 be such that ξi1 · · · ξin dµC (ξ ) ≤ (b/2)n
(II.3)
for all n ≥ 0 and all i1 , · · · , in . Then b is an integral bound for the covariance C. In Appendix B we give criteria under which (II.3) is fulfilled. When dim V = ∞, it is not a priori clear whether or not the renormalization group map C (W )(ψ) = log Z1 eW (ψ+ξ ) dµC (ξ ) of Def. II.3 makes sense. However, in the case of interest, we can define it as a formal power series in W . The Taylor expansion of eW (ψ+ξ ) dµC (ξ ) is ∞ n=1 Gn (W, · · · , W ), where the nth term is the n–linear map 1 Gn (W1 , · · · , Wn ) = n! W1 (ψ + ξ ) · · · Wn (ψ + ξ ) dµC (ξ ) from A V × · · · × A V to A V , restricted to the diagonal. Explicit evaluation of the Grassmann integral expresses Gn as the sum of all graphs with vertices W1, · · · , Wn and lines C. The (formal) Taylor coefficient ddt1 · · · ddtn C (t1 W1 +· · ·+tn Wn ) of C (W ) is similar, but with only connected graphs.
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Definition II.27. Let · be a family of symmetric seminorms and let C be a covariance on V with a finite integral bound and a contraction bound c obeying c0 < ∞. Then d d · · · (t W + · · · + t W ) n n dt1 dtn C 1 1 t1 =···=tn =0
interpreted as a sum of graphs, as above, is a bounded n–linear map . Then C is defined to be the formal Taylor series associated to the sequence of these multilinear maps. Theorem II.28. Let · be a family of symmetric seminorms and let C be a covariance on V with contraction bound c and integral bound b. Then the formal Taylor series 2 C (:W :) converges to an analytic map3 on W ∈ A V W even, N W ; 8α 0 < α4 . Furthermore, if W (ψ) ∈ A V is an even Grassmann function such that N W ; 8α 0 < then
N C (:W :) − W ; α ≤
α2 4 ,
N(W ;8α)2 2 . α 2 1− 42 N(W ;8α) α
This theorem is proven in §III.3.
II.6. Contraction and Integral Bounds. In this subsection we investigate properties of contraction and integral bounds as introduced in Def. II.25. These properties will be used in §III to prove Theorem II.28. Lemma II.29. Assume that c is a contraction bound and b an integral bound for C. (i) If nr , nr+1 ≥ 1, then for f1 (ξ (1) , · · · , ξ (r−1) , ξ (r) ) ∈ Am [n1 , · · · , nr−1 , nr , 0], f2 (ξ (1) , · · · , ξ (r−1) , ξ (r+1) ) ∈ Am [n1 , · · · nr−1 , 0, nr+1 ], one has
ConC f1 (ξ (1) , · · · , ξ (r−1) , ξ (r) ) f2 (ξ (1) , · · · , ξ (r−1) , ξ (r+1) )
ξ (r) →ξ (r+1)
≤ nr+1 c f1 (ξ (1) , · · · , ξ (r−1) , ξ (r) ) f2 (ξ (1) , · · · , ξ (r−1) , ξ (r+1) ). Observe that the contraction
ConC ξ (r) →ξ (r+1) Am+m [n1 + n1 , · · ·
ξ (r−1) , ξ (r+1) ) lies in (ii) Let 0 ≤ s ≤ t ≤ r and let
f1 (ξ (1) , · · · , ξ (r−1) , ξ (r) ) f2 (ξ (1) , · · · , , nr−1 + nr−1 , nr − 1, nr+1 − 1].
f (ξ (1) , · · · , ξ (s) , ξ (s+1) , · · · , ξ (t) , ξ (t+1) , · · · , ξ (r) ) ∈ Am [n1 , · · · , ns , ns+1 , · · · , nt , nt+1 , · · · , nr ]. 3 For an elementary discussion of analytic maps between Banach spaces see, for example, Appendix A of [PT].
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Set f˜(ξ (1) , · · · , ξ (s) , ξ (s+1) , · · · , ξ (t) , · · · , ξ (r) ) = :f (ξ (1) , · · · , ξ (s) , ξ (s+1) , · · · , ξ (t) , · · · , ξ (r) ):ξ (1) ,··· ,ξ (s) , where : · :ξ (1) ,··· ,ξ (s) denotes Wick ordering, with respect to the covariance C, in the variables ξ (1) , · · · , ξ (s) separately. Then f˜(ξ, · · · , ξ, ξ (t+1) · · · , ξ (r) ) dµC (ξ ) ≤ bn1 +···+nt f (ξ (1) , · · · , ξ (r) ). Observe that f˜(ξ, · · ·, ξ, ξ (t+1) · · · , ξ (r) ) dµC (ξ ) ∈ Am [0, · · · , 0, nt+1 , · · · , nr ]. Proof.
i) By definition ConC f1 (ξ (1) , · · · , ξ (r−1) , ξ (r) ) f2 (ξ (1) , · · · , ξ (r−1) , ξ (r+1) )
ξ (r) →ξ (r+1)
= ± nr+1 Ant n1 +n1 ,··· ,nr−1 +nr−1 ,nr −1,nr+1 −1
ConC f1 ⊗ f2
π
,
µ→ν
r r r−1 where 1 + r−1 i=1 ni , 1 + i=1 ni + i=1 ni ≤ µ ≤ i=1 ni ≤ ν ≤ nr+1 + r−1 r i=1 ni + i=1 ni and π is the permutation that maps the sequence 1, · · · , n1 , · · · , n1 + · · · + nr − 1, · · · , n1 + · · · + nr + n1 + · · · + nr+1 − 2 to the sequence 1, · · · , n1 , n1 + · · · + nr − 1 + 1, · · · , n1 + · · · + nr − 1 + n1 , n1 + 1, · · · . The desired estimate now follows from Defs. II.25, II.18 and II.9. ii) Set g(ξ, ξ (t+1) · · · , ξ (r) ) = f˜(ξ, · · · , ξ, ξ (t+1) · · · , ξ (r) ). In the case s = 0 one has f˜ = f and hence g ≤ f˜ ≤ f by part (ii) of Lemma II.22. Therefore, by Def. II.25, f˜(ξ, · · · , ξ, ξ (t+1) · · · , ξ (r) ) dµC (ξ ) ≤ (b/2)n1 +···+nt g ≤ (b/2)n1 +···+nt f . In the general case, we have by part (i) of Prop. A.2 and part (iii) of Lemma II.22, ˜ f = f (ξ (1) + ıζ (1) , · · · , ξ (s) +ıζ (s) , ξ (s+1) , · · · , ξ (t) , · · · , ξ (r) ) dµC (ζ (1) , · · · , ζ (s) ) fk1 ,··· ,ks (ξ (1) , ζ (1) , · · · , ξ (s) , ζ (s) , ξ (s+1) , · · · , = ki =1,··· ,ni i=1,··· ,s
ξ (t) , · · · , ξ (r) )dµC (ζ (1) , · · · , ζ (s) ) with
fk1 ,··· ,ks ∈ Am [n1 − k1 , k1 , · · · , ns − ks , ks , ns+1 , · · · , nr ]
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fulfilling fk1 ,··· ,ks ≤ f
s ni i=1
ki
.
By the special case discussed above fk1 ,··· ,ks (ξ (1) , ζ (1) , · · · , ξ (s) , ζ (s) , ξ (s+1) , · · · , ξ (t) , ξ (t+1) , · · · , ξ (r) ) dµC (ζ (1) , · · · , ζ (s) ) ≤ (b/2)k1 +···+ks fk1 ,··· ,ks . Therefore, again by the special case discussed above, fk1 ,··· ,ks (ξ, ζ (1) , · · · , ξ, ζ (s) , ξ, · · · , ξ, ξ (t+1) , · · · , ξ (r) ) dµC (ζ (1) , · · · , ζ (s) ) dµC (ξ ) ≤ (b/2)n1 +···+nt fk1 ,··· ,ks . Consequently f˜(ξ, · · · , ξ, ξ (t+1) · · · , ξ (r) ) dµC (ξ ) ≤ bn1 +···+nt f
s
ki =1,··· ,ni i=1,··· ,s
i=1
1 ni 2ni ki
≤ bn1 +···+nt f . Remark II.30. Let C1 , C2 be covariances on V and λ1 , λ2 ∈ C. i) If c1 is a contraction bound for C1 and c2 is a contraction bound for C2 then |λ1 |c1 + |λ2 |c2 is a contraction bound for λ1 C1 + λ2 C2 . √ √ ii) If b1 and b2 are integral bounds for C1 and C2 , then |λ1 | b1 + |λ2 | b2 is an integral bound for λ1 C1 + λ2 C2 . Proof. Part (i) follows from Remark II.6. To prove part (ii), let n ≤ n. For I ⊂ n −|I | |I | V ⊗A V ⊗(n−n ) {1, · · · , n } let AntI be the map from A ⊗ V ⊗n to A V ⊗A A characterized by AntI (v1 ⊗ · · · ⊗ vn ⊗ w1 ⊗ · · · ⊗ wn−n ) = I vi ⊗ vi ⊗ w1 ⊗ · · · ⊗ wn−n , i∈I
j ∈{1,··· ,n }\I
where I is the sign of the permutation that puts the sequence I, {1, · · · , n } \ I back in increasing order. Then AntI (f )(ξ, ξ ) dµλ1 C1 (ξ ) dµλ2 C2 (ξ ), Antn (f ) dµλ1 C1 +λ2 C2 = I ⊂{1,··· ,n } |I | even
Convergence of Perturbation Expansions in Fermionic Models. Part 1
and consequently n Antn (f )dµC ≤
r=0
I ⊂{1,··· ,n } |I |=r
r=0
≤
AntI (f )(ξ, ξ ) dµλ1 C1 (ξ ) dµλ2 C2 (ξ )
√ n √ |λ1 | b1 r |λ2 | b2 n −r n
≤
2
r
1 n 2
213
|λ1 | b1 +
2
|λ2 | b2
n
f
f .
(1) · · , ξ (s) , ξ (s+1) , · · · , ξ (t) , ξ (t+1) , · · · , ξ (r) ) be a Grassmann Lemma II.31. Let f (ξ , · function in A V ⊗ · · · ⊗ A V ∼ = A (V ⊕ · · · ⊕ V ). Set
f˜(ξ (1) , · · · , ξ (s) , ξ (s+1) , · · · , ξ (t) , · · · , ξ (r) ) = :f (ξ (1) , · · · , ξ (s) , ξ (s+1) , · · · , ξ (t) , · · · , ξ (r) ):ξ (1) ,··· ,ξ (s) . Let 0 < ε ≤ α. If εb is is an integral bound for the covariance C, then
N f˜(ξ, · · · , ξ, ξ (t+1) , · · · , ξ (r) ) dµC (ξ ) ; α ≤ N (f ; α). If f (0, · · · , 0, ξ (t+1) , · · · , ξ (r) ) = 0 then
N f˜(ξ, · · · , ξ, ξ (t+1) , · · · , ξ (r) ) dµC (ξ ) ; α ≤ Proof. Set
g(ξ
(t+1)
,··· ,ξ
(r)
Write
)=
ε2 α2
N (f ; α).
f˜(ξ, · · · , ξ, ξ (t+1) , · · · , ξ (r) ) dµC (ξ ).
f =
fm;n1 ,··· ,nr
m,n1 ,··· ,nr ≥0
with fm;n1 ,··· ,nr ∈ Am [n1 , · · · , nr ] and set f˜m;n1 ,··· ,nr (ξ (1) , · · · , ξ (t) , · · · , ξ (r) ) = :fm;n1 ,··· ,nr (ξ (1) , · · · , ξ (s) , · · · , ξ (t) , · · · , ξ (r) ):ξ (1) ,··· ,ξ (s) gm;n1 ,··· ,nr (ξ (t+1) , · · · , ξ (r) ) = f˜m;n1 ,··· ,nr (ξ, · · · , ξ, ξ (t+1) , · · · , ξ (r) ) dµC (ξ ). Then g =
and therefore, by part (ii) of Lemma II.29, nt+1 +···+nr (εb)n1 +···+nt fm;n1 ,··· ,nr αb
m;n1 ,··· ,nr ≥0 gm;n1 ,··· ,nr
N(g; α) ≤
1 b2
c
m;nt+1 ,··· ,nr
=
1 b2
c
m;n1 ,··· ,nt+1 ,··· ,nr
n1 ,··· ,nt
n1 +···+nt ε α
α |n| b|n| fm;n1 ,··· ,nr .
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Since αε ≤ 1, this implies that N (g; α) ≤ N (f ; α). If f (0, · · · , 0, ξ (t+1) , · · · , ξ (r) ) = 0 then fm;n1 ,··· ,nr = 0 for n1 = · · · = nt = 0 and gm;n1 ,··· ,nr = 0 for n1 + · · · + nt ≤ 1, since ξi dµC (ξ ) = 0. Consequently
n1 +···+nt 2 ε α |n| b|n| fm;n1 ,··· ,nr ≤ αε 2 N (f ; α). N (g; α) ≤ b12 c α m;n1 +···+nt ≥2 nt+1 ,··· ,nr
Corollary II.32. Let f (ξ ) ∈ A V . (i) If αb is an integral bound for the covariance C, then N (:f :C ; α) ≤ N (f ; 2α), N (f ; α) ≤ N (:f :C ; 2α). (ii) Let C1 , C2 be two covariances and 0 < ε ≤ √α . Assume that εb is an integral 2 bound for C1 − C2 . Set :f (ξ ):C2 = :f (ξ ):C1 . Then 2 N (f − f ; α) ≤ 2ε N (f ; 2α). α2 (iii) Let, for C0 and κ in a neighbourhood of zero, Cκ be a covariance on V . Assume. that . d dκ Cκ κ=0 have integral bounds b and b , respectively. Define fκ by . fκ . Cκ = f . Then b 2 1 N ddκ fκ κ=0 ; α ≤ (α−1) N (f ; 2α). 2 b Proof. (i) By part (i) of Prop. A.2, Remark II.30, the lemma above with s = 0 and ε = α, and part (i) of Remark II.24,
N (:f :C ; α) = N f (ξ + ξ ) dµ−C (ξ ); α ≤ N f (ξ + ξ ); α ≤ N (f ; 2α). The proof of the other bound is similar. (ii) Define fz by :f (ξ ):C2 = :fz (ξ ):C2 +z(C1 −C2 ) . By part (i) of Lemma A.4, fz (ξ ) = .. f (ξ ) .. −z(C −C ) . 1 2 √ By Remark II.30.ii, −z(C1 − C2 ) has integral bound |z| ε b, which is bounded α 2 by αb for all |z| ≤ ε . Hence, by part (i), N (fz ; α) ≤ N (f ; 2α) 2 for all |z| ≤ ε . Let C be the contour |ζ | = αε in the complex plane, with standard orientation. Then 1 1 1 1 1 f − f = f1 − f0 = 2πı f − dζ = ζ ζ −1 ζ 2πı ζ (ζ −1) fζ dζ. α 2
C
Hence,
N f − f;α ≤
1 ( αε )2 −1
C
2 sup N fζ ; α ≤ 2 αε 2 N (f ; 2α).
|ζ |=( αε )2
Convergence of Perturbation Expansions in Fermionic Models. Part 1
215
(iii) Set Dz = C0 + z ddκ Cκ κ=0 and define gz by .. gz .. D = f . Since Cκ and Dκ agree z to order κ, ddκ fκ κ=0 = ddz gz z=0 . By Remark II.30.ii, Dz has integral bound 2 √ b + |z| b . For all |z| ≤ (α − 1)2 bb this is bounded by αb. By part (i), N (gz ; α) ≤ N (f ; 2α) 2 for all |z| ≤ (α − 1)2 bb . Hence, by the Cauchy integral formula b 2 1 sup N gz ; α N ddκ fκ κ=0 ; α = N ddz gz z=0 ; α ≤ (α−1) 2 b ≤
1 (α−1)2
b 2 b
|z|=(α−1)2 ( bb )2
N (f ; 2α)
The following proposition is the key to our estimate on the renormalization group map and will be used in the proof of Prop. III.7. Proposition II.33. Assume that c is a contraction bound and b an integral bound for C. Let ≥ 1 and r ≥ s ≥ 1. Let fi (ξ (1) , · · · , ξ (s) , ξ (s+1) , · · · , ξ (r) ), 1 ≤ i ≤ and f (ξ (1) , · · · , ξ (r) ) be Grassmann functions. Assume that for each 1 ≤ i ≤ there is a 1 ≤ ji ≤ s such that fi vanishes when ξ (ji ) = 0. Set (s+1) (r) g(ξ , · · · , ξ ) = .. fi (ξ (1) , · · · , ξ (s) , · · · , ξ (r) ) .. ξ (1) ,··· ,ξ (s) i=1
s . f (ξ (1) , · · · , ξ (s) , · · · , ξ (r) ) . dµC (ξ (i) ). . . ξ (1) ,··· ,ξ (s) i=1
Then, for α ≥ 2, we have the two bounds N (g; α) ≤ N (g; α) ≤
! N (f ; α) α N (f ; α) α 2
N (fi ; α),
i=1
N (fi ; α).
i=1
Proof. We first prove the statement in the case that f and all the fi ’s are homogeneous. That is f ∈ Am [n(1) , · · · , n(s) , · · · , n(r) ], (1)
(r)
fi ∈ Ami [ni , · · · , ni ] (ji )
By hypothesis ni
1 ≤ i ≤ .
≥ 1 for each 1 ≤ i ≤ . Set, for each 1 ≤ i ≤ , Coni =
ConC (ji )
ξi
→ζ (ji )
and g (ξ (1) , · · · , ξ (r) ; ζ (1) , · · · , ζ (r) ) (1) (r) = Coni fi (ξi , · · · , ξi ) f (ζ (1) , · · · , ζ (r) ) gw (ξ
i=1 (1)
i=1 (r)
,··· ,ξ
(j )
ξi =ξ (j )
,
; ζ (1) , · · · , ζ (r) ) = .. g (ξ (1) , · · · , ξ (r) ; ζ (1) , · · · , ζ (r) ) .. ξ (1) ,··· ,ξ (s) . ζ (1) ,··· ,ζ (s)
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J. Feldman, H. Kn¨orrer, E. Trubowitz
The
(j )
(j )
ξi =ξ (j )
signifies that ξi
is to be evaluated at ξ (j ) for each 1 ≤ j ≤ r and
1 ≤ i ≤ . By Lemma II.13, times, s (ξ (1) , · · · , ξ (r) ; ξ (1) , · · · , ξ (r) ) dµC (ξ (i) ). g = gw i=1
(j ) Observe that g = 0 unless n(j ) = i=1 ni for all 1 ≤ j ≤ s and then g is of degree s s (j ) ni − 2 = 2n(j ) − 2 n(j ) + j =1
j =1
i=1
in ξ (1) , · · · , ξ (s) . Hence (1) +···+n(s) −)
g (ξ (1) , · · · , ξ (r) ; ξ (1) , · · · , ξ (r) ) by Lemma II.29.ii (1) (s) (1) (r) ≤ b2(n +···+n −) Coni fi (ξi , · · · , ξi ) f (ζ (1) , · · · , ζ (r) )
g ≤ b2(n
i=1
i=1
by Lemma II.22.ii. Set, for each 1 ≤ j ≤ s,
pj = # i 1 ≤ i ≤ , ji = j . Note that pj ≤ n(j ) and sj =1 pj = . Therefore, by Lemma II.29.i, g ≤ = =
s
pj !
j =1 s j =1 s j =1
n(j ) pj
pj ! pj !
As g is of degree
(1) +···+n(s) −)
c b2(n
n(j )
1
pj
(j ) α 2n
n(j )
1
pj
(j ) α 2n
f
fi
i=1
2(n(1) +···+n(s) )
αb
f
b−2 cfi
i=1
n(1) +···+n(s)
αb
f
n(1) +···+n(s) i f . (II.4) b−2 c αb i i
i=1
(j ) (j ) + i=1 ni in ξ (s+1) , · · · , ξ (r) , Def. II.23 implies that j =s+1 n
r
N (g; α) = ≤
j (n(j ) +i n ) 1 i c αb g 2 b s (j ) N (fi ; α). pj ! npj 2n1(j ) N (f ; α) α j =1 i=1 (j )
(II.5)
To complete the proof in the homogeneous case, we derive two bounds on p! pn α12n for all p ≤ n. The first is n p! pn α12n ≤ p! α22n ≤ p! α1n ≤ p! α1p since α ≥ 2. It yields the first bound of the proposition, since s 1 1 1 pj ! pj ≤ j pj ! p = ! . j j α α α j =1
Convergence of Perturbation Expansions in Fermionic Models. Part 1
217
The second is, setting n = p + m, 1 p 1 1 (p+m)! 1 1 p! p+m p α 2(p+m) = α 2p m! α 2m ≤ α 2p (p + m) α 2m = p m p p p ≤ αp2p em/p α12m = αp2p αe2 ≤ αp2p ,
p 1 pp 1+ m p α 2p α 2m
since α 2 > e. It yields the second bound of the proposition, since s p pj j j =1 α
2pj
p ≤ j pj j j
1 α 2j pj
=
1 . α 2
The general case now follows by decomposing f and the fi ’s into homogeneous pieces. III. The Schwinger Functional III.1. Description of the Schwinger Functional. Let A be a superalgebra and V be a complex vector space with generators {ξi }. Furthermore let C be an antisymmetric bilinear form (covariance) on V . First, suppose that V is finite dimensional. Let U (ξ ) ∈ A V be an even Grassmann function such that Z(U, C) = eU (ξ ) dµC (ξ ) is invertible in A. For any f (ξ ) ∈
AV,
S(f ) = SU,C (f ) =
set
1 Z(U,C)
f (ξ ) eU (ξ ) dµC (ξ ).
S is called the Schwinger functional. If V is infinite dimensional, we define S as a formal power series in U , as in Def. II.27. The coefficient that is of order n in U is a sum of connected graphs that has n vertices U , one vertex f and lines C. Remark III.1. (i) The renormalization group map can be expressed in terms of the Schwinger functional. Recall that
W (ψ+ξ ) 1 C (W )(ψ) = log Z e dµC (ξ ), where Z = Z eW (ξ ) dµC (ξ ) , where the ψi are the generators of a vector space V , which is a second copy of V . As C (0) = 0, for even Grassmann functions W (ψ), 1 d C (W )(ψ) = dt C (tW ) dt 0 1 W (ψ + ξ ) etW (ψ+ξ ) dµC (ξ ) = dt − log Z etW (ψ+ξ ) dµC (ξ ) 0 1 StU,C (U ) dt − log Z, = 0
∼ V and the where in the integral U (ψ; ξ ) = W (ψ + ξ ) ∈ A (V ⊕ V ) = AV Schwinger functional is taken in the Grassmann algebra over V with coefficients in the algebra A V .
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(ii) More generally, if W1 and W2 are even Grassmann functions and W2 = W1 + W , then 1 d C (W2 )(ψ) − C (W1 )(ψ) = dt C (W1 + tW ) dt
0
1
= 0
Z1 SU1 +tU,C (U ) dt − log Z , 2
where U1 (ψ; ξ ) = W1 (ψ + ξ ) and U (ψ; ξ ) = W (ψ + ξ ). In [FKT1] we gave a representation for Schwinger functionals which we repeat in the present context. Choose an additional copy V of the vector space V . We denote the canonical isomorphism from V to V by σ and set ηi = σ (ξi ). The antisymmetric bilinear form C on V corresponding to C is given by C (v, w) = C(σ −1 v, σ −1 w). Using the canonical isomorphisms A
(V ⊕ V ) =
r A
r,r
V ⊗
r A
V ∼ =
A
V ⊗
A
V ∼ =
AV
V ,
C defines the Grassmann Gaussian integral dµC (η) as a map from (V ⊕ V ) to A V →V ⊕V A V → A (V ⊕V ) and A V . The diagonal embedding v →v⊕σ (v) induces an embedding the isomorphism
V →V v →σ (v)
induces an isomorphism
define the map R = RU,C : by
f −→
A
V −→
AV→ AV
f (ξ ) →f (ξ +η)
f (ξ ) →f (η)
A
. With this notation we
V
:eU (ξ +η)−U (ξ ) − 1:η f (η) dµC (η).
Once again, if dimV = ∞, R, is, a priori, defined as a formal power series in U , i.e. as a sequence of multilinear maps. In this case, it is easy to explicitly find maps. The nth map is n . [U (ξ + η) − U (ξ )] . f (η) dµ (η). 1 (U1 , U2 , · · · , Un , f ) → n! . .η i i C i=1
For all Grassmann functions As in [FKT1] we have Theorem III.2. f ∈ A V , −1 SU,C (f ) = − RU,C (f ) dµC . Proof. We first prove U (ξ ) f (ξ ) e dµC (ξ ) = f (η) dµC (η) eU (ξ ) dµC (ξ ) + RU,C (f )(ξ ) eU (ξ ) dµC (ξ ).
(III.1)
Convergence of Perturbation Expansions in Fermionic Models. Part 1
219
Inserting the definition of RU,C (f ) into the right-hand side, U (ξ ) f (η) dµC (η) e dµC (ξ ) + RU,C (f )(ξ ) eU (ξ ) dµC (ξ ) . eU (ξ +η)−U (ξ ) . f (η) dµ (η) eU (ξ ) dµ (ξ ) = . .η C C . eU (ξ +η) . f (η) dµ (η) dµ (ξ ), = . .η C C since .. eU (ξ +η)−U (ξ ) .. η = .. eU (ξ +η) .. η e−U (ξ ) . Continuing, f (η) dµC (η) eU (ξ ) dµC (ξ ) + RU,C (f )(ξ ) eU (ξ ) dµC (ξ ) . eU (ξ +η) . f (η) dµ (η) dµ (ξ ) by Prop. A.2.ii = . .ξ C C = f (η) eU (η) dµC (η) by Prop. A.2.i = f (ξ ) eU (ξ ) dµC (ξ ). This completes the proof of (III.1). Now we prove the theorem itself. For all g(ξ ) ∈ AV, ( − RU,C )(g) eU (ξ ) dµC (ξ ) = Z(U, C) g(ξ ) dµC (ξ ) by (III.1). Since R0,C = 0, the map − RU,C trivially has a formal power series inverse and we may choose g = ( − RU,C )−1 (f ). So U (ξ ) f (ξ ) e dµC (ξ ) = Z(U, C) ( − RU,C )−1 (f )(ξ ) dµC (ξ ). The left-hand side does not vanish for all f ∈ A V (for example, for f = e−U ) so Z(U, C) is nonzero and U (ξ ) 1 f (ξ ) e dµC (ξ ) = ( − RU,C )−1 (f )(ξ ) dµC (ξ ). Z(U,C) This construction is functorial in the following sense: Remark III.3. Let πA : A˜ → A be a homomorphism of superalgebras, and πV : V˜ → V a linear map of complex vector spaces. Define the antisymmetric bilinear C˜ form on ˜ V˜ by C(v, w) = C(πV (v), πV (w)) . πA and πV induce an algebra homomorphism π∗ : A˜ V˜ → A V . Let U˜ ∈ A˜ V˜ with π∗ (U˜ ) = U . Then for all even f˜ ∈ A˜ V˜ , πA SU˜ ,C˜ (f˜) = SU,C (π∗ f ) and
π∗ RU˜ ,C˜ (f˜) = RU,C (π∗ f˜).
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J. Feldman, H. Kn¨orrer, E. Trubowitz
In our context the Grassmann functions will all be Wick ordered with respect to the covariance C. We give a description of the map R of Theorem III.3 adapted to Wick ordering. We use a further copy of the vector space V with generators {ξi } corresponding to the generators {ξi } of V . Definition III.4. For any Grassmann function K(ξ, ξ , η) define the map RK,C : A V → A V by . . e:K(ξ,ξ ,η):ξ − 1 . f (η) dµ (ξ ) dµ (η) . . RK,C (f ) = . .η . C C .ξ Yet again, when dim V = ∞, RK,C is a formal power series in K. Proposition III.5. Assume that U (ξ ) = :Uˆ (ξ ):. Set K(ξ, ξ , η) = Uˆ (ξ + ξ + η) − Uˆ (ξ + ξ ). Then RU,C = RK,C . Proof. By part (ii) of Prop. A.2, U (ξ + η) − U (ξ ) = :Uˆ (ξ + η) − Uˆ (ξ ):ξ . Hence by part (iv) of Prop. A.2, eU (ξ +η)−U (ξ ) = ..
ˆ
ˆ
e:U (ξ +ξ +η)−U (ξ +ξ ):ξ dµC (ξ ) .. ξ .
Consequently
. eU (ξ +η)−U (ξ ) − 1 . f (η) dµ (η) . .η C . . . :Uˆ (ξ +ξ +η)−Uˆ (ξ +ξ ):ξ . = . . e dµC (ξ ) . ξ − 1 . η f (η) dµC (η) :K(ξ,ξ ,η): . . ξ − 1 dµ (ξ ) . = . .. e . ξ . η f (η) dµC (η) C . . e:K(ξ,ξ ,η):ξ − 1 . f (η) dµ (ξ ) dµ (η) . . = . . .η C C .ξ
RU,C (f ) =
To perform estimates we expand the exponential on the right-hand side of Def. III.4. For even Grassmann functions K1 (ξ, ξ , η), · · · , K (ξ, ξ , η) define the map RC (K1 , · · · , K ) : V −→ V A
A
by . f −→ .
. :K (ξ, ξ , η): . f (η) dµ (ξ ) dµ (η) . . . i C C ξ .η .ξ i=1
Observe that RC (K1 , · · · , K ) is multilinear and symmetric in K1 , · · · , K .
(III.2)
Convergence of Perturbation Expansions in Fermionic Models. Part 1
221
ξ
K1 ξ η
K2 ξ
f
K3 η ξ
K Expanding the exponential gives
Remark III.6. For any even Grassmann function K(ξ, ξ , η), RK,C =
∞ =1
1 !
RC (K, · · · , K).
That is, the th order term in the formal Taylor expansion of RK,C is the –linear map 1 ! RC (K1 , · · · , K ). III.2. Norm Estimates on the Schwinger Functional. Again we assume that A is a graded superalgebra and that we are given a family of symmetric seminorms on the spaces Am ⊗ V ⊗n . Assume that c is a contraction bound and b an integral bound for the covariance C. (See Def. II.25.) Fix α > 1. We write N (f ) for the N (f ; α) of Def. II.23. Proposition III.7. Let K (1) (ξ, ξ , η), · · · , K () (ξ, ξ , η) be even Grassmann functions that obey K (i) (ξ, ξ , 0) = 0. Furthermore let f (ξ ) be a Grassmann function and ≥ 1. Set (1) () 1 ! RC (K , · · · , K )(:f :)(ξ ) = :f (ξ ):. (i) Assume that f ∈ Am [n] for some index m and some n ≥ 0. Then f = 0 if > n, and N (f ) ≤ α12n n N (f ) N (K (i) ). i=1
(ii) In general, if α ≥ 2, then N (f ) ≤
1 α
N (f )
N (K (i) ).
i=1
Proof. Set (i) (i) G1 (ξ ; ξ (1) , · · · , ξ () ; η) = .. K (ξ, ξ , η) .. η :f (η):η , i=1 G2 (ξ ; ξ (1) , · · · , ξ () ) = G1 (ξ ; ξ (1) , · · · , ξ () ; η) dµC (η),
G3 (ξ ; ξ (1) , · · · , ξ () ) = :G2 (ξ ; ξ (1) , · · · , ξ () ):ξ (1) ,··· ,ξ () .
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Then
f (ξ ) =
G3 (ξ ; ξ , · · · , ξ ) dµC (ξ ).
1 !
By Lemma II.31,
N (f ) ≤
1 ! N (G2 ).
By Prop. II.33, with s = 1 and r = + 2, N (G2 ) ≤
! α
N (f )
N (K (i) ).
i=1
This proves part ii. Part i follows from (II.5) with s = 1, p1 = and n(1) = n.
Lemma III.8. Let f (ξ ) be a Grassmann function over A. The formal Taylor series RK,C (:f :) converges to an analytic map on K(ξ, ξ , η) K even, K(ξ, ξ , 0) = 2 0, N(K)0 < α2 . Furthermore, if K(ξ, ξ , η) is an even Grassmann function over A α2 2
with K(ξ, ξ , 0) = 0 and N (K)0 <
N (f ) ≤
and if :f : = RK,C (:f :) then
N(K) 2 α 2 1− 22 N(K)
N (f ).
α
Proof. Write
f (ξ ) =
fm;n (ξ )
m,n≥0
with fm;n ∈ Am [n] and set : = :fm;n
By part (i) of Prop. III.7,
1 !
∞ =1
RC (K, · · · , K)(:fm;n :).
1 !
RC (K, · · · , K)(:fm;n :) = 0 for > n and
N fm;n ≤
1 α 2n
N (fm;n )
=1 n
≤ N (fm;n )
=1
≤
2 α2
n n
N (fm;n )
2 α 2
N (K)
N (K)
N(K) . 1− 22 N(K) α
Consequently N (f ) ≤
N fm;n
m≥0, n≥1
≤
N(K) 2 α 2 1− 22 N(K) α
≤
N(K) 2 α 2 1− 22 N(K)
N (fm;n )
m≥0, n≥1
N (f ).
α
This also proves that the formal Taylor series expansion of RK,C converges to an analytic function.
Convergence of Perturbation Expansions in Fermionic Models. Part 1
223
Corollary III.9. Let f (ξ ) and K(ξ, ξ , η) be Grassmann functions over A with K even 2 and K(ξ, ξ , 0) = 0. Assume that N (K)0 < α4 . If :f : =
1 (:f :) − :f :, − RK,C
then N (f ) ≤
N(K) 2 α 2 1− 42 N(K)
N (f ).
α
Proof. Set β =
N(K) 2 . Observe that β0 α 2 1− 22 N(K)
=
α
set
N(K)0 2 α 2 1− 22 N(K)0
<
α
1 2
1− 21
= 1. For ≥ 0
RK,C (:f :) = :f :. By Lemma III.8 above Since f =
N (f ) ≤ β N (f ).
∞
N (f ) ≤
=1 f , ∞ =1
N (f ) ≤ N (f )
∞
β =
=1
β 1−β
N (f ) =
N(K) 2 α 2 1− 42 N(K)
N (f ).
α
Proposition III.10. Let f (ξ ) ∈ A V . The formal Taylor series S:U :,C (:f :) converges 2 to an analytic map on U (ξ ) ∈ A V U even, N (U ; 4α)0 < α4 . Furthermore, if U (ξ ) is an even Grassmann function with coefficients in A and N (U ; 4α)0 <
N S:U :,C (:f :) − f (0) ; α ≤
N(U ;4α) 2 α 2 1− 42 N(U ;4α)
α2 4 ,
then
N (f ).
α
Proof. Set K(ξ, ξ , η) = U (ξ + ξ + η) − U (ξ + ξ ). By Remark II.24, N (K; α) ≤ N U (ξ + ξ + η); α ≤ N (U ; 4α). By Prop. III.5, R:U :,C = RK,C . Therefore, by Theorem III.3 and part (i) of Prop. A.2, 1 (:f :)(ξ ) − :f (ξ ): dµC (ξ ). S:U :,C (:f :) − f (0) = −RK,C Consequently, by Lemma II.31 and Cor. III.9 N S:U :,C (:f :) − f (0) ≤
N(K) 2 α 2 1− 42 N(K) α
N (f ) ≤
N(U ;4α) 2 α 2 1− 42 N(U ;4α) α
N (f ).
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III.3. Proof of Theorem II.28. Recall that the renormalization group map C is defined by
C (:W :)(ψ) = log Z1 e:W :(ψ+ξ ) dµC (ξ ), where Z = Z e:W :(ξ ) dµC (ξ ) . Grassmann in A. As Let A = A V be the algebra in the variables ψ with coefficients ⊕ V) ∼ , is map W (ψ + ξ ) ∈ A (V V and · dµ (ξ ) maps V to A = C C A A from (a subset of ) A V to A = A V ∼ = A V (since V is a copy of V ). Set U (ψ; ξ ) = W (ψ + ξ ) ∈ V. A
Then U (ψ, 0) = W (ψ). By part (ii) of Prop. A.2, :U (ψ, ξ ):ξ = :W :(ψ + ξ ), and by Remark III.1.i,
1
C (:W :)(ψ) − W (ψ) =
S:tU :,C (:U :) − U (ψ; 0) dt − log Z.
0
We now wish to apply Prop. III.10, with (U, f, A) replaced by (tU, U, A ), to bound C (:W :)(ψ) − W . We have been given, in the statement of Theorem II.28, a system · of symmetric seminorms on the spaces Am ⊗ V ⊗n . Any f ∈ Am ⊗ V ⊗n may be uniquely expressed as f = fm ,m with fm ,m ∈ Am ⊗
m +m =m
m
V ⊗ V ⊗n . We define α m bm fm ,m . f = m +m =m
Then · is a system of symmetric seminorms on the spaces Am ⊗ V ⊗n and the covariance bound c and integral bound b with respect to these norms. C has contraction For f ∈ A V ∼ and α > 1, let N (f ; α) be the quantity of Def. II.23, = A (V ⊕ V ) considering f as an element of A (V ⊕ V ) and using the seminorms · ; and let N (f ; α) be defined viewing f as an element of A V and using the seminorms · . Then N (f ; α) = N (f ; α), while for α > α, N (f ; α ) ≤ N (f ; α ). By Remark II.24 N U ; α ≤ N W ; 2α . Prop. III.10 now implies that N C (:W :)(ψ) − W ; α) = N C (:W :)(ψ) − W ; α) ≤ sup N S:tU :,C (:U :) − U (0) 0≤t≤1
≤
N (U ;4α) 2 α 2 1− 42 N (U ;4α)
N (U ; α) ≤
N(W ;8α) 2 α 2 1− 42 N(W ;8α)
N (W ; 2α) ≤
α
≤
α
N(U ;4α) 2 α 2 1− 42 N(U ;4α)
N (U ; α)
α
N(W ;8α)2 2 . α 2 1− 42 N(W ;8α) α
Convergence of Perturbation Expansions in Fermionic Models. Part 1
225
In the last step we used Remark II.24, part (ii). The hypotheses of Prop. III.10 are fulfilled, since, by hypothesis 2 N U ; 4α 0 ≤ N W ; 8α 0 < α4 . Remark III.11. Suppose that · bc is a norm on the space of antisymmetric bilinear forms on V and that there is a κ > 0 such that every C with Cbc < κ has integral bound b and contraction bound c0 + δ=0 ∞t δ . Then C (:W :) is jointly analytic in C 2 and W on (W, C) W even, N W ; 8α 0 < α4 , Cbc < κ . IV. More Estimates on the Renormalization Group Map In the situation of [FKTf1, FKTf2, FKTf3], the effective interaction is Wick ordered both with respect to the covariance that is integrated out at the renormalization group step and a covariance that is approximately the sum of the covariances for all future renormalization group steps. In this section we modify the construction of the previous two sections to accommodate the second “output” Wick ordering. Furthermore, we estimate the derivative of C :W :ψ,C+D with respect to the effective interaction W and the covariances C and D. Let again V be a complex vector space with generators {ξi }, let A be a graded superalgebra, and let · be a system of symmetric seminorms. Furthermore let C and D be two covariances on V . Theorem IV.1. Let W (ψ) be an even Grassmann function with coefficients in A. Let :W (ψ):ψ,D = C (:W :ψ,C+D ). Let α ≥ 1 and assume that c is a contraction bound for the covariance C and b is an integral bound for C and for D. If N (W ; 32α)0 < α 2 , then N W − W; α ≤
N (W ;32α)2 1 . 2α 2 1− 12 N(W ;32α) α
Remark IV.2. By Remark II.4.ii, Z :W (ψ):ψ,D = 0. In general Z W = 0. If one defines 1 eW (ψ+ξ ) dµC (ξ ) where ZC,D C,D (W )(ψ) = log ZC,D
=Z eW (ψ+ξ ) dµC (ξ )dµD (ψ) ,
then :W (ψ):ψ,D = C,D (:W :ψ,C+D ) obeys the normalization condition Z(W ) = 0. Furthermore, W and W differ only by a constant, so that N W − W ; α ≤
N(W ;32α)2 1 . 2α 2 1− 12 N(W ;32α) α
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J. Feldman, H. Kn¨orrer, E. Trubowitz
Proof of Theorem IV.1. Define U and U by U (ψ) = :W (ψ):ψ,D ,
U (ψ) = :W (ψ):ψ,D .
Then, by Lemma A.4.i, :U :ψ,C = :W :ψ,C+D , U = C (:W :ψ,C+D ) = C (:U :ψ,C ), U − U = :W − W :ψ,D By Cor. II.32 N U ; 16α 0 = N :W :ψ,D ; 16α 0 ≤ N W ; 32α 0 < α 2 , so that by Cor. II.32, followed by Theorem II.28 (with W = U and 2α replacing α) N W − W ; α ≤ N U − U ; 2α ≤
N(U ;16α)2 1 2α 2 1− 12 N(U ;16α)
≤
α
N(W ;32α)2 1 2α 2 1− 12 N(W ;32α) α
Remark IV.3. Suppose that · b and · bc are norms on the space of antisymmetric bilinear forms on V and that there are κ, κ > 0 such that every C with Cbc < κ has integral bound b and contraction bound c0 + δ=0 ∞t δ and every D with Db < κ has integral bound b. Then W is jointly analytic in C, D and W on (W, C, D) W even, N W ; 32α 0 < α 2 , Cbc < κ, Db < κ . The derivatives of C (:W :ψ,C+D ) with respect to W , C and D are bounded in the following theorem, which is an amalgam of Lemmas IV.5, IV.7 and IV.8 below. Theorem IV.4. Let, for κ in a neighbourhood of 0, Wκ (ψ) be an even Grassmann function and Cκ , Dκ be covariances on V . Assume that α ≥ 1 and N (W0 ; 32α)0 < α 2 , and that C0 has contraction bound c, d Cκ has contraction bound c , dκ
κ=0
and that c ≤
1 2 µc .
1 2 b is an integral bound for C0 , D0 , 1 d 2 b is an integral bound for dκ Dκ κ=0 ,
Set :W˜ κ (ψ):ψ,Dκ = Cκ (:Wκ :ψ,Cκ +Dκ ).
Then N
d
˜ − Wκ ]κ=0 ; α ≤
dκ [Wκ
N(W0 ;32α) 1 N ddκ Wκ κ=0 ; 8α 2α 2 1− 12 N(W0 ;32α) α
2 0 ;32α) + 2α1 2 N(W 1− 12 N(W0 ;32α)
1 4µ c
+
b 2 b
.
α
Lemma IV.5. Let C be a covariance on V with contraction bound c and integral bound b. Let, for κ in a neighbourhood of 0, Wκ (ψ) ∈ A V be an even Grassmann function.
Convergence of Perturbation Expansions in Fermionic Models. Part 1
i) Set
227
W˜ κ (ψ) = C (:Wκ :ψ,C ).
2 If N W0 ; 8α 0 < α4 , then N ddκ [W˜ κ − Wκ ]κ=0 ; α ≤
N(W0 ;8α) 2 N ddκ Wκ κ=0 ; 2α . α 2 1− 42 N(W0 ;8α) α
ii) Let D be a covariance on V with integral bound b. Set :W˜ κ (ψ):ψ,D = C (:Wκ :ψ,C+D ).
If N W0 ; 32α 0 < α 2 , then N ddκ [W˜ κ − Wκ ]κ=0 ; α ≤
N(W0 ;32α) 1 N ddκ Wκ κ=0 ; 8α . 2α 2 1− 12 N(W0 ;32α) α
Proof. Set Uκ (ψ, ξ ) = Wκ (ψ + ξ ),
Uκ (ψ, ξ ) =
d dκ Wκ (ψ
+ ξ ).
By Prop. A.2.ii, :Uκ (ψ, ξ ):ξ = .. ddκ Wκ .. (ψ + ξ ).
:Uκ (ψ, ξ ):ξ = :Wκ :(ψ + ξ ), By Remark II.24, N Uκ ; α ≤ N Wκ ; 2α ,
N Uκ ; α ≤ N ddκ Wκ ; 2α .
Differentiating Def. II.3, .d . . dκ Wκ . (ψ + ξ ) e:Wκ :(ψ+ξ ) dµC (ξ ) d : = :W = S:Uκ :,C (:Uκ :) κ dκ C e:Wκ :(ψ+ξ ) dµC (ξ ) so that
d dκ
C :Wκ : − Wκ = S:Uκ :,C (:Uκ :) − Uκ (ψ, 0)
and the norm N
mod A0
mod A0 . f ; α as in the proof
Define the system of symmetric seminorms · of Theorem II.28. Prop. III.10 now implies that N ddκ [W˜ κ − Wκ ] ; α) = N ddκ [W˜ κ − Wκ ] ; α) = N S:Uκ :,C (:Uκ :) − Uκ (ψ, 0) ; α N (Uκ ;4α) 2 α 2 1− 42 N (Uκ ;4α)
N (Uκ ; α)
≤
N(Uκ ;4α) 2 α 2 1− 42 N(Uκ ;4α)
N (Uκ ; α)
≤
N(Wκ ;8α) 2 α 2 1− 42 N(Wκ ;8α)
N
≤
α
α
d
dκ Wκ ; 2α
.
α
The hypotheses of Prop. III.10 are fulfilled, at κ = 0, since, by hypothesis 2 N U0 ; 4α 0 ≤ N W0 ; 8α 0 < α4 . ii) Part (ii) follows from part (i) as Theorem IV.1 follows from Theorem II.28.
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J. Feldman, H. Kn¨orrer, E. Trubowitz
Lemma IV.6. Let c be a contraction bound for C and c be a contraction bound for C . If c ≤ µ1 c2 , then N
i,j
Cij
∂f ∂ξi
∂g ∂ξj
;α ≤
1 c µα 2
N (f ; 2α) N (g; 2α).
Proof. Write
f (ξ ) =
g(ξ ) =
fm;n (ξ )
gm ;n (ξ )
m ,n ≥0
m,n≥0
with fm;n ∈ Am [n] and gm ;n ∈ Am [n ]. Then, by Lemma II.10 and Def. II.9, ∂fm;n ∂ξi
i,j
∂gm ;n ∂ξj
Cij
= −n ConC fm;n (ξ )gm ;n (ζ ) ξ →ζ
= −nn ConC 1→n+1
fm;n ⊗ gm ;n
ζ =ξ
ζ =ξ
so that, by Lemma II.22.ii and Def. II.25, ∂fm;n ∂gm ;n fm;n ⊗ gm ;n ≤ nn Con C C ij ∂ξi ∂ξj 1→n+1
i,j
≤ nn c fm;n gm ;n .
Hence, by Def. II.23, N
∂f ∂g 1 ∂ξ Cij ∂ξ ; α ≤ 2 c i,j
i
b
j
m,m ,n,n ≥0
≤
1 c 1 c µα 2 b2
≤ =
∂fm;n ∂gm ;n (αb)n+n −2 ∂ξ Cij ∂ξ i,j
m,n,≥0
i
j
1 c n(αb)n fm;n 2 b
1 c 1 c (2αb)n f 1 c m;n 2 2 2 b
µα
1 c µα 2
b
m,n,≥0
m ,n ,≥0
m ,n ,≥0
n (αb)n gm ;n
(2αb)n gm ;n
N(f ; 2α) N(g; 2α).
on V . Assume that Lemma IV.7. Let, for κ in a neighbourhood of 0, Cκ be a covariance C0 has contraction bound c and integral bound b, that ddκ Cκ κ=0 has contraction bound c and that c ≤ µ1 c2 . Let W (ψ) ∈ A V be an even Grassmann function. i) Set W˜ κ (ψ) = Cκ (:W :ψ,Cκ ). If N W ; 8α 0 <
α2 4
N
, then d
˜
dκ Wκ κ=0 ; α
≤
N(W ;8α)2 1 1 c. 2α 2 1− 42 N(W ;8α) µ α
Convergence of Perturbation Expansions in Fermionic Models. Part 1
229
ii) Let D be a covariance on V with integral bound b. Set
If N W ; 32α
0
:W˜ κ (ψ):ψ,D = Cκ (:W :ψ,Cκ +D ). < α 2 , then N
˜
d
dκ Wκ κ=0 ; α
≤
N(W ;32α)2 1 1 c. 8α 2 1− 12 N(W ;32α) µ α
Proof. Set U (ψ, ξ ) = W (ψ +ξ ) . By Prop. A.2.ii, :U (ψ, ξ ):ξ,Cκ = :W :Cκ (ψ +ξ ) and by Remark II.24, N U ; α ≤ N W ; 2α . By Lemma A.8.iv, setting C (κ) = ddκ Cκ , :W : (ψ+ξ ) d e Cκ dµCκ (ξ ) dκ d ˜ :W : (ψ+ξ ) mod A0 dκ Wκ = C κ e dµCκ ξ ) . ∂U .
:U (ψ,ξ ): . ∂U . ξ,Cκ dµ − 21 Cκ (ξ ) i,j . ∂ξi . ξ,Ct Cij (κ) . ∂ξj . ξ,Cκ e = e:U (ψ,ξ ):ξ,Cκ dµCκ ξ )
. ∂U . . ∂U . = − 21 S:U :ξ,Cκ ,Cκ . ∂ξi . ξ,Cκ Cij (κ) . ∂ξj . ξ,Cκ i,j
= S:U :ξ,Cκ ,Cκ (:Vκ :ξ,Cκ ), where
:Vκ :ξ,Cκ = − 21
. ∂U . . ∂U . . ∂ξi . ξ,Cκ Cij (κ) . ∂ξj . ξ,Cκ . i,j
As in Lemma IV.7, Prop. III.10 now implies that N ddκ W˜ κ κ=0 ; α) = N ddκ W˜ κ κ=0 ; α) ≤ N ddκ W˜ κ κ=0 − V0 (ψ, 0) ; α) + N V0 (ψ, 0) ; α) = N S:U :ξ,C0 ,C0 (:V0 :ξ,C0 ) − V0 (ψ, 0) + N V0 (ψ, 0) ; α) ≤ α22 N4 (U;4α) N (V0 ; α) + N V0 ; α) ≤ ≤ ≤
1− 2 N (U ;4α) α N(U ;4α) 4 N (V0 ; α) + N (V0 ; α) α 2 1− 42 N(U ;4α) α 1 N (V0 ; α) 1− 42 N(U ;4α) α 1 N V0 ; α . 1− 42 N(W ;8α) α
By Prop. A.2.i (three times) V0 (ψ, ξ ) = :V0 :ξ,C0 (ψ, ξ + ξ ) dµC0 (ξ ) 1 ∂U = −2 ∂ξi (ψ, ξ + ξ + ζ ) Cij (0) i,j
(ψ, ξ + ξ + ζ ) dµ−C0 (ζ ) dµ−C0 (ζ ) dµC0 (ξ ) ∂W = − 21 ∂ξi (ψ + ξ + ξ + ζ ) Cij (0) ×
∂U ∂ξj
×
∂W ∂ξj (ψ
i,j
+ ξ + ξ + ζ ) dµ−C0 (ζ ) dµ−C0 (ζ ) dµC0 (ξ ).
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By Lemma II.31, with s = 0, followed by Lemma IV.6 and then Remark II.24, ∂W N V0 ; α ≤ 21 N ∂W (ψ + ξ + ξ + ζ ) C (0) (ψ + ξ + ξ + ζ ) ; α ij ∂ξi ∂ξj ≤ ≤
i,j 1 c N W (ψ + ξ 2µα 2 2 1 c N W ; 8α . 2µα 2
+ ξ + ζ ) ; 2α N W (ψ + ξ + ξ + ζ ) ; 2α
ii) Part (ii) follows from part (i) as Theorem IV.1 follows from Theorem II.28.
Lemma IV.8. Let C and, for κ in a neighbourhood of 0, Dκ be covariances on V . Assume that C has contraction bound c and integral bound 21 b, that D0 has integral bound 21 b and that ddκ Dκ κ=0 has integral bound 21 b . Let W (ψ) ∈ A V be an even Grassmann function. Set :W˜ κ (ψ):ψ,Dκ = C (:W :ψ,C+Dκ ). If N W ; 32α 0 < α 2 , then N
d
˜
dκ Wκ κ=0 ; α
N(W ;32α)2 b 2 1 . 2α 2 1− 12 N(W ;32α) b
≤
α
Proof. Set Ez =
D0 + z ddκ Dκ κ=0
and define Vz by
:Vz (ψ):ψ,Ez = C (:W :ψ,C+Ez ). As Dκ and Eκ agree to order κ, Remark II.30,
d ˜ dκ Wκ κ=0 1 1 2b + 2
is an integral bound for Ez for all |z| ≤
1 εd .
N Vz − W ; α ≤
=
d dz Vz z=0 .
Set εd =
b 2 b
. Then, by
|z| b ≤ b By Theorem IV.1,
N(W ;32α)2 1 2α 2 1− 12 N(W ;32α) α
for all |z| ≤ ε1d . By the Cauchy integral formula, if f (z) is analytic and bounded in absolute value by Q on |z| ≤ r, then f (ζ ) 1 f (z) = 2πı dζ (ζ −z)2 |ζ |=r
and
f (0) ≤
1 Q 2π r 2 2πr
= Q 1r .
Hence N
d
˜
dκ Wκ κ=0 ; α
=N =
d
dz Vz − W z=0 ; α ≤
N(W ;32α)2 b 2 1 . 2α 2 1− 12 N(W ;32α) b α
N(W ;32α)2 εd 2α 2 1− 12 N(W ;32α) α
Convergence of Perturbation Expansions in Fermionic Models. Part 1
231
Appendix A. Wick–Ordering Let A be a superalgebra, V be a complex vector space and C an antisymmetric bilinear form (covariance) on V . Wick ordering with respect to a covariance C is the A–linear map on A V , f (ξ ) → :f (ξ ):ξ,C characterized by :e ξi ζi :ξ,C = e1/2 ζi Cij ζj e ξi ζi for any set {ζi } of odd Grassmann variables. If the context admits, we delete the Wick ordering covariance C or the variable ξ (or both) from the symbol : · :ξ,C for Wick ordering. Also recall that the Grassmann Gaussian integral with covariance C is characterized by e ξi ζi dµC (ξ ) = e−1/2 ζi Cij ζj . Lemma A.1. For n, m ≥ 0, !
(:ξi1 ξi2 · · · ξin :) (:ξjm ξjm−1 · · · ξj1 :) dµC (ξ ) =
det Cik j if m = n . 0 if m = n
Proof. This follows easily from the definitions by induction. See, for example, Prop. I.31 of [FKTffi]. Choose a copy of V with generating system (ξi ) corresponding to the generating system (ξi ) of V . Proposition A.2. (i) Let f (ξ ) = :fˆ(ξ ):ξ ∈ A V . Then f (ξ ) = fˆ(ξ + iξ ) dµC (ξ ) = fˆ(ξ + ξ ) dµ−C (ξ ), fˆ(ξ ) = f (ξ + ξ ) dµC (ξ ). In particular, f (0) = fˆ(iξ ) dµC (ξ ) = fˆ(ξ ) dµ−C (ξ ). (ii) Let f (ξ ) = :fˆ(ξ ):ξ ∈ A V . Then f (ξ + ξ ) = :fˆ(ξ + ξ ):ξ = :fˆ(ξ + ξ ):ξ . (iii) For f1 (ξ ), · · · , f (ξ ) ∈ A V , :f1 (ξ ):ξ · · · :f (ξ ):ξ = .. :f1 (ξ + ξ ):ξ · · · :f (ξ + ξ ):ξ dµC (ξ ) .. ξ . (iv) For f (ξ ) ∈
AV,
e:f (ξ ):ξ = ..
e:f (ξ +ξ ):ξ dµC (ξ ) .. ξ .
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J. Feldman, H. Kn¨orrer, E. Trubowitz
Proof.
(i) It suffices to prove the identities for fˆ(ξ ) = e ξj ζj . In this case ˆ f (ξ + iξ ) dµC (ξ ) = e ξj ζj ei ξj ζj dµC (ξ ) = e ξj ζj e1/2 ζi Cij ζj = :fˆ(ξ ):C = f (ξ )
and
fˆ(ξ + ξ ) dµ−C (ξ ) =
e ξj ζj e ξj ζj dµ−C (ξ )
= e ξj ζj e1/2 ζi Cij ζj = :fˆ(ξ ):C = f (ξ ). The statement about fˆ(ξ ) is proven in the same way, and the formula for f (0) follows immediately. (ii) Again we may assume that fˆ(ξ ) = e ξi ζi . Then f (ξ ) = e1/2 ζi Cij ζj eξi ζi , fˆ(ξ + ξ ) = e ξi ζi e ξi ζi , and 1
:fˆ(ξ + ξ ):ξ = e 2 ζi Cij ζj + ξi ζi e ξi ζi = e1/2 ζi Cij ζj + (ξi +ξi )ζi = f (ξ + ξ ).
Similarly one shows that :fˆ(ξ + ξ ):ξ = f (ξ + ξ ). (iii) Let :g(ξ ): = :f1 (ξ ): · · · :f (ξ ):. By part (ii) :g(ξ + ξ ):ξ = :f1 (ξ + ξ ):ξ · · · :f (ξ + ξ ):ξ . Therefore
g(ξ ) = =
:g(ξ + ξ ):ξ dµC (ξ ) :f1 (ξ + ξ ):ξ · · · :f (ξ + ξ ):ξ dµC (ξ ).
(iv) By part (iii) e:f (ξ ): = =
∞ =0 ∞ =0
= ..
1 ! 1 !
:f (ξ ): . :f (ξ + ξ ): dµ (ξ ) . .ξ . C ξ
e:f (ξ +ξ ):ξ dµC (ξ ) .. ξ .
Convergence of Perturbation Expansions in Fermionic Models. Part 1
Corollary A.3. For f1 (ξ ), · · · , f (ξ ), g(ξ ) ∈
233
AV,
:f1 (ξ ):ξ · · · :f (ξ ):ξ :g(ξ ):ξ . . . =. . :fi (ξ + ξ + η):ξ .. η :g(ξ + η):η dµC (ξ ) dµC (η) . ξ . i=1
Proof. By iterated application of part (iii) of Prop. A.2, :f1 (ξ ):ξ · · · :f (ξ ):ξ :g(ξ ):ξ = .. :fi (ξ + ξ ):ξ dµC (ξ ) .. ξ :g(ξ ):ξ i=1
. . = . .. :fi (ξ + ξ + η):ξ dµC (ξ ) .. η :g(ξ + η):η dµC (η) . ξ . i=1
Lemma A.4. Let D be a second covariance on V , and let f (ξ ) ∈ (i)
AV.
:f (ξ ):ξ,C+D = .. :f (ξ ):ξ,C .. ξ,D
(ii)
:f : C+D (ξ + ξ ) = .. :f (ξ + ξ ):ξ,C .. ξ ,D ,
(iii)
:f (ξ ):ξ,C+D = :f (ξ ):ξ,C ,
where
f (ξ ) =
f (ξ + iξ ) dµD (ξ ) =
f (ξ + ξ ) dµ−D (ξ ).
Proof. Again we may assume that f (ξ ) = e ξi ζi . Set fˆ(ξ ) = :f (ξ ):ξ,C = e1/2 ζi Cij ζj e ξi ζi . (i) :fˆ(ξ ):ξ,D = e1/2 ζi Cij ζj :e ξi ζi :ξ,D = e1/2 ζi (Cij +Dij )ζj e ξi ζi = :f (ξ ):ξ,C+D , (ii) . :f (ξ + ξ ): . 1/2 ζi Cij ζj (ξi +ξi )ζi :e :ξ ,D . ξ,C . ξ ,D = e
= e1/2 ζi (Cij +Dij )ζj e (ξi +ξi )ζi = :f : C+D (ξ + ξ ).
(iii) By part (i) and parts (i),(ii) of Prop. A.2, :f (ξ ):ξ,C+D = :fˆ(ξ ):ξ,D = fˆ(ξ + iξ ) dµD (ξ ) . = :f (ξ + iξ ):ξ,C dµD (ξ ) = . f (ξ + iξ ) dµD (ξ ) .. ξ,C .
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Lemma A.5. Let f (ξ ), g(ξ ), h(ξ ) ∈ A V . Then :f (ξ )g(ξ ):ξ h(ξ ) dµC (ξ ) = :f (ξ ):ξ :g(ξ ):ξ :h(ξ + ξ ):ξ dµC (ξ ) dµC (ξ ) Proof. We may assume that f (ξ ) = e ξi ζi ,
g(ξ ) = e ξi ζi ,
h(ξ ) = e ξi ηi ,
with additional Grassmann variables ζi , ζi , ηi . Then :f (ξ )g(ξ ):ξ h(ξ ) dµC (ξ ) = e1/2 (ζi +ζi )Cij (ζj +ζj ) e ξi (ζi +ζi +ηi ) dµC (ξ )
= e1/2 [(ζi +ζi )Cij (ζj +ζj )−(ζ i +ζi +ηi )Cij (ζj +ζj +ηj )] = e−1/2 ηi Cij ηj e− (ζi +ζi )Cij ηj . On the other hand :g(ξ ):ξ :h(ξ + ξ ):ξ dµC (ξ ) = .. e1/2 ζi Cij ζj e ξi (ζi +ηi ) e ξi ηi dµC (ξ ) .. ξ = .. e1/2 [ζi Cij ζj −(ζi +ηi )Cij (ζj +ηj ) e ξi ηi .. ξ
= e1/2 [ζi Cij ζj +ηi Cij ηj −(ζi +ηi )Cij (ζj +ηj ) e ξi ηi = e− ζi Cij ηj e ξi ηi . Therefore, also
:f (ξ ):ξ :g(ξ ):ξ :h(ξ + ξ ):ξ dµC (ξ ) dµC (ξ ) = e1/2 ζi Cij ζj e− ζi Cij ηj e ξi (ζi +ηi ) dµC (ξ )
= e1/2 [ ζi Cij ζj −(ζi +ηi )Cij (ζ j +ηj )] e− ζi Cij ηj = e−1/2 ηi Cij ηj e− (ζi +ζi )Cij ηj .
Corollary A.6. Let f (ξ ), h(ξ ) ∈ A V . Then . . f (ξ )h(ξ ) dµC (ξ ) = f (ξ ) . h(ξ + ξ ) dµC (ξ ) . ξ dµC (ξ ). Proof. Set g(ξ ) = 1 and replace :f (ξ ):ξ by f (ξ ) in Lemma A.5. Lemma A.7. Let f (ξ ) be a homogeneous Grassmann polynomial of degree two. Write f (ξ + ξ ) = f (ξ ) + f (ξ ) + fmix (ξ , ξ ). Furthermore let g(ξ ), h(ξ ) ∈ A V . Then :f (ξ ):ξ :g(ξ )h(ξ ):ξ dµC (ξ ) = fmix (ξ , ξ ) :g(ξ ):ξ :h(ξ ):ξ dµC (ξ ) dµC (ξ ) +h(0) :f (ξ ):ξ :g(ξ ):ξ dµC (ξ ) +g(0) :f (ξ ):ξ :h(ξ ):ξ dµC (ξ ).
Convergence of Perturbation Expansions in Fermionic Models. Part 1
235
Proof. By Lemma A.5 and part (ii) of Prop. A.2, :f (ξ ):ξ :g(ξ )h(ξ ):ξ dµC (ξ ) . :f (ξ + ξ ): . :g(ξ ): :h(ξ ): dµ (ξ ) dµ (ξ ) = . C C ξ . ξ ξ ξ
= : f (ξ ) + f (ξ ) + fmix (ξ , ξ ) :ξ ,ξ :g(ξ ):ξ :h(ξ ):ξ dµC (ξ ) dµC (ξ ) Since fmix has degree one in ξ and in ξ :fmix (ξ , ξ ):ξ ,ξ :g(ξ ):ξ :h(ξ ):ξ dµC (ξ ) dµC (ξ ) = fmix (ξ , ξ ) :g(ξ ):ξ :h(ξ ):ξ dµC (ξ ) dµC (ξ ). Clearly :f (ξ ):ξ ,ξ :g(ξ ):ξ :h(ξ ):ξ dµC (ξ ) dµC (ξ ) = h(0) :f (ξ ):ξ :g(ξ ):ξ dµC (ξ ), :f (ξ ):ξ ,ξ :g(ξ ):ξ :h(ξ ):ξ dµC (ξ ) dµC (ξ ) = g(0) :f (ξ ):ξ :h(ξ ):ξ dµC (ξ ). Lemma A.8. Let C(κ) be a C 1 family of antisymmetric bilinear forms (covariances). Then i)
d dκ
f (ξ + tξ ) dµC(κ) (ξ ) dµC (κ) (ξ ) t=0 1 ∂ ∂ = −2 Cij (κ) ∂ξi ∂ξj f (ξ ) dµC(κ) (ξ ).
f (ξ ) dµC(κ) (ξ ) =
1 d2 2 dt 2
i,j
ii) d dκ :f (ξ ):C(κ)
. .
=
1 d2 2 dt 2
=
1 2
. f (ξ + tξ ) dµ−C (κ) (ξ ) . ξ,C(κ)
Cij (κ) .. ∂∂ξi
∂ ∂ξj
t=0
f (ξ ) .. C(κ) .
i,j
iii) If f is even, ∂f f (ξ ) d 1 e dµ (ξ ) = − C (κ) C(κ) ij dκ 2 ∂ξi
∂f ∂ξj
+
∂ ∂ ∂ξi ∂ξj
f (ξ ) ef (ξ ) dµC(κ) (ξ ).
i,j
iv) If f is even,
e:f (ξ ):C(κ) dµC(κ) (ξ ) . ∂f . . ∂f . :f (ξ ):C(κ) 1 = −2 Cij (κ) dµC(κ) (ξ ). . ∂ξi . C(κ) . ∂ξj . C(κ) e
d dκ
i,j
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Proof.
i) d dκ
f (ξ ) dµC(κ) (ξ ) =
1 d2 2 dt 2
=
1 d2 2 dt 2
=
1 d2 2 dt 2
=
− 21
f (ξ ) dµC(κ+t 2 ) (ξ )
t=0
f (ξ ) dµC(κ)+t 2 C (κ) (ξ )
t=0
f (ξ + ξ ) dµC(κ) (ξ ) dµt 2 C (κ) (ξ ) t=0 1 d2 f (ξ + tξ ) dµC(κ) (ξ ) dµC (κ) (ξ ) = 2 dt 2 t=0 ∂ ∂ = − 21 ξi ξj ∂ξ f (ξ ) dµC(κ) (ξ ) dµC (κ) (ξ ) i ∂ξj
i,j
Cij (κ)
∂ ∂ ∂ξi ∂ξj
f (ξ ) dµC(κ) (ξ ).
i,j
ii) d dκ
. f (ξ ) . . . C(κ) =
f (ξ + ξ ) dµ−C(κ) (ξ ) d2 f (ξ + ξ + tξ ) dµ−C(κ) (ξ ) dµ−C (κ) (ξ ) = 21 dt 2 t=0 1 d2 = f (ξ + tξ + ξ ) dµ−C (κ) (ξ ) dµ−C(κ) (ξ ) 2 dt 2 t=0 . d2 . = . 21 dt f (ξ + tξ ) dµ (ξ ) . C(κ) −C (κ) 2 t=0 = 21 Cij (κ) .. ∂∂ξi ∂∂ξj f (ξ ) .. C(κ) . d dκ
i,j
iii) By the first part d ef (ξ ) dµC(κ) (ξ ) dκ 1 = −2 Cij (κ) i,j
=
− 21
=
− 21
Cij (κ)
i,j
Cij (κ)
∂ ∂ ∂ξi ∂ξj
∂ ∂ξi
ef (ξ ) dµC(κ) (ξ )
∂ f (ξ ) dµC(κ) (ξ ) ef (ξ ) ∂ξ j
ef (ξ )
∂f ∂f ∂ξi (ξ ) ∂ξj
(ξ ) +
∂ ∂ ∂ξi ∂ξj
f (ξ ) dµC(κ) (ξ ).
i,j
iv) By parts (iii) and (ii), d e:f (ξ ):C(κ) dµC(κ) (ξ ) dκ ∂:f : ∂:f : = − 21 Cij (κ) ∂ξi ∂ξj + +
i,j
d
dκ :f (ξ ):C(κ)
∂ ∂ ∂ξi ∂ξj
:f (ξ ): e:f (ξ ): dµC(κ) (ξ )
:f (ξ ): C(κ) dµ e C(κ) (ξ )
Convergence of Perturbation Expansions in Fermionic Models. Part 1
= − 21
Cij (κ)
i,j
+ 21
Cij (κ)
i,j
=
− 21
Cij (κ)
∂:f : ∂:f : ∂ξi ∂ξj
∂
∂ ∂ξi ∂ξj
+
∂ ∂ ∂ξi ∂ξj
237
:f (ξ ): e:f (ξ ): dµC(κ) (ξ )
:f (ξ ): . f (ξ ) . C(κ) dµ . . C(κ) e C(κ) (ξ )
. ∂f . . ∂f . :f (ξ ):C(κ) dµC(κ) (ξ ). . ∂ξi . C(κ) . ∂ξj . C(κ) e
i,j
Appendix B. Gram Bounds Let C be an antisymmetric bilinear form (covariance) on the vector space V , and let {ξi } be a system of generators for V . Proposition B.1. Suppose that V is the direct sum of two subspaces Va and Vc such that each of the generators ξi lies either in Va or Vc and C(ξ, ξ ) = 0
if both ξ, ξ ∈ Va or both ξ, ξ ∈ Vc .
Assume furthermore that there is a Hilbert space H, and that there is associated to each generator ξi a vector wi ∈ H. Set S = sup wi . i
i) If
" # C(ξi , ξj ) = wi , wj H
for all i, j , then
if ξi ∈ Va , ξj ∈ Vc
ξi1 · · · ξim dµC (ξ ) ≤ S m
for all i1 , · · · , im . ii) Assume that, for each generator ξi , there exists a real number τi such that ! " # e−(τi −τj ) wi , wj H if τi > τj C(ξi , ξj ) = 0 if τi ≤ τj for all i, j with ξi ∈ Va , ξj ∈ Vc . Then again ξi1 · · · ξim dµC (ξ ) ≤ S m for all i1 , · · · , im . In both cases, 2S is an integral bound for the covariance C with respect to the norms of Example II.20.
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Proof of part (i). If the integral does not vanish, we may reorder the factors in the integrand ξi1 · · · ξim = ±ξj1 ξ1 · · · ξjn ξn so that ξjp ∈ Va and ξp ∈ Vc for all 1 ≤ p ≤ n = m 2 . Then
ξi1 · · · ξim dµC (ξ ) = ± det C(ξjp , ξq ) 1≤p,q≤n . Part (i) now follows by Gram’s inequality. Part (ii) shall be proven following Lemma B.4. Example B.2. In [FKTo3], part (i) of Prop. B.1 is applied to covariances of the form d d+1 k ı− χ (k) e , C(ξi , ξj ) = δσi ,σj (2π)d+1 ık0 − e(k) where σi , σj ∈ {↑, ↓} are spins, xi = (τi , xi ), xj = (τj , xj ) ∈ Rd+1 are points in space– (imaginary)time, k = (k0 , k) ∈ R×Rd and < k, xi −xj >− = −k0 (τi −τj )+k·(xi −xj ). The function e(k) is the dispersion relation, minus the chemical potential, and χ (k) is a nonnegative cutoff function. In this case, we may take H = L2 (Rd+1 × C2 ), $ χ(k) 1 if ξi ∈ Vc −ı− $ (2π)d+1 ık0 −e(k) wi (k, σ ) = δσ,σi e χ(k) 1 if ξi ∈ Va (2π)d+1 ık0 −e(k) for any single–valued square root, and % S=
d d+1 k χ(k) . (2π)d+1 ık0 −e(k)
Part (ii) of Prop. B.1 will be used in [FKTo2]. It is designed to deal with covariances of the form d d+1 k ı− U (k) C(ξi , ξj ) = δσi ,σj e (2π)d+1 ık0 − 1 d d+1 k in which the nonnegative cutoff function U (k) is independent of k0 . In this case (2π) d+1 U (k) ık0 −1 diverges. As
! dk0 −ık0 (τi −τj ) U (k) −U (k)e−(τi −τj ) e = 0 2π ık0 − 1
if τi > τj if τi ≤ τj
(actually, the case τi = τj is defined by the limit τj → τi +), we may take H = L2 (Rd × C2 ), ! $ −1 if ξi ∈ Va −ık·xi 1 U (k) wi (k, σ ) = δσ,σi e (2π)d 1 if ξi ∈ Vc %
and then S=
dd k U (k). (2π)d
Convergence of Perturbation Expansions in Fermionic Models. Part 1
239
n To prepare for the proof of part (ii) of Prop. B.1, let H = H be the Grassn≥0 0 mann algebra over H. The element H is also denoted by 0 (the ground n1 ∈ C = H there is an inner product such that state). On each of the summands ' " # & v1 · · · vn , v1 · · · vn = det vi , vj i,j =1,··· ,n for all v1 , · · · , vn , v1 , · · · , vn ∈ H. On H there is an inner product determined by the n requirement that H be an orthogonal direct sum. For v ∈ H, let a † (v) be the n≥0 operator on H that maps f ∈ n H to vf ∈ n+1 H, and a(v) its adjoint. We have the standard Lemma B.3. For all v, w ∈ H, i) {a(v), a(w)} = {a † (v), a † (w)} = 0, {a(v), a † (w)} = v, w . ii) The operator norms a(v), a † (v) are bounded by vH . Proof. Let ej j ∈J be an orthonormal basis for H, indexed by a totally ordered set J . For each finite subset J of J set eJ = ej1 · · · ejn
when J = {j1 , · · · , jn } with j1 ≺ · · · ≺ jn . The elements eJ , |J | = n are an orthonormal basis for n H. Then !
J,j eJ ∪{j } if j ∈ /J a (ej )eJ = , 0 if j ∈ J !
J \{j },j eJ \{j } if j ∈ J a(ej )eJ = , 0 if j ∈ /J †
where, for j ∈ / J , J,j is the sign of the permutation that brings the sequence j, J to standard order. In the case v = ej , w = ej , part (i) of the lemma follows directly from this description. The general case follows from this special case, since all terms are bilinear in v and w. To prove part (ii) of the lemma observe that, for all u ∈ H and f ∈ H, " # a(u)f 2∧H + a † (u)f 2∧H = {a(u), a † (u)}f, f = u2H f 2∧H so that a(u)f ∧H ≤ uH f ∧H
a † (u)f ∧H ≤ uH f ∧H .
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Let N be the number operator on H. By definition, its restriction to n H is multiplication by n. For each index i, labeling a generator ξi , set ! eτi N a(wi )e−τi N = e−τi a(wi ) if ξi ∈ Va . ai = τi N † e a (wi )e−τi N = eτi a † (wi ) if ξi ∈ Vc A sequence i1 , · · · , im is called time ordered if τi1 ≥ · · · ≥ τim and for 1 ≤ k < ≤ m, τik = τi , ξik ∈ Va
implies ξi ∈ Va .
Lemma B.4. Let i1 , · · · , im be a time ordered sequence. Then, under the assumptions of part (ii) of Prop. B.1, # " ξi1 · · · ξim dµC (ξ ) = 0 , ai1 · · · aim 0 . Proof. The proof is by induction on m. The cases m = 0 and m = 1 are trivial. We perform the induction step m − 2 → m. Assume first that ξi1 ∈ Va . Then ! " # e−τi1 +τik {a(wi1 ), a † (wik )} = e−(τi1 −τik ) wi1 , wik if ξik ∈ Vc {ai1 , aik } = 0 if ξik ∈ Va and ai1 0 = 0. Therefore m " # " # 0 , ai1 · · · aim 0 = (−1)k e−(τi1 −τik ) wi1 , wik k=2 ξi ∈Vc k
" # × 0 , ai2 · · · aik−1 aik+1 · · · aim 0 m = (−1)k C(ξi1 , ξik ) ξi2 · · · ξik−1 ξik+1 · · · ξim dµC (ξ ) k=2 = ξi1 · · · ξim dµC (ξ ). For the second equality we used the assumption on C, the fact that the sequence i1 , · · · , im is time ordered, and the induction hypothesis. The third equality is the integration by parts formula. Now assume that ξi1 ∈ Vc . Then # " # " 0 , ai1 · · · aim 0 = eτi1 a(wi1 )0 , ai2 · · · aim 0 = 0 and, by the integration by parts formula ξi1 · · · ξim dµC (ξ ) =
m
(−1)k C(ξi1 , ξik )
ξi2 · · · ξik−1 ξik+1 · · · ξim dµC (ξ ) = 0,
k=2
since C(ξi1 , ξik ) = −C(ξik , ξi1 ) = 0 for k = 2, · · · , m.
Convergence of Perturbation Expansions in Fermionic Models. Part 1
241
Proof of part (ii) of Prop. B.1. We may assume that the sequence i1 , · · · , im is time ordered. If #{ν ξiν ∈ Vc } = #{ν ξiν ∈ Va } then ξi1 · · · ξim dµC (ξ ) = 0. Otherwise, by Lemma B.4 and Lemma B.3, " # ξi1 · · · ξim dµC (ξ ) = 0 , ai1 · · · aim 0 & ' = 0 , a (†) (wi1 )e−(τi1 −τi2 )N a (†) (wi2 ) · · · e−(τim−1 −τim )N a (†) (wim )0 (†) −(τ −τ )N (†) −(τ ≤ a (wi1 ) e i1 i2 a (wi2 ) · · · e im−1 −τim )N a (†) (wim ) m a (†) (wi ) ≤ S m . ≤ k k=1
Here, we used that ai1 · · · aim 0 ∈ 0 H and that the restriction of the number operator N to 0 H is identically zero.
Notation Not’n Z (f ) V AV A m [nξ1 ζ, · · · , nr ] e i i dµC (ξ ) C (W )(ψ) S (f ) R RC (K1 , · · · , K ) RK,C (f )
:e ξi ζi :ξ,C C onC , ConC i→j
Nd c b N(f ; α)
ξ →ξ
Description degree zero component of f Grassmann algebra over V Grassmann algebra over V with coefficients in A partially antisymmetric elements of Am ⊗ V ⊗(n1 +···+nr ) ζi Cij ζj = e−1/2 Grassmann Gaussian integral = log Z1 eW (ψ+ξ ) dµC (ξ ) renormalization group map 1 = Z(U,C) f (ξ ) eU (ξ ) dµC (ξ ) Schwinger functional R–operator th Taylor coefficient of R . . :K(ξ,ξ ,η):ξ . − 1 .. η f (η)dµC (ξ ) dµC (η) . ξ . .e = e1/2 ζi Cij ζj e ξi ζi Wick ordering contractions
Reference Def. II.1.iii Example II.2 Example II.2 Def. II.21 before Def. II.3 Definitions II.3, II.27 before Remark III.1 before Theorem III.2 (III.2)
norm domain contraction bound integral bound 1 c |n| |n| m,n1 ,··· ,nr ≥0 α b fm;n1 ,··· ,nr 2
Def. II.14 Def. II.25.i Def. II.25.ii Def. II.23
b
Def. III.4 after Remark II.4 Definitions II.5, II.9
References [BS]
Berezin, F., Shubin, M.: The Schr¨odinger Equation. Amsterdam: Kluwer, 1991. Supplement 3: D.Le˘ites, Quantization and supermanifolds [DR] Disertori, M., Rivasseau, V.: Continuous constructive fermionic renormalization. Annales Henri Poincar´e 1, (2000) [FKLT1] Feldman, J., Kn¨orrer, H., Lehmann, D., Trubowitz, E.: Fermi Liquids in Two-Space Dimensions. In: Constructive Physics V. Rivasseau, (ed.), Springer Lecture Notes in Physics 446, Berlin-Heidelberg-New York: Springer, 1995, pp. 267–300 [FKLT2] Feldman, J., Kn¨orrer, H., Lehmann, D., Trubowitz, E.: Are There Two Dimensional Fermi Liquids? In: Proceedings of the XIth International Congress of Mathematical Physics, D. Iagolnitzer, (ed.), Cambridge, MA: International Press, 1995, pp. 440–444 [FKT1] Feldman, J., Kn¨orrer, H., Trubowitz, E.: A Representation for Fermionic Correlation Functions. Commun. Math. Phys. 195, 465–493 (1998)
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[FKTf1] Feldman, J., Kn¨orrer, H., Trubowitz, E.: A Two Dimensional Fermi Liquid, Part 1: Overview. Commun. Math. Phys. 247, 1–47 (2004) [FKTf2] Feldman, J., Kn¨orrer, H., Trubowitz, E.: A Two Dimensional Fermi Liquid, Part 2: Convergence. Commun. Math. Phys. 247, 49–111 (2004) [FKTf3] Feldman, J., Kn¨orrer, H., Trubowitz, E.: A Two Dimensional Fermi Liquid, Part 3: The Fermi Surface. Commun. Math. Phys. 247, 113–177 (2004) [FKTffi] Feldman, J., Kn¨orrer, H., Trubowitz, E.: Fermionic Functional Integrals and the Renormalization Group, CRM Monograph Series 16, American Mathematical Society (2002) [FKTo1] Feldman, J., Kn¨orrer, H., Trubowitz, E.: Single Scale Analysis of Many Fermion Systems, Part 1: Insulators. Rev. Math. Phys. 15, 949–993 (2003) [FKTo2] Feldman, J., Kn¨orrer, H., Trubowitz, E.: Single Scale Analysis of Many Fermion Systems, Part 2: The First Scale. Rev. Math. Phys. 15, 995–1037 (2003) [FKTo3] Feldman, J., Kn¨orrer, H., Trubowitz, E.: Single Scale Analysis of Many Fermion Systems, Part 3: Sectorized Norms. Rev. Math. Phys. 15, 1039–1120 (2003) [FKTo4] Feldman, J., Kn¨orrer, H., Trubowitz, E.: Single Scale Analysis of Many Fermion Systems, Part 4: Sector Counting. Rev. Math. Phys. 15, 1121–1169 (2003) [FT] Feldman, J., Trubowitz, E.: Perturbation Theory for Many Fermion Systems. Helv. Phys. Acta 63, 156–260 (1990) [PT] P¨oschel, J., Trubowitz, E.: Inverse Spectral Theory, New York-London: Academic Press, 1987 [S] Salmhofer, M.: Improved Power Counting and Fermi Surface Renormalization. Rev. Math. Phys. 10, 553–578 (1998) [SW] Salmhofer, M., Wieczerkowski, C.: Positivity and Convergence in Fermionic Quantum Field Theory. J. Stat. Phys. 99, 557–586 (2000) Communicated by J.Z. Imbrie
Commun. Math. Phys. 247, 243–319 (2004) Digital Object Identifier (DOI) 10.1007/s00220-004-1040-8
Communications in
Mathematical Physics
Convergence of Perturbation Expansions in Fermionic Models. Part 2: Overlapping Loops Joel Feldman1, , Horst Kn¨orrer2 , Eugene Trubowitz2 1 2
Department of Mathematics, University of British Columbia, Vancouver, B.C., Canada V6T 1Z2. E-mail: [email protected] Mathematik, ETH-Zentrum, 8092 Z¨urich, Switzerland. E-mail: [email protected]; [email protected]
Received: 21 September 2002 / Accepted: 12 August 2003 Published online: 6 April 2004 – © Springer-Verlag 2004
Abstract: We improve on the abstract estimate obtained in Part 1 by assuming that there are constraints imposed by ‘overlapping momentum loops’. These constraints are active in a two dimensional, weakly coupled fermion gas with a strictly convex Fermi curve. The improved estimate is used in another paper to control everything but the sum of all ladder contributions to the thermodynamic Green’s functions. Contents V. VI.
Introduction to Part 2 . . . . . . . . . . . . . . . . . . . . . . . Overlapping Loops . . . . . . . . . . . . . . . . . . . . . . . . VI.1 Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . VI.2 Ladders . . . . . . . . . . . . . . . . . . . . . . . . . . VI.3 Overlapping Loops for the Schwinger Functional . . . . VI.4 Configurations of Norms with Improved Power Counting VII. Finding Overlapping Loops . . . . . . . . . . . . . . . . . . . . VII.1 Overlapping Loops Created by the Operator RK,C . . . VII.2 Tails . . . . . . . . . . . . . . . . . . . . . . . . . . . . VIII. The Enlarged Algebra . . . . . . . . . . . . . . . . . . . . . . . VIII.1 Definition of the Enlarged Algebra . . . . . . . . . . . . VIII.2 Norm Estimates for the Enlarged Algebra . . . . . . . . VIII.3 Schwinger Functionals over the Extended Algebra . . . VIII.4 A Second Proof of Theorem IV.1 . . . . . . . . . . . . . VIII.5 The Operator Q . . . . . . . . . . . . . . . . . . . . . . IX. Overlapping Loops Created by the Second Covariance . . . . . IX.1 Implementing Overlapping Loops . . . . . . . . . . . .
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Research supported in part by the Natural Sciences and Engineering Research Council of Canada and the Forschungsinstitut f¨ur Mathematik, ETH Z¨urich.
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IX.2 Tails . . . . . . . . . . . . . . . . . . . . . . IX.3 Proof of Theorem VI.10 in the General Case X. Example: A Vector Model . . . . . . . . . . . . . . . Appendix C. Ladders Expressed in Terms of Kernels . . . Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . .
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V. Introduction to Part 2 In the perturbative analysis of many fermion systems with weak short–range interaction in two or more space dimensions, the presence of an overlapping loop in a Feynman diagram introduces a volume effect in momentum space that leads to an improvement to “naive power counting”. For a detailed discussion of this effect and its consequences, see [FST1, FST2, FST3, FST4]. For a short description, see Subsect. 4 of [FKTf1, §II]. In [FKTo3], we use nonperturbative bounds for systems, in two space dimensions, that are based on the cancellation scheme between diagrams developed in Part 1 of this paper. In this second part, we modify the construction so that we can exploit enough overlapping loops to get improved power counting for the two point function and the non–ladder part of the four point function. As in Part 1, the treatment is in an abstract setting, formulated using systems of seminorms. The postulated volume improvement effects are expressed in terms of these seminorms (Def.VI.1). The main result for the renormalization group map is Theorem VI.6. It follows from Theorem VI.10, which is the main result on the Schwinger functional. The discussion of the renormalization group map in the first part of the paper is based on the representation developed in [FKT1] (which in turn evolved out of the representation developed in [FMRT]). The representation of [FKT1] decomposes Feynman diagrams into annuli. The first annulus consists of all interaction vertices directly connected to the external vertices. The second annulus consists of all interaction vertices directly connected to the first annulus but not to the external vertices. And so on. See the introduction to [FKT1]. Overlapping loops that only involve vertices of neighbouring annuli are relatively easy to exploit. It turns out, that for the analysis of the two point function and the non–ladder part of the four point function, it suffices to use overlapping loops that involve only vertices of at most three adjacent annuli. A special case of Theorem VI.10, for which this combinatorial fact is easier to see, is proven at the end of §VII. After some preparation in §VIII, the general case is proven at the end of §IX. In §X, we apply Theorem VI.6 to a simple vector model. We also describe, by drawing an analogy with the vector model, how sectors can be used to nonperturbatively implement overlapping loops for many fermion systems. A notation table is provided at the end of the paper. VI. Overlapping Loops VI.1. Norms. Again, let A be a graded superalgebra and A = A V the Grassmann algebra in the variable ψ over A. Also fix two covariances C and D on V . Definition VI.1. Let · and · impr be two families of symmetric seminorms on the spaces Am ⊗ V ⊗n such that · impr ≤ · and f impr = 0 if f ∈ Am ⊗ V ⊗n with m ≥ 1. We say that (C, D) have improved integration constants c ∈ Nd , b, J ∈ R+ for the families · and · impr of seminorms if c is a contraction bound for the covariance
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C for both seminorms · and · impr , b is an integral bound for C and D for both seminorms and the following triple contraction estimate holds: Let n, n ≥ 3; 1 ≤ i1 , i2 , i3 ≤ n and 1 ≤ j1 , j2 , j3 ≤ n with i1 , i2 , i3 all different and j1 , j2 , j3 all different. Also let the covariances C1 , C2 , C3 each be either C or D with at least one of these covariances equal to C. Then for f ∈ A0 ⊗ V ⊗n , f ∈ A0 ⊗ V ⊗n , ConC ConC ConC (f ⊗ f ) ≤ J b4 c f f . 1 2 3 impr i1 →j1
i2 →j2
i3 →j3
Observe that ConC1 ConC2 ConC3 (f ⊗ f ) ∈ A0 ⊗ V ⊗(n+n −6) . i1 →j1
i2 →j2
i3 →j3
Lemma VI.2. Assume that (C, D) have improved integration constants c, b, J for the families · and · impr of seminorms. Let n1 , · · · , nr , nr+1 , · · · , nr+s ≥ 0, let f1 (ξ (1) , · · · , ξ (r) ) ∈ A0 [n1 , · · · , nr ], f2 (ξ (r+1) , · · · , ξ (r+s) ) ∈ A0 [nr+1 , · · · , nr+s ], and let i1 , i2 , i3 ∈ {1, · · · , r} and j1 , j2 , j3 ∈ {r + 1, · · · , r + s}. Also let the covariances C1 , C2 , C3 each be either C or D with at least one of these covariances equal to C. Then ConC2 ConC3 (f1 f2 ) ≤ J nj1 nj2 nj3 b4 c f1 f2 , ConC1 impr
ξ (i1 ) →ξ (j1 ) ξ (i2 ) →ξ (j2 ) ξ (i3 ) →ξ (j3 )
ξ (1) , · · · , ξ (r)
C1 C2
f1
ξ (r+1) , · · · , ξ (s)
f2
C3 Proof. The proof is analogous to that of Lemma II.29.i.
We define, for a Grassmann function f the improved norm Nimpr (f ) as in Def. II.23. That is, α |n| b|n| f0;n1 ,···nr impr . Nimpr (f ; α) = bc2 n1 ,···nr ≥0
An abstract example of such norms is described at the end of this section, and this abstract example is made concrete in §X. Definition VI.3. Let f (ξ (1) , · · · , ξ (r) ) be a Grassmann function and I ⊂ {1, · · · , r}. We say that f has degree d in the variables ξ (i) , i ∈ I if f ∈ Am [n1 , · · · , nr ], m;n1 ,··· ,nr
i∈I ni =d where Am [n1 , · · · , nr ] was defined in Def. II.21. We say that f has degree at least d in the variables ξ (i) , i ∈ I if f = d ≥d fd , where each fd has degree d in the variables i ∈ I . Similarly we say that f has degree at most d in the variables ξ (i) , i ∈ I if ξ (i) , f = d ≤d fd , where each fd has degree d in the variables ξ (i) , i ∈ I .
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Proposition VI.4. Let (C, D) have improved integration constants c, b, J . Let r ≥ t ≥ s ≥ 1, and let f1 (ξ (1) , · · · , ξ (s) , ξ (s+1) , · · · , ξ (t) , ξ (t+1) , · · · , ξ (r) ) and f2 (ξ (1) , · · · , ξ (r) ) be Grassmann functions. Set g(ξ (t+1) , · · · , ξ (r) ) . :f (ξ (1) , · · · , ξ (s) , · · · , ξ (t) , · · · , ξ (r) ): (1) . = . 1 ξ ,··· ,ξ (s) ,C . ξ (s+1) ,··· ,ξ (t) ,D × .. :f2 (ξ (1) , · · · , ξ (r) ):ξ (1) ,··· ,ξ (s) ,C .. ξ (s+1) ,··· ,ξ (t) ,D
s
dµC (ξ (i) )
i=1
t j =s+1
dµD (ξ (j ) ).
If f1 has degree at least one in the variables ξ (1) , · · · , ξ (s) and degree at least three in the variables ξ (1) , · · · , ξ (t) then Nimpr (g; α) ≤ 27 αJ6 N (f1 ; α) N (f2 ; α) for α ≥ 2. Proof. Set f˜i = .. :fi :ξ (1) ,··· ,ξ (s) ,C .. ξ (s+1) ,··· ,ξ (t) ,D . We first prove the statement in the case that f1 and f2 are both homogeneous, that is f1 ∈ A0 [n1 , · · · , ns , · · · , nt , · · · , nr ], f2 ∈ A0 [n1 , · · · , nr ]. Then g = 0 unless ni = ni for 1 ≤ i ≤ t, and g ∈ A0 [nt+1 + nt+1 , · · · , nr + nr ]. By hypothesis n1 + · · · + ns ≥ 1 and n1 + · · · + nt ≥ 3. Therefore it is possible to choose i1 ∈ {1, · · · , s} with ni1 ≥ 1, and to choose i2 , i3 ∈ {1, · · · , t} such that 1 if i2 = i1 ni2 ≥ , 2 if i2 = i1 i1 , i2 1 if i3 = ni3 ≥ 2 if i3 ∈ {i1 , i2 } but i1 = i2 . 3 if i = i = i 3 2 1 Set Cν
C = D
Clearly, C1 = C. Also set Conν =
if 1 ≤ iν ≤ s . if s + 1 ≤ iν ≤ t
ConCν ξ (iν ) →ζ (iν )
and
g (ξ (1) , · · · , ξ (r) ; ζ (1) , · · · , ζ (r) ) = Con1 Con2 Con3 f1 (ξ (1) , · · · , ξ (r) )
g (ξ
(1)
,··· ,ξ
(r)
;ζ
(1)
,··· ,ζ
(r)
×f2 (ζ (1) , · · · , ζ (r) ), ) = Con1 Con2 Con3 f˜1 (ξ (1) , · · · , ξ (r) ) ×f˜2 (ζ (1) , · · · , ζ (r) ).
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Observe that g ∈ A0 [n1 − (δ1 i1 + δ1 i2 + δ1 i3 ), · · · , nr − (δr i1 + δr i2 + δr i3 ), n1 − (δ1 i1 + δ1 i2 + δ1 i3 ), · · · , nr − (δr i1 + δr i2 + δr i3 )] and g (ξ (1) , · · · , ξ (r) ; ζ (1) , · · · , ζ (r) ) = .. :g (ξ (1) , · · · , ξ (r) ; ζ (1) , · · · , ζ (r) ): ξ (1) ,··· ,ξ (s) ,C .. ξ (s+1) ,··· ,ξ (t) ,D ζ (1) ,··· ,ζ (s) ,C
ζ (s+1) ,··· ,ζ (t) ,D
by Remark II.12. By Lemma II.13, g=
g (ξ (1) , · · · , ξ (r) ; ξ (1) , · · · , ξ (r) )
s
dµC (ξ (i) )
i=1
t j =s+1
dµD (ξ (j ) ).
By Lemma II.29 and Lemma VI.2, gimpr ≤ b2(n1 +···+nt −3) g impr ≤ J
ni1 ni2 ni3 c b2
b2(n1 +···+nt ) f1 f2 .
(VI.1)
Therefore Nimpr (g) = bc2 α nt+1 +···+nr +nt+1 +···+nr bnt+1 +···+nr +nt+1 +···+nr gimpr n +···+nr +n +···+nr ni1 ni2 ni3 J c2 1 ≤ α 2(n αb 1 f1 f2 1 +···+nt ) b4
n n n
i1 i2 i3 ≤ J α 2(n N (f1 ) N (f2 ) 1 +···+nt )
≤ 27 αJ6 N (f1 ) N (f2 ). The general case now follows by decomposing f1 and f2 into homogeneous pieces.
VI.2. Ladders. Theorem VI.6, below, which is the main result of this paper, shows that under appropriate assumptions on an effective interaction W (ψ), the two
point and non–ladder four point parts of the effective interaction :W (ψ):ψ,D = C :W :ψ,C+D , constructed using the Grassmann Gaussian integral with covariance C, obeys estimates that are better by a factor J than those one would expect from Theorem IV.1. To formulate this precisely, we first give the definition of ladders.
In a ladder, neighbouring four legged vertices are connected by two covariances. Since ladders result from integrating with covariance C, at least one of the connecting covariances is equal to C. The other connecting covariance may be C or D. In the rest of the paper, we will systematically use ξ, ξ , ξ , · · · for fields associated to the covariance C. We will use ζ, ζ , ζ , · · · for fields associated to the covariance D and ψ for the external fields.
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Definition VI.5.
(i) A rung is a Grassmann function
ρ(ζ, ξ ; ζ , ξ ) ∈ A[0, 2, 0, 2] ⊕ A[1, 1, 0, 2] ⊕ A[0, 2, 1, 1] ⊕ A[1, 1, 1, 1]. We think of ζ, ξ as the D resp. C fields on the left side of the rung and of ζ , ξ as the D resp. C fields on the right side of the rung. ζ , ξ
ρ
ζ, ξ An end is a Grassmann function
E(ψ; ζ, ξ ) ∈ A[2, 0, 2] ⊕ A[2, 1, 1]. We think of ψ as the external fields at the end of the ladder and of ζ, ξ as the D resp. C fields going into the ladder. ψ
ζ, ξ
E
(ii) If E is an end and ρ is a rung, we define the end E ◦ ρ by . E(ψ; ζ, ξ ) . ξ,C . ρ(ζ, ξ ; ζ , ξ ) . ξ,C dµ (ξ )dµ (ζ ) E ◦ ρ(ψ; ζ , ξ ) = . . . . C D ζ,D
ψ
ζ,D
ρ
E
ζ, ξ
ζ , ξ
If E1 , E2 are ends, we define the ladder E1 ◦ E2 by . E (ψ; ζ , ξ ) . . E (ψ; ζ , ξ ) . dµ (ξ )dµ (ζ ) E1 ◦ E2 (ψ) = . 1 . ξ ,C . 2 . ξ ,C C D ζ ,D
ψ (iii) Let F (ξ ) ∈ A[4]. Write F (ξ (1) + ξ (2) + ξ (3) ) =
E1
ζ ,D
ζ , ξ
E2
ψ
Fn1 ,n2 ,n3 (ξ (1) , ξ (2) , ξ (3) ),
n1 +n2 +n3 =4
F (ξ
(1)
+ξ
(2)
+ξ
(3)
+ξ
(4)
)=
Fn1 ,n2 ,n3 ,n4 (ξ (1) , ξ (2) , ξ (3) , ξ (4) )
n1 +n2 +n3 +n4 =4
with Fn1 ,n2 ,n3 ∈ A[n1 , n2 , n3 ] and Fn1 ,n2 ,n3 ,n4 ∈ A[n1 , n2 , n3 , n4 ]. The rung associated to F is ρ(F )(ζ, ξ ; ζ , ξ ) = F0,2,0,2 + F1,1,0,2 + F0,2,1,1 + F1,1,1,1 . The end associated to F is E(F )(ψ; ζ , ξ ) = F2,0,2 + F2,1,1 .
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The ladder of length r ≥ 1 with vertex F is defined as Lr (F )(ψ) = E(F ) ◦ ρ(F ) ◦ ρ(F ) ◦ · · · ρ(F ) ◦ E(F ) with (r − 1) copies of ρ(F ). ψ
E(F )
ρ(F )
ρ(F )
ρ(F )
E(F )
ψ
In Appendix C, we describe ladders in terms of kernels. The main result of this paper is Theorem
VI.6. Let W (ψ) be an even Grassmann function with coefficients in A. Assume that N W ; 64α 0 < 18 α, and that α ≥ 8. Set
: W (ψ) :ψ,D = C : W :ψ,C+D . If (C, D) have improved integration constants c, b, J , then (i)
N W − W; α ≤
Nimpr W − W ; α ≤
N(W ;32α)2 1 , 2α 2 1− 12 N(W ;32α) α
N(W ;32α)2 1 . 2α 2 1− 12 N(W ;32α) α
(ψ) with W (ii) Write W (ψ) = m;n Wm;n (ψ), W (ψ) = m;n Wm;n m;n , Wm;n ∈ Am [n]. If W0;2 = 0, then
;α ≤ Nimpr W0,2
− W0,4 − Nimpr W0,4
1 2
∞ r=1
Lr (W0,4 ); α ≤
210 J N(W ;64α)2 , α 6 1− α8 N(W ;64α) 210 J N(W ;64α)2 . α 6 1− α8 N(W ;64α)
Remark VI.7. i) Part (i) of the theorem follows directly from Theorem IV.1. For the , W and L (W ) proof of part (ii) one can replace the algebra A by A0 , since W0,2 r 0,4 0,2 ∞ depend only on n=0 W0;n . ii) The hypothesis that W0;2 = 0 in part (ii) of Theorem VI.6 prevents strings of two– legged vertices from appearing in diagrammatic expansions. The expansion used in the proof of part (ii) cannot detect certain overlapping loops containing such strings. In practice a nonzero W0;2 can be absorbed in the propagator.
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The proof of part (ii) of Theorem VI.6 is based on an analysis of VI.3. Overlapping Loops for the Schwinger Functional. We first generalise the concept of a ladder. If U (ψ; ξ ) is a Grassmann function we write Un0 ;p1 ,p2 ;n1 ,n2 (ψ; ζ, ζ ; ξ, ξ ) U (ψ + ζ + ζ ; ξ + ξ ) = p1 ,p2 n0 ,n1 ,n2
with Un0 ;p1 ,p2 ;n1 ,n2 ∈ A[n0 ; p1 , p2 , n1 , n2 ]. Definition VI.8. Let U be as above. (i) The rung associated to U is Rung(U )(ζ, ξ ; ζ , ξ ) = U0;0,2;0,2 + U0;1,1;0,2 + U0;0,2;1,1 + U0;1,1;1,1 . (ii) The tail T (U ) associated to U is recursively defined as T1 (U )(ψ; ζ , ξ ) = U2;0,0;0,2 + U2;0,1;0,1 , T +1 (U )(ψ; ζ , ξ ) = T (U ) ◦ Rung(U ). Observe that T (U )(ψ; ζ , ξ ) lies in A[2, 0, 2] ⊕ A[2, 1, 1]. Later we need Remark VI.9. Let E1 , E2 be ends whose coefficients are even elements of A and let g(ψ; ξ ) be a Grassmann function. Set . . E (ψ; ζ , ξ ) E (ψ; ζ , ξ ) . . h(ψ) = . ζ ,D . g(ψ + ζ ; ξ ) . ζ ,D dµD (ζ ) dµC (ξ ), . 1 2 ξ ,C
h(ψ) =
∞
hn (ψ)
ξ ,C
with hn ∈ A[n].
n=4
Then
h4 = E1 ◦ Rung(g) ◦ E2 .
Proof. By Lemma A.5, . E (ψ; ζ, ξ ) . ζ,D . g(ψ + ζ + ζ ; ξ + ξ ) . h(ψ) = . 1 . . . ζ,ζ ,D ξ,C
ξ,ξ ,C
. E (ψ; ζ , ξ ) . dµ (ζ, ζ ) dµ (ξ, ξ ). . 2 . ζ ,D D C ξ ,C
As E1 is of degree at most one in ζ and E2 is of degree at most one in ζ , :E1 (ψ; ζ, ξ ):ξ,C :Rung(g)(ζ, ξ ; ζ , ξ ):ξ,ξ ,C h4 (ψ) = :E2 (ψ; ζ , ξ ):ξ ,C dµD (ζ, ζ ) dµC (ξ, ξ )
= E1 ◦ Rung(g) ◦ E2 (ψ).
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The main estimate on the Schwinger functional is: Theorem VI.10. Let A be a superalgebra, with all elements having degree zero (that is A = A0 ), · and · impr be two families of symmetric seminorms on the spaces Am ⊗ V ⊗n and let (C, D) have improved integration constants c, b, J . Let Uˆ (ψ, ξ ), fˆ(ψ, ξ ) be Grassmann functions with coefficients in A of degree at least four and Uˆ even. Set U (ψ, ξ ) = .. Uˆ (ψ, ξ ) .. ψ,D , ξ,C
f (ψ, ξ ) = .. fˆ(ψ, ξ ) .. ψ,D . ξ,C
Assume that α ≥ 8 and N (Uˆ ; 32α)0 < 18 α. By Proposition III.10, :f (ψ):ψ,D = SU,C (f ) exists. Write
fˆ(ψ, ξ ) =
n0 ,n1
fˆn0 ,n1 (ψ, ξ ),
f (ψ) =
n
fn (ψ)
with fˆn0 ,n1 ∈ A[n0 , n1 ], fn ∈ A[n]. Then
N(Uˆ ;32α) 210 J N fˆ; 32α α 6 1− 8 N(Uˆ ;32α) α
Nimpr (f2 ; α) ≤
and there exists a Grassmann function g(ψ) such that f4 = fˆ4,0 +
∞
T (Uˆ ) ◦ T1 (fˆ) +
=1
and
1 2
, ≥1
T (Uˆ ) ◦ Rung(fˆ) ◦ T (Uˆ ) + g
N(Uˆ ;32α) 210 J ˆ; 32α . N f 6 8 ˆ α 1− α N(U ;32α)
Nimpr (g; α) ≤
In the case D = 0 this theorem is proven in Sect. VI; in the general case it is proven in Sect. VIII. Proof that Theorem VI.10 implies Theorem VI.6. By part (i) of Remark VI.7 we may . Set assume that A = A0 . We write Wn for W0;n and Wn for W0;n Uˆ (ψ, ξ ) = W (ψ + ξ ) ∈ V, A . . U (ψ, ξ ) = . Uˆ (ψ, ξ ) . ψ,D , ξ,C
:
Ut (ψ) :ψ,D
= StU,C (U ).
By Remark II.24, N (Uˆ ; α) ≤ N (W ; 2α). As in the proof of Theorem II.28, 1
:W (ψ):ψ,D − :W (ψ):ψ,D = :Ut (ψ):ψ,D − :Uˆ (ψ, 0):ψ,D dt mod A0 0
so W (ψ) − W (ψ) =
1 0
Ut (ψ) − Uˆ (ψ, 0) dt
mod A0 .
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In particular, for n = 2, 4, Wn − Wn =
1 0
− Uˆ n,0 dt. Ut,n
Therefore, by Theorem VI.10,
Nimpr (W2 ) ≤ max Nimpr Ut,2 0≤t≤1
≤
N(Uˆ ;32α) 210 J α 6 1− 8 N(Uˆ ;32α) α
≤
210 J N(W ;64α)2 . α 6 1− α8 N(W ;64α)
N (Uˆ ; 32α)
Observe that Rung(Uˆ ) = ρ(W4 ), T (Uˆ ) = E(W4 ) ◦ ρ(W4 ) ◦ · · · ◦ ρ(W4 ) with − 1 copies of ρ(W4 ). Therefore T (t Uˆ ) ◦ T1 (Uˆ ) = t L (W4 ), T (t Uˆ ) ◦ Rung(Uˆ ) ◦ T (t Uˆ ) = t + L + (W4 ). Hence, by Theorem VI.10, 1 ∞ W4 = W4 + t L (W4 ) dt + =1 0
with Nimpr (g) ≤ Now ∞ =1 0
1
t L (W4 ) dt +
N(Uˆ ;32α) 210 J α 6 1− 8 N(Uˆ ;32α) α
1 2
, ≥1
1
1 2
, ≥1 0
1
t + L + (W4 ) dt + g
N (Uˆ ; 32α) ≤
t + L + (W4 ) dt =
0
=
210 J N(W ;64α)2 . α 6 1− α8 N(W ;64α) ∞
1 2
1
(r + 1) t r Lr (W4 ) dt
r=1 0 ∞ 1 Lr (W4 ). 2 r=1
VI.4. Configurations of Norms with Improved Power Counting. Definition VI.11. Let q be an even natural number. For p = 1, 2, 3, · · · , q, let · p be a system of symmetric seminorms on the spaces Am ⊗ V ⊗n . We say that (C, D) have integration constants c, b for the configuration · 1 , · 2 , · · · , · q of seminorms if the following estimates hold: Let m, m ≥ 0 and 1 ≤ i ≤ n, 1 ≤ j ≤ n . Also let f ∈ Am ⊗V ⊗n , f ∈ Am ⊗V ⊗n . Then for all natural numbers p ≤ q the simple contraction estimate ConC (f ⊗ f ) ≤ c f p f p i→n+j
holds.
p
p1 +p2 =p+1 at least one odd
1
2
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Furthermore, if C2 , C3 ∈ {C, D}, m = m = 0, 1 ≤ i1 , i2 , i3 ≤ n with i1 , i2 , i3 all different and 1 ≤ j1 , j2 , j3 ≤ n with j1 , j2 , j3 all different, the improved contraction estimate ConC ConC ConC (f ⊗ f ) ≤ b4 c f p f p 2
3
p
i1 →n+j1 i2 →n+j2 i3 →n+j3
1
p1 +p2 =p+3 at least one odd
2
holds for p ≤ q − 2. For every f ∈ Am ⊗ V ⊗n and every n ≤ n the modified integral bound (f )dµ Antn (f )dµD ≤ 21 (b/2)n f p + f p−(−1)p Ant C , n p
p
(VI.2) holds. The partial antisymmetrization Antn was defined in Def. II.25.ii. Lemma VI.12. Let q be an even natural number. Assume that (C, D) have integration constants c, b for the configuration · 1 , · 2 , · · · , · q of seminorms and let J > 0. For f ∈ Am ⊗ V ⊗n , set f =
q p=1
f impr =
J −[(p−1)/2] f p = f 1 + f 2 + J1 f 3
1 + J1 f 4 + · · · + J (q−2)/2 f q , q−2 −[(p−1)/2] J f p if m = 0
.
p=1
if m = 0
0
Here [(p − 1)/2] is the integer part of p−2 2 . Then (C, D) have improved integration constants c, b, J for the families · and · impr of seminorms. Proof. Clearly · impr ≤ · . To verify that c is a contraction bound for C, let f ∈ Am ⊗ V ⊗n , f ∈ Am ⊗ V ⊗n and 1 ≤ i ≤ n, 1 ≤ j ≤ n . Observe that if p1 + p2 = p + 1 with at least one of p1 and p2 odd, then 1
1 J
[(p1 −1)/2]
J
[(p2 −1)/2]
=
1 J
[(p1 +p2 −2)/2]
=
1 J
[(p−1)/2]
.
Consequently, q ConC (f ⊗ f ) = J −[(p−1)/2] ConC f ⊗ f p i→n+j
≤
p=1 q
i→n+j
c
p=1
≤
q p=1
c
p1 +p2 =p+1 at least one odd
J −[(p−1)/2] f p1 f p2
p1 +p2 =p+1
≤ c f f .
J −[(p1 −1)/2] f p1 J −[(p2 −1)/2] f p2
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Replacing q by q − 2 gives the corresponding bound for · impr . To verify the triple contraction estimate of Def. VI.1, let C2 , C3 ∈ {C, D}, m = m = 0, 1 ≤ i1 , i2 , i3 ≤ n with i1 , i2 , i3 all different and 1 ≤ j1 , j2 , j3 ≤ n with j1 , j2 , j3 all different. Then ConC
ConC2
q−2 −[ p−1 ] 2 ConC ConC3 (f ⊗ f )impr = J
i1 →n+j1 i2 →n+j2 i3 →n+j3
p=1
× ConC3 (f i3 →n+j3
≤ J b4 c
ConC2 i1 →n+j1 i2 →n+j2 ⊗ f ) p
q−2
q
p=1
p1 +p2 =p+3 at least one odd
J −[(p+1)/2]
×f p1 f p2 q ≤ J b4 c J −[(p1 −1)/2] f p1 p1 ,p2 =1
×J
−[(p2 −1)/2]
f p2
= J b4 c f f . We verify that b is an integral bound for C for the norm · . The other cases are similar. Let f ∈ Am ⊗ V ⊗n and n ≤ n. Then q −[(p−1)/2] J Antn (f )dµC = Antn (f )dµC p
p=1
=
1 2
(b/2)n
q
J −[(p−1)/2] f p + f p−(−1)p
p=1 n
≤ (b/2) f .
(VI.3)
In our main application, we use a special case of Def. VI.11 in which only norms · p , with p odd, appear. Definition VI.13. Let q be an odd natural number. For p = 1, 3, 5, · · · , q, let · p be a system of symmetric seminorms on the spaces Am ⊗ V ⊗n . We say that (C, D) have integration constants c, b for the configuration · 1 , · 3 , · · · , · q of seminorms if b is an integral bound for both C and D and all of the seminorms · p (see Def. II.25.ii) and the following contraction estimates hold: Let m, m ≥ 0 and 1 ≤ i ≤ n, 1 ≤ j ≤ n . Also let f ∈ Am ⊗V ⊗n , f ∈ Am ⊗V ⊗n Then for all odd natural numbers p ≤ q, ConC (f ⊗ f ) ≤ c f p f p . p
i→n+j
p1 +p2 =p+1 p1 ,p2 odd
1
2
Furthermore, if C2 , C3 ∈ {C, D}, m = m = 0, 1 ≤ i1 , i2 , i3 ≤ n with i1 , i2 , i3 all different and 1 ≤ j1 , j2 , j3 ≤ n with j1 , j2 , j3 all different, then, for all odd p ≤ q − 2, ConC ConC ConC (f ⊗ f ) ≤ b4 c f p f p . 2
3
i1 →n+j1 i2 →n+j2 i3 →n+j3
p
p1 +p2 =p+3 p1 ,p2 odd
1
2
Convergence of Perturbation Expansions in Fermionic Models. Part 2
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Remark VI.14. If, in the setting of Def. VI.13, the norm · p is defined to be zero for all even p, then the conditions of Def. VI.11 are fulfilled, except that the factor of 21 in (VI.2) does not appear in Def. II.25.ii of integral bound. Lemma VI.15. Let q be an odd natural number. Assume that (C, D) have integration constants c, b for the configuration · 1 , · 3 , · · · , · q of seminorms and let J > 0. For f ∈ Am ⊗ V ⊗n , set f =
q
J (1−p)/2 f p ,
p=1 podd
f impr =
q−2 (1−p)/2 J f p if m = 0 p=1 podd
0
. if m = 0
Then (C, D) have improved integration constants c, b, J for the families · and ·impr of seminorms. Proof. By Remark VI.14, Lemma VI.12 implies all of the conditions of Def. VI.1, except that b be an integral bound for C and D for both seminorms. However, the proof of this condition is virtually identical to (VI.3). Remark VI.16. Lemma VI.15 holds for all J > 0. In applications, J is chosen so that f p ≤ const J (p−1)/2 f 1
(VI.4)
for all f of interest. If J satisfying (VI.4) can be chosen sufficiently small, Lemma VI.15 can be used in conjunction with Prop. VI.4 to obtain improved bounds, as the following example illustrates.
For simplicity, we assume that q = 3. Let f ξ (1) , ξ (2) ∈ A0 [n1 , n2 ] with n2 ≥ 3 and set 2
(2) . f ξ (1) , ξ (2) . g ξ (1) = . . . ξ (2) ,C dµC ξ The standard bound, without improvement, follows from (VI.4) in the proof of Prop. II.33: g1 ≤ n2 c b2(n2 −1) f 21 . On the other hand, by (VI.1), in the proof of Prop. VI.4,
2 g1 = gimpr ≤ n32 J c b2(n2 −1) f 2 = n32 J c b2(n2 −1) f 1 + J1 f 3 . If f 3 ≤ const J f 1 , g1 ≤ const n32 J c b2(n2 −1) f 21 .
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VII. Finding Overlapping Loops In this chapter, we give the1 proof of Theorem VI.10 in the case D = 0, using the representation SU,C = dµC of Theorem III.2. We assume that the coefficient −RU,C algebra A contains only elements of degree zero, that is A = A0 . Let · and · impr be two families of symmetric seminorms on the spaces Am ⊗ V ⊗n such that (C, 0) has improved integration constants c, b, J for these families of seminorms. Recall from Remark III.6 that the operator RK,C is written as a sum of operators RC (K1 , · · · , K ) with even Grassmann functions K1 (ξ, ξ , η), · · · , K (ξ, ξ , η). If one of these Grassmann functions, say K1 has degree at least three in the variables ξ , η then, for any Grassmann function f (ξ ), there is a pair of overlapping loops in each Feynman diagram contributing to RC (K1 , · · · , K )(: f :). The way these overlapping loops can occur is indicated in the figures below. K1 K1 f
K2 or
K1
K1 or
f
or
f
f K
K
K
K
VII.1. Overlapping Loops Created by the Operator RK,C . In this subsection, we suppress the external fields ψ by working in the Grassmann algebra A V with coefficients in the algebra A = A V generated by the fields ψ. This algebra was defined in Subsect. III.2. Recall that · and · impr induce a family of symmetric seminorms on the spaces Am ⊗ V ⊗n , which we here denote by the same symbols. We split up the operators RC (K1 , · · · , K ) of (III.2) in order to exhibit possible overlapping loops. For Grassmann functions K2 (ξ, ξ , η), · · · , K (ξ, ξ , η) and f (ξ ) we define . :K (ξ, ξ + ξ , η ): . ˜ RC (K2 , · · · , K )(f ) = . i ξ . η i=2
×f (η + η ) dµC (ξ ) dµC (η ).
(VII.1)
This is a Grassmann function of ξ, ξ , η that is schematically represented in the figure below. η f
η
ξ η
η
ξ K2
ξ
ξ K3
ξ
Convergence of Perturbation Expansions in Fermionic Models. Part 2
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Proposition VII.1. For even Grassmann functions K1 (ξ, ξ , η), · · · , K (ξ, ξ , η) and f (ξ ) RC (K1 , · · · , K )(f ) = ..
:K1 (ξ, ξ , η):ξ ,η :R˜ C (K2 , · · · , K )
×(f )(ξ, ξ , η):ξ ,η dµC (ξ ) dµC (η) .. ξ , η f
K1
η
ξ η
η
ξ
ξ K2
ξ
ξ K3
ξ
Proof. If :f :ξ = RC (K1 , · · · , K )(f ), then by part (iii) of Prop. A.2 (applied to the variable ξ ) and Lemma A.5 (applied to the variable η) f (ξ ) =
=
. :K (ξ, ξ , η): :K (ξ, ξ , η): . f (η) dµ (ξ ) dµ (η) . 1 i C C ξ ξ .η i=2
. . . :Ki (ξ, ξ + ξ , η):ξ dµC (ξ ) .. ξ . η . :K1 (ξ, ξ , η):ξ . i=2
×f (η) dµC (ξ ) dµC (η) = :K1 (ξ, ξ , η):ξ ,η .. :Ki (ξ, ξ + ξ , η ):ξ dµC (ξ ) .. ξ ,η i=2
:f (η + η ):η dµC (η ) dµC (ξ ) dµC (η) = :K1 (ξ, ξ , η):ξ ,η :R˜ C (K2 , · · · , K )(f )(ξ, ξ , η):ξ ,η ×dµC (ξ ) dµC (η). Remark VII.2. Set Kˆ (i) (ξ, ξ ; ξ , η ) = K (i) (ξ, ξ + ξ , η ), fˆ(ξ, ξ ) = f (ξ + ξ ). Then the map f → :R˜ C (K2 , · · · , K )(f ):ξ over the algebra A agrees with the map f → RC (Kˆ 2 , · · · , Kˆ )(fˆ) of (III.2) over the algebra A˜ of Grassmann functions in the variables ξ , η with coefficients in A . Therefore we can use the results of §III to obtain estimates on R˜ C .
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Lemma VII.3. Let K(ξ, ξ , η) be an even Grassmann function with K(ξ, ξ , 0) = 0. Decompose K(ξ, ξ , η) = K (ξ, ξ , η) + K (ξ, ξ , η), where K has degree at most two in the variables ξ , η and K has degree at least three in the variables ξ , η. Let each of the functions K (1) , · · · , K ( ) beone of K , K or K, and assume that at least one of them is equal to K . Let f (ξ ) ∈ A V , and set 1 !
Then, if α ≥ 2,
RC (K (1) , · · · , K ( ) )(:f :)(ξ ) = :f (ξ ):.
Nimpr (f ; α) ≤
25 J α +5
N (f ; 2α) N (K; 2α) .
Proof. We may assume that K (1) = K . Set g(ξ, ξ , η) = R˜ C (K (2) , · · · , K ( ) )(:f :)(ξ, ξ , η). ˜ By Remark VII.2, in the algebra A, :g:ξ = RC (Kˆ (2) , · · · , Kˆ ( ) )(:fˆ:). Therefore, by part (ii) of Prop. III.7 and Remark II.24, 1 ( −1)!
N (g) ≤ ≤ ≤
By Prop. VII.1, f (ξ ) =
1 !
1 α −1 1 α −1 1 α −1
N (fˆ)
N (Kˆ (i) )
i=2
N (f ; 2α)
N (K (i) ; 2α)
i=2
N (f ; 2α) N(K; 2α) −1 .
:K (ξ, ξ , η):ξ ,η :g(ξ, ξ , η):ξ ,η dµC (ξ ) dµC (η).
Proposition VI.4 implies that Nimpr (f ; α) ≤ ≤
27J N (K ; α) N (g; α) ! α 6 25 J N (f ; 2α) N (K; 2α) α +5
Proposition VII.4. Let K(ξ, ξ , η) be an even Grassmann function. Decompose K(ξ, ξ , η) = K (ξ, ξ , η) + K (ξ, ξ , η), where K has degree at most two in the variables ξ , η and K has degree at least three in the variables ξ , η. Let f (ξ ) ∈ A V , and set
:g (ξ ): = RK ,C (:f :).
:g(ξ ): = RK,C (:f :), Then, if α ≥ 2 and N(K; 2α)0 ≤ α, Nimpr (g − g ; α) ≤
25 J α6
N (f ; 2α)
N(K;2α) . 1− α1 N(K;2α)
Convergence of Perturbation Expansions in Fermionic Models. Part 2
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Proof. By Remark III.6, g=
∞
∞
g =
g ,
=1
=1
g ,
where :g : = :g :
=
1 ! 1 !
RC (K, · · · , K)(:f :), RC (K , · · · , K )(:f :).
Since RC (K, · · · , K) − RC (K , · · · , K ) = RC (K − K , K, · · · , K) +RC (K , K − K , K, · · · , K) + · · · + RC (K , · · · , K − K ) = RC (K , K, · · · , K) + RC (K , K , K, · · · , K) + · · · + RC (K , · · · , K , K ) it follows from Lemma VII.3 that Nimpr (g − g ) ≤
25 J α +5
N (f ; 2α) N (K; 2α) .
Therefore Nimpr (g − g ) ≤
∞ =1
Nimpr (g − g ) ≤
25 J α6
N (f ; 2α)
N(K;2α) . 1− α1 N(K;2α)
Corollary VII.5. Under the hypotheses of Prop. VII.4, set :h: =
1 1 (:f :) − (:f :). − RK,C − RK ,C
If N (K; 2α)0 < α6 , then Nimpr (h; α) ≤
N(K;2α) 25 J α 6 1− α6 N(K;2α)
N (f ; 2α).
Proof. Since
−1 −1 :h: = − RK,C (:f :) RK,C − RK ,C − RK ,C it follows from Cor. III.9, with Nimpr in place of N, and Prop. VII.4 that Nimpr (h) ≤ 1 + ≤ 1+
Nimpr (K) 2 α 2 1− 42 Nimpr (K) α
Nimpr (K) 2 α 2 1− 42 Nimpr (K) α
Nimpr
−1 RK,C − RK ,C − RK ,C (:f :) −1 N(K;2α) N − R (:f :); 2α . K ,C 1
25 J α 6 1− α N(K;2α)
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By Def. VI.1 and Remark II.24, Nimpr (K) ≤ N (K) ≤ N (K; 2α) so that Cor. III.9 implies 5 N(K;2α) 2 J Nimpr (h) ≤ 1 + α22 N(K;2α) 4 α 6 1− α1 N(K;2α) 1− 2 N(K;2α) α ;2α) × 1 + 2α1 2 N(K N (f ; 2α) 1 N(K ;2α)
1−
α2 2 N(K;2α) α2 1− 42 N(K;2α) α
1−
≤
25 J α6
≤
25 J α 6 1−
≤
N(K;2α) 25 J α 6 [1− α2 N(K;2α)]3
≤
N(K;2α) 25 J α 6 1− α6 N(K;2α)
1 4 N(K;2α) α2
N(K;2α) 1− α1 N(K;2α)
1−
N(K;2α) 1− α1 N(K;2α) 1−
1 N(K;2α) 2α 2 1− 12 N(K;2α) α
1 1 N(K;2α) α2
N (f ; 2α)
N (f ; 2α)
N (f ; 2α)
N (f ; 2α)
since, for X ∈ Nd , 1 [1−X]3
=
∞
−3 r
(−X)r =
r=0
∞ 3·4·5···(3+r−1) 1·2·3···r r=0
Xr ≤
∞
(3X)r . =
1 1−3X .
r=0
Proposition VII.4 exploits overlapping loops that are created by one application of the operator RK,C . There are additional overlapping loops created by the composite operator RK,C ◦ RK,C which we shall exploit now. Lemma VII.6. Let B(η , η ) ∈ A [1, 1] and let H (ξ, ξ , η), K(ξ, ξ , η) be even Grassmann functions that vanish for η = 0. Assume that H or K has degree at least two in the variables ξ , η. Let f (ξ ) ∈ A V and set g(ξ ) =
. B(η , η ) :H (ξ + η , ξ , η):η ,ξ :K(ξ + η , ξ , η):η ,ξ . ×dµC (η , η , ξ ) .. η :f (η):η dµC (η). ξ η
η
H ξ
η
B η
K ξ
Then, if α ≥ 2, Nimpr (g; α) ≤
25 J α 10
N (f ; 2α) N(H ; 2α) N (K; 2α) N (B; α).
Convergence of Perturbation Expansions in Fermionic Models. Part 2
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Proof. We may assume that K has degree at least two in the variables ξ , η. Set h(ξ, η, ξ , η ) = B(η , η ) :H (ξ + η , ξ , ζ ):η ,ξ,ζ :f (η + ζ ):ζ dµC (η )dµC (ζ ). By Lemma A.5, applied to the variable η, g(ξ ) = B(η , η ) :H (ξ + η , ξ , ζ ):η ,ξ ,ζ :K(ξ + η , ξ , η):η ,ξ ,η :f (η + ζ ):η,ζ dµC (η , η , ξ , η, ζ ) = :h(ξ, η, ξ , η ):ξ ,η :K(ξ + η , ξ , η):η ,ξ ,η dµC (η, η , ξ ). Since B is of degree one in η and H is of degree at least one in ζ , iterated application of Prop. II.33 and Remark II.24 yields N (h) ≤
1 N (B) N (H ; 2α) N (f ; 2α). α4
Since h is of degree one in η , only the part of K that is of degree at least three in the variables η , ξ , η can contribute. Hence Prop. VI.4 implies that Nimpr (g) ≤ ≤
27 J α6 25 J α 10
N (h) N (K; 2α) N (B) N (H ; 2α) N (f ; 2α) N (K; 2α).
Proposition VII.7. Let K(ξ, ξ , η) be a Grassmann function of degree at most two in the variables ξ , η and of degree at least one in η. Write K(ξ, ξ , η) = Kn1 ,n2 ,n3 (ξ, ξ , η) with Kn1 ,n2 ,n3 ∈ A [n1 , n2 , n3 ]. n1 ,n2 ,n3
Also set K·,n2 ,n3 =
n1
Kn1 ,n2 ,n3 . Furthermore let T (η) ∈ A [2] and write
T (η + η ) = T (η) + T (η ) + Tmix (η, η ) with Tmix ∈ A [1, 1]. Set ˜ K(ξ, ξ , η) =
:T (η ):η :K(ξ + η , ξ , η):η dµC (η )
η
η K
T
ξ ξ ˆ K(ξ, ξ , η) = Tmix (η , η ):K·,0,1 (ξ + η , ξ , η):η :K·,0,1 (ξ + η , ξ , η):η dµC (η , η ) ξ η
K·,0,1
η T
η
η
K·,0,1 ξ
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Observe that Kˆ is independent of ξ . Finally let f (ξ ) ∈
A
V and set
f (η) = R˜ C (T )RC (K, K)(:f :), ˜ K)(:f :), :f˜(ξ ):ξ = 2 RC (K, ˆ :fˆ(ξ ):ξ = RC (K)(:f :). Then
Nimpr (f − (f˜ + fˆ); α) ≤
Proof. By definition f (η) =
26 J α 10
N (f ; 2α) N (K; 2α)2 N (T ; 2α).
:T (η ):η RC (K, K)(:f :)(η + η ) dµC (η ).
Consequently, by part (ii) of Prop. A.2 and Lemma A.7, in the variable η , f (ξ ) = :T (η ):η RC (K, K)(:f :)(ξ + η ) dµC (η ) . . = :T (η ):η . .. :K(ξ + η , ξ , η):ξ :K(ξ + η , ξ , η):ξ .. η :f (η):η . η ×dµC (ξ , η, η ) . . = 2 . .. :T (η ):η :K(ξ + η , ξ , η):η dµC (η ) .. ξ :K(ξ, ξ , η):ξ . η :f (η):η dµC (ξ , η) . + . Tmix (η , η ):K(ξ + η , ξ , η):ξ ,η :K(ξ + η , ξ , η):ξ ,η . ×dµC (ξ , η , η ) . η :f (η):η dµC (η) ˜ = 2 .. :K(ξ, ξ , η):ξ :K(ξ, ξ , η):ξ .. η :f (η):η dµC (ξ , η) ˆ + :K(ξ, ξ , η):ξ ,η :f (η):η dµC (ξ , η) + .. Tmix (η , η ) :K(ξ + η , ξ , η):ξ ,η :K(ξ + η , ξ , η):ξ ,η −:K·,0,1 (ξ + η , ξ , η):ξ ,η :K·,0,1 (ξ + η , ξ , η):ξ ,η ×dµC (ξ , η , η ) .. η :f (η):η dµC (η) = f˜(ξ ) + fˆ(ξ ) + g (ξ ) + g (ξ ) with
g (ξ ) = .. Tmix (η , η ) :H (ξ + η , ξ , η):ξ ,η :K(ξ + η , ξ , η):ξ ,η ×dµC (ξ , η , η ) .. η :f (η):η dµC (η), g (ξ ) = .. Tmix (η , η ):K·,0,1 (ξ + η , ξ , η):ξ ,η :H (ξ + η , ξ , η):ξ ,η ×dµC (ξ , η , η ) .. η :f (η):η dµC (η),
Convergence of Perturbation Expansions in Fermionic Models. Part 2
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where H = K − K·,0,1 . In the last equality, we used the fact that K·,0,1 (ξ + η , ξ , η) and hence Kˆ is independent of ξ , so that we are free to drop the Wick ordering with respect to ξ in the expression yielding fˆ(ξ ). By Lemma VII.6 and the observations that N (K·,0,1 ; 2α), N(K − K·,0,1 ; 2α) ≤ N (K; 2α), N (Tmix ; α) ≤ N (T (η + η ); α) ≤ N (T ; 2α), we have Nimpr (f − f˜ − fˆ) = Nimpr (g + g ) ≤
26 J α 10
N (f ; 2α) N (K; 2α)2 N (T ; 2α).
VII.2. Tails. In this subsection let K(ψ; ξ, ξ , η) be an even Grassmann function with coefficients in A that has degree at least four in the variables ψ, ξ, ξ , η and degree at least one in the variable η. We always write K= Kn0 ,n1 ,n2 ,n3 with Kn0 ,n1 ,n2 ,n3 ∈ A[n0 , n1 , n2 , n3 ]. n0 ,··· ,n3
For f (ψ; ξ ) :∈ A (V ⊕ V ) we are interested in the two and four legged contributions 1 to (:f :ξ ). Therefore we make the following −RK,C Definition VII.8. The projection P : (V ⊕ V ) −→ V A
A
:f (ψ; ξ ):ξ −→ f4,0 (ψ, 0) + f2,0 (ψ, 0),
where f (ψ; ξ ) =
n0 ,n1
fn0 ,n1 (ψ; ξ ) with fn0 ,n1 ∈ A[n0 , n1 ].
Definition VII.9. (i) An n–legged tail is a Grassmann function T (ψ; η) ∈ A[d, n]. We say that an n–legged tail T has at least d external legs if d≥2 T ∈ d ≥d A[d , n]. (ii) If T is a two–legged tail we define the two–legged tail T ◦ K by (T ◦ K)(ψ; η) = :T (ψ; η ):η :K0,2,0,2 (ψ; η , ξ , η):η dµC (η ) η
η K
T
ψ .
Observe that T ◦ K depends only on the part of K that has degree at most two in the variables ξ , η. Remark VII.10. A two–legged tail with two external legs is an end in the sense of Def. VI.5.i. If K(ψ; ξ, ξ , η) = U (ψ; ξ + ξ + η) − U (ψ; ξ + ξ ) for some even Grassmann function U (ψ; ξ ) then T ◦ K agrees with T ◦ Rung(U ) of Def. VI.5.iii.
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Lemma VII.11. Assume that K has degree at most two in the variables ξ , η. Let T be a two–legged tail. Then there exists a one–legged tail t1 with at least three external legs and a two–legged tail t2 with at least three external legs such that for any f (ψ; ξ ) ∈ A (V ⊕ V ) the following holds: Set f (ψ) = P RC (T )RK,C (:f :) − RC (T ◦ K)(:f :) − RC (T ◦ K, K2,0,0,2 )(:f :) −RC (t1 + t2 )(:f :) , where :f : is shorthand for :f :ξ . Then Nimpr (f ; α) ≤
26 J α 10
N (f ; 2α) N(K; 2α)2 N (T ; 2α).
If T has at least three external legs then P RC (T )RK,C (:f :) = P RC (T ◦ K)(:f :) + P RC (t1 + t2 )(:f :). Proof. By assumption, K(ψ; ξ, ξ , η) is of degree at least two in ψ, ξ , so RC (K, · · · , K) (:f :) (with K’s) is of degree at least 2 in ψ, ξ . Since T is two–legged, RC (T ) RC (K, · · · , K)(:f :) is of degree at least 2 − 2 + 2 = 2 (with the last +2 coming from the d ≥ 2 external legs of T ) in ψ and is independent of ξ . So, by Remark III.6, P RC (T )RK,C (:f :) = P RC (T )RC (K)(:f :) + 21 P RC (T )RC (K, K)(:f :) . Set
t11 (ψ; η) = t21 (ψ; η) =
:T (ψ; η ):η :K·,2,0,1 (ψ; η , ξ , η):η dµC (η ),
:T (ψ; η ):η .. K·,2,0,2 (ψ; η , ξ , η) −K0,2,0,2 (ψ; η , ξ , η) .. η dµC (η ).
Since K has degree at least four overall and T has degree at least two in ψ, t11 is a one–legged tail and t21 is a two–legged tail, both having at least three external legs. As K has degree at most two in the variables ξ , η and degree at least one in η and T has degree two in η and the definition of RC (K) Wick orders K with respect to ξ , P RC (T )RC (K)(:f :) = P RC (T )RC (K·,2,0,1 )(:f :) + P RC (T )RC (K·,2,0,2 )(:f :) = P RC (t11 + t21 )(:f :) + P RC (T ◦ K)(:f :) . Define the projection P : (V ⊕ V ) −→ V A
A
.f (ψ; ξ ) −→ f4,0 (ψ, 0) + f2,0 (ψ, 0).
As P and P only differ by Wick ordering in the ξ –argument, P RC (T )RC (K, K)(:f :) = P RC (T )RC (K, K)(:f :) = P R˜ C (T )RC (K, K)(:f :)(ψ; 0, 0, ξ ) ,
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so that we can apply Prop. VII.7. Modulo a term whose improved norm Nimpr is bounded 6 by 2α 10J N (f ; 2α) N (K; 2α)2 N (T ; 2α), ˜ K)(:f :) + P RC (K)(:f ˆ P R˜ C (T )RC (K, K)(:f :)(ψ; 0, 0, ξ ) = 2P RC (K, :) with ˜ K(ψ; ξ, ξ , η) = ˆ K(ψ; ξ, ξ , η) =
:T (ψ; η ):η :K(ψ; η , ξ , η):η dµC (η ), Tmix (ψ; η , η ) :K·,1,0,1 (ψ; η , ξ , η):η
:K·,1,0,1 (ψ; η , ξ , η):η dµC (η , η ). Here we have used the projection P to set ξ = 0 and we used that Tmix (ψ; η , η ) is of degree one in η and in η . Again, since K has degree at least four overall and T is of ˆ ˆ degree at least two in ψ, the tail K has at least six external legs, so that P RC (K)(:f :) = 0. Finally, since ˜ K)(:f :) to zero. • P sets the ξ ’s in the argument K of P RC (K, • K˜ is of degree at least two in ψ. • K·,0,0,1 is of degree at least three in ψ. • K is of degree at most two in ξ , η and degree at least one in η, we have ˜ K)(:f :) = P RC (K˜ ·,0,·,· , K·,0,0,2 )(:f :) + P RC (K˜ ·,0,·,· , K·,0,1,1 )(:f :) P RC (K, = P RC (K˜ ·,0,0,· , K·,0,0,2 )(:f :) + P RC (K˜ ·,0,1,· , K·,0,1,1 )(:f :) , and since • K is of degree at least four overall and K˜ is of degree at least two in ψ, we have ˜ K)(:f :) = P RC (K˜ 2,0,0,· , K2,0,0,2 )(:f :) P RC (K, +P RC (K˜ 2,0,1,· , K2,0,1,1 )(:f :) = P RC (K˜ 2,0,0,2 , K2,0,0,2 )(:f :) +P RC (K˜ 2,0,1,1 , K2,0,1,1 )(:f :) = P RC (T ◦ K, K2,0,0,2 )(:f :) + P RC (t22 )(:f :) , where
t22 (ψ; η) =
K˜ 2,0,1,1 (ψ; ξ, ξ , η) K2,0,1,1 (ψ; ξ, ξ , η) dµC (ξ )
is a two–legged tail with at least four external legs. Setting t1 = t11 and t2 = t21 +2t22 yields the main result. If T has at least three external legs, then RC (T )RC (K, · · · , K)(:f :) (with K’s) is of degree at least 2 − 2 + 3 = 2 + 1, so P RC (T )RK,C (:f :) = P RC (T )RC (K)(:f :) = P RC (t11 + t21 )(:f :) + P RC (T ◦ K)(:f :) .
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Lemma VII.12. Assume that K has degree at most two in the variables ξ , η. Let T1 , T2 be two–legged tails. Then there exists a one–legged tail t1 anda two–legged tail t2 , each with at least four external legs, such that for any f (ψ; ξ ) ∈ A (V ⊕ V ) the following holds: Set f (ψ) = P RC (T1 , T2 )RK,C (:f :) − RC (T1 ◦ K, T2 ◦ K)(:f :) − RC (t1 + t2 )(:f :) . Then, if α ≥ 2, Nimpr (f ; α) ≤
26 J α 12
N (f ; 2α) N(K; 2α)2 N (T1 ; 2α) N (T2 ; 2α).
Proof. Again, since K has degree at most two in the variables ξ , η and degree at least four in all variables, P RC (T1 , T2 )RK,C (:f :) = P RC (T1 , T2 )RC (K)(:f :) + 21 P RC (T1 , T2 )RC (K, K)(:f :) and P RC (T1 , T2 )RC (K)(:f :) = P RC (T1 , T2 )RC (K0,4,0,· )(:f :) = P RC (t11 + t21 )(:f :) , where
t11 (ψ; η) = t21 (ψ; η) =
:T1 (ψ; η )T2 (ψ; η ):η K0,4,0,1 (ψ; η , ξ , η) dµC (η ), :T1 (ψ; η )T2 (ψ; η ):η K0,4,0,2 (ψ; η , ξ , η) dµC (η )
are one– resp. two–legged tails with at least four external legs. Similarly P RC (T1 , T2 )RC (K, K)(:f :) = P RC (T1 , T2 )RC (K0,2,0,2 , K0,2,0,2 )(:f :) +P RC (T1 , T2 )RC (K0,2,1,1 , K0,2,1,1 )(:f :) . The second term is P RC (t22 )(:f :), where t22 (ψ; η) = :T1 (ψ; η )T2 (ψ; η ):η :K0,2,1,1 (ψ; η , ξ , η) ×K0,2,1,1 (ψ; η , ξ , η):η dµC (η ) is a two–legged tail with at least four external legs. To deal with the first term we use Prop. VII.1 to see that RC (T1 , T2 )RC (K0,2,0,2 , K0,2,0,2 )(:f :) = :T1 (ψ, η ):η :g(ψ; η ):η dµC (η ), where
g(ψ; η) = R˜ C (T2 )RC (K0,2,0,2 , K0,2,0,2 )(:f :).
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By Prop. VII.7, :g(ψ; ξ ):ξ is – modulo terms whose improved norm can be bounded by 26 J N (f ; 2α)N (K; 2α)2 N (T2 ; 2α) – equal to 2 RC (K0,2,0,2 , T2 ◦ K)(:f :). Therefore α 10 by Prop. II.33, modulo terms whose improved norm can be bounded by N(K; 2α)2 Nimpr (T1 ; α) N(T2 ; 2α), 1 2
26 J α 12
N (f ; 2α)
P RC (T1 , T2 )RC (K0,2,0,2 , K0,2,0,2 )(:f :) = P :T1 (ψ, η ):η RC (K0,2,0,2 , T2 ◦ K)(:f :)(ψ; η ) dµC (η ) = P RC (T1 ◦ K, T2 ◦ K)(:f :) .
Since Nimpr (T1 ; α) ≤ N (T1 ; α) ≤ N (T1 ; 2α), the lemma follows.
Lemma VII.13. Assume that K has degree at most two in the variables ξ , η. Let T be a one–legged tail with at least three external legs. Then there is a two–legged tail t2 with at least four external legs such that for all f (ψ; ξ ) ∈ A (V ⊕ V ), P RC (T )RK,C (:f :) = P RC (t2 )(:f :) . Proof. Since T is one–legged and K has degree at least four P RC (T ) RK,C = P RC (T ) RC (K)
= P RC (T ) RC Kd,1,0,1 + Kd,1,0,2 d≥2
d≥1
= P RC (T ) RC (K1,1,0,2 ) = P RC (t2 ) with
t2 (ψ; η) =
T (ψ; η ) K1,1,0,2 (ψ; η , ξ , η) dµC (η ).
Definition VII.14. The two–legged tails T (K) are recursively defined by T1 (K)(ψ; η) = K2,0,0,2 (ψ; ξ, ξ , η), T +1 (K) = T ◦ K for ≥ 1. Remark VII.15. (i) Clearly T (K) has two external legs. Using Prop. II.33 one proves by induction that for α ≥ 2,
N T (K) ≤
1 N (K) . α 2 −2
(ii) If K(ψ; ξ, ξ , η) = U (ψ; ξ + ξ + η) − U (ψ; ξ + ξ ) for some even Grassmann function U (ψ; ξ ) then, by Remark VII.10, T (K) = T (U ), where T (U ) was defined in Def. VI.8.ii. (iii) T (K) depends only on the part of K that has degree at most two in the variables ξ , η.
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Proposition VII.16. Assume that K has degree at most two in the variables ξ , η. For each ≥ 1 there exists a one–legged tail t1 and a two–legged tail t2 , each with at least three external legs such that for any f (ψ; ξ ) ∈ A (V ⊕ V ) the following holds: Set
RC T (K), T (K) (:f :) f (ψ) = P R K,C (:f :) − RC T (K) (:f :) − 21 , ≥1 max{ , }=
−RC (t1 + t2 )(:f :) . Then, if α ≥ 2 and N(K; 2α)0 < α 2 , Nimpr (f ; α) ≤ J
27 α 2 +6
N f ; 2α
N(K;2α) +1 1−
1 N(K;2α) α2
−1 .
Proof by induction on . Set NK = N (K; 2α). Since K has degree at most two in the variables ξ , η and degree at least four overall, for = 1, P RK,C = P RC (K·,0,0,2 + K·,0,0,1 ) +
1 2
P RC (K2,0,1,1 , K2,0,1,1 )
+ 21
P RC (K2,0,0,2 , K2,0,0,2 )
= P RC T1 (K) + P RC (t1 + t2 ) +
1 2
P RC T1 (K), T1 (K)
with t1 = K3,0,0,1 + K4,0,0,1 ,
t2 = K3,0,0,2 + K4,0,0,2 +
1 2
K2,0,1,1 (ψ; ξ, ξ , η)
×K2,0,1,1 (ψ; ξ, ξ , η) dµC (ξ ). Now assume that the statement of the lemma is true for . By the induction hypothesis there exist a one–legged tail t1 and a two–legged tail t2 , each with at least three external legs, such that P R +1 K,C (:f :) = P RK,C RK,C (:f :) differs from P RC T (K) RK,C (:f :) +
+RC (t1 + t2 ) RK,C (:f :)
1 2
RC T (K), T (K) RK,C (:f :)
, ≥1 max{ , }=
by a function g0 (ψ) with 27
Nimpr (g0 ) ≤
α 2 +6
J
≤
27 α 2 +6
J
≤
26 α 2 +8
J
+1 NK
−1 N f˜; 2α
1 NK α2 +1 NK −1 1− 12 NK α
1−
+2 NK 1− 12 NK α
NK 1 N 2α 2 1− 12 NK 2α
N f ; 2α ,
f ; 2α
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˜ where
:f : = RK,C (:f :). Here, we used Lemma III.8, with α replaced by 2α, to estimate N f˜; 2α . By Lemma VII.11, there exists a one–legged tail t11 and a two–legged tail t21 , with at least three external legs each, such that P RC T (K) RK,C (:f :) = P RC T +1 (K) (:f :) +P RC T +1 (K), T1 (K) (:f :) +P RC t11 + t21 )(:f :) + g1 (ψ) = P RC T +1 (K) (:f :) + 21 P RC T +1 (K), T1 (K) (:f :) + 21 P RC T1 (K), T +1 (K) (:f :) +P RC t11 + t21 )(:f :) + g1 (ψ) with Nimpr (g1 ) ≤ ≤
26 J N (f ; 2α) NK2 N (T (K); 2α) α 10 26 J N (f ; 2α) NK +2 . α 2 +8
1 Here, we used Remark VII.15 to bound N (T (K); 2α) by α 2 −2 NK . Similarly, for , ≥ 1 with max{ , } = , by Lemma VII.12, there exists a ( , ) ( , ) one–legged tail t12 and a two–legged tail t22 with at least three external legs such that P RC T (K), T (K) RK,C (:f :) = P RC T +1 (K), T +1 (K) (:f :) ( , ) ( , ) +P RC t12 + t22 )(:f :) + g2, , (ψ)
with Nimpr (g2, , ) ≤
26 J α 12
≤ J
N (f ; 2α) NK2 N (T (K); 2α) N (T (K); 2α)
26 α 2( + )+8
+2
N (f ; 2α) NK +
.
By Lemma VII.13, there exists a two–legged tail t23 with at least four external legs such that
P RC (t1 ) RK,C (:f :) = P RC (t23 )(:f :) . Finally, by Lemma VII.11
P RC (t2 ) RK,C (:f :) = P RC (t2 ◦ K + t14 + t24 )(:f :) , where t14 , t24 are one– resp. two–legged tails with at least three external legs. Combining the results above, we see that P R +1 K,C (:f :) = f +1 (ψ) + P RC T +1 (K) (:f :)
+ 21 RC T (K), T (K) (:f :) + RC (t1 + t2 )(:f :) , ≥1 max{ , }= +1
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with one– resp. two–legged tails t1 , t2 with at least three external legs, and f +1 = g0 + g1 + 21 g2, , . , ≥1 max{ , }=
By the triangle inequality Nimpr (f +1 ) ≤ Nimpr (g0 ) + Nimpr (g1 ) +
≤
26 α 2 +8
1 2
, ≥1 max{ , }=
J N (f ; 2α) NK +2
1 1−
26
≤
α 2 +8
≤
α 2 +8
27
J N (f ; 2α) NK +2 J N (f ; 2α)
1 1−
+2 NK
1−
1 NK α2
1 NK α2
1 NK α2
Nimpr (g2, , )
+ 1 + +
=1
NK 2 α
∞ NK =0
α2
.
Corollary VII.17. Let K(ψ; ξ, ξ , η) be an even Grassmann function that has degree at most two in the variables ξ , η, degree at least one in η and degree at least four in the variables ψ, ξ, ξ , η. Furthermore let f (ψ; ξ ) be a Grassmann function of degree at least four in the variables ψ, ξ . Set ∞
1 h(ψ) = P (:f :) − RC T (K) (:f :) − RK,C =0
1 −2 RC T (K), T (K) (:f :) . , ≥1
If α ≥ 2 and N (K; 2α)0 <
α2 2 ,
then
Nimpr (h) ≤
27 J α8
N (f ; 2α)
N(K;2α)2 . 1− 22 N(K;2α) α
Proof. By Prop. VII.16, Nimpr (h) ≤
∞ =1
Nimpr (f ) ∞
≤
27 J α8
=
27 J α8
N (f ; 2α) N(K; 2α)2 1 −
=
27 J α8
1− 1 N(K;2α) N (f ; 2α) N(K; 2α)2 α22 1− 2 N(K;2α)
≤
27 J α8
N(K;2α)2 . 1− 22 N(K;2α)
N (f ; 2α) N(K; 2α)2
=1
N(K;2α) 1 α 2 1− 12 N(K;2α) α
N(K;2α) 1 α 2 1− 12 N(K;2α) α
α
N (f ; 2α)
α
−1 −1
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Proof of Theorem VI.10 in the case D = 0. Set K(ψ; ξ, ξ , η) = Uˆ (ψ; ξ + ξ + η) − Uˆ (ψ; ξ + ξ ). By Theorem III.2 and Prop. III.5, 1 (:fˆ:) . Pf = P SU,C (:fˆ:) = P −RK,C Observe that K has degree at least one in η and four overall and that, by part i of Remark II.24 (twice), N (K; 2α)0 ≤ N (Uˆ (ψ; ξ + ξ + η); 2α)0 ≤ N (Uˆ (ψ; ξ ); 8α)0 < α8 . Decompose K(ψ; ξ, ξ , η) = K (ψ; ξ, ξ , η) + K (ψ; ξ, ξ , η), where K has degree at most two in the variables ξ , η and K has degree at least three in the variables ξ , η. By Cor. VII.5 and Cor. VII.17 there exists a Grassmann function g(ψ) with Nimpr (g) ≤
25 J α6
≤
25 J α6
≤
25 J α6
≤
25 J α6
≤
25 J α6
N fˆ; 2α N (K; 2α)
1 1− α6 N(K;2α)
+
4 N(K;2α) α2 1− 22 N(K;2α) α
1+ 2 N(K;2α) N fˆ; 2α N (K; 2α) α6 1− α N(K;2α)
1 N fˆ; 2α N (K; 2α) 6 1 1− α N(K;2α) 1− α2 N(K;2α)
N fˆ; 2α N (K; 2α) 8 1
N fˆ; 2α
1− α N(K;2α)
N(Uˆ ;8α) 1− α8 N(Uˆ ;8α)
and ∞
Pf = P fˆ(ψ, 0) + RC T (K) (:fˆ:) + =1
1 2
,
RC T (K), T (K) (:fˆ:) + g.
We have used that P :fˆ(ψ, ξ ): = P fˆ(ψ, 0) and T (K ) = T (K). Since fˆ and K have degree at least four overall,
P RC T (K) (:fˆ:) = T (K) ◦ T1 (fˆ) (where the ◦ composition was defined in Def. VI.5.ii) and by Remark VI.9, P RC T (K), T (K) (:fˆ:) = T (K) ◦ Rung(fˆ) ◦ T (K). Furthermore, by part (ii) of Remark VII.15, T (K) = T (Uˆ ). Thus Pf = P fˆ(ψ, 0) +
∞ =1
T (Uˆ ) ◦ T1 (fˆ) +
1 2
, ≥1
T (Uˆ ) ◦ Rung(fˆ) ◦ T (Uˆ ) + g.
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VIII. The Enlarged Algebra The estimate of Theorem IV.1 on W (ψ), defined by :W (ψ):ψ,D = C (:W :ψ,C+D ) was proven in the following way. We applied the results of Theorem II.28, combined with the estimates on Wick ordering (Cor. II.32) to get estimates on W (ψ) = C (:W :ψ,C+D ) = C ( .. :W :ψ,D .. ψ,C ) in terms of the norm of W . Then we used Cor. II.32 again to estimate the norm of W in terms of the norm of W . The transition from W to W = :W :ψ,−D creates new two and four legged vertices, whose improved norm cannot be estimated by the technique of Sect. VI. This transition also creates new ladder diagrams. The same difficulty would occur if we tried to reduce the general case of Theorem VI.10 to the special case D = 0 by Wick ordering SU,C (f ) at the end of the construction. Therefore we monitor the Wick ordering with respect to D throughout the construction. To do this we introduce fields ζ, ζ , ϕ for these Wick contractions and use an analogue of the operator RK,C of Def. III.4. Definition VIII.1. (i) Let K¯ 1 (ψ; ζ, ζ , ϕ; ξ, ξ , η), · · · , K¯ (ψ; ζ, ζ , ϕ; ξ, ξ , η) be even Grassmann functions with K¯ i (ψ; ζ, ζ , ϕ; ξ, ξ , 0) = 0. For a Grassmann function g(ψ; ζ ; ξ ) we define . Q(K¯ 1 , · · · , K¯ )(g)(ψ; ζ ; ξ ) = .
. . K¯ (ψ; ζ, ζ , ϕ; ξ, ξ , η) . . ϕ,D . i . ζ ,D . . ξ ,C
i=1
:g(ψ; ζ + ϕ; η):ϕ,D
η,C
. dµD (ζ , ϕ) dµC (ξ , η) . ξ,C .
The structure of Q is illustrated in the figure ζ K¯ 1 ξ ζ ϕ η
g
K¯ 2
ϕ η
η
K¯
ζ ξ ψ ζ ξ ψ ζ ξ ψ ψ
¯ ¯ (ii) For a Grassmann function K(ψ; ζ, ζ , ϕ; ξ, ξ , η) with K(ψ; ζ, ζ , ϕ; ξ, ξ , 0) = 0 we define the operator QK¯ by QK¯ (g) =
∞ =1
1 !
¯ · · · , K)(g). ¯ Q(K,
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In this definition, the fields ξ, ξ , η involving the covariance C are treated in the same way as in Sect. III. The fields ϕ are analogous to the fields η and describe Wick contractions between g and the Ki . The fields ζ are analogous to the fields ξ and describe Wick contractions among Ki . Similar to the ξ fields, the fields ζ are not integrated out and are later used for Wick contractions in further applications of QK¯ . In contrast to the ξ –fields, however, also Wick contractions between fields of g and kernels appearing in future applications of QK¯ have to be allowed. This is the reason for the field ζ that appears in the term g(ψ; ζ + ϕ; η):ϕ,D in Def. VIII.1.i. Definition VIII.1.i of Q was chosen so that :Q(K¯ 1 , · · · , K¯ )(g):ζ,D = R(:K1 :ζ,D , · · · , :K :ζ,D )(:g:ζ,D ) when K¯ i (ψ; ζ, ζ , ϕ; ξ, ξ , η) = Ki (ψ; ζ + ζ + ϕ; ξ, ξ , η). This formula is proven in Lemma VIII.12.i.1 Consequently :QnK¯ (h):ζ,D = Rn:K:ζ,D (:h:ζ,D ) for n = 0, 1, 2, · · · . From this one can deduce, as in Prop. VIII.9, that . SU,C (:f :ψ,D ) = .
∞
. QnK¯ (h)(ψ; 0; ξ ) dµC (ξ ) . ψ,D ,
n=0
where, given an even Grassmann function Uˆ (ψ; ξ ) and a Grassmann function f (ψ; ξ ), U (ψ; ξ ) = .. Uˆ (ψ; ξ ) ..
ξ,C ψ,D
,
h(ψ; ζ ; ξ ) = f (ψ + ζ ; ξ ), ¯ K(ψ; ζ, ζ , ϕ; ξ, ξ , η) = Uˆ (ψ + ζ + ζ + ϕ; ξ + ξ + η) −Uˆ (ψ + ζ + ζ + ϕ; ξ + ξ ).
However, this formula would not be good enough to get an estimate on SU,C (:f :C+D ). At each application of QK¯ , the passage from g(ψ; ζ ; ξ ) to g(ψ; ζ + ϕ; η), that is the separation between the D- fields that are to be Wick contracted at the present step and those that are to be Wick contracted at a future step, leads to a deterioration in the norm:
N g(ψ; ζ + ϕ; η); α ≤ N g(ψ; ζ ; ξ ); 2α by Remark II.24. In iterated applications of QK¯ this deterioration in the norm would build up excessively. For this reason, we avoid Wick contractions at most steps of the construction and perform them only before and during steps in which overlapping loops are exploited. For bookkeeping of the partially Wick ordered fields in intermediate steps we introduce an enlarged algebra. 1
To see this, choose f = :g:ζ,D .
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VIII.1. Definition of the Enlarged Algebra. Definition VIII.2. (i) A Z2 –graded vector space is a complex vector space E, together with a decomposition E = E+ ⊕ E− . The elements of E+ are called even, the elements of E− odd. A graded ∞vector space is a complex vector space E, together with a decomposition E = m=0 Em . Every graded vector space is considered as a Z2 –graded vector space with E+ =
Em
E− =
r even
Em .
r odd
(ii) If E is a (Z2 –) graded vector space, the tensor algebra T (E) has a natural (Z2 –) grading. The symmetric superalgebra over E is denoted S(E) and is defined as the quotient of T (E) by the two sided ideal I (E) generated by a⊗b−b⊗a a⊗b+b⊗a
with a ∈ E+ or b ∈ E+ , with a, b ∈ E− .
It is a superalgebra, and it is a graded superalgebra if E is a graded vector space (see [BS]). Example VIII.3. Let E be a complex vector space. Setting E+ = {0}, E− = E, we give E the structure of a Z2 –graded vector space in which all elements are odd. Then the symmetric superalgebra over E is the Grassmann algebra E over E. Setting E+ = E, E− = {0}, S(E) is the classical symmetric algebra S(E) over E. Remark VIII.4. There is a natural isomorphism ι between S(E) and the algebra S(E+ )⊗ E− , constructed in the following way: Observe that T (E) ∼ =
⊗m1 ⊗n1 ⊗mr ⊗nr E+ ⊗ E− ⊗ · · · ⊗ E+ ⊗ E− .
m1 ,n1 ,··· ,mr ,nr ≥0 n1 ,m2 ,··· ,nr−1 ,mr ≥1
Define the algebra homomorphism ι : T (E) → S(E+ ) ⊗
E− by
ι v1+ ⊗ v1− ⊗ · · · ⊗ vr+ ⊗ vr− = v1+ · . . . · vr+ ⊗ v1− · . . . · vr− ⊗mi ⊗mi for vi+ ∈ E+ , vi− ∈ E− . Clearly I (E) lies in the kernel of ι , so ι induces an algebra homomorphism ι : S(E) → S(E+ ) ⊗ E− . One can check that ι is an isomorphism.
As S(E) and E are subalgebras of the tensor algebra T (E), S(E) can also be viewed as a subalgebra of T (E) ⊗ T (E) ∼ = T (E). Definition VIII.5. (i) Let E be a complex vector space. The symmetric superalgebra S( E) over E (considered as a graded vector space) is called the enlarged algebra E over E. It is a graded superalgebra.
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(ii) Multiplication f1 ⊗ · · · ⊗ fr −→ f1 · · · fr
defines an algebra homomorphism from the tensor algebra T ( E) to E. The ideal I ( E) of part (ii) of Def. VIII.2 lies in the kernel of this homomorphism. Therefore it induces an algebra homomorphism Ev : E −→ E called the evaluation map. It is graded, that is m
Ev(x) ∈ E when x ∈ E m. (ii) If A is a superalgebra, we define the enlarged algebra over E with coefficients in A as the tensor product E AE =A⊗ in the sense of Def. II.1.iv. The evaluation map extends by A-linearity to an algebra homomorphism Ev : E. A E −→ A
Remark VIII.6. (i) E is identified with the subspace ( V ) of V. 1
(ii) Ev x · y = Ev x · Ev(y) = Ev Ev(x) · y for allx, y ∈ V .
(iii) By Remark VIII.4, E = S E as a graded
there is a natural inclusion of subalgebra of T E ⊂ T T (E) ∼ T (E). = VIII.2. Norm Estimates for the Enlarged Algebra. As in Subsect. III.3, let A = A V be the Grassmann algebra in the variables ψi with coefficients in A. Furthermore let E be a copy of V with generators ζi corresponding to the fields ψi . We will use the enlarged algebra A= A E over E with coefficients in A . Elements of A will be written as Grassmann functions f (ψ; ζ ). They are linear combinations of monomials of the form ψi1 · · · ψin (ζj (1) · · · ζj (1) ) ⊗ (ζj (2) · · · ζj (2) ) ⊗ · · · ⊗ (ζj (r) · · · ζj (r) ). 1
p1
p2
1
1
pr
∼ (V ⊕ E) maps such a monomial to A E = A . As in Remark VIII.6, the Grassmann algebra
The evaluation map Ev : A → ψi1 · · · ψin ζj (1) · · · ζj (1) ζj (2) · · · ζj (r) p1 pr 1 1 A E of Grassmann functions g(ψ; as a subspace of A. ζ ) is viewed The evaluation map Ev : A → A E ∼ = A (V ⊕ E) extends to a map Ev : (V (1) ⊕ · · · ⊕ V (r) ) −→ (E ⊕ V (1) ⊕ · · · ⊕ V (r) ) A A ∼ (V ⊕ E ⊕ V (1) ⊕ · · · ⊕ V (r) ), = A
where V (j ) , j = 1, · · · , r are copies of V with generators ξ (j ) .
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Definition VIII.7. By Remark VIII.6.iii, A can be viewed as a graded subalgebra of A ⊗ T (E) ∼ = A ⊗ T (V ). Therefore the family · of seminorms on the spaces ⊗n Am ⊗ V , introduced in Subsect. III.3, induces a family of symmetric seminorms on the spaces Am ⊗ V ⊗n , which we again denote by · . This definition extends the definition of · given in Subsect. III.3. Also, c is a contraction bound and b an integral bound for the covariance C with respect to these norms. Lemma VIII.8. Let f ∈
A (V1
⊕ · · · ⊕ Vr ). Then
N Ev(f ); α ≤ N (f ; α). Proof. Ev(f ) is obtained from f by antisymmetrization.
VIII.3. Schwinger Functionals over the Extended Algebra. Recall that for any even U (ψ; ξ ) ∈ V , the Schwinger functional with respect to U and C is the map from A A V to A given by SU,C (f )(ψ) =
1 Z
e
U (ψ;ξ )
Z=
f (ψ; ξ ) dµC (ξ ),
eU (ψ;ξ ) dµC (ξ ).
Also recall that for an even Grassmann function K(ψ, ζ ; ξ, ξ , η) in the variables ξ, ξ , η over the extended algebra A and any Grassmann function f (ψ; ζ ; ξ ) ∈ A V , . RK,C (f ) = .
. . e:K(ψ,ζ ;ξ,ξ ,η):ξ ,C − 1 . . . η,C f (ψ; ζ ; η) dµC (ξ ) dµC (η) . ξ,C .
Proposition VIII.9. Let U (ψ; ξ ) = .. Uˆ (ψ; ξ ) .. ψ,D ξ,C
be even and set K(ψ, ζ ; ξ, ξ , η) = :Uˆ (ψ + ζ ; ξ + ξ + η):ζ,D − :Uˆ (ψ + ζ ; ξ + ξ ):ζ,D . Let f (ψ; ξ ) ∈
A
V and set f˜(ψ; ζ ; ξ ) = :f (ψ + ζ ; ξ ):ζ,D ∈
. SU,C :f :ψ,D (ψ) = . Observe that
1
−RK,C
(f˜) ∈
AV,
Ev
1
−RK,C
so that
A V . Then
. f˜ dµC (ξ ) dµD (ζ ) .
1
−RK,C
(f˜) dµC (ξ ) ∈ A.
For the proof of this proposition we use Lemma VIII.10. Let f (ψ; ξ ) be a Grassmann function and set f˜(ψ; ζ ; ξ ) = :f (ψ + ζ ; ξ ):ζ,D . Furthermore, let n ≥ 1.
ψ,D
.
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(i) Let 1 , · · · , n ≥ 1 and let Kˆ j i (ψ; ξ, ξ , η),
j = 1, · · · n, i = 1, · · · , j
be even Grassmann functions with coefficients in A. Let Kj i (ψ; ζ ; ξ, ξ , η) = :Kˆ j i (ψ + ζ ; ξ, ξ , η):ζ,D considered as Grassmann functions in the variables ξ, ξ , η with coefficients in the enlarged algebra A. Then n
RC :Kˆ j 1 :ψ,D , · · · , :Kˆ j j :ψ,D :f :ψ,D j =1
. = .
Ev
n j =1
. RC Kj 1 , · · · , Kj j (f˜) dµD (ζ ) . ψ,D .
On the left-hand side of the equation above, the operator RC (defined in (III.2) ) is considered over the algebra A , while on the right-hand side it is considered over the extended algebra A. ˆ (ii) Let K(ψ; ξ, ξ , η) be an even Grassmann function and ˆ K(ψ; ζ ; ξ, ξ , η) = :K(ψ + ζ ; ξ, ξ , η):ζ,D . Then
Rn ˆ
:K:ψ,D ,C
Proof.
. :f :ψ,D = .
. Ev RnK,C (f˜) dµD (ζ ) . ψ,D .
(i) Let h(ψ; ξ ) = :g(ψ; ξ ):ψ,D =
n j =1
RC :Kˆ j 1 :ψ,D , · · · , :Kˆ j j :ψ,D :f :ψ,D .
By part (ii) of Prop. A.2, in the Grassmann algebra over A with variables ψ, ζ, ξ, ξ , η, n
h(ψ + ζ ; ξ ) = :g(ψ + ζ ; ξ ):ζ,D = RC Kj 1 , · · · , Kj j :f˜:ζ,D . j =1
Hence, by the construction of A and Ev, h(ψ + ζ ; ξ ) = :g(ψ + ζ ; ξ ):ζ,D = Ev Therefore
n j =1
RC Kj 1 , · · · , Kj j :f˜:ζ,D .
g(ψ; ξ ) =
:g(ψ + ζ ; ξ ):ζ,D dµD (ζ )
=
Ev
n j =1
RC Kj 1 , · · · , Kj j (f˜) dµD (ζ ).
(ii) follows from part (i) and Remark III.6.
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Proof of Proposition VIII.9. Set ˆ :K(ψ; ξ, ξ , η):ψ,D = :Uˆ (ψ; ξ + ξ + η):ψ,D − :Uˆ (ψ; ξ + ξ ):ψ,D . By Prop. III.5, with Uˆ (ξ ) replaced by :Uˆ (ψ; ξ ):ψ,D and K(ξ, ξ , η) replaced by ˆ :K(ψ; ξ, ξ , η):ψ,D , and part ii of Lemma VIII.10,
n :f :ψ,D = Rn ˆ :f :ψ,D RU,C :K:ψ,D ,C
. . = . Ev RnK,C (f˜) dµD (ζ ) . ψ,D . By Theorem III.2,
SU,C :f :ψ,D (ψ) =
n≥0
. = . . = .
n :f :ψ,D dµC (ξ ) RU,C
Ev
Ev
n≥0
. RnK,C (f˜) dµC (ξ ) dµD (ζ ) . ψ,D 1
−RK,C
. f˜ dµC (ξ ) dµD (ζ ) . ψ,D .
We use the extended algebra to give VIII.4. A Second Proof of Theorem IV.1. Proposition VIII.11. Let Uˆ (ψ; ξ ), fˆ(ψ; ξ ) ∈
A
V with Uˆ even. Set
U (ψ; ξ ) = .. Uˆ (ψ; ξ ) .. ψ,D , ξ,C
f (ψ; ξ ) = .. fˆ(ψ; ξ ) .. ψ,D . ξ,C
Assume that c is a contraction bound for the covariance C and b is an integral bound 2 for C and for D and that N (Uˆ ; 8α)0 < α4 . Then SU,C (f ) exists. If SU,C (f ) = :f (ψ):ψ,D , then
N f (ψ) − fˆ(ψ; 0); α ≤
2 α2
N (fˆ; 4α)
N(Uˆ ;8α) . 1− 42 N(Uˆ ;8α) α
Proof. As above, set K(ψ; ζ ; ξ, ξ , η) = :Uˆ (ψ + ζ ; ξ + ξ + η):ζ,D − :Uˆ (ψ + ζ ; ξ + ξ ):ζ,D , f˜(ψ; ζ ; ξ ) = :fˆ(ψ + ζ ; ξ ):ζ,D . By Remark II.24 and Cor. II.32, N (K) ≤ N (Uˆ ; 8α), N (f˜) ≤ N (fˆ; 4α).
