Communications in Mathematical Physics - Volume 190

Commun. Math. Phys. 190, 1 – 56 (1997)

Communications in

Mathematical Physics c Springer-Verlag 1997

Twisted Wess-Zumino-Witten Models on Elliptic Curves Gen Kuroki1 , Takashi Takebe2 1

Mathematical Institute, Tohoku University, Sendai 980, Japan Department of Mathematical Sciences, University of Tokyo, Komaba, Tokyo 153, Japan. E-mail: [email protected]

2

Received: 21 January 1997/ Accepted: 1 April 1997

Abstract: Investigated is a variant of the Wess-Zumino-Witten model called a twisted WZW model, which is associated to a certain Lie group bundle on a family of elliptic curves. The Lie group bundle is a non-trivial bundle with flat connection and related to the classical elliptic r-matrix. (The usual (non-twisted) WZW model is associated to a trivial group bundle with trivial connection on a family of compact Riemann surfaces and a family of its principal bundles.) The twisted WZW model on a fixed elliptic curve at the critical level describes the XYZ Gaudin model. The elliptic Knizhnik-Zamolodchikov equations associated to the classical elliptic r-matrix appear as flat connections on the sheaves of conformal blocks in the twisted WZW model. Contents 0 1 1.1 1.2 1.3 1.4 2 3 3.1 3.2 3.3 4 4.1 4.2 5 5.1 5.2

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Spaces of Conformal Coinvariants and Conformal Blocks . . . . . . . . . . . Group bundles and their associated Lie algebra bundles . . . . . . . . . . . . . Definition of the spaces of conformal coinvariants and conformal blocks Correlation functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Action of the Virasoro algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Critical Level and the XYZ Gaudin Model . . . . . . . . . . . . . . . . . . . . . . . Sheaves of Conformal Coinvariants and Conformal Blocks . . . . . . . . . . Family of pointed elliptic curves and Lie algebra bundles . . . . . . . . . . . Sheaf of affine Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Definition of the sheaves of conformal coinvariants and conformal blocks Sheaf of the Virasoro Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Definition of the sheaf of the Virasoro algebras . . . . . . . . . . . . . . . . . . . Action of the sheaf of Virasoro algebras . . . . . . . . . . . . . . . . . . . . . . . . . Flat Connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Construction of flat connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Elliptic Knizhnik-Zamolodchikov equations . . . . . . . . . . . . . . . . . . . . . .

2 5 5 7 10 15 16 20 20 22 23 26 26 29 35 35 38

2

G. Kuroki, T. Takebe

5.3 Modular invariance of the flat connections . . . . . . . . . . . . . . . . . . . . . . . 6 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix A. Theta functions with characteristics . . . . . . . . . . . . . . . . . . . . Appendix B. The Kodaira-Spencer map of a family of Riemann surfaces . . Appendix C. On a formulation for higher genus Riemann surfaces . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41 47 48 50 51 54

0. Introduction In this paper, we deal with a variant of the (chiral) Wess-Zumino-Witten model (WZW model, for short) on elliptic curves, which shall be called a twisted WZW model. The usual (non-twisted) WZW model on a compact Riemann surface X gives rise to the sheaves of vector spaces (of conformal blocks and of conformal coinvariants) on any family (or the moduli stack) of principal G-bundles, where G is a semisimple complex algebraic group. Note that the notion of principal G-bundles is equivalent to that of Gnt -torsors, where Gnt denotes the trivial group bundle G × X on X. (The symbol (·)nt stands for “non-twisted”.) This suggests that there exists a model associated to a nontrivial group bundle Gtw with a flat connection on X, which gives sheaves of conformal blocks and conformal coinvariants on a family of Gtw -torsors. (The symbol (·)tw stands for “twisted”.) We call such a model a twisted WZW model associated to Gtw . The aim of this work is not to establish a general theory of the twisted WZW models but to describe certain interesting examples of the twisted WZW models related to the elliptic classical r-matrices ([BelD]). In this introduction, we explain our motivations and clarify the relationship between the twisted WZW models and various problems in mathematics and physics. One of the motivations is the viewpoint of representation theory where the WZW model is formulated as an analogue of a theory of automorphic forms due to Langlands. We list corresponding ingredients of both theories in Table 1. Notations shall be explained in the main part of this paper. There is a theory of automorphic forms for arbitrary (possibly non-split) reductive groups over a global field as well as over a local field. But so far only the WZW model associated to the trivial group bundle has been considered and the counterpart of the non-split reductive group over a global field has been absent. The twisted WZW models fills this blank. The second motivation comes from the geometric Langlands program over C and its relation with quantum integrable systems. A geometric analogue of the Langlands correspondence over C is described by using the WZW model at the critical level, where the centers of the completed enveloping algebras of affine Lie algebras are sufficiently large so that we can consider analogues of the infinitesimal characters of finite-dimensional semisimple Lie algebras ([Hay, GW]). For introduction to the original Langlands program we refer to [Bo] and [Ge]. For a general formulation of the geometric Langlands correspondence over C related to the non-twisted WZW models at the critical level, see [Bei] and [BeiD] and for the analogue of the local Langlands correspondence to affine Lie algebras at the critical level, see [FF3 and Fr1]. The twisted WZW model at the critical level shall give a geometric analogue of the Langlands correspondence for a non-split reductive group over a global field. To study this model at the critical level is important not only in this context of the geometric Langlands program but also in the theory of the quantum integrable spin chains. B. Feigin, E. Frenkel and N. Reshetikhin found in [FFR] that the non-twisted

Twisted WZW Models on Elliptic Curves

3

Table 1. Analogy between automorphic forms and conformal blocks Theory of automorphic forms

Theory of the WZW models

a global field, i.e., a number field or the function field of an algebraic curve over a finite field

the function field of a compact Riemann surface X

a local field

C((ξ)), a field of formal Laurent series

a reductive group over the global field

a semisimple group bundle on X with flat connection or the associated Lie algebra bundle gtw

a non-split reductive group over the global field

a semisimple group bundle with flat connection on X which is not locally trivial under the Zariski topology

the ad`ele group associated to the reductive group

the affine Lie algebra LL (g⊕L )∧ = g ⊗ C((ξi )) ⊕ Ckˆ i=1

the principal ad`ele subgroup of the ad`ele group

the subalgebra gD˙ X (g⊕L )∧

a unitary representation of the ad`ele group

a representation M of (g⊕L )∧ or its algebraic dual M ∗

the space of automorphic forms in the representation space, i.e., the invariant subspace of the representation space with respect to the principal ad`ele subgroup

the space CB(M ) of conformal blocks, i.e., the invariant subspace of M ∗ with respect to gD ˙ X

=

H 0 (X, gtw (∗D)) of

WZW model at the critical level on the Riemann sphere is closely related to a spin chain model called the Gaudin model. See also [Fr2]. Its Hamiltonian is described as an insertion of a singular vector of the vacuum representation at a point and the diagonalization problem turns out to be equivalent to a description of a certain space of conformal blocks. This “Gaudin” model is, however, merely a special case of the model introduced by M. Gaudin [Ga1, Ga2, Ga3] as a quasi-classical limit of the XYZ spin chain model. Let us call this general model the XYZ Gaudin model, following [ST1] where the diagonalization problem of this model was studied by the algebraic Bethe Ansatz. In order to extend the results of [FFR], we need the twisted WZW model on a elliptic curve at the critical level as we shall see in Sect. 2. We remark that the non-twisted WZW models at the critical level on an elliptic curve is related to quantum integrable systems on root systems. In fact those systems defined by the trigonometric (dynamical) r-matrices are described by the non-twisted WZW model on a degenerated elliptic curve with only one ordinary double point. The nontwisted WZW model at the critical level on an elliptic curve leads to a system called the Gaudin-Calogero model ([ER, N]) which was defined as Hitchin’s classical integrable system ([Hi]) on the moduli space of semistable principal bundles on an elliptic curve.1 The reason why root systems appear in the non-twisted WZW models is explained as follows. Let G be a complex semisimple group and T its maximal torus. Let a and b denote generators of the fundamental group π1 (X) of an elliptic curve X. Then, for g ∈ T , the homomorphism from π1 (X) into G sending a and b to 1 and g respectively induces a semistable principal G-bundle on X. This defines the covering by T of the moduli space of semistable G-bundles on X. Furthermore the universal covering of T is identified with its Lie algebra, on which the root system structure exists. Namely, the 1 This relation of the non-twisted WZW model and the Gaudin-Calogero model is due to B. Enriquez and A. Stoyanovsky. T.T. thanks Enriquez for communicating their unpublished result.

4

G. Kuroki, T. Takebe

root system appears as a covering space of the moduli space of semistable principal G-bundles on X. The third motivation is a geometric interpretation of Etingof’s elliptic KZ equations. As is well-known, the Knizhnik-Zamolodchikov equation is a system of differential equations satisfied by matrix elements of products of vertex operators ([KZ, TK]) and is a flat connection over the family of pointed Riemann spheres. Similarly from the nontwisted WZW model over elliptic curves arises the elliptic Knizhnik-ZamolodchikovBernard equations (KZB equations, for short), which Bernard found in [Be1] by computing traces of products of vertex operators twisted by g ∈ G. The interpretation of the elliptic KZB equations as flat connections on sheaves of conformal blocks, which are defined without use of the traces, was found in [FW]. Using the same idea as [Be1], Etingof computed in [E] a twisted trace of a product of vertex operators and found that it obeys linear differential equations of KZ type defined by the elliptic classical r-matrices. We call these equations the elliptic KZ equations. In the present paper it is shown that the elliptic KZ equations also has an interpretation as flat connections on sheaves of conformal blocks. Let us explain now the contents of this paper. In Sect. 1, we give a definition of the conformal coinvariants and the conformal blocks of the twisted WZW model on an elliptic curve. The definition of the non-trivial group bundle Gtw (1.3) and the associated Lie algebra bundle gtw (1.4) is given in Sect. 1.1 and their fundamental properties are studied. This bundle gtw was used by I. Cherednik [C] for an algebro-geometric interpretation of classical elliptic r-matrices. An important point is that the cohomology groups of gtw vanish in all degrees. Since the 1-cohomology H 1 (X, gtw ) can be canonically identified with the tangent space of the moduli space of Gtw -torsors at the equivalence class consisting of trivial ones, the trivial Gtw -torsor can not be deformed. Thus non-trivial Gtw -torsors do not appear in the twisted WZW model associated to Gtw . Conformal coinvariants and conformal blocks of this model are defined in Sect. 1.2. We introduce correlation functions of current and the energy-momentum tensor in Sect. 1.3, following mostly [TUY]. An action of the Virasoro algebra on the conformal coinvariants and the conformal blocks is defined in Sect 1.4. This model at the critical level for G = SL2 (C) describes the XYZ Gaudin model and the case for G = SLN (C) is related to the higher rank generalizations of the XYZ Gaudin model, as shown in Sect. 2. Away from the critical level, we can define a connection on the sheaves of conformal coinvariants and blocks over the family of pointed elliptic curves. Sect. 3 and Sect. 4 are a sheaf version of Sect. 1 over a family of pointed elliptic curves. By extending the tangent sheaf of the base space of the family (4.3) and constructing its action on the sheaves of conformal coinvariants and conformal blocks, we can introduce the D-module structure on them in Sect. 5, which naturally implies the existence of flat connections. The explicit formulae in Sct. 5.2 show that our connections are identical with Etingof’s elliptic KZ equations. This connection has modular invariance which Etingof proved by his explicit expressions of the equations. We give a geometric proof of this fact in Sect. 5.3. Useful properties of theta functions are listed in Appendix A. An algebro-geometric meaning of the extension (4.3) is explained in Appendix B. Higher-genus generalization of the theory is discussed in Appendix C.

Twisted WZW Models on Elliptic Curves

5

1. Spaces of Conformal Coinvariants and Conformal Blocks In this section we define the space of conformal coinvariants and conformal blocks associated to a twisted Group bundle. 1.1. Group bundles and their associated Lie algebra bundles. In this section we define a group bundle Gtw and an associated Lie algebra bundle gtw on an elliptic curve X = Xτ = C/(Z + τ Z), where τ belongs to the upper half plane H := {Im τ > 0}. We fix a global coordinate t on X which comes from that of C. Let G be the Lie group SLN (C) and g be its Lie algebra, slN (C) = {A ∈ MN (C) | tr A = 0}. We fix an invariant inner product of g by (A|B) := tr(AB) Define matrices α and β by 0 1 .  0 ..  α :=  .. . 1

0  ,  1 0

for A, B ∈ g. 

 β :=  

(1.1)

1

0 ε ..

0

.

  , 

(1.2)

N −1

ε

where ε = exp(2πi/N ). Then we have αN = β N = 1 and αβ = εβα. We define the group bundle Gtw and its associated Lie algebra bundle gtw by Gtw := (C × G)/∼, gtw := (C × g)/≈,

(1.3) (1.4)

where the equivalence relations ∼ and ≈ are defined by (t, g) ∼ (t + 1, αgα−1 ) ∼ (t + τ, βgβ −1 ), (t, A) ≈ (t + 1, αAα−1 ) ≈ (t + τ, βAβ −1 ).

(1.5) (1.6)

(Because of αβ = εβα, the group bundle Gtw is not a principal bundle.) The fibers of Gtw are isomorphic to G and those of gtw are isomorphic to g, but there are not canonical isomorphisms. The twisted Lie algebra bundle gtw has a natural connection, ∇d/dt = d/dt, and is decomposed into a direct sum of line bundles: M La,b , (1.7) gtw ∼ = (a,b)6=(0,0)

where the indices (a, b) runs through (Z/N Z)N r {(0, 0)}. Here the line bundle La,b on X is defined by (1.8) La,b := (C × C)/≈a,b , where ≈a,b is an equivalence relation defined by (t, x) ≈a,b (t + 1, εa x) ≈a,b (t + τ, εb x). We regard La,b as a line subbundle of gtw through the injection given by

(1.9)

6

G. Kuroki, T. Takebe

La,b 3 (t, x) 7→ (t, xJa,b ) ∈ gtw ,

(1.10)

where Ja,b is the element of g defined by Ja,b := β a α−b .

(1.11)

We remark that { Ja,b | (a, b) ∈ (Z/N Z)N r {(0, 0)} } is a basis of g = slN (C). The space of meromorphic sections of La,b over X pulled back to C can be canonically identified with Ka,b = { f ∈ M(C) | f (t + 1) = εa f (t), f (t + τ ) = εb f (t) }.

(1.12)

Here M(C) is the space of meromorphic functions on C. (K0,0 is the space of elliptic functions and corresponds to the trivial line bundle on X.) The mapping f 7→ (t, f (t)) modulo ≈a,b

(1.13)

gives a canonical isomorphism from Ka,b onto H (X, La,b ⊗ KX ). The Liouville theorem implies that the only holomorphic function in Ka,b is zero when (a, b) 6= (0, 0). This is equivalent to H 0 (X, La,b ) = 0. Since L∗a,b ∼ = L−a,−b and the canonical line bundle of X is trivial, it follows from the Serre duality that H 1 (X, La,b ) = 0. Thus we obtain a simple vanishing result H p (X, La,b ) = 0 and therefore from the decomposition (1.7) we obtain the following result. 0

Lemma 1.1. H 0 (X, gtw ) = H 1 (X, gtw ) = 0. Example 1.2. For N = 2, matrices α and β are nothing but the Pauli matrices σ 1 and σ 3 . The Jacobian elliptic functions sn, cn, and dn are meromorphic functions in K1,0 , K1,1 , and K0,1 and can be regarded as meromorphic sections of the line bundles L1,0 , L1,1 , and L0,1 respectively. Example 1.3. For general N and each (a, b) ∈ (Z/N Z)2 r {(0, 0)}, we define the function wa,b by 0 θ[0,0] θ[a,b] (t; τ ) . (1.14) wa,b (t) = wa,b (τ ; t) := θ[a,b] θ[0,0] (t; τ ) (See (A.8) in Appendix A for the notation.) The function wa,b (t) on C is uniquely characterized by the following properties: 1. The function wa,b (t) is a meromorphic function in Ka,b and hence can be regarded as a global meromorphic section of La,b ; 2. The poles of wa,b (t) are all simple and contained in Z + Zτ ; 3. The residue of wa,b (t) at t = 0 is equal to 1. Because of these properties, it will play an important role in concrete computations in later sections. For convenience of those computations, let us list several other properties of wa,b (t): • The Laurent expansion at t = 0 is equal to ∞

wa,b (t) =

1 X 1 + wa,b,ν tν = + wa,b,0 + wa,b,1 t + · · · , t t

(1.15)

ν=0

where the coefficients are written in the following forms: wa,b,0 =

0 θ[a,b] , θ[a,b]

wa,b,1 =

00 000 θ[a,b] θ[0,0] − 0 , 2θ[a,b] 6θ[0,0]

... .

(1.16)

Twisted WZW Models on Elliptic Curves

7

• Formulae (A.6) and (A.7) imply wa,b (t) = wa0 ,b0 (t) if a ≡ a0 and b ≡ b0 mod N . (1.17) • Lemma A.1 will be used in the following form: X wa,b,1 = 0, (1.18) w−a,−b (t) = −wa,b (−t),

(a,b)6=(0,0)

where the summation is taken over all (a, b) ∈ (Z/N Z)2 r {(0, 0)}. 1.2. Definition of the spaces of conformal coinvariants and conformal blocks. In this section we define a conformal block associated to the twisted Lie algebra bundle gtw defined in Sect. 1.1. First let us introduce notation of sheaves. As usual, the structure sheaf on X = Xτ is denoted by OX and the sheaf of meromorphic functions on X by KX . A stalk of a sheaf F on X at a point P ∈ X is denoted by FP . When F is a OX -module, we denote its fiber FP /mP FP by F|P , where mP is the maximal ideal of the local ring OX,P . Denote by FP∧ the mP -adic completion of FP . We shall use the same symbol for a vector bundle and for a locally free coherent OX -module consisting of its local holomorphic sections. For instance, the invertible sheaf associated to the line bundle La,b is also denoted by the same symbol La,b . Denote by 1X the sheaf of holomorphic 1-forms on X, which is isomorphic to OX since X is an elliptic curve. The fiberwise Lie algebra structure of the bundle gtw induces that of the associated sheaf gtw over OX . Define the invariant OX -inner product on gtw by (A|B) :=

1 tr (ad A ad B) ∈ OX 2N gtw

for A, B ∈ gtw ,

(1.19)

where the symbol ad denotes the adjoint representation of the OX -Lie algebra gtw . Then the inner product on gtw is invariant under the translations with respect to the connection ∇ : gtw → gtw ⊗OX 1X : d(A|B) = (∇A|B) + (A|∇B) ∈ 1X

for A, B ∈ gtw .

(1.20)

Under the trivialization of gtw defined by the construction (1.4), the connection ∇ and the inner product (·|·) on gtw respectively coincide with the exterior derivative by t and the inner product defined by (1.1). For any point P on X with t(P ) = z, we put gP := (gtw ⊗OX KX )∧ P, which is a topological Lie algebra non-canonically isomorphic to the loop algebra g((t− tw ∧ ∼ z)). Its subspace gP + := (g )P = g[[t − z]] is a maximal linearly compact subalgebra P of g under the (t − z)-adic linear topology. Let us fix mutually distinct points Q1 , . . . , QL on X whose coordinates are t = z1 , . . . , zL and put D := {Q1 , . . . , QL }. We shall also regard D as a divisor on X (i.e., L ˙ The Lie algebra gD := L gQi has the D = Q1 + · · · + QL ). Denote X r D by X. i=1 natural 2-cocycle defined by ca (A, B) :=

L X i=1

Res(∇Ai |Bi ), t=zi

(1.21)

8

G. Kuroki, T. Takebe

L D where A = (Ai )L i=1 , B = (Bi )i=1 ∈ g and Rest=z is the residue at t = z. (The symbol “ca ” stands for “Cocycle defining the Affine Lie algebra”.) We denote the central extension of gD with respect to this cocycle by gˆ D :

ˆ gˆ D := gD ⊕ Ck, where kˆ is a central element. Explicitly the bracket of gˆ D is represented as ˆ [A, B] = ([Ai , Bi ]0 )L i=1 ⊕ ca (A, B)k

for A, B ∈ gD ,

(1.22)

where [Ai , Bi ]0 are the natural bracket in gQi . The Lie algebra gˆ P for a point P is noncanonically isomorphic to the affine Lie algebra gˆ of type A(1) N −1 (a central extension
of the loop algebra g((t − z)) = slN C((t − z)) ). If P = Qi for i = 1, . . . , L, then Qi tw ∧ gˆ P = gˆ Qi can be regarded as a subalgebra of gˆ D . Put gP + := (g )P as above. Then g+ Q D can be also regarded as a subalgebra of gˆ i and gˆ . tw Let gD ˙ be the space of global meromorphic sections of g which are holomorphic X ˙ on X: tw gD ˙ := 0(X, g (∗D)). X D There is a natural linear map from gD ˙ into g which maps a meromorphic section of X tw g to its germ at Qi ’s. As in the non-twisted case (e.g., Sect. 2.2 of [TUY]), the residue theorem implies that this linear map is extended to a Lie algebra injection from gD ˙ into X D ˆ gˆ D , which allows us to regard gD as a subalgebra of g . ˙ X

Definition 1.4. The space of conformal coinvariants CC(M ) and that of conformal blocks CB(M ) associated to gˆ Qi -modules Mi with the same level kˆ = k are defined to NL be the space of coinvariants of M := i=1 Mi with respect to gD ˙ and its dual: X CC(M ) := M/gD ˙ M, X

∗ CB(M ) := (M/gD ˙ M) . X

(1.23)

(In [TUY], CC(M ) and CB(M ) are called the space of covacua and that of vacua respectively.) In other words, the space of conformal coinvariants CC(M ) is generated by M with relations (1.24) AX˙ v ≡ 0 for all AX˙ ∈ gD ˙ and v ∈ M , and a linear functional 8 on M belongs to the space of X conformal blocks CB(M ) if and only if it satisfies that 8(AX˙ v) = 0

for all AX˙ ∈ gD ˙ and v ∈ M . X

(1.25)

These Eqs. (1.24) and (1.25) are called the Ward identities. The most important conformal blocks for our purpose are constructed from Weyl modules (or generalized Verma modules) which are determined from the following data: – A parameter k ∈ C which is called the level of the model. – Finite-dimensional irreducible representations Vi of the fiber Lie algebra gtw |Qi isomorphic to g.

Twisted WZW Models on Elliptic Curves

9

i i i ˆ ˆQ ˆ Qi . The Put gQ := (gtw )∧ := gQ + + + ⊕ Ck, which are subalgebras of g Qi and g Qi Qi Q i subalgebra gˆ + := g+ ⊕ Ckˆ of gˆ acts on Vi through the linear map kˆ 7→ k idVi and i tw ˆ Qi -module induced from Vi is the natural projection gQ + → g |Qi , A 7→ A(Qi ). The g called a Weyl module or a generalized Verma module:

gˆ Qi

Mk (Vi ) := Ind Vi = U (ˆgQi ) ⊗U (gˆ Qi ) Vi Qi

(1.26)

+

gˆ +

See [KL] for properties of Weyl modules. The space of conformal coinvariants and that of conformal blocks associated to the data (Q, V ) = ({Qi }, {Vi }) are defined to be the space of conformal coinvariants and that of conformal blocks associated to the gˆ D -module Mk (V ) :=

L O

Mk (Vi ),

i=1

on which the center kˆ acts as multiplication by k. Namely we define them as follows: CCk (Q, V ) = CCk ({Qi }, {Vi }) := CC(Mk (V )) = Mk (V )/gD ˙ Mk (V ), X CBk (Q, V ) = CBk ({Qi }, {Vi }) := CB(Mk (V )) = (Mk (V

)/gD ˙ Mk (V X

(1.27)

∗

)) . (1.28)

Hereafter we use the word “conformal coinvariants” and “conformal block” for this kind of conformal coinvariants and conformal blocks, namely those associated to Weyl modules, unless otherwise stated. It is easy to see that the spaces of conformal coinvariants and conformal blocks are NL determined by the finite-dimensional part V = i=1 Vi of Mk (V ) as is the case with the space of conformal coinvariants and conformal blocks on P1 (C) (e.g., Lemma 1 of [FFR]), because of the cohomology vanishing. In fact, Lemma 1.1 implies a decomposition, ˆD (1.29) gˆ D = gD ˙ ⊕g + , X L L Q i D ˆ where gˆ D ˙ -modules, + = i=1 g+ ⊕ Ck. Hence we have, as left gX gˆ D

V = U (gD Mk (V ) = Ind V = U (ˆgD ) ⊗U (gˆ D ˙ ) ⊗C V, X + )

(1.30)

gˆ D +

NL ˆ where V := action of gˆ D + is defined by the mapping k 7→ k · id and i=1 Vi and the Q L D tw the natural projection g+ → i=1 (g |Qi ). Therefore, due to the Ward identity (1.25) and the definition of the Weyl module (1.26), the space of conformal coinvariants is canonically isomorphic to the tensor product of gtw |Qi -modules by the natural inclusion NL NL NL map V = i=1 Vi ,→ Mk (V ), i=1 vi 7→ i=1 (1 ⊗ vi ). (In the following we shall identify vi ∈ Vi with 1 ⊗ vi ∈ Mk (Vi ).) Proposition 1.5. The inclusion map V ,→ Mk (V ) and the induced restriction map Mk (V )∗  V ∗ induce the following isomorphisms respectively: ∼

CCk (Q, V ) ← V =

L O i=1

Vi

and

∼

CBk (Q, V ) → V ∗ =

L O i=1

Vi∗ .

10

G. Kuroki, T. Takebe

For any point P ∈ X = Xτ , let us denote the 1-dimensional trivial representation of gtw |P by CP = CuP . Then the proposition above readily leads to the following corollary. Corollary 1.6. Let P be a point of X distinct from Qi ’s. Then the canonical inclusion Mk (V ) ,→ Mk (CP ) ⊗ Mk (V ), v 7→ uP ⊗ v, induces an isomorphism ∼

CBk ({P, Qi }, {CP , Vi }) → CBk ({Qi }, {Vi }).

(1.31)

The property above is called propagation of vacua in [TUY]. In our case the proof is far simpler due to Proposition 1.5, as in the case of P1 (cf. Sect. 3 of [FFR].) 1.3. Correlation functions. The current and the energy-momentum correlation functions are defined as in Sect. 2 of [TUY], but we must take twisting into account and use the decomposition (1.7). First we consider the current correlation functions. Let 8 be a conformal block in CBk (Q, V ) and v a vector in Mk (V ). There exists a unique ωi ∈ (L∗a,b ⊗OX KX ⊗OX 1X )∧ Qi with the property that Rest=zi hfi , ωi i := 8(ρi (fi Ja,b )v)

for all fi ∈ (KX )∧ Qi ,

(1.32)

∗ ∧ where h·, ·i is the canonical pairing of (La,b )∧ P and (La,b )P , fi Ja,b can be regarded as an element of gQi by means of (1.10) and its action on the ith component of v (namely, the Mk (Vi )-component) is denoted by ρi (fi Ja,b ). Thus we obtain a linear functional L X i=1

ResQi h·, ωi i :

L M

(La,b ⊗OX KX )∧ Qi → C

i=1

∧ which maps (fi )L i=1 ∈ (La,b ⊗OX KX )Qi to The Ward identity (1.25) implies that L X i=1

ResQi hfQi , ωi i =

PL

L X

i=1

ResQi hfi , ωi i ∈ C.

8(ρi (fQi Ja,b )v) = 0

i=1

for any meromorphic section f ∈ H 0 (X, La,b (∗D)), where fQi is the germ of f at Qi . Since H 0 (X, La,b (∗D)) and H 0 (X, L∗a,b ⊗OX 1X (∗D)) are orthogonal complements LL LL ∗ to each other under the residue pairing of i=1 (La,b ⊗OX KX )∧ Qi and i=1 (La,b ⊗OX 1 ∧ KX ⊗ X )Qi (cf. [Tat] or Theorem 2.20 of [I]), we have a meromorphic 1-form ω with values in L∗a,b such that the germ of ω at Qi gives ωi and is holomorphic outside of {Q1 , . . . , QL }: ω ∈ H 0 (X, L∗a,b ⊗OX 1X (∗D)),

(ω)Qi = ωi .

(1.33)

In order to define the correlation functions, we need explicit expression of the action of (gtw )∧ P which we can identify with the affine Lie algebra by fixing a trivialization of gtw around P . Let P be any point of X and z(P ) a point of C whose image in X = C/(Z + τ Z) is equal to P . The description (1.4) of gtw naturally determines a local trivialization of gtw at P , once we fix the coordinate t = z(P ) of P . By means of this trivialization, we fix

Twisted WZW Models on Elliptic Curves

11

isomorphisms gP ∼ = gˆ , gtw |P ∼ = g, and so on. The induced trivialization of La,b at P is the same as the trivialization defined by the isomorphism (1.13): ∼

∼

(La,b )P → OC,z(P ) ← OX,P

for (a, b) ∈ (Z/N Z)2 ,

(1.34)

which corresponds ∇-flat sections of La,b to constant functions on X. The decomposition (1.7) of gtw induces a decomposition of its stalk at P and is consistent with the trivializations above: M M Ja,b (La,b )P ∼ (1.35) Ja,b OX,P = g ⊗ OX,P , gtw = P = where the indices (a, b) run through (Z/N Z)2 r {(0, 0)}. Let ξ be a local coordinate at P . For A ∈ g and m ∈ Z, we denote by A[m] the ˆ P which is represented by Aξ m under the trivialization (1.35). element of (gtw )∧ P ⊂ g Since A[0] is ∇-flat for A ∈ g (i.e., ∇A[0] = 0), the bracket of gˆ P is represented as: [A[m], B[n]] = [A, B][m + n] + (A|B)mδm+n,0 kˆ

for A, B ∈ g and m, n ∈ Z,

which coincides with the usual commutation relation of the affine Lie algebra. ˜ Lemma 1.7. Under the situation above, let P be in X˙ (i.e., distinct from Qi ’s) and 8 the conformal block in CBk ({P, Qi }, {CP , Vi }) corresponding to 8 by the isomorphism (1.31). Then we have the following: (i) Take x ∈ La,b |P and let fx be an element of (La,b )∧ P with a principal part x/ξ (i.e., ˜ x Ja,b uP ⊗ v) dξ does not depend on the choice of ξ and fx = (x/ξ + regular)). Then 8(f ˜ a,b [−1]uP ⊗ v) dξ ∈ (L∗a,b ⊗OX 1X )|P by fx . Thus we can define 8(J ˜ a,b [−1]uP ⊗ v)i dξ := 8(f ˜ x Ja,b uP ⊗ v) dξ. hx, 8(J (ii) The following equation holds at any point P in X˙ = X r {Q1 , . . . , QL }: ˜ a,b [−1]uP ⊗ v) dξ ∈ (L∗a,b ⊗OX 1X )|P . ω(P ) = 8(J

(1.36)

Proof. The statement (i) can be shown by the same argument as the proof of Claim 1 of Theorem 2.4.1 of [TUY]. Using the Riemann-Roch theorem (or the function wa,b (t) defined by (1.14)), we can choose fx in (i) from H 0 (X, La,b (∗D + P )). Then we have ˜ x Ja,b uP ⊗ v) dξ ˜ a,b [−1]uP ⊗ v)i dξ = 8(f hx, 8(J =−

L X

˜ P ⊗ ρi (fx Ja,b )v) dξ 8(u

i=1

=−

L X

8(ρi (fx Ja,b )v) dξ = −

i=1

= ResP hfx , ωi dξ = ResP

L X

ResQi hfx , ωi dξ

i=1

hx, ωi dξ = hx, ω|P i. ξ

Here we have used the Ward identity (1.25) and the residue theorem. Thus we have proved Eq. (1.36). 

12

G. Kuroki, T. Takebe

˜ a,b [−1]uP ⊗ v) dξ a correlation function Definition 1.8. We call this 1-form ω = 8(J of the current Ja,b (ξ) and v under 8, or a current correlation function for short, and denote it by 8(Ja,b (P )v) dP or 8(Ja,b (ξ)v) dξ when we fix a local coordinate ξ. We now proceed to the definition of the energy-momentum correlation functions. Lemma 1.9. Let P be in X˙ and ξ a local coordinate defined on an open neighborhood U of P . Then the following expression gives a holomorphic section of 2X (∗D) = (1X )⊗2 (∗D) on sufficiently small U : X 1 lim 8(Ja,b (ξ(P )) J a,b (ξ(P 0 ))v) 8(S(P )v) (dξ(P ))2 := 2 P 0 →P (a,b) (1.37)  k dim g dξ(P ) dξ(P 0 ). − (ξ(P ) − ξ(P 0 ))2 Here the indices (a, b) run through (Z/N Z)2 r {(0, 0)}, and J a,b is the dual basis of Ja,b with respect to (·|·), namely, J a,b = ε−ab J−a,−b /N . Proof. The argument in the proof of Lemma 1.7 and the Hartogs theorem of holomorphy show that the current correlation function 8(Ja,b (ξ)J a,b (ζ)v) dξ dζ defines a global section on X ×X of the sheaf F := (L∗a,b ⊗OX 1X (∗D))(L∗−a,−b ⊗OX 1X (∗D))(∗1), where 1 is the diagonal divisor of X × X. We define 8(S(P, P 0 )v) dξ(P ) dξ(P 0 ) ∈ H 0 (U × U, F ) by 8(S(P, P 0 )v) dξ(P ) dξ(P 0 )  X 8(Ja,b (ξ(P )) J a,b (ξ(P 0 ))v) − :=  a,b

 k dim g  dξ(P ) dξ(P 0 ). (ξ(P ) − ξ(P 0 ))2

In order to show Lemma 1.9, first take a local coordinate ξ on a sufficiently small neighborhood U of P with ξ(P ) = 0 and a local trivialization of La,b on U and choose a meromorphic section f ∈ H 0 (X, La,b ⊗OX KX ) whose Laurent expansion has the form f (ξ) = ξ −1 + (regular at ξ = 0). (1.38) The inclusion (1.10) and the Ward identity (1.25) imply that 8(Ja,b (ξ(P ))J a,b (ξ(P 0 ))v) is equal to a,b ˜ 8(J [−1]uP 0 ⊗ v) a,b [−1]uP ⊗ J

˜ = −8((f Ja,b )P 0 J a,b [−1]uP 0 ⊗ v) −

L X i=1

˜ a,b [−1]uP 0 ⊗ ρi ((f Ja,b )Qi )v), 8(J

(1.39) ˜ ∈ CB ({P, P 0 , Q }, {C , C 0 , V }), and 8 0 ˜ 0 where 8 ∈ CB ({P , Q }, {C , V }) cork i P P i k i P i respond to 8 through the isomorphism (1.31). The second term of (1.39) is regular as a function of P at P 0 as shown in Lemma 1.7 and the first term is rewritten as

Twisted WZW Models on Elliptic Curves

˜ − 8((f Ja,b )P 0 J a,b [−1]uP 0 ⊗v) =

13

k 8(v)+(regular at P 0 = P ). (1.40) (ξ(P ) − ξ(P 0 ))2

(Details of the computation is the same as that of the proof of the assertion (4) of Theorem 2.4.1 of [TUY]. Note that [Ja,b , J a,b ] = 0.) Equations (1.39) and (1.40) mean that 8(S(P, P 0 )v) dξ(P ) dξ(P 0 ) is a holomorphic section of (L∗a,b ⊗OX 1X (∗D))  (L∗−a,−b ⊗OX 1X (∗D)) on U × U for the coordinate ξ that satisfies (1.38). Restricting it to the diagonal of U × U , we obtain a local holomorphic section 8(S(P )v) (dξ(P ))2 ∈ H 0 (U, 2X (∗D)). Note that since L∗a,b = L−a,−b , the factors L∗a,b and L∗−a,−b cancel out on the diagonal. Thanks to this fact, the trivialization of La,b ’s which we implicitly fixed in the argument above does not affect the result.  Definition 1.10. Put κ := k + h∨ , where h∨ is the dual Coxeter number of g (i.e., h∨ = N because g = slN ). We call 8(S(ξ)v) (dξ)2 in (1.37) a correlation function of the Sugawara tensor S(ξ) and v under 8, or a Sugawara correlation function for short, and (1.41) 8(T (ξ)v) (dξ)2 := κ−1 8(S(ξ)v) (dξ)2 a correlation function of the energy-momentum tensor T (ξ) and v under 8, or a energymomentum correlation function for short when κ 6= 0. Let us calculate the coordinate transformation law of the correlation function of the Sugawara tensor. Let ζ be another coordinate on U . The differential dξ(P )dξ(P 0 )/(ξ(P )− ξ(P 0 ))2 transforms under the coordinate change ξ 7→ ζ = ζ(ξ) as dξ(P ) dξ(P 0 ) {ζ, ξ}(P 0 ) dζ(P ) dζ(P 0 ) dξ(P ) dξ(P 0 ) + O(ξ(P ) − ξ(P 0 )), = + (ζ(P ) − ζ(P 0 ))2 (ξ(P ) − ξ(P 0 ))2 6 (1.42) where {ζ, ξ} = ζ 000 /ζ 0 − 3/2(ζ 00 /ζ 0 )2 (ζ 0 = dζ/dξ) is the Schwarzian derivative of ζ = ζ(P ) with respect to ξ = ξ(P ). Hence the correlation function of the Sugawara tensor transforms with respect to a coordinate change ξ 7→ ζ = ζ(ξ) not as 2-differentials but as k dim g {ζ, ξ}8(v). (1.43) 8(S(ζ)v)dζ 2 = 8(S(ξ)v)dξ 2 + 12 This means that the family {8(S(ξ)v)dξ 2 } defines a meromorphic projective connection on X. (For the notion of projective connections on Riemann surfaces, see [Gu].) The Schwarzian derivative {ζ, ξ} vanishes identically if and only if ζ is a fractional linear transformation of ξ (i.e., ζ = (aξ + b)/(cξ + d)). From this fact it follows that {8(S(ξ)v) dξ 2 } behaves like a 2-differential under fractional linear coordinate changes. For later use, we compute the local expression of the energy-momentum correlation function around Qi . We take a holomorphic local chart (U, ζ) with Qi ∈ U and ζ(Qi ) = 0, a local trivialization of La,b by (1.34), and a trivialization of gtw Qi by (1.35). Under these trivializations, we have, due to (1.33), 8(Ja,b (P1 )J a,b (P2 )v) dP1 dP2 X L X = ζ1−m−1 ζ2−n+m−1 8(ρi ( ◦◦ J a,b [n − m]Ja,b [m] ◦◦ )v) i=1 n,m∈Z

+

k8(v) (ζ2 − ζ1 )2

 dζ1 dζ2

(1.44)

14

G. Kuroki, T. Takebe

if |ζ1 | > |ζ2 |. Here P1 , P2 ∈ U , ζ1 = ζ(P1 ), ζ2 = ζ(P2 ), and ◦◦ ◦◦ denotes the normal ordered product defined by ( A[m]B[n], if m < n, ◦ ◦ 1 (A[m]B[n] + B[n]A[m]), if m = n, A[m]B[n] = (1.45) 2 ◦ ◦ B[n]A[m], if m > n. Using (1.44), we obtain an expression of the correlation function (1.37) around the point Qi : X 8(S(ζ)v)(dξ)2 = ζ −m−2 8(ρi (S[m])v) (dζ)2 , (1.46) m∈Z

where S[m] are the Sugawara operators defined by: S[m] =

1 XX ◦ a,b [n] ◦◦ , ◦ Ja,b [m − n] J 2

(1.47)

a,b n∈Z

which satisfy the following commutation relations: [S[m], A[n]] = −κnA[m + n] for A ∈ g, (1.48)   k dim g 3 (m − m)δm+n,0 id . (1.49) [S[m], S[n]] = κ (m − n)S[m + n] + 12 In particular the Sugawara operators S[m] commute with gP if κ = 0 (i.e., the level k is critical). When κ = k + h∨ 6= 0, the usual Virasoro operators are defined by normalizing S[m]: (1.50) T [m] := κ−1 S[m], which satisfy the well-known commutation relations: [T [m], A[n]] = −nA[m + n]

for A ∈ g, ck [T [m], T [n]] = (m − n)T [m + n] + (m3 − m)δm+n,0 id, 12

(1.51) (1.52)

where ck = k dim g/κ. Later we fix the local coordinate at Qi to ξi = t − zi and the one at P ∈ X to ξ = t − z with t(P ) = z, where t is the global coordinate of C (cf. (1.4)) and zi = t(Qi ). Lemma 1.11. In this coordinate 8(S(ξ)v) (dξ)2 and 8(T (ξ)v)(dξ)2 can be extended to global 2-differentials 8(S(t)v)(dt)2 and 8(T (t)v) (dt)2 . Proof. Under fractional linear coordinate changes, {8(S(ξ)v) (dξ)2 } behaves like a 2differential due to (1.43). Since the coordinate changes between two ξ’s are merely translations, if X is covered by these coordinates, then {8(S(ξ)v) (dξ)2 } gives a meromorphic 2-differential on X.  Remark 1.12. Using Weierstraß’ ℘-function, we can prove the lemma above in a more explicit manner. In fact, since   1 2 + O((t1 − t2 ) ) dt1 dt2 ℘(t1 − t2 ) dt1 dt2 = (t1 − t2 )2

Twisted WZW Models on Elliptic Curves

15

is a global meromorphic 2-form on X × X with a pole along the diagonal, we can equivalently replace definition (1.37) by X 1 2 lim 8(Ja,b (t(P ))J a,b (t(P 0 ))v) 8(S(P )v) (dt(P )) := 2 P 0 →P a,b . (1.53)  0 0 − k dim g · ℘(t(P ) − t(P )) dt(P ) dt(P ) This definition is meaningful globally on X and coincides with that of the proof above. 1.4. Action of the Virasoro algebra. When the level k is not −h∨ , the Lie algebras of formal meromorphic vector fields at Qi are projectively represented on Mk (V ) through the energy-momentum tensor. We denote by TX the tangent sheaf of X = Xτ (i.e., the sheaf of vector fields on X). Let us fix a local coordinate at Qi to ξi = t − zi , and denote by T D the direct sum of the Lie algebra of formal meromorphic vector fields at Qi for i = 1, . . . , L: T D :=

L M

T Qi ,

T Qi := (TX ⊗OX KX )∧ Qi =

i=1

L M

C((ξi ))

i=1

∂ . ∂ξi

(1.54)

The Virasoro algebra Vir D at D is defined to be the central extension of T D by Cˆc: Vir D := T D ⊕ Cˆc,

(1.55)

whose Lie algebra structure is defined by h
L
L i θi (ξi )∂ξi i=1 , ηi (ξi )∂ξi i=1 L L  cˆ X ⊕ Res(θi000 (ξi )ηi (ξi ) dξi ), = (θi (ξi )ηi0 (ξi ) − ηi (ξi )θi0 (ξi ))∂ξi ξi =0 12 i=1

(1.56)

i=1

where θi (ξi ), ηi (ξi ) ∈ C((ξi )) and ∂ξi := ∂/∂ξi . When L = 1, this Virasoro algebra Vir D is the usual one defined as a central extension of the Lie algebra of vector fields on a circle. D on The action of θi = θi (ξi )∂ξi ∈ T Qi and that of θ = (θi (ξi )∂ξi )L i=1 ∈ T v ∈ Mk (V ) and 8 ∈ (Mk (V ))∗ are given by Ti {θi }v := −

P m∈Z

ρi (θi,m T [m])v,

T {θ}(v) :=

L X

Ti {θi }v,

(1.57)

Ti∗ {θi }8,

(1.58)

i=1

(Ti∗ {θi }8)(v) := −8(Ti {θi }v), where θi (ξi )∂ξi = Note that we have

P

T ∗ {θ}8 :=

L X i=1

m+1 ∂ ξi m∈Z θi,m ξi

is the Laurent expansion (cf. (1.47) and (1.50)).

(Ti∗ {θi }8)(v) = Resh8(T (ξi )v) (dξi )2 , θi (ξi )∂ξi i, ξi =0

where h·, ·i is the contraction of a 2-differential and a tangent vector.

(1.59)

16

G. Kuroki, T. Takebe

L D Proposition 1.13. Let θ = (θi ∂ξi )L i=1 and η = (ηi ∂ξi )i=1 be elements of T . Then the operators T {θ}, T {η} acting on Mk (V ) and the operators T ∗ {θ}, T ∗ {η} acting on (Mk (V ))∗ satisfy

ck X Res(θi000 (ξi )ηi (ξi ) dξi ) id, ξi =0 12 L

[T {θ}, T {η}] = T {[θ, η]} +

(1.60)

i=1

[T ∗ {θ}, T ∗ {η}] = T ∗ {[θ, η]} −

ck X Res(θi000 (ξi )ηi (ξi ) dξi ) id, ξi =0 12 L

(1.61)

i=1

where ck = k dim g/κ and κ = k + h∨ . Namely, the definitions (1.57) and (1.58) define representations of Vir D on Mk (V ) with central charge ck and on (Mk (V ))∗ with central charge −ck respectively. This is a direct consequence of the definitions (1.57), (1.58) and the Virasoro commutation relations (1.52). As shown in Lemma 1.11, we have a global meromorphic 2-differential 8(T (t)v)(dt)2 for 8 ∈ CBk (Q, V ) and v ∈ Mk (V ). Therefore (1.59) and the residue theorem imply the following lemma. Lemma 1.14. Let 8 be a conformal block in CBk (Q, V ), θ(t)∂t in H 0 (X, TX (∗D)), and θi (ξi )∂ξi the Laurent expansion of θ(t)∂t in ξi for each i = 1, . . . , L. Denote (θi (ξi )∂ξi )L i=1 by θ. Then T ∗ {θ}8 = 0. 2. Critical Level and the XYZ Gaudin Model In this section we restrict ourselves to the case k = −h∨ = −N , namely the case when the level is critical. We showed in Sect. 1.2 that the conformal block is determined by its finitedimensional part. (See Proposition 1.5.) We shall see in this section that the Sugawara tensor is expressed by integrals of motion of the XYZ Gaudin model on this finitedimensional space. Indeed, it shall be shown that determining certain spaces of conformal blocks at the critical level is equivalent to solving the XYZ Gaudin model. First let us recall the definition of the XYZ Gaudin model, generalizing the definition in [Ga1, ST1] to slN case. Keeping in mind that we will show the relation of the conformal field theory and the XYZ Gaudin model, we use the same notation for slN modules, points Qi on a elliptic curve, etc. as in the previous section Sect. 1, and fix local coordinates at each point Qi to ξi = t − zi , where t is the global coordinate of C. The Hilbert space of the model is a tensor product of the finite-dimensional irreNL ducible representation spaces of slN (C): V := i=1 Vi . The generating function τˆ (u) of the integrals of motion of the model is defined as the trace of square of the quasiclassical limit T (u) of the monodromy matrix of the spin chain model associated with the Baxter-Belavin’s elliptic R-matrix: T (u) :=

L X

X

i=1 (a,b)6=(0,0)

τˆ (u) :=

1 tr(T (u))2 2

wa,b (u − zi )Ja,b ⊗ ρi (J a,b ),

(2.1)

Twisted WZW Models on Elliptic Curves L 1X = 2

X

17

w−a,−b (u − zi )wa,b (u − zj )ρi (Ja,b )ρj (J a,b ),

(2.2)

i,j=1 (a,b)6=(0,0)

where the indices of the summations over (a, b) run through a = 0, . . . , N − 1, b = 0, . . . , N − 1, (a, b) 6= (0, 0), and wa,b are functions defined by (1.14). As before, ρi is the representation of g on the ith factor Vi of V . The integrals of motion are encoded here in the following way: τˆ (u) =

L X

Ci ℘(u − zi ) +

i=1

L X

Hi ζ(u − zi ) + H0 ,

(2.3)

i=1

where ζ and ℘ are Weierstraß’ ζ and ℘ functions, Ci is the Casimir operator of g acting on Vi , i.e., 1 X ρi (Ja,b )ρi (J a,b ), (2.4) Ci = 2 (a,b)6=(0,0) PL and Hi (i = 1, . . . , L) and H0 are integrals of motion. The operators Hi satisfy i=1 Hi = 0, and hence there are L independent integrals of motion. Example 2.1. When N = 2, the Casimir operator Ci is equal to li (li + 1)idVi , where li = (dim Vi − 1)/2, and Hi are expressed as (cf. [ST1]): X X wa,b (zi − zj )ρi (Ja,b )ρj (J a,b ), Hi = j6=i (a,b)=(0,1),(1,1),(1,0) L  X 1X −ea,b ρi (Ja,b )ρi (J a,b ) H0 = (2.5) 2 i=1 (a,b)=(0,1),(1,1),(1,0)  ω     X ωa,b  a,b + −ζ ρi (Ja,b )ρj (J a,b ) , wa,b (zi − zj ) ζ zi − zj + 2 2 j6=i

where ωa,b = aτ + b and ea,b = ℘(ωa,b /2). We interpret this system as a twisted WZW model at the critical level. Let us come back to the situation in Sect. 1 and put u = t(P ). The slN -module Vi is assigned to the point Qi and regarded as the gtw |Qi module by the trivialization (1.35). Assign the vac˜ ∈ CBk ({P, Qi }, {CP , Vi }) uum module Mk (CP ) at P (k = −h∨ = −N ). As before 8 corresponds to a conformal block 8 ∈ CBk ({Qi }, {Vi }) through the isomorphism 2 ˜ (1.31). The correlation function of the Sugawara tensor 8(S(t)(u P ⊗ v)) (dt) has an expansion (1.46) at Qi and at P , X 2 ˜ P ⊗ ρi (S[m])v)(t − zi )−m−2 (dt)2 ˜ 8(u 8(S(t)(u P ⊗ v)) (dt) = m∈Z

=

X

m∈Z

−m−2 ˜ (dt)2 , 8(S[m]u P ⊗ v)(t − u)

(2.6)

P −m−2 ˜ (dt)2 , respectively. The right-hand side of (2.6) is m≤−2 8(S[m]u P ⊗ v)(t − u) since S[m]uP = 0 for all m = −1. Hence, evaluating (2.6) at t = u, we have X ˜ 8(ρi (S[m])v)(u − zi )−m−2 = 8(S[−2]u (2.7) P ⊗ v). m∈Z

18

G. Kuroki, T. Takebe

Lemma 2.2. Let v be a vector in V =

NL i=1

S[−2]uP ⊗ v ≡ uP ⊗ τˆ (u)v

Vi . Then we have in CCk ({P, Qi }, {CP , Vi }).

Hence the right-hand side of (2.7) is equal to 8(τˆ (u)v). Proof. First note that S[−2]uP =

1 2

X

Ja,b [−1]J a,b [−1]uP .

(a,b)6=(0,0)

The key step is to exchange the operators Ja,b [−1] and J a,b [−1] with operators acting on v by using the Ward identity (1.24). Recall that the functions wa,b (t − u) (1.14) in Ka,b define meromorphic sections Ja,b,P (t) := wa,b (t − u)Ja,b ,

J a,b,P (t) := w−a,−b (t − u)J a,b ,

(2.8)

of gtw through the inclusion (1.13) and (1.10). These sections belong to π∗ gtw (P ). Since Ja,b,P (t) has a Laurent expansion Ja,b,P (t) =

Ja,b + wa,b,0 Ja,b + wa,b,1 Ja,b · (t − u) + O((t − u)2 ) t−u

(2.9)

at P (see (1.15)), and J a,b (t) has a similar expansion, we have Ja,b [−1]J a,b [−1]uP = (Ja,b,P (t))P (J a,b,P (t))P uP − kwa,b,1 uP ,

(2.10)

where k = −h∨ = −N . Summing up (2.10) for (a, b) and using (1.18), we obtain 1 X S[−2]uP = (Ja,b,P (t))P (J a,b,P (t))P uP . (2.11) 2 (a,b)6=(0,0)

Substituting (2.11) into S[−2]uP ⊗ v and swapping Ja,b,P (t) and then J a,b,P (t) by the Ward identity (1.24), we obtain S[−2]uP ⊗ v ≡ uP ⊗

L 1X 2

X

wa,b (zi − u)w−a,−b (zj − u)ρi (Ja,b )ρj (J a,b )v,

i,j=1 (a,b)6=(0,0)

which proves the lemma because of (1.17).

(2.12)



Corollary 2.3. [τˆ (u), τˆ (u0 )] = 0 for any u and u0 . In particular, Hi (i = 0, 1, . . . , L) commute with each other. Proof. Since the Sugawara operators S[m] commute with the affine Lie algebra at the critical level due to (1.48), we have A[n]S[m]uP = 0

for A ∈ g and n = 0.

Hence we can find the following formula in the similar way as the proof of Lemma 2.2: S[−2]uP 0 ⊗ S[m]uP ⊗ v ≡ uP 0 ⊗ S[m]uP ⊗ τˆ (u0 )v, where t(P ) = u, t(P 0 ) = u0 , and v ∈ V . Using this formula and Lemma 2.2, we obtain uP 0 ⊗ uP ⊗ τˆ (u)τˆ (u0 )v ≡ uP 0 ⊗ S[−2]uP ⊗ τˆ (u0 )v ≡ S[−2]uP 0 ⊗ S[−2]uP ⊗ v ≡ S[−2]uP 0 ⊗ uP ⊗ τˆ (u)v ≡ uP 0 ⊗ uP ⊗ τˆ (u0 )τˆ (u)v. This proves the corollary in view of Proposition 1.5.



Twisted WZW Models on Elliptic Curves

19

Once the correspondence of the Hamiltonians of the XYZ Gaudin model and the correlation functions of the twisted WZW model is established, the eigenvalue problem of the XYZ Gaudin model is rewritten in terms of the conformal block of the twisted WZW model, as is the case with the (XXX) Gaudin model. (See [Fr2].) We sketch below how it goes, restricting ourselves to the sl2 case. For the general slN case, we should introduce higher order Sugawara operators, whose constructions are found in [Hay] and [GW]. Let us introduce a meromorphic (single-valued) function on X of the form q(t) =

L X

li (li + 1)℘(t − zi ) +

i=1

L X

µi ζ(t − zi ) + µ0 ,

(2.13)

i=1

where li = (dim Vi − 1)/2 (cf. Example 2.1), µi and µ0 are parameters satisfying PL P −n−2 be the Laurent expansion of n∈Z qi,n (t − zi ) i=1 µi = 0. Let qi (t − zi ) = qi q(t) at Qi . Denote by K (Vi ) the submodule of M−2 (Vi ) generated by the vectors (S[m] − qi,m )vi for vi ∈ Vi , m ∈ Z and put M qi (Vi ) := M−2 (Vi )/K qi (Vi ). Theorem 2.4. The space of conformal coinvariants and that of conformal blocks asNL qi sociated to the module M q (V ) := i=1 M (Vi ) are isomorphic to the quotient of NL µ V := i=1 Vi by the subspace J (V ) spanned by vectors of the form (Hi − µi )v for i = 0, 1, . . . , L and v ∈ V and its dual: CCk (M q (V )) ∼ = V /J µ (V ),

CBk (M q (V )) ∼ = (V /J µ (V ))∗ .

Proof. We prove the statement for the conformal blocks. The statement for the space of conformal coinvariants follows from this since it is finite-dimensional and dual to the space of conformal blocks. NL Let 8 be any linear functional on M−2 (V ) = i=1 M−2 (Vi ). A necessary and sufficient condition for 8 to be a conformal block in CBk (M q (V )) is that it vanishes on gD ˙ M−2 (V ) and on the subspaces X K q (V ) :=

L X

M−2 (V1 ) ⊗ · · · ⊗ K qi (Vi ) ⊗ · · · ⊗ M−2 (VL ).

i=1

First we show that this condition implies 8((Hi − µi )v) = 0 for i = 0, 1, . . . , L and v ∈V. The assumption is encapsulated in the following expression by a generating function, X 8(ρi (S[m] − qi,m )v)(u − zi )−m−2 = 0, (2.14) m∈Z

which means 8(τˆ (u)v) = q(u)8(v) because of (2.7) and Lemma 2.2. Thus (2.3) and (2.13) shows that 8(Hi v) = µi 8(v). We prove the converse statement next. Assume that 8 vanishes on the subspace J µ (V ). Let v be an arbitrary vector in M−2 (V ). We want to show that 8(ρi (S[m] − qi,m )v) vanishes for any m and i, but for this purpose we may assume v ∈ V without loss of generality. Indeed any v can be written in the form v = gX˙ v 0 by the decomposition 0 (1.29), where gX˙ ∈ U (gD ˙ ), v ∈ V , and therefore X ρi (S[m] − qi,m )v = gX˙ ρi (S[m] − qi,m )v 0 ,

20

G. Kuroki, T. Takebe

b since S[m] belongs to the center of U−2 (sl(2)). The Ward identity (1.25) implies that 8(ρi (S[m] − qi,m )v) = 0 if 8(ρi (S[m] − qi,m )v 0 ) = 0. For v ∈ V , we can prove 8(ρi (S[m] − qi,m )v) = 0 by tracing back the first part of this proof.  3. Sheaves of Conformal Coinvariants and Conformal Blocks So far we have fixed the modulus τ of an elliptic curve and marked points on it. In this section we introduce sheaves of conformal coinvariants and conformal blocks on a family of pointed elliptic curves. 3.1. Family of pointed elliptic curves and Lie algebra bundles. In this subsection we construct a family of elliptic curves with marked points, a group bundle, and the associated Lie algebra bundle over this family. The fiber at a point of the base space of the family gives the group bundle Gtw and the Lie algebra bundle gtw on a pointed elliptic curve defined in Sect. 1.1. e and S by Recall that H denotes the upper half plane. We define X S := { (τ ; z) = (τ ; z1 , . . . , zL ) ∈ H × CL | zi − zj 6∈ Z + τ Z if i 6= j }, e := S × C. X e Let π˜ = πX e/S be the projection from X onto S along C and q˜i the section of π˜ given by zi : e for (τ ; z) = (τ ; z1 , . . . , zL ) ∈ S. q˜i (τ ; z) := (τ ; z; zi ) ∈ X A family of L-pointed elliptic curves π : X  S is constructed as follows. Define e by the action of Z2 on X (m, n) · (τ ; z; t) := (τ ; z; t + mτ + n)

e for (m, n) ∈ Z2 , (τ ; z; t) ∈ X.

(3.1)

e by the action of Z2 : Let X be the quotient space of X e X := Z2 \X.

(3.2)

e Let πX e/X be the natural projection from X onto X and π = πX/S the projection from X onto S induced by π. ˜ We put qi := πX e/X ◦ q˜i ,

Qi := qi (S),

D :=

L [

Qi ,

˙ := X r D. X

i=1

PL Here qi is the section of π induced by q˜i and D is also regarded as a divisor i=1 Qi on X. The fiber of π at (τ, z) = (τ ; z1 , . . . , zL ) ∈ S is an elliptic curve with modulus τ and marked points z1 , . . . , zL . tw A group bundle Gtw X and a Lie algebra bundle gX on X are defined as follows. Due to e the definition of X, the Galois group of the covering πX e/X : X  X is naturally identified e is given by (τ ; z; t) · (m, n) := (−m, −n) · (τ ; z; t). with Z2 . Its natural right action on X 2 e Then the covering πX e/X : X  X is regarded as a principal Z -bundle on X. The actions of the Galois group Z2 on G and g are defined by

Twisted WZW Models on Elliptic Curves

21

(m, n) · g := (β m αn )g(β m αn )−1 (m, n) · A := (β m αn )A(β m αn )−1

for g ∈ G and (m, n) ∈ Z2 , for A ∈ g and (m, n) ∈ Z2 .

(3.3) (3.4)

These actions produce the associated group bundle Gtw X and the associated Lie algebra on X: bundle gtw X e ×Z2 G, gtw := X e ×Z2 g. Gtw := X (3.5) X

X

Their fibers at a point (τ ; z) ∈ S can be identified with Gtw and gtw in Sect. 1.1. We denote the OX -Lie algebra associated to the Lie algebra bundle gtw X by the same tw , as mentioned in Sect. 1.2. The sheaf g can be written in the following symbol gtw X X form: e p ∈ Z2 }. ˜ = p · A(x) ˜ for x˜ ∈ X, gtw X = { A ∈ (πX e/X )∗ (g ⊗ OX e ) | A(p · x)

(3.6)

e which does not intersect (m, n) · U 0 for any Hence if we take an open subset U 0 of X 2 (m, n) ∈ Z r {(0, 0)} and denote by U the image of U 0 on X, then the restriction of gtw X on U can be canonically identified with g ⊗ OU : ∼ gtw X |U = (πU 0 /U )∗ (g ⊗ OU 0 ) = g ⊗ OU ,

(3.7)

∼

where πU 0 /U is the natural biholomorphic projection U 0 → U . Denote by 1X the sheaf of 1-forms on X and by 1X/S the sheaf of relative diftw ferentials on X over S. It follows from the definition of gtw X that gX possesses a nat1 tw tw ural connection ∇ : gX → gX ⊗OX X , which is induced by the trivial connection 1 id ⊗ d : g ⊗ OX e → g ⊗ X e through the identification (3.6). This means that, under the trivialization (3.7), the connection ∇ is identified with id ⊗ dU where dU is the exterior derivation on U . The relative connection ∇X/S along the fibers is defined to be the composite of the connection ∇ and the natural homomorphism 1X → 1X/S . Under the trivialization (3.7) and the coordinate (τ ; z; t), the relative connection ∇X e/S is equal to the exterior derivation by t. Define the invariant OX -inner product on gtw X by (A|B) :=

1 1 tr (ad A ad B) ∈ OX tr (ad A ad B) = 2h∨ gtwX 2N gtwX

for A, B ∈ gtw X,

(3.8)

where the symbol ad denotes the adjoint representation of the OX -Lie algebra gtw X . Under is equal to the inner product defined by the trivialization (3.7), the inner product on gtw X (1.1) and hence it is invariant under the translation along the connection ∇. Recall that ε = exp(2πi/N ). For (a, b) ∈ (Z/N Z)2 , the 1-dimensional representation (m, n) 7→ εbm+an of Z2 defines the associated flat line bundle La,b on X. We obtain the decomposition of gtw X into line bundles: M Ja,b La,b . (3.9) gtw X = (a,b)6=(0,0)

This is a sheaf version of (1.7). Lemma 3.1.

Rp π∗ gtw X = 0 for all p.

22

G. Kuroki, T. Takebe

Proof. Since L∗a,b ⊗OX 1X/S is isomorphic to L−a,−b , it follows that R1 π∗ La,b ∼ = HomOS (π∗ L−a,−b , OS ) by the Serre duality. Therefore, because of the decomposition (3.9), it is enough to show π∗ La,b = 0 for (a, b) 6= (0, 0). Let U be any open subset of S and put V := π −1 (U ). For each s = (τ ; z) ∈ U , the restriction La,b |Xs of La,b on the fiber Xs := π −1 (s) can be identified with the line bundle La,b on Xτ defined in Sect. 1.1. Hence we obtain H 0 (Xs , La,b |Xs ) = 0 for each s ∈ S. In particular, for every f ∈ H 0 (V, La,b ) = H 0 (U, π∗ La,b ), the restriction f |Xs of f on the fiber vanishes for each s ∈ S and hence f vanishes itself. This means that H 0 (U, π∗ La,b ) = 0. We have proved the lemma.  3.2. Sheaf of affine Lie algebras. In this section we define a sheaf version of the Lie algebras gˆ D , gD ˙ , etc. on the base space S of the family. X For an OX -module F and a closed analytic subset W of X, the restriction F |W of F on W and the completion Fˆ |W = (F)∧ W of F at W are defined by n Fb|W = (F)∧ W := projlim (F/IW F ),

F|W := F /IW F,

(3.10)

n→∞

where IW is the defining ideal of W in X. Qi i D D D We define the OS -Lie algebras gQ ˙ as follows: S , gS,+ , gS , gS,+ , and gX tw ∧ i gQ S := π∗ (gX (∗Qi ))Qi , tw ∧ gD S := π∗ (gX (∗D))D =

tw ∧ i gQ S,+ := π∗ (gX )Qi , L M

i gQ S ,

tw ∧ gD S,+ := π∗ (gX )D =

i=1

L M

i gQ S,+ ,

(3.11)

i=1

tw gD ˙ := π∗ (gX (∗D)). X

The 2-cocycle of gD S is defined by ca (A, B) :=

L X i=1

ResQi (∇Ai |Bi ) =

L X

ResQi (∇X/S Ai |Bi ),

(3.12)

i=1

L D where A = (Ai )L i=1 , B = (Bi )i=1 ∈ gS and ResQi is the residue along Qi . Using the D 2-cocycle ca (·, ·), we define a central extension gˆ D S of gS : D ˆ gˆ D S := gS ⊕ OS k,

(3.13)

i where its Lie algebra structure is defined by the formula similar to (1.22). Put gˆ Q S := Qi Q ˆ which is a OS -Lie subalgebra of gˆ D . We call gˆ D (resp. gˆ i ) the sheaf of gS ⊕ OS k, S S S affine Lie algebras at D (resp. Qi ). D The diagonal embedding of gD ˙ into gS is defined to be the mapping which sends X L D A ∈ gD ˙ to (Ai )i=1 ∈ gS , where each Ai is the image of A given by the natural embedding X tw tw D D D ˆD gX (∗D) ,→ (gX (∗Qi ))∧ ˙ with its image in gS and g ˙ , Qi . We identify gX S . For A, B ∈ gX 1 we can regard (∇X/S A|B) as an element of π∗ X/S (∗D). Hence, using the residue

Qi theorem, we obtain that ca (A, B) = 0. Thus gD ˙ , as well as gS,+ , is an OS -Lie subalgebra X of gˆ D S. D ˆ Put gˆ D S,+ := gS ⊕ OS k. Then Lemma 3.1 implies the sheaf version of (1.29).

Twisted WZW Models on Elliptic Curves

23

D ˆD Lemma 3.2. gˆ D ˙ ⊕g S = gX S,+ . D Proof. We can calculate Rp π∗ gtw ˙ ⊕ X for p = 0, 1 as the kernel and the cokernel of gX D D gS,+ → gS , which sends (aX˙ ; a+ ) to aX˙ − a+ . But then Lemma 3.1 means that both the D D kernel and the cokernel vanish and hence gD ˙ ⊕ gS,+ = gS . We have proved the lemma. X 

e i := q˜i (S) which does not intersect (m, n)·U 0 Choose any open neighborhood U 0 of Q 2 for any (m, n) ∈ Z r {(0, 0)}. Then, applying the trivialization (3.7) to U 0 , we obtain a natural isomorphism i ∼ b (3.14) gQ S = g ⊗ π∗ (OX|Qi (∗Qi )), which does not depend on the choice of U 0 and is defined globally on S. Furthermore, using the coordinate (τ ; z; ξi ) with ξi = t − zi , we have the following isomorphism defined over S: i ∼ (3.15) gQ S = g ⊗ OS ((ξi )). Qi i Under this trivialization, gQ S,+ is identified with g ⊗ OS [[ξi ]] and the connections on gS induced by ∇ and ∇X/S are written in the following forms:

X ∂A ∂A ∂A dτ + dzi + dξi ∈ g ⊗ 1S ((ξi )) ⊕ g ⊗ OS ((ξi )) dξi , (3.16) ∂τ ∂zi ∂ξi L

∇A =

i=1

∂A dξi ∈ g ⊗ OS ((ξi )) dξi , ∇X/S A = ∂ξi

(3.17)

Q where A ∈ g ⊗ OS ((ξi )) ∼ = gS i . We also obtain the induced global trivialization of the sheaf of affine Lie algebras on S:

∼ gˆ D S =

L M

ˆ g ⊗ OS ((ξi )) ⊕ OS k.

(3.18)

i−1

Under this trivialization, the bracket of gˆ D S is represented in the following form: 

L X 
L L ˆ , (B ⊗ g ) , B ] ⊗ f g + k (Ai |Bi ) Res(dfi · g), (3.19) (Ai ⊗ fi )L = [A i i i i i i i=1 i=1 i=1 i=1

ξi =0

where Ai , Bi ∈ g and fi , gi ∈ OS ((ξi )). 3.3. Definition of the sheaves of conformal coinvariants and conformal blocks. For any ˆD OS -Lie algebra a = gD S,+ , g S , etc., we denote by US (a) the universal OS -enveloping algebra of a and define the category of a-modules to be that of US (a)-modules. Definition 3.3. For any gˆ D S -module M, we define the sheaf CC(M) of conformal coinvariants and the sheaf CB(M) of conformal blocks by CC(M) := M/gD ˙ M, X CB(M) := HomOS (CC(M), OS ). Namely, the OS -module CC(M) is generated by M with relations

(3.20) (3.21)

24

G. Kuroki, T. Takebe

AX˙ v ≡ 0

(3.22)

for all AX˙ ∈ gD ˙ , v ∈ M, and 8 ∈ CB(M) means that 8 belongs to HomOS (M, OS ) X and satisfies (3.23) 8(AX˙ v) = 0 for all AX˙ ∈ gD ˙ , v ∈ M. These equations, (3.22) and (3.23), are also called the Ward X identities. We can regard CC(·) as a covariant right exact functor from the category of gˆ D S -modules to that of OS -modules and similarly CB(·) as a contravariant left exact functor. The gˆ D S -modules of our concern are the sheaf version Mk (V ) of Mk (V ) in Sect. 1.2. We give two equivalent definitions of Mk (V ). First definition of Mk (V ). Fix an arbitrary complex number k. For each i = 1, . . . , L, ˆ let Mi be a representation with level k of the affine Lie algebra gˆ i := g ⊗ C((ξi )) ⊕ Ck. ˆ (Here gˆ i -modules are said to be of level k if the canonical central element k acts on them as k · id.) Assume the smoothness of Mi , namely, for any vi ∈ Mi , there exists m = 0 such that, for Ai1 , . . . , Aiν ∈ g, m1 , . . . , mν = 0, and ν = 0, 1, 2, . . ., (Ai1 ⊗ ξim1 C[[ξi ]]) · · · (Aiν ⊗ ξimν C[[ξi ]])vi = 0 if m1 + . . . + mν = m. (3.24) NL Put M := i=1 Mi and M := M ⊗ OS . Then M is a representation with level k of the
LL affine Lie algebra (g⊕L )∧ := i=1 g ⊗ C((ξi )) ⊕ Ckˆ associated to g⊕L . We can define the gˆ D S -module structure on M by (Ai ⊗ fi (ξi ))L i=1 (v ⊗ a) :=

L X X

(ρi (Ai ⊗ ξim )v) ⊗ (fi,m a),

ˆ := kv, kv

(3.25)

i=1 m∈Z

P where Ai ∈ g, fi (ξi ) = m fi,m ξim ∈ OS ((ξi )), fi,m , a ∈ OS , v ∈ M , and ρi (Ai ⊗ξim ) denotes the action of Ai ⊗ ξim on the ith factors in v. If each Mi is the Weyl module Mk (Vi ) induced up from a finite-dimensional irreducible representation Vi of g, then we NL NL put V := i=1 Vi , M := Mk (V ) := i=1 Mk (Vi ), and M := Mk (V ) := Mk (V ) ⊗ OS and denote CC(M) and CB(M) by CCk (V ) and CBk (V ) respectively. Second definition of Mk (V ). Let Vi be a finite-dimensional irreducible representation NL of g and put V := i=1 Vi . Denote the constant sheaf associated to V by the same ˆ symbol V . Using the trivialization (3.18), we can define the action of gˆ S,+ = gD S,+ ⊕ OS k on V ⊗ OS by X ˆ := kv (Ai ⊗ fi (ξi ))L (ρi (Ai )v) ⊗ (fi (0)a), kv (3.26) i=1 (v ⊗ a) := i

where Ai ∈ g, fi (ξi ) ∈ OS [[ξi ]], a ∈ OS , v ∈ V , and ρi (Ai ) is the action of Ai on the ith factors in v. The gˆ D S -module MS,k (V ) induced from V ⊗ OS is defined by gˆ D S

MS,k (V ) := Ind (V ⊗ OS ) = US (ˆgD ) (V ⊗ OS ). S ) ⊗US (gˆ D S,+ gˆ D S,+

Using the decomposition

(3.27)

Twisted WZW Models on Elliptic Curves

ÿ gˆ D S =

L M

25

! g⊗

ξi−1 OS [ξi−1 ]

⊕ gˆ D S,+ ,

i=1

we can show that MS,k (V ) has the following OS -free basis: ρi1 (As1 [m1 ]) · · · ρiν (Asν [mν ])vj ,

(3.28)

where ν = 0, 1, 2, . . ., in = 1, . . . , L, {As } is a basis of g, {vj } is a basis of V , and m1 5 · · · 5 mν < 0. This is also an OS -free basis of Mk (V ) and hence MS,k (V ) is isomorphic to Mk (V ) as a gˆ D S -module. In the following we identify Mk (V ) with MS,k (V ). This identification of the two definitions and Lemma 3.2 prove the sheaf version of Proposition 1.5. Proposition 3.4. Let Vi be a finite-dimensional irreducible representation of g for each NL i and put V := i=1 Vi . Then the natural inclusion V ⊗ OS ,→ Mk (V ) induces the following isomorphisms: ∼

CCk (V ) ← V ⊗ OS

and

∼

CBk (V ) → V ∗ ⊗ OS .

Proof. From the second definitions of Mk (V ) and Lemma 3.2, it follows that ∼

D Mk (V ) = US (ˆgD ˙ ) ⊗OS (V ⊗ OS ). ) (V ⊗ OS ) ← US (gX S ) ⊗US (gˆ D S,+

Namely, Mk (V ) is freely generated by V ⊗ OS over US (gD ˙ ). Hence we obtain the X formulae
∼ D D D CCk (V ) = Mk (V )/gD ˙ Mk (V ) ← US (gX ˙ )/gX ˙ US (gX ˙ ) ⊗OS (V ⊗ OS ) X ∼

← OS ⊗OS (V ⊗ OS ) = V ⊗ OS , ∼

CBk (V ) = HomOS (CCk (V ), OS ) → HomOS (V ⊗ OS , OS ) = V ∗ ⊗ OS . We have completed the proof of the proposition.



Corollary 3.5. For each i = 1, . . . , L, let Vi be a finite-dimensional irreducible representation of g and Mi a quotient module of the generalized Verma module Mk (Vi ) of the NL affine Lie algebra gˆ i . Put M := i=1 Mi and M := M ⊗ OS . Then the sheaf CC(M) of conformal coinvariants and the sheaf CB(M) of conformal blocks are OS -coherent. Proof. Since CB(M) = HomOS (CC(M), OS ), it suffices for the proof to see that CC(M) is coherent. The right exactness of the functor CC(·) and the fact that M is a quotient gˆ D S -module of Mk (V ) imply that CC(M) is a quotient OS -module of CCk (V ) = CC(Mk (V )), which is OS -coherent due to Proposition 3.4. Hence CC(M) is  also OS -coherent.

26

G. Kuroki, T. Takebe

4. Sheaf of the Virasoro Algebras This section provides the sheaf version of the Virasoro algebras and its actions on representations of the sheaf of the affine Lie algebra, which will be used in Sect. 5 to endow the sheaf of conformal coinvariants and the sheaf of conformal blocks with DS -module structures, when the level is not critical (i.e., κ = k + h∨ 6= 0). 4.1. Definition of the sheaf of the Virasoro algebras. We define the sheaf of the Virasoro algebras by L M OS ((ξi ))∂ξi ⊕ OS cˆ, (4.1) VirSD := TS ⊕ i=1

where TS is the tangent sheaf of S. The Lie algebra structure which we shall give to this OS -sheaf below reduces to the Virasoro algebra structure on Vir D (1.55), when S is replaced with a point. In order to define a Lie algebra structure on VirSD , we introduce the following notation: – For µ, ν ∈ TS , the symbol [µ, ν] denotes the natural Lie bracket in TS ; LL L – For θ = (θi )L O ((ξ ))∂ , the symbol [θ, η]0 = ([θi , ηi ]0 )L i=1 , η = (ηi )i=1 ∈ i=1 i=1 LLS i ξi denotes the natural Lie bracket in i=1 OS ((ξi ))∂ξi given by
[θi (ξi )∂ξi , ηi (ξi )∂ξi ]0 = θi (ξi )ηi0 (ξi ) − ηi (ξi )θi0 (ξi ) ∂ξi .
PL 000 – cV (θ, η) := i=1 Resξi =0 θi (ξi )ηi (ξi ) dξi (the symbol cV stands for “Cocycle defining the Virasoro algebra”); LL – For θ ∈ the symbols µ(θ) and µ(f ) denote the i=1 OS ((ξi ))∂ξi and f ∈ OS , L L natural actions of a vector field µ ∈ TS on i=1 OS ((ξi ))∂ξi and OS respectively. We define the Lie algebra structure on VirSD by [(µ; θ; f cˆ), (ν; η; g cˆ)] := ([µ, ν]; µ(η) − ν(θ) + [θ, η]0 ; (µ(g) − ν(f ) + cV (θ, η))ˆc),

(4.2)

where (µ; θ; f cˆ), (ν; η; g cˆ) ∈ VirSD . Note that VirSD is not an OS -Lie algebra but a CS -Lie algebra. We call VirSD the sheaf of Virasoro algebras on S. Remark 4.1. Later representations of VirSD shall be interpreted as representations of an D extension VirX ˙ of TS defined below and thus shall be given a DS -module structure. Let TX denote the tangent sheaf of the total space X and TX/S the relative tangent sheaf of the family π : X → S (i.e., the sheaf of vector fields along the fibers of π on X). Since π : X → S is smooth, we have the following short exact sequence: 0 → TX/S (∗D) → TX (∗D) → (π ∗ TS )(∗D) → 0. Note that π ∗ TS = OX ⊗π−1 OS π −1 TS does not possess a natural Lie algebra structure, but π −1 TS ⊂ π ∗ TS does. Defining TX,π (∗D) to be the inverse image of π −1 TS in TX (∗D), we obtain the following Lie algebra extension: 0 → TX/S (∗D) → TX,π (∗D) → π −1 TS → 0.

Twisted WZW Models on Elliptic Curves

27

The direct image of this sequence by π is also exact and gives the following Lie algebra extension: D (4.3) 0 → TX˙D → VirX ˙ → TS → 0, where we put

D VirX ˙ := π∗ TX,π (∗D),

TX˙D := π∗ TX/S (∗D).

Remark 4.2. The exact sequence (4.3) is essential in the constructions of connections on the sheaf CC(M) of conformal coinvariants and the sheaf CB(M) of conformal blocks. Generally, a connection is defined to be an action of the tangent sheaf satisfying certain axioms. Using the exact sequence (4.3), we can obtain a connection if we have actions D D of VirX ˙ whose restriction on TX ˙ is trivial (cf. Lemma 4.10, Lemma 4.11, Lemma 4.12, Lemma 4.13, Lemma 4.14, and Theorem 5.1). D e is of ˜ X˙ to X Lemma 4.3. For a local section aX˙ of VirX ˙ = π∗ TX,π (∗D), its pull-back a the form: L X a˜ X˙ = µ0 (τ ; z)∂τ + µi (τ ; z)∂zi + θt (τ ; z; t)∂t , i=1

where µi = µi (τ ; z) ∈ OS and θt (τ ; z; t) is a meromorphic function globally defined along the fibers of π˜ with the following properties: 1. The poles of θt (τ ; z; t) are contained in π −1 (D); e/X X 2. The quasi-periodicity: θt (τ ; z; t + mτ + n) = θt (τ ; z; t) + mµ0 (τ ; z) for (m, n) ∈ Z2 .

(4.4)

e given by (τ ; z; t) 7→ Proof. Let (m, n) be in Z2 and fm,n denote the action of (m, n) on X (τ ; z; t + mτ + n). Then its derivative dfm,n sends ∂τ , ∂zi , and ∂t to ∂τ + m∂t , ∂zi , and ∂t respectively. Since a˜ X˙ induces the vector field aX˙ in π∗ TX,π (∗D), we obtain a formula µ + θt (t + mτ + n)∂t = dfm,n (˜aX˙ ), which is equivalent to θt (t + mτ + n) = θt (t) + mµ0 , which proves the lemma.



PL The local section aX˙ is mapped to µ = µ0 ∂t + i=1 µi ∂zi ∈ TS by the projection along π in (4.3) and belongs to TX˙D = π∗ TX/S (∗D) if and only if µ = 0. Under the local coordinate ξi = t − zi , the local section aX˙ is uniquely represented in the following form: aX˙ = µ + θi (ξi )∂ξi ∈ TS ⊕ OS ((ξi ))∂ξi , where θi (ξi ) is the Laurent expansion of θt (τ ; z; t) in ξi = t − zi . Thus we obtain the LL D D embedding of VirX ˙ into TS ⊕ i=1 OS ((ξi ))∂ξi ⊂ VirS given by D VirX ˙ ,→ TS ⊕

L M

OS ((ξi ))∂ξi ⊂ VirSD ,

aX˙ 7→ (µ; θ) = (µ; (θi (ξi )∂ξi )L i=1 ). (4.5)

i=1 D D D We identify VirX ˙ with its image in VirS . For instance, (µ; θ; 0) ∈ VirX ˙ means that LL D D µ ∈ TS , θ ∈ i=1 OS ((ξi ))∂ξi , and (µ; θ; 0) belongs to the image of VirX ˙ in VirS . We D D D also identify the subsheaf TX˙ ⊂ VirX˙ with its image in VirS .

28

G. Kuroki, T. Takebe

Remark 4.4. These formulations are essentially an application of the Beilinson-Schechtman theory in [BS] to our situation. The theory contains a natural construction of the Kodaira-Spencer map of a family of compact Riemann surfaces and its generalization to Virasoro algebras. For a brief sketch, see Appendix B. D D A natural question is whether or not the embeddings of VirX ˙ into VirS is a Lie algebra homomorphism. D D Lemma 4.5. The embedding VirX ˙ ,→ VirS is a Lie algebra homomorphism. D D Proof. Let aX˙ and bX˙ be in VirX ˙ = π∗ TX,π (∗D). Denote their images in VirS by e by a˜ ˙ and b˜ ˙ respectively. It suffices for (µ; θ; 0) and (ν; η; 0) and their pull-backs to X X X the proof to show that cV (θ, η) = 0. e the vector fields a˜ ˙ and b˜ ˙ are represented as Under the coordinate (τ ; z; t) of X, X X

b˜ X˙ = ν + η t (t)∂t ,

a˜ X˙ = µ + θt (t)∂t ,

(4.6)

where we write µ, ν ∈ TS in the following forms: µ = µ0 ∂τ +

L X

µi ∂ zi ,

ν = ν0 ∂ τ +

i=1

L X

ν i ∂ zi .

(4.7)

i=1

Here we omit the arguments (τ ; z) for simplicity: µi = µi (τ ; z), θt (t) = θt (τ ; z; t), etc. Because of Lemma 4.3, we have θt (t + mτ + n) = θt (t) + mµ0 ,

η t (t + mτ + n) = η t (t) + mν0 .

Hence we can define the relative meromorphic 1-form ω ∈ π∗ 1X/S (∗D) by ω = ω(t) dt :=

∂ 2 θt (t) ∂η t (t) dt. ∂t2 ∂t

(4.8)

Here, the well-definedness of ω as a 1-form in π∗ 1X/S (∗D) follows from the fact that the definition of ω(t) implies ω(t + mτ + n) = ω(t) for m, n ∈ Z. On the other hand, under the local coordinate (τ ; z; ξi ) of X around Qi given by ξi = t − zi , the vector fields aX˙ and bX˙ are represented in the following forms: aX˙ = µ + θi (ξi )∂ξi ,

bX˙ = ν + ηi (ξi )∂ξi ,

where µ and ν are the same as those in (4.6) and θi and ηi are given in terms of θt , η t , µi and νi in (4.6) and (4.7) by θi (ξi ) = θt (zi + ξi ) − µi ,

ηi (ξi ) = η t (zi + ξi ) − νi .

Thus by (4.8), we have ω=

∂ 2 θi (ξi ) ∂ηi (ξi ) dξi ∂ξi ∂ξi2

around Qi .

Hence the residue theorem leads to  2  L L X X ∂ θi (τ ; z; ξi ) ∂ηi (τ ; z; ξi ) Res dξ ResQi ω = 0. = − cV (θ, η) = − i ξi =0 ∂ξi ∂ξi2 i=1 i=1 This proves the lemma.



Twisted WZW Models on Elliptic Curves

29

D D Remark 4.6. The same question about the embedding VirX ˙ ,→ VirS can be answered under a more general formulation for higher genus compact Riemann surfaces with a D D projective structure. However, in the higher genus case, the embedding VirX ˙ ,→ VirS is not always a Lie algebra homomorphism. The case of genus 1 is very special. See Appendix C for a short sketch of a formulation.

– – – – –

D ˆD In order to define the action of VirX ˙ on g S , let us introduce the following notation: L L L A = (Ai )L i=1 , B = (Bi )i=1 ∈ i=1 g ⊗ OS ((ξi )); 0 0 L ∼ LL g ⊗ OS ((ξi )) [A, B] = ([Ai , Bi ] )i=1 denotes the natural Lie bracket in gD S = i=1 given by the base extension of the Lie algebra g; ˆ (B; g k) ˆ ∈ gˆ D = gD ⊕ OS k; ˆ (A; f k), S S For µ ∈ TS , the symbol µ(A) denotes the natural actions of TS on gD S; LL L D The natural action of θ = (θi )i=1 ∈ i=1 OS ((ξi ))∂ξi on A ∈ gS is defined by

θi (Ai ⊗ fi (ξi )) := Ai ⊗ (θi (ξi )fi0 (ξi ))

θ(A) := (θi (Ai ⊗ fi ))L i=1 ,

where θi = θi (ξi )∂ξi and A = Ai ⊗ fi (ξi ) ∈ g ⊗ OS ((ξi )). D ˆD Then the action of VirX ˙ on g S is defined by D ˆ ˆD for (µ; θ; 0) ∈ VirX ˙ and (A; g k) ∈ g S , (4.9)

ˆ := [(µ; θ; 0), (A; g k)] ˆ (µ; θ; 0) · (A; g k)

D D where VirX ˙ is identified with its image in VirS by (4.5) and the bracket of the right-hand side is a Lie bracket in the semi-direct product Lie algebra VirSD n gˆ D S defined by

ˆ := (µ(A) + θ(A); µ(g)k) ˆ [(µ; θ; f cˆ), (A; g k)] ˆ ∈ gˆ D . for (µ; θ; f cˆ) ∈ Vir D and (A; g k) S

(4.10)

S

D D ˆD Lemma 4.7. The action of VirX ˙ on g ˙ . S preserves gX

Proof. Because of (3.16), under the identifications above, the restriction of the action D D tw of VirX ˙ = π∗ TX,π (∗D) on gX ˙ comes from the action of TX,π (∗D) on gX (∗D) via the tw D D connection ∇ on gX . Namely, if aX˙ ∈ VirX˙ , AX˙ ∈ gX˙ , and their images in VirSD and gˆ D S are denoted by (µ; θ; 0) and (A; 0) respectively, then
[(µ; θ; 0), (A; 0)] = the image of ∇aX˙ AX˙ ∈ π∗ gtw (∗D) . D D D Thus we obtain [VirX ˙ , gX ˙ ] ⊂ gX ˙ .



4.2. Action of the sheaf of Virasoro algebras. In this subsection we define an action of ˆD the Lie algebra VirSD n gˆ D S on g S modules. Fix an arbitrary complex number k. For each i = 1, . . . , L, let Mi be a representation with level k of the affine Lie algebra gˆ i satisfying the smoothness condition (3.24). Put NL M := ⊗ O . Then M is a representation with level k of the i=1 Mi and M := M
LL S affine Lie algebra (g⊕L )∧ = i=1 g ⊗ C((ξi )) ⊕ Ckˆ and M is a gˆ D S -module. The Sugawara operators S[m] acting on Mi ’s are given by the formula (1.47) and its action on the ith factor Mi in M is denoted by ρi (S[m]). Define the Sugawara tensor field by

30

G. Kuroki, T. Takebe

S(ξ)(dξ)2 :=

X

ξ −m−2 S[m] (dξ)2 ,

m∈Z

and its action on Mi is denoted by ρi (S(ξi ))(dξ)2 . Then, by the same way as Lemma 1.11, we can prove the following lemma. Lemma 4.8. For any s ∈ S, 8 ∈ CB(M)s and v ∈ Ms , there exists a unique ω ∈ (π∗ 2X/S (∗D))s such that the expression of ω under the coordinate ξi coincides with 8(ρi (S(ξi ))v) (dξi )2 for each i = 1, . . . , L. We denote ω in Lemma 4.8 by 8(S(ξ)v) (dξ)2 or 8(S(P )v) (dP )2 , which is called a correlation function of the Sugawara tensor S(ξ) and v under 8, or a Sugawara correlation function for short. Assume that κ = k + h∨ 6= 0 and put ck := k dim g/κ. Define the Virasoro operators T [m] and the energy-momentum tensor T (ξ) by T [m] := κ−1 S[m],

T (ξ)(dξ)2 := κ−1 S(ξ)(dξ)2 ,

(4.11)

as in (1.50) and the energy-momentum correlation function 8(T (ξ)v) (dξ)2 to be κ−1 8(S(ξ)v) (dξ)2 as in (1.41). The action ρi (T [m]) of the Virasoro operators on Mi defines a representation of the Virasoro algebraPwith central charge ck = k dim g/κ (Lemma 1.13). For vi ⊗ g ∈ Mi ⊗ OS and θi = m∈Z θi,m ξim+1 ∂ξi ∈ OS ((ξi ))∂ξi , put X ρi (T {θi })(vi ⊗ g) = (ρi (−T [m])vi ) ⊗ (θi,m g). (4.12) m∈Z

For example, ρi (T {ξim+1 ∂ξi }) = ρi (−T [m]). For θ = (θi )L i=1 ∈ operator T {θ} acting on M is defined by T {θ} :=

L X

LL i=1

OS ((ξi ))∂ξi , the

ρi (T {θi }),

(4.13)

i=1

where we consider ρi (T {θi }) as an operator acting on the ith factor in M. Define the action of (µ; θ; f cˆ) ∈ VirSD on M by (µ; θ; f cˆ) · (v ⊗ g) := v ⊗ µ(g) + T {θ}(v ⊗ g) + ck v ⊗ (f g)

(4.14)

for v ⊗ g ∈ M = M ⊗ OS . The dual actions on M∗ := HomOS (M, OS ) are defined by (µ8)(v) := µ(8(v)) − 8(µ(v)), (ρ∗i (T ∗ {θi }8))(v) := −8(ρi (T {θi })v), T ∗ {θ} :=

L X

ρ∗i (T {θi }),

(4.15) (4.16) (4.17)

i=1

((µ; θ; f cˆ) · 8)(v) := µ(8(v)) − 8((µ; θ; f cˆ) · v),

(4.18)

where 8 ∈ M∗ , and v ∈ M. Since we have (µ; θ; f cˆ) · 8 = µ8 + T ∗ {θ}8 − ck f 8, ∗

VirSD

(4.19)

on M defines a representation of with central charge −ck . the action of The Virasoro operators T [m] satisfy the commutation relations (1.52). Therefore a straightforward calculation proves the following lemma. VirSD

Twisted WZW Models on Elliptic Curves

31

D Lemma 4.9. The action of gˆ D S (3.25) and that of VirS (4.14) on M induce a represen, whose semi-direct product Lie algebra structure tation of the Lie algebra VirSD n gˆ D S is given by (4.10). D ∗ D D Define the actions of VirX ˙ on M and M through the embedding VirX ˙ ,→ VirS D and the actions of VirS . Then Lemma 4.5 and Lemma 4.9 immediately lead to the following lemma. D ∗ Lemma 4.10. These actions of VirX ˙ on M and M are representations of the Lie D algebra VirX˙ . D D Lemma 4.11. The action of VirX ˙ on M preserves gX ˙ M and hence defines a repreD sentation on CC(M) of the Lie algebra VirX˙ . D D Proof. Assume that αX˙ ∈ VirX ˙ ∈ gX ˙ , AX ˙ , and v ∈ M. Lemma 4.9 implies that

αX˙ AX˙ v = [αX˙ , AX˙ ]v + AX˙ αX˙ v, D and Lemma 4.7 means that [αX˙ , AX˙ ] ∈ gD ˙ AX ˙ v belongs to gX ˙ . Hence αX ˙ M and X D D D VirX˙ gX˙ M is included in gX˙ M. From Lemma 4.10 it follows that the induced action of D D D VirX  ˙ on CC(M) = M/gX ˙ M defines a representation of the Lie algebra VirX ˙ .

As a result of Lemma 4.10, Lemma 4.11 and (4.15), we obtain the following lemma. D ∗ ∗ Lemma 4.12. The action of VirX ˙ on M preserves the subsheaf CB(M) of M and D defines a representation on CB(M) of the Lie algebra VirX˙ .

The actions of TX˙D on M and M∗ are also defined through the embedding TX˙D ,→ VirSD . Then, as in the proof of Lemma 1.14, we can show the following lemma from Lemma 4.8 thanks to the existence of the energy-momentum correlation function (4.11). Lemma 4.13. The action of TX˙D on M∗ satisfies TX˙D · CB(M) = 0. Using the exact sequence (4.3), Lemma 4.12, and Lemma 4.13, we can construct a flat connection on the sheaf CB(M) of conformal blocks in Sect. 5 (Remark 5.2). However, for the construction of a flat connection on the sheaf CC(M) of conformal coinvariants, we shall need the following lemma, as well as the exact sequence (4.3) and Lemma 4.11. Lemma 4.14. The action of TX˙D on M satisfies TX˙D M ⊂ gD ˙ M. X We remark that Lemma 4.14 implies Lemma 4.13, but the converse does not hold. The key point in the proof of Lemma 4.13 is the notion of the energy-momentum correlation function, which is not useful for the proof of Lemma 4.14. Hence we must find a direct proof of Lemma 4.14 without using the energy-momentum correlation functions. The rest of this subsection is devoted to the proof of this lemma along the course similar to that of [Ts]. ∼ LL g ⊗ OS ((ξi )) by We define the OS -inner product ( . , . ) on gD S = i=1 L
X L (Ai |Bi ) Res(fi gi dξi ) (Ai ⊗ fi )L i=1 , (Bi ⊗ gi )i=1 = i=1

ξi =0

(4.20)

32

G. Kuroki, T. Takebe

for Ai , Bi ∈ g and fi , gi ∈ OS ((ξi )). This inner product is non-degenerate and allows us to regard gD S as the topological dual of itself under the ξi -adic topologies. Putting R := { (a, b, m, i) | (a, b) ∈ (Z/N Z)2 r {(0, 0)}, m ∈ Z, i = 1, . . . , L },(4.21) m L m L em a,b,i = (ea,b,i,j (ξj ))j=1 := (δi,j Ja,b ⊗ ξj )j=1 ,

ea,b,i = m

L (ea,b,i m,j (ξj ))j=1

(4.22)

:= (δi,j J a,b ⊗ ξj−m−1 )L j=1 ,

(4.23)

we obtain the following topological dual OS -bases of gD S with respect to the inner product: F0 := { em a,b,i | (a, b, m, i) ∈ R },

F 0 := { ea,b,i | (a, b, m, i) ∈ R }. m

L D For A = (Ai )L i=1 , B = (Bi )i=1 ∈ gS , we introduce the following notation:

ρ(A) :=

L X

ρi (Ai ),

(4.24)

i=1 ◦ ◦ ◦ ρ(A)ρ(B) ◦

:=

L X

ρi ( ◦◦ Ai Bi ◦◦ ) +

i=1

X

ρi (Ai )ρj (Bj ).

(4.25)

i6=j

D Recall that, for θ = (θi (ξi )∂ξi )L i=1 ∈ TS , the Virasoro operator T {θ} acting on M is defined by (4.13), (4.12), (4.11) and (1.47). Using the dual bases above, we can represent the Virasoro operator T {θ} in the following form:

T {θ} = −

1 2κ

X

◦ a,b,i ◦ ρ(em

◦ ◦ θ)ρ(em a,b,i ) ◦ ,

(4.26)

(a,b,m,i)∈R

where we put L D ◦ θ := (ea,b,i ea,b,i m m,j (ξj )θj (ξj ))j=1 ∈ gS .

The formula (4.26) follows from the special cases with θ = (δi,j ξjn+1 ∂ξj )L j=1 for n ∈ Z, which are obtained by straightforward calculations. The bases which we really need later are however not these naively defined bases, F0 and F 0 , but “good” dual frames in the sense of [Ts]. See Lemma 4.15 and Remark 4.16 n below. In order to construct such dual bases, we define the meromorphic functions wa,b on H × C for n = 0, 1, 2, . . . by derivatives of wa,b (1.14): n n wa,b (t) = wa,b (τ ; t) :=

(−1)n ∂ n wa,b (τ ; t). n! ∂tn

(4.27)

n e can be regarded as a global section Then the meromorphic function wa,b (τ ; t − zi ) on X of the line bundle La,b (∗Qi ) on X and its Laurent expansion in ξi is written in the following form: n (ξi ) = ξi−n−1 + (−1)n wa,b,n + O(ξi ), (4.28) wa,b

where wa,b,n is the coefficient of tn in the Laurent expansion (1.15) of wa,b (t). For m ∈ OS ((ξj )) (a, b) ∈ (Z/N Z)2 r {(0, 0)}, i, j = 1, . . . , L, and m ∈ Z, we define fa,b,i,j by  if m = 0, δi,j ξjm m fa,b,i,j (ξj ) := (4.29) −m−1 wa,b (zj − zi + ξj ) if m < 0,

Twisted WZW Models on Elliptic Curves

33

and put m m m L = (Ja,b,i,j )L Ja,b,i j=1 := (Ja,b ⊗ fa,b,i,j (ξj ))j=1 , a,b,i m −m−1 a,b,i Jm = (Jm,j )j=1 := (J a,b ⊗ f−a,−b,i,j (ξj ))L j=1 .

(4.30)

Then the following topological OS -bases of gD S: m F1 := { Ja,b,i | (a, b, m, i) ∈ R },

a,b,i F 1 := { Jm | (a, b, m, i) ∈ R },

(4.31)

are dual to each other by virtue of the residue theorem. Changing the bases of expansion from F0 and F 0 to F1 and F 1 , we obtain another expression of T {θ} from (4.26): T {θ} = −

1 2κ

X

◦ a,b,i ◦ ρ(Jm

m ◦ θ)ρ(Ja,b,i ) ◦◦ ,

(4.32)

(a,b,m,i)∈R

where we put, as above, a,b,i a,b,i D ◦ θ := (Jm,j (ξj )θj (ξj ))L Jm j=1 ∈ gS .

The following lemma immediately follows from the definitions above and the decomposition (3.9) of gtw X to the direct sum of the line bundles La,b . Lemma 4.15. Under the notation above, we have the following: m m is equal to the image of Ja,b wa,b (τ ; t − zi ) ∈ gD 1. If m is negative, then Ja,b,i ˙ . X 2. If m is not negative and θ is a local section of TX˙D = π∗ TX/S (∗D) with θ = −m−1 a,b,i θt (τ ; z; t)∂t , then Jm ◦θ is equal to the image of J a,b w−a,−b (τ ; t−zi )θt (τ ; z; t) ∈ gD ˙ . X

Remark 4.16. This lemma means that the topological dual bases F1 and F 1 given by (4.31) are good dual frames of gD S in the sense of Tsuchimoto [Ts]. For fi ∈ OS ((ξi )), the regular part fi,+ and the singular part fi,− are uniquely defined by the conditions fi = fi,+ + fi,− ,

fi,+ ∈ OS [[ξi ]],

fi,− ∈ ξi−1 OS [ξi−1 ].

Note that the differentiation ∂ξi commutes the operations fi 7→ fi,± : ∂ξi (fi,± (ξi )) = (fi0 (ξi ))± , 0 which shall be denoted by fi,± (ξi ).

Lemma 4.17. For fi , gi ∈ OS ((ξi )), the normal product ◦◦ (J a,b ⊗ fi )(Ja,b ⊗ gi ) ◦◦ can be represented in the following forms: ◦ a,b ◦ (J

0 ⊗ fi )(Ja,b ⊗ gi ) ◦◦ = (J a,b ⊗ fi )(Ja,b ⊗ gi ) + kˆ Res(fi,+ (ξi )gi,− (ξi ) dξi ) ξi =0

= (Ja,b ⊗ gi )(J

a,b

0 ⊗ fi ) − kˆ Res(fi,− (ξi )gi,+ (ξi ) dξi ). ξi =0

34

G. Kuroki, T. Takebe

Proof. From the commutativity of Ja,b and J a,b and the definition of the normal product (1.45), we can find that ◦ a,b ◦ (J

⊗ fi )(Ja,b ⊗ gi ) ◦◦ = (J a,b ⊗ fi )(Ja,b ⊗ gi,+ ) + (Ja,b ⊗ gi,− )(J a,b ⊗ fi ).

Using the definition (3.19) of the Lie algebra structure on gˆ D S , we obtain the formulae
0 (ξi )fi (ξi ) dξi [J a,b ⊗ gi,− , Ja,b ⊗ fi ] = kˆ Res gi,− ξi =0
0 (ξi ) dξi , (4.33) = kˆ Res fi,+ (ξi )gi,− ξi =0
[J a,b ⊗ fi , Ja,b ⊗ gi,+ ] = kˆ Res fi0 (ξi )gi,+ (ξi ) dξi ξi =0
0 (ξi ) dξi . (4.34) = −kˆ Res fi,− (ξi )gi,+ ξi =0

These formulae prove the lemma. For the brevity of notation, we introduce the sets R± by R+ := { (a, b, m, i) ∈ R | m = 0 },

R− := { (a, b, m, i) ∈ R | m < 0 },

where R is defined by (4.21). LL Lemma 4.18. For θ = (θi (ξi )∂ξi )L i=1 ∈ i=1 OS ((ξi ))∂ξi , we have the following expressions of the Virasoro operator T {θ}:  X 1 X a,b,i m m a,b,i T {θ} = − ρ(Jm ◦ θ)ρ(Ja,b,i )+ ρ(Ja,b,i )ρ(Jm ◦ θ) , (4.35) 2κ R+

where the symbols

P

R−

denote the summations over (a, b, m, i) ∈ R± . Moreover, substi-

R±

tuting the definition (4.30) to this formula, we obtain 1 X X m ρj (J a,b ⊗ w−a,−b (zj − zi + ξj )θj (ξj )) ρi (Ja,b ⊗ ξim ) 2κ R+ j=1 (4.36)  X m a,b,i ρ(Ja,b,i )ρ(Jm ◦ θ) . + L

T {θ} = −

R−

Proof. From the definitions (4.27), (4.29), and (4.30), we can derive the following formulae: m ) = ρi (Ja,b ⊗ ξim ) ρ(Ja,b,i a,b,i ρ(Jm

◦ θ) = ρi (J

a,b

⊗

for (a, b, m, i) ∈ R+ , ξi−m−1 θi (ξi ))

−m−1 m Ja,b,i,i = Ja,b ⊗ wa,b (ξi ) −m−1 −m ∂ξi (wa,b,+ (ξ)) = mwa,b,+ (ξi )

for (a, b, m, i) ∈ R− , for (a, b, m, i) ∈ R− , for (a, b, m, i) ∈ R− .

Therefore, applying Lemma 4.17 to the formula (4.32), we obtain ˆ T {θ} = (the right-hand side of (4.35)) + kR(θ),

Twisted WZW Models on Elliptic Curves

35

where R(θ) is defined by X
−m Res (ξi−m−1 θi (ξi ))− (−m)wa,b,+ (ξi ) dξi , R(θ) := R−

ξi =0

which is a finite sum and hence is a local section of OS . Hence, for the proof of (4.35), it is enough to show that R(θ) = 0 in the case of θ = (δi,j ξj−l ∂ξj )L j=1 for l = 1, 2, 3, . . . and i = 1, . . . , L. Using the formula (4.27) and the Laurent expansion (1.15) of wa,b (t), we can find that X
−m Res (ξi−m−1 ξi−l )− (−m)wa,b,+ (ξi ) dξi m<0

=

ξi =0

l X n=1

ÿ =



Res ξin−l−1 n(−1)n ξi =0

l X n=1

X ν  ν=n

n

 wa,b,ν ξiν−n dξi 

 ! n l n (−1) n wa,b,l = −wa,b,1 0 n

if l = 1, if l > 2.

Hence we obtain R(θ) = 0 by (1.18). We have proved the first expression (4.35) of T {θ}.  Remark 4.19. The second expression (4.36) of T {θ} is very useful for the explicit calculations of the elliptic Knizhnik-Zamolodchikov connections (Sect. 5.2). We are ready to prove Lemma 4.14. Let v be in M and θ in TX˙D . Then, applying Lemma 4.15 to the first expression (4.35) of T {θ} in Lemma 4.18 shows that T {θ}v ∈ gD ˙ M. Hence we have proved Lemma 4.14. X 5. Flat Connections We keep all the notation in the previous section. In this section, we assume κ = k+h∨ 6= 0 and shall construct DS -module structures on the sheaf CC(M) of conformal coinvariants and on the sheaf CB(M) of conformal blocks. We can show as a direct consequence that, under the assumption of Lemma 3.5, the sheaves CC(M) and CB(M) are locally free coherent OS -modules (i.e., vector bundles on S) with flat connections and their fibers at s ∈ S are canonically isomorphic to the space of conformal coinvariants and that of conformal blocks respectively. In Sect. 5.2, we shall show that the connections on CCk (V ) and CBk (V ) coincide with the elliptic Knizhnik-Zamolodchikov equations introduced by Etingof [E]. In Sect. 5.3, we shall obtain a proof of the modular property of the connections without referring to the explicit expressions of them. 5.1. Construction of flat connections. Recall that we have the Lie algebra extension D D D VirX ˙ of the tangent sheaf TS by TX ˙ in (4.3). Since the action of TX ˙ maps M into D D gX˙ M due to Lemma 4.14, the representation of the Lie algebra VirX˙ on CC(M) given by Lemma 4.11 induces the Lie algebra action of TS on CC(M), which shall be denoted by (5.1) TS × CC(M) → CC(M), (µ, φ) 7→ Dµ φ. Moreover it immediately follows from (4.14) that

36

G. Kuroki, T. Takebe

Df µ v = f (Dµ v),

Dµ (f v) = µ(f )v + f (Dµ v)

for f ∈ OS , µ ∈ TS , and v ∈ CC(M). Thus we obtain the flat connection D on the sheaf CC(M) of conformal coinvariants. Because of CB(M) = HomOS (CC(M), OS ), we obtain the dual connection D∗ on CB(M): (Dµ∗ 8)(v) := µ(8(v)) − 8(Dµ v).

(5.2)

We can summarize the results as follows. Theorem 5.1. For each i = 1, . . . , L, let Mi be a representation with level k of the NL affine Lie algebra gˆ i satisfying the smoothness condition (3.24). Put M := i=1 Mi ⊕L ∧ and M := M ⊗ OS . Then M is a representation with level k of (g ) and M is a ∨ D gˆ D ˙ on M induces the S -module. Assume that κ = k + h 6= 0. Then the action of VirX DS -module structures on the sheaf CC(M) of conformal coinvariants and on the sheaf CB(M) of conformal blocks. Remark 5.2. We have another description of the dual connection D∗ on CB(M). D Lemma 4.13 shows that the representation of the Lie algebra VirX ˙ on CB(M) given by Lemma 4.12 induces the Lie algebra action of TS on CB(M), which coincides with the dual connection D∗ . Corollary 5.3. For each i = 1, . . . , L, let Vi be a finite-dimensional irreducible representation of g and Mi a quotient module of the Weyl module Mk (Vi ) over the affine Lie NL algebra gˆ i . Put M := i=1 Mi and M := M ⊗ OS . Assume that κ = k + h∨ 6= 0. Then the sheaf CC(M) of conformal coinvariants and the sheaf CB(M) of conformal blocks are locally free coherent OS -modules with flat connections on S and dual to each other. Proof. We have already shown the OS -coherencies of CC(M) and CB(M) in Corollary 3.5 and the existence of DS -module structures on CC(M) and on CB(M) in Theorem 5.1. It is well-known that any OS -coherent DS -module is OS -locally free. (See Theorem 6.1 in Chapter I of [Ho], Proposition 1.7 in Chapter VI of [BEGHKM], or Theorem 1.1.25 of [Bj].)  Corollary 5.4. For each i = 1, . . . , L, let Mi be a representation with level k of the NL affine Lie algebra gˆ i satisfying the smoothness condition (3.24). Put M := i=1 Mi and M := M ⊗ OS . Then the fiber of CC(M) at s = (τ ; z) ∈ S is canonically isomorphic L to the space of conformal coinvariants for (X; {Qi }L i=1 ) = (Xτ ; {qi (s)}i=1 ): CC(M)|s ∼ (5.3) = CC(M ). Moreover, under the assumptions of Corollary 5.3, the fiber of CB(M) at s ∈ S is canonically isomorphic to the space of conformal blocks: (5.4) CB(M)|s ∼ = CB(M ). −1 Proof. We can identify the restriction of gtw (s) with gtw in Sect. 1. Put X on X = π Q := D ∩ X. Then we can find the following canonical isomorphisms without using Corollary 5.3: (5.5) M|s = (M ⊗ OS )|s ∼ = M ⊗ C = M, tw 0 tw ∼ (5.6) (π∗ gX (mD)) s = H (X, g (mQ)) for m = 0, Q D tw 0 tw gX˙ s = (π∗ gX (∗D)) s ∼ (5.7) = H (X, g (∗Q)) = gX˙ , ∼ Q M. (5.8) (gD ˙ M) s = gX ˙ X

Twisted WZW Models on Elliptic Curves

37

The Riemann-Roch theorem shows that dimC H 0 (π −1 (s), gtw (mD)|π−1 (s) ) is a constant function of s ∈ S if m = 0. Therefore the existence of the isomorphism (5.6) follows from the Grauert theorem. (For the Grauert theorem, see, for example, Corollary 12.9 in Chapter III of [Har] and Theorem in Chapter 10 Sect. 5.3 (p. 209) of [GR].) The isomorphism (5.7) is obtained by the inductive limit of (5.6). The isomorphism (5.8) is obtained by using (5.5), (5.7), and applying the right exact functor (·)|s to the exact D sequence gD ˙ ⊗ OS M → g X ˙ M → 0. Similarly the isomorphism (5.3) is obtained from X (5.5) and (5.8). Under the assumption of Corollary 5.3, the sheaves CC(M) and CB(M) are locally free OS -modules of finite rank and dual to each other. Then we have CB(M)|s = HomC (CC(M)|s , C). This formula together with the isomorphism (5.3) gives the isomorphism (5.4).  Let us describe the connections D and D∗ more explicitly. For any vector field µ ∈ TS , the action of Dµ on CC(M) and that of Dµ∗ on CB(M) are described as follows. D Using the short exact sequence (4.3), we can lift µ to an element (µ; θ; 0) ∈ VirX ˙ at D least locally on S, and the ambiguity in the choice of (µ; θ; 0) is equal to TX˙ . But, since the actions of TX˙D on CC(M) and CB(M) are trivial, the actions of (µ; θ; 0) on CC(M) and CB(M) do not depend on the choice of the lift and give Dµ and Dµ∗ : Dµ v = (µ; θ; 0) · v = µ(v) + T {θ}v, Dµ∗ 8 = (µ; θ; 0) · 8 = µ(8) + T ∗ {θ}8

(5.9) (5.10)

D for (µ; θ; 0) ∈ VirX ˙ and v ∈ CC(M), and 8 ∈ CB(M). The following lemma provides D us with the explicit formulae of (µ; θ; 0) ∈ VirX ˙ .

Lemma 5.5. The following expressions in the coordinate (τ ; z; t) define vector fields in D VirX ˙ = π∗ TX,π (∗D) and are lifts of ∂zi and ∂τ respectively: aX˙ (∂zi ) = ∂zi ,

aX˙ (∂τ ) = ∂τ + Z(τ ; z; t)∂t ,

(5.11)

e satisfying the where the function Z(τ ; z; t) is a global meromorphic function on X following properties: 1. The poles of Z(τ ; z; t) are contained in π −1 (D); e/X X 2. The quasi-periodicity: Z(τ ; z; t + mτ + n) = Z(τ ; z; t) + m for (m, n) ∈ Z2 .

(5.12)

e sends ∂τ , ∂z , and ∂t to ∂τ + n∂t , ∂z , and Proof. Since the action of (m, n) ∈ Z2 on X i i ∂t respectively, the expressions (5.11) are vector fields in π∗ TX,π (∗D).  Example 5.6. We can use the following function as Z(τ ; z; t) for any i0 ∈ {1, . . . , L} (cf. Sect. 3.1 of [FW]): Z(τ ; z; t) = Z1,1 (τ ; t − zi0 ),

Z1,1 (τ ; t) = −

(See Appendix A for the notation of the theta functions.)

0 (t) 1 θ[0,0] . 2πi θ[0,0] (t)

(5.13)

38

G. Kuroki, T. Takebe

Lemma 5.7. The connections D on CC(M) and D∗ on CB(M) possess the following expressions: D∂/∂zi = ∂zi − ρi (T {∂ξi }),

(5.14)

D∂/∂τ = ∂τ + T {Z(τ ; z; t)∂t } = ∂τ +

L X

ρi (T {Z(τ ; z; zi + ξi )∂ξi }),

(5.15)

i=1

∗ = ∂zi − ρ∗i (T {∂ξi }), D∂/∂z i

(5.16)

∗ = ∂τ + T ∗ {Z(τ ; z; t)∂t } = ∂τ + D∂/∂τ

L X

ρ∗i (T {Z(τ ; z; zi + ξi )∂ξi }). (5.17)

i=1

Proof. In the local coordinate (τ ; z; ξj ) with ξj = t − zj , the vector fields given by (5.11) can be represented in the following forms around Qi : aX˙ (∂zi ) = ∂zi − δi,j ∂ξj ,

aX˙ (∂τ ) = ∂τ + Z(τ ; z; zj + ξj )∂ξj .

(5.18)

Namely, their images in VirSD are of the following forms: aX˙ (∂zi ) = (∂zi ; (−δi,j ∂ξj )L j=1 ; 0),

aX˙ (∂τ ) = (∂τ ; (Z(τ ; z; zj + ξj )∂ξj )L j=1 ; 0), (5.19)

whose actions on CC(M) and CB(M) can be written in the forms (5.14), (5.15), (5.16), and (5.17).  Remark 5.8. The explicit formulae (5.16) and (5.17) of the flat connection coincide with the expressions of ∇τ and ∇zi in Sect. 3.1 of [FW]. In order to prove the flatness of their connection, Felder and Wieczerkowski use the explicit expression of ∇zi , which corresponds to (5.24) in our case. However in our construction the flatness of the connection is a priori obvious. Lemma 4.5 is the key lemma for the proof of the flatness. Our proof of the flatness can be also applied to Proposition 3.4 of [FW]. 5.2. Elliptic Knizhnik-Zamolodchikov equations. In this subsection, we show that the flat connection on the sheaf of conformal blocks defined in Sect. 5.1 is nothing but the elliptic Knizhnik-Zamolodchikov equation introduced by Etingof [E], when M = Mk (V ). Let Vi be a finite-dimensional irreducible representation of g = slN (C) for each i NL and put V := i=1 Vi . In view of Proposition 3.4, the connections D on CCk (V ) and D∗ on CBk (V ) are regarded as connections on V ⊗ OS and on V ∗ ⊗ OS respectively and are dual to each other. In order to give the explicit expressions of these connections, we define the function Za,b (t) = Za,b (τ ; t) by wa,b (t) Za,b (t) := 4πi



0 0 (t) θ[a,b] θ[a,b] − θ[a,b] (t) θ[a,b]

 .

(5.20)

The point t = 0 is an apparent singularity of this function and we can analytically continue Za,b (t) to t = 0 by ÿ 1 Za,b (0) := 4πi

00 θ[a,b] − θ[a,b]



0 θ[a,b] θ[a,b]

2 ! .

(5.21)

Twisted WZW Models on Elliptic Curves

39

Theorem 5.9. As operators acting on V ⊗ OS and V ∗ ⊗ OS , the operators Dµ and Dµ∗ for µ = ∂/∂zi , ∂/∂τ possess the following expressions: D∂/∂zi =

∂ 1X − ∂zi κ

X

wa,b (zj − zi )ρj (Ja,b )ρi (J a,b ),

(5.22)

Za,b (zj − zi )ρj (Ja,b )ρi (J a,b ),

(5.23)

wa,b (zj − zi )ρ∗j (Ja,b )ρ∗i (J a,b ),

(5.24)

Za,b (zj − zi )ρ∗j (Ja,b )ρ∗i (J a,b ),

(5.25)

j6=i (a,b)6=(0,0)

D∂/∂τ =

L 1 X ∂ + ∂τ κ

X

∂ 1X + ∂zi κ

X

L 1 X ∂ − ∂τ κ

X

i,j=1 (a,b)6=(0,0)

∗ D∂/∂z = i

j6=i (a,b)6=(0,0)

∗ = D∂/∂τ

i,j=1 (a,b)6=(0,0)

where ρi (Ai ) and ρ∗i (Ai ) for Ai ∈ g act on the ith factor Vi of V and on the ith factor Vi∗ of V ∗ respectively. Here, for each (a, b) 6= (0, 0), the function wa,b (t) = wa,b (τ ; t) is defined by (1.14) and the function Za,b (t) = Za,b (τ ; t) by (5.20) and (5.21). Proof. Since the connections D and D ∗ are dual to each other, for the proof of the proposition it suffices to obtain either the formulae for Dµ (i.e., (5.22) and (5.23)) or those for Dµ∗ (i.e., (5.24) and (5.25)). The formulae (5.24) and (5.25) can be proved in the same way as Lemma 2.2 or the statements in Sect. 6 of [FFR]. But we shall give the proof of (5.22) and (5.23) using the expression (4.36) of T {θ} in Lemma 4.18. Let us fix v ∈ V ∗ ⊗ OS ⊂ M := Mk (V ). We rewrite ρi (T {∂ξi })v and T {Z(τ ; z; t)∂t }v in Lemma 5.7 modulo gD ˙ M in terms of operators acting on V , apX plying the Ward identity (3.22) in the following form: X n n (ξi ))v 0 ≡ − w−a,−b (zj − zi )ρj (J a,b )v 0 , (5.26) ρi (J a,b ⊗ w−a,−b j6=i

ρi (J a,b ⊗ ξ −n−1 )v 0 ≡ (−1)n+1 w−a,−b,n ρi (J a,b )v 0 X n − w−a,−b (zj − zi )ρj (J a,b )v 0 ,

(5.27)

j6=i

for v 0 ∈ V ⊗OS and n = 0. Here we used the Laurent expansion (4.28) and Lemma 4.15. First we prove (5.22) from (5.14). The formula (4.36) for θ = (δi,j ∂ξj )L j=1 together with Lemma 4.15 shows 1 X ρi (J a,b ⊗ w−a,−b (zi − zi0 + ξi ))ρi0 (Ja,b )v. (5.28) ρi (T {∂ξi })v ≡ − 2κ (a,b)6=(0,0) i0 =1,...,L

Applying the Ward identity (5.26) (n = 0) to the terms with i0 = i in the right-hand side of (5.28), we find that X X 1 w−a,−b (zj 0 − zi )ρj 0 (J a,b )ρi (Ja,b )v ρi (T {∂ξi })v ≡ 2κ (a,b)6=(0,0) j 0 6=i (5.29) X X 1 w−a,−b (zi − zi0 )ρi (J a,b )ρi0 (Ja,b )v. − 2κ 0 (a,b)6=(0,0) i 6=i

40

G. Kuroki, T. Takebe

Renumbering the indices of the first sum by (a, b) → 7 (−a, −b) and applying (1.17) to the second, we conclude that 1 X X wa,b (zj − zi )ρj (Ja,b )ρi (J a,b )v. ρi (T {∂ξi })v ≡ κ (a,b)6=(0,0) j6=i

This means that the operator D∂/∂zi acting on V ⊗ OS is represented as (5.22). The expression (5.23) of D∂/∂τ can be deduced from (5.15) in the similar manner, which is however more involved. Let us use here the function Z(t) = Z(τ ; z; t) given by Example 5.6. Then Z(t) is regular at t 6= zi0 and the Laurent expansion of Z(t) in ξi0 = t − zi0 is represented as Z(zi0 + ξi0 ) = −

1 −1 (ξ + Z1 ξi0 + O(ξi30 )), 2πi i0

Z1 :=

000 θ[0,0] . 0 3θ[0,0]

(5.30)

The formula (4.36) and Lemma 4.15 imply T {Z(τ ; z; t)∂t }v ≡ −

1 2κ

X

L X

va,b,i,j ,

(5.31)

(a,b)6=(0,0) i,j=1

where the vectors va,b,i,j are defined by va,b,i,j := ρj (J a,b ⊗ w−a,−b (zj − zi + ξj )Z(zj + ξj ))ρi (Ja,b )v. Using the Laurent expansions (1.15), (5.30), and the Ward identities (5.26) (n = 0) and (5.27) (n = 0, 1), we can find the following expressions for the vectors va,b,i,j : • If j 6= i and j 6= i0 , then va,b,i,j = w−a,−b (zj − zi )Z(zj )ρj (J a,b )ρi (Ja,b )v. • If j = i and j 6= i0 , then va,b,i,i ≡ Z 0 (zi )ρi (J a,b )ρi (Ja,b )v −

X

w−a,−b (zj 0 − zi )Z(zi )ρj 0 (J a,b )ρi (Ja,b )v.

j 0 6=i

• If j 6= i and j = i0 , then 1 0 (−w−a,−b,0 w−a,−b (zi0 − zi ) + w−a,−b (zi0 − zi ))ρi0 (J a,b )ρi (Ja,b )v 2πi 1 X w−a,−b (zi0 − zi )w−a,−b (zj 0 − zi0 )ρj 0 (J a,b )ρi (Ja,b )v. + 2πi 0

va,b,i,i0 ≡ −

j 6=i0

• If j = i and j = i0 , then 1 (2w−a,−b,1 − (w−a,−b,0 )2 + Z1 )ρi0 (J a,b )ρi0 (Ja,b )v va,b,i0 ,i0 ≡ − 2πi 1 X 0 (−w−a,−b (zj 0 − zi0 ) + w−a,−b,0 w−a,−b (zj 0 − zi0 ))ρj 0 (J a,b )ρi0 (Ja,b ) v. + 2πi 0 j 6=i0

1 0 Note that w−a,−b (t) = −w−a,−b (t). Substituting these expressions into (5.31), we obtain an expression for T {Z(τ ; z; t)∂t }v like

Twisted WZW Models on Elliptic Curves

T {Z(τ ; z; t)∂t }v = −

1 2κ

X

41

Za,b,i,j ρj (Ja,b )ρi (J a,b )v

with Za,b,j,i = Z−a,−b,i,j .

(a,b)6=(0,0) i,j=1,...,L

(5.32) Note that ρj (J a,b )ρi (Ja,b ) = ρj (J−a,−b )ρi (J −a,−b ). It is possible to compute all Za,b,i,j directly, but we take a short cut. Since the coefficients Za,b,i,j do not depend on the representations Vi and the choice of Z(t) = Z(τ ; z; t), for the determination of all Za,b,i,j , we have only to calculate Za,b,i0 ,j for j = 1, . . . , L. Picking up the terms which should be contained in Za,b,i0 ,j from the expressions for va,b,i,j , we obtain 1 (2wa,b,1 − (wa,b,0 )2 + Z1 ), 2πi = wa,b (zj − zi0 )Z(zj ) 1 0 (−wa,b,0 wa,b (zj − zi0 ) + wa,b − (zj − zi0 )), 2πi

Za,b,i0 ,i0 = − Za,b,i0 ,j

(5.33)

(5.34)

where we assume j 6= i0 . (Here we have used the formulae (1.17) and w−a,−b,ν = (−1)ν+1 wa,b,ν .) Note that we have 0 (t) wa,b

0

= wa,b (t)(log wa,b (t)) = wa,b (t)



0 0 (t) θ[0,0] θ[a,b] (t) − θ[a,b] (t) θ[0,0] (t)

 .

(5.35)

Substituting (1.14), (1.16), (5.35), and the expression of Z1 in (5.30) to (5.33) and (5.34), we can find the following results: Za,b,i0 ,i0 = −2Za,b (0) = −2Za,b (zi0 − zi0 ), Za,b,i0 ,j = −2Za,b (zj − zi0 ). These formulae complete the proof of (5.23).

(5.36) (5.37)



Remark 5.10. Etingof found in [E] that certain twisted traces F of vertex operators of ∗ b N satisfy D∗ sl ∂/∂zi F = 0 and D∂/∂τ F = 0, namely, that they are flat sections of the ∗ connection D . 5.3. Modular invariance of the flat connections. In [E], Etingof proved the modular invariance of the elliptic Knizhnik-Zamolodchikov equations by explicit computation. In this subsection, we give a more geometric proof of this fact without use of the explicit formulae of the connections. A similar property was proved for the non-twisted WZW model on elliptic curves in [FW]. For the brevity of notation, we introduce the following symbols: – Put 0 := SL2 (Z); – In the following, we assume that the symbol γ always denote an arbitrary matrix ( a bc d ) ∈ 0; – For s = (τ ; z) ∈ S, t ∈ C, and i = 1, . . . , L, put  z L z aτ + b i , zˇ = (zˇi )L = := , i=1 cτ + d cτ + d cτ + d i=1 t ξi tˇ := , ξˇi := , sˇ := (τˇ ; z). ˇ cτ + d cτ + d τˇ :=

42

G. Kuroki, T. Takebe

As is well-known, γ ∈ 0 acts on S by γ · s := sˇ = (τˇ ; z) ˇ

for s = (τ ; z) ∈ S.

(5.38)

tw 2 In order to define actions of 0 on X, Gtw X and gX , we first extend the actions of Z e on X, G, and g defined by (3.1), (3.3), and (3.4) respectively to those of the semi-direct e := Z2 o 0 defined by product group 0

(m, n; γ)(m0 , n0 ; γ 0 ) := (m + m0 d − n0 c, n − m0 b + n0 a; γγ 0 ) for (m, n; γ), (m0 , n0 ; γ 0 ) ∈ Z2 × 0. Then we have (m, n; 1)(0, 0; γ) = (0, 0; γ)(ma + nc, mb + nd; 1)

(5.39)

for (m, n) ∈ Z2 and γ ∈ 0. e is a standard one. An element γ of 0 acts on X e as e on X The action of 0 γ · (τ ; z; t) := (τˇ ; z; ˇ tˇ)

e for (τ ; z; t) ∈ X.

e e on X. This action together with (3.1) induces an action of 0 e The actions of 0 on G and g are defined via the action of 0 on the Heisenberg group HN , the central extension of (Z/N Z)2 defined by HN := C× × (Z/N Z)2 . Here the group structure of HN is given by 0

(r; m, n)(r0 ; m0 , n0 ) := (rr0 εnm ; m + m0 , n + n0 ) for (r; m, n), (r0 ; m0 , n0 ) ∈ HN . The Heisenberg group HN is isomorphic to the group ˆ and rˆ for r ∈ C× with defining relations generated by the symbols α, ˆ β, ˆ αˆ N = βˆ N = 1, αˆ βˆ = εβˆ α, ˆ ˆ rˆαˆ = αˆ r, ˆ rˆβ = β r, ˆ rd 1 r2 = rˆ1 rˆ2

for r, ri ∈ C× ,

(5.40)

where the identification of the group with HN is given by (r; m, n) = rˆβˆ m αˆ n .

(5.41)

Thus the matrices α and β given by (1.2) define a representation of HN on CN by αv ˆ = αv,

ˆ = βv, βv

rv ˆ = rv

for v ∈ CN and r ∈ C× .

Note that this representation is irreducible. For γ ∈ 0 = SL2 (Z), we can define the group automorphism (·)γ of HN as follows: – If N is odd, then we put (r; 0, 0)γ := (r; 0, 0), (1; 1, 0)γ := (1; a, b), for r ∈ C× .

(1; 0, 1)γ := (1; c, d)

(5.42)

Twisted WZW Models on Elliptic Curves

43

– If N is even, then we put (r; 0, 0)γ := (r; 0, 0), √ √ (1; 1, 0)γ := (( ε)ab ; a, b), (1; 0, 1)γ := (( ε)cd ; c, d) √ for r ∈ C× , where ε = exp(πi/N ), a primitive (2N )−th root of unity.

(5.43)

The fact that this defines an automorphism of HN follows from the presentation (5.40) and (5.41). Note that this action of 0 on HN induces that of 0 on (Z/N Z)2 given by (m, n)γ = (ma + nc, mb + nd) for (m, n) ∈ (Z/N Z)2 and γ ∈ 0 (cf. (5.39)). Twisting the representation of HN on CN by γ ∈ 0, we obtain another irreducible representation of HN on CN : HN × CN → CN ,

(h, v) 7→ hγ v.

Since the Heisenberg group HN has a unique irreducible representation, up to isomorphism, in which rˆ ∈ HN for r ∈ C× acts as multiplication by r (the theorem of von Neumann and Stone), using the Schur lemma, we can find that there is xγ ∈ GLN (C), uniquely determined up to scalar multiplications, such that hxγ v = xγ hγ v

for h ∈ HN and γ ∈ 0.

(5.44)

The mapping γ 7→ xγ induces a group homomorphism from 0 = SL2 (Z) into P GLN (C), which does not depend on the choices of xγ ’s. In the following, we take xγ from G = SLN (C), which is uniquely determined up to factor ±1 by γ.   0 1 Example 5.11. For γ = , xγ ∝ (ε−(a−1)(b−1) )N a,b=1 . −1 0 Example 5.12. When N is odd, we can choose xγ = 1 for γ ∈ 0(N ) = { γ ∈ 0 | γ ≡ 1 mod N }. e on G and g are now defined by The desired actions of 0 (m, n; γ) · g := (β m αn xγ )g(β m αn xγ )−1 (m, n; γ) · A := (β m αn xγ )A(β m αn xγ )−1

e and g ∈ G, for (m, n; γ) ∈ 0 e and A ∈ g. for (m, n; γ) ∈ 0

These are extensions of the actions of Z2 given by (3.3) and (3.4). e × G and X e × g are defined by e on X The diagonal actions of 0 γ˜ · (x; ˜ g) := (γ˜ · x; ˜ γ˜ · g) γ˜ · (x; ˜ A) := (γ˜ · x; ˜ γ˜ · A)

e × G and γ˜ ∈ 0, e for (x; ˜ g) ∈ X e × g and γ˜ ∈ 0. e for (x; ˜ A) ∈ X

Note that these actions also do not depend on the choice of xγ . e defined above, we obtain the induced actions of 0 on X, Gtw , From the actions of 0 X tw and gX defined by γ · x := πX ˜ e/X (γ · x)

for x = πX ˜ ∈ X, e/X (x)

γ · g tw := [γ · (x; ˜ g)]

for g tw = [(x; ˜ g)] ∈ Gtw X,

γ · Atw := [γ · (x; ˜ A)]

for Atw = [(x; ˜ A)] ∈ gtw X,

44

G. Kuroki, T. Takebe

e (x; e × g are representatives of x ∈ X, where x˜ ∈ X, ˜ g) ∈ X × G, and (x; ˜ A) ∈ X tw tw e ˜ A) ∈ X × g respectively and γ ∈ 0 is identified with (0, 0; γ) ∈ 0. g ∈ GX , and (x; tw tw e (For the definitions of X, GX , and gX , see (3.2) and (3.5).) The projections X  S, gtw X  X, etc. are equivariant with respect to the actions of 0. Moreover we obtain the following induced equivariant actions of γ ∈ 0 on TS , gD S, and VirSD : – The biholomorphic map γ : S → S induces the Lie algebra isomorphism (·)γ : ∼ γ −1 TS → TS of vector fields: ÿ ! L L X X ˇ (cτ + d)2 ∂τ + c(cτ + d)zi ∂zi + µi (s)(cτ ˇ + d)∂zi (5.45) µγ := µ0 (s) i=1

i=1

PL

for µ = µ0 (s)∂τ + i=1 µi (s)∂zi ∈ γ −1 TS . (Formally µγ is obtained by the substitution of sˇ = (τˇ ; z) ˇ in s = (τ ; z).) ∼ D ˆ S is defined by – The Lie algebra isomorphism (·)γ : γ −1 gˆ D S →g

ˆ γ := (x−1 (Ai (s; ξi ))L ˇ ξˇi )xγ )L ˇ kˆ (5.46) i=1 ; f (s)k γ Ai (s; i=1 ; f (s) for Ai (s; ξi ) ∈ γ −1 (g ⊗ OS ((ξi ))) and f (s) ∈ γ −1 OS . ∼ – The Lie algebra isomorphism (·)γ : γ −1 VirSD → VirSD is defined by
γ
c := µγ ; (µ0 (s)c(cτ µ; (θi )L ˇ + d)ξi ∂ξi + θi (s; ˇ ξˇi )(cτ + d)∂ξi )L ˇc i=1 ; f (s)ˆ i=1 ; f (s)ˆ (5.47) PL for µ = µ0 (s)∂τ + i=1 µi (s)∂zi ∈ γ −1 TS , θi = θi (s; ξi )∂ξi ∈ γ −1 (OS ((ξi ))∂ξi ), and f (s) ∈ γ −1 OS . The formula (5.47) reflects the fact that the vector field ˇ τˇ + µ0 (s)∂

L X

µi (s)∂ ˇ zˇi + θi (s; ˇ ξˇi )∂ξˇi

i=1

represented in the local coordinate (τˇ ; z; ˇ ξˇi ) is equal to ÿ ! L X 2 ˇ (cτ + d) ∂τ + c(cτ + d)zi ∂zi + c(cτ + d)ξi ∂ξi µ0 (s) i=1

+

L X

µi (s)(cτ ˇ + d)∂zi + θi (s; ˇ ξˇi )(cτ + d)∂ξi

i=1

represented in the local coordinate (τ ; z; ξi ). Therefore the following lemma is a direct consequence of the definitions above. Lemma 5.13. For γ ∈ 0, the isomorphisms above satisfy the following: 1. The isomorphisms induce the following Lie algebra isomorphism:
∼ D ˆD (·)γ : γ −1 VirSD n gˆ D S → VirS n g S. γ D 2. (γ −1 gD ˙ ) = gX ˙ . X

Twisted WZW Models on Elliptic Curves

45

D γ D −1 D γ 3. (γ −1 VirX TX˙ ) = TX˙D . ˙ ) = VirX ˙ and (γ

Under these preparations, we show the modular property of the sheaf of conformal coinvariants CC(M) and the sheaf of conformal blocks CB(M) coming from quotients of the Weyl modules. For each i = 1, . . . , L, let Vi be a finite-dimensional irreducible representation of g NL and Mi a quotient of the Weyl module Mk (Vi ). Put M := i=1 Mi and M := M ⊗ OS . We denote by ci the eigenvalue of the Casimir operator Ci acting on Vi given by (2.4) and put 1i := κ−1 ci . The Virasoro operator ρi (T [0]) acting on M is diagonalizable and each of its eigenvalues is of the form 1i + m, where m is a non-negative integer. Thus, fixing branches of the holomorphic functions (cτ + d)Ci /κ on the upper half plane H, QL we obtain an operator (cτ + d)ρ(T [0]) = i=1 (cτ + d)ρi (T [0]) acting on M, where we put PL ρ(T [0]) := i=1 ρi (T [0]). ∼ For γ ∈ 0, define the isomorphism (·)γ : γ −1 M → M by ρ(T [0]) v(s) ˇ v(s)γ := x−1 γ (cτ + d)

for v(s) ∈ γ −1 M,

(5.48)

regarding x−1 γ as an automorphism of M through the natural diagonal action of G = ∼ SLN (C) on M . This isomorphism (5.48) induces the isomorphism (·)γ : γ −1 (M∗ ) → ∗ M given by ρ 8(s)γ := x−1 γ (cτ + d)

where we put ρ∗ (T [0]) :=

PL i=1

∗

(T [0])

8(s) ˇ

for 8(s) ∈ γ −1 (M∗ ),

(5.49)

ρ∗i (T [0]).

Lemma 5.14. For P ∈ VirSD n gˆ D S , v ∈ M, and γ ∈ 0, we have (P · v)γ = P γ · v γ . Proof. Since ρi (T {ξi ∂ξi }) = −ρi (T [0]), we have the following identity of operators acting on M (cf. (5.47)): µ0 (s)(cτ ˇ + d)2 ∂τ +

L X

ρi (T {µ0 (s)c(cτ ˇ + d)ξi ∂ξi })

i=1

(5.50)

ˇ + d)2 ∂τ ) · (cτ + d)−ρ(T [0]) . = (cτ + d)ρ(T [0]) · (µ0 (s)(cτ Similarly, the definition of VirSD n gˆ D S (4.10) leads to the following identities: e−rρi (T [0]) ρi (Ai ⊗ fi (ξi ))erρi (T [0]) = ρi (Ai ⊗ fi (er ξi )), e−rρi (T [0]) ρi (θi (ξi )∂ξi )erρi (T [0]) = ρi (θi (er ξi )e−r ∂ξi ), g ρi (θi (ξi )∂ξi ) = ρi (θi (ξi )∂ξi ) g

(5.51) (5.52) (5.53)

for Ai ⊗ fi (ξi ) ∈ g ⊗ OS ((ξi )), r ∈ C, θi (ξi )∂ξi ∈ O((ξi ))∂ξi , and g ∈ G. Hence we have
ρ(T [0]) ˆ γ · x−1 (cτ + d)−ρ(T [0]) xγ · (Ai (s; ξi ))L i=1 ; f (s)k γ (cτ + d) (5.54)
= (Ai (s; ˇ ξi ))L ˇ kˆ , i=1 ; f (s) for Ai (s; ξi ) ∈ γ −1 (g ⊗ OS ((ξi ))), f (s) ∈ γ −1 OS because of (5.46), and

46

G. Kuroki, T. Takebe

(cτ + d)−ρ(T [0]) xγ · µ; (θi (s; ξi ))L c i=1 ; g(s)ˆ
γ L = µ ; (θi (s; ˇ ξi )∂ξi )i=1 ; g(s)ˆ ˇc


γ

ρ(T [0]) · x−1 γ (cτ + d)

(5.55)

PL for µ = µ0 (s)∂τ + i=1 µi (s)∂ξi ∈ γ −1 TS , θi (s; ξi )∂ξi ∈ γ −1 (OS ((ξi ))∂ξi ), g(s) ∈ γ −1 OS because of (5.47). These formulae prove the lemma.  From Lemma 5.13 and Lemma 5.14 follow the modular invariance of CC(M) and CB(M), and the modular transformations of connections D (5.1) and D∗ (5.2). The results are summarized in the following theorem. Theorem 5.15. For each i = 1, . . . , L, let Vi be a finite-dimensional irreducible repreNL sentation of g and Mi a quotient of the Weyl module Mk (Vi ). Put M := i=1 Mi and ∼ M := M ⊗ OS . Let γ be in 0 = SL2 (Z). Then the isomorphisms (·)γ : γ −1 M → M γ −1 ∗ ∼ ∗ and (·) : γ (M ) → M defined by (5.48) and (5.49) induce the isomorphisms ∼

ρ(T [0]) v(s) 7→ x−1 v(s), ˇ γ (cτ + d)

∼

ρ 8(s) 7→ x−1 γ (cτ + d)

(·)γ : γ −1 CC(M) → CC(M), (·)γ : γ −1 CB(M) → CB(M),

∗

(T [0])

8(s), ˇ

where xγ is an element of G satisfying (5.44). Furthermore these isomorphisms correspond local flat sections with respect to the connections D and D∗ to local flat sections. Namely, denoting the subsheaf of local flat sections of CC(M) and CB(M) by CC(M)D D∗ and CB(M) respectively, we obtain the following induced isomorphisms: ∼

(·)γ : γ −1 (CC(M)D ) → CC(M)D , ∗

∼

∗

(·)γ : γ −1 (CB(M)D ) → CB(M)D . The modular transformations of the connections are represented as D µγ = µγ +

L X

ρi (T {θi (s; ˇ ξˇi )∂ξˇi }) − µ0 (s) ˇ c(cτ + d)ρ(T [0])

i=1

  ˇ c(cτ + d)ρ(T [0]), = Dµ s7→sˇ − µ0 (s) Dµ∗ γ = µγ +

L X

(5.56)

ρ∗i (T {θi (s; ˇ ξˇi )∂ξˇi }) − µ0 (s) ˇ c(cτ + d)ρ∗ (T [0])

i=1

  ˇ c(cτ + d)ρ∗ (T [0]), = Dµ∗ s7→sˇ − µ0 (s)

(5.57)

PL where µ = µ0 (s)∂τ + i=1 µi (s)∂zi ∈ γ −1 TS , θi = θi (s; ξi )∂ξi ∈ γ −1 (OS ((ξi ))∂ξi ), −1 D (µ; (θi )L (VirX ˇ in s, namely, ˙ ), and [ · ]s7→sˇ denotes the substitution of s i=1 ; 0) ∈ γ γ ˇ for µ ∈ γ −1 TS and A(s) ∈ γ −1 EndOS (CC(M)) or A(s) ∈ [µ + A(s)]s7→sˇ = µ + A(s) γ −1 EndOS (CB(M)). Indeed (5.56) and (5.57) follow from the explicit expressions (5.9) and (5.10) and the definition (5.47). Using Proposition 3.4, we can identify CCk (V ) and CBk (V ) with V ⊗ OS and PL V ∗ ⊗ OS respectively. Put 1 := i=1 1i . Then the theorem above implies the following corollary.

Twisted WZW Models on Elliptic Curves

47

Corollary 5.16. Then, under the identifications CCk (V ) = V ⊗OS and CBk (V ) = V ∗ ⊗ ∼ ∼ OS , the isomorphisms (·)γ : γ −1 CCk (V ) → CCk (V ) and (·)γ : γ −1 CBk (M) → CBk (V ) are of the following forms: ∼

(·)γ : γ −1 (V ⊗ OS ) → V ⊗ OS , ∼

(·)γ : γ −1 (V ∗ ⊗ OS ) → V ∗ ⊗ OS ,

1 v(s) 7→ x−1 ˇ γ (cτ + d) v(s), −1 F (s) 7→ x−1 F (s). ˇ γ (cτ + d)

The modular transformations of the connections expressed as in Theorem 5.9 are represented as   ˇ + d)1, (5.58) Dµγ = Dµ s7→sˇ − µ0 (s)c(cτ  ∗ ∗ Dµγ = Dµ s7→sˇ + µ0 (s)c(cτ ˇ + d)1, (5.59) PL where µ = µ0 (s)∂τ + i=1 µi (s)∂ξi ∈ γ −1 TS and [ · ]s7→sˇ denotes the substitution of sˇ in s, namely, [µ + A(s)]s7→sˇ = µγ + A(s) ˇ for µ ∈ γ −1 TS and A(s) ∈ γ −1 EndOS (V ⊗ OS ) −1 ∗ or A(s) ∈ γ EndOS (V ⊗ OS ). Example 5.17. Applying (5.59) to µ = ∂zi and µ = ∂τ , we obtain i h ∗ ∗ D(∂/∂z = D , γ ∂/∂zi i) h i s7→sˇ ∗ ∗ D(∂/∂τ + c(cτ + d)1, )γ = D∂/∂τ s7→sˇ

which were found by Etingof [E], Sect. 4.

6. Concluding Remarks We have examined a twisted WZW model on elliptic curves which gives the XYZ Gaudin model at the critical level and Etingof’s elliptic KZ equations at the off-critical level. We make several comments on the related interesting problems to be solved. Factorization. We have studied the twisted WZW model only on a family of smooth pointed elliptic curves, but as in [TUY] we can also consider the model on a family of stable pointed elliptic curves. By extending the connections acting on the sheaves of conformal blocks to those with regular singularities at the boundary of the family, we shall be able to establish the equivalence between our geometric approach and Etingof’s approach by means of twisted traces of the products of twisted vertex operators on the Riemann sphere. Furthermore we shall be able to obtain a dimension formulae for the spaces of conformal blocks. A detailed investigation shall be given in a forthcoming paper. Generalization to higher genus. Bernard generalized the KZB equations to higher genus Riemann surfaces in [Be2]. In [Fe], Felder established the geometric interpretation of the KZB equations on Riemann surfaces by the non-twisted WZW models clarifying the notion of the dynamical r-matrices in higher genus cases. Our formulation for the twisted WZW model is also valid for arbitrary Riemann surfaces. See Appendix C for details.

48

G. Kuroki, T. Takebe

Discretization. Felder calls his interpretation of the KZB equations in [Fe] “the first step of the ‘St. Petersburg q-deformation recipe’ in higher genus cases”. We hope that our twisted WZW model on elliptic curves can also be q-deformed. The resulting “elliptic q-KZ equations”, for example, would be related to the difference equations proposed in [Tak2]. Intertwining vectors. Note that the Boltzmann weights of the A(1) N −1 face model can be expressed by the elliptic quantum R-matrix and the intertwining vectors ([Ba, JMO, DJKMO]). Therefore it can be expected that there exists a quasi-classical limit of the intertwining vectors by means of which the relation of the non-twisted and twisted WZW models on elliptic curves will be clarified. In addition, the intertwining vectors play an important role in constructing Bethe vectors of the XYZ spin chain models ([Ba, Tak1]) and the integral solution of the difference equations in [Tak2]. They are introduced as a kind of technical tool there, but our approach from the twisted WZW model, i.e., from the classical limit should reveal their algebro-geometric meaning.

Appendix A. Theta functions with characteristics Here we collect properties of theta functions of one variable used in this paper. Following [M], we use the notation: X 2 0 eπi(n+κ) τ +2πi(n+κ)(t+κ ) θκ,κ0 (t; τ ) =

(A.1)

n∈Z

for the theta functions with characteristics. Here t is a complex number, τ belongs to the upper half plane H and κ, κ0 are rational numbers. They are related with each other by shifts of t: 0

0

θκ1 +κ2 ,κ01 +κ02 (t; τ ) = eπiκ2 τ +2πiκ2 (t+κ1 +κ2 ) θκ1 ,κ01 (t + κ2 τ + κ02 ; τ ). 2

(A.2)

Since the zero set of θ00 (t; τ ) is {1/2 + τ /2} + Z + Zτ , the zero set of θκ,κ0 (t; τ ) is     1 1 − κ0 + − κ τ + Z + Zτ, (A.3) 2 2 because of (A.2). Fundamental properties of the function θκ,κ0 are the quasi-periodicity with respect to t: θκ,κ0 (t + 1; τ ) = e2πiκ θκ,κ0 (t; τ ), 0

θκ,κ0 (t + τ ; τ ) = e−πiτ −2πi(t+κ ) θκ,κ0 (t; τ ),

(A.4)

and the automorphic property: θκ,κ0 (t; τ + 1) = e−πiκ(κ+1) θκ,κ+κ0 + 1 (t; τ ), 2   1 t 1/2 2πiκκ0 πit2 /τ ;− = (−iτ ) e e θκ0 ,−κ (t; τ ). θκ,κ0 τ τ The formulae

(A.5)

Twisted WZW Models on Elliptic Curves

49

θ−κ,−κ0 (t; τ ) = θκ,κ0 (−t; τ ), θκ+m,κ0 +n (t; τ ) = e

2πiκn

(A.6)

θκ,κ0 (t; τ ),

(A.7)

are easily deduced from the definition (A.1), where m and n are integers. We use mostly the following special characteristics. Let N = 2 be a natural number and a, b arbitrary integers. We denote θ[a,b] (t; τ ) := θ a − 1 ,− b + 1 (t; τ ). N

The standard abbreviations, 0 θ[a,b]

θ[a,b] := θ[a,b] (0; τ ),

N

2

(A.8)

2

d θ[a,b] (t; τ ) , := dt t=0

00 etc. likewise defined are also used. and θ[a,b]

Lemma A.1.