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By Cor. III.9, applied with A replaced by A, 1
−RK,C
(:f˜:ξ,C ) − :f˜:ξ,C = :g(ψ, ζ ; ξ ):ξ,C
with N (g) ≤
2 α2
N (f˜)
N (K) 1− 42 N (K)
=
α
2 α2
N (f˜)
N(K) 1− 42 N(K)
≤
α
2 α2
N (fˆ; 4α)
N(Uˆ ;8α) . 1− 42 N(Uˆ ;8α) α
Consequently, by Lemma II.31, 1 ˜ ˜ :g(ψ; ζ ; ξ ):ξ,C dµC (ξ ) N (:f :ξ,C ) − :f :ξ,C dµC (ξ ) = N − RK,C
≤ N g(ψ; ζ ; ξ ) ≤ Observe that
2 α2
N (fˆ; 4α)
N(Uˆ ;8α) . 1− 42 N(Uˆ ;8α) α
:f˜:ξ,C dµC (ξ ) = f˜(ψ; ζ ; 0) = :fˆ(ψ + ζ ; 0):ζ,D .
Hence, by Prop.VIII.9, with f (ψ; ξ ) replaced by :fˆ(ψ; ξ ):ξ,C , Lemma II.31 and Lemma VIII.8,
N f − fˆ(ψ, 0) = N f − fˆ(ψ, 0) 1 Ev = N (:f˜:ξ,C ) dµC (ξ ) −RK,C −:fˆ(ψ + ζ, 0):ζ,D dµD (ζ ) 1 ≤ N Ev (:f˜:ξ,C ) dµC (ξ ) − :fˆ(ψ + ζ, 0):ζ,D −RK,C 1 ≤ N (:f˜:ξ,C ) − :f˜:ξ,C dµC (ξ ) −RK,C ≤
2 α2
N (fˆ; 4α)
N(Uˆ ;8α) . 1− 42 N(Uˆ ;8α) α
Proof of Theorem IV.1, again. By Remark III.1.i, with W replaced by :W :C+D , :W (ψ) − W (ψ):ψ,D = C (:W :C+D )(ψ) − :W (ψ):ψ,D 1
= StU,C (U ) − :W (ψ):ψ,D dt 0
where
U (ψ; ξ ) = .. W (ψ + ξ ) ..
ξ,C ψ,D
.
mod A0 ,
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Define ft and fˆ by fˆ(ψ; ξ ) = W (ψ + ξ ), = StU,C (U ).
:ft (ψ):ψ,D Then W (ψ) − W (ψ) =
1 0
ft (ψ) − fˆ(ψ, 0) dt
mod A0 .
By Prop. VIII.11, with Uˆ t (ψ; ξ ) = tW (ψ + ξ ), N(W − W ; α) ≤ max N (ft (ψ) − fˆ(ψ, 0); α) 0≤t≤1
N (fˆ; 4α)
≤
2 max α 2 0≤t≤1
≤
N(W ;16α)2 2 α 2 1− 42 N(W ;16α)
N(Uˆ t ;8α) 1− 42 N(Uˆ t ;8α) α
α
for all W with N(W ; 16α)0 <
α2 4 .
Replacing α by 2α gives
N (W − W ; 2α) ≤
N(W ;32α)2 1 2α 2 1− 12 N(W ;32α) α
for all W with N(W ; 32α)0 < α 2 .
VIII.5. The Operator Q. Lemma VIII.12. Let f (ψ, ζ ; ξ ) be a Grassmann function over the extended algebra A and set :f (ψ, ζ ; ξ ):ζ,D = Ev(f ). (i) Let Ki (ψ, ζ ; ξ, ξ , η), i = 1, · · · , be Grassmann functions with Ki (ψ, ζ ; ξ, ξ , 0) = 0. Set K¯ i (ψ; ζ, ζ , ϕ; ξ, ξ , η) = Ki (ψ; ζ + ζ + ϕ; ξ, ξ , η). Then Ev RC (:K1 :ζ,D , · · · , :K :ζ,D )(f ) = .. Q(K¯ 1 , · · · , K¯ )(f ) .. ζ,D . (ii) Let K(ψ, ζ ; ξ, ξ , η) be a Grassmann function with K(ψ, ζ ; ξ, ξ , 0) = 0. Set ¯ K(ψ; ζ, ζ , ϕ; ξ, ξ , η) = K(ψ; ζ + ζ + ϕ; ξ, ξ , η). Then, when R:K:ζ,D ,C is considered as an operator over A, Ev R:K:ζ,D ,C (f ) = .. QK¯ (f ) .. ζ,D .
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Proof. Applying Remark VIII.6 and Cor. A.3 to the variable ζ we see that
Ev :K1 (ψ, ζ ; ξ, ξ , η):ζ,D · · · · · :K (ψ, ζ ; ξ, ξ , η):ζ,D · f (ψ, ζ ; η)
= Ev :K1 :ζ,D · · · · · :K :ζ,D · Evf (ψ, ζ ; η)
= Ev :K1 :ζ,D · · · · · :K :ζ,D · :f :ζ,D . = . .. :Ki (ψ, ζ + ζ + ϕ; ξ, ξ , η):ζ ,D .. ϕ,D :f (ψ, ζ i=1
. +ϕ; η):ϕ,D dµD (ζ , ϕ) . ζ,D . Applying each of the following operations to both sides of this equation: • Wick order each :Ki :ζ,Dwith respect to C in the ξ variable • Wick order the product i .. Ki .. ξ ,C with respect to C in the η variable ζ,D • Integrate using · dµC (ξ , η) • Wick order the result with respect to C in the ξ variable yields Ev RC (:K1 :ζ,D , · · · , :K :ζ,D )(f ) on the left and .. Q(K¯ 1 , · · · , K¯ )(f ) .. ζ,D on the right. (ii) follows from part (i), the definition of QK¯ and Remark III.6. As in Subsect. VII.1, we define for Grassmann functions K¯ 2 (ψ; ζ, ζ , ϕ; ξ, ξ , η),· · · , ˜ K¯ 2 , · · · , K¯ ) by K¯ (ψ; ζ, ζ , ϕ; ξ, ξ , η) and f (ψ, ζ ; ξ ) the Grassmann function Q( ˜ K¯ 2 , · · · , K¯ )(f )(ψ; ζ, ζ , ϕ; ξ, ξ , η) Q( . . K¯ (ψ; ζ, ζ + ζ , ϕ ; ξ, ξ + ξ , η ) . . = . i . ζ ,D . ϕ ,D . ξ ,C
i=2
:f (ψ, ζ + ϕ + ϕ ; η + η ):
ϕ ,D
η ,C
dµD (ζ , ϕ ) dµC (ξ , η ).
We have, as in Prop. VII.1 Proposition VIII.13. Let K¯ 1 (ψ; ζ, ζ , ϕ; ξ, ξ , η), · · · , K¯ (ψ; ζ, ζ , ϕ; ξ, ξ , η) be Grassmann functions with K¯ i (ψ; ζ, ζ , ϕ; ξ, ξ , 0) = 0. Then for any Grassmann function f (ψ, ζ, ξ ), Q(K¯ 1 , · · · , K¯ )(f )(ψ, ζ ; ξ ) . . K¯ (ψ; ζ, ζ , ϕ; ξ, ξ , η) . = . . 1 . ζ ,ϕ;D ξ ,η;C
. . Q( . . ˜ K¯ 2 , · · · , K¯ )(f )(ψ; ζ, ζ , ϕ; ξ, ξ , η) . ζ ,ϕ;D dµD (ζ , ϕ) dµC (ξ , η) . ξ,C . ξ ,η;C
Lemma VIII.14. Let K¯ i (ψ; ζ, ζ , ϕ; ξ, ξ , η), i = 1, · · · , be Grassmann functions such that K¯ i (ψ; ζ, ζ , ϕ; ξ, ξ , 0) = 0. Furthermore let f (ψ, ζ, ξ ) be any Grassmann function. Set :f1 (ψ, ζ, ξ ):ξ,C =
1 !
Q(K¯ 1 , · · · , K¯ )(:f :ξ,C )(ψ, ζ, ξ ).
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Then N (f1 ; α) ≤ N (f1 (ψ, 0, ξ ); α) ≤ N Nimpr
1 ( −1)!
˜ K¯ 2 , · · · , K¯ )(:f :ξ,C ); α) ≤ Q(
1 ( −1)!
˜ K¯ 2 , · · · , K¯ )(:f :ξ,C ); α) ≤ Q(
1 α
N (f ; 2α)
1 α
N (f ; α)
N (K¯ i ; α),
i=1
N (K¯ i ; α),
i=1
1 α −1 1 α −1
N (f ; 4α)
N (K¯ i ; 2α),
i=2
Nimpr (f ; 4α)
Nimpr (K¯ i ; 2α).
i=2
Proof. Set f˜(ψ; ζ, ϕ , ξ ) = f (ψ; ζ + ϕ , ξ ) and :f˜1 (ψ; ζ, ζ , ϕ, ϕ ; ξ ):ξ,C = :f˜2 (ψ; ζ, ζ , ϕ, ϕ ; ξ ):ξ,C =
1 ! 1 !
. R (:K¯ : , · · · , :K¯ : ) :f˜: . . C 1 ζ ,D ζ ,D ξ,C . ϕ,ϕ ;D ,
RC (K¯ 1 , · · · , K¯ ) :f˜:ξ,C ,
where RC is considered as an operator over the Grassmann algebra with coefficients in A, ˜ generated by ψi , ζi , ζi , ϕi , ϕi . Then f1 (ψ, ζ, ξ ) = f1 (ψ; ζ, ζ , ϕ, ϕ; ξ ) dµD (ζ , ϕ) , and consequently by Lemma II.31 (twice, with C replaced by D), Prop. III.7 and Remark II.24, N(f1 ) ≤ N (f˜2 ) ≤
1 N (f˜) α
N (K¯ i ) ≤
i=1
1 α
N (f ; 2α)
N (K¯ i ; α).
i=1
When ζ is set to zero,
RC (K¯ 1 , · · · , K¯ ) :f˜:ξ,C ζ =0
1 = ! RC (K¯ 1 , · · · , K¯ ) :f :ξ,C
:f˜2 (ψ; 0, ζ , ϕ, ϕ ; ξ ):ξ,C =
1 !
ζ =0
so
N f1 (ψ, 0, ξ ) ≤ N (f˜2 (ψ, 0, ξ )) ≤ The proofs of the other inequalities are similar.
1 N (f ) α
N (K¯ i ).
i=1
IX. Overlapping Loops Created by the Second Covariance In this section we prove Theorem VI.10 in the general case. We assume that A is a superalgebra in which all elements have degree zero and that · and · impr are two systems of symmetric seminorms on the spaces A ⊗ V ⊗n such that the covariances (C, D) have improved integration constants c, b, J for these families of seminorms.
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IX.1. Implementing Overlapping Loops. Proposition IX.1. Let K(ψ; ζ, ζ , ϕ; ξ, ξ , η) be an even Grassmann function such that K(ψ; ζ, ζ , ϕ; ξ, ξ , 0) = 0. Decompose K = K + K , where K has degree at most two in the variables ζ , ξ , ϕ, η and K has degree at least three in these variables. Let f (ψ; ζ ; ξ ) be a Grassmann function, and set :g :ξ,C = QK (:f :ξ,C ).
:g:ξ,C = QK (:f :ξ,C ), Then, if α ≥ 2 and N (K; 2α)0 < α, Nimpr (g − g ; α) ≤
25 J α6
N (f ; 4α)
N(K;2α) . 1− α1 N(K;2α)
The proof is analogous to the proof of Prop. VII.4 and is given following Lemma IX.2. Let K(ψ; ζ, ζ , ϕ; ξ, ξ , η) be an even Grassmann function that satisfies K(ψ; ζ, ζ , ϕ; ξ, ξ , 0) = 0. Decompose K = K + K , where K has degree at most two in the variables ζ , ξ , ϕ, η and K has degree at least three in these variables. Let each of the functions K (1) , · · · , K ( ) be either K or K , and assume that at least one of them is equal to K . Let f (ψ; ζ ; ξ ) be a Grassmann function, and set 1 !
Q(K (1) , · · · , K ( ) )(:f :)(ψ; ζ ; ξ ) = :f (ψ; ζ ; ξ ):ξ,C ,
where :f : is shorthand for :f (ψ; ζ ; ξ ):ξ,C . Then, if α ≥ 2, Nimpr (f ; α) ≤
25 α +5
J N (f ; 4α) N (K; 2α) .
Proof. We may assume that K (1) = K . Set ˜ (2) , · · · , K ( ) )(:f :). g(ψ; ζ, ζ , ϕ; ξ, ξ , η) = Q(K By Lemma VIII.14 1 ( −1)!
By Prop. VIII.13, f (ψ; ζ ; ξ ) =
N (g) ≤
1 !
1 α −1
N (f ; 4α) N (K; 2α) −1 .
. K (ψ; ζ, ζ , ϕ; ξ, ξ , η) . . . ζ ,ϕ;D ξ ,η;C
. g(ψ; ζ, ζ , ϕ; ξ, ξ , η) . . . ζ ,ϕ;D dµD (ζ , ϕ) dµC (ξ , η). ξ ,η;C
Proposition VI.4 implies that Nimpr (f ) ≤ ≤
27J N (K ) N (g) ! α 6 25 J N (f ; 4α) N (K; 2α) . α +5
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Proof of Prop. IX.1. By Def. VIII.1 ii g=
∞
g =
g ,
=1
∞ =1
g ,
where :g : = :g :
=
1 ! 1 !
Q(K, · · · , K)(:f :), Q(K , · · · , K )(:f :).
Since Q(K, · · · , K) − Q(K , · · · , K ) = Q(K − K , K, · · · , K) + Q(K , K − K , K, · · · , K) + · · · + Q(K , · · · , K − K ) = Q(K , K, · · · , K) + Q(K , K , K, · · · , K) + · · · + Q(K , · · · , K ), it follows from Lemma IX.2 that Nimpr (g − g ) ≤
25 α +5
J N (f ; 4α) N (K; 2α) .
Therefore Nimpr (g − g ) ≤
∞ =1
Nimpr (g − g ) ≤
25 J α6
N (f ; 4α)
N(K;2α) . 1− α1 N(K;2α)
Proposition IX.3. Let K(ψ; ζ, ζ , ϕ; ξ, ξ , η) be an even Grassmann function of degree at most two in the variables ζ , ξ , ϕ, η and of degree at least one in η. Write
K(ψ; ζ, ζ , ϕ; ξ, ξ , η) =
n0 ,n1 ,n2 ,n3 p1 ,p2 ,p3
Kn0 ( pn1 pn2 np3 )(ψ; ζ, ζ , ϕ; ξ, ξ , η) 1 2 3
with Kn0 ( pn1 pn2 np3 ) ∈ A[n0 , p1 , p2 , p3 , n1 , n2 , n3 ]. Let 1 2 3
K·,1 =
n0 ,n1 ,p1
Kn0
p1 0 0 n1 0 1
be the part of K that has degree precisely one in ζ , ξ , ϕ, η. Furthermore let T (ψ; ϕ; η) be a Grassmann function that has degree two in the variables ϕ, η and degree at least one in the variable η. Write T = T11 + T02 , where T11 has degree one in ϕ and η, and T02 has degree two in η. Set Tmix (ψ; ϕ , ϕ ; η , η ) = T11 (ψ; ϕ ; η ) + T11 (ψ; ϕ ; η ) + T02 (ψ; 0; η + η ) − T02 (ψ; 0; η ) − T02 (ψ; 0; η ) .
Convergence of Perturbation Expansions in Fermionic Models. Part 2
Furthermore set
K˜ 1 (ψ; ζ, ζ , ϕ; ξ, ξ , η) =
285
:T (ψ; ϕ ; η ):η ,C .. K(ψ; ζ
+ϕ , ζ , ϕ; ξ + η , ξ , η) .. ϕ ,D dµD (ϕ ) dµC (η ), η ,C K˜ 2 (ψ; ζ, ζ , ϕ; ξ, ξ , η) = T11 (ψ; ϕ; η ) :K(ψ; ζ, ζ , ϕ; ξ +η , ξ , η):η ,C dµC (η ), ξ ζ
η K˜ 1= ϕ
ξ ζ ϕ, η
K ζ
ξ
η K˜ 2= ϕ
ψ,
T
η
K
ψ ζ
ξ
ψ,
T11
ψ
ϕ and K˜ = K˜ 1 + K˜ 2 , ˆ K(ψ; ζ, ϕ; ξ, η) =
Tmix (ψ; ϕ , ϕ ; η , η ) .. K·,1 (ψ; ζ
+ϕ , ζ , ϕ; ξ + η , ξ , η) .. ϕ ,D η ,C
. K (ψ; ζ + ϕ , ζ , ϕ; ξ + η , ξ , η) . . ·,1 . ϕ ,D η ,C
×dµD (ζ , ϕ , ϕ ) dµC (ξ , η , η ), ζ ξψ η
K·,1
ζ ξψ η
ϕ
Kˆ =
T11 η
K·,1
ψ
K·,1
+
η
T11 η
K·,1
ζ ξψ
ζ ξψ η
K·,1
ζ ξψ η T02
+ η
K·,1 ζ ξψ
η
η
ψ
ϕ
ψ
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Finally let f (ψ; ζ ; ξ ) be a Grassmann function and set ˜ ) Q(K, K)(:f :ξ,C )(ψ; ζ, 0, ϕ; 0, 0, ξ ), f (ψ; ζ, ϕ; ξ ) = Q(T ˜ K)(:f :ξ,C )(ψ; ζ + ϕ; ξ ), :f˜(ψ; ζ, ϕ; ξ ):ξ,C = 2 Q(K, ˆ :fˆ(ψ; ζ, ϕ; ξ ):ξ,C = Q(K)(:f :ξ,C )(ψ; ζ + ϕ; ξ ). Then, if α ≥ 2, Nimpr (f − (f˜ + fˆ); α) ≤
29 J α 10
N (f ; 4α) N (K; 4α)2 N (T ; 2α).
The proof is similar to the proof of Prop. VII.7 and is given following Lemma IX.4. Let B(ψ; ϕ , ϕ ; η , η ) be a Grassmann function that has degree one in the variables ϕ , η , degree one in the variables ϕ , η and degree at least one in the variables η , η . Furthermore let H (ψ; ζ, ζ , γ ; ξ, ξ , η), K(ψ; ζ, ζ , γ ; ξ, ξ , η) be even Grassmann functions that vanish for η = 0. Assume that H or K has degree at least two in the variables ζ , γ , ξ , η. Let f (ψ; ζ ; ξ ) be any Grassmann function and set . g(ψ; ζ, ϕ; ξ ) = B(ψ; ϕ , ϕ ; η , η ) .. H (ψ; ζ + ϕ . +ϕ , ζ , γ ; ξ + η , ξ , η) .. ζ ,ϕ ;D η ,ξ ;C
. K(ψ; ζ +ϕ +ϕ , ζ , γ ; ξ +η , ξ , η) . . . ζ ,ϕ ;D ξ ,η ;C . ×dµD (ϕ , ϕ , ζ ) dµC (η , η , ξ ) . γ ,D η,C . f (ψ; ζ + ϕ + γ ; η) . γ ,D dµ (γ ) dµ (η). . . D C η,C
Then, if α ≥ 2, Nimpr (g; α) ≤
25 J α 10
N (B; α) N(H ; 4α) N (K; 4α) N (f ; 4α).
Proof. In the proof we suppress the variable ψ. We assume that K has degree at least two in the variables ζ , γ , ξ , η. First we discuss the case that B has degree one in the variable η . Set h(ζ, ϕ, γ , ζ , ϕ ; ξ, η, ξ , η ) = B(ϕ , ϕ ; η , η ) .. H (ζ + ϕ + ϕ , ζ , ζ ; ξ + η , ξ , ξ ) .. ϕ ,ζ ;D . f (ζ + ϕ + γ + ζ ; η + ξ ) . dµ (ϕ , ζ ) dµ (η , ξ ). . . ζ ,D D C
η ,ξ ;C
ξ ,C
By Lemma A.5, twice . H (···, γ ···, η)K(···, γ ···, η) . γ ,D . f (···, γ ; η) . γ ,D dµ (η)dµ (γ ) . . . . C D η,C η,C = .. H (···, ζ ···, ξ ) .. ζ ,D .. K(···, γ ···, η) .. γ ,D .. f (···, γ + ζ ; η + ξ ) .. γ ,ζ ;D ξ ,C
η,C
×dµC (η, ξ )dµD (γ , ζ )
η,ξ ;C
(IX.1)
Convergence of Perturbation Expansions in Fermionic Models. Part 2
so
g(ζ, ϕ; ξ ) =
287
. h(ζ, ϕ, γ , ζ , ϕ ; ξ, η, ξ , η ) . . ζ ,γ ;D . ξ ,η;C
. K(ζ + ϕ + ϕ , ζ , γ ; ξ + η , ξ , η) . . . ϕ ,ζ ,γ ;D η ,ξ ,η;C
×dµD (γ , ϕ , ζ ) dµC (η, η , ξ ). By of Lemma II.31 combined with Prop. II.33, first integrating iterated application · dµC (η )dµD (ϕ ) and then integrating · dµC (ξ )dµD (ζ ), and Remark II.24, several times, N (h) ≤
1 N(B) N (H (ζ +ϕ+ϕ ,ζ ,ζ ;ξ +η ,ξ ,ξ ); α) N (f (ζ +ϕ+γ +ζ ;η+ξ ); α) α4 ≤ α14 N(B) N (H (ζ + ϕ, ζ , ζ ; ξ, ξ , ξ ); 2α) N (f (ζ + γ ; η); 2α) ≤ α14 N(B) N (H ; 4α) N(f ; 4α). Since h, through B, has degree one in ϕ , η , only the part of K(ζ + ϕ + ϕ , ζ , γ ; ξ + η , ξ , η) that has degree at least one in ϕ , η can contribute to g(ζ, ϕ; ξ ). As K also has degree at least two in the variables ζ , γ , ξ , η and degree at least one in η, Prop. VI.4
implies that Nimpr (g) ≤ ≤
27 J N (h) N (K; 4α) α6 5 2 J N (B) N (H ; 4α) N (f ; 4α) N (K; 4α). α 10
Next we discuss the situation that B has degree zero in the variable η . This implies that B has degree one in ϕ and degree one in η . Set . H (ζ + ϕ + ϕ , ζ , ζ ; ξ + η , ξ , ξ ) . h(ζ, ϕ, γ , ζ , ϕ ; ξ, ξ , η, η ) = . . ζ ;D ξ ;C
. f (ζ + ϕ + γ + ζ ; η + ξ ) . dµ (ζ ) dµ (ξ ), . . ζ ;D D C ξ ;C B(ϕ , ϕ ; η , η ) .. K(ζ + ϕ + ϕ , ζ , γ ; ξ k(ζ, ϕ, γ , ζ , ϕ ; ξ, ξ , η, η ) = +η , ξ , η) .. ϕ ;D dµD (ϕ ) dµC (η ). η ;C
Again by (IX.1), . h(ζ, ϕ, γ , ζ , ϕ ; ξ, ξ , η, η ) . g= . . ϕ ,ζ ,γ ;D η ,ξ ,η;C
. k(ζ, ϕ, γ , ζ , ϕ ; ξ, ξ , η, η ) . . . ϕ ,ζ ,γ ;D dµD (ϕ , ζ , γ ) dµC (η , ξ , η). η ,ξ ,η;C
By Lemma II.31, Prop. II.33 and Remark II.24, N (h) ≤ N (k) ≤
1 N (H ; 4α) N(f ; 4α), α2 1 N (B; α) N(K; 4α). α2
Since k has degree at least one in η and degree at least three in ϕ , ζ , γ , η , ξ , η we have Nimpr (g) ≤
27J α6
N (h) N (k) ≤
25 J α 10
N (H ; 4α) N (f ; 4α) N (K; 4α) N (B).
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Proof of Prop. IX.3. We again suppress ψ in the proof. By definition f (ζ, ϕ; η) = :T (ϕ , η ):η ,C :Q(K, K)(:f :ξ,C )(ζ + ϕ + ϕ ; η + η ):ϕ ,D ×dµD (ϕ ) dµC (η ) so that, by Lemma A.7, recalling that T (ϕ , η ) is of degree two in ϕ , η , f (ζ, ϕ; ξ ) = :T (ϕ , η ):η ,C :Q(K, K)(:f :)(ζ + ϕ + ϕ ; ξ + η ):ϕ ,D ×dµD (ϕ ) dµC (η ) . = :T (ϕ , η ):η ,C . .. .. K(ζ + ϕ + ϕ , ζ , γ ; ξ + η , ξ , η) .. ζ ,D ξ ,C . K(ζ + ϕ + ϕ , ζ , γ ; ξ + η , ξ , η) . . . . ζ ,D . γ ,D η,C ξ ,C . f (ζ + ϕ + ϕ + γ ; η) . γ ,D . dµ (ζ , ϕ , γ ) dµ (ξ , η, η ) . . D C . ϕ ,D η,C η ,C . = 2 :T (ϕ , η ):η ,C . .. .. K(ζ + ϕ + ϕ , ζ , γ ; ξ + η , ξ , η) .. ζ ,ϕ ;D ξ ,η ;C . K(ζ + ϕ, ζ , γ ; ξ, ξ , η) . . . . ζ ,D . γ ,D η,C ξ ,C . f (ζ + ϕ + ϕ + γ ; η) . γ ,D . . . . ϕ ,D dµD (ζ , ϕ , γ ) dµC (ξ , η, η ) η,C . . . + . Tmix (ϕ , ϕ ; η , η ) . K(ζ + ϕ + ϕ , ζ , γ ; ξ + η , ξ , η) . ϕ ,ζ ;D η ,ξ ;C . . K(ζ + ϕ + ϕ , ζ , γ ; ξ + η , ξ , η) . . . ϕ ,ζ ;D . γ ,D η,C η ,ξ ;C . f (ζ + ϕ + γ ; η) . γ ,D dµ (ϕ , ϕ , ζ , γ ) dµ (η , η , ξ , η). (IX.2) . . D C η,C
The Tmix term in (IX.2) above differs from . K(ζ . . . . ˆ + ϕ, γ ; ξ, η) . γ ,D . f (ζ + ϕ + γ ; η) . γ ,D dµD (γ ) dµC (η) = fˆ(ζ, ϕ; ξ ) η,C
by
η,C
. . . . Tmix (ϕ , ϕ ; η , η ) . H (ζ + ϕ + ϕ , ζ , γ ; ξ + η , ξ , η) . ϕ ,ζ ;D η ,ξ ;C . . H (ζ + ϕ + ϕ , ζ , γ ; ξ + η , ξ , η) . . . ϕ ,ζ ;D . γ ,D η,C η ,ξ ;C . f (ζ + ϕ + γ ; η) . γ ,D dµ (ϕ , ϕ , ζ , γ ) dµ (η , η , ξ , η) . . D C η,C . . . +2 . Tmix (ϕ , ϕ ; η , η ) . H (ζ + ϕ + ϕ , ζ , γ ; ξ + η , ξ , η) . ϕ ,ζ ;D η ,ξ ;C . . K (ζ + ϕ + ϕ , ζ , γ ; ξ + η , ξ , η) . . ·,1 . ϕ ,ζ ;D . γ ,D η,C η ,ξ ;C . f (ζ + ϕ + γ ; η) . γ ,D dµ (ϕ , ϕ , ζ , γ ) dµ (η , η , ξ , η), . . D C η,C
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where K = K·,1 + H . By Lemma IX.4, the improved norm of this difference is bounded by 5
3 2α 10J N (Tmix ; α) N(H ; 4α) N(K; 4α) N (f ; 4α) ≤
29 J α 10
N (T ; 2α) N(K; 4α)2 N (f ; 4α).
We apply Lemma A.5 to the variable ϕ in the 2 :T (ϕ , η ):η ,C term of (IX.2). We get . 2 :T (ϕ + ϕ ; η ):η ,C . .. K(ζ + ϕ + ϕ , ζ , γ ; ξ + η , ξ , η) .. ζ ,ϕ ;D ξ ,η ;C . . K(ζ + ϕ, ζ , γ ; ξ, ξ , η) . . . ζ ,D . γ ,D η,C ξ ,C . f (ζ + ϕ + ϕ + γ ; η) . . . ϕ ,γ ;D dµD (ζ , ϕ , ϕ , γ ) dµC (ξ , η, η ). η,C
As T has degree at most one in ϕ + ϕ , this is equal to . . . 2 . :T (ϕ ; η ):η ,C . K(ζ + ϕ + ϕ , ζ , γ ; ξ + η , ξ , η) . ζ ,ϕ ;D ξ ,η ;C . . K(ζ + ϕ, ζ , γ ; ξ, ξ , η) . . . ζ ,D . γ ,D η,C ξ ,C . f (ζ + ϕ + γ ; η) . γ ,D dµ (ζ , ϕ , γ ) dµ (ξ , η, η ) . . D C η,C . +2 :T11 (ϕ ; η ):η ,C . .. K(ζ + ϕ, ζ , γ ; ξ + η , ξ , η) .. ζ ,D ξ ,η ;C . . K(ζ + ϕ, ζ , γ ; ξ, ξ , η) . . . ζ ,D . γ ,D η,C ξ ,C . f (ζ + ϕ + ϕ + γ ; η) . . . ϕ ,γ ;D dµD (ζ , ϕ , γ ) dµC (ξ , η, η ). η,C
Applying Lemma A.5, with ξ replaced by γ and ξ replaced by ϕ , to the second term, this is equal to . . . 2 . :T (ϕ ; η ):η ,C . K(ζ + ϕ + ϕ , ζ , γ ; ξ + η , ξ , η) . ζ ,ϕ ;D ξ ,η ;C . . K(ζ + ϕ, ζ , γ ; ξ, ξ , η) . . . ζ ,D . γ ,D η,C ξ ,C . f (ζ + ϕ + γ ; η) . γ ,D dµ (ζ , ϕ , γ ) dµ (ξ , η, η ) . . D C η,C . . . +2 . T11 (γ ; η ) . K(ζ + ϕ, ζ , γ ; ξ + η , ξ , η) . ζ ,D ξ ,η ;C . . K(ζ + ϕ, ζ , γ ; ξ, ξ , η) . . . ζ ,D . γ ,D η,C ξ ,C . f (ζ + ϕ + γ ; η) . γ ;D dµ (ζ , γ ) dµ (ξ , η, η ) . . D C η,C . . . ˜ . . K(ζ + ϕ, ζ , γ ; ξ, ξ , η) . (ζ + ϕ, ζ , γ ; ξ, ξ , η) =2 K ,D . . . . ζ ,D ζ ,D 1 . γη,C . ξ ,C ξ ,C . f (ζ + ϕ + γ ; η) . γ ,D dµ (ζ , γ ) dµ (ξ , η) . . D C η,C
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J. Feldman, H. Kn¨orrer, E. Trubowitz
+2
. . . ˜ . . . ,D . K(ζ + ϕ, ζ , γ ; ξ, ξ , η) . ζ ,D K (ζ + ϕ, ζ , γ ; ξ, ξ , η) ,D . . ζ 2 . γη,C . ξ ,C
ξ ,C
. f (ζ + ϕ + γ ; η) . γ ,D dµ (ζ , γ ) dµ (ξ , η). . . D C η,C
So the 2 :T (ϕ , η ):η ,C term of (IX.2) equals f˜1 (ζ, ϕ; ξ ) + f˜2 (ζ, ϕ; ξ ), where :f˜1 (ζ, ϕ; ξ ):ξ,C = 2 Q(K˜ 1 , K)(:f :)(ζ + ϕ; ξ ), :f˜2 (ζ, ϕ; ξ ):ξ,C = 2 Q(K˜ 2 , K)(:f :)(ζ + ϕ; ξ ). As f˜ = f˜1 + f˜2 , the proposition follows.
IX.2. Tails. In this subsection let K(ψ; ζ, ζ , ϕ; ξ, ξ , η) be an even Grassmann function that has degree at least four in the variables ψ, ζ, ζ , ϕ, ξ, ξ , η such that K(ψ; ζ, ζ , ϕ; ξ, ξ , 0) = 0 . We always write K= with Kn0 ( pn1 pn2 np3 ) ∈ A[n0 , p1 , p2 , p3 , n1 , n2 , n3 ]. Kn0 ( pn1 pn2 np3 ) 1 2 3 1 2 3 n0 ,n1 ,n2 ,n3 p1 ,p2 ,p3
Definition IX.5. function
(i) An n–legged tail with at least e external legs is a Grassmann T (ψ; ϕ; η) ∈
d≥e
A[d, n1 , n2 ].
n1 +n2 =n n2 ≥1
A n–legged tail is a n–legged tail with at least two external legs. (ii) If T is a two–legged tail we define the two–legged tail T ◦ K by (T ◦ K)(ψ; ϕ; η) = :T (ψ; ϕ ; η ):η ,C . . . p ,n K0 ( pn11 00 pn33 )(ψ; ϕ , ζ , ϕ; η , ξ , η) . ϕ ,D 1 1 p3 +n3 =2
η ,C
×dµD (ϕ ) dµC (η ). Remark IX.6. Again, a two–legged tail with two external legs is an end in the sense of Def. VI.5. If K(ψ; ζ, ζ , ϕ; ξ, ξ , η) = U (ψ + ζ + ζ + ϕ; ξ + ξ + η) − U (ψ + ζ + ζ + ϕ; ξ + ξ ) for some Grassmann function U (ψ; ξ ) then T ◦ K agrees with T ◦ Rung(U ) of Defs. VI.5 and VI.8. Recall that we are interested in the two– and four–legged contributions to the Grassmann function f (ψ) of Theorem VI.10. As in Def. VII.8 such contributions are extracted by Definition IX.7. The operator P maps :f (ψ; ζ ; ξ ):ξ,C to f4,0,0 (ψ; 0; 0) + f2,0,0 (ψ; 0; 0), when f = n0 ,n1 ,n2 fn0 ,n1 ,n2 with fn0 ,n1 ,n2 ∈ A[n0 , n1 , n2 ]. Definition IX.8. Let T (ψ; ϕ; η) be an n–legged tail.
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i) An element ν of the norm domain Nd is said to be an effective bound for T if
N :T (ψ; ϕ; η):η,C h ψ; ϕ; η; ξ (1) , · · · , ξ (r) dµC (η) ; α ≤ αν2 N (h; α)
Nimpr :T (ψ; ϕ; η):η,C h ψ; ϕ; η; ξ (1) , · · · , ξ (r) dµC (η) ; α ≤ αν2 Nimpr (h; α)
for all Grassmann functions h ψ; ϕ; η; ξ (1) , · · · , ξ (r) . ii) For ν ∈ Nd , we write Neff (T ; α) ≤ ν
⇐⇒
ν is an effective bound for T .
Remark IX.9. Proposition II.33 and the fact that Nimpr (T ; α) ≤ N (T ; α) imply Neff (T ; α) ≤ N (T ; α). Lemma IX.10. Assume that K(ψ; ζ, ζ , ϕ; ξ, ξ , η) has degree at most two in the variables ζ , ϕ, ξ , η and that α ≥ 2. Let T be a two–legged tail with at least e external legs. If e ≥ 4, then P Q(T )QK (:f :) = P Q(T ◦ K)(:f :), where :f : is shorthand for :f :ξ,C . More generally, if e ≥ 2, there exists a one–legged tail t1 with at least e + 1 external legs, a two–legged tail t2 with at least e + 1 external legs and a two–legged tail τ (ψ; ϕ; η) with at least e + 2 external legs and degree two2 in η such that for any Grassmann function f (ψ; ζ ; ξ ) the following holds: Set
f (ψ) = P Q(T )QK (:f :) − Q(T ◦ K)(:f :) − Q T ◦ K, T1 (K) (:f :) −Q(t1 + t2 + τ )(:f :) , where
T1 (K) = K2 ( 00 00 02 ) + K2 ( 00 00 11 ).
Then Nimpr (f ; α) ≤
29 J α8
N (f ; 4α) N(T ; 2α) N(K; 4α) 1 +
1 N (K; 4α) α2
.
Furthermore N (t1 ) ≤ N (t2 ) ≤ Neff (τ ) ≤
1 α2 2 α2 4 α4
N (T ) N (K), N (T ) N (K), N (T ) N (K)2 .
Proof. The proof is similar to that of Lemma VII.11. If :f :ξ,C = Q(T )Q(K, · · · , K) (:f :), with K’s, then, as K is of degree at least two in ψ, ζ, ξ and T is 2–legged with e external legs, f has degree at least e + 2 − 2. Hence P Q(T )QK (:f :) = P Q(T )Q(K)(:f :) if e ≥ 4, 1 P Q(T )QK (:f :) = P Q(T )Q(K)(:f :) + 2 P Q(T )Q(K, K)(:f :) if e ≥ 2. 2
Hence τ is independent of ϕ.
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The contribution P Q(T )Q Kn0 ( pn1 pn2 np3 ) vanishes unless 1 2 3 • p1 + n1 ≤ 2, since otherwise Q(T )Q K· ( pn1 pn2 np3 ) is of degree at least one in ζ and 1 2 3 P sets ζ to zero. • n1 ∈ {1, 2}, since these fields must connect to η fields of T and T has degree one or two in η. • p2 = n2 = 0, since there is only a single K in P Q(T )Q Kn0 ( pn1 pn2 np3 ) . 1 2 3 • p3 + n3 ≤ 2 because K is of degree at most two in ζ , ϕ, ξ , η. • n3 ≥ 1 because K is of degree at least one in η. • n0 + n1 + p1 + n3 + p3 ≥ 4 because K is of degree at least four overall. Hence P Q(T )Q(K) = P Q(T ) Q
p1 +n1 =2 p3 +n3 =2
K· ( p1 0 p3 ) + P Q(T ) Q n1 0 n3
p3 +n3 =2
K· ( 0 0 p 3 ) 1 0 n3
K· ( p1 0 0 ) + P Q(T ) Q K· ( 01 00 01 ) +P Q(T ) Q n1 0 1 p1 +n1 =2 = P Q(T ◦ K) + Q(k) + Q(t1 + t21 + t22 ) , where
T11 (ψ; ϕ; η )
k(ψ; ϕ; η) = t1 (ψ; ϕ; η) =
p3 +n3 =2 n0 ≥1
:T (ψ; ϕ ; η ):η
Kn0 ( 0 0 p3 )(ψ; 0, 0, ϕ; η , 0, η) dµC (η ),
p1 +n1 =2 n0 ≥1
1 0 n3
Kn0 ( p1 0 0 )
n1 0 1
× (ψ; ϕ , 0, 0; η , 0, η) dµD (ϕ ) dµC (η ) is a one–legged tail with at least e + 1 external legs,
t21 (ψ; ϕ; η) = :T (ψ; ϕ ; η ):η Kn0 ( p1 0 p3 ) n0 ≥1
n1 0 n3
p1 +n1 =2 p3 +n3 =2
× (ψ; ϕ , 0, ϕ; η , 0, η) dµD (ϕ ) dµC (η ) is a two–legged tail with at least e + 1 external legs,
t22 (ψ; ϕ; η) = T11 (ψ; ϕ; η ) Kn0 ( 01 00 01 )(ψ; 0, 0, 0; η , 0, η) dµC (η ) n0 ≥2
is a two–legged tail with at least e + 2 external legs. If e ≥ 4,
P Q(k) = P Q(t1 ) = P Q(t21 ) = P Q(t22 ) = 0.
If e ≥ 2, by Lemma II.31 and Prop. II.33, N (t1 ), N (t21 ), N (t22 ) ≤
1 α2
N (T ) N (K).
Furthermore k has degree at least one in the variable η and degree three in the variables ϕ, η. By Lemma IX.2 and Prop. II.33
5 Nimpr P Q(k)(:f :) ≤ α2 6 J N (k; 2α) N (f ; 4α) ≤
23 α8
J N (T ; 2α)N (K; 2α) N (f ; 4α).
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We define the projection P by P f (ψ; ζ ; ξ ) = f4 (ψ; 0; 0) + f2 (ψ; 0; 0). Observe that ˜ )Q(K, K)(:f :)(ψ; ζ, 0, 0; ξ, 0, 0) P Q(T )Q(K, K)(:f :) = P Q(T ˜ )Q(K, K)(:f :)(ψ; 0, 0, 0; 0, 0, 0) = P Q(T so that we can apply Prop. IX.3. Modulo a term whose improved norm Nimpr is bounded 9 by 2α 10J N (f ; 4α) N(K; 4α)2 N (T ; 2α) ˜ )Q(K, K)(:f :)(ψ; ζ, 0, 0; 0, 0, ξ ) P Q(T ˆ = 2 P Q(K˜ 1 , K)(:f :) + 2 P Q(K˜ 2 , K)(:f :) + P Q(K)(:f :) , where K˜ 1 , K˜ 2 , Kˆ are as in Prop. IX.3. Since K·,1 has degree at least three in ψ, ζ, ξ , ˆ the tail Kˆ has at least six external legs, so that P Q(K)(:f :) = 0. Similarly, as K˜ 2 has degree at least three in ψ, ζ, ξ and K has degree at least two in ψ, ζ, ξ , P Q(K˜ 2 , K)(:f :) = 0. As K˜ 1 and K have degree at most two in ζ , ϕ, ξ , η and degree at least four overall, P Q(K˜ 1 , K)(:f :) = P Q T1 (K˜ 1 ), T1 (K) (:f :) + P Q(τ )(:f :) = P Q(T ◦ K, T1 (K))(:f :) + P Q(τ )(:f :) , where τ (ψ; ϕ; η) =
p2 +n2 =1
(K˜ 1 )2 ( 0 p2 0 )(ψ; 0, ζ , 0; 0, ξ , η) 0 n2 1
× K2 ( 0 p2 0 )(ψ; 0, ζ , 0; 0, ξ , η) dµD (ζ ) dµC (ξ ) 0 n2 1
is a two–legged tail with at least e + 2 external legs and degree two in η. By Prop. II.33, N (K˜ 1 )2 ( 0 p2 0 )(ψ; 0, ζ , 0; 0, ξ , η) 0 n2 1 p2 +n2 =1 =N :T11 (ψ; ϕ ; η ):η ,C K2 ( 1 p2 0 )(ψ; ϕ , ζ , 0; η , ξ , η) × dµD (ϕ ) dµC (η ) ≤
p2 +n2 =1
1 n2 1
1 N(T ) N (K). α2
Therefore, by Prop. II.33, for any Grassmann function h,
N :τ (ψ; ϕ; η):η,C h ψ; ϕ; η; ξ (1) , · · · , ξ (r) dµC (η) ≤ α44 N (K˜ 1 )2 ( 0 p2 0 )(ψ; 0, ζ , 0; 0, ξ , η) N (K) N (h) 0 n2 1
p2 +n2 =1
≤
4 α6
2
N (T ) N (K) N (h).
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Similarly,
Nimpr ≤ ≤
:τ (ψ; ϕ; η):η,C h ψ; ϕ; η; ξ (1) , · · · , ξ (r) dµC (η)
4 N (T ) Nimpr (K)2 Nimpr (h) α 6 impr 4 N (T ) N (K)2 Nimpr (h). α6
Therefore
Neff (τ ) ≤
4 α4
N (T ) N (K)2 .
Lemma IX.11. Assume that K(ψ; ζ, ζ , ϕ; ξ, ξ , η) has degree at most two in the variables ζ , ϕ, ξ , η. Let T1 , T2 be two–legged tails. Then there exists a one–legged tail t1 , a two–legged tail t2 and a two–legged tail τ (ψ; ϕ; η) of degree two in η, each with at least four external legs, such that for any Grassmann function f (ψ; ζ ; ξ ) the following holds: Set f (ψ) = P Q(T1 , T2 )QK (:f :) − Q(T1 ◦ K, T2 ◦ K)(:f :) − Q(t1 + t2 + τ )(:f :) . Then, if α ≥ 2, Nimpr (f ) ≤
29 J α 10
N (f ; 4α) N(T1 ; 2α) N(T2 ; 2α) N (K; 4α) 1 +
N(K;4α) , α2
and N (t1 ) ≤ N (t2 ) ≤ Neff (τ ) ≤
4 α4 12 α4 8 α6
N (T1 ) N(T2 ) N (K), N (T1 ) N(T2 ) N (K), N (T1 ) N (T2 ) N (K)2 .
Proof. The proof is similar to that of Lemma VII.12. Again, if :f :ξ,C = Q(T1 , T2 )Q(K, · · · , K)(:f :) with K’s, then, as K is of degree at least two in ψ, ζ, ξ and T1 , T2 are 2–legged with at least two external legs, f has degree at least 4 + 2 − 4. Hence P Q(T1 , T2 )QK (:f :) = P Q(T1 , T2 )Q(K)(:f :) + 21 P Q(T1 , T2 )Q(K, K)(:f :) . The contribution P Q(T1 , T2 )Q Kn0 ( pn1 pn2 np3 ) vanishes unless 1 2 3 • n0 = 0 because T1 and T2 are each of degree at least two in ψ. • p1 + n1 ≤ 4, since otherwise Q(T1 , T2 )Q K0 ( pn1 pn2 np3 ) is of degree at least one in ζ 1 2 3 and P sets ζ to zero. • n1 ≥ 2 since these fields must connect to η fields of T1 , T2 which have combined degree at least two in η. • p2 = n2 = 0 since there is only a single K in P Q(T1 , T2 )Q Kn0 ( pn1 pn2 np3 ) . 1 2 3
• p3 + n3 ≤ 2 because K is of degree at most two in ζ , ϕ, ξ , η. • n3 ≥ 1 because K is of degree at least one in η. • n1 + p1 + n3 + p3 ≥ 4 because K is of degree at least four overall.
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Hence P Q(T1 , T2 )Q(K)(:f :) = P Q(t11 + t21 + t22 + t23 )(:f :) + P Q(k)(:f :) , where
t11 (ψ; ϕ; η) = ×
t21 (ψ; ϕ; η) = ×
t22 (ψ; ϕ; η) = ×
t23 (ψ; ϕ; η) = ×
k(ψ; ϕ; η) = ×
. T (ψ; ϕ ; η )T (ψ; ϕ ; η ) . . 1 . ϕ ,D 2 η ,C
K0 ( p1 0 0 )(0; ϕ , 0, 0; η , 0, η) dµD (ϕ ) dµC (η ),
p1 +n1 =4
n1 0 1
. T (ψ; ϕ ; η )T (ψ; ϕ ; η ) . . 1 . ϕ ,D 2 p1 +n1 =4 p3 +n3 =2
η ,C
K0 ( p1 0 p3 )(0; ϕ , 0, ϕ; η , 0, η) dµD (ϕ ) dµC (η ), n1 0 n3
. T (ψ; ϕ; η )T (ψ; ϕ ; η ) . . 1 . η ,C 2 p1 +n1 =3
K0 ( p1 0 0 )(0; ϕ , 0, ϕ; η , 0, η) dµD (ϕ ) dµC (η ), n1 0 1
. T (ψ; ϕ ; η )T (ψ; ϕ; η ) . . 1 . η ,C 2 p1 +n1 =3
K0 ( p1 0 0 )(0; ϕ , 0, ϕ; η , 0, η) dµD (ϕ ) dµC (η ), n1 0 1
. T (ψ; ϕ; η )T (ψ; ϕ; η ) . . 1 . η ,C 2 p3 +n3 =2 n3 ≥1
K0 ( 0 0 p3 )(0; 0, 0, ϕ; η , 0, η) dµC (η ). 2 0 n3
Here, each tij is a i–legged tail with at least four external legs, that, by Lemma II.31 and Prop. II.33, with = 2, fulfills N (tij ) ≤
4 α4
N (T1 ) N(T2 ) N (K).
Furthermore k has degree at least one in the variable η and degree three in the variables ϕ, η. By Lemma IX.2 and Prop. II.33 with = 2 and α replaced by 2α,
5 Nimpr P Q(k)(:f :) ≤ α2 6 J N (k; 2α) N(f ; 4α) ≤
23 α 10
J N (T1 ; 2α)N(T2 ; 2α)N (K; 2α) N (f ; 4α).
By Prop. VIII.13. Q(T1 , T2 )Q(K, K)(:f :) . . =. :T1 (ψ; ϕ; η):η,C .. g(ψ; ζ, 0, ϕ; ξ, 0, η) .. ϕ,D dµD (ϕ) dµC (η) . ξ,C , η,C
˜ 2 )Q(K, K)(:f :). In particular where g = Q(T P Q(T1 , T2 ) Q(K, K)(:f :) = P :T1 (ψ; ϕ; η):η,C .. g(ψ; 0, 0, ϕ; 0, 0, η) .. ϕ,D dµD (ϕ) dµC (η). η,C
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By Prop. IX.3, ˆ ϕ; η) + h(ψ; ϕ; η), g(ψ; 0, 0, ϕ; 0, 0, η) = g˜ 1 (ψ; ϕ; η) + g˜ 2 (ψ; ϕ; η) + g(ψ; where g˜ 1 (ψ; ϕ; η) = 2Q(K˜ 1 , K)(:f :)(ψ; ϕ; η), g˜ 2 (ψ; ϕ; η) = 2Q(K˜ 2 , K)(:f :)(ψ; ϕ; η), ˆ g(ψ; ˆ ϕ; η) = Q(K)(:f :)(ψ; ϕ; η), and Nimpr (h) ≤
29 J α 10
N (f ; 4α) N(K; 4α)2 N (T2 ; 2α).
By Lemma II.31 and Prop. II.33, the improved norm of :T1 (ψ; ϕ; η):η,C .. h(ψ; ϕ; η) .. ϕ,D dµD (ϕ) dµC (η) η,C
is bounded by 1 N (T )Nimpr (h) α 2 impr 1
≤
29 J α 12
N (f ; 4α) N(K; 4α)2 N (T1 ; α) N (T2 ; 2α).
Observe that gˆ has degree at least six, so that P :T1 (ψ; ϕ; η):η,C .. g(ψ; ˆ ϕ; η) .. ϕ,D dµD (ϕ) dµC (η ) = 0. η,C
Similarly, g˜ 2 has degree at least five, so that :T1 (ψ; ϕ; η):η,C .. g˜ 2 (ψ; ϕ; η) .. ϕ,D dµD (ϕ) dµC (η ) = 0. P η,C
Finally, the contribution to g˜ 1 with ψ–degree two and overall degree four is g˜ 11 + g˜ 12 , where g˜ 11 = 2Q T2 ◦ K, p1 +n1 =2 K0 ( p1 0 p3 ) (:f :), n1 0 n3 p3 +n3 =2 g˜ 12 = 2Q p, p1 +n1 =2 K0 ( p1 p2 0 ) (:f :), n1 n2 1 p2 +n2 =1
where
p(ψ; ζ , ϕ; ξ , η) = × Then P
:T2 (ψ; ϕ ; η ):η ,C p1 +n1 =2 p2 +n2 =1
K0 ( p1 p2 0 )(ψ; ϕ , ζ , ϕ; η , ξ , η) dµD (ϕ ) dµC (η ). n1 n2 1
:T1 (ψ; ϕ; η):η,C .. g˜ 11 (ψ; ϕ; η) .. ϕ,D dµD (ϕ) dµC (η) η,C
= 2P Q(T1 ◦ K, T2 ◦ K)(:f :),
Convergence of Perturbation Expansions in Fermionic Models. Part 2
while P
297
:T1 (ψ; ϕ; η):η,C .. g˜ 12 (ψ; ϕ; η) .. ϕ,D dµD (ϕ) dµC (η) = P Q(τ )(:f :) η,C
with the two–legged tail, with four external legs and degree two in η, τ = 2 p(ψ; ζ , ϕ; ξ , η) p (ψ; ζ , ϕ; ξ , η) dµD (ζ ) dµC (ξ ), where p was defined above and p (ψ; ζ , ϕ; ξ , η) = :T1 (ψ; ϕ , η ):η ,C × K0 ( p1 p2 0 )(ψ; ϕ , ζ , ϕ; η , ξ , η) dµD (ϕ )dµC (η ). n1 n2 1 p1 +n1 =2 p2 +n2 =1
By Lemma II.31 and Prop. II.33, N (p) ≤ N (p ) ≤
1 N (T2 )N (K), α2 1 N (T1 )N (K). α2
By Lemma II.31 and Prop. II.33, with = 2,
N :τ (ψ; ϕ; η):η,C h ψ; ϕ; η; ξ (1) , · · · , ξ (r) dµC (η) ≤ ≤
8 N (p)N (p )N (h) α4 8 N (T1 )N (T2 )N (K)2 N (h) α8
for all Grassmann functions h. A similar estimate applies for the Nimpr norm so that Neff (τ ) ≤
8 N (T1 )N (T2 )N (K)2 . α6
Lemma IX.12. Assume that K has degree at most two in the variables ζ , ϕ, ξ , η. Let T be a one–legged tail with at least three external legs. Then there is a two–legged tail t2 with at least four external legs such that for all Grassmann functions f (ψ; ζ, ξ ), P Q(T )QK (:f :) = P Q(t2 )(:f :) and
N (t2 ) ≤
2 α2
N (T ) N (K).
Proof. The proof is similar to that of Lemma VII.13. Since T is one–legged and K has degree at least four P Q(T ) QK = P Q(T ) Q(K) = P Q(T ) Q
p3 +n3 =2
with
t2 (ψ; ϕ; η) =
K1 ( 0 0 p 3 ) 1 0 n3
= P Q(t2 )
T (ψ; ϕ; η ) K1 ( 01 00 11 )(ψ; 0, 0, ϕ; η , 0, η) +K1 ( 01 00 20 )(ψ; 0, 0, 0; η , 0, η) dµC (η )
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Definition IX.13. The two–legged tails T (K) are recursively defined as follows: T1 (K) was defined in Lemma IX.10, and T +1 (K) = T ◦ K
for ≥ 1
Remark IX.14. (i) Clearly T (K) has two external legs. Using Lemma II.31 and Prop. II.33 one proves by induction that for α ≥ 2,
1 N T (K) ≤ α 2 −2 N (K) . (ii) If K(ψ; ζ, ζ , ϕ; ξ, ξ , η) = U (ψ +ζ +ζ +ϕ; ξ +ξ +η)−U (ψ +ζ +ζ +ϕ; ξ +ξ ) for some even Grassmann function U (ψ; ξ ) then, by Remark IX.6, T (K) = T (U ). Recall that T (U ) was defined in Definition VI.8. IX.3. Proof of Theorem VI.10 in the General Case. First, we prove the analog of Prop. VII.16 for the enlarged algebra. Proposition IX.15. Let K(ψ; ζ ; ξ, ξ , η) be an even Grassmann function that vanishes for η = 0 and has degree at least four overall. Set ¯ K(ψ; ζ, ζ , ϕ; ξ, ξ , η) = K(ψ; ζ + ζ + ϕ; ξ, ξ , η). ¯ 4α)0 < 2α . Furthermore let f (ψ; ζ ; ξ ) be a Grassmann Assume that α ≥ 8 and N (K; 3 function. For each n ≥ 1 there exists a Grassmann function hn (ψ; f ), a one–legged tail t1 , a two–legged tail t2 , each with at least three external legs and a two–legged tail τ (ψ; ϕ; η) with at least four external legs and of degree two in η such that
¯ (:f :ξ,C ) P Ev Rn:K:ζ,D ,C :f : ξ,C dµD (ζ ) = P Q Tn (K) ζ,D
¯ T (K) ¯ (:f :ξ,C ) + 21 Q T (K), , ≥1 max{ , }=n
+Q(t1 + t2 + τ )(:f :ξ,C ) + hn (ψ; f ) and N(t1 ), N (t2 ) ≤ 4
Nimpr hn (ψ; f ); α ≤ where
n ¯ 8 n−1 N(K;α) , ¯ α2 1− 22 N(K;α)
210 J α5
n−1 m=0
α
1
α n−m
n−m ¯ N(K;4α) 3 ¯ 1− 2α N(K;4α)
n+1 ¯ N(K;α) 8 , ¯ α 2n 1− 12 N(K;α) α
N Fm (ψ; f ); 4α ,
:Fm (ψ; f ): ξ,C = Ev Rm :K:ζ,D ,C :f : ξ,C . ζ,D
Proof. Decompose
Neff (τ ) ≤
ζ,D
K¯ = K + K ,
where K has degree at most two in the variables ζ , ξ , ϕ, η and K has degree at least three in these variables. We perform induction on n. For n = 1, by Lemma VIII.12.ii,
Ev R:K:ζ,D ,C :f : ξ,C = :QK¯ (:f :ξ,C ):ζ,D , ζ,D
Convergence of Perturbation Expansions in Fermionic Models. Part 2
and therefore
299
Ev R:K:ζ,D ,C :f : ξ,C dµD (ζ ) = QK¯ (:f :ξ,C )(ψ, 0; ξ ). ζ,D
Set
h1 (ψ; f ) = P (QK¯ − QK )(:f :ξ,C ).
By Prop. IX.1,
Nimpr h1 (ψ; f ) ≤
25 J α6
N f ; 4α
¯ N(K;2α) . ¯ 1− α1 N(K;2α)
Since K has degree at most two in the variables ζ , ϕ, ξ , η and degree at least four overall P QK = P Q K· ( 00 00 01 ) + K· ( 00 00 02 ) + K· ( 00 00 11 ) + 21 P Q K2 ( 00 01 01 ) + K2 ( 00 10 01 ) , K2 ( 00 01 01 ) + K2 ( 00 10 01 ) + 21 P Q K2 ( 00 00 02 ) + K2 ( 00 00 11 ) , K2 ( 00 00 02 ) + K2 ( 00 00 11 )
¯ + P Q(t1 + t2 ) + 1 P Q T1 (K), ¯ T1 (K) ¯ = P Q T1 (K) 2
with t1 = K3 ( 00 00 01 ) + K4 ( 00 00 01 ),
t2 = K3 ( ) + K4 ( ) + K3 ( ) + K4 ( 1 τ = 2 K2 ( 00 10 01 ) K2 ( 00 10 01 ) dµD (ζ ). 000 002
000 002
001 001
001 001
)+
1 2
K2 ( 00 01 01 ) K2 ( 00 01 01 ) dµC (ξ ),
In particular ¯ N (t1 ) ≤ N (K), ¯ + N (t2 ) ≤ N (K) Neff (τ ) ≤
1 4 2 α2
1 2α 2 ¯ 2
¯ 2≤ N (K)
N (K) ≤
2 α2
¯ N(K) ¯ , 1− 12 N(K) 2α 2
¯ . N (K)
Proposition, II.33, with = 1, was used to bound the last term of t2 . Lemma II.31 and Prop. II.33, with = 2, were used to bound :τ (ψ; ϕ; η):η,C h ψ; ϕ; η; ξ (1) , · · · , ξ (r) dµC (η) as in Definition IX.8. Now assume that the statement of the lemma is true for n. By Remark VIII.6.ii,
n :F . Ev Rn+1 :f : = Ev R (ψ; f ): ξ,C ξ,C 1 :K:ζ,D ,C :K:ζ,D ,C ζ,D
By Lemma VIII.12.ii,
ζ,D
:F1 (ψ; f ):ξ,C = QK¯ (:f :ξ,C ).
The induction hypothesis therefore implies that
P Ev Rn+1 ξ,C dµD (ζ ) :K:ζ,D ,C :f : ζ,D
= P Ev Rn:K:ζ,D ,C :QK¯ (:f :ξ,C ):ζ,D dµD (ζ )
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¯ Q ¯ (:f :ξ,C ) = hn (ψ; F1 (ψ; f )) + P Q Tn (K) K
¯ T (K) ¯ Q ¯ (:f :ξ,C ) + 1 P Q T (K), K
2
, ≥1 max{ , }=n
+P Q(t1 + t2 + τ ) QK¯ (:f :ξ,C ) with N (t1 ), N (t2 ) ≤ 4
¯ n 8 n−1 N(K) ¯ , α2 1− 22 N(K)
Neff (τ ) ≤
α
¯ n+1 N(K) 8 ¯ α 2n 1− 12 N(K) α
and
Nimpr hn (ψ; F1 (ψ; f )) ≤
210 J α5
n−1 m=0
n−m ¯ N(K;4α) 1 ¯ α n−m 1− 3 N(K;4α) 2α
N Fm ψ; F1 (f ; ψ) ; 4α .
By Remark VIII.6.ii, . F (ψ; f ) . ξ,C . F ψ; F (ψ; f ) . ξ,C = Ev Rm . . . . m 1 1 :K: ,C ζ,D ζ,D ζ,D
m = Ev R:K:ζ,D ,C Ev R:K:ζ,D ,C .. f .. ξ,C ζ,D
. . = Ev Rm+1 f ξ,C :K:ζ,D ,C . . ζ,D = .. Fm+1 (ψ; f ) .. ξ,C . ζ,D
Therefore
Nimpr hn (ψ; F1 (ψ; f )) ≤
210 J α5
n m=1
n+1−m ¯ N(K;4α) 1 N Fm (ψ; f ); 4α . 3 ¯ α n+1−m 1− 2α N(K;4α)
Let g0 be the difference between ¯ Q ¯ (:f :ξ,C ) + 1 Q T (K), ¯ T (K) ¯ Q ¯ (:f :ξ,C ) P Q Tn (K) K K 2 , ≥1 max{ , }=n
+Q(t1 + t2 + τ ) QK¯ (:f :ξ,C ) and
¯ QK (:f :ξ,C ) + 1 Q T (K), ¯ T (K) ¯ QK (:f :ξ,C ) P Q Tn (K) 2 , ≥1 max{ , }=n
+Q(t1 + t2 + τ ) QK (:f :ξ,C ) .
Let :f :ξ,D = QK¯ (:f :ξ,C ) − QK (:f :ξ,C ). By Lemma VIII.14 and Def. IX.8
¯ +1 ¯ N T (K) ¯ N T (K) Nimpr (g0 ) ≤ 1 Nimpr f ; α N Tn (K) α
+N (t1 ) + N (t2 ) + ν
α
, ≥1 max{ , }=n
Convergence of Perturbation Expansions in Fermionic Models. Part 2
301
if ν is any effective bound for τ . By Prop. IX.1, Remark IX.14 and the induction hypothesis, Nimpr (g0 ) ≤
25 J α7
¯ N(K;2α) ¯ 1− α1 N(K;2α)
N (f ; 4α)
+8
¯ n 8 n−1 N(K) 2 ¯ α 1− 22 N(K)
+
α
n ¯ N(K) ¯ n 1+2 N (K) 2 −1 α =1 ¯ n+1 N(K) 1 ¯ . 1
α 2n−2
8 α 2n 1−
α2
N(K)
Since 1+2
n =1
¯ N(K) α 2 −1
= 1 + 2α
n
¯ N(K) α 2
=1
≤8 ≤8
1−
1 1 ¯ ¯ N(K)+ 4α N(K) 8α 2 ¯ 1− 12 N(K) α
1−
≤ 1 + 2α
1
1 7 4α + 8α 2
¯ N(K)
≤
1 ¯ N(K) α2 1 ¯ 1− 2 N(K) α
=
1−
1 2 ¯ ¯ N(K)+ α N(K) α2 1 ¯ 1− 2 N(K) α
≤8
1 1 1 ¯ 1+ 1 N(K)− ¯ ¯ 1− 12 N(K) 2 4α N(K) α 8α 1 8 1 ¯ , 1− 2α N(K)
we have Nimpr (g0 ) ≤
25 J α7
≤ ≤
n+1 ¯ N(K;2α) ¯ 1− α1 N(K;2α)
n−1 1 1 8 1 + 8 α82 1 ¯ ¯ 1− 1 N(K) ¯ α 2n−2 1− 2α N(K) 1− 22 N(K) 2 α α
n+1 5 ¯ N(K;2α) 1 2 J 8 1 8 n−1 N (f ; 4α) + 8 7 1 2n−2 1 2 1 ¯ ¯ ¯ α α α 1− α N(K;2α) 1− 2α N(K) 1− 2α N(K) n+1 ¯ N(K;2α) 25 J N (f ; 4α) [8 + 8] 3 ¯ α n+6 1− 2α N(K;2α) n+1 9 ¯ 2 J N (f ; 4α) N(3K;2α)¯ . α n+6 1− N(K;2α)
× ≤
N (f ; 4α)
2α
By Lemma IX.10 and Remark II.24, there exists a one–legged tail t11 , a two–legged tail t21 , each with at least three external legs, and a two–legged tail τ1 with at least four external legs and of degree two in η such that ¯ QK (:f :ξ,C ) = P Q Tn+1 (K) ¯ (:f :ξ,C ) P Q Tn (K) ¯ T1 (K) ¯ (:f :ξ,C ) +P Q Tn+1 (K), +P Q(t11 + t21 + τ1 )(:f :ξ,C ) + g1 (ψ) with
≤
29 J ¯ 2α) N (K; ¯ 4α) N (f ; 4α) N(Tn (K); α8 9 ¯ K;4α) 2 J ¯ 2α)n N( N (f ; 4α) N(K; ¯ α 2n+6 1− 1 N(K;4α)
≤
29 J α 2n+6
Nimpr (g1 ) ≤
N (f ; 4α)
n+1 ¯ N(K;4α) 1 ¯ 1− 2 N(K;4α)
1+
1 ¯ 4α) N (K; α2
α2
α
and N(t11 ), N (t21 ) ≤ Neff (τ1 ) ≤
2 ¯ N (K) ¯ ≤ 22n N (K) ¯ n+1 N T (K) α2 n α 2 4 4 ¯ N (K) ¯ ≤ 2n+2 N (K) ¯ n+2 . N Tn (K) α4 α
302
J. Feldman, H. Kn¨orrer, E. Trubowitz ( )
Similarly, for 1 ≤ ≤ n, by Lemma IX.11, there exists a one–legged tail t12 and a ( ) two–legged tail t22 , each with at least three external legs, and a two–legged tail τ ( ) with at least four external legs and of degree two in η such that ¯ Tn (K) ¯ QK (:f :ξ,C ) = P Q T +1 (K), ¯ Tn+1 (K) ¯ (:f :ξ,C ) P Q T (K), ( ) ( ) +P Q(t12 + t22 + τ ( ) )(:f :ξ,C ) + g2, (ψ) with ¯ 2α) N (T (K); ¯ 2α) N (f ; 4α) N(Tn (K);
¯ 4α) ¯ 4α) 1 + 12 N (K; × N (K; α
¯ K;4α) 29 J ¯ 2α)n 12 N (K; ¯ 2α) N( ≤ α 2n+6 N (f ; 4α) N(K; ¯ α 1− 12 N(K;4α) α
n+1 9 ¯ J ¯ 2α) N(1K;4α) ≤ α22n+6 N (f ; 4α) α12 N (K; ¯
Nimpr (g2, ) ≤
29 J α 10
1−
α2
N(K;4α)
and
( )
( ) N t12 , N t22 ≤
( ) ≤ Neff τ
12 12 ¯ N Tn (K) ¯ N T (K) ¯ ≤ 2(n+ ) ¯ n+ +1 , N (K) N (K) α4
α 8 2 8 ¯ N Tn (K) ¯ N T (K) ¯ ≤ 2(n+ )+2 N (K) ¯ n+ +2 . N (K) α6 α
In particular n
Nimpr g2, ≤ Nimpr g1 + =1
29 J α 2n+6
×
N (f ; 4α)
α
1 ¯ 1− 12 N(K;2α) α
≤
N t11 +
n =1
N
( ) t12 ,
n
( ) N t21 + N t22 ≤
Neff τ1 +
n−1
=1
τ
( )
+
=1
1 (n) 2τ
29 J α 2n+6
n+1 ¯ N(K;4α) ¯ 1− 12 N(K;4α)
N (f ; 4α)
n+1 ¯ N(K;4α) , ¯ 1− 22 N(K;4α) α
¯ n+1 12 N(K) ¯ , α 2n 1− 12 N(K) α
≤
¯ n+2 N(K) ¯ . α 2n+2 1− 12 N(K) 8
α
By Lemma IX.12, there exists a two–legged tail t23 with at least four external legs such that
P Q(t1 ) QK (:f :ξ,C ) = P Q(t23 )(:f :ξ,C ) and, by the induction hypothesis, N (t23 ) ≤
2 ¯ N (t1 ) N (K). α2
Also, by Lemma IX.10,
P Q(t2 ) QK (:f :ξ,C ) = P Q(t14 + t24 + t25 )(:f :ξ,C ) + g3 , where t14 , t24 are one– resp. two–legged tails with at least three external legs, fulfilling
N (t14 ), N (t24 ) ≤ α22 N (t2 ) N K¯ .
Convergence of Perturbation Expansions in Fermionic Models. Part 2
303
t25 = t2 ◦ K is a two–legged tail with at least four external legs fulfilling
N (t25 ) ≤ α12 N (t2 ) N K¯ , and the g3 term obeys
Nimpr g3 ≤
¯ 4α) 1 + N (f ; 4α) N(t2 ; 2α) N(K;
n−1 N(K;2α) n ¯ N (f ; 4α) 4 α22 1 ¯
≤
29 J α8 29 J α8
≤
2n+10 J α 2n+6
1−
N (f ; 4α)
2α
1 ¯ 4α) N (K; α2 ¯ N(K;4α) 1 ¯ 2 N(K;2α) 1− 2 N(K;4α)
n+1 ¯ N(K;4α) . ¯ 1− 22 N(K;4α)
α
α
Here we have used that Q t2 ◦ K, T1 (K) (:f :) has at least five external legs so that
P Q t2 ◦ K , T1 (K) (:f :) = 0. For the same reason, the term “P Q(τ )(:f :)” of Lemma IX.10 also vanishes. Finally, by Lemma IX.10,
P Q(τ ) QK (:f :ξ,C ) = P Q(t26 )(:f :ξ,C ) , where t26 = τ ◦ K is a two–legged tail with at least four external legs. By Def. IX.8, N(t26 ) ≤ α12 ν N K¯ for any effective bound ν for τ . Hence, by the induction hypothesis,
¯ n+1 ¯ n+2 N(K) 8 N(t26 ) ≤ α12 α82n N(1K) ¯ N K¯ = α 2n+2 1 ¯ . 1−
α2
1−
N(K)
α2
N(K)
Combining the results above, we see that
¯ P Ev Rn+1 :K:ζ,D ,C :f : ξ,C dµD (ζ ) = P Q Tn+1 (K) (:f :ξ,C ) ζ,D
+ 21
, ≥1 max{ , }=n+1
¯ T (K) ¯ (:f :ξ,C ) P Q T (K),
+P Q(t1 + t2 + τ )(:f :ξ,C ) + hn+1 (ψ; f ) with hn+1 (ψ; f ) = hn (ψ; F1 (ψ; f )) + g0 (ψ) + g1 (ψ) + 21 g2,n (ψ) +
n−1 =1
one– resp. two–legged tails with at least three external legs, t1 = t11 + 21 t12 + (n)
t2 = t21 + 21 t22 + (n)
n−1 =1 n−1 =1
( )
t12 + t14 , ( )
t22 + t23 + t24 + t25 + t26 ,
and the two–legged tail with four external legs and degree two in η τ = τ1 + 21 τ (n) +
n−1 =1
τ ( ) .
g2, (ψ) + g3
304
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By the estimates obtained above N(t1 ), N (t2 ) ≤
¯ n+1 12 N(K) ¯ α 2n 1− 12 N(K)
+
α
1 ¯ N (K) α2
2N (t1 ) + 2N (t2 ) + N (t2 )
¯ n+2 N(K) ¯ 1− 12 N(K) α ¯ n+1 12 N(K) 2 ¯ ¯ N ( K) + α52 N (K) 1 + ¯ α 2n 1− 12 N(K) 3α 2 α
n+1 ¯ n+1 ¯ 12 N(K) 5 8 n N(K;α) ¯ + 2 α2 ¯ α 2n 1− 2 N(K) 1− 2 N(K;α)
+ α 2n+2
8
≤ ≤ ≤ ≤ Neff (τ ) ≤
α2
n+1 ¯ 8 n N(K;α) ¯ α2 1− 22 N(K;α) α n+1 ¯ N(K;α) 4 α2 , ¯ 1− 22 N(K;α) α n+2 ¯ N(K) 8 ¯ , α 2n+2 1− 1 N(K) + 25
8 n
12 8n
4
n ¯ 8 n−1 N(K;α) ¯ α2 1− 22 N(K;α) α
α2
α2
and Nimpr (g0 ) + Nimpr (g1 ) + ≤
29 J α n+6
N (f ; 4α)
≤
210 J α n+6
N (f ; 4α)
n
Nimpr (g2, ) + Nimpr (g3 ) =1 n+1 ¯ N(K;4α) 1 2n+1 1 + + 3 αn αn ¯ 1− 2α N(K;4α) n+1 ¯ N(K;4α) , 3 ¯ 1− 2α N(K;4α)
so that
Nimpr hn+1 (ψ; f ) ≤ Nimpr hn (ψ; F1 (ψ; f )) + Nimpr (g0 ) + Nimpr (g1 ) n + Nimpr (g2, ) + Nimpr (g3 ) =1
≤
≤ since F0 (ψ; f ) = f .
210
J α5
n
n+1−m ¯ N(K;4α) 1 N Fm (ψ; f ); 4α 3 ¯ α n+1−m 1− 2α N(K;4α)
m=1 210 J + α n+6 N (f ; 4α) n 210 J 1 5 α n+1−m α m=0
n+1 ¯ N(K;4α) 3 ¯ 1− 2α N(K;4α)
n+1−m ¯ N(K;4α) N Fm (ψ; f ); 4α , 3 ¯ 1− 2α N(K;4α)
Corollary IX.16. Let K(ψ; ζ ; ξ, ξ , η) be a Grassmann function that vanishes for η = 0 and has degree at least four overall. Furthermore let f (ψ; ζ ; ξ ) be a Grassmann function of degree at least four in the variables ψ, ζ, ξ . Set 1 (:f : ξ,C ) dµD (ζ ) − :f (ψ; 0; ξ ):ξ,C h(ψ) = P Ev ζ,D − R:K:ζ,D ,C ∞
¯ (:f :ξ,C ) − 1 ¯ T (K) ¯ (:f :ξ,C ) . − Q T (K) Q T ( K), 2 , ≥1
=1
If α ≥ 8 and N (K; 16α)0 < 13 α, then Nimpr (h) ≤
210 J α6
N (f ; 16α)
N(K;16α) . 1− α3 N(K;16α)
Convergence of Perturbation Expansions in Fermionic Models. Part 2
305
If K(ψ; ζ ; ξ, ξ , η) = Uˆ (ψ +ζ ; ξ +ξ +η)− Uˆ (ψ +ζ ; ξ +ξ ), α ≥ 8 and N (Uˆ ; 32α)0 < 13 α, then Nimpr (h) ≤
210 J α6
N (f ; 16α)
N(Uˆ ;16α) . 1− α3 N(Uˆ ;32α)
Proof. By Prop. IX.15 1 (:f : ξ,C ) − :f : ξ,C dµD (ζ ) P Ev ζ,D ζ,D − R:K:ζ,D ,C ∞ P Ev R :K:ζ,D ,C (:f : ξ,C ) dµD (ζ ) = =
=1 ∞
ζ,D
¯ (:f :ξ,C ) + P Q T (K)
=1 ∞
+
1 2
, ≥1
¯ T (K) ¯ (:f :ξ,C ) P Q T (K),
h (ψ; f ) + P Q(t¯1 + t¯2 )(:f :ξ,C )
=1
tail t¯2 , each external legs. As with a one–legged tail t¯1 and a two–legged with at least three the degree of f is at least four, P Q(t¯1 + t¯2 )(:f :ξ,C ) = 0. Set h(ψ) = ∞ =1 h (ψ; f ). Then,
−m ∞ −1 ¯
210 J 1 N K;4α N Fm (ψ; f ); 4α . Nimpr (h) ≤ α 5 3 ¯ α −m 1− 2α N(K;4α)
=1 m=0
Define F˜m (ψ; f ) by
:F˜m (ψ; f ):ξ,C = Ev Rm :K:ζ,D ,C :f : ξ,C . ζ,D
Then, by Remark II.24, followed by Lemma VIII.8 and Lemma III.8,
N Fm (ψ; f ); 4α ≤ N F˜m (ψ; f ); 8α m N(:K:ζ,D ;8α) 1 ≤ 32α N :f :ζ,D ; 8α 2 1 1−
32α 2
N(:K:ζ,D ;8α)
so that Nimpr (h) ≤
210 J α5
=
210 J α5
=
∞ −1 =1 m=0 ∞ ∞
1 α −m
−m ¯ N(K;4α) 3 ¯ 1− 2α N(K;4α)
p ¯ N(K;4α) 1 α p 1− 3 N(K;4α) ¯ 2α
p=1 m=0 ¯ N(K;4α) 1 210 J 3 ¯ ¯ α 6 1− α1 N(K;4α) 1− 2α N(K;4α)
1 1−
≤
¯ N(K;4α) 1 210 J 3 ¯ ¯ α 6 1− α1 N(K;4α) 1− 2α N(K;4α) 1−
≤
N(K;16α) 210 J 1 3 α 6 1− α1 N(K;16α) 1− 2α N(K;16α) 1−
≤
N(K;16α) 210 J N f ; 16α . α 6 1− α3 N(K;16α)
1 m N(:K:ζ,D ;8α) 32α 2 1− 1 2 N(:K:ζ,D ;8α) 32α 1 m N(:K:ζ,D ;8α) 2 32α 1 1− 2 N(:K:ζ,D ;8α) 32α 1 N(:K:ζ,D ;8α) 32α 2 1− 1 2 N(:K:ζ,D ;8α) 32α
1 1
N f ; 16α
N f ; 16α
N f ; 16α
N f ; 16α
1 N(:K:ζ,D ;8α) 16α 2 1 N(K;16α) 16α 2
N f ; 16α
306
If
J. Feldman, H. Kn¨orrer, E. Trubowitz
K(ψ; ζ ; ξ, ξ , η) = Uˆ (ψ + ζ ; ξ + ξ + η) − Uˆ (ψ + ζ ; ξ + ξ )
so that ¯ K(ψ; ζ, ζ , ϕ; ξ, ξ , η) = K(ψ; ζ + ζ + ϕ; ξ, ξ , η) = Uˆ (ψ + ζ + ζ + ϕ; ξ + ξ + η) −Uˆ (ψ + ζ + ζ + ϕ; ξ + ξ ) and
:K(ψ; ζ ; ξ, ξ , η):ζ,D =
Uˆ (ψ +ζ +ζ ; ξ +ξ +η)− Uˆ (ψ +ζ +ζ ; ξ +ξ ) dµ−D (ζ )
then, by Lemma II.31 and Remark II.24, ¯ 4α) ≤ N (Uˆ ; 16α), N (K; N (:K:ζ,D ; 8α) ≤ N (Uˆ ; 32α). Consequently Nimpr (h) ≤ ≤
¯ N(K;4α) 1 1 210 J N f ; 16α 3 ¯ ¯ α 6 1− α1 N(K;4α) 1− 2α N(K;4α) 1− 1 2 N(:K:ζ,D ;8α) 16α
N(Uˆ ;16α) 210 J 1 1 N f ; 16α α 6 1− 1 N(Uˆ ;32α) 1− 3 N(Uˆ ;32α) 1− 1 N(Uˆ ;32α) α
≤
N(Uˆ ;16α) 210 J . α 6 1− 3 N(Uˆ ;32α) α
16α 2
2α
Proof of Theorem VI.10. Set K(ψ; ζ ; ξ, ξ , η) = Uˆ (ψ + ζ ; ξ + ξ + η) − Uˆ (ψ + ζ ; ξ + ξ ) and ¯ K(ψ; ζ, ζ , ϕ; ξ, ξ , η) = K(ψ; ζ + ζ + ϕ; ξ, ξ , η) = Uˆ (ψ + ζ + ζ + ϕ; ξ + ξ + η) −Uˆ (ψ + ζ + ζ + ϕ; ξ + ξ ), ˜ f (ψ; ζ ; ξ ) = fˆ(ψ + ζ ; ξ ). By Prop. VIII.9
SU,C (f ) = SU,C :f˜(ψ; 0; ξ ): ξ,C ψ,D . 1 (:f˜: ξ,C ) dµC (ξ ) dµD (ζ ) .. ψ,D = . Ev −R :K:ζ,D ,C
so that P f = P
Ev
1
−R:K:ζ,D ,C
ζ,D
(:f˜: ξ,C ) dµD (ζ ). ζ,D
Convergence of Perturbation Expansions in Fermionic Models. Part 2
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Therefore, by Cor. IX.16 and Remark II.24, there is g(ψ) with ∞
¯ (:f˜:ξ,C ) + 1 ¯ T (K) ¯ (:f˜:ξ,C ) + g Pf = P fˆ(ψ; 0) + Q T (K) Q T ( K), 2 , ≥1
=1
and Nimpr (g) ≤ ≤
N(Uˆ ;32α) 210 J N f˜; 16α α 6 1− 3 N(Uˆ ;32α) α
N(Uˆ ;32α) 210 J N fˆ; 32α . α 6 1− 3 N(Uˆ ;32α) α
Since fˆ and K have degree at least four overall,
¯ (:f˜: ξ,C ) = T (K) ¯ ◦ T1 (fˆ), P Q T (K) ζ,D
and by Remark VI.9,
¯ T (K) ¯ (:f˜: ξ,C ) = T (K) ¯ ◦ Rung(fˆ) ◦ T (K). ¯ P Q T (K), ζ,D
¯ = T (Uˆ ). Thus Furthermore, by part (ii) of Remark IX.14, T (K) Pf = P fˆ(ψ, 0) +
∞
T (Uˆ ) ◦ T1 (fˆ) +
=1
1 2
, ≥1
T (Uˆ ) ◦ Rung(fˆ) ◦ T (Uˆ ) + g.
X. Example: A Vector Model We consider a model, which while simple, still captures the main features of one scale of a many fermion model. Let F be a finite set of at least two “colours” (Farben) and X be a finite set of points in “space–time”. Let V be the complex vector space with basis ξc,x c ∈ F, x ∈ X and V be the complex vector space with basis ψc,x c ∈ F, x ∈ X . Let C(x, x ), x, x ∈ X be a skew symmetric matrix and define the covariance
C ξc,x , ξc ,x = δc,c C(x, x ). An antisymmetric function W on (F × X )n , with n even, is said to be colour preserving if it is of the form
W (c1 , x1 ), · · · (cn , xn ) = Ant δc1 ,c2 · · · δcn−1 ,cn w(x1 , · · · , xn ) , (X.1) where Ant means antisymmetrization. An example, with n = 2, is the function δc,c C(x, x ). An even element W of the Grassmann algebra V is said to be colour preserving if it is of the form
W= Wn (c1 , x1 ), · · · (cn , xn ) ξc1 ,x1 · · · ξcn ,xn n∈2N
c1 ,··· ,cn ∈F x1 ,··· ,xn ∈X
with each coefficient function Wn colour preserving.
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J. Feldman, H. Kn¨orrer, E. Trubowitz
For p odd, we define the (0–dimensional) norm ϕp of a complex valued function ϕ on (F × X )n by ϕp = max sup 1≤i1 <···
× sup
xk ∈X
ci1 ,··· ,cip ∈F
ci ∈F i =i1 ,··· ,ip
ϕ (c1 , x1 ), · · · (cn , xn ) . xj ∈X j =k
Also, for a colour preserving function W on (F × X )n , we define ! " |||W ||| = inf max sup w(x1 , · · · , xn ) w satisfies (X.1) . 1≤k≤n xk ∈X
xj ∈X j =k
Then n−p−1
W p ≤ |F| 2 |||W |||, |||W ||| ≤ (n − 1)!! W n−1
if
|F| ≥ n2 .
(X.2)
In particular, if n = 4, W 1 ≤ |F| |||W ||| ≤ 3|F| W 3 . Every element f ∈ V ⊗n has a unique representation
f = ϕ (c1 , x1 ), · · · (cn , xn ) ξc1 ,x1 ⊗ · · · ⊗ ξcn ,xn c1 ,··· ,cn ∈F x1 ,··· ,xn ∈X
with ϕ a complex valued function on (F × X )n . We set f p = ϕp . Observe that, for each odd p, · p is a family of symmetric norms on the spaces V ⊗n , in the sense of Def. II.18. Proposition X.1. Suppose that X is a disjoint union of two subsets Xa and Xc such that C(x, x ) = 0
if both x, x ∈ Xa or both x, x ∈ Xc .
Assume furthermore that there is a Hilbert space H and vectors wx ∈ H, x ∈ X such that for all x ∈ Xa , x ∈ Xc . C(x, x ) = wx , wx H Set b = 2 supx∈ X wx , c = supx∈X |C(x, x )|. x ∈X
Then (C, 0) has integration constants c, b for the configuration · p of seminorms, in the sense of Def. VI.13.
Convergence of Perturbation Expansions in Fermionic Models. Part 2
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Proof. We first verify that b is an integral bound for C with respect to each family · p of seminorms. Set H = L2 (F) ⊗ H, wc,x = δc ⊗ wx , where δc is the function on F which vanishes except at c and takes the value one there. Then, for all c, c ∈ F, if both x, x ∈ Xa or both x, x ∈ Xc
C(ξc,x , ξc ,x ) = 0 and
$ # C(ξc,x , ξc ,x ) = δc,c wx , wx H = wc,x , wc ,x
H
Furthermore
for all x ∈ Xa , x ∈ Xc .
wc,x H ≤ wx H ≤ 2b .
Let Vc (respectively Va ) be the subspace of V generated by ξc,x x ∈ Xc , c ∈ F (respectively ξc,x x ∈ Xa , c ∈ F ). By Prop. B.1, m ξc1 ,x1 · · · ξcm ,xm dµC (ξ ) ≤ 2b . As in Example II.26, b is an integral bound for C with respect to each family · p of seminorms. Also 2 sup C(ξc,x , ξc ,x ) = sup |C(x, x )| ≤ sup wx wx ≤ b4 . (X.3) c,c ∈F x,x ∈X
x,x ∈X
x,x ∈X
We now verify the contraction estimates of Def. VI.13. Let
f = ϕ (c1 , x1 ), · · · (cn , xn ) ξc1 ,x1 ⊗ · · · ⊗ ξcn ,xn ∈ V ⊗n , f =
c1 ,··· ,cn ∈F x1 ,··· ,xn ∈X
ϕ (c1 , x1 ), · · · (cn , xn ) ξc1 ,x1 ⊗ · · · ⊗ ξcn ,xn ∈ V ⊗n ,
c1 ,··· ,cn ∈F x1 ,··· ,xn ∈X
and, for 1 ≤ k ≤ n and 1 ≤ k ≤ n , k (c1 , · · · , cn ) = sup
xk ∈X
k (c1 , · · · , cn ) = sup
xk ∈X
ϕ (c1 , x1 ), · · · (cn , xn ) , xi ∈X i =k
ϕ (c , x ), · · · (c , x ) . 1
1
n
n
xi ∈X i =k
Observe that, for all 1 ≤ k ≤ n, all 1 ≤ i1 < · · · < ip ≤ n and all ci1 , · · · , cip , k (c1 , · · · , cn ) ≤ ϕp = f p , ci ∈F i =i1 ,··· ,ip
and similarly for .
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For the first contraction estimate of Def. VI.13, let 1 ≤ i ≤ n and 1 ≤ j ≤ n . By the symmetry of the norms, it suffices to consider i = j = 1. Set γ ((c2 ,x2 ),···(cn ,xn ),(c2 ,x2 ),···(cn ,xn )) ϕ((c1 ,x1 ),··· ,(cn ,xn )) δc1 ,c1 C(x1 ,x1 ) ϕ ((c1 ,x1 ),··· ,(cn ,xn )). = c1 ,c1 ∈F x1 ,x1 ∈X
Then ⊗ f )
ConC (f 1→n+1
=
c1 ,··· ,cn+n −2 ∈F x1 ,··· ,xn+n −2 ∈X
γ (c1 , x1 ), · · · , (cn+n −2 , xn+n −2 ) ξc1 ,x1 ⊗ · · · ⊗ ξcn+n −2 ,xn+n −2 .
In particular ConC (f ⊗ f )p = γ p .
i→n+j Fix any 1 ≤ i1 < · · · < ip ≤ n + n − 2 and colours c , · · · , c . Set q = max ν iν ≤ i i p 1 n − 1 with the convention that if ν iν ≤ n − 1 = ∅, then q = 0. Set j1 = i1 + 1, · · · , jq = iq + 1; j1 = iq+1 − n + 2, · · · , jp−q = ip − n + 2 and
djν = ciν ν = 1, · · · , q, dj = ciq+ν ν = 1, · · · , p − q. ν
n+n −2. First assume that k
Also fix 1 ≤ k ≤ ≤ n−1. We have, for all c1 , · · · , cn+n −2 ∈ F, γ (c1 , x1 ), · · · , (cn+n −2 , xn+n −2 ) sup xk
xi ,i =k
≤
k+1 (c, c1 , · · · , cn−1 ) δc,c c 1 (c , cn , · · · , cn+n −2 ).
c,c ∈F
Now assume that q is odd. Then p − q is even and γ (c1 , x1 ), · · · , (cn+n −2 , xn+n −2 ) sup ci ∈F i =i1 ,··· ,ip
≤c
xk
xi ,i =k
dj ∈F d ∈F j =1,j1 ,··· ,jq =1,j ,··· ,j p−q 1
≤c
k+1 (c, d2 , · · · , dn ) δc,c 1 (c , d2 , · · · , dn )
c,c ∈F
k+1 (d1 , · · · , dn ) sup
dj ∈F j =j1 ,··· ,jq
≤ c f q f p−q+1 .
d1
1 (d1 , d2 , · · · , dn )
d ∈F =1,j1 ,··· ,jp−q
In the event that q is even, one obtains in a similar way that γ (c1 , x1 ), · · · , (cn+n −2 , xn+n −2 ) ≤ c f q+1 f p−q . sup ci ∈F i =i1 ,··· ,ip
xk
xi ,i =k
The case that k ≥ n follows by interchanging the roles of f and f .
Convergence of Perturbation Expansions in Fermionic Models. Part 2
311
For the second contraction estimate of Def. VI.13 it suffices to consider, by the sym metry of the norms, ConC ConC ConC (f ⊗ f )p . Set 1→n+1 2→n+2 3→n+3
γ˜ ((c4 ,x4 ),···(cn ,xn ),(c4 ,x4 ),···(cn ,xn )) =
c1 ,c1 ,c2 ,c2 ,c3 ,c3 ∈F x1 ,x1 ,x2 ,x2 ,x3 ,x3 ∈X
ϕ((c1 ,x1 ),··· ,(cn ,xn ))δc1 ,c1 C(x1 ,x1 )δc2 ,c2
× C(x2 ,x2 )δc3 ,c3 C(x3 ,x3 )ϕ ((c1 ,x1 ),··· ,(cn ,xn )). Then ConC ConC ConC (f 1→n+1 2→n+2 3→n+3 =
c1 ,··· ,cn+n −6 ∈F x1 ,··· ,xn+n −6 ∈X
⊗ f )
γ˜ ((c1 ,x1 ),··· ,(cn+n −6 ,xn+n −6 )) ξc1 ,x1 ⊗ · · · ⊗ ξcn+n −6 ,xn+n −6 .
Fix any 1 ≤ i1 < · · · < ip ≤ n + n − 6 and colours ci1 , · · · , cip . Set q = max ν iν ≤ = ip − n + 6 n − 3 . Set j1 = i1 + 3, · · · , jq = iq + 3; j1 = iq+1 − n + 6, · · · , jp−q and djν = ciν , ν = 1, · · · , q, dj = ciq+ν , ν = 1, · · · , p − q. ν
Also fix 1 ≤ k ≤ n+n −6. First assume that k ≤ n−3. We have, for all c1 , · · · , cn+n −6 ∈ F, γ˜ (c1 , x1 ), · · · , (cn+n −6 , xn+n −6 )
sup xk
xi ,i =k
≤
d1 ,d2 ,d3 ∈F d1 ,d2 ,d3 ∈F
k+1 (d1 , d2 , d3 , c1 , · · · , cn−3 ) δd1 ,d1 δd2 ,d2 δd3 ,d3 c
2 sup |C(y, y )|
y,y ∈X
× 1 (d 1 , d2 , d3 , cn−2 , · · · , cn+n −6 ) 4 ≤b c k+1 (d1 , d2 , d3 , c1 , · · · , cn−3 )1 (d1 , d2 , d3 , cn−2 , · · · , cn+n −6 ) d1 ,d2 ,d3 ∈F
by (X.3). Now assume that q is odd. Then
sup
ci ∈F i =i1 ,··· ,ip
≤ b4 c
xk
γ˜ (c1 , x1 ), · · · , (cn+n −6 , xn+n −6 ) xi ,i =k
k+1 (d1 , · · · , dn )
d1 ,d2 ,d3
d ∈F =1,2,3,j1 ,··· ,jp−q
dj ∈F j =j1 ,··· ,jq
≤ b4 c f q f p−q+3 . The remaining cases are similar.
sup
1 (d1 , d2 , · · · , dn )
312
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Theorem X.2. Let W be a colour preserving function on (F × X )n and
W= W (c1 , x1 ), · · · (c4 , x4 ) ψc1 ,x1 · · · ψc4 ,x4 c1 ,··· ,c4 ∈F x1 ,··· ,x4 ∈X
be the associated interaction. Let
W (ψ) = C :W: = log Z1 e:W :(ψ+ξ ) dµC (ξ ) where Z = e:W :(ξ ) dµC (ξ ) and C is the covariance of Prop. X.1. Since C and W are colour preserving, we can write
W = Wn (c1 , x1 ), · · · (cn , xn ) ψc1 ,x1 · · · ψcn ,xn n∈2N
c1 ,··· ,cn ∈F x1 ,··· ,xn ∈X
with colour preserving coefficients Wn . If |||W ||| ≤ ∞
1 , 238 b2 c |F |
then
α0n−6 bn−6 Wn 1 ≤ 248 c |F|2 |||W |||2
n=6
with α0 = √ 3
1 , 229 b2 c |F | |||W |||
W − W + 4
1 4
and
∞
|||W2 ||| ≤ 261 b4 c |F| |||W |||2 , (−12)r+1 W ◦ (C ◦ W )r ≤ 257 b2 c |||W |||2 ,
r=1
where W ◦ (C ◦ W )r is the ladder W
W
W
W
W
r + 1 W ’s with r + 1 rungs W and propagator C, defined in Appendix C. Proof. Set J = |F1 | . For f ∈ V ⊗n , set f = f 1 + |F| f 3 + |F|2 f 5 , f impr = f 1 + |F| f 3 . By Lemma VI.15, (C, 0) have improved integration constants c, b, J for the families · and · impr of seminorms. By (X.2), W ≤ 2 |F| |||W |||. Let α1 = 8 and α ∈ {α0 , α1 }. Then α ≥ 8 and, using the notation of Def. II.23, N(W ; 64α) = 224 α 4 b2 cW ≤ 225 α 4 b2 c |F| |||W ||| ≤
1 16 α
Convergence of Perturbation Expansions in Fermionic Models. Part 2
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since b2 c |F| |||W ||| ≤ 2138 . Therefore the hypotheses of Theorem VI.6 are fulfilled. By part (i) of Theorem VI.6, N (W − W; α) ≤
N(W ;32α)2 1 2α 2 1− 12 N(W ;32α)
≤ 248 α 6 b4 c2 |F|2 |||W |||2 .
α
In particular c b2
∞
α n bn Wn 1 ≤ 248 α 6 b4 c2 |F|2 |||W |||2
n=6
so that ∞
α n−6 bn−6 Wn 1 ≤ 248 c |F|2 |||W |||2 .
n=6
By part (ii) of Theorem VI.6 and Prop. C.4, both Nimpr W2 ; α and Nimpr W4 − W + 1 ∞ r+1 W ◦ (C ◦ W )r ; α are bounded by r=1 (−12) 4 210 J N(W ;64α)2 α 6 1− α8 N(W ;64α)
≤ 261 α 2 b4 c2 |F| |||W |||2 .
Using (X.2), |||W2 ||| ≤ W2 impr = Similarly, setting F = W4 − W +
1 α2 c 1 4
Nimpr W2 ; α ≤ 261 b4 c |F| |||W |||2
∞
|||F ||| ≤ 3F 3 ≤ |F3 | F impr =
r=1 (−12) 3
α 4 b2 c |F |
With α = α1 = 8 , we get the desired bound.
r+1 W
◦ (C ◦ W )r ,
Nimpr F ; α ≤ 263 α12 b2 c |||W |||2 .
The j th shell, j ≥ 1, of the many fermion model of [FKTf1] behaves qualitatively like the vector model that we have just discussed. The covariance for the j th shell of that many fermion model is C (j ) (k) =
ν (j ) (k) ık0 −e(k) ,
where ν (j ) (k) is approximately the characteristic function with support Sj =
k = (k0 , k) ∈ R × Rd
1 Mj
≤ |ık0 − e(k)| <
1 M j −1
.
To define seminorms in position space that mimic well, in our context, the supremum in momentum space, we introduce sectorizations. We choose a projection k → πF (k) onto the Fermi curve F and a decomposition of the Fermi curve into disjoint intervals I1 , · · · , IN each of length l between M1j and M1j/2 . The sectorization is the collection of “sectors” s = πF−1 (I ) ∩ Sj , = 1, · · · , N .
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s3
s2
sN−2
sN
s1
F Sj
Let V be the complex vector space with basis ψs,x , ψ¯ s,x s ∈ , x ∈ R × R2 and define the covariance3 ¯ ¯ ¯ C(ψs,x , ψs ,x ) = C(ψs,x , ψs ,x ) = 0, C(ψs,x , ψs ,x ) = δs,s d 3 k eık·x C (j ) (k). s
The norms on V ⊗n are similar to those of the vector model. In this case b2 = const Ml j ,
c = const M j ,
J =
1 ||
= const l,
and the conclusions of Theorem X.2 become W − W + 4
1 4
∞
|||W2 ||| ≤ const Ml j |||W |||2 , (−12)r+1 W ◦ (C ◦ W )r ≤ const l |||W |||2
r=1
for all four–legged interactions W whose norm |||W ||| is bounded by a sufficiently small, but j –independent, constant.
Appendix C. Ladders Expressed in Terms of Kernels Let V be a complex vector space and {ξi } a system of generators for V that is indexed by i in a set X . Definition C.1. i) Depending on the context, a complex valued function on X 4 is called a four legged kernel over X or a bubble propagator over X . ii) To a four legged kernel f over X we associate the Grassmann function Gr(ξ ; f ) =
f (i1 ,i2 ,i3 ,i4 ) ξi1 ξi2 ξi3 ξi4
i1 ,i2 ,i3 ,i4 ∈X
in the complex Grassmann algebra
V.
3 We are deliberately ignoring many technical fine points; in particular, we ignore spin, smoothness of partitions of unity, factors of 2π.
Convergence of Perturbation Expansions in Fermionic Models. Part 2
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iii) If A and B are covariances on V then the tensor product A ⊗ B(i1 , i2 , i3 , i4 ) = A(ξi1 , ξi3 )B(ξi2 , ξi4 ) is a bubble propagator over X . In particular we shall consider the bubble propagator C = C ⊗ C + C ⊗ D + D ⊗ C, where C and D are the covariances of Sect. V.2. iv) Let f1 and f2 be functions on X 4 . The operator product of f1 and f2 is (f1 ◦ f2 )(i1 , i2 , i3 , i4 ) = f1 (i1 , i2 , j1 , j2 ) f2 (j1 , j2 , i3 , i4 ) j1 ,j2 ∈X
whenever the sum is well–defined. v) Let f be a four legged kernel over X . The antisymmetrization of f is the four legged kernel
1 Antf (i1 , i2 , i3 , i4 ) = 4! sgn(π ) f (iπ(1) , iπ(2) , iπ(3) , iπ(4) ). π ∈S4
Clearly Gr(ξ ; f ) = Gr(ξ ; Antf ). The kernel f is called antisymmetric if f = Antf . Lemma C.2. Let E= i1 ,i2 ,i3 ,i4 ∈X
e202 (i1 ,i2 ,i3 ,i4 ) ψi1 ψi2 ξi3 ξi4 + 2e211 (i1 ,i2 ,i3 ,i4 ) ψi1 ψi2 ζi3 ξi4
be an end as in Def. VI.5, where e202 and e211 are four legged kernels over X that are antisymmetric under permutations of their four arguments. i) If E =
i1 ,i2 ,i3 ,i4 ∈X
e202 (i1 ,i2 ,i3 ,i4 ) ψi1 ψi2 ξi3 ξi4 + 2e211 (i1 ,i2 ,i3 ,i4 ) ψi1 ψi2 ζi3 ξi4
and e211 then is another end with antisymmetric four legged kernels e202
. E ◦ E = −2 Gr ψ; e202 ◦ (C ⊗ C) ◦ e202 + e211 ◦ (C ⊗ D + D ⊗ C) ◦ e211
ii) If ρ=
i1 ,i2 ,i3 ,i4 ∈X
ρ0202 (i1 ,i2 ,i3 ,i4 )ξi1 ξi2 ξi3 ξi4 + 2 ρ1102 (i1 ,i2 ,i3 ,i4 )ζi1 ξi2 ξi3 ξi4
+2 ρ0211 (i1 ,i2 ,i3 ,i4 )ξi1 ξi2 ζi3 ξi4 + 4 ρ1111 (i1 ,i2 ,i3 ,i4 )ζi1 ξi2 ζi3 ξi4
is a rung with antisymmetric kernels ρ0202 , ρ1102 , ρ0211 , ρ1111 , then E ◦ ρ = −2 202 (i1 ,i2 ,i3 ,i4 ) ψi1 ψi2 ξi3 ξi4 + 2 211 (i1 ,i2 ,i3 ,i4 ) ψi1 ψi2 ζi3 ξi4 i1 ,i2 ,i3 ,i4 ∈X
with 202 = e202 ◦ (C ⊗ C) ◦ ρ0202 + e211 ◦ (C ⊗ D + D ⊗ C) ◦ ρ1102 , 211 = e202 ◦ (C ⊗ C) ◦ ρ0211 + e211 ◦ (C ⊗ D + D ⊗ C) ◦ ρ1111 .
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Proof.
i) E ◦ E =
e202 (i1 ,i2 ,i3 ,i4 ) ψi1 ψi2 :ξi3 ξi4 :C + 2e211 (i1 ,i2 ,i3 ,i4 ) ψi1 ψi2 ζi3 ξi4
i1 ,i2 ,i3 ,i4 1 , 2 , 3 , 4
× e202 ( 1 , 2 , 3 , 4 ) ψ 1 ψ 2 :ξ 3 ξ 4 :C + 2e211 ( 1 , 2 , 3 , 4 ) ψ 1 ψ 2 ζ 3 ξ 4
× dµC (ξ ) dµD (ζ ) =− ψi1 ψi2 ψ 1 ψ 2 4e211 (i1 ,i2 ,i3 ,i4 ) e211 ( 1 , 2 , 3 , 4 ) D(ζi3 ,ζ 3 )C(ξi4 ,ξ 4 ) i1 ,i2 ,i3 ,i4 1 , 2 , 3 , 4
+e202 (i1 ,i2 ,i3 ,i4 ) e202 ( 1 , 2 , 3 , 4 ) C(ξi3 ,ξ 3 ) C(ξi4 ,ξ 4 ) − C(ξi3 ,ξ 4 ) C(ξi4 ,ξ 3 ) =− ψi1 ψi2 ψ 1 ψ 2 4e211 (i1 ,i2 ,j1 ,j2 ) e211 ( 1 , 2 ,k1 ,k2 ) D(ζj1 ,ζk1 )C(ξj2 ,ξk2 ) i1 ,i2 ,j1 ,j2 1 , 2 ,k1 ,k2
+e202 (i1 ,i2 ,j1 ,j2 ) e202 ( 1 , 2 ,k1 ,k2 )C(ξj1 ,ξk1 ) C(ξj2 ,ξk2 )
−e202 (i1 ,i2 ,j1 ,j2 ) e202 ( 1 , 2 ,k2 ,k1 )C(ξj1 ,ξk1 ) C(ξj2 ,ξk2 ) =− (i1 ,i2 , 1 , 2 ) ψi1 ψi2 ψ 1 ψ 2 4e211 ◦ (C ⊗ D) ◦ e211 i1 ,i2 , 1 , 2
(i1 ,i2 , 1 , 2 ) +2 e202 ◦ (C ⊗ C) ◦ e202 = −2 ψi1 ψi2 ψi3 ψi4 e211 ◦ (C ⊗ D + D ⊗ C) ◦ e211 (i1 ,i2 ,i3 ,i4 ) i1 ,i2 ,i3 ,i4
(i1 ,i2 ,i3 ,i4 ) . + e202 ◦ (C ⊗ C) ◦ e202
For the last two equalities we used the antisymmetry of the kernels. ii) Similarly, E◦ρ =
ψi1 ψi2
e202 (i1 ,i2 ,i3 ,i4 ) :ξi3 ξi4 :C + 2e211 (i1 ,i2 ,i3 ,i4 ) ζi3 ξi4
i1 ,i2 ,i3 ,i4 1 , 2 , 3 , 4
× ρ0202 ( 1 , 2 , 3 , 4 ) :ξ 1 ξ 2 :C + 2 ρ1102 ( 1 , 2 , 3 , 4 ) ζ 1 ξ 2 ξ 3 ξ 4 + 2 ρ0211 ( 1 , 2 , 3 , 4 ) :ξ 1 ξ 2 :C + 4 ρ1111 ( 1 , 2 , 3 , 4 ) ζ 1 ξ 2 ζ 3 ξ 4 × dµC (ξ ) dµD (ζ ) = −2 ψi1 ψi2 ξi3 ξi4 e211 ◦ (C ⊗ D + D ⊗ C) ◦ ρ1102 (i1 ,i2 ,i3 ,i4 )
+ e202 ◦ (C ⊗ C) ◦ ρ0202 (i1 ,i2 ,i3 ,i4 ) −4 ψi1 ψi2 ζi3 ξi4 e211 ◦ (C ⊗ D + D ⊗ C) ◦ ρ1111 (i1 ,i2 ,i3 ,i4 ) i1 ,i2 ,i3 ,i4
+ e202 ◦ (C ⊗ C) ◦ ρ0211 (i1 ,i2 ,i3 ,i4 ) .
i1 ,i2 ,i3 ,i4
Lemma C.3. Let f be an antisymmetric four legged kernel and n ≥ 0. Set F = Gr(f ; ξ ) and En (F ) = E(F ) ◦ ρ(F ) ◦ ρ(F ) · · · ρ(F )
Convergence of Perturbation Expansions in Fermionic Models. Part 2
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with n copies of ρ(F ), where E(F ) and ρ(F ) were defined in Def. VI.5.iii. Then
En (F ) =
i1 ,i2 ,i3 ,i4 ∈X
fn (i1 ,i2 ,i3 ,i4 ) ψi1 ψi2 ξi3 ξi4 + 2 ψi1 ψi2 ζi3 ξi4
with fn =
(−1)n n+1 f 2 (12)
◦ (C ◦ f )n .
Proof. Observe that 4 f (i1 ,i2 ,i3 ,i4 ) ψi1 ψi2 ξi3 ξi4 + 2 ψi1 ψi2 ζi3 ξi4 , 2 2 ,i3 ,i4 i1 ,i 4 ρ(F ) = f (i1 ,i2 ,i3 ,i4 ) 2 i ,i ,i ,i 1 2 3 4 × ξi1 ξi2 ξi3 ξi4 + 2 ζi1 ξi2 ξi3 ξi4 + 2 ξi1 ξi2 ζi3 ξi4 + 4 ζi1 ξi2 ζi3 ξi4 .
E(F ) =
(C.1)
Since E(F ) = E0 (F ), this proves the case n = 0. We now perform induction on n. By Lemma C.2 and the induction hypothesis En+1 (F ) =
i1 ,i2 ,i3 ,i4 ∈X
202 (i1 ,i2 ,i3 ,i4 ) ψi1 ψi2 ξi3 ξi4 + 2 211 (i1 ,i2 ,i3 ,i4 ) ψi1 ψi2 ζi3 ξi4
with 202 = 211 = −2
4
fn ◦ (C ⊗ C) ◦ f + fn ◦ (C ⊗ D + D ⊗ C) ◦ f
2 = −12 fn ◦ C ◦ f = fn+1 .
Proposition C.4. Let f be an antisymmetric four legged kernel. Set F = Gr(f ; ξ ) and let Lr (F ) = E(F ) ◦ ρ(F ) ◦ · · · ◦ ρ(F ) ◦ E(F ) the ladder of length r ≥ 1 with r − 1 rungs ρ(F ). Then Lr (F ) =
(−1)r r+1 Gr ψ; 2 (12)
f ◦ (C ◦ f )r .
Proof. By Lemma C.2, Lemma C.3 and (C.1), Lr (F ) = Er−1 (F ) ◦ E(F )
4 = −2 Gr ψ; fr−1 ◦ (C ⊗ C) ◦ f + fr−1 ◦ (C ⊗ D + D ⊗ C) ◦ f 2 = −12 Gr ψ; fr−1 ◦ C ◦ f
r r+1 = (−1) Gr ψ; f ◦ (C ◦ f )r . 2 (12)
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Notation Not’n Z(f ) V AV A · · , nr ] m [n1 , · E, AE Ev ξ ζ e i i dµC (ξ ) C (W )(ψ) S (f ) R RC (K1 , · · · , K )
RK,C (f ) R˜ C (K2 , · · · , K )(f ) :e ξi ζi :ξ,C C onC , ConC i→j ξ →ξ
Nd c b c, b, J N(f ; α) ρ(F ) E(F ) Lr (F ) · p P Q(K¯ 1 , · · · , K¯ ) Gr(ξ ; f ) ◦
Description
Reference
degree zero component of f Grassmann algebra over V Grassmann algebra over V with coefficients in A partially antisymmetric elements of Am ⊗ V ⊗(n1 +···+nr ) enlarged algebra evaluation map −1/2 ζi Cij ζj =e Grassmann Gaussian integral = log Z1 eW (ψ+ξ ) dµC (ξ ) renormalization group map 1 = Z(U,C) f (ξ ) eU (ξ ) dµC (ξ ) Schwinger functional R–operator th Taylor coefficient of R . . :K(ξ,ξ ,η):ξ . − 1 .. η f (η) dµC (ξ ) dµC (η) . ξ . .e . :Ki (ξ, ξ + ξ , η ):ξ .. η f (η + η ) dµC (ξ ) dµC (η ) .
Definition II.1.iii Example II.2 Example II.2 Definition II.21 Definition VIII.5 Definition VIII.5 before Definition II.3 Definitions II.3, II.27 before Remark III.1 before Theorem III.2 (III.2)
=e contractions
after Remark II.4 Definitions II.5, II.9
i=2 1/2 ζi Cij ζj
e ξi ζi
Wick ordering
norm domain contraction bound integral bound improved integration constants 1 c m,n1 ,··· ,nr ≥0 α |n| b|n| fm;n1 ,··· ,nr b2 rung end ladder of length r configuration of seminorms projects :f (ψ; ξ ):ξ to f4,0 (ψ, 0) + f2,0 (ψ, 0) enlarged algebra analog of RC (K1 , · · · , K ) Grassmann function convolution
Definition III.4 (VII.1)
Definition II.14 Definition II.25.i Definition II.25.ii Definition VI.1 Definition II.23 Definition VI.5 Definition VI.5 Definition VI.5 Definition VI.13 Definition VII.8 Definition VIII.1 Definition C.1.ii Definition C.1.iv
References [BS] [FKLT1] [FKLT2] [FKT1] [FKTf1] [FKTo3] [FMRT] [FST1] [FST2]
Berezin, F., Shubin, M.: The Schr¨odinger Equation. Dordrecht: Kluwer, 1991. Supplement 3: D.Le˘ites, Quantization and supermanifolds Feldman, J., Kn¨orrer, H., Lehmann, D., Trubowitz, E.: Fermi Liquids in Two-Space Dimensions. In: Constructive Physics V. Rivasseau, (ed.), Springer Lecture Notes in Physics 446, Berlin-Heidelberg-New York: Springer, 1995, pp. 267–300 Feldman, J., Kn¨orrer, H., Lehmann, D., Trubowitz, E.: Are There Two Dimensional Fermi Liquids? In: Proceedings of the XIth International Congress of Mathematical Physics, D. Iagolnitzer, (ed.), Cambridge, MA: International Press, 1995, pp. 440–444 Feldman, J., Kn¨orrer, H., Trubowitz, E.: A Representation for Fermionic Correlation Functions. Commun. Math. Phys. 195, 465–493 (1998) Feldman, J., Kn¨orrer, H., Trubowitz, E.: A Two Dimensional Fermi Liquid, Part 1: Overview. Commun. Math. Phys. 247, 1–47 (2004) Feldman, J., Kn¨orrer, H., Trubowitz, E.: Single Scale Analysis of Many Fermion Systems, Part 3: Sectorized Norms. Rev. Math. Phys. 15, 1039–1120 (2003) Feldman, J., Magnen, J., Rivasseau, V., Trubowitz, E.: An Infinite Volume Expansion for Many Fermion Green’s Functions. Helv. Phys. Acta 65, 679–721 (1992) Feldman, J., Salmhofer, M., Trubowitz, E.: Perturbation Theory around Non-nested Fermi Surfaces, I: Keeping the Fermi Surface Fixed. J. Stat. Phys. 84, 1209–1336 (1996) Feldman, J., Salmhofer, M., Trubowitz, E.: Regularity of the Moving Fermi Surface: RPA Contributions. Commun. Pure and Appl. Math. LI, 1133–1246 (1998)
Convergence of Perturbation Expansions in Fermionic Models. Part 2 [FST3] [FST4]
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Feldman, J., Salmhofer, M., Trubowitz, E.: Regularity of Interacting Nonspherical Fermi Surfaces: The Full Self–Energy. Commun. Pure and Appl. Math. LII, 273–324 (1999) Feldman, J., Salmhofer, M., Trubowitz, E.: An inversion theorem in Fermi surface theory. Commun. Pure and Appl. Math. LIII, 1350–1384 (2000)
Communicated by J.Z. Imbrie
Commun. Math. Phys. 247, 321–351 (2004) Digital Object Identifier (DOI) 10.1007/s00220-004-1077-8
Communications in
Mathematical Physics
Second Quantization of the Elliptic Calogero-Sutherland Model Edwin Langmann Mathematical Physics, Department of Physics, KTH, AlbaNova, 106 91 Stockholm, Sweden Received: 15 February 2001 / Accepted: 30 January 2004 Published online: 7 April 2004 – © Springer-Verlag 2004
Abstract: We construct a quantum field theory model of anyons on a circle and at finite temperature. We find an anyon Hamiltonian providing a second quantization of the elliptic Calogero-Sutherland model. This allows us to prove a remarkable identity which is a starting point for an algorithm to construct eigenfunctions and eigenvalues of the elliptic Calogero-Sutherland Hamiltonian. 1. Introduction This is the first of two papers on the elliptic Calogero-Sutherland model, providing details and proofs of various results announced in Ref. [L1] and culminating in an algorithm to solve this model [L2]. The results here are based on loop group techniques: we relate this model to a quantum field theory model of anyons which, as we believe, is interesting also in its own right. We have attempted to keep our discussion reasonably self-contained (and in particular explain all the physics terminology we use) and make it accessible also for mathematicians. In this chapter we specify what we mean by finite temperature anyons and second quantization, and we also provide some background to the elliptic Calogero-Sutherland model. We then give the plan for the rest of the paper by summarizing our results. Background. By anyons we mean quantum fields φ(x) parametrized by a coordinate x on the circle, −π ≤ x < π , obeying exchange relations φ(x)φ(y) = e±iπλ φ(y)φ(x),
(1)
where λ > 0 is the so-called statistics parameter. These anyons generalize bosons and fermions which correspond to the special cases where the phase factor e±iπλ is +1 and
Work supported by the Swedish Natural Science Research Council (NFR).
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−1, respectively.1 Mathematically, these anyons are operator valued distributions on some Hilbert space F, and our construction amounts to giving a precise mathematical meaning to these objects and defining and computing anyon correlation functions. The latter are given by a linear functional ω on the ∗-algebra generated by the anyons (the ∗ is the Hilbert space adjoint). We construct a particular representation of the anyons which is such that the anyon correlation functions are given by elliptic functions with the nome q = exp (−β/2) ,
β > 0,
(2)
where β can be interpreted as inverse temperature. For example, the simplest non-trivial anyon correlation function which we obtain is ω(φ(x)∗ φ(y)) = const. θ (x − y)−λ ,
(3)
where θ (r) = sin(r/2)
∞
(1 − 2q 2n cos(r) + q 4n )
(4)
n=1
is equal, up to a multiplicative constant, to the Jacobi Theta function ϑ1 (r/2). This function, or rather its regularized version defined in Eq. (45), will play a prominent role in this paper. We also mention a particular element in the anyon Hilbert space F which can be interpreted as the thermal vacuum since it allows to compute the anyon correlation functions as vacuum expectation value, i.e., ω(·) = , · (·, · is the inner product in the anyon Hilbert space). Our construction provides another example of a quantum field theory model which can be made mathematically precise using the representation theory of loop groups (see e.g. [PS]). A main difficulty is what physicists call ultra-violet divergences: the anyons φ (∗) (x) are operator valued distributions and thus products of them need to be defined with care. For example, the relation in Eq. (1) becomes problematic for x = y, and this difficulty also manifests itself in the anyon correlation function in Eq. (3) which is singular for x → y. The approach we use provides a particularly simple solution to this problem (we will describe the main idea in the Summary below). It is interesting to note that this very same quantum field theory model of anyons, for odd integers λ, has been used in the theory of the fractional quantum Hall effect [W]. We also note that different constructions of finite temperature anyons on the real line were recently given in [IT, LMP], and (many of) the results there can be (formally) obtained from ours by rescaling variables x → 2πx/L and β → 2πβ/L (which changes the circumference of the circle from 2π to an arbitrary length L > 0) and taking the limit L → ∞. We now give some background to the elliptic Calogero-Sutherland (eCS) model. The eCS model is defined by the differential operator HN = − 1
N ∂2 + γ V (xj − xk ) ∂xj2 1≤j
The name anyons is due to the fact that this can be any phase in general.
(5)
Second Quantization of the Elliptic Calogero-Sutherland Model
323
with −π ≤ xj ≤ π coordinates on the circle S 1 , N = 2, 3, . . . , γ > −1/2, and V (r) = −
∂2 log θ(r) , ∂r 2
(6)
θ as in Eq. (4). This function V (r) is equal, up to an additive constant, to Weierstrass’ elliptic function ℘ (r) with periods 2π and iβ (see Eq. (A6) in Appendix A for the precise formula). This differential operator defines a selfadjoint operator on the Hilbert space of square integrable functions on [−π, π]N which provides a quantum mechanical model of N identical particles moving on a circle of length 2π and interacting with a two body potential proportional to V (r) where γ is the coupling constant.2 This model is a prominent integrable many body system (a standard review is Ref. [OP]). In particular the limiting case q = 0 where the interaction potential becomes a trigonometric function, V (r) = (1/4) sin−2 (r/2), is the celebrated Sutherland model whose complete solution was found about 30 years ago [Su]. This explicit solution plays a central role in remarkably many different topics in theoretical physics including matrix models, quantum chaos, QCD, and two dimensional quantum gravity (for review see, e.g., Ref. [GMW], Sect. 7). There is also an interesting relation between the Sutherland model and the theory of the fractional quantum Hall effect (see e.g. [IR, W, YZZ]) which will be discussed in more detail below. We note that eigenfunctions of the eCS differential operator in Eq. (5) √ are known only for N = 2 and/or integer values of the coupling parameter3 (1 + 1 + 2γ )/2: For N = 2 these are classical results on Lam´e’s equation (see e.g. [WW]) which recently were generalized [EK, R] and extended to N > 2 [FV1, FV2]. Of course, the differential operator in Eq. (5) does not define a unique self-adjoint operator, but our approach will automatically specify a particular self-adjoint extension [L2] (which for q = 0 is identical with the one solved by Sutherland [Su]; we note that some of the known eigenfunctions of the eCS differential operator mentioned are singular and do not correspond to that particular self-adjoint extension). By a second quantization of the eCS model we mean one operator H = H∗ on the anyon Hilbert space F which accounts for the eCS Hamiltonians HN in Eq. (5) for all particle numbers N . To be more specific, the commutator of this operator H with a product of N anyons
N (x) := φ(x1 ) · · · φ(xN ) is essentially equal to the eCS Hamiltonian applied to this very product, i.e., H, N (x) = HN N (x) ,
(7)
(8)
where is the above-mentioned thermal vacuum, and the coupling constant of the eCS model is determined by the statistics parameter of the anyons as follows, γ = 2λ(λ − 1) .
(9)
Such a second quantization was previously known in the trigonometric limit (corresponding to zero temperature) [MP, AMOS, I, MS, CL], and in this paper we generalize 2 As will be discussed in more detail in [L2], this model corresponds to a particular self-adjoint extension [KT] which, for positive γ , is the Friedrich’s extension of HN ; see e.g. Theorem X.23 in Ref. [RS]. 3 If we write γ as in Eq. (9) below then the coupling parameter is equal to λ.
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it to the elliptic case. It is remarkable that this generalization is most natural from a physical point of view: it amounts to going from zero- to finite temperature. In the trigonometric limit (corresponding to zero temperature), this second quantization has provided an interesting direct link between the Sutherland model and the theory of the fractional quantum Hall effect, and we expect that our finite temperature generalization should be interesting in this context, too. However, our main motivation and emphasis is mathematical: In Ref. [CL] the second quantization of the Sutherland model was used to derive an algorithm for constructing eigenvalues and eigenfunctions of the Sutherland model and thus recover the solution of Sutherland [Su]. We will use the second quantization to derive a remarkable identity for the anyon correlation function FN (x, y) = , N (x)∗ N (y) (see Proposition 3) which, as we will outline in the conclusions of the paper, is the starting point for a novel algorithm for solving the eCS model [L2]. To obtain this identity we will need4 (10) , [H, N (x)∗ N (y)] = 0 which is the second important property of H and in fact the one which restricts us to interactions V (r) which are Weierstrass elliptic functions (Eq. (8) actually holds true in more generality). Summary of results. In Sect. 2 we construct anyons, i.e., give a precise mathematical meaning to the quantum fields φ(x) and compute all anyon correlation functions. The basic idea of our construction is to use vertex operators similar to the ones used in string theory (see e.g. [P]) and which we make mathematically precise using the representation theory of the loop group of U(1) (in the spirit of Ref. [Se]). We deviate from a similar previous construction of zero temperature anyons [CL] in that we use a somewhat unusual class of reducible representation of this loop group which, in special cases of particular interest to us, can be interpreted as finite temperature representations (the precise statement and proof of this is given in Appendix B.3). Technically, we account for the distributional nature of the quantum fields φ(x) by using a regularization which, roughly speaking, is a generalization of the idea to represent distributions as limits of smooth functions (e.g. the delta distribution on the circle as limit ε ↓ 0 of the smooth function δε (x) = 1/(2π) n∈Z exp (inx − |n|ε)). In a similar manner, we will construct regularized anyons φε (x) which for ε > 0 can be multiplied without ambiguities and obey Eq. (1) only in the limit ε ↓ 0. We perform that latter limit at a later point where it can be taken without difficulty.5 For simplicity, we will regard all quantum fields only as sesquilinear forms (using results in the literature, e.g. from Refs. [CR, GL], one can prove that many sesquilinear forms which we construct can be extended to well-defined operators, but since we actually do not need these results we will only mention them in passing). In Sect. 2.1 representations of the loop group of U(1) are defined, and we also collect some important technical results which we will use throughout the paper (Lemma 1). The definition and main properties of these regularized anyons, and in particular explicit formulas for all anyon correlation functions, are given in Sect. 2.2 (Proposition 1). In Sect. 3 the second quantization of the eCS model is constructed. We will give an explicit formula for H and prove that it indeed obeys Eqs. (8) and (10), or rather, a generalization of these relations to regularized anyons, i.e., with the parameters ε 4 5
[a, b] := ab − ba One can interpret 1/ε as ultra violet cut-off.
Second Quantization of the Elliptic Calogero-Sutherland Model
325
inserted. In particular, we will get, instead of the eCS differential operator HN in Eq. (5), a regularized operator HN2ε which, roughly speaking, is obtained by replacing the singular potential V (r) by a potential V2ε (r), where the 1/r 2 -singularity of the Weierstrass ℘-function is regularized to 1/(r +iε)2 . We will explicitly give the ε-corrections to Eq. (8). These results are summarized in Proposition 2. The proof is by explicit, lengthy computations which we divide in lemmas and partly defer to Appendix C. We note that it is precisely this ε-regularization which determines the self-adjoint extension of the eCS differential operator for us (however, this will only become important for us in the second paper [L2]). In Sect. 4 we derive a remarkable identity (Proposition 3) providing the starting point for constructing eigenvalues and eigenfunctions of the eCS model. In the conclusions (Sect. 5) we only state the theorem underlying this algorithm and outline its proof based on Propositon 3 (this algorithm will be elaborated in Ref. [L2]). Identities about (regularized) elliptic functions which we need are collected and proven in Appendix A. Appendix B contains a self-contained discussion of the relation of our quantum field theory techniques to the representation theory of loop groups. The proofs of various lemmas are collected in Appendix C. Notation. All Hilbert spaces considered are separable, and Hilbert space inner products ·, · are linear in the second and anti-linear in the first argument. We denote as C, R, Z the complex, real and integer numbers, N are the positive integers, and N0 = N ∪ {0}. √ We denote as c¯ the complex conjugate of c ∈ C, and |c| = cc. ¯ We identify elements in U(1) with phases, i.e., c ∈ C such that |c| = 1. 2. Finite Temperature Anyons on the Circle 2.1. Reducible representations of the loop group of U(1). In this subsection we set the stage for our construction of anyons. ˆ n integer, We consider the ∗-algebra A0 with identity 1 generated by elements ρ(n), and R obeying the following relations,
ρ(m), ˆ ρ(n) ˆ = mδm,−n 1 , ρ(n), ˆ R = δn,0 R (11) and R ∗ = R −1 ,
ρ(n) ˆ ∗ = ρ(−n) ˆ
(12)
for all integers m, n. We will also use the notation Q = ρ(0). ˆ Note that Eq. (11) implies eiαQ R w = eiwα R w eiαQ
∀α ∈ R, w ∈ Z
(13)
which will be useful for us later on. Remark 2.1. As explained in Appendix B, the algebra A0 defines essentially a central extension of the loop group of U(1). We now construct a class of representations of A0 using a standard highest weight representation of the auxiliary ∗-algebra A = A0 ⊗ A0 with identity 1 generated by elements RA and ρˆA (n), A = 1, 2 and n ∈ Z, defined by the relations
ρˆA (m), ρˆB (n) = mδm,−n δA,B 1 , ρˆA (n), RB = δn,0 δA,B RA (14)
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and −1 ∗ = RA , RA
ρˆA (n)∗ = ρˆA (−n)
(15)
for all integers m, n and A, B = 1, 2. The representation of A is on a Hilbert space F with inner product ·, · and completely characterized by the following conditions, ρˆA (n) = 0
∀n ≥ 0 , A = 1, 2 ,
(16)
and , R1w1 R2w2 = δw1 ,0 δw2 ,0
∀w1,2 ∈ Z,
(17)
where ∈ F is the highest weight vector and ∗ is the Hilbert space adjoint. Indeed, it is easy to check that the rules above imply that the elements η=
∞ ρˆA (−n)mA,n w1 w2 R R2 , mA,n 1 m !n A,n A=1,2 n=1
mA,n ∈ N0 ,
∞
mA,n < ∞ , wA ∈ Z
(18)
A=1,2 n=1
are orthonormal, and the set D of all finite linear combinations of such elements η is a pre-Hilbert space carrying a ∗-representation of the algebra A. The Hilbert space F is defined as the norm completion of D. We now observe that ˆ := ρˆ1 (0), π(R) := R1 π(Q) = π(ρ(0)) π(ρ(n)) ˆ := cn ρˆ1 (n) + sn ρˆ2 (−n) ∀n ∈ Z\{0}
(19)
(sn , cn ∈ C) obviously defines a unitary representation π of the ∗-algebra A0 provided that |cn |2 − |sn |2 = 1
(20)
and c−n = cn ,
s−n = sn
(21)
for all non-zero integers n. As explained in Remark 2.4 below, it is natural to also require that |sn |4 < ∞ . (22) n∈Z
One choice of particular interest for us is
cn =
1 1 − q 2|n|
1/2
,
sn =
q 2|n| 1 − q 2|n|
1/2 ∀n ∈ Z\{0}
(23)
with |q| < 1 and q 2 real, even though many of our results hold true more generally.
Second Quantization of the Elliptic Calogero-Sutherland Model
327
Remark 2.2. It is interesting to note that the representation π with sn and cn as in Eq. (23) and q = exp (−β/2) is the finite temperature representation of the ∗-algebra A0 with temperature 1/β and the Hamiltonian ∞
H =
a 2 Q + ρ(−n) ˆ ρ(n) ˆ 2
(24)
n=1
in the limit a → ∞. The interested reader can find a precise formulation and proof of this statement in Appendix B.3. Since there is no danger of confusion we simplify notation and write R, Q and ρ(n) ˆ short for π(R), π(Q) and π(ρ(n)) ˆ in the following. We now collect some (standard) technical results which we will need. We define × × normal ordering × · × as the linear map on the algebra generated by monomials M in the ρˆA (n) and R1 by the following inductive rules, × ×
×
∀w ∈ Z,
R1w × := R1w
× × ×M × ρˆA (n) × × × × × × × 1 × ×M × ρ × × × Mρ ˆA (n) × = × ρˆA (n)M × := ˆ (0) + ρ ˆ (0) M A A 2 × × ρˆA (n) × M ×
if n > 0 if n = 0 . (25) if n < 0
Note that theserules and Eqs. (14)–(17) imply that for arbitrary fixed vectors η, η ∈ D, × × the expression η, × ρˆA1 (n1 ) · · · ρˆAk (nk ) × η is non-zero only for a finite number of different combinations n1 , . . . , nk ∈ Z\{0} and A1 , . . . , Ak ∈ {1, 2}. This implies that for arbitrary complex numbers vkA1 ,... ,Ak (n1 , . . . , nk ), the expression
V := v0 1 +
∞
k=1 A1 ,... ,Ak =1,2 n1 ,... ,nk ∈Z\{0}
×
×
vkA1 ,... ,Ak (n1 , . . . , nk ) × ρˆA1 (n1 ) · · · ρˆAk (nk ) × (26)
is a well-defined sesquilinear form on D, i.e., η, Vη is finite for all η, η ∈ D (since it is always a finite sum there is never any problem with convergence). Moreover, for all × × w ∈ Z and a ∈ C and V = × V × as above, × ×
×
×
×
R w eaQ V × = × eaQ R w V × = eaQ/2 R w eaQ/2 V
(27)
is a sesquilinear form on D. In particular, these rules imply the following Lemma 1. For arbitrary complex αn , ×
×
:= × R w eiJ (α) ×
with J (α) :=
αn ρ(−n) ˆ
n∈Z
is a well-defined sesquilinear form on D equal to
= eiα0 Q/2 R w eiα0 Q/2 eiJ
+ (α)
eiJ
− (α)
,
and w ∈ Z
(28)
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E. Langmann
J ± (α) =
∞
α±n c±n ρˆ1 (∓n) + α∓n s∓n ρˆ2 (∓n) ,
(29)
n=1
where J + (α) and J − (α) are the creation-and annihilation parts of J (α), i.e., J − (α) = J + (α)∗ = 0. Moreover, the forms with αn such that |n||αn |2 < ∞ (30) n∈Z
generate a ∗-algebra of forms on D, with ∗× ×
(× R w eiJ (α) ×)∗ = × R −w e−iJ (α) ×
×
×
J (α)∗ =
,
α−n ρ(−n), ˆ
(31)
n∈Z
and the multiplication rule obtained with the Hausdorff formula eA eB = ec/2 eA+B = ec eB eA
if [A, B] = c1, c ∈ C
(32)
and Eqs. (11)–(13), i.e., × ×
R w eiJ (α) ×× R w eiJ (β) ×= e−[J ××
×
− (α),J + (β)]+i(α w −β w)/2 × 0 0 ×
×
R w+w eiJ (α+β) ×, (33)
where [J − (α), J + (β)] =
∞ n |cn |2 αn β−n + |sn |2 α−n βn .
(34)
n=1
Moreover,
× × , × R w eiJ (α) × = δw,0 .
(35)
Remark 2.3. As discussed in Appendix B, for αn obeying the conditions in Eq. (B2), the forms R w eiJ (α) define unitary operators and provide a unitary representation of the loop group of U(1) on F. Note that the Hausdorff formula implies × ×
eiJ (α) × = e−[J ×
− (α),J + (α)]/2
eiJ (α) ,
(36)
i.e., the forms in this case are proportional to unitary operators. We will not make use of these facts in this paper and will regard such ’s only as sesquilinar forms. Remark 2.4. Equation (34) explains the condition in Eq. (22) above: recalling that |cn |2 = 1 + |sn |2 and Cauchy’s inequality, we see that this is the natural condition ensuring that [J − (α), J + (β)] is well-defined for all α and β obeying the condition in Eq. (30). Proof of Lemma 1. The l.h.s. of Eq. (33) equals eiα0 Q/2 R w eiα0 Q/2 eiJ
+ (α)
eiJ
− (α)
eiβ0 Q/2 R w eiβ0 Q/2 eiJ
− J − (α),J + (β) +i(α w −β w)/2
+ (β)
eiJ
− (β)
0 0 =e ei(α0 +β0 )Q/2 R w+w ei(α0 +β0 )Q/2 − − + + × ei(J (α)+J (β)) ei(J (α)+J (β))
which obviously is equal to the r.h.s. of Eq. (33).
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329
2.2. Construction of anyons. Definition. The anyons associated with a real, non-zero parameter ν are, φεµ (x) := e−iµν
2 Qx/2
R µ e−iµν
2 Qx/2 × ×
e−iµνKε (x) × , ×
µ∈Z
(37)
with Kε (x) :=
n∈Z\{0}
1 inx−|n|ε e ρ(n), ˆ in
(38)
where −π ≤ x ≤ π is a coordinate on the circle and ε > 0 a regularization parameter. In particular, we define φε (x) := φε1 (x) .
(39)
The main properties of these anyons are summarized in the following µ
Proposition 1. The φε (x) generate a ∗-algebra of sesquilinar forms on D and obey φεµ (x)∗ = φε−µ (x)
(40)
and the exchange relations µ
2 sgn
φεµ (x)φε (y) = e−iπµµ ν where
ε+ε (x−y)
µ
φε (y)φεµ (x),
1 1 x+ einx−|n|ε , sgnε (x) := π in
(41)
(42)
n∈Z\{0}
a regularized sign function on the circle, i.e., it is C ∞ and converges to sgn(x) in the limit ε ↓ 0. Moreover, 2 , φεµ11 (x1 ) · · · φεµM1 (xM ) = δµ1 +···+µN ,0 (43) bεj +εk (xj − xk )µj µk ν 1≤j
with bε (r) := e−ir/2−Cε (r) ,
Cε (r) :=
∞ 1 2 inr−nε |cn | e + |sn |2 e−inr−nε . n
(44)
n=1
In particular, for cn and sn as in Eq. (23), bε (x) := 2e3π i/2−ε/2 sin 21 (x + iε)
∞
(1 − 2q 2n e−ε cos(x) + q 4n e−2ε ) .
(45)
n=1
Remark 2.5. The relations in Eq. (41) show that the φε (x) are regularized anyons with statistics parameter λ = ν2 as described in the Introduction.
(46)
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E. Langmann
Remark 2.6. We stress that we introduce the parameter ε > 0 as a convenient technical µ tool: the φε (x) are regularized quantum fields, i.e., in the limit ε ↓ 0 they become operator valued distributions whereas for ε > 0 they are well-defined operators and, in particular, can be multiplied with each other. Indeed, it follows from Remark 2.3 above µ that φε (x) is proportional to the unitary operator e−iµν
2 Qx/2
R µ e−iµν
2 Qx/2
e−iµνKε (x)
with the proportionality constant b2ε (0)−(µν) /2 which diverges in the limit ε ↓ 0. Eventually we are interested in this singular limit ε ↓ 0, but we will be able to take this limit at a later point without difficulty. 2
Proof. The proof is by straightforward computations using Lemma 1 above. In particular, Eq. (40) is a trivial consequence of Eq. (31), and to prove Eq. (41) it is sufficient to note the following identity, [Kε (x), Kε (y)] =
1 ein(x−y)−|n|(ε+ε ) 1 , n
n∈Z\{0}
Eq. (13) and the Hausdorff formula Eq. (32). To see that sgnε (x) is a regularized sign function one only needs to check that it is an odd function in x, and its x-derivative divided by 2 equals δε (x) :=
1 inx−|n|ε e 2π
(47)
n∈Z
which obviously is a regularized delta function. To compute the normal ordering of products of anyons we determine the creationand annihilation parts Kε± (x) of the form Kε (x), Kε± (x) = ∓
∞ 1 c±n e∓inx−nε ρˆ1 (∓n) − s∓n e±inx−nε ρˆ2 (∓n) in
(48)
n=1
and compute
Kε− (x), Kε+ (y) = Cε+ε (x − y)1
(49)
with Cε (r) given in Eq. (44). The normal ordering of an arbitrary product of anyons is defined as follows, × µ1 × µM × φε1 (x1 ) · · · φεM (xM ) ×
2 2 = e−iν Q(µ1 x1 +···+µM xM )/2 R µ1 +...+µM e−iν Q(µ1 x1 +···+µM xM )/2 +
+
−
−
×e−iν[µ1 Kε1 (x1 )+···+µM KεM (xM )] e−iν[µ1 νKε1 (x1 )+···+µM KεM (xM )] ,
and by repeated application of Eq. (33) we obtain 2 × × φεµ11 (x1 ) · · · φεµMM (xM ) = bεj +εk (xj − xk )µj µk ν × φεµ11 (x1 ) · · · φεµMM (xM ) × 1≤j
(50) with bε (r) in Eq. (44). This and Eq. (35) prove Eq. (43). Equation (45) is proven by straightforward computations given in Appendix A.
Second Quantization of the Elliptic Calogero-Sutherland Model
331
Remark 2.7. In computations it is sometimes convenient to write φεµ (x) = × e−iνµXε (x) × ×
×
(51)
with Xε (x) = q + px +
n∈Z\{0}
1 inx−|n|ε e ρ(n) ˆ in
with q = νQ and p = “(i/ν) log(R)” formally obeying
p, ρ(n) ˆ = q, ρ(n) ˆ =0 [q, p] = i1 ,
∀n = 0 ,
(52)
(53)
and therefore [Xε (x), Xε (y)] = −iπ sgnε+ε (x − y). This makes many computations simpler, but it is important to remember that p itself is not well-defined but only its exponentials e−iµνp ≡ R µ and only for integers µ. This makes clear that our anyons are essentially vertex operators used, e.g., in string theory (see e.g. [P]). Proposition 1 above summarizes the main properties of anyons. In particular, it gives explicit formulas for all so-called anyon correlation functions which are defined by the l.h.s. of Eq. (43) and which, according to physics folklore, capture the physical properties of the anyon system. For the eCS model the following anyon correlation function will play a central role, FNε ,ε (x; y) := , φε (xN )∗ · · · φε (x1 )∗ φε (y1 ) · · · φε (yN ) ,
(54)
and with Proposition 1 we obtain the following explicit formula, FNε,ε (x; y)
=
1≤j
b2ε (xk − xj )λ 1≤j
(55)
which will be important for us later on.
3. Second Quantization of the eCS Model In this section we construct the second quantization of the eCS model, i.e., find a selfadjoint form H on F such that the commutator of H with a product of anyons,
N ε (x) := φε (x1 ) · · · φε (xN ) ,
(56)
φε (x) defined in Eqs. (38)–(39), is essentially equal to the eCS Hamiltonian applied to this product. To be precise Proposition 2. Let ρ(n) ˆ be as in Eqs. (19)–(21) and 1 H := νW 3 + (1 − ν 2 )C + 2ν(ν − 1)W 2 Q + ν(ν − 1)2 (2 + ν)Q3 − ν 4 cε Q 3
(57)
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E. Langmann
with W 2 :=
1× × × ρ(−n) ˆ ρ(n) ˆ ×, 2 n∈Z
1 W 3 := 3 C :=
× ×
×
ρ(−m) ˆ ρ(−n) ˆ ρ(m ˆ + n) ×,
m,n∈Z ∞
n ρˆ1 (−n)ρˆ1 (n) + ρˆ2 (−n)ρˆ2 (n) ,
(58)
n=1
and the constant ∞ ∞ 1 cε = n|sn |2 e−2|n|ε − 2|sm |2 |sn |2 e−(m+n)ε e−|m−n|ε − e−|m+n|ε . −2 12 n=1
m,n=1
(59) Then
2ε N N H, N ε (x) = HN ε (x) + ενε (x) ,
(60)
where HNε := −
N ∂2 + 2ν 2 (ν 2 − 1) Vε (xj − xk ) 2 ∂xj j =1 1≤j
(61)
with Vε (r) := −
∞ n |cn |2 einr−nε + |sn |2 e−inr−nε
(62)
n=1
is a regularized many particle Hamiltonian, and εN (x) :=
N
×
×
φε (x1 ) . . . φε (xj −1 ) × φε (xj )Rε (xj ) × φε (xj +1 ) · · · φε (xN )
(63)
j =1
with Rε (x) :=
∞
× × inx −inx 2ν ν|sm |2 ρ(n)e − ρ(−n)e ˆ ˆ ρ(n) ˆ × ˆ − × ρ(−m)
m,n=1
1 −|m−n|ε − e−|m+n|ε e−|m|ε . e ε In particular, if cn and sn are as in Eq. (23) then ∞ ∂2 1 − 2q 2m e−ε cos(r) + q 4m e−2ε , Vε (r) = − 2 log sin 21 (r + iε) ∂r ×
(64)
(65)
m=1
i.e., HNε is a regularized version of the eCS differential operator in Eq. (5) with λ = ν 2 . Moreover, if cn and sn are as in Eq. (23) then also the following identity holds true, ∗ N (66) , H, N ε (x) ε (y) = 0 .
Second Quantization of the Elliptic Calogero-Sutherland Model
333
Remark 3.1. Note that the sesquilinear form Rε (x) in Eq. (64) is non-singular in the limit ε ↓ 0, hence the second term on the r.h.s. of Eq. (60) vanishes in this limit. It is therefore natural to write Eq. (60) in the following suggestive form: H, N (x) HN2ε N (67) ε ε (x) , where the symbol ‘’ means ‘equal in the limit ε ↓ 0’. Remark 3.2. As we will see, this proposition implies a remarkable identity (Proposition 3 in the next section) which is the starting point of our solution algorithm for the eCS model. We note that Eq. (60) comes for free in the Sutherland case q = 0 where H = 0, but this no longer holds true in general. It is also interesting to note that a form H obeying the relations in Eq. (60) exists for a much larger class of models: in the first part of this proposition we do not need to assume Eq. (23). However, to prove Eq. (66) we need Eq. (23): it is this relation which restricts this proposition to interaction potentials which are equal to Weierstrass’ ℘-function. Remark 3.3. We note that only the first two terms on the r.h.s. of Eq. (57) are really important, and dropping the other terms can be compensated by adding trivial terms to the eCS differential operator: simpler variants of our proof of Lemma 2 yield N Q, N ε (x) = N ε (x),
N ∂ N W 2 + 21 (ν − 1)(ν − 3)Q2 , N i
(x) . ε (x) = ∂xj ε
(68)
j =1
This shows that we have a relation ˜ 2ε N νW 3 + (1 − ν 2 )C, N ε (x) HN ε (x) ,
(69)
where H˜ N2ε equals HN2ε up to a constant and a multiple of the total momentum operator 3 2 PN := − N j =1 i∂/∂xj . Note also that the change H → νW + (1 − ν )C does not affect the identity in Eq. (66). Proof of Proposition 1. We prove this proposition in several steps. We start by stating two lemmas which provide the essential properties of the building blocks of H. The proofs of these lemmas are by straightforward but tedious computations which we defer to Appendix C. Lemma 2. The sesquilinear form 1 W˜ 3 := lim ε↓0 6π
π −π
with ρ˜ε (x) = νQ +
dx
× ×
×
ρ˜ε (x)3 × −ν 3 cε Q
(70)
inx−|n|ε ρ(n)e ˆ
(71)
n∈Z\{0}
and cε defined in Eq. (59), obeys the following relations 1 ∂2 × × × × W˜ 3 , φε (x) = − φε (x) + i(ν 2 − 1) × ρ˜ε (x)φε (x) × +ε × Rε (x)φε (x) × (72) ν ∂x 2 with Rε (x) defined in Eq. (64) and ρ˜ε (x) = ∂ ρ˜ε (x)/∂x.
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(Proof in Appendix C.1). s Remark 3.4. It is useful to note that W˜ 3 accounts for all but one term of H: the forms π W× s (s = 2, 3) defined in Eq. (58) can also be computed as follows, W = limε↓0 −π dx × × inx−|n|ε . Since ρ˜ (x) = ρ (x) + (ν − 1)Q, ρε (x)s × /(2π s), with ρε (x) = n∈Z ρ(n)e ˆ ε ε this implies
1 W˜ 3 = W 3 + 2(ν − 1)W 2 Q + (ν − 1)2 (ν + 2)Q3 − ν 3 cε Q , 3
(73)
which shows that H in Eq. (57) is identical with H = ν W˜ 3 + (1 − ν 2 )C .
(74)
Lemma 3. For all sesquilinear forms as in Eq. (28), the sesquilinear form C defined in Eq. (58) obeys × × × × C + C = 2 × C × −i × J (α ) × , with J (α ) := − n2 αn ρ(−n) ˆ . (75) n∈Z
(Proof in Appendix C.2). In particular, this lemma implies ×
×
×
×
×
×
N N C × N ε (x) × + × ε (x) × C = 2 × ε (x)C × + ×
×
×
N j =1
iν × ρ˜ε (xj ) N ε (x) × . ×
×
(76)
×
N iJ (α) × with To see that, recall that × N ε (x) × = × R e N 1 inxj −|n|ε νxj Q + J (α) = −ν , ρ(n)e ˆ in j =1
n∈Z\{0}
and therefore in this case J (α ) = −ν
N
inxj −|n|ε inρ(n)e ˆ ≡ −ν
j =1 n∈Z\{0}
N j =1
ρ˜ε (xj ) .
We now show how Lemmas 2 and Eq. (76) imply Eq. (60). From the definition in Eqs. (56) we get by a simple computation, using repeatedly Eq. (72), N N 1 ∂2 3 N 3 ˜ ˜ W , ε (x) = φε (x1 ) · · · W , φε (xj ) · · · φε (xN ) = −
ε (x) ν ∂xj2 j =1 j =1
+i(ν 2 − 1)
N j =1
φε (x1 ) · · · × ρ˜ε (xj )φε (xj ) × · · · φε (xN ) + εεN (x) ×
×
with εN (x) given in Eq. (63). Moreover, Eq. (76) together with 2 × × 2ε b2ε (xj − xk )ν
ε (x) = 2ε N (x) × ε (x) × , N (x) := 1≤j
(77)
Second Quantization of the Elliptic Calogero-Sutherland Model
335
(which is a special case of Eq. (50)) and the definition of normal ordering imply N × × × × 2ε N iν × ρ˜ε (xj ) N [C, ε (x)] = −2 N ε (x)C + N (x) 2 × ε (x)C × + ε (x) × j =1
=
−2 N ε (x)C N
+
j =1
+ 22ε N (x)
× ×
×
N ε (x)C ×
N − iν (ρ˜ε )− (xj ) N ε (x) + ε (x)(ρ˜ε ) (xj )
± 2 ± 2 with (ρ˜ )± ε (y) = ∂ ρ˜ε (y)/∂y = ∂ Kε (y)/∂y the creation- and annihilation parts of ρ˜ε (y). We thus get
3 N ˜ H, N (x) = ν W ,
(x) + (1 − ν 2 ) [C, ε (x)] ε ε N 2 ∂ = −
(x) + iν(ν 2 − 1)(·) + ενεN (x) 2 ε ∂x j j =1
(78)
with (·) :=
N j =1
φε (x1 ) · · · × ρ˜ε (xj )φε (xj ) × · · · φε (xN ) ×
×
N − −(ρ˜ε )− (xj ) N (x) −
(x)( ρ ˜ ) (x ) , j ε ε ε
×
×
where we used C = × N ε (x)C × = 0. We now recall Eq. (49) which implies
∓ ∂2
∂2 (ρ˜ )ε (y), Kε± (x) = 2 Kε∓ (y), Kε± (x) = ± 2 Cε+ε (∓(x − y))1 , ∂y ∂y and with Eqs. (44) and (62),
∓ (ρ˜ )ε (y), Kε± (x) = ±V2ε (∓(x − y))1 . With that we obtain
∓ ± (ρ˜ )ε (y), φε (x) = −iν (ρ˜ )∓ ε (y), Kε (x) φε (x) = ∓iνV2ε (∓(x − y))φε (x) , − and with ρ˜ε (xj ) = (ρ˜ )+ ε (xj ) + (ρ˜ )ε (xj ) and repeated application of these latter identities we get
φε (x1 ) · · · × ρ˜ε (xj )φε (xj ) × · · · φε (xN ) N N − = (ρ˜ )+ ε (xj ) ε (x) + ε (x)(ρ˜ )ε (xj ) j −1 N −iν V2ε (xk − xj ) + V2ε (xj − xk ) N ε (x) , ×
k=1
×
k=j +1
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E. Langmann
and thus
j −1 N N (·) = −iν V2ε (xk − xj ) + V2ε (xj − xk ) N ε (x) j =1
≡ −2iν
k=j +1
k=1
V2ε (xj − xk ) N ε (x) .
1≤j
If we insert this in Eq. (78) we obtain Eqs. (60)–(61). Finally, we note that validity of Eq. (66) under the condition Eq. (23) is obviously a special case of the following more general result: Lemma 4. For all sesquilinear forms as in Eq. (28) and for the form H defined in Eqs. (57)–(58), the relation , [H, ] = 0
(79)
|cm |2 |cn |2 |sm+n |2 = |sm |2 |sn |2 |cm+n |2 ∀m, n ∈ N .
(80)
holds true provided that
(Proof in Appendix C.3). Remark 3.5. Equation (79) is non-trivial only for w = 0. This completes the proof of Proposition 2.
Remark 3.6. It is obvious that cn and sn as in Eq. (23) fulfill the condition in Eq. (80). It is interesting to note that Eq. (23) actually accounts for all possibilities: simple computations shows that the conditions in Eqs. (20) and (80) are equivalent to: Qn := 1 −
1 |cn |2
obey
Qm+n = Qm Qn
∀m, n ∈ N ,
and all solutions of the latter are of the form Qn = q 2n for some complex q. If we also require Eq. (21) then q 2 must be real, and Eq. (22) further restricts to |q 2 | < 1. 4. A Remarkable Identity From the results in the previous section we now obtain the following Proposition 3. The anyon correlation function in Eq. (55) obeys the following identity, " # (81) HN2ε (x) − HN2ε (y) FNε,ε (x; y) = R ε,ε (x; y), where the regularized eCS differential operators are defined in Eqs. (61)–(62) and act on different arguments as indicated, and ∗ N N ∗ N R ε,ε (x; y) = −ε , N (x) (y) + ε , (x)
(y) , (82) ε ε ε ε εN (x) defined in Eqs. (63)–(64), are correction terms vanishing pointwise in the limit ε, ε ↓ 0.
Second Quantization of the Elliptic Calogero-Sutherland Model
337
Remark 4.1. Note that the complex conjugation in Eq. (81) is important as it affects the regularization: since Vε (r) = Vε (−r) it amounts to replacing the regularized singularity ∼ 1/(r + iε)2 by ∼ 1/(r − iε)2 . Remark 4.2. One can express this result as follows,
HN2ε (x)FNε,ε (x; y) HN2ε (y)FNε,ε (x; y) .
(83)
Remark 4.3. In the limits ε, ε ↓ 0 the function FNε,ε (x; y) becomes equal, up to a constant, to the function λ λ 1≤j
(85)
1≤j
with V (r) in Eq. (6). Recalling that θ (r) = const. ϑ1 (r/2) and V (r) = ℘ (r) + const., we see that this is a remarkable identity for elliptic functions. Proof of Proposition 3. We start with the following (trivial) identity ∗ N N ∗ N N ∗ N [H, N ε (x) ε (y)] = ε (x) [H, ε (y)] − [H, ε (x)] ε (y)
(where we used H = H∗ ) and compute its vacuum expectation value using Eq. (66), ∗ N N ∗ N , N (x) [H,
(y)] − , [H,
(x)]
(y) = 0. ε ε ε ε Inserting now Eqs. (66) and recalling Eqs. (54)–(55) we obtain the result.
5. Conclusion As stated in the beginning of Sect. 1, the results of this paper provide the starting point for an algorithm to construct eigenfunctions and eigenvalues of the eCS model, and this algorithm will be elaborated in [L2]. It is interesting to note that in the limiting case q = 0 this algorithm is different from the one of Sutherland, even though it is equivalent to it in the sense that it yields the same solutions and is equally simple (a detailed comparison of these algorithms is given in [L3]). For the convenience of the reader we outline here how Proposition 3 is used to obtain this algorithm. The idea is to take the Fourier transform of the identity in Eq. (81) with respect to the variables y, and then take the limits ε, ε ↓ 0. One thus obtains the following
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E. Langmann
Theorem ([L2]). Let Fˆ (x; n) = P(n; x)(x) , with
(x) =
n ∈ ZN
(86)
θ (xk − xj )λ
(87)
1≤j
and P(n; x) = lim ε↓0
π −π
dy1 in1 y1 ··· e 2π
π −π
dyN inN yN e 2π
ˇ
1≤j
λ
,
(88)
where ∞ bˇε (r) = 1 − eir−ε 1 − q 2m eir−ε 1 − q 2m e−ir−ε .
(89)
m=1
Then the eCS differential operator defined in Eqs. (5), (6), (4) and (9) obeys HN Fˆ (x; n) = E0 (n)Fˆ (x; n) − γ
∞ n
1≤j
+
q 2n 1−q
Fˆ (x; n − nEj k ), 2n
1 Fˆ (x; n + nEj k ) 1 − q 2n
(90)
where 2 E0 (n) = nj + 21 λ(N − 1 − 2j )
(91)
and (Ej k ) = δj − δk for j, k, = 1, . . . , N. Outline of Proof. We use Eq. (55) and insert bε (r) = e−ir/2 bˇε (r) which yields F 0,ε (x; y) = const. eiNλ
N
j =1 (xj −yj )/2
e−iλ
1≤j
Pˇ ε (x; y)(x),
where Pˇ ε (x; y) =
ˇ
1≤j
λ
is periodic in the yj . It is not difficult to show that Eq. (83) implies " # N 2ε HN (x) − HN (y) e−iNλ j =1 yj (N+1−2j )/2 Pˇ ε (x; y)(x) 0,
(92)
(93)
where we dropped the phase factor describing a center-of-mass motion which is the same in x and y and thus does not contribute,6 and we used (y j
See Ref. [L2] for the (trivial) details.
Second Quantization of the Elliptic Calogero-Sutherland Model
339
+ 1 − 2j )yj . We now can take the Fourier transform of Eq. (93), i.e., apply to it (2π )−N d N y eiP·y , where the Fourier variables need to be chosen as j (N
Pj = nj + 21 λ(N + 1 − 2j ) ,
nj ∈ Z
(94)
so as to compensate the non-periodicity in the yj . With that we obtain Eq. (90): the term on the l.h.s. is obvious, the first term on the r.h.s. comes from the derivative terms in HN2ε (y) and partial integration (note that E0 = j Pj2 ), and the other terms come from the potentials V2ε (yj − yk ), where we used Eq. (A3).
Remark 5.1. To complete this proof one needs to show that the correction term R ε,ε (x; y) in Eq. (81) does not contribute, i.e., that for this term the limits ε, ε ↓ 0 commute with Fourier transformation. This is plausible, and from our explicit formulas one can prove this by straightforward computations which, however, are somewhat tedious. In Ref. [L2] we will give an alternative proof avoiding this technicality. This theorem yields a recursive procedure to construct eigenfunctions as linear combinations of the functions Fˆ (x; n) [L2]. Remark 5.2. The functions Fˆ (x; n) in Ref. [L1] differ from the ones here by the phase factor exp(iNλ j xj /2) describing a center-of-mass motion. It is easy to see that this (trivial) change accounts for the different formulas for E0 (n) (the formula given here agrees with the eigenvalues of the Sutherland model given in Ref. [Su]). Appendix A. Regularized Elliptic Functions In this appendix we derive various identities related to the functions defined in Eqs. (44) and (23), i.e.,
∞ 1 q 2n 1 inr−nε −inr−nε Cε (r) = (A1) e + e n 1 − q 2n 1 − q 2n n=1
and bε (r) = e−ir/2−Cε (r) . In particular we prove Eq. (45) and derive three different representations for the function Vε (r) = −
∂2 ∂2 log bε (r) = 2 Cε (r) 2 ∂r ∂r
(A2)
which also prove that it is a regularization of the interaction potential defined in Eq. (6), V0 (r) = V (r). We also make precise the relation of V (r) to Weierstrass’ ℘ function. From the formulas given above we get immediately our first representation, # " ∞ 1 q 2n −inr −nε inr Vε (r) = − e n e + e . (A3) 1 − q 2n 1 − q 2n n=1
By expanding 1/(1 − q 2n ) in geometric series we obtain % $ ∞ ∞ 1 inr−nε 2nm inr−nε −inr−nε Cε (r) = + q +e e , e n n=1
m=1
340
E. Langmann
and interchanging the summation we can do the n-sum and obtain
Cε (r) = − log 1 − e
ir−ε
−
∞
log 1 − q 2m eir−ε + log 1 − q 2m e−ir−ε ,
m=1
i.e., Cε (r) = − log eir/2 bε (r)
(A4)
with ∞ bε (r) = e−ir/2 1 − eir−ε 1 − q 2m eir−ε 1 − q 2m e−ir−ε m=1
identical with bε (r) in Eq. (45). This also yields our second representation Eq. (65), which proves that V0 (r) is identical with V (r) defined in Eq. (6). Inserting q = exp (−β/2) we can also write Cε (r) = −
∞
log 1 − e
i(r+iβm+iε)
−
m=0
∞
log 1 − e−i(r−iβm−iε) ,
m=1
and using & ' ∂2 ∂2 log 1 − e±i(r±iβm±iε) = 2 log sin 21 (r ± iβm ± iε) 2 ∂r ∂r 1 =− , 1 2 4 sin 2 (r ± iβm ± iε) we obtain Vε (r) =
∞
1
4 sin2 21 (r m=0
+ iβm + iε)
+
∞
1
4 sin2 21 (r m=1
− iβm − iε)
.
(A5)
From this representation it is obvious that the function V (z) ≡ V0 (z), z ∈ C, is doubly periodic with periods 2π and iβ, it has a single pole of order 2 in each period−2 parallelogram, ∞ V (z) − z−2 is analytic in some neighborhood of z = 0 and equal to 1/12 − m=1 (1/2) sinh (βm/2) in z = 0. These facts imply (see e.g. [EMOT], Sect. 13.12) ∞
V (r) = ℘ (r|π, 21 iβ) +
1 1 − . 2 12 2 sinh 21 (βm) m=1
(A6)
Appendix B. On the Loop Group of U(1) In this appendix we clarify the relation of our construction of the quantum field theory model of anyons and the representation theory of loop groups. We also explain and prove the physical interpretation of the representations which we are using.
Second Quantization of the Elliptic Calogero-Sutherland Model
341
B.1 Loop group of U(1). Let G =Map(S 1 ,U(1)) be the set of all C 1/2 functions S 1 → U(1), i.e., it contains all functions of the form eif with f (x) = wx + αn einx , (B1) n∈Z
−π ≤ x ≤ π, w an integer called winding number, and complex αn such that αn = α−n ,
∞
n|αn |2 < ∞ .
(B2)
n=1
This set has an obvious group structure, i.e., the product is by point-wise multiplication and the inverse by point-wise complex conjugation. The group G has an interesting central extension Gˆ := U(1) × G = {(c, eif )|c ∈ U(1), eif ∈ G} with the group structure given by (c, eif )−1 := (c, ¯ e−if ) and
(c, eif )(c , eif ) := (cc e−iS(f,f
)/2
where S(f, f ) := wα0 − w α0 − i
, ei(f +f ) ) ,
n∈Z
nα−n αn
(B3)
(B4)
is a 2-cocycle of the group G, i.e., it obeys relations such that Eq. (B3) indeed defines ˆ Note that the conditions in Eq. (B2) and Cauchy’s inequality a group structure on G. ensure that S(f, f ) is always well-defined. B.2. Representations of the loop group of U(1). Given a representation of the algebra A0 defined in Eqs. (11) and Eq. (12), one has, at least formally, also a unitary representation ˆ This can be seen as follows: Defining of the group G. (eif ) := eiα0 Q/2 R w eiα0 Q/2 eiJ (α) with J (α) :=
αn ρ(−n) ˆ ,
(B5)
(B6)
n∈Z\{0}
a simple computation using the Hausdorff formula Eq. (32) together with
J (α), J (α ) = nα−n αn 1 n∈Z
(which follows from Eq. (11)) and Eq. (13) yields
(eif )(eif ) = e−iS(f,f
)/2
(ei(f +f ) ) .
(B7)
Moreover, it is easy to see that (eif )∗ = (e−if ). Thus (c, eif ) → c(eif ) is a unitary ˆ representation of G.
342
E. Langmann
B.3 Zero- and finite temperature representations. The standard representation π0 of the algebra A0 is on a Hilbert space F0 and completely characterized by the following conditions, ρˆ0 (n) 0 = 0 ∀n ≥ 0 , 0 , R0w 0 = δw,0 ∀w ∈ Z, (B8) where we write ρˆ0 (n) ≡ π0 (ρ(n)) ˆ and similarly for R and Q, and 0 ∈ F0 is the highest weight state. These conditions imply that the states ηw (m) =
∞ ρˆ0 (−n)mn w R , √ mn !nmn 0 n=1
mn ∈ N 0 ,
∞
mn < ∞ , w ∈ Z
(B9)
n=1
provide a complete orthonormal basis in F0 , and it is easy to check that these states all are eigenstates of the Hamiltonian ∞
H0 =
a 2 Q + ρˆ0 (−n)ρˆ0 (n) 2 0
(B10)
n=1
with corresponding eigenvalues ∞
Ew (m) =
a 2 mn n . w + 2
(B11)
n=1
This also shows that H0 defines a self-adjoint operator. Moreover, for a > 0 the Hamiltonian H0 is non-negative and 0 is the ground-state, i.e., the unique eigenstate of H0 with minimum eigenvalue. The representation π0 is therefore interpreted as zero temperature representation, and for elements A in A0 , < A >∞ := 0 , A0 0 is interpreted as the ground state expectation value of A (A0 ≡ π0 (A)). More generally, the finite temperature expectation value is defined as 1 Tr F0 (e−βH0 A0 ) , Z := Tr F0 (e−βH0 ) (B12) Z with a normalization constant Z called the partition function and Tr F0 the Hilbert space trace in F0 . The parameter β > 0 is interpreted as the inverse temperature. The representation discussed in the main text is on the Hilbert space F = F0 ⊗ F0 with = 0 ⊗ 0 and < A >β :=
ρˆ1 (n) = ρˆ0 (n) ⊗ 1 ,
ρˆ2 (n) = 1 ⊗ ρˆ0 (n)
(B13)
and similarly for R1,2 . We now can formulate the precise meaning of the statement in Remark 2.2 in the main text: Proposition 4. Let A0 be the dense set in A0 containing all finite linear combinations of elements A = R u Qv
∞
kn ρ(−n) ˆ ρ(n) ˆ n ,
(B14)
n=1
where u ∈ Z and v, kn , n ∈ N0 and such that ∞ n=1 (kn + n ) < ∞. For the representation π defined in Eqs. (19) and (23) and for all A ∈ A0 , the following holds true: , π(A) F = < A >β if q = exp (−β/2) and a → ∞.
(B15)
Second Quantization of the Elliptic Calogero-Sutherland Model
343
Proof. It is obviously enough to prove Eq. (B15) for all A in Eq. (B14). We do this by explicit computations. We first compute Tr F0 (e
−βH0
A0 ) = =
∞
ηw (m), e−βH0 A0 ηw (m)
w∈Z m1 ,m2 ,...=0 ∞
e−β Ew (m) δu,0 w v
w∈Z m1 ,m2 ,...=0
= δu,0
e−βaw
∞
δkn ,n
n=1
2 /2
wv
w∈Z
∞
δkn ,n nkn
(mn + kn )!nmn +kn mn !nmn
∞ (mn + kn )! −βmn n e , mn !
mn =0
n=1
and with ∞ ∞ (m + k)! m d k m+k dk 1 1 x = x = = k! k m! dx dx k 1 − x (1 − x)k+1
m=0
m=0
for x = e−βn , we get Tr F0 (e−βH0 A0 ) = δu,0
e−βaw
2 /2
∞
wv
w∈Z
δkn ,n nkn kn !
n=1
1 (1 − e−βn )kn +1
.
In particular,
Z=
e
−βaw2 /2
w∈Z
∞ n=1
1 (1 − e−βn )
(B16)
and
w∈Z e
∞ −βaw2 /2 w v
w∈Z
n=1
< A >β = δu,0
2 e−βaw /2
δkn ,n nkn kn !
1 . (1 − e−βn )kn
(B17)
We now compute the l.h.s. of Eq. (B15) using Eqs. (19) and (14)–(17), ( ∞ , π(A) F = , R1u Qv1 (c−n ρˆ1 (−n) + s−n ρˆ2 (n))kn n=1
)
×(cn ρˆ1 (n) + sn ρˆ2 (−n)) n
F
= δu,0 δv,0
∞
δkn ,n |cn |kn nkn kn ! ,
n=1
and with Eq. (23), , π(A) F = δu,0 δv,0
∞ n=1
1 δkn ,n n kn ! 1 − q 2n kn
kn (B18)
.
This obviously coincides with Eq. (B17) for q = exp (−β/2) and a → ∞.
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E. Langmann
Remark 5.3. From a physics point of view, the more natural Hamiltonian for the anyon system would be H0 as in Eq. (B10) with a = 1/2 (and not a → ∞). It is therefore interesting to note that the finite temperature state for this Hamiltonian and arbitrary a > 0 is 1 2 2 = 1/2 (R1 R2 )w e−aw /2 , z := e−aw /2 , (B19) z w∈Z
w∈Z
which is easily proven by generalizing our argument above. It is also easy to compute the anyon correlation functions with that state. However, we find that this state is useful for the eCS system only if we choose a → ∞. Appendix C. Proofs C.1. Proof of Lemma 2. We will prove this lemma by direct computation. For that we will need several identities which we derive first. We will need [Q, φε (x)] = φε (x),
(C1)
which follows trivially from the relations in Eq. (11) and the definition of φε (x) ≡ φε1 (x) in Eqs. (38)–(37). It is useful to write ρ˜ε (x) = νQ + ρε+ (x) + ρε− (x) with the creationand annihilation parts ρε± (x) :=
c±n e∓inx−nε ρˆ1 (∓n) + s∓n e±inx−nε ρˆ2 (∓n) .
∞
(C2)
n=1
We note that ρε± (x) = ∂Kε± (x)/∂x, and it follows therefore from Eq. (44) that ∂ × × × × φε (x) = −i × ν 2 Q + νρε+ (x) + νρε− (x) φε (x) × ≡ −iν × ρ˜ε (x)φε (x) ×, ∂x
(C3)
which implies ∂2 ∂ φε (x) = −iν ∂x 2 ∂x
× ×
ρ˜ε (x)φε (x) × = −iν × ρ˜ε (x)φε (x) × −ν 2 × ρ˜ε (x)2 φε (x) × . ×
×
×
×
×
(C4) Moreover, a simple computation using Eqs. (48), (14), (20) and (21) yields ∞ ρ ∓ (y)ε , Kε± (x) = i (1 + |sn |2 ) e∓inr−nε − |sn |2 e±inr−nε ≡ 2π iδε∓ ˜ (r) ± jε˜ (r),
n=1
where δε± (r) :=
∞ 1 ±inr−nε e 2π
(C5)
n=1
and jε (r) := 2
∞ n=1
|sn |2 sin(nr)e−nε .