000 1 N 2 − 1 θ[0,0] − 0 6 θ[0,0] 2

X (a,b)6=(0,0)

00 θ[a,b] = 0, θ[a,b]

(A.9)

where the indices in the second term run through a = 0, . . . , N − 1, b = 0, . . . , N − 1 and (a, b) 6= (0, 0). This is a generalization of the well-known formula 000 θ1/2,1/2 0 θ1/2,1/2

=

00 θ1/2,0

θ1/2,0

+

00 00 θ0,1/2 θ0,0 + , θ0,0 θ0,1/2

(A.10)

which is the case N = 2 of Lemma A.1. Proof. It is easy to show that N

−1 N −1 N Y Y a=0

 θ[a,b] (t; τ ) = 

b=0

Y

 θ[a,b]  θ[0,0] (N t; τ ).

(A.11)

(a,b)6=(0,0)

In fact we have only to compare the periodicity and zeros of both sides by using (A.4) and (A.3). The over-all coefficient can be determined by the first term (namely the coefficient of t) in the Taylor expansion around t = 0. The coefficients of t2 of the Taylor expansion of (A.11) give X (a,b)6=(0,0)

0 θ[a,b] = 0, θ[a,b]

and using this equality, we can rewrite the terms of order t3 in (A.11) as follows: 000 1 N 2 − 1 θ[0,0] − 0 6 θ[0,0] 2

X (a,b)6=(0,0)

00 θ[a,b] =− θ[a,b]

X (a,b)6=(0,0)



0 θ[a,b] θ[a,b]

2 .

(A.12)

Therefore, in order to prove Lemma A.1, we have to show that the right-hand side of (A.12) is zero. Let us denote it by f (τ ) as a function of τ . It has the following properties:

50

G. Kuroki, T. Takebe

– f (τ ) is a holomorphic function on the upper half plane H. ((A.3)) – f (τ ) is bounded when Im τ → +∞. ((A.1)) – f (τ + 1) = f (τ ), f (−1/τ ) = τ 2 f (τ ). ((A.5), (A.7)) Hence f (τ ) is an integral modular form of weight 2 and level 1, which is nothing but zero. (See, for example, Th´eor`em 4 (i), Sect. 3 Chapitre VII of [Se1], Proposition 2.26 of [Sh] or Theorem 14 in Chapter II of [Sc].) This proves the lemma. 

Appendix B.

The Kodaira-Spencer map of a family of Riemann surfaces

Let π : X → S be a family of compact Riemann surfaces over a complex manifold S and qi : S → X a holomorphic section of π for each i = 1, . . . , L. Put Qi := qi (S) and SL assume that Qi ∩ Qj = ∅ if i 6= j. Then D := i=1 Qi is a divisor of X e´ tale over S. We call (π : X → S; q1 , . . . , qL ) a family of pointed compact Riemann surfaces. Denote by TX (− log D) the sheaf of vector fields tangent to D. As in Sect. 4.1, let TX,π (− log D) be the inverse image of π −1 TS ⊂ π ∗ TS in TX (− log D). Then we obtain the following short exact sequence: 0 → TX/S (−D) → TX,π (− log D) → π −1 TS → 0, which is a Lie algebra extension. The derived direct image of this sequence produces the following long exact sequence: · · · → π∗ TX,π (− log D) → TS → R1 π∗ TX/S (−D) → R1 π∗ TX,π (− log D) → · · · . The connecting homomorphism TS → R1 π∗ TX/S (−D) is called the Kodaira-Spencer map of the family (π : X → S; q1 , . . . , qL ). For an OX -module F and a closed analytic subspace Z of X, denote by FZ∧ the completion of F at Z. Consider the following exact sequences: ∧ ∗ ∧ 0 → (TX/S (−D))∧ D → (TX (− log D))D → (π TS )D → 0, ∧ ∗ ∧ 0 → (TX/S (∗D))∧ D → (TX (∗D))D → (π TS )D → 0.

∧ ∧ As above, the inverse images of π −1 TS |D ⊂ (π ∗ TS )∧ D in (TX (− log D))D and (TX (∗D))D is denoted by Tπ (− log D) and Tπ (∗D) respectively. Then we obtain the Lie algebra extensions below: −1 TS |D → 0, 0 → (TX/S (−D))∧ D → Tπ (− log D) → π −1 0 → (TX/S (∗D))∧ TS |D → 0. D → Tπ (∗D) → π

The direct images of these exact sequences to S are also exact. Then we have the following commutative diagram:

Twisted WZW Models on Elliptic Curves

0   y 0 −→

−D π∗ TX/S   y

0   y −→

TX∗D ⊕ T −D ˙   y

51

0   y

(])   y

−D −→ T ∗D −→ R1 π∗ TX/S −→ 0,   y

− log D − log D ∗D −→ Tπ, −→ Tπ∗D 0 −→ π∗ TX,π ˙ ⊕ Tπ X       y y y

0 −→

TS   y

−→

(])

TS ⊕ (TS )L   y 0

(B.13)

−→ (TS )L   y 0

where we put −D := TX/S (−D), TX/S

− log D TX,π := TX,π (− log D),

TX∗D := π∗ TX/S (∗D), ˙

T −D := π∗ (TX/S (−D))∧ D,

T ∗D := π∗ (TX/S (∗D))∧ D,

Tπ− log D := π∗ Tπ (− log D),

Tπ∗ := π∗ Tπ (∗D).

∗D Tπ, ˙ := π∗ TX,π (∗D), X

The horizontal and vertical sequences in the diagram (B.13) are all exact and the arrow −D from TS to R1 π∗ TX/S = R1 π∗ TX/S (−D) through (]) is nothing but the Kodaira-Spencer map, which is described as follows. For µ ∈ TS , chasing the diagram above, we can choose (aX˙ , a+ ), α and [α] in order: − log D ∗D , whose image in TS ⊕ (TS )L is equal to (µ; (µ)L 1. (aX˙ , a+ ) ∈ Tπ, ˙ ⊕ Tπ i=1 ) ∈ X L TS ⊕ (TS ) ; 2. α ∈ T ∗D , whose image in Tπ∗D is equal to aX˙ − a+ ∈ Tπ∗D ; −D −D , which is the image of α in R1 π∗ TX/S . 3. [α] ∈ R1 π∗ TX/S

Then the cohomology class [α] ∈ R1 π∗ TX/S (−D) does not depend on the choice of (aX˙ , a+ ) and α. The Kodaira-Spencer map sends µ ∈ TS to [α] ∈ R1 π∗ TX/S (−D). ∗D D D and Tπ, Recall that TX∗D ˙ are denoted by TX ˙ ˙ and VirX ˙ respectively in Sect. 5. The short X exact sequence (4.3) is included in the second vertical exact sequence in (B.13) and the D lifting from TS to VirX ˙ (cf. Sect. 5.1) essentially corresponds to the operation 1 above. Hence we can see that the constructions in Sect. 5 originate in the above description of the Kodaira-Spencer map. S by OS cˆ. This is However we considered not only Tπ∗D but also its extension VirD a difference between the description of the Kodaira-Spencer map and the constructions in Sect. 5. We remark that Beilinson and Schechtman give the intrinsic (i.e., coordinatefree) description of the Virasoro algebras in [BS]. Appendix C. On a formulation for higher genus Riemann surfaces In this appendix, we shall comment on a formulation of twisted Wess-Zumino-Witten models on higher genus Riemann surfaces.

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G. Kuroki, T. Takebe

P Let π : X → S, qi : S → X, Qi = qi (S), and D = Qi be the same as Appendix B. Suppose that, for each i, we can take a holomorphic function ξi on an open neighborhood Ui of Qi with the property that the mapping Ui → S ×ξi (Ui ) given by x 7→ (π(x), ξi (x)) is biholomorphic and ξi (Qi ) = {0}. Then, in precisely the same way as Sect. 5, we can define TX,π (∗D), TX/S (∗D), D D VirSD , TX˙D , VirX ˙ , etc. We define the action of π∗ TX,π (∗D) on VirS by aX˙ · α := [aX˙ , α]

D D for aX˙ ∈ VirX ˙ , α ∈ VirS ,

(C.14)

D D where VirX ˙ is identified with its image in VirS and the bracket of the right-hand side D is the Lie algebra structure of VirS . D D D We remark that the embedding VirX ˙ and that of TX ˙ into VirS are not always Lie D D algebra homomorphisms and the action of VirX˙ on VirS does not always preserve TX˙D . Thus we must add a supplementary structure on the Riemann surface. Suppose that we can take an open covering {Uλ }λ∈3 of X and a family {ξλ : Uλ → C}λ∈3 of holomorphic functions satisfying the following properties:

1. For each λ ∈ 3, the mapping Uλ → S × C given by x 7→ (π(x), ξλ (x)) is a biholomorphic mapping from Uλ onto an open subset of S × C. 2. For any λ, λ0 ∈ 3 with Uλ ∩Uλ0 6= ∅, there exists a, b, c, d ∈ OS (S) with the property that ξλ0 = (aξλ + b)/(cξλ + d) on Uλ ∩ Uλ0 . We call {ξλ }λ∈3 a projective structure on the family π : X → S of Riemann surfaces. Moreover assume that {ξλ }λ∈3 ∪ {ξi }L i=1 is also a projective structure on the family. D D D Lemma C.2. Under the assumption above, the action of VirX ˙ on VirS preserves TX ˙ and in particular the embedding TX˙D ,→ VirSD is a Lie algebra homomorphism. D D Proof. It suffices to show that cV (θ, η) = 0 for (µ; θ; 0) ∈ VirX ˙ and (0; ν; 0) ∈ TX ˙ . For this purpose, as in the proof of Lemma 4.5, it is enough to show that, for 1 D D (µ; θ; 0) ∈ VirX ˙ and (0; ν; 0) ∈ TX ˙ , we can take ω ∈ π∗ X/S (∗D) with the property 000 that ω = θ (ξi )η(ξi ) dξi near Qi . To do this, we calculate the transformation property of θ000 (ξλ )η(ξλ ) dξλ under coordinate changes. Take any ξ, ζ from {ξλ }λ∈3 ∪ {ξi }L i=1 . Then there is a, b, c, d ∈ OS (S) with ζ = (aξ +b)/(cξ +d). Fix a local coordinate s = (si )M i=1 on S. Then we obtain two local coordinates (s; ξ) and (s; ζ) on X. Let aX˙ be in π∗ TX,π (∗D) and α in π∗ TX/S (∗D). Then we can represent aX˙ and α in the two local coordinates (s; ξ) and (s; ζ):   ξ µ + θξ (s; ξ)∂ξ in (s; ξ), η (s; ξ)∂ξ in (s; ξ), α= aX˙ = µ + θζ (s; ζ)∂ζ in (s; ζ), η ζ (s; ζ)∂ζ in (s; ζ),

where µ =

PM i=1

µi (s)∂si . Then a straightforward calculation shows that ∂ 3 θζ (s; ζ) ζ ∂ 3 θξ (s; ξ) ξ η (s; ξ) dξ = η (s; ζ) dζ. ∂ξ 3 ∂ζ 3

Hence there is a unique ω ∈ π∗ 1X/S (∗D) such that the representation of ω under the coordinate (s; ξ) is equal to (∂ξ3 θξ (s; ξ))η ξ (s; ξ) dξ for any ξ ∈ {ξλ }λ∈3 ∪ {ξi }L i=1 . Thus we have completed the proof. 

Twisted WZW Models on Elliptic Curves

53

Example C.3. Assume that π : X → S denotes the family of elliptic curves defined in Sect. 3.1. Then the local coordinate t along the fibers gives a projective structure on the family. For each holomorphic section q : S → X of π : X  S, put ξq := t − q, which is regarded as a holomorphic function on a sufficiently small open neighborhood of q(S). Then the family {ξq } is a projective structure on the family and contains {ξi }L i=1 = {ξqi }L i=1 . Example C.4. A family of compact Riemann surfaces given by the Schottky parametrization possess a natural projective structure, because each compact Riemann surface in that family is represented as the quotient space of the punctured Riemann sphere by the action of a Schottky group, which is generated by a certain finite set consisting of fractional linear transformations. The Schottky parametrization is used in [Be2]. We can generalize the statement of Lemma 4.7 for the family of Riemann surfaces with projective structures. Let gtw X denote an OX -Lie algebra which is locally OX -free of finite rank with holomorphic flat connection ∇ and suppose that the action of a vector field on gtw X via the connection is a Lie algebra derivation: ∇[A, B] = [∇A, B] + [A, ∇B]

for A, B ∈ gtw X.

Assume that the fibers of gtw X are (non-canonically) isomorphic to a simple Lie algebra g over C. The OX -inner product (.|.) is defined by (A|B) :=

1 tr (ad A ad B) 2h∨ gtwX

for A, B ∈ gtw X,

where h∨ denotes the dual Coxeter number of g. Then the inner product is non-degenerate ∼ and invariant under the translations along ∇. We can take a local trivialization gtw X = g ⊗ OX , under which the connection is represented as the exterior derivative on X (i.e., the trivial connection). We assume that we can take such a trivialization of gtw X on some neighborhood of the divisor D. Under this situation, the constructions in Sect. 3 go through in precisely the same way and then Lemma 4.7 also holds. D However, Lemma 4.10 does not always hold. The action of VirX ˙ on M is not a representation but a projective representation in general, because the embedding D D VirX ˙ ,→ VirS is not always a Lie algebra homomorphism but so is the composition of the embedding and the natural projection VirSD  VirSD /Ocˆ. Nevertheless Lemma 4.11 D and Lemma 4.12 also hold in our case. Namely, the Lie algebra VirX ˙ acts on both the sheaf CC(M) of conformal coinvariants and the sheaf CB(M) of conformal blocks. Furthermore Lemma 4.13 can be proved in the same way. Therefore we conclude that CB(M) possess a projectively flat connection. For the non-twisted Wess-Zumino-Witten models, the coordinate-free version of Lemma 4.18 for θ ∈ TX˙D is used in the proof of the main theorem 4.2 in [Ts]. Since the analogue of Lemma 4.18 for θ ∈ TX˙D can be also proved in our situation, we can obtain the projectively flat connections on CC(M). We can generalize the setting above in various ways: 1. We can replace the family of pointed compact Riemann surfaces by that of stable pointed curves in the course of [TUY, U], and [Ts]. Then we shall be able to show the factorization property of conformal blocks under appropriate assumptions. 2. We can consider not only deformations of pointed Riemann surfaces but also deformations of Gtw X -torsors (or principal bundles). For example, the KnizhnikZamolodchikov-Bernard equations on Riemann surfaces (cf. [Be1, Be2, FW, Fe]) can be formulated on a family of pairs of compact Riemann surfaces and principal G-bundles on them, where G is a finite-dimensional simple algebraic group over C.

54

G. Kuroki, T. Takebe

3. Furthermore we can also consider Borel subgroup bundles which are subbundles of the restriction of Gtw X on D and their deformations. (More generally, we can consider a family of quasi parabolic structures on Gtw -torsors.) Then we can define the notion of highest weight representations of the sheaf of affine Lie algebras with respect to the Borel subgroup bundles. Anyway a choice of a Borel subgroup is required by the definition of the category O of representations, which contains the Verma modules, their irreducible quotients, and especially the Wakimoto modules. Note that the constructions of the Wakimoto modules (cf. [FF1, FF2], and [K]) essentially depend on the choice of a triangular decomposition (equivalently that of a Borel subalgebra) of a finite-dimensional semi-simple Lie algebra over C. 4. We can replace the holomorphic flat connection on gtw X by a meromorphic flat connection with regular singularity along the divisor D. Assume that the local monodromy group of the connection around D is finite. Then we can construct a sheaf of twisted affine Lie algebras at D and can define the notion of conformal blocks for representations of the twisted affine Lie algebras. Detailed expositions shall be given in forthcoming papers. Acknowledgement. TT is supported by a Postdoctoral Fellowship for Research abroad of the Japan Society for the Promotion of Science. He expresses his gratitude to Benjamin Enriquez, Giovanni Felder, Edward Frenkel, Ian Grojnowski, Takeshi Ikeda, Feodor Malikov, Hirosi Ooguri, Nicolai Reshetikhin for comments and discussions. The authors also express their gratitude to Yoshifumi Tsuchimoto, who explained to them a detail of the proof of the main theorem in [Ts]. Parts of this work were done while T.T. was visiting the Department of Mathematics of the University of California at Berkeley, the Department of Mathematics of Kyoto University, the Erwin Schr¨odinger Institute for Mathematical Physics, the Landau Institute for Theoretical Physics and Centre Emile Borel - Institut Henri Poincar´e - UMS 839, CNRS/UPMC. He thanks these institutes for hospitality.

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G. Kuroki, T. Takebe

Serre,J.-P.: Cours d’arithm´etique. Paris: Press Universitaires de France, 1970 (in French); A course in arithmetic. Graduate Texts in Mathematics 7, New York-Heidelberg: Springer-Verlag, 1973 (English transl.) Serre, J.-P.: Groupes alg´ebriques et corps de classes. 2nd ed. Paris: Hermann, 1975 (in French); Algebraic groups and class fields. Graduate Texts in Mathematics 117, New York-Berlin: Springer-Verlag, 1988 (English transl.). Shimura,G.: Introduction to the arithmetic theory of automorphic functions. Publication of the Mathematical Society of Japan 11, Tokyo: Iwanami and Princeton: Princeton University Press, 1971 Sklyanin, E. K.: Separation of variables in the Gaudin model. Zap. Nauch. Sem. LOMI 164, 151–169 (1987) (in Russian); J. Sov. Math. 47, 2473–2488 (1989) (English transl.) Sklyanin,E.K., Takebe, T.: Algebraic Bethe Ansatz for XYZ Gaudin model. Phys. Lett. A 219, 217–225 (1996) Sklyanin, E. K., Takebe, T.: In preparation Takebe,T.: Generalized Bethe Ansatz with the general spin representations of the Sklyanin algebra. J. Phys. A 25, 1071–1084 (1992); Bethe ansatz for higher spin eight-vertex models. J. Phys. A 28, 6675–6706 (1995); Corrigendum. ibid. 29, 1563–1566 (1996); Bethe Ansatz for Higher Spin XYZ Models, – Low-lying Excitations –. J. Phys. A 29, 6961–6966 Takebe, T.: A system of difference equations with elliptic coefficients and Bethe vectors. q-alg/9604002, to appear in Commun. Math. Phys. ´ Norm. Sup., 4e s´erie, 1, 149–159 Tate,J.: Residue of differentials on curves. Ann. Scient. Ec. (1968) Tsuchiya, A., Kanie, Y.: Vertex operators in conformal field theory on P1 and monodromy representations of braid group. In: Conformal field theory and solvable lattice models (Kyoto, 1986), Adv. Stud. Pure Math. 16, 297–372 (1988); Errata. In: Integrable systems in quantum field theory and statistical mechanics Adv. Stud. Pure Math., 19, 675–682 (1989) Tsuchimoto, Y.: On the coordinate-free description of the conformal blocks. J. Math. Kyoto Univ. 33-1, 29–49 (1993) Tsuchiya, A., Ueno, K., Yamada, Y.: Conformal field theory on universal family of stable curves with gauge symmetries. In: Integrable systems in quantum field theory and statistical mechanics, Adv. Stud. Pure Math. 19, 459–566 (1989) Ueno, K.: On conformal field theory. In: Vector Bundles in Algebraic Geometry, Durham 1993, ed. by N. J. Hitchin, P. E. Newstead and W. M. Oxbury, London Mathematical Society Lecture Note Series 208, 283–345 (1995)

Communicated by G. Felder

Commun. Math. Phys. 190, 57 – 111 (1997)

Communications in

Mathematical Physics c Springer-Verlag 1997

Affine Orbifolds and Rational Conformal Field Theory Extensions of W1+∞ ? Victor G. Kac, Ivan T. Todorov?? International Erwin Schr¨odinger Institute for Mathematical Physics and Department of Mathematics, MIT, Cambridge, MA 02139, USA. E-mail: [email protected], [email protected] Received: 5 December 1996 / Accepted: 1 April 1997

Abstract: Chiral orbifold models are defined as gauge field theories with a finite gauge group 0. We start with a conformal current algebra A associated with a connected compact Lie group G and a negative definite integral invariant bilinear form on its Lie algebra. Any finite group 0 of inner automorphisms or A (in particular, any finite subgroup of G) gives rise to a gauge theory with a chiral subalgebra A0 ⊂ A of local observables invariant under 0. A set of positive energy A0 modules is constructed whose characters span, under some assumptions on 0, a finite dimensional unitary representation of SL(2, Z). We compute their asymptotic dimensions (thus singling out the nontrivial orbifold modules) and find explicit formulae for the modular transformations and hence, for the fusion rules. As an application we construct a family of rational conformal field theory (RCFT) extensions of W1+∞ that appear to provide a bridge between two approaches to the quantum Hall effect. Contents 0 1 1.1 1.2 1.3 2 2.1 2.2 3

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chiral Algebras Associated with Connected Compact Lie Groups . . . . . Definition of a chiral algebra. Current algebras . . . . . . . . . . . . . . . . . . . . Lattice vertex algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Current chiral algebras associated to simple Lie algebras . . . . . . . . . . . . Twisted Modules of a Current Chiral Algebra . . . . . . . . . . . . . . . . . . . . . Positive energy irreducible A(G)-modules . . . . . . . . . . . . . . . . . . . . . . . ZN -twisted current chiral algebra modules . . . . . . . . . . . . . . . . . . . . . . . Twisted Characters and Modular Transformations . . . . . . . . . . . . . . . . .

58 60 60 62 64 66 66 66 68

? Supported by the Federal Ministry of Science and Research, Austria, NSF grants DMS-9103792 and DMS-9622870 and the Bulgarian NFSR under contract F-404. ?? On leave of absence from the Institute for Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, Tsarigradsko Chauss´ee 72, BG-1784, Sofia, Bulgaria

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Victor G. Kac, Ivan T. Todorov

3.1 Kac-Moody and lattice characters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 3.2 Modular transformations of twisted characters . . . . . . . . . . . . . . . . . . . . 70 3.3 Small τ asymptotics of twisted characters of A(G) . . . . . . . . . . . . . . . . . 72 4 Affine Orbifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 4.1 Projection on a centralizer’s irreducible representation. Asymptotic dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 4.2 Affine orbifold models for non-exceptional 0. Action of Z. Modular transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 4.3 Fusion rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 5 U (l) orbifolds as RCFT extensions of W1+∞ . . . . . . . . . . . . . . . . . . . . . 85 6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 6.1 Lattice current algebras for c = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 6.2 SU(2) orbifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 6.3 A Level 1 SU(3) Orbifold. Charge Conjugation Associated with a NonAbelian Centralizer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 Appendix A. Action of the Center of a Simply Connected Simple Lie Group on the Coroots and Fundamental Weights . . . . . . . . . . . . . . . . . . . . . . . . 105 A.1 Simply laced algebras (αi∨ = αi , a∨ i = ai ) . . . . . . . . . . . . . . . . . . . . . . . 105 A.2 Z2 action on Bl and Cl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 Appendix B. Exceptional Elements of a Compact Lie Group . . . . . . . . . . . . 108 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 0. Introduction Given a chiral conformal field theory (CFT) – i.e., a chiral algebra A and a family of positive energy A-modules (closed under “fusion”) – there are two ways of constructing other CFT with the same stress-energy tensor T (z) and associated central charge c. First, one can, in some cases, extend A by adjoining to it local primary fields. The stress energy tensor generates an RCFT for the minimal models [BPZ] corresponding to central charge c < 1. For c ≥ 1 one needs in addition a chiral current algebra or a W -algebra to construct an RCFT (for special rational values of c). All RCFT extensions of the (c = 1)u(1)-current algebra have been classified in [BMT]; all local extensions of the su(2) current algebras have been described in [MST]. The second path goes in the opposite direction: one restricts A to a distinguished subalgebra of “observables” including T (z); we shall be concerned here with the case in which the subalgebra A0 consists of all elements of A invariant under a finite automorphism group 0. The resulting CFT is called a 0-orbifold. Examples of orbifolds (first in the context of a “Gaussian model” [G,H]) have been studied in detail in [DV3 ] where some general properties of arbitrary orbifold models have also been pointed out. Non unitary models of c = 1 have been considered in [F]. The present paper provides a systematic approach to orbifold RCFT. Our starting point is a chiral algebra A = A(G) associated with a connected compact Lie group G whose Lie algebra g is equipped with a negative definite integral invariant bilinear form. It appears as a tensor product of a lattice chiral algebra A(L) and (chiral) affine Kac-Moody algebras (corresponding to the simple components gj of g): A(G) = A(L) ⊗ (⊗sj=1 Akj (gj )),

(0.1)

where kj (∈ Z+ ) is the level of the vacuum gˆ j - module. The lattice L consists of all vectors ω in the direct sum g0 of u(1)-components of the centre of g such that e2πiω = 1.

Affine Orbifolds and Rational CFT of W1+∞

59

To each ω of length square 2 we can associate a “charge shift” operator E ω providing a non-abelian extension of the Lie algebra g0 . Let Gc be the corresponding maximal compact group extension of G. Its significance stems from the fact that each finite order inner automorphism of A(G) is given by (the adjoint action of) an element of Gc . An orbifold chiral algebra is the fixed point set A0 of a finite group of automorphisms 0 of a chiral algebra A. For any “non-exceptional” finite subgroup 0 ⊂ Gc we construct a finite family of A0 -modules V , which is complete in the sense that their characters transform among themselves under the modular group SL(2, Z). Each V is labeled by a weight 3 (characterizing an A(G)-module), a conjugacy class b¯ ⊂ 0, and an irreducible representation σ = σ b of the centralizer 0b of an element b ∈ b¯ in 0. It involves a choice of “phases” β(b) ∈ g, for non-exceptional conjugacy classes, satisfying the following two conditions: (i) b = e2πiβ(b) ,

¯ g∈0. (ii) β(gbg −1 ) = Adg β(b) for b ∈ b,

(0.2)

Condition (ii) implies that the centralizer of b should stabilize β. Two β’s satisfying (0.2) differ by a coroot m which is also stabilized by 0b . Any such m gives rise to a 1-dimensional representation σm of 0b . The change β → β + m can be compensated by a change in the representation σ: β+m β = V3, V3, ∗ . ¯ ¯ b,σ b,σ⊗σ m

(0.3)

Thus the family of A0 -modules is independent of the choice of β (allowing us to skip the superscript β on V ). Knowing the character χ3 of an A(G)-module V3 [K1] we are able to calculate of V3,b,σ the A0 - characters χ3,b,σ ¯ ¯ . Similarly, the modular transformation properties and hence the orbifold fusion rules. We point of χ3 [KP2] determine those of χ3,b,σ ¯ out that the group factors of fusion coefficients Nb¯ 1 σ1 ,b¯ 2 σ2 ,b¯ 3 σ3 (σi ∈ 0ˆ bi ) of an affine orbifold differ from those of the associated Grothendieck ring (see [Lus] as well as the discussion in Sect. 4 of [DV3 ]) due to multipliers µ(h|Σβi ) which define (for b1 b2 b3 = 1) 1-dimensional representations of the intersection 0b1 ∩ 0b2 (3 h). This difference shows up already for (finite) subgroups of SU(2). For higher rank G it may yield a change of charge conjugation, as displayed in the example of a 1080 element subgroup of SU(3) which admits a conjugacy class of involutive elements with a non-abelian centralizer. We compute (in Sect. 4.1) the asymptotic dimensions of orbifold characters singling out, in particular, the non-trivial orbifold modules. If G is a simple simply-connected Lie group then the non-trivial elements of Z = Z(G) ∩ 0 ,

(0.4)

where Z(G) is the center of G, are exceptional – they cannot be written in the form (0.2) (with β satisfying (ii)). Each element of Z (different from the group unit) is associated with a fundamental weight 3j satisfying (3j |θ) = 1 where θ is the highest root. We associate with it (in Sect. 4) a permutation of the orbifold modules which maps, in ˜ j and thus cannot be viewed as an ˜ 0 into 3 particular, the (affine) vacuum weight 3 automorphism (“gauge transformation”) of the (vacuum) chiral algebra. Knowing the action of e2πi3j on {V3bσ ¯ } we can extend our treatment to all exceptional elements of a 0 ⊂ SU(n). The treatment of Ad-exceptional elements (described in Appendix A), which are encountered in other simple Lie groups, remains however, outside the scope of the present paper.

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Note an essential difference between coset models and orbifold models. For the construction of a modular invariant family of characters of coset modules it suffices to take characters of isotypic components of all (untwisted) modules of the chiral current algebra with respect to its chiral current subalgebra [KP0, KP2, KW, K1]. In a sharp contrast, for an orbifold model one has to take in addition decompositions into isotypic components of twisted chiral current algebra modules which become untwisted when restricted to the orbifold chiral subalgebra. As an application we construct (in Sect. 5) a family of RCFT extensions of W1+∞ -one for each value l(∈ N) of the central charge and for each finite subgroup 0 of U (l). It is designed to provide a bridge between two current attempts to understand the fractional quantum Hall effect in terms of chiral conformal algebras (see [FT] and [CTZ]).

1. Chiral Algebras Associated with Connected Compact Lie Groups We shall first recall the general notion of a chiral algebra and will then introduce a class of such algebras which appear to be of paramount importance in the study of RCFT. 1.1. Definition of a chiral algebra. Current algebras. The mathematical concept of a vertex or chiral algebra was introduced by R. Borcherds [Bor] and later developed by a number of authors (see, e.g. [FLM, Go, DGM, FZ, LZ, FKRW, KR2]). The version adopted here is a specialization of [K2] to Z-graded algebras (restricting from the outset attention to fields of a given conformal dimension). Let V be a Z+ -graded inner product space with a unique vacuum state, (n) , V = ⊕∞ n=0 V

dim V (0) = 1,

dim V (n) < ∞ ;

(1.1)

the gradation defines (and can be, conversely, defined by) a distinguished hermitian operator L0 called the (chiral) energy operator such that (L0 − n)V (n) = 0 .

(1.2)

The unique (up to a phase factor) vector |0i ∈ V (0) normalized by h0|0i = 1 is called the vacuum. A chiral field Y (s) of dimension s is a power series X Y (s) (z) = Yn z −n−s , s ∈ Z+ (1.3) n∈Z

with Yn (= Yn(s) ) ∈ EndV satisfying the commutation relations (CR),   d + s Y (s) (z) , [Yn , L0 ] = nYn ⇔ [L0 , Y (s) (z)] = z dz Yn |vm i = 0 for vm ∈ V (m) ,

n>m.

(1.4) (1.5)

Equation (1.5) expresses the postulate that the vacuum is the lowest energy state in V . In physical terms V is the vacuum space of finite energy states. A chiral (vertex) algebra structure on V is a linear map, called the state-field correspondence, from V (s) to the space of fields of dimension s: V (s) 3 vs → Y (vs , z) = P −n−s , defined for all s ∈ Z+ and satisfying the following three axioms: n Yn (vs )z V1. Vacuum axioms: the vacuum vector corresponds to the identity operator in V :

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Y (|0i, z) = 1V ;

(1.6)

the field Y (vs , z) allows to recuperate the vector vs : lim Y (vs , z)|0i = vs ,

z→0

i.e.Y−s (vs )|0i = vs and Yn−s (vs )|0i = 0 for n > 0 . (1.7)

V2. The translation operator L−1 : V → V defined by L−1 vs = Y−s−1 (vs )|0i satisfies the translation covariance condition: [L−1 , Y (vs , z)] =

d Y (vs , z) . dz

(1.8)

V3. The chiral fields are local: (z − w)n [Y (vs , z), Y (vs0 , w)] = 0 for n ≥ s + s0 .

(1.9)

Note that the inner product is not logically necessary in this generality. It is, however, present in all CFT (being indefinite for non-unitary theories) and gives rise to a distinguished (anti-involutive) star operation ([DGM]). We shall be concerned with (orbifolds of) chiral current algebras described below. Let G be a compact Lie group of the form G = G0 ×G1 ×. . .×Gs where G0 = U (1)r , and Gj , j = 1, . . . , s, are simple simply-connected groups. (Every compact Lie group can be viewed as a product of the above form factored by a finite central subgroup.) Let gj denote the Lie algebra of Gj (j = 0, . . . , s) and let L = {ω ∈ g0 | exp 2πiω = 1}. We assume that g is equipped with a symmetric integral negative definite invariant bilinear form. A bilinear form on g is called integral if the length square of any ω ∈ igj (j = 1, . . . , s) such that exp 2πiω = 1 (resp. of any ω ∈ L) is an even integer (respectively an integer). When restricted to a simple gj , the integrality property means that the bilinear form is equal to kj (v|v 0 ), where kj ∈ N will be identified with the level of the affine Kac-Moody algebra gˆ j and 1 (v|v 0 ) = ∨ trgj (adv adv0 ) 2gj (gj∨ is the dual Coxeter number of gj ; recall that with such a normalization (α|α) = 2 for long roots α). In what follows we let also k0 = 1. Remark 1.1. Admitting lattice vectors α with odd square lengths requires, as it will become clear shortly, extending the Z+ gradation of the vacuum space (1.1) to a 21 Z+ gradation. In physical terms it amounts to admitting locally anti-commuting (Fermi) fields of half-integer conformal dimensions in the chiral algebra. Such fields do not describe local observables (in the strict sense of the word) and could alternatively be incorporated in the positive energy representations of a chiral Bose algebra corresponding to an even integral lattice. A way to get rid of Fermi fields is to go to a double cover of the group G, which makes the lattice L even. Given the above data one can construct a chiral algebra A(G) = A(L) ⊗ (⊗sj=1 Akj (gj )) , called an affine (or current) chiral algebra as follows. For each gj (j = 0, 1, . . . , s) consider its affinization [K1]: gˆ j = C[t, t−1 ] ⊗R gj + CKj .

(1.10)

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d It is a Z-graded algebra, the energy operator L0 acting on it as −t dt . j Let V0 (gj , kj ) denote the unique irreducible gˆ -module which admits a non-zero vector |0i such that (C[t] ⊗ gj )|0i = 0 and Kj |0i = kj |0i. Given an element v ∈ gj and n ∈ Z we let vn denote the operator on V0 (gj , kj ) corresponding to tn ⊗ v. Let v(z) = Σn∈Z vn z −n−1 be the current corresponding to v. Then the chiral algebra structure Akj (gj ) on the vacuum space V0 (gj , kj ) is defined for each j = 1, . . . , s by the following state-field correspondence: 1 n . . . v−i |0ij , z) =: ∂ i1 v 1 (z) . . . ∂ in v n (z) : /i1 ! . . . in ! Y (v−i 1 −1 n −1

(with appropriately defined normal products, [K2]). The vacuum space V is given by V = V (L) ⊗ ⊗sj=1 V0 (gj , kj ) .

(1.11)

In the next section we describe the first factor in (1.11) and the corresponding chiral algebra structure A(L) (cf. [K2], Sect. 5.4). 1.2. Lattice vertex algebras. Let Cε [L] be the twisted group algebra of the lattice L with basis eω (ω ∈ L) and multiplication rule 0

0

eω eω = ε(ω, ω 0 )eω+ω ,

ω, ω 0 ∈ L ,

(1.12)

where ε(ω1 , ω2 ) is a ±1-valued cocycle: ε(ω, 0) = ε(0, ω) = 1 , ε(ω, ω 0 )ε(ω + ω 0 , ω 00 ) = ε(ω, ω 0 + ω 00 )ε(ω 0 , ω 00 ) .

(1.13a) (1.13b)

Equation (1.13a) means that e0 = 1 and Eq. (1.13b) is equivalent to associativity. Let S = V0 (g0 , 1). This is the symmetric algebra over the positive energy subspace gˆ 0(+) = ⊕n<0 Ctn g0 .

(1.14)

(Here and below we omit the tensor product sign between tn and g.) The space V (L) is then defined as the tensor product V (L) = S ⊗ Cε [L] .

(1.15)

It is an infinite direct sum (over the lattice) of irreducible positive energy gˆ 0 modules with k0 = 1: (1.16) V (L) = ⊕ω∈L S ⊗ eω . The corresponding ground state vectors are 1 ⊗ eω ; in particular, the V (L) vacuum is |0i = 1 ⊗ 1. The chiral subalgebra A(S ⊗ 1) is generated by currents. The ground state vector |ωi ≡ 1 ⊗ eω of each term in (1.16) is characterized by being an eigenvector of gˆ 0(−) = ⊕n≥0 Ctn g0 : (v0 − (v|ω))|ωi = 0 = vn |ωi, n = 1, 2, . . .

(1.17)

To display the full chiral algebra A(L) it remains to recall the Frenkel-Kac construction for the charged fields [FK]: Y (eω , z) = eω eϕ+ (z,ω) z ω0 eϕ− (z,ω) ,

(1.18)

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where ϕ± (z, ω) = ±

∞ X n=1

ω∓n

z ±n . n

(1.19)

Y (eω , z) is a primary field with respect to the current subalgebra A(S ⊗ 1): [v(z), Y (eω , w)] = (v|ω)δ(z − w)Y (eω , w) .

(1.20) 0

Let p(ω) ∈ Z/2Z denote the parity of (ω|ω), ω ∈ L. The fields Y (eω , z) and Y (eω , z) are local if and only if 0

0

ε(ω, ω 0 ) = (−1)p(ω)p(ω )+(ω|ω ) ε(ω 0 , ω) .

(1.21)

The state-field correspondence for the chiral algebra A(L) is defined by 1 n . . . v−i ⊗ eω , z) =: ∂ i1 v 1 (z) . . . ∂ in v n (z)Y (eω , z) : /i1 !...in ! . Y (v−i 1 −1 n−1

(1.22)

Note that a 2-cocycle ε(ω, ω 0 ) satisfying (1.21) always exists and the chiral algebra A(L) is independent of the choice of this cocycle. The conformal properties of Y (eω , z) are given by ∂ |ω|2 ∂ Y (eω , w) + Y (eω , w) δ(z − w), |ω|2 ≡ (ω|ω) . ∂w 2 ∂w (1.23) Here T is the stress energy tensor X Ln z −n−2 ([vm , Ln ] = mvm+n ) (1.24) T (z) =

[T (z), Y (eω , w)] = δ(z − w)

n

expressed in terms of an orthonormal basis v i (z) of u(1) currents, (v i |v j ) = δij , by the Sugawara formula r 1X : (v i (z))2 : , (1.25) T (z) = 2 i=1

where the normal product can be thought of as a limit   1 i i lim v (z1 )v (z2 ) − 2 z1,2 →z z12 (see the comment following Eq. (1.36) below). The fusion rules for the A(L) vertex operators have the form 0

0

0

lim {(z − w)−(ω|ω ) Y (eω , z)Y (eω , w)} = ε(ω, ω 0 )Y (eω+ω , w) ;

z→w

(1.26)

the operator product expansion for oppositely charged fields can be written in more detail as Z z1  |ω|2 ω(z)dz : , z12 = z1 − z2 . (1.27) z12 Y (eω , z1 )Y (e−ω , z2 ) =: exp z2

(The normal ordered exponential is defined in such a way that the nth term of its Taylor expansion is an integral over a single quasiprimary field of dimension n – cf. [FST].)

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1.3. Current chiral algebras associated to simple Lie algebras. The CR between two currents Y (t−1 v i , zi ), i = 1, 2, for two arbitrary elements v 1 and v 2 of gˆ are given by 
 −1 1 Y (t v , z1 ), Y (t−1 v 2 , z2 ) = Y t−1 [v 1 , v 2 ], z2 δ(z12 ) − (v 1 |v 2 )k δ 0 (z12 ) . (1.28) Here and further z12 = z1 − z2 . We shall write down for later reference these relations for the Chevalley-Cartan basis of a simple component gj of g. We shall set (v1 |v2 )k = kj (v1 |v2 ) for v1 , v2 ∈ gj .

(1.29)

Restricting attention to a simple component we skip the index j on g. We choose a Cartan subalgebra h in g and a basis αi , i = 1, . . . , l, of simple roots in its dual thus introducing a standard partial order in h∗ , which from now on we shall identify with h using the bilinear form (.|.). To each positive root α > 0 we associate a certain current representing the corresponding coroot α∨ : H α (z) =

X

Hnα z −n−1 , H0α = α∨ :=

n

2α , |α|2 = (α|α) . |α|2

(1.30)

We shall use the positive integer marks ai (and a∨ i ) of the Dynkin diagram of g which enter the expression for the highest root θ=

l X i=1

ai αi =

l X

∨ ∨ a∨ i αi = θ

(1.31)

i=1

(see [K1], Chap. 4, Tables). Their ratio relates the Cartan matrix aij of g to the symmetric Gram matrix of the coroots,
ai (αi∨ |αj∨ ) = aij ∨ (aij = αi∨ |αj ) , aj while the sum of check marks of the extended Dynkin diagram gives the dual Coxeter number

∨ (1.32) tr adv1 adv2 = 2g ∨ (v1 |v2 ) . g ∨ = 1 + a∨ 1 + . . . + al The set of indices (j ∈)J for which the exponentials e2πi3j of the corresponding fundamental weights 3j generate the center Z(G) of the simply connected group G with Lie algebra g is given by J = {j = 1, . . . , l|

aj = 1} .

(1.33)

Let E α be a raising or a lowering operator, depending on the sign of α. Then the current CR (1.28) assume the form: [H α (z1 ), E β (z2 )] = (α∨ |β)E β (z2 )δ(z12 ) ∨

∨

(α, β roots) ,

0

[H (z1 ), H (z2 )] = −k(α |β )δ (z12 ), α

β

[E αi (z1 ), E −αj (z2 )] = 0 for i 6= j, [E α (z1 ), E −α (z2 )] = H α (z2 )δ(z12 ) −

(1.34) 2k 0 δ (z12 ) . |α|2

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The affine chiral algebra Ak (g) contains the Sugawara stress energy tensor (see e.g. [K2] Sect. 5.7.): ( 1 X (α|α) : (E α (z)E −α (z) + E −α (z)E α (z)) : T (z) = 2h 2 α>0 (1.35) ) l X i ∨ : Hi (z)H (z) : , h = k + g . + i=1

Here H i and Hi correspond to dual bases in the Cartan subalgebra: H i = αi∨ ,

Hi = 3i ,

(αi∨ |3j ) = δij .

(1.36)

The normal product :: can be defined by either subtracting the singular in z12 part of an ordinary product Ja (z1 )J a (z2 ) or by ordering the frequency parts of the currents (inequivalent definitions of the normal product used in [FST] and [K2] yield the same expression for the stress energy tensor). Equations (1.34), (1.35) imply the Virasoro CR   c 3 ∂ , (1.37) [T (z1 ), T (z2 )] = δ(z12 )∂2 T (z2 ) + 2T (z2 )∂2 δ(z12 ) + ∂2 δ(z12 ) ∂2 ≡ 12 ∂z2 where the Virasoro central charge exceeds the rank l of g. Denoting by d(g) the dimension of g, we have k (1.38) c = ck (g) = d(g) ≥ l . h The positive integer h entering (1.35) and (1.38) (the sum of the level and the dual Coxeter number) is called the height. The last inequality in (1.38) follows from the fact that T can be split into a sum of two commuting terms, the stress tensor TH of the Cartan subalgebra and a remainder TR : T = TH + TR ,

TH (z) =

l 1 X : Hi (z)H i (z) : . 2k

(1.39)

i=1

We find, as a consequence of (1.34), (1.35) and (1.36), [TH (z1 ), H i (z2 )] = ∂2 (δ(z12 )H i (z2 )) = [T (z1 ), H i (z2 )],

(1.40a)

[TR (z1 ), H i (z2 )] = 0 = [TR (z1 ), TH (z2 )].

(1.40b)

and hence For a level k > 1 simply laced (A-D-E) simple Lie algebra the RCFT with stress energy tensor TR correspond to (generalized) G/H parafermions – see [KP0 and Gep]. For a simply laced level 1 gˆ we have c1 (ˆg) = l and hence TR = 0. Note that the lattice chiral algebra A(L) could also contain a level 1 simply laced current subalgebra. In fact, each even (integral) lattice Lr has a sublattice Wr−ν ⊕ Lν ⊂ Lr of the same dimension r. Here Wr−ν is the root lattice of a direct sum of A-D-E (simple) Lie algebras, generated by vectors of length square 2, and Lν is its orthogonal complement (with no vector of length square 2), so that L/(Wr−ν ⊕ Lν ) is a finite abelian group, the glue group. The stress energy tensor T (z) of the chiral algebra A(G) is defined as the sum of the stress energy tensors of the factors of A(G).

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2. Twisted Modules of a Current Chiral Algebra 2.1. Positive energy irreducible A(G)-modules. Let A(G) = A(L) ⊗ (⊗sj=1 Akj (gj )) be a current chiral algebra. Its positive energy irreducible modules are tensor products of such modules for each factor. Let L∗ = {µ ∈ g0 |(µ|ω) ∈ Z for all ω ∈ L} be the dual lattice. It is easy to see that the positive energy irreducible modules over A(L) are labeled by the elements of the finite abelian group L∗ /L as follows. Extend the cocycle ε(ω1 , ω2 ) to L∗ in such a way that (1.13) holds for ω, ω 0 ∈ L and ω 00 ∈ L∗ . We choose a vector µ of a coset of L∗ mod L, and let X S ⊗ eω . (2.1) Vµ (L) = ω∈µ+L

Then Eqs. (1.18), (1.19) and (1.22) define an irreducible positive energy module over A(L). As a consequence of the Sugawara formula (1.25), the ground state energy 1(µ) of the module Vµ (L) is given by 1(µ) =

(µ|µ) , if µ is a minimal length vector in µ + L . 2

(2.2)

Let g be the Lie algebra of a simple connected compact Lie group and let k be a non-negative integer. Then the integrable positive energy irreducible modules over gˆ of level k are labeled by the highest weight 3 of g in the lowest energy subspace (which is a finite-dimensional irreducible g-module). We denote these modules by V3 (g, k). Recall that 3 then satisfies the integrability condition ([K1], Chap. 12): (3|αi∨ ) ∈ Z+ for i = 1, . . . , l ,

(3|θ) ≤ k .

(2.3)

Each of the gˆ -modules V3 (g, k) extends to a Ak (g)-module and all positive energy irreducible Ak (g)-modules are obtained in this way [FZ]. As a consequence of Eq. (1.35), the ground state energy (= conformal dimension) 1(3) of the module V3 (g, k) is given by: 1(3) =

X (3 + 2ρ|3) , where h = k + g ∨ , 2ρ = α. 2h

(2.4)

α>0

2.2. ZN -twisted current chiral algebra modules. Let G0 be the connected compact Lie group whose maximal torus is U (1)r = Rr /L, i.e. L is the coroot lattice of G0 . (G0 contains the torus U (1)r but can, in general, be larger due to the presence of ω’s in L of length square 2; the semi-simple part of G0 is a product of simply laced compact simple Lie groups). Let (2.5) Gc = G0 × G1 × . . . × Gs , the corresponding decomposition of Lie algebras being g = g0 ⊕ g1 ⊕ . . . ⊕ gs .

(2.6)

Let Z j ⊂ Gj denote the center of Gj , j = 1, . . . , s, and let Z 0 = L∗ /L (Z 0 is a central subgroup of G0 ). The following finite subgroup of Gc will play an important role in the sequel: (2.7) Z(Gc ) = Z 0 × Z 1 × . . . × Z s .

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Recall (see (1.33)) that the center of a simple connected simply connected compact Lie group consists of 1 and the elements (2.8) exp 2πi3j , where j ∈ J . P Recall that if Y (v1 , z) = n∈Z Yn (v1 )z −n−1 is a field of conformal dimension 1 of a chiral algebra, then Y0 (v1 ) is a derivation of A and exp Y0 (v1 ) converge in any positive energy A-module (see e.g. [K2], Sect. 4.9.). Since such derivations of the chiral algebra A(G) form the Lie algebra gC (the complexification of g), the group Gc acts on A(G) by automorphisms, and moreover, acts on each positive energy A(G)-module U in a consistent way (i.e. g(au) = g(a)g(u) for g ∈ Gc , a ∈ A(G), u ∈ U ) preserving the Hilbert metric. It follows from the usual properties of the Casimir operator that the stress energy tensor T (z) is a Gc -invariant observable: T (z) ∈ A(G)Gc .

(2.9)

Now we recall briefly the notion of a twisted module U over a chiral algebra A. Let b be an automorphism of order N of A; then we get a Z/N Z-grading A = ⊕m∈Z/N Z Am , where Am is the exp 2πim/N eigenspace of b. A b-twisted A-module U is a linear map a 7→ π(a) from A to the space of fields with values in End U such that the twisted Borcherds identity holds (see e.g. [KR2]), in particular all the CR are preserved and e2πiL0 π(a)e−2πiL0 = (−1)p(a) e

2πim N

π(a) for a ∈ Am .

(2.10)

If A = A0 , we get a usual (untwisted) A-module. Returning to A(G), fix β ∈ ig such that the corresponding element b = exp 2πiβ ∈ Gc has finite order N and choose a Cartan subalgebra of g containing iβ. Given a positive energy representation π of A(G) in a vector space U , we construct a b-twisted representation πβ in U as follows. First, due to the decomposition (1.10) of A(G) and the corresponding decomposition U = ⊗sj=0 U j , it suffices to construct the P bj -twisted representation πβ j in U j for each j, where β = j β j is the decomposition (2.6) and bj = exp 2πiβ j . Next, for a positive energy representation π of A(g) we let X α En+(α|β) z −n−(α|β)−1 , (2.11) πβ (E α (z)) = π(E α (z))z −(α|β) = n∈Z

and extend to the whole A(g) using the twisted Borcherds identity. In order to preserve CR we should have πβ (H α (z)) = π(H α (z)) −

k(α∨ |β) . z

(2.12)

Similarly, for a positive energy representation π of A(L) we let πβ (Y (eω , z)) = π(Y (eω , z))z −(ω|β) , πβ (ω(z)) = π(ω(z)) −

(ω|β) , z

(2.13) (2.14)

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Victor G. Kac, Ivan T. Todorov

and extend to A(L) using the twisted Borcherds identity. The constructed b-twisted A(G)-module will be denoted by U (β) . Going to the stress tensor, which is a sum of a torus part, TL (1.25), and a contribution of type (1.35), (1.39) for each simple factor in G, we shall see that only the Cartan part 1X : vi (z)2 : , 2 r

T h = TL + T H ,

TL =

TH =

i=1

l 1 X : Hj H j : (z) 2k

(2.15)

j=1

changes following (2.12), (2.14) while the remainder TR in (1.39) is left unchanged. Proposition 2.1. If we set 1 1 T˜h = Th − β(z) + 2 (β|β)k , z 2z

T˜R = TR

(2.16a)

implying for the Laurent modes of T˜ 1 L˜ n = Ln − βn + (β|β)k δn0 , 2

(2.16b)

where (β|β)k = k|β|2 for each simple component of (2.6), then T˜ and J˜ satisfy the same CR as T and J (J standing for any of the g-currents, H α , E α , v i ) e.g.
 d ˜ ˜ . L˜ n , J(z) z n+1 J(z) = dz



(2.17)

Proof. It is straightforward to verify that the commutator of L˜ n with E˜ α ≡ πβ (Eα ) (2.11) reproduces (2.17). One further uses the fact that πβ defines a Lie algebra homomorphism on the currents, preserving their CR. The constant term in L˜ 0 is obtained by  computing [L˜ 1 , L˜ −1 ]. 3. Twisted Characters and Modular Transformations The complete character of a positive energy A(G)-module V is defined on the product of the upper half plane τ and the group G as follows:   ck (3.1a) χV (τ, z, u) = e2πi(k,u) trV q L0 − 24 e2πiz . Here q = e2πiτ (|q| < 1), z ∈ ig, (k, u) = u0 +

s X

kj uj ,

(3.1b)

j=1

uj are auxiliary (complex) parameters, L0 is the chiral energy operator (1.2), (1.4) (the zero mode of the stress energy tensor (1.24)), ck is the Virasoro central charge (cf. (1.38)): ck = r +

s X

ckj (gj ) .

(3.1c)

j=1

If V is irreducible then χV splits into a product of Kac-Moody and lattice characters; we reproduce their expressions and transformation properties separately. This will allow us to write down the general orbifold characters.

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3.1. Kac-Moody and lattice characters. Let now G be a connected simply connected compact Lie group with a simple Lie algebra g. We shall use the following notation: M ∗ is the weight lattice dual to the coroot lattice M ; the set of level k dominant weights is [K1] (3.2) P+k = {3 ∈ M ∗ |(3|αi∨ ) ≥ 0, i = 1, . . . , l; (3|θ∨ ) ≤ k} ; Q ⊂ M ∗ is the root lattice; the quotient M ∗ /kM is a finite abelian group of order |M ∗ /kM | = k l |M ∗ /M |. The values of |M ∗ /M | may be found e.g. in [KW] (in the simply laced case |M ∗ /M | is the determinant of the Cartan matrix). The Kac-Moody character χ3 (τ, z, u) ≡ χV3 (g,k) (τ, z, u) can be expressed in terms of classical Θ functions of weight l/2 and certain almost holomorphic modular forms c3 λ , the string functions, of opposite weight ([K1], Eq. (12.7.12)): X M c3 (3.3a) χ3 (τ, z, u) = λ (τ )Θλk (τ, z, u) , λ∈M ∗ /kM 3−λ∈Q

M Θλk (τ, z, u) = e2πiku

X

k

q 2 (γ|γ) e2πik(γ|z) .

(3.3b)

γ∈M + λ k

We assume here that iz is an element of g and choose a Cartan subalgebra containing iz. The modular transformation law for Θ is given by (see Theorem 13.5 of [K1]):     (z|z) 1 z 0 −1 M M S= : Θλk (τ, z, u) → Θλk − , , u − 1 0 τ τ 2τ (3.4) X (λ|λ0 ) e−2πi k ΘλM0 k (τ, z, u) , = (−iτ )l/2 |M ∗ /kM |−1/2 λ0 ∈M ∗ /kM

 T =

1 0

 (λ|λ) 1 M M M (τ, z, u) → Θλk (τ + 1, z, u) = eiπ k Θλk (τ, z, u) . : Θλk 1

(3.5)

The characters χ3 span a finite dimensional representation of SL(2, Z) as well (see [KP] or Theorem 13.8 of [K1]):   X (z|z) 1 z = S330 χ30 (τ, z, u) ; (3.6) χ3 − , , u − τ τ 2τ k 0 3 ∈P+

here the S330 are given by the Kac-Peterson formula:   X (3 + ρ|w(30 + ρ)) |1 | ∗ −1/2 + S330 = i |M /kM | , ε(w) exp −2πi h

(3.7)

w∈W (g)

where |1+ | is the number of positive roots, W (g) is the Weyl group of g, ε(w) = ± according to the parity of w, 2ρ and h are defined in (2.4), χ3 (τ + 1, z, u) = e2πim3 χ3 (τ, z, u) ,

(3.8)

ck (g) , (3.9) 24 where 1(3) is the conformal dimension (2.4), ck (g) is the Virasoro central charge (1.38). In the special case of g = su(2) we have m3 = 1(3) −

70

Victor G. Kac, Ivan T. Todorov

Sλλ0 =

p (λ + 1)(λ0 + 1) , 2/h sin π h

h=k+2,

(3.10a)

c λ(λ + 2) 3k − , c = ck (su(2)) = . (3.10b) 4h 24 h Note that for a simply laced affine algebra at level 1 (so that c = l) there is only one non-zero string function, which is a negative power of the Dedekind η-function: −l c3 3 (τ )|k=1 = (η(τ )) . Recall the transformation properties of the η-function: mλ =

1 η(− ) = (−iτ )1/2 η(τ ) , τ

η(τ + 1) = eπi/12 η(τ ) .

(3.11)

The matrix S simplifies in this case as it coincides with that for the lattice characters (see (3.14) below). It is clear from the construction that the lattice character χµ of the module Vµ (L) is given by L (τ, z, u) . (3.12) χµ (τ, z, u) = (η(τ ))−r Θµ1 Here, as before, z is an element of g0 and we choose a Cartan subalgebra containing z. (The expression (3.12) has, of course, the same form as the level 1 simply laced Kac-Moody character; it coincides with (3.3), (3.11) for L = M, r = l.) The modular transformation law for χµ can be read off (3.4), (3.5) and (3.11) (the expression for S in the counterpart of (3.6) being simpler than (3.7)):   X 1 2 1 z |z| = Sµµ0 χµ (τ, z, u), (3.13) χµ − , , u − τ τ 2τ 0 ∗ µ ∈L /L

where

0

Sµµ0 = |L∗ /L|−1/2 e−2πi(µ|µ ) ;

(3.14)

1 r , 1(µ) = |µ|2 . (3.15) 24 2 As mentioned above, an irreducible positive energy A(G)-module V is the tensor product of the A(L)-module Vµ (L) and A(gj )-modules V3j (gj ). Hence positive energy irreducible A(G)-modules are parameterized by the set χµ (τ + 1, z, u) = e2πimµ χµ (τ, z, u) ,

mµ = 1(µ) −

P+k = (L∗ /L) × P+k1 × . . . × P+ks . Ps We let µ = 30 , call 3 = j=0 3j the highest weight of V , and write V = V3 . The character of V3 , 3 ∈ P+k , is the product χ3 (τ, z, u) =

s Y

χ3j (τ, z j , uj ) .

(3.16)

j=0

3.2. Modular transformations of twisted characters. Recall that the affine chiral algebra A(G) is defined by the data consisting of a compact group G and an invariant bilinear form on its Lie algebra g. This invariant bilinear form looks as follows: (x|y)k ≡

s X j=0

kj (xj |y j ) .

(3.17a)

Affine Orbifolds and Rational CFT of W1+∞

71

We will also use the normalized invariant bilinear form (x|y) =

s X

(xj |y j ) .

(3.17b)

j=0

Let now β ∈ ig be such that b = exp 2πiβ ∈ G has finite order and choose a Cartan subalgebra of g containing iβ. It follows from (2.16b) and (3.1) that the value of the character of the b-twisted A(G)-module V3(β) at e2πiα ∈ G is given by the following formula: c L0 −β+ 21 (β|β)k − 24k 2πiα e χα,β 3 (τ ) ≡ trV3 q   (3.18) 1 = eπi(α|β)k χ3 τ, α − τ β, − (α − τ β|β) . 2 Each factor in (3.18) can be written in a similar form for the Kac-Moody and the lattice case (assuming that α and β lie in the same Cartan subalgebra): X

χα,β 3 (τ ) =

α,β c3 λ (τ )Θλk (τ )

(3.19)

λ∈M ∗ /kM

3−λ∈Q

α,β = [η(τ )]−r Θµ1 (τ )

χα,β µ (τ )

(3.20)

where in both cases 

 α,β (τ ) Θλk

=e

iπk(α|β)

1 τ, α − βτ, (βτ − α|β) 2

M Θλk

=

X

k

q 2 |γ−β| e2πik(γ|α) 2

γ∈M + λ k

(3.21) (We can read off the lattice Θ-function from (3.21) setting M = Q = L, λ = µ, k = 1.) The modular transformation law for twisted characters is deduced from the known transformation properties of Kac-Moody and lattice characters (3.6–3.9) and (3.13–3.15) using the following lemma (cf. [KP2] and [K1]). Lemma 3.1. Let the finite set of functions {Fi (τ, z, u), i ∈ I} be closed under modular transformations:  Fi  for

a c

z c(z|z) aτ + b , ,u − cτ + d cτ + d 2(cτ + d)

 =

X

Aij ∈ C ,

Aij Fj (τ, z, u) ,

(3.22)

j∈I

 b ∈ SL(2, Z). Define d  Fiα,β (τ )

Then

 Fiα,β

= Fi

aτ + b cτ + d

1 τ, α − τ β, − (α − τ β|β) 2

 =

X j∈I

 .

(3.23)

Aij Fjdα−bβ,aβ−cα (τ ) .

(3.24)

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Victor G. Kac, Ivan T. Todorov

Proof. If we set α−β then

aτ + b z˜ = , with z˜ = dα − bβ − (aβ − cα)τ, cτ + d cτ + d 

Fiα,β

aτ + b cτ + d



 = Fi

z˜ c(z| ˜ z) ˜ aτ + b , , u˜ − cτ + d cτ + d 2(cτ + d)

where u˜ = 21 (z|cα − aβ). The law (3.24) then follows from (3.22).

 , 

It is now straightforward to apply Lemma 3.1 to (3.18) to find the transformation formula of twisted A(G)-characters χα,β 3 using the transformation formula for complete characters from the previous section. Introduce the following notation: S3,30 = m3 =

s Y j=0 s X

S3j ,30j ,

(3.25a)

m3j ,

(3.25b)

j=0

where the S3j ,30j are given by (3.7) and (3.14) and the m3j are given by (3.9) and (3.15). Then we have   X 1 = e2πi(α|β)k S330 χβ,−α (τ ) , − χα,β 3 30 τ 0

(3.26)

3

1

2πi(m3 + 2 (β|β)k ) α−β,β χα,β χ3 (τ ) . 3 (τ + 1) = e

(3.27)

3.3. Small τ asymptotics of twisted characters of A(G). The small τ asymptotics will be used in the sequel for singling out non-trivial orbifold modules. Since the parameter β = 2π i τ (which has a positive real part) can be interpreted as inverse temperature, the small τ asymptotics can be interpreted as the high temperature behaviour. α,β ck /24 α,β (τ ), q ck /24 c3 χ3 (τ ) involve Lemma 3.2. (a) The q-expansions of Θλk λ (τ ) and q only non-negative powers of q. α,β (τ ) has a non-zero constant term iff λ − kβ ∈ kM . This (b) The q-expansion of Θλk constant term equals e2πi(α|β)k . (c) The q-expansion of q ck /24 c3 λ (τ ) has a non-zero constant term iff 3 = k3j with j ∈ J (see (1.33)) or 3 = 0, and λ − 3 ∈ kM . This constant term equals 1. (Recall that 3j are fundamental weights.)

(d) The q-expansion of q ck /24 χα,β (τ ) has a non-zero constant term iff 3 = k3j with j ∈ J or 3 = 0, and 3 − kβ ∈ kM . This constant term equals e2πi(α|β)k . Proof. (a) and (b) are clear. (c) is proved in [KW]. (d) follows from (b) and (c) by making use of (3.19). 

Affine Orbifolds and Rational CFT of W1+∞

73

The modular inversion S relates low temperature to high temperature behaviour and is a key to computing small τ asymptotics. πick 1 By Lemma 3.2(a) and (d) each term in the expansion of e− 12τ χα,β 3 (− τ ) vanishes exponentially for τ ↓ 0 unless 3 = k3j with j ∈ J or 3 = 0, and 3 − kβ ∈ kM , hence, by Lemma 3.2(d):   πick 1 lim e− 12τ χα,β − 3 τ ↓0 τ  (3.28) 2πi(α|β)k for 3 = k3j , j ∈ J, or 3 = 0, and 3 − kβ ∈ kM , = e 0 otherwise. Similarly, we have: lim e− τ ↓0

πick 12τ

   2πi(α|β) 1 = e − χα,β µ τ 0

for β − µ ∈ L, otherwise.

(3.29)

Substituting τ by − τ1 in (3.26): χα,β 3 (τ )

=e

2πi(α|β)k

X 30

 S330 χβ,−α 30

1 − τ

πick

 ,

α.β − ∗ −1/2 2πi(3 |α) we eP Qs find, using (3.28) and (3.29), that e 12τ χ3 (τ j) ∼ |L /L| s j S , as τ ↓ 0, where γ = k 3 with i ∈ J or γ = 0, if α + j j i j 3 ,γj j=1 j=1 γj s j ∈ L ⊕ (⊕j=1 M ), and tends to 0 otherwise. Recalling that [KW]

S3,k3j = S3,0 e−2πi(3|3j ) if j ∈ J ,

0

(3.30)

we arrive at the following result. Proposition 3.1. The high temperature asymptotics of twisted A(G) characters is given by  πic 2πi(3|β)k − 12τk α,β if exp 2πiα ∈ Z(G) χ3 (τ ) = S3,0 e lim e τ ↓0 0 otherwise. Here Z(Gc ) is the finite central subgroup of Gc defined by (2.7) and we use (2.8).

4. Affine Orbifolds 4.1. Projection on a centralizer’s irreducible representation. Asymptotic dimension. Let as before β ∈ ig be such that b = exp 2πiβ ∈ Gc has finite order. Given a positive energy A(G)-module U , we have the b-twisted module U (β) constructed in Sect. 2.2. Consider the chiral subalgebra A(G)b of fixed elements of A(G) with respect to Adb . When restricted to A(G)b , U (β) becomes an untwisted A(G)b -module. This simple, but important observation allows one to construct in many cases all untwisted modules of a chiral algebra (see e.g. [KR2]). We shall use in the sequel the following orthogonality relations of irreducible characters of a finite group 0 :

74

Victor G. Kac, Ivan T. Todorov

σ(g) 1 X ∗ δσ,σ0 , σ, σ 0 ∈ 0ˆ , σ (h)σ 0 (hg) = |0| σ(1) h∈0 1 X ∗ σ (g)σ(h) = δg, g, h ∈ 0 . ¯ , ¯ h |0g |

(4.1) (4.2)

σ∈0ˆ

Here and further 0ˆ denotes the set of all irreducible characters (= representations) of 0, σ ∗ stands for the complex conjugate character, 0g stands for the centralizer of g ∈ 0. We shall also denote by g¯ the conjugacy class of g in 0. Recall that |0| = |0g ||g|. ¯ Let 0 be a finite subgroup of the compact group Gc . We shall consider 0 as the gauge group of our CFT and define the chiral subalgebra A0 of gauge invariant observables as the set of Ad0 -invariant elements of A(G). This is called an orbifold chiral algebra. One can ensure that A0 only contains local Bose fields (even when A(L) involves fermionic vertex operators) replacing L by Leven (the maximal even sublattice of L) and 0 by its extension by L/Leven . It will be the objective of this section to construct a set of positive energy representations of A0 which again give rise to an RCFT. That will be demonstrated in the next section by displaying the SL(2, Z) properties of their characters. (This is, in general, not the case if the subgroup 0 of G is infinite.) The A0 -modules in question are obtained by splitting the twisted A(G)-modules into A0 -invariant parts. Remark 4.1. It is clear that A0 = A0Z(Gc ) , where 0Z(Gc ) is the finite subgroup of Gc generated by 0 and Z(Gc ). Hence the orbifold model does not change if we enlarge 0 by the central group Z(Gc ) and in principle we may assume that 0 contains Z(Gc ) (but we shall not do that). Pick b ∈ 0 and write it in the form b = exp 2πiβ, where iβ ∈ g. Let 0β be the stabilizer of β in 0 with respect to the adjoint action of 0 on g. Then the twisted A(G)module U (β) becomes untwisted with respect to the chiral subalgebra A(G)0β of fixed elements with respect to 0β . It follows from the construction that the group 0β acts on U (β) . Let σ be an irreducible character of the group 0β . It follows from (4.1) that the projector on the σ-isotypic component of a representation of 0β is given by Pσ =

σ(1) X ∗ σ (h)h . |0β |

(4.3)

h∈0β

The subspace Pσ U (β) is irreducible with respect to the pair (0β , A(G)0β ). This can be proved in the same way as Theorem 1.1 from [KR2]. It follows that the A(G)0β -module Pσ U (β) is isomorphic to the sum of σ(1) copies of an irreducible module which we denote by Uσ(β) . Since the affine orbifold A(G)0 is contained in A(G)0β , we obtain a A(G)0 -module Uσ(β) by restriction. Take now U = V3 . It follows from (3.18) and (4.3) that the character ck 1 (β) χβ3,σ = trq L0 −β+ 2 (β|β)k − 24 of the A(G)0 -module V3,σ is given by χβ3,σ (τ ) =

1 |0β |

X

σ ∗ (h)χα,β 3 (τ ) .

h∈0β h=e2πiα ,[α,β]=0

Applying the orthogonality relation (4.2), we can invert (4.4):

(4.4)

Affine Orbifolds and Rational CFT of W1+∞

χα,β 3 (τ ) =

X

75

σ(h)χβ3,σ (τ ) for h = e2πiα .

(4.5)

σ∈0ˆ β

Let Z = 0 ∩ Z(Gc ) denote the small center of the subgroup 0 of Gc . Theorem 4.1. The orbifold character χβ3,σ (τ ) is nontrivial iff 3 and σ agree on Z: 3|Z = σ|Z .

(4.6)

lim e− 12τ χβ3,σ (τ ) = S3,0 σ(1)|Z|/|0β | .

(4.7)

Provided that (4.6) holds, one has: πic

τ ↓0

(β) = 0 if (4.6) fails. Furthermore, by Proof. It is clear from the construction that V3,σ Proposition 3.1 and (4.4) we have: πic

lim e− 12τ χβ3,σ (τ ) = τ ↓0

S3,0 X ∗ σ (h)e2πi(3|α)k . |0β | h∈Z h=e2πiα

It follows from the orthogonality (4.1) of characters of the group Z that this is zero unless (4.6) holds, in which case it is given by the right-hand side of (4.7). The latter is  positive since S3,0 is a positive real number (see the discussion below). An important characteristic of a chiral algebra module V is its asymptotic dimension [KP2, KW] and Sect. 13.13 of [K1]. It is defined as the coefficient a(V ) of the leading term of the small τ (or high temperature) expansion of the specialized character χV : c

πic

χV (τ ) = trV q (L0 − 24 ) ≈ a(V )e 12τ .

(4.8)

For example Theorem 4.1 states that the asymptotic dimension of the orbifold module (β) is given by the right hand side of (4.7) provided that condition (4.6) holds. The V3,σ positive reals a(V ) have multifold interpretations. If A(V1 ) ⊂ A(V2 ) are two chiral algebras (with V1 ⊂ V2 ) then a(V2 )/a(V1 ) gives the index of embedding of the associated von Neumann algebras (see [R, LR] and [RST] and references therein). If V3k is an affine algebra module and V0k the corresponding vacuum module of height h then ah (3)/ah (0) is the “quantum dimension” of V3 [V]. In the case at hand the knowledge of a(V ) appears as an efficient tool for singling out non-trivial orbifold modules, and, as we shall see, for handling the splitting of reducible modules into irreducible components. An A(G)-module V3 appears as an outer product of representations of the chiral algebras A(L) and Akj (gj ). (We use the term outer tensor product to be distinguished from the tensor product of representations of a group G that is again regarded as a representation of G rather than as a representation of the direct product G × G.) The asymptotic dimension of an outer product of representations obviously equals the product of asymptotic dimensions of factors. Hence the asymptotic dimension a(V3 ) of a A(G)-module V3 is equal to the product of a(Vµ (L)) and a(V3j (gj )), j = 1, . . . , s. The asymptotic dimension of lattice modules is independent of µ: a(Vµ (L)) = Sµ,0 = |L∗ /L|− 2 . 1

(4.9)

The asymptotic dimension of Kac-Moody modules is given by (see [KP,KW] or [K1] (13.8.10)):

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Victor G. Kac, Ivan T. Todorov

a(V3 (g)) = S3,0 = |M ∗ /hM |− 2

1

Y

2 sin

α>0

π(3 + ρ|α) . h

(4.10)

This number is positive since (3|α) ≤ k and (ρ|α) ≤ (ρ|θ) = g ∨ −1, so that (3+ρ|α) < h = k + g∨ . 4.2. Affine orbifold models for non-exceptional 0. Action of Z. Modular transformations. In order to construct a modular invariant family of 0-orbifold modules we need to impose some restrictions on the subgroup 0 of Gc . Let Z be the small center of 0. Definition 4.1. An element b ∈ 0 is called non-exceptional if there exists β(b) ∈ ig such that b = exp 2πiβ(b) and 0b = 0β . The subgroup 0 of the compact group Gc is called a non-exceptional subgroup if for any g ∈ 0 there exists ζ ∈ Z such that ζg is a non-exceptional element of 0. An element g ∈ Gc is called an Ad-exceptional element of Gc if it cannot be written in the form g = bζ, where b is a non-exceptional element of Gc and ζ ∈ Z(G). Obviously, a subgroup 0 of Gc containing Z(Gc ) (recall that, due to Remark 4.1, we may assume that 0 ⊃ Z(Gc )) which does not contain Ad-exceptional elements of Gc is a non-exceptional subgroup of Gc . We shall describe Ad-exceptional elements of a compact group G in Appendix B. Here we only note that U (n) contains no exceptional elements and SU(n) contains no Ad-exceptional elements. Any connected simple compact Lie group other than SU(n) does contain Ad-exceptional elements. From now on let 0 be a non-exceptional finite subgroup of the compact Lie group Gc . It follows from the definition that for each g ∈ 0 there exists a ζ ∈ Z such that b = ζ −1 g is non-exceptional. Moreover for each g of a conjugacy class g¯ we can choose the same ζ ∈ Z and a map β : b¯ → ig satisfying b = e2πiβ(b) ,

β(hbh−1 ) = Adh β(b)

¯ h∈0. for all b ∈ b,

(4.11)

Note that a choice of β(b) such that 0b = 0β(b) , determines uniquely the map β satisfying (4.11). A quadruple (3, b, β, σ), where 3 ∈ P+k , b is a non-exceptional element of 0, β is a map satisfying (4.11) and σ ∈ 0ˆ β is called an admissible quadruple if the compatibility (β(b)) is nontrivial for any condition (4.6) holds. Due to Theorem 4.1 the A0 -module V3,σ β admissible quadruple (3, b, β, σ); we shall denote it by V3,b,σ . We have for any g ∈ 0 the identity Adg β β (4.12a) V3,gbg −1 ,σ g = V3,b,σ , where σ g ∈ 0ˆ gbg−1 is defined by σ g (h) = σ(g −1 hg) .

(4.12b)

We thus obtain the first equivalence of admissible quadruples: (3, b, β, σ) ∼ (3, gbg −1 , Adg β, σ g ) .

(4.13)

Recalling that (4.11) defines a map β : b¯ → ig and dropping the superscript g on σ we may denote the character of the module (4.12a) by χβ3,b,σ ¯ . Furthermore, if β(b) is replaced by β(b) + m, where

Affine Orbifolds and Rational CFT of W1+∞

e2πim = 1 ,

[β(b), m] = 0 ,

77

0β(b)+m = 0b ,

(4.14)

then β+m β = V3,b,σ , V3,b,σ⊗σ m

(4.15a)

where σm is a 1-dimensional representation of 0b defined by σm (h) = e2πi(m|α)k for h = e2πiα ∈ 0b .

(4.15b)

Here and further we are using the following simple fact. Lemma 4.1. Let G be a connected compact Lie group with Lie algebra g and let λ ∈ ig be a weight, i.e. (4.16a) e2πi(λ|m) = 1 if e2πim = 1 and [λ, m] = 0 . Then λ defines a 1-dimensional representation σλ of its stabilizer Gλ by the formula σλ (g) = e2πi(λ|γ) for g = e2πiγ ∈ Gλ , γ ∈ igλ .

(4.16b)

Proof. Since the group Gλ is connected, it is generated by elements g of the form (4.16b). The map σλ is independent of the choice of γ representing g due to (4.16a). If gj = e2πiγj ∈ Gλ where γj ∈ igλ , j = 1, 2, then the Cambell-Hausdorff formula implies σλ (g1 g2 ) = exp{2πi[(λ|γ1 +γ2 )+(λ|γ)]} where γ is a linear combination of commutators [γ1 , γ2 ], . . . , [[γi1 , γi2 ], . . . , ], for i1 , i2 , . . . ∈ {1, 2}. But (λ|[γ1 , γ2 ]) = ([λ, γ1 ]|γ2 ) = 0 and the same holds for multifold commutators of γj . Thus (4.16b) does indeed define a  1-dimensional representation of Gλ . The isomorphism (4.15) gives a second equivalence relation for admissible quadruples: (4.17) (3, b, β(b), σ) ∼ (3, b, β(b) + m, σ ⊗ σm ) provided that m ∈ ig satisfies (4.14). In deriving the equality of the corresponding characters we use the identity e−2πi(m|α)k χα,β+m (τ ) = χα,β 3 3 (τ ) .

(4.18)

The least obvious equivalence relation appears when two non-exceptional elements of 0 are obtained from each other by multiplication with an element ζ ∈ Z. Every element of Z can be written in the form , . . . , ζj(s) ) ∈ Z 0 × · · · × Zs , ζ = (ζj(0) 0 s

(ν)

ζj(ν) = e2πi3j or 1 .

∗ Here {3(0) j } generate the finite abelian group L /L; for each simple component g the fundamental weight 3j belongs to the set J of indices with aj = 1, see (1.33). If both b and ζj b are non-exceptional we can write

kβ(ζj b) = kβ(b) + k3j + m, Ad0b (k3j + m) = k3j + m ,

e

2πim

(4.19a) =1.

(4.19b)

We proceed to define the action ζj on σ and 3. According to Lemma 4.1 the phase factor 0 0 (4.20) σj (b0 ) = e2πi(k3j +m|β ) for b0 = e2πiβ , Ad0b β 0 = β 0

78

Victor G. Kac, Ivan T. Todorov

gives rise to a 1-dimensional representation σj of 0b . The transformation 3 → ζj (3) of a lattice weight 3 ∈ L∗ is given by ζj (3) = (3 + 3j ) mod L. If g is a simple rank ` Lie algebra and 3 ∈ P+k , then ζj (3) is defined by ζj (3) = k3j + wj 3,

(4.21a)

where wj is the unique element of the Weyl group W of g that permutes the set {−θ, α1 , . . . , α` } and satisfies (4.21b) −wj θ = αj . Theorem 4.2. The pair of non-exceptional quadruples ! ÿ s X ν ¯ β, σ) 3 and x = (3, b, 3= ÿ ζ(x) =

ν=0

X

(wjν 3 + kν 3jν ), ζb, β + ν

ν

X

3j ν

ν

mν + kν

!

 , σ ⊗ ⊗ ν σjν




(4.22) gives rise to the same orbifold module leaving the corresponding character invariant. The action of the center on non-exceptional quadruples for which b and ζb belong to the same conjugacy class b¯ has no fixed points for level k = 1 in the simply laced case, but may have a fixed point for higher levels. For G = SU(2) this happens for even k and 3 = 1 2 k. An example of this type is provided in Sect. 6 (see Example 6.4). The corresponding twisted orbifold module turns out to be reducible in this case. Understanding its splitting into irreducible components requires more work and will be postponed to a subsequent publication. Here we shall restrict our attention to the case when Z acts on the admissible quadruples without fixed points (thus including all level 1 orbifolds, all SU(p) orbifolds (with p prime) for levels not divisible by p, as well as all 0 ⊂ G orbifolds with a trivial small center). We denote by X the set of equivalence classes of all admissible quadruples with equivalence relations (4.13), (4.17) and (4.22). One may use the following description of X . Consider the action of Z × 0 on 0 for which Z acts by multiplication and 0 by conjugation. Choose a subset B ⊂ 0 consisting of non-exceptional representatives of orbits of this action, and for each b ∈ B choose β(b) ∈ ig satisfying (4.11). We call such B an admissible subset of 0. Then X may be identified with the set of admissible quadruples (3, b, β(b), σ), where 3 ∈ P+k , b ∈ B, σ ∈ 0ˆ b , with the equivalence relation that occurs only if ζb = gbg −1 for some ζ ∈ Z and g ∈ 0 . Then we let (cf. (4.22)): ÿ X ∗ (wjν 3ν + k3jν ), b, β(b), σ ⊗ σP (3, b, β(b), σ) ∼ ν

(4.23a) !

ν

mν

⊗ σ(1−Adg−1 )β(b)

.

(4.23b) We can state now our main result. Theorem 4.3.

Affine Orbifolds and Rational CFT of W1+∞

79

(a) Under the modular inversion S the characters χx (x ∈ X ) transform among themselves:   X X X X 1 β = − S330 Sbσ, χβ3,b,σ ¯ ¯ b¯ 0 σ 0 χ30 ,b¯ 0 ,σ 0 (τ ), (4.24a) τ 0 0 g=ζ ¯ 0 b0 ⊂0

b=e2πiβ ∈b¯

σ ∈0ˆ b 3

0 b0 =e2πiβ ∈b0 [β,β 0 ]=0

where S330 is the affine Kac-Moody S-matrix (3.25a), and the “group theoretic” factor looks as follows: β Sbσ, ¯ b¯ 0 σ 0 =

1 |0|

X

0

σ 0 (b)σ(b0 )e−2πi(β(b)|β(b ))k .

(4.25)

¯ 0 ∈b0 b∈b,b bb0 =b0 b

For levels and groups 0 ⊂ G for which the small center Z acts without fixed points each equivalence class of quadruples in X is encountered |Z| times and we can write   X 1 β β β0 = − |Z|S330 Sbσ, (4.24b) χ3,b,σ ¯ ¯ b¯ 0 σ 0 χ30 ,b¯ 0 ,σ 0 (τ ) . τ 0 0 0 0 (3 ,b ,β ,σ )∈X

(b) If the lattice L is even then the characters χx are eigenfunctions of the modular translation T :    ∗
1 σ (b) β β 0 β(b)|β(b ) k χ ¯ (τ ) . (4.26) χ3,b,σ ¯ (τ + 1) = exp 2πi m3 + 2 σ(1) 3,b,σ They are eigenfunctions of T 2 also for odd lattices. (c) The inverse matrix S −1 is complex conjugate to S. The matrix S in (4.24b) is manifestly symmetric and hence also unitary. (d) The matrix elements of S and T remain unchanged under the equivalence relations (4.13), (4.17), (4.22), (4.23). (e) The charge conjugation operator C = S 2 gives rise to the following involutive permutation of the set X : C : (3, b, β(b), σ) 7−→ (3c , b−1 , β(b−1 ), σ c ),

(4.27a)

where 3c = −3 in the lattice case, 3c is the highest weight of the contragredient to 3 representation of g in the affine case, and σ c (h) = σ ∗ (h)e2πi(β(b)+β(b

−1

)|α)k

for h = e2πiα ∈ 0b .

(4.27b)

Proof of Theorem 4.2. We shall content ourselves with verifying the equality of characters for admissible quadruples (4.22). The crux of the argument is the proof of the relation α,β+3 +m

χk3j +wjj 3 (τ ) = e2πi(3j +m|α)k χα,β 3 (τ )

(4.28)

(for an appropriate choice of m ∈ M ) in the case of a (rank `) simple Lie algebra g. To prove it we use the Weyl-Kac formula for the affine characters ([K1] Chap. 10). We first extend the coroot and weight spaces of g by introducing the central element

80

Victor G. Kac, Ivan T. Todorov

K=

` X

∨ ∨ ∨ a∨ ν αν ↔ α0 = K − θ




(4.29)

ν=0

and the gradation operator d(↔ −L0 ) (see Chap. 7 of [K1]). The bilinear form (.|.) is extended to the resulting ` + 2 dimensional space by (K|K) = (d|d) = 0 = (K|αi ) = (d|αi ) ,

i = 1, . . . , ` ;

(K|d) = 1 .

(4.30)

The Weyl-Kac formula then gives:  

P χα,β kd+3 (τ ) =

w˜

ε(w)e ˜

P w˜

2πi τ

ε(w)e ˜

|β|2 2

  2πi τ





K−β−d +α|w(kd+3+ ˜ ρ) ˜ |β|2 2





,

(4.31)

K−β−d +α|w˜ ρ˜

where the sum is over the affine Weyl group W (ˆg), ρ˜ is defined by ρ˜ = g ∨ d + ρ ,

ρ=

` X

3i ,

(4.32)

i=1

and ε(w) ˜ = ±1 according to the parity of w. ˜ We define the element w˜ j of the extended ˆ as follows (cf. Sect. 1 of [FKW] and Appendix B below): affine Weyl group W w˜ j = tj wj , wj d = d ,

tj d = d + 3j −

|3j |2 K, 2

tj v = v − (v|3j )K(v ∈ h) ,

(4.33)

w˜ j K = K ,

(where wj ∈ W (g) is defined on h as above). We shall use the following three properties of w˜ j : (i) it preserves the extended Killing form; (ii) it leaves ρ˜ invariant; (iii) it normalizes W (ˆg). They allow us to write down the exponent in the numerator of (4.31) as       2 |β| K − β − d + α |w {w˜ j (kd + 3) + ρ} ˜ = w˜ j τ 2

k τ |wj β + 3j |2 − k wj α|3j ) + (wj α − τ d + 3j + wj β |w {ρ˜ + w˜ j (kd + 3)} . 2 It follows that

w α,w β+3j

j χw˜ jj (kd+3)

(τ ) = e2πik(wj α|3j ) χα,β kd+3 (τ ) .

(4.34)

Observing on the other hand the invariance relation w−1 α,wj−1 β

j χkd+3

(τ ) = χα,β kd+3 (τ )

and the fact that w˜ j (kd + 3) can be substituted by ζj (kd + 3) in the expression (3.18) for the character, we complete the proof of (4.28). It remains to insert the result into (4.4) in order to conclude that

Affine Orbifolds and Rational CFT of W1+∞

81

β+3j +m (τ ) ¯ ¯ j 3+3j ,ζj b,σ⊗σj

χw

= χβ3,b,σ ¯ (τ ) ,

(4.35)



thus proving Theorem 4.2.

Proof of Theorem 4.3. We use the assumption that 0 is a non-exceptional subgroup of G in order to express h in the formula (4.4) for the orbifold character by a non-exceptional element b0−1 : 0 (4.36a) h = ζb0−1 = e2πi(αζ +β− ) , where

ζ = e2πiαζ ∈ Z ,

0

b0−1 = e2πiβ− .

[αζ , β(b)] = 0 ,

This allows to rewrite (4.4) in the form 1 X χβ3,b,σ ¯ (τ ) = |0| ¯ b∈b

X

β 0 ,β

σ(b0 )χ3− (τ ),

(4.36b)

(4.37)

h=ζb0−1 ∈0b [β(b),β 0 ]=0 −

where we have used the relation 0 αζ +β− ,β

χ3

β 0 ,β

(τ ) = e2πi(3|αζ )k χ3− (τ )

(4.38a)

for e2πi(m|αζ ) = 1 whenever m ∈ M , [αζ , m] = 0, implying 0 αζ +β− ,β

σ ∗ (h)χ3

∗

β 0 ,β

(τ ) = σ(b0 )χ3− (τ )

0−1

(4.38b)

0

for σ|Z = 3|Z (we have also used σ (b ) = σ(b )). Inserting the modular inversion law (3.26) into (4.37) we find   X X 0 1 X 1 β,−β 0 χβ3,b,σ = σ(b0 )e2πi(β|β− )k S330 χ30 − (τ ) , (4.39a) − ¯ τ |0| ¯ 0 0−1 b∈b

3

h=ζb b0 ∈0 bβ,−β 0 −

where, in view of (4.5), we can write 0 β,−β−

χ3

(τ ) =

X

0

σ −β− (b)χ

−β 0 σ − ∈0ˆ b

0 −β−

30 ,b¯ 0 ,σ

−β 0 −

.

(4.39b)

Finally, we would like to substitute the upper index of χ by the phase β 0 of b0 which 0 by a coroot: differs from −β− 0

b0 = e2πiβ ⇒ e−2πi(β Applying (4.15) we obtain   1 β χ3,b,σ − ¯ τ X 1 X = |0| g⊂ ¯ 0

b=e2πiβ ∈b

0 g=ζ ¯ 0 b0 b0 =e2πiβ ∈b0 bb0 =b0 b

0

X X σ 0 ∈0ˆ b 30

0 +β− )

=1

0 ([β 0 + β− , β] = 0) .

σ(b0 )σ 0 (b)e−2πi(β|β

0

)k

0

(4.40)

S330 χβ30 ,b¯ 0 ,σ0 (τ ), (4.41)

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Victor G. Kac, Ivan T. Todorov

where

0

σ 0 (b) = σ −β− (b)e2πi(β

0

0 +β− |β)k

.

(4.42)

If the small center Z acts on admissible quadruples for which ζb ∈ b¯ without fixed points, then each term in the sum is encountered exactly |Z| times and we end up with (4.24b), (4.25). The T -transformation law (4.26) follows from Eq. (3.27): X 2πi{m3 + 21 (β|β)k } σ ∗ (h)χα−β,β (τ ) χβ3,b,σ ¯ (τ + 1) = e 3 h∈0b h=e2πiα

(4.43)

[α,β]=0

= e2πi{m3 + 2 (β|β)k −(σ|β)} χβ3,b,σ ¯ (τ ) . 1

Here we have used the fact that b is in the center of 0b and σ(∈ 0ˆ b ) is irreducible, so that σ ∗ (b) , (4.44a) σ ∗ (h) = σ ∗ (hb−1 ) σ(1) where the last factor is a complex number of absolute value 1 which can be written as σ ∗ (b) =: e−2πi(σ|β) . σ(1)

(4.44b)

(Equation (4.44b) thus defines a linear functional (σ|β) in β whose exponential agrees with the value of 3 on Z.)   0 1 Using once more Lemma 3.1 for the inverse transformation to (3.4) we −1 0 −1 ∗ derive S = S , where ∗ stands for complex conjugate. The symmetry of S is manifest β from the expressions for S330 and Sbσ, ¯ b¯ 0 σ 0 . To prove the invariance of S-matrix elements with non-exceptional entries under the equivalence relation (4.23) we use an extension of (3.30): 0

Sζj (3),30 = e−2πi(3j |3 ) S330 ,

(4.45)

(cf. [KW]) and the fact that σ 0 and 30 coincide on the central element 3j . To verify that T is also invariant under ζj one uses (σ|3j + m) = (3|3j + m) and (wj 3|3j ) = (3|wj−1 3j ) = (3|3j + m0 ) (m0 ∈ M ) to prove that the phase φ(kd + 3, β, σ) =

k 1 (kd + 3 + 2ρ|kd ˜ + 3) − (σ|β) + |β|2 2k 2

changes by an integer: k (|3j |2 − |3j + m|2 ) + (3|m0 − m) + (3 − σ|3j + m) 2 1 = (3|m0 − m) − k{(3j |m) + |m|2 } ∈ Z(⇒ e2πi1φ = 1) . 2

1φ =

We finally proceed to prove (4.27). To this end we compute C = S 2 by applying Lemma 3.1 to the central element of SL(2, Z).

Affine Orbifolds and Rational CFT of W1+∞



−1 0

0 1

83



2 =

−1 0

0 −1

 .

This gives 1 X ∗ 0 −β(b0 ),−β(b) σ (b )χ30 (τ ) |0b | 0 b ∈0b 30 X X β β0 β0 = C330 Cbσ, ¯ b¯ 0 σ 0 χ30 ,b¯ 0 ,σ 0 (τ ) ,

(Cχ)β3,b,σ ¯ (τ ) =

X

30

C330

b¯ 0 σ 0

where C330 = δ3c 30 is known from the modular properties of affine Kac-Moody characters ([K1] Chap. 13), while the second factor is computed to be 0

β β Cbσ, ¯ b¯ 0 σ 0 = δb−1 ,b0 δσ c ,σ 0 δβ,−β 0 .

(4.46) 

We note that the equivalence class v of the vacuum admissible quadruple, i.e. that corresponding to the vacuum A(G)0 -module, is selfconjugate: v := class of (0, 1, 0, 1) = Cv. Note also the following formula for any x = (3, b, β(b), σ) ∈ X : Sx,v = S3,0

¯ |b| σ(1) . |0|

Remark 4.2. It follows from Lemma 3.2d that the eigenvalues of L0 are strictly positive in all A(G)-modules Vx , x ∈ X , except for the vacuum module Vv . The 0th eigenspace of L0 in Vv is C|0i. Remark 4.3. The A(G)0 -modules Vx and VCx (x ∈ X ) are contragredient. 4.3. Fusion rules. We can summarize the most important features of the outcome of the previous section as follows. Starting with a compact Lie group G = (Rr /L) × G0 , where G0 is simply connected, and a negative definite integral invariant bilinear form on its Lie algebra which is even on the lattice L, we have constructed for every non-exceptional finite subgroup 0 of G a collection of A(G)0 -modules parametrized by a finite set X . This set is equipped with an involutive permutation C (corresponding to taking a contragredient module) and a distinguished element v (corresponding to the vacuum module) such that Cv = v. We have also matrices S = (Sxy )x,y∈X and T = (Txy )x,y∈X satisfying the following three properties, provided that the small center Z acts on X without fixed points: (a) S is symmetric and T is diagonal,      0 −1 1 1 −1 (b) the map ⇒ S, ⇒ T, 1 0 0 1 0 sentation of the group SL2 (Z).

0 −1



⇒ C gives a unitary repre-

84

Victor G. Kac, Ivan T. Todorov

(c) Sxv > 0 for all x ∈ X . Following Verlinde [V], introduce the fusion algebra A(X ) = ⊕x∈X Cx by the formula: X Nxyz Cz , (4.47a) xy = z∈X

where the fusion coefficients Nxyz are defined by X Sax Say Saz /Sav . Nxyz =

(4.47b)

a∈X

It follows from the above properties of S that the fusion algebra A(X ) is a finitedimensional commutative associative semisimple algebra with identity element v and involutive automorphism C. All homomorphisms of the algebra A(X ) to C are labeled by elements y ∈ X and given by chy (x) = Sxy /Svy (x ∈ X ) .

(4.48)

The positive real number chv (x) is the relative (= quantum) dimension. The basic observation of [V] is that the fusion algebras arising in a RCFT have the following fundamental property: (d) Nxyz ∈ Z+ . Denote by X af the set P+k labeling all positive energy irreducible representations of the chiral algebra A(G) with vacuum element v = 0, conjugation C3 = 3c , S-matrix S af = (S330 ) and T -matrix T af = e2πim3 δ330 . It follows from [KP2] that the properties (a)-(c) hold, and it is a very difficult theorem established by the efforts of many people that (d) holds as well. Denote by N330 300 (∈ Z+ ) the fusion coefficients. ¯ σ), where g¯ is a conjugacy class of Similarly, let X gr denote the set of all pairs (g, 0 and σ is an irreducible character of 0g . Let v = (1, 1) be the vacuum element and let gr be the matrix defined C(g, ¯ σ) = (g −1 , σ c ), where σ c is defined by (4.27b). Let Sbσ,g 0 σ0 ¯ by the right-hand side of (4.25) and let (cf. (4.26)): gr = e2πi(β|β)k Tgσ,g 0 σ0 ¯

σ ∗ (b) . σ(1)

(4.49)

It follows from the remarks of the previous section that the properties (a), (b) and (c) hold. It can be demonstrated by an appropriate example of an SU(2) subgroup of level 1 (see Example 6.5), that property (d) does not hold in general. Lusztig [Lus] studied the “limiting” case of our X gr when in (4.26), (4.27b) and (4.49) one sets all β(b) equal zero and b = g. In this case (d) holds due to his interpretation of the fusion algebra as the Grothendieck ring of the category of 0-equivariant vector bundles. Whenever the center of G is trivial like in the case of E8 the fusion rules factorize: Nxx0 x00 = N330 300 Ng00 σ,g0 σ0 ,g00 σ00 . In particular, for a level 1 orbifold like A1 (E8 )0 they coincide with the group theoretic fusion rules which we proceed to compute. The following cubic sum rule tells us that the fusion coefficient Ng¯ 1 σ1 ,g¯ 2 σ2 ,g¯ 3 σ3 =

X Sg¯ σ ,hσ ¯ Sg¯ 1

¯ hσ

1

¯

2 σ2 ,hσ

Sg¯ 3 σ3 ,hσ ¯

S11,hσ ¯

vanishes unless there are triples gj ∈ g¯ j , j = 1, 2, 3 such that g1 g2 g3 = 1.

(4.50)

Affine Orbifolds and Rational CFT of W1+∞

85

Proposition 4.1 ([Gor] Theorem 2.12). Let g¯ i , i = 1, 2, 3, be three conjugacy classes in a finite group 0. The number n123 of triples gi ∈ g¯ i such that g1 g2 g3 = 1 is given by n123 =

|g¯ 1 ||g¯ 2 ||g¯ 3 | X 1 σ(g1 )σ(g2 )σ(g3 ) . |0| σ(1) σ∈0ˆ

In deriving the fusion rules we follow [DV3 ], but compute explicitly the phase factors. Theorem 4.4. The fusion rules (4.50) can be expressed in either of the two forms: Nb¯ 1 σ1 ,b¯ 2 σ2 ,b¯ 3 σ3 =

1 X |0|

h∈0

Nb¯ 1 σ1 ,b¯ 2 σ2 ,b¯ 3 σ3 =

X O12

X

X

1 |0b1 ,b2 |

Here the multiplier µ is given by µ(h|Σβi ) = e2πi(α|

σ1 (h)σ2 (h)σ3 (h)µ(h|Σβi ),

(4.51a)

bi ∈bi ∩0h b1 b2 b3 =1

P

σ1 (h)σ2 (h)σ3 (h)µ(h|Σβi ) .

(4.51b)

h∈0b1 ,b2

β i )k

,

βi = β(bi ) ,

h = e2πiα .

(4.52)

The outer sum in (4.51b) is over different orbits O12 of pairs (b1 , b2 ) under the adjoint action of 0; the number |O12 | of such orbits is determined from the relation |O12 ||012 | = |0| . The proof uses the form Sg¯ j σj ,hσ ¯ =

1 |0h |

X

σj (h)σ(bj )e−2πi(α|βj )k

(4.53)

bj ∈b¯ j ∩0h

of (4.25) for the three factors in the numerator of (4.47) and reduces to a straightforward application of Proposition 4.4 (noting the conjugation invariance of µ). (For x3 = v (the ¯ σ3 = 1 (β3 = 0) we reproduce as a special case the charge vacuum module) b¯ 3 = 1, conjugation matrix (4.46): Cg¯ 1 σ1 ,g¯ 2 σ2 = Ng¯ 1 σ1 ,g¯ 2 σ2 ,11 ¯ .) The multiplier (4.52) does not depend on the choice of the phase α of h provided it belongs to the stabilizer gb1 ,b2 of the pair (b1 , b2 ) in g; µ thus defines a representation of 0b1 ,b2 according to Lemma 4.1 applied to G = Gb1 , λ = β2 . 5. U (l) orbifolds as RCFT extensions of W1+∞ What is now called W1+∞ first appeared as the (unique nontrivial) central extension Dˆ of the Lie algebra D of differential operators on the circle [KP1]. Its representation theory (including the classification of quasi-finite positive energy representations) was developed in [KR2] and [FKRW]. It has also attracted the attention of physicists, in particular, the most degenerate ‘minimal series’ of unitary representations of W1+∞ of [FKRW] are being applied in the study of quantum Hall fluids [CTZ]. (More reference to both physical applications and related mathematical developments are cited in the above ˆ papers and in the bibliography to [AFMO].) The vacuum D-module (corresponding for unitary representation to a positive integer central charge c = l) was shown [FKRW] to

86

Victor G. Kac, Ivan T. Todorov

(l) carry a canonical chiral (vertex) algebra structure. The resulting chiral algebra W1+∞ was described in [BGT] in terms of a series of quasi primary fields of dimension ν +1, ν = 0, 1, . . .: X Vnν z −n−ν−1 , [Lm , V ν (z)] V ν (z) =   (5.1a) d m =z + (m + 1)(ν + 1) V ν (z) , m = 0 , ±1, z dz

satisfying local CR such that µ+ν−1 + ... + c [Vmµ , Vnν ] = (νm − µn)Vm+n

(ν!)4  (2ν)!

 m+ν δ , δ m − ν − 1 m,−n µν

c=l.

(5.1b) (l) The (quasi finite) irreducible positive energy modules V~r of W1+∞ are characterized by l exponents (see [KR1,2]) ~r = (r1 , . . . , rl ) that take real values for unitary representations. Each V~r has a cyclic minimal energy vector |~ri such that Vnν |~ri = 0 for n = 1, 2, . . . , {V0ν − vν (~r)}|~ri = 0,

(5.2)

where v0 (~r) =

l X i=1

ri , ν−1 

(ν − 1)!ν! X vν (~r) = (2ν)! j=0

ν j



ν j+1

X l

(5.3) (ri − j) . . . (ri + ν − j − 1)ri .

i=1

In particular V 1 (z) = T (z) so that the ground state energy eigenvalue is v1 (~r) = 21 ~r2 = Pl 2 i=1 ri : 

 l 1 1X 3 1X 4 1 2 L0 − ~r2 |~ri = 0 (L0 = V01 ) , v2 (~r) = ri , v3 (~r) = (r + r ) . (5.4) 2 3 4 i i 5 i i=1

ˆ The vacuum D-module contains for c = l ∈ N a unique singular vector of degree l + 1 such that the quotient by the submodule generated from this singular vector is irreducible [KR1]. This irreducible quotient (together with its chiral algebra structure) is isomorphic to a (level l) W (ul ) vacuum module – see [FKRW], Sect. 5. As a result, any irreducible (l) representation of W1+∞ has a canonical structure of an irreducible representation of W (ul ) of level l, and all irreducible representations of W (ul ) with central charge l arise in this way. Any V~r splits into a tensor product of a W (su(l)) module of central charge l − 1 (1) -module. To see this we rescale the u(1) current and split the stress energy and a W1+∞ tensor as in (1.39): 1 J(z) = √ V 0 (z) , T (z) = TJ (z) + Tsu (z) , l   1 1 2 0 2 TJ = : J : = : (V ) : , 2 2l

(5.5a)

Affine Orbifolds and Rational CFT of W1+∞

87

[Tsu (z), J(w)] = 0 .

(5.5b)

The minimal eigenvalue of the energy of the second term, Lsu 0 is then given by the difference ÿ r !2 1 X 1 X 1 2 ~r − ri = (ri − rj )2 =: ωl (~r). (5.6) 2 2l 2l i<j i=1

(l) -module V~r is degenerate if some of the differences ri − rj are integers. It A W1+∞ is maximally degenerate if all ri − rj are integers (such representations are termed minimal [CTZ]); the representation of the second (su)-factor is indeed then a limit of the Zamolodchikov-Fateev-Lukyanov Wl (p)-models of central charge c = (l − 1) o n l(l+1) 1 − p(p+1) as observed in [CTZ]. Since every V~r can be viewed as a tensor product of maximally degenerate (including c = 1) modules we shall turn our attention to the case of integer ri − rj . Assume that ri − rj ∈ Z, we then arrange the ri ’s in a decreasing order and denote the set of such ~r’s as P + :

P + = {~r ∈ Rl |r1 ≥ r2 ≥ . . . ≥ rl , ri − rj ∈ Z} .

(5.7)

If we interpret the ordered set 3 = (λ1 , . . . , λl−1 ) of differences λi = ri − ri+1 ,

i = 1, 2, . . . , l − 1

(5.8)

as defining a highest weight of SU(l), then for the fundamental weights 31 = (1, 0, . . . , 0), . . . , 3l−1 = (0, . . . , 0, 1) the ground state energy eigenvalues (5.6) coincide with the level 1 eigenvalues of the su ˆ l current algebra A1 (su(l)): ωl (~r(i)) =

(3i + 2ρ|3i ) (i) for rj(i) − rj+1 = δij 2(l + 1)

(5.9)

(which can be verified by a direct computation). It is natural to expect that the W (sul ) representations of such weights obey fusion rules given by the tensor product expansion formulae for SU(l) (see Conjecture 6.1 of [FKRW]). It follows that a CFT with chiral algebra W (su(l)) and a highest weight module V~r with ri −rj non-zero integers has an infinite number of sectors and hence is not a rational CFT. (We are using here the basic property of any quantum field theory to be closed under fusion.) This ‘irrationality’ can also be seen from an analysis of the characters of these representations (computed in [FKRW]). The orbifold construction of the previous sections allows to define a large class of RCFT extensions of W1+∞ with the same stress energy tensor. (l) into the Fock space Fl of l In fact the embedding of the vacuum module of W1+∞ free complex fermion fields, used from the outset in [FKRW] and [KR2], does provide one such (chiral superalgebra) extension. So does its even (bosonic) part which coincides with the level 1 current algebra of the rank l (even) orthogonal group A1 (so(2l)). (Indeed, if we separate the real and imaginary part of the free fermions writing them as 1 ψj = √ (ϕ2j−1 − iϕ2j ) , j = 1, . . . , l , then Jjk (z) = iϕj (z)ϕk (z) (j < k) (5.10) 2 satisfy the commutation relations of level 1 so(2l) currents. The complex structure selects a Cartan subalgebra that includes V 0 :

88

Victor G. Kac, Ivan T. Todorov

j

H (z) =:

ψj∗ (z)ψj (z)

:= J2j−1,2j (z) ,

0

V (z) =

l X

H j (z) .

(5.11)

j=1 (l) as the U (l)-invariant subalgebra of A1 (so(2l)) (u(l)) and Then we can define W1+∞ (l) so(2l) sharing the same Cartan subalgebra.) A more general RCFT extension of W1+∞ is provided by the chiral algebra associated with the compact group U (l), equipped with a lattice structure Q (see Sect. 1). Here Q is an l-dimensional even integral lattice whose sublattice of vectors of length square 2 includes the (rank l − 1) su(l) lattice. (The root lattices of rank l semi-simple Lie algebras- so(2l), su(l + 1), su(l) ⊕ su(2)) – appear then as special cases. Note that the su(l) Cartan currents are orthogonal to V 0 (5.11) (or J (5.5)); they are

H αi (z) = H i (z) − H i+1 (z) ,

i = 1, . . . , l − 1 ,

(5.12)

(l) , α1 , . . . , αl−1 being the simple roots of su(l). Any of the extensions A(Q) of W1+∞ where Q is a (rank l) lattice with the above properties admits a finite set of positive energy CFT representations whose characters span a (finite dimensional) representation of SL(2, Z). All these extensions involve, in particular, l commuting u(1) currents and can be thus related to the approach of Fr¨ohlich, Thiran et al. to the quantum Hall effect (see [FT] and references therein). A large family of intermediate observable algebras is provided by 0 orbifolds of A(Q) where 0 is any finite subgroup of U (l). If 0 is not contained in any proper Lie subgroup of SU(l), then A0 only involves a single u(1) (l) current – the one belonging to W1+∞ . Such A0 could be viewed as RCFT extensions of (l) minimal W1+∞ models (exploited in [CTZ]). We proceed to state the precise results for the Fock space Fl of l free (complex) fermions and its orbifolds.

Theorem 5.1. [FRKW] The fermion Fock space Fl viewed as a representation of the l ) splits into an infinite direct sum of tensor products pair (U (l), W1+∞ Fl = ⊕~r∈P+ F (~r) ⊗ L(~r) ,

(5.13)

where P+ = {~r = (r1 , . . . , rl ) ∈ Zl |r1 ≥ · · · ≥ rl }, F (~r) is the finite dimensional (l) irreducible U (l)-module of highest weight ~r, L(~r) is a unitary W1+∞ positive energy module with exponents ~r and specialized character Y l 1 2 (1 − q ri −rj +j−i ) . (5.14) χ~r (τ ) = trL(~r) q L0 − 24 = q 2 ~r η −l (τ ) 1≤i<j≤l

The following result is a specialization of Eq. (4.4) and Theorem 4.3 applied to the chiral algebra A(Zl )0 , where Zl is the integral lattice with the standard bilinear form, and 0 is a finite subgroup of U (l). Recall that A(Zl ) has a unique irreducible representation, hence we may skip the index 3. Theorem 5.2. Let 0 be a finite subgroup of U (l). Write each b ∈ 0 in the form b = exp 2πiβ, where iβ ∈ u(l) is fixed by Ad0b . Let {βi (~r)} denote the set of eigenvalues of β in F (~r). Given an irreducible character σ of 0b , let X 1 m~r,σ,βi (~r) q −βi (~r) , m~r,σ,β (q) = q 2 (β|β) i

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where m~r,σ,βi (~r) is the multiplicity of σ in the βi (~r)-eigenspace of β in F (~r). Then the A(Zl )0 -characters can be written in the following form: X χβb,σ m~r,σ,β (q)χ~r (τ ) . (5.15) ¯ (τ ) = ~ r ∈P +

All these characters are modular functions and their C-span is invariant under the transformation τ 7→ − τ1 . ¯ we have β = 0 and all m~r,σβ (q) ∈ Z+ , and we find the In particular, for b¯ = 1, characters of untwisted orbifold modules, which, unlike χ~r are modular functions of τ . This special case of Theorem 5.2 provides a family of solutions to the following problem: find non negative integers n(~r) such that X n(~r)χ~r (τ ) ~ r ∈P +

is a modular function of τ . Each pair 0 ⊂ U (n) (0 finite subgroup), σ ∈ 0ˆ gives a solution to this problem with n(~r) = nσ0 (~r) being the multiplicity of σ in F (~r) viewed as a 0-module. Proof of Theorem 5.2. In view of (4.4) and (3.18) we can write X 1 1 σ ∗ (a)χ(τ, α − βτ, (β|βτ )) , χβb,σ ¯ (τ ) = |0b | 2 a ∈ 0b a = exp 2πiα [α, β] = 0 where, due to (5.13), χ(τ, z, u) = e2πiu

X

χ~r (τ )trF (~r) e2πiz .

(5.16)

(5.17)

~ r ∈P +

Hence we have: χβb,σ ¯ (τ ) =

X

1

χ~r (τ )q 2 (β|β)

~ r ∈P +

X

σ ∗ (a)trF (~r) (aq −β ) .

(5.18)

a∈0b

Since 0b fixes β, each eigenspace of β in F (~r) is 0b -invariant. The contribution of the βi (~r)-eigenspace to the inner sum of (5.18) is clearly equal m~r,σ,βi (~r) q −βi (~r) . This proves (5.15).  Remark 5.1. Theorem 5.2 can be generalized to any simply laced simple Lie algebra g of rank l and 3 ∈ P+1 . Namely, formula (5.15) holds for any non-exceptional element b, where the sum is taken over λ ∈ (3 + Q) ∩ P+1 , and (see [K1], Exercise 12.17):  Y 1 1 − q (λ+ρ|α) . χλ (τ ) = q 2 (λ|λ) η −l (τ ) α>0

We have:

χβ3,b,σ ¯ (τ ) =

X

mλ,σ,β (q)χλ (τ ) .

(5.19)

λ∈(3+Q)∩P+

The character χβ3,σ,b (τ ) is a modular function and their C-span is SL2 (Z)-invariant provided that 0 is a non-exceptional finite subgroup of our simple Lie group.

90

Victor G. Kac, Ivan T. Todorov

Remark 5.2. Taking 0 = {1} in Remark 5.1 we arrive at the following curious identity by comparing two expressions for 0-orbifold characters for each weight 3 and real number m: X Y (λ + ρ|α) . |{λ ∈ 3 + Q|(λ|λ) = m}| = (ρ|α) λ∈3+Q α>0

(λ|λ)=m

6. Examples 6.1. Lattice current algebras for c = 1. The simplest (c = 1) case of a lattice current algebra is worth singling out for at least two reasons: (1) the basic Θ-functions encountered here also appear in the SU(2) affine orbifold model; (2) the lattice part of a U (l) orbifold encountered in a W1+∞ theory is of this (U (1)−)type. A 1-dimensional lattice L = Zω is characterized by a single natural number m = |ω|2 ; we shall denote A(L; |ω|2 = m) by A(m). Note that m is twice the dimension of the basic 1 charged fields Y (e±ω , z), while v(z) = m− 2 ω(z) is the corresponding u(1) current (see 1 . The factor Sect. 1.1). The dual lattice is L∗ = Zω ∗ , where (ω ∗ |ω) = 1 ⇒ |ω ∗ |2 = m group L∗ /L is the cyclic group of order m; there are, correspondingly, m untwisted modules whose weights will be labeled by minimal length representatives µω ∗ ∈ L∗ /L ,

m m−1 ≤µ≤ , 2 2

µ∈Z.

(6.1)

The specialized character of the positive energy A(m)-module Vµ (of ground state |µω ∗ i) is given by (see [DFSZ, PT]) Kµ (τ, m) =

1 1 X m (n+ µ )2 L m Θµ1 (τ, 0, 0) = q2 . η(τ ) η(τ )

(6.2)

n∈Z

This set spans a representation of SL(2, Z) in the case of a bosonic algebra (m even) and requires supplementing it with Ramond sector (Z2 twisted) modules corresponding to half-odd integer µ’s in the interval (6.1) for m odd and splitting each integer µ character into two (corresponding to summing over even and odd n’s in (6.2)). For m = 2s even the modular transformation law for Kµ is given, according to (3.15), by µ2

Kµ (τ + 1, 2s) = eiπ( 2s − 12 ) Kµ (τ, 2s),   s 1 X −iπ µν 1 s K (τ, 2s). Kµ − , 2s = √ e ν τ 2s 1

(6.3) (6.4)

ν=1−s

Example 6.1. A ZN -orbifold of A(m) is given by the chiral algebra A(N 2 m) (and its positive energy modules). If indeed we introduce the inner automorphism A(m) 3 A → U AU −1 , U eω U −1 = e2πi/N eω

∗

U = e2πiω0 /N ,

(U J(z)U −1 = J(z))

(t−1 √ωm , z),

ω0∗

√1 J0 , m

(6.5a) (6.5b)

(J being the u(1) current J(z) = Y = cf. Sect. 1.1), then the ±N ω vertex operators Y (e , z) generate the gauge invariant subalgebra

Affine Orbifolds and Rational CFT of W1+∞

91

A(m)ZN = A(N 2 m) .

(6.6)

The Z2 -orbifold of A(m) with m odd has an even gauge invariant subalgebra A(4m). The representation theory of A(m), m = 2ρ + 1, ρ ∈ Z+ , can be deduced from this remark. Example 6.2. Modular properties of characters of A(m = 2ρ + 1) derived from those for A(4m). The characters Kµ (τ, m), m odd, µ = 21 Z mod m are expressed in terms of Kν (τ, 4m) as follows: Kµ (τ, m) = K2µ (τ, 4m) + K2µ+2m (τ, 4m) .

(6.7)

The periodicity relation Kν+m (τ, m) = Kν (τ, m)

(6.8)

allows to replace (if necessary) the indices in the right hand side of (6.7) by equivalent ones in the canonical interval (6.1). The resulting SL2 (Z) transformation properties of Kµ (τ, m) then read   2 1 iπ µ  m − 12 (6.9a) Kµ (τ + 1, m) = e K2µ (τ, 4m) + (−1)2µ+m K2µ+2m (τ, 4m)   m 1 X −2πiµν/m 1 e K2ν (τ, 4m) Kµ − , m = √ τ m ν=1−m (ν∈Z) (6.9b) 1 X − 2πiµν e m Kν (τ, m) . =√ m νmod m

Thus, for m odd, only the entire set of 4m characters Kν (τ, 4m) is closed under SL2 (Z). The original set {Kµ (τ, m), µ ∈ Z/mZ}, corresponding to the Neveu-Schwarz sector of the supersymmetric theory, is however invariant under the subgroup of the modular group generated by T 2 (τ → τ + 2) and S. It is remarkable that the diagonal partition function (in which we restore the dependence on the u(1) variable z), X χµm (τ, z)χ¯ µm (τ, z), (6.10) Z(τ, z) = µ mod m

where χµm (τ, z) =

X

1

2

q 2m (mn+µ) e2πiz

mn+µ m

(6.11)

n

is related to the Laughlin plateaus of the quantum Hall effect (corresponding to filling µ µ2 1 , charge m and fractional spin J = 2m , µ ∈ Z – see [CZ]). (The characters factor ν = m 2

π (Imz) used in [CZ] differ from (6.11) by a non-analytic factor, exp{− m Imτ } corresponding to a modified Hamiltonian and ensuring invariance under z → z + τ .)

Example 6.3. Charge conjugation orbifolds. The involutive lattice conjugation CL : eω → e−ω ,

J → −J

(6.12)

provides, for m 6= 2, an example of an outer automorphism of the chiral algebra A(m). Our construction of orbifold modules does not apply, strictly speaking, to this case. Nevertheless, it is easy to construct a modular invariant set of CL -orbifold characters. We shall write them down for the bosonic (m = 2s, s ∈ N) case.

92

Victor G. Kac, Ivan T. Todorov

The CL -orbifold chiral algebra A(2s)CL is generated by a single primary field φ = φ(z, ω) with respect to its A(S ⊗ 1)CL subalgebra, the real part of the vertex operator Y (eω , z): 1  φ(z, ω) = √ Y (eω , z) + Y (e−ω , z) . 2 Here A(S ⊗ 1) is the u(1) chiral current subalgebra corresponding to the subspace S ⊗ 1 (1.16). The operator product expansion of two φ’s involves the stress energy tensor T and the Virasoro primary field : J 4 (z) : that generates A(S ⊗ 1)CL . The chiral algebra splits into a CL -even and a CL -odd parts. The vacuum module character splits, accordingly, into two pieces: (6.13a) K0 (τ, 2s) = K0+ (τ, 2s) + K0− (τ, 2s), where K0± (τ, 2s) =

1 {K0 (τ, 2s) ± (K0 (τ, 8) − K4 (τ, 8))} . 2

(6.13b)

The difference of Z2 twisted level 1 A(1) 1 characters (that appears in parentheses) can be written in the form 2 1 X (−q)n . (6.13c) K0 (τ, 8) − K4 (τ, 8) = η(τ ) n 1 ) for 1 ≤ µ ≤ s − 1 Each pair of representations of weights ±µω ∗ of A(2s)(|ω ∗ |2 = 2s gives rise to a single representation of the gauge invariant subalgebra A(2s)CL . The characters K0± (6.13), being expressed in terms of Kµ , have known modular transformation properties; in particular, 1 n + 1 K0 (τ, 2s) + K0− (τ, 2s) + Ks (τ, 2s) K0± (− , 2s) = √ τ 2 2s s (6.14) o X 1 Kµ (τ, 2s) ± √ (K1 (τ, 8) + K3 (τ, 8)) . +2 2 µ=1

Analyzing this relation together with the unitarity requirement for the S-matrix one concludes that there are altogether s+7 inequivalent representations of A(2s)CL (see [DV3 ]) corresponding to s+3 untwisted and 4 twisted orbifold modules. The µ = s A(2s)-module splits, in particular, into two A(2s)CL -modules with the same specialized character ∞ 1 X s(n+ 1 )2 1 2 . Ks (τ, 2s) = q 2 η

(6.15)

n=0

Similarly, there are two pairs of twisted representations with characters Ki (τ, 8), i = 1, 3, 1 ) in (6.14). each Ki appearing twice (with a coefficient ± 2√ 2 For s = 1 the model reduces to a Z2 affine orbifold. For s = 2, 3, 4 and 6 it has been identified with known models in [DV 3 ]. We conjecture that these CL -orbifolds can be shown to exist for all values of s using the vertex operator construction of Sect. 1.1. 6.2. SU(2) orbifolds. The finite subgroups of SU(2) being thoroughly studied,1 the Ak (su(2)) orbifold characters and their modular properties can be worked out quite 1

For a modern treatment based on the McKay correspondence – see [Kos].

Affine Orbifolds and Rational CFT of W1+∞

93

explicitly. Noting that the Cartan subalgebra of su(2) is 1-dimensional we can express its elements α, β, γ, λ by (rational) numbers identifying each of them with the coefficient 1 to 3∨ 1 = 2 σ3 (σj are the Pauli matrices – see (6.23)); then λ 1 (2n + − β)2 , n ∈ Z , 2 k  λ (γ|α) = n + α , α, β ∈ Q . 2k

|γ − β|2 =

λ = 1 − k, . . . , 0, 1, . . . , k , (6.16)

The character (4.36), (3.18), (3.3) can be written in the form χβ3,b,σ ¯ (τ ) = where β (τ ) = Θλ,k,σ

k X

β c3 λ (τ )Θλ,k,σ (τ ) ,

(6.17)

λ=1−k 3−λ∈2Z

X

k

λ

q 4 (2n+ k −β) σ2kn+λ , 2

(6.18)

n∈Z

σ2kn+λ =

1 |0b |

X

σ ∗ (h)eiπ(2kn+λ)α .

(6.19)

h∈0b tr h=2 cos πα

For b 6= 1 and non-exceptional, 0b is abelian and h can be assumed diagonal. We have treated in Sects. 2, 3 and 6.1 the case of a ZN orbifold (as an automorphism group of A(SU(2)), ZN appears as a subgroup of SO(3); 0 in this case should be identified with its double cover Z2N ⊂ SU(2)). Each ZN automorphism group leaves a u(1) (Cartan) current invariant. The remaining non-abelian subgroups of SO(3) can be described as groups on two generators, s and t, obeying three relations: sn1 = tn2 = (st)n3 = 1 ,

1 1 1 2 >1 + + =1+ n 1 n2 n 3 |Ad 0|

(6.20)

(n1 , n2 , n3 are natural numbers and we denote the group unit by 1). The double cover 0(⊂ SU(2)) of Ad 0 is again generated by two elements s and t but the group unit in the first relation (6.20) is replaced by the non-trivial central element ε of SU(2): s, t ∈ 0 ⇒ sn1 = tn2 = (st)n3 = ε ,

ε2 = 1(|0| = 2|Ad 0|) .

(6.21)

Example 6.4. The H8 ⊂ SU(2) orbifold. The abstract group of quaternion units has 8 elements, {1, ε, qi , εqi , i = 1, 2, 3}; they obey multiplication rules qi2 = ε, q1 q2 = q3 which fit (6.21) with n1 = n2 = n3 = 2. It corresponds (according to McKay) to the affine Dynkin diagram D4(1) (see [K1] Chap. 4, Table Aff 1). The dimensions of its nontrivial representations coincide with the coefficients aj in the expansion of the highest root θ of D4 in terms of simple roots: θ = α1 + 2α2 + α3 + α4 .

(6.22)

We shall denote the (equivalence classes of) irreducible representations (IR) of IH8 by the simple roots αν of D4(1) (α0 corresponding to the trivial representation). Then α2 maps IH8 into a subgroup of SU(2):

94

Victor G. Kac, Ivan T. Todorov Table 1. 0 = IH8 : characters and centralizers



CC

IR α0 α1 α2 α3 α4

0g

α2 (qj ) =

1

ε

{q1 , εq1 }

{q2 , εq2 }

{q3 , εq3 }

1 1 2 1 1

1 1 −2 1 1

1 1 0 −1 −1 Z4

1 −1 0 1 −1 Z4

1 −1 0 −1 1 Z4

0

0

  1 0 σj , j = 1, 2, 3 σ1 = 1 i

1 0



 , σ3 =

1 0

0 −1



 , σ2 = iσ1 σ3 .

(6.23) We reproduce in Table 1, for the reader’s convenience, the character table for 0 = IH8 also indicating the centralizer 0g of an element in each conjugacy class (CC). Using Table 1 and symmetrizing with respect to 2kn + λ we compute the sum (6.19) for the untwisted characters (i.e., for 0g = 0, β = 0): 1 [1 + (−1)λ ][1 + 3(−1)kn iλ ] , 8 1 = [1 + (−1)λ ][1 − (−1)kn iλ ] , j = 1, 3, 4 , 8 1 = [1 − (−1)λ ] . 4

(α0 )2kn+λ = (αj )2kn+λ (α2 )2kn+λ

(6.24)

Inserting these expressions in (6.17), (6.18) we recover for k = 1 the characters (6.13) of the CL -orbifold for s = 4: 2 1 X [1 + 3(−1)n ]q n = K0+ (τ, 8) , 4η(τ ) n 1 X (2n+1)2 1 q = K4 (τ, 8) , j = 1, 3, 4 , m = 0, 1 , χ0,1,αj (τ ) = 2η(τ ) n 2   m X 1 (−1) 1 (2n+1)2 4 χ1,1,α2 (τ ) = q = K2 (τ, 8) = K1 (τ, 2) , 2η(τ ) n 2 (6.25a)

k = 1 : χ0,1,α0 (τ ) =

where 1 K0+ (τ, 8) = K0 (τ, 8) − K4 (τ, 8) . 2

(6.25b)

The characters of the Z2 -twisted orbifolds are also computed from (6.17), (6.18) for β = 21 and σ(qjµ ) = iσµ (qjµ , µ ∈ Z/4Z is the general form of an element of the centralizer Z4 of qj ). Equation (6.19) then gives σ2kn+λ =

1 X 2kn+λ−σ)µ 1 + (−1)λ−σ [1 + (−1)kn iλ−σ ] , i = 4 4 µmod 4

reproducing, for k = 1 the CL -twisted characters of A(8):

(6.26)

Affine Orbifolds and Rational CFT of W1+∞

χ0,q¯j ,0 (τ ) =

95

1 X 1 (4n− 1 )2 2 q4 = K1 (τ, 8) = χ1,q¯j ,1 (τ ) , η(τ ) n

j = 1, 2, 3 , (6.27)

χ0,q¯j ,2 (τ ) = K3 (τ, 8) = χ1,q¯j ,−1 . (We label throughout the irreducible representations of Z4 – and their characters – by the exponents σ = 0, ±1, 2.) The number of inequivalent orbifold modules of a level 1 current algebra (for a simple g) is 1 X ˆ |0g | . (6.28) N (0 ⊂ G; k = 1) = |Z| g⊂0 ¯ In the case at hand it is 21 (5 + 5 + 3 × 4) = 11 thus coinciding with the number s + 7 of CL -orbifold modules for s = 4. Equations (6.24) and (6.26) also allow to compute orbifold characters for higher levels; in particular, for k = 2, g = 1, we obtain (expressing the string functions c3 λ in 3 l 3 terms of the branching coefficients bλ = η cλ , for a rank l g – see [K1] Sect. 12.12): ( ) X 2 1 X 1 1 3 2n 3 (2n+1)2 2 χ3,1,α0 (τ ) = b0 (τ ) q − b2 (τ ) q η(τ ) 2 n n (6.29a) 1 3 3 = b0 (τ )K0 (τ, 4) − b2 (τ )K2 (τ, 4) , 3 = 0, 2 , 2 χ3,1,αj (τ ) =

1 3 b (τ )K2 (τ, 4) , 2 2

j = 1, 3, 4 ,

3 = 0, 2 ,

3 χ1,1,α2 (τ ) = b11 (τ )K1 (τ, 4) (since b3 λ = b−λ ) .

(6.29b) (6.29c)

Similarly, using (6.26), we can evaluate the twisted characters. For those permuted by the action of the center we find (see Example 6.6 below) χ0,q¯j ,0 (τ ) = b00 (τ )K1 (τ, 4) = χ2,q¯j ,2 (τ ) , χ2,q¯j ,0 (τ ) = b20 (τ )K1 (τ, 4) = χ0,q¯j ,2 (τ ) ,

j = 1, 2, 3 .

(6.30a)

The remaining twisted characters are split by the action of the center, and we only obtain their sums: 1 χ+1,q¯j ,1 (τ ) + χ− 1,q¯j ,1 (τ ) = b1 (τ )K0 (τ, 4) , (6.30b) 1 χ+1,q¯j ,−1 (τ ) + χ− 1,q¯j ,−1 (τ ) = b1 (τ )K2 (τ, 4) . Here the branching coefficients can be expressed in terms of the Virasoro characters 1 1 , 2 ): χ1 (τ, c) of the Ising model (corresponding to c = 21 , 1 = 0, 16   1 , b00 (τ ) = b22 (τ ) = χ0 τ, 2 1 (6.30c) b20 (τ ) = b02 (τ ) = χ 1 (τ, ) , 2 2   1 b11 (τ ) = b1−1 (τ ) = χ 1 τ, . 16 2 It follows from (6.29) and (6.30) that there are 2 × 4 + 1 = 9 untwisted and 3 × 6 = 18 twisted level 2 orbifold modules or altogether 27 A2 (su(2))IH8 -representations.

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Example 6.5. Group theoretic S-matrix and fusion rules for IH8 ⊂ SU(2) and for IH8 ⊂ SU(2) ⊂ E8 . The simply connected compact group E8 is singled out (among the Lie groups with simple simply laced Lie algebras) for having a trivial center. The corresponding current algebra has a single level 1 representation, the vacuum A1 (E8 ) module; the modular S-matrix is then the identity operator (multiplication by 1). Hence, if 0 is a (non-exceptional) finite subgroup of E8 then the 0 ⊂ E8 group theoretic Smatrix coincides with the A1 (E8 )0 orbifold S-matrix. The possibility to embed the pair IH8 ⊂ SU(2) in E8 thus provides an additional justification for the study of the group theoretic S-matrix per-se. We observe that the S-matrix elements depend on both the Lie group G containing the pair IH8 ⊂ SU(2) and on the level of embedding of SU(2) in G which is defined as follows. Let the bases in su(2) and g be chosen in such a way that the Cartan generator H of su(2) is expressed as a linear combination of the Cartan generators H i with nonPl negative integer coefficients mi : H = i=1 mi H i . Then the integers mi satisfy the quadratic relation l l X 1X mi aij mj = mj =: N , 2 i,j=1

j=1

where, for a simply laced g, (aij ) is its Cartan matrix. The positive integer N is the level of embedding of su(2) in g. For a level 1 embedding the S-matrix elements involving at least one non-exceptional entry are independent of G. In the case of H8 the phase factor in (4.25) for a nonexceptional b and an arbitrary g is only non-trivial if both b and g belong to the same conjugacy class qj . We shall then set (−1)m−1 σ3 ⇒ exp{−2πik(β(εm qj )|β(εn qj ))} 4   m+n kπ = exp (−1) . 4i

β(εm q3 ) =

(6.31)

Omitting the upper index β on S (for this fixed choice) we obtain 4Sεm αµ ,q¯j σ = (−1)mσ αµ (qj ),

(6.32a)

0 o 0 kπ iσ+σ n −i kπ e 4 + (−1)σ+σ ei 4 2    k π = cos σ + σ0 − . 2 2

2Sq¯j σ,q¯j σ0 =

(6.32b)

(In computing the sum in the 2 elements b = ±qj of the conjugacy class q¯j in the expression (4.25) for S it is important to change at the same time σ according to (4.12). This yields (6.32b).) The only G dependence appears if the central element ε of 0 is present in both entries: (−1)nδµ2 +mδν2 2δµ2 +δν2 , 8Sεm αµ ,εn αν = pmnk ε

pε := e−2πi|β(ε)| , 2

(6.33a)

where β(ε) = 0 if G = SU(2), or, more generally, if it is an exceptional element of 0 ⊂ G, while (6.33b) pε = −1 if 0β(ε) = 0ε

Affine Orbifolds and Rational CFT of W1+∞

97

(in a level 1 embedding). It turns out that the fusion rules involving a pair of qj and an ε are integer iff ε is a regular element of 0 ⊂ G (i.e., if (6.33b) takes place). Indeed we have Nq¯j σ1 ,q¯j σ2 ,1αµ =

σ1 − σ2 1 + (−1)σ1 +σ2 +δµ2 αµ (qj ) αµ (1) + cos π , 4 2 2

which is a k independent non-negative integer, but Nq¯j σ1 ,q¯j σ2 ,εαµ

1 + pkε (−1)σ1 +σ2 +δµ2 αµ (qj ) αµ (1) + cos = 4 2



 k − σ1 − σ2 π , 2

which is integer only for odd k if pε = −1. Remark 6.1. Equation (6.33b) always takes place for a level 1 embedding SU(2) ⊂ E8 . In spite of the fact that ε is an involution (ε2 = 1) and every involution in E8 is exceptional (as a consequence of the description of finite order automorphisms of a simple Lie algebra presented in Appendix B) ε is not exceptional in 0 ⊂ SU(2) ⊂ E8 whenever SU(2) is generated by a pair of opposite roots of E8 – which is always the case (up to conjugation) for a level 1 embedding. In other words (E8 )β(ε) is strictly smaller than (E8 )ε but SU(2) ∩ (E8 )β(ε) = SU(2) ∩ (E8 )ε . By contrast, for the maximal embedding SU(2) ⊂ E8 given by

E=

8 X i=1

E αi ,

H = 2ρ =

8 X

bi αi∨ ,

F =

i=1

8 X

bi F α i ,

i=1

where bi are positive integers,2 ε is exceptional in 0 ⊂ E8 . However by the strange formula, the level of this embedding, N = 2(ρ|ρ) = g ∨ dim E8 /6 = 1240 (=

X

bi ) ,

is divisible by 4, hence the group theoretic fusion rules (with β(ε) = 0 = 1− pε ) coincide with those of the Grothendieck ring proven to be non-negative integers in [Lus1]. We have an exceptional subgroup 0 ⊂ SO(3) ⊂ E8 in this case. The image of any 4th ˜ ⊂ SU(2) is an involution whose centralizer in SO(3) is disconnected order element of 0 (see the discussion at the end of Appendix B). It is likely that at least in the case when orders of all elements of 0 divide N the corresponding twisted orbifold modules do exist and the resulting modular S-matrix coincides with the one for the Grothendieck ring. To compute the fusion rules for the Ak (SU(2))0 orbifold we shall use the (non-factorizable) |X | × |X | S-matrix of the full theory. ¯ σ) = (0, 1, αν ), ν = 0, 1, 3, 4, For the level 1 orbifold ordering the states as (3, b, ¯ α2 ), (0, q¯j , 0)(' (1, q¯j , 1)), (1, q¯j , −1)(' (0, q¯j , 2)), j = 1, 2, 3, we can write the (1, 1, 11 × 11 S-matrix as 2 Note that for the simple roots labeling of Table 2 below we have: b = 2m = 58, b = 6m = 114, 1 1 2 2 b3 = 24m3 = 168, b4 = 20m4 = 220, b5 = 270m5 = 270, b6 = 14m6 = 182, b7 = 4m7 = 92, b8 = 8m8 = 136, where m1 , . . . , m8 are the Coxeter exponents of E8 .

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Victor G. Kac, Ivan T. Todorov

1 2 1 2 1 2 1 2

      1 √  2 2S =  1  1  1  1 1 1

1 2 1 2 1 2 1 2

1 2 1 2 1 2 1 2

1 2 1 2 1 2 1 2

1 1 1 1 −1 −1 1 −1 −1 −1 1 −1 −1 1 −1 −1 −1 1 −1 −1 1

1 1 1 1 −2 0 0 0 0 0 0

1 1 −1 −1 √0 √2 − 2 0 0 0 0

1 1 −1 −1 0 √ −√ 2 2 0 0 0 0

1 −1 1 −1 0 0 √0 √2 − 2 0 0

1 −1 1 −1 0 0 0 √ −√ 2 2 0 0

1 −1 −1 1 0 0 0 0 √0 √2 − 2

 1 −1   −1   1   0   . 0    0  0   0  √ −√ 2  2

The resulting fusion rules differ, in general, from the group theoretic ones even for admissible entries. We have, for instance, N0q¯j 0,3q¯j −3,11α2 = 1 for 3 = 0, 1 , while Nq¯j 0,q¯j −3,1α ¯ 2 = N0q¯j 0,1q¯j −1,01α ¯ µ =

1 + (−1)1−3 , 2

1 − αµ (qj ) for µ 6= 2 , while Nq¯j 0,q¯j −1,1αµ = 0 for µ 6= 2 . 2

Example 6.6. The A2 (su(2))H8 orbifold and its Clifford algebra extension. The study of level 2 SU(2)-orbifolds is simplified by the observation that A2 ≡ A2 (su(2)) is the even part of the Clifford algebra Cl3 of 3 anticommuting Majorana-Weyl spinor fields ψj (z), j = 1, 2, 3. Indeed, the 3 = 2 A2 -module is generated by an “isotopic triplet” 1 of primary fields of dimension 13 = 4h 3(3 + 2) (for 3 = k = 2, h = k + 2 = 4), k the Virasoro central charge being c = 3 h = 23 . The fields ψj (z) are single-valued in the vacuum (Neveu-Schwarz) sector and satisfy the canonical anticommutation relations (and hermiticity) [ψi (z), ψj (w)]+ = δij δ(z − w) ,

ψj∗ = ψj ,

i, j = 1, 2, 3 .

The Z2 -graded algebra Cl3 (with odd generators ψj (z)) provides a superconformal extension of A2 whose SU(2) invariant subalgebra is generated by the 1 = 23 partner G(z) = iψ1 (z)ψ2 (z)ψ3 (z)(= G∗ (z))

(6.34a)

of the stress energy tensor T (z) = T1 (z) + T2 (z) + T3 (z) ,

Tj (z) =

1 : [∂ψj (z), ψj (z)]: 4

(6.34b)

which can be viewed as a composite of two G-fields. The generator G(z) of the superVirasoro algebra is a primary field with respect to T but not with respect to A2 ; its commutator with a Cartan current is [J(z), G(w)] = δ 0 (z − w)ψ3 (w) for J(z) = −iψ1 (z)ψ2 (z) . It intertwines the 3 = 0 and 3 = 2 Neveu-Schwarz modules mapping the 3 = 1 Ramond sector into itself.

Affine Orbifolds and Rational CFT of W1+∞

99

Each subgroup 0 of SU(2) acts on Cl3 by automorphisms which form the adjoint group Ad0 = 0/Z2 ⊂ SO(3) ; for 0 = H8 , Ad0 = Z2 × Z2 .

(6.35)

In the (orthonormal SO(3)) basis {ψj } the non-trivial elements Ej = α2 (qj ) of Z2 × Z2 act as diagonal matrices: ÿ E1 =

1 0 0

0 −1 0

0 0 −1

ÿ

! ,

E2 =

−1 0 0 1 0 0

0 0 −1

! ,

E3 = E1 E2 .

(6.36)

The Ad0 (= Z2 × Z2 ) invariant subalgebra Cl30 (0 = H8 ) of the Cl3 superalgebra is generated by G and by the individual stress-tensors Tj of the 3 “Ising models” (associated with each ψj ) – see (6.34b). The 3 commuting (1 = 2) field operators Tj (z) give rise to the even part A02 of this superalgebra. Its positive energy representations are tensor products of irreducible representations of the 3 (minimal) Ising models. There are, as expected, 33 = 27 such A02 orbifold modules. In particular, the characters of the fixed point modules split into a sum of two irreducible characters:  χ1,q¯j ,1 (τ ) = b11 (τ )K0 (τ, 4) = b11 (τ ) [b00 (τ )]2 + [b20 (τ )]2 , χ1,q¯j ,−1 (τ ) = b11 (τ )K2 (τ, 4) = 2b11 (τ )b20 (τ )b00 (τ ) .

(6.37)

2 The asymptotic dimensions of b11 (b3 0 ) for 3 = 0, 2 indeed coincide, (the quantum 1 1 dimension of the (c = 2 , 1 = 2 ) module being 1). Here we have used the expression (6.30) of the Ising model characters in terms of the branching coefficients. The remaining 1 1 , 2 ) of three orbifold modules are identified in the tensor product (11 , 12 , 13 )(1i = 0, 16 Ising modules as follows:   1 1 1 ¯ ll(0, 1, α0 ) = (0, 0, 0), , , , (2, 1, α0 ) = 2 2 2  
1 ¯ α1 ) = 0, 1 , 1 , (0, 1, , 0, 0 , (2, 1, α1 ) = 2 2 2  
1 ¯ α3 ) = 1 , 0, 1 , (0, 1, , 0 , ) = 0, (2, 1, α 3 2 2 2  
1 ¯ α4 ) = 1 , 1 , 0 , (0, 1, , (2, 1, α ) = 0, 0, 4 2 2 2
(6.38a) ¯ α2 ) = 1 , 1 , 1 , (1, 1, 16 16 16  
1 1 1 1 1 , , , , 16 , (2, q¯1 , 0) = (0, q¯1 , 0) = 0, 16 2 16 16  
1 1 1 1 1 (0, q¯2 , 0) = 16 , , , , 0, 16 , (2, q¯2 , 0) = 16 2 16  
1 1 1 1 1 (0, q¯3 , 0) = 16 , , , 16 , 0 , ; (2, q¯3 , 0) = 16 16 2

the reducible (fixed point) modules with characters (6.37) split according to the law

100

Victor G. Kac, Ivan T. Todorov

   1 1 1 1 , 0, 0 + , , , (1, q¯1 , 1) = 16 16 2 2     1 1 1 1 (1, q¯1 , −1) = , ,0 + , 0, , etc. 16 2 16 2 

(6.38b)

8 The AH 2 S-matrix is the tensor product of 3 Ising model S-matrices of the form √ ÿ ! 2 1 √ 1 √1 2 0 − 2 . SIsing = (6.39) √ 2 1 1 − 2

We note that while S1q¯1 σ,1q¯2 σ0 = 0 according to (4.25) (since the conjugacy classes q¯1 and q¯2 do not contain commuting elements) the corresponding split S-matrix elements do not vanish:   1 −1 −1 1  1  1 −1   −1 1 S 1 12 13 ,10 1 10 =  , 16 1 16 3 −1 1 1 −1  4 (6.40) 1 −1 −1 1      1 1 1 1 , , ,0 0, . (1i , 1j ) = (0, 0), 2 2 2 2 Note that the sum of A02 -modules in each line of Eq. (6.38) is irreducible with respect to the conformal superalgebra Cl30 . The characters of the subset of Neveu-Schwarz modules spanned by the direct sum of 3 = 0 and 3 = 2 representations give rise to a 7-dimensional representation of the subgroup 00 (2) of SL2 (Z) generated by T 2 and S. In particular, the Neveu-Schwarz S-matrix is 1 1 1 1 2 2 2  SN S

1  1 1 1 =  4 2  2 2

1 1 1 2 −2 −2

1 1 1 −2 2 −2

1 1 1 −2 −2 2

2 −2 −2 0 0 0

−2 2 −2 0 0 0

−2  −2   2   . 0   0 0

(6.41)

The importance of this example stems from the fact that it has a bearing on other SU(2) orbifold models. The three conjugacy classes of imaginary quaternion units {±qj , j = 1, 2, 3} of H8 combine in a single 6-element conjugacy class in the binary tetrahedral group A˜ 4 which in turn is a part of a 12-element conjugacy class of the binary octahedral group S˜ 4 and of a 30 element class of the binary icosahedral group A˜ 5 . Here Sn is the permutation group of n letters, An is its alternating invariant subgroup, G˜ ⊂ SU(2) denotes, in general, the double cover of a subgroup G of SO(3). In all three cases the centralizer 0qj of an element qj of this conjugacy class is Z4 . Hence, the reducible character χ1q¯j σ is the same for all three orbifold modules and splits in the same way – according to (6.37) – for all three binary polyhedral groups. There are no other conjugacy ¯ Furthermore, for all finite classes b¯ in either A˜ 4 or A˜ 5 such that both b and εb belong to b. SU(2) subgroups 0 the Neveu-Schwarz module of Cl30 contain no fixed points and give rise to a 00 (2)-invariant subset of characters. Furthermore, a similar argument extends to a level n representation of SU(n) which also involves fixed points of the action of the center. Indeed, there is a conformal embedding

Affine Orbifolds and Rational CFT of W1+∞

101





1 An ≡ An (su(n)) ⊂ A1 (spin(n − 1)) c = (n2 − 1) 2 2

allowing to extend an A0n orbifold to a Cln02 −1 -orbifold. The k = 1 tetrahedral (A˜ 4 ⊂ SU(2)) orbifold and its fusion rules are displayed in [DV3 ]. The octahedral (S˜ 4 ⊂ SU(2)) and the icosahedral (A˜ 5 ⊂ SU(2)) orbifolds can be studied with equal ease. We shall reproduce in Table 2 for a later reference the character table for the 120 element binary icosahedral group A˜ 5 (associated with E8(1) under the McKay correspondence). Table 2. Characters of A5 = A˜ 5 /Z2 and of its double cover 0 = A˜ 5 .