(C6)
Second Quantization of the Elliptic Calogero-Sutherland Model
345
Using that we compute (t ∈ C) × ×
+
−
et ρ˜ε (y) × φε (x) = etνQ et ρ˜ε (y) et ρ˜ε (y) e−iν ×
2 Qx/2
−itν ρ˜ − (y),K + (x)
R e−iν
2 Qx/2
2
ε ε = etν/2 e+ e(−iν x+νt)Q/2 R e(−iν t ρ˜ε (y) −iνKε+ (x) t ρ˜ε− (y) −iνKε− (x) ×e e e e
≡e
tν− (x−y) ε+ε
× ×
+ (x)
e−iνKε
− (x)
e−iνKε
2 x+νt)Q/2
et ρ˜ε (y) φε (x) ×, ×
− where − ε (r) = 1/2 + 2πδε (r) − ijε (r). Similarly,
φε (x) × et ρ˜ε (y) × = e−tν/2 e ×
×
≡e
−itν Kε− (x),ρ˜ε+ (y)
−tν+ (x−y) ε+ε
× ×
× ×
et ρ˜ε (y) φε (x) × ×
et ρ˜ε (y) φε (x) ×, ×
+ where + ε (r) = 1/2 + 2πδε (r) + ijε (r). Thus + × t ρ˜ (y) × × × tν− ε˜ (r) − e−tνε˜ (r) × et ρ˜ε (y) φε (x) × ×e ε ×, φε (x) = e
(C7)
with ε˜ = ε + ε , r = x − y, and ± ε (r) =
1 + 2πδε± (r) ± ijε (r) . 2
(C8)
We will also need the following three identities, + − ε (r) + ε (r) = 2πδε (r),
2 + 2 − ε (r) − ε (r) = 2π iδε (r) − 4π ijε (r)δε (r), 1 3 + 3 2 − ε (r) + ε (r) = −πδε (r) + πδε (r) + 6πjε (r)iδε (r) − 6πjε (r) δε (r),(C9) 2
where δε (r) is the regularized δ-function defined in Eq. (47) and the prime indicates differentiation with respect to the argument r. The first of these identities is obvious, and the second and third identities can be proven by straightforward computations using 1 1 + e±ir−ε i + 2πδε± (r) = = ± cot 21 (r ± iε) ±ir−ε 2 2(1 − e ) 2 and cot 2 ( 21 z) = −2
d cot( 21 z) − 1 , dz
cot 3 ( 21 z) = 2
d2 cot( 21 z) − cot( 21 z) . dz2
For the convenience of the reader we give some details of these computations, " #2 2 2 1 1 ± cot (r) = (±i) (r ± iε) + j (r) ε ε 2 2 1 = − cot2 21 (r ± iε) − jε (r) cot 21 (r ± iε) − jε (r)2
4 1 ∂ 1 − jε (r) cot 21 (r ± iε) + − jε (r)2 = 2 ∂r 4
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E. Langmann
which together with cot 21 (r − iε) − cot 21 (r + iε) = 4π iδε (r) proves the second identity in (C9), and similarly #3 1 cot 21 (r ± iε) + jε (r) 2 1 3 3 = cot 3 21 (r ± iε) + jε (r) cot2 21 (r ± iε) + jε (r)2 cot 21 (r ± iε) + jε (r)3 8 4 2 3 3 1 ∂2 1 ∂ 3 2 + jε (r) cot 21 (r ± iε) − jε (r) − jε (r)3 = − − jε (r) 4 ∂r 2 8 2 ∂r 2 4
3 4 ±i± ε˜ (r) = (±i)
"
implies the third identity. Differentiating Eq. (C7) three times with respect to t and setting t = 0 yields
× ×
× + − + 3 3 2 2 2 φ ρ˜ε (y)3 ×, φε (x) = ν 3 − (r) + (r) (x) + 3ν (r) − (r) ε ε˜ ε˜ ε˜ ε˜
− × × × × + 2 × × ρ˜ε (y)φε (x) × +3ν ε˜ (r) + ε˜ (r) × ρ˜ε (y) φε (x) × ,
and by inserting the identities in Eq. (C9) we obtain "
1 6π
#
× 3× × ρ˜ε (y) ×, φε (x)
# " 1 δε˜ (r) + jε˜ (r)δε˜ (r) − jε˜ (r)2 δε˜ (r) φε (x) = ν 3 − 16 δε˜ (r) + 12 × × × × 2 +ν iδε˜ (r) − 2ijε˜ (r)δε˜ (r) × ρ˜ε (y)φε (x) × +νδε˜ (r) × ρ˜ε (y)2 φε (x) × .
With that we compute 3 ˜ W , φε (x) = lim
=
π
"
1 6π
× ×
ε ↓0 −π ν 3 aε φε (x)+iν 2
# ρ˜ε (y) ×, φε (x) − ν 3 cε [Q, φε (x)] 3×
× ×
ρ˜ε (x)φε (x) × +ν × ρ˜ε (x)2 φε (x) × + × restε (x)φε (x) × ×
with 1 − j2ε (0) − lim aε := 12 ε ↓0
π −π
×
dy δε˜ (r)jε˜ (r)2 − cε ,
where we used Eq. (C1) and the following relations,
π
−π π
−π π −π
dy δε˜ (x − y) = 0, dy δε˜ (x − y) = 1,
dy jε˜ (x − y)δε˜ (x − y) = −j2ε+2ε (0), π dy δε˜ (x − y)ρ˜ε (y) = ρ˜ε+2ε (x), −π
×
×
×
Second Quantization of the Elliptic Calogero-Sutherland Model
347
(note the sign in the last relation!) which all are simple consequences of the definitions, and a rest term π restε (x) := lim dy δε˜ (x − y)[−2iν 2 jε˜ (x − y)ρ˜ε (y) + ν ρ˜ε (y)2 ] − ν ρ˜ε (x)2 , ε ↓0 −π
(C10)
which we expect to vanish in the limit ε ↓ 0 (recall that jε (0) = 0). For the following computations we find it convenient to write (cf. Eq. (C6)) (C11) jˆ(n) einx−|n|ε , jˆ(|n|) = −i|sn2 | = −jˆ(−|n|) . jε (r) = n∈Z
With that,
π
dy δε˜ (r)jε˜ (r)2
lim
ε ↓0 −π
= lim
ε ↓0
= =
n1 ,n2 ,n3 ∈Z
1 2π
π
−π
dr ein1 r−|n1 |˜ε jˆ(n2 )ein2 r−|n2 |˜ε jˆ(n3 )ein3 r−|n3 |˜ε
jˆ(m)jˆ(n)e−(|m|+|n|+|m+n|)ε
m,n∈Z ∞
2|sm |2 |sn |2 e−(m+n)ε e−|m−n|ε − e−|m+n|ε ,
m,n=1
∞ (0) = 2 2 −2|n|ε (cf. (C6)) and Eq. (59) we get a ≡ 0. Using and with j2ε ε n=1 n|sn | e then Eq. (C4) to eliminate the term with ρ˜ε (x)2 we obtain 1 ∂2 × × × × φε (x) + i(ν 2 − 1) × ρ˜ε (x)φε (x) × + × restε φε (x) ×, W˜ 3 , φε (x) = − 2 ν ∂x
which coincides with Eq. (72) provided that restε (x) = εRε (x)
(C12)
with Rε (x) defined in Eq. (64). We thus are left to prove Eq. (C12). For that we compute the rest term explicitly. Firstly, recalling Eqs. (C11) and (71) and writing ρ(0) ˜ = νQ and ρ(n) ˜ = ρ(n) ˆ for n = 0, π (∗)1 := lim dy δε˜ (x − y)jε˜ (x − y)ρ˜ε (y) ε ↓0 −π π 1 dy ein1 (x−y)−|n1 |(ε+ε ) = lim 2π −π ε ↓0 n1 ,n2 ,n3 ∈Z
× jˆ(n2 ) ein2 (x−y)−|n2 |(ε+ε ) ρ(n ˜ 3 ) ein3 y−|n3 |ε inx −(|m|+|m+n|)ε = e jˆ(m)ρ(n)e ˜
=
m,n∈Z ∞ m,n=1
inx −inx ˆ e−|m−n|ε − e−|m+n|ε e−|m|ε . i|sm |2 ρ(n)e − ρ(−n)e ˆ
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E. Langmann
Secondly,
(∗)2 := lim
π
dy δε+ε (x − y)ρ˜ε (y)2 π 1 dy ein1 (x−y)−|n1 |(ε+ε ) ρ(n ˜ 2 ) ein2 y−ε |n2 | ρ(n ˜ 3 ) ein3 y−ε |n3 | 2π −π
ε ↓0 −π
= lim
ε ↓0
=
n1 ,n2 ,n3 ∈Z
i(m+n)x−ε|m+n| ρ(m) ˜ ρ(n)e ˜ ,
m,n∈Z
and recalling ρ˜ε (x)2 = × ×
˜ ρ(n)e ˜ m,n∈Z ρ(m)
×
[(∗)2 − ρ˜ε (x)2 ] × =
× ×
m,n∈Z ∞
=2
i(m+n)x−ε(|m|+|n|) ,
we obtain
ρ(m) ˜ ρ(n) ˜ × ei(m+n)x e−ε(|m+n|) − e−(|m|+|n|)ε ×
× ×
× ρ(−m) ˆ ρ(n) ˆ × e−i(m−n)x e−|m+n|ε − e−|m−n|ε .
m,n=1
We therefore obtain (cf. Eq. (64)) ×
×
restε (x) = −2iν 2 (∗)1 + ν × [(∗)2 − ρ˜ε (x)2 ] × ≡ εRε (r), which completes the proof of Lemma 2.
C.2. Proof of Lemma 3. We write J (α) = α0 Q + A− + A+ where A± = J ± (α) and recall from the definition of normal ordering that +
−
= eiα0 Q/2 R w eiα0 Q/2 eiA eiA and × ×
×
C × =
∞
n ρˆ1 (−n) ρˆ1 (n) + ρˆ2 (−n) ρˆ2 (n) . n=1
To compute the l.h.s. of Eq. (75) we use
ρˆ1 (±n), A± = ±nc∓n α±n 1,
ρˆ2 (±n), A± = ±ns±n α∓n 1 ∀n ≥ 0, following from Eqs. (19) and (14). Moreover, we get for n > 0, * * ∂ t ρˆ1 (n) ** ∂ it [ρˆ1 (n),A+ ] t ρˆ1 (n) ** e e
* =
e ρˆ1 (n) = * ∂t ∂t t=0
t=0 = ρˆ1 (n) + i ρˆ1 (n), A+ = ρˆ1 (n) + inc−n αn 1 ,
(C13)
and similarly,
ρˆ1 (−n) = ρˆ1 (−n) − i ρˆ1 (−n), A− = ρˆ1 (−n) + incn α−n 1,
ρˆ2 (n) = ρˆ2 (n)i + i ρˆ2 (n), A+ = ρˆ2 (n) + insn α−n 1,
ρˆ2 (−n) = ρˆ2 (−n) − i ρˆ2 (−n), A− = ρˆ2 (−n) + ins−n αn 1 .
(C14)
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349
With that we obtain ×
×
C + C − 2 × C × =
∞
×
n × inc−n αn ρˆ1 (−n) + incn α−n ρˆ1 (n)
n=1
× +insn α−n ρˆ2 (−n) + ins−n αn ρˆ2 (n) × ×
× =× in2 αn c−n ρˆ1 (−n) + s−n ρˆ2 (n) × n∈Z × ×
× ×
equal to − J (α ) .
C.3. Proof of Lemma 4. We introduce the notations 2π 1 W±3 := lim dy ρε± (y)3 ε↓0 6π 0 with ρε± given in Eq. (C2), and a simple computation yields W±3 = lim ε↓0
1 6π
2π
dy 0
∞
c∓n1 ρˆ1 (∓n1 )e∓in1 y + s±n1 ρˆ2 (∓n1 )e±in1 y
n1 ,n2 ,n3 =1
× (n1 ↔ n2 ) (n2 ↔ n3 ) e−(n1 +n2 +n3 )ε ∞ & = c∓m c∓n s±(m+n) ρˆ1 (∓m)ρˆ1 (∓n)ρˆ2 (∓m ∓ n)
(C15)
m,n=1
' +s±m s±n c∓(m+n) ρˆ2 (∓m)ρˆ2 (∓n)ρˆ1 (∓m ∓ n) ,
(C16)
where ‘(n1 ↔ n2 )’ means the same term as the previous one but with n2 instead of n1 . We observe that C = Q = ρε− (x) = 0 and thus H = νW+3 ,
(W+3 )∗ = W−3 ,
which implies that (79) is true if and only if (∗)+ − (∗)− := , W+3 − , W−3 vanishes. We now compute (∗)− using Eqs. (C13)–(C14) which imply (for n > 0) ρˆ1 (n) = inc−n αn ,
ρˆ2 (n) = insn α−n ,
and therefore (∗)− =
∞
, [cm cn s−(m+n) ρˆ1 (m)ρˆ1 (n)ρˆ2 (m + n)
m,n=1 ∞ (i)3 mn(m + n) +s−m s−n cm+n ρˆ2 (m)ρˆ2 (n)ρˆ1 (m + n)] = δw,0
m,n=1
× |cm |2 |cn |2 |sm+n |2 αm αn α−(m+n) + |sm |2 |sn |2 |cm+n |2 α−m α−n αm+n ,
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E. Langmann
where we used , = δw,0 . The equations in (C14) also imply (for n > 0) < , ρˆ1 (−n) = incn α−n < , ,
< , ρˆ2 (−n) = ins−n αn < , ,
and thus (∗)+ =
∞
, [c−m c−n sm+n ρˆ1 (−m)ρˆ1 (−n)ρˆ2 (−m − n)
m,n=1 ∞ (i)3 mn(m + n) +sm sn c−(m+n) ρˆ2 (−m)ρˆ2 (−n)ρˆ1 (−m − n)] = δw,0
m,n=1
× |cm | |cn | |sm+n | α−m α−n αm+n + |sm | |sn | |cm+n | αm αn α−(m+n) . 2
2
2
2
2
2
We thus obtain (∗)+ − (∗)− = δw,0
∞
(i)3 mn(m + n) |cm |2 |cn |2 |sm+n |2 − |sm |2 |sn |2 |cm+n |2
m,n=1
& ' × α−m α−n αm+n + αm αn α−(m+n) , which shows explicitly that the relations in Eq. (80) imply (∗)+ − (∗)− = 0.
(C17)
Acknowledgement. I thank Alan Carey and Alexios Polychronakos for their interest and helpful discussions.
References [AMOS] Awata, H., Matsuo, Y., Odake, S., Shiraishi, J.: Collective field theory, Calogero-Sutherland model and generalized matrix models. Phys. Lett. B347, 49 (1995) [C] Calogero, F.: Exactly solvable one-dimensional many body problems. Lett. Nuovo Cim. 13, 411 (1975) [CHa] Carey, A.L., Hannabuss, K.C.: Temperature states on the loop groups, theta functions and the Luttinger model. J. Func. Anal. 75, 128 (1987) [CL] Carey, A.L., Langmann, E.: Loop groups, anyons and the Calogero-Sutherland model. Commun. Math. Phys. 201, 1 (1999) [CR] Carey, A.L., Ruijsenaars, S.N.M.: On fermion gauge groups, current algebras and Kac-Moody algebras. Acta Appl. Mat. 10, 1 (1987) [EMOT] Erd´elyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.G.: Higher transcendental functions. Vol. 2. New York-Toronto-London: McGraw-Hill Book Company, Inc., 1953 [EK] Etingof, P.I., Kirillov, A.A.: Representation of affine Lie algebras, parabolic differential equations and Lame functions. Duke Math. J. 74, 585 (1994) [FV1] Felder, G., Varchenko, A.: Integral representation of solutions of the elliptic Knizhnik-Zamolodchikov-Bernard equations. Int. Math. Res. Notices No. 5, 221 (1995) [FV2] Felder, G., Varchenko, A.: Three formulas for eigenfunctions of integrable Schroedinger operators. hep-th/9511120 [GL] Grosse, H., Langmann, E.: A super-version of quasi-free second quantization. I. Charged particles. J. Math. Phys. 33, 1032 (1992) [GMW] Guhr, T., Muller-Groeling, A., Weidenmuller, H.A.: Random matrix theories in quantum physics: Common concepts. Phys. Rept. 299, 189 (1998) [IT] Ilieva, N., Thirring, W.: Laughlin type wave function for two-dimensional anyon fields in a KMS-state. Phys. Lett. B 504, 201 (2001) [I] Iso, S.: Anyon basis in c = 1 conformal field theory. Nucl. Phys. B443[FS], 581 (1995) [IR] Iso, S., Rey, S.J.: Collective field theory of the fractional quantum hall edge state and the Calogero-Sutherland model. Phys. Lett. B352, 111 (1995)
Second Quantization of the Elliptic Calogero-Sutherland Model [KT] [L1] [L2] [L3] [LMP] [McD] [MS] [MP] [Mo] [OP] [P] [PS] [RS] [R] [Se] [St] [Su] [W] [WW] [YZZ]
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Komori, Y., Takemura, K.: The perturbation of the quantum Calogero-Moser-Sutherland system and related results. Commun. Math. Phys. 227, 93 (2002) Langmann, E.: Solution algorithm for the elliptic Calogero-Sutherland model. Lett. Math. Phys. 54, 279 (2000) Langmann, E.: A perturbative algorithm to solve the (quantum) elliptic Calogero-Sutherland system. math-ph/0401029, (to appear in Commun. Math. Phys.) Langmann, E.: Algorithms to solve the (quantum) Sutherland model. J. Math. Phys. 42, 4148 (2001) Liguoria, A., Mintchevb, M., Piloc, L.: Bosonization at finite temperature and anyon condensation. Nucl. Phys. B569, 577 (2000) Macdonald, I.G.: Symmetric functions and Hall polynomials. Oxford Mathematical Monographs. Oxford: Clarendon Press, 1979 Marotta, V., Sciarrino, A.: From vertex operators to Calogero-Sutherland models. Nucl. Phys. B476, 351 (1996) Minahan, J.A., Polychronakos, A.P.: Density correlation functions in Calogero-Sutherland models. Phys. Rev. B50, 4236 (1994) Moser, J.: Three integrable Hamiltonian systems connected with isospectral deformations. Adv. Math. 16, 197 (1975) Olshanetsky, M.A., Perelomov, A.M.: Quantum integrable systems related to Lie algebras. Phys. Rept. 94, 313 (1983) Polchinski, J.: String theory. Vol. I. Cambridge: Cambridge University Press, 1998 Pressley, A., Segal, G.: Loop Groups. Oxford: Oxford Mathematical Monographs, 1986 Reed, M., Simon, B.: Methods of modern mathematical physics. Vol. II. New York, London: Academic Press, 1975 Ruijsenaars, S.N.M.: Generalized Lam´e functions. I. The elliptic case. J. Math. Phys. 40, 1595 (1999) Segal, G.B.: Unitary representations of some infinite dimensional groups. Commun. Math. Phys. 80, 301 (1981) Stanley, R.P.: Some properties of Jack symmetric functions. Adv. in Math. 77, 76 (1989) Sutherland, B.: Exact results for a quantum many body problem in one-dimension. I and II. Phys. Rev. A4, 2019 (1971); and ibid. A5, 1372 (1972) Wen, X.G.: Chiral Luttinger liquid and the edge excitations in the fractional Quantum Hall states. Phys. Rev. B41, 12838 (1990) Whitaker, E.T., Watson, G.N.: Course of modern analysis. 4th edition. Cambridge: Cambridge University Press, 1958 Yu,Y., Zheng, W., Zhu, Z.: Microscopic picture of a chiral Luttinger liquid: Composite fermion theory of edge states. Phys. Rev. B56, 13279 (1997)
Communicated by N. Nekrasov
Commun. Math. Phys. 247, 353–390 (2004) Digital Object Identifier (DOI) 10.1007/s00220-004-1083-x
Communications in
Mathematical Physics
Semidensities on Odd Symplectic Supermanifolds Hovhannes M. Khudaverdian1,2 1
Department of Mathematics, University of Science and Technology (UMIST), Manchester, M60 1QD, UK. E-mail: [email protected] 2 Laboratory of Computing Technique and Automation, Joint Institute for Nuclear Research, Dubna 141980, Russia Received: 29 May 2001 / Accepted: 21 January 2004 Published online: 7 April 2004 – © Springer-Verlag 2004
Abstract: We consider semidensities on a supermanifold E with an odd symplectic structure. We define a new -operator action on semidensities as the proper framework for the Batalin-Vilkovisky (BV) formalism. We establish relations between semidensities on E and differential forms on Lagrangian surfaces. We apply these results to Batalin-Vilkovisky geometry. Another application is to (1.1)-codimensional surfaces in E. We construct a kind of “pull-back” of semidensities to such surfaces. This operation and the -operator are used for obtaining integral invariants for (1.1)-codimensional surfaces.
Contents 1. 2. 3. 4.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . -Operator on Semidensities . . . . . . . . . . . . . . . . . . . . . . . . Differential Forms on Cotangent Bundle and Semidensities . . . . . . . . Semidensities on E and Differential Forms on Even Lagrangian Surfaces 4.1 Identifying symplectomorphisms for even Lagrangian surfaces . . . 4.2 Relation between semidensities and differential forms on a Lagrangian surface . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Application to BV-geometry . . . . . . . . . . . . . . . . . . . . . 5. Invariant Densities on Surfaces . . . . . . . . . . . . . . . . . . . . . . . 6. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 1. -Points on Supermanifolds . . . . . . . . . . . . . . . . . . . Appendix 2. A Simple Proof of the Darboux Theorem for Odd Symplectic Structure . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 3. Hamiltonians of Adjusted Canonical Transformations . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . .
354 357 362 365 366
. . . . .
372 376 377 382 383
. . .
384 387 389
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H.M. Khudaverdian
1. Introduction A density of weight σ is a function on a manifold (supermanifold) subject to the condition that under change of coordinates it is multiplied by the σ th power of the determinant (Berezinian) of the transformation. A density of weight σ = 1 is a volume form. (We avoid discussion of orientation here.) In this paper we study semidensities (densities of weight σ = 1/2) on a supermanifold provided with an odd symplectic structure (an odd symplectic supermanifold). We introduce a differential operator , which acts on semidensities. Our considerations lead to a straightforward geometrical interpretation of the Batalin-Vilkovisky master equation. On the other hand, we elaborate a new outlook for the invariant semidensity defined on (1.1)-codimensional surfaces embedded in an odd symplectic supermanifold [12] and construct integral invariants for these surfaces. The concept of an odd symplectic supermanifold and a -operator on it appeared in mathematical physics in the pioneer works of I.A.Batalin and G.A.Vilkovisky [4, 5], where these objects were used for constructing a covariant Lagrangian version of the BRST quantization (BV formalism). The geometrical meaning of these objects and interpretation of the BV master equation in its terms were studied in [11, 14, 15] and most notably by A.S.Schwarz in [22]. Let us briefly sketch the results of [11, 14, 15, 22]. If an odd symplectic supermanifold is provided with a volume form dv, then one can consider an operator dv such that its action on a function on this supermanifold is equal (up to a coefficient) to the divergence of the Hamiltonian vector field corresponding to this function w.r.t. the volume form dv [11]. This second order differential operator is not trivial because transformation preserving odd symplectic structures do not preserve any volume form (the Liouville theorem does not hold in the case of an odd symplectic structure). We call the coordinates zA = {x 1 , . . . , x n , θ1 , . . . , θn } in an odd symplectic supermanifold Darboux coordinates if in these coordinates the Poisson bracket corresponding to the symplectic structure has the canonical form: {x i , θj } = δji , {x i , x j } = 0. Consider a special case, where a volume form in some Darboux coordinates is just the coordinate volume form: dv = D(x, θ ),
(D(x, θ ) = dx 1 . . . dx n dθ1 . . . dθn ) .
(1.1)
In the following we shall refer to it as to a particular condition for a volume form. Then in this case the operator dv is given by the following explicit formula: dv =
n i=1
∂2 , ∂x i ∂θi
and it obeys the condition 2dv = 0 .
(1.2)
(See Sect. 2 for details.) The concept of an odd symplectic supermanifold provided with a volume form is crucial in the geometrical interpretation of the BV formalism. Let f be an even function on an odd symplectic supermanifold with a coordinate volume form (1.1) in some Darboux coordinates and let dv = f dv be a new volume form on it. In general, for the new volume form dv neither condition (1.1) in some Darboux
Semidensities on Odd Symplectic Supermanifolds
355
coordinates, nor condition (1.2) are true. The main essence of the geometrical formulation of the BV formalism can be shortly expressed in the following two statements [14, 22, 15]: Statement 1 (see [14, 22, 15]). Consider the following three conditions on the volume form dv = f dv and the corresponding -operator: a) there exist Darboux coordinates such that the volume form dv = f dv has the appearance (1.1) in these coordinates , (1.3a) b)
√ dv f = 0 , √ (the BV master-equation for the master-action S = log f ) ,
(1.3b)
c) 2dv = 0 .
(1.3c)
The implications a) ⇒ b) ⇒ c) hold. The conditions a), b), c) are equivalent under some assumptions (see details below). Statement 2 (see [22]). The integrand of the BV partition function is a semidensity √ f dv, which is a natural integration object over Lagrangian surfaces in odd symplectic supermanifolds. In the case if condition (1.3b) is fulfilled, the corresponding integral does not change under small variations of the Lagrangian surface (the gauge-independence condition). The analysis of these statements in [14, 22, 15] is particularly based on the following geometrical observations. Let T ∗ M be the supermanifold associated with the cotangent bundle T ∗ M for an arbitrary manifold M. (T ∗ M is obtained by changing the parity of fibers in T ∗ M.) Functions on T ∗ M correspond to multivector fields on M. The supermanifold T ∗ M is provided with a canonical odd symplectic structure. The Schouten bracket of multivector fields on M corresponds to the odd Poisson bracket of functions on T ∗ M. The manifold M is a Lagrangian surface in T ∗ M. If dv is a volume form on M, then the odd symplectic supermanifold T ∗ M provided with the volume form dv = dv 2 satisfies conditions (1.1) and (1.2). In this case the action of operator dv on the function on T ∗ M corresponds to the divergence operator on multivector fields on M. The most profound and detailed analysis of these constructions and their relations with Statements 1 and 2 was performed in the paper [22]. Particularly in this paper some important relations were established between differential forms on M and volume forms in T ∗ M and it was observed that the square root of an arbitrary volume form in an odd symplectic supermanifold is a natural integration object over arbitrary Lagrangian surfaces in this supermanifold. In this paper we consider an odd symplectic supermanifold E = E n.n . We consider semidensities on E. We define a new operator which acts on semidensities. Our new operator is related with the operator considered above, but it does not require any additional structure on E. We see that semidensities in an odd symplectic supermanifold, not
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volume forms (densities) are naturally related with differential forms on even Lagrangian surfaces. In particular the action of a -operator on semidensities corresponds to the action of the exterior differential on differential forms. A detailed analysis of the group of canonical transformations for an arbitrary odd symplectic supermanifold E leads us to establishing relations between the calculus of semidensities on E and a calculus of differential forms on even Lagrangian surface. Our considerations have the following two applications. In terms of semidensities the BV master equation (1.3b) gets an invariant formulation and the difference between conditions (1.3a,1.3b,1.3c) can be formulated exactly. (In papers [15] and [22] it was stated that conditions (1.3a), (1.3b) and (1.3c) are equivalent, in spite of the fact that a difference between these conditions implicitly follows from Theorem 5 of the paper [22].) Note also that symmetry transformations in BV formalism considered in the paper [23] receive their proper place in the semidensities framework. Also we come to a new approach for obtaining invariant densities and the corresponding integral invariants on surfaces embedded in an odd symplectic supermanifold with a volume form. (The problem of constructing integral invariants for an odd symplectic structure drastically differs from the corresponding problem for the usual symplectic structure (see details in Sect. 5)). The exposition is organized as follows. In Sect. 2 we recall the basic definitions of an odd symplectic supermanifold and the properties of the -operator acting on functions and defined when a volume form is chosen. Then we consider semidensities and give an intrinsic definition of the -operator acting on semidensities. Using this operator we formulate the BV master equation in an invariant way. In Sect. 3 we analyze these objects in terms of the underlying even geometry considering as the basic example the supermanifold T ∗ M associated with the cotangent bundle of a usual manifold M. We establish a correspondence between differential forms on M and semidensities on the supermanifold T ∗ M and analyze the basic formulae of the calculus of differential forms in terms of semidensities. We also come to new algebraic operations on differential forms which naturally appear in terms of semidensities. In Sect. 4 we consider even ((n.0)-dimensional) Lagrangian surfaces in an odd symplectic supermanifold E and study the correspondence between differential forms on these Lagrangian surfaces and semidensities on E. For any given even Lagrangian surface L this correspondence depends on a symplectomorphism identifying T ∗ L with E. We prove the existence of an identifying symplectomorphism, study identifying symplectomorphisms and corresponding subgroups of canonical transformations, and investigate in detail to what extent the correspondence between semidensities and differential forms depends on a choice of a Lagrangian surface and identifying symplectomorphism. On the basis of these considerations we come to statements that generalize results of the paper [22] and we formulate exactly differences between conditions (1.3a), (1.3b) and (1.3c) in the BV formalism geometry. In Sect. 5 we provide a natural interpretation of the odd invariant semidensity on (1.1)-codimensional surfaces that was constructed in [13, 12]. We show that this semidensity can be considered as a kind of “pull-back” of a semidensity in the ambient odd symplectic supermanifold. This leads us to a construction of another semidensity and two densities (integral invariants), even and odd, of rank k = 4 on (1.1)-codimensional surfaces. These densities seem to be the simplest (having the lowest rank) non-trivial integral invariants on surfaces in an odd symplectic supermanifold provided with a volume form.
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The paper contains also three appendices. In Appendix 1 we briefly sketch the definition of a supermanifold as a functor from the category of Grassmann algebras to the category of sets, suggested and elaborated by A.S. Schwarz [21] (see also [19]), and which we use throughout this paper. This definition makes it possible to use the language of points for supermanifolds. (For basic definitions and constructions of supermathematics, see the books [6, 19, 25].) In Appendix 2 we give a simple proof of the Darboux theorem for odd symplectic supermanifolds. In Appendix 3 we prove a technical result about canonical transformations generated by Hamiltonians. 2. ∆-Operator on Semidensities In this section we recall the definitions and properties of odd symplectic supermanifold and of the -operator on functions. Then we consider semidensities in an odd symplectic supermanifold and define the action of the -operator on semidensities. Compared to functions this definition is intrinsic and does not require any additional structures (like volume form). Let E n.n be an (n.n)-dimensional supermanifold and zA = {x 1 , . . . , x n , θ1 , . . . , θn } be local coordinates on it (p(x i ) = 0, p(θj ) = 1, where p is a parity). We say that this supermanifold is an odd symplectic supermanifold if it is endowed with an odd symplectic structure, i.e. an odd closed non-degenerate 2-form = AB (z)dzA dzB (p() = 1, d = 0) is defined on it [6, 18, 19]. In the same way as in the standard symplectic calculus one can relate to the odd symplectic structure the odd Poisson bracket (Buttin bracket) [8, 6, 18, 19]: {f, g} =
∂f ∂g A A (−1)p(f )p(z )+p(z ) AB B , A ∂z ∂z
(2.1)
where AB = {zA , zB } is the inverse matrix to AB : AC CB = δBA . The Hamiltonian vector field Df = {f, zA }
∂ , ∂zA
Df (g) = {f, g},
(Df , Dg ) = −{f, g}
(2.2)
corresponds to every function f . The condition of the closedness of the form defining symplectic structure implies the Jacoby identity: {f, {g, h}}(−1)(p(f )+1)(p(h)+1) + cycl. permutations = 0 .
(2.3)
Using the analog of the Darboux Theorem [24, 22] (see also Appendix 2) one can consider in a vicinity of an arbitrary point coordinates zA = {x 1 , . . . , x n , θ1 , . . . , θn } such that in these coordinates symplectic structure and the corresponding odd Poisson bracket have locally the canonical expressions ∂ ∂ A B = IAB dz dz : , j = 0, i ∂x ∂x ∂ ∂ ∂ ∂ , j = 0, , j = −δij , i i ∂θ ∂θ ∂x ∂θ
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and respectively {x i , x j } = 0, {θi , θj } = 0, {x i , θj } = −{θj , x i } = δji , n ∂f ∂g p(f ) ∂f ∂g {f, g} = . + (−1) ∂x i ∂θi ∂θi ∂x i
(2.4)
i=1
These coordinates are called Darboux coordinates. Transformation of Darboux coordinates to another Darboux coordinate is called canonical transformation of coordinates. Respectively transformation of the supermanifold that transforms Darboux coordinates to another Darboux coordinate is called canonical transformation. We consider also an odd symplectic supermanifold provided additionally with a volume form: dv = ρ(z)Dz = ρ(x, θ )D(x, θ ) ,
(p(ρ) = 0) .
(2.5)
Dz = D(x, θ ) is a coordinate volume form (D(x, θ ) = dx 1 ...dx n dθ1 . . . dθn ). Coordinate volume forms in different coordinates are related by a Berezinian (superdeterminant) of coordinate transformation [6]: −1 I10 ) det (I00 − I01 I11 D z˜ ∂ z˜ I I . (2.6) = Ber , where Ber 00 01 = I10 I11 Dz ∂z det I11 We suppose that the volume form (2.5) is non-degenerate, i.e. for the every point z0 the number part m(ρ(z0 )) of ρ(z0 ) is not equal to zero. In the paper [11] we show that an odd symplectic structure (in fact an odd Poisson bracket structure, which might be degenerate) and a volume form allow to define the -operator (or Batalin-Vilkovisky operator; this is the invariant formulation of the operator introduced in BV-formalism [4]). The construction is as follows. The action of -operator on an arbitrary function in an odd symplectic supermanifold provided with a volume form is equal (up to coefficient) to the divergence w.r.t. volume form (2.5) of the Hamiltonian vector field corresponding to this function. Using (2.2) we come to the formula 1 (−1)p(f ) divdv Df 2 ∂ ∂ log ρ(z) 1 A A . = (−1)f (−1)p(Df z +z ) A {f, zA } + DfA 2 ∂z ∂zA
dv f =
(2.7)
In Darboux coordinates: 1 dv f = 0 f + {log ρ, f } , 2
(2.8)
where ρ(z) is given by (2.5) and 0 f =
n ∂ 2f . ∂x i ∂θi
(2.9)
i=1
The -operator on functions satisfies the relations [5, 14] : dv {f, g} = {dv f, g} + (−1)p(f )+1 {f, dv g} , dv (f · g) = dv f · g + (−1)p(f ) f · dv g + (−1)p(f ) {f, g} .
(2.10)
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The Operator 0 in (2.9) is not an invariant operator on functions (i.e. it depends on the choice of Darboux coordinates). It can be considered as the dv operator for coordinate volume form D(x, θ ) in the chosen Darboux coordinates zA = {x 1 , . . . , x n , θ1 , . . . , θn }. If z˜ A = {x˜ 1 , . . . , x˜ n , θ˜1 , . . . , θ˜n } is another Darboux coordinate then from (2.8) it follows that ∂z 1 0 f + {log Ber , f } , 0 f = 2 ∂ z˜
(2.11)
0 is operator (2.9) in Darboux coordinates z˜ A = {x˜ 1 , . . . , x˜ n , θ˜1 , . . . , θ˜n }. where Now we consider semidensities on an odd symplectic √ supermanifold. √ In local coordinates zA = {x i , θj } they have the appearance s = s(z) Dz = s(x, θ ) D(x, θ). Under coordinate transformation zA = zA (˜z) the coefficient s(z) is multiplied by the square root of the Berezinian of the corresponding transformation: s(z) → s(z(˜z))Ber 1/2 (∂z/∂ z˜ ). We shall define a new operator, which we denote # , and which will act on the space of semidensities. √ Definition. Let s be a semidensity and s(z) Dz be its local expression in some Darboux coordinates zA = {x 1 , . . . , x n , θ1 , . . . , θn }. The local expression for the semidensity # s in these coordinates is given by the following formula: n √ ∂ 2s s = (0 s(z)) Dz = D(x, θ) . ∂x i ∂θi #
(2.12)
i=1
The semidensity # s is odd (even) if semidensity s is an even (odd) semidensity, thus # is an odd operator. Contrary to the operator dv on functions, the operator # on semidensities does not need any volume structure. To prove that the # -operator is well-defined by formula (2.12), one has to check that the r.h.s. of (2.12) indeed defines semidensity, i.e. if {˜zA } = {x˜ 1 , . . . , x˜ n , θ˜1 , . . . , θ˜n } another Darboux coordinate then n i=1
∂2 s(z) ∂x i ∂θi
z(˜z)
∂z(˜z) · Ber ∂ z˜
1/2 =
n
∂2
i=1
∂ x˜ i ∂ θ˜i
s(z(˜z)) · Ber
z) 1/2 ∂z(˜ ∂ z˜
.
(2.13) First of all we check this condition for infinitesimal canonical transformations. They are generated by an odd function (Hamiltonian) via the corresponding Hamiltonian vector field. To an odd Hamiltonian Q(z) corresponds the infinitesimal canonical transformation z˜ a = zA + ε{Q, zA } generated by the vector field DQ in (2.2). To the √ action (s Dz) = of this transformation on the semidensity s corresponds the differential δ Q √ √ 0 Q · s Dz− {Q, s} Dz, because δs = −ε{Q, s} and δDz = εδBer(∂z/∂ z˜ )Dz = 20 QDz. Using that 20 = 0 and relation (2.10) we come to commutation relations 0 δQ = δQ 0 . Thus we come to condition (2.13), for infinitesimal transformations.
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To check condition (2.13) for arbitrary canonical transformation we need the following A A Lemma 1. 1. Every canonical transformation of Darboux coordinates z˜ = F (z) can be decomposed into canonical transformations F(z) = Fs Fp Fadj (z) , where 1) canonical transformation z˜ = Fadj (z), has the following form:
x˜ i (x, θ ) θ=0 = x i , (i = 1, . . . , n) , (2.14a) θ˜i (x, θ ) θ=0 = 0 ,
we later call this canonical transformation of Darboux coordinates the adjusted canonical transformation; 2) canonical transformation z˜ = Fp (z) has the form
x˜ i = x i (x) m (x) (i, j = 1, . . . , n) , (2.14b) ˜ θ˜i = ∂x∂ xi ˜ θm we later call this canonical transformation of Darboux coordinates, which is generated by transformation x˜ i = x i (x) a “point”-canonical transformation; 3) canonical transformation z˜ = Fs (z) has the following form:
∂ i (x) ∂ j (x) x˜ i = x i − = 0 , (i = 1, . . . , n) , such that θ˜i = θi + i (x) ∂ xj ∂x i (2.14c) we later call this canonical transformation of Darboux coordinates a special canonical transformation. 2. Berezinian of adjusted canonical transformation (2.14a) obeys the condition Ber ∂∂zz˜ θ=0 = 1, the Berezinian of “point” canonical transformation (2.14b) is equal ∂ x˜ to det 2 ∂x , and Berezinian of special canonical transformation (2.14c) is equal to one. In particular, the numerical part of the Berezinian of arbitrary canonical transformation is positive. 3. Adjusted canonical and special canonical transformations of Darboux coordinates (2.14a), (2.14c) are canonical transformations generated by Hamiltonian, i.e. they can be included in the one-parametric family of canonical transformations of Darboux coordinates generated by an odd Hamiltonian: ∃Q(z, t) :
dzt = {Q, zt }, 0 ≤ t ≤ 1 such that z0 = z , z1 = z˜ . dt
(2.15)
The Special canonical transformation (2.14c) is generated locally by Hamiltonian Q = Q(x), such that ∂i Q(x) = i (x). There exists a unique “time”-independent Hamiltonian Q = Q(z) obeying the condition Q(x, θ ) = Qik θi θk + . . . , i.e. Q = O(θ 2 ) that generates the given adjusted canonical transformation. Proof of this lemma. Let z˜ = F(z) be an arbitrary canonical transformation: x˜ i = f i (x, θ ) = f0i (x)+O(θ ) and θ˜i = i (x)+O(θ ). Consider coordinates {¯zA } = {x¯ i , θ¯i } that are related with coordinates {˜zA } by the following special canonical transformation: z˜ A = Fs (¯z) such that x˜ i = x¯ i and θ˜i = θ¯i + i (g(x)), ¯ where g ◦ f0 = id. Then
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x¯ i = f i (x, θ ) and θ¯i = O(θ ). Now consider coordinates {zA } = {x i , θi } that are related with coordinates {¯zA } by the “point” canonical transformation z¯ A = Fp (z ) generated by functions x¯ i = f0i (x ). Then it is easy to see that initial coordinates {zA } are related with coordinates {zA } by the adjusted canonical transformation zA = Fadj (z): x i = x i + O(θ ), θi = O(θ ). The second statement of the lemma can be proved by an easy straightforward calculation of Berezinian (2.6) for transformations (2.14a), (2.14b), (2.14c). We perform the proof of Statement 3 of the lemma for adjusted canonical transformations in Appendix 3.
Now we return to the proof of relation (2.13). First we note that from second statement of Lemma it follows that square root operation in (2.13) is well-defined. From decomposition (2.14) it follows that it is sufficient to check condition (2.13) separately for adjusted, “point”, and special canonical transformations. From the third statement of the lemma it follows that for adjusted and special canonical transformations the condition (2.13) can be checked only infinitesimally and this is performed already. For the “point” canonical transformation (2.14b) the condition (2.13) can be easily checked straightforwardly using (2.10), (2.11) and the fact that Berezinian of this transformation does not depend on θ .
The action of the differential δQ corresponding to an infinitesimal canonical transformation on semidensities can be rewritten in an explicitly invariant way: δQ s = Q · # s + # (Qs) = [Q, # ]+ s .
(2.16)
On an odd symplectic supermanifold provided with a volume form dv (density of the weight σ = 1) we can construct new invariant objects, expressing them via the semidensity related with volume form and the the operator # : √ s = dv semidensity (σ = 21 ) , (2.17a) √ # # 1 s = dv semidensity (σ = 2 ) , (2.17b) √ √ s# s = dv# dv density (σ = 1) , (2.17c) √ 1 # 1 # function (σ = 0) . (2.17d) s = √ dv s dv From definition (2.12) of # -operator and relations (2.10) it follows that the obeys the following properties:
# -operator
# 2 ( √ √ ) = 0, √ (f · dv) = (dv f ) · dv + (−1)f f · # dv , #
(2.18)
and √ 1 2dv f = { √ # dv, f } . dv
(2.19)
We call the semidensity s a closed semidensity if # s = 0 and we call s exact if there exists another semidensity r such that s = # r.
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In the case if an odd symplectic supermanifold is provided with a volume form dv ˜ in some Darsuch that this volume form is equal to a coordinate volume form ˜ θ) √ D(x, boux coordinates {x˜ 1 , . . . , x˜ n , θ˜1 , . . . , θ˜n } then evidently # dv = 0. Considering this relation in another Darboux coordinate zA = {x 1 , . . . , x n , θ1 , . . . , θn } we come to the formula n ˜ ˜ θ) ∂2 1/2 ∂ z˜ 1/2 ∂(x, 0 Ber = = 0. (2.20) Ber ∂z ∂x i ∂θi ∂(x, θ) i=1
We note that formulae (2.9) and (2.11) for the 0 operator were first studied by I.A.Batalin and G.A.Vilkovisky ( [4, 5]). In particular they obtained formula (2.20). These results receive its clear geometrical interpretation in terms of semidensities and action of the # -operator on them. √ We say that semidensity s = s(x, θ ) D(x, θ ) is non-degenerate if a number part m(s(x, θ )) of s(x, θ ) is not equal to zero at any x. Every volume form defines non-degenerate even semidensity by relation (2.17a) and respectively volume form corresponds to every non-degenerate even semidensity. We say that an even non-degenerate semidensity s obeys the BV-master equation if it is closed and we denote by Bdeg a set of these densities, Bdeg = {s :
# s = 0 ,
p(s(x, θ )) = 0,
m(s(x, θ )) = 0} .
(2.21)
The BV-master equation (condition (1.3b)) was not formulated invariantly in [14, 22]. Condition # s = 0 (closedness of semidensity s) gives invariant formulation to the BV master-equation. 3. Differential Forms on Cotangent Bundle and Semidensities We consider in this section a basic example of an odd symplectic supermanifold yielded by a cotangent bundle of the usual manifolds. We clarify the geometrical meaning of previous constructions and establish relations between differential forms on the manifold and semidensities on this odd symplectic supermanifold. In the standard symplectic calculus cotangent bundle of any manifold can be provided with canonical symplectic structure and it can be considered as a basic example of the symplectic manifold [10]. A basic example of an odd symplectic supermanifold is constructed in the following way. Let M be an arbitrary n-dimensional manifold and T ∗ M be its cotangent bundle. Consider a supermanifold T ∗ M associated with the cotangent bundle T ∗ M, changing the parity of fibers of cotangent bundle T ∗ M. Let {x 1 , . . . , x n , p1 , . . . , pn } be canonical coordinates on T ∗ M corresponding to arbitrary local coordinates {x 1 , . . . , x n } on M, i.e. for a form w ∈ T ∗ Mpi (w) = w( ∂x∂ i ). Canonical coordinates zA = (x 1 , . . . , x n , θ1 , . . . , θn ) on T ∗ M, (p(θi ) = 1) correspond to the canonical coordinates {x 1 , . . . , x n , p1 , . . . , pn } on T ∗ M. Odd coordinates {θ1 , . . . , θn } transform via the differential of the corresponding transformation of coordinates {x i } of underlying space M in the same way as coordinates {p1 , . . . , pn } on T ∗ M: x˜ i = x˜ i (x),
θ˜i =
n ∂x k (x) ˜ k=1
∂ x˜ i
θk ,
(i = 1, . . . , n) .
(3.1)
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We define the canonical odd symplectic structure on T ∗ M considering
these coordii nates as Darboux coordinates (2.4). Thus we assign to every atlas {x(α) } of coordinates
i ,θ on the manifold M an atlas {x(α) j (α) } of Darboux coordinates on the supermanifold T ∗ M. Pasting formulae (3.1) ensure us that this canonical symplectic structure is well-defined. Later on we call Darboux coordinates (3.1) on T ∗ M induced by coordinates on M Darboux coordinates adjusted to the cotangent bundle structure. Unless otherwise stated we assume further that Darboux coordinates in a supermanifold associated with the cotangent bundle are Darboux coordinates adjusted to cotangent bundle structure. (Canonical transformations (3.1), induced by coordinate transformations on the manifold M are “point” canonical transformations (2.14b).) The relations between the cotangent bundle structure on T ∗ M and the odd canonical symplectic structure on T ∗ M reveal in the properties of the following canonical map τM between multivector fields on M and functions on T ∗ M: ∂ i1 ...ik ∂ τM T ∧ · · · ∧ i = T i1 ...ik θi1 . . . θik . (3.2) ∂x i1 ∂x k This map transforms the Schoutten bracket of multivector fields to the odd canonical Poisson bracket (Buttin bracket) (2.2) of corresponding functions [19, 8]: τM ([T1 , T2 ]) = {τM (T1 ) , τM (T2 )} .
(3.3)
Now we construct a map that establishes correspondence between differential forms on M and semidensities on T ∗ M. We consider arbitrary Darboux coordinates {x 1 , . . . , x n , θ1 , . . . , θn } on T ∗ M adjusted to cotangent bundle structure and define this map in these Darboux coordinates in the following way: τM# (1) = θ1 . . . θn D(x, θ ), τM# (dx i ) = (−1)i+1 θ1 . . . θi . . . θn D(x, θ ) , τM# (dx i ∧ dx j ) = (−1)i+j θ1 . . . θi . . . θj . . . θn D(x, θ) , (i < j ), ... τM# (dx i1 ∧ · · · ∧ dx ik ) = (−1)i1 +···+ik +k θ1 . . . θi1 . . . θik . . . θn D(x, θ ), (i1 < · · · < ik ), τM# (f (x)w) = f (x)τM# (w) , for every function f (x) on M , (3.4) where the sign means omitting the corresponding term. For example if M is two√ dimensional space, then τM# (f (x)) = f (x)θ1 θ2 D(x, θ ), τM# (w1 dx 1 + w2 (x)dx 2 ) = √ √ (w1 θ2 − w2 (x)θ1 ) D(x, θ ), τM# (wdx 1 ∧ dx 2 ) = −w D(x, θ ). One can rewrite (3.4) in a more compressed way: # i n τM (w) = w(x, ξ ) exp(θi ξ )d ξ D(x, θ ) , (3.4a) where w(x, ξ ) is a function corresponding to the differential form w in the supermanifold T M associated to the tangent bundle T M: w(x, ξ ) = wi1 ...ik ξ i1 . . . ξ ik . Odd coordinates {ξ i } of the fibers in T M transform as differentials {dx i }: x˜ i = x˜ i (x) ∂ x˜ i k # 2 → ξ˜ i = ∂x k ξ . The square of the map (3.4a) w → (τ (w)) transforms differential ∗ forms on M to density (volume form) on T M and this map was considered in [22].
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To prove that (3.4) is well-defined for arbitrary Darboux coordinates adjusted to cotangent bundle structure we note that under arbitrary coordinate transformation (3.1) the integral in the r.h.s. of (3.4a) is multiplied on the det(∂ x/∂x) ˜ and coordinate volume √ form D(x, θ ) is divided on the module of this determinant, because for transformation (3.1): i i ∂ x˜ (x) ˜ ∂ x˜ r ∂ 2 x m ˜ θ˜ ) ∂ x˜ (x) θ m k k r i 1/2 ∂(x, 1/2 ∂x ∂x ∂ x˜ ∂ x˜ = det Ber . (3.5) = Ber ∂x k ∂(x, θ ) ∂x k 0 i ∂ x˜
Remark. Map (3.4) establishes correspondence only up to a sign factor because the r.h.s. of (3.5) is positive for the canonical transformation induced by any coordinate transformation of M. In the case if M is an orientable manifold considering only Darboux coordinates such that Jacobian of coordinate transformations is positive one comes to a globally defined map. The sign factor depends on the orientation of M. We say that the semidensity s corresponds to the differential form w (to the linear # (w) = τ # ( combination of differential forms wk ) if s = τM wk ). M In the end of this section using correspondence between semidensities and differential forms we consider some standard constructions of differential forms calculus in terms of semidensities and two geometrical operations on differential forms, which naturally arise in terms of semidensities. 1) From (3.4a) it is easy to see that an action of the operator ξ i ∂x∂ i on the function w(x, ξ ) corresponds to the action of the exterior differential d on a differential form and to the action of the # -operator on a semidensity, i.e. the action of the # -operator corresponds to the action of the exterior differential: # ◦ τM# = τM# ◦ d .
(3.6)
Closed (exact) semidensity corresponds to closed (exact) differential form. 2) If the semidensity s in T ∗ M corresponds to volume form (a differential top-degree form w on M) and an odd sympelctic supermanifold T ∗ M is provided with volume form such that it is equal to the square of this semidensity then the action of operator dv corresponds to the divergence w.r.t. to the volume form w on M: dv ◦ τM = τM ◦ divw
if
# dv = s2 and s = τM w.
(3.7)
(See also [15, 22].) 3) From (3.2) and (3.4) it follows that τM# (Tw) = τM (T) · τM# (w) ,
(3.8)
where Tw is the inner product of the multivector field T with differential form w. 4) The meaning of relation (2.16) in terms of differential forms is the following. In the special case if the Hamiltonian Q corresponds to vector field T i (x) ∂x∂ i (Q = T i (x)θi ), then this Hamiltonian induces an infinitesimal canonical transformation that corresponds to the infinitesimal transformation of M induced by the vector field T i (x) ∂x∂ i . From (3.6), (3.8) it follows that in this case the standard formula for the Lie derivative of differential forms (LT w = dwT + d(wT )) corresponds to relation (2.16). In a general case canonical transformations of T ∗ M destroy the cotangent bundle structure and mix forms of different degrees. For example if we consider the action of the Hamiltonian Q = Lθ1 . . . θn on a semidensity corresponding to form w = dx 1 ∧ · · · ∧ dx n then we obtain using (2.16) that δw = dL.
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5) If a = ai (x)dx i is a 1-form on M then one can see that τ # (a ∧ w) = ai
∂s D(x, θ ), ∂θi
where
τ # (w) = s .
(3.9)
It is more natural from the point of view of semidensities to consider the following relation between 1-forms on M and semidensities in T ∗ M. Let a = ai dx i be an odd-valued one-form on M with coefficients in an arbitrary Grassmann √ algebra (see Appendix 1). For this form and arbitrary semidensity s = s(x, θ ) D(x, θ ) consider a new semidensity s , which we denote by a s such that it is given by relation (3.10) s = a s = s(x, θi + ai ) D(x, θ ) . Respectively if semidensity s corresponds to differential form w = wk then we denote by a w the differential form such that semidensity a s corresponds to a w. From (3.10) and (3.4a) it follows that aw =
k 1 a ∧ · · · ∧ a ∧wk−p , p!
p=0
p
(k = 0, . . . , n) .
(3.11)
times
Relations (3.10) and (3.11) define an action of the abelian supergroup of differential odd valued one-forms on semidensities and differential forms. 6) Consider also the following algebraic operation on differential forms that seems very natural from the point of view of semidensity calculus. Let w = wk and w = wk be differential forms on M n such that top-degree forms wn and wn are not equal to zero. Then we consider a new form # (w ) · τ # (w ) . w˜ = w ∗ w : τ # w˜ = τM (3.12) 1 2 M The condition wn = 0, wn = 0 for top-degree forms makes well-defined a square root operation on corresponding semidensities. 4. Semidensities on E and Differential Forms on Even Lagrangian Surfaces In the previous section we analyzed relations between differential forms on the manifold M and semidensities on the supermanifold T ∗ M using Darboux coordinates in T ∗ M that are adjusted to cotangent bundle structure of T ∗ M. (Relations (3.4) are not invariant with respect to an arbitrary canonical transformation of Darboux coordinates.) In this section we analyze a more general situation. We consider relations between semidensities on an arbitrary odd symplectic supermanifold and differential forms on even Lagrangian surfaces in this supermanifold. Then we apply these results for analyzing relations between conditions (1.3a), (1.3b) and (1.3c) for Batalin-Vilkovisky formalism geometry. A Lagrangian surface in (n.n)-dimensional odd symplectic supermanifold E = E n.n is a (k.n − k)-dimensional surface embedded in this supermanifold such that the restriction of the symplectic form on it is equal to zero. We call an (n.0)-dimensional Lagrangian surface an even Lagrangian surface. For an odd symplectic supermanifold T ∗ M an initial underlying n-dimensional manifold M can be considered as an even Lagrangian
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surface embedded in this supermanifold. (Note that in a case if we consider -supermanifolds, the underlying manifold is not necessarily a Lagrangian surface.) If L is an even Lagrangian surface in an odd symplectic manifold E and T ∗ L is a supermanifold associated with the cotangent bundle of L, then one can consider correspondence between semidensities on E and differential forms on L provided there is an identifying symplectomorphism between supermanifolds T ∗ L and E: (4.1) symplectomorphism ϕL : T ∗ L → E and ϕL L = id . In this case the pull-back ϕ ∗ s of semidensity s corresponds to differential forms on L via map (3.4): τL# (wn + wn−1 + · · · + w1 + w0 ) = ϕL∗ s ,
(4.2)
where wk is a differential k-form on L. This correspondence depends on a choice of the identifying symplectomorphism (4.1) Thus at first we study properties of identifying symplectomorphisms. 4.1. Identifying symplectomorphisms for even Lagrangian surfaces. In the usual symplectic calculus if L is a Lagrangian surface in a symplectic manifold N then there exists a symplectomorphism between tubular neighborhoods of L in T ∗ L and in N that is identical on L [10]. In the general case there is no Lagrangian surface L such that T ∗ L is symplectomorphic to N . The nilpotency of odd variables leads to the fact that odd symplectic supermanifolds have simpler structure. Particularly, any (n.0)-dimensional surface in (n.n)-dimensional supermanifold can be expressed locally by equations θi − i (x) = 0, i = 1, . . . , n in any coordinates {x i , θj }. Hence for every even Lagrangian surface in an odd symplectic supermanifold E = E n.n its underlying n-dimensional manifold M = M n (L) is an open submanifold in the underlying manifold M of E. If E is a corresponding restriction of the supermanifold E with underlying manifold M then one can prove that there exists a symplectomorphism ϕ that identifies T ∗ L with E . We suppose later that M coincides with M. For example this is the case if M is a closed connected manifold and M (L) is also closed. We call such Lagrangian surfaces closed. Proposition 1. Let L be an arbitrary closed even Lagrangian surface in an odd symplectic supermanifold E. Then there exists an identifying symplectomorphism (4.1) between T ∗ L and E. Proof of this proposition. An identifying symplectomorphism can be constructed for every even Lagrangian surface in terms of suitable Darboux coordinates.
i ,θ Namely, consider an arbitrary atlas A(E) = {x(α) j (α) } of Darboux coordinates on a supermanifold E with closed connected underlying manifold M. (Every coordinate i ,θ ˆ α with underlying domain Uα . {x(α) j (α) } of this atlas is defined on the superdomain U
i i that are numerical parts of the functions xαi define an atlas {x0(α) } on Functions x0(α) the underlying manifold M.) We say that Darboux coordinates {x i , θj } in E are adjusted to the Lagrangian surface L if θ1 = · · · = θn = 0 on L. Respectively we say that an atlas of Darboux coordinates is adjusted to the Lagrangian surface L if all coordinates from this atlas are adjusted to this surface.
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Suppose that there already exists an identifying symplectomorphism
ϕL (4.1) for a
i ,η given closed even Lagrangian surface L. Let A(T ∗ L) = {y(α) j (α) } be an atlas of ∗ Darboux coordinates in T L adjusted to the cotangent bundle structure of T ∗ L (see
i (3.1)). Consider an atlas A(E) = {x(α) , θj (α) } of Darboux coordinates on E defined by relations i i ϕL∗ x(α) = y(α) ,
ϕL∗ θj (α) = ηj (α) .
(4.3)
This atlas is adjusted to the Lagrangian surface L. Moreover from definition of this atlas and (3.1) it follows that all transition functions αβ of A(E) on superdomains Uˆ αβ with underlying domains Uαβ = Uα ∩ Uβ are “point”-like canonical transformations (2.14b). We also call this atlas on E an atlas adjusted to cotangent bundle structure of Lagrangian surface L. It is easy to see that arbitrary atlas of Darboux coordinates on E adjusted to cotangent bundle structure of a given Lagrangian surface L defines some identifying symplectomorphism for this Lagrangian surface via relations (4.3). (Darboux coordinates in r.h.s. of (4.3) adjusted to cotangent bundle structure of T ∗ L i } in E.) Thus Proposition 1 follows are generated by restriction on L of coordinates {x(α) from the following Lemma Lemma 2. For arbitrary even Lagrangian surface L in an odd symplectic supermanifold E there exists an atlas of Darboux coordinates in E adjusted to cotangent bundle structure of this surface. Proof of this lemma. Considering in a vicinity of an arbitrary point of E arbitrary i ,θ Darboux coordinates (see Appendix 2 for details) we come to some atlas {x(α) j (α) } n.n of Darboux coordinates on E . If the Lagrangian surface L is defined in this atlas by equations θi(α) − i(α) (xα ) = 0, then the condition that the surface L is Lagrangian implies that ∂i j − ∂i j = 0. Hence changing θiα → θi(α) − i(α) (xα ) we come to the atlas Aadj of Darboux coordinates adjusted to the surface L (θi(α) |L = 0). We show that it is possible to change coordinates in every superdomain Uα for an atlas Aadj in a way that all transition functions become “point”-like canonical transformations (2.14b). We prove it by induction. Without loss of generality consider a case when the number of charts is countable (α = 1, . . . , n, . . . ). Suppose that we already changed coordinates in the required way for the first k charts: all transition functions z(α) = αβ (z(β) ) are already “point”-like canonical transformations for α, β = 1, . . . , k. A } = {x i , θ } on the superdomain U ˆ α (with Consider Darboux coordinates {z(α) (α) (α) underlying domain Uα ) for α = k + 1. For every β ≤ k consider the transition function A = (z ) in the superdomain U ˆ βα (canonical transformation of coordinates) z(β) βα (α) (with underlying domain Uβα = Uβ ∩ Uα ). All coordinates are adjusted to the Lagrangian surface (θi(α) |L = 0), hence from statement 1 of Lemma 1 it follows that one can consider in every superdomain Uˆ βα (α = A } such that zA = (z ) = F ◦ F (z ) = k + 1, β ≤ k) new coordinates {˜zαβ βα (α) p adj (α) (β) A (˜ A (˜ z(β) z(βα) (z(α) )), where z(β) z(βα) ) is the point-like canonical transformation and A (z )) is the adjusted canonical transformation. z˜ (βα) (α) To complete the proof of the lemma we have to define in the superdomain Uˆ α (α = A } such that restrictions of these coordinates on superdomains k +1) new coordinates {˜z(α)
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A } constructed above. From statement 3 of Lemma 1 Uβα coincide with coordinates {˜z(βα) it follows that there exist Hamiltonians Q(βα) in Uˆ βα that generate an adjusted canonical A (α ≤ k + 1, β ≤ k). From an transformation from coordinates zαA to coordinates z˜ (βα) inductive hypothesis and uniqueness of these Hamiltonians it follows that Q(αβ) = Q(αγ ) in superdomains Uˆ αβγ . Hence one can consider an odd Hamiltonian obeying the condition Q = O(θ 2 ) on a superdomain Uˆ α (α = k + 1) such that restriction of this Hamiltonian on superdomains Uˆ βα is equal to Q(βα) . This Hamiltonian generates an A } to the new required Darboux adjusted canonical transformation from coordinates {z(α) A coordinates {˜z } on superdomain Uˆ α .
(α)
Certainly, the identifying symplectomorphism (4.1) for a given closed even Lagrangian surface L is not unique. To study this point consider the (infinite-dimensional) supergroup Can(E) of canonical transformations of supermanifold E n.n . Every canonical transformation is an -point (element) of this supergroup (see Appendix 1). The supergroup Can(E) acts transitively on the superspace of closed even Lagrangian surfaces. Denote by Can(L) the stationary subgroup of supergroup Can(E) for L and consider the subgroup Canadj (L) of supergroup Can(L) such that -points of Canadj (L) are canonical transformations that are identical on the surface L: Canadj (L) F ⇔ F |L = id. It is easy to see that canonical transformations obeying this condition have the following appearance in arbitrary Darboux coordinates adjusted to the surface L:
x˜ i = x i + f i (x, θ ), where f i (x, θ ) = O(θ) (4.4) , if θi |L = 0 . θ˜i = θi + gi (x, θ ), where gi (x, θ ) = O(θ 2 ) Later we call canonical transformations obeying the condition F |L = id canonical transformations adjusted to the Lagrangian surface L. The adjusted canonical transformation (2.14a) corresponds to transformation (4.4) in adjusted coordinates. Now consider the superspace (L) of identifying symplectomorphisms for a given closed even Lagrangian surface L. (Every identifying symplectomorphism ϕL is an -point (element) of this superspace). The supergroup Canadj (L) acts free on the superspace (L) of identifying symplectomorphisms: two arbitrary identifying symplectomorphisms ϕL and ϕL differ on the canonical transformation adjusted to the surface L: ϕL = F ◦ ϕL , where F = id . (4.5) L
Consider the supergroup Can0 (E) that is a unity connectivity component of the supergroup Can(E), i.e. the canonical transformation F belongs to Can0 (E) if it can be included in a one-parametric (continuous) family Ft of canonical transformations (0 ≤ t ≤ 1) such that F0 = id and F1 = F . Consider also the subgroup CanH (E) of Can0 (E) such that canonical transformation F belongs to CanH (E) if it can be included in a one-parametric family Ft of canonical transformations (0 ≤ t ≤ 1) generated by some Hamiltonian Q(x, θ, t): F˙t = {Q, F }, F0 = id and F1 = F . We call canonical transformations belonging to CanH (E) canonical transformations generated by the Hamiltonian. Consider the Lie superalgebra Gadj (L) such that -points (elements) of this superalgebra are odd functions on E (“time”-independent Hamiltonians Q(x, θ )) that obey the following condition: Q = Qik (x, θ )θi θk ,
i.e.
Q = O(θ 2 )
(4.6)
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in Darboux coordinates adjusted to Lagrangian surface L. (Lie algebra structure is defined via the odd Poisson bracket (2.4).) One can show that the superalgebra Gadj (L) corresponds to the supergroup Canadj (L). Indeed it is easy to see that the arbitrary Hamiltonian obeying condition (4.6) generates a one-parametric family of canonical transformations Ft = Exp tQ (0 ≤ t ≤ 1) adjusted to the surface L and Exp tQ1 = Exp tQ2 if Q1 = Q2 . (see Appendix 3 for details). Thus the map Exp Q : Gadj (L) → Canadj (L) is a well-defined injection. Moreover this exponential map is a bijective map. To find the Hamiltonian Q ∈ Gadj (L) that generates a given transformation F ∈ Canadj (L) (F = Exp Q) consider the transformation F in arbitrary atlas A of Darboux coordinates adjusted to the cotangent bundle structure of the surface L. In every coordinate from this atlas transformation F has the appearance (4.4), hence according to statement 3 of Lemma 1 in every coordinate from the atlas A there exists a unique “time”-independent Hamiltonian Q(α) obeying condition (4.6) such that this Hamiltonian generates locally this transformation (Q(a) = −θi f i (x, θ ) + O(θ 3 )). One can see that local Hamiltonians {Q(α) } do not depend on a choice of coordinates from this atlas. Hence they define uniquely a global Hamiltonian Q in the superalgebra Gadj (L). We come to Proposition 2. For a given closed even Lagrangian surface L in E two arbitrary identifying symplectomorphisms are related with each other by the canonical transformation adjusted to the Lagrangian surface. This canonical transformation is generated by a “time”-independent Hamiltonian that is defined uniquely by condition (4.6). In other words the supergroup Canadj (L) acts freely on superspace (L) of identifying symplectomorphisms. The exponential map Exp from Lie superlalgebra Lie Gadj (L) to Canadj (L) is a bijection. For later considerations we need to study the difference between supergroups Can0 (E) (unity connectivity component in Can(E)) and supergroup CanH (E) of canonical transformations generated by the Hamiltonian. For this purpose we consider decomposition of the group Can(E) of all canonical transformations on subgroups that are isomorphic to Canadj (L), the supergroup Diff (L) of diffeomorphisms of Lagrangian surface L, and the supergroup that acts freely on the superspace of all even Lagrangian surfaces. To describe this latter supergroup consider the abelian supergroup Z 1 (L) of closed differential one-forms on L, where Z 1 (L) is the superspace of closed differential forms on L and is parity reversing functor. (-points of the supergroup Z 1 (L) (Z 1 (L)) are closed one-forms with odd (even) coefficients from the Grassmann algebra ). The supergroup Z 1 (L) is a subgroup of the abelian supergroup of odd-valued differential one-forms considered in Sect. 3 (see (3.10)). The superspace Z 1 (L) can be identified with a superspace of even closed Lagrangian surfaces in T ∗ L, because every odd valued differential one-form i dx i can be identified with an (n.0)-dimensional surface embedded in T ∗ L given by equations θi − i (x) = 0. Under this identification closed even Lagrangian surfaces in T ∗ L correspond to closed forms. There is a natural monomorphism of the supergroup Z 1 (L) in the supergroup Can(T ∗ (L)) of all canonical transformations of the supermanifold T ∗ L: the special canonical transfori i , θ mation (2.14c) x(α) → x(α) j (α) → θj (α) + i(α) (x(α) ) corresponds to an element i i (x)dx of the supergroup Z 1 (L) in an atlas of Darboux coordinates on T ∗ L adjusted to the cotangent bundle structure of L. The abelian supergroup Z 1 (L) acts freely on the superspace of closed even Lagrangian surfaces in T ∗ L. The action of this
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supergroup on semidensities in T ∗ L and arbitrary differential forms on L is defined by the operation (3.10). There is also a natural monomorphism of the supergroup Diff (L) in the supergroup Can(T ∗ (L)) of all canonical transformations of the supermanifold T ∗ L corresponding to the point-canonical transformation (see (2.14b) and (3.1)). Now for supergroups Diff (L) and Z 1 (L) we consider an affine supergroup Z 1 (L) Diff (L) such that the semidirect product is induced by an action of diffeomorphisms of L on forms: [ 1 , f1 ] ◦ [ 2 , f2 ] = [ 1 + (f1−1 )∗ 2 , f1 ◦ f2 ], where 1 , 2 ∈ Z 1 (L) are closed odd valued one-forms and f1 , f2 ∈ Diff (L) are diffeomorphisms of L. Monomorphisms of the supergroups Z 1 (L) and Diff (L) in the supergroup Can(T ∗ L) considered above define a monomorphism ι of the affine supergroup Z 1 (L) Diff (L) in the supergroup Can(T ∗ L). Thus every identifying symplectomorphism ϕL defines a monomorphism ιϕL = ϕL ◦ ι ◦ ϕL−1 of the supergroup Z 1 (L) Diff (L) in the supergroup Can(E). On the other hand consider for an arbitrary canonical transformation F ∈ Can(E) the = ϕ −1 ◦ F (L) in T ∗ L and a closed odd valued one-form on Lagrangian surface L L Then the canonical transformation F = L corresponding to the Lagrangian surface L. ιϕL ([− , id]) ◦ F of the supermanifold E belongs to the supergroup Can(L) of canonical transformations that transform the Lagrangian surface L to itself. The restriction of the canonical transformation F on L defines the diffeomorphism f = F |L ∈ Diff (L). Thus we define a projection map pϕL :
Can(E) → Z 1 (L)Diff (L)
(4.7)
that depends on the identifying symplectomorphism. This map projects the subgroup Canadj (L) of canonical transformations adjusted to the surface L to the unity element and obeys the condition pϕL ◦ιϕL = id. We come to the following result: for a given Lagrangian surface L and identifying symplectomorphism ϕL the arbitrary canonical transformation F ∈ Can(E) can be decomposed uniquely in the following way: F = ιϕL ([ , f ]) ◦ Fadj = Fs ◦ Fp ◦ Fadj , where [ , f ] = pϕL (F ) , Fs = ιϕL ([ , id]) , Fp = ιϕL ([0, f ]) , Fadj ∈ Canadj (L) . (4.8) (The decomposition (2.14) in Lemma 1 corresponds to this decomposition.) One can check the following property of the projection map (4.7): If ϕL , ϕL are two arbitrary identifying symplectomorphisms for a given Lagrangian surface L and pϕL (F ) = [ , f ], pϕ (F ) = [ , f ] then L
− = d ,
f = f0 ◦ f ,
(4.9)
where f0 is a diffeomorphism that can be included in the one-parametric continuous family ft of diffeomorphisms such that f1 = f and f0 = id. In other words f0 belongs to the group Diff0 (L) that is a unity connectivity component of the group Diff (L). According to Proposition 2 conditions (4.9) have to be checked only for infinitesimal canonical transformations (4.4) adjusted to the Lagrangian surface L and generated by the Hamiltonian (4.6). This can be done by easy straightforward calculations. From (4.9) it follows that for a given Lagrangian surface L the projection map (4.7) defines a map pL :
Can(E) → H 1 (L)π0 (Diff (L)) ,
(4.10a)
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where H 1 (L) is an abelian group of cohomology classes of one-forms on L (with reversed parity) and π0 (Diff (L)) = Diff (L)/Diff0 (L) is the discrete group of connectivity components of Diff (L). Using decomposition (4.8), relations (4.9) and the fact that the supergroup Canadj is a normal subgroup in Can(L) one can show that (4.10a) is a epimorphism. (The projection map (4.7) is not an epimorphism, because the supergroup Canadj is not a normal subgroup in Can(E).) One can consider also a composition of an epimorphism (4.10a) with a natural epimorphism of H 1 (L) π0 (Diff (L)) on π0 (Diff (L)): pˆ L :
pL
Can(E)−→H 1 (L)π0 (Diff (L)) → π0 (Diff (L)) .
(4.10b)
Epimorphisms (4.10a) and (4.10b) allow to check the difference between supergroups Can0 (E) and CanH (E) because ker pL = CanH (E)
and
ker pˆ L = Can0 (E) .
(4.11)
Namely consider an arbitrary canonical transformation F that belongs to the kernel of the epimorphism (4.10a). Then for the projection map (4.7) pϕL (F ) = [ , f ], where = d and f ∈ Diff0 (L). Consider the decomposition (4.8) for this canonical transformation F . Then the canonical transformation Fs = ρ([ , id]) is generated by the Hamiltonian Q = (x). The canonical transformation Fs = ρ([0, f ]) is generated by the Hamiltonian Q = K i (t, x)θi , where the “time”-dependent vector field K i (t, x) is equal to ft−1 ◦ f˙t for a family ft of diffeomorphisms that connects the diffeomorphism f with the identity diffeomorphism. The canonical transformation Fadj is generated by some Hamiltonian Q(x, θ ) according to Proposition 2. Hence the kernel of the epimorphism (4.10a) belongs to CanH (E). To prove the converse implication consider the one-parametric family Ft of canonical transformations generated by an arbitrary Hamiltonian Q(x, t). Decompose for every t the transformation Ft by formula (4.8) for an arbitrary identifying symplectomorphism ϕL : Ft = Fs (t) ◦ Fp (t) ◦ Fadj (t). Transformations Fp (t) and Fadj (t) are generated by Hamiltonians, hence the transformation Fs (t) is generated by the Hamiltonian Q also. Hence = dQ ,where pϕL Fp (t) = [ t , 1] and pL (F ) = [0, 1] in H 1 (L) π0 (Diff (L)). The proof of the second relation in (4.11) is analogous. We come to Proposition 3. Let L be a closed even Lagrangian surface in an odd symplectic supermanifold E. Let Can0 (E) be the unity connectivity component of the supergroup Can(E) of canonical transformations of E and CanH (E) be the supergroup of canonical transformations generated by the Hamiltonian. Then the following relations between supergroups Can(E), Can0 (E) and CanH (E) are obeyed: Can(E)/Can0 (E) = π0 (Diff (L)), Can(E)/CanH (E) = H 1 (L) π0 (Diff (L)), Can0 (E)/CanH (E) = H 1 (L) . In particular the supergroup Can0 (E) is equal to the supergroup CanH (E) if H 1 (L) = 0. Groups Diff (L), π0 (Diff (L)), Z 1 (L) and H (L) are isomorphic to groups Diff (M), π0 (Diff (M)), Z 1 (M) and H (M) respectively, where M is an underlying supermanifold, but isomorphisms are not canonical.
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4.2. Relation between semidensities and differential forms on a Lagrangian surface. Now we return to relation (4.2) between semidensities on an odd symplectic supermanifold E = E n.n and differential forms on even Lagrangian surfaces. We assume that the underlying manifold is orientable (see the remark after (3.5)) and its orientation is fixed. This fixes the orientation on even Lagrangian surfaces. We note also that if we consider the points of the supermanifold as -points where is an arbitrary Grassmann algebra, then one has to consider differential forms with coefficients in this algebra (see Appendix 2). It follows from (3.4) that if s is an even (odd) semidensity then the k-form in l.h.s. of relation (4.2) has coefficients in the Grassmann algebra with parity p = (−1)n−k (p = (−1)n−k+1 ). More precisely denote by S a the superspace of semidensities in √ E. -points of the superspace S are even semidensities, i.e. semidensities s = s(x, θ ) D(x, θ ), such that s(x, θ ) are even functions with coefficients in the Grassmann algebra . Denote by k the superspace of differential k-forms on an even Lagrangian surface L and consider also the superspace k , where is a parity reversing functor. -points of the superspace k are differential k-forms with even coefficients from the Grassmann algebra , -points of superspace k are differential k-forms, with odd coefficients from the Grassmann algebra . Consider a superspace ∗ (L) = n ⊕ n−1 ⊕ n−2 ⊕ n−3 ⊕ n−4 . . . .
(4.12)
Relation (4.2) defines a map: w(L, ϕL , s) = (τL# )−1 ϕL∗ s
(4.13)
between superspace S and superspace ∗ (L). (Here and later where it will not lead to confusion we denote by w a linear combination of differential forms wn + wn−1 + · · · + w0 .) At what extent does the map (4.13) depend on a choice of identifying symplectomorphism and on a choice of even Lagrangian surface? If F is an arbitrary canonical transformation of E and ϕL = F ◦ ϕL , then for map (4.13) w(L, ϕL , s) = w(L, F ◦ ϕL , s) = w(L, ϕL , F ∗ s) .
(4.14)
Thus bearing in mind Proposition 2 we study the action of the supergroup Canadj (L) of canonical transformations on semidensities. Proposition 4. a) Let s be an arbitrary semidensity on an odd symplectic supermanifold E = E n.n with closed connected underlying manifold M n and F be an arbitrary canonical transformation of E n.n adjusted to a given even Lagrangian surface L in E (F L = id, i.e.F ∈ Canadj (L)). Then (F ∗ s − s) L = 0. b) an arbitrary canonical transformation F generated by the Hamiltonian (F ∈ CanH (E)) changes an arbitrary closed semidensity on an exact form: if # s = 0 then F ∗ s − s = # r. In the case if this transformation is adjusted to Lagrangian surface L (F ∈ Canadj (L) ⊆ CanH (L)), then condition (F ∗ s − s) L = 0 is obeyed also. c) If s and s1 are arbitrary even closed non-degenerate semidensities (s, s1 ∈ Bdeg (see # (2.21)), differ on an exact semidensity: s1 − s = r, and for an even Lagrangian surface the L condition (s1 − s) L = 0 is obeyed, then there exists a canonical transformation F adjusted to L such that s = F ∗ s1 .
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(We say that semidensity s is equal to zero on an even Lagrangian surface L (s L = 0) √ if in Darboux coordinates adjusted to L s = s(x, θ ) D(x, θ ) with s(x, θ)|θ=0 = 0). Statement a) follows from the explicit expression (4.4) for the transformation F adjusted to the Lagrangian surface L. According to Proposition 2 statement b) has to be checked only for infinitesimal transformations generated by the Hamiltonian. For these transformations this statement follows from formula (2.16). To prove statement c) we consider the following “time”-depending Hamiltonian: Q(t) =
−r , s + t# r
0≤t ≤1
(4.15)
for any one-parameter family st = s0 + t# r, 0 ≤ t ≤ 1 of even closed non-degenerate semidensities (st ∈ Bdeg at any t). It is easy to check that during “time” t this Hamiltonian generates the canonical transformation Ft that transforms st to s (Ft∗ st = s). Indeed according to (4.15) and (2.16) if the transformation Ft obeys the conditions F˙t = {Q, Ft } and F0 = id then d ∗ Ft st = Ft∗ # r + Ft∗ # (Q(t)st ) = 0 ⇒ Ft∗ st = s0 . dt Consider Hamiltonian (4.15) for semidensities s, s1 ∈ Bdeg with # r = s1 − s choosing r in such a way that r = O(θ 2 ) in coordinates adjusted to L. Then Hamiltonian (4.15) leads to the canonical transformation Ft that is adjusted to L at any t and the transformation F = F1 transforms s1 to s.
Now we use this proposition for analyzing relation (4.2) for a given even ((n.0)dimensional) Lagrangian surface L. 1. According to Proposition 2 two identifying symplectomorphisms for a given even Lagrangian surface differ on the canonical transformation adjusted to this surface. Hence from statement a) of Proposition 4 and condition (4.14) for map (4.13) it follows that the top-degree form wn in (4.13) does not depend on the choice of identifying symplectomorphism ϕL : for a given even Lagrangian surface L relations (4.13) induce a well-defined map def
V (L, s) = wn (L, s) = wn (L, ϕL , s) ,
(4.16)
where ϕL is the arbitrary identifying symplectomorphism ϕL for Lagrangian surface L. Formula (4.16) defines the map from the superspace S of semidensities in E = E n.n to the superspace n (L) of top-degree forms on L. This means that the semidensity can be considered as a well-defined integration object over an even Lagrangian surface. This corresponds to the general result that semidensity can be considered as an integration object over an arbitrary (n − k.k)-dimensional Lagrangian surface. (See [22] for the corresponding construction and [15, 3] for explicit formulae.) 2. Consider restriction of the map (4.13) on the superspace B of closed semidensities. If semidensity s is closed then it follows from statement b) of Proposition 4 and relation (3.6) that for a given even Lagrangian surface L under changing of identifying symplectomorphism ϕL,E → ϕL,E = Fadj ◦ ϕL,E , corresponding differential forms in (4.13) change on exact forms: if # s = 0 then wk (L, ϕL , s) = wk (L, F ◦ ϕL , s) = wk (L, ϕL , F ∗ s) = wk (ϕL , s) + dwk−1 (L, ϕL , r) .
(4.17)
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In particular w0 (constant) as well as wn do not depend on the identifying symplectomorphism. Projection of superspaces k in (4.12) on a superspace H k of cohomology classes for k ≤ n − 1 induces projection of the superspace ∗ on superspace: n (L) ⊕ H n−1 (L) ⊕ H n−2 (L) ⊕ H n−3 (L) ⊕ H n−4 (L) . . . .
(4.18)
From (4.17) it follows that considering map (4.13) on closed semidensities for arbitrary identifying symplectomorphisms and projecting the value of this map on the superspace (4.18) we come to a well-defined map Vˆ (L, s) = wn (L, s) + [wn−1 ](L, s) + · · · + [w0 ](L, s),
if # s = 0
(4.19)
([wk ] is the cohomology class of form wk ). The map Vˆ (L, s) is the linear surjection map from the space B of closed semidensity on superspace (4.18). By definition Vˆ (L, s) is the Canadj -invariant map: it does not change under arbitrary canonical transformation adjusted to the surface L: if s1 = F ∗ s2 where F ∈ Canadj (L) then Vˆ (L, s1 ) = Vˆ (L, s2 ) .
(4.20)
The opposite implication is obeyed for closed non-degenerate semidensities. Namely consider the map (4.19) for the subset Bdeg of closed even non-degenerate semidensities (see2.21). If Vˆ (L, s1 ) = Vˆ (L, s2 ) for two arbitrary closed non-degenerate semidensities, then from statement c) of Proposition 4 it follows that there exists a canonical transformation F adjusted to surface L such that s1 = F ∗ s2 . Now we analyze dependence of the map (4.19) under a changing of the even Lagrangian surface L. We study this point from a more general point of view considering an action of the group Can(E) of all canonical transformations on maps (4.13) and (4.19). Projecting the superspace n (L) of top-degree forms on the superspace H n (L) of corresponding cohomology classes we come from the map Vˆ (L, s) to the map Hˆ (L, s) = [w](L, s) = [wn ](L, s) + · · · + [w0 ](L, s)
(4.21)
that is defined on superspace B of closed semidensities and takes values in a superspace H ∗ (L) = H n (L) ⊕ H n−1 (L) ⊕ H n−2 (L) ⊕ H n−3 (L) ⊕ . . . . From statement b) of Proposition 4 it follows that this map is CanH (E)-invariant: it does not change under arbitrary canonical transformations generated by the Hamiltonian: if s1 = F ∗ s2 where F ∈ CanH (E) then Hˆ (L, F ∗ s) = Hˆ (L, s) .
(4.22)
The opposite implication is obeyed under the following restriction. Let s1 and s2 be two closed non-degenerated semidensities (s1 , s2 ∈ Bdeg ) such that for the map (4.22) Hˆ (L, s1 ) = Hˆ (L, s2 ). In this case s2 = s1 + # r. If the one-parametric family of closed semidensities st = s1 + t# r (0 ≤ t ≤ 1) belongs also to Bdeg then there exists a canonical transformation F generated by the Hamiltonian (F ∈ CanH (E)) such that F ∗ s1 = s2 . One comes to this transformation considering the Hamiltonian (4.15). We note that in the special case when Can0 (E) = CanH (E), i.e. H 1 (M) = 0 (see Proposition 3), these considerations lead to the statement of Theorem 5 in the paper [22].
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Consider now the action of an arbitrary canonical transformation on the map (4.21). From Proposition 3 and (4.22) it follows that under arbitrary canonical transformation the map Hˆ (L.s) has to be transformed under the action of the group Can(E)/CanH (E) = H 1 (L) π0 (Diff (L)). Namely (4.23) Hˆ (L, F ∗ s) = [f ]∗ [ ] Hˆ (L, s) , where [[ ], [f ]] is an element of the supergroup H 1 (L) π0 (Diff (L)) defined by the action of the epimorphism (4.10a) on the canonical transformation F and the operation is defined for semidensities and the corresponding differential forms by operations (3.10) and (3.11). The pull-back [f ]∗ of the equivalence class [f ] is welldefined, because the pull-back f0∗ of the diffeomorphism f0 ∈ Diff0 (L) acts identically on cohomological classes of differential forms. On the other hand it follows from (4.14) that s) , Hˆ (L, F ∗ s) = (F |L )∗ Hˆ (L,
(4.24)
is an image of the Lagrangian surface L under canonical transformation F . where L One can easily derive formulae (4.23) and (4.24) from (4.13) performing calculations in arbitrary Darboux coordinates adjusted to the cotangent bundle structure of the Lagrangian surface L (i.e. choosing an arbitrary identifying symplectomorphism ϕL ) and using the decomposition formula (4.8). It is useful to rewrite formulae (4.23) and (4.24) in components: s) )∗ [wk ](L, [wk ](L, F ∗ s) = (F |L k 1 = [f ]∗ (4.25) [ ] ∧ · · · ∧ [ ] ∧[wk−p ] . p! p=0
p
times
of even Lagrangian surfaces the canonical We note that if for a given pair (L, L) then the cohomological class of an odd valued onetransformation F transforms L to L form corresponding to the pair (F, L) by the epimorphism (4.10a) is the well-defined function of the pair (L, L): . H 1 (L) [ ] = [ ](L, L)
(4.26)
)∗
The pull-back (F |L of restriction F |L of the canonical transformation F on the Lagrangian surface L induces a bijective map between differential forms and cor Using (4.25) we can comresponding cohomological classes on surfaces L and L. pare cohomological classes of differential forms corresponding to a given closed semidensity for two different even Lagrangian surfaces. In particular, from (4.25) it follows that for an arbitrary closed semidensity s and for an arbitrary pair of closed Lagrangian surfaces (L, L), s) = 0 if [w0 ](L, s) = · · · = [wk ](L, s) = 0 , [wk ](L, ˜ in (4.26) is equal to zero, and in the case if cohomological class of one-form [ ](L, L) then s) = 0 iff [wk ](L, s) = 0 . (4.27) [wk ](L, The simple but important consequence of these considerations is the following: a [w0 ]-component of the function Hˆ (L, s) (4.21) does not depend on canonical transformation and it is an invariant constant on all Lagrangian surfaces.
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Corollary 1. To every closed semidensity s (# s = 0) corresponds a positive constant c(s). If in arbitrary Darboux coordinates s = s(x, θ ) D(x, θ ) = (ρ(x) + bi (x)θi + · · · + cθ1 θ2 . . . θn ) D(x, θ ) then c(s) = |c|. This constant does not depend on the choice of Darboux coordinates and on the changing of density under arbitrary canonical transformation. This constant is equal (up to a sign) to the cohomological class [w0 ] of zeroth order differential form corresponding to semidensity s on an arbitrary even Lagrangian surface L. (The sign of c(s) depends on orientation.) Note that c(s) can be considered as integral of semidensity s over the Lagrangian (0.n)-dimensional surface x 1 = x01 , . . . , x n = x0n : c = L s = s(x0 , θ)d n θ . 4.3. Application to BV-geometry. Now using results obtained in this section we analyze Statement 1 (see the Introduction) of the Batalin-Vilkovisky master equation. Let s be an arbitrary closed semidensity in Bdeg , i.e. non-degenerate semidensity that obeys the BV-master equation (1.3b), (2.21). For the arbitrary -point α in the odd symplectic supermanifold E = E n.n consider an arbitrary closed even Lagrangian surface L such that this point belongs to this surface and choose an arbitrary identifying symplectomorphism ϕL , corresponding to this surface, i.e. the atlas of Darboux coordinates adjusted to the cotangent bundle structure of this Lagrangian surface. Consider on L the differential form w(L, ϕL , s) = wn + wn−1 + · · · + w0
(4.28)
defined by the map (4.13). Locally all closed differential forms except zeroth-forms are exact and [w0 ] = ±c(s) is an invariant constant according to Corollary 1. Hence using statement c) of Proposition 4 one can find the canonical transformation adjusted to this Lagrangian surface and correspondingly another identifying symplectomorphism ϕL such that in (4.28) all differential forms wk for k = 1, . . . , n − 1 vanish in the vicinity of the point α. Consider the Darboux coordinates zA = {x 1 , . . . , x n , θ1 , . . . , θn } on E in the vicinity of this point from the atlas of Darboux coordinates corresponding to the identifying symplectomorphism ϕL and by a suitable “point” canonical transformation (2.14b) choose them in a way that the differential form wn is equal to dx 1 ∧ · · · ∧ dx n in these coordinates. Thus we come to Darboux coordinates in the vicinity of the point α such that in these Darboux coordinates semidensity s has the following appearance: (4.29) s = s(x, θ ) D(x, θ ) = (1 + cθ1 θ2 . . . θn ) D(x, θ ) , where c is equal up to sign to the invariant constant c(s) corresponding to the semidensity s (see Corollary 1). The condition c(s) = 0 is the obstacle to condition (1.3a). Consider now the value of the map (4.19) on this Lagrangian surface: Vˆ (L, s) = wn + [wn−1 ] + · · · + [w0 ] .
(4.30)
If Vˆ (L, s) = wn + c, i.e. all cohomological classes [wk ] for k = 1, . . . , n − 1 in (4.30) vanish on the surface L, then one can consider the identifying symplectomorphism ϕL such that τL# (wn + c) = ϕL∗ s for the map (4.13). It means that there exists
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i ,θ n.n adjusted to the cotangent bunan atlas {x(α) j (α) } of Darboux coordinates on E dle structure of the Lagrangian surface L such that in arbitrary coordinates from this atlas √ semidensity s is expressed by relation (4.29). Semidensity s has the appearance D(x, θ ) in any Darboux coordinates from this atlas if the invariant constant c(s) = 0. In other words in this case the supermanifold can be identified with T ∗ L with the volume form on T ∗ L induced by the volume form on L. It follows from (4.23) – (4.27) that this statement holds for another even Lagrangian ˜ surface L˜ iff the cohomological class [ ] of the one-form corresponding to a pair (L, L) of Lagrangian surfaces (see 4.26) is equal to zero. In particular this statement is irrelevant to a choice of Lagrangian surface if H 1 (M) = 0. (M is an underlying manifold for √E.) Now we analyze condition (1.3c) for even-nondegenerate semidensity s = dv. From (2.19) it follows that condition (1.3c) means that the function (2.17d) is equal to an odd constant ν, and # s = νs. One can see using correspondence between semidensities and differential forms that all solutions to this equation are the following: s = # h − νh, where h is an arbitrary semidensity. The odd constant ν = 0 is the obstacle to condition (1.3b), if condition (1.3c) is obeyed. We come to the Corollary 2. Let E = E n.n be an odd symplectic supermanifold with connected orientable underlying manifold M and this supermanifold is provided with a volume form dv, such that 2dv = 0. Then √ √ √ 1) To the volume form dv corresponds the odd constant ν: # dv = ν dv and dv = # h − νh for some odd semidensity h. √ # 2) If the odd constant √ ν is equal to zero, then the master-equation dv = 0 holds for semidensity √ dv. In this case to the volume form dv corresponds a non-negative constant c = c( dv) and there exists an atlas of Darboux coordinates on E n.n such that dv = (1 ± 2c)D(x, θ ) in any coordinates from this atlas. 3) In the case if the constant c(s) = 0 then there exists an atlas of Darboux coordinates on E such that dv = D(x, θ ) in any coordinates from this atlas. 4) In the case if all cohomological classes of differential forms of degree less than n √ corresponding to the semidensity dv on an even Lagrangian surface L are equal to zero also, then there exists an atlas of Darboux coordinates on an E n.n adjusted to the cotangent bundle structure of L such that dv = D(x, θ ) in any coordinates from this atlas, i.e. E can be identified with T ∗ L with the volume form on T ∗ L induced by the volume form on L. if a cohoThis statement holds for another (n.0)-dimensional Lagrangian surface L is equal to zero. mological class of odd valued one-form [ ](L, L) This corollary removes the error in the considerations about equivalence of conditions (1.3a), (1.3b) and (1.3c) in Statement 1 of Introduction, which was done in [15] and [22]. On the other hand some statements of this proposition in a non explicit way were contained in the statements of Lemma 4 and Theorem 5 of the paper [22]. 5. Invariant Densities on Surfaces First we recall shortly the problem of construction of invariant densities in sympelctic (super)manifolds. Then we consider explicit formulae for the odd invariant semidensity on non-degenerate surfaces of codimension (1.1) embedded in an odd symplectic supermanifold E provided with a volume form dv ( [12, 13]). We consider this semidensity as
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a kind of pull-back of the semidensity s from the ambient odd symplectic supermanifold √ on embedded (1.1)-codimensional surfaces in the case if s = dv. Using this con√ struction for the semidensity # dv we will construct the new densities on embedded non-degenerate surfaces. In the case if we consider a volume form not only on the space (superspace) but on arbitrary embedded surfaces we come to the concept of densities on embedded surfaces.The kz ∂z , . . . , ∂ζ∂...∂ζ ) density of weight σ and rank k on embedded surfaces is a function A(z, ∂ζ that is defined on parameterized surfaces z(ζ ), depends on first k derivatives of z(ζ ) and is multiplied on the σ th power of the determinant (Berezinian) of surface reparametrization. A density of weight σ defines on every given the surface the σ th power of volume form. Such a concept of density is very useful in supermathematics where the notion of differential forms as integration objects is ill-defined. It was elaborated by A.S.Schwarz, particularly for analyzing supergravity Lagrangians [20, 9, 21]. In usual mathematics, for every 2k-dimensional surface C 2k embedded in a symplectic space, the so called Poincar´e-Cartan integral invariants (invariant volume forms on embedded surfaces) are given by the formula µ ∂x ν (ξ ) 2k ∂x (ξ ) d ξ, s˙ ∧ w = det wµν (5.1) w ∧ ∂ξ i ∂ξ j C 2k k−times
where a non-degenerate closed two-form w = wµν dx µ ∧ dx ν defines the symplectic structure, and functions x µ = x µ (ξ i ) define some parameterizations of the surface C 2k . In supermathematics one can consider even and odd symplectic structures on the supermanifold generated by even and odd non-degenerate closed two-forms respectively [6, 18, 19]. In the case of an even symplectic supermanifold, the l.h.s. of (5.1) is ill-defined but the r.h.s. of this formula can be straightforwardly generalized, by changing the determinant on the Berezinian (superdeterminant). The properties of the integral invariant do not change drastically. In particular one can prove that the integrand in (5.1) (the density of the weight σ = 1 and of the rank k = 1) is locally the total derivative and all invariant densities on surfaces are exhausted by (5.1) as well as in the case of the usual symplectic structure [16, 1]. The situation is less trivial in the case of an odd symplectic supermanifold. Formula (5.1) cannot be generalized in this case because transformations preserving odd symplectic structure do not preserve any volume form. One can consider invariant densities only in an odd symplectic supermanifold provided with a volume form. The problem of the existence of invariant densities on non-degenerate surfaces embedded in an odd symplectic supermanifold provided with a volume form was studied in [12, 13]. In particular it was proved that there are no invariant densities of the rank k = 1 (except of the volume form itself), and invariant semidensity of the rank k = 2 that is defined on non-degenerate surfaces of the codimension (1.1) was obtained. We briefly expose its construction here. The surface embedded in a symplectic supermanifold is called non-degenerate if the symplectic structure of the supermanifold generates the symplectic structure on the embedded surface also, i.e. if the pull-back of the symplectic 2-form on the surface is a non-degenerate 2-form. This symplectic structure on an embedded surface is called an induced symplectic structure. Let {zA } be Darboux coordinates on an odd symplectic supermanifold E = E n.n provided with volume form dv = ρ(z)Dz. It is convenient in this section to use for
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Darboux coordinates notations zA = (x µ , θµ ), (µ = (0, i) = (0, 1, . . . , n − 1), i = (1, . . . . , n − 1)). Let z(ζ ) be an arbitrary parameterization of an arbitrary non-degenerate surface of codimension (1.1), embedded in E. (ζ = (ξ i , ηj ), ξ i and ηj are even and odd parameters respectively, (i, j = 1, . . . , n − 1)). The invariant semidensity of the rank k = 2 (depending on the first and second derivatives of z(ζ )) is given by the following formula [12]: ∂z ∂ 2 z A z(ζ ), Dζ , ∂ζ ∂ζ ∂ζ ∂ 2 zB A ∂ log ρ(z) A αβ p(zB (ζ α +ζ β )+ζ α = − AB α β (z(ζ ))(−1) Dζ , A ∂z ∂ζ ∂ζ (5.2) where AB dzA dzB is the two-form defining the odd symplectic structure on E n.n and αβ is the tensor inversed to the two-form that defines the induced symplectic structure on the surface. The vector field = A ∂z∂A is defined as follows: one has to consider the pair of vectors (H, ), H even and odd that are symplectoorthogonal to the surface and obey the following conditions: (H, ) = 1, (, ) = 0 dv
! ∂z , H, = 1 ∂ζ
(symplectoorthonormality conditions) ,
(volume form normalization conditions) .
(5.3)
(5.4)
These conditions fix uniquely the vector field . (See [12] for details). The explicit expression for this semidensity was calculated in [13] in terms of dual densities: If the (1.1)-codimensional surface C is given not by parameterization, but by the equations f = 0, ϕ = 0, where f is an even function and ϕ is an odd function, then to the semidensity (5.2) there corresponds the dual semidensity: {f, f } 1 {f, f } {f, {f, ϕ}} dv f − A˜ dv ϕ − − =√ {ϕ, {f, ϕ}} . f =ϕ=0 2{f, ϕ} {f, ϕ} 2{f, ϕ}2 {f, ϕ} (5.5) One can check that the r.h.s. of (5.5) restricted by conditions f = ϕ = 0 is multiplied by the square root of the corresponding Berezinian (superdeterminant) under the transformation f → af + αϕ, ϕ → βf + bϕ, which does not change the surface C [13]. This invariant semidensity takes odd values. It is an exotic analogue of the Poincar´e–Cartan invariant: the corresponding density (the square of this odd semidensity) is equal to zero, so it cannot be integrated nontrivially over surfaces. On the other hand this semidensity can be considered as an analog of the mean curvature of hypersurfaces in the Riemanian space [12]. This odd semidensity in an odd symplectic supermanifold is unique (up to multiplication by a constant) in the class of densities of the rank k = 2 that are defined on non-degenerate surfaces of arbitrary dimension [13]. This means that one has to search non-trivial integral invariants (invariant densities of weight σ = 1) in higher order derivatives (rank k ≥ 3). Tedious calculations, which lead to the construction of the odd invariant semidensity in the papers [12, 13], did not give hope to go further for finding them, using the technique used in these papers.
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Now we develop√ another approach rewriting the semidensity (5.2) straightforwardly via the semidensity dv on the ambient odd symplectic supermanifold E = E n.n . Consider for every given non-degenerate surface C of codimension (1.1) embedded in an odd symplectic supermanifold E Darboux coordinates such that in these Darboux coordinates the surface C locally is given by equations x 0 = θ0 = 0 .
(5.6)
We call these Darboux coordinates adjusted to the surface C. (The existence of Darboux coordinates obeying these conditions can be proved using the technique considered in Appendices 2 and 3). If {x µ , θν } are Darboux coordinates in E adjusted to the surface C, then {x i , θj } are Darboux coordinates on the surface C n−1.n−1 w.r.t. the induced symplectic structure (µ, ν = 0, . . . , n − 1, i, j = 1, . . . , n − 1). Consider a semidensity (5.2) on the arbitrary non-degenerate surface C = C n−1.n−1 of codimension (1.1) in Darboux coordinates (5.6) adjusted to this surface. Conditions of symplectoorthonormality in (5.3) give that H = a1 ∂x∂ 0 + β ∂θ∂ 0 and = a ∂θ∂ 0 , where a is even and β is odd. The condition (5.4) of the volume form normalization gives that i √ 1/2 ∂(x , θj ) a = ρ Ber , ∂(ξ i , ηj ) where a volume form dv is equal to ρ(x, θ )D(x, θ ) and ζ = (ξ i , ηj ) are parameters (x 0 = θ0 = 0, x i = x i (ξ, η), θj = θj (ξ, η)). Hence the semidensity (5.2) on a surface (5.6) is reduced to √ ∂ ρ ∂z ∂ 2 z ∂ log ρ A z(ζ ), , Dζ = a Dζ = 2 D(x i , θj ) . ∂ζ ∂ζ ∂ζ ∂θ0 ∂θ0 We come to the following statement Theorem. To every semidensity s in the odd symplectic supermanifold E corresponds the semidensity K(s) of an opposite parity defined on non-degenerate (1.1)-codimensional surfaces embedded in this supermanifold. √ If semidensity s is given by expression s = s(x, θ ) D(x, θ) in Darboux coordinates {x µ , θν } = {x 0 , x i , θ0 , θj } adjusted to the given non-degenerate surface C of codimension (1.1) (x 0 |C = θ0 |C = 0), then semidensity K(s) on this surface in these Darboux coordinates is given by the following expression: ∂s(x µ , θν ) K(s) C = D(x i , θj ) , (5.7) 0 x =θ0 =0 ∂θ0 where D(x µ , θν ) is the coordinate volume form on the supermanifold E and D(x i , θj ) is the coordinate volume form on the surface C. The considerations above lead to the statement of this theorem for √ semidensities related with a volume form on an odd symplectic supermanifold (s = dv), i.e. for even non-degenerate even semidensities. Continuity considerations lead to the fact that the formula (5.7) is well-defined for an arbitrary semidensity e.g. for an odd semidensity, when the corresponding volume form is equal to zero.
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Alternatively one can prove this theorem checking in the same way as for (2.12) that the semidensity in ther.h.s. of (5.7) is well defined. For example consider the canonical transformation that has the following appearance in Darboux coordinates adjusted to the surface C:
x˜ 0 = x˜ 0 (x 0 , θ0 ) , θ˜0 = θ0 (x 0 , θ0 ) . x˜ i = x i , θ˜i = θi One can see that these canonical transformations are exhausted by transformations x˜ 0 = f (x 0 ), θ˜0 = β(x 0 ) + θ0 /fx , x˜ i = x i , θi = θi , where f (x) and β(x) are even-valued and odd valued functions on x respectively. Hence for transformation of adjusted coordinates x˜ 0 = f (x 0 ), θ˜0 = θ/fx . Obviously the r.h.s. of formula (5.7) transforms as a semidensity under this transformation. This is the central point of the construction (5.7) and also of (5.2) (see [12] for details).) We can consider a semidensity K(s) in (5.7) as a kind of pull-back of semidensity s on C, but this construction does not obey a condition of transitivity for the pull-back: consider an arbitrary (k.k)-dimensional non-degenerate surface embedded in E n.n and include it in a flag of non-degenerated surfaces: Y k.k → Y k+1.k+1 · · · → Y n−1.n−1 → E n.n .
(5.8)
Then one can consider semidensity K(. . . K(s) . . . ) on Y k.k corresponding to the semidensity s depending on flag (5.8). The statement of the theorem allows us to construct the semidensity on embedded surfaces via odd semidensities on the ambient supermanifold, which cannot be yielded from volume forms. In an odd symplectic supermanifold provided with a volume form √ dv on (1.1)-codimensional non-degenerate surfaces except an odd semidensity K( dv), √ that is nothing # dv) correspondbut semidensity (5.2), one can consider also an even semidensity K( √ √ # dv) cannot be represented ing to an odd semidensity # dv. The semidensity K( √ (5.2)-like because the square of the odd semidensity # dv is equal to zero. We note that for semidensities K(# s) and K(s) for arbitrary (1.1)-codimensional surface C the following condition is obeyed: "# K(s) , (5.9) K(# s) = − C
C
"# is the # -operator on surface C w.r.t. induced symplectic structure. This relawhere tion can be immediately checked in Darboux coordinates (5.6) adjusted to the surface C. √ √ The semidensities K( dv) and K(# dv) can be integrated over Lagrangian subsurfaces in C, according (4.16). On the other hand one can consider the√following non-trivial densities of weight √ σ = 1 constructed via the semidensities K( dv) and K(# dv): √ √ √ (5.10) P0 = K2 (# dv) and P1 = K( dv)K(# dv) . The density P0 takes even values, the density P1 takes odd values. In general case these densities give non-trivial integration objects (volume forms) over non-degenerate (1.1)-codimensional surfaces embedded in an odd symplectic supermanifold with volume form dv.
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The densities P0 and P1 have rank k = 4 (i.e. depend on derivatives of the parameterization z(ζ ) up√to fourth order). It follows from the fact that according √ to (5.9) the semidensity K(# dv) has the rank k = 4, because the semidensity K( dv) has the rank k = 2. This is hidden in representation (5.7), where the function ρ(z) corresponding to the volume form in adjusted coordinates depends non-explicitly on derivatives of the surface parameterization z(ζ ). Finally we consider a simple example of these constructions and their relations with differential forms. Let E 3.3 be a superspace associated to the cotangent bundle of 3-dimensional space 3 E , E 3.3 = T ∗ E 3 . We assume that coordinates x 0 , x 1 , x 2 are globally defined on E 3 . We consider on E 3 the differential form w = −dx 0 ∧ dx 1 ∧ dx 2 + b0 dx 0 + b1 dx 1 + b2 dx 2 . According to (3.4) a semidensity
s = τ # (w) = (1 + b0 θ1 θ2 + b1 θ2 θ0 + b2 θ0 θ1 ) D(x 0 , x 1 , x 2 , θ0 , θ1 , θ2 )
in T ∗ E corresponds to this differential form. Let C be a surface in E 3.3 that is defined by equations x 0 = θ0 = 0 and E 2 is a subspace in E 3 defined by equation x 0 = 0. The (2.2)-dimensional superspace C provided with coordinates x 1 , x 2 , θ1 , θ2 can be identified with superspace T ∗ E 2 associated with cotangent bundle T ∗ E 2 of subspace E2. ∗ 2 Then the value of the odd semidensity K(s) on C = T E is equal to (b2 θ1 − 1 2 b1 θ2 ) D(x , x , θ1 , θ2 ). This semidensity corresponds to the differential form b1 dx 1 + b2 dx 2 , the pull-back of w on E 2 . The value of the even semidensity K(# s) on C is equal to (∂2 b1 − ∂1 b2 ) D(x 1 , x 2 , θ1 , θ2 ). This corresponds to the differential form d(b1 dx 1 + b2 dx 2 ) = (∂2 b1 − ∂1 b2 )dx 1 ∧ dx 2 , the pull-back of dw on E 2 . The even density (volume form) on M is equal to P0 = (∂2 b1 −∂1 b2 )2 D(x 1 , x 2 , θ1 , θ2 ) and the odd density P1 = (∂2 b1 − ∂1 b2 )(b1 θ2 − b2 θ1 )D(x 1 , x 2 , θ1 , θ2 ). 6. Discussion The definition (2.7) of the dv -operator is applicable not only for a symplectic supermanifold but also for a Poisson supermanifold (provided with a volume form) even if the corresponding Poisson bracket is degenerate. Is it possible to define the # -operator on semidensities in an odd Poisson supermanifold? This question is studied in [17]. It turns out that in a general odd Poisson supermanifold there is a canonical -operator on semidensities depending only on the class of a volume form modulo the action of a natural “master” groupoid (see [17]). We note also that from relations (2.10) it follows that one can express an odd Poisson bracket via the operator dv . Moreover every second order odd differential operator Aˆ on functions on a supermanifold obeying the condition Aˆ 2 = 0 defines the Poisson structure via relations (2.10). This approach was elaborated by I.A. Batalin and I.V. Tyutin [2]. (The exhaustive study of these questions giving in particular a complete description of the BV operators can be found in [26].) On odd symplectic supermanifolds the integration theory on surfaces interplays with symplectic geometry. Using our approach one can consider integrands (differential forms) in terms of semidensities corresponding to these differential forms. In this case symmetry transformations of the corresponding functionals are not exhausted by
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transformations induced by diffeomorphisms of underlying space. General canonical transformations of supermanifold induce mixing of the corresponding differential forms with different degrees. In Sects. 3 and 4 we investigated relations between semidensities on an odd symplectic supermanifold and differential forms on the even Lagrangian surfaces. It is interesting to generalize these results to the case when the Lagrangian surface is (n−k.k)-dimensional for k = 0 using the analysis of arbitrary Lagrangian surfaces performed in the paper [22]. In this case there exist analogies of differential forms considered as integration objects. (See [7, 1] and for a more detailed analysis [27, 25].) For example if T ∗ M is the (r + s.r + s)-dimensional supermanifold associated with the cotangent bundle of an (r.s)-dimensional supermanifold M then considering an analogue of formula (3.4a) we obtain relations between semidensities in T ∗ M and the so-called pseudodifferential forms: functions on the supermanifold T M. Pseudodifferential forms are well-defined integration objects over the supermanifold and embedded surfaces [7, 1, 27, 25]. In this way it is possible to come to an analogue of the map (4.16) (see [22, 15]). It is interesting to construct analogues of the maps (4.19) and (4.21) for (n − k.k)-dimensional Lagrangian surfaces. We note that our considerations in Subsects. 4.2 and 4.3 overlap partially with some results of the paper [22]. A distinctive feature of our approach is the use of semidensities where a calculus analogous to the calculus of differential forms arises. In particularly this leads to the statements in Corollary 2. Also, by using -points we come to the difference between supergroups Can0 (E) and CanH (E). We hope that considerations presented in Sect. 5 of this paper can be generalized for constructing densities depending on higher order derivatives for surfaces of arbitrary dimension embedded in an odd symplectic supermanifold provided with a volume form and for finding a complete set of local invariants of this geometry. In particular, from considerations which lead to theorem it follows that if k(p) is the rank of non-trivial invariant densities on non-degenerate surfaces of codimension (p.p), then k(2) ≥ 5 and k(p + 1) > k(p). In [12] some relations of the semidensity (5.2, 5.7) with mean curvature in Riemanian geometry were indicated. It is interesting to analyze these relations in terms of geometry of semidensities presented in this paper. Densities presented in formula (5.10) need to be investigated in greater detail. Particularly one has to present explicit formulae for them and consider the corresponding functionals over surfaces. These functionals are equal to zero in the special case if the volume form in the ambient odd symplectic supermanifold obeys the BV-master equation. Are Euler-Lagrange equations for these functionals satisfied identically in the general case, as for the usual Poincar´e-Cartan integral invariants (5.1)? Results presented in Sect. 5 strongly indicate that there exists non-trivial geometry in an odd symplectic supermanifold provided with a volume form.
Appendix 1. Λ-Points on Supermanifolds
i } be a smooth atlas of coordinates on m-dimensional manifold M m , where Let {x(α) i } are defined on domain U and x i = i (x ) are transition funccoordinates {x(α) α αβ (β) (α)
j j i tions. Consider an atlas {x(α) , θ(α) } , where odd variables {θ(α) } (j = 1, . . . , n) are generators of the Grassmann algebra and transition functions
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i = i (x(β) , θ(β) ) x(α) αβ j
j
θ(α) = αβ (x(β) , θ(β) )
(A.1.1)
obey the following properties: αβ ) = 0, p(αβ ) = 1, where p(x i ) = 0, p(θ j ) = 1) they are parity preserving, i.e. p( 1, αβ (x(β) , θ(β) ) j = αβ (x(β) ) and ∂j /∂θ i are inverting matrices. 2) (β) θ =0 i , θ j } define the (m.n)-dimensional superdomain U ˆ m.n with underCoordinates {x(α) (α) (α) m lying domain U(α) . Pasting in formulae (A.1.1), define the (m.n)-dimensional supermanifold with underlying manifold M m . In this definition of the supermanifold which belongs to F.Berezin and D.Leites (see [6] and [19]) a supermanifold “has no points”. If E is the supermanifold and is an arbitrary Grassmann algebra one can construct a set E of -points of the supermanifold E. For example if E m.n is the superdomain with underlying domain M m , we define E as a set of rows (a 1 , . . . , a m , α 1 , . . . , α n ), where a 1 , . . . , a m are arbitrary even elements and α 1 , . . . , α n are arbitrary odd elements of the Grassmann algebra and (m(a 1 ), . . . , m(a m )) ∈ M m , where m is a standard homomorphism of on IR. A map of superdomains generates a map of the corresponding sets of -points. Thus one comes to the definition of a set E for the arbitrary supermanifold E. To every parity preserving homomorphism ρ : → of Grassmann algebras one can naturally assign a map ρ˜E : E → E . If ρ : → and ρ : → are two parity preserving homomorphisms, then (ρ ◦ ρ )E = ρ˜E ◦ ρ˜E . The supermanifold can be considered as a functor on the category of Grassmann algebras taking values in the category of sets. This definition of supermanifolds is used in the paper. It was suggested and widely used by A.S. Schwarz [21]. It makes possible to use a language of “points” and is more convenient for supergeometry and in applications in theoretical physics. In terms of -points one can easily generalize the standard geometrical definitions on the supercase [21]. For example
1. A map F from the supermanifold E in the supermanifold N can be considered as a functor from category {} of Grassmann algebras to category {F }, where for every Grassmann algebra , F is a map from the set E to the set N such that F ◦ ρ˜E = ρ˜N ◦ F for every parity preserving homomorphism ρ : → . 2. The action of the supergroup G on the supermanifold E can be considered as a functor that assigns to every Grassmann algebra the pair [G , E ] where G is a group of -points of the supergroup G, that acts on the set E of -points of supermanifold E.
Appendix 2. A Simple Proof of the Darboux Theorem for Odd Symplectic Structure Using nilpotency of odd variables one can directly prove the Darboux theorem for an odd symplectic supermanifold presenting a finite recurrent procedure for constructing Darboux coordinates starting from arbitrary coordinates. Let { , } be an odd non-degenerate Poisson bracket (2.1) corresponding to the symplectic structure. According to (2.1) for arbitrary two functions f and g, {f, g} =
∂f i j ∂g ∂f ∂f ∂g ∂g {x , x } j + i {x i , θj } + (−1)p(f )+1 {θi , x j } j i ∂x ∂x ∂x ∂θj ∂θi ∂x
Semidensities on Odd Symplectic Supermanifolds
+(−1)p(f )+1
385
∂f ∂g {θi , θj } , ∂θi ∂θj
(A.2.1)
and Jacoby identities (2.3) are obeyed. For given arbitrary coordinates {x 1 , . . . , x n , θ1 , . . . , θn } denote by E ij (x, θ ) = {x i , x j } , Fij (x, θ ) = {θi , θj } , Aij (x, θ ) = δji + Pji (x, θ ) = {x i , θj } . (A.2.2) From the definition of symplectic structure it follows that E ij = E j i , Fij = −Fj i are odd-valued matrices taking values in the Grassmann algebra and Aij (x, θ ) is an even non-degenerate matrix taking values in the Grassmann algebra . In Darboux coordinates matrices E ij , Fij and Pji have to be equal to zero. First of all we note that in the case if for coordinates {x i , θj } the conditions E ik (x, θ ) = 0 ,
Pki (x, θ ) = 0
(A.2.3)
are obeyed then Jacoby identities {x m , {θi , θj }} + cycl. permut. = 0 imply that Fij do not depend on θ and Jacoby identities {θi , {θj , θm }} + cycl. permut. = 0 imply the condition ∂i Fj m (x) + ∂j Fmi (x) + ∂m Fij (x) = 0. (In other words the two-form Fij (x)dx i ∧ dx j is closed.) Locally it means that there exist functions Ai (x) such that Fij (x) = ∂i Aj (x) − ∂j Ai (x). Under the transformation θi → θi + Ai (x), Fij (x) transform to zero also and we come to Darboux coordinates. Thus we have to find the transformation from arbitrary coordinates to new coordinates such that in new coordinates conditions (A.2.3) will be obeyed. Consider a set M of all coordinates {x i , θj }. Denote by M(p.q) a subset of M such that for coordinates {x i , θj } belonging to the subset M(p.q) the following conditions are obeyed for matrices E ij (x, θ ) and Pji (x, θ ) in (A.2.2): E ij (x, θ ) = O(θ p ) ,
Pji (x, θ ) = O(θ q ) .
(A.2.4)
M0.0 = M and condition {x i , θj } ∈ Mn+1.n+1 mean that relations (A.2.3) are obeyed for these coordinates, because θi1 . . . θik = 0 if k ≥ n + 1. Consider four maps F1 , F2 , F3 , F4 defined on the set M of coordinates, such that these maps obey the following conditions: F1 F2 F3 F4
maps Mr.0 in Mr.1 for r = 0, 1, . . . , maps M0.1 in M1.0 , maps Mr.1 in Mr+1.1 for r ≥ 1 , maps Mn+1.r in Mn+1.r+1 for r ≥ 1 .
(A.2.5)
Provided conditions (A.2.5) are obeyed the map F4n ◦ F3n ◦ F1 ◦ F2 ◦ F1 transforms arbitrary coordinates to coordinates that belong to the subset Mn+1.n+1 , i.e. conditions (A.2.3) are obeyed for transformed coordinates. Now we present maps F1 , F2 , F3 , F4 obeying conditions (A.2.5). 1. Definition of the map F1 : F1 ({x i , θj }) = {x˜ i , θ˜j },
where
x˜ i = x i , θ˜j = θm (A−1 )m j ,
(A.2.6)
where the matrix A−1 is inverse to the matrix A defined by relations (A.2.2) for coordinates {x i , θj }. It is easy to see from (A.2.1) that the map (A.2.6) obeys condition (A.2.5).
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2. Definition of the map F2 : F2 ({x i , θj }) = {x˜ i , θ˜j },
where
x˜ i = x i − θm R mi ,
θ˜j = θj ,
(A.2.7)
where the symmetrical odd-valued matrix R is the solution to the matrix equation 2R + RF R = E ,
(R ij = R j i )
(A.2.8)
and the matrices E and F for coordinates {x i , θj } are defined by (A.2.2). The solution to this equation is well-defined because elements of the symmetric matrix E and antisymmetric matrix F take odd values in the Grassmann algebra . 2 R is given by the finite power series R = E2 − EF8 E + . . . containing less than [ n2 ] terms. One can present an explicit solution to Eq. (A.2.8): n(n−1) 2
R=
k
ck (EF ) E,
k=0
where
∞ k=0
√ 1+t −1 . ck t = t k
(A.2.9)
Now it follows from (A.2.1) and (A.2.8) that under transformation (A.2.7) matrix E ij = {x i , x j } transforms to the matrix E˜ ij = {x˜ i , x˜ j } such that ˜ = O(θ) ˜ , E˜ ij = E ij − 2R ij − R im Fmk R mj + O(θ)
(A.2.10)
if coordinates {x i , θj } belong to M0.1 (i.e. Aij = δji + O(θ )) and matrix R obeys Eq. (A.2.8). Hence map (A.2.7) obeys condition (A.2.5). 3. Definition of the map F3 : F3 ({x i , θj }) = {x˜ i , θ˜j },
where
1
x˜ i = x i − θm
τ E mi (x, τ θ )dτ,
θ˜j = θj .
0
(A.2.11) From (A.2.1) it follows that transformation (A.2.11) maps Mr.1 in Mr.1 if r ≥ 1. Matrix E ij (x, θ ) transforms to matrix ∂E mj 2E ij 1 E − + (i ↔ j ) + O(θ r+1 ) . + θm r +2 r +2 ∂θi ij
(A.2.12)
On the other hand from the Jacoby identity (2.3): {x i {x j , x m }} + {x j {x m , x i }} + {x m {x i , x j }} = 0 and from (A.2.1) it follows that θm
∂E mj ∂E ij + (i ↔ j ) = −θm + O(θ r+1 ) = −rE ij (x, θ ) + O(θ r+1 ) . ∂θi ∂θm (A.2.13)
Hence (A.2.12) is equal to zero up to O(θ˜ r+1 ) and condition (A.2.5) is obeyed for transformation (A.2.11).
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4. Definition of the map F4 : F3 ({x , θj }) = {x˜ , θ˜j }, i
i
where
x˜ = x , θ˜j = θj − θm i
1
i
0
Pjm (x, τ θ )dτ . (A.2.14)
We prove that (A.2.10) maps Mn+1.r in Mn+1.r+1 analogous to the proof for (A.2.11). Suppose that coordinates {x i , θj } belong to Mn+1.r (r ≥ 1). Then transformation (A.2.14) maps matrix Pji (x, θ ) to matrix Pji −
Pji r +1
m i Pji θm ∂Pj θm ∂Pj − + O(θ r+1 ) = Pji − + O(θ r+1 ) r + 1 ∂θi r + 1 r + 1 ∂θm = O(θ r+1 ) ,
+
because of the Jacoby identity {x i , {x m , θj }} + {x m , {x i , θj }} + {θj , {x i , x m }} = 0. Hence condition (A.2.5) is obeyed for transformation (A.2.14).
Appendix 3. Hamiltonians of Adjusted Canonical Transformations In this appendix we prove that for any given adjusted canonical transformation {x i , θj } → {x˜ i , θ˜j } (2.14a) there exists a time-independent Hamiltonian Q(x, θ ) that generates this transformation via differential equations (2.15) and this Hamiltonian is defined uniquely by the condition Q(x, θ ) = Qik θi θk + . . . ,
i.e. Q = O(θ 2 ) .
(A.3.1)
For every Hamiltonian (odd function) Q(x, θ ) obeying condition (A.3.1) consider the one-parametric family of functions (Darboux coordinates) {y i (t), ηj (t)} (i, j = 1, . . . , n) that are solutions to the differential equation (2.15): dy i (t) = {Q(y, η), y i } = − ∂Q(y,η) , dt ∂ηi (0 ≤ t ≤ 1) , (A.3.2) dηj (t) = {Q(y, η), ηj } = ∂Q(y,η) , i dt
with initial conditions
∂y
y i (t) t=0 = x i , ηi (t) t=0 = θi .
It is easy to see from the explicit expression (2.4) for the odd Poisson bracket that if {x i , θj } and {x˜ i , θ˜j } are Darboux coordinates such that x˜ i = x i and θ˜j = O(θ), then θ˜j = θj also. Hence every adjusted canonical transformation {x i , θj } → {x˜ i , θ˜j } is uniquely defined by functions {f i (x, θ )} that obey the conditions: {x i + f i (x, θ ), x j + f j (x, θ )} = 0 and
f i (x, θ ) ∈ O(θ) .
(A.3.3)
Statement 3 of Lemma 1 follows from the lemma: Lemma 3. For every set of functions {f i (x, θ )} (i = 1, . . . , n) obeying conditions (A.3.3) there exists a unique Hamiltonian Q obeying condition (A.3.1) such that functions {y i (t)}, solutions to differential equation (A.3.2) obey conditions y i (t)|t=1 = x i + f i (x, θ ) (i = 1, . . . , n).
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Proof of this lemma. Consider a ring A of functions on coordinates (x 1 , . . . , x n , θ1 , . . . , θn ). (As always functions take values in an arbitrary Grassmann algebra . Consider in A the following gradation: A(p) is a space of functions that are linear com ∂f = pf . binations of p th order monoms on variables {θ1 , . . . , θn }: f ∈ A(p) iff k θk ∂θ k A(p) = 0 for p ≥ n + 1. For every function f ∈ A we denote by f(p) its component in A(p) : f = f(0) + f(1) + · · · + f(n) . It is evident that for the canonical Poisson bracket (2.4), {f, g}(p) =
n
{f(i) , g(p+1−i) } .
(A.3.4)
i=0
Consider also a corresponding filtration: 0 = A(n+1) ⊂ A(n) ⊂ · · · ⊂ A(1) ⊂ A(0) = A , where A(p) = ⊕k≥p A(k) . We denote by A+ (A− ) a subspace of even-valued (odd valued) functions in A. ± ±(k) = A(k) ∩ A± . Respectively we denote by A± (k) = A(k) ∩ A and A We note first that condition (A.3.1) implies that solutions to Eqs. (A.3.2) are well defined. Indeed consider an arbitrary function ϕ(x, θ ), the odd Hamiltonian Q ∈ A−(2) and the differential equation ϕ˙ = {Q, ϕ}. Projecting this differential equation on the subspace A(p) , using (A.3.4) we come to equations ϕ˙(p) = {Q(p+1) , ϕ(0) } + · · · + {Q(2) , ϕ(p−1) }. Function ϕ(0) does not depend on t (ϕ˙0 = 0) and these equations can be solved recurrently: ϕ(p) t=a = ϕ(p) t=0 + a{Q(p+1) , ϕ(0) } + . . . , (A.3.5) where we denote by dots terms depending on Q(2) , . . . , Q(p) and functions ϕ(0) , ϕ(1) |t=0 , . . . , ϕ(p−1) |t=0 . Denote by N a space of sets of even-valued functions {f i (x, θ )} (i = 1, . . . , n) such that these functions obey condition (A.3.3). Consider a map that assigns to every Hamiltonian Q ∈ A−(2) the solutions {y i (t)|t=1 } = x i + f i (x, θ ) to differential equations (A.3.2). Thus we define map U : A−(2) → N . Relations (A.3.5) for ϕ = x i imply that i f(p) =−
∂Qp+1 + terms depending on Q(2) , . . . , Q(p) . ∂θi
(A.3.6)
Consider also the following map δ : N → A−(2) such that for every {f i } ∈ N : 1 i (x, θ ) n n f(p) i δ({f (x)}) = − =− θi θi f i (x, τ θ )dτ . (A.3.7) p+1 0 i=1,p=1
i=1
From condition (A.3.3) for functions {f i } and (2.4) it follows that 1 ˜ ∂Q ˜ = δ({f i }) . fi = − i + θm {f i , f m }|x,τ θ dτ if Q ∂θ τ =0 m Projection of this equation on subspaces A(p) implies i =− f(p)
˜ (p+1) ∂Q i ,...fi + terms depending on f(1) (p−1) . ∂θi
(A.3.8)
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Hence δ is injection. Comparing this relation with relation (A.3.6) we see that the map δ◦U : A−(2) → A−(2) is bijection. Hence the map U is also bijection. For every {f i } ∈ N the odd function Q = (δ ◦ U)−1 ◦ δ({f i })) is the unique Hamiltonian in A(2) required by the lemma.
Acknowledgement. This work was highly stimulated by very illuminating discussions with S.P. Novikov during my talk on his seminar in Moscow in August 1999. I am deeply grateful to him. I want to express my deep gratitude to my teacher A.S.Schwarz, and to I.A.Batalin, V.M.Buchstaber, and I.V.Tyutin for encouraging me to do this work. I am very grateful to T. Voronov. Continuous discussions with him were very useful in the final part of this work. He also did enormous work by reading a draft of this paper and giving much valuable advice. I am deeply grateful for hospitality and support of the Abdus Salam International Centre for Theoretical Physics in Trieste and the Max-Plank-Institut f¨ur Mathematik in Bonn, where I began a work on this paper. The work was partially supported by grant EPSRC GR/N00821.
References 1. Baranov, M.A., Schwarz, A.S.: Characteristic Classes of Supergauge Fields. Funkts. Analiz i ego pril. 18(2), 53–54 (1984); Cohomologies of Supermanifolds. Funkts. analiz i ego pril. 18(3), 69–70 (1984) 2. Batalin, I.A., Tyutin, I.A.: On possible generalizations of field–antifield formalism. Int. J. Mod. Phys., A8, 2333–2350 (1993) 3. Batalin, I.A., Tyutin, I.V.: On the Multilevel Field–Antifield Formalism with the Most General Lagrangian Hypergauges. Mod. Phys. Lett. A9, 1707–1712 (1994) 4. Batalin, I.A., Vilkovisky, G.A.: Gauge algebra and Quantization. Phys. Lett. 102B, 27–31 (1981) 5. Batalin, I.A., Vilkovisky, G.A.: Closure of the Gauge Algebra, Generalized Lie Equations and Feynman Rules. Nucl. Phys. B234, 106–124 (1984) 6. Berezin, F.A.: Introduction to Algebra and Analysis with Anticommuting Variables. Moscow: MGU, 1983 (in English, – Introduction to Superanalysis. Dordrecht–Boston: D.Reidel Pub. Co. 1987) 7. Bernstein, J.N., Leites, D.A.: Integral forms and the Stokes formula on supermanifolds. Funkts. Analiz i ego pril. 11(1), 55–56 (1977); How to integrate differential forms on supermanifolds. Funkts.. Analiz i ego pril. 11(3), 70–71 (1977) 8. Buttin, C.: C.R. Acad. Sci. Paris, Ser. A–B 269, A–87 (1969) 9. Gayduk, A.V., Khudaverdian, O.M., Schwarz A.S.: Integration on Surfaces in Superspace. Teor. Mat. Fiz. 52, 375–383 (1982) 10. Guillemin, V., Sternberg, S.: Geometric Asymptotics. Math. Surveys 14, Providence, RI: AMS, 1977 11. Khudaverdian, O.M.: Geometry of Superspace with Even and Odd Brackets. J. Math. Phys. 32, 1934–1937 (1991) (Preprint, Geneva University UGVA-DPT 1989/05–613) 12. Khudaverdian, O.M.: Odd Invariant Semidensity and Divergence-like Operators in an Odd Symplectic Superspace. Commun. Math. Phys. 198, 591–606 (1998) 13. Khudaverdian, O.M., Mkrtchian R.L.: Integral Invariants of Buttin Bracket. Lett. Math. Phys. 18, 229–234 (1989) 14. Khudaverdian, O.M., Nersessian, A.P.: On Geometry of Batalin- Vilkovisky Formalism. Mod. Phys. Lett. A8(25), 2377–2385 (1993) 15. Khudaverdian, O.M., Nersessian, A.P.: Batalin–Vilkovisky Formalism and Integration Theory on Manifolds. J. Math. Phys. 37, 3713–3724 (1996) 16. Khudaverdian, O.M., Schwarz, A.S., Tyupkin, Yu.S.: Integral Invariants for Supercanonical Transformations. Lett. Math. Phys. 5, 517–522. (1981) 17. Khudaverdian, H.M., Voronov, T.: On odd Laplace operators. Lett. Math. Phys. 62, 127–142 (2002) 18. Leites, D.A.: The new Lie Superalgebras and Mechanics. Docl. Acad. Nauk SSSR 236, 804–807 (1977) 19. Leites, D.A.:The Theory of Supermanifolds. Karelskij Filial AN SSSR, 1983 20. Schwarz, A.S.: Are the Field and Space Variables on an Equal Footing? Nucl. Phys. B171, 154–166 (1980) 21. Schwarz, A.S.: Supergravity, Complex Geometry and G-structures. Commun. Math. Phys. 87, 37–63 (1982) 22. Schwarz, A.S.: Geometry of Batalin-Vilkovisky Formalism. Commun. Math. Phys. 155, 249–260 (1993)
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23. Schwarz, A.S.: Symmetry transformations in Batalin-Vilkovisky formalism. Lett. Math. Phys. 31, 299–302 (1994) 24. Shander, V.N.: Analogues of the Frobenius and Darboux Theorems for Supermanifolds. Comptes rendus de l’ Academie Bulgare des Sciences 36(3), 309–311 (1983) 25. Voronov, Th.: Geometric Integration Theory on Supermanifolds. Sov. Sci. Rev. C Math. 9, 1–138 (1992) 26. Voronov, T.T., Khudaverdian, O.M.: Geometry of differential operators, and odd Laplace operators. Russ. Math. Surv. 58(1), 197–1982 (2003) 27. Voronov, Th.Th., Zorich, A.V.: Complexes of forms on supermanifold. Funkts. Analiz i ego pril. 20(2), 58–59 (1986); Integral transformations of pseudodifferential forms. Usp. Mat. Nauk, 41(6), 167–168 (1986) Communicated by N. Nekrasov
Commun. Math. Phys. 247, 391–419 (2004) Digital Object Identifier (DOI) 10.1007/s00220-004-1064-0
Communications in
Mathematical Physics
A Semiclassical Egorov Theorem and Quantum Ergodicity for Matrix Valued Operators Jens Bolte, Rainer Glaser Abteilung Theoretische Physik, Universit¨at Ulm, Albert-Einstein-Allee 11, 89069 Ulm, Germany. E-mail: [email protected]; [email protected] Received: 18 April 2002 / Accepted: 14 November 2003 Published online: 16 March 2004 – © Springer-Verlag 2004
Abstract: We study the semiclassical time evolution of observables given by matrix valued pseudodifferential operators and construct a decomposition of the Hilbert space L2 (Rd ) ⊗ Cn into a finite number of almost invariant subspaces. For a certain class of observables, that is preserved by the time evolution, we prove an Egorov theorem. We then associate with each almost invariant subspace of L2 (Rd ) ⊗ Cn a classical system on a product phase space T∗ Rd × O, where O is a compact symplectic manifold on which the classical counterpart of the matrix degrees of freedom is represented. For the projections of eigenvectors of the quantum Hamiltonian to the almost invariant subspaces we finally prove quantum ergodicity to hold, if the associated classical systems are ergodic. Introduction The relation between dynamical properties of a quantum system and its classical limit is a central subject in the field of quantum chaos. In this context quantum ergodicity is a well-established concept [Zel87, CdV85, HMR87, Zel96]. It states for quantisations of ergodic classical systems that the phase space lifts of almost all eigenfunctions of the quantum Hamiltonian converge in the semiclassical limit to an equidistribution on the level surfaces of the classical Hamiltonian. The principal goal of this paper is to establish quantum ergodicity in systems whose degrees of freedom can be divided into two classes such that they are represented in the Hilbert space L2 (Rd ) ⊗ Cn . The semiclassical limit shall be performed in terms of a parameter → 0 which is primarily linked to the (translational) degrees of freedom that are described by the infinite-dimensional factor L2 (Rd ). The finite dimension n of the other factor is fixed. Examples for systems where this description can be applied are relativistic particles with spin 1/2 in slowly varying external fields governed by a Dirac-Hamiltonian, or adiabatic situations modelled with a Born-Oppenheimer Hamiltonian. This setting leads to a representation of quantum mechanical observables as matrix valued pseudodifferential operators acting on L2 (Rd ) ⊗ Cn , whose symbols are suitable
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matrix valued functions on the phase space T∗ Rd = Rd × Rd with an expansion in . In particular, the principal symbol H0 of the selfadjoint quantum Hamiltonian H is a hermitian matrix valued function on T∗ Rd . Its spectral resolution requires to introduce several classical dynamics on T∗ Rd , each of them generated by one eigenvalue of H0 . Lifted to the quantum level, this structure results in a decomposition of the Hilbert space L2 (Rd ) ⊗ Cn into almost invariant subspaces with respect to the dynamics generated by the quantum Hamiltonian H that is directly associated with the spectral resolution of H0 . Recently the case of matrix valued operators for certain quantum Hamiltonians with scalar principal symbol, such that on the classical side one still has to deal with a single system, has been considered in [BG00, BGK01]. Here we extend this approach to the general setting of matrix valued operators where one has to define suitable classical systems corresponding to each almost invariant subspace of the Hilbert space. So far it appears that the semiclassical limit has only been performed with respect to one type of the degrees of freedom. For a complete (semi-)classical description of the quantum systems under consideration one would also require the second type of degrees of freedom, that are represented by the factor Cn of the Hilbert space L2 (Rd ) ⊗ Cn , to be transferred to a classical level. It turns out, however, that for this purpose no further semiclassical parameter is needed and the dimension n of the second factor can be held fixed. Indeed, a suitable Stratonovich-Weyl calculus [Str57] allows to map the principal symbols (with respect to the parameter ) of observables and their dynamics in a one-to-one manner to genuinely classical systems associated with the decomposition of L2 (Rd ) ⊗ Cn into almost invariant subspaces. On this classical level the hierarchy of the two types of degrees of freedom is reflected in the structure of the classical dynamics: these are skew-product flows built over the Hamiltonian dynamics generated by the eigenvalues of H0 . Apart from classical ergodicity the proof of quantum ergodicity typically requires two essential inputs. The first one is a suitable version of an Egorov theorem [Ego69] that allows to express the time evolution of quantum observables in the semiclassical limit in terms of a classical dynamics of principal symbols. We achieve this in two steps: beginning with matrix valued principal symbols, we proceed to a completely classical level by exploiting the Stratonovich-Weyl calculus in the form developed in [FGV90]. It is in the last step where the skew-product flows become relevant. The second input is a Szeg¨o-type limit formula that relates averaged expectation values of observables to classical phase space averages. This can be obtained by a straightforward generalisation of previous results [HMR87, BG00]. Our main results are the Egorov theorem in Sect. 3 and the quantum ergodicity theorem in Sect. 6. In order to formulate the Egorov theorem we first identify a subalgebra in the class of bounded semiclassical pseudodifferential operators that is invariant under the time evolution. The operators in this subalgebra have to be block-diagonal with respect to the projections onto the almost invariant subspaces of L2 (Rd ) ⊗ Cn . Theorem 3.2 then asserts that the (matrix valued) principal symbol of each block is evolved with the Hamiltonian flow associated with that block. In addition, it is conjugated with unitary transport matrices that describe the time evolution of the matrix degrees of freedom along the trajectories of the Hamiltonian flow. We next identify the dynamics provided by the transport matrices with a coadjoint action of a certain Lie group. Kirillov’s method of orbits [Kir76] then enables us to connect the apparently quantum mechanical dynamics with a genuinely classical dynamics on a certain coadjoint orbit O, which is a symplectic manifold. This relation can be constructed explicitly with the help of the Stratonovich-Weyl calculus developed in [FGV90]. As a result we obtain that after a Stratonovich-Weyl transform the principal
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symbol of each block of an observable is evolved with a skew-product dynamics on the combined symplectic phase space T∗ Rd × O. This observation restores the general picture behind Egorov-type theorems: in leading semiclassical order the quantum mechanical time evolution is determined by classical dynamics. The decomposition of L2 (Rd ) ⊗ Cn into almost invariant subspaces and the corresponding set of classical flows force quantum ergodicity to be concerned with projections of the eigenvectors of H to the subspaces, since only these are associated with unique classical systems. The projected eigenvectors, however, are no longer genuine eigenvectors of H, but only provide approximate solutions to the eigenvalue problem and thus yield, after normalisation, quasimodes (see [Laz93]). For the latter we prove quantum ergodicity to hold in the usual sense. In this context the relevant version of the Egorov theorem introduces on the classical side the skew-product flow associated with the given subspace as described above. We show that if this flow is ergodic, the phase space lifts of almost all normalised projected eigenvectors converge to equidistribution on the product phase space. 1. Background on Matrix Valued Pseudodifferential Operators In this section we provide the principal notions and conventions for the matrix valued pseudodifferential calculus that will be used in the sequel. The results are well known for operators with scalar symbols, see e.g. [Rob87, DS99], and carry over to the setting with matrix valued symbols almost immediately, see also [BG00]. Let us briefly review the basic definitions. We will be concerned with Weyl operators B = opW [B] whose action on ψ ∈ S (Rd ) ⊗ Cn is given by x + y W i 1 (x−y)ξ B e , ξ ψ(y) dy dξ, op [B]ψ (x) = (2π)d 2 T ∗ Rd where B ∈ S (Rd ) ⊗ Mn (C) is the corresponding matrix valued Weyl symbol. We will, however, restrict the class of admissible symbols by employing order functions m : T∗ Rd → (1, ∞). These are required to fulfill a growth property in the sense that there are positive constants C, N > 0 such that N/2 m(x, ξ ) ≤ C 1 + (x − y)2 + (ξ − η)2 m(y, η) for all (x, ξ ), (y, η) ∈ T∗ Rd . This notion allows to define symbol classes according to Definition 1.1. Let m : T∗ Rd → (1, ∞) be an order function. For ∈ (0, 0 ], with 0 > 0, let B(·, ·; ) be in C ∞ (T∗ Rd ) ⊗ Mn (C). We say that B ∈ Sq (m), q ∈ R, if for all (x, ξ ) ∈ T∗ Rd and for all α, β ∈ Nd0 there exists Cα,β > 0 with ∂ξα ∂xβ B(x, ξ ; )n×n ≤ Cα,β −q m(x, ξ ) uniformly in ∈ (0, 0 ].
q0 An asymptotic expansion B ∼ ∞ j =0 Bj of a symbol B ∈ S (m) is defined by a q j (unique) sequence {Bj ∈ S (m)}j ∈N0 , with qj → −∞ monotonically, fulfilling B−
N j =0
Bj ∈ SqN +1 (m)
for all
N ∈ N0 .
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We will often use the set Scl (m) of classical symbols, whose elements have asymptotic expansions in integer powers of , B∼
∞
−q+j Bj ,
(1.1)
j =0
where Bj ∈ S0 (m) =: S(m). A Weyl operator B = opW [B] with classical symbol q B ∈ Scl (m) is called a semiclassical pseudodifferential operator. The leading order term B0 in (1.1) is then referred to as the principal symbol and the subsequent term B1 as the subprincipal symbol of B. Let us also introduce the notations S∞ (m) := ∪q∈R Sq (m) and S−∞ (m) := ∩q∈R Sq (m). In the discussion below we will basically encounter two types of operators: quantum Hamiltonians H = opW [H ] with symbols H ∈ S0cl (m) generating the quantum q mechanical time evolution, and observables B = opW [B], where B ∈ Scl (1). It is well known that the set of Weyl operators with symbols in the class S∞ (m) is stable under multiplication: The product of two Weyl operators is again a Weyl operator whose symbol can be calculated from the symbols of the two factors, see e.g. [Rob87, DS99]. More precisely, there is a map S(m1 )×S(m2 ) → S(m1 m2 ), (B1 , B2 ) → B1 #B2 , such that opW [B1 ] opW [B2 ] = opW [B1 #B2 ]. Furthermore, according to a result of Calder´on and Vaillancourt [CV71], which immediately carries over to the case of matrix valued symbols, Weyl operators with symbols in S(1) are bounded as operators on L2 (Rd ) ⊗ Cn . If a Weyl operator A = opW [A] with symbol A ∈ S(m) is elliptic, in the sense that A−1 exists in S(m−1 ), one can construct an asymptotic inverse opW [Q] with Q ∈ S(m−1 ). The symbol Q is called a parametrix for A and fulfills A#Q ∼ Q#A ∼ 1. Its construction is based on the observation opW [A]# opW [A−1 ] ∼ 1 − opW [R] with R ∈ S(m). An asymptotic expansion of Q then follows from the Neumann series Q ∼ A−1 + (A−1 #R) + 2 (A−1 #R#R) + · · · .
(1.2)
2. Semiclassical Projections We motivate the following construction of semiclassical projection operators by considering the time evolution generated by a quantum Hamiltonian H, i.e., the Cauchy problem i
∂ ψ(t) = Hψ(t) ∂t
(2.1)
for an essentially selfadjoint operator H on D(H) ⊂ L2 (Rd ) ⊗ Cn . If one introduces the strongly continuous one-parameter group of unitary operators U(t) := exp(− i Ht), t ∈ R, a solution of (2.1) can be obtained by defining ψ(t) := U(t)ψ0 for ψ0 ∈ D(H). Therefore the time evolution B(t) := U(t)∗ BU(t) of a bounded operator B ∈ B(L2 (Rd )⊗Cn ) in the Heisenberg picture has to fulfill the (Heisenberg) equation of motion ∂ i B(t) = [H, B(t)]. ∂t
(2.2)
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If one assumes B and H to be semiclassical pseudodifferential operators with symbols q in the classes Scl (1) and S0cl (m), respectively, Eq. (2.2) yields in leading semiclassical order an equation for the principal symbols: ∂ i B0 (t) = [H0 , B0 (t)] + O(0 ), ∂t
→ 0.
If one now requires the time evolution to respect the filtration of the algebra S∞ cl (1) := q S (1) then, in particular, the principal symbol B (t) should stay in its class that 0 q∈Z cl q
q−1
derives from the associated grading Scl (1)/Scl (1), q ∈ Z. One thus has to restrict to operators whose principal parts B0 commute with the principal symbol H0 of the operator H. This condition is equivalent to a block-diagonal form of B0 , B0 (x, ξ ) =
l
Pµ,0 (x, ξ )B0 (x, ξ )Pµ,0 (x, ξ ),
(2.3)
µ=1
with respect to the projection matrices Pµ,0 : T∗ Rd → Mn (C), µ = 1, . . . , l, onto the eigenspaces corresponding to the eigenvalue functions λµ : T∗ Rd → R of the hermitian principal symbol matrix H0 : T∗ Rd → Mn (C). Since (2.3) is the semiclassical limit of q the symbol of the operator q lµ=1 opW [Pµ,0 ] opW [B] opW [Pµ,0 ], with B ∈ Scl (1), one can ask how the symbols Pµ,0 are related to projection operators onto (almost) invariant subspaces of L2 (Rd ) ⊗ Cn with respect to H = opW [H ]. We are hence looking for quantisations P˜ µ of symbols Pµ ∈ S0cl (1), with principal symbols Pµ,0 , which are (almost) orthogonal projections, i.e., P˜ µ P˜ µ = P˜ µ = P˜ µ∗
mod O(∞ )
(2.4)
in the operator norm. Moreover, in order that these operators map to almost invariant subspaces of L2 (Rd ) ⊗ Cn with respect to the time evolution U(t) = exp(− i Ht) generated by H, we require them to fulfill [H, P˜ µ ]L2 (Rd )⊗Cn = 0
mod O(∞ ).
(2.5)
As it will turn out, it is even possible to modify the operators P˜ µ in such a way that they satisfy the relation (2.4) exactly, i.e., not only mod O(∞ ). The above requirements lead us to consider (formal) asymptotic expansions for the symbols Pµ , Pµ (x, ξ ) ∼
∞
j Pµ,j (x, ξ ),
j =0
which satisfy (2.4) and (2.5) on a (formal) symbol level: Pµ #Pµ ∼ Pµ ∼ Pµ∗ ,
and
[Pµ , H ]# := Pµ #H − H #Pµ ∼ 0.
(2.6)
The solutions of the above equations will be called semiclassical projections and can be constructed by two different methods. The first one is based on solving the recursive problem arising from (2.6) by using asymptotic expansions of Pµ and H in S0cl (1) and S0cl (m), respectively, and finally equating equal powers of the semiclassical parameter .
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For this procedure cf. [EW96, BN99]. The second method employs the Riesz projection formula in the context of pseudodifferential calculus [HS88, NS01]. In the following we will pursue the latter method. To this end we consider the matrix valued hermitian principal symbol H0 ∈ S(m) of the operator H, and in the following we assume: (H0) The (real) eigenvalues λµ , µ = 1, . . . , l, of H0 have constant multiplicities k1 , . . . , kl and fulfill the hyperbolicity condition |λν (x, ξ ) − λµ (x, ξ )| ≥ Cm(x, ξ )
for
ν = µ and
|x| + |ξ | ≥ c.
This requirement is analogous to a condition imposed in [Cor82] on the eigenvalues of the symbol of an operator in a strictly hyperbolic system, i.e., where the eigenvalues are non-degenerate. In particular, the problem of mode conversion that arises from points where multiplicities of eigenvalues change is avoided. Since the eigenvalues are solu tions of the algebraic equation det H0 (x, ξ ) − λ = 0, they are smooth functions on T∗ Rd . Moreover, since H0 is supposed to be hermitian, the eigenvalues are bounded by the matrix norm of H0 . Using the smoothness of the eigenvalues and the hyperbolicity condition (H0), one obtains: Proposition 2.1. Let H ∈ S0cl (m) be hermitian and let the hyperbolicity condition (H0) be fulfilled. Then there exist symbols Pµ ∈ S0cl (1) with asymptotic expansions Pµ ∼
∞
j Pµ,j ,
µ = 1, . . . , l,
(2.7)
j =0
that fulfill the conditions (2.6). In particular, the coefficients Pµ,j , j ∈ N0 , are unique, i.e., the symbols Pµ are uniquely determined modulo S−∞ (1). Furthermore, the corresponding almost projection operators P˜ µ = opW [Pµ ] provide a semiclassical resolution of the identity on L2 (Rd ) ⊗ Cd , P˜ 1 + · · · + P˜ l = idL2 (Rd )⊗Cn mod O(∞ ). Proof. We use the technique of [HS88, NS01] and consider the Riesz projections 1 Q(x, ξ, z) dz, (2.8) Pµ (x, ξ ) := 2πi µ (x,ξ ) where µ (x, ξ ) is a simply closed and positively oriented regular curve in the complex plane enclosing the, and only the, eigenvalue λµ (x, ξ ) ∈ R of H0 (x, ξ ). Moreover, Q(x, ξ, z) denotes a parametrix for H − z to be constructed below, i.e., (H − z)#Q ∼ Q#(H −z) ∼ 1. For technical considerations one may choose the contour as µ (x, ξ ) = {λµ (x, ξ ) + ρµ (x, ξ ) eiϕ , 0 ≤ ϕ ≤ 2π } with 0 < c ≤ ρµ < 21 minν=µ {|λµ − λν |}. Since H0 is hermitian with eigenvalues λν , ν = 1, . . . , l, one can estimate the matrix norm of (H0 − z)−1 for z ∈ µ (x, ξ ) from above by Cρµ−1 . The condition (H0) then allows to choose ρµ (x, ξ ) ≥ cm(x, ξ ), so that H0 − z is elliptic for z ∈ µ . If then is sufficiently small, also H − z = H0 − z + O() is elliptic and the relation −1 = 1 − R H − z # H0 − z
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enables one to construct a parametrix Q(x, ξ, z) ∈ S0cl (m−1 ) for H − z with asymptotic expansion Q(x, ξ, z) ∼
∞
j Qj (x, ξ, z)
j =0
in the same manner as in (1.2), see also [Rob87, DS99]. Plugging this expansion into (2.8) one obtains Pµ (x, ξ ) ∼
∞ j =0
j
1 2πi
Qj (x, ξ, z) dz =:
µ (x,ξ )
∞
j Pµ,j (x, ξ ).
(2.9)
j =0
According to the properties of the Riesz integral the symbols Pµ fulfill (2.6). Since these equations have unique solutions modulo O(∞ ) [EW96], the coefficients Pµ,j are unique. We now consider more general z ∈ C, and by inspecting the above construction notice that the parametrix Q(z) is well-defined as long as z has a sufficiently large distance from the eigenvalues of H0 . According to Eq. (1.2) its asymptotic expansion then reads Q(z) ∼ (H0 − z)−1 + (H0 − z)−1 #R(z)#(idCn + R(z) + 2 R(z)#R(z) + · · · ). Since R(z) =
1 1 − (H − z)#(H0 − z)−1 ,
it follows according to the composition formula for Weyl operators that R(z) contains a factor (H0 − z)−1 , and therefore the only singularities of Q(z) are caused by the eigenvalues of H0 . Thus, according to the Cauchy formula in the expression 1 P 1 + · · · + Pl = Q(z) dz 2πi lµ=1 µ the contour can be replaced by (r) which has a minimal distance r from the origin in C and encloses all eigenvalues of H0 while keeping a sufficient distance from them. The value of the above integral does not depend on the particular choice of (r) so that one can take the limit r → ∞ and hence obtains 1 1 lim Q(z) dz = lim (H0 − z)−1 dz = idCn r→∞ 2π i (r) r→∞ 2πi (r) mod O(∞ ).
The so constructed symbols Pµ yield semiclassical almost projection operators P˜ µ := on-Vaillancourt theorem are bounded and satµ ] which according to the Calder´ isfy the relation (2.4). That (2.5) holds true can be seen by using (2.6) and employing the semiclassical time evolution of pseudodifferential operators, see [BR02] and also Sect. 3. Following [Nen02] one can even construct pseudodifferential operators Pµ that opW [P
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are semiclassically equivalent to P˜ µ in the sense that P˜ µ − Pµ = O(∞ ), and which fulfill (2.4) exactly. To see this, consider the operator 1 Pµ := (P˜ µ − z)−1 dz, 2πi |z−1|= 21 which is well-defined since the spectrum of P˜ µ is concentrated near 0 and 1. Thus Pµ is an orthogonal projector acting on L2 (Rd )⊗Cn , with [Pµ , H] ≤ c[P˜ µ , H] = O(∞ ). Since Pµ is close to P˜ µ in operator norm, Beals’ characterisation of pseudodifferential operators (see e.g. [HS88, DS99]) yields that Pµ is again a pseudodifferential operator with symbol in the class S0 (1). This has already been noticed in [NS01] and follows from the fact that (P˜ µ − z)−1 for |z − 1| = 1/2 is a pseudodifferential operator according to the parametrix construction. Having projectors available, one can also construct (pseudodifferential) unitary transformations of L2 (Rd ) ⊗ Cn which convert H by conjugation in an almost block-diagonal form, see [Cor83b, LF91, BR99, NS01, PST03]. Such unitary transformations are obviously not unique, and since for most purposes it suffices to work with the projectors, we hence refrain from using the unitary operators here. In view of the fact that Pµ is an orthogonal projector on the Hilbert space L2 (Rd )⊗Cn , one can ask if it is possible to satisfy also the relation (2.5) exactly. To this end we wish to inquire to what extent Pµ can be related to a spectral projection of H. (See [HS88, Cor00, Cor01] for examples.) We illustrate this question in the case where the principal symbol H0 of H possesses two well-separated eigenvalues λν < λν+1 with constant multiplicities kν and kν+1 , respectively, among the eigenvalues λ1 , . . . , λl . For l = 2 this is exactly the situation that occurs in the case of a Dirac-Hamiltonian. We also assume that there exists λ ∈ R separated from the spectrum of H such that λ − λν (x, ξ ) > Cm(x, ξ ) and λν+1 (x, ξ ) − λ > C m(x, ξ )
(2.10)
∗ d for all (x, ξ ) ∈ T R . It follows from these assumptions that one can replace the contour
< := νµ=1 µ in
P< (x, ξ ) :=
ν µ=1
Pµ (x, ξ ) =
1 2πi
Q(x, ξ, z) dz,
(2.11)
<
see (2.8), by a straight line + := {z ∈ C; z = λ + it, t ∈ R} that avoids the eigenvalues of the principal symbol H0 as well as the spectrum of H. Correspondingly,
> := lµ=ν+1 µ is deformed into − given by + with reversed orientation. Thus P˜ < = opW [P< ] is semiclassically equivalent to the spectral projection of H onto the interval (−∞, λ) given by 1 ?(−∞,λ) (H) = (H − z)−1 dz, 2πi + whereas P˜ > corresponds to ?(λ,∞) (H). We therefore have Proposition 2.2. If the eigenvalues λ1 , . . . , λl of the principal symbol H0 are separated according to (H0) and the condition (2.10) is fulfilled, the almost projection operators
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399
P˜ <> := opW [P<> ], whose symbols are defined in (2.11), can be semiclassically identified with the spectral projections ?(−∞,λ) (H) and ?(λ,∞) (H) of the operator H to the intervals (−∞, λ) and (λ, ∞), respectively. This means P˜ < − ?(−∞,λ) (H) = O(∞ ) and P˜ > − ?(λ,∞) (H) = O(∞ ). A corresponding statement holds for the related orthogonal projectors P<> , P< − ?(−∞,λ) (H) = O(∞ ) and P> − ?(λ,∞) (H) = O(∞ ). 3. Invariant Algebra and Egorov Theorem In this section our aim is to identify a suitable class of operators that is left invariant by the time evolution. Recalling the reasoning from the beginning of Sect. 2, we are interested is respected by the time evolution generated in a subalgebra of S∞ cl (1) whose filtration by the one-parameter group U(t) = exp − i Ht , where H is an essentially selfadjoint pseudodifferential operator with symbol H in the class S0cl (m). The following assumptions on the symbol H guarantee the essential selfadjointness of H on S (Rd ) ⊗ Cn (see [DS99]): (H1) H ∈ S0cl (m) is hermitian, (H2) H0 + i is elliptic in the sense that (H0 (x, ξ ) + i)−1 n×n ≤ cm(x, ξ )−1 . With these assumptions U(t) defines a strongly-continuous unitary one-parameter group. q We now consider the time evolution of an operator B = opW [B] with B ∈ Scl (1), which according to the Calder´on-Vaillancourt theorem is a bounded operator. Then B(t) := U(t)∗ BU(t) is again bounded on L2 (Rd )⊗Cn . A conjugation with lµ=1 Pµ = idL2 (Rd )⊗Cn +O(∞ ) yields B(t) =
l ν,µ=1
Pµ e HPµ t Be− HPν t Pν = i
i
l
e HPµ t Pµ BPν e− HPν t i
i
(3.1)
ν,µ=1
mod O(∞ ) in the operator norm. Here we have used the property e− Ht Pν = i e− HPν t Pν modulo O(∞ ) that follows from the Duhamel principle. Now, the principal symbol1 of HPµ is a scalar multiple of the identity in the eigenspace Pµ,0 Cn of H0 corresponding to λµ , i.e., H0 Pµ,0 = λµ Pµ,0 . Thus, for µ = ν the operator exp i HPµ t B exp − i HPν t is a pseudodifferential operator with symbol in the class S0 (1), see [Ivr98, BG00]. But when µ = ν the corresponding expressions are semiclassical Fourier integral operators. In that case the semiclassical limit at time t = 0 is different in nature from that at time zero. For a Dirac-Hamiltonian this phenomenon is related to the so-called “Zitterbewegung” which we will discuss in more detail in q [BG]. Therefore, we are here interested in operators B with symbols in B ∈ Scl (1) for which U ∗ (t)BU(t) is again a semiclassical pseudodifferential operator with symbol q B(t) ∈ Scl (1). We hence introduce the following notion: i
1 We remark that before transferring operators to symbol level one can replace P by P˜ and employ µ µ the classical asymptotic expansion of the symbol Pµ . This will only amount to an error of order ∞ .
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J. Bolte, R. Glaser
Definition 3.1. A symbol B ∈ Scl (1) is in the invariant subalgebra S∞ inv (1) of the algebra S∞ (1), if and only if for all finite t the (bounded) operator B(t) = U ∗ (t)BU(t), cl q W B = op [B], is a semiclassical pseudodifferential operator with symbol B(t) ∈ Scl (1), i.e.,
q q W ∗ S∞ inv (1) := B ∈ Scl (1) ; symb [U (t)BU(t)] ∈ Scl (1) for t ∈ [0, T ], q ∈ Z . q
This means that the invariant algebra S∞ inv (1) has a filtration, induced by the filtration of S∞ cl (1), which is invariant under conjugation of the corresponding operators with U(t). Due to the results of [BG00] we expect that operators which are block-diagonal with respect to the projections Pµ are in the associated invariant operator algebra. This statement is made precise in Theorem 3.2 which is a variant of the Egorov theorem [Ego69] for general hyperbolic systems. q Let us first consider an operator B with symbol B ∈ Scl (1) that is block-diagonal with respect to the semiclassical projections, i.e., B∼
l
Pµ #B#Pµ
in
q
Scl (1).
µ=1
According to the Heisenberg equation of motion (2.2) its time evolution B(t) is governed by ∂ i B(t) ∼ [H, B(t)]# . ∂t
(3.2)
Suppose now that B(t) has a (formal) asymptotic expansion B(t) ∼
∞
−q+j B(t)j
j =0
and use the composition formula for Weyl operators together with the fact that the blockdiagonal form of an operator B is preserved under the time evolution, see (3.1). On the symbol level the diagonal blocks Pν B(t)Pν then obey the following equation: ∞ ∂ −q+j B(t)νν,j ∂t j =0
∼
∞
(β) γ (α, β)−q+l+j +|α|+|β|−1 B(t)νν,l (α) Hν,j
(α) (β)
l,j =0 |α|+|β|≥0
−(−1)|α|−|β| Hν,j
(β) (α) (β) B(t)νν,l (α)
.
Here we introduced the notation F(β) := ∂ξα ∂x F for F ∈ C ∞ (T∗ Rd ) ⊗ Mn (C), as well as (α)
β
Quantum Ergodicity for Matrix Valued Operators
γ (α, β) :=
401
i|α|−|β|−1 , 2|α|+|β| |α|!|β|!
Hν := Pν #H #Pν ∼ H #Pν ∼
∞
j Hν,j ,
j =0
B(t)νν := Pν #B(t)#Pν ∼
∞
−q+j B(t)νν,j .
j =0
One hence has to solve, by taking [Hν,0 , B(t)νν,0 ] = 0 into account, [Hν,0 , B(t)νν,n+1 ] 1 ∂ {B(t)νν,n , Hν,0 } − {Hν,0 , B(t)νν,n } − i[B(t)νν,n , Hν,1 ] = − B(t)νν,n − ∂t 2 (β) (β) (α) (α) + γ (α, β) B(t)νν,l (α) Hν,j (β) − (−1)|α|−|β| Hν,j (β) B(t)νν,l (α) . 0≤l≤n−1 j +|α|+|β|=n−l+1
(3.3) Upon multiplying this commutator equation with the projection matrices Pµ,0 from both sides one first realises that it is only solvable, if the diagonal blocks of the right-hand side, that we denote by Rn,ν (t), vanish. The off-diagonal blocks on both sides of the relation (3.3) then yield the general structure of the solution, which reads B(t)νν,n+1 =
l µ=1
Pµ,0 B(t)νν,n+1 Pµ,0 +
Pµ,0 Rn,ν (t)Pη,0 , λµ − λ η
(3.4)
µ=η
see also [Cor95]. This demonstrates that one obtains the off-diagonal parts of B(t)νν,n+1 with respect to the projection matrices Pµ,0 from the preceding coefficients of the asymptotic expansion of B(t)νν . The diagonal parts then have to be determined by the condition that the commutator equation (3.3) must possess a (non-trivial) solution with initial value B(t)νν,n+1 t=0 = Bνν,n+1 . Starting with n = 0, where the sum in (3.3) is empty, one has to solve ∂ 1 Pµ,0 B(t)νν,0 Pµ,0 + Pµ,0 {B(t)νν,0 , Hν,0 } − {Hν,0 , B(t)νν,0 } ∂t 2
+ i[B(t)νν,0 , Hν,1 ] Pµ,0 = 0. Expressions of this type have already been considered in [Spo00], where it was shown that the above equation is equivalent to ∂ Pµ,0 B(t)νν,0 Pµ,0 − δνµ {λν , Pµ,0 B(t)νν,0 Pµ,0 } ∂t −i[H˜ νµ,1 , Pµ,0 B(t)νν,0 Pµ,0 ] = 0.
(3.5)
Here we have defined the hermitian n × n matrix λν H˜ νµ,1 := i(−1)δνµ Pµ,0 {Pν,0 , Pν,0 }Pµ,0 − iδνµ [Pν,0 , {λν , Pν,0 }] + Pµ,0 Hν,1 Pµ,0 . 2 (3.6)
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Now, Eq. (3.5) is trivially fulfilled for ν = µ. The case ν = µ has already been considered in [Ivr98, BN99, BG00], where it was shown that the solution reads −1 B(t)νν,0 (ξ, x) = dνν (x, ξ, t)Bνν,0 tν (x, ξ ) dνν (x, ξ, t). In this expression tν : T∗ Rd → T∗ Rd denotes the Hamiltonian flow generated by the eigenvalue λν of H0 , and the transport matrix dνν is determined by the equation d˙νν (x, ξ, t) + iH˜ νν,1 tν (x, ξ ) dνν (x, ξ, t) = 0, dνν (x, ξ, 0) = idCn . (3.7) One has thus fixed the coefficients B(t)νν,0 = Pν,0 B(t)0 Pν,0 , i.e., the principal symbol of B(t), since the off-diagonal terms B(t)νµ,0 = Pν,0 B(t)0 Pµ,0 vanish and therefore trivially fulfill (3.3). According to (3.4) we hence have also determined the off-diagonal parts of the sub-principal term B(t)νν,1 , which vanish as well. The diagonal contributions Pµ,0 B(t)νν,1 Pµ,0 with respect to the projection matrices obey [Pη,0 , Pµ,0 B(t)νν,1 Pµ,0 ] = 0 and thus can be determined from the relation (3.3). As in [Ivr98, BG00], we hence obtain a recursive Cauchy problem for the coefficients B(t)νν,n and are now in a position to state: Theorem 3.2. Let H ∈ S0cl (m) be hermitian with the property Hj
(α) (β) (x, ξ )n×n
≤ Cα,β for all (x, ξ ) ∈ T∗ Rd and |α| + |β| + j ≥ 2 − δj 0 , (3.8)
and such that the conditions (H0) and (H2) are fulfilled. Furthermore, suppose that q B ∈ Scl (1) is block-diagonal with respect to the semiclassical projections defined in (2.7), B∼
l
Pµ #B#Pµ .
µ=1
Then B is in the invariant algebra S∞ inv (1) introduced in Definition 3.1, i.e., B(t) is again q a semiclassical pseudodifferential operator with symbol B(t) ∈ Scl (1). Furthermore, the principal symbol of B(t) is given by B(t)0 (x, ξ ) =
l
∗ dνν (x, ξ, t)Bνν,0 tν (x, ξ ) dνν (x, ξ, t),
(3.9)
ν=1
where tν is the Hamiltonian flow generated by the eigenvalue λν of H0 , and dνν is a unitary n × n matrix which is determined by the transport equation (3.7). Proof. As in [Ivr98, BG00] we start by rewriting (3.3) for the diagonal block of B(t)νν,n with respect to Pµ,0 in the form d −1 −tδ dνµ (x, ξ, −t)(Pµ,0 B(t)νν,n Pµ,0 ) ◦ ν νµ (x, ξ )dνµ (x, ξ, −t) dt (β) (α) γ (α, β)Pµ,0 B(t)νν,l (α) Hν,j (β) = 0≤l≤n−1 j +|α|+|β|=n−l+1
−(−1)|α|−|β| Hν,j
(β) (α) (β) B(t)νν,l (α)
Pµ,0 ,
(3.10)
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where dνµ is determined by the transport equation tδ d˙νµ (x, ξ, t) + iH˜ νµ,1 (ν νµ (x, ξ ))dνµ (x, ξ, t),
dνµ (x, ξ, 0) = idCn ,
that generalises (3.7) also to the off-diagonal transport. And since H˜ νµ,1 is hermitian, the solution dνµ is a unitary n × n matrix, which in the case ν = µ is obviously given by ˜
dνµ (x, ξ, t) = e−iHνµ,1 (x,ξ )t . In order to obtain estimates on the derivatives of the symbols Pµ,0 B(t)νν,n (t)Pµ,0 one has to control the behaviour of the flow tν generated by the eigenvalue λν of H0 . To this end we first notice that H0 ∈ S(m) implies the bound |λν (x, ξ )| ≤ cm(x, ξ ) on its eigenvalues. Furthermore, due to the hyperbolicity condition (H0) the projections Pν,0 onto the eigenspaces of H0 are in S(1). We then consider the first order derivatives (|α| + |β| = 1) of the relation H0 (x, ξ )Pν,0 (x, ξ ) = λν (x, ξ )Pν,0 (x, ξ ) to obtain
(α) (α) (α) λν (β) (x, ξ )Pν,0 (x, ξ ) = H0 (x, ξ )Pν,0 (x, ξ ) (β) − λν (x, ξ )Pν,0 (β) (x, ξ ). (α)
Now, since Pν,0 (x, ξ )Pν,0 (β) (x, ξ )Pν,0 (x, ξ ) = 0, a multiplication of the above equation with Pν,0 (x, ξ ) from both sides yields (α)
(α)
λν (β) (x, ξ )Pν,0 (x, ξ ) = Pν,0 (x, ξ )H0 (x, ξ )(β) Pν,0 (x, ξ ),
(3.11)
and hence (α) (α) (α) λν = cλν (α) Pν,0 = cPν,0 H0 (β) Pν,0 n×n ≤ c˜H0 (β) n×n . (β) (β) n×n H0 ∈ S(m) therefore implies that the first order derivatives of λν are bounded by the order function m. One can continue this argument by successively differentiating Eq. (3.11), and concludes that λν idCn ∈ S(m) for all ν = 1, . . . , l. In particular, the property (α) H0 (x, ξ ) ≤ Cα,β for |α| + |β| ≥ 1, (β) n×n which follows from (3.8), transfers to a corresponding growth property of the eigenvalues of H0 : (α) λν (x, ξ ) ≤ Cα,β for |α| + |β| ≥ 1. (β) One hence concludes that the Hamiltonian flows tν exist globally on T∗ Rd such that (α) |tν (β) (x, ξ )| ≤ Cα,β for all α, β ∈ Nd0 and for all finite times t ∈ [0, T ], see [Rob87]. This property guarantees that B ◦ tν ∈ S(1) for all B ∈ S(1). Concerning the unitary matrices dνµ the following is true: (α)
Lemma 3.3. If the subprincipal symbol H1 of H satisfies H1 (β) n×n ≤ Cα,β for all |α| + |β| ≥ 1, then dνµ (β) (x, ξ, t)n×n ≤ Cα,β for all t ∈ [0, T ], |α| + |β| ≥ 1 and ν, µ = 1, . . . , l. (α)
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For the proof of this lemma see [Ivr98]. With these properties at hand one can integrate Eq. (3.10) and solve for Pµ,0 B(t)νν,n Pµ,0 by conjugating with dνµ (x, ξ, −t) and δ t shifting the arguments by ννµ (which only amounts to an actual shift in the case ν = µ). For the principal symbol of B(t) one thus obtains −1 B(t)νν,0 (x, ξ ) = dνν (tν (x, ξ ), −t)Bνν,0 (tν (x, ξ ))dνν (tν (x, ξ ), −t),
which is the only block of B(t)νν,0 with respect to Pµ,0 , µ = 1, . . . , l, that is different from zero. Using −1 ∗ dνµ (ν νµ (x, ξ ), −t) = dνµ (x, ξ, t) = dνµ (x, ξ, t), tδ
(3.12)
see [BN99], one finally obtains (3.9). For the higher coefficients B(t)νν,n , n ≥ 1, one employs the Duhamel principle and uses that fact that the sum in (3.10) is taken over indices with |α| + |β| + j ≥ 2, and thus involves terms in S(1), in order to conclude q that B(t)νν,n ∈ S(1). This shows that one has found an asymptotic expansion in Scl (1) ∗ for the symbol of U (t)BU(t) that can be summed with the Borel method to yield a complete symbol. This theorem shows that, for finite times t, one can associate to a (semiclassicalq q ly) block-diagonal symbol B ∈ Scl (1) a symbol B(t) ∈ Scl (1) whose quantisation opW [B(t)] is semiclassically close to B(t) = U ∗ (t)BU(t), i.e., B(t) − opW [B(t)] = O(∞ )
for all
t ∈ [0, T ].