 CC

IR

α0 α2 α4 α6 α8 (A 5 )g CC

IR α1 α3 α5 α7

1

{p, p4 }

{p2 , p3 }

{t, t2 }

E = α2 (q)

1 3 5 4 3 A5

1 x+ 0 −1 x− Z5

1 x− 0 −1 x+ Z5

1 0 −1 1 0 Z3

1 −1 1 0 −1 Z 2 × Z2

ε

1

−2 −4 −6 −2 0 = A˜ 5

2 4 6 2

0g

p

p4

x+ 1 −1 x−

−x+ −1 1 −x−

√ 1± 5 , 2

Z10

p2 −x− −1 1 −x+ Z10

A5 = α2 (A˜ 5 ), x± = 0 = α1 (A˜ 5 ) θE8 = 2(α1 + α7 ) + 3(α2 + α8 ) + 4(α3 + α6 ) + 5α4 + 6α5 .

p3

t

t2

q

x− 1 −1 x+

1 −1 0 1

−1 1 0 −1

0 0 0 0 Z4

'

Z6 A˜ 5 , p5

=

t3

=

q2

=

ε,

Equation (6.28) implies: N (A˜ 5 ⊂ SU(2); k = 1) = 21 (9×2+10×4+6×2+4) = 37. It ˜ is a straightforward exercise to write down, using Table 2, the characters of A1 (su(2))A5 . 6.3. A Level 1 SU(3) Orbifold. Charge Conjugation Associated with a Non-Abelian Centralizer. We shall consider the subgroup 0 of SU(3) of order |0| = 1080 which is a non-trivial central extension of the simple alternating group A6 : 1 → Z3 → 0 → A6 → 1. It is generated by the (60 element) icosahedral group A5 ⊂ SO(3) and by one more element of order 2. In a basis in which a selected Z2 × Z2 subgroup of A5 (see Table 2) is generated by any two of the matrices Ei (= α2 (qi ), i = 1, 2, 3) given by (6.36) while the generators of its Z3 and Z5 subgroups are chosen as ÿ ÿ ! ! 1 −x+ 1 −x− 1 x− 1 −x+ 1 x+ −x− , p = −1 x+ −x− , (6.42) t= 2 x 2 x −1 x −x 1 −

+

+

−

where t = p = (tp) = 1, tp = E2 , and the additional involutive generator E4 of 0 is given by (ω 2 + ω + 1 = 0): ÿ ! 0 ω 0 (6.43) E4 = − ω¯ 0 0 , E42 = 1 = (E3 E4 )2 . 0 0 1 3

5

2

102

Victor G. Kac, Ivan T. Todorov

It is the 360 element factor group A6 = 0/Z3 that acts by non-trivial automorphisms on the su(3) current algebra. There are 17 conjugacy classes of 0 versus 7 of A6 . Both are listed in the combined character table below (see Table 3). We observe that to each of the first 5 conjugacy classes in A6 correspond 3 such classes (of the same size) in 0 while the last two are mapped into classes of triple size: |t¯0 | = 3|t¯A6 | = 3 × 40(= |t¯00 |). The essential difference between A6 = 0/Z3 and ¯ with a non-abelian the subgroups 0/Z2 of SO(3) is the presence of elements E(∈ E) centralizer 08 . Table 4 is its character table (E5 = E3 E4 , q = E4 E2 , q 3 = E2 E4 ). a normal subgroup of 08 . We note that the centralizer Z4 of q in A6 isP |0ˆ g | = 17 + 3.15 + 12 + 3 + 3 = 80 There are (according to (6.28)) altogether 13 g⊂0 ¯ level 1 0 ⊂ SU(3) orbifold modules. Although it is not practical to write down the 80 × 80 S-matrix, one can Pextract the relevant information about E-twisted orbifolds. The multipliers µ(h| βi ) give rise to a new notion of conjugation whenever the class E¯ of involutions labels a sector. To display this fact we first observe that the set of (45)2 pairs (E, E 0 ) splits into 9 different orbits displayed in Table 5. Table 3. Aˆ 6 ⊂0ˆ : Zero versus non-zero triality representations Table 3a. Aˆ 6

 cc

IR

1

E(E 2 = 1)

q(q 2 ∈ E)

p(p5 = 1)

p2

t(t3 = 1 = t03 )

t0

1

1

1

1

1

1

1

1

5

5

1

−1

0

0

2

−1

50

50

1

−1

0

0

−1

2

8

8

0

0

x+

x−

−1

−1

80 9 10

80 9 10

0 1 −2

0 1 0

x− −1 0

x+ −1 0

−1 0 1

−1 0 1

(A6 )g

A6

08

Z4

Z5

Z5

Z23

Z23

The stabilizer 0E,E 0 E of the pair E, E 0 E in 0 is the direct product of the central subgroup Z3 with the above 0(0) E,E 0 E ⊂ A6 . To verify the data of Table 5 one needs to construct a representative pair in each orbit. The number of elements of such an orbit is |A6 | = 360 divided by |0(0) E,E 0 E |. For instance, the orbit Op¯ is obtained by conjugation of the pair (Ep , E1 ), where ÿ 1 Ep = p−1 E3 p = 2 E1 Ep = E2 p−1 E2 ∈ p¯

! x+ −x− , −1 √ ! 1± 5 . x± = 2

−x− 1 x+ ÿ

1 −x+ −x−

We shall now prove that the oppositely ordered pairs (E2 , E4 ) and (E4 , E2 ) belong to different orbits Oq¯ although they belong to the same SU(3) orbit. To this end we construct the most general u ∈ SU(3) such that uE2 u∗ = E4 ,

uE4 u∗ = E2 ;

(6.44a)

Affine Orbifolds and Rational CFT of W1+∞ Table 3b. 0ˆ

 IR

103

cc 1

ω

E ωE ω 2 E

q

3ω 3ω 2 −1 −ω −ω 2

1

ω

3ω −1 −ω 2 −ω

1

ω2

3ω

3

3∗ω

3 3ω 2

30ω 30∗ ω

3

3ω

3

3ω 2

ω2

3ω 2

3ω −1

6ω

6

6ω

6ω 2

6∗ω

6 6ω 2

6ω

9ω

9ω 2

9

9ω

9

9ω 2

9ω

15ω 15 15ω

15ω 2

9∗ω

−1 −ω

−ω 2

ωq ω 2 q

1

ω

1

ω2

p

ω 2 x+

ω2

x−

−ω

2ω

2ω 2

0

0

0

1

2 2ω 2

2ω

0

0

0

1

1

ω

ω2

1

ω

ω2

1

ω2

ω

1

ω2

−1 −ω

−ω 2

−1 −ω

t

t0

ωx+ ω 2 x+ x− ωx− ω 2 x−

0

0

0

0

+

0

0

+

ωx+

0

0

ω x+ ω 2 x+

−ω 2

2

ωp2 ω 2 p2

ωp

ω x−

ω 2 p p2

ωx+ x− ω 2 x− ωx−

ωx− ω 2 x− ω2 x

−

x+

ωx− x+

ωx+ ω2 x

ω2 x

ω

ω2

1

ω

ω2

0

0

ω2

ω

1

ω2

ω

0

0

−1

−ω

−ω 2

−ω

−ω 2

0

0

ω −1

−ω 2

−ω

−ω 2

0

0

−1

−ω −1

−ω 2

0

0

0

0

0

0

0

0

15∗ω 15 15ω 2 15ω −1 −ω 2 −ω −1 −ω 2 −ω

0

0

0

0

0

0

0

0

Z23

Z23

0g

0

Z3 × 08

Z12

Z15

Z15

|g|

1

45

90

72

72

120 120

it is given by a 2(real) parameter family, ÿ u=

u1 −ζ u¯ 2 0

u2 ζ u¯ 1 0

0 0 ζ¯

! with |ζ|2 = 2|u1 |2 = 2|u2 |2 = 1 ,

¯ 1 u2 = −ω . (6.44b) 2ζu

It remains to prove that this family of 3 × 3 matrices does not intersect our group 0. √ Comparing the relation | tr(u + uE1 )| = |2u1 | = 2 (implied by (6.44b) with the first row of Table 3B, we deduce that if u ∈ 0 then (tr uE1 )3 = 1, (tr u)6 = 1 + 4(tr u)3 . The resulting set of equations for the parameters u1 , u2 and ζ in Eq. (6.44b) has no solutions with |ζ| = 1. It turns out that the same 2-parameter family of u’s is the most general subset of SU(3) elements that transforms the two OE¯ orbits among themselves: uE3 u∗ = E3 ⇒ uE1 u∗ = uE2 E3 u∗ = E4 E3 = E5 .

(6.45)

This completes the proof that each of the two pairs of representatives in the last column of Table 5 belongs to a different 0-orbit. We finally note that the sum of all |Og¯ |(4.360 + 2.180 + 2.90 + 45) adds up, as it should, to (45)2 = 2025. Proposition 6.1. The charge conjugation matrix (4.27) for the A1 (su(3))0 orbifold involves a non-trivial involution σ → σ c for b ∈ E, σ ∈ 0ˆ E : C30 E3 σ,30 E3 σ0 = δσ0 σc ,

σc = σ ∗ ⊗ σ E ,

σ E = e2πi(2β3 |β(h)) ,

where 0E = Z3 × 08 , σ ∗ = σ (i.e., σ(ωh) = σ(h) for h ∈ 0E ),

(6.46)

104

Victor G. Kac, Ivan T. Todorov

ÿ 1 1 0 βi = β(Ei ) : β3 = 2 0 ÿ ! 1 −2 0 0 0 1 0 , β1 = 2 0 0 1 ÿ −1 ±3ω 1 ±3ω¯ −1 β 4 == 5 4 0 0

0 1 0

0 0 2

0 0 −2

! ⇒ ÿ

1 β2 = 2 !

1 0 0

0 −2 0

0 0 1

! ,

(6.47)

.

Proof. The statement is a straightforward consequence of (4.27) (Theorem 4.3e) and of the observation that β3 = β(E3 ) = β(E3−1 ). The representation σ is trivial on Z3 (and hence, selfconjugate; see Table 4), since it has to agree with the representation 30 = 0 of SU(3) on the small center.  Table 4. Characters of 08 =0E3 ⊂A6

 IR 10 11 12 13 2

cc

1 1 1 1 1 2

E3 1 1 1 1 −2

0E3 ,g 08 08

Table 5. Orbits O(i)0

E E

CC of E 0 E

E1 , E 2 1 1 −1 −1 0 Z 2 × Z2

0(0) E,E 0 E

⊂ A6

08

E 0 E = EE 0 ∈ E¯

Z 2 × Z2

E0E

p¯n

∈ , n = 1, 2 E 0 E ∈ t¯ or t¯0

q, q 3 1 −1 −1 1 0 Z4

of pairs (E, E 0 ) ⊂ E¯ and their stabilizers (i = 1, 2)

E 0 E = 1(E = E 0 )

E 0 E ∈ q¯

E4 , E 5 1 −1 1 −1 0 Z 2 × Z2

Z2 {1} {1}

O(i) E0 E

Representative pairs

45

(i) O ¯ = 90, E

i = 1, 2

(1) OE ¯ = O(E1 , E2 ) , (2) OE ¯ = O(E5 , E4 )

(i) Oq¯ = 180 ,

Oq(1) ¯ = O(E2 , E4 ) ,

i = 1, 2

Oq(2) ¯ = O(E4 , E2 )

360 360

Remark 6.2. The appearance of a non-trivial conjugation depends on the choice of a representative in a class of equivalent quadruples. Had we chosen instead of the involution ¯ 3 ∈ ω 2 E¯ for which element E3 ∈ E¯ a representative of a minimal phase like ωE ÿ !   1 1 −1 0 0 1 ˜ 0 −1 0 so that |β˜3 |2 = β3 := β(ωE = |β3 |2 , ¯ 3) = (6.48) 6 6 9 0 0 2 then we would have dealt with complex representations since

Affine Orbifolds and Rational CFT of W1+∞

105

˜

β3 χβ33 ,E,σ (τ ) with σ2 (h) = σ(h)e2πi(32 |α) , ¯ ¯ (τ ) = χ3 ,ω 2 E,σ 0

2

2

(6.49a)

where 32 is the fundamental weight of the “antiquark” representation 3∗ , ÿ ! 2 −1 0 0 0 −1 0 = β˜3 −β3 , h = e2πiα , [α, 32 ] = 0(= [α, β3 ]) . (6.49b) 32 = 3 0 0 2 The charge conjugation matrix in these new labels would assume its usual form with non-zero entry C32 bσ2 ,3∗2 b−1 σ2∗ = 1 ,

b ∈ ω 2 E¯ ,

b−1 ∈ ω E¯ .

(6.50)

Appendix A. Action of the Center of a Simply Connected Simple Lie Group on the Coroots and Fundamental Weights We shall display the action of wj for the classical Lie algebras as well as for E6 and E7 (the simply connected groups with Lie algebras G2 , F4 and E8 have a trivial center). We let J˜ = J ∪ {0}, a0 = a∨ 0 = 1. A.1. Simply laced algebras (αi∨ = αi , a∨ i = ai ). The center Zl+1 of SU(l + 1) acts on both the (co) roots and weights of A(1) via cyclic permutations: l wj = w1j ,

w1 (α0 , α1 , . . . , αl ) = (α1 , α2 , . . . , αl , α0 ) , ˜ 0, 3 ˜ 1, . . . , 3 ˜ l ) = (3 ˜ 1, 3 ˜ 2, . . . , 3 ˜ l, 3 ˜ 0) , w˜ 1 (3

w1l+1 = 1 .

(A.1)

˜ ν are the extended fundamental weights Here 3 ˜ ν = d + 3ν + κν K 3

(A.2)

chosen to have equal norm squares: ˜ ν |2 = 2κν + |3

ν(l + 1 − ν) = 2κ0 . l+1

(A.3)

The set J˜ consists of all indices 0, 1, . . . , l. The element w1 is a Coxeter element of the finite Weyl group W (Al ) = Sl+1 . In terms of the elementary Weyl reflections si it is written as: (A.4) w1 = s1 . . . sl ⇒ w1 3j = 3j − α1 − . . . − αj . The center of the simply connected group Spin (2l) with Lie algebra Dl is Z2 ×Z2 for l even and Z4 for l odd. To exhibit its action on roots and weights of Dl(1) it is convenient to use an orthonormal basis {ei } in the l dimensional root space of Dl setting αi = ei − ei+1 , 3i =

i X s=1

i = 1, . . . , l − 1 ,

αl = el−1 + el ,

α 0 = K − e1 − e 2 ,

(A.5)

1X ei . 2

(A.6)

l

es

for i ≤ l − 2,

3l−1 = 3l − el ,

3l =

i=1

The set J˜ of indices µ for which aµ = 1 consists of 4 elements: 0, 1, l − 1, l. Writing again

106

Victor G. Kac, Ivan T. Todorov

˜ ν = aν d + 3ν + κν K, 3 ˜ coincide: ˜ µ (µ ∈ J) we restrict κν demanding that the norm squares of 3 ˜ 0 |2 = 2κ0 = |3 ˜ j |2 = 1 + 2κ1 = |3

(A.7a)

l l + 2κl−1 = + 2κl . 4 4

(A.7b)

We shall first determine the finite part wl of w˜ l defined by wl α0 = wl (−θ) = αl , and ˜ wl α1 = αl−1 . As a consequence of invariance hence (being a permutation of αµ , µ ∈ J), of inner products we further deduce wl αi = αl−i , i = 1, . . . , l − 2; hence, in view of (A.5), (A.8a) wl ei = −el+1−i , i = 1, . . . , l − 1 ; wl el is then determined from the condition that an element of W (Dl ) should involve an even number of reflections: (A.8b) wl el = −(−1)l e1 . As a result, we have wl2 = w1 for l odd, wl2 = 1 for l even; in both cases w12 = 1; w1 (e1 , e2 , . . . , el−1 , el ) = (−e1 , e2 , . . . , el−1 , −el ).

(A.9)

The corresponding permutations of fundamental weights are  ˜ for l even 3 ˜ ˜ ˜ ˜ ˜ ν = 0, 1 ; w˜ l 30 = 3l , w˜ l−1 31 = 3l−1 , w˜ l 3l−ν = ˜ ν 31−ν for l odd ˜ 1 , w˜ 1 3 ˜ l−1 , w˜ 12 = 1 , w˜ l−1 = w˜ 1 w˜ l . ˜0 =3 ˜l =3 w˜ 1 3 (A.10) The center of the group E6 is Z3 . Choosing a basis of simple roots of E6 in such a way that the highest root is θ = α2 +α4 +2(α1 +α3 +α5 )+3α6 , we have J˜ = {0, 2, 4}. The center acts on an arbitrary weight 3 according to the law w˜ j 3 = k3j + wj 3, j = 2, 4, where w2 (−θ, α1 , α2 , α3 , α4 , α5 , α6 ) = (α2 , α3 , α4 , α5 , −θ, α1 , α6 ) ,

w22 = w4 ,

(A.11a)

1 ˜2 =3 ˜ 4 , w23 = 1 . (A.11b) (α5 − α3 + 2α4 − 2α2 ) ⇒ w˜ 2 3 3 Here we have used the expressions for the fundamental weights in terms of simple roots: w2 32 = 34 − 32 =

1 32 = α1 + (4α2 + 5α3 + 2α4 + 4α5 + 6α6 ) , 3 1 34 = α1 + (2α2 + 4α3 + 4α4 + 5α5 + 6α6 ) , 3

 |32 |2 = |34 |2 =

4 3



˜ ν = d + 3ν + κν K with as well as the relations 3 ˜ 2 |2 = | 3 ˜ 4 |2 ⇒ 2κ0 = ˜ 0 |2 = |3 |3

4 4 2 + 2κ2 = + 2κ4 or κ2 = κ4 = κ0 − . 3 3 3

The center of E7 is Z2 . Choosing a basis of simple roots of E7 such that the highest root is θ = α6 + 2(α1 + α5 + α7 ) + 3(α2 + α4 ) + 4α3 , we have J˜ = {0, 6}. The non-trivial element of the center is w˜ 6 = t6 w6 where w6 (−θ, α1 , α2 , α3 , α4 , α5 , α6 , α7 ) = (α6 , α5 , α4 , α3 , α2 , α1 , −θ, α7 ),

(A.12a)

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˜6 =3 ˜ 0, w6 36 = −36 ⇒ w˜ 6 3 1 36 = α1 + 2α2 + 3α3 + 2α5 + (5α4 + 3α6 + 3α7 ) . 2 ˜ ν = d + 3ν + κν K, where |3 ˜ 6 |2 = 2κ6 + 3 = 2κ0 . Here again 3

(A.12b)

2

A.2. Z2 action on Bl and Cl . The simple roots, the highest root and the fundamental weights of Bl can be written in an orthonormal basis {ei } as αi = ei − ei+1 , i = 1, . . . , l − 1 , αl = el , θ = α1 + 2(α2 + . . . + αl ) = e1 + e2 , 3i =

i X

(A.13) es ,

i = 1, . . . , l .

s=1

The center Z2 of the simply connected group Spin (2l + 1) acts on (α0 = K − θ, αi ) and on (30 , 3i ) as w˜ 1 = t1 w1 , where w1 (e1 , e2 , . . . , el ) = (−e1 , e2 , . . . , el ), t1 α∨ = α∨ − (α∨ |31 )K for α∨ ∈ M ,

(A.14a)

˜ ν + (3 ˜ ν |K)3 ˜ 1 for 3 ˜ ν ∈ M∗ ; ˜ν =3 t1 3 (A.14b)

thus t1 w1 α0 = t1 (K + α1 ) = α1 , w1 αi = αi = t1 w1 αi for i = 2, . . . , l , t1 w1 α1 = t1 (−θ) = −θ + K = α0 ; ˜0+3 ˜ 0 = t1 3 ˜0 =3 ˜1 , t1 w1 3 ˜0+3 ˜ 1 ) = t1 (3 ˜0−3 ˜ 1 ) = 30 − 31 + 31 = 30 . t1 w 1 ( 3 The simple roots, the highest root and the fundamental weights for Cl are expressed as

1 αi = √ (ei − ei+1 ) , i = 1, . . . l − 1 , 2 l−1 X √ √ αi + αl = 2e1 , αl = 2el , θ = 2

(A.15)

i=1 i √ X es , 3i = 2

1 X 3l = √ es . 2 s=1 l

i = 1, . . . , l − 1 ,

s=1

The non-trivial element w˜ l = tl wl of the center Z2 of Sp(2l) acts on these orthonormal basis ei as wl (e1 , e2 , . . . , el−1 , el ) = (−el , −el−1 , . . . , −e2 , −e1 ) ; (A.16) hence wl (−θ, α1 , . . . , αl ) = (αl , αl−1 , . . . , α1 , −θ) , ˜l =3 ˜0 wl 3l = −3l ⇒ w˜ l 3  ˜ l |2 = 2κl + l . ˜ 0 |2 = 2κ0 = |3 |3 2

(A.17a)

˜0 =3 ˜ l (= d + 3l + κl K) , w˜ l 3

(A.17b)

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Appendix B. Exceptional Elements of a Compact Lie Group Let G be a connected compact Lie group with a simple Lie algebra g of rank l, and let AdG denote the adjoint group. An element g ∈ G is called ad-exceptional if it cannot be written in the form g = exp 2πiβ, where β ∈ ig is such that Adg x = x iff [β, x] = 0 for all x ∈ g. Note that an element g ∈ G is Ad-exceptional iff it is ad-exceptional or its centralizer in G is not connected. (Recall that in a simply connected G the centralizer of any element is connected.) In this Appendix we classify ad-exceptional elements of finite order of the group AdG . The finite order inner automorphisms of the simple Lie algebra g belong to AdG and can be described as follows (see Theorem 8.6 and Proposition 8.6b of [K1]). Proposition B.1. Each order N inner automorphism of g is conjugate to Adb(s) ,

b(s) = exp 2πiβ(s) ,

where β(s) =

l 1 X sj 3∨ j N

(B.1)

(B.2)

j=1

and s0 , sj , j = 1, . . . , l are relatively prime non-negative integers such that: s0 +

l X

aj sj = N .

(B.3)

j=1

Here 3∨ j are the fundamental coweights: ∨ (αi |3∨ j ) = (αi |3j ) = δij ,

i, j = 1, . . . , l .

(B.4)

Proposition B.2. The centralizer of Adb(s) in g is generated by the E ±αν , ν = 0, 1, . . . , l, for which sν = 0 and by the Cartan subalgebra. According to Definition 4.1 an element b ∈ 0 is exceptional if there is no β ∈ g such that (B.5) b = e2πiβ and 0b = 0β . As noted, G = U (l) has no exceptional elements. By contrast, for each partition of the positive integer n ≥ 2 of the type n = k1 + . . . + kρ ,

kmin = min(k1 , . . . , kρ ) = 2,

(B.6)

there are exceptional elements of SU(n) conjugate to diagonal matrices with kj eigenν conditions: (i) values exp(2πi Nj ), j = 1, . . . , ρ, where the νj are subject to the P (ν1 , . . . , νρ , N ) = 1 (i.e. these ρ+1 integers have no common factor) and j kj νj = kN with 1 ≤ k < kmin . For n = 2, 3 all such elements belong to the center Zn of SU(n). More generally, for any n, one can find an element ζ ∈ Zn such that g = bζ is nonexceptional. (In the above example it suffices to choose ζ = exp(−2πi nk ).) This proves the statement (of Sect. 4.2) that SU(n) contains no exceptional subgroups. Recall that an element b ∈ G is Ad-exceptional if bζ is exceptional for any choice of ζ ∈ Z(G). The following theorem describes all finite order ad-exceptional elements of AdG (for a simple g), and hence all finite order Ad-exceptional elements of a simply connected G.

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Proposition B.3. The finite order automorphism Adb(s) is ad-exceptional iff the marks aν with sν > 0 have a non-trivial common factor. Proof. It follows from Proposition B.2 that it suffices to study the commutator of β(s) with E αν for those ν(= 0, . . . , l) for which sν = 0. This commutator is trivial for j = 1, . . . , l such that sj = 0 since Eqs. (B. 1-4) imply [β(s), E αj ] = (αj |β(s))E αj ,

(αj |β(s)) = sj = 0 .

(B.7)

Thus Adb can only be ad-exceptional if s0 = 0; in this case  s 0 − 1 E α0 = −E α0 . [β(s), E α0 ] = [β(s), E −θ ] = N

(B.8)

This is still not sufficient to assert that Adb is ad-exceptional since β(s) is not unique: Pl we can add to it i=1 mi 3∨ i for mi ∈ Z without changing the automorphism. That would give   [β(s) +

X i

X   −θ mi 3∨ ] = −1 − ai mi  E −θ , i ,E i si 6=0

which can be made zero iff the ai in the sum have no common factor.



Proposition B.3 shows that SU(l) has no Ad-exceptional elements, while all other simple simply connected compact groups do. Examples of Ad-exceptional b are provided by the special elements with β(s) = a1j 3∨ j for aj > 1, corresponding to sν = δνj . Such is, for instance, the diagonal symplectic matrix   −1 0 0 0  0 1 0 0  b1 = e2πi31 =  ∈ Sp(4) = {g ∈ SU(4)|t gCg = C} , 0 0 1 0  0 0 0 −1     1 0 0 0 1 (B.9) 1 1 0 0 1 0  0  =  C= , 31 = 3∨   1 0 0 −1 0 0 2 2 −1 −1 0 0 0 1 (= α1 + α2 ) . 2 (31 is only stabilized by U (2) while the centralizer of b1 in Sp(4) is SU(2) × SU(2)). If 01 ⊂ SU2 is the binary icosahedral group, then 0 =< b1 , −1 > ×01 ×01 ⊂ Sp(4) is clearly an exceptional subgroup containing the center of Sp(4). The simplest example of a non-special Ad-exceptional element is provided by the simply laced Lie algebra D5 (corresponding to the simply connected group Spin (10)). If we label the nodes of the affine diagram D5(1) so that a2 = a3 = 2 (while a0 = a1 = a4 = a5 = 1) then the non-special Ad-exceptional element of Spin (10) correspond to ∨ β = 41 (3∨ 2 + 33 ). An example of an element of SO(3) = AdSU(2) with a disconnected centralizer is provided by either of the diagonal matrices Ei , i = 1, 2, 3 of Eq. (6.36). Indeed, there is no Cartan subalgebra of SO(3) containing the infinitesimal generators of both E1 and E2 . Note that the preimages of Ei in the simply connected double cover SU(2) of SO(3)

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do not commute (in fact, they anticommute). This example extends to the n3 element Heisenberg subgroup Hn of SU(n) generated by the n × n matrices a and b satisfying an = bn = 1 ,

ab = e2πi/n ba .

(B.10)

Clearly, Ada and Adb commute but their infinitesimal generators do not. This happens since AdSU(n) (unlike SU(n)) is not simply connected and the centralizer of either Ada or Adb is disconnected. Acknowledgement. I.T. acknowledges the support of a Fulbright grant 19684 and the hospitality of the Department of Mathematics at M.I.T. during the course of this work. Both authors acknowledge the hospitality of the Erwin Schr¨odinger International Institute for Mathematical Physics where this paper was completed. The authors thank Bojko Bakalov who took part in the computations of the S matrix and the associated fusion rules presented in Sect. 6.2.

References [AFMO] Awata, J.H., Fukuma, M., Matsuo, Y., Odake,S.: Representation theory of the W1+∞ algebra. Prog. Theor. Phys., Proc. Suppl.118, 343–373 (1995) [BGT] B.N. Bakalov, L.S. Georgiev, I.T. Todorov: A QFT approach to W1+∞ . New Trends in Quantum Field Theory In: Proc. of the 1995 Razlog (Bulgaria) Workshop, A. Ganchev et al. (eds.), Sofia: Heron Press, 1996 pp. 147–158 [BPZ] Belavin, A.A., Polyakov, A.M., Zamolodchikov, A.B.: Infinite conformal symmetry in twodimensional quantum field theory. Nucl. Phys. B241, 333–381 (1984) [Bor] Borcherds, R.: Vertex algebras, Kac-Moody algebras and the Monster. Proc. Natl. Acad. Sci. USA 83, 3068–3071 (1986); Monstrous moonshine and monstrous Lie Superalgebras. Invent. Math. 109, 405–444 (1992) [BMP] Bouwknegt, P., McCarthy, J., Pilch, K.: Semi-infinite cohomology and w-gravity. J. Geom. Phys. 11, 225–249 (1993) [BMT] Buchholz, D., Mack, G., Todorov, I.T.: The current algebra on the circle as a germ of local field theory. Nucl. Phys. B (Proc. Suppl.) 5B, 20–56 (1988) [CTZ] Cappelli, A., Trugenberger, C.A., Zemba, G.R.: Stable hierarchical quantum Hall fluids on W1+∞ minimal models. Nucl. Phys. B448, [FS] 470–504 (1995); W1+∞ dynamics of edge excitations in the quantum Hall effect, Ann. Phys. (NY) 246, 86–120 (1996) [CZ] Cappelli, A., Zemba, G.: Modular invariant partition functions in the quantum Hall effect. hepth/9605127 [DFSZ] Di Francesco, P., Saleur, H., Zuber, J.-B.: Modular invariance in non-minimal two-dimensional conformal theories. Nucl. Phys. B285, [FS19] 454–480 (1987) Dijkgraaf, R., Vafa, C., Verlinde, E., Verlinde, H.: The operator algebra of orbifold models. Commun. [DV3 ] Math. Phys. 123, 485–526 (1989) [DGM] Dolan, L., Goddard, P., Montague, P.: Conformal field theory of twisted vertex operators. Nucl. Phys. B338, 529–601 (1990) [FFK] Fairbairn, W.M., Fulton, T., Klink, W.H.: Finite and disconnected subgroups of SU3 and their application to the elementary particle spectrum. J. Math. Phys. 5, 1038–1051 (1964) [F] Flohr, M.: W -algebras, new rational models and completeness of the c = 1 classification. Commun. Math. Phys. 157, 179–212 (1993) [FK] Frenkel, I.B., Kac, V.G.: Basic representations of affine Lie algebras and dual resonance models. Invent. Math. 62, 23–66 (1980) [FKRW] Frenkel, E., Kac, V., Radul, A., Wang, W.: W1+∞ and W (glN ) with central charge N . Commun. Math. Phys. 170, 337–357 (1995) [FKW] Frenkel, E., Kac, V., Wakimoto, M.: Characters and fusion rules for W -algebra via quantized Drinfield-Sokolov reduction. Commun. Math. Phys. 147, 295–328 (1992) [FLM] Frenkel, I.B., Lepowsky, J., Meurman, A.: Vertex Operator Algebras and the Monster. New York: Academic Press, 1988

Affine Orbifolds and Rational CFT of W1+∞ [FZ] [FT] [FST] [Gep] [G] [Go] [Gor] [H] [K1] [K2] [KP0] [KP1] [KP2] [KR1] [KR2] [KW] [Kos]

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[MST]

[PT] [R] [RST] [V]

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Frenkel, I.B., Zhu, Y.: Vertex operator algebras associated to representations of affine and Virasoro algebra. Duke Math. J. 66, 123–168 (1992) Fr¨ohlich, J., Thiran, E.: Integral quadratic forms, Kac-Moody algebras, and fractional quantum Hall effect. An ADE-O classification. J. Stat. Phys. 76, 209–283 (1994) Furlan, P., Sotkov, G.M., Todorov, I.T.: Two-dimensional conformal quantum field theory. Riv. Nuovo Gim 12:6, 1–202 (1989) Gepner, D.: New conformal field theories associated with Lie algebras and their partition functions. Nucl. Phys. B285, [FS20] 10–24 (1987) Ginsparg, P.: Curiosities at c = 1. Nucl. Phys. B295, [FS21] 153–170 (1988) Goddard, P.: Meromorphic conformal fields theory. In: Infinite-dimensional Lie algebras and groups. Adv. Ser. Math. Phys. 7 ed. V. Kac, Singapore: World Sci., 1989 pp. 556–587 Gorenstein, D.: it Finite Groups. New York: Harper & Row, 1968 Harris, G: SU(2) current algebra orbifolds of the Gaussian model. Nucl. Phys. B300, (FS22) 588– 610 (1988) Kac, V.: Infinite Dimensional Lie Algebras. Third edition, Cambridge: Cambridge Univ. Press, 1990 Kac, V.: Vertex Algebras. In: New Trends in Quantum Field Theory, Proceedings of the 1995 Razlog (Bulgaria)-Workshop, A. Ganchev et al. editors, Sofia: Heron Press, 1996 pp. 261–358 Kac, V.G., Peterson, D.H.: Affine Lie algebras and Hecke modular forms. Bull. Amer. Math. Soc. 3, 1057–1061 (1980) Kac, V.G., Peterson, D.H.: Spin and wedge representations of infinite dimensional Lie algebras and groups. Proc. Nat. Acad. Sci. USA 78, 3308–3312 (1981) Kac, V.G., Peterson, D.H.: Infinite dimensional Lie algebras, theta-functions and modular forms. Adv. in Math. 53, 125–264 (1984) Kac, V.G., Radul, A.: Quasi-finite highest weight modules over the Lie algebra of differential operators on the circle. Commun. Math. Phys. 157, 429–457 (1993) Kac, V.G., Radul, A.: it Representation theory of the vertex algebra W1+∞ . Transformation groups 1, 41–70 (1996) Kac, V.G., Wakimoto, M.: Modular and conformal invariance constraints in representation theory of affine algebras. Advances in Math. 70, 156–236 (1988) Kostant, B.: The McKay correspondence, the Coxeter element and representation theory. In: The Mathematical Heritage of Elie Cartan. Soc. Math. de France, Asterisque, hors series, 1985 pp. 209–255 Lian, B.H., Zuckerman, G.: Commutative quantum operator algebras. J. Pure Appl. Alg. 100, 117– 139 (1995) Longo, R., Rehren, K.-H.: Nets of subfactors. Rev. Math. Phys. 7, 567–597 (1995) Lusztig, G.: Unipotent representations of finite Chevalley groups of type E8 . Quart. J. Math. Oxford (2) 30, 315–338 (1979) Lusztig, G.: Leading coefficients of character values of Hecke alebras. In: Arcata Conference of Representations of Finite Groups In: Proceedings of Symposium in Pure Math (P. Fong, ed.)47, (1987) Michel, L., Stanev, Ya.S., Todorov, I.T.: D − E-classification of the local extensions of the su2 current algebras. Teor Mat. Fiz. 92, 507–521 (1992); (American edition: Theor. Math. Phys. 92, 1063 (1993)) Paunov, R.R., Todorov, I.T.: Modular invariant QFT models of u(1) conformal current algebra. Phys. Lett. B196, 519–526 (1987) Rehren, K.-H.: A new view of the Virasoro algebra. Lett. Math. Phys. 30, 125–130 (1994) Rehren, K.-H., Stanev, Ya.S., Todorov, I.T.: Characterizing invariants for local extensions of current algebras. hep-th/9409165 Commun. Math. Phys. 174, 605–633 (1996) Verlinde, E.: Fusion rules and modular transformations in 2D conformal field theory. Nucl. Phys. B300, [FS22] 360–376 (1988)

Communicated by G. Felder

Commun. Math. Phys. 190, 113 – 132 (1997)

Communications in

Mathematical Physics c Springer-Verlag 1997

Huygens’ Principle in Minkowski Spaces and Soliton Solutions of the Korteweg–de Vries Equation Yuri Yu. Berest1 , Igor M. Loutsenko2 1 Department of Mathematics, University of California, Berkeley, CA 94720, USA. E-mail: beresty@math. berkeley.edu 2 Universit´ e de Montr´eal, Centre de Recherches Math´ematiques, C.P. 6128, succ. Centre-Ville, Montreal (Quebec), H3C 3J7, Canada. E-mail: [email protected], [email protected]

Received: 3 April 1996 / Accepted: 1 April 1997

Abstract: A new class of linear second order hyperbolic partial differential operators satisfying Huygens’ principle in Minkowski spaces is presented. The construction reveals a direct connection between Huygens’ principle and the theory of solitary wave solutions of the Korteweg–de Vries equation.

I. Introduction The present paper deals with the problem of describing all linear second order partial differential operators for which Huygens’ principle is valid in the sense of “Hadamard’s minor premise”. Originally posed by J.Hadamard in his Yale lectures on hyperbolic equations [26], this problem is still far from being completely solved1 . The simplest examples of Huygens’ operators are the ordinary wave operators   2  2 2 ∂ ∂ ∂ − − . . . − (1) n+1 = ∂x0 ∂x1 ∂xn in an odd number n ≥ 3 of space dimensions and those ones reduced to (1) by means of elementary transformations, i.e. by local nondegenerate changes of coordinates x 7→ f (x) ; gauge and conformal transformations of a given operator L 7→ θ(x) ◦ L ◦ θ(x)−1 , L 7→ µ(x)L with some locally smooth nonzero functions θ(x) and µ(x). These operators are usually called trivial Huygens’ operators, and the famous “Hadamard’s conjecture” claims that all Huygens’ operators are trivial. Such a strong assertion turns out to be valid only for (real) Huygens’ operators with a constant principal symbol in n = 3 [33]. Stellmacher [40] found the first 1 Hadamard’s problem, or the problem of diffusion of waves, has received a good deal of attention and the literature is extensive (see, e.g., [8, 12, 15, 21, 22, 24, 27, 28, 35], and references therein). For a historical account we refer the reader to the articles [19, 25].

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non-trivial examples of hyperbolic wave-type operators satisfying Huygens’ principle, and thereby disproved Hadamard’s conjecture in higher dimensional Minkowski spaces. Later Lagnese & Stellmacher [31] extended these examples and even solved [32] Hadamard’s problem for a restricted class of hyperbolic operators, namely L = n+1 + u(x0 ) ,


0

(2)

where u x is an analytic function (in its domain of definition) depending on a single variable only. It turns out that the potentials u(z) entering into (2) are rational functions which can be expressed explicitly in terms of some polynomials2 Pk (z):  2 d log Pk (z) , k = 0, 1, 2, . . . , (3) u(z) = 2 dz the latter being defined via the following differential-recurrence relation: 0 0 Pk−1 − Pk−1 Pk+1 = (2k + 1)Pk2 , Pk+1

P 0 = 1 , P1 = z .

(4)

Since the works of Moser et al. [2, 3] the potentials (3) are known as rational solutions of the Korteweg–de Vries equation decreasing at infinity3 . A wide class of Huygens’ operators in Minkowski spaces has been discovered recently by Veselov and one of the authors [9, 10] (see also the review article [8]). These operators can also be presented in a self-adjoint form L = n+1 + u(x)

(5)

with a locally analytic potential u (x) depending on several variables. More precisely, u (x) belongs to the class of so-called Calogero-Moser potentials associated with finite reflection groups (Coxeter groups): u(x) =

X mα (mα + 1)(α, α) . (α, x)2

(6)

α∈<+

In formula (6) <+ ≡ <+ (G) stands for a properly chosen and oriented subset of normals to reflection hyperplanes of a Coxeter group G. The group G acts on Mn+1 in such a way that the time direction is preserved. The set {mα } is a collection of non-negative integer labels attached to the normals α ∈ < so that mw(α) = mα for all w ∈ G. Huygens’ principle holds for (5), (6), provided n is odd, and n≥3+2

X

mα .

(7)

α∈<+

In the present work we construct a new class of self-adjoint wave-type operators (5) satisfying Huygens’ principle in Minkowski spaces. As we will see, this class provides a natural extension of the hierarchy of Huygens’ operators associated to Coxeter groups. On the other hand, it turns out to be related in a surprisingly simple and fundamental way to the theory of solitons. 2 This remarkable class of polynomials seems to have been found for the first time by Burchnall and Chaundy [11]. 3 The coincidence of such rational solutions of the KdV-hierarchy with the Lagnese-Stellmacher potentials has been observed by Schimming [38, 39].

Huygens’ Principle in Minkowski Spaces and Soliton Solutions KdV Equation

115

To present the construction we consider a (n + 1)-dimensional Minkowski space Mn+1 ∼ = R1,n with the metric signature (+, −, −, . . . , −) and fixed time direction θ ∈ n+1 M . We write Gr⊥ (n + 1, 2) ⊂ Gr(n + 1, 2) for a set of all 2-dimensional space-like linear subspaces in Mn+1 orthogonal to θ. Every 2-plane E ∈ Gr⊥ (n + 1, 2) is equipped with the usual Euclidean structure induced from Mn+1 . To define the potential u(x) we fix such a plane E and introduce polar coordinates (r, ϕ) therein. 4 Let (ki )N i=1 be a strictly increasing sequence of integer positive numbers : 0 ≤ k1 < k2 < . . . < kN −1 < kN , and let {9i (ϕ)} be a set of 2π-periodic functions on R1 : 9i (ϕ) := cos(ki ϕ + ϕi ) , associated to (ki ). The Wronskian of this set  91 (ϕ)  901 (ϕ)  W [91 , 92 , . . . , 9N ] := det  ..  .

ϕi ∈ R ,

92 (ϕ) 902 (ϕ) .. .

(8)

... ... .. .

9N (ϕ) 90N (ϕ) .. .

−1) −1) −1) (ϕ) 9(N (ϕ) . . . 9(N (ϕ) 9(N 1 2 N

does not vanish identically since 9i (ϕ) are linearly independent. Let ( Ξ :=

x ∈ Mn+1 | r|k| W [91 , 92 , . . . , 9N ] = 0 , |k| :=

N X

    

(9)

) ki

i=1

be an algebraic hypersurface of zeros of the Wronskian (9) in the Minkowski space Mn+1 , and let  ⊂ Mn+1 \ Ξ be an open connected part in its complement. We define u(x) in terms of cylindrical coordinates in Mn+1 with polar components in E :  2 2 ∂ log W [91 (ϕ), 92 (ϕ), . . . , 9N (ϕ)] . (10) u = uk (x) := − 2 r ∂ϕ It is easy to see that in a standard Minkowskian coordinate chart u(x) is a real rational function on Mn+1 having its singularities on Ξ. In particular, it is locally analytic in . Our main result reads as follows. Theorem. Let Mn+1 ∼ = R1,n be a Minkowski space, and let L(k) := n+1 + uk (x)

(11)

be a wave-type second order hyperbolic operator with the potential (10) associated to an arbitrary strictly monotonic partition (ki ) of height N : 0 ≤ k1 < k2 < . . . < kN ,

ki ∈ Z ,

i = 1, 2, . . . , N .

Then operator L(k) satisfies Huygens’ principle at every point ξ ∈ , provided n is odd, and n ≥ 2 kN + 3 .

(12)

4 Using the terminology adopted in the group representation theory we will call such integer monotonic sequences partitions.

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Remark I. A similar result is also valid if one takes an arbitrary Lorentzian 2-plane H ∈ Grk (n+1, 2) in the Minkowski space Mn+1 containing the time-like vector θ. More precisely, in this case the potential uk (x) associated to the partition (ki ) is introduced in terms of pseudo-polar coordinates (%, ϑ) in H:  2 2 ∂ log W [ψ1 , ψ2 , . . . , ψN ] , (13) uk (x) := − 2 % ∂ϑ where x0 = % sinh ϑ , and, say, x1 = % cosh ϑ . The functions ψi involved in (13) are given by (14) ψi = cosh(ki ϑ + ϑi ) , ϑi ∈ R . The theorem formulated above holds when the potential (10) is replaced by (13). Remark II. The potentials (10) considerably extend the class of Calogero-Moser potentials (6) related to Coxeter groups of rank 2. Indeed, in R2 any Coxeter group G is a dihedral group I2 (q), i.e. the group of symmetries of a regular 2q-polygon. It has one or two conjugacy classes of reflections according to wether q is odd or even. The corresponding potential (6) can be rewritten in terms of polar coordinates as follows (see [36]): m(m + 1)q 2 , when q odd , u(r, ϕ) = r2 sin2 (q ϕ) and
2
2 m(m + 1) q/2 m1 (m1 + 1) q/2 , when q even . u(r, ϕ) = + r2 cos2 (q/2) ϕ r2 sin2 (q/2) ϕ It is easy to verify that formula (10) boils down to these forms if we fix N := m ; ϕi := (−1)i π/2 , i = 1, 2, . . . , N , and choose k := (q, 2q, 3q, . . . , mq) , when q is odd, and 3q q q q q q , . . . , (m − m1 ) , q + (m − m1 ) , 2q + (m − m1 ) , . . . , (m + m1 ) ) , k := ( , q, 2 2 2 2 2 2 when q is even and m > m1 , respectively. Remark III. Let us set ϕi = 4ki3 t+ϕ0i and ϑi = −4ki3 t+ϑ0i , i = 1, 2, . . . , N ; ϕ0i , ϑ0i ∈ R. The angular parts of potentials (10), (13), i.e.  2 ∂ v(ϕ) = −2 log W [91 (ϕ), 92 (ϕ), . . . , 9N (ϕ)] , (15) ∂ϕ  2 ∂ log W [ψ1 (ϑ), ψ2 (ϑ), . . . , ψN (ϑ)] , (16) v(ϑ) = −2 ∂ϑ are known (see, e.g., [18], [34]) to be respectively singular periodic and proper N-soliton solutions of the Korteweg–de Vries equation vt = −vϕϕϕ + 6vvϕ .

(17)

It is also well-known that N -soliton potentials (16) constitute the whole class of socalled reflectionless real potentials for the one-dimensional Schr¨odinger operator L = −∂ 2 /∂ϑ2 + v(ϑ) (see, e.g., [1]). To conclude this section we put forward the following conjecture5 . 5

Note added in the proof. This conjecture has been proved recently by one of the authors in [6].

Huygens’ Principle in Minkowski Spaces and Soliton Solutions KdV Equation

117

Conjecture. The wave-type operators (11) with potentials of the form (10) give a complete solution of Hadamard’s problem in Minkowski spaces Mn+1 within a restricted class of linear second order hyperbolic operators  L=

∂ ∂x0



2 −

∂ ∂x1



2 −

∂ ∂x2



2 − ... −

∂ ∂xn

2 + u(x1 , x2 )

with real locally analytic potentials u = u(x1 , x2 ) depending on two spatial variables and homogeneous of degree (−2): u(αx1 , αx2 ) = α−2 u(x1 , x2 ) , α > 0 .

II. Huygens’ Principle and Hadamard–Riesz Expansions The proof of the theorem stated above rests heavily on the Hadamard theory of Cauchy’s problem for linear second order hyperbolic partial differential equations. Here, we summarize briefly some necessary results from this theory following essentially M. Riesz’s approach [37] (see also [21, 24]). Let Mn+1 ∼ = R1,n be a Minkowski space, and let  be an open connected part in n+1 M . We consider a (formally) self-adjoint scalar wave-type operator L = n+1 + u(x) ,

(18)

defined in , the scalar field (potential) u(x) being assumed to be in C ∞ (). For any ξ ∈ , we define a cone of isotropic (null) vectors in Mn+1 with its vertex at ξ: γ(x, ξ) := (x0 − ξ 0 )2 − (x1 − ξ 1 )2 − . . . − (xn − ξ n )2 = 0 , and single out the following sets :  C± (ξ) := x ∈ Mn+1 | γ(x, ξ) = 0 , ξ 0 ≶ x0 ,  J± (ξ) := x ∈ Mn+1 | γ(x, ξ) > 0 , ξ 0 ≶ x0 .

(19)

(20)

Definition. A (forward) Riesz kernel of operator L is a holomorphic (entire analytic) 6 0 mapping λ 7→ 8 λ (x, ξ) , λ ∈ C, with values in the space of distributions D (), such that for any ξ ∈ : (i) supp 8 λ (x, ξ) ⊆ J+ (ξ) ,    (21) (ii) L 8λ (x, ξ) = 8 λ−1 (x, ξ) , (iii) 8 0 (x, ξ) = δ(x − ξ) . The value of the Riesz kernel 8 1 (x, ξ) := 8+ (x, ξ) at λ = 1 is called a (forward) fundamental solution of the operator L: L[8+ (x, ξ)] = δ(x − ξ) ,

supp 8+ (x, ξ) ⊆ J+ (ξ) .

(22)

Such a solution is known to exist for any u(x) ∈ C ∞ (), and it is uniquely determined. 6 By a distribution f ∈ D 0 () we mean, as usual, a linear continuous form on the space D() of C ∞ functions with supports compactly imbedded in  (cf., e.g., [23]).

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Definition. The operator L defined by (18) satisfies Huygens’ principle in a domain 0 ⊆  in Mn+1 if (23) supp 8+ (x, ξ) ⊆ C+ (ξ) = ∂J+ (ξ) . for every point ξ ∈ 0 . The analytic description of singularities of Riesz kernel distributions (and, in particular, fundamental solutions) for second order hyperbolic differential operators is given in terms of their asymptotic expansions in the vicinity of the characteristic cone by a graded scale of distributions with weaker and weaker singularities. Such “asymptotics in smoothness”, usually called Hadamard–Riesz expansions, turn out to be very important for testing Huygens’ principle for the operators under consideration. In order to construct an appropriate scale of distributions (Riesz convolution algebra) in Minkowski space Mn+1 we consider (for a fixed ξ ∈ Mn+1 ) a holomorphic D0 -valued mapping C → D0 (Mn+1 ) , λ 7→ Rλ (x, ξ), such that Rλ (x, ξ) is an analytic continuation (in λ) of the following (regular) distribution: Z n+1 n−1 γ(x, ξ)λ− 2 g(x) dx , Re λ > , (24) hRλ (x, ξ), g(x)i = Hn+1 (λ) 2 J+ (ξ)

where dx = dx0 ∧ dx1 ∧ . . . ∧ dxn is a volume form in Mn+1 , g(x) ∈ D(Mn+1 ), and Hn+1 (λ) is a constant given by
n−1 (25) Hn+1 (λ) = 2π 2 4λ−1 0(λ)0 λ − (n − 1)/2 . The following properties of this family of distributions are deduced directly from their definition. For all λ ∈ C and ξ ∈ Mn+1 we have supp Rλ (x, ξ) ⊆ J+ (ξ)

(26)

n+1 Rλ = Rλ−1 ,

(27)

Rλ ∗ Rµ = Rλ+µ ,

µ∈C,

(x − ξ, ∂x )Rλ = (2λ − n + 1)Rλ ,
γ Rλ = 4 (λ)ν λ − (n − 1)/2 ν Rλ+ν , ν ∈ Z≥0 , ν

ν

(28) (29) (30)

where (κ)ν := 0(κ + ν)/0(κ) is Pochhammer’s symbol, and γ = γ(x, ξ) is a square of the geodesic distance between x and ξ in Mn+1 . In addition, when n is odd, one can prove that Rλ (x, ξ) =

1 2π

n−1 2

( n−1 −λ)

δ+ 2 (γ) 4λ−1 (λ − 1)!

for

λ = 1, 2, . . . , (n − 1)/2 ,

(31)

where δ+(m) (γ) stands for the mth derivative of Dirac’s delta-measure concentrated on the surface of the future-directed characteristic half-cone C+ (ξ). Another important property of Riesz distributions is that R0 (x, ξ) = δ(x − ξ) .

(32)

Formulas (26), (27), (32) show that Rλ (x, ξ) is a Riesz kernel for the ordinary wave operator n+1 . The property (31) means precisely that in even-dimensional Minkowski

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119

spaces Mn+1 (n is odd) Huygens’ principle holds for sufficiently low powers of the wave operator d , d ≤ (n − 1)/2. Now we are able to construct the Hadamard–Riesz expansion for the Riesz kernel of a general self-adjoint wave-type operator (18) on Mn+1 . First, we have to find a sequence of two-point smooth functions Uν := Uν (x, ξ) ∈ C ∞ ( × ) , ν = 0, 1, 2 . . ., as a solution of the following transport equations:  1  (x − ξ, ∂x ) Uν (x, ξ) + νUν (x, ξ) = − L Uν−1 (x, ξ) , 4

ν ≥1.

(33)

It is well-known (essentially due to [26]) that the differential-recurrence system (33) has a unique solution provided each Uν is required to be bounded in the vicinity of the vertex of the characteristic cone and U0 (x, ξ) is fixed for a normalization, i.e. U0 (x, ξ) ≡ 1 ,

Uν (ξ, ξ) ∼ O(1) ,

∀ ν = 1, 2, 3, . . . .

These functions Uν are called Hadamard’s coefficients of the operator L. In terms of Uν the required asymptotic expansion can be presented as follows: 8 λ (x, ξ) ∼

∞ X

4ν (λ)ν Uν (x, ξ) Rλ+ν (x, ξ) .

(34)

ν=0

One can prove that for a hyperbolic differential operator L with locally analytic coefficients the Hadamard–Riesz expansion is locally uniformly convergent. From now on we will restrict our consideration to this case. For λ = 1 formula (34) provides an expansion of the fundamental solution of the operator L in a neighborhood of the vertex x = ξ of the characteristic cone: ∞ X

8+ (x, ξ) =

4ν ν! Uν (x, ξ) Rν+1 (x, ξ) .

(35)

ν=0

When n is even, we have supp Rν+1 (x, ξ) = J+ (ξ) for all ν = 0, 1, 2, . . ., and therefore Huygens’ principle never occurs in odd-dimensional Minkowski spaces M2l+1 . On the other hand, in the case of an odd number of space dimensions n ≥ 3, we know due to (31) that for ν = 0, 1, 2, . . . , (n − 3)/2, supp Rν+1 (x, ξ) = C+ (ξ) . Hence, using (30), we can rewrite the series (35) in following form: 8+ (x, ξ) =


1 V (x, ξ) δ+(p−1) (γ) + W (x, ξ) η+ (γ) , p 2π

(36)

where p := (n − 1)/2 , η+ (γ) is a regular distribution characteristic for the region J+ (ξ): Z hη+ (γ), g(x)i = g(x) dx , g(x) ∈ D(Mn+1 ) , J+ (ξ)

and V (x, ξ) , W (x, ξ) are analytic functions in a neighborhood of the vertex x = ξ which admit the following expansions therein: V (x, ξ) =

p−1 X ν=0

1 Uν (x, ξ) γ ν , (1 − p) . . . (ν − p)

(37)

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Y. Y. Berest, I. M. Loutsenko

W (x, ξ) =

∞ X ν=p

1 Uν (x, ξ) γ ν−p , (ν − p)!

p=

n−1 . 2

(38)

The function W (x, ξ) is usually called a logarithmic term of the fundamental solution7 . It follows directly from the representation formula (36) that operator L satisfies Huygens’ principle in a neighborhood of the point ξ, if and only if, the logarithmic term W (x, ξ) of its fundamental solution vanishes in this neighborhood identically in x: W (x, ξ) ≡ 0 . The function W (x, ξ) is known to be a regular solution of the characteristic Goursat problem for the operator L : L [W (x, ξ)] = 0 (39) with a boundary value given on the cone surface C+ (ξ) . Such a boundary problem has a unique solution, and hence, the necessary and sufficient condition for L to be Huygens’ operator becomes W (x, ξ) , 0 , (40) where the symbol , implies that the equation in hand is satisfied only on C+ (ξ) . By definition (38), the latter condition is equivalent to the following one Up (x, ξ) , 0 ,

p=

n−1 . 2

(41)

In this way, we arrive at the important criterion for the validity of Huygens’ principle in terms of coefficients of the Hadamard–Riesz expansion (34). Equation (41) is essentially due to Hadamard [26]. It will play a central role in the proof of our main theorem.

III. Proof of the Main Theorem We start with some remarks concerning the properties of the one-dimensional Schr¨odinger operator  2 ∂ + vk (ϕ) (42) L(k) := − ∂ϕ with a general periodic soliton potential  vk (ϕ) := −2

∂ ∂ϕ

2 log W [91 , 92 , . . . , 9N ] .

(43)

Here, as already discussed in the Introduction, W [91 , 92 , . . . , 9N ] stands for a Wronskian of the set of periodic functions on R1 : 9i (ϕ) := cos(ki ϕ + ϕi ) ,

ϕi ∈ R ,

(44)

associated to an arbitrary strictly monotonic sequence of real positive numbers ("soliton amplitudes"): 0 ≤ k1 < . . . < kN −1 < kN . 7 Such a terminology goes back to Hadamard’s book [26], where the function W (x, ξ) is introduced as a coefficient under the logarithmic singularity of an elementary solution (see for details [15], pp. 740–743).

Huygens’ Principle in Minkowski Spaces and Soliton Solutions KdV Equation

121

It is well-known (see, e.g., [34]) that any such operator L(k) (as well as its proper solitonic counterpart (16)) can be constructed by a successive application of DarbouxCrum factorization transformations ([17, 16]) to the Schr¨odinger operator with the identically zero potential:  2 ∂ . (45) L0 := − ∂ϕ To be precise, let L be a second order ordinary differential operator with a sufficiently smooth potential:  2 ∂ + v(ϕ) . (46) L := − ∂ϕ We ask for formal factorizations of the operator L − λ I = A∗ ◦ A ,

(47)

where I is an identity operator, λ is a (real) constant, and A , A∗ are the first order operators adjoint to each other in a formal sense. According to Frobenius’ theorem (see, e.g., [29]), the most general factorization (47) is obtained if we take χ(ϕ) as a generic element in Ker(L − λ I) \ {0} and set     ∂ ∂ ◦ χ−1 , A∗ := −χ−1 ◦ ◦χ. (48) A := χ ◦ ∂ϕ ∂ϕ Indeed, A∗ ◦ A is obviously a self-adjoint second order operator with the principal part −∂ 2 /∂ϕ2 . Hence, it is of the form (46). Moreover, since A[χ] = 0 , we have χ ∈ Ker A∗ ◦ A , so that (47) becomes evident. Note that for every λ ∈ R we actually get a one-parameter family of factorizations of L − λ I . This follows from the fact that dim Ker(L − λ I) = 2, whereas χ(ϕ) and C χ(ϕ) give rise to the same factorization pair (A, A∗ ) . By definition, the Darboux-Crum transformation maps an operator L = λ I +A∗ ◦A into the operator (49) L˜ := λ I + A ◦ A∗ , ∗ in which A and A are interchanged. The operator L˜ is also a (formally) self-adjoint second-order differential operator  2 ∂ ˜ + v(ϕ) ˜ , (50) L := − ∂ϕ where v(ϕ) ˜ is given explicitly by  v(ϕ) ˜ = v(ϕ) − 2

∂ ∂ϕ

2 log χ(ϕ) .

(51)

The initial operator L and its Darboux-Crum transform L˜ are obviously related to each other via the following intertwining indentities: L˜ ◦ A = A ◦ L

,

L ◦ A∗ = A∗ ◦ L˜ .

(52)

The Darboux-Crum transformation has a lot of important applications in the spectral theory of Sturm-Liouville operators and related problems of quantum mechanics [30]. In particular, it is used to insert or remove one eigenvalue without changing the rest of the

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spectrum of a Schr¨odinger operator (for details see the monograph [34] and references therein). The explicit construction of the family of operators (42) with periodic soliton potentials (43) is based on the following Crum’s lemma: Lemma ([16]). Let L be a given second order Sturm-Liouville operator (46) with a sufficiently smooth potential, and let {91 , 92 , . . . , 9N } be its eigenfunctions corresponding to arbitrarily fixed pairwise different eigenvalues {λ1 , λ2 , . . . , λN } , i.e. 9i ∈ Ker(L − λi I) , i = 1, 2, . . . , N . Then, for arbitrary 9 ∈ Ker(L − λ I) , λ ∈ R , the function W [91 , 92 , . . . , 9N , 9] χN (ϕ) := (53) W [91 , 92 , . . . , 9N ] satisfies the differential equation # "   2 ∂ + vN (ϕ) χN (ϕ) = λ χN (ϕ) − ∂ϕ

(54)

with the potential  vN (ϕ) := v(ϕ) − 2

∂ ∂ϕ

2 log W [91 , 92 , . . . , 9N ] .

(55)

Given a sequence of real positive numbers (ki )N i=1 : 0 ≤ k1 < k2 < . . . < kN , the Darboux-Crum factorization scheme: 2 2 I → Li+1 := Ai ◦ A∗i + ki+1 I, Li := Ai−1 ◦ A∗i−1 + ki2 I = A∗i ◦ Ai + ki+1

(56)

starting from the Schr¨odinger operator (45) with a zero potential  2 ∂ L0 ≡ − = A∗0 ◦ A0 + k12 I , ∂ϕ produces the required operator L(k) ≡ LN with the general periodic potential (43). Now we proceed to the proof of our main theorem formulated in the Introduction. When N = 0 , the statement of the theorem is evident, since the operator L0 is just the ordinary wave operator in an odd number n of spatial variables. Using the Darboux-Crum scheme as outlined above we will carry out the proof by induction in N . Suppose that the statement of the theorem is valid for all m = 0, 1, 2, . . . , N . Consider an arbitrary integer monotonic partition (ki ) of height N : 0 < k1 < k2 < . . . < kN , ki ∈ Z. By our assumption, the wave-type operator LN := L(k) = n+1 + uk (x) ,

(57)

associated to this partition, satisfies Huygens’ principle in the (n + 1)-dimensional Minkowski space Mn+1 with n odd, and n ≥ 2 kN + 3 . We fix the minimal admissible number of space variables, i.e. n = 2 kN + 3 , and denote p :=

n−1 = kN + 1 . 2

(58)

Huygens’ Principle in Minkowski Spaces and Soliton Solutions KdV Equation

123

By construction, the operator LN can be written explicitly in terms of suitably chosen cylindrical coordinates in Mn+1 : ÿ   "  !# 2 2 1 1 ∂ ∂ ∂ − − + + vN (ϕ) , (59) LN = n−1 − ∂r r ∂r r2 ∂ϕ where (r, ϕ) are the polar coordinates in some Euclidean 2-plane E orthogonal to the time direction in Mn+1 , i.e. E ∈ Gr⊥ (n + 1, 2) ; n−1 is a wave operator in the orthogonal complement E ⊥ ∼ = Mn−1 of E in Mn+1 ; and vN (ϕ) is a 2π-periodic potential given by (43). Let k := kN +1 be an arbitrary positive integer such that k > kN .

(60)

We apply the Darboux-Crum transformation (56) with the spectral parameter k to the angular part of the Laplacian in E. For this we rewrite LN in the form "  # 2
1 1 ∂ ∂ ∗ 2 − + AN ◦ AN + k , (61) LN = n−1 − ∂r r ∂r r2 "

and set LN +1 := n−1 −

∂ ∂r

2

#
1 1 ∂ ∗ 2 − + A N ◦ AN + k , r ∂r r2

(62)

where AN := AN (ϕ) and A∗N := A∗N (ϕ) are the first order ordinary differential operators of the form (48). According to (52), we have LN +1 ◦ AN = AN ◦ LN

,

LN ◦ A∗N = A∗N ◦ LN +1 .

(63)

N +1 Let 8N λ (x, ξ) and 8λ (x, ξ) be the Riesz kernels of hyperbolic operators LN and LN +1 respectively. Then, by virtue of (63) we must have the relation     +1 A∗N (ϕ) 8N − AN (φ) 8N (64) λ λ = 0 for all λ ∈ C ,

where AN (φ) is the differential operator AN written in terms of the variable φ conjugated to ϕ . Indeed, if identity (64) were not valid, one could define a holomorphic ˜N ˜ N : C → D0 , λ 7→ 8 mapping 8 λ (x, ξ) , such that  N +1   
N ∗ ˜N 8 − AN (φ) 8N . (65) λ (x, ξ) := 8λ (x, ξ) + a AN (ϕ) 8λ λ N

˜ λ (x, ξ) , depending on an arbitrary complex parameter a ∈ C, would The distribution 8 also satisfy all the axioms (21) in the definition of a Riesz kernel for the operator LN . In this way, we would arrive at the contradiction with the uniqueness of such a kernel. In particular, when λ = 1, the identity (64) gives the relation between the fundamental N N +1 +1 (x, ξ) ≡ 8N (x, ξ) of operators LN and solutions 8N + (x, ξ) ≡ 81 (x, ξ) and 8+ 1 LN +1 . In accordance with (36), we have 8N + (x, ξ) = and


1 VN (x, ξ) δ+(p−1) (γ) + WN (x, ξ) η+ (γ) p 2π

(66)

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Y. Y. Berest, I. M. Loutsenko


1 (67) VN +1 (x, ξ) δ+(p−1) (γ) + WN +1 (x, ξ) η+ (γ) , p 2π where γ is a square of the geodesic distance between the points x and ξ in Mn+1 . Substituting (66), (67) into (64), we get the relation between the logarithmic terms WN (x, ξ) and WN +1 (x, ξ) of operators LN and LN +1 +1 8N (x, ξ) = +

A∗N (ϕ) [WN +1 (x, ξ)] − AN (φ) [WN (x, ξ)] = 0 .

(68)

By our assumption, LN is a Huygens’ operator in Mn+1 , so that WN (x, ξ) ≡ 0. Hence, Eq. (68) implies A∗N (ϕ) [WN +1 (x, ξ)] = 0 . On the other hand, as discussed in Sect. II, the logarithmic term WN +1 (x, ξ) is a regular solution of the characteristic Goursat problem for LN +1 , i.e. in particular, LN +1 [WN +1 (x, ξ)] = 0 .

(69)

Taking into account definition (62) of the operator LN +1 , we arrive at the following equation for WN +1 (x, ξ) : ! ÿ  2 k2 1 ∂ ∂ n−1 WN +1 (x, ξ) = − 2 WN +1 (x, ξ) . + (70) ∂r r ∂r r According to (38), the logarithmic term WN +1 admits the following expansion: WN +1 (x, ξ) =

∞ X

Uν (x, ξ)

ν=p

γ ν−p , (ν − p)!

p=

n−1 , 2

(71)

where Uν (x, ξ) are the Hadamard coefficients of the operator LN +1 . Since the potential of the wave-type operator LN +1 depends only on the variables r, ϕ , its Hadamard coefficients Uν must depend on the same variables r, ϕ and their conjugates ρ, φ only: for all ν = 0, 1, 2, . . . . (72) Uν = Uν (r, ϕ, ρ, φ) This follows immediately from the uniqueness of the solution of Hadamard’s transport equations (33). On the other hand, since γ = s2 − r2 − ρ2 + 2r ρ cos(ϕ − φ) ,

(73)

where s is a geodesic distance in the space E ⊥ ∼ = Mn−1 orthogonally complementary to the 2-plane E, we conclude that WN +1 is actually a function of five variables: WN +1 = WN +1 (s, r, ρ, ϕ, φ) . On the space of such functions the wave operator n−1 in E ⊥ acts in the same way as its “radial part”, i.e. ! ÿ  2 n−2 ∂ ∂ WN +1 . + n−1 WN +1 = ∂s s ∂s Hence, Eq. (70) becomes ÿ  !  2 2 1 ∂ k2 n−2 ∂ ∂ ∂ + − 2 WN +1 = 0 . − − ∂r ∂s s ∂s r ∂r r Now we substitute the expansion (71)

(74)

Huygens’ Principle in Minkowski Spaces and Soliton Solutions KdV Equation

WN +1 =

∞ X

Uν (r, ϕ, ρ, φ)

ν=p

γ ν−p , (ν − p)!

p=

125

n−1 , 2

(75)

into the left-hand side of the latter equation and develop the result into the similar power series in γ , taking into account formula (73). After simple calculations we obtain  ∞  X 1 k2 0 − (76) Uν00 + Uν0 − 2 Uν − 4 (r − ρ cos(ϕ − φ)) Uν+1 r r ν=p  ν−p  ρ γ 2 2 =0, − 2 2 (ν + 1) − cos(ϕ − φ) Uν+1 − 4 ρ sin (ϕ − φ) Uν+2 r (ν − p)! 

where the prime means differentiation with respect to r . Since the functions Uν do not depend explicitly on γ , Eq. (76) can be satisfied only if each coefficient under the powers of γ vanishes separately. In this way we arrive at the following differential-recurrence relation for the Hadamard coefficients of the operator LN +1 :   1 k2 4 ρ2 sin2 (ϕ − φ) Uν+2 = Uν00 + Uν0 − 2 Uν + r r

2ρ 0 0 + cos(ϕ − φ) 2 r Uν+1 + Uν+1 − 4 r Uν+1 + (ν + 1) Uν+1 , (77) r where ν runs from p : ν = p, p + 1, p + 2, . . .. To get a further simplification of Eq. (77) we notice that all the Hadamard coefficients of the operators under consideration (11), (10) are homogeneous functions of appropriate degrees. More precisely, they have the following specific form Uν (r, ϕ, ρ, φ) =

1 σν (ϕ, φ) , (r ρ)ν

ν = 0, 1, 2, . . . ,

(78)

where σν (ϕ, φ) = σν (φ, ϕ) are symmetric 2π-periodic functions depending on the angular variables only. In order to prove Ansatz (78) we have to go back to the relation (64) between the Riesz kernels of operators LN and LN +1 :    N  +1 λ∈C, (79) A∗N (ϕ) 8N λ (x, ξ) − AN (φ) 8λ (x, ξ) = 0 , If we substitute the Hadamard–Riesz expansions (34) of the kernels 8N λ (x, ξ) and +1 8N (x, ξ) into (79) directly and take into account that A and its adjoint A∗N are the N λ first order ordinary differential operators of the following form (cf. (48)): AN (ϕ) =

∂ − fN (ϕ) , ∂ϕ

A∗N (ϕ) = −

∂ − fN (ϕ) , ∂ϕ

(80)

where fN (ϕ) = (∂/∂ϕ) log χN (ϕ) , we obtain  ∞ X
N +1 N 4ν (λ)ν 2rρ sin(ϕ − φ) Uν+1 − Uν+1 −  −

ν=0



∂ + fN (ϕ) ∂ϕ

 UνN +1

−

∂ − fN (φ) ∂φ



 UνN

Rλ+ν = 0 ,

(81)

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where UνN (r, ϕ, ρ, φ) and UνN +1 (r, ϕ, ρ, φ) are the Hadamard coefficients of operators LN and LN +1 respectively; Rλ := Rλ (x, ξ) is the family of Riesz distributions in Mn+1 . The same argument as above (see the remark before formula (77)) shows that all the coefficients of the series (81) under the Riesz distributions of different weights must vanish separately. So we arrive at the recurrence relation between the sequences of Hadamard’s coefficients of operators LN and LN +1 :      ∂ 1 ∂ N +1 N N +1 N Uν+1 = Uν+1 + + fN (ϕ) Uν + − fN (φ) Uν , 2rρ sin(ϕ − φ) ∂ϕ ∂φ (82) where U0N +1 = U0N ≡ 1 and ν = 0, 1, 2, . . . Now it is easy to conclude from (82) by induction in N that the Ansatz (78) really holds for Hadamard’s coefficients of all wave-type operators (11) with potentials (10). Returning to Eq. (77) and substituting (78) therein, we obtain the following threeterm recurrence relation for the angular functions σν (ϕ, φ) : 4 sin2 (ϕ − φ) σν+2 = (ν 2 − k 2 ) σν − 2(2ν + 1) cos(ϕ − φ) σν+1 ,

(83)

where ν = p, p + 1, p + 2, . . . . In order to analyze Eq. (83) it is convenient to introduce a formal generating function for the quantities { σν } : F (t) :=

∞ X

σν (ϕ, φ)

ν=p

tν−p . (ν − p)!

(84)

The recurrence relation (83) turns out to be equivalent to the classical hypergeometric differential equation for the function F (t), 4 (1 − ω 2 ) + 4ωt − t2


d2 F dF + (k 2 − p2 ) F = 0 , + (2p + 1) (2ω − t) dt2 dt

(85)

where ω := cos(ϕ − φ) . The general solution to (85) is given in terms of Gauss’ hypergeometric series: F (t) = C 2 F1 (p − k; p + k; p + 1/2 | z) + C1 z −p+1/2 2 F1 (1/2 − k; 1/2 + k; 3/2 − p | z) , (86) where z := (t − 2ω + 2)/4 and 2 F1 is defined by 2

F1 (a; b; c | z) :=

∞ X (a)µ (b)µ z µ . (c)µ µ!

(87)

µ=0

As discussed in Sect.II, the Hadamard coefficients Uν (x, ξ) must be regular in a neighborhood of the vertex of the characteristic cone x = ξ . When x → ξ , we have ω → 1 and Up (ξ, ξ) ∝ σp (φ, φ) = F (0)|ω=1 is not bounded unless C1 = 0 . In this way, setting C1 = 0 in (86), we obtain ∞ X ν=p

σν (ϕ, φ)


tν−p = C 2 F1 p − k; p + k; p + 1/2 | (t − 2ω + 2)/4 . (ν − p)!

(88)

Now it remains to recall that by our assumption (60) k ∈ Z and k > kN . Since p = (n − 1)/2 = kN + 1 , we have k ≥ p . So the hypergeometric series in the right-hand

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127

side of Eq. (88) is truncated. In fact, the generating function (84) is expressed in terms (p−1/2,p+1/2) of the classical Jacobi polynomial Pk−p (ω − t/2) of degree k − p . Hence, th σk+1 (ϕ, φ) ≡ 0 , and the (k + 1) Hadamard coefficient of the operator LN +1 vanishes identically: Uk+1 (x, ξ) ≡ 0 .

(89)

According to Hadamard’s criterion (41), it means that the operator LN +1 satisfies Huygens’ principle in Minkowski space Mn+1 , if n is odd and n ≥ 2k +3 , Thus, the proof of the theorem is completed.

IV. Concluding Remarks and Examples In the present paper we have constructed a new hierarchy of Huygens’ operators in higher dimensional Minkowski spaces Mn+1 , n > 3. However, the problem of complete description of the whole class of such operators for arbitrary n still remains open. As mentioned in the Introduction, the famous Hadamard’s conjecture claiming that any Huygens’ operator L can be reduced to the ordinary d’Alembertian n+1 with the help of trivial transformations is valid only in M3+1 . Recently, in the work [4] one of the authors put forward the relevant modification of Hadamard’s conjecture for Minkowski spaces of arbitrary dimensions. Here we recall and discuss briefly this statement. Let  be an open set in Minkowski space Mn+1 ∼ = Rn+1 , and let F () be a ring of partial differential operators defined over the function space C ∞ () . For a fixed pair of operators L0 , L ∈ F () we introduce the map adL,L0 : F() → F () ,

A 7→ adL,L0 [A] ,

(90)

such that adL,L0 [A] := L ◦ A − A ◦ L0 .

(91)

Then, given M ∈ Z>0 , the iterated adL,L0 -map is determined by adM L,L0 [A]









:= adL,L0 adL,L0 . . . adL,L0 [A] . . . =

M X

 k

(−1)

k=0

 M LM −k ◦ A ◦ Lk0 . k (92)

Definition. The operator L ∈ F() is called M-gauge related to the operator L0 ∈ F () , if there exists a smooth function θ(x) ∈ C ∞ () non-vanishing in , and an integer positive number M ∈ Z>0 , such that adM L,L0 [θ(x)] ≡ 0

identically in F () .

(93)

In particular, when M = 1 , the operators L and L0 are connected just by the trivial gauge transformation L = θ(x) ◦ L0 ◦ θ(x)−1 .

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The modified Hadamard’s conjecture claims: Any Huygens’ operator L of the general form L = n+1 + (a(x), ∂) + u(x) ,

(94)

in a Minkowski space Mn+1 (n is odd, n ≥ 3) is M -gauge related to the ordinary wave operator n+1 in Mn+1 . For Huygens’ operators associated to the rational solutions of the KdV-equation (2), (3) and to Coxeter groups (5), (6) this conjecture has been proved in [4] and [7]. In these cases the required identities (93) are the following: Mk +1 [Pk (x0 )] = 0 , adL k ,L0

Mk :=

k(k + 1) , 2

(95)

where Lk is given by (2) with the potential (3) for k = 0, 1, 2, . . . and Mm +1 adL [πm (x)] = 0 , m ,L0

Mm :=

X

mα ,

(96)

α∈<+

Q where Lm is defined by (5), (6) and πm (x) := α∈<+ (α, x)mα . It is remarkable that for the operators constructed in the present work the modified Hadamard’s conjecture is also verified. More precisely, for a given wave-type operator  2 2 ∂ log W [91 (ϕ), 92 (ϕ), . . . , 9N (ϕ)] , (97) L(k) = n+1 − 2 r ∂ϕ associated to a positive integer partition (ki ) : 0 ≤ k1 < k2 < . . . < kN , we have the identity |k|+1 (98) adL(k) ,L0 [Θ(k) (x)] = 0 , P N where Θ(k) (x) := r|k| W [91 , 92 , . . . , 9N ] and |k| := i=1 ki is a weight of the partition (ki ) . We are not going to prove (98) in the present paper. A more detailed discussion of this identity and associated algebraic structures will be the subject of our subsequent work. Here, we only mention that such type identities naturally appear [4]–[5] in connection with a classification of overcomplete commutative rings of partial differential operators [13, 14, 41], and with the bispectral problem [20]. We conclude the paper with several concrete examples illustrating our main theorem. 1. As a first example we consider the dihedral group I2 (q), q ∈ Z>0 , acting on the Euclidean plane E ∼ = R2 ⊂ Gr⊥ (n + 1, 2) and fix the simplest partition k = (q) and the phase ϕ = π/2. According to Remark II, in this case our theorem gives the wavetype operator with the Calogero-Moser potential related to the Coxeter group I2 (q) with m = 1: 2 q2 . L(k) = n+1 + r2 sin2 (q ϕ) This operator satisfies Huygens’ principle in Mn+1 if n is odd and n ≥ 2 q + 3. The Hadamard coefficients of L(k) can be presented in a simple closed form in terms of polar coordinates on E: U0 = 1 ,

Huygens’ Principle in Minkowski Spaces and Soliton Solutions KdV Equation

Uν =

1 Tq(ν) (cos(ϕ − φ)) , (2rρ)ν sin(q ϕ) sin(q φ)

129

ν ≥1,

where Tq (z) := cos(q arccos(z)), z ∈ [−1, 1], is the q th Chebyshev polynomial, and Tq(ν) (z) is its derivative of order ν with respect to z. These formulas are easily obtained with the help of recurrence relation (82). 2. Now we fix N = 2, k1 = 2, k2 = 3 and ϕ1 = π/2, ϕ2 = 0 . The corresponding wave-type operator

10 x21 + x22 15 x22 − x21 , L(k) = n+1 +
2 5x22 + x21 x21 satisfies Huygens’ principle for odd n ≥ 9. The nonzero Hadamard coefficients of this operator are given explicitly by the formulas: U0 = 1, U1 =

40 x2 ξ1 ξ2 x1 + 15 ξ1 2 x2 2 + 75 ξ2 2 x2 2 + 15 ξ2 2 x1 2 − 5 ξ1 2 x1 2  ,
 2 ξ1 x 1 5 x 2 2 + x 1 2 5 ξ 2 2 + ξ 1 2

U2 =

120 x2 ξ1 ξ2 x1 + 15 ξ2 2 x1 2 − 5 ξ1 2 x1 2 + 15 ξ1 2 x2 2 + 75 ξ2 2 x2 2  ,
 4 ξ1 2 x 1 2 5 x 2 2 + x 1 2 5 ξ 2 2 + ξ 1 2 U3 = −

2

ξ1 x1 2

15 x2 ξ2 .
 5 x 2 2 + x 1 2 5 ξ2 2 + ξ 1 2

3. Now we take N = 3, the partition k = (1, 3, 4) , and the phases ϕ1 = ϕ2 = ϕ3 = π/2 . The corresponding operator
12 49x41 + 28x21 x22 − x42 L(k) = n+1 + ,
2 x22 7x21 + x22 is a Huygens operator in Mn+1 when n is odd and n ≥ 11 . The nonzero Hadamard’s coefficients are U0 = 1, U1 =

U2 =

U3 =

−21 ξ2 2 x1 2 − 42 x2 ξ1 ξ2 x1 − 21 ξ1 2 x2 2 + 3 ξ2 2 x2 2 − 147 ξ1 2 x1 2  ,
 ξ 2 x 2 7 x 1 2 + x 2 2 7 ξ1 2 + ξ 2 2 735 ξ1 2 x1 2 + 504 x2 ξ1 ξ2 x1 + 105 ξ1 2 x2 2 − 21 ξ2 2 x2 2 + 105 ξ2 2 x1 2  ,
 4 ξ2 2 x 2 2 7 x 1 2 + x 2 2 7 ξ 1 2 + ξ 2 2

−1260 x2 ξ1 ξ2 x1 + 21 ξ2 2 x2 2 − 105 ξ1 2 x2 2 − 105 ξ2 2 x1 2 − 735 ξ1 2 x1 2  ,
 8 ξ2 3 x 2 3 7 x 1 2 + x 2 2 7 ξ 1 2 + ξ 2 2 U4 =

3

4 ξ2 x 2 3

315 x1 ξ1 .
 7 x 1 2 + x 2 2 7 ξ1 2 + ξ 2 2

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4. The last example illustrates Remark I following the theorem (see Introduction). In this case we consider the operator (11) with the potential (13) associated with the proper N -soliton solution of the KdV equation. We take N = 2 and fix k1 = 1, k2 = 2 . The real phases are chosen as follows ϑ1 = arctanh (1/2), ϑ2 = arctanh (1/4) . The corresponding operator L(k) reads
2 (2x0 − 3x1 ) 3x31 − 6x0 x21 + 4x1 x20 + 8x30 . L(k) = n+1 +
2 x21 4x20 − 2x0 x1 − x21 According to the theorem, it is huygensian provided n is odd and n ≥ 7. The nonzero Hadamard coefficients are given by the following formulas: U0 = 1, U1 = +

2x1

4ξ02 x21 + 9ξ12 x21 − 16ξ02 x20 + 8ξ02 x0 x1

+ 4x20 − 2x0 x1 − x21 ξ1 4ξ02 − 2ξ0 ξ1 − ξ12

8ξ0 ξ1 x20 − 12ξ12 x0 x1 + 4ξ12 x20 − 12ξ0 ξ1 x21 + 16ξ0 ξ1 x0 x1

, 2x1 4x20 − 2x0 x1 − x21 ξ1 4ξ02 − 2ξ0 ξ1 − ξ12

U2 = −

4 x1

5 (2 ξ0 − ξ1 ) (2 x0 − x1 ) .
 4 x 0 2 − 2 x 0 x 1 − x 1 2 ξ 1 4 ξ 0 2 − 2 ξ0 ξ 1 − ξ 1 2

Acknowledgement. One of the authors (Yu. B.) is grateful to Prof. A. Veselov (Loughborough University, UK) and his collaborators Dr. O. Chalykh and M. Feigin who have kindly informed him [42] about their recent results showing the existence of new algebraically integrable Schr¨odinger operators which are not related to Coxeter root systems. In fact, this observation was a motivation for the present work. We would like also to thank Prof. P. Winternitz (CRM, Universit´e de Montr´eal) for his encouragement and highly stimulating discussions. The work of Yu. B. was partially supported by the fellowship from Institut des Sciences Math´ematiques (Montr´eal) which is gratefully acknowledged. The second author (I. L.) is grateful to Prof. L. Vinet for his support.

References 1. Ablowitz, M.J. and Clarkson, P.A.: Solitons, Nonlinear Evolution Equations and Inverse Scattering. Cambridge: Cambridge University Press, 1991 2. Adler, M. and Moser, J. On a class of polynomials connected with the Korteweg–de Vries Equation. Commun. Math. Phys. 61, 1–30 (1978) 3. Airault, H., McKean, H. and Moser, J.: Rational and elliptic solutions of the KdV-equation and related many body problems. Commun. Pure Appl. Math. 30, 95–148 (1977) 4. Berest, Yu.Yu.: Hierarchies of Huygens’ operators and Hadamard’s conjecture. Preprint CRM-2296, to be published in Acta Math Appl. (1997) 5. Berest, Yu.Yu.: Lacunae of hyperbolic Riesz kernels and commutative rings of partial differential operators. Lett. Math. Phys. 41, 227–235 (1997) 6. Berest, Yu.Yu.: Solution of a restricted Hadamard’s problem in Minkowski spaces. Commun. Pure Appl. Math. 50, 1021–1054 (1997) 7. Berest, Yu.Yu. and Molchanov, Yu.A.: Fundamental solutions for partial differential equations with reflection group invariance. J. Math. Phys. 36 (8), 4324–4339 (1995)

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8. Berest, Yu.Yu. and Veselov, A.P.: Huygens’ principle and integrability. Uspekhi Mat. Nauk, 49 (6), 5–77 (1994) 9. Berest, Yu.Yu. and Veselov, A.P.: Huygens’ principle and Coxeter groups. Uspekhi Mat. Nauk, 48 (2), 181–182 (1993) 10. Berest, Yu.Yu. and Veselov, A.P.: Hadamard’s problem and Coxeter groups: new examples of Huygens’ equations. Funct. Anal. Appl. 28 (1), 3–15 (1994) 11. Burchnall, J.L. and Chaundy, T.W.: A set of differential equations which can be solved by polynomials. Proc. London Math. Soc. 30, 401–414 (1929–1930) 12. Carminati, J. and McLenaghan, R.G.: On the validity of Huygens’ principle for the wave equation on curved space-time. In: Huygens’ principle 1690-1990: Theory and Applications. (H.Blok, H.A.Ferwerda, H.K.Kuiken, Eds.), New York: Elsevier Publishers B.V.,1992, pp. 435–442 13. Chalykh, O.A. and Veselov, A.P.: Commutative rings of partial differential operators and Lie algebras. Commun. Math. Phys. 126, 597–611 (1990) 14. Chalykh, O.A. and Veselov, A.P.: Integrability in the theory of the Schr¨odinger operator and harmonic analysis. Commun. Math. Phys. 152, 29–40 (1993) 15. Courant, R. and Hilbert, D.: Methods of Mathematical Physics II, New York: Interscience Publ., 1962 16. Crum, M.: Associated Sturm-Liouville systems. Quart. J. Math. 6 (2), 121–127 (1955) 17. Darboux, G.: Sur la representation sph´erique des surfaces. Compt. Rend. 94, 1343–1345 (1882) 18. Deift, P.A.: Application of a commutation formula. Duke Math. Journal, 45 (2), 267–310 (1978) 19. Duistermaat, J.J.: Huygens’ principle for linear partial differential equations. In Huygens’ principle 1690-1990: Theory and Applications, (H.Blok, H.A.Ferwerda, H.K.Kuiken, Eds.), New York: Elsevier Publishers B.V. 1992, pp. 273–297 20. Duistermaat, J.J. and Gr¨unbaum, F.A.: Differential equations in the spectral parameter. Commun. Math. Phys. 103, 177–240 (1986) 21. Friedlander, F.G.: The Wave Equation on a Curved Space-time, Cambridge: Cambridge University Press, 1975 22. Gabrielov, A.M. and Palamodov, V.P.: Huygens’ principle and its generalizations. In: I. G. Petrovskii, Collected papers, v. 1, Nauka, Moscow, 1986 (in Russian) pp. 449–456 23. Gel’fand, I.M. and Shilov, G.E.: Generalized Functions, V. 1, New York: Acad. Press, 1964 24. P.G¨unther, P.: Huygens’ Principle and Hyperbolic Equations, Boston: Acad. Press, 1988 25. G¨unther, P.: Huygens’ principle and Hadamard’s conjecture. The Mathematical Intelligencer, 13(2), 56–63 (1991) 26. Hadamard, J.: Lectures on Cauchy’s Problem in Linear Partial Differential Equations. New Haven: Yale University Press, 1923 27. Helgason, S.: Wave equations on homogeneous spaces. Lect. Notes in Math. 1077, Berlin: SpringerVerlag, (1984) pp.252–287 28. Ibragimov, N.H.: Transformation Groups Applied to Mathematical Physics. Dordrecht: Reidel, 1985 29. Ince, E.L.: Ordinary Differential Equations. New-York: Dover, 1956 30. Infeld, L. and Hull, T.E.: The factorization method. Rev. Mod. Phys. 23 (1), 21–70 (1951) 31. Lagnese, J.E. and Stellmacher, K.L.: A method of generating classes of Huygens’ operators. J. Math. Mech. 17 (5), 461–472 (1967) 32. Lagnese, J.E.: A solution of Hadamard’s problem for a restricted class of operators. Proc. Amer. Math. Soc. 19, 981–988 (1968) 33. Mathisson, M.: Le probl`eme de M. Hadamard relatif a` la diffusion des ondes. Acta Math., 71, 249–282 (1939) 34. Matveev, V.B. and Salle, M.A.: Darboux Transformations and Solitons. Berlin–Heidelberg–New York: Springer-Verlag, 1991 35. McLenaghan, R.G.: Huygens’ principle. Ann. Inst. Henri Poincar´e, 37 (3), 211–236 (1982)

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36. Olshanetsky, M.A. and Perelomov, A.M.: Quantum integrable systems related to Lie algebras. Phys. Reps. 94, 313–404 (1983) 37. Riesz, M.: L’int´egrale de Riemann-Liouville et le probl`eme de Cauchy. Acta Math. 81, 1–223 (1949) 38. Schimming, R.: Korteweg–de Vries-Hierarchie und Huygenssches Prinzip. Dresdener Seminar f¨ur Theoretische Physik, Sitzungsberichte, N 26 (1986) 39. Schimming, R.: Laplace-like linear differential operators with a logarithmic-free elementary solution. Math. Nachr. 148, 145–174 (1990) 40. Stellmacher, K.: Ein Beispiel einer Huygensschen Differentialgleichung. Nachr. Akad. Wiss. G¨ottingen, Math.-Phys. K1, IIa, Bd. 10, 133–138 (1953) 41. Veselov, A.P., Styrkas, K.L. and Chalykh, O.A.: Algebraic integrability for the Schr¨odinger operator and reflection groups. Theor. Math. Phys. 94, 253–275 (1993) 42. Veselov, A.P., Feigin, M.V. and Chalykh, O.A.: New integrable deformations of quantum CalogeroMoser problem. Usp. Mat. Nauk. 51 (3), 185–186 (1996) Communicated by T. Miwa

Commun. Math. Phys. 190, 133 – 141 (1997)

Communications in

Mathematical Physics c Springer-Verlag 1997

Existence of the Density of States for Multi-Dimensional Continuum Schr¨odinger Operators with Gaussian Random Potentials? 2 ¨ Werner Fischer1 , Thomas Hupfer1 , Hajo Leschke1 , Peter Muller 1

Institut f¨ur Theoretische Physik, Universit¨at Erlangen-N¨urnberg, Staudtstraße 7, D-91058 Erlangen, Germany 2 Institut f¨ ur Theoretische Physik, Georg-August-Universit¨at, D-37073 G¨ottingen, Germany Received: 13 November 1996 / Accepted: 30 April 1997

Abstract: A Wegner estimate is proved for quantum systems in multi-dimensional Euclidean space which are characterized by one-particle Schr¨odinger operators with random potentials that admit a certain one-parameter decomposition. In particular, the Wegner estimate applies to systems with rather general Gaussian random potentials. As a consequence, these systems possess an absolutely continuous integrated density of states, whose derivative, the density of states, is locally bounded. An explicit upper bound is derived. 1. Introduction The integrated density of states is a quantity of primary interest in the theory and applications of one-particle random Schr¨odinger operators [SE, BEE+, LGP, CL, PF]. For example, the topological support of the associated measure coincides with the almostsure spectrum of the infinite-volume operator. Moreover, its knowledge allows to compute the free energy and hence all basic thermostatic quantities of the corresponding non-interacting many-particle system. An important method to prove the absolute continuity of the integrated density of states, that is, the existence of the density of states, dates back to an idea of Wegner [W]. The aim is to provide a suitable upper bound on the averaged number of eigenvalues of the finite-volume operator in a given energy interval. Nowadays bounds of this genre go under the name Wegner estimates. Starting with the fundamental work [FS] Wegner estimates have also turned out to play a key rˆole in rigorous proofs of Anderson localisation in multi-dimensional space, see e.g. [MH, KS, Sp, vDK, AM, CH, Kl, Ki3]. The so-called spectral-averaging technique [Ko1, Ko2, KS, CH, CHM, BCH, Si2] has been very efficient in deriving Wegner estimates for random Schr¨odinger operators in continuous space. To our knowledge however, if the random potential does not possess ?

Research supported by the Deutsche Forschungsgemeinschaft.

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W. Fischer, T. Hupfer, H. Leschke, P. M¨uller

an underlying lattice structure the estimates obtained so far are not stringent enough as to deduce the absolute continuity of the integrated density of states. Our goal here is to use spectral averaging to prove this property for a truly continuum random potential. More precisely, in building on [CH] we will establish the absolute continuity of the integrated density of states for a Schr¨odinger operator with a rather general Gaussian random potential in d-dimensional Euclidean space Rd for all d ≥ 1. As a by-product we get an explicit upper bound on the density of states. We would also like to remark that the present paper accomplishes one part of the localisation proofs announced in [FLM1] and [FHLM]. The remaining part is deferred to [FLM2]. This paper is organized as follows. In the next section we will formulate our main result. The final section shows that this result is a consequence of a Wegner estimate which is proved to hold for all continuum Schr¨odinger operators whose random potential admits a certain one-parameter decomposition. 2. Main Result Let (Ω, A, P ) be a complete probability space and let V : Ω × Rd → R , (ω, x) 7→ V (ω) (x) be a random field. We list four properties which V may have or not: (M) V is jointly measurable with respect to the sigma-algebra A and the Borel sigmaalgebra of Rd . (F) For all bounded open (hyper-) cubes Λ ⊂ Rd the realisations of V , when viewed as multiplication operators, are P -almost surely relatively form-bounded with respect to both − 21 ∆Λ,D and − 21 ∆Λ,N with relative bounds strictly smaller than 1. Here ∆Λ,D , respectively ∆Λ,N , denotes the Dirichlet Laplacian, respectively the Neumann Laplacian on the Hilbert space L2 (Λ) of complex-valued functions on Λ which are square-integrable over Λ with respect to the d-dimensional Lebesgue measure dd x. (E) V is ergodic with respect to translations in Rd . (G) V is a Gaussian random fieldR which is homogeneous with respect to translations in Rd . ItRhas zero mean, Ω dP (ω) V (ω) (0) = 0, and its covariance function x 7→ C(x) := Ω dP (ω) V (ω) (x)V (ω) (0) is continuous at the origin where it obeys 0 < C(0) < ∞. Remark 1. Consider a random field V with the Gaussian property (G). Then by Bochner’s theorem its covariance function C is bounded and uniformly continuous on Rd . Consequently, [F, Thm. 3.2.2] implies the existence of a separable version of V which has the measurability property (M). From now on it is tacitly assumed that only this version will be dealt with when we refer to a random field with the property (G). We also remark that (G) implies the form-boundedness property (F). This may be checked with the help of [KM2, Lemma 2.1], since V is P -almost surely in the Lebesgue space Lp (Λ) for any 1 ≤ p < ∞ as follows from Z Z Z 2 |Λ| d (ω) p dP (ω) d x |V (x)| = √ dv e−v /2C(0) |v|p < ∞ , (1) 2πC(0) R Ω Λ where we have used Fubini’s theorem and the homogeneity of V . Here |Λ| denotes the volume of Λ measured by dd x. A simple sufficient criterion for a Gaussian random field to have the ergodicity property (E) is provided by the mixing condition lim|x|→∞ C(x) = 0.

Density of States for Gaussian Random Potentials

135

Definition 1. Let Λ ⊂ Rd be a bounded open cube and let V be a random field having the properties (M) and (F). Then the finite-volume Schr¨odinger operator HΛ,X (V ) := − 21 ∆Λ,X + V ,

X = D or X = N ,

(2)

with random potential V is defined on L2 (Λ) as a sum of quadratic forms. Remark 2. It follows from [KM1] and [KM2] that HΛ,X (V ) is measurable and P -almost surely self-adjoint. Its spectrum is P -almost surely discrete and therefore its finitevolume integrated density of states
(3) NΛ,X (E) := Tr FΛ,X ] − ∞, E[ exists P -almost surely. Here Tr denotes the trace over the Hilbert space L2 (Λ) and FΛ,X (ω) the spectral family of HΛ,X (V ). Consequently, NΛ,X (E) is just the number of eigenvalues (ω) of HΛ,X (V ), counting multiplicity, which are strictly smaller than E. After these preparations and writing χB for the indicator function of a subset B ⊂ Rd , we now state a Wegner estimate for NΛ,X in Theorem 1. Let HΛ,X (V ) be defined as in (2) with a random field V having the Gaussian property (G). Moreover, assume that there exist a finite signed Borel measure µ on Rd , a cube Γ ⊂ Rd with |Γ | > 0 and a constant γ > 0 such that the covariance function obeys Z dµ(y) C(x − y) ≥ γ C(0)χΓ (x) (4) Rd

for all x ∈ R . Then for every energy E ∈ R there is a constant 0 < W (E) < ∞, independent of Λ and X, such that the finite-volume integrated density of states (3) obeys Z (ω) (ω) dP (ω) NΛ,X (E1 ) − NΛ,X (E2 ) ≤ |Λ| |E1 − E2 | W (E) (5) d

Ω

for all E1 , E2 ≤ E and all bounded open cubes Λ ⊂ Rd with |Λ| ≥ |Γ |. The proof of Theorem 1 will be deferred to the next section. Remark 3. (i) In what follows it will be assumed, without loss of generality, that the measure µ is normalized according to Z Z dµ(x) dµ(y) C(x − y) = C(0) . (6) Rd

Rd

(ii) If C(x) ≥ 0 for all x ∈ Rd , one may simply choose the signed measure µ to be Dirac’s point measure at the origin. Due to the continuity of C and since C(0) > 0, Condition (4) is then fulfilled with some sufficiently small cube Γ containing the origin and γ = inf x∈Γ C(x)/C(0). With other choices of µ it is also possible to satisfy (4) for certain covariance functions which R take on negative values. An example for the case d = 1 is given by C(x) := R dy w(x + y)w(y) with w := χ[−3,3] − 45 χ[−1,1] . A suitable choice is dµ(x) = m χ[−6,6] (x)dx , Γ =] − 1, 1[ and γ = 8m, where m > 0 is determined by the normalization (6).

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W. Fischer, T. Hupfer, H. Leschke, P. M¨uller

(iii) The Lipschitz continuity (5) of the averaged finite-volume integrated density of states implies by the Chebyshev-Markov inequality that the probability of finding the spectrum of HΛ,X (V ) near a given energy E0 is controlled by the inequality  o n 
≤ 2|Λ| ε W (E) . (7) P ω : dist spec HΛ,X (V (ω) ) , E0 < ε It is valid for all bounded open cubes Λ ⊂ Rd with |Λ| ≥ |Γ | and for all energies E0 ∈ R and ε > 0 such that E0 +ε ≤ E. Actually it is (7) – or some weakened form of (7), where, for example, |Λ| is replaced by |Λ|α with α > 1 – that plays a key rˆole in proofs of Anderson localisation for multi-dimensional random Schr¨odinger operators. (iv) Theorem 1 can be generalised to situations where the random potential is Gaussian but not Rd -homogeneous, the extension being essentially a matter of notation. This allows in particular to treat correlated Gaussian alloy-type potentials. Now we assume that the random field V has the ergodicity property (E) in addiR tion to the Gaussian property (G). Due to this and since Ω dP (ω) exp{−tV (ω) (0)} = exp{t2 C(0)/2} < ∞ for all t > 0, standard techniques show the existence of a nonrandom left-continuous distribution function N on R , called the integrated density of states (per volume) in the macroscopic limit, such that N (E) = lim

(ω) NΛ,X (E)

|Λ|

Λ↑Rd

.

(8)

More precisely, there is a set Ω0 ∈ A of full probability, P (Ω0 ) = 1, such that (8) holds for all ω ∈ Ω0 , for both boundary conditions X and for all E ∈ R except for the at most countably many discontinuity points of N , see [KM2, Ki1] or [Ki2, Sect. 7.3]. The non-randomness of N and Fatou’s lemma imply Z (ω) (ω) (E1 ) − NΛ,X (E2 ) (9) |N (E1 ) − N (E2 )| ≤ lim inf |Λ|−1 dP (ω) NΛ,X Λ↑Rd

Ω

for all those E1 , E2 ∈ R for which (8) is true. Accordingly, Theorem 1 and the leftcontinuity of N yield the Lipschitz continuity of N , namely
(10) |N (E1 ) − N (E2 )| ≤ |E1 − E2 | W max{E1 , E2 } for all E1 , E2 ∈ R . These arguments are summarized in Corollary 1. Under the assumptions of Theorem 1 and supposing property (E), the integrated density of states in the macroscopic limit is absolutely continuous on any bounded interval and its derivative, the density of states, is locally bounded in the sense that dN (E) ≤ W (E) (11) 0≤ dE for Lebesgue-almost all E ∈ R . Remark 4. We infer from the proof of Theorem 1, which is presented in the next section, that the Wegner constant may be taken as W (E) =

d exp{βE + β 2 CE /2}  −1 √ 2`E + (2πβ)−1/2 , 2πC(0) γ

(12)

Density of States for Gaussian Random Potentials

137

where β > 0 is arbitrary and may be considered as a variational parameter. Furthermore, 2 − b2E ) and the constants `E , BE and bE are defined below by (16), CE := C(0)(1 + BE (21) and (22), respectively. To get (12) we have made use p of the normalization (6). For
the simple (but not optimal) choice β = (2CE )−1 − E + E 2 + 2CE /π one obtains the leading asymptotic low- and high-energy behaviour 1 ln W (E) , =− 2 E→−∞ E 2C(0) 3d e1/2π W (E) = √ , lim d/2 E→∞ E 2πC(0) γ lim

(13) (14)

which coincides with the known asymptotics [PF, Thms. 5.29, 9.3] of N except for the value of the constant on the right-hand side of (14).

3. A Wegner Estimate for Continuum Schr¨odinger Operators In this section we prove Theorem 1 by tracing it back to a Wegner estimate that holds for a wider class of random fields than the ones considered so far. The price one has to pay for the increased generality is a couple of more technical notions. From now on the random field V will only be required to fulfil the measurability property (M) and the form-boundedness property (F). We follow the line of reasoning laid down in [CH]. The strategy there is based on a one-parameter decomposition of the random field from which we abstract Definition 2. A random field V : Ω × Rd → R admits a (U, λ, u, %)-decomposition if there exists a random field U : Ω × Rd → R , a random variable λ : Ω → R and a Borel-measurable function u : Rd → R such that (i) V (ω) = U (ω) + λ(ω) u for P -almost all ω, (ii) the conditional probability measure of λ relative to the sub-sigma-algebra generated by {U (x)}x∈Rd has a jointly measurable density % : Ω × R → R+ with respect to the Lebesgue measure on R . Here R+ is the set of the non-negative reals. As a variant of the Wegner estimate in [CH] we state int S J Λ be the interior of the closure of a finite Theorem 2. Let the cube Λ = j j=1 union of pairwise disjoint open cubes Λj . For a random field V , which has the properties (M) and (F), define HΛ,X (V ) as in (2). Assume that for all 1 ≤ j ≤ J the random field V admits a (Uj , λj , uj , %j )-decomposition subject to the following three conditions: there exist five strictly positive and finite constants ν, v, β, R, Z such that for all 1 ≤ j ≤ J: (i) (ii)

νχΛj (x) ≤ uj (x) and uj (x)χΛj (x) ≤ v for all x ∈ Rd ,   −βνξ −βvξ (ξ) max{e , e } ≤ R for P -almost all ω, ess sup %(ω) j Z

ξ∈R

(iii) Ω

  n o dP (ω) Tr χΛj exp − β HΛj ,N (Uj(ω) ) χΛj ≤ |Λj | Z.

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Then the finite-volume integrated density of states (3) obeys Z RZ β max{E1 ,E2 } (ω) (ω) e dP (ω) NΛ,X (E1 ) − NΛ,X (E2 ) ≤ |Λ| |E1 − E2 | ν Ω

(15)

for both boundary conditions X and all E1 , E2 ∈ R . In Condition (iii) of Theorem 2 the indicator function χΛj is understood as a multiplication operator and the random Schr¨odinger operator HΛj ,N (Uj ) is well-defined due to the boundedness of uj on Λj which in its turn follows from Condition (i) of Theorem 2. Before we prove Theorem 2 we show how it can be exploited to deduce Theorem 1. Proof (of Theorem 1). First we recall that the properties (M) and (F) are implied by the Gaussian property (G), cf. Remark 1. Then, given E ≥ max{E1 , E2 } we partition the cube Λ into J < ∞ disjoint cubes Λj with edges of length obeying `E /2 ≤ |Λj |1/d ≤ `E , where we have introduced the energy-dependent length `E := min {|E|−1/2 , |Γ |1/d } .

(16)

It is only because `E becomes small with increasing |E| that one obtains the low-energy behaviour (13) of the Wegner constant W (E). Next we pick a cube ΓE ⊆ Γ with volume |ΓE | = `dE . In what follows, we assume without loss of generality that the edges of ΓE are parallel to those of the Λj ’s and that the measure µ obeys (6). The main point is that V admits a (Uj , λj , uj , %j )-decomposition for all j as can be inferred from the definitions Z −1/2 λ(ω) := (C(0)) dµj (y) V (ω) (y) , (17) j d R Z dµj (y) C(x − y) , (18) uj (x) := (C(0))−1/2 Uj(ω) (x) %(ω) j (ξ)

:= V

(ω)

(x) −

Rd (ω) λj uj (x) ,

−1/2 −ξ 2 /2

:= (2π)

e

.

(19) (20)

The signed measures µj are defined by suitably translating √ µ such √ the signed measure that Condition (i) of Theorem 2 is fulfilled with v = C(0) BE and ν = C(0) bE , where Z −1 sup dµ(y) C(x − y) ≤ |µ|(Rd ) , (21) BE := (C(0)) x∈ΓE Rd Z dµ(y) C(x − y) ≥ γ . (22) bE := (C(0))−1 inf x∈ΓE

Rd

Note that the integral in (17) is well-defined for P -almost all ω because of C(0) < ∞. A decomposition similar to (17) – (20) is used for other purposes in [U, p. 185]. The random field Uj is non-homogeneous and Gaussian with zero mean and covariance function (23) Dj (x, y) := C(x − y) − uj (x)uj (y) , which obeys |Dj (x, y)| ≤ C(0)(1 − b2E ) for all x, y ∈ Λj . Moreover, Uj is stochastically independent of the standardized Gaussian random variable λj as is consistently taken into account by (20). Hence, Condition (ii) of Theorem 2 is obviously satisfied with R =

Density of States for Gaussian Random Potentials

139

2 (2π)−1/2 exp{β 2 C(0)BE /2} for any choice of β > 0. It remains to check Condition (iii). To do so we claim that Z   n o dP (ω) Tr χΛj exp −β HΛj ,N (Uj(ω) ) χΛj Ω Z    (ω) dP (ω) e−β Uj (x) . (24) ≤ Tr χΛj exp −β HΛj ,N (0) χΛj sup x∈Λj

Ω

If Uj is P -almost surely continuous, and hence bounded on Λj , this inequality follows from the Golden-Thompson inequality (see e.g. [RS, p. 320]) and Fubini’s theorem. To show (24) under our assumptions on the covariance function, which allow for noncontinuous realisations Uj(ω) with non-zero probability, we adapt the approximation result [Si1, Thm. B.10.1] to the present finite-volume situation with Neumann boundary conditions by using the appropriate Feynman-Kac formula [BR, Thm. 6.3.12] and the fact that P -almost surely Uj ∈ Lp (Λj ) , exp{−βUj } ∈ L1 (Λj ) for arbitrary finite p ≥ 1 and β > 0. Explicit computations show that the supremum in (24) does not exceed exp{β 2 C(0)
−1/2 d < ∞ so that the (1 − b2E )/2} and that there is a constant z ≤ 2`−1 E + (2πβ) trace on the right-hand side of (24), the free Neumann partition function, is bounded by |Λj | z. Putting Z := z exp{β 2 C(0)(1 − b2E )/2} gives Condition (iii) of Theorem 2. Hence the proof is complete, because Theorem 1 now follows from Theorem 2 and the Wegner constant may be taken as in (12). The rest of the paper is devoted to a proof of Theorem 2. Proof (of Theorem 2). Let E2 ≤ E1 and I := [E2 , E1 [. By definition of NΛ,X and by the spectral theorem one has Z Z     (ω) (ω) (ω) (ω) βE1 dP (ω) NΛ,X (E1 ) − NΛ,X (E2 ) ≤ e dP (ω) Tr FΛ,X (I) e−β HΛ,X (V ) . Ω

Ω

(25)

To bound the trace in (25) from above by the sum J X

  n o (ω) (ω) (ω) (26) Tr max{e−βνλj , e−βvλj } FΛ,X (I) χΛj exp − β HΛj ,N (Uj(ω) ) χΛj

j=1

we employ the operator inequality HΛ,X (V

(ω)

)≥

J X

  (ω) χΛj HΛj ,N (Uj(ω) ) + min{νλ(ω) j , vλj } χΛj

(27)

j=1

LJ 2 on L2 (Λ) = j=1 L (Λj ). It is valid for P -almost all ω and follows from DirichletNeumann bracketing, the fact that introducing “Neumann surfaces” lowers eigenvalues (see e.g. [RS, Ch. XIII.15, Prop. 4]), the (Uj , λj , uj , %j )-decomposition of V and Condition (i) of Theorem 2. Of course, in the case X = N Dirichlet-Neumann bracketing is not needed. Evaluating the trace in (25) in an orthonormal eigenbasis of HΛ,X (V (ω) ), using (27) and the Jensen-Peierls inequality [B] one obtains the bound (26). Let us now estimate the conditional expectation relative to the sub-sigma-algebra generated by {Uj (x)}x∈Rd of the j-th term in the sum (26). Since dominated convergence allows to interchange the trace with the emerging ξ-integration, this conditional expectation may be written as

140

W. Fischer, T. Hupfer, H. Leschke, P. M¨uller

Z dξ

Tr R

(ω) %˜(ω) j (ξ) Fj,ξ,X (I) χΛj

n exp

−β



o

HΛj ,N (Uj(ω) )

χ Λj

.

(28)

(ω) −βνξ −βvξ Here we have introduced the abbreviation %˜(ω) ,e } and j (ξ) := %j (ξ) max{e (ω) (ω) the spectral family Fj,ξ,X of the operator HΛ,X (Uj + ξuj ). Cyclic invariance of the trace and the H¨older inequality with the usual operator norm k . k and the trace norm bound (28) from above by

Z  n o 

dξ %˜(ω) (ξ) χ F (ω) (I) χ Tr χ exp − β HΛj ,N (U (ω) ) χ . (29) j j Λj j,ξ,X Λj Λj Λj

R

Thanks to Conditions (i) and (ii), one may now estimate the operator norm in (29) with the help of the spectral-averaging technique. To do so we apply Corollary 4.2 in [CH] since its validity extends from nonand remark that one may choose there g = %˜(ω) j negative bounded functions with compact support to non-negative bounded functions by a monotone-convergence argument. In this way we arrive at the ω- and j-independent upper bound |I| R/ν for the operator norm in (29). Hence, according to Condition (iii) the expectation of (28) with respect to dP is bounded by |Λj | |I|RZ/ν. By virtue of (25) and (26) the proof is complete. Acknowledgement. We are grateful to P. D. Hislop for stimulating discussions.

Note added in proof After completion of this work we have been able to extend our results to random Schr¨odinger operators with rather general magnetic fields (including constant ones). The details shall be published in an Addendum to the present paper.

References [AM] [BCH] [B] [BEE+]

[BR] [CL] [CH] [CHM] [vDK] [F]

Aizenman, M., Molchanov, S.: Localization at large disorder and at extreme energies: An elementary derivation. Commun. Math. Phys. 157, 245–278 (1993) Barbaroux, J.-M., Combes, J.M., Hislop, P.D.: Landau Hamiltonians with unbounded random potentials. Lett. Math. Phys. 40, 355–369 (1997) Berezin, F.A.: Convex operator functions. Math. USSR. Sbornik 17, 269–277 (1972). Russian original: Mat. Sbornik 88, 268–276 (1972) Bonch-Bruevich, V.L., Enderlein, R., Esser, B., Keiper, R., Mironov, A.G., Zvyagin, I.P.: Elektronentheorie ungeordneter Halbleiter. Berlin: VEB Deutscher Verlag der Wissenschaften, 1984. Russian original: Moscow: Nauka, 1981 Bratteli, O., Robinson, D.W.: Operator algebras and quantum statistical mechanics 2. 2nd ed. Berlin: Springer, 1997 Carmona, R., Lacroix, J.: Spectral theory of random Schr¨odinger operators. Boston: Birkh¨auser, 1990 Combes, J.-M., Hislop, P.D.: Localization for some continuous, random Hamiltonians in ddimensions. J. Funct. Anal. 124, 149–180 (1994) Combes, J.M., Hislop, P.D., Mourre, E.: Spectral averaging, perturbation of singular spectrum, and localization. Trans. Am. Math. Soc. 348, 4883–4894 (1996) von Dreifus, H., Klein, A.: A new proof of localization in the Anderson tight binding model. Commun. Math. Phys. 124, 285–299 (1989) Fernique, X.M.: Regularit´e des trajectoires des fonctions al´eatoires Gaussiennes. In: Hennequin, P.-L. (ed.) Ecole d’Et´e de Probabilit´es de Saint-Flour IV - 1974. Lecture Notes in Mathematics vol. 480, Berlin: Springer, 1975, pp. 1–96

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[FHLM] Fischer, W., Hupfer, T., Leschke, H., M¨uller, P.: Rigorous results on Schr¨odinger operators with certain Gaussian random potentials in multi-dimensional continuous space. In: Demuth, M., Schulze, B.-W. (eds.) Differential equations, asymptotic analysis, and mathematical physics, Berlin: Akademie Verlag, 1997, pp. 105–112 [FLM1] Fischer, W., Leschke, H., M¨uller, P.: Towards localisation by Gaussian random potentials in multidimensional continuous space. Lett. Math. Phys. 38, 343–348 (1996) [FLM2] Fischer, W., Leschke, H., M¨uller, P.: In preparation, to be submitted to J. Stat. Phys. [FS] Fr¨ohlich, J., Spencer, T.: Absence of diffusion in the Anderson tight binding model for large disorder or low energy. Commun. Math. Phys. 88, 151–184 (1983) [Ki1] Kirsch, W.: Random Schr¨odinger operators and the density of states. In: Albeverio, S., Combe, Ph., Sirugue-Collin, M. (eds.) Stochastic aspects of classical and quantum systems. Lecture Notes in Mathematics vol. 1109, Berlin: Springer, 1985, pp. 68–102 [Ki2] Kirsch, W.: Random Schr¨odinger operators, a course. In: Holden, H., Jensen, A. (eds.) Schr¨odinger operators. Lecture Notes in Physics vol. 345, Berlin: Springer, 1989, pp. 264–370 [Ki3] Kirsch, W.: Wegner estimates and Anderson localization for alloy-type potentials. Math. Z. 221, 507–512 (1996) [KM1] Kirsch, W., Martinelli, F.: On the ergodic properties of the spectrum of general random operators. J. Reine Angew. Math. 334, 141–156 (1982) [KM2] Kirsch, W., Martinelli, F.: On the density of states of Schr¨odinger operators with a random potential. J. Phys. A 15, 2139–2156 (1982) [Kl] Klopp, F.: Localization for some continuous random Schr¨odinger operators. Commun. Math. Phys. 167, 553–569 (1995) [Ko1] Kotani, S.: Lyapunov indices determine absolutely continuous spectra of stationary random onedimensional Schr¨odinger operators. In: Itˆo, K. (ed.) Stochastic analysis, Amsterdam: NorthHolland, 1984, pp. 225–247 [Ko2] Kotani, S.: Lyapunov exponents and spectra for one-dimensional random Schr¨odinger operators. In: Cohen, J.E., Kesten, H., Newman, C.M. (eds.) Random matrices and their applications. Contemporary Mathematics vol. 50, Providence, RI: American Mathematical Society, 1986, pp. 277–286 [KS] Kotani, S., Simon, B.: Localization in general one-dimensional random systems II. Continuum Schr¨odinger operators. Commun. Math. Phys. 112, 103–119 (1987) [LGP] Lifshits, I.M., Gredeskul, S.A., Pastur, L.A.: Introduction to the theory of disordered systems. New York: Wiley, 1988. Russian original: Moscow: Nauka, 1982 [MH] Martinelli, F., Holden, H.: On absence of diffusion near the bottom of the spectrum for a random Schr¨odinger operator on L2 (Rν ). Commun. Math. Phys. 93, 197–217 (1984) [PF] Pastur, L., Figotin, A.: Spectra of random and almost-periodic operators. Berlin: Springer, 1992 [RS] Reed, M., Simon, B.: Methods of modern mathematical physics IV: Analysis of operators. New York: Academic, 1978 [SE] Shklovskii, B.I., Efros, A.L.: Electronic properties of doped semiconductors. Berlin: Springer, 1984. Russian original: Moscow: Nauka, 1979 [Si1] Simon, B.: Schr¨odinger semigroups. Bull. Am. Math. Soc. (N.S.) 3, 447–526 (1982). Erratum: ibid. 11, 426 (1984) [Si2] Simon, B.: Spectral averaging and the Krein spectral shift. Preprint mp arc 96–492, appear in Proc. Am. Math. Soc. [Sp] Spencer, T.: Localization for random and quasi-periodic potentials. J. Stat. Phys. 51, 1009–1019 (1988) [U] Ueki, N.: On spectra of random Schr¨odinger operators with magnetic fields. Osaka J. Math. 31, 177–187 (1994) [W] Wegner, F.: Bounds on the density of states in disordered systems. Z. Phys. B 44, 9–15 (1981) Communicated by B. Simon

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Commun. Math. Phys. 190, 143 – 172 (1997)

Communications in

Mathematical Physics c Springer-Verlag 1997

Le Groupe Quantique Compact Libre U(n) Teodor Banica? Alg`ebres d’op´erateurs et repr´esentations - URA 747 du CNRS, Universit´e de Paris Jussieu, 4 place Jussieu, 75005 Paris, France Received: 1 March 1996 / Accepted: 4 April 1997

Abstract: The free analogues of U(n) in Woronowicz’ theory [Wo2] are the compact matrix quantum groups {Au (F ) | F ∈ GL(n, C)} introduced by Wang and Van Daele. We classify here their irreducible representations. Their fusion rules turn to be related to the combinatorics of Voiculescu’s circular variable. If F F ∈ RIn we find an embedding Au (F )red ,→ C(T) ∗red Ao (F ), where Ao (F ) is the deformation of SU(2) studied in [B2]. We use the representation theory and Powers’ method for showing that the reduced algebras Au (F )red are simple, with at most one trace. Introduction L’une des constructions de base de l’analyse harmonique est la dualit´e de Pontryagin: elle associe a` un groupe ab´elien le groupe ab´elien de ses caract`eres et permet d’´etudier cette correspondance auto-duale. Cette dualit´e a e´ t´e e´ tendue aux groupes non-commutatifs, mais l’objet dual (l’alg`ebre de convolution du groupe) n’est plus de mˆeme nature. Afin d’obtenir un cadre g´en´eralisant a` la fois les groupes et leur objets duaux, on est amen´e a` d´efinir de nouveaux objets dans la cat´egorie des alg`ebres de Hopf qu’on appelle des “groupes quantiques”. Un certain nombre de familles d’exemples ont e´ t´e e´ tudi´ees au niveau des alg`ebres d’op´erateurs. Ainsi, Woronowicz [Wo2] a d´efini en 1987 la classe des “groupes quantiques compacts matriciels”: un groupe quantique compact matriciel est une paire (G, u) form´ee d’une C∗ -alg`ebre unif`ere G et d’une matrice u ∈ Mn (G) telle que: (a) les coefficients {uij } de u engendrent une ∗-alg`ebre Gs dense dans PG. uik ⊗ ukj . (b) il existe un C∗ -morphisme δ : G → G ⊗min G qui envoie uij 7→ (c) il existe une application lin´eaire antimultiplicative κ : Gs → Gs telle que κ(κ(a∗ )∗ ) = a pour tout a ∈ Gs et telle que (Id ⊗ κ)(u) = u−1 . ? Present adress: Institut de Math´ ematiques de Luminy, case 930, F-13288 Marseille Cedex 9, France. E-mail: [email protected]

144

T. Banica

Cette d´efinition recouvre e´ galement le cas “quantique compact” (obtenu par des limites projectives) et le cas “quantique discret” (par dualit´e). Le cas “quantique localement compact” a e´ t´e trait´e dans un cadre g´en´eral par Baaj et Skandalis [BS]. Pour tout n ∈ N, la C∗ -alg`ebre universelle Au (In ) engendr´ee par les coefficients d’une matrice n × n unitaire, telle que sa transpos´ee soit aussi unitaire, est un groupe quantique compact matriciel [W1, W2, VDW]. Au (In ) est un analogue de U(n) dans la th´eorie de Woronowicz. Cette alg`ebre, ainsi que ses versions “d´eform´ees” {Au (F ) | F ∈ GL(n, C)} constitue l’objet d’´etude de ce papier. Je tiens a` exprimer ma profonde reconnaissance a` mon directeur de th`ese, G. Skandalis. Je voudrais aussi remercier E. Blanchard pour de nombreuses discussions sur les C∗ -alg`ebres de Hopf, ainsi que S.Z. Wang pour plusieurs commentaires utiles sur ce papier. ´ 1. D´efinitions et Enonc´ es des R´esultats Dans cette section on d´efinit les groupes quantiques compacts matriciels Au (F ) (d’une mani`ere l´eg´erement diff´erente que dans l’article de Wang et Van Daele [VDW]) et on e´ nonce les resultats principaux. La fin de cette section contient le plan de l’article, ainsi que des rappels et notations. 1) Il existe plusieures d´efinitions pour les morphismes entre les groupes quantiques compacts matriciels, auxquelles correspondent des diff´erentes notions d’isomorphisme. Sans rentrer dans les d´etails (dans ce papier on dira que (G, u) = (H, v) si G = H en tant que C∗ -alg`ebres et si u = v), rappelons la d´efinition [Wo2] de la similarit´e: Deux groupes quantiques compacts matriciels (G, u) et (H, v) avec u ∈ Mn (G), v ∈ Mm (H) sont dits similaires (on e´ crira G ∼sim H) si n = m et s’il existe une matrice Q ∈ GL(n, C) et un C∗ -isomorphisme f : G → H tel que (Id ⊗ f )(u) = QvQ−1 . 2) Soit (G, u) un groupe quantique compact matriciel. On appelle repr´esentation de (G, u) toute matrice inversible r ∈ Mk (G) telle que X (Id ⊗ δ)(r) = r12 r13 := eij ⊗ rik ⊗ rkj . La th´eorie de “Peter-Weyl” de Woronowicz [Wo2] montre que toute repr´esentation est e´ quivalente a` une repr´esentation unitaire. En particulier, v = Q−1 uQ est unitaire pour une certaine matrice Q ∈ GL(n, C). Quitte a` remplacer (G, u) par un groupe quantique compact matriciel similaire, on peut supposer que u est unitaire. 3) Soit (G, u) un groupe quantique compact matriciel avec u ∈ Mn (G) unitaire. Alors la repr´esentation u := (u∗ij ) est e´ quivalente a` une repr´esentation unitaire, donc il existe une matrice F ∈ GL(n, C) telle que F uF −1 soit unitaire. Il en r´esulte que G est un quotient de la C∗ -alg`ebre Au (F ), o`u: D´efinition 1. Pour tout n ∈ N et toute matrice F ∈ GL(n, C) on d´efinit la C∗ -alg`ebre Au (F ) avec g´en´erateurs {uij }1≤i,j≤n et les relations qui rendent unitaires les matrices u = (uij ) et F uF −1 . Remarquons que Au (F ) est bien d´efinie: si J est l’id´eal bilat`ere engendr´e dans l’alg`ebre libre sur 2n2 variables L := C < uij , u∗ij > par les relations qui rendent unitaires les matrices u := (uij ) et F uF −1 := F (u∗ij )F −1 , alors les images des g´en´erateurs uij , u∗ij dans le quotient L/J sont de norme ≤ 1 pour toute C∗ -norme sur L/J. Donc L/J admet une C∗ -alg`ebre enveloppante, qu’on peut noter Au (F ).

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(Au (F ), u) est un groupe quantique compact matriciel. En effet, on a v unitaire =⇒ v12 v13 unitaire, ce qui appliqu´e a` v = u et a` v = F uF −1 (avec la remarque que F u12 u13 F −1 = (F uF −1 )12 (F uF −1 )13 ) permet de d´efinir δ par propri´et´e universelle. Enfin, par [Wo4] l’existence de l’antipode κ est e´ quivalente au fait que u, u soient inversibles, ce qui est e´ vident dans le cas de Au (F ). Remarquons que pour tout groupe compact G ⊂ U(n), C(G) est un quotient de C(U(n)). Par ce qui pr´ec`ede, l’analogue de U(n) parmi les groupes quantiques compacts est la famille {Au (F ) | F ∈ GL(n, C)}. Remarque. Les relations qui d´efinissent Au (F ) sont: uu∗ = u∗ u = (F ∗ F )u(F ∗ F )−1 ut = ut (F ∗ F )u(F ∗ F )−1 = I. On en d´eduit des e´ galit´es entre les Au (F ): √ Au (F ) = Au ( F ∗ F ) = Au (λF ), ∀ F ∈ GL(n, C), λ ∈ C∗ . Il existent aussi d’autres similarit´es entre les Au (F ) - si V, W ∈ U(n) et F ∈ GL(n, C) 6). On pourrait donc utiliser d’autres alors Au (F ) ∼sim Au (V F W ) (voir la Proposition√ param`etres √ pour les Au (F ) - par exemple F ∗ F , ou F ∗ F , ou encore la liste des valeurs propres de F ∗ F etc., voir [W2, VDW]. Bien-sˆur, le choix du param`etre n’est pas un probl`eme s´erieux: on obtient toujours les mˆemes objets, au moins modulo la similarit´e. Le quotient de Au (F ) par les relations u = F uF −1 pourrait eˆ tre consid´er´e comme e´ tant une “version orthogonale de Au (F )”. Remarquons que la condition u = F uF −1 −1 implique u = F uF , donc u = (F F )u(F F )−1 . Il en r´esulte que si F F n’est pas un multiple scalaire de l’identit´e de Mn (C), alors u est r´eductible dans ce quotient. Remarquons e´ galement que F F = cIn avec c ∈ C implique F F = cIn , donc c = c. D´efinition 2. Pour tout n ∈ N et pour toute matrice F ∈ GL(n, C) telle que F F = cIn avec c ∈ R on note Ao (F ) le quotient de Au (F ) par les relations u = F uF −1 . Les repr´esentations irr´eductibles de Ao (F ) sont ind´ex´ees par N, et leur formules de fusion sont exactement les formules connues pour les repr´esentations de SU(2) ([B2], voir le Th´eor`eme 4 ci-dessous). Notations. N ∗ N est le coproduit dans la cat´egorie des mono¨ıdes de deux copies de N ayant α, β comme g´en´erateurs ; e est l’´el´ement neutre de N ∗ N ; − est l’involution antimultiplicative de N ∗ N d´efinie par e = e, α = β et β = α. Le r´esultat principal de ce papier est le suivant: Th´eor`eme 1. Soit n ∈ N et F ∈ GL(n, C). Alors: (i) Les repr´esentations irr´eductibles de (Au (F ), u) sont index´ees par N ∗ N, avec re = 1, rα = u, rβ = u. Pour tous les x, y ∈ N ∗ N on a les formules rx = rx et: X rab . r x ⊗ ry = {a,b,g∈N ∗N |x=ag,y=gb}

(ii) La sous-alg`ebre de Au (F ) engendr´ee par les caract`eres de toutes les repr´esentations est la ∗-alg`ebre libre sur le caract`ere χ(u) de la repr´esentation fondamentale. (iii) χ(u)/2 est une variable circulaire dans Au (F ), munie de la mesure de Haar. (iv) Si F F ∈ RIn alors Au (F )red se plonge dans C(T) ∗red Ao (F ) par uij 7→ zvij (o`u v est la repr´esentation fondamentale de Ao (F ) et z est le g´en´erateur canonique de C(T)).

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Le point (i) montre que la famille F = {Au (F ) | n ∈ N, F ∈ GL(n, C)} a la propri´et´e remarquable suivante: Si G, H ∈ F alors il existe une bijection ψ entre les classes d’´equivalence de repr´esentations de G et celles de H qui pr´eserve les sommes et les produits tensoriels et qui envoie l’ensemble des repr´esentations irr´eductibles de G sur l’ensemble des repr´esentations irr´eductibles de H, ainsi que la repr´esentation fondamentale de G sur celle de H. Un r´esultat important de ce type, pour la famille (`a un param`etre r´eel positif) de groupes quantiques compacts matriciels associ´es a` une alg`ebre de Lie classique, a e´ t´e d´emontr´e par Rosso [R1, R2]. Un autre r´esultat dans cette direction, mais cette fois-ci de “rigidit´e”, est celui de [B2] - la famille {Ao (F ) | n ∈ N, F ∈ GL(n, C), F F ∈ RIn } v´erifie la propri´et´e ci-dessus, mais de plus est maximale. Le r´esultat suivant est du mˆeme type: Th´eor`eme 2. Si les repr´esentations irr´eductibles d’un groupe quantique compact matriciel (G, u) sont index´ees par N ∗ N, avec re = 1, rα = u, rβ = u et rx ⊗ ry = P x=ag,y=gb rab , alors il existe un n ∈ N et une matrice F ∈ GL(n, C) tels que Gp ∼sim Au (F ). La th´eorie des repr´esentations de Au (F ) fait l’objet de la premi`ere partie (Sects. 2, 3, 4) de ce papier. Dans la deuxi`eme partie (Sects. 6, 7, 8) on utilise la th´eorie des repr´esentations pour r´esoudre certaines questions topologiques li´ees aux C∗ -alg`ebres Au (F ) et Au (F )red . Rappelons que pour un groupe quantique compact matriciel (G, u) la mesure de Haar h n’est pas forc´ement une trace, mais elle v´erifie la formule h(xy) = h(y(f1 ∗ x ∗ f1 )), ∀ x, y ∈ Gs o`u ∗ est la convolution au dessus de Gs et {fz }z∈C est une famille canonique de caract`eres de Gs (voir le Th´eor`eme 5.6 de [Wo2]). Th´eor`eme 3. Soit n ∈ N et F ∈ GL(n, C). Alors Au (F )red est simple. Soient s, t ∈ R et soit ψ un e´ tat de Au (F )red tel que ∀ x, y ∈ Au (F )s on ait ψ(xy) = ψ(y(fs ∗ x ∗ ft )). Alors ψ est la mesure de Haar de Au (F )red . En particulier, si F est un multiple scalaire d’une matrice unitaire, alors Au (F )red est simple a` trace unique ; sinon, Au (F )red est simple sans trace. Parmi les autres r´esultats sur Ao (F ) et Au (F ), citons: - un r´esultat de commutation dans Au (I2 ). - l’´egalit´e de facteurs Au (I2 )"red = W ∗ (F2 ). - la non-moyennabilit´e de Ao (F ) et Au (F ). - la non-nucl´earit´e de Ao (In )red et Au (In )red . - des remarques sur les mesures de Haar de Ao (F ) et Au (F ). Une partie de ces r´esultats sont des cas particuliers d’´enonc´es plus g´en´eraux sur les groupes quantiques compacts. Citons ici le r´esultat de simplicit´e (la Proposition 9), dont la d´emonstration pour G = C∗red (Fn ) contient une simplification par rapport aux d´emonstrations classiques [P, H, HS] de la simplicit´e de C∗red (Fn ).

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L’organisation de ce travail est la suivante: 2eme section: on rappelle les r´esultats de [B2] sur Ao (F ) et on donne une description (en termes de certaines partitions non-crois´ees) de l’espace des vecteurs fixes de la repr´esentation u⊗k de Ao (F ). 3eme section: on construit l’alg`ebre abstraite PA engendr´ee par des {rx | x ∈ N ∗ N} qui se multiplient par les formules rx ry = x=ag,y=gb rab et on montre que A ' C < X, X ∗ >. En utilisant cette alg`ebre, ainsi que la mˆeme methode que dans le cas “orthogonal” [B2], on voit que la d´emonstration du Th´eor`eme 1 est e´ quivalente au calcul des dimensions des commutants des repr´esentations de la forme u⊗m1 ⊗ u⊗n1 ⊗ u⊗m2 ⊗ u⊗n2 ⊗ ...

(?).

Ces dimensions sont des ∗-moments du caract`ere χ(u) de la repr´esentation fondamentale de Au (F ) par rapport a` la mesure de Haar, et en fait on voit que χ(u)/2 doit eˆ tre une variable circulaire. 4eme section: si F F ∈ RIn on combine les r´esultats sur Ao (F ) avec un r´esultat de probabilit´es non commutatives pour demontrer le Th´eor`eme 1. On utilise ensuite des r´esultats de reconstruction de [Wo3] pour trouver un syst`eme de g´en´erateurs des espaces des vecteurs fixes des repr´esentations de la forme (?). Les dimensions de ces espaces sont exactement les ∗-moments de χ(u), et en utilisant cette remarque on passe du cas F F ∈ RIn au cas g´en´eral F ∈ GL(n, C). 5eme section: on decrit les Ao (F ) et Au (F ) pour F ∈ GL(2, C). Le point (iv) du Th´eor`eme 1 permet d’identifier Au (I2 )red comme une sous-C∗ -alg`ebre de C(T) ∗red C(SU(2)), et on en d´eduit deux plongements (de C∗ -alg`ebres de Hopf) de C(SO(3)) dans Au (I2 ), ainsi que l’´egalit´e de facteurs Au (I2 )"red = W ∗ (F2 ). 6eme section: on utilise les caract`eres {fz } de [Wo2] pour “perturber” la repr´esentation adjointe d’un groupe quantique compact matriciel. 7eme section: on g´en´eralise aux groupes quantiques compacts la “Propri´et´e de Powers” de de la Harpe [H], ainsi que la d´emonstration de simplicit´e de [HS]. L’id´ee est de remplacer les automorphismes int´erieurs x 7→ ug xu∗g du cas discret par les applications b compl`etement positives de la forme x 7→ ad(r)(x), avec r ∈ A. 8eme section: Au (F )red n’a pas la propri´et´e de Powers, mais en utilisant les calculs de la 6eme et 7eme section on arrive a` d´emontrer le Th´eor`eme 3. Rappels et Notations: A) matrices: on note {e1 , ..., en } la base canonique de Cn , et eij le syst`eme d’unit´es matricielles de Mn (C), qui v´erifie Si A est une ∗-alg`eP bre et u ∈ Mn (A), P eij : e∗j 7→tei . P P eij ⊗ uij , u = eij ⊗ uji , u∗ = eij ⊗ u∗ji . u = eij ⊗ uij , on note u = B) repr´esentations: pour tout groupe quantique compact G on note Rep(G) l’enb ⊂ Rep(G) semble des classes d’´equivalence de repr´esentations unitaires de G et G l’ensembleP des classes de repr´esentations Punitaires irr´eductibles. Si u = eij ⊗ uij ∈ Mn (G) etP v = eij ⊗ vij ∈ Mm (G) sont des repr´esentations, on note u ⊗ v la matrice u13 v23 := eij ⊗ ekl ⊗ uij vkl , et u + v la matrice diag(u, v). Alors u ⊗ v et u + v sont des repr´esentations. L’application (u, v) 7→ u ⊗ v induit une structure de mono¨ıde sur Rep(G).PDe mˆeme pour l’application (u, v) 7→ u + v Le caract`ere de u est χ(u) := uii ∈ G. On a χ(u + v) = χ(u) + χ(v) et χ(u ⊗ v) = χ(u)χ(v). (voir [Wo2, Wo3]). C) th´eorie de “Peter-Weyl” de Woronowicz: on note Gcentral l’espace lin´eaire (donc ∗-alg`ebre) engendr´e dans la ∗-alg`ebre “des coefficients” Gs par les caract`eres de toutes les repr´esentations. On utilisera souvent, sans r´ef´erence, le r´esultat fondamental suivant (Th. 5.8 de [Wo2]):

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b est une La mesure de Haar est une trace sur Gcentral . L’ensemble {χ(u) | u ∈ G} base de Gcentral , orthonorm´ee par rapport au produit scalaire associ´e a` la mesure de Haar. D) version pleine et r´eduite: la version r´eduite d’un groupe quantique compact matriciel (G, u) est Gred = G/{x | h(xx∗ ) = 0} (h e´ tant la mesure de Haar de G). La version pleine est Gp = C ∗ (Gs ) (la C∗ -alg`ebre enveloppante de Gs ). Alors Gp et Gred sont des groupes quantiques compacts matriciels (cf. [Wo2, BS]). G est dit moyennable si la projection Gp → Gred est un isomorphisme. Il est dit plein (resp. r´eduit) si la projection Gp → G (resp. G → Gred ) est un isomorphisme. On a Gs = Hs ⇐⇒ Gred = Hred ⇐⇒ Gp = Hp . Notons aussi que Ao (F ) et Au (F ) sont pleins. E) libert´e: si (A, φ) est une ∗-alg`ebre unif`ere munie d’une forme lin´eaire unitale, une famille de sous-alg`ebres 1 ∈ Ai ⊂ A (i ∈ I) est dite libre si aj ∈ Aj ∩ ker(φ) avec ij 6= ij+1 , 1 ≤ j ≤ n − 1 implique a1 a2 ...an ∈ ker(φ) (voir [VDN]). Deux e´ l´ements a, b ∈ A sont dits ∗-libres si les deux ∗-alg`ebres unif`eres qu’ils engendrent dans A sont libres. Exemple fondamental: soient (A, φ) et (B, ψ) deux C∗ -alg`ebres unif`eres munies d’´etats et A ∗ B le produit libre (= coproduit dans la cat´egorie des C∗ -alg`ebres unif`eres) de A et B. Si on note φ ∗ ψ le produit libre de φ et ψ et πφ∗ψ la repr´esentation GNS associ´ee, alors πφ∗ψ (A) et πφ∗ψ (B) sont libres dans πφ∗ψ (A ∗ B) (voir [A, VDN]). F) produits libres: si (G, u) et (H, v) sont deux groupes quantiques compacts alors (G ∗ H, diag(u, v)) est un groupe quantique compact matriciel plein, et sa mesure de Haar est le produit libre h ∗ k des mesures de Haar h de G et k de H (voir [W2]). Le produit libre r´eduit πh∗k (G ∗ H) sera not´e G ∗red H ; c’est un groupe quantique compact matriciel r´eduit. Notons que h, k e´ tant fid`eles sur Gred , Hred respectivement, on a des plongements canoniques de Gred et Hred dans G ∗red H. G) ∗-distribution: pour tout e´ l´ement a ∈ (M, φ) d’une ∗-alg`ebre munie d’une forme lin´eaire, sa ∗-distribution est la fonctionnelle sur C < X, X ∗ > donn´ee par P 7→ φ(P (a, a∗ )), i.e. la compos´ee: X7→a

φ

C < X, X ∗ > −→ M −→ C. Les ∗-moments de a sont les valeurs de µa sur les monomes non-commutatifs en X, X ∗ , i.e. sur le mono¨ıde engendr´e dans (C < X, X ∗ >, ·) par X et X ∗ . Si (M, φ) est une C∗ -alg`ebre munie d’un e´ tat fid`ele et a = a∗ , alors la ∗-distribution µa peut eˆ tre vue (par restriction a` C[X], ensuite en compl`etant) comme une mesure de probabilit´e sur le spectre de a. H) √ variables circulaires: la loi semicirculaire (centr´ee) est la mesure γ0,1 = 2/π 1 − t2 dt sur [−1, 1]. Tout hermitien ayant cette distribution est dit semicirculaire.√Un quart-circulaire est un e´ l´ement positif ayant comme distribution la mesure 4/π 1 − t2 dt sur [0, 1]. Un Haar-unitaire est un unitaire u tel que µu (X k ) = 0 pour tout k 6= 0. Une variable g est dite circulaire si 2−1/2 (g + g ∗ ) et −i2−1/2 (g − g ∗ ) sont semicirculaires et libres (voir [VDN]). 2. Compl´ements sur Ao (F )  On voit facilement a` partir de la d´efinition de Ao (F ) que Ao mˆeme plus, qu’on a une e´ galit´e (modulo la similarit´e)

0 1 −1 0

 = C(SU(2)), et

{Ao (F ) | F ∈ GL(2, C), F F ∈ RI2 } = {Sµ U(2) | µ ∈ [−1, 1] − {0} }

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o`u Sµ U(2) sont les d´eformations de S1 U(2) := C(SU(2)) d´efinies par Woronowicz dans [Wo1, Wo2] (voir 5eme section). Si F ∈ GL(n, C) avec n arbitraire on a le r´esultat suivant. Th´eor`eme 4. [B2] Soit n ∈ N et F ∈ GL(n, C) avec F F ∈ RIn . Alors les repr´esentations irr´eductibles de Ao (F ) sont auto-adjointes et index´ees par N, avec r0 = 1, r1 = u et rk ⊗ rs = r|k−s| + r|k−s|+2 + ... + rk+s−2 + rk+s (i.e. les mˆemes formules que pour les repr´esentations de SU(2)). Rappelons bri` Pevement la d´emonstration: la condition F F ∈ RIn montre que la projection sur C Fji ei ⊗ ej , qui entrelace u⊗2 , d´efinit pour tout k une repr´esentation de l’alg`ebre de Jones Aβ,k dans M or(u⊗k , u⊗k ). En utilisant les r´esultats de [Wo3] on voit que cette repr´esentation est surjective, et l’in´egalit´e dim(M or(u⊗k , u⊗k )) ≤ dim(Aβ,k ) ≤ Ck ainsi obtenue permet de construire (par r´ecurrence sur k) des repr´esentations irr´eductibles rk de Ao (F ) qui v´erifient les mˆemes formules de multiplication que celles de SU(2). Un corollaire de la d´emonstration (voir Remarque (ii) de [B2]) est l’´egalit´e dim(M or(u⊗k , u⊗k )) = dim(Aβ,k ) = Ck =

(2k)! . k!(k + 1)!

Notons h la mesure de Haar de Ao (F ). On a (cf. Rappel C): dim(M or(u⊗k , u⊗k )) = h(χ(u)⊗2k ) = dim(M or(1, u⊗2k )). Les nombres de Catalan Ck ont une autre propri´et´e remarquable - ce sont les moments de la loi semicirculaire de Wigner et Voiculescu. En effet, on peut calculer les moments de γ0,1 a l’aide de 3.3 et 3.4. de [VDN], de la formule des r´esidus et celle du binˆome: Z (2k)! 2k −1 −1 γ0,1 (X ) = (2k + 1) (2πi) . (z −1 + z/4)2k+1 = 4−k k!(k + 1)! T En combinant toutes ces e´ galit´es, on en d´eduit que:  Proposition 1. χ(u)/2 ∈ (Ao (F ), h) est une variable semicirculaire.   0 1 Remarque. Si F = , alors la caract`ere de la repr´esentation fondamentale −1 0   a b de Ao (F ) = C(SU(2)) est χ(u) = 2Re(a). Le fait que Re(a) soit u = −b a semicirculaire par rapport a` la mesure de Haar de SU(2) peut eˆ tre vu g´eometriquement, en identifiant SU(2) avec la sph`ere S3 , et sa mesure de Haar avec la mesure uniforme sur cette sph`ere. Corollaire 1. (G. Skandalis) Si F ∈ GL(2, C) alors Ao (F ) est moyennable. Si F ∈ GL(n, C) et n > 2 alors Ao (F ) n’est pas moyennable. D´emonstration. Le support de la loi semicirculaire e´ tant [−1, 1] et h e´ tant fid`ele sur Ao (F )red , on obtient que Sp(χ(u)/2) = [−1, 1] dans Ao (F )red (voir Rappels). Donc si n ≥ 3, alors n − χ(u) est inversible dans Ao (F )red . Mais la co¨unit´e de Ao (F ) est un ∗-morphisme unital qui envoie n − χ(u) sur 0, donc Ao (F ) 6= Ao (F )red . Enfin, si F ∈ GL(2, C), alors Ao (F ) est similaire a` un certain Sµ U(2) (voir 5eme section), qui est moyennable, cf. [N, Bl]. 

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Remarque. Une partie des r´esultats classiques sur la moyennabilit´e a e´ t´e e´ tendue aux groupes quantiques localements compacts dans [BS, Bl] (voir aussi la Proposition 10 ci-dessous). La d´emonstration ci-dessus de la non-moyennabilit´e de Ao (F ) peut eˆ tre e´ tendue a` des groupes quantiques compacts matriciels quelconques - on d´emontre par la mˆeme m´ethode que (G, u) avec u ∈ Mn (G) est moyennable si et seulement si le support de la loi de Re(χ(u)) par rapport a` la mesure de Haar contient n. Rappelons que pour toute repr´esentation r ∈ B(Hr )⊗G d’un groupe quantique compact G, les vecteurs fixes de r sont les x ∈ Hr tels que r(x ⊗ 1) = (x ⊗ 1). Ces vecteurs forment un sous-espace vectoriel de Hr qui s’identifie naturellement avec M or(1, r). On va donner maintenant une description des vecteurs fixes de la repr´esentation u⊗k de Ao (F ). Les r´esultats qui suivent seront utilis´es dans la 4eme section, pour d´emontrer le Th´eor`eme 1 pour les matrices F qui ne v´erifient pas (!) la condition F F ∈ RIn . Ainsi, le lecteur int´eress´e uniquement par les alg`ebres Au (F ) avec F F ∈ RIn (e.g. par Au (In )) pourra passer directement a` la section suivante. Lemme 1 ([B2]). Soit n ∈ N et F ∈ GL(n, C) avec F F ∈ RIn . Soit H = Cn , avec la base orthonormale {ei }. P (i) L’op´erateur E ∈ B(C, H ⊗2 ), x 7→ x ei ⊗ F ei est dans M or(1, u⊗2 ). (ii) On a (E ∗ ⊗ IdH )(IdH ⊗ E) ∈ CIdH . (iiii) Pour r, s ∈ N, on d´efinit les ensembles M or(r, s) ⊂ B(H ⊗r , H ⊗s ) des combinaisons lin´eaires de produits (composables) d’applications de la forme IdH ⊗k ou IdH ⊗k ⊗ E ⊗ IdH ⊗p ou IdH ⊗k ⊗ E ∗ ⊗ IdH ⊗p . Alors la W ∗ -cat´egorie concr`ete mono¨ıdale des repr´esentations de Ao (F ) est la compl´etion (dans le sens de [Wo3]) de la W ∗ -cat´egorie concr`ete mono¨ıdale W (F ) := {N, +, {H ⊗r }r∈N , {M or(r, s)}r,s∈N }



En gardant toutes les notations, on a: Lemme 2. (i) On note I(p) = IdH ⊗p et V (p, q) = I(p) ⊗ E ⊗ I(q). Alors tout morphisme de W (F ) est une combinaison lin´eaire d’applications de la forme I(.) ou de la forme V (., .) ◦ ... ◦ V (., .) ◦ V (., .)∗ ◦ ... ◦ V (., .)∗ . (ii) Pour tout k ≥ 0, les applications de la forme M ⊗I(1)⊗N avec M ∈ M or(0, 2x), N ∈ M or(0, 2y) et x + y = k engendrent M or(1, 2k + 1). (i) Pour tout k ≥ 0, les applications de la forme (I(1) ⊗ M ⊗ I(1) ⊗ N ) ◦ E avec M ∈ M or(0, 2x), N ∈ M or(0, 2y) et x + y = k engendrent M or(0, 2k + 2). D´emonstration. Le point (i), i.e. le fait qu’on “peut passer les ∗ a` droite”, r´esulte du point (ii) du Lemme 1. On d´emontre (ii) par r´ecurrence sur k. Pour k = 0 le point (i) montre que M or(1, 1) = {CI(1)}. Soit donc k ≥ 1 et A ∈ M or(1, 2k + 1). Par le point (i), A est une combinaison lin´eaire d’applications de la forme V (k1 , s1 ) ◦ ... ◦ V (km , sm ), avec k1 + s1 = 2k − 1 et T := V (k2 , s2 ) ◦ ... ◦ V (km , sm ) dans M or(1, 2k − 1). Par l’hypoth`ese de r´ecurrence T est une combinaison lin´eaire d’applications de la forme B = (M ⊗ I(1) ⊗ N ) pour certains M ∈ M or(0, 2x) et N ∈ M or(0, 2y), avec x + y = k − 1, donc: – soit k1 ≥ 2x+1, et alors V (k1 , s1 )◦B = M ⊗I(1)⊗((I(k1 −2x−1)⊗E ⊗I(s1 ))◦N ). – soit k1 ≤ 2x, et alors V (k1 , s1 ) ◦ B = ((I(k1 ) ⊗ E ⊗ I(2x − k1 )) ◦ M ) ⊗ I(1) ⊗ N .

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On d´emontre maintenant (iii): soit A ∈ M or(0, 2k + 2). Par le point (i), A est une combinaison lin´eaire d’applications de la forme B = V (k1 , s1 ) ◦ ... ◦ V (km , sm ). Remarquons que V (km , sm ) = V (0, 0), donc on peut consid´erer le plus petit p tel que kp = 0. Alors B = (I(1) ⊗ G) ◦ (E ⊗ I(sp )) ◦ K, avec G = V (k1 − 1, s1 ) ◦ ... ◦ V (kp−1 − 1, sp−1 ) et K = V (kp+1 , sp+1 ) ◦ ... ◦ V (km , sm ). Il en r´esulte que: B = (I(1) ⊗ G) ◦ (I(2) ⊗ K) ◦ E = (I(1) ⊗ (G ◦ (I(1) ⊗ K))) ◦ E. Mais G◦(I(1)⊗K) ∈ M or(1, 2k +1) est, par le point (ii), de la forme M ⊗I(1)⊗N , pour certains M ∈ M or(0, 2x) et N ∈ M or(0, 2y), et (iii) en r´esulte.  Proposition 2. On d´efinit pour tout k ∈ N la partie W2k (F ) ⊂ M or(0, 2k) par W0 (F ) = 1 et (par r´ecurrence) par: W2k+2 (F ) = ∪k=x+y {(I(1) ⊗ M ⊗ I(1) ⊗ N ) ◦ E | M ∈ W2x (F ), N ∈ W2y (F )} Alors W2k (F ) est une base de M or(0, 2k), ∀ k ≥ 0. D´emonstration. Les nombres Dk := Card(W2k (F )) v´erifient D0 = D1 = 1 et X Dx Dy . Dk+1 = k=x+y

Ce P sontk donc les nombres de Catalan (classique, consid´erer le carr´e de la s´erie Dk z ...). Il en r´esulte que Card(W2k (F )) = dim(M or(0, 2k)). D’autre part, le point (iii) du Lemme 2 montre que W2k (F ) engendre M or(0, 2k).  ` ` Remarque. Soit P = P1 ... Pk une partition non-crois´ee en parties a` deux e´ l´ements de {1, ..., 2k}, i.e. une partition telle que si on note Pm = {im , jm } avec im < jm pour chaque 1 ≤ m ≤ k, alors: ∀ m 6= n, im < in < jm =⇒ jn < jm . On associe a` P le vecteur suivant de (Cn )⊗2k : X Fsj1 si1 ...Fsjk sik es1 ⊗ ... ⊗ es2k . v(P ) = 1≤s1 ...s2k ≤k

On peut montrer par r´ecurrence sur k, en utilisant la Proposition 2, que l’ensemble de ces v(P ) coinc¨ıde avec l’ensemble {X(1) | X ∈ W2k (F )}, donc est une base de l’espace des vecteurs fixes de la repr´esentation u⊗2k de Ao (F ). Ceci permet en principe de calculer la mesure de Haar de Ao (F ) - pour toute representation r d’un groupe quantique compact matriciel on a (Id ⊗ h)(r) = projecteur sur l’espace des vecteurs fixes de r (voir [Wo2]). 3. Reconstruction de Au (F )central On construit et on e´ tudie dans cette section l’alg`eP bre A engendr´ee par des {rx | x ∈ N ∗ N} qui se multiplient par les formules rx ry = x=ag,y=gb rab . Notations. N ∗ N est le produit libre (i.e. coproduit dans la cat´egorie des mono¨ıdes) de deux copies de N, not´ees multiplicativement {e, α, α2 , ...} et {e, β, β 2 , ...}. On consid`ere l’ensemble A de fonctions N ∗ N → C avec support fini. On va identifier N ∗ N ⊂ A,

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comme masses de Dirac. Avec l’addition et la multiplication des fonctions A est l’alg`ebre des polynˆomes non commutatifs en deux variables, c’est a` dire on a un isomorphisme: (C < X, X ∗ >, +, ·) ' (A, +, ·) par X 7→ α , X ∗ 7→ β. On d´efinit sur N ∗ N une involution antimultiplicative par e = e, α = β et β = α. Cette involution s’´etend par antilin´earit´e en une involution de A, not´ee encore −. On note En ⊂ A l’espace lin´eaire engendr´e par les e´ l´ements de N ∗ N de longeur ≤ n. Soit l2 (N ∗ N) la compl´etion de A pour la norme 2. Notons que les e´ l´ements de N ∗ N (vus comme e´ l´ements de A, donc de l2 (N ∗ N), voir les identifications ci-dessus) forment une base orthonorm´ee de l2 (N ∗ N). Soit τ0 : x 7→< x(e), e > l’´etat canonique sur B(l2 (N ∗ N)). On d´efinit S, T ∈ 2 B(l (N ∗ N)) par lin´earit´e et S(x) = αx, T (x) = βx pour tout x ∈ N ∗ N. Rappel. A tout espace de Hilbert H on peut associer [VDN] l’espace de Fock plein F (H): si {hi }i∈I une base orthonormale de H, alors {hi1 ⊗ hi2 ⊗ ... ⊗ hik | k ≥ 0} est une base orthonormale de F (H) (on fait la convention que pour k = 0, le vecteur correspondant est celui du vide). Notons {fi }i∈I les g´en´erateurs du mono¨ıde libre N∗I . Alors la base orthonormale canonique de l2 (N∗I ) est la famille {δm }, avec m ∈ N∗I , donc de la forme m = fi1 ...fik . On a donc une isometrie: l2 (N∗I ) ' F (H) par δfi1 ...fik 7→ hi1 ⊗ hi2 ⊗ ... ⊗ hik . L’op´erateur de cr´eation l(hi ) correspond ainsi a` λN ∗I (fi ), o`u λN ∗I est la repr´esentation r´eguli`ere gauche (par isometries !) du mono¨ıde N∗I . Pour I = {1, 2} on a S = λN ∗N (α) et T = λN ∗N (β), donc en identifiant l2 (N ∗ N) = F (H), avec H de base orthonormale {h1 , h2 }, on a: S = l(h1 ), T = l(h2 ). Lemme 3. On d´efinit l’application : N ∗ N × N ∗ N → A par X ab. x y = x=ag,y=gb

(i) s’´etend par lin´earit´e en une multiplication associative sur A. (ii) Si P : (A, +, ·) → (B(l2 (N ∗ N)), +, ◦) est le ∗-morphisme d´efini par α 7→ S + T ∗ et J : A → A est l’application f 7→ P (f )e, alors (J − Id)En ⊂ En−1 pour tout n. (iii) J est un isomorphisme de ∗-alg`ebres (A, +, ·) ' (A, +, ). D´emonstration. (i) Notons que est bien d´efinie, car la somme est finie. Montrons qu’elle est associative. Soient x, y, z ∈ N ∗ N. Alors (x y) z = X {g,a,b∈N ∗N |x=ag,y=gb}

ab z =

X {g,h,a,b,c,d∈N ∗N |x=ag,y=gb,ab=ch,z=hd}

cd.

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Remarquons que pour a, b, c, h ∈ N ∗ N l’´egalit´e ab = ch est e´ quivalente a` une d´ecomposition de la forme b = uh, c = au avec u ∈ N ∗ N, ou de la forme a = cv, h = vb pour un certain v ∈ N ∗ N. Donc (x y) z =

X

X

aud +

{g,h,a,d,u∈N ∗N,x=ag,y=guh,z=hd}

cd

{g,b,c,d,v∈N ∗N,x=cvg,y=gb,z=bvd}

Un calcul similaire montre que x (y z) est donn´e par la mˆeme formule, donc est associative. (ii) Soit f ∈ A. On peut v´erifier facilement que P (α)f = (S + T ∗ )f = α f . Donc J(αg) = P (α)J(g) = α J(g) = J(α) J(g) pour toute g ∈ A, et par le mˆeme argument on obtient J(βg) = J(β) J(g), pour toute g ∈ A. (A, +, ·) e´ tant engendr´ee par α et β, il en r´esulte que J est un morphisme d’alg`ebres: J : (A, +, ·) → (A, +, ). On d´emontre par r´ecurrence sur n ≥ 1 que (J − Id)En ⊂ En−1 . Pour n = 1 on a J(α) = α, J(β) = β et J(e) = e, et comme E1 est engendr´ee par e, α, β on a J = Id sur E1 . Supposons que c’est vrai pour n et soit k ∈ En+1 . On ecrit k = αf + βg + h avec f, g, h ∈ En (a noter que cette d´ecomposition n’est pas unique). Alors: (J − Id)k = J(αf + βg + h) − (αf + βg + h) = = [(S + T ∗ )J(f ) + (S ∗ + T )J(g) + J(h)] − [Sf + T g + h] = = S(J(f ) − f ) + T (J(g) − g) + T ∗ J(f ) + S ∗ J(g) + (J(h) − h). En appliquant l’hypoth`ese de r´ecurrence a` f, g, h on trouve que En contient tous les termes de la somme, donc contient (J − Id)k et on a fini. Enfin, pour d´emontrer (iii) il nous reste a` voir que J pr´eserve l’involution ∗ et qu’il est une bijectif. On a J∗ = ∗J sur les g´en´erateurs {e, α, β} de A, donc J pr´eserve l’involution. Aussi par (ii), la restriction de J −Id a` En est un endomorphisme nilpotent, donc J est bijectif.  Lemme 4. Soit (G, u) un groupe quantique compact matriciel et soit 9G : (A, +, ) → G l’unique morphisme d´efini par α 7→ χ(u), β 7→ χ(u) (cf. point (iii) du Lemme 3). Soit n ≥ 1 et supposons que 9G (x) est le caract`ere d’une repr´esentation irr´eductible rx de G, pour tout x ∈ N ∗ N de longeur ≤ n. Alors 9G (x) est le caract`ere d’une repr´esentation (non nulle) de G, pour tout x ∈ N ∗ N de longeur n + 1. D´emonstration. Pour n = 1 c’est clair. Supposons n ≥ 2 et soit x ∈ N ∗ N de longeur n + 1. Si x contient une puissance ≥ 2 de α ou de β, par exemple si x = zα2 y, alors on pose rx := rzα ⊗ rαy et on a fini. Supposons donc que x est un produit des α alternant avec des β. On peut supposer que x commence avec α. Alors x = αβαy, avec y ∈ N ∗ N de longeur n − 2. Notons que l’´egalit´e 9G (z) = 9G (z)∗ est vraie sur les g´en´erateurs {e, α, β} de N∗N, donc elle est vraie pour tout z ∈ N ∗ N. Si <, > est le produit scalaire sur G associ´e a` la mesure de Haar, alors (voir Rappels) < χ(rα ⊗ rβαy ), χ(rαy ) >=< χ(rβαy ), χ(rβ ⊗ rαy ) >= < χ(rβαy ), 9G (β αy) >=< χ(rβαy ), 9G (βαy) + 9G (y) >= < χ(rβαy ), χ(rβαy ) + χ(ry ) > ≥ 1.

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Comme rαy est irr´eductible, il en r´esulte qu’elle est une sous-repr´esentation de rα ⊗ rβαy . Donc χ(rα ⊗ rβαy ) − χ(rαy ) = 9G (α βαy − αy) = 9G (x) est le caract´ere d’une repr´esentation de G.  Soit (G, u) un groupe quantique compact matriciel et notons fx = 9G (x) pour tout x ∈ N ∗ N (notations du LemmeP4). Alors la famille {fx | x ∈ N ∗ N} v´erifie fe = 1, fα = χ(u), fβ = χ(u) et fx fy = x=ag,y=gb fab . On veut montrer: – le Th´eor`eme 1 (i): pour G = Au (F ), les fx sont exactement les caract`eres des repr´esentations irr´eductibles de Au (F ). – le Th´eor`eme 2: si les fx sont les caract`eres des repr´esentations irr´eductibles de G, alors Gp ∼sim Au (F ) pour une certaine matrice F . Il est commode de consid´erer, pour tout n ∈ N et F ∈ GL(n, C) et pour tout groupe quantique compact matriciel (G, u) avec u, F uF −1 unitaires, le diagramme suivant: N∗N

N∗N

∩

∩

% 9u

(A, +, )

G −→

τ↓

(?)

C

C −→

C < X, X ∗ >= (A, +, ·)

J

−→

P ↓ B(l2 (N ∗ N))

τ

0 −→

Au (F )

9

Id

↓8 Gp ↓h C

o`u: – τ0 , J, P ont d´ej`a e´ t´e d´efinies et τ (f ) = f (e) = coefficient de e dans f . La commutation du carr´e est e´ vidente. – Gp est la version pleine de G (voir Rappels) et 8 est la surjection canonique, d´efinie par la propri´et´e universelle de Au (F ). – 9G (resp. 9u ) est l’unique ∗-morphisme (voir Lemme 3 (iii)) qui envoie α sur le caract`ere de la repr´esentation fondamentale de Gp (resp. de Au (F )). La commutation du triangle est e´ vidente. – h est la mesure de Haar de G. Notons que les inclusions (voir Notations du d´ebut) N ∗ N ⊂ A ne commutent pas avec J. Proposition 3. Soit (G, u) un groupe quantique compact matriciel avec u et F uF −1 unitaires. Alors les assertions suivantes sont e´ quivalentes: 1) Les repr´esentations irr´P eductibles de G sont index´ees par N ∗ N, avec re = 1, rα = u, rβ = u et rx ⊗ ry = x=ag,y=gb rab . 2) Le diagramme (?) commute. 3) χ(u)/2 est une variable circulaire dans (G, h). 4) Tous les ∗-moments de χ(u)/2 ∈ (G, h) sont plus petits que les ∗-moments d’une variable circulaire, i.e. µχ(u)/2 (M ) ≤ µc (M ) pour tout monome non commutatif M (o`u µc est la ∗-distribution de la variable circulaire). De plus, si ces conditions sont v´erifi´ees, alors:

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5) 8 : Au (F ) → Gp est un isomorphisme. Note. On verra dans la section suivante que la condition 5) est en fait equivalente a` 1)-4). D´emonstration. (1 ⇒ 2) Il est clair que N ∗ N est un syst`eme orthonormal dans ((A, +, ), τ ). Si 1) est vraie, alors 9G (N ∗ N) = {χ(rx ) | x ∈ N ∗ N} est un syst`eme orthonormal dans (Gs , h), d’o`u la commutativit´e de (?). (2 ⇒ 3) Remarquons que (?) commute ⇐⇒ h9G J = τ0 P . En identifiant (C < X, X ∗ >, +, ·) = (A, +, ·) on a (voir Rappel G): – la ∗-distribution de χ(u) ∈ (G, h) est la fonctionnelle h9G J. – la ∗-distribution de S + T ∗ ∈ (B(l2 (N ∗ N)), τ0 ) est la fonctionnelle τ0 P . D’autre part, les identifications du d´ebut de cette section montrent que (S + T ∗ )/2 a la mˆeme ∗-distribution que la variable (l(h1 ) + l(h2 )∗ )/2 sur l’espace de Fock plein, qui est l’exemple standard de variable circulaire (voir 1ere section de [V]). (3 ⇒ 4) est trivial. Montrons (4 ⇒ 1). Toujours en identifiant (C < X, X ∗ > , +, ·) = (A, +, ·), les monˆomes non-commutatifs en X, X ∗ correspondent aux e´ l´ements de N∗N ⊂ A. Donc l’hypoth`ese sur les ∗-moments de χ(u)/2 se traduit tout simplement par: (i) h9G J ≤ τ J sur N ∗ N. On d´emontre par r´ecurrence sur n ≥ 0 que pour tout z ∈ N ∗ N de longeur n, 9G (z) est le caract`ere d’une repr´esentation irr´eductible rz de G. Pour n = 0 on a 9G (e) = 1, qui est le caract`ere de la repr´esentation triviale. Supposons donc c’est vrai pour n ≥ 0 et soit x ∈ N ∗ N de longeur n + 1. Le point (ii) du Lemme 3 implique J(x) = x + z avec z ∈ En . Notons AN ⊂ A l’ensemble des fonctions f telles que f (x) ∈ N pour tout x ∈ N ∗ N. Alors J(α), J(β) ∈ AN , donc par multiplicativit´e, J(N ∗ N) ⊂ AN . En particulier, J(x) ∈ AN . Il en r´esulte qu’il existe des entiers positifs m(z) tels que: X J(x) = x + m(z)z. z∈N ∗N,l(z)≤n

Calculons a` l’aide de cette formule h9G J(xx) et τ J(xx): a) Il est clair que si a, b ∈ N ∗ N, alors τ (a b) = δa,b . On obtient donc: X X X m(z)z) (x + m(z)z)) = 1 + m(z)2 . τ J(xx) = τ ((x + b) Par l’hypoth`ese de r´ecurrence et par le Lemme 4, 9G (x) est P le caract`ere d’une repr´esentation rx de G. Donc 9G J(x) est le caract`ere de rx + z∈N ∗N,l(z)≤n m(z)rz . Par les formules d’orthogonalit´e des caract`eres on a: X h9G J(xx) ≥ h(χ(rx )χ(rx )∗ ) + m(z)2 . En utilisant (i), a) et b) on conclut que rx est irr´eductible, ce qui termine la r´ecurrence. Le fait que les rx ainsi construites soient distinctes r´esulte de (i). En effet, N ∗ N e´ tant une base orthonorm´ee de ((A, +, ), τ ), on obtient que pour tous les x, y ∈ N ∗ N, x 6= y on a τ (x y) = 0, donc que: h(χ(rx ⊗ ry )) = h9G J(xy) ≤ τ J(xy) = τ (x y) = 0.

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(1 ⇒ 5) On montre par r´ecurrence sur n ≥ 0 que pour tout x ∈ N ∗ N de longeur n, 9u (x) est le caract`ere d’une repr´esentation irr´eductible px de Au (F ). Pour n = 0 c’est trivial - 9u (e) est le caract`ere de la repr´esentation triviale. Supposons-le pour n et soit x ∈ N ∗ N de longeur n + 1. Par le Lemme 4, 9u (x) est le caract`ere d’une repr´esentation px . Comme 8 envoie px 7→ rx , qui est irr´eductible, px est aussi irr´eductible, donc on a fini. La surjection 8 envoie donc les (classes de) repr´esentations irr´eductibles de Au (F ) sur les (classes de) repr´esentations irr´eductibles de Gp . On conclut en utilisant un argument standard (Th. 5.7 de [Wo2]): soit {ci } une base de Au (F )s form´ee des coefficients des repr´esentations irr´eductibles ; alors {8(ci )} est une base de Gs form´ee des coefficients des repr´esentations irr´eductibles. Il en r´esulte que 8 : Au (F ) → Gp est bijective.  4. Repr´esentations de Au (F ) Cette section est consacr´ee a` la d´emonstration des Th´eor`emes 1 et 2. On verra que la Proposition 3 implique facilement le Th´eor`eme 2, ainsi que (modulo un r´esultat de probabilit´es libres) le Th´eor`eme 1 pour des matrices v´erifiant F F ∈ RIn . Le Th´eor`eme 1 sera ensuite d´emontr´e pour des matrices F ∈ GL(n, C) arbitraires, en utilisant les r´esultats obtenus pour F = In . D´emonstration du Th´eor`eme 2. On se donne un groupe quantique compact matriciel (G, u) avec u ∈ Mn (G) tel que ses repr´esentations irr´eductibles sont index´ees par N ∗ N, avec re = 1, rα = u, rβ = u et telles que ∀ x, y, X rab . r x ⊗ ry = x=ag,y=gb

On peut supposer (modulo la similarit´e) que u est unitaire. Il existe donc F ∈ GL(n, C) telle que F uF −1 soit unitaire, et le th´eor`eme r´esulte alors de l’implication (1 ⇒ 5) de la Proposition 3.  D´emonstration du Th´eor`eme 1 dans le cas F F ∈ RIn . Soit n ∈ N et F ∈ GL(n) telle que F F ∈ RIn . On note u la repr´esentation fondamentale de Au (F ), v la repr´esentation fondamentale de Ao (F ) et z ∈ C(T) la fonction x 7→ x. Soit G la sous-C∗ -alg`ebre de C(T) ∗red Ao (F ) engendr´ee par les coefficients de la matrice zv = (zvij )ij . Alors: • χ(v)/2 est semicirculaire par rapport a` la mesure de Haar de Ao (F ) (cf. Prop. 1). • z est un Haar-unitaire dans C(T) muni de sa mesure de Haar (´evident). • χ(v)/2 et z sont ∗-libres dans C(T) ∗red Ao (F ) par rapport a sa mesure de Haar (cf. Rappel F ). Ces trois conditions impliquent que le produit zχ(v)/2 est circulaire dans C(T) ∗red Ao (F ) (ceci est une version connue du th´eor`eme de Voiculescu [V] de d´ecomposition polaire des variables circulaires, voir par exemple [B1] ou [NS]). Mais zχ(v) = χ(zv) est le caract`ere de la repr´esentation fondamentale de (G, zv), donc on peut appliquer la Proposition 3: (3 ⇒ 5) implique Au (F ) = Gp , donc que Au (F )red = G, d’o`u le point (iv) du Th´eor`eme 1.  (3 ⇒ 1) classifie les repr´esentations de Gp = Au (F ), d’o`u (i,ii,iii).

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Remarque. On aurait pu d´emontrer le Th´eor`eme 1 dans le cas F F ∈ RIn de la mani`ere suivante. On consid`ere le groupe quantique compact G ⊂ C(T)∗red Ao (F ) engendr´e par les coefficients de zv, v e´ tant la repr´esentation fondamentale de Ao (F ). En combinant la th´eorie des repr´esentations de Ao (F ) de [B2] avec la th´eorie des repr´esentations des produits libres de [W2], on peut classifier les repr´esentations de G. On applique ensuite le Th´eor`eme 2, pour voir que Au (F )red = G. Notons que cette d´emonstration ne fournit aucun outil pour aborder le cas g´en´eral. D´emonstration du Th´eor`eme 1 dans le cas g´en´eral. Soit n ∈ N et F ∈ GL(n, C) quelconque. Notons u la repr´esentation fondamentale de Au (F ). On doit estimer les ∗-moments du caract`ere χ(u), i.e. les nombres: h(χ(u)a1 χ(u)∗b1 χ(u)a2 ...) = dim(M or(1, u⊗a1 ⊗ u⊗b1 ⊗ u⊗a2 ⊗ ...)). En utilisant le fait que N ∗ N est un mono¨ıde libre: – on associe a` tout espace de Hilbert H une famille d’espaces de Hilbert {Hx }x∈N ∗N de la mani`ere suivante: He = C, Hα = H, Hβ = H (l’espace conjugu´e de H), et Hab = Ha ⊗ Hb , ∀ a, b ∈ N ∗ N. – on d´efinit une famille {ux }x∈N ∗N de repr´esentations unitaires de Au (F ) de la mani`ere n suivante: ue = 1, uα = u, uβ = F uF −1 (agissant sur C ) et uab = ua ⊗ ub , ∀ a, b ∈ N ∗ N. Notons que ux ∈ B(Cnx ) ⊗ Au (F ) pour tout x. Les ∗-moments de χ(u) sont ainsi les nombres {dim(M or(1, uk )) | k ∈ N ∗ N}. On va les estimer en appliquant le Th. 1.3. de [Wo3]: Lemme 5. Soit n ∈ N et F ∈ GL(n, C). Soit H = Cn , avec la base orthonormale {ei }. P On consid`ere les applications lin´eaires E1 : He → Hαβ d´efinie par 1 7→ F (ei ) ⊗ ei P −1 et E2 : He → Hβα d´efinie par 1 7→ ei ⊗ F (ei ). (i) E1 ∈ M or(1, uαβ ) et E2 ∈ M or(1, uβα ). (ii) (E2∗ ⊗ IdHβ )(IdHβ ⊗ E1 ) ∈ CIdHβ et (E1∗ ⊗ IdHα )(IdHα ⊗ E2 ) ∈ CIdHα . (iii) Pour r, s ∈ N ∗ N, on d´efinit les ensembles M or(r, s) ⊂ B(Hr , Hs ) des combinaisons lin´eaires de produits (composables) d’applications de la forme IdHk ou IdHk ⊗E1 ⊗IdHp ou IdHk ⊗E2 ⊗IdHp ou IdHk ⊗E1∗ ⊗IdHp ou IdHk ⊗E2∗ ⊗IdHp . Alors la W ∗ -cat´egorie concr`ete mono¨ıdale des repr´esentations de Au (F ) est la compl´etion (dans le sens de [Wo3]) de la W ∗ -cat´egorie concr`ete mono¨ıdale, Z(F ) := {N ∗ N, ·, {Hr }r∈N ∗N , {M or(r, s)}r,s∈N ∗N }. D´emonstration. (i) Si matrice unitaire a` coefficients dans une Pw ∈ Mn (B)n est une ∗-alg`ebre B et si ζ = ei ⊗ ei ∈ C ⊗ Cn alors X X ∗ (w13 w23 )(ζ ⊗ 1) = ei ⊗ ek ⊗ wia wka = ei ⊗ ek ⊗ δik 1 = (ζ ⊗ 1). En particulier: – pour B := Au (F ) et w := u cela montre que (1 ⊗ F )ζ = fixe de (1 ⊗ F )(u ⊗ u)(1 ⊗ F )−1 = u ⊗ (F uF −1 ).

P

ei ⊗ F ei est un vecteur

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T. Banica −1

– pour B := Au (F ) et w := F uF −1 cela montre que (1 ⊗ F )ζ = un vecteur fixe de (1 ⊗ F )−1 (w ⊗ w)(1 ⊗ F ) = (F uF −1 ) ⊗ u.

P

ei ⊗ F

−1

ei est

Par d´efinition des ux on a uαβ = (idCn ⊗ φ ⊗ idAu (F ) )(u ⊗ (F uF −1 )) et uβα = n (φ ⊗ idCn ⊗ idAu (F ) )((F uF −1 ) ⊗ u), o`u φ : B(Cn ) → B(C ) est l’isomorphisme canonique, et (i) en r´esulte. (ii) est un calcul facile. Pour (iii), notons que Z(F ) est par construction une W ∗ -cat´egorie mono¨ıdale concr`ete. Soit j : Hα → Hβ l’application antilin´eaire d´efinie par ei → F (ei ). Alors (avec les notations de [Wo3], p. 39) on a tj = E1 et (tj )∗ = tj −1 = E2 , donc tj ∈ M or(e, αβ) et tj ∈ M or(1, βα), donc α = β dans Z(F ). Par [Wo3], la paire universelle Z(F )-admissible est un groupe quantique compact (G, v). Le point (i) montre que (Au (F ), u) est une paire Z(F )-admissible, donc qu’on a un C∗ -morphisme f : G → Au (F ) tel que (id ⊗ f )(v) = u. D’autre part la propri´et´e universelle de Au (F ) permet de construire un C∗ -morphisme g : Au (F ) → G tel que  (id ⊗ g)(u) = v. Il en r´esulte que (G, v) = (Au (F ), u). La d´emonstration du Lemme suivant est similaire a` celle du Lemme 2. Lemme 6. (i) On note I(p) = IdHp et Vi (p, q) = I(p) ⊗ Ei ⊗ I(q) pour i = 1, 2. Alors tout morphisme de Z(F ) est une combinaison lin´eaire d’applications de la forme I(.) ou de la forme V· (., .) ◦ ... ◦ V· (., .) ◦ V· (., .)∗ ◦ ... ◦ V· (., .)∗ . (ii) L’ensemble des applications de la forme M ⊗ I(α) ⊗ N avec M ∈ M or(e, x), N ∈ M or(e, y) et xαy = k engendre M or(α, k). L’ensemble des applications de la forme M ⊗ I(β) ⊗ N avec M ∈ M or(e, x), N ∈ M or(e, y) et xβy = k engendre M or(β, k). (iii) L’ensemble des applications de la forme (I(α) ⊗ M ⊗ I(β) ⊗ N ) ◦ E1 avec M ∈ M or(e, x), N ∈ M or(e, y) et αxβy = k, ou des applications de la forme (I(β) ⊗ M ⊗ I(α) ⊗ N ) ◦ E2 avec M ∈ M or(e, x), N ∈ M or(e, y) et βxαy = k engendre M or(e, k).  Proposition 4. On d´efinit pour tout k ∈ N ∗ N la partie Zk (F ) ⊂ M or(e, k) par Ze (F ) = 1, Zα (F ) = Zβ (F ) = ∅ et (par r´ecurrence) par Zk (F ) = ∪k=αxβy {(I(α) ⊗ M ⊗ I(β) ⊗ N ) ◦ E1 | M ∈ Zx (F ), N ∈ Zy (F )} si k commence par α et Zk (F ) = ∪k=βxαy {(I(β) ⊗ M ⊗ I(α) ⊗ N ) ◦ E2 | M ∈ Zx (F ), N ∈ Zy (F )} si k commence par β. Alors: (i) Zk (F ) engendre M or(e, k). (ii) Zk (In ) est une base de M or(e, k) (pour F = In ). D´emonstration. (i) r´esulte du point (iii) du Lemme 6. Rappelons que Hk est un certain produit tensoriel entre H et son conjugu´e. Soit ψ : H → H l’isometrie donn´ee par ei 7→ ei . En identifiant H avec H a` l’aide de ψ, on obtient une isometrie ψk : Hk → H ⊗l(k) , o`u l(k) est la longeur du mot k. En regardant les d´efinitions, il est clair que ψk envoie l’ensemble {X(1) | X ∈ Zk (In )} sur une partie de l’ensemble {X(1) | X ∈ Wl(k) (In )}. En utilisant la Proposition 2, l’ensemble {X(1) | X ∈ Wl(k) (In )} est form´e de vecteurs lin´eairement ind´ependents de H ⊗l(k) . Ceci implique que {X(1) | X ∈ Zk (In )}

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est form´e de vecteurs lin´eairement ind´ependents, donc que Zk (In ) est une base de M or(e, k).  Fin de la d´emonstration du Th´eor`eme 1. Rappelons que n ∈ N et F ∈ GL(n, C) Ck , o`u les nomsont arbitraires. Par construction de Zk (F ) on a Card(Zk (F )) = P bres {Cx }x∈N ∗N sont d´efinis par Ce = 1, Cα = Cβ = 0 et Ck = k=αxβy Cx Cy + P k=βxαy Cx Cy pour k ∈ N ∗ N. Notons u(F ) la repr´esentation fondamentale de Au (F ). En utilisant les points (iii) du Lemme 5 et (i) de la Proposition 4 on a pour tout k ∈ N ∗ N: dim(M or(1, u(F )k )) = dim(M or(e, k)) ≤ Card(Zk (F )) = Ck . La Proposition 4 (ii) dit que pour F = In on a e´ galit´e. Donc: dim(M or(1, u(F )k )) ≤ dim(M or(1, u(In )k )). Mais les nombres dim(M or(1, u(F )k )) sont les ∗-moments du caract`ere de u(F ) et les nombres dim(M or(1, u(In )k )) sont les ∗-moments du caract`ere de u(In ), donc les ∗-moments d’une variable circulaire (par le point (iii) du Th´eor`eme 1 appliqu´e a F = In , cas d´ej`a r´esolu). On conclut a` l’aide de la Proposition 3.  Remarque. Il est clair maintenant que pour toute matrice F , {X(1) | X ∈ Zk (F )} est une base de l’ensemble des vecteurs fixes de la repr´esentation u(F )k de Au (F ). Si F F ∈ RIn , l’application ψk qu’on a construit dans la d´emonstration de la Proposition 4 permet d’identifier ces vecteurs comme une partie de la base des vecteurs fixes de la repr´esentation u⊗l(k) de Ao (F ) de la fin de la Sect. 2 (en fait, cette identification est celle qui correspond a` la fl`eche canonique Au (F ) → Ao (F )). Il est facile a` d´et´erminer les partitions non-crois´ees en parties a` deux e´ l´ements de {1...l(k)} qui correspondent aux vecteurs fixes de la repr´esentation uk de Au (F ). En fait, on peut obtenir une base de l’ensemble des vecteurs fixes de la repr´esentation uk de Au (F ) en ecrivant `k= `x1 ...xl(k) avec xi ∈ {α, β} et en associant a` toute partition noncrois´ee P = P1 ... Pl(k)/2 de {1, ..., k} avec des parties de la forme Ps = {is , js } telles que (xis , xjs ) soit e´ gale a` (α, β) ou a` (β, α) pour tout s le vecteur suivant: X

v=

1≤s1 ...sl(k) ≤l(k)

Fsj1 si1 ...Fsjl(k)/2 sil(k)/2 es1 ⊗ ... ⊗ esl(k) .

5. Le cas n = 2 Rappelons que pour µ ∈ [−1, 1] − {0} le groupe quantique compact matriciel Sµ U(2) est d´efini avec g´en´erateurs α, γ et relations α∗ α + γ ∗ γ = 1, αα∗ + µ2 γγ ∗ = 1, γγ ∗ = γ ∗ γ, µγα = αγ, µγ ∗ α = αγ ∗ . On a S1 U(2) = C(SU(2)) (voir [Wo1] et les formules (1.33) de [Wo2]).  Proposition 5. Ao

0 1 −µ−1 0

 = Sµ U(2).

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   u11 u12 0 1 la repr´esentation fondamentale de Ao . u21 u22 −µ−1 0  0 1 sont uu∗ = u∗ u = 1 et: Les relations qui d´efinissent Ao −µ−1 0      ∗  u11 u12 0 1 0 −µ u11 u∗12 = . −1 ∗ ∗ u21 u22 0 u21 u22 −µ 1 0     −µu∗21 u∗22 u11 u12 = . Si α := u11 et γ := u21 , alors En multipliant, −1 ∗ ∗ −µ u12 u11  u21 u22   0 1 α −µγ ∗ et les relations qui d´efinissent Ao sont celles donn´ees u= ∗ −1 γ α 0 −µ ∗ ∗ par uu = u u = 1, i.e.:   ∗     ∗   γ∗ γ∗ α −µγ ∗ α 1 0 α α −µγ ∗ = = . ∗ ∗ γ α −µγ α −µγ α γ α 0 1 D´emonstration. Notons

En calculant on obtient les relations qui d´efinissent Sµ U(2).



Proposition 6. Pour tous les n ∈ N, F ∈ GL(n, C), λ ∈ C∗ et V, W ∈ U(n) on a les similarit´es suivantes: (i) Ao (F ) ∼sim Ao (λV F V t ) (si F v´erifie F F ∈ RIn ). (ii) Au (F ) ∼sim Au (λV F W ). D´emonstration. (i) Notons u (resp. v) la repr´esentation fondamentale de Ao (F ) (resp. Ao (λV F V t )). Alors v = (λV F V t )v(λV F V t )−1 est unitaire =⇒ v = V F V t vV F −1 V ∗ est unitaire . Il en r´esulte que V ∗ vV = F V ∗ vV F −1 est unitaire, donc on peut d´efinir f : Ao (F ) → Ao (λV F V t ) par la propri´et´e universelle et par (Id ⊗ f )(u) = V ∗ vV . Par les mˆemes arguments, il existe g : Ao (λV F V t ) → Ao (F ) avec (Id ⊗ g)(v) = V uV ∗ . Donc f et g sont des bijections inverses, donc f est un isomorphisme, donc une similarit´e. (ii) Notons u (resp. v) la repr´esentation fondamentale de Au (F ) (resp. Au (λV F W )). Alors v et (λV F W )v(λV F W )−1 sont unitaires, donc W vW t et F W vW ∗ F −1 sont unitaires, donc W vW t et F W vW t F −1 sont unitaires. On peut donc d´efinir f : Au (F ) → Au (λV F W ) par la propri´et´e universelle et par (Id ⊗ f )(u) = W vW t . Par les mˆemes arguments, il existe g : Au (λV F W ) → Au (F ) avec (Id ⊗ g)(v) = W t uW . Alors f et g sont des bijections inverses.  Proposition 7. Pour tout µ ∈ [−1, 1] − {0} soit Gµ la C∗ -alg`ebre engendr´ee dans C(T) ∗red Sµ U(2) par les coefficients de la matrice zuµ , uµ e´ tant la repr´esentation fondamentale de Sµ U(2). Alors on a (modulo la similarit´e) les e´ galit´es suivantes: (i) {Ao (F ) | F ∈ GL(2, C) } = {Sµ U(2) | µ ∈ [−1, 1] − {0} }, (ii) {Au (F )red | F ∈ GL(2, C) } = {Gµ | µ ∈ [−1, 1] − {0} }. D´emonstration. (i) Il suffit de montrer que pour chaque F ∈ GL(2, C) telle que F F ∈ R, t il  existe V ∈ GL(2, C) multiple scalaire d’une matrice unitaire tel queλV F V = 0 1 x y pour un certain µ (cf. Propositions 5 et 6). Soit donc F = . Si −µ−1 0 z t   α 1 , qui est un multiple x 6= 0, soit α une solution de α2 x + α(y + z) + t = 0 et V = −1 α t er 2 1 coefficient α x + α(y 0, scalaire d’une matrice unitaire ; alors V F V a le   + z) + t =   yt yz 0 y . donc on peut supposer x = 0. Dans le cas F = on a F F = tz zy + tt z t

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  0 1 Comme F F ∈ R, t = 0 et yz = zy, donc F = y avec k = z/y ∈ R∗ . Si | k |≥ 1 k 0   0 −1 on a fini. Sinon, on pose V = . 1 0 (ii) Il suffit de montrer que pour chaque F ∈ GL(2, C) il existent V, W ∈ U(2) telles (par que V F W V F W ∈ R (cf. point (i) et Proposition 6). En effet, on peut supposer   x 0 d´ecomposition polaire) que F > 0; en la diagonalisant, on peut supposer F = , 0 y   0 −1 avec x, y > 0, et dans ce cas on pose V = et W = 1.  1 0   a b , (z) et u les repr´esentations fondamentales de C(SU(2)), Lemme 7. Notons −b a C(T) et Au (I2 ) respectivement. Alors il existe un plongement:   za zb Au (I2 )red ,→ C(T) ∗red C(SU(2)) , u 7→ . −zb za   0 1 D´emonstration. On a Ao = S1 U(2) = C(SU(2)) (Prop. 5), Au (I2 ) = −1 0   √ 0 1 Au (car Au (F ) = Au ( F ∗ F ) pour toute F ), et par le Th´eor`eme 1 on a un −1 0     0 1 0 1 plongement Au ,→ C(T) ∗red Ao .  −1 0 −1 0 Th´eor`eme 5. Les coefficients de la repr´esentation rβα = u⊗u−1 de Au (I2 ) commutent entre eux et engendrent une C∗ -alg`ebre e´ gale a` C(SO(3)). De mˆeme pour les coefficients de rαβ = u ⊗ u − 1. D´emonstration. En utilisant le Lemme 7 on voit que la repr´esentation u⊗u de Au (I2 )red est la representation 

       za zb a b za zb a b ⊗ = ⊗ . −zb za −b a −zb za −b a

de C(T) ∗red C(SU(2)). Il en r´esulte que la repr´esentation u ⊗ u − 1 de Au (I2 )red correspond a` la repr´esentation de dimension 3 de SU(2), i.e. a` la repr´esentation fondamentale de SO(3). Donc les coefficients de la repr´esentation rβα = u ⊗ u − 1 de Au (I2 )red commutent entre eux et engendrent une C∗ -alg`ebre commutative e´ gale a` C(SO(3)). On conclut en remarquant que C(SO(3)) est la C∗ -alg`ebre enveloppante de C(SO(3))s .  On va montrer maintenant que l’alg`ebre de von Neumann Au (I2 )"red (le bicommutant de l’image de Au (I2 ) par sa repr´esentation r´eguli`ere gauche sur l2 (Au (I2 ), h), h e´ tant la mesure de Haar) est isomorphe au facteur W ∗ (F2 ) associ´e au groupe libre a deux g´en´erateurs F2 . Rappelons que si (M, φ) est une ∗-alg`ebre munie d’une forme lin´eaire unitale et si A ⊂ M est une ∗-alg`ebre unif`ere, alors un e´ l´ement x ∈ M est dit ∗-libre par rapport a` A si la ∗-alg`ebre engendr´ee par x dans M est libre par rapport a` A. On va utiliser le lemme technique suivant. Lemme 8. Soit (M, φ) une ∗-alg`ebre munie d’une trace, 1 ∈ A ⊂ M une sous-∗alg`ebre, d ∈ A un unitaire tel que φ(d) = φ(d∗ ) = 0 et u ∈ M un Haar-unitaire ∗-libre par rapport a` A. Alors ud est un Haar-unitaire ∗-libre par rapport a` A.

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D´emonstration. ud est clairement un Haar-unitaire. Pour montrer que ud est ∗-libre par rapport a` A il suffit de v´erifier que si ai ∈ Z∗ et fi ∈ A ∩ ker(φ) (1 ≤ i ≤ n), alors P := (ud)a1 f1 (ud)a2 f2 ... est dans ker(φ). P est un produit de termes de la forme u ou u∗ alternant avec des termes de la forme ∗ d, d , fi , dfi , fi d∗ ou dfi d∗ . Remarquons que φ(d) = φ(d∗ ) = φ(fi ) = φ(dfi d∗ ) = 0 et que les termes de la forme dfi ou fi d∗ apparaissent dans P entre u et u ou entre u∗ et u∗ . On ecrit chaque dfi sous la forme [φ(dfi )1] + [dfi − φ(dfi )1] et chaque fi d∗ sous la forme [φ(fi d∗ )1] + [fi d∗ − φ(fi d∗ )]. En developpant P , on obtient une combinaison lin´eaire de termes, chaqun e´ tant un produit d’´el´ements de la forme uk avec k ∈ Z∗ alternant avec des e´ l´ements de A ∩ ker(φ). Il en r´esulte que φ(P ) = 0.  Th´eor`eme 6. Au (I2 )"red = W ∗ (F2 ). D´emonstration. Par le Lemme 7 on a un plongement: Au (I2 )"red ,→ L∞ (T) ∗ L∞ (SU(2)) , u 7→



 za zb . −zb za

Notons d = sgn ◦ (a + a), i.e. la compos´ee de a + a : SU(2) → R avec la fonction signe sgn : R → {−1, 0, 1}. Alors d ∈ L∞ (SU(2)) est un unitaire tel que d2 = 1. La partie polaire de za + za est zd, donc zd ∈ Au (I2 )"red . Il en r´esulte que dz ∗ ∈ Au (I2 )"red , et en multipliant a gauche par dz ∗ les g´en´erateurs za, zb, za, zb de Au (I2 )"red , on obtient que Au (I2 )"red est engendr´ee par zd, da, db, da, db. En utilisant le Lemme 8 on obtient que W ∗ (da, db, da, db) et W ∗ (zd) sont des sous-alg`ebres ab´eliennes diffuses libres qui engendrent Au (I2 )"red , ce qui implique  Au (I2 )"red = W ∗ (F2 ) (voir Th. 2.6.2 de [VDN]). Remarque. Soit Unnc la C∗ -alg`ebre universelle engendr´ee par les coefficients d’une nc matrice de taille n unitaire u. En combinant la formule Un,red ⊗ Mn = Mn ∗red C(T) de McClanahan [MC1] avec la formule Mn ∗ W ∗ (Fs ) = W ∗ (Fn2 s ) ⊗ Mn de Dykema nc," [D] on obtient U2,red = W ∗ (F4 ). 6. Remarques sur la repr´esentation adjointe La repr´esentation adjointe d’un groupe discret 0 est un outil important pour traiter plusieurs probl`emes li´es aux C∗ -alg`ebres C∗ (0) et C∗red (0): moyennabilit´e de 0, nucl´earit´e de C∗red (0), simplicit´e de C∗red (0) etc. On va se poser les mˆemes questions pour les C∗ -alg`ebres Au (F ) et Au (F )red , qui, du point de vue de la dualit´e de Pontryagin, sont les “C∗ (Fn ) et C∗red (Fn ) quantiques”. Si G est un groupe quantique compact matriciel tel que sa mesure de Haar soit une trace, on peut d´efinir sur Gp les repr´esentations r´eguli`eres gauche λ et droite ρ par λ(x)(y) = xy et ρ(x)(y) = yκ(x), et la repr´esentation adjointe comme e´ tant la compos´ee: δ

λ⊗ρ

Gp −→Gp ⊗max Gp −→B(l2 (Gred )). Le cas g´en´eral est plus subtil, et on va utiliser le th´eor`eme suivant de Woronowicz (voir aussi les Rappels 5.1 de [BS]). Th´eor`eme 7. ([Wo2]) A tout groupe quantique compact matriciel G on peut associer une (unique) famille de caract`eres (fz )z∈C de Gs qui v´erifient les formules suivantes:

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163

h(ab) = h(b(f1 ∗ a ∗ f1 )), pour tous les a ∈ Gs et b ∈ G. κ2 (a) = f−1 ∗ a ∗ f1 , pour tout a ∈ Gs . f0 = e (la co¨unit´e de Gs ) et fz+t = fz ∗ ft , pour tous les z, t ∈ C. fz ∗ κ(a) = κ(a ∗ f−z ) et κ(a) ∗ fz = κ(f−z ∗ a) pour tous les a ∈ Gs et z ∈ C.

b et si F est l’unique De plus, les fz sont d´efinis de la mani`ere suivante: si u ∈ G ∗ 2 matrice positive qui entrelace u et (I ⊗ κ)(u ) = (I ⊗ κ )(u), normalis´ee telle que T r(F ) = T r(F −1 ) (cf. Th. 5.4 de [Wo2]), alors: (Id ⊗ fz )(u) = F z . Notations. L(Gred ) est l’alg`ebre des op´erateurs born´es Gred → Gred . L’application b → Gs sera not´ee χ (voir Rappels B et C). caract`ere G Corollaire 2. Soit G un groupe quantique compact matriciel et a, b, c, d ∈ C. L’application x 7→ fa ∗ x ∗ fb est un automorphisme de Gs . L’application λa,b : Gs → L(Gred ) donn´ee par λa,b (x)(y) = (fa ∗ x ∗ fb )y est un morphisme de C-alg`ebres unif`eres. (ii) L’application x 7→ fc ∗ κ(x) ∗ fd est un antiautomorphisme de Gs . L’application ρc,d : Gs → L(Gred ) donn´ee par ρc,d (x)(y) = y(fc ∗ κ(x) ∗ fd ) est un morphisme de C-alg`ebres unif`eres.

(i)

δ

λa,b ⊗ρc,d

(iii) L’application Gs −→Gs ⊗ Gs −→ L(Gred ) est un morphisme de C-alg`ebres unif`eres.  L’application d´efinie au point (iii) permet d’associer a` tous les a, b, c, d ∈ C une b → L(Gred ). L’int´erˆet du Lemme suivant apparaˆıtra dans les application ad : G d´emonstrations de simplicit´e, quand on va utiliser plusieures applications ad de ce type. Lemme 9. Soit G un groupe quantique compact matriciel et a, b, c, d ∈ R. Soit: b → L(Gred ). ad = (λa,b ⊗ ρc,d ) ◦ δ ◦ χ : G b et soit F la matrice d´efinie dans le Th´eor`eme 7. Alors: Soit u ∈ G P (i) ad(u)(z) = i,k (F b uF a )ik z(F −c u∗ F −d )ki

P (ii) Si a + c = b + d = 0, alors ad(u) est une application de la forme z 7→ ak za∗k avec ak ∈ Gs (somme finie). (iii) Si a = c, alors ad(u)(1) = K · 1 avec K ∈ R+∗ . (iv) Soient s, t ∈ R et supposons que t + b − d = 0 ou que s + a − c + 1 = 0. Alors il existe M ∈ R+∗ tel que ad(u)/M pr´eserve tout e´ tat φ de Gred tel que φ(xy) = φ(y(fs ∗ x ∗ ft )), ∀ x, y ∈ Gs . D´emonstration. (i) En utilisant le Th´eor`eme 7, on a X fa ∗ uij ∗ fb = (fb ⊗ Id ⊗ fa )( uis ⊗ usk ⊗ ukj ) = (F b uF a )ij . Par Th. 7 (iv) et en utilisant (I ⊗ κ)(u) = u∗ on obtient fc ∗ κ(uij ) ∗ fd = κ(f−d ∗ uij ∗ f−c ) = (F −c u∗ F −d )ij .

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On a donc (λa,b ⊗ ρc,d )(uik ⊗ ulj )(z) = (F b uF a )ik z(F −c u∗ F −d )lj , d’o`u: X (F b uF a )ik z(F −c u∗ F −d )kj . (λa,b ⊗ ρc,d )δ(uij )(z) = k

(ii) On a (F −c u∗ F −d )ki = [(F −c u∗ F −d )∗ ]∗ik = (F −d uF −c )∗ik (rappelons que F > 0), donc si a + c = b + d = 0, alors: X (F b uF a )ik z(F b uF a )∗ik . ad(u) : z 7→ i,k

P (iii) Si a = c alors ad(u)(1) = i,k (F b uF a )ik 1(F −a u∗ F −d )ki = T r(F b−d )1. P (iv) On a φ(ad(u)(z)) = φ[ (fa ∗ uik ∗ fb )z(fc ∗ κ(uki ) ∗ fd )] = φ(zM ), o`u X (fc ∗ κ(uki ) ∗ fd )(fs+a ∗ uik ∗ ft+b ). M := i,k

Si t + b − d = 0 alors en utilisant les formules de la d´emonstration de (i) on a X (F −c u∗ F −d )ki (F t+b uF s+a )ik = T r(F s+a−c ) > 0. M= i,k

Supposons maintenant que s + a − c + 1 = 0. En utilisant le point (ii) du Th´eor`eme 7 on a uik = f1 ∗ κ2 (uik ) ∗ f−1 . En utilisant cette formule, ainsi que le point (iv) du Th´eor`eme 7 et les formules de la d´emonstration de (i) on obtient P M = i,k (fc ∗ κ(uki ) ∗ fd )(fs+a+1 ∗ κ2 (uik ) ∗ ft+b−1 ) = P = κ[ i,k (f−t−b+1 ∗ κ(uik ) ∗ f−s−a−1 )(f−d ∗ uki ∗ f−c )] = = κ[(I ⊗ T r)(F t+b−1 u∗ F s+a+1−c uF −d )] = T r(F t+b−1−d ) > 0.



7. La Propri´et´e de Powers Soit (A, τ ) une C∗ -alg`ebre unif`ere munie d’une trace fid`ele. Haagerup et Zsido ont montr´e [HZ] que A est simple a` trace unique si et seulement si elle a la propri´et´e de Dixmier: Pour tout a ∈ A, l’enveloppe convexe ferm´ee de {uau∗ | u unitaire } contient un multiple scalaire de 1A . La simplicit´e de C∗red (Fn ) a e´ t´e d´emontr´ee en [P], et la m´ethode de Powers a e´ t´e e´ tendue par de nombreux auteurs aux produits libres [A, MC2] ou aux C∗ -alg`ebres de groupes discrets [H, HS, BCH]. Ces d´emonstrations de simplicit´e utilisent des estimations techniques dans B(l2 (A, τ )), qui “bougent” vers 0 tout e´ l´ement de trace 0, en utilisant des sommes d’automorphismes int´erieurs, i.e. qui prouvent la propri´et´e de Dixmier. Dans le cas des C∗ -alg`ebres r´eduites de groupes discrets 0, l’estimation dans l2 (0) est obtenue en utilisant des propri´et´es combinatoires, g´eom´etriques etc. de 0. Une d’entre elles est la propri´et´e de Powers, d´efinie dans [H]: Pour tout ensemble fini F ⊂ 0 − {1}, il existe des e´ l´ements g1 , g2 , g3 ∈ 0 et une ` partition 0 = D E telles que F · D ∩ D = ∅ et gs · E ∩ gk · E = ∅, ∀ s 6= k.

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On voit facilement que les groupes libres Fn ont la propri´et´e de Powers. Cette propri´et´e apparait dans beaucoup d’autres contextes - voir [H] - par exemple toute action fortement hyperbolique, minimale et fortement fid`ele de 0 sur un espace de Hausdorff ` fournit une partition 0 = D E et des (en fait, une infinit´e de) e´ l´ements gi comme en haut. La preuve de la simplicit´e de C∗red (0) de [HS] a deux e´ tapes: P I. Si x ∈ l2 (F ) est hermitien de trace 0, alors || 1/3 ugs xu∗gs ||≤ 0.98 || x ||. II: C∗red (0) a la propri´et´e de Dixmier, donc est simple a` trace unique. On va e´ tendre cette d´emonstration aux groupes quantiques compacts “de Powers”: b des D´efinition 3. Soit G un groupe quantique compact. On munit l’ensemble P (G) b avec l’involution A = {a | a ∈ A} et la multiplication ◦ d´efinie par: parties de G b | ∃ a ∈ A, ∃ b ∈ B avec r ⊂ a ⊗ b}. A ◦ B = {r ∈ G b − {1}, il existe On dira que G a la propri´et´e de Powers si pour toute partie finie F ⊂ G ` b et une partition G b = D E telles que F ◦ D ∩ D = ∅ et des e´ l´ements r1 , r2 , r3 ∈ G rs ◦ E ∩ rk ◦ E = ∅, ∀ s 6= k. Le but de cette derni`ere partie du papier est de montrer que les alg`ebres Au (F )red sont simples (et avec au plus une trace). En principe on doit r´esoudre trois questions: a) Etendre la d´emonstration de simplicit´e de [HS] aux groupes quantiques compacts de Powers ayant une mesure de Haar traciale. b) Etendre a) aux groupes quantiques compacts de Powers quelconques. c) Etendre b) a` Au (F ), qui n’a pas la Propri´et´e de Powers - le Th´eor`eme 1 montre que \ la partie F = {rαβ , rβα } a la propri´et´e rx ∈ F ◦ {rx }, pour toute 1 6= rx ∈ A u (F ). En fait, toutes les d´emonstrations de simplicit´e et de non-existence d’´etats KMS qu’on va donner seront bas´ees sur la mˆeme estimation (Prop. 8). Remarquons que: – si la mesure de Haar de G est une trace, l’´enonc´e de la Prop. 8 se simplifie consid`erablement. De mˆeme pour sa d´emonstration - on ne doit pas utiliser la 6eme section. ∗ ∗ (0) (e.g. Cred (F2 )), alors la Proposition 8 est le lemme technique – si de plus G = Cred de [HS], mais l’estimation qu’on obtient ici est plus forte √ 1 X 2 2 || || x || . ugs xu∗gs ||≤ 3 3 s=1,2,3

Notons que les m´ethodes qu’on developpe ici n’ont aucune chance de s’appliquer a` Ao (F ), car Ao (F )central est commutative. b l’ensemble des repr´esentations Notation. Pour tout x ∈ Gs on note supp(x) ⊂ P (G) irr´eductibles qui ont des coefficients qui apparaissent dans x (rappelons que l’espace des coefficients de r ∈ B(Hr ) ⊗ Gs est G(r) := {(φ ⊗ Id)(r) | φ ∈ B(Hr )∗ } ; par [Wo2] on a Gs = ⊕r∈G bG(r)). Proposition 8. Soit (G, u) un groupe quantique compact matriciel r´eduit et soient s, t ∈ b telles que rl ◦ E ∩ rk ◦ E = ∅, ∀ l 6= k. b = D ` E une partition et r1 , r2 , r3 ∈ G R. Soit G Alors il existe une application lin´eaire unitale T : G → G telle que:

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P a) il existe une famille finie {ai } d’´el´ements de Gs tels que T : z 7→ ai za∗i . b) T pr´eserve les e´ tats φ ∈ G∗red v´erifiant φ(xy) = φ(y(fs ∗ x ∗ ft )), ∀ x, y ∈ Gs . c) pour tout z = z ∗ ∈ Gs tel que supp(z) ◦ D ∩ D = ∅, on a || T (z) ||≤ 0.95 || z || et supp(T (z)) ⊂ ∪i ri ◦ supp(z) ◦ ri . D´emonstration. Le Lemme 9 appliqu´e avec a = c = 0 et d = −b = t/2 fournit une b → L(G) (rappelons que Gred = G). Notons que le choix de certaine application ad : G a, b, c, d implique a + c = b + d = 0, a = c et t + b − d = 0, donc on peut appliquer les points (ii-iv) du Lemme 9 avec u := ri , pour i = 1, 2, 3. On obtient trois familles finies {ai,k }k d’´el´ements de Gs et six r´eels positifs non nuls Ki , Mi (i = 1, 2, 3) tels que pour tout i: P (i) ad(ri )(z) = k ai,k za∗i,k . (ii) ad(ri )(1) = Ki · 1. (iii) ad(ri )/Mi pr´eserve φ. Comme φ(ad(ri )(1)) = Mi (par (iii)) et φ(ad(ri )(1)) = Ki (par (ii)), on a Ki = Mi pour tout i. Posons   1 ad(r1 ) ad(r2 ) ad(r3 ) + + . T := 3 M1 M2 M3 Il nous reste a` v´erifier la condition (c). Notons h la mesure de Haar de G (qui est fid`ele par hypoth`ese) et (H, π) la construction GNS associ´ee a` (G, h). Pour i = 1, 2, 3 notons: X π(ai,k )P π(a∗i,k ). Ti0 : B(H) → B(H), P 7→ Mi−1 k

Soit T 0 = (T10 + T20 + T30 )/3. C’est une application compl`etement positive unitale. Soit p (resp. q) la projection dans H sur la fermeture de l’espace lin´eaire engendr´e par les b = D ` E et supp(z)◦D∩D = ∅, coefficients des repr´esentations de D (resp. E). On a G d’o`u: p + q = 1 , pπ(z)p = 0. Si t 6= s ∈ {1, 2, 3}, alors rt ◦ rs ◦ E ∩ E = ∅: en effet, si r, p ∈ E sont telles que r ⊂ rt ⊗ rs ⊗ p, alors on a h(χ(rt ⊗ r ⊗ rs ⊗ p)) ≥ 1, donc rt ⊗ r et rs ⊗ p ont une composante irr´eductible commune, qui doit eˆ tre dans rs ◦ E ∩ rt ◦ E = ∅, contradiction (cf. Rappel C). Il en r´esulte que si a (resp. b) sont des coefficients arbitraires de rt (resp. rs ), alors qπ(a∗ b)q = 0. Les ai,k e´ tant des coefficients de ri pour tout i (cf. Lemme 9 (i)), on a: X Tt0 (q) Ts0 (q) = (Mt Ms )−1 π(at,k )qπ(a∗t,k as,h )qπ(a∗s,h ) = 0. k,h 0

Il en r´esulte que la norme de T (q) est || T 0 (q) ||= lim || T 0 (q)n || n = 1

X X 1 1 1 1 lim || ( Ti0 (q))n || n = lim || (Ti0 (q))n || n 3 3 i i

donc plus petite que 13 (car les (Ti0 )n sont des applications compl`etement positives unitales). L’assertion de (c) sur supp(T (z)) est e´ vidente. L’in´egalit´e || T (z) ||≤ 0.95 || z || r´esulte du lemme suivant (avec f = T 0 , x = π(z) et δ = 1/3) et du fait que la repr´esentation GNS π est isometrique:

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Lemme 10. Soit H un espace de Hilbert, x = x∗ ∈ B(H), p + q = 1 projections dans H, et f : B(H) → B(H) une application √ compl`etement positive unitale. Si pxp = 0 et || f (q) ||≤ δ < 1/2, alors || f (x) ||≤ 2 δ − δ 2 || x ||. D´emonstration (G. Skandalis). Soit ζ ∈ H arbitraire de norme un. On veut montrer que √ | < f (x)ζ, ζ >| ≤ 2 δ − δ 2 || x ||. Par le th´eor`eme de Stinespring on peut supposer que f (z) = ω ∗ zω avec ω ∗ ω = 1. En posant ξ = ωζ, il suffit de d´emontrer l’´enonc´e suivant: Si H est un espace de Hilbert, x = x∗ ∈ B(H), p + q = 1 sont des projections dans H avec pxp = 0,√et ξ ∈ H est de norme 1 et tel que < qξ, ξ > ≤ δ < 1/2, alors | < xξ, ξ > | ≤ 2 δ − δ 2 || x ||. Notons E ∈ B(H) la projection sur Cpξ ⊕ Cqξ. Alors on peut remplacer dans l’´enonc´e ci-dessus H, p, q, x, ξ par E(H), EpE, EqE, ExE, ξ. En effet, on a < qξ, ξ > =< EqEξ, ξ >, < xξ, ξ >=< ExEξ, ξ > et || ExE ||≤|| x ||. On peut aussi supposer que || x ||= 1.         0 0 m 1 0 a b et ξ = Soient donc H = C2 , p = ,q = avec ,x= b 0 0 1 n 0 0 a ∈ R et b, m, n ∈ C. On a:     m am + bn , >= a | m |2 +2Re(b n m). < xξ, ξ >=< bm n On a | m |2 =< qξ, ξ > ≤ δ et | m |2 + | n |2 =|| ξ ||2 = 1, donc: p | < xξ, ξ > |≤ δ | a | +2 δ(1 − δ) | b | . 2 2 On p peut supposer que a ≥ 0. Les racines de det(x − zI) = z − az− | b | sont 2 2 (a±p a + 4 | b | )/2. Mais || x ||= 1, donc ces racines sont dans [−1, 1], ce qui implique que a2 + 4 | b |2 ≤ 2 − a, d’o`u a ≤ 1− | b |2 . On a donc: p √ √ | < xξ, ξ > |≤ δ(1− | b |2 ) + 2 δ(1 − δ) | b |= 1 − ( 1 − δ − δ | b |)2 . √ √ On a δ < 1/2, donc la fonction b 7→ 1 √ − ( 1 − δ − δ | b |)2 atteint son maximum sur [−1, 1] en b = ±1. Ce maximum est 2 δ − δ 2 . 

Enfin, on utilisera le lemme suivant au lieu de la propri´et´e de Dixmier: Lemme 11. Soit (A, φ) une C∗ -alg`ebre munie d’un e´ tat fid`ele, soit ψ ∈ A∗ un e´ tat, soit As ⊂ A une ∗-alg`ebre dense et soit 0 < δ < 1. Supposons que pour tout hermitien x ∈ ker(φ) ∩ As il existe une famille finie d’´el´ements ai ∈ A telle que l’application P z 7→ ai za∗i soit unitale, pr´eserve φ et ψ et envoie x sur un e´ l´ement de norme ≤ δ || x ||. Alors A est simple et ψ = φ. D´emonstration. On peut supposer As = A. En appliquant plusieures fois l’hypoth`ese, on peut supposer que δ > 0 est aussi petit que l’on veut. Soit J ⊂ A un id´eal bilat`ere. Soit y ∈ J et z = yy ∗ /φ(yy ∗ ). Alors on peut trouver des ai avec || Σai (1 − z)a∗i ||< 1, i.e. avec Σai za∗i inversible. Mais Σai za∗i ∈ J, donc J = A. Soit x = x∗ ∈ ker(φ) quelconque et  > 0 petit. On peut trouver un y = Σai xa∗i de norme plus petite que , donc | ψ(x) |=| ψ(y) |≤ . On obtient ψ(x) = 0, donc que ψ = φ sur les e´ l´ements hermitiens. Tout op´erateur e´ tant une combinaison lin´eaire finie de 1 et de hermitiens de ker(φ), on a ψ = φ. 

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Proposition 9. Si G a` la propri´et´e de Powers alors Gred est simple. Supposons de plus qu’on se donne un e´ tat ψ de Gred tel que ∀ x, y ∈ Gs on ait ψ(xy) = ψ(y(f1 ∗ x ∗ f1 )). Alors ψ = h (la mesure de Haar de Gred ). Donc si h est une trace, alors elle est la trace unique de Gred . D´emonstration. Soit x ∈ ker(h) ∩ Gs un hermitien. Remarquons que 1 n’est pas dans F := supp(x). G ayant la propri´et´e de Powers, on peut appliquer la Proposition 8 avec s = t = 1. On obtient donc une application unitale P f de la forme z 7→ ai za∗i qui laisse invariantes h et ψ (par le point (b) de la Prop. 8), telle que || f (x) ||≤ 0.95 || x || (normes de Gred ). On conclut a` l’aide du Lemme 11  (avec A = Gred , As = Gs et φ = h). 8. Simplicit´e de Au (F )red Au (F )red n’a pas la propri´et´e de Powers (prendre F = {rαβ , rβα }), mais on va montrer qu’elle est simple en utilisant la Proposition 8. On identifie les objets d´efinis dans la section pr´ec´edente pour G = Au (F ). En utilisant la d´escription des repr´esentations de \ Au (F ), on peut identifier A ee u (F ) = N ∗ N. La multiplication ◦ sur P (N ∗ N) est donn´ (pour x, y ∈ N ∗ N) par la formule suivante: {x} ◦ {y} = {ab | ∃ g ∈ N ∗ N avec x = ag, y = gb}. Notation. Pour w ∈ N ∗ N on note {w...} (resp. {...w}) l’ensemble des mots de N ∗ N qui commencent (resp. finissent) avec w. Pour w, y ∈ N ∗ N on note {w...y} = {w...} ∩ {...y}. On note (βα)N le mot βαβα...βα (N fois). Lemme 12. On consid`ere les ensembles D = {α...}, E = {β...} ∪ {e}, F ` = {β...α} et les e´ l´ements r1 = βαβ, r2 = βα2 β et r3 = βα3 β. Alors N ∗ N = D E est une partition, F ◦ D ∩ D = ∅ et rs ◦ E ∩ rk ◦ E = ∅, ∀ s 6= k.  Notation. On fixe n ∈ N, F ∈ GL(n, C) et on note G = Au (F )red et h sa mesure de Haar. Corollaire 3. Soient s, t ∈ R et  > 0. Soit ψ un e´ tat de G tel que ψ(xy) = ψ(y(fs ∗ x ∗ ft )), ∀ x, y ∈ Gs . P Alors il existe une application lin´eaire unitale V : G → G de la forme z 7→ ai za∗i ∗ (somme finie, avec ai ∈ Gs ) qui pr´eserve ψ telle que pour tout x = x ∈ Au (F )s avec supp(x) ⊂ {β...α} on a || V (x) ||≤  || x || et supp(V (x)) ⊂ {β...α}. D´emonstration. On applique la Proposition 8 aux parties d´efinies dans P le Lemme 12. On obtient ainsi une certaine application T : G → G de la forme z 7→ bi zb∗i (somme finie, avec bi ∈ Gs ) qui pr´eserve ψ. En posant z = x dans le point (c) de la Prop. 8 on obtient || T (x) ||≤ 0.95 || x ||, supp(T (x)) ⊂ ∪i ri ◦ supp(x) ◦ ri ⊂ {β...β} ◦ {β...α} ◦ {α...α} ⊂ {β...α}. La condition (a) de la Prop. 8 implique que T (x) = T (x)∗ ∈ Gs , donc on peut appliquer le point (c) de la Prop. 8 avec z = T (x), puis avec z = T 2 (x) etc. On choisit  m ∈ N tel que 0.95m ≤  et on pose V = T m . Lemme 13. ∀ F ⊂ N ∗ N finie, (βα)N ◦ F ◦ (βα)N ⊂ {β...α} ∪ {e} pour N grand.

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D´emonstration. Si Y ⊂ N ∗ N est l’ensemble des mots altern´es (i.e. qui ne contiennent ni α2 ni β 2 ), on voit facilement que (a) Y ◦ {...α} ∩ {...β} = ∅ ; (b) Y ◦ {...β} ∩ {...α} = ∅ (c) {α...} ◦ Y ∩ {β...} = ∅ ; (d) {β...} ◦ Y ∩ {α...} = ∅. Il suffit d´emontrer le lemme pour les parties de cardinal 1. Soit F = {z} une telle partie. • Supposons z ∈ Y . Par (d), (βα)N ◦ z est e´ gal a` e (et dans ce cas on a fini), ou commence par β. En utilisant encore une fois (d) on voit que (βα)N ◦ z ◦ (βα)N est e´ gal a` e ou commence par β. De mˆeme, en appliquant deux fois (a), on voit que (βα)N ◦ z ◦ (βα)N est e´ gal a` e ou finit par α. Donc (βα)N ◦ z ◦ (βα)N est dans {β...α} ∪ {e}. • Supposons z ∈ N ∗ N − Y , par exemple que z = xα2 y. Alors (βα)N ◦ xα ⊂ {...α} ∪ {e} par (a). Pour N ≥ l(x), il est clair que (βα)N ◦ xα ⊂ {β...α}. Par les mˆemes arguments, αy ◦ (βα)N ⊂ {α...α} pour N ≥ l(y). Donc pour N grand: (βα)N ◦ (xα2 y) ◦ (βα)N = [(βα)N ◦ xα] ◦ [αy ◦ (βα)N ] ⊂ {β...α}.



Corollaire 4. Soit x = x∗ ∈ Gs tel que h(x) = 0.

P (i) Il existe une application lin´eaire unitale W : G → G de la forme z 7→ bi zb∗i (somme finie, avec bi ∈ Gs ), qui pr´eserve h et telle que supp(W (x)) ⊂ {β...α}. (ii) Soient v, w ∈ R. Alors L ∈ R+∗ et une application lin´eaire U : G → G P il existe de la forme z 7→ ci zc∗i (somme finie, avec ci ∈ Gs ), qui pr´eserve h, telle que supp(U (x)) ⊂ {β...α} et telle que U/L pr´eserve tout e´ tat ψ de G v´erifiant ψ(pq) = ψ(q(fv ∗ p ∗ fw )), ∀ p, q ∈ Gs . D´emonstration. Fixons K ∈ N tel que (βα)K ◦ supp(x) ◦ (βα)K ⊂ {β...α} ∪ {e} (cf. Lemme 13). Notons r = r(βα)K . (i) Le Lemme 9 appliqu´e avec a = c = 0 et d = −b = 1/2 fournit une certaine b → L(G). Remarquons que le choix de a, b, c, d permet d’appliquer application ad : G (avec u := r) les points (ii) et (iii) du Lemme 9, ainsi que le point (iv) avec s = t = 1 et φ = h. On obtient deux r´eels positifs non-nuls K, M , qui sont e´ videmment e´ gaux. En posant W = ad(r)/M , il nous reste a` v´erifier que supp(W (x)) ⊂ {β...α}. En utilisant r ◦ supp(x) ◦ r ⊂ {β...α} ∪ {e} et la formule de ad (Lemme 9 (i)), ainsi que l’´egalit´e r = r on obtient supp(W (x)) ⊂ {β...α} ∪ {e}. Mais h(W (x)) = h(x) = 0, donc e n’est pas dans supp(W (x)). (ii) Le Lemme 9 appliqu´e avec c = −a = (v + 1)/2 et d = −b = 1/2 fournit une b → L(G). On peut appliquer (avec u := r) le point (ii) du certaine application ad : G Lemme 9, ainsi que le point (iv) avec s = t = 1 et φ = h. On obtient ainsi un M ∈ R+∗ P tel que si on note U = ad(r)/M , alors U est de la forme z 7→ ci zc∗i et pr´eserve h. Appliquons de nouveau le point (iv) du Lemme 9, pour les mˆemes a, b, c, d, mais avec s = v, t = w et φ = ψ cette fois-ci. On obtient un M1 ∈ R+∗ tel que ad(r)/M1 pr´eserve ψ. On pose alors L = M1 /M . Enfin, l’assertion sur supp(U (x)) se d´emontre comme au point (i).  D´emonstration du Th´eor`eme 3. Rappelons qu’on a fix´e n ∈ N et F ∈ GL(n, C) et on a not´e G = Au (F )red . Soit x = x∗ ∈ Gs arbitraire tel que h(x) = 0 et soit 1 > 0 arbitraire.

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I.) En appliquant le Corollaire 4 (i) avec le x ci-dessus on obtient une certaine application W : G → G. En appliquant le Corollaire 3 avec s = β = 1, ψ := h et  := 1 on obtient une certaine application V : G → G. Remarquons que l’application V W a les propri´et´es suivantes: P • V W est unitale de la forme z 7→ s as za∗s (somme finie). • V W pr´eserve h. • || (V W )(x) ||≤ 1 || x ||. Le Lemme 11 (avec A = G, As = Gs et ψ = φ = h) montre alors que G est simple. II.) Soient s, t ∈ R et φ un e´ tat de G v´erifiant φ(pq) = φ(q(fs ∗ p ∗ ft )), ∀ p, q ∈ Gs . En appliquant le Corollaire 4 (ii) avec v = s, w = t et ψ = φ on obtient une application U : G → G et un L ∈ R+∗ . En appliquant le Corollaire 3 avec ψ = φ et avec un  > 0 tel que  || U (x) ||< L1 on obtient une certaine application V : G → G. Remarquons que l’application V U/L a les propri´et´es suivantes: (a) V U/L pr´eserve φ. (b) || (V U/L)(x) ||≤ L−1 || U (x) ||≤ 1 . En utilisant (a) et en faisant 1 → 0 dans (b) on obtient φ(x) = 0. Donc φ = h sur les e´ l´ements hermitiens, d’o`u φ = h. III.) Enfin, par le Th´eor`eme 7, la mesure de Haar de Au (F ) est une trace si et seulement  si F F ∗ ∈ C1. Proposition 10. Soit (G, u) un groupe quantique compact tel que sa mesure de Haar soit une trace. Alors (G, u) est moyennable si et seulement si Gred est nucl´eaire. D´emonstration. Notons J le noyau de la projection π : Gp → Gred . On analyse les extensions a` Gp et Gred des applications λ ⊗ ρ, δ, e d´efinies sur Gs : • La repr´esentation gauche-droite λ ⊗ ρ : Gs ⊗ Gs → B(l2 (Gred )) est un ∗morphisme, qui s’´etend donc en une application (λ ⊗ ρ)p : Gp ⊗max Gp → B(l2 (Gred )) (voir 6eme section). Le noyau de la projection π ⊗ I : Gp ⊗max Gp → Gred ⊗max Gp e´ tant J ⊗max Gp (voir [Wa]), on voit que (λ⊗ρ)p se factorise par π⊗I en une application (λ ⊗ ρ)r : Gred ⊗max Gp → B(l2 (Gred )). • La comultiplication δ : Gs → Gs ⊗Gs est un ∗-morphisme qui s’´etend a` Gp en une application δp : Gp → Gp ⊗max Gp . En composant avec la projection Gp ⊗max Gp → Gred ⊗max Gp on obtient une application δ1 : Gp → Gred ⊗max Gp . • La comultiplication δr : Gred → Gred ⊗min Gred se rel`eve en une application δ2 : Gred → Gred ⊗min Gp (voir Corollaire A.6 de [BS]). • La co¨unit´e e : Gs → C s’´etend en une application ep : Gp → C. En notant τ : T 7→< T 1, 1 > l’´etat canonique de B(l2 (Gred )), on a le diagramme commutatif suivant : Gred δ2 ↓

π

←−

Gp

ep

−→

δ1 ↓

C ↑τ

(λ⊗ρ)r

−→ B(l2 (Gred )) Gred ⊗min Gp ←− Gred ⊗max Gp (o`u la commutation du carr´e de gauche r´esulte de la construction de δ1 , δ2 , et celle du carr´e de droite se v´erifie sur les g´en´erateurs uij ). Il en r´esulte que si Gred est nucl´eaire, alors ker(π) ⊂ ker(ep ), donc G est moyennable (cf. Prop. 5.5 de [Bl]). Pour l’autre implication, voir les Remarques A.13 de [BS]. 

Free Unitary Compact Quantum Group

Corollaire 5. Au (In )red n’est pas nucl´eaire. De mˆeme pour Ao (In )red si n ≥ 3.

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Remarque. La Proposition 6 montre qu’on a Au (F ) ∼sim Au (F 0 ), avec F 0 diagonale. La propri´et´e universelle de Au (F 0 ) implique l’existence d’une surjection Au (F 0 ) → C∗ (Fn ), et en composant avec l’application de similarit´e on obtient une surjection Au (F ) → C∗ (Fn ). Le noyau de la surjection Au (F ) → C∗ (Fn ) e´ tant un id´eal non-trivial de Au (F ), on a Au (F ) 6= Au (F )red . References [A] [BS] [B1] [B2] [BCH] [Bl] [D] [HZ] [H] [HS] [J] [MC1] [MC2] [N] [NS] [P] [R1] [R2] [VDW] [V] [VDN] [W1] [W2] [Wa] [Wo1] [Wo2]

Avitzour, D.: Free products of C ∗ -algebras. Trans. Am. Math. Soc. 271, 423–465 (1982) Baaj, S., Skandalis, G.: Unitaires multiplicatifs et dualit´e pour les produits crois´es de C ∗ -alg`ebres. Ann. Sci. Ec. Norm. Sup., 4eme serie, t.26, 425–488 (1993) Banica, T.: On the polar decomposition of circular variables. Int. Eq. and Op. Th. 24, 372–377 (1996) Banica, T.: Th´eorie des repr´esentations du groupe quantique compact libre O(n). C. R. Acad. Sci. Paris 322, 241–244 (1996) Bekka, M., Cowling, M., de la Harpe, P.: Some groups whose reduced C ∗ -algebra is simple. Publ. Math. IHES 80, 117–134 (1995) Blanchard, E.: D´eformations de C ∗ -alg`ebres de Hopf. Bull. Soc. Math. Fr. 124, 141-215 (1996) Dykema, K.: On certain free product factors via an extended matrix model. J. Funct. Anal. 112, 31–60 (1993) Haagerup, U., Zsido, L.: Sur la propri´et´e de Dixmier pour les C ∗ -alg`ebres. C. R. Acad. Sci. Paris 298, 173–176 (1984) de la Harpe, P.: Reduced C ∗ -algebras of discrete groups which are simple with unique trace. Lect. Notes Math. 1132, Berlin–Heidelberg–New York: Springer, 1985, pp. 230–253 de la Harpe, P., Skandalis, G.: Powers’ property and simple C ∗ -algebras. Math. Ann. 273, 241–250 (1986) Jones, V.F.R.: Index for subfactors. Invent. Math. 72, 1–25 (1983) McClanahan, K.: C ∗ -algebras generated by elements of a unitary matrix. J. Funct. Anal. 107, 439–457 (1992) McClanahan, K.: Simplicity of reduced amalgamated free products of C ∗ -algebras. Canad. J. Math. 46, 793–807 (1994) Nagy, G.: On the Haar measure of the quantum SU (N ) group. Commun. Math. Phys. 153, 217–228 (1993) Nica, A., Speicher, R.: R-diagonal pairs - a common approach to Haar unitaries and circular elements. Preprint (1995) Powers, R.: Simplicity of the C ∗ -algebra associated with the free group on two generators. Duke Math. J. 42, 151–156 (1975) Rosso, M.: Finite dimensional representations of the quantum analog of the enveloping algebra of a complex semisimple Lie algebra. Commun. Math. Phys. 117, 581–593 (1988) Rosso, M.: Alg`ebres enveloppantes quantifi´ees, groupes quantiques compacts de matrices et calcul differentiel non-commutatif. Duke Math. J. 61, 11–40 (1990) Van Daele, A., Wang, S.Z.: Universal quantum groups. International J. of Math. Vol. 7, No. 2, 255–264 (1996) Voiculescu, D.: Circular and semicircular systems and free product factors. Progress in Math. 92, Boston: Birkh¨auser, 1990 pp. 45–60 Voiculescu, D., Dykema, K., Nica, A.: Free random variables. CRM Monograph Series n◦ 1, AMS (1993) Wang, S.Z.: General constructions of compact quantum groups. Ph. D. Thesis, Berkeley (1993) Wang, S.Z.: Free products of compact quantum groups. Commun. Math. Phys. 167, 671-692 (1995) Wassermann, S.: Exact C ∗ -algebras and related topics. Lecture Notes Series no 19, Seoul National Univ. (1994) Woronowicz, S.L.: Twisted SU (2) group. An example of a non-commutative differential calculus. Publ. RIMS Kyoto 23, 117–181 (1987) Woronowicz, S.L.: Compact matrix pseudogroups. Commun. Math. Phys. 111, 613–665 (1987)

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[Wo3] Woronowicz, S.L.: Tannaka-Krein duality for compact matrix pseudogroups. Twisted SU (n) groups. Invent. Math. 93, 35–76 (1988) [Wo4] Woronowicz, S.L.: A remark on compact matrix quantum groups. Lett. Math. Phys. 21, 35–39 (1991) Communicated by A. Connes

Commun. Math. Phys. 190, 173 – 211 (1997)

Communications in

Mathematical Physics c Springer-Verlag 1997

Geometric Stability Analysis for Periodic Solutions of the Swift-Hohenberg Equation Jean-Pierre Eckmann1,2 , C. Eugene Wayne3 , Peter Wittwer3 1 2 3

D´ept. de Physique Th´eorique, Universit´e de Gen`eve, CH-1211 Gen`eve 4, Switzerland Section de Math´ematiques, Universit´e de Gen`eve, CH-1211 Gen`eve 4, Switzerland Dept. of Mathematics, The Pennsylvania State University, University Park, PA 16802, USA

Received: 30 January 1997 / Accepted: 6 April 1997

Abstract: In this paper we describe invariant geometrical structures in the phase space of the Swift-Hohenberg equation in a neighborhood of its periodic stationary states. We show that in spite of the fact that these states are only marginally stable (i.e., the linearized problem about these states has continuous spectrum extending all the way up to zero), there exist finite dimensional invariant manifolds in the phase space of this equation which determine the long-time behavior of solutions near these stationary solutions. In particular, using this point of view, we obtain a new demonstration of Schneider’s recent proof that these states are nonlinearly stable.

1. Introduction In this paper, we study the non-linear stability of space-periodic, time-independent solutions of the Swift-Hohenberg equation
(1.1) ∂t u = ε2 − (1 + ∂x2 )2 u − u3 . Here, u(x, t) is defined on R × R+ and takes real values and ε ≥ 0 is a small parameter. Equation(1.1) has stationary solutions u(x, t) = uε,ω (x) which are of the form uε,ω (x) =

X

uε,ω,n eiωnx .

(1.2)

n∈Z

The non-linear stability problem addresses the question of the time evolution of initial data which are close to uε,ω , and stability in this context means that the solution converges to uε,ω as t → ∞. The range of possible values of ω is given by ε2 > (1 − ω 2 )2 when ω is close to 1. To simplify the exposition we shall concentrate on the case ω = 1, and omit henceforth the index ω.

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In a very interesting paper, G. Schneider [Sch] has solved this problem, and the present work relies heavily on his ideas. Our aim is to simplify somewhat the exposition of [Sch] and to extend the result by giving a more precise asymptotic analysis, using the description of the asymptotic behavior in terms of a continuous renormalization group and invariant manifolds as introduced in [W], see below. The existence of solutions of the form Eq. (1.2) is a well-established fact, (see e.g. [CE]) and we repeat here only those points of the discussion which are needed in the sequel. The equation for the stationary solution is F (u, ε) = 0, where
(1.3) F (u, ε) ≡ ε2 − (1 + ∂x2 )2 u − u3 . The equation F = 0 has the trivial solution u = 0, ε = 0. Linearizing around this solution, we see that DF equals DF = −(1 + ∂x2 )2 ⊕ 0 , acting on some weighted subspace of L2 (R) ⊕ R. The null space of DF is spanned by {cos x, sin x} ⊕ 0

and

0⊕1,

(1.4)

and thus, bifurcation theory suggests the existence of solutions of the form of Eq. (1.2), when ε 6= 0. This is indeed what happens (cf. [CR, CE]), and the higher frequency terms in Eq. (1.2) are generated from the basis Eq. (1.4) by the non-linearity u3 . The method clearly extends to similar polynomial non-linearities. An explicit calculation shows that F (uε , ε) = 0 for 2 (1.5) uε (x) = ε √ cos(x) + ε2 hε (x) , 3 and hε (x) = hε (x + 2π). Thus, the function uε equals uε,1 of Eq. (1.2). We have broken the translation invariance of the problem by the choice of cos in Eq. (1.5), instead of, say, sin. We next pass to the linear stability analysis of the solution uε . This is again a classical subject, initiated by Eckhaus [E], which we summarize for convenience, see also [CE]. Linearizing
Eq. (1.1) around the solution uε we are led to study the operator Lε = ε2 − (1 + ∂x2 )2 − 3u2ε , that is,

Lε v (x) = ε2 − 3u2ε (x) v(x) − (1 + ∂x2 )2 v(x) . Because uε is a 2π periodic function, it is most convenient to work in Floquet coordinates (i.e., with Bloch waves). To fix the notation, we give some details: Begin by introducing the following representation for f ∈ L2 (R): Z X Z 1/2 −ikx ˆ f (k) = d` e−imx e−i`x fˆ(m + `) f (x) = dke Z

m∈Z 1/2

= −1/2

where

−1/2

d` e−i`x f˜` (x) ,

f˜` (x) =

X

e−imx fˆ(m + `) .

(1.6)

m∈Z

Properties of f˜. Note first that f˜` is 2π periodic. Furthermore, the definition of f˜` (x) can be extended to all ` ∈ R by the definition

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175

f˜`+1 (x) = e−ix f˜` (x) . We next observe that if f has a smooth, rapidly decaying Fourier transform, then f˜` (x) will also be a smooth function of ` and x. If f , g are in L2 (R), then it follows from the definition of f˜` that Z 1/2 ∼ dk f˜`−k (x) g˜ k (x) . (1.7) (f g)` (x) = −1/2

We finally note that if s is a 2π periodic function, then s˜` (x) = δ(`)s(x) .

(1.8)

It is now easy to see that
∼
Lε v ` (x) = ε2 − (1 + (i` + ∂x )2 )2 v˜ ` (x) − 3(u2ε v)∼ ` (x) . In the language of condensed matter physics, ` is the quasi-momentum in the “Brillouin zone” [− 21 , 21 ] and Lε leaves the subspace F` of functions with quasi-momentum ` invariant. Using the properties just described, we get
∼

Lε v ` (x) = ε2 − (1 + (i` + ∂x )2 )2 v˜ ` (x) − 3u2ε (x) · v˜ ` (x) ≡ Lε,` v` (x) . (1.9) To fix the notation, we repeat the calculation done by Eckhaus, cf. also [CE, M]. We denote c(x) = cos(x), s(x) = sin(x). The method of Eckhaus consists in projecting the eigenvalue problem for Lε,` onto the subspace spanned by the “bifurcating directions” c and s. Observe that, modulo higher frequency terms, we have c3 = 43 c, c2 s = 41 s, and therefore the projection of Lε,` onto this subspace is described by the matrix     −4`2 − `4 − 2ε2 + O(ε4 ) −4i`3 O(`2 ) O(`) 4 + O(ε ) . 4i`3 −4`2 − `4 O(`) O(`2 ) The eigenvalues of this matrix are
λ0`,0 = − 4 + O(ε2 ) `2 + O(`3 ) ,

λ0`,1 = −2 ε2 + O(ε4 ) − 4 + O(ε2 ) `2 + O(`3 ) + O(`4 + ε4 ) . Thus, the restriction of Lε,` on the subspace spanned by c and s has its spectrum in the left half-plane. Note that the corresponding eigenvectors are s + O(|`| + ε) and c + O(|`| + ε). Extending this calculation to the full space, one shows in the same way [E, CE, M] that Theorem 1.1. For sufficiently small ε > 0 the operators Lε,` , with ` ∈ [− 21 , 21 ] are selfadjoint on the Sobolev space H 4 , have compact resolvent and a spectrum satisfying
λ`,0 (ε2 ) = − 4 + O(ε2 ) `2 + O(`3 ) ≡ −c0 (ε2 )`2 + O(`3 ) ,

(1.10) λ`,1 (ε2 ) = −2 ε2 + O(ε4 ) − 4 + O(ε2 ) `2 + O(`3 ) , λ`,j ≤ −(1 − j 2 )2 + O(ε2 ) ,

j = 2, 3, . . . .

Notation. Since we mostly concentrate on the branch 0, we shall abbreviate λ` = λ`,0 (ε2 ). The eigenfunction corresponding to λ` is
(1.11) ϕε,` (x) = const. u0ε (x) + i`gε (x) + hε,` (x)`2 ,

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J.-P. Eckmann, C.E. Wayne, P. Wittwer

where uε is the stationary solution, and both gε and h`,ε are 2π periodic. If we choose the constant to normalize the L2 norm of ϕε,` to 1, then ϕε,` = π −1/2 sin(x) + O(ε + |`|).We can now formulate the main question of this paper: Having seen that the solution uε is linearly (marginally) stable, is it true that this solution is stable under the non-linear evolution? The answer will be affirmative. As pointed out by Schneider [Sch], the result is not obvious, since the leading non-linear term does not have a sign. Indeed, the nonlinear evolution equation for a (small) perturbation of uε is ∂t v = −(1 + ∂x2 )2 v + ε2 v − 3u2ε v − 3uε v 2 − v 3 , where we recall that uε is of order ε, and approximately equal to O(ε) cos(x). Reducing again to quasi-momentum `, and using Eq. (1.8), we get the equation 3 ∼ ∂t v˜ ` = Lε,` v˜ ` − 3uε (v 2 )∼ ` − (v )` ,

(1.12)

and it is the term 3uε (v 2 )∼ ` which does not have a sign. The saving grace will be the diffusive behavior suggested by the spectrum (in particular the branch λ` ). At first sight, the non-linearities seem to be too singular for diffusion to dominate a potential divergence. Indeed, it is well known that, e.g., the equation ∂t u = ∂x2 u + u3 , has solutions which blow up in finite time [L], and the quadratic term makes things even worse. The beautiful observation of Schneider [Sch] is, however, that the problem Eq. (1.12) is rather of a form reminiscent of ∂t v = ∂x2 v − ∂x2 v 2 − ∂x v 3 ,

(1.13)

which is good enough for convergence [CEE, BK, BKL]. In later sections we examine in detail the form of the non-linear terms in Eq. (1.12), but here we explain briefly why these terms are similar to the non-linear terms in Eq. (1.13). The derivatives in the non-linearity have their origin in the symmetries of the problem, and they are easier to understand in momentum space. In fact, Eq. (1.13) is a good approximation to Eq. (1.12) only in the low-momentum (small `) regime, but this is sufficient since for ` outside a neighborhood of ` = 0, the stationary solutions are linearly stable, (and not only marginally stable) and the form of the non-linearity is unimportant. To understand the low-momentum behavior of Eq. (1.1), note first that the SwiftHohenberg equation Eq. (1.1) – and, incidentally, other equations with coordinate independent right-hand side – has a circle of fixed points generated by translations. If we now study Eq. (1.12) at ` = 0, this corresponds to studying the Swift-Hohenberg equation in the space of functions of period 2π. In this space, say L2 ([0, 2π]), the linear operator in Eq. (1.12) has pure point spectrum with a simple eigenvalue at 0 and all other eigenvalues real and strictly negative. In this case, as Schneider notes, the center manifold theorem can be applied, and there exists a 1-dimensional center manifold. We also see immediately that the eigenvector corresponding to the 0 eigenvalue is ∂x uε , i.e., it is tangent to the circle of fixed points generated by translations. In fact, since any fixed point sufficiently close to the origin must lie in the center manifold, we see that the center manifold coincides with the 1-dimensional circle of fixed points. Thus the non-linearity in the equation, when restricted to the center manifold, must vanish. This shows that the effective non-linearity in Eq. (1.12), when evaluated at ` = 0, must vanish

Geometric Stability Analysis for Solutions of Swift-Hohenberg Equation

177

and this accounts for one derivative in Eq. (1.13). More precisely, we see that the effective non-linearity in Eq. (1.12) is bounded by O(`), as is the non-linearity in Eq. (1.13). The second derivative of the non-linearity in Eq. (1.13) arises because of “momentum conservation.” Since ϕε,` is a smooth function of `, the linear term in Eq. (1.11) must of the form i`gε , with gε independent of `. Since the interaction is local in x, one sees upon working out the integrals that all terms proportional to ` in the non-linearity cancel exactly, see Eq.(A.3). Thus, the low momentum behavior of Eq. (1.12) is as if the non-linearity was differentiated twice – i.e., exactly as in Eq. (1.13). Our main result is that this intuitive argument correctly predicts that the leading order asymptotics are diffusive, and that furthermore, the higher order asymptotics are controlled by a sequence of finite dimensional invariant manifolds. Thus, our approach provides some insight into how finite dimensional geometrical structures can arise from a problem with continuous spectrum. Stability Theorem 1.2. Fix n ≥ 1 and δ > 0. There exists a Hilbert space, H(n), such that there is an n + 1 dimensional, invariant manifold for (1.12) in the extended phase space P (n) = R+ × H(n) of this equation. Any “sufficiently small” solution of (1.12) will either lie on this manifold, or approach it at a rate O(t−(n+1−δ)/2 ). In particular, if n = 1, small solutions of (1.12) have the asymptotic form: 1 A − x2 v(x, t) = √ e 4c0 (ε2 )t + O( 3/4−δ ) , t t where c0 (ε2 ) = 4 + O(ε2 ). Remark. In Sects. 2 and 3, we will make clear precisely what the Hilbert spaces H(n) are and what we mean by “sufficiently small.” The remainder of the paper is devoted to a proof of the Stability Theorem 1.2. 2. Formulating the Stability Theorem 1.2 in Terms of Scaling Variables In this section, we transform the problem to a rescaled dynamical system. In the next section, we will cast the dynamical system thus obtained into an invariant manifold problem. The idea of the proof is to focus on the “central branch” of the spectrum, λ` = λ`,0 (ε2 ), which is only marginally stable. The relevant part of the spectrum for the long-time asymptotics is only the part in a small neighborhood of ` = 0, a fact we exhibit by an appropriate rescaling of the dependent and independent variables. This rescaling has the disadvantage that it introduces a singular perturbation in the variables corresponding to the “stable branches” of the spectrum, λ`,n (ε2 ), n ≥ 1, because the corresponding modes decay extremely fast, when rescaled (at least on a linear level). However, invariant manifold theory has long been used to treat singular perturbation problems, and we are able to use it for that purpose here as well. In addition, these invariant manifolds will provide us with a geometric description of the long-time asymptotics of solutions near the stationary states. Our method generalizes to other problems of similar spectral nature, see the example of a cylindrical domain given in [W2]. Henceforth, we fix ε > 0, and omit it from most subscripts. Since L` = Lε,` is selfadjoint, we can define the (orthogonal) spectral projections P` and P`⊥ , which project onto the central branch and its complement.

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J.-P. Eckmann, C.E. Wayne, P. Wittwer

Remark. We know that for |`| sufficiently small, say |`| < `0 /2, one has spec(P` L` P` ) = −c0 (ε2 )`2 + O(`3 ) , and that this is the eigenvalue closest to 0 in spec(L` ). We continue this projection smoothly to larger ` even if it cannot be guaranteed to be a projection onto the highest eigenvalue. But note that for those values of ` the spectrum of L` can be shown to be strictly bounded away from 0, see, e.g., [CE, p. 102]. To study the non-linearity, and to show the mechanism leading to the result which is analogous to Eq. (1.13), we write Eq. (1.12) in more detail:


∂t v˜ ` (x) = Lε,` v˜ ` (x) − 3uε (x) Z −

Z

1/2

dk v˜ `−k (x)v˜ k (x)

−1/2

1/2

dk1 dk2 v˜ `−k1 −k2 (x)v˜ k1 (x)v˜ k2 (x)

(2.1)

−1/2




˜ ` (x) − F3 (v) ˜ ` (x) . ≡ L` v˜ ` (x) − F2 (v) We now decompose Eq. (2.1) by projecting onto P` and P`⊥ . If f ∈ L2 , we let f˜`c = P` f˜` , ⊥ ⊥ and f˜`⊥ = P`⊥ f˜` . Similarly, Lc` = P` L` P` and L⊥ ` = P` L` P` . Then we get

∂t v˜ `c (x) = Lc` v˜ `c (x) − P` F2 (v) ˜ ` (x) − P` F3 (v) ˜ ` (x) , (2.2) and a similar equation for v˜ `⊥ :



∂t v˜ `⊥ (x) = L⊥ ˜ `⊥ (x) − P`⊥ F2 (v) ˜ ` (x) − P`⊥ F3 (v) ˜ ` (x) . ` v

(2.3)

We next split the first equation into a piece corresponding to small |`|, i.e., |`| < `0 and another corresponding to large `. Since we want to construct invariant manifolds, we need some smoothness in this construction and we choose a smooth cutoff χ satisfying  1, if |`| ≤ `0 , χ(`) = 0, if |`| > 2`0 , and of course `0 < 21 . In fact, we shall choose `0 > 0 so small that P` is the projection onto the central eigenspace for all ` ∈ [−`0 , `0 ]. Let ϕ` denote the normalized eigenvector which spans the range of P` (for |`| < `0 , and smoothly continued for ` beyond that value). Then v˜ `c can be written as v˜ `c = V (`)ϕ` , where it is understood that V is really a function of v. We also let Π` denote the operation Π` f` = hϕ` |f` i, where h·i is the scalar product in F` . This operation extracts the coefficient V and therefore Eq. (2.2) can be written as ˜ ` − Π` P` F3 (v) ˜ `. ∂t V (`) = λ` V (`) − Π` P` F2 (v)

(2.4)

Defining V < (`) = χ(`)V (`), and V > (`) = (1 − χ(`))V (`), Eq. (2.4) can be rewritten as
∂t V < (`) = λ` V < (`) − f c (V < , V > , v˜ ⊥ ) (`) , (2.5)
∂t V > (`) = λ` V > (`) − f s (V < , V > , v˜ ⊥ ) (`) , where

Geometric Stability Analysis for Solutions of Swift-Hohenberg Equation

179

 

f (V , V , v˜ ) (`) = χ(`) Π` P` F2 v˜ ` + Π` P` F3 v˜ ` ,  



s < > ⊥ f (V , V , v˜ ) (`) = 1 − χ(`) Π` P` F2 v˜ ` + Π` P` F3 v˜ ` , c

<

and

>




⊥

v˜ ` (x) = (V < (`) + V > (`)) · ϕ` (x) + v˜ `⊥ (x) .

Note that since V > is supported outside [−`0 , `0 ], both it and v˜ ⊥ decay exponentially (at least at the linear level) and hence will be irrelevant for the asymptotics of V < , as we shall show. With this in mind, we introduce a new coordinate, V s , which combines the “irrelevant” pieces, V s = (V > , v˜ ⊥ ). Then the Eq. (2.5) combined with Eq. (2.3) takes the more suggestive form
∂t V < (`) = λ` V < (`) − f (V < , V s ) (`) , (2.6) s < s ∂t V s = L(0) b V + g(V , V ) , s and we know that the spectrum of the linear operator L(0) b is contained in (−∞, −σ ), for some σ s > 0. In order to proceed further, we analyze the non-linear terms in Eq. (2.6) in more detail. In particular, we concentrate on the most critical terms, namely those in f of Eq. (2.6) which depend only on V < . We decompose f (V < , V s ) = f2(0) (V < )+f3(0) (V < )+ f4(0) (V < , V s ), where f2(0) collects the terms which are homogeneous of degree 2 in V < and f3(0) those of degree 3. One gets


f2(0) (V < ) (`) = 3χ(`)

Z

Z dx ϕ` (x)uε (x)

Z ≡ 3χ(`)
f3(0) (V < ) (`)

Z = χ(`)

1/2

dk ϕk (x)ϕ`−k (x)V < (k)V < (` − k)

−1/2

1/2

dk K2 (`, k)V < (k)V < (` − k) ,

−1/2

dx ϕ` (x)

Z

1/2

dk1 dk2 ϕk1 (x)ϕk2 (x)ϕ`−k1 −k2 (x) −1/2 < (k2 )V < (` − k1 − k2 )

× V < (k1 )V Z 1/2 ≡ χ(`) dk1 dk2 K3 (`, k1 , k2 )V < (k1 )V < (k2 )V < (` − k1 − k2 ) . −1/2

(2.7) At this point, we make use of the diffusive nature of the problem for V < , by introducing scaling variables as in [W]. This will give us a more precise description of the convergence process than the one obtained in [Sch]. We rescale the variables in Eq. (2.6) as follows: We first fix, once and for all, a (large) constant t0 > 0. Then we define p
V < (`, t) = wc sign(`) |3` |(t + t0 ) , log(t + t0 ) , (2.8) p
V s (`, t) = ws sign(`) |3` |(t + t0 ) , log(t + t0 ) /(t + t0 )1/2 , where 3` = λ` for |`| < `0 /2 and is monotonically extended beyond that region in such a way that it is parabolic for large |`|. (This artifact is needed because we have no guarantee that λ` itself is monotone.) Note that if λ` were equal to −const. `2 , this scaling would

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J.-P. Eckmann, C.E. Wayne, P. Wittwer

amount to the usual “diffusive” rescaling. Our choice takes into accountphigher order corrections produced by higher order terms in λ` . If we let now p = sign(`) |3` |(t + t0 ), and τ = log(t + t0 ), then Eq. (2.6) implies that wc and ws obey the following equations: ∂τ wc = (−p2 − 21 p∂p )wc   + eτ f2 (wc , e−τ /2 ) + f3 (wc , e−τ /2 ) + f4 (wc , ws e−τ /2 , e−τ /2 ) , e−τ ∂τ ws = Mexp(−τ /2) ws + 21 e−τ ws − 21 e−τ p∂p ws + eτ /2 g(wc , ws e−τ /2 , e−τ /2 ) , (2.9) where f2 , f3 , f4 and M in Eq. (2.9) are defined below. If p pe−τ /2 = p(t + t0 )−1/2 = sign(`) |3` | , and if we denote the inverse transformation by ` = 8(pe−τ /2 ) , p where 8 is the inverse function of x 7→ sign(x) |3x |, then, given a function w = w(`, t), we define the nonlinearity     f2 (w, e−τ /2 ) (p) = f2(0) (w(·, eτ )) (8(pe−τ /2 ))   = f2(0) (w(·, t + t0 )) (8(p(t + t0 )−1/2 )) . (Note that 8(x) = x 1 + O(x)).) Analogous definitions apply to f3 and f4 . The operator M will be described in detail in Eq.(2.13). Remark. The non-linearities f2 ,. . . depend on the choice of t0 . If we consider the initial value problem for the Swift-Hohenberg equation, the “smallness” assumption on the perturbation of the periodic state is to be understood with respect to a choice of a (sufficiently large) t0 . As we will see, however, the nonlinear terms can be bounded, independent of t0 , for all t0 ≥ T > 0. To this change of variables will correspond the following (non-exhaustive) list of substitutions in the integrals in Eq. (2.7): Let a, b ∈ [− 21 , 21 ]. Then Z χ(`)

b


dk → χ 8(pe−τ /2 ) e−τ /2

Z

a

eτ /2 8−1 (b) eτ /2 8−1 (a)

dq 80 (qe−τ /2 ) ,

ϕ` → ϕ8(pe−τ /2 ) , ϕk−` → ϕ0(p,q,τ ) , V (k, t) → w(p, τ ) , V (` − k, t) → w(1(p, q, τ )) .

(2.10)

Here, we define 0(p, q, τ ) = 8(pe−τ /2 ) − 8(qe−τ /2 ) ,


1(p, q, τ ) = eτ /2 8−1 8(pe−τ /2 ) − 8(qe−τ /2 ) . It follows at once from the definition of 8 that

(2.11)

Geometric Stability Analysis for Solutions of Swift-Hohenberg Equation


0(p, q, τ ) = e−τ /2 (p − q) · 1 + γ(p, q, τ ) ,
1(p, q, τ ) = (p − q) · 1 + κ(p, q, τ ) ,

181

(2.12)

where κ and γ are bounded and smooth. We next discuss in detail the spectrum of Mexp(−τ /2) , which is just the rescaled linear operator for the “stable” part of w, cf. Eq. (2.6). Recall first that V s = (V > , v˜ ⊥ ). This introduces a natural decomposition of ws = (w1s , w2s ), as well as of Mexp(−τ /2) = Mexp(−τ /2),1 ⊕ Mexp(−τ /2),2 . From the definition of the first component, we get  


2 Mexp(−τ /2),1 f1s (p, τ ) = ε2 − 1 + (i + i8(pe−τ /2 ))2 − K 8(pe−τ /2 ) f1s (p, τ ) , where K(`) is a kernel given by K(`) = 3

(2.13) Z dx ϕ` (x)u2ε (x)ϕ` (x) .

(Recall that ϕ` really depends on ε as well and should be written ϕε,` .) Since V s has s support bounded paway from ` = 0, say |`| > `0 /2, we see that w1 (p, τ ) will have support −τ /2 > |3`0 /3 |, and the spectrum of Mexp(−τ /2),1 is seen to be contained in in |p|e {σ|Re σ ≤ σ0 < 0}, for some σ0 and for all τ > 0. A very similar argument detailed in Appendix B shows that the spectrum of Mexp(−τ /2),2 is also contained in such a set. Thus, the linear evolution generated by Mexp(−τ /2) contracts exponentially. See Lemma B.6 below for details. We next consider the operator L = (−p2 − 21 p∂p ), which appears in the first component of Eq. (2.9). The detailed study of the semi-group generated by L will be given in Appendix B. Here, we discuss its properties on an informal level. The Fourier transform of L is ∂x2 + 21 x∂x + 21 , which is conjugate to the harmonic oscillator H0 = ∂x2 − x2 /16 + 1/4 by the (unbounded!) transformation T , of multiplication by exp(x2 /8). In formulas: L = T −1 H0 T . Therefore, H0 has (say, on L2 ), discrete spectrum µj = −j/2, j = 0, 1, . . . . It is this spectrum which leads to a nice interpretation of the convergence properties of the Swift-Hohenberg equation. The eigenvalues of L are unchanged by the transformation T , (and the eigenfunctions are multiplied by a Gaussian), so to each eigenvalue µ of L there corresponds a decay rate eτ µ in the linear problem. Because of the transformation of variables from t to τ , this decay rate becomes (t+t0 )µ in the original problem Eq. (2.6). In other words: Neglecting the non-linearities in Eq. (2.9) and setting ws = 0, (and ignoring potential problems related to the unbounded operator T ) we have a solution wc (p, τ ) =

∞ X

wm e−τ m/2 Hm (2p) ,

(2.14)

m=0

where Hm is the mth eigenfunction of L. In the original variables, this means that V < (`, t) =

∞ X


wm (t + t0 )−m/2 Hm 2`(t + t0 )1/2 (1 + O(|`|1/2 )) .

(2.15)

m=0

Thus, to each m there corresponds a specific rate (µm = −m/2) of decay for a part of the function V < . Note that a change of t0 just corresponds to a rearrangement of the series.

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(This is not contradictory, since a change of t0 also changes the initial condition, and hence the solution whose asymptotics we are computing.) In particular, the slowest rate of decay is associated with H0 , which is Gaussian, and thus, at least at the linear level, a “generic” perturbation of the stationary state will decay like exp(−c`2 t), for some c > 0. In terms of the original independent variables (x, t), it decays like t−1/2 exp(−x2 /(4tc)), as t → ∞. This means that at this level, the periodic stationary states are stable, and that perturbations of them decay like solutions of the linear heat equation. The invariant manifold theory guarantees that this behavior persists in the non-linear problem, and in fact it tells us more. We will see that in suitable spaces we can construct a sequence of manifolds Mj of dimension j = 1, 2, . . ., such that any solution of Eq. (2.9) approaches a solution on Mj at a rate eτ µj−1 , or again reverting to the original (x, t) variables, at a rate O((t + t0 )µj−1 ). In the case at hand, this is O((t + t0 )−j/2 ). Thus, in principle, we can analyze finer and finer details of the asymptotics of perturbations of the stationary state by considering the behavior of the solution on these finite dimensional manifolds. 3. Casting the Stability Theorem 1.2 into an Invariant Manifold Theorem At the end of the preceding section, we have seen that the spectrum of the linear part of Eq. (2.9) has the following nature: The component wc satisfies a differential equation whose linear part has eigenvalues µj = −j/2, j = 0, 1, . . . , N , provided we work on a space of sufficiently smooth and rapidly decaying functions. The evolution of ws is governed by an equation with an even more stable spectrum. The invariant manifold theorem will show in which sense the built-in scalings of Eq. (2.14) survive the addition of non-linearities. While this presents no conceptual problems at all – and this is the beauty of the present approach – some care is of course needed in the application of the invariant manifold theorem. Another point which might be overlooked is the following: The invariant manifold theorem does not say that the representation of the full non-linear problem is the same as in Eq. (2.15), but with slowly varying wj . Rather, we will show that on the complement of a dimension j − 1 surface in the function space, the solutions decay at least like t−j/2 , (for every j ≥ 1), provided the initial data are sufficiently small and smooth. In order to apply the invariant manifold method to the problem, we need bounds on the non-linearities and bounds on the semi-group generated by L. While the factor of t = exp(τ ) in front of f2 in Eq. (2.9) might look like a disaster, we will see that by working in appropriate function spaces, and taking advantage of the nature of the nonlinear term, this factor will disappear. Its presence is in part due to the fact that we chose to work in “momentum” space, rather than “position” space, because the linear problem is most naturally studied in Floquet variables. If we rewrote these terms in position space (i.e., in the original (x, t) variables), they would look much less singular. We will work in Sobolev spaces, and we define Hq,r = {v | (1 − ∂p2 )r/2 (1 + p2 )q/2 v ∈ L2 } ,

(3.1)

equipped with the corresponding norm k · kq,r . The function wc will be an element of Hq,r . The function ws has two components. The first component comes from the central branch of the spectrum of the linear operator (1.9), and will also be in Hq,r . The second component comes from the stable branches of the spectrum, and it depends on both p, and x. It will be an element of the space:

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Hq,r,ν = {w = w(p; x) | w(p; x) = w(p; x + 2π), (1 − ∂x2 )ν/2 (1 − ∂p2 )r/2 (1 + p2 )q/2 w ∈ L2 (R × [−π, π])} . By a slight abuse of notation, we will denote by kws kHq,r,ν the sum of the Hq,r norm of the first component of ws and the Hq,r,ν norm of the second component, and by kws kq,r,ν , we will mean the Hq,r,ν norm of just the second component. We will also use Hq,r,ν to denote the space of all functions with finite Hq,r,ν norm. The non-linearities satisfy the following bounds: Proposition 3.1. For every q ≥ 2 and every r ≥ 0 there is a constant C for which keτ f2 (w, e−τ /2 )kq−1,r ≤ Ckwk2q,r ,

(3.2)

keτ f3 (w, e−τ /2 )kq,r ≤ Ckwk3q,r , for all τ > 0.

Proposition 3.2. For every q ≥ 2 and every r ≥ 0 there is a constant C for which keτ f4 (wc , ws e−τ /2 , e−τ /2 )kq,r ≤ Ceτ /2 kws kHq,r,ν e−τ /2 kwc kq,r + e−τ kws kHq,r,ν





× 1 + e−τ /2 kwc kq,r + e−τ kws kHq,r,ν , (3.3)

keτ /2 g(wc , ws e−τ /2 , e−τ /2 )kHq,r,ν ≤ Ceτ e−τ /2 kwc kq,r + e−τ kws kHq,r,ν


2


× 1 + e−τ /2 kwc kq,r + e−τ kws kHq,r,ν ,

(3.4)

for all τ > 0. Remark. Note that every factor of kwc kq,r is multiplied by e−τ /2 and every factor of kws kHq,r,ν is multiplied by e−τ . Remark. As we pointed out above, the nonlinear terms depend on the constant t0 . However, the bounds in the two preceding propositions are independent of t0 . More precisely, for any T > 0, the constants C in both propositions can be chosen so that the estimates in (3.2)–(3.4) hold for all t0 ≥ T . The proofs will be given in Appendix A. Note that one loses a power of p in the first estimate of Eq. (3.2), but of course, one “gains” the square of the function. We will regain the “lost” power of p by examining in detail the semi-group generated by L. We denote by PN the projection onto the space spanned by the N eigenvalues {µj = −j/2}j=0,...,N −1 of L. We define QN = 1 − PN . (We verify in Appendix B that these projections are defined.) On the space corresponding to QN , we expect the norm of the semi-group generated by L to decay like exp(τ µN ). This is indeed the case. Theorem 3.3. For every δ > 0, there are a constant N0 and a function r(N, q) such that for every N ≥ N0 , every q ≥ 1 and every r ≥ r(N, q), there is a C = C(q, r, N ) < ∞ such that

τL C(q, r, N ) −τ (|µN |−δ)

e QN v ≤ √ kvkq−1,r , (3.5) e q,r a(τ ) where a(τ ) = 1 − e−τ and L = −p2 − 21 p∂p The proof will be given in Appendix B.

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We also need an estimate on the linear evolution generated by Mexp(−τ /2) . Let Uτ be the solution of e−τ ∂τ Uτ = Mexp(−τ /2) Uτ , with initial condition U0 = 1. (Compare with the linear part of (2.9).) Then, in Appendix B, we prove Theorem 3.4. If w0 ∈ Hq,r,ν , then there exists c0 > 0, such that for all τ ≥ 0, kUτ w0 kHq,r,ν ≤ exp(−ec0 τ /2 )kw0 kHq,r,ν .

With the help of the bounds Proposition 3.1–Theorem 3.4, we can now reformulate the problem in terms of invariant manifolds. Equation(2.9) can be written as an autonomous system by defining η = (t + t0 )−1/2 = e−τ /2 :
∂τ wc = Lwc + η −2 f2 (wc , η) + f3 (wc , η) + f4 (wc , ws η, η) , η 2 ∂τ ws = Mη ws + η −1 g(wc , ws η, η) , ∂τ η =

− 21 η

(3.6)

.

We will construct an invariant manifold tangent at the origin to the eigenspace corresponding to the N largest eigenvalues of L, and the η direction. We subdivide the center variable wc according to the projection QN defined earlier, where N is fixed once and for all. Define (3.7) x1 = (1 − QN )wc , x2 = QN wc , x3 = ws . Note that the variable x1 is in a finite dimensional space, while x2 and x3 are in infinite dimensional Hilbert spaces. The system of equations Eq. (3.6) now takes the form ∂τ x1 = A1 x1 + N1 (x1 , η, x2 , x3 ) , ∂τ η = − 21 η , ∂τ x2 = A2 x2 + N2 (x1 , η, x2 , x3 ) ,

(3.8)

η 2 ∂τ x3 = A3,η x3 + N3 (x1 , η, x2 , x3 ) . Here A1 = (1 − QN )L, A2 = QN L, and A3,η = Mη . Remark. In view of later developments, we consider x1 and η to be the “interesting” variables and x2 and x3 the “slaved” variables, hence the new order of the variables. Remark. Equation(3.8) is a very singular perturbation problem, because of the factor of η 2 in front of the derivative of x3 . What is more, since η(τ ) = e−τ /2 , it becomes steadily more singular in precisely the limiting regime in which we are interested. Nonetheless, we will see that the invariant manifold theorem provides just the tool we need to understand this limit. Singular perturbation problems of this type do not seem to have been studied much, but they do arise naturally in other contexts, such as the study of parabolic equations in cylindrical domains ([W2]). We shall call Eq. (3.8) the full system. To simplify the notation, we shall omit the dependence on η in A3,η . Consider the spectra of A1 , A2 , A3 . From what we have seen earlier, we find that

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185

spec(A1 ) = {0, −1/2, −1, . . . , −(N − 1)/2} , spec(A2 ) ⊆ [−∞, −N/2] , spec(η

−2

(3.9)

A3 ) = [−∞, −c/η ] , 2

where c is some positive constant. Thus, we expect to apply a pseudo center manifold theorem to “slave” the variables x2 , x3 to the variables x1 and η. While there are certain technical difficulties associated with the very singular perturbation, in Appendix C, we demonstrate the following proposition: Proposition 3.5. Fix N > 0. There exist r > 0, q ≥ 1, and ν > 1/2, such that the system of equations (3.8) has an invariant, N + 1-dimensional manifold, given in a neighborhood of the origin by the graph of a pair of functions h∗2 : RN × R → Hq,r , h∗3 : RN × R → Hq,r,ν . We next turn to the task of showing that the invariant manifold we found for Eq. (3.6) actually attracts solutions at an exponential rate. Notation. It is useful to introduce the notation
ξ = (x1 , η) for the two relevant variables.
Consider a solution of the form wc (τ ), ws (τ ) of Eq. (3.6), with wc (τ ) = x1 (τ ), x2 (τ ) as in Eq. (3.7), and ws (τ ) = x3 (τ ). We wish to show that

ξ(τ ) , x2 (τ ) , x3 (τ ) −→ ξ(τ ) , h∗2 (ξ(τ )) , h∗3 (ξ(τ )) , as τ → ∞, and furthermore, that it does so at an exponential rate, given essentially by the least negative eigenvalue, µN , of the operator A2 . Proposition 3.6. Fix N > 0. For every positive δ there is a ρ0 > 0 such that if the solution of Eq. (3.6) remains in a neighborhood of the origin of size ρ0 one has the following bound: There is a C ∗ < ∞ for which kx2 (τ ) − h∗2 (ξ(τ ))kq,r + kx3 (τ ) − h∗3 (ξ(τ ))kHq,r,ν ≤ C ∗ e−(|µN |−δ)τ , as τ → ∞. Proof. This proof is relatively standard, see e.g., Carr [C]. Let     x2 (τ ) − h∗2 (ξ(τ )) z2 (τ )
≡ . z(τ ) = z3 (τ ) x3 (τ ) − h∗3 (ξ(τ ) ÿ

Then we have z˙ =

A2 z2 + Nˆ 2 (ξ, z2 , z3 ) −2 η A3 z + η −2 Nˆ 3 (ξ, z2 , z3 )

! ,

(3.10)

where, with the notation of Eq. (3.8), Nˆ j (ξ, z2 , z3 ) = Nj (ξ, z2 + h∗2 (ξ), z3 + h∗3 (ξ)) − Nj (ξ, h∗2 (ξ), h∗3 (ξ)) , for j = 2, 3. The only novelty in Eq. (3.10) w.r.t. [C] is the factor of η −2 in the “3”component which is the reason for our repeating his arguments. But we can integrate Eq. (3.10) explicitly and get

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Z z2 (τ ) = e

τ A2

τ

z2 (0) +


dσ e(τ −σ)A2 Nˆ 2 ξ(σ), z2 (σ), z3 (σ) ,

0 −2

z3 (τ ) = e(η(τ ) Z τ dσ + 0

−η(0)−2 )A3

z3 (0)


1 (η(τ )−2 −η(σ)−2 )A3 ˆ N3 ξ(σ), z2 (σ), z3 (σ) . e η(σ)2 −1/2

We assume η(0) > 0, since we are interested in the case η(0) = t0 , and we have chosen the scaling factor t0 to be a positive, finite constant. Note also that ξ remains in a neighborhood of the origin, as τ → ∞. From the bounds on the non-linear terms we see that if the solution satisfies kx2 (τ )kq,r + kx3 (τ )kHq,r,ν ≤ ρ , for all τ ≥ 0, then, with νN = N/2, the modulus of the N th eigenvalue µN of L, we have kz2 (τ )kq,r ≤ e−τ νN kz2 (0)kq,r Z τ
+ Cε dσe−(τ −σ)νN kz2 (σ)kq,r + kz3 (σ)kHq,r,ν , 0 −2

kz3 (τ )kHq,r,ν ≤ e(η(τ ) Z + Cε

−η(0)−2 )νN

kz3 (0)kHq,r,ν


1 −(η(τ )−1 −η(σ)−1 )νN e kz2 (σ)kq,r + kz3 (σ)kHq,r,ν . 2 η(σ) 0 (3.11) In deriving these inequalities, we used the inequalities τ

dσ

keτ A2 Nˆ 2 (ξ, z2 , z3 )kq,r ≤ e−τ νN kNˆ 2 (ξ, z2 , z3 )kq−1,r , keρA3 Nˆ 3 (ξ, z2 , z3 )kHq,r,ν ≤ e−ρνN kNˆ 3 (ξ, z2 , z3 )kHq,r,ν , which follow from the bounds of Appendix B. If we now fix δ > 0 and define C2 (τ ) = C3 (τ ) =

sup eτ

0

(νN −δ)

kz2 (τ 0 )kq,r ,

0

(νN −δ)

kz3 (τ 0 )kHq,r,ν ,

0≤τ 0 ≤τ

sup eτ

0≤τ 0 ≤τ

then Eq. (3.11) leads to the inequality C2 (τ ) ≤ K1 + K2 ε C2 (τ ) + C3 (τ ) C3 (τ ) ≤ K3 + K4 ε C2 (τ ) + C3 (τ )







Z Z

τ

dσ e−(τ −σ)δ ,

0 τ

dσ 0

1 (η(τ )−2 −η(σ)−2 )νN (τ −σ)(νN −δ) e e . η(σ)2

If we insert into these integrals the definitions η(σ) = exp(−σ/2)η(0) ,

η(τ ) = exp(−τ /2)η(0) ,

we find that both integrals are uniformly bounded in τ ≥ 0 if η(0) is in a compact subinterval of (0, 1). The proof of Proposition 3.6 is complete. Thus, all solutions near the invariant manifold approach it exponentially fast in τ .

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187

One can now show without difficulty that every solution approaches exponentially quickly a particular solution on the (approximate) invariant manifold
x1 (τ ), η = 0, h∗2 (x1 (τ ), 0), h∗3 (x1 (τ ), 0) . This consists simply in translating pp.21–24 of [C] into the present setting and thus there is no need to repeat this argument here. If we combine these results with Proposition 3.5, we arrive finally at a description of the invariant manifolds which exist close to the origin for (3.8). Theorem 3.7. Fix N > 0 and δ > 0. There exist r > 0, q ≥ 1, and ν > 1/2, such that the system of equations (3.8) has an invariant, N + 1-dimensional manifold, given in a neighborhood of the origin by the graph of a pair of functions h∗2 : RN × R → Hq,r , and h∗3 : RN × R → Hq,r,ν . Any solution of (3.8) which remains in a neighborhood of the origin for all τ ≥ 0 approaches a solution of the N + 1-dimensional system of ordinary differential equations ∂τ x1 = A1 x1 + N1 (x1 , η, h∗2 (x1 , η), h∗3 (x1 , η)) , ∂τ η = − 21 η ,

(3.12)

which results from restricting (3.8) to this invariant manifold. Furthermore, the rate of approach to this manifold is O(exp(−τ (N/2 − δ))). Remark. This theorem almost suffices to prove Stability Theorem 1.2 . In particular, it emphasizes that in a neighborhood of the periodic solutions of (1.1) there exists a family of invariant manifolds, M2 , M3 , . . ., described in that theorem. The one remaining piece of the puzzle is to describe the behavior of solutions restricted to the invariant manifold, and that we do in the next section.

4. The Projection of the Non-Linearity onto Zero Momentum We have already shown that there exists a (smooth) invariant manifold, parameterized by (ξ, h∗2 (ξ), h∗3 (ξ)), where ξ = (x1 , η). This manifold satisfies Eq. (3.8), which, in the case of N = 1, i.e., in the case of a two-dimensional invariant manifold amounts to
∂τ x1 = N1 x1 , η, h∗2 (ξ), h∗3 (ξ) , ∂τ η = − 21 η ,

∂τ h∗2 (ξ) = A2 h∗2 (ξ) + N2 x1 , η, h∗2 (ξ), h∗3 (ξ) ,
η 2 ∂τ h∗3 (ξ)) = A3 h∗3 (ξ) + N3 x1 , η, h∗2 (ξ), h∗3 (ξ) .

(4.1)

Note that because N = 1 the operator A1 equals zero (which is the highest eigenvalue of L). To understand the dynamics inside this invariant manifold, we now state and prove the following proposition, which is based on Schneider’s beautiful observation: Let N˜ 1 (x1 , η) be the r.h.s. of the first equation in (4.1), i.e., ∂τ x1 = N˜ 1 (x1 , η). Proposition 4.1. There is an x1,0 > 0 such that N˜ 1 (x1 , 0) = 0, for all |x1 | < x1,0 .Thus, the non-linearity vanishes identically at “infinite time,” which corresponds to η = 0. Before proving Proposition 4.1, we show that it implies the following important

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Theorem 4.2. If x1 (0) is sufficiently close to 0, then there are a constant C < ∞ and an x∗1 such that (4.2) |x1 (τ ) − x∗1 | < Ce−τ /2 . Proof. Using the fact that η(τ ) = e−τ /2 , we can rewrite the equation for x1 as ∂τ x1 = N˜ 1 (x1 , e−τ /2 ) .

(4.3)

Since N˜ 1 is a smooth (at least C 1+α ) function with N˜ 1 (x1 , 0) = 0 in some neighborhood of the origin, there exists a constant CN > 0, such that |N˜ 1 (x1 , e−τ /2 )| ≤ CN exp(−τ /2), for |x1 | sufficiently small. Integrating (4.3) and applying this estimate yields: Z τf dσ N˜ 1 (x(σ), e−σ/2 ) |x1 (τf ) − x1 (τi )| = τi Z τf dσ e−σ/2 = 2CN e−τi /2 (1 − e(τi −τf )/2 ) . ≤ CN τi

This estimate immediately implies the behavior claimed in Theorem 4.2. Proof of Proposition 4.1. The basic idea is to relate N˜ 1 (x1 , 0) to the non-linear term of another problem, which is known to be 0. This other problem is the center manifold equation for the perturbations of a stationary solution of Eq. (1.1) restricted to a space of 2π-periodic functions. In this case, the equation analogous to Eq. (1.12) is ∂t v = Lper v + F (v) , where F (v) collects the non-linear terms in v. The spectrum of Lper is pure point, with a simple zero eigenvalue, and all others negative, and bounded away from 0. The eigenvector with 0 eigenvalue is u0ε , where uε is given by Eq. (1.5). If we call x1,per the coordinate in the u0ε direction, then there exists a one-dimensional center manifold, tangent to this direction and given as the graph of a function H(x1,per ). A very nice observation by Schneider is that this center manifold must coincide with the translates of the stationary state uε , which is formed of fixed points of the Swift-Hohenberg Eq. (1.1). Hence, on this center manifold we must have x˙ 1,per = 0. Using this information, the equations for this center manifold take a particularly simple form. Let Pper denote the projection onto u0ε and let Qper = 1 − Pper . Then the preceding discussion implies that the flow ψt,per is the identity on x1,per , and hence the equations for the invariant manifold read: x˙ 1,per = Pper F (x1,per , H(x1,per )) = 0 , Z 0 dτ e−Qper Lper Qper F (x1,per , H(x1,per )) H(x1,per ) = −∞

= − Qper Lper


−1

Qper F (x1,per , H(x1,per )) .

(4.4)

(4.5)

We now wish to use this information to prove Proposition 4.1. The rough idea is to show that (4.6) N˜ 1 (x1 , 0) = Pper F (x1,per , H(x1,per )) , and this quantity vanishes by Eq. (4.4). More precisely, we shall show:

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189

Proposition 4.3. The cubic term in x1 of N˜ 1 (x1 , η) coincides in the limit η → 0 with the cubic term in x1 of Pper F (x1,per , H(x1,per )). All other terms in N˜ 1 go to 0 as η → 0.
Remark. Since Pper F x1,per , H(x1,per ) = 0, this proves Eq. (4.6) and thus Proposition 4.1. Proof. The proof of Proposition 4.3 will be given in Appendix D. 5. Completion of the Proof of Stability Theorem 1.2 We now consider exactly how the results of the previous two sections about the behavior of solutions in, and near, the invariant manifold translate back into statements about solutions in terms of the original variables. We will focus specifically on the case considered in the previous section in which the invariant manifold is two-dimensional, with coordinates (x1 , η), but the results can be immediately extended to the case of a manifold of arbitrary dimension. Suppose we have a solution wτ = wτc + wτs , of the system (3.6), which remains in a neighborhood of the origin for all τ ≥ 0. This will be the case if its initial condition is sufficiently small in Hq,r ⊕ Hq,r,ν . We measure the size of w in the norm ||| · |||, which is the sum of the Hq,r norm of wc , and the Hq,r,ν norm of ws . By the results of Theorem 3.7, we know that there exists a solution, wτinv , on the invariant manifold such that (5.1) |||wτ − wτinv ||| ≤ Ce−τ (1/2−δ) , with δ > 0. In addition, from Theorem 4.2, we know that there exists some w∗ , which lies in the invariant manifold for which |||wτinv − w∗ ||| ≤ Ce−τ /2 .

(5.2)

Here, w∗ is the function whose coordinates in the invariant manifold representation
is just the limiting point x∗1 in Theorem 4.2, i.e., w∗ = x∗1 , 0, h∗2 (x∗1 , 0), h∗3 (x∗1 , 0) . Combining (5.1) and (5.2), we see that for solutions that remain near the origin, there exists a function w∗ , for which |||wτ − w∗ ||| ≤ Ce−τ (1/2−δ) .

(5.3)

Our final task is now to untangle the various changes of variables which we made in the original equation. If we first “undo” the rescaling in (2.8), we see that the solution v(`, t), corresponding to w(·, τ ) = wτ is p v(`, t) = wc (sign(`) |3` |(t + t0 ), log(t + t0 )) p 1 s (4.4) w (sign(`) |3` |(t + t0 ), log(t + t0 )) + (t + t0 )1/2 ≡ v c (`, t) + v s (`, t) . One can make a corresponding decomposition of v ∗ , the solution corresponding to w∗ . First consider v c . From (5.3), one has Z c ∗,c 2 dp |(1−∂p2 )r/2 (1+p2 )q/2 (wc (p, τ )−w∗,c (p, τ ))|2 ≤ Ce−τ (1−2δ) . kwτ −wτ kq,r = (4.5)

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According to (4.4), wc (`, τ ) = v c (8−1 (pe−τ /2 ), t), so substituting this expression – and the analog for w∗,c – into (4.5) one finds that the left hand side of that inequality is equal to: Z
dp |(1 − ∂p2 )r/2 (1 + p2 )q/2 v c (8−1 (pe−τ /2 ), t) − v ∗,c (8−1 (pe−τ /2 ), t) |2 Z
(4.6) ≥ dp |(1 + p2 )q/2 v c (8−1 (pe−τ /2 ), t) − v ∗,c (8−1 (pe−τ /2 ), t) |2 Z
≥ d` (t + t0 )1/2 80 (`) |(1 + (t + t0 )(8(`))2 )q/2 v c (`, t) − v ∗,c (`, t) |2 , where in the last integral we changed the integration variable to ` = 8−1 (pe−τ /2 ) = 8−1 (p(t + t0 )−1/2 ). Remark. We dropped the derivatives with respect to p in the second line of (4.6) for simplicity – one could retain them at the expense of complicating the following expressions. Since 8(x) ≈ x, for x small, and is equal to a constant times x for |x| large (due to the definition of 3` ), we see that combining (4.5) and (4.6) and recalling that t0 > 0, one finds: Z (4.7) d` |(1 + `2 )q/2 (v c (`, t) − v ∗,c (`, t))|2 ≤ Ct−3/2(1−2δ) . Analogous estimates hold for the “stable” part of the solution. Proceeding as above, one can show that Z X 2 ν d` |(1 + `2 )q/2 (v s (`, t) − v ∗,s (`, t))|2 ≤ Ct−5/2(1−2δ) . (1 + n ) (4.8) n

Thus, the “stable” part of a solution near the origin approaches the solution v ∗ on the invariant manifold faster than the “center” part of the solution. (An effect that is entirely in accord with one’s intuition.) We next take a closer look at the solution w∗ (or v ∗ ) on the invariant manifold. From the computation in the previous section, we know that since the eigenfunction in the x1 direction is exp(−p2 ), cf. Eq(2.14), we have w∗ (p) = c∗ exp(−p2 ) + h∗3 (c∗ exp(−p2 )). If we now rewrite this in terms of the v(`, t) variables, we find v ∗ (`, t) = c∗ e−3` t + t−1/2 h∗3 (c∗ e−3` t ) .

(4.9)

Thus, if v(`, t) is a solution of (1.12) (in the unscaled variables), we see from (4.7)–(4.9) that in the L2 ((1 + `2 )q/2 d`) norm, v(`, t) = c∗ e−3` t + O(t−1/2(1−2δ) ) .

(4.10)

But we know from Sect. 2 that 3` = c0 (ε2 )`2 + O(`3 ) for ` small, and 3` = c`2 , for |`| large, so one finds by an easy and explicit estimate that Z 2 2 (4.11) d` |(1 + `2 )q/2 (e−3` t − e−c0 (ε )` t )|2 ≤ Ct−1/2 . Combining (4.10) and (4.11) one has Proposition 4.4. If v is a solution of (1.12) with sufficiently small initial condition (in Hq,r ⊕ Hq,r,ν ), then

Geometric Stability Analysis for Solutions of Swift-Hohenberg Equation

Z (

|(1 + `2 )q/2 (v(`, t) − c∗ e−c0 (ε

2

191

)| d`)1/2 ≤ Ct−1/4(1−2δ) .

)`2 t 2

Note that if we transform back to the (x, t) variables, this implies the asymptotic estimate in Stability Theorem 1.2 , and hence the proof of that theorem is complete. A. Bounds on the Non-Linearities In this section, we prove Proposition 3.1 and Proposition 3.2. We begin by studying the kernels K2 (`, k), and K3 (`, k) introduced in (2.7). Lemma A.1. There is a constant C such that


|K2 (`, k)| ≤ Cε min (|k|2 + |`|2 ), 1 .

Proof. By the definition of Eq. (2.7), we have Z K2 (`, k) = dx ϕ` (x)uε (x)ϕk (x)ϕ`−k (x) .

(A.1)

Since uε and ϕk are both uniformly bounded, we have immediately that |K2 (k, `)| ≤ Cε. The crucial observation of Schneider[Sch] is that because of Eq. (1.11), repeated here for convenience (A.2) ϕε,` (x) = u0ε (x) + i`gε (x) + hε,` (x)`2 , (with real gε ), K2 has an expansion Z
2
dx uε (x)(u0ε (x))3 + uε (x) u0ε (x) gε (x) −i` + ik + i(` − k) + εO(`2 + k 2 ) . (A.3) Note that the first term vanishes because u is a symmetric function and hence u(u0 )3 is odd, and the term which is linear in k and ` vanishes as well, because of momentum conservation, so the proof of Lemma A.1 is complete. Remark. Note that a similar calculation immediately shows that the kernel K3 satisfies: |K3 (`, k1 , k2 )| ≤ C . We now need the following auxiliary result: Lemma A.2. If ρ2 and ρ3 are in Hq,r , and if ρ1 = ρ1 (p, p0 ) is a C r function, then Z Ξ(p) = dp0 ρ1 (p, p0 )ρ2 (1(p, p0 , τ ))ρ3 (p0 ) is in Hq,r and

kΞkq,r ≤ Ckρ1 kC r kρ2 kq,r kρ3 kq,r .

Proof. Recall from Eq. (2.12) that 1(p, p0 , τ ) ≈ p − p0 , so we are really estimating a slightly distorted convolution. If 1(p, p0 , τ ) = (p − p0 ), the proof is easy using the

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definition of the norms. In the present case, where 1(p, p0 , τ ) is not trivial, the result follows in a similar way by “undoing” part of the variable transformation which led from the variables `, k to the variables p, p0 . To simplify matters, we consider only the somewhat easier problem of bounding Z (A.4) dp0 80 (pe−τ /2 )ρ2 (1(p, p0 , τ ))ρ3 (p0 ) . Using the definition of 1(p, p0 , τ ) this is equal to Z



dp0 80 (pe−τ /2 )ρ2 eτ /2 8−1 8(pe−τ /2 )−8(p0 e−τ /2 ) ρ3 eτ /2 8−1 8(p0 e−τ /2 ) . Changing variables to k = 8(e−τ /2 p) and ` = 8(e−τ /2 p0 ), we get Z

d` eτ /2 ρ2 eτ /2 8−1 (k − `) ρ3 eτ /2 8−1 (`) .

(A.5)

(A.6)

We now define a function 9τ by 9τ (eτ /2 x) = eτ /2 8−1 (x) ,

and note that from 8(x) = x · 1 + O(x) it follows that 9τ (y) = y · 1 + O(e−τ /2 y) . We can rewrite Eq. (A.6) as Z

d` eτ /2 ρ2 9τ (eτ /2 (k − `)) ρ3 9τ (eτ /2 `) . (A.7) We define next ρˆj (k) = ρj ◦ 9τ , and we see that Eq. (A.7) is equal to Z

d` ρˆ2 k − ` ρˆ3 ` .

(A.8)

Thus, we can bound the Hq,r norm of Eq. (A.4) by kρˆ2 kq,r kρˆ3 kq,r , and, since 9τ is uniformly close to the identity for all τ , this is in turn bounded by const. kρ2 kq,r kρ3 kq,r . This proves Lemma A.2 in this special case. The extension to the general case is easy and is left to the reader. We now have the necessary tools to attack the proofs of Proposition 3.1 and Proposition 3.2. Proof of Proposition 3.1. If we write out the transformation leading to f2 , i.e., from Eq. (2.7) to Eq. (2.9), we get, using Eq. (2.10), Z

P (τ ) 0 −τ /2 0 0 −τ /2 τ τ −τ /2 −τ /2 ) (p) = e 3χ 8(pe ) dp e 8 (p e ) e · f2 (w, e (A.9) −P (τ )
−τ /2 0 −τ /2 0 0 ×K2 8(pe ), 8(p e ) w(1(p, p , τ ))w(p ) , where

P (τ ) = 8−1 ( 21 )eτ /2 ≈ 21 eτ /2 .
We bound |K2 8(pe−τ /2 ), 8(p0 e−τ /2 ) | by Cε|8(pe−τ /2 )2 + 8(p0 e−τ /2 )2 | using Lemma A.1. Since the expressions 8(pe−τ /2 ), and 8(p0 e−τ /2 ), in Eq. (A.9) are bounded, and 8(x) = x(1 + O(x)), we can extract another factor of e−τ /2 and get a bound on eτ f2 of the form

Geometric Stability Analysis for Solutions of Swift-Hohenberg Equation




Z

193


dp0 |8(pe−τ /2 )| + |8(p0 e−τ /2 )| · |w(1(p, p0 , τ ))w(p0 )| −P (τ ) Z
∞ 0 dp |p + p0 ||w(1(p, p0 , τ ))w(p0 )| . ≤ const. χ 8(pe−τ /2 )

const. eτ /2 χ 8(pe−τ /2 )

P (τ )

−∞

(A.10) If w is in Hq,r , then with the aid of Lemma A.2, we can estimate the Hq−1,r norm of Eq. (A.9) by Ckwk2q,r . Note further, that from the above discussion it is also clear that eτ f2 (wc , e−τ /2 )(p) is also a smooth function of e−τ /2 . Remark. The factors |p|, |p0 | are responsible for the loss of one power in the norm estimate of Proposition 3.1. It is only in the study of the flow within the invariant manifold that we will need the second order bound of Lemma A.1. Remark. Note that the nonlinear terms depend (implicitly) on the constant t0 which entered the definition of the new temporal variable τ . However, all the estimates above (as well as those which follow in the proof of Proposition 3.2) are independent of this constant. The bound on f3 is similar, but no additional regularization is needed, since there are two integrations, each of which contributes a factor e−τ /2 . We leave this to the reader. The proof of the asserted bounds of Eq. (3.2) is complete. We now turn to the estimates of the nonlinear terms f4 and g. Because these terms involve the ws , we begin with a discussion of the appropriate function space for these components. These were defined in Sect. 3, but we repeat them here for convenience. Recall that wc ∈ Hq,r , while ws ∈ Hq,r ⊕ Hq,r,ν , where Hq,r,ν = {w = w(p; x) | w(p; x) = w(p; x + 2π), (1 − ∂x2 )ν/2 (1 − ∂p2 )r/2 (1 + p2 )q/2 w ∈ L2 (R × [−π, π])} . The fact that ws is an element of the direct sum of two spaces reflects the fact (see the paragraph preceding (2.6), and then (2.8) ) that it has two components, the first of which comes from the central branch of the spectrum of L` , but with ` localized away from zero, and the second component coming from the stable branches of the spectrum of L` . In a slight abuse of notation we will denote by kws kHq,r,ν the sum of the Hq,r norm of the first component of ws and the Hq,r,ν norm of the second component, and by kws kq,r,ν , we will mean the Hq,r,ν norm of just the second component. Remark. An easy fact which will be useful later is that if we expand w(p; x) ∈ Hq,r,ν in a Fourier series with respect to x, w(p; x) =

∞ X

einx wˆ n (p) ,

n=−∞

then the Hq,r,ν norm of w is equivalent to the norm kwk2Hq,r,ν =

∞ X

(1 + n2 )ν kwˆ n k2q,r .

(A.11)

n=−∞

Thus we will use the two norms interchangeably. Now consider eτ f4 (wc , ws e−τ /2 , e−τ /2 ) .

(A.12)

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We shall concentrate on the most “dangerous” piece which is the quadratic term with one factor of wc and one of ws . Other terms are “less dangerous” in the sense that they contain either more factors of ws each of which contributes a small factor of e−τ /2 , or more convolutions which again contribute a factor of e−τ /2 . The quadratic piece of (A.12) has the form Z
dx ϕ¯ 8(pe−τ /2 ) (x) uε (x) eτ 3χ 8(pe−τ /2 ) Z P (τ )
(A.13) dp0 e−τ /2 80 (p0 e−τ /2 )wc 1(p, p0 , τ ) × −P (τ )

× ϕ0(p,p0 ,τ ) (x) e−τ /2 ws (p0 ; x) . As we mentioned above, ws has two components – one in Hq,r , and one in Hq,r,ν . The contribution from the component in Hq,r is bounded by the same techniques used to control f3 – note that it is not necessary to extract any additional factors of e−τ /2 , since we get one from the integration, and one from the fact that each factor of ws is multiplied by e−τ /2 . Thus, we restrict our attention to the component of ws in Hq,r,ν , which is where the new ingredients are necessary. Interchanging the order of the x and p0 integrals, we use Lemma A.2, with
ρ1 (p, p0 ) = sup |3χ 8(pe−τ /2 ) 80 (p0 e−τ /2 ) ϕ8(pe−τ /2 ) (x) ϕ0(p,p0 ,τ ) (x)uε (x)| , x

ρ2 (r) = |wc (r)| , Z ρ3 (p0 ) = | dx ws (p0 ; x)| . Since ϕ` (x) and uε (x) are smooth, 2π-periodic functions of x, and kρ1 kC r is bounded, Lemma A.2 implies that the Hq,r norm of (A.13) is bounded by Z c Ckw kq,r k dx ws (·; x)kq,r . (A.14) The Hq,r norm of the integral can be bounded by sup kws (·; x)kq,r ≤ Ckws kHq,r,ν ,

(A.15)

x

provided ν > 1/2, where we used Sobolev’s inequality to estimate the supremum over x. Inserting (A.15) into (A.14) yields the bound claimed in (3.3). The remaining terms in f4 can be bounded in a similar fashion, but as noted above, they will tend to 0 as τ → ∞. In fact, they will be bounded by Cεe−τ /2 . Proof of Eq. (3.4) of Proposition 3.2. We finally bound the non-linear term eτ /2 g(wc , ws e−τ /2 , e−τ /2 ) .

(A.16)

In bounding eτ /2 g(wc , ws e−τ /2 ; e−τ /2 ), recall that just as ws did, this expression will have two components – one in Hq,r , and one in Hq,r,ν . The component in Hq,r is bounded using exactly the same techniques used to control the term f4 above, so we concentrate here on explaining the new ingredients necessary to bound the component in Hq,r,ν .

Geometric Stability Analysis for Solutions of Swift-Hohenberg Equation

195

As in the bound on f4 , the potentially largest terms are those of minimal order, because each additional order provides a factor of e−τ /2 . So we look at the terms which are quadratic and which are of order wc wc , wc ws , and ws ws , respectively. The first term leads us to study   Z 1/2 τ /2 ⊥ 0 c 0 c 0 0 0 dp V (p − p )V (p )ϕp (x) ϕp−p (x) . (A.17) e Pp uε (x) −1/2

Rescaling as in (2.8), we see we must bound  Z P (τ ) ⊥ dp0 80 (p0 e−τ /2 ) P8(pe−τ /2 ) uε (x) −P (τ )

0

0



(A.18)

× ϕ8(p0 e−τ /2 ) (x) ϕ0(p,p0 ,τ ) (x) w (p ) w (1(p, p , τ )) . c

c

Note that the prefactor of eτ /2 has disappeared due to the factor of e−τ /2 which we gain as usual from the change of variables. Since the projection P`⊥ has bounded norm and is a smooth function of `, we can discard this factor at the price of introducing an overall constant in the estimate. Note next that the square of the Hq,r,ν norm of the remaining expression is equal to:

Z P (τ )

dp0 80 (p0 e−τ /2 )wc (p0 )wc (1(p, p0 , τ ))

−P (τ ) (A.19)

2

2 0 , × kuε (x) ϕ8(p0 e−τ /2 ) (x) ϕ0(p,p ,τ ) (x) kH ν (dx) Hq,r (dp)

where the H ν norm is the H ν -Sobolev norm of the quantity uε (x) ϕ8(p0 e−τ /2 ) (x) ϕ0(p,p0 ,τ ) (x) , considered as a function of x, and the Hq,r norm is the norm of the resulting function of p. Since uε (x) ϕ8(p0 e−τ /2 ) (x) ϕ0(p,p0 ,τ ) (x) is a smooth function of x, p0 , and p, there exists a smooth, bounded function ψ(p, p0 ), such that ψ(p, p0 ) = kuε (x) ϕ8(p0 e−τ /2 ) (x) ϕ0(p,p0 ,τ ) (x) kH ν (dx) .

(A.20)

But now, by Lemma A.2, we can conclude that (A.19) is bounded by Z P (τ ) dp0 80 (p0 e−τ /2 )wc (p0 )wc (1(p, p0 , τ ))ψ(p, p0 )k2Hq,r (dp) ≤ Ck80 ψk2C r kwc k4q,r . k −P (τ )

(A.21) We next consider the quadratic term in g which contains one factor of wc and one factor of ws . In this case, the analog of (A.18) is  Z P (τ ) ⊥ u (x) dp0 80 (p0 e−τ /2 ) e−τ /2 P8(pe −τ /2 ) ε −P (τ ) (A.22)  × ϕ0(p,p0 ,τ ) (x) wc (1(p, p0 , τ ))ws (p0 ; x) . Note that in this case, we pick up an extra factor of e−τ /2 , in comparison with (A.18), since each factor of ws is multiplied by this exponential.

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Once again, we must contend with the fact that ws has two components. However, the component in Hq,r behaves exactly as in the estimates leading to (3.2), so we concentrate on the component in Hq,r,ν . As above, the projection operator can be dropped at the cost of an overall constant, and we are left with the task of bounding the Hq,r,ν norm of the remainder. The square of this norm is equal to

Z P (τ )

dp0 80 (p0 e−τ /2 )wc (1(p, p0 , τ ))

−P (τ )

2

× kuε (x) ϕ8(p0 e−τ /2 ) (x) ϕ0(p,p0 ,τ ) (x) ws (p0 ; x)k2H ν (dx)

Hq,r (dp)

0

kC r kwc k2Hq,r

≤ Ck8

2

× kuε (x) ϕ8(p0 e−τ /2 ) (x) ϕ0(p,p0 ,τ ) (x) ws (p0 ; x)k2H ν (dx)

Hq,r (dp0 )

,

(A.23) by Lemma A.2. Note that the pair of norms on the last factor is equivalent to computing the square of the Hq,r,ν norm of uε (x) ϕ8(p0 e−τ /2 ) (x) ϕ0(p,p0 ,τ ) (x) ws (p0 ; x) .

(A.24)

Since uε , 8, ϕ0 , and 1 are all smooth, bounded functions, we see just by writing out the definition of the norm that this is bounded by Ckws k2Hq,r,ν .

(A.25)

If we estimate the term quadratic in ws in a similar fashion, and combine this estimate with that in (A.21) we see that the quadratic terms in e−τ /2 g(wc , ws e−τ /2 ; e−τ /2 ) are bounded in Hq,r,ν , by C(kwc kHq,r + e−τ /2 kws kHq,r,ν )2 .