This is a semiclassical version of the Egorov theorem [Ego69], which was originally formulated for the case of scalar symbols. We will now show (generalising results of Cordes [Cor83a, Cor00, Cor01]) that the semiclassical block-diagonal operators exhaust all operators with symbols in the invariant algebra S∞ inv (1). ∞ Proposition 3.4. The invariant subalgebra S∞ inv (1) of Scl (1) consists of precisely those q B ∈ Scl (1) which are semiclassically block-diagonal with respect to the projections Pµ , µ = 1, . . . , l, defined in (2.9) of Proposition 2.1, i.e., ∞ B ∈ S∞ inv ⊂ Scl (1)
⇔
B∼
l
Pµ #B#Pµ .
µ=1
Proof. Consider an operator B with symbol B ∈ S∞ cl (1), whose equation of motion is given by (3.2). For the symbol of the time-evolved operator we now assume an asymptotic expansion B(t) ∼
∞
−q+j B(t)j
j =0 q
in Scl (1). Furthermore, one can use (2.6) to separate (3.2) into blocks with respect to Pµ , µ = 1, . . . , l. For the off-diagonal blocks (ν = µ) one therefore obtains ∂ i B(t)νµ ∼ [H, B(t)νµ ]# , ∂t
(3.13)
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405
−q+j B(t) where B(t)νµ := Pν #B(t)#Pµ ∼ ∞ νµ,j . In leading semiclassical order j =0 −1 the factor on the right-hand side of Eq.(3.13) enforces the condition [H0 , B(t)νµ,0 ] = (λν − λµ )B(t)νµ,0 = 0. Since λµ = λν for µ = ν, this immediately yields B(t)νµ,0 = 0. Furthermore, ∞ ∞ ∂ −q+j B(t)νµ,j ∼ i H, −q+j −1 B(t)νµ,j . # ∂t j =1
j =1
Again the leading order on the right-hand side has to vanish, i.e., [H0 , B(t)νµ,1 ] = 0. This means that the symbol B(t)νµ,1 must be block-diagonal with respect to the projection matrices Pµ,0 ∈ S(1). But −1 Pλ,0 B(t)νµ,1 Pλ,0 = symbW #(B(t) − B(t) )#P P = 0, λ νµ νµ,0 λ P since B(t)νµ,0 = 0 for ν = µ. Iterating the above procedure we see that if B ∈ Sinv (1), then it has to be block-diagonal with respect to Pµ , µ = 1, . . . , l. This proves one direction asserted in the proposition. The other direction, that the block-diagonal operators form a subset of the invariant algebra, is contained in the Egorov Theorem 3.2. 4. Dynamics in the Eigenspaces According to the Egorov Theorem 3.2, the semiclassical calculus outlined above results not only in a transport of the principal symbols of observables by the Hamiltonian flows tν , but also in a conjugation by the (unitary n × n) transport matrices dνν . The latter define the dynamics of those degrees of freedom that on the quantum mechanical level are described by the factor Cn of the total Hilbert space L2 (Rd ) ⊗ Cn . Our intention in this section now is to develop combined classical dynamics of both types of degrees of freedom, i.e., those described by the Hamiltonian flows and those that are represented by the conjugations. In this context the fact that the conjugations enter along integral curves of the Hamiltonian flows introduces a hierarchy among the two types of degrees of freedom. In a first step we confirm that the dynamics represented by the transport matrices dνν take place in the eigenspaces of the principal symbol H0 in Cn . To this end we notice that since at every point (x, ξ ) ∈ T∗ Rd the projection matrices Pν,0 (x, ξ ) yield an orthogonal splitting of Cn and have constant rank kν , they define kν -dimensional subbundles πν : E ν → T∗ Rd of the trivial vector bundle T∗ Rd × Cn over phase space. The fibre ν −1 ∗ d E(x,ξ ) = πν (x, ξ ) over (x, ξ ) ∈ T R is given by the range of the projection, i.e., ν n E(x,ξ ) = Pν,0 (x, ξ )C . Furthermore, the canonical hermitian structure of Cn induces a hermitian structure on the fibres. We now intend to interpret the conjugation by dνν as a dynamics in the eigenvector bundle E ν , and for this purpose notice: Lemma 4.1. The restricted transport matrices dνν (x, ξ, t)Pν,0 (x, ξ ) provide unitary ν ν maps between the fibres E(x,ξ ) and Et (x,ξ ) . ν
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ν ν Proof. In order to see that dνν (x, ξ, t)Pν,0 (x, ξ ) maps E(x,ξ ) into Et (x,ξ ) we show ν
Pν,0 (tν (x, ξ ))dνν (x, ξ, t)Pν,0 (x, ξ )
= dνν (x, ξ, t)Pν,0 (x, ξ ).
(4.1)
This is certainly true for t = 0. Moreover, the derivative with respect to t of the left-hand side can be brought into the form −iH˜ νν,1 (tν (x, ξ ))Pν,0 (tν (x, ξ ))dνν (x, ξ, t)Pν,0 (x, ξ ). This requires the identity [Pν,0 , H˜ νν,1 ] = −i{λν , Pν,0 } that can be obtained from (3.6). Thus, Pν,0 (tν (x, ξ ))dνν (x, ξ, t)Pν,0 (x, ξ ) fulfills the same differential equation with respect to t as dνν (x, ξ, t)Pν,0 , and this finally implies (4.1). In order to see the unitarity, one shows that dνν (x, ξ, t)Pν,0 (x, ξ ) is an isometry with ν n range E t (x,ξ ) . The first point is clear since dνν is unitary on C and the fibres inherit ν n their hermitian structures from C . The second point follows from the observation that the transport provided by dνν can be reversed: Given v(tν (x, ξ )) ∈ Etν (x,ξ ) , the vector ν t Pν,0 (x, ξ )dνν (tν (x, ξ ), −t)v(tν (x, ξ )) lies in E(x,ξ ) and is mapped to v(ν (x, ξ )) by dνν (x, ξ, t)Pν,0 (x, ξ ), see (3.12). According to the above, the action of dνν (x, ξ, t) on a section in E ν can be viewed as a parallel transport along the integral curves of the flow tν . If one now introduces ν sections of E ν that yield orthonormal bases {e1 (x, ξ ), . . . , ekν (x, ξ )} of the fibres E(x,ξ ), the representations of dνν (x, ξ, t) in these bases are unitary kν × kν matrices Dν (x, ξ, t). Since the principal symbol H0 of the operator H is hermitian (on Cn ), a preferred choice for the sections {e1 , . . . , ekν } would consist of orthonormal eigenvectors of H0 . However, this choice is obviously not unique because it amounts to fixing an isometry Vν (x, ξ ) : ν ∗ ∗ Ckν → E(x,ξ ) , such that Vν (x, ξ )Vν (x, ξ ) = Pν,0 (x, ξ ) and Vν (x, ξ )Vν (x, ξ ) = idCkν . Here one still has the freedom to change the isometry by an arbitrary unitary automorν phism of Ckν . Having chosen an isometry Vν (x, ξ ) for every fibre E(x,ξ ) in a smooth way, the n × n transport matrices dνν (x, ξ, t) are mapped to unitary kν × kν matrices, Dν (x, ξ, t) := Vν∗ tν (x, ξ ) dνν (x, ξ, t)Vν (x, ξ ). (4.2) Their dynamics follows from the transport equation (3.7) as D˙ ν (x, ξ, t) + iH˜ ν tν (x, ξ ) Dν (x, ξ, t) = 0 with Dν (x, ξ, 0) = idCkν , where the hermitian kν × kν matrix H˜ ν is derived from (3.6) for µ = ν, λν H˜ ν = −i Vν∗ {Pν,0 , Pν,0 }Vν + i{λν , Vν∗ }Vν + Vν∗ Hν,1 Vν . 2 What is of more importance for later purposes than the non-uniqueness of this representation, however, is the fact that the above construction allows to introduce a skew-product flow over the Hamiltonian flow tν , thus reflecting the hierarchy of the two types of degrees of freedom. See [CFS82] for a definition of skew-product flows and cf. [BK99b] where these occur in the context of a semiclassical trace formula for matrix valued operators. At first sight the cocycle relation Dν (x, ξ, t + t ) = Dν (tν (x, ξ ), t )Dν (x, ξ, t) suggests to introduce the flow Yˆνt (x, ξ, g) := (tν (x, ξ ), Dν (x, ξ, t)g) on T∗ Rd × U(kν ). This construction might, however, turn out too general because its fibre part does not necessarily require the complete group U(kν ). E.g., in
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407
[BGK01] a situation was considered where kν = 2j + 1, j ∈ 21 N, and the transport matrices Dν were operators in a 2j + 1-dimensional unitary irreducible representation of SU(2). This fact could be identified by the observation that when (x, ξ ) ranges over T∗ Rd , the skew-hermitian matrices iH˜ ν (x, ξ ) generate a Lie subalgebra of u(2j + 1) which is isomorphic to su(2). We therefore now consider the Lie subgroup of U(kν ) that is generated by the transport matrices Dν . In order to identify this group we consider the Lie subalgebra (4.3) iH˜ ν (x, ξ ); (x, ξ ) ∈ T∗ Rd ⊂ u(kν ) generated by the skew-hermitian matrices iH˜ ν (x, ξ ). Via exponentiation of this subalgebra one hence obtains a Lie subgroup G ⊂ U(kν ) that is compact and connected. To be more precise, the result of the exponentiation is a kν -dimensional unitary representation ρ of G. Its Lie algebra g then is embedded in (4.3) via the derived representation dρ. In this setting the transport matrices Dν are operators in the representation ρ, i.e., Dν (x, ξ, t) = ρ(gν (x, ξ, t)). Hence we are now in a position to define the skew-product flows Y˜νt : T∗ Rd × G → T∗ Rd × G through Y˜νt (x, ξ, g) = tν (x, ξ ), gν (x, ξ, t)g .
(4.4)
These flows leave the product measure dx dξ dg on T∗ Rd × G invariant, which consists of Lebesgue measure dx dξ on T∗ Rd and the normalised Haar measure dg on G. Moreover, if one restricts the Hamiltonian flows tν to compact level surfaces of the eigenvalue functions λν at non-critical values E, ∗ d ν,E := λ−1 ν (E) = (x, ξ ) ∈ T R ; λν (x, ξ ) = E , the restrictions of the skew-product flows Y˜νt to ν,E ×G leave the measures d(x, ξ ) dg invariant, where d(x, ξ ) denotes the normalised Liouville measure on ν,E . Below we are interested in the question under which conditions imposed on suitable classical dynamics quantum ergodicity holds, see Sect. 6. In analogy to [BG00] one approach to this problem would be to consider the restriction of the skew-product flow Yˆνt to ν,E × U(kν ): Its ergodicity with respect to the product measure that consists of Liouville measure on ν,E and Haar measure on U(kν ) implies quantum ergodicity. Since, however, the dynamics in the eigenspaces is completely fixed by a restriction to the group G, the dynamical behaviour of the flow Yˆνt is determined by that of Y˜νt . One hence concludes that in order to prove quantum ergodicity one requires the following condition (see Remark 6.3): (Irrν ) The representation ρ : G → U(kν ) is irreducible. In the sequel we always assume this to be the case. Our intention now is to relate the dynamics in the eigenspaces, given by the conjugation with the transport matrices dνν , to proper classical dynamics. To this end we require a symplectic manifold with the dynamics realised in a Hamiltonian fashion. For this purpose Kirillov’s orbit method [Kir76] provides the necessary tools: It relates the unitary irreducible representation (ρ, Ckν ) to a coadjoint orbit Oλ := {Ad∗g λ; g ∈ G} of G.
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Here Ad∗ is the coadjoint action of G on its dual Lie algebra g∗ and λ is a highest (real) weight corresponding to (ρ, Ckν ). The coadjoint orbit Oλ possesses a natural symplectic structure, yielding the product phase space T∗ Rd ×Oλ for the combined dynamics. As in the case of G = SU(2) considered in [BGK01], this setting then also allows to introduce a Moyal-type quantisation such that hermitian matrix valued symbols can be uniquely related to real valued functions on T∗ Rd × Oλ . The classical setting related to the coadjoint orbit Oλ with sympletic form σ is based on observables given as integrable functions on Oλ with respect to the volume form that arises as the maximal exterior power dη of σ . This form is normalised by the requirement vol(Oλ ) = kν . On the quantum side observables are hermitian endomorphisms of the representation space V = Ckν . A Moyal quantiser now assigns to a hermitian A ∈ L(V ) a function a ∈ L1 (Oλ ) in a certain way (see [FGV90]): Proposition 4.2. To each A ∈ L(V ) one can assign symbSW [A] ∈ L1 (Oλ ), called the Stratonovich-Weyl symbol, such that the map A → symbSW [A] has the following properties: (i) It is linear and one-to-one. (ii) symbSW [A∗ ] = symbSW [A]. (iii) symbSW [idV ] = 1. (iv) symbSW [ρ(g)Aρ(g −1 )](η) = symbSW [A](Ad∗g −1 η) for all η ∈ Oλ , g ∈ G. (v) symbSW [A](η) symbSW [B](η)dη = tr(AB). Oλ
With this formalism at hand one can now transfer the dynamics of a (hermitian) B ∈ L(V ) given by a conjugation with D(t) = ρ(g(t)), B → B(t) = D −1 (t)BD(t), to the coadjoint action of g(t) on the symplectic manifold Oλ via the relation symbSW [B(t)](η) = symbSW [B](Ad∗g(t) η). The symplectic structure on Oλ defined by the form σ , furthermore, allows to identify the dynamics η → Ad∗g(t) η as being Hamiltonian, with the Hamiltonian given by the momentum map of the coadjoint action. As an ultimate outcome of the above formalism we are now in a position to introduce a skew-product flow on the symplectic phase space T∗ Rd × Oλ that completely determines the time evolution of the ν th diagonal block of an observable on the level of its principal symbol. Explicitly, this flow is given by Yνt : T∗ Rd × Oλ → T∗ Rd × Oλ
(4.5)
Yνt (x, ξ, η) := tν (x, ξ ), Ad∗gν (x,ξ,t) η ;
(4.6)
with
it leaves the product measure dx dξ dη invariant. Consider now a semiclassical pseudodifferential operator B with symbol B ∈ S∞ cl (1). Mod O(∞ ) the quantum dynamics preserves the diagonal structure of its blocks Pν BPν . According to the Egorov Theorem 3.2, together with the definition (4.2), the principal symbol of Pν B(t)Pν hence reads Vν (x, ξ )Dν∗ (x, ξ, t) Vν∗ B0 Vν tν (x, ξ ) Dν (x, ξ, t)Vν∗ (x, ξ ). (4.7)
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We now exploit the possibility to uniquely represent the value of Vν∗ B0 Vν : T∗ Rd → L(Ckν ) in terms of a Stratonovich-Weyl symbol, b0,ν (x, ξ, η) := symbSW (Vν∗ B0 Vν )(x, ξ ) (η). (4.8) The dynamics of the principal symbol in this representation is now summarised in the following variant of the Egorov theorem: Proposition 4.3. The Stratonovich-Weyl symbol b(t)0,ν associated with the principal symbol of the operator Pν B(t)Pν is the time evolution of b0,ν under the skew-product flow Yνt defined in Eq. (4.5)–(4.6), i.e., b(t)0,ν (x, ξ, η) = b0,ν Yνt (x, ξ, η) . Proof. According to (4.7) and (4.8), b(t)0,ν is given by b(t)0,ν (x, ξ, η) = symbSW ρ(gν−1 (x, ξ, t)) Vν∗ B0 Vν tν (x, ξ ) ρ(gν (x, ξ, t)) (η), which due to the covariance property (iv) of Proposition 4.2 reads b(t)0,ν (x, ξ, η) = symbSW Vν∗ B0 Vν tν (x, ξ ) (Ad∗gν (x,ξ,t) η) = b0,ν tν (x, ξ ), Ad∗gν (x,ξ,t) η . 5. Trace Asymptotics and a Limit Formula for Averaged Expectation Values A fundamental ingredient in the asymptotics of eigenvectors we are aiming at is a semiclassical limit formula for the expectation values of bounded operators B on L2 (Rd )⊗Cn . Below we will obtain a Szeg¨o-type formula which connects semiclassically averaged expectation values with objects that can be calculated from the principal symbol B0 of the operator B and therefore allow for a classical interpretation. On the so defined classical side we fix a value E for all eigenvalue functions λν , ν = 1, . . . , l, of the principal symbol H0 with the following properties: ∗ d (H3ν ) There exists some ε > 0 such that all λ−1 ν ([E − ε, E + ε]) ⊂ T R are compact. (H4ν ) The functions λν shall possess no critical values in [E − ε, E + ε]. (H5ν ) Among the level surfaces ν,E = λ−1 ν (E), ν = 1, . . . , l, at least one is non-empty.
In addition to (H1) and (H2), which imply the essential selfadjointness of the operator H, these conditions guarantee as in the scalar case [DS99] that for sufficiently small the spectrum of H is discrete in any open interval contained in [E − ε, E + ε]. This setting now allows us to generalise the constructions made in [BG00] to Hamiltonians with non-scalar principal symbols: The expectation values of an operator B will be considered in normalised eigenvectors ψj of H with corresponding eigenvalues Ej in an interval I (E, ) = [E − ω, E + ω], ω > 0, such that I (E, ) ⊂ [E − ε, E + ε] if is small enough. On the classical side the Hamiltonian flows tν generated by the eigenvalue functions λν will enter on the level surfaces ν,E . Regarding these we assume: (H6ν ) The periodic points of tν with non-trivial periods form a set of Liouville measure zero in ν,E .
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The quantities appearing on the classical side of the limit formula turn out to be averages of smooth matrix valued functions B ∈ C ∞ (T∗ Rd ) ⊗ Mn (C) over ν,E with respect to Liouville measure, for which we introduce the notation ν,E (B) := B(x, ξ ) d(x, ξ ). ν,E
The main result of this section is now summarised in the following Szeg¨o-type limit formula: Proposition 5.1. Let H be a semiclassical pseudodifferential operator with symbol H ∈ S0cl (m), such that the principal symbol H0 satisfies the assumptions (H0)–(H2) and (H3ν )–(H6ν ) for all ν = 1, . . . , l. Furthermore, let B be an operator with symbol B ∈ S0cl (1) and principal symbol B0 . Then the limit formula l vol ν,E tr ν,E (Pν,0 B0 Pν,0 ) 1 lim ψj , Bψj = ν=1 l (5.1) →0 NI ν=1 kν vol ν,E E ∈I (E,) j
holds. Proof. Adapted to the spectral localisation mentioned above we choose a smooth and compactly supported function g ∈ C0∞ (R) such that g(λ) = λ on a neighbourhood of [E − ε, E + ε]. Furthermore, we apply the semiclassical splitting of the Hilbert space L2 (Rd ) ⊗ Cn given by the projection operators Pν , L2 (Rd ) ⊗ Cn = ran P1 ⊕ · · · ⊕ ran Pl mod ∞ , (5.2) l and the corresponding decomposition H = ν=1 HPν (mod O(∞ )) of the Hamiltonian. By employing the generalisation of the Helffer-Sj¨ostrand formula to matrix valued operators developed in [Dim93, Dim98], we represent g(H) = lν=1 g(HPν )Pν (mod O(∞ )) with 1 ∂z g(z)(H ˜ − z)−1 Pν dz, g(HPν )Pν = − π C where g˜ is an almost-analytic extension of g. Since the principal symbol H0 Pν,0 of HPν is scalar, H0 Pν,0 = λν Pν,0 , when considered to act on sections in the eigenvector bundle E ν , one can use the methods of [DS99] to show that on λ−1 ν ([E−ε, E+ε]) the asymptotic W W expansions of symb [g(HPν )] and of symb [HPν ] coincide. Below we will always employ the spectral localisation to the interval I (E, ), and since symbW [g(HPν )] ∈ S0cl (1), one can therefore now assume that H ∈ S0cl (1). Furthermore, the decomposition (5.2) allows us to employ the techniques of [DS99] in the same manner as in [BG00]. Hence, if χ ∈ C0∞ (R) with χ ≡ 1 on I (E, ) and supp χ ⊂ [E − ε, E + ε], the operator Uχ (t) := e− Ht χ(H) i
l ν=1
Pν =
l
e− HPν t χ (HPν )Pν i
mod
O(∞ ),
ν=1
has a pure point spectrum. Moreover, each of the operators e− HPν t χ (HPν ) can be approximated in trace norm up to an error of O(∞ ) by a semiclassical Fourier integral operator with a kernel of the form i 1 Kν (x, y, t) = aν (x, y, t, ξ )e (Sν (x,ξ,t)−ξy) dξ. d (2π) Rd i
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Here, as in [BK99a], the phases Sν have to fulfill the Hamilton-Jacobi equations λν x, ∂x Sν (x, ξ, t) + ∂t Sν (x, ξ, t) = 0, Sν (x, ξ, 0) = xξ. j The amplitudes aν ∈ S0cl (1) with asymptotic expansions aν ∼ ∞ j =0 aν,j are determined as solutions of certain transport equations [BK99a] with initial conditions aν |t=0 = χ(λν )Pν,0 + O(). Now, employing the techniques used in [BG00] for each ν immediately proves the proposition and gives a semiclassical expression for the number NI of eigenvalues of H in I (E, ), NI := #{Ej ∈ I (E, )} =
l ω vol ν,E kν + o(1−d ). π (2π )d−1
(5.3)
ν=1
Let us add two comments: 1. Under the additional assumption (Irrν ) the Stratonovich-Weyl calculus discussed in Sect. 4 can be applied. It allows to express tr(Pν,0 B0 Pν,0 ) = tr(Vν∗ B0 Vν ) in terms of the symbol b0,ν introduced in (4.8). This then leads to the representation 1 1 tr ν,E (Pν,0 B0 Pν,0 ) = b0,ν (x, ξ, η) dη d(x, ξ ) vol Oλ ν,E Oλ kν =: ME,ν,λ b0,ν (5.4) as an integral over the product space ν,E × Oλ . Here the relation kν = vol Oλ , introduced in Sect. 4, enables one to give the right-hand side of (5.1) a genuinely classical interpretation. 2. The operators B considered in the limit formula (5.1) have not been restricted to those with symbols in the invariant subalgebra S0inv (1) ⊂ S0cl (1). Nevertheless, only the diagonal blocks of their principal symbols B0 with respect to the projection matrices Pν,0 enter on the right-hand side of (5.1). In particular, this implies that for an operator B with a purely off-diagonal principal symbol the semiclassical average vanishes. Thus one can replace an operator B with symbol B ∈ S0cl (1) by its diagonal part µ P˜ µ B P˜ µ , whose symbol is in the invariant algebra S0inv (1), without changing the value of the limit on the right-hand side of (5.1). So far we have considered expectation values in normalised eigenvectors of H. Our intention now is to discuss the projections Pν ψj of the eigenvectors of H to a fixed almost invariant subspace of L2 (Rd ) ⊗ Cn . One thus expresses averaged expectation values in the projected eigenvectors in terms of classical quantities related to the single Hamiltonian flow tν . In order to achieve this one applies Proposition 5.1 to operators Pν BPν and exploits the selfadjointness of Pν . This results in Corollary 5.2. Under the assumptions stated in Proposition 5.1, for each ν ∈ {1, . . . , l} the restricted limit formula 1 →0 NI lim
holds.
Ej ∈I (E,)
Pν ψj , BPν ψj =
vol ν,E tr ν,E (Pν,0 B0 Pν,0 ) l µ=1 kµ vol µ,E
(5.5)
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Thus the semiclassical average of the projected eigenvectors Pν ψj , with Ej ∈ I (E, ), localises on the corresponding level surface ν,E ⊂ T∗ Rd . If one considers (5.5) for different ν, the relative weights of the corresponding projections are determined by the relative volumes of the associated level surfaces and the dimensions of the eigenspaces E ν , which equal the volumes of the coadjoint orbits Oλ . In general, however, the projected eigenvectors Pν ψj are neither normalised, nor are they genuine eigenvectors of H. We therefore now introduce the normalised vectors φj,ν :=
Pν ψj . Pν ψj
(5.6)
Since the projectors Pν only commute with H up to a term of O(∞ ), the pairs (Ej , φj,ν ) are quasimodes with discrepancies rj,ν , i.e.,
[H, Pν ]ψj H − Ej φj,ν = Pν ψj
and rj,ν =
[H, Pν ]ψj . Pν ψj
This observation only ensures the (trivial) existence of an eigenvalue of H in the interval [Ej − rj,ν , Ej + rj,ν ]. It does not imply that φj,ν is close to an eigenvector of H, see [Laz93]. Of somewhat more interest is to consider the operator HPν , whose spectrum inside the interval I (E, ) is also purely discrete. Following the above reasoning, one concludes that (Ej , φj,ν ) is a quasimode with discrepancy rj,ν also for this operator. Thus, if Pν ψj ≥ cN for some N ≥ 0 and hence rj,ν = O(∞ ), the operator HPν has an eigenvalue with distance O(∞ ) away from Ej . Since there are NI eigenvalues Ej ∈ I (E, ) one finds as many quasimodes for HPν . But this operator has only NIν =
kν ω vol ν,E + o(1−d ) π (2π)d−1
eigenvalues in I (E, ), compare (5.3). This observation might suggest that only approximately NIν of the NI projected eigenvectors Pν ψj are of considerable size, such that the discrepancies of the associated quasimodes are smaller than the distance of Ej to neighbouring eigenvalues of H. This expectation can be strengthened by an application of the limit formula (5.5) with the choice B = id, which implies that NIν =
Pν ψj 2 + o(1),
→ 0.
(5.7)
Ej ∈I (E,)
One could thus expect that roughly NIν of the projected eigenvectors Pν ψj are close to ψj , and the rest is such that Pν ψj is semiclassically small. However, (5.7) does not rule out the other extreme situation, provided by projected eigenvectors Pν ψj , ν = 1, . . . , l, equidistributing in the sense that their squared norms are asymptotic to NIν /NI as → 0. In that case the discrepancies of the associated quasimodes for the operators HPν can be estimated as rj,ν = O(∞ ). In order now that these quasimodes do not produce more than NIν eigenvalues of HPν in I (E, ), a finite fraction of the eigenvalues Ej of H must possess spacings to their nearest neighbours of the order ∞ . Since in general there exist no sufficient lower bounds on eigenvalue spacings, none of the two extreme situations discussed above can be excluded so far.
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What is possible, however, is to derive from (5.5) an upper bound for the fraction of the projected eigenvectors Pν ψj that are close in norm to ψj , lim
→0
1 kν vol ν,E # Ej ∈ I (E, ); Pν ψj − ψj = o(1) ≤ l , NI µ=1 kµ vol µ,E
see also [Sch01]. To obtain lower bounds is notoriously more difficult. The limit formula (5.5) only allows to estimate the fraction of projected eigenvectors with norms that tend to a finite limit as → 0. One conveniently measures this fraction in units of the value k vol ν,E that is expected for equidistributed projections. Therefore, with δ := δ˜ l ν , µ=1 kµ vol µ,E
we consider δ Nν,I := # Ej ∈ I (E, ); Pν ψj 2 ≥ δ . Since 1 NI
Pν ψj 2 ≤
Ej ∈I (E,)
≤
1 NI
Ej ∈I (E,) Pν ψj 2 ≥δ
δ Nν,I
NI
+
1+
1 NI
Pν ψj 2
Ej ∈I (E,) Pν ψj 2 <δ
δ δ (NI − Nν,I ), NI
the relative fraction of projected eigenvectors with finite semiclassical limit can be estimated from below as lim
→0
δ Nν,I
NI
˜ ν vol ν,E (1 − δ)k ≥ l . µ=1 kµ vol µ,E
(5.8)
6. Quantum Ergodicity Our intention in this section is to consider quantum ergodicity for the normalised eigenvectors ψj , Ej ∈ I (E, ), of the quantum Hamiltonian H. In the case of scalar pseudodifferential operators one denotes by quantum ergodicity a weak convergence of the phase space lifts of almost all eigenfunctions to Liouville measure on the level surface E = H0−1 (E), and proves this to hold if the flow generated by the principal symbol H0 of the quantum Hamiltonian is ergodic on E . In the present situation of operators with matrix valued symbols, however, each eigenvalue λν of H0 defines its own classical dynamics. One hence can only expect quantum ergodicity to be concerned with statements about the projections Pν ψj of the eigenvectors to the different almost invariant subspaces of L2 (Rd ) ⊗ Cn in relation to the behaviour of the associated classical systems. In the preceding section we discussed the question of identifying those projected eigenvectors whose norms are not semiclassically small. Since presently this problem cannot be resolved directly, quantum ergodicity can only be formulated by restricting to those eigenvectors whose squared norms exceed a value of δ in the semiclassical limit, without specifying them further. Conventionally the convergence of quantum states determined by the eigenvectors ψj of H is discussed in terms of expectation values of observables in these states. Explicit
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lifts of the eigenfunctions to phase space are then, e.g., provided by their Wigner transforms. The choice of the projected eigenvectors Pν ψj leads to consider expectation q values of diagonal blocks Pν BPν of operators B with symbols B ∈ Scl (1). On the symbol level the time evolution of these blocks is covered by the Egorov Theorem 3.2. Representing then the blocks of the principal symbols by Stratonovich-Weyl symbols as described in Sect. 4, according to Proposition 4.3 we are faced with the skew-product flows Yνt on the product phase spaces T∗ Rd × Oλ . Since the Stratonovich-Weyl symq bols b0,ν defined in Eq. (4.8) that are associated with symbols B ∈ Scl (1) are clearly integrable with respect to the measures d dη on the (compact) manifolds ν,E × Oλ , the (assumed) ergodicity of the flow Yνt implies that 1 1 T t b0,ν ◦ Yν (x, ξ, η) dt = b0,ν (x , ξ , η ) dη d(x , ξ ) lim T →∞ T 0 vol Oλ ν,E Oλ = ME,ν,λ (b0,ν ) (6.1) holds for almost all initial conditions (x, ξ, η) ∈ ν,E × Oλ . In particular, one immediately realises that the supposed ergodicity of Yνt implies ergodicity for the flow tν on ν,E with respect to Liouville measure d. As a consequence the condition (H6ν ) is automatically fulfilled. For the subsequent formulation and proof of quantum ergodicity we choose to follow in principle the approach of [Zel96, ZZ96]. This means that we investigate the variance of expectation values about their mean in the semiclassical limit. In order to avoid the problem of explicitly estimating the norms of projected eigenvectors we here consider the normalised vectors φj,ν , defined in (5.6), which have been identified as quasimodes for both the operators H and HPν . Moreover, we concentrate on vectors corresponding to projected eigenvectors with norms that do not vanish semiclassically, i.e., with Pν ψj 2 ≥ δ for some fixed δ ∈ (0, 1). This approach is similar to the one introduced by Schubert [Sch01] in the context of local quantum ergodicity, where an equidistribution was shown for quasimodes associated with ergodic components of phase space. In δ /N of the associated eigenvectors among Sect. 5 we estimated the relative number Nν,I I all eigenvectors of H in the semiclassical limit from below, see (5.8). A non-trivial bound could only be obtained for δ˜ < 1 corresponding to δ < δν := l
kν vol ν,E
µ=1 kµ vol µ,E
.
Therefore, from now on we confine δ to the interval δ ∈ (0, δν ), and are thus in a position to state our main result. Theorem 6.1. Let H be a semiclassical pseudodifferential operator with hermitian symbol H ∈ S0cl (m) whose principal part H0 fulfills the conditions (H1) and (H2) of Sect. 3. The eigenvalues λ1 , . . . , λl of H0 are required to have constant multiplicities and shall obey the conditions (H3ν )–(H5ν ) of Sect. 5 for all ν ∈ {1, . . . , l}. Moreover, they shall be separated according to the hyperbolicity condition (H0), |λν (x, ξ ) − λµ (x, ξ )| ≥ Cm(x, ξ ) for ν = µ and |x| + |ξ | ≥ c. j Assume now that the symbol H ∼ ∞ j =0 Hj satisfies the growth condition Hj
(α) (β) (x, ξ )n×n
≤ Cα,β for all (x, ξ ) ∈ T∗ Rd and |α| + |β| + j ≥ 2 − δj 0 , (3.8)
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and that the condition (Irrν ) of Sect. 4 holds. If then the flow Yνt defined in (4.6) is ergodic on ν,E × Oλ with respect to the invariant measure d dη, in every sequence of normalised projected eigenvectors {φj,ν }j ∈N , with Pν ψj 2 ≥ δ, δ ∈ (0, δν ) fixed, one finds a subsequence {φjα ,ν }α∈N of density one, i.e., #{α; Pν ψjα 2 ≥ δ} = 1, →0 #{j ; Pν ψj 2 ≥ δ} lim
such that for every operator B with symbol B ∈ S0cl (1) and principal symbol B0 , lim φjα ,ν , Bφjα ,ν = ME,ν,λ (b0,ν ),
→0
(6.2)
where b0,ν denotes the Stratonovich-Weyl symbol associated with Pν,0 B0 Pν,0 . Furthermore, the density-one subsequence {φjα ,ν }α∈N can be chosen to be independent of the operator B. Proof. We start with considering expectation values of the operator B taken in the quasimodes {φj,ν } and denote their variance about the mean ME,ν,λ (b0,ν ) of the corresponding Stratonovich-Weyl symbol b0,ν defined in (4.8) as δ (E, ) := S2,ν
1 δ Nν,I
φj,ν , Bφj,ν − ME,ν,λ (b0,ν ) 2 .
Ej ∈I (E,) Pν ψj 2 ≥δ
Due to the definition (5.6) of the normalised vectors φj,ν , this can also be written as δ (E, ) = S2,ν
1 δ Nν,I
2 Pν ψj −2 ψj , Pν B − ME,ν,λ (b0,ν ) Pν ψj .
Ej ∈I (E,) Pν ψj 2 ≥δ
Allowing for an error of O(∞ ), in this expression the expectation values can be replaced by those of the operator P˜ ν (B − ME,ν,λ (b0,ν ))P˜ ν , whose symbol is in the invariant subalgebra S0inv (1) ⊂ S0cl (1). Therefore, since all further requirements are also met, the Egorov Theorem 3.2 applies and yields that for finite times t ∈ [0, T ] the evolution U ∗ (t)P˜ ν (B − ME,ν,λ (b0,ν ))P˜ ν U(t) of this operator is again a pseudodifferential operator with symbol in the class S0cl (1). Taking into account that the ψj s are eigenvectors of H with eigenvalues Ej , the above expression can be rewritten as δ S2,ν (E, ) =
1 δ Nν,I
ψj , Bν,T ψj 2 Pν ψj −2 ,
Ej ∈I (E,) Pν ψj 2 ≥δ
where we have defined the auxiliary operator Bν,T :=
1 T
0
T
U ∗ (t)P˜ ν B − ME,ν,λ (b0,ν ) P˜ ν U(t) dt.
(6.3)
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Furthermore, by using using the Cauchy-Schwarz inequality and the lower bound on the norms Pν ψj 2 ≥ δ > 0 we obtain as an upper bound δ S2,ν (E, ) ≤
1 NI 1 δ N δ Nν,I I
2 ψj , Bν,T ψj .
Ej ∈I (E,)
One now requires the principal symbol Bν,T ,0 of the auxiliary operator Bν,T , which follows from Theorem 3.2 as 1 T ∗ Bν,T ,0 = d (Pν,0 B0 Pν,0 ) ◦ tν dνν dt − ME,ν,λ (b0,ν )Pν,0 . T 0 νν δ yield Given this, the limit formula (5.1) and the estimate (5.8) for the factor NI /Nν,I δ lim S2,ν (E, ) ≤
→0
1 1 ME,ν,λ (symbSW [Bν,T ,0 ])2 , δ 1 − δ˜
(6.4)
when employing the tracial property (v) of Proposition 4.2. According to Proposition 4.3 the Stratonovich-Weyl symbol of Bν,T ,0 can now be easily calculated as 1 T SW symb [Bν,T ,0 (x, ξ )](η) = b0,ν ◦ Yνt (x, ξ, η) dt − ME,ν,λ (b0,ν ). T 0 Since we assume the skew-product flow Yνt to be ergodic with respect to d dη, the relation (6.1) implies that symbSW [Bν,T ,0 (x, ξ )](η) vanishes in the limit T → ∞ for almost all points (x, ξ, η) ∈ ν,E × Oλ . Now, on the right-hand side of (6.4) the square of symbSW [Bν,T ,0 ] enters integrated over ν,E × Oλ , so that this expression vanishes as T → ∞. We hence conclude that δ lim S2,ν (E, ) = 0.
→0
This, in turn, is equivalent to the existence of a subsequence {φjα ,ν }α∈N ⊂ {φj,ν }j ∈N of density one, such that Eq. (6.2) holds. Finally, by a diagonal construction as in [Zel87, CdV85] one can extract a subsequence of {φjα ,ν }α∈N that is still of density one in {φj,ν }j ∈N , such that (6.2) holds independently of the operator B. The version of quantum ergodicity asserted in Theorem 6.1 means that in the semiclassical limit the lifts of almost all quasimodes φj,ν to the phase space T∗ Rd × Oλ equidistribute in the sense that suitable Wigner functions (weakly) converge to an invariant measure on ν,E × Oλ that is proportional to d dη. However, since in φj,ν the normalisation of Pν ψj is hidden, an equivalent equidistribution for the lifts of the projected eigenvectors is only shown up to a constant. In analogy to the discussion in [Sch01] this means that in the sequence {ψj ; Ej ∈ I (E, )} there exists a subsequence {ψjα } of density one such that as → 0, ψjα , Pν BPν ψjα = Pν ψjα 2 ME,ν,λ (b0,ν ) + o(1). Notice that the factor Pν ψjα 2 is independent of the operator B so that the subsequence can again be chosen independently of B. Therefore, a non-vanishing semiclassical limit only exists for those subsequences along which the norms Pν ψjα do not tend to zero
Quantum Ergodicity for Matrix Valued Operators
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as → 0. These subsequences are excluded in the formulation of Theorem 6.1 since δ is fixed and positive. The difficulties with estimating norms of the projected eigenvectors Pν ψj arise from the presence of several level surfaces ν,E on which the lifts of eigenfunctions potentially condense in the semiclassical limit. The situation simplifies considerably, if at the energy E all of the l level surfaces except one are empty. Corollary 6.2. If under the conditions stated in Theorem 6.1 only the level surface ν,E ⊂ T∗ Rd is non-empty, there exists a subsequence {ψjα } of density one in {ψj ; Ej ∈ I (E, )}, independent of the operator B, such that lim ψjα , Pµ BPµ ψjα = δµν ME,ν,λ (b0,ν ).
→0
In this situation the norms Pµ ψjα converge to one for µ = ν and to zero otherwise as → 0 along the subsequence. The lifts of the eigenvectors therefore condense on the only available level surface in T∗ Rd , as one clearly would have expected. Remark 6.3. As a condition for quantum ergodicity to hold we have assumed the skewproduct flow Yνt on ν,E × Oλ to be ergodic. The reason for introducing this flow was to formulate a genuinely classical criterion in terms of a dynamics on the symplectic phase space T∗ Rd × Oλ . The formulation will be somewhat simpler, if one refrains from insisting on a completely classical description and employs the skew-product flow Y˜νt defined on T∗ Rd × G, see (4.4), instead. Then the use of the Stratonovich-Weyl calculus can be avoided. Such a formulation is based on a hybrid of the classical Hamiltonian flow tν on T∗ Rd and the dynamics represented by the conjugation with the unitary matrices Dν , which appears to be quantum mechanical in nature. Both formulations, however, are equivalent in the sense that, first, the Stratonovich-Weyl calculus relates the quantum dynamics in the eigenspace to a classical dynamics on the coadjoint orbit in a one-to-one manner. Furthermore, we have Lemma 6.4. The flow Y˜νt : ν,E × G → ν,E × G is ergodic with respect to d dg, if and only if the associated flow Yνt : ν,E × Oν → ν,E × Oν is ergodic with respect to d dη. Proof. We employ that for λ ∈ g∗ we have a diffeomorphism κ : G/Gλ → Oλ between the coadjoint orbit and the coset space which is the image of the canonical projection π : G → G/Gλ . This immediately shows that Y˜νt -invariant sets A ⊂ ν,E × G project to Yνt -invariant subsets (idT∗ Rd ×κ ◦ π )(A) ⊂ ν,E × Oλ . Furthermore, the normalised Haar measure dg on G projects to the volume measure dη on Oλ under κ ◦ π , which is an easy consequence of Fubini’s theorem. This gives the “if” part of the above lemma. For the reverse direction note that Yνt -invariant subsets can be lifted to Y˜νt -invariant sets and use the above argument. One can therefore formulate Theorem 6.1 without recourse to the Stratonovich-Weyl calculus once the limit ME,ν,λ (b0,ν ) is expressed in the form (5.4). Up to Eq. (6.4) the proof of Theorem 6.1 proceeds in the same manner as shown. From this point on one can then basically follow the method of [BG00], and to this end represent the principal symbol Bν,T ,0 of the auxiliary operator (6.3) in terms of the isometries Vν ,
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Vν∗ Bν,T ,0 Vν
1 = T
0
T
1 Dν∗ (Vν∗ B0 Vν ) ◦ tν Dν dt − tr ν,E (Vν∗ B0 Vν ). kν
We now suppose that the flow Y˜νt is ergodic on ν,E × G and choose the function F (x, ξ, g) := ρ(g)∗ (Vν∗ B0 Vν )(x, ξ )ρ(g) ∈ L1 (ν,E × G) ⊗ Mkν (C) to exploit the ergodicity. This yields for almost all initial values (x, ξ, g) ∈ ν,E × G that lim ρ(g)∗ Vν∗ (x, ξ )Bν,T ,0 (x, ξ )Vν (x, ξ )ρ(g) 1 ρ(h)∗ (Vν∗ B0 Vν )(y, ζ )ρ(h) dh d(y, ζ ) − tr ν,E (Vν∗ B0 Vν ). = k ν ν,E G
T →∞
Furthermore, since the representation (ρ, Ckν ) is assumed to be irreducible and the integral in the above expression is invariant under conjugation with arbitrary elements of U (kν ), Schur’s lemma implies that this integral is a multiple of the identity in Ckν , leading to 1 ρ(h)∗ (Vν∗ B0 Vν )(y, ζ )ρ(h) dh d(y, ζ ) = tr ν,E (Vν∗ B0 Vν ). k ν ν,E G Due to the way the principal symbol Bν,T ,0 enters on the right-hand side of (6.4), the conjugation with Vν (x, ξ )ρ(g) as well as the restriction to almost all (x, ξ, g) is inessential, δ (E, ) as → 0. so that again one concludes a vanishing of S2,ν Acknowledgement. We would like to thank M. Klein for drawing our attention to the paper [Sim80], as well as S. Teufel and G.-L. Panati for comments on an earlier version of the manuscript. Financial support by the Deutsche Forschungsgemeinschaft (DFG) under contracts no. Ste 241/15-1 and /15-2 is gratefully acknowledged.
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Bolte, J., Glaser, R.: Zitterbewegung and semiclassical observables for the Dirac equation. Preprint, 2004, available at arXiv:quant-ph/0402154 [BG00] Bolte, J., Glaser, R.: Quantum ergodicity for Pauli Hamiltonians with spin 1/2. Nonlinearity 13, 1987–2003 (2000) [BGK01] Bolte, J., Glaser, R., Keppeler, S.: Quantum and classical ergodicity of spinning particles. Ann. Phys. (NY) 293,1–14 (2001) [BK99a] Bolte, J., Keppeler, S.: A semiclassical approach to the Dirac equation. Ann. Phys. (NY) 274, 125–162 (1999) [BK99b] Bolte, J., Keppeler, S.: Semiclassical form factor for chaotic systems with spin 1/2. J. Phys. A: Math. Gen. 32, 8863–8880 (1999) [BN99] Brummelhuis, R., Nourrigat, J.: Scattering amplitude for Dirac operators. Commun. Part. Diff. Eq. 24, 377–394 (1999) [BR99] Bruneau, V., Robert, D.: Asymptotics of the scattering phase for the Dirac operator: High energy, semi-classical and non-relativistic limits. Ark. Mat. 37, 1–32 (1999) [BR02] Bouzouina, A., Robert, D.: Uniform semiclassical estimates for the propagation of quantum observables. Duke Math. J. 111, 223–252 (2002) [CdV85] Colin de Verdi`ere, Y.: Ergodicit´e et fonctions propres du laplacien. Commun. Math. Phys. 102, 497–502 (1985) [CFS82] Cornfeld, I.P., Fomin, S.V., Sinai, Ya.G.: Ergodic Theory, Grundlehren der mathematischen Wissenschaften, Vol. 245, Berlin, Heidelberg, New York: Springer-Verlag, 1982 [Cor82] Cordes, H.O.: A version of Egorov’s theorem for systems of hyperbolic pseudo-differential equations. J. Funct. Anal. 48, 285–300 (1982) [Cor83a] Cordes, H.O.: A pseudo-algebra of observables for the Dirac equation. Manuscripta Math. 45, 77–105 (1983)
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Cordes, H.O.: A pseudodifferential-Foldy-Wouthuysen transform. Commun. Part. Diff. Eqs. 8, 1475–1485 (1983) [Cor95] Cordes, H.O.: The Technique of Pseudodifferential Operators. London Mathematical Society Lecture Note Series, No. 202, Cambridge: Cambridge University Press, 1995 [Cor00] Cordes, H.O.: On Dirac observables. Progr. Nonlinear Diff. Eqs. Appl. 42, 61–77 (2000) [Cor01] Cordes, H.O.: Dirac algebra and Foldy-Wouthuysen transform. In: Evolution Equations and their Applications in Physical and Life Sciences, Lecture Notes in Pure and Applied Mathematics, Vol. 215, New York: Dekker, 2001, pp. 335–346 [CV71] Calder´on, P., Vaillancourt, R.: On the boundedness of pseudo-differential operators. J. Math. Soc. Japan 23, 374–378 (1971) [Dim93] Dimassi, M.: D´eveloppements asymptotiques des perturbations lentes de l’op´erateur de Schr¨odinger p´eriodique. Commun. Part. Diff. Eq. 18, 771–803 (1993) [Dim98] Dimassi, M.: Trace asymptotics formulas and some applications. Asymptot. Anal. 18, 1–32 (1998) [DS99] Dimassi, M., Sj¨ostrand, J.: Spectral Asymptotics in the Semi-Classical Limit. London Mathematical Society Lecture Notes, Vol. 268, Cambridge: Cambridge University Press, 1999 [Ego69] Egorov, Y.V.: The canonical transformations of pseudodifferential operators. Usp. Mat. Nauk 25, 235–236 (1969) [EW96] Emmrich, C., Weinstein, A.: Geometry of the transport equation in multicomponent WKB approximations. Commun. Math. Phys. 176, 701–711 (1996) [FGV90] Figueroa, H., Gracia-Bond´ıa, J.M., V´arilly, J.C.: Moyal quantization with compact symmetry groups and noncommutative harmonic analysis. J. Math. Phys. 31, 2664–2671 (1990) [HMR87] Helffer, B., Martinez, A., Robert, D.: Ergodicit´e et limite semi-classique. Commun. Math. Phys. 109, 313–326 (1987) [HS88] Helffer, B., Sj¨ostrand, J.: Analyse semi-classique pour l’´equation de Harper (avec application a` l’´equation de Schr¨odinger avec champ magn´etique). M´em. Soc. Math. France (N.S.) 116(34), (1988) [Ivr98] Ivrii, V.: Microlocal Analysis and Precise Spectral Asymptotics. Springer Monographs in Mathematics, Berlin, Heidelberg, New York: Springer-Verlag, 1998 [Kir76] Kirillov, A.A.: Elements of the Theory of Representations. Grundlehren der mathematischen Wissenschaften, Vol. 220, Berlin, Heidelberg, New York: Springer-Verlag, 1976 [Laz93] Lazutkin, V.F.: KAM Theory and Semiclassical Approximation to Eigenfunctions. Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 24, Berlin, Heidelberg, New York: SpringerVerlag, 1993 [LF91] Littlejohn, R.G., Flynn, W.G.: Geometric phases in the asymptotic theory of coupled wave equations. Phys. Rev. A 44, 5239–5256 (1991) [Nen02] Nenciu, G.: On asymptotic perturbation theory for quantum mechanics: Almost invariant subspaces and gauge invariant magnetic perturbation theory. J. Math. Phys. 1, 1273–1298 (2002) [NS01] Nenciu, G., Sordoni, V.: Semiclassical limit for multistate Klein-Gordon systems: almost invariant subspaces and scattering theory. Preprint, 2001 [PST03] Panati, G., Spohn, H., Teufel, S.: Space-adiabatic perturbation theory. Adv. Theor. Math. Phys. 7, (2003) [Rob87] Robert, D.: Autour de l’Approximation Semi-Classique. Progress in Mathematics, Vol. 68, Boston, Basel, Stuttgart: Birkh¨auser, 1987 [Sch01] Schubert, R.: Semiclassical localization in phase space. Ph.D. thesis, Universit¨at Ulm, 2001 [Sim80] Simon, B.: The classical limit of quantum partition functions. Commun. Math. Phys. 71, 247–276 (1980) [Spo00] Spohn, H.: Semiclassical limit of the Dirac equation and spin precession. Ann. Phys. (NY) 282, 420–431 (2000) [Str57] Stratonovich, R.L.: On distributions in representation space. Soviet Physics JETP 4, 891–898 (1957) [Zel87] Zelditch, S.: Uniform distribution of eigenfunctions on compact hyperbolic surfaces. Duke Math. J. 55, 919–941 (1987) [Zel96] Zelditch, S.: Quantum ergodicity of C ∗ dynamical systems. Commun. Math. Phys. 177, 502–528 (1996) [ZZ96] Zelditch, S., Zworski, M.: Ergodicity of eigenfunctions for ergodic billiards. Commun. Math. Phys. 175, 673–682 (1996) Communicated by P. Sarnak
Commun. Math. Phys. 247, 421–445 (2004) Digital Object Identifier (DOI) 10.1007/s00220-004-1066-y
Communications in
Mathematical Physics
G-Structures and Wrapped NS5-Branes Jerome P. Gauntlett1,2 , Dario Martelli1 , Stathis Pakis1 , Daniel Waldram1 1 2
Department of Physics, Queen Mary, University of London, Mile End Rd, London E1 4NS, U.K. Isaac Newton Institute for Mathematical Sciences, University of Cambridge, 20 Clarkson Road, Cambridge, CB3 0EH, U.K.
Received: 5 July 2002 / Accepted: 24 October 2003 Published online: 16 March 2004 – © Springer-Verlag 2004
Abstract: We analyse the geometrical structure of supersymmetric solutions of type II supergravity of the form R1,9−n × Mn with non-trivial NS flux and dilaton. Solutions of this type arise naturally as the near-horizon limits of wrapped NS fivebrane geometries. We concentrate on the case d = 7, preserving two or four supersymmetries, corresponding to branes wrapped on associative or SLAG three-cycles. Given the existence of Killing spinors, we show that M7 admits a G2 -structure or an SU (3)-structure, respectively, of specific type. We also prove the converse result. We use the existence of these geometric structures as a new technique to derive some known and new explicit solutions, as well as a simple theorem implying that we have vanishing NS three-form and constant dilaton whenever M7 is compact with no boundary. The analysis extends simply to other type II examples and also to type I supergravity. 1. Introduction Solutions of type II supergravity corresponding to NS fivebranes wrapping supersymmetric cycles provide an interesting arena for studying the holographic duals of supersymmetric Yang-Mills (SYM) theory [1]. Solutions, in the “near-horizon limit”, have now been found for a number of different cases [2]–[10]. In each case the final geometry is of the form R1,5−d × M4+d , where d is the dimension of the cycle on which the fivebrane is wrapped, and the NS three-form H and the dilaton are non-trivial. In [7] various aspects of the geometry of such supersymmetric solutions was elucidated. The key input is the existence of Killing spinors describing the preserved supersymmetries. In type II supergravity, the vanishing of the supersymmetry variation of the gravitino implies that any Killing spinor is parallel with respect to one of two connections ∇ ± with totally anti-symmetric torsion ± 21 H . This implies that M4+d admits certain geometric structures and the vanishing of the variation of the dilatino imposes additional conditions on the structures. It was shown in [7], following [11], that the resulting structures give rise to generalised calibrations [12].
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Here we would like to continue the investigations of [7], again assuming only the existence of particular sets of Killing spinors. Thus while motivated by considering solutions for wrapped NS five-branes, the results apply generally to supersymmetric backgrounds with non-trivial NS-flux H and dilaton. For definiteness we will focus on seven-dimensional geometries M7 , though it is clear that the analysis generalises to all cases discussed in [7]. Two distinct types of geometry arise. The first is when the Killing spinors are all parallel with respect to the same connection, say ∇ + . Geometrical structures of this type have been discussed in [13–18] as well as [7]. The second, new, type [7] is when there are some Killing spinors parallel with respect to ∇ + and some with respect to ∇ − . For our particular example, seven-dimensional geometries arise when fivebranes wrap associative three-cycles or SLAG three-cycles . The geometries are distinguished by the fact that the former is of the first type with a single Killing spinor parallel with respect to ∇ + . The latter case, on the other hand, is of the second type, preserving twice as much supersymmetry, with two Killing spinors ± , parallel with respect to ∇ ± respectively. For the generic case with a single Killing spinor, the seven-dimensional geometry admits a G2 -structure of a specific type, to be reviewed below. On the other hand, the seven-dimensional geometries with two Killing spinors ± admit two G2 -structures, or more precisely an SU (3) structure, again of a specific type to be discussed. Note that because of the non-vanishing intrinsic torsion the SU (3)-structure does not imply that the manifold is a direct product of a six-dimensional geometry with a one-dimensional geometry. However, there is a product structure which does allow the metric to be put into a canonical form with a six-one split as we shall discuss. The geometrical structures defined by the preserved supersymmetries can equivalently be specified by tensor fields satisfying first-order differential equations. These give a set of necessary conditions imposed by preservation of supersymmetry and the equations of motion. We will show that for the particular cases of G2 - and SU (3)-structures mentioned above, these conditions are also sufficient. From the derivation, it is clear that it should always be possible to find such a set of conditions. Note that, for the cases of single G2 - and Spin(7)-structures, this issue was also analysed in [16, 17] and [18] respectively, although these works did not consider the relationship between Killing spinors and the equations of motion, as we shall here. One of the motivations for this work was to see if these sufficient conditions just mentioned could provide a new method for constructing supersymmetric solutions describing wrapped fivebranes. This would provide an alternative to the “standard” two-step construction [19] of first finding a solution of D = 7 gauged supergravity and then uplifting to D = 10. We shall show that some known solutions can be recovered in this way. In addition to providing a direct D = 10 check of the solutions, this makes their underlying geometry manifest. We will also use the method to construct a new solution that describes a fivebrane wrapping a non-compact associative three-cycle. It is a co-homogeneity one solution with principle orbits given by SU (3)/U (1) × U (1). Recall that the solution of [6, 7] describing fivebranes wrapping SLAG three-cycles was argued to be dual to pure N = 2, D = 3 SYM. By analogy with [2, 3] it seems likely that there are more general solutions that would be dual to N = 2, D = 3 SYM with a Chern–Simons term. These seem difficult to find using gauged supergravity. Unfortunately they also seem to be difficult to obtain using the methods to be described here. In particular, to recover the known solution of [6, 7] one is first naturally led to partial differential equations, and it is somewhat miraculous that there is change of variables that leads one to the relatively simple solution obtained in [6, 7] using the gauged supergravity approach.
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The type II supergravity solutions for wrapped branes that are dual to quantum field theories are non-compact. As somewhat of an aside we also prove a simple vanishing theorem for compact manifolds. In particular, we show that the expression for the three-form H in terms of the G2 -structure allows one to prove that on a compact manifold without boundary given dH = 0, the three-form must necessarily vanish and the dilaton is constant. There are analogous expressions for H in terms of generalised calibrations for other fivebrane geometries [7], and hence this result generalises easily. The plan of the rest of the paper is as follows. We begin in Sect. 2 with some general discussion of G-structures and G-invariant tensors using G2 as our example. In Sect. 3 we review and extend what is known about the geometry with G2 structure that arises when type II fivebranes wrap associative three-cycles. We also prove the vanishing theorem for compact manifolds. Section 4 discusses the geometry that arises when fivebranes wrap SLAG three-cycles. In this case there are two G2 structures or equivalently an SU (3) structure. Section 5 uses the necessary and sufficient conditions for G2 -structures admitting Killing spinors as a technique to rederive some known solutions as well as deriving a new solution that describes a fivebrane wrapping a non-compact associative three-cycle. In Sect. 6 we use the analogous conditions for SU (3)-structures to derive BPS equations for solutions corresponding to fivebranes wrapped on SLAG three-cycles. We conclude in Sect. 7 by discussing how the results would extend to fivebranes wrapping other supersymmetric cycles as discussed in [7] and we also briefly comment on the extension to type I supergravity. 2. G-Structures In this section we review the notion of G-structure and G-invariant tensors on a Riemannian manifold M and the relation to intrinsic torsion and holonomy. Though the discussion is general, our examples will concentrate on the case relevant here of G2 -structure on a seven-manifold. Further details can be found, for example, in [20] and [21]. In general the existence of a G-structure on an n-dimensional Riemannian manifold means that the structure group of the frame bundle is not completely general but can be reduced to G ⊂ O(n). Thus, for G2 -structures on a seven-manifold, the structure group reduces to G2 ⊂ SO(7) ⊂ O(7). An alternative and sometimes more convenient way to define G-structures is via G-invariant tensors. A non-vanishing, globally defined tensor η is G-invariant if it is invariant under G ⊂ O(n) rotations of the orthonormal frame. Since η is globally defined, by considering the set of frames for which η takes the same fixed form, one can see that the structure group of the frame bundle must then reduce to G (or a subgroup of G). Thus the existence of η implies we have a G-structure. Typically, the converse is also true. Recall that, relative to an orthonormal frame, tensors of a given type form the vector space, or module, for a given representation of O(n). If the structure group of the frame bundle is reduced to G ⊂ O(n), this module can be decomposed into irreducible modules of G. Typically there will be some type of tensor that will have a component in this decomposition which is invariant under G. The corresponding vector bundle of this component must be trivial, and thus will admit a globally defined non-vanishing section η. To see how this works in the case of G2 -structures, consider the three-form on R7 given by φ0 = dx 136 + dx 235 + dx 145 − dx 246 − dx 127 − dx 347 − dx 567 ,
(1)
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where dx ij k = dx i ∧ dx j ∧ dx k and let g0 = dx12 + ... + dx72 denote the standard Euclidean metric. The group G2 can be defined as the subgroup of the O(n) symmetries of g0 which leaves φ0 invariant. A seven-dimensional manifold M7 then admits a G2 -structure if and only if there is a globally defined three-form φ on M7 which is G2 -invariant. That is, at each point on M7 we can consistently identify the three-form φ with the standard G2 -invariant three-form φ0 . Note that given φ0 we also have the metric g0 , an orientation dx 1...7 and the Hodge-dual four-form ∗φ 0 . Thus given a G2 -invariant φ on M7 we actually also get an associated metric g and four-form ∗φ on M7 such that (φ, ∗φ, g) are identified under the map to R7 with (φ0 , ∗φ 0 , g0 ). It will be useful to give explicitly some of the tensor decompositions in the G2 case. For instance, for two-forms one finds 2 = 27 ⊕ 214 ,
(2)
27 = {α ∈ 2 : ∗(φ ∧ α) = −2α} = {iβ φ : β ∈ T M},
(3)
where 214
= {α ∈ : ∗(φ ∧ α) = α}. 2
Recall that the space of two-forms 2 is isomorphic to the Lie algebra or adjoint representation so(7). Thus this decomposition is just the decomposition of so(7) under G2 , namely 21 → 14 + 7, where 14 is the Lie algebra g2 ∼ = 214 of G2 . There is, similarly, a decomposition of three-forms 3 = 31 ⊕ 37 ⊕ 327
(4)
following 35 → 1 + 7 + 27 under G2 ⊂ SO(7). Note that elements of the singlet 31 module are simply multiples of the G2 -invariant three-form φ. Riemannian manifolds with G2 -structures have been classified some time ago by Fernandez and Gray [22]. The idea is the same for any G-structure on a Riemannian manifold, as discussed for example in [21]. Given some G-invariant form η defining a G-structure, the derivative of η with respect to the Levi–Civita connection, ∇η, can be decomposed into G-modules. The different types of G-structures are then specified by which of these modules are present, if any. In more detail, one first uses the fact that there is no obstruction to finding some connection ∇ such that ∇ η = 0. Choosing one, then ∇ − ∇ is a tensor which has values in 1 ⊗ 2 . Since 2 ∼ = so(n) = g ⊕ g ⊥ , where g ⊥ is the orthogonal complement of the Lie algebra g in so(n), we conclude that ∇η = (∇ − ∇ )η can be identified with an element K of 1 ⊗ g ⊥ . Furthermore, this element is a function only of the particular G-structure, independent of the choice of ∇ . It is in one-to-one correspondence with what is known as the intrinsic torsion T0 . Explicitly, we have in components, acting on a q-form, ∇m ηn1 ...nq = −Km n1 p ηpn2 ...nq − Km n2 p ηn1 p...nq − · · · − Km nq p ηn1 ...nq−1 p ,
(5)
where for Km n p ∈ 1 ⊗ g ⊥ , the m and antisymmetric n, p indices label the one-form 1 and two-form g ⊥ ⊂ 2 modules respectively. In the G2 case, from the decomposition (2), we see that g2⊥ ∼ = 27 , while 1 is simply the 7 representation of G2 . Thus specifying ∇φ is equivalent to giving elements in the four G2 -modules in the decomposition of K, 7 × 7 → 1 + 7 + 14 + 27.
(6)
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Given the general relation (5) with η = φ, we see that dφ and d † φ ≡ − ∗ d ∗ φ pick out different parts in this decomposition. For example, following (2), the two-form d † φ contains the 7 + 14 pieces as follows: d † φ = iθ φ + α14 .
(7)
Here α14 ∈ 214 and θ , which corresponds to the 7, is called the Lee form and is given by 3θ ≡ ∗(d † φ ∧ ∗φ),
(8)
or in components θa = − 16 φabc ∇e φ ebc . Similarly, the four-form dφ can be decomposed into 1 + 7 + 27 pieces, and so contains all but the 14 in (6). Note, in particular, since it is derived from the same general expression (5), the 7 in this decomposition must be proportional to the Lee form defined in the decomposition of d † φ. It is clear from the above discussion that ∇ 0 ≡ ∇ + K canonically defines a connection for which ∇ 0 η = 0. It is the unique connection with torsion given by the intrinsic torsion T0 . Since the holonomy of this connection1 , and any connection ∇ for which ∇ η = 0, must stabilise η, we conclude that its holonomy, Hol(∇ 0 ), must be contained within G. On the other hand demanding that specific types of connection have holonomy in G, in general restricts the G-structure. For example, for the Levi–Civita connection to have Hol(∇) ⊆ G we require that all the elements in the decomposition of K vanish so that ∇η = 0. The G-structure is then said to be “torsionfree”. This is probably the most familiar case of G2 -structure. With torsion-free structure, so ∇φ = 0, (M7 , φ, g) is said to be a “G2 manifold”. It means that the Levi–Civita connection ∇ has holonomy contained in G2 and g is a Ricci-flat metric. Given the preceding discussion it is clear that the condition ∇φ = 0 is equivalent to requiring dφ = d∗φ = 0,
(9)
since all the relevant G2 -modules in ∇φ = 0 appear either in dφ or d∗φ. This equivalence has been exploited in [23] to provide a method for finding G2 holonomy metrics for manifolds of co-homogeneity one. The strategy is the following. Write down an ansatz for the associative three form φ in terms of several arbitrary functions of one radial variable. Find the associated metric and impose the conditions (9) to obtain a system of first-order differential equations for the arbitrary functions. Solving these one obtains a G2 holonomy metric. For type II supergravity solutions describing NS fivebranes wrapping supersymmetric three-cycles one finds seven manifolds with G2 structures of a different type [7] since the connection with holonomy in G2 is not the Levi–Civita connection ∇. This is reviewed in the next section. We can derive an analogous pair of necessary and sufficient conditions to (9). We will then exploit these, generalising [23], to find new solutions in a later section. 1 Here, and in the rest of the paper, when we are discussing holonomy we will assume that M is d simply connected, otherwise we consider the universal covering space.
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3. G2 -Structure and NS Fivebranes on Associative Three-Cycles The action for the bosonic NS-NS fields of type IIA or type IIB supergravity is given by 1 1 2 10 √ −2 2 S = 2 d x −ge R + 4(∇) − H , (10) 2κ 12 with H = dB. The corresponding equations of motion read 1 Rµν − Hµρσ Hν ρσ + 2∇µ ∇ν = 0, 4 1 2 ∇ − 2(∇)2 + Hµνρ H µνρ = 0, 12
(11)
∇ µ e−2 Hµνρ = 0.
As shown in [2], the IIB supergravity solution describing fivebranes wrapped on an associative three-cycle in a manifold of G2 -holonomy is of the form R1,2 × M7 , where M7 admits a single Spin(7) spinor satisfying 1 + np ∇m ≡ ∇m + Hmnp γ = 0, 8 (12) Hmnr γ mnr = −12∂n γ n , where ∇ + is a connection with totally antisymmetric torsion 21 H . Here the Spin(7) Dirac matrices γ m are imaginary, antisymmetric and satisfy {γm , γn } = 2δmn
(13)
with γ 1 · · · γ 7 = −i. From the existence of a single solution to the first equation in (12) it immediately follows that we have a G2 -structure on M7 and the Hol(∇ + ) = G2 . To see this, we first define a three form φ in terms of the Killing spinor by: φmnr = i T γmnr ,
(14)
where we have normalized the spinors to satisfy T = 1. This G2 -structure then satisfies ∇ +φ = 0
(15)
and hence Hol(∇ + ) ⊆ G2 . Using the second equation in (12) we find that the G2 -structure also satisfies − ∗ e2 d(e−2 φ) = H, d(e−2 ∗φ) = 0, dφ ∧ φ = 0.
(16)
That is, for a solution to the equations of motion to preserve supersymmetry, M7 must admit a G2 structure which satisfies the conditions (16) with H closed. This form of the conditions [7] naturally displays the connections with generalised calibrations [12]. A converse result has also been proved in [16, 17]. Let us summarise the idea behind it before extending it. One assumes the existence of a G2 -structure satisfying the last
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two equations in (16). Recalling the definition of the Lee-form introduced in (7) it is easy to see that d(e−2 ∗ φ) = 0 is equivalent to the statement that (i) the Lee form is given by θ = −2d and (ii) that α14 vanishes. It was shown in [16] that the second condition is the necessary and sufficient condition for the existence of a unique connection ∇ + = ∇ + 21 H , with totally anti-symmetric torsion 21 H , that preserves the three form φ and admits parallel spinors. The idea behind this is rather simple. First recall from (4) that H , which is in the 35 of SO(7), decomposes under G2 as 35 → 1 + 7 + 27. On the other hand as we discussed in Sect. 2, the different types of G2 structure correspond to the modules given in the decomposition (6) of K. It is thus clear that totally anti-symmetric torsion is associated with vanishing α14 in K. Moreover, it was shown in [17] that the G2 singlet piece in H is proportional to ∗(dφ ∧ φ) and vanishes if and only if the supersymmetry variation of the dilatino vanishes. The point is that the Clifford action of the 27 piece of H on vanishes. In other words it was proved in [17] that a manifold with a G2 structure (M7 , g, φ) admits solutions to (12) with varying dilaton and non vanishing NS three form H providing that the G2 structure satisfies: dφ ∧ φ = 0, ∗ (φ ∧ d † φ) = −2d † φ, θ = −2d,
(17)
or equivalently dφ ∧ φ = 0, d(e−2 ∗ φ) = 0.
(18)
The torsion of the unique connection with totally anti-symmetric torsion preserving the G2 structure is then given by H = − ∗ e2 d(e−2 φ). Note that supersymmetry alone is not sufficient to ensure that we have a solution to the type II field equations. We also need to impose at least the closure of H . In fact this is all we need as we now show using the integrability conditions of the Killing spinor equations (12). As shown in the Appendix these imply that 1 Rmn − Hmpq Hn pq + 2∇m ∇n γ n 4 1 1 = dHmnpq γ npq + e2 ∇ p e−2 Hpmn γ n (19) 12 2 and
1 Hmnp H mnp 12 1 1 = − dHmnpq γ mnpq − e2 ∇ m e−2 Hmnp γ np . 48 4
∇ 2 − 2[∇]2 +
(20)
The assumptions on the G2 structure (18) mean that the H equation of motion (11) is automatically satisfied. We thus immediately conclude from (20) that, if we also impose dH = 0, then the dilaton equation of motion is satisfied. The other equation (19) is then of the form Amn γ n = 0 which implies Amn Amn = 0. On a Riemannian manifold we then deduce Amn = 0 which is precisely the Einstein equations.
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In summary, we have shown that a solution of the equations of motion of the form R1,2 × M7 admits a single Killing spinor if and only if dφ ∧ φ = 0, d(e−2 ∗ φ) = 0, dH = 0,
(21)
where H = − ∗ e2 d(e−2 φ). This result is the analog of (9) for G2 -manifolds and in principle provides a method for finding new supersymmetric solutions with non-zero H . One starts with an ansatz for φ, finds the associated metric and imposes these equations to obtain, in the case of a metric of co-homogeneity one, ordinary differential equations for the arbitrary metric functions. We give examples of this technique in Sect. 5. It is interesting to note that the expression for H implies a simple vanishing theorem2 : on a compact manifold without boundary the only solutions to (21) have H = 0 and constant, that is, M7 is a G2 -manifold. To see this, we first note, given the expression for H , −2 e H ∧ ∗H = − H ∧ d(e−2 φ) = 0, (22) M7
M7
where in the final equation we integrate by parts and use dH = 0. Since the first integrand is positive definite, we conclude that H = 0. Integrating the dilaton equation of motion then implies by a similar argument that d = 0, so is constant. The conditions (21) then reduce to dφ = d∗φ = 0 which imply that M7 is a G2 -manifold. In [7] we derived analogous expressions for H in terms of the generalised calibrations for other geometries in dimensions six and eight arising when fivebranes wrap calibrated cycles. Since only this expression and the equation of motion entered the above argument, clearly this theorem easily generalises. For all compact supersymmetric manifolds M without boundary, the flux vanishes H = 0 and is constant. 4. SU (3)-Structure and NS Fivebranes on SLAG Three-Cycles It was shown in [7] that the type II supergravity solutions describing fivebranes wrapping SLAG three-cycles are also of the form R1,2 × M7 , where now M7 admits a pair of Spin(7) spinors ± satisfying 1 ± ± np ∇m ≡ ∇m ± Hmnp γ ± = 0, 8 (23) Hmnr γ mnr ± = ∓12∂n γ n ± , where ∇ ± are two connections with anti-symmetric torsion ± 21 H . From the discussion of the previous section, it follows that the two spinors ± define two distinct G2 -structures of the type characterised by the conditions (21). The two connections ∇ ± have holonomy contained in two different G2 subgroups of SO(7). The G2 -invariant three forms are constructed from the Killing spinors ± φmnr = i ±T γmnr ± ,
where we have again chosen the normalization ±T ± = 1, as we can always do. 2
A different vanishing theorem was proved in [24], which assumed vanishing dilaton.
(24)
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The appearance of two G2 -structures is again quite general, depending only on the requirement that there are two distinct solutions + and − . We can analyse this structure further, making only one further assumption, as in [7], that the Killing spinors are orthogonal to each other, i.e. +T − = 0. We expect that this should cover the general class of supersymmetric solutions describing fivebranes wrapped on SLAG three-cycles. This is because we can deduce what projections we expect to be imposed on the preserved supersymmetries by considering the supersymmetries preserved by a fivebrane probe wrapped on a SLAG three-cycle, as described in [7]. Note in particular, that this condition was satisfied for the specific supersymmetric supergravity solutions of [6, 7]. It is equivalent to the statement that the two G2 structures satisfy + r1 r2 − φ n]r1 r2 = 0, φ[m
(25)
as can be shown by Fierz rearrangement. Apart from the two G2 three-forms we can also construct various other forms using the Killing spinors Kn = i +T γn − , ωmn = +T γmn − ,
(26)
χmnr = i +T γmnr − . These are the basic objects in the sense that the two G2 structures can be constructed from K, ω and χ as follows: φ ± = ±iK ∗ χ − K ∧ ω.
(27)
K, ω, χ satisfy a series of algebraic relations that follow from Fierz rearrangements. First, given the normalization of the Killing spinors, we find that: Km K m = 1, ωmn ωmn = 6, χmnp χ mnp = 24,
(28)
and also ωm r ωr n iK χ iK ω (iK ∗ χ )mnr
n = −δm + Km K n , = 0, = 0, = ωm l χnrl = ω[m l χnr]l .
(29)
In addition, we can calculate the covariant derivatives of these forms using the first Killing spinor equation to get 1 Hml1 l2 χn l1 l2 , 4 1 = Hml1 l2 ∗χ n1 n2 l1 l2 , 4 3 1 = − Hm[n1 n2 Kn3 ] − Hml1 l2 ∗ωn1 n2 n3 l1 l2 . 2 4
∇m Kn = ∇m ωn1 n2 ∇m χn1 n2 n3
(30)
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From the dilatino equations we then deduce the following relations for the exterior derivatives of the forms d(e− K) = 0, d(e− ω) = 0, e d(e− χ ) = −H ∧ K,
(31)
as well as for the G2 -structures as in (16), e2 d(e−2 φ ± ) = ∓ ∗ H.
(32)
It is also not difficult to show in addition that d(iK ∗ χ ) ∧ iK ∗ χ = 0, d(iK ∗ χ ) ∧ K ∧ ω = 0.
(33)
4.1. SU (3) structure. Let us now discuss what the presence of these three invariant forms K, ω and χ implies about the type of G-structure we have on M7 . Recall that the existence of φ ± , or equivalently ± , implied that there were two distinct G2 -structures. Of course there can only be one actual structure group G of the frame bundle, so the implication is that G must be a common subgroup of these two distinct embeddings of G2 in SO(7). The largest possible such group is SU (3). This is consistent with the existence of two Killing spinors since the 8 spinor of Spin(7) includes two singlets in its decomposition 1 + 1 + 3 + 3¯ under SU (3). Thus we might expect that in fact there is actually an SU (3)-structure on M7 . By considering each of the invariant forms K, ω and χ in turn we will see that this is indeed the case. Each will further restrict the G-structure until we are left with SU (3). We start with K. Clearly, a fixed vector is left invariant by SO(6) ⊂ SO(7) rotations of the orthonormal frame. Thus we see that K defines an SO(6)-structure. Equivalently we can introduce m n = 2Km K n − δm n
(34)
satisfying 2 = δ, since K 2 = 1, and hence defining an almost product structure. It is metric compatible in the sense that gT = g, or equivalently mn = m r grn is symmetric. It is also integrable in that its Nijenhuis tensor defined by Nmn r = m k ∂[n k] r − n k ∂[m k] r
(35)
vanishes using d(e− K) = 0. This implies that we in fact have a product structure. It follows that we can find coordinates such that is diagonal, or equivalently K = e dx 7 . In these coordinates the seven-dimensional metric takes the form ds72 = gab dx a dx b + e2 dx72 .
(36)
Note that the geometry is not a direct product since gab and are allowed to depend on all the coordinates. The metrics of the solutions presented in [6, 7] indeed have this form, as we shall show in Sect. 6. Now consider ω. The pair (K, ω) define what is known as an almost contact metric structure (see for example [25]). This means, in general, that the structure group of the frame bundle on a manifold M2k+1 reduces from SO(2k + 1) to U (k), implying here
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that we have an U (3)-structure. It is the analog of an almost hermitian structure for odd-dimensional manifolds. A manifold M2k+1 is said to have an almost contact metric structure if it admits a (1, 1) tensor ωm n and a one-form K satisfying the first equation of (29), and furthermore ω is metric compatible so that ωgωT = g, or equivalently ωmn = ωm r grn is a two form. Note that this implies iK ω = 0. Essentially, the existence of K allows one to consistently decompose the tangent space into 2k-dimensional piece and a one-dimensional piece. The two-form ω then defines an almost hermitian structure on the 2k-dimensional piece, so that the corresponding complexified tangent space splits into the sum of a k-dimensional complex space and its complex conjugate. Thus, in general, we have the decomposition T M2k+1 ⊗ C = T 1,0 ⊕ T 0,1 ⊕ (TK ⊗ C). There is an integrability condition similar to that of an almost hermitian structure and if it is integrable the almost contact metric structure is called normal. In the geometries we are interested in the structure is not integrable in general. It is interesting to note that restricting to the six-dimensional part of the metric (36) by setting x7 constant we obtain a conventional almost hermitian structure. However this is again not integrable and the six-dimensional manifold is not a complex manifold. This is perhaps surprising since such solutions describe fivebranes wrapped on SLAG three cycles in Calabi-Yau manifolds and hence one might naively have expected that the six-dimensional part of the geometry is complex. Finally we come to χ . We first note that we can define a complex three-form ϑ = χ − i(iK ∗χ ).
(37)
We see that ϑ is normal to K, and also, from the third equation of (29), that it is a (3, 0)-form with respect to the almost contact structure (K, ω). In other words it is a section of 3 T 1,0 . In this sense it is the analog of the holomorphic three-form on the original Calabi–Yau manifold. In fact, in an exactly analogous way, it is easy to see that the subgroup of U (3) which preserves ϑ is SU (3). Thus we conclude that we do indeed have an SU (3)-structure on M7 . Since d(e− K) = d(e− ω) = 0 it follows that ∗H = e2 d(e−2 Im ϑ)
(38)
which shows that ϑ is a generalized calibration. This mirrors the fact that for a CalabiYau 3-fold, one can always choose the phase of the holomorphic three-form such that the imaginary part calibrates a given SLAG three-cycle.
4.2. The necessary and sufficient conditions. We have shown that given two spinors ± satisfying (23), with +T − = 0, M7 necessarily has an SU (3)-structure given by (K, ω, χ). The structure is not general, but as in the case of G2 -structure above is restricted. We have seen already that, for instance, the almost product structure defined by K is integrable, though the almost contact structure (K, ω) is not. In general, we showed that we have the conditions dK = d ∧ K, dω = d ∧ ω, dχ ∧ K = d ∧ χ ∧ K, d(iK ∗ χ ) ∧ iK ∗ χ = 0.
(39)
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These are also sufficient conditions for the existence of a solution to (23). To see this one needs to show that φ ± defined by (27) satisfy (25), which follow from the algebraic properties of (K, ω, χ ) and also that each satisfy the conditions (21). The latter is straightforward to show using the fact that ∗φ ± = ±χ ∧ K − 21 ω ∧ ω. In other words, a spacetime of the form R1,2 × M7 admits two orthogonal Killing spinors ± satisfying (23), if and only if we have an SU (3)-structure on M7 satisfying the above conditions (39). In addition, the antisymmetric torsion ± 21 H is given by (38). Furthermore, this gives a supersymmetric solution of type II supergravity if and only if we impose in addition the closure of H as before. Once again we can in principle use this result as a method for finding solutions. In Sect. 6 we shall recover the solutions that were found in [6, 7] using gauged supergravity techniques. 5. Constructing Solutions with a Single G2 Structure In this section we will use the results of Sect. 3 to construct examples of the geometries with a single G2 -structure that were described there. These correspond to fivebranes wrapping associative three-cycles. We focus on co-homogeneity one manifolds. We generalise the method presented in [23]. One first makes an ansatz for the G2 structure φ satisfying appropriate symmetries and then finds the associated metric from the expression: gij = (det sij )−1/9 sij , (40) 1 φin1 n2 φj n3 n3 φn5 n6 n7 n1 n2 n3 n4 n5 n6 n7 , 1234567 = 1. sij = 144 The three form φ must be stable in the sense of [31] to ensure that it is generic enough to make the metric non-degenerate. We then impose Eq. (18). If these are satisfied we have a solution to (12). One must then impose the closure of H to obtain a solution to the full supergravity theory. 5.1. The example of [2]. Let us first demonstrate this method by recovering the example presented in [2]. This is a co-homogeneity one example with principle orbits S 3 × S 3 . Our starting point is an ansatz for the three-form that has appeared in constructions of G2 holonomy metrics in [26, 27]: 3 1 a3 a − σa ∧ σa + abc σa ∧ σb ∧ σc φ = ab dt ∧ 2 3! a=1 ab2 1 1 abc σa ∧ b − σb ∧ c − σc , (41) − 2! 2 2 where (a , σa ) are left invariant one-forms on SU (2) × SU (2), satisfying dσ1 = −σ2 ∧ σ3 , d1 = −2 ∧ 3 plus cyclic permutations, and t is a radial variable. The two arbitrary functions a and b depend on the radial variable only. The associated metric is given by 2 3 3 1 a − σa + a 2 (σa )2 . (42) ds72 = dt 2 + b2 2 a=1
a=1
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Introduce a frame et = dt, 1 a e = b a − σa , 2 e˜a = a σa .
(43)
˜˜˜
In this frame the three form φ and its dual are given by (with t123123 = −1) φ = et ∧ ea ∧ e˜a +
1 1 abc e˜a ∧ e˜b ∧ e˜c − abc e˜a ∧ eb ∧ ec , 3! 2!
(44) 1 1 1 a t a b c t a b c a b ∗φ = − abc e ∧ e ∧ e ∧ e + abc e ∧ e ∧ e˜ ∧ e˜ + e˜ ∧ e ∧ e˜ ∧ eb , 3! 2! 2! and it is straightforward to calculate: 1 b dφ = (log a) − 2 abc et ∧ e˜a ∧ e˜b ∧ e˜c 2 4a 1 1 2 − (log b a) − abc et ∧ e˜a ∧ eb ∧ ec , 2 b b 1 d ∗ φ = (log ab) − − 2 et ∧ e˜a ∧ ea ∧ e˜b ∧ eb . 2b 8a
(45)
First note that dφ ∧φ = 0 is automatically satisfied. Also we have d ∗φ = d(2)∧∗φ with:
1 b 2 2 d(2) = (log a b ) − − 2 dt. (46) b 4a So all the conditions (17) are satisfied and we have a solution to the Killing spinor equations (12). Note that the two functions a, b are still arbitrary. This is because we started with a very special ansatz which guaranteed from the beginning that all the conditions were satisfied. However we still need to impose the closure of H . This will give us second-order equations in principle but as we shall see, in this case they are trivially integrated once. The torsion H is constructed from (16) and we find that: H = where
1 1 abc F ea ∧ eb ∧ ec + abc G ea ∧ e˜b ∧ e˜c , 3! 2! 1 b F = (log b2 a −1 ) − + 2 , b 2a b G = −(log a) + 2 . 4a
(47)
(48)
Imposing dH = 0 we get the equations F + F (log b3 ) = 0,
(49)
(50)
2
G + G(log ba ) = 0, 1 b G − F 2 = 0. 2b 8a
(51)
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The first two are trivially integrated to give F b3 = C1 and Gba 2 = C2 while the third implies that C1 = 4C2 ≡ −µ. Using the definitions of F and G we thus arrive at a system of first order equations for the metric functions a, b: b2 µ 1 b = (52) 1− 2 , 1− 2 2 b 4a µ b a = (53) 1+ 2 . 4a b These equations are precisely those derived at the end of Sect. 3.1.1 of [2]. For the special case µ = 0 the torsion and the dilaton vanish and the equations can be integrated to recover the √ G2 holonomy metric on the spin bundle of S 3 [28]. The solution with b2 = µ, a 2 = µt was found in [2] using gauged supergravity methods. It corresponds to fivebranes wrapped on the associative three-sphere of the G2 -holonomy manifolds of [28], in the near horizon limit. The general solution of these equations remains an outstanding problem. One can extend this analysis in a relatively straightforward way to recover the solution first presented in [3], but the formulae are rather lengthy so we shall not present the details here.
5.2. New solution. Another example is to start with a cohomogeneity one manifold with principal orbits SU (3)/U (1) × U (1). Such G2 structures have appeared in [29, 30], and solutions have been found for a G2 metric on the R3 bundle over CP2 [28]. Here we use the results of [29] to find and solve the BPS equations for solutions that describe fivebranes wrapped on the R3 fibres, which are non-compact associative three cycles, in such G2 manifolds. Let {ea } be the left invariant one forms on SU (3). We define ω1 = e12 , ω2 = e34 , ω3 = e56 ,
(54)
and also a basis for the SU (3) invariant three forms: α = e246 − e235 − e145 − e136 , β = e135 − e146 − e236 − e245 ,
(55)
where e12 ≡ e1 e2 , etc., and the exterior product of forms is understood. It then follows that these satisfy [29] dω1 = dω2 = dω3 =
1 α, 2
dα = 0, dβ = −2(ω1 ∧ ω2 + ω2 ∧ ω3 + ω3 ∧ ω1 ), d(ωi ∧ ωj ) = 0, i = j.
(56)
The G2 structure and its associated metric are given by: φ = (f12 ω1 + f22 ω2 + f32 ω3 ) ∧ dt + f1 f2 f3 (cos θ α + sin θ β),
(57)
ds72 = dt 2 + f12 g1 + f22 g2 + f32 g3 ,
(58)
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where fi , θ are arbitrary functions of t and g1 = e12 + e22 , g2 = e32 + e42 , g3 = e52 + e62 .
(59)
∗φ = f22 f32 ω2 ∧ ω3 + f32 f12 ω3 ∧ ω1 + f12 f22 ω1 ∧ ω2 + f1 f2 f3 (cos θ β − sin θ α) ∧ dt
(60)
We find that
and hence
dφ =
1 2 (f1 + f22 + f32 ) − (f1 f2 f3 cos θ) α ∧ dt 2 − (f1 f2 f3 sin θ ) β ∧ dt − 2f1 f2 f3 sin θ (ω1 ∧ ω2 + ω2 ∧ ω3 + ω3 ∧ ω1 ),
d ∗ φ = (f22 f32 ) − 2f1 f2 f3 cos θ ω2 ∧ ω3 ∧ dt
+ (f32 f12 ) − 2f1 f2 f3 cos θ ω3 ∧ ω1 ∧ dt
+ (f12 f22 ) − 2f1 f2 f3 cos θ ω1 ∧ ω2 ∧ dt.
(61)
(62)
Let us first consider the equation d ∗ φ = d(2) ∧ ∗φ. This gives: = log(fi fj ) −
fk cos θ, i = j = k fi fj
(63)
which defines the dilaton and also imposes: fk log fi + cos θ
fj fi = fk log fj + cos θ , i = j = k fj fi
(64)
which gives two independent equations. Next we impose φ ∧ dφ = 0 and we conclude that: θ = − sin θ
f12 + f22 + f32 . f1 f2 f3
(65)
Having satisfied these conditions we need to impose the closure of H . We find that the torsion is given by: f32 f12 f22 H = 2f1 f2 f3 sin θ ω3 + 2 2 ω1 + 2 2 ω2 ∧ dt f12 f22 f2 f3 f1 f3
− 2 f1 f2 f3 sin θ − (f1 f2 f3 sin θ ) α
1 − (f1 f2 f3 cos θ ) − 2 f1 f2 f3 cos θ − (f12 + f22 + f32 ) β. (66) 2 For H to be closed we thus need to impose the following: 1 (f1 f2 f3 cos θ ) − 2 f1 f2 f3 cos θ − (f12 + f22 + f32 ) = 0, 2
(67)
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[2 f1 f2 f3 sin θ − (f1 f2 f3 sin θ ) ] + f1 f2 f3 sin θ
f32 f12 f22
+
f12 f22 f32
+
f22 f12 f32
= 0. (68)
Now (68) is a second-order equation and also we have five equations for four unknown functions. However the second-order equation follows from the four first-order equations so there is no inconsistency. To see this we first rearrange Eq. (64),(65),(67), and write them as: fi2 1 f12 + f22 + f32 − cos θ , i = 1, 2, 3, 2 cos θ f1 f2 f 3 f 1 f2 f 3 f 2 + f22 + f32 θ = − sin θ 1 . f 1 f2 f3
(log fi ) =
(69)
Then we note that we can write (68) as : 1 (tan θ (f12 + f22 + f32 )) + f1 f2 f3 sin θ 2
f32 f12 f22
+
f12 f22 f32
+
f22 f12 f32
= 0. (70)
This is satisfied given (69). So indeed we have arrived at a system of BPS equations for the four unknown functions. To solve the BPS equations we first define a new radial variable by dt = f1 f2 f3 dλ. In terms of this the equations become: 1 2 d fi − cos θfi2 , (log fi ) = dλ 2 cos θ i dθ fi2 . = − sin θ dλ
(71)
i
Define ui = fi2 tan θ, then using the above we find: d(u−1 cos2 θ i ) =2 , dλ sin θ dθ ui . = − cos θ dλ
(72)
i
Now define another radial variable by dρ = (2 cos2 θ/ sin θ)dλ. Now in terms of this we can solve for ui and then for sin θ. We find that: 1 , ρ − αi sin θ = (Mq(ρ, αi ))−1/2 , ui =
(73)
where q(ρ, αi ) ≡ i (ρ − αi ).The αi and M are four arbitrary integration constants. By rescaling the radial coordinate we find that the solution takes the form:
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q − M2 ds = + gi , ρ − αi 4 q − M2 i M2 e2 = e20 1 − , q ωi 1 1 H =M ∧ dρ − α . (ρ − αi )2 2 ρ − αi 2
dρ 2
i
(74)
i
In the limit M → 0 the torsion vanishes, the dilaton tends to a constant and we recover the metric of [30]. This is a G2 holonomy metric with a conical singularity for generic values of the αi but is regular when two of these constants are equal. In this case one obtains the G2 holonomy metric on the R3 bundle over CP2 [28]. For non-zero M the torsion is non-vanishing and in the large ρ limit the solution approaches the one in [30]. In the interior we see that the radial variable is constrained by ρ ≥ ρ0 , where ρ0 is the solution of q − M 2 = 0. Note that we always have ρ0 ≥ αi . At ρ = ρ0 the metric is singular for all values of αi . When M = 0 the G2 holonomy manifolds do not have any compact associative three-cycles on which to wrap a fivebrane, but they do have non-compact associative three-cycles. In the example of the G2 -holonomy metric on the R3 bundle over CP2 there is a co-associative CP2 bolt, and the R3 fibres are non-compact and associative. It is thus natural to interpret the solutions with M = 0 as describing fivebranes wrapping such a non-compact associative three-cycle, in the near horizon limit. Finally, we point out that it should be very straightforward to generalise the solutions in this section to co-homogeneity one solutions where the principle orbits are CP3 . These include the G2 holonomy metric on the R3 bundle over S 4 [28].
6. Recovering the Solution of [6, 7] In this section we will use a similar procedure to recover the solutions of [6, 7]. In so doing we will explicitly demonstrate the SU (3) structure of these solutions. We first introduce a set of one-forms as in [7]: ν 1 = dθ, ν 2 = sin θ dφ, σ1 σ2 S 1 = cos φ − sin φ , 2 2 3 σ1 σ2 σ S 2 = sin θ − cos θ sin φ + cos φ , 2 2 2 σ3 σ1 σ2 S 3 = − cos θ − sin θ sin φ + cos φ , 2 2 2
(75)
where θ, φ are angles on a two-sphere and σ a are the usual left-invariant one-forms on SU (2) satisfying dσ a = 21 abc σ b ∧ σ c . These satisfy the exterior algebra:
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dS 1 dS 2 dS 3 dν 1 dν 2
= 2S 2 ∧ S 3 + ν 2 ∧ S 3 + A ∧ S 2 , = 2S 3 ∧ S 1 − ν 1 ∧ S 3 − A ∧ S 1 , = 2S 1 ∧ S 2 + ν 1 ∧ S 2 − ν 2 ∧ S 1 , = 0, = −A ∧ ν 1 ,
(76)
where A = cos θ dφ. We introduce a frame: er ea e˜a e3 e7
= a(r, x7 )dr, = b(r, x7 )S a = c(r, x7 )(ν a + S a ), = b(r, x7 )S 3 , = e dx7 ,
(77)
where a = 1, 2, and make an ansatz for the SU (3) invariant forms: K
= e7 ,
ω
= er ∧ e3 + e1 ∧ e˜2 − e2 ∧ e˜1 ,
χ
= er ∧ (−e1 ∧ e2 + e˜1 ∧ e˜2 ) − e3 ∧ (e1 ∧ e˜1 + e2 ∧ e˜2 ),
(78)
iK ∗ χ = er ∧ (e1 ∧ e˜1 + e2 ∧ e˜2 ) − e3 ∧ (e1 ∧ e2 − e˜1 ∧ e˜2 ) corresponding to the metric: ds 2 = a 2 dr 2 + b2 d23 + c2
(ν a + S a )2 + e2 dx72 ,
(79)
a=1,2 ˜ ˜
where the orientation is taken to be r311227 = −1. The following identities are useful: 1 dα = − dβ = S 3 ∧ γ , 2 dγ = 2S 3 ∧ β,
(80)
α = S1 ∧ S2, β = S1 ∧ ν2 − S2 ∧ ν1 + ν1 ∧ ν2, γ = S1 ∧ ν1 + S2 ∧ ν2.
(81)
where
Using these we can write ω
= ab dr ∧ S 3 + bc dS 3 ,
cb2 χ = a(c2 − b2 )dr ∧ α + ac2 dr ∧ β + dβ, 2 bc2 iK ∗ χ = abc dr ∧ γ + b(c2 − b2 )S 3 ∧ α + dγ . 2
(82) (83) (84)
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Imposing the necessary and sufficient conditions discussed in Sect. 4.2 we find the following equations must be imposed: ∂x7 (e− ab) = 0, ∂x7 (e− bc) = 0, ∂r (e− bc) = e− ab, ac , ∂r b = b
(85)
and that the torsion is given by H = F1 dα + F2 dr ∧ α + F3 dr ∧ β + F4 dx7 ∧ α + F5 dx7 ∧ β,
(86)
where F1 = −b2 ce− ∂x7 log(abc e−2 ), F2 = −ae− [b2 ∂x7 log(bc2 e−2 ) − c2 ∂x7 log(b3 e−2 )], F3 = ac2 e− ∂x7 log(b3 e−2 ), 2
b c2 2b2 2c3 F4 = e ∂r log(bc2 e−2 ) − ∂r log(b3 e−2 ) − + 2 , a a c b 2
3 c 2c F5 = e − ∂r log(b3 e−2 ) + 2 . a b
(87)
Now imposing closure of H we find that ∂r F1 + 2F3 − F2 = 0, ∂x7 F1 + 2F5 − F4 = 0, ∂x7 F2 − ∂r F4 = 0,
(88)
∂x7 F3 − ∂r F5 = 0. Using the first-order equations (85) we find that the above equations reduce to the single second-order equation abe− ∂x27 b + ∂r (ce ) = 0.
(89)
Now (85) imply that abe− = h(r) and by choosing the radial coordinate appropriately we can set h ≡ 1. Then the rest of the equations determine the dilaton and a, c in terms of b via: b ∂r b, r c2 = a 2 r 2 , e = ab a2 =
(90)
where b satisfies the second-order non-linear pde: ∂2 3 ∂2 b +3 b = 0. 2 ∂r ∂x7 2
(91)
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We recover the solution of [7] by making a change of variables to (z, ψ) such that : √ r = zB(z) sin ψ, x7 = A(z) cos ψ, (92) where A(z) and B(z) satisfy A (z) = B(z), B (z) = A(z) −
B(z) , 2z
(93)
so that z1/4 A(z) = √ (I−1/4 (z) + µ K1/4 (z)), 2 z1/4 B(z) = √ (I3/4 (z) − µ K3/4 (z)). 2
(94)
Here µ is an integration constant. The solution is then just b2 = z.
(95)
Thus we have explicitly demonstrated the SU (3) structure of the solution found in [6, 7]. It seems a formidable challenge to find the general solution of (91). Let us just note that it is easy to construct solutions which do not depend on x7 . We then have b = (λ1 r + λ2 )1/3 .
(96)
These solutions might be interpreted as solutions corresponding to wrapped NS fivebranes that are smeared over the x 7 direction. Note in particular, that the torsion is non-vanishing for any choice of the constants λ1 , λ2 so that we do not recover the pure geometry CY3 × S 1 , where CY3 is the deformed conifold, as one might have expected. The reason for this is simply that a more general ansatz for the SU (3) structure is required. Enlarging our ansatz would also allow one in principle to obtain more general wrapped NS fivebrane solutions as well, but we expect that the pdes will be intractable without further insight. 7. Discussion We have analysed supersymmetric type II geometries of the form R1,5−d × Md+4 with non-trivial NS three-form flux and dilaton, motivated by the fact that the near-horizon limits of wrapped NS fivebranes geometries are of this type. In particular, we considered the examples of seven-dimensional manifolds arising from branes wrapped on associative or SLAG three-cycles. These geometries admit a G2 or an SU (3) structure, respectively, of a specific type that we determined. We also proved a converse result, namely that given such a geometric structure then one obtains a supersymmetric solution to the equations of motion. We used the converse result as a method to construct solutions. Note that for both cases the group G in the G-structure is exactly the same as that of the underlying special holonomy group of the manifold containing the supersymmetric cycle on which fivebrane is wrapped.
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It is straightforward to extend the results for these specific examples to the geometries arising when type II NS5-branes wrap other supersymmetric cycles. In [7] we analysed the holonomy of the connections ∇ ± that would arise in each case, and a summary appeared in Table 1 of that reference. In n dimensions, in the cases where just one of the connections ∇ ± has special holonomy G ⊂ SO(n), the geometry is specified by a G-structure of a type that can be easily specified, by following the discussion of Sect. 2 (for related work see [13–15, 18]). In the cases where both ∇ ± have special holonomy contained in G ⊂ SO(d) say, we find that the manifolds admit a G ⊂ G -structure. For both cases, one finds that the group G of the G-structure that appears in the final geometry is exactly the same as the special holonomy group of the manifold, just as for the examples explicitly discussed in this paper. For example, D = 6 geometries can arise when NS fivebranes wrap K¨ahler two-cycles in Calabi-Yau three-folds or two-folds. In the former case one of the connections ∇ ± has special holonomy SU (3) while the other has general holonomy SO(6). The resulting geometry has an SU (3)-structure which was discussed in [13–15]. Examples of this geometry were presented in [1]. On the other hand when NS-fivebranes wrap two-cycles in a Calabi-Yau two-fold we find that both connections ∇ ± have SU (3)-holonomy. In this case the structure group of the six-manifold is in fact SU (2). This structure includes a product structure which allows one to choose co-ordinates with a four-two split to the metric, but it is not a direct product. Examples of this kind of geometry were presented in [4, 5]. Finally, it is also worth mentioning that much of the discussion applies to type I supergravity. The action and supersymmetry transformations are recorded in the Appendix. For this case there is only a single connection with totally anti-symmetric torsion ∇ + and so supersymmetry will just give rise to a single G-structure. Consider for example the D = 7 case. Since the variation of the dilatino and gravitino for the type I theory are the same as for + we deduce that the G2 structure is exactly the same as that discussed in Sect. 3. In addition we need to ensure the vanishing of the supersymmetry variation of the gaugino Fmn γ mn = 0.
(97)
This implies, following [32] that F must satisfy the G2 instanton equation Fmn = 1 pq 2 ∗φ mn Fpq , i.e. the two-form F is in the 14 in the decomposition (2). This is the type of geometry dictated by supersymmetry that would arise when type I fivebranes wrap associative three-cycles and also SLAG three-cycles. To obtain a solution to the equations of motion for type I supergravity we have to solve dH = 2α Tr F ∧ F . Using the integrability conditions given in the appendix it is clear that these conditions are also sufficient to obtain a supersymmetric solution to the equations of motion, by generalising the argument in Sect. 2. As yet there are no known solutions of this kind with non-vanishing F . Such solutions would have the geometry naturally expected for type I or heterotic “gauge” fivebranes [33] wrapping associative three-cycles. For the case of type I fivebranes wrapping SLAG three-cycles the interesting possibility arises that there will in fact be an SU (3) structure, despite the fact that it is not dictated by supersymmetry alone. It is interesting to note that the type II solutions give rise to type I solutions with F = 0. These correspond to type I or heterotic “neutral” fivebranes [33] wrapping associative three-cycles. The type II solutions corresponding to fivebranes wrapping SLAG three-cycles thus give rise to type I solutions describing type I fivebranes wrapping SLAG three-cycles that have an SU (3) structure. This is some evidence that this will also occur for wrapped gauge-fivebranes.
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In type I or heterotic string theory anomaly cancellation implies that the Bianchi identity is modified by higher-order corrections in α . To leading order this is most informatively written as dH = 2α Tr[F ∧F −R(− )∧R(− )], where − = ω−H /2 [34]. This should be viewed as implicitly defining H . One can ask whether one can solve this for wrapped branes by identifying A with − as this would be the analogue of “symmetric fivebranes” [33]. For supersymmetric fivebranes wrapping associative three cycles, only + = ω + H /2 has holonomy contained in G2 and hence identifying A with − would not be supersymmetric. Interestingly, for supersymmetric fivebranes wrapping SLAG three-cycles both ± have holonomy contained in G2 and hence one can obtain supersymmetric solutions for these cases. More explicitly, the solution constructed in [7] automatically gives a solution of the heterotic or type I string theory with non-vanishing gauge-fields if we simply identify A = − . This argument equally applies to the solutions found in [4, 5] corresponding to fivebranes wrapping two-cycles in Calabi-Yau two-folds. For the type II theory these are holographically dual to a slice of the Coulomb branch of pure N = 2 super-Yang-Mills theory [4]. The corresponding type I solution has half the supersymmetry and so should holographically encode information about the N = 1 gauge theories arising on type I or heterotic fivebranes wrapping two-cycles in Calabi-Yau two-folds. It would be interesting to study this further. Acknowledgements. We would like to thank Gary Gibbons and Chris Hull for helpful discussions. All authors are supported in part by PPARC through SPG #613. DW also thanks the Royal Society for support.
8. Appendix We will derive the integrability conditions in the context of type I SUGRA. The bosonic fields are the same as the NS sector of the type II supergravity supplemented by a gauge field in the adjoint of some gauge group, with field strength F . For gauge group SO(32) or E8 × E8 this is part of the low-energy effective action of type I or heterotic string theory. The action is given by: S=
1 2κ 2
√ 1 d 10 x −ge−2 R + 4(∇)2 − H 2 − α Tr F 2 12
(98)
with dH = 2α Tr F ∧ F.
(99)
The equations of motion are given by 1 Rµν − Hµρσ Hν ρσ + 2∇µ ∇ν − 2α Tr Fµ ρ Fνρ = 0, 4 1 α ∇ 2 − 2(∇)2 + Hµνρ H µνρ + Tr Fµν F µν = 0, 12 2 ∇µ (e−2 H µνρ ) = 0, 2e2 D µ (e−2 Fµν ) − F ρσ Hρσ ν = 0.
(100)
G-Structures and Wrapped NS5-Branes
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Supersymmetric configurations have vanishing variation of the gravitino, dilatino and gaugino: 1 δψµ ∼ ∇µ + Hµνρ νρ = 0, 8 1 µ δλ ∼ ∂µ + Hµνρ µνρ = 0, 12 δχ ∼ Fµν µν = 0,
(101)
where is a Majorana-Weyl spinor of Spin(1, 9). Note that the first two conditions are half of the conditions arising in the type II theories. We now deduce some consequences of the integrability conditions of these equations. First, take the covariant derivative of the variation of the gravitino and antisymmetrise, to get 1 Rµνσ1 σ2 σ1 σ2 = −∇[µ Hν]σ1 σ2 σ1 σ2 − H[µ σ1 |ρ| Hν] ρσ2 . 2
(102)
Next multiply this expression by ν and use a Bianchi identity to obtain an expression for Rµν ν . Then use Hµνρ νρ times the dilatino variation, the covariant derivative of the dilatino, as well as Fµν ν times the variation of the gaugino to get 1 (Rµν − Hµρσ Hν ρσ + 2∇µ ∇ν − 2α Tr Fµ ρ Fνρ ) ν = 4 1 1 (dH − 2α Tr F ∧ F )µνρσ νρσ + e2 ∇ ρ (e−2 Hρµν ν ). 12 2
(103)
Similar manipulations on µ ∇µ acting on the variation of the dilatino implies 1 α Hµνρ H µνρ + Tr Fµν F µν ) 12 2 1 1 = − (dH − 2α Tr F ∧ F )µνρσ µνρσ − e2 ∇ µ (e−2 Hµνρ ) νρ 4 48
(∇ 2 − 2(∇)2 +
(104) while µ ∇µ acting on the variation of the gaugino yields (2e2 D µ (e−2 Fµν ) − F ρσ Hρσ ν ) ν = 3DFµνρ µνρ .
(105)
We next note that if the Bianchi identities for H and F are satisfied as well as the H equation of motion then we deduce the dilaton equation of motion. Equation (105) is of the form Aµ µ = 0 which implies Aµ Aµ = 0. Similarly (103) is of the form Bµν ν = 0 which implies Bµν B µν = 0. If we assume that we have a solution of the form R1,9−n × Mn then we can deduce that Am = Bmn = 0 which give the gauge and Einstein equations of motion. In other words the Killing spinor equations, combined with the Bianchi identities for H and F , plus the H equations of motion imply all equations of motion are satisfied.
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References 1. Maldacena, J.M., Nunez, C.: Towards the large N limit of pure N = 1 super Yang Mills. Phys. Rev. Lett. 86, 588 (2001) [arXiv:hep-th/0008001] 2. Acharya, B.S., Gauntlett, J.P., Kim, N.: Fivebranes wrapped on associative three-cycles. Phys. Rev. D 63, 106003 (2001) [arXiv:hep-th/0011190] 3. Maldacena, J., Nastase, H.: The supergravity dual of a theory with dynamical supersymmetry breaking. JHEP 0109, 024 (2001) [arXiv:hep-th/0105049] 4. Gauntlett, J.P., Kim, N., Martelli, D., Waldram, D.: Wrapped fivebranes and N = 2 superYang-Mills theory. Phys. Rev. D 64, 106008 (2001) [arXiv:hep-th/0106117] 5. Bigazzi, F., Cotrone, A.L., Zaffaroni, A.: N = 2 gauge theories from wrapped five-branes. Phys. Lett. B 519, 269 (2001) [arXiv:hep-th/0106160] 6. Gomis, J., Russo, J.G.: D = 2 + 1 N = 2 Yang-Mills theory from wrapped branes. JHEP 0110, 028 (2001) [arXiv:hep-th/0109177] 7. Gauntlett, J.P., Kim, N., Martelli, D., Waldram, D.: Fivebranes wrapped on SLAG three-cycles and related geometry. JHEP 0111, 018 (2001) [arXiv:hep-th/0110034] 8. Gomis, J.: On SUSY breaking and χ SB from string duals. Nucl. Phys. B 624, 181 (2002) [arXiv:hep-th/0111060] 9. Apreda, R., Bigazzi, F., Cotrone, A.L., Petrini, M., Zaffaroni, A.: Some Comments on N = 1 Gauge Theories from Wrapped Branes. Phys. Lett. B 536, 161 (2002) arXiv:hep-th/0112236 10. Hori, K., Kapustin, A.: Worldsheet descriptions of wrapped NS five-branes. JHEP 0211, 038 (2002) arXiv:th/020314 11. Gauntlett, J.P., Kim, N., Pakis, S., Waldram, D.: Membranes wrapped on holomorphic curves. Phys. Rev. D 65, 026003 (2002) [arXiv:hep-th/0105250] 12. Gutowski, J., Papadopoulos, G., Townsend, P.K.: Supersymmetry and generalized calibrations. Phys. Rev. D 60, 106006 (1999) [arXiv:hep-th/9905156] 13. Strominger, A.: Superstrings With Torsion. Nucl. Phys. B 274, 253 (1986) 14. Hull, C.M.: Superstring Compactifications With Torsion And Space-Time Supersymmetry. In Turin 1985, Proceedings, Superunification and Extra Dimensions, pp. 347–375 15. Ivanov, S., Papadopoulos, G.: A no-go theorem for string warped compactifications. Phys. Lett. B 497, 309 (2001) [arXiv:hep-th/0008232] 16. Friedrich, T., Ivanov, S.: Parallel spinors and connections with skew-symmetric torsion in string theory. arXiv:math.dg/0102142 17. Friedrich, T., Ivanov, S.: Killing spinor equations in dimension 7 and geometry of integrable G2 -manifolds. arXiv:math.dg/0112201 18. Ivanov, S.: Connection with torsion, parallel spinors and geometry of Spin(7) manifolds. arXiv:math.dg/0111216 19. Maldacena, J., Nunez, C.: Supergravity description of field theories on curved manifolds and a no go theorem. Int. J. Mod. Phys. A 16, 822 (2001) [arXiv:hep-th/0007018] 20. Joyce, D.D.: Compact Manifolds with Special Holonomy. Oxford Mathematical Monographs, Oxford: Oxford University Press, 2000 21. Salamon, S.: Riemannian Geometry and Holonomy Groups. Vol. 201 of Pitman Research Notes in Mathematics, Harlow: Longman, 1989 22. Fernandez, M., Gray, A.: Riemannian Manifolds with Structure Group G2 . Ann. Mat. Pura Appl. 32, 19–45 (1982) 23. Brandhuber, A.: G2 Holonomy Spaces from Invariant Three-Forms. Nucl. Phys. B 629, 393 (2002) hep-th/0112113 24. Agricola, I.: Connections on naturally reductive spaces, their Dirac operator and homogeneous models in string theory. Commun. Math. Phys. 232, 535 (2003) math.DG/0202094 25. Blair, D.E.: Contact manifolds in Riemannian geometry. Lect. Notes No. 509, Berlin-HeidelbergNew York: Springer-Verlag, 1976 26. Cvetic, M., Gibbons, G.W., Lu, H., Pope, C.N.: A G2 Unification of the Deformed and Resolved Conifold. Phys. Lett. B 534, 172 (2002) hep-th/0112138 27. Brandhuber, A., Gomis, J., Gubser, S.S., Gukov, S.: Gauge Theory at Large N and New G2 Holonomy Metrics. Nucl. Phys. B 611, 179 (2001) hep-th/0106034 28. Bryant, R.L., Salamon, S.: On the construction of some complete meetrics with exceptional holonomy. Duke Math. J. 58, 829 (1989) 29. Cleyton, R., Swann, A.: Cohomogeneity-one G2 -structures. J. Geom. Phys. 44, 202 (2002) math.DG/0111056 30. Cvetic, M., Gibbons, G.W., Lu, H., Pope, C.N.: Cohomogeneity One Manifolds of Spin(7) and G2 Holonomy. Phys. Rev. D 65, 106004 (2002) hep-th/0108245 31. Hitchin, N.: Stable Forms and special metrics. math.DG/0107101
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32. Gunaydin, M., Nicolai, H.: Seven-dimensional octonionic Yang-Mills instanton and its extension to an heterotic string soliton. Phys. Lett. B 351, 169 (1995) [Addendum-ibid. B 376, 329 (1996)] [arXiv:hep-th/9502009] 33. Strominger, A.: Heterotic Solitons. Nucl. Phys. B 343, 167 (1990) [Erratum-ibid. B 353, 565 (1991)]; Callan, C.G., Harvey, J.A., Strominger, A.: World Sheet Approach To Heterotic Instantons And Solitons. Nucl. Phys. B 359, 611 (1991) 34. Bergshoeff, E.A., de Roo, M.: Nucl. Phys. B 328, 439 (1989) Communicated by R.H. Dijkgraaf
Commun. Math. Phys. 247, 447–466 (2004) Digital Object Identifier (DOI) 10.1007/s00220-004-1065-z
Communications in
Mathematical Physics
Higher Selberg Zeta Functions Nobushige Kurokawa1 , Masato Wakayama2 1
Department of Mathematics, Tokyo Institute of Technology, Oh-okayama Meguro-ku, Tokyo, 152-8551 Japan. E-mail: [email protected] 2 Faculty of Mathematics, Kyushu University, Hakozaki Higashi-ku, Fukuoka, 812-8581 Japan. E-mail: [email protected] Received: 23 July 2002 / Accepted: 14 November 2003 Published online: 7 April 2004 – © Springer-Verlag 2004
Abstract: In the paper [KW2] we introduced a new type of Selberg zeta function for establishing a certain identity among the non-trivial zeroes of the Selberg zeta function and of the Riemann zeta function. We shall call this zeta function a higher Selberg zeta function. The purpose of this paper is to study the analytic properties of the higher Selberg zeta function z (s), especially to obtain the functional equation. We also describe the gamma factor of z (s) in terms of the triple sine function explicitly and, further, determine the complete higher Selberg zeta function with having a discussion of a certain generalized zeta regularization.
0. Introduction Around 1952, Atle Selberg discovered his celebrated zeta function which is known as the Selberg zeta function Z (s) on his way of establishing the trace formula for a discrete co-compact (later, co-finite) subgroup of SL2 (R) (see [Se1] and the paper “Harmonic analysis" with the Introduction to the G¨ottingen lecture notes in [Se2]). Since then, various Selberg type zeta functions and L-functions were studied from the arithmetic, the spectral and the geometric point of interests, especially related to the distribution of closed prime geodesics, etc. (see [He] and also the references in [SW]). Apart from these, in [KW2] we introduced a new type of Selberg zeta function z (s) which we called a higher Selberg zeta function as follows: Let Prim() be the set of hyperbolic primitive conjugacy classes of , where a hyperbolic element P (and hence also its conjugacy class) is said to be primitive when P is a generator of an infinite cyclic group Z (P ), the centralizer of P in . Then the higher Selberg zeta function z (s) is defined by the Work in part supported by Grant-in Aid for Scientific Research (B) No.11440010, and by Grant-in Aid for Exploratory Research No.13874004, Japan Society for the Promotion of Science
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Euler product as z (s) =
∞
(1 − N (P )−s−m )−m .
(0.1)
m=1 P ∈Prim()
Here the norm N(P ) (> 1) of P ∈ Prim() is defined by N (P ) = max{|αP |2 , |βP |2 }, where αP and βP are the eigenvalues of a representative matrix of P . Since the Selberg zeta function Z (s) is given by Z (s) =
∞
(1 − N (P )−s−n ),
(0.2)
n=0 P ∈Prim()
it is easy to see that Z (s) = z (s)/z (s − 1),
(0.3)
in other words z (s) =
∞
Z (s + n)−1
(0.4)
n=1
holds. In this paper, we investigate several properties which z (s) is expected to possess. We are particularly concerned with a study of the functional equation. A similar procedure developed in [KW5] provides (see §1) the meromorphic continuation to the whole complex plane by the relation (0.3). Nevertheless, in order to discover a possible functional equation of z (s) we shall give a proof of the analytic continuation of the second derivative of log z (s) based on the trace formula. We firstly study the meromorphic continuation and secondly examine what test functions should be taken for the analytic continuation of the higher Selberg zeta function in order to find a functional equation. We also discuss a complete form of the higher Selberg zeta function in terms of the triple sine functions S3 (s) and a zeta regularized product in the sense of [I] and [KW4] (see also [KiKSW1]) which is a generalization of the one introduced in [D1] with respect to the spectrum of the Laplacian . 1. Meromorphic Continuation Throughout the paper, we assume that is a discrete, co-compact and torsion free subgroup of SL2 (R), that is, a strictly hyperbolic group. Note that there exists a positive constant ε such that N(P ) > eε . Let H = {z = x+iy ; x ∈ R, y > 0} = SL2 (R)/SO(2) be the upper half plane with the Poincar´e metric. acts on H by fractional linear transformations. Then the assumption implies that \H is a compact Riemann surface of genus g with g > 1. We now check the fact that the product (0.1) converges absolutely for Re(s) > 0. It is sufficient to show that the series log(1 − N (P )−σ −m )−m (1.1) m=1 P ∈Prim()
Higher Selberg Zeta Functions
449
converges absolutely for σ > 1. Since − log(1 − x) < 2x for 0 < x < 21 , the series above is dominated by C mN (P )−σ −m for some positive constant C.
m=1 P −m Since ∞ m=1 mx
log(1 − N (P )−σ −m )−1 ≤ C
=
m=1 P
for |x| < 1, we have
x (x−1)2
N (P )−σ
P
≤ 2C
N (P )−1 (N (P )−1 − 1)2
N (P )−σ − 2
P
N (P ) 2 − N (P )− 2
1
1
1
.
We recall here the series which gives the logarithmic derivative of the Selberg zeta function Z (s): d log Z (s) = ds
∞
δ∈Prim() k=1
log N (δ) k 2
N (δ) − N (δ)
1
− 2k
N (δ)−k(s− 2 ) .
(1.2)
Since it is well-known that this series converges absolutely for Re(s) > 1 the aforementioned estimate shows that (1.2) converges absolutely for the prescribed region. This in particular says that z (s) is holomorphic and non-zero for the region Re(s) > 0. In order to describe the analytic properties of z (s) we recall now the corresponding properties of Z (s) ([Se1, He]). Z (s) is an entire function of order two with the following properties: (1) (2) (3) (4)
s = −k (k ≥ 1) is a trivial zero of order 2(g − 1)(2k + 1), s = 0 is a zero of order 2g − 1, s = 1 is a simple zero, there is a non-trivial zero of order mn at s = 21 ± irn (n ≥ 1). Here λn = rn2 + 1 2 ∂ 2 + ∂ 2 on (n = 1, 2, 3, . . . ) denote the non-zero eigenvalues of = −y 2 2 4 ∂x ∂y
L2 (\H ) and mn denotes the corresponding multiplicity of λn , (5) those described above exhaust all the zeros of Z (s), (6) Z (s) has the functional equation. s− 1 2 Z (s) = exp 4π(g − 1) r tan(π r)dr Z (1 − s).
(1.3)
0
Note also that using the gamma-factor S 2 (s) given by (see [K2])
1 1 = {2 (s)2 (s + 1)}−1 , S 2 + + s − S 2 (s) = det 4 2 the functional equation (1.3) can be expressed as Z (s) S 2 (s)2g−2 = . Z (1 − s) S 2 (1 − s)2g−2
(1.4)
Here S 2 is the Laplacian of the 2-dimensional sphere S 2 , which is the compact dual space of the Riemannian symmetric space H = SL2 (R)/SO(2) and 2 (s) denotes the double gamma function:
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2 (z + 1)
−1
= (2π)
z/2 − (1+γ 2)z
e
2 +z
∞
z k −z+ z2 2k 1+ , e k
k=1
where γ is the Euler constant. From this, one can easily describe the poles of z (s) by the relation inherited from (0.3): z (s) = z (s + 1)/Z (s + 1).
(1.5)
Theorem 1.1. The higher Selberg zeta function z (s) is non-zero, holomorphic in Re(s) > 0 and meromorphically continued to the whole complex plane. Moreover the inverse z (s)−1 has both trivial zeros and non-trivial zeros as follows. The trivial zeros consist of the zero at s = 0 of order 1 and at s = −k (k ≥ 1) of order 2(g −1)k 2 +2. The non-trivial zeros are located at s = − 21 − k ± irn (n = 1, 2, 3, . . . , k = 0, 1, 2, . . . ) of order mn . The inverse z (s)−1 has no zeros other than described above and no poles. 2. Some Observation of Test Functions In order to find a possible functional equation we need the trace formula. Suppose that the function h(r) is even, holomorphic and h(r) = O((1 + |r|2 )−1−δ ), δ > 0 in the strip | Im r| < 21 + ε for some ε > 0. Let λn = rn2 + 41 (n ≥ 1) be the eigenvalues of and put λ0 = 0 for convenience. Then the trace formula reads as follows: ∞ Vol(\H ) ∞ h(rn ) = h(r)r tanh(π r)dr 2π 0 n=0
+
∞
log N (δ) N (δ) 2 − N (δ)− 2 k
δ∈Prim() k=1
k
G(k log N (δ)).
Note that Vol(\H ) = 4π(g − 1). Here G(v) denotes the inverse of the Euclidean Fourier transform F of h(r) as ∞ 1 −1 G(v) = (F h)(v) = h(t)eivt dt. 2π −∞ We shall first make an experimental computation for choosing a test function suitably to investigate the logarithmic derivative of the higher Selberg zeta function. By the Euler product expression of z (s) we see that d log z (s) = − ds =−
∞ m(log N (δ))N (δ)−s−m 1 − N (δ)−s−m
δ∈Prim() m=1 ∞
m log N (δ)
δ∈Prim() k=1
=−
γ ∈C()
log N (δ)N (δ)−s
∞
n=1 ∞
N (δ)−(s+m)n mN (δ)−m ,
m=1
where C() denotes the set of all hyperbolic conjugacy classes of . We used the fact that the centralizer Z (γ ) is an infinite cyclic group and is generated by a primitive
Higher Selberg Zeta Functions
451
element γ0 in Z (γ ). The relation to write
∞
m=1 mx
−m
=
x (x−1)2
for |x| < 1 permits us again
N (δ) d log z (s) = − log N (δ)N (δ)−s ds (N (δ) − 1)2 γ ∈C()
∞
=−
δ∈Prim() k=1
N (δ)−ks
log N (δ) k 2
N (δ) − N (δ)
In view of this expression, though one wants to take
− 2k
N (δ) 2 − N (δ)− 2 k
e−s|u| 2 sinh |u| 2
k
.
(2.1)
as a test function while it
is impossible because of its existence of a singularity at 0. To remove this singularity, −s|u| −s|u| we shall take gs (u) = 2uesinh u in place of e |u| . 2 sinh
2
2
−s|u|
Suppose that Re s > 0. We now give the Fourier image of gs (u) = 2uesinh u as fol2 lows. Although the result can be derived by a pretty well-known result, since the way employed now could make our further study easy, we give the proof using by the zeta regularization method (see e.g. [KW3, KW4]). Lemma 2.1. Put gs (u) =
∞
−∞
ue−s|u| 2 sinh u2
. Then we have
∞ 2r d 1 dr (s + + n)2 + r 2 2 n=0 1 d2 1 = 2 log s + + ir s + − ir . ds 2 2
gs (u)eiru du =
Proof. First note that
∞
−∞
∞ ue−su iru ue−su −iru e du + e du u 2 sinh 2 2 sinh u2 0 0 ∞ −su ∞ −su d e e d iru = −i e du + i e−iru du. dr 0 2 sinh u2 dr 0 2 sinh u2
gs (u)eiru du =
∞
Let ε > 0. Consider the integral ε
∞
e−su iru e du = 2 sinh u2 =
∞ ε
1
∞
1
e−(s+ 2 +n−ir)u du
n=0 ε
1 ∞ e−(s+ 2 +n−ir)ε
n=0
∞
e−(s+ 2 )u iru e du = −u 1−e
s+
1 2
+ n − ir
.
The manipulation made above is clearly legitimate. Actually the last series converges absolutely and uniformly for any compact subset of the region Re s ≥ 21 + τ for τ > 0. By making a similar computation for the integral over the interval [−∞, ε], we see that
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N. Kurokawa, M. Wakayama
∞ −∞
gs (u)e
iru
d du = −i dr
lim
ε→0
∞ n=0
1
e−(s+ 2 +n+ir)ε s+
1 2
+ n + ir
1
−
e−(s+ 2 +n−ir)ε s+
1 2
+ n − ir
∞ d 2r . dr (s + n + 21 )2 + r 2 n=0
=
(2.2)
In order to evaluate the last sum, let us now recall the zeta regularized product (see [D1, D2, JL, and as a survey see also [M]). Let {an }n=0,1,2,... be a sequence. Assume KW3] −s that ∞ n=0 an is analytically continued to a holomorphic function around at s = 0. Then the regularized product of {an }n=0,1,2,... is defined by ∞ ∞ d −s . (2.3) an = exp − an ds s=0 n=0
n=0
As an example we recall first the following famous formula due to Lerch [L]: √ ∞ 2π (n + x) = , (x) n=0
where (x) is the gamma function. Lerch proved also ∞
(n + x)2 + y 2 =
n=0
2π . (x + iy)(x − iy)
Let us now consider the following function: −x ∞ 1 2 ζs (x, r) = s+n+ + r2 . 2
(2.4)
(2.5)
n=0
This obviously converges absolutely for Re x > 21 . Its analytic continuation is accomplished by the following relation: ∂ ζs (x, r) = −2rxζs (x + 1, r). ∂r The formula (2.4) means nothing but the relation ∞ 1 2 ∂ 2 s+n+ = − log +r . ζs (x, r) ∂x 2 x=0
(2.6)
(2.7)
n=0
Take then a partial derivative of (2.6) with respect to x and change the order of differentiation for r and for x. It then follows that ∂ ∂ ∂ ζs (x, r) = −2rζs (x + 1, r) − 2rx ζs (x + 1, r). ∂r ∂x ∂x Letting x → 0 one has ∞ 1 2 ∂ 2 s+n+ = −2rζs (1, r). − log +r ∂r 2 n=0
Higher Selberg Zeta Functions
453
Hence by (2.2) we have ∞ ∞ 2r d gs (u)eiru du = dr (s + n + 21 )2 + r 2 −∞ n=0 ∞ d2 1 2 2 = 2 log s+n+ . +r dr 2 n=0
Using the formula (2.4) it follows immediately that ∞ 2π d2 iru gs (u)e du = 2 log 1 dr (s + 2 + ir)(s + −∞ = This completes the proof.
1 2
− ir)
∂2 1 1 log (s + + ir)(s + − ir). 2 ∂s 2 2
We note that the following well-known identity can be seen in the proof above: ∞
d2 1 log (s) = . ds 2 (s + n)2
(2.8)
n=0
From the first expression, it is unfortunate to see that the Fourier transform (Fgs )(r) does not satisfy the required condition of a test function. In fact, it violates the condition for a rapidly decreasing property. So we shall overcome such an obstacle by employing the Paley-Wiener type discussion for the G(u)-side and make a necessary improvement of this gs (u). 3. The Functional Equation In view of the form of the logarithmic derivative of the higher Selberg zeta function it is crucial to consider the following even function: G(u) = gs (u) =
ue−s|u| , 2 sinh u2
h(u) = hs (r) = (Fgs )(r) =
∂2 1 1 log s + + ir s + − ir . ∂s 2 2 2
However as indicated in the preceding remark we have to improve this choice. At least it is necessary to impose a smoothness of the second derivative. Let us hence define the following function: Gs (u) = gs (u) − e−s|u| .
(3.1)
Then the following lemma holds. Lemma 3.1. For Re(s) > − 21 , the function Gs (u) decays exponentially, even and Gs (u) ∈ C 2 (R). Further, put Hs (r) = (FGs )(r), the Euclidean Fourier transform of Gs (u). If Re s > 21 , then Hs (r) satisfies the required condition of the trace formula.
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Proof. As to the first assertion, the only fact we have to check is a C 2 -ness at 0. The −2s|u| remains are obvious. Put f (u) = uesinh u = gs (2u) for fixed s. Then for real u, the Taylor expansions of the denominator and the numerator respectively around u = 0 show that f (u) =
u(1 − 2s|u| + 21 (2su)2 −
1 3 3! (2s|u|)
+ ···)
u+ + ··· 1 1 1 = (1 − 2s|u| + (2su)2 − (2s|u|)3 + O(u5 )) × (1 − u2 + O(u4 )) 3! 2 3! 1 2 2 3 = 1 − 2s|u| + 2s − u + O(u ). (3.2) 6 1 3 3! u
This implies that G s (0) = 0,
G s (0) =
4 2 1 s − . 3 6
We next prove the second assertion concerning the function Hs (r). First note that ∞ ∂2 1 1 1 1 log z + = − , 1 2 2 ∂s 2 z 4 (z + m + 2 ) (z + m)(z + m + 1) m=0
(3.3)
simply because of (2.8) and by the following elementary identity: ∞ 1 1 1 = − . z z+m z+m+1 m=0
Since we have ∂2 1 1 2s log s + + ir s + − ir − 2 2 ∂s 2 2 s + r2 2 2 ∂ 1 1 1 ∂ 1 = 2 log s + + ir − + 2 log s + − ir − , ∂s 2 s + ir ∂s 2 s − ir
Hs (r) =
we see from (3.3) that ∞ 1 Hs (r) = − 4 (s + m + m=0
−
1 4
∞ m=0
1 1 2
+ ir)2 (s
+ m + ir)(s + m + 1 + ir) 1
(s + m +
1 2
− ir)2 (s + m − ir)(s + m + 1 − ir)
.
(3.4)
Since the denominator of each term which appeared in the summation above is of degree 4 with respect to m, we immediately see again that the series above indeed converges absolutely and hence Hs (r) satisfies the required growth condition. Also from this expression, we see that the possible poles of Hs (r) are r = ±(s + m)i (m ≥ 0) and ±(s + m + 21 )i (m ≥ 0). Hence there is no pole of Hs (r) in | Im r| < 21 + ε since Re s > 21 . Thus Hs (r) is holomorphic in the strip | Im r| < 21 + ε. This shows that Hs (r) satisfies the required condition of the trace formula.
Higher Selberg Zeta Functions
455
Note here that the possible poles of the Hs (r) are located as follows: Simple poles are r = ±(s + m)i (m ≥ 0). Though there exists also a double pole at ±(s + m + 21 )i for each m ≥ 0, it is easy to see that no residue appears at this point, since the denominator of the first series of Hs (r) can be expressed as 2 2 1 1 1 s + m + ± ir s + m + ± ir − . 2 2 4 In order to seek a possible functional equation of the derivative of the logarithmic derivative of the higher Selberg zeta function z (s), we observe first the analytic continuation of it. By virtue of (2.1) we have ∂2 log z (s) = ∂s 2
∞
δ∈Prim() k=1
log N (δ) k 2
N (δ) − N (δ)
− 2k
·
k{log N (δ)}N (δ)−ks N (δ) 2 − N (δ)− 2 k
k
.
(3.5)
Using this expression and the one for the Selberg zeta function given by (1.2), it is immediate to see by the trace formula that ∞
Hs (rn )
n=0
=
Vol(\H ) 2π
∞
Hs (r)r tanh(π r)dr +
0
1 ∂2 ∂ s + log z (s) − log Z . ∂s 2 ∂s 2 (3.6) 2
∂ In view of the expression (3.4) we have the meromorphic continuation of ∂s 2 log z (s) ∂ 1 to the whole complex plane, since ∂s log Z (s + 2 ) has it. To be explicit more, we have the following lemma.
Lemma 3.2. The formula holds for Re s > 0,
∞
Hs (r)r tanh(π r)dr = −
0
∞ n=0
2n + 1 (2n + 1)φs − i . 2
(3.7)
Here we put φs (r) = −
∞ 1 4 (s + m + m=0
1 1 2
+ ir)2 (s
+ m + ir)(s + m + 1 + ir)
.
This gives a meromorphic continuation of the integral above to the whole s-plane. Proof. By (3.4) one has Hs (r) = φs (r) + φs (−r).
456
N. Kurokawa, M. Wakayama
We consider the integral Is±
∞
:=
φs (±r)r tanh(π r)dr. 0
Note first the poles of the integrand of Is+ on the lower half plane Im r < 0 are all simple and located as follows: r=−
2n + 1 i (n ≥ 0). 2
We shift the integration from the positive real axis to the path along the following δ,R and LR : For each non-negative integer N , consider a large positive number R satisfying 2N +1 < R < 2N2+3 . Take a small δ > 0, 2 1 1 3 1 − δ ∪ Cδ − i ∪ − i + δ , −i + δ ∪ ··· δ,R : 0, −i 2 2 2 2 2N + 1 2N + 1 · · · ∪ Cδ − i ∪ −i + δ , −iR 2 2 and LR = {r = t − iR ; t ≥ 0}. Here we put π 3π ≤θ ≤ , Cδ (a) = r = a + δeiθ ; 2 2 the semi-circle around the center a with the positive direction. Since φs (r) is rapidly decreasing with respect to the direction | Re r| → ∞, we have −iR+∞ + Is = φs (r)r tanh(π r)dr + φs (r)r tanh(π r)dr −iR
δ,R
=
2n+1 δ,R \{∪N n=0 Cδ (− 2 i)}
−
N
2n+1 n=0 Cδ (− 2 i)
φs (r)r tanh(π r)dr
φs (r)r tanh(π r)dr +
−iR+∞
φs (r)r tanh(π r)dr.
−iR
Note here that there are no poles arising from φs (r) in the region in question since we are assuming Re s > 0. Now the residue theorem asserts that lim φs (r)r tanh(π r)dr = πi Resr=− 2n+1 i φs (r)r tanh(π r) δ↓0
Cδ (− 2n+1 2 i)
2
=
2n + 1 2n + 1 φs − i . 2 2
Furthermore, since one has ∞ 1 4 (Re s + m + m=0 ∞ 1 < 4 0 (Re s + x +
|φs (r)| ≤
1 1 2
+ R)2 {(Re s
+m+
1 2
+ R)2 − 41 }
dx 1 2
+ R)2 {(Re s + x +
1 2
+ R)2 − 41 }
=O
1 , R3
Higher Selberg Zeta Functions
457
and | tanh(π r)| is bounded, it follows immediately that −iR+∞ →0 φ (r)r tanh(π r)dr s −iR
when R → ∞. Since N + < R < N + Is+ = lim lim 1 2
n=0
we hence obtain
2n+1 2n+1 δ,R \{∪∞ n=0 Cδ (Cδ (− 2 i)− 2 i)}
R→∞ δ↓0 ∞
−
3 2
φs (r)r tanh(π r)dr
2n + 1 2n + 1 φs − i . 2 2
If a similar argument applies to the integral Is− in the upper-right domain one has ∞ 2n + 1 2n + 1 Is− = lim lim φs (−r)r tanh(π r)dr − φs − i , R→∞ δ↓0 \{∪∞ Cδ ( 2n+1 i)} 2 2 n=0 δ,R 2 n=0
where δ,R
is defined similarly as δ,R in the upper imaginary axis. Since it is clear that lim lim φs (−r)r tanh(π r)dr R→∞ δ↓0
+
we have
∞
0
\{∪∞ C (− 2n+1 i)} δ,R n=0 δ 2
2n+1 δ,R \{∪∞ n=0 Cδ ( 2 i)}
φs (r)r tanh(π r)dr = 0,
Hs (r)r tanh(π r)dr = Is+ + Is− = −
∞ n=0
This proves the lemma.
2n + 1 (2n + 1)φs − i . 2
(3.8)
Plugging (3.7) into (3.6) we immediately have ∞
2n + 1 Vol(\H ) i (2n + 1)φs − 2π 2 ∞
Hs (rn ) = −
n=0
n=0
∂2 1 ∂ + 2 log z (s) − log Z s + . ∂s ∂s 2
Note here that ∞ 2n + 1 (2n + 1)φs − i 2 n=0
∞
∞
=−
1 1 (2n + 1) 2 (s + m + n + 1 )(s + m + n + 3 ) 4 (s + m + n + 1) 2 2 n=0 m=0
=−
1 2n + 1 2 (s + m + n + 1 )(s + m + n + 3 ) 4 (s + m + n + 1) 2 2 k=0 n=0
=−
1 (k + 1)2 . 4 (s + k + 1)2 (s + k + 21 )(s + k + 23 ) k=0
∞
∞
k
(3.9)
458
N. Kurokawa, M. Wakayama
Hence we have ∞
2n + 1 i (2n + 1)φs − 2 n=0 ∞ (s + 21 )2 (s − 21 )2 s2 2s = . − + − (s + k + 1)2 s + k + 1 s + k + 21 s + k + 23 k=0
(3.10)
It follows that ∞ n=0
∞ (s − 21 )2 (s + 21 )2 s2 Vol(\H ) 2s − Hs (rn ) = − − + 2π (s + k + 1)2 s + k + 1 s + k + 21 s + k + 23 k=0
∂2 ∂ 1 + 2 log z (s) − log Z s + . ∂s ∂s 2 From this equation we have the
(3.11)
Corollary 3.3. ∞ n=0
Vol(\H ) {Hs (rn ) + H−s (rn )} = 2π +
∂ 2 πs cot(π s) + 2π s tan(π s) ∂s
Z (s + 21 ) ∂2 ∂ log z (s)z (−s) − log . ∂s 2 ∂s Z (−s + 21 )
(3.12)
Proof. By the formula (3.10) we see that 2n + 1 (2n + 1)φ−s − i 2 n=0 ∞ (s − 21 )2 (s + 21 )2 s2 2s = . + + − (−s + k + 1)2 −s + k + 1 −s + k + 21 −s + k + 23 k=0 ∞
(3.10 )
Adding (3.10) and (3.10 ) we see that ∞ 2n + 1 2n + 1 (2n + 1) φs − i + φ−s − i 2 2 n=0
= 2s 2
∞ ∞ s 2 + (k + 1)2 1 2 − 4s {s 2 − (k + 1)2 }2 s 2 − (k + 1)2 k=0
1 + 2(s + )3 2
k=0
∞ k=0
∞
1 1 − 2(s − )3 . 1 2 1 2 2 2 (s + 2 ) − (k + 1) (s − 2 ) − (k + 1)2 k=0 1
We now recall the partial fractional expansions of trigonometric functions: 2
∞ 1 s 2 + n2 π2 − 2, = 2 (s 2 − n2 )2 s sin (π s) n=1
2s
∞ n=1
s2
1 1 = π cot(π s) − . 2 s −n
Higher Selberg Zeta Functions
459
Using these expansions it is not hard to see that 2n + 1 2n + 1 i + φ−s − i (2n + 1) φs − 2 2 n=0 1 π2 1 2 =s − − 2s π cot(π s) − s sin2 (π s) s 2 2 1 1 2 1 1 π tan(π s) + + s− π tan(π s) + − s+ 2 2 s + 21 s − 21 ∞
π 2s2 − 2πs cot(π s) − 2πs tan(π s) sin2 (π s) ∂ 2 π s cot(π s) − 2πs tan(π s). =− ∂s =
Hence the assertion follows from (3.11). This completes the proof the corollary.
Recall the functional equation (1.3) of Z (s). It follows immediately from (1.3) that Z (s + 21 ) ∂ log = 4π(g − 1)s tan(π s) = Vol(\H )s tan(π s). ∂s Z (−s + 21 )
(3.13)
Note also that Hs (r) + H−s (r) (3.14) 2 1 1 1 1 ∂ = 2 log s + + ir s + − ir − s + + ir − s + − ir ∂s 2 2 2 2 ∂2 1 1 = − 2 log sin π s + + ir sin π s + − ir ∂s 2 2 ∂2 ∂2 = − 2 log cos π(s + ir) cos π(s − ir) = − 2 log(cos 2π s + cosh 2π r) ∂s ∂s 1 1 2 + , =π cosh2 π(r − is) cosh2 π(r + is) because of the formula (z)(1 − z) = π/ sin πz. Using (3.13) and (3.14) we conclude that d2 log z (s)z (−s) ds 2
∞ 1 Vol(\H ) d 2 1 2 =π + − s cot(π s) . 2 ds cosh2 π(rn − is) cosh2 π(rn + is) n=0 (3.15) In order to integrate and exponentiate the series which appeared in (3.15) we now recall the zeta regularized product in the sense of [I] and [KW4] (see also [KiKSW1, KiKSW2]). It is a relaxed form of the standard zeta regularized product [D1, D2] discussed in §2.
460
N. Kurokawa, M. Wakayama
Let a = {an }n=0,1,... be a sequence of non-zero number. Assume that the complex −s defines a holomorphic funcattached zeta function (Dirichlet series) ζa (s) = ∞ a n=0 n tion for sufficiently large Re(s) and meromorphically continued to the origin s = 0. Then we define a generalized zeta regularized product of the sequence {an }n=0,1,... by ∞ ζa (s) • an = exp − Ress=0 2 . s
(3.16)
n=0
Using this notion we now have the Theorem 3.4. There exists a zeta regularized product defined by ∞ Ch (s) := • cosh π(rn − is)
(3.17)
n=0
for any s ∈ C. The regularized product Ch (s) defines an entire and quasi periodic function. The zeros of Ch (s) are given by s = 21 − irn + m (n ∈ Z≥0 , m ∈ Z). Furthermore, we have the formula, ∞
d2 1 . log Ch (s) = 2(iπ s + log 2)(g − 1) − π 2 2 2 ds cosh π(rn − is)
(3.18)
n=0
To prove the theorem we need the following result due to [CV]. Lemma 3.5. The function (w) defined by the series e−πwrn (w) =
(3.19)
n≥0
is holomorphic in the region Re w > 0 and has a meromorphic continuation to the whole plane C. This (w) has double poles at w = −2m (m = 0, 1, 2, . . . ) and simple poles at w = ± ik π log N (P ) (P ∈ Prim(), k = 1, 2, . . . ). These exhaust the singularities of (w). In particular, if we write the Laurent expansion of (w) at w = 0 as (w) =
∞
bj w j
j =−2
then b−2 = 2(g − 1)/π 2 , b−1 = 0 and b2n =
π 2n (−1)n (2−2n−1 − 1)B2n+2 · (g − 1) (n + 1)(2n)!
where Bn is the nth Bernoulli number.
(n ∈ Z≥0 ),
Proof of Theorem 3.4. Put L(x : s) =
∞ n=0
cosh π(rn − is)−x .
(3.20)
Higher Selberg Zeta Functions
461
1 Since one knows ∞ n=0 |rn |x converges for Re x > 2, (or by the lemma above), for Re x > 0, the defining series of L(x : s) converges absolutely and uniformly for each compact set which does not contain any −irn + 2m+1 2 , (n ≥ 0, m ∈ Z) in the s-space. Hence L(x : s) defines a meromorphic function with respect to x (Re x > 0) for each s provided s = −irn + 2m+1 2 . We first show that L(x : s) is meromorphic around x = 0 for s in a generic position. Suppose now −1 ≤ Re s < 1. We use the binomial expansion to see the behavior of L(x : s) around x = 0. In fact, since we have −x ∞ eπ(rn −is) + e−π(rn −is) L(x : s) = 2 n=0
= 2x
∞
−x e−πx(rn −is) 1 + e−2π(rn −is)
n=0
∞ −x −2π(rn −is) e =2 e n=0 =0 ∞ ∞ −x −2π(rn −is) x iπxs −πxrn =2 e 1+ e e n=0 =1 ∞ ∞ −x 2πis −π(2+x)rn x iπxs e (x) + e =2 e n=0 =1 ∞ −x e2πis (x + 2) , = 2x eiπxs (x) + x
∞
−πx(rn −is)
=1
the desired meromorphy around x = 0 follows immediately from Lemma 3.6. The assertion concerning the location of the zeros of Ch (s) follows from [I] (see also [KiW] and the corollary below). In order to show the second assertion we first note the differential-difference equation for L(x : s); ∂2 L(x : s) = −π 2 x 2 L(x : s) + π 2 x(x + 1)L(x + 2 : s). ∂ 2s
(3.21)
This means that ∂2 L(x : s) = −π 2 Resx=0 L(x : s) + π 2 L(2 : s). Resx=0 ∂ 2s x2
(3.22)
We now compute the residue Resx=0 L(x : s). Since 2x eiπ xs = e(iπs+log 2)x
(3.23) 1 1 2 2 3 3 = 1 + (iπ s + log 2)x + (iπ s + log 2) x + (iπ s + log 2) x + · · · ., 2 6
it follows from the Laurent expansion of (x) in Lemma 3.5 that Resx=0 L(x : s) =
2(iπ s + log 2)(g − 1) . π2
(3.24)
462
N. Kurokawa, M. Wakayama
It is hence immediate from (3.22) and (3.24) that ∂2 L(x : s) = −2(iπ s + log 2)(g − 1) + π 2 L(2 : s). (3.25) Resx=0 ∂ 2s x2 , when the Dirichlet series L(x : s) is meromorSince Ch (s) = exp − Resx=0 L(x:s) x2 phic at the origin, we see that d2 log Ch (s) = 2(iπ s + log 2)(g − 1) − π 2 L(2 : s). ds 2 This completes the proof of the theorem.
(3.26)
Remark 3.1. One may write symbolically the regularized product Ch (s) also as
1 − − is . Ch (s) = det cosh π 4 Thus, it is interesting to have the determinant expression of z (s) as Z (s) has, developed in [S, D’HP] (see also [V]). Furthermore, when we try to study the non-uniform lattice in SL2 (R), for instance, like the modular group SL2 (Z) and its congruence subgroups, it is known in [H1, H2] that there arises a log-branch at the origin x = 0 for the theta function (x) defined in (3.19). In these cases, in order to overcome such difficulty for defining a regularized product Ch (s) adequately we need further generalization of a zeta regularized product which allows us to handle the log singularity. This generalization has been accomplished in [KiW]. See also [HKW] for some intimately related analysis. Remark 3.2. As in the case of the q-analogue of the ring sine function for Z discussed in [KiKSW1] (see also [KMOW]) the Laurent expansion of the attached Dirichlet series around x = 0 depends on the place of s. Hence the coefficients cj (s) of the Laurent k expansion at the origin L(x : s) = ∞ k=−N ck (s)x are not periodic in general. In particular, the corresponding zeta regularized product is not necessarily periodic though it is quasi periodic (in the sense that it is periodic up to a factor of the form eP (x) , where P (x) is a polynomial). In order to describe the quasi periodicity of Ch (s) we introduce a periodic function with period 1 by ϕ (s) =
∞
(1 + e−2π(rn −is) ).
(3.27)
n=0
This infinite product indeed converges absolutely and uniformly for every compact set in C, and hence define the entire function having obvious zeros at 21 + m − irn . Since ∞ (−1) 2πis L(x : s) = 2x eiπxs (x) + x e (2) =1 2(g − 1) 1 − +{periodic term with period 2} , (g − 1)+x b = 2x eiπxs 1 π 2x2 12
the expansion (2.24) yields the
Higher Selberg Zeta Functions
463
Corollary 3.6. We have 1 1 3 (g − 1)(iπ s + log 2) + ϕ (s). (g − 1)(iπ s + log 2) − b Ch (s) = exp − 1 3π 2 12 (3.28) From (3.18) and (3.28) it follows that ∞
d2 1 2 . log ϕ (s) = −π 2 ds 2 cosh π(rn − is)
(3.29)
n=0
Hence by (3.15) we obtain d2 Vol(\H ) d 2 d2 log z (s)z (−s) = − log{ϕ (s)ϕ (−s)} − cot(π s) . s ds 2 ds 2 2 ds (3.30) Furthermore, to determine the gamma factor of the higher Selberg zeta function z (s) it is necessary to employ the multiple sine functions (see [K2, KKo]). Recall the multiple sine functions Sr (z) (r ≥ 2). (S1 (z) = 2 sin π z). It is also characterized by the differential equation Sr (z) = πzr−1 cot(π z) with Sr (z)
Sr (0) = 1.
(3.31)
Especially, S3 (z) has the product and the integral representations.
∞ z2 n2 z2 z2 /2 S3 (z) = e 1− 2 e n n=1 z 2 = exp πt cot(π t)dt ,
(3.32) (3.33)
0
where the contour lies in C\{±1, ±2, . . . }. Note that S3 (−z) = S3 (z). This shows that 1 d2 d 2 s cot(π s) = log S3 (s). ds π ds 2 Since Vol(\H ) = 4π(g − 1) it follows hence from (3.30) that d2 2(g−1) log z (s)z (−s)ϕ (s)ϕ (−s)S (s) = 0. (3.34) 3 ds 2 There is hence a non-zero constant α such that z (s)z (−s)ϕ (s)ϕ (−s)S3 (s)2(g−1) = α, since the left-hand side is even with respect to s. Since z (s) has a simple pole at s = 0 with residue 1 and ϕ (s) has a zero of order 1 at s = 0 corresponding to r0 = i 21 (the space of constant functions in L2 (\H )), we have lim z (s)ϕ (s) = Ress=0 z (s) × ϕ (0) = z (1)Z (1)−1 ϕ (0).
s→0
Since S3 (0) = 1, this formula implies that α = {z (1)−1 Z (1)ϕ (0)}2 . Summarizing these observation we conclude that
464
N. Kurokawa, M. Wakayama
Theorem 3.7. Define the complete higher Selberg zeta function zˆ (s) by zˆ (s) = z (1)−1 Z (1)ϕ (0)−1 ϕ (s)S3 (s)g−1 z (s). (3.35) ∞ Here z (1) = n=2 Z (n)−1 . Then zˆ (s) is meromorphic in the whole complex plane and satisfies the functional equation zˆ (s)ˆz (−s) = 1.
(3.36)
4. Concluding Remarks Like the Euler product of the Riemann zeta function, it is sometimes convenient to consider the following “lower" Selberg zeta function ζ (s). In fact, this ζ (s) is considered as a zeta function of geodesic flows, usually called by the Ruelle type zeta function. The properties of this ζ (s) is described, for instance, in [K1]. ζ (s) = (1 − N (P )−s )−1 . (4.1) P ∈Prim()
In particular, it has a remarkable functional equation ζ (s)ζ (−s) = (2 sin πs)−2(2g−2) .
(4.2)
Concerning those three Selberg zeta function, the situation is described as z (s) → Z (s) =
z (s) Z (s + 1) z (1 + s)z (s − 1) . → ζ (s) = = z (s − 1) Z (s) z (s)2
Obviously the last relation asserts that {z (−s + 1)z (s − 1)}{z (s + 1)z (−s − 1)} = (2 sin π s)−2(2g−2) . {z (s)z (−s)}2
(4.3)
From this, one has the following relation: Lemma 4.1. For any integer n, we have 1 1 1 1 z n + z −n − = 2−n(n+1)(2g−2) z z − . 2 2 2 2 Proof. It suffices to show the relation 1 1 1 1 z n + z −n − = 2−2n(2g−2) z n − z −n + . 2 2 2 2
(4.4)
(4.5)
In fact, if (4.5) is true, we have easily 1 1 1 1 −2(2g−2) nk=0 k z n + z −n − =2 z − , z 2 2 2 2 whence the result follows. We prove (4.5) by induction on positive n. Put s = 21 at (4.3). Then we see that 3 3 1 1 −2(2g−2) z z − =2 z − , z 2 2 2 2 whence it is in fact true for n = 1. Assume (4.4) holds for n. Then, again by (4.3) for s = n + 21 , the induction assumption shows (4.5) for n + 1.
Higher Selberg Zeta Functions
465
In the definition of the complete higher Selberg zeta function zˆ (s) we find the triple sine function S3 (s). This S3 (s) is important because the mysterious value ζ (3) can be −1 1 2 2 4 (see [KW1, KOW]). We remark here that expressed as ζ (3) = 8π7 log S3 21 the relation m+1 1 1 S3 m + = (−1) 2 2m(m+1) S3 , (m ∈ Z) (4.6) 2 2 follows from (4.5) immediately. Here [x] denotes the integral part of x ∈ R. In fact, since zˆ (s)ˆz (−s) = 1 by Theorem 3.6 and S3 (−s) = S3 (s) we have in particular 1 1 1 1 1 2g−2 z m + z −m − ϕ m + ϕ −m − S3 m + 2 2 2 2 2 = Constant. S3 Since ϕ (s) is periodic with period 1, (4.4) implies
m+ 21
S3
1 2
= 2m(m+1) . Hence the
sign in (4.6) is determined by the product expression (3.32) of S3 (m + 21 ). Remark 4.1. There is another multiple sine function Sr (s) which we call a normalized multiple trigonometric function. It is defined by Sr (s) = r (s)−1 r (r − s)(−1) , r
where r (s) denotes the Barnes Gamma function. Note that S1 (s) = S1 (s) = 2 sin π s. It is known in [KKo] that Sr (s + 1) = Sr (s)Sr−1 (s)−1 and the functional equations S3 (3 − s) = S3 (s), S2 (2 − s) = S2 (s)−1 . Using these we have the expressions; S3 (s) = e2ζ
(−2)
S3 (s)2 S2 (s)−3 S1 (s) and
S2 (s) = S2 (s)−1 S1 (s).
By these formulas and (4.7) for m = 1, for instance, we verify that the value S2 √ the “division point” 21 is equal to 2.
1 2
at
It would be quite interesting to continue the present procedure of higherlization of the Selberg zeta functions. Namely, if we define Z (s) = ζ (s − 1)−1 , (0)
(n−1)
(1)
Z (s) = Z (s), (n)
(n)
(2)
Z (s) = z (s), . . . (n)
successively such as Z (s) = Z (s)/Z (s − 1), does Z (s) have the functional equation? What happens if one considers the other zeta functions, for instance, such as the Dedekind zeta functions for number fields? We treat the problem (for the one step version) in the case of the Riemann zeta function in [KMW]. The functional equation for such a high Riemann zeta function is quite similar to the present one for the higher Selberg zeta function. On the other hand, we note that, quite recently in [G], the higher Selberg zeta function has come up naturally in the study of the first variation of the Selberg zeta function.
466
N. Kurokawa, M. Wakayama
References [CV]
Cartier, P., Voros, A.: Une nouvelle interpr´etation de la formule des traces de Selberg. C. R. Acad. Sci. Paris 307, 143–148 (1988) [D1] Deninger, Ch.: On the -factors attached to motives. Invent. Math. 104, 245–261 (1991) [D2] Deninger, Ch.: Local L-factors of motives and regularized determinants. Invent. Math. 107, 135–150 (1992) [G] Gon, Y.: First variation of Selberg zeta functions and variational trace formulas. J. Ramanujan Math. Soc. 18, 257–280 (2003) [He] Hejhal, D.: The Selberg trace formula for P SL(2, R), Volumes 1 and 2. Lecture Note in Math. Vols. 548, 1001, Berlin-Heidelberg-New York: Springer Verlag, 1976, 1983 [H1] Hirano, M.: On Cartier-Voros type Selberg trace formula for congruence subgroups of P SL(2, R). Proc. Japan Acad. 71, 144–147 (1995) [H2] Hirano, M.: On theta type functions associated with the zeros of the Selberg zeta functions. Manuscripta Math. 92, 87–105 (1997) [HKW] Hirano, M., Kurokawa, N., Wakayama, M.: Half zeta functions. J. Ramanujan Math. Soc. 18, 195–209 (2003) [D’HP] D’Hoker, E., Phong, D.H.: On determinants of Laplacians on Riemann surfaces. Commun. Math. Phys. 104, 537–545 (1986) [I] Illies, G.: Regularized products and determinants. Commun. Math. Phys. 220, 69–94 (2001) [JL] Jorgenson, J., Lang, S.: Basic Analysis of Regularized Series. Lecture Note in Math., Vol. 1564, Berlin-Heidelberg-New York: Springer Verlag, 1993 [KiKSW1] Kimoto, K., Kurokawa, N., Sonoki, C., Wakayama, M.: Zeta regularizations and q-analogue of ring sine functions. Kyushu Math. J. 57, 197–215 (2003) [KiKSW2] Kimoto, K., Kurokawa, N., Sonoki, C., Wakayama, M.: Some examples of generalized zeta regularized products. Preprint 2002 [KiW] Kimoto, K., Wakayama, M.: Remarks on zeta regularized products. Intern. Math. Res. Notices, 17, 855–875 (2004) [K1] Kurokawa, N.: Special values of Selberg zeta functions. Contemp. Math. 83, 133–150 (1989) [K2] Kurokawa, N.: Gamma factors and Plancherel measures. Proc. Japan Acad. 68A, 256–260 (1992) [KKo] Kurokawa, N., Koyama, S.: Multiple sine functions. Forum. Math. 15, 839–876 (2003) [KMOW] Kurokawa, N., M¨uller-St¨uler, E-M., Ochiai, H., Wakayama, M.: Kronecker’s Jugendtraum and ring sine functions. J. Ramanujan Math. Soc. 17, 211–220 (2002) [KMW] Kurokawa, N., Matsuda, S., Wakayama, M.: Gamma factors and functional equations of higher Riemann zeta functions. Preprint 2003 [KOW] Kurokawa, N., Ochiai, H., Wakayama, M.: Zetas and multiple trigonometry. J. Ramanujan Math. Soc. 17, 101–113 (2002) [KW1] Kurokawa, N., Wakayama, M.: On ζ (3). J. Ramanujan Math. Soc. 16, 205–214 (2001) [KW2] Kurokawa, N., Wakayama, M.: A comparison between the sum over Selberg’s zeroes and Riemann’s zeroes (corrections, ibid. 18, 415–416 (2003)). J. Ramanujan Math. Soc. 18, 221–236 (2003) [KW3] Kurokawa, N., Wakayama, M.: A generalization of Lerch’s formula. Czech. Math. J. To appear [KW4] Kurokawa, N., Wakayama, M.: Generalized zeta regularizations, quantum class number formulas, and Appell’s O-functions. The Ramanujan J. To appear [KW5] Kurokawa, N., Wakayama, M.: Zeta extensions. Proc. Japan. Acad. 78A, 126–130 (2002) ˇ e Akad. 3(28), 1–61 [L] Lerch, M.: Dalˇsi studie v oboru Malmst´enovsk´ych rˇad. Rozpravy Cesk´ (1894) [M] Manin, Yu-I.: Lectures on zeta functions and motives (according to Deninger and Kurokawa). Ast´erisque 228, 121–163 (1995) [S] Sarnak, P.: Determinants of Laplacians. Commun. Math. Phys. 110, 113–120 (1987) [SW] Sarnak, P., Wakayama, M.: Equidistribution of holonomy about closed geodesics. Duke Math. J. 100, 1–57 (1999) [Se1] Selberg, A.: Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series. J. Indian Math. Soc. 20, 47–87 (1956) [Se2] Selberg, A.: Collected Papers I. Berlin-Heidelberg-New York: Springer-Verlag, 1989 [V] Voros, A.: Spectral functions, special functions and the Selberg zeta functions. Commun. Math. Phys. 110, 439–465 (1987) Communicated by P. Sarnak
Commun. Math. Phys. 247, 467–512 (2004) Digital Object Identifier (DOI) 10.1007/s00220-004-1067-x
Communications in
Mathematical Physics
All Loop Topological String Amplitudes from Chern-Simons Theory Mina Aganagic, Marcos Marino, ˜ Cumrun Vafa Jefferson Physical Laboratory, Harvard University, Cambridge, MA 02138, USA Received: 8 September 2002 / Accepted: 8 December 2003 Published online: 9 April 2004 – © Springer-Verlag 2004
Abstract: We demonstrate the equivalence of all loop closed topological string amplitudes on toric local Calabi-Yau threefolds with computations of certain knot invariants for Chern-Simons theory. We use this equivalence to compute the topological string amplitudes in certain cases to very high degree and to all genera. In particular we explicitly compute the topological string amplitudes for P2 up to degree 12 and P1 × P1 up to total degree 10 to all genera. This also leads to certain novel large N dualities in the context of ordinary superstrings, involving duals of type II superstrings on local Calabi-Yau three-folds without any fluxes. 1. Introduction In [1] it was conjectured that U (N ) Chern-Simons theory on S3 , which describes the topological A-model of N D-branes on X = T ∗ S3 , is dual at large N to topological closed string theory on Xt = O(−1) ⊕ O(−1) → P1 . There it was shown that the ’t Hooft expansion of Chern-Simons free energy agrees with topological string amplitudes on Xt to all genera. The conjecture was further tested in [2], where computations of certain Wilson loop observables in Chern-Simons theory were shown to match the corresponding quantities on Xt . Various aspects of the duality were studied in [3–10] from different points of view. The topological string duality was embedded in the superstring theory in [11]. In [12] the target space derivation of the superstring duality of [11] was found by lifting up to M-theory [12, 13]. This was further studied in [14, 15], and also in a related context in [16–23]. Recently, [24] gave a world-sheet proof of the topological string duality based on some earlier ideas in [1]. In [25] a large class of new large N dualities was proposed which generalizes the conjecture of [1] to more general backgrounds, employing the philosophy of [1] that the large N dualities are geometric transitions. On the open string side, replacing T ∗ S3 with
This research is supported in part by NSF grants PHY-9802709 and DMS-0074329
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a more general Calabi-Yau manifold X led one to incorporate large open string instantons whose contributions deform Chern-Simons theory [26]. In the spirit of ’t Hooft’s original large N conjecture, the holes in open string Riemann surfaces fill up at large N, and the complicated open string instanton sums that arise in a general Calabi-Yau X get related to a complicated structure of instantons on the dual closed string side. Some important aspects of how this works were clarified in [27]. For one of the examples of [25], where both sides of the duality are explicitly computable to all orders [27] verifies the correspondence at the level of the partition functions. However, in a general setting, the descriptions of the theory in terms of open and closed strings are at the same level of complexity, and the duality was not easy to check (beyond the leading disk amplitude). In this paper, by combining all of the ideas above together with several new technical ingredients, we show that Chern-Simons theory with product gauge groups and topological matter in bifundamental representations computes all loop topological string amplitudes on non-compact toric Calabi-Yau manifolds. Namely, it is shown that open string duals of a certain class of local toric Calabi-Yau manifolds involve D-branes on chains of Lagrangian submanifolds that are coupled only via annuli. In terms of ChernSimons theory this is related to computations of appropriate combinations of Wilson loop observables associated with knots that are the boundaries of the annuli. The duality is local in the sense that, as in [1], the three-manifolds wrapped by D-branes get replaced by P1 ’s in the dual. However in this case, open string theories build very complicated closed string geometries: in fact any noncompact toric Calabi-Yau manifold arises in some limit of this. The paper is organized as follows. In Sect. 2 we review the relevant geometries for open and closed strings that are related by large N duality. In Sect. 3, we discuss the physics of open string theories, and explain why the model simplifies dramatically using a deformation argument. In Sect. 4 we explain what is the relevant Chern-Simons computation in terms of three-manifolds glued with annuli. In Sect. 5 we propose the large N dualities and we argue that the results of [24] should be applicable to derive them. In Sect. 6 we discuss the relation between the predictions of this duality to localization in the A-model closed string computation. In Sect. 7 we present explicit evaluations of the amplitudes and provide predictions for the integer invariants for some examples including P2 and P1 × P1 , and we show that they agree with the known results when they are available [28–31]. In Sect. 8 we consider embedding of this in the superstring context. Results of previous sections give open string duals of closed string geometries with no RR flux. Moreover, we show that some local geometries in IIB string theory have dual description in terms of gauge theory alone. The work in Sect. 7.2 was done in collaboration with P. Ramadevi, to whom we are very grateful. Also, our work has some overlap with the work of [32], and we thank the authors for discussing their work prior to publication. In particular we learned of their result that only a limited number of holomorphic curves contributes to the amplitudes before we found the general argument presented in Sect. 3. The argument discussed in [32] (in the context of dP2 ) uses the localization principle, whereas our argument that only annuli contribute for all toric 3-folds is based on complex structure deformation invariance. 2. Geometry 2.1. Open string geometry: T 2 fibrations and their degenerations. In this paper, the relevant Calabi-Yau manifolds are non-compact and admit a description as a special
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Lagrangian T 2 × R fibration over R3 . The T 2 fibers degenerate over loci in the base. The geometry of the manifold is encoded in the one dimensional graphs in R3 that correspond to the discriminant of the fibration. A very familiar example of a Calabi-Yau manifold of this type is X = T ∗ S3 . The complex structure of X is given by xy = z,
uv = z + µ.
(2.1)
The two-torus is visible in the above equation as it is generated by two U (1) isometries of X acting as x, y, u, v → xeiα , ye−iα , ueiβ , ve−iβ . The α and β actions above can be taken to generate the (1, 0) and (0, 1) cycle of the T 2 . The local type of the singularity has a T 2 fiber that degenerates to S1 by collapsing one of its one-cycles. In the equation above, the U (1)α action fixes x = 0 = y and therefore fails to generate a circle there. In the total space, the locus where this happens, i.e. the x = 0 = y = z subspace of X, is another cylinder uv = µ. The projection to the base space forgets the circle of this cylinder and is a line in R3 . Such a geometry locally looks like a Taub-Nut (TN) space times a cylinder C∗ = R × S1 . Here, the TN space itself is thought of as a cylinder xy = const which is fibered over the z plane and which degenerates at z = 0. Analogous considerations apply to the U (1)β action. The locus of degenerate fibers in the base R3 of the deformed conifold is given in the figure below. In this and similar figures below, two of the directions of the base are the axes of the two cylinders, and the third direction represents the real axis of the z−plane. In general, any (p, q) cycle of the T 2 can degenerate in this way. As long as the degenerate loci do not intersect, the local geometry is that of Taub-Nut space, as an SL(2, Z) transformation on the T 2 fiber can be used to relate it to the degenerations discussed above. In what follows, it will be important that the orientation of the locus where the T 2 fiber degenerates in the base R3 is correlated with the (p, q) type of degenerating cycle. We have seen an example of this above in the case of T ∗ S3 , where the α and β cycles degenerated along orthogonal directions in the base in Fig. 1. The origin of this is
α
re(z)
β
Fig. 1. The figure depicts the discriminant locus of the T 2 × R fibration in the base R3 . The α and β cycles of the T 2 degenerate over lines z = 0, z = −µ
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α
β
α+β Fig. 2. The degeneration locus of the T 2 fibration in the base specifies the Calabi-Yau geometry. The orientation of the lines are related to the (p, q) type of the 1-cycle that degenerates over it. In the type IIB language, this corresponds to different (p, q) fivebranes
the fact that the Calabi-Yau manifold is a complex manifold and the fibration is special Lagrangian. We will not go into detail here in this language as it is cumbersome for physicists, and explained in the literature (see for example [25])1 , especially because there is a string theory duality that provides excellent intuition about the geometry, which we would like to explain instead. 2.2. Relation to (p, q) fivebranes in IIB. In this section we connect the description of Calabi-Yau geometry by a duality to the web of (p, q) fivebranes [33]. This will be helpful for us for an intuitive picture of holomorphic curves in the geometry. The connection was derived in [34] and we will now review it. Recall that M-theory on T 2 is related to type IIB string theory on S1 . Since the Calabi-Yau manifolds we have been considering are T 2 fibered over B = R4 , we can relate geometric M theory compactification on the Calabi-Yau manifold X to type IIB on flat space on B × S1 . However, due to the fact that T 2 is not fibered trivially, this is not related to the vacuum type IIB compactification. The local type of singularity, as we have seen above, is the Taub Nut space T Np,q , where the (p, q) label denotes which cycle of the T 2 corresponds to the S1 of the TaubTo give an idea of the more general situation, let xˆα,β be the single valued holomorphic coordinates on C∗ × C∗ , and let z be a coordinate on R2 . If a (p, q) cycle of the T 2 degenerates at a point in z, then the fixed point locus which is invariant under xˆα → xˆα eipθ , xˆβ → xˆβ eiqθ . In terms of periodic variables xˆα,β = exp(xα,β ) we can write the degeneration locus as 1
qxα − pxβ = const.