(A.26)

Analogous estimates of the cubic terms lead to a bound Ce−τ /2 (kwc kHq,r + e−τ /2 kws kHq,r,ν )3 ,

(A.27)

where the additional factor of e−τ /2 comes from the additional convolution. Combining (A.26) and (A.27) leads to the estimate in (3.4) and completes the proof of Proposition 3.2. B. Bounds on the Linear Operators In this Appendix, we give bounds on the semi-group generated by L and on the linear evolution defined by Mexp(−τ /2) . B.1. Bound on the semi-group generated by L. We consider the semi-group whose generator is L = ∂x2 + 21 x∂x + 21 . Note that in this section, for ease of use, we define L in the Fourier transformed variables, compared to Sect. 2. Fourier transformation is an isomorphism from Hq,r (in the p-variables) to Hr,q (in the x-variables), so establishing estimates on the semigroup associated to ∂x2 + 21 x∂x + 21 in the space Hr,q (dx) will

Geometric Stability Analysis for Solutions of Swift-Hohenberg Equation

197

immediately imply estimates on the representation of L in the p-variables in the space Hq,r (dp). In order to avoid confusion, in what follows we will denote by | · |q,r the norm on Hr,q (dx). With this notation, the norms k · kq,r and | · |q,r resp. the spaces Hq,r (dp) and Hr,q (dx) are equivalent. The integral kernel of the semigroup generated by L is given by [GJ] Z 2 1 τL τ /2 dz e−z /(4a(τ )) v(eτ /2 (x + z)) , e e v)(x) = √ 4πa(τ ) where a(τ ) = 1 − e−τ . If we denote by T the operator of multiplication by exp(x2 /8) and by H0 the harmonic oscillator Hamiltonian H0 = ∂x2 − x2 /16 + 1/4, (note the unconventional sign!), then L = T −1 H0 T . Thus, the two operators L and H0 are “the same,” but they act on two quite different spaces. If the {ϕj }j≥0 are the eigenfunctions of H0 , then the ψj = T −1 Pϕj are the eigenfunctions of L, with the same eigenvalues µj = −j/2. We let Pn f = j≤n ψj (ψj , f )q , where (·, ·)q is the scalar product (f, g)q = (T f, T (1 − L0 )q g) = (T f, (1 − H0 )q T g) . We next show that for n < q − 2, the operator Pn is bounded in Hr,q (dx). First of 2 all, the eigenfunctions ϕj are bounded by O(1)|x|j e−x /8 at large x. Therefore, we also have ψj = T −1 ϕj ∈ Hr,q (dx), since it decays exponentially. Finally, (ψj , f )q = (T ψj , (1 − H0 )q T f ) = |1 − µj |q (ϕj , T f ) , and the last scalar product is bounded if f ∈ Hr,q (dx) when r > j + 2, since, with a weight function W (x) = (1 + x2 )1/2 , |(ϕj , T f )| ≤ C|(W j , f )| ≤ C|(W −1 , W j+1 f )| ≤ CkW j+1 f k2 ≤ C|f |0,r . Thus Pn is defined. We let Qn = 1 − Pn (in Hr,q (dx)). Theorem B.1. For every δ > 0, there exists an m0 and a function r(m, q) such that for every m ≥ m0 , every q ≥ 1 and every r ≥ r(m, q), there is a C = C(q, r, m) < ∞ such that C(q, r, m) −τ (|µm |−δ) |v|q−1,r . (B.1) e |eτ L Qm v|q,r ≤ √ a(τ ) Remark. The function r(m, q) is of order O(m + q). Proof. To explain the strategy of the proof, we need some notation. Let Pn(0) denote the (0) projection in H0,q (dx) onto the subspace spanned by {ϕj }j≤n and let Q(0) n = 1 − Pn . (0) −1 (0) Then, formally, T Qn = Qn T , and LQn = T H0 Qn T . This suggests that L restricted to Qn has no spectrum in the half-plane {z | Re z > −|µn+1 |}, and thus one can understand the decay in Eq. (B.1). The square-root singularity at τ = 0 is related to our gain in smoothness. The problem is that T Qn = Q(0) n T is ill-defined. However, it will be well defined if we localize near x = 0. In that region, the heuristic argument will be seen to be valid, whereas in the complement of such a region, when |x| > R, decay will be shown by direct methods, using the explicit form of the integral kernel.

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We study first the quantity χR eτ L , where χR is a smooth characteristic function which vanishes for |x| < R and is equal to 1 for |x| > 4R/3. Thus we study a region far from the origin. Our bound is Proposition B.2. For every q ≥ 1 and every r ≥ 0 there exists a C(q, r) < ∞ such that for all v ∈ Hr,q (dx) one has  2 C(q, r) τ q/2  −τ r/2 |χR eτ L v|q,r ≤ √ + e−3R /16 |v|q−1,r , e e a(τ )   2 τL |χR e v|q,r ≤ C(q, r)eτ q/2 e−τ r/2 + e−3R /16 |v|q,r .

(B.2) (B.3)

Corollary B.3. For every q ≥ 1 and every r ≥ 0 there exists a C(q, r) < ∞ such that for all v ∈ Hr,q (dx) one has C(q, r) τ q/2 |eτ L v|q,r ≤ √ |v|q−1,r , e a(τ )

(B.4)

|eτ L v|q,r ≤ C(q, r)eτ q/2 |v|q,r .

(B.5)

Remarks. The improvement over [W] is that we “gain” a derivative in x. The corollary follows easily by repeating the proof of Proposition B.2 with R = 0. Proof. We let D = ∂x and denote, as before, by W the operator of multiplication by (1 + x2 )1/2 . Then X 0 |χR eτ L w|2q,r and kW r Dq χR eτ L wk22 q 0 ≤q

are equivalent. We shall only consider the term with the highest derivative, because only there is the issue of regularization important. Thus we are led to bound X 2 = kW r Dq χR eτ L wk22 . Since L = ∂x2 + 21 x∂x + 21 , a quick calculation shows that Dq eτ L = eτ q/2 eτ L Dq . The diverging factor exp(τ q/2) will appear in the final bound. Note now that Z

2 1 τL q τ /2 dz e−z /(4a(τ )) Dq v (eτ /2 (x + z)) , e D v (x) = √ e 4πa(τ ) which upon integrating by parts becomes Z
2 z 1 √ dz e−z /(4a(τ )) Dq−1 v (eτ /2 (x + z)) . 2a(τ ) 4πa(τ ) Use now the Schwarz inequality in the form (for positive f and g),

(B.6)

Geometric Stability Analysis for Solutions of Swift-Hohenberg Equation

Z kf ∗ gk22 =

199

Z dx

Z

dz1 dz2 f (z1 )f (z2 )g(x − z1 )g(x − z2 )

dz1 dz2 f (z1 )f (z2 )kg(· − z1 )k2 kg(· − z2 )k2

≤ Z

2 dz f (z)kg(· − z)k2

=

.

This leads to a bound Z
eτ q/2 |z| −z2 /(4a(τ )) X ≤ √ e dz kW r χR Dq−1 w (eτ /2 (· + z))k2 4πa(τ ) R1 ∪R2 2a(τ ) ≡ X1 + X2 ,

(B.7)

where we let R1 = {x : |x| < 7R/8} and R2 = R\R1 . To be more precise, we define χR by the scaling of a fixed function: χR (x) = χ(x/R). If R → ∞, then ∂x χR (x) = O(R−1 ) and therefore it is uniformly bounded. Lemma A.2 of [W] B.4. One has the bounds  −rτ 2 Ce |v|0,r , kW r χR (·)v(eτ /2 (· + z))k22 ≤ C(1 + z 2 )r |v|20,r ,

if |z| ≤ 7R/8, if |z| > 7R/8.

(B.8)

Proof of Lemma B.4. Consider first the case |z| ≤ 7R/8. Since |x| > R on the support of χR , we have |x + z| ≥ |x|/8 and hence
(1 + x2 )/ 1 + (eτ /2 |x + z|)2 ≤ const. e−τ . Using this, we bound Z dx (1 + x2 )r |χR (x)v(eτ /2 (x + z))|2 R1

Z dx

= R1


r (1 + x2 )r
r · 1 + (eτ /2 |x + z|)2 |v(eτ /2 (x + z))|2 τ /2 2 1 + (e |x + z|)

≤ const. e−τ r e−τ /2 |v|20,r ≤ const. e−τ r |v|20,r . In the second case, we get Z dx (1 + x2 )r |χR (x)v(eτ /2 (x + z))|2 R2

= e

−τ /2

Z

1 + (e−τ /2 y − z)2 dy (1 + y 2 )r


r (1 + y 2 )r |v(y)|2

≤ const. e−τ /2 (1 + z 2 )r |v|20,r ≤ const. (1 + z 2 )r |v|20,r . The proof of Lemma B.4 is complete. Continuing the proof of Proposition B.2, we first bound the integral over R1 in Eq. (B.7). We get from the first alternative of Lemma B.4,

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J.-P. Eckmann, C.E. Wayne, P. Wittwer

Z
2 1 |z| τ q/2 dz √ e e−z /(4a(τ )) kW r χR Dq−1 w (e−τ /2 (· + z))k2 4πa(τ ) 2a(τ ) R1 Z 2 1 |z| τ q/2 ≤ const. √ dz √ e e−z /(4a(τ )) e−τ r/2 |w|q−1,r 4πa(τ ) 2a(τ ) R1 1 τ (q/2−r/2) ≤ const. √ |w|q−1,r . e 4πa(τ )

X1 = √

Similarly, using the second alternative in Eq. (B.8), we get Z
2 1 |z| dz √ X2 = √ eτ q/2 e−z /(4a(τ )) kW r χR Dq−1 w (eτ /2 (· + z))k2 4πa(τ ) 2a(τ ) R2 Z 2 1 |z| ≤ const. √ dz (1 + z 2 )r/2 √ eτ q/2 e−z /(4a(τ )) |w|q−1,r . 4πa(τ ) 2a(τ ) R2 1 τ q/2 −3R2 /16 e |w|q−1,r , ≤ const. √ e 4πa(τ ) since 3/16 < (7/8)2 /4. Note that the constants above depend on r and q, but can be chosen uniformly for all R ≥ 1. The proof of Eq. (B.2) is complete. Omitting the integration by parts in Eq. (B.6), the assertion Eq. (B.3) follows in the same way. The proof of Proposition B.2 is complete. We next study eτ L Qn (1 − χR )w. We have the following bound Proposition B.5. For every δ > 0, q ≥ 1, and every r ≥ 0 there is a C(δ, q, r) < ∞ such that |eτ L Qn (1 − χR )w|q,r ≤

C(δ, q, r) −(|µn+1 |−δ)τ R2 /6 √ e |w|q−1,r . e a(τ )

(B.9)

Proof. Recall that T = ex /8 and that L = T −1 H0 T . The operator T (1 − χR ) is bounded 2 and kT (1 − χR )k ≤ const. eR /6 . Therefore we have 2

Qn T (1 − χR ) = (1 − Pn )T (1 − χR ) = T (1 − χR ) − T Pn(0) (1 − χR ) = T (1 − Pn(0) )(1 − χR ) = T Q(0) n (1 − χR ) , where Q(0) n is the orthogonal projection onto the complement of the subspace spanned by the first n eigenvalues of H0 in Hq,0 . It is easy to see that on Hr,q (dx), the operator (1 + x2 )1/2 (1 − H0 )−1/2 is bounded. Thus, we get, using the spectral properties of H0 (on Q(0) n ), |eτ H0 T Qn (1 − χR )w|q,r = τ −1/2 × |(1 − H0 )−1/2 eτ H0 (τ (1 − H0 ))1/2 Q(0) n T (1 − χR )w|q,r ≤ const. τ −1/2 |eτ H0 (τ (1 − H0 ))1/2 Q(0) n T (1 − χR )w|q−1,r ≤ const. τ −1/2 e−τ (|µn+1 |−δ) |T (1 − χR )w|q−1,r ≤ const. τ −1/2 e−τ (|µn+1 |−δ) eR

2

/6

|w|q−1,r . (B.10)

The proof of Proposition B.5 is complete.

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201

End of proof of Theorem B.1. We first rewrite eτ L Qn as eτ L Qn = eτ L/2 Qn eτ L/2 = eτ L/2 Qn χR eτ L/2 + eτ L/2 Qn (1 − χR )eτ L/2 . The second term can be bounded by Proposition B.5 and Eq. (B.5) as 2 C eR /6−τ |µn+1 |/4 |eτ L/2 w|q−1,r a(τ ) 2 C ≤ √ eR /6−τ |µn+1 |/4 eτ q/4 |w|q−1,r . a(τ )

|eτ L/2 Qn (1 − χR )eτ L/2 w|q,r ≤ √

This quantity is bounded by √

C e−τ n/8 |w|q−1,r , a(τ )

(B.11)

provided n is much larger than q and R2 /6 < τ n/16. The first term can be bounded by Eq. (B.5) and Eq. (B.2) as C eτ q/4 |χR eτ L/2 w|q−1,r a(τ )   2 C ≤ √ eτ q/2 e−τ r/2 + e−3R /16 |w|q−1,r a(τ ) C ≤ √ e−τ n/8 |w|q−1,r , a(τ )

|eτ L/2 Qn χR eτ L/2 w|q,r ≤ √

(B.12)

provided r ≥ n/4 + q and 3R2 /16 ≥ τ (n/8 + q/2). Note that the conditions on R from the first and second term are compatible Combining Eqs.(B.11)–(B.12), we get |eτ L Qn w|q,r ≤ √

C e−τ n/8 |w|q−1,r . a(τ )

(B.13)

It remains to improve the decay rate from n/8 to |µm+1 |. The idea is to just take n = 8(m + 1). Then we find eτ L Qm = eτ L Qn Qm + eτ L Pm Qm + eτ L (Pn − Pm )Qm .

(B.14)

The first term is bounded by Eq. (B.13), and m/8 > −|µn+1 |. The second term vanishes and the third is diagonalized explicitly: (0) )T Qm . eτ L (Pn − Pm )Qm = T −1 e−τ H0 T (Pn − Pm )Qm = T −1 e−τ H0 (Pn(0) − Pm

We are operating here on the finite dimensional subspace spanned by the eigenvectors ϕm+1 , . . . , ϕn , and there the technique of Eq. (B.10) yields a bound √

C p τ |µm+1 |e−|µm+1 |τ . a(τ )

Combining this with the bound on the first term in Eq. (B.14), we complete the proof of Theorem B.1.

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B.2. The linear evolution generated by Mη,2 . In this section, we deal with the problem of giving bounds on the linear evolution generated by the operator Mη,2 , which is defined by ∞

Mη,2 = Mη,2,0 ⊕ ⊕ Mη,2,n , n=2

where

 Mη,2,n =


2 2

ε − (1 + (in + i8(pη)) 2






− K 8(pη)

− η 2 21 p∂p .

We want to bound the solution Un,τ of the equation e−τ ∂τ Un,τ = Mexp(−τ /2),2,n Un,τ ,

(B.15)

with Un,0 = 1. Recall the definition of L = −p2 − 21 p∂p , and rewrite Mexp(−τ /2),2,n as Mexp(−τ /2),2,n  
2 = ε2 − 1 + (in + i8(pe−τ /2 ))2 − K(8(pe−τ /2 )) − e−τ 21 p∂p  
2 = ε2 − 1 + (in + i8(pe−τ /2 ))2 − K(8(pe−τ /2 )) + e−τ p2 + e−τ L = Xn (pe−τ /2 ) + e−τ L , where Xn (ξ) = ε2 − 1 + (in + i8(ξ))2


2

− K(8(ξ)) + ξ 2 . We want to solve Eq. (B.15):

e−τ ∂τ Un,τ = (e−τ L + Xn (pe−τ /2 ))Un,τ , with initial condition Un,0 = 1. Observe now that Xn is an operator of multiplication m by a function of pη. Since the commutator [pm , −p2 − 21 p∂p ] is equal to m 2 p , we find 1 0 [h(p), L] = 2 ph (p), and, furthermore, eh(p) L = (L + 21 ph0 (p))eh(p) . It follows that the solution of Eq. (B.15) is τ

Un,τ = e(e

−1)Xn (pe−τ /2 ) τ

e L,

as one can check by explicit computation. From the explicit form of Xn , (in particular, the factor of −n4 ), and the estimates derived in Theorem B.1, we see that for any xn ∈ Hq,r , we have kUn,τ xn kq,r ≤ C exp(−c0 (eτ − 1)n4 )eτ q/2 kxn kq,r . Combining this with the Remark of (A.11), we immediately obtain Lemma B.6. If Uτ satisfies e−τ ∂τ Uτ = Mexp(−τ /2),2 Uτ , with U0 = 1, then there exist a C(r, q, ν) > 0, and a c0 > 0 such that for any w ∈ Hq,r,ν , kUτ wkq,r,ν ≤ C exp(−ec0 τ /2 )kwkq,r,ν .

(B.16)

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203

To complete the proof of Theorem 3.4, we also need an estimate of the semigroup generated by Mη,1 . This is simply obtained, however, because from (2.13) we see that Mη,1 = Mexp(−τ /2),2,1 , restricted to functions whose Fourier transform is supported away from the origin. Using this fact, and the explicit formula given above for Mexp(−τ /2),2,n , we see immediately that for any w1 ∈ Hq,r , there exists a constant c1 > 0, such that if Uτ,1 is the semigroup generated by Mη,1 one has kUτ,1 w1 kq,r ≤ Ce−c1 τ kw1 kq,r .

C. The Pseudo Center Manifold Theorem for the Singular System Eq. (3.8) In this section, we prove Theorem 3.7. Before we start with the proof, we wish to point out in which sense we are here confronted with a new problem, which does not allow for a straightforward application of results from the literature. If we write the system Eq. (3.8) in the form ∂τ x1 = A1 x1 + N1 (x1 , η, x2 , x3 ) , ∂τ η = − 21 η , ∂τ x2 = A2 x2 + N2 (x1 , η, x2 , x3 ) ,

(C.1)

∂τ x3 = η −2 A3,η x3 + η −2 N3 (x1 , η, x2 , x3 ) , then, in view of the spectral properties of Eq. (3.9), there is a “gap” between the “central” part (corresponding to x1 and η) and the “stable” part (corresponding to x2 , x3 ). The problem is that we are really dealing with a singular perturbation because the nonlinearity in the equation for x3 also diverges as η ↓ 0. This problem would be more easily overcome if A2 were bounded. In that case, for sufficiently small η, the spectra of A2 and η −2 A3 would not overlap, and we could define first an invariant manifold by “eliminating” x3 , and then the true invariant manifold by eliminating x2 from the equations obtained after elimination of x3 . However, since the spectra overlap for all values of η, we resort to a strategy which consists of a converging sequence of alternate eliminations of x2 and x3 . To define these successive eliminations, we consider two equivalent representations of Eq. (3.8), one being Eq. (C.1) above and the other being
∂t x1 = η 2 A1 x1 + N1 (x1 , η, x2 , x3 ) , ∂t η = − 21 η 3 ,


∂t x2 = η 2 A2 x2 + N2 (x1 , η, x2 , x3 ) ,

(C.2)

∂t x3 = A3,η x3 + N3 (x1 , η, x2 , x3 ) . We shall again omit the index η from A3 . We obtain Eq. (C.2) from Eq. (C.1) by rescaling the evolution parameter of the autonomous system as t + t0 = exp(τ ). (Note that time is really given by 1/η 2 − t0 , while we view t and τ as the evolution parameters of the vector fields.) We will call 8center the flow corresponding to Eq. (C.1) and 8stable the τ t flow corresponding to Eq. (C.2). A simple inspection of the definition of these flows yields the useful identity:

204

J.-P. Eckmann, C.E. Wayne, P. Wittwer stable 8center τ =log (y+t0 ) (ξ, x) = 8t=y (ξ, x) ,

(C.3)

where ξ = (x1 , η),

x = (x2 , x3 ) .

(C.4)

We shall use the relations (C.4) throughout. The identity (C.3) holds for all x1 , x2 , x3 and for η ≥ 0. Note that the initial conditions are given for the parameter t = 0 and the −1/2 parameter τ = log(t0 ), and that η(0) = t0 . Thus, η(0) is small if the parameter t0 has been chosen sufficiently large. (The bounds on the nonlinearities are uniform in t0 ≥ t∗0 as follows from the calculations.) Let h0 be a function of ξ. This function will always be an approximate invariant manifold for one of two problems. To define these problems, we first introduce two effective non-linearities
Fj (h0 ; ξ, x2 ) = Nj x1 , η, x2 , h0 (ξ) , for j = 1, 2 ,
Gj (h0 ; ξ, x3 ) = Nj x1 , η, h0 (ξ), x3 , for j = 1, 3 . We then define two equations (corresponding to the two different time scales Eq. (C.1) and Eq. (C.2) of the same problem Eq. (3.8)): The first equation will be called the center system: ∂τ x1 = A1 x1 + F1 (h0 ; ξ, x2 ) , ∂τ η = − 21 η ,

(C.5)

∂τ x2 = A2 x2 + F2 (h0 ; ξ, x2 ) . Similarly, we define the stable system ∂t x1 = η 2 A1 x1 + η 2 G1 (h0 ; ξ, x3 ) , ∂t η = − 21 η 3 ,

(C.6)

∂t x3 = A3 x3 + G3 (h0 ; ξ, x3 ) . Assume now that h2 and h3 are two given functions of x1 and η. We define a map  0   h2 h2 7→ , F : h3 h03 through the following construction: We let h02 (ξ) be the function whose graph is the invariant manifold for the center system Eq. (C.5) with non-linearity Fj (h3 ; ξ, x2 ), and similarly we let h03 (ξ) be the function whose graph is the invariant manifold for the stable system Eq. (C.6) with non-linearity Gj (h2 ; ξ, x3 ). Our main result here is Proposition C.1. The map F has a fixed point (h∗2 , h∗3 ). This fixed point provides an invariant manifold for the system Eq. (3.8). Remark. We shall in fact show that F is a contraction in a suitable function space. In particular, we show that F n (0, 0), the n-fold iterate of F , converges to the limit (h∗2 , h∗3 ). The intuitive approach behind this construction is that the F n (0, 0) provide a sequence of successive approximations to invariant manifolds for Eqs.(C.6) and (C.5), in which the non-linearities at the nth step are given by the approximate solutions for the invariant ; ξ, x2 ) manifold problem of the other equation: The non-linearities are then Fj (h(n−1) 3 ; ξ, x ) (in Eq. (C.6)). (in Eq. (C.5)) and Gj (h(n−1) 3 2

Geometric Stability Analysis for Solutions of Swift-Hohenberg Equation

205

Proof. That the systems of equations (C.5) and (C.6) have invariant manifolds follows from our estimates (given in Appendix B) on the semi-group generated by the linear operators A2 and A3 , and our estimates on the non-linear terms. (For expositions of this theory that are particularly relevant in the present context, see e.g., [H, M, G].) The functions h∗2 and h∗3 whose graphs define the invariant manifolds satisfy well known integral equations, see below. Fix h = (h2 , h3 ) and consider Eq. (C.5). We want to find the function h02 (h; ξ) which eliminates x2 . To construct h02 , we first consider the equation
∂τ x1 = A1 x1 + F1 h3 ; ξ, h2 (ξ) , (C.7) ∂τ η = − 21 η . This is a differential equation on a finite dimensional space and we let 92τ (ξ; h) denote the corresponding flow. (Of course, the η-component of this problem can be explicitly integrated.) We can then formulate the problem of finding the invariant manifold which eliminates x2 from Eq. (C.6) by looking at the map defined by h 7→ F2 (h) where Z 0
dτ e−A2 τ F2 h3 ; 92τ (ξ; h), h2 (92τ (ξ; h)) . (C.8) F2 (h) = −∞

(A particularly clear derivation of these equations can be found in [G].) In a similar way, we define the flow 93τ (ξ; h) for the equation
∂t x1 = η 2 A1 x1 + η 2 G1 h2 ; x1 , h3 (ξ) , (C.9) ∂t η = − 21 η 3 , and the map Z F3 (h) =

0 −∞


dt e−A3 t G3 h2 ; 93t (ξ; h), h3 (93t (ξ; h)) .

(C.10)

We now specify the function spaces in which we work. Recall that x1 ∈ RN , η ∈ R and that ξ ∈ RN +1 . We let E c = RN ⊕ R with the usual Euclidean norm. We also assume that E 2 and E 3 are the Banach spaces in which the x2 and x3 live. In our problem, these Banach spaces are the Hilbert spaces Hq,r and Hq,r,ν , but since we believe the present theory of singular vector fields may have further applications, we consider the more general case for the moment (see, for example, [W2]). These Banach spaces should have the C k extension property [BF]. The functions h2 and h3 will be Lipshitz functions from a ball of radius r in E 2 and E 3 , respectively. They satisfy hj (0) = 0 and are tangent at the origin to E j , for j = 2, 3. Thus, we define the metric spaces, for j = 2, 3: ˜ E j ≤ σkξ − ξk} ˜ . Hj,σ = {hj : E c → E j | hj (0) = 0, khj (ξ) − hj (ξ)k We also define a distance khj (ξ) − h˜ j (ξ)kEj , ρHj,σ (hj , h˜ j ) = sup kξk ξ6=0 and introduce the notation ˜ = ρH2,σ (h2 , h˜ 2 ) + ρH3,σ (h3 , h˜ 3 ) . ρHσ (h, h)

(C.11)

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J.-P. Eckmann, C.E. Wayne, P. Wittwer

Standard results about the existence and uniqueness of solutions of systems of differential equations now imply that ˜ h)k ≤ Ceβ2 |τ | kξ − ξk ˜ , k92τ (ξ; h) − 92τ (ξ; while

(C.12)

˜ ≤ Ceβ2 |τ | ρHσ (h, h) ˜ , k92τ (ξ; h) − 92τ (ξ; h)k

(C.13)

for any β2 > (N − 1)/2. Analogous estimates hold for the flow 9 , though in that case one can choose any exponential growth rate β3 > 0, provided |η| is sufficiently small. This is due to the presence of the factor of η 2 A1 in the first equation of Eq. (C.6). With this in mind we define two more metric spaces (for j = 2, 3): 3

Kj,βj ,Dj = {9τ : R+ × E c × H2,σ × H3,σ → E c | 90 (ξ; h) = ξ, 9τ (0; h) = 0, 9τ is C 1 in τ, ˜ h)k ˜ k9τ (ξ, h) − 9τ (ξ,


˜ ˜ + ρHσ (h, h)kξk ≤ Dj eβj |τ | kξ − ξk },

with a corresponding Lipshitz metric ˜ = k9 − 9k ˜ Kj , dj (9, 9) where k9kKj = sup sup t≥0

ξ∈E c ξ6=0

(C.14)

eβj t k9t (ξ)k . kξk

These spaces are modeled on those used in [EW]. Remark. Since we are interested in local invariant manifolds, we will assume that the non-linear terms have been cut off outside a ball of radius r in each of their arguments. Since in the applications of this paper all our functions are elements of Hilbert spaces, we can assume that there exist smooth cut-off functions which are equal to 1 inside a ball of radius r/2 and are equal to zero outside a ball of radius r, and we multiply each of the non-linear terms in Eq. (3.8) by such a cutoff. For example, in Eq. (C.6), we certainly need to cutoff the function η 2 by η 2 χ(η) (where χ is the cutoff function) to avoid blowup problems. Given this setup, we show that the map F is a contraction of H2,σ × H3,σ .
In terms of the notation given above F is now defined as F (h) = F2 (h), F3 (h) . One must first show that F maps this space to itself. This step is however an easy variant of the argument which shows that F is a contraction, and we leave it as an exercise to the reader. To show that F is a contraction, we use the maps (C.8) and (C.10). Then we see that the “j” component, j = 2, 3, of F(h2 , h3 )(ξ) − F (h˜ 2 , h˜ 3 )(ξ) is given by   Z 0 −Aj τ ˜ (C.15) Uj (h; ξ, τ ) − Uj (h; ξ, τ ) , dτ e 1j = −∞

where



U2 (h; ξ, τ ) = F2 h3 ; 92τ (ξ; h), h2 92τ (ξ; h)


= N2 92τ (ξ; h), h2 92τ (ξ; h) , h3 92τ (ξ; h) ,

U3 (h; ξ, τ ) = G3 h2 ; 93τ (ξ; h), h3 93τ (ξ; h)


= N3 93τ (ξ; h), h2 93τ (ξ; h) , h3 93τ (ξ; h) ,

Geometric Stability Analysis for Solutions of Swift-Hohenberg Equation

207

cf. Eqs.(C.5), (C.6). Consider now 12 . From the estimates on the non-linear term N2 in Eq. (3.8), we see that F2 is a multi-linear function of its arguments. Thus, we can estimate the difference in the integrand of 12 by the sum of the differences in the arguments of F2 , multiplied by the Lipshitz constant of F2 . Because we have cutoff F2 outside a ball of radius r, this Lipshitz constant can be made arbitrarily small by making r sufficiently small. Thus, calling this Lipshitz constant `2 (r), we see from the estimates on eA2 t which follow from the results of Appendix B and from Eqs.(C.11)–(C.14) that  Z ∞ C ˜ dτ √ e−N τ /2 `2 (r) ρHσ (h, h) k12 kE 2 ≤ τ 0 

˜ + kh2 92τ (ξ, h) − h˜ 2 92τ (ξ, h) ˜ kE 2 + k92τ (ξ, h) − 92τ (ξ, h)k  Z ∞ C ˜ ≤ dτ √ e−N τ /2 `2 (r) ρHσ (h, h) τ 0  β2 τ β2 τ ˜ ˜ ˜ . + ρHσ (h, h)Ce ≤ const. `2 (r)ρHσ (h, h) + ρHσ (h, h)Ce Thus, we have shown that F is a contraction. We next consider the manifold M given by (ξ, h∗2 (ξ), h∗3 (ξ)) – where x1 is in a small neighborhood of 0 and η is in a small positive interval 0 ≤ η ≤ η0 . We want to show that M is indeed an invariant manifold for the full system Eq. (3.8). From this it follows, since the flows 8stable and 8center are equivalent, up to rescaling of time, that M is also an invariant manifold for Eqs.(C.1) and (C.2). If we set x2 = h∗2 (ξ) and x3 = h∗3 (ξ), then the third equation of Eq. (C.2) is satisfied because the third equation, when restricted to the manifold x2 = h∗2 is just the second equation of the stable system Eq. (C.6). with non-linearity G3 (h∗2 ; . . .). To see that the remaining equations are satisfied just note that the first, second and fourth equations in the full system Eq. (3.8) become, after rescaling of time, x˙ 1 = A1 x1 + N1 (x1 , η, x2 , x3 ) , η˙ = − 21 η 3 , x˙ 2 = A2 x2 + N2 (x1 , η, x2 , x3 ) , h∗2

and if we set x2 = and x3 = h∗3 , we see that we are just on the invariant manifold for the center system Eq. (C.5). Hence, we have found the invariant manifold for the full system Eq. (3.8).

D. The Vanishing of the Non-Linearity at Zero Momentum In this Appendix, we prove Proposition 4.3. This proof is essentially a scaling argument. We shall study the nonlinearity N1 (x1 , η, x2 , x3 ) and we restrict it to the invariant manifold, i.e., we replace it by N˜ 1 (x1 , η) and let η go to 0. In particular, we shall show that only one term survives, namely the one which is cubic in x31 , and all others go to 0 as η → 0. To prove this, we will analyze the nonlinearities Nj term by term, using their definitions as given in Eqs.(3.6) and (3.8). Recall again that A1 = 0 since we are considering here the projection onto the first eigenvalue of L. In Eq. (3.6), the nonlinearities are given by the terms f2 , f3 , f4 , and g, and these have been bounded in Proposition 3.1 and

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Proposition 3.2. Recall finally that every factor of wc contributes a factor of e−τ /2 = η and every factor of ws contributes a factor of e−τ = η 2 to these bounds. Begin by considering the contribution from f2 . According to Eq. (A.9), we can extract another factor of η from Eq. (A.10), by using the quadratic nature of K2 , cf. Lemma A.1. Using Proposition 3.1 and Proposition 3.2, we see that the only contributions from f3 , f4 , and g which do not vanish as η → 0 are those of the type (wc )3 in f3 , of the type ws (wc )2 in f4 , and of the type (wc )2 in g. We start by analyzing f3 . If we write it out, we find Z

dx ϕ¯ 8(pη) (x) η −2 f3 (wc ) (p) = η −2 χ 8(pη) Z ×η

2

η −1 8(1/2)

dp1 dp2 η −1 8(−1/2)

80 (p1 η)80 (p2 η)

× ϕ8(p1 η) (x) ϕ8(p2 η) (x) ϕ8(pη)−8(p1 η)−8(p2 η) (x)


× wc (p1 ) wc (p2 ) wc η −1 8−1 (8(pη) − 8(p1 η) − 8(p2 η)) ,

cf. Eq. (2.11). Upon taking η → 0, this converges to Z Z χ(0) dx ϕ¯ 0 (x)ϕ30 (x) dp1 dp2 wc (p1 )wc (p2 )wc (p − p1 − p2 ) .

(D.1)

Analogous arguments can be used to discuss the “surviving” terms of f4 and g. We just summarize the steps analogous to the calculation of f3 . One gets, as η → 0, Z Z
−2 c s f4 (w , w η, η) (p) → 6χ(0) dx ϕ¯ 0 (x)uε (x)ϕ0 (x) dp0 wc (p0 )ws (p − p0 ; x) , η (D.2) and
η −1 g(wc , ws η, η) (p) → −3uε (x)ϕ20 (x)

Z

dp0 wc (p − p0 )wc (p0 ) .

(D.3)

We next study these limiting expressions in the basis {ψn (p)}∞ n=0 of eigenfunctions of L = −p2 − 21 p∂p . Then we can write wc (p) as x1 ψ0 (p) +

∞ X

x(n) 2 ψn (p) .

(D.4)

n=1 ∗,(n) The crucial remark is now that on the invariant manifold, x(n) , 2 will be replaced by h2 s ∗ ∗ and similarly w will be equal to h3 . We now compute the limiting forms of h2 and h∗3 , and then we substitute these values in Eqs.(D.1)–(D.3). Consider the equation for h∗3 . Then from Eq. (C.9), we have

∂t x1 = η 2 G1 (h∗2 ; x1 , h∗3 (ξ)) , ∂t η = − 21 η 3 , because we are considering the case N = 1 where the linear part vanishes. We also have from Eq. (C.10), Z 0
dt e−A3,η t G3 h∗2 ; 93t (x1 , η; h∗ ), h∗3 (93t (x1 , η; h∗ )) . (D.5) h∗3 (x1 , η) = −∞

Geometric Stability Analysis for Solutions of Swift-Hohenberg Equation

209

Now, when η = 0, we have 93t (ξ; h) = 93t (x1 , 0; h) = x1 , and this reduces to h∗3 (x1 , 0)

Z

0

dt e−A3,0 t G3 h∗2 ; x1 , 0, h∗3 (x1 , 0) −∞
∗ ∗ = −A−1 3,0 G3 h2 ; x1 , 0, h3 (x1 , 0) . =


(D.6)

Note next that for η = 0 we have A3,0 = M0 , cf. Eq. (3.8), and this means A3,0 = Qper Lper . We denote by ξn (x) the eigenfunctions and by σn the eigenvalues of Qper Lper . Using Eq. (1.9) and Theorem 1.1, we see that σn = λ`=0,n−1 and therefore they are given by σ1 = −O(ε2 ) and σn ≈ −(1 − (n − 1)2 )2 , when n 6= 1. Then the nth component (in this basis) of h∗3 (at η = 0) is given by Z Z
dp0 wc (p − p0 )wc (p0 ) , (D.7) h3∗,(n) (p) = −σn−1 · −3 dx ξ¯n (x)uε (x)ϕ20 (x) since all other terms vanish in the limit η → 0. We next substitute the value Eq. (D.4) for wc and set x2 = h∗2 in Eq. (D.7), and get Z
∗,(n) 2 −1 h3 (p) = −x1 σn · −3 dx ξ¯n (x)uε (x)ϕ20 (x)  Z × dp0 ψ0 (p0 ) ψ0 (p − p0 ) + O(x1 h∗2 + (h∗2 )2 ) . Next, we replace ws in Eq. (D.2) with h∗3 , and in that same equation make the substitution for wc that we used above, and we find: 18x31

∞ X

σn−1

n=0

Z

Z




dx ξ¯n (x)uε (x)ϕ20 (x)




dx0 ϕ¯ 0 (x0 )uε (x0 )ξn (x0 )

Z ×

dp1 dp2 ψ0 (p1 )ψ0 (p2 )ψ0 (p − p1 − p2 ) +

O(x1 h∗2



+

(h∗2 )2 )

(D.8) .

Thus we see that the only terms which survive in N1 and N2 in the limit η → 0 result from adding together Eqs.(D.8) and (D.1). We obtain Z X = x31 Z × Z ×

dx ϕ¯ 0 (x)ϕ30 (x) + 18

∞ X n=0

σn−1 (

Z






dx00 ϕ¯ 0 (x00 ) uε (x00 ) ξn (x00 )




dx0 ξ¯n (x0 )uε (x0 )ϕ20 (x0 )

(D.9)

 dp1 dp2 ψ0 (p1 ) ψ0 (p2 ) ψ0 (p − p1 − p2 ) .

This coefficient will turn out to be exactly the same as that which appears below as the coefficient of the cubic terms in the center manifold in the periodic case, and since we know that in the periodic case this coefficient (and indeed, the entire nonlinear term) is zero, it must vanish in the present case as well. The only remaining point in the proof of Proposition 4.3 is the computation of the coefficient of the cubic term in the equation in the center manifold in the periodic case, and we do that in the following subsection.

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J.-P. Eckmann, C.E. Wayne, P. Wittwer

Remark. The above argument might seem incomplete since it ignores the O(x1 h∗2 +(h∗2 )2 ) error terms in (D.8). In fact, those terms vanish for x1 small. To see why, note that our computations of the η → 0 limit of f2 , f3 , f4 and g apply also to the nonlinear term N2 (x1 , η, h∗2 (ξ), h∗3 (ξ)) in the equation for h∗2 in (4.1). Thus, in the η → 0 limit h∗2 satisfies: ∂x1 h∗2 (x1 , 0)N˜ 1 (x1 , 0) = A2 h∗2 (x1 , 0) + N2 (x1 , 0, h∗2 (x1 , 0), h∗3 (x1 , 0)) . Using the estimates on h∗2 and h∗3 derived above, we see that this equation implies h∗2 (x1 , 0) = 0 for all x1 sufficiently small, and hence the error terms in (D.8) vanish. D.1. The non-linearity in the periodic case. In this subsection we compute the explicit form of the non-linearity (which we know to be 0 because the invariant manifold is made up of fixed points in this case). But this explicit form will allow us to compare it with the expression obtained in Eq. (D.9) so that the proof of Proposition 4.3 will be complete. We start from the equation ∂τ v = Lper v − 3uε v 2 − v 3 .

(D.10)

Let y0 be the component of v in the direction of the highest eigenvalue, σ0 = 0, of Lper , and yn , the projection onto the directions ξn , defined after Eq. (D.6), associated to the eigenvalues σn . Then the invariant manifold can be written in the form yn = Yn (y0 ) ,

n = 1, 2, . . . .

(D.11)

Using the fact that the eigenfunction with eigenvalue 0 is u0ε , we can decompose v as: v(x) = y0 u0ε (x) +

∞ X

ξn (x)Yn (y0 ) ,

(D.12)

n=1

the projection of Eq. (D.10) onto the invariant manifold leads to Z
∂τ y0 = − dx u0ε (x) 3uε (x)v(x)2 + v(x)3 .

(D.13)

Note that there is no linear term because σ0 = 0. We are interested in the exact form of the cubic term in y0 on the r.h.s. of Eq. (D.13). There are two contributions, one from v 3 , leading to Z 3 −y0 dx u0ε (x)4 , (D.14) and one from the quadratic non-linearity: Z ∞ X Yn(2) (y0 ) dx u0ε (x) uε (x) u0ε (x) ξn (x) . Y = −6y0

(D.15)

n=1

Here, Yn(2) (y0 ) is the quadratic term in y0 of Yn . Substituting Eq. (D.13) into the equation for Yn , we find the perturbative result: Z (2) 2 −1 Yn (y0 ) = y0 · 3σn dx ξ¯n (x) uε (x) u0ε (x)2 .

Geometric Stability Analysis for Solutions of Swift-Hohenberg Equation

Inserting into Eq. (D.15), it is seen to become Z Z ∞ X σn−1 dx u0ε (x)2 uε (x) ξn (x) dx0 ξ¯n (x0 ) uε (x0 ) u0ε (x0 ) . Y = −y03 18

211

(D.16)

n=1

Combining Eqs.(D.14) and (D.16), we get the desired result, namely that the cubic nonlinearity in the periodic case coincides with the quantity X of Eq. (D.9), provided we recall that ϕ0 = u0ε . This completes the proof of Proposition 4.3. Acknowledgement. This work was begun while J-P E was a visitor at the Pennsylvania State University. It was completed during a visit of CEW to the University of Geneva. The support of the Shapiro Visitors Fund at Penn State and the hospitality of the Department of Theoretical Physics at the University of Geneva are gratefully acknowledged. In addition, the authors’ research is supported in part by the Fonds National Suisse and the National Science Foundation Grant DMS-9501225.

References [BF]

Bonic, R. and Frampton, J.: Smooth functions on Banach manifolds. J. Math. Mech. 15, 877–898 (1966) [BK] Bricmont, J. and Kupiainen, A.: Stable Non-Gaussian Diffusive Profiles. Nonlinear Analysis 26, 583– 593 (1995) [BKL] Bricmont, J., Kupiainen, A., and Lin G.: Renormalization Group and Asymptotics of Solutions. Comm. Pure Appl. Math. 47, 893–922 (1994) [C] Carr, J.: The Centre Manifold Theorem and its Applications. New York: Springer-Verlag, 1983 [CE] Collet, P. and Eckmann, J.-P.: Instabilities and Fronts in Extended Systems. Princeton, NJ: Princeton University Press 1990 [CEE] Collet, P., Eckmann, J.-P., and Epstein, H.: Diffusive repair for the Ginzburg-Landau equation. Helv. Phys. Acta 65, 56–92 (1992) [CR] Crandall, M. and Rabinowitz, P.: Bifurcation from simple eigenvalues. J. Funct. Analysis 8, 321–340 (1971) [E] Eckhaus, W.: Studies in non-linear stability theory. Springer tracts in Nat. Phil. 6, Berlin, Heidelberg, New York: Springer. (1965) [EW] Eckmann, J.-P. and Wayne, C.E.: Propagating Fronts and the Center Manifold Theorem. Commun. Math. Phys. 136, 285–307 (1991) [G] Gallay, Th.: A center-stable manifold theorem for differential equations in Banach spaces. Commun. Math. Phys. 152, 249–268 (1993) [GJ] Glimm, J. and Jaffe, A.: Quantum Physics, A Functional Integral Point of View. New York: SpringerVerlag 1981 [H] Henry, D.: Geometric Theory of Semilinear Parabolic Equations. Lecture Notes in Mathematics 840, New York: Springer-Verlag 1981 [L] Levine, H.A.: The role of critical exponents in blow up theorems. SIAM Review 32, 262–288 (1990) [M] Mielke, A.: A new approach to sideband-instabilities using the principle. In: Nonlinear Dynamics and Pattern Formation in the Natural. Doelman, A., van Harten, A., eds. UK: Longman 1995, pp. 206–222 [Sch] Schneider, G.: Diffusive stability of spatial periodic solutions of the Swift-Hohenberg Equation. Commun. Math. Phys. 178, 679–702 (1996) [W] Wayne, C.E.: Invariant manifolds for parabolic partial differential. Arch. Rat. Mech., To appear [W2] Wayne, C.E.: Invariant manifolds and the asymptotics of parabolic. Proceedings of the China/US conference on differential equations and applications, Hangzhou, PRC, July 1996. To appear Communicated by A. Kupiainen

This article was processed by the author using the LaTEX style file pljour1 from Springer-Verlag.

Commun. Math. Phys. 190, 213 – 245 (1997)

Communications in

Mathematical Physics c Springer-Verlag 1997

Melnikov Potential for Exact Symplectic Maps Amadeu Delshams, Rafael Ram´ırez-Ros Departament de Matem`atica Aplicada I, Universitat Polit`ecnica de Catalunya, Diagonal 647, 08028 Barcelona, Spain. E-mail: [email protected]; [email protected] Received: 6 June 1996 / Accepted: 16 April 1997

Abstract: The splitting of separatrices of hyperbolic fixed points for exact symplectic maps of n degrees of freedom is considered. The non-degenerate critical points of a real-valued function (called the Melnikov potential) are associated to transverse homoclinic orbits and an asymptotic expression for the symplectic area between homoclinic orbits is given. Moreover, if the unperturbed invariant manifolds are completely doubled, it is shown that there exist, in general, at least 4 primary homoclinic orbits (4n in antisymmetric maps). Both lower bounds are optimal. Two examples are presented: a 2n-dimensional central standard-like map and the Hamiltonian map associated to a magnetized spherical pendulum. Several topics are studied about these examples: existence of splitting, explicit computations of Melnikov potentials, transverse homoclinic orbits, exponentially small splitting, etc. 1. Introduction In a previous work [DR96], the authors were able to develop a general theory for perturbations of an integrable planar map with a separatrix to a hyperbolic fixed point. The splitting of the perturbed invariant curves was measured, in first order with respect to the parameter of perturbation, by means of a periodic Melnikov function M defined on the unperturbed separatrix. In case of area preserving perturbations, M has zero mean and therefore there exists a periodic function L (called the Melnikov potential) such that M = L0 . Consequently, if L is not identically constant (respectively, has nondegenerate critical points), the separatrix splits (respectively, the perturbed curves cross transversely). Moreover, under some hypothesis of meromorphicity, the Melnikov potential is elliptic and there exists a Summation Formula (see the Appendix) to compute it explicitly. The aim of this paper is to develop a similar theory for more dimensions. The natural frame is to consider exact symplectic perturbations of a 2n-dimensional exact map with a n-dimensional separatrix associated to a hyperbolic fixed point.

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Exact symplectic maps F : P → P are defined on exact manifolds, i.e., 2n-dimensional manifolds P endowed with a symplectic form ω which is exact: ω = − dφ; and they are characterized by the equation F ∗ φ − φ = dS for some function S : P → R, called the generating function of F . The typical example of an exact symplectic manifold is provided by a cotangent bundle T ∗ M, together with the canonical forms φ0 , ω0 , which in cotangent coordinates (x, y) read as φ0 = y dx, ω0 = dx ∧ dy. Typical exact symplectic maps are the socalled twist maps, which satisfy F ∗ (y dx) − y dx = Y dX − y dx = dL(x, X), where (X, Y ) = F (x, y). The fact that the generating function S can be written in terms of old and new coordinates: S(x, y) = L(x, X), is the twist condition that gives the name to these maps. The function L is called a twist generating function. As in [Eas91], we will not restrict ourselves to this typical case, since the results to be presented in this paper are valid on arbitrary exact symplectic manifolds and the twist condition is not needed. The exact symplectic structure plays a fundamental role in our construction, since it allows us to work neatly with geometric objects. For example, it is used to introduce two homoclinic invariants: the action of a homoclinic orbit and the symplectic area between two homoclinic orbits, called simply the homoclinic area. Namely, let p∞ ∈ P be a hyperbolic fixed point of F , which lies in the intersection of the n-dimensional invariant manifolds W u,s . Given a homoclinic orbit O = (pk )k∈Z of F , i.e., O ⊂ (W u ∩ W s ) \ {p∞ } and F (pk ) = pk+1 , we define the homoclinic action of the orbit O as X W [O] := S(pk ), k∈Z

where, in order to get an absolutely convergent series, the generating function S has been determined by imposing S(p∞ ) = 0. Given another homoclinic orbit O0 of F , the homoclinic area between the two homoclinic orbits O, O0 is defined as the difference of homoclinic actions 1W [O, O0 ] := W [O] − W [O0 ]. These two objects are symplectic invariants, i.e., they neither depend on the symplectic coordinates used, nor on the choice of the one-form φ. It is worth noting that in the planar case, the homoclinic area is the standard (algebraic) area of the lobes between the invariant curves [MMP84, Mat86, Eas91] and also measures the flux along the homoclinic tangle, which is related to the study of transport [MMP84, RW88, Mei92]. The unperturbed role will be played by an exact symplectic diffeomorphism F0 : P → P, defined on a 2n-dimensional exact manifold P, which possesses a hyperbolic fixed point p∞ and a n-dimensional separatrix 3 ⊂ W0u ∩ W0s , where W0u,s denote the invariant manifolds associated to p∞ . Consider now a family of exact symplectic diffeomorphisms {Fε }, as a general perturbation of the situation above, and let Sε = S0 + εS1 + O(ε2 ) be the generating function of Fε . The main analytical results of this paper are stated and proved in Sect. 2. There, the Melnikov potential is introduced as the real-valued smooth function L : 3 → R given by X pk = F0k (p), Sb1 (pk ), L(p) := k∈Z

where Sb1 : P → R is defined as Sb1 (p) = S1 (p) − φ(F0 (p))[F1 (p)], and F1 is the first order variation in ε of the family {Fε }, that is, F1 (p) = [∂Fε (p)/∂ε]|ε=0 . Obviously, S1 is determined by imposing Sb1 (p∞ ) = 0, in order to get an absolutely convergent series. In Theorem 2.1 it is established that

Melnikov Potential for Exact Symplectic Maps

215

(i) the Melnikov potential L is F0 -invariant: L ◦ F0 = L, (ii) if L ≡ 6 constant, the perturbed invariant manifolds Wεu,s split for 0 < |ε|  1, (iii) the non-degenerate critical points of L are associated to transverse intersections of the perturbed invariant manifolds, (iv) the above-mentioned homoclinic invariants are given in first order by L. As a matter of fact, the perturbed homoclinic orbits detected by the Melnikov potential are all of them primary homoclinic orbits Oε of Fε , i.e., they are smooth in ε for |ε| small enough. The Melnikov potential admits several reformulations. For example, if Fε is a twist map on a cotangent bundle T ∗ M, with twist generating function Lε = L0 + εL1 + O(ε2 ), Sb1 has the simple form Sb1 (p) = L1 (π(p), π(F0 (p))), where π : T ∗ M → M is the natural projection. Consequently, the Melnikov potential reads as [DRS97] X L1 (xk , xk+1 ), xk = π(pk ), L(p) = k∈Z

where L1 is determined by imposing L1 (x∞ , x∞ ) = 0, and x∞ = π(p∞ ). Another interesting situation, that allows us to compare the continuous and discrete frames, is to consider Hamiltonian maps. Let Hε : P × R → R be a time-periodic Hamiltonian of period T , and Fε = 9Tε , where 9tε (p) is the solution of the associated Hamiltonian equations with initial condition p at t = 0. If Hε = H0 + εH1 + O(ε2 ), then RT Sb1 (p) = − 0 H1 (9t0 (p), t) dt, so the Melnikov potential takes the form (already known to Poincar´e) Z L(p) = − H1 (9t0 (p), t) dt, R

where H1 is determined by imposing H1 (9t0 (p∞ ), t) ≡ 0, or simply H1 (p∞ , t) ≡ 0, if H0 is autonomous. An essential ingredient for the proof of Theorem 2.1 is the fact that the invariant manifolds Wεu,s are exact Lagrangian immersed submanifolds of P and therefore can be expressed in terms of generating functions Lu,s ε . The Lagrangian property of the invariant manifolds was already noticed by Poincar´e [Poi99] for flows, although we learned it for maps from E. Tabacman [Tab95], as well as the expression for the invariant manifolds given in Proposition 2.1, in the twist frame. The relationship between Lu,s 1 and S1 , the first order variations in ε of the generating functions Lu,s ε and Sε , gives then the formula for the Melnikov potential. The tools utilized are very similar to those of D. Treschev [Tre94]. However, D. Treschev considers autonomous Hamiltonian flows, and the conservation of energy makes easier the deduction of the continuous version of Eq. (2.5). In that frame (Hamiltonian-Lagrangian flows), it is worth noting that a variational approach to the Melnikov method was carried out by S. Angenent [Ang93] for Hamiltonian systems with 1 21 degrees of freedom, and that a mechanism for finding homoclinic orbits in positively definite symplectic diffeomorphisms is due to S. Bolotin [Bol95], based on interpolating them by Hamiltonian flows. Section 2 contains also some remarks on the non-symplectic case: a vector-valued Melnikov function M is then defined, whose non-degenerated zeros are associated to transverse homoclinic orbits. The last part of Sect. 2 is devoted to gain information on the number of primary homoclinic orbits after perturbation. Since the Melnikov potential L is F0 -invariant, it can be defined on the reduced separatrix 3∗ := 3/F0 , which is the quotient of the

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A. Delshams, R. Ram´ırez-Ros

separatrix by the unperturbed map. The reduced separatrix is a compact manifold without boundary, provided that the unperturbed invariant manifolds are completely doubled, i.e., W0u = W0s and W0u,s \ {p∞ } is a submanifold of P and not only an immersed submanifold of P. This is equivalent to require that the separatrix is 3 = W0u,s \ {p∞ }. Several dynamical consequences of this fact can be pointed out using topological tools. In particular, Morse theory gives lower bounds on the number of primary transverse homoclinic orbits, under conditions of generic position: in Theorem 2.2 it is stated that the number of primary homoclinic orbits is at least 4. Moreover, if the maps Fε have a common symmetry I : P → P (Fε ◦I = I ◦Fε , and Fε (p∞ ) = I(p∞ ) = p∞ ) such that the one-form φ is preserved by I: I ∗ φ = φ, then the Melnikov potential is I-invariant (see Lemma 2.6). Consequently, it can be considered as a function over the quotient manifold 3∗I := 3/{F0 , I}. If, in addition, I is an involution (I 2 = Id) such that DI(p∞ ) = −Id, the family {Fε } will be called antisymmetric. In this case, in Theorem 2.2 it is stated that the number of primary homoclinic orbits is at least 4n and that they appear coupled in (anti)symmetric pairs: Oε is a primary homoclinic orbit if and only if I(Oε ) also is. It is worth mentioning that any family of odd maps Fε : R2n → R2n (with the standard symplectic structure) is antisymmetric. To prove Theorem 2.2, it is enough to check that the sum of the Z2 -Betti numbers of 3∗ and 3∗I are 4 and 2n, respectively. This is accomplished by computing the Z2 homology of 3∗ and 3∗I . Both lower bounds are optimal, as it is shown in several perturbations of maps with a central symmetry, so that the unperturbed invariant manifolds are completely doubled. It is important to notice that the invariant manifolds of a product of uncoupled planar maps with double loops are not completely doubled, see Remark 2.3, and hence, the topological results do not hold in this case. Indeed, the number of primary homoclinic orbits may be rather different under perturbation; for instance, it is possible to construct explicitly perturbations with an infinite number of primary homoclinic orbits, all of them being transverse. The study of this kind of phenomena is currently being researched. In Sect. 3, as a first example, we consider the family of twist maps on R2n : ÿ Fε (x, y) =

y, −x +

!

2µy 1 + |y|

2

+ ε∇V (y) ,

µ > 1, ε ∈ R,

with V : Rn → R determined by imposing V (0) = 0. The map above is a perturbation of the McLachlan map [McL94], which is a multi-dimensional generalization of the McMillan map [McM71], which in its turn is a particular case of the standard-like Suris integrable maps [Sur89]. The McLachlan map has a central symmetry that makes the dynamics over the separatrix essentially one-dimensional. This is the key fact that allow us to perform a complete analysis, since the natural parametrizations (3.2) can be introduced. If the potential V is entire and not identically zero, in Theorem 3.1 it is proved that the manifolds Wεu,s of the map Fε split, for 0 < |ε|  1. This result is obtained simply by checking that the Melnikov potential is not constant. Moreover, if V is a polynomial, the Melnikov potential can be computed explicitly. In particular, if V is a quadratic form: V (y) = y > By for some symmetric n×n matrix B, in Proposition 3.1 it is stated that under generic conditions on B (det(B) 6= 0 and B does not have multiple eigenvalues), the perturbed invariant manifolds are transverse along exactly 4n primary homoclinic orbits.

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217

If V is linear: V (y) = b> y for some vector b ∈ Rn \ {0}, in Proposition 3.2 it is stated that the perturbed invariant manifolds are transverse along exactly 4 primary homoclinic orbits. The difference between both kinds of perturbations is that quadratic potentials V give rise to odd maps, whereas linear ones do not. Moreover, propositions 3.1 and 3.2 give the unperturbed homoclinic orbits that survive and the first order (in ε) of the homoclinic areas between the different primary homoclinic orbits. The weakly hyperbolic case 0 < h  1, cosh(h) := µ, is also studied for the case of a quadratic potential V , and asymptotic expressions for the homoclinic areas are given at the end of Sect. 3. It turns out that, for some distinguished pairs, interlaced in the same way as in the case of 1 degree of freedom, the homoclinic area predicted by the Melnikov potential is exponentially small with respect to the hyperbolicity parameter h. Of course, this does not prove that the splitting size is exponentially small in singular cases, i.e., when ε and h tend simultaneously to zero. The last section is devoted to the study of the Hamiltonian maps arising from timeperiodic perturbations of an (undamped) magnetized spherical pendulum. This model was introduced by J. Gruendler [Gru85] as a first example of application of the Melnikov method for high-dimensional (continuous) systems. The Hamiltonians considered have the form [Gru85] Hε : R2n × R → R,

Hε (x, y, t) = v 2 /2 + (r4 − r2 )/2 + εV (x, t/h),

h > 0, ε ∈ R,

where v = |y|, r = |x|, and V = V (x, ϕ) is 1-periodic in ϕ. We determine V by imposing V (0, ϕ) ≡ 0. Note that small values of h correspond to a quick forcing. General perturbations, and not only symplectic ones, are considered in [Gru85]. As a consequence, the homoclinic orbits are given in the general case by non-degenerate zeros of a vector-valued Melnikov function, instead of non-degenerate critical points of the real-valued Melnikov potential. We have computed the Melnikov potential for the Hamiltonian perturbations studied in [Gru85], and have verified that his Melnikov function is the gradient of our Melnikov potential. Most of the results stated above for the McLachlan map also hold for this Hamiltonian map. There is, however, a significant difference. One cannot deduce a priori that the Melnikov potential is not identically constant without computing it. This has to do with the fact that the Melnikov potential is simply periodic and regular for the polynomial perturbations considered, in contrast with the complex period and singularities that the Melnikov potential has for the entire perturbations of the McLachlan map. To finish the account of results, let us point out that a similar Melnikov analysis for perturbed ellipsoidal billiards has not been included for the sake of brevity and will appear elsewhere. Such billiards are a high-dimensional version of perturbed elliptic billiard tables, which have already been studied in several papers [LT93, Tab94, DR96, Lom96a]. After this research was complete, we became aware of some recent papers [Lom97, Lom96b] of H. Lomel´ı for twist maps on the annulus An = T ∗ Tn = Tn × Rn that resemble our method. However, they do not contain explicit computations (i.e., in terms of known functions) of the Melnikov potential, since complex variable methods are not used. Besides, in those papers it is assumed that the separatrix is globally horizontal, a condition that does not hold for homoclinics in R2n , since the separatrix must fold to go back to the fixed point. Other related papers are [Sun96, BGK95], but their approach is rather different, since they deal, like [Gru85], with the general case, with no symplectic structure, and therefore a vector-valued Melnikov function is needed. This makes an important difference not

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only from a computational point of view (there are not explicit (analytic) computations in these works), but also from a theoretical point of view, since Morse theory cannot be applied in the general situation. We also want to mention the work [BF96], where perturbations of n-dimensional maps having homo-heteroclinic connections to compact normally hyperbolic invariant manifolds are considered.

2. Main Results For the sake of simplicity, we will assume that the objects here considered are smooth. For a general background on symplectic geometry we refer to [Arn76, GS77, AM78]. The basic properties of immersed submanifolds can be found in [GG73, pages 6–11]. 2.1. Exact objects. A 2n-dimensional manifold P together with an exact non-degenerate two-form ω over it, is called an exact symplectic manifold. Then, ω = − dφ for some one-form φ, usually called Liouville form, symplectic potential or action H form.H A map F : P → P is called exact symplectic (or simply, exact) if γ φ = F γ φ for all closed path γ ⊂ P or, equivalently, if F ∗ φ − φ = dS for some function S : P → R, called generating function of F . A n-dimensional submanifold 3H ⊂ P is called an exact Lagrangian submanifold (or simply, an exact submanifold) if γ φ = 0 for all closed path γ ⊂ 3 or, equivalently, if ı∗3 φ = dL for some function L : 3 → R, called generating function of 3. Here ı3 : 3 ,→ P stands for the inclusion map. Unfortunately, the invariant manifolds that we will deal with are not submanifolds, but just immersed submanifolds. Thus, the introduction of some technicalities seems unavoidable in order to give a rigorous exposition of the subject, and more precisely, to introduce the notion of separatrix, where the distance between the perturbed invariant manifolds will be measured. Given a manifold N , we recall that a map g : N → P is called an immersion when its differential dg(z) has maximal rank at any point z ∈ N . If g is one-to-one onto its image W = g(N ), there is a natural way to make W a smooth manifold: the topology on W is the one which makes g a homeomorphism and the charts on W are the pull-backs via g −1 of the charts on N . The manifold W constructed in this way is called an immersed submanifold of P and its dimension is equal to the dimension of N . It is important to notice that the topology of the immersed manifold need not be the same as the induced one via the inclusion W ⊂ P or, in other words, that W need not be a submanifold of P in the usual sense. Fig. 1 shows an example of a double loop W = g(R) to p∞ = limz→±∞ g(z) for an immersion g : R → R2 . At p∞ , the induced topology on W via the inclusion W ⊂ R2 is not the same as the induced one via g. Both g(B), for all open bounded intervals B ⊂ R, and W \ {p∞ } are submanifolds, but not W. This situation is a particular case of the following elementary result [GG73, p. 11]. Lemma 2.1. Let g : N → P be a one-to-one immersion and set W = g(N ). (i) Let B be an open subset of N with compact closure. Then, g|B : B → P is an embedding, that is, a homeomorphism onto its image g(B). Thus, g(B) is a submanifold of P, which will be called an embedded disk in W. (ii) Let Σ ⊂ W be the set of points where the two topologies on W (the one induced by the inclusion W ⊂ P and the one that makes g a homeomorphism) differ. Then,

Melnikov Potential for Exact Symplectic Maps

219

R

W R 2

g

1 = g(0)

p

Fig. 1. g = (g1 , g2 ) : R → R2 , where g1 (z) = 23 z/(1 + z 2 ), g2 (z) = g1 (2z)

3 = W \ Σ is a submanifold of P. Indeed, W is not a submanifold of P just at the points of Σ. For the sake of clearness, submanifolds and immersed submanifolds will be denoted by different letters, namely 3 and W, respectively. For immersed submanifolds W, the map ıW : W → P stands for the inclusion map, as before. It should be noted that ıW is smooth, even when W is not a submanifold of P, because of the differential structure given to W. Moreover, if γ ⊂ P is a (closed) path, we will say that γ is a (closed) path in the immersed submanifold W if and only if γ is contained in W and it is continuous in the topology of W. For example, if γ is one loop of Fig. 1, it is a closed path in R2 but not in W. With these notations and definitions, we are naturally led to define exact immersed submanifolds in the same way as exact H submanifolds. A n-dimensional immersed submanifold W ⊂ P is called exact if γ φ = 0 for all closed path γ in W or, equivalently, if ı∗W φ = dL for some function L : W → R, called a generating function of W. The symplectic potential φ is determined except for the addition of a closed zeroform, and the generating functions of maps or (immersed) submanifolds are determined Rq except for an additive constant. Henceforth, the symbol W p φ, with p, q ∈ W, will denote the integral of φ along an arbitrary path from p to q in W. It only makes sense for an exact immersed submanifold W, since then the integral does not depend on the path. The difference of values of L can be expressed as an integral of this kind: Z q Z q dL = φ, ∀p, q ∈ W. (2.1) L(q) − L(p) = p

W p

Lemma 2.2. Let W be a connected exact immersed submanifold of P, invariant under 1 an exact map F . Let L and S be their respective generating functions. Then, S(p) + constant = L(F (p)) − L(p),

∀p ∈ W.

(2.2)

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Moreover, if p∞ ∈ W is a fixed point of F , the constant is −S(p∞ ). Proof. From dS = F ∗ φ − φ and dL = ı∗W φ we get


∗ d S|W = ı∗W dS = F|W dL − dL = d L ◦ F|W − L , where S|W = S ◦ ıW and F|W = (ıW )−1 ◦ F ◦ ıW are the restrictions of S and F to W. Thus, S − L ◦ F + L is constant over W by connectedness and (2.2) is proved. To end  the proof we only need to evaluate Eq. (2.2) at p = p∞ .

Fig. 2. The invariant manifolds W u and W s are different as smooth manifolds, and are not submanifolds of R2 . There exist no paths γ u,s in W u,s from p to p0 such that γ u = γ s

Let p∞ ∈ P be a hyperbolic fixed point of F . The point p∞ lies in the intersection of the n-dimensional unstable and stable invariant manifolds of the map F associated to p∞ :     W s := p ∈ P : lim F k (p) = p∞ . W u := p ∈ P : lim F k (p) = p∞ , k→−∞

k→+∞

The manifolds W u,s need not be submanifolds of P, but just connected immersed submanifolds, see Fig. 2. In fact, W u,s = g u,s (Rn ) for some one-to-one immersions g u,s : Rn → P, such that g u,s (0) = p∞ and dg u,s (0)[Rn ] is the tangent space to W u,s at p∞ [PM82, II §6]. Since F is exact, they submanifolds: if γ H H are exact immersed H is a closed path in W u (W s ), then γ φ = F k γ φ −→ p∞ φ = 0, when k → −∞ (k → +∞). It should be noted that if γ ⊂ P is closed and contained in W u (resp. W s ), but it is not a path in W u (resp. W s ), the above argument fails. (For instance, if γ is one loop of Fig. 2.) We denote by Lu,s the generating functions of W u,s and we determine the generating functions S, Lu,s by imposing S(p∞ ) = Lu,s (p∞ ) = 0. The next proposition gives a nice interpretation of the generating functions of the stable and unstable invariant manifolds in terms of the generating function of the map. k u,s Proposition 2.1. Given pu,s ∈ W u,s , let us denote pu,s k = F (p ), for k ∈ Z. Then,

Lu (pu ) =

X k<0

S(puk ),

Ls (ps ) = −

X k≥0

S(psk ).

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u,s u,s u,s u,s Proof. From Lemma 2.2, one has S(pu,s k ) = L (pk+1 ) − L (pk ), for all k ∈ Z. To get the formulae above, we simply consider the telescopic sums X X [Lu (puk+1 ) − Lu (puk )] = S(puk ), Lu (pu ) = k<0

s

s

L (p ) =

X

k<0

[L

s

(psk )

−L

s

(psk+1 )]

=−

k≥0

X

S(psk ).

k≥0

These series are absolutely convergent, since S(p∞ ) = 0 and puk (psk ) tends to p∞ at an exponential rate as k tends to −∞ (+∞).  Let O = (pk )k∈Z be a homoclinic orbit of F , i.e., O ⊂ (W u ∩W s )\{p∞ } and F (pk ) = pk+1 . We define the homoclinic action of the orbit O as W [O] := Lu (pk ) − Ls (pk ). This definition does not depend on k, since a direct application of Proposition 2.1 with pu,s k = pk yields an equivalent k-independent definition X S(pk ). (2.3) W [O] := k∈Z

Let O0 be another homoclinic orbit of F . The homoclinic area between the two homoclinic orbits O, O0 is defined as the difference of homoclinic actions 1W [O, O0 ] := W [O] − W [O0 ]. For a motivation of this name, consider p ∈ O, p0 ∈ O0 , γ u,s a path from p to p0 in W u,s , γ = γ u − γ s , and suppose that D is an oriented 2-chain such that ∂D = γ. Then, by Eq. (2.1) and Stokes’ formula, we have I ZZ φ=− ω. (2.4) 1W [O, O0 ] = γ

D

This formula shows clearly that the homoclinic area is a symplectic invariant, i.e., it neither depends on the symplectic coordinates used, nor on the choice of the symplectic potential φ. The homoclinic action can be considered as the homoclinic area between the homoclinic orbit at hand and the “orbit” of the fixed point p∞ . Thus, it is a symplectic invariant, too. In particular, if P = R2 with the standard area as the symplectic structure, and p ∈ O, 0 p ∈ O0 are consecutive intersections of the invariant manifolds, then the homoclinic area 1W [O, O0 ] is simply the (algebraic) area of the associated lobe. Remark 2.1. Set W = W u ∩ W s and let p, p0 be two points of the same connected u,s u,s 0 component of W. When it is possible to R choose the paths γ in W from p to p such u s 0 that γ = γ − γ = 0, 1W [O, O ] = γ φ = 0, i.e., the actions coincide. They can be 0 different if p and p0 are not in the RR same component of W.RRFor instance, if p, p , D and D0 are as in Fig. 2, W [O] = − D ω > 0 and W [O0 ] = D0 ω < 0. 2.2. Families of exact objects. Now, we carry out the generalization of Lemma 2.2 and Proposition 2.1 for families of exact immersed submanifolds and maps, depending (in a smooth way) on a small parameter ε. First, let us recall the following standard fact from symplectic geometry [Wei73, GS77]. Lemma 2.3. In any point p of any Lagrangian submanifold 3 of P there exists a neighbourhood p ∈ U ⊂ P and local coordinates (x, y) over U such that φ = y dx (i.e., ω = dx ∧ dy) and the set 3 ∩ U is given by the equation y = 0.

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We recall that a n-dimensional submanifold 3 is Lagrangian if ı∗3 ω = 0. In particular, exact submanifolds are Lagrangian. The coordinates above are called cotangent coordinates since they give a symplectic change of variables from the neighbourhood U onto a neighbourhood V of p in the cotangent space T ∗ 3. Let gε : N → P be one-to-one immersions and set Wε = gε (N ). We will say that the family of immersed submanifolds {Wε } is smooth (at ε = 0) when for any embedded disk 3 ⊂ W0 there exists a smooth family of embedded disks {3ε } such that 3ε ⊂ Wε and 30 = 3. We remember that embedded disks are submanifolds of P, and they are exact if the immersed submanifolds are. Lemma 2.4. Let {Wε } be a smooth (at ε = 0) family of connected exact immersed submanifolds. (i) Let p ∈ W0 and 3 ⊂ W0 be an embedded disk containing p. Let {3ε } be a smooth family of embedded disks such that 3ε ⊂ Wε and 30 = 3. Let U be a neighbourhood of p in P, where cotangent coordinates (x, y) exist for 3. Thus, the set 3ε ∩ U has the form y = ε∂Lε (x)/∂x, for some function Lε , since 3ε is an exact submanifold. We can write Lε = L1 + O(ε). Then, the function L1 : W0 → R is well-defined, that is, it neither depends on the family {3ε }, nor on the cotangent coordinates. (Of course, L1 is determined except for an additive constant.) (ii) Assume that Wε is invariant under some exact map Fε . Let Sε = S0 + εS1 + O(ε2 ) be the generating function of Fε , and F1 (p) = [∂Fε (p)/∂ε]|ε=0 be the first order variation in ε of the family {Fε }. Then, S1 (p) − φ(F0 (p))[F1 (p)] + constant = L1 (F0 (p)) − L1 (p),

∀p ∈ W0 .

(2.5)

Besides, the constant is φ(p∞ )[F1 (p∞ )] − S1 (p∞ ), if p∞ ∈ W0 is a fixed point of F0 . Remark 2.2. It is clear that 3ε ∩ U has the equation y = ε∂L1 (x)/∂x + O(ε2 ). From (i), the function L1 : W0 → R is a geometrical object associated to the family {Wε }, and therefore its differential gives the first order variation at ε = 0 of the family along the coordinate y in any cotangent coordinates (x, y). We will call L1 the infinitesimal generating function of the family {Wε }. Proof. (i) On the one hand, any two families {3ε }, {30ε } coincide on a small neighbourhood of the point p. This proves the independence on the family. On the other hand, the independence on the cotangent coordinates for a fixed family is proved in [Tre94], using coordinates. A geometric interpretation of L1 , useful in order to prove below (ii) (and consequently another proof of the fact that L1 : W0 → R is well-defined), is given now. It is inspired in a similar construction that can be found in [AA89, p. 238]. Let E ⊂ R be the small neighbourhood of 0 where ε runs. Given p ∈ W0 , we denote by pb : E → P any smooth curve such that pb(ε) ∈ Wε which has a non-tangent contact with W0 at p for ε = 0. Moreover, σ(p, ε) will denote the path pb(τ ), 0 ≤ τ ≤ ε. Given p, q ∈ W0 , let D(p, q, ε) be any oriented 2-chain of P such that ∂D(p, q, ε) = γ(p, q, ε) − γ(p, q, 0) + σ(p, ε) − σ(q, ε), where γ(p, q, ε) is any path from pb(ε) to qb(ε) in Wε . Such a construction is possible, provided that |ε| is small enough. Let us set

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223

ZZ 1(p, q, ε) := −

D(p,q,ε)

ω = ε11 (p, q) + O(ε2 ).

This integral neither depends on the symplectic coordinates, nor on the choice of the paths γ(p, q, ε). In addition, its first order term 11 (p, q) does not depend on the choice of the curves pb and qb, since such different choices only affect second order terms of 1(p, q, ε). Now, it will be shown that L1 (q) − L1 (p) = 11 (p, q), if p, q ∈ W0 are close enough over W0 , that is, if there exist an embedded disk 30 ⊂ W0 and an open U ⊂ P, where cotangent coordinates (x, y) are defined, such that p, q ∈ 30 = {y = 0} ∩ U . We denote by π : U → 30 the projection π(z) = p, if z = (x, y) and p = (x, 0) are the cotangent coordinates of z and p, respectively. We determine the curves pb, qb by imposing π ◦ pb ≡ p, π ◦ qb ≡ q, and we choose γ(p, q, ε) in such a way that they are contained in U . Then, Z Z Z ∂Lε (x) dx y dx = y dx = ε 1(p, q, ε) = γ(p,q,ε)−γ(p,q,0)+σ(p,ε)−σ(q,ε) γ(p,q,ε) π(γ(p,q,ε)) ∂x = ε[Lε (π(b q (ε))) − Lε (π(b p(ε)))] = ε[L1 (q) − L1 (q)] + O(ε2 ). Finally, if p, q ∈ W0 are arbitrary, we consider a chain of points (rj )0≤j≤J such that r0 = p, rJ = q, and two consecutive points of the chain are close enough so that L1 (rj ) − L1 (rj−1 ) = 11 (rj−1 , rj ) holds. Then, a trivial argument with telescopic sums shows that L1 (q) − L1 (p) = 11 (p, q), since 11 (r, s) + 11 (s, t) = 11 (r, t) holds for all r, s, t ∈ W0 . (ii) Given p ∈ W0 , we set q = F0 (p). For any curve pb like the previous ones, let p(ε)). If v = ( db p/ dε)(0), then w = ( db q / dε)(0) = dF0 (p)[v] + F1 (p), so v qb(ε) = Fε (b (i.e., pb) can be chosen in such a way that qb is not tangent to W0 at q, due to the fact that the map v 7→ w is bijective. Using (i), we get ZZ Z ω = lim ε−1 φ. L1 (q) − L1 (p) = − lim ε−1 ε→0

D(p,q,ε)

ε→0

γ(p,q,ε)−γ(p,q,0)+σ(p,ε)−σ(q,ε)

Now, by equations (2.1) and (2.2), there exist constants c(ε) (independent of the point p) such that ( Z Z Z q ) q(ε) ˆ φ = lim ε−1 φ− φ lim ε−1 ε→0

ε→0

γ(p,q,ε)−γ(p,q,0)

= lim ε ε→0

Wε p(ε) ˆ

−1

W0 p

[Lε (b q (ε)) − Lε (b p(ε)) − L0 (q) + L0 (p)]

p(ε)) + c(ε) − S0 (p) − c(0)] = lim ε−1 [Sε (b ε→0

= S1 (p) + dS0 (p)[v] + ( dc/ dε)(0). Finally, we use that F0∗ φ − φ = dS0 and consequently, Z Z ε    
lim ε−1 φ = lim ε−1 φ(b p(τ )) ( db p/ dε)(τ ) − φ(b q (τ )) ( db q / dε)(τ ) dτ

ε→0

ε→0

σ(p,ε)−σ(q,ε)

0

= φ(p)[v] − φ(q)[w] = − dS0 (p)[v] − φ(F0 (p)[F1 (p))], and the proof follows.



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Let F0 : P → P be an exact symplectic diffeomorphism with a hyperbolic fixed point p∞ and invariant manifolds W0u,s . Let us consider a family of exact symplectic diffeomorphisms {Fε }, as a general perturbation of the situation above, and let Sε = S0 + εS1 + O(ε2 ), be the generating function of Fε . In order to simplify some formulae later, we introduce the function Sb1 : P → R,

Sb1 (p) = S1 (p) − φ(F0 (p))[F1 (p)],

(2.6)

where F1 (p) = [∂Fε (p)/∂ε]|ε=0 . From the invariant manifold theory for maps [PM82, II §6], it follows that for small |ε| there exists a hyperbolic fixed point p∞ (ε) of the perturbed map Fε near p∞ . Moreover, p∞ (ε) lies in the intersection of two (connected) exact immersed submanifolds Wεu,s , and the families {Wεu,s } are smooth (at ε = 0). We denote by Lu,s 1 their infinitesimal b1 (p∞ ) = Lu,s (p∞ ) = 0. generating functions and, as usual, we determine S1 , Lu,s by S 1 1 k u,s Proposition 2.2. Given pu,s ∈ W0u,s , let pu,s k = F0 (p ), for k ∈ Z. Then, X X Ls1 (ps ) = − Lu1 (pu ) = Sb1 (puk ), Sb1 (psk ). k<0

k≥0

Proof. Identical to the proof of Proposition 2.1, but using Eq. (2.5) instead of Eq. (2.2).  2.3. Melnikov potential. Assume now that the invariant manifolds W0u,s are doubled, that is, W := W0u = W0s . Then, we can consider three topologies on W: the one induced by the inclusion W ⊂ P, and the two ones induced by the inclusions W ⊂ W0u,s . We define the bifurcation set Σ and the separatrix 3 of this problem as the subset of W of points where the three topologies do not coincide, and 3 := W \ Σ, respectively. Lemma 2.5. The bifurcation set and the separatrix have the following properties: (i) 3 is an exact submanifold of P and p∞ ∈ Σ. (ii) 3 and Σ are F0 -invariant. (iii) Let p, p0 be points on the same connected component of 3. Then, the unperturbed homoclinic orbits O and O0 generated by p and p0 , have the same action. Proof. (i) On the one hand, using (ii) of Lemma 2.1, 3 is a submanifold. It must be exact, since it is contained in the exact immersed submanifolds W0u , W0s . On the other hand, W0u and W0s have a transverse intersection at p∞ , so their topology at p∞ as immersed submanifolds can not coincide and p∞ ∈ Σ. (Indeed, Σ is just formed by the points of W where this set has self-intersections, considered as a subset of P.) (ii) Since W is F0 -invariant, it is enough to see that Σ is invariant, and this follows from the fact that F0 is a diffeomorphism. (iii) This is clear from Remark 2.1. We can connect p and p0 by a path in 3, and so  in W0u,s , since their topologies coincide on 3. Remark 2.3. In the planar case with a double loop (∞), the bifurcation set is just the hyperbolic fixed point. In general, for more dimensions the situation is not so simple. For example, let F0 : R2n → R2n be the product of n planar maps fj : R2 → R2 , each one with a double loop 0j = {pj∞ } ∪ 3j , where pj∞ ∈ R2 stands for the fixed point of fj and

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225

3j are the two components of 0j \{pj∞ }, for j = 1, . . . , n. Then, 3 = 31 ×· · ·×3n has 2n connected components and Σ = (01 × · · · × 0n ) \ 3 contains strictly the hyperbolic fixed point p∞ = (p1∞ , . . . , pn∞ ) ∈ R2n . In particular, 3 6= W0u,s \ {p∞ }. Remark 2.4. As the case of a planar map with a single loop (∝) shows, the situation W0u 6= W0s does not exclude that W0u ∩ W0s can contain n-dimensional submanifolds. For the sake of simplicity, we have defined the notion of separatrix only if the invariant manifolds are doubled and then, from the arguments above, the separatrix 3 satisfies: (a) 3 is a doubly asymptotic exact submanifold, invariant by F0 , and (b) the three topologies on 3 coincide (the ones induced by the inclusions 3 ⊂ P, 3 ⊂ W0u , and 3 ⊂ W0s ). Since these properties are the only ones needed in this section, they can be taken as a definition for a separatrix when W0u,s are partially doubled: W0u 6= W0s . Thus, with this definition, the analytical results of this paper also apply to this case. u,s By Remark 2.2, the differential of Lu,s 1 gives the first order variation of Wε at u,s u,s u,s ε = 0. Besides, since L1 is defined over W0 and 3 ⊂ W0 , the perturbed invariant manifolds Wεu,s can be compared over the separatrix 3. For this purpose, we introduce the real-valued function X pk = F0k (p), (2.7) L : 3 → R, L(p) := Lu1 (p) − Ls1 (p) = Sb1 (pk ), k∈Z

called the Melnikov potential of the problem. The series above is absolutely convergent since any orbit in the manifold 3 tends to p∞ at an exponential rate as |k| → ∞ and Sb1 (p∞ ) = 0. Thus, L is well-defined, and its differential gives the first order distance, along the coordinate y in any cotangent coordinates (x, y), between the perturbed invariant manifolds. This geometric interpretation is the fundamental point to find conditions for the splitting of the separatrices. It still remains to check the smoothness of L on 3. It is clear that Lu,s 1 are smooth over W0u,s , but since the smooth structures on W0u , W0s do not coincide, Lu1 − Ls1 could be defined over the whole intersection W but need not be smooth on the bifurcation set Σ. Thus, it is necessary to restrict ourselves to a subset of W where the two smooth structures coincide, and because of this, we have defined the separatrix 3 as the set W \ Σ to get a smooth L on 3. Before stating our main analytical result, we must introduce the kind of perturbed homoclinic orbits that can be detected by “Melnikov methods”. A primary homoclinic orbit of the perturbed problem is a perturbed homoclinic orbit Oε of Fε , defined for |ε| small enough and depending in a smooth way on ε. This is a perturbative definition, since in the multi-dimensional case (contrary to the planar case, see [Wig91]), it seems difficult to give a geometric definition. Non-primary homoclinic orbits are invisible for the standard Melnikov techniques. (However, a new Melnikov-like theory has been recently developed in [Rom95], to study secondary homoclinic orbits for time-periodic perturbations of integrable planar differential equations.) Theorem 2.1. Under the above notations and hypothesis: (i) L is F0 -invariant (i.e., L ◦ F0 = L). (ii) If L is not locally constant, the manifolds Wεu,s split for 0 < |ε|  1, i.e., the separatrix 3 is not preserved by the perturbation. (iii) If p ∈ 3 is a non-degenerate critical point of L, the manifolds Wεu,s are transverse along a primary homoclinic orbit Oε of Fε for 0 < |ε|  1, with O0 = (F0k (p))k∈Z .

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Moreover, when all the critical points of L are non-degenerate, all the primary homoclinic orbits arising from 3 are found in this way. (iv) Let Oε be a primary homoclinic orbit such that O0 = (F0k (p))k∈Z for some p ∈ 3. Then, the homoclinic action admits the asymptotic expression W [Oε ] = W [O0 ] + εL(p) + O(ε2 ). Given another orbit Oε0 such that O00 = (F0k (p0 ))k∈Z for some p0 in the same connected component of 3 as p, the homoclinic area is given by 1W [Oε , Oε0 ] = ε[L(p) − L(p0 )] + O(ε2 ). Proof. (i) A shift in the index of the sum does not change its value, so L is F0 -invariant. (ii) If dL is not zero, the perturbed invariant manifolds do not coincide at first order, so they split. (iii) This result follows directly from the geometric interpretation of the Melnikov potential and the Implicit Function Theorem. pk (ε))k∈Z , pk = pbk (0) = F0k (p), and vk = ( db pk / dε)(0). From (iv) Let Oε = (b ∗ Eq. (2.3), dS0 = F0 φ − φ, and dF0 (pk )[vk ] = vk+1 − F1 (pk ), we obtain: X W [Oε ] = Sε (b pk (ε)) k∈Z

=

X

{S0 (pk ) + ε(S1 (pk ) + dS0 (pk )[vk ]) + O(ε2 )}

k∈Z

=

X

S0 (pk ) + ε

k∈Z

= W [O0 ] + ε

X

{S1 (pk ) + φ(pk+1 ) [ dF0 (pk )[vk ]] − φ(pk )[vk ]} + O(ε2 )

k∈Z

Xn

o Sb1 (pk ) + φ(pk+1 )[vk+1 ] − φ(pk )[vk ] + O(ε2 )

k∈Z

= W [O0 ] + εL(p) + O(ε2 ). Finally, the asymptotic formula for the homoclinic area follows from its definition, using (iii) of Lemma 2.5.  Remark 2.5. The actions of homoclinic orbits arising from different connected components of the separatrix need not be equal at ε = 0, see Remark 2.1, whereas the splitting size is of order O(ε). Thus, it seems inappropriate to measure the splitting comparing the action of homoclinic orbits arising from different components of 3. For instance, in the planar case with a double loop, the geometric sense of the area between primary homoclinic orbits arising from different loops is very unclear. Remark 2.6. If L has some non-degenerate critical point, the perturbed invariant manifolds of Fε have a transverse intersection and, in particular, a topological crossing. Thus, using some recent results contained in [BW95], the perturbed maps have positive topological entropy, for 0 < |ε|  1. Let us see now that the Melnikov potential is invariant under additional diffeomorphisms, if the family {Fε } has suitable symmetries. We recall that given a diffeomorphism I : P → P the family {Fε } is called I-symmetric if Fε ◦ I = I ◦ Fε and Fε (p∞ ) = I(p∞ ) = p∞ , for all ε. Lemma 2.6. Assume that the family {Fε } is I-symmetric, and that the symplectic potential is preserved by the symmetry: I ∗ φ = φ. Then, the Melnikov potential L is I-invariant: L ◦ I = L.

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227

Proof. Let p ∈ W = W0u,s and q = I(p). Using that F0k ◦ I = I ◦ F0k for all k ∈ Z, we get   k k k lim F0 (q) = lim I(F0 (p)) = I lim F0 (p) = I(p∞ ) = p∞ . k→∞

k→∞

k→∞

This proves that W is I-invariant. Thus, the separatrix 3 also is, by the same argument as in (ii) of Lemma 2.5, and the expression L ◦ I makes sense on 3. From Fε∗ φ − φ = dSε , I ∗ φ = φ, and Fε ◦ I = I ◦ Fε we have d(Sε ◦ I) = I ∗ ( dSε ) = I ∗ Fε∗ φ − I ∗ φ = Fε∗ I ∗ φ − φ = Fε∗ φ − φ = dSε . Hence, Sε ◦ I − Sε is a constant function that evaluated at p∞ vanishes, so Sε (and in particular S1 ) are I-invariant. The first order terms of Fε ◦I = I ◦Fε give F1 ◦I = DI(F0 )[F1 ]. Using this equality, we see that the function φ(F0 )[F1 ] is also I-invariant: φ(F0 ◦ I)[F1 ◦ I] = φ(I ◦ F0 )[DI(F0 )[F1 ]] = I ∗ φ(F0 )[F1 ] = φ(F0 )[F1 ]. Thus, the difference Sb1 = S1 − φ(F0 )[F1 ] is I-invariant, too. P P P Finally, L ◦ I = k∈Z (Sb1 ◦ F0k ◦ I) = k∈Z (Sb1 ◦ I ◦ F0k ) = k∈Z (Sb1 ◦ F0k ) = L.  As we have seen, the differential of L measures the distance between invariant manifolds and thus M = dL is called the Melnikov function of the problem. It can be also constructed in the non-symplectic case, although it is not longer the differential of a function. We recall now this construction, but we will not go further in this direction, since the non-symplectic framework is out of the spirit of this paper. For the sake of simplicity, we only consider P = R2n . Assume that a diffeomorphism F0 : R2n → R2n has a separatrix 3 and n first integrals H1 , . . . , Hn , independent over the separatrix (but not necessarily in involution, since this concept requires a symplectic structure), and let Fε = F0 + εF1 + O(ε2 ) be a general perturbation of F0 . Given p ∈ 3, let Πp be the n-dimensional linear variety spanned by the point p and the vectors ∇Hj (p) (1 ≤ j ≤ n). Since Πp is transverse to 3 at p, there exist pu,s (ε) ∈ Wεu,s ∩ Πp , depending in a smooth way on ε, such that pu,s (0) = p. A natural measure of the distance between the invariant manifolds is given by the difference of first integrals (“energies”) 1(p, ε) = H(pu (ε)) − H(ps (ε)) = εM (p) + O(ε2 ),

H = (H1 , . . . , Hn )> ,

where M : 3 → Rn is the vector-valued Melnikov function of the problem. It is easy to generalize (actually, rewrite) the proof given in [DR96] for the planar case to see that X DH(pk+1 )[F1 (pk )], pk = F0k (p). (2.8) M (p) = k∈Z

Remark 2.7. Some similar results can be found in [BGK95], although with a less geometrical (and more functional) setting. They only can prove that a necessary condition for the existence of primary homoclinic orbits is the existence of zeros for M . Our geometrical construction shows that the existence of non-degenerate zeros for M is a sufficient condition for the existence of transverse primary homoclinic orbits, even in the non-symplectic case. However, it should be noted that [BGK95] deals with a broader range of maps; for example, the existence of first integrals is not needed.

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2.4. Twist maps. Now, we present another formulation of the method that is useful for the physical problems that verify the twist condition, since the formula for the Melnikov potential is simpler. For more details on twist maps, the reader is referred to [Gol94a, Gol94b, BG96]. We follow closely the notations and definitions of the later reference. An exact symplectic twist map (or simply, twist map) F is a map from a connected subset U of the cotangent bundle of a manifold M (which can be non-compact) into U , which comes equipped with a twist generating function L : M × M → R that satisfies F ∗ (y dx) − y dx = Y dX − y dx = dL(x, X),

(X, Y ) = F (x, y),

where (x, y) are any cotangent coordinates on T ∗ M, that is, x are coordinates on M, extended to coordinates (x, y) in the obvious way. The canonical form φ0 on T ∗ M reads as φ0 = y dx in cotangent coordinates. This can also be written in a coordinate free manner. Given L, one can retrieve the map (at least implicitly) from y = −∂1 L(x, X), and Y = ∂2 L(x, X). This can be done globally (i.e., U = T ∗ M) only when M is diffeomorphic to a fiber of T ∗ M, for example when M is the covering space of Tn or a manifold of constant negative curvature. The form F ∗ φ0 −φ0 is exact, so F is exact. Let S : U → R be the generating function of F , in the geometric sense of the previous definitions. Then, S(x, y) = L(x, X). The fact that S can be written in terms of old and new coordinates: (x, X), is the twist condition. In a coordinate free formulation it reads as S(p) = L(π(p), π(F (p))),

∀p ∈ U,

(2.9)

where π : T ∗ M → M is the canonical projection. Now, we carry out the generalization of (2.9) for families of twist maps, depending (in a smooth way) on a small parameter ε. That is, we search for the relationship between the first order variations in ε of the twist and geometric generating functions. Lemma 2.7. Let {Fε } be a smooth family of twist maps. Let Lε (resp. Sε ) be the twist (resp. geometric) generating function of Fε . Set Lε = L0 + εL1 + O(ε2 ) and Sε = S0 + εS1 + O(ε2 ). Then, Sb1 (p) = L1 (π(p), π(F0 (p))),

∀p ∈ U,

(2.10)

where Sb1 is the function given in (2.6). Proof. Fix p ∈ U and let (x, y) be cotangent coordinates in a neighbourhood of p. If we denote (Xε , Yε ) = Fε (x, y) = (X0 , Y0 ) + ε(X1 , Y1 ) + O(ε2 ), the O(ε) terms of the equality Sε (x, y) = Lε (x, Xε ) give S1 (x, y) = L1 (x, X0 ) + ∂2 L0 (x, X0 )X1 = L1 (x, X0 ) + Y0 X1 . Thus, from the definition of Sb1 and using φ0 = y dx we get Sb1 (x, y) = L1 (x, X0 ).



Assume now that F0 has a hyperbolic fixed point p∞ with a separatrix 3 ⊂ U and that Fε : U → U are exact diffeomorphisms. The choice Sb1 (p∞ ) = 0 reads as L1 (x∞ , x∞ ) = 0 in the twist frame, where x∞ = π(p∞ ). From Eq. (2.10), it follows directly that the Melnikov potential (2.7) can be written as X L1 (xk , xk+1 ), xk = π(pk ), pk = F0k (p). (2.11) L(p) = k∈Z

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This formula is simpler than (2.7), since only the first order term of the twist generating function Lε appears in it. 2.5. Hamiltonian maps. One of the main ideas in dynamical systems is to study maps in order to understand flows. For example, the description of Hamiltonian systems can be carried out considering the time-T maps of their flows, which are exact maps. Thus, it is interesting to present the previous results from the Hamiltonian point of view. Besides, this allows us to compare the discrete and continuous frameworks. Recall that a non-autonomous Hamiltonian system over an exact symplectic manifold (P, ω = − dφ) is given by a real-valued function (called the Hamiltonian) H : P × R → R. Then, the equations of motion have the form p˙ = XH (p, t), p ∈ P, t ∈ R, where for every fixed t, XH (·, t) is the Hamiltonian field generated by H(·, t): dH(p, t) = ω(p)(XH (p, t), ·), ∀p ∈ P. In symplectic coordinates (x, y) on P, we have φ = y dx, ω = dx ∧ dy and the Hamiltonian equations take the canonical form x˙ =

∂H (x, y, t), ∂y

y˙ = −

∂H (x, y, t). ∂x

It is clear that XH does not change if a function depending only on time is added to the Hamiltonian H. We will restrict ourselves to Hamiltonians such that generate a Hamiltonian flow, i.e., all the trajectories of XH are defined for all time. A Hamiltonian map F is the time-T map of some Hamiltonian H and for some T > 0, i.e., F = 9T : P → P, where 9t (p) stands for the solution of the Hamiltonian equations of H, with initial condition p at t = 0. Obviously, Hamiltonian maps are diffeomorphisms isotopic to the identity. Besides, they are exact over exact manifolds; if i(X)ω stands for the inner product of a form ω by a field X, and 8 : P × R → P × R is given by 8(p, t) = (9t (p), t), we get F ∗ φ − φ = (9T )∗ φ − (90 )∗ φ =

Z 0

Z

T

=

T

d [(9t )∗ φ] dt dt

∗

"Z

T

8 {i(XH ) dφ + d(i(XH )φ)} dt = d

0

# ∗

8 (i(XH )φ − H) dt .

0

Thus, the generating function S of F is given by Z

(F (p),T )

S(p) =

λ,

λ = φ − H dt,

(2.12)

(p,0)

where the one-form λ is the so-called Poincar´e-Cartan invariant integral, defined on the (extended) phase space P × R, and the path of integration is the trajectory 8(p, t), 0 ≤ t ≤ T , of the (extended) flow. Now, we carry out the generalization of Eq. (2.12) for families of Hamiltonian maps, depending (in a smooth way) on a small parameter ε. That is, we look for the relationship between the first order variations in ε of the Hamiltonians and the generating functions of their Hamiltonian maps. Lemma 2.8. Let Hε be a smooth family of non-autonomous Hamiltonians, and 9tε (p) the solution of its Hamiltonian equations with 90ε (p) = p. Let Fε and Sε be the Hamiltonian map 9Tε and its generating function, respectively. Set Hε = H0 + εH1 + O(ε2 ) and Sε = S0 + εS1 + O(ε2 ). Then

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Sb1 (p) = −

Z

T

H1 (9t0 (p), t) dt,

∀p ∈ P,

(2.13)

0

where Sb1 is the function given in (2.6). Proof. Let γ(p, ε) be the path in the (extended) phase space (9tε (p), t), 0 ≤ t ≤ T . Set ˙ tε (p)] − H0 (9tε (p), t), where the dot means the derivative with Aε (p, t) = φ(9tε (p))[9 respect to the time t. We will use through the proof the following notations for the first variation of the considered objects: ∂Fε ∂9tε ∂Aε t (p) , 91 (p) = (p) , A1 (p, t) = (p, t) . F1 (p) = ∂ε ∂ε ∂ε ε=0 ε=0 ε=0 Besides, we will prove below that B1 (p, t) = φ(9t0 (p))[9t1 (p)]. A1 (p, t) = B˙ 1 (p, t), R From Sε (p) = γ(p,ε) [φ − Hε dt], A1 = B˙ 1 , 9T1 = F1 and 901 ≡ 0, we get Z Z Sε (p) = [φ − H0 dt] − ε H1 dt + O(ε2 ) Z

γ(p,ε) T

γ(p,ε)

Z

T

Aε (p, t) dt − ε

= 0

0

Z

T

= S0 (p) + ε

(2.14)

H1 (9tε (p), t) dt + O(ε2 ) Z

T

B˙ 1 (p, t) dt − ε

0

H1 (9t0 (p), t) dt + O(ε2 )

0

Z

T

= S0 (p) + εφ(F0 (p))[F1 (p)] − ε

H1 (9t0 (p), t) dt + O(ε2 ),

0

and the terms O(ε) in this equation give (2.13). To end the proof, it only remains to check that (2.14) holds. For simplicity, we prove it using symplectic coordinates. Given p ∈ P and t ∈ R, let (x, y) be symplectic coordinates in a neighbourhood of 9t0 (p). We denote the coordinates of 9tε (p) by (xε , yε ) = (x0 , y0 ) + ε(x1 , y1 ) + O(ε2 ). Thus, Aε (p, t) = yε x˙ ε − H0 (xε , yε , t) = A0 (p, t) + ε[y0 x˙ 1 + y1 x˙ 0 − ∂x H0 (x0 , y0 , t)x1 − ∂y H0 (x0 , y0 , t)y1 ] + O(ε2 ) = A0 (p, t) + ε d[y0 x1 ]/ dt + O(ε2 ), where we have used the canonical form of Hamiltonian equations in symplectic coordi nates. Finally, since in these coordinates B1 = y0 x1 , Eq. (2.14) follows. Henceforth, we restrict ourselves to time-periodic Hamiltonians Hε , with T their period. Assume now that F0 has a hyperbolic fixed point p∞ with a separatrix 3. In the RT Hamiltonian frame, the choice Sb1 (p∞ ) = 0 becomes 0 H1 (9t0 (p∞ ), t) dt = 0. Indeed, it is possible (and more usual) to determine the Hamiltonian in such a way that it verifies the stronger condition H1 (9t0 (p∞ ), t) ≡ 0. From Eq. (2.13), it follows easily that the Melnikov potential (2.7) can be written as Z (2.15) L(p) = − H1 (9t0 (p), t) dt, R

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since 9t0 (F0k (p)) = 9t+kT (p), for all integer k and real t, and H1 is T -periodic in t. (This 0 is the reason to consider only periodic Hamiltonians.) We want to emphasize that the Hamiltonian version of the Melnikov potential can be deduced directly in the continuous frame, without appealing to discrete tools. However, taking into account the theory already developed in this paper, it has been easier to work directly on Hamiltonian maps. Remark 2.8. Usually, the unperturbed Hamiltonian H0 is time independent. In fact, in most of the applications it is Liouville integrable. Remark 2.9. Using the Lagrangian formalism instead of the Hamiltonian one, a similar formula to (2.15) can be obtained for Lagrangian maps (i.e., time-T maps of some Euler-Lagrangian flow), but with −H1 replaced by the first order in ε of the Lagrangian. 2.6. Lower Bounds. Along this subsection, we will assume without explicit mention that: (a) the invariant manifolds are doubled, that is, W0u = W0s , and (b) the bifurcation set is minimal, i.e., Σ = {p∞ }. (Remember that the hyperbolic fixed point p∞ is always contained in the bifurcation set Σ, see (i) of Lemma 2.5.) These hypotheses are equivalent to require that the separatrix is 3 = W0u,s \{p∞ }. We will say that the invariant manifolds are completely doubled in this case. Besides, we also assume n > 1, to avoid trivial degenerate cases. (In particular, the separatrix is connected.) To avoid a tedious exposition, several standard computations about Betti numbers are omitted. The expert reader in differential and algebraic topology will be able to fill in the gaps without difficulty, and we prefer to give the appropriate references for the novice one, instead of writing here a treatise. Thus, for a general discussion of Morse theory we refer to [Hir76], and for thorough discussions of homology the reader is urged to consult [Swi75, GH81]. The quotient manifold 3∗ := 3/F0 , consisting of unperturbed homoclinic orbits of 3, will be called the reduced separatrix (of the unperturbed map). It is shown below that 3∗ is a compact manifold without boundary. Since the Melnikov potential L is invariant under F0 , we can consider it defined over the reduced separatrix. (The new function is called L, too.) We search for lower bounds of the number of homoclinic orbits and the main idea is to apply Morse’s inequalities to the map L : 3∗ → R. The presence of symmetries and/or reversions usually leads to better results concerning the existence of homoclinic orbits. Let us introduce the (anti)symmetries that allow us to improve the lower bounds. We will say that the family {Fε } is antisymmetric if {Fε } is I-symmetric, for some involution I preserving the symplectic potential such that DI(p∞ ) = −Id. As it is well-known, involutions are locally conjugate to their linear parts at fixed points. Thus, there exist coordinates z = (z1 , . . . , z2n ) in some neighbourhood of p∞ such that I(z) = −z, that is, the maps Fε are odd in some coordinates defined close to p∞ . The definition above of antisymmetric maps is intended to translate the main features of odd maps on (R2n , dx ∧ dy) to maps on general exact manifolds. Under these hypotheses, Lemma 2.6 claims that the Melnikov potential is I-invariant. Thus, we can consider L defined over the quotient manifold 3∗I := 3/{F0 , I}, which has a richer topological structure than 3∗ , in the sense that Morse theory gives better lower bounds of the number of homoclinic orbits. We recall that a real-valued smooth function over a compact manifold without boundary is called a Morse function when all its critical points are non-degenerate. It is very well-known that the set of Morse functions is open and dense in the set of real-valued smooth functions [Hir76, p. 147]. Thus, to be a Morse function is a condition of generic

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position. Now we can state a result about the number of primary homoclinic orbits that persist under a general perturbation. In Sect. 3, we will verify the optimality of this result for specific examples. Theorem 2.2. Assume that L : 3∗ → R is a Morse function. Then the number of primary homoclinic orbits is at least 4. If the family {Fε } is antisymmetric, there exist at least 2n antisymmetric pairs of primary homoclinic orbits, and so at least 4n primary homoclinic orbits. Proof. From the celebrated Morse inequalities, a Morse functionPover a n-dimensional n compact manifold without boundary X has at least SB(X; R) := q=0 βq (X; R) critical points, where βq (X; R) are the R-Betti numbers of X and R is any field. Let us recall that βq (X; R) is the dimension of the q-th singular homology R-vector space of X, noted Hq (X, R). In the antisymmetric case, I 2 (p) = p 6= I(p), for all p ∈ 3. Thus (3∗ , Π) is a covering space of 3∗I of two sheets, where Π : 3∗ → 3∗I is the canonical projection onto the quotient of 3∗ by the antisymmetry I. In particular, L : 3∗I → R is a Morse function if and only if the same happens to L : 3∗ → R, and each critical point Q of L : 3∗I → R corresponds to an antisymmetric pair of critical points Π −1 (Q) = {O, I(O)} of L : 3∗ → R, for some unperturbed homoclinic orbit O ∈ 3∗ . Now the theorem follows from the formulae SB(3∗ ; Z2 ) = 4 and SB(3∗I ; Z2 ) = 2n. The rest of the proof is devoted to check that these formulae hold. Since Betti numbers are topological invariants, we look for topological spaces homeomorphic to 3∗ and 3∗I whose homologies can be easily computed. To accomplish it, let us consider the restriction f u,s of F0 to W0u,s , and denote B u,s = Df u,s (p∞ ). Since F0 is symplectic, det(B u ) · det(B s ) = 1, so det(B u ) and det(B s ) have the same sign. When these signs are positive (resp. negative) the map F0 preserves (resp. reverses) the orientation of 3, and we denote by σ = + (resp. σ = −) the so-called index of orientation. In the following lemma it is shown that the topological classification of f u only depends on σ. This will allow us to classify 3∗ and 3∗I just in terms of σ. Lemma 2.9. Let A± : Rn → Rn be the linear isomorphisms given by: A± (x) = 2x± ,

x = (x1 , . . . , xn ),

x± = (±x1 , x2 , . . . , xn ).

Then, there exists a global topological conjugation between f u and Aσ , that is, a homeomorphism g : Rn → W0u such that f u ◦ g = g ◦ Aσ . In the antisymmetric case, the conjugation g can be chosen in such a way that g(−x) = I(g(x)). Proof. We note that p∞ is a hyperbolic fixed point of f u , and all the eigenvalues of B u have modulus greater than one. From [PM82, Th. 5.5, II §5], we get that f u is locally conjugated at p∞ to A+ (resp. A− ) in the orientation-preserving (resp. orientationreversing) case. This local conjugation can be extended to a global one, using that f u and Aσ are global repulsors. The existence of an antisymmetric conjugation (certainly, a very intuitive fact) follows the same lines. We omit the details.  Thanks to Lemma 2.9, we now easily introduce time-energy coordinates (t, a) on 3. First, we give some notations. We denote by Sn , Tn , and Pn , the n-dimensional sphere, the n-dimensional torus, and the n-dimensional projective space, respectively. Besides, we introduce the n-dimensional manifold Xn := R × Sn−1 ,

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233

and the homeomorphism η : Xn → Rn \ {0}, η(t, a) = 2t a, whose inverse is given t(x), b a(x)) = (log2 |x| , x/ |x|). Then, b t(A± x) = b t(2x± ) = b t(x) + 1 and by η −1 (x) = (b a(2x± ) = (b a(x))± , so A± ◦ η = η ◦ ρ± , where the map ρ± : Xn → Xn is b a(A± x) = b ρ± (t, a) = (t + 1, a± ),

a = (a1 , . . . , an ),

a± = (±a1 , a2 , . . . , an ).

Thus, F0 : 3 → 3 and ρσ : Xn → Xn are topologically conjugated by g ◦ η, where g is the conjugation given in Lemma 2.9. This proves that 3∗ = 3/F0 and Xnσ := Xn /ρσ are homeomorphic. Hence, SB(3∗ ; Z2 ) = SB(Xnσ ; Z2 ). Concerning the antisymmetric case, we note that η ◦  = −η, where  : Xn → Xn ,

(t, a) = (t, −a).

Thus, the pairs of maps F0 , I : 3 → 3 and ρσ ,  : Xn → Xn are simultaneously topologically conjugated by g ◦ η. This proves that 3∗I = 3/{F0 , I} and Ynσ := Xn /{ρσ , } are homeomorphic. Hence, SB(3∗I ; Z2 ) = SB(Ynσ ; Z2 ). Consequently, it only remains to prove that SB(Xn± ; Z2 ) = 4 and SB(Yn± ; Z2 ) = 2n. First, we consider the case σ = +. In this case, Xn+ = S1 × Sn−1 and Yn+ = S1 × Pn−1 , since S1 = R/{t = t + 1} and Pn−1 = Sn−1 /{a = −a}. Therefore, from the well-known Z2 -homologies Hq (Sm ; Z2 ) ∼ =



Z2 if q = 0, m 0 otherwise

and K¨unneth’s Formula Hq (X × Y ; Z2 ) ∼ =   Z2 if q = 0, 2 Hq (X2± ; Z2 ) ∼ = Z2 ⊕ Z2 if q = 1 , 0 otherwise

Hq (Pm ; Z2 ) ∼ = Lq

p=0



Z2 if 0 ≤ q ≤ m , 0 otherwise

Hp (X; Z2 ) ⊗ Hq−p (Y ; Z2 ), we get

Hq (Xn± ; Z2 ) ∼ =



Z2 if q = 0, 1, n − 1, n 0 otherwise

for all n > 2, and   Z2 if q = 0, n Hq (Yn± ; Z2 ) ∼ = Z2 ⊕ Z2 if q = 1, . . . , n − 1 , 0 otherwise for all n > 1. Adding dimensions, we get SB(Xn+ ; Z2 ) = 4 and SB(Yn+ ; Z2 ) = 2n. Finally, a standard Mayer-Vietoris sequence argument shows that the Z2 -homologies  of Xnσ and Ynσ do not depend on σ, so SB(Xn− ; Z2 ) = 4 and SB(Yn− ; Z2 ) = 2n. Remark 2.10. Since the case σ = − is more intricate, one could believe that it is better to replace the maps with their squares to get σ = +. However, it should be noted that the lower bounds obtained in this way are worse since a single homoclinic orbit consist of two different ones for the square map: one gets 2 and 2n, instead of 4 and 4n, as the number of homoclinic orbits. Thus, the case σ = − deserves its own separate study. We also remark that this case cannot appear in the continuous frame, since the maps generated by a flow are isotopic to the identity.

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3. Standard-like Maps As a first example we deal with standard-like maps over the symplectic manifold (P, ω) = (R2n , dx ∧ dy), n > 1, which are ones of the most celebrated examples of twist maps. Among them, we consider perturbations of maps with central symmetry, since then the dynamics over the unperturbed separatrix is essentially one-dimensional and gives rise to explicit computations, as already announced in [DR97c]. In the sequel, given √ x, y ∈ Rn , Pn > x y and |x| stand for the scalar product i=0 xi yi and the Euclidean norm x> x. 3.1. Central standard-like maps. Let V : Rn → R be a function. The map F : R2n → R2n with equations F (x, y) = (y, −x + ∇V (y)) is called the standard-like map with potential V . It is immediate to check that L(x, X) = −x> X +V (X) is a twist generating function of F , so F is a twist map. When V is even, F is odd. It is worth mentioning that standard-like maps are also expressed in the literature as F (x0 , y 0 ) = (x0 + y 0 + ∇U (x0 ), y 0 + ∇U (x0 )), for some function U . The symplectic linear change of variables (x0 , y 0 ) = (y, y − x) is the bridge between these two equivalent 2 formulations, and the relation between the potentials is given by V (y) = |y| + U (y). Thus, it makes no difference which formulation is used, since we deal with symplectic invariants. A central standard-like map is a standard-like map with a central potential, i.e., 2 V (y) = Vc (|y| ) for some function Vc : [0, ∞) → R. Central standard-like maps are odd and have the “angular momenta” Aij (x, y) = xi yj − xj yi as first integrals. We denote = {(x, y) : Aij (x, y) = 0} the (n + 1)-dimensional manifold in R2n of zero by An+1 0 = {(qa, pa) : a ∈ Sn−1 , (q, p) ∈ R2 }. angular momenta. Clearly, An+1 0 Let F be a central standard-like map with potential V , and f : R2 → R2 the standardlike area preserving map defined by f (q, p) = (p, −q + 2Vc0 (p2 )p). We will call f the reduced map (in An+1 0 ) of F . This definition becomes clear when it is noted that f (q, p) = (Q, P ) ⇐⇒ F (qa, pa) = (Qa, P a),

∀(q, p) ∈ R2 , a ∈ Sn−1 .

(3.1)

Our interest in central standard-like maps is motivated by the following lemma, which follows easily from (3.1). Lemma 3.1. Let F be a central standard-like map and f its reduced map. Assume that Spec[Df (0)] = {e±h }, for some h > 0, and hence that the origin is a hyperbolic fixed point of f . Then: (i) The origin is a hyperbolic fixed point of F . Moreover, Spec[DF (0)] = {e±h }. (ii) Suppose now that f has a separatrix 0. Then, the invariant manifolds of F are completely doubled, giving rise to the separatrix 3 = {(qa, pa) : (q, p) ∈ 0, a ∈ Sn−1 }. (iii) Let σ = (q, p) : R → 0 be a natural parametrization of the separatrix 0, i.e., σ is a diffeomorphism that satisfies f (σ(t)) = σ(t + h), for all t ∈ R. Then, the diffeomorphism λ : R × Sn−1 → 3 defined by λ(t, a) := (q(t)a, p(t)a) satisfies F (λ(t, a)) = λ(t + h, a),

∀t ∈ R, a ∈ Sn−1 .

(3.2)

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235

We note that f is odd, so when it has a separatrix, it has in fact a double (symmetric) loop. The separatrix 3 is analytically diffeomorphic to R × Sn−1 , by means of λ. Thus, from now on, the functions defined over 3 will be expressed as functions of the timeenergy coordinates (t, a) ∈ R × Sn−1 . Now, we introduce the McLachlan map [McL94] as the central standard-like map 2 with potential V0 (y) = µ ln(1 + |y| ) (µ ∈ R). It has the expression ÿ ! 2µy F0 (x, y) = y, −x + . (3.3) 2 1 + |y| It is easy to check that for µ > 1 the reduced map of (3.3) – usually called the McMillan map – has a separatrix to the origin. (See Fig. 2 for a representation of the invariant curves.) In addition, the following natural parametrization of its separatrix can be found in [GPB89, DR96]: σ(t) = (q(t), p(t)), where q(t) = p(t − h) and p(t) = sinh(h) sech(t). Thus, using Lemma 3.1, the McLachlan map has its invariant manifolds completely doubled, and the function λ given by λ(t, a) = (p(t − h)a, p(t)a),

p(t) = sinh(h) sech(t),

cosh(h) = µ(> 1),

(3.4)

verifies Eq. (3.2). Remark 3.1. The McLachlan map has n first integrals Hj (j = 1, . . . , n), independent 2 2 2 2 over its separatrix: H1 (x, y) = |x| + |y| + |x| |y| − 2µx> y, and the angular momenta Hj = A1j (j = 2, . . . , n). This is not important for our purposes, but it would be essential for the study of non-symplectic perturbations with the Melnikov function (2.8). 3.2. Standard-like perturbations. Let us consider a general perturbation of (3.3) that preserves the standard character, i.e., ÿ ! 2µy + ε∇V (y) , µ > 1, ε ∈ R, (3.5) Fε (x, y) = y, −x + 2 1 + |y| where V : Rn → R. We determine V by imposing V (0) = 0. Then, the twist generating function of Fε that vanishes at the origin is Lε = L0 + εL1 , where L0 (x, X) = −x> X + 2 µ ln(1 + |X| ) and L1 (x, X) = V (X). Using formulae (2.7), (3.2) and (3.4), the Melnikov potential of the problem is L : R × Sn−1 → R,

L(t, a) =

X k∈Z

V (p(t + hk)a),

p(t) =

sinh(h) . cosh(t)

(3.6)

Obviously, L is h-periodic in t, so we can consider t defined modulo h and L as a function over S1 × Sn−1 . Henceforth it will be assumed that h > 0, cosh(h) = µ. Now, we focus our attention on entire perturbations, i.e., maps (3.5) with V an entire function. The result about the splitting in this case is the following one. Theorem 3.1. If V is entire but not identically zero, then the manifolds Wεu,s of the map (3.5) split, for 0 < |ε|  1.

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Proof. By Theorem 2.1, it is sufficient to check that the Melnikov potential (3.6) is not constant. First, we note that the only singularities of p(t) are simple poles at any point tp ∈ π i/2 + π iZ, and therefore it is analytic at tp + hk for k ∈ Z \ {0}. Now, let Va , fa : R → R be the functions defined by Va (t) := V (ta) and fa (t) = Va (p(t)) = V (p(t)a) a ∈ Sn−1 such that (a ∈ Sn−1 ). Since V is a non-zero entire function, there exists b Vba is a non-zero entire function. Thus, fba has non-removable singularities at any point tp ∈ π i/2 + π iZ, and P however it is analytic at tp + hk for k ∈ Z \ {0}. Consequently, a) = k∈Z fba (t + kh) has a non-removable singularity at any point in Lba (t) := L(t, b π i/2 + hZ + π iZ. This proves that L is not constant.  Remark 3.2. The assumption of the entire function on V has only been used to ensure a ∈ Sn−1 , such that fba (t) has an isolated singularity that there exist tp ∈ π i/2+π iZ and b at tp , and however is analytic on tp + hk for k ∈ Z \ {0}. Thus, this assumption on V can be relaxed, although the entire case is the simplest case to study. We observe that for even V , the maps Fε are odd and hence the family {Fε } is antisymmetric. Therefore, Theorem 2.2 gives the following corollary. Corollary 3.1. Assume that the function L given in (3.6) is a Morse function. Then, the map (3.5) has at least 4 primary homoclinic orbits, for 0 < |ε|  1. If, in addition, the potential V is an even function, there exist at least 2n antisymmetric pairs of primary homoclinic orbits, and so at least 4n primary homoclinic orbits. 3.3. Polynomial perturbations: Explicit computations. We show here that explicit computations of Melnikov potentials can be performed, for any polynomial perturbations of PN the McLachlan map, i.e., maps (3.5) with V (y) = `=1 V` (y), for some finite N , where V` denotes a homogeneous polynomial of order `. In this case, the Melnikov potential (3.6) turns out to be a linear combination of products of certain elliptic functions Σ` in the variable t ∈ C (of periods h, 2π i) and the homogeneous polynomials V` restricted to Sn−1 : L(t, a) =

N X `=1

sinh` (h)V` (a)Σ` (t),

Σ` (t) =

X

[sech(t + hk)]` .

(3.7)

k∈Z

Using the Summation Formula of the Appendix, all the elliptic functions Σ` (and consequently, the Melnikov potentials) can be explicitly computed. However, using the Summation Formula to find Σ` for big values of ` is rather tedious. It is better to use an idea contained in [GPB89]. The point is to note that the odd (respectively, even) powers of the hyperbolic function sech can be expressed as a linear combination, with rational coefficients, of the even derivatives of sech (respectively, sech2 ). This allows us to write Σ` as a linear combination, with rational coefficients, of the even derivatives of Σ1 (if ` is odd) or Σ2 (if ` is even). For example, sech3 = (sech − sech00 )/2 and sech4 = [4 sech2 −(sech2 )00 ]/6, so Σ3 = (Σ1 − Σ100 )/2 and Σ4 = (4Σ2 − Σ200 )/6. Consequently, it is enough to compute Σ` for ` = 1, 2. This is done in Lemma A.1 (see the Appendix) and the result is:       √ 2K2π 4K2π t 4K2π t m2π + dn m2π m2π cn , Σ1 (t) = h h h 2  0    Eπ 2Kπ t 2Kπ 2 mπ , − 1 + dn Σ2 (t) = h Kπ0 h

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237

where, if K(m) and E(m) are the elliptic integrals of the first and second kind, the parameter m = mT (T = π, 2π) of the Jacobian elliptic functions is determined by the equation K(1 − mT )/K(mT ) = T /h; and KT = K(mT ), KT0 = K(1 − mT ), ET0 = E(1 − mT ). It is equivalent to choose q = qT = e−πT /h as the nome of the elliptic functions. For the notations about elliptic functions we refer again to the Appendix. Assume now that V = V2 , i.e., V is a quadratic form or, in other words, the perturbation ∇V is linear. We can write V (y) = y > By, for some symmetric n×n matrix B. Then, there exists an orthogonal matrix Q = (q1 · · · qn ) such that diag(b1 , . . . , bn ) = Q> BQ, where bi are the eigenvalues of B and qi are their respective (normalized) eigenvectors. Proposition 3.1. Suppose det(B) 6= 0 and that B does not have multiple eigenvalues. Then: 1. The invariant manifolds Wεu,s are transverse along exactly 4n primary homoclinic orbits Oσ,±i (ε) (σ ∈ {0, 1}, i ∈ {1, . . . , n}), for 0 < |ε|  1. These perturbed homoclinic orbits are created from the unperturbed ones Oσ,±i (0) = λ σh/2 + kh, ±qi



k∈Z

,

σ ∈ {0, 1}, i ∈ {1, . . . , n}.

2. The homoclinic area between the primary homoclinic orbits Oσ,±i (ε) and Oτ,±j (ε) is given by the asymptotic expression   1W Oσ,±i (ε), Oτ,±j (ε) = ε1σ,τ,i,j + O(ε2 ), where

  1σ,τ,i,j = 1σ,τ,i,j (h) = sinh2 (h)(2Kπ /h)2 bi δσ − bj δτ ,

with δ0 = Eπ0 /Kπ0 and δ1 = Eπ0 /Kπ0 − mπ . Proof. We note that Q(Sn−1 ) = Sn−1 P,nso we can perform the change of variables a ↔ Qa in Sn−1 and then, V (Qa) = i=1 bi (ai )2 , where bi 6= 0, for all i, and bi 6= bs , for all i 6= s. It is easy to check that the only critical points of the restriction of V to Sn−1 are {±qi : 1 ≤ i ≤ n}, all of them being non-degenerate. Moreover, from the properties of the Jacobian elliptic function dn(u|m), the real critical points of Σ2 are {kh/2 : k ∈ Z}, that are also non-degenerate. Consequently, L is a Morse function over (R/hZ) × Sn−1 and its critical points are (σh/2, ±qi ), for σ ∈ {0, 1}, i ∈ {1, . . . , n}. Now the first part of the proposition follows from Theorem 2.1. For the second part, it is enough to observe that   1W Oσ,±i (ε), Oτ,±j (ε) = ε[L(σh/2, ±qi ) − L(τ h/2, ±qj )] + O(ε2 ), 2 2 and L(σh/2, ±qi ) = sinh2 (h)V (±qi )Σ2 (σh/2) √ = sinh (h)bi (2Kπ /h) δσ , where we have used that dn(0|m) = 1 and dn(K|m) = 1 − m. 

Finally, we study the linear potentials (constant perturbations ∇V ), that is, V = V1 . Thus, V (y) = b> y, for some vector b ∈ Rn \ {0}, and the critical points of V in Sn−1 are ±q, where q = b/ |b|. Of course, they are non-degenerate. Then, using the same arguments as in the proof of the preceding proposition, we get the following result.

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Proposition 3.2. With the previous notations and assumptions: 1. The invariant manifolds Wεu,s are transverse along exactly 4 primary homoclinic b±σ (ε) (σ ∈ {0, 1}), for 0 < |ε|  1. These perturbed homoclinic orbits are orbits, O created from the unperturbed ones
b±σ (0) = λ(σh/2 + kh, ±q) , σ ∈ {0, 1}. O k∈Z b±τ (ε) is b±σ (ε) and O 2. The homoclinic area between the primary homoclinic orbits O given by the asymptotic expression i h b±τ (ε) = ε1 b±σ (ε), O b ±σ,±τ + O(ε2 ), 1W O where

h i b ±σ,±τ = 1 b ±σ,±τ (h) = sinh(h) |b| (2K2π /h) δb±σ − δb±τ , 1

with δb±σ = ±δbσ , (σ ∈ {0, 1}), and δb0 = 1 +

√

m2π , δb1 = 1 −

√

m2π .

The conditions det(B) 6= 0, B without multiple eigenvalues (for the quadratic potentials) and b 6= 0 (for the linear ones) are the conditions of generic position for L to be a Morse function. The condition B without multiple eigenvalues is equivalent to the complete breakdown of the central symmetry. The examples of this subsection show that the lower bounds on the number of homoclinic orbits provided by Theorem 2.2 are optimal. 3.4. Polynomial perturbations: weakly hyperbolic cases. It is a very well-known fact that the splitting size for analytic area preserving maps in the plane is exponentially small in the hyperbolicity parameter h, for families of maps which degenerate to the identity when h = 0 [FS90]. Here, e±h stands for the eigenvalues of the differential of the perturbed map on the perturbed weakly hyperbolic fixed point. Then, there arises the natural question about whether a similar result holds for analytic and symplectic maps in higher dimensions. We show here some results that lead us to believe that the answer is affirmative. For the sake of brevity, we restrict ourselves to the case V (y) = y > By, but the same study can be carried out polynomial perturbation. Using that p for any concrete P 2 2 qπ = e−π /h and the formula 2Km1/2 /π = 2 k≥0 q (k+1/2) [WW27, p. 479], we get 10,1,i,i (h) = 16π 2 bi h−2 sinh2 (h)e−π

2

/h

 X 

k≥0

exp[−π 2 k(k + 1)/h]

4  

.

Thus, the homoclinic area between O0,±i (ε) and O1,±i (ε) (i ∈ {1, . . . , n}), is a priori exponentially small in h. A priori means that the first order term in ε is exponentially small in h. Of course, this does not imply that the higher order terms are also exponentially small in h. All the other homoclinic areas are not a priori exponentially small, or are trivially zero because of the odd character of Fε . It is important to remark that this is only a partial result: we have assumed that h is small enough, but fixed, and ε → 0. If ε and h tend simultaneously to zero, then one is confronted with the difficult problem of justifying that some errors that seem to be

Melnikov Potential for Exact Symplectic Maps

239

O(ε2 ) can be neglected in front of the main term that is O(e−π /h ). Thus, the question is whether some asymptotic formulae like   2 1W O0,±i (ε), O1,±i (ε) ∼ ε10,1,i,i (h) ∼ 16π 2 bi εe−π /h , 2

hold, when ε and h tend to zero in any independent way. At the present moment, we do not have an analytical proof of these asymptotic formulae, but, concerning the planar case (n = 1), in [DR97a] we have succeeded in proving that the Melnikov method gives the correct asymptotic exponentially small behaviour under a generic assumption on the perturbative potential V (y), for ε = O(hp ) and p > 6. Besides, there is numerical evidence that the hypothesis ε = O(hp ), p > 6, can be improved up to ε = o(1) [DR97b]. (It is important to remark here that such numerical experiments require an expensive multiple-precision arithmetic in order to detect the exponentially small size of the splitting.) Nevertheless, from the computations above, it turns out that the exponentially small splitting can only take place along the direction of the t coordinate over 3, since a directional derivative of L is exponentially small only in the t direction. (Recall that the differential of L measures the distance between the perturbed invariant manifolds.) This leads us to propose an affirmative answer about the exponentially small character of the splitting of the separatrices, at least in one direction. To give a dynamical interpretation of this distinguished direction, we note that if h → 0 the action of the unperturbed map over 3 tends to a flow whose orbits are the coordinate curves {a = constant} of the parametrization λ(t, a). It is important to observe that this direction does not depend on the perturbation. Moreover, the computations above show that the distinguished pairs of homoclinic orbits which give a priori exponentially small splittings are just the interlaced pairs, i.e., the pairs created from unperturbed orbits situated on the same coordinate curve {a = constant} (in a interlaced way) of the separatrix 3. Finally, we want to stress that a priori exponentially small asymptotic expressions can be computed for the splitting angles in the t-direction over 3. However, it seems better to work with the homoclinic area since it is an homoclinic invariant, whereas the splitting angles are not. 4. A Magnetized Spherical Pendulum Finally, as a second example, we focus our attention on Hamiltonian maps that arise from perturbations of a central field. The exact manifold is the same as in the previous example. 4.1. Unperturbed problem. First, we give some well-known definitions and results. Let 2 T : Rn → R be the so-called kinetic energy T (y) = 21 |y| and let V : Rn × R → R be the potential energy. The Hamiltonians H : R2n × R → R of the form H(x, y, t) = T (y) + V (x, t), are called natural. The Hamiltonian equations can be written as x¨ = −∂V (x, t)/∂x. Notice that if V (x, t) is even in the spatial variable x, the Hamiltonian map is odd. 2 When V (x, t) = Vc (|x| ), for some function Vc : [0, ∞) → R, the Hamiltonian field is an (autonomous) central field, and hence the angular momenta are preserved. Let = {(ra, ra) ˙ : a ∈ Sn−1 , (r, r) ˙ ∈ R2 } be the manifold of zero angular momenta. An+1 0 the Hamiltonian system to one Using the central symmetry, we can reduce on An+1 0

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degree of freedom: r¨ = −2Vc0 (r2 )r; that is, if r(t) is a solution of the reduced system, then λ(t, a) = (r(t)a, r(t)a) ˙ is a solution of the original system, for all a ∈ Sn−1 . In [Gru85], one of the first papers on the generalization of the Melnikov method for high-dimensional (continuous) systems, an (undamped) magnetized spherical pendulum was considered. It is given by the (autonomous) central field with Vc (r2 ) = (r4 − r2 )/2. Obviously, the cases n > 2 have no real physical meaning and the cited reference does not deal with them, but the generalization is trivial and it is interesting in order to compare with the section before. The following lemma follows from a straightforward computation on the reduced system r¨ = r − 2r3 , i.e., a Duffing equation. Lemma 4.1. Let 9t0 (p) be the solution of the Hamiltonian equations of this magnetized spherical pendulum, with initial condition p at t = 0. Given h > 0, let F0 be the Hamiltonian map 9h0 : R2n → R2n . Then: (i) The origin is a hyperbolic fixed point of F0 . Moreover, Spec[DF0 (0)] = {e±h }. (ii) The invariant manifolds of F0 are completely doubled, giving rise to the separatrix 3 = {(ra, ra) ˙ : r˙2 = r2 − r4 , r 6= 0, a ∈ Sn−1 }. (iii) The diffeomorphism λ : R × Sn−1 → 3 defined by λ(t, a) = (r(t)a, r(t)a), ˙ verifies

9s0 (λ(t, a)) = λ(t + s, a),

r(t) = sech t, ∀t, s ∈ R, a ∈ Sn−1 .

(4.1) (4.2)

4.2. Perturbed problem. Let us consider a perturbation that preserves the natural character, i.e., the perturbed Hamiltonians are 4

2

Hε (x, y, t) = T (y) + (|x| − |x| )/2 + εV (x, t/h),

h > 0, ε ∈ R,

where V = V (x, ϕ) is 1-periodic in ϕ. We determine V by imposing V (0, ϕ) ≡ 0. Small values of h correspond to a rapidly forced pendulum of angular frequency (radians per second) ω = 2π/h. We denote by Fε the Hamiltonian map 9hε , where 9tε (p) is the solution of the Hamiltonian equations of Hε , with initial condition p. (The dependence on the parameter h is omitted to simplify the notation.) Using Eqs. (2.15), (4.2) and (4.1), the Melnikov potential L : R × Sn−1 → R of the problem turns out to be Z Z L(t, a) = − V (r(t + s)a, s/h) ds = − V (r(s)a, (s − t)/h) ds, r(s) = sech s. R

R

(4.3) Now, we consider polynomial perturbations, that is, we assume that the TaylorFourier expansion of the potential V has a finite number of terms. We write X [Ck,` (x) cos(2πkϕ) + Sk,` (x) sin(2πkϕ)], (4.4) V (x, ϕ) = (k,`)∈K

where K is a finite subset of {(k, `) ∈ Z2 : k ≥ 0, ` ≥ 1} and Ck,` , Sk,` are homogeneous polynomials of degree `. In this case, the Melnikov potential can be explicitly computed. The result is summarized in the following lemma, whose proof is straightforward.

Melnikov Potential for Exact Symplectic Maps

241

Lemma 4.2. Let P` (ω) (` ≥ 0) be the polynomials generated by the recurrences P0 (ω) = 1,

P1 (ω) = ω,

P`+1 (ω) =

ω 2 + `2 P`−1 (ω). `(` + 1)

(4.5)

Then, the Melnikov potential (4.3) with V given in (4.4) is X L(t, a) = π {sech(πkω/2)P`−1 (kω)[Ck,` (a) cos(kωt) − Sk,` (a) sin(kωt)]}, (k,`)∈K

(4.6) where ω = 2π/h is the frequency of the perturbation. A typical difference between the continuous and discrete frames is revealed here: the Melnikov potential (4.6) is an entire periodic function in the complex variable t, whereas the Melnikov potential (3.6) is a doubly periodic one with singularities. Another difference is that a theorem like 3.1 does not hold for the pendulum, since there exist perturbative potentials V (x, ϕ) such that the Melnikov potential (4.6) vanishes identically. 2 We also notice that sech(πkω/2) = sech(kπ 2 /h) ∼ e−π /h , when h → 0. Thus, a discussion on a priori exponentially small splittings for this rapidly forced magnetized pendulum, along the lines of the previous section, can be given for any polynomial perturbation. As in the previous section, the exponentially small asymptotic expressions predicted by the Melnikov method are far from being proved for n > 1. However, it is well-known that for some perturbations of the rapidly forced planar pendulum [DS92], the Melnikov method gives the right answer. Finally, we consider the perturbative potential V (x1 , x2 , ϕ) =

2 x2 (x21 + x22 ) cos(2πϕ), ω2 + 1

which was already studied in [Gru85]. In that paper, the general (non-Hamiltonian) case is considered, and consequently the symplectic structure is not taken into account, even in the examples where it was possible, like the one above. Using the formula (4.6), we get 1 the Melnikov potential L(t, a) = π sech πω 2 sin ϑ cos ωt, where a = (cos ϑ, sin ϑ) ∈ S . Its gradient is just the vector-valued Melnikov function used in [Gru85] to measure the splitting. Obviously, it is easier to compute a real-valued function than a vector-valued one. For higher dimensional cases, the saving of work is even more. Appendix: Elliptic Functions A function that plays an important role in the computation of the infinite sums that appear in Melnikov potentials, is a complex function χ satisfying the following properties, where T, h > 0 are given parameters: (C1) χ is meromorphic on C. (C2) χ is T i-periodic and its derivative is h-periodic. (C3) The set of poles of χ is hZ + T iZ, and all of them are simple and of residue 1. Remark A.1. Conditions (C1)–(C3) determine a function except for an additive constant: if χ1 satisfies also (C1)-(C3), (χ − χ1 )0 is an entire doubly periodic function, and it must be a constant; thus, χ(z) − χ1 (z) = az + b, but a = 0 due to the T i-periodicity.

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The function χ can be expressed in terms of Jacobian elliptic functions, Theta functions, or Weierstrassian functions. The Jacobian elliptic functions are well adapted to pencil-and-paper computations, whereas the Theta functions are the best from the numerical point of view, and the Weierstrassian functions are the natural choice for theoretical work on account of their symmetry in the periods. Here, we deal with pencil-and-paper computations, so our choice are the Jacobian elliptic functions. For a general background on elliptic functions of any kind, we refer to [AS72, WW27]. We follow the notation of the first reference. Given the parameter m ∈ [0, 1], we recall that Z π/2 Z π/2 −1/2 (1 − m sin ϑ) dϑ, E = E(m) := (1 − m sin ϑ)1/2 dϑ, K = K(m) := 0

0

are the complete elliptical integrals of the first and second kind and that Z u E(u) = E(u|m) := dn2 (v|m) dv, 0

is the incomplete elliptic integral of the second kind, where dn is one of the wellknown Jacobian elliptic functions. Moreover, introducing K 0 = K 0 (m) := K(1 − m), E 0 = E 0 (m) := E(1 − m), we also recall that the nome q, |q| < 1, is defined by 0 q = q(m) := e−πK /K . If any of the numbers m, q, K, K 0 , E, E 0 or K 0 /K is given, all the rest are determined. From a numerical point of view, it is better to fix first the nome q, and after compute the rest of parameters and elliptic functions, since the q-series are rapidly convergent. It is not difficult to check (see [DR96]) that χT (z) = (2KT /h)2 (ET0 /KT0 − 1)z + (2KT /h)E(2KT z/h + KT0 i|mT ) verifies (C1)-(C3), where the nome is determined by q = qT = e−πT /h , and mT , KT , KT0 , ET , ET0 are the associated parameters. (The dependence on h is not explicitly written.) Thus, (A.1) KT0 /KT = π −1 log(1/qT ) = T /h. Given an isolated singularity z0 ∈ C of a function f , let us denote a−j (f, z0 ) the coefficient of (z−z0 )−j in the Laurent expansion of f around z0 . Obviously, a−j (f, z0 ) = 0 if z0 is a pole of f and j is greater than its order. Proposition A.1 (Summation Formula). Let f be a function verifying: (P1) f is analytic in R and has only isolated singularities on C. (P2) f is T i-periodic for some T > 0. (P3) |f (t)| ≤ Ae−c|
X z∈SingT (f )

res(χT (· − t)f (·), z) = −

X

X a−(j+1) (f, z)

z∈SingT (f ) j≥0

j!

χ(j) T (z − t),

where SingT (f ) is the set of singularities of f in IT = {z ∈ C : 0 < =z < T }.

(A.2)

Melnikov Potential for Exact Symplectic Maps

Proof. See [DR96, Prop. 3.1].

243



If f is meromorphic in C, the same happens to Σ, and then Σ is elliptic. From a computational point of view, this is the interesting case, since then (A.2) is a finite sum and can be explicitly computed, as the following lemma, used in Sect. 3, shows. P Lemma A.1. Let Σ` (t) = k∈Z f ` (t + kh), where f = sech. Then:      √ 4K2π t 2K2π 4K2π t m2π + dn m2π Σ1 (t) = m2π cn , h h h 2  0    2Kπ Eπ 2Kπ t 2 m − 1 + dn . Σ2 (t) = π h Kπ0 h 

Proof. Clearly, f = sech satisfies properties (P1)-(P3) with T = 2π. Moreover, the singularities of f in I2π = {z ∈ C : 0 < =z < 2π} are simple poles: π i/2 and 3π i/2, with a−1 (f, π i/2) = −a−1 (f, 3π i/2) = − i. Thus, from (A.2) we get   Σ1 (t) = i χ2π (π i/2 − t) − χ2π (3π i/2 − t) . From Eq. (A.1) with T = 2π, and using that E(u + 2K 0 i) − E(u) is a constant, and that E(−u) = −E(u), we have 0 0 0 − E2π ) − E(v/2 + K2π i|m2π ) + E(v/2|m2π )], Σ1 (t) = i(2K2π /h)[ i(K2π 0 i and u = 4K2π t/h. where v = u − K2π In [WW27, pp. 520 and 508] we find the following formulae

E(v + K 0 i) − E(v) = i(K 0 − E 0 ) + cn(v) ds(v), dn(v) + cn(v) cn(v/2) ds(v/2) = = ds(v) + cs(v). sn(v) Therefore, we arrive at the following expression for Σ1 0 0 Σ1 (t) = − i(2K2π /h)[ds(u − K2π i|m2π ) + cs(u − K2π i|m2π )], √ and the formula for Σ1 follows from ds(u − K 0 i) = i m cn(u) and cs(u − K 0 i) = i dn(u). The formula for Σ2 is easier, since f 2 = sech2 also verifies the properties (P1)-(P3), but with T = π instead of T = 2π. It has only one singularity in Iπ : π i/2. Moreover, π i/2 is a double pole with a−1 (f 2 , π i/2) = 0 and a−2 (f 2 , π i/2) = −1. Thus, by (A.2) we get Σ2 (t) = χ0π (π i/2 − t). But E 0 (u) = dn2 (u) is an even 2K 0 i-periodic function, so the formula for Σ2 follows from (A.1) for T = π. 

Acknowledgement. This work has been partially supported by the EC grant ERBCHRXCT-940460 and the NATO grant CRG950273. Research by Amadeu Delshams is also supported by the Spanish grant DGICYT PB94-0215 and the Catalan grant CIRIT 1996SGR–00105. Research by Rafael Ram´ırez-Ros is also supported by the U.P.C. grant PR9409. Both authors wish to express our appreciation to J. Amor´os, M.A. Barja and P. Pascual for their help on the topics related to algebraic topology. It is also a pleasure to thank V. Gelfreich, A. Haro, R. de la Llave, J. Ortega, C. Sim´o, E. Tabacman and D. Treschev for very stimulating discussions and fruitful remarks.

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References [AA89] [AM78] [Ang93]

[Arn76] [AS72] [BF96] [BG96] [BGK95] [Bol95]

[BW95] [DR96] [DR97a] [DR97b] [DR97c] [DRS97] [DS92] [Eas91] [FS90] [GG73] [GH81] [Gol94a] [Gol94b] [GPB89] [Gru85] [GS77] [Hir76] [Lom96a] [Lom96b] [Lom97]

Arnold, V.I. and Avez, A.: Ergodic Problems of Classical Mechanics: Advanced Book Classics. Addison-Wesley, 1989 Abraham, R.H. and Marsden, J.E.: Foundations of Mechanics Reading, Mass.: Benjamin/Cummings, 1978 Angenent, S.: A variational interpretation of Melnikov’s function and exponentially small separatrix splitting. In: D. Salamon, editor, Symplectic geometry, volume 192 of London Math. Soc. Lecture Note Ser., Cambridge: Cambridge Univ. Press, 1993 pp. 5–35 Arnold, V.I.: Les m´ethodes math´ematiques de la m´ecanique classique. Moscou: Mir, 1976 Abramowitz, M. and Stegun, I.: Handbook of Mathematical Functions. New York: Dover, 1972 Baldom`a, M. and Fontich, E.: Poincar´e-Melnikov theory for n-dimensional diffeomorphisms. Preprint, 1996. To appear in Applicationes Mathematicae Boyland, P. and Gol´e, C.: Lagrangian systems on hyperbolic manifolds. Preprint 1996/1, Institute for Mathematical Sciences, Stony Brook, January 1996 Bountis, T.C., Goriely, A. and Kollmann, M.: A Melnikov vector for n-dimensional mappings. Phys. Lett. A 206, 38–48 (1995) Bolotin, S.V.: Homoclinic orbits to invariant tori of Hamiltonian systems. In: Kozlov, V.V., editor, Dynamical systems in classical mechanics. volume 168 of Amer. Math. Soc. Transl. Ser. 2, Providence, RI: Amer. Math. Soc., 1995, pp. 21–90 Burns, K. and Weiss, H. A geometric criterion for positive topological entropy. Comm. Math. Phys. 172, 95–118 (1995) Delshams, A. and Ram´ırez-Ros, R.: Poincar´e-Melnikov-Arnold method for analytic planar maps. Nonlinearity 9 (1), 1–26 (1996) Delshams, A. and Ram´ırez-Ros, R.: Exponentially small splitting of separatrices for perturbed integrable standard-like maps. Preprint, 1997. To appear in J. Nonlinear Sci. Delshams, A. and Ram´ırez-Ros, R.: Singular separatrix splitting and Melnikov method: An experimental approach. Preprint, 1997 Delshams, A. and Ram´ırez-Ros, R.: Poincar´e-Melnikov-Arnold method for symplectic maps. In: [Sim97] below Delshams, A. Ram´ırez-Ros, R., and Seara, T.M.: Splitting of separatrices in Hamiltonian systems and symplectic maps. In: [Sim97] below Delshams, A. and Seara, T.M.: An asymptotic expression for the splitting of separatrices of the rapidly forced pendulum. Comm. Math. Phys. 150, 433–463 (1992) Easton, R.W.: Transport through chaos. Nonlinearity, 4, 583–590 (1991) Fontich, E. and Sim´o, C.: The splitting of separatrices for analytic diffeomorphisms. Ergodic Theory Dynamical Systems 10, 295–318 (1990) Golubitsky, M. and Guillemin, V.: Stable Mappings and Their Singularities, volume 14 Graduate Texts in Math. New York: Springer, 1973 Greenberg.M.J. and Harper, J.R.: Algebraic Topology – A First Course, volume 58 Math. Lecture Note Ser. Reading, MA: Addison-Wesley, 1981 Gol´e, C.. Optical Hamiltonians and symplectic twist maps. Phys. D 71, 185–195 (1994) Gol´e, C.: Periodic orbits for Hamiltonian in cotangent bundles. Trans. Am. Math. Soc. 343 (1), 327–347 (1994) Glasser, M.L., Papageorgiou, V.G. and Bountis, T.C.: Melnikov’s function for two-dimensional mappings. SIAM J. Appl. Math. 49 (3), 692–703 (1989) Gruendler, J.: The existence of homoclinic orbits and the method of Melnikov for systems in Rn . SIAM J. Math. Anal. 16 (5), 907–931 (1985) Guillemin, V. and Sternberg, S. Geometric Asymptotics. Am. Math. Soc., Providence, R.J., 1977 Hirsch, M.W.: Differential Topology, volume 33 Graduate Texts in Math. New York, Heidelberg, Berlin: Springer-Verlag, 1976 Lomel´ı, H.E.: Perturbations of elliptic billiards. Phys. D 99, 59–80 (1996) Lomel´ı, H.E.: Saddle connections and heteroclinic orbits for standard maps. Nonlinearity 9, 649– 668 (1996) Lomel´ı, H.E.: Applications of the Melnikov method to twist maps in higher dimensions using the variational approach. Ergodic Theory Dynam. Systems 17 (2), 445–462 (1997)

Melnikov Potential for Exact Symplectic Maps

[LT93]

[Mat86] [McL94] [McM71]

[Mei92] [MMP84] [PM82] [Poi99] [Rom95] [RW88] [Sim97]

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Levallois, P. and Tabanov, M.B.: S´eparation des s´eparatrices du billard elliptique pour une perturbation alg´ebrique et sym´etrique de l’ellipse. C. R. Acad. Sci. Paris S´er. I Math. 316, 589–592 (1993) Mather, J.N.: A criterion for non-existence of invariant circles. Publ. Math. I.H.E.S. 63, 153–204 (1986) McLachlan, R.I.: Integrable four–dimensional maps of standard type. Phys. Lett. A 177, 211–214 (1994) McMillan, E.M.: A problem in the stability of periodic systems. In: E. Brittin and H. Odabasi, editors, Topics in modern physics, a tribute to E.V. Condon, Boulder, CO: Colorado Assoc. Univ. Press, 1971, pp. 219–244 Meiss,J.D.: Symplectic maps, variational principles, and transport. Rev. Modern Phys. 64 (3), 795–848 (1992) MacKay, R.S., Meiss, J.D. and Percival, I.C.: Transport in Hamiltonian systems. Phys. D 13, 55–81 1984 Palis, J., Jr., and de Melo, W.: Geometric Theory of Dynamical Systems. New York: SpringerVerlag, 1982 Poincar´e, H.: Les m´ethodes nouvelles de la m´ecanique c´eleste, volume 1, 2, 3 Paris: GauthierVillars, 1892–1899 Rom-Kedar, V.: Secondary homoclinic bifurcations theorems. Chaos 5 (2), 385–401 (1995) Rom-Kedar, V. and Wiggins, S.: Transport in two-dimensional maps. Arch. Rational Mech. Anal. 109, 239–298 (1988) Sim´o, C. ed.: Hamiltonian Systems with Three or More Degrees of Freedom. NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. Held in S’Agar´o, Spain, 19–30 June 1995. Dordrecht: Kluwer Acad. Publ., Holland, to appear in 1997 Sun, J.H.: Melnikov vector function for high-dimensional maps. Phys. Lett. A 216, 47–52 (1996) Suris, Yu.B.: Integrable mappings of the standard type. Funct. Anal. Appl. 23, 74–76 (1989) Switzer, R.M.: Algebraic Topology – Homotopy and Homology, volume 212 Grundlehren Math. Wiss. New York–Heidelberg–Berlin: Springer-Verlag, 1975 Tabanov, M.B.: Separatrices splitting for Birkhoff’s billiard in a symmetric convex domain, close to an ellipse. Chaos 4(4), 595–606 (1994) Tabacman, E.: Variational computation of homoclinic orbits for twist maps. Phys. D 85, 548–562 (1995) Treschev, D.V.: Hyperbolic tori and asymptotic surfaces in Hamiltonian systems. Russ. J. Math. Phys. 2 (1), 93–110 (1994) Weinstein, A.: Lagrangian submanifolds and Hamiltonian systems. Ann. Math. (2) 98 377–410 1973 Wiggins, S.: Introduction to applied nonlinear dynamical systems and chaos, volume 3 of Texts Appl. Math. New York: Springer, 1991 Whittaker, E.T. and Watson, G.N. A course of modern analysis. Cambridge: Cambridge Univ. Press, 1927

Communicated by Ya. G. Sinai

Commun. Math. Phys. 190, 247 – 278 (1997)

Communications in

Mathematical Physics c Springer-Verlag 1997

Integrable Structure of Conformal Field Theory II. Q-operator and DDV equation Vladimir V. Bazhanov1 , Sergei L. Lukyanov2,4 , Alexander B. Zamolodchikov3,4,? 1 Department of Theoretical Physics and Center of Mathematics, and its Applications, IAS, Australian National University, Canberra, ACT 0200, Australia. E-mail: [email protected], and Saint Petersburg Branch of Steklov Mathematical Institute, Fontanka 27, Saint Petersburg, 191011, Russia. 2 Newman Laboratory, Cornell University, Ithaca, NY 14853-5001, USA. E-mail: [email protected] 3 Laboratoire de Physique Th´ eorique de l’Ecole Normale Sup´erieure, 24 rue Lhomond - 75231 Paris Cedex 05, France 4 L.D. Landau Institute for Theoretical Physics, Chernogolovka, 142432, Russia

Received: 2 December 1996 / Accepted: 11 March 1997

Abstract: This paper is a direct continuation of [1] where we began the study of the integrable structures in Conformal Field Theory. We show here how to construct the operators Q± (λ) which act in the highest weight Virasoro module and commute for different values of the parameter λ. These operators appear to be the CFT analogs of the Q - matrix of Baxter [2], in particular they satisfy Baxter’s famous T − Q equation. We also show that under natural assumptions about analytic properties of the operators Q(λ) as the functions of λ the Baxter’s relation allows one to derive the nonlinear integral equations of Destri-de Vega (DDV) [3] for the eigenvalues of the Q-operators. We then use the DDV equation to obtain the asymptotic expansions of the Q - operators at large λ; it is remarkable that unlike the expansions of the T operators of [1], the asymptotic series for Q(λ) contains the “dual” nonlocal Integrals of Motion along with the local ones. We also discuss an intriguing relation between the vacuum eigenvalues of the Q operators and the stationary transport properties in the boundary sine-Gordon model. On this basis we propose a number of new exact results about finite voltage charge transport through the point contact in the quantum Hall system.