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Nut geometry. Under the duality, this local degeneration of X is mapped to the (p, q) five-brane that wraps the discriminant locus in the base space B, and lives on a point on the S1 . The fact that the (p, q) type of the five brane is correlated with its orientation in the base is a consequence of the BPS condition. More precisely, configurations of five branes that preserve supersymmetry and 4 + 1-dimensional Lorentz invariance are pointlike in a fixed R2 subspace of the base that we called the z plane above. In the two remaining directions of the base, the five branes are lines where the equation of the (p, q) five brane is pxα + qxβ = const. 2.3. Geometric transitions. Consider a pair of lines in the base space over which two one-cycles of the T 2 degenerate. Any path in the base space ending on the two lines, together with the T 2 fiber over it, gives rise to a closed three-manifold in the total space. This is because a cycle of the T 2 degenerates over the start and the end point of the path, so the three-manifold has no boundaries. If the two lines intersect in the base space, the three-cycle obtained in this way can be shrunken to a point. If they don’t, it generates a homology class in H3 (X, Z). Let n be the number of five-branes. If the five branes are in generic positions and do not intersect, the manifold is smooth, and it is easy to see that the dimension of third homology is b3 (X) = n − 1. In the superstring context, among Lagrangian three-cycles in the Calabi-Yau manifold, special Lagrangian three-cycles are of particular interest as they are supersymmetric, i.e. D-branes wrapped on them preserve some supersymmetry of the theory. These cycles are volume-minimizing and project to paths of shortest length in the base (the Lagrangian condition can always be satisfied with some choice of symplectic form on X). In the non-compact situation we are discussing, the meaning of this is particularly transparent in the IIB string theory as the five-branes live in R4 with flat metric. The number of supersymmetric cycles, for five-branes in generic positions, is easily counted by doing the projection of the base R3 → R2 that suppresses the z-direction and counts the number of intersections. Generically there will be n(n−1)/2 such intersection points (unless some (p, q) 5-branes are of the same type). The Calabi-Yau manifolds we have been discussing have geometric transitions where three-cycles in geometry shrink and the resulting singularity is smoothed to a manifold Xt of different topology. This was explained in some detail in [25]. In the examples we will be studying in this paper, the local geometry of the singularity will be T ∗ S3 , so the geometric transition in question involves an S3 shrinking and a P1 growing. The geometric transitions do not spoil the fact that the manifold is T 2 fibered, however they do change the locus of singular fibers. After the transition that shrinks all the three-cycles (and these always exist in the family of X’s we consider), the resulting manifolds are toric varieties. Toric varieties admit a group of U (1) isometries whose rank is the complex dimension of the manifold. In our case this is U (1)3 , and the symmetry enhancement comes from the fact that the transition which gets rid of all the three-cycles requires all the loci of singular fibers to coincide in the z-plane, and the extra U (1) is the group of rotations about this point. While the reader might get an impression from the above discussion that the manifold after transition gains new cycles only in H2 (X t , Z), this is in fact not the case. In fact, in the generic case the number of shrinking minimal threecycles is larger than the number of classes in H3 (X). Then, since not all three-cycles are independent in homology, there are four-chains with boundaries on some of them corresponding to the relations which they satisfy. After the transition, the four-chains close off because their boundaries shrink. As a consequence, the dual geometry does involve compact cycles in H4 (X, Z).
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α+β
α
α
2α+β
α β
β
α−β
α+β β
α+2β
β
β α+β α
α+β
α+β
Fig. 3. This shows the geometric transition of the Calabi-Yau in the previous figure. In the leftmost geometry there are three minimal 3-cycles. The lengths of the dashed lines are proportional to their sizes. The intermediate geometry is singular, and the figure on the right is the base of the smooth toric Calabi-Yau after the transition. This Calabi-Yau is related to B3 by flopping three P1 ’s
In the language of (p, q) five branes, the geometric transition corresponds to a phase transition in the five-dimensional theory. Namely, the configuration of intersecting (1, 0) and (0, 1) five-branes is a phase transition point: the Higgs phase with five-branes separated in the z plane2 meets the Coulomb phase, where a piece of (1, 1) brane resolves the singularity. In the geometry, there is a T 2 fiber whose (1, 1) cycle degenerates over this interval, and the cylinder is capped off to a P1 by all the cycles of the T 2 degenerating over the boundaries of the interval. The singularity can also be resolved with a (1, −1) brane, which corresponds to the flopped P1 .
2.4. Geometry of holomorphic curves. Calabi-Yau manifolds generally come with families of embedded curves. In the topological A-model only holomorphic curves are relevant, as the A-model string amplitudes localize on them. In the presence of D-branes wrapping Lagrangian submanifolds Mi in X, we must also consider holomorphic curves with boundaries on the Mi ’s. Holomorphic curves have a very simple description in the toric base, or equivalently, in the (p, q) five brane language. Let us first consider closed string geometries, the family of Calabi-Yau manifolds we have called X t above. In this case, it can be shown that all the compact holomorphic curves in a non-compact toric Calabi-Yau manifold wrap a 1-cycle in the T 2 fiber direction. Holomorphic curves project to lines in the toric base, and locally the direction of the curve in the base is correlated with its direction in the fiber. For the compact curves, the direction in the fiber is the (p, q) 1-cycle of the T 2 . This is most transparent in the (p, q) five-brane language. Namely, consider an M-theory membrane, wrapping a holomorphic curve on X t . By M-theory/type IIB duality, a membrane wrapping a (p, q) cycle of the T 2 is dual to a (p, q) string, therefore membranes on holomorphic curves in X t that are along the T 2 in the fiber are dual to webs of (p, q) strings in type IIB string theory that are BPS. 2 Recall that this is the complex structure modulus of the geometry. The five dimensional hyper-multiplet contains a compact scalar from the period of the C-field through the S3 (or the positions of the five-branes on the S1 in type IIB). It also has a non-compact scalar from the “period” of the C-field through the non-compact three-cycle dual to S3 or the “Wilson line on R” for the five-brane. Topology-changing transitions of toric manifolds are discussed in [35].
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Fig. 4. The figure on the left depicts a genus one holomorphic curve with three holes ending on three minimal three-cycles. The figure on the right is after the transition, and also depicts a genus one curve, but without boundaries
Moreover, the compact curves are related to webs ending on (p, q) five-branes [33]. An example of such a curve is given in the right portion of Fig. 4. Much of the same considerations are clearly true in X as well. There is however an important distinction from the point of view of topological string amplitudes. Namely, in a generic situation, there are no compact holomorphic curves. This is easy to see in the (p, q) five-brane picture because a condition for the web to be supersymmetric is that it lives in the plane parallel to the web of the five branes, and is therefore pointlike in the z plane (there is an additional condition that fixes the orientation of the (p, q) string depending on its charge, which comes from the balance of tensions, by requiring it to be orthogonal to a (p, q) five brane. We refer the reader to [33] for a detailed discussion). However, a given (p, q) string can only end on the five brane of the same charge, and so for five-branes at generic locations in the z plane, the string webs are never compact. This situation changes if strings can end elsewhere. For example, if there are M-theory five-branes wrapped on Lagrangian cycles in X, membranes can end on them. The M-theory five-brane is replaced with a D3-brane in IIB string theory ending on the various (p, q) five-branes. There are then compact string webs ending on the D3-branes, corresponding to holomorphic curves with boundaries. In relation to large N dualities in the superstring context it is more natural to consider type IIA string on X instead, with D6-branes on the Mi ’s. This is related, via duality of IIA/M-theory on S1 , to IIB string theory with Kaluza-Klein monopoles ending on the (p, q) five brane web [7]. Namely, the D6 branes in Lagrangian submanifolds of X lift to M-theory on a G2 holonomy manifold. To obtain this manifold, we need to consider an extra S1 which is fibered over the corresponding CY. This is related to IIB on B × S1 , where we exchange the 11th circle with the T 2 that fibers X. What used to be the 11th circle is now fibered nontrivially over B. In particular, the circle vanishes over a 2-dimensional subspace of the type IIB 5-dimensional geometry. It vanishes along the line in B ending on the (p, q) five branes as well as on the S1 which is dual to the T 2 of M-theory. This line in B corresponds to the line in the base of X over which there is the three-manifold that the D6 brane wraps. The IIB 5-brane web is in a background of ALF geometry dictated by the location of the Lagrangian submanifold in X.
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2.5. Geometry of three-cycles. In this paper, we will wrap D-branes on the minimal three-manifolds in X. The physics of the D-branes depends on both X and the threemanifold M it is wrapped on, so we will describe the geometries of the latter. In our context, M is obtained by pinching the cycles of the T 2 fibers over the endpoints of an interval in the base. Clearly, if the same cycle vanishes at both ends, the topology of the three-manifold is S2 ×S1 , as there is a cycle of the T 2 that never vanishes on M. An example where the manifold is S3 arises in the familiar context of T ∗ S3 . This S3 comes from a (1, 0) cycle of the T 2 vanishing at one end, and (0, 1) cycle vanishing on the other. To see that this is an S3 , note that at x = y¯ and u = −v¯ Eq. (2.1) defining T ∗ S3 becomes |x|2 + |u|2 = µ, and µ is real and positive, so this is a three-sphere. In view of the discussion above, we can regard this as a real interval, together with the (1, 0) one-cycle that corresponds to the phase of x = y, ¯ degenerating at the end with x = 0, and the (0, 1) cycle that is the phase of u = −v¯ degenerating over the u = 0 endpoint 3 . More generally, we have the following. For our current purpose, by an SL(2, Z) transformation of the T 2 we can make (1, 0) be the vanishing cycle over one of the boundaries, and let (q, p) be the cycle that vanishes over the other. The 3−manifold itself is a Lens space L(p, q). Remember that lens spaces are defined as quotients of S3 by a Zp action. The space L(p, q) is given by |x|2 + |u|2 = 1,
(x, u) ∼ (exp(2iπ/p)x, exp(2iπ q/p)u).
(2.2)
To see that, consider an S3 which, as explained above, is a T 2 fibration over an interval, where the cycles of the T 2 are generated by phases of x, u. If the complex structure of the T 2 corresponding to S3 is τ , then an SL(2, Z) transformation that takes this T 2 to a T 2 with (1, 0) and (q, p) cycles vanishing over the endpoints will take τ to τ = τ +q p . But the T 2 with the new complex structure is precisely a quotient of the original one by the Zp action specified in (2.2). For our later considerations in this paper it is important to have another view on this construction of a three-manifold M as a T 2 fiber over interval. The construction is as follows: we are gluing two solid tori over (say) the midpoint of the interval, up to an SL(2, Z) transformation VM that corresponds to a diffeomorphism identification of their boundaries. Let us call the two tori on each side of the midpoint by TL2 and TR2 . The embedding of this in the Calabi-Yau geometry provides a canonical choice of VM . In the Calabi-Yau geometry, there is a natural choice of basis of cycles α, β of the T 2 that fibers X, which is provided by the choice of complex structure on X. We can identify the one-cycles of the T 2 fiber that shrink over the left and the right sides of the interval with the shrinking 1-cycles of TL and TR . The diffeomorphism map VM is the SL(2, Z) transformation that relates one of the shrinking cycles of the fiber of X to the other one. Let us now explain the construction of the gluing matrices that will suit our purpose. Let (pL , qL ) be the cycle of the T 2 fiber that degenerates over the left half on M, and let (pR , qR ) be the cycle that degenerates over the right half. The gluing matrix VM can be written as VM = VL−1 VR ,
(2.3)
3 This is in fact the minimal S3 in X, as it is a fixed point set of the real involution on (2.1) given by x → y¯ and u → −v. ¯
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pL,R sL,R ∈ SL(2, Z). Clearly, VM is unique up to a homeomorqL,R tL,R phism that changes the “framing” of three-manifold [36] and takes
where VL,R =
VL,R → VL,R T nL,R , 11 where T is a generator of SL(2, Z), T = . This is a consequence of the fact that 01 there is no natural choice of the cycle that is finite on the solid torus. In the case of M = S3 above, since(1, 0) degenerates in the left half of M and (0, 1) 0 −1 . As a small modification, we could make in the right half, VM = S, where S = 1 0 (p, 1) degenerate over the left half instead, so that VL = T p S is a lens space L(p, 1) and V is S −1 T −p S. For most considerations in this paper we will be considering the cases L(1, 1) or L(1, 0), which are homeomorphic to S3 . 3. Open String Theory We are interested in the topological A-model on the Calabi-Yau geometries described above, with D-branes wrapping special Lagrangian three-spheres. The local geometry in some neighborhood of a Lagrangian three-manifold M is T ∗ M and it was shown in [26] that the topological A-model corresponding to N D-branes on M is a U (N ) Chern-Simons theory on three-manifold M, Z = DAeSCS (A) , where
ik SCS (A) = 4π
2 Tr A ∧ dA + A ∧ A ∧ A 3 M
is the Chern-Simons action. The basic idea of this equivalence is as follows: the pathintegral of the topological A-model localizes on holomorphic curves. When there are D-branes, this means holomorphic curves with boundaries ending on them. In the T ∗ M geometry with D-branes wrapping M there are no honest holomorphic curves, however there are degenerate holomorphic curves that look like trivalent ribbon graphs and come from the boundaries of the moduli space. This leads to a field theory description in target space, which is equivalent to topological Chern-Simons theory (as the abstract open string field theory formulation demonstrates [37]). In this map, the level k would be naively related to the inverse of the string coupling constant gs . However, quantum corrections [36] shift this identification to 2πi = gs . k+N More globally, however, the geometry is generally not that of the cotangent space to any manifold, and there can be D-branes wrapping other minimal three-spheres in X. In this case the topological open strings will have contributions from degenerate holomorphic curves, which are captured by Chern-Simons theories, as well as some honest holomorphic curves, which lead to insertion of some Wilson loop observables for the Chern-Simons theory [26]. If we have a number of Mi ’s distributed in some way inside
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a Calabi-Yau, with Ni D-branes wrapped over Mi , then we can trade the degenerate holomorphic curves by including the corresponding Chern-Simons theories Si = SCS (Ai ) coupled in an appropriate way with the honest holomorphic curves. Namely, we have eFall = DAi eSi +Fndg (Ui (γi )) , (3.1) i
where Fall denotes the full topological A-model amplitude, and Fndg denotes the contribution of the non-degenerate holomorphic curves to the topological amplitudes. These holomorphic curves give rise to Wilson loops on the D-branes: each holomorphic curve with area A ending on Mi over the knot γi leads to the contribution e−A i TrUi (γi ) to Fndg , where Ui (γi ) denotes the holonomy of the Chern-Simons gauge connection around the knot γi . Notice that all these Chern-Simons theories have the same coupling constant. More precisely, 2πi = gs . ki + N i In the toric examples we will consider in this paper it turns out that only holomorphic annuli contribute to Fndg and thus this connection with Chern-Simons theory is a useful way to compute the topological A-model amplitudes as some particular correlation function in a system of coupled Chern-Simons theories. 3.1. The annulus amplitude. As an example, let us consider the Calabi-Yau manifold X with b3 = 2, b2 = 1 in Fig. 5, whose complex structure is described by xy = z, x y = (z − µ1 )(z − µ2 ),
(3.2)
and was studied in a physics context in [38]. In X, the α cycle of the T 2 degenerates over the point z = 0, but the β cycle degenerates twice, over z = µ1 and µ2 . The cycles over [µ1 , 0] and [0, µ2 ] are three–spheres M1,2 that generate H3 (X) . The base space of X, with loci with degenerate fibers – is pictured in Fig. 5, where we have taken µ’s to be real, and µ1 < 0 < µ2 (the reader should keep in mind that only one of the dimensions of the z plane is visible in the base). As is clear from the picture, there is an additional parameter visible in the base: the relative distance of two β-branes. This is a K¨ahler parameter corresponding to the one compact 2-cycle in X (since X contains an S2 × S1 , as we discussed above, it certainly contains an S2 that cannot be contracted – we simply pick a point on the S1 in S2 × S1 )4 . As we discussed above, at the level of topological strings the theory is a U (N1 ) × U (N2 ) Chern-Simons theory, but since there are two stacks of D-branes, there is a new open string sector where one end of the string is on the D-branes wrapping M1 and the other on M2 . The ground states of this string correspond to constant maps to the S1 that the three-manifolds “intersect” over. Correspondingly, there are two states in the Ramond sector of the topological string, a real scalar and a one form, with U (1)R charges −1/2 and 1/2. Only the scalar is physical, and taking into account both orientations of the string, we get a complex scalar φ in (N1 , N 2 ). This complex scalar 4 The interpretation of this K¨ahler parameter is obvious in the type IIB dual, as it is the scalar field of the six dimensional N = (1, 1) supersymmetric theory on the two parallel (0, 1) five-branes. Duality relates this to a Calabi-Yau three-fold in M-theory containing a curve of A1 singularities [38].
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Fig. 5. Calabi-Yau geometry with b2 = 1, b3 = 2 and two minimal S3 ’s as the dashed lines
is generically massive, and its mass is proportional to the “distance” between M1 and M2 given by the complexified K¨ahler parameter r. We will show below that the only modification of the topological string we need to make in this geometry is to include the minimally coupled complex scalar in this sector. Because of the topological invariance ofthe theory the action of a charged scalar with minimal coupling is of the form Lφ ∼ γ Tr φ¯ (d − A1 + A2 ) φ. Note that the scalar field gets a mass from turning on a Wilson line on the S1 it propagates on. We will pick its “background” value which we denote by r below. The path integral involving φ is Gaussian so it can be easily evaluated [2] and gives:
−1/2 1/2 1/2 −1/2 O(U1 , U2 ; r) = exp −Tr log er/2 U1 ⊗ U2 − e−r/2 U1 ⊗ U2 ∞
e−nr −n n = exp , (3.3) TrU1 TrU2 n n=1
where U1,2 are the holonomies of the corresponding gauge fields around a loop5 Ui = P exp Ai ∈ U (Ni ), i = 1, 2. γ
Note that the operator O is the amplitude for a primitive annulus of size r with boundaries on M1 and M2 , together with its multicovers [2, 5, 8]. This annulus is depicted in 5
1/2
−1/2
In going from the first to the second line in (3.3), we have dropped a factor of det(U1 )det(U2 ) √ √ N1 N2 in O. This factor, which equals exp TrA − TrA , can be absorbed away in a redefi1 2 γ γ 2 2 nition of r. It is likely that this is related to the holomorphic anomaly of topological strings [39], and this clearly deserves further investigation.
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M2
M1
Fig. 6. There is one holomorphic annulus connecting the two S3 ’s. This corresponds to a one-loop computation with a bifundamental string running around the loop
Fig. 6, and it is a piece of the holomorphic curve that is wrapped by the (1, 0) brane. This curve is obtained by setting x = 0 = y = z in (3.2) and is given by x y = µ1 µ2 . The Chern-Simons path integral in this geometry is therefore defined with the insertion of the above operator. The path integral of the A-model string field theory in this background is therefore given by:
∞ e−nr −n SCS (A1 )+SCS (A2 ) n (3.4) TrU1 TrU2 . exp Z = DA1 DA2 e n n=1
To recapitulate, we have Chern Simons theory on two three-manifolds M1 and M2 connected via an annulus. The boundaries of the annulus look like S1 ’s in both of them, i.e. we have one knot in each Mi . The topological theory is computing the expectation value of the operator O(U1 , U2 ; r), which involves Wilson loop operators around the two knots. This obviously extends to more general configurations: for every pair of three-manifolds M1 and M2 that are connected by a holomorphic annulus, we will get a bifundamental complex scalar. Integrating this field out, we find we need to insert an operator (3.3), where r is the size of the corresponding annulus diagram in spacetime. As an example, consider Fig. 7. There are Ni D-branes wrapping a chain of four minimal spheres Mi , i = 1, . . . 4 connecting two (1, 0) branes and two (0, 1) branes. For every pair of spheres intersecting over an S1 we get a bifundamental scalar field, so we have matter in representations (Ni , N i+1 ), where i = 5 corresponds to the first sphere again. Note that in the (N1 , N 2 ) and (N3 , N 4 ) sector, the bifundamental scalar is not localized, and correspondingly in Fig. 7 there is a family of annuli. In fact, a careful reader has probably noticed that this could have happened in the two-sphere case as well, had we not chosen judiciously the ordering of (1, 0) and (0, 1) branes in the z direction of the base. In other words, we could have picked µ1 , µ2 > 0 in (3.2) , and we would have found a family of annuli. See Fig. 7. This objection is in fact its own cure. Namely,
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M1
M2
M3
Fig. 7. In the left figure, it looks like there is a family of holomorphic annuli between M1 and M2 , and a holomorphic annulus connecting M1 with M3 . However, by moving in the complex structure moduli space we get to the figure on the right, where it is clear that there is an isolated holomorphic annulus connecting M1 and M2 and no holomorphic curve between M1 and M3
by changing the complex structure of X we could go from one configuration to the other. In fact, using the other direction in the z-plane we can do this in a smooth way, as the µ’s are complex, without passing through a singularity of the three manifold. On the other hand, the topological A-model amplitudes cannot depend on the complex structure moduli. As a consequence, the value of operators O(Ui , Ui+1 ; ri ) cannot change in passing between the two configurations, and they are given by the annulus computation we already outlined. This idea is rather powerful and it leads to the fact that, in all the toric cases, the only holomorphic curves are annuli connecting pairs of S3 ’s along lines on the toric base (i.e. along loci of (p, q) 5-branes). The argument for this is extremely simple: as we explained before, we can deform the theory to a generic point in the complex structure moduli space, where it is manifest that the only holomorphic curves are annuli. By the fact that topological A-model amplitudes do not depend on complex structure moduli, we can immediately conclude that only annuli contribute to topological string amplitudes. Let us now explain why this preserves only annuli. At a generic point of the complex structure moduli space (which we can always choose), the T 2 fibers degenerate over a set of points in the z plane, and the three-cycles Mi in X project to lines connecting them, which are generically not aligned. Recall that the holomorphic curves project to points in the z plane. This means that the “large” holomorphic curves must project to points in the z-plane (as discussed in Sect. 2) where different Mi ’s intersect, and for a generic choice of complex structure these are the points where the T 2 fibers degenerate. In other words, we are left only with annuli over loci where the T 2 fibration degenerates, as we wished to show6 . 4. D-branes on Chains of Three-Manifolds and Knot Invariants In order to evaluate the A-model partition functions in these backgrounds we need a few additional pieces of data. Namely, we need to know how the different knots are 6 Note that at a generic point in complex structure moduli space, M are not mutually supersymmetric, i and when we add D-branes supersymmetry is broken. However, for topological A-model amplitudes to make sense we do not require supersymmetry, and D-branes need only be Lagrangian, which holds for any complex structure.
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linked, in particular their linking numbers lk(γi , γj ), and also what is their framing – the self linking number of each of the γi ’s. As it is explained in [36, 40], the framing is a rather subtle effect from the point of view of Chern-Simons theory, having to do with the fact that in evaluating expectation values of the Wilson loop operator associated to the knot, one encounters certain ambiguities in the calculation. These are akin to a choice of point-splitting regularisation, since to calculate the self-linking number in a way that is consistent with topological invariance one must choose a “framing” by thickening a knot into a ribbon. Different framings differ by adding twists to the ribbon, the framing itself being defined as the linking number of the two edges of the ribbon. The Wilson loop operators are not invariant under the change of framing. We will show below that different choices of framing correspond in the present context to different target space geometries. The role of framing in topological string theory was discovered in [7] in a closely related context and studied subsequently in [8, 41, 42]. 4.1. Rewriting O. Before we proceed, it is key to note that there is an illuminating way to write the operator O(U1 , U2 ; r), by using the techniques of [3]. If we expand the exponential explicitly, we get: 1+
∞ h=1 n1 ,··· ,nh
h
1 e−r i=1 ni TrU1n1 · · · TrU1nh TrU2−n1 · · · TrU2−nh . h! n1 · · · nh
(4.1)
as in [3]: ki is the number Now we write the h-uples (n1 , · · · , nh ) in terms of a vector k, of i’s in (n1 , · · · , nh ). Taking into account that there are h!/ j kj ! h-uples that give k and that n1 · · · nh = j j j , we find that (4.1) equals the same vector k, 1+ where zk =
j
e− r k
zk
j
(4.2)
kj !j kj , ϒk (U ) =
and =
ϒk (U1 )ϒk (U2−1 ),
∞
Tr U j
kj
,
(4.3)
j =1
j kj . Now, the Frobenius formula tells us that Tr R (U ) =
1 χR (C(k))ϒ k (U ). zk
(4.4)
k
Using this together with orthonormality of the characters gives immediately that O(U1 , U2 ; r) = Tr R U1 e− r Tr R U2−1 , (4.5) R
where is the number of boxes in the Young tableau of R and the sum is a sum over all representations, including the trivial one. We remind the reader that Ui is a Wilson line in the three-manifold Mi . Notice that this operator is the cylinder propagator for a two-dimensional gauge theory [43, 44], in which r plays the role of time and the Hamiltonian is given by the first Casimir of U (N ) (which counts precisely the number of boxes of a representation).
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4.2. Framing. One of the key ideas used in [36] is that one can cut the manifold up into pieces on which one can solve the theory, and then glue them back together. Central to the story is also the relation of the Hilbert space of Chern-Simons theory with the space of conformal blocks of WZW models. Recall that all manifolds Mi in our geometry can be obtained by gluing together solid tori with a diffeomorphism identification of the boundary. Associated to the boundary T 2 we have a finite dimensional Hilbert space, and a basis of states is labeled by the representations of the affine Lie algebra [36]. We will denote this basis by |R . The dual Hilbert space has a basis R|, where R denotes the representation conjugate to R. The dual pairing is simply R 1 |R2 = δR1 R2 . Notice that |R can be computed by the path integral on a solid torus with insertion of a Wilson line in representation R around the cycle that is non-trivial in homology. The corresponding state in the dual Hilbert space R| is obtained by doing the same path integral but over the manifold with opposite orientation. In the context of Chern-Simons theory with no insertions, because the diffeomorphism of the boundary induces a linear transformation of the Hilbert space, one can think about the path integral on M in terms of the path integrals on two solid tori, that are then glued together with an SL(2, Z) matrix VM that specifies M. Since we are making no insertions, the state associated to each of the solid tori is the vacuum |0 (corresponding to the trivial representation), and the partition function of Chern-Simons theory on M is Z(M) = 0|VM |0 . In the problem at hand, we are interested in the Chern-Simons amplitude not in the vacuum but in the presence of Wilson lines. The gluing procedure that gives the partition function can be generalized to this setting, since the role of the insertions will simply be that the solid tori give rise to states |R with arbitrary R. In our problem we have insertions of operators O(Ui , Uj ; r) corresponding to annuli connecting the two manifolds. Each annulus is attached to the Mi ’s either on its left or the right “half”, and by (4.5) we can regard it as carrying a Wilson line in an arbitrary representation R of the gauge group on the right half, and the conjugate representation R in the left half. We also have to sum over all representations. For example, in the two-sphere case, there is a knot on the right half of M1 and the left half of M2 . We thus have Z(M1 , Tr R U ) = 0|VM1 |R , and
Z(M2 , Tr R U −1 ) = R|VM2 |0 ,
where the Tr R U , Tr R U −1 mean that we do the path integral with the insertion of these operators. Thus, by using (4.5) and the above gluing techniques, the full amplitude (3.4) can be written as: Z=
0|VM1 |R e− r R|VM2 |0 . (4.6) R − r Note that were it not for the weight of e , we could use the resolution of the identity R |R R| = 1 and the operator insertion in (4.6) would correspond to a surgery operation that glues together M1 and M2 . The resulting manifold M1 #M2 would have gluing matrix VM1 #M2 = VM1 VM2 .
This corresponds to the geometric fact that, when r = 0, the two special Lagrangian three-spheres that we called M1 and M2 are exactly degenerate with M1 #M2 = S2 × S1 that is their sum in homology. But instead we have a finite r-time propagation with
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a Hamiltonian that counts the numbers of boxes. Namely, insertion of the operator O corresponds to cutting off the right half of M1 in the vacuum and the left half of M2 , and gluing in instead O(U1 , U2 ; r) = |R e− r R|. R
with canonical framing. Since α = (1, 0) In the case of Fig. 6, M1 and M2 are degenerates in the left half of M1 and β = (0, 1) in its right half, and α and β are 0 −1 −1 . Note exchanged for M2 , the gluing matrices are VM1 = S = VM2 , where S = 1 0 that standard surgery gives S2 × S1 with partition function equal to one, as expected. The amplitude (4.6) receives contributions from unknots on M1 and M2 in representation R. Note that the transformation that changes the framing of the three-manifold affects the Wilson loop amplitudes. The diffeomorphism by T n on the boundary of the solid torus with a Wilson loop in representation R in the center, adds n twists to the “ribbon” that frames the knot. The change of framing acts on the Wilson loop amplitude by S3 ’s
T |R = e2πi(hR −c/24) |R , where c is the central charge of the current algebra, and hR is the conformal weight of the WZW primary field in representation R. Recall that hR is given by hR =
R · ( R + ρ) , 2(k + N )
where R is the highest weight of R and ρ is the Weyl vector. The numerator CR =
R · ( R + ρ) is the quadratic Casimir of the representation. While there is no natural choice of framing, there is a canonical choice at least on S3 and this corresponds to zero self-linking number. In the above example, both unknots on S3 were framed canonically. Below, we will see that other choices of framing arise as well. 4.3. Linking. The considerations above completely determine the gluing matrices, up to irrelevant framings that do not affect the amplitude by other than renormalization of r. Therefore, the linking of the different knots should be determined as well. Let us discuss this issue with a concrete example. The three-spheres in Fig. 8 form a necklace where each is intersecting the other two over an S1 , so we get scalar fields in the bifundamental, and integrating them out leaves us with the annuli shown in the figure. The path integral of the A-model in this background involves four Chern-Simons theories on a chain of four three-spheres connected with annuli: 3 Z= DAi eSCS (Ai ) O(U1 , U2 ; r1 )O(U2 , U3 ; r2 )O(U3 , U1 ; r3 ). (4.7) i=1
There are two unknots on each three-sphere, and the amplitude will depend on their linking, in addition to their framing. Just as above, we can use (4.5) to write this in a more transparent form
R 1 |VM3 R3 e− 3 r3 R 3 |VM2 R2 e− 2 r2 R 2 |VM1 R1 e− 1 r1 . (4.8) Z= R1 ,R2 ,R3
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Since the D-branes go around the loop from one degeneration locus to the other and then back to the first one, we must have VM3 VM2 VM1 = 1. This will hold generally whenever there are closed loops with D-branes in the toric diagram. Looking at Fig. 8 we can read off, VM1 = S −1 ,
VM2 = ST −1 S,
VM3 = T S −1 ,
(4.9)
α
R1 R3
R1
R3 β
R2 R2
α+β Fig. 8. In the figure there is a chain of three minimal spheres connecting (1, 0), (0, 1) and (1, −1) branes, with branes wrapped on each sphere
Fig. 9. The Hopf link with linking number lk = +1
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so that the last factor of (4.8) is given by . Z(M1 , L(R2 , R1 )) = R 2 |S −1 |R1 = SR−1 1 R2
(4.10)
This means that M1 is a three-sphere with two unknots that are linked into a Hopf link L(R2 , R1 ) of linking number lk = 1 (see Fig. 9). Namely, as we have seen previously, S3 is obtained by identifying two solid tori up to an S transformation that exchanges the α and the β cycles of the T 2 . If β is the nontrivial cycle of the solid T 2 , and along this cycle one has a knot in representation R1 and another one in representation R2 , then the S −1 transformation results in two unknots with zero framing (this does not add any twists to the ribbons that frame the knots), but which are linked in a Hopf link with linking number lk = 1 (an S transformation would give a Hopf link with linking number −1). Similarly, using ST −1 S = T S −1 T , we see that M2 has a Hopf link with two knots of framing +1, Z(M2 , L(R3 , R2 )) = R 3 |T S −1 T |R2 .
(4.11)
Finally, M3 is a three-sphere with a Hopf link whose components are an unknot carrying representation R1 and with framing +1, and an unknot carrying representation R3 with canonical framing: Z(M3 , L(R1 , R3 )) = R 1 |T S −1 |R3 .
(4.12)
Fig. 10. The amplitude associated to this geometry can be interpreted in terms of a lattice model. The annuli correspond to states of the lattice. The 3-manifolds correspond to the interaction vertices. The figure shows the annuli on the “primitive” edges and some “non-primitive” ones
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4.4. Lattice model interpretation. The models discussed above clearly generalize to more complicated geometries like the one depicted in Fig. 10, where we have suppressed one direction of the base. The rules for computing the amplitudes should be clear from the previous discussion: 1) The model has states associated to all the edges of the lattice. Some of the edges are “primitive” (connecting nearest neighbor nodes) and some are “non-primitive” (connecting other nodes but always along the straight lines of the lattice). The states are labeled by representations of the affine Lie algebra (i.e. a state in the Hilbert space of T 2 , H). To each state on the ith edge we associate a weight e− ri , where ri is the length of the corresponding edge and the is the number of boxes in the corresponding representation. 2) To every vertex we associate a linear operator. This linear operator is obtained by computing matrix elements like the ones depicted below:
R1
R4
~ R3
< R 1R 2 |V| R 3 R 4 >
R2 Fig. 11. The four-point vertex
Here, R1 , R2 and R3 , R4 are the two pairs of representations corresponding to the collinear edges. A state |R, R is obtained by doing the Chern-Simons path integral over the solid torus with two parallel Wilson lines inserted along its nontrivial cycle, in representations R, R . V is the gluing matrix, as explained before. As is well known, using the fusion rules of the WZW theory, we can write R |R, R = NRR |R , R
where the fusion coefficients are given by the Verlinde formula [45]
R NRR =
SRQ SR Q SR−1 Q S0Q
Q
,
Using this we can write the four-point vertex as Q Q
R 1 , R 2 |V |R3 , R4 = NR1 R2 VQQ NR3 R4 ,
(4.13)
(4.14)
Q,Q
where V denotes the corresponding modular transformation matrix. Notice that, although there are four primitive edges ending on each vertex, one can have many non-primitive edges ending on the same vertex. In that case, we will have matrix elements of the form ,
R 1 , · · · , R n |V |R1 , · · · , Rm
(4.15)
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where the in- and out-states can be evaluated by a repeated use of the fusion rules. to the In (4.15), R 1 , · · · , R n correspond to a set of collinear edges, and R1 , · · · , Rm other set. As explained before the solid tori are glued together by an SL(2, Z) matrix V that is computed as in (2.3). 3) Since there are edges that go off to infinity, there are boundary conditions: the state on these edges is always the trivial representation7 . 4) The amplitude is the product of the linear operators over all the vertices, together with the weights associated to the connecting edges. At the end we sum over all representations on each link. 5. Large N Duality As was recently demonstrated [24], one can derive the large N duality conjecture of Chern-Simons on S3 with topological strings [1]. In this derivation one starts with the linear sigma model description of the closed string side and finds that in some limit the theory develops the Coulomb and Higgs branch. The Coulomb branch plays the role of holes in the dual Chern-Simons description. The models we are considering here all admit a linear sigma model description [46], as discussed in [47] . Thus one can start from the gravity side, and go to the point on moduli for each U (1) gauge factor and repeat the analysis of [24], which should lead to a topological open string description with Ni D-branes wrapped around S3i . The analysis we did for the open string demonstrated that this open string can in turn be written in terms of some link observables in the product of U (Ni ) Chern-Simons theories. Thus we find the general prediction that Fclosed (ti , ra ) = Fopen (Ni gs , ra ), where by Fopen we mean the open string amplitude with link observables inserted, and the ra correspond to sizes of the annuli. In this equation, the ti on the closed string side are the K¨ahler moduli of the blowups corresponding to where the S3i were, and ti = Ni gs . As we will see later, ra = ra − 21 (ta1 + ta2 ), where tai denote the K¨ahler moduli associated to the two ends of the annulus a. It would be interesting to repeat in detail the analysis of [24] for the case at hand and thus obtain these shifts directly. 6. Closed String Localization In this section we argue that the large N duality proposal given in the previous section is in accord with localization ideas in computation of the closed string invariants. It was suggested in [48] that one can use circle actions to localize closed topological string amplitudes. The final answer takes the form of a sum over certain graphs with nodes corresponding to genus g Riemann surfaces. This idea has been further developed [49] and applied to some concrete examples in [30] (at genus zero) and [29] (for higher genus). The geometry of the localization is very much related to the (p, q) 5-brane graphs we have in our setup. Basically one ends up with sums over graphs whose links correspond to intervals in the (p, q) 5-brane web connecting adjacent vertices. These correspond to rational curves in the closed string side, and in the gauge theory setup they 7 It would also be interesting to put periodic boundary conditions and interpret them as partially compact Calabi-Yau models.
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correspond to annuli. Moreover one is instructed to consider all genera computation on each node, which corresponds to mapping the whole Riemann surface to that point on the toric geometry. This ends up with a particular computation of a characteristic class on the moduli space of Riemann surfaces, that in particular depends on which links have been used in the graph. This seems to match naturally with the Chern-Simons computation, where each node is replaced by open string Riemann surfaces captured by Chern-Simons, coupled to each other through the Wilson loop expectation values coming from annuli. It is as if the Chern-Simons theory is computing directly the relevant characteristic classes on moduli of Riemann surfaces. This is not at all surprising, in light of the observations in [8] where one can use the framing dependence of unknot in Chern-Simons theory to compute all intersection numbers of Mumford classes with up to three Hodge classes, which is what the closed string side computes [41]. It would be extremely interesting to make this connection with Kontsevich integral more precise and reduce the statement of the equivalence to some concrete computation at each node, which is being done using Chern-Simons gauge theory. Incidentally this is in the same spirit of the current methods of computation of these invariants where one uses Kontsevich’s results on matrix realization of Mumford classes [50] together with certain results of Faber [51]. However, the Chern-Simon gauge theory is a more natural realization of this computation. 7. Closed String Invariants from Chern-Simons Theory In this section we will show that the large N duality proposed in Sect. 5 is a powerful way to compute closed string topological A-model amplitudes for local Calabi-Yau manifolds, in terms of Chern-Simons amplitudes. In particular we will consider examples of P2 blown up at three points (B3 del Pezzo), and P1 × P1 blown up at four points. Since the size of the blow ups are proportional to the rank of the corresponding dual gauge group, we can also consider the limit where the blown up P1 ’s have infinite size by considering the Ni → ∞ limit. This in particular leads to computation of topological strings for P2 and P1 × P1 inside a Calabi-Yau threefold. 7.1. Chern-Simons invariants of unknots and Hopf links. The toric geometries that we have described involve framed unknots and Hopf links, therefore in the evaluation of the Chern-Simons amplitudes we will need the invariants of the unknot and the Hopf link in arbitrary representations of SU (N ). In this section we give precise formulae for these invariants. Our notation is as follows: WR1 ,R2 (L) denotes the vacuum expectation value in Chern-Simons theory corresponding to the link L with components K, K : WR1 ,R2 (L) = Tr R1 (U1 )Tr R2 (U2 ) ,
(7.1)
where U1 , U2 are the holonomies of the gauge field around the knots K and K , respectively. If, say R2 = · is the trivial representation, the vev (7.1) becomes the vev of the knot K (the second knot disappears), and we will denote this vev by WR (K). The vacuum expectation values denoted by · are normalized, so that they denote the path integral with insertions and divided by the partition function (in other words, the vev of the identity operator is one). Of course the duality also cares for the overall normalization (i.e., the vacuum energy) and this we will put in at the end of the computation. We also recall our notation for the Chern-Simons variables:
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q = exp
2πi , λ = qN . k+N
(7.2)
It is well-known that the Chern-Simons invariant of the unknot in an arbitrary representation R is given by the quantum dimension of R: WR =
S0R = dimq R. S00
(7.3)
The explicit expression for dimq R is as follows. Let R be a representation corresponding to a Young tableau with row lengths {µi }i=1,··· ,d(µ) , with µ1 ≥ µ2 ≥ · · · , and where d(µ) denotes the number of rows. Define the following q-numbers: [x] = q 2 − q − 2 , x
1
x
1
[x]λ = λ 2 q 2 − λ− 2 q − 2 . x
x
(7.4)
Then, the quantum dimension of R is given by µi −i d(µ) [v]λ [µi − µj + j − i] . dimq R = µi v=−i+1 [j − i] [v − i + d(µ)] v=1 i=1 1≤i<j ≤d(µ)
(7.5)
1
The quantum dimension is a Laurent polynomial in λ± 2 whose coefficients are rational 1 functions of q ± 2 . In what follows in some cases we will also be interested in the leading power of λ in the above expression. It is easy to see that this power is /2, where = i µi is the total number of boxes in the representation R, and the coefficient of 1 this power is the rational function of q ± 2 , q κR /4
1≤i<j ≤d(µ)
d(µ) µi [µi − µj + j − i] 1 , [j − i] [v − i + d(µ)]
(7.6)
i=1 v=1
where κR = +
d(µ)
µi (µi − 2i).
(7.7)
i=1
This quantity is related to the quadratic Casimir of the representation. In fact, one has
R · ( R + ρ) = κR + N − 2 /N . Let us now consider the Hopf link with linking number 1. Its invariant for representations R1 , R2 is given by WR1 ,R2 = q 1 2 /N
SR−1 1 R2 S00
,
(7.8)
where i is the total number of boxes in the Young tableau of Ri , i = 1, 2. The prefactor q 1 2 /N in (7.8) is a correction which was pointed out in [8], and is due to the fact that the vev WR1 ,R2 has to be computed in the theory with gauge group U (N ). Although the expression for the S-matrix is explicitly known, it is not straightforward to write it in terms of q and λ, which is what we need. We can use the Verlinde formula (4.13) giving
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the fusion coefficients in terms of the S-matrix elements, as well as the well-known identity (ST )3 = S 2 = C, to obtain the expression WR1 ,R2 =
1
NRR1 ,R2 q 2 (κR −κR1 −κR2 ) dimq R.
(7.9)
R
This can be also derived by using the formalism of knot operators [52, 53], and it was used in [5] to obtain the integral invariants associated to the Hopf link (we must observe, however, that in [5] the Hopf link with linking number −1 was considered). Notice that the fusion coefficients NRR1 ,R2 become, in the large k limit, the Littlewood-Richardson coefficients for the tensor product R1 ⊗ R2 = R NRR1 R2 R, and since we are evaluating the invariants at large k, N , to compute (7.9) we have to use these tensor product coefficients. Another expression for the Chern-Simons invariant of the Hopf link in arbitrary representations has been recently obtained by Morton and Lukac by using skein theory [54, 55]. Let us briefly describe their result, which turns out to be very useful in order to compute the invariants. Let µ be a Young tableau, and let µ∨ denote its transposed tableau (remember that this tableau is obtained from µ by exchanging rows and columns). The Schur polynomial in the variables (x1 , · · · , xN ) corresponding to µ (which is the character of the diagonal SU (N ) matrix (x1 , · · · , xN ) in the representation corresponding to µ), will be denoted by sµ . They can be written in terms of elementary symmetric polynomials ei (x1 , · · · , xN ), i ≥ 1, as follows [56]: sµ = detMµ ,
(7.10)
where Mµij = (eµ∨i +j −i ), Mµ is an r × r matrix, with r = d(µ∨ ). To evaluate sµ we put e0 = 1, ek = 0 for k < 0. The expression (7.10), known sometimes as the Jacobi-Trudy identity, can be formally extended to give the Schur polynomial sµ (E(t)) associated to any formal power series i E(t) = 1 + ∞ n=1 ai t . To obtain this, we simply use the Jacobi-Trudy formula (7.10), but where ei denote now the coefficients of the series E(t), i.e. ei = ai . Morton and Lukac define the series E∅ (t) as follows: E∅ (t) = 1 +
∞
cn t n ,
(7.11)
n=1
where the coefficients cn are defined by cn =
n 1 − λ−1 q i−1 i=1
qi − 1
.
(7.12)
They also define a formal power series associated to a tableau µ, Eµ (t), as follows: Eµ (t) = E∅ (t)
d(µ) j =1
1 + q µj −j t . 1 + q −j t
(7.13)
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M. Aganagic, M. Mari˜no, C. Vafa
One can then consider the Schur function of the power series (7.13), sµ (Eµ (t)), for any pair of tableaux µ, µ , by expanding Eµ (t) and substituting its coefficients in the JacobiTrudy formula (7.10). It turns out that this Schur function is essentially the invariant we were looking for. More precisely, one has 2
WR1 ,R2 = (dimq R1 )(λq) 2 sµ2 (Eµ1 (t)),
(7.14)
where µ1,2 are the tableaux corresponding to R1,2 , and 2 is the number of boxes of µ2 . More details and examples can be found in [54]. It is easy to see from (7.14) that the leading power in λ of WR1 ,R2 is ( 1 + 2 )/2, and its coefficient is given by the leading 1 coefficient of the quantum dimension, (7.6), times a rational function of q ± 2 that can be easily computed by taking λ → ∞ in E∅ (t). As a simple example of the Morton-Lukac formula, we can compute W( , ) . In this case, s{1} = e1 , and it is enough to expand E{1} (t) at first order, 1 − λ−1 E{1} (t) = 1 + 1 − q −1 + t + ··· , q −1
(7.15)
so that we obtain W(
, )
=
1
1
1
1
λ 2 − λ− 2
2
q 2 − q− 2
+ λ − 1.
(7.16)
In the same way, one can easily find: W(
3
, )
1 1 − q2 + q3 q −1 + 1 + q 3 − λ2 3 1 1 1 1 3 q 2 − q − 2 (q + 1) q 2 − q − 2 (q + 1)
= λ2
3 q −1 + 1 + q 2 1 − λ− 2 , 1 1 3 1 1 3 q 2 − q − 2 (q + 1) q 2 − q − 2 (q + 1)
1
+λ− 2
W
,
1 q −2 − q −1 + q q −2 + q + q 2 − λ2 =λ 3 1 1 1 1 3 q 2 − q − 2 (q + 1) q 2 − q − 2 (q + 1)
(7.17)
3 2
1
3 q −1 + q + q 2 q − λ− 2 , 1 1 3 1 1 3 q 2 − q − 2 (q + 1) q 2 − q − 2 (q + 1)
+λ− 2
and so on. In the computations that give the invariants of P2 , we only need the rational 1 function of q ± 2 which multiplies the highest power in λ. The above results are for knots and links in the standard framing. The framing can be incorporated as in [8], by simply multiplying the Chern-Simons invariant of a link with components in the representations R1 , · · · , RL , by the factor (−1)
L
α=1 pα α
1
q2
L
α=1 pα κRα
,
(7.18)
where pα , α = 1, · · · , L are integers labeling the choice of framing for each component.
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7.2. Evaluation of the two-sphere example8 . The simplest example of how to compute a closed string amplitude from Chern-Simons theory comes from the geometry depicted in Fig. 6. As explained there, this involves computing the vacuum expectation value of the operator (4.5): Tr R U1 e− r Tr R U2−1 , (7.19) O(U1 , U2 ; r) = R
where U1 and U2 are the holonomies of dynamical gauge fields (in other words, we are computing the vev in a U (N1 ) × U (N2 ) theory, but with the same coupling constant). We are going to discuss the operator (7.19) in a more general setting, so that U1 is the holonomy around an arbitrary knot. We are going to assume however that U2 is the holonomy around an unframed unknot. We now have to take the vev of this expression by doing the functional integral over both the U (N1 ) field A1 and the U (N2 ) field A2 . Since we are assuming that U2 is the holonomy around an unknot with zero framing, we have that [2]
Tr R U2−1 A2 = Tr R U0−1 ,
(7.20)
where U0 is the element in the Cartan subalgebra that corresponds to exp(2π iρ/(k2 + N2 )). Therefore, the vev with respect to the field A2 gives Tr R U1 e− r Tr R U0−1 . (7.21)
O(U1 , U2 ; r) A2 = R
Notice that we can regard this vev as the generating functional considered in [2], where the source takes the particular value U0−1 . We can now take the vev with respect to the A1 field, and use the results of [2] to write: ∞ 1 (7.22) fR (q d , λd1 )e−d r Tr R U0−d .
O(U1 , U2 ; r) A1 ,A2 = exp d d=1
R
This expression is valid for any framed knot along which we take the holonomy U1 of the gauge field A1 . In this equation, λ1 = q N1 is the exponential of the ’t Hooft coupling for the U (N1 ) Chern-Simons theory. We can write the exponent of the right-hand side as follows,
fR (q d , λd1 )e− r Tr R U0−d =
1 f (q d , λd1 )e− r ϒk (U0−d ), zk k
(7.23)
k
R
where fk was introduced in [5] and is simply the character transform of the fR . The last factor can be easily computed to be ϒk (U0−d ) =
j
dj
− dj 2
dj 2
− q− 2
λ22 − λ2 q
dj
kj ,
where λ2 = q N2 . 8
The results in this section have been obtained in collaboration with P. Ramadevi.
(7.24)
492
M. Aganagic, M. Mari˜no, C. Vafa
The vev (7.22) can be written in a very suggestive way by using the results of [5] on Chern-Simons vevs. In particular, fk has the structure: fk (q, λ1 ) =
j
j
q− 2 − q 2
kj
j
1 1 2g−2 Q nk,g,Q λ1 . q− 2 − q 2
(7.25)
g,Q
Therefore, one finds for (7.22): log O(U1 , U2 ; r) A1 ,A2 kj ∞ dj − dj d 2g−2 nk,g,Q 1 −d dQ λ2 2 − λ22 = e−d r λ1 . q 2 −q2 d zk k
d=1 g,Q
(7.26)
j
This has the structure of the free energy for a closed string [57]: ∞
g
nm
d=1 g,m
dgs 2g−2 −d m· 1 2 sin e t , d 2
(7.27)
provided one finds the appropriate relation between the closed string K¨ahler parameters t , and the K¨ahler parameters for the open string appearing in (7.26). We will find the precise relation in the examples below. We also have to show that the expansion in (7.26) involves integers in a manifest way, since in (7.26) we are dividing the integers nk,g,Q by zk . The way to fix that is to recall that, as shown in [5], the “primitive” integer invariants R,g,Q . They are related by a linear transformation involving the , but N are not nk,g,Q characters of the symmetric group, N R,g,Q . = χR (C(k)) (7.28) nk,g,Q R
Using again the results of [5], one can show that
j
j
λ− 2 − λ 2
kj
1 1 R (λ−1 ), = λ− 2 − λ 2 χR (C(k))S
j
(7.29)
R
where SR (λ) is the monomial defined in [5]: if R is not a hook representation, it is zero, and if R is a hook of boxes with − s boxes in the first row, then SR (λ) = (−1)s λ−
−1 2 +s
.
Using this, we find nk,g,Q k
− dj 2
λ2
zk
dj
− λ22
kj
d d − = λ2 2 − λ22 SR (λ−d 2 )NR,g,Q ,
j
R
and this would lead to the identification 1 g 1 −2 Q −m· t 2 nm e λ1 N = λ2 − λ 2 e− r SR λ−1 R,g,Q . 2 m
R,Q
(7.30)
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493
From this expression, together with a suitable linear map between K¨ahler parameters t and (r, N1 gs , N2 gs ) (which will lead to only negative exponents for t and which will depend on the choice of knot), the integral structure on both sides is compatible and g one can express the closed string integral invariants nd in terms of the open string inteR,g,Q . In the examples we will encounter in this paper, both knots will gral invariants N be unknots and the relation will be rather simple. It is interesting to notice that, when N = M, i.e. both gauge groups coincide, and r = 0, (7.19) is the partition function of the three-manifold obtained after performing surgery on the knot where U is supported (for finite k and N) [36, 58, 59]. The above result can be easily generalized to more complicated situations. For example, one can consider L arbitrary knots, Kα , where α = 1, · · · , L, and suppose that in each of the components we have a U (Nα ) Chern-Simons theory. Let us also consider L unknots at zero framing Kα with a U (Mα ) Chern-Simons theory in each of them. If we denote by Uα , Vα the holonomies of the U (Nα ), U (Mα ) fields, respectively, one can construct the operator L ∞
O(U1 , · · · , UL ; V1 , · · · , VL ) = exp
α=1 n=1
e−nrα n −n TrUα TrVα . n
(7.31)
Again it can be easily shown that log Z(U1 , · · · , UL ; V1 , · · · , VL ) has the structure of the free energy for a closed topological string. As an application of the above computation, we can evaluate (4.6) when p = 0, corresponding to Fig. 6. In this case, U1 is the holonomy around an unknot with trivial framing, and the only nontrivial fR corresponds to the fundamental representation. We then find, Z(M1 , M2 ) = exp(−F (M1 ) − F (M2 ) − F (M1 , M2 ; r)), where F (Mi ) is the free energy of the three-sphere Mi , and F (M1 , M2 ; r) = log O(U1 , U2 ; r) A1 ,A2 =
∞ −dr e (1 − e−dt1 )(1 − e−dt2 ) d=1
d(2 sin(dgs /2))2
. (7.32)
Notice that, in writing (7.32), we have defined: r = r −
t1 + t2 , 2
(7.33)
i.e. the parameter r that appears in (7.19) has to be renormalized in order to match the closed string K¨ahler parameter r . We will see below other examples of this, in which the same structure (7.33) appears. This shift was first observed in [27]. The dual closed string geometry is depicted in Fig. 12. The two S3 ’s, whose local neighborhood are T ∗ S3 ’s are replaced by two P1 ’s with normal bundle O(−1)⊕O(−1). In the total geometry, each of them intersects a P1 with O ⊕ O(−2) normal bundle. It is not difficult to calculate the genus zero amplitude of the closed string background using mirror symmetry, as in [25], and we find Fg=0 (r , t1 , t2 ) =
∞ −dti e i=1,2 d=1
d3
+
∞ −dr e (1 − e−dt1 )(1 − e−dt2 ) d=1
d3
.
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M. Aganagic, M. Mari˜no, C. Vafa
Fig. 12. After the transition, the two S3 ’s that we have wrapped D-branes on disappear, and with them all of H3 (X), so we are left with b2 (X) = 3
This has contribution from a single primitive curve in each of the classes [r ], [ti ], [r + ti ], [r + t1 + t2 ]. In fact, the only primitive curves in this geometry are rational curves, which are enumerated by the genus zero amplitude. One easy way to see this is as follows. Note that under the duality of M-theory on X/ type IIB theory on B × S 1 that we discussed in the previous sections, M2 branes wrapping holomorphic curves in X that have components along the T 2 directions map to (p, q) string web in type IIB string theory ending on the five-brane web (recall that M2 brane wrapping (p, q) cycle of the T 2 maps to a (p, q) string in type IIB string theory). The requirement for supersymmetry is that the (p, q) string must be parallel to (p, q) five brane, and that strings must be parallel to the plane defined by the five-branes. Compact curves in X correspond to string webs ending on the five-branes, but not any string can end on any five brane – a (p, q) string can only end on the (p, q) five brane. These are conditions for holomorphic curves in X, rephrased in the IIB language (to be complete, there is also the zero force condition and string string charge conservation that must be conserved at each vertex, in addition to the junctions with five-branes). Stated this way, it is clear from Fig. 12 that there are no BPS strings of finite length in the IIB dual, and correspondingly, no curves in X other than the ones counted above. This, together with the prediction for integrality properties of the amplitude (7.27), implies that the all genus partition function of the closed string theory is given by F =
∞ −dt1 e + e−dt2 + e−dr (1 − e−dt1 )(1 − e−dt2 ) . d(2 sin(dgs /2))2
(7.34)
d=1
This agrees exactly with the Chern-Simons answer (7.32). Note that ti = Ni gs are two new K¨ahler parameters, the sizes of two-spheres that grow with N , replacing the S3 ’s. The K¨ahler parameter r was already present in the open string geometry as the size of the holomorphic annulus with boundaries on the two spheres, but we have seen that their precise relation is given by (7.33). 7.3. O(−3) → P2 . The geometric transition in the three-sphere case is similar, and we have depicted it in Fig. 3. The dual closed string geometry contains a P2 with three exceptional P1 ’s touching it at three points. We can send the size of these P1 ’s to infinity by taking Ni → ∞ in our calculation to recover the P2 amplitude. The novelty in this
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495
case is that the closed string geometry has curves of arbitrarily high genus contributing to the topological A-model amplitudes, and infinitely many integer invariants are non-zero, as we will see. We now focus on the Chern-Simons computation that gives the invariants for O(−3) → P2 . According to what we discussed before, the Chern-Simons scenario involves three different gauge groups, with ’t Hooft parameters t1 , t2 , t3 , and with the same coupling constant gs . Accordingly, the quantum invariants will be a rational function of q and λi = eti , i = 1, 2, 3. As we saw in Sect. 5, for each Chern-Simons theory we have a Hopf link, and we will denote them by Li , with components Ki and Ki , i = 1, 2, 3. The framings can be read from (4.10), (4.12) and (4.11), and are as follows: K1 , K1 and K3 have framing zero, while the remaining knots have framing p = 1. This means that L1 is in the canonical framing, in L2 both components are framed, while in L3 only one of the components, K3 , is framed. The free energy at all genus for the topological closed string is given by 3 F = log e− i=1 i ri WR1 ,R2 (L1 )WR2 ,R3 (L2 )WR3 ,R1 (L3 ) . (7.35) R1 ,R2 ,R3
The sum in (7.35) is over all possible representations, including the trivial one (and that will be denoted by ·). In this equation, ri are “bare” K¨ahler parameters that will lead to a “renormalized” K¨ahler parameter r for the K¨ahler class of O(−3) → P2 . The relation between ri and r can be obtained by requiring a consistent limit ti → ∞ (which corresponds to the local P2 limit of the original, more complicated geometry). We will discuss this relation in a moment. Of course, there is also a piece F (M1 ) + F (M2 ) + F (M3 ) given by the sum of the free energies of the spheres that should be added to (7.35). Once the “renormalized” parameter has been restored, and the limit ti → ∞ taken, the free energy has the structure: ∞ ∞ (c) − r = a (q)e a (q)e− r , F = log 1 + =1
(7.36)
=1
(c)
and we will refer to a (q) as to the connected coefficients. If we keep the ti finite, we find the free energy for a closed string propagating in P2 blown up at 3 points, which is known as B3 (a particular case of del Pezzo). The 1 inside the brackets in (7.36) corresponds to R1 = R2 = R3 = ·, i.e. all the representations being the trivial one. In order to extract the integral invariants from this expression, we have to recall the general structure of the closed string topological amplitudes (7.27). One can use this 0
0
K1
K’1
1
1
K2
K’2
0
1
K3
K’3
Fig. 13. The figure shows the Hopf links in the manifolds M1 , M2 and M3 respectively. The numbers indicate the framing of each knot
496
M. Aganagic, M. Mari˜no, C. Vafa
formula to write the integral invariants in terms of closed string amplitudes by using the M¨obius function µ(n) [60]. Recall that µ(n) = 0 if n is not square-free, and it is (−1)f otherwise, where f is the number of factors in the prime decomposition of n. One finds: g≥0
1 µ(k) (c) 1 2g−2 g (−1)g−1 nd q − 2 − q 2 = a (q k ). k d/k
(7.37)
k|d
(c)
Therefore, if we know the coefficients a for = 1, · · · , d, we can extract the integral g invariants n for degrees = 1, · · · , d, and for all genera. Of course, from the point of (c) view of Chern-Simons theory, it is highly nontrivial that the coefficients a extracted from (7.35) have the structure required by (7.37). In the examples discussed in the previous section, the properties of the Chern-Simons invariants derived in [2, 5] guaranteed that one obtained a closed string expansion at all degrees and genera. Here we do not have a general proof, but we will explicitly show at low degrees that again, the rather constraining structural properties of the invariants of knots and links guarantee that one finds the right structure (7.37). Let us now look at the expansion of (7.36) at low degrees, which will also fix the relation between ri and r. First notice that, at every given degree d, we have a combinatorial problem of finding all possible representations R1 , R2 , R3 such that 1 + 2 + 3 = d. At degree one, we have three possibilities ( · · and two permutations), and we find: e−r a1 = e−r1 W (K1 )W (K3 ) + e−r2 W (K2 )W (K1 ) +e−r3 W (K3 )W (K2 ).
(7.38)
The knots involved in this computation are just framed unknots. Since the invariant of an unknot with framing p in the fundamental representation is (−1)p
1
1
1
1
λ− 2 − λ 2
q− 2 − q 2
,
(7.39)
we see that in order to have a finite limit as ti → ∞, we must have: r = r1 −
t1 + t3 t1 + t2 t2 + t3 = r2 − = r3 − , 2 2 2
(7.40)
so we take the limit ti → ∞ and at the same time ri → ∞ in such a way that the above combinations remain finite and equal to the closed string K¨ahler parameter r. The relation (7.40) is identical to the one we found before, in (7.33). Notice that, from the point of view of the Chern-Simons computation, this means that we have to renormalize every vev in the representations R, R as follows, − + 2
WR,R (Li ) → λi
WR,R (Li ),
(7.41)
where , are the number of boxes in the representations R, R . We have assumed that the renormalized K¨ahler parameters corresponding to the different annuli are all equal to the single K¨ahler parameter of the local P2 geometry. As we will see later, one can take the renormalized parameters to be different to obtain a refined version of the invariants.
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The conclusion of the above analysis is that, in order to recover the local P2 geometry, we have to take the limit λi → ∞ after the renormalization factor has been introduced. We then find that, for the local P2 geometry, 3
a1 (q) = − q
− 21
1
−q2
2 ,
(7.42)
which gives immediately the integral invariants at degree one: g
n01 = 3,
n1 = 0 for g > 0,
(7.43)
indeed the right result [29, 31]. Let us do the computation at degree two. There are nine possible choices of representations that lead to this degree: · ·, · ·, ·, and their permutations. This gives −2 a2 = λ−1 W( , ) (L1 )W (K3 )W (K2 ) 1 (λ2 λ3 ) +(λ1 λ3 )−1 W (K1 )W (K3 ) + W (K1 )W (K3 ) + perms, 1
(7.44)
where the permutations act cyclically as follows: L1 → L2 → L3 , K1 → K2 → K3 , K1 → K2 → K3 . The connected coefficient can be easily computed to be, −2 a2 = λ−1 f( , ) (L1 )W (K3 )W (K2 ) + perms 1 (λ2 λ3 ) + (λ2 λ3 )−1 fR (K2 )WR (K3 ) + (λ1 λ3 )−1 fR (K3 )WR (K1 ) 1
(c)
R=
,
+(λ1 λ2 )−1 fR (K2 )WR (K1 ) 1 (2) (2) + (λ1 λ3 )−1 W (K3 )W (K2 ) + perms. 2
(7.45)
(2)
We have denoted WR (q, λ) = WR (q 2 , λ2 ). The invariant f(R1 ,R2 ) (L) was introduced in [5], and we recall that for R1 = R2 = , one has f(
, ) (L)
= W(
, ) (L) − W
(7.46)
(K1 )W (K2 ),
where K1,2 are the components of L. One can also prove [5] that f( structure: 1 1 2g ( , ),g,Q q − 2 − q 2 f( , ) (L) = λQ , N
, ) (L)
has the (7.47)
g,Q 1
1
so it is an even polynomial in q − 2 − q 2 . We can now show that the structure of (7.45) is 1 1 compatible with (7.37): the first three lines give a Laurent polynomial in q − 2 − q 2 with even powers ≥ −2. For the first line, this is guaranteed by (7.47), while for the second and third lines we just notice that they have the structure of (7.23) (with d = 1), therefore we can use (7.25), (7.29) to write it in the desired form. In fact, the last three lines of (7.45) are precisely what we would obtain if the knots were all unlinked, and therefore the arguments of the last section guarantee that they have the right structure. Indeed, the very last line gives very precisely the two-cover of the degree one contribution, in agreement with (7.37).
498
M. Aganagic, M. Mari˜no, C. Vafa
In the above computation we have considered the most general geometry with four K¨ahler classes. In order to recover the local P2 case, we have to take ti → ∞. It is easy to see that in this limit the only relevant integral invariants of the links are ( N
, ),g=0,Q=1 (L1 )
( =N
, ),g=0,Q=1 (L2 )
( = −N
, ),g=0,Q=1 (L3 )
= 1. (7.48)
These invariants can be computed from (7.46) and (7.16), after including the framing corrections9 , while for the framed unknots we only need, in this limit, the invariant [8] N
,g=0,Q=1 (p
= 1) = 1.
(7.49)
The relevant integral invariants with g > 0 all vanish. We then find, 6
(c)
a2 (q) = q
− 21
1 2 2 + a1 (q ), 2
(7.50)
n2 = 0 for g > 0,
(7.51)
−q
1 2
therefore: n02 = −6,
g
again in agreement with the A and B-model computations [29, 31]. The procedure is now clear: to any given degree d, one has to compute the coefficient ad as a sum of different contributions given by the combinatorics of Young tableaux, compute the connected piece, and finally extract the multicovering contributions. Notice that, if one is just interested in the local P2 results, one can take the limit ti → ∞ at the beginning of the computation. In this limit we only have to keep the leading term in λ in the Chern-Simons invariant of the Hopf link presented in Sect. 5.1. In this way we have a very powerful method to compute the integral invariants of the local P2 geometry that can be easily implemented in a symbolic manipulation program. The results, up to degree 12 and at all genera, are presented in the following tables. It is interesting to compare this procedure to obtain the integer invariants with the ones based in the A and the B-model. As in the A-model computations based on localization, our procedure has to proceed degree by degree, and as the degree is increased the number of terms that contribute to ad grows very rapidly: to evaluate the integer invariants up to degree 12, one has to find a1 , · · · , a12 , and this involves evaluating 18239 terms in total. Degree 20 involves 943304 terms (there are 341649 terms contributing just to a20 ). However, the number of terms seems to be substantially lower than in a localization computation (compare for example with [29]), and of course the crucial advantage of the Chern-Simons approach is that one gets the invariants for all genera. This is also its main advantage with respect to the B-model computations, which also become more and more difficult as the genus is increased. The B-model results for higher genera are in fact determined only up to some unknown constants, due to the holomorphic ambiguity [39], and in order to find the actual value of the invariants one has to provide the value of the integral invariants coming from A-model computations. Therefore, the computation via Chern-Simons provides another way of fixing the holomorphic ambiguity of the B-model. Some comments on the results listed in Tables 1–3 are in order. First observe that, g for a given degree d, nd vanishes for g > (d − 1)(d − 2)/2. Indeed, (d − 1)(d − 2)/2 9 In fact, many of the Chern-Simons invariants of the Hopf link with trivial framing can be read from Sect. 6.2 of [5], after changing λ, q → λ−1 , q −1 , due to the fact that the Hopf link considered there has the opposite linking number to the one considered here.
All Loop Topological String Amplitudes from Chern-Simons Theory
499
g
Table 1. The integral invariants nd for the local P2 case g d=1 2 3 4 5 6 7 8 9 0 3 −6 27 −192 1695 −17064 188454 −2228160 27748899 1 0 0 −10 231 −4452 80948 −1438086 25301295 −443384578 2 0 0 0 −102 5430 −194022 5784837 −155322234 3894455457 3 0 0 0 15 −3672 290853 −15363990 649358826 −23769907110 4 0 0 0 0 1386 −290400 29056614 −2003386626 109496290149 5 0 0 0 0 −270 196857 −40492272 4741754985 −396521732268 6 0 0 0 0 21 −90390 42297741 −8802201084 1156156082181 7 0 0 0 0 0 27538 −33388020 12991744968 −2756768768616 8 0 0 0 0 0 −5310 19956294 −15382690248 5434042220973 9 0 0 0 0 0 585 −9001908 14696175789 −8925467876838 10 0 0 0 0 0 −28 3035271 −11368277886 12289618988434 11 0 0 0 0 0 0 −751218 7130565654 −14251504205448 12 0 0 0 0 0 0 132201 −3624105918 13968129299517 13 0 0 0 0 0 0 −15636 1487970738 −11600960414160 14 0 0 0 0 0 0 1113 −490564242 8178041540439 15 0 0 0 0 0 0 −36 128595720 −4896802729542 16 0 0 0 0 0 0 0 −26398788 2489687953666 17 0 0 0 0 0 0 0 4146627 −1073258752968 18 0 0 0 0 0 0 0 −480636 391168899747 19 0 0 0 0 0 0 0 38703 −120003463932 20 0 0 0 0 0 0 0 −1932 30788199027 21 0 0 0 0 0 0 0 45 −6546191256 22 0 0 0 0 0 0 0 0 1138978170 23 0 0 0 0 0 0 0 0 −159318126 24 0 0 0 0 0 0 0 0 17465232 25 0 0 0 0 0 0 0 0 −1444132 26 0 0 0 0 0 0 0 0 84636 27 0 0 0 0 0 0 0 0 −3132 28 0 0 0 0 0 0 0 0 55
is the genus of a nondegenerate curve of degree d in P2 . As shown in [31], one has in this case (−1)d(d+3)/2 (d + 1)(d + 2), (7.52) 2 in full agreement with the corresponding entries in Table 1 for d = 1, · · · , 12. For d > 2, we have contributions from curves with one node (therefore g = d(d − 3)/2), and the arguments of [31] give d d(d−3)/2 nd (d 2 + d − 3), = −(−1)d(d+3)/2 (7.53) 2 (d−1)(d−2)/2
nd
=
again in full agreement with the results that we have obtained. Curves with two nodes start contributing at d > 3, and one finds: (−1)d(d+3)/2 (d − 1)(d 5 − 2d 4 − 6d 3 + 9d 2 + 36), (7.54) 4 which reproduces our results for 4 ≤ d ≤ 12. For curves with three nodes, the integral invariant is given by (d 2 −3d−2)/2
nd
=
(d 2 −3d−4)/2
nd
(−1)d(d+3)/2 (−96 + 222 d − 323 d 2 + 54 d 3 − 34 d 4 12 +36 d 5 + 2 d 6 − 6 d 7 + d 8 ), (7.55)
=−
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Table 2. The integral invariants nd for the local P2 case (continuation) g 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33
d = 10 −360012150 7760515332 −93050366010 786400843911 −5094944994204 26383404443193 −111935744536416 395499033672279 −1177301126712306 2978210177817558 −6445913624274390 12001782164043306 −19310842755095748 26952467292328782 −32736035592797946 34693175820656421 −32151370513161966 26099440805196660 −18580932613650624 11609627766170547 −6367395873587820 3064262549419899 −1292593922494452 477101143946277 −153692555590206 43057471189239 −10441089412308 2177999212647 −387688567518 58269383541 −7292193288 745600245 −60650490 3773652
11 4827935937 −135854179422 2145146041119 −24130293606924 210503102300868 −1485630816648252 8698748079113310 −42968546119317066 181202644392392127 −658244675887405242 2074294284130247058 −5702866358492557440 13744538465609779287 −29157942375100015002 54641056077839878893 −90735478019244786786 133885726253316075984 −175976406401479949154 206477591201198965488 −216671841840838260606 203674311322868998065 −171730940091766865658 130015073789764141299 −88451172530198637924 54098277648908454123 −29751302949160261398 14709694749741501501 −6535189635435373326 2606677300588276035 −932238829973577348 298408032566091294 −85297647759486510 21708810999461607 −4901354114590566
12 −66537713520 2380305803719 −48109281322212 698473748830878 −7935125096754762 73613315148586317 −572001241783007370 3786284014554551293 −21609631514881755756 107311593188998164015 −466990545532708577390 1791208287019324701495 −6085017394087513680618 18384612378910358924791 −49578782776769125835658 119723947998685791289164 −259634731498425150837576 506961721474582218552270 −893407075206205808615238 1424048002136300951108030 −2057099617415644933602618 2697839037217627321703085 −3217397468483821476968358 3494176460021369389735746 −3460084190968494003073062 3127576636374963802648718 −2582938330708242629937150 1950461493734929553600580 −1347524558332336039964082 852109374825775079556606 −493309207337589509893062 261477149328500781917776 −126876156355185161374314 56339101711825399890960
which reproduces our results for 5 ≤ d ≤ 12. For d = 4 there are reducible curves with three nodes, and in order to reproduce n04 one has to introduce a correction, as explained in detail in [31]. We then see that the results obtained from Chern-Simons theory are in full agreement with what is expected from the geometric interpretation of the integral invariants. Notice that we have been able to check results for very high genus, which is not easy to do in the A or B model computations.
7.4. B3 . In the previous subsection we have seen how to recover the integer invariants for P2 by taking the limit ti → ∞. Keeping the blow up parameters ti finite we obtain the integer invariants of the local del Pezzo B3 (we recall that B3 is the rational surface obtained from P2 by blowing up three points). We will write the generating functional for the integer invariants at genus g as Fg (r, t1 , t2 , t3 ) =
e− r F (t1 , t2 , t3 ), g
(7.56)
All Loop Topological String Amplitudes from Chern-Simons Theory
501
g
Table 3. The integral invariants nd for the local P2 case (continuation) g 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55
d = 10 −168606 4815 −66 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
11 977233475777499 171090302865948 26117674453665 −3445690553358 388460380746 −36878620320 2891025822 −182125500 8859513 −312270 7095 −78 0 0 0 0 0 0 0 0 0 0
12 −22881258328195868502320 8492649924309368930964 −2877665040430021956492 888968505074075552261 −249952226921825722236 63836429603183934921 −14772524364719546808 3088415413809592461 −581271967556317272 98073062075574517 −14758388168491098 1968679573589997 −231043750764510 23635158339861 −2082988758060 155790863415 −9693024822 488072208 −19105426 545391 −10098 91
where g
Fd (t1 , t2 , t3 ) =
g
nd1 ,d2 ,d3 q1d1 q2d2 q3d3 ,
(7.57)
d1 ,d2 d3
and we have written qi = e−ti (these shouldn’t be confused with the Chern-Simons variable introduced before). We present the results for these generating functionals up to degree four in P2 : F10 = 3 − 2(q1 + q2 + q3 ) + q1 q2 + q1 q3 + q2 q3 , F20 = − 6 + 5(q1 + q2 + q3 ) − 4(q1 q2 + q1 q3 + q2 q3 ) + 3 q1 q2 q3 , F30 = 27 − 32(q1 + q2 + q3 ) + 35(q1 q2 + q1 q3 + q2 q3 ) + 7(q12 + q22 + q32 ) − 6(q1 q22 + q1 q32 + q2 q12 + q2 q32 + q3 q12 + q3 q22 ) − 36 q1 q2 q3 + 5(q1 q22 q32 + q2 q12 q32 + q3 q12 q22 ), F31 F40
= −10 + 9(q1 + q2 + q3 ) − 8(q1 q2 + q1 q3 + q2 q3 ) + 7 q1 q2 q3 , = −192 + 286(q1 + q2 + q3 ) − 400(q1 q2 + q1 q3 + q2 q3 ) − 110(q12 + q22 + q32 ) + 135(q1 q22 + q1 q32 + q2 q12 + q2 q32 + q3 q12 + q3 q22 ) + 531 q1 q2 q3 − 9(q13 + q23 + q32 ) − 8(q1 q23 + q1 q33 + q2 q13 + q2 q33 + q3 q13 + q3 q23 ) − 32(q12 q22 + q12 q32 + q22 q32 ) − 160(q1 q2 q32 + q1 q3 q22 + q2 q3 q12 ) + 35(q1 q22 q32 + q2 q33 q22 + q3 q12 q22 ) + 7(q1 q2 q33 + q1 q3 q23 + q2 q3 q13 ) − 6 q12 q22 q32 ,
(7.58)
502
M. Aganagic, M. Mari˜no, C. Vafa
and finally, F41 = 231 − 288(q1 + q2 + q3 ) + 344(q1 q2 + q1 q3 + q2 q3 ) + 68(q12 + q22 + q32 ) − 72(q1 q22 + q1 q32 + q2 q12 + q2 q32 + q3 q14 + q3 q22 ) + 396 q1 q2 q3 + 9(q12 q22 + q12 q32 + q22 q32 ) + 74(q1 q2 q32 + q1 q3 q22 + q2 q3 q12 ) − 8(q1 q22 q32 + q2 q33 q22 + q3 q12 q22 ), F42
=
−102 + 108(q1 + q2 + q3 ) − 112(q1 q2 + q1 q3 + q2 q3 ) − 12(q12 + 11(q1 q22 + q1 q32 + q2 q12 + q2 q32 + q3 q14 + q3 q22 ) + 114 q1 q2 q3 − 10(q1 q2 q32 + q1 q3 q22 + q2 q3 q12 ),
+ q22
(7.59) + q32 )
F43 = 15 − 14(q1 + q2 + q3 ) + 13(q1 q2 + q1 q3 + q2 q3 ) − 12 q1 q2 q3 . We can take the limit in which one of the qi ’s, say q3 , goes to zero. The corresponding results for the g = 0 amplitudes agree with those presented in [30] for the B2 local del Pezzo, after relabeling t1,2 → −t1,2 , r → r + t1 + t2 .
7.5. O(K) → P1 × P1 . We now consider the geometry that leads to local P1 × P1 . In Fig. 14 there are Ni D-branes, i = 1, . . . , 4, wrapping a chain of four minimal spheres connecting two (1, 0) branes and two (0, 1) branes. For every pair of spheres “intersecting” over an S1 we get a bifundamental scalar field, so we have matter in representation (Ni , N i+1 ), where i = 5 corresponds to the first sphere again. The path integral of the A-model in this background can be written as:
M2
M3 M1 M4
Fig. 14. The figure depicts four S3 ’s connected with annuli
All Loop Topological String Amplitudes from Chern-Simons Theory
Z=
4
DAi eSCS (Ai ) O(U1 , U2 )O(U2 , U3 )O(U3 , U4 )O(U4 , U1 ).
503
(7.60)
i=1
There are two unknots on each three-sphere and the amplitude will depend on their linking numbers, in addition to framing. As before, we can use (4.5) to write this in a more transparent form, Z=
V1 R1 |V4 R4 e− 4 r4 V4 R4 |V3 R3 e− 3 r3 R1 ,R2 ,R3 ,R4
· V3 R3 |V2 R2 e− 2 r2 V2 R2 |V1 R1 e− 1 r1 .
(7.61)
As in the previous case, the requisite diffeomorphism are determined by the geometry. From the figure, we have, V1 = S,
V2 = C,
V3 = SC,
V4 = 1.
This gives four S3 ’s, each of which has a Hopf link with linking number +1 and whose components have zero framing. The geometric transition is represented in Fig. 15. The resulting dual closed string geometry contains a P1 × P1 , together with four exceptional P1 ’s. As in the previous case, we can take the limit ti → ∞ in order to extract the integer invariants of the local P1 × P1 geometry. The Chern-Simons computation that gives the invariants is very similar to the one we discussed in the previous section, so we won’t give all the details. According to the geometric picture, we have four Chern-Simons theories with ’t Hooft parameters t1 , t2 , t3 , and t4 , and with the same coupling constant gs . Since all knots and links are identical, it is sufficient to label the vevs by indicating explicitly the corresponding ’t Hooft parameter. The free energy at all genus for the topological closed string is then given by F = log e− 1 r1 − 3 r2 − 2 s1 − 4 s2 WR1 ,R2 (t1 )WR2 ,R3 (t2 ) R1 ,R2 ,R3 ,R4
×WR3 ,R4 (t3 )WR4 ,R1 (t4 ) .
(7.62)
Fig. 15. The figure shows a geometric transition of four S3 in the previous figure. The dual geometry is related by four flops of the external P1 ’s to a non-generic del Pezzo B5
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M. Aganagic, M. Mari˜no, C. Vafa
Again, ri and si are “bare” K¨ahler parameters that will lead to two “renormalized” K¨ahler parameter r, s. The relation between them can be obtained as in the previous case, and one easily finds: t1 + t4 t2 + t3 = r2 − , 2 2 t1 + t2 t3 + t4 = s2 − , s = s1 − 2 2
r = r1 −
(7.63)
and we have to rescale the Chern-Simons vevs as before, − + 2
WR,R (ti ) → λi
(7.64)
WR,R (ti ).
Once we have done that, the free energy is given by: ∞ ∞ (c) − 1 r− 2 s = a 1 , 2 (q)e a 1 , 2 (q)e− 1 r− 2 s , F = log 1 + 1 , 2 =1
(7.65)
1 , 2 =1
g
and from here we can again extract the integral invariants n 1 , 2 by subtracting multicovering effects. Let us present some explicit results at lower degree. For degrees ( 1 , 2 ) = (1, 0) and (0, 1), we find: 1
1
1
1
a1,0 = (λ1 λ4 )− 2 W (t1 )W (t4 ) + (λ2 λ3 )− 2 W (t2 )W (t3 ), a0,1 = (λ1 λ2 )− 2 W (t1 )W (t2 ) + (λ3 λ4 )− 2 W (t3 )W (t4 ).
(7.66)
In general, the coefficients an,m and am,n are related by exchanging t2 ↔ t4 . By taking the limit ti → ∞, we find 2
a1,0 = a0,1 = q
− 21
1
−q2
2 ,
(7.67)
therefore n01,0 = n00,1 = −2,
(7.68)
and the invariants for higher genus all vanish. This is indeed the right result [28, 30]. (c) For a1,1 we find: −2 a1,1 = λ−1 f( 1 (λ2 λ4 ) (c)
1
, ) (t1 )W
(t2 )W (t4 ) + perms,
(7.69)
where perms stands for three terms that are obtained from the first one by permuting ti → ti+1 . Due to (7.47), this has the structure of the degree (1, 1) term in a closed string free energy. After taking the limit ti → ∞, one finds 4
(c)
a1,1 = q
− 21
−q
1 2
2 ,
(7.70)
All Loop Topological String Amplitudes from Chern-Simons Theory
505
therefore n01,1 = −4,
(7.71)
while the invariants for higher genera vanish.Again this is the right value for the invariant. We can again easily implement the computation of these invariants. In the following tables we present most of the results up to total degree 10 and genus 8 (the non-trivial invariants for total degree 10 go all the way to genus 16, which we have obtained, but have not included here for the economy of space). These results are in full agreement with the ones presented in [28, 30, 31]. Again, we can verify many of these numbers with the geometric formulae of [31]. For a given g bidegree (a, b), n(a,b) vanishes for g > (a − 1)(b − 1), which is indeed the arithmetic genus of a curve of bidegree (a, b) in P1 × P1 . One finds, (a−1)(b−1)
n(a,b)
= −(−1)(a+1)(b+1) (a + 1)(b + 1),
(7.72)
which reproduces the corresponding results listed in the tables above. For curves with one node, one finds: (a−1)(b−1)−1
n(a,b)
= 2(−1)(a+1)(b+1) (a + b + ab − a 2 − b2 + a 2 b2 ),
Table 4. The integral invariants n0d for the local P1 × P1 case d2 0 1 2 3 4 5 6
d1 =0
1 −2 −4 −6 −8 −10 −12 −14
−2 0 0 0 0 0
2 0 −6 −32 −110 −288 −644 −1280
3 0 −8 −110 −756 −3556 −13072 −40338
4 0 −10 −288 −3556 −27264 −153324 −690400
5 0 −12 −644 −13072 −153324 −1252040
6 0 −14 −1280 −40338 −690400
Table 5. The integral invariants n1d for the local P1 × P1 case d2 2 3 4 5 6
d1 =2 9 68 300 988 2698
3 68 1016 7792 41736 172124
4 300 7792 95313 760764 4552692
5 988 41376 760764 8695048
6 2698 172124 4552692
Table 6. The integral invariants n2d for the local P1 × P1 case d2 2 3 4 5 6
d1 =2 0 −12 −116 −628 −2488
3 −12 −580 −8042 −64624 −371980
4 −116 −8042 −167936 −1964440 −15913228
5 −628 −64624 −1964440 −32242268
6 −2488 −371980 −15913228
(7.73)
506
M. Aganagic, M. Mari˜no, C. Vafa Table 7. The integral invariants n3d for the local P1 × P1 case d2 2 3 4 5 6
d1 =2 0 0 15 176 1130
3 0 156 4680 60840 501440
4 15 4680 184056 3288688 36882969
5 176 60840 3288688 80072160
6 1130 501440 36882969
Table 8. The integral invariants n4d for the local P1 × P1 case d2 2 3 4 5 6
d1 =2 0 0 0 −18 −248
3 0 −16 −1560 −36408 −450438
4 0 −1560 −133464 −3839632 −61250176
5 −18 −36408 −3839632 −144085372
6 −248 −450438 −61250176
Table 9. The integral invariants n5d for the local P1 × P1 case d2 2 3 4 5 6
d1 =2 0 0 0 0 21
3 0 0 276 13888 276144
4 0 276 64973 3224340 75592238
5 0 13888 3224340 195035824
6 21 276144 75592238
Table 10. The integral invariants n6d for the local P1 × P1 case d2 3 4 5 6
d1 =3 0 −20 −3260 −115744
4 −20 −20936 −1969710 −70665312
5 −3260 −1969710 −202598268
6 −115744 −70665312
Table 11. The integral invariants n7d for the local P1 × P1 case d2 3 4 5 6
d1 =3 0 0 428 32568
4 0 4266 873972 50501308
5 428 873972 163185964
6 32568 50501308
Table 12. The integral invariants n8d for the local P1 × P1 case d2 3 4 5 6
d1 =3 0 0 −24 −5872
4 0 −496 −277880 −27655024
5 −24 −277880 −102321184
6 −5872 −27655024
All Loop Topological String Amplitudes from Chern-Simons Theory
507
again in full agreement with the tables. For curves with two nodes (extending the derivation in [31] ) we have: (a−1)(b−1)−2 n(a,b) = −(−1)(a+1)(b+1) −14 + 9(a + b) − 3ab − 3(a 2 + b2 ) + 3a 2 b2 + 2(a 3 + b3 + a 2 b + b2 a) − 2(a 3 b + b3 a) − 2(a 3 b2 + b3 a 2 ) + 2a 3 b3 ,
(7.74)
which reproduces for example n2(3,3) = −580, n8(3,6) = −5872 and n7(4,4) = 4266. For the invariants corresponding to bidegrees (2, n), where 3 ≤ n ≤ 6, and curves with two nodes, one has to introduce corrections associated to reducible curves. For example, for bidegrees (2, 6), (7.74) gives the value 1166, but there are reducible curves of type (5, 2) ∪ (1, 0) with two nodes. Since n0(1,0) = −2, and n4(5,2) = −18, the subtraction scheme proposed in [31] gives n3(2,6) = 1166 − (−2)(−18) = 1130, in agreement with the result of Table 7. g
7.6. Refined integral invariants. The integral invariants nd defined in [57] denote the (net) number of wrapped M2 branes in 4 + 1 dimensional effective theory, obtained by compactification of M-theory on the corresponding Calabi-Yau, where d ∈ H2 (X) denotes the class the M2 brane is wrapped and g denotes a basis for the SU (2)L rotation subgroup of SO(4) (see [57] for details). If the Calabi-Yau space has global symmetries, then these states also form representations of this group. Compact Calabi-Yau manifolds do not admit global symmetries, so this does not arise in that context. However for local g toric 3-folds there always are extra global symmetries and one can ask how the nd decompose in representations of this symmetry algebra. Thus it is natural to ask whether we can use our techniques to also compute these refined invariants. For example, consider the linear sigma model describing O(−3) → P2 , which contains three matter fields of charge +1. In this case there are two extra U (1) global symmetries, which for some metric in P2 could give rise to the Cartan of SU (3). This can be implemented in terms of the toric diagram, by assigning different sizes to the different edges. In the local P2 case, we should assign different sizes to the triangle describing the base of the P2 , i.e. we should introduce three K¨ahler parameters instead of one. Notice that this is perfectly natural from the point of view of the Chern-Simons description, because in the limit where we took the Ni → ∞ we had to tune r’s. Nothing prevents us from tuning the three edges to different values by considering a suitable limit. Let us consider the computation of the refined integral invariants in some detail, in the case of local P2 . We have to introduce three parameters associated to the three different edges, and we will denote them by ri , with i = 1, 2, 3, where we view e−ri as forming a Cartan torus of U (3). Notice that, if we write U (3) = U (1) × SU (3), the overall U (1) quantum number is precisely the degree d. We then have to further decompose the spectrum with respect to the SU (3). This goes as follows: due to the underlying symmetry, the closed string free energy will be now of the form: ∞ m=1 g,d
g nd (x1m , x2m , x3m )
1 mgs 2g−2 , 2 sin m 2
(7.75)
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where xi = e−ri , i = 1, 2, 3, and nd (x1 , x2 , x3 ) is now a symmetric polynomial of degree d in the xi , with integer coefficients. Therefore, we can expand it in terms of Schur polynomials sR in three variables and of degree d, which are labeled by representations R of SU (3) with d boxes. We then write: g g nd (x1 , x2 , x3 ) = nd,R sR (x1 , x2 , x3 ), (7.76) g
R g
where the sum is over representations of SU (3) with d boxes, and nd,R denote the number of M2 branes of degree d with SU (2)L representation g and transforming as representation R of the SU (3) global symmetry. These are the refined invariants of the local P2 . Notice that, if we put x1 = x2 = x3 = 1 in (7.76), we recover the usual integer invariants, therefore one has g g nd = (dimR) nd,R . (7.77) R
The computation of the refined invariants can be easily done in the Chern-Simons setting, by taking the renormalized sizes of the annuli to be different. The renormalized sizes will give in this way the parameters ri appearing in the closed string side, in other words: t1 + t2 r1 = r − 2 and so on. The refined invariants for the first few degrees can be easily computed. At degree one we find, n01 (x1 , x2 , x3 ) = x1 + x2 + x3 ,
(7.78)
therefore n01, = 1. At degree two, one has: = −1,
n02,
n02, = 0.
(7.79)
At degree three, we find: n03,
= 2,
n13,
= −1,
= 1,
n0
3,
3,
= 0,
n1
3,
= −1,
n0
n1
3,
(7.80)
= 0,
We finally list the results for degree four: n04,
= −7,
n0
= −6,
n14,
= 11,
n1
= 5,
n24,
= −6,
n2
= −1,
n34,
= 1,
4,
4,
4,
n3
4,
= 0,
n0
4,
= 1,
n1
4,
n2
4,
n3
4,
= −2,
4,
= −5,
n1
4,
= 0,
= 0,
= 5,
n0
n2
4,
n3
4,
= −1,
= 0.
(7.81)
All Loop Topological String Amplitudes from Chern-Simons Theory
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For P1 × P1 one can similarly decompose the invariants with respect to the SU (2) × SU (2) global symmetry of the model. Note that from a mathematical point of view, these refined integer invariants should be related to the equivariant Gromov-Witten invariants associated to the group action on the manifold, as was studied in the Fano case in [61]. Moreover, one could use the techniques of [61] to obtain the mirror of these deformations for the toric Calabi-Yau manifolds we have discussed and check the results obtained here against the predictions of mirror symmetry (at least for genus 0). 8. Embedding in Superstrings It is natural to ask what kind of dualities these geometric transitions lead to, once we embed them in superstrings, as was done in [11] for the original Chern-Simons duality [1]. Embedding these dualities for topological strings in type IIA strings is easily done by replacing the branes with D6 branes wrapping S3 ’s and filling 4 dimensional spacetime. Thus we end up, at low energy with a system involving N = 1 U (Ni ) gauge symmetry. Moreover for each annulus contribution we end up with a bifundamental matter “hypermultiplet” in the superstring context. Of course this is only the low energy limit of the brane system. The high energy aspects of this theory differ from that of pure Yang-Mills. This can be deduced by considering the superpotential for this theory, as was done in [11]. In the IR the gaugino condensation will take place where the S3 ’s are replaced by blown up S2 ’s with RR fluxes through them. There is no RR flux through S2 ’s which come from matter bifundamentals. In the applications we have looked at, we have also considered the interesting limit where the sizes of blown up S2 ’s go to infinity, while keeping the effective masses of the bifundamental fields finite. This was, for example, how we got the full answer for P2 in topological strings. In the gauge theory setup the size of the blown up S2 ’s correspond to the size of the gaugino condensate getting large, which can be adjusted by increasing the corresponding gauge coupling. Note that in this limit we will have no RR flux left in the type IIA superstring theory. Since we have fixed total RR flux through the S2 ’s which get infinitely large, in the limit we are considering the flux per unit volume goes to zero. Moreover, the finite S2 ’s in this limit correspond to where the bifundamental matter came from, and there is no flux through them. Thus we end up with a novel large N duality, where the bifundamental matter structure dictates the geometry of the dual and this geometry has no RR flux in it. The statement of the above dualities correspond to gauge theories with all the string interactions on them. One would naturally ask if there are any large N dualities along these lines for pure gauge theories. For this purpose it is convenient to go to the type IIB mirror setup. To illustrate the idea let us first consider a simple example. Consider the N = 2, U (2N) gauge theory deformed by the addition of superpotential 1 3 2 W = gtr − m , 3 where is the adjoint field. There are two classical values for the eigenvalues of , namely = ±m. Let us choose N eigenvalues of to be at +m and N to be at −N . Then the large N dual of this system in type IIB is proposed in [17] (and further elaborated recently in [62]) to be given by propagation in the non-compact Calabi-Yau given by the hypersurface in I C4 : uv + y 2 + g 2 (x 2 − m2 )2 + g 2 4 = 0,
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M. Aganagic, M. Mari˜no, C. Vafa
where is related to the scale of the original N = 2 theory. Note that for small we have two conifold points centered near x = m and x = −m. In this dual gravitational geometry, there is RR flux of N units through each of the corresponding S3 ’s. However there is no RR flux through the compact S3 which runs between these two S3 ’s (and intersects both at 1 point). It is convenient to rewrite the above geometry as uv + y 2 + g 2 P (x) = 0, where
P (x) = (x 2 − M 2 )(x 2 − a 2 ),
and we identify m2 =
4 =
1 2 (M + a 2 ), 2 −(M 2 − a 2 )2 . 4
In this parameterization the two S3 ’s with RR flux project in the x-plane to the intervals −M ≤ x ≤ −a and a ≤ x ≤ M. In particular there is no flux through the S3 which projects to the interval −a ≤ x ≤ a. We consider the situation where (M/a) >> 1. In this limit the two S3 ’s have become big. In particular as M → ∞, keeping a and α = −g 2 M 2 fixed the geometry reduces to uv + y 2 + α(x 2 − a 2 ) = 0 which is the ordinary conifold. Moreover, in this limit the RR fields per unit volume go to zero everywhere. Thus we have found a gauge theory/gravity duality where the geometry is free of RR flux. To be precise we have to note that we need to complete the duality by going farther in the UV of gauge system, which forces a cascade structure [19] generalizing the construction of [16] to the case at hand. Namely, we will end up with a U (2N + M) × U (M) gauge system, with two bifundamental hypermultiplets, as M → ∞. Moreover we have superpotentials W1 (1 ) and W2 (2 ) which have the same functional form as the superpotential W () discussed before, namely W1 = −W2 = W , where the coefficients of W are carefully tuned, as discussed above. Thus we have a proposal for a gauge dual description of the standard conifold with no flux through it. Clearly this example can be generalized. In fact a large class of local Calabi-Yau threefolds were constructed in [18] as duals to gauge systems, which were analyzed in [19]. Applying a similar kind of reasoning as the above example we end up describing a rather large class of local threefold without fluxes, as duals to some limits of N = 2 gauge systems deformed to N = 1 by superpotential terms. It would be very interesting to study the physical implications of these dualities. Acknowledgements. We would like to thank E. Diaconescu, A. Grassi, S. Katz and P. Ramadevi for valuable discussions.
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3. Labastida, J.M.F., Mari˜no, M.: Polynomial invariants for torus knots and topological strings. hepth/0004196, Commun. Math. Phys. 217, 423 (2001) 4. Ramadevi, P., Sarkar, T.: On link invariants and topological string amplitudes, hep-th/0009188, Nucl. Phys. B 600, 487 (2001) 5. Labastida, J.M.F., Mari˜no, M., Vafa, C.: Knots, links and branes at large N. hep-th/0010102, JHEP 0011, 007 (2000) 6. Labastida, J.M., Mari˜no, M.: A new point of view in the theory of knot and link invariants. math.QA/0104180, J. Knot Theory Ramifications 11, 173 (2002) 7. Aganagic, M., Klemm, A., Vafa, C.: Disk instantons, mirror symmetry and the duality web. hepth/0105045, Z. Naturforsch. A 57, 1 (2002) 8. Mari˜no, M., Vafa, C.: Framed knots at large N. hep-th/0108064 9. Sinha, S., Vafa, C.: SO and Sp Chern-Simons at large N. hep-th/0012136 10. Acharya, B., Aganagic, M., Hori, K., Vafa, C.: Orientifolds, mirror symmetry and superpotentials. hep-th/0202208 11. Vafa, C.: Superstrings and topological strings at large N. hep-th/0008142, J. Math. Phys. 42, 2798 (2001) 12. Atiyah, M., Maldacena, J.M., Vafa, C.: An M-theory flop as a large N duality. hep-th/0011256, J. Math. Phys. 42, 3209 (2001) 13. Acharya, B.: On realising N = 1 super Yang-Mills in M theory. hep-th/0011089 14. Aganagic, M., Vafa, C.: Mirror symmetry and a G(2) flop. hep-th/0105225 15. Atiyah, M., Witten, E.: M-theory dynamics on a manifold of G(2) holonomy. hep-th/0107177 16. Klebanov, I.R., Strassler, M.J.: Supergravity and a confining gauge theory: Duality cascades and chiSB-resolution of naked singularities. hep-th/0007191, JHEP 0008, 052 (2000) 17. Cachazo, F., Intriligator, K.A., Vafa, C.: A large N duality via a geometric transition. hep-th/0103067, Nucl. Phys. B 603, 3 (2001) 18. Cachazo, F., Katz, S., Vafa, C.: Geometric transitions and N = 1 quiver theories. hep-th/0108120 19. Cachazo, F., Fiol, B., Intriligator, K.A., Katz, S., Vafa, C.: A geometric unification of dualities. hep-th/0110028, Nucl. Phys. B 628, 3 (2002) 20. Edelstein, J.D., Oh, K., Tatar, R.: Orientifold, geometric transition and large N duality for SO/Sp gauge theories. hep-th/0104037, JHEP 0105, 009 (2001) 21. Dasgupta, K., Oh, K., Tatar, R.: Geometric transition, large N dualities and MQCD dynamics. hepth/0105066, Nucl. Phys. B 610, 331 (2001); Open/closed string dualities and Seiberg duality from geometric transitions in M-theory. hep-th/0106040; Geometric transition versus cascading solution. hep-th/0110050, JHEP 0201, 031 (2002) 22. Fuji, H., Ookouchi, Y.: Confining phase superpotentials for SO/Sp gauge theories via geometric transition. hep-th/0205301 23. Giveon, A., Kehagias, A., Partouche, H.: Geometric transitions, brane dynamics and gauge theories. hep-th/0110115, JHEP 0112, 021 (2001) 24. Ooguri, H., Vafa, C.: Worldsheet derivation of a large N duality. hep-th/0205297 25. Aganagic, M., Vafa, C.: G(2) manifolds, mirror symmetry and geometric engineering. hepth/0110171 26. Witten, E.: Chern-Simons gauge theory as a string theory. hep-th/9207094. In: The Floer memorial volume, H. Hofer, C.H. Taubes, A. Weinstein, E. Zehner (eds.), Baset Boston: Birkh¨auser, 1995, p. 637 27. Diaconescu, E., Florea, B., Grassi, A.: Geometric transitions and open string instantons. hepth/0205234 28. Katz, S., Klemm, A., Vafa, C.: Geometric engineering of quantum field theories. hep-th/9609239, Nucl. Phys. B 497, 173 (1997) 29. Klemm, A., Zaslow, E.: Local mirror symmetry at higher genus. hep-th/9906046. In: Winter School on Mirror Symmetry, Vector bundles and Lagrangian Submanifolds, Providence, RI: American Mathematical Society, 2001, p. 183 30. Chiang, T.M., Klemm, A., Yau, S.T., Zaslow, E.: Local mirror symmetry: Calculations and interpretations. hep-th/9903053, Adv. Theor. Math. Phys. 3, 495. (1999) 31. Katz, S., Klemm, A., Vafa, C.: M-theory, topological strings and spinning black holes. hepth/9910181, Adv. Theor. Math. Phys. 3, 1445 (1999) 32. Diaconescu, D.-E., Florea, B., Grassi, A.: Geometric transitions, del Pezzo surfaces and open string instantons. hep-th/0206163, Adv. Theor. Math. Phys. 6, 643 (2003) 33. Aharony, O., Hanany, A.: Branes, superpotentials and superconformal fixed points. hep-th/9704170, Nucl. Phys. B 504, 239 (1997); O. Aharony, A. Hanany, B. Kol, Webs of (p,q) 5-branes, five dimensional field theories and grid diagrams. hep-th/9710116, JHEP 9801, 002 (1998) 34. Leung, N.C., Vafa, C.: Branes and toric geometry. hep-th/9711013, Adv. Theor. Math. Phys. 2, 91 (1998)
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35. Aganagic, M., Karch, A., Lust, D., Miemiec, A.: Mirror symmetries for brane configurations and branes at singularities. hep-th/9903093, Nucl. Phys. B 569, 277 (2000) 36. Witten, E.: Quantum field theory and the Jones polynomial. Commun. Math. Phys. 121, 351 (1989) 37. Witten, E.: Noncommutative geometry and string field theory. Nucl. Phys. B 268, 253 (1986) 38. Bershadsky, M., Vafa, C., Sadov, V.: D-strings on D-manifolds. hep-th/9510225, Nucl. Phys. B 463, 398 (1996) 39. Bershadsky, M., Cecotti, S., Ooguri, H., Vafa, C.: Kodaira-Spencer theory of gravity and exact results for quantum string amplitudes. hep-th/9309140, Commun. Math. Phys. 165, 311 (1994) 40. Guadagnini, E., Martellini, M., Mintchev, M.: Wilson lines in Chern-Simons theory and link invariants. Nucl. Phys. B 330, 575 (1990) 41. Katz, S., Liu, M.: Enumerative geometry of stable maps with Lagrangian boundary conditions and multiple covers of the disc. math.AG/0103074, Adv. Theor. Math. Phys. 5, 1 (2002) 42. Graber, T., Zaslow, E.: Open-string Gromov-Witten invariants: calculations and a mirror ‘theorem’. hep-th/0109075 43. Witten, E.: On quantum gauge theories in two-dimensions. Commun. Math. Phys. 141, 153 (1991) 44. Cordes, S., Moore, G.W., Ramgoolam, S.: Lectures on 2-d Yang-Mills theory, equivariant cohomology and topological field theories. hep-th/9411210, Nucl. Phys. Proc. Suppl. 41, 184 (1995) 45. Verlinde, E.: Fusion rules and modular transformations in 2-D conformal field theory. Nucl. Phys. B 300, 360 (1988) 46. Witten, E.: Phases of N = 2 theories in two dimensions. hep-th/9301042, Nucl. Phys. B 403, 159 (1993) 47. Katz, S., Mayr, P., Vafa, C.: Mirror symmetry and exact solution of 4D N = 2 gauge theories. I. hep-th/9706110, Adv. Theor. Math. Phys. 1, 53 (1998) 48. Kontsevich, M.: Enumeration of rational curves via torus actions. hep-th/9405035, In: The moduli space of curves, Baset Boston: Birkh¨auser, 1995, p. 335 49. Graber, T., Pandharipande, R.: Localization of virtual classes. alg-geom/9708001, Invent. Math. 135, 487 (1999) 50. Kontsevich, M.: Intersection theory on the moduli space of curves and the matrix Airy function. Commun. Math. Phys. 147, 1 (1992) 51. Faber, C.: Algorithms for computing intersection numbers of curves, with an application to the class of the locus of Jacobians. alg-geom/9706006. In: New trends in algebraic geometry, Cambridge: Cambridge Univ. Press, 1999 52. Labastida, J.M.F., Llatas, P.M., Ramallo, A.V.: Knot operators in Chern-Simons gauge theory. Nucl. Phys. B 348, 651 (1991) 53. Isidro, J.M., Labastida, J.M.F., Ramallo, A.V.: Polynomials for torus links from Chern-Simons gauge theory. hep-th/9210124, Nucl. Phys. B 398, 187 (1993) 54. Morton, H.R., Lukac, S.G.: The HOMFLY polynomial of the decorated Hopf link. math.GT/0108011 55. Lukac, S.G.: HOMFLY skeins and the Hopf link. Ph.D. Thesis, June 2001, in http://www.liv. ac.uk/ su14/knotgroup.html 56. Macdonald, I.G.: Symmetric functions and Hall polynomials. 2nd edition, Oxford: Oxford University Press, 1995 57. Gopakumar, R., Vafa, C.: M-theory and topological strings, II. hep-th/9812127 58. Kaul, R.K.: Chern-Simons theory, knot invariants, vertex models and three-manifold invariants. hep-th/9804122, In: Frontiers of field theory, quantum gravity and strings, Hauppauge, NY: Nova Science, 1999, p. 45 59. Kaul, R.K., Ramadevi, P.: Three-manifold invariants from Chern-Simons field theory with arbitrary semi-simple gauge groups. hep-th/0005096, Commun. Math. Phys. 217, 295 (2001) 60. Bryan, J., Pandharipande, R.: BPS states of curves in Calabi-Yau threefolds. math.AG/0009025, Geom. Topolo. 5, 287 (2001) 61. Hori, K., Vafa, C.: Mirror symmetry. hep-th/0002222 62. Cachazo, F., Vafa, C.: N = 1 and N = 2 geometry from fluxes. hep-th/0206017 Communicated by N.A. Nekrasov
Commun. Math. Phys. 247, 513–526 (2004) Digital Object Identifier (DOI) 10.1007/s00220-004-1068-9
Communications in
Mathematical Physics
A Semi-Classical Trace Formula at a Totally Degenerate Critical Level Contributions of Local Extremum Brice Camus Mathematisches Institut der Ludwig-Maximilians-Universit¨at M¨unchen, Theresienstraße 39, 80333 Munich, Germany. E-mail: [email protected] Received: 10 December 2002 / Accepted: 4 November 2003 Published online: 9 April 2004 – © Springer-Verlag 2004
Abstract: We study the semi-classical trace formula at a critical energy level for an h-pseudo-differential operator on Rn whose principal symbol has a totally degenerate critical point for that energy. This problem is studied for a large time behavior and under the hypothesis that the principal symbol of the operator has a local extremum at the critical point. 1. Introduction The semi-classical trace formula for a self-adjoint h-pseudo-differential operator Ph , or more generally h-admissible (see [17]), studies the asymptotic behavior, as h goes to 0, of the spectral function : γ (E, h, ϕ) =
|λj (h)−E|≤ε
ϕ(
λj (h) − E ), h
(1)
where the λj (h) are the eigenvalues of Ph and where we suppose that the spectrum is discrete in [E − ε, E + ε], some sufficient conditions for this are given below. If p0 is the principal symbol of Ph we recall that an energy E is regular when ∇p0 (x, ξ ) = 0 on the energy surface E = {(x, ξ ) ∈ T Rn / p0 (x, ξ ) = E} and critical when it is not regular. A classical property is the existence of a link between the asymptotics of (1), as h tends to 0, and the closed trajectories of the Hamiltonian flow of p0 on E lim γ (E, h, ϕ) {(t, x, ξ ) ∈ R × E / t (x, ξ ) = (x, ξ )},
h→0
This work was supported by the IHP-Network Analysis and Quantum, reference number HPRN-CT2002-00277. We thank Raymond Brummelhuis for encouragement and for advice concerning this work. We also thank Bernard Helffer for constructive remarks on this work.
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where t = exp(tHp0 ) and Hp0 = ∂ξ p0 .∂x − ∂x p0 .∂ξ . A non-exhaustive list of references concerning this subject is Gutzwiller [10], Balian and Bloch [1] for the physic literature and for a mathematical point of view Brummelhuis and Uribe [3], Paul and Uribe [15], and more recently Combescure, Ralston and Robert [7], Petkov and Popov [16], Charbonnel and Popov [6]. The case of a non-degenerate critical energy of the principal symbol p0 (x, ξ ), that is such that the critical-set C(p0 ) = {(x, ξ ) ∈ T ∗ Rn / dp0 (x, ξ ) = 0} is a compact C ∞ manifold with a Hessian d 2 p0 transversely non-degenerate along this manifold, has been investigated first by Brummelhuis, Paul and Uribe in [2]. They treated this question for quite general operators but for some ”small times”, that is it was assumed that 0 is the only period of the linearized flow in supp(ϕ) ˆ when it is small. Later, Khuat-Duy in [13] and [14] has obtained the contributions of the non-zero periods of the linearized flow with the assumption that supp(ϕ) ˆ is compact, but for Schr¨odinger operators with symbol ξ 2 + V (x) and a non-degenerate potential V . Our contribution to this subject was to compute the contributions of the non-zero periods of the linearized flow for some more general operators, always with ϕˆ of compact support and under some geometrical assumptions on the flow (see [4 or 5]). Basically, the asymptotics of (1) can be expressed in terms of oscillatory integrals whose phases are related to the classical dynamic of p0 on E . For (x0 , ξ0 ) a critical point of p0 , it is well known that the relation : Ker(dx,ξ t (x0 , ξ0 ) − Id) = {0},
(2)
leads to the study of degenerate oscillatory integrals. What is new here is that we examine the case of a totally degenerate energy, that is such that the Hessian matrix at our critical point is zero. Hence, the linearized flow for such a critical point satisfies dx,ξ t (x0 , ξ0 ) = Id, for all t ∈ R and the oscillatory integrals we have to consider are totally degenerate. The results obtained here are global in time, that is we only assume that supp(ϕ) ˆ is compact. The core of the proof lies in establishing suitable normal forms for our phase functions and in a generalization of the stationary phase formula for these normal forms. 2. Hypotheses and Main Result Let Ph = Ophw (p(x, ξ, h)) be a h-pseudodifferential operator in the class of the h admissible operators with symbol p(x, ξ, h) ∼ hj pj (x, ξ ), i.e. there exist sequences (pj )j ∈ 0m (T ∗ Rn ) and (RN (h))N such that hj pjw (x, hDx ) + hN RN (h), ∀N ∈ N, Ph = j
where RN (h) is a bounded family of operators on L2 (Rn ), for h ≤ h0 , and 0m (T ∗ Rn ) = {a : T ∗ Rn → C, sup |∂xα ∂ξ a(x, ξ )| < Cα,β m(x, ξ ), ∀α, β ∈ Nn }, β
where m is a tempered weight on T ∗ Rn . For a detailed exposition on h-admissible operators we refer to the book of Robert [17]. Let us note p0 (x, ξ ) the principal symbol of Ph , p1 (x, ξ ) the sub-principal symbol and t = exp(tHp0 ) : T ∗ Rn → T ∗ Rn , the Hamiltonian flow of Hp0 = ∂ξ p0 .∂x − ∂x p0 .∂ξ .
A Semi-Classical Trace Formula at a Totally Degenerate Critical Level
515
If Ec is a critical value of p0 , we study the asymptotics of the spectral function λj (h) − Ec γ (Ec , h) = ϕ( ), (3) h λj (h)∈[Ec −ε,Ec +ε]
under the hypotheses (H1 ) to (H4 ) given below. (H1 ) The symbol of Ph is real. There exists ε0 > 0 such that p0−1 ([Ec − ε0 , Ec − ε0 ])is compact. Then, by Theorem 3.13 of [17] the spectrum σ (Ph ) ∩ [Ec − ε, Ec + ε] is discrete and consists in a sequence λ1 (h) ≤ λ2 (h) ≤ ... ≤ λj (h) of eigenvalues of finite multiplicities, if ε < ε0 and h is small enough. To simplify notations we write z = (x, ξ ) for any point of the phase space. (H2 ) On the energy surface EC = p0−1 ({Ec }), p0 has a unique critical point z0 = (x0 , ξ0 )and near z0 : p0 (z) = Ec +
N
pj (z) + O(||(z − z0 )||N+1 ), k > 2,
j =k
where the functions pj are homogeneous of degree j in z − z0 . (H3 ) We have ϕˆ ∈ C0∞ (R). With (H2 ) the oscillatory-integrals we will have to consider are totally degenerate. Hence, they cannot be treated by the classical stationary phase method. To solve this problem we impose the following condition on the symbol : (H4 ) The critical point z0 is a local extremum of p0 . Remark 1. (H4 ) implies that the first non-zero homogeneous component pk is even and is positive or negative definite and also that z0 is isolated on Ec . Since we are interested in the contribution to the trace formula of the fixed point z0 , to understand the new phenomenon, it suffices to study tEc 1 i w γz0 (Ec , h) = ˆ (x, hDx )exp(− tPh )(Ph )dt. (4) Tr ei h ϕ(t)ψ 2π h R
Here is a function of localization near the critical energy surface Ec and ψ ∈ C0∞ (T ∗ Rn ) has an appropriate support near z0 . Rigorous justifications are given in Sect. 3 for the introduction of (Ph ) and in Sect. 4 for ψ w (x, hDx ). Then, the new contribution to the trace formula is given by : Theorem 1. Under hypotheses (H1 ) to (H4 ) we obtain : 2n
γz0 (Ec , h) ∼ h k
−n
(
N
j
j,k (ϕ)h k + O(h
N +1 k
)), as h → 0,
j =0
where the j,k are some distributions and the leading coefficient is given by 2n−k 2n 1 1 0,k (ϕ) = ϕ(t + p1 (z0 )), tz0 k |pk (θ )|− k dθ, n k (2π ) S2n−1
with tz0 = max(t, 0) if z0 is a minimum and tz0 = max(−t, 0) for a maximum. Remark 2. One can derive a full asymptotic expansion as is shown in Lemma 5.
(5)
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3. Oscillatory Representation Let be ϕ ∈ S(R) with ϕˆ ∈ C0∞ (R). We recall that λj (h) − Ec γ (Ec , h) = ), Iε = [Ec − ε, Ec + ε], ϕ( h λj (h)∈Iε
with p0−1 (Iε0 ) compact in T ∗ Rn the spectrum of Ph is discrete in Iε for ε < ε0 and h small enough. Now, we localize near the critical energy Ec with a cut-off function ∈ C0∞ (]Ec − ε, Ec + ε[), such that = 1 near Ec and 0 ≤ ≤ 1 on R. The associated decomposition is γ (Ec , h) = γ1 (Ec , h) + γ2 (Ec , h), with
γ1 (Ec , h) =
(1 − )(λj (h))ϕ(
λj (h)∈Iε
γ2 (Ec , h) =
(λj (h))ϕ(
λj (h)∈Iε
(6)
λj (h) − Ec ), h
λj (h) − Ec ). h
The asymptotic behavior of γ1 (Ec , h) is given by Lemma 1. γ1 (Ec , h) = O(h∞ ) as h → 0. Proof. Since ϕ ∈ S(R), ∀k ∈ N, ∃Ck such that |x k ϕ(x)| ≤ Ck on R. By Theorem 3.13 of [17] the number of eigenvalues N (h) lying in Iε ∩ supp(1 − ) is of order O(h−n ), for h small enough. This gives the estimate: λj (h) − Ec −k | . h But on the support of (1 − ) we have |λj (h) − Ec | > ε0 > 0. This leads to |γ1 (Ec , h)| ≤ N (h)Ck |
|γ1 (Ec , h)| ≤ N (h)CN ε0−k hk ≤ cN hk−n . Since the property is true for all k ∈ N, this ends the proof.
O(h∞ ),
we have only to consider Consequently, for the study of γ (Ec , h) modulo the quantity γ2 (Ec , h). By inversion of the Fourier transform we have : Ph − Ec 1 )= (Ph )ϕ( h 2π
ei R
tEc h
i ϕ(t)exp(− ˆ tPh )(Ph )dt. h
Since the trace of the left hand-side is exactly γ2 (Ec , h), we obtain : tEc 1 i Tr ei h ϕ(t)exp(− tPh )(Ph )dt, ˆ γ2 (Ec , h) = 2π h R
and with Lemma 1 this gives : tEc 1 i Tr ei h ϕ(t)exp(− tPh )(Ph )dt + O(h∞ ). ˆ γ (Ec , h) = 2π h R
(7)
A Semi-Classical Trace Formula at a Totally Degenerate Critical Level
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Remark 3. Another interest of this formulation is that, under the geometrical condition to have a “clean” flow, γ (Ec , h) can be expressed, at the first order, as the composition of two Fourier integral-operators. Let Uh (t) = exp(− ith Ph ) be the evolution operator. For each integer L we can approximate Uh (t)(Ph ), modulo O(hL ), by a Fourier integral-operator, or FIO, depending on a parameter h. To give a precise formulation of this approximation we recall briefly the principal notions on FIO. Let N ∈ N, ϕ ∈ C ∞ (Rn × RN ) and a ∈ C0∞ (Rn × RN ). An oscillatory-integral with phase ϕ and amplitude a is i i ϕ − N2 h I (ae ) = (2πh) a(x, θ)e h ϕ(x,θ) dθ. (8) RN
To attain our objectives, it suffices to consider amplitudes a with compact support. We suppose, as usual, that ϕ = ϕ(x, θ ) is a non-degenerate phase function, i.e. d(∂θ1 ϕ) ∧ ... ∧ d(∂θN ϕ) = 0 on C(ϕ) = {(x, θ ) ∈ Rn × RN / dθ ϕ(x, θ) = 0}. This implies that C(ϕ) is a sub-manifold of class C ∞ of Rn × RN and that C(ϕ) → T ∗ (Rn ), iϕ : (x, θ ) → (x, dx ϕ(x, θ )), is an immersion. In this situation one says that ϕ parameterizes the Lagrangian manifold ϕ = iϕ (C(ϕ)). Conversely, if ⊂ T ∗ Rn is a Lagrangian sub-manifold we can always find locally some non-degenerate phase functions parameterizing , see e.g. [8]. Now if ϕ1 and ϕ2 , ϕj ∈ C ∞ (Rn × RNj ), are two non-degenerate phase functions parameterizing locally the same Lagrangian manifold, i.e. ϕ1 ∩ U = ϕ2 ∩ U , for an open U ⊂ T ∗ Rn , then it is well-known that there exists a constant c ∈ R such that for all a1 ∈ C0∞ (Rn × RN1 ) with iϕ1 (supp(a1 ) ∩ C(ϕ1 )) ⊂ U , small enough, there exists a sequence a2,j ∈ C0∞ (Rn × RN2 ), j ∈ N, such that for all L ∈ N : i i i hj I (a2,j e h ϕ2 ) + hL rL (h), I (a1 e h ϕ1 ) = e h c j
with rL (h) uniformly bounded in L2 (Rn ), since a1 ∈ C0∞ , and c comes from Sϕ1 −Sϕ2 = c on ϕ1 ∩ U = ϕ2 ∩ U . This recalls that the fundamental object associated to an oscillatory-integral is the Lagrangian manifold parameterized by the phase function. The next definition follows H¨ormander. Definition 1. Let ⊂ T ∗ Rn be a Lagrangian sub-manifold of class C ∞ . The class I (Rn , ) of oscillatory functions associated to is given by i uh ∈ I (Rn , ) ⇔ uh = I (av (h)e h ϕv ), v
where the sum is locally finite and the functions av (h) are of the form : av (h) = hj aj , aj ∈ C ∞ (Rn × RNv ), −dv ≤j <Jv
with dv , Jv and Nv in N and where ϕv is a non-degenerate phase function parameterizing ∩ iϕv (supp(av (h))), where supp(av (h)) = supp(av,j ). j
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Now let ⊂ T ∗ (Rn × Rl ) be a C ∞ Lagrangian sub-manifold of T ∗ (Rn × Rl ). Definition 2. The family of operators (Fh ) : L2 (Rn ) → L2 (Rl ), 0 < h ≤ 1, is a h-FIO, (N) (N) associated to , if there exist two families Fˆh and Rh , N ∈ N, such that : (N) i) Each Fˆh has an integral kernel in I (Rn × Rl , ). (N) ii) For each N the operator Rh is bounded on L2 and there exists CN > 0 such that (N) ||Rh ||L(L2 (Rn ),L2 (Rl )) ≤ CN , uniformly in h, 0 < h ≤ 1. (N) (N) iii) For each N : Fh = Fˆ + hN R . h
h
After these definitions, we recall the theorem that gives the approximation of Uh (t) (Ph ). Let be the Lagrangian manifold associated to the flow of p0 : = {(t, τ, x, ξ, y, η) ∈ T ∗ R × T ∗ Rn × T ∗ Rn : τ = p0 (x, ξ ), (x, ξ ) = t (y, η)}.
(N)
Theorem 2. The operator Uh (t)(Ph ) is an h-FIO associated to , there exists U,h (t) (N)
with integral kernel in I (R2n+1 , ) and Rh (t) bounded, with a L2 -norm uniformly bounded for 0 < h ≤ 1 and t in a compact subset of R, such that Uh (t)(Ph ) = (N) (N) U,h (t) + hN Rh (t). We refer to Duistermaat [8] for a proof of this theorem. Remark 4. By a theorem of Helffer and Robert, see e.g. [17], Theorem 3.11 and Remark 3.14, (Ph ) is an h-admissible operator with a symbol of compact support in p0−1 (Iε ). This allows us to consider only oscillatory-integrals with compact support. For our goal the following corollary is crucial. Corollary 1. Let 1 ∈ C0∞ (R) be such that 1 = 1 on supp() and supp(1 ) ⊂ Iε , then ∀N ∈ N : i Ph − Ec 1 (N) N h tEc U Tr((Ph )ϕ( ˆ )) = Tr ϕ(t)e ,h (t)1 (Ph )dt + O(h ). h 2π R
Proof. By cyclicity of the trace, for all N ∈ N we have : 1 Ph − E c )) = Tr Tr((Ph )ϕ( h 2π
i
(N)
(N)
N h tEc (U ϕ(t)e ˆ ,h (t) + h Rh (t))1 (Ph )dt.
R (N)
But 1 (Ph ), and a fortiori 1 (Ph )Rh (t) by the ideal property, is a class-trace operator (see e.g. [17], Sect. 2.5), with norm : (N)
(N)
||1 (Ph )Rh (t)||Tr ≤ ||Rh (t)|| ||1 (Ph )||Tr ≤ Cϕˆ ||1 (Ph )||Tr ; since it is true for all t ∈ supp(ϕ) ˆ the corollary is proven.
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The geometrical approach associated to the H¨ormander class I (R2n+1 , ) gives a great degree of freedom in the choice of the phase function; this will be exploited below. In fact if (x0 , ξ0 ) ∈ and if ϕ = ϕ(x, θ ) ∈ C ∞ (Rn × RN ) parameterizes in a neighborhood U, small enough, of (x0 , ξ0 ) then for each uh ∈ I (Rn , ) and χ ∈ C0∞ (T ∗ Rn ), supp(χ ) ⊂ U, there exists a sequence of amplitudes aj = aj (x, θ ) ∈ C0∞ (Rn × RN ) such that for all integer L : i χ w (x, hDx )uh = hj I (aj e h ϕ ) + O(hL ). (9) −d≤j
We will use this remark with the following result of H¨ormander (see [11], tome 4, Prop. 25.3.3). Let (T , τ, x0 , ξ0 , y0 , −η0 ) ∈ flow , η0 = 0, then near this point there exists, after perhaps a change of local coordinates in y near y0 , a function S(t, x, η) such that φ(t, x, y, η) = S(t, x, η) − y, η ,
(10)
parameterizes flow . In particular this implies that {(t, ∂t S(t, x, η), x, ∂x S(t, x, η), ∂η S(t, x, η), −η)} ⊂ flow , and that the function S is a generating function of the flow, i.e. t (∂η S(t, x, η), η) = (x, ∂x S(t, x, η)).
(11)
Moreover, S satisfies the Hamilton-Jacobi equation : ∂t S(t, x, η) + p0 (x, ∂x S(t, x, η)) = 0, S(0, x, ξ ) = x, ξ . Now, we apply this result with (x0 , ξ0 ) = (y0 , η0 ), our unique fixed point of the flow on i i the energy surface Ec . If ξ0 = 0 we can replace the operator Ph by e h x,ξ1 Ph e− h x,ξ1 with ξ1 = 0. This will not change the spectrum since this new operator has the symbol p(x, ξ − ξ1 , h) and the critical point is now (x0 , ξ1 ) with ξ1 = 0. Consequently, the localized trace γ2 (Ec , h), defined by Eq. (6), can be written for all N ∈ N and modulo O(hN ) as i (2πh)−d+j e h (S(t,x,ξ )−x,ξ +tEc ) aj (t, x, ξ )ϕ(t)dtdxdξ. ˆ (12) γ2 (Ec , h) = j
R×R2n
To obtain the right power −d of h occurring in Eq. (12) we apply results of Duistermaat [8] (following here H¨ormander for the FIO, see [12] tome 4, for example) concerning the order. An h-pseudodifferential operator obtained by Weyl quantization : i x+y − N2 (2πh) , ξ )e h x−y,ξ dξ, a( 2 RN
is of order 0 w.r.t. 1/ h. Now, since the order of Uh (t)(Ph ) is − 41 , we find that i (2πh)−n+j aj (t, x, y, η)e h (S(t,x,η)−y,η) dy. (13) ψ w (x, hDx )Uh (t)(Ph ) ∼ j
Rn
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h tEc and passing to the trace we find (12) with d = n, Multiplying Eq. (13) by ϕ(t)e ˆ where we write again aj (t, x, η) for aj (t, x, x, η). i
To each element uh of I (Rn , ) we can associate a principal symbol e h S σprinc (uh ), i
where S is a function on such that ξ dx = dS on . If uh = I (ae h ϕ ), then we have 1 S = Sϕ = ϕ ◦iϕ−1 and σprinc (uh ) is a section of || 2 ⊗M(), where M() is the Maslov 1
vector-bundle of and || 2 the bundle of half-densities on . If p1 is the sub-principal symbol of Ph , the half-density of the propagator Uh (t) is easily expressed in the global coordinates (t, y, η) on flow via t
1
p1 (s (y, −η))ds)|dtdydη| 2 .
exp(i
(14)
0
This expression is related to the resolution of the first transport equation for the propagator. For a proof we refer to Duistermaat and H¨ormander [9]. 4. Classical Dynamic Near the Critical Point A critical point of the phase function of Eq. (12) leads to the equations : Ec = −∂t S(t, x, ξ ), p0 (x, ξ ) = Ec , x = ∂ξ S(t, x, ξ ), ⇔ t (x, ξ ) = (x, ξ ), ξ = ∂ S(t, x, ξ ), x where the right-hand side defines a closed trajectory of the flow inside Ec . Since we are interested in the contribution of the critical point, we choose a function ψ ∈ C0∞ (T ∗ Rn ), with ψ = 1 near z0 , hence tEc 1 i w ˆ (x, hDx )exp(− tPh )(Ph )dt Tr ei h ϕ(t)ψ γ2 (Ec , h) = 2π h R tEc 1 i + Tr ei h ϕ(t)(1 ˆ − ψ w (x, hDx ))exp(− tPh )(Ph )dt. 2π h R
The contribution on supp(1 − ψ) is non-singular and can be treated by the regular trace ˆ formula. If supp(ψ) is small enough only points (t, x, ξ ) = (t, z0 ), for t ∈ supp(ϕ), will give a contribution, since z0 is isolated on Ec . Now, we restrict our attention to the contribution of the critical point. Until further notice, the derivatives d will be taken with respect to initial conditions. Since z0 is totally degenerate, we obtain : dt (z0 ) = exp(0) = Id, ∀t.
(15)
With (H2 ), the next homogeneous components of the flow are given by : d j t (z0 ) = 0, ∀t, ∀j ∈ {2, .., k − 2}.
(16)
To obtain the next non-zero terms of the Taylor expansion of the flow, we will use the following technical result :
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Lemma 2. Let z0 be an equilibrium of the C ∞ vector field X and t the flow of X. Then for all m ∈ N∗ , there exists a polynomial map Pm , vector valued and of degree at most m, such that t d t (z0 )(z ) = dt (z0 ) m
m
d−s (z0 )Pm (ds (z0 )(z), ..., d m−1 s (z0 )(zm−1 ))ds. 0
(17) Proof. We note x l ∈ (Rn )l the image of x under the diagonal mapping, with the same convention for any vector. If f, g are two C ∞ applications, by the “Faa di Bruno formula”, we have :
d m (f og)(x0 )(x m ) =
C(p)d r f (g(x0 ))(d n1 g(x0 )(x n1 ), ..., d nr g(x0 )(x nr )),
p∈P (m)
where P(m) is the set of partitions of the integer m and where it is assumed that the partition p of m is given by m = n1 + ... + nr and C(p) are some universal integers. Hence, for our fixed point z0 we obtain :
d m (Xos )(z0 )(zm ) =
C(p)d r X(z0 )(d n1 s (z0 )(zn1 ), ..., d nr s (z0 )(znr )).
p∈P (m)
For Y = (Y1 , ..., Ym ), we can define : C(p)d r X(z0 )(Yn1 , ..., Ynr ) − dX(z0 )(Ym ); Pm (Y ) =
(18)
p∈P (m)
this leads to the differential equation, operator valued d m (d s (z0 ))(zm ) = dX(z0 )d m s (z0 )(zm )+Pm (ds (z0 )(z), ..., d m−1 s (z0 )(zm−1 )). ds With the initial condition d m 0 (z0 ) = 0, the solution is given by Eq. (17).
Since dt (z0 ) = Id, ∀t, the first non-zero term of the Taylor expansion is t d
k−1
t (z0 )(z
k−1
)=
d k−1 Hp0 (z0 )(zk−1 )ds = td k−1 Hpk (z0 )(zk−1 ),
(19)
0
where the last identity is obtained using that d k−1 Hp0 (z0 ) = d k−1 Hpk (z0 ). Moreover, with d 2 p0 (z0 ) = 0, for the next term Lemma 2 gives t d t (z0 )(z ) = k
d k Hp0 (z0 )(zk )ds = td k Hpk+1 (z0 )(zk ).
k
0
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Remark 5. For the derivatives d j t (z0 ), with j > k there are two different kinds of terms, namely t d j Hp0 (z0 )(zj )ds = td j Hp0 (z0 )(zj ) = O(||z||j ), 0
and terms involving powers of t. For example, we have : t
d j +2−k Hp0 (z0 )(zj +1−k , d k−1 t (z0 )(zk−1 ))
0
=
t 2 j +2−k Hp0 (z0 )(zj +1−k , d k−1 Hpk (z0 )(zk−1 )). d 2
This term is simultaneously O(t 2 ) and O(||z||j ) near (0, z0 ). A similar result holds for other terms, which are O(t d ) for d ≥ 2, by an easy recurrence. Lemma 3. Near z0 , here supposed to be 0 to simplify, we have : S(t, x, ξ ) − x, ξ + tEc = −t (pk (x, ξ ) + Rk+1 (x, ξ ) + tGk+1 (t, x, ξ )),
(20)
where Rk+1 (x, ξ ) = O(||(x, ξ )||k+1 ) and Gk+1 (t, x, ξ ) = O(||(x, ξ )||k+1 ), uniformly for t in a compact subset of R. Proof. By Taylor and under hypothesis (H2 ) we obtain : t (x, ξ ) = (x, ξ ) +
1 d k−1 t (0)(zk−1 ) + O(||z||k ). (k − 1)!
(21)
Now, we search our local generating function as S(t, x, ξ ) = −tEc + x, ξ +
N
Sj (t, x, ξ ) + O(||(x, ξ )||N+1 ),
j =3
where the functions Sj are time dependent and homogeneous of degree j w.r.t. (x, ξ ). With the implicit relation : t (∂ξ S(t, x, ξ ), ξ ) = (x, ∂x S(t, x, ξ )) and using Eq. (21) we have : S(t, x, ξ ) = −tEc + x, ξ + Sk (t, x, ξ ) + O(||(x, ξ )||k+1 ), and comparing terms of degree k − 1 gives : J ∇Sk (t, x, ξ ) = −
1 d k−1 t (0)((x, ξ )k−1 ), (k − 1)!
where J is the matrix of the usual symplectic form. By homogeneity and with Eq. (19) we obtain that 1 Sk (t, x, ξ ) = (x, ξ ), tJ d k−1 Hpk (x, ξ )k−1 = −tpk (x, ξ ). k!
A Semi-Classical Trace Formula at a Totally Degenerate Critical Level
523
It remains now to treat the remainder. Since S(0, x, ξ ) = x, ξ , we have : S(t, x, ξ ) − x, ξ = tF (t, x, ξ ), with F smooth in a neighborhood of (x, ξ ) = 0. Now, the Hamilton-Jacobi equation imposes that F (0, x, ξ ) = −p0 (x, ξ ) and we obtain : Rk+1 (x, ξ ) = p0 (x, ξ ) − Ec − pk (x, ξ ) = O(||(x, ξ )||k+1 ). Finally, the time dependent remainder can be written : S(t, x, ξ ) − S(0, x, ξ ) − t∂t S(0, x, ξ ) = O(t 2 ); since by construction this term is of order O(||(x, ξ )||k+1 ) we get the result.
5. Normal Forms of the Phase Function Since the contribution we study is local, we can work with some coordinates and identify locally T ∗ Rn with R2n near the critical point. We define (t, z) = (t, x, ξ ) = S(t, x, ξ ) − x, ξ + tEc , z = (x, ξ ) ∈ R2n .
(22)
Lemma 4. If Ph satisfies conditions (H2 ) and (H4 ) then, in a neighborhood of z = z0 , there exist local coordinates χ such that (t, z) χ0 χ1k if z0 is a maximum and (t, z) −χ0 χ1k if z0 is a minimum. Proof. We can here assume that z0 is the origin and we use polar coordinates z = (r, θ ), θ ∈ S2n−1 (R). With Lemma 3, near the critical point we have : (t, z) (t, rθ ) = −tr k (pk (θ ) + rRk+1 (θ ) + tGk+1 (t, rθ )), where pk (θ ) is the restriction of pk on S2n−1 . We define new coordinates : (χ0 , χ2 , ..., χ2n )(t, r, θ) = (t, θ1 , ..., θ2n−1 ), 1
χ1 (t, r, θ) = r|pk (θ ) + rRk+1 (θ ) + tGk+1 (t, rθ )| k . In these coordinates the phase becomes −χ0 χ1k for a minimum and χ0 χ1k for a maximum. Now, we observe that 1 ∂χ1 (t, 0, θ) = |pk (θ )| k , ∀t, ∂r
hence, the corresponding Jacobian satisfies the relation 1
|J χ |(t, 0, θ) = |pk (θ )| k = 0, ∀t, and this defines a local system of coordinates near z0 for all t.
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This change of coordinates leads to i i k e h (t,r,θ) a(t, rθ )r 2n−1 dtdrdθ = e± h χ0 χ1 A(χ0 , χ1 )dχ0 dχ1 , R×R+ ×S2n−1
with a new amplitude A defined by : A(χ0 , χ1 ) = χ ∗ (a(t, rθ )r 2n−1 |J χ |)dχ2 ...dχ2n .
(23)
1
Remark 6. Since χ1 (t, r, θ) = r|pk (θ ) + rRk+1 (θ ) + tGk+1 (t, rθ )| k , our new amplitude satisfies A(χ0 , χ1 ) = O(χ12n−1 ), near χ1 = 0. This will play a major role since distributions of Lemma 5 below are supported in {χ1 = 0}. We end this section with a lemma on asymptotics of oscillatory integrals. Lemma 5. There exists a sequence (cj )j of distributions, whose support is contained in the set {χ1 = 0}, such that for all functions a ∈ C0∞ (R+ ×R) : ∞ ∞ j +1 k ( eiλχ0 χ1 a(χ0 , χ1 )dχ0 )dχ1 ∼ λ− k cj (a), j =0
R
0
(24)
asymptotically for λ → ∞, with cj =
j +1−k 1 1 (j ) (F(x− k )(χ0 ) ⊗ δ0 (χ1 )), x− = max(−x, 0). k j!
ˆ r) = Ft (a(t, r))(τ ), where Ft is Proof. We note (t, r) for (χ0 , χ1 ) and we define g(τ, the partial Fourier transform with respect to t. Then, we obtain : ∞ ∞ ∞ iλtr k k − k1 k r ( e a(t, r)dt)dr = g(−λr ˆ , r)dr = λ g(−r ˆ , 1 )dr. λk 0
R
0
0
A Taylor expansion with respect to r for g(τ, ˆ r) at the origin gives : k g(−r ˆ ,
r
)= 1
λk
l N λ− k
l=0
l!
rl
N +1 ∂ l gˆ (−r k , 0) + λ− k RN+1 (r, λ), ∂r l
where RN+1 (r, λ) is integrable with respect to r, with L1 norm uniformly bounded in λ. By a new change of variable we obtain : − k1
∞
0 l N l+1−k N +1 ∂ gˆ 1 1 − 1+l λ k g(−r ˆ , 1 )dr = (r, 0)|r| k dr + O(λ− k ). l k l! ∂r λk l=0 −∞ k
λ
0
r
l+1−k k
If we introduce x− = max(−x, 0) and F(x−
)(r) the result follows.
Remark 7. A similar result holds for the phase −χ0 χ1k if we change χ0 into −χ0 and we then have simply to replace terms x− by x+ = max(x, 0).
A Semi-Classical Trace Formula at a Totally Degenerate Critical Level
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6. Proof of the Main Result We can assume that z0 is a maximum. Lemma 5 shows that the first non-zero coefficient is obtained for l = 2n − 1 (see Remark 6) and is given by: 2n−k 2n−k 1 1 1 (2n−1) ˜ 0 , 0)dχ0 , ), A(χ0 , χ1 ) = (F(x− k ) ⊗ δ0 (F(x− k )(χ0 )A(χ k (2n − 1)! k where, by construction, the new amplitude is : 2n−1 ˜ 0 , χ1 ) = χ ∗ (a(t, rθ )|J χ ||pk (θ ) + rRk+1 (θ ) + tGk+1 (t, rθ )|− k )dχ2 ...dχ2n . A(χ ˜ 0 , 0) via Now, we use an oscillatory representation of A(χ ˜ 0 , 0) = 1 ˜ 0 , χ1 )eizχ1 dzdχ1 . A(χ A(χ 2π We return to the initial coordinates and, since χ0 = t, we obtain : 2n−1 ˜ 0) = 1 A(t, a(t, rθ )|pk (θ ) + rRk+1 (θ ) + tGk+1 (t, rθ )|− k eizχ1 (t,r,θ) dzdrdθ. 2π Inserting the definition of χ1 , we use the change of variable : 1
y = z|pk (θ ) + rRk+1 (θ ) + tGk+1 (t, rθ )| k . Since in r = 0 we have Gk+1 (t, 0) = 0, by integration in (y, r) we obtain : 2n ˜ 0 , 0) = A(χ a(χ0 , 0)|pk (θ )|− k dθ.
(25)
S2n−1 2n−k
The distribution F(x− k )(χ0 ) of Lemma 5 acts on this function via 2n−k 2n−k 2n ˜ 0 , 0) = F(x− k ), a(., 0) |pk (θ )|− k dθ. F(x− k )(χ0 ), A(χ S2n−1
Hence, the top-order contribution to the trace formula is given by 2n−k 2n 2n 2n+1 1 h−n k k γ (Ec , h) = F(x h ), a(., 0) |pk (θ )|− k dθ + O(h k −n ). − k (2π)n+1 S2n−1
With a(t, 0) = ϕ(t) ˆ exp(itp1 (z0 )), see (14), by Fourier inversion we have : 2n−k 2n−k 1 k F(x− ), a(., 0) = ϕ(t + p1 (z0 ))t− k dt. 2π R
Finally, if z0 is a local minimum we must simply replace t− by t+ = max(t, 0) and this completes the proof of Theorem 1.
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Commun. Math. Phys. 247, 527–551 (2004) Digital Object Identifier (DOI) 10.1007/s00220-004-1078-7
Communications in
Mathematical Physics
A∞ -Structures on an Elliptic Curve A. Polishchuk Department of Mathematics, University of Oregon, Eugene, OR 97403-1221, USA Received: 8 May 2001 / Accepted: 23 January 2004 Published online: 28 April 2004 – © Springer-Verlag 2004
Abstract: The main result of this paper is the proof of the “transversal part” of the homological mirror symmetry conjecture for an elliptic curve that states an equivalence of two A∞ -categories: one is built using holomorphic vector bundles on an elliptic curve and another is a subcategory in the Fukaya A∞ -category of a torus. The proof is based on the study of A∞ -structures on the category of line bundles over an elliptic curve satisfying some natural restrictions (in particular, m1 should be zero, m2 should coincide with the usual composition). The key observation is that such a structure is uniquely determined up to equivalence by certain triple products. Introduction Let E be an elliptic curve over a field k. Let us denote by Vect(E) the category of algebraic vector bundles on E, where as a space of morphisms from V1 to V2 we take the graded space Hom(V1 , V2 ) ⊕ Ext 1 (V1 , V2 ) with the natural composition law. In this paper we study extensions of this (strictly associative) composition to A∞ -structures on Vect(E) (see Sect. 1 for the definition). The motivation comes from the homological mirror symmetry for elliptic curves formulated by Kontsevich (see [9]) that provides two such extensions in the case k = C and states that they should be equivalent. We recall the definitions of these A∞ -structures in Sect. 1.5. One of them is given by an A∞ -version of the derived category of vector bundles while another comes from a general construction in symplectic geometry due to Fukaya. Roughly speaking, one can associate to an indecomposable vector bundle on E a geodesic circle on the torus R2 /Z2 with a local system on it. Then the second A∞ -structure is defined using generating series counting holomorphic maps from the disk bounding given geodesic circles. In [19] we checked that double products defined in this way coincide with the standard composition law on Vect(E). In this paper we use this together with some calculations
This work is partially supported by NSF grant
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of triple products (see [15]) to prove the essential part of the homological mirror conjecture for E. Namely, we construct a homotopy between transversal products given by these two A∞ -structures. This means that we are looking only at the products such that the corresponding geodesic circles form a transversal configuration. The precise formulation of our main result will be given in Sect. 1 (see Theorem 1.3). We leave for a future investigation more subtle points of defining non-transversal products in the Fukaya category and extending the above homotopy to the entire derived categories. In general the equality of double products and triple Massey products in A∞ -categories corresponding to a mirror dual pair (symplectic torus, abelian variety) was established by Fukaya in [3]. In the case of elliptic curves, as we show in the present paper, this is enough for (the transversal part of) the homological mirror conjecture. For abelian varieties of higher dimensions, a version of this conjecture was proved by Kontsevich and Soibelman in [11]. 1 The main point of our paper is that in the case of elliptic curves we can formulate a result on A∞ -structures on the category L of line bundles on E which is valid over an arbitrary field k. More precisely, we axiomatize the notion of transversality and prove that if one imposes some natural restrictions on a transversal A∞ -structure on L (in particular m1 = 0, m2 is equal to the standard composition), then such a structure is uniquely determined up to a unique homotopy by triple products of simple kind. In fact, these are certain univalued well-defined triple Massey products (that are invariant under any homotopy). We apply this result to two A∞ -structures arising in the homological mirror symmetry and then use isogenies between elliptic curves (as in [19]) to construct the required homotopy on the category of vector bundles on E. Note that the advantage of working with transversal structures is that the homotopy in question is unique provided it is compatible with push-forwards under isogenies and with direct sums. The natural framework for the generalization of our result which is valid over any field k should involve the notion of a triangulated A∞ -category (as sketched in [10]). Our result seems to imply that there exists a unique up to homotopy triangulated A∞ -structure on the derived category of an elliptic curve, compatible with the standard double products and with Serre duality (see Sect. 1.3 for the definition of the latter compatibility). Indeed, the triple products appearing in our statement are univalued Massey products that are uniquely determined by the double products in the case when A∞ -structure is triangulated. One may hope that such a uniqueness of A∞ -structure on the derived category holds also for other varieties. The main reason why in the case of A∞ -structures on elliptic curves only triple products matter is the absense of non-trivial univalued well-defined k-tuple Massey products for k > 3 2 . Conventions. We always work over a ground field k; we specialize to k = C when talking about homological mirror symmetry. To shorten the formulas sometimes we denote the tensor product of vector spaces V1 and V2 over k simply by V1 V2 omitting the sign of the tensor product. We use the same abbreviation for tensor products of vector bundles. By a bundle we always mean an algebraic vector bundle (or a holomorphic vector bundle if k = C). When working with A∞ -categories it is convenient to denote the n-tuple products of composable morphisms a1 : X0 → X1 , a2 : X1 → X2 , ..., an : Xn−1 → Xn 1 The A -equivalence established in [11] deals with certain full subcategories in symplectic and ho∞ lomorphic A∞ -categories. In fact, both sides are slightly modified: Fukaya category is replaced by its degeneration, while on the holomorphic side the ground field is changed from C to C((q)). Also, only transversal products are considered. 2 with the exception of certain quadruple Massey products that can still be expressed in terms of triple products.
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by mn (a1 , a2 , . . . , an ). In particular, we denote the double composition by m2 (a1 , a2 ); often we abbreviate this to a1 a2 . This contradicts the usual convention of going from right to left when considering composition in the usual categories. To avoid confusion we will use the notation a2 ◦ a1 for the composition in the usual categories. 1. A∞ -Structures and Their Homotopies In the following definitions we use the sign convention of [4] which is different from the one in the original definition of [20]. 1.1. A∞ -algebras. A (Z-graded) A∞ -algebra is a Z-graded vector space A equipped with linear maps mn : A⊗n → A for n ≥ 1 of degree 2 − n satisfying for every n ≥ 1 the following A∞ -constraint Axn :
k
(−1)l(a˜ 1 +...+a˜ j −1 )+j (l+1) mk (a1 , . . . , aj −1 , ml (aj , . . . , aj +l−1 ), aj +l , . . . , an ) = 0,
k+l=n+1 j =1
where a˜ i = deg(ai ) mod (2). For example, Ax1 says that m21 = 0, Ax2 gives the Leibnitz identity for m1 and m2 , etc. One can consider mn as components of a coderivation d of the coalgebra T (sA), where s denotes the suspension. The elements of T (sA) are denoted traditionally as follows: [a1 |a2 | . . . |an ] = (sa1 ) ⊗ (sa2 ) ⊗ . . . (san ). The coderivation d has a component dn : (sA)⊗n → sA given by s ◦ mn ◦ (s −1 )⊗n , so that d([a1 | . . . |an ]) =
k
(−1)a˜ 1 +...+a˜ j −1 +j −1+µ(aj ,... ,aj +l−1 )
k+l=n+1 j =1
× [a1 | . . . |aj −1 |ml (aj , . . . , aj +l−1 )|aj +l | . . . |an ], where for every collection of elements (a1 , . . . , an ) in A we denote µ(a1 , . . . , an ) = (n − 1)a˜ 1 + (n − 2)a˜ 2 + . . . + a˜ n−1 +
n(n − 1) . 2
The A∞ -constraints are equivalent to the condition d 2 = 0. For a pair of A∞ -algebras (A, mA ) and (B, mB ) there is a natural notion of a A∞ morphism from A to B. Namely, such a morphism consists of the data (fn , n ≥ 1) where fn : A⊗n → B is a linear map of degree 1 − n such that (−1)L mB i (fk1 (a1 , . . . , ak1 ), fk2 −k1 1≤k1
× (ak1 +1 , . . . , ak2 ), . . . , fn−ki−1 (aki−1 +1 , . . . , an )) =
k
k+l=n+1 j =1
(−1)R fk (a1 , . . . , aj −1 , mA l (aj , . . . , aj +l−1 ), aj +l , . . . , an ),
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where the signs L and R are defined as follows: L = µ(a1 , . . . , ak1 ) + µ(ak1 +1 , . . . , ak2 ) + . . . + µ(aki−1 +1 , . . . , an ) +µ(fk1 (a1 , . . . , ak1 ), . . . , fn−ki−1 (aki−1 +1 , . . . , an )), R = a1 + . . . + aj −1 + j − 1 + µ(aj , . . . , aj +l−1 ) +µ(a1 , . . . , aj −1 , mA l (aj , . . . , aj +l−1 ), aj +l , . . . , an ). Again one can consider (fn ) as components of a coalgebra homomorphism F : T (sA) → T (sB), so that the above equation is equivalent to F ◦ d A = d B ◦ F, where dA (resp. d B ) is the coderivation on A (resp. B) defined by mA (resp. mB ). In particular, there is a natural composition of A∞ -morphisms defined as follows: (f ◦ g)n (a1 , . . . , an ) = (−1) fi (gk1 (a1 , . . . , ak1 ), gk2 −k1 1≤k1
× (ak1 +1 , . . . , ak2 ), . . . , gn−ki−1 (aki−1 +1 , . . . , an )), where = µ(a1 , . . . , ak1 ) + µ(ak1 +1 , . . . , ak2 ) + . . . + µ(aki−1 +1 , . . . , an ) +µ(gk1 (a1 , . . . , ak1 ), . . . , gn−ki−1 (aki−1 +1 , . . . , an )). In the case when B and A have the same underlying spaces and f1 = id we will call the data f = (fn , n ≥ 2) a homotopy3 between two A∞ -structures m = mA and m = mB on the same space. Note that for homotopic m and m we necessarily have m1 = m1 . If f is a homotopy between m and m , g is a homotopy between m and m , then g ◦ f is a homotopy between m and m . Lemma 1.1. Let m = (mn ) be an A∞ -structure on A, (fn : A⊗n → A, n ≥ 2) be an arbitrary family of maps, deg fn = 1 − n. Then there exists a unique A∞ -structure m on A such that f = (fn ) (where f1 = id) is a homotopy between m and m . Proof. This follows immediately from the fact that the coalgebra homomorphism T (sA) → T (sA) defined by (fn ) is an isomophism.
We denote the A∞ -structure m constructed in the above lemma by m + δf (note that it depends non-linearly on f ). 1.2. A∞ -categories. The definition of an A∞ -category is similar to that of an A∞ algebra (see [1, 8, 12]). Namely, an A∞ -category C consists of a class of objects Ob C, for every pair of objects X and X , a graded space of morphisms Hom(X, X ), and a collection of linear maps (compositions) mn : Hom(X0 , X1 ) ⊗ Hom(X1 , X2 ) ⊗ . . . ⊗ Hom∗ (Xn−1 , Xn ) → Hom(X0 , Xn ) of degree 2−n for all n ≥ 1. The associativity constraint is that these compositions define a structure of A∞ -algebra on ⊕ij Hom(Xi , Xj ) for every collection X0 , . . . , Xn ∈ Ob C. 3
In [18] the same notion is referred to as strict A∞ -isomorphism.
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An A∞ -functor (see [2, 8, 12]) φ : C → C between A∞ -categories consists of a map φ : Ob C → Ob C and of a collection of linear maps fn : HomC (X0 , X1 ) ⊗ HomC (X1 , X2 ) ⊗ . . . ⊗ HomC (Xn−1 , Xn ) → HomC (φ(X0 ), φ(Xn ))
(1.1)
of degree 1 − n for n ≥ 1, that define A∞ -morphisms ⊕ij HomC (Xi , Xj ) → ⊕ij HomC (φ(Xi ), φ(Xj )). Now assume that we are given two structures of A∞ -category with the same class of objects C and with the same morphism spaces. Let m = (mn ) and m = (mn ) be the collections of the corresponding composition maps. A homotopy between m and m is an A∞ -functor φ : (C, m) → (C, m ) such that the corresponding map on objects is identity and such that f1 is the identity map on morphisms. The analogue of Lemma 1.1 is valid in this situation. In the case when m1 = 0 for an A∞ -category C the products m2 define a structure of the usual category on C (without units). If we have two such A∞ -categories C and C and a functor φ0 : C → C between them considered as usual categories then we say that φ0 is strictly compatible with A∞ -structures if it extends to an A∞ -functor with fn = 0 for n > 1. Let X be an object of an A∞ -category that has m1 = 0 equipped with a decomposition X = X1 ⊕ X2 into a direct sum. By definition (here we deal with the usual category structure without units) this means that we have functorial in Y isomorphisms Hom(X, Y ) Hom(X1 , Y ) ⊕ Hom(X2 , Y ) and Hom(Y, X) Hom(Y, X1 ) ⊕ Hom(Y, X2 ). We say that the decomposition X = X1 ⊕ X2 is strictly compatibile with an A∞ -structure if every composition mn involving the spaces Hom(X, Y ) or Hom(Y, X) is a direct sum of the corresponding compositions with the spaces Hom(Xi , Y ) and Hom(Y, Xi ).
1.3. Cyclic A∞ -structures. We will consider a special class of A∞ -algebras, namely, those equipped with a cyclic symmetry. Definition 1.2. Let A be an A∞ -algebra equipped with a bilinear form b : A ⊗ A → k. We will call A cyclic if for every n ≥ 1 the following identity is satisfied: b(mn (a1 , . . . , an ), an+1 ) = (−1)n(a˜ 1 +1) b(a1 , mn (a2 , . . . , an+1 )). Remark. Assume in addition that b satisfies the following symmetry: b(a1 , a2 ) = (−1)a1 a2 b(a2 , a1 ). Then (1.2) can be rewritten as follows: n+1 ) b(m (a , . . . , a b(mn (a1 , . . . , an ), an+1 ) = (−1)n+a1 (a2 +...+a n 2 n+1 ), a1 ).
(1.2)
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Using b we can define a linear functional ξ on T (sA) by setting ξ = b ◦ (s −1 )⊗2 on (sA)⊗2 while ξ = 0 on (sA)⊗n for n = 2. Thus, we have ξ([a1 |a2 ]) = (−1)a1 +1 b(a1 , a2 ). Then Eq. (1.2) is equivalent to the condition ξ ◦ d = 0, where d is the coderivation defined by (mn ). The collection of maps f = (fn : A⊗n → A, n ≥ 1), deg fn = 1 − n, f1 = id is called a cyclic homotopy if (−1)(l+1)(a1 +...+ak )+nk b(fk (a1 , . . . , ak ), fl (ak+1 , . . . , an )) = 0 (1.3) k+l=n
for n ≥ 3. This is equivalent to the condition ξ ◦ F = ξ , where F : T (sA) → T (sA) is the coalgebra homomorphism defined by (fn ). Let m be a cyclic A∞ -structure, f be a cyclic homotopy. Then the A∞ -structure m + δf is cyclic. If f and g are cyclic homotopies then f ◦ g is also cyclic. Remark. Assume that fk = 0 unless k = 1 or k = n for some n ≥ 2. Then f is a cyclic homotopy if and only if b(fn (a1 , . . . , an ), an+1 ) = (−1)(n+1)a1 +n b(a1 , fn (a2 , . . . , an+1 )) and b(fn (a1 , . . . , an ), fn (an+1 , . . . , a2n )) = 0. The definition of cyclic A∞ -categories follows the same pattern. We assume that there is a bilinear form b : Hom(X, Y ) ⊗ Hom(Y, X) → k for every pair of objects (X, Y ). Then an A∞ -category is called cyclic if the identity (1.2) is satisfied whenever a1 ∈ Hom(X1 , X2 ), . . . , an ∈ Hom(Xn , Xn+1 ), an+1 ∈ Hom(Xn+1 , X1 ). Similarly we define cyclic homotopy between two cyclic A∞ -structures with the same objects and morphism spaces (and the same bilinear form b). 1.4. Transversal A∞ -structures. Assume that we are given a class of objects and a symmetric binary relation R on objects, so that pairs of objects (X, X ) belonging to R are called transversal. We will call an n-tuple of objects (X1 , . . . , Xn ) transversal if for every 1 ≤ i < j ≤ n the pair (Xi , Xj ) is transversal. Then the structure of transversal (cyclic) A∞ -category on this class of objects consists of the following data. For every pair of transversal objects (X, Y ) a graded space of morphisms Hom(X, Y ) is given. For every transversal collection (X0 , . . . , Xn ), n ≥ 1 we have linear maps mn : Hom(X0 , X1 ) Hom(X1 , X2 ) . . . Hom(Xn−1 , Xn ) → Hom(X0 , Xn ) of degree 2 − n such that the axioms Axn and the identity (1.2) are satisfied whenever the objects involved in it form a transversal collection. Similarly we define a notion of homotopy between transversal A∞ -structures and the cyclic analogues of these notions. If C is an A∞ -category, C is a transversal A∞ -category, then an A∞ -functor is a map φ : Ob C → Ob C together with a collection of maps (1.1) for all k + 1-tuples (X0 , . . . , Xk ) of objects of C such that (φ(X0 ), . . . , φ(Xk )) is transversal. These maps should satisfy the same axioms as for the usual A∞ -functors.
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The motivating example is that of Fukaya category (see [1]) where objects are Lagrangain submanifolds in a symplectic manifold with some additional structure. Then we have the standard notion of transversality for pairs of Lagrangians. However, note that the notion of transversality for n-tuples we use is weaker than the standard one: we just require every pair of them to intersect transversally but, for example, we allow three Lagrangians to intersect in one point. 1.5. Two A∞ -structures. The first A∞ -structure (or rather a class of equivalent structures) can be defined on the category Vect(M), where M is a variety over k or a complex manifold as follows. Let us choose some functorial acyclic resolution V → R · (V ) for every vector bundle V on M such that for every pair of bundles there are functorial morphisms R · (V1 ) ⊗ R · (V2 ) → R · (V1 ⊗ V2 ) inducing the identity map on V1 ⊗ V2 and satisfying the natural associativity condition (e.g. one can take Cech complexes of acyclic covering or in the case k = C Dolbeault complexes). Then we can define a dg-category whose objects are vector bundles with Hom(V1 , V2 ) = R · (V1∨ ⊗ V2 ). By homotopy invariance of the notion of A∞ algebra there exists an equivalent A∞ -category structure on Vect(M) with morphisms Hom∗ (V1 , V2 ) = ⊕i Ext i (V1 , V2 ) that has m1 = 0 (see [5–7, 13, 14]). We will use a particular representative of this class of A∞ -structures in the case when M is a compact complex manifold equipped with a hermitian metric. This A∞ -structure appears naturally on the category Vecth (M) of holomorphic vector bundles equipped with a hermitian metric. Starting with the dg-category given by Dolbeault complexes one can use metrics to write an explicit formula for higher compositions involving some Hodge theory operators (see [16]). We will denote this A∞ -structure by mH = (mH n ). Note that since mH is the standard composition (while m = 0) the choices of hermi1 2 tian metrics on bundles are not really important. More precisely, if we choose a hermitian metric on every holomorphic vector bundle then the corresponding embedding Vect(M) → Vecth (M) of A∞ -categories is an A∞ -equivalence. Indeed, this follows from the fact that an A∞ -functor inducing an equivalence on the cohomology level is itself an A∞ -equivalence (see [12], Theorem 9.2.0.4). Assume in addition that ωM is trivialized. Then the Serre duality gives a non-degenerate pairing Hom∗ (V1 , V2 ) ⊗ Hom∗ (V2 , V1 ) → C. The main feature of our particular choice of an A∞ -structure is the cyclic symmetry (1.2) of mH n with respect to the Serre duality (see [16]). Note also that the higher products mH n are compatible with Massey products when the latter are well-defined. Another A∞ -category we are interested in (or rather, a transversal A∞ -category) is a full A∞ -subcategory of the Fukaya A∞ -category of the torus T = R2 /Z2 with the (complexified) symplectic form ω = −2πiτ dx ∧ dy, where τ is an element of the upper half-plane. We give here a very concrete version of the general definition that can be found in [1, 9, 19]. The objects of our (transversal) A∞ -category are pairs (L, A), where L = p(L) is the image of a non-vertical line L with rational slope under the natural projection p : R2 → T (a geodesic circle), A : V → V is an operator on a finite dimensional complex vector space V with real eigenvalues 4 . We call a pair of 4 The slight difference with [19] is that we do not attach to L an integer and do not consider vertical lines.
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objects (L1 , A1 ) and (L2 , A2 ) transversal if L1 and L2 are different. For such a pair the morphism space is Hom((L1 , A1 ), (L2 , A2 )) = Hom(V1 , V2 ) ⊗ Hom(L1 , L2 ), where Hom(L1 , L2 ) = ⊕P ∈L1 ∩L2 C[P ] ([P ] is a basis vector attached to a point P ). Note that there is a natural pairing Hom((L1 , A1 ), (L2 , A2 )) ⊗ Hom((L2 , A2 ), (L1 , A1 )) → C
(1.4)
induced by the natural duality between Hom(V1 , V2 ) and Hom(V2 , V1 ) and by the selfduality of Hom(L1 , L2 ) = Hom(L2 , L1 ) (such that the basis ([P ]) is autodual). Let λi be the slope of the line Li (i = 1, 2). Then Hom((L1 , A1 ), (L2 , A2 )) = 0 only if λ1 = λ2 . This space has grading 0 if λ1 < λ2 and grading 1 if λ1 > λ2 . By definition the differential m1 is zero. The compositions mn for n ≥ 2 are defined as follows. Let (Li , Ai ), i = 0, . . . , n be objects of the Fukaya categories such that the corresponding circles are pairwise different. Below it will be convenient to identify the set of indices [0, n] with Z/(n + 1)Z. For every i ∈ Z/(n + 1)Z, let di ∈ [0, 1] be the grading of Hom((Li , Ai ), (Li+1 , Ai+1 )). The composition mFn : Hom((L0 , A0 ), (L1 , A1 )) ⊗ . . . ⊗ Hom((Ln−1 , An−1 ), (Ln , An )) → Hom((L0 , A0 ), (Ln , An )) is non-zero only if ni=0 di = n − 1. Let Pi,i+1 be some intersection points of Li and Li+1 (i = 0, . . . , n − 1). For every i = 0, . . . , n − 1 let Mi,i+1 be an element of Hom(Vi , Vi+1 ). Then mFn (M0,1 [P0,1 ], M1,2 [P1,2 ], . . . , Mn−1,n [Pn−1,n ]) = ± exp(2πiτ · dx ∧ dy exp(2π i(x(pn ) − x(pn−1 ))An ) ◦ Mn−1,n ◦ P0,n ,
exp(2π i(x(pn−1 ) − x(pn−2 ))An−1 ) . . . ◦ M1,2 ◦ exp(2π i(x(p1 ) − x(p0 ))A1 ) ◦ M0,1 ◦ exp(2π i(x(p0 ) − x(pn ))A0 )[P0,n ], where the sum is taken over points of intersections P0,n of L0 with Ln and over all (n + 1)-gons (considered up to translation by Z2 ) with vertices pi ≡ Pi,i+1 mod Z2 , i ∈ Z/(n + 1)Z, such that the edge [pi−1 , pi ] belongs to p −1 (Li ) (the restriction on degrees di implies that is convex). We also require that the path formed by the edges [p0 , p1 ], [p1 , p2 ], . . . , [pn , p0 ] goes in the clockwise direction. The sign in the RHS is given by the following rule. If n is even then all signs are “plus”. If n is odd then the sign is equal to the sign of x(p0 ) − x(pn ) (recall that we do not allow vertical lines). It is not difficult to check that the products mF = (mFn ) give a structure of a transversal A∞ -category, cyclic with respect to the pairing (1.4). This A∞ -structure is strictly compatible with any decomposition of an operator A into a direct sum of operators. Let us denote by F(T , ω) this A∞ -category enlarged by allowing formal direct sums of objects supported on distinct lines (the products are strictly compatible with such direct sums).
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The main theorem of [19] identifies the corresponding graded category given by mF2 with Vect(E), where E = C/Z + Zτ . More precisely, the construction of [19] gives an equivalence Vect(E) →H ∗ F(T , ω)
(1.5)
of graded cyclic categories, where the cyclic structure on Vect(E) is induced by a trivialization of ωE . Since the A∞ -structure naturally appears on the category of hermitian bundles we will replace Vect(E) with Vect h (E). Note that if we view Vecth (E) as a usual category then the forgetful functor Vect h (E) → Vect(E) is an equivalence. Composing it with (1.5) we get an equivalence φ : Vecth (E) → H ∗ F(T , ω). Theorem 1.3. The functor φ lifts to an A∞ -functor φ : Vecth (E) → F(T , ω). We will recall some details of the correspondence between vector bundles on E and objects of the Fukaya category later. Let us only mention here that the slope of a line corresponding to an indecomposable bundle V is equal to the slope of V (the ratio of the degree and the rank). Stable bundles correspond to objects (L, A), where A ∈ R, is a real number (considered as an operator on a one-dimensional space). 2. Transversal A∞ -Structures on the Category of Line Bundles over an Elliptic Curve 2.1. Transversality and admissibility. Let E be an elliptic curve over a field k. Let L be the full subcategory in Vect(E) consisting of line bundles. One can consider extensions of the (strictly associative) composition m2 on L to A∞ -structures. The following definition gives some natural restrictions one can impose on such an extension. Let us fix a trivialization of ωE . Then the Serre duality gives a non-degenerate pairing Hom∗ (V1 , V2 ) ⊗ Hom∗ (V2 , V1 ) → k. Definition 2.1. Let us call a cyclic (with respect to the Serre duality) A∞ -structure m on the category L admissible if m1 = 0, m2 is the standard composition, and the functor of tensor multiplication by a line bundle is strictly compatible with m. Note that if m1 = 0 then for any A∞ -structure m which is homotopic to m one has = m2 . So it makes sense to try to classify admissible A∞ -structures on L up to cyclic homotopy, strictly compatible with tensor multiplication by any line bundle. We refer to such homotopies as admissible ones. We also define an admissible transversal A∞ -structure on L by similar restrictions provided that we have some notion of transversality for pairs of line bundles. We assume that such a notion is given and that it has the following properties: m2
(i) (ii) (iii) (iv)
(L, M) is transversal if and only if (M, L) is transversal; (L, M) is transversal if and only if (L−1 , M −1 ) is transversal; (L1 , L2 ) is transversal if and only if (L1 M, L2 M) is transversal; for every finite collection of line bundles (L1 , . . . , Ln ) and every integer d there exists an infinite number of isomorphism classes of line bundles L of degree d such that L and L2 are transversal to all Li ; (v) if (L, M) is transversal then L M.
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For example, assume that E(k) is infinite (this is necessary for the property (iv) to hold). Then one can call (L, M) transversal if L M. Another example arises from the correspondence between line bundles and objects of Fukaya category defined in [19]. In this example the complex parameter describing an isomorphism class of a line bundle splits into two real parameters: one describes the position of the corresponding geodesic circle and another specifies the connection on it. Then the pair (L, M) is transversal if the first real parameter takes different values at L and M (see Sect. 3.2 for details). The data of an admissible transversal A∞ -structure are encoded in the sequence of maps mn : H i1 (L1 )H i2 (L2 ) . . . H in (Ln ) → H i1 +i2 +...+in +2−n (L1 L2 . . . Ln ) for line bundles (Li ) such that the collection (O, L1 , L1 L2 , . . . , L1 . . . Ln ) is transversal.
2.2. Construction of the homotopy. Theorem 2.2. Let m and m be admissible transversal A∞ -structures on the category of line bundles on E. Assume that for every triple of line bundles (L1 , M, L2 ), where deg(L1 ) = deg(L2 ) = 1, deg(M) = −1, such that (O, L1 , L1 M, L1 ML2 ), is transversal, the maps H 0 (L1 ) ⊗ H 1 (M) ⊗ H 0 (L2 ) → H 0 (L1 L2 M)
(2.1)
given by m3 and m3 coincide. Then there exist a unique admissible homotopy between m and m . The following lemma is the main ingredient of the proof. Lemma 2.3. Let L be a line bundle of degree ≥ 3 on E, S ⊂ Pic(E) be a subset such that for every d and all isomorphism classes [L1 ], [L2 ] ∈ Pic(E) there exists an infinite number of [M] ∈ S such that deg(M) = d, 2[M] ∈ S, [L1 ] + [M] ∈ S and [L2 ] − [M] ∈ S. Then the following sequence is exact: α
⊕L1 L2 L3 =L,[L3 ]∈S,[L2 L3 ]∈S H 0 (L1 )H 0 (L2 )H 0 (L3 ) → ⊕L1 L2 =L,[L2 ]∈S H 0 (L1 )H 0 (L2 ) β
→ H 0 (L) → 0,
(2.2)
where Li denote line bundles of positive degrees, the map α sends s1 ⊗ s2 ⊗ s3 to s1 s2 ⊗ s3 − s1 ⊗ s2 s3 , β sends s1 ⊗ s2 to s1 s2 . Proof. Clearly, β is surjective. Thus, it suffices to prove the following statement. Assume that for every pair of line bundles of positive degree (L1 , L2 ) such that [L2 ] ∈ S and L1 L2 L we are given a linear map bL1 ,L2 : H 0 (L1 )H 0 (L2 ) → k such that for every triple (L1 , L2 , L3 ) such that deg(Li ) > 0, i = 1, 2, 3, L1 L2 L3 L, [L3 ] ∈ S, [L2 L3 ] ∈ S, one has bL1 ,L2 L3 (s1 , s2 s3 ) = bL1 L2 ,L3 (s1 s2 , s3 ),
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where si ∈ H 0 (Li ), i = 1, 2, 3. Then there exists a functional φ on H 0 (L) such that for every (L1 , L2 ) (with [L2 ] ∈ S) bL1 ,L2 (s1 , s2 ) = φ(s1 s2 ).