1. Introduction Existence of an infinite set of mutually commuting local Integrals of Motion (IM) is the characteristic feature of an integrable quantum field theory (IQFT). Therefore simultaneous diagonalization of these local IM is the fundamental problem of IQFT. In the case of an infinite-size system this problem reduces to finding mass spectrum and factorizable S-matrix associated with IQFT; much progress in this direction has been made during the last two decades (see e.g. [4] for a review). On the other hand, for a ? On leave of absence from Department of Physics and Astronomy, Rutgers University, Piscataway, NJ 08855-0849, USA, and Guggenheim Fellow

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finite-size system (say, with the spatial coordinate compactified on a circle of circumference R) this problem becomes highly nontrivial and so far its solution is known to a very limited extent. The most important progress here has been made with the help of the so-called Thermodynamic Bethe Ansatz (TBA) approach [5], [6]. TBA allows one to find the eigenvalues associated with the ground state of the system (in particular the ground-state energy) in terms of solutions of the nonlinear integral equation (TBA equation). However it is not clear how the combination of thermodynamic and relativistic ideas which is used in the traditional derivation of the TBA equation can be extended to include the excited states. The above diagonalization problem is very similar to that treated in solvable lattice models. In the lattice theory very powerful algebraic and analytic methods of diagonalization of the Baxter’s families of commuting transfer-matrices are known [2], [7]; these methods are further developed in the Quantum Inverse Scattering Method (QISM) [8], [9]. Of course many IQFT can be obtained by taking continuous limits of solvable lattice models and the method based on commuting transfer-matrices can be used to solve these QFT. This is essentially the way IQFT are treated in the QISM. However, for many IQFT (notably, for most of IQFT defined as perturbed CFT [10]) the associated solvable lattice models are not known. Besides, it seems to be conceptually important to develop the above methods directly in continuous QFT, in particular, to find continuous QFT versions of the Baxter’s commuting transfer-matrices. This problem was addressed in our recent paper [1] where we concentrated attention on the case of Conformal Field Theory (CFT), more specifically on c < 1 CFT. We should stress here that although the structure of the space of states and the energy spectrum in CFT are relatively well understood the diagonalization of the full set of the local IM remains a very nontrivial open problem. In [1] we have constructed an infinite set of operator valued functions Tj (λ), where j = 0, 21 , 1, 23 , ... and λ is a complex variable. These operators (we will exhibit their explicit form in Sect. 2) act invariantly in the irreducible highest weight Virasoro module V1 and they commute between themselves for any values of λ, i.e. Tj (λ) : V1 → V1 , (1.1) [Tj (λ), Tj 0 (λ0 )] = 0. The operators Tj (λ) are defined in terms of certain monodromy matrices associated with 2j + 1 dimensional representations of the quantum algebra Uq (sl2 ) where q = eiπβ , 2

(1.2)

and β is related to the Virasoro central charge as
c = 13 − 6 β 2 + β −2 .

(1.3)

Evidently, the operators Tj (λ) are CFT versions of the commuting transfer-matrices of the Baxter’s lattice theory. We will still call these operators “transfer-matrices” although the original meaning of this term [7] apparently is lost. As we have shown in [1], in CFT the operators Tj (λ) enjoy particularly simple analytic properties, namely they are entire functions of λ2 with an essential singularity at λ2 = ∞ and their asymptotic behavior near this point is described in terms of the local IM. Therefore the operators Tj (λ) can be thought of as the generating functions for the local IM since all the information about their eigenvalues is contained in the eigenvalues of Tj (λ). The operators Tj (λ) are shown to obey the “fusion relations” which for any rational value of β 2 in (1.2) provide a finite

Integrable Structure of Conformal Field Theory II.

249

system of functional equations for the eigenvalues of these operators. For the groundstate eigenvalue (in CFT it corresponds to a primary state) these functional equations turn out to be equivalent to the TBA equations; in the general case they provide modified TBA equations suitable for the excited states. An interesting but somewhat inconvenient feature of this approach is that the resulting TBA equations depend on c in a very irregular manner (they depend on the arithmetic properties of the rational number β 2 ) whereas the resulting eigenvalues of Tj (λ) are expected to be smooth functions of c. Another powerful method known in the lattice theory is based on the so-called Q -operator. This method was introduced by Baxter in his original study of the 8-vertex model [2]. One of its advantages is that it is not limited to the cases when q is a root of unity. In this paper (which is a sequel to [1]) we introduce the analog of the Q-operator directly in CFT and study its properties. The Q-operators (in fact we will define two Q-operators, Q± (λ)) are defined again as the traces of certain monodromy matrices, this time associated with infinite-dimensional representations of the so-called “q-oscillator algebra”. The operators Q± (λ) obey the Baxter’s functional relation T(λ)Q(λ) = Q(qλ) + Q(q −1 λ),

(1.4)

where T(λ) ≡ T 1 (λ). This construction is presented in Sect. 2 where also the most 2 important properties of the Q-operators are discussed. In the lattice theory the Baxter’s relation (1.4) is known to be a powerful tool for finding the eigenvalues of the transfer-matrices [7], the knowledge about analytic properties of the Q-operator being a key ingredient in this approach. Our construction of the Q-operators as the traces of the monodromy matrices makes it natural to assume that they enjoy very simple analytic properties similar to those of the operators Tj (λ): up to overall power-like factors they are entire functions of λ2 with the following asymptotic at λ2 → −∞: 1 log Q± (λ) ∼ M (−λ2 ) 2−2β2 , (1.5) where M is a constant (which will be actually calculated in Sect. 3). Using these properties we show that the eigenvalues of the Q-operators satisfy the Destry-de Vega (DDV) equation [3]. This is done in Sect. 3. The DDV equation can be solved exactly in the limit 1 → +∞, where 1 is the Virasoro highest weight in (1.1) , and we analyze in Sect. 3 the vacuum eigenvalues of the operators Q(λ) in this limit. In Sect. 4 we further study the properties of the Q-operators and formulate our basic conjectures about their analytic characteristics. The exact asymptotic expansions of the Q and T operators at λ2 → ∞ are proposed here. We observe that unlike the asymptotic expansion of T(λ) the large λ2 expansion of Q(λ) contains both local and nonlocal IM and that the operators Q(λ) obey remarkable duality relation with respect to the substitution β 2 → β −2 . Although the results of this section have somewhat conjectural status we support them by explicit study of the eigenvalues of the Q operators at the “free fermion point” β 2 = 1/2. In Sect. 5 we discuss the relation of the Q operators to the characteristics of stationary non-equilibrium states in the so-called boundary sine-Gordon model [11] (see also [12]); these states attracted much attention lately in relation to the finite-voltage current through the point contact in a quantum Hall system [13], [14], [15], [16]. Possible directions of further studies are discussed in Sect. 6. 2. The Q-operators In this section we will introduce the operators Q± (λ) which satisfy (1.4) . We start with a brief review of the definitions and results of [1].

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V.V. Bazhanov, S.L. Lukyanov, A.B. Zamolodchikov

Let ϕ(u) be a free chiral Bose field, i.e. the operator-valued function X a−n ϕ(u) = iQ + iP u + einu . n

(2.1)

n6 =0

Here P, Q and an , n = ±1, ±2, . . ., are operators which satisfy the commutation relations of the Heisenberg algebra [Q, P ] =

i 2 β ; 2

[an , am ] =

n 2 β δn+m,0 2

(2.2)

with real β. The variable u is interpreted as a complex coordinate on the 2D cylinder of a circumference 2π. As follows from (2.1) the field ϕ(u) is a quasi-periodic function of u, i.e. ϕ(u + 2π) = ϕ(u) + 2πiP. (2.3) Let Fp be the Fock space, i.e. the space generated by a free action of the operators an with n < 0 on the vacuum vector | pi which satisfies an | pi = 0, for P | pi = p | pi.

n > 0;

(2.4)

The composite field −β 2 T (u) =: ϕ0 (u)2 : +(1 − β 2 )ϕ00 (u) +

β2 24

(2.5)

is called the energy-momentum tensor; it is a periodic function of u and its Fourier modes   Z π c inu du T (u) + e (2.6) Ln = 24 −π 2π generate the Virasoro algebra with the central charge (1.3) [17], [18]. It is well known that for generic values of the parameters β and p the Fock space Fp realizes an irreducible highest weight Virasoro module V1 with the highest weight 1 related to p as  p 2 c − 1 . (2.7) + 1= β 24 For particular values of these parameters, when null-vectors appear in Fp , V1 is obtained from Fp by factoring out all the invariant subspaces. The space Fˆ p = ⊕∞ n=−∞ Fp+nβ 2

(2.8)

admits the action of the exponential fields V± (u) =: e±2ϕ(u) : .

(2.9)
Also, let E, F and H be canonical generating elements of the algebra Uq sl(2) [19], i.e. q H − q −H [H, E] = 2E, [H, F ] = −2F, [E, F ] = , (2.10) q − q −1 where q is given by (1.2) . Let j be a non-negative integer or half-integer number. We
denote πj an irreducible 2j + 1 dimensional matrix representation of Uq sl(2) so that

Integrable Structure of Conformal Field Theory II.

251

Ej ≡ πj (E), Fj ≡ πj (F ) and Hj ≡ πj (H) are (2j + 1) × (2j + 1) matrices which satisfy the relations (2.10) . The “transfer-matrices” Tj (λ), j = 0, 21 , 1, 23 , ... are defined as [1]  Z Tj (λ) = trπj e2πiP Hj Pexp λ  ≡ trπj e

2πiP Hj

∞ X n=0

λ

n

2π

Kj (u)du




0

Z 2π≥u1 ≥u2 ≥...≥un ≥0



Kj (u1 )Kj (u2 )...Kj (un )du1 du2 ...dun . (2.11)

Here Kj (u) = V− (u)q

Hj 2

Ej + V+ (u)q −

Hj 2

Fj

and P denotes the operator ordering along the integration path. Obviously, T0 (λ) = I,

(2.12)

where I is the identity operator. Although the exponentials in (2.11) act from one component of the sum (2.8) to another it is easy to see that the operators (2.11) invariantly act in the Fock space Fp . Another obvious property of Tj is that these operators are in fact the functions of λ2 as the traces of the odd-order terms in (2.11) vanish. As is explained in [1] the operators (2.11) form the commuting family, i.e. they satisfy the relations (1.1) . They also obey the “fusion relations” T(λ)Tj (q j+ 2 λ) = Tj− 1 (q j+1 λ) + Tj+ 1 (q j λ). 1

2

2

(2.13)

Together with (2.12) these relations allow one to express recurrently any of the “higherspin” operators Tj (λ) with j = 1, 23 , 2, ... in terms of the basic one T(λ) ≡ T 1 (λ).

(2.14)

2

In the case j = expansion

1 2

it is easy to evaluate the traces in (2.11) and obtain the power-series T(λ) = 2 cos(2πP ) +

∞ X

λ2n Gn ,

(2.15)

n=1

where the coefficients define an infinite set of basic nonlocal IM, Z e2iπP V− (u1 )V+ (u2 )V− (u3 ) · · · V+ (u2n )+ Gn = q n 2π≥u1 ≥u2 ≥···≥u2n ≥0


e−2iπP V+ (u1 )V− (u2 )V+ (u3 ) · · · V− (u2n ) du1 du2 · · · du2n .

(2.16)

It follows from (1.1) that these nonlocal IM commute among themselves [Gn , Gm ] = 0. The operator-product expansion V+ (u)V− (u0 ) = (u − u0 )−2β

2


1 + O(u − u0 ) , u − u0 → 0

shows that the expressions (2.11) and (2.16) can be taken literally only if

(2.17)

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V.V. Bazhanov, S.L. Lukyanov, A.B. Zamolodchikov

0 < β2 <

1 , 2

(2.18)

for otherwise the integrals in (2.16) diverge. In what follows we will call the region (2.18) the Semi-classical Domain (SD). In fact one can define the operators Gn outside the SD by analytic continuation in β 2 . A convenient way to do that is to transform the ordered integrals in (2.16) to contour integrals. For example G1 can be written as  Z Z 2π
−1 2π
G1 = q 2 − q −2 du1 du2 qe−2πiP − q −1 e2πiP × 0 0  (2.19)
2πiP −1 −2πiP V− (u1 + i0)V+ (u2 − i0) + qe −q e V+ (u1 + i0)V− (u2 − i0) . Obviously, (2.19) does not contain divergent integrals and for β 2 in SD coincides with the ordered integral in (2.16) . Similar representation exists for higher Gn , and thus the operator T(λ) can be defined outside SD through (2.15) . Of course the operators Gn and T(λ) obtained this way exhibit singularities at β 2 = βn2 , βn2 =

2n − 1 , 2n

n = 1, 2, 3, ...,

(2.20)

where the integrals (2.16) develop logarithmic divergences. In order to define the operators Tj (λ) at the singular points (2.20) some renormalization is needed but we do not discuss it here (see, however, our analysis of the case β 2 = 1/2 in Sect. 4). As is mentioned in the Introduction, the power series (2.15) defines T(λ) as an entire function of λ2 with an essential singularity at λ2 → ∞. Its asymptotic expansion near this essential singularity can be expressed in terms of local IM as 1

log T(λ) ' mλ 1−β2 I −

∞ X

1−2n

Cn λ 1−β2 I2n−1 ,

(2.21)

n=1

where I2n−1 is the basic set of commutative local IM as defined in [1]; the operators I2n−1 can be written as the integrals Z 2π du T2n (u), (2.22) I2n−1 = 2π 0 where the local densities T2n (u) are particular normal ordered polynomials of ∂u ϕ(u),..., ∂u2n ϕ(u) (see [1]for details and for our convention about the normalization of I2n−1 ). The numerical coefficients m and Cn in the expansion (2.21) will be calculated exactly in Sect. 3. As follows from (1.1) all these local IM commute with the nonlocal IM Gn . As is known [20], [21], [22] the local IM I2n−1 defined in [1] do not change under the substitution β 2 → β −2 , if we simultaneously make the replacement ϕ(u) → β −2 ϕ(u), I2n−1 {β 2 , ϕ(u)} = I2n−1 {β −2 , β −2 ϕ(u)}.

(2.23)

Evidently the nonlocal IM do change and so there exists an infinite set of “dual” nonlocal ˜ n which are obtained from (2.16) by just this substitution, i.e. IM G Z ˜ n = q˜n G q˜2P U− (u1 )U+ (u2 )U− (u3 )...U+ (u2n )+ (2.24) 2π≥u1 ≥u2 ≥...≥u2n ≥0
−2P U+ (u1 )U− (u2 )U+ (u3 )...U− (u2n ) du1 du2 ...du2n , q˜

Integrable Structure of Conformal Field Theory II.

253

where U± (u) =: e and q˜ = e

±

i

2 β2

π β2

ϕ(u)

:

.

(2.25) ˜ n. Of course if β < 2 the analytic continuation described above is needed to define G ˜ n thus defined commute among However it is possible to check that the operators G themselves and commute with all the nonlocal IM Gn and local IM I2m−1 2

˜ n, G ˜ m ] = [G ˜ n , Gm ] = [G ˜ n , I2m−1 ] = 0. [G

(2.26)

Now we are in a position to define the Q-operators and to describe their basic properties. Let E+ , E− and H be operators which satisfy the commutation relations of the so-called “q-oscillator algebra”, q E+ E− − q −1 E− E+ =

1 , q − q −1

[H, E± ] = ±2E± ,

(2.27)

and let ρ be any representation of this algebra such that the trace Z(p) = trρ [e2πipH ]

(2.28)

exists for complex p belonging to the upper half plane, =mp > 0. Then one can define two operators Z 2π
  H H du(V− (u)q ± 2 E± + V+ (u)q ∓ 2 E∓ ) , A± (λ) = Z −1 (±P )trρ e±2iπP H Pexp λ 0

(2.29) where again the symbol P denotes the operator ordering along the integration domain. As we are interested in the action of these operators in Fp the operator P in (2.29) can be substituted for its eigenvalue p. Strictly speaking the definition (2.29) makes sense only if =m p > 0 for A+ and if =m p < 0 for A− . However these operators can be defined for all complex p (except for some set of singular points on the real axis) by analytic continuation in p. Then it is easy to see that A− can be obtained from A+ by substitution, P → −P, ϕ(u) → −ϕ(u). (2.30) The operators (2.29) can be written as the power series A± (λ) = 1 +

∞ X

X

n=1

{σi =±1} σ1 +···+σ2n =0

λ2n a2n (σ1 , . . . , σ2n | ± P )J2n (∓σ1 , ∓σ2 , . . . , ∓σ2n ), (2.31)

where J2n (σ1 , . . . , σ2n ) = q n

Z 2π≥u1 ···u2n ≥0

and an (σ1 , . . . , σ2n |P ) = Z

−1

Vσ1 (u1 )Vσ2 (u2 ) · · · Vσ2n (u2n )du1 . . . du2n 

(P )trρ e

2πiP H

 Eσ1 Eσ2 · · · Eσ2n .

(2.32)

It is easy to see that the coefficients (2.32) are completely determined by the commutation relations (2.27) and the cyclic property of the trace, and so the operators (2.29) do not

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V.V. Bazhanov, S.L. Lukyanov, A.B. Zamolodchikov

depend on the particular choice of the representation ρ. The operator coefficients in the power series (2.31) can be expressed in terms of the basic nonlocal IM (2.16) . It is not difficult to calculate first few terms of this series  G1 G2 2 4 −λ − A± (λ) = 1 − λ 4 sin a sin(a ± x) 4 sin 2a sin(2a ± x)   G12 G3 − λ6 + (2.33) 16 sin a sin 2a sin(a ± x) sin(2a ± x) 4 sin 3a sin(3a ± x)
 G13 − 4G1 G2 sin a sin(a ± x) + sin 2a sin(2a ± x) + O(λ8 ), 64 sin a sin 2a sin 3a sin(a ± x) sin(2a ± x) sin(3a ± x) where a = πβ 2 .

x = 2πP,

For further references it is convenient to introduce an alternative set of nonlocal IM defined as coefficients in the expansion log A+ (λ) = −

∞ X

y 2n Hn ,

(2.34)

n=1

where

y = β −2 0(1 − β 2 )λ.

(2.35)

These coefficients are, of course, algebraically dependent on those in (2.16) . For example, β 4 0(β 2 ) G1 . (2.36) H1 = 4π0(1 − β 2 ) sin(2πP + πβ 2 ) e n, Define also a new set of “dual” nonlocal IM H e n {β 2 , ϕ(u)} = Hn {β −2 , β −2 ϕ(u)}, H

(2.37)

obtained from Hn by the analytic continuation, as described above. The operators Q± (λ) are defined as Q± (λ) = λ±2P/β A± (λ). 2

(2.38)

Like the operators Tj (λ) above the operators (2.38) act in a Fock space Fp , Q± (λ) :

Fp → Fp .

(2.39)

The operators Q± (λ) exhibit remarkable properties. Here we simply list some of them leaving the proofs to another paper. i. The operators Q± (λ) commute among themselves and with all the operators Tj (λ), [Q± (λ), Q± (λ0 )] = [Q± (λ), Tj (λ0 )] = 0.

(2.40)

ii. The operators Q± (λ) satisfy the equation (1.4) , i.e. T(λ)Q± (λ) = Q± (qλ) + Q± (q −1 λ).

(2.41)

Equation (1.4) can be thought of as the finite-difference analog of the second order differential equation, so we expect it to have two linearly independent solutions. As T(λ)

Integrable Structure of Conformal Field Theory II.

255

is a single-valued function of λ2 , i.e. it is a periodic function of log λ2 , the operators Q± (λ) are just two “Bloch-wave” solutions to Eq. (1.4) . The operators Q± (λ) satisfy the “quantum Wronskian” condition Q+ (q 2 λ)Q− (q − 2 λ) − Q+ (q − 2 λ)Q− (q 2 λ) = 2i sin(2πP ). 1

1

1

1

(2.42)

iii. The “transfer-matrices” Tj (λ) can be expressed in terms of Q± (λ) as 2i sin(2πP )Tj (λ) = Q+ (q j+ 2 λ)Q− (q −j− 2 λ) − Q+ (q −j− 2 λ)Q− (q j+ 2 λ). 1

1

1

1

(2.43)

In view of this equation the operators Q± (λ) appear more fundamental than the transfermatrices Tj (λ). We will see more support for this idea below. Let us just briefly sketch the derivation [23] of the functional relation of this section. The main idea is to consider more general T-operators T+j (λ) defined as in (2.11) , but
associated with the infinite dimensional representation π + of Uq sl(2) with arbitrary (complex) highest weight 2j. Note that if j takes a non-negative integer or half-integer value then the matrices πj+ (E), πj+ (F ) and πj+ (H) have a block-triangular form with two diagonal blocks, one equivalent to the (2j + 1)-dimensional representation πj and the + . In this way we obtain the following other being the highest weight representation π−j−1 simple relation T+j (λ) = Tj (λ) + T+−j−1 (λ),

j = 0, 1/2, 1, 3/2, . . . .

(2.44)

Next, the operator T+j (λ) enjoy the following remarkable factorization property 2i sin(2πP )T+j (λ) = Q+ (q j+ 2 λ) Q− (q −j− 2 λ), 1

1

(2.45)

which is proved explicitly by using decomposition properties of the tensor product of two representations of the q-oscillator algebra (the latter are also representations of the

d with respect to the co-multiplication from Uq sl(2) d . Borel sub-algebra of Uq sl(2) Then the functional relations (2.42) , (2.43) trivially follow from (2.44) and (2.45) above, while the remaining relations (2.40) , (2.41) are simple corollaries of these two. Although our considerations here were specific to the continuous theory similar results hold for the Q-matrix of the lattice theory as well. This is quite obvious since the structure of the functional equations is determined merely by the decomposition
d associated with the monodromy properties of products of representations of Uq sl(2) matrices. The details of these calculations will be given in [23]. The operators Q± (λ) take particularly simple form when applied to the space Fp with 2p equal to some integer which we denote N . In these cases the “quantum Wronskian” (2.42) is equal to zero and the solutions A+ (λ) and A− (λ) (2.38) coincide. For 2p = N all the coefficients (2.32) are readily calculated a2n (σ1 , . . . , σ2n |N/2) = (q − q −1 )−2n ,

(2.46)

and the operators (2.38) can be written as h Z 2π ∞ in h Z 2π du in  X µ2n n du V+ (u) = q P V− (u) , A± (λ) (n!)2 2π 2π p=N/2 p=N/2 0 0 n=0 (2.47) where

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µ = −i

πλ sin(πβ 2 )

(2.48)

and, as before, the symbol P denotes “u-ordering”, i.e. the operators V± (u) with greater u are placed to the left. For p = N/2 the exponentials V± (u) are 2π-periodic functions of u and so the integration contours in (2.47) close. The series (2.47) is closely related to the boundary state in the so-called Boundary Sine-Gordon model (with the bulk mass (λ), defined equal to zero) [11]. In particular, its vacuum-vacuum matrix element A(vac) ± as (λ) | pi, (2.49) A± (λ) | pi = A(vac) ± coincides for p = N/2 with the one-dimensional Coulomb gas partition function Z 2π Z 2π Z Z 2π ∞ X µ2n du1 dun 2π dv1 dvn . . . . . . × 2 (n!) 0 2π 2π 0 2π 2π 0 0 n=0 2 2 Pn Y vi − vj
2β Y u i − uj
ui − vj
−2β sin . eiN i=1 (vi −ui ) 4 sin 2 sin 2 2 2 i,j A(vac) (λ)|p=N/2 ≡ ZN (µ) =

i6=j

(2.50) As was shown in [24] the series (2.50) defines an entire function of µ2 with the asymptotic 1

behavior log ZN (µ) ∼ const(µ2 ) 2−2β2 . In fact, it is easy to show that this result implies not only to the vacuum eigenvalue (2.49) but also to any matrix element of the operator A. We conclude that for p = N/2 the operators A± (λ) are entire functions of λ2 and they enjoy the asymptotic form 1

log A± (λ) ∼ M (−λ2 ) 2−2β2 , λ2 → −∞.

(2.51)

We should stress again that the relations (2.47) and (2.50) hold only for 2p = N . For non-integer 2p they do not hold. Nonetheless we find it natural to assume that the above analytic properties of the operators A± (λ) as the functions of λ2 hold for any p. In the following sections we use the functional Eq. (1.4) together with this analyticity assumption to derive various asymptotic expansions for the operators A± (λ). 3. Destri-de Vega Equation Now let us turn to the eigenvalue problem for the operators Q± (λ). Q± (λ) | αi = Qα ± (λ) | αi,

(3.1)

where | αi ∈ Fp . In this section we consider only the case of β 2 in SD (2.18) . Let us concentrate attention on one of the operators (2.38) , say Q(λ) ≡ Q+ (λ); the problem for Q− (λ) can be solved then by the substitution (2.30) . As follows from (2.41) any eigenvalue Q(λ) of Q(λ) satisfies the Baxter’s functional equation (3.2) T (λ)Q(λ) = Q(qλ) + Q(q −1 λ), where T (λ) is the corresponding eigenvalue of the operator T(λ). Denoting A(λ) the eigenvalue of the operator A+ (λ) in (2.38) one can rewrite (3.2) as T (λ)A(λ) = e2πip A(qλ) + e−2πip A(q −1 λ).

(3.3)

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Although this equation looks just like a relation between two unknown functions in fact it imposes severe restrictions on these functions provided their analytic properties are known1 . Motivated by the relation to the Coulomb gas partition function (2.50) discussed in Sect. 2 we accept here the following assumptions about analytic properties of the eigenvalues A(λ). Let β 2 be restricted to SD and let =mp = 0. Then (i) Analyticity. The functions A(λ) and T (λ) are entire functions of the complex variable2 λ2 . (ii) Location of zeroes. Zeroes of the function A(λ) in the λ2 -plane are either real or occur in complex conjugated pairs. For any given eigenvalue A(λ) there are only a finite number of complex or real negative zeroes. Real zeroes accumulate toward +∞ in the λ2 -plane. For the vacuum eigenvalues all zeroes are real and if 2p > −β 2 they are all positive. (iii) Asymptotic behavior. The leading asymptotic behavior of A(λ) for large λ2 is given by (2.51) with some constant M . If 0 < β 2 < 21 an entire function A(λ) with the asymptotic behavior (2.51) is completely determined by its zeroes λ2k , k = 0, 1, . . . ; it can be represented by a convergent product  ∞  Y λ2 1− 2 , (3.4) A(λ) = λk k=0 where the normalization condition A(0) = 1

(3.5)

is taken into account. Actually, one important remark about the normalization (2.38) , (3.5) of the operators A must be made here. For a given eigenvalue A(λ) the positions of its zeroes λk depend on p and for special values of this parameter one of these zeroes, say k0 , can happen (λ) this happens to be at zero, λk0 = 0. For example, for the vacuum eigenvalue A(vac) + at 2p = −β 2 − n; n = 0, 1, 2, . . ., as brief inspection of (2.33) and (3.21) below shows. Obviously, at these values of 2p the normalization of the operators A(λ) as in (2.38) , (3.5) P is not suitable as under this normalization all the terms in the power series ∞ A(λ) = 1 + n=1 λ2n An except the first one would diverge. Of course this formal singularity can be eliminated by renormalization of A with an appropriate p-dependent factor. As our results below are not sensitive to this subtlety, in what follows we use the normalization (2.38) , (3.5) (which we find technically very convenient) and just assume that p does not take these “dangerous” values (i.e. that all λk 6= 0). Introduce the function A(λq) . (3.6) a(λ) = e4πip A(λq −1 ) Setting λ2 = λ2k in (3.3) and remembering that T (λ) has no singularities at finite λ2 one obtains the Bethe-Ansatz type equations for the positions of the zeroes λ2k a(λk ) = −1. 1

(3.7)

In lattice theory Eq. (3.3) (which usually appears there “decorated” with some non-universal factors in the right-hand side) leads to the Bethe Ansatz equations which completely determine the eigenvalues [7]. 2 In fact one can prove this property for T (λ), see [1]. A similar asumption for Q(λ) has been used in [25].

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This is an infinite set of transcendental equations for the infinite set of unknowns λ2k . It can be transformed into a coupled system of a single non-linear integral equation and a finite set of Eq. (3.7) only for those zeroes which do not belong to the positive real axis in the λ2 -plane. This system is known as the Destry-de Vega (DDV) equation [3]. In the context of the lattice XXZ model it was originally derived by Kl¨umper Batchelor and Pearce [26]. In Appendix A we show that under the assumptions (i)-(iii) given above the infinite set of Eqs. (3.7) is equivalent to  X0
πξ 2πp  i log a(θ) = − + 2M cos eθ +i logS(θ−θa )−2G ∗ =m log 1 + a(θ−i0) 2 β 2 a  a(θa ) = −1, (3.8) where we have used new variables θ

λ = e 1+ξ ,

θa

λa = e 1+ξ ,

β2 =

ξ 1+ξ

(3.9)

and λ2a denote zeroes of A(λ) lying outside the positive real axis of λ2 . The star in (3.8) denotes the convolution Z ∞ dθ0 A(θ − θ0 )B(θ0 ) (3.10) A ∗ B(θ) = −∞

and



Z

S(θ) = exp −i 0

∞


 sinh πν(1 + ξ)/2 dν sin(νθ) , ν cosh( πν/2 ) sinh( πνξ/2 )

(3.11)

1 ∂θ logS(θ). 2πi Note that the function S(θ) coincide with the soliton-soliton scattering amplitude for the Sine-Gordon model [27]. The equation (3.8) for the vacuum eigenvalues (in fact, more general equation obtained by replacing eθ in the r.h.s. by 2 sinh θ) was obtained from the Bethe ansatz equations associated with XXZ lattice model by taking appropriate continuous limit which leads to the Sine-Gordon field theory [3]. Here we obtain (the “mass-less” version of) this equation directly in continuous field theory, bypassing any direct reference to the lattice theory. Given a solution of (3.8) the function logA(λ) can be calculated as [3] Z
iν(1+ξ)/2 g(ν) − λ2 dν , (3.12) logA(λ) = −i cosh πν/2 sinh πνξ/2 Cν G(θ) = δ(θ) +

where the function g(ν) is defined as Z +∞
dθ =m log 1 + a(θ − i0) e−iνθ . g(ν) = −∞ 2π

(3.13)

The integration contour Cν in (3.12) goes along the line =mν = −1 −  with arbitrary small positive . It is instructive to study the vacuum eigenvalue A(vac) (λ) of A(λ), defined by (2.49), in the limit p → +∞. Note that the parameter p enters the DDV equation (3.8) exactly the same way the external gauge field (which couples to the soliton charge) appears in the sine-Gordon DDV equation [28]. The DDV Eq. (3.8) is greatly simplified in the

Integrable Structure of Conformal Field Theory II.

259

limit p → +∞. According to assumption (ii) above all zeroes λ2k in this case are real P 0 and positive and so one can drop the term i a logS(θ − θa ) in the r.h.s. of the integral Eq. (3.8) . Moreover, in the limit p → +∞ this equation reduces to a linear equation of Winer-Hopf type (see Appendix A for some details) Z B(p) 0
πξ dθ πp ∂θ logS(θ − θ0 )=m log 1 + a(vac) (θ0 ) = 0, (3.14) − 2 + M cos eθ − β 2 −∞ 2πi provided one assumes that B(p) =

1+ξ log λ20 ∼ const log p, 2

p → +∞,

(3.15)

where λ20 is a minimal of the zeroes λ2k . Equation (3.14) can be solved by the standard technique [29]. As a result one obtains
√
−iν(1+ξ)/2 0 1 − iν(1 + ξ)/2 ip π (vac) eiδν λ20 (ν)|p→+∞ ∼ g , (3.16) 4ν 0(3/2 − iν/2)0(1 − iνξ/2) where g (vac) (ν) is related to a(vac) (θ) through (3.13) λ20


(1+ξ)/2

p→+∞

∼

0( ξ2 )0( 21 − ξ2 ) δ √ e p, πM

(3.17)

and 2δ = (1 + ξ)log(1 + ξ) − ξlogξ. Note that (3.17) supports the assumption (3.15) . Equation (3.12) gives logA(vac) (λ)|p→+∞ ∼ Z  λ2 iν(1+ξ)/2
p 1 dν iδν − + iν/2)e 0 1 − iν(1 + ξ)/2 0(1 + iνξ/2)0(− , − 2 3 2 λ0 2π 2 ξ Cν ν 2 (3.18) where the integration contour Cν go along the line =mν = −1 − . The integrand σ(ν) in (3.18) has the simple asymptotic behavior in the whole complex ν-plane except the positive imaginary axis p  λ2 iν(1+ξ)/2 p β2
− 2 , ν→∞ σ(ν) ∼ λ0 sinh πν(1 + ξ)/2 and we can work out the integral (3.18) for |λ2 | < |λ20 | by closing the integration contour Cν at the infinity in the lower half plane =mν < −1 and calculating the residues of the poles at ν = −2in/(1 + ξ), n = 1, 2, . . .. The result can be written in the form log A(vac) (λ)|p→+∞ = −

∞ X

y 2n Hn(vac) |p→∞ ,

(3.19)

n=1

where y is given by (2.35) and the above residue calculations give the coefficients Hn(vac) |p→∞ explicitly. According to (2.34) these coefficients coincide with the large p asymptotic of the vacuum eigenvalues of the nonlocal IM Hn . In particular, we find

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V.V. Bazhanov, S.L. Lukyanov, A.B. Zamolodchikov

H1(vac) |p→+∞ ∼

β 4 0(β 2 )0(1/2 − β 2 ) 2β 2 −1 √ . p 2 π

(3.20)

On the other hand an exact expression for H1(vac) for all values of p is known explicitly from (2.36) β 4 0(β 2 )0(1 − 2β 2 ) 0(β 2 + 2p) , (3.21) H1(vac) = 0(1 − β 2 ) 0(1 − β 2 + 2p) where we have used the known vacuum eigenvalue = G(vac) 1

4π 2 0(1 − 2β 2 ) 0(1 − β 2 − 2p)0(1 − β 2 + 2p)

(3.22)

of the nonlocal IM G1 [1]and (2.36) . Comparing the asymptotic of (3.21) with (3.20) one obtains exactly the coefficients M which enters the leading asymptotic behaviors (2.51) of the operators A(λ): M=


1+ξ 0( ξ2 )0( 21 − ξ2 )  √ . 0 1 − β2 π

(3.23)

Calculating further coefficients in (3.19) we find the large p asymptotic of the eigenvalues Hn(vac) ,

2n 1−2n+2nβ 2 0(nβ 2 )0 − 21 + n(1 − β 2 ) (vac) √ Hn |p→+∞ ∼ p . (3.24) β2 2 πn! It is not obvious at all how this asymptotic can be obtained directly from the definition of nonlocal IM Hn in terms of the ordered integrals (2.16) through (2.15) and (2.34) . The large p asymptotic of vacuum eigenvalues of the dual nonlocal IM (2.37) is obtained from (3.24) by the substitution β 2 → β −2 , p → β −2 p,
1 −2 −2
− 2n −1 1−2n+ 2n2 0(nβ )0 − + n(1 − β ) (vac) 2 e β . √ H |p→+∞ ∼ (3.25) β 2 β2 p n 2 πn! For completeness we also present here the large p asymptotic of the eigenvalues of the local IM,
−n 2n (vac) |p→+∞ ∼ β 2 p , (3.26) I2n−1 which follow directly from the definition of the operators I2n−1 , see [1]. Note that this asymptotic hold for any eigenvalues (not just the vacuum ones) of the local IM. Let us now consider the large λ behavior of (3.18) . To obtain the large λ asymptotic expansion of A(vac) (λ) one can close the integral in (3.18) in the upper half-plane =mν ≥ −1. Then (3.18) can be represented as a sum of infinitely many terms associated with the residues of the poles of the integrand located at =mν ≥ −1, plus the integral over the large circle. Note that the contribution of the large circle in fact diverges. Correspondingly, the sum of the residues gives only asymptotic series expansion for A(vac) (λ), with zero radius of convergence. This asymptotic expansion has the form  X ∞

− p
1−2n (vac) A(vac) (λ) ' C (vac) β 2 , p − y 2 β2 exp × Bn − y 2 2−2β2 I2n−1 n=0

  X +∞
n n 2 − β 2 e (vac) (−1) − y , Hn exp − n=1

(3.27)

Integrable Structure of Conformal Field Theory II.

261

where the variable y is related to λ as in (2.35) ,

− 2p  2p  β2p2 −2p C (vac) β 2 , p |p→+∞ ∼ β 2 β2 , e

(3.28)

and 2  2n − 1   2n − 1 
nβ2 +n−β (−1)n+1 2 β 2 −1 0 β . 0 Bn = √ 2 − 2β −2 2 π(1 − β 2 )n! 2 − 2β 2

(3.29)

Strictly speaking, here we have derived the expansion (3.27) in the limit p → +∞, (vac) ˜ n(vac) in (3.27) denote the p → +∞ asymptotic of the and H so the quantities I2n−1 corresponding vacuum eigenvalues given by (3.26) and (3.25) . However in the next section we adopt additional analyticity assumptions about the operators A(λ) which allow for derivation of the same asymptotic expansion (3.27) for arbitrary p. Moreover, it is natural to expect that the expansion (3.27) holds not only for the vacuum eigenvalues (vac) but for the whole operator A(λ); in that case of course the vacuum eigenvalues I2n−1 (vac) ˜ n themselves. and H˜ n in (3.27) must be replaced by the operators I2n−1 and H 4. Conjectures: Exact Asymptotic Expansions and Duality Consider the following operator-valued function of the complex variable ν: 3 −iν(1+ξ) √  2 π 0(1 − β 2 )


× 9(ν) = 0 iνξ/2 0 − 1/2 + iν/2 0 − iν(1 + ξ)/2 Z 0
−iν(ξ+1)/2 dλ2 log A(λ). − λ2 2 −∞ λ

(4.1)

The function 9(ν) provides a continuous set of IM parameterized by the variable ν. Indeed, the operators A(λ) with different values of λ as well as their arbitrary linear combinations commute among themselves and with all local and nonlocal IM. As we shall see below the function 9(ν) is remarkable in many respects, in particular, it can be thought of as an analytic continuation of the local IM I2n−1 to arbitrary complex values of their index n. The integral (4.1) converges only for 2β 2 −2 < =m ν < −1, however the definition of 9(ν) can be extended to the whole ν-plane by means of the analytic continuation. The latter can be done in various ways. For instance, using the product representation (3.4) which holds for any eigenvalue A(λ) of the operator A(λ) one can write the corresponding eigenvalue of 9(ν) as a generalized Dirichlet series of the form4 3 In writing (4.1) we make a technical assumption that all zeroes λ of all eigenvalues of A are away from k zero, see the remark in Sect. 3. 4 The definition (4.1) contains an ambiguity in the choice of the branches of the logarithm. In (4.2) this translates into the choice of the phases of the roots λ2k . This ambiguity, however, does not affect the following arguments. For definiteness one may assume that | arg λ2k | < π when λ2k lies outside the negative real axis of λ2 , while for real negative λ2k the corresponding entry in the sum (4.2) is replaced by cosh πν(1 +




ξ)/2 |λk |−iν(1+ξ) .

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V.V. Bazhanov, S.L. Lukyanov, A.B. Zamolodchikov

−iν(1+ξ)
 2 ∞ X
−iν(1+ξ)/2 √ 0 iν(1 + ξ)/2 0(1 − β )

9(ν) = 2 π . λ2k 0 iνξ/2 0 − 1/2 + iν/2 k=0

(4.2)

For 0 < β 2 < 1/2 this series converges absolutely for =mν < −1 so that (4.2) defines an analytic function of ν in the half plane =mν < −1 which can be then analytically continued to the whole complex ν-plane by the standard technique of the analytic continuation of Dirichlet series [30]. We expect that the function 9(ν) defined in this way enjoy the following remarkable analytic properties Conjecture 1. For real p and 0 < β 2 ≤ complex variable ν.

1 2

the function 9(ν) is an entire function of the

These simple analytic properties makes 9(ν) extremely convenient for studying eigenvalues of the A and T operators. Converting the integral transform in (4.1) one expresses A(λ) as Z
dν i 0 1 − iν(1 + ξ)/2 0(iνξ/2)0(−1/2 + iν/2)× logA(λ) = − 3 ν 4π 2 Cν (4.3)  iν(1+ξ)
2 2 iν(1+ξ)/2 0(1 − β ) 9(ν) − λ , where the integration contour Cν is the same as in (3.12) . At the same time, the corresponding eigenvalue of the transfer matrix T(λ) reads T(λ) = 3(q 2 λ) + 3−1 (q − 2 λ), 1

1

(4.4)

where Z log3(λ) = 2πiP + Cν


dν 0 1 − iν(1 + ξ)/2 0(−1/2 + iν/2) √ × ν 2 π0(1 − iνξ/2)  iν(1+ξ)
iν(1+ξ)/2 0(1 − β 2 ) 9(ν) − λ2 .

(4.5)

The values of the function 9(ν) at special points on the imaginary ν-axis (where the Gamma-functions in (4.3) and (4.5) display poles) are of
particular interest. For example, it is not difficult to see that the values 9 − 2in(1 − β 2 ) , n = 1, 2, . . . , ∞, are related to the eigenvalues of the nonlocal IM Hn . Using (3.4) and (2.34) one obtains for the latter ∞  −2n X Hn = n−1 β −2 0(1 − β 2 ) λ−2n . k

(4.6)

k=0

Then it follows from (4.2) that
−2n √ 2 πn! β 2
Hn . 9 − 2in(1 − β ) = 0(nβ 2 )0 − 21 + n(1 − β 2 ) 2




For other special values on the imaginary ν-axis we will adopt the following

(4.7)

Integrable Structure of Conformal Field Theory II.

263

Conjecture 2. For real p and 0 < β 2 ≤ 21 the operator 9(ν) has the following special values on the imaginary ν-axis:
n
(4.8) 9 (2n − 1)i = β 2 I2n−1 , n = 0, 1, . . . , 9 2in(β

−2


2n +1 √ 2 πn! β 2 β2 ˜ n,
H − 1) = 0(nβ −2 )0 − 21 + n(1 − β −2 )


n = 1, 2, . . . ,

9(0) = P,

(4.9) (4.10)

e n denote the local IM (2.22) and the dual nonlocal IM (2.37) respecwhere I2n−1 and H tively. Note that (4.8) with n = 0 reads 9(−i) = I−1 ≡ I,

(4.11)

where I is the identity operator. The relations (4.8) -(4.10) together with the conditions (2.23) , (2.37) makes it natural to assume that the operator 9 satisfy the following duality condition: Conjecture 3. 9{ν, β 2 , ϕ(u)} = β 2 )1−iν 9{ν, β −2 , β −2 ϕ(u)}.

(4.12)

An initial motivation for the above conjectures came from the study of the large p asymptotic of the vacuum eigenvalues of A(λ) in the previous section. Indeed, comparing (3.18) and (4.3) one obtains 9(vac) (ν)|p→+∞ ∼ p1−iν ,

(4.13)

which taking into account (3.26) and (3.25) obviously satisfies all the statements of Conjectures 1–3 5 . Further motivations and justifications of the these conjectures are discussed below. We are now ready to derive exact asymptotic expansions of the eigenvalues for large λ2 . This is achieved by calculating the integrals (4.3) and (4.5) as formal sums over residues in the upper half plane =m ν ≥ −1. Using (4.8) one thus has from (4.5) for λ2 → ∞, log3(λ) ' imI − λ2




1 2−2β 2

−i

∞ X

(−1)n Cn − λ2


1−2n2 2−2β

I2n−1 ,

(4.14)

n=1

where m= and √ Cn = 5

√ 2 π0( 21 − ξ2 )  0(1 − ξ2 )

0 1 − β2


1+ξ


−(2n−1)(1+ξ) π(1 + ξ) 2
n 0 (n − 21 )(1 + ξ)  2
β ) . 0(1 − β n! 0 1 + (n − 21 )ξ

Like in (3.26) , the asymptotic (4.13) holds for all eigenvalues of 9, not just the vacuum ones.

(4.15)

(4.16)

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V.V. Bazhanov, S.L. Lukyanov, A.B. Zamolodchikov

It follows then that the large λ asymptotic behavior of the operator T is given by (2.21) , where the numerical coefficients m and Cn are given by the formulas (4.15) and (4.16). The asymptotic expansion (2.21) holds for −π < arg λ2 < π.

(4.17)

Similarly one can get the asymptotic expansion for the operator A(λ) (4.3) . Define the operator  2P2 −2P
− 2P  β , (4.18) C = β 2 β2 2ei∂ν log9(0)−1 which does not depend on the spectral parameter λ. It is also convenient to use the variable y given by (2.35) instead of the variable λ and exhibit all arguments of A(iy): ∞  X
n  (−1)n y 2 Hn . A{iy, β 2 , ϕ(u)} ≡ exp − n=1

Then, from (4.3) one has for |y| → ∞, −π < arg y 2 < π, X  ∞ 1−2n
P 2 2 2 − β2 2 2−2β 2 exp Bn ( y ) I2n−1 × A{iy, β , ϕ(u)} 'C{β , ϕ(u)} y n=0

A{iy

− 12 β

(4.19)

, β −2 , β −2 ϕ(u)},

where the coefficients Bn are given by (3.29) and A{iy

−

1 β2

∞  X 
− n en . , β −2 , β −2 ϕ(u)} ≡ exp − (−1)n y 2 β2 H

(4.20)

n=1

The explicit form of the operator C in (4.19) is not determined by the above calculations. For the vacuum eigenvalue its p → +∞ asymptotic is given by (3.28) . Moreover, one can show that [24] p (4.21) C (vac) (β 2 , 0) = β 2 . We have the following conjecture about the exact form of this coefficient for all values of p and 0 < β 2 ≤ 21 :
p 0 1 + 2pβ −2
. (4.22) C (vac) (β 2 , p) = β 2 0 1 + 2p The formula for the asymptotic expansion (4.19) is essentially equivalent to the Conjectures 1,2 given above. In fact, the coefficients in front of powers of λ in (4.19) are in one-to-one correspondence with the values (4.8) while any singularity of 9(ν) in the upper half plane would bring in some additional terms in (4.19) . The asymptotic expansions (2.21) and (4.19) are in remarkable agreement with the numerical calculation through the (modified) TBA equation. We postpone a detailed description of these calculations to a separate publicationbut just mention some of the results here. In [1]we calculated numerically a few coefficients in (2.21) for two vacuum states (1 = −1/5 and 1 = 0) in the M2,5 CFT (c = −22/5) and found an excellent agreement with the corresponding exact eigenvalues of the local IM given explicitly in [1] (up to I7 inclusive). Another numerical result concerning the part of (4.19) containing the dual nonlocal IM is mentioned in Sect. 5.

Integrable Structure of Conformal Field Theory II.

265

As an additional support to our conjectures consider the case β2 =

1 , 2

where the eigenvalues of T and A-operators can be calculated explicitly. Moreover, the eigenvalues of the local IM can be independently found using the fermionic representation. The value β 2 = 21 does not lie in SD (2.18) , therefore the results of Sect. 2 do not apply directly. In particular, the definitions (2.11) and (2.29) for T(λ) and A(λ) requires a renormalization since the nonlocal IM G2n in (2.15) and (2.33) diverge logarithmically at β 2 = 21 . It turns out that this renormalization affects the functional Eq. (3.3) . To see this consider the expressions for the vacuum eigenvalues to within the first order in λ2 , T (vac) (λ) = 2 cos(2πp) + λ2 G(vac) + O(λ4 ), (4.23) 1 A(vac) (λ) = 1 −

  λ2 G(vac) − 2π 2 sin(2πp) + O(λ4 ), 1 4 cos(2πp)

(4.24)

where



 1 1 + 2p − πψ − 2p , (4.25) G(vac) = 2 cos(2πp) 2C − πψ 1 2 2 ψ(x) = ∂x log 0(x) is the logarithmic derivative of the gamma-function and C is a (nonuniversal) renormalization constant depending on the ultraviolet cutoff. A simplest way to obtain these expressions is to set β 2 = 21 − ,  → 0 in (3.21) and (3.22) ; that gives an analytic regularization of the divergent integrals with the value of C, C=

π + πψ(1). 2

(4.26)

With the above accuracy in λ2 the eigenvalues (4.23) (4.24) satisfy a “renormalized” functional equation T (λ)A(λ) = e2πip−iπ

2

λ2

A(qλ) + e−2πip+iπ

2

λ2

A(q −1 λ),

(4.27)

where q = exp(iπ/2). Using the lattice regularization (i.e., considering discrete approximations to the P-exponents in (2.11) and (2.29) and then tending the number of partitions to infinity) one can show that the functional Eq. (4.27) is, in fact, exact in the sense that it is valid for arbitrary eigenvalues T (λ) and A(λ) to all order in λ2 . The functional Eq. (4.27) completely determine the eigenvalues T (λ) and A(λ) provided one assumes them to be entire functions of λ2 . One can obtain T (λ) = T (vac) (λ)

L Y k=1

− + F (iλ, p, n+k , n− k )F (iλ, −p, nk , nk ),

A(λ) = A(vac) (λ)

L Y k=1

where F (λ, p, n+ , n− ) =

F (λ, p, n+k , n− k ),

(2p − n− + 21 − πλ2 )(2p + n+ − 21 ) , (2p + n+ − 21 − πλ2 )(2p − n− + 21 )

(4.28)

(4.29)

(4.30)

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V.V. Bazhanov, S.L. Lukyanov, A.B. Zamolodchikov

± ± ± and n± 1 , n2 , · · · , nL are two finite sequences of non-negative integers, 1 ≤ n1 < · · · < 6 n± L , which uniquely specifies a certain vector in the Fock space Fp ; this vector has the Virasoro weight − (vac) (p) + 1{p; n+1 , . . . , n+L ; n− 1 , . . . , nL } = 1

L X k=1

(n+k + n− k − 1).

(4.31)

For the vacuum eigenvalues L = 0 and 2

2πe2Cλ

, 0 1/2 + 2p + πλ2 0 1/2 − 2p + πλ2
0 2p + 1/2 (vac) −Cλ2
. (λ) = e A 0 2p + 1/2 − πλ2

T (vac) (λ) =

(4.32)

(4.33)

It follows then from (4.2) that n o 1
9(ν) = 2iν−1 (iν − 1) ζ iν, 2p + + EL (iν, p) , 2 L n X

−s o 1 1 −s 2p + − n− − 2p − + n+k , EL (s, p) = k 2 2

(4.34)

k=1

where ζ(s, α) is the generalized zeta function defined as the analytic continuation of the series ∞ X (α + n)−s , <e s > 1 (4.35) ζ(s, α) = n=0

to the whole complex plane of the variable s. This function is analytic everywhere in the s-plane except the point s = 1, where it has a simple pole. Therefore the eigenvalues (4.34) are entire functions of ν in agreement with our conjecture. The values of (4.34) at the integer points on the imaginary ν-axis follow from the formula Bm+1 (α) , m = 0, 1, . . . , (4.36) ζ(−m, α) = − m+1 where Bm (α) are the Bernoulli polynomials [33]. Note, in particular, that 9(−i) = 1,

9(0) = p,

(4.37)

in agreement with (4.11) and (4.10) . Further, the values of (4.34) at ν = i(2n − 1), n = 0, 1 . . ., conjectured in (4.8) , perfectly match the eigenvalues of the local IM, which for β 2 = 21 can be independently obtained from the explicit expression for the local IM through free fermion fields (vac) = 2−n B2n 2p + I2n−1

1
. 2

(4.38)

e n are Equation (4.9) can not be tested since the eigenvalues of the dual nonlocal IM H (vac) e not generally known independently except for the vacuum eigenvalue of H1 . For the latter one we have 6

The basis of this kind in the bosonic Fock space is usually referred to as a “fermionic basis” [31], [32].

Integrable Structure of Conformal Field Theory II.

9(vac) (2i) = 2−3 B3 2p +

267

1
1
1
=p p− p+ , 2 4 4

(4.39)

in precise agreement with (4.9) and the relation which is dual to (3.21) . Finally note that the asymptotic expansions of (4.28) and (4.29) agree with (2.21) and (4.19) provided one calculate the latter in the limit β 2 = 21 − ,  → 0, and identifies the divergent part of the coefficient in the leading asymptotic of (2.21) and (4.19) with the renormalization constant C in (4.26) , (4.32) and (4.33) . 5. Vacuum Eigenvalues of the Q-Operators and Non-Equilibrium States in Boundary sine-Gordon Model As is explained in Sect. 2 the vacuum eigenvalue A(vac) (λ) for integer 2p = N coincides with the Coulomb gas partition function (2.50) . In fact it was already observed in [34] that the Coulomb gas partition function (2.50) with p = 0 satisfies the functional Eq. (1.4)7 . The Coulomb gas partition function is obviously related to the finite temperature theory of the massless bose field 8(t, x) on the half plane x < 0 with interaction at the boundary; its action is Z ∞ Z 0 Z ∞

κ 1 2 2 dt dx 8 − 8 dt cos 8(t, 0) + V t . (5.1) + A= t x 2 2 4πβ −∞ β −∞ −∞ Here β 2 , κ and V are parameters 8 . This model finds interesting applications in dissipative quantum mechanics [35], [36], [37], [38]. As was discussed in [13, 14] it also describes the universal current through the point contact in the quantum Hall system. At nonzero driving potential V and arbitrary temperature T the system (5.1) develops a stationary non-equilibrium state with
(5.2) JB ≡ (2π)h 8x (t, x) i = −κ h sin 8(t, 0) + V t i 6= 0. This quantity is interpreted as the backscattering current through the point contact 9 . In the general case (5.2) is a non-equilibrium expectation value and as such it requires non-equilibrium methods for its computation [13]. Note that if the driving potential is continued to pure imaginary values V = VN ≡ 2πiN β −2 T

(5.3)

with integer N , one can make the Wick rotation t → −iτ in (5.1) and formally define the Matsubara partition function 7 We obtained this result independently, along with more the general statement (3.3) , before the paper [34] appeared. 8 The composite field cos(8 + V t) in (5.1) is assumed to be canonically normalized with respect to its








−2β 2

short-distance behavior, i.e. cos 8(t)+V t cos 8(t0 )+V t0 ∼ 2−1 i(t−t0 )+0 , as is conventional in conformal perturbation theory. This is why no ultraviolet cutoff will appear in the matrix elements below. 9 The voltage V and current J (phys) and total B (5.2) differs in normalization from the real voltage V current J (phys) in the Hall system V (phys) = e−1 V, J (phys) =


e 2 β V + JB , h

where e and h are the electron charge and Plank’s constant. Also, β 2 coincides with the fractional filling of the Luttinger state in a Hall bar and the temperature is measured in energy units.

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V.V. Bazhanov, S.L. Lukyanov, A.B. Zamolodchikov

Z ZN =

[D8]exp(−AM ),

(5.4)

where AM

1 = 4πβ 2

Z

Z

β 2 /T

0

dτ 0

dx −∞

82τ +82x




κ − 2 β

Z

β 2 /T

dτ cos 8(0, τ )+2πN β −2 T τ




0

(5.5) and the functional integral is taken over Euclidean fields 8 which satisfy the Matsubara condition 8(τ + β 2 /T, x) = 8(τ, x). It is easy to see that up to an overall constant (the partition function (5.4) with κ = 0) (5.4) coincides with the series (2.50) with µ=

2 κ  β 2 −β . 2T 2πT

(5.6)

Moreover, it is possible to show, using non-equilibrium methods, that for V = VN (5.3) the expectation values of e±i8(t,0) in (5.2) can be calculated in the Matsubara theory (5.5) as Z . ±i8 i = [D8]e±i8(τ,0) e−AM ZN (µ). (5.7) he Of course the equilibrium state can not support a nonzero current and indeed it is not difficult to show that {eV τ h ei8 i − e−V τ h e−i8 i}|V =VN = 0.

(5.8)

Nonetheless it is natural to expect that the non-equilibrium expectation value (5.2) can be obtained by some kind of analytic continuation of (5.7) back to real V . Needless to say this analytic continuation is ambiguous. In [15] the current (5.2) is calculated exactly (for integer values of β −2 ) using the Boltzmann equation with the distribution function of charge carriers determined through a Thermodynamic Bethe Ansatz technique. A conjecture is proposed about the exact current for arbitrary β 2 and V in the recent paper [16]. It was suggested there to define the “partition function” Z2p (µ) for V = 4iπpβ −2 T as a certain analytic continuation of ZN which uses infinite sum expressions for the coefficients in (2.50) obtained with the help of Jack polynomials [24]10 . It is conjectured in [16] that the current (5.2) can be expressed in terms of Z2p (µ) as JB (V, µ, β 2 ) = iπT µ∂µ log

Z2p (µ) , Z−2p (µ)

p=−

iβ 2 V . 4πT

(5.9)

This conjecture agrees with the earlier conjecture for the linear conductance in [13]. In fact, various checks performed in [16] suggest that Z2p thus defined satisfies the functional relation (3.3) . It is therefore very plausible that Z2p coincides with the vacuum (λ). eigenvalue A(vac) + Indeed, we found that the result of [15] is in complete agreement with the following formula 11 10 We acknowledge a private communication with P. Fendley, F. Lesage and H. Saleur who explained to us how the definition of Z2p in [16] must be understood. 11 Again, we have arrived at (5.10) independently, before the paper [16] appeared. However we were significantly influenced by the conjecture about the universal conductance proposed in [24] and by the results of [15].

Integrable Structure of Conformal Field Theory II.

269

JB (V, µ, β 2 ) = iπT λ∂λ log

A(vac) (λ) + A(vac) (λ) −

,

(5.10)

where

sin(πβ 2 ) iβ 2 V µ, p = − . π 4πT Namely, the expression for JB obtained in [39] can be written in the form (5.10) and with A± (λ) solving the functional Eq. (3.3) . Note that for 2p = N both A(vac) + A(vac) coincide with Z , so each of them defines a certain analytic continuation of ZN N − (vac) to complex N . It is suggestive to note though that analytic properties of A+ and (vac) 2 as the functions of p are different, namely A is analytic at <e 2p > −β A(vac) + − (vac) 2 whereas A− is analytic at <e 2p < β . There is no clear notion of partition function (and A(vac) as well) for a non-equilibrium state, and therefore it is remarkable that A(vac) + − admits interpretation as an “equilibrium-state” partition function of the system similar to (5.1) but with an additional boundary degree of freedom described by the “q-oscillator” E± , H in (2.29) . According to (4.19) the conjecture (5.10) implies in particular that the backscattering current (5.2) satisfies the following “strong-weak barrier” duality relation λ=i



− 1 JB V, µ, β 2 ' −V − β −2 JB β 2 V, Cµ β2 , β −2 ,

(5.11)

(remarkably, the factor containing the local IM in (4.19) cancels in the ratio A+ /A− ) which generalizes a similar relation obtained in [15] for the case T = 0 12 . The constant C = C(β 2 ) in (5.11) reads explicitly: C(β 2 ) =



0 1 + β2


 β12


0 1 + β −2 .

(5.12)

Conventionally, one introduces the renormalized coupling parameter X [13, 24] X 2 = 21−2β

2

√

π

0(1 + β 2 ) 2 µ , 0(1/2 + β 2 )

(5.13)

defined in such a way that
∂V JB V, X, β 2 |V =0 = −X 2 + O(X 4 ), X 2 → 0.

(5.14)

According to the formula (5.11) the current admits the following decompositions: ∞ X

Fn V, β 2 X 2n JB V, X, β 2 = − n=1

' −V + β

−2

∞ X

2

Fn β V, β

−2




β

−2

K


n

(5.15) X

− 2n2 β

,

n=1 12 In fact, exactly this duality relation (for the nonlinear mobility in the associated dissipative quantum mechanics problem) was proposed a while ago in [37]. The arguments in [37] are based on an “instanton” description of the hopping amplitudes in the strong barrier limit; from a general point of view this description could be regarded as an approximation. Therefore it looks quite remarkable to us that this relation is indeed exact. See also the discussion in [15].

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V.V. Bazhanov, S.L. Lukyanov, A.B. Zamolodchikov

where Fn are certain functions and the constant K = K(β 2 ) is
1 3
  21 + 1 2  3 2β β2 0 1 + β −2 0 1 + β2 2 2 π

. K(β ) = β 4 0 1/2 + β 2 0 1/2 + β −2

(5.16)

Note that this constant determines the leading X → ∞ asymptotic of the backscattering current,

V V 0 β −2 − i 2πT V 0 β −2 + i 2πT − 2 −2 2
X β2 + JB ' − V + 2β T K(β ) sinh 2 −2 2T 0 β − 42
O X β . (5.17) We should stress that the formula (5.10) is a conjecture and therefore any checks would be valuable. As was mentioned above this formula agrees for the integer β −2 with the TBA solution obtained in [15]. Although the calculations in [15] are also based on a conjecture (exact applicability of the Boltzmann approximation to integrable systems), one could check (5.10) against numerical results obtained from the TBA solution. We have done that for β −2 = 3 and found an excellent agreement with (5.17) . In particular our numerical value for the constant K is Knum (1/3) = 3.35485280612 . . . ,

(5.18)

(note that this number is slightly off from the numerical result for the same constant given in [39]) which is in agreement with the exact value √ 3  15 3 2π 4 2 0(7/6) 2 √ = 3.35485280611990 . . . . (5.19) K(1/3) = 5 3 A completely rigorous check can be made in the case β −2 = 2, where the formula (5.10) agrees with the explicit calculations in the free-fermion theory [13, 15] JB V, X,

4T X 2  1 2X 2 1 2X 2 iV
iV
 1
=− ψ + 2 + −ψ + 2 − , (5.20) 2 iπ 2 π 4πT 2 π 4πT

where ψ(x) = ∂x log 0(x). Another interesting limiting case is the classical limit β 2 → 0. (λ) reduce to (see Appendix B): In this limit the eigenvalues A(vac) ± √ ρ (λ)|β 2 →0 → 2± 2 0(1 ± ρ)X ∓ρ J±ρ ( 2X), (5.21) A(vac) ± where the variables X=

√ −2 2β λ,

ρ = 2β −2 p

are kept fixed when β 2 → 0. Substituting this into (5.10) one obtains after a little algebra
JB V, X, 0 = −V +

V 2T sinh 2T √ √ , Iρ ( 2X)I−ρ ( 2X)

ρ=−

iV , 2πT

(5.22)

where Iρ (x) is a modified Bessel function. As is known the field theory (5.1) can be interpreted in terms of dissipative quantum mechanics of a single particle in the periodic potential [35], [36], [37], [38]. As β 2 plays the role of the Planck constant, in the limit β 2 → 0 this reduces to the theory of the classical Brownian particle at finite temperature

Integrable Structure of Conformal Field Theory II.

271

T . In Appendix C we study the associated Fokker-Planck equation and calculate the classical current. This gives additional support to the conjecture (5.10) . As was explained above, the vacuum eigenvalue A(vac) (λ) at p = 0 coincides with the partition function of (5.1) at the temperature T and zero voltage V = 0. Therefore the asymptotic expansion (4.19) , (4.20) specialized for the vacuum eigenvalue with p = 0 is essentially the low temperature expansion for the associated
free energy. In particular, the leading asymptotic of the heat capacitance C X, β 2 of the point contact in the quantum Hall system at T ∼ const X


−

1 1−β 2

→ 0 and zero voltage V reads


√


 1 2−2β 2 π03 1 + β 2 2 − 1
X 1−β2 , 0 < β 2 < , C X, β ∼ 1 4 2 3 2β 0 2 + β

2
1 √   (1−β 2 )2 02 1+ β12 0 3β2β−2 β2
π03 1+β 2 2 2 − 2

< β 2 < 1. X β2 , C X, β 2 ∼ √ 4 1 1 2 3 πβ 0 2− β 2 20 2 + β (5.23) Likewise, (2.21) determines the low temperature expansion for the impurity free energy in the s = 1/2 anisotropic Kondo problem. 2




0

2−3β 2 2−2β 2 √ 6 πβ 2

1 2−2β 2

0

6. Discussion In this paper we have studied further how the powerful apparatus of the Yang-Baxter theory of integrability can be brought about directly in continuous Quantum Fields Theory. We have constructed the operators Q± (λ), which are field theoretic versions of the Qmatrix of Baxter, directly in Conformal Field Theory. The Q-operators are constructed as the traces of certain monodromy matrices associated with the infinite-dimensional spaces – the representation spaces of “q-oscillator algebra” (2.27) . It is worth mentioning that our construction is not specific for the continuous theory – the Q-matrix of lattice theory admits a similar representation (we will give the details elsewhere). We also found that the Q-operators thus constructed satisfy the remarkable relations (2.40), (2.41) , (2.42) and (2.43) . These relations allow one to employ the powerful machinery of nonlinear integral DDV equations to study the eigenvalues of the Q-operators in the highest weight Virasoro module. We have used the DDV approach to derive (under some analyticity conjectures) various asymptotic expansions for both Q and T operators. We also observed a remarkable (although somewhat puzzling) relation between the vacuum eigenvalues of the Q-operators and the stationary transport characteristics in the boundary sine-Gordon model (the later also relate to the kinetic properties of a one-dimensional quantum particle coupled to a dissipative environment). In this paper we did not present the derivation of our basic relations (2.40) , (2.41) , (2.42) and (2.43) ; this gap will be filled in the forthcoming paper [23]. Clearly, further study of the Q and T operators is desirable. First, almost all the discussion in this paper concerns the “Semi-classical Domain” β 2 < 1/2 (the case β 2 = 1/2 is studied explicitly through the free fermion theory in Sect. 4). However, the most interesting CFT (notably, the unitary CFT) lay outside this domain. We have argued in Sect. 2 that the Q operators can be defined for 1 > β 2 > 1/2 as well, but there are reasons to believe that outside the SD the analytic properties of both Q and T operators undergo significant (and very interesting) changes. We are planning to extend our analysis to the domain 1 > β 2 > 1/2 in the future.

272

V.V. Bazhanov, S.L. Lukyanov, A.B. Zamolodchikov

The analysis in this paper is concerned explicitly with the conformal field theories (more precisely, the chiral sectors of a CFT). One can argue however that the operators Q± (λ) (as well as the T operators of [1]) can be defined in non-conformal integrable QFT obtained by perturbing the CFT with the local operator 91,3 , the most important relations (2.40) , (2.41) , (2.42) and (2.43) remaining intact, and the most significant modifications being in analytic properties of these operators (they develop essential singularities at λ2 → 0 in the perturbed case). The Q operators and associated (modified) DDV equations can be used then to study the finite-size spectra of these non-conformal theories. We will report some preliminary results in this direction in the forthcoming paper [40]. And finally it seems extremely desirable to get more understanding about the relation of the Q-operators to the non-equilibrium properties of the boundary sine-Gordon model discussed in Sect. 5 above. In particular it seems important to find a physical interpretation to the “q-oscillator” degrees of freedom which evidently play central role in our construction of the Q-operators.

7. Appendix A. In this Appendix we give details of the derivation of the DDV equation (3.8) [3]. For any eigenvalue A(λ) satisfying (3.3) consider the function a(λ) defined by (3.6) . For real values of p the property (ii) of the function A(λ) (given in Sect. 3 above the formula (3.4) ) implies (7.1) a(λ)∗ = a(λ∗ )−1 , where the star denotes the complex conjugation. Next, it follows from the leading asymptotic (4.16) of A(λ) at large λ that log a(λ) ∼ −2i M cos(πξ/2)λξ+1 , where β 2 =

ξ 1+ξ .

λ2 → ∞,

| arg λ2 | < 2πβ 2 ,

(7.2)

Moreover, the function a(λ) obviously remains finite for small λ2 , a(λ) = 4πip + O(λ2 ),

λ2 → 0.

(7.3)

The function a(λ) satisfies the Bethe-Ansatz type Eqs. (3.7) . The product representation (3.4) implies ∞ X F (λλ−1 (7.4) log a(λ) − 4πip = k ), k=0

where F (λ) = log

1 − λ2 q 2 . 1 − λ2 q −2

The sum in the RHS of (7.4) can be written as a contour integral Z
dµ F (λµ−1 )∂µ log 1 + a(µ) , log a(λ) − 4πip = f (λ) + 2πi C where f (λ) =

X0 a

F (λλ−1 a )

(7.5)

(7.6)

(7.7)

Integrable Structure of Conformal Field Theory II.

273

denotes a finite sum including only those zeroes λ2a which do not lie on the positive real axis in the λ2 -plane. The contour C goes from +∞ to zero above the positive real axis, then winds around zero and returns to infinity below the positive real axis in the λ2 -plane. Integrating by parts (boundary terms vanish due to (7.2) , since 0 < ξ < 1 for β 2 in SD (2.18) ) one obtains Z ∞

dµ (7.6) =f (λ) − λ∂λ F (λµ−1 ){log 1 + a(µ + i0) − log 1 + a(µ − i0) } 2πiµ 0 Z ∞ Z ∞
dµ dµ −1 λ∂λ F (λµ )=m log 1 + a(µ − i0) − λ∂λ F (λµ−1 ) log a(µ), = πµ 2πiµ 0 0 (7.8) where we have used (7.1) . Introducing new variables θ, θ0 and θk as θ

λ = e 1+ξ ,

θ0

µ = e 1+ξ ,

θk

λk = e 1+ξ

(7.9)

iπξ

and recalling that q = e 1+ξ , one obtains Z Z ∞
dθ0 R(θ −θ0 )=m log 1+a(θ0 −i0) + (7.6) = f (θ)−2i −∞

∞

−∞

dθ0 R(θ −θ0 ) log a(θ0 ), (7.10)

where

i λ∂λ F (λ). (7.11) 2π(1 + ξ) With the standard notation for the convolution (3.10) , the Eq. (7.10) can be written as
K ∗ log a(θ) = 4πip + f (θ) − 2iR ∗ =m log 1 + a(θ − i0) , (7.12) R(θ) =

where

K(θ) = δ(θ) − R(θ).

(7.13)

Let us now apply the inverse of the integral operator K to both sides of (7.12) ; in this one has to add an appropriate zero mode of K to the r.h.s. of the resulting equation to make it consistent with the asymptotic conditions (7.2) and (7.3) . In this way one obtains the integral Eq. (3.8) in the main text. As is noted there, this integral equation has to be complemented by a finite number of the transcendental equations (3.7) for the roots λ2a lying outside the positive real axis of λ2 . For the vacuum eigenvalues with 2p > −β 2 all the roots λ2k are real and positive and P0 the term i a log S(θ − θa ) in the r.h.s. of (3.8) is absent. Let λ20 be the minimal of the zeroes λ2k . Introduce the function a(vac) (λ) related to A(vac) (λ) as in (3.6) and denote B(p) =

1+ξ log λ20 . 2

(7.14)


The function log 1 + a(vac) (θ) is analytic in the strip −πξ < =m θ < πξ with the branch cut along the positive real axis from B(p) to infinity. So it is analytic for real θ < B(P ), where it obeys the relation
log a(θ) = 2=m log 1 + a(θ − i0) , (7.15) which follows from (7.1) . Note that the infinitesimal shift −i0 here is not essential. Taking (7.15) into account one can rewrite the (3.8) for θ < B(p) as

274

V.V. Bazhanov, S.L. Lukyanov, A.B. Zamolodchikov

−

Z B(P ) 0
πp πξ θ dθ e ∂θ logS(θ − θ0 )=m log 1 + a(vac) (θ0 ) + M cos − 2 β 2 2πi −∞ Z +∞ 0
dθ ∂θ logS(θ − θ0 )=m log 1 + a(vac) (θ0 − i0) . = B(p) 2πi

(7.16)

Let us make a technical assumption, B(p) ∼ const log p

p → +∞.

as

(7.17)

Then one can show that the r.h.s. of (7.16) decreases at the large positive p and therefore can be dropped in the leading approximation at p → +∞. This brings (7.16) to the linear integral Eq. (3.14) of the Winer-Hopf type. 8. Appendix B In this appendix we consider the functional equations for the eigenvalues of T and Q operators in the classical limit β 2 → 0, where they reveal a remarkable connection with the theory of the classical Liouville equation13 . We start with the functional equations for the eigenvalues Tj (λ) of the operators (2.11) which follow from (2.13) ; they can be written in the form [1] Tj (q 2 λ)Tj (q − 2 λ) = 1 + Tj− 1 (λ)Tj+ 1 (λ). 1

1

2

2

(8.1)

Consider limiting values of the eigenvalues Tj (λ) and Q± (λ), when 2λ = β 2 eσ ,

j=

τ , πβ 2

2p = β 2 ρ,

β 2 → 0,

(8.2)

and the variables σ, τ and ρ are kept fixed. We assume that in this limit Tj (λ) →

2 −φ(σ,τ ) e , πβ 2

Q± (λ) → β 2


±ρ

Q± (σ),

(8.3)

where φ(σ, τ ) and Q± (σ) are smooth functions of their arguments. Then it is easy to see that the functional Eq. (8.1) becomes the classical Liuoville equation (∂σ2 + ∂τ2 )φ(σ, τ ) = e2φ(σ,τ )

(8.4)

for the field φ(σ, τ ) in the Euclidean space. It should be complemented by the periodic boundary conditions φ(σ + iπ, τ ) = φ(σ, τ ), (8.5) since Tj (λ) in (8.3) is a single-valued function of λ2 . The limiting form of the functional Eq. (2.43) reads
(8.6) e−φ(σ,τ ) = (4iρ)−1 Q+ (σ + iτ )Q− (σ − iτ ) − Q+ (σ − iτ )Q− (σ + iτ ) , whereas (2.42) becomes the ordinary Wronskian condition ∂σ Q+ (σ)Q− (σ) − Q+ (σ)∂σ Q− (σ) = 2ρ.

(8.7)

13 The fact that the functional relations associated with the TBA equations have many features in common with the classical Liouville equation is well known to experts [28].

Integrable Structure of Conformal Field Theory II.

275

for the functions Q± (σ). One can recognize in (8.6) , (8.7) the general local solution of the Liouville equation (8.4) satisfying the condition e−φ(σ,τ ) = τ + O(τ 2 ),

τ → 0.

The Baxter’s relation (1.4) reduces to the second order differential equation
∂σ2 + w(σ) Q(σ) = 0,

(8.8)

(8.9)

with a periodic potential w(σ + iπ) = w(σ),

(8.10)

which is determined by the leading asymptotic of the eigenvalue T (λ) ≡ T 1 (λ) in the 2 limit (8.2) , T (λ) = 2 + (πβ)2 w(σ) + O(β 4 ). (8.11) The functions Q± (u) are just two linearly independent Bloch-wave solutions Q± (σ + iπ) = e±πiρ Q± (σ).

(8.12)

to the second order differential Eq. (8.9) . Note that (8.12) imply the periodicity (8.5) of the solutions (8.6) for the Liuoville equation. (8.4) . It is illustrative to apply the above limiting procedure to the vacuum eigenvalues of T and Q operators. Notice that all the vacuum eigenvalues of the nonlocal IM (2.16) entering the series expansion (2.15) remain finite in the limit (8.2) . Therefore only the two first terms of the expansion + O(λ4 ). T (vac) (λ) = 2 cos(2πp) + λ2 G(vac) 1

(8.13)

contribute to (8.11) . Using (3.22) one thus obtains w(σ) = e2σ − ρ2 .

(8.14)

The solutions to the differential equation (∂σ2 + e2σ − ρ2 )Q± (σ) = 0, satisfying the conditions (8.7) , (8.12) have the forms Q± (ρ) = 0(1 ± ρ)J±ρ (eσ ), where Jσ (x) is the conventional Bessel function. The eigenvalues Tj(vac) (λ) in the limit (8.2) are then expressed from (8.3) through the corresponding solution (8.6) of the Liouville equation Tj(vac) (λ) →

2 −φ(σ,τ ) e = πβ 2
1 Jρ (eσ+iτ )J−ρ (eσ−iτ ) − Jρ (eσ−iτ )J−ρ (eσ+iτ ) . 2iβ 2 sinπρ

(8.15)

It would be interesting to check this formula against the solutions to the TBA equations in the limit β 2 → 0.

276

V.V. Bazhanov, S.L. Lukyanov, A.B. Zamolodchikov

9. Appendix C As is discussed in [35] the field theory (5.1) describes a dissipative quantum mechanics of one dimensional particle if one interprets the boundary value 8(t) ≡ 8(t, 0) as the position of the particle, the bulk degrees of freedom 8(t, x), x 6= 0 playing the role of thermostat. As β 2 enters (5.1) as the Planck’s constant, in the limit β 2 → 0 this problem reduces to the one of the classical Brownian particle which is described by the Langevin equation
1 ˙ 8(t) = −κ sin 8(t) + V t + ξ(t), (9.1) 2π ˙ = ∂t 8(t), and ξ(t) is the white noise where 8 h ξ(t)ξ(t0 ) i =

T δ(t − t0 ). π

(9.2)

We are interested in the limiting value of the average velocity ˙ it→∞ . JB (V ) = h 8

(9.3)

The time dependence of the potential term in (9.1) can be eliminated by the substitution z(t) = 8(t) + V t,

(9.4)

z˙ = V − 2πκ sin(z) + 2πξ.

(9.5)

which transforms (9.1) to the form

The driving force V appears here because the transformation (9.4) brings us to the frame which moves with the velocity −V with respect to the thermostat. The current (9.3) then is J(V ) = h z˙ it→∞ . (9.6) JB (V ) = −V + J(V ), The probability distribution P (z) associated with the stochastic process (9.5) satisfies the Fokker-Planck equation √ (9.7) ∂t P = 2πT ∂z { (−ν + 2X sin z)P + ∂z P }, where

√ V κ , (9.8) 2X = . 2πT T The suitable solution of (9.7) , which describes the stationary drift has the form (see e.g. [41]) Z z+2π √ dy −√2X cos(y)−νy e , (9.9) P (z) = N −1 P0 (z), P0 (z) = e 2X cos(z)+νz 2π z ν=

where

Z

2π

N =

dzP0 (z)

(9.10)

0

is the normalization constant. The current J can be then expressed as
J = 2πT N −1 1 − e−2πν .

(9.11)

Integrable Structure of Conformal Field Theory II.

277

The integral (9.10) is calculated explicitly with the result J=

2T sinh( πν ) √ √ , Iiν ( 2X)I−iν ( 2X)

(9.12)

where Iρ (x) is the modified Bessel function. With the account (9.6) , (9.8) , (5.6) , (5.13), this agrees perfectly with (5.22) . Acknowledgement. V.B. would like to thank R. Baxter for stimulating interest to this work. S.L. is grateful to the Department of Theoretical Physics, Australian National University for the hospitality, and would like to thank A. Belavin, V. Drinfel’d, J. Jimbo, P. Fendley, E. Frenkel, A. LeClair, T. Miwa and N. Reshetikhin for interesting discussions. The work of S.L. is supported in part by anNSF grant. A.Z. is pleased to acknowledge warm hospitality extended to him at LPM, University of Montpellier and at the Laboratoire de Physique Theorique, ENS, Paris, during the last stages of this work, and very useful discussions with Al. Zamolodchikov and V. Kazakov and N. Sourlas. The research of A.Z. is supported by a Guggenheim Fellowship and by DOE grant #DE-FG05-90ER40559.

References 1. Bazhanov, V.V., Lukyanov, S.L. and Zamolodchikov, A.B.: Integrable structure of conformal field theory, quantum KdV theory and thermodynamic Bethe ansatz. Commun. Math. Phys. 177, 381–398 (1996) 2. Baxter, R.J.: Eight-vertex model in lattice statistics and one-dimensional anisotropic Heisenberg chain; 1. Some fundamental eigenvectors. Ann. Phys. (N.Y.) 76, 1–24 (1973); 2. Equivalence to a generalized ice-type model. Ann. Phys. (N.Y.) 76, 25–47 (1973); 3. Eigenvectors of the transfer matrix and Hamiltonian. Ann. Phys. (N.Y.) 76, 48–71 (1973) 3. de Vega, H.J. and Destri., C.: Unified approach to thermodynamic Bethe Ansatz and finite size corrections for lattice models and field theories. Nucl. Phys. B438, 413–454 (1995) 4. Mussardo, G.: Off-critical statistical models, factorized scattering theories and bootstrap program. Phys. Rep. 218, 215–379 (1992) 5. Yang, C.N. and Yang, C.P.: Thermodynamics of one-dimensional system of bosons with repulsive deltafunction potential. J. Math. Phys. 10, 1115–1123 (1969) 6. Zamolodchikov, Al.B.: Thermodynamic Bethe ansatz in relativistic models: Scaling 3-state Potts and Lee-Yang models. Nucl. Phys. B342, 695–720 (1990) 7. Baxter, R.J.: Exactly Solved Models in Statistical Mechanics. London: Academic Press, 1982 8. Faddeev, L.D., Sklyanin, E.K. and Takhtajan, L.A.: Quantum inverse scattering method I. Theor. Math. Phys. 40, 194–219 (1979) (in Russian) 9. Bogoliubov, N.M., Izergin, A.G. and Korepin, V.E.: Quantum Inverse Scattering Method and Correlation Functions. Cambridge: Camabridge University Press, 1993 10. Zamolodchikov, A.B.: Integrable field theory from conformal field theory. Adv. Stud. in Pure Math. 19, 641–674 (1989) 11. Fendley, P., Saleur, H. and Warner, N.P.: Exact solution of a massless scalar field with a relevant boundary interaction. Nucl. Phys. B430, 577–596 (1994) 12. Ghosal, S. and Zamolodchikov, A.B.: Boundary S-matrix and boundary state in two-dimensional integrable quantum field theory. Int. J. Mod. Phys. A9, 3841–3885 (1994) 13. Kane, C.L. and Fisher, M.P.A.: Transmission through barriers and resonant tunneling in an interacting one-dimensional electron gas. Phys. Rev. B46, 15233–15262 (1992) 14. Moon, K., Yi, H., Kane, C.L., Girvin, S.M. and Fisher, M.P.A.: Resonant tunneling between quantum Hall edge states. Phys. Rev. Lett. 27, 4381–4384 (1993) 15. Fendley, P., Ludwig, A.W.W. and Saleur H.: Exact Conductance through point contacts in the ν = 1/3 fractional quantum Hall effects. Phys. Rev. Lett. 74, 3005–3008 (1995) 16. Fendley, P., Lesage, F. and Saleur H.: A unified framework for the Kondo problem and for an impurity in a Luttinger liquid. Preprint USC-95-20, #cond-mat/9510055 (1995) 17. Feigin, B.L. and Fuchs, D.B.: Representations of the Virasoro algebra. In: Faddeev, L.D., Mal’cev, A.A. (eds.) Topology. Proceedings, Leningrad 1982. Lect. Notes in Math. 1060. Berlin–Heidelberg–New York: Springer 1984

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18. Dotsenko, Vl.S. and Fateev, V.A.: Conformal algebra and multipoint correlation functions in 2d statistical models. Nucl. Phys. B240 [FS12], 312–348 (1984); Dotsenko, Vl.S. and Fateev, V.A.: Four-point correlation functions and the operator algebra in 2d conformal invariant theories with central charge c ≤ 1. Nucl. Phys. B251 [FS13], 691–734 (1985) 19. Kulish, P.P., Reshetikhin, N.Yu. and Sklyanin, E.K.: Yang-Baxter equation and representation theory. Lett. Math. Phys. 5, 393–403 (1981) 20. Sasaki, R. and Yamanaka, I.: Virasoro algebra, vertex operators, quantum Sine-Gordon and solvable Quantum Field theories. Adv. Stud. in Pure Math. 16, 271–296 (1988) 21. Eguchi, T. and Yang, S.K.: Deformation of conformal field theories and soliton equations. Phys. Lett. B224, 373–378 (1989) 22. Feigin, B. and Frenkel, E.: Integrals of motion and quantum groups. Proceeding of C.I.M.E. Summer School on “Integrable systems and Quantum groups”, #hep-th/9310022 23. Bazhanov, V.V., Lukyanov, S.L. and Zamolodchikov, A.B.: Integrable Structure of Conformal Field Theory III. The Yang-Baxter Relations. To appear 24. Fendley, P., Lesage, F. and Saleur, H.: Solving 1d plasmas and 2d boundary problems using Jack polynomial and functional relations. Preprint USC-94-16, SPhT-94/107, #hepth 9409176 (1994) 25. Sklyanin, E.K.: The quantum Toda chain. In: Non-linear equations in classical and quantum field theory. Ed. by N. Sanchez, Lecture Notes in Physics, 226, NY: Springer, 1985, pp. 196–233 26. Kl¨umper A., Batchelor M.T., Pearce P.A.: Central charges of the 6- and 19-vertex models with twisted boundary conditions. J. Phys. A24, 3111–3133 (1991) 27. Zamolodchikov, A.B. and Zamolodchikov, Al.B.: Factorized S-matrices in two dimensions as the exact solutions of certain relativistic quantum field theory models. Ann. Phys. (N.Y.) 120, 253–291 (1979) 28. Zamolodchikov, Al.B.: Private communication. 29. Morse, M.P. and Feshbach, H.: Methods of theoretical physics. New York, Toronto, London: McGrawHill Book Company, Inc., 1953 30. Mumford, D.: Tata Lectures on Theta I, II. Progress in Mathematics. Vol. 28, Boston, Basel, Stuttgart: Birkh¨auser, 1983, 1984, p. 43 31. Kedem, R., Klassen, T.R., McCoy, B.M. and Melzer, E.: Fermionic sum representations for conformal field theory characters. Phys. Lett. B307, 68–76 (1993) 32. McCoy, B.M.: The connection between statistical mechanics and Quantum Field Theory. In: Bazhanov, V.V. and Burden, C.J. (eds.) Field Theory and Statistical Mechanics. Proceedings 7th Physics Summer School at the Australian National University. Canberra. January 1994, Singapore: World Scientific, 1995 33. Abramowitz, M. and Stegun, I.: Handbook of mathematical function. New York: Dover Publications, Inc., 1970 34. Fendley, P. and Saleur, H.: Exact perturbative solution of the Kondo problem. Phys. Rev. Lett. 75, 4492–4495 (1995) 35. Caldeira, A.O. and Legget, A.J.: Influence of dissipation on quantum tunneling in macroscopic systems. Phys. Rev. Lett. 46, 211–214 (1981); Caldeira, A.O. and Legget, A.J.: Path integral approach to quantum Brownian motion. Physica A121, 587–616 (1983) 36. Schmid, A.: Diffusion and localization in a dissipative quantum system. Phys. Rev. Lett. 51, 1506–1509 (1983) 37. Fisher, M.P.A. and Zwerger, W.: Quantum Brownian motion in a periodic potential. Phys. Rev. B32, 6190–6206 (1985) 38. Callan, C.G. and Thorlacius L.: Open string theory as dissipative quantum mechanics. Nucl. Phys. B329, 117–138 (1990) 39. Fendley, P., Ludwig, A.W.W. and Saleur, H.: Exact non-equilibrium transport through point contacts in quantum wires and fractional quantum Hall devices. Phys. Rev. B52, 8934–8950 (1995) 40. Bazhanov, V.V., Lukyanov, S.L. and Zamolodchikov, A.B.: In preparation. 41. Stratonovich, R.L.: Topics in the Theory of Random Noise. Vol. 2, Chapter 9. New York: Gordon and Breach, 1967 Communicated by T. Miwa

Commun. Math. Phys. 190, 279 – 285 (1997)

Communications in

Mathematical Physics c Springer-Verlag 1997

SRB States and Nonequilibrium Statistical Mechanics Close to Equilibrium Giovanni Gallavotti? , David Ruelle?? Mathematics Dept., Rutgers University, New Brunswick, NJ 08903, USA Received: 2 December 1996 / Accepted: 13 March 1997

Abstract: Nonequilibrium statistical mechanics close to equilibrium is studied using SRB states and a formula [10] for their derivatives with respect to parameters. We write general expressions for the thermodynamic fluxes (or currents) and the transport coefficients, generalizing the results of [4, 5]. In this framework we give a general proof of the Onsager reciprocity relations. There is currently a strong revival of nonequilibrium statistical mechanics, based on taking seriously the nonlinear (chaotic) microscopic dynamics. One natural idea in this direction is to use nonequilibrium steady states, which are defined to be the SRB states for the dynamics (see below). The idea of using SRB states eventually led to useful results only recently, when it was combined in [6] with reversibility of the dynamics to obtain a nontrivial fluctuation formula for the entropy production. In [5] the same ideas were applied to prove the Onsager reciprocity relations, and the fluctuation-dissipation formula for a rather special class of models. The analysis dealt with examples rather than the general situation, and relied on an unproven conjecture on Anosov systems. In this note we generalize [5] and, using [10], give results that can be proved rigorously for Axiom A diffeomorphisms (this is a strong chaoticity assumption, see [12], here we skip technical details). To be discussed is the classical microscopic description of the time evolution of a physical system. (No large system assumption will be made). We let the accessible phase space be a compact manifold M . The time evolution is given by iterates of a diffeomorphism f of M (discrete time case) or by integrating a vector field F on M (continuous time case). We choose a Riemann metric on M , and let dx be the corresponding volume element. ? ??

Permanent address: Fisica, Universit´a di Roma “La Sapienza”, 00185 Roma, Italia. Permanent address: IHES, 91440 Bures sur Yvette, France.

280

G. Gallavotti, D.Ruelle

In the idealization of nonequilibrium statistical mechanics which we want to consider, the time evolution typically does not preserve any measure which has a density with respect to dx. Let m(dx) = m(x)dx be a probability measure (with density m), and f ∗k m the direct image of m by f ∗k ; any weak limit for n → ∞ of n−1

1 X ∗k f m n k=0

is an f -invariant probability measure ρf on M . If furthermore ρf is ergodic we may say that it is a natural nonequilibrium state (the SRB states are special examples of this, [3]). Typically, there are only few natural nonequilibrium states, for instance one SRB state on an attractor: in the latter case, as a consequence, ρg does not depend on the initial m(dx). An infinitesimal change δf of f corresponds to an (infinitesimal) vector field X = δf ◦ f −1 on M and, see [10]: δρf (8) =

∞ X

ρf hgrad(8 ◦ f n ), Xi,

(1)

n=0

where 8 is a smooth test function and h· , · i is the scalar product of a tangent vector and a cotangent vector to M . Equation (1) expresses the change in the expectation value of the observable 8 when the system is subjected to a force X. Replacing X by a time dependent force Xt we have for the change of expectation value of the observable 8 at time s the analogous formula X ρf hgrad(8 ◦ f s−t ), Xt i. (2) δs ρf (8) = t≤s

Note that the condition t ≤ s corresponds to the "causality principle" and (2) can be used to derive Kramers-Kronig dispersion relations. For Axiom A diffeomorphisms, (1) and (2) can be proved rigorously1 : the left-hand side is a derivative, and the right-hand side a convergent series. In the case of continuous time systems described by a differential equation x˙ = F (x) and by the corresponding flow (f t ), an infinitesimal variation δF = X generates a variation in the expectation value ρF (8) (for the natural nonequilibrium state) given, as shown by an easy formal calculation2 along the lines of [10], by: 1

See [10]; the proof given in [10] for the time dependent case assumes that the pertubation has finite support in time. 2 We have

Z

δ[(f ∗t m)(8)] = δm(8 ◦ f t ) =

m(dx) δ 8(f t x)

Z

Z m(dx)h(grad8)(f t x), δf t xi =

=

Z

Z

t

Z

Z

0 t

dτ

= 0

t 0

((f t−τ )∗ m)(dy)h(grad8)(f τ y), (T f τ )δf (y)i

dτ

=

Z m(dx)h(grad8)(f t x),

((f t−τ )∗ m)(dy)h(grad(8 ◦ f τ ))(y), X(y)i.

dτ (T f τ )δf (f t−τ x)i

SRB States and Nonequilibrium Statistical Mechanics near Equilibrium

Z

Z

+∞

δρF (8) =

ρF (dx)h grad x (8 ◦ f t ), X(x)i.

dt

281

(10 )

0

(A rigorous proof for Axiom A flows has not been given yet.) The entropy production associated with the diffeomorphism f is defined by3 ef = ρf (σf )

σf = − log Jf ,

,

(3)

where Jf is the absolute value of the Jacobian of f with respect to the Riemann volume element dx. In the continuous time systems, we let eF = ρF (σF )

,

σF = −divF

(30 )

(see [5]). From now on we shall fix f such that ρf (dx) = dx, i.e., ρf is the Riemann volume element (if ρf has smooth density this can be achieved by a change of metric). Note that in particular this implies that (1) can be rewritten as δρf (8) =

∞ X

ρf (8 ◦ f n . (−divX)).

(4)

n=0

In this nondissipative situation we have log Jf = 0, hence ef = 0. If we write as before X = δf ◦ f −1 we obtain to second order in X, using (3) and (1), ef +δf = (ρf + δρf )(σf +δf ) 1 ρf ((divX)2 ) − δρf (divX ◦ f ) 2 ∞ X 1 ρf hgrad((divX) ◦ f n ), Xi = ρf ((divX)2 ) − 2 =

n=1

∞

=

X 1 ρf ((divX)2 ) + ρf ((divX) ◦ f n . divX) 2

(5)

n=1

=

∞ 1 X ρf ((divX) ◦ f n . divX). 2 n=−∞

The same analysis leads, in the continuous time case (with δF = X, and ρF (dx) = dx) to: Z 1 +∞ dt ρf (σf ◦ f t . σf ) eF +δF = 2 −∞ Z Z 1 +∞ dt dx div X(f t x) . div X(x), (50 ) = 2 −∞ see [4, 5].

Z

∞

When t → ∞ we obtain formally δρf (8) = 3

0

For a discussion of entropy production see [9].

dτ ρf hgrad(8 ◦ f τ ), Xi as asserted.

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G. Gallavotti, D.Ruelle

We shall now relate the above expressions for the entropy production to the definition of the thermodynamic forces Xα , and the conjugated thermodynamic fluxes Jα as they appear for instance in [8]. We begin by an informal discussion, and assume, as usual in applications, that f or F depends on parameters Eα , so that we may write (to first order) X X= Vα δEα . α

We identify the thermodynamic forces Xα with the parameters Eα . Considering first the continuous time case, we follow [4, 5], and we define the thermodynamic flux conjugated to Eα as ∂ σF +δF ). Jα = ρF +δF ( ∂Eα Since ∂σ/∂Eα is a divergence, ρF (∂σ/∂Eα ) = 0 and we have Z

Z

+∞

dt

=

Jα = δρF (−divVα )

+ h.o. =

ρF (dx)hgradx ((−divVα ) ◦ f t ), X(x)i

+ h.o.

0

From now on we neglect higher order terms and (using integration by parts, since ρF (dx) = dx) we write Z +∞ Z dt ρF (dx) (divx X)(divf t x Vα ). Jα = 0

In the discrete time case we define the thermodynamic flux only to leading order in the Eα by ∞

Jα =

X 1 ρf ((divVα ). (divX)) + ρf ((divVα ) ◦ f n . (divX)). 2 n=1

With these definitions Jα depends only on the application of X in the past (causality, cf. (2)) and the entropy production (to second order) is X or eF +δF = Xα J α . ef +δf α

These conditions uniquely determine the Jα . Notice that the formulae for Jα involve only the divergences of X and Vα . To continue the discussion, we assume that there is a (sufficiently large) Banach space B of functions 8 : M → R such that ρf (8) = 0

if

8 ∈ B.

and for some constant C X |ρf (9 ◦ f k . 8)| ≤ Ck8kB k9kB

if

8, 9 ∈ B.

(6)

k∈Z

[This is the discrete time case, the continuous time case is similar. If f is an Anosov diffeomorphism4 we can take for B a space of H¨older continuous functions on M , similarly for Anosov flows.] 4

An Axiom A diffeomorphism f preserving dx is an Anosov diffeomorphism.

SRB States and Nonequilibrium Statistical Mechanics near Equilibrium

283

From now on, we assume that divX is in the Banach space B just introduced, and we may define X ∈ B and J ∈ B ∗ (the dual of B) as follows: X = −divX ∈ B,

(7)

∞

(J , 8) =

X 1 ρf (−divX. 8) + ρf ((−divX) ◦ f n . 8) 2

(8)

n=1

for discrete time, and 1 (J , 8) = 2

Z

∞

ρF ((−divX) ◦ f t . 8)

(80 )

0

for continuous time, where 8 ∈ B, and (· , · ) is the pairing B ∗ × B → C. Note that if divVα ∈ B we have Jα = (J , −divVα ) and that the entropy production is ef +δf

or

eF +δF = (J , X ).

With the above notation and assumptions we may write J = LX , where L is, in view of (6), a continuous linear mapRB → B ∗ . If we define a unitary operator U on L2 (ρf ) π by U 8 = 8 ◦ f and write U = −π eiα dP(α) we have X

e−ikα ρf (8 ◦ f k . 8) = 2π

k∈Z

d (8, dP(α)8)L2 dα

so that the quadratic form associated with L satisfies (L8, 8) =

d 1X ρf (8 ◦ f k . 8) = π (8, dP(α)8)L2 |α=0 ≥ 0. 2 dα k∈Z

In particular, this quadratic form is ≥ 0. The formulae obtained up to now hold under the only assumption of closeness to equilibrium. If we make the further assumption of (microscopic) reversibility, we shall obtain a symmetry property of L called Onsager reciprocity. For simplicity we discuss only the discrete time case. We say that the dynamics is reversible if there exists a diffeomorphism i : M → M such that i2 = identity, i ◦ f = f −1 ◦ i. We have then also i∗ ρf = ρf . [Note that ρf , i.e., the Riemann volume, is mixing by (6), hence f -ergodic. Since i∗ ρf is absolutely continuous with respect to ρf and satisfies f ∗ (i∗ ρf ) = i∗ (f −1 )∗ ρf = (i∗ ρf ) we have i∗ ρf = ρf by ergodicity.] Assuming reversibility we may define 8 = 8 ◦ i for 8 ∈ B, and we find ∞

(L9, 8) =

∞

X X 1 1 ρf (9. 8) + ρf (9 ◦ f n . 8) = ρf (9. 8) + ρf (9 ◦ f −n . 8) 2 2 n=1

= The relation

1 ρf (8. 9) + 2

n=1

∞ X n=1

ρf (8 ◦ f n . 9) = (L(8), 9).

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(L9, 8) = (L(8), 9) is a form of the Onsager reciprocity relation as we shall see in a moment. Note that reversibility was assumed only for f (i.e., at equilibrium), the perturbation δf is arbitrary. To obtain a more familiar form of the entropy production formula (see [8]), we assume that B has a basis (8α ) with a corresponding system (φα ) of elements of B ∗ such that (φα , 8β ) = δαβ (see [11]; more modestly we may assume that there are finite families (8α ) in B and (φα ) in B ∗ such that (φα , 8β ) = δαβ and −divX is a linear combination of the φα ). Write Xα = (φα , X ) = (φα , −divX), ∞

Jα = (J , 8α ) =

X 1 ρf (−divX. 8α ) + ρf ((−divX) ◦ f n . 8α ). 2 n=1

In particular −divX =

X X (φα , −divX). 8α = Xα . 8 α α

and

α

ef +δf = (J , X ) =

X

Jα . Xα ,

α ∞

Jα =

XX 1X Xβ ρf (8β 8α ) + Xβ ρf (8β ◦ f n . 8α ). 2 n=1

β

β

To avoid convergence problems suppose that finitely many Xβ only are nonzero. Then X Lαβ Xβ , Jα = β

where

∞

Lαβ =

X 1 ρf (8β 8α ) + ρf (8β ◦ f n . 8α ). 2 n=1

Suppose again reversibility of the dynamics, and suppose that the map f varies in a i–invariant finite dimensional subspace of B. Let (8α ) of B be such that 8α = 8α ◦ i = α 8α with α = ±1. Then Lαβ = α β Lβα , which is the usual form of the Onsager reciprocity relation. We conclude by sketching an example, see [1], of the formalism described above. Let Σ be a surface of constant negative curvature, and genus g, with the automorphic forms φα (z)dz. We normalize these forms so that they are orthonormal in the space L2 (T Σ) in the natural scalar product, [7]. We can then consider the hamiltonian equations of the motion of a particle on Σ subject P to the external force generated by the “electric” field E such that Ex dx + Ey dy = Re Eα ϕα (z)dz. We also impose, via Gauss’ principle, [5], that there is a thermostat force that keeps the kinetic energy constant (and equal to 1/2) in spite of the field’s action. Thus the “only” effect that the fields will have on the flow is that currents flowing “around” the

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g “holes” of the surface will be established. But the flow being a mixing Anosov flow on a compact surface (because of the gaussian constraint) it will result that a stationary state will be reached and the latter will be the SRB distribution, [2]. The equations of motion can be written explicitly and one finds in particular that the entropy creation rate at the point ~q, p~ of the phase space is σ(~q, p~) =

E~ · p~ . p~2

The transport coefficient can also be explicitly computed, and one finds Lαβ = 21 δαβ , see [1]. References 1. Bonetto, F., Gentile, G., Mastropietro, V.: Electric fields on a surface of constant negative curvature. IHES preprint, 1997 2. Bowen, R., Ruelle, D.: Ergodic theory of Axiom A flows. Invent. Math. 29, 181–202 (1975) 3. Eckmann J.-P. and Ruelle, D.: Ergodic theory of chaos and strange attractors. Rev. Mod. Phys. 57, 617–656 (1985) 4. Gallavotti, G.: Extension of Onsager’s reciprocity to large fields and the chaotic hypothesis. Phys. Rev. Letters. 77, 4334 -4337 (1996) 5. Gallavotti, G.: Chaotic hypothesis: Onsager reciprocity and fluctuation-dissipation theorem. J. Statist. Phys. 84, 899–926 (1996) 6. Gallavotti, G. and Cohen, E.G.D.: Dynamical ensembles in nonequilibrium statistical mechanics. Phys. Rev. Letters 74, 2694–2697 (1995); Dynamical ensembles in stationary states. J. Statist. Phys. 80, 931– 970 (1995) 7. Gel’fand, I.M., Graev, M.I, Pyateckii-Shapiro, I.I.: Representation theory and automorphic functions. Philadelphia: Saunders, 1969 8. de Groot, S.R. and Mazur, P.: Non-equilibrium statistical thermodynamics. New York: Dover, 1984 9. Ruelle, D.: Positivity of entropy production in nonequilibrium statistical mechanics. J. Statist. Phys. 85, 1–25 (1996) 10. Ruelle, D.: Differentiation of SRB states. To be published; the preprint is archived in mp [email protected], 96, #499 11. Singer, I.: Bases in Banach spaces I. Berlin: Springer, 1970 12. Smale, S.: Differentiable dynamical systems. Bull. Am.. Math. Soc. 73, 747–817 (1967) Communicated by J. L. Lebowitz

Commun. Math. Phys. 190, 287 – 307 (1997)

Communications in

Mathematical Physics c Springer-Verlag 1997

A Random Matrix Model From Two Dimensional Yang-Mills Theory Feng Xu Department of Mathematics, UCLA, Los Angeles, CA 90024, USA. E-mail: [email protected] Received: 10 June 1996 / Accepted: 6 April 1997

Abstract: We use fat graphs to give a unified treatment of various asymptotic freeness results. In particular, a random matrix model from two dimensional Yang-Mills Theory on the plane is presented.

1. Introduction The purpose of this work is to give a mathematical proof of the well-known fact that “the non-overlapping families of Wilson loop operators in two dimensional Yang-Mills theory on the plane form a multiplicative free family”. (The precise statement is given in Sect. 3.) Although this fact has been known to physicists since the work of V. Kazakov and I. Kostov [4], and more recently the work of R. Gopakumar and D. Gross [3] (inspired by [12]), a mathematical proof has not appeared. Although one might assume that one might use the techniques of [1] together with various “freeness” arguments to prove this theorem, we have been unable to do so. We have therefore turned to the notion of “fat graphs” to complete the proof. It turns out that the same ideas also give a different proof of some D. Voiculescu’s results. Thus we are able to give a unified treatment of asymptotic freeness. This paper is organized as follows: The basic properties of fat graphs are introduced in Sect. 2.0 following [9]. An important class of fat graphs is also studied. In the remainder of Sect. 2 we give a proof of the factorization hypothesis in the Gaussian measure, Haar measure and heat kernel cases. Although proofs for the Gaussian measure and Haar measure cases are well-known, we have included these cases in order to illustrate the diagram approach. Lemma 2.4 plays an important role in Sect. 3. After the proof of Lemma 3.2 in Sect. 3, we present a unified proof for the various freeness results, using the heat kernel case as an example (see Theorem 3.4). In Sect. 4 we complete the arguments, and we consider some interesting questions.

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2. Factorization hypothesis Let us suppose that we are given a theory in which the dynamical variables are N × N dimensional Hermitian matrices Mi , with an action S[Mi ] that is invariant under the global U (N ) transformation Mi → U Mi U + . We consider the correlation functions of U (N ) invariant observables. If we have a denumerable set of variables, then the most general invariant is a function of traces of products of the matrices, i.e., O = 1 1 N T r[Mi1 , MI2 · · · Min ], where N T r is the normalized trace on Hermitian matrices. We denote the expectation value of O as Z Y   1 −1 hOi ≡ Z dMi e−S[Mi ] T r Mi1 Mi2 · · · Min , N i RQ where Z = dMi exp(−S[Mi ]). i

In this context we have the important “factorization hypothesis” that in the large N limit, the expectation value of a product of such invariant observables factorizes ([5, 8]): + * k   k Y Y 1 Oi = hOi i + O . (1) N2 i=1

i=1

This asymptotic property is suggested by an analysis of the Feynman diagrams in perturbation theory. It should be noted, however, that (1) is not true for arbitrary S[Mi ] (see [7]). In the following we will prove this hypothesis for various cases. We begin by introducing the fat graphs which will be important in Sect. 3. 2.1. Preliminaries on fat graphs. Fat graphs were introduced in [9]. We shall present some of their basic properties. The reader may find more details and some elegant applications of fat graphs to the theory of Riemann surfaces in [9]. By a graph G we simply mean a one-dimensional CW-complex, so that G consists of vertices and open edges. The boundary of an edge may consist of a single vertex, and two distinct edges may have the same vertices. An edge of the first barycentric subdivision of G is called a half-edge of G. The valence of a vertex v is the number of half-edges incident on it, and we say v is k-valent if this number is k. A fattening of G is a specification of cyclic order on the half-edges of G incident on v for each vertex v of G, and a graph together with a fattening is called a fat graph. In practice, we may specify a fat graph by drawing a planar projection of a graph embedded in space: a neighborhood of the vertex set is required to be embedded in the plane, and the collection of half-edges incident on a vertex inherits a cyclic (say counter-clockwise) order from the orientation of the plane; away from the vertices, we introduce under/over crossings. We illustrate these notions below. We note that the first two examples correspond to distinct fattenings of a common underlying graph. Any fat graph may be so represented.

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A fat graph G determines an oriented surface F (G) with spine G ⊂ F (G) (i.e., the inclusion is a homotopy equivalence) as follows. Suppose G is represented by a planar projection and let V denote a regular neighborhood of the vertex set in the plane. V inherits an orientation from that of the plane, and if e is an edge of G, then we attach a band to V between the endpoints of e preserving orientation to construct the surface. The following are examples of F (G) corresponding to the above examples of G.

After filling the boundaries of F (G) (a union of circles) with disks, we obtain an oriented surface without boundary which is denoted by F 0 (G). Next we shall introduce an important class of fat graphs. A conjugacy class of symmetric group Sq on q letters is specified by a sequence of nonnegative P integers m1 , . . . , mq , where mk (1 ≤ k ≤ q) is the number of k cycles. We have 1≤i≤q imi = q. Let m = (m1 , m2 , . . . , mq ). For σ ∈ Sq , denote by [σ] the conjugacy classes of σ

[σ] = (σ1 , . . . , σq ),

(σ) =

q X

σi .

i=1

Notice that (σ −1 ) = (σ). Let σ, τ ∈ Sq . Fix Z = (123 · · · q) ∈ Sq and two sets of indices A1 , A2 , ...Aq and B1 , B2 , ...Bq . We label 2q points clockwise on a unit circle in this order by B1 , A1 , B2 , A2 , . . . , Bq , Aq . Each vertex is endowed with a small disk centered at the vertex such that disks with different centers do not intersect. Inside each such disk we draw two different vectors which are outside the unit circle with initial points at the center, and the vector to the left (resp. to the right) is called the left vector (resp. the right vector). Thus each vertex is endowed with an additional structure: a left vector and a right vector. Outside the circle, we use τ −1 ∈ Sq to connect B1 , B2 , ...Bq in the following way: first represent τ −1 uniquely in terms of products of different cycles; for each cycle (i1 ...ik ) of τ −1 , we draw an edge e1 connecting the right vector of Bi1 with the left vector of Bi2 , then e2 connecting the right vector of Bi2 with the left vector of Bi3 ,..., and finally ek connecting the right vector of Bik with the left vector of Bi1 . When k = 1 we simply connect the right vector of Bi1 to its left vector. The following is an example for the case q = 4, τ −1 = (1423).

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Similarly we use σ ◦ Z to connect A1 , A2 , . . . , Aq . All the edges intersect only at the vertices A’s, B 0 s . The following is an example for the case q = 2, τ = (12) and σ = (12).

These graphs, with a clockwise cyclic order from the orientation of the plane, are fat graphs defined above. Denote by Gσ,τ the corresponding oriented surface without boundary . It is easy to see, for the case q = 2, τ = (12) and σ = (12), Gσ,τ is a sphere. Let us denote the number of vertices, edges, and circles of the fat graph by V, E, F respectively. The genus g(Gσ,τ ) of the Riemann surface Gσ,τ is given by 2 − 2g(Gσ,τ ) = V − E + F Clearly V = 2q. There are two different kinds of edges. There are 2q edges coming from the boundary of the circle. There are also edges coming from connecting A’s and B’s. Each i-cycle of τ (resp. σ ◦ Z) gives i (resp. i) edges. So the number of edges from connecting B’s are P P i iτi = q, and the number of edges from connecting A’s are given by i i(σ ◦ Z)i = q, hence E = 4q. Due to our construction, there is one circle inside the unit circle. There are two kinds of circles outside the unit circle which are discussed below. Consider a regular neighborhood Xi (resp. Yi ) of a vertex Bi (resp. Ai ), and label the left-most boundary line and the right-most boundary line of Xi outside the unit circle by i0 and i. The left-most boundary line (resp. the right-most boundary line) is on the left (resp. the right) of the left (resp. the right) vector. Then the left-most boundary line and the right-most boundary line of Yi outside the unit circle inherited labels i and

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291

(i + 1)0 respectively. If a circle pass through the right-most boundary line of some Xi , it will be joined to τ −1 (i) which is necessarily the left-most boundary line of Xτ −1 (i) . The left-most boundary line of Xτ −1 (i) is connected to the right-most boundary line of YZ −1 ◦τ −1 (i) , and it will be connected to the left-most boundary line of Yσ◦τ −1 (i) , which is connected to the right-most boundary line of Xσ◦τ −1 (i) . Hence such circles are in one-to-one correspondence to different cycles in σ ◦ τ −1 . If a circle does not pass through the right-most boundary line of any Xi , then it is easy to see that it will not pass through the right-most or left-most boundary line of any Xi ’s and Yi ’s. Such circles are in one-to-one correspondence with different cycles of τ −1 and σ ◦ Z. It follows from the above that (recall (τ −1 ) = (τ )) F = 1 + (σ −1 ◦ τ ) + (τ ) + (σ ◦ Z). Putting everything together, we have proved the following lemma: Lemma 2.0. 2 − 2g(Gσ,τ ) = −2q + 1 + (σ −1 ◦ τ ) + (τ ) + (σ ◦ Z). 2

2.2. Gaussian case. The set of N × N Hermitian matrices M is a Euclidean space RN , on which one introduces a translation invariant measure dM normalized so that   Z T rM 2 =1 dM exp − 2 In this expression T r is the usual trace with T r(id) = N (id denotes the identity matrix). We denote trace by hM i = N1 T r(M ). We will denote the measure   the normalized 2

by d1 M ; d1 M is called a Gaussian measure. dM exp − T rM 2 Let ki ≥ 0 (i = 1, . . . , s). The following lemma is due to ’t Hooft. We shall present the proof as in [8]. Such a proof contains important diagrammatic ideas which have important applications to the theory of Riemann surfaces (see [9]). Lemma 2.1. * *  ki + Y  ki + Z s s Z Y M M √ √ d1 M · = + N −2 · f{ki } d1 M N N i=1 i=1 where limN →+∞ f{ki } exists and is bounded. Proof. To begin with, by completing the square, one notices that Z d1 M exp(M J) = exp(T r(J 2 )/2) for any N × N hermitian matrix Hence

J.



Z d 1 M M i1 j 1 · · · M in j n

 Z ∂ ∂ d1 M exp(M J) = ··· ∂Jj1 i1 ∂Jjn in   ∂ ∂ 2 = ··· exp(T r(J )/2) . ∂Jj1 i1 ∂Jjn in J=0

(2)

Each derivative ∂J∂ij either “creates” or “annilates” a factor of Jji . Since one is to set J = 0 at the end of the computation, every factor of Jij that is created must be annihilated, so that (2) involves a sum over pairings.

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It is natural to represent such a pairing by a double line (a band) connecting two vertices. The two edges of the band correspond to the two indices of the matrix Jij , and the N possible labels of each edge correspond to the N possible values of the corresponding index.

Let us use these to calculate the left hand side (LHS) of the lemma. Notice T rM k gives a k-valent vertex. For example, T rM 4 = Mij Mjk Mk` M`i , in double line notation, this corresponds to:

Hence a graph corresponding to the LHS of the lemma consists of s vertices of valence k1 , . . . , kP s respectively. This is a fat graph as defined in Sect. 2.0. The number s of edges is E = i=1 ki /2. Each circle in the graph contributes a factor of N . Upon filling the circles with discs, we obtain a surface S with a simplicial decomposition. Denote by F the number of circles. Suppose S is the union of ` disjoint connected surfaces Si of genus gi (1 ≤ i ≤ ` ≤ s), then: s−

s X

ki /2 + F =

i=1

` X

(2 − 2gi )

(3)

i=1

such a graph contribute NF−

P

ki /2−s

P` =N

i=1

(2−2gi )−2s

.

P`

Notice i=1 (2 − 2gi ) − 2s ≤ 0 (remember ` ≤ s) with equality if and only if gi = 0 for all i and ` = s. Hence the dominant contribution (the N 0 part) to the LHS comes from graphs which consists of s disconnected planar graphs each of which contains exactly one vertex. It is easy to see that this implies * *  ki + Y  ki + Z s s Z Y M M √ √ d1 M − = N −2 f{ki } . d1 M N N i=1 i=1 The fact f{ki } has a bounded limit as N → +∞ follows from the fact that it is a finite Q.E.D. sum of polynomials of N −2 with fixed coefficient (depending on {ki }). R 2.3. Haar case. Let dU be the Haar measure on U (N ) normalized such that dU = 1. We will be interested in the following integral: Z Ip,q = dU Ui1 j1 Ui2 j2 · · · Uiq jq Uj∗0 i0 · · · Uj∗p0 i0p . 1 1

Since dU is invariant under action of U (N ), we may replace U by eiθ U where eiθ is a fixed element of the unit circle without changing the value of Ip,q . Hence Ip,q is zero unless p = q.

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The asymptotic expansion of Ip,q in Lemma 2.2 below is essentially due to D. Weingarten (see [2] where the leading order term is stated). We shall give a proof following his elegant argument for the SO(N ) case. Let us first do some preparations. Recall from Sect. 2.0 that a conjugacy class of integers symmetric group Sq on q letters is specified by a sequence of nonnegative P m1 , . . . , mq , where mk (1 ≤ k ≤ q) is the number of k cycles. We have 1≤i≤q imi = q. The set of such sequences is denoted by Cq and weP use |Cq | to denote the number of elements in Cq . Let ξi be a basis in Rq . Define m = 1≤i≤q mi ξi . We let Vs (resp. V¯s , Vs0 , V¯s0 ) be a complex N dimensional vector space, 1 ≤ s ≤ q. We define Ws = Vs ⊗ V¯s , Ws0 = Vs0 ⊗ V¯s0 . and we label a basis of Ws (resp. Ws0 ) by eis ⊗ e¯i0s , 1 ≤ is , i0s ≤ N (resp. e0js ⊗ e¯0js0 , 1 ≤ js , js0 ≤ N ). We define a right action of U ∈ U (N ) on Ws and a left action of U ∈ U (N ) on Ws0 as follows: πs (U )(eis ⊗ e¯i0s ) = πs0 (U )(e0js ⊗ e¯0js0 ) =

X ik ,i0t

X

jk ,jt0

Uis ik eik ⊗ Ui∗0t i0s e¯i0t , Ujk js e0jk ⊗ Uj∗s0 jt0 e¯0jt0 .

We let V = W1 ⊗ ... ⊗ Wq , V 0 = W10 ⊗ ... ⊗ Wq0 and W (q) = V ⊗ V 0 . For U ∈ U (N ), we define: πV (U ) = π1 (U ) ⊗ ... ⊗ πq (U ), πV0 (U ) = π10 (U ) ⊗ ... ⊗ πq0 (U ) πW,r (U ) = πV (U ) ⊗ id, πW,l (U ) = id ⊗ πV0 (U ). πW,r (resp.πW,l ) defines a right (resp.left) action of U (N ) on W. We define the action of U (N ) on C to be the trivial action. We will use Hom(W, C) to denote those linear maps from W to C which commutes with the left and the right action of U (N ). For σ, τ ∈ Sq , define g(σ, τ ) : W → C by: g(σ, τ )(ei1 ⊗ e¯i01 ⊗ ...eiq ⊗ e¯i0q ⊗ e0j1 ⊗ e¯0j 0 ⊗ ... ⊗ e0jq ⊗ e¯0jq0 ) 1

= δi1 i0σ(1) · · · δiq i0σ(q) · δj1 jτ0 (1) · · · δjq jτ0 (q) . By the Frobenius-Schur duality between tensor product representations of U (N ) and the symmetric group (see, for example, Chapter 4 of [13]) , it follows that if N > q, the set {g(σ, τ ), σ, τ ∈ Sq } is a basis in Hom(W, C) . Let us define Iq : W → C by Iq (ei1 ⊗ e¯i01 ⊗ ...eiq ⊗ e¯i0q ⊗ e0j1 ⊗ e¯0j 0 ⊗ ... ⊗ e0jq ⊗ e¯0jq0 ) = Iq,q . 1

It follows from the invariance of Haar measure with respect to the left and right action on U (N ) that Iq ∈ Hom(W, C). Hence there exists f (σ, τ ) ∈ C such that Iq =

X

f (σ, τ )g(σ, τ ).

σ,τ ∈Sq

To determine f (σ, τ ), let us define two actions of Sq on W (q) as follows: for any δ ∈ Sq ,

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δ(ei1 ⊗ e¯i01 ⊗ ...eiq ⊗ e¯i0q ⊗ e0j1 ⊗ e¯0j 0 ⊗ ... ⊗ e0jq ⊗ e¯0jq0 ) 1

= eiδ(1) ⊗ e¯i01 ⊗ ...eiδ(q) ⊗ e¯i0q ⊗ e0jδ(1) ⊗ e¯0j 0 ⊗ ... ⊗ e0jδ(q) ⊗ e¯0jq0 1

δ 0 (ei1 ⊗ e¯i01 ⊗ ...eiq ⊗ e¯i0q ⊗ e0j1 ⊗ e¯0j 0 ⊗ ... ⊗ e0jq ⊗ e¯0jq0 ) 1

= ei1 ⊗ e¯i0δ(1) ⊗ ...eiq ⊗ e¯i0δ(q) ⊗ e0j1 ⊗ e¯0j 0 ⊗ ... ⊗ e0jq ⊗ e¯0j 0 . δ(1)

δ(q)

It follows directly from the definition of Iq and g(σ, τ ) that Iq ◦ δ = I q , I q ◦ δ 0 = I q , g(σ, τ ) ◦ δ = g(σ ◦ δ, τ ◦ δ), g(σ, τ ) ◦ δ 0 = g(δ −1 ◦ σ, δ −1 ◦ τ ). Since g(σ, τ ) is a basis in Hom(W, C) for N > q , we have for N > q and any σ, τ, δ ∈ Sq , f (σ, τ ) = f (σ ◦ δ, τ ◦ δ) = f (δ ◦ σ, δ ◦ τ ). So f (σ, τ ) depends only on [τ ◦ σ −1 ] = m. We define fq (m) = f (σ, τ ) and X Lq (m) = g(σ, τ ). σ,τ ∈Sq ,[τ ◦σ −1 ]=m

˜ = W1 ⊗ ... ⊗ Wq ⊗ W 0 ⊗ ... ⊗ W 0 . To further determine fq (m), let us We let W 1 q−1 ˜ → W (q) and i : W ˜ → W (q − 1) as follows: for any introduce linear maps j : W 0 0 ˜ , a ∈ W , x ∈ W1 ⊗ ... ⊗ Wq−1 , y ∈ W1 ⊗ ... ⊗ Wq−1 X j(a) = a ⊗ ( e0js ⊗ e¯0js ), i(x ⊗ eis ⊗ e¯i0s ⊗ y) = δis i0s x ⊗ y. js

It is straightforward to check that Iq ◦ j = Iq−1 ◦ i. ˜ , C). Moreover, Iq−1 ◦ i ∈ Hom(W Let us suppose that we are given σ ∈ Sq , τ ∈ Sq−1 . We will denote by h(σ, τ ) : ˜ → C the mapping defined as follows: W h(σ, τ )(ei1 ⊗ e¯i01 ⊗ ...eiq ⊗ e¯i0q ⊗ e0j1 ⊗ e¯0j 0 ⊗ ... ⊗ e0jq−1 ⊗ e¯0j 0 ) 1

q−1

= δi1 i0σ(1) · · · δiq i0σ(q) · δj1 jτ0 (1) · · · δjq−1 jτ0 (q−1) . ˜ , C) if N > q (see Chapter 4 of h(σ, τ ), σ ∈ Sq , τ ∈ Sq−1 forms a basis of Hom(W [13]). Expanding Iq ◦ j and Iq−1 ◦ i in terms of h(σ, τ ), we have: X X Iq ◦ j = (N fq (m) + fq (m00 ))h(σ, τ ) bσ,τ ∈Sq ,τ (q)=q, [τ ◦σ −1 ]=m∈Cq

Iq−1 ◦ i =

X

1≤k1 ≤q−1, [(k1 q)τ ◦σ −1 ]=m"

fq−1 (m0 )h(σ, τ ),

σ,τ ∈Sq ,σ(q)=q [τ ◦σ −1 ]=m0 ∈Cq−1

where any element τ ∈ Sq , τ (q) = q is also considered as an element of Sq−1 in the usual way.

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Now let us fix m ∈ Cq . If m1 > 0, we may choose σ, τ ∈ Sq , σ(q) = τ (q) = q such that [τ ◦ σ −1 ] = m. Comparing the coefficients of h(σ, τ ) from above we have: X fq (m00 )a(m00 ) = fq−1 (m − ξ1 ), (a) N fq (m) + m00

where a(m00 ) is the number of (k1 q)τ (k1 6= q) such that [(k1 q)τ ◦ σ −1 ] = m00 . The explicit form of a(m00 ) is not important to us. It is important, however, that a(m00 ) is a finite number which is independent of N . Recall that every element in Sq can be written uniquely as a product of disjoint cycles. Since τ σ −1 = (q)(k1 ...x)..., where the dots indicate the cycles which do not contain any elements that have already appeared, we have (k1 q) ◦ τ σ −1 = (qk1 ...x)... . If (k1 ...x) is a k − 1, (k > 1)-cycle, then k1 appears in a k-cycle in (k1 q) ◦ τ ◦ σ −1 . Meanwhile a 1-cycle (q) and a (k − 1)-cycle (k1 ...x) disappear in (k1 q) ◦ τ σ −1 . So m00 = m − ξ1 − ξk−1 + ξk . We define P fq (m) = N −2q+ i mi f¯q (m). Equation (a) can now be written as X f¯q (m) + N −1 a(m00 )f¯q (m − ξ1 − ξk−1 + ξk ) = f¯q−1 (m − ξ1 ).

(a0 )

2≤k≤q

If m1 = ... = ml−1 = 0, ml > 0, l > 1, choose σ, τ ∈ Sq , τ (q) = q such that [τ ◦ σ −1 ] = m, and the cycle in τ ◦ σ −1 which contains q is a l-cycle. By comparing the coefficients of h(σ, τ ) above we have: X b(m00 )fq (m00 ) = 0, (b) N fq (m) + m00

where b(m00 ) is some finite number independent of N . We would like to express m00 in terms of m. There are two cases to consider. If k1 is in a k-cycle different from the l-cycle containing q in τ ◦ σ −1 , then (k1 q) ◦ τ ◦ σ −1 = (k1 q) ◦ (k1 ...x)(q...y)... = (k1 ...xq...y). It is clear that

m00 = m − ξl − ξk + ξk+l .

If k1 , q are in the same l-cycle of τ ◦ σ −1 , it follows from (k1 q)(k1 ...q) = (q)(k1 ...),(k1 q)(q...k1 ) = (k1 )(q...) (k1 q)(k1 ...xq...y) = (k1 ...x)(q...y) that m00 = m − ξl + ξk + ξl−k for some 1 ≤ k ≤ l − 1. We can now write (b) in terms of f¯ as follows: X X bk f¯q (m−ξl −ξk +ξk+l )+ bk f¯q (m−ξl +ξk +ξl−k ) = 0. (b0 ) f¯q (m)+N −1 l≤k≤q−l

1≤k≤l−1

Equations (a’) and (b’) can be written as linear equations:

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X

Amp f¯q (p) = Bm ,

p∈Cq

where Bm is a linear combination of f¯q−1 ’s with known coefficients and (Amp ) is a |Cq | × |Cq | matrix with coefficients which are polynomials in N −1 . We claim det(Amp ) is nonzero if we set N −1 = 0. In fact this statement is equivalent to saying that for any given f¯q−1 (m − ξ1 ), the Eqs. (a’) and (b’) have a unique solution if we set N −1 = 0. When N −1 = 0, (a’) and (b’) become: f¯q (m) = f¯q−1 (m − ξ1 ), X f¯q (m) = − bk f¯q (m − ξl + ξk + ξl−k ),

(c) (d)

1≤k≤l−1

where (d) is valid for m1 = ...ml−1 = 0, ml > 0, l > 1. By (c) we can determine f¯q (m) uniquely for any m1 > 0. By (d) we can determine f¯q (m) uniquely when m1 = ...ml−1 = 0, ml > 0, l > 1 from fq (m − ξl + ξk + ξl−k ), 1 ≤ k ≤ l − 1. It follows that (c) and (d) uniquely determine f¯q (m) for any given f¯q−1 (m − ξ1 ). We conclude that det(Amp ) is nonzero if N is sufficiently large , and we can inductively determine f¯q (m) by using (a’) and (b’) starting from f¯1 (1) = 1. It is clear that limN →+∞ f¯q (m) exists for all q, m. Slightly abusing our notation, we shall also use Lq (m) to denote the following: X δi1 i0σ(1) · · · δiq i0σ(q) · δj1 jτ0 (1) · · · δjq jτ0 (q) , Lq (m) = σ,τ ∈Sq [σ −1 ◦τ ]=m

where the δ’s are Kronecker deltas and we have suppressed the dependence on the indices i, i0 , j, j 0 ’s. We have therefore proved the following lemma: Lemma 2.2. Let N be sufficiently large, then X −2q+Pq m i ¯ i=1 fq (m)Lq (m). Iq,q = N m

Here f¯q (m) is independent of i1 , ...iq , i01 , ...i0q , j1 , ...jq and j10 , ...jq0 . Moreover limN →+∞ f¯q (m) exists for all q, m. Remark 1. It is mentioned in [2] that analogous results can be proved for SP (N ) by essentially the same arguments used for SO(N ). R R R We note that dU N1 T r(U m ) = dU N1 T r((eiθ U )m ) = eimθ · dU N1 T r(U m ) for R any eiθ ∈ S 1 . Hence dU N1 T r(U m ) = δm,0 . We are now ready to prove the following: P Lemma 2.3. Let p1 , · · · , pt and q1 , · · · , qs be positive integers such that 1≤i≤t pi = P 1≤j≤s qj = q. Then: Z

1 1 1 1 1 dU T r (U p1 ) · T r (U p2 ) · · · T r (U pt ) · T r U −q1 · · · T r U −qs N N N N N = N −2 g(p1 , · · · pt ; q1 , · · · qs ), where limN →∞ g(p1 , · · · pt ; q1 , · · · qs ) exists and is bounded.

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297

Proof. We can expand the LHS of the above lemma using the definition of trace. Each term in the expansion consists of q U 0 s and q U ∗ ’s. By using Lemma 2.2, each such term can be further expanded as sum of the following terms: Pq N −2q+ i=1 mi f¯q (m)Lq (m), where Lq (m) is defined as in Sect. 2.2. The index structure of the Kronecker deltas in Lq (m) can be conveniently represented by a diagram. Each such diagram g consists of t white vertices of valence p1 , · · · pt and s black vertices of valence q1 , · · · qs , and the edges from each white vertex is connected to a black vertex. The edges of each vertex are labeled by matrix indices consistent with the structure of the vertex as in the proof of Lemma 2.1, and there is an index (the index takes value from 1 to N ) running through every circle in the diagram. The way that edges are connected is specified by the permutations σ and τ in Lq (m). The following is an example of such a diagram in the case t = 1, s = 1, p1 = q1 = 2, σ = (12), τ = identity with one white vertex and one black vertex.

In the above diagram the conditions j1 = i2 , j2 = i1 , j10 = i02 , j10 = i02 are required by the consistency of the vertex structure. We note that such a diagram is in general not a fat graph. The number of edges in the diagram is 2q. We denote by F the number of circles in the diagram. Since each edge connects two different vertices, it follows that each circle in the diagram contains at least two edges. Therefore F ≤ q since different circles pass through different edges. Notice each vertex contribute a factor N −1 , and each circle contribute a factor of N . By Lemma 2.2, the highest power of N in such a term is N −q−s−t+F . Since F ≤ q, it follows that the highest power of N on the expansion of the left hand side of Lemma 2.3 is −2. Therefore the LHS of the above lemma is equal to N −2 g(p1 , . . . , ps ; q1 , . . . , qt ). That limN →+∞ g(p1 , . . . , ps ; q1 , . . . , qt ) exists and is bounded follows from the fact that g(p1 , . . . , ps ; q1 , . . . , qt ) is a finite polynomial in N −1 with fixed coefficients which depend on (p1 , . . . , ps ; q1 , . . . , qt ).  Remark 2. By using analogous results Lemma 2.2 for SO(N ), SP (N ) cases (see Remark 1), one can prove the factorization hypothesis for SO(N ), SP (N ) cases. Remark 3. For much more stronger results than Lemma 2.3, see [11]. However, lemma 2.3 will be sufficient to prove asymptotic freeness in the Haar measure case in Sect. 3 (see Remark 8 in Sect. 3). The next lemma will play an important role in Subsect. 3.2. We begin by introducing some additional notations. Let us suppose that we are given σ ∈ Sq with [σ] = (σ1 , . . . , σq ) the conjugacy classes of σ (see Subsect. 2.0). We may represent σ as a unique product of different cycles:

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σ=

q Y σk Y

aj,k

k=1 j=1

with each aj,k a k-cycle. For a set of N ×N matrices M1 , ...Mq and a k-cycle a = (i1 ...ik ) with 1 ≤ i1 , i2 , ...ik ≤ q, we define hM ia = hMi1 ...Mik i and hM iσ =

q Y σk Y

hM iaj,k .

k=1 j=1

For an example, if q = 3 and σ = (1)(23), then hM iσ = hM1 ihM2 M3 i. Now we are ready to prove the following lemma: Lemma 2.4. Let A1 , A2 , ...Aq and B1 , B2 , ...Bq be N × N matrices, and let dU be the normalized Haar measure on U (N ) as in Subsect. 2.2. Assume N is large enough, then Z dU hU B1 U ∗ A1 U B2 U ∗ A2 · · · U Bq U ∗ Aq i X X f¯q (m) N −2g(Gσ,τ ) · hBiτ −1 · hAiσ◦Z , = σ,τ ∈Sq ,[σ −1 ◦τ ]=m

m

where g(Gσ,τ ) is as in Lemma 2.0 and f¯q (m) is as in Lemma 2.2. Proof. By definition, hU B1 U ∗ A1 U B2 U ∗ A2 · · · U Bq U ∗ Aq i 1 X Ui1 j1 (B1 )j1 j10 Uj∗0 i0 (A1 )i01 i2 Ui2 j2 (B2 )j2 j 0 362 Uj∗0 i0 (A2 )i02 i3 · · · (Aq )i0q i1 , = 1 1 2 2 N 0 0 i,j,i ,j

P where we have used i,j,i0 ,j 0 to denote a summation over all i1 , ...iq , j1 , ...jq , i01 , ...i0q , j10 , ...jq0 . After integration and using Lemma 2.2, we have: Z dU hU B1 U ∗ A1 U B2 U ∗ A2 · · · U Bq U ∗ Aq i X −2q−1+P m X i ¯ i fq (m) = N Lq (m)(B1 )j1 j10 · · · (Aq )i0q i1 m

=

X m

N −2q−1+

i,j,i0 ,j 0

P i

mi

f¯q (m)·

X

i,j,i0 ,j 0 ,σ,τ ∈Sq ,[σ −1 ◦τ ]=m

Let us compute

δi1 ,i0σ(1) · · · δjq jτ0 (q) (B1 )j1 j10 · · · (Aq )i0q i1 .

Random Matrix Model from 2D Yang-Mills Theory

N −2q−1+

P i

X

mi

i,j,i0 ,j 0

299

δi1 i0σ(1) · · · δjq jτ0 (q) (B1 )j1 j10 · · · (Aq )i0q i1 −

By using the notation introduced in this section, we have: P X δi1 i0σ(1) · · · δjq jτ0 (q) (B1 )j1 j10 · · · (Aq )i0q i1 N −2q−1+ i mi =N

−2q−1+

i,j,i0 ,j 0

P

P

i

mi +

i

P

τi +

i

σi

hBiτ −1 hAiσ◦Z . 

By using Lemma 2.0 , the proof of the lemma immediately follows.

c2 (α)A P 2.4. Heat kernel case. Let k(U ; A) = dα χα (U )e− 2N be the heat kernel on U (N ), where dα is the dimension of the irreducible representation α of U (N ) and C2 (α) is the the measure on U (N ). We will use casmir of α, A > 0. Denote by d2 A = k(U ; A)dU R hW i to denote N1 T rW and hhW ii to denote d2 AhW i.

Lemma 2.5. Let k0 , k1 , . . . , kn be positive integers . Then (1) hhU k1 ihU k2 i · · · hU kn ii = hhU k1 ii · hhU k2 ii · · · hhU kn ii + f (k1 , . . . , kn ; N, A), where f (k1 , . . . kn ; N, A) converges uniformly to 0 with respect to A as N goes to +∞. (2) hhU k0 ii converges uniformly to Pk0 (A) with respect to A as N goes to +∞, where Pk0 (A) is uniquely determined by the following recursion relation: kX 0 −2 k0 d Pk (A) = Pk (A) + (k0 − ` − 1)P`+1 (A)Pk0 −`−1 (A) − dA 0 2 0 `=0

P0 (A) = 1, Pk0 (0) = 1 for all k0 ≥ 0, P1 (A) = e

−A/2

(4)

.

Proof. Let λα be a basis of N × N Hermitian matrices such that T rλα λβ = 21 δ αβ , it follows X δad δbc . (5) (λα )ab (λα )cd = 2 α For an arbitrary function f (U ) on U (N ), the Laplacian operator 1 is defined by    X X ∂2 β    x λ U exp i (1f )(U ) = β . ∂x2 x=0 α

(6)

β

d 1 d By using the heat equation − dA k(U ; A) = 2N 1k(U ; A), one has − dA hhU k1 i · · · 1 hU ks ii = 2N h1(hU k1 i · · · hU ks i)i. Let us first calculate 1hU k1 i. By using (5) and (6), one has

1hU k1 i = k1 N hU k1 i +

kX 1 −2 `=0

Hence

(k1 − ` − 1)N hU `+1 ihU k1 −`−1 i.

(7)

300

F. Xu s
X
1 hU k1 i · · · hU ks i = hU k1 i · · · 1hU ki i · · · hU ks i i=1

X

+

hU i · · · hU ki λα i · · · hU kj λα i · · · hU ks i. k1

1≤i6=j≤s,α

We note that

X

hU ki λα i · hU kj λα i

α

= =

1 N2

X

(U ki )i1 i2 (λα )i2 i1 · (U kj )i3 i4 (λα )i4 i3

(i1 ,i2 ,i3 ,i4 ,α)

1 X ki (U )i1 i2 (U kj )i2 i1 2N 2 i ,i 1

(8)

2

1 hU ki +kj i. = 2N Therefore s 1 X d hhU k1 i · · · hU ks ii = hhU k1 i · · · (1hU ki i) · · · hU ks ii dA 2N i=1 X 1 k1 + hhU i · · · hU ki +kj i · · · hU ks ii 2N 1≤i6=j≤s ÿ * ! kX s i −2 X ki k i k1 `+1 ki −`−1 hU i + hU i · · · (ki − ` − 1)hU i · hU i ··· = 2 i=1 `=0  X 1 hhU k1 i · · · hU ki +kj i · · · hU ks ii. hU ks i + 2N

−

Ps

(9)

1≤i6=j≤s

For fixed i=1 ki = m, (9) givesPa system of first order ordinary differential equations s in variables (hhU k1 i · · · hU ks ii; i=1 ki = m is fixed) which we denote by vector µ(A). Assume µ(A) has `(m) components. Then (9) can be written as dµ = 0µ, dA where 0 is a `(m) × `(m) matrix whose entries are finite polynomials of N −1 . It follows limn→+∞ 0 = 01 . We notice that µ(0) = (1, 1, . . . , 1), and µ(A) = e0A µ(0). Therefore µ(A) converges uniformly with respect to A to µ1 (A) = e01 A µ(0). Let us denote the components of µ1 (A) by hhU k1 i · · · hU ks ii∞ . It follows from (9) that: d hhU k1 i · · · hU ks ii∞ dA * ÿ ! kX s i −2 X k i hU ki i + hU k1 i · · · = (ki − ` − 1)hU `+1 ihU ki −`−1 i 2 i=1 `=0  ks · · · hU i .

−

(10)

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301

Let µ2 (A) be a vector with components Pk1 (A) · · · Pks (A). It follows from (4) that d Pk (A) · · · Pks (A) dA 1 ÿ ! kX s i −2 X ki Pk (A) + Pk1 (A) · · · (ki − ` − 1)P`+1 (A) · Pki −`−1 (A) . = 2 i

−

i=1

`=0

i µ1 (A), µ2 (A) satisfies the same equation dµ dA = 01 µi . Notice µ2 (0) = (1, . . . , 1) = µ1 (0). Hence µ1 (A) = µ2 (A). We have thus proved that for any non-negative integers k1 , . . . , ks hhU k1 i · · · hU ks ii converges uniformly (with respect to A) to BPk1 (A) · · · Pks (A). It is easy to see that this is equivalent to (1) and (2) of the lemma. 

Remark 4. The differential equation in part (2) of the lemma above appeared in [4]. One can prove similar statements (with different Pk (A)’s) for SO(N ), SP (2N ) cases by the same method.

3. Basic concepts We introduce some basic concepts about free random variables. The reader is encouraged to consult the excellent exposition [1]. Definition. Let (Gi )i∈I be a family of sets. An alternating word W in (Gi )i∈I is a monomial g1 g2 ...gn with gj ∈ Gij and i1 6= i2 6= · · · 6= in . The gi ’s are called the components of W . Definition. Let (G, φ) be a noncommutative probability space. A family, (Gi )i∈I of unital subalgebras of G is called free if φ(W ) = 0 for any alternating word g1 g2 ...gn in (Gi )i∈I with φ(gi ) = 0 for all 1 ≤ i ≤ n. We will also need the definition of limit distribution and asymptotic freeness. Denote by ChXi | i ∈ Ii the set of non-commutative polynomials. Definition. For each n ∈ N , let (Ti(n) )i∈I be a family of noncommutative random variables in a non-commutative probability space (Gn , φn ). Then the sequence of joint distributions µ(Ti(n) )i∈I converges as n → +∞, if there exists a distribution µ such that µ(Ti(n) )i∈I (p) → µ(p) as n → +∞ for every non-commutative polynomial p in (Ti(n) )i∈I . We call µ the limit distribution. Definition. Let I = ∪j∈J Ij be a partition of I. A sequence of families ({Ti(n) | i ∈ Ij })j∈J of sets of noncommutative random variables is said to be asymptotically free as n → +∞ if it converges and if ({Xi | i ∈ Ij })j∈J is a free family of sets of random variables in (ChXi | i ∈ Ii, µ). We will be interested in the following S-transform. Let σ ∗ = {µ : C[x] → C | µ linear, µ(1) = 1, µ(x) 6= 0}. For µ ∈ σ ∗ , consider the formal power series ψµ , χµ , and Sµ such that

302

F. Xu

ψµ (z) =

∞ X

µ(xk )z k ,

χµ (ψµ (z)) = ψµ (χµ (z)) = z,

k=1

Sµ (z) = χµ (z)z −1 (1 + z). Sµ is called the S-transform of µ. S-transform has the property that for free multiplicative convolution µ  ν (for definition, see Chapter 3 of [1]), Sµν = Sµ Sν . A It is proved in [10] that Sµa (z) = e 2 (1+2z) for A > 0 is the S-transform of an infinite divisible measure µA on the unit circle. Such a measure is an analogue of Gaussian measure on the circle. Notice µA (X −n ) = µA (X n ) for n ≥ 0. Let us calculate the A A z e 2 (1+2z) . Since ψµ (z) = moments of µA . (See [3].) From SµA (z) = e 2 (1+2z) , χA (z) = 1+z P∞ k k k=1 µ(x )z , it follows: Z 1 k ψµ (e−iθ )einθ dθ. µA (X ) = 2π We have, with χA (z) = e−iθ , I dz z[χA (z)]−(n+1) χ0A (z) µA (X k ) = 2πi I 1 dz [χA (z)]−n = n 2πi  n I A 1 1 dz 1+ = e−n· 2 (1+2z) . n 2πi z In particular, µ0 (X k ) = 1. P∞ To find a recursion relation between µA (X k ),let us define R(eiθ , A) = n=1 µA (X n ) ∂F e−i(n+1)θ . Then F (θ, A) = i eiθ R(eiθ , A) − 21 and F (θ, A) satisfies ∂F ∂A + F ∂θ = 0 (see p. 44 of [3]). Expanding this equation in powers of eiθ , we obtain the following recursion relations: X n dµA (X n ) =− mµA (X m )µA (X n−m ) + µA (X n ). dA 2 n−1

m=1

n

We notice that µ0 (X ) = 1. Compared to the equations and initial conditions for Pn (A) in Lemma 2.4, we have Pn (A) = µA (X n ) for n ≥ 0. But we also have hhU −n ii = hhU n ii. It follows from Lemma 2.4 that the limit distribution of {(U )n } is µA . We record this result in the following corollary. Corollary 3.1. For A > 0, hh(U )n ii converges to µA (X n ) for all integer n, where µA A is the infinite divisible measure on the unit circle with S-transform SµA (z) = e 2 (1+2z) . 3.1. Relative angular integrals. Let us first introduce some notation. We let I = {0, 1, 2, ..., n} with n a non-negative integer. Let us suppose that we are given two sets of N × N matrices M1 , M2 , ...Ms and P1 , P2 , ...Pn and a set of N × N unitary matrices U1 , U2 , ...Un . We define Gi to be the linear span of the following set: {Ui (Pi )j Ui∗ , j ≥ 0, j ∈ Z}, where i = 1, 2, ...n. We let G0 to be the set of all noncommutative polynomials in M1 , M2 , ...Ms . By slight abuse of notation, we enumerate the elements of Gi by the

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303

same letter g(i, j) with 0 ≤ i ≤ n, 0 ≤ j in the following, even though the g(i, j) may have different properties. It should be noted that they always belong to Gi , and the appropriate interpretation should be evident from the context. Recall from Sect. 2 that hg(i, j)i = N1 T r(g(i, j)) and they are independent of Ui when i > 0. Lemma 3.2. (a) Let W be an alternating word in (Gi )i∈I . Then Z dU1 dU2 ...dUn hW i is a finite polynomial Q(W ) in hg(i, j)i, i.e., Q(W ) =

X

Qa,b

a,b

bi n Y Y

hg(i, j)i,

i=0 j=ai

where only a finite number of Qa,b ’s are nonzero (we have suppressed the subscript of a, b). Moreover, Qa,b have bounded limits as N → +∞. (b) Assume hXi i has a bounded limit as N → +∞ for all Xi ∈ Gi . If each component g of the alternating word W has the property that limN →+∞ hgi = 0, then lim Q(W ) = 0.

N →+∞

Proof. Let us prove both parts (a) and (b) by induction on n. If n = 0, Z dU1 ...dUn hg(0, j)i = hg(0, j)i, the lemma is trivial. Assume the lemma is true for all n < k (k ≥ 1). Let us prove it for the case of n = k. By using the cyclicity of the trace, we may, without loss of generality, assume hW i takes either one of the following forms:

or

hU1 g(1, 1)U1∗ A1 U1 g(1, 2)U1∗ A2 · · · U1 g(1, q)U1∗ Aq i

(12)

hU1 g(1, 1)U1∗ A1 U1 g(1, 2)U1∗ A2 · · · U1 g(1, q)U1∗ i,

(13)

where A1 , ...Aq are alternating words in Gi with i = 0, 2, 3, ...k. Let us consider the case that hW i is of the form (12). By Lemma 2.4 ( with Bj replaced by g(1, j)), we have: Z dU1 hU1 g(1, 1)U1∗ A1 U1 g(1, 2)U1∗ A2 · · · U1 g(1, q)U1∗ Aq i X X (14) f¯q (m) N −2g(Gσ,τ ) · hgiτ · hAiσ◦Z . = m

σ,τ ∈Sq ,[σ −1 ◦τ ]=m

Since g(Gσ,τ ) ≥ 0 and limN →+∞ f¯q (m) has bounded limit by Lemma 2.2, it follows that lim N −2g(Gσ,τ ) f¯q (m) N →+∞

has a bounded limit. After integration with respect to the remaining U2 , ...Uk and use of the induction hypothesis , we conclude that for n = k and hW i is of the form (12), part

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F. Xu

(a) of the lemma is true. The case when hW i takes the form (13) follows from the case of (12) if we set Aq = identity. Hence by induction, part (a) is proved. To prove part (b) in the case when hW i is of the form (12), assume the condition of part (b) is satisfied. As N → +∞, it is clear from (14) that the only possible nonzero limit comes from Gσ,τ such that g(Gσ,τ ) = 0, namely the fat graph which is planar. Let G be any such planar graph and let a, b be two points on G such that a, b are connected by an edge outside the unit circle and clockwise on the circle the number of vertices between a and b is minimal. If a = b, then a is a vertex which is connected to itself. If a 6= b, since a, b are not adjacent by the construction of G and G is planar, we conclude that between a and b there is always one vertex on the unit circle which is not connected to any other point except to itself by edges outside the circle. Hence we conclude that there is always a vertex on G which is only connected to itself by an edge outside the unit circle. If this vertex is g(1, j), the corresponding expression hgiτ −1 contains a term hg(1, j)i which goes to 0 as N → +∞ by assumption. If this vertex is Aj , then hAiσ◦Z contains a term hAj i which after integration with respect to U2 , ...Uk goes to 0 as N → +∞ by the induction hypothesis. Thus we have proved part (b) when hW i takes the form (12). To prove (b) in the case (13), by using the cyclicity of trace and the fact part (b) is true when hW i takes the form (12), we have Z dU1 ...dUk hU1 g(1, 1)U1∗ · · · Aq−1 U1 g(1, q)U1∗ i lim N →+∞ Z dU1 ...dUk hU1 g(1, q)g(1, 1)U1∗ A2 · · · Aq−1 i = lim N →+∞ Z = lim hg(1, q)g(1, 1)i dU1 ...dUk hA2 U1 g(1, 2)U1∗ · · · Aq−1 i N →+∞ Z = lim hg(1, q)g(1, 1)i dU1 ...dUk hU1 g(1, 2)U1∗ · · · Aq−1 A2 i N →+∞ Z = lim hg(1, q)g(1, 1)ihAq−1 A2 i dU1 ...dUk hU1 g(1, 2)U1∗ · · · N →+∞

U1 g(1, q)U1∗ i = · · · = constant ×

Z lim

N →+∞

dU1 ...dUk hM i,

where M is U1 g(1, j)U1∗ for q odd or Ai for q even. It follows by assumption and the induction hypothesis that Z dU1 ...dUk hW i = 0. lim N →+∞

By induction, part (b) is proved.



Remark 5. One can prove the same results for SO(N ), SP (N ) group cases (see Remark 1) by the same method above. 3.2. Asymptotic freeness. We are now ready to prove the asymptotic freeness results for Gaussian, Haar and heat kernel cases. We will only give the proof for the heat kernel case (see Subsect. 2.3). The other two cases are proved in exactly the same way after one makes the appropriate changes of notation and uses Lemma 2.1 and Lemma 2.2 instead of Lemma 2.5.

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305

We will use the notation introduced in Subsect. 3.1 and Subsect. 3.2. For clarity, let us define Z hhW ii = dU1 · · · dUn K(U1 ; A1 ) · · · K(Un ; An )hW i Z

and hhg(i, j)ii =

dUi K(Ui ; Ai )hg(i, j)i.

When i = 0, we understand hhg(0, j)ii as hg(0, j)i. Slightly abusing our notation, we let Gi (i > 0) denote the set of all polynomials in Ui and again we use the symbols g(i, j) to enumerate the elements of Gi . Corollary 3.3. (a) Let us assume that W is an alternating word in G0 , G1 , ...Gn and that for any elements M ∈ G0 , limN →+∞ hM i exists. Then lim hhW ii

N →+∞

exists. (b) Suppose that each component g of W has the property that lim hhgii = 0.

N →+∞

Then

lim hhW ii = 0.

N →+∞

Proof. Since K(U ; A)dU = K(T U T + ; A)d(T U T + ) for any fixed T ∈ U (N ), by [7] we can write N Y dαk 12 (α) · K(αk ; A)dT K(U ; A)dU = CN · k=1

12 (α) =

α i − αj , sin2 2 i<j Y

where CN is a constant and dT is the normalized Haar measure on U (N ). Hence hhW ii

Z

Z

Y

= (CN )P ·

dαi,k 12 (αi,k )K(αi,k ; Ai )

dT1 · · · dTn · hW i.

1≤i≤n,1≤k≤N

By Lemma 3.2, we have Z dT1 · · · dTn · hW i =

X

Qa,b

a,b

Hence

bi n Y Y i=0 j=ai

hhW ii =

X a,b

Qa,b

bi n Y Y h( hg(i, j)i)i. i=0

j=ai

As N → +∞, by (1) of Lemma 2.5 we have:

hg(i, j)i.

306

F. Xu

lim h(

N →+∞

bi Y

hg(i, j)i)i = lim

N →+∞

j=ai

bi Y

hhg(i, j)ii.

j=ai

It follows from (2) of Lemma 2.5 that lim hhg(i, j)ii

N →+∞

exists for i > 0. When i = 0, the above limits exist by our assumption in the lemma. It follows that lim hhW ii N →+∞

exists. The proof of the second part of the corollary follows exactly the same as the proof of part (b) in Lemma 3.2: one simply replaces hg(i, j)i there by hhg(i, j)ii  Remark 6. The same method, combined with Remarks 1, 2, 3, 4, 5, can be used to prove the same results as Corollary 3.2 for SO(N ), SP (N ) cases. Remark 7. If µN is any U (N ) invariant measure on N × N Hermitian matrices which satisfies the factorization hypothesis (1) in Sect. 2 and has a limit distribution as N → +∞, then it is easy to see from the proof presented above that Corollary 3.3 is true for such a measure. In particular, Corollary 3.3 is true for Gaussian case and Haar case by Lemma 2.1 and Lemma 2.3. In the Haar case this gives a direct proof of Theorem 4.3.2. in [1]. 3.3. A random matrix model. Following p. 52 of [1], we will consider now families (V (s, n))s∈S in Mn of random matrices such that each V (s, n) is unitary. Such a family defines a map 8n :  → (U (n))S given by φn (x) = (V (s, n)(x))s∈S , where  is a space with measure µ. A (As )s∈S measure on U (n)S is defined to be the product of measures K(Us ; As )dUs for s ∈ S. The classical joint distribution of the family (V (s, n))s∈S is the push forward of the measure µ by 8n . Theorem 3.4. For each n, let (V (s, n))s∈S be a family of unitary random matrices such that its classical joint distribution is the (As )s∈S measure on U (n)S , and let W (n; t)t∈T be a family of constant n × n matrices such that W (n; t)t∈T has a limit distribution as N → ∞. Then the sequence of families of sets of random variables (W (n; t)t∈T , (V (s, n))s∈S ) is asymptotically ∗-free as n → +∞. Moreover, for every s ∈ S, the limit distribution of {V (s, n), V (s, n)∗ } as n → +∞ is µAs measure on the As circle whose S-transform is SµAs (z) = e 2 (1+2z) . Proof. The theorem follows immediately from Corollary 3.3 and 3.1.



Remark 8. Theorem 3.4 implies the following statement about two dimensional YangMills theory with gauge group U (N ) on the plane when N → +∞, namely, ”the nonoverlapping families of Wilson loop operators” form a multiplicative free family (see [3]). Even though there has been much progress in constructing functional integral measure in two dimensional Yang Mills theory, we take the approach of using heat kernel measure which is more convenient for our purposes.

Random Matrix Model from 2D Yang-Mills Theory

307

4. Conclusions and Questions In this paper we have given a unified treatment of asymptotic freeness by using fat graphs. With this we have demonstrated Theorem 3.4 which may be viewed as a “Fourier transform” of the Gaussian random matrix models in [1]. It remains to see if one can produce more random matrix models from our methods and if these models can be used to solve some questions in operator algebras. Acknowledgement. I’d like to thank the referee for his very useful suggestions. I’d also like to thank Professor E. Effros and Mr. David Kan for proof reading this paper. This work is partially supported by NSF grant DMS-9500882.

Note added in proof After this work was completed and submitted for publication, the author is informed by a Referee about a preprint of Ph.Biane (to appear in Fields Institute Communications) where some of the results of this paper are proved by different methods. References 1. Voiculescu, D., Dykema, K. and Nica, A.: Free random variables. CRM Monograph series 1, Providence, RI: AMS, 1992 2. Weingarten, D.: Asymptotic behavior of groups integrals in the limit of infinite rank. J. Math. Phys. 19, 5, (1978) 3. Gopakumar, R. and Gross, D.: Mastering the master field. hep-th/9503126, to appear in Nucl. Phys. B 4. Kazakov, V. and Kostov, I.: Nucl. Phys. B. 176, 199 (1980) 5. Witten, E.: In recent developments in Gauge theories eds. G.’t Hooft et al., New York and London: Plenum Press, 1980 6. Voiculescu, D.: Limit laws for random matrices and free products. Invent. Math. 104, 201–220 (1991) 7. Itzykson, C. and Zuber, J.: The planar approximation: II. J. Math. Phys. 21, 411 (1980) 8. Witten, E.: Two-dimensional gravity and intersection theory on moduli space. Surveys in Diff. Geom. 1, 243 (1993) 9. Penner, R.: Perturbative series and moduli space of Riemann surfaces. J. Diff. Geom. 27, 35–53 (1988) 10. Bercovici, H. and Voiculescu, D.: L´evy-Hinˇcin type theorems for multiplicative and additive free convolution. Pacific J. Math. 153, no. 2, 217–248 (1992) 11. Diaconis, P. and Shahshahani, M. : On the eigenvalues of random matrices. J. Appl. Prob. Special Vol. 31A, 49–62 (1994) 12. Singer, I.: On the master field in two dimensions. In: Functional analysis on the eve of the 21st century in honor of the 80th birthday of I.M. Gelfand. Progress in Mathematics Vol. 131 13. Weyl, H.: The Classical Groups. Princeton, NJ: Princeton University Press, 1946 Communicated by H. Araki

Commun. Math. Phys. 190, 309 – 330 (1997)

Communications in

Mathematical Physics c Springer-Verlag 1997

Stability of Ultraviolet-Cutoff Quantum Electrodynamics with Non-Relativistic Matter ¨ Fr¨ohlich2 , Gian Michele Graf2 Charles Fefferman1 , Jurg 1 2

Department of Mathematics, Fine Hall, Princeton University, Princeton, New Jersey 08544, USA Theoretical Physics, ETH-H¨onggerberg, CH–8093 Z¨urich, Switzerland

Received: 4 September 1996/ Accepted: 9 April 1997

Abstract: We prove that the quantum-mechanical ground state energy of a system consisting of an arbitrary number, M , of static nuclei of atomic number ≤ Z and of an arbitrary number, N , of Pauli electrons interacting with the quantized, ultraviolet-cutoff radiation field is bounded below by −K · M , where K is a finite constant depending on Z, on the finestructure constant α and on the ultraviolet cutoff 3, with K ≤ K 0 3, as 3 → ∞, and K 0 independent of 3.

1. Introduction and Survey of Results In this paper we prove that the quantum electrodynamics (QED) of non-relativistic quantum-mechanical matter interacting with the quantized radiation field is stable (more precisely, H-stable), provided an ultraviolet cutoff is imposed on the quantized electromagnetic vector potential. A typical physical system described by this theory consists of an arbitrary number, N , of non-relativistic electrons with electric charge −e, bare mass m > 0, spin 1/2 and a bare gyro-magnetic factor g = 2, an arbitrary number, M , of nuclei of nuclear charge ≤ Ze, for some positive integer Z < ∞ (e.g. Z < 150), and an arbitrary, variable number of photons; (see refs. [1–6]). H-stability is the statement that in the ground state of the system, the energy per charged particle (electron or nucleus) remains finite, as N and/or M tend to ∞; see [7]. Since the masses of nuclei are much larger than the electron mass m, we shall treat the nuclei as static. This would yield a lower bound on the ground state energy of the system if the gyro-magnetic factors of nuclei were less than or equal to 2, [1]. However, there are plenty of nuclei (including the proton) with a gyro-magnetic factor > 2. For systems containing such nuclei, H-stability will not hold, unless the interactions between the nuclear magnetic moments and the quantized radiation field are neglected or regularized by a hard- or soft-core form factor of nuclei. However, for a size of nuclei much smaller than the Bohr radius of an atom, the contribution of the nuclear Zeeman energies to

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the ground state energy of the system is much smaller than typical atomic energies, provided an ultraviolet cutoff at an energy at least as large as the rest energy of an electron is imposed on the quantized radiation field. It is then justified to treat nuclei as static point particles in estimates of the ground state energies of atomic and molecular systems. This approximation is made throughout our paper. Much of atomic, molecular and condensed matter physics is the study of physical properties of the systems just described. Throughout this paper, we impose the Coulomb (or radiation) gauge condition on the electromagnetic potentials. In this gauge, electrons and nuclei interact through instantaneous Coulomb two-body potentials, and the electrons are coupled to the transverse degrees of freedom of the radiation field by minimal substitution (i.e., by replacing ordinary derivatives by covariant ones). Treating the nuclei as static implies that they do not couple to the transverse degrees of freedom of the radiation field. The field quanta of the transverse degrees of freedom of the electromagnetic field are the photons. A typical electron energy in an atom is of order −mc2 (Zα)2 , where c is the velocity of light, α = e2 /4π~c ≈ 1/137 is the dimensionless finestructure constant, and ~ is Planck’s constant. Interactions between photons with energy large compared to typical atomic electron energies and the electrons are turned off by means of an ultraviolet cutoff imposed on the electromagnetic vector potential. The Hilbert space of pure state vectors of a system of N -electrons and arbitrarily many photons is given by (1.1) H = (H1 )3N ⊗ F , where H1 = L2 (E3 , d3 x) ⊗ C2 is the one-electron Hilbert space, 3 denotes an antisymmetric tensor product, and F is the bosonic Fock space over the one-photon Hilbert space L2 (R3 , d3 k) ⊗ C2 , i.e., F is the symmetric tensor algebra over L2 (R3 , d3 k) ⊗ C2 . The factors C2 describe the spin states of an electron and the helicities of photons, respectively. The choice of an antisymmetric tensor product in (H1 )3N , on the r.s. of (1.1), corresponds to the Pauli principle, i.e., to the Fermi statistics of electrons. Photons are bosons, and therefore symmetric tensor products are used in the definition of F. Next, we describe the Hamiltonian that generates the dynamics of a system of N electrons, M nuclei and arbitrarily many photons. We work in units, where ~ = c = 1, and we impose the Coulomb gauge condition on the electromagnetic potentials, as announced. Then the Hamiltonian of the system is given by e ph , e = H e el + 1I ⊗ H H where e el = H

N X j=1



 1  (j) ˜ (3) xj 2 + VeC , σ · −i∇j + eA 2m

(1.2)

(1.3)

and VeC =

X 1≤i<j≤N

+

M N X X α α Zl − |xi − xj | |xi − yl |

X

1≤k
i=1 l=1

α Zk Zl , |yl − yk |

with Zl ≤ Z < ∞, for all l = 1, · · · , M and arbitrary M .

(1.4)

Stability of Ultraviolet-Cutoff QED with Non-Relativistic Matter

311

We write σβ(j) for 1I ⊗ · · · ⊗ 1I ⊗ σβ ⊗ 1I ⊗ · · · ⊗ 1I on spin space (C2 )⊗N , where σ1 , σ2 and σ3 are the three Pauli matrices, and the factor σβ appears in the j th factor of the tensor product. An explicit matrix representation of the Pauli matrices is given by       01 0 −i 1 0 , σ2 = , σ3 = . σ1 = 10 i 0 0 −1 The point xj ∈ E3 denotes the position of the j th electron, −i∇j is its momentum operator (in the Schr¨odinger representation), and yl ∈ E3 denotes the position of the lth nucleus; {y1 , · · · , yM } describes an arbitrary, but fixed configuration of nuclei of nuclear charges Z1 e, · · · , ZM e. Furthermore, VeC is the electro-static Coulomb potential. ˜ The quantized electromagnetic vector potential, A(x), in the Coulomb gauge is given by the following operator-valued distributions on Fock space.
˜ + (x), A ˜ + (x) = A ˜ − (x) ∗ , ˜ ˜ − (x) + A (1.5) A(x) = A where ˜ − (x) = A

1 (2π)3/2

Z

d3 k X p cλ (k) ελ (k)eik·x . 2|k| λ=±

(1.6)

The integral on the r.s. of (1.6) may be understood as a weak integral on F × F. In these k formulae, the direction of propagation, kˆ = |k| , where k denotes a wave vector, and the polarization vectors ε+ (k), ε− (k) form an orthonormal basis of R3 ⊗ C. For concreteness, we identify ε + and ε− with right- and left circular polarization, respectively. Furthermore, the operators cλ (k)∗ and cλ (k) are standard creation- and annihilation operators on F satisfying canonical commutation relations:   cλ (k)# , cλ0 (k0 )# = 0,   (1.7) cλ (k), cλ0 (k)∗ = δλλ0 δ (3) (k − k0 ). Here cλ (k)# = cλ (k) or cλ (k)∗ , and cλ (k)∗ is the adjoint (operator-valued distribution) of cλ (k). Fock space F contains a vector  (unique, up to a complex phase), the vacuum, with the property that (1.8) cλ (k)  = 0 , for all λ = ±, k ∈ R3 . The vector  is cyclic for the algebra of polynomials in cλ and c∗λ (smeared out with test functions). From this fact and Eqs. (1.7) and (1.8), it follows that
⊗ s n ∞ , (1.9) F∼ = ⊕ L2 (R3 ) ⊗ C2 n=0

where ⊗s denotes a symmetric tensor product, (as claimed above). See e.g. [8, 9] for back ground material on Fock space. Let 3(k) be a function satisfying |3(k)| ≤ 1,

supp 3 ⊆ {k ∈ R3 | |k| ≤ 3},

(1.10)

for some constant 3 < ∞ (the ultraviolet cutoff). The electromagnetic vector potential ˜ (3) (x), is given by with ultraviolet cutoff 3, A  ∗ ˜ (3) (x) = A ˜ (3) ˜ (3) ˜ (3) ˜ (3) A A A , + (x) + A− (x), + (x) = − (x)

312

C. Fefferman, J. Fr¨ohlich, G.-M. Graf

where ˜ (3) A − (x) =

1 (2π)3/2

Z

X d3 k p 3(k) cλ (k) ελ (k) eik·x . 2|k| λ=±

(1.11)

It is well known (and easy to prove) that, for 3 < ∞, A˜ (3) β (x) , β = 1, 2, 3, are three commuting selfadjoint operators on F, for all x ∈ E3 , which are smooth in x; (see Proposition 1, Sect. 3). Finally, we define the free photon Hamiltonian H˜ ph XZ ˜ ph = H d3 k cλ (k)∗ |k|cλ (k). (1.12) λ=±

This operator is positive and selfadjoint on its dense domain of definition. Its spectrum is absolutely continuous, except for a single, simple eigenvalue at 0. The corresponding eigenvector is the vacuum . The main results proven in this paper are Theorems 1 and 2. Theorem 1 is stated below, while Theorem 2 is stated in Sect. 4 and proven in Sect. 5 and in [12]. Theorem 2 and arguments first described in [6] are the basic ingredients used in the proof of Theorem 1; but Theorem 2 is of independent interest. Theorem 1. (1) For an arbitrary number, N , of electrons, an arbitrary number, M , of nuclei of charges ≤ Ze, for arbitrary values of Z and of the finestructure constant α, and for an arbitrary ultraviolet cutoff 3 (see (1.10)), there exists a finite constant E(α, Z) (only depending on α and Z) such that e ≥ − E(α, Z) (3 + 1) · M. H (2) For arbitrary N , M , and Z (as above), for an ultraviolet cutoff 3 ≤ 2 m Z 2 α5/4 ,

(1.13)

and for sufficiently small values of αZ and α5/2 Z 3 , there exists a finite constant ε(Z) ≤ const. Z 3/2 independent of α, m, N and M such that e ≥ − ε(Z) m(αZ)2 · M. H

(1.14)

Remarks. (i) Part (1) says that the ground state energy of a system of M static nuclei and N nonrelativistic electrons interacting with the quantized radiation field cutoff at an energy 3 tends to −∞ no worse than linearly in M and in 3, as M → ∞, or 3 → ∞. The dependence on 3 of our estimate on the ground state energy is related to the fact that we are studying the unrenormalized Hamiltonian of quantum electrodynamics with non-relativistic matter; (see the discussion at the end of ref. [6]). One expects that the ground state energy of the correctly renormalized Hamiltonian grows linearly in M , but is independent of 3, (for α and Z small enough). However, at present, there is no proof in sight of this conjecture. (ii) As remarked before, typical electron energies in a system of the kind we are considering are of order −m(Zα)2 . Thus the bound imposed on the ultraviolet cutoff 3 in (1.13) is physically acceptable, since, in nature, α = 1/137  1. Furthermore, our estimate on ground state energies in part (2), inequality (1.14), has the right asymptotic behavior in α, as α → 0.

Stability of Ultraviolet-Cutoff QED with Non-Relativistic Matter

313

(iii) It is shown in [6], by methods relying on [5], that, for 3 as in (1.13), α ≤ 1/132 and Z ≤ 6, there is a finite constant ε such that e ≥ − ε m α2 (N + M ). H 2. Dimensionless Variables and Natural Units Before we start with the proof of our main results we choose natural, dimensionless variables that will simplify our formulae. We set xj = β ξ j , y l = β η l ,

and k = µ κ,

(2.1)

with 0 < β, µ < ∞, and β and µ−1 have the dimension of length. Then ξ j , j = 1, · · · , N , η l , l = 1, · · · , M, and the rescaled wave vectors κ are dimensionless variables. We introduce rescaled creation- and annihilation operators aλ (κ)# := µ3/2 cλ (k)# ,

(2.2)

 aλ (κ)# , aλ0 (κ)# = 0,   aλ (κ), aλ0 (κ0 )∗ = δλλ0 δ (3) (κ − κ0 ).

(2.3)

and note that 

It is easy to verify that there exist unitary operators U (µ) on F , corresponding to a dilatation of photon wave vectors by a factor µ > 0, such that U (µ) U (µ0 ) = U (µ · µ0 ) and aλ (κ)# = U (µ) cλ (κ)# U (µ)−1 , and hence the commutation relations (1.7) and (2.3) are equivalent. One now checks that XZ e ph = µ d3 κ aλ (κ)∗ |κ| aλ (κ) H λ=±

=: µ Hph ,

(2.4)

e ph U (µ)−1 . where Hph = U (µ) H Furthermore Z X 1 d3 k ˜ (3) p 3(k) (x) = cλ (k) ελ (k) eik·x A − 3/2 (2π) 2|k| λ=± Z 3 X µ d κ p 3(µκ) = aλ (κ) ελ (µκ) eiκκ ·(µx) 3/2 (2π) 2|κ| λ=± (3 )

=: µ A− µ (µx),

(2.5)

where 3µ (κ) = 3(µκ), with 3 as in Eq. (1.10). Note that, for our choice of polarization vectors ε± (·), we have that ε± (µκ) = ε± (κ). We set  ∗ (3 ) (3 ) A+ µ (x) = A− µ (x) ,

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and

(3µ )

A(3µ ) (x) = A+

(3 )

(x) + A− µ (x).

e is given by We conclude that, in dimensionless variables ξ j , η l , κ, the Hamiltonian H N X

e = H

j=1



2 1  (j) σ · −i∇j + βµ eA(3µ ) βµξ j 2mβ 2

α + VC ⊗ 1I + µ 1I ⊗ Hph , β

(2.6)

where X

VC =

1≤i<j≤N

M N X X 1 Zl − |ξ i − ξ j | |ξ i − η l |

X

+

1≤l
i=1

l=1

Zl Zk . |η l − η k |

(2.7)

Equations (2.4), (2.6) and (2.7) suggest to choose β and µ as follows: 2mβ 2 i.e., and hence


−1

= α β −1 = µ ,

(2.8)

β = (2mα)−1 , µ = 2mα2 ,

(2.9)

√ βµ = α , βµe = 2 π α3/2 .

(2.10)

We set H :=

1 e 1 e, H = H µ 2mα2

(2.11)

so that energy eigenvalues of H are given in units where the ground state energy (1 Rydberg) of a hydrogen atom is = −4. See [10] for further details. In what follows, we rename variables: ξ j → xj , η l → yl , κ → k and 3µ → 3 , with |3(k)| ≤ 1, and supp 3 ⊆



k | |k| ≤ (2mα2 )−1 3 .

(2.12)

We find that H = Hel (α) + 1I ⊗ Hph ,

(2.13)

where Hel (α) =

N h X

σ (j) ·




i2 √ −i ∇j + 2 π α3/2 A(3) αxj + VC ⊗ 1I .

(2.14)

j=1

In the following, it will be convenient to make use of field variables A and B given by

Stability of Ultraviolet-Cutoff QED with Non-Relativistic Matter

√ A(x) := 2 π α3/2 A(3) (αx), B(x) := curl A(x) . Then, for an arbitrary function h(x), Z Z x d3 x , |B(x)|2 h(x)d3 x = 4πα2 |curl A(3) (x)|2 h α and

315

(2.15)

(2.16)

Z |(∇ ⊗ B)(x)|2 h(x) d3 x = Z x 4 d3 x . |∇ ⊗ (curl A(3) )(x)|2 h 4π α α

(2.17)

3. The Schr¨odinger Representation of Fock Space In the proof of our main result it is convenient to make use of the Schr¨odinger representation of Fock space F , which we briefly review next (see also [9]). We define the Sobolev space H−1/2 as the complex, linear space of distributions, h, ˆ have the property that on E3 whose Fourier transforms, h, Z 3 dk ˆ |h(k)|2 < ∞ . (3.1) |k| For h ∈ H−1/2 and β = 1, 2, 3, we define Z d3 k X n ˆ p Aβ (h) := aλ (k) ελ,β (k) h(k) 2|k| λ=± o ˆ , + aλ (k)∗ ελ,β (k) h(−k)

(3.2)

where ελ,β is the β th component of ελ , and z¯ denotes the complex conjugate of z. If h is real then ˆ ˆ h(−k) = h(k) , and hence Aβ (h) is a densely defined, symmetric operator on F , (see, e.g., [8, 9]). Actually, much more is true: Let Z X aλ (k)∗ aλ (k) (3.3) N = d3 k λ=±

denote the photon number operator on F. This is a positive, selfadjoint operator on F, and its spectrum consists of all non-negative integers. The eigenspace of N corresponding to

⊗ s n of F in the representation a non-negative integer n is the subspace L2 R3 ⊗ C2 of F given in Eq. (1.9). We recall the following well known result.

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C. Fefferman, J. Fr¨ohlich, G.-M. Graf

Proposition 1 ([8, 9, 11]). (1) For h ∈ H−1/2 , h real, β = 1, 2, 3, the operator Aβ (h) defined in (3.2) is essentially selfadjoint on any core for N . (2) The operators + * 3 X Aβ (hβ ), hβ ∈ H−1/2 , hβ real, ∀ β = 1, 2, 3 exp iA(h) | A(h) := β=1

(3.4) generate an abelian ∗ algebra of bounded operators on F . The vacuum  is a cyclic and separating vector for this algebra. (3) For real functions h1 , h2 , h3 in H−1/2 we have that   1 h, exp i A(h)iF = exp − Q(h) , (3.5) 2 where Q(h) :=

  3 Z X kβ kγ d3 k hβ (k) δβ γ − hγ (k). |k| |k|2

(3.6)

β,γ=1

Note that the quadratic form Q is positive semi-definite. More precisely, on the space  T H−1/2 = h = (h1 , h2 , h3 ) | hβ ∈ H−1/2 , β = 1, 2, 3, ∇ · h = 0 , Q is positive-definite. This implies that   1 exp − Q(h) 2 is the characteristic functional of a Gaussian measure, dµQ (A), of mean 0 and covariance Q with support contained in the Schwartz distribution space of transverse vector potentials,  A ≡ S T (R3 )0 = A = (A1 , A2 , A3 ) | Aβ ∈ S(R3 )0 , ∀β = 1, 2, 3, ∇ · A = 0 , (3.7) (Minlos’ theorem, see e.g. [8]). Remarks concerning the proof of Proposition 1. The proofs of (1) and (2) can be inferred from [9, 11]. [The proof of the fact that  is cyclic for the algebra of operators defined in (3.4) follows from explicit calculations, see [9]; the rest follows from general results in [11].] Part (3) follows from a straightforward calculation. For further discussion and references see [8, 9, 11]. As a corollary of Proposition 1 and Minlos’ theorem (see e.g. [8]), we conclude that Fock space F is isomorphic to the L2 -space L2 (A, dµQ ). In the representation of F as F ∼ = L2 (A, dµQ ),

(3.8)

called the Schr¨odinger representation of F , all the operators affiliated with the weak closure of the ∗ algebra defined in (3.4), in particular A(h), for h1 , h2 and h3 functions in H−1/2 , are multiplication operators. In particular, they are “simultaneously diagonalized”, (see [8, 9] for further details).

Stability of Ultraviolet-Cutoff QED with Non-Relativistic Matter

317

4. Beginning of the Proof of Theorem 1 In this section we reduce the proof of Theorem 1 to a result on H-stability of systems in arbitrary external magnetic fields (Theorem 2, below), making use of the Schr¨odinger representation of Fock space reviewed in the last section. The proof of Theorem 2 is then outlined in Sect. 5 (technical details can be found in [12]). Our strategy and most of the arguments in Sect. 4 follow those in [6]. We define (4.1) D(x) = min |x − yk | . 1≤k≤M

We employ the natural, dimensionless variables introduced in Sect. 2. Thus x and yk are dimensionless, and the Bohr radius of an electron orbit bound to a nucleus of charge Ze is of order Z −1 . We decompose the Hamiltonian H defined in Eq. (2.13) into a sum of two terms: H = HI + 1I ⊗ HII , where HI =


i2 √ σ (j) · −i ∇j + 2 π α3/2 A(3) αxj

N h X j=1

 Z −1 + VC ⊗ 1I + 1I ⊗ 4πα2 0 | curl A(3) (x)|2 e−L D(x/α) d3 x  Z −1 + 4π α4 C(0, Z)L2 |∇ ⊗ (curl A(3) )(x)|2 e−L D(x/α) d3 x and

Z

(4.2)

−1

| curl A(3) (x)|2 e−L D(x/α) d3 x Z −1 4 2 |∇ ⊗ (curl A(3) )(x)|2 e−L D(x/α) d3 x , (4.3) −4π α C(0, Z)L

HII = Hph − 4π α2 0

where 0 and C(0, Z) are finite, positive constants yet to be chosen (see Theorem 2 and Sect. 5). On the r.s. of (4.2) we insert the fields A and B defined in Eq. (2.15). Then we find that HI =

N X 


2

σ (j) · −i ∇j + A(xj )

+ VC ⊗ 1I

j=1

 Z −1 + 1I ⊗ 0 |B(x)|2 e−L D(x) d3 x  Z −1 + C(0, Z) L2 |(∇ ⊗ B)(x)|2 e−L D(x) d3 x .

(4.4)

It follows from the definition of A and B = curl A, see Eq. (2.15), and Proposition 1 of Sect. 3 that the components Aβ (x), Bβ (x) and ∂α Bβ (x) (with ∂α = (∂/∂xα ), α, β = 1, 2, 3) are commuting, selfadjoint operators on F, for arbitrary x and an arbitrary choice of the ultraviolet cutoff 3 < ∞; (see Eq. (2.12)). These operators are multiplication operators in the Schr¨odinger representation (3.8) of Fock space. We conclude that, in

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C. Fefferman, J. Fr¨ohlich, G.-M. Graf

order to prove a lower bound on the Hamiltonian HI , it is enough to establish a lower bound on the following energy functional, E:  + * N  X 
 2 + VC ψ σ (j) · −i∇j + A(xj ) E(ψ, A) := ψ,   j=1 Z  −1 + 0|B(x)|2 + C (0, Z) L2 |(∇ ⊗ B)(x)|2 e−L D(x) d3 x , (4.5) for ψ ∈ (H1 )3N , with k ψ k= 1, (where, recall, H1 = L2 (E3 , d3 x) ⊗ C2 is the oneelectron Hilbert space), and for an arbitrary external vector potential A (belonging to a suitable Sobolev space), with B = curl A . More precisely, ) ( HI ≥

inf

ψ : kψk = 1, A

E(ψ, A)

1I .

(4.6)

The functional E(ψ, A) is studied in Sect. 5 and in ref. [12]. Our main result on this functional is described in the following theorem. Theorem 2. Given 0 > 0 and Z < ∞, there are finite, positive constants c(0, Z), C(0, Z) and C 0 (0, Z) such that, for E(ψ, A) as in (4.5), k ψ k= 1, and L ≤ c(0, Z), E(ψ, A) ≥ − C 0 (0, Z) L−1 M ,

(4.7)

where M is the number of nuclei. If 0 ≥ Z 2 there are finite, positive constants c and C such that, for c (0, Z) = c Z −1 , 0

C(0, Z) = C Z 2 ,

(4.8)

C (0, Z) ≤ C Z . 2

Remarks. (1) In [12], explicit expressions for c (0, Z), C(0, Z) and C 0 (0, Z) are derived in the three regimes: (i) 0 ≥ Z 2 , (ii) Z 2 ≥ 0 ≥ 1, (iii) 0 ≤ 1. 1 Below we shall choose 0 = 16π α2 , where α is the finestructure constant. Since α ∼ = 1/137, regime (i) is the one most relevant for atomic physics. We therefore do not present explicit expressions for c (0, Z) and C 0 (0, Z) in regimes (ii) and (iii), but refer the reader to [12]. (2) We recall that the results proven in [5, 6] imply that, for 0 = (8π α2 )−1 , α ≤ 1/132 and Z ≤ 6, we have that  + * N  X 
 2 + VC ψ σ (j) · −i ∇j + A (xj ) ψ,   j=1 Z 2 (4.9) + 0 | (∇ ⊗ A)(x) | d3 x ≥ − CZ 2 (N + M ), 

for a finite constant C. Here   = x ∈ E3 | |x − yl | ≤ (Z + 1)−1 , for some l = 1, · · · , M . As shown in [6], this implies part (2) of Theorem 1, (for Z ≤ 6).

Stability of Ultraviolet-Cutoff QED with Non-Relativistic Matter

319

Our next task is to prove a lower bound on the operator HII , and here we follow arguments described in Sect. 3 of [6]. Recall that ( ) Z X 1 d3 k (3) i k·x p 3(k) aλ (k) ελ (k) e , A− (x) = (2π)3/2 2|k| λ=±  ∗ (3) A(3) , A(3) (x) = A(3) (4.10) A(3) + (x) = + (x) + A− (x), − (x) and 3(k) is a (smooth) function, with |3(k)| ≤ 1

and

supp 3 ⊆





k | |k| ≤ (2m α2 )−1 3

;

see Eq. (2.12). We set B± (x) := curl A(3) ± (x), Clearly Bβ+ (x) =



∗

Bβ− (x)

We define F ± (x) =

Π ± (x) = ∇ ⊗ B± (x) .

+ Πβδ (x) =

, 



∗

− Πβδ (x)

.

Bβ± (x), or ± (x), Πβδ

(4.11)

(4.12)

(4.13)

β, δ = 1, 2, 3, and F (x) = F + (x) + F − (x). Let f (x) be a positive function on E3 . Then   f (x) F (x)2 = f (x) F + (x) F − (x) + F − (x) F + (x) + F + (x)2 + F − (x)2   ≤ 2f (x) F + (x) F − (x) + F − (x) F + (x) (4.14)   1 = 4f (x) F + (x) F − (x) + k F + (x)  k2 , 2 where  is the vacuum vector in F. To prove (4.14), we use the general inequality | hψ, T 2 ψi | ≤ k T ∗ ψ k k T ψ k 1 ≤ hψ, (T T ∗ + T ∗ T ) ψi, 2 for an arbitrary operator T and ψ in the intersection of the domain of T and the domain of T ∗ . We apply this inequality to the special choice of T = F ± (x). Next, we note that   F − (x) F + (x) = F + (x) F − (x) + F − (x), F + (x) = F + (x) F − (x) + h, F − (x) F + (x)i = F + (x) F − (x) + k F + (x)  k2 , which is a simple consequence of Eqs. (4.10)-(4.13); (recall the commutation relations (2.3) and use that aλ (k) = 0, for λ = ± and all k ∈ R3 , where  is the vacuum vector in F). It is obvious that k F + (x) k2 is independent of x. Thus inequality (4.14) yields the bound

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C. Fefferman, J. Fr¨ohlich, G.-M. Graf

Z

Z f (x) F (x)2 d3 x ≤ 4 k f k∞

F + (x) F − (x) d3 x + 2 k f k1 k F + (0) k2 .

(4.15) We apply inequality (4.15) to the examples in (4.13) and sum over β, δ = 1, 2, 3. This yields Z 2 f (x) | curl A(3) (x) | d3 x ( ) Z X 2 a∗λ (k)|k|aλ (k) (4.16) ≤ 2 k f k∞ d3 k | 3(k) | λ=±

+ 2 k f k1 C3 , where

3 X

C3 =

k Bβ+ (0)  k2 ,

(4.17)

β=1

and

Z


2 f (x) | ∇ ⊗ curl A(3) (x) | d3 x ( Z 2

d3 k | 3(k) |

≤ 2 k f k∞

X

) a∗λ (k)|k|3 aλ (k)

(4.18)

λ=±

+ 2 k f k1 D3 , where D3

=

3 X

k Πβ+ δ (0)  k2 .

(4.19)

β,δ=1

Comparing these expressions with Eq. (2.4) and using the definition of 3(k), Eq. (2.12), we find that Z 2 (4.20) f (x) | curl A(3) (x) | d3 x ≤ 2 k f k∞ Hph + 2 k f k1 C3 , and

Z


2 f (x) | ∇ ⊗ curl A(3) (x) | d3 x ≤ 2 k f k∞ (2m α2 )−2 32 Hph + 2 k f k1 D3 .

(4.21)

We are now prepared to prove a lower bound on the operator HII defined in (4.3). We choose
(4.22) f (x) = exp −L−1 D(x/α) , so that

k f k∞ = 1,

k f k1 ≤ 8π α3 L3 M .

We choose 0 and L so small that



8π α 0 + 8π C(0, Z) L 2

2

3 2m

(4.23)

2 ≤ 1.

(4.24)

Stability of Ultraviolet-Cutoff QED with Non-Relativistic Matter

Then

321


HII ≥ − 64π 2 α5 L3 M 0 C3 + α2 C(0, Z) L2 D3 .

Next, we calculate the constants C3 and D3 . The result is that  4 Z 1 π 3 3 C3 ≤ d k|k| = , 2 2 2m α2

(4.25)

(4.26)

B3

and 1 ≤ 2

D3

Z

π d k|k| = 3 3

B3



3

3 2m α2

6 ,

(4.27)

where B3 is the ball of radius (2m α2 )−1 3 centered at the origin of R3 ; (refer to Eq. (2.12)). Plugging (4.26) and (4.27) into the r.s. of (4.25) and using (4.24), we obtain the bound 4  3 M. (4.28) HII ≥ − 4π 2 α−5 L3 2m Choosing



ÿ

L ≤ min c (0, Z), 16π C(0, Z)



3 2m

2 !−1/2

  ,

(4.29)

see Theorem 2 and (4.24), and combining inequalities (4.7) and (4.28), we find that ! ÿ 4  C 0 (0, Z) 3 −5 3 M. (4.30) + H ≥ − 4π α L 2m L This bound enables us to complete the proof of part (1) of Theorem 1: By (4.24) and (4.29), we can choose 0 =

16π α2


−1

00

, L = C (α, Z)

2m , 3

(4.31)

00

for some finite constant C (α, Z). Then inequality (4.30) shows that there is a finite constant ε(α, Z) depending on α and Z, but independent of 3, such that H ≥ − ε(α, Z)

3 · M. 2m

(4.32)

e = 2m α2 H, part (1) of Theorem 1 is proven. Hence, since H In order to prove part (2) of Theorem 1, we set 0 = (16π α2 )−1 and assume that 0 ≥ Z 2 , i.e., 16π(αZ)2 ≤ 1. Furthermore we make the following choices: L = c0 Z −3/2 , for some c0 ≤ c,

(4.33)

3 = 2m Z α

(4.34)

2

For these choices

5/4

.

L ≤ c (0, Z) = c Z −1 , for Z ≥ 1 ,

so that Theorem 2 applies, and

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C. Fefferman, J. Fr¨ohlich, G.-M. Graf

 8π C(0, Z) L2

3 2m

2

 2 1 = C 000 Z −1 Z 2 α5/4 < 2

(4.35)

for Z 3 α5/2 small enough, so that inequality (4.24) holds. Then inequality (4.30) implies that (4.36) H ≥ − ε Z 7/2 · M , e = 2m α2 H, part (2) of Theorem for some constant ε independent of α and Z. Since H 1 follows from (4.36). It remains to prove Theorem 2. This is the task of Sect. 5, (see also [12]).

5. Outline of Proof of Theorem 2 Before beginning with the proof of Theorem 2 we briefly recall its statement. Let A(x) denote a vector potential on E3 , and let B(x) = curl A(x). We fix a configuration of nuclei at positions y1 , · · · , yM in E3 and with atomic numbers Z1 , · · · , ZM . We assume that max Zl ≤ Z, for an arbitrary, but fixed Z < ∞. 1≤l≤M

Let HPauli =

N X 


2

σ (j) · i ∇j − A(xj )

+ VC ,

j=1

with VC as in (2.7). For a smooth, bounded vector potential A, HPauli is a densely defined, selfadjoint operator on (H1 )3N , with H = L2 (E3 , d3 x) ⊗ C2 (see Sect. 1) whose spectrum is bounded below. In Sect. 4, the following result, Theorem 2, has been stated. Theorem. Given positive constants 0 and Z, there are finite, positive constants c (0, Z), C(0, Z) and C 0 (0, Z) ≤ C(0, Z) such that, for L ≤ c (0, Z), Z hψ, HPauli ψi + 0 |B(x)|2 e−D(x)/L d3 x Z C 0 (0, Z) M, + C(0, Z) L2 |(∇ ⊗ B)(x)|2 e−D(x)/L d3 x ≥ − L for any ψ ∈ (H1 )3N of norm 1. Remark. The constants c (0, Z), C(0, Z) and C 0 (0, Z) only depend on 0 and Z. If we are not interested in optimal estimates we can set C(0, Z) = C 0 (0, Z). In [12] it is shown that, for 0 ≥ Z 2 , there are finite, positive constants c and C such that c (0, Z) = c Z −1 , C(0, Z) = C Z 2 , C 0 (0, Z) ≤ C Z 2 . We sketch the proof of the above theorem. Our goal is to convey the spirit of the proof, sacrificing technical accuracy for the sake of simplicity. A complete proof of the theorem appears in [12]. The proof combines tricks from [13] and [14] with a new lower bound for the Pauli kinetic energy. We explain first the old tricks, then the new kinetic energy bound, and finally how to mix the ingredients and prove the theorem. Without sacrificing any ideas, we just treat the case Z1 = · · · = ZM = 1, but see [12].

Stability of Ultraviolet-Cutoff QED with Non-Relativistic Matter

323

To explain the old tricks, we first note that one can associate local kinetic and potential energies to a cube Q ⊂ E3 . If A is a vector potential and ψ ∈ (H1 )3N , then we write N Z X
2 | σ (k) · i ∇xk − A(xk ) ψ | χQ (xk ) d3 x1 · · · d3 xN TPauli (ψ, A, Q) = k=1

and T (ψ, A, Q) =

E3N

N Z X k=1

(5.1)


2 | i ∇xk − A(xk ) ψ | χQ (xk ) d3 x1 · · · d3 xN ,

(5.2)

E3N

where χQ denotes the characteristic function of Q. To localize the potential energy to Q, we use the following identity from [13]. ZZ 1 d3 z dR z R , where (5.3) VC = 2π R5 E3 ×R+

z R = Nz R (Nz R − 1) + Mz R (Mz R − 1) − 2 Nz R Mz R , with Nz R = number of electrons in the ball B(z, R), and Mz R = number of nuclei in B(z, R) .

(5.4) (5.5) (5.6)

Here, B(z, R) denotes the ball in E3 with center at z and radius R. Identity (5.3) lets us define the localized potential energy as follows. If Q ⊂ E3 is a cube of side δ, then we set  ZZ X  1 zR d3 z dR ψ, ψ . (5.7) P E(ψ, Q) = 2π MzR R5 yk ∈ Q B(z, R) 3 yk 1 R < 10 δ Since zR ≥ −CMzR always, and, in particular, zR ≥ 0 when MzR = 0, the following remarks are trivial consequences of the above definitions. (i) If E3 is partitioned into cubes {Qν } of sides {δν }, then  X N X
2  (k) TPauli (ψ, A, Q), and (5.8) σ · i ∇xk − A(xk ) ψ, ψ = k=1

hVC ψ, ψi ≥

X

P E(ψ, Qν ) − C

ν

X ν

ν

MQν δν−1 .

(5.9)

In (5.9) and throughout our discussion, NQ denotes the number of electrons and MQ the number of nuclei in Q. Similarly: (ii) If Q is partitioned into subcubes {Qν } with sides {δν }, then X T (ψ, A, Q) = T (ψ, A, Qν ) , (5.10) ν

TPauli (ψ, A, Q) = P E(ψ, Q) ≥

X ν

X

TPauli (ψ, A, Qν ) , and

ν

P E(ψ, Qν ) − C

X ν

δν−1 MQν .

(5.11) (5.12)

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C. Fefferman, J. Fr¨ohlich, G.-M. Graf

R R R The standard integration by parts, E3 |σ·(i∇−A)ϕ|2 d3 x = E3 |(i∇−A)ϕ|2 d3 x± E3 (σ · B)ϕ · ϕ¯ d3 x trivially implies the following; (we write Q∗ for the double of Q). KE Comparison Lemma. Let Q be a cube of side δ. Suppose the magnetic field B is constant on Q∗ . Then TPauli (ψ, A, Q∗ ) ≥ c T (ψ, A, Q) − C {|B| + δ −2 }hNQ∗ ψ, ψi .

(5.13)

One of the main technical results in [14], proven also in [12], is as follows. Local Stability of Matter. Let Q be a cube of side δ. Suppose the magnetic field B satisfies |B| ≤ C δ −2 on Q∗ . Then δ · T (ψ, A, Q∗ ) + P E(ψ, Q) ≥ − C δ −1 hNQ∗ ψ, ψi .

(5.14)

Here, we have oversimplified slightly the statement of local stability. We will also oversimplify below by neglecting the distinction between Q and Q∗ , and by supposing where necessary that NQ and MQ are of the same order of magnitude. (If not, then Q is far from electrically neutral, so that P E(ψ, Q) will be large positive, and whatever lower bounds we claim for the energy will therefore be trivial.) The new ingredient in the proof of our theorem is the following estimate. KE Lower Bound. Let Q be the unit cube, and suppose the magnetic field B is constant on Q∗ . If K > 1, then TPauli (ψ, A, Q∗ ) ≥ c KhNQ ψ, ψi − C K 5/2 {|B| + 1} − ChNQ∗ ψ, ψi .

(5.15)

We sketch the proof of the KE Lower Bound, then return to the proof of Theorem 2. The main point in establishing (5.15) is to prove an estimate for one-electron wave functions ϕ ∈ H1 , from which (5.15) will follow by separation of variables. Without significant loss of generality, we may suppose that the magnetic field B points parallel to the x3 -axis. In a suitable gauge we then have A(x) = (A1 (x), A2 (x), 0), with Aν (x) independent of x3 , and |∇Aν | ≤ C|B|. Integration by parts yields the identities Z Z Z 2 3 2 3 | σ · (i∇ − A)ϕ | d x = | (i∇ − A)ϕ | d x± (σ · B)ϕ · ϕd ¯ 3 x, (5.16) E3

Z

E3

E3

2 Z 3 ∂ 2 | σ · (i∇ − A)ϕ | d3 x = ϕ ∂x3 d x

E3

E3

  2 Z X ∂ σν i − Aν ϕ d3 x. + ∂xν E3

(5.17)

ν=1,2

Applying (5.16) and (5.17) with ϕ replaced by θϕ, for a suitable cutoff θ, we derive the estimate Z 2 | σ · (i∇ − A)ϕ | d3 x ≥ (5.18) Q∗

   2  Z Z  X  ∂ ∂ϕ 2 −1 3 2 3 + (|B| + 1) c i ∂xν − Aν ϕ  d x − C |ϕ| d x.  ∂x3 Q

ν=1,2

Q∗

Stability of Ultraviolet-Cutoff QED with Non-Relativistic Matter

325

In deriving (5.18), we had to multiply (5.16) by (|B| + 1)−1 to reduce the effect of R ¯ 3 x. Next, we cut the unit cube Q into thin rectangular the error term E3 (σ · B)ϕ · ϕd tubes Tα = Iα1 × Iα2 × Iα3 , where |Iα3 | = 1 and Iα1 × Iα2 is a small square of size ∼ (|B| + 1)−1/2 . Thus the Tα are parallel to the magnetic field. The number of distinct Tα ’s is ∼ (|B| + 1). Since |Iα1 |, |Iα2 | ∼ (|B| + 1)−1/2 , while |∇Aν | ≤ C|B| and Aν (x) −1/2 , for x ∈ Tα , where is independent of x3 , it follows that |Aν (x) − Aα ν | ≤ C(|B| + 1) α Aν is the value of Aν at the center of Tα . Hence, for x ∈ Tα , we have    2  2 i ∂ − Aν (x) ϕ ≥ 1 i ∂ − Aα ϕ −C(|B| + 1)|ϕ|2 , ν ∂xν 2 ∂xν and therefore (5.18) implies that ( Z X Z ∂ϕ 2 2 3 |σ · (i∇ − A)ϕ| d x ≥ c ∂x3 Q∗ α Tα    2  Z X 3 i ∂ − Aα +(|B| + 1)−1 x − C |ϕ|2 d3 x . (5.19) ϕ d ν ∂xν  ∗ Q ν=1,2

This is the basic one-electron estimate in the proof of (5.15). By rescaling and making a gauge transformation, we may regard the r.s. of (5.19) as the kinetic energy of a free particle in ∼ (|B| + 1) copies of the unit cube. Thus, the KE Bound (5.15) follows easily from (5.19) by separation of variables (reflecting the fact that there are ∼ K 3/2 eigenstates of a free particle in the unit cube having energy less than K). In using (5.15), we will drop the last term hNQ∗ ψ, ψi. Roughly speaking, if we neglect the distinction between Q∗ and Q, then the error term hNQ∗ ψ, ψi may be absorbed in the main term c KhNQ ψ, ψi if K is large. This is of course only a heuristic justification for dropping the last term in (5.15). In the full proof we keep the last term which complicates the analysis. We can start to combine the above ingredients to prove our theorem. The first main point is to prove the following local stability bound. Local Pauli Stability (Constant Fields). Let K, B0 be large constants, to be picked later, and let Q be a cube of side δ. Suppose the magnetic field B = curl A is constant on Q, and satisfies there |B| ≤ CB0 δ −2 . Then 1/2

3/2

CB0 δ · TPauli (ψ, A, Q) + P E(ψ, Q) ≥ −C K 5/2 B0

δ −1 + c

1/2

B0 K hNQ ψ, ψi. δ (5.20)

An essential point here is that B0 appears on the right side in (5.20), raised to a power −1/2 less than 2. To prove (5.20), we subdivide Q into subcubes {Qˆ α } of side ∼ B0 δ. Applying local stability of matter (5.14) to the Qˆ α , and summing over α using (5.10) and (5.12), we find that −1/2

C B0

δ · T (ψ, A, Q) + P E(ψ, A, Q)

≥ −C B0 δ −1 hNQ ψ, ψi − C B0 δ −1 MQ . 1/2

1/2

(5.21)

We drop the MQ term in (5.21) for expository purposes. The rough justification for this is that we may suppose (as explained earlier) that MQ and NQ are comparable, so

326

C. Fefferman, J. Fr¨ohlich, G.-M. Graf

that the two terms on the right side in (5.21) are also comparable. In the complete proof of our theorem, we do not drop the MQ term from (5.21), and matters become more complicated. The KE Comparison Lemma yields (roughly speaking) −1/2

C 0 B0

−1/2

δ · TPauli (ψ, A, Q) ≥ C B0

δ · T (ψ, A, Q) − C 0 B0 δ −1 hNQ , ψ, ψi. (5.22) 1/2

Putting (5.22) into (5.21), we find that −1/2

C B0

δ · TPauli (ψ, A, Q) + P E(ψ, Q) ≥ −C 0 B0 δ −1 hNQ ψ, ψi. 1/2

(5.23)

Also, the KE Lower Bound (5.15) shows, roughly speaking, that C B0 δ · TPauli (ψ, A, Q) ≥ c K B0 δ −1 hNQ ψ, ψi − C K 5/2 B0 δ −1 . 1/2

1/2

3/2

(5.24)

(Here, we have rescaled (5.15) from the unit cube to the cube Q.) Adding (5.23) and (5.24), we obtain 1/2

C B0 δ · TPauli (ψ, A, Q) + P E(ψ, Q) ≥ (c K − C 0 ) B0 δ −1 hNQ ψ, ψi − C K 5/2 B0 δ −1 . 1/2

3/2

(5.25)

If we take K to exceed a large enough universal constant, then (5.25) implies (5.20). The proof of local Pauli Stability is complete, modulo polite fictions. So far, we have worked with a constant magnetic field B. If we suppose instead that B varies only slightly on Q, then we can write the vector potential A in the form A = Ac + A# on Q, where Ac has constant magnetic field, and A# is small. We may then apply local Pauli stability (5.20) to the vector potential Ac , and regard A# as a small perturbation. In this spirit, we can extend (5.20) as follows. Local Pauli Stability 1 (Varying Magnetic Fields). Let K, B0 be large constants to be picked later, and let Q be a cube with side δ. Suppose the magnetic field B = curl A, satisfies Z

|B(x)|2 d3 x ≤ B02 δ −1 and

(5.26)

|B(x) − meanQ B|2 d3 x ≤ B0−1 δ −1 .

(5.27)

Q

Z

Q

Then we have that 1/2

C B0 δ · TPauli (ψ, A, Q) + P E(ψ, Q) ≥ −C K 5/2 B0 δ −1 + c 3/2

1/2

K B0 hNQ ψ, ψi. δ

(5.28)

Stability of Ultraviolet-Cutoff QED with Non-Relativistic Matter

that

327

We omit all discussion of the proof, but just point out that (5.26) and (5.27) suggest Z Z |B(x) − meanQ B|2 d3 x  |B(x)|2 d3 x, Q

Q

so that B varies only slightly on Q. The proof of our theorem proceeds by making a Calder´on-Zygmund decomposition of E3 into cubes {Qν } that satisfy (5.26) and (5.27). We apply Local Pauli Stability (5.28) to each Qν , and sum the result over ν. To construct the Qν , we begin by partitioning E3 into cubes {Q0α } of side L, where L is as in the statement of our theorem. We retain all the Q0α that satisfy (5.26), (5.27), and we cut all the other Q0α into subcubes of side L/2. Thus, we obtain a collection {Q1α } of cubes of side L/2. We retain all the Q1α that satisfy (5.26), (5.27), and we cut all the other Q1α into subcubes of side L/4. Thus, we obtain a collection {Q2α } of cubes of side L/4. We continue in this way, and let {Qν } denote the collection of all the cubes Qjα (j, α arbitrary) retained during the above construction. Thus E3 is partitioned into cubes {Qν } that satisfy (5.26) and (5.27). Let δν = side (Qν ), and note that δν ≤ L. Let Qν be one of the Calder´on-Zygmund cubes with side δν strictly less than L. Then Qν arose by cutting a cube Q+ν with side 2δν . We know that Q+ν cannot satisfy (5.26) and (5.27), since we cut Q+ν to arrive at Qν . If (5.26) fails for Q+ν , then, roughly speaking, we have Z |B(x)|2 d3 x > c B02 δν−1 . (5.29) Qν

If (5.27) fails for Q+ν , then, roughly speaking, we have that Z |B(x) − meanQν B|2 d3 x > c B0−1 δν−1 , Qν

so that

Z

δν2 | (∇ ⊗ B)(x) | d3 x > c B0−1 δν−1 , 2

(5.30)

Qν

by the Poincar´e inequality. Hence, for every Qν , we have ν ∈ S1 ∪ S2 ∪ S3 , where S1 = {ν : (5.29) holds},

S2 = {ν : (5.30) holds},

S3 = {ν : δν = L} . (5.31) Next, we bring in the function D(x) = min {|x − yk | | 1 ≤ k ≤ M } from the statement of our theorem. We set θν = maxx ∈ Qν e− D(x)/L .

(5.32)

Since Qν has side δν ≤ L, definition (5.32) shows at once that (for a constant c) c θν ≤ e− D(x)/L ≤ θν for all x ∈ Qν .

(5.33)

The basic estimate for our Calder´on-Zygmund cubes, apart from (5.26) and (5.27), is the following bound.

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X ν

θν δν−1 ≤ C B0−2 Z

Z

|B(x)|2 e−D(x)/L d3 x +

E3

L2 | (∇ ⊗ B) (x) | e−D(x)/L d3 x + 2

C B0

C M. L

(5.34)

E3

To prove (5.34), we argue as follows. If ν ∈ S1 , then (5.29) and (5.33) imply that Z 2 | B(x) | e−D(x)/L d3 x . θν δν−1 ≤ C B0−2 Qν

Summing over ν ∈ S1 , we obtain the bound Z X 2 −2 −1 θν δ ν ≤ C B0 | B(x) | e−D(x)/L d3 x . ν ∈ S1

(5.35)

E3

Similarly, if ν ∈ S2 , then (5.30) and (5.33) imply that Z 2 θν δν−1 ≤ C B0 L2 | (∇ ⊗ B) (x) | e−D(x)/L d3 x,

since δν ≤ L .

Qν

Summing over ν ∈ S2 , we get X

θν δν−1 ≤ C B0

ν ∈ S2

Z

L2 | (∇ ⊗ B) (x) | e−D(x)/L d3 x . 2

(5.36)

E3

For ν ∈ S3 , we have that θν δν−1

= θν L

−1

≤ CL

−4

Z e

−D(x)/L

d x ≤ CL 3

−4

M Z X

e−|x−yk |/L d3 x,

k=1 Q ν

Qν

by (5.33) and the definition of D(x). Summing over ν ∈ S3 , we find that X

θν δν−1 ≤ C L−4

M Z X

ν ∈ S3

k=1

e−|x−yk |/L d3 x =

C0 M. L

(5.37)

E3

Combining estimates (5.35), (5.36) and (5.37), we obtain the desired result (5.34). At last we are ready to prove our theorem. For each ν, we have C B0 δν · TPauli (ψ, A, Qν ) + P E(ψ, Qν ) ≥ − C B0 δν−1 θν K 5/2 + 1/2

3/2

1/2

c B0

θν δν−1 KhNQν ψ, ψi .

(5.38)

On one hand, (5.38) reduces to local Pauli stability (inequality (5.28)) if Qν contains at least one nucleus, since then we have θν = 1 by (5.32). On the other hand, if Qν contains no nuclei, then a glance at definition (5.7) gives P E(ψ, Qν ) = 0, so that (5.38) is an immediate consequence of the KE Lower Bound (5.15). (Recall, we are ignoring the last term on the right side in (5.15).) Thus, (5.38) holds in all cases. Summing (5.38) over all ν, and invoking (5.8) and (5.9), we obtain the estimate

Stability of Ultraviolet-Cutoff QED with Non-Relativistic Matter

* C

1/2 B0 L

N X 


2

σ (k) · i∇xk − A(xk )

k=1 3/2

+ hVC ψ, ψi ≥ − C K 5/2 B0 ( +

* cK

1/2 B0

X ν

X ν

θν δν−1

329

+ ψ, ψ θν δν−1

+

NQν ψ, ψ

− C

X ν

) θν δν−1

M Qν

, (5.39)

since δν ≤ L and θν = 1 when MQν 6= 0. We have agreed to suppose that NQν and MQν are comparable. Therefore, the term in curly brackets in (5.39) is positive, since K and B0 are large constants. Let us restrict the parameter L, by demanding that 1/2

C B0 Then (5.34), (5.39) and (5.40) imply −1/2

Z

(5.40)

| B(x) | e−D(x)/L d3 x + 2

hHPauli ψ, ψi + CK B0

E3

Z 5/2 C K B0

L ≤ 1.

2

L | (∇ ⊗ B)(x) | e 2

3/2

C K B0 d x ≥ − L

−D(x)/L 3

M,

(5.41)

E3

where CK depends on K. Our theorem follows easily from (5.41). We first take K to be a large enough universal −1/2 ≤ 0. constant. Then, with 0 as in our theorem, we pick B0 so large that CK B0  −1 1/2 Finally we take c (0, Z) = C B0 , with C as in (5.40). Thus, for L < c (0, Z), (5.40) holds; hence, (5.41) holds and tells us that Z 2 hHPauli ψ, ψi + 0 | B(x) | e−D(x)/L d3 x Z

E3

L2 | (∇ ⊗ B)(x) | e−D(x)/L d3 x ≥ − C (0, Z) 2

+ C (0, Z)

M . (5.42) L

E3

This is the conclusion of our theorem. Of course we have repeatedly oversimplified the argument. For full details, see [12]. References 1. Fr¨ohlich, J., Lieb, E.H., Loss, M.: Stability of Coulomb systems with magnetic fields I: The one-electron atom. Commun. Math. Phys. 104, 251–270 (1986) 2. Lieb, M., Loss, M.: Stability of Coulomb systems with magnetic fields II: The many-electron atom and the one-electron molecule. Commun. Math. Phys. 104, 271–282 (1986) 3. Loss, M., Yau, H.T.: Stability of Coulomb systems with magnetic fields III: Zero energy bound states of the Pauli operator. Commun. Math. Phys. 104, 283–290 (1986) 4. Fefferman, C.: Stability of Coulomb systems in a magnetic field. Proc. Natl. Acad. Sci. 92, 5006–5007 (1995)

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5. Lieb, E.H., Loss, M., Solovej, J.P.: Stability of matter in magnetic fields. Phys. Rev. Lett. 75, 985–989 (1995) 6. Bugliaro, L., Fr¨ohlich, J., Graf, G.M.: Stability of quantum electrodynamics with non-relativistic matter. Phys. Rev. Lett.77, 3494–3497 (1996) 7. Lieb, E.H.: The stability of matter: From atoms to stars: Selecta of Elliott H. Lieb. W. Thirring ed., Berlin–Heidelberg–New York: Springer, 1991 8. Glimm, J., Jaffe, A.: Quantum physics: A functional integral point of view. 2nd ed. , Berlin–Heidelberg– New York: Springer, 1987 9. Glimm, J., Jaffe, A.: Collected papers, Boston: Birkh¨auser 1985 10. Bach, V., Fr¨ohlich, J., Sigal, I.M.: Mathematical theory of non-relativistic matter and radiation. Lett. Math. Phys. 34, 183–201 (1995) 11. Fr¨ohlich, J.: Application of commutator theorems to the integration of representations of Lie algebras and commutation relations. Commun. Math. Phys. 54, 135–150 (1977) 12. Fefferman, C.: On electrons and nuclei in a magnetic field. Adv. Math. 124, 100–153 (1996) 13. Fefferman, C., de la Llave, R.: Relativistic stability of matter-I. Revista Matematica Iberoamericana 2, 119–161 (1986) 14. Minicozzi, W.: Notes to Lectures by C. Fefferman at Stanford University (1994) Communicated by B. Simon

Commun. Math. Phys. 190, 331 – 373 (1997)

Communications in

Mathematical Physics c Springer-Verlag 1997

Bispectral Algebras of Commuting Ordinary Differential Operators B. Bakalov? , E. Horozov, M. Yakimov?? Department of Mathematics and Informatics, Sofia University, 5 J. Bourchier Blvd., Sofia 1126, Bulgaria. E-mail: [email protected] Received: 8 April 1996 / Accepted: 9 April 1997

Abstract: We develop a systematic way for constructing bispectral algebras of commuting ordinary differential operators of any rank N . It combines and unifies the ideas of Duistermaat–Gr¨unbaum and Wilson. Our construction is completely algorithmic and enables us to obtain all previously known classes or individual examples of bispectral operators. The method also provides new broad families of bispectral algebras which may help to penetrate deeper into the problem. Contents 0 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332 1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336 1.1 Sato’s Grassmannian and KP–hierarchy . . . . . . . . . . . . . . . . . . . . . . . . . 336 1.2 Darboux transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338 1.3 Bessel operators, Bessel planes and related objects . . . . . . . . . . . . . . . . . 339 1.4 Involutions in Sato’s Grassmannian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340 2 Polynomial Darboux Transformations of Bessel Wave Functions . . . . . 342 3 Bispectrality of Polynomial Darboux Transformations . . . . . . . . . . . . . . 350 4 Polynomial Darboux Transformations of Airy Planes . . . . . . . . . . . . . . 355 5 Explicit Formulae and Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360 5.1 Monomial Darboux transformations of Bessel planes . . . . . . . . . . . . . . . 361 5.2 Polynomial Darboux transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . 367 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371 ? Present address: Department of Mathematics, MIT, Cambridge, MA 02139. E-mail: [email protected] ?? Present address: Department of Mathematics, University of California, Berkeley, CA 94720. E-mail: [email protected]

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0. Introduction In this paper we reconsider the bispectral problem. As stated in [DG], it asks for which ordinary differential operators L(x, ∂x ) there exists a family of eigenfunctions 9(x, z) that are also eigenfunctions for another differential operator 3(z, ∂z ), but this time in the “spectral parameter” z, to wit L(x, ∂x )9(x, z) = f (z)9(x, z), 3(z, ∂z )9(x, z) = Θ(x)9(x, z)

(0.1) (0.2)

for some functions f (z), Θ(x). Both operators L and 3 are called bispectral. This problem first appeared in [G1] in connection with “limited angle tomography” (see also [G2, G3, DG]). Later it turned out to be related to several, seemingly far from it, topics and in particular, to soliton mathematics. To be more specific, we have to mention the deep connection with some very actively developing areas of research in mathematics and theoretical physics like the Calogero–Moser particle system [W2, K] (see also [R]), additional symmetries of KdV and KP hierarchies [MZ, BHY4], representation of the theory W1+∞ –algebra [BHY4], etc. These studies not only revealed the rich mathematical structure behind the bispectral problem, but also (if we use a remark by G. Wilson [W2]) “deepened the mystery” around it. Thus, not only applications, but also purely mathematical questions motivated the great activity in the past few years in the bispectral problem. In the present paper we construct new families of bispectral operators. In order to explain better our contribution, we need to review some of the achievements in the subject. The first general result in the direction of classifying bispectral operators belongs to J. J. Duistermaat and F. A. Gr¨unbaum [DG]. They determined all second order operators L admitting an operator 3 such that the pair (L, 3) solves the bispectral problem (0.1, 0.2). Their answer is as follows. If we write the operator L in the standard Schr¨odinger form d2 L = 2 + u(x), dx the bispectral potentials u(x) are given (up to translations and rescalings of x and z) by the following list: u(x) = x (Airy); −2 u(x) = cx , c ∈ C (Bessel); u(x), which can be obtained by finitely many rational Darboux transformations from u(x) = 0; u(x), which can be obtained by finitely many rational Darboux 1 transformations from u(x) = − 2 . 4x

(0.3) (0.4) (0.5) (0.6)

The family (0.5) has previously appeared in [AMM, AM] and is known as “rational solutions of KdV ”. They can be obtained also by applying “higher KdV flows” to potentials vk (x) = k(k + 1)x−2 , k ∈ N. The second family (0.6) was interpreted by F. Magri and J. Zubelli [MZ] as potentials invariant under the flows of the “master symmetries” or Virasoro flows. Besides the classification of the bispectral operators by their order, another scheme has been suggested in [DG] and used in [W1]. Below we explain it in a general context as

Bispectral Algebras of Commuting Ordinary Differential Operators

333

it will be used throughout this paper. One may consider an operator L(x, ∂x ) as an element of a maximal algebra A of commuting ordinary differential operators [BC]. Following G. Wilson [W1], we call such an algebra bispectral if there exists a joint eigenfunction 9(x, z) for the operators L in A that satisfies also Eq. (0.2). The dimension of the space of eigenfunctions 9(x, z) is called rank of the commutative algebra A (see e.g. [KrN]). This number coincides with the greatest common divisor of the orders of the operators in A. For example, the operators with potentials (0.5) belong to rank 1 algebras and those with potentials (0.3, 0.4, 0.6) to rank 2 algebras [DG]. All rank 1 maximal bispectral algebras were recently found by G. Wilson [W1]. These algebras do not necessarily contain an operator of order two. The methods of the above mentioned papers [DG] and [W1] may seem quite different. Indeed, while in [DG] the “rational” Darboux transformations play a decisive role, G. Wilson [W1] uses planes in Sato’s Grassmannian obtained from the standard H+ = span{z k }k≥0 by imposing a number of conditions on it. One of our main observations is that both methods, appropriately modified, can be looked upon as the two sides of one general theory. From this new point of view in the present paper we construct nontrivial maximal bispectral algebras of any rank N , thus extending the results from [DG, W1]. For example, for any positive integer k we obtain bispectral algebras of rank N with the lowest order of the operators equal to kN . Our method allows us to obtain all classes and single examples of bispectral operators known to us by a unique method. At the same time we suggest an effective procedure for constructing bispectral operators, despite the fact that the theory involves highly transcendental functions like Airy or Bessel ones. The point is that the latter are used in the proofs while the algorithm given at the end of Sect. 3 performs arithmetic operations and differentiations only on explicit rational functions. In the rest of the introduction we describe in more detail the main results of the paper together with some of the ideas behind them. The framework of our construction is Sato’s theory of KP–hierarchy [S, DJKM, SW, vM]. In particular, our eigenfunctions are Baker or wave functions 9V (x, z) corresponding to planes V in Sato’s Grassmannian Gr and our algebras of commuting differential operators are the spectral algebras AV . We obtain our bispectral algebras by applying a version of Darboux transformations, introduced in our previous paper [BHY3], on specific wave functions which we call Bessel (and Airy) wave functions (see Sects. 1 and 4). As both notions are fundamental for the present paper we focus the attention of the reader on them. Bessel wave functions are the simplest functions which solve the bispectral problem (see [Z] where they were introduced and [F]). They can be defined as follows. For β = (β1 , . . . , βN ) ∈ CN , 9β (x, z) is the unique wave function satisfying x∂x 9β (x, z) = z∂z 9β (x, z) (i.e. 9β (x, z) depends only on xz) and Lβ (x, ∂x )9β (x, z) = z N 9β (x, z), where Lβ (x, ∂x ) = x−N (x∂x − β1 ) · · · (x∂x − βN ) is the Bessel operator. Obviously, the above equations lead to Lβ (z, ∂z )9β (x, z) = xN 9β (x, z). Similarly, for α = (α0 , α2 , . . . , αN −1 ) ∈ CN −1 consider the (generalized) Airy wave function (see [KS, Dij]) satisfying:

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B. Bakalov, E. Horozov, M. Yakimov

ÿ ∂xN +

N −1 X

! αi ∂xN −i − α0 x 9α (x, z) = z N 9α (x, z).

i=2

It depends only on α0 x + z N and again gives a simple solution to the bispectral problem. The Airy case is in many respects similar to the Bessel one. As we find the latter case richer in properties, we pay more attention to it, contenting ourselves only with a sketch of the former. Classically, a Darboux transformation [BC, Da] of a differential operator L, presented as a product L = QP , is defined by exchanging the places of the factors, i.e. L = P Q. Obviously, if 9(x, λ) is an eigenfunction of L, i.e. L(x, ∂x )9(x, λ) = λ9(x, λ), then P 9(x, λ) is an eigenfunction of L. Here we introduce Darboux transformations not only on individual operators but also on the entire spectral algebra corresponding to a Bessel (or Airy) plane. In other words, we apply them on operators L which are polynomials h(Lβ ) of Bessel (or Airy) operators. These transformations may be considered as B¨acklund–Darboux transformations on the corresponding wave functions [AvM]. Such Darboux transformation is completely determined by a choice of a ZN -invariant operator P (x, ∂x ) with rational coefficients normalized appropriately by a factor g −1 (z) to ensure that 9W (x, z) =

1 P (x, ∂x )9β (x, z) g(z)

is a wave function. We call 9W (x, z) (respectively W ) a polynomial Darboux transformation of 9β (respectively Vβ ). The definition of polynomial Darboux transformations of Airy planes is similar to that in the Bessel case with only minor modifications: P is not necessarily ZN -invariant and g(z) has to belong to C[z N ]. Thus we come to our main result. Theorem 0.1. If the wave function 9W (x, z) is a polynomial Darboux transformation of a Bessel or Airy wave function 9β (x, z), then it is a solution to the bispectral problem, i.e. there exist differential operators L(x, ∂x ), 3(z, ∂z ) and functions f (z), Θ(x) such that (0.1) and (0.2) hold. Note the difference between the classical definition and the definition introduced here. In contrast to [DG] where the authors make a finite number of “rational” Darboux transformations, we perform only one polynomial Darboux transformation to achieve the same result. Our definition of polynomial Darboux transformation is constructive as P (x, ∂x ) is determined by the finite dimensional space KerP . For this reason one can explicitly present at least one operator L ∈ AW ; it can be given by P h(Lβ )P −1 . Usually it is of high order. But as it is only one element of the whole bispectral algebra AW there can be eventually operators of a lower order. For example, the bispectral operators of [DG] are of order two. There is a simple procedure (see [BHY3]) to produce the entire bispectral algebra AW of commuting differential operators. In addition, one can show that the spectral curve SpecAW (see e.g. [AMcD] for definition) is rational, unicursal and ZN –invariant. In the course of our work we have widely used important ideas introduced by G. Wilson [W1]. Among them we mention first the idea of explicitly writing conditions on vectors of a plane V ∈ Gr which define the new plane obtained by a Darboux transformation. Second is the notion of involutions on Sato’s Grassmannian. In particular, we extend the bispectral involution b introduced in [W1] to the manifolds of polynomial

Bispectral Algebras of Commuting Ordinary Differential Operators

335

Darboux transformations. More precisely, we prove the following theorem, from which Theorem 0.1 is an obvious consequence. Theorem 0.2. (i) The bispectral involution is defined for planes W which are polynomial Darboux transformations of Bessel or Airy planes. (ii) The image bW of such a plane W is again a polynomial Darboux transformation of the corresponding Bessel (respectively) Airy plane. Our main concern in the present paper is to prove Theorem 0.2. Our second goal is to provide explicit formulae and examples (see Sect. 5), which are not only an illustration of our method but also show the existence of new families of bispectral operators with particular properties. Some of them generalize directly the well known ones like Duistermaat–Gr¨unbaum’s “even case” (0.6) [DG]. Other families exhibit quite different properties from the well known examples. In this respect Sect. 5 has also the role to supply diverse experimental material for new insights into the theory of bispectral algebras. We draw the attention of the reader also to the explicit formulae for the action of the bispectral involution on an important class of Darboux transformations (which we call monomial) of Bessel operators. As a particular case, we obtain such formulae for all second order bispectral operators found in [DG]. The class of monomial Darboux transformations has also other remarkable properties, e.g. they are connected to representation theory of W1+∞ –algebra. We do not touch this matter here for lack of space. The interested reader can learn about it in [BHY4]. A natural question is if the operators found in this paper form the entire class of bispectral operators. The answer is negative as recently shown in [BHY5]. Finally, for the reader’s convenience we now give a brief description of the organization of the paper. Section 1 is intended only for reference. It reviews results connected with Sato’s theory, which we need for the treatment of the bispectral problem. Besides the general notions (see e.g. [S, DJKM, SW]) we recall the involutions, introduced by G. Wilson [W1] and in particular, the bispectral involution. In Sect. 2 we introduce our (N ) of polynomial Darboux transformations of Bessel planes. We give two manifolds GrB equivalent definitions (Definition 2.5 and the one provided by the statement of Theorem 2.7). Section 3 contains our main results – Theorems 0.1 and 0.2. for the Bessel case. Section 4 deals with the analogs of Sect. 2 and 3 for the Airy case (although in different order). The last Sect. 5 is devoted to explicit examples of bispectral operators, which have been studied in other papers [DG, W2], as well as new families (which we have not seen elsewhere). The emphasis in Sect. 5 is rather on the simple algorithmic way of constructing bispectral operators (wave functions, etc.) than on the novelty of the examples. For readers who wish to see the main results as soon as possible we propose another plan of reading the paper. They can start with Sect. 2 and read it up to the statement of Theorem 2.7, returning to Sect. 1 for reference when needed. Then skipping the (technical) proof of Theorem 2.7, they can go Sect. 3. After that, taking for granted the proof of Theorem 3.2, they can look at the examples of bispectral operators, originating from Bessel ones in Sect. 5. Thus they will have a complete picture of the results in the Bessel case, and having this experience, they can easily go through the Airy case. More detailed information about the material included in each section can be found in its beginning. The present paper is a part of our project on the bispectral problem [BHY2]–[BHY5]. The main results contained here were announced at the conference of Geometry and Mathematical Physics, Zlatograd 95 (see [BHY1]).

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After this paper was written, we got a paper [KR] where some of the results about the Airy case were obtained independently.

1. Preliminaries In this section we have collected results about Sato’s theory, relevant to the bispectral problem. For the reader’s convenience we have divided the section into 4 subsections, whose titles, hopefully, give an idea of their content. The reader, who is acquainted with Sato’s theory may even skip this section and return to it for reference when needed. More detailed account of the material of the subsections can be found in their beginnings. 1.1. Sato’s Grassmannian and KP–hierarchy. We shall recall some facts and notation from Sato’s theory of KP-hierarchy needed in the paper. The survey below cannot be used as a systematic study. There are several complete texts on Sato’s theory, starting with the original papers of M. Sato and his collaborators [S, DJKM] (see also [SW, vM]). Consider the space of formal series o nX ak vk ak = 0 for k  0 . V= k∈Z

Sato’s Grassmannian Gr [S, DJKM, SW] consists of all subspaces (planes) W ⊂ V which have an admissible basis X wik vi , k = 0, 1, 2, . . . . wk = vk + i
In Sato’s theory V is most often realized as the space of formal Laurent series in z −1 via vk = z k . The Baker (or wave function) 9W (x, z) of the plane W contains all the information about W as the vectors wk = ∂xk 9W (x, z)|x=0 form an admissible basis of W . We can expand 9W (x, z) in a formal series ÿ ! ∞ X xz −k 9W (x, z) = e ak (x)z 1+ . (1.1) k=1

The wave function 9W (x, z) can also be written in terms of the so-called wave operator KW . This is a pseudo-differential operator defined by KW (x, ∂x ) = 1 +

∞ X

ak (x)∂x−k .

(1.2)

k=1

Then obviously 9W (x, z) = KW (x, ∂x )exz . Introduce also the pseudo-differential operator −1 . P = KW ◦ ∂x ◦ KW

(1.3)

For the treatment of the bispectral problem the following identity is crucial: P 9W (x, z) = z9W (x, z).

(1.4)

Bispectral Algebras of Commuting Ordinary Differential Operators

337

When it happens that some power of P , say P N , is a differential operator L we get that 9W (x, z) is an eigenfunction of an ordinary differential operator: L9W (x, z) = z N 9W (x, z).

(1.5)

It is easy to show that P N is differential iff the plane W is invariant under the multiplication by z N : (1.6) z N W ⊂ W. The submanifold of Gr consisting of planes W satisfying (1.6) is denoted by Gr(N ) . A very important object connected to the plane W is the algebra AW of polynomials f (z) that leave W invariant: AW = {f (z)|f (z)W ⊂ W }.

(1.7)

For each f (z) ∈ AW one can show that there exists a unique differential operator Lf (x, ∂x ), the order of Lf being equal to the degree of f , such that

Explicitly we have

Lf (x, ∂x )9W (x, z) = f (z)9W (x, z).

(1.8)

−1 . Lf = KW ◦ f (∂x ) ◦ KW

(1.9)

We denote the commutative algebra of these operators by AW , i.e. AW = {Lf |Lf 9W = f 9W , f ∈ AW }.

(1.10)

Obviously, AW and AW are isomorphic. We call AW a spectral algebra corresponding to the plane W . Following I. Krichever (see e.g. [KrN]), we introduce a rank of AW to be the dimension of the space of eigenfunctions 9W ; this number is equal to the greatest common divisor of the orders of the operators Lf . At the end we define the spectral curve corresponding to the plane W to be SpecAW (for definition, see e.g. [AMcD]). It is known that SpecAW is an algebraic curve (see [BC, I, KrN]). Remark 1.1. If 9W (x, z) is well defined for x = x0 we set vk = ex0 z z k and consider the subspace W x0 of V with an admissible basis wk = ∂xk 9W (x, z)|x=x0 . The wave functions of W x0 and W are connected by 9W x0 (x, z) = e−x0 z 9W (x + x0 , z) and obviously KW x0 (x, ∂x ) = KW (x + x0 , ∂x ). These shifts are inessential for the bispectral problem and for our proofs. Throughout the paper we shall sometimes work with 9W x0 calling it by abuse of notation a wave function of W and denoting it 9W . Remark 1.2. In Sato’s theory of KP hierarchy one usually considers the wave function ÿ ! ∞ P∞ X k 9W (t, z) = e k=1 tk z 1 + ak (t)z −k k=1

depending on all times t1 , t2 , . . . (here t1 = x). Then the operators KW (t, ∂x ) and P (t, ∂x ) are given again by formulae (1.2, 1.3) with ak (x) replaced with ak (t) and P satisfies the following infinite system of non-linear differential equations: ∂ P = [P+k , P ], ∂tk

(1.11)

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where P+k stands for the differential part of the k th power P k of the operator P , [ . , . ] is the standard commutator of pseudo-differential operators. The equations (1.11) are called the KP hierarchy. If the plane W lies in Gr(N ) , then (1.11) can be written in the form ∂ k/N L = [L+ , L] (1.12) ∂tk (L = P N ) and is called an N th reduction of the KP-hierarchy or an N th Gelfand–Dickey hierarchy [GD]. 1.2. Darboux transformations. In this subsection we recall the notion of Darboux transformations on objects connected to points of Sato’s Grassmannian, introduced in our recent paper [BHY3]. Recall that a Darboux transformation [Da] of an ordinary differential operator L is obtained by presenting it as a product and exchanging the places of the factors: L = QP 7→ L = P Q. A (monic) operator L is completely determined by its kernel: if {80 , . . . , 8n−1 } is a basis of KerL then (see e.g. [I]) L8 =

Wr(80 , . . . , 8n−1 , 8) , Wr(80 , . . . , 8n−1 )

(1.13)

where Wr denotes the Wronski determinant. The next lemma answers the question when the factorization L = QP is possible (see e.g. [I]). Lemma 1.3. (i) For a given basis {80 , . . . , 8n−1 } of KerL set q1 (x) = ∂x log 80 , Wr(80 , . . . , 8k−1 ) , qk (x) = ∂x log Wr(80 , . . . , 8k−2 )

(1.14) 2 ≤ k ≤ n.

(1.15)

Then the operator L can be factorized as follows: L = (∂x − qn )(∂x − qn−1 ) · · · (∂x − q1 ).

(1.16)

(ii) L can be factorized as L = QP iff KerP ⊂ KerL.

(1.17)

KerQ = P (KerL).

(1.18)

In this case A slightly more general construction is the following one. For operators L and P such that the kernel of P is invariant under L, i.e. L(KerP ) ⊂ KerP,

(1.19)

L 7→ L = P LP −1 .

(1.20)

we consider the transformation The fact that L is a differential operator follows from Lemma 1.3 (ii). Indeed, L(KerP ) ⊂ KerP is equivalent to KerP ⊂ Ker(P L). In [BHY3] we defined a version of Darboux transformation on points in Sato’s Grassmannian and on related objects – wave functions, tau-functions and spectral algebras.

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Definition 1.4. We say that a plane W (or the corresponding wave function 9W (x, z)) is a Darboux transformation of the plane V (respectively wave function 9V (x, z)) iff there exist monic polynomials f (z), g(z) and differential operators P (x, ∂x ), Q(x, ∂x ) such that 1 P (x, ∂x )9V (x, z), g(z) 1 9V (x, z) = Q(x, ∂x )9W (x, z). f (z) 9W (x, z) =

(1.21) (1.22)

An equivalent definition is that W is a Darboux transformation of V iff fV ⊂ W ⊂

1 V g

(1.23)

for some polynomials f (z), g(z). Simple consequences of Definition 1.4 are the identities P Q9W (x, z) = f (z)g(z)9W (x, z), QP 9V (x, z) = f (z)g(z)9V (x, z).

(1.24) (1.25)

The operator L = P Q ∈ AW is a Darboux transformation of L = QP ∈ AV . Having in mind applications to the bispectral problem, the most important for our study is the case when (1.26) AV = C[z N ], AV = C[LV ] for some natural number N and a differential operator LV of order N . (This is the simplest case of rank N spectral algebra with a spectral curve C.) Then due to (1.25) we have f (z)g(z) = h(z N ), QP = h(LV )

(1.27) (1.28)

for some polynomial h(z). In [BHY3] we connected the spectral algebra AW (respectively AW ) with AV (respectively AV ). Proposition 1.5. [BHY3] (i) The Darboux transformations preserve the rank of the spectral algebras, i.e. if W is a Darboux transformation of V then rankAW = rankAV . (ii) If AV = C[LV ], ordLV = N then  (1.29) AW = u ∈ C[z N ] | u(LV )KerP ⊂ KerP and

 AW = P u(LV )P −1 | u ∈ AW .

(1.30)

1.3. Bessel operators, Bessel planes and related objects. Now we define the planes of the Sato’s Grassmannian on which we shall perform the Darboux transformations. For β ∈ CN such that N X N (N − 1) (1.31) βi = 2 i=1

we introduce the ordinary differential operator

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Pβ (Dz ) = (Dz − β1 )(Dz − β2 ) · · · (Dz − βN ),

(1.32)

where Dz = z∂z , and consider the differential equation Pβ (Dz )8β (z) = z N 8β (z).

(1.33)

For every sector S with a center at the irregular singular point z = ∞ and an angle less than 2π, Eq. (1.33) has a solution 8β with an asymptotics 8β (z) ∼ 9β (z) = e

z

 1+

∞ X

ak (β)z −k

 (1.34)

k=1

for |z| → ∞, z ∈ S (see e.g. [Wa]). Here ak (β) are symmetric polynomials in βi . The function 8β (z) can be taken to be (up to a rescaling) the Meijer’s G-function  −z N 1  0 (1.35) 8β (z) = GN β , 0N N N see [BE], §5.3. The next definition is fundamental for the present paper. Definition 1.6. Bessel wave function is called the function 9β (x, z) = 9β (xz) (cf. [F, Z]). The Bessel operator Lβ is defined as Lβ (x, ∂x ) = x−N Pβ (Dx ).

(1.36)

A Bessel wave function 9β defines a plane Vβ ∈ Gr (called Bessel plane) by the standard procedure: Vβ = span{∂xk 9β (x, z)|x=1 }. (In fact 9Vβ (x, z) = e−z 9β (x + 1, z), cf. Remark 1.1; we took x0 = 1 because 9β (x, z) is singular at x = 0, arbitrary x0 6= 0 will do.) Because 9β (x, z) depends only on xz, i.e. Dx 9β (x, z) = Dz 9β (x, z),

(1.37)

it gives the simplest solution to the bispectral problem: Lβ (x, ∂x )9β (x, z) = z N 9β (x, z),

(1.38)

Lβ (z, ∂z )9β (x, z) = x 9β (x, z).

(1.39)

N

1.4. Involutions in Sato’s Grassmannian. In this subsection, following G. Wilson [W1], we define several involutions on points of Sato’s Grassmannian and on related objects – wave functions and wave operators. Besides the general properties of the involutions taken from [W1], we specify their action on Bessel planes. We introduce after [DJKM] the non-degenerate form in V (realized as the space of formal Laurent series in z −1 ) B(f, g) = −Res∞ f (z)g(−z) dz,

f, g ∈ V.

If V ∈ Gr is a plane, define aV ∈ Gr to be the plane orthogonal to V with respect to the form B( . , . ), to wit aV = {g(z) | B(f, g) = 0, for all f ∈ V }.

(1.40)

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Obviously, a(aV ) = V , i.e. the map a is an involution. Following [W1], we call it the adjoint involution. On the wave operator KW (1.2) the involution a acts as [W1] KaV = KV∗


−1

,

(1.41)

where ∗ is the formal conjugation on pseudo-differential operators, i.e. the antiautomorphism defined by ∂x∗ = −∂x , x∗ = x. For our purposes the most important property of the involution a is that it inverses inclusions, i.e.: if W ⊂ V, then aW ⊃ aV.

(1.42)

The following proposition will be used in the description of the action of a on Darboux transformations. Its simple proof is similar to that of Corollary 7.7 from [W1]. Proposition 1.7. (i) If 9W (x, z) =

1 P (x, ∂x )9V (x, z), then g(z)

9aV (x, z) =

1 ∗ P (x, ∂x )9aW (x, z), g(z) ˇ

(1.43)

where g(z) ˇ = g(−z). (ii) [W1] Let AV be the algebra of operators (1.10). Then AaV consists of the conjugated operators of AV . Proof. We have 9W (x, z) = g −1 (z)P 9V (x, z) = g −1 (z)P KV exz = P KV g −1 (∂x )exz , which implies KW = P KV g −1 (∂x ). Applying the involution a we obtain ∗ KaW = KW


−1

= (P ∗ )−1 KV∗

yielding P ∗ KaW = KaV g(∂ ˇ x ) and hence (1.43).


−1

g(∂ ˇ x ),



The sign involution s [W1] is defined on the wave functions by the property 9sV (x, z) = 9V (−x, −z).

(1.44)

On the wave operators KV (x, ∂x ) (1.44) translates into KsV (x, ∂x ) = KV (−x, −∂x ).

(1.45)

sV = {f (−z) | f (z) ∈ V }.

(1.46)

The plane sV is defined by

In the subsequent chapters we shall need the action of a and s on the Bessel planes. We describe them in the next proposition. Proposition 1.8. The involutions s and a act on Bessel planes Vβ (β ∈ CN ) as follows sVβ = Vβ , aVβ = Va(β) , where a(β) = (N − 1)δ − β, δ = (1, 1, . . . , 1).

(1.47) (1.48)

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Proof. We compute the action of the involutions on the corresponding wave functions. Obviously, 9sVβ (x, z) = 9β (−x, −z) = 9β (x, z), showing (1.47). Proposition 1.7, Eq. (1.38) and the fact that L∗β = (−1)N La(β) imply 9aVβ (x, z) = γ(z)9Va(β) (x, z) for some formal power series γ(z) in z. To show that γ(z) = 1 we notice that 9aVβ (x, z) depends on xz. Indeed, (1.37) implies KVβ (cx, c−1 ∂x ) = KVβ (x, ∂x )  for all c 6= 0 and the same is true for KaVβ = (KV∗β )−1 . We end this section by recalling the bispectral involution b which Wilson [W1] introduced for the purpose of the bispectral problem. Contrary to a and s, the bispectral involution b is not defined on the entire Grassmannian. Whenever one can define b, put 9bV (x, z) = 9V (z, x),

(1.49)

i.e. the involution b interchanges the places of the arguments x and z. A simple example of a point V ∈ Gr, where the involution b is well defined is the Bessel plane Vβ . It immediately follows from the definition that 9β (x, z) = 9β (z, x),

i.e. bVβ = Vβ .

In terms of the bispectral involution our approach to the bispectral problem can be formulated geometrically as follows: Find points V ∈ Gr such that 1) g(z)V ⊂ V for some nontrivial polynomial g(z); 2) bV exists and f (z)bV ⊂ bV for some nontrivial polynomial f (z). A very important general property of b, which we intend to use, is its connection to the other involutions [W1]: ab = bas. (1.50) For completeness we also point out that the involution s commutes with a and b: as = sa,

bs = sb.

(1.51)

2. Polynomial Darboux Transformations of Bessel Wave Functions The main purpose of this section is to introduce the submanifolds (denoted below by (N ) ) on which, as we prove in the next section, the bispectral involution b is wellGrB (N ) defined, and whose points correspond to bispectral operators. The points of GrB are obtained by a version of B¨acklund–Darboux transformation performed on Bessel wave functions (or equivalently, on polynomials h(Lβ ) of Bessel operators). Below we call these transformations polynomial Darboux transformations. Definition 2.5, where this is done and the statement of Theorem 2.7, where we provide an equivalent definition, form the heart of the present section. Definition 2.5 has the advantage to be more natural and to supply an algorithmic procedure for constructing bispectral operators. The second definition (from Theorem 2.7) is more suitable for the proof of our bispectrality theorem in Sect. 3. The reader who wishes to see as soon as possible the main results of the paper can use the second definition, the one from the statement of Theorem 2.7, skipping its technical proof, which occupies half of the section.

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In the first half of this section we describe the kernel of the operator P from Definition 1.4 and (which is equivalent) – the conditions of the type as in [W1], imposed on a Bessel plane, which define the corresponding Darboux transformation. To do so, we first need a description of the kernels of the operators h(Lβ ) which are polynomials of Lβ . Fix β ∈ CN satisfying (1.31) and let Vβ be the corresponding Bessel plane (see Subsect. 1.3). Throughout this section W will be a Darboux transformation of Vβ (we shall use the notation of Definition 1.4 with V = Vβ ). We shall need a lemma describing the kernel of the operator h(Lβ ) for an arbitrary polynomial h. Lemma 2.1. Let h(z) be a polynomial
d 1
d r · · · z − λN , h(z) = z d0 z − λN 1 r

N λN i 6= λj , λ0 = 0, di ≥ 0.

(2.1)

Then we have
d i Lr . (i) Kerh(Lβ ) = i=0 Ker Lβ − λN r (ii) (Lβ )d = Lβ d , where β d = (β1 , β1 + N, . . . , β1 + (d − 1)N, . . . , βN , . . . , βN + (d − 1)N ).

(2.2)

(iii) If {β1 , . . . , βN } = {α1 , . . . , α1 , . . . , αs , . . . , αs } with distinct α1 , . . . , αs , then | {z } | {z } k1

ks



KerLβ = span xαi (ln x)k 1≤i≤s, 0≤k≤ki −1 . (iv) For λ 6= 0 Ker Lβ − λN


d

 = span ∂zk 9β (x, z)|z=λεj 0≤k≤d−1, 0≤j≤N −1 ,

where ε = e2πi/N is an N th root of unity. The proof being obvious is omitted (cf. Lemma 1.3).  Let us consider the simplest example of a Darboux transformation. Set h(z) = z d ,

g(z) = z n ,

f (z) = z dN −n

(2.3)

d

and γ = β , i.e. γ(k−1)d+j := βk + (j − 1)N,

1 ≤ k ≤ N, 1 ≤ j ≤ d.

(2.4)

For an n-element subset I of {1, . . . , dN } such that γi 6= γj for i 6= j ∈ I, we put KerP = span{xγi }i∈I .

(2.5)

Such P corresponds to a Darboux transformation 9I (x, z) of 9β (x, z). The following simple fact will be useful in the sequel. Lemma 2.2. 9I (x, z) is again a Bessel wave function: 9I (x, z) = 9γ+dN δI −nδ (x, z). Here and further we use the vectors δI , δ defined by  1, if i ∈ I (δI )i = 0, if i 6∈ I and

δi = 1 for all i ∈ {1, . . . , dN }.

(2.6)

(2.7)

(2.8)

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Proof. By definition

9I (x, z) = z −n LγI 9γ (x, z),

(2.9)

where γI = {γi }i∈I . Then 9I (x, z) is an eigenfunction of the differential operator  LγI Lγ L−1 γI , which is straightforwardly computed to be equal to Lγ+dN δI −nδ . Our next step is to study the spectral algebra Aβ ≡ AVβ of a Bessel plane Vβ (see (1.7, 1.10)). Lemma 2.3. If L(x, ∂x )9β (x, z) = u(z)9β (x, z) for some operator L ∈ Aβ and some polynomial u(z) ∈ Aβ , then L is a linear combination of Bessel operators Lα , α ∈ Ck such that Lα 9β (x, z) = z k 9β (x, z). P Proof. Let u(z) = uk z k , uk 6= 0, 0 ≤ k ≤ M for some M . Then for arbitrary c 6= 0 we have u(cz)9(x, z) = u(cz)9(c−1 x, cz) = L(c−1 x, c∂x )9β (x, z). This implies that u(cz) ∈ Aβ and thus z k ∈ Aβ . On the other hand Dz Vβ ⊂ Vβ and the compatibility condition is of the form (Dz − α1 ) · · · (Dz − αk )9β (1, z) = z k 9β (1, z), which implies Lα 9β (x, z) = z k 9β (x, z).  Let us introduce the following terminology. We say that β ∈ CN is generic if Vβ 0 is not a Darboux transformation of another Bessel plane Vβ 0 with β 0 ∈ CN , N 0 < N . The next proposition seems obvious but we do not know a simpler proof. Proposition 2.4. For a generic β ∈ CN we have Aβ = C[z N ],

Aβ = C[Lβ ].

(2.10)

Proof. We shall prove that if rankAβ = r < N then there exists β 0 ∈ C r (with Aβ 0 = C[z r ]) such that Vβ is a Darboux transformation of Vβ 0 . The main idea is to apply to Vβ Darboux transformations which lead again to Bessel planes but reduce the order of the operator Lβ . Note that according to Proposition 1.5 they preserve the rank r of the spectral algebra. Split the set β into congruent modN Z classes   (p) (p) (1) , . . . , β , . . . , β β = β1(1) , . . . , βN 1 Np 1 such that

βs(i) − βt(j) 6∈ N Z

for i 6= j andall s, t

(N = N1 + · · · + Np ). By a Darboux transformation this β can be changed to


β 0 = β (1) , . . . , β (1) , . . . , β (p) , . . . , β (p) | {z } | {z } N1

such that

|Re(β (i) − β (j) )| < N

Np

and β (i) 6= β (j) for i 6= j

(2.11)

(see Lemma 2.2). Suppose that Aβ 0 6= C[z N ]. Then by Lemma 2.3 there exists a Bessel operator Lα such that (2.12) Lα 9β 0 (x, z) = z M 9β 0 (x, z), α ∈ CM

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and Lα 6= Lkβ 0 for any k. It is clear that Lα L β 0 = L β 0 L α ,

(2.13)

which is equivalent to  α1 + N, α2 + N, . . . , αM + N, β (1) , . . . , β (1) , . . . , β (p) , . . . , β (p)  = β (1) + M, . . . , β (1) + M, . . . , β (p) + M, . . . , β (p) + M, α1 , . . . , αM . Now if M > N this implies that β 0 ⊂ α and therefore there exists a Bessel operator Lα0 such that Lα = Lα0 Lβ 0 and Lα0 Lβ 0 = Lβ 0 Lα0 . Repeating the same argument with α0 , we obtain that there exists Lα satisfying (2.13) with M < N . But then (2.13) is equivalent to Vβ 0 = Vα . By Proposition 1.5 r = rankAβ = rankAβ 0 = rankAα divides M and N . If Vα = Vβ 0 = C[z r ] this finishes the proof. Otherwise we can repeat the above argument with Vα instead of Vβ 0 .  Now we come to the main purpose of this section: the definition of manifolds of Darboux transformations, which will give solutions to the bispectral problem. To get some insight we shall consider, following Wilson [W1], the geometrical meaning of Darboux transformations, provided by the so-called conditions C. Proposition 2.4 implies that for generic β ∈ CN (1.27, 1.28) hold with V = Vβ and KerP is a subspace of Kerh(Lβ ). Each element f of KerP corresponds to a condition c (a linear functional on Vβ ), such that f (x) = hc, 9β (x, z)i,

(2.14)

c acts on the variable z. These linear functionals form an n-dimensional linear space C (space of conditions) where n = ordP . In this terminology the definition of Darboux transformation can be reformulated as o 1 n v ∈ Vβ hc, vi = 0 for all c ∈ C W = g(z) (see [W1, BHY3]). Following Wilson [W1], we call the condition c supported at λ iff it is of the form (cf. Lemma 2.1 (iv)) X ak ∂zk |z=λ (2.15) c= k

(the sum is over k ∈ Z≥0 and only a finite number of ak 6= 0). For Bessel wave functions this definition does not make sense when λ = 0 (since 9β (x, z) has a singularity at z = 0 for N > 1). In this case we say that c is supported at z = 0 iff it is of the form (cf. Lemma 2.1 (ii, iii)) XX bαj xα (ln x)j . hc, 9β (x, z)i = α

SN

j

The sums are over α ∈ i=1 {βi + N Z≥0 } and 0 ≤ j ≤ mult(α) − 1, where mult(α) is the multiplicity of α in the above union (only a finite number of bαj 6= 0). The space of conditions C is called homogeneous iff it has a basis of homogeneous conditions c (i.e. the support of c is a point).

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It is easy to see that if C is homogeneous then the spectral curve SpecAW is rational and unicursal [W1] (i.e. its singularities can be only cusps) – the condition c supported at λ “makes” a cusp at λ. For rank one algebras rationality and unicursality of SpecAW are necessary and sufficient for bispectrality [W1]. For rank N > 1 another necessary condition is that SpecAW be ZN -invariant, i.e. AW ⊂ C[z N ].

(2.16)

When W is a Darboux transformation of a Bessel plane Vβ , with generic β ∈ CN , this condition is satisfied because of Propositions 2.4, 1.5. It is natural to demand that the space of conditions C (or equivalently KerP ) also be ZN -invariant. The ZN -invariance of KerP simply means that f (x) ∈ KerP ⇒ f (εx) ∈ KerP,

ε = e2πi/N .

(2.17)

It is easy to see that C is homogeneous and ZN -invariant iff KerP has a basis which is a union of: (i) Several groups of elements supported at 0 of the form: ∂yl

k0 mult(βX i +kN )−1 X k=0

 bkj xβi +kN y j

j=0

, y=ln x

0 ≤ l ≤ j0 ,

(2.18)

where j0 = max{j|bkj 6= 0 for some k}; (ii) Several groups of elements supported at the points εi λ (0 ≤ i ≤ N − 1, λ 6= 0) of the form: k0 X ak εki ∂zk 9β (x, z)|z=εi λ , 0 ≤ i ≤ N − 1. (2.19) k=0

Instead of (2.19) we can also take k0 X

ak Dzk 9β (x, z)|z=εi λ ,

0 ≤ i ≤ N − 1.

(2.20)

k=0

Denote by n0 the number of conditions c supported at 0 (i.e. the number of elements of the form (2.18) in the above basis of KerP ). For 1 ≤ j ≤ r denote by nj the number of conditions c supported at each of the points εi λj , 0 ≤ i ≤ N − 1 (i.e. the number of groups of elements of the form (2.19) with λ = λj ). We have at last arrived at our fundamental definition. Definition 2.5. We say that the wave function 9W (x, z) is a polynomial Darboux transformation of the Bessel wave function 9β (x, z), β ∈ CN , iff (1.21) holds (for V = Vβ ) with P (x, ∂x ) and g(z) satisfying: (i) The corresponding space of conditions C is homogeneous and ZN -invariant, or equivalently KerP has a basis of the form (2.18, 2.19). (ii) The polynomial g(z) is given by
n1
nr · · · z N − λN , (2.21) g(z) = z n0 z N − λN 1 r where nj are the numbers defined above. (N ) = set of planes W satisfying (i), (ii) by GrB (β) and put GrB S We denote the N Gr (β), β ∈ C -generic. B β

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We point out that the form (2.21) of g(z) was introduced for N = 1 by Wilson [W1]. Qr QN −1 (Note that g(z) = z n0 j=1 i=0 (z−εi λj )nj .) We make this normalization in order that 9bW (x, z) = 9W (z, x) be a wave function; for the bispectral problem it is inessential. Definition 2.6. We say that the polynomial Darboux transformation 9W (x, z) of 9β (x, z) is monomial iff g(z) = z n0 (i.e. iff all conditions c are supported at 0). Denote the set of the corresponding planes S (N ) N W by GrM B (β) and put GrM B = β GrM B (β), β ∈ C -generic. The next theorem provides another equivalent definition of GrB (β) and is used essentially in the proof of the bispectrality in the next section. Theorem 2.7. The wave function 9W (x, z) is a polynomial Darboux transformation of the Bessel wave function 9β (x, z), β ∈ CN , iff (1.21, 1.22, 1.27, 1.28) hold (for V = Vβ ) and (i) The operator P has the form P (x, ∂x ) = x−n

n X

pk (xN )(x∂x )k ,

(2.22)

k=0

where pk are rational functions, pn ≡ 1. (ii) There exists the formal limit lim e−xz 9W (x, z) = 1.

(2.23)

x→∞

The proof will be split into three lemmas. Before giving it we shall make a few comments. The rationality of P is always necessary for bispectrality [DG, W1], (2.22) also imposes the ZN -invariance. The condition (2.23) is necessary in order that 9bW (x, z) = 9W (z, x) be a wave function. The limit in (2.23) is formal in the sense that it is taken in the coefficient at any power of z in the formal expansion (1.1) separately, i.e. lim aj (x) = 0

x→∞

for all j ≥ 1.

(2.24)

Our first lemma is similar to Proposition 5.1 ((i) ⇒ (ii)) from [W1]. Lemma 2.8. If P has rational coefficients and is ZN -invariant (see (2.22)) then the conditions C are homogeneous and ZN -invariant (see (2.18, 2.19)) Proof. If KerP = span{f0 , . . . , fn−1 }, the second coefficient of P is −∂x log Wr(f0 , . . . , fn−1 ) and is rational. Lemma 2.1 implies that Wr(f0 , . . . , fn−1 ) is of the form xα eλx × (Laurent series in x−1 ). In particular, each element of KerP is a sum of terms of the form eλx × (Laurent series in x−1 )

or

xα (ln x)k .

We order the (finite) set of all such eλx and xα (ln x)k occurring in KerP . The highest term in Wr(fi ) is just the Wronskian of the highest terms of the fi . If it vanishes then the

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highest terms of the fi are linearly dependent, so by a linear combination we can obtain a new basis with lower highest terms. So we can suppose that the highest term of Wr(fi ) is non-zero. Repeating the same argument with the lowest term, we shall finally obtain a basis whose elements consist of only one term, i.e. are homogeneous (cf. [W1]). Because the coefficients of P are rational, (1.13) implies that it does not matter which branch of the functions xα (ln x)k in KerP we take for x ∈ C. Let j0 X

fj (x)(ln x)j ∈ KerP

j=0

P

P with fj (x) = α bαj xα . Then fj (x)(ln x + 2lπi)j ∈ KerP for arbitrary l ∈ Z and also for l ∈ C since it is polynomial in l. Taking the derivative with respect to l we obtain that j0 X fj (x)j(ln x)j−1 j=0

also belongs to KerP . P On the other hand the ZN -invariance of P (see (2.17)) implies fj (εx)(ln x + 2πi/N )j ∈ KerP and j0 X fj (εx)(ln x)j ∈ KerP j=0

for ε = e2πi/N . Now it is obvious that KerP has a basis of the form (2.18, 2.19).



Lemma 2.9. If KerP has a basis of the form (2.18, 2.19) then P has rational coefficients and is ZN -invariant (see (2.22)). Proof. Consider first the case when the basis of KerP is X ak ∂zk 9β (εi x, z)|z=λ , 0 ≤ i ≤ N − 1, λ 6= 0. fi (x) = k


n We shall show that det ∂x j fi (x) 0≤i,j≤N −1 is a rational function of x for arbitrary nj ∈ Z≥0 . Using (1.37, 1.38) we can express all derivatives of 9β (x, z) (both with respect to z and x) only by ∂xk 9β (x, z), 0 ≤ k ≤ N − 1, to obtain ∂xnj fi (x) =

N −1 X

αkj (x, λ)∂xk 9β (εi x, λ)

(2.25)

k=0

with rational coefficients αkj . Therefore


det ∂xnj fi (x) = det αkj (x, λ) det ∂xk 9β (εi x, λ) .
But det ∂xk 9β (εi x, λ) = const because the second coefficient of Lβ −λ is 0. If the basis f0 , . . . , fmN −1 of KerP contains m groups of the type considered above (i.e. (2.19)) we can represent the matrix
∂xnj fi (x) 0≤i,j≤mN −1 , nj ∈ Z≥0 ,

Bispectral Algebras of Commuting Ordinary Differential Operators

in the block-diagonal form



W1



0

  0 

349

..  .  

W2 ...

Wm where each block Ws has the form already considered above. This can be achieved by columns and rows operations, using the representation (2.25). If in addition there are some groups of elements of the form (2.18), we kill the logarithms by columns operations and then cancel the powers xβi from the numerator and the denominator of (1.13).  Lemma 2.10. If C is homogeneous and h, g are as in (2.1), (2.21), then (2.23) is satisfied. Conversely, (2.23) implies (2.21). Proof. The second part of the lemma is an obvious consequence of the first one. For a basis {8i (x)}0≤i≤dN −1 of Kerh(Lβ ) (d = deg h) we consider the basis of KerP , dN −1 X fk (x) = aki 8i (x), 0 ≤ k ≤ n − 1. (2.26) i=0

Formulae (1.21, 1.13) imply Wr f0 (x), . . . , fn−1 (x), 9β (x, z)
9W (x, z) = g(z)Wr f0 (x), . . . , fn−1 (x)
P det AI Wr 8I (x) 9I (x, z)
P = . det AI Wr 8I (x)


(2.27) (2.28)

The sum is taken over all n-element subsets I = {i0 < i1 < . . . < in−1 } ⊂ {0, 1, . . . , dN − 1}, and here and further we use the following notation: A is the matrix from (2.26) and A I =  (ak,il )0≤k, l≤n−1 is the corresponding minor of A, 8I (x) = 8i0 (x), . . . , 8in−1 (x) is the corresponding subset of the basis {8i (x)} of Kerh(Lβ ) and
Wr 8I (x), 9β (x, z)
9I (x, z) = (2.29) g(z)Wr 8I (x) is a Darboux transformation of 9β (x, z) with a basis of KerP fk = 8ik . Using (2.28) it is sufficient to prove (2.23) for 9I (x, z), hence we can take KerP consisting of functions fi (x) = ∂zki 9β (x, z)|z=λi , αi

li

fi (x) = x (ln x) ,

0 ≤ i ≤ p − 1,

p ≤ i ≤ n − 1.

(2.30) (2.31)

We shall consider the case when λi 6= λj for i 6= j. The general case can be reduced to this by taking a limit. In the formula (2.27) we expand the determinants in the last n − p columns (using the Laplace rule):

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B. Bakalov, E. Horozov, M. Yakimov

Wr(f, 9β ) = Wr(f ) =

X X

± det ∂xjs fi (x), ∂xjs 9β (x, z)


± det ∂xjs fi (x)

0≤s,i≤p−1







0≤s≤p 0≤i≤p−1

· det ∂xjs fi (x)


. det ∂xjs fi (x)

p≤s,i≤n−1

p+1≤s≤n p≤i≤n−1

,

;

(2.32) (2.33)

where the sums are over the permutations (j0 , . . . , jn ) (resp. (j0 , . . . , jn−1 )) of (0, . . . , n) (resp. (0, . . . , n−1)) such that j0 < . . . < jp and jp+1 < . . . < jn (resp. j0 < . . . < jp−1 and jp < . . . < jn−1 ). We extract the terms with the highest power of x in the numerator and in the denominator of (2.27). Obviously, Pn−1 P
α − js s RJ (ln x) (2.34) det ∂xjs fi (x) p≤i≤n−1 = const · x i=p i for some polynomials RJ (ln x) (J is the permutation (js )). On the other hand for 0 ≤ i≤p−1   (2.35) ∂xjs fi (x) = ∂xjs ∂zki 9β (x, z)|z=λi = xki eλi x λji s + O(x−1 ) and


∂xjs 9β (x, z) = exz z js + O(x−1 ) .

(2.36)

Now it is easy to see that the leading terms are obtained for the permutations (n − p, n − p + 1, . . . , n, 0, 1, . . . , n − p − 1), respectively

(n − p, n − p + 1, . . . , n − 1, 0, 1, . . . , n − p − 1).

Substituting (2.34, 2.35, 2.36) in (2.32, 2.33) and canceling the determinant (2.34) for J = (0, 1, . . . , n − p − 1), we derive that lim e−xz P (x, ∂x )9β (x, z)

x→∞

is a fraction of two van der Monde determinants and therefore is equal to g(z).



3. Bispectrality of Polynomial Darboux Transformations In this section we prove the main result of the paper, Theorem 3.3, claiming that polynomial Darboux transformations (see Definition 2.5), performed on Bessel operators, produce bispectral operators. On one hand Theorem 3.3 is an almost obvious consequence of Theorem 3.2 in which we prove that the bispectral involution is well-defined on the submanifolds GrB (β) and maps them into themselves. The importance of Theorem 3.2 is not only to provide a proof of our main result (Theorem 3.3) but also to enlighten the bispectral involution. Its proof uses only the definition of polynomial Darboux transformation from Theorem 2.7 (i.e. it does not use Definition 2.5). On the other hand, the proof is completely constructive and together with Definition 2.5 it provides an algorithmic procedure to compute bispectral wave functions and the corresponding bispectral operators. This procedure is described at the end of the section. Many examples computed by making use of it are presented in Sect. 5. Let Vβ be a Bessel plane for a generic β ∈ CN (i.e. Vβ is not a Darboux transforma0 tion of Vβ 0 with β 0 ∈ CN , N 0 < N ). In this section W will be a polynomial Darboux transformation of Vβ , i.e.

Bispectral Algebras of Commuting Ordinary Differential Operators

351

W ∈ GrB (β). We use the notation from (1.21, 1.22) with V = Vβ . In the next proposition we show that the manifold of polynomial Darboux transformations is preserved by the involutions a and s (introduced in Subsect. 1.4). Proposition 3.1. If W ∈ GrB (β), then (i) sW ∈ GrB (β); (ii) aW ∈ GrB (a(β)), where a(β) = (N − 1)δ − β, δ = (1, 1, . . . , 1). Proof. First recall that (Proposition 1.8) sVβ = Vβ and aVβ = Va(β) . We shall study the action of the involutions on 9W (x, z) and check that the conditions of Theorem 2.7 are satisfied. (i) is trivial because 9sW (x, z) = 9W (−x, −z) =

1 P (−x, −∂x )9β (x, z). g(−z)

To prove (ii) we note that the ZN -homogeneity of P (see (2.22)) is equivalent to P (εx, ε−1 ∂x ) = ε−n P (x, ∂x ),

(3.1)

for n = ordP , ε = e2πi/N . It follows from (1.28) that the operator Q (from (1.22)) has the same property and also that Q = h(Lβ )P −1 has rational coefficients. Proposition 1.7 implies that 9aW is a Darboux transformation of 9a(β) with 9aW (x, z) =

1 Q∗ (x, ∂x )9a(β) (x, z). g(z) ˇ

Obviously, Q∗ also satisfies (3.1). To check (2.23), we set KW = 1 +

∞ X

aj (x)∂x−j

j=1

(see (1.1, 1.2)). Recalling that ∗ =1+ KW

∞ X

(−∂x )−j aj (x)

j=1

and KaW = 1 +

∞ X

∗ −1 bj (x)∂x−j = (KW ) ,

j=1

we compute the coefficients bj (x) inductively and find that all of them are polynomials in aj (x) and their derivatives. But by Theorem 2.7 all aj (x) are rational functions of x and limx→∞ aj (x) = 0, which leads to limx→∞ bj (x) = 0 for all j ≥ 1. This proves (2.23) for aW (cf. (2.24)).  (N ) . The central Proposition 3.1 shows that the involutions a and s preserve GrB result of the present paper is that the bispectral involution b has the same property. It (N ) give solutions to the immediately implies that wave functions 9W with W ∈ GrB bispectral problem. Our next theorem addresses this issue.

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Theorem 3.2. If W ∈ GrB (β) then bW exists and bW ∈ GrB (β). Proof. Before proving the existence of bW , we shall find an analog of (1.21) for 9bW (x, z) = 9W (z, x), i.e. we shall show the existence of an operator Pb (x, ∂x ) and a polynomial gb (z) such that 9bW (x, z) =

1 Pb (x, ∂x )9β (x, z). gb (z)

(3.2)

From (2.22) it follows that the operator P can be written as X 1 P (x, ∂x ) = n pk (xN )(x∂x )k , x pn (xN ) n

(3.3)

k=0

where now pk (xN ) are polynomials. Use (1.37–1.39) to obtain X 1 pk (xN )(x∂x )k 9β (x, z) xn pn (xN )g(z) X
1 = n (z∂z )k pk Lβ (z, ∂z ) 9β (x, z). N x pn (x )g(z)

9W (x, z) =

This implies (3.2) with
1 X (x∂x )k pk Lβ (x, ∂x ) , g(x) n

Pb (x, ∂x ) =

(3.4)

k=0

gb (z) = z n pn (z N ).

(3.5)

Now we can prove the existence of bW , i.e. that 9bW (x, z) is a wave function (see (1.1)). Indeed, using (3.2) we can differentiate the formal expansion (1.34) of 9β (x, z) = 9β (xz); expanding gb−1 (z) at z = ∞ we obtain X bk (x)z −k 9bW (x, z) = exz k≥k0

for some finite k0 . Note that the coefficients bk (x) are rational. On the other hand X aj (z)x−j 9bW (x, z) = 9W (z, x) = exz j≥0

with rational aj (z) such that (see (2.24)) lim aj (z) = 0, j ≥ 1;

z→∞

a0 (z) ≡ 1.

These two (formal) expansions of 9bW (x, z) are connected by X bkj z −k , aj (z) = k≥k0

where bk (x) =

X j

bkj x−j ,

bkj = 0 for j < 0.

(3.6)

Bispectral Algebras of Commuting Ordinary Differential Operators

Now (3.6) implies This shows that

353

bkj = 0 for k < 0 , j ≥ 1.   X 9bW (x, z) = exz 1 + bk (x)z −k k≥1

is a wave function. It is clear that it satisfies (2.23) as well. To show an analog of (1.22), i.e. that 1 Qb (x, ∂x )9bW (x, z) fb (z)

9β (x, z) =

(3.7)

with an operator Qb and a polynomial fb , we shall use the above proven identity (3.2) with asW instead of W . It follows from Proposition 1.7 that 9asW (x, z) =

1 Q∗ (−x, −∂x )9a(β) (x, z). f (z)

(3.8)

Proposition 3.1 and Theorem 2.7 (i) allow us to present Q∗ (−x, −∂x ) in the form Q∗ (−x, −∂x ) =

1

m X

xm q m (xN )

s=0

q s (xN )(x∂x )s

(3.9)

with polynomials q s (xN ). Then X
1 (x∂x )s q s La(β) (x, ∂x ) 9a(β) (x, z). m N f (x)z q m (z ) m

9basW (x, z) =

(3.10)

s=0

The identity ab = bas [W1] and Proposition 1.7 now lead to (3.7) with ÿ !∗ m
1 X s (x∂x ) q s La(β) (x, ∂x ) Qb (x, ∂x ) = f (x) s=0

=

m X s=0

and


1 q s (−1)N Lβ (x, ∂x ) (−x∂x − 1)s f (x)


fb (z) = (−z)m q m (−z)N .

(3.11)

(3.12)

From (2.21) and (3.4) it is obvious that Pb is ZN -homogeneous. This completes the proof of Theorem 3.2.  An immediate corollary is the following result, which we state as a theorem because of its fundamental character. (N ) then the wave function 9W (x, z) solves the bispectral Theorem 3.3. If W ∈ GrB problem, i.e. there exist operators L(x, ∂x ) and 3(z, ∂z ) such that

L(x, ∂x )9W (x, z) = h(z N )9W (x, z), 3(z, ∂z )9W (x, z) = Θ(xN )9W (x, z),

(3.13) (3.14)

rankAW = rankAbW = N.

(3.15)

Moreover,

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B. Bakalov, E. Horozov, M. Yakimov

Proof. Equations (3.13, 3.14) follow from (1.21, 1.22, 3.2, 3.7) if we set L(x, ∂x ) = P (x, ∂x )Q(x, ∂x ), h(z N ) = f (z)g(z); 3(z, ∂z ) = Pb (z, ∂z )Qb (z, ∂z ), Θ(xN ) = fb (x)gb (x).

(3.16) (3.17)



Equation (3.15) follows from Propositions 1.5 (i) and 2.4.

Example 3.4. All bispectral algebras of rank 1 are polynomial Darboux transformations of the plane H+ = {z k }k≥0 (see [W1]). This corresponds to the N = 1 Bessel with β = (0),

L(0) = ∂x ,

V(0) = H+ = {z k }k≥0 ,

ψ(0) (x, z) = exz .

Every linear functional on H+ is a linear combination of e(k, λ) = ∂zk |z=λ
and h L(0) = h(∂x ) is an operator with constant coefficients. The “adelic Grassman(1) ). In our nian” Grad , introduced by Wilson [W1], coincides with GrB ((0)) (= GrB terminology the result of [W1] can be reformulated as follows. All bispectral operators belonging to rank one bispectral algebras are polynomial Darboux transformations of operators with constant coefficients.  Remark 3.5. The eigenfunction 9W (x, z) from Eq. (1.21) is a formal series. Let 8β (x, z) = 8β (xz), where 8β (z) is the Meijer’s G-function (1.35) (or any convergent solution of (1.33) in arbitrary domain) and set 8W (x, z) =

1 P (x, ∂x )8β (x, z). g(z)

(3.18)

8W (x, z) =

1 Pb (z, ∂z )8β (x, z) gb (x)

(3.19)

Then

because of (1.33) and x∂x 8β (x, z) = z∂z 8β (x, z). The equations QP = h(Lβ ) and Qb Pb = Θ(Lβ ) imply 1 Q(x, ∂x )8β (x, z), f (z) 1 8β (x, z) = Qb (z, ∂z )8W (x, z). fb (x) 8β (x, z) =

(3.20) (3.21)

So, we proved that 8W (x, z) is a convergent bispectral eigenfunction of the same operators L(x, ∂x ) and 3(z, ∂z ) as 9W (x, z). The involutions a, s and b can be defined on the manifold of “convergent” polynomial Darboux transformations (3.18) by Eqs. (1.43, 1.44, 1.49) in which 9 is replaced by 8 and they preserve it (Proposition 1.7 (i) now becomes a definition). The validity of the equation ab = bas in the “convergent” case is a consequence of that in the “formal” one (see the proof of Theorem 3.2). The rationality of the coefficients of the operator P (x, ∂x ) implies that its kernel has one and the same form (see Eqs. (2.18, 2.19)) in 9- and in 8-bases.

Bispectral Algebras of Commuting Ordinary Differential Operators

355

It is not difficult to provide an explicit algorithm for producing bispectral pairs L(x, ∂x ), 3(z, ∂z ). Although obvious we have collected the steps of this algorithm as they are scattered in the present and the previous sections. Step 1. Choose an arbitrary set of conditions based in some points λ0 = 0, λ1 , . . . , λr of the form (2.18, 2.19), i.e. a basis of KerP . The proof of Lemma 2.9 provides an explicit computation of the coefficients of P in terms of the coefficients ak , bkj in KerP . The polynomial g(z) is given by Definition 2.5 (ii).
d j Qr with high enough powers d0 , . . . , dr Step 2. Take h(z N ) = z d0 N j=1 z N − λN j such that KerP ⊂ Kerh(Lβ ) (cf. Lemma 2.1). The minimal such dj ’s can be computed as follows. (i) For a condition, supported at 0, of the form (2.18) set j(k) = max{j|bkj 6= 0}, 0 ≤ k ≤ k0 . Let βi + kN = βis + ps N for 0 ≤ s ≤ mult(βi + kN ) − 1 with 0 ≤ p0 ≤ . . . ≤ pmult(βi +kN )−1 and is 6= it for s 6= t. Then set d0 = 1 + max pj(k) , the maximum is over all k and all conditions of the form (2.18). (ii) For a condition, supported at λj 6= 0, of the form (2.19) let k0 = max{k|ak 6= 0}. Then set dj = 1 + max k0 , the maximum is over all conditions of the form (2.19) supported at λj . Then put f (z) = h(z N )/g(z). Step 3. Find the coefficients of the operator Q(x, ∂x ) recursively out of the equation Q(x, ∂x )P (x, ∂x ) = h(Lβ (x, ∂x )). Then L(x, ∂x ) = P (x, ∂x )Q(x, ∂x ). A lower order operator L can be constructed using Proposition 1.5, i.e. find u(Lβ ) such that KerP is invariant under u(Lβ ) and then L out of the equation LP = P u(Lβ ). Step 4. Compute by (3.4) Pb (x, ∂x ) and by (3.5) gb (z). Also (3.11) and (3.12) give Qb (x, ∂x ) and fb (z). All expressions are explicit in terms of the coefficients of the operators P and Q. Then 3(z, ∂z ) = Pb (z, ∂z )Qb (z, ∂z ) and Θ(x) = fb (x)gb (x). 4. Polynomial Darboux Transformations of Airy Planes This section contains analogs of the results from Sects. 2 and 3 but here the building blocks are (generalized) Airy operators (see [KS, Dij]) instead of Bessel ones. There is a minor difference in the organization of the present section compared to that of Sects. 2 and 3. Here we give the definition of polynomial Darboux transformations on Airy wave functions (see Definitions 4.2, 4.3) in the spirit of the one provided by Theorem 2.7. Then we prove our main result Theorem 4.5 (which is an analog of Theorem 3.2). As in Sect. 2, it automatically implies bispectrality of the polynomial Darboux transformations. At the end, in Proposition 4.9 we show that Definition 4.3 is equivalent to a second one (analog of Definition 2.5) in terms of conditions on Airy planes. This is again important for algorithmic computations, some of which are presented in the next section. First we recall the definition of (generalized higher) Airy functions. For α = (α0 , α2 , α3 , . . . , αN −1 ) ∈ CN −1 , α0 6= 0, consider the Airy operator

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B. Bakalov, E. Horozov, M. Yakimov

Lα (x, ∂x ) =

∂xN

− α0 x +

N −1 X

αi ∂xN −i ≡ Pα0 (∂x ) − α0 x,

(4.1)

i=2

where α0 = (α2 , α3 , . . . , αN −1 ). The Airy equation is Lα (x, ∂x )8(x) = 0, i.e.

Pα0 (∂x )8(x) = α0 x8(x).

(4.2)

Example 4.1. When α0 = 1, α0 = 0, Eq. (4.2) becomes the classical higher Airy equation (cf. [KS]) (4.3) ∂xN 8(x) = x8(x). In every sector S with a center at x = ∞ and an angle less than N π/(N + 1), it has a solution with an asymptotics of the form (see e.g. [Wa]) 8(x) ∼ x−

N −1 2N

N

e N +1 x

N +1 N

 1+

∞ X

 ai x−i/N ,

|x| → ∞, x ∈ S.

(4.4)

i=1

Similarly, in each sector S as in Example 4.1 Eq. (4.2) has a solution with an asymptotics of the form ∞   X 1/N (4.5) ai x−i/N , |x| → ∞, x ∈ S 8(x) ∼ 9α (x) := xd/N eQ(x ) 1 + i=1

for some d ∈ C and a polynomial Q(x) of degree N + 1 with leading coefficient µ0 NN+1 xN +1 , where α0 = µN 0 . The solution 8 is by no means unique, but d, Q and all ai are uniquely determined and do not depend on S. In the sequel we shall deal only with 9α , which is a formal solution of Eq. (4.2). Definition 4.2. For each α ∈ CN −1 we call an Airy wave function the following function −1

ψα (x, z) := µd0 z −d e−Q(µ0 where

z)

9α (x, z),

(4.6)

9α (x, z) := 9α (α0−1 z N + x).

It is easy to see that ψα is indeed a wave function if we expand 9α (α0−1 z N + x) at x = 0: X −i/N  −1 N −i/N −i−kN k (α0 z + x) = x (4.7) (µ−1 0 z) k k≥0

(we shall always use µ0 as an N th root of α0 ). The plane in Sato’s Grassmannian corresponding to ψα (x, z) will be called an Airy plane and will be denoted by Vα . Obviously, 9α (x, z) solves the bispectral problem Lα (x, ∂x )9α (x, z) = z N 9α (x, z), Lα (α0−1 z N , ∂α−1 zN )9α (x, z) = α0 x9α (x, z),

(4.8) (4.9)

∂x 9α (x, z) = ∂α−1 zN 9α (x, z).

(4.10)

0

because

0

Bispectral Algebras of Commuting Ordinary Differential Operators

357

It is clear that ψα satisfies (4.8) and analogs of (4.9, 4.10) obtained by conjugating −1 by z −d e−Q(µ0 z) . (Up to this conjugation 9α and ψα give one and the same solution to the bispectral problem.) We shall define polynomial Darboux transformations of Airy planes as in the Bessel case (see Definition 2.5 and Theorem 2.7). Before that we shall define a bispectral involution b1 on them. Note that the involution b from [W1] (see Subsect. 1.4) is not well defined on Vα (i.e. ψα (z, x) is not a wave function). The properties of b we would like b1 to have, are: 1) it has to interchange the roles of x and z; 2) it has to preserve Airy planes. Therefore we define b1 9α (x, z) := 9α (x, z) = 9α (α0−1 z N , µ0 x1/N ),

(4.11)

or equivalently, b1 ψα (x, z) := ψα (x, z) = µd0 xd/N z −d eQ(µ0 x

1/N

)−Q(µ−1 z) 0

ψα (α0−1 z N , µ0 x1/N ). (4.12)

For a Darboux transformation W of Vα we define ψb1 W and 9b1 W in a similar way. (We still do not know whether b1 W ∈ Gr, the notation ψb1 W is still formal.) Definition 4.3. A Darboux transformation W of an Airy plane Vα is called polynomial iff (in the notation of Definition 1.4) (i) the operator P has rational coefficients; (ii) g(z) = g1 (z N ), g1 ∈ C[z]; (iii) lim e−xz ψb1 W (x, z) = 1. z→∞

(The limit is formal and has the same meaning as in (2.23).) S (N ) Denote the set of all such W ∈ Gr by GrA (α) and put GrA = α∈CN −1 GrA (α). Remark 4.4. The parts (i) and (ii) of the above definition remain the same if we substitute ψα and ψW by 9α and 9W , where −1

ψW (x, z) := µd0 z −d e−Q(µ0

z)

9W (x, z).

(4.13)

The main result of this section is that GrA (α) is preserved by the involution b1 . Theorem 4.5. (i) If W ∈ GrA (α), then ψb1 W (x, z) is a wave function corresponding to a plane b1 W ∈ GrA (α). (ii) For α ∈ CN −1 the spectral algebra AVα is C[Lα ]. An immediate corollary is that the planes W ∈ GrA (α) give solutions to the bispectral problem of rankN : rankAW = rankAb1 W = N. The proof of Theorem 4.5 is completely parallel to that of Theorem 3.2. We shall be very brief, indicating only the major differences and the most important steps. We start with a lemma illuminating the purpose of the constraints (i) and (ii) in Definition 4.3 (cf. (3.2)).

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Lemma 4.6. If W ∈ GrA (α), then 9b1 W (x, z) =

1 Pb (x, ∂x )9α (x, z), gb (z)

(4.14)

Pb is with rational coefficients and gb is polynomial in z N . Proof. We compute 9b1 W (x, z) = 9W (α0−1 z N , µ0 x1/N ) =

P (α0−1 z N , ∂α−1 zN )9α (α0−1 z N , µ0 x1/N ) 0

g(µ0

where if

x1/N )

1 X pk (x)∂xk , pn (x)

=

1 Pb (x, ∂x )9α (x, z), gb (z)

n

P (x, ∂x ) =

g(z) = g1 (z N )

(4.15)

k=0

with polynomials pk and g1 , then (using (4.8, 4.10)) X 1 ∂xk pk (α0−1 Lα (x, ∂x )), Pb (x, ∂x ) = g1 (α0 x)

(4.16)

gb (z) = pn (α0−1 z N ).

(4.17)

n

k=0

 The proof that ψb1 W (x, z) is a wave function is the same as in the Bessel case, using the above lemma and the condition (iii) of Definition 4.3. Now the identity ab = bas [W1] is modified in the following way. Introduce the maps p and p−1 as follows 9pW (x, z) := 9W (α0−1 xN , µ0 z 1/N ), 9p−1 W (x, z) := 9W (µ0 x1/N , α0−1 z N ). The notation pW , p−1 W is formal – these are not planes in Gr. But 9b1 W (x, z) = 9bpW (x, z) = 9p−1 bW (x, z) corresponds to the wave function ψb1 W (x, z) and to b1 W ∈ Gr. Multiplying the identity ab = bas on the right by p, we obtain ab1 = b1 a1 ,

where a1 = p−1 asp.

(4.18)

Note that for W ∈ GrA (α) aW, b1 W and hence a1 W are planes in Gr. The next lemma gives the action of the involutions on the Airy planes (the proof is the same as that of Proposition 1.8). Lemma 4.7. (i) sVα = Vs(α) , where s(α) = ((−1)N +1 α0 , α2 ,−α3 , . . . , (−1)N −1 αN −1 ); (ii) aVα = a1 Vα = Va(α) , where a(α) = ((−1)N α0 , α2 , −α3 , . . . , (−1)N −1 αN −1 ). We also need an analog of Proposition 3.1. Lemma 4.8. If W ∈ GrA (α), then aW and a1 W belong to GrA (a(α)).

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For the proof we need an analog of Proposition 1.7 for a1 . A simple computation shows that if 1 Q(x, ∂x )9W (x, z) 9V (x, z) = f (z) for V, W ∈ GrA (α) and X Q(x, ∂x ) = qk (x)∂xk , then 9a1 W (x, z) = with

1 Q∗1 (x, ∂x )9a1 V (x, z) f (z)

X

1 − N −1 k x (−1)(N −1)k qk ((−1)N x). N The rest of the proof is left to the reader.  The proof of part (i) of Theorem 4.5 is completed exactly as in the Bessel case. For part (ii), we note that while the Bessel wave functions are “multiplication invariant”, the Airy ones are “translation invariant”. More precisely, for arbitrary c ∈ C, Q ∗1 =

∂x +

9α (x + c, (z N − α0 c)1/N ) = 9α (α0−1 (z N − α0 c) + x + c) = 9α (α0−1 z N + x) = 9α (x, z)
1−kN P (−α0 c)k ). Let u(z) ∈ Aα , L(x, ∂x ) ∈ Aα (expand (z N − α0 c)1/N = k≥0 1/N k z and L(x, ∂x )9α (x, z) = u(z)9α (x, z) (this is equivalent to Lψα (x, z) = uψα (x, z)). Then L(x + c, ∂x )9α (x, z) = u((z N − α0 c)1/N )9α (x, z) and L(x + c, ∂x ) ∈ Aα , u((z N − α0 c)1/N ) ∈ Aα . But Aα ⊂ C[z], therefore u((z N − α0 c)1/N ) ∈ C[z] for all c and u(z) ∈ C[z N ]. This completes the proof of Theorem 4.5. At the end of this section we note that an equivalent definition of GrA (α) can be given in terms of conditions C (cf. Sect. 2). Using the translation invariance of 9α we can suppose that none of the conditions C is supported at 0. Then we have an analog of Theorem 2.7. Proposition 4.9. The Darboux transformation W of Vα is polynomial iff (i) The space of conditions C is homogeneous and ZN -invariant. Equivalently, KerP has a basis of the form X aki ∂zk ψα (x, εj z)|z=λi fij (x) = k

=

X

aki ε−jk ∂zk ψα (x, z)

k

z=εj λi

,

(4.19)

0 ≤ j ≤ N − 1, 1 ≤ i ≤ r (for some r), λi 6= 0. (ii) The polynomial g(z) has the form (2.21), i.e. n1 N N nr g(z) = (z N − λN 1 ) · · · (z − λr ) ,

where ni is the number of conditions C supported at each of the points εj λi , 0 ≤ j ≤ N − 1.

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The proof of the “if” part is the same as in the Bessel case and will be omitted. (In fact, most of the proofs in Sect. 2 are valid in a more general situation.) The “only if” part is also similar to the corresponding result in the Bessel case but some more explanation is needed. For fixed λ 6= 0 we shall use representations of the kernel of the operator Lα − λN in three different linear spaces of formal power series. First set y = α0−1 λN + x, (4.20) ϕα (x, λ) = µd0 λ−d e−Q(λ) 9α (y), considered as a formal power series in y −1/N , (where 9α is from (4.5)). The Airy wave function ψα (x, λ) (see (4.6)) is given by the same formula after expanding y −1/N at x = 0 as in (4.7). The other possibility is to expand y −1/N at x = ∞: X −i/N  x−i/N −k (α0−1 λN )k . (4.21) (x + α0−1 λN )−i/N = k k≥0

(j) Inserting (4.21) in (4.20), we obtain another formal series χα (x, λ). Denote by ϕ(j) α , ψα , the images of ϕ , ψ , χ under the transformations χ(j) α α α α

y 1/N 7→ εj y 1/N ,

λ 7→ εj λ,

x1/N 7→ εj x1/N ,

(j) respectively (ε = e2πi/N ). Then ψα(j) and χ(j) α are obtained by expanding ϕα and in the N d corresponding spaces of formal series Ker(Lα − λ ) has bases k (j) {∂λk ψα(j) }, {∂λk ϕ(j) α }, {∂λ χα },

0 ≤ k ≤ d − 1, 0 ≤ j ≤ N − 1.

Our observation is that if KerP has a basis X fi (x) = aikj ∂zk ψα(j) (x, z)|z=λ , k,j

then the same formula gives a basis of KerP when ψ’s are substituted by ϕ’s or χ’s and vice versa. Indeed, this follows from (1.13) and the fact that P has rational coefficients. We complete the proof of Proposition 4.9 noting that while P depends rationally on x, 1/N and the same argument as in the Bessel case gives that χ(j) α are formal series in x KerP has a χ-basis of the form (4.19).  5. Explicit Formulae and Examples In this section we have collected several classes of examples. We wanted at least to include all previously known examples (unless by ignorance we miss some of them) – see [DG, W1, Z, G3, LP]. We hope that we have elucidated and unified them. For monomial transformations we derive formulae expressing the operators L and 3, solving the bispectral problem, only in terms of the matrix A and the vector γ (see Proposition 5.1 below). This explicit expression for 3 (though possibly of high order) to the best of our knowledge is new even for N = 2 (see [DG]). In other examples we illustrate the properties of the operator of minimal order from a bispectral algebra: when does its order coincide with the rank of the algebra and when this operator is a Darboux transformation of a power of a Bessel operator. We also point out that the classical Bessel potentials u(x) = cx−2 [DG] can produce new solutions of the bispectral problem for any c.

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361

We describe in detail the polynomial Darboux transformations from (Lα − λN )2 , where Lα is an arbitrary Airy or Bessel operator of order N . We do not want simply to show that our procedure of constructing bispectral operators works but to point out that the involutions a and b (b1 in the Airy case) possess some very interesting properties which deserve further study. 5.1. Monomial Darboux transformations of Bessel planes. Let β ∈ CN and W ∈ GrM B (β). We use the notation from (1.21, 1.22) (with V = Vβ ) and from (3.2, 3.7). When the Darboux transformation is monomial g(z) = z n ,

h(z) = z d

(5.1)

for some n, d. We shall consider only the case when there are no logarithms in the basis (2.18) of KerP . The general case can be reduced to this one by taking a limit in all formulae (see Example 5.2 below). Now KerP has a basis of the form

fk (x) =

dN X

0 ≤ k ≤ n − 1,

(5.2)

if aki akj 6= 0, i 6= j,

(5.3)

aki xγi ,

i=1

such that γi − γj ∈ N Z \ 0

where γ = β d is from (2.4). Let A be the matrix (aki ). We shall use multi-index notation for subsets I = {i0 < . . . < in−1 } of {1, . . . , dN } and δI from (2.7). We also put γI = {γi }i∈I , AI = (ak,il )0≤k, l≤n−1 and 1I =

Y

(γir − γis ).

r<s

Let Imin be the subset of {1, . . . , dN } with n elements such that det AImin 6= 0 and P i∈Imin γi be the minimum of all such sums, and set pI =

X i∈I

γi −

X

γi .

i∈Imin

Equation (5.3) implies that these numbers are divisible by N . Finally, for a subset I of {1, . . . , dN } denote by I 0 its complement. In the following proposition we express everything entering (1.21, 1.22, 3.2, 3.7) only in terms of the matrix A and the vector γ. Therefore for each A and β ∈ CN satisfying (5.3) (with γ = β d ) we give an explicit solution to the bispectral problem (cf. (3.16, 3.17)). Proposition 5.1. In the above notation the operators and the polynomials from (3.16, 3.17) are given by the following formulae:

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(a) g(z) = z n , X −1 X  P = det AI 1I xpI det AI 1I xpI LγI . (b) f (z) = z dN −n , X X −1 Q= det AI 1I LγI 0 −nδI 0 xpI det AI 1I xpI . P (c) gb (z) = z n det AI 1I z pI , X Pb = det AI 1I LγI (Lβ )pI /N . P (d) fb (z) = z dN −n det AI 1I z pI , X Qb = det AI 1I (Lβ )pI /N LγI 0 −nδI 0 . Proof. Note that (c) and (d) follow from (a) and (b) (see the proof of Theorem 3.2). To prove (a) we note that
Wr f0 (x), . . . , fn−1 (x), 9β (x, z)
9W (x, z) = z n Wr f0 (x), . . . , fn−1 (x)
P det AI Wr xγI 9I (x, z)
P = . (5.4) det AI Wr xγI The sum is taken over all n-element subsets I = {i0 < i1 < . . . < in−1 } ⊂ {0, 1, . . . , dN − 1}, xγI = {xγi }i∈I and 9I (x, z) are the Bessel wave functions (2.6). Using (2.9) and the simple fact P γ − n(n−1) 2 Wr(xγI ) = 1I x i∈I i , (5.5) we obtain (a). To prove (b) we shall apply the involution a directly on the tau-function τW of the plane W . Recall that [S, SW]
P∞ −1 ] tk z k τ t − [z , (5.6) 9W (t, z) = e k=1 τ (t)
where [z −1 ] is the vector z −1 , z −2 /2, . . . . The action of a is given by [W1] τaW (t1 , t2 , . . . , tk , . . .) = τW (t1 , −t2 , . . . , (−1)k−1 tk , . . .). We shall need the formulae [BHY3]

P det AI 1I τI (t) τW (t) = P , det AI 1I

and

(5.7)

1 Wr (xγI ) τγ (x), (5.8) 1I where τI (t) is the tau-function corresponding to the wave function 9I (x, z) and τ (x) = τ (x, 0, 0, . . .). Applying a to both sides of (5.7) and using (5.6) and (2.6) we obtain τI (x) =

Bispectral Algebras of Commuting Ordinary Differential Operators

P 9aW (x, z) = We compute

363

det AI 1I τa(γ+dN δI −nδ) (x)9a(γ+dN δI −nδ) (x, z) P . det AI 1I τa(γ+dN δI −nδ) (x)

a(γ + dN δI − nδ) = a(γ) + dN δI 0 − (dN − n)δ,

(5.9)

(5.10)

which is a Darboux transformation of a(γ). Equations (5.8) and (5.5) imply τI (x) xpI = . τJ (x) xpJ

(5.11)

To apply (5.11) in (5.9) we have to compute pI 0 but for a(γ) instead of γ. It is a simple exercise to see that pI 0 (a(γ)) = pI (γ) ≡ pI . Using this we obtain P 9aW (x, z) = z

−dN +n

det AI 1I xpI L(a(γ))I 0 P 9a(γ) (x, z). det AI 1I xpI

Now Proposition 1.7 gives (b) because (Lβ )∗ = (−1)N La(β) and

for β ∈ CN

  a (a(γ))I 0 = γI 0 − nδI 0 .



In the following example we consider the case when there are logarithms in the basis (2.18) of KerP . Example 5.2. Let d = 2, β = (1, 1, 1), γ = β 2 = (1, 1, 1, 4, 4, 4) and KerP has a basis f0 (x) = x4 , f1 (x) = a1 x + 2a2 x4 ln x, f2 (x) = a0 x + a1 x ln x + a2 x4 ln2 x. Using that lnk x = ∂k x |=0 we approximate the above functions with f0 (x, ) = x4 , f1 (x, ) = a1 x1+ + 2a2 −1 (x4+ − x4 ), f2 (x, ) = a0 x1+2 + a1 −1 (x1+2 − x1+ ) + a2 −2 (x4+2 − 2x4+ + x4 ). Consider the Darboux transformation W () of Vβ() , where β() = (1, 1 + , 1 + 2), with a basis of the operator P () consisting of the functions fk (x, ). After changing the basis this corresponds to a matrix (cf. (5.2)) ! ÿ 0 0 0 1 0 0 0 0 . A() = 0 0 a1 2a2 0 (a + a ) a 0 0 0 0

1

2

We apply (5.7) for τW () . To make the limit  → 0 we note that the numerator and the denominator depend polynomially on  and that (in the notation of (5.7)) both τ{2,3,6}

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and τ{2,4,5} tend to one and the same Bessel tau-function. So after canceling 3 and setting  = 0 we obtain that τW is a linear combination of 3 Bessel tau-functions: τW =

9a21 τ(−2,1,1,4,4,7) + 18a2 (a0 − a1 )τ(−2,−2,1,4,7,7) + 4a22 τ(−2,−2,−2,7,7,7) . 9a21 + 18a2 (a0 − a1 ) + 4a22

(5.12)

As in the proof of Proposition 5.1 from this formula one can compute the operators P , Q, Pb and Qb . It is clear that they also can be obtained by taking the limit  → 0 directly in the corresponding expressions for W ().  From here to the end of the subsection we shall restrict ourselves to the case when β d = γ has different coordinates. We choose the following basis of KerLdβ (cf. [MZ]) 8(k−1)d+j (x) := µkj xβk +(j−1)N , where µk,1 := 1,

µkj := µk,j−1 ·

N Y

1 ≤ k ≤ N, 1 ≤ j ≤ d,

(5.13)

(βi − βk − (j − 1)N )−1 .

i=1

In this basis the action of Lβ is quite simple:  8(k−1)d+j−1 , Lβ 8(k−1)d+j = 0,

for 2 ≤ j ≤ d for j = 1.

(5.14)

Let a basis of KerP be fk (x) =

dN X

aki 8i (x),

k = 0, . . . , n − 1.

(5.15)

i=1

Example 5.3. Let n = d, βi − βj ∈ N Z for all i, form:  (1) ) . . . t(N t0 0 )  t(1) t(1) . . . t(N  1 0 1  (1) ) A= t(1) t(1) . . . t(N 1 0 2  t2  . .. .. ..  .. . . . (1) (1) (N ) t(1) t . . . t . . . t n−1 n−2 0 n−1

j and the matrix A = (aki ) has the     .   