(2.3)
We will consider separately several cases. (i) deg(L) = 3. Let p1 , p2 ∈ E be a pair of distinct points such that [O(pi )] ∈ S, i = 1, 2, and [O(p1 + p2 )] ∈ S. Let spi ∈ H 0 (O(pi )), i = 1, 2, be non-zero sections. Then we have the following exact sequence: β
α
0 → H 0 (L(−p1 − p2 )) → H 0 (L(−p1 )) ⊕ H 0 (L(−p2 )) → H 0 (L) → 0, on where α (s) = (ssp2 , −ssp1 ), β (t1 , t2 ) = t1 sp1 + t2 sp2 . Let us define a functional φ H 0 (L(−p1 )) ⊕ H 0 (L(−p2 )) by the formula (t1 , t2 ) = bL(−p1 ),O(p1 ) (t1 , sp1 ) + bL(−p2 ),O(p2 ) (t2 , sp2 ). φ vanishes on the image of α . Indeed, we have Note that φ b(ssp2 , sp1 ) = b(s, sp2 sp1 ) = b(s, sp1 sp2 ) = b(ssp1 , sp2 ). = φ ◦ β . We are going to Therefore, there exists a functional φ on H 0 (L) such that φ show that this functional is the one we are looking for. Let L1 and L2 be line bundles of degrees 1 and 2 respectively such that L1 L2 L, [L2 ] ∈ S. Assume in addition that L2 O(p1 + p2 ). Then we claim that bL1 ,L2 (s, t) = φ(st) for any s ∈ H 0 (L1 ), t ∈ H 0 (L2 ). Indeed, the space H 0 (L2 ) is a direct sum of subspaces H 0 (L2 (−p1 ))sp1 and H 0 (L2 (−p2 ))sp2 . Thus, it suffices to prove that bL1 ,L2 (s, t) = φ(st) for t in any of these subspaces. For example, let t = t sp1 , where t ∈ H 0 (L2 (−p1 )). Then we have, b(s, t) = b(s, t sp1 ) = b(st , sp1 ) = φ(st sp1 ) as required. Now we claim that if M1 and M2 are arbitrary line bundles of degrees 2 and 1 respectively such that M1 M2 L and [M2 ] ∈ S, then bM1 ,M2 (s, t) = φ(st) for s ∈ H 0 (M1 ), t ∈ H 0 (M2 ). Indeed, let us choose points q1 , q2 ∈ E such that M1 O(q1 + q2 ), M2 (qi ) O(p1 + p2 ) and [M2 (qi )] ∈ S for i = 1, 2. Then H 0 (M1 ) is a direct sum of H 0 (M1 (−q1 ))H 0 (O(q1 )) and H 0 (M1 (−q2 ))H 0 (O(q2 )), so we can assume that s is in one of these subspaces. For example, assume that s = s sq1 , where sq1 ∈ H 0 (O(q1 )), s ∈ H 0 (M1 (−q1 )). Then we have b(s, t) = b(s sq1 , t) = b(s , sq1 t). Applying the previous part of the proof to L1 = M1 (−q1 ), L2 = M2 (q1 ) we obtain that b(s , sq1 t) = φ(s sq1 t) = φ(st) as required. Finally, a similar argument shows that for arbitrary line bundles L1 and L2 of degrees 1 and 2 one has bL1 ,L2 (s, t) = φ(st), where s ∈ H 0 (L1 ), t ∈ H 0 (L2 ).
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(ii) deg(L) = 4. Let us choose a pair of line bundles L1 and L2 both of degree 2 such that L1 L2 , [L2 ] ∈ S, and L1 L2 L. Then the product map H 0 (L1 ) ⊗ H 0 (L2 ) → H 0 (L) is an isomorphism, so we can define φ by setting φ(s1 s2 ) = bL1 ,L2 (s1 , s2 ), where s1 ∈ H 0 (L1 ), s2 ∈ H 0 (L2 ). Let L1 , L2 be line bundles of degree 2 such that L1 L2 L and [L2 ] ∈ S. Let p, q ∈ E be a pair of points such that O(p + q) L1 , [L2 (−q)] ∈ S. Then we claim that the equality bL1 ,L2 (s, t) = φ(st)
(2.4)
holds whenever s ∈ H 0 (L1 (−p)), t ∈ H 0 (L2 (−q)). Indeed, assume s = s sp , t = t sq , where sp ∈ H 0 (O(p)), sq ∈ H 0 (O(q)), s ∈ H 0 (L1 (−p)), t ∈ H 0 (L2 (−q)). Then b(s, t) = b(s sp , t sq ) = b(s sp sq , t ) = b(sp sq , s t ) = φ(sp sq s t ) = φ(st). Now let p1 , p2 ∈ E be a pair of distinct points such that L1 O(p1 + p2 ) and for q1 , q2 ∈ E defined by O(p1 + q1 ) O(p2 + q2 ) L1 one has [L2 (−q1 )] ∈ S, [L2 (−q2 )] ∈ S. Note that we have L1 (q1 + q2 ) L21 L since L2 L1 . Hence, O(q1 + q2 ) L2 and H 0 (L2 ) has a basis (t1 , t2 ) such that t1 vanishes at q1 and t2 vanishes at q2 . Therefore, if s is a section of L1 vanishing at p1 and p2 then by the previous part of the proof we have b(s, ti ) = φ(sti ) for i = 1, 2. Repeating this argument for another pair of points (p1 , p2 ) as above we get a similar statement for a section of L1 linearly independent from s. Thus, we conclude that (2.4) holds for all s ∈ H 0 (L1 ), t ∈ H 0 (L2 ). Now let M1 be a line bundle of degree 1, M2 be a line bundle of degree 3 such that M1 M2 L, [M2 ] ∈ S. Let s1 ∈ H 0 (M1 ), s2 ∈ H 0 (M2 ), p be a point in the divisor of s2 . Then assuming that [M2 (−p)] ∈ S we can write s2 = sp s2 where sp ∈ H 0 (O(p)), s2 ∈ H 0 (M2 (−p)) and b(s1 , s2 ) = b(s1 , sp s2 ) = b(s1 sp , s2 ) = φ(s1 sp s2 ) = φ(s1 s2 ). Since H 0 (M2 ) is spanned by H 0 (M2 (−p)) and H 0 (M2 (−p )) for two distinct points p, p ∈ E, this proves that bM1 ,M2 (s1 , s2 ) = φ(s1 s2 ) for all s1 and s2 . The case when deg(M1 ) = 3, deg(M2 ) = 1 is completely analogous. (iii) deg(L) = d ≥ 5. Let us fix a line bundle L2 of degree 2 such that [L2 ] ∈ S and [L22 ] ∈ S. Then there is an exact sequence 0 0 → L−1 2 → H (L2 ) ⊗ O → L2 → 0,
that induces for every line bundle M of degree ≥ 3 an exact sequence 0 0 0 0 → H 0 (ML−1 2 ) → H (M)H (L2 ) → H (ML2 ) → 0.
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Let L1 = LL−1 2 . Consider the following diagram with exact rows α2
0 0 0 0 0 0 → ⊕H 0 (O (p))H 0 (L1 L−1 2 (−p)) → ⊕H (O (p))H (L1 (−p))H (L2 ) → ⊕H (O (p))H (L(−p)) → 0
γ
α1
?
0→
?
H 0 (L1 L−1 2 )
→
H 0 (L1 )H 0 (L2 )
β
→
? H 0 (L)
→0 (2.5)
where the direct sums in the first row are taken over all p ∈ E such that [L(−p)] ∈ S. Notice that γ is surjective. Indeed, if d ≥ 6 this is clear, while for d = 5 we have to check that for the unique point p such that O(p) L1 L−1 2 one has [L(−p)] ∈ S. But this followsfrom our assumptions of L2 , since L(−p) L22 for such p. We have bL1 ,L2 ◦ α1 = p bO(p),L(−p) ◦ α2 . From this by an easy diagram chasing (using the surjectivity of γ ) we obtain that bL1 ,L2 vanishes on the kernel of β, hence, there exists a functional φ on H 0 (L) such that bL1 ,L2 = φ ◦ β. It follows that for any p ∈ E such that [L(−p)] ∈ S one has bO(p),L(−p) (s, t) = φ(st) for s ∈ H 0 (O(p)), t ∈ H 0 (L(−p)). Indeed, we can assume that t = t1 t2 with t1 ∈ H 0 (L1 (−p)), t2 ∈ L2 , in which case b(s, t) = b(s, t1 t2 ) = b(st1 , t2 ) = φ(st). Now we can deduce (2.3) in the general case using the same argument as in the end of case (ii).
Remark. It is easy to see from the proof that our assumptions on the set S ∈ Pic(E) can be weakened. Let us denote by Sd the subset of elements of S of degree d. Then it suffices to require that: (1) for any [L] ∈ Pic(E) and any d one has |Sd ∩ (S + [L])| > 4, |Sd ∩ ([L] − S)| > 5; (2) there exists [L] ∈ S2 such that 2[L] ∈ S. Proof of Theorem 2.2. Let us prove the existence first. Clearly we can replace m by m + δf , where f = (fn , n ≥ 2) is an admissible homotopy. Therefore, we can argue by induction: for every n ≥ 3, assuming that mk = mk for k < n we will construct an admissible homotopy f n such that fkn = 0 for k < n − 1 and (m + δf n )n = mn (this implies that (m + δf n )k = mk for all k ≤ n). Using the cyclic symmetry we can reduce various types of non-zero transversal ntuple products to the following two types: (i) mn : H 0 (L1 )H 1 (M1 ) . . . H 1 (Mi )H 0 (L2 )H 1 (Mi+1 ) . . . H 1 (Mn−2 ) → H 0 (L1 L2 M1 . . . Mn−2 ), where 1 ≤ i ≤ n − 2, (ii) mn : H 0 (L1 )⊗H 0 (L2 ) ⊗ H 1 (M1 ) ⊗ . . . ⊗ H 1 (Mn−2 ) → H 0 (L1 L2 M1 . . . Mn−2 ).
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Let us call w = deg(L1 ) + deg(L2 ) the weight of the corresponding n-tuple product type. Note that we have w ≥ 2. The first observation is that any n-tuple product of type (i) of weight > 2 can be expressed via k-tuple products with k < n and via n-tuple products of smaller weight. Indeed, if deg(L1 ) + deg(L2 ) > 2 then either deg(L1 ) > 1 or deg(L2 ) > 1. Assume for example that deg(L1 ) > 1. Then H 0 (L1 ) is spanned by various products sp s, where sp ∈ O(p), s ∈ L1 (−p), a point p ∈ E is such that the collection (O, O(p), L1 , L1 M1 , . . . , L1 M1 . . . Mi , L1 L2 M1 . . . Mi , L1 L2 M1 . . . , Mi+1 , . . . , L1 L2 M1 . . . Mn−2 ) is transversal. Now for any collection of elements ej ∈ H 1 (Mj ), j = 1, . . . , n − 2, t ∈ H 0 (L2 ) and any 1 ≤ j < n − 2 we have mn (sp s, e1 , . . . , ei , t, ei+1 , . . . , en−2 ) = mn (sp , se1 , . . . , ei , t, ei+1 , . . . , en−2 ) ±mn (sp , s, e1 , . . . , ei t, ei+1 , . . . , en−2 ) ±mn (sp , s, e1 , . . . , ei , tei+1 , . . . , en−2 ) ±sp mn (s, e1 , . . . , ei , t, ei+1 , . . . ) + . . . , where the unwritten terms contain only mk with k < n, while the weights of the n-tuple products in the RHS are smaller than w. If j = n − 2 then there is an additional term mn (sp , s, e1 , . . . , en−2 )t that does not affect our argument. Similarly one considers the case when deg(L2 ) > 1. On the other hand, the only non-zero transversal products of type (i) and weight 2 are those of type (2.1). As we will see below this allows us to restrict our attention to products of type (ii). To construct the homotopy f n we again apply induction. Namely, assuming that mn = mn for all products (of types (i) and (ii)) of weight < w (and mk = mk for k < n) we will construct a homotopy f n,w such that (m + δf n,w )n = mn for all products of weight ≤ w and such that the only non-zero component f n,w (other than f1n,w = id) reduces by cyclic symmetry to the following type: n,w fn−1 : H 0 (L) ⊗ H 1 (M1 ) ⊗ . . . ⊗ H 1 (Mn−2 ) → H 0 (LM1 . . . Mn−2 ),
where deg(L) = w. Note that f n,w is automatically cyclic. Indeed, any non-zero value of f n,w is an element of H i (M), where the degree of M is either −w or d, such that 0 < d < w. On the other hand, by definition of Serre duality b(H i (M), H j (M )) = 0 unless deg(M) + deg(M ) = 0. It follows that one has n,w n,w (a1 , . . . , an−1 ), fn−1 (an , . . . , a2n−2 )) = 0, b(fn−1
so f n,w is cyclic. By the above observation it will be sufficient to check the relation (m + δf n,w )n = mn only for products of type (ii) (and weight w). Assume first that w = 2. Then we necessarily have n = 3. Let us fix line bundles L and M, deg(L) = 2, deg(M) = 1, such that the triple (O, L, LM) is transversal. We want to construct a map f23,2 : H 0 (L) ⊗ H 1 (M) → H 0 (LM) such that for every pair of line bundles L1 , L2 of degree 1, where L1 L2 L and the quadruple (O, L1 , L, LM) is transversal, the map m3 − m3 : H 0 (L1 )H 0 (L2 )H 1 (M) → H 0 (LM) is a composition of the product map H 0 (L1 )H 0 (L2 ) → H 0 (L) with −f23,2 .
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Let us fix line bundles M and L such that deg(M ) = −2, deg(L ) = 1, M L M and the quadruple (O, L, LM , LM) is transversal. Let e ∈ H 1 (M) be a non-zero element. Then e = e s for some e ∈ H 1 (M ), s ∈ H 0 (L ). Now for every line bundle L1 and L2 such that L1 L2 L, where the quintuple (O, L1 , L, LM , LM) is transversal, and every s1 ∈ H 0 (L1 ) and s2 ∈ H 0 (L2 ) we have m3 (s1 , s2 , e) = m3 (s1 , s2 , e s ) = m3 (s1 , s2 e , s ) − m3 (s1 s2 , e , s ) and the similar equality holds for m3 . Note that we have m3 (s1 , s2 e , s ) = m3 (s1 , s2 e , s ) by the assumption of the theorem. Therefore, (m3 − m3 )(s1 , s2 , e) = −(m3 − m3 )(s1 s2 , e , s ).
(2.6)
Let us define the linear map fe ,s : H 0 (L) ⊗ H 1 (M) → H 0 (LM) by the formula fe ,s (s, e) = (m3 − m3 )(s, e , s ). We claim that fe ,s doesn’t depend on a choice of (M , L ) and e , s such that e s = e. Indeed, H 0 (L) is generated by sections of the form s = s1 s2 , where s1 and s2 are as above and the equality (2.6) shows that for such sections fe ,s (s, e) doesn’t depend on (e , s ). Thus, we can set f23,2 = fe ,s . Now the same equality shows that for any line bundles L1 , L2 such that deg(Li ) = 1, L1 L2 L and the quadruple (O, L1 , L, LM) is transversal one has (m3 − m3 )(s1 , s2 , e) = −f23,2 (s1 s2 , e). Now assume that w ≥ 3. Let us fix line bundles M1 , . . . , Mn−2 and elements ei ∈ H 1 (Mi ) for i = 1, . . . , n − 2. Let us also fix a line bundle L of degree w, such that the collection (O, L, LM1 , . . . , LM1 . . . Mn−2 ) is transversal. Then for every pair of line bundles L1 and L2 of positive degree such that L1 L2 L and the collection (O, L2 , L2 M1 , . . . , L2 M1 . . . Mn−2 ) is transversal, consider the map bL1 ,L2 : H 0 (L1 )H 0 (L2 ) → H 0 (L1 L2 M1 . . . Mn−2 ) : (s1 , s2 ) → (mn − mn )(s1 , s2 , e1 , . . . , en−2 ). We claim that these maps satisfy the condition b(s1 s2 , s3 ) = b(s1 , s2 s3 ) for any sections si ∈ Li , i = 1, 2, 3, where L1 L2 L3 L, deg(Li ) > 0, the collection (O, L2 , L2 L3 , L2 L3 M1 , . . . , L2 L3 M1 . . . Mn−2 ) is transversal. Indeed, the constraint Axn implies that mn (s1 s2 , s3 , e1 , . . . , en−2 ) − mn (s1 , s2 s3 , e1 , . . . , en−2 )
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is a linear combination of terms either involving only mk with k < n or involving products mn of weight < w. The same is true for m , so our claim follows from the induction assumptions on m and m . Therefore, we can apply Lemma 2.3 to the line bundle L and the set of isomorphism classes S = {[M] : (O, M, MM1 , . . . , MM1 . . . Mn−2 ) is transversal}. We conclude that there exists a linear map fe1 ,... ,en−2 : H 0 (L) → H 0 (LM1 . . . Mn−2 ) satisfying mn (s1 , s2 , e1 , . . . , en−2 ) − mn (s1 , s2 , e1 , . . . , en−2 ) = (−1)n fe1 ,... ,en−2 (s1 s2 , e1 , . . . , en−2 ). One can see from this defining property that the map n,w fn−1 : H 0 (L)H 1 (M1 ) . . . H 1 (Mn−2 ) → H 0 (LM1 . . . Mn−2 ) : s ⊗ ⊗e1 . . . ⊗ en−2 → fe1 ,... ,en−2 (s)
is linear and gives the required homotopy. The proof of uniqueness is also achieved by induction. It suffices to check that an admissible transversal homotopy f = (fn ) from m to m such that fk = 0 for 2 ≤ k < n has also fn = 0. By cyclic symmetry it suffices to consider the maps fn : H 0 (L)H 1 (M1 ) . . . H 1 (Mn−1 ) → H 0 (LM1 . . . Mn−1 ), where (O, L, LM1 , . . . , LM1 . . . Mn−1 ) is transversal. Now we use the induction in degree of L. If deg(L) = 1 then such a map is automatically zero. If deg(L) > 1 then it suffices to consider elements of H 0 (L) of the form ssp , where sp ∈ H 0 (O(p)), s ∈ H 0 (L(−p)) (where O(p) is transversal to all the relevant bundles). Then we can use the identity for fn and the induction assumption to prove the desired vanishing.
2.3. Uniqueness of homotopy for vector bundles. The part of Theorem 2.2 regarding uniqueness of homotopy is a special case of a more general uniqueness statement that we will need later for the proof of Theorem 1.3. Assume that we are given a pair of transversal cyclic A∞ -structures m and m on the category of vector bundles over an elliptic curve E, such that m1 = m1 = 0 and m2 = m2 is the standard product. Let C be a class of semistable vector bundles on E with the following two properties: (i) if V1 , V2 ∈ C are transversal and have the same slope then Hom(V1 , V2 ) = 0; (ii) for every V , V1 , . . . , Vn ∈ C such that V is transversal to Vi for i = 1, . . . , n, and every nonzero integer d there exists a point ξ ∈ Pic0 (E) such that for all points η of order d in Pic0 (E) the object V ξ η belongs to C and is transversal to Vi for i = 1, . . . , n. Proposition 2.4. Under the above assumptions let us consider the restrictions of the A∞ -structures m and m to the full subcategory C ⊂ Vect(E). Assume that f, g : (C, m) → (C, m ) is a pair of cyclic homotopies between these structures. Then f = g.
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Proof. It suffices to check that every homotopy from m to m on C is trivial. Let us prove by induction that fn = 0 for all n ≥ 2. Assume that we already know that fk = 0 for 2 ≤ k < n and let us deduce from this that fn = 0. By cyclic symmetry it suffices to consider the maps fn : Hom(V0 , V1 ) Ext1 (V1 , V2 ) . . . Ext1 (Vn−1 , Vn ) → Hom(V0 , Vn ) for a transversal collection (V0 , . . . , Vn ) of objects in C. By property (i) above we can assume that µ(V0 ) < µ(Vn ) < . . . < µ(V2 ) < µ(V1 ), where µ(Vi ) denotes the slope of a bundle Vi . According to Proposition 3.1 of [17] there exists a nonzero integer d such that for every semistable vector bundle W of slope µ(W ) = µ(Vn ) the composition map ⊕η∈Pic0 (E)d Hom(V0 , W η) Hom(W η, V1 ) → Hom(V0 , V1 ) is surjective5 . By property (ii) there exists a point ξ ∈ Pic0 (E) such that all the bundles Vn ξ η, where η ∈ Pic0 (E)d , belong to C and are transversal to Vi , i = 0, . . . , n. Applying the above surjectivity for W = Vn ξ we see that it suffices to check that for every η ∈ Pic0 (E)d one has fn (s1 s2 , e1 , . . . , en−1 ) = 0, where s1 ∈ Hom(V0 , Vn ξ η), s2 ∈ Hom(Vn ξ η, V1 ), ei ∈ Ext 1 (Vi , Vi+1 ). Since fk = 0 for 2 ≤ k < n, we have fn (s1 s2 , e1 , . . . , en−1 ) = ±s1 fn (s2 , e1 , . . . , en−1 ) ± fn (s1 , s2 e1 , e2 , . . . , en−1 ). But fn (s2 , e1 , . . . , en−1 ) is an element of Hom(Vn ξ η, Vn ) = 0, hence the first term in the RHS is zero. On the other hand, s2 e1 ∈ Ext1 (Vn ξ η, V2 ) = 0, hence, the second term is also zero.
2.4. An identity between triple products. Assume that we are given a transversal admissible A∞ -structure on the category of line bundles on E. Let (L1 , M, L2 ) be a triple of line bundles such that deg(L1 ) = deg(L2 ) = n > 0, deg(M) = −n, and the collection (O, L1 , L1 M, L1 ML2 ) is transversal. Then the triple products m3 : H 0 (L1 )H 1 (M)H 0 (L2 ) → H 0 (L1 ML2 ) are invariant under any homotopy. However, in Theorem 2.2 only such triple products with n = 1 appear. The reason is that one can express all triple products as above in terms of those with n = 1. This is done by induction in n using the identity below. Assume that Li = Li Li for i = 1, 2, where deg(Li ) = n , deg(Li ) = n for some positive integers n , n such that n = n + n . Assume also that the collection (O, L1 , L1 , L1 M, L1 ML2 , L1 ML2 ) is transversal. 5 The result of Prop. 3.1 of [17] is formulated for stable bundles, however the similar statement for semistable bundles is an immediate consequence.
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Proposition 2.5. One has the following identity: m3 (s1 s1 , e, s2 s2 ) = m3 (s1 , s1 e, s2 )s2 + s1 m3 (s1 , es2 , s2 ), where si ∈ H 0 (Li ), si ∈ H 0 (Li ), e ∈ H 1 (M). Proof. Applying the A∞ -constraint Ax3 we get m3 (s1 s1 , e, s2 ) = m3 (s1 , s1 e, s2 s2 ) + s1 m3 (s1 , e, s2 s2 ). Applying Ax3 again we obtain the following expressions for the terms in the RHS: m3 (s1 , s1 e, s2 s2 ) = m3 (s1 , s1 e, s2 )s2 + s1 m3 (s1 e, s2 , s2 ), m3 (s1 , e, s2 s2 ) = m3 (s1 , es2 , s2 ) − m3 (s1 e, s2 , s2 ). Substituting these expressions in the above equality we get the result.
3. Application to Homological Mirror Symmetry 3.1. Tensoring with unipotent bundles. By a unipotent bundle we mean a vector bundle that has a filtration by subbundles such that the associated graded bundle is trivial. Let LU = LU(E) be the full subcategory in Vect(E) consisting of bundles of the form LU , where L is a line bundle, U is a unipotent bundle. Note that a decomposition of LU into a tensor product of a line bundle and a unipotent bundle is unique up to an isomorphism. Assume that we are given a notion of transversality for pairs of line bundles satisfying the conditions listed in 2.1. We can extend it to the category LU by calling a pair (LU, L U ) transversal if and only if (L, L ) is transversal. Then we define an admissible transversal A∞ -structure on LU as a transversal A∞ -structure on LU that is cyclic with respect to Serre duality and is strictly compatible with tensor multiplication by a line bundle, has m1 = 0 and m2 equal to the standard product. One defines a notion of admissible homotopy between admissible A∞ -structures on LU similarly to the case of the category L. The proof of the following theorem is very similar to that of Theorem 2.2 so we omit it. Theorem 3.1. Let m and m be admissible transversal A∞ -structures on the category LU. Assume that for every triple of line bundles (L1 , M, L2 ) such that deg(L1 ) = deg(L2 ) = 1, deg(M) = −1 and such that (O, L1 , L1 M, L1 ML2 ) is transversal, and for every quadruple of unipotent bundles U0 , U1 , U2 and U3 the maps Hom(U0 , L1 U1 ) ⊗ Ext 1 (L1 U1 , L1 MU2 ) ⊗ Hom(L1 MU2 , L1 ML2 U3 ) → Hom(U0 , L1 ML2 U3 ) (3.1) given by m3 and m3 coincide. Then there exist a unique admissible homotopy between m and m .
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3.2. Connection with the Fukaya category. Let τ ∈ C be an element in the upper halfplane, E = C/Z + Zτ , be the corresponding elliptic curve. Then as shown in [19] the (graded) category LU is equivalent to the subcategory in the Fukaya category (with compositions mF2 ) consisting of objects (L, λ · Id +N ), where L has an integer slope, λ ∈ R, N is a nilpotent operator. Below we briefly recall the construction of this equivalence. Let L(0) be the line bundle on E such that the theta-function θ (z) = θ (z, τ ) = exp(π iτ n2 + 2π inz) n∈Z
is the pull-back of a section of L. So L(0) OE (z0 ), where z0 = τ +1 mod (Z + Zτ ). 2 For every u ∈ E let us denote L(u) = tu∗ L(0), where tu : E → E is the translation by u. Then every line bundle of degree n is isomorphic to a line bundle of the form L(0)⊗(n−1) ⊗ L(u). For a nilpotent operator N : V → V we denote by VN the unipotent bundle on E, such that the sections of VN correspond to V -valued functions on C satisfying the quasi-periodicity equations f (z + 1) = f (z), f (z + τ ) = exp(2π iN )f (z). Then every unipotent bundle is isomorphic to a bundle of the form VN . Let us denote by LU st ⊂ LU the full subcategory of LU consisting of standard bundles, i.e. bundles of the form L(0)⊗(n−1) ⊗ L(u) ⊗ VN . Clearly, LU st is equivalent to LU. The correspondence between standard bundles and objects of the Fukaya category constructed in [19] associates to a bundle V = L(0)⊗(n−1) ⊗ L(u) ⊗ VN the object O = (L, −u1 Id +N ), where u = u1 +τ u2 , ui ∈ R, L = {(u2 +x, (n−1)u2 +nx), x ∈ R/Z}. This correspondence extends to a functor from LU st to the Fukaya category (with F m2 as a composition) as follows. Let V = L(0)⊗(n −1) ⊗L(u )⊗VN be another bundle in LU, where n ∈ Z, u = u1 + τ u2 ∈ C, N : V → V is a nilpotent operator. Let O = (L , −u1 Id +N ) be the corresponding object in the Fukaya category. Note that O and O are transversal if and only if either n = n, or n = n and u2 − u2 ∈ Z. In the latter case Hom(V, V ) = Hom(O, O ) = 0 so we can assume that n = n . Assume first that n < n . Then Hom(O, O ) = Hom(V , V ) ⊗ Hom(L, L ) has degree zero. We can enumerate the points of intersection L ∩ L by residues k ∈ Z/(n − n)Z. Namely, this intersection consists of the points Pk = (
k + u2 − u2 nk + nu2 − n u2 , ), n − n n − n
where k ∈ Z/(n − n)Z. On the other hand, we have
Hom(V, V ) = H 0 (E, L(0)n −n−1 ⊗ L(u − u) ⊗ VN −N ∗ ), where we consider N ∗ and N as operators on V ∗ ⊗ V (acting trivially on one component). Note that if M is a line bundle on E of the form L(0)⊗(m−1) ⊗ L(u), where m = 0 and N : V → V is a nilpotent operator, then there is a natural isomorphism between Dolbeault complexes of bundles M ⊗ V and M ⊗O VN . Indeed, using the trivialization of the pull-backs of M and VN to C we can define the map from the Dolbeault complex of M ⊗ V to that of M ⊗O VN by sending η(z) to η(z − N/m) where (f ⊗ v)(z −
N N ) = exp(−∂z )(f ) · v. m m
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In particular, we can identify Hom(V, V ) with the space Hom(V , V ) ⊗ H 0 (E, L(0)n −n−1 ⊗ L(u − u)). The space of global sections of the line bundle L(0)n −n−1 ⊗ L(u − u) has a natural basis of theta functions θk (z) =
m∈(n −n)Z+k
exp(
1 (π iτ m2 + 2π im((n − n)z + u − u))), n − n
where k ∈ Z/(n −n)Z. Now we can identify Hom(V, V ) with Hom(O, O ) by sending T ⊗ [Pk ] (where T ∈ Hom(V , V )) to exp(
n
1 (−π iτ (u2 − u2 )2 Id +2πi(u2 − u2 )(N − N ∗ − (u1 − u1 ) Id))) · T ⊗ θk . −n
To construct a similar identification in the case n > n we use Serre duality and its natural analogue on the Fukaya category to reduce to the case considered above. As shown in [19] this identification is compatible with compositions m2 . Using it we can consider mF as a transversal A∞ -structure on LU st . Furthermore, it is easy to see that mF is admissible. The main point is that the functor of tensoring with a standard line bundle on LU st corresponds to an automorphism of the Fukaya category given by some symplectic automorphism of the torus. Let LU h,st be the category of vector bundles from LU st equipped with hermitian metrics. Then LU h,st has a natural A∞ -structure mH . On the other hand, we can also view mF as a transversal A∞ -structure on LU h,st via the forgetful functor LU h,st → LU st . As we will see in Sect. 3.4 the assumptions of Theorem 3.1 are satisfied in this case so there exists a homotopy between the structures mF and mH on LU h,st . The equivalence of LU with a subcategory of the Fukaya category (with mF2 as a composition) is extended to all bundles in [19] using the construction of vector bundles on E as push-forwards of objects in LU under isogenies. Below we consider the corresponding extension of the equivalence between A∞ -structures. 3.3. Proof of Theorem 1.3. Recall that we have to construct an A∞ -functor φ : Vecth (E) → F(T , ω) lifting φ. It is enough to construct an A∞ -functor from any full subcategory Vecth (E) of Vect h (E) such that every vector bundle is isomorphic to an object of Vect h (E) . Indeed, this follows from the fact that the natural embedding of Vecth (E) into Vect h (E) is an A∞ -equivalence (by Theorem 9.2.0.4 of [12]). We are going to construct a subcategory Vect h,st (E) ⊂ Vecth (E) with the above property and the functor φ : Vecth,st (E) → F(T , ω) using isogenies between elliptic curves. For every positive integer r we consider the elliptic curve Er = C/Z + Zrτ . Then we have a natural isogeny π r : Er → E1 = E of degree r. More generally, for every s|r there is an isogeny πsr : Er → Es such that π r = π s ◦ πsr . The corresponding isogenies on the symplectic side are r πsr : (T , rω) → (T , sω) : (x, y) → ( x, y). s Replacing τ by rτ we can consider A∞ -categories (Vecth (Er ), mH ) and (F(T , rω), mF ). An important observation is that both families of products mH and mF are strictly compatible with the pull-back functors with respect to isogenies πsr , i.e., the following
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diagrams are commutative: Hom∗ (V0 , V1 ) . . . Hom∗ (Vn−1 , Vn )
mn
-
Hom∗ (V0 , Vn )
π∗
π∗
? Hom∗ (π ∗ V0 , π ∗ V1 ) . . . Hom(π ∗ Vn−1 , π ∗ Vn )
mn
? - Hom∗ (π ∗ V , π ∗ V ) 1 n (3.2)
where π = πsr and either Vi ∈ Vect h (Es ) and mn = mH n , or Vi ∈ F(T , sω) and F mn = mn . Now we define Vect h,st (E) ⊂ Vecth (E) as the full subcategory consisting of hermitian vector bundles of the form V = ⊕ni=1 π∗ri (Vi ) for some Vi ∈ LU h,st , where the hermitian metric on V is induced by the hermitian metrics on Vi . Let us call these hermitian vector bundles standard. Note that every vector bundle on E is isomorphic to a standard one: it suffices to observe that every stable bundle of rank r can be represented as a push-forward of a line bundle under π r and that every indecomposable bundle is a tensor product of a stable bundle with a unipotent bundle. The construction of an A∞ -functor φ : Vect h,st (E) → F(T , ω) is done in two steps: Step 1. Construct a family of A∞ -functors LU h,st (Er ) → F(T , rω), compatible with pull-backs. Step 2. Descend these functors to a functor Vect h,st (E) → F(T , ω). Step 1 essentially reduces to Theorem 3.1. Namely, using the isomorphism of LU st (Er ) with the full subcategory in H ∗ F(T , rω) described before we can view mF as an admissible transversal A∞ -structure on LU h,st (Er ) extending the usual category structure. Then we can construct an admissible homotopy f r between products mH and mF on LU h,st (Er ) using Theorem 3.1 (we leave for the reader to make appropriate changes in the formulation of this theorem associated with the fact that LU is now replaced by LU h,st ). We just have to check that the Massey triple products of the type (3.1) coincide for these two structures. This is done by an explicit calculation that we postpone until Sect. 3.4 (at this point it will be important to use a specific trivialization of ωE on which the construction of equivalence in [19] depends). We claim that the homotopies f r and f s for s|r are strictly compatible with the pull-back functors π ∗ : LU h,st (Es ) → LU h,st (Er ) and π ∗ : F(T , sω) → F(T , rω), where π = πsr . Indeed, we observe that the restrictions of the A∞ -structures mH and mF to the full subcategory C = π ∗ LU h,st (Es ) ⊂ LU h,st (Er ) are strictly compatible with the natural action of the group ker(π ) on C (this action is trivial on objects but is nontrivial on morphisms). Now Theorem 2.4 implies that the homotopy f r is compatible with the action of ker(π ) on C. In other words, for every V0 , . . . , Vn ∈ LU h,st (Es ) the map f r : Hom(π ∗ V0 , π ∗ V1 ) . . . Hom(π ∗ Vn−1 , π ∗ Vn ) → Hom(π ∗ V0 , π ∗ Vn )
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is ker(π )-equivariant. It follows that f r sends Hom(V0 , V1 ) . . . Hom(Vn−1 , Vn ) to ker(π )-invariants in Hom(π ∗ V0 , π ∗ Vn ), i.e., to Hom(V0 , Vn ). Therefore, f r restricts to a homotopy between mH and mF on LU h,st (Es ) considered as a subcategory of LU h,st (Er ) via π ∗ . Applying the uniqueness part of Theorem 3.1 we conclude that this homotopy coincides with f s as we claimed. Step 2. First, we define the functor φ on objects by setting φ(⊕ni=1 π∗ri (Vi )) = ⊕ni=1 π∗ri φ(Vi ), where Vi ∈ LU h,st (Eri ) and φ is defined on LU h,st as in Step 1. It is easy to check that these maps on objects (defined for all curves Er ) are compatible with pull-backs under isogenies πsr . Next, we should define our A∞ -functor φ on morphisms. Let (V0 , . . . , Vn ) be a transversal collection of indecomposable standard hermitian bundles on E. Let us choose r divisible by the ranks of all bundles Vi . Then (π r )∗ (Vi ) is a semistable bundle and is an orthogonal direct sum of objects of LU h,st (Er ). Thus, we can define a map fnr : Hom((π r )∗ V0 , (π r )∗ V1 ) ⊗ . . . ⊗ Hom((π r )∗ Vn−1 , (π r )∗ Vn ) → Hom(φ(π r )∗ V0 , φ(π r )∗ Vn )
Hom((π r )∗ φ(V0 ), (π r )∗ φ(Vn )) by extending the construction of Step 1 to orthogonal direct sums in a strictly compatible way. Applying Theorem 2.4 to the subcategory C ∈ Vect h (E r ) consisting of bundles of the form (π r )∗ V , where V is an indecomposable standard hermitian bundle on E with rk V |r, we derive that the above maps fnr are ker(π r )-equivariant. Hence, they restrict to maps fn : Hom(V0 , V1 ) ⊗ . . . ⊗ Hom(Vn−1 , Vn ) → Hom(φ(V0 ), φ(Vn )). The compatibility checked in Step 1 implies that the obtained maps fn do not depend on a choice of r. To prove that these maps satisfy the axioms of the A∞ -functor we choose r sufficiently divisible and use A∞ -axioms for the maps (fnr ). Finally, we define similar maps fn for arbitrary transversal collections of standard hermitian bundles (V0 , . . . , Vn ) by imposing strict compatibility with direct sums.
3.4. Massey products. It remains to compute explicitly the products mF3 and mH 3 of the type (3.1). Let us trivialize ωE in such a way that the Serre duality induces the pairing b : Hom(V1 , V2 ) ⊗ Ext 1 (V2 , V1 ) → C given by the formula
b(f, gdz) =
dz ∧ Tr(f ◦ gdz), E
where f ∈ Hom(V1 , V2 ), gdz ∈ 0,1 (Hom(V2 , V1 )). First let us compute mH 3 . We start with the case when all Ui are trivial of rank 1. Then we have to compute the product 0 1 0 0 mH 3 : H (L1 )H (M)H (L2 ) → H (L1 ML2 ),
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where L1 M O, L2 M O. Using a translation on E we can assume without loss of generality that M = L(0)−1 . Let L1 = L(t), L2 = L(u), where t, u ∈ C. Let z1 and z2 be the real components of the complex variable z defined by the equality z = z1 + τ z2 . The transversality condition means that t2 , u2 ∈ Z. We will compute the above product under the weaker assumption t, u ∈ Z + Zτ . It is easy to check that the (0, 1)-form with values in L(0)−1 α(z) = √
i θ (z) exp(−2π Im(τ )(z22 ))dz 2 Im(τ )
is a representative of the class in H 1 (L(0)−1 ) dual to the class in H 0 (L(0)) given by θ(z). Now for every u ∈ C, such that u ∈ Z + Zτ there exists a unique section h(z, u) of L(0)−1 L(u) such that θ (z + u)α(z) = ∂h(z, u), where ∂ = ∂ z . Indeed, this follows from the fact that all the cohomologies of L(0)−1 L(u) vanish. One can write an explicit formula for h(z, u)(see [16]): h(z, u) = −
π (|γ |2 + 2γ u + u2 ) + 2π i(mz + (n − u)z )) exp(− 2 Im(τ 1 2 1 ) , (−1)mn 2π i γ +u m,n∈Z
where γ = mτ − n. Now we have mH 3 (θ (z + t), α, θ(z + u)) = h(z, t)θ (z + u) − h(z, u)θ (z + t). As a function of z up to a constant factor this should be equal to θ(z + u + v), so we have h(z, t)θ (z + u) − h(z, u)θ (z + t) = H (t, u)θ (z + t + u)
(3.3)
for some meromorphic function H . We have H (t, u) = −H (u, t). Also it is easy to see that the function H (t, u) satisfies the following quasi-periodicity equations: H (t + 1, u) = H (t, u), H (t + τ, u) = exp(2π iu)H (t, u). The only poles of H (t, u) are poles of order 1 along the divisors t = γ and u = γ , where γ ∈ Z + Zτ . It follows that H (t, u) is equal up to a constant to the function F (t, u) =
θ ( τ +1 2 )θ (t − u + 2πiθ (t +
τ +1 2 ) . τ +1 τ +1 2 )θ (−u + 2 )
Furthermore, comparing the residues at t = 0 we conclude that H (t, u) = −F (t, u). Now let us compute the product ∨ 1 ∨ 0 ∨ 0 ∨ 0 mH 3 : H (L1 U0 U1 )H (MU1 U2 )H (L2 U2 U3 ) → H (L1 ML2 U0 U3 ),
where Ui are unipotent bundles. As before we can take M = L(0)−1 , L1 = L(t), L2 = L(u). Let Ui = VNi where Ni : Vi → Vi are nilpotent operators. Then Ui∗ Ui+1
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VNi+1 −Ni∗ , where Ni+1 −Ni∗ is an operator on Vi∗ Vi+1 . As in Sect. 3.2 we use the isomorphisms between the Dolbeault complexes of bundles LV and LVN , where L is one of the line bundles of degree 1 above, N : V → V is the corresponding nilpotent operator, sending η(z) to η(z − N ). Similarly, we have an isomorphism between the Dolbeault complexes of L(0)−1 V1∗ V2 and L(0)−1 VN2 −N1∗ given by η(z) → η(z + N2 − N1∗ ). Let vi,i+1 ∈ Vi∗ ⊗ Vi+1 be some elements. Then we have (α(z + N2 − N1∗ )v1,2 ) ◦ (θ (z + t − N1 + N0∗ ))v0,1 ) = Tr V1 (∂h(z + N2 − N1∗ , t − N2 + N1∗ − N1 + N0∗ ))v0,1 v1,2 ) = Tr V1 (∂h(z + N2 − N1∗ , t − N2 + N0∗ )v0,1 v1,2 ), since we can replace N1∗ by N1 under the sign of Tr V1 . Similarly, we get (θ (z + u − N3 + N2∗ ))v2,3 ) ◦ (α(z + N2 − N1∗ )v1,2 ) = Tr V2 (∂h(z + N2 − N1∗ , u − N3 + N1∗ )v1,2 v2,3 ). Hence, ∗ ∗ ∗ mH 3 (θ (z + t − N1 + N0 ))v0,1 , α(z + N2 − N1 )v1,2 , θ (z + u − N3 + N2 )v2,3 ) = Tr V1 V2 ((θ (z + u − N3 + N2∗ )h(z + N2 − N1∗ , t − N2 + N0∗ ) −h(z + N2 − N1∗ , u − N3 + N1∗ )θ (z + t − N1 + N0∗ ))v0,1 v1,2 v2,3 ).
Making a substitution z → z + N2 − N1∗ , t → t − N2 + N0∗ , u → u − N3 + N1∗ in the identity (3.3) and using the equality H = −F we can rewrite the above formula as follows: ∗ ∗ ∗ mH 3 (θ (z + t − N1 + N0 )v0,1 , α(z + N2 − N1 )v1,2 , θ (z + u − N3 + N2 )v2,3 ) ∗ ∗ ∗ = Tr V1 V2 (F (t − N2 + N0 ), u − N3 + N1 ))θ (z + t + u − N3 + N0 )v0,1 v1,2 v2,3 ), (3.4)
Now let us compute the corresponding product mF3 . The objects of the Fukaya category corresponding to our four bundles U0 = VN0 , L1 U1 = L(t)VN1 , L1 MU2 = L(0)−1 L(t)VN2 and L1 ML2 U3 = L(t +u)VN3 are ((x, 0), N0 ), ((x +t2 , x), −t1 +N1 ), ((x, −t2 ), −t1 + N2 ) and ((x + t2 + u2 , x), −t1 − u1 + N3 ), where t = t1 + τ t2 , u = u1 + τ u2 , t2 , u2 ∈ Z. Note that any two of these circles either do not intersect or intersect at a unique point. So we can identify morphisms between these objects with spaces Hom(V0 , V1 ), Hom(V1 , V2 ), etc. Now we have −mF3 (v0,1 [P0,1 ], v1,2 [P 1,2 ], v2,3 [P2,3 ]) = Tr V1 V2 sign(m − t2 ) exp(2π iτ (m − t2 )(n + u2 ) (m,n)∈Z2 ,(m−t2 )(n+u2 )>0
+2π i(m − t2 )(−t1 + N2 − N0∗ ) + 2πi(n + u2 ) N3 + N1∗ ))v0,1 v1,2 v2,3 )[P0,3 ] × (u1 − = Tr V1 V2 ( sign(m − t2 ) exp(2πiτ mn + 2π im(u − N3 + N1∗ ) +2π in(−t + N2 − N0∗ )) · C · v0,1 v1,2 v2,3 ), where C = exp(−2πiτ t2 u2 − 2πit2 (u1 − N3 + N1∗ ) + 2π iu2 (−t1 + N2 − N0∗ )). At this point we need the following identity (which essentially coincides with the formula (2.3.4) of [15]): sign(m − t2 ) exp(2πiτ mn + 2π i(mu − nt)) = F (t, u) (m,n)∈Z2 ,(m−t2 )(n+u2 )>0
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for arbitrary t = t1 + τ t2 , u = u1 + τ u2 such that t2 , u2 ∈ Z. This identity which is due to Kronecker can be proven as follows: first, one has to check that the left-hand side extends to a meromorphic function of u and t with poles at the lattice points, then one has to compare its quasi-periodicity properties and residues at poles with those of F . Hence, we get mF3 (v0,1 [P0,1 ], v1,2 [P1,2 ], v2,3 [P2,3 ]) = − Tr V1 V2 (F (t − N2 + N0∗ , u − N3 + N1∗ ) · C · v0,1 v1,2 v2,3 )[P0,3 ]. (3.5) Now it easy to see that the exponential factors involved in the identification of morphisms in LU with morphisms in the Fukaya category (see Sect. 3.2) kill the factor C and we F get mH 3 = m3 on the products of the type (3.1). References 1. Fukaya, K.: Morse homotopy, A∞ -category, and Floer homologies. In: Proceedings of GARC Workshop on Geometry and Topology ’93 (Seoul, 1993), Seoul: Seoul Nat. Univ., 1993, pp. 1–102 2. Fukaya, K.: Floer homology for 3-manifolds with boundary. Topology, geometry and field theory, River Edge, NJ: World Sci. Publishing, 1994, pp. 1–21 3. Fukaya, K.: Mirror symmetry of abelian varieties and multi-theta functions. J. Alg. Geom. 11, 393– 512 (2002) 4. Getzler, E., Jones, J.D.S.: A∞ -algebras and the cyclic bar complex. Illinois J. Math. 34, 256–283 (1990) 5. Gugenheim, V.K.A.M., Stasheff, J.D.: On perturbations and A∞ -structures. Bull. Soc. Math. Belg. Sir. A 38, 237–246 (1986) 6. Gugenheim, V.K.A.M., Lambe, L.A., Stasheff, J.D.: Perturbation theory in differential homological algebra. II. Illinois J. Math. 35(3), 357–373 (1991) 7. Kadeishvili, T.V.: The category of differential coalgebras and the category of A∞ -algebras (in Russian). Trudy Tbiliss. Mat. Instituta 77, 50–70 (1985) 8. Keller, B.: Introduction to A-infinity algebras and modules. Homology Homotopy Appl. 3, 1–35 (2001) 9. Kontsevich, M.: Homological algebra of mirror symmetry. In: Proc. of ICM (Z¨urich, 1994), Basel: Birkh¨auser, 1995, pp. 120–129 10. Kontsevich, M.: Talk at the conference on non-commutative geometry. MPIM, June 1999 11. Kontsevich, M., Soibelman, Y.: Homological mirror symmetry and torus fibrations. In: Symplectic geometry and mirror symmetry (Seoul, 2000), River Edge, NJ: World Sci. Publishing, 2001, pp. 203–263 12. Lef`evre-Hasegawa, K.: Sur les A∞ -cat´egories. Th`ese de doctorat, Paris: Universit´e 7, 2003 13. Markl, M.: Homotopy algebras are homotopy algebras. Preprint math.AT/9907138 14. Merkulov, S.A.: Strong homotopy algebras of a K¨ahler manifold. Int. Math. Res. Notices 3, 153–164 (1999) 15. Polishchuk, A.: Massey and Fukaya products on elliptic curves. Adv. Theor. Math. Phys. 4, 1187– 1207 (2000) 16. Polishchuk, A.: Homological mirror symmetry with higher products. In: Winter School on Mirror Symmetry, Vector Bundles and Lagrangian Submanifolds (Cambridge, MA, 1999), Providence, RI: AMS, 2001, pp. 247–259 17. Polishchuk, A.: Indefinite theta series of signature (1, 1) from the point of view of homological mirror symmetry. Preprint math.AG/0003076 18. Polishchuk,A.: Extensions of homogeneous coordinate rings to A∞ -algebras. Homology, Homotopy and Appl. 5, 407–421 (2003) 19. Polishchuk, A., Zaslow, E.: Categorical mirror symmetry in the elliptic curve. In: Winter School on Mirror Symmetry, Vector Bundles and Lagrangian Submanifolds (Cambridge, MA, 1999), Providence, RI: AMS, 2001, pp. 275–295 20. Stasheff, J.D.: Homotopy associativity of H -spaces II. Trans. AMS 108, 293–312 (1963) Communicated by M.R. Douglas
Commun. Math. Phys. 247, 553–599 (2004) Digital Object Identifier (DOI) 10.1007/s00220-004-1069-8
Communications in
Mathematical Physics
Averaging Versus Chaos in Turbulent Transport? Houman Owhadi LATP, UMR CNRS 6632, CMI, Universit´e de Provence, 39 rue Joliot-Curie, 13453 Marseille Cedex 1, France. E-mail: [email protected] Received: 18 October 2002 / Accepted: 15 November 2003 Published online: 28 April 2004 – © Springer-Verlag 2004
Abstract: In this paper we analyze the transport of passive tracers by deterministic stationary incompressible flows which can be decomposed over an infinite number of spatial scales without separation between them. It appears that a low order dynamical system related to local Peclet numbers can be extracted from these flows and it controls their transport properties. Its analysis shows that these flows are strongly self-averaging and super-diffusive: the delay τ (r) for any finite number of passive tracers initially close to separate till a distance r is almost surely anomalously fast (τ (r) ∼ r 2−ν , with ν > 0). This strong self-averaging property is such that the dissipative power of the flow compensates its convective power at every scale. However as the circulation increases in the eddies the transport behavior of the flow may (discontinuously) bifurcate and become ruled by deterministic chaos: the self-averaging property collapses and advection dominates dissipation. When the flow is anisotropic a new formula describing turbulent conductivity is identified. 1. Introduction In this paper we study the passive transport in Rd (d ≥ 2) of a scalar T by a divergence free steady vector field v characterized by the following partial differential equation (κ > 0 being the molecular conductivity): ∂t T + v∇T = κT
(1)
We will assume v to be given by an infinite (or large) number of spatial scales without any assumption of self-similarity [Ave96]. It will be shown that one can extract from the flow a low order dynamical system related to local Peclet tensors which controls the transport properties of the flow. Based on the analysis of this dynamical system we will show that the transport is almost surely super-diffusive, that is to say, the time of separation of any finite number of passive tracers driven by the same flow and independent thermal noise behave like r 2−ν with ν > 0. Similar programs of investigations have shown that
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the mean squared displacement of a single particle is anomalously fast when averaged with respect to space, time and the randomness of the flow ([Pit97, KO02, Fan02]). The point here is to show that the transport is strongly self-averaging: the diffusive properties are anomalously fast (before being averaged with respect to the thermal noise, or a probability law of the flow), moreover the pair separation is also anomalously fast. The fast behavior of the transport of a single particle can be created by long distance correlations in the structure of the velocity field but this is not sufficient to produce fast pair separation. In this paper non-asymptotic estimates will be given, showing that the transport is controlled by a never-ending averaging phenomenon ([Owh01a, Owh01b, BO02a, BO02b]). The analysis of the low order dynamical system allows to obtain a formula linking the minimal and maximal eigenvalues of the turbulent eddy diffusivity. It will be shown that the transport properties depend only on the power law in v and not on its particular geometry (which is not a priori obvious since we consider a quenched model). However, depending on the geometrical characteristics of the eddies at each scale, as the flow rate is increased in these eddies we observe that the super-diffusive behavior may bifurcate towards a Chaotic transport: the multi-scale averaging picture collapses and the flow becomes highly unstable, sensitive to the characteristics of the microstructure and dominated by convective terms. 2. The Model We want to analyze the properties of the solutions of the following stochastic differential equation which is the Lagrangian formulation of the passive transport equation (1): √ dyt = 2κdωt + ∇.(yt ) dt. (2) Here κ > 0 is the molecular conductivity of the flow, ωt a standard Brownian Motion on Rd related to the thermal noise, is a skew-symmetric matrix on Rd called the stream matrix of the flow and ∇. its divergence. Thus ∇. is the divergence free drift defined by (∇.)i = dj =1 ∂j ij . We assume that is given by an infinite sum of periodic stream matrices with (geometrically) increasing periods and increasing amplitude, ∞ x . (3) γk E k = Rk k=0
In the formula (3) we have three important ingredients: the stream matrices E k (also called eddies), the scale parameters Rk and the amplitude parameters γk (the stream matrices E k are dimensionless and the parameters γk have the dimension of a conductivity). We will now describe the hypothesis we make on these three items of the model. Let us write T d := Rd /Zd the torus of dimension d and side one and for α ∈ [0, 1], S α (T d ) the space of d × d skew-symmetric matrices on T d with α-Holder continuous coefficients and .α the norm associated to that space. For E ∈ S α (T d ), Eα :=
sup
sup |Eij (x) − Eij (y)|/|x − y|α .
i,j ∈{1,... ,d} x=y
(4)
I Hypotheses on the stream matrices E k . There exists 0 < α ≤ 1 such that for all k ∈ N, E k ∈ S α (Td ).
(5)
Averaging Versus Chaos in Turbulent Transport?
555
The S α -norm of the E k are uniformly bounded, i.e. Kα := sup E k α < ∞.
(6)
E k (0) = 0.
(7)
k∈N
Moreover for all k,
Observe that the S 0 -norms of the E k are uniformly bounded and we will write K0 := sup
sup
k∈N i,j ∈{1,... ,d}
E k 0 .
(8)
II Hypotheses on the scale parameters Rk . Rk is a spatial scale parameter growing exponentially fast with k; more precisely we will assume that R0 = r0 = 1 and that the ratios between scales defined by rk = Rk /Rk−1 ∈ R∗
(9)
for k ≥ 1, are reals uniformly bounded away from 1 and ∞: we will denote by ρmin := inf rk k∈N∗
and ρmax := sup rk ,
(10)
and ρmax < ∞.
(11)
k∈N∗
and assume that ρmin ≥ 2
III Hypotheses on the flow rates γk . γk is an amplitude parameter (related to the local rate of the flow) growing exponentially fast with the scale k; more precisely we will assume that γ0 = 1 and that their ratios γk /γk−1 for k ≥ 1, are positive reals uniformly bounded away from 1 and ∞: we will denote by γmin := inf (γk /γk−1 ) k∈N∗
and γmax := sup (γk /γk−1 ), k∈N∗
(12)
and assume that γmin > 1
α and γmax < ρmin .
(13)
Remark 2.1. The uniform α-Holder continuity of the stream matrices E k is sufficient to obtain a well defined α-Holder continuous stream matrix , however is not differentiable in general. In this case the stochastic differential equation (2) is formal. For the simplicity of the presentation and to start with, when referring to the SDE (2) we will assume that α=1
and that the stream matrices E k are uniformly C 1 .
(14)
It follows from the Hypothesis I, II and III that is a well defined uniformly C 1 skewsymmetric matrix on Rd , thus the Stochastic Differential Equation 2 is well defined and admits a unique solution. The differentiability hypothesis (14) though convenient in order to define the process yt is in fact useless, the theorems are also meaningful and true for 0 < α < 1 (since they will refer to the diffusion associated to the weakly defined operator ∇.(κ + )∇).
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Fig. 1. A simple example of the multiscale flow
Remark 2.2. Observe that the power law of the flow in this paper is not Kolmogorov. Indeed if v(l) is the velocity of the eddies of size l and E(k) the kinetic energy distribution in the Fourier modes then with the Kolmogorov law one should have 1
v(l) ∼ l 3
5
and E(k) ∼ k − 3 .
In our Model we have ln γ
v(l) ∼ l ln ρ −1
ln γ
E(k) ∼ k 1−2 ln ρ . 4
Thus to be consistent with a Kolmogorov spectrum one should have γ = ρ 3 ; this case will be analyzed in a forthcoming paper. As an example, we have illustrated in Fig. 1 the contour lines of a two scale flow with stream function h20 (x, y) = 2k=0 γ k h( ρxk , ρyk ) , with ρ = 3, γ = 1.1 and h(x, y) = 2 sin(2π x + 3 cos(2πy − 3 sin(2πx + 1))) sin(2πy + 3 cos(2π x − 3 sin(2πy + 1))). 3. A Reminder on the Eddy Conductivity We write Md,sym the space of d × d symmetric elliptic constant matrices and SL∞ (TdR ) the space of skew-symmetric matrices with coefficients in L∞ (TdR ) (TdR := RTd stands for the torus of dimension d and side R). For a ∈ Md,sym and E a skew symmetric matrix with bounded coefficients the heat kernel associated to the passive transport operator ∇.(a + E)∇ (defined in a weak sense) is Gaussian by Aronson estimates [Nor97]. We will now assume E to be periodic: E ∈ SL∞ (TdR ). In this case the process associated to the operator L = ∇.(a + E)∇ exhibits self-averaging properties and we will note σsym (a, E) the effective conductivity associated to the homogenization of that operator [BLP78, JKO91]. Writing p(t, x, y) the heat kernel associated to L, it is well known that σsym (a, E) is a d × d elliptic symmetric matrix satisfying, for all x, l ∈ Rd , |l|2σsym (a,E) =
1 p(t, x, y)(y.l)2 dy. lim t −1 2 t→∞ Rd
(15)
Averaging Versus Chaos in Turbulent Transport?
557
We have used the notation |l|2a := t lal. If zt is the process generated by L, then as ↓ 0,
zt/ 2 converges in law to a Brownian motion with covariance matrix D(a, E) called effective diffusivity and proportional to the effective conductivity, D(a, E) = 2σsym (a, E).
(16)
Let us recall that σsym (a, E) is given by the following variational formula ([Nor97] Lemma 3.1): for ξ ∈ Rd , 2 −d |ξ |σ −1 (a,E) = inf R |ξ − ∇.H + (a + E)∇f |2a −1 dx, (17) (f,H )∈C ∞ (TdR )×S (TdR )
sym
TdR
where we have written S(TdR ) the space of skew symmetric matrices with coefficients in C ∞ (TdR ). The symmetric tensor σsym (a, E) is also called eddy conductivity: after averaging the information on particular geometry of the eddies associated to E is lost and the conductivity of the flow is replaced by an increased conductivity σsym (a, E). Let us define for P ∈ SL∞ (Td ) and ρ ∈ R∗ , Sρ P ∈ SL∞ (Td1 ) by ρ
Sρ P (x) := P (ρx).
(18)
It is important to note that the effective conductivity is invariant by scaling, i.e. σsym (a, Sρ P ) = σsym (a, P ); thus we can assume for simplicity that R = 1 and E ∈ SL∞ (Td ). When E is smooth σsym (a, E) is given [BLP78] by solving the following cell problem: (19) ∇.(a + E)(l − ∇χla,E ) = 0, where l ∈ Rd , χla,E ∈ C ∞ (Td ) and Td χla,E (x) dx = 0. Write Fla,E = l.x − χla,E (x), observe that Fla,E is linear in l, thus we will write F a,E the vector field (F a,E )i := Fea,E i and ∇F a,E the matrix (∇F a,E )ij := ∂i Fea,E . The eddy conductivity is then given by j t ∇F a,E (x)a∇F a,E (x) dx. (20) σsym (a, E) = Td
Let us recall that the matrix σ (a, E) defined by σ (a, E) := (a + E(x))∇F a,E (x) dx Td
(21)
is called the flow effective conductivity [FP94] and is also given by the following variational formula [Nor97]: for ξ, l ∈ Rd , |ξ − σ (a, E)l|2σ −1 (a,E) sym
:=
inf
(f,H )∈C ∞ (Td )×S (Td ) Td
|ξ − ∇.H − (a + E)(l − ∇f )|2a −1 dx.
(22)
It is easy to check that σsym (a, E) is the symmetric part of σ (a, E) which implies the following variational formulation for the eddy conductivity: |l|2σsym = inf |ξ − ∇.H − (a + E)(l − ∇f )|2a −1 dx, (23) ξ ⊥l,(f,H )∈C ∞ (Td )×S (Td ) Td
where we have written ξ ⊥ l is the subspace of ξ ∈ Rd orthogonal to the vector l: ξ.l = 0.
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4. Main Results 4.1. Averaging with two scales. Let a ∈ Md,sym , P ∈ SL∞ (Td ) and K ∈ S α (Td ). We will prove in Subsect. 5.1 the following estimate of σsym (a, SR P + K), the effective conductivity for a two-scale medium when R is an integer (and SR is the scaling operator (18)). Theorem 4.1. There exists a function f : R2 → R+ increasing in each of its arguments such that for a ∈ Md,sym , R ∈ N∗ , P ∈ SL∞ (Td ) and K ∈ S α (Td ), (1 + (R))−4 σsym σsym (a, P ), K ≤ σsym (a, SR P + K) ≤ σsym σsym (a, P ), K (1 + (R))4 , (24) with (R) =
Kα 21 a + P ∞
. f d, λmin (a) min (a)
Rα λ
Remark 4.2. Theorem 4.1 implies obviously that σsym σsym (a, P ), K = lim σsym (a, SR P + K). R→∞
(25)
(26)
Thus σsym σsym (a, P ), K should be interpreted as the effective conductivity of the two-scale flow with a complete separation of scales. So we will also write it σsym (a, S∞ P +K). Naturally σsym σsym (a, P ), K is also computable from an explicit cell problem (see (20)). Averaging versus chaotic coupling. Equation (24) basically says that when (R) is small, the mixing length of the smaller scale P (Rx) is smaller than the scale at which the fluctuations of the larger scale K(x) start to be felt. Now it is very important to observe that as λmin (a) ↓ 0, (R) explode towards infinity and this collapse of the two-scale averaging is not an artefact, it is easy to see that the estimate (25) is sharp. What happens is a transition from averaging to a chaotic coupling between the two scales. More precisely as λmin (a) ↓ 0, the mixing length of the smaller scale explode well above the visibility length of the larger scale, the two scales are no longer separated in the averaging and their particular geometry can no longer be ignored (collapse of the averaging paradigm). Moreover writing for y ∈ [0, 1]d , y the translation operator acting on functions f of Rd by y f (x) = f (x + y), observe that in the limit of complete separation between scales the two-scale averaging is invariant with respect to a relative translation of one scale with respect to another: lim σsym (a, SR y P + K) = lim σsym (a, SR P + K).
R→∞
R→∞
(27)
But the limit λmin (a) ↓ 0 is singular and this invariance by translation is lost: for l ∈ Rd ,
−1 t (28) lσsym (a, SR y P + K)l − t lσsym (a, SR P + K)l t lσsym (a, SR P + K)l may explode towards infinity. Indeed it is easy to see that for any R ∈ N∗ , there exist P , K ∈ S 1 (Td ) with P 1 ≤ Cd , K1 ≤ Cd such that there exists y ∈ [0, 1]d and l ∈ Rd with
−1 = ∞. lim t lσsym (ζ Id , SR y P + K)l t lσsym (a, SR P + K)l (29) ζ ↓0
Averaging Versus Chaos in Turbulent Transport?
559
(a)
(b)
Fig. 2a,b. Two scales flow. (a) Stream lines of SR P and K; (b) Stream lines of SR y P and K
We have illustrated this symmetry breaking in Fig. 2 representing a two scale flow. In Fig. 2(a) the larger eddies are surrounded by a non-void region where the flow is null and asymptotic behavior of the effective conductivity at vanishing molecular conductivity is given by σsym (ζ Id , SR P + K) ∼ C1 ζ Id .
(30)
In Fig. 2(b) we have operated a small translation of the smaller scale with respect to the larger one. The result of this relative translation is the percolation of stream lines of the flow: a particle driven by the flow can go to infinity by following them. It follows after this small perturbation that asymptotic behavior of the effective conductivity of the two scale flow at vanishing molecular conductivity is given by (31). 1
σsym (ζ Id , SR y P + K) ∼ C2 ζ 2 Id .
(31)
We call this sensibility with respect to the relative translation y , chaotic coupling between scales. The asymptotic (31) can be understood from a boundary layer analysis (see [Chi79] and [FP94]). 4.2. Multiscale eddy conductivity and the renormalization core. Let us write 0,n =
n k=0
γk E k
x Rk
.
(32)
For this subsection we will use the following hypothesis IV. Hypothesis on the ratios between scales: For all k ∈ N, rk ∈ N∗ . objective is to obtain quantitative estimates the multi-scale eddy viscosities Our σ ( 0,n ) n∈N ; observe that under the hypothesis IV, 0,n is periodic, thus its effective conductivity is well defined by Eq. (17) (that is its only utility, we will not need this hypothesis to prove super-diffusion). These estimates (Theorem 4.4) will be proven by
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H. Owhadi
induction on the number of scales. The basic step in this induction is the estimate (24) on the effective conductivity for a two scale periodic medium. We will need to introduce a dynamical system called the renormalization core which will play a central role in the transport properties of the stochastic differential equation (2). Definition 4.3. We propose to call “renormalization core” the dynamical system (An )n∈N of d × d symmetric strictly elliptic matrices defined by A0 =
κ γn Id and An+1 = σsym (An , E n ). γ0 γn+1
(33)
For B a d × d symmetric coercive matrix let us define the function g(B) by g(B) :=
Kα α ) λmin (B)(1 − γmax /ρmin
1 2
f d, (λmax (B) + K0 )/λmin (B) ,
(34)
where f is the function appearing in Theorem 4.1. We will prove the following theorem in Subsect. 5.2. Theorem 4.4. Under Hypotheses I, II, III and IV for all n ∈ N∗ , γn+1 An+1
n
(1 + p )−4 ≤ σsym (κId , 0,n ) ≤ γn+1 An+1
p=1
n
(1 + p )4
(35)
p=1
with
p =
γp γp−1 rpα
1 2
g(Ap−1 ),
(36)
An being the renormalization core (33). Observe that γn+1 An+1 is the estimate given by reiterated homogenization under the assumption of complete separation between scales, i.e. ρmin → ∞ and the error term n 4 k p=1 (1 + p ) controlled by the renormalization core A which reflects the interaction between the scales k and k + 1. As λmin (Ak ) ↓ 0 one passes from a separation of the scales k and k + 1 to a chaotic coupling between these two scales. Moreover it is easy to obtain from Theorem 4.4 that lim
r1 ,... ,rn−2 →∞
σsym (κId , 0,n ) = γn−1 σsym (An−1 , Srn E n−1 +
γn n E ). γn−1
(37)
Assume that the multi-scale averaging scenario holds and ρmin < ∞. In that scenario, σsym (κId , 0,n ) can be approximated by its limit at asymptotic separation between scales. We obtain a contradiction if lim inf n→∞ λmin (An ) = 0 from (37) and the collapse of the two-scale averaging scenario given in Subsect. 4.1 and Fig. 2. In other words if lim inf n→∞ λmin (An ) = 0, then the self-averaging property of the flow collapses towards a chaotic coupling between all the scales which is characterized by the breaking of the invariance by relative translation between the scales.
Averaging Versus Chaos in Turbulent Transport?
561
4.2.1. What is the renormalization core? First observe that it is a dimensionless tensor. At the limit of infinite separation between scales the eddy conductivity created by the scales 0, . . . , n − 1 is limρmin →∞ σsym (κId , 0,n−1 ). The typical scale length associated to the scale n is Rn and the velocity of the flow at this scale is of the order of γn Rn−1 (we assume K1 to be of order one). Thus at the scale n one can define a local renormalized Peclet tensor Pen by: Pen := Rn ×
−1 γn × lim σsym (κId , 0,n−1 ) . ρmin →∞ Rn
(38)
But at the limit of complete separation between scales (An )−1 is equal to the ratio between the convective strength γn of the scale n and the local turbulent conductivity at the scale n − 1: −1 (39) (An )−1 = lim γn σsym (κId , 0,n−1 ) . ρmin →∞
It follows that (An )−1 = Pen .
(40)
Thus one can interpret the renormalization core as the inverse of the Peclet tensor of the flow at the scale n assuming that all the smaller scales have been completely averaged. Definition 4.5. We call the "local renormalized Peclet tensor" (Pen )n∈N the inverse of the renormalization core Pen = (An )−1 .
(41)
4.2.2. Pathologies of the renormalization core. Definition 4.6. We call stability of the renormalization core (33) the sequence p λ− n := inf λmin (A ).
(42)
− − n λ− ∞ := lim λn and λ := lim inf λmin (A ).
(43)
0≤p≤n
We write n→∞
n→∞
The renormalization core is said to be stable if and only if λ− > 0. Definition 4.7. We call anisotropic distortion of the renormalization core (33) the sequence µn = sup λmax (Ap )/λmin (Ap ) . (44) 0≤p≤n
We write µ∞ := lim µn and µ := lim sup λmax (An )/λmin (An ) . n→∞
(45)
n→∞
The renormalization core (33) is said to have unbounded (bounded) anisotropic distortion if and only if µ = ∞ (µ < ∞).
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Definition 4.8. We call ubiety of the renormalization core (33) the sequence p λ+ n := sup λmax (A ).
(46)
+ + n λ+ ∞ := lim λn and λ := lim sup λmax (A ).
(47)
0≤p≤n
We write n→∞
n→∞
The renormalization core is said to be vanishing if and only if λ+ = 0. Definition 4.9. The renormalization core (33) is said to be bounded if and only if λ+ < ∞. The renormalization core is gifted with remarkable properties which will be analyzed in detail in Subsect. 4.4. Before proceeding to super-diffusion we will give a first theorem stressing the role of the stability of the renormalization core, that is to say the fact that the local renormalized Peclet tensor stays bounded away from infinity. Indeed, it follows from Theorem 4.4 that the averaging paradigm for our model is valid if the renormalization core is stable, and has bounded anisotropic distortion. We may naturally wonder whether the fact that the local renormalized Peclet tensor stays bounded away from infinity is sufficient; the answer is positive as shown by the following theorem which will be proven in Subsect. 5.3. Theorem 4.10. Writing C = Cd K02 (1 − 1/γmin )−1 we have 1. If the renormalization core is not bounded (λ+ = ∞) then it is not stable (λ− = 0) 2. If the renormalization core is stable (λ− > 0) then it is bounded and λ+ ≤
C . λ−
(48)
3. The renormalization core has unbounded anisotropic distortion (µ = ∞) if and only if it is not stable (λ− = 0) 4. If the renormalization core is stable (λ− > 0) then it has bounded anisotropic distortion (µ < ∞) and µ≤
C . (λ− )2
(49)
Combining Theorem 4.10 and 4.4 we obtain that if the renormalization core is stable then the local turbulent eddy conductivity diverges towards infinity like γn independently of the geometry of the eddies (if it is not stable the behavior of the local turbulent eddy conductivity depends on the geometry of the eddies). More precisely we have the following theorem. Theorem 4.11. Under Hypotheses I, II, III and IV, if the renormalization core is stable α > Cγ then there exists C such that for ρmin max one has
ln λmax σsym (κId , 0,n ) (50) lim sup ≤1+ ln γn n→∞
Averaging Versus Chaos in Turbulent Transport?
and
563
ln λmin σsym (κId , 0,n ) lim inf
ln γn
n→∞
≥1−
(51)
1 with := 0.5 Cγmax /(ρmin ) 2 < 0.5 and C := Kα h(d, K0 /λ− ), h being a finite increasing positive function in each of its arguments. Remark 4.12. For a real flow, call σ (r) the local turbulent diffusivity of the flow at the scale r and v(r) the magnitude of the vector velocity field at that scale. Then the key relation implying that the distortions created at the scale r are dissipated by the mixing power of the smaller scales is the relation σ (r) ∼ rv(r).
(52)
This relation is at the core of the Kolmogorov (K41) analysis and the analysis of fully developed turbulence by Landau-Lifschitz [LL84]. The result given in Theorem 4.11 corresponds to the relation (52) obtained and used from a heuristic point of view (dimension analysis) by physicists. 4.3. Super-diffusion. Anomalous fast exit times. We write τ (r) the exit time of the process yt (2) from the ball B(0, r). We write Ex the expectation associated to the process yt started from the point x. We write Vol B(0, r) the Lebesgue measure of B(0, r). We define n(r) as the number of (smaller) scales which will be considered as averaged at the scale r, n(r) := sup{p ∈ N : Rp ≤ r}.
(53)
Let mr be the Lebesgue probability measure on the ball B(0, r) defined by mr (dx) :=
dx B(0,r) dx
1B(0,r) .
(54)
We will consider the mean exit time for the process started with initial distribution mr , i.e.
1 Emr τ (r) = Ex τ (r) dx. (55) Vol B(0, r) B(0,r) We will prove in Subsect. 5.4 the following theorem. Theorem 4.13. Under Hypotheses I, II, and III with α = 1, if the renormalization core is stable (λ− > 0) then there exists a constant Q such that for ρmin > Qγmax one has
1 lim sup ln Emr τ (r) < 2. (56) r→∞ ln r More precisely for r > R1 one has
Emr τ (r) = r 2−ν(r)
(57)
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H. Owhadi
with C(r) ln γn(r) 1 + (r) + , ln r ln r and |C(r)| ≤ C(d, K0 , γmax ) + | ln λ− ∞ |, 1 Qγmax 2 | (r)| < 0.5 ≤ 0.5, ρmin ν(r) =
(58)
(59)
and there exists a finite increasing positive function in each of its arguments F such that 1 (1 + K0 )2 (1 − 1/γmin )−1 + κ Q := (1 + K1 ). F d, (60) (ln γmin )2 λ− ∞ Remark 4.14. Equation (58) shows that the anomalous constant is directly related to the number of effective scales. Observe that Qγmax 1 C(d, K0 , γmax ) ln γmax 2 ν(r) ≤ 1 + 0.5 + (61) ln ρmin ρmin ln r and Qγmax 1 C(d, K0 , γmax ) ln γmin 2 ν(r) ≥ 1 − 0.5 − (62) ln ρmax ρmin ln r and ν(r) > 0.4(ln γmin / ln ρmax ) for r large enough. The anomalous parameter ν(r) is not a constant because the model is not self-similar; in a self similar case (γmin = γmax = γ and ρmin = ρmax = ρ) one would have at a logarithmic approximation E[τ (r)] ∼ r 2−ν
with ν ∼
ln γ . ln ρ
The error terms in ν(r) are explained by the interaction between the scales which are sensitive to the particular geometry of the eddies. We recall that we consider a quenched model and it is not a priori obvious that the transport should depend only on the power law in velocity field and not on its particular geometry. Sufficient (and necessary) conditions for the stability of the renormalization core (λ− > 0) will be given in Subsect. 4.4; we refer to Theorems 4.25, 4.26 and 4.30. In particular if d = 2 and if for all k, E k = E, where E corresponds to the cellular flow (E12 (x, y) := sin(2πx) cos(2πy)) then the renormalization core is stable (λ− > 0). We have illustrated the contour lines of the superposition of 4 scales of cellular flows in Fig. 3.
Fig. 3. Superposition of cellular flows
Averaging Versus Chaos in Turbulent Transport?
565
There exists an important literature on the fast transport phenomenon in turbulence addressed (from both the heuristic and rigorous point of view) by using the tools of homogenization or renormalization; we refer to [KS79, AM90, AM87, FGL+91, GLPP92, GZ92, Zha92, GK98, IK91, Gau98, Ave96, Bha99, FK01, BO02b, CP01, AC02] and this panorama is far from being complete; we refer to [MK99] and [Woy00] for a survey. For non-exactly solvable models (non-shear flows) asymptotic fast scaling in the transport behavior have been obtained in the framework of spectral averaging in turbulence. Along this axis L. Piterbarg has obtained [Pit97] fast asymptotic scaling after averaging the transport with respect to the law of the velocity field and the thermal noise and rescaling with respect to space and time. More recently S. Olla and T. Komorowski [KO02] have observed the asymptotic anomalously fast behavior of the mean squared displacement averaged with respect to the thermal noise, the law of the velocity field and time. A. Fannjiang [Fan02] has studied a model where the law of separation of two particles is postulated to be the transport law of a single one as studied in [KO02] and [Pit97]. Fast mixing. In order to show that the phenomenon presented in Theorem 4.13 is super-diffusion and not mere convection, we must compute the rate at which particles do separate and show that this rate follows the same fast behavior. More precisely we will consider (yt , zt ) ∈ Rd × Rd , where yt is the solution of (2) and zt follows the following stochastic differential equation: √ (63) dzt = 2κd ω¯ t + ∇.(zt ) dt, where ω¯ t is a standard Brownian motion independent of ωt . Thus yt and zt can be seen as two particles transported by the same drift but with independent identically distributed noise. Let us write B(0, r, l) the following subset of Rd × Rd : (64) B(0, r, l) := (y, z) ∈ Rd × Rd : |y − z| < r and y 2 + z2 < l 2 .
We write Ey,z τ (r, l) the expectation of the exit time of the diffusion (yt , zt ) from B(0, r, l) with (y0 , z0 ) = (y, z). Let mr,l be the Lebesgue probability measure on the set B(0, r, l) defined by mr,l (dy dz) :=
dy dz (y,z)∈B(0,r,l) dy dz
1B(0,r,l) .
(65)
We will consider the mean exit time for the process (yt , zt ) started with initial distribution mr,l , i.e.
1 Ey,z τ (r, l) dx. Emr,l τ (r, l) = (66) Vol B(0, r, l) B(0,r,l) We have the following theorem proven in Subsect. 5.4. Theorem 4.15. Under Hypotheses I, II, and III with α = 1, if the renormalization core is stable then there exists a constant Q such that for ρmin > Qγmax one has
1 ln Emr,l τ (r, l) < 2. lim sup lim (67) r→∞ l→∞ ln r More precisely for r > R1 one has
lim Emr,l τ (r, l) = r 2−ν(r) ,
l→∞
where ν(r) is given by (58) and Q by (60).
(68)
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H. Owhadi
Remark 4.16. It is easy to extend this theorem to any finite number of particles driven by the same flow but independent thermal noise. Strong self-averaging property. A trivial consequence of Theorem 4.13 and 4.15 is the fact that fast mixing is an almost sure event. More precisely, let us write H (r) and H (r, l) the events H (r) := τ (r) ≤ r 2−δ and H (r, l) := τ (r, l) ≤ r 2−δ with δ = 0.9 ln γmin / ln ρmax . Observe that δ > 0 and we have the following theorem Theorem 4.17. Under Hypotheses I, II and III with α = 1, if the renormalization core is stable then there exists a constant Q such that for ρmin > Qγmax one has
(69) lim Pmr H (r) = 1 and lim lim Pmr,l H (r, l) = 1. r→∞
r→∞ l→∞
In this theorem Q is given by (60), by slightly modifying the constants. 4.4. Diagnosis of renormalization core’s pathologies. With Subsect. 4.3 we have seen that our model is super-diffusive if the renormalization core is stable. With this subsection we will give necessary and sufficient conditions for the stability of the renormalization core by analyzing in detail its dynamic. The results given here will be proven in Subsect. 5.3. Diffusive properties of the eddies at vanishing molecular conductivity. We will need the following functions V and W describing the effective behavior of the eddies E k of the renormalization core at vanishing molecular conductivity. For E ∈ SL∞ (Td ) and ζ > 0 we write λmin σsym (ζ Id , E) V (ζ, E) := . (70) ζ Observe that by the variational formulation (17) one has 1/V (ζ, E) = sup inf |ξ − ∇.H + E∇f |2 dx ∞ d d ξ ∈Sd−1 (f,H )∈C (T )×S (T ) Td
+ζ2
Td
|∇f (x)|2 dx,
(71)
where we have written Sd−1 the unit sphere of Rd centered on 0. Observe that if for ξ ∈ Sd−1 , ∇.E.ξ ≡ 0 one has ∀ζ > 0 V (ζ, E) > 1. Moreover V is continuous and decreasing in ζ . Let us define Definition 4.18. V (ζ ) := inf V (ζ, E n ),
(72)
V (0) := lim V (ζ ).
(73)
n∈N
ζ ↓0
Averaging Versus Chaos in Turbulent Transport?
567
Observe that V (ζ ) is a decreasing function in ζ thus the limit (73) is well defined and belongs to [1, ∞]. We define for x ∈ 1, V (0) , the inverse function V −1 as V −1 (x) := sup{y > 0 : V (y) > x}. Observe that if V (0) > 1, V −1 (x) is a decreasing function of x in 1, V (0) . Similarly we introduce λmax σsym (ζ Id , E) W (ζ, E) := . ζ
(74)
(75)
Observe that by the variational formulation (23) one has W (ζ, E) = 1 + sup inf ζ −2 |ξ − ∇.H − E(l − ∇f )|2 dx +
∞ d d l∈S d−1 ξ ⊥l,(f,H )∈C (T )×S (T )
Td
Td
|∇f |2 dx.
(76)
Observe that if for l ∈ Sd−1 , ∇.E.l ≡ 0 one has ∀ζ > 0 W (ζ, E) > 1. Moreover W is continuous and decreasing in ζ . Let us define Definition 4.19. W (ζ ) := sup W (ζ, E n ),
(77)
W (0) := lim W (ζ ).
(78)
n∈N
ζ ↓0
Observe that W (ζ ) is a decreasing function in ζ , thus the limit (78) is well defined and belongs to [1, ∞]. We recall that for E ∈ SL∞ (Td ) and ζ > 0, one has
−2 t 1 ≤ V (ζ, E) ≤ 1 + Cd ζ λmin E(x)E(x) dx (79) Td
and 1 ≤ W (ζ, E) ≤ 1 + Cd ζ −2 λmax
Td
t
E(x)E(x) dx .
(80)
Moreover the behavior of V (ζ, E) and W (ζ, E) at vanishing molecular conductivity (as ζ ↓ 0) and their connections with the stream lines of the eddies has been widely studied in the literature (we refer to [IK91, FP94] and the references therein). Thus it has been obtained [FP94] that for any β ∈ [− 21 , 0] there exist E ∈ SL∞ (Td ) such that V (ζ, E) = W (ζ, E) and as ζ ↓ 0, V (ζ, E)) ∼ c∗ ζ β ,
(81)
where c∗ can be calculated explicitly in several cases. A particular example with V (ζ, E)) ∼ −c∗ ln ζ is also given in [FP94]. For anisotropic cases, for any δ ∈ [0, 1/2) there exist E ∈ SL∞ (Td ) such that [FP94] W (ζ, E) ∼ c1∗ ζ 3δ−2
V (ζ, E) ∼ c2∗ ζ −δ .
(82)
568
H. Owhadi
The stability of the renormalization core and its anisotropy. Theorem 4.20 shows that the anisotropy of the local turbulent conductivity is one of the causes of the instability of the renormalization core. It is natural to wonder whether the converse is true; the answer is positive at low flow rate as shown by the following theorem and corollary. Theorem 4.20. If the renormalization core has bounded anisotropic distortion (µ < ∞) then 1. if γmax < V (0) then the renormalization core is stable (λ− > 0). Moreover if the monotony of V is strict then V −1 (γmax ) . γmax 2. If W (0) < ∞ then the renormalization core is bounded from above and 1 1 λ+ ≤ µ 2 Cd K0 (γmin − 1)− 2 1 + W (0) . 1
λ− ≥ µ− 2
(83)
(84)
Corollary 4.21. If γmax < V (0) and the renormalization core is not stable (λ− = 0) then it has unbounded anisotropic distortion (µ = ∞). Definition 4.22. The flow is said to be isotropic if for all ζ > 0, k ∈ N, σsym (ζ Id , E k ) is a multiple of the identity matrix. Definition 4.23. The renormalization core is said to be isotropic if for all k, Ak is a multiple of the identity matrix. We then write Ak = λ(Ak )Id . Observe that if the flow is isotropic then so is the renormalization core, µ = 1 and from Theorem 4.20 we obtain the following corollary. Corollary 4.24. If the flow is isotropic then the renormalization core is stable for γmax < V (0). Combining Theorem 4.20 and 4.10 we obtain that for γmax < V (0) the renormalization core is not stable if and only if it has unbounded anisotropic distortion. Moreover we have the following theorem Theorem 4.25. If the renormalization core has bounded anisotropic distortion, the monotony of V is strict, γmax < V (0) then the renormalization core is stable (λ− > 0) and
2
C1 ≤ λ− λ+ ≤ C2 ,
(85)
with C1 = V −1 (γmax )/γmax and C2 = Cd K02 (1 − 1/γmin )−1 . We believe that Eq. (85) could be at the origin of the isotropy of turbulence at small scales. Let us observe that if V (0) = ∞ then the stability of the renormalization core is equivalent to the fact that it has bounded anisotropic distortion. It is easy to build from Theorem 4.25 and the analysis of V (ζ, E) given above, examples of flows with stable renormalization core and thus a strongly-super-diffusive behavior. In particular we have the following theorem Theorem 4.26. If V (0) = ∞ and the renormalization core is isotropic then under hypotheses I, II and III with α = 1 the flow is strongly super-diffusive for ρmin > 11Qγmax (where Q is given by (60)) and Theorems 4.13, 4.15 and 4.17 are valid (with λ− > 0). Observe that if d = 2 and if for all k, E k = E, where E corresponds to the cellular flow (E12 (x, y) := sin(2πx) cos(2πy)), then V (0) = ∞ ([FP94]) and the renormalization core is stable (λ− > 0).
Averaging Versus Chaos in Turbulent Transport?
569
Viscosity implosion. It is easy to obtain that if there exists δ > 0 such that for all k the drift ∇.E k is null on [0, 1]d \ [δ, 1 − δ]d then W (0) < ∞. Moreover we have the following theorem. Theorem 4.27. If γmin > W (0) then the renormalization core is vanishing with exponential rate and ln λmax (An ) W (0) . (86) lim sup ≤ ln n γmin n→∞ It follows from Theorem 4.27 the renormalization core can be isotropic and not stable at the same time. Now it is natural to wonder whether a renormalization core (and thus the transport properties of the flow) may undergo a brutal alteration. Definition 4.28. We call viscosity implosion the bifurcation from a stable renormalization core to a vanishing renormalization core. We will now analyze this phenomenon. Definition 4.29. The flow is said to be self-similar if and only if γmax = γmin = γ , ρmax = ρmin = ρ and for all k, E k = E 0 = E. Let us recall that a real turbulent flow has a non self-similar multi-scale structure, we refer to [DC97]. Observe that if the flow is self-similar then V (ζ ) = W (ζ ). In this case we will write γc := V (0).
(87)
Theorem 4.30. Assume the flow to be self-similar and isotropic. 1. If γ < γc then the renormalization core is stable (λ− > 0) and lim An = ζ0 Id ,
n→∞
(88)
where ζ0 is the unique solution of V (ζ0 ) = γ . 2. If γ = γc and (V (0)−V (x))x −p admits a non-null limit as x ↓ 0 with p > 0 then the renormalization core is vanishing with polynomial rate (in particular λ+ = λ− = 0): 1 ln λ(An ) =− . n→∞ ln n p lim
(89)
3. If γ > γc then the renormalization core is vanishing with exponential rate (in particular λ+ = λ− = 0) γc 1 ln λ(An ) = ln . n→∞ n γ lim
(90)
It follows from Eq. (88) that if the flow is self-similar and the renormalization core isotropic and E non-constant then V (0) > 1 and for 1 < γ < V (0) the flow is strongly super-diffusive and Theorems 4.13, 4.15 and 4.17 are valid (with λ− > 0). The viscosity implosion of the renormalization core implies that the strong self-averaging property of the flow collapses towards a chaotic coupling between the scales. Let us give a particular example to illustrate what we mean by such bifurcation. The flow is assumed to be self-similar and isotropic and the stream lines of the eddy E over a period [0, 1]3 are given in Fig. 4(a). Since there exists δ > 0 such that the drift ∇.E is
570
H. Owhadi
(a)
(b)
(c)
Fig. 4a–c. Viscosity implosion. (a) An implosive eddy geometry; (b) Stable renormalization core; (c) Vanishing renormalization core
null on [0, 1]d \ [δ, 1 − δ]d we have γc < ∞ with the eddy illustrated in Fig. 4(a)Now imagine that one puts a drop of dye in such a flow and observe its transport at very large spatial scale. We have illustrated in Fig. 4 a metaphorical illustration of what one could see; it would be interesting to run numerical simulations to analyze the behavior of a drop of dye at the transition between a stable and vanishing renormalization core. For γ < γc dye is transported by strong super-diffusion, and the density of its colorant in the flow is homogeneous (Fig. 4(b)). Moreover in the domain (0, γc ) an increase of the flow rate γ in the eddies is compensated by an increase of the diffusive (dissipative) power of the smaller eddies. The picture undergoes a brutal transformation at γ ≥ γc ; in this domain an increase of γ results in the growth of the advective power of the eddies but their diffusive power remains bounded and can no longer compensate convection. The diffusive power of the smaller scales becomes dominated by the convective power of the eddy at the observation scale (Fig. 4(c)). The drop dye is then transported by advection and presents high density gradients. Variational formulae for γc . Assume the flow to be self-similar and isotropic. Thus from Eq. (71) it is easy to obtain that γc−1
=
inf
(f,H )∈C ∞ (Td )×S (Td ) Td
|ξ − div H + E∇f |2 dx;
(91)
from Eq. (76) it is also easy to obtain that for any unit vector l in Rd , inf ζ −2 |ξ − div H − E(l − ∇f )|2 dx γc = 1 + lim ζ ↓0 ξ ⊥l,(f,H )∈C ∞ (Td )×S (Td ) Td + |∇f |2 dx. (92) Td
Write G the set of f ∈ H 1 (Td ) such that there exists ξ, l ∈ Rd and H a skew symmetric matrix with coefficients in H 1 (Td ) with ξ ⊥ l and ξ − div H − E(l − ∇f ) = 0.
(93)
Averaging Versus Chaos in Turbulent Transport?
571
Then if G = ∅ it is easy to obtain from (92) that γc = ∞. If G = ∅ then one has γc = 1 + inf
f ∈G Td
|∇f |2 dx.
(94)
Equation (93) is degenerate, thus it is not easy to prove a solution for that equation in a general case and actually most of the time it has no solution which means that γc = ∞. It would be interesting to obtain non-trivial criteria ensuring the existence of a solution for (93). The most trivial example of a stream matrix E such that γc < ∞ is the following one. Take d = 2 and E a skew symmetric matrix with E1,2 = h, where h over the period [0, 1]2 is equal to
h(x1 , x2 ) = sin(2πx1 ) sin(2πx2 )g 4 (x1 − 0.5)2 + (x2 − 0.5)2 ,
(95)
where g is any smooth function on [0,1] such that g = 1 on [0, 1/3] and g = 0 on [2/3, 1]. Then it is easy to check that 1 < γc < ∞ and estimate it from the variational formulae given above. For instance write G the set of smooth Td periodic function f such that ∇f = e2 on {x : (x1 − 0.5)2 + (x2 − 0.5)2 ≤ 1/6}, then it is easy to check that |∇f |2 dx. (96) γc ≤ 1 + inf f ∈G
Td
The renormalization core with a finite number of scales. The results given above were related to the asymptotic behavior of the renormalization core. When the flow has only a finite number of scales we will give below quantitative estimates controlling the renormalization core. Theorem 4.31. The ubiety of the renormalization core is bounded from above by the inverse of its stability. Writing C = Cd K02 (1 − 1/γmin )−1 we have λ+ n ≤κ+
C λ− n−1
and µn ≤
κ C . − + 2 λn (λ− n)
Theorem 4.32. We have
−1 (γ max ) − 21 V . ≥ min λ (κ), (µ ) λ− min n n γmax
(97)
Theorem 4.33. We have for n ∈ N,
1 − 21 2 λ+ 1 + W (0) . ≤ max λ (κ), C K µ (γ − 1) max d 0 min n n
(98)
In particular, observe that if γmax < ∞ and λ− = 0 then the stability of the renormalization core should decrease according to the following relation ln λ− n ∼ −0.5 ln µn and its ubiety should increase like ln λ+ ∼ 0.5 ln µ . n n
572
H. Owhadi
5. Proofs 5.1. Averaging with two scales: Proof of Theorem 4.1. There are two strategies to prove Theorem 4.1; the first one is based on the relative translation method introduced in [Owh01a] and the variational formulations of the effective conductivity; this is the strategy used in [BO02a]. The second one is new and based only on the relative translation method. Although the first strategy in the case considered here would give (the proof is rather long) a sharper estimate of the error term: (1 + (R))2 with (R) = Kα a+P ∞ R α λmin (a) f (d, λmin (a) ) instead of (24), we have preferred to write here the second one for its simplicity and the fact that it allows to obtain a lower and an upper bound at once without the need of any variational formulation. Let us now give this new alternative strategy. By the variational formulation (23) the effective conductivity σsym (a, SR P + K) is continuous in L∞ (Td ) norm with respect to the stream matrices P and K and by density it is sufficient to prove the estimate (24) assuming that P and K are smooth and belong to S(Td ). First we will prove the following proposition where we have used the notation introduced in Sect. 3 (we write E := SR P + K). Proposition 5.1. Let l ∈ Rd , l = 0,
t
lσsym (a, SR P + K)l
1 2
−
t
21 lσsym σsym (a, P ), K l
2 ≤ J1 + J2 + J3 + J4 (99)
with J1 =
(x,y)∈Td ×[− 21 , 21 ]d σsym (a,P ),K
∇Fl
∇Fla,E (x +
(100)
(x) dx dy,
J2 = −
y y ) a − E(x + ) ∇χ a,P (Rx + y) R R
Td ×[− 21 , 21 ]d
∇Fla,E (x +
y ) (a + P (Rx + y))∇F a,P (Rx + y) R
σ (a,P ),K −σsym (a, P ) ∇Fl sym (x) dx dy, J3 = −
∇Fla,E (x +
Td ×[− 21 , 21 ]d σ (a,P ),K ∇Fl sym (x) dx dy,
J4 =
Td ×[− 21 , 21 ]d
∇Fla,E (x +
(101)
y y ) K(x + ) − K(x) ∇F a,P (Rx + y) R R (102)
y σ (a,P ),K (x) dx dy. )K(x)∇χ a,P (Rx + y)∇Fl sym R (103)
Averaging Versus Chaos in Turbulent Transport?
Proof. Let us write I=
(x,y)∈Td ×[− 21 , 21 ]d
∇Fla,E (x +
573
y σ (a,P ),K (x) dx dy. )a∇F a,P (Rx + y)∇Fl sym R (104)
Using the Cauchy-Schwartz inequality and formula (20) one obtains that I≤
t
lσsym (a, SR P + K)l
1 2
t
21 lσsym σsym (a, P ), K l .
(105)
Now, writing E = SR P + K observe that I = (I1 + I2 )/2 with
I1 =
(x,y)∈Td ×[− 21 , 21 ]d σsym (a,P ),K
×∇Fl and
(x,y)∈Td ×[− 21 , 21 ]d σsym (a,P ),K
×∇Fl Using
(x,y)∈Td ×[− 21 , 21 ]d
y y
∇Fla,E x + a+E x+ ∇F a,P (Rx + y) R R
(x) dx dy
I2 =
(106)
y y
∇Fla,E x + a−E x+ ∇F a,P (Rx + y) R R
(x) dx dy.
y y
a−E x+ l = t lσsym (a, SR P + K)l ∇Fla,E x + R R
and the fact that ∇Fla,E (x + obtains that
y R) a
− E(x +
y R)
is a divergence free vector field one
I2 = tlσsym (a, SR P + K)l − J1 with J1 given by (100). Moreover I 1 = G0 − J 2 − J 3 − J 4 with
(107)
y σ (a,P ),K σsym (a, P ) + K(x) ∇Fl sym ∇Fla,E x + (x) dx dy 1 1 d d R (x,y)∈T ×[− 2 , 2 ] = t lσsym σsym (a, P ), K l, (108)
G0 =
σ (a,P ),K where we have used in the last equality the fact that σsym (a, P )+K(x) ∇Fl sym (x) is divergence free. And J2 , J3 , J4 are given by (101), (102) and (103). Thus combining (105) and (106) we have obtained (99), which proves the proposition.
574
H. Owhadi
Now we will show that J1 , J2 , J3 and J4 act as error terms in the homogenization process. Using div (a + E)∇Fla,E = 0 and observing that, ∇χ a,P (Rx + y) = ∇y χ a,P (Rx + y) and integrating by parts in y one obtains (writing ∂ i ([− 21 , 21 ]d ) = {x ∈ [− 21 , 21 ]d : xi = − 21 }) J 1 = G1 + G 2
(109)
with (writing (ei )1≤i≤d the orthonormal basis of Rd compatible with the axis of periodicity of Td ) G1 =
d i=1 t
(x,y i )∈Td ×∂ i ([− 21 , 21 ]d )
∇Fla,E (x + (y i + ei )/R) − ∇Fla,E (x + y i /R)
σsym (a,P ),K
×(a − P (Rx + y i )).ei χ a,P (Rx + y i )∇Fl G2 = −
d i=1
t (x,y i )∈Td ×∂ i ([− 21 , 21 ]d )
(x) dx dy i
(110)
∇Fla,E (x + (y i + ei )/R)K(x + (y i + ei )/R)
σ (a,P ),K − ∇Fla,E (x + y i /R)K(x + y i /R) .ei χ a,P (Rx + y i )∇Fl sym (x) dx dy i , (111) t
Now we will need the following lemma which says that the solution of the two-scale cell problem keeps in its structure a signature of the fast period. Lemma 5.2. For i ∈ {1, . . . d} one has |∇Fla,E (x + ei /R) − ∇Fla,E (x)|2a ≤ |l|2σ (a,E) Cd x∈Td
Kα 2 . R α λmin (a)
(112)
Proof. Observe that ∇.(a + E(x))∇ Fla,E (x + ei /R) − Fla,E (x)
= ∇. K(x) − K(x + ei /R) ∇Fla,E (x + ei /R) .
(113)
It follows that |∇Fla,E (x + ei /R) − ∇Fla,E (x)|2a = ∇Fla,E (x + ei /R) − ∇Fla,E (x) x∈Td x∈Td K(x) − K(x + ei /R) ∇Fla,E (x + ei /R), (114) thus using Cauchy-Schwartz inequality one obtains |∇Fla,E (x + ei /R) − ∇Fla,E (x)|2a d x∈T 2 ≤ (K(x) − K(x + ei /R))∇F a,E (x + ei /R) −1 , x∈Td
and Eq. 112 follows easily.
a
(115)
Averaging Versus Chaos in Turbulent Transport?
575
It follows from Lemma 5.2 Eq. 110 and Cauchy-Schwartz inequality that
1 Kα a − P ∞ t 2 lσsym (a, SR P + K)l χ a,P ∞ G 1 ≤ Cd α R λmin (a) λmin (a) 21 × t lσsym σsym (a, P ), K l .
(116)
Now we will use the following lemma which is a consequence of Stampacchia estimates [Sta66, Sta65] for elliptic operators with discontinuous coefficients (see [Owh01a], Appendix B, Theorem B.1.1) (we recall that χ a,P is uniquely defined by the cell problem a,P and Td χ (x) dx = 0). Lemma 5.3. χ a,P ∞ ≤ Cd
a + P 3d+2 ∞
λmin (a)
.
(117)
Thus one obtains from (117) and (116) that
1 Kα a + P ∞ 3d+3 t 2 t lσsym (a, SR P + K)l lσsym G1 ≤ Cd α R λmin (a) λmin (a) 21 × σsym (a, P ), K l (118) similarly, observing that divy K(x)∇Fla,E (x + Ry ) = 0 and integrating by part in y in Eq. (103) one obtains d t J4 = ∇Fla,E (x +(y i +ei )/R)− t ∇Fla,E (x +y i /R) K(x).ei (x,y i )∈Td ×∂ i ([− 21 , 21 ]d )
i=1
σsym (a,P ),K
χ a,P (Rx + y i )∇Fl
(x) dx dy i . (119)
Adding Eq. (111) to Eq. (119) we obtain J4 + G2 =
d i=1
t (x,y i )∈Td ×∂ i ([− 21 , 21 ]d )
∇Fla,E (x +(y i +ei )/R) K(x)−K(x +(y i +ei )/R) .ei
σsym (a,P ),K
× χ a,P (Rx + y i )∇Fl d + i=1
t
(x,y i )∈Td ×∂ i ([− 21 , 21 ]d )
(x) dx dy i
∇Fla,E (x + y i /R) K(x + y i /R) − K(x) .ei
σsym (a,P ),K
× χ a,P (Rx + y i )∇Fl
(x) dx dy i , (120)
and by Cauchy Schwartz inequality and Lemma 117 one obtains that
1 Kα a + P ∞ 3d+3 t 2 lσsym (a, SR P + K)l J4 + G2 ≤ Cd α R λmin (a) λmin (a) 21 × t lσsym σsym (a, P ), K l .
(121)
576
H. Owhadi
Moreover from Eq. (102) and Cauchy Schwartz inequality one easily obtains J3 ≤ Cd
1 21 Kα t 2 t lσ (a, S P + K)l lσ (a, P ), K l . σ sym R sym sym R α λmin (a)
(122)
Now we will need the following lemma
Lemma 5.4. If V ∈ (C ∞ (Td ))d is such that div V = 0 and Td V (x)dx = 0 then for p > d, there exists a skew symmetric Td -periodic matrix M such that M∞ ≤ Cd,p V Lp (Td ) and V = ∇.M. Proof. From the proof of Lemma 4.7 of [BO02a] one obtains that there exists a Td -periodic smooth skew-symmetric matrix M such that Vi =
d
(123)
Mij
j =1
and M is given by j
Mij = Bi − Bji ,
(124)
where Bji are the smooth Td periodic solutions of Bji = ∂i Vj
(125)
with 0 mean Lebesgue measure. Using Theorem 5.4 of [Sta66] one obtains that for p>d Bji ∞ ≤ Cd,p Vj Lp (Td )
(126)
which proves the lemma. Let us now prove the following lemma. Lemma 5.5.
a + P (Rx + y) ∇F a,P (Rx + y) − σ (P )
kj
=
d
P ∂i Mkij (Rx + y),
(127)
i=1
P = −M P and where M P is a Td periodic d × d × d tensor such that Mikj kij
M P ∞ ≤ Cd,a+P ∞ /λmin (a) a + P ∞ . Proof. From Lemma 5.4 one obtains that for p > d, M P ∞ ≤ Cd,p a + P (.))∇F a,P (.) − σ (P ) Lp (Td ) .
(128)
(129)
Using Meyers argument [Mey63] one obtains that there existsp(a+E∞ /λmin (a)) > d such that ∇χ a,P (.)Lp (Td ) ≤ Cd,a+P ∞ /λmin (a) , which implies Eq. (128).
(130)
Averaging Versus Chaos in Turbulent Transport?
577
Using Eq. (127) and integrating by part in y in (101) one obtains J2 = −
d
i,j,k=1 P Mkij (Rx
t (x,y i )∈Td ×∂ i ([− 21 , 21 ]d ) σsym (a,P ),K
+ y i )ej .∇Fl
∇Fla,E (x +(y i +ei )/R)− t ∇Fla,E (x +y i /R) .ek (x) dx dy i .
(131)
Combining this with (128) and (112) one obtains from Cauchy-Schwartz inequality that J2 ≤ Cd,a+P ∞ /λmin (a)
1 21 Kα t 2 t lσ (a, S P +K)l lσ (a, P ), K l . σ sym R sym sym R α λmin (a) (132)
In conclusion we have obtained from Eq. (99), (109), (118), (121), (122) and (132) that
1 21 2 2 t lσsym (a, SR P + K)l − t lσsym σsym (a, P ), K l ≤ Cd,a+P ∞ /λmin (a)
1 21 Kα t 2 t lσ (a, S P + K)l lσ (a, P ), K l . σ sym R sym sym R α λmin (a) (133)
Now we will use the following lemma whose proof is trivial √ Lemma 5.6. If (X − Y )2 ≤ δXY then X/Y ≤ (1 + 8 δ)2 . And the estimate (24) is a simple consequence of (133) and Lemma 5.6 which proves the theorem.
5.2. Averaging with n scales: Proof of Theorem 4.4. The proof of Theorem 4.4 is based on Theorem 4.1 and a reverse induction. It is important to note that contrary to reiterated homogenization, here the larger scales are homogenized first; this reversion in the inductive process is essential to obtain sharp estimates. Observe that by the variational formula 17 one has for γ > 0, B ∈ Md,sym and K ∈ SL∞ (Td ), B σsym (B, γ K) = γ σsym ( , K). γ
(134)
From this we deduce that for p ∈ {0, . . . , n − 1},
γ γ
1 1 p+1 p σsym σsym (B, E p ), p+1,n . σsym σsym (B, E p ), p+1,n = γp γp γp+1 γp+1 (135) Combining this with Theorem 4.1 one obtains that for p ∈ {0, . . . , n − 1},
γ γ
1 1 p+1 p σsym B, p,n ≤ σsym σsym (B, E p ), p+1,n (1 + p+1 (B))4 , γp γp+1 γp γp+1 (136)
578
H. Owhadi
γ γ
1 1 p+1 p σsym B, p,n ≥ σsym σsym (B, E p ), p+1,n (1 + p+1 (B))−4 γp γp γp+1 γp+1 (137) with
p (B) =
1 γp Kα 2 f (d, (λmax (B) + K0 )/λmin (B)) . α ) γp−1 rpα λmin (B)(1 − γmax /ρmin (138)
Then one obtains by a simple induction that γn+1 An+1
n−1
(1 + p+1 (Ap ))−4 ≤ σsym (a, 0,n ) ≤ γn+1 An+1
p=0
n−1
(1 + p+1 (Ap ))4 ,
p=0
(139) where Ak , is the renormalization coreization sequence given in Definition 4.3 which proves Theorem 4.4. 5.3. Diagnosis of renormalization core’s pathologies: Proofs. Let a ∈ Md,sym and E ∈ SL∞ (Td ); it is well known ([AM91]) and a simple consequence of (17) and (23) that t a ≤ σsym (a, E) ≤ a + E(x)a −1 E(x) dx. (140) Td
Then the following proposition follows from (33), (140) and a simple induction on the number of scales. Proposition 5.7. For all n ∈ N, γn−1 Id ≤ An ≤ (κ/γn )Id +
n−1
(γp /γn )
p=0
t T1d
E p (x)(Ap )−1 E p (x) dx.
(141)
Theorems 4.20 and 4.31 are straightforward consequences of Proposition 5.7. We will need the following proposition giving isotropic estimates on anisotropic viscosities. Proposition 5.8. For a ∈ Md,sym and E ∈ SL∞ (Td ), one has for all l ∈ (Rd )∗ , λ (a) 1 t lσ λmax (a) − 1 2 sym (a, E)l max 2 ≤ ≤ .
1 (142) λmin (a) λ (a) min t lσ 2I ,E l λ (a)λ (a) sym max min d Proof. By the variational formula 17 one has for ξ ∈ Rd , 2 2 |ξ |σ −1 (a,E) = inf |ξ − ∇.H + E∇f |a −1 dx + sym
(f,H )∈C ∞ (Td )×S (Td ) Td
Td
|∇f (x)|2a dx
1 − 1 ≤ λmax (a)/λmin (a) 2 inf λmax (a)λmin (a) 2 (f,H )∈C ∞ (Td )×S (Td ) 1 |ξ − ∇.H + E∇f |2 dx + λmax (a)λmin (a) 2 |∇f (x)|2 dx. Td
Td
(143)
Averaging Versus Chaos in Turbulent Transport?
579
It follows that
− 1 1 σsym (a, E) ≥ λmax (a)/λmin (a) 2 σsym λmax (a)λmin (a) 2 Id , E .
(144)
Similarly from the variational formula 23 one obtains that for l ∈ Rd , 2 inf |ξ − ∇.H − E(l − ∇f )|2a −1 dx |l|σsym = ξ ⊥l,(f,H )∈C ∞ (Td )×S (Td ) Td + |l − ∇f |2a dx Td 1 inf ≤ λmax (a)/λmin (a) 2 ξ ⊥l,(f,H )∈C ∞ (Td )×S (Td ) − 1 1 λmax (a)λmin (a) 2 |ξ − ∇.H − E(l − ∇f )|2 dx + λmax (a)λmin (a) 2 Td |l − ∇f |2a dx, (145) Td
which leads us to
1 1 σsym (a, E) ≤ λmax (a)/λmin (a) 2 σsym λmax (a)λmin (a) 2 Id , E .
(146)
A direct consequence of Proposition 5.8 is the following corollary which controls the minimal and maximal enhancement of the conductivity in the flow associated to the 1 stream matrix E by the geometric mean λmax (a)λmin (a) 2 of the maximal and minimal eigenvalues of a. Corollary 5.9.
λmin σsym (a, E) ≥ λmin (a)
λmax σsym (a, E) ≤ λmax (a)
1 λmin σsym λmax (a)λmin (a) 2 Id , E
λmax (a)λmin (a)
1
,
1 λmax σsym λmax (a)λmin (a) 2 Id , E
λmax (a)λmin (a)
(147)
2
1
.
(148)
2
It is then a simple consequence of Corollary 5.9 that Proposition 5.10. 1
λmin (σsym (a, E)) ≥ V λmin (a)λmax (a) 2 , E λmin (a)
(149)
1
λmax (σsym (a, E)) ≤ W λmin (a)λmax (a) 2 , E . λmax (a)
(150)
and
580
H. Owhadi
From Proposition 5.10 one obtains that for n ∈ N, 1 n
γn λmin (An+1 ) n n 2 ≥ V λ (A )λ (A ) ,E . min max λmin (An ) γn+1
(151)
It follows from Eq. (151) and the monotony of V that n 1 λmin (An+1 ) n λmax (A ) 21 ≥ V λ (A )( ) min λmin (An ) γmax λmin (An )
(152)
it follows from (152) that λmin (An ) is increasing if it belongs to (An ) − 21 −1 0, ( λλmax V (γmax ) ; which implies Eq. (83) of Theorem 4.20 and Eq. (97) of n ) min (A ) Theorem 4.32. Now, observe that from the variational formulation (76) one obtains that W (ζ, E) ≤ 1 + ζ −2 λmax
Td
E(x)E(x)dx .
t
(153)
It follows from Proposition 5.10 that −1 λmax (σsym (a, E)) ≤ 1 + Cd K02 λmin (a)λmax (a) . λmax (a)
(154)
Thus one obtains for all n ∈ N, n
λmax (An+1 ) −1 2 λmax (A ) n −2 K (A ) 1 + C . ≤ γ λ d max 0 min λmax (An ) λmin (An )
(155)
It follows from (155) that λ max (An ) is decreasing if it belongs to 1 (An ) −1 2 , ∞ ; which implies Eq. (84) of Theorem 4.20 and Cd K02 λλmax n (γmin − 1) min (A ) Eq. (98) of Theorem 4.33. Now observe that by Proposition 5.10 one has λmax (An ) 1 1 λmax (An+1 ) W λmin (An )( ≤ )2 , n λmax (A ) γmin λmin (An )
(156)
which proves Theorem 4.27 since W is decreasing. Now if the flow is self-similar and isotropic, Theorem 4.30 is a simple consequence of the following recursive relation: λ(An+1 ) 1 = V λ(An ) . n λ(A ) γ
(157)
Averaging Versus Chaos in Turbulent Transport?
581
5.4. Super diffusion: Proofs. 5.4.1. A variational formula for the exit times. Let be a smooth subset of Rd , we ¯ write for a ∈ Md,sym and E a skew symmetric matrix with coefficients in L∞ (), ψ a,E = Ex [τ a,E ()],
(158)
the expectation of the exit time from of the diffusion associated to the generator ∇.(a + E)∇ started from x. Observe that ψ a,E can be defined as the weak solution of the following equation with null Dirichlet boundary condition on ∂,
(159) ∇. a + E(x) ∇ψ a,E (x) = −1. We will need the following variational formulation for the mean exit times. Theorem 5.11. Ex [τ a,E ()] dx
=
sup
¯ f ∈C0∞ (),H ∈S ()
2 2 f (x)dx − |∇f |a dx − |∇.H + E∇f |2a −1 dx ,
(160) where the minimization (160) is done over smooth functions f on , null on ∂ and ¯ From Theorem 5.11 we deduce the following smooth skew symmetric matrices H on . corollary Corollary 5.12. Ex [τ a+λId ,0 ()] dx ≤ Ex [τ a,E ()] dx ≤ Ex [τ a,0 ()] dx
(161)
with λ := sup λmax t E(x)a −1 E(x) .
(162)
x∈
Let us now prove Theorem 5.11. By density we can first assume E to be smooth. Our purpose is to show that Ex [τ a,E ()] dx 1 = −2 inf |∇.H + (a − E)∇f |2a −1 dx − f (x)dx . (163) ¯ 2 f ∈C0∞ (),H ∈S () By considering variations around the minimum one obtains that ∇.H0 + (a − E)∇f0 = a∇ψ(x)
(164)
∇.(a + E)∇ψ(x) = −1.
(165)
with ψ = 0 on ∂ and
582
H. Owhadi
From which one obtains = Ex [τ E ()] and f0 (x) = that ψ(x) E −E Ex [τ ()] + Ex [τ ()] /2. Thus at the minimum 1 − inf |∇.H + (a − E)∇f |2a −1 dx − f (x)dx H,f 2 1 t =− ∇f0 (a + E)∇ψdx − f0 (x)dx = −1/2 f0 (x)dx, (166) 2 since
t
∇f0 (a + E)∇ψ(x)dx =
but also
|a∇ψ(x)|2a −1 dx
|a∇ψ(x)|2a −1 dx =
(167)
t
∇ψ(x)(a + E)∇ψ(x)dx =
ψ(x)dx
(168)
which leads to the result, which can be written as (160). 5.4.2. Averaging with two scales the exit times. We will use the notation of Subsect. 5.4.1 and assume that E = P (Rx) + K(x),
(169)
where x ∈ , R ∈ [2, ∞), P belongs to SL∞ (Td ) and K is a Lipschitz-continuous skew symmetric matrix on Rd (α = 1). Our purpose is to obtain sharp quantitative estimates on the mean exit time. ψ a,E (x) dx. (170)
It follows from Theorem 5.11 that the mean exit time (170) is continuous in L∞ norm with respect to E, thus we can by density assume E, P and K to be smooth and ψ a,E shall be a strong solution of (159). To estimate (170) we will need to introduce a relative translation with respect to the fast scale associated to the medium E, i.e. we introduce for x, y ∈ × [0, 1]d , E(x, y) as E(x, y) := P (Rx + y) + K(x).
(171)
We will write for y ∈ [0, 1]d , ψ a,E (x, y) the strong solution of the following equation with null Dirichlet boundary condition on ∂,
∇x a + E(x, y) ∇x ψ a,E (x, y) = −1. (172) Let us define J :=−
∇ψ a,E (x, y) (a + P (Rx + y))∇F a,P (Rx + y) − σ (a, P ) x∈,y∈[0,1]d t ×∇ψ σsym (a,P ),K (x) dx dy + ∇ψ a,E (x, y) a − P (Rx + y)
×∇χ
t
a,P
(Rx + y)∇ψ
x∈,y∈[0,1]d σsym (a,P ),K
(x) dx dy.
(173)
Now we will show that J controls the multi-scale homogenization associated to ψ(x, y)
Averaging Versus Chaos in Turbulent Transport?
583
Proposition 5.13. One has 1 ψ a,E (x, y) dx dy 2 −
1 2
≤ J.
(174)
∇ψ a,E (x, y)a∇F a,P (Rx + y)∇ψ σsym (a,P ),K (x) dx dy.
(175)
x∈,y∈[0,1]d
ψ σsym (a,P ),K (x) dx
2
x∈
Proof. Let us write I=
t
x∈,y∈[0,1]d
Observing that t x∈,y∈[0,1]d
∇ψ a,E (x, y)a∇ψ a,E dx dy
=
t x∈,y∈[0,1]d
∇ψ a,E (x, y) a + E(x, y) ∇ψ a,E dx dy
= and
ψ a,E (x, y) dx dy
(176)
x∈,y∈[0,1]d
∇ψ σsym (a,P ),K (x)t ∇F a,P (Rx + y)a∇F a,P (Rx + y) σsym (a,P ),K ×∇ψ (x) dx dy = ψ σsym (a,P ),K (x) dx, t
x∈,y∈[0,1]d
(177)
x∈
one obtains by Cauchy-Schwartz inequality from (175) that
1 2 I≤ ψ a,E (x, y) dx dy ψ σsym (a,P ),K (x) dx . x∈,y∈[0,1]d
Now let us polarize I as
with I1 =
t x∈,y∈[0,1]d
(178)
x∈
I = I1 + I2 /2
(179)
∇ψ a,E (x, y) a −E(x, y) ∇F a,P (Rx +y)∇ψ σsym (a,P ),K (x) dx dy (180)
and
I2 =
t
∇ψ a,E (x, y) a + E(x, y) ∇F a,P (Rx + y)
x∈,y∈[0,1]d σsym (a,P ),K
∇ψ
(x) dx dy. (181) Using ∇. a + E(x + y/R) ∇ψ a,E (x, y) = −1 one obtains that t I1 = ψ σsym (a,P ),K (x) dx − ∇ψ a,E (x, y) a − E(x, y)
x∈
×∇χ
a,P
(Rx + y)∇ψ
x∈,y∈[0,1]d σsym (a,P ),K
(x) dx dy.
(182)
584
H. Owhadi
Moreover I2 =
t
x∈,y∈[0,1]d
∇ψ a,E (x, y) σ (a, P ) + K(x) ∇ψ σsym (a,P ),K (x) dx dy
+
t x∈,y∈[0,1]d σsym (a,P ),K
∇ψ −
∇ψ a,E (x, y) (a + P (Rx + y))∇F a,P (Rx + y) − σ (a, P )
(x) dx dy t
x∈,y∈[0,1]d
∇ψ a,E (x, y)K(x)∇χ a,P (Rx + y)∇ψ σsym (a,P ),K (x) dx dy, (183)
and observing that t x∈,y∈[0,1]d
∇ψ a,E (x, y) σ (a, P ) + K(x) ∇ψ σsym (a,P ),K (x) dx dy
=
ψ a,E (x, y) dx dy,
(184)
x∈,y∈[0,1]d
one obtains from the combination of (179), (182) and (183) that 2I = ψ a,E (x, y) dx dy + ψ σsym (a,P ),K (x) dx − J x∈,y∈[0,1]d
(185)
x∈
with J given by Eq. (173). Next one easily obtains (174) from (185) and (178).
We will now show that J acts as an error term. We will need the following lemmas. Lemma 5.14. Let σ be a positive definite symmetric constant matrix. There exists a constant Cd depending only on the dimension d such that for any function f ∈ C02 () one has d
2
∂i ∂j f (x)
−2 dx ≤ Cd λmin (σ )
i,j =1
2
∇σ ∇f (x)
dx.
(186)
Proof. When = Rd and f ∈ C0∞ (R2 ), the inequality (186) is standard, we refer to Theorem 1.7 of [Sim72]. When is a bounded open subset of Rd with smooth boundary the proof follows trivially from the density of C0∞ (R2 ) in C02 (). ¯ d ¯ ξ ∈ C ∞ () We write T () the set of smooth d-dimensional vector field on , such that ∀z ∈ ∂,
ξ(z).n(z) = 1,
(187)
where ∂ is the boundary of and n(z) the exterior orthonormal vector at the point z of the boundary. For a bounded open subset of Rd with smooth boundary we write () the following isoperimetric constant associated to : () := inf max ξ ∞ , ∇ξ ∞ . (188) ξ ∈T ()
Averaging Versus Chaos in Turbulent Transport?
585
Lemma 5.15. We have t ∇ψ a,E (z, y)a∇ψ σsym (a,P ),K (z) dz dy ≤ Cd,a+P ∞ /λmin (a) () z∈∂,y∈[0,1]d
1 K1 2 a,E 1+ ψ (x, y) dx dy ψ σsym (a,P ),K (x) dx d λmin (a) x∈,y∈[0,1] x∈ Vol() 1
1 2 2 . + ψ σsym (a,P ),K (x) dx + ψ a,E (x, y) dx dy λmin (a) x∈ x∈,y∈[0,1]d (189)
¯ we will use the Proof. Let f and v be a smooth function and a smooth vector field on following Green formula: f (x) div v(x) dx = − ∇f (x)v(x) dx + f (z) v(z).n(z) dz, (190)
∂
where dz is the measure surface at the boundary. Let ξ ∈ T (). Let us write G=
d d i,j,k=1 x∈,y∈[0,1]
∂i ψ a,E (x, y)
× a − P (Rx + y) ij ∂j ξk (x)∂k ψ σsym (a,P ),K (x) dx dy.
(191)
Applying formula (190) to Eq. (191) with ∇f = ∇ξk (x) we obtain that G = G1 + G2 + G3
(192)
with (using the skew symmetry of Pij in ij ) G1 = −
d d i,j,k=1 x∈,y∈[0,1]
∂j
a + P (Rx + y)
∂ ψ a,E (x, y) ji i
×ξk (x)∂k ψ σsym (a,P ),K (x) dx dy,
G2 = −
d d i,j,k=1 x∈,y∈[0,1]
∂i ψ a,E (x, y) a − P (Rx + y) ij ξk (x)
(193)
(194)
∂j ∂k ψ σsym (a,P ),K (x) dx dy,
G3 =
d i,j,k=1
z∈∂,y∈[0,1]d
∂i ψ a,E (z, y) a − P (Rz + y) ij ξk (z)
(195)
× nj (z)∂k ψ σsym (a,P ),K (z) dz dy, where nj are the coordinates of the exterior orthonormal vector n(z). Using the fact that ∇ψ a,E (z, y) and ∇ψ σsym (a,P ),K (z) are parallel to n(z) at the boundary of and both
586
H. Owhadi
heading towards the opposite direction of n, we obtain that (using the skew symmetry of Pij in ij ) t G3 = ∇ψ a,E (z, y)a∇ψ σsym (a,P ),K (z) ξ(z).n(z) dz dy. (196) z∈∂,y∈[0,1]d
Thus by Eq. (187), G3 =
t
z∈∂,y∈[0,1]d
∇ψ a,E (z, y)a∇ψ σsym (a,P ),K (z) dz dy.
(197)
Now, by Cauchy-Schwartz inequality we obtain from (191) |G| ≤ Cd a + P ∞ (λmin (a))−1 ∇ξ ∞ a,E ψ (x, y) dx dy x∈,y∈[0,1]d
ψ σsym (a,P ),K (x) dx
1 2
(198)
.
x∈
Using Cauchy-Schwartz inequality and ∇.(a + P (Rx + y))∇ψ a,E (x, y) = −1 − ∇.K(x)∇ψ a,E (x, y) we obtain from Eq. (193) that − 1 1 2 ξ ∞ ψ σsym (a,P ),K (x) dx 2 |G1 | ≤Cd λmin (σsym (a, P ))
Vol() + K21 (λmin (a))−1
x∈
ψ a,E (x, y) dx dy
1 2
(199) .
x∈,y∈[0,1]d
Using Cauchy-Schwartz inequality, Lemma 5.14 and ∇.σsym (a, P )∇ψ σsym (a,P ),K (x) = −1 − ∇.K(x)∇ψ σsym (a,P ),K (x), we obtain from Eq. (194) that −1 1 |G2 | ≤ Cd a + P ∞ (λmin (a))− 2 λmin (σsym (a, P )) ξ ∞ 1 × ψ a,E (x, y) dx dy 2 x∈,y∈[0,1]d
×
Vol() + K21
−1 λmin (σsym (a, P ))
ψ σsym (a,P ),K (x) dx
1 2
. (200)
x∈
Combining (192), (197), (198), (199) and (200) we obtain that t ∇ψ a,E (z, y)a∇ψ σsym (a,P ),K (z) dz dy z∈∂,y∈[0,1]d
K1 ≤ Cd,a+P ∞ /λmin (a) (1 + ) λ min (a) ψ a,E (x, y) dx dy ξ ∞ + ∇ξ ∞ x∈,y∈[0,1]d
Vol() 1 2 +Cd,a+P ∞ /λmin (a) ξ ∞ λmin (a) σsym (a,P ),K ψ (x) dx + x∈
ψ σsym (a,P ),K (x) dx
1 2
x∈
ψ a,E (x, y) dx dy
x∈,y∈[0,1]d
which proves the lemma by optimization on the vector field ξ .
1 2
,
(201)
Averaging Versus Chaos in Turbulent Transport?
587
Proposition 5.16. We have |J | ≤ R −1 Cd,a+P ∞ /λmin (a) (() + 1)
1 K1 2 1+ ψ a,E (x, y) dx dy ψ σsym (a,P ),K (x) dx λmin (a) x∈,y∈[0,1]d x∈ Vol() 1 1 2 + ψ σsym (a,P ),K (x) dx 2 λmin (a) x∈ 1
a,E 2 + . (202) ψ (x, y) dx dy x∈,y∈[0,1]d
Proof. Using formulae (173) and (127) one obtains that J =
d
∂i ψ a,E (x, y)Bi,j,k (x, y)∂k ψ σsym (a,P ),K (x) dx dy
(203)
Bi,j,k (x, y) = −∂j MijP k (Rx + y) + a − P (Rx + y) ij ∂j χka,P (Rx + y).
(204)
i,j,k=1
x∈,y∈[0,1]d
with
Applying formula (190) to Eq. (203) first with div ξ = with ∇f =
∇χka,P (Rx
d
P j =1 −∂j Mij k (Rx
+ y), next
+ y), we obtain that J = J 1 + J2 + J3
(205)
with (using the skew symmetry of MijP k in ij ) J1 = −R
d
−1
d i,j,k=1 x∈,y∈[0,1]
∂j a + P (Rx + y) j i ∂i ψ a,E (x, y)
×χka,P (Rx + y)∂k ψ σsym (a,P ),K (x) dx dy,
J2 = R
−1
d d i,j,k=1 x∈,y∈[0,1]
(206)
∂i ψ a,E (x, y)
× MijP k (Rx + y) − a − P (Rx + y) ij χka,P (Rx + y) ∂j ∂k ψ σsym (a,P ),K (x) dx dy,
J3 = R
−1
d d i,j,k=1 z∈∂,y∈[0,1]
(207)
∂i ψ a,E (z, y)
× − MijP k (Rz + y) + a − P (Rz + y) ij χka,P (Rz + y) nj (z)∂k ψ σsym (a,P ),K (z) dz dy.
(208)
588
H. Owhadi
Using the fact that ∇ψ a,E (z, y) and ∇ψ σsym (a,P ),K (z) are parallel to n(z) at the boundary of and both heading towards the opposite direction of n, we obtain that (using the skew symmetry of MijP k and Pij in ij ) t −1 J3 = R ∇ψ a,E (z, y)a∇ψ σsym (a,P ),K (z) χ.a,P (Rz + y).n(z) dz dy. z∈∂,y∈[0,1]d
(209) Thus by Lemma 5.15 and Eq. (117) t ∇ψ a,E (z, y)a∇ψ σsym (a,P ),K (z) dz dy |J3 | ≤R −1 χ.a,P ∞ z∈∂,y∈[0,1]d
−1
≤R Cd,a+P ∞ /λmin (a) ()
1 K1 2 a,E ψ (x, y) dx dy ψ σsym (a,P ),K (x) dx 1+ d λmin (a) x∈,y∈[0,1] x∈ Vol() 1
1 2 2 . + ψ σsym (a,P ),K (x) dx + ψ a,E (x, y) dx dy d λmin (a) x∈ x∈,y∈[0,1] (210) Using Cauchy-Schwartz inequality and ∇.(a + P (Rx + y))∇ψ a,E (x, y) = −1 − ∇.K(x)∇ψ a,E (x, y) we obtain from Eq. (206) and (117) that 1 Vol() 21 −1 σsym (a,P ),K 2 ψ (x) dx |J1 | ≤ Cd,a+P ∞ /λmin (a) R λmin (a) x∈
1 K1 2 a,E . (211) + ψ (x, y) dx dy λmin (a) x∈,y∈[0,1]d Using Cauchy-Schwartz inequality, Lemma 5.14 and ∇.σsym (a, P )∇ψ σsym (a,P ),K (x) = −1 − ∇.K(x)∇ψ σsym (a,P ),K (x), we obtain from Eq. (207), (117) and (128) that 1 Vol() 21 ψ a,E (x, y) dx dy 2 |J2 | ≤ Cd R −1 Cd,a+P ∞ /λmin (a) λmin (a) x∈,y∈[0,1]d
1 K1 2 . (212) + ψ σsym (a,P ),K (x) dx λmin (a) x∈ Proposition (5.16) is then a straightforward combination of (205), (210), (211) and (212). We will now need the following lemma whose proof is trivial algebra Lemma 5.17. Assume X, Y, δ, η > 0 and (X − Y )2 ≤ δXY + η(X + Y ), then 1
1
X 2 ≤ Y 2 (1 + and
√
δ) +
√
η
√ 1 √ 1 X 2 ≥ Y 2 − η (1 + δ)−1 .
(213)
(214)
(215)
Averaging Versus Chaos in Turbulent Transport?
589 √
Proof. The upper root of Eq. (213) is X0 = Y (1 + 2δ ) + η2 + 2 with = Y 2 δ(δ + √ 4) + Y η(8 + 2δ) + η2 . Then by applying the Minkowski inequality to we obtain X ≤ Y (1 +
√ √ √ √ 2 δ) + 2 Y η(1 + δ) + η
(216)
which leads to (214). Equation (215) is then obtained by the symmetry of (213) in X and Y . Combining Proposition 5.13 and 5.16 with Lemma 5.17 we obtain Theorem 5.18: Theorem 5.18. There exists a finite function h : (R+ )2 → R+ increasing in each argument such that the following inequalities are valid: X ≤ Y (1 + δ) + η and X ≥ Y − η (1 + δ)−1
(217)
with X :=
1 Vol()
ψ a,E (x, y) dx dy
1 4
,
(218)
x∈,y∈[0,1]d
1 Y := Vol()
ψ σsym (a,P ),K (x) dx
1 4
,
(219)
x∈
1 + a + P
1 1 1 2 ∞ δ := R − 2 h d, (() + 1) 2 1 + K1 , λmin (a)
(220)
1 + a + P
1 1 ∞ (() + 1) 2 . η := R − 2 h d, λmin (a)
(221)
and
5.4.3. Effect of relative translation on averaging. For a bounded open subset of Rd with smooth boundary and E a skew symmetric matrix with smooth coefficients in a,E a,E d L∞ loc (R ) and a ∈ Md,sym , let ψ (x) be the solution of ∇.(a + E)∇ψ = −1 in d . For y ∈ [0, 1] let us introduce the operator θy such that for any function f on Rd , θy f (x) = f (x + y). Using the notation (172), let us observe that for y ∈ [0, 1]d , a,θy/R (SR P )+K
ψ
(x) = ψ (x, y).
(222)
Lemma 5.19. For y ∈ Rd one has x∈
a,θy/R (SR P +K)
|∇ψ
(x) − ∇ψ (x, y)|2a ≤ Cd
K 2 1 a,E ψ (x, y) dx. Rλmin (a) x∈ (223)
590
H. Owhadi
Proof. Observe that a,θ (S P +K) a,E (x) − ψ (x, y) ∇.(a + E(x + y/R))∇ ψ y/R R a,E = ∇. K(x + y/R) − K(x) ∇ψ (x, y) .
(224)
It follows that a,θ (S P +K) a,E |∇ψ y/R R (x) − ∇ψ (x, y)|2a x∈ a,θ (S P +K) a,E a,E ∇ψ y/R R = (x) − ∇ψ (x, y) K(x + y/R) − K(x) ∇ψ (x, y), x∈
(225)
thus by Cauchy-Schwartz inequality a,θ (S P +K) a,E |∇ψ y/R R (x) − ∇ψ (x, y)|2a x∈ 2 a,E ≤ (K(x + y/R) − K(x))∇ψ (x, y) −1 , a
x∈
and Eq. (223) follows easily.
(226)
Now we will need the following lemma Lemma 5.20. For y ∈ [0, 1]d , a,θ (S P +K) ψ y/R R (x) dx ≤
K1 2 ψ (x, y) dx 1 + Cd Rλmin (a) x∈
(227)
K1 −2 ψ (x, y) dx 1 + Cd . Rλmin (a) x∈
(228)
x∈
and
x∈
a,θy/R (SR P +K)
ψ
(x) dx ≥
Proof. Combining the identity a,θ (S P +K) ψ y/R R (x) dx = x∈
a,θy/R (SR P +K)
x∈
|∇ψ
(x)|2a dx
(229)
with Minkowski inequality we obtain that
1
1 a,θ (S P +K) 2 2 a,E ψ y/R R (x) dx ≤ |∇ψ (x, y)|2a dx x∈
x∈
+
a,θy/R (SR P +K)
x∈
|∇ψ
(x) − ∇ψ (x, y)|2a
1 2
,
(230) and Eq. (227) follows from Lemma 5.19. The proof of inequality (228) is similar. We write −1 X(, a, E) := Vol()
x∈
a,E ψ (x) dx.
(231)
For y ∈ Rd we write θy := {x + y : x ∈ }. From Lemma 5.20 we obtain the following proposition
Averaging Versus Chaos in Turbulent Transport?
591
Proposition 5.21. K1 2 X(θ y , a, SR P + K) ≤ X(, a, θ y (SR P ) + K) 1 + Cd R R Rλmin (a)
(232)
K1 −2 . X(θ y , a, SR P + K) ≥ X(, a, θ y (SR P ) + K) 1 + Cd R R Rλmin (a)
(233)
and
5.4.4. Reverse iteration to obtain supper-diffusion. It is easy to obtain from Theorem (5.11), that for any γ > 0, X(, a, γ E) = γ −1 X(, γ −1 a, E).
(234)
Moreover for R > 0, writing SR := {x ∈ Rd : R −1 x ∈ } it is easy to obtain by scaling that X(SR , a, E) = R 2 X(, a, SR E).
(235)
Let us write for 0 ≤ p ≤ n − 1, Z(p, B) :=
n−1
1 1 4 X SRn θ yk Rk , B, p,∞ dyp . . . dyn−1 . γp Rn (yp ,... ,yn−1 )∈[0,1]d×(n−p) k=p
(236) We will need the following proposition Proposition 5.22. There exists a finite increasing function F : (R + )2 → R+ such that for ρmin > γmax Qn 1 +
Rn
2 Z(n, An )
one has
R 2 1 Z n, An
Z(0, A0 ) ≥ 0.5 and Z(0, A0 ) ≤ 2
n
Q γ 1 −n n max 2 ρmin
(238)
1+
Q γ 1 n n max 2 ρmin
(239)
1 2
Rn
R 2 1 Z n, An n
γn
4
1
Rn2
(237)
1+
4
γn
with 1 + K0 Qn := F d, + µn (() + 1)(1 + K1 ), − λn
(240)
n where λ− n is the stability of the renormalization core (A )n∈N and µn its anisotropic distortion.
592
H. Owhadi
Proof. From Proposition 5.21 and Eq. (235) we obtain that Z(p, B) ≤
X (yp ,... ,yn−1
)∈[0,1]d×(n−p)
n−1
θ yk Rk , B, S Rn θyp E p
k=p+1
Rn
Rp
1 1 γ1p SRn p+1,∞ 1 Rp 21 1 4 p+1,∞ 2 dyp . . . dyn−1 Rn 1 + Cd + SRn γp Rn λmin (B) (241) and Z(p, B) ≥
X (yp ,... ,yn−1
)∈[0,1]d×(n−p)
n−1
θ yk Rk , B, S Rn θyp E p
k=p+1
Rn
Rp
1 1 γ1p SRn p+1,∞ 1 Rp − 21 1 4 p+1,∞ 2 + SRn . dyp . . . dyn−1 Rn 1 + Cd γp Rn λmin (B) (242) Now let us observe that SRn
∞ 1 p+1,∞ Rn 1 ≤ K1 (γk /γp )(Rp+1 /Rk ) γp Rp+1 k=p+1
(243)
Rn γp+1 ≤ K1 (1 − γmax /ρmin )−1 . Rp+1 γp Combining (241) and (242) with Theorem 5.18 (with R = Rn /Rp , P = E p , K = SRn p+1,∞ /γp ) and (234) one obtains that Z(p, B) ≤
γ 1 γp 4 p Z p + 1, σsym (B, E p ) 1 + δp (B) + ηp (B) γp+1 γp+1
(244)
and Z(p, B) ≥
γ 1
−1 γp p 4 Z p + 1, σsym (B, E p ) − ηp (B) 1 + δp (B) (245) γp+1 γp+1
with δp (B) :=
Rp γp+1 1 1 + λmax (B) + K0 1 1 2 h d, (() + 1) 2 (1 + K1 ) 2 Rp+1 γp λmin (B) ×(1 − γmax /ρmin )−1
(246)
and 1 1 1 1 + λmax (B) + K0 ηp (B) := Rp2 h d, (() + 1) 2 (1 − γmax /ρmin )− 2 , λmin (B)
(247)
Averaging Versus Chaos in Turbulent Transport?
593
where, in (246) we have used the inequality (243) and we have integrated the error terms involving B appearing in (241) and (242) in the function h and used the assumption K1 γmax ≤ ρmin . Then one obtains from (244) by a simple induction that for n ≥ 2, Z(0, A0 ) ≤
γ 1 n−1 0 4 1 + δk (Ak ) Z n, An γn k=0
+
n−2 p=0
p γ 1 4 0 1 + δk (Ak ) ηp+1 (Ap+1 ) γp+1 k=0
+ η0 (A0 ).
(248)
Similarly one obtains from (245) by a simple induction that for n ≥ 2, Z(0, A0 ) ≥
γ 1 −1 n−1 0 4 Z n, An 1 + δk (Ak ) γn k=0
p γ 1 −1 4 0 1 + δk (Ak ) γp+1 p=0 k=0 −1 −η0 (A0 ) 1 + δ0 (A0 ) ,
−
n−2
ηp+1 (Ap+1 )
(249) k where A k∈N , is the renormalization core (33). Now combining (249) and (235) we obtain that R 2 1 Z n, An n−1
−1 γ
1 2 k+1 n 4 δ(Ak ) 1 + Z(0, A0 ) ≥ 1 γn rk+1 γk k=0 Rn2 p n−2 1 γ
1 γ 1
−1 4 2 0 k+1 p+1 2 1+ − Rp+1 δ(A ) δ(Ak ) γp+1 rk+1 γk p=0 k=0 −1 −δ(A0 ) 1 + δ(A0 ) (250) with
1 + λ (B) + K
1 1 max 0 δ(B) := h d, (() + 1) 2 (1 + K1 ) 2 (1 − γmax /ρmin )−1 . λmin (B) (251)
Thus
R 2 1 Z n, An n−1
Z(0, A0 ) ≥ (1 − ζi )
n
γn
4
1+
1
Rn2
k=0
γ
1
−1 2 k+1 δ(Ak ) rk+1 γk
(252)
with 1 n−2 R 2 γ 1 n−1
γ
1 Rn2 2 p+1 n 4 k+1 p+1 ζi ≤ δ(A ) δ(A ) 1 + k rk+1 γk Z n, An p=0 Rn2 γp+1 k=p+1 γ 1 n−1
γ
1 2 n 4 k+1 + 2 δ(Ak ) . (253) 1+ Rn rk+1 γk
k=0
594
H. Owhadi
Thus writing
1+K 1 1 0 + µn (() + 1) 2 (1 + K1 ) 2 (1 − γmax /ρmin )−1 , Wn := h d, − λn
(254)
2 >γ 2 we obtain from (253) under the assumption ρmin max (16 + Wn ) that 1
γ 1 Rn2 max 4 4Wn 2 ζi ≤ . Z n, An ρmin
(255)
Thus for 1 2
ρmin > γmax 64 1 + Wn
1
2 Rn2 , 1+ Z n, An
(256)
ζi < 0.5 and ζi acts as an error term in the inequality (252). Then combining (252), (256) and (254) we obtain the control (238). Moreover, we obtain from (235) and (248) that γ 1 Z n, An n−1
γ
1 2 0 4 k+1 Z(0, A0 ) ≤ Rn2 δ(A ) 1 + k 1 γn rk+1 γk k=0 Rn2 p n−2 1 γ
1 γ 1
4 2 0 k+1 2 1+ Rp+1 δ(Ap+1 ) δ(Ak ) + δ(A0 ). + γp+1 rk+1 γk p=0
k=0
(257) From this point the proof of Eq. (239) is similar to the one of Eq. (238)
We will need the following lemma Lemma 5.23. We have 1 1 Z(n, An ) ≤ Rn2 X(, An , 0) 4
(258)
1 1 − 1 Z(n, An ) ≥ Rn2 X(, Id , 0) 4 λmax (An ) 4 − 41 . × 1 + γn−2 sup λmax t n,∞ (x) n,∞ (x)
(259)
and
x∈SRn
Proof. Equations (258) and (259) are an easy application of Theorem 5.11, Corollary 5.12 and Eq. (235). For r > √ 0 we write T [r] the set of x ∈ Rd such that there exists y ∈ with |x − y| ≤ r √ d. We write T [−r] the set of x ∈ such that there exists y ∈ with −1 |x − y| > r d. From Eq. (236), using n−1 k=0 θ yk Rk ⊂ T [(ρmin − 1) ] we obtain that
Rn
1 X SRn T [(ρmin − 1)−1 ], A0 , 0,∞ ≥ Z(0, A0 ) 4 .
(260)
Averaging Versus Chaos in Turbulent Transport?
595
Similarly we obtain that 1 X SRn T [−(ρmin − 1)−1 ], A0 , 0,∞ ≤ Z(0, A0 ) 4 .
(261)
We will need the following proposition Proposition 5.24. There exists a finite increasing function F : (R + )2 → R+ such that for ρmin > γmax Qn and r > R1
(262)
one has Q γ 1 −4n Cd,K r2 n max 2 X B(0, r), A0 , 0,∞ ≥ 2 0 1+ γmax γn λmax (An ) ρmin
(263)
and X B(0, r), A0 , 0,∞ ≤
r2 γn λmax (An )
1+
Q γ 1 4n n max 2 ρmin
(264)
with n = sup{p ∈ N : Rp ≤ r},
(265)
1+K
0 + µ (1 + K1 ), Qn := F d, n λ− n
(266)
where λ− n is the stability of the renormalization core (An )n∈N and µn its anisotropic distortion. √ Proof. Taking := B 0, (r/Rn ) − d(ρmin − 1)−1 in Eq. (260) we obtain from Eq. (238) of Proposition 5.22 and (259) that Q γ 1 −4n Rn2 n max 2 X B(0, r/Rn ), Id , 0 1 + γn λmax (An ) ρmin −1 × 1 + γn−2 sup λmax t n,∞ (x) n,∞ (x) ,
X B(0, r), A0 , 0,∞ ≥(0.5)4
x∈B(0,r)
(267) which leads to (263) by (262), incorporating the new constants in Qn (observing that for r ≥ 1, (B(0, r)) is uniformly bounded away from infinity by a constant depending only on the dimension) and using γmax −1 r 2 sup γn−2 λmax t n,∞ (x) n,∞ (x) ≤ Cd K0 + γmax 1 − . ρmin Rn+1 x∈B(0,r) (268) √ The proof of (264) follows similarly by taking := B 0, (r/Rn ) + d(ρmin − 1)−1 in Eq. (261) and using Eq. (239) of Proposition 5.22 and (258).
596
H. Owhadi
Let us write n(r) := sup{p ∈ N : Rp ≤ r}.
(269)
From Proposition (5.24) we easily deduce the following theorem Theorem 5.25. There exists a finite increasing function F : (R + )2 → R+ and a function |C(r)| ≤ C(d, K0 , γmax ) such that for ρmin > Q(r) and r > R1 (270) γmax one has 1 Vol B(0, r)
Ex τ (r) dx =
B(0,r)
r 2−ν(r) λmax (An(r) )
(271)
with ν(r) =
C(r) ln γn(r) 1 + (r) + ln r ln r
(272)
and | (r)| ≤ with
Q(r)γ
max
1 2
(273)
ρmin
1+K
1 0 F d, − + µn(r) (1 + K1 ). 2 (ln γmin ) λn(r)
Q(r) :=
(274)
Then Theorem 4.13 is a simplified version of Theorem 5.25 (using Theorem 4.31). Now we will show that the anomalous fast behavior of the exit times from B(0, r) is a super-diffusive phenomenon and not a convective phenomenon. We will consider
Emr,l τ (r, l) defined by Eq. (66). The following theorem implies Theorem 4.15. Theorem 5.26. There exists a finite increasing function F : (R + )2 → R+ such that for ρmin > γmax 10Q(r) and r > R1
(275)
one has
−1 lim Vol B(0, r, l)
Ey,z τ B(0, r, l) =
l→∞
(y,z)∈B(0,r,l)
r 2−ν(r) λmax (An(r) )
(276)
where ν(r) is given by (272) and Q(r) by (274). Proof. Let us observe that ˆ B(0, r, l) ⊂ B(0, r, l) and
ˆ B(0, r, l) ⊂ B(0, r, l + r).
(277)
ˆ Thus, it is sufficient to control exit times from B(0, r, l) in order to prove Theorem 5.26. Now let us observe that the diffusion (yt , zt ) is associated to the following generator L acting on f ∈ C0∞ (Rd × Rd ): Lf (y, z) := ∇y .(κId + (y) − (z))∇y + ∇z .(κId + (z) − (y))∇z .
(278)
Averaging Versus Chaos in Turbulent Transport?
597
ˆ Thus one can apply Proposition 5.22 with = B(0, r, l). Let us observe that the renormalization core associated to (yt , zt ) is Ak 0 . (279) 0 Ak ˆ Moreover it is easy to observe that B(0, r, l) is bounded uniformly away from infinity on r ≤ l and that
n,∞
(y) −
n,∞
(z) ≤ Cd
∞
Cd K1 γk Rk−1 |y − z|.
(280)
k=n
From this point the proof of Theorem 5.26 is trivially similar to the one of Theorem 5.25. Super-diffusion as a common event. Let us write G(r) as the set of points of B(0, r) such that if yt starts from those points, its exit time from B(0, r) is anomalously fast with probability asymptotically close to one. We also write G(r, l) as the set of points of B(0, r, l) such that if (yt , zt ) starts for those points, their separation time is anomalously fast with probability asymptotically close to one. More precisely let us write δ(r) =
Cd,K0 ,γmax ln γn(r) 1 − 3C(r) − ln r ln r
(281)
with C(r) =
Q(r)γ
max
1 2
ρmin
(282)
,
where Q(r) is given by (274). Let us write
2 (r) = exp − ln γn(r) C(r) .
(283)
We will consider G(r) := x ∈ B(0, r) : Px τ (r) ≤
r 2−δ(r) (r) ≥ 1 −
2 λmax (An(r) )
and G(r, l) := (y, z) ∈ B(0, r, l) : Py,z τ B(0, r, l) ≤
r 2−δ(r) (r) . ≥ 1 −
2 λmax (An(r) )
Let us write mr , mr,l the Lebesgue probability measure defined on B(0, r) and B(0, r, l) by G(r) dx mr (G(r)) := , (284) B(0,r) dx (y,z)∈G(r,l) mr,l (G(r, l)) := (y,z)∈B(0,r,l)
dy dz dy dz
.
A trivial consequence of Theorems 5.25 and 5.26 is the following theorem.
(285)
598
H. Owhadi
Theorem 5.27. There exists a finite increasing function F : (R + )2 → R+ such that for ρmin > 10γmax Q(r) and r > R1 ,
(286)
mr (G(r)) ≥ 1 − 2 (r),
(287)
mr,l (G(r, l)) ≥ 1 − 2 (r).
(288)
Theorem 4.17 is a particular case of Theorem 5.27. Acknowledgements. Part of this work was supported by the Aly Kaufman fellowship. The author would like to thank F. Castell, Y. Velenik and G. Ben Arous for reading the manuscript and A. Majda, T. Hou and P.E. Dimotakis for useful discussions.
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Commun. Math. Phys. 247, 601–611 (2004) Digital Object Identifier (DOI) 10.1007/s00220-004-1072-0
Communications in
Mathematical Physics
On Depletion of the Vortex-Stretching Term in the 3D Navier-Stokes Equations Anastasia Ruzmaikina1 , Zoran Gruji´c2 1 2
Department of Statistics and Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA. E-mail: [email protected] Department of Mathematics, University of Virginia, Charlottesville, VA 22903, USA. E-mail: [email protected]
Received: 18 January 2003 / Accepted: 20 November 2003 Published online: 30 April 2004 – © Springer-Verlag 2004
Abstract: Certain new cancellation properties in the vortex-stretching term are detected leading to new geometric criteria for preventing finite-time blow-up in the 3D NavierStokes equations. 1. Introduction There is an extensive literature on formulating sufficient conditions for regularity of solutions of the 3D Navier-Stokes equations. The most classical ones are the space-time integrability conditions on the velocity u [P] 2q/(q−3)
u(·)q
∈ L1 (0, T )
for some q ∈ (3, ∞]. A localized version was given in [S] and a weak L3 -version can be found in [Ko]. There are analogous space-time integrability conditions for the vorticity ω = curl u – in particular the Beale-Kato-Majda ω(·)∞ ∈ L1 (0, T ) (originally derived for the 3D Euler equations [BKM]). This condition has been recently weakened [KoT] to ω(·)BMO ∈ L1 (0, T ) via a sharp BMO bilinear estimate. Constantin [C2] discovered an integral representation of the stretching factor in the evolution of the vorticity magnitude α revealing that local alignment of the vorticity directions, a geometric condition, depletes α. (It should be noted that coherent vortex structures, e.g., vortex sheets and vortex tubes, exhibit local alignment of the vorticity directions.) This phenomenon was then exploited in [CF] where it was shown that as
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long as the vorticity direction satisfies a Lipschitz-like regularity no blow-up can occur. More recently the Lipschitz condition was scaled down to a 1/2-H¨older-like regularity [BdVB]. In a work just completed, the authors formulated a more general sufficient condition for the regularity including the 1/2-H¨older condition and the Beale-Kato-Majda condition as the ‘end-point cases’ [GR]. A different geometric condition controlling the growth of the vorticity magnitude was presented in [G] – essentially, local existence of a sparse direction in the regions of high vorticity magnitude on the scales comparable to a localized vorticity version of the Kolmogorov dissipation scale. The proof relied on certain estimates on the complexified solutions in L∞ and a plurisubharmonic measure maximum principle in C3 , and a minimal scale was derived from a lower bound on the uniform radius of spatial analyticity. In this paper we study a representation of the L2 -product of the vortex stretching term with ω induced by the aforementioned integral representation of α. Rewriting the integral in a suitable form and utilizing some cancellation properties we first show that small scales in a convolution integral are harmless. More precisely, the convolution integral restricted to the balls Br , where r ≤ r ∗ is bounded by the (viscous) positive term appearing in the evolution of enstrophy (it is quite intriguing that a minimal scale r ∗ coincides with the minimal scale that appeared in [G] although the techniques are completely different). Since the convolution integral restricted to large scales can be controlled via known a priori estimates on ω we immediately see that if the regions of high fluid activity are comprised of sparsely populated small-scale structures no blowup can occur. Next, more cancellation properties are detected leading to a proof that a certain isotropy condition effectively controls the evolution of enstrophy preventing finite-time blow-up. This is interesting since the previous conditions – regularity, i.e. the alignment of the vorticity directions – are essentially anisotropic. 2. Preliminaries We consider the vorticity formulation of the 3D NSE, ωt − νω + (u · ∇)ω = (ω · ∇)u,
(1)
where u is the velocity of the fluid, div u = 0, ω = curl u is the vorticity and ν > 0 is the viscosity. The spatial domain will be the whole space R3 . The right-hand-side term (ω · ∇)u is the vortex-stretching term and is absent in the 2D case. The other nonlinear term is only virtually there – since u is divergence free, it vanishes when multiplied with ω|ω|2p−2 in any Lp , p ≥ 1. ω . The strain tensor (or the deformation Denote by ξ the vorticity direction, ξ = |ω|
tensor) S is given by S = 21 (∇u + ∇uT ), and a key quantity α is defined by α = Sξ · ξ . A direct calculation shows that the vortex-stretching term (ω · ∇)u is equal to Sω and that Sω · ω = α|ω|2 (and more generally Sω · ω|ω|2p−2 = α|ω|2p ). Hence α effectively controls the evolution of the enstrophy ω22 (and more generally any Lq -norm of ω, q ≥ 2). Constantin [C2] derived an integral representation of α which explicitly revealed geometric depletion of α. Denoting by yˆ the unit vector in the y-direction, dy 3 P .V . D(y, ˆ ξ(x + y), ξ(x))|ω(x + y)| 3 , α(x) = (2) 4π |y|
Depletion of Vortex-Stretching Term in 3D N-S Equations
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where the geometric factor D is proportional to the volume spanned by the unit vectors y, ˆ ξ(x + y), and ξ(x). More precisely, D(e1 , e2 , e3 ) = (e1 · e3 )Det(e1 , e2 , e3 )
(3)
for any triplet of unit vectors e1 , e2 , and e3 . It is easily seen that |D| ≤ | sin ϕ| ≤ |ξ(x + y) − ξ(x)| (ϕ denotes the angle between ξ(x + y) and ξ(x)) – hence regularity of ξ , i.e., local alignment of the vorticity directions, will deplete the α-singularity. Since geometric conditions will be assumed only in the regions of high fluid activity we introduce notation for the appropriate super-level sets of ω and curl ω. For M > 0, t > 0 define t (M) = {x ∈ : |ω(x, t)| ≥ M} and t (M) = {x ∈ : | curl ω(x, t)| ≥ M}. Utilizing the geometric depletion of α, Constantin and Fefferman [CF] proved that as long as the vorticity direction satisfies Lipschitz-like regularity no blow-up can occur. Theorem 1 ([CF]). Assume that there exist constants c, M > 0 such that | sin ϕ| ≤ c|y| for all x ∈ t (M), t ∈ (0, T ). Then limt↑T ω(t)2 < ∞. Following the approach of [CF], Beirao da Veiga and Berselli [BdVB] showed that 1/2-H¨older-like regularity of ξ suffices. Theorem 2 ([BdVB]). Assume that there exist constants c, M > 0 such that | sin ϕ| ≤ c|y|1/2 for all x ∈ t (M), t ∈ (0, T ). Then limt↑T ω(t)2 < ∞. The above theorems were proved controlling the enstrophy via the geometric depletion of α. The following theorem was obtained controlling Lq -norms (q ≥ 2) via the geometric depletion of α. It can be viewed as an interpolation result between the 1/2H¨older condition and the Beale-Kato-Majda condition. Theorem 3 ([GR]). Assume that there exist absolute constants c, M > 0 such that for some q ≥ 2, q/(q−1)
i) ωq ∈ L1 (0, T ), ii) | sin ϕ| ≤ c|y|1/q for all x ∈ t (M), t ∈ (0, T ). Then limt↑T ω(t)q < ∞.
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3. Small Scales and Large Scales In this section we obtain an explicit (and rigorous) manifestation of a basic principle of turbulence: viscosity dominates the nonlinearity on small scales. A minimal scale coincides with the one derived in [G] and is a localized ω-version of the Kolmogorov dissipation scale. Throughout the paper we consider solutions smooth (regular) on an open interval (0, T ) and are interested in a possible loss of regularity as t ↑ T . Local-in-time existence of smooth (in fact analytic in both space and time) solutions is guaranteed by, e.g., ω0 = ω(0) ∈ L2 . In addition, assuming u0 = u(0) ∈ L2 and ω0 ∈ L1 provides a num1/2 ber of useful a priori estimates [C1, FGT]: ω(·)1 ∈ L∞ (0, τ ), ω(·)∞ ∈ L1 (0, τ ) 4/(3+ ) and ∇ω(·)4/(3+ ) ∈ L1 (0, τ ) (any 0 < ≤ 21 ) for any τ > 0. Multiplying (1) by ω in L2 yields the following expression for the evolution of enstrophy 1 d (4) |ω|2 dx + ν |∇ω|2 dx = Sω · ω dx = α|ω|2 dx. 2 dt Utilizing the integral representation of α (2), the nonlinear term can be written as 1 dy dx. (5) I≡ (ω(x) · y)(ω(x ˆ + y) × ω(x) · y) ˆ |y|3 (The y-integral is to be understood in the P .V .-sense.) First, we derive some estimates on I restricted to small and large y-scales. For 0 < r < ∞ and 0 < r1 < r2 < ∞ define Ir , Ir C and Ir1 ,r2 by 1 (ω(x) · y)(ω(x ˆ + y) × ω(x) · y) ˆ dy dx, Ir = |y|3 |y|≤r 1 Ir C = (ω(x) · y)(ω(x ˆ + y) × ω(x) · y) ˆ dy dx, |y|3 |y|≥r and
Ir1 ,r2 =
r1 ≤|y|≤r2
(ω(x) · y)(ω(x ˆ + y) × ω(x) · y) ˆ
1 dy dx. |y|3
Remark 4. In the following lemmas and propositions ω is a smooth vector field for which the manipulations performed in a proof are legal and yield finite quantities. 1 ν 1/2 . Then Proposition 5 (Control on small scales). Let ν > 0 and r ≤ r ∗ ≡ √ 2 π ω1/2 ∞ ν |Ir | ≤ |∇ω|2 dx. 4 Proof. Using the properties of the cross product and changing some variables we obtain 1 (ω(x) · y)(ω(x ˆ + y) × ω(x) · y) ˆ dy dx Ir = 3 |y| |y|≤r 1 = ((ω(x) − ω(x + y)) · y)(ω(x ˆ + y) 2 |y|≤r 1 ×(ω(x) − ω(x + y)) · y) ˆ dy dx. (6) |y|3
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By the Fundamental Theorem of Calculus
1
ω(x) − ω(x + y) = −|y|
∇ω(x + sy)yˆ ds,
0
and thus 1 1 1 dydx ds1 ds2 (∇ω(x + s1 y)yˆ · y)(ω(x ˆ + y) × ∇ω(x + s2 y)yˆ · y) ˆ |y| 0 0 |y|≤r 1 1 1 ≤ ds1 |∇ω(x + s1 y)|2 ω∞ dydx ≤ π r 2 ω∞ |∇ω(z)|2 dz. (7) 2 |y| 0 |y|≤r
|Ir | =
1 2
The nonlinearity restricted to large scales can be bounded in terms of a priori estimates on ω. We will use the following simple inequality. Proposition 6 (Control on large scales). Let R > 0. Then |IR C | ≤
1 ω1 R3
|ω|2 dx.
Proof. 1 |IR C | ≤ 3 R
|ω(x)|
2
1 |ω(x + y)| dy dx = 3 ω1 R
|ω|2 dx.
Henceforth, for two positive functions f and g on (0, T ), f (t) ∼ g(t) means there 1 is a constant c > 1 such that g(t) ≤ f (t) ≤ cg(t) for all t in (0, T ). c Theorem 7. Let ω be a smooth solution of (1) on (0, T ) for some T > 0 corresponding 1/2 to the initial data satisfying u0 ∈ L2 and ω0 ∈ L1 ∩ L2 . Let r ∗ (t) = 1/2ν 1/2 , 2π
R ∗ (t) ∼
1 1/6 ω(t)∞
ω(t)∞
and M ∗ (t) such that M ∗ (t)(log M ∗ (t))3 ∼ ω(t)∞ , and assume 1/2
that for every t in (0, T ) the super-level set t (M ∗ (t)) is a union of small-scale vortex structures {Vβ (t)} (β in some index set) satisfying the following two properties: i) diamVβ (t) ≤ r ∗ (t), ii) dist(Vβ (t), Vγ (t)) ≥ R ∗ (t), β = γ . Then limt↑T ω(t)2 < ∞, i.e. T is not a singular time, and thus ω can be extended smoothly beyond T . Remark 8. The extension ω is actually analytic (both in space and time). Namely, ω(T )2 < ∞ and since ω(·)2 is equivalent to ∇ u(·)2 (by the CalderonZygmund Theorem), we can simply invoke local-in-time space-time analytic smoothing of the NSE given in [FT].
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Proof. Consider the enstrophy equation (4) 1 d |ω|2 dx + ν |∇ω|2 dx = Sω · ω dx = α|ω|2 dx = I 2 dt and write I = Ir ∗ + Ir ∗ ,R ∗ + I(R ∗ )C . The first term is bounded using Proposition 5 and the last term is bounded using Proposition 6 in conjunction with the fact that ω1 is estimated in terms of the initial data uniformly in time [C1]. The x-integral in the middle term is then split in an integral over t (M ∗ ) and an integral over (t (M ∗ ))C . The first integral is 0 (utilizing i) and ii)) and the second is bounded in the same way as I over (t (M ∗ ))C (using the fact that M ∗ (logM ∗ )3 = ω1/2 ∈ L1 , see [GR]). Inserting these bounds in the enstrophy equation yields d |ω|2 dx ≤ c0,ν M ∗ (log M ∗ )3 |ω|2 dx dt on (0, T ). Hence limt↑T ω(t)2 < ∞ and we can extend ω to (0, T ] – denote the extension by ω. Since ω(T )2 < ∞, local-in-time well-posedness of (1) in L2 yields the (smooth) extension beyond T .
If we could handle the sub-level sets ((M ∗ ))C for M ∗ larger than in Theorem 7, we could take smaller R ∗ . That follows from the following simple estimate. Proposition 9. Let x be restricted to ( 1c ωδ∞ ) for some δ in (0, 1] and some constant −δ/3 c > 1. Then for R ≥ ω∞ , |IR C | ≤ cω22 |ω|2 dx. Proof. |ω(x + y)|2 |ω(x)|2 dy dx 3 |y|≥R R |ω(x + y)| ≤ c |ω(x)|2 |ω(x + y)|2 dy dx |y|≥R ≤ cω22 |ω|2 dx.
|IR C | ≤
In particular, Proposition 9 implies that if (M ∗ ) ⊆ ( 1c ω∞ ), we can take R ∗ = −1/3 ω∞ in Theorem 7. The following theorem will be proved in the last section. Theorem 10. Let ω be a smooth solution of (1) on (0, T ) for some T > 0 correspond1/2 ing to the initial data satisfying u0 ∈ L2 and ω0 ∈ L1 ∩ L2 . Let r ∗ (t) ∼ ν 1/2 , ω(t)∞
M1∗ (t) ∼ ω(t)2 and M2∗ (t) ∼ curl ω(t)4/(3+ ) for some 0 < ≤ 21 , and assume that for every t in (0, T ) a region of high fluid activity t (M1∗ (t)) ∩ t (M2∗ (t)) consists of small-scale structures {Wβ (t)} (β in some index set) satisfying the following two properties: 4/(3+ )
Depletion of Vortex-Stretching Term in 3D N-S Equations
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i) diamWβ (t) ≤ 1c r ∗ (t), ii) dist(Wβ (t), Wγ (t)) ≥ c, β = γ , for an appropriate constant c > 2. Then limt↑T ω(t)2 < ∞, i.e. T is not a singular time, and thus ω can be extended smoothly beyond T . Remark 11. A sparseness condition here is imposed only in a region where both ω and curl ω are large. 4. Isotropy A special case of anisotropy, i.e. 1/2-H¨older-like regularity of the vorticity direction suffices to prevent finite-time blow-up. It will be shown in this section that a certain isotropy condition controls the evolution of enstrophy. We start by introducing another piece of notation. For 0 < r < ∞ and 0 < r1 < r2 < ∞ define Jr , Jr C , Jr1 ,r2 and Sr by 1 Jr = (ω(x) · y)( ˆ curl ω(x + y) · ω(x)) 2 dy dx, |y| |y|≤r 1 Jr C = (ω(x) · y)( ˆ curl ω(x + y) · ω(x)) 2 dy dx, |y| |y|≥r 1 Jr1 ,r2 = (ω(x) · y)( ˆ curl ω(x + y) · ω(x)) 2 dy dx, |y| r1 ≤|y|≤r2 and
Sr =
|y|=r
(ω(x) · y)(ω(x ˆ + y) × ω(x) · y) ˆ
1 dSy dx. |y|2
Lemma 12. Let 0 < r1 < r2 < ∞. Then 1 1 1 Jr ,r − Sr + Sr . 3 1 2 3 2 3 1 Proof. Computing the y-surface integrals via the Divergence Theorem gives 1 (ω(x) · y)(ω(x ˆ + y) × ω(x) · y) ˆ − dSy 2 |y| |y|=r2 |y|=r1 ω(x) · y = − ω(x + y) × ω(x) · yˆ dSy |y|3 |y|=r2 |y|=r1 ω(x) · y = div (ω(x + y) × ω(x)) |y|3 r1 ≤|y|≤r2 ω(x) · y +(ω(x + y) × ω(x)) · ∇ dy |y|3 ω(x) · yˆ ω(x + y) × ω(x) · ω(x) = ( curl ω(x + y) · ω(x)) + 2 |y| |y|3 r1 ≤|y|≤r2 (ω(x) · y)(ω(x ˆ + y) × ω(x) · y) ˆ dy. −3 |y|3 Ir1 ,r2 =
Noting that the second term in the last integral is zero, and integrating in x yields the desired identity.
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Control of the surface terms on integral and small scales is given in the following two lemmas. Lemma 13. Let 0 < ≤ 21 . Then 4/(3+ ) |ω|2 dx. |S1 | ≤ c 1 + ω1 + curl ω4/(3+ ) Proof. The following manipulations are elementary.
1 (ω(x) · y)(ω(x ˆ + y) × ω(x) · y) ˆ dSy dx |y|2 |y|=1 = − ω(x) · ˆ dSy dx ((ω(x) · y)ω(x + y)) × y) |y|=1 = ω(x) · curl ((ω(x) · y)ω(x + y)) dy dx |y|≤1 = ω(x) · ((ω(x) · y) curl ω(x + y) − ω(x + y) × ∇(ω(x) · y)) dy dx |y|≤1 = ω(x) · ((ω(x) · y) curl ω(x + y) − ω(x + y) × ω(x)) dy dx |y|≤1 2 2 ≤ |ω(x)| | curl ω(x + y)| dy dx + |ω(x)| |ω(x + y)| dy dx. |y|≤1
|y|≤1
Lemma 14. Let ν > 0 and r ≤
r∗ 2.
Then
|Sr | ≤
ν 4
|∇ω|2 dx.
Proof. As in the proof of Proposition 5 we can show that Sr = 1 = 2
|y|=r
(ω(x) · y)(ω(x ˆ + y) × ω(x) · y) ˆ
|y|=r
1 dSy dx |y|2
((ω(x) − ω(x + y)) · y)(ω(x ˆ + y)
×(ω(x) − ω(x + y)) · y) ˆ
1 dSy dx, |y|2
(8)
and thus 1 1 |Sr | = ds1 ds2 (∇ω(x + s1 y)yˆ · y)(ω(x ˆ + y) × ∇ω(x + s2 y)yˆ · y)dS ˆ y dx 0 0 |y|=r 1 ≤ ds1 |∇ω(x + s1 y)|2 ω∞ dSy dx ≤ 4πr 2 ω∞ |∇ω(z)|2 dz. (9) 0
|y|=r
Depletion of Vortex-Stretching Term in 3D N-S Equations
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For x in (M ∗ ) consider an orthonormal triplet {e1 , e2 , e3 }, where e1 = ξ(x), and define the fluxes of the orthogonal projections of curl ω by Iri (x) = [ curl ω(x + y) · ei )ei ] · yˆ dSy |y|=r
for r > 0, i = 1, 2, 3. Then we have the following theorem. Theorem 15. Let ω be a smooth solution of (1) on (0, T ) for some T > 0 corresponding to the initial data satisfying u0 ∈ L2 and ω0 ∈ L1 ∩ L2 . Assume that for every t in (0, T ), every x in t (M ∗ (t)), and all r ∗ (t) ≤ r(t) ≤ 1, j
i |Ir(t) (x) − Ir(t) (x)| ≤ c
for some absolute constant c. Then limt↑T ω(t)2 < ∞, and thus ω can be extended smoothly beyond T . Proof. Similarly as in the proof of Theorem 7 decompose I as I = Ir ∗ + Ir ∗ ,1 + I1C . Hence only the restriction of the middle term to t (M ∗ ) needs to be estimated. Observing that Lemma 12 holds on any x-subset of R3 , write Ir∗∗ ,1 = 13 Jr∗∗ ,1 − 13 S1∗ + 13 Sr∗∗ , where a superscript ∗ denotes a restriction of the corresponding integral to t (M ∗ ). Since S1∗ ≤ S1 , Sr∗∗ ≤ Sr ∗ and they can be bounded in the same way as S1 and Sr ∗ (cf. Lemmas 13 and 14), we are left with Jr∗∗ ,1 which can be written as t (M ∗ )
|ω(x)|
2
1 r∗
1 ρ2
|y|=ρ
[( curl ω(x + y) · ξ(x))ξ(x)] · yˆ dSy dρ dx.
(10)
Notice that by the choice of e1 the surface integral is equal to I 1 . We claim that I 1 + I 2 + I 3 = 0. Observe that curl ω(x + y) · yˆ dSy = div curl ω(x + y) dy = 0. |y|=ρ
|y|≤ρ
Expanding yˆ as yˆ = (e1 · y)e ˆ 1 + (e2 · y)e ˆ 2 + (e3 · y)e ˆ 3 yields the claim. Combining the cancellation relation I 1 + I 2 + I 3 = 0 with the isotropy assumption |I i − I j | ≤ c gives |I 1 | ≤ c. Inserting this in (10) finishes the proof.
Remark 16. It is worth noticing that curl ω = −u, and since the Laplacian is a rotationally invariant operator, the condition in the theorem can be viewed as an isotropy condition on the velocity field u. 5. Proof of Theorem 10 Proof. Consider the enstrophy equation (4) 1 d |ω|2 dx + ν |∇ω|2 dx = Sω · ω dx = α|ω|2 dx = I 2 dt and write |I | ≤ |Ir ∗ | + |Sr ∗ | + |Jr ∗ ,1 | + |S1 | + |I1C |. As in the previous proofs, only Jr ∗ ,1 remains to be estimated.
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To this effect, decompose both ω and curl ω in the low and high magnitude parts. Let A be a smooth cut-off function over a set A. We write ω(t) = ω1 (t) + ω2 (t), where ω1 (t) = (1 − t (M1∗ (t)) )ω(t) and ω2 (t) = t (M1∗ (t)) ω(t), and similarly, curl ω(t) = ( curl ω)1 (t) + ( curl ω)2 (t), where ( curl ω)1 (t) = (1 − t (M2∗ (t)) ) curl ω(t) and ( curl ω)2 (t) = t (M2∗ (t)) curl ω(t). Then
1 | curl ω(x + y)| |ω(x)|2 dy dx |y|2 r∗ 1 |( curl ω)1 (x + y)| dy dx ≤2 |ω1 (x)|2 |y|2 r∗ 1 |( curl ω)2 (x + y)| dy dx + |ω1 (x)|2 ∗ |y|2 r 1 |( curl ω)1 (x + y)| 2 dy dx + |ω2 (x)| ∗ |y|2 r 1 |( curl ω)2 (x + y)| 2 dy dx + |ω2 (x)| |y|2 r∗ = 2 J 1,1 + J 1,2 + J 2,1 + J 2,2 .
|Jr ∗ ,1 | ≤
For J 1,1 and J 1,2 the integral is bounded (via H¨older and Hardy-Littlewood-Sobolev inequalities) by | curl ω(x + y)| (M1∗ )1/3 |ω(x)|5/3 dy dx |y|2 5/6 ∗ 1/3 2 ≤ c(M1 ) |ω| dx curl ω2 5/3 c ν |ω|2 dx + (M1∗ )2/3 |∇ω|2 dx ν 2 c ν ≤ ω22 |ω|2 dx + |∇ω|2 dx. ν 4
≤
For J 2,1 the integral is bounded by 4/(3+ ) M2∗ (1 − r ∗ ) |ω|2 dx ≤ c curl ω4/(3+ ) |ω|2 dx. Finally, for J 2,2 , we are in a region of high fluid activity and according to our geometric assumptions i) and ii) the integral is equal to 0. Collecting all the estimates, 1 4/(3+ ) I ≤ c 1 + ω1 + ω22 + curl ω4/(3+ ) |ω|2 dx + ν |∇ω|2 dx. ν Since the expression in the parantheses is in L1 (0, T ), inserting this inequality in the enstrophy equation finishes the proof.
Depletion of Vortex-Stretching Term in 3D N-S Equations
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Remark 17. For simplicity of the exposition the cut-off for the large scales in the proofs was taken to be O(1). The same line of reasoning applies for a cut-off of the order of R(t), where R(t) satisfies 1 R(·)
9+3
1−
∈ L1 (0, T )
for some 0 < ≤ 21 . Acknowledgements. We thank the referee for a number of comments and suggestions that led to the present version of the paper.
References [BKM] Beale, J.T., Kato, T., Majda, A.: Remarks on the breakdown of smooth solutions for the 3-D Euler equations. Commun. Math. Phys. 94, 61–66 (1984) [BdVB] Beirao da Veiga, H., Berselli, L.C.: On the regularizing effect of the vorticity direction in incompressible viscous flows. Diff. Int. Eq. 15, 345–356 (2002) [C1] Constantin, P.: Navier-Stokes equations and area of interfaces. Commun. Math. Phys. 129, 241–266 (1990) [C2] Constantin, P.: Geometric statistics in turbulence. SIAM Rev. 36, 73–98 (1994) [CF] Constantin, P., Fefferman, C.: Direction of vorticity and the problem of global regularity for the Navier-Stokes equations. Indiana Univ. Math. J. 42, 775–789 (1993) [FGT] Foias, C., Guillope, C., Temam, R.: New a priori estimates for Navier-Stokes equations in dimension 3. Commun. PDE 6, 329–359 (1981) [FT] Foias, C., Temam, R.: Gevrey class regularity for the solutions of the Navier-Stokes equations. J. Funct. Anal. 87, 359–369 (1989) [G] Gruji´c, Z.: The geometric structure of the super-level sets and regularity for 3D Navier-Stokes equations. Indiana Univ. Math. J. 50, 1309–1317 (2001) [GR] Gruji´c, Z., Ruzmaikina, A.: Interpolation between algebraic and geometric conditions for smoothness of the vorticity in the 3D NSE. Indiana Univ. Math. J., to appear [Ko] Kozono, H.: Removable singularities of weak solutions to the Navier-Stokes equations. Commun. PDE 23, 949–966 (1998) [KoT] Kozono, H., Taniuchi, Y.: Bilinear estimates in BMO and the Navier-Stokes equations. Math. Z. 235, 173–194 (2000) [P] Prodi, G.: Un teorema de unicita per le equazioni di Navier-Stokes. Annali di Mat. 48, 173–182 (1959) [S] Serrin, J.: On the interior regularity of weak solutions of the Navier-Stokes equations. Arch. Rat. Mech. Anal. 9, 187–195 (1962) Communicated by P. Constantin
Commun. Math. Phys. 247, 613–654 (2004) Digital Object Identifier (DOI) 10.1007/s00220-004-1073-z
Communications in
Mathematical Physics
Phase Turbulence in the Complex Ginzburg-Landau Equation via Kuramoto–Sivashinsky Phase Dynamics Guillaume van Baalen D´epartement de Physique Th´eorique, Universit´e de Gen`eve, Switzerland. E-mail: [email protected] Received: 10 February 2003 / Accepted: 12 November 2003 Published online: 30 April 2004 – © Springer-Verlag 2004
Abstract: We study the Complex Ginzburg-Landau initial value problem ∂t u = (1 + iα) ∂x2 u + u − (1 + iβ) u |u|2 ,
u(x, 0) = u0 (x) ,
(CGL)
for a complex field u ∈ C, with α, β ∈ R. We consider the Benjamin–Feir linear instability αβ = −ε2 with ε 1 and α 2 < 1/2. We show that for all √ region 1 +−32/37 2 ε ≤ O( 1 − 2α L0 ), and for all initial data u0 sufficiently close to 1 (up to a global phase factor ei φ0 , φ0 ∈ R) in the appropriate space, there exists a unique (spatially) periodic solution of space period L0 . These solutions are small even perturbations of the traveling wave solution, u = (1 + α 2 s) ei φ0 −iβ t eiα η , and s, η have bounded norms in various Lp and Sobolev spaces. We prove that s ≈ − 21 η apart from O(ε 2 ) corrections whenever the initial data satisfy this condition, and that in the linear instabil−32/37 ity range L−1 ), the dynamics is essentially determined by the motion 0 ≤ ε ≤ O(L0 of the phase alone, and so exhibits ‘phase turbulence’. Indeed, we prove that the phase η satisfies the Kuramoto–Sivashinsky equation 1+α 2
∂t η = −
2
−32/5
for times t0 ≤ O(ε −52/5 L0
2 η − ε 2 η − (1 + α 2 ) (η )2
(KS)
), while the amplitude 1 + α 2 s is essentially constant.
Contents 1. 2. 3. 4.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . The Phase Equation . . . . . . . . . . . . . . . . . . . . . . . The Amplitude Equation . . . . . . . . . . . . . . . . . . . . The Condition 2.11, Properties of µ → F (µ) and µ → r2 (µ) Supported in part by the Fonds National Suisse.
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A. B. C. D. E. F. G.
G. van Baalen
Coercive Functional for the Phase . . . . . . . . . . . . . . . . . . . Properties of the Spaces Wσ . . . . . . . . . . . . . . . . . . . . . . Bounds on Nonlinear Terms . . . . . . . . . . . . . . . . . . . . . . Proof of Proposition 2.12 . . . . . . . . . . . . . . . . . . . . . . . . Further Properties of the Amplitude Equation . . . . . . . . . . . . . Coercive Functionals and Other Properties for the Amplitude Equation Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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638 642 645 650 651 652 654
1. Introduction 1.1. Generalities about the Ginzburg-Landau equation. The Complex GinzburgLandau equation (CGL) admits explicit traveling wave solutions of the form u(x, t) = c(p) exp (i(φ0 + p x − ω(p) t)) ,
(1.1)
with φ0 ∈ R, p ∈ [−1, 1], c(p) = 1 − p 2 and ω(p) = α p2 + β (1 − p 2 ). For all α, β with 1 + α β > 0, there exists a parameter pE = pE (α, β), with pE → 0 as 1 + α β → 0+ such that traveling wave solutions (1.1) with |p| ≥ pE (α, β) are linearly unstable, a phenomenon called ‘sideband’ or ‘Eckhaus’ instability, while those with |p| ≤ pE are linearly stable (see e.g. [CH93] and the references therein). When 1 + αβ < 0, all traveling wave solutions are linearly unstable, a phenomenon called ‘Benjamin–Feir’ or ‘Benjamin–Feir–Newell’ instability (see e.g. [BF67] and [New74]). In this paper, we consider the case 1+α β = −ε2 . When ε is small enough, numerical simulations on finite domains (see e.g. [MHAM97] and the references therein) indicate that the dynamics of the phase is turbulent, the phase evolving irregularly, (with fluctuations of order ε2 around the global phase φ0 ), while the amplitude of u is constant up to O(ε4 ) corrections. This type of behavior is called ‘phase turbulence’. The persistence of phase turbulence on infinite domains is not known, while its existence on finite domains is, to our knowledge, not proven rigorously. As ε increases (or the domain is larger), ‘amplitude’ or ‘defect’ turbulence occurs, the amplitude of u vanishing at some instants and places, called ‘defects’ or ‘phase slips’ (see also [EGW95]). Note that ‘phase’ and ‘amplitude’ turbulence may coexist at the same time in the α, β parameter space, depending on initial conditions, in which case one speaks of ‘bichaos’. The ‘amplitude’ turbulence regime is technically difficult because the phase is not well defined when the amplitude vanishes. In this paper, we concentrate on the easier phase turbulence regime and prove that for the particular case1 p = 0, phase turbulence occurs for small initial perturbations of the traveling wave ei φ0 −iβ t on domains of size L0 for all α 2 < 1/2 and for all ε ≤ ε0 (L0 , α) with ε0 (L0 , α) → 0 as L0 → ∞ or α 2 → 1/2, see Fig. 1.1. We restrict ourselves to even perturbations for concision, though general perturbations could be treated as well (see Remark 2.4 below). We believe that the restriction α 2 < 1/2 could be weakened to some extent (see the discussion at the end of Sect. 1.4), at the price of unwanted additional technical difficulties. 1
The case p = 0 should give a similar result but is more challenging.
Phase Turbulence in the Complex Ginzburg-Landau Equation
615
√β 2 1 + α β = −ε0 (L0 , α)2
1+α β =0 √ 2α
1 −1
1
−1
1 + α β = −ε0 (L0 , α)2
Fig. 1.1. Parameter space for (CGL). Linear instability occurs for 1 + α β < 0, and phase turbulence is shown in this paper to occur in shaded region
1.2. Setting. We consider perturbations of the solution ei φ0 −iβ t of (CGL) which are of the form2 u(x, t) = (1 + α 2 s(x, t)) ei φ0 −iβt eiα η(x,t) ,
(1.2)
for (small) s, η ∈ R. To state our results, we introduce the following scalings3 εˆ 2 η( ˆ x, ˆ tˆ) , 4 s(x, t) = εˆ sˆ (x, ˆ tˆ) ,
η(x, t) =
1 4
(1.3) (1.4)
4 with χ = 1+α ˆ = χ2 ε, xˆ = εˆ x and tˆ = χ2 εˆ 4 t. 2, ε We consider the initial value problem (CGL) with η(x, 0) = η0 (x) and s(x, 0) = s0 (x), where η0 and s0 are even periodic functions of period L0 , or equivalently, in terms of the ‘hat’ variables, ηˆ 0 and sˆ0 are even periodic functions of period L = εˆ L0 . To state our conditions on the initial data sˆ0 and ηˆ 0 , we introduce the Banach space W0,σ obtained ∞ ([−L/2, L/2], R) under the norm · = · by completing Cper σ L2 ([−L/2,L/2]) + · W ,σ , where · W ,σ is a sup norm with algebraic weight (going like |k|σ at infinity) on the Fourier transform, see Sect. 2.3 for details. Essentially W0,σ consists of functions in L2 ([−L/2, L/2], R), whose Fourier transform decays (at least) like |k|−σ as |k| → ∞ 2 3
The α factors in front of s and η are only a convenient normalization. They will be justified in the next subsection.
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(this is a regularity assumption). Since we consider only real valued functions, we will from now on write L2 ([−L/2, L/2]) instead of L2 ([−L/2, L/2], R). We will also often use the shorthand notation L2 for L2 ([−L/2, L/2]), while we will always write L2 ([−L0 /2, L0 /2]) to avoid confusion. We postpone the precise definition of the class C of admissible initial conditions to the end of Sect. 2.3 (see Definition 2.8). At this point, we will only say that if ηˆ 0 and sˆ0 are admissible initial conditions, then ηˆ 0 ∈ W0,σ and sˆ0 ∈ W0,σ −1 , and ηˆ 0 (0) = 0 ,
εˆ 2 sˆ0 ≤ cs0 ρ 3 , ˆs0 − 2 σ −1
ηˆ 0 σ ≤ cη0 ρ ,
(1.5)
for ρ = K L8/5 , and εˆ 2 sˆ0 ηˆ 0 εˆ 2 (ηˆ 0 )2 + + sˆ0 − 2 8 32
≤ λ2,0
L2
εˆ 2 ε 2 cs0 ρ 3 . εˆ 0
(1.6)
The class C of admissible initial conditions is characterized by the different parameters in (1.5) and (1.6), which we now describe. The parameter L is the (space) period (in the scaled variables) of the solution. The constant K is essentially the same as that of [CEES93] in their discussion of the Kuramoto–Sivashinsky equation, 1 ∂tˆηˆ c = −2 ηˆ c − ηˆ c − (ηˆ c )2 , 2
(1.7)
where it appears in the bound lim ηˆ c (·, t) L2 ≤ K L8/5 for symmetric periodic solut→∞
tions. Therefore, K is independent of α, ε and L. The parameters α and εˆ are those of β (CGL), with εˆ 2 = −2 1+α , while εˆ 0 is the maximal value of εˆ for which our results 1+α 2 hold. The parameters cη0 and cs0 measure the size of the initial perturbation. Note that only ηˆ 0 and ηˆ 0 (0) appear in the conditions. We can motivate this by noting that (CGL) has a U (1) symmetry (the global phase factor ei φ0 ). Expressing all constraints in terms of ηˆ 0 and ηˆ 0 (0) is a convenient way to take this invariance into account. The condition η0 (0) = 0 can always be satisfied, up to a redefinition of the global phase φ0 . Furthermore, this condition is preserved by the evolution (see e.g. (1.17)). We will prove that if ηˆ 0 and sˆ0 are in the class C, the (CGL) dynamics (which has a complex function as initial condition) is increasingly well approximated as εˆ → 0 by the Kuramoto–Sivashinsky dynamics (1.7), which has a real function as initial condition. For this to hold, sˆ0 and ηˆ 0 have to be tightly related as εˆ → 0. This relation is quantified by (1.6), which says that, up to O(ε 4 ) corrections, sˆ0 and ηˆ 0 are related by 1 ˆ εˆ 2 ˆ 2 ηˆ 0 − G (ηˆ 0 ) , sˆ0 = − G 8 32 ˆ is the operator with symbol G(k) ˆ where G = (1 + (positive) operator 1 −
εˆ 2 2
∂x2ˆ .
εˆ 2 2
k 2 )−1 , i.e. the inverse of the
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617
1.3. Main results and their physical discussion. Our main results are twofold. We first have an existence and uniqueness result for the solutions of (CGL), see Theorem 1.1 below, and then an approximation result in Theorem 1.2. From now on, we will denote generic constants by the letters C and c. We will use the letter c with different labels to recall the quantity on which the bound is. By constants, we mean quantities which do not depend on α, εˆ , L and σ in the ranges 0 ≤ εˆ ≤ 1 , α 2 < 1/2 , L > 2π for some finite σ0 >
and σ ≤ σ0
11 2 .
2 1−2 α 2 Theorem 1.1. Let α 2 < 1/2, σ > 11 2 , cs0 > 0, cη0 > 0, λ1,0 < min 3 , 1−α 2 , λ2,0 > 0 and√ L > 2 π. There exist constants K and cε such that for all mε ≥ 4, for any εˆ ≤ εˆ 0 = cε 1 − 2α 2 ρ −mε and for all ηˆ 0 and sˆ0 in the class C, the solution of (CGL) 2 ) εˆ 2 with parameters α and β = − 2+(1+α exists for all times, is of the form (1.2) and 2α satisfies ˆ tˆ) σ ≤ cη ρ , sup η(·,
sup ˆs (·, tˆ) σ −1 ≤ cs ρ 3 ,
(1.8)
εˆ 4 1 ˆ εˆ 2 ˆ 2 ˆ ˆ ˆ ≤ G (ηˆ ) (·, t ) cη ρ , sup sˆ (·, t ) + G ηˆ (·, t ) + 8 32 εˆ 0 L2 tˆ≥0
(1.9)
tˆ≥0
tˆ≥0
with ρ = K L8/5 , cη > 1 + cη0 and cs > cs0 . This solution is unique among functions satisfying (1.8). √ Our results are valid for any εˆ ≤ εˆ 0 = cε 1 − 2α 2 ρ −4 and for any L > 2 π . Since L = εˆ L0 and ρ = K L8/5 , we see that the applicability range is −32/37 C L−1 ≤ ε ≤ C 1 − 2α 2 L0 . 0 The lower bound is the linear instability condition. In terms of the original variables, Theorem 1.1 shows that solutions of (CGL) of the form (1.2) exist, and that (see Appendix G for details) sup η(·, t) L2 ([−L0 /2,L0 /2]) ≤ C ε 5/2−1/mε ,
(1.10)
sup s(·, t) L2 ([−L0 /2,L0 /2]) ≤ C ε 7/2−3/mε ,
(1.11)
t≥0
t≥0
sup
sup
|η(x, t)| ≤ C ε 2−13/(8 mε ) ,
(1.12)
|s(x, t)| ≤ C ε4−4/mε .
(1.13)
t≥0 x∈[−L0 /2,L0 /2]
sup
sup
t≥0 x∈[−L0 /2,L0 /2]
The inequalities (1.12) and (1.13) quantify the ‘physical intuition’ η = O(ε 2 ) and s = O(ε4 ), see Sect. 1.1. Inequalities (1.8) or (1.10)–(1.13) also show that the solutions belong to a (local) attractor, while (1.9) shows that on that attractor, the ‘amplitude’ s satisfies s = − 21 η + O(ε2 ). The attractor is thus well approximated by the graph s = − 21 η in the s, η space. This result was discovered at a heuristic level by Kuramoto and Tsuzuki in [KT76].
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G. van Baalen
4
3
ηˆ (·, tˆ) L2 2
1
ˆs (·, tˆ) L2
0
200 tˆ
100
εˆ 2 1
0
100
200
ηˆ (·,tˆ)−ηˆ c (·,tˆ) L2
ηˆ (·,tˆ) L2 0 1 Fig. 1.2. Numerical results for εˆ =
2 10−3 ,
α=
3
4 tˆ
10−2
and L0 = 104 · 2π
We do not expect the bounds (1.8) and (1.9) to be optimal. Numerical simulations show that ηˆ and sˆ are uniformly bounded in space and time, at least for a large range of √ L = ε L0 . This suggests that ηˆ L2 and ˆs L2 should both scale with L like L and √ not like L8/5 and L24/5 , hence we should have ρ ∼ L. In the upper panel of Fig. 1.2, we display as a function of tˆ ∈ [0, 200] (by decreasing size) the typical behavior of ˆ ηˆ (·, tˆ) + εˆ 2 G ˆ (ηˆ )2 (·, tˆ) L2 in units
ηˆ (·, tˆ) L2 , ˆs (·, tˆ) L2 and εˆ −4 ˆs (·, tˆ) + 18 G 32 √ proportional to εL0 . We now show that the dynamics of the phase on the attractor is well approximated by the Kuramoto–Sivashinsky equation.
Theorem 1.2. Under the assumptions of Theorem 1.1, there exists a constant ct such that if tˆ1 ≤ ct ρ −4 , then for all tˆ0 ≥ 0,
Phase Turbulence in the Complex Ginzburg-Landau Equation
sup η(·, ˆ tˆ0 + tˆ) − ηˆ c (·, tˆ) L2 ≤
0≤tˆ≤tˆ1
619
εˆ 2 cη ρ , εˆ 0
(1.14)
where ηˆ c satisfies the Kuramoto–Sivashinsky equation (1.7) with ηˆ c (x, ˆ 0) = η( ˆ x, ˆ tˆ0 ). In physical terms, Theorem 1.2 says that on each time interval [t0 , t0 +t1 ], the distance between η and the solution of the Kuramoto–Sivashinsky equation with initial condition η(t0 ) is small compared to the size of the attractor (see (1.14)), at least for time intervals −32/5 of length t1 of order ε −4 ρ −4 = ε−52/5 L0 . This result gives a rigorous foundation to the heuristic derivation in [KT76]) of the Kuramoto–Sivashinsky equation as a phase equation for the Complex Ginzburg-Landau equation near the Benjamin–Feir line (see also [Man90]). Furthermore, if ε is sufficiently small, the amplitude 1 + α 2 s does not vanish by (1.13). This proves that the solution exhibits phase turbulence for all times, the solutions of the Kuramoto–Sivashinsky equation being believed to be chaotic. The bound (1.14) for tˆ1 ≤ ct ρ −4 is again certainly not optimal.√Numerical simulations show that tˆ1 scales like L−2 (this is in agreement with ρ ∼ L). In the lower panel of Fig. 1.2, we show in the large plot
ηˆ (·,tˆ)−ηˆ c (·,tˆ) L2
ηˆ (·,tˆ) L2
in units of εˆ 2 for short times
(large times are displayed in small inserted plot in absolute units). In the remainder of this section, we derive the dynamical equations for sˆ and η, ˆ then we discuss informally these equations to motivate the analytical treatment that we will present in the next sections. In particular, we will explain the particular choice of the scalings (1.3) and (1.4). We will treat the phase dynamical equation in Sect. 2, while the treatment of the dynamical equation for s is postponed to Sect. 3, s being ‘slaved’ to η by that equation. 1.4. Derivation of the amplitude and phase equations. The ansatz (1.2) leads, after separation of the real and imaginary parts of equation (CGL), to ∂t s = s − 2s − η − (η )2 − α 2 3s 2 + α 2 s 3 + 2s η + sη + s(η )2 , (1.15) 2s η α 2 ss ∂t η = η + α 2 s − 2αβs − α 2 (η )2 + αβs 2 − + . (1.16) 1 + α2 s 1 + α2 s Since these equations preserve the subspace of functions that are even in the space var2 iable, we restrict ourselves to that particular case. We also use α, − 1+ε α as parameters instead of α, β as it allows to emphasize the dependence on the small parameter ε. Finally, as the right-hand sides of (1.15) and (1.16) contain only (space) derivatives of the function η, we introduce the odd function µ (the phase derivative) by x η(x, t) = dy µ(y, t) , (1.17) 0
and obtain ∂t s = s − 2 s − µ − µ2 − α 2
3 s 2 + 2 s µ + s µ + s µ2 + α 2 s 3
∂t µ = µ + α 2 s + 2 (1 + ε 2 ) s − α 2 (µ2 )
α 2 s s 2s µ +α 2 (1 + ε 2 ) s 2 + − . 1 + α2 s 1 + α2 s
, (1.18)
(1.19)
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G. van Baalen
We expect ∂t s, s s, µ µ 1 when ε 1. We then have ∂t s = s − 2 s − µ − µ2 + fs (s, µ) , ∂t µ = − s − 2 s − µ − µ2 + (1 + α 2 ) s + 2 ε 2 s −2 (1 + α 2 ) µ µ + fµ (s, µ) ,
(1.20)
(1.21)
where fs (s, µ), respectively fµ (s, µ), is defined as the function appearing in the second line of (1.18) resp. (1.19). The −2s term in (1.20) strongly damps s, which therefore is ‘slaved’ to µ. Indeed, as we will show in Sect. 3, for given µ satisfying appropriate bounds, the map µ → s(µ) defined by the (global and strong) solution of (1.20) is well defined and Lipschitz in µ. Furthermore, to third order in ε, the map is given by the solution s1 of s1 − 2 s1 − µ − µ2 = 0, which can be represented as 1 s1 (µ) = − G µ + µ2 , (1.22) 2 where G is the operator of convolution with the fundamental solution G of G(x) − 1 2 G (x) = δ(x). Note that G acts multiplicatively in Fourier space, with symbol (1 + k 2 −1 2) ,
in particular, G f has two more derivatives than f . As we will also show in Sect. 3, s(µ) will have the same structure as s1 (µ), that is, the G–convolution of another map with the same regularity as µ . As such, s(µ) is once more differentiable than µ, due to the regularizing properties of G, and s(µ) = s(η ) is as regular as η. This is reasonable, since from u = (1 + α 2 s) e−iβ t+iαη we see that s and η should have both the same degree of regularity as u. Inserting (1.22) into (1.21) and neglecting fµ leads to the (modified) Kuramoto– Sivashinsky equation for the phase ∂t µ = − 1+α − ε 2 Gµ − 2(1+α 2 )µµ − ε 2 G(µ2 ) − 2 Gµ 2
1+α 2 2
G(µ2 ) , (1.23)
from which we recover the Benjamin–Feir linear instability criterion 1 + αβ < 0. Namely, linear stability analysis in Fourier space (set µ = ε0 eikx+λ(k)t with ε0 1) gives the dispersion relation 2 2 ε2 k 2 − k 4 1+α −(1 + αβ) k 2 − k 4 1+α 2 2 λ(k) = = . k2 k2 1+ 2 1+ 2 This shows that there are linearly unstable modes for |k| ≤ ε 1, growing at most like 4 eε t . This suggests that the dynamics of (1.23) should be dominated by the dynamics of the Fourier modes in the small |k| region, the high |k| modes being slaved to them. For |k| 1, we have G ≈ 1, and neglecting the last two terms of (1.23), we get the Kuramoto–Sivashinsky equation in derivative form ∂t µ ≈ − 1+α µ − ε 2 µ − 2(1+α 2 ) µµ . 2 2
(1.24)
Defining µ(x, t) = with χ =
4 , 1+α 2
εˆ =
√χ 2
1 4
εˆ 3 µ( ˆ x, ˆ tˆ) ,
ε, xˆ = εˆ x, and tˆ =
2 χ
(1.25)
εˆ 4 t, we get from (1.24)
∂t µˆ = −µˆ − µˆ − µˆ µˆ ,
(1.26)
Phase Turbulence in the Complex Ginzburg-Landau Equation
621
which is the original Kuramoto–Sivashinsky equation in derivative form. This justifies the scalings (1.3). Equation (1.26) possesses an universal attractor of finite radius in L2 ([−L/2, L/2]) with periodic boundary conditions (see e.g. [CEES93]), hence we can expect µ to be of size ε 3 times a typical solution in that attractor. From (1.22), we get (the µ–dependence of s1 is implicit here for concision) s1 (x, t) = −
εˆ 4 ˆ ˆ tˆ) + εˆ 2 µ( ˆ x, ˆ tˆ)2 ≡ εˆ 4 sˆ1 (x, ˆ tˆ) , G 4µˆ (x, 32
(1.27)
ε2 ˆ ˆ ˆ is the convolution operator with the fundamental solution Gˆ of G(x)− where G 2 G (x) = ˆ acts multiplicatively in Fourier space, with symbol G(k) ˆ δ(x). As above, G = (1 + ε2 k 2 −1 4 ˆ ˆ t ) we introduced for 2 ) . Equation (1.27) motivates the scalings s(x, t) = εˆ sˆ (x, s in (1.4). We now apply (1.25) and (1.4) to (1.20) and (1.21). From now on we drop the hats. Then s and µ satisfy the following equations: 2 χ χ Ls s + 4 r1 (µ) − α 8 χ (2s µ + sµ ) + F3 (s, µ) , ε4 ε ∂t µ = −Lµ µ − µµ + ε 2 F0 (s, µ) + ε 2 χ Lµ,r r2 ,
∂t s = −
(1.28) (1.29)
where Ls , Lµ and Lµ,r are multiplicative operators in Fourier space, with symbols given by ε2 k 2 , 2 4 2 k −k Lµ (k) = , 2 2 1 + ε 2k Ls (k) = 1 +
Lµ,r (k) = 2
(1.30)
2 + ε 2 (1 + α 2 ) − α 2 ε 2 k 2 1+
ε2 k 2 2
,
(1.31)
while r1 , r2 , F3 and F0 are defined by 1 (4µ + ε 2 µ2 ) , 32 r r1 (µ) r2 = 4 − 4 , ε ε
r1 (µ) = −
F3 (s, µ) = −χ α 2
3 2 2s
(1.32) (1.33)
1 + ε 2 32 sµ2 +
α2 4 3 2 ε s
,
s µ − 2 ε 2 α 2 ss 2 + ε 2 (1 + α 2 ) s 2 + 1 + ε4 α 2 s 1 1 2 2 − Gµ − G(µ ) , 4 4
(1.34)
F0 (s, µ) = χ α 2
(1.35)
where the auxiliary variable r and the operator G are defined by G(k) =
1 2 2 1+ ε 2k
r =s− and satisfy s = G r.
(1.36)
,
ε2 2s
,
(1.37)
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G. van Baalen
We will prove that (1.28) defines a map µ → s(µ) for all µ in an open ball of Wσ , and that this map has indeed ‘the same properties’ as G r1 (µ), e.g. in terms of regularity. This is so essentially because for ε 1, we have εχ4 Ls 1, so that by Duhamel’s 4 4 formula, s ∼ L−1 s r1 (µ) + O(ε ) = G r1 (µ) + O(ε ) (see Sect. 3). At the same time, as a dynamical variable, r2 satisfies ∂ t r2 = −
χ χ 1 G Lr r2 + µ Lµ,r r2 + 4 F6 (s, µ) , ε4 16 ε
(1.38)
where Lr is the multiplicative operator in Fourier space with symbol 2 2 2 1−α 2 ε4 k 4 , ε k + Lr (k) = 1 + 23 + ε 2 1+α 4 4 and
F6 (s, µ) = Ls F3 (s, µ) + F4 (s, µ) + F7 (s, µ) + F8 (µ) , 2
F4 (s, µ) = − α 8 χ (2s µ + sµ ) , ε2 ε 2 µ F7 (s, µ) = ∂x + F0 (s, µ) , 8 2 1 ε 2 µ F8 (s, µ) = − ∂x + Lµ µ + µ µ . 8 2
(1.39) (1.40) (1.41)
Once s is considered as a given map µ → s(µ), (1.38) defines the map µ → r2 (µ) through a linear equation for r2 . By the same mechanism as for s, we have r2 ∼ (G Lr )−1 F6 (s, µ) ∼ G F6 (s, µ) if α 2 < 1 (see Sect. 4). The restriction α 2 < 1 is necessary here to make Lr positive definite. For technical reasons, we have in fact to restrict α 2 < 1/2 to prove theorems 1.1 and 1.2. We believe that the results of these theorems could be extended to part of the α 2 > 1/2 region by exploiting the following argument. If α 2 > 1, Eq. (1.38) for r2 is linearly unstable at high frequencies. However, the linear coupling of r2 to µ through (1.29) stabilizes r2 . To see this, we introduce the vector v = (µ, r2 ), and consider (1.29) and (1.38) simultaneously, as a vector dynamical system of the form ∂t v = LM v + f (v) ,
(1.42)
for a (nonlinear) vector map f , where LM is the operator with (matrix) symbol
ε 2 χ Lµ,r (k) ik −Lµ (k) LM (k) = . − 8 1ε4 Lµ (k) ik − εχ4 GLr (k) The stability of (1.42) at high frequency is then determined by the eigenvalues λ± (k) of LM (k) for large k. Since4 λ± (k) → −(1 ± i|α|)
k2 εˆ 2
as k → ∞, (1.42) is stable at high frequency, the real part of the eigenvalues λ± (k) of L(k) being negative for large k. However to exploit this would force us to solve (1.29) and (1.38) simultaneously, which is technically (and notationally) more difficult, see 4
This is the analogon of (1 + iα) u in (CGL).
Phase Turbulence in the Complex Ginzburg-Landau Equation
623
[GvB02] for a similar problem. Instead, in our approach the system (1.28), (1.29) and (1.38) is considered as a ‘main’ equation, (1.29), of the form ∂t µ = −Lµ µ − µµ + ε 2 F (µ) ,
(1.43)
supplied with two ‘auxiliary’ equations, (1.28) and (1.38), which can be solved independently. We will first study (1.43) for a general class of map F (µ) in Sect. 2 below, because it explains the choice of the functional space, and which properties of the solutions of the amplitude equations (1.28) and (1.38) are needed. Then, in Sect. 3 and 4, we will show that the solutions of the amplitude equations (1.28) and (1.38) exist and satisfy the ‘right’ properties. 2. The Phase Equation 2.1. Strategy. Having argued that r2 = r2 (µ), we rewrite (1.29) as ∂t µ = −Lµ µ − µµ + ε 2 F (µ) ,
µ(x, 0) = µ0 (x) ,
(2.1)
where µ0 is a given (odd) space periodic function of period L for some given L. Since (2.1) preserves the mean of µ over [−L/2, L/2], and since µ0 is the space derivative of a space periodic function, we restrict ourselves to µ0 which have zero mean over [−L/2, L/2]. We will show that the term ε2 F (µ) is in some sense negligible. If ε = 0, then Lµ = ∂x4 + ∂x2 ≡ Lµ,c , and (2.1) is the Kuramoto–Sivashinsky equation. If F = 0 and ε > 0, (in this case, Lµ is of smaller order than ∂x4 + ∂x2 ), this situation can still be easily handled by the techniques of [CEES93] or [NST85], which show that equation (2.1) possesses a universal attractor of finite radius in L2 ([−L/2, L/2]) if F = 0. A key ingredient of that proof is the observation that form dx µ2 µ vanishes the trilinear 2 for periodic functions. However, in general, ε dx µ F (µ) will not vanish, and might even not exist at all for µ ∈ L2 . We will explain precisely below how we circumvent this, but the mechanism is indeed quite simple. If the nth Fourier coefficients of µ were vanishing for all n ≥ qδ 2 with 1 δ 1/ε, we would have e.g. µ L2 ≤ δ µ L2 , which would (presumably) give ε2 dx µ F (µ) ∼ ε 2 δ 2 µ 2L2 . For ε sufficiently small, this would only give a small blur to the attractor of the true Kuramoto–Sivashinsky equation. Evidently, we cannot expect the high–n Fourier modes to vanish, so we will have to treat them separately. On that matter, we want to point out that contrary to the ‘true’ Kuramoto–Sivashinsky equation (1.26), where the linear operator Lµ,c acting on µ on the r.h.s. is of fourth order, Lµ is only of second order due to the regularizing properties of G. From the point of view of derivatives of µ, it is easy to see that ε2 sˆ1 and ε2 sˆ1 contain at most first derivatives of µ, hence we expect ε 2 F (µ) to contain at most second order derivatives of µ, and we see that at high frequencies, (1.43) is more similar to the well studied equation u˙ = u + f (u, u , u ) (see e.g. [BKL94]) than to the Kuramoto–Sivashinsky equation. Note that the term F (µ) is ‘irrelevant’ due to its prefactor ε 2 , while µ µ is certainly not. Indeed, it would be catastrophic to solve (2.1) by successive approximations, beginning with the solution of the equation with −µµ + ε 2 F (µ) = 0, inserting that solution into the nonlinear terms and solving again the linear inhomogeneous problem. This
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would lead to (apparently) exponentially growing modes, because the linear operator Lµ is not positive definite at small frequencies. Solving (2.1) iteratively as ∂t µn+1 = −Lµ µn+1 − µn+1 µn+1 + ε 2 F (µn ) , for n ≥ 0 is a much better choice. We therefore consider the following class of equations ∂t µ = −Lµ µ − µµ + ε 2 g ,
µ(x, 0) = µ0 (x) ,
(2.2)
for some given time dependent and spatially periodic perturbation g and periodic initial data µ0 . From this (informal) discussion, we see that we should treat the small n Fourier coefficients with an L2 –like norm as in [CEES93] or [NST85], and the high n modes as in e.g. [BKL94]. In the next three subsections, we implement this idea. We first show L2 estimates for (2.2) in Subsect. 2.2. Then in Subsect. 2.3 we define functional spaces similar to those of [BKL94], and prove inequalities in these spaces, which will allow us to prove the ‘high frequency estimates’ in Subsect. 2.4. In Subsect. 2.5, we will prove that the full phase equation has a solution if µ → F (µ) is a well behaved Lipschitz map, and finally, in Subsect. 2.6, we will show how the phase equation relates to the Kuramoto–Sivashinsky equation. 2.2. Coercive functional method, L2 estimates. The initial value problem (2.2) is globally well posed in L2 ([−L/2, L/2]) if the perturbation g is periodic and in L2 for all t ≥ 0. The local uniqueness/existence theory follows from standard techniques (see e.g. [Tem97]), whereas the global existence follows from the a priori estimate
µ(·, t) 2L2 ≤ et µ(·, 0) 2L2 + 2 ε 4 et − 1 sup g(·, s) 2L2 . (2.3) 0≤s≤t
Namely, denoting by the integral over [−L/2, L/2], using Young’s inequality, integration by parts and the fact that µ2 µ = 0 by periodicity of µ, we have 1 (µ )2 + 2 ε 4 g 2 ≤ µ2 + 2 ε 4 g 2 , ∂t µ2 ≤ −2 µ Lµ µ + 2 from which (2.3) follows immediately. As a much stronger result, we can in fact prove that the L2 –norm of the solution stays bounded for all t ≥ 0. To do this, we adapt the strategy of [NST85] and [CEES93] to our setting. We first need a technical result. Proposition 2.1. Let (v, w) = vw, (v, w)γ φ = v(Lµ + γ φ )w and 4 . (2.4) Lv (k) = 13 1+k ε2 k 2 1+
2
For all L ≥ 2π , there exist a constant K and an antisymmetric periodic function φ such that for all γ ∈ [ 41 , 1] and ε ≤ L−2/5 , and for every antisymmetric periodic function v, one has 3 (Lv v, Lv v) ≤ (v, v)γ φ ≤ φ ∞ (v, v) + (v , v ) , 4 (φ, φ)γ φ ≤ K L16/5 and (φ, φ) ≤ 43 L3 .
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625
The proof, which follows closely [CEES93] is relegated to Appendix A. We then have the Theorem 2.2. There exists a constant K such that the solution µ of (2.2) is periodic, antisymmetric, and satisfies sup µ(·, t) L2 ≤ ρ + µ0 L2 + 4 ε 2 sup L−1 v g(·, t) L2 , t≥0
t≥0
where ρ = K L8/5 , if µ0 and g are antisymmetric (spatially) periodic functions of period L, 2 Proof. Note first that L−1 v ∂x is a bounded operator on L with norm ≤ 2 (see Lemma F.1 2 in Appendix F), then local existence in L follows from the above argument. Next, following [NST85] with the modifications of [CEES93], we write µ(x, t) = v(x, t)+φ(x) for some constant periodic function φ to be chosen later on. Denoting by the integral over [−L/2, L/2], using integration by parts, that v 2 v vanishes because v is periodic and the inner products defined in Proposition 2.1, we get from (2.2)
1 ∂t (v, v) = −(v, v)φ/2 − (v, φ)φ + ε 2 (v, g ) . (2.5) 2 √ 4 Next, we use that (Lv v, Lv v) ≥ 43 ε ε+4−2 (v, v) ≡ cv2 (v, v), Young’s inequality 4 and Proposition 2.1 to get from (2.5), ∂t (v, v) ≤ −2 (v, v)φ/2 + 23 (v, v)φ + 23 (φ, φ)φ + 2 ε 2 (v, g ) ≤ − 43 (v, v)φ/4 + 23 (φ, φ)φ + 2 ε 2 (v, g ) ≤ −(Lv v, Lv v) + 23 (φ, φ)φ + 2 ε 2 (Lv v, L−1 v g) c2 2 ≤ − v (v, v) + 23 (φ, φ)φ + 2 ε 4 L−1 v g L 2 . 2
(2.6)
Since v(x, t) = µ(x, t) − φ(x) we conclude that
µ(·, t) − φ(·) 2L2 ≤ µ0 − φ 2L2 + Finally, since
2 cv
3 4 ε4 2 (φ, φ) + sup L−1 φ v g(·, t) L2 . cv2 cv2 t≥0
≤ 4, we have
sup µ(·, t) L2 ≤ µ0 L2 + ρ + 4 ε 2 sup L−1 v g(·, t) L2 , t≥0
t≥0
where ρ = 2 φ L2 + 4 (φ, φ)φ . Furthermore, by Proposition 2.1, we have ρ < ∞, √ since φ L2 = (φ, φ) < ∞ and (φ, φ)φ < ∞. This completes the proof of the theorem. Corollary 2.3. The antisymmetric solution of the Kuramoto–Sivashinsky equation with periodic boundary conditions on [−L/2, L/2] ∂t µ = −µ − µ − µ µ ,
µ(x, 0) = µ0 (x) ,
stays in a ball of radius O(L8/5 ) in L2 as L → ∞.
(2.7)
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G. van Baalen
Proof. This result was already established in [NST85] and [CEES93]. To prove it, we only have to note that (2.7) corresponds to (2.2) with ε = 0, and that Theorem 2.2 is uniformly valid in ε ≤ 1. Remark 2.4. The proof of Theorem 2.2 is the only point in this paper where we need s, respectively µ, to be spatially even, resp. odd, functions. The theorem holds also in the general (non symmetric) case. The proof can be obtained as a straightforward extension of the result of [CEES93] for the Kuramoto–Sivashinsky equation in the non symmetric case. If ε = 0, Theorem 2.2 shows that the solution of (2.2) stays in a ball in L2 , centered on 0 and of radius µ0 L2 + ρ for all t ≥ 0, with ρ = O(L8/5 ) as L → ∞. When ε = 0, the radius of the ball widens to lowest order like ε2 sup g(·, t) L2 . t≥0
2.3. Functional spaces, definitions and properties. In this section, we explain how to treat the high frequency part of the solution of (2.2). This development is inspired by [BKL94] (see also [GvB02] for similar definitions). Let L ≥ 2π and q ≡ 2π L ≤ 1. We define the Fourier coefficients fn of a function f : [−L/2, L/2] → R by fn =
1 L
L/2
dx e−iqnx f (x) ,
−L/2
so that
f (x) =
eiqnx fn ,
n∈Z
and P< , P> , the projectors on the small/high frequency part by
P< f (x) = eiqnx fn , P> f (x) = eiqnx fn , |n|≤ qδ
|n|> qδ
where the parameter δ ≥ 2 will be chosen later. We also define the Lp and l p norms as p
f Lp
=
L/2
dx |f (x)|p ,
−L/2
p
f l p =
|fn |p ,
f l ∞ = sup |fn | ,
n∈Z
n∈Z
and f L∞ = ess sup |f (x)|. We will use Plancherel’s equality f L2 = x∈[−L/2,L/2]
√ L f l 2
without notice. Finally, for σ ≥ 0, δ ≥ 2, we define the norm · N ,σ by
f N ,σ =
√ δ q
σ 2 2 sup 1 + ( qn |fn | . δ )
n∈Z
With a different normalization, the norm · N ,σ was introduced in [BKL94] to study the long time asymptotics of solutions of u˙ = u + f (u, u , u ), where f is some (polynomial) nonlinearity. From the point of view of the nonlinearity, our situation is similar to the case treated there, but our linear operator Lµ is not positive definite as − was in their case. The potentially exponentially growing modes correspond to |n| ≤ q1 , and we saw in Sect. 2.2 that their l 2 norm was bounded. Since there are only a finite (but large) number of linearly unstable modes, changing the definition of the · N ,σ –norm
Phase Turbulence in the Complex Ginzburg-Landau Equation
627
on these modes to an l 2 –like norm will give an equivalent norm which is better suited to our case. Thus we define the norms · W ,σ and · σ by √ σ 2 2
f W ,σ = qδ sup 1 + ( qn |fn | , (2.8) δ ) |n|> qδ
f σ = f L2 + f W ,σ .
(2.9)
While · W ,σ is clearly not a norm, · σ is a norm which √ is equivalent to · N ,σ for σ ≥ 1. Indeed, easy calculations lead to f σ ≤ 1 + π 2 f N ,σ and √ √
f N ,σ ≤ 2σ L δ f L2 + f W ,σ ≤ 1 + 2σ L δ f σ . We point out also that if σ > 21 , the · W ,σ –semi–norm is a decreasing function of δ. Indeed, we have (here the norms carry an additional index to specify the value of δ) σ − 1 σ 2
f W ,σ,δ1 ≤ 2 2 δδ01
f W ,σ,δ0 , (2.10) for all δ1 ≥ δ0 ≥ 2. As δ will be fixed later on, the additional index is suppressed to simplify the notation. On the other hand, · σ is an non–decreasing function of σ , since, for all σ1 ≥ σ0 ,
f σ0 ≤ f σ1 .
(2.11)
We now define the functional spaces ∞ ([−L/2, L/2], R) the set of infinitely differentiable Definition 2.5. Denoting by C0,per periodic real valued functions on [−L/2, L/2], we define the (Banach) space W0,σ as ∞ ([−L/2, L/2], R) under the norm · , and B the completion of C0,per σ 0,σ (r) ⊂ W0,σ the open ball of radius r centered on 0 ∈ W0,σ .
Up to now, we considered functions depending on the space variable only. We extend the definition 2.9 to functions f : [−L/2, L/2] × [0, ∞) → R by |||f |||σ = sup f (·, t) σ . t≥0
The same convention applies for definition.
Lp
and l p norms. Finally, we make the following
∞ (, R) denote the set of infinitely Definition 2.6. Let = [−L/2, L/2] × R+ and Cper differentiable functions on compactly supported on R+ and satisfying f (−L/2, t) = f (L/2, t) for all t ∈ R+ . We define the (Banach) space Wσ as the completion of ∞ (, R) under the norm ||| · ||| , and B (r) ⊂ W the open ball of radius r centered Cper σ σ σ on 0 ∈ Wσ .
The spaces Wσ satisfy nice properties under derivation and multiplication. Space deri-vation maps Wσ to Wσ −1 essentially with a factor √ δ on the norms, while multiplication maps Wσ × Wσ to Wσ with essentially a factor δ. Furthermore, in the spaces Wσ , it is very easy to quantify the regularising effects of the evolution equation (2.2) on the inhomogeneous term g (or the nonlinearity F (µ) ). For precise statement on these results, see Lemma B.1, and Propositions B.2 and B.3 in Appendix B. The following proposition, which follows directly from Lemma B.1 relate Wσ to more well known spaces:
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G. van Baalen
Proposition 2.7. For all σ > 25 , Wσ ⊂ L∞ (R+ , W2,2 ([−L/2, L/2])), the Banach space of functions on which are (together with their space derivatives up to order 2) uniformly (in time) bounded in L2 ([−L/2, L/2]). We can now define the class C of initial conditions for which Theorems 1.1 and 1.2 hold. Definition 2.8. We say that η0 and s0 are in the class C if η0 ∈ W0,σ and s0 ∈ W0,σ −1 , if η0 (0) = 0 ,
η0 σ ≤ cη0 ρ ,
εˆ 2 s0 ≤ cs0 ρ 3 , s0 − 2 σ −1
(2.12)
for ρ = K L8/5 , cη0 > 0 and cs0 > 0 and if 2 −1 4ε2 χ Lµ,r L−1 (2.13) v r2,0 L2 + ε χ Lµ,r Lµ r2,0 W ,σ ≤ λ1,0 cη0 ρ , ε 2 ε2 r2,0 L2 + ε 2 χ Lµ,r L−1 cs0 ρ 3 , (2.14) v r2,0 L2 ≤ λ2,0 ε0 2 α2 ε2 2 (η0 ) . and r2,0 = ε14 s0 − ε2 s0 + 18 η0 + 32 with λ2,0 > 0, λ1,0 < min 23 , 1−2 1−α 2
Note that (2.14) is stronger than (2.13) as ε → 0, while it is the contrary as ε → ε0 . 2.4. High frequency estimates. By Theorem 2.2, the solution µ of (2.2) exists and is bounded in L2 for all t ≥ 0 if µ0 L2 + |||L−1 v g |||L2 < ∞. We will now show that upon further restrictions on µ0 and g, the solution has bounded · σ –norm for all t ≥ 0. Namely, setting 2 ||| + ε2 g m c0 ρ c0 = 1 + µρ0 σ + 4ερ |||L−1 g and ξ = C√ , (2.15) 2 L v ρ Lµ W ,σ
2δ
we have the following theorem. Theorem 2.9. Let c0 and ξ be defined by (2.15), and assume that the initial condition µ0 and the function g satisfy ξ < 41 . Then the solution µ of (2.2) satisfies √
1 − 1 − 4ξ |||µ|||σ ≤ c0 ρ . (2.16) 2ξ Remark 2.10. Note that c0 is implicitly dependent of δ (because the norm · σ is). If µ0 and g are given, c0 is a non-increasing function of δ (see (2.10)). Hence we can surely satisfy ξ < 41 by taking δ sufficiently large. Proof of Theorem 2.9. Let d0 = |||µ|||L2 , σ0 = 0, σ1 = σn+1 =
1 2
and, for all n ≥ 1, define
σn + 1 if σn + 1 < σ . σ if σn + 1 ≥ σ
We will now show inductively that dn ≡ |||µ|||σn are bounded for all n ≥ 1, and that |||µ|||σ = lim dn satisfies (2.16). The first step is to note that by Theorem 2.2, we have n→∞
d0 = |||µ|||L2 ≤ ρ + µ0 L2 + 4 ε 2 |||L−1 v g |||L2 ≤ c0 ρ .
(2.17)
Phase Turbulence in the Complex Ginzburg-Landau Equation
629
To bound |||µ|||W ,σ , we use Duhamel’s representation formula for the solution of (2.2), t µ(x, t) = e−Lµ t µ0 + ε 2 ds e−Lµ (t−s) g (x, s) + T (µ)(x, t) , 0 t T (µ)(x, t) = − ds e−Lµ (t−s) (µµ )(·, s) .
(2.18)
0
Since µ µ = 21 (µ2 ) , using Propositions B.2 and B.3, we get for all n ≥ 1 the bound 1 Cm |||T (µ)|||W ,σn +1 ≤ √ |||µ2 |||W ,σn ≤ √ dn2 . 2δ 2δ Using again Proposition B.2 and the definitions (2.15), we get for all n ≥ 0 the bound d 2 dn+1 n , ≤1+ξ c0 ρ c0 ρ
2 g because ρ + µ0 σn + 4ε 2 |||L−1 v g |||L2 + ε ||| Lµ |||W ,σn ≤ c0 ρ for all n ≥ 0. Note that since ξ < 1 , the (infinite) sequence d˜n+1 = 1 + ξ d˜n2 , d˜0 = 1, is increasing and satisfies 4
d˜n ≤ lim d˜n = d˜∞ ≡ n→∞ immediately.
√ 1− 1−4ξ , 2ξ
hence |||µ|||σ ≤ d˜∞ c0 ρ, from which (2.16) follows
2.5. Existence and uniqueness of the solution of the phase equation. Let µ˜ ∈ Wσ and µ0 ∈ B0,σ (cη0 ρ) ⊂ W0,σ . We consider the equation ˜ , ∂t f = −Lµ f − f f + ε 2 F (µ)
f (x, 0) = µ0 (x) .
(2.19)
By Theorem 2.9, f exists if µ0 σ + |||L−1 ˜ |||L2 + |||L−1 ˜ |||W ,σ < ∞, in v F (µ) µ F (µ) which case, we define the map (µ, ˜ µ0 ) → F(µ, ˜ µ0 ), by F(µ, ˜ µ0 ) ≡ f . We will show that for fixed µ0 , µ˜ → F(µ, ˜ µ0 ) is a contraction in the ball Bσ (cη ρ) if the following condition holds. c +1
η0 Condition 2.11. There exist constants λ1 < 1 and λ2 > 0 such that for all cη > 1−λ , 1 there exists a constant ε0 such that for all ε ≤ ε0 and for all µi ∈ Bσ (cη ρ) the following bounds hold:
2 F (µi ) 4 ε2 |||L−1 F (µ ) ||| + ε ≤ λ1 c η ρ , 2 i L v W ,σ Lµ 2 F F ||| + ε ≤ λ1 |||µ1 − µ2 |||σ , ε2 |||L−1 2 L v Lµ W ,σ ε 2 ε2 |||r2 (µi )|||L2 + ε 2 |||L−1 F (µ ) ||| ≤ λ cs ρ 3 , 2 i 2 L v ε0 where F = F (µ1 ) − F (µ2 ).
(2.20) (2.21) (2.22)
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G. van Baalen
We prove that this condition holds in Sect. 4. The proof requires bounds on s and r2 . We will now motivate briefly why this condition is a natural one. We recall that F (µ) = F0 (s(µ), µ) + χ Lµ,r r2 . If we consider (2.20)–(2.22) only at t = 0, and set F0 = 0, we see that by Definition 2.8, the bounds (2.20)–(2.22) are satisfied with λ1 = λ1,0 < 1 and λ2 = λ2,0 > 0. We will see in Sect. 4 that r2 satisfies the same kind of bounds as those of Definition 2.8 for any time t > 0. On the other hand, if s = s1 (µ), or equivalently r2 = 0, we have F (µ) = F0 (s1 (µ), µ), and (see Appendix C or the beginning of Sect. 4) we can satisfy Condition 2.11 for any λ1 < 1 and ε0 = cε δ −5/4 ρ −1/2 if cε is sufficiently small (depending on λ1 ). To apply Theorem 2.9, we need ξ = Cm√ c0 ρ < 41 , and from (2.20), we have c0 < cη , hence we can satisfy 2δ
ξ < 41 by choosing δ = cδ ρ 2 for some constant cδ . This implies also that we should take (at least) ε0 = cε ρ −mε with mε ≥ 3. We then have the following proposition 1+c
Proposition 2.12. Let cη > 1−λη10 , and assume that Condition 2.11 holds with ε0 sufficiently small. Then there exists a constant cδ sufficiently large such that if δ = cδ ρ 2 and ε ≤ ε0 , then for all µ0 ∈ B0,σ (cη0 ρ), it holds |||F(µ˜ i , µ0 )|||σ < cη ρ .
(2.23)
Proof. The proof follows from Theorem 2.9. We first note that F(µ˜ i , µ0 ) satisfies (2.2) with g = F (µ), ˜ and define c0 (µ) ˜ and ξ(µ) ˜ as in (2.15). Then by Condition 2.11, for all 1+c µ ∈ Bσ (cη ρ) and µ0 ∈ B0,σ (cη0 ρ), we have c0 (µ) ˜ < λ cη with λ = λ1 + cηη0 < 1. Choosing cδ sufficiently large, we have ξ(µ) ˜ < 41 . The proof is then completed noting
√ 1− 1−4ξ(µ) ˜ that by Theorem 2.9, we have |||F(µ, ˜ µ0 )|||σ ≤ c0 (µ) ˜ ρ < cη ρ. 2ξ(µ) ˜ Proposition 2.13. Let cη , cδ and ε0 be given by Proposition 2.12, and assume that for all µ˜ 1 , µ˜ 2 ∈ Bσ (cη ρ) we have |||F(µ˜ i , µ0 )|||σ < cη ρ for all µ0 ∈ B0,σ (cη ρ). Then there exists a time t0 such that sup F(µ˜ 1 , µ0 )(·, t) − F(µ˜ 2 , µ0 )(·, t) σ < sup µ˜ 1 (·, t) − µ˜ 2 (·, t) σ . 0≤t≤t0
0≤t≤t0
Proof. The proof, being very similar to the estimates leading to (2.23), can be found in Appendix D. Note that here we only asked for µ0 ∈ B0,σ (cη ρ) and not for µ0 ∈ B0,σ (cη0 ρ). We now deduce from Propositions 2.12 and 2.13 existence, uniqueness, and estimates for the solution of the phase equation. Theorem 2.14. Let cη , cδ and ε0 be given by Proposition 2.12. Then for all T ≥ 0, the solution µ of (2.1) exists for all 0 ≤ t ≤ T and satisfies sup µ(·, t) σ ≤ cη ρ , 0≤t≤T
for all µ0 ∈ B0,σ (cη0 ρ).
(2.24)
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631
Proof. Let F(µ, ˜ µ0 ) be defined by the solution of (2.19). By Proposition 2.12, we know that |||F(µ)||| ˜ σ < cη ρ if |||µ||| ˜ σ ≤ cη ρ. Hence, we can apply Proposition 2.13 and get that µ˜ → F(µ, ˜ µ0 ) is a contraction for 0 ≤ t ≤ t0 in the ball of radius cη ρ. Thus µ˜ → F(µ, ˜ µ0 ) has a unique fixed point µ in that ball. By easy arguments (see e.g. [GvB02]), this fixed point is the unique strong solution of (2.1) for 0 ≤ t ≤ t0 . Furthermore, since the image of µ˜ → F(µ, ˜ µ0 ) is in a ball of radius cη ρ, µ satisfies (2.24) with T = t0 . We can now show inductively that µ exists for all t ≥ 0 and satisfies (2.24) for all T ≥ 0. Define tn = (n + 1)t0 for n ≥ 1, and suppose that µ exists on 0 ≤ t ≤ tn−1 and satisfies (2.24) with T = tn−1 . By Proposition 2.12, we know that for tn−1 ≤ t ≤ tn , the solution F(µ, ˜ µ (·, tn−1 )) of ∂t µ = −Lµ µ − µ µ + ε 2 F (µ) ˜ ,
µ(x, t0 ) = µ (x, tn−1 )
(2.25)
is in a ball of size cη ρ if µ˜ is in a ball of size cη ρ for tn−1 ≤ t ≤ tn , because it is the continuation of a solution of (2.19), beginning with µ0 in t = 0, with µ(x, ˜ t) = µ (x, t) for 0 ≤ t ≤ tn−1 . Shifting the origin of time to tn−1 and replacing µ0 by µ (·, tn−1 ), we see that the conditions of Proposition 2.13 are satisfied, hence µ˜ → F(µ, ˜ µ (·, tn−1 )) is a contraction for tn−1 ≤ t ≤ tn . As above, this implies that there exists an unique fixed point µ which is the unique strong solution of 2.1 on 0 ≤ t ≤ tn and satisfies (2.24) with T = tn .
2.6. Consequences. Up to now, we did not use (2.22) of Condition 2.11. This inequality has two important consequences which are proved in Theorems 2.15 and 2.16 below. The first one is that s (if it exists) and η are related by s = − 18 η = − 18 µ up to corrections of order ε2 and the second one concerns the relation with the Kuramoto–Sivashinsky equation. Once these theorems are proved, we will only have to prove the bound on sˆ to complete the proof of Theorems 1.1 and 1.2. Theorem 2.15. There exists a constant cε > 0 sufficiently small such that if Condition 2.11 is satisfied with ε0 ≤ cε ρ −4 , δ is given by Proposition 2.12 and ε ≤ ε0 , then it holds |||s +
1 8
ε2 ε 4 G µ + 32 G (µ)2 |||L2 ≤ . cη ρ ε0
Proof. The proof is very simple. We use that s + that by assumption (see (2.22)), we have ε4 r2 L2 ≤ λ2
1 32 G
(4µ + ε 2 µ2 ) = ε 4 G r2 , and
ε 4 c λ c ε 2 ε 2 s ε 2 cs ρ 3 ≤ cη ρ , ε0 ε0 cη
choosing cε sufficiently small achieves the proof.
We next show that the solution µc of the Kuramoto–Sivashinsky equation (in derivative form) captures the dynamics of the (derivative of the) phase for short times (then − 18 µc captures the dynamics of the amplitude by Theorem 2.15). To state the result, we introduce the operator Lµ,c = ∂x4 + ∂x2 . We have the following theorem.
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Theorem 2.16. Let µ and µc be the solutions of ∂t µ = −Lµ µ − µµ + ε 2 F (µ) , µ(x, 0) = µ0 (x) , ∂t µc = −Lµ,c µc − µc µc ,
µc (x, 0) = µ0 (x).
There exist constants cε and ct such that if Condition 2.11 holds with ε0 ≤ cε ρ −4 , then sup 0≤t≤t0
ε 2
µ(·, t) − µc (·, t) L2 , ≤ cη ρ ε0
(2.26)
for all t0 ≤ ct ρ −4 and for all ε ≤ ε0 if cε and ct are sufficiently small. Although this theorem compares µ with µc on the time interval [0, ct ρ −4 ], it is also valid on any interval of the form [t0 , t0 + ct ρ −4 ] if µ and µc are equal at time t0 , and thus implies directly Theorem 1.2 (see the remark after the proof). Proof of Theorem 2.16. Let µ± = µ ± µc and L− ≡ Lµ − Lµ,c = µc exists and satisfies |||µc |||σ ≤ cη ρ. Furthermore, µ− satisfies
ε2 2 2 Lµ ∂x .
Note that
Lµ 1 1 ∂t (µ− , µ− ) = −(µ− , Lµ,c µ− ) − (µ− , µ+ µ− ) + ε 2 µ− , F (µ) − µ . 2 4 2
Next, we define the operator Lv,c by Lv,c (k) = 7 6,
the Cauchy–Schwartz inequality and
1 1 2 2 4 3 (1 + k ). Using 2 Lv,c +Lv −2Lµ,c
µ ∞ defining ζ = 76 + +2 L , we get
≤
2 −1 2 ∂t (µ− , µ− ) ≤ ζ (µ− , µ− ) + ε 4 L−1 L µ
+
L F (µ)
µ 2 2 v,c v L L ε 4 C cε4 ρ 2 + C ρ 6 , ≤ ζ (µ− , µ− ) + ε0 for some constants C, C √ . The second inequality follows from Condition 2.8, ε0 ≤ cε ρ −4 4 and L−1 v,c Lµ µ L2 ≤ 3 µ L2 ≤ C δ µ σ (see Lemma B.1). 7 −4 Let t0 ≤ ct ρ , since ζ ≤ 6 + C cη ρ δ 3/2 = cζ ρ 4 , we have sup 0≤t≤t0
ε 2 √ c
µ(·, t) − µc (·, t) L2 ≤ eζ C cε2 + C cη ρ ε0
if cε and ct are sufficiently small.
ct
−1≤
ε 2 , ε0
Note that in the proof we only used global bounds on the solutions and that the initial condition is absent from the estimations, thus the theorem generalizes immediately to intervals of the form [t0 , t0 + ct ρ −4 ].
Phase Turbulence in the Complex Ginzburg-Landau Equation
633
3. The Amplitude Equation This section is devoted to the study of the ‘amplitude’ equation (1.28). Using the definitions and properties of the norms ||| · |||σ of Sects. 2.3, we will show that for given µ with |||µ|||σ not too large, the solution of (1.28) is determined by a well defined Lipschitz map of µ. As in Sect. 2, Eq. (1.28) suggests that we study ∂t s = −
χ ε 2 s − s − ε4 2
α2 χ 8 (2s ν
+ sν ) +
χ f , ε4
s(x, 0) = s0 (x) ,
(3.1)
for given s0 , ν and f . Since |||ν|||σ < ∞, (3.1) is a linear (in s) inhomogeneous heat equation with bounded coefficients, hence the local existence and uniqueness of the solution in L2 is known by classical arguments (see e.g. [Tem97]). For later reference, we state the Condition 3.1. There exist constants cδ , cε , cs0 , cη and cf such that δ = cδ ρ 2 , ε ≤ ε0 = cε ρ −3 , s0 ∈ W0,σ −1 , ν ∈ Bσ (cη ρ) and G1/2 f ∈ Wσ −1 . Proposition 3.2. If Condition 3.1 holds with cε sufficiently small, then there exist a constant λ > 1 such that the solution s of (3.1) satisfies (3.2) |||s|||σ −1 ≤ λ s0 σ −1 + G1/2 f σ −1 . Proof. As in the proof of Theorem 2.9, let d0 ≡ |||s|||L2 , σ0 = 0, σ1 = n ≥ 1, define dn ≡ |||s|||σn , where σn + 1 if σn + 1 < σ − 1 σn+1 = . σ if σn + 1 ≥ σ − 1
1 2
and, for all
Multiplying (3.1) with s, integrating over one period, using Young’s inequality, noting that s(2s ν + sν ) = (s 2 ν) = 0 because s and ν are periodic, and finally integrating the differential inequality, we get immediately that |||s|||L2 ≤ s0 L2 +
ε4 |||G1/2 f |||L2 . χ
(3.3)
From Duhamel’s representation formula, we get s(x, t) = e−Lt s0 (x) + T (s, f )(s, t), where L = εχ4 Ls and T is given by t 2 T (s, f )(s, t) = dτ e−L(t−τ ) α 8 χ 2∂x sν − s∂x ν + f (x, τ ) . 0
Using (3.3) and the inequalities t dτ e−L(t−τ ) f (x, τ ) 0
W ,σ
≤
ε4 ε4 |||G f |||W ,σ ≤ |||G1/2 f |||W ,σ , χ χ
(3.4)
we get that for any n ≥ 1, we have (recall that dn = |||s|||σn ) dn ≤ s0 σ −1 + G1/2 f σ −1 + Cε 4 |||G(sν) |||W ,σn + |||G(sν )|||W ,σn . (3.5) Using ε2 Gf σn ≤ 2 f σn −1 and ε Gf σn ≤ 2 f σn −1 , we see that the r.h.s. of (3.5) involves only dn−1 , which shows that the dn are bounded for all n ≥ 1, which gives
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G. van Baalen
|||s|||σ −1 < ∞. Using ε Gf σ −1 ≤ 2 f σ −1 , Gf σ −1 ≤ f σ −1 and Proposition B.3, we get from (3.5) and ν ∈ Bσ (cη ρ) the inequality √ |||s|||σ −1 ≤ s0 σ −1 + G1/2 f σ −1 + C ε 3 α 2 δ (1 + ε δ) ρ |||s|||σ −1 .
Choosing cε sufficiently small in Condition 3.1 completes the proof.
We are now in position to prove that the solution of (1.28) exists if ε0 is sufficiently small. Theorem 3.3. Let cr1 and cη be given by Proposition C.1 and Theorem 2.14, and cs0 > 0. There exist constants cs > cr1 + cs0 and cε such that for all ε ≤ cε ρ −3 , for all µ ∈ Bσ (cη ρ) and for all s0 ∈ B0,σ −1 (cs0 δ ρ), the solution s of (1.28) with s(x, 0) = s0 (x) exists and is unique in Bσ −1 (cs δ ρ). As such, it defines the map µ → s(µ), which, for all µi ∈ Bσ (cη ρ), satisfies |||s(µi )|||σ −1 ≤ cs δ ρ , |||s(µ1 ) − s(µ2 )|||σ −1 ≤ cs δ |||µ1 − µ2 |||σ .
(3.6) (3.7)
Proof. For all s˜ ∈ Wσ −1 , define T (˜s , µ) as the solution of (3.1) with ν = µ and 4 f = r1 (µ) + εχ F3 (˜s , µ). By Proposition 3.2, T (˜s , µ) is well defined if s0 σ −1 + |||r1 (µ) σ −1 + |||F3 (˜s , µ) σ −1 < ∞. To show that s(µ) exists, is unique and satisfies (3.6), we only have to show that if ε is sufficiently small, s˜ → T (˜s , µ) is a contraction in Bσ −1 (cs δ ρ) ⊂ Wσ −1 . Using Propositions 3.2 and C.1 and the assumption on s0 , we have |||T (s, µ)|||σ −1 ≤ λ (cr1 + cs0 ) δ ρ + |||T (s1 , µ) − T (s2 , µ)|||σ −1 ≤
λ ε4 |||F3 (s, µ)|||σ −1 , χ
λ ε4 |||F3 (s1 , µ) − F3 (s2 , µ)|||σ −1 . χ
(3.8) (3.9)
The contraction property follows immediately from Proposition C.4 if cε is sufficiently small and cs > λ(cr1 + cs0 + ζ ). Hence, the map s → T (s, µ) has a unique fixed point s (µ). This fixed point satisfies (3.6) and is a strong solution of (1.28) (see also [GvB02]). For (3.7), we define µ± = µ1 ± µ2 , s1 = s (µ1 ), s2 = s (µ2 ) and s± = s1 ± s2 . First, we note that s− satisfies (3.1) with ν = µ+ and f = F3 (s1 , µ1 ) − F3 (s2 , µ2 ) − α 2 χ ε4 16 (2s+ µ− + s+ µ− ). Next, for all 0 < ζ1 < 1, we have the estimations √ ε4 |||G1/2 (s+ µ− ) )|||σ −1 ≤ Cε3 δ|||s+ |||σ −1 |||µ− |||σ ≤ ζ1 δ|||µ− |||σ ,
ε4 |||G1/2 s+ µ− |||σ −1 ≤ Cε 4 δ 3/2 |||s+ |||σ −1 |||µ− |||σ ≤ ζ1 δ|||µ− |||σ ,
if ε ≤ cε ρ −3 with cε sufficiently small. Finally, writing F3 (s1 , µ1 ) − F3 (s2 , µ2 ) = F3 (s1 , µ1 ) − F3 (s1 , µ2 ) + F3 (s1 , µ2 ) − F3 (s2 , µ2 ), using Propositions 3.2, C.1 and C.4, we conclude that |||s− |||σ −1 ≤ λ (cr1 + 2ζ1 + ζ ) δ |||µ− |||σ + ζ |||s− |||σ −1 . Since ζ < 1, the proof is completed choosing cs sufficiently large.
Phase Turbulence in the Complex Ginzburg-Landau Equation
635 2
We end this section by proving that µ → r(µ) = s(µ)− ε2 s(µ) satisfies essentially the same bounds as µ → s(µ). Corollary 3.4. Assume that r0 ∈ B0,σ −1 (cr1 δ ρ). Then there exists a constant cr > cr1 + cs0 such that µ → r(µ) satisfies |||r(µ)|||σ −1 ≤ cr δ ρ , |||r(µ1 ) − r(µ2 )|||σ −1 ≤ cr δ |||µ1 − µ2 |||σ ,
(3.10) (3.11)
if the conditions of Theorem 3.3 are satisfied. Proof. The proof, being very similar to the ones of Proposition 3.2 and Theorem 3.3 is outlined in Appendix E. 4. The Condition 2.11, Properties of µ → F (µ) and µ → r2 (µ) We recall that F (µ) = F0 (s(µ), µ) + χ Lµ,r r2 (µ), where r2 is defined in (1.33). If r2 = 0, Condition 2.11 can be satisfied if ε0 ≤ cε ρ −3 with cε sufficiently small. Namely, from Theorem C.2, Appendix C, using also L−1 v f L2 ≤ 2 f L2 ≤ 2 f σ −2 , we have 2 F0 (µi ) 4ε2 |||L−1 F (µ ) ||| + ε ≤ Cε 2 δ 5/2 ρ 2 , (4.1) 2 0 i L v W ,σ Lµ 2 F0 ε2 |||L−1 F ||| + ε ≤ Cε 2 δ 5/2 ρ|||µ1 − µ2 |||σ , (4.2) 2 v 0 L Lµ W ,σ where F0 = F0 (µ1 ) − F0 (µ2 ). Since δ = cδ ρ 2 , we see that for ε0 = cε ρ −3 , the contribution of F0 to the bounds (2.20)–(2.22) can be made arbitrarily small, choosing cε sufficiently small, independently of ρ, or of the size of the system L. So what we need is more detailed information on r2 . Note that r2 inherits the bounds of r and r1 , but with a factor ε−4 , so that we have to work a little more to show that the bounds on r2 are finite as ε → 0, and that (2.20)–(2.22) are also satisfied when the contribution of r2 is taken into account. The essential input will be that as a dynamical variable, r2 0) . Since we know that s(µ) exists, satisfies (1.38) with r2 (x, 0) = r2,0 (x) ≡ εr04 − r1 (µ ε4 we can view (1.38) as a linear inhomogeneous equation for r2 and derive bounds from it. These bounds are proved in the four following lemmas, where, for convenience, we write F6 (s(µ), µ) = F6 (µ). Lemma 4.1. If r2 solves (1.38) with r2 (x, 0) = r2,0 (x), then for all µ ∈ Bσ (cη ρ), one has |||r2 (µ)|||L2 −1 |||χLµ,r Lv r2 (µ) |||L2
≤ r2,0 L2 + |||F6 (µ)|||L2 , (4.3) √ −1 −1 ≤ |||χ Lµ,r Lv r2,0 |||L2 + 2|||Lµ,r Lv F6 (µ) |||L2 . (4.4)
Proof. Let r4 = χLµ,r L−1 v r2 , then using Young’s inequality and Proposition F.2 (see Appendix F) we get that r2 and r4 satisfy
χ 2 (r2 , r2 ) +
F6 (µ) 2L2 , ε4 χ ε4 χ 2χ 2 ∂t (r4 , r4 ) ≤ − 4 (r4 , r4 ) + 4 Lµ,r L−1 v F6 (µ) L2 . ε ε Integrating these differential inequalities completes the proof. ∂t (r2 , r2 ) ≤ −
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G. van Baalen
Lemma 4.2. Assume that the solution r2 of (1.38) satisfies ε 2 |||χ Lµ,r L−1 v r2 (µ) |||L2 ≤ cη ρ. Then for all γ > 1, there exist constants C and cε such that for all µ ∈ Bσ (cη ρ) and for all ε ≤ cε with cε sufficiently small, one has L F (µ) χ L χL µ,r µ,r 6 µ,r r2 (µ) ≤γ r2,0 W ,σ + + Cε 2 cη ρ . W ,σ Lµ Lµ Lµ GLr W ,σ
Proof. We define r3 = P>
χL
µ,r r Lµ 2
(4.5)
,
and we note that |||r3 |||W ,σ < ∞, because χ L C C µ,r |||r3 |||W ,σ = r2 ≤ |||r2 |||σ −1 ≤ 4 |||r(µ)|||σ −1 + |||r1 (µ)|||σ −1 . σ Lµ δ δε On the other hand, r3 satisfies ∂t r3 = − εχ4 GLr r3 +
Lµ,r χ 16 P> Lµ µLµ r3
+
χ F (µ, P< r2 ) ε4 9
,
(4.6)
with r3 (x, 0) = r3,0 (x) and F9 (µ, P< r2 ) =
Lµ,r Lµ,r χ ε4 F6 (µ) . µLµ,r P< r2 + P> P> 16 Lµ Lµ
Using that |||r3 |||L2 ≤ cη ρ by hypothesis on r2 , and ε 2 Lµ f σ −2 ≤ 2δ 2 f σ , we get ε2 δ −3 |||µLµ r3 |||W ,σ −3 ≤ Cδ −5/2 |||µ|||σ |||r3 |||σ ≤ Ccη ρ + C|||r3 |||W ,σ . Using this estimate, Duhamel’s formula for the solution of (4.6) and Lemma F.1, we get F (µ, P r ) < 2 9 |||r3 |||W ,σ ≤ r3,0 W ,σ −1 + + Cε 2 cη ρ + Cε 2 |||r3 |||W ,σ . W ,σ GLr This gives an estimation on |||r3 |||W ,σ if ε ≤ cε with cε sufficiently small. Using Lemma F.1, P< Lv f σ = P< Lv f L2 ≤ 4δ 2 f L2 and P< f σ = P< f L2 , we get χ ε4 Lµ,r ≤ Cε 4 δ −1/2 ρ|||χ P< Lµ,r L−1 µLµ,r P< r2 v r2 |||σ W ,σ 16 Lµ GLr 2 = Cε 4 |||χ Lµ,r L−1 v r2 |||L2 ≤ Cε cη ρ .
Choosing cε sufficiently small completes the proof.
Lemma 4.3. Let r2 (µi ) be the solution of (1.38) with µ = µi , and define r2 = r2 (µ1 ) − r2 (µ2 ) and F6 = F6 (µ1 ) − F6 (µ2 ). Assume that 2 χ Lµ,r 4ε2 |||χ Lµ,r L−1 r (µ ) ||| + ε r (µ ) ≤ cη ρ . (4.7) 2 2 i 2 i L v W ,σ Lµ Then for all µi ∈ Bσ (cη ρ), there exists a constant C such that −1 5/2 ρ|||µ − µ ||| . |||χ Lµ,r L−1 1 2 σ v r2 |||L2 ≤ |||χ Lµ,r Lv F6 |||L2 + Cδ
(4.8)
Phase Turbulence in the Complex Ginzburg-Landau Equation
637
Proof. The proof follows from Lemma 4.1, with the replacements r2,0 = 0, r2 ↔ r2 , . We estimate the additional term by F6 ↔ F6 + µLµ,r r+ ε4
χ Lµ,r L−1/2 (µLµ,r r+ ) L2 ≤ 8ε 4 µLµ,r r+ L2 . v 8 Using (4.7), defining r3 in terms of r+ as in (4.5) in terms of r2 , we have 4 ε4 µLµ,r r+
L2 ≤ ε 4 µP< Lµ,r Lv L−1 v r+ L2 + ε µLµ r3 L2
4 ≤ Cε 4 δ 5/2 µ σ Lµ,r L−1 v r+ L2 + ε µLµ r3 σ −2
≤ Cε 2 δ 5/2 µ σ cη ρ + Cε 2 δ 5/2 µ σ r3 σ , since P< Lv f L2 ≤ 3δ 2 f L2 , ε 2 Lµ f σ −2 ≤ 2δ 2 f σ and ε 2 r3 σ ≤ cη ρ and the proof is completed. Lemma 4.4. Let r2 (µi ) be the solution of (1.38) with µ = µi , and define r2 = r2 (µ1 ) − r2 (µ2 ) and F6 = F6 (µ1 ) − F6 (µ2 ). Assume that 2 χ Lµ,r r2 (µi ) ≤ cη ρ , 4ε2 |||χ Lµ,r L−1 v r2 (µi ) |||L2 + ε W ,σ Lµ
(4.9)
ε2 |||χ Lµ,r L−1 v r2 (µi ) |||L2 ≤ |||µ1 − µ2 |||σ . (4.10)
Then for all γ > 1 and for all µi ∈ Bσ (cη ρ) and for all ε ≤ cε sufficiently small, there exists a constant C such that χ L Lµ,r µ,r r2 ≤ γ Lµ G F + C|||µ − µ ||| 1 2 σ . 6 W ,σ Lr W ,σ Lµ
(4.11)
Proof. The proof follows from Lemma 4.2, with the replacements of the proof of Lemma 4.3 for F6 and µLµ,r r2 , we omit the details. √ We can now show that Condition 2.11 is satisfied if ε ≤ ε0 ≤ cε 1 − 2 α 2 ρ −4 with cε sufficiently small, α 2 < 21 and if the initial data µ0 and s0 are in the class C. Proposition 4.5. Let α 2 < 21 , δ = cδ ρ 2 , and assume that r2,0 is an admissible initial condition. For√all γ > 1, there exist a constant cε sufficiently small such that for all ε ≤ ε0 ≤ cε 1 − 2 α 2 ρ −4 there exist constants λ1 < 1 and λ2 < 1 such that for all µ ∈ Bσ (cη ρ), one has 2 F (µi ) 4 ε 2 |||L−1 F (µ ) ||| + ε ≤ λ1 c η ρ , 2 i L v W ,σ Lµ 2 F F ||| + ε ≤ λ1 |||µ1 − µ2 |||σ , 4 ε2 |||L−1 2 L v Lµ W ,σ ε 2 F (µ) ||| ≤ λ cs ρ 3 . ε2 |||r2 (µ)|||L2 + ε 2 |||L−1 2 2 L v ε0
(4.12) (4.13) (4.14)
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G. van Baalen
Proof. We recall that F = F0 + χ Lµ,r r2 . The logical order of the proof is to start with (4.14), which follows from Definition 2.8, Lemma 4.1, Proposition C.6 and (4.1) if cε is sufficiently small. Using the same results, we get ε 2 |||χ Lµ,r L−1 v r2 (µ) |||L2 ≤ cη ρ (this is needed in Lemma 4.2), and (4.12) then follows from Definition 2.8, Lemma 4.2, Proposition C.6 and (4.1), again if cε is sufficiently small. Now, since λ1 < 1, we get (4.7) from (4.12), so the hypothesis of Lemma 4.3 is fulfilled, which in turn shows that both hypotheses (4.9) and (4.10) of Lemma 4.4 are fulfilled, and then (4.13) follows from Definition 2.8, Lemmas 4.3 and 4.4, Proposition C.6 and (4.2), again if cε is sufficiently small. We end this paper by noting that the class C is almost preserved by the time evolution, in the sense that the solution of the Complex Ginzburg Landau equation with corresponding initial data exists for all times and is for all times in a (larger) class C characterized by the same constants as those of C, except for cs , cη , λ1 , λ2 which are larger than cs0 , cη0 , λ1,0 , λ2,0 . Acknowledgement. The author would like to express his gratitude to Jean-Pierre Eckmann and Pierre Collet for proposing the problem. Jean–Pierre Eckmann’s suggestions and advice during the elaboration of the results and the redaction of the paper were invaluable. Finally, the author would also like to thank the referee for encouraging him to give more concise versions of the proofs, and Pierre Collet, Martin Hairer, Thierry Gallay, Serge¨i Kuksin and Emmanuel Zabey for helpful discussions.
A. Coercive Functional for the Phase Proposition A.1. Let (v, w) = vw, (v, w)γ φ = v(Lµ + γ φ )w and 4 Lv (k) = 13 1+k . ε2 k 2 1+
(A.1)
2
For all L ≥ 2π, there exist a constant K and an antisymmetric periodic function φ such that for all γ ∈ [ 41 , 1] and ε ≤ L−2/5 , and for every antisymmetric periodic function v, one has 3 (Lv v, Lv v) ≤ (v, v)γ φ ≤ φ ∞ (v, v) + (v , v ) , 4 (φ, φ)γ φ ≤ K L16/5 and (φ, φ) ≤ 43 L3 . Proof. The proof is based on a similar result of [CEES93] for Lµ,c = ∂x4 + ∂x2 . We will need some technical alterations of their proof to take into account that Lµ is of lower order than Lµ,c . However, by (1.31) and (1.36), the two operators are equal in the limit ε → 0, so we will recover their result as a particular case. As we will see, in the statement of Proposition 2.1, the restriction ε ≤ (π L2/5 )−1 is a convenient one because then we can use the same function φ as that defined in [CEES93]. We will see later that we need a much stronger restriction on ε anyway. The proof really amounts to construct the function φ. Let q ≡ 2π L ≤ 1 and M be the 1 7/5 smallest integer (strictly) larger than 2 L . We define φ by φ(x) =
n∈Z
eiqnx φn ,
Phase Turbulence in the Complex Ginzburg-Landau Equation
639
where the Fourier coefficients φn are given by
φn =
0,
n=0
4i 1 ≤ |n| ≤ 2M , qn , 4 i f (|n|/2M−1) , otherwise qn
where f is a non-increasing C 1 function satisfying f (0) = 1, f (0) = 0 and
f ≥ 0, sup |f | < 1,
∞
dk (1 + k)2 |f (k)|2 < ∞ .
0
The proof then follows from the three technical lemmas below.
Lemma A.2. There exists a constant K such that the function φ defined above satisfies (φ, φ) ≤
4 3
L3 , (φ, φ)γ φ ≤ K L16/5 and (v, v)γ φ ≤ K L7/5 v 2L2 + v 2L2
for all periodic antisymmetric functions v. Proof. For the first inequality, we have
∞ ∞ 43 π 1 1 4π 3 4 4π 2 |φn | ≤ 3 = = L3 . (φ, φ) = q q n2 6 q 3 n=1
n=1
For the second inequality, we use that φ is periodic, so that (φ, φ)γ φ = (φ, Lµ φ) =
φ 2 φ = 0, giving
∞ 4π Lµ (qn) |φn |2 , q n=1
where Lµ is defined in (1.30). Since Lµ (qn) ≤ (qn)4 and M < L7/5 , we get (φ, φ)γ φ = (φ, Lµ φ) = ≤4 πq 3
≤ C L16/5
∞ 4π Lµ (qn) |φn |2 q n=1
n 2 n 2 n + (2M) f 1+ 2M 2M n=1 n=1
∞ 1+ dk (1 + k)2 f (k)2 . 2M
2
2
∞
0
Finally, using again Lµ (qn) ≤ (qn)4 , we have (v, v)γ φ ≤ φ L∞ v 2L2 + v 2L2 .
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G. van Baalen
Using the Cauchy–Schwartz inequality, we have
φ
L∞
∞
n ≤2 |qn| |φn | ≤ 16M + 2 f 2M n=1 n=1 ∞ 1 + k |f (k)| dk ≤ 16M + 4M 1+k 0 ∞
∞
≤ C L7/5 1 +
dk (1 + k)2 |f (k)|2
.
0
This completes the proof of the lemma.
Lemma A.3. For all L ≥ 2π, for all γ ∈ [ 41 , 1] and for all ε ≤ (v, v)γ φ ≥
1 Mq ,
one has
3 (Lv v, Lv v) . 4
(A.2)
Proof. Following [CEES93] one shows first that (v, v)γ φ
2 = 2L (Lµ (qn) + γ ψ2n )vn + 2γ vk vm (ψ|k+m| − ψ|k−m| ) , n>0
k>m>0
where ψn = −iqn φn . Then one notices that for 0 ≤ ε ≤ 1, one has Lµ (qn) + γ ψ2n ≥ τ (qn)2 ≡
1 1 + (qn)4 1 (qn)4 ≥ τ1 (qn)2 ≡ . 2 2 2 1 + ε (qn) 2 1 + ε2 (qn)2 2 2
The definition of τ here is different from that of [CEES93], except in the ε = 0 limit.
We now define wn = vn τn (in particular w = (w, w) = L
0<m
wk
3 2
Lv v), and the operator by
ψ|k+m| − ψ|k−m| wm , τk τm
(A.3)
and get (v, v)γ φ ≥ (w , (Id + 2γ ) w) ≥
3 1 (w, w) = (Lv v, Lv v) , 2 4
since (see Lemma A.4 below), the Hilbert–Schmidt norm of 2γ is less than 21 .
Lemma A.4. Let the operator be defined by (A.3). For all L ≥ 2π , for all γ ∈ [ 41 , 1] 1 and for all ε ≤ Mq , the Hilbert–Schmidt norm of 2γ .
Phase Turbulence in the Complex Ginzburg-Landau Equation
641
1 Proof. Note again that ε ≤ Mq = (π L2/5 )−1 is only a convenient restriction (see remark above). To prove this lemma, it is sufficient (see [CEES93]) to show that
2HS
ψ|k+m| − ψ|k−m| 2 < 1 . ≡ τk τm 16
(A.4)
0<m
By definition of φ, for all k > m > 0, we have |ψk−m − ψk+m | = 0 , if k + m ≤ 2M , m , for all k > m . |ψk−m − ψk+m | ≤ 4 min 1, M We distinguish two sets of summation indices S = SI ∪ SII in the sum (A.4), SI = (m, k) ∈ N2 s.t. M + 1 ≤ m and m + 1 ≤ k , SII = (m, k) ∈ N2 s.t. 1 ≤ m ≤ M and 2M − m + 1 ≤ k , and write 2HS = TI + TII accordingly. In the region I, we have m > M, and using ε ≤ ∞
TI ≤ 16
m=M+1
1 Mq
and
1 τ (k)
≤
1 τ1 (k) ,
we get
∞ ∞ ∞
1 1 1 1 ≤ 16 dm dk , 2 2 2 τ (qm) τ (qk) τ1 (qm) m τ1 (qk)2 M k=m+1
whereas in the region II, we have m ≤ M and k ≥ M + 1, and using again ε ≤ 1 τ (k)
≤
1 τ1 (k) ,
TII ≤
1 Mq
and
we get
M 16 m2 M2 τ (qm)2 m=1
∞
k=2M−m+1
∞ M 1 16 m2 1 ≤ dk τ (qk)2 M2 τ (qm)2 M τ1 (qk)2 m=1
M 160 1 1 q 2 m2 q 4 m4 1 ≤ + 3 M5 q4 q 2 1 + m4 q 4 2 M 2 q 4 1 + m4 q 4 m=1 ∞
160 1 dm 1 1 ≤ . + 3 M 5 q 4 q 2 0 1 + q 2 m2 2 M q4 Collecting these results, we get 2HS ≤
1 80 π 3 q7 M5
+
1 440 9 q 8 M 6 . This bound is worse than that of [CEES93] by numerical factors only (in their bound 803 π is replaced by 128 3 16 1 1 and 440 9 by 3 ), but is uniform in ε ≤ Mq . This motivates the restriction ε ≤ Mq . The proof is then completed using M > 21 L7/5 .
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G. van Baalen
B. Properties of the Spaces Wσ Lemma B.1. Let σ ≥ 23 . There exists a constant C such that for all n ≤ σ − all m ≤ σ − 1, we have
f (m) σ −m + G f (m) σ −m ≤ C δ m
f
(n)
L∞ + G f
(n)
L ∞
≤ Cδ
n+ 21
3 2
and for
f σ ,
(B.1)
f σ ,
(B.2)
where f (m) is the mth order spatial derivative of f . Proof. Throughout the proof, we use that G acts multiplicatively in Fourier space, (G f )n = G(qn) fn with G(k) ≤ 1 (see 1.36), so that G is a bounded operator in the l p and · σ norms. Using that f L∞ ≤ f l 1 , and that the space derivative commutes with G, we see that we need only prove (B.1) and (B.2) for the terms without G, and with L∞ replaced by l 1 in (B.2). In the sequel, we denote by K the operator with symbol K(k) = |k|. √ √ For (B.1), we use that |x| ≤ 1 + x 2 and that · L2 = L · l 2 to show that
f (m) σ −m = f (m) W ,σ −m + f (m) L2 ≤ δ m f W ,σ + f (m) L2 √ ≤ δ m f W ,σ + P< f L2 + L δ m (1 + (K/δ)2 )m/2 P> f l 2 ∞ 1/2 2π dx ≤ δ m f σ + δ m
f W ,σ ≤ Cδ m f σ . 2 σ −m −∞ (1 + x ) For (B.2), using the Cauchy–Schwartz inequality, we have
P< f l 1 ≤
2 δ 1/2 q
P< f l 2 ≤
√ √ δ L P< f l 2 ≤ δ P< f L2 ,
(B.3)
so that 1
f (n) l 1 ≤ K n P< f l 1 + K n P> f l 1 ≤ δ n+ 2 f L2 + K n P> f l 1 ∞ 1 1 dx n+ 21 ≤ δ n+ 2 f L2 + δ n+ 2
f σ . σ −n f W ,σ ≤ C δ 2 2 −∞ (1 + x ) The proof is completed.
Proposition B.2. Let δ ≥ 2, then there exist constants C1 and C2 such that −Lµ t f (·) ≤ f (·) W ,σ , e t W ,σ g (·, s) ds e−Lµ (t−s) g (·, s) ≤ sup , L µ 0≤s≤t 0 W ,σ W ,σ f + G g C1 C2
f W ,σ −1 + 3 g W ,σ −3 , ≤ L δ δ µ
(B.4)
W ,σ
where Lµ is defined in (1.30) and e−Lµ t is the propagation Kernel associated with ∂t f = −Lµ f .
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Proof. The propagation Kernel e−Lµ t acts as e−Lµ t f n = e−Lµ (qn)t fn in Fourier space. For ε ≤ 1 and |k| ≥ δ ≥ 2, by (1.30), one has Lµ (k) ≥ Lµ (δ) ≥ 4, which gives ≤ e−Lµ (δ) t f (·) W ,σ ≤ f (·) W ,σ . sup e−Lµ t f (·) W ,σ
t≥0
Next, we use that t ds e−Lµ (t−s) g (·, s)
W ,σ
0
(·, s) ≤ sup Lµ (qn) ds e L µ 0 W ,σ |n|≥ qδ g (·, s) sup ≤ sup 1 − e−Lµ (qn) t . L µ 0≤s≤t W ,σ |n|≥ δ
t
−Lµ (qn) (t−s) g
q
Since 1 − e−Lµ (qn) t ≤ 1 for qn ≥ δ ≥ 2, the proof of (B.4) is completed. Finally, we have √ f qn 2 σ/2 δ ε 2 (qn)2 |qn| |fn | ≤ 1+ sup 1 + L 4 − (qn)2 q δ 2 (qn) δ µ W ,σ |n|≥ q √ 1 + x 2 1 + x2 1 ≤ f W ,σ −1 sup sup 2 , δ x x≥1 x≥2 x − 1 √ G g qn 2 σ/2 |qn| |gn | δ ≤ sup 1 + L q δ (qn)4 − (qn)2 δ µ
W ,σ
|n|≥ q
≤
1 sup
g W ,σ −3 δ3 x≥1
This completes the proof.
√
1 + x 2 3 x4 sup 4 . 2 x x≥2 x − x
Proposition B.3. Let u σ1 < ∞, v σ2 < ∞ and σ = min(σ1 , σ2 ) ≥ 23 . Then there exists a constant Cm depending only on σ such that √
uv σ ≤ Cm δ u σ1 v σ2 , (B.5) u
u σ1 (B.6) 1 + v ≤ 1 − C √δ v , σ m σ2 √ provided Cm δ v σ2 < 1. If σ ≤ 1, we have the two particular cases √ √
uv W , 1 ≤ Cm δ u 1 v 1 and uv W ,0 ≤ Cm δ u L2 v L2 . (B.7) 2
Proof. We first note that if σ = min(σ1 , σ2 ) ≥ 23 , by Lemma B.1, we have √
uv L2 ≤ u L∞ v L2 ≤ C δ u σ1 v σ2 . So the L2 part of (B.5) is proved. For the · W ,σ part of (B.5) and for (B.7), we write u = u< + u> , where u< = P< u and u> = P> u and the same for v. Then we have
uv W ,σ ≤ uv N ,σ ≤ u< v< N ,σ + u< v> N ,σ + u> v< N ,σ + u> v> N ,σ .
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Clearly, P> f N ,σ ≤ f W ,σ ≤ f σ , so that we can apply Lemma B.4 below to the last term (see [BKL94] for the original version of the lemma). The first three terms in turn are bounded using Lemma B.5. 1 To prove (B.6), we write a geometric series for 1+v and use (B.5) inductively, getting m
√ u
uv m σ ≤ u σ1 . Cm δ v σ2 1 + w ≤ σ m≥0
Summing the series since Cm
√
m≥0
δ v σ2 < 1 completes the proof.
Lemma B.4. Let σ1 , σ2 ≥ 23 and σ = min(σ1 , σ2 ) ≥ 23 , then there exists a constant cb depending only on σ such that √
uv N ,σ ≤ cb δ u N ,σ1 v N ,σ2 , (B.8) and if σ < 1, we have the two particular cases √ √
uv N , 1 ≤ cb δ u N ,1 v N ,1 and uv N ,0 ≤ cb δ u L2 v L2 . (B.9) 2
Proof. We begin with the second inequality of (B.9). We have √
√ √ δ
uv N ,0 = |un | |vm−n | ≤ δ L u l 2 v l 2 ≤ δ u L2 v L2 . sup q n∈Z m∈Z
Next, we define p = qδ , x = pn and y = pm. We have σ (1 + (pn)2 ) 2 1 |um ||vn−m |
uv N ,σ ≤ √ sup p δ n∈Z m∈Z ∞ √ 1 + x 2 σ2 1 ≤ δ u N ,σ1 v N ,σ2 sup dy , (B.10) 2 2 x∈R −∞ 1 + y 1 + (x − y) 1 ∞ √ (1 + x 2 ) 4 1
uv N , 1 ≤ δ u N ,1 v N ,1 sup dy . (B.11) 1 1 2 x∈R −∞ (1 + y 2 ) 2 (1 + (x − y)2 ) 2
To see that both integrals in (B.10) and (B.11) are uniformly bounded in x ∈ R, we can assume without loss of generality that x ≥ 0, and use the uniform bounds 1 1 1 + x2 ≤ 4 and ≤ , 2 2 1 + (x − y) 1 + (x − y) 1 + y2 1 + x2 1 1 y ∈ [x/2, ∞) ⇒ ≤ 4 and ≤ , 2 2 1+y 1+y 1 + (x − y)2
y ∈ (−∞, x/2] ⇒
from which we get the desired result.
Lemma B.5. Let σ ≥ 0, then there exists a constant C depending only on σ such that √
(P> u) (P< v) N ,σ ≤ C δ u σ v σ , (B.12) √ (B.13)
(P< u) (P< v) N ,σ ≤ C δ u σ v σ .
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Proof. Let p = qδ . Using that |p(n − m)| ≤ 1 implies 1 + (pn)2 ≤ 3(1 + (pm)2 ) we get easily √
σ |vn−m | qδ (1 + (pm)2 ) 2 |um | .
(P> u)(P< v) N ,σ ≤ 3 sup n∈Z |pm|≥1, |p(n−m)|≤1
∞ ∞ The proof of (B.12) √ then follows since v u l ≤ v l 1 u l , using u W ,σ ≤ u σ and P< v l 1 ≤ δ v L2 (see (B.3) above). Similarly, since |pm| ≤ 1 and |p(m−n)| ≤ 1 implies |pn| ≤ 2, we have
σ √ |um | |vn−m | .
(P< u)(P< v) N ,σ ≤ 5 2 δ L sup
n∈Z |pm|≤1, |p(m−n)|≤1
The proof is completed since L u v l ∞ ≤ L u l 2 v l 2 = u L2 v L2 ≤ u σ v σ . C. Bounds on Nonlinear Terms C.1. Bounds on r1 . Proposition C.1. Assume that µ1 σ ≤ cη ρ, µ2 σ ≤ cη ρ, δ = cδ ρ 2 and ε ≤ 1, and let r1 be defined by (1.32). Then there exists a constant cr1 such that
r1 (µ) σ −1 ≤ cr1 δ ρ ,
r1 (µ1 ) − r1 (µ2 ) σ −1 ≤ cr1 δ µ1 − µ2 σ −1 .
(C.1) (C.2)
Proof. Using Lemma B.1, Proposition B.3 and µ21 − µ22 = (µ1 − µ2 )(µ1 + µ2 ), we have Cm
r1 (µi ) σ −1 ≤ δ µi σ + µ2i σ ≤ δ µ σ 1 + √ µ σ , δ Cm
r1 (µ1 ) − r1 (µ2 ) σ ≤ δ µ1 − µ2 σ 1 + √ µ1 + µ2 σ . δ The proof is completed noting that µi σ ≤ cη ρ and δ = cδ ρ 2 .
C.2. Bounds on F0 . Theorem C.2. Let δ = cδ ρ 2 , and suppose that for all µ, µi ∈ Bσ (cη ρ), we have r(µ) ∈ Bσ −1 (cr δ ρ) and r(µ1 )−r(µ2 ) σ −1 ≤ cr δ µ1 −µ2 σ . Let F0 = F0 (µ1 )−F0 (µ2 ). Then there exist constants cε and cF0 such that for all ε ≤ cε ρ −1 we have F0 (µ)
F0 (µ) σ −2 + ≤ cF0 δ 5/2 ρ 2 , (C.3) Lµ W ,σ F0
F0 L2 + ≤ cF0 δ 5/2 ρ µ1 − µ2 σ . (C.4) L µ
W ,σ
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Proof. We first note that since s = G r, from the definition of G (see (1.36)) we have
s σ −1 + εs σ −1 + ε 2 s
σ −1
≤ 5 r σ −1 ≤ C δ ρ .
(C.5)
Since r σ −1 ≤ cr δ ρ and µ ∈ Bσ (cη ρ), for all ε ≤ cε ρ −1 with cε sufficiently small, we have 1 − Cm ε 4 α 2
√
δ r σ −1 ≥ 1/2
and ε 3
√
δ µ σ ≤ C .
(C.6)
Let σi = σ − i, i = 1, 2, 3. Using (C.5), (C.6) and Proposition B.3, we easily show that 2 s
σ1
s(ε 2 s) + 1 + ε4 α 2 s
µ2 σ3 + (µ2 ) σ3 + µr σ3
µs(ε 4 s) + 4α2 s 1 + ε σ1 σ1 µs 1 + ε 4 α 2 s σ2 + µ r σ3 + µ s σ3 2 µ (ε s ) σ3
≤ Cδ 5/2 ρ 2 ,
(C.7)
≤ Cδ 5/2 ρ 2 ,
(C.8)
≤ Cδ 5/2 ρ 2 ,
(C.9)
≤
Cδ 7/2 ρ 2
.
(C.10)
Applying Lemma B.1 for the two last terms of F0 (µ), and using (C.7) and (C.8), we see that F0 (µ) σ −2 ≤ Cδ 5/2 ρ 2 , which proves the first part of (C.3). To prove the remainder of (C.3), we use that 1 s µ = G µ r + 2µ r − 2µ s − µ (ε 2 s ) , 2 which follows from easy algebra5 , to get F0 (s, µ) = F1 (s, µ) + G F2 (s, r, µ), where
α 2 (ε 4 s )sµ 2α 2 s(ε 2 s ) 2 + ε 2 (1 + α 2 ) s 2 − − , 1 + ε4 α 2 s 1 + ε4 α 2 s 1 1 χ α2 F2 (s, r, µ) = − µ2 − (µ2 ) + 2µr + 4µ r − 4µ s − µ (ε 2 s ) . 4 4 2 F1 (s, µ) = χ α 2
We then use Proposition B.2 which gives F0 (µ) C1 C2
F1 (µ) W ,σ −1 + 3 F2 (µ) W ,σ −3 . ≤ L δ δ µ W ,σ Using (C.9) and (C.10) for the F2 –term and (C.7) for the F1 –term completes the proof of (C.3), while equalities like a1 b1 − a2 b2 = (a1 − a2 )b1 + (b1 − b2 )a2 and ab11 − ab22 = a1 −a2 b1
1 + ab22 b2b−b show that the estimates needed to prove (C.4) are similar to those for 1 (C.3), we omit the details.
5
2 act on both sides of this equation with 1− ε2 ∂x2 and use that ε2 s = 2s −2r and ε2 s = 2s −2r .
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2 2 Corollary C.3. Let α < 1 and F7 (µ) = ε8 ∂x + ε 2 µ F0 (µ) as in (1.41). Then there exist constants cε and cF7 such that for all ε ≤ cε ρ −2 the following bounds hold: Lµ,r + F7 (µi ) ≤ cF7 δ 5/2 ρ 2 , (C.11) L2 Lµ GLr W ,σ Lµ,r F + F ≤ cF7 δ 5/2 ρ µ σ , (C.12) Lµ,r L−1 v 7 2 7 L Lµ GLr W ,σ
Lµ,r L−1 v F7 (µi )
where F7 = F7 (µ1 ) − F7 (µ2 ) and µ = µ1 − µ2 . Proof. We first use that Lµ,r L−1 v f L2 ≤ 16 f L2 (see Lemma F.1 in Appendix F), 2 so that for the L bounds in (C.11) and (C.12), we need only bound F7 (µi ) L2 and
F7 (µ1 ) − F7 (µ2 ) L2 . Then we have also f L2 ≤ f σ for any σ > 0, from which we get
√
F7 (µi ) L2 ≤ ε2 F0 (µi ) σ −4 + ε 4 C δρ F0 (µi ) σ −3 . Using f σ −3 ≤ δ f σ −2 , F0 (µi ) σ −2 ≤ cF0 δ 5/2 ρ 2 and ε ≤ cε ρ −2 gives the desired result. Similarly, since µ1 F0 (µ1 ) − µ2 F0 (µ2 ) =
1 1 µ F+ + (µ1 + µ2 ) F− , 2 2
(C.13)
where µ = µ1 − µ2 and F± = F0 (µ1 ) ± F0 (µ2 ), we also have
F7 L2 ≤ C1 F− σ −2 + C2 ε 2 F+ σ −2 µ1 − µ2 σ . The proof of (C.12) is completed noting that F− σ −2 ≤ cF0 δ 5/2 ρ µ1 − µ2 σ , and using again ε ≤ cε ρ −2 . For the remainder of the proof of (C.11), we use Lemma F.1 to get Lµ,r Lµ,r ε2 f ≤ C
f
and f ≤ Cδ −3 f W ,σ −3 W ,σ GL L GL r µ r W ,σ W ,σ for some constant C, and we conclude that Lµ,r L GL F7 (µi ) µ
r
W ,σ
F0 (µi ) 4 −3 ≤C + ε δ µi F0 (µi ) σ −3 Lµ W ,σ F0 (µi ) ≤C + F0 (µi ) σ −2 , L µ W ,σ
which gives the desired result. The proof of the remainder of (C.12) is very similar (use e.g. (C.13) and proceed as above), we omit the details.
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C.3. Bounds on F3 and F4 . Proposition C.4. Let cη , cs > 0 and δ = cδ ρ 2 > 2. For all 0 < ζ < 1, there exists a constant cε such that for all ε ≤ cε ρ −3 , for all µi ∈ Bσ (cη ρ) and for all si ∈ Bσ −1 (cs δ ρ) the following bounds hold: ε4
F3 (si , µi ) σ −1 ≤ ζ δρ , χ ε4
F3 (s1 , µi ) − F3 (s2 , µi ) σ −1 ≤ ζ s1 − s2 σ −1 , χ ε4
F3 (si , µ1 ) − F3 (si , µ2 ) σ −1 ≤ ζ δ µ1 − µ2 σ −1 . χ Proof. The proof is very easy. For instance, we have ε4 si2 σ −1 + ε 6 si µ2i σ −1 + ε 8 si3 σ −1 ≤ C ε 4 δ 3/2 ρ + ε 6 δρ 2 + ε 8 δ 3 ρ 2 δρ , which can be made arbitrarily small choosing cε sufficiently small.
Proposition C.5. Let F5 (s, µ) = F3 (s, µ) + F4 (s, µ). There exist constants cε and cF5 such that for all ε ≤ cε ρ −2 , for all µi ∈ Bσ (cη ρ) and for all maps s satisfying
Ls s(µi ) σ −1 ≤ cs δ ρ and Ls (s(µ1 ) − s(µ2 )) σ −1 ≤ cs δ µ1 − µ2 the following bounds hold:
Ls F5 (s(µi ), µi ) σ −3 ≤ cF5 δ 5/2 ρ 2 ,
Ls F5 (s(µ1 ), µ1 ) − Ls F5 (s(µ2 ), µ2 ) σ −3 ≤ cF3 δ Proof. We first note that since Ls = 1 −
ε2 2 2 ∂x ,
5/2
(C.14)
ρ µ1 − µ2 σ . (C.15)
we have
Ls f σ −3 ≤ 1 + C ε 2 δ 2 f σ −1 ≤ C f σ −1 . Then, as in the proof of Proposition C.4, for the contribution of F3 , we have ρ
si2 σ −1 + ε 2 si µ2i σ −1 + ε 4 si3 σ −1 ≤ Cδ 5/2 ρ 2 1 + ε 2 √ + ε 4 δ 3/2 ρ , δ where s(µi ) = si . For the contribution of F4 , we have √
Ls (s(µi )µi ) σ −3 ≤ C δ s(µi ) σ −1 µi σ −1 ≤ Cδ 5/2 ρ 2 , and for the other term in F4 , we use Ls (s µ) = µ(Ls s ) + 2µ (Ls s) + s (Ls µ) − 2sµ − s µ , and get Ls (s(µi ) µi ) σ −3 ≤ Cδ 5/2 ρ 2 . The proof of (C.15) is similar, we omit the details.
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C.4. Bounds on F6 . Proposition C.6. Let α 2 < 1, δ = cδ ρ 2 and F6 = F6 (µ1 ) − F6 (µ2 ). For all ζ < 1, there exist constants cε and cF6 such that for all ε ≤ cε ρ −2 and for all µi ∈ Bσ (cη ρ), the following bounds hold: 2 Lµ,r α2 ε F6 (µi ) ≤ max 13 , 1−α (C.16) 2 + ζ cη ρ , Lµ GLr W ,σ Lµ,r 1 α2 ε2 + ζ
µ1 − µ2 σ , F ≤ max , (C.17) 6 L GL 3 1−α 2 µ r W ,σ
F6 (µi ) L2 ≤ cF6 δ 5 ρ + δ 5/2 ρ 2 , (C.18) 4 5/2 2 (C.19) ρ , Lµ,r L−1 v F6 (µi ) 2 ≤ cF6 δ ρ + δ L 4 5/2 ρ µ1 − µ2 σ . (C.20) Lµ,r L−1 v F6 2 ≤ cF6 δ + δ L
Proof. It is crucial for the phase equation that the prefactor of cη ρ in (C.16) and of |||µ1 − µ2 |||σ in (C.17) is smaller than 1. Using Lemma F.1 of Appendix F, we see that the term 18 Lµ µ in F6 gives the leftmost contribution in (C.16)–(C.20), since 1 α2 ε2 Lµ,r µ ≤ max
µ W ,σ , , 8 G Lr 3 1 − α2 W ,σ 4 4 4
Lµ,r L−1 v Lµ µ L2 ≤ C (1 + ∂x ) µ L2 ≤ C δ µ σ ≤ C δ ρ ,
Lµ µ L2 ≤ C (1 − ∂x2 )5/2 µ L2 ≤ C µ σ ≤ C δ 5 ρ , while using Lµ,r L−1 v f L2 ≤ 16 f L2 , Corollary C.3 and Proposition C.5 above, it is easy to see that F3 , F4 and F7 give a contribution to ζ in (C.16) and (C.17) which can be made arbitrarily small choosing cε sufficiently small, and part of the rightmost contribution in (C.18)–(C.20). It remains to bound the contribution of F10 (µ) = F8 (µ) − 18 Lµ µ . For the proof of (C.16), we first note that by inequality (F.5) of Lemma F.1, it is sufficient to bound
F10 (µi ) W ,σ −3 , on which we have 2 √ ε µi Lµ µi ≤ Cm δ µi σ ε 2 Lµ µi σ −2 ≤ C δ 5/2 ρ 2 , 2 W ,σ −3 2 ∂x + ε µi µi µ ≤ C1 δ 5/2 ρ 2 + ε 2 C2 δ 2 ρ 3 ≤ C δ 5/2 ρ 2 , i 2 W ,σ −3
which gives an arbitrarily small contribution to ζ in (C.16) if cε is sufficiently small. To get the contribution of F10 to (C.18) and (C.19), we use
F10 (µi ) L2 ≤ ε 2 µi Lµ µi L2 + (µi µi ) L2 + ε 2 µ2i µi L2 √ ≤ C1 δ µi σ ε 2 Lµ µi σ −2 + C2 δ 5/2 µi 2σ + ε 2 C3 δ 2 µi 3σ
µi σ ≤ Cδ 5/2 µi 2σ 1 + √ ≤ Cδ 5/2 ρ 2 . δ The proof of (C.20) and (C.17) are similar to the above, we omit the details.
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D. Proof of Proposition 2.12 Before proving Proposition 2.12, we prove a simpler lemma. Lemma D.1. Let δ and ε0 be given by Proposition 2.12, and let F(µ, ˜ µ0 ) be given by the solution of (2.19). Assume that |||F(µ, ˜ µ0 )|||σ ≤ cη ρ, and that (2.21) holds with λ1 < 1 for all ε ≤ ε0 , for all µ˜ ∈ Bσ (cη ρ), and for all µ0 ∈ B0,σ (cη ρ). Then for all 0 < cλ < 1, there exists a t1 > 0 such that for all µ˜ i ∈ Bσ (cη ρ), it holds sup F(µ˜ 1 , µ0 )(·, t) − F(µ˜ 2 , µ0 )(·, t) L2 ≤ cλ λ1 sup µ˜ 1 (·, t) − µ˜ 2 (·, t) σ .
0≤t≤t1
0≤t≤t1
Proof. Let µi = F(µ˜ i , µ0 ), i = 1, 2 and µ± = F(µ˜ 1 , µ0 ) ± F(µ˜ 2 , µ0 ). We have 1 ∂t µ− = −Lµ µ− − (µ+ µ− ) + ε 2 F , 2
µ− (x, 0) = 0 ,
(D.1)
where F = F (µ˜ 1 ) − F (µ˜ 2 ). Multiplying (D.1) by µ− , integrating over [−L/2, L/2], using Young’s inequality and L2v − 2Lµ ≤ 1, we get 1 ∂t (µ− , µ− ) = −2(µ− , Lµ µ− ) − (µ− , µ+ µ− ) + 2 ε 2 (µ− , F ) 2 1 2 ≤ (1 + µ+ L∞ ) (µ− , µ− ) + ε 4 L−1 v F L2 . 2 ˜ 1 − µ˜ 2 σ with λ1 < 1, so that By (2.21), we have ε2 L−1 v F L2 ≤ λ1 µ
sup µ− (·, t) L2 ≤ λ1
0≤t≤t1
eζ t1 −1 ζ
sup µ˜ 1 (·, s) − µ˜ 2 (·, s) σ , 0≤s≤t1
where ζ = 1 + 21 µ+ L∞ ≤ 1 + C cη δ 3/2 ρ. Setting t1 = the proof.
1 ζ
ln 1 + cλ2 ζ completes
Proposition 2.12 is then an easy consequence of the following proposition. Proposition D.2. There exist constants cδ sufficiently large and cλ sufficiently small such that if t1 is given by Lemma D.1, and F(µ, ˜ µ0 ) (the solution of (2.19)) satisfies |||F(µ, ˜ µ0 )|||σ ≤ cη ρ, and (2.21) holds with λ1 < 1 for all ε ≤ ε0 , for all µ˜ ∈ Bσ (cη ρ), and for all µ0 ∈ B0,σ (cη ρ), then there exists a constant 0 < λ < 1 such that for all µ˜ i ∈ Bσ (cη ρ), it holds sup F(µ˜ 1 , µ0 )(·, t) − F(µ˜ 2 , µ0 )(·, t) σ ≤ λ sup µ˜ 1 (·, t) − µ˜ 2 (·, t) σ . 0≤t≤t1
0≤t≤t1
Proof. We will use the same definitions as in Lemma D.1 above, and F = F (µ˜ 1 ) − F (µ˜ 2 ). We first note that we have sup µ± (·, t) σ = sup F(µ˜ 1 , µ0 )(·, t) ± F(µ˜ 2 , µ0 )(·, t) σ ≤ 2cη ρ .
0≤t≤t1
0≤t≤t1
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Then we use that Duhamel’s representation formula for the solution of (D.1) gives t 1 t −Lµ (t−s) 2 µ− (x, t) = − ds e (µ− µ+ ) (x, s) + ε ds e−Lµ (t−s) F (x, s) , 2 0 0 from which we get, using Condition 2.8, Propositions B.2 and B.3, and Lemma D.1 that Cρ sup µ− (·, t) σ ≤ λ1 (1 + cλ ) sup µ˜ − (·, t) W ,σ + √ sup µ− (·, t) σ , δ 0≤t≤t1 0≤t≤t1 0≤t≤t1 for some λ1 < 1. Since δ = cδ ρ 2 , choosing cδ sufficiently large and cλ sufficiently small completes the proof. E. Further Properties of the Amplitude Equation Corollary E.1. Assume that r0 σ −1 ≤ cs0 δ ρ. Then there exist constants cr > cs and cε such that for all ε ≤ cε ρ −2 and for all µ ∈ Bσ (cη ρ), we have |||r(µ)|||σ −1 ≤ cr δ ρ , |||r(µ1 ) − r(µ2 )|||σ −1 ≤ cr δ |||µ1 − µ2 |||σ .
(E.1) (E.2)
Proof. As a first step, we note that |||r(µ)|||σ −3 is finite, because |||r(µ)|||σ −3 ≤ |||s(µ)|||σ −3 + ε 2 |||s(µ) |||σ −3 ≤ 1 + ε 2 δ 2 cs δρ , since |||s(µ)|||σ −1 . Using ε ≤ cε ρ −2 , we also have |||r(µ)|||σ −1 ≤ |||r(µ)|||σ −3 + |||r(µ)|||W ,σ −1 ≤ ζ1 cs δρ + |||r(µ)|||W ,σ −1 ,
(E.3)
for some ζ1 > 1, while using Propositions C.1 and C.4, we have
r0 σ −1 + r1 σ −1 +
ε4 |||F3 (s, µ)|||σ −1 ≤ (cs0 + cr1 + ζ )δρ ≤ ζ2 cs δρ , χ
for some ζ2 > 1. Then, as in the proof of Proposition 3.2, we have that for all σ ≤ σ −1, |||r(µ)|||σ ≤ ζ3 cs δρ + ε 4 |||sµ |||σ −1 + |||(sµ) |||σ ≤ ζ4 cs δρ + ε 4 |||(sµ) |||σ , for some ζ4 > 1, since ε4 cs δ 3/2 µ σ is arbitrarily small if cε is sufficiently small. And 2 now, we use that sµ = G rµ − ε2 s µ − ε2 sµ , from which we get + sµ ) |||σ −1 + ε4 |||G(r(µ)µ) |||σ |||r(µ)|||σ ≤ ζ4 cs δρ + ε 6 |||G(2s µ 4 ≤ ζ4 cs δρ + ε 2|||s µ |||σ −2 + |||sµ |||σ −2 + ε 4 |||G(r(µ)µ) |||σ ≤ ζ5 cs δρ + ε 4 |||G(r(µ)µ) |||σ , (E.4)
for some ζ5 > 1, since ε 4 δ 5/2 |||µ|||σ is arbitrarily small if cε is sufficiently small. Since |||G(r(µ)µ) |||σ ≤ 2|||r(µ)µ|||σ −1 , we use (E.4) with σ = σ − 2, and then with σ = σ − 1 to conclude that |||r(µ)|||σ −1 is finite, and then we have √ |||r(µ)|||σ −1 ≤ ζ5 cs δρ + ε 3 |||r(µ)µ|||σ −1 ≤ ζ5 cs δρ + Cε 3 δ|||µ|||σ |||r(µ)|||σ −1 . √ Since ε3 δ|||µ|||σ is arbitrarily small if cε is sufficiently small, the proof of (E.1) is completed. The proof of (E.2) is similar, we omit the details.
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F. Coercive Functionals and Other Properties for the Amplitude Equation We begin with a preliminary lemma. Lemma F.1. For all ε 2 ≤ 1 and α 2 < 1/2, there exist a constant C such that 1 α2 ε2 Lµ,r f ≤ max
f W ,σ , , 8 G Lr 3 1 − α2 W ,σ
Lµ,r f σ ≤ C f σ ,
L−1 v f L 2
(1 − ∂x2 )−1 Lv f L2
Lµ,r L G L f µ
W ,σ
r
(F.1) (F.2)
≤ C f L2 ,
(F.3)
≤ f L2 ,
(F.4)
≤ C δ −3 f W ,σ −3 .
(F.5)
Proof. In terms of the Fourier coefficients, we have
2 Lµ,r (qn) ε (qn)2 ε2 Lµ,r fn , f =− 8 G Lr 8 G(qn) Lr (qn) n and
with ξ = −
ε2 k 2 2
ξ 2 (λ2 − α 2 ξ 2 ) ε2 k 2 Lµ,r (k) = , 8 G(k) Lr (k) 1 + (2 + λ2 ) ξ 2 + (1 − α 2 ) ξ 4 2 and λ2 = 1 + ε 2 1+α . Then as a function of ξ , we have 2
ξ 2 (λ2 − α 2 ξ 2 ) λ4 1 α2 ≤ ≤ ≤ , 1 − α2 1 + (2 + λ2 ) ξ 2 + (1 − α 2 ) ξ 4 λ4 + 4 λ2 + 4α 2 3
2 2 where the last inequality comes from the fact that ε ≤ 1 and α < 1 imply that 2 1 ≤ λ ≤ 2. This proves (F.1). For (F.2), we have Lµ,r f n = Lµ,r (qn) fn , and
|Lµ,r (k)| = 4
|λ2 − α 2 ξ 2 | ≤ 4 max(α 2 , λ2 ) ≤ 8 , 1 + ξ2
using the same notations. For (F.3) and (F.4), we use that 2 Lv (k) 3 k 2 1 + k2 −1 ≤1. ≤ 2 and |ik Lv (k) | ≤ 1 + k4 1 + k2 For (F.5), we have
Lµ,r f Lµ GLr
= i qn
n
Lµ,r (qn) Lµ (qn)G(qn)Lr (qn)
fn ,
and for |qn| = |k| ≥ δ ≥ 2, we have k4 Lµ,r (k) (1 + ξ 2 )(λ2 − α 2 ξ 2 ) 8 L (k)G(k)L (k) ≤ k 4 sup k 4 − k 2 sup 1 + (2 + λ2 )ξ 2 + (1 − α 2 )ξ 4 . µ r |ξ |≥0 |k|≥δ The second supremum is finite if α 2 < 1. Now, let (K −4 f )n ≡ (qn)−4 fn . We have
K −4 f W ,σ = δ −3 f W ,σ −3 , which completes the proof of (F.5).
Phase Turbulence in the Complex Ginzburg-Landau Equation
653
Proposition F.2. Let δ = cδ ρ 2 , cη > 0 and α 2 < 1. There exists a constant cε such that for all ε ≤ cε ρ −2 and for all µ ∈ Bσ (cη ρ), we have ε4 3 r2 µLµ,r r2 ≥ r22 , (F.6) r2 GLr r2 − 16 4 3 ε4 r4 GLr r4 − r (F.7) µL r4 Lµ,r L−1 r42 . v 4 ≥ v 16 4 2
Proof. We notice first that Lµ,r r2 = a1 Gr2 − a2 ε2 Gr2 with a1 = 4 + 2ε 2 (1 + α 2 ) and a2 = 4α 2 . Since (by Fourier transform) Gf L2 ≤ f L2 and ε 2 Gf L2 ≤ 2 f L2 , there exists a constant C such that √ µr2 Lµ,r r ≤ µ L∞ r2 L2 r L2 ≤ Cρ δ r2 2 2 + r 2 2 , 2 2 2 L L and we get 4
√ ε ε2 2 2 2 . r2 + (r2 ) 16 r2 µLµ,r r2 ≤ Cε ρ δ 2 2 Let now a3 = 3 + ε 2 1+α and a4 = 1 − α 2 . We have 2 r2 GLr r2 − where
ε4 16
r2 µLµ,r r2 ≥ γ
r22 ,
√ 1 + a 3 ξ 2 + a4 ξ 4 γ = min − Cε 2 ρ δ(1 + ξ 2 ) 2 ξ ∈R 1+ξ
.
Since a3 ≥ 3 and a4 > 0, choosing cε sufficiently small completes the proof of (F.6). The proof of (F.7) is similar. We first use 1/2 −1 r4 Lµ,r L−1 r = − L L r µL r = f µ(1 − ∂x2 )g µL 4 µ,r v 4 v 4 v v = f µg + f µg + f µ g , 2 −1 (m) be the mth order spatial where f = Lµ,r L−1 v r4 and g = (1 − ∂x ) Lv r4 . Let f (m) (m) (m) derivative of f . Then we have f L2 ≤ 16 r4 L2 and g (m) L2 ≤ r4 L2 . Furthermore, we have µ L∞ ≤ Cδ 3/2 ρ and µ L∞ ≤ Cδ 1/2 ρ ≤ Cδ 3/2 ρ. Using these inequalities, we get ε4 −1 1/2 r4 Lµ,r Lv µLv r4 ≤ Cε4 δ 3/2 ρ r4 2L2 + r4 L2 r4 L2 + r4 2L2 16
ε2 ≤ Cε2 δ 3/2 ρ r42 + (r4 )2 . 2
As above, choosing cε sufficiently small completes the proof of (F.7).
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G. Discussion The proofs of this section follow from definitions and proofs of Sect. 2.3 which should be read first. By (1.17), we have εˆ x εˆ 2 dz µ(z, ˆ tˆ) , η(x, t) = 4 0 and if δ = cδ ρ 2 and εˆ ≤ cε ρ −mε with mε ≥ 4, we get εˆ L0 |||µ||| ˆ L∞ ≤ cη εˆ 2 L ρ ≤ C ε2−13/(8 mε ) , 2 ≤ εˆ 4 |||ˆs |||L∞ ≤ C ε 4 δ 3/2 ρ ≤ C ε4−4/mε ,
|||η|||L∞ ([−L0 /2,L0 /2]) ≤ εˆ 2 |||s|||L∞ ([−L0 /2,L0 /2])
since ρ ≤ cε εˆ −1/mε . We also have ˆ L2 ≤ C ε 5/2 ρ ≤ C ε5/2−1/mε , |||η |||L2 ([−L0 /2,L0 /2]) ≤ εˆ 5/2 |||µ||| √ ˆ L∞ ≤ C ε 3 δ ρ ≤ C ε3−2/mε , |||η |||L∞ ([−L0 /2,L0 /2]) ≤ εˆ 3 |||µ||| |||s|||L2 ([−L0 /2,L0 /2]) ≤ εˆ
7/2
|||ˆs |||L2 ≤ C ε
7/2
δρ≤Cε
7/2−3/mε
(G.1) (G.2) .
(G.3)
Various other estimates, e.g. on higher order derivatives can be obtained in a similar way. References [BKL94]
Bricmont, J., Kupiainen, A., Lin, G.: Renormalization Group and Asymptotics of Solutions of Nonlinear Parabolic Equations. Commun. Pure Appl. Math. 47(6), 893–922 (1994) [GvB02] van Baalen, G.: Stationary solutions of the Navier-Stokes equations in a half-plane downstream of an obstacle: ‘universality’ of the wake. Nonlinearity 15(2), 315–366 (2002) [CH93] Cross, M., Hohenberg, P.: Rev. Mod. Phys. 65, 851 (1993) [CEES93] Collet, P., Eckmann, J.-P., Epstein, H., Stubbe, J.: A global attracting set for the Kuramoto-Sivashinsky equation. Commun. Math. Phys. 152(1), 203–214 (1993) [KT76] Kuramoto, Y., Tsuzuki, T.: Persistent Propagation of Concentration Waves in Dissipative Media Far from Thermal Equilibrium. Prog. Theor. Phys. 55, 356–369 (1976) [EGW95] Eckmann, J.-P., Gallay, Th., Wayne, C.E.: Phase slips and the Eckhaus instability. Nonlinearity 8(6), 943–961 (1995) [Man90] Manneville, P.: Dissipative structures and weak turbulence. Boston, MA: Academic Press, 1990 [MHAM97] Montagne, R., Hern´andez-Garc´ıa, E., Amengual, A., San Miguel, M.: Wound-up phase turbulence in the complex Ginzburg-Landau equation. Phys. Rev. E (3) 56(1), part A, 151–167 (1997) [New74] Newell, A.: Lect. Appl. Math. 15, 157 (1974) [NST85] Nicolaenko, B., Scheurer, B., Temam, R.: Phys. D 16(2), 155–183 (1985) [Tem97] Temam, R.: Infinite-dimensional dynamical systems in mechanics and physics. Second edition, New York: Springer, 1997 [BF67] Benjamin, T.B., Feir, J.: J. Fluid Mech. 27, 417 (1967) Communicated by A. Kupiainen
Commun. Math. Phys. 247, 655–695 (2004) Digital Object Identifier (DOI) 10.1007/s00220-004-1075-x
Communications in
Mathematical Physics
Propagation of Oscillations in Real Vanishing Viscosity Limit Christophe Cheverry IRMAR, Universit´e de Rennes I, Campus de Beaulieu, 35042 Rennes cedex, France. E-mail: [email protected] Received: 4 May 2003 / Accepted: 21 November 2003 Published online: 28 April 2004 – © Springer-Verlag 2004
Abstract: In this paper, we study viscous perturbations of quasilinear hyperbolic systems in several space dimensions. The equations involve a singular parameter ε which goes to zero. They arise in realistic models of compressible flow: the large-scale motions in the atmosphere [12]. They come also from theoretical considerations in non-linear geometric optics [4], [5]. We prove that solutions uε exist on a domain of space time independent on ε ∈ ]0, ε0 ], where ε0 > 0. Contents 0. Introduction . . . . . . . . . . . . . . . . . . . . . . . . 1. Two Models . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Large-scale motions in the atmosphere . . . . . . . 1.2 Strong non-linear geometric optics . . . . . . . . . 2. The Linearized Equations . . . . . . . . . . . . . . . . . 2.1 A class of singular equations . . . . . . . . . . . . 2.2 A first reduction . . . . . . . . . . . . . . . . . . . 2.3 Transformation of the equations . . . . . . . . . . 2.4 Analysis of turbulences . . . . . . . . . . . . . . . 3. The Nonlinear Equations . . . . . . . . . . . . . . . . . 3.1 Proof of Theorem 1 . . . . . . . . . . . . . . . . . 3.2 Proof of Theorem 2 . . . . . . . . . . . . . . . . . 3.3 Compressible equations of isentropic gas dynamics
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655 659 659 665 667 667 671 673 680 687 689 690 693
0. Introduction This article deals with quasilinear hyperbolic systems involving large nonlinear terms. Let us first specify the general framework. It allows to understand the specific structure
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of the singularities which can be marked below by looking at the coefficients with ε −2 and ε −1 in factor. The state variables are: u = t (v, w), v = t (v1 , · · · , vp ) ∈ Rp , w = t (w1 , · · · , wq ) ∈ Rq , whereas the space time variables are: (t, θ, y) = (t, θ, y1 , · · · , yd ) ∈ R+ × T × Rd , T := R/Z . Note: ∂t := ∂/∂t , ∂θ := ∂/∂θ , ∂j := ∂/∂yj , 1≤j ≤d. := ∂θ2 + ∂12 + · · · + ∂d2 . Let (ε, µ) ∈ := ]0, 1] × R+ . Consider the system of n = p + q equations: (Sµε )
0 (v) X(ε, u; ∂t,θ,y ) u + θ (ε, u) ∂θ u + y (u; ∂y ) u = g(ε, u) + µ u
with the structure conditions: Assumption 1. A.1.1) We can isolate the singular parts of the expressions according to: X(ε, u; ∂t,θ,y ) = ε−2 Xsθ (v) ∂θ + X(u; ∂t,θ,y ), θ (ε, u) = ε−2 sθ (v) + θ (u), g(ε, u) = ε−2 gs (v) + g(u). Above X(u; ∂t,θ,y ) and y (u; ∂y ) are first order operators: X(u; ∂t,θ,y ) = ∂t + X θ (u) ∂θ +
d
X j (u) ∂j ,
j =1
y (u; ∂y ) =
d
j (u) ∂j ,
j =1
A.1.2) The letters and represent scalars whereas 0 , sθ , θ and j are n × n symmetric matrices. The quantities Xsθ , Xθ , Xj , 0 , sθ , θ , j , gs and g are C ∞ functions of u in a neighbourhood of 0 ∈ Rn . A.1.3) The source terms gs (v) and g(u) can be decomposed in Rp × Rq according to: g(u) = t g1 (v), g2 (u) . gs (v) = t gs1 (v), 0 , Xsθ ,
Xθ
Xj
A.1.4) The p × p matrix := (Dv gs1 )(0) is skew symmetric. A.1.5) The matrices 0 , sθ , θ and j have the p × q block structures: 0 θ 11 (v) 0 s11 (v) 0 0 θ (v) = . 0 (v) , s (v) = 0 0 22 0 θ j j 11 (u) 12 (u) 11 (u) 0 θ j . , (u) = t j (u) = 0 0 12 (u) 0 A.1.6) We can adjust u and (t, θ, y) such that: gs (0) = 0, g(0) = 0 ; 0 (0) = Id , Xsθ (0) = 0 , ∇v Xsθ (0) = (a, 0, · · · , 0) ∈ Rp , Xθ (0) = 0 , Xj (0) = 0, 1 ≤ j ≤ d .
a ∈ R∗ ;
Propagation of Oscillations in Real Vanishing Viscosity Limit
Let h ∈ H ∞ (T; Rq ) be such that ∂θ h ≡ 0. The Cauchy problem: ∂t wεµ + X θ (0, ε wεµ ) ∂θ wεµ = ε−1 g2 (0, ε wεµ ) + µ ∂θ2 wεµ wεµ (0, θ) = h(θ )
657
(0.1)
is locally well-posed: we can find smooth solutions wεµ defined on [0, T˜ ]×T, where T˜ > 0 does not depend on (ε, µ) ∈ ]0, 1] × R+ . We observe that the expression t (0, ε wεµ ) is a
solution to (Sµε ). We fix µ ≥ 0 and we study the stability of the family t (0, ε wεµ ) ε∈]0,1] . ε = t (vε , wε ) of t (0, ε wε ) which More precisely, we want to construct perturbations uµ µ µ µ ε ε ε does depend on y are still solutions of (Sµ ). We require that vµ is non-trivial, that uµ ε does not shrink to zero as ε goes to 0. and that the life span Tµε of uµ ε By linearizing the system (Sµ ) along the expressions t (0, ε wεµ ), one gets: (Lεµ ) ∂t u˙ = ε −2 A u˙ + ε−1 B u˙ + C u˙ + µ u˙ . The operator A is skew-symmetric. The contribution due to ε−2 A u˙ when performing L2 −estimates can be managed by the usual tools (just follow the procedure explained in [8]). When ε goes to 0, the term ε −1 B u˙ becomes singular. Now B is non-trivial because of the hypothesis a = 0. Moreover the operator B is not skew-symmetric. A term like B does not arise in weakly, diffractive or dispersive nonlinear geometric optics. It is not involved by the analysis of [7] or [13]. Such a B appears for the first time in [4] which is concerned with the case µ = 0. It induces new difficulties which are the core of the present article. ε of t (0, ε wε ) can be conWhen µ = 0 it is not sure that non-trivial perturbations uµ µ structed. Obstructions come from instabilities induced by the fact that B ≡ 0. In [4] (see ε − t (0, ε wε ) can undergo an also [11]) we prove that the L2 − norm of the difference uµ µ γ t/ε exponential amplification e with γ > 0. This phenomenon explains why we are not ε which are defined on [0, T ] with T ∈ ]0, T˜ ] independent on able to find solutions uµ ε ∈ ]0, 1]. The hyperbolic situation is still open with regard to the problem of existence. The goal of this paper is to examine if the situation is changed when µ > 0. Two questions are raised. When µ ≥ 1 is fixed, what are the conditions which allow to get the stability? When these conditions are satisfied, what is the critical viscosity µ(ε) at the level of which there is a transition from stability to instability? The analysis relies on the absorption of the singular terms contained in (Sµε ). This can be done by reductions and blow up. These procedures induce a defect of hyperbolicity. The matter is to compensate this lack of hyperbolicity by adding a suitable viscosity µ. We start the discussion by looking at large-scale motions in the atmosphere [12]. It concerns a slightly compressible and slightly adiabatic version of the Navier-Stokes equations. It has important applications in meteorology. It lies at the extreme end of the class of equations which we consider since it can be managed by a simple reduction. We can assert: Theorem 1. Suppose that Assumptions 1, 2 and 3 are satisfied. Consider a family {u ε0 }ε∈]0,1] which is bounded in H σ (T × Rd ; Rn ) with σ > d+3 2 . We fix any ν > 0. ˜ Then there is T ∈ ]0, T ] such that for all (ε, µ) ∈ εν := ]0, 1] × [ν ε 2 , +∞[ the Cauchy problem for (Sµε ) with initial data: ε (0, θ, y) := t ε 2 vε (θ, y), ε wε (0, θ) + ε 2 w ε (θ, y) , uµ u ε0 = t (vε0 , wε0 ) µ 0 0 ε d has a unique solution uµ (t, θ, y) defined on [0, T ] × T × R . Moreover we can find
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C. Cheverry
c > 0 such that: ε t uµ − (0, ε wεµ ) (t, ·) H σ (T×Rd ) ≤ c ε 2 ,
∀ (ε, µ, t) ∈ εν × [0, T ] .
(0.2)
In general Assumption 3 is not verified. Then we need other constraints to get the stability. To cover the many interesting examples, we set a hierarchy of conditions: Assumption 2 ⊃ Assumption 4 ⊃ Assumption 5 ⊃ Assumption 6 . These conditions are detailed in Subsect. 2.2 and 2.3. They involve the operators P and M whose definitions and meanings become clear when performing the linear analysis of Chapter 2. They are organized so that the viscosity µ can be diminished more and more (but it will still remain bigger than ε2 ): Theorem 2. Suppose that Assumptions 1 and 2 are satisfied. Consider a family {u ε0 }ε∈]0,1] ˜ which is bounded in H σ (T × Rd ; Rn ) with σ > d+3 2 . There is ν ≥ 1 and T ∈ ]0, T ] ε such that for all parameters (ε, µ) belonging to the domain ν which is: a) εν := ]0, 1] × [ν; +∞[ under Assumption 4, b) εν := ]0, 1] × [ν ε; +∞[ under Assumption 5, c) εν := ]0, 1] × [ν ε 2 ; +∞[ under Assumption 6, the Cauchy problem for (Sµε ) with initial data: ε uµ (0, θ, y) := t (0, ε wεµ )(0, θ) + ε4
u ε0 (θ, y),
u ε0 = t (vε0 , wε0 )
ε (t, θ, y) defined on [0, T ] × T × Rd . Moreover we can find has a unique solution uµ c > 0 such that: ε t uµ − (0, ε wεµ ) (t, ·) H σ (T×Rd ) ≤ c ε 3 , ∀ (ε, µ, t) ∈ εν × [0, T ] . (0.3)
The power ε4 in front of u ε0 is here for convenience. It can be improved. The point is ε − t (0, ε wε ) is multiplied by the factor ε −1 instead of eγ t/ε with that the difference uµ µ γ > 0. It means that the amplification is less strong than initially expected. The case c) is relevant when dealing with compressible equations of isentropic gas dynamics (see Theorem 3 in Sect. 3.3). It allows to take a fresh look at turbulences in the continuation of [3]. Our paper is organized as follows. In Sect. 1, we present two models belonging to the class of equations under consideration. We first focus on large-scale motions in the atmosphere [12]. This part 1.1 is long with several changes of variables, adimensionalization and physical considerations. It can be avoided by a reader who is not interested in physical applications. Then we are concerned in strong oscillations [4]. This second subject is issued from theoretical considerations in non-linear geometric optics. It includes a large class of situations which allow to test Assumptions 4, 5 and 6. At the extreme end it contains Euler’s equations. Section 2 works on the equations (Lεµ ) obtained by linearizing the system (Sµε ) along the expressions t (0, ε wεµ ). We examine carefully how the L2 −norm of u˙ is amplified. In other words we seek a function g(ε, µ; t) as minimal as possible such that: ˙ ˙ ˙ u(t) L2 ≤ g(ε, µ; t) u(0) L2 , ∀ u(0) ∈ L2 (T × Rd ) . Since A is skew symmetric whereas B is not, we expect to find: 2 ∃ (c, C) ∈ (R+ ∗) ;
ec t/ε ≤ g(ε, µ; t) ≤ eC t/ε ,
∀ (t, ε) ∈ R+ ×]0, 1] .
(0.4)
Propagation of Oscillations in Real Vanishing Viscosity Limit
659
In Chapters 2.2 and 2.3, we transform (Lεµ ) by changes of variables. We exhibit restrictions 2, · · · , 6 on A, B and C which allow to remove the coefficient B. This is the central part of the work. It follows that the rough control (0.4) can be changed into: ∃ C ∈ R+ ∗ ;
g(ε, µ; t) ≤ C t/ε,
∀ (t, ε) ∈ R+ ×]0, 1] .
(0.5)
In paragraph 2.4, we test Assumptions 2, · · · , 6 with regard to the models introduced in Sect. 1. The question is to determine concretely the minimal viscosity µ which allows to get the stability. It corresponds to the analysis of turbulences. In the last section, we consider the system (Sµε ). We prove that the procedure of Sect. 2 can be implemented in the nonlinear situation. It gives access to Theorem 1 and Theorem 2. 1. Two Models In this paragraph, we present two models and we check that the structure conditions listed in Assumption 1 are verified. 1.1. Large-scale motions in the atmosphere. We are here interested in atmospheric phenomena. We refer to the book of J. Pedlosky [12] for a detailed presentation of the physical model. The equations and the numerical data which follow are directly extracted from [12]. For the convenience of the reader, we will give precise references in the form [12]. where the star indicates the pages in [12] which contain the pertinent information. We first present the structure of the equations. We note t ∈ R+ the time variable, x1 ∈ R the vertical variable and y = (x2 , x3 ) ∈ R2 the horizontal variables. Let g and be the gravitational acceleration and the angular velocity of the earth. The component of the planetary vorticity normal to the earth’s surface is the Coriolis parameter ω. In the vicinity of the latitude π4 , we can take ([12].3): π 7, 3 × 10−5 s −1 , g 10 m s −2 .
10−4 s −1 , 4 The physical quantities are the pressure p, the fluid density , the vertical velocity v1 , the horizontal vector velocity (v2 , v3 ), and the potential temperature s ([12].10 and 338). We introduce the total derivative: d := ∂t + v1 ∂x1 + v2 ∂x2 + v3 ∂x3 , v = t (v1 , v2 , v3 ) . dt ω = 2 sin
The condition of mass conservation is expressed by the continuity equation ([12].10 and 338): d + (∂x1 v1 + ∂x2 v2 + ∂x3 v3 ) = 0 . dt
(1.1)
Newton’s law of motion gives rise to the following averaged equations for the large-scale flow ([12].179–184 and 339): d dt v1 + ∂x1 p = − g + F10 , (1.2) d v2 + ∂x2 p = ω v3 + F20 , dt d v3 + ∂x3 p = − ω v2 + F30 , dt
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C. Cheverry
where F10 , F20 and F30 are the three components of the frictional forces acting on the fluid. We complete these equations with the first thermodynamic law which takes the form ([12].339): d s = Fs0 , dt
(1.3)
where Fs0 contains viscosity terms expressing the thermal diffusion. In the absence of motion, the equations in (1.2) reduce to: ∂x1 p = − g ,
∂x2 p = ∂x3 p = 0 .
(1.4)
We see that the function p (and hence ) depends only on the variable x1 . These quantities must be related by the state relation ([12].355): 1 ln p(x1 ) − ln (x1 ) − ln s(x1 ) = 0 . γ
(1.5)
Here γ 1, 4 is the ratio of the specific heats of air at constant pressure and density respectively. Let D and L be characteristic scales of x1 and y. We work with the nondimensional variables (t, x1 , x2 , x3 ) defined by: y = (x2 , x3 ) = L (x2 , x3 ) = L y,
x 1 = D x1 ,
ωt = t.
We introduce: (p, ρ, s)(x1 ) = (p, , s)(D x1 ),
x1 ∈ R .
We suppose that the expressions ∂x1 ln p(x1 ) and ∂x1 ln ρ(x1 ) are equal and do not depend on x1 . Then the length D can be adjusted such that: ∂x1 ln p(x1 ) = ∂x1 ln ρ(x1 ) = −1,
∀ x1 ∈ R .
(1.6)
It follows from (1.4), (1.5) and (1.6) that: Dgρ p
= 1,
−
γ −1 ∂x1 ln s(x1 ) =
0, 3 = 10−1 . ∂x1 ln p(x1 ) γ
(1.7)
We remark that the hypothesis (1.6) and (1.7) imply that the fluid is almost adiabatic. This is consistent with the observations ([12].351–354). We can think of the data (p, ρ, s) as defining a standard atmosphere upon which fluctuations due to the motion occur. We seek perturbations which are realistic. So we have to measure the relative importance of the small physical forces acting on the fluid. Let V be a typical horizontal velocity scale. We introduce the small independent parameters ([12].202 and 342): := V/ω L, δ := D/L, σ 2 := ω2 L2 /g D .
the Rossby number, the aspect ratio,
We choose the state variables (s, p, v) and ρ such that: v = ω L v / σ α, = (x1 ) (1 + ρ),
p = p(x1 ) + (x1 ) D g p . s = s(x1 ) (1 + σ s) .
Propagation of Oscillations in Real Vanishing Viscosity Limit
661
We suppose that: p = α 2 ρ + β s,
α > 0,
β > 0.
(1.8)
The coefficients α and β are fixed constants. They will be determined later. In (1.10), we will find α β 1. We can interpret (1.8) as the linear approximation of the state relation linking p, ρ and s. Since p, and s will finally be small perturbations of p, and s, the identity (1.5) will finally be roughly preserved: | γ1 ln p − ln − ln s| 0 . We introduce the substantial derivative: d 1 1 1 := ∂t + 2 2 σ α v1 ∂x1 + v2 ∂x2 + v3 ∂x3 . dt σ α δ σα σα We can put the equations in the following symmetric form: d 1 1 dt s = σ 2 α 2 δ fs + Fs , α α α 1 d 1 −1 (1 + ρ) dt p + σ δ ∂x1 v1 + σ ∂x2 v2 + σ ∂x3 v3 = σ 2 α 2 δ fp + Fp , d α 1 (1 + ρ) dt v1 + σ δ ∂x1 p = σ 2 α 2 δ f1 + F11 , d (1 + ρ) dt v2 + σα ∂x2 p = f2 + F21 , d (1 + ρ) dt v3 + σα ∂x3 p = f3 + F31 , where fs (s, v1 ) = − α ∂x1 ln s × v1 (1 + σ s), s fp (s, p, v1 ) = − α 3 σ ∂x1 ln ρ + α β ∂x1 ln s 1+σ × v1 , 1+ρ 3 f1 (s, p) = − α σ ∂x1 ln ρ + α σ p + α β σ s, f2 (s, p, v3 ) = (1 + ρ) v3 , f3 (s, p, v2 ) = − (1 + ρ) v2 , and Fs1 = Fs0 / ω σ s, Fp1 = β Fs0 / (1 + ρ) ω σ s, Fj1 = α Fj0 / σ δ g ρ, ∀ j ∈ {1, 2, 3}. To this point the only approximation concerns the relation (1.6) which takes into account in a realistic way the dependence on x1 of both p and ρ. Otherwise the equations have simply been scaled so that the relative order of each term is clearly measured by the nondimensional parameters multiplying it. To go further, we collect some information furnished by the observations ([12].10 and 345, 352–354): (v1 , v2 ) V = 10 m s −1 , x1 D = 104 m, (x2 , x3 ) L = 106 m,
horizontal velocity size. vertical scale. horizontal scale.
We compute: = 10−1 ,
δ = 10−2 2 ,
σ 0, 3 .
662
C. Cheverry
We remark that the amplitude of the horizontal vector velocity (v2 , v3 ) must be of the size: (v2 , v3 ) ε := σ α V / ω L σ α 10−3/2 .
(1.9)
Each dependent variable s, p and v is a function not only of x and t but also of , δ and σ . In order to progress we must exploit the smallness of and δ. In expanding the equations and the solutions we must, of necessity, choose particular ordering relations between the parameters , δ and σ . We take ε as a small parameter describing the size of these indices. We impose the relationship ε2 = σ 2 α 2 2 and we consider the limit ε goes to zero, with σ chosen to be of order one. This model is suggested by the observational evidence described before. It distinguishes natural balances that are physically relevant to the phenomena of interest [12]. The determination of the Fj0 is a delicate question. For an incompressible homogeneous fluid, the Fj0 are specified in terms of the large-scale data (v1 , v2 , v3 ) in order to find a closed set of equations. A simple model gives rise to the anisotropic action ([12].183): Fj0 = (AV ∂x21 + AH ∂x22 + AH ∂x23 ) vj ,
j ∈ {1, 2, 3} .
The coefficients AH and AV are empiric constants. Estimates for AH and AV in the atmosphere suggest ([12].184): AH 105 m2 s −1 ,
AV 10 m2 s −1 .
The components Fj0 are modified by the preceding manipulations. They are transformed according to: Fj1 = (1 + ρ) (EV ∂x21 + EH ∂x22 + EH ∂x23 ) vj ,
j ∈ {1, 2, 3} .
The coefficients EH and EV are called the horizontal and vertical Ekman numbers. We find: EH = AH /ω L2 10−3 ,
EV = AV /ω D 2 10−3 .
These computations indicate that: Fj1 = ν ε 2 vj ,
:= ∂x21 + ∂x22 + ∂x23 ,
ν 1,
1 ≤ j ≤ 3.
On the analogy, we suppose that Fs1 and Fp1 share the same structure: Fs1 = ν ε 2 s,
Fp1 = ν ε 2 p .
We explain now how to put the equations into the formalism of the introduction. We split the state variable into: u = t (v, w) ∈ R5 ,
v := t (v1 , s, p) ∈ R3 ,
w = t (v2 , v3 ) ∈ R2 .
We take p = 3, q = 2, n = 5 and: Xsθ (v) = σ α v1 , Xj (u) = σ1α wj ,
a = ασ = 0, θ (u) = 0,
X θ (u) = 0 . j 11 (u) = 0 .
Propagation of Oscillations in Real Vanishing Viscosity Limit
663
We define: 0 (v) = (1 + ρ) Id 22 3×3 ,
1+ρ 0 0 0 (v) = 0 , 1 0 11 0 0 (1 + ρ)−1
t 2 (u) 12
=
α σ
001 , 000
j
22 (u) = 0,
001 θ (v) = α 3 σ 0 0 0 , s11 100 t 3 (u) 12
=
α σ
000 , 001
where ρ is the following function of v : ρ = ρ(v) = α −2 (v3 − β v2 ) . We fix: g1 (v) = 0,
t
g2 (u) = f2 (u), f3 (u) = 1 + ρ(v) (w2 , −w1 ) .
We obtain:
α β σ v2 + α σ (α 2 − 1) v3 f1 (v) γ −1 gs1 (v) = fs (v) = − α γ v1 (1 + σ v2 ) . 3 fp (v) α σ − α β γ −1 1+σ v2 v1
γ
We compute:
1+ρ(v)
0 α β σ α σ (α 2 − 1) 0 0 − α γ γ−1 = . γ −1 3 α σ −α β γ 0 0
With these conventions the equations are in the form (Sµε ) with µ = ν ε 2 . The properties A.1.1), A.1.2), A.1.3), A.1.5) and A.1.6) are obviously verified. It remains to look at A.1.4). The restriction A.1.4) is a stability condition. It prevents the formation of exponential amplifications eγ t/ε with γ > 0. It is usual in physics [9]. In particular it occurs in the description of atmospheric phenomena [12] and it has already led to specific developments (see for instance [2]). In our context, it allows to determine the scalars α and β. We have to impose: 1 (γ − 1)2 1/2 α = √ 1+
1, γ2 σ2 2
β =
γ −1
1. γσ
(1.10)
We see here a posteriori that α and β are of the order 1 and that: 0 α˜ β˜ 0 σ 0 = − α˜ 0 0 − σ 0 0 = − ∗ . 0 00 − β˜ 0 0 In that manner the equations describing large-scale motions in the atmosphere can be put in the form: (Aεν )
0 (v) X(ε, u; ∂t,x1 ,y ) u + θ (ε, u) ∂x1 u + y (u; ∂y ) u = g(ε, u) + ν ε 2 u,
664
C. Cheverry
where 0 , X, θ , y and g are subjected to Assumption 1. We have now to discuss the size of the data. We must also explain the hypothesis of periodicity in x1 . We are interested in physically relevant solutions uε = t (vε , wε ) of (Aεν ). We take into account (1.9) by imposing wε = (ε). Observations indicate that the component vε has an amplitude which is much smaller than ε. It means that the leading term in uε is of the form t (0, ε wε ), where wε must be subjected to: ∂t wε = t (wε2 , −wε1 ) + ν ε2 ∂x21 wε . Now the component vε is non-trivial though it is small. Therefore we have to seek exact solutions which are non-trivial perturbations of t (0, ε wε ) : uε (t, x) = t ε 2 vε , ε wε + ε 2 wε )(t, x), u ε = t (vε , wε ) . We suppose that wε (and hence u ε and uε ) is periodic in x1 : wε (t, x1 + P , y) = wε (t, x),
uε (t, x1 + P , y) = uε (t, x),
P > 0.
There are physical reasons behind these boundary conditions. Oscillations of the fluid element about the equilibrium level are observed ([12].352). They form high frequency nonlinear waves around the more ponderous large-scale geostrophic motion. They concern mainly the component vε of the perturbation uε . They occur in the direction x1 with the Brunt-V¨ais¨al¨a frequency N defined by ([12].353): √ N := (g ∂x1 ln s / D)1/2 3 × 10−2 s −1 . To determine P , we remark that information concerning vε travel a distance x1 α/σ δ in times t of order 1. So we have to fix P such that: P = α ω / σ δ N = (1) . This is coherent with Fig. 6.4.2 in [12].353. We multiply (t, x) by a scalar (of the order 1) in order to have P = 1 and x1 ≡ θ ∈ T. Now the horizontal vector velocity wε undergoes variations when changing the altitude. It is natural to impose ∂θ wε ≡ 0. We suppose for convenience that wε is smooth. By this way, we recover exactly the problem which we set in the introduction. In Sect. 2, we will see that Assumptions 2 and 3 are satisfied. In Sect. 3, we will deduce from this fact Theorem 1. The construction of uε and the description of its asymptotic behaviour when ε goes to 0 are known as the problem of geostrophic approximation. In fact the difficulty is to understand how vε is coupled with wε . This is a subtle issue which has already received much attention. In Chapter 6 of [12], this problem is tackled by a BKW method. Formal ε−expansions are plugged into the equations. The (1) terms provide the geostrophic approximation [12].347. It does not allow to determine the quantities p0 , u0 and w0 ([12].348). To go further, it is necessary to consider the higher-order dynamics that is the (ε) terms which lead to the vorticity equation (6.3.17)–[12]. To solve (6.3.17), w1 must be related to p0 . The theory is completed in [12] by formal arguments using the thermodynamic equation (1.3) (see Sect. 6.4 and 6.5 in [12]). There is a main drawback with the approach presented in [12]. The expansions are obtained by using a non-symmetric version of the equations. Nothing guarantees that
Propagation of Oscillations in Real Vanishing Viscosity Limit
665
this procedure respects the good hierarchy between the terms. At all events this lack of symmetry prevents to justify rigorously the computations. As in [12], we take into account the precise physical information inherent to largescale motions in the atmosphere. We prove the existence of exact solutions uε which are non-trivial perturbations of t (0, ε wε ). As a corollary we know that the BKW construction made in [4] is adapted to the situation. It allows to explain in a new way how the small large-scale vertical velocity vε is coupled with the geostrophic fields. By this way, we reply to a concrete question raised but not solved in [12].353–354.
1.2. Strong non-linear geometric optics. In [4] and [5], we consider systems of conservation laws which can be put in the symmetric form: 0 (u) ∂t u + 1 (u) ∂x1 u +
d
j (u) ∂j u = 0,
† (u) = t † (u) .
j =1
We suppose that this system has a linearly degenerate eigenvalue. It follows that the matrices † (u) inherit special properties. These properties are listed in [4 and 5]. In most of the applications, there is a good symmetrizer (see Sect. 5.2 in [4]). It means that we can decompose u into t (v, w) ∈ Rp × Rq so that the equations can be reduced to: 0 (v) X(u; ∂t,x1 ,y ) u + sθ (v) ∂x1 u +
d
j (u) ∂j u = 0,
(1.11)
j =1
where X is the vector field: X(u; ∂t,x1 ,y ) = ∂t + Xsθ (v) ∂x1 +
d
X j (u) ∂j
j =1
and the † (u) have the p × q block structures : 0 θ 11 (v) 0 θ (v) = s11 (v) 0 , 0 (v) = , 0 s 0 0 0 22 (v) j j Id (u) (u) 0 p×p j 0 11 12 (u) = t j , , (0) = 0 Idq×q 12 (u) 0 θ (v) is invertible for all v. By a linear change on u and (t, x), we can adjust where s11 the data such that:
Xsθ (0) = 0, Xj (0) = 0,
∇v Xsθ (0) = (a, 0, · · · , 0) , ∀ j ∈ {1, · · · , d} .
The link with (Sµε ) is established by seeking solutions to (1.11) which are periodic in x1 with period ε2 > 0. We define θ := ε−2 x1 so that we have to deal with: 0 (v) [ ∂t + ε −2 Xsθ (v) ∂θ + dj =1 X j (u) ∂j ] u (S˜0ε ) + ε−2 sθ (v) ∂θ u + y (u; ∂y ) u = 0 .
666
C. Cheverry
We recover here the equations of the introduction. Assumption 1 is clearly satisfied. In [4], we construct strong oscillations: uεa (t, x) = t vaε (t, x), waε (t, x) ∞ ϕ (t, x) ε = u¯ 0 (t, x) + ε k uk t, ,x , ∂θ u1 ≡ 0 ε2 k=1
which solve (1.11) on [0, T˜ ] × Rd with infinite accuracy: 0 (vaε ) [ ∂t + ε−2 Xsθ (vaε ) ∂θ + dj =1 X j (uεa ) ∂j ] uεa + ε−2 sθ (vaε ) ∂θ uεa + y (uεa ; ∂y ) uεa = raε = (ε∞ ) . In particular, we can choose: u¯ 0 (t, x) = 0, ϕε (t, x) = x1 , uk (t, θ, x) = uk (t, θ, y) ∈ H ∞ ([0, T˜ ] × T × Rd ),
∀ k ∈ N∗ .
It furnishes approximate solutions uεa (t, θ, y) to (S˜0ε ). If a = 0, there are exact solutions uε corresponding to the uεa on the interval [0, T˜ ] (this is Theorem 3.2 in [4]). We suppose that a = 0. It implies that the stability condition D ≡ 0 (see Theorem 3.2 and Proposition 3.2 in [4]) is violated. Consequently, the family {uεa }ε∈]0,1] is unstable. The difficulties come from the small oscillations contained in the remainder raε . The interactions of uεa with auxiliary oscillations of the form: εk U t, θ, y, ψ(t, y)/ε2 , k1 (1.12) can be organized in a coherent way to affect uεa . Therefore the perturbations are amplified by a factor eγ t/ε with γ > 0 (Sect. 6 in [4]). Such instabilities prevent to construct (by way of classical tools) exact solutions uε which are non-trivial perturbations of uεa and which exist on a domain of space time independent on ε ∈ ]0, 1]. The question is to know if the situation is different when we incorporate some viscosity µ on the right side of (S˜0ε ). On the one hand, the dissipation µ must be small enough so that strong oscillations can propagate. It leads to the choice of some anisotropic viscosity µ (ε4 ∂x21 + ∂y21 + · · · + ∂y2d ) in the variables x, or simply µ in the variables (θ, y). On the other hand, the dissipation µ must be sufficiently large so that the hyperbolic instabilities are kept under control. The oscillations (1.12) must be damped by the viscosity. From this point of view, the condition µ ≥ ν ε2 is convenient. Let h ∈ H ∞ (T; Rq ) be such that ∂θ h ≡ 0. We consider the expression wµ obtained by solving: ∂t wµ = µ ∂θ2 wµ ,
wµ (0, θ) = h(θ ) .
(1.13)
We will see in Sect. 2 that Assumptions 2 and 4 are satisfied. The other Assumptions 5 and 6 depend on structure conditions which will be given explicitly. We can apply Theorem 2 to the system: (S˜µε )
0 (v) [ ∂t + ε−2 Xsθ (v) ∂θ + dj =1 X j (u) ∂j ] u + ε−2 sθ (v) ∂θ u + y (u; ∂y ) u = µ u .
Propagation of Oscillations in Real Vanishing Viscosity Limit
667
Then we can interpret the result in the original scales (t, x). For (ε, µ) ∈ εν , it furnishes strong oscillations: ε ε ε uµ (t, x) = t vµ (t, x), wµ (t, x) = t ε4 vεµ (t, ε−2 x1 , y) , ε wµ (t, ε−2 x1 ) + ε 4 wεµ (t, ε−2 x1 , y) which are periodic in x1 of period ε 2 and which are exact solutions to: 0 (v) X(u; ∂t,x1 ,y ) u + θ (v) ∂x1 u + y (u; ∂y ) u = µ (ε4 ∂x21 + ∂y21 + · · · + ∂y2d )u . 2. The Linearized Equations We work under Assumption 1. We select some expression wεµ (t, θ ) which is subjected to (0.1). 2.1. A class of singular equations. In the discussion of the existence of non-trivial perε of t (0, ε wε ) which are still solutions to (S ε ), the main step is to study the turbations uµ µ µ linear stability of t (0, ε wεµ ). The linearized equations are: θ (0, ε wε ) ∂ v ∂t v˙ − ε−2 A11 v˙ + Xθ (0, ε wεµ ) ∂θ v˙ + 11 µ θ˙ d d j j ε ε )∂ v ˙ v + X (0, ε w ) ∂ (0, ε w + µ j µ j˙ j =1 11 jd =1 j ε ˙ − G11 v˙ = µ ˙v . + j =1 12 (0, ε wµ ) ∂j w (Lεµ ) −1 B v θ (0, ε wε ) ∂ w ˙ ˙ ∂ w − ε + X t 21 µ θ ˙ d j j ε ˙ + dj =1 t 12 (0, ε wεµ ) ∂j v˙ + j =1 X (0, ε wµ ) ∂j w ˙ − ε R21 v˙ − ε R22 w ˙ = µ w ˙. − G21 v˙ − G22 w We have to specify the notations. The symbol A11 is for the skew symmetric operator: θ A11 := − s11 (0) ∂θ + ,
A∗11 = − A11 .
The letter B21 is for the q × p matrix: B21 = B21 (ε, µ; t, θ ) := − a t ∂θ wεµ (t, θ ) ⊗ (1, 0, · · · , 0), where
(e1 , · · · , eq ) ⊗ (f1 , · · · , fp ) = (ei fj )1≤i≤q , 1≤j ≤p .
We define also:
G := Du g(0) =
G11 0 G21 G22
=
Dv g1 (0) 0 , Dv g2 (0) Dw g2 (0)
0 (0) ∂t wεµ + X θ (0, ε wεµ ) ∂θ wεµ R21 v˙ := − (˙v · ∇v )22 − (˙v · ∇v )X θ (0, ε wεµ ) ∂θ wεµ 1 ε + (wµ · ∇w )Dv g2 (0, s ε wεµ ) ds v˙ , 0
668
C. Cheverry
˙ := − (w ˙ · ∇w )X θ (0, ε wεµ ) ∂θ wεµ R22 w 1 ε ˙. + (wµ · ∇w )Dw g2 (0, s ε wεµ ) ds w 0
In abbreviated form, the equation (Lεµ ) can be formulated as: ∂t u˙ = ε −2 A u˙ + ε−1 B u˙ + C u˙ + µ u˙
(Lεµ ) if we set:
A :=
A11 0 , 0 0
B :=
0 0 , B21 0
C :=
C11 C12 C21 C22
,
where by construction the blocks C are: θ (0, ε wεµ ) ∂θ C11 = − Xθ (0, ε wεµ ) ∂θ − 11
−
d
X j (0, ε wεµ ) ∂j −
j =1
C12 = −
d
d
j
11 (0, ε wεµ ) ∂j + G11 ,
j =1 j
12 (0, ε wεµ ) ∂j ,
j =1
C21 = −
d
t
j
12 (0, ε wεµ ) ∂j + G21 + ε R21 ,
j =1
C22 = − Xθ (0, ε wεµ ) ∂θ −
d
X j (0, ε wεµ ) ∂j + G22 + ε R22 .
j =1
We use the abbreviations: < k > = (1 + k 2 )1/2 ,
< η > = (1 + |η|2 )1/2 ,
(k, η) ∈ Z × Rd .
We consider the space Pp of trigonometric polynomials whose coefficients depend on y ∈ Rd and take their values in Cp :
ck (y) ei k θ ; ck (y) ∈ Cp and K ⊂ Z is a finite set . Pp := u(θ ) = k∈K
Let (t, s) ∈ R2 . We define on Pp the actions: < Dθ >s < Dy >t u := < k >s −1 < η >t cˆk (η) ei k θ k∈K
and the norms: u Hps,t := < Dθ >s < Dy >t u L2 (T×Rd ;Cp ) ,
u Hps := u Hps,s .
We introduce the Hilbert spaces Hps and Hps,t obtained by looking at the closure of Pp in D (T × Rd ) for the norms · Hps and · Hps,t . We note L(Hp† ; Hq‡ ) the space of linear continuous applications from Hp† to Hq‡ . We use the notation [ · ; · ] for the commutator [T ; T ] := T ◦ T − T ◦ T .
Propagation of Oscillations in Real Vanishing Viscosity Limit
669
s,t Definition 1. We say that T is in the class p,q if: s+s ,t+t s ,t ∞ ∞ ˜ T ∈ L ; C [0, T ]; L(Hp ; Hq ) , ∀ (s , t ) ∈ R2 , [∂θ ; T ] ∈ L∞ ; C ∞ [0, T˜ ]; L(Hps+s ,t+t ; Hqs ,t ) , ∀ (s , t ) ∈ R2 , and for all j ∈ {1, · · · , d} : [∂j ; T ] ∈ L∞ ; C ∞ [0, T˜ ]; L(Hps+s ,t+t ; Hqs ,t ) , ∀ (s , t ) ∈ R2 .
For example: s,t < Dθ >s < Dy >t ∈ p,q ,
∀ (s, t) ∈ R2 .
We define:
||| T |||s ,ts,t := T L∞ (×[0,T˜ ];L(H s+s ,t+t ;H s ,t )) , p,q
p
s,t T ∈ p,q .
q
We retain the composition rules:
s+s ,t+t T ◦ T ∈ p,r ,
s,t s ,t ∀ (T , T ) ∈ p,q × q,r .
The actions which we will consider are built by composing differential operators in (θ, y) and Fourier multipliers in θ . From this remark and Definition 1 we can deduce special properties. In particular we will deal with operators T and T which satisfy: s+l−1,t [< Dθ >l ; T ] ∈ n,n ,
l ∈ Z,
s,t T ∈ n,n ,
s,t+l−1 [< Dy >l ; T ] ∈ n,n ,
l ∈ Z,
s,t T ∈ n,n .
Moreover, for scalar operators, we have:
s+s −1,t+t s+s ,t+t −1 + n,n , [T ; T ] ∈ n,n
s,t s ,t (T , T ) ∈ n,n × n,n .
These informations will be implicitly used in the computations which follow. We observe now that the operators A, B and C introduced before satisfy: Property 1. 1,0 P.1.1) A ∈ n,n is skew symmetric and we have: ∂t A = 0,
[∂θ ; A] = 0,
[∂j ; A] = 0,
∀ j ∈ {1, · · · , d} .
0,0 P.1.2) B ∈ n,n and we have [∂j ; B] = 0 for all j ∈ {1, · · · , d}. 1,1 can be decomposed into: P.1.3) C ∈ n,n
C = Cθ + Cy ,
C θ = C0θ + ε C1θ + ε 2 C2θ ,
y
y
with the constraints: 0,0 (Cjθ )∗ + Cjθ ∈ n,n ,
1,0 Cjθ ∈ n,n ,
0,0 (Cj )∗ + Cj ∈ n,n ,
0,1 Cj ∈ n,n ,
y
y
y
[∂j ; C0 ] = 0,
y
y
C y = C0 + ε C 1 + ε 2 C 2
∀ j ∈ { , 0, 1, 2}, ∀ j ∈ { , 0, 1, 2},
∀ j ∈ {1, · · · , d}.
670
C. Cheverry
The condition P.1.1) is obvious since A is a skew symmetric differential operator of order 1 in θ , with constant coefficients. The restriction P.1.2) is due to the fact that B is a matrix which does not depend on y. Since the action C comes from a symmetric hyperbolic system, its principal symbol is skew symmetric. Concerning the decomposition of C, we just have to set: Cθ = Cy
θ (0,ε wε ) ∂ +G − 11 0 θ 11 µ G22 +ε R22 G21 +ε R21
= −
d
j =1
j
−
j
11 (0,ε wεµ ) 12 (0,ε wεµ ) t j (0,ε wε ) µ 12
Xθ (0,ε wεµ )
0
∂j −
Id ∂θ ,
d
j ε j =1 X (0,ε wµ )
Id ∂j .
Then we can develop C θ and C y in powers of ε. In particular, we find: C0θ =
θ (0) ∂ +G − 11 θ 11 0 G22 G21
,
y
C0 = −
d j =1
j
j
11 (0) 12 (0) t j (0) 12
0
∂j .
We define: y
Cj := Cjθ + Cj ,
j ∈ {0, 1, 2},
C1ε := C1 + ε C2 ,
∈ { , θ, y} .
We complete (Lεµ ) with initial data: ˙ 0 )(θ, y) ∈ H s (T × Rd ; Rn ) . u˙ 0 (θ, y) = t (˙v0 , w
(I)
According to Property 1, we have: (ε−2 A + ε−1 B + C)∗ + (ε −2 A + ε−1 B + C) 0,0 = ε−1 (B ∗ + B) + (C ∗ + C) ∈ n,n .
(2.1)
Thus the Cauchy problem (Lεµ )−(I) can be solved on [0, T˜ ]. At the time t > 0, the Hns − norm of its solution u˙ is multiplied by some factor. We control this amplification through the quantity:
˙ g s (ε, µ; t) := sup u(t) Hns ; u˙ 0 Hns ≤ 1 ,
s ∈ R.
(2.2)
In view of (2.1), energy estimates on (Lεµ ) yield: ∀ s ∈ R,
∃ cs > 0 ;
g s (ε, µ; t) ≤ ec
s
t/ε
,
∀ (ε, µ, t) ∈ × [0, T˜ ] .
(2.3)
In the two next Sects. 2.2 and 2.3, we work with general operators A, B and C satisfying Property 1. We introduce conditions on A, B and C which allow to improve (2.3). Then, in Sect. 2.4, we test these restrictions in the case of the two models under consideration.
Propagation of Oscillations in Real Vanishing Viscosity Limit
671
2.2. A first reduction. In this paragraph, we work under special conditions on the operators A, B and C : Assumption 2. 0,0 −1,0 A.2.1) There is a projector P = P∗ = P2 ∈ n,n and some operator Q ∈ n,n such that: Q P = P Q = 0, A P = P A = 0, A Q = Q A = Id − P . A.2.2) The actions P and Q do not depend on (t, θ, y) ∈ [0, T˜ ] × T × Rd : [∂ ; P] = 0,
[∂ ; Q] = 0,
∀ ∈ {t, θ, y} .
A.2.3) We have (Id − P) B = 0. We impose also: Assumption 3. We have P B P = 0. Under Assumptions 2 and 3, and when the viscosity µ is chosen conveniently, the rough estimate (2.3) can be improved: Proposition 2.1. Fix any ν > 0. If the applications A, B and C satisfy the Assumptions 2 and 3, we have: ∀ s ∈ R,
∃ cs > 0 ;
g s (ε, µ; t) ≤ cs ,
∀ (ε, µ, t) ∈ εν × [0, T˜ ],
(2.4)
where εν := ]0, 1] × [ν ε 2 ; +∞[. Proof of Proposition 2.1. In what follows, we drop the˙on u. We seek the normal form associated with (Lεµ ). To this end, we define: up := u − ε D u,
−1,0 D := P B Q ∈ n,n .
We observe that: D 2 = A D = (∂t D) D = D [∂θ2 ; D] = 0, [∂j2 ; D] = 0,
(Id − ε D)−1 = Id + ε D .
∀ j ∈ {1, · · · , d} .
Therefore, the equation (Lεµ ) is equivalent to: (Lεµp )
∂t up = ε−2 Ap up + ε −1 Bp up + Cp up + µ up
with: 1,0 , Ap = A = −A∗p ∈ n,n 0,0 Bp = B − D A = B − P B (Id − P) = P B P ∈ n,n , 1,1 Cp = C + [B; D] + ε ([C; D] − D B D − ∂t D + µ [∂θ2 ; D]) − ε 2 D C D ∈ n,n .
We can decompose Cp into: y
Cp = Cpθ + Cp ,
θ θ θ Cpθ = Cp0 + ε Cp1 + ε 2 Cp2 ,
y
y
y
y
Cp = Cp0 + ε Cp1 + ε 2 Cp2
672
C. Cheverry
with: y
y
Cp0 = C0 ,
Cp0 = C0 + [B; D],
∈ { , θ} ,
θ 1,0 = C1θ + [C0θ ; D] − D B D − ∂t D + µ [∂θ2 ; D] ∈ n,n , Cp1 y
y
y
0,1 Cp1 = C1 + [C0 ; D] ∈ n,n , = C2 + [C1ε ; D] − D C D, Cp2
∈ {θ, y} .
We define as before: y
θ Cpj := Cpj + Cpj ,
j ∈ {0, 1, 2},
Cp1ε := Cp1 + ε Cp2 ,
∈ { , θ, y} .
We remark that: θ ∗ θ 0,0 (Cp0 ) + Cp0 = (C0θ )∗ + C0θ + n,n , θ 0,0 = C†θ + n,n , Cp†
(Cp0 )∗ + Cp0 = (C0 )∗ + C0 , y
y
0,1 Cp† = C† + n,n ,
y
y
( , †) ∈ { , y} × {0, 1, 2} .
Under Assumptions 2 and 3, we get also: B D = D B = Bp = 0 and we have the simplification: 1,1 Cp = C + ε ([C; D] − ∂t D + µ [∂θ2 ; D]) − ε 2 D C D ∈ n,n .
On the one hand, the singular part ε−1 Bp has disappeared in (Lεµp ). On the other hand, the symmetry is destroyed since we have only: 0,0 0,1 (Cp )∗ + Cp ∈ n,n + ε n,n . y
y
This lack of hyperbolicity is compensated by the presence of a viscosity µ of the order ε 2 . For the sake of completeness, we detail below the energy estimates which allow to justify this affirmation. The expression: upl,m := < Dθ >l < Dy >m up ,
(l, m) ∈ N2
is subjected to: θ θ ∂t upl,m = ε −2 Ap upl,m + Cp upl,m + Tl,m up + δm0 Tl+1,m−1 up y
y
+ Tl,m up + δl0 Tl−1,m+1 up + µ upl,m , where θ θ + δm0 Tl+1,m−1 , [< Dθ >l < Dy >m ; Cpθ ] = Tl,m
y
y
y
[< Dθ >l < Dy >m ; Cp ] = Tl,m + δl0 Tl−1,m+1 ,
˜
l,m ˜ Tl,˜θm˜ ∈ n,n , y l,m ˜
T˜
˜
l,m ˜ ∈ n,n .
We note (·, ·) the scalar product in Hn0 . We multiply the equation by upl,m . Then we integrate over T × Rd to find:
Propagation of Oscillations in Real Vanishing Viscosity Limit
673
∂t upl,m 2H 0 ≤ C upl,m 2H 0
+ C upl,m Hn0 δm0 upl+1,m−1 Hn0 + δl0 upl−1,m+1 Hn0 θ∗ θ + Cp1ε ) < ∂θ >−1 upl+1,m , upl,m + ε (Cp1ε y∗ y + ε (Cp1ε + Cp1ε ) < Dy >−1 upl,m+1 , upl,m − C µ upl+1,m 2H 0 + upl,m+1 2H 0 .
n
n
n
n
N2 ;
Let j ∈ N. We define J := {(l, m) ∈ l + m = j } and 2 l,m 2 up ˜ j := up H 0 ∼ up 2 j . Hn
Hn
n
(l,m)∈J
We take the sum on (l, m) ∈ J of the preceding inequalities to get: ∂t up 2˜ j ≤ C up 2˜ j Hn
H
n + ε C up H˜ j up H j +1,j + up H j,j +1 n n n − C µ up 2 j +1,j + up 2 j,j +1 .
Hn
Hn
Since µ ≥ ν ε2 , it follows that: ∃ cj > 0 ;
∀ t ∈ [0, T˜ ] .
up (t) H j ≤ cj up (0) H j , n
n
0,0 , we have: Since D ∈ n,n 2 ∃ (C1 , C2 ) ∈ (R+ ∗) ;
C1 u H j ≤ up H j ≤ C2 u H j . n
n
n
It gives (2.4) if s = j ∈ N. By interpolation, we deduce (2.4) for all s ∈ R+ . Now the adjoint of (Lεµp ) inherits the same characteristics as (Lεµp ). Therefore an argument of duality leads to (2.4) for all s ∈ R. It can happen in the applications that Assumption 3 is not satisfied. In this case an estimation as (2.4) is not reachable. However, under supplementary constraints on A, B and C, we can obtain bounds which are intermediate between (2.3) and (2.4). This is the goal of the next paragraph. 2.3. Transformation of the equations. To cover the many interesting examples, we set a hierarchy of constraints: Assumption 2 ⊃ Assumption 4 ⊃ Assumption 5 ⊃ Assumption 6 which are defined in the following way: 0,0 Assumption 4. We have Assumption 2 and we can find a projector M ∈ n,n satisfying:
M = P M P = M∗ = M2 , [∂θ ; M] = 0, M C0 (P − M) = 0,
I m (P − M) ⊂ {0} × Rq ,
[∂j ; M] = 0,
[∂t ; M] = 0 ,
B = P B (Id − P) + (P − M) B M .
674
C. Cheverry
Assumption 5. We have Assumption 4 and y
0,0 M (C1 − B Q C0 ) (P − M) ∈ n,n .
(2.5)
Assumption 6. We have Assumption 5 and y
y
0,0 , Q C0 (P − M) C0 (P − M) ∈ n,n
y
y
0,0 C0 Q C0 (P − M) ∈ n,n .
(2.6)
These conditions are organized so that the viscosity µ can be diminished more and more (but still remains bigger than ε2 ): Proposition 2.2. We can find a constant ν ≥ 1 so that for all parameters (ε, µ) belonging to the domain εν defined according to : a) εν := ]0, 1] × [ν; +∞[ under Assumption 4, b) εν := ]0, 1] × [ν ε; +∞[ under Assumption 5, c) εν := ]0, 1] × [ν ε 2 ; +∞[ under Assumption 6, we have: ∀ s ∈ R,
∃ cs > 0 ;
g s (ε, µ; t) ≤ ε −1 cs ,
∀ (ε, µ, t) ∈ εν × [0, T˜ ] .
(2.7)
Let t ∈ ]0, T˜ ] be fixed. When ε goes to zero, the majoration (2.7) gives informas tions which are much more precise than (2.3). Indeed the amplification factor ec t/ε is replaced by the polynomial growth ε−1 cs . Proof of the Proposition 2.2. To treat the cases a), b) and c), we follow the same strategy. We transform the equation (Lεµ ) into: ∂t uf = ε −2 Af uf + Cf uf + µ uf ,
(Lεµf )
where uf ∈ Rnf with nf > n. Then we study the quantity:
gfs (ε, µ; t) := sup uf (t) Hns ; uf (0) Hns ≤ 1 , f
f
s ∈ R.
The link between the vector valued functions u and uf is achieved by way of an application T which is adjusted such that: Property 2. P.2.1) ε T ∈ n0,0 f ,n . P.2.2) The restriction T (ε, µ; 0) has a right inverse. More precisely: 0,0 ∃ T −1 ∈ n,n T (ε, µ; 0) ◦ T −1 (ε, µ) = Id, ∀ (ε, µ) ∈ . f ; ˜ P.2.3) For all (ε, µ, t) ∈ × [0, T ], we have the commutative diagram: (Lεµ )
u(0) −−−−→ u(t) T (ε,µ;t) T −1 (ε,µ) uf (0) −−−−→ uf (t) (Lεµf )
with the uniform estimates: ∀ s ∈ R,
∃ cfs > 0 ;
gfs (ε, µ; t) ≤ cfs ,
∀ (ε, µ, t) ∈ εν × [0, T˜ ] .
(2.8)
Propagation of Oscillations in Real Vanishing Viscosity Limit
675
We can exploit Property 2 in order to obtain: u(t) Hns ≤ ε −1 ||| ε T |||s,s0,0
nf ,n
≤ε
−1
≤ε
−1
uf (t) Hns
f
||| ε T
|||s,s0,0 nf ,n
cfs
||| ε T
|||s,s0,0 nf ,n
cfs
uf (0) Hns
f
||| T
−1
|||s,s0,0 n,nf
u(0) Hns .
We recover here (2.7) with: cs = ||| ε T |||s,s0,0 cfs ||| T −1 |||s,s0,0 nf ,n
n,nf
< ∞.
In order to implement this method, we will proceed in five stages. The three first are devoted to the construction of T . The fourth verifies that T inherits the required properties. The fifth shows that the equation (Lεµf ) is well-posed in Hnsf in the sense of (2.8). • Step 1. Blow up. Our starting point is the equation (Lεµp ). In the absence of the relation P B P = 0, we do not have as good simplifications as before. It remains: 0,0 , Bp = (P − M) B M ∈ n,n 1,1 Cp = C + B D + ε ([C; D] − ∂t D + µ [∂θ2 ; D]) − ε 2 D C D ∈ n,n .
We define the projectors: E := Id − P + M = E∗ = E2 ,
F := P − M = F∗ = F2 .
We remark that E + F = Id,
EF = FE = 0.
The equation (Lεµp ) has a special structure. In particular, we can isolate inside (Lεµp ) the singular contribution: 1 0 FB M F up F up = . (2.9) ∂t M up M up ε 0 0 Suppose in a first approximation that F B M is a constant matrix. Then the solution of (2.9) is given by F up (t) = F up (0) + ε−1 t F B M up (0),
M up (t) = M up (0) .
We see here that the components F up and M up play separated parts. Only F up is amplified by the factor ε−1 . This remark is still right when B has variable coefficients. We take it into account by putting up in the form: up = us1 + ε −1 us2 + us3 ,
j
us ∈ Hn0 ,
j ∈ {1, 2, 3},
where the us are adjusted so that: us1 = E us1 ,
us2 = t (0, ws2 ) = F us2 ,
us3 = t (0, ws3 ) = F us3 .
(2.10)
676
C. Cheverry
We plug (2.10) into (Lεµp ) to get after simplifications: ∂t us1 + ε −1 ∂t us2 + ∂t us3 = ε −2 Ap us1 + ε −1 Bp us1 + (Cp + µ ) (us1 + ε −1 us2 + us3 ) .
(2.11)
We compose (2.11) on the left with E. It gives: ∂t us1 = ε −2 Ap us1 + ε −1 E Cp us2 + E Cp us1 + E Cp us3 + µ us1 . We compose (2.11) on the left with the projector F. The resulting equation is underdetermined. We remedy this difficulty by distributing the terms according to: ∂t us2 = F Bp us1 + F Cp us2 + µ us2 , ∂t us3 = F Cp us1 + F Cp us3 + µ us3 . We introduce: us := t (us1 , us2 , us3 ) ∈ Hn0s ,
ns := 3 n .
We observe that the equation for us can be formulated as: ∂t us = ε−2 As us + ε −1 Bs us + Cs us + µ us
(Lεµs ) if we set:
A00 As := 0 0 0 , 0 00
0 (Id − P) Cp F + M C0 F 0 0 0, Bs := 0 0 0 0
E Cp E M Cp1ε F E Cp F 0 . Cs := F Bp E F Cp F F Cp E 0 F Cp F We introduce:
P 0 0 Ps := 0 Id 0 , 0 0 Id
Q00 Qs := 0 0 0 . 0 00
With these conventions, we recover A.2.1) and A.2.2) with the suffix s. On the other hand, we have no more P.1.2) and P.1.3) since: Bs ∈ n0,0 , s ,ns
(Cs )∗ + Cs ∈ n0,0 . s ,ns
We decide to complete (Lεµs ) with the initial data: us (0) =
t
us1 (0), us2 (0), us3 (0) = t E up (0), 0, F up (0) .
This choice is coherent with the formula (2.10) since: us1 (0) + ε −1 us2 (0) + us3 (0) = E up (0) + ε −1 × 0 + F up (0) = up (0) .
Propagation of Oscillations in Real Vanishing Viscosity Limit
677
• Step 2. Reduction. We want to suppress in (Lεµs ) the artificial singularities contained at the level of the coefficient ε −1 Bs . We define: uc := us − ε Ds us ,
Ds := − Qs Bs Ps ∈ n0,1 . s ,ns
We can decompose uc according to: uc = (uc1 , uc2 , uc3 ) := (us1 + ε Q Cp F us2 , us2 , us3 ) . In particular, we still have: uc1 = E uc1 ,
uc2 = F uc2 ,
uc3 = F uc3 .
We observe that Ds2 = Ds As = (∂t Ds ) Ds = Ds [; Ds ] = 0,
(Id − ε Ds )−1 = Id + ε Ds .
Therefore, the equation (Lεµs ) is equivalent to (Lεµc )
∂t uc = ε −2 Ac uc + ε −1 Bc uc + Cc uc + µuc
with Ac = As = −A∗c ∈ n1,0 ,n , s s 0 M C0 F 0 Bc = Bs + As Ds = 0 0 0
0 0 ∈ n1,0 + n0,1 , s ,ns s ,ns 0
Cc = Cs + [Bs ; Ds ] + ε ([Cs ; Ds ] − Ds Bs Ds − ∂t Ds + µ [; Ds ]) − ε2 Ds Cs Ds ∈ n1,3 . s ,ns Cancellations give rise to [Bs ; Ds ] = Ds Bs Ds = Ds Cs Ds = 0 . Under Assumption 4, we have Bc = 0. Moreover: Q Cp F Bp E Q Cp F Cp F − E Cp Q Cp F 0 0 0 0, [Cs ; Ds ] = 0 − F Cp Q Cp F 0
0 − Q [; Cp ] F 0 0 0. [; Ds ] = 0 0 0 0 It follows that 0,1 0,2 Cc ∈ n1,0 s ,ns + ns ,ns + ε ns ,ns . ε We complete (Lµc ) with the initial data: uc (0) =
t
uc1 (0), uc2 (0), uc3 (0) = t us1 (0), 0, us3 (0) .
678
C. Cheverry
• Step 3. Blow up. We define: t
(uf1 , uf2 , uf3 ) := t (uc1 , uc2 , uc3 ) = t (E uf1 , F uf2 , F uf3 ) ∈ Hn0 × Hn0 × Hn0 .
We can isolate in (Lεµc ) the equation governing the component uf2 : ∂t uf2 = F Bp E uf1 + F Cp F uf2 + µ uf2 .
(2.12)
0,0 . It gives the idea We observe that uf2 is coupled with uf1 by way of an operator in n,n to introduce in the machinery the new unknowns: j
j
uf := ε ∂j −3 uf2 = F uf ,
j ∈ {4, · · · , d + 3},
which can be interpreted at the level of the linearized equation (Lεµ ) as ε − transversal j
derivatives of some specific components. The expressions uf are subjected to: j
j
j
∂t uf = F Cp uf + ε ∂j −3 F Bp uf1 + ε [∂j −3 ; F Cp ] uf2 + µ uf . We introduce the function: uf := t (uf1 , · · · , ufd+3 ) ∈ Hn0f ,
nf := (d + 3) n .
We remark that the equation for uf can be formulated as: ∂t uf = ε −2 Af uf + Cf uf + µ uf
(Lεµf ) with:
A 0 ··· 0 0 0 ··· 0
Af := .. .. . . .. = − A∗f ∈ n1,0 , f ,nf . . .. 0 0 ··· 0 0 ··· ··· ···
0 0 0
ε [∂1 ;F Cp ] 0 Cf 11 ···
0
0 0
Cc
Cf := ε ∂1 F Bp E .. .
.. .
ε ∂d F Bp E
.. .
.. .
..
ε [∂d ;F Cp ] 0
0
···
.
+ n0,1 + ε n0,2 , ∈ n1,0 f ,nf f ,nf f ,nf .. .
Cf dd
where Cfjj := F Cp F,
∀ j ∈ {1, · · · , d} .
We complete (Lεµf ) with the initial data: uf (0) =
t
uf1 (0), · · · , ufd+3 (0) = t uc1 (0), 0, uc3 (0), 0, · · · , 0 .
Propagation of Oscillations in Real Vanishing Viscosity Limit
679
• Step 4. The change of variables T . We have to express u in terms of uf . By construction, we know that: u = (Id + ε D) up = (Id + ε D) (us1 + ε −1 us2 + us3 ) = (Id + ε D) (uc1 − ε Q Cp uc2 + ε −1 uc2 + uc3 ) = T uf = ε −1 uf2 + uf1 + uf3 + ε D uf1 − ε (Id + ε D) Q Cp < Dy >−2 uf2 −
d
j +3
(Id + ε D) Q Cp < Dy >−2 ∂j uf
.
j =1 0,0 We multiply T by ε. Coefficients remains which are operators in n,n . It implies that the property P.2.1) is true. To obtain P.2.2), it suffices to make explicit some right inverse T −1 . We can choose: 0,0 T −1 (ε, µ) u = t (Id − ε D) E u, 0, F u, 0, · · · , 0 , T −1 ∈ n,n . f
• Step 5. The equation (Lεµf ). The singular factor ε−1 is absent from (Lεµf ). All the difficulties are transfered to Cf . The operator Cf is of the order 2. It contains parts which come from the two reductions and the two blow ups. The question is to know if these contributions can be absorbed by the viscosity µ . We need a spectral analysis on (Lεµf ). We adopt below a basic point of view. We just perform energy estimates by multiplying (Lεµf ) by uf and then by integrating over T × Rd . We find : ∂t (uf , uf ) = (Cf + Cf∗ ) uf , uf − 2 µ (∇θ,y uf , ∇θ,y uf ) .
(2.13)
The condition µ ≥ ν ε2 with ν big enough allows to compensate the terms belonging to: := n0,0 + ε n1,0 + ε n0,1 + ε 2 n0,2 . f ,nf f ,nf f ,nf f ,nf The discussion works modulo the space . We have to look at: Sf = (Cf + Cf∗ ) mod . The non-trivial parts in Sf are only due to Cc + Cc∗ . We have to consider: Sc = (Cc + Cc∗ ) mod . We can decompose Sc into Sca + ε Scb with: Sca ∈ (n1,0 + n0,1 ) mod , s ,ns s ,ns
Scb ∈ n0,2 mod . s ,ns
Computations using the preceding informations lead to: b a 0 0 Sc12 0 0 Sc12 a∗ b∗ , b∗ 0 0, Sca = Sc12 Scb = Sc12 0 Sc32 b 0 0 0 0 Sc32 0
680
C. Cheverry
where: y
y
a 1,0 0,1 Sc12 := M C1θ F + M (C1 − B Q C0 ) F ∈ n,n + n,n , y
y
y
y
b 0,2 Sc12 := Q C0 F C0 F − E C0 Q C0 F ∈ n,n , y
y
b 0,2 Sc32 := − F C0 Q C0 F ∈ n,n .
The relation (2.13) yields: ∂t uf 2H 0 ≤ C uf 2H 0 +(C1 + ε C) uf H 1,0 uf Hn0 nf
nf
nf
f
+(C2 + ε C) uf H 0,1 uf Hn0 nf
uf 2 0,1 Hnf
+ε C3 −C µ uf 2
Hn1,0 f
f
+ ε2 C uf 2 0,1 Hnf 2 + uf 0,1 Hnf
with C1 := M C1θ F < Dθ >−1 0,00,0 , n,n
C2 := M C3 :=
y (C1
b Sc12
<
−B Q
y C0 )
F < Dy >−1 0,00,0 , n,n
Dy >−2 0,00,0 n,n
+
b Sc32
< Dy >−2 0,00,0 . n,n
The majoration: ∃ cf0 > 0 ;
gf0 (ε, µ; t) ≤ cf0
can be deduced from the preceding inequality if: µ ≥ ν when C1 = 0 or C2 = 0 , µ ≥ ν ε when C1 = C2 = 0 , µ ≥ ν ε2 when C1 = C2 = C3 = 0 . We recover here the hierarchy of constraints corresponding to Assumptions 4, 5 and 6. We have proved (2.8) when s = 0. The other cases s ∈ R∗ can be obtained by arguments similar to these given in the proof of Proposition 2.1. The property P.2.3) is true. 2.4. Analysis of turbulences. We show in this paragraph that the preceding approach can be applied when dealing with the concrete operators issued from (Sµε ). We first have to check Assumption 2. The operators P and Q involved by Assumption 2 are defined as follows. Let Pk be the orthogonal projector on the kernel of A(i k) : Pk A(i k) = A(i k) Pk = 0,
A(i k) := − i k sθ (0) + .
We denote Qk a partial inverse of A(i k) : Qk A(i k) = A(i k) Qk = Id − Pk ,
Pk Qk = Qk Pk = 0 .
Propagation of Oscillations in Real Vanishing Viscosity Limit
681
Let P be the orthogonal projector on the kernel of A(∂θ ). Let Q be a partial inverse of A(∂θ ). We have: 0,0 ck (y) ei k θ = Pk ck (y) ei k θ , P ∈ n,n , P k∈Z
Q
ck (y) e
ikθ
k∈Z
=
k∈Z
Qk ck (y) ei k θ .
k∈Z
With these conventions, we are sure that the algebraic conditions in A.2.1) and A.2.2) −1,0 are verified. The other restrictions Q ∈ n,n , A.2.3) and Assumptions 3, · · · , 6 depend on special features of the selected system. We study separately the two models which we have introduced. 2.4.1. Large-scale motions in the atmosphere. We consider the model presented in Subsect. 1.1. By construction: A(i k) =
˜ α3 σ k α˜ β−i 0 0 0 0 0 0 0 0
0 − α˜ − β−i ˜ α3 σ k 0 0
0 0 0 0 0
0 0 0 0 0
σ
0 −1 −i k 0 0
1 −i k 0 0 0 0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
.
We compute: P0 u = t (0, 0, v3 , w1 , w2 ),
Q0 u = t (−v2 , v1 , 0, 0, 0),
and, for all k ∈ Z∗ :
Pk u = < k >−2 t 0, k 2 v2 + i k v3 , −i k v2 + v3 , w1 , w2 , (Id − Pk ) u = t v1 , < k >−2 (v2 − i k v3 ), < k >−2 (i k v2 + k 2 v3 ), 0, 0 , Qk u = σ −1 < k >−2 t −v2 + i k v3 , v1 , i k v1 , 0, 0 .
We see that P and Q are Fourier multipliers on T. These operators do not depend on (ε, µ, t, θ, y). They commute with ∂t , ∂θ and ∂j . In view of the increasing of Pk and Qk 0,0 −1,0 with respect to k, we have P ∈ n,n and Q ∈ n,n . Moreover, we observe that:
Im B ⊂
u = t (v, w) ; v = 0
⊂ ker (Id − P) ,
I m P ⊂ ker B = u = (v, w) ; v1 = 0 . t
These inclusions give access to Assumptions 2 and 3. We can apply Proposition 2.1. 2.4.2. Strong oscillations We test our Assumptions in the case of Subsect. 1.2. We introduce: v(θ ) dθ, v∗ (θ ) := v(θ ) − v¯ , v¯ := ∂θ−1 v∗ (θ ) :=
T θ
0
v∗ (s) ds −
T
θ 0
v∗ (s) ds dθ .
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C. Cheverry
θ (0) is invertible, we have: By construction, A(∂θ ) = −sθ (0) ∂θ . Since the block s11
P u = t (¯v, w),
Qu =
t
θ −s11 (0)−1 ∂θ−1 v∗ , 0 .
The applications P and Q do not depend on (ε, µ, t, θ, y). They commute with ∂t , ∂θ and ∂j . We see that:
Q ∈ L(Hns −1,t ; Hns ,t ),
P ∈ L(Hns ,t ; Hns ,t ),
∀ (s , t ) ∈ R2 .
Moreover: (Id − P) B u = (Id − P) t ( 0 , a v¯ 1 ∂θ wεµ ) = 0,
(Id − P) u = t (v∗ , 0) ,
which gives Assumption 2. 0,0 ¯ Obviously M is a projector in n,n We take M u = t (¯v, w). which does not depend on (ε, µ, t, θ, y) and which commutes with ∂t , ∂θ and ∂j . We compute:
M C0 (P − M) u = −
d
M
j =1
We remark that: PB Pu =
0 a v¯ 1 × ∂θ wεµ
12 (0) ∂j w∗ 0 j
= 0.
≡ (P − M) B M u .
Assumption 4 is established. We can apply Proposition 2.2. The inequalities (2.4) are justified for µ ≥ ν > 0. In order to identify exactly the Reynold’s number µ at the level of which there is a transition between a stable and instable regime, we can test Assumptions 5 and 6, or we can study directly (Lεµf ). We examine below what happens in the case of Euler’s equations. 2.4.3. Euler’s equations of isentropic gas dynamics. This well-known model is a particular case of Sect. 1.2. The state variables are the pressure p ∈ R and the vector velocity v = t (v1 , · · · , vd+1 ). We distribute these components according to: u := t (v, w),
v := t (v1 , p),
w := t (v2 , · · · , vd+1 ) .
We see that p = 2, q = d and n = d + 2. We set: σ (p) 0 0 , 0 e0 (v) := 0 1 0 0 σ (p) Idd×d where σ ∈ C ∞ (R; R) is such that σ (0) = 1. We impose: θ Xes (v) = v1 = v1 , θ es (v) ∂x1 u j e (u) ∂j u
j
Xe (u) = wj = vj +1 ,
∀ j ∈ {1, · · · , d} ,
:= ( ∂x1 p , ∂x1 v1 , 0 , · · · , 0 ) , t
:= t ( 0 , ∂j +1 vj +1 , δ2(j +1) ∂2 p , · · · , δ(d+1)(j +1) ∂d+1 p ) .
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We look at: Eµε (u)
:=
e0 (v)
Xe (u; ∂t,x1 ,y ) u
θ + es (v) ∂x1 u
+
d
j
e (u) ∂j u,
j =1
−µ
(ε4 ∂x21
+ ∂y21
+ · · · + ∂y2d ) u,
where Xe is the vector field: θ (v) ∂x1 + Xe (u; ∂t,x1 ,y ) = ∂t + Xes
d
j
Xe (u) ∂j .
j =1
Let h ∈ H ∞ (T; Rd ) be such that ∂θ h ≡ 0 and T h(θ ) dθ = 0. Let wµ be the expression obtained by solving (1.13). We follow the procedure of Subsect. 2.1. We consider the linearized equations along t (0, ε wµ ). As in (2.2), we define the expression ges (ε, µ; t) which measures (in the case of Euler’s equations) how the Hns − norm is amplified. Proposition 2.3. We can find a constant ν ≥ 1 so that: ∀ s ∈ R, ∃ cs > 0 ; ges (ε, ν ε2 ; t) ≤ ε−1 cs , ∀ (ε, t) ∈ ]0, 1] × [0, T˜ ] .
(2.14)
Proof of Proposition 2.3. We give the demonstration in space dimension two, when d = 1 and y = y1 . The other cases d ≥ 2 can be obtained by similar arguments. We note: hε (t, θ ) := wν ε2 (t, θ ) ∈ C ∞ ([0, T˜ ] × T; R),
T˜ < ∞ .
By construction, we have: ∂t hε = ν ε 2 ∂θ2 hε ,
h ∈ H ∞ (T; R) .
hε (0, θ) = h(θ),
We can decompose hε into k ε (t, θ ) ∈ C ∞ ([0, T˜ ] × T; R)
hε (t, θ ) = h(θ ) + ε 2 k ε (t, θ ), where k ε is subjected to
∂t k ε = ν ε 2 ∂θ2 k ε + ν ∂θ2 h,
k ε (0, θ) = 0 .
The discussion involves the following operators: v v1 0 −∂θ 0 1 p v A = −∂θ 0 0 , u = 2 = v , w
2
C = C = y
with
y
C0 =
0
0 0
− ε hε ∂1 0 0 − ∂1 0 − ε hε ∂1 ε 0 − ∂1 −ε σ (0) ∂t h − ε hε ∂1
0 0 0 0 0 − ∂1 0 − ∂1 0
,
y
C1 =
B =
0 0 0 0 0 0 , − ∂θ hε 0 0
y
y
= C0 + ε C1 ,
− hε ∂1 0 0 0 0 − hε ∂1 ε 0 − σ (0) ∂t h − hε ∂1
.
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C. Cheverry
We compute: − ∂θ−1 v2∗ Q u = − ∂θ−1 v1∗ , 0
v¯ 1 P u = v¯ 2 , w
v¯ 1 M u = v¯ 2 . ¯ w
We apply the procedure of Subsect. 2.3. We obtain nf = 6 by identifying uf with the vector: ¯ w1∗ , w2∗ , w3∗ ) ∈ R6 , uf = t (uf1 , uf2 , uf3 , uf4 ) = t (v1 , v2 , w, where uf2 = w1∗ ,
¯ uf1 = t (v1 , v2 , w),
uf3 = w2∗ ,
uf4 = w3∗ .
We recall that the equation governing the component uf2 is: ∂t uf2 = F Bp E uf1 + F Cp F uf2 + ν ε 2 uf2 . By construction, we have: F Bp E uf1 = F B M uf1 ≡ − ∂θ hε v¯ 1 , F Cp F uf2 = F C F uf2 − ε F B Q C F uf2 −ε (hε ∂1 w1∗ )∗ + ε (∂θ hε ∂θ−1 ∂1 w1∗ )∗ . Therefore, we have to deal with ∂t w1∗ = − ∂θ hε v¯ 1 − ε hε ∂1 w1∗ + ε ∂θ hε ∂θ−1 ∂1 w1∗ +2 ε
T
hε ∂1 w1∗ dθ + ν ε 2 w1∗ .
(2.15)
We test Assumption 5 to find y
M (C1 − B Q C0 ) F u(t) = where ∗
L(t) w (y) :=
T
t
0, 0, − 2 L(t) w∗ (t) ,
hε (t, θ ) ∂1 w∗ (θ, y) dθ,
0,1 L(t) ∈ 1,1 .
In general this quantity is not equal to zero. For example: hε (t, θ )2 dθ = 0 . L(t) y hε (t, ·) =
T
At first sight we have to take µ ≥ ν case a) of Proposition 2.2 . However a more careful examination of the problem reveals that we can go further. Indeed, the operator L(t) is tested at the level of Cf only against uf2 (t) which is subjected to (2.15) with initial data w1∗ (0, ·) ≡ 0. The point is that the solutions w1∗ (t, ·) so obtained are approximately polarized in the kernel of L(t). To understand what happens, we first present the argument when the source term v¯ 1 is smooth, when the function hε (t, θ ) is replaced by h(θ ) and when the contribution ν ε2 ∂θ2 w1∗ is removed:
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685
Lemma 2.1. Let v¯ 1 ∈ C ∞ ([0, T˜ ] × T × R; R). The solution w1∗ of the following Cauchy problem: ∂t w1∗ = − ∂θ h v¯ 1 − ε h ∂1 w1∗ + ε ∂θ h ∂θ−1 ∂1 w1∗ (2.16) + 2 ε T h ∂1 w1∗ dθ + ν ε 2 ∂y2 w1∗ ∗ w1 (0, θ, y) = 0 is subjected to L(0) w1∗ (t, y)
=
h(θ ) ∂1 w1∗ (t, θ, y) dθ = 0,
T
∀ t ∈ [0, T˜ ] .
Proof of Lemma 2.1. We introduce the vector space: V :=
N
cn (y) h(θ )n ∂θ h(θ ) ; cn ∈ C ∞ (R; R) for all n .
n=0
We observe that: ∂θ h v¯ 1 ∈ V , h ∂1 w ∈ V , ∂ θ h ∂θ−1 ∂1 w ∈ V , T h(θ ) ∂1 w(θ, y) dθ = 0,
∀ v¯ 1 ∈ C ∞ (R; R) , ∀w ∈ V , ∀w ∈ V , ∀w ∈ V .
It follows that the solution w1∗ of (2.16) is such that: h(θ ) ∂1 w1∗ (t, θ, y) dθ = 0, w1∗ (t, ·) ∈ V¯ , T
where V¯ is the closure of V in L2 (T × R; R).
∀ t ∈ [0, T˜ ],
Now we have to adapt this reasoning in the case of the full Eq. (2.15). We show below y that the contribution M (C1 − B Q C0 ) F uf2 is in fact negligible so that we have no need in Assumption 5. We define the constant:
ι := 1 + 2 sup hε L∞ ([0,T˜ ]×T) ; ε ∈ ]0, 1] < ∞, the smooth functions: fnε (t, θ ) := [ n (n − 1) (hε )n−2 (∂θ hε )2 + (n + 1) (hε )n−1 ∂θ2 hε ](t, θ ), ∞ ε −n g (t, θ ) := ι hε (t, θ )n dθ ei n θ , T
n=0
and the operators:
cn (w) := T52 w := ν
T
e− i n θ w(θ, y) dθ,
∞
ι
−n
n=0
T55 w := − ε ι e
−i θ
T
n ∈ N,
fnε (t, θ ) w∗ (θ, y) dθ ei n θ ,
∂1 w − i ε ι e− i θ ∂1 ∂θ−1 w + 2 ε ι g ε ∂1 c1 (w) .
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C. Cheverry
The choice of ι implies that: −∞,0 , cn ∈ 1,1
−∞,0 T52 ∈ 1,1 ,
0,1 T55 ∈ ε 1,1 .
We introduce the function: =
ud5
F ud5
:= ε
−2
∞
ι
−n
n=0
T
hε (t, θ )n w1∗ (t, θ, y) dθ ei n θ .
Simple computations using the identities:
T
hε (t, θ )n ∂θ hε (t, θ ) dθ = 0,
T
w1∗ (t, θ, y) dθ = 0
indicate that the quantity ud5 is subjected to ∂t ud5 = T52 uf2 + T55 ud5 + ν ε 2 ∂y2 ud5 ,
uf2 ≡ w1∗ .
We do again a blow up. We consider the new unknown: ud = t (ud1 , ud2 , ud3 , ud4 , ud5 ) := t (uf1 , uf2 , uf3 , uf4 , ud5 ) . In view of the characteristics of the operators T∗ , the introduction of ud5 is compatible a as with energy estimates. In other respects, it allows to interpret the block Sc12 y
a Sc12 = M (C1 − B Q C0 ) F uf2 =
t
0, 0, − 2 ι ε2 ∂1 c1 (ud5 ) .
We remark that 0,1 − 2 ι ε2 ∂1 c1 ∈ ε 2 1,1 ,
− 2 ι ε2 ∂1 c1 ≡ 0 mod .
Therefore we have Sca ≡ 0 mod . It remains to study Scb . Now C0 (P − M) u = t (0, − ∂1 w∗ , 0) y
which leads to y
(P − M) C0 (P − M) u = 0 , C0 Q C0 (P − M) u = C0 t (∂1 ∂θ−1 w∗ , 0, 0) = 0 . y
y
y
We find Scb ≡ 0 mod . Assumption 6 is satisfied. We have (2.8) for µ = ν ε 2 . It gives access to (2.14).
Propagation of Oscillations in Real Vanishing Viscosity Limit
687
3. The Nonlinear Equations Consider the two Cauchy problems which are set in Theorem 1 and Theorem 2. Since ε on the system (Sµε ) is symmetric hyperbolic, there is Tµε > 0 and a unique solution uµ C 0 ([0, Tµε ]; Hnσ ). We note T¯µε the maximal time of existence of such a smooth solution. Classical results (see for example [8]) guarantee that ∀ µ > 0,
∃ cµ > 0 ;
T¯µε ≥ cµ ε,
∀ ε ∈ ]0, 1] .
Moreover, if T¯µε is finite, we must have (see [10]): lim
t −→ T¯µε
ε uµ (t, ·) W 1,∞ (T×Rd ) = +∞ .
ε (t, ·) is unbounded as t tends to T¯ ε < ∞. In particular, it implies that the Hnσ − norm of uµ µ In view of this remark, statements 1 and 2 are a consequence of the existence of T > 0 such that
ε sup uµ (3.1) (t, ·) Hnσ ; (t, ε, µ) ∈ [0, min(T¯µε , T )] × εν < ∞ .
To get this information, we look at the corrector: ε ε ε cµ (t, θ, y) = t (aµ , bεµ )(t, θ, y) := ε −2 uµ (t, θ, y) − t (0, ε wεµ )(t, θ ) . ε . The equation for c can be formulated We drop the parameters ε and µ at the level of cµ as
˜ 0 ∂t c = ε −2 A c + ε−1 B c + C˜ c + µ c with
(3.2)
0 (ε 2 a) 11 0 := 0 (ε 2 a) , 0 22 C11 + RC11 C12 + RC12 ˜ . C = C + RC = C21 + RC21 C22 + RC22
˜0 =
˜0 0 11 ˜0 0 22
We make explicit the operators RC . Let f ∈ C ∞ (Rn ; Rn ) with f (0) = 0. We introduce the notations: 1 (Du f )(t u) dt , f (u) = Lf (u) u, Lf (u) = 0
Lf (u) = Lv f (u) + Lw f (u), 1 (D f )(t u) dt, ∈ {v, w} , L f (u) = 0
˜ + Lf (u, ˜ u) u, f (u˜ + u) = f (u) 1 ˜ u) = (Du f )(u˜ + t u) dt , Lf (u, 0
f (u) = (Du f )(0) u + Qf (u)(u, u), 1 (1 − t) (Du2 f )(t u) dt , Qf (u) = 0
688
C. Cheverry
˜ X(ε, ε 2 a, ε wεµ + ε 2 b; ∂t,θ,y ) = ∂t + X˜ + a a1 ∂θ + ε 2 LX(c; ∂θ,y ) , X˜ = Xθ (0, ε wεµ ) ∂θ +
d
X j (0, ε wεµ ) ∂j ,
j =1
˜ LX(c; ∂θ,y ) = Lv˜X(a; ∂θ,y ) + Lw˜ X(b; ∂θ,y ) , Lv˜X(a; ∂θ,y ) = QXsθ (ε 2 a)(a, a) ∂θ + Lv X θ (ε wεµ , ε2 c) a ∂θ , +
d
Lv X j (ε wεµ , ε2 c)a ∂j ,
j =1
Lw˜ X(b; ∂θ,y ) = Lw X θ (ε wεµ , ε2 c)b ∂θ +
d
Lw X j (ε wεµ , ε2 c) b ∂j ,
j =1
Now we can set: θ (ε 2 a) a ∂θ a + Qgs1 (ε 2 a)(a, a) RC11 a = − Ls11 0 ˜ −ε2 L11 (ε 2 a) a X˜ a − a a1 ∂θ a − ε2 LX(c; ∂θ,y ) a d j θ −ε2 L11 (ε wεµ , ε2 c) c ∂θ a − ε2 L11 (ε wεµ , ε2 c) c ∂j a j =1
+ε Qg1 (ε a)(a, a) − ε 0 ˜ −ε4 L11 (ε 2 a) a LX(c; ∂θ,y ) a , 2
RC12 b = − ε2
2
d
2
0 L11 (ε 2 a) a
a a1 ∂θ a
j
L12 (ε wεµ , ε2 c) c ∂j b ,
j =1
RC21 a = − ε
2
d
j Lt 12 (ε wεµ , ε2 c) a ∂j a − ε Lv˜X(a; ∂θ,y ) wεµ
j =1
0 ˜ ∂θ,y ) wεµ −ε L22 (ε 2 a) a ∂t + X˜ + a a1 ∂θ + ε 2 LX(c; 0 −ε 2 L22 (ε 2 a) a X˜ b − ε R21 a + Lv g2 (ε wεµ , ε2 c) − Lv g2 (0, 0) a , RC22 b = − a a1 ∂θ b − ε Lw˜ X(b; ∂θ,y ) wεµ + Lw g2 (ε wεµ , ε2 c) − Lw g2 (0, 0) b − ε R22 b 0 ˜ −ε2 LX(c; ∂θ,y ) b − ε2 L22 (ε 2 a) a a a1 ∂θ b 0 ˜ −ε4 L22 (ε 2 a) a LX(c; ∂θ,y ) b .
Propagation of Oscillations in Real Vanishing Viscosity Limit
689
Equation (3.2) is not exactly of the same form as (Lεµ ). The derivative ∂t is replaced by ˜ 0 ∂t and the operator C is changed into C˜ = C + RC, where RC depends on c. We have to adjust the procedure of Sect. 2 in order to incorporate these modifications. First, we generalize Definition 1 in the case of non-linear operators just by looking at the extra variables (like a, b and c) as parameters. With this convention, we find: 0,1 1,0 RC ∈ n,n + n,n ,
∀ ∈ {11, 12, 21, 22} .
Then, we distinguish the rˆole of c when it occurs as a parameter or as a variable. We use the notation c instead of c in the first case and we keep c in the second case. It follows some flexibility when computing the order of operators. Take for example D = (c) ∂θ , where is a symmetric matrix, and compute D + D ∗ . We can consider that (D + D ∗ ) c = [(∂θ c · ∇c )(c)] c
⇒
1,0 (D + D ∗ )(c) ∈ n,n ,
or that (D + D ∗ ) c = [(∂θ c · ∇c )(c)] c
⇒
0,0 (D + D ∗ )(c, ∇θ,y c) ∈ n,n .
There will be no possible confusion between these two choices since they will be implicitly specified by putting (c) or (c, ∇θ,y c) in front of the operators. The second interpretation will be allowed when the contribution ∇θ,y c can be managed by classical tools in the hyperbolic situation (the fact that σ > d+3 2 is important at this level). We explain in the next paragraphs how to adapt the procedure of sect. 2 in this new context. 3.1. Proof of Theorem 1. Assumptions 2 and 3 are still verified since A and B are as before. The state variable: cp := c − ε D c = t (ap , bp ),
ap ∈ Rp ,
bp ∈ Rq
is subjected to ˜ p0 ∂t cp = ε −2 A cp + C˜ p cp + µ cp ,
(3.3)
where: ˜ p0 (ε 2 cp ) = ˜ 0 + ε 3 [Lv 0 (ε 2 a) a; D] − ε4 D [Lv 0 (ε 2 a) a] D , ˜ D] − ˜ 0 ∂t D + µ [∂θ2 ; D]) + ε 2 (D ˜ 0 ∂t D − D C˜ D) . C˜ p = C˜ + ε ([C; We remark that C˜ p = Cp + RCp with: RCp = RC + ε [RC; D] − ε 2 D RC D −ε3 [Lv 0 (ε 2 a) a] ∂t D + ε 4 D [Lv 0 (ε 2 a) a] ∂t D . Equation (3.3) has hyperbolic and parabolic features. The analysis of (3.3) groups these two aspects. The symmetric parts of the non linear coefficients are handled as usual in the hyperbolic situation [1]. The remaining terms are compared to the viscosity. It is important to explain how the different contributions are distributed when performing energy estimates on (3.3). The strategy is the same when dealing with cp or
690
C. Cheverry
with the spatial derivatives of cp (up to the order σ ). Therefore, it suffices to describe the case of L2 − estimates. We look at ˜ p0 cp , cp ) = (Cp + Cp∗ ) cp , cp + (RCp + RCp∗ ) cp , cp ∂t ( ˜ p0 ) cp , cp + cp , ( ˜ p0∗ − ˜ p0 ) ∂t cp + (∂t − 2 µ (∇θ,y cp , ∇θ,y cp ) . We deal with Cp + Cp∗ as in Subsect. 2.2. It remains to discuss the other terms. We observe that 0,0 ˜ p0∗ − ˜ p0 )(ε 2 cp ) ∂t cp ∈ n,n ( · ε 3 ∂t c p , 0,0 ˜ p0 (ε 2 cp ) ∈ n,n ˜ p0 (ε 2 cp ) = (ε2 ∂t cp · ∇cp ) ∂t · ε 2 ∂ t cp , 0,0 1,0 0,1 (RCp + RCp∗ )(cp , ∇θ,y cp ) ∈ n,n + ε (n,n + n,n ).
We use (3.3) in order to exchange the time derivative ε2 ∂t cp for the space derivatives ∇θ,y cp and ε 2 cp . We make an integration by parts to replace ε 2 cp by a term ε2 Q with Q quadratic in ∇θ,y cp . It follows that
˜ p0∗ − ˜ p0 ) ∂t cp = cp , (cp ) cp + ε2 Q(∇θ,y cp , ∇θ,y cp ), cp , ( ˜ p0 ) cp , cp = (cp , ∇θ,y cp ) cp , cp + ε 2 Q(∇θ,y cp , ∇θ,y cp ), (∂t
where: 1,0 0,1 (cp ) ∈ ε (n,n + n,n ),
0,0 (cp , ∇θ,y cp ) ∈ n,n .
The non-linearities in cp and ∇θ,y cp can be controlled by way of the inclusion Hnσ → W 1,∞ . All the other contributions can clearly be absorbed by the viscosity. It implies that we can find some T > 0 and cσ > 0 such that cp (t) Hnσ ≤ cσ cp (0) Hnσ ,
∀ (t, ε, µ) ∈ [0, min(T¯µε , T )] × εν .
We recall that ε uµ = t (0, ε wεµ ) + ε2 (Id + ε D) cp .
The majorations (3.1) and (0.2) follow directly.
3.2. Proof of Theorem 2. Since cp (0) = (ε 2 ), we can take dp = t (ep , hp ) := ε −2 cp ,
ep := ε −2 ap ,
hp := ε −2 bp
as new unknowns. When Assumption 3 is not satisfied. Equation (3.3) must be replaced by ˜ p0 ∂t dp = ε −2 A dp + C˜ p dp + ε −1 Bp dp + µ dp .
(3.4)
Propagation of Oscillations in Real Vanishing Viscosity Limit
691
We define Cp as in Step 1 of Subsect. 2.3. We have again C˜ p = Cp + RCp . The quanti˜ p0 and RCp are obtained from the preceding definitions by expressing a and b in ties terms of ap and bp , and then by replacing ap and bp by ε 2 ep and ε 2 hp . It yields: RCp = ε 2 Cp (ε, t, θ, y, ep , ε2 hp ; ∂θ,y ) + ε ϒCp (ε, t, θ, y, ep , ε2 hp ) . Here Cp is a differential operator whose symbol is homogeneous of order one with respect to the dual variables (k, η) ∈ Z × Rd . The matrix ϒCp and the coefficients in Cp are smooth functions of their arguments ε ∈ ]0, 1], t ∈ [0, T¯µε ], θ ∈ T, y ∈ Rd , ep ∈ Rp and ε 2 hp ∈ Rq . We remark that 1,0 0,1 0,0 RCp ∈ ε 2 (n,n + n,n ) + ε n,n .
It follows that we can decompose C˜ p into C˜ p = C˜ p0 + ε C˜ p1 + ε 2 C˜ p2 ,
C˜ p0 = Cp0 ,
0,0 C˜ p1 ≡ Cp1 mod n,n .
Assumptions 4, 5 and 6 are not changed since they are tested with the operators C˜ p0 and 0,0 . Therefore, we can take up the procedure of Sect. 2.3 as it C˜ p1 modulo the space n,n was. It leads to some non-linear equation (NLεµf ) instead of (Lεµf ). To understand the structure of (NLεµf ), we follow step by step what happens when transforming Eq. (3.4). Step 1. We make a blow up similar to (2.10): dp = ds1 + ε −1 ds2 + ds3 ,
ds1 = E ds1 ,
ds2 = F ds2 ,
ds3 = F ds3 .
The equations are distributed according to ˜ p0 E ∂t ds1 + ε −1 E ˜ p0 F ∂t ds2 + E ˜ p0 F ∂t ds3 E = ε −2 A ds1 + ε −1 E C˜ p ds2 + E C˜ p ds1 + E C˜ p ds3 + µ ds1 , ∂t ds2 = F Bp ds1 + F Cp ds2 + µ ds2 , ˜ p0 E ∂t ds1 + ε −1 (F ˜ p0 F − Id) ∂t ds2 + F ˜ p0 F ∂t ds3 F −1 2 1 3 = ε F RCp F ds + F C˜ p ds + F C˜ p ds + µ ds3 . The expression ds := t (ds1 , ds2 , ds3 ) is solution to (NLεµs )
˜ s0 ∂t ds = ε −2 As ds + ε −1 Bs ds + C˜ s ds + µ ds .
We find ˜ s0
:=
E ˜ p0 E ε−1 E (˜ p0 −Id) F E ˜ p0 F 0 Id 0 F ˜ p0 E ε−1 F (˜ p0 −Id) F F ˜ p0 F
˜ s0 , = Id + ε 3 R
Moreover C˜ s = Cs + RCs with E RCp E ε−1 E RCp F E RCp F RCs := . 0 0 0 F RCp E ε−1 F RCp F F RCp F
˜ s0 ∈ n0,0,n . R s s
692
C. Cheverry
The blow up costs ε −1 in front of the component hp . Since RCp depends on ε 2 hp , it remains RCs = ε Cs (ε, t, θ, y, ds ; ∂θ,y ) + ε ϒCs (ε, t, θ, y, ds ) with 1,0 0,1 Cs ∈ n,n + n,n ,
0,0 ϒCs ∈ n,n .
Steps 2, 3 and 4. Since As , Bs and also the equation for ds2 (in comparison with this on us2 ) are not changed, there is no modification here. Step 5. We define: dc = t (dc1 , dc2 , dc3 ) = t (df1 , df2 , df3 ) := ds − ε Ds ds , j
df := t (df1 , · · · , dfd+3 ),
j ∈ {4, · · · , d + 3} .
df := ε ∂j −3 df2 ,
The expression df is subjected to the nonlinear parabolic equation: ˜ f0 ∂t df = ε −2 Af df + C˜ f df + µ df .
(NLεµf ) ˜ 0 is given by Here f 0 ˜ 0 f := . .. 0
˜0 f 11
0 0 0 0 Id
··· ··· ··· ···
0 0 0 0
.. .
, .. .. . . .. . . . .
0
0 0 ··· Id
0
˜ s0 + ε [ ˜ s0 ;Ds ] − ε2 Ds ˜ s0 Ds ˜ 0 := f 11
Simple computations lead to: ˜ p0 − Id) Q Cp F 0 E ( 0 ˜ s ; Ds ] = 0 0 [ ˜ p0 − Id) Q Cp F 0 − F ( Moreover
C˜ f
:= ε ∂1 F Bp E .. . ε ∂d F Bp E
0 0 0
C˜ c
0 0 , 0
··· ··· ···
ε [∂1 ;F Cp ] 0 Cf 11 ···
.. .
.. .
.. .
..
ε [∂d ;F Cp ] 0
0
···
.
0 0 0
.
˜ s0 Ds = 0 . Ds
0 , .. .
Cf dd
where C˜ c = Cc + RCc with ˜ s0 (∂t Ds ) . RCc = RCs + ε [RCs ; Ds ] − ε 2 Ds RCs Ds + ε 4 (Id − ε Ds ) R We remark that
0 −E RCp Q Cp F 0 [RCs ; Ds ] = 0 0 − F RCp Q Cp F
0 0 , 0
Ds RCs Ds = 0 .
Propagation of Oscillations in Real Vanishing Viscosity Limit
693
In particular [RCs ; Ds ] ∈ ε n0,0 + ε 2 (n1,0 + n0,1 ) + ε 2 n0,2 . s ,ns s ,ns s ,ns s ,ns It follows that C˜ f + C˜ f∗ = Cf + Cf∗ mod When performing energy estimates on (NLεµf ), the only contributions which possibly can raise a difficulty are 0∗ ˜ f0 ) ∂t df , df , ˜f − ˜ f0 ) df , df . ( (∂t ˜ s0 are The parts due to ˜ s0∗ − R ˜ s0 ) ∂t dc , dc , ε 3 (R
˜ s0 ) dc , dc . ε3 (∂t R
˜ s0 ; Ds ] can be interpreted as The parts coming from ε [ 0,1 ε5 (n,n ∂t df2 , dfl ) = ε 5
d+3
j
0,0 (n,n ∂t df , dfl ),
l ∈ {1, 2} .
j =4
All these contributions can be managed by the method explained in the Proof of Theorem 1. As in Subsect. 2.3, we recover: ∀ (t, ε, µ) ∈ [0, min(T¯µε , T )] × εν .
dp (t) Hnσ ≤ ε −1 cσ dp (0) Hnσ , We recall that
ε = t (0, ε wεµ ) + ε4 (Id + ε D) dp . uµ
It follows that we have (3.1) and (0.3).
Remark. The power ε4 in front of u ε0 can be relaxed but we need supplementary assumptions to do that. It implies technicalities which we will not describe here. 3.3. Compressible equations of isentropic gas dynamics. The context is as in paragraph 2.4.3. Consider a family {u ε0 }ε∈]0,1] which is bounded in H σ (T × Rd ; Rd+1 ) with σ > d+3 2 . Introduce the data: x x 1 1 ε ∈ ]0, 1] + ε4 u ε0 2 , y , uε0 (x) := t 0 , ε h 2 ε ε Theorem 3. There is ν ≥ 1 and T > 0 such that for all ε ∈ ]0, 1], the Cauchy problem: ε Eν ε2 (uε ) = 0 , uε (0, x) = uε0 (x) has a unique solution uε (t, x) defined on [0, T ] × Rd+1 . Moreover fix any m ∈ R+ ∗. There is some constant cm such that: x 1 uε (t, x1 , y) − ε t (0, wν ε2 ) t, 2 L2 ([−m,m]×Rd ) ≤ CK ε 3 ε for all (t, ε) ∈ [0, T ]×]0, 1].
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Proof of Theorem 3. Since Case c) of Theorem 2 is relevant when dealing with compressible gas dynamics, we can find uε (t, θ, y) = t (vε , wε )(t, θ, y) defined on the strip [0, T ] × T × Rd such that: d ε 0 ε ε −2 ε ε ε e (v ) ∂t u + ε v1 ∂θ u + j =1 wj ∂j u ] j d θ (vε ) ∂ uε + + ε−2 es (uε ) ∂j uε = ν ε 2 uε , θ j =14 εe ε t u (0, θ, y) = 0 , ε h(θ ) + ε u 0 (θ, y) . Moreover uε (t, ·) − t (0, ε wν ε2 )(t, ·) H σ (T×Rd ) ≤ c ε 3 ,
∀ (ε, t) ∈ ]0, 1] × [0, T ]
which gives the expected result since
x 1 uε (t, x1 , y) − t (0, ε wν ε2 ) t, 2 L2 ([−m,m]×Rd ) ε = uε (t, θ, y) − t (0, ε wν ε2 )(t, θ ) L2 (T×Rd ) + (ε5 ) .
It is interesting to draw a parallel between Theorem 3 and another actual problem related to Navier-Stokes equations. Let ϕ ∈ C0∞ (R). There is some C ∈ R+ ∗ such that (1 + |ξ1 |2 )1/2 | ϕ(x1 ) uε0 (x) (ξ )|2 dξ ≤ C , ∀ ε ∈ ]0, 1] . Rd+1
1/2
Let E be the space of functions which are Hloc in x1 and L2 in y. The family {uε0 (x)}ε∈ ]0,1] is contained in a ball of E. Since the family {h ( xε1 )}ε∈ ]0,1] converges weakly (but not strongly) to 0 when ε goes to 0, the family {uε0 (x)}ε∈ ]0,1] is included in no compact of E. Therefore the solutions uε given by Theorem 3 belong to a space for which the results exposed in [3] can no more be applied. The other difference with [3] comes from the fact that the horizontal viscosity goes to 0 with ε (whereas it is fixed in [3]). References [1]
Alinhac, S., Gerard, P.: Op´erateurs pseudo-diff´erentiels et th´eor`eme de Nash-Moser. Savoirs Actuels, e´ ditions du CNRS, 1991, 190 p [2] Chemin, J-Y.: Syst`eme primitif de l’oc´ean-atmosph`ere et limite quasi-g´eostrophique. Pr´epublication de l’Ecole Polytechnique, expos´e VII, 1999–1996 [3] Chemin, J-Y., Desjardins, B., Gallagher, I., Grenier, E.: Fluids with anisotropic viscosity. Mathematical Modelling and Numerical Analysis, M2AN, 34(2), 315–335 (2000) [4] Cheverry, C., Gu`es, O., M´etivier, G.: Oscillations fortes sur un champ lin´eairement d´eg´en´er´e. Annales Scientifiques de l’ENS 5, (2003), 70 p [5] Cheverry, C., Gu`es, O., M´etivier, G.: Large amplitude high frequency waves for quasilinear hyperbolic systems, to appear in Advances in Differential equations [6] Friedlandler, S., Strauss, W., Vishik, M.: Nonlinear instability in an ideal fluid. Ann. Inst. Henri Poincar´e 14(2), 187–209 (1997) [7] Gallagher, I., Saint-Raymond, L.: Asymptotic results for pressureless magneto-hydrodynamics. Preprint de l’Ecole Polytechnique, n. 2003-11 [8] Gu`es, O.: D´eveloppement asymptotique de solutions exactes de syst`emes hyperboliques quasilin´eaires. Asymptotic Anal. 6(3), 241–269 (1993) [9] Joly, J-L., M´etivier, G., Rauch, J.: Transparent nonlinear geometric optics and Maxwell-Bloch equations. J. Differ. Eq. 166, 175–250 (2000) [10] Majda, A.: Compressible Fluid Flow and Systems of Conservation Laws in Several space variables. New-York: Springer Verlag, 1984
Propagation of Oscillations in Real Vanishing Viscosity Limit
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[11] M´etivier, G.: Exemples d’instabilit´es pour des e´ quations d’ondes non lin´eaires. Expos´e S´eminaire Bourbaki, Novembre 2002 [12] Pedlosky, J.: Geophysical fluid dynamics. Berlin-Heidelberg-New York: Springer (1979) [13] Gerard-Varet, D.: A geometric optics type approach to fluid boundary layers. Accepted for publication in Communications in Partial Differential Equations Communicated by P. Constantin
Commun. Math. Phys. 247, 697–712 (2004) Digital Object Identifier (DOI) 10.1007/s00220-004-1054-2
Communications in
Mathematical Physics
An Ergodic Theorem for the Quantum Relative Entropy Igor Bjelakovi´c, Rainer Siegmund-Schultze Technische Universität Berlin, Fakultät II - Mathematik und Naturwissenschaften, Institut für Mathematik MA 7-2, Strasse des 17. Juni 136, 10623 Berlin, Germany. E-mail: [email protected]; [email protected] Received: 8 July 2003 / Accepted: 3 November 2003 Published online: 5 March 2004 – © Springer-Verlag 2004
Abstract: We prove the ergodic version of the quantum Stein’s lemma which was conjectured by Hiai and Petz. The result provides an operational and statistical interpretation of the quantum relative entropy as a statistical measure of distinguishability, and contains as a special case the quantum version of the Shannon-McMillan theorem for ergodic states. A version of the quantum relative Asymptotic Equipartition Property (AEP) is given. 1. Introduction This paper can be seen as an extension of the article [1] by Bjelakovi´c, Krüger, Siegmund-Schultze and Szkoła, where instead of the von Neumann mean entropy of an ergodic quantum state ψ on a quasilocal algebra A∞ , dim A < ∞, the mean relative entropy s(ψ, ϕ) of ψ with respect to some stationary product state ϕ is the basic quantity. If we choose ϕ to be the tracial state, the results here reduce to the quantum Shannon-McMillan theorem of [1]. It turns out that the quantum mean relative entropy specifies the maximum exponential order, at which a typical subspace for the ergodic state ψ becomes untypical for the product state ϕ. Typical subspaces which asymptotically attain this exponential order might be called maximally separating subspaces. In particular, it turns out that projectors onto maximally separating subspaces can be chosen in a way that an extended version of the quantum AEP is fulfilled: Each one-dimensional projector dominated by the maximally separating projector has an expected value with respect to ψ which is of an exponential order given by the von Neumann mean entropy of ψ (in accordance with the quantum Shannon-McMillan theorem). The exponential order with respect to the reference state ϕ is larger exactly by the relative entropy. This is a complete analogue to the classical situation, which is included in the result since A may be chosen abelian. We point to the fact, that already in the classical situation the i.i.d. assumption concerning the reference state cannot be weakened very substantially, since there are examples
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with a reference process of very good mixing properties (B-process), but where a mean relative entropy does not even exist, see [8]. The methods used to derive the results are mainly based on the techniques developed in the paper Hiai/Petz [3] and in [1]. The Hiai/Petz paper considered the class of completely ergodic states ψ, and only an (exact) upper bound for the separation order was derived. Hiai and Petz were able to asymptotically reduce the problem to the abelian situation by constructing abelian sub-algebras with the remarkable property that they simultaneously allow a restriction of the two states ψ, ϕ with asymptotically vanishing distortion of their mean entropies and mean relative entropy. With some extension of the methods used already in [1] it was possible to drop the assumption of complete ergodicity as well as to show that the upper bound of the separation order is really a limit. This was conjectured already in Hiai/Petz [3] and proved for the case that both states are product (i.i.d.) states by Ogawa and Nagaoka in [4]. 2. Asymptotics of the Quantum Relative Entropy In this section we shall state our main result. We consider the lattice Zν . To each lattice site x ∈ Zν we assign a finite dimensional C ∗ -algebra Ax being ∗-isomorphic to a fixed ∗ -algebra A. Recall that each finite dimensional C ∗ -algebra A can finite dimensional C n be thought of as i=1 B(Hi ) up to a ∗-isomorphism where the Hi are finite dimensional Hilbert spaces and B(Hi ) is the algebra of linear operators on Hi . For a finite ⊂ Zν the algebra of local observables associated to is defined by A := Ax . x∈
Clearly, this definition implies that for ⊂ we have A = A ⊗ A \ . Hence there is a canonical embedding of A into A given by a → a ⊗ 1 \ for a ∈ A , where 1 \ denotes the identity ofA \ . The quasilocal C ∗ -algebra A∞ is the norm completion of the normed algebra ⊂Zν A , where the union is taken over all finite subsets . For a precise definition of the quasilocal algebra we refer to [2 and 7]. The group Zν acts in a natural way on the quasilocal algebra A∞ as translations by x ∈ Zν . A translation T (x) by x associates to an a ∈ A the corresponding element T (x)a ∈ A+x . This mapping T (x) extends canonically to a ∗-automorphism on A∞ (cf. [2, 7]). A state ψ on A∞ is a linear, positive and unital mapping of A∞ into C. Each state ψ on A∞ corresponds to a compatible family {ψ () }⊂Zν ,#<∞ , where each ψ () denotes the restriction of ψ to the local algebra A . The compatibility means that ψ ( ) A = ψ () for ⊂ . A state ψ is called stationary or translationally invariant if ψ = ψ ◦ T (x) for all x ∈ Zν . The convex set of stationary states is denoted by T (A∞ ) and is compact in the weak∗ -topology. The extremal points in T (A∞ ) are the ergodic states. The basic theory of ergodic states is described in [2 and 7]. In this article the local sets will mainly be boxes in Zν which are defined for n = (n1 , n2 , . . . , nν ) ∈ Nν by (n) := {0, . . . , n1 − 1} × . . . × {0, . . . , nν − 1}, and the cubic boxes of edge length n ∈ N will be denoted by (n). The local algebras associated with (n) will be denoted by A(n) . In a similar fashion we will denote the
An Ergodic Theorem for the Quantum Relative Entropy
699
restriction of a given state ψ on A∞ to the local algebra A(n) by ψ (n) . A state ϕ is called a stationary product state if ϕ (n) = (ϕ (1) )⊗(n) holds for each n ∈ N. The mean relative entropy of a stationary state ψ with respect to the stationary product state ϕ is defined by s(ψ, ϕ) :=
lim
(n) Nν
1 1 S(ψ (n) , ϕ (n) ) = sup S(ψ (n) , ϕ (n) ). #(n) #(n) (n)
(1)
The last equation is an easy consequence of the superadditivity of the relative entropy of the state ψ (n) with respect to the product state ϕ (n) . For the convenience of the reader we recall the definition of the relative entropy of the state σ on a finite dimensional C ∗ -algebra C with respect to the state τ : tr(Dσ (log Dσ − log Dτ )) if supp(σ ) ≤ supp(τ ) S(σ, τ ) := , ∞ otherwise where Dσ respectively Dτ denote the density operators of σ respectively τ and supp(σ ) is the support projector of the state σ . The important properties of the quantum relative entropy are summed up in the monograph [5] by Ohya and Petz. Later on we will need the notion of the mean relative entropy of the stationary state ψ with respect to the stationary product state ϕ on the lattice Gl := l · Zν , l ≥ 1 an integer, given by s(ψ, ϕ, Gl ) :=
lim
(n) Nν
1 S(ψ (ln) , ϕ (ln) ) = l ν s(ψ, ϕ). #(n)
(2)
The basic quantity in this article is given by βε,n (ψ, ϕ) := min{log ϕ (n) (q) : q ∈ A(n) projector and ψ (n) (q) ≥ 1 − ε},
(3)
where ε ∈ (0, 1) and ψ, ϕ ∈ T (A∞ ). The quantities appearing in (3) have a natural interpretation from the point of view of quantum statistical hypothesis testing. Assume that under the hypothesis H1 the state of the system is given by ψ (n) and under the hypothesis H2 the state is given by ϕ (n) . A projector q ∈ A(n) can be thought of as a decision rule: If a measurement of the projector q on the system under consideration has outcome 1, then the hypothesis H1 is accepted and the measurement outcome 0 implies that H2 is accepted. The quantities ψ (n) (1 − q) and ϕ (n) (q) respectively describe the probabilities for the occurrence of the eigenvalue 0 respectively 1 if the hypothesis H1 respectively H2 was accepted, i.e. the error probabilities. The following theorem had been conjectured by Hiai and Petz in [3]. Theorem 1. Let ψ be an ergodic state and let ϕ be a stationary product state on A∞ with mean relative entropy s(ψ, ϕ) < ∞. Then for all ε ∈ (0, 1) we have lim
(n) Nν
1 βε,n (ψ, ϕ) = −s(ψ, ϕ). #(n)
The fact that in the case where both states are stationary product states the limit of
1 #(n) βε,n (ψ, ϕ) exists and equals −s(ψ, ϕ), had been shown by Ogawa and Nagaoka in
[4]. Their proof requires some deeper properties of so called quasi-entropies which are quantities defined analogously to the relative entropy where an arbitrary operator convex function is used in their definition instead of the function − log x. We shall derive now the existence of the limit in the general case. As an instrument we will use some abelian approximation of all relevant quantities.
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Projectors q in (3) for which ϕ (n) (q) is of the order e−#(n)s(ψ,ϕ) and for which ψ(q) ≥ 1 − ε could be called maximally separating projectors and their range could be denoted as maximally separating subspace. The proof of this theorem reveals some additional information about maximally separating projectors, which is collected in the following Theorem 2 (quantum relative AEP). Let ψ be an ergodic state with mean entropy s(ψ) and let ϕ be a stationary product state on A∞ with mean relative entropy s(ψ, ϕ) < ∞. Then for all ε > 0 there is an Nε ∈ Nν such that for all n ∈ Nν with (n) ⊇ (Nε ) there exists an orthogonal projector pn (ε) ∈ A(n) such that 1. ψ (n) (pn (ε)) ≥ 1 − ε; 2. for all minimal projectors p ∈ A(n) with p ≤ pn (ε) we have e−#((n))(s(ψ)+ε) < ψ (n) (p) < e−#((n))(s(ψ)−ε) , and consequently e#((n))(s(ψ)−ε) < trn (pn (ε)) < e#((n))(s(ψ)+ε) . 3. for all minimal projectors p ∈ A(n) with p ≤ pn (ε) we have e−#((n))(s(ψ)+s(ψ,ϕ)+ε) < ϕ (n) (p) < e−#((n))(s(ψ)+s(ψ,ϕ)−ε) , and consequently e−#((n))(s(ψ,ϕ)+ε) < ϕ (n) (pn (ε)) < e−#((n))(s(ψ,ϕ)−ε) . In the case that the state ϕ is the tracial state on A∞ Theorem 2 above is equivalent to the quantum version of the Shannon-McMillan theorem proved in [1] (cf. Theorem 2.1 in [1]). The proof of Theorem 1 will make use of a classical law of large numbers for the (classical) mean relative entropy. Theorem 3. Let A be a finite setν and νP respectively Q be an ergodic respectively an ν i.i.d. probability measure on [AZ , AZ ], where AZ is the σ -algebra generated by the cylinder sets. We have P (n) (ωn ) 1 log (n) = DM (P , Q) Q (ωn ) (n) Nν #(n) lim
P − almost surely,
(4)
entropy of P with respect to Q and ωn ∈ where DM (P , Q) denotes the mean relative ν A(n) are the components of ω ∈ AZ corresponding to the box (n). The proof of this classical assertion is an elementary application of the ShannonMcMillan-Breiman theorem and the individual ergodic theorem. The higher dimensional versions of these theorems needed in the present situation can be found in the article [6] by Ornstein and Weiss. 3. Proof of the Main Theorems We start with the remark that due to the assumption s(ψ, ϕ) < ∞ without any loss of generality we may assume the state ϕ (1) to be faithful. The following theorem and the subsequent Lemma 1 provide the tools to drop the condition of complete ergodicity that was originally used by Hiai and Petz to prove the assertion of Lemma 2 below. Then
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we make use of the Hiai/Petz approximation Theorem 5 to construct two stochastic processes which approximate the two states in the sense of entropy and relative entropy. The classical result Theorem 3 immediately yields a sequence of projectors which are separating with a rate given by the mean relative entropy. We still have to prove that this rate cannot be beaten. This is done in Lemma 4. The idea can be summarized as follows: Assume the existence of a better (higher rate) separating subspace. It is easy to see that projecting this subspace into the constructed typical subspace would yield another separating subspace which would be “better” than the constructed one. Now we make use of the classical relative AEP valid for the constructed subspace to exclude this possibility. Theorem 4. Let ψ be an ergodic state on A∞ . Then for every subgroup Gl := l · Zν , with l > 1 an integer, there exists a k(l) ∈ Nν and a unique convex decomposition of ψ into Gl -ergodic states ψx : 1 ψ= ψx . (5) #(k(l)) x∈(k(l))
The Gl -ergodic decomposition (5) has the following properties: 1. kj (l) ≤ l and kj (l)|l for all j ∈ {1, . . . , ν}. 2. {ψx }x∈(k(l)) = {ψ0 ◦ T (−x)}x∈(k(l)) . 3. For every Gl -ergodic state ψx in the convex decomposition (5) of ψ the mean entropy with respect to Gl , s(ψx , Gl ), is equal to the mean entropy s(ψ, Gl ), i.e. s(ψx , Gl ) = s(ψ, Gl )
(6)
for all x ∈ (k(l)). 4. For each Gl -ergodic state ψx in the convex decomposition (5) of ψ and for every stationary product state ϕ the mean relative entropy with respect to Gl , s(ψx , ϕ, Gl ), fulfills s(ψx , ϕ, Gl ) = s(ψ, ϕ, Gl )
(7)
for all x ∈ (k(l)). Proof of Theorem 4. The first three items have been established in [1], Theorem 3.1. The proof of the last item is based on the monotonicity of the relative entropy and the ((ln)) usage of item 2 of the theorem. For each n ∈ Nν and x ∈ (k(l)) we have ψx = ((ln)−x) ˜ by the second item. We consider the box (n) containing (ln) and each ψ0 (ln) − x defined by ˜ (n) := {−l, . . . , ln1 − 1} × . . . × {−l, . . . , lnν − 1}, ˆ and the box (n) contained in (ln) and (ln) − x given by ˆ (n) := {0, . . . , l(n1 − 1) − 1} × . . . × {0, . . . , l(nν − 1) − 1}. The volumes of these boxes are asymptotically equivalent in the sense that the quotient tends to one. Hence using the observation above and twice the monotonicity of the relative entropy we obtain ˜ ((n))
S(ψ0
˜
((ln)−x)
, ϕ ((n)) ) ≥ S(ψ0
, ϕ ((ln)−x) )
= S(ψx((ln)) , ϕ ((ln)−x) ) ˆ
ˆ
≥ S(ψx((n)) , ϕ ((n)) ).
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After dividing by #(n) and taking the limit (n) Nν this inequality chain shows that s(ψ0 , ϕ, Gl ) ≥ s(ψx , ϕ, Gl ) ((ln)+x)
holds. A similar argument using ψx is also valid. Hence we established
((ln))
= ψ0
shows that the reverse inequality
s(ψ0 , ϕ, Gl ) = s(ψx , ϕ, Gl ). Since the mean relative entropy is affine in its first argument on the set of Gl -invariant states this implies s(ψx , ϕ, Gl ) = s(ψ, ϕ, Gl ) = l ν s(ψ, ϕ). An important ingredient in our proof of Theorem 1 is a result proved by Hiai and Petz in [3]. The starting point is the spectral decomposition of the density operator Dϕ (1) corresponding to the state ϕ (1) on A: Dϕ (1) =
d
λ i ei ,
i=1
where ei are one-dimensional projectors. Clearly, the spectral representation of the tensor product of Dϕ (1) over a box (y) can be written as Dϕ (y) =
d
λi1 · · · λiN (y) ei1 ⊗ . . . ⊗ eiN (y) ,
i1 ,... ,iN (y) =1
where we have chosen some enumeration {1, . . . , N(y)} of the points belonging to the box (y). We have N (y) = y1 · . . . · yν . Collecting all one dimensional projectors ei1 ⊗ . . . ⊗ eiN (y) which correspond to the same eigenvalue of Dϕ (y) we can rewrite the last expression as d n k λk pn1 ,...nd , (8) Dϕ (y) = n1 ,...nd :n1 +...+nd =N(y)
with pn1 ,...nd :=
k=1
ei1 ⊗ . . . ⊗ eiN (y) ,
(i1 ,... ,iN (y) )∈In1 ,...nd
where In1 ,...nd := {(i1 , . . . , iN(y) ) : #{j : ij = k} = nk for 1 ≤ k ≤ d}. We define the conditional expectation with respect to the trace by pn1 ,... ,nd A(y) pn1 ,... ,nd , Ey : A(y) → n1 ,... ,nd : n1 +...+nd =N(y)
Ey (a) :=
pn1 ,... ,nd a pn1 ,... ,nd .
(9)
n1 ,... ,nd : n1 +...+nd =N(y)
We are prepared to state the announced important result of Hiai and Petz, Lemma 3.1 and Lemma 3.2 in [3].
An Ergodic Theorem for the Quantum Relative Entropy
703
Theorem 5. If ψ is a stationary state on A∞ and Dy is the abelian subalgebra of A(y) generated by {pn1 ,...nd Dψ (y) pn1 ,...nd }n1 ,... ,nd ∪ {pn1 ,...nd }n1 ,... ,nd then S(ψ (y) , ϕ (y) ) = S(ψ (y) Dy , ϕ (y) Dy ) + S(ψ (y) ◦ Ey ) − S(ψ (y) ), and S(ψ (y) ◦ Ey ) − S(ψ (y) ) ≤ d log(#(y) + 1).
(10)
Consequently we have lim
(y) Nν
1 S(ψ (y) Dy , ϕ (y) Dy ) = s(ψ, ϕ). #(y)
(11)
Remark. This theorem had been proved by Hiai and Petz in [3] for the one-dimensional lattice. However, their proof extends canonically to the present situation. Any abelian algebra Dy in Theorem 5 can be represented as Dy =
ay
C · fy,i ,
(12)
i=1 a
y where {fy,i }i=1 is the set of orthogonal minimal projectors in Dy . For any y we introduce a maximally abelian refinement By of Dy by splitting each fy,i into a sum of orthogonal and minimal (in the sense of the algebra A(y) ) projectors gy,i,j which leads to the representation
By =
ay by,i
C · gy,i,j .
(13)
i=1 j =1
By the monotonicity of the relative entropy we get S(ψ (y) Dy , ϕ (y) Dy ) ≤ S(ψ (y) By , ϕ (y) By ) ≤ S(ψ (y) , ϕ (y) ),
(14)
from which we deduce lim
(y) Nν
1 S(ψ (y) By , ϕ (y) By ) = s(ψ, ϕ), #(y)
(15)
by (11). We choose a positive integer l and consider the decomposition
of ψ into states ψx 1 being ergodic with respect to the action of Gl , i.e. ψ = #(k(l)) x∈(k(l)) ψx in accordance with Theorem 4. Moreover, we consider a stationary product state ϕ. Note that this state is Gl -ergodic for each l ∈ Z. In order to keep our notation transparent we agree on the following abbreviations: s := s(ψ, ϕ, Zν ) = s(ψ, ϕ), i.e. the mean relative entropy of the state ψ computed with respect to Zν . We write Bl for B(l,l,... ,l) and set sx(l) :=
1 S(ψx((l)) Bl , ϕ ((l)) Bl ) #(l)
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and s (l) :=
1 S(ψ ((l)) Bl , ϕ ((l)) Bl ). #(l)
From Theorem 4 above we know that s(ψx , ϕ, Gl ) = s(ψ, ϕ, Gl ) = l ν · s(ψ, ϕ),
∀x ∈ (k(l)).
(16)
For an arbitrarily chosen η > 0 let us define the set Al,η := {x ∈ (k(l)) : sx(l) < s − η}.
(17)
By Acl,η we denote its complement. In the next lemma we shall show that the essential part of Gl -ergodic components of ψ have the entropy per site close to s as l becomes large, even if we restrict them to the abelian algebras Bl . Lemma 1. If ψ is a Zν -ergodic state and ϕ a stationary product state on A∞ and if s(ψ, ϕ) < ∞, then #Al,η =0 #(k(l))
lim
l→∞
holds for every η > 0. Proof of Lemma 1. We suppose, on the contrary, that there exists some η0 > 0 such that #Al,η0 lim supl #(k(l)) = a > 0. Then we can select a subsequence, which we denote again by (l) for simplicity, with the property lim
l→∞
#Al,η0 = a. #(k(l))
Using joint convexity of the relative entropy we have the following estimate: #(k(l)) · s (l) ≤ sx(l) x∈(k(l))
=
sx(l) +
x∈Al,η0
sx(l)
x∈Acl,η 0
(17)
< #Al,η0 · (s − η0 ) + #Acl,η0 · max sx(l) . c x∈Al,η
0
Employing that for the mean entropy sx(l) ≤
1 S(ψx((l)) , ϕ ((l)) ) #(l)
(by the monotonicity)
and 1 S(ψx(lm) , ϕ (lm) ) #(m) 1 = sup S(ψx(lm) , ϕ (lm) ) #(m) (m)
s(ψx , ϕ, Gl ) =
lim
(m) Nν
(18)
An Ergodic Theorem for the Quantum Relative Entropy
705
are fulfilled, we obtain an upper bound for the last term in (18): #Acl,η0 · max sx(l) ≤ #Acl,η0 · max c c x∈Al,η
x∈Al,η
0
=
#Acl,η0
0
1 s(ψx , ϕ, Gl ) lν
· s(ψ, ϕ)
(by (16)).
Inserting this in (18) and dividing by #(k(l)) we obtain s (l) <
#Acl,η0 #Al,η0 (s − η0 ) + s. #(k(l)) #(k(l))
And after taking limits we arrive at the following contradictory inequality: s ≤ a(s − η0 ) + (1 − a)s = s − aη0 < s, since liml→∞ s (l) = s by Theorem 5 and s < ∞. Hence a = 0.
Lemma 2. Let ψ be an ergodic state on A∞ and let ϕ be a stationary product state on A∞ fulfilling s(ψ, ϕ) < ∞. Then for every ε ∈ (0, 1), lim sup
(n) Nν
1 βε,n (ψ, ϕ) ≤ −s(ψ, ϕ). #(n)
Proof of Lemma 2. We fix ε > 0 and choose arbitrary η, δ > 0. Consider the Gl -ergodic decomposition ψ=
1 #(k(l))
ψx
x∈(k(l))
of ψ for integers l ≥ 1. By Lemma 1 there is an integer L ≥ 1 such that for any l ≥ L, ε 1 ≥ #Al,η ≥ 0 2 #(k(l)) holds, where Al,η is defined by (17). This inequality implies 1 ε #Acl,η · (1 − ) ≥ 1 − ε. #(k(l)) 2
(19)
Recall that by definition of Al,η we have 1 S(ψx((l)) Bl , ϕ ((l)) Bl ) ≥ s(ψ, ϕ) − η #(l)
for all x ∈ Acl,η .
(20)
We fix an l ≥ L and consider the abelian quasi-local C ∗ -algebra Bl∞ built up from Bl . Bl∞ is clearly a C ∗ -subalgebra of A∞ . We set mx := ψx Bl∞ and m(n) x := ψx Bl . (n)
Moreover we define p := ϕ Bl∞ and p (n) := ϕ Bl . (n)
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The state p is a Gl -stationary product state. On the other hand, Theorem 4.3.17. in [2] shows that the states mx are Gl -ergodic. Due to the Gelfand isomorphism and the Riesz representation theorem we can (and shall) identify all the states above with the probability measures on the corresponding maximal ideal space of Bl∞ . Since the algebra Bl is abelian and finite dimensional this compact maximal ideal space can be thought ν of as BlZ for an appropriately chosen finite set Bl . This is essentially the well-known Kolmogorov representation of a classical dynamical system. By the definition of the measures mx and p, monotonicity of the relative entropy and the fourth item in Theorem 4 we have DM (mx , p) ≤ s(ψx , ϕ, Gl ) = l ν s(ψ, ϕ) < ∞,
(21)
where DM (mx , p) denotes the mean relative entropy of mx with respect to p. Using Theorem 3 we see that (n)
1 mx (ωn ) log (n) = DM (mx , p) =: DM,x , p (ωn ) (n) Nν (n) lim
(22)
mx -almost surely for all x ∈ (k(l)), where ωn ∈ Bl are the components of ω ∈ BlZ c corresponding to the box (n). For each n and x ∈ Al,η let (n) mx (ωn ) 1 (n) (n) Cx := ωn ∈ Bl : | log (n) − DM,x | < δ #(n) p (ωn ) (n) mx (ωn ) (n) #(n)·(DM,x −δ) #(n)·(DM,x +δ) = ωn ∈ Bl | e . < (n) <e p (ωn )
ν
(n)
(n)
(n)
(n)
The indicator function of Cx corresponds to the projector qx ∈ Bl . The definition of (n) the set Cx implies a bound on the probability of this set with respect to the probability measure p: (n) −#(n)·(DM,x −δ) p (n) (Cx(n) ) = ϕ(qx(n) ) ≤ m(n) x (Cx )e
≤ e−#(n)·(DM,x −δ) ≤ e−#(n)·(#(l)(s(ψ,ϕ)−η)−δ) ,
(23)
because for all x ∈ Acl,η we have by (1)
(1) ((l)) Bl , ϕ ((l)) Bl ) DM,x ≥ D(m(1) x , p ) = S(ψx ≥ #(l)(s(ψ, ϕ) − η),
(24)
by (20). The limit assertion (22) implies the existence of an N ∈ N such that for any n ∈ Nν with (n) ⊃ (N ), ε (n) m(n) ∀ x ∈ Acl,η . (25) x (Cx ) ≥ 1 − , 2 For each y ∈ Nν with yi ≥ N l let yi = ni l + ji , where ni ≥ N and 0 ≤ ji < l. We set
qx(n) qln := x∈Acl,η
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and denote by qy the embedding of qln in A(y) . By (25) and (19) we obtain 1 (y) ψx (qy )
(y) (qy ) = #(k(l)) x∈(k(l))
1 ≥ #(k(l)) ≥
(y)
ψx (qx(n) )
x∈(k(l))
1 ε #Ac · (1 − ) ≥ 1 − ε. #(k(l)) l,η 2
Thus the condition in the definition of βε,y (ψ, ϕ) is satisfied. We are now able to estimate βε,y (ψ, ϕ) using (23): βε,y (ψ, ϕ) ≤ log ϕ (y) (qy ) e−#(n)·(DM,x −δ) ≤ log x∈Acl,η
≤ log(#Acl,η · e−#(n)·(#(l)(s(ψ,ϕ)−η)−δ) ) = log(#Acl,η ) − #(ln)(s(ψ, ϕ) − η −
δ ). #(ł)
This leads to 1 βε,y (ψ, ϕ) ≤ −s(ψ, ϕ) + η + δ, (y) Nν #(y) lim sup
since #Acl,η does not depend on n and (y) Nν if and only if (n) Nν . Since η, δ > 0 were chosen arbitrarily we have lim sup
(n) Nν
1 βε,n (ψ, ϕ) ≤ −s(ψ, ϕ). #(n)
Suppose we are given a sequence (pn ) of projectors in A(n) and a stationary state ψ on A∞ with mean entropy s(ψ). We consider the positive operators pn Dψ (n) pn =
d(n)
λn,i rn,i ,
d(n) := tr(pn ),
(26)
i=1
where the numbers λn,i are the eigenvalues and the rn,i form a complete set of eigenprojectors of pn Dψ (n) pn . We set Tn,δ := {i ∈ {1, . . . , d(n)} : λn,i ≤ e−#(n)(s(ψ)−δ) }
for δ > 0,
(27)
c the complement of this set. and denote by Tn,δ
Lemma 3. Let ψ be an ergodic state on A∞ with mean entropy s(ψ) and let (pn ) be c is the projector corresponding to a sequence of projectors in A(n) , respectively. If pTn,δ c then the set Tn,δ lim
(n) Nν
for all δ > 0.
c ) = 0 ψ (n) (pTn,δ
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Proof of Lemma 3. We have c c ), λn,i > e−#(n)(s(ψ)−δ) #Tn,δ = e−#(n)(s(ψ)−δ) tr(pTn,δ 1≥ c i∈Tn,δ
and, consequently 1 c )) < s(ψ) − δ. log(tr(pTn,δ #(n) c ) ≥ a > 0 for infinitely many n and some a > 0 there If we would have ψ ((n)) (pTn,δ 0 would be a contradiction to Proposition 2.1 in [1], which implies that there is no sequence 1 (qn ) of projectors in A(n) with ψ (n) (qn ) ≥ a > 0 and lim sup(n) Nν #(n) log(tr(qn )) < s(ψ).
Lemma 4. Let ψ be an ergodic state on A∞ and let ϕ be a stationary product state on A∞ . Suppose that s(ψ, ϕ) < ∞ holds, then for every ε ∈ (0, 1), lim inf
(n) Nν
1 βε,n (ψ, ϕ) ≥ −s(ψ, ϕ). #(n)
Proof of Lemma 4. Let (ty ) be a sequence of projectors, ty ∈ A(y) , with (y) Nν and lim inf
(y) Nν
1 log ϕ (y) (ty ) < −s(ψ, ϕ). #(y)
Then there exists an a > 0 and a subsequence, which we denote by (ty ) for notational simplicity, fulfilling ϕ (y) (ty ) < e−#(y)(s(ψ,ϕ)+a) .
(28)
We consider an integer l ≥ 1 and the Gl −ergodic decomposition of ψ: ψ=
1 #(k(l))
ψx .
x∈(k(l))
As in the proof of Lemma 2 for each ε, η > 0 we can choose l in such a way that we have #Acl,η #(k(l))
ε (1 − ) ≥ 1 − ε, 2
(29)
where the set Al,η was defined by (17). Recall that we have by definition, 1 S(ψx(l) Bl , ϕ (l) Bl ) ≥ s(ψ, ϕ) − η #(l)
for all x ∈ Acl,η .
We consider again the abelian quasi-local algebra Bl∞ , which will be identified with the ν algebra of continuous functions on the maximal ideal space Bl∞ := BlZ , bearing in mind that the restrictions of the Gl −ergodic components of ψ and ϕ to this algebra are
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Gl −ergodic. We denote those restrictions by mx , x ∈ (k(l)), and p. As in the proof of Lemma 2 we can show that for δ > 0 the sets (n)
(n)
Cx,δ := {ωn ∈ Bl
: e#(n)(DM,x − 2 ) < δ
(n)
δ mx (ωn ) < e#(n)(DM,x + 2 ) } (n) p (ωn )
fulfill (n)
lim
(n) Nν
m(n) x (Cx,δ ) = 1
for all x ∈ (k(l)).
(30)
In a similar way, by employing the classical Shannon-McMillan theorem, we can see that for (n)
(n)
Fx,δ := {ωn ∈ Bl
−#(n)(hx − 2 ) : e−#(n)(hx + 2 ) < m(n) }, x (ωn ) < e δ
δ
we have lim
(n)
(n) Nν
m(n) x (Fx,δ ) = 1
for all x ∈ (k(l)),
(31) (n)
where hx denotes the Shannon entropy rate of mx . Each ωn ∈ Bl corresponds to a (n) (n) (n) minimal projector qn ∈ Bl ⊂ A(ln) . So, for all x ∈ Acl,η and for any ωn ∈ Cx,δ ∩ Fx,δ and corresponding qn we have ϕ (ln) (qn ) = p (n) (ωn ) > e−#(n)(DM,x +hx +δ) ν ≥ e−#(n)(s(ψ,ϕ)l +hx +δ) .
(32) (n)
(n)
In fact, the first inequality follows from the definitions of the sets Cx,δ and Fx,δ and (n)
(n)
the assumption that ωn ∈ Cx,δ ∩ Fx,δ . The second inequality is a consequence of (21). Moreover, by (24) we get the upper estimate ϕ (ln) (qn ) = p(n) (ωn ) < e−#(n)(DM,x +hx −δ) ν ≤ e−#(n)((s(ψ,ϕ)−η)l +hx −δ) .
(33)
Next, observe that the representation (13) of Bl implies that Dψ (l)B = ql,i Dψ (l) ql,i , x
l
x
where (ql,i ) is a complete set of minimal eigen-projectors of Dϕ (l) . Observe that from the fact that the maximal abelian algebra Bl is generated by eigen-projectors of Dϕ (l) it follows that Dϕ (l)Bl = Dϕ (l) . We get ql,i Dψ (l) ql,i log Dϕ (l) ) tr(Dψ (l)B log Dϕ (l)Bl ) = tr( l x x ql,i (log Dϕ (l) )ql,i ) = tr(Dψ (l) x
= tr(Dψ (l) log Dϕ (l) ). x
(34)
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Finally, by our assumption that s(ψ, ϕ) < ∞ holds we know that s(ψ, ϕ) = −s(ψ) − tr(Dψ (1) log Dϕ (1) ). Using the product structure of the state ϕ and the fact just derived we obtain (n)
(n)
S(ψx(ln) Bl , ϕ (ln) Bl )
=
−H (m(n) x ) − tr(Dψ (ln)Bn log Dϕ (ln) )
=
−H (m(n) x ) − #(n)tr(Dψ (l)B log Dϕ (l) )
(by (34))
=
x
l
x
−H (m(n) x ) − #(n)(tr(Dψx(l)
l
log Dϕ (l) ).
(35)
On the other hand we have for all x ∈ Acl,η , (n)
(n)
S(ψx(ln) Bl , ϕ (ln) Bl ) ≥ #(n)(−H (m(1) x ) − tr(Dψ (l)B log Dϕ (l) )) l
x
(by the subadditivity of the entropy) = #(n)S(ψx(l) Bl , ϕ (l) Bl ) ≥ #(n)l ν (s(ψ, ϕ) − η) (by the choice of algebra Bl ) = #(n)l ν (−s(ψ) − tr(Dψ (1) log Dϕ (1) ) − η).
(36)
The equation chain (35) and the inequality chain (36) imply that −H (m(n) x ) − #(n)(tr(Dψ (l) log Dϕ (l) )) x
≥ #(n)l ν (−s(ψ) − tr(Dψ (1) log Dϕ (1) ) − η).
(37)
Note that the third and the fourth item of Theorem 4 show that tr(Dψ (l) log Dϕ (l) ) = l ν tr(Dψ (1) log Dϕ (1) ), x
in the case where s(ψ, ϕ) < ∞. Dividing (37) by −#(n) and taking the limit (n)
Nν leads to l ν s(ψ) ≤ hx ≤ l ν (s(ψ) + η),
(38)
where the lower bound simply follows from the fact that the entropy on a maximally abelian subalgebra is not less than the entropy on the algebra. This inequality, (32) and (33) imply: e−#(ln)(s(ψ,ϕ)+s(ψ)+η+δ/ l Let pln
≤ ϕ (ln) (qn ) ≤ e−#(ln)(s(ψ,ϕ)+s(ψ)−η−δ/ l ) . (39) (n) (n) denote the projector corresponding to the set x∈Ac Cx,δ ∩ Fx,δ and pn,x ν)
ν
l,η
be the projector which corresponds to
(n) Cx,δ
ψx (pn,x ) ≥ 1 −
ε 2
(n) ∩ Fx,δ .
For sufficiently large Nν we have
by (30) and (31),
and hence by (29), ψ(pln ) ≥
1 ψx (pn,x ) #(k(l)) c x∈Al,η
≥
1 ε #Ac (1 − ) ≥ 1 − ε. #(k(l)) l,η 2
(40)
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Any ty fulfilling (28) can be embedded in an appropriately chosen A(ln) . Indeed, choose the unique Nν such that (ni − 1)l ≤ yi < ni l for all i ∈ {1, . . . , ν} and set eln := ty ⊗ 1(ln)\(y) . Then we have ψ (ln) (eln ) = ψ (y) (ty ),
(41)
and ψ (ln) (eln ) = ψ (ln) (eln pln ) + ψ (ln) (eln (1 − pln )) ≤ ψ (ln) (eln pln ) + ε
(by (40)).
Applying this argument once more we obtain ψ (ln) (eln ) ≤ ψ (ln) (pln eln pln ) + 2ε = tr(pln Dψ (ln) pln eln ) + 2ε.
(42)
In the final step we will prove that the first term in the last line above can be made arbitrarily small. Using the notation from (26), (27) and applying Lemma 3 we know that tr(rln,i eln ) + ε tr(pln Dψ (ln) pln eln ) ≤ e−#(ln)(s(ψ)−δ) i∈Tln,δ
≤ e−#(ln)(s(ψ)−δ)
tr(p ln )
tr(rln,i eln ) + ε
i=1
= e−#(ln)(s(ψ)−δ) tr(pln eln ) + ε,
(43)
for sufficiently large Nν . We represent the projector pln as a sum of unique minimal (n) projections in Bl : pln =
tr(p ln )
qn,i .
i=1
Hence by (39) we have tr(pln eln ) ≤ e =e
#(ln)(s(ψ,ϕ)+s(ψ)+η+δ/ l ν )
tr(p ln )
i=1 #(ln)(s(ψ,ϕ)+s(ψ)+η+δ/ l ν ) (ln)
ϕ
ϕ (ln) (qn,i )tr(qn,i eln )
(pln eln )
ν
≤ e#(ln)(s(ψ,ϕ)+s(ψ)+η+δ/ l ) ϕ (y) (ty ) ≤ e#(ln)(s(ψ,ϕ)+s(ψ)+η+δ/ l ) e−#(y)(s(ψ,ϕ)+a) (by (28)), ν
(44)
since the minimal projectors qn,i correspond to the eigen-vectors of Dϕ (ln) . Inserting this into (43) we obtain #(y)
#(y)
tr(pln Dψ (ln) pln eln ) ≤ e−#(ln)(( #(ln) −1)s(ψ,ϕ)+ #(ln) a−δ−η− l ν ) + ε. δ
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#(y) Note that lim(y) Nν #(ln) = 1, and that a > 0. Hence if we choose δ, η small enough and n large enough we can achieve that the exponent in the last inequality becomes negative eventually. This inequality together with (42) and (41) shows that
lim
(y) Nν
ψ(ty ) = 0.
The combination of Lemma 2 and Lemma 4 yields the assertion of Theorem 1. To derive Theorem 2, consider the projectors pln constructed in the proof of Lemma 4. They can be written as the sum of minimal projectors qn fulfilling (39). From the fact, that these are minimal eigen-projectors for Dϕ (n) we derive that (39) is valid even if we replace qn by any minimal projector which is dominated by pln . So we see that, if we would choose the projectors pn (ε) to be specified for Theorem 2 just as pln for boxes with edge lengths being multiples of l, item 3 would be satisfied for large n, supposed l is (fixed but) large enough. Item 1 is fulfilled for large n in view of (40). Observe now that embedding the projectors pln into A(y) for 0 ≤ y − ln < (l, l, . . . , l) leads us to a family of projectors (pn ) still fulfilling Item 1. Item 3 is satisfied by this family, too, since ϕ (1) was supposed faithful. In order to ensure that Item 2 is fulfilled, the constructed family of projectors (pn ) has to be modified. Represent pn as a sum of eigen-projectors (n) of the operator pn Dψ (n) pn . Now from the definition of the sets Fx,δ we easily conclude #(n)(s(ψ)+ε) that tr(pn ) < e can be guaranteed for large n. So the asymptotic contribution to ψ (n) (pn ) of eigen-values of pn Dψ (n) pn of magnitude exponentially smaller than e−#(n)(s(ψ)+ε) can be neglected. The asymptotic contribution to ψ (n) (pn ) of eigen-values of pn Dψ (n) pn of magnitude exponentially larger than e−#(n)(s(ψ)−ε) can be neglected, too, because of Lemma 3. So we may omit the eigen-projectors corresponding to either too large or too small eigen-values from the sum, getting a modified family (pn ), which additionally fulfills Item 2. This proves Theorem 2. Acknowledgements. We are very thankful to our colleagues Tyll Krüger, Ruedi Seiler and Arleta Szkoła for their constant interest and encouragement during the preparation of this article and for many very helpful comments and improvements. This work was supported by the DFG via SFB 288 ‘Differentialgeometrie und Quantenphysik’ at TU Berlin.
References 1. Bjelakovi´c, I., Krüger, T., Siegmund-Schultze, Ra., Szkoła, A.: The Shannon-McMillan Theorem for Ergodic Quantum Lattice Systems. arXiv.org, math.DS/0207121. Invent. Math. 155, 203–222 (2004) 2. Bratteli, O., Robinson, D.W.: Operator Algebras and Quantum Statistical Mechanics I. New York: Springer, 1979 3. Hiai, F., Petz, D.: The Proper Formula for Relative Entropy and its Asymptotics in Quantum Probability. Commun. Math. Phys. 143, 99–114 (1991) 4. Ogawa, T., Nagaoka, H.: Strong Converse and Stein’s Lemma in Quantum Hypothesis Testing. IEEE Trans. Inform. Theo. 46(7), 2428–2433 (2000) 5. Ohya, M., Petz, D.: Quantum Entropy and its Use. Berlin: Springer, 1993 6. Ornstein, D., Weiss, B.: The Shannon-McMillan-Breiman Theorem for a Class of Amenable Groups. Israel J. Math. 44(1), 53–60 (1983) 7. Ruelle, D.: Statistical Mechanics. New York: W.A. Benjamin, 1969 8. Shields, P.C.: Two divergence-rate counterexamples. J. Theor. Prob. 6, 521–545 (1993) Communicated by M.B. Ruskai
Commun. Math. Phys. 247, 713–742 (2004) Digital Object Identifier (DOI) 10.1007/s00220-004-1058-y
Communications in
Mathematical Physics
Nonsemisimple Fusion Algebras and the Verlinde Formula J. Fuchs1 , S. Hwang1 , A.M. Semikhatov2 , I.Yu. Tipunin2 1 2
Karlstad University, Karlstad, Sweden Lebedev Physics Institute, Moscow, Russia
Received: 14 July 2003 / Accepted: 15 October 2003 Published online: 2 March 2004 – © Springer-Verlag 2004
Abstract: We find a nonsemisimple fusion algebra Fp associated with each (1, p) Virasoro model. We present a nonsemisimple generalization of the Verlinde formula which allows us to derive Fp from modular transformations of characters. 1. Introduction Fusion algebras [1–5] describe basis-independent aspects of operator products and thus provide essential information about conformal field theory. They can in principle be found by calculating coinvariants, but the most practical derivation, which at the same time is of fundamental importance, is from the modular transformation properties of characters, via the Verlinde formula [1]. The relation between fusion and modular properties can be considered a basic principle underlying consistency of CFT. A fusion algebra F is a unital commutative associative algebra over C with a distinguished basis (the one corresponding to the “sectors,” or primary fields, of the model) in which the structure constants are nonnegative integers (we refer to this basis as the canonical basis of F in what follows). For rational CFTs, which possess semisimple fusion algebras, the Verlinde formula is often formulated as the motto that “the matrix S diagonalizes the fusion rules.” This involves two statements at least. The first is merely a lemma of linear algebra and can be stated as follows: there exists a matrix P that relates the canonical basis in the fusion algebra to the basis of primitive idempotents. This property is not specific to fusion algebras originating from conformal field theories, and in fact applies to any association scheme [6]; we borrow the terminology from [6] and call P the eigenmatrix. The second, nontrivial, statement contained in the Verlinde formula is that the eigenmatrix P is related to the matrix S that represents the modular group element S = 01 −1 on 0 the characters of the chiral algebra; this relation is given by P = S Kdiag , where Kdiag (the denominator in the Verlinde formula) is a diagonal matrix whose entries are the inverse of the distinguished row of the S matrix. With P expressed this way, we arrive
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at the statement that the matrices of the regular representation of the fusion algebra are diagonalized by the S matrix. This cannot apply to nonsemisimple fusion algebras, however, for which the regular representation matrices cannot be diagonalized. The relation between modular transformations and the structure of nonsemisimple fusion algebras is therefore more difficult to identify, which considerably complicates attempts to build a nonsemisimple Verlinde formula. Nonsemisimple fusion algebras are expected to arise in logarithmic models of conformal field theory [7–15], where irreducible representations of the chiral algebra allow nontrivial (indecomposable) extensions. In what follows, we generalize the Verlinde formula and derive nonsemisimple fusion algebras for the series of (1, p) Virasoro models with integer p ≥ 2. The (1, p) models provide an excellent illustration of complications involved in generalizing the Verlinde formula to the nonsemisimple case. Unlike the (p , p) minimal models with coprime p , p ≥ 2, the (1, p) model is defined not as the cohomology, but as the kernel of a screening, and the first question that must be answered in constructing its fusion, as well as the fusion beyond minimal models in general, is: Q1: How to organize the Virasoro representations into a finite number of families? That is, which chiral algebra, extending the Virasoro algebra, is to be used to classify representations? Assuming that such an algebra has been chosen, the fusion algebra can in principle be derived using different means, e.g., by directly finding coinvariants (if, against expectations, this is feasible). Another possibility is via a Verlinde-like formula, starting with characters of representations of the chosen chiral algebra. Compared to the semisimple case, the basic problems with constructing an analogue of the Verlinde formula are then as follows. Q2: The matrices implementing modular transformations of the characters of chiral algebra representations involve the modular parameter τ and do not therefore generate a finite-dimensional representation of SL(2, Z). How to extract a τ -independent matrix S representing S ∈ SL(2, Z) on characters? Q3: With fusion matrices that are not simultaneously diagonalizable, it is not a priori known which “special” (instead of diagonal) matrix form is to be used in a Verlinde-like formula. In other words, how to define the eigenmatrix P that performs the transformation to a “special” basis in a nonsemisimple fusion algebra? Q4: Assuming the matrix S is known and the matrix P that performs the transformation to the chosen “special” basis has been selected, how are S and P related? The most nontrivial part of the nonsemisimple generalization of the Verlinde formula is the answer to Q4. We also note that with a chiral algebra chosen in Q1, we face yet another problem of a “nonsemisimple” nature, originating in the structure of the category of representations of the chosen chiral algebra: Q5: With indecomposable representations of the chiral algebra involved, how many generators should the fusion algebra have? More specifically, whenever there is a nonsplittable exact sequence 0 → V0 → R → V1 → 0, should there be fusion algebra generators corresponding to one (i.e., R), two (i.e., V0 and V1 ), or three representations? (This becomes critical, e.g., when V0 corresponds to the unit element of the fusion algebra, cf. [9]).
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We also note the following complications that are already apparent in nonunitary semisimple fusion (see the relevant remarks in [16]), but become more acute for nonsemisimple fusion: R1: Whenever the S matrix is not symmetric, the sought generalization of the Verlinde formula is sensitive to the choice between S and St . This is essential, in particular, in selecting the distinguished row/column of S whose elements determine eigenvalues of the fusion matrices (the denominator of the Verlinde formula).1 R2: The sector with the minimal conformal weight is different from the vacuum sector. It is therefore necessary to decide which of these two distinguished sectors is to play the “reference” role in the Verlinde formula. (That is, as a continuation of the previous remark, the distinguished row of the S matrix to be used in the denominator of the Verlinde formula must be identified properly). In answering Q5, one must be aware that fusion algebras only provide a “coarsegrained” description of conformal field theory, and there can be several degrees of neglecting the details. The concept of nonsemisimple fusion advocated in [8, 9] aims at accounting for the “fine” structure given by the different ways in which simple (irreducible) modules can be arranged into indecomposable representations. (Such a detailed fusion will be needed, e.g., in studying boundary conditions in conformal field theory models and for a proper interpretation of modular invariants.) In that setting, a natural basis in the fusion algebra would be given by all indecomposable representations (the irreducible ones included).2 A coarser description is to think of the fusion algebra as the Grothendieck ring of the representation category of the chiral algebra, i.e., as the K0 functor, not distinguishing between different compositions of the same subquotients. This fusion is sufficient for the construction of the generalized Verlinde formula. Indeed, the appropriately generalized Verlinde formula should relate the matrix S that represents S ∈ SL(2, Z) on a collection of characters of the chiral algebra to the fusion algebra structure constants. But the character of an indecomposable representation R is the sum of the characters of its simple subquotients, independently of how the algebra action “glues” them into R. Therefore, for the fusion functor defined for the purpose of constructing the generalized Verlinde formula, an indecomposable representation R as in Q5 is indistinguishable from the direct sum of V0 and V1 (as well as from R in 0 → V1 → R → V0 → 0). In other words, the element of the fusion algebra corresponding to R is the sum of those corresponding to V0 and V1 . In this paper, we only deal with this particular concept of fusion that corresponds to the K0 functor. Thus, the number of elements in a basis of the fusion algebra associated with a collection {Vj , Ri } of chiral algebra representations must be given by the number of all simple subquotients of all the indecomposable representations Ri and all simple Vj (with each irreducible representation occurring just once). But the fact that no linearly independent elements of the fusion algebra correspond to indecomposable representations does not mean that “nonsemisimple effects” are neglected: the existence of a nontrivial extension of any two representations V0 and V1 already makes the fusion algebra nonsemisimple, giving rise to all of the problems listed above. 1 In addition, it becomes essential whether a representation or an antirepresentation of SL(2, Z) is considered as the modular group action (in most of the known semisimple examples, this point can safely be ignored). 2 The p = 2 fusion in [8, 9] is “intermediate” in that not all of the indecomposable representations are taken into account. But it is certainly sufficient for extracting the coarser, “K0 ”-fusion that follows from Theorem 5.3.1 below for p = 2.
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The answer to Q1 can be extracted from the literature [17, 9]: we take the maximal local subalgebra in the (nonlocal) chiral algebra that is naturally present in the (1, p) model. This W algebra, denoted by W(p) for brevity, has 2p irreducible representations in the (1, p) model. As regards Q2, the answer amounts to the use of matrix automorphy “factors,” as explained below (cf. [18]). The answer to Q3 is related to the structure of associative algebras [19] and, once a canonical basis is fixed, to nonsemisimple generalizations of some notions from the theory of association schemes [6]. Any finitely generated associative algebra F (with a unit) is the vector-space sum of a distinguished ideal R, called the radical (the intersection of all maximal left ideals, or equivalently, of all maximal right ideals), and a semisimple algebra (necessarily isomorphic to a direct sum of matrix algebras over division algebras over the base field) [19]. This implies that in any commutative associative algebra, there is a basis (eA , wα ),
A = 1, . . . , n ,
α = 1, . . . , n
(with n +n = n = dim F), composed of primitive idempotents eA and elements wα in the radical. In the semisimple case, the radical is zero, and “diagonalization of the fusion” can equivalently be stated as the transformation to the basis (λ1 e1 , . . . , λn en ) of “rescaled idempotents,” where λa are scalars read off from the distinguished row of the S matrix (the row corresponding to the vacuum representation). Let XI , I = 1, . . . , n, denote the elements of the canonical basis in the fusion algebra. Even for semisimple algebras, it is useful to distinguish between the S matrix that transforms the canonical basis X• to the basis (λ1 e1 , . . . , λn en ) and the matrix P that transforms the canonical basis to the basis of primitive idempotents, even though S and P are related by multiplication with a diagonal matrix. In the nonsemisimple case, the eigenmatrix P that maps the canonical basis to the basis consisting of primitive idempotents and elements in the radical, X1 e ... = P A , wα X n
is related to the S matrix in a more complicated way. The resolution of Q4, which is the heart of the nonsemisimple Verlinde formula, is the construction, from the entries of S, of a (nondiagonal) interpolating matrix K (which plays the role of the denominator in the Verlinde formula) such that P = S K. The points raised in R2 and R1 can be clarified as follows. The rows and columns of S are labeled by chiral algebra representations in the model under consideration. The S matrix has a distinguished row that corresponds to the vacuum representation and a distinguished column that corresponds to the minimum-dimension representation of the chiral algebra (the entries in this column are related to the asymptotic form of the characters labeled by the respective rows of S). The columns of the P matrix are labeled by the elements (eA , wα ) of the basis consisting of primitive idempotents and elements in the radical, and its rows correspond to elements of the canonical basis in the fusion algebra; the distinguished row of P then corresponds to the unit element of the algebra. (The choice of rows vs. columns in P, S, and other matrices is of course conventional, but the replacement rows ↔ columns must be made consistently with other transpositions and change of the order in matrix multiplication.)
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We now summarize our strategy to construct the (1, p) fusion via a nonsemisimple generalization of the Verlinde formula and also describe the contents of the paper: 1. In the (1, p) model, we identify the maximal local algebra W(p) as the chiral algebra of the model. There then exist only 2p irreducible W(p) representations in the model, which solves Q1. (The algebra is introduced in Sect. 2.2, and its category of representations is described in Sect. 2.3 and 2.4.) 2. We then evaluate the 2p characters χ of these representations and find (τ -dependent) 2p × 2p matrices J (γ , τ ) such that χ (γ τ ) = J (γ , τ ) χ (τ ) for γ ∈ SL(2, Z). (The characters are evaluated in Sect. 3.1 and their modular transformation properties are derived in Sect. 3.2.) 3. Next, we find a 2p × 2p automorphy “factor” j (γ , τ ), satisfying the cocycle condition, such that γ → ρ(γ ) = j (γ , τ )J (γ , τ ) is a finite-dimensional representation of SL(2, Z). This solves Q2 (Sect. 4.1 and 4.2) and gives the S matrix (Sect. 4.3). 4. From the entries of the distinguished row of S = ρ(S), we build the interpolating matrix K and use it to construct the eigenmatrix P of the fusion algebra as P = S K. This solves Q4 (Sect. 5.2). 5. From the eigenmatrix P, we uniquely reconstruct the fusion algebra Fp in the canonical basis whose elements are labeled by the rows of P, via a recipe that involves answering Q3 (Sect. 5.3). For the impatient, we here present the answer for the structure constants expressed through the entries of the S matrix: arranged into matrices NI in the standard way, the structure constants of the fusion algebra are given by NI = SOI S−1 , where S = ρ(S) acts in a finite-dimensional (in (1, p) models, 2p-dimensional) representation of SL(2, Z) and OI = OI 0 ⊕ OI 1 ⊕ . . . ⊕ OI,p−1 are block-diagonal matrices with the 2 × 2 blocks S 1 S 2 given by OI 0 = diag SI 1 , SI 2 and
OIj =
1 2j +1 S
2j +2
− S
2j +1 2j +1 2j +2 2j +1 2j +2 2j +2 2j +2 2j +1 2j +1 2j +2 S SI − 2S SI + S SI −S SI + S SI , × 2j +2 2j +1 2j +1 2j +2 2j +1 2j +1 2j +1 2j +2 2j +2 2j +2 S SI − S SI S SI − 2S SI + S SI
where j = 1, . . . , p−1 and SIJ with I, J = 1, 2, . . . , 2p are entries of the S matrix, with S• being its row corresponding to the vacuum representation. Thus written, these formulas may seem messy (and the labeling of SIJ involves a convention on ordering the representations in accordance with their linkage classes), but they in fact have a clear structure (Eqs. (5.16), (5.2) – (5.14), and (5.8) – (5.10)), to be explained in what follows. The resulting (1, p) fusion algebra is given in Theorem 5.3.1. A posteriori, it turns out to have positive integral coefficients, although we do not derive the proposed recipe for the generalized Verlinde formula from first principles such that this property would be guaranteed in advance. 2. The Maximal Local W Algebra in the (1, p) Model 2.1. Energy-momentum tensor, screening operators, and resolutions. For the (p , p) minimal Virasoro models with coprime p , p ≥ 2, the Kac table of size (p −1)× (p−1)
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J. Fuchs, S. Hwang, A.M. Semikhatov, I.Yu. Tipunin
(after suitable identifications of boxes) contains those modules that do not admit nontrivial extensions among themselves. The extended Kac table of size p × p then corresponds to a logarithmic extension. The Kac table is selected as the cohomology, and the extended Kac table as the kernel, of an appropriate screening. We consider the models with p = 1, where the Kac table is empty, while the extended Kac table has size 1 × p, with its boxes corresponding to Virasoro representations Vs , s = 1, . . . , p. Similarly to the logarithmically extended (p , p) models, the (1, p) model is also defined as the kernel of the corresponding screening operator (this does not automatically yield its chiral algebra, however, which has then to be found, see below). The conformal dimensions (weights) of the primary fields corresponding to the irreducible modules Vs , s = 1, . . . , p, are given by (1, s), where for future use we define (r, s) :=
r2 − 1 s2 − 1 1 − rs p + + 2 . 4 4p
In the free-field realization through a scalar field ϕ(z) with the OPE ϕ(z) ϕ(w) = log(z−w), the corresponding primary fields are represented by the vertex operators ej (1,s)ϕ , where j (r, s) :=
1−r 1−s α+ + 2 α− 2
with α+ = 2p,
2
α− = − p .
Because pα− = −α+ , we have j (r, s+np) = j (r−n, s), n ∈ Z. The energy-momentum tensor is given by 1
α0
T = 2 ∂ϕ ∂ϕ + 2 ∂ 2 ϕ
(2.1)
(here and in similar formulas below, normal ordering is implied in the products), where α0 = α+ + α− , and the central charge is c = 13 − 6(p + p1 ). There then exist two screening operators α+ ϕ S+ = e , S− = eα− ϕ , satisfying [S± , T (z)] = 0. Let Fj (r,s) denote the Fock module generated from (the state corresponding to) the vertex operator ej (r,s)ϕ by elements of the Heisenberg algebra generated by the modes of the current ∂ϕ. Set F[s] = Fj (1,s) , and let the corresponding Feigin–Fuchs module over the Virasoro algebra (2.1) be denoted by the same symbol. For each 1 ≤ s ≤ p−1, F[s] is included into the acyclic Felder complex p−s
S−
s S−
p−s
S−
s S−
. . . ← F[s−2p] ←−−− F[−s] ←− F[s] ←−−− F[−s+2p] ←− F[s+2p] ← . . . , (2.2) where F[±s+2np] = Fj (1−2n,±s) .
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We define a (nonlocal) algebra A(p) as the kernel A(p) := Ker S− F of the S− screening on the direct sum F :=
Fj (r,s)
r∈Z s=1,...,p
of Fock modules. The algebra A(p) is generated by a − := e−
α+ 2
ϕ
and
a + := [S+ , a − ]
and is therefore determined by the lattice 21 α+ Z. It is slightly nonlocal: the scalar products of lattice vectors are in 21 Z.
2.2. The maximal local algebra. We next consider the W algebra that is the maximal local subalgebra in A(p) and use the notation W(p) for it for brevity. It is generated by the three currents W − , W 0 , and W + given by W − (z) := e−α+ ϕ (z),
W 0 (z) := [S+ , W − (z)],
W + (z) := [S+ , W 0 (z)].
We note that W 0 is a (free-field) descendant of the identity operator, while W + is a descendant of eα+ ϕ . The fields W − , W 0 , and W + are Virasoro primaries; their conformal dimensions are given by 2p−1. √ 6ϕ ,
Example 1. For p = 3, the W(3) generators are given by W − = e−
W
0
1 1 3 = 2 ∂ 3 ϕ ∂ 2 ϕ + 4 ∂ 4 ϕ ∂ϕ + 2
3 + 3 ∂ ϕ ∂ϕ ∂ϕ ∂ϕ + 5 2
3 2 ∂ ϕ ∂ 2 ϕ ∂ϕ + 2
3 3 ∂ ϕ ∂ϕ ∂ϕ 2
3 1 ∂ϕ ∂ϕ ∂ϕ ∂ϕ ∂ϕ + √ ∂ 5 ϕ, 2 20 6
and W
+
3 = − 2 ∂ 4 ϕ − 39 ∂ 2 ϕ ∂ 2 ϕ + 18 ∂ 3 ϕ ∂ϕ √ √ + 12 6 ∂ 2 ϕ ∂ϕ ∂ϕ − 18 ∂ϕ ∂ϕ ∂ϕ ∂ϕ e 6ϕ
(in the last formula, despite the brackets introduced for compactness of notation, the √ 2 6ϕ )))). nested normal ordering is from right to left, e.g., ∂ ϕ(∂ϕ(∂ϕ(e
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J. Fuchs, S. Hwang, A.M. Semikhatov, I.Yu. Tipunin Fj (1,s) •6
6 a+ 666 2s−3p 66 4 6 Fj (2,s) ◦ ◦- -- + − a 2s−5p -- a 2s−5p 4 -- 4 - - Fj (3,s) • •) )) )) )) )) a + a− 2s−7p )) 2s−7p 4 4 )) )) )) a− 2s−3p 4
Fj (4,s) ◦
◦
Fig. 1. The and modules. Filled (open) dots denote Virasoro representations that are combined in (s) (respectively, (s)). The A(p) generators a + and a − act between these representations with noninteger modes, but W (p) generators (not indicated) connecting dots of the same type are integermoded
2.3. W(p) representations. The W(p) generators change the “momentum” x of a vertex exϕ by nα+ with integer n, which corresponds to changing r in ej (r,s)ϕ by an even integer. It therefore follows that for each fixed s = 1, . . . , p, the sum
F(s) := Fj (r,s) r∈Z
of Fock modules contains two W(p) modules, to be denoted by (s) and (s), where
(s) is the W(p) representation generated from ej (1,s)ϕ (the highest-weight vector in Fj (1,s) ), while (s) is the W(p) representation generated from ej (2,s)ϕ (the highestweight vector in Fj (2,s) ), see Fig. 1. The dimensions of the corresponding highest-weight vectors are given by c
(s) − 24 =
(p − s)2 1 − 24 , 4p
c
(s) − 24 =
(2p − s)2 1 − 24 . 4p
(2.3)
A somewhat more involved analysis shows that the corresponding kernel of the screening S− , K(s) := Ker S− F (s) , s = 1, . . . , p, is precisely the direct sum K(s) = (s) ⊕ (s).
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2.4. Extensions among the representations. We next describe the nontrivial extensions allowed by the W(p) representations. The category of representations of W(p) in the (1, p) model decomposes into linkage classes of representations, which are full subcategories of the representation category.3 The representation category of the algebra W(p) associated with the (1, p) model has p + 1 linkage classes; we denote them as LC, LC , and LC(s) with 1 ≤ s ≤ p−1. The indecomposable representations in each linkage class are as follows. The classes LC and LC contain only a single indecomposable (hence, irreducible) representation each, (p) and (p) respectively. For 1 ≤ s ≤ p−1, the linkage class LC(s) contains two irreducible representations (s) and (p−s), as well as the following set of other indecomposable representations: N0± (s),
N0 (s),
N1± (s),
N1 (s),
R0 (s),
R1 (s).
There are nontrivial extensions 0 → (s) → N0± (s) → (p−s) → 0,
0 → (p−s) → N1± (s) → (s) → 0, and in addition, 0 → (s) → N0 (s) → (p−s) ⊕ (p−s) → 0, 0 → (p−s) → N1 (s) → (s) ⊕ (s) → 0. We note that L0 is diagonalizable in each of these representations. The “logarithmic” modules (those with a nondiagonalizable action of L0 ) appear in the extensions 0 → N0 (s) → R0 (s) → (s) → 0,
0 → N1 (s) → R1 (s) → (p−s) → 0.
It follows that N0+ (s) ∩ N0− (s) = (s) and N1+ (s) ∩ N1− (s) = (p−s). Thus we have towers of indecomposable representations given by R0 (s) ⊃ N0 (s) ⊃ N0± (s) ⊃ (s),
R1 (s) ⊃ N1 (s) ⊃ N1± (s) ⊃ (p−s)
for each s = 1, . . . , p − 1. The detailed structure of these representations will be considered elsewhere (see more comments in the Conclusions, however). Example 2. For p = 2, the four irreducible representations are V− 1 = (2), V 3 = (2), 8 8 V0 = (1), and V1 = (1). The “logarithmic” modules are R0 = R0 (1) and R1 = R1 (1) [8]. In addition, there are six other indecomposable representations N0± , N0 , N1± , and N1 . 3. Modular Transformations of the W(p) Characters In this section, we evaluate the characters of the W(p) representations introduced above and find their modular transformation properties. 3 The term “linkage class” is borrowed from the theory of finite-dimensional Lie algebras. The linkage classes of an additive category C are additive full subcategories Ci such that there are no (nonzero) morphisms between objects in two distinct linkage classes, every object of C is a direct sum of objects of the linkage classes, and none of the Ci can be split further in the same manner.
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3.1. Calculation of the W(p) characters. The route from representations to fusion starts
with ∈ { , } for the charwith the characters of W(p) representations. We write χs,p acter of (s) in the (1, p) model,
(q) = q L0 − 24 (s) . χs,p c
Proposition 3.1.1. The W(p) characters are given by
χs,p (q) = (q) χs,p
s 1 θ (q) + 2 θ (q) , p−s,p p−s,p η(q) p
s 1 = θ (q) − 2 θ (q) , s,p s,p η(q) p
1 ≤ s ≤ p.
(3.1)
Here, we use the eta function
1
η(q) = q 24
∞
(1 − q n )
n=1
and the theta functions
θs,p (q, z) =
2
q pj zj ,
|q| < 1, z ∈ C ,
s j ∈Z+ 2p
(q) := z ∂ θ (q, z) . and set θs,p (q) := θs,p (q, 1) and θs,p ∂z s,p z=1 Proof. Formulas (3.1) (which also have been suggested in [20]) can be derived by standard calculations, which we outline here for completeness. The characters of (s) and (s) are found by summing the characters of the kernels of S− on the appropriate Fock modules,
χs,p = char K(1, s) + 2 χs,p
=2
char K(2a+1, s),
a≥1
char K(2a, s),
a≥1
where K(r, s) := Ker S− F
j (r,s)
.
The character of K(r, s), in turn, is easily calculated from a “half” of the complex (2.2), i.e., from the one-sided resolution, as either the kernel or the image of the corresponding differential, which amounts to taking the alternating sum of characters of the modules in the left or right part of the complex. A standard calculation (with some care to be taken in rearranging double sums) then gives the formulas in the proposition.
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3.2. S and T transformations of the characters. With the characters of (s) and (s) expressed through theta functions, it is straightforward to find their modular properties. We resort to the standard abuse of notation by writing θs,p (τ ) for θs,p (e2iπτ ), for τ in the upper complex half-plane h. Proposition 3.2.1. Under the S transformation of τ , the W(p) characters transform as 1 1 s
√ χs,p (− τ ) = (τ ) + (−1)p−s χp,p (τ ) χp,p p 2p
+2
p−1
(p−s)
cos π p (τ ) χp−,p (τ ) + χ,p
=1 p−1
−2iτ
=1
and 1
1 2p
χs,p (− τ ) = √
+2
(p−s)
p− sin π p χ (τ ) − p χ,p (τ ) p p−,p
s
χp,p (τ ) + (−1)s χp,p (τ )
p
p−1
s
cos π p χp−,p (τ ) + χ,p (τ )
=1 p−1
+ 2iτ (with i =
=1
√ −1).
s
p− sin π p p χp−,p (τ ) − p χ,p (τ )
are modular forms of different weights ( 1 and 3 respecThe functions θs,p and θs,p 2 2 tively) and do not therefore mix in modular transformations. In contrast, the characters are linear combinations of modular forms of weights 0 and 1 and hence involve explicit occurrences of τ in their S transformation.
Proof. The formulas in the proposition are shown by directly applying the well-known relations
p−1 s 1 −iτ s θs,p (− τ ) = 2p θ0,p (τ ) + (−1) θp,p (τ ) + 2 cos π p θ,p (τ ) , 1
θs,p (− τ ) = −2iτ
=1
−iτ 2p
p−1 =1
s sin π p θ,p (τ ).
3.2.2. The S p (τ ) matrix. We now write the S transformation in a matrix form. To this end, we order the representations as
(p), (p), (1), (p−1), . . . , (p−1), (1), and arrange the characters into a column vector χ p ,
χ tp = ( χp,p , χp,p , χ1,p , χp−1,p , . . . , χp−1,p , χ1,p ).
(3.2)
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J. Fuchs, S. Hwang, A.M. Semikhatov, I.Yu. Tipunin
This order is chosen such that representations in the same linkage class are placed next to each other; it is one of the ingredients that make the block structure explicit in what follows. The above S transformation formulas then become 1
χ p (− τ ) = S p (τ ) χ p (τ ),
(3.3)
where S p (τ ) is most conveniently written using the 2 × 2 block notation A0,0 A0,1 . . . A0,p−1 A1,1 . . . A1,p−1 A S p (τ ) = 1,0 . . . . . . . . . . . . . . . . . . . . . . . . . . . Ap−1,0 Ap−1,1 . . . Ap−1,p−1 with
2 1 1 1 1 √ , A0,j = , 2p (−1)p−j (−1)p−j 1 (−1)p s p+s s (−1) p 1 p = √ p−s , p−s 2p p+s (−1) p p
1 A0,0 = √ 2p
As,0
(3.4)
and
As,j =
2 (−1)p+j +s p
s sj p−j sj sj j sj s cos π − iτ sin π cos π + iτ sin π p p p p p p p p × sj p−j sj p−s sj j sj p−s cos π + iτ sin π cos π − iτ sin π p p p p p p p p
for 1 ≤ s, j ≤ p−1. 3.2.3. The T p matrix. We next find the T transformation of the W(p) characters. For the vector χ p introduced above, we have χ p (τ +1) = T p χ p (τ ),
(3.5)
where T p is a block-diagonal matrix that can be compactly written as a direct sum of 2 × 2 blocks, T p = T0 ⊕ T1 ⊕ · · · ⊕ Tp−1
(3.6)
with π e−i 12 0 , T0 = p 1 0 eiπ( 2 − 12 )
2
Ts = e
1 iπ( (p−s) 2p − 12 )
12×2 ,
s = 1, . . . , p−1.
(3.7)
Starting from the W(p) algebra in (1, p) models, we have thus arrived at the S p (τ ) and T p matrices that implement modular transformations on the characters. Problem Q1 in the Introduction has thus been solved. With the resulting S p (τ ) involving a dependence on τ , we next face problem Q2, to be addressed in the next section.
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4. A Finite-Dimensional SL(2, Z) Representation from Characters 4.1. Matrix automorphy factors. The modular group action on characters generated by (3.3) and (3.5) fits the following general scheme. It is well known (or easily checked) that (γ · f )(τ, ν) := j (γ ; τ, ν) f (γ τ, γ ν),
(4.1)
with j (γ ; τ, ν) an n × n -matrix satisfying the cocycle condition j (γ γ ; τ, ν) = j (γ ; τ, ν) j (γ ; γ τ, γ ν),
j (1; τ, ν) = 1n×n ,
(4.2)
furnishes an action (actually, an antirepresentation) of the modular group SL(2, Z) on the space of functions f : h × C → Cn . We use the standard SL(2, Z) action on h × C , aτ + b ν a b γ = : (τ, ν) → (γ τ, γ ν) := cτ + d , cτ + d c d (the notation γ ν is somewhat loose, because this action depends on τ ). The matrix j (γ ; τ, ν) is called the (matrix) automorphy factor. An example of a scalar automorphy factor is given by the following classic result in the theory of theta functions [21]: the Jacobi theta function ϑ(τ, ν) is invariant under the action of 1,2 ⊂ SL(2, Z) (the subgroup of SL(2, Z) matrices γ = ac db with ab and cd even) on functions f : h × C → C given by (4.3) (γ · f )(τ, ν) = j ( ac db ; τ, ν) f (γ τ, γ ν) with the automorphy factor a b
j(
c d
cν 2
−1 (cτ +d)− 2 e−iπ cτ +d , ; τ, ν) = ζc,d 1
where ζc,d is an eighth root of unity (see [21]; its definition, which is far from trivial, ensures the cocycle condition for j ). 4.2. Constructing a finite-dimensional SL(2, Z) representation. The W(p) characters that we study here do not involve the ν dependence. Because S = 01 −1 and T = 01 11 0 generate SL(2, Z), Eqs. (3.3) and (3.5) uniquely determine a 2p × 2p matrix J p (γ , τ ) such that χ p (γ τ ) = J p (γ , τ ) χ p (τ ) for all γ ∈ SL(2, Z). It then follows that J p satisfies the condition J p (γ γ , τ ) = J p (γ , γ τ ) J p (γ , τ ),
γ , γ ∈ SL(2, Z).
(4.4)
Given this SL(2, Z) action, we now seek an SL(2, Z) action on χ p : h → C2p with a 2p × 2p matrix automorphy factor jp , γ · χ p (τ ) = jp (γ , τ ) χ p (γ τ ) = jp (γ , τ ) J p (γ , τ ) χ p (τ ), such that ρ(γ ) := jp (γ , τ ) J p (γ , τ )
(4.5)
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J. Fuchs, S. Hwang, A.M. Semikhatov, I.Yu. Tipunin
is a finite-dimensional representation of SL(2, Z) (in particular, the left-hand side must be independent of τ ). This condition is reformulated as the condition that ρ and jp “strongly” commute, i.e., that ρ(γ ) jp (γ , τ ) = jp (γ , τ ) ρ(γ ),
γ , γ ∈ SL(2, Z).
(4.6)
It is easy to verify that for a given J p ( · , · ), each jp that satisfies both the commutation property (4.6) (with ρ defined by (4.5)) and the cocycle condition (4.2) provides a (finite-dimensional) SL(2, Z) representation ρ. Indeed, ρ(γ γ ) = jp (γ γ , τ ) J p (γ γ , τ ) = jp (γ , τ ) ρ(γ ) J p (γ , τ ) = ρ(γ ) jp (γ , τ ) J p (γ , τ ) = ρ(γ ) ρ(γ ). 4.3. SL(2, Z) representation in (1, p) models. We now find a matrix automorphy factor jp that “converts” the action in (3.3) – (3.6) into a representation. As noted above, J p ( · , τ ) is uniquely determined on all of SL(2, Z) by Eqs. (4.4) from J p (T , τ ) = T p (τ ) and J p (S, τ ) = S p (τ ). With S p (τ ) and T p (τ ) = T p given by (3.4) and (3.6), we define the automorphy factor jp ( · , · ) as a block-diagonal matrix consisting of 2 × 2 blocks that we compactly write as jp (γ , τ ) = 12×2 ⊕ B1 (γ , τ ) ⊕ · · · ⊕ Bp−1 (γ , τ ),
(4.7)
where for γ = S,
s p−s s s + i − i τp p τp p Bs (S, τ ) = , p−s p−s s p−s − i + i p τp p τp
s = 1, . . . , p − 1,
(4.8)
s = 1, . . . , p − 1,
(4.9)
and for γ = T ,
s p−s s s + t − t p p p p , Bs (T , τ ) = p−s p−s p−s s − t + t p p p p
with t 3 = −i (we can set t = i). The structure in (4.8) is easily discernible by subjecting all matrices to the similarity transformation that relates the basis of characters to the . The automorphy factor is then diagonalized, as shown basis provided by θs,p and θs,p explicitly in the proof of the next proposition. Proposition 4.3.1. The matrix automorphy factor defined in (4.7) – (4.9) satisfies the cocycle condition (4.2). Proof. The proof amounts to a direct verification of the formulas (ST )3 = (T S)3 = S 2 reformulated for jp (γ , τ ). That is, in proving that jp (S 2 , τ ) = jp ((ST )3 , τ ), we have, in accordance with (4.2), 1
jp (S 2 , τ ) = jp (S, τ ) jp (S, − τ ),
−1
jp ((ST )3 , τ ) = jp (ST , τ ) jp (ST , τ +1 ) jp (ST ,
−τ −1 ), τ
(4.10)
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727
where in turn, jp (ST , τ ) = jp (T , τ )jp (S, τ +1). The calculation reduces to a separate computation for each of the 2 × 2 blocks given above; further, each block can be diagonalized as 1 0 L−1 Bs (γ , τ ) = Ls s , 0 ζ (γ )α(γ , τ ) where ζ (γ ) is the character of SL(2, Z) defined by the relations ζ (S) = i,
t 3 = −i,
ζ (T ) = t,
(4.11)
and a b
α(
c d
1
, τ ) = cτ + d
is already an automorphy factor [21]. Equations (4.11) immediately imply that ζ (S 2 ) = ζ ((ST )3 ), and Eqs. (4.10) are therefore proved. With this jp , we evaluate S(p) = jp (S, τ )S p (τ ) as S(p) = jp (S, τ ) S p (τ ) = S p (i).
(4.12)
That is, S(p) has a block form similar to that of S p in Sect. 3.2, with the 2 × 2 blocks Si,j given by S0,0 = A0,0 , S0,j = A0,j , Ss,0 = As,0 , and
Ss,j =
s sj p−j sj sj j sj s cos π + sin π cos π − sin π p p p p p p p 2 p (−1)p+j +s . p sj p−j sj p−s sj j sj p−s cos π − sin π cos π + sin π p p p p p p p p
Similarly, T(p) = jp (T , τ ) T p (where as we have seen, jp (T , τ ) is actually independent of τ ). We do not write the blocks of T(p) explicitly because they are simply given by multiplication of the blocks in (4.9) with matrices (3.7). Proposition 4.3.2. The matrices S(p) and T(p) generate a finite-dimensional representation of SL(2, Z). Proof. The proof consists in verifying (4.6) for (γ , γ ) being any of the pairs (S, T ), (T , S), (S, S), and (T , T ), which is straightforward. Together with the cocycle condition, this then implies that (S(p))2 = (T(p)S(p))3 = (S(p)T(p))3 . The above construction of the numeric (τ -independent) matrix S(p) representing S ∈ SL(2, Z) solves problem Q2 in the Introduction.
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4.4. Some properties of the S(p) matrix. The vacuum representation (1) is the third in the order of representations chosen in (3.2). This distinguishes the third row of the S matrix; we let σ (p) ≡ σ denote this distinguished row of S(p). Explicitly, σ (p) is given by √
σ (p) =
2 (−1)p √
(−1)p 2
p p
1
, −2,
π
π
π
π
cos p + (p−1) sin p , cos p − sin p , 2π
2π
2π
2π
− cos p − (p−2) sin p , − cos p + 2 sin p , ..., jπ jπ jπ jπ j +1 (−1) cos p + (p−j ) sin p , (−1)j +1 cos p − j sin p , ..., (p−1)π jπ (p−1)π (p − 1)π p (−1) cos p + sin p , (−1)p cos p − (p−1) sin . p (4.13) Next, it follows from (3.3) that (S(p))2 χ p (i) = χ p (i). In fact, we have the following result. Proposition 4.4.1. (S(p))2 = 12p×2p .
(4.14)
Proof. Indeed, we evaluate (S(p))2 as (4.5)
(4.6)
(4.5)
ρ(S) ρ(S) = ρ(S) jp (S, τ )J p (S, τ ) = jp (S, τ ) ρ(S) J p (S, τ ) = (4.4)
jp (S, τ ) jp (S, Sτ ) J p (S, Sτ ) J p (S, τ ) = jp (S, τ ) jp (S, Sτ ) J p (S 2 , τ ). Next, J p (S 2 , τ ) = 12p×2p because S 2 τ = τ . Finally we have jp (S, τ ) jp (S, Sτ ) = 12p×2p , which is obtained by a direct calculation similar to the one in the proof of Prop. 4.3.1. Equation (4.14) thus follows. Remark 4.4.2. With the explicit form of S(p) given above, Prop. 4.4.1 can also be shown directly, which gives a good illustration of a typical calculation with the matrices encountered throughout this paper. Writing C = (S(p))2 in the 2 × 2-block form, C0,0 C0,1 . . . C0,p−1 C1,1 . . . C1,p−1 C , C = 1,0 . . . . . . . . . . . . . . . . . . . . . . . . . . . Cp−1,0 Cp−1,1 . . . Cp−1,p−1
Nonsemisimple Fusion Algebras and the Verlinde Formula
729
we concentrate on the more involved blocks Cs,j with 0 < s, j < p. Assuming that p is odd for brevity (in order to avoid extra sign factors) we find that 2 p2 p−1 p−1 (p−j ) (p−s) (p−j ) (p−s) s cos π cos π s cos π cos π p p p p =[s+j ]2 =[s+j ]2 p−1 p−1 ) ) + (p−j ) sin π (p−j sin π (p−s) −j sin π (p−j sin π (p−s) p p p p =1 =1 × , p−1 p−1 (p−j ) (p−s) (p−j ) (p−s) (p−s) cos π p cos π p (p−s) cos π p cos π p =[s+j ]2 =[s+j ]2 p−1 p−1 ) ) − (p−j ) sin π (p−j sin π (p−s) +j sin π (p−j sin π (p−s) p p p p
Cs,j =
=1
=1
where [a]2 := a mod 2. Using elementary trigonometric rearrangements (expressing cos α sin β through the sine and cosine of α+β and α−β), we see that all entries in the matrices above vanish, with the exception of the diagonal entries of Cs,s , which (for 0 < s ≤ p) are given by p−1 p−1 2 (p−s) 2 (p−s) 2 + (p−s) s cos π sin π = 1. p p p2 =0
=1
Together with similar (and in fact, simpler) calculations for the other blocks, this shows Eq. (4.14). We also note that S(p) is not symmetric, S(p) = S(p)t . It admits a different symmetry S(p)∨ = S(p),
(4.15)
where for a matrix r = (ri,j )i,j =1,...,2p with i and j considered modulo 2p, we define the involutive operation (r ∨ )m,n := (−1)p(1−δm,1 −δn,1 )+(m+n+1)/2+mn r2p−m+3,2p−n+3 . For example, with r = (rij )i,j =1,...,6 , we have −r
22
r12 −r62 r∨ = −r52 r42 r32
r21 −r11 r61 r51 −r41 −r31
−r26 r16 −r66 −r56 r46 r36
−r25 r15 −r65 −r55 r45 r35
r24 −r14 r64 r54 −r44 −r34
r23 −r13 r63 . r53 −r43 −r33
The symmetry (4.15) originates in the existence of a simple current, as we see below.
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Example 3. For p = 2 and p = 3, the S(p) matrices can be evaluated as 1 1 1 1 2 2 1 1 −1 −1 2 2 S(2) = 1 − 1 1 − 1 ,
S(3) =
4 1 4
4
2
− 41 − 21
1 2
2
2 2 2 2 √1 √1 6 3 3 3 3 6 √1 2 2 − √1 − 23 − 23 3 3 6 6 √ √ √ √ 1 −(6+ −(6+ 3−√ 3 3−√ 3 1 √ √ √ 3) √ 3) 3 6 3 6 √ 9 2√ 9 2 9 2 9 2 √ √ . √ √ √2/3 √2/3 2(3− 3) −(3+2 3) 3) 2(3− 3) −(3+2 √ √ 3 3 9 9 9 2 9 2 √ √ √ √ √ √ √ 2/3 −√2/3 2(3− 3) −(3+2 3) 2( 3−3) 3+2 3 √ √ 3 3 9 9 9 2 9 2 √ √ √ √ −(6+ 3−√ 3 6+√ 3 3−3 −1 1 √ √ √ 3) √ 3 6 3 6 9 2 9 2 9 2 9 2
5. Constructing the Eigenmatrix P and the Fusion Having extracted a finite-dimensional SL(2, Z) representation from the SL(2, Z) action on characters, we now address problems Q3 and Q4 in the Introduction. We use the S(p) matrix found in the previous section in the construction of the eigenmatrix P of the fusion algebra. From the eigenmatrix, we then find the fusion. In Sect. 5.1, we first describe the role of the P matrix in a commutative associative algebra in a slightly more general setting than we actually need in (1, p) models. In Sect. 5.2, we formulate the generalized Verlinde formula and use it to find the eigenmatrix P(p) in the (1, p) model. In Sect. 5.3, we then obtain the fusion following the recipe in Sect. 5.1. 5.1. Fusion constants from the eigenmatrix. A fusion algebra is a finite-dimensional commutative associative algebra F over C with a unit element 1, together with a canonical basis {XI }, I = 1, . . . , n = dimC F (containing 1), such that the structure constants NIKJ defined by X I XJ =
n
NIKJ XK
K=1
are nonnegative integers. As any finitely generated associative algebra with a unit, F is a vector-space sum of the radical R and a semisimple algebra [19]. The algebra contains a set of primitive idempotents satisfying eA eB = δA,B eB and
eA = 1.
(5.1) (5.2)
all primitive idempotents
The primitive idempotents characterize the semisimple quotient up to Morita equivalence. A commutative associative algebra has a basis given by the union of a basis in the radical and the primitive idempotents eA .
Nonsemisimple Fusion Algebras and the Verlinde Formula
731
The primitive idempotents can be classified by the dimensions νA of their images. For the purposes of (1, p) models, we only need to consider the case where all νA ≤ 2.4 The structure of the algebra F is then conveniently expressed by its quiver •
...
•
•
•
...
•
r
Here, the dots are in one-to-one correspondence with primitive idempotents. The quiver is disconnected because the algebra is commutative. A vertex eA has a self-link if νA = 2, and has no links if νA = 1. Each link can be associated with an element in the radical, and moreover, these elements constitute a basis in the radical. We let eα denote the primitive idempotents with να = 2 and let wα ∈ R be the corresponding element, defined modulo a nonzero factor, represented by the link of eα with itself. Then eα wβ = δα,β wβ .
(5.3)
The other primitive idempotents, to be denoted by ea , satisfy ea wβ = 0.
(5.4)
The elements wα can be chosen such that they constitute a basis in the radical and satisfy wα wβ = 0.
(5.5)
Let Y• be the basis consisting of ea , eα , and wα ; with r introduced in the quiver above (as r = dimC R), we have a = 1, . . . , n−2r and α = n−2r+1, . . . , n−r. We order the elements in this basis as Y1 = e1 , . . . , Yn−2r = en−2r , Yn−2r+1 = en−2r+1 , Yn−2r+2 = wn−2r+1 , ......, Yn−1 = en−r , Yn = wn−r .
(5.6)
This ordering may seem inconvenient in that labeling of wα starts with wn−2r+1 , but it is actually very useful in what follows, because it makes the 2 × 2 block structure explicit by placing each element wα in the radical next to the primitive idempotent eα that satisfies eα wα = wα ; the primitive idempotents that annihilate the radical are given first. It may be useful to rewrite (5.6) as I = 1, 2, . . . , n−2r, eI , YI = e(I +n+1)/2−r , I = n−2r+2i+1, 0 ≤ i ≤ r−1, w I = n−2r+2i, 1 ≤ i ≤ r. (I +n)/2−r , 4 The fusion algebra for a general logarithmic conformal field theory can involve primitive idempotents with arbitrary νA . We restrict our attention to the particular case where νA ≤ 2 because of the lack of instructive examples of higher-“rank” logarithmic theories; the definitions may need to be refined as further examples are worked out. When the set of idempotents with νA = 2 is empty, we recover the semisimple case [16] (we do not impose conditions F2 and F3 in [16] because they imply semisimplicity of the fusion algebra).
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The multiplication table of Y• , Eqs. (5.1) – (5.5), defines an associative algebra. But it does not define a fusion algebra structure, because the latter involves a canonical basis. The canonical basis X• in F is specified by a nondegenerate n × n matrix P, called the eigenmatrix, that contains a row entirely consisting of 0 (r times) and 1 (n−r ≥ r times). We let π denote this row, and order the columns of P in accordance with (5.6), such that π = ( 1, . . . , 1 1, 0, 1, 0, . . . , 1, 0 ). 2r
n−2r
Elements of the canonical basis are given by XI =
n J =1
P I J YJ
(5.7)
and are therefore in one-to-one correspondence with the rows of P; permuting the rows of P is equivalent to relabeling the elements of the canonical basis. The order of the columns of P is fixed by the assignments of Y• in (5.6), i.e., by the order chosen for the elements of the basis consisting of idempotents and elements in the radical, and is therefore conventional. Each column corresponding to an element in the radical (that is, containing zero in the intersection with the row π ) is defined up to a factor, because wα in the radical cannot be canonically normalized. In view of (5.2), it follows that X = 1. We now express the structure constants of the fusion algebra in the canonical basis through a given eigenmatrix P. We organize the structure constants into matrices NI with the entries (NI )JK := NIKJ . Let πI = (PI 1 , . . . , PI n ) be the I th row of P. For each I = 1, . . . , n, we define the n × n matrix 1 PI . .. n−2r PI n−2r+2 n−2r+1 PI PI 0 PI n−2r+1 , MI := n−2r+3 n−2r+4 PI PI n−2r+3 0 PI .. . PI n−1 PI n 0 PI n−1
0
(5.8)
0
which is the direct sum of a diagonal matrix and r upper-triangular 2 × 2 matrices. These matrices relate the rows of P as πI = π MI ,
I = 1, . . . , n.
(5.9)
They can be characterized as the upper-triangular 2 × 2 -block-diagonal matrices that satisfy (5.9).
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733
The next result answers the problem addressed in Q3. Proposition 5.1.1. The structure constants are reconstructed from the eigenmatrix as NI = P MI P−1 .
(5.10)
Proof. The regular representation λ: F → End F of the algebra F, where F is the underlying vector space, is faithful because 1 ∈ F; therefore, F is completely determined by its regular representation. By definition, the matrices NI represent the elements XI ∈ F in the basis X• : λ(XI ) = NI . On the other hand, using relations (5.1) – (5.5), we calculate J J = 1, 2, . . . , n−2r, PI YJ , XI YA = PI J YJ + PI J +1 YJ +1 , J = n−2r+2i+1, i ≥ 0, P J −1 Y , J = n−2r+2i, i ≥ 1 J I (no summation over J ). This implies that the matrices MI in (5.8) represent the elements XI ∈ F in the basis Y• , and hence (5.10) follows. Remark 5.1.2. The eigenmatrix P of the fusion algebra is different from the modular transformation matrix S even in the semisimple case. The most essential part of the semisimple Verlinde formula consists in the relation between the eigenmatrix P, which maps the canonical basis of the fusion algebra to primitive idempotents, and the matrix S, which represents S ∈ SL(2, Z) on characters, P = S Kdiag ,
(5.11)
with Kdiag in turn expressed through the elements (S1 , S2 , . . . , Sn ) of the vacuum row of S, 1 1 1 Kdiag := diag 1 , 2 , . . . , S n . S
S
(5.12)
In the nonsemisimple case, a relation between S and P generalizing (5.11) – (5.12) gives the nontrivial part of the corresponding generalized Verlinde formula. This is studied in the next subsection. 5.2. From S to P. We now construct P from S via a generalization of the Verlinde formula to nonsemisimple fusion algebras described in (5.1) – (5.5). The first step is to construct the interpolating matrix K generalizing Kdiag ; the diagonal structure present in the semisimple case is replaced by a 2 × 2 block-diagonal structure. We recall that the rows and columns of S are labeled by representations, and that the distinguished row 2p−1
σ = ( S1 , S2 , . . . , S
2p
, S )
of S(p) corresponds to the vacuum representation. Then K is the block-diagonal matrix K := K0 ⊕ K1 ⊕ · · · ⊕ Kp−1 ,
Ki ∈ Mat2 (C)
(5.13)
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J. Fuchs, S. Hwang, A.M. Semikhatov, I.Yu. Tipunin
with 1 S 1 K0 := 0
0
1
S 2j +1 − S 2j +2 Kj := −1
1 ,
2j +1 S
S2
2j +2 − S
2j +2
−S
2j +1 S
(5.14)
for j = 1, . . . , p−1. This matrix relates the distinguished rows of P and S as π = σ K.
(5.15)
It can be characterized as the block-diagonal matrix of form(5.13), with diagonal K0 ki ∗ and with each Ki , i = 1, . . . , p−1, of the form Ki = −k defined up to a normalii ∗ zation of the second column, that satisfies (5.15). In (5.14), we chose the normalization such that det Kj = 1; the freedom in this (nonzero) normalization factor is related to the freedom in rescaling each element in the radical, and hence rescaling the corresponding columns of P. For a given S, we set (restoring the explicit dependence on the parameter p that specifies the model) P(p) := S(p) K(p).
(5.16)
The above prescription for the interpolating matrix K and the resulting expression (5.16) for the eigenmatrix P solve problem Q4 in the Introduction. Remark 5.2.1. Combining formulas (5.10) and (5.16), we can write the generalized Verlinde formula as SI ) S−1 , NI = S (K "
(5.17)
where " SI := MI K−1 . In the semisimple case, this reduces to the ordinary Verlinde forSdiag,I ) S−1 , with diagonal matrices Kdiag given by (5.12) mula written as NI = S (Kdiag " K K K and (" Sdiag,I )J = SI δJ . In the (1, p) model, we use the S(p) matrix obtained in Sect. 4 and its distinguished row (4.13) to derive 1 0 , K0 = p 2p 0 (−1)p+1 −1 2 jπ jπ cos p − j sin p sin j π 2 p p p , j = 1, . . . , p−1. Kj = (−1)p+j 2 1 −2 jπ j π 2 cos p + (p−j ) sin p jπ sin p
p
A straightforward calculation then shows that P0,0 P0,1 . . . P0,p−1 P1,1 . . . P1,p−1 P , P(p) = 1,0 . . . . . . . . . . . . . . . . . . . . . . . . . . Pp−1,0 Pp−1,1 . . . Pp−1,p−1
(5.18)
Nonsemisimple Fusion Algebras and the Verlinde Formula
with the 2 × 2 blocks P0,0 =
p (−1)p+1 p , p −p
s (−1)s+1 s = , p−s (−1)s+1 (p−s)
and
Ps,j
2
jπ
− p sin p
0
Ps,0
735
P0,j = , 2 jπ j +p 0 −(−1) sin p p
(5.20)
2 sj π jπ sj π jπ sin + sin cos −s cos p p p p p2
sin sjpπ
− j π sin p = (−1) sin sj π p
(5.19)
jπ
s
2 sj π jπ sj π −(p−s) cos p sin p − sin p cos p p2
sin jpπ
(5.21)
for s, j = 1, . . . , p−1. The first column of P(p) contains the quantum dimensions of all the irreducible representations in the model. They are given by ( p, p, 1, p−1, 2, p−2, . . . , p−1, 1 ), listed in the order (3.2), i.e., qdim( (s)) = s = qdim( (s)),
s = 1, . . . , p.
(5.22)
Remarkably, all these quantum dimensions are integral. This points to an underlying quantum-group structure, such that the quantum dimensions are the dimensions of certain quantum group modules. This quantum-group structure will be considered elsewhere (see more comments in the Conclusions, however). As noted above, the normalization of each even column of P starting with the fourth can be changed arbitrarily because wα in the radical cannot be canonically normalized. Example 4. For p = 2, 3, 4, the eigenmatrices found above are evaluated as follows: √1 3 3 0 − √1 0 3 3 3 −3 0 − √13 0 − √13 2 −2 0 1 2 −2 0 −1 1 1 1 0 1 0 P(2) = , P(3) = 1 −1 , 1 1 1 0 2 2 −1 2√3 −1 2√3 1 1 −1 0 1 1 2 −2 −1 √ √ 1 2 3 2 3 1 −1 1 0 −1 0 1 1 √ √ 4 −4 0 0 − 21 0 2 2 2 2 −1 √ √ 0 − 21 0 4 −4 0 2−1 2 2 2 1 1 1 0 1 0 1 0 1 3 3 −1 −1 0 −1 − 41 4 √ √ P(4) = . 1 1 2 −2 − 2 √ 0 41 2 √ 8 2 8 2 √ 2 −2 √2 −1 √ √ 0 41 − 2 −1 8 2 8 2 1 3 3 1 − 1 −1 0 1 1 1
−1
4
0
1
0
−1
4
0
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5.3. The fusion algebra Fp . From S p (τ ) in (3.4), we have arrived at the eigenmatrix P(p) in (5.18) – (5.21). As we saw in Sect. 5.1, the fusion is reconstructed from the eigenmatrix. We now perform this reconstruction for the (1, p) model. Theorem 5.3.1. For each p ≥ 2, the fusion algebra Fp determined by the eigenmatrix P(p) is described by the following multiplication table of the 2p canonical basis elements (p), (p), (1), (p−1), (2), (p−2), . . . , (p−1), (1):
(s) (t) =
s+t−1
"
(r),
(s) (t) =
r=|s−t|+1 step=2
(s) (t) =
s+t−1
s+t−1
" (r),
r=|s−t|+1 step=2
"
(r),
s, t = 1, . . . , p,
r=|s−t|+1 step=2
where # " := (r),
(r)
(2p−r) + 2 (r−p), # " := (r), (r) (2p−r) + 2 (r−p),
1 ≤ r ≤ p, p+1 ≤ r ≤ 2p−1, 1 ≤ r ≤ p, p+1 ≤ r ≤ 2p−1.
Proof. We first evaluate the matrices MI in accordance with (5.8). For each s = 0, . . . , p−1, the matrix M2s+1 corresponds to the (2s+1)th row of the eigenmatrix P(p), and hence to the representation (s). For s = 1, . . . , p−1, we have M2s+1 ≡ M( (s))
=
s
(−1)s+1 s
... (−1)s+1
sin sjpπ sin jpπ
0
2(−1)s sj π sin p cos jpπ − s cos sjpπ sin jpπ p2 (−1)s+1
sin sjpπ sin jpπ
...
,
where the dots denote the 2 × 2 block of the indicated structure written p−1 times, for j = 1, . . . , p−1. (In particular, M3 = 1; the matrices M1 and M2 have a simple form and are not written here for brevity.) The matrices M2s+2 , s = 0, . . . , p−1, have a similar structure, which can be written most compactly by first noting that
Nonsemisimple Fusion Algebras and the Verlinde Formula
737
1
(−1)p .. . M4 ≡ M( (1)) = 0 (−1)p+j p+j 0 (−1)
..
.
(where the block is again to be written p−1 times, for j = 1, . . . , p−1) and then M2s+2 ≡ M( (s)) = M( (1)) M( (s)).
(5.23)
With the MI matrices thus found, we can reconstruct the structure constants from (5.10). But it is technically easier to find the same structure constants from the algebra satisfied by the matrices MI , 2p M I MJ = NIKJ MK , K=1
which (just by (5.10)) furnish an equivalent representation of the fusion algebra. From (5.23), we conclude that (1) (s) = (s); it immediately follows that (1) (s) = (s), s = 1, . . . , p. By associativity, it therefore remains to prove only the (s) (t) fusion, that is, to show the matrix identities (assuming s ≥ t for definiteness) M2s+1 M2t+1 =
t−1
" M( (s−t+1+2a)),
a=0
where we extend the mapping (s) → M( (s)), (s) → M( (s)) by linearity, such that " M( (r)) = M( (2p−r)) + 2 M( (r−p)) = M2(2p−r)+1 + 2 M2(r−p)+2 for r ≥ p + 1. But elementary calculations with the matrices explicitly given above show that " M( (r)) = M( (r)) " (which may be rephrased by saying that M( (r)) “continues” M( (r)) to r ≥ p+1). Therefore, the statement of the theorem reduces to the matrix identity M2s+1 M2t+1 =
t−1
M2(s−t+1+2a)+1 ,
a=0
which can be verified directly. For the upper-left 2 × 2 blocks, this is totally straightforward,
s 0 0 (−1)s+1 s
t 0 0 (−1)t+1 t
t−1 s − t + 1 + 2a 0 , = 0 (−1)s+t (s − t + 1 + 2a) a=0
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and for the other blocks the calculation amounts to evaluating sums of the form t−1 a=0
sin
sin αt sin r+t−1 r + 2a α = α sin α1
and their derivatives. Remark 5.3.2. We see that (1) is a simple current of order two, acting without fixed points; it underlies the symmetry (4.15). This simple current symmetry is analogous to the one present in rational CFTs. The permutations of the entries of S(p) correspond to the action of the simple current (1) by the fusion product, while the sign factors are exponentiated monodromy charges, which are combinations of conformal weights. We also note that the quantum dimensions (5.22) furnish a one-dimensional representation of the fusion algebra. Example 5. For p = 2, the F2 algebra coincides with the fusion obtained in [8], written in terms of linearly independent elements corresponding to the irreducible subquotients, as explained above. For p = 3 and 4, we write the fusion algebras explicitly. To reduce the number of formulas, we note that for all p, (1) is the unit element and (1) is an order-2 simple current that acts as (1) (s) = (s),
(1) (s) = (s).
Further, (s) (t) = (s) (t) and (s) (t) = (t) (s). The remaining relations are now written explicitly. For p = 3, the remaining F3 relations are given by
(2) (2) = (1) + (3),
(2) (3) = 2 (2) + 2 (1),
(2) (2) = (1) + (3),
(2) (3) = 2 (2) + 2 (1),
(3) (3) = 2 (1) + 2 (2) + (3),
(3) (3) = 2 (2) + 2 (1) + (3). For p = 4, the remaining F4 relations are
(2) (2) = (1) + (3),
(2) (3) = (2) + (4),
(2) (4) = 2 (1) + 2 (3),
(3) (3) = (1) + 2 (3) + 2 (1),
(3) (4) = 2 (2) + 2 (2) + (4),
(4) (4) = 2 (1) + 2 (3) + 2 (3) + 2 (1),
(2) (2) = (1) + (3),
(2) (3) = (2) + (4),
(2) (4) = 2 (3) + 2 (1),
(3) (3) = (1) + 2 (3) + 2 (1),
(3) (4) = 2 (2) + 2 (2) + (4),
(4) (4) = 2 (1) + 2 (3) + 2 (3) + 2 (1).
Nonsemisimple Fusion Algebras and the Verlinde Formula
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6. Conclusions To summarize, our proposal for a nonsemisimple generalization of theVerlinde formula is given by (5.16), with the interpolating matrix K built in accordance with (5.2) – (5.14) from S constructed in (4.12). From the matrix P that is provided by the generalized Verlinde formula (5.16), the structure constants of the fusion algebra are reconstructed via (5.8) and (5.10). In (1, p) models, this leads to the fusion in Theorem 5.3.1. The rest of this concluding section is more a to do list than the conclusions to what has been done. First, we have used a generalization of the Verlinde formula to derive the fusion in (1, p) models, see Theorem 5.3.1, but we have not presented a systematic “first-principle” proof of the proposed recipe. The relevant first principles are the properly formulated axioms of chiral conformal field theory. The situation is thus reminiscent of the one with the ordinary (semisimple) Verlinde formula, whose proof could be attacked only after those axioms had been formulated [22] (see also [23, 24]) for rational conformal field theory. In the semisimple case, the structure constants are expressed through the defining data of the representation category, which is a modular tensor category, and thus through the matrices of the basic B and F operations of [22] as + k j+ B jj kk B j + k + 00 Sij Sj l = Nikl , Fk j
where Fk = F00
k+ k k k
.
These formulas are to be related to the above construction of the fusion algebra constants expressed as NK = S OK S, with the matrices OI = KMI K−1 (already given in the Introduction) whose structure readily follows from Sect. 5. The necessary modifications of the RCFT axioms are then to lead to a block-diagonal structure, with nontrivial blocks being in one-to-one correspondence with the linkage classes, with the size of a block given by the number of irreducible representations in the relevant linkage class. Another obvious task is to place the structures encountered here into their proper categorical context. For rational CFT, the representation category C of the chiral algebra — a rational conformal vertex algebra — is a modular tensor category, and can thus in particular be used to associate a three-dimensional topological field theory to the chiral CFT. For instance, the state spaces of the three-dimensional TFT are the spaces of chiral blocks of the CFT, and the modular S matrix (or, to be precise, the symmetric matrix that diagonalizes the fusion rules) is, up to normalization, the invariant of the Hopf link in the three-dimensional TFT. Also, a full (nonchiral) CFT based on a given chiral CFT corresponds to a certain Frobenius algebra in the category C, and the correlation functions of the full CFT can be determined by combining methods from three-dimensional TFT and from noncommutative algebra in monoidal categories [27, 28]. In the nonrational case, C is no longer modular, in particular not semisimple, but in any case it should still be an additive braided monoidal category. In addition, other properties of C, as well as the relevance of noncommutative algebra in C to the construction of full from chiral CFT, can be expected to generalize from the rational to the nonrational case.
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It is, however, an open (and complicated) problem to make this statement more precise. For instance, it is not known how to generalize the duality structure. (We note that the fusion rule algebra Fp does not share the duality property familiar from rational fusion algebras: evaluation at the unit element does not furnish an involution of the algebra.) On the other hand, the fact that we are able to identify a finite-dimensional representation of the modular group in each of the (1, p) models indicates that the chiral blocks of these models should nevertheless possess the basic covariance properties under the relevant mapping class group. This suggests, in turn, that they can still be interpreted as the state spaces of a suitable three-dimensional TFT. (For one proposal on how to associate a three-dimensional TFT to a nonrational CFT, see [29]. However, the S matrix is generically not symmetric, which certainly complicates the relation to three-dimensional TFT.) Furthermore, we expect that this also applies to many other nonrational CFTs, at least to those for which C has a finite number of (isomorphism classes of) simple objects (and thus in particular finitely many linkage classes), with all of them having finite quantum dimensions. A first step in developing the categorical context could consist in finding the “fine” fusion, where each indecomposable W(p) representation corresponds to a linearly independent generator in the fusion algebra. This fusion would define the monoidal structure of the category C. It should therefore be important for finding modular invariants and possible boundary conditions in conformal field theory. For example, one can imagine that a boundary condition involves only an indecomposable representation, but not its subquotients (cf. [25, 26]). A preliminary analysis shows that for p = 2, 3, invariants χ †p H(p)χ p under the SL(2, Z) action on the characters of irreducible W(p) representations are given by 1 0 0 0 4 (h1 + h2 ) 1 0 4 (h1 + h2 ) 0 0 H(2) = 0 0 h1 h2 0 0 h 2 h1 and 1 H(3) =
6 (h1
+ 2h2 ) 0 1 (h + 2h ) 0 1 2 6 0 0 0 0 0 0 0 0
0 0 0 0 0 0 h2 0 h1 h2 21 (h1 + h2 ) 0 1 0 0 2 (h1 + h2 ) 0 0 h2
0 0 0 , 0 h2 h1
where in each case, the coefficients h1,2 must be chosen such that the matrix entries are integers, for example, h1 = h2 = 2 for p = 3. The “fine” fusion is needed precisely here in order to correctly interpret the result. It allows distinguishing between inequivalent representations that possess identical characters and is therefore needed for interpreting the result for the modular invariant as a proper partition function not only at the level of characters, but also at the level of representations (or, rephrased in CFT terms, not just describing the dimensions of spaces of states of the full CFT, but completely telling which bulk fields result from combining the two chiral parts of the theory). We also note that behind the scenes in Theorem 5.3.1 is a quantum group of dimension 2p3 . Its representation category is equivalent to the category of W(p) representations described in Sect. 2.4, and the quantum dimensions (5.22) are the dimensions of
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its representations. The close relation between this quantum group and the fusion will be studied elsewhere. Next, the structure of the indecomposable W(p) modules in Sect. 2.4 should be studied further. This can be done by traditional means, but a very useful approach is in the spirit of [15] (which provides the required description for p = 2). The idea is to add extra modes to the algebra of a + and a − in Sect. 2.1 such that the W(p) action in the indecomposable modules is realized explicitly. With these extra modes added, some states that are not singular vectors in the module in Fig. 1 become singular vectors built on new states, and the construction of these new states can be rephrased as the “inversion” of singular vector operators, similarly to how the operator of the simplest singular vector L−1 was inverted in [15] (where both the singular vector operator was the simplest possible and the a ± operators were actually fermions). Finally, it is highly desirable, but apparently quite complicated, to extend the analysis in this paper to logarithmic extensions of the (p , p) models with coprime p , p ≥ 2. The extended Kac table of size p × p is then selected as the kernel of the appropriate screening operator. Already the (2, 3) model (which is trivial in its nonlogarithmic version) is of interest because of its possible relation to percolation. However, it is not obvious how to describe the kernel of the screening in reasonably explicit terms; in particular, we do not know good analogues of the operators a + and a − . Acknowledgements. We are grateful to B. Feigin for useful discussions. This work was supported in part by a grant from The Royal Swedish Academy of Sciences. AMS and IYuT were also supported in part by the RFBR (grants 01-01-00906 and LSS-1578.2003.2), INTAS (grant 00-00262), and the Foundation for Support of Russian Science. JF and SH are supported in part by VR under contracts no. F 5102 – 2000-5368 and 621 – 2002-4226.
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