) t(N 0 ) t(N 1 .. . ) t(N n−2

) t(N 0 .. .

...

(5.16)

) t(N 0

The type of the matrix is tantamount to the identities Lβ f0 = 0, Lβ fk+1 = fk , k = 1, . . . , n − 1. Then KerP is invariant under the action of Lβ and by Proposition 1.5 the operator L = P Lβ P −1 is differential of order N and solves the bispectral problem. For a generic β ∈ CN the spectral algebra has rank N (i.e. it is C[L]). This family can be considered as the most direct generalization of the “even case” of J. J. Duistermaat and F. A. Gr¨unbaum [DG] (see also [MZ]). When N = 2 our example coincides with it but for N > 2 here we present a completely new class of bispectral operators.  In connection with the above example we prove the following proposition. Proposition 5.4. Let W ∈ GrB (β) (β ∈ CN –generic) be such that AW contains an operator of order N . Then W is a monomial Darboux transformation of Vβ , i.e. W ∈ GrM B (β) ∩ Gr(N ) .

Bispectral Algebras of Commuting Ordinary Differential Operators

365

Proof. Proposition 1.5 implies that W ∈ GrB (β) belongs to Gr(N ) iff Lβ (KerP ) ⊂ KerP.

(5.17)

If we suppose that W 6∈ GrM B (β) then KerP would contain some elements of the form (2.20). The action of Lβ on them is easily computed:


Lβ Dλk 9β (x, εi λ) = Dλk Lβ 9β (x, εi λ) = Dλk λN 9β (x, εi λ) = λN (Dλ + N )k 9β (x, εi λ).

 Thus the linear space span Dλk 9β (x, εi λ) 0≤k≤m can be identified with the space of polynomials in D of degree ≤ m, with the action of Lβ corresponding to P (D) 7→ λN P (D + N ). It is clear that all Lβ -invariant subspaces are of the form  span Dλk 9β (x, εi λ) 0≤k≤k0 for some k0 . The corresponding polynomial Darboux transformation is trivial in the sense that it leads again to the same plane Vβ (the operator P = (Lβ − λ)k0 commutes  with Lβ ). Therefore W ∈ GrM B (β). In the same manner as in Example 5.3, one can build for arbitrary k rank N bispectral algebras with the lowest order of the operators equal to kN . It is clear that when the matrix A is not of the form (5.16) (or a direct sum of such matrices) then KerP (given by (5.15)) is not invariant under the action of Lβ . Proposition 1.5 implies that in this case the spectral algebra does not contain operators of order N . The following example is one of the simplest of this type. Example 5.5. Let N = 2, β = (β1 , β2 ), β1 + β2 = 1, d = n = 2. We take KerP with a basis (5.15) where   1 a 0 0 A= 0 0 1 b for some a, b ∈ C, i.e. a xβ1 +2 , 2(β1 − β2 + 2) b xβ2 +2 . + 2(β2 − β1 + 2)

f0 (x) = 81 (x) + a82 (x) = xβ1 + f1 (x) = 83 (x) + b84 (x) = xβ2 Then Lβ f0 (x) = axβ1 ,

Lβ f2 (x) = bxβ2

and KerP is not invariant under Lβ when ab 6= 0. The spectral algebra AW = P L2β C[Lβ ]P −1 consists of operators of orders 4, 6, 8, 10, . . .. This example is also interesting for the fact that it does not require β1 − β2 ∈ 2Z. The generalization for arbitrary N is obvious.  Another example illustrating Proposition 1.5 is the following one.

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Example 5.6. Let N = 2, β = (β1 , β2 ) ∈ C2 , β1 + β2 = 1, β1 − β2 ∈ 2Z, d = 4, n = 2. We take KerP with a basis (5.15) where   λ 0 0 0 λa + b λb 0 0 A= 0 0 1 0 0 0 a b for some a, b, λ ∈ C. Then it is easy to see that KerP is invariant under the operator L3β + λL2β but it is not invariant under any polynomial of Lβ of degree ≤ 2. On the other hand KerP ⊂ KerL4β obviously implies L4+k β KerP ⊂ KerP for k ≥ 0. Therefore the spectral algebra AW is the linear span of the operators
−1 , k ≥ 0. P L3β + λL2β P −1 , P L4+k β P This example is interesting for the fact that (for λ 6= 0) the operator of minimal order in the spectral algebra is not a Darboux transformation of a power of Lβ , although the Darboux transformation is monomial.  In the last example of this subsection we show that for d = n = 1 our results agree with those of [Z]. Example 5.7. Let d = n = 1, KerP = Cf0 , f0 (x) =

N X

P = ∂x −

ai xβi ,

i=1

f00 (x) , f0 (x)

and βi − βj ∈ N Z if ai aj 6= 0. Then

  −1 X X βi L = P Lβ P −1 = P Q = × a i xp i ∂ x − ai xp i x X  −1 Pβ (Dx + N ) pi −N +1 X × ai ai xpi x , D x + N − βi   X  βi 3 = Pb Qb = (Lβ )pi /N × ai ∂ z − z X  Pβ (Dz + 1) × ai (Lβ )pi /N z −N +1 , Dz + 1 − βi

where pi = βi − βmin , βmin = min βi , ai 6=0

Pβ (D) =

N Y

(D − βi ),

Dx = x∂x .

i=1

We have Θ(x) = xN

X

a i xpi

2 ,

deg Θ = N + 2(βmax − βmin ),

where βmax = max βi . When f0 (x) = txβ1 + xβ2 , β2 − β1 = N α, α ∈ Z≥0 , ai 6=0

Θ(x) = xN (t + xN α )2 , and we obtain the operator 3 from [Z].



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5.2. Polynomial Darboux transformations. In this subsection we shall consider the simplest case of polynomial Darboux transformation of an operator of order N , namely when the polynomial h(z) from (1.27) is equal to (z − λN )2 for some λ ∈ C \ 0. Using the kernels of the operators P , Q∗ , Pb and Q∗b from (1.21, 1.22, 3.2, 3.7), we describe the action of the involutions a and b (b1 in the Airy case). Propositions 5.10, 5.12 below raise some interesting questions and conjectures. The Bessel and Airy cases are very similar. We shall consider first the Airy one since it is simpler. Let W ∈ GrA (α), α ∈ CN −1 . Set h(z) = (z − λN )2 ,

g(z) = f (z) = z N − λN .

Then Kerh(Lα ) has a basis of the form n o ∂xk 9α (x, εj λ)

(5.18)

(5.19)

0≤j≤N −1, k=0,1

and KerP has a basis fj (x) = 9α (x, εj λ) + a∂x 9α (x, εj λ),

0≤j ≤N −1

(5.20)

for some a ∈ C. We shall start with the case N = 2. The following example is due to [G3, LP]. We shall obtain it as the simplest special case of Theorem 4.5. Example 5.8. Let N = 2 and α = (α0 ) ∈ C1 . For fixed α0 , a ∈ C \ 0 we take the basis (5.20) of KerP : (5.21) fk (x) = ψk (x) + a∂x ψk (x), k = 0, 1, where ψk (x) = 9α (x, (−1)k λ). Using that ∂x fk = a(α0 x + λ2 )ψk + ∂x ψk , ∂x2 fk = (aα0 + α0 x + λ2 )ψk + a(α0 x + λ2 )∂x ψk , we compute P from Wr(f0 , f1 , ϕ) Wr(f0 , f1 ) 1 a 1 a(α0 x + λ2 ) aα + α x + λ2 a(α x + λ2 ) 0 0 0 = 1 a 2 a(α0 x + λ ) 1

Pϕ =

ϕ ∂x ϕ 2 ∂x ϕ

.

The result is P = ∂x2 +

a2 α0 a2 (α0 x + λ2 )2 − (α0 x + λ2 ) − aα0 ∂x + . 2 1− 1 − a2 (α0 x + λ2 ) 0x + λ ) a2 (α

This expression coincides with that given in [G3] if we set α0 =

2 2 = , 2 + 3t s

a=

s , 2y

λ = 0.

We compute the operators P , Q and Q∗ as follows. If we write

(5.22)

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B. Bakalov, E. Horozov, M. Yakimov

P = ∂x2 + p1 (x)∂x + p0 (x), Q = ∂x2 + q1 (x)∂x + q0 (x),

Q∗ = ∂x2 + qe1 (x)∂x + qe0 (x),

then the identity QP = h(Lα ) imply q1 + p1 = 0, 2p01 + q1 p1 + p0 + q0 = −2(α0 x + λ2 ) and

qe1 = −q1 ,

qe0 = −q10 + q0 .

Our observation is that because P ∗ Q∗ = h(La(α) ) and 9aW = f −1 Q∗ 9a(α) , the operator Q∗ has a basis of the form (5.21) with some b ∈ C instead of a and a(α) instead of α. Comparing the above expressions for Q∗ with (5.22) we obtain that b = −a. By Theorem 4.5 the operator Pb also has a basis (5.19) with some c instead of a and µ instead of λ. On the other hand we can compute it directly using Eqs. (4.15, 4.16). Then gb (z) = 1 − a2 (z 2 + λ2 ) which on the other hand is up to a constant z 2 − µ2 . This gives µ2 =

1 − a 2 λ2 . a2

(5.23)

The other coefficients give a surprising result: c = a. In conclusion, if we denote the operator P from (5.22) with P (a, λ) then P = P (a, λ),

Q = P ∗ (−a, λ),

Pb = P (a, µ),

Qb = P ∗ (−a, µ)

(5.24)



where µ and λ are connected by (5.23).

The next example is completely analogous to the above one but to the best of our knowledge it is new. Example 5.9. For N = 3 the Airy operator is Lα = ∂x3 + α2 ∂x − α0 x, α = (α0 , α2 ) ∈ C2 , α0 6= 0. We take P with a basis (5.20) (N = 3). Then using Eq. (4.8) we compute a3 α 0 ∂2 a3 (α0 x + λ3 ) + (1 + a2 α2 ) x a3 α2 (α0 x + λ3 ) + (1 + a2 α2 )α2 + a2 α0 ∂x + a3 (α0 x + λ3 ) + (1 + a2 α2 ) a3 (α0 x + λ3 )2 + aα0 (1 + a2 α2 )(1 + a2 α2 )(α0 x + λ3 ) . − a3 (α0 x + λ3 ) + (1 + a2 α2 )

P = ∂x3 −

A direct computation using Proposition 1.7, Theorem 4.5 and QP = h(Lα ) leads to P = P (a, λ),

Q = −P ∗ (−a, −λ),

Pb = P (a, µ),

Qb = −P ∗ (−a, −µ) (5.25)

1 + a2 α 2 . a3

(5.26)

with µ given by µ3 + λ 3 = − 

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The above examples can be generalized for arbitrary N as follows. Proposition 5.10. Denote by P = P (a, λ) the operator P with a basis (5.20). Then in the above notation we have Q = (−1)N P ∗ (−a, −λ),

Pb = P (a, µ),

Qb = (−1)N P ∗ (−a, −µ)

(5.27)

with λ and µ connected by λN + µN = Pα0 (−1/a),

(5.28)

where Pα0 is the polynomial from (4.1). The spectral algebras
2 AW = P Lα − λN C[Lα ]P −1 ,
2 Ab1 W = Pb Lα − µN C[Lα ]Pb−1

(5.29) (5.30)

consist of operators of orders 2N, 3N, 4N, . . .. Proof. Because (−1)N P ∗ (−1)N Q∗ = (La(α) − (−λ)N )2 , Qb Pb = (Lα − µN )2 ,

9bW (x, z) =

9aW (x, z) =

Pb 9α (x, z) , z N − µN

(−1)N Q∗ 9a(α) (x, z) , z N − (−λ)N

we see that Q = (−1)N P ∗ (b, −λ) and Pb = P (c, µ) for some b, c, µ. Using the equation Lα (x, ∂x )9α (x, εj λ) = λN 9α (x, εj λ) we compute pN (x) = Wr(f0 , f1 , . . . , fN −1 ) as in the proof of Lemma 2.9. We obtain pN (x) = −(−a)N (α0 x + λN − Pα0 (−1/a)). Equation (4.17) leads to (5.28) because gb (z) = const · (z N − µN ). Applying (5.28) for Pb instead of P , we obtain Pα0 (−1/a) = Pα0 (−1/c). Note that the map a 7→ c is an automorphism of CP1 since it is an involution. The only solution of the above equation with this property is c = a. To compute (−1)N Q∗ , we note that its second coefficient is equal to that of P which is equal to −p0N (x)/pN (x). This, Proposition 1.7 and Lemma 4.7 imply a(α)0 α0 = , α0 x + λN − Pα0 (−1/a) a(α)0 x + (−λ)N − Pa(α)0 (−1/b) which leads to a polynomial equation for b in terms of a and α. Because a 7→ b is an automorphism of CP1 we obtain that b = −a. Equations (5.29, 5.30) follow from Proposition 5.4.  We shall find the analog of Proposition 5.10 in the Bessel case. We use the notation from the beginning of the subsection with β ∈ CN instead of α and Eq. (5.20) modified as follows (cf. (2.20)) (5.31) fj (x) = 9β (x, εj λ) + aDx 9β (x, εj λ) (j = 0, . . . , N − 1, Dx = x∂x ). In the next example we shall study the simplest case N = 2.

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Example 5.11. For N = 2, β = (1 − ν, ν) the corresponding Bessel operator is Lβ = x−2 (Dx − (1 − ν))(Dx − ν) = ∂x2 +

ν(1 − ν) , x2

Dx = x∂x .

Using (1.38) we compute the operator P from f0 f1 ϕ D x f0 D x f1 D x ϕ 1 Dx2 f0 Dx2 f1 Dx2 ϕ . Pϕ = 2 f1 f0 x D x f0 D x f1 The answer is the following. If we set µ2 = then P =

a + 1 + a2 ν(1 − ν) , a 2 λ2

n o 1 2 2 2 2 p (x )D + p (x )D + p (x ) 2 1 x 0 x x2 p2 (x2 )

with p2 (x2 ) = x2 − µ2 , p1 (x2 ) = µ2 − 3x2 and p0 (x2 ) = −λ2 x4 + (2λ2 µ2 + (a + 1)(2a − 1)a−2 )x2 + ((a + 1)a−2 − λ2 µ2 )µ2 . The operator Pb is (cf. (3.4)) o 1 n 2 Pb = Dx p2 (Lβ ) + Dx p1 (Lβ ) + p0 (Lβ ) g(x) and gb (z) = z 2 (z 2 − µ2 ). A straightforward computation shows that if we set P = P (a, λ, µ) then Q = P ∗ (−a/(a+1), λ, µ), Pb = P (a, µ, λ)Lβ , Qb = Lβ P ∗ (−a/(a+1), µ, λ). (5.32) Therefore we can take Pb = P (a, µ, λ),

Qb = P ∗ (−a/(a + 1), µ, λ),

(5.33)

i.e. the involution b acts simply by exchanging λ with µ and vice versa, while the involution a acts as a 7→ −a/(a + 1).  The action of the involutions for arbitrary N is given in the next proposition. Proposition 5.12. Denote by P = P (a, λ) the operator P with a basis (5.31). Then we can take Pb and Qb such that Q = (−1)N P ∗ (b, −λ),

Pb = P (a, µ),

Qb = (−1)N P ∗ (b, −µ)

(5.34)

with λ, µ and a, b connected by λN µN = Pβ (−1/a),

1 1 + + N − 1 = 0, a b

where Pβ is the polynomial from (1.32). The spectral algebras
2 AW = P Lβ − λN C[Lβ ]P −1 ,
2 AbW = Pb Lβ − µN C[Lβ ]Pb−1 , consist of operators of orders 2N , 3N , 4N, . . ..

(5.35)

(5.36) (5.37)

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Proof. We have gb (z) = const · z N (z N − µN ) for some µ ∈ C. Using (3.5) we compute gb (z) = z N det(Dzi fj (z))i,j=0,...,N −1 = (−a)N z N (z N λN − Pβ (−1/a)) which gives the value of µ. We have to prove that Pb given by (3.4) (which is of order 2N ) is divisible by Lβ from the right. Indeed, it is easy to see that Pb (x, ∂x )xβi = P (x, ∂x )xβi |λ=0 and the proof of Lemma 2.9 implies P |λ=0 = Lβ . Thus we can take Pb = P (c, µ) for some c ∈ C. Now (5.35) implies Pβ (−1/a) = Pβ (−1/c) leading to c = a. Finally, as in the Airy case, if Q = (−1)N P ∗ (b, −λ) for some b ∈ C then Pβ (−1/a) = (−1)N Pa(β) (−1/b) = Pβ (1/b + N − 1), showing that a−1 + b−1 + N − 1 = 0. Equations (5.36, 5.37) follow from Proposition 5.4.



In conclusion we want to make some comments. In the case N = 1 the adjoint involution a has a simple and beautiful geometric interpretation (see [W1]): in terms of Krichever’s construction it preserves the spectral curve and maps the “sheaf of eigenfunctions” into some kind of a dual sheaf. In [W1] G. Wilson also posed the problem of describing the action of the bispectral involution on Grad . We think that in the general case the study of the action of the involutions a and b on the bispectral manifolds of polynomial Darboux transformations of Bessel and Airy planes is an equally interesting and difficult task. The above examples lead us to the conjecture that the involutions a and b (b1 in the Airy case) possess some universality property. Any polynomial Darboux transformation W of a Bessel plane Vβ (respectively an Airy plane Vα ) is determined by the points λ1 , . . . , λN (6= 0) at which the conditions C are supported (see (2.1)), by the matrix A defined by (2.20) (resp. (2.19)), and of course by the vector β (resp. α). Then the corresponding matrices for aW and bW (resp. b1 W ) depend only on the matrix A. The point is that they do not depend on the points λ1 , . . . , λN at which the conditions C are supported nor on the vector β (resp. α). Acknowledgement. We are grateful to F. A. Gr¨unbaum and G. Wilson for their interest in the paper and for suggestions which led to improving the presentation of our results. We also thank the referee who proposed important changes towards making the text more “reader friendly”. This work was partially supported by Grant MM–523/95 of Bulgarian Ministry of Education, Science and Technologies.

References [AM]

Adler, M., Moser, J.: On a class of polynomials connected with the Korteweg–de Vries equation. Commun. Math. Phys. 61, 1–30 (1978) [AMM] Airault, H., McKean, H.P., Moser, J.: Rational and elliptic solutions of the Korteweg–de Vries equation and a related many-body problem. Comm. Pure Appl. Math. 30, 95–148 (1977) [AMcD] Atiyah, M.F., Macdonald, I.G.: Introduction to commutative algebra. Reading, MA: Addison– Wesley, 1969 [AvM] Adler, M., van Moerbeke, P.: Birkhoff strata, B¨acklund transformations, and regularization of isospectral operators. Adv. Math. 108, 140–204 (1994)

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[BHY1] Bakalov, B., Horozov, E., Yakimov, M.: Highest weight modules of W1+∞ , Darboux transformations and the bispectral problem. To appear in Proc. Conf. Geom. and Math. Phys., Zlatograd 95, Bulgaria, in a special volume of Serdica Math. J., q-alg/9601017 [BHY2] Bakalov, B., Horozov, E., Yakimov, M.: Tau-functions as highest weight vectors for W1+∞ algebra. J. Phys. A: Math. Gen. 29, 5565–5573 (1996), hep-th/9510211 [BHY3] Bakalov, B., Horozov, E., Yakimov, M.: B¨acklund–Darboux transformations in Sato’s Grassmannian. Serdica Math. J. 22, no. 4, 571–588 (1996), q-alg/9602010 [BHY4] Bakalov, B., Horozov, E., Yakimov, M.: Highest weight modules over W1+∞ algebra and the bispectral problem. To appear in Duke Math. J., q-alg/9602012 [BHY5] Bakalov, B., Horozov, E., Yakimov, M.: General methods for constructing bispectral operators. Phys. Lett. A222, 59–66 (1996), q-alg/9605011 [BE] Bateman, H., Erd´elyi, A.: Higher transcendental functions. New York: McGraw-Hill, 1953 [BC] Burchnall, J.L., Chaundy, T.W.: Commutative ordinary differential operators. Proc. Lond. Math. Soc. 21, 420–440 (1923); Proc. Royal Soc. London (A) 118, 557–583 (1928); Proc. Royal Soc. London (A) 134, 471–485 (1932) [Da] Darboux, G.: Lec¸ons sur la th´eorie g´en´erale des surfaces. 2`eme partie, Paris: Gauthiers–Villars, 1889 [DJKM] Date, E., Jimbo, M., Kashiwara, M., Miwa, T.: Transformation groups for soliton equations. In:Proc. RIMS Symp. Nonlinear integrable systems – Classical and Quantum theory. (Kyoto 1981), M. Jimbo, T. Miwa (eds.), 39–111, Singapore: World Scientific, 1983 [Dij] Dijkgraaf, R.: Intersection theory, integrable hierarchies and topological field theory. Lecture Notes at Cargese Summer School (1991), hep-th/9201003 [DG] Duistermaat, J.J., Gr¨unbaum, F.A.: Differential equations in the spectral parameter. Commun. Math. Phys. 103, 177–240 (1986) [F] Fastr´e, J.: B¨acklund–Darboux transformations and W -algebras. Doctoral Dissertation, Univ. of Louvain, 1993 [GD] Gelfand, I.M., Dickey, L.A.: Fractional powers of operators and Hamiltonian systems. Funct. Anal. Appl. 10, 13–39 (1976) [G1] Gr¨unbaum, F.A.: The limited angle reconstruction problem in computer tomography. Proc. Symp. Appl. Math. 27, AMS, L. Shepp (ed.), 43–61 (1982) [G2] Gr¨unbaum, F.A.: The Kadomtsev–Petviashvilii equation: an alternative approach to the “rank two” solutions of Krichever and Novikov. Phys. Lett. A 139, 146–150 (1989) [G3] Gr¨unbaum, F.A.: Time-band limiting and the bispectral problem. Comm. Pure Appl. Math. 47, 307–328 (1994) [I] Ince, E.L.: Ordinary Differential Equations. New York: Dover, 1944 [KS] Kac, V.G., Schwarz, A.: Geometric interpretation of the partition function of 2D gravity. Phys. Lett. B257, 329–334 (1991) [KV] Kac, V.G., van de Leur, J.W.: The n–component KP hierarchy and representation theory. In: Important developments in soliton theory. A. Fokas, V. Zakharov (eds.), Springer series in nonlinear dynamics, New York: Springer 1993, pp. 302–343 [K] Kasman, A.: Bispectral KP solutions and linearization of Calogero–Moser particle systems. Commun. Math. Phys. 172, 427–448 (1995) [KR] Kasman, A., Rothstein, M.: Bispectral Darboux transformations: The generalized Airy case. To appear in Physica D, q-alg/9606018 [KrN] Krichever, I., Novikov, S.: Holomorphic bundles over algebraic curves and nonlinear equations. Russian Math. Surveys 35, 53–79 (1980) [LP] Latham, G., Previato, E.: Higher rank Darboux transformations. In: NATO ARW Lyon 91 Singular limits of dispersive waves, N. Ercolani, D. Levermore (eds.), New York: Plenum, 1994, pp. 117–134 [MZ] Magri, F., Zubelli, J.: Differential equations in the spectral parameter, Darboux transformations and a hierarchy of master equations for KdV. Commun. Math. Phys. 141, 329–351 (1991) [R] Rothstein, M.: Calogero-Moser pairs and the Airy and Bessel bispectral involutions. Preprint (1996), q-alg/9611027 [S] Sato, M.: Soliton equations as dynamical systems on infinite dimensional Grassmann manifolds. RIMS Kokyuroku 439, 30–40 (1981) [SW] Segal, G., Wilson, G.: Loop Groups and equations of KdV type. Publ. Math. IHES 61, 5–65 (1985) [vM] van Moerbeke, P.: Integrable foundations of string theory. CIMPA–Summer school at Sophia– Antipolis (1991), In: Lectures on integrable systems. O. Babelon et al. (eds.), Singapore: World Scientific, 1994 pp.163–267

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Communicated by T. Miwa

Commun. Math. Phys. 190, 375–394 (1997)

Communications in

Mathematical Physics c Springer-Verlag 1997

Towards a Kneading Theory for Lozi Mappings. II: Monotonicity of the Topological Entropy and Hausdorff Dimension of Attractors Yutaka Ishii? Laboratoire de Topologie et Dynamique, D´epartement de Math´ematiques, Universit´e de Paris–Sud, Bˆatiment 425, 91405 Orsay, France Received: 1 September 1996 / Accepted: 16 April 1997

D´edi´e au Professeur A. Douady pour son 60`eme anniversaire Abstract: We construct a kneading theory a` la Milnor–Thurston for Lozi mappings (piecewise affine homeomorphisms of the plane). In the first article a two-dimensional analogue of the kneading sequence called the pruning pair is defined, and a topological model of a Lozi mapping is constructed in terms of the pruning pair only. As an application of this result, in the current paper we show the partial monotonicity of the topological entropy and of bifurcations for the Lozi family near horseshoes. Upper and lower bounds for the Hausdorff dimension of the Lozi attractor are also given in terms of parameters. Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375 2 The Pruning Pair: Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379 3 Monotonicity when b is Close to Zero . . . . . . . . . . . . . . . . . . . . . . . . . . . 382 4 Monotonicity near Horseshoes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387 5 Hausdorff Dimension of the Attractors . . . . . . . . . . . . . . . . . . . . . . . . . . 388 5.1 Upper bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389 5.2 Lower bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392 1. Introduction In his paper [Loz] R. Lozi studied the interesting dynamical behavior of the following two parameter family of piecewise affine homeomorphisms of the plane which is now called the Lozi family:     x 1 − a|x| + by 7−→ a, b ∈ R, b 6= 0, L = La,b : y x ? Current address: Department of Mathematical Sciences, University of Tokyo, Komaba, Tokyo 153, Japan. E-mail: [email protected]

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similar to the H´enon family. He observed numerically that this simple mapping may give rise to very complicated dynamics, i.e. he found a strange attractor for the parameter values a = 1.7 and b = 0.5 (see the following Fig. 1.1). Later, this kind of observation was mathematically justified by Misiurewicz for suitable choices of parameters [Mis]. Actually he proved that under some conditions on the parameters, there exists a nonempty compact invariant set F = FL (let us call it the Lozi attractor of L) which satisfies the following three conditions: There exists a neighborhood G of F such that the distance between F and Lna,b (X) tends to zero when n goes to +∞ for every X ∈ G. (ii) The unstable manifold of the hyperbolic fixed point in the first quadrant is dense in F . (iii) La,b on F is topologically mixing, i.e. for all non-empty open sets U and V in F there exists an integer N such that, for any n ≥ N , (i)

Lna,b (U ) ∩ V 6= ∅. Ergodic studies of the dynamics on the attractor can be found in papers of Collet– Levy, Young, etc. [CL, Yo2], where they showed the existence of an SRB measure for some Lozi mappings.

Fig. 1.1. Lozi attractor for a = 1.7, b = 0.5

In the first paper [Ish], inspired by an article of P. Cvitanovi´c et al. [CGP], we constructed symbolic dynamics for Lozi mappings similar to Milnor and Thurston’s

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kneading theory for unimodal maps [MT]. More precisely, we defined a two-dimensional analogue of the kneading sequence called the pruning front PL and the primary pruned region DL , gave a solution of the pruning front conjecture (a characterization of the set of all admissible sequences), and constructed a topological model of the dynamics of L in terms of the pruning pair (PL , DL ) only. As a consequence, we gave a solution to the so-called first tangency problem, and proved that the boundary of the set of all horseshoes: H ≡ {(a, b) | La,b on Ka,b is equivalent to the full shift and a > 1 + |b|} is described as a graph of an algebraic curve a = g(b), where Ka,b denotes the set of points whose forward and backward orbits by La,b are bounded. We remark that Lozi mappings are homeomorphisms, so do not have critical points in the usual sense. See [CGP, Cvit] for more on the general discussion about the pruning front. The purpose of the current article is to consider some applications of the method developed in the previous paper. The first main result establishes the partial monotonicity of the topological entropy of the Lozi family with fixed Jacobian. To consider the topological entropy of a Lozi mapping, we take the one-point compactification of R2 and extend the map continuously by putting L(∞) = ∞. Theorem 1.1 (Partial Monotonicity of the Topological Entropy). For every b 6= 0 there exists a∗ = a∗ (b) strictly smaller than g(b) such that the topological entropy of La,b is a monotone increasing and non-constant function of a on [a∗ , +∞). Moreover, we have that h(La,b ) < log 2 when a∗ ≤ a < g(b). Besides, the difference g(b) − a∗ (b) is uniformly bounded from below by a positive constant if b runs over a bounded region. When b is sufficiently close to zero, a∗ can be taken approximately 1.97. Here, “monotone increasing and non-constant” in the statement of Theorem 1.1 (and other statements) does not mean “strictly increasing” (remark that, when a is sufficiently large, then the dynamics of La,b on Ka,b is equivalent to the full shift on the two symbols, and thus the topological entropy is always log 2 there). See the remark prior to the proof of Theorem 1.1. This result comes from the monotonicity of the pair (P, D) with respect to the parameter a ≥ a∗ . Roughly speaking, what the monotonicity of the pruning pair (P, D) means is that, if we increase the parameter a a little bit, then the dynamics corresponding to the smaller parameter is realized as a quotient of sub-dynamics of the one corresponding to the larger parameter. See Sect. 3 for the complete definition and statement. From this fact, we also prove that: Theorem 1.2 (Partial Monotonicity of the Bifurcations). Under the same conditions as in Theorem 1.1, (i)

The bifurcations of the periodic orbits of such Lozi family are monotone increasing, i.e. there are only orbit creations but no annihilation when a increases. (ii) A periodic point of minimal period d whose orbit intersects with the x-axis at r points (d ≥ r ≥ 1) at some parameter value a = a0 bifurcates precisely to 2nr periodic points of period nd (not necessarily minimal) for all n ∈ N when a increases. Before that bifurcation the corresponding periodic orbit does not exist, and it never bifurcates after that bifurcation. (iii) All new periodic points are created by this procedure in this monotone region.

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So there are only saddle-node type bifurcations in this monotone region and no period-doubling bifurcation. We can observe bifurcation phenomenon even for a nonperiodic orbit (Corollary 3.4) and determine precisely when it bifurcates (Corollary 3.5). Moreover, one can prove these monotonicity results not only for the a-direction with b fixed but also for every family along a sufficiently short C 1 -curve in the parameter plane which is transverse to ∂H from the outside to the inside of H (see Sect. 4). This monotonicity property may be interpreted as a counterpart of a surprising result by Kanet et al. [KKY] for diffeomorphisms of the plane. There, they showed that one can find both infinitely many orbit-creation and orbit-annihilation parameter values in an arbitrary neighborhood of a non-degenerate homoclinic tangency of a one-parameter family of dissipative C 3 -diffeomorphisms of the plane. Moreover, they conjectured that the H´enon family with a fixed Jacobian −0.3 has many parameter values of such tangency. Our observations on the monotonicity seem to reveal a difference between the Lozi family and the H´enon family or, in other words, between the piecewise hyperbolic category and the smooth non-uniform one (compare the difference of bifurcations between the tent map and the quadratic map in one-dimension; the quadratic family has period doubling bifurcations but tent maps have only saddle-node bifurcations). Actually in the Lozi case, when (a, b) ∈ ∂H and b < 0, the Lozi mapping La,b has a homoclinic tangency (see Theorem 1.3 of [Ish]) but there is no concurrence of orbit-creation and orbit-annihilation in any one-parameter family stated in the previous theorem. Our method gives another application; an upper bound for the Hausdorff dimension of the Lozi attractors. As one sees in Fig. 1.1, the attractor seems to have a fractal-like structure; locally it looks like (segment) × (Cantor set). But if the mappings defining a fractal set have both expanding and contracting directions, then (unlike the conformal case) it becomes a hard task to estimate the Hausdorff dimension of the fractal set even if they are affine (see the book of Falconer [Fal]). Our method, however, gives an upper bound for the Hausdorff dimension of the set K = Ka,b of all points which have bounded orbit both in forward and backward time when a > 1 + |b| and 2a > 1 + 4|b| (see Theorem 5.5). Combining this with Young’s equality [Yo1] for the lower bound, we obtain the following explicit estimates of the Hausdorff dimension of the attractors when an SRB measure exists. √ Corollary 1.3 (Hausdorff Dimension of the Attractors). Assume that 2a > b + 2, b < 4 − 2a and b > 0 small. Then, we have √ log 2 log(a + a2 − 4b) − log 2 √ √ ≤ dimH F ≤ 1 + . 1+ log(a + a2 − 4b) − log 2b log(a + a2 − 4b) − log 2b This estimate becomes sharper when b > 0 goes to 0. Actually the ratio: (the upper bound) − (the lower bound) (the upper bound) − 1 tends to 1 − log a/ log 2 which is close to zero. For example, if we choose a = 1.7 and b = 0.1, then the estimate gives 1.176669 · · · ≤ dimH F ≤ 1.247848 · · · , and for the choice a = 1.7 and b = 0.01 we get 1.102712 · · · ≤ dimH F ≤ 1.135055 · · · .

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Note. In recent work with D. Sands [IS], the author has proved the monotonicity in a of the Lozi family for all a > 1 and |b| sufficiently small, extending a part of Theorems 1.1 and 1.2.

2. The Pruning Pair: Review In this section we summarize some results established in the first paper [Ish] in a more convenient form for our purpose. In the following, we always assume that a > 1 + |b|. Let K = KL be the set of all points whose forward and backward orbits remain bounded. It is not difficult to see that K is completely invariant under the application of L, non-empty and compact. We develop the symbolic dynamics attached to the Lozi mapping L ≡ La,b on this set K. For a point X ∈ K, we put πL (X) ≡ · · · ε−2 ε−1 · ε0 ε1 ε2 · · · , where

  +1 εi ≡ ∗  −1

Li (X)x > 0, Li (X)x = 0, Li (X)x < 0.

Here ∗ plays a role of “joker”, i.e. we substitute both +1 and −1 for ∗; Yx refers to the x-component of Y . We call an element of π(X) an itinerary of X. So, if an orbit of X lands on the y-axis n times, π(X) consists of 2n itineraries. We sometimes use the notion of an itinerary with joker in which ∗ is not replaced by +1 or −1. Remark that this multi-valued map π conjugates the Lozi mapping on K and the shift map on a subset of {+1, −1}Z . In the following, we write [ π(X) π(B) ≡ X∈B

for any subset B of K. The partial order on the symbol space is given as follows. For every element ε = · · · ε−2 ε−1 · ε0 ε1 · · · in {+1, −1}Z , we call εu ≡ · · · ε−2 ε−1 · the tail of ε, and εs ≡ ·ε0 ε1 · · · the head of ε. Let C u (resp. C s ) be the set of all tails (resp. heads) of the elements of {+1, −1}Z equipped the standard topology. Definition 2.1. Let −1 < +1 be the order on the letters. (i) Take two distinct sequences εs and δ s in C s . Then, we can find the smallest number i ≥ 0 such that εi 6= δi . We say εs <s δ s if one of the following conditions is satisfied: • the number of +1’s in ε0 · · · εi−1 is even and εi < δi , • the number of +1’s in ε0 · · · εi−1 is odd and εi > δi . (ii) Take two distinct sequences εu and δ u in C u . Then, we can find the largest number i < 0 such that εi 6= δi . When b > 0 (resp. b < 0), we say εu δi .

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Fig. 2.1. Partial orders in the symbol space

It is quite convenient to “visualize” the symbol space {+1, −1}Z as a product of two middle third Cantor sets which preserves the orders defined above (see Fig. 2.1). Then, the shift map on the symbol space becomes a Smale horseshoe. Now let us define our two-dimensional analogue of the Milnor–Thurston kneading sequence. The idea is, as in [Cvit, CGP], to regard a Lozi mapping as an “incomplete horseshoe” and measure its incompleteness compared with the full shift. To do this, we first consider the following continued fraction: sn = s(· · · εn−2 εn−1 εn ·) ≡

and

1

,

(2.1)

p(· · · ε−2 ε−1 ·) ≡ 1 − bs−2 + b2 s−2 s−3 − b3 s−2 s−3 s−4 + · · · .

(2.2)

−aεn +

b −aεn−1 +

b −aεn−2 + .

..

Also we consider q(·ε0 ε1 · · ·) ≡ b−1 r0 − b−2 r0 r1 + b−3 r0 r1 r2 − · · · ,

(2.3)

where rn is defined as rn = r(·εn εn+1 εn+2 · · ·) ≡

b

.

b

aεn + aεn+1 +

(2.4)

b aεn+2 + .

..

It was shown in the previous paper that p, sn , q and rn (and their partial derivatives as well) are holomorphic functions of (a, b) ∈ C2 , and continuous with respect to the three variables (a, b, ε). See Lemma 6.1 in [Ish].

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Now our two-dimensional analogue of the kneading sequence for a Lozi mapping is: Definition 2.2. We call PL ≡ {ε ∈ {+1, −1}Z | p(εu ) − q(εs ) = 0} the pruning front of L and DL ≡ {ε ∈ {+1, −1}Z | p(εu ) − q(εs ) < 0} the primary pruned region of L. The pair (PL , DL ) is called the pruning pair of L. The dynamical interpretation of p and q is the following. In [Ish] we constructed the map: x0,−1 : {+1, −1}Z −→ R2 , which gave the inverse of the itinerary map π on π(K). What we showed there was that each sequence ε (even for a sequence outside π(K)) with a fixed tail · · · ε−2 ε−1 · (resp. head ·ε0 ε1 · · ·) is mapped into a line: x − p(εu ) = y/s−1 (resp. x − q(εs ) = r0 y) which is denoted by Luε (resp. Lsε ). The dynamical interpretation of these lines is that, if ε is in π(K) and if the orbit of X ≡ x0,−1 (ε) never lands on the y-axis, then Luε (resp. Lsε ) expresses the expanding (resp. contracting) direction for the iteration of L at X (see Fig. 2.2).

s Fig. 2.2. Dynamical interpretation of Lu ε and Lε

One of the main results of the previous paper was to give a solution of the “pruning front conjecture” [Cvit, CGP] in this setting. We say that ε is admissible if it is realized as an itinerary of a point X ∈ KL . Let us denote the set of all admissible sequences of L by AL , i.e. π(KL ) = AL . Let σ be the shift map on {+1, −1}Z : σ(· · · ε−2 ε−1 · ε0 ε1 ε2 · · ·) ≡ · · · ε−2 ε−1 ε0 · ε1 ε2 · · · . Then, Theorem 1.2 of [Ish] (the pruning front conjecture) tells us that the set of all admissible symbol sequences is completely characterized by the primary pruned region. More precisely, we have shown that

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AL = {+1, −1}Z \

[

σ n DL .

(2.5)

n∈Z

Next let us define an equivalence relation in AL to construct a topological model of the dynamics of L on KL . Definition 2.3. Let L be a Lozi mapping and P be its pruning front. For an admissible sequence ε (resp. ε0 ), let M (resp. M0 ) be the set of all integers m such that σ m ε ∈ P (resp. σ m ε0 ∈ P). We write ε ∼P ε0 if M = M0 and εn−1 = ε0n−1 for all n ∈ Z \ M. Let ι be the natural projection with respect to this relation. We set π/∼P ≡ ι ◦ π. Let σ/∼P be the factor of σ by the projection, i.e. σ/∼P ≡ ι ◦ σ ◦ ι−1 . Theorem 2.4 (Combinatorics, [Ish] Theorem  5.5). π/∼P gives a topological conjugacy between L on KL and σ/∼P on AL ∼ , i.e. π/∼P is a homeomorphism onto P  AL ∼ such that the following diagram commutes: P

KL   π/∼P y  AL ∼ 

L

−−−−→

P

KL  π/ y ∼P  σ/∼ −−−−P→ AL ∼ . P

We remark that σ/∼P on AL ∼ is determined by the pruning pair (PL , DL ) of P L only. Thus, we can analyze any topological properties of the dynamics of L on KL through the investigation of its pruning pair. Using this theorem, we have solved the so-called first tangency problem, and as a consequence, one observed that the boundary of H is algebraic. Corollary 2.5 (Boundary of Horseshoes, [Ish] Corollary 1.4). The boundary of H forms an algebraic curve. Moreover, the boundary becomes a graph of a function a = g(b). When (a, b) ∈ ∂H, the primary pruned region is empty, and the pruning front consists of two points · · · + + + − + · + − − − · · · and · · · + + + − − · + − − − · · · when b > 0 (· · · − − − + + · + − − − · · · and · · · − − − + − · + − − − · · · when b < 0). 3. Monotonicity when b is Close to Zero In this section we study dependence of the Lozi family with respect to the parameter a for fixed Jacobian −b sufficiently close to zero. Let L and L0 be two Lozi mappings. Define a partial order between their pruning pairs as follows. Definition 3.1. We write (PL , DL ) < (PL0 , DL0 ) if DL ⊃ DL0 and DL ⊃ PL0 . This definition means that the dynamics corresponding to the smaller pair (PL , DL ) is realized as a quotient of sub-dynamics of the one corresponding to the pair (PL0 , DL0 ). The next proposition plays a central role in this section which establishes the partial monotonicity of the pruning pair.

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Proposition 3.2. There exist b∗ > 0 and a∗ ≈ 1.97 such that, for any fixed b with 0 < |b| < b∗ , the pair (PL , DL ) is a monotone increasing function of a ≥ a∗ . Moreover, this function is non-constant in the sense that DL contains a cylinder set when a∗ ≤ a < 2. Remark that, when a is large, both PL and DL are empty, i.e. the pruning pair is maximal. First, we show that a Lozi mapping is “close” to a horseshoe if a is not small. Let us put Cl ≡ {ε ∈ {+1, −1}Z | ε0 = 1, ε1 = −1, · · · , εl = −1}. Recall that the biggest head is · + − − − · · ·. See Fig. 3.1.

Fig. 3.1. P and D close to a horseshoe

Lemma 3.3. Let (Pa,b , Da,b ) denote the pair corresponding to La,b . (i) For a fixed aˆ > 1, let n be a number which satisfies 2/ˆan+1 > (2 − aˆ )/(ˆa − 1). Then, there exists bˆ > 0 such that, for every (a, b) satisfying a > aˆ and bˆ > |b| > 0, we have Pa,b ∪ Da,b ⊂ Cn . (ii) For a fixed a˜ < 2, let m be a number which satisfies 2/˜am+1 < 2 − a˜ . Then, there exists b˜ > 0 such that, for any b satisfying b˜ > |b| > 0, we have Cm ⊂ Da,b ˜ . Proof. The following specific calculation was suggested by H. H. Rugh. Suppose that b = 0. Then, by (2.2) and (2.3), (p − q) equals 1−

1 1 1 + − 3 + ···. aε0 a2 ε0 ε1 a ε0 ε 1 ε 2

Given n as above, one easily sees that (p − q) is positive outside Cn for b = 0 and a ≥ aˆ . We know that (p − q) is a continuous function of (ε, a, b). So, in a neighborhood of b = 0 and aˆ ≤ a ≤ 3, (p − q) is positive outside Cn . Given m as above, one can see that (p − q) is negative in Cm for b = 0, a = a˜ . The rest of the proof of (ii) is now similar. 

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Proof of Proposition 3.2. Take first b = 0. We remark that, in this case, p ≡ 1. Then, an easy calculation shows that the a-derivative of (p − q) at the head · + − − − · · · becomes
0 −1/(a − 1) = 1/(a − 1)2 , which is always positive. The continuity of ∂q/∂a with respect to the head asserts that the derivative is also positive on the cylinder set CN for sufficiently large N when b = 0. By the same argument as in the previous lemma, we conclude that (p − q) is a monotone increasing function of a on CN for b 6= 0 sufficiently close to 0. Take a∗ ≈ 1.97 so that the cylinder set CN contains D and P as in the previous lemma (actually, one can take N = 5). This proves the monotonicity of the pruning pair (P, D).  Here we will give the proofs of Theorems 1.1 and 1.2 when b is sufficiently close to zero. The complete proofs will be given in the next section. From this proposition we can first observe the bifurcations of the Lozi family. To do this, for a fixed symbol sequence ε in {+1, −1}Z we consider the parameter dependence of X ≡ x0,−1 (ε). Under the assumption of Proposition 3.2 (i.e. for a fixed b sufficiently close to 0 and a ≥ a∗ ), we say such a point X appears at a = a0 if ε is not admissible for all a < a0 sufficiently close to a0 and is admissible for a = a0 . This definition is well-defined, i.e. it does not depend on the choice of ε such that X = x0,−1 (ε). In the same way, we say such a point X = x0,−1 (ε) disappears at a = a0 if ε is not admissible for a > a0 sufficiently close to a0 and is admissible for a = a0 . One gets the next corollary on the bifurcations of non-periodic orbits. Corollary 3.4. Let b∗ > 0 and a∗ ≈ 1.97 be as in Proposition 3.2 and fix b with 0 < |b| < b∗ . Then, every point in K moves continuously, and does not disappear when a ≥ a∗ increases. Moreover, a non-periodic point in K whose orbit intersects with the x-axis at r points (+∞ ≥ r ≥ 0) bifurcates to precisely 2r points as a ≥ a∗ increases, and it never bifurcates after that bifurcation. Proof. Take a0 and a1 such that a∗ ≤ a0 < a1 . Then, by Proposition 3.2, any itinerary ε in π(X) of a point X ∈ K for a = a0 is admissible for a = a1 . This proves the first half of the corollary. Now suppose that the orbit of a point X intersects with the x-axis at r points when a = a0 . This means that π(X) contains r jokers. Thus, the map x0,−1 is 2r to one on π(X). When one takes a = a1 , then no element of π(X) is contained in the shifted images of the pruning front, so now the map x0,−1 is one to one on each element of π(X). Thus, the second half of the corollary follows from Theorem 2.4.  Proof of the second part of Theorem 1.2. Suppose that we have a periodic point X of minimal period d at a = a0 whose orbit intersects with the x-axis at r points (d ≥ r ≥ 1; if r = 0, then there is no more bifurcation by Proposition 3.2). Then, its itinerary with joker ∗ is a symbol sequence of period d which contains r jokers in every successive d letters. For any n ∈ N, there are precisely 2nr ways to make a symbol sequence of period nd (not necessarily minimal) by replacing each joker by +1 or −1. When one takes a > a0 , the map x0,−1 sends such periodic sequences to distinct periodic points in the dynamical space of period nd (not necessarily minimal). This proves the second part of Theorem 1.2.  One can sharpen these results on the occurrence of bifurcations in the following way: Corollary 3.5. X ∈ K appears at a = a0 if and only if {Ln (X)}n∈Z ∩ {x-axis} is non-empty.

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Proof. Let X ≡ x0,−1 (ε) and put un (a) = un (ε, a) ≡ (p − q)(σ n ε) at a. Then, X appears at a = a0 if and only if un (a0 ) ≥ 0 for all n ∈ Z, and there exists a sequence ai < a0 (i > 0) converging to a0 and a sequence of integers n(i) such that un(i) (ai ) < 0. So zeros of un(i) (a) (which will be denoted by zn(i) ) accumulate to a0 . The a-derivative of un (a) is uniformly bounded with respect to n near a0 so un(i) tends to zero, which shows the accumulation of Ln (X) to the x-axis. Conversely if Ln (X) accumulates to the x-axis, then there exists a sequence n(i) such that un(i) (a0 ) tends to 0. This means that, when i is big, σ n(i) (ε) is in the cylinder set CN as in (ii) of Lemma 3.3. Here the a-derivative of un(i) (a) is uniformly away from 0 with respect to n(i) near a0 so this implies that zn(i) accumulates to a0 , which proves the converse.  Proof of the third part of Theorem 1.2. This proceeds almost in the same way as the previous corollary. Suppose that there exists a periodic point X of period d at a = a1 and assume that its itineraries π(X) are not admissible at a = a2 < a1 . This inadmissibility is independent of the choice of a sequence in π(X). In fact, if π(X) has more than one sequence (this means that each element of π(X) is in the (pre)image of the pruning front), then all sequences in π(X) are inadmissible when a = a2 < a1 by Proposition 3.2. Because X is periodic of period d, there exists a periodic symbol sequence ε ∈ π(X) of period d. As in the argument of the previous corollary, we can find a3 between a1 and a2 and some number 0 ≤ m ≤ d − 1 such that (p − q)(a3 , b, σ m ε) = 0 and ε is admissible at a = a3 by the intermediate value theorem. This means that Lm a3 ,b (X) lands on the x-axis. Hence we have proven Theorem 1.2.  Remark. This proof shows that the (pre)images of “tangencies” of the stable line Lsε and the unstable line Luε on the x-axis create all new periodic points in the monotonicity region. Here, a “tangency” means a contact of an unstable manifold with a stable manifold (which are broken lines) at its corner. See Fig. 3.2.

Fig. 3.2. “Tangency” on the x-axis

The following Fig. 3.3 shows the typical bifurcation diagram in which we summarize the statements in Theorem 1.2. Another consequence of Proposition 3.2 is the dependence of the topological entropy with respect to the parameters. It is natural to consider the topological entropy for a

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Fig. 3.3. Monotonicity of the bifurcation diagram

continuous map on a compact space. So we take the one-point compactification of R2 and extend the Lozi maps continuously by putting L∗ (∞) = ∞. Because the topological entropy on the whole compact space is equal to that on the non-wandering set Ω, one easily obtains hL∗ (R2 ∪ {∞}) = hL∗ (Ω∗ ) ≤ hL∗ (K ∪ {∞}) = hL (K), where Ω∗ = Ω(L∗ ). But hL (K) ≤ hL∗ (R2 ∪ {∞}), so we have equality everywhere. Thus we may calculate the topological entropy on the K-set, i.e. on the topological model in the symbol space due to Theorem 2.4. Remark. For any a˜ < 2, we may pick m as in Lemma 3.3 (ii). This implies that the corresponding symbol space lacks at least sequences which contain a certain word of finite length m. If we regard each word of length m as a new symbol and σ m as a new shift map, then we have a new full shift with 2m symbols. When a = a˜ , there are at most 2m − 1 admissible symbols, so the topological entropy of this new shift map at a = a˜ is at most log(2m − 1). Hence the topological entropy at a = a˜ is at most 1 log(2m − 1) m due to the fact that the entropy of σ m is equal to the entropy of σ times m. Thus, we conclude that h(La,b ) < log 2 for any a < 2 (see also Corollary 1.5 of [Ish]). Proof of Theorem 1.1. Take b with 0 < |b| < b∗ and a∗ ≤ a < a0 . Put L ≡ La,b and L0 ≡ La0 ,b , and let P and P 0 be their pruning fronts respectively. From (2.5) and Proposition 3.2, one sees that AL ⊂ AL0 . Again by the proposition, there is a natural projection (this is just ι induced by P):   ι : AL ∼ 0 −→ AL ∼ , P

P

which conjugates the dynamics. So we get 

hσ/∼P AL ∼ ≤ hσ/∼ 0 AL ∼ 0 ≤ hσ/∼ P

P

P

P0

A L0




∼P 0

.

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By Theorem 2.4, this implies that hL (KL ) ≤ hL0 (KL0 ). This completes the proof of Theorem 1.1 when b is close to zero.



4. Monotonicity near Horseshoes In this section, we consider the monotonicity properties of the Lozi family along several directions near horseshoes. To state this, let nX ∈ TX R2 be an inner normal vector of ∂H at X ∈ ∂H \ {(2, 0)}, and let us put M ≡ {(X, v) ∈ T R2 | X ∈ ∂H \ {(2, 0)}, v ∈ TX R2 such that (nX , v) > 0}. Proposition 4.1. Let

f (t) = f : (−1, 1) −→ R2

be any C 1 -curve from the open interval to the parameter space such that (f (0), Dt f |t=0 ) ∈ M (see Fig. 4.1). Then, there exists δ > 0 such that the mapping: t 7−→ (Pf (t) , Df (t) ) is a monotone increasing function on [−δ, δ]. Moreover, this function is non-constant in the sense that Df (t) contains a cylinder set when −δ ≤ t < 0.

Fig. 4.1. C 1 -curves transverse to ∂H

Proof. The idea is first to show that the a-derivative of (p − q) is positive at (a, b, ε), where (a, b) ∈ ∂H and ε satisfying (p − q)(a, b, ε) = 0. For simplicity, we consider only the case b > 0. Then, when (a, b) ∈ ∂H, the primary pruned region is empty and the pruning front consists of two points · · ·+++−+·+−−− · · · and · · ·+++−−·+−−− · · · by Corollary 2.5. At these special points, we see p(· · · + + + − ± ·) − q(· + − − − · · ·) = 1 −

b b − , (a + x)(1 + x) (a + x)(b + x)

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where x = (a −

√ a2 + 4b)/2. So, one has

n ∂x (2x + a + 1) + (x + 1) ∂ (p − q) = b ∂a + ∂a (a + x)2 (1 + x)2

+ a + 1) + (x + b) o . (a + x)2 (b + x)2

∂x ∂a (2x

It is easy to see that all of ∂x ∂a , (2x + a + 1), (x + 1) and (x + b) are positive. Because the boundary of H is described as a graph of a function a = g(b) which is monotone increasing when b > 0, the derivative of (p − q) in the direction Dt f |t=0 is also positive at f (0) ∈ ∂H and at the two special points as above. Due to the continuity of the derivative of (p−q) in the direction Dt f |t=0 , the derivative is still positive in a neighborhood of the special two points and near t = 0. Thus, Df (t) contains a cylinder set when t is negative and close to zero. Take a smaller neighborhood of t = 0, if necessary, so that all the corresponding pruning fronts Pf (t) are in the neighborhood of the two points. This finishes the proof.  Remark. The length of the C 1 -curve can be chosen uniformly for every choice of compact subset of M . However, it may shrink to zero when b tends to zero. In particular, we know nothing about the monotonicity of b 7−→ (P2,b , D2,b ). The proofs of the following two corollaries are same as in Sect. 3, so we omit them. Corollary 4.2. Under the conditions of Proposition 4.1, there exists δ > 0 such that the mapping: t 7−→ h(Lf (t) ) is a monotone increasing function on [−δ, δ]. Moreover, we have h(Lf (t) ) < log 2 when −δ ≤ t < 0. Corollary 4.3. Under the conditions of Proposition 4.1, the bifurcations of Lf (t) are monotone as in the statements of Theorem 1.2. Combining these corollaries with the results in the previous section, we have the complete proofs of Theorems 1.1 and 1.2. Finally we summarize the results above to get a monotone picture in the parameter space near ∂H as Fig. 4.2. 5. Hausdorff Dimension of the Attractors The Lozi attractor is a typical example of a fractal. As mentioned in the introduction, locally it looks like (Cantor set)×(segment). If the mappings defining a fractal set are contractions, then one can get some estimates of the Hausdorff dimension of the fractal using the contraction constants. But if they have both expanding and contracting direction, then the estimation becomes very difficult even if they are affine maps. This is because the Hausdorff dimension depends largely not only on the contraction (or expanding) ratios but also on the relative position of the miniatures of the fractal (see [Fal], p. 126). In the Lozi case, we can estimate the Hausdorff dimension of the set K from above using our method. This upper bound is given when a > 1 + |b| and 2a > 1 + 4|b|. The lower bound is given when an SBR measure exists due to Young’s equality between entropy, Lyapunov exponents and Hausdorff dimension of an invariant ergodic measure.

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Here, −→ means the direction of increase of the topological entropy and the bifurcations Fig. 4.2. Global picture of the monotonicity near ∂H

5.1. Upper bound. As we have mentioned in the introduction, the Lozi attractor F densely contains an unstable manifold which is a broken line. So, there is little hope that the upper bound of the Hausdorff dimension of F in the “unstable direction” would be better than 1, and thus, we should analyze the Hausdorff dimension of F in the “Cantor direction”. More precisely, due to the inclusion: [ [
Luε , Lsε ∩ Luε ⊂ K⊂ ε∈{+1,−1}Z

ε∈{+1,−1}Z

we want to calculate the Hausdorff dimension of the distribution of p(εu ) in the x-axis: Λa,b ≡ {p(εu ) ∈ R | εu ∈ C u }. Recall first a basic fact on Hausdorff dimension (see the book [Fal], for example, for the definition of Hausdorff dimension and the following lemma). Lemma 5.1. Let X and Y be two metric spaces and f be a Lipschitz map from X to Y . Then we have dimH f (X) ≤ dimH X, where dimH D denotes the Hausdorff dimension of D. Here, we prepare a standard model of the Cantor set as;  X κn · · · ε−3 ε−2 ∈ {+1, −1}N Γ (κ) ≡ ε−2 · ε−3 · · · ε−n n≥2

for every κ < 1/2. It is easy to see that the Hausdorff dimension of Γ (κ) is − log 2/ log κ. The central claim in this subsection is: Proposition 5.2. The set Λa,b in the x-axis has Hausdorff dimension not greater than log 2 p . log 2 − log(a − a2 − 4|b|)

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We remark that Λa,b is not a dynamically defined Cantor set in the sense of Palis– Takens. Proof of Proposition 5.2. We put p p 0     −(a − a2 − 4|b|) a − a2 − 4|b| 1 , x∈ κ1 ≡ max −aε + bx 2|b| 2|b| =

4|b| p , (a + a2 − 4|b|)2

and u

u

u

κ2 ≡ max{|bs(ε )| | ε ∈ C } =

a−

p a2 − 4|b| . 2

Recall that the dynamics: x 7−→

1 −aε + bx

(ε = ±1)

(5.1)

is associated to the definition of sn . It is easy to check that κ1 < κ2 . By the previous lemma, it is enough to construct a Lipschitz map from Γ (κ) onto Λa,b for every κ > κ2 . For εu = · · · ε−2 ε−1 · ∈ C u and n ≥ 0, we introduce the notation: (σ n εu )u ≡ · · · ε−n−2 ε−n−1 · . Then, one can rewrite u

p(ε ) = 1 +

∞ X

(−b)n s((σεu )u ) · · · s((σ n εu )u ).

n=1

Lemma 5.3. We have the following two recursion equations: s((σ n−1 εu )u ) = and

1 −aε−n + bs((σ n εu )u )

p((σ n−1 εu )u ) = 1 − bs((σ n εu )u )p((σ n εu )u ).

(5.2)

(5.3)

Now, fix an integer N > 0 and consider two elements in C u : εu = · · · ε−N −3 ε−N −2 ε−N −1 · · · ε−1 · and

δ u = · · · δ−N −3 δ−N −2 ε−N −1 · · · ε−1 · .

By (5.1), the first recursion Eq. (5.2) and the definition of κ1 , one gets −n |bs((σ n εu )u ) − bs((σ n δ u )u )| ≤ C · κN 1

(5.4)

for all n > N . If 2a > 1 + 4|b|, then |bs(εu )| < 1/2, which implies that each value of p(εu ) is in the open interval (0, 2). So, we start with |p(· · · ε−N −2 ε−N −1 ·) − p(· · · δ−N −2 ε−N −1 ·)| = |p((σ N εu )u ) − p((σ N δ u )u )| ≤ 2.

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From the second recursion Eqs. (5.3), (5.4) and the previous estimate, we see |p((σ N −1 εu )u )−p((σ N −1 δ u )u )| = |bs((σ N εu )u )p((σ N εu )u ) − bs((σ N δ u )u )p((σ N δ u )u )| ≤ 2Cκ1 + 2κ2 ≤ (2C)2κ2 . Again, in the same way, one obtains (see Fig. 5.1) |p((σ N −2 εu )u ) − p((σ N −2 δ u )u )| = |bs((σ N −1 εu )u )p((σ N −1 εu )u ) − bs((σ N −1 δ u )u )p((σ N −1 δ u )u )| ≤ ((2C)2κ2 )κ2 + 2Cκ21 ≤ (2C)3κ22 .

Fig. 5.1. Recursion Eq. (5.3)

Inductively, we can get |p(εu ) − p(δ u )| ≤ (2C)(N + 1)κN 2 . Thus, we have shown that the mapping: v=

X n≥2

κn 7−→ p = p(· · · ε−2 ε−1 ) ε−2 · ε−3 · · · ε−n

from Γ (κ) onto Λa,b is Lipschitz for any κ > κ2 . This finishes the proof of Proposition 5.2.  Next we work for the unstable direction. Let pt (εu ) be the x-coordinate of the intersection of Luε and {y = t}.

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Lemma 5.4. The mapping εu 7−→ pt (εu ) from Γ (κ) to the line {y = t} is Lipschitz for all κ > κ2 . Proof. By the definition, one sees pt (εu ) = p(εu ) − t/s−1 . So, we get t u u |pt (εu ) − pt (δ u )| ≤ |p(εu ) − p(δ u )| + u u |s(ε ) − s(δ )| . s(ε )s(δ ) The first term in the right hand side is Lipschitz. |s(εu )| is bounded from below inde pendent of εu . So, we obtain the result from (5.4). Thus, we have shown that the surjective map: [ Γ (κ) × R −→

Luε

ε∈{+1,−1}Z

defined by (εu , t) 7−→ (pt (εu ), t) is Lipschitz for every κ > κ2 as promised. This implies that Theorem 5.5 (Hausdorff Dimension of the K-sets). Suppose that a > 1 + |b| and 2a > 1 + 4|b|. Then we have dimH K ≤ 1 +

log 2 p . a2 − 4|b|)

log 2 − log(a −

Remark. When K is a horseshoe, we can repeat a similar argument for an estimate of the Hausdorff dimension in the “unstable direction”, and get a better bound for dimH K, i.e. the number “1” in the upper bound of Theorem 5.5 can be replaced by a better bound. 5.2. Lower bound. The lower bound for the Hausdorff dimension is essentially given by Young’s celebrated equality [Yo1]. To explain this, we first recall some definitions. Let f be a C 2 -diffeomorphism on a compact surface (or a Lozi mapping) and ν be an invariant ergodic Borel probability measure. Then, the Hausdorff dimension of ν is defined by dimH ν ≡ inf {dimH B | ν(B) = 1} . By the ergodic theorem, the numbers: λ± f ≡ lim

n→±∞

1 log kDX f n k n

exist and constant for ν-almost everywhere X. We call them the ν-Lyapunov exponents of f . Now suppose that the Lozi mapping has a strange attractor F as in the introduction, and let b > 0 be small. Then it is shown by Collet–Levy and Young [CL, Yo2] that there exists an SRB measure µ for La,b . The following is an extended version of the Young’s equality [Yo1] for Lozi mappings established in [CL]. Proposition 5.6. For an SRB measure µ for La,b , we have   1 1 , − dimH µ = hµ (L) λ+L λ− L where hµ (L) is the µ-entropy of L.

Kneading Theory for Lozi Mappings II

that

393

Using Pesin’s entropy formula hµ (L) = λ+L for maps with singularity, we conclude dimH F ≥ 1 −

λ+L . λ− L

(5.5)

Lemma 5.7. We have the following estimates of the exponents of La,b : √ a + a2 − 4b + λL ≥ log > 0, 2 and λ− L

≥ − log

a+

√ a2 − 4b < 0. 2b

Proof. It is proven by Misiurewicz [Mis] that, if La,b satisfies a > 1 + |b|, then there is an invariant splitting: u s ⊕ EX TX R2 = EX u s of the tangent space TX R2 into one dimensional subspaces EX and EX at every point X ∈ R2 , where Lm is differentiable for all m ∈ Z, so that !n ÿ √ a + a2 − 4b n u kvk for all v ∈ EX kDX L vk ≥ 2

ÿ

and n

kDX L vk ≤

a−

!n √ a2 − 4b kvk 2

s for all v ∈ EX

for n ≥ 0. From these inequalities, we obtain the estimates for λ± L.



Proof of Corollary 1.5. Equation (5.5) combined with the previous lemma gives the desired estimates.  Acknowledgement. The author would like to express his gratitude to P. Cvitanovi´c, H. H. Rugh, D. Sands and M. Shishikura for interesting discussions and encouragements. This paper was finished during the author’s stay at Laboratoire de Topologie et Dynamique, Universit´e de Paris–Sud. He is very grateful to the hospitality of this institute. This work is partially supported by the Research Fellowships of the Japan Society for the Promotion of Science for Young Scientists no. 3079.

References [CL]

Collet, P., Levy, Y.: Ergodic properties of the Lozi mappings. Commun. Math. Phys. 93, 461–481 (1984) [Cvit] Cvitanovi´c, P.: Periodic orbits as the skeleton of classical and quantum chaos. Physica D 51, 138–151 (1991) [CGP] Cvitanovi´c, P., Gunaratne, G.H., Procaccia, I.: Topological and metric properties of H´enon-type strange attractors. Phys. Rev. A 38, 1503–1520 (1988) [Fal] Falconer, K.: Fractal Geometry; Mathematical Foundations and Applications. New York: John Wiley and Sons, 1990 [Ish] Ishii, Y.: Towards a kneading theory for Lozi mappings I: A solution of the pruning front conjecture and the first tangency problem. To appear in Nonlinearity (1997) [IS] Ishii, Y., Sands, D.: The Lozi family is monotone near the tent-maps. Preprint, Paris (1996)

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[KKY] Kan, I., Koc¸ak, H., Yorke, J.: Antimonotonicity: Concurrent creation and annihilation of periodic orbits. Ann. Math. 136, 219–252 (1992) [Loz] Lozi, R.: Un attracteur e´ trange(?) du type attracteur de H´enon. J. Phys. (Paris) 39 (Coll. C5), 69–77 (1978) [MT] Milnor, J., Thurston, W.: On iterated maps of the interval. Preprint Princeton University (1977) and published in: Dynamical Systems, J.C. Alexander (ed.). Lecture Notes in Math., Vol. 1342. Berlin– Heidelberg–New York: Springer, 1988, pp. 465–563 [Mis] Misiurewicz, M.: Strange attractors for the Lozi mappings. In: Nonlinear Dynamics, R.G. Helleman (ed.). New York: The New York Academy of Sciences, 1980, pp. 348–358 [Yo1] Young, L.-S.: Dimension, entropy, and Lyapunov exponents. Erg. Th. Dyn. Syst. 2, 109–124 (1982) [Yo2] Young, L.-S.: Bowen–Ruelle measures for certain piecewise hyperbolic maps. Trans. Amer. Math. Soc. 287, 41–48 (1985) Communicated by Ya.G. Sinai

Commun. Math. Phys. 190, 395 – 410 (1997)

Communications in

Mathematical Physics c Springer-Verlag 1997

N = 2 KP and KdV Hierarchies in Extended Superspace F. Delduc, L. Gallot Laboratoire de Physique Th´eorique ENSLAPP, URA 14-36 du CNRS, associ´ee a` l’ENS de Lyon et au LAPP, Groupe de Lyon: ENS Lyon, 46 All´ee d’Italie, 69364 Lyon, France Received: 6 March 1997 / Accepted: 18 April 1997

Abstract: We give the formulation in extended superspace of an N = 2 supersymmetric KP hierarchy using chirality preserving pseudo-differential operators. We obtain two quadratic hamiltonian structures, which lead to different reductions of the KP hierarchy. In particular we find two different hierarchies with the N = 2 classical super-Wn algebra as a hamiltonian structure. The relation with the formulation in N = 1 superspace and the bosonic limit are carried out. Introduction There has been recently an important activity in the study of N = 2 supersymmetric hierarchies (KP [1, 2, 3, 4, 6], generalizations of KdV [5, 7], Two Bosons [8], NLS [9, 10, 11], etc..). The most usual tools in this field are the algebra of N = 1 pseudodifferential operators and Gelfand-Dickey type Poisson brackets [12]. Although these systems have N = 2 supersymmetry, only for very few of them with very low number of fields is a formulation in extended superspace known. It is the purpose of this paper to partially fill this gap. The formalism which we shall present here partly originates from the article [13]. It turns out that in order to construct the Lax operators of N = 2 supersymmetric hierarchies, one should not use the whole algebra of N = 2 pseudo-differential operators, but rather the subalgebra of pseudo-differential operators preserving chirality. These operators were first considered in [14]. They will be defined in Sect. 1, where we also study the KP Lax equations and the two associated Hamiltonian structures. It turns out that the first (linear) bracket is associated with a non-antisymmetric r matrix [15]. Because of that, the second (quadratic) bracket is not of pure Gelfand-Dickey type. The main result of this paper is that we find two possibilities for this quadratic bracket. In fact, we show that there is an invertible map in the KP phase space which sends one of the quadratic Poisson structures into the other. However, this map does not preserve the Hamiltonians.

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In Sect. 2, we study the possible reductions of the KP hierarchy by looking for Poisson subspaces in the phase space. These are different depending on the quadratic bracket which is used. Among these reductions, there are two different hierarchies with the N = 2 classical super-Wn algebra [16] as a hamiltonian structure. In particular, two of the three known N = 2 supersymmetric extensions of the KdV hierarchy [17] are found. They correspond to a = −2 and a = 4 in the classification of Mathieu. These and some other examples are described in Sect. 3. Notice that from the known cases with a low number of fields [17, 18, 19, 20, 21, 22], one expects for any n three hierarchies with super-Wn as a hamiltonian structure. So our construction does not exhaust the possible cases. We also found two hierarchies whose Poisson structure is the classical “small" N = 4 superconformal algebra. In one case the evolution equations are N = 4 supersymmetric, while in the other they are only N = 2 supersymmetric. In Sect. 4 we give the relation of our formulation with the usual formulation of the N = 2 supersymmetric KP Lax equations in N = 1 superspace [23, 3, 6]. Finally, in Sect. 5 we study the bosonic limit of the N = 2 hierarchies. It will turn out that the bosonic limit of the two N = 2 Poisson brackets contains the Gelfand-Dickey bracket, together with two non-standard brackets, found by Oevel in [24], and associated with non-standard KP. We can also check that for the hierarchies studied in this article, the conjecture formulated in [5] about the bosonic limit of N = 2 KdV-type hierarchies applies. 1. N = 2 KP Hierarchy N = 2 supersymmetry. We shall consider an N = 2 superspace with space coordinate ¯ We shall use the notation x for the triple of x and two Grassmann coordinates θ, θ. ¯ The supersymmetric covariant derivatives are defined by coordinates (x, θ, θ). ∂≡

∂ 1¯ ∂ ¯ = ∂ + 1 θ∂, D2 = D ¯ 2 = 0, {D, D} ¯ =∂ , D= + θ∂, D ∂x ∂θ 2 ∂ θ¯ 2

(1.1)

Besides ordinary superfields H(x) depending arbitrarily on Grassmann coordinates, one ¯ can also define chiral superfields ϕ(x) satisfying Dϕ = 0 and antichiral superfields ϕ(x) ¯ ϕ¯ = 0. We define the integration over the N = 2 superspace to be satisfying D Z Z 3 ¯ ¯ θ=θ=0 ¯ θ, θ)| (1.2) d x H(x, θ, θ) = dxDDH(x, ¯ . The elements of the associative algebra of N = 2 pseudo-differential operators (9DOs) are the operators X ¯ + αi D + βi D)∂ i , (ai + bi [D, D] (1.3) P = i<M

where ai , bi and αi , βi are respectively even and odd N = 2 superfields. However, this algebra is not very manageable. In particular, the set of strictly pseudo-differential operators (M = 0 in(1.3)) is not a proper subalgebra, but only a Lie subalgebra. Also, there are too many fields in these operators. We expect the phase space of the N = 2 KdV hierarchies to consist of the supercurrents of the N = 2 Wn algebras. In extended superspace, these supercurrents are bosonic superfields, and there is one such superfield for a given integer dimension. But in (1.3), each power of ∂ corresponds to four superfields, two even ones of integer dimension and two odd ones of half-integer dimension.

N = 2 KP and KdV Hierarchies in Extended Superspace

397

It is thus clear that one has to restrict suitably the form of the N = 2 operators. It turns out that a possible restriction is to define the set Cˇ of pseudo-differential operators Lˇ preserving chirality of the form1 X ¯ Lˇ = DLD, L= ui ∂ i (1.4) i<M

The coefficient functions ui are bosonic N = 2 superfields. These operators satisfy ¯ = 0. The product of two chiral operators is again a chiral operator. The DLˇ = Lˇ D explicit product rule is easily worked out
¯ 0 ) D, ¯ (1.5) Lˇ Lˇ 0 = D L∂L0 + (D.L)(D.L where we have used the notation (D.L) =

X

(Dui )∂ i .

(1.6)

i<M

¯ is the unit of the algebra C. ˇ We could have used as well the Notice that I = D∂ −1 D ¯ ¯ ˇ ˇ algebra C of 9DOs satisfying DL = LD = 0. Notice that the product of an element in Cˇ ¯ tD ∈ by an element in C¯ vanishes. In fact Cˇ and C¯ are related by transposition, Lˇ t = −DL ¯ ˇ ¯ C. Although the transposition leads from C to C, there exists an anti-involution which ˇ It is given by acts inside C. ˇ = DLˇ t ∂ −1 D, ¯ τ (Lˇ 1 Lˇ 2 ) = τ (Lˇ 2 )τ (Lˇ 1 ). τ (L)

(1.7)

Notice that it does not make sense in the algebra Cˇ to multiply a 9DO by a function. However, it is possible to multiply on the left by a chiral function φ, Dφ = 0, ¯ ¯ = λ(φ)L, ˇ λ(φ) ≡ Dφ∂ −1 D, φLˇ = DφLD

(1.8)

¯ D ¯ φ¯ = 0, and on the right by an antichiral function φ, ¯ φ), ¯ λ( ¯ φ) ¯ ≡ D∂ −1 φ¯ D. ¯ = Lˇ λ( ¯ Lˇ φ¯ = DLφ¯ D

(1.9)

We define the residue of the pseudo-differential operator Lˇ by resLˇ = u−1 [17]. The ¯ The trace of Lˇ is ˇ Lˇ 0 ] = Dω¯ + Dω. residue of a commutator is a total derivative, res[L, the integral of the residue Z ˇ ˇ Tr[L, ˇ Lˇ 0 ] = 0. TrL = d3 x resL, (1.10) Cˇ can be divided into two proper subalgebras Cˇ = Cˇ+ ⊕ Cˇ− , where Lˇ is in Cˇ+ if L is a differential operator and Lˇ is in Cˇ− if L is a strictly pseudo-differential operator (M = 0 in (1.4)). We shall note ¯ ∈ Cˇ+ , Lˇ − = DL− D ¯ ∈ Cˇ− . Lˇ = Lˇ + + Lˇ − , Lˇ + = DL+ D

(1.11)

Here an important difference with the usual bosonic and N =R 1 cases occurs. For ˇ res(Lˇ 0 ) 6= 0. any two 9DOs Lˇ and Lˇ 0 in Cˇ one has Tr(Lˇ − Lˇ 0− ) = d3 x res(L) While Cˇ+ is an isotropic subalgebra, Cˇ− is not. One important consequence of this fact 1

Operators of this type were first considered in [14]

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ˇ = 1 (Lˇ + − Lˇ − ), then R is a is that if one defines the endomorphism R of Cˇ by R(L) 2 non-antisymmetric classical r matrix, Z 0 0 ˇ ˇ ˇ ˇ (1.12) Tr(R(L)L + LR(L )) = − d3 x resLˇ resLˇ 0 ,
ˇ Yˇ ]. ˇ Yˇ ] + [X, ˇ R(Yˇ )] = [R(X), ˇ R(Yˇ )] + 1 [X, R [R(X), 4

(1.13)

Notice that a non-antisymmetric r matrix in the context of bosonic KP Lax equations first appeared in [25].

KP equations. Let us now write the evolution equations of the N = 2 supersymmetric ¯ in Cˇ of the form KP hierarchy. We consider operators Lˇ = DLD L = ∂ n−1 +

∞ X

Vi ∂ n−i−1 .

(1.14)

¯ Wi ∂ −i )D,

(1.15)

i=1

Lˇ has a unique nth root in Cˇ of the form 1 Lˇ n = D(1 +

∞ X i=1

and we are led to consider the commuting flows k ∂ ˇ ˇ = [R(Lˇ nk ), L]. ˇ L = [(Lˇ n )+ , L] ∂tk

(1.16)

There are symmetries of these equations which may be described as follows. Let us first introduce a chiral, Grassmann even superfield ϕ which satisfies k ∂ ϕ = (Lˇ n )+ .ϕ, ∂tk

(1.17)

where the right-hand side is the chiral field obtained by acting with the differential k operator (Lˇ n )+ on the field ϕ. Then the transformed operator ˇ ˇ = λ(ϕ−1 )Lλ(ϕ) s(L)

(1.18)

ˇ satisfies an evolution equation of the same form (1.16) as that of L. We may also consider an antichiral, Grassmann odd superfield χ¯ which satisfies k ∂ χ¯ = −(Lˇ n )t+ .χ. ¯ ∂tk

(1.19)

ˇ ˇ = (−1)n λ((Dχ) ¯ −1 )τ (L)λ(D χ) ¯ σ(L)

(1.20)

Then the transformed operator

ˇ with the direction satisfies an evolution equation of the same form (1.16) as that of L, of time reversed.

N = 2 KP and KdV Hierarchies in Extended Superspace

399

Poisson brackets. The Lax Eq. (1.16) are bi-hamiltonian with respect to two compatible Poisson brackets which we now exhibit. Let Xˇ be some 9DO in Cˇ with coefficients ˇ = independent of the phase space fields {Vi }, then define the linear functional lXˇ (L) ˇ ˇ Tr(LX). The generalization of the first Gelfand-Dickey bracket is obvious and reads
ˇ = Tr L[ ˇ Xˇ + , Yˇ+ ] − L[ ˇ Xˇ − , Yˇ− ] . (1.21) {lXˇ , lYˇ }(1) (L) This is nothing but the linear bracket associated with the matrix R. Now we turn to the construction of the second bracket. It will turn out more complicated than the standard Gelfand-Dickey bracket because of the non-antisymmetry of the r matrix. An analogous situation in the bosonic case is studied in [26, 24]. We finally found two different possibilities. In order to write them down, we need to be able to separate the residue of a 9DO in Cˇ into a chiral and an antichiral part. For an arbitrary superfield H(x), we define ¯ 8[H] ¯ ¯ H = 8[H] + 8[H], D8[H] = 0, D = 0.

(1.22)

ˇ An explicit form may be chosen as This is not a local operation in C. Z Z ¯ ¯ ¯ = DD d3 x0 1(x − x0 )H(x0 ), (1.23) 8[H] = DD d3 x0 1(x − x0 )H(x0 ), 8[H] where 1 is the distribution 1(x − x0 ) = (θ − θ0 )(θ¯ − θ¯0 )(x − x0 ), ∂(x − x0 ) = δ(x − x0 ), (x − x0 ) = −(x0 − x).

(1.24)

ˇ X]] ˇ In the following, we shall use the short-hand notations 8[ res[L, = 8Xˇ , ˇ ˇ ¯ ¯ 8[ res[L, X]] = 8Xˇ . In general, 8Xˇ will not satisfy the same boundary conditions as the phase space fields do. However, we noted earlier that in the case of a commutator, ˇ X] ˇ = Dω¯ + Dω. ¯ Here ω and ω¯ are differential the residue is a total derivative, res[L, ¯ − α, ¯ ˇ = Dω polynomials in the fields. Then one easily shows that 8Xˇ = Dω¯ + α, 8 X ¯ Up to this where α is a constant reflecting the arbitrariness in the definition of 8, 8. constant, 8Xˇ will respect the boundary conditions. We are now in a position to write the two possibilities for the second bracket as
ˇ = Tr Lˇ X( ˇ Lˇ Yˇ )+ − Xˇ L( ˇ Yˇ L) ˇ + + 8 ˇ Lˇ Xˇ + Xˇ Lˇ 8 ¯ ˇ , (1.25) {lXˇ , lYˇ }a(2) (L) Y Y and
ˇ = Tr Lˇ X( ˇ Lˇ Yˇ )+ − Xˇ L( ˇ Yˇ L) ˇ + + 8 ˇ Xˇ Lˇ + Lˇ Xˇ 8 ¯ ˇ . {lXˇ , lYˇ }b(2) (L) Y Y

(1.26)

These expressions do not depend on the arbitrary constant α. Checking the antisymmetry of the Poisson brackets and the Jacobi identity can be done with a little effort. As usual, the first bracket is a linearization of the two quadratic ones, that is to say −1 ¯ ˇ ˇ ˇ D) = {lXˇ , lYˇ }a,b {lXˇ , lYˇ }a,b ˇ , lYˇ }(1) (L), (2) (L + zD∂ (2) (L) + z{lX

(1.27)

and the linear bracket is compatible with each of the two quadratic brackets. k Introducing the hamiltonians Hk = nk Tr(Lˇ n ), the KP evolution Eq. (1.16) may be written as
ˇ = {l ˇ , Hk+n }(1) (L) ˇ = {l ˇ , Hk }a,b (L). ˇ (1.28) ∂tk lXˇ (L) (2) X X

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The Poisson brackets (1.25,1.26) may be put in the general abcd form introduced in [27]
ˇ = Tr Lˇ Xa( ˇ Lˇ Yˇ ) + Xˇ Lb( ˇ Lˇ Yˇ ) − Lˇ Xc( ˇ Yˇ L) ˇ − Xˇ Ld( ˇ Yˇ L) ˇ , (1.29) {lXˇ , lYˇ }a(2) (L)
ˇ = Tr Lˇ Xd( ˇ Lˇ Yˇ ) + Xˇ Lc( ˇ Lˇ Yˇ ) − Lˇ Xb( ˇ Yˇ L) ˇ − Xˇ La( ˇ Yˇ L) ˇ . (1.30) {l ˇ , l ˇ }b(2) (L) X

Y

ˇ Indeed However, the price to pay is that a, b, c, d are non-local endomorphisms of C. their expressions are ¯ 8[ ˇ = λ( ˇ ¯ resX]), a = R + c, b(X) ˇ = λ(8[ resX]), ˇ c(X) d = R + b.

(1.31) (1.32)

Using (1.12,1.13), one easily checks that a, d are non-local antisymmetric r matrices and that the following two compatibility relations hold: ˇ c(Yˇ )] − a([X, ˇ c(Yˇ )]) + c([b(X), ˇ Yˇ ]) = 0, [a(X), ˇ ˇ ˇ ˇ ˇ Yˇ ]) = 0. [d(X), b(Y )] − d([X, b(Y )]) + b([c(X),

(1.33)

These are sufficient conditions for {, }a(2) and {, }b(2) to be Poisson brackets. Poisson maps. Before turning to the study of the reductions of the KP hierarchies, let us exhibit some relations between the two quadratic brackets. Due to the abcd structure ˇ = Lˇ −1 is an anti-Poisson map of the brackets, the inversion I(L) {lXˇ ◦ I, lYˇ ◦ I}a(2) = −{lXˇ , lYˇ }b(2) ◦ I.

(1.34)

ˇ Let us now consider the invertible map in C, ¯ ˇ = ∂ −1 τ (L) ˇ = D∂ −1 Lˇ t ∂ −1 D. p(L)

(1.35)

A straightforward calculation leads to {lXˇ ◦ p, lYˇ ◦ p}a(2) = −{lXˇ , lYˇ }b(2) ◦ p,

(1.36)

which gives another equivalence relation between (1.25) and (1.26). However there is k k ˇ n−1 no relation between the hamiltonians Tr(Lˇ n ) and Tr(p(L) ). There is another relation between the two brackets, which involves the chiral superfield R ϕ satisfying the evolution Eq. (1.17). Let us introduce the linear functional lt = d3 x(tϕ), where t(x) is a Grassmann even superfield. We consider an enlarged phase space including ϕ, and extend the Poisson bracket (1.25) to this phase space by Z ˇ ϕ) = d3 xt((Lˇ Yˇ )+ .ϕ + 8 ˇ ϕ), {lt , lt0 }a(2) = 0. {lt , lYˇ }a(2) (L, (1.37) Y Then one finds

{lXˇ ◦ s, lYˇ ◦ s}a(2) = {lXˇ , lYˇ }b(2) ◦ s,

(1.38)

where the transformation s has been defined in (1.18). Notice that the hamiltonians are k ˇ nk ). invariant functions for the transformation s, Tr(Lˇ n ) = Tr(s(L) A last relation uses the antichiral R superfield χ¯ satisfying the evolution (1.19). Let us ¯ where t¯(x) is a Grassmann odd superfield. introduce the linear functional lt¯ = d3 x(t¯χ),

N = 2 KP and KdV Hierarchies in Extended Superspace

401

We consider an enlarged phase space including χ, ¯ and extend the Poisson bracket (1.25) to this phase space by R ˇ t+ .χ¯ + 8 ˇ χ) ¯ ˇ χ), ¯ = d3 xt¯(−(Yˇ L) (1.39) {lt¯, lYˇ }a(2) (L, Y ¯ R a 3 ¯ ¯ {lt¯1 , lt¯2 }(2) = −2 d xt1 χ¯ 8[t¯2 χ], ¯ (1.40) ¯ are defined in Eqs. (1.23,1.24). Notice that this is a non-local Poisson where 8, 8 bracket. One finds (1.41) {lXˇ ◦ σ, lYˇ ◦ σ}a(2) = −{lXˇ , lYˇ }b(2) ◦ σ, where the transformation σ has been defined in (1.20).

2. Reductions of the KP Hierarchy In order to obtain consistent reductions of the KP hierarchy, we need to find Poisson submanifolds of the KP phase space. Considering first the quadratic bracket (1.25), we rewrite it as ˇ = TrXξ ˇ a , {lXˇ , lYˇ }a(2) (L) lYˇ ˇ Yˇ L) ˇ + + 8 ˇ Lˇ + Lˇ 8 ¯ ˇ. ξlaYˇ = (Lˇ Yˇ )+ Lˇ − L( Y Y

(2.1)

ξlaYˇ is the hamiltonian vector field associated with the function lYˇ . One easily checks P ¯ It is that if Lˇ has the form (1.14), then for any Yˇ , ξlaYˇ has the form D( i
the constraint Lˇ = Lˇ +

(2.2)

defines a Poisson submanifold. The hierarchies obtained in this way are the N = 2 supersymmetric KdV hierarchies studied by Inami and Kanno [23], and the Lax operators (2.2) already appeared in [14]. The lowest order cases will be presented in the next section. Another possible reduction is to take Lˇ of the form ¯ Lˇ = Lˇ + + D ϕ∂ −1 ϕ¯ D,

¯ ϕ¯ = 0, Dϕ = D

(2.3)

where ϕ and ϕ¯ are Grassmann even or odd chiral superfields. With Lˇ of the form (2.3) and Yˇ arbitrary, one finds ˇ t+ .ϕ¯ + 8 ¯ ¯ ˇ ϕ)) (ξlaYˇ )− = D((Lˇ Yˇ )+ .ϕ + 8Yˇ ϕ)∂ −1 ϕ¯ + ϕ∂ −1 (−(Yˇ L) Y ¯ D,

(2.4)

ˇ t+ .ϕ¯ an antichiral superfield, it is Noticing that (Lˇ Yˇ )+ .ϕ is a chiral superfield and (Yˇ L) a easily checked that ξlYˇ is indeed tangent to the submanifold defined by the constraints (2.3). It is possible to consider an enlarged phase space which coordinates are the fields in Lˇ and ϕ, ϕ. ¯ Let us introduce the linear functionals Z Z 3 lt = d x(ϕt), lt¯ = d3 x(t¯ϕ), ¯ (2.5)

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where t and t¯ are general superfields, of the same Grassmann parity as ϕ and ϕ. ¯ In this enlarged phase space, the second Poisson bracket, in the case when ϕ and ϕ¯ are Grassmann even, is defined by (1.25) and R ˇ ϕ, ϕ) ¯ = d3 x((Lˇ Yˇ )+ .ϕ + 8Yˇ ϕ)t, (2.6) {lt , lYˇ }a(2) (L, R ˇ t+ .ϕ¯ + 8 ˇ ϕ, ϕ) ¯ ˇ ϕ), {lt¯, l ˇ }a (L, ¯ = d3 x t¯(−(Yˇ L) ¯ Y

(2)

Y

and ˇ ϕ, ϕ) ¯ = {lt , lt¯}a(2) (L, {lt1 , lt2 }a(2)

= 0,

R

d3 x (Lˇ + .t¯)t,

{lt¯1 , lt¯2 }a(2)

(2.7)

= 0.

In the case when ϕ and ϕ¯ are Grassmann odd, the last two lines should be modified to R ˇ ϕ, ϕ) ¯ = d3 x((Lˇ + .t¯)t − 2ϕt8[t¯ϕ]), ¯ (2.8) {lt , lt¯}a(2) (L, R 3 R 3 a a ¯ ¯ ¯ {lt1 , lt2 }(2) = 2 d x ϕt1 8[ϕt2 ], {lt¯1 , lt¯2 }(2) = −2 d x t1 ϕ¯ 8[t2 ϕ], ¯ ¯ have been defined in (1.23). The lowest order case is where the applications 8 and 8 ¯ Then if ϕ and ϕ¯ are odd, the equation d Lˇ = [Lˇ 2+ , L] ˇ is the Lˇ = D(1 + ϕ∂ −1 ϕ) ¯ D. dt N = 2 supersymmetric extension of the NLS Eq. [28]. The next-to-lowest order case is ¯ If ϕ and ϕ¯ are even, the hamiltonian structure (1.25) reduces in Lˇ = D(∂ +H +ϕ∂ −1 ϕ) ¯ D. this case to the classical version of the “small” N = 4 superconformal algebra. Although the Poisson algebra contains 4 supersymmetry generators, the evolution Eq. (1.16) have only N = 2 supersymmetry. This case was first obtained by another method which will be given, as part of a detailed study, in [29]. We now turn to the second quadratic bracket (1.26). We rewrite it as ˇ = TrXξ ˇ b , {lXˇ , lYˇ }b(2) (L) lYˇ ¯ ¯ ˇ ˇ Yˇ L) ˇ + + Lλ(8 ˇ ξlbYˇ = (Lˇ Yˇ )+ Lˇ − L( Yˇ ) + λ(8Yˇ )L.

(2.9)

It is easily seen that neither the condition (2.2), nor the more complicated condition (2.3) are admissible reductions in this case. The easiest way to find Poisson subspaces for the bracket (1.26) is to apply the map (1.35) to the Poisson subspaces of the first quadratic bracket. From (2.2), we are then lead to the restriction: ¯ −1 H∂ −1 DD. ¯ Lˇ = Lˇ + + DD∂

(2.10)

With Lˇ of the form (2.10) and Yˇ arbitrary, one finds ¯ −1 ((Lˇ Yˇ )+ .H − (Yˇ L) ˇ t+ .H + res[L, ˇ Yˇ ]H)∂ −1 DD, ¯ (ξlbYˇ )− = DD∂

(2.11)

which directly shows that condition (2.10) defines a Poisson submanifold for the Poisson bracket (1.26). It turns out that (2.10) also defines a Poisson submanifold for the linear Poisson bracket (1.21). To show this we rewrite the linear bracket as ¯ 8 ˇ = TrXη ˇ l ˇ , ηl ˇ = [L, ˇ Yˇ ]+ − [L, ˇ Yˇ+ ] + λ(8 ˇ ) + λ( ¯ ˇ ). {lXˇ , lYˇ }(1) (L) Y Y Y Y

(2.12)

With Lˇ of the form (2.10) and Yˇ arbitrary, one finds ¯ −1 ((Yˇ+ − Yˇ+t ).H + res[L, ˇ Yˇ ])∂ −1 DD. ¯ (ηlYˇ )− = DD∂

(2.13)

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403

Thus the reduced hierarchies defined by condition (2.10) are bi-hamiltonian. The lowest order cases will be studied in the next section. Notice that the transformation (1.18) maps the systems satisfying the condition (2.3) with Grassmann even fields ϕ and ϕ¯ into systems satisfying condition (2.10) with H = ϕϕ¯ + ϕ−1 Lˇ + .ϕ.

(2.14)

Analogously, the transformation (1.20) maps the systems satisfying the condition (2.3) with Grassmann odd fields ϕ and ϕ¯ into systems satisfying condition (2.10) with
¯ . (2.15) ¯ + (Dϕ) ¯ −1 D(Lˇ t+ .ϕ) H = (−1)n ϕϕ Such transformations may be found in [9, 5]. Finally we may consider the image of the Poisson subspace defined by (2.3) under the map p. One finds the condition ¯ −1 (H + ϕ∂ ¯ ¯ −1 ϕ)∂ −1 DD. Lˇ = Lˇ + + DD∂

(2.16)

It turns out that (2.16) also defines a Poisson submanifold for the linear Poisson bracket (1.21), as may be shown using relation (1.27). ¯ The hamiltonian structure (1.26) reduces The lowest order case is when Lˇ + = DD. in this case to the classical version of the “small” N = 4 superconformal algebra. The ∂ ˇ ˇ becomes, after suitable redefinitions, the N = 4 supersymL = [(Lˇ 3 )+ , L] equation ∂t metric extension of the KdV equation derived in [30] and written in N = 2 superspace in [31]. One can again consider an enlarged phase space which coordinates are the fields in Lˇ and ϕ, ϕ. ¯ The second quadratic bracket in this phase space is easily obtained from the first one by applying the map p to the first quadratic bracket. p acts as the identity on ϕ and ϕ. ¯ As a consequence the Poisson brackets (2.7) and (2.8) keep the same form, whereas (2.6) should be modified to
R ˇ ϕ, ϕ) ˇ + )λ(ϕ) + 8 ˇ ϕ)t, ¯ = d3 x( res τ ((Yˇ L) (2.17) {lt , lYˇ }b(2) (L, Y
R ¯ ϕ)∂ ˇ ϕ, ϕ) ¯ ˇ ϕ). {lt¯, l ˇ }b (L, ¯ −1 τ ((Lˇ Yˇ )+ )∂ + 8 ¯ = d3 x t¯(− res λ( ¯ Y

(2)

Y

3. Examples and Comparison with Other Works This paragraph is devoted to the presentation of the simplest integrable equations obtained using our formalism. Considering first the condition (2.2), the simplest example is the lax operator Lˇ = ¯ Then the evolution equation D(∂ + W )D. 3 ∂ ˇ ˇ L = [Lˇ +2 , L], ∂t

(3.1)

leads to the equation 8



∂ W = 2Wxxx + 6 (DW )(DW ) x − W 3 x , ∂t

(3.2)

which coincide after the redefinition W = 2i8 with the a = −2, N = 2 extension of the KdV equation in the classification of Mathieu [17, 18]. The Lax operator given in [17] may be obtained from Lˇ in the following way. Let us consider the operator

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F. Delduc, L. Gallot

¯ + (DW )D ¯ − (DW ¯ )D. Lˇ −2 = Lˇ + Lˇ t = ∂ 2 + W [D, D]

(3.3)

¯ If we remember that the product of an element in Cˇ and an Lˇ is in Cˇ and Lˇ t is in C. element in C¯ always vanishes, we immediately get that a square root of Lˇ −2 with h ighest 1 1 1 derivative term equal to ∂ is (Lˇ −2 ) 2 = Lˇ 2 − (Lˇ 2 )t . From this we deduce the relation 3 3 3 (Lˇ −2 ) 2 = Lˇ 2 − (Lˇ 2 )t . As a consequence Lˇ −2 satisfies the evolution equation 3 3 ∂ ˇ ˇ + ([Lˇ +2 , L]) ˇ t = [(Lˇ −2 ) 23 , Lˇ −2 ], L−2 = [Lˇ +2 , L] ∂t

(3.4)

which is thus an equivalent Lax representation for Eq. (3.3). As the next example, we consider the Lax operator ¯ Lˇ = D(∂ 2 + V ∂ + W )D.

(3.5)

2/3 ˇ ∂ ˇ L = [Lˇ + , L] should coincide, after suitable redefinitions, Then the evolution equation ∂t with one of the three N = 2 supersymmetric extensions of the Boussinesq equations derived in [21]. Indeed one can check that the Lax operator given there for the α = −1/2 ¯ 2 D. Then one easily obtains (Lˇ (1) ) 23 = Lˇ 23 + equation may be written as Lˇ (1) = Lˇ + D∂ ¯ ˇ D∂D, and the evolution equation for Lˇ (1) is easily deduced from that of L, 2 ∂ ˇ (1) ∂ ˇ L = L = [(Lˇ (1) ) 3 , Lˇ (1) ]. ∂t ∂t

(3.6)

Turning now to condition (2.10), the lowest order case corresponds to the Lax oper¯ + DD∂ ¯ −1 W ∂ −1 DD. ¯ Then the equation ∂ Lˇ = [(Lˇ 3 )+ , L] ˇ becomes, after ator Lˇ = DD ∂t suitable redefinitions, the N = 2 supersymmetric extension of the KdV equation with parameter a = 4,

3 ∂ W = Wxxx + [D, D]W 2 x − 3 (DW )(DW ) x + (W 3 )x . ∂t 2

(3.7)

Notice that all integer powers of Lˇ define conserved charges in this case (an alternative Lax operator with the same property was derived in [10]). The last example that we shall study is the Lax operator Lˇ = D (∂ + V ) D + DD∂ −1 W ∂ −1 DD.

(3.8)

∂ ˇ ˇ L = [Lˇ + , L] ∂t

(3.9)

Then the equation

explicitly reads ∂ V = 2Wx , ∂t ∂ W = [D, D]Wx + V Wx + (DV )(DW ) + (DV )(DW ). ∂t

(3.10) (3.11)

This equation is identical, up to a rescaling of time, to the N = 2 supersymmetric extension of the Boussinesq equation with parameter α = −2 derived in [21].

N = 2 KP and KdV Hierarchies in Extended Superspace

405

4. From N = 2 to N = 1 Superspace N = 2 extensions of the KP and KdV hierarchies have been studied in several articles [23, 4, 3, 6] using an N = 1 superspace formalism. In this section we wish to relate the KP hierarchies that we described in Sect. 1 to those given in the literature. The first step will be to relate the N = 2 algebra Cˇ of pseudo-differential operators to the ¯ in Cˇ should be N = 1 algebra of pseudo-differential operators. An operator Lˇ = DLD considered as acting on a chiral object 9, D9=0, and this action writes ˇ = DLD.9 ¯ ¯ L.9 = L∂.9 + (D.L)D.9.

(4.1)

We shall use the following combinations of the chiral derivatives: ¯ D2 = −D + D, ¯ D12 = −D22 = ∂, {D1 , D2 } = 0. D1 = D + D,

(4.2)

Then the action of Lˇ on 9 is ˇ = (L∂ + (D.L)D1 ).9. L.9

(4.3)

We then choose to associate to the N = 2 pseudo-differential operator Lˇ the N = 1 pseudo-differential operator L given by L = L|θ2 =0 ∂ + (D.L)|θ2 =0 D1 .

(4.4) 0

0

It is easily checked that this correspondence respects the product, LL = L L . It also has the property (4.5) L+ = L>0 . That is to say that the image of an N = 2 differential operator is a strictly differential N = 1 operator, without the non-derivative term. Notice also the useful relations ˇ θ2 =0 , res(L) = (D. res(L))| R 2 R 2 R ˇ = Tr(L) ≡ d x res(L), d x ≡ dxdθ1 , Tr(L)

(4.6) (4.7)

where the residue of the operator L is the coefficient of D1−1 ≡ D1 ∂ −1 . From now on, all expressions will be written in N = 1 superspace, and we drop the index of D1 and θ1 . The KP hierarchy described in Sect. 1 may be described in N = 1 superspace as follows. We consider an operator L of the form L = D2n +

∞ X

wp D2n−p−1

(4.8)

p=1

and consider evolution equations k ∂ n L = [L>0 , L]. ∂tk

(4.9)

This is nothing but the non-standard supersymmetric KP hierarchy described in [4, 3]. p The evolution Eqs. (4.9) admit the conserved quantities Hp = Tr(L n ), and they are bi-hamiltonian. The first Poisson bracket is easily deduced from its N = 2 counterpart (1.21). With lX = Tr(L X), we have {lX , lY }1 (L) = TrL([X >0 , Y >0 ] − [X ≤0 , Y ≤0 ]).

(4.10)

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F. Delduc, L. Gallot

As in the N = 2 formalism, this is a standard bracket associated with a non-antisymmetric r matrix. As a consequence, the two quadratic brackets deduced from (1.25) and (1.26) are quite complicated. They involve the quantity ψX defined up to a constant by DψX = res[L , X]. The first one is R {lX , lY }a2 (L) = Tr(L X(L Y )+ − X L(Y L)+ ) + d2 x(−ψY res[L, X] + res[L , Y ] res(X L D−1 ) − res[L , X] res(Y L D−1 )).

(4.11)

The second one becomes {lX , lY }b2 (L) = Tr(L X(L Y )+ − X L(Y L)+ ) + + res[L , Y ] res(L X D

−1

R

d2 x(ψY res[L, X]

) − res[L , X] res(L Y D−1 )),

(4.12)

and already appeared in [6]. It is not a difficult task to obtain the N = 1 restrictions which correspond to the N = 2 conditions (2.2,2.3,2.10,2.16). Some of the lax operators obtained in this way are already known, in particular those satisfying (2.2) from [23] and the lowest order operator coming from (2.3) with odd ϕ and ϕ, ¯ which is the super-NLS Lax operator obtained in [3]. 5. Bosonic Limit of N = 2 KP Hierarchies In this section we wish to study the bosonic limit of the N=2 KP hierarchies described before. A conjecture on this limit for N=2 KdV hierarchies has been given in [5] and we shall verify that it holds in our case. Before starting, let us recall some basic facts about the Poisson bracket structures of KP and non-standard KP hierarchies. We denote by C the algebra of (bosonic) pseudo-differential operators. The trace operation in C will be R denoted by tr(L) = dx res(L). In the study of the KP hierarchy one considers operators of the form ∞ X vi ∂ n−i . (5.1) L = ∂n + i=1

The commuting flows are defined with the help of the antisymmetric r-matrix R defined by R(L) = 21 (L+ − L− ). These flows are hamiltonian with respect to the one parameter family of quadratic Poisson structures {lX , lY }λGD (L) = tr (LX(LY )+ − XL(Y L)+ ) R +λ dx(D−1 res[L, X]) res[L, Y ],

(5.2)

where ∂(D−1 f ) = f . These brackets are local since D−1 is applied to the residue of a commutator, which is a total derivative. They are of the general abcd form with non-local operators aλ = R + λr, bλ = −λr, cλ = λr, dλ = R − λr, where r(L) = D−1 res(L). The Gelfand-Dickey bracket corresponds to λ = 0, and the interchange abcd → dcba simply reverses the sign of λ. For the particular value λ = n1 , and when L has the form (5.1), the field v1 is central in the Poisson algebra and may be set to zero. There is a Poisson map which leads from a generic value of λ to the special value λ = n1 . It reads g(L) = e8 Le−8 , 8 = n1 D−1 v1 . Indeed one finds 1

n ◦ g. {lX ◦ g, lY ◦ g}λGD = {lX , lY }GD

(5.3)

N = 2 KP and KdV Hierarchies in Extended Superspace

407

For any value of λ, the restriction L = L+ defines a Poisson submanifold. The hierarchy thus obtained is the nth KdV hierarchy. We now turn to the non-standard KP hierarchy. It is constructed with the help of the non-antisymmetric r-matrix R − P0 , where P0 is the projector on the non-differential term in L: P0 (L) ≡ L0 = vn . The equations of the non-standard KP hierarchy are hamiltonian with respect to two quadratic Poisson brackets given by Oevel in [24], {lX , lY }aO (L) = tr(LXa(LY ) + XLb(LY ) − LXc(Y L) − XLd(Y L)),

(5.4)

{lX , lY }bO (L) = tr(LXd(LY ) + XLc(LY ) − LXb(Y L) − XLa(Y L)),

(5.5)

with a = R + r, b = −∂ −1 r∂, c = P0 + r, d = R − P0 − ∂ −1 r∂. The two brackets admit different Poisson subspaces. In the case of bracket (5.4), one may take L = L> and in the case of bracket (5.5), L = L+ + ∂ −1 h. There is a relation between the three brackets (5.2), (5.4), (5.5), which will come out of the bosonic limit of the N = 2 case that we now study. ¯ in Cˇ satisfying the conditions From now on, we restrict to operators Lˇ = DLD ¯ DL|0 = DL|0 = 0, where the limit |0 means that θ and θ¯ are set to zero. This defines a subspace CˇB of Cˇ which is closed under the product. To an operator Lˇ in CˇB we can associate two ordinary operators in C by ˇ = L|0 ∂, µ1 (L) ˇ = L|0 ∂ + (DDL)| ¯ µ2 (L) 0.

(5.6) (5.7)

ˇ = L1 , From now on, whenever possible we shall use the short-hand notations µ1 (L) ˇ = L2 . It is easily checked that µ1 and µ2 are morphisms from CˇB to C and have µ2 (L) the properties µ1 (Lˇ + ) = (L1 )> , µ1 (Lˇ − ) = (L1 )≤ , µ2 (Lˇ + ) = (L2 )+ − (L1 )0 , µ2 (Lˇ − ) = (L2 )− + (L1 )0 TrLˇ = trL2 − trL1 .

(5.8) (5.9) (5.10)

Using this, one easily sees that the N=2 KP Eqs. (1.16) imply k ∂ L1 = [(L1n )> , L1 ], ∂tk k k ∂ L2 = [(L2n )+ − (L1n )0 , L2 ]. ∂tk

(5.11) (5.12)

While one recognizes the non-standard KP flows in (5.11), the interpretation of (5.12) requires more care. From the definitions (5.6,5.7), one finds that L1 and L2 have the same leading and next-to-leading order terms ∂ n + v∂ n−1 + . . .. The flow equation of v, deduced from Eq. (5.11) or equivalently from (5.12), is k ∂ v = n∂(L1 )0n . ∂tk

(5.13)

We may then transform L2 according to g(L2 ) = e8 L2 e−8 , 8 =

1 −1 D v. n

(5.14)

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F. Delduc, L. Gallot

Then Eq. (5.12) implies for the image the standard KP equations k ∂ g(L2 ) = [(g(L2 ))+n , g(L2 )]. ∂tk

(5.15)

We shall now see how the quadratic Poisson structures of Eqs. (5.2) and (5.4,5.5) emerge from the quadratic structures (1.25,1.26) of the N=2 KP hierarchy. Using (5.10) we ˇ on CˇB as decompose the linear function lXˇ (L) ˇ = TrXˇ Lˇ = l2 (L2 ) − l1 (L1 ), lXˇ (L) X2 X1 1 lX (L1 )

= trXL1 ,

2 lX (L2 )

(5.16)

= trXL2 .

(5.17)

Then one finds for the first bracket 1 1 , lY1 }a(2) (L1 , L2 ) = −{lX , lY1 }aO (L1 ), {lX 2 2 {lX , lY2 }a(2) (L1 , L2 ) = {lX , lY2 }+1 GD (L2 ),

(5.18)


2 , lY1 }a(2) (L1 , L2 ) = tr (Y L1 )0 + D−1 res[Y, L1 ] [X, L2 ], {lX

(5.19) (5.20)

and for the second bracket 1 1 , lY1 }b(2) (L1 , L2 ) = −{lX , lY1 }bO (L1 ), {lX 2 {lX , lY2 2 , lY1 {lX

2 = {lX , lY2 }−1 GD (L2 ),
−1 L1 )0 + D res[Y, L1 ] [X, L2 ].

}b(2) (L1 , L2 )

}b(2) (L1 , L2 )

= tr (Y

(5.21) (5.22) (5.23)

It is not hard to check that these brackets generate the flows (5.11,5.12) by using the k

k

hamiltonian Hk = nk (trL1n − trL2n ). One can extract the Poisson brackets of v with 1 2 functionals lX and lX , 2 a }(2) (L1 , L2 ) = (n − 1) res[X, L2 ], {v, lX 1 a {v, lX }(2) (L1 , L2 )

= n (∂(XL1 )0 + res[X, L1 ]) .

(5.24) (5.25)

The form of the Poisson brackets simplifies by using the map g defined in (5.14). A direct calculation gives 1 1 1 a,b , lY1 }a,b {lX (2) (L1 , L2 ) = −{lX , lY }O (L1 ), 1 n

2 2 2 {lX ◦ g, lY2 ◦ g}a,b (2) (L1 , L2 ) = {lX , lY }GD (g(L2 )), 2 ◦ {lX

g, lY1 }a,b (2) (L1 , L2 )

= 0.

(5.26) (5.27) (5.28)

These are the quadratic Poisson structures generating the non-standard (5.11) and standard (5.15) KP hierarchies, which are now completely decoupled. Analogous considerations yield the corresponding result for the linear Poisson bracket. In Sect. 1, we have shown that there exists an anti-Poisson map p, eq.(1.35), between the brackets a and b. It is natural to investigate the consequences of this property in the bosonic limit. Let us define two invertible maps u and v of C by u(L) = ∂ −1 Lt and v(L) = Lt ∂. Notice that u−1 = −v. Then the following properties hold µ2 ◦ p = u ◦ µ1 , µ1 ◦ p = u ◦ µ2 .

(5.29)

N = 2 KP and KdV Hierarchies in Extended Superspace

409

It is just a matter of calculations to show, using (5.20) and (5.23), that they lead to the following Poisson properties between the bosonic brackets2 : b {lX ◦ u, lY ◦ u}+1 GD = {lX , lY }O ◦ u,

{lX ◦ v, lY ◦

v}−1 GD

=

{lX , lY }aO

◦ v.

(5.30) (5.31)

Let us now examine the consequences of these results. They clearly imply that the bosonic limit of the N=2 KP hierarchy is composed of two decoupled bosonic hierarchies, namely the standard and non-standard KP hierarchies. The bosonic limit of the N=2 nth KdV corresponding to the reduction (2.2) is composed of the usual nth KdV, g(L2 ) = ∂ n +

n−2 X

un−k ∂ k ,

(5.32)

k=0

the quadratic Poisson structure of which is the Wn algebra, and of a reduction of the non-standard KP hierarchy L1 = ∂ n +

n−1 X

vn−k ∂ k ,

(5.33)

k=1

the quadratic Poisson structure of which is the Wn−1 ⊕ U (1) algebra. The bosonic limit of the N=2 nth KdV corresponding to the reduction (2.10) is composed again of the nth KdV, and of a reduction of the non-standard KP hierarchy L1 = ∂ n +

n−1 X

vn−k ∂ k + ∂ −1 vn+1 ,

(5.34)

k=0

the quadratic Poisson structure of which is Wn+1 ⊕ U (1). This is exactly the conjecture of [5] for the cases considered here. 6. Conclusion An easy generalization of the hierarchies presented in this article would be to consider multi-components KP hierarchies, that is to say replace the fields ϕ and ϕ¯ in (2.3) and (2.16) by a set of n + m fields ϕi and ϕ¯ i , n of them being Grassmann even and the other m being Grassmann odd. For the lowest order case of Eq. (2.3), such a generalization has been considered in [5]. The Lax representation that we propose for such hierarchies has the advantage that one does not need to modify the definition of the residue. For the next to lowest order case of Eq. (2.3), and the lowest order case of Eq. (2.16), it should be possible to obtain in this way hierarchies based on W-superalgebras with an arbitrary number of supersymmetry charges. Little is known about the matrix Lax formulation of the hierarchies presented here. In the case of operators satisfying condition (2.2), such a matrix Lax formulation was constructed in N = 1 superspace by Inami and Kanno [23, 32]. It involves the loop 2 Using these properties and the Poisson property of the transposition T (T (L) = Lt ), {l ◦T, l ◦T }λ X Y GD = −1 Lt ∂, satisfies ˜ ˜ {lX , lY }λ GD ◦ T , it is easily shown that T , the anti-involution of G defined as T (L) = ∂ a,b ˜ {lX ◦ T˜ , lY ◦ T˜ }a,b O = {lX , lY }O ◦ T . Thus Poisson structures are obtained for Kuperschmidt reductions [25].

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F. Delduc, L. Gallot

superalgebra based on sl(n|n). What we know about the matrix Lax formulation in N = 2 superspace for hierarchies based on Lax operators satisfying conditions (2.2) or (2.3) will be reported elsewhere. Notice that we obtained the form (2.2) of the scalar Lax operators from a matrix Lax representation, and only later became aware of reference [14] where these operators first appeared. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32.

Popowicz, Z.: J. Phys. A29, 1281 (1996) Aratyn, H. and Rasinariu, C.: Preprint UICHEP-TH/96-15 and hep-th/9608107 Brunelli,J.C. and Das, A.: Rev. Math. Phys. 7, 1181 (1995) Ghosh, S. and Paul, S.: Phys. Lett. B 341, 293 (1995) Bonora, L., Krivonos, S. and Sorin, A.: Preprint SISSA-56-96-EP and hep-th/9604165 Das, A. and Panda, S.: Mod. Phys. Lett. A 11, 723 (1996) Ivanov, E., Krivonos, S. and Malik, R.P.: Int. J. Mod. Phys. A 10, 253 (1995) Das, A. and Brunelli, J.C.: Phys. Lett. B 337, 303 (1994); Phys. Lett. B 354, 307 (1995); Int. J. Mod. Phys. bf A 10, 4563 (1995); Preprint hep-th/9506096 Krivonos, S. and Sorin, A.: Phys. Lett. B 357, 94 (1995) Krivonos, S., Sorin, A. and Toppan, F.: Phys. Lett. A 206, 146 (1995) Das, A. and Brunelli, J.C.: Mod. Phys. Lett. A 10, 2019 (1995); J. Math. Phys. 36, 268 (1995) Gelfand, I.M. and Dikii, L.A.: Funct. Anal. Appl. 10, 259 (1976) Delduc, F. and Magro, M.: J. Phys. A : Math. Gen. 29, 4987 (1996) Popowicz, Z.: Phys. Lett. B 319, 478 (1993) Semenov-Tian-Shansky, M.A.: Funct. Anal. Appl. 17, 259 (1983) Lu, H., Pope, C.N., Romans, L.J., Shen, X. and Wang, X.J.: Phys. Lett. B 264, 91 (1991) Laberge, C.A., Mathieu, P.: Phys. Lett. B 215, 718 (1988) Labelle, P. and Mathieu, P.: J. Math Phys. 32, 923 (1991) Popowicz, Z.: Phys. Lett. A 174, 411 (1993) Yung, C.M.: Phys. Lett. B 309, 175 (1993) Bellucci, S., Ivanov, E., Krivonos, S. and Pichugin, A.: Phys. Lett. B 312, 463 (1993) Yung, C.M. and Warner, R.C.. J. Math. Phys. 34, 4050 (1993) Inami, T. and Kanno, H.: Int. J. Mod. Phys. A 7,, Suppl. 1A, 419 (1992) Oevel, W.: Phys. Lett. A 186, 79 (1994) Kupershmidt, B.A.: Commun. Math. Phy. 99, 51 (1985) Oevel, W. and Strampp, W.: Commun. Math. Phys. 157, 51 (1993) Freidel, L., Maillet, J.M.: Phys. Lett. B 262, 278 (1991) Roelofs, G. and Kersten, P.: J. Math Phys. 33, 2185 (1992) Delduc, F., Ivanov, E. and Gallot, L.: In preparation Delduc F. and Ivanov, E.: Phys. Lett. B 309, 312 (1993) Delduc, F., Ivanov, E. and Krivonos, S.: J. Math. Phys. 37, 1356 (1996) Inami, T. and Kanno, H.: J. Phys. A 25, 3729 (1992)

Communicated by T. Miwa

Commun. Math. Phys. 190, 411 – 457 (1997)

Communications in

Mathematical Physics c Springer-Verlag 1997

Quantization of Coset Space σ -Models Coupled to Two-Dimensional Gravity D. Korotkin? , H. Samtleben?? II. Institut f¨ur Theoretische Physik, Universit¨at Hamburg, Luruper Chaussee 149, D-22761 Hamburg, Germany Received: 25 July 1996 / Accepted: 23 April 1997

Abstract: The mathematical framework for an exact quantization of the two-dimensional coset space σ-models coupled to dilaton gravity, that arise from dimensional reduction of gravity and supergravity theories, is presented. Extending previous results [49] the two-time Hamiltonian formulation is obtained, which describes the complete phase space of the model in the isomonodromic sector. The Dirac brackets arising from the coset constraints are calculated. Their quantization allows to relate exact solutions of the corresponding Wheeler–DeWitt equations to solutions of a modified (Coset-)KnizhnikZamolodchikov system. On the classical level, a set of observables is identified, that is complete for essential sectors of the theory. Quantum counterparts of these observables and their algebraic structure are investigated. Their status in alternative quantization procedures is discussed, employing the link with Hamiltonian Chern–Simons theory.

1. Introduction It is an important class of physical theories, that admit the formulation as a gravity coupled coset space σ-model after dimensional reduction to two dimensions. Including pure gravity and Kaluza-Klein theories as well as extended supergravity theories, in 3+1 dimensions they are described by a set of scalar and vector fields coupled to gravity, where the scalar fields already form a non-linear σ-model. Further reduction is achieved by imposing additional symmetries – manifest by assuming two additional commuting Killing vector fields, for example corresponding to the study of axisymmetric stationary models. ?

On leave of absence from Steklov Mathematical Institute, Fontanka, 27, St.Petersburg 191011, Russia Present address: Max-Planck-Institut f¨ur Gravitationsphysik, Albert-Einstein-Institut, Schlaatzweg 1, D14473 Potsdam, Germany ??

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This reduction to effectively two dimensions leads to a non-linear σ-model in an enlarged coset space, coupled to two-dimensional gravity and a dilaton field. The arising additional scalar fields that contribute to parametrizing the coset space are a remnant of the original vector fields and of components of the former higher-dimensional metric. For general reason, related to boundedness of the energy, it is the maximal compact subgroup H of G that is divided out in the coset. The first reduction of this type, discovered for pure gravity [33], leads to the simplest coset space SL(2, R)/SO(2). It was generalized up to the case of maximally extended N = 8 supergravity, where the E8(+8) /SO(16) arises [40, 41]. The general proceeding was analyzed in [13, 55]. In [47–49] a program was started to perform an exact quantization of these dimensionally reduced gravity models. Progress has been achieved using methods and techniques similar to those developed in the theory of flat space integrable systems [24, 26, 46]. Despite the fact that dimensional reduction via additional symmetries represents an essential truncation of the theory, these so-called midi-superspace models under investigation are sufficiently complicated to justify the hope that their exact quantization might provide insights into fundamental features of a still outstanding quantized theory of gravitation. In particular and in contrast to previously exactly quantized minisuperspace models, they exhibit an infinite number of degrees of freedom, which is broadly accepted to be a sine qua non for any significant model of quantum gravity (compare [52, 5] for a discussion of this point in the context of related models). One of the final purposes of this approach is the identification of exact quantum states, whose classical limit corresponds to the known classical solutions. For pure gravity this includes the quantum analogue of the Kerr solution describing the rotating black hole; for extended supergravities recently discovered corresponding solutions have been of particular interest exhibiting fundamental duality symmetries [17, 16], such that their exact quantum counterparts should shed further light onto the role of these symmetries in a quantized theory. The main ideas of the new framework are the following: Exploiting the integrability of the model, new fundamental variables have been identified (certain components of the flat connection of the auxiliary linear system continued into the plane of the spectral parameter), in terms of which the “right” and “left” moving sectors have been completely decoupled [47]. The quantization is further performed in the framework of a generalized “two-time” Hamiltonian formalism, i.e. these sectors are quantized independently. The whole procedure has been established in that sector of the theory, where the new fundamental connection exhibits simple poles at fixed singularities. In the present paper we achieve the consistent general formulation of the desired coset-models in this approach. So far the formalism was mainly elaborated in the technically simplified principal model, where the coset G/H had been replaced by the group G itself. For the coset model the phase space spanned by the new variables is too large and must be restricted by proper constraints. Their canonical treatment requires a Dirac procedure, which effectively reduces the degrees of freedom. It leads to a consistent analogous Hamiltonian formulation of the coset model allowing canonical quantization. Exact quantum states are shown to be in correspondence to solutions of a modified (Coset-)Knizhnik-Zamolodchikov system. Moreover, the formalism is kept general as long as possible, without restricting to the simple pole sector. In particular, we completely extend it to the case of connections with poles of arbitrary high order at fixed singularities, which span the isomonodromic sector of the theory. Generalization of the scheme to the full phase space is sketched in Appendix A. The other main result of this paper is the identification of classical and quantum observables. For the above mentioned simple pole sector, these sets are complete. Natural

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candidates for classical observables are the monodromies of the fundamental connection in the plane of the variable spectral parameter. We determine their (quadratic) Poisson structure. After quantization of the connection quantum counterparts of these monodromy matrices are identified as monodromies of certain higher-dimensional KZ systems. Following Drinfeld [22] their algebraic structure may be determined to build some quasi-associative braided bialgebra. The classical limit of this structure coincides with the Poisson algebra of the classical monodromies found above. In this sense, complete consistency of the picture is established. The weakened coassociativity leads to a quantum algebra of observables with operator-valued structure constants. This might have been avoided by directly quantizing the regularized classical algebra of monodromies, as is common in Chern–Simons theory [2, 3], instead of recovering quantum monodromies in the picture of the quantized connection. We discuss this link and its consequences. The treatment of observables is performed in great detail for the simplified principal model mentioned above. This is for the sake of clarity of the presentation, since the arising difficulties in the coset case deserve an extra study in the sequel. However, the main tools and strategies that will finally be required can already and more clearly be developed and used in this context. The modifications required for the coset model are clarified afterwards. The paper is organized as follows. In Chap.2 we start by introducing the known linear system associated to the model and describe the related on-shell conformal symmetry. A short summary and generalization of the results from [47, 49] about the classical treatment of the principal model is given without restricting to the simple pole sector. The link to Hamiltonian Chern–Simons theory is discussed, where the same holomorphic Poisson structure is obtained by symplectic reduction of the complexified phase space in a holomorphic gauge fixing. This link in particular enables us to relate the status of observables in both theories. Observables in terms of monodromy matrices are identified; their Poisson structure is calculated and discussed. The technical part of the calculation is shifted into Appendix B. Chapter 3 treats the quantization of the principal model. We first briefly repeat the quantization of the simple pole sector of this model [48, 49]. Quantum analogues of the monodromy matrices are defined. Their algebraic structure and its classical limit are determined and shown to be consistent with the classical results. The alternative treatment in Chern–Simons theory and the identification of quantum observables in these approaches are discussed. In Chap.4 we finally present the generalization of the formalism to the coset models. A Hamiltonian formulation in terms of modified fundamental variables is provided. The coset constraints are explicitly solved by a Dirac procedure. Furthermore, we quantize the simple pole sector of the coset model, showing that solutions of a modified Knizhnik-Zamolodchikov system identify physical quantum states, i.e. exact solutions of the Wheeler–DeWitt equations. We close with a sketch of how to employ the whole machinery to the simplest case of pure four-dimensional axisymmetric stationary gravity. In particular, the existence of normalizable quantum states is shown. Chapter 5 briefly summarizes the open problems for future work. 2. Principal σ-Model Coupled to Two-Dimensional Dilaton Gravity The model to be studied in this paper is described by the two-dimensional Lagrangian   (2.1) L = eρ R + hµν tr[∂µ gg −1 ∂ν gg −1 ] .

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p Here, hµν is the 2D (“worldsheet”) metric, e = | det h|, R is the Gaussian curvature of hµν , ρ ∈ R is the dilaton field and g takes values in some real coset space G/H, where H is the maximal compact subgroup of G. The currents ∂µ gg −1 therefore live in a fixed faithful representation of the algebra g on some auxiliary d0 -dimensional space V0 . It is well known that this type of model arises from the dimensional reduction of higher dimensional gravities [13, 55], e.g. from 4D gravity in the presence of two commuting Killing vectors [12]. In the latter case which describes axisymmetric stationary gravity, the relevant symmetric space is G/H = SL(2, R)/SO(2). Let us first briefly describe further reduction of the Lagrangian (2.1) by means of gauge fixing and state the resulting equations of motion. The residual freedom of coordinate transformations can be used to achieve conformal gauge of the 2D metric hµν : ¯ z¯ , hµν dxµ dxν = h(z, z)dzd with world-sheet coordinates z, z, ¯ which reduces the Lagrangian to   L = ρ hR + tr[gz g −1 gz¯ g −1 ] .

(2.2)

In this gauge the Gaussian curvature takes the form R = (log h)zz¯ /h. The equation of motion for ρ derived from (2.2) (2.3) ρzz¯ = 0 is solved by ρ(z, z) ¯ = Im ξ(z), where ξ(z) is a (locally) holomorphic function. Then the equations of motion for g coming from (2.2) read

¯ z¯ g −1 = 0 . ¯ z g −1 + (ξ − ξ)g (2.4) (ξ − ξ)g z¯ z We can further specialize the gauge by identifying ξ, ξ¯ with the worldsheet coordinates. Then (2.4) turns into

¯ ξ¯ g −1 = 0 . ¯ ξ g −1 ¯ + (ξ − ξ)g (2.5) (ξ − ξ)g ξ ξ The equations of motion for the conformal factor are derived from the original Lagrangian (2.1): ξ − ξ¯ tr(gξ g −1 )2 and c.c. (2.6) (log h)ξ = 4 Throughout this whole chapter we will for above mentioned reasons of clarity investigate the simplified model, where the symmetric space G/H is replaced by the group G itself. We will refer to this plainer model as the principal model. 2.1. Linear system and on-shell conformal symmetry of the model. The starting point of our treatment is the following well-known linear system associated to Eqs. (2.5) [10, 54]: d9 gξ g −1 = 9, dξ 1−γ

d9 gξ¯ g −1 9, = 1+γ dξ¯

where γ is the spacetime-coordinates dependent “variable spectral parameter”   q ξ + ξ¯ 2 ¯ ± (w − ξ)(w − ξ) , w− γ= 2 ξ − ξ¯

(2.7)

(2.8)

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or alternatively w ∈ C may be interpreted as a hidden “constant spectral parameter”; ¯ is a GC -valued function. The variable spectral parameter γ lives on the twofold 9(w, ξ, ξ) covering of the complex w-plane, the transition between the sheets being performed by γ 7→ γ1 . It satisfies ∂γ γ 1+γ = , ∂ξ ξ − ξ¯ 1 − γ

∂γ γ 1−γ , = ¯ ¯ ∂ξ ξ − ξ 1 + γ

(2.9)

such that in (2.7) it is ∂ γ 1+γ ∂ d = + , dξ ∂ξ ξ − ξ¯ 1 − γ ∂γ

d ∂ γ 1−γ ∂ . = ¯+ ¯ ¯ dξ ∂ ξ ξ − ξ 1 + γ ∂γ

(2.10)

The linear system (2.7) exists due to the following on-shell M¨obius symmetry of equations of motion.1 Theorem 2.1. Let g(z, z), ¯ ρ(z, z) ¯ = Imξ(z) and h(z, z) ¯ be some solution of (2.3), (2.4), (2.6) and 9 be the related solution of the linear system (2.7). Then   wξ(z) 1 9(γ) , σ w [ξ] ≡ , σ w [h] ≡ h , (2.11) σ w [g] ≡ 9−1 γ w − ξ(z) also solve (2.4), (2.6). Proof. We have s

    w − ξ¯ −1 1 1 −1 9 gξ g 9 , σ [gξ g ] = w−ξ γ γ s     w − ξ −1 1 1 w −1 −1 σ [gξ¯ g ] = gξ¯ g 9 . 9 ¯ γ γ w−ξ w

−1

Now fulfillment of (2.4), (2.6) may be checked by straightforward calculation.



The transformations σ w form a one-parametric abelian subgroup of the group SL(2, R) of conformal transformations. We have σ w1 σ w 2 = σ w3 ,

1 1 1 + = . w1 w2 w3

The full M¨obius group may be obtained combining transformations σ w with the (essentially trivial) transformations ξ(z) 7→ aξ(z) + b ,

g(z) 7→ g(z) ,

which obviously leave the equations of motion invariant. As a result the action of an arbitrary SL(2, R) M¨obius transformation σ on a solution of the equations of motion is   wξ(z) 1 +b, g(z, z) ¯ 7→ σ[g] ≡ 9−1 9(γ) , (2.12) ξ(z) 7→ σ[ξ] ≡ a w − ξ(z) γ 1

A similar symmetry exists in the theory of Bianchi surfaces [11].

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leaving h invariant. In addition to the M¨obius symmetry (2.12) the model possesses the symmetry corresponding to an arbitrary holomorphic change of the worldsheet coordinate z (this symmetry disappears if we identify z with ξ). Combining this symmetry with (2.12) reveals the following M¨obius symmetry of Eq. (2.5)   ¯ ¯ 7→ σ[g] w(ξ − b) , w(ξ − b) , (2.13) g(ξ, ξ) aw + ξ − b aw + ξ¯ − b   w(ξ¯ − b) w(ξ − b) ¯ , h(ξ, ξ) 7→ h . (2.14) aw + ξ − b aw + ξ¯ − b Infinitesimally, the symmetry (2.13) is a subalgebra of the Virasoro symmetry of (2.5) [42]. Note 2.1. It is known that the Ernst equation (2.4) for SL(2, R)/SO(2) may be rewritten as a fourth order differential equation in terms of the conformal factor h. The transformation (2.14) shows that this equation is, in contrast to the Ernst equation itself, M¨obius ¯ invariant in the ξ, ξ-plane. 2.2. Two-time Hamiltonian formulation of the principal model. Here we present a generalized version of the “two-time” Hamiltonian formalism of the principal σ-model proposed in [47, 48]. It is the strategy to define a new set of fundamental variables by means of exploiting the corresponding linear system. These variables may be equipped with a Poisson structure such that a two-time Hamiltonian formulation of the model is achieved. 2.2.1. New fundamental variables and the isomonodromic sector. The main objects we are going to consider as fundamental variables in the sequel are certain components of the following one-form: ¯ be a solution of the linear system (2.7). Then the g-valued Definition 2.1. Let 9(γ, ξ, ξ) one-form A is defined as (2.15) A := d99−1 . In particular, we are interested in the components ¯ ¯ A = Aγ dγ + Aξ dξ + Aξ dξ¯ = Aw dw + A˜ ξ dξ + A˜ ξ dξ¯ ,

(2.16)

¯ and (w, ξ, ξ) ¯ respectively are considered to be independent variables. In where (γ, ξ, ξ) the sequel we shall use the shortened notation A ≡ Aγ . Moreover, we will restrict our study to that sector of the theory, where A is a singlevalued meromorphic function of γ, i.e. that also A is single-valued and meromorphic in γ. A solution 9 of (2.7) with this property is called isomonodromic, as its monodromies in the γ-plane then have no w-dependence due to (2.15). Further on, we immediately get the following relations: Lemma 2.1. The relation of the original field g to A is given by 2 2 −1 ¯ ¯ , g g = , A(γ, ξ, ξ) A(γ, ξ, ξ) gξ g −1 = ¯ ξ ξ − ξ¯ ξ − ξ¯ γ=1 γ=−1

(2.17)

as a corollary of (2.7) and (2.10). Moreover, the linear system (2.7) and definition (2.16) imply

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∂γ A, ∂w 2A(1) 2A(−1) ¯ A˜ ξ = A˜ ξ = ¯ − γ) , ¯ + γ) , (ξ − ξ)(1 (ξ − ξ)(1 2A(−1) + γ(1 − γ)A(γ) 2A(1) − γ(1 + γ)A(γ) ¯ Aξ = , Aξ = . ¯ − γ) ¯ + γ) (ξ − ξ)(1 (ξ − ξ)(1

Aw =

(2.18)

 Note 2.2. In the sequel A(γ) will be exploited as the basic fundamental variable. At this point we should stress the difference between the real group G (with algebra g) entering the physical models and the related complexified group GC (with algebra gC ). Namely, it is A(γ ∈ C) ∈ gC , whereas we will additionally impose the “imaginary cut” iA(γ ∈ iR) ∈ g. Since A(γ) is a (locally) holomorphic function, this implies A(γ) ¯ = −A∗ (−γ) ,

(2.19)

where ∗ denotes the anti-linear conjugation on gC defined by the real form g. Together with (2.17) this ensures g ∈ G. Note 2.3. The linear system (2.7) admits the normalization 9(γ = ∞) = I ,

(2.20)

which implies regularity of A at infinity: A∞ := lim γA(γ) = 0 .

(2.21)

γ→∞

Furthermore, (2.7) implies an additional relation between the original field g and the 9-function: (2.22) 9(γ = 0) = gC0 , where C0 is a constant matrix in the isomonodromic sector. The definition of A as pure gauge (2.15) implies integrability conditions on its components, which in particular give rise to the following closed system for A(γ): ∂Aξ ∂A = [Aξ , A] + , ∂ξ ∂γ

¯

∂A ∂Aξ ξ¯ . , A] + = [A ∂γ ∂ ξ¯

(2.23)

The main advantage of the system (2.23) in comparison with the original equations of motion in terms of g (2.5) is that the dependence on ξ and ξ¯ is now completely decoupled. Once the system (2.23) is solved, it is easy to check that Eqs. (2.17) are compatible and the field g restored by means of them satisfies (2.5). The remaining set of equations of the principal model (2.6), which concern the conformal factor h, may be rewritten taking into account (2.17) as the following constraints: C ξ := −(log h)ξ +

1 trA2 (1) = 0 , ξ − ξ¯

1 ¯ C ξ := −(log h)ξ¯ + ¯ trA2 (−1) = 0 . (2.24) ξ−ξ

2.2.2. Poisson structure and Hamiltonians. The described decoupling of ξ and ξ¯ dependence allows to treat the system (2.23), (2.24) in the framework of a manifestly covariant

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two-time Hamiltonian formalism, where the field A(γ), the “times” ξ, ξ¯ and the fields (log h)ξ , (log h)ξ¯ are considered as new basic variables. The spirit of the generalized “several-times” Hamiltonian formalism is described for example in [44, 18]. ¯ Poisson structure: For this purpose we equip A(γ) with the following (equal ξ, ξ) Definition 2.2. Define the Poisson bracket on A(γ) ≡ Aa (γ)ta as: Ac (γ) − Ac (µ) , Aa (γ) , Ab (µ) = −f abc γ−µ



(2.25)

f abc being the structure constants of g.2 The relations

  ξ  1 2 (1) = A (γ) , A(γ) , trA ξ − ξ¯   h i 1 ¯ A(γ) , ¯ trA2 (−1) = Aξ (γ) , A(γ) , ξ−ξ  A(γ) ,

(2.26)

compared with the equations of motion (2.23) give rise to ¯ Definition 2.3. We call the (ξ, ξ)-dynamics that is generated by H ξ :=

1 trA2 (1) , ξ − ξ¯

1 ¯ H ξ := ¯ trA2 (−1) , ξ−ξ

(2.27)

¯ the implicit time dependence of the fields. The remaining (ξ, ξ)-dynamics is referred to as explicit time dependence. In fact, the motivation for this definition arises from [47, 48], where it has been shown that in essential sectors of the theory (simple pole singularities in the connection A), it is possible to identify a complete set of explicitly time-independent variables. They may be ¯ treated as canonical variables then, such that H ξ and H ξ serve as complete Hamiltonians. This will be illustrated and generalized in the next subsections for the isomonodromic sector of the theory, where A(γ) is assumed to be a meromorphic function of γ. The extension of this framework to the whole phase space of arbitrary connections A, that is strongly inspired from the treatment of the simple pole case, is sketched in Appendix A. The variables A(γ) themselves are explicitly time-dependent in general according to (2.23) and (2.26). Note 2.4. The quantities B(w) = Aw (γ) + Aw

     ∂γ 1 1 1 ≡ A(γ) − 2 A γ ∂w γ γ

(2.28)

build a rather simple set of explicitly time-independent variables, carrying half of the degrees of freedom of the full phase space. This may be checked by straightforward calculation. Moreover, (2.25) implies {B a (w), B b (v)} = −f abc

B c (w) − B c (v) . w−v

(2.29)

2 Assuming g to be semisimple, the existence of the symmetric Killing-form enables us to arbitrarily pull up and down the algebra indices.

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Note 2.5. From the mathematical point of view, (2.25) is a rather natural structure [26], even though it is not canonically derived from the Lagrangian (2.1). It may however be obtained from an alternative Chern–Simons Lagrangian formulation of the model, as is sketched in the following section. Comparison to the conventional Poisson structure of (2.1) should be worked out on the space of observables, where due to spacetimediffeomorphism invariance no principal difference between one- and two-time structures appears. ¯ In order to gain a Hamiltonian description for the total (ξ, ξ)-dependence of the fields, we employ a full covariant treatment by additionally introducing conjugate momenta ¯ for the canonical “time” variables ξ and ξ. ¯ Poisson bracket Definition 2.4. Define the (equal ξ, ξ) o n o n ¯ −(log h)ξ¯ = 1 , ξ, −(log h)ξ = ξ,

(2.30)

where in the sense of a covariant theory only the explicit appearance of ξ, ξ¯ (compare Def. 2.3) is covered by treating these previous “times” as additional canonical variables, which obey the bracket (2.30). This identification of the conjugate momenta for the explicitly appearing times with the logarithmic derivatives of the conformal factor is motivated from the Lagrangian (2.2) [56]. It implies that the dynamics in ξ and ξ¯ directions is completely given by the ¯ Hamiltonian constraints C ξ and C ξ defined in (2.24), i.e. for any functional F we have dF = {F, C ξ } , dξ

dF ¯ = {F, C ξ } . dξ¯

(2.31)

The remaining equations of motion (2.24) mean weak vanishing of the Hamiltonians. This phenomena always arises in the framework of covariant Hamiltonian formalism when time is treated as canonical variable in its own right canonically conjugated to the Hamiltonian [35]; it is a standard way to take into account possible reparametrization of the time variable. 2.2.3. First order poles. In this simplest case considered in [47, 49] we assume that A(γ) has only simple poles, i.e. N X ¯ Aj (ξ, ξ) , (2.32) A(γ) = γ − γj j=1

¯ wj ∈ C. Then where according to (2.7) all γj should satisfy (2.9), i.e. γj = γ(wj , ξ, ξ), the equations of motion (2.23) yield 2 X ∂Aj ∂Aj 2 X [Ak , Aj ] [Ak , Aj ] = , , = ¯ ¯ ¯ ∂ξ (1 − γk )(1 − γj ) (1 + γk )(1 + γj ) ξ−ξ ξ−ξ ∂ξ k6=j

k6=j

(2.33) and the Poisson brackets (2.25) and (2.30) reduce to {Aai , Abj } = δij f abc Aj ,

(2.34)

{Aj , (log h)ξ } = {Aj , (log h)ξ¯ } = 0 , {γj , (log h)ξ } = −∂ξ γj , {γj , (log h)ξ¯ } = −∂ξ¯ γj ,

(2.35)

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i.e. in this case, the residues Aj together with the set of (hidden constant) positions of the singularities {wj } give the full set of explicitly time-independent variables. 2.2.4. Higher order poles. We can also generalize the described formulation to the case, where A(γ) has higher order poles in the γ-plane: A(γ) =

rj N X ¯ X Akj (ξ, ξ) . (γ − γj )k

(2.36)

j=1 k=1

The Poisson structure (2.25) in terms of Akj has the following form:  )c for k + l − 1 ≤ rj δij f abc (Ak+l−1 j , {(Aki )a , (Alj )b } = 0 for k + l − 1 > rj

(2.37)

building a set of mutually commuting truncated half affine algebras. However, it turns out that for rj > 1 the variables Akj for k = 1, . . . rj − 1 have non-trivial Poisson brackets with (log h)ξ and (log h)ξ¯ , and, therefore, are not explicitly time-independent. The problem of identification of explicitly time-independent variables can be solved in the following way. Consider Aw (γ) =

∂γ A(γ) , ∂w

which as a function of w is meromorphic on the twofold covering of the w-plane. Parametrize the local expansion of Aw around one of its singularities γj as w

A (γ) =

rj X k=1

A(w)k j + O((w − wj )0 ) (w − wj )k

for

γ ∼ γj .

(2.38)

We can now formulate Theorem 2.2. The coefficients A(w)k of the local expansion of Aw have no explicit time j dependence, i.e. = {A(w)k , Hξ} , ∂ξ A(w)k j j

¯

∂ξ¯ A(w)k = {A(w)k , Hξ} . j j

They satisfy the same Poisson structure as the Akj (2.37): n o  δij f abc (Aj(w)k+l−1 )c for k + l − 1 ≤ rj (w)l b a (A(w)k ) , (A ) . = i j 0 for k + l − 1 > rj

(2.39)

(2.40)

Proof. Let us first prove (2.39). From (2.25) and the definition of H ξ it follows that   2trA2 (1) {Aw (γ), H ξ } = ∂w γA(γ) , ¯ (ξ − ξ)   ∂w γ 2A(1) , A(γ) = [A˜ ξ (γ), Aw (γ)] , = ¯ 1−γ (ξ − ξ) whereas from (2.15) the ξ-dynamics of Aw is determined to be 2A(1) . ∂ξ Aw = [A˜ ξ (γ), Aw (γ)] + ∂w A˜ ξ (γ) = [A˜ ξ (γ), Aw (γ)] + ∂w γ (1 − γ)2

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As the last term is regular in γ = γj , comparison of the two previous lines shows that the ξ-dependence of the coefficients in the w-expansion around these points is completely generated by H ξ , which proves (2.39). To show the Poisson structure (2.40), one has to consider the corresponding coefficients of singularities in (2.25). For i 6= j, the result follows directly from (2.37), as A(w)k j is a function of Alj , l = 1, . . . , rj only, such that locality remains. For i = j, one may first extract from (2.25) the behavior of {Aw (γ), Aw (µ)} around γ ∼ γj : {(Aw )a (γ), (Aw )b (µ)} = −∂w γ∂v µf abc

Ac (γ) − Ac (µ) (Aw )c (γ) ∼ f abc ∂v µ , γ−µ µ−γ

to then further study the asymptotical behavior µ ∼ γ: {(Aw )a (γ), (Aw )b (µ)} ∼ f abc

(Aw )c (γ) , v−w

such that (2.40) for i = j follows in the same way, as does (2.37) from (2.25).



Thus, also in this case we have succeeded in identifying a complete set of canonical explicitly time-independent variables. are related to the Alj by Note 2.6. Comparing (2.36) with (2.38) shows that the A(w)k j means of explicit recurrent relations that may be derived, expanding (2.36) in (w−wj ). Then A(w)k is a function of Alj with k ≤ l ≤ rj . In particular, the residues of highest j order are related by r −1  ∂γj j (w)r r Aj j = Aj j , ∂wj which explains for example, why this difference was not relevant in the case of simple poles in the last subsection. 2.3. The link to Hamiltonian Chern–Simons theory. The treatment of the principal model of dimensionally reduced gravity in the previous section was inspired by the fact that the equations of motion were obtained as compatibility conditions (2.23) of special linear systems. The interpretation of these equations as zero curvature conditions suggests a link with Chern–Simons theory whose equations of motion also state the vanishing of some curvature. The Chern–Simons gauge connection then lives on a space locally parametrized simultaneously by the spectral parameter γ and one of the true space time coordinates playing the role of time. The relevant Chern–Simons action reads Z 2 k tr[AdA − A3 ] , (2.41) S= 4π M 3 where A is a connection on a trivial G principal bundle over the 3-dimensional manifold M . In the case of interest here, the manifold M is the direct product of the Riemann surface Σ, on which the spectral parameter γ lives, and the real axis, which is interpreted as time. For this configuration, Chern–Simons theory is known to have a Hamiltonian formulation. Choosing proper boundary conditions on the connection, the action may be rewritten in the form

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S=−

k 4π

Z tr[A∂t A]dt + M

k 2π

Z tr[A0 (dA − A2 )]dt .

(2.42)

M

The connection has been split A = A + A0 dt into spatial and time components, where A0 now plays the role of a Lagrangian multiplier for the constraint F = dA − A2 = 0 .

(2.43)

Usually, A0 is gauged to zero which leads to static components A. In particular, any singularities of the connection are time-independent in this case and treated by inserting static Wilson lines in the action (2.42) [61, 23]. A nontrivial and somewhat singular gauge for A0 must be chosen, to derive the equations of motion of the described principal model of dimensionally reduced gravity. The further required holomorphic reduction of Chern–Simons theory can still be described for arbitrary gauge fixing of A0 , as the results will be valid in any gauge. 2.3.1. Holomorphic reduction and Poisson bracket of the connection. For the following we first complexify the phase space and thereby also the gauge group. This enlarged gauge freedom may be used for a holomorphic gauge fixing then. Denoting the spatial coordinates which locally parametrize Σ by γ = x+iy, γ¯ = x−iy, k κ dxdy ≡ −2iκ ¯ and splitting the remaining defining the measure as 4π 4π dxdy = 4π dγdγ γ γ¯ dynamical parts of A into A = A dγ + A dγ, ¯ the action (2.42) implies the Poisson structure iπ ¯ ¯ Aγ,b (µ, µ)} ¯ = − δ ab δ (2) (γ − µ) , (2.44) {Aγ,a (γ, γ), κ where here and in the following the δ-function is understood two-dimensional R as a real ¯ (2) (γ) = 1. δ-function: δ (2) (x + iy) ≡ 2i δ(x)δ(y), normalized such that dγdγδ This Poisson structure corresponds to the Atiyah-Bott symplectic form on the space of smooth connections on the Riemann surface Σ [6]: Z k tr δA ∧ δA . = 4π Σ The flatness constraints (2.43) are of the first class with respect to this bracket: {F a (γ, γ), ¯ F b (µ, µ)} ¯ =

iπ abc c f F (γ)δ (2) (γ − µ) , κ

where f abc are the total antisymmetric structure constants of gC . These constraints generate the canonical gauge transformations A 7→ gAg −1 + dgg −1 ,

(2.45)

which leave the symplectic structure invariant. The phase space of the original theory is therefore reduced to the space of flat connections A(γ, γ) ¯ modulo the action of the complex gauge group (2.45). If the singularities of the connection A are restricted to simple poles, this phase space is for instance completely described by the monodromies of the connection. As a first step to explicitly reduce the number of degrees of freedom, we will fix the gauge freedom (2.45) in A, by demanding (2.46) Aγ¯ = 0 , which makes flatness of A(γ, γ) ¯ turn into holomorphy of the surviving component Aγ (γ).

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Note 2.7. The existence of corresponding gauge transformations is a nontrivial problem. In general, when Aγ¯ is gauged away, Aγ dγ becomes a connection on a nontrivial bundle over Σ. On Riemann surfaces of higher genus, this form of gauge generically leads to multivalued holomorphic quantities exhibiting certain twist properties [50]. On the Riemann sphere the gauge transformations preserving single-valuedness of Aγ dγ at least exist on a dense subspace of connections [6, 31]. For the purpose here, strictly speaking we a priori restrict the phase space to the class of functions on the punctured sphere that allow this gauge fixing. This includes e.g. all the connections with the curvature exhibiting δ-function singularities treated in [23] (gauge fixed to holomorphic connections with simple poles) as well as connections with higher order derivatives of δ-functions in the curvature. This gauge fixing of first-class constraints changes the Poisson structure according to Dirac [19], leading to Theorem 2.3. Let the Poisson structure (2.44) for the connection ¯ ¯ a dγ + Aγ,a (γ, γ)t ¯ a dγ¯ A(γ, γ) ¯ ≡ Aγ,a (γ, γ)t

be restricted by the constraints (2.43) and (2.46). Then the Dirac bracket for the surviving holomorphic components Aa (γ) ≡ Aγ,a (γ) is given by {Aa (γ), Ab (µ)}∗ =

1 abc Ac (γ) − Ac (µ) f . 2κ γ−µ

(2.47)

In this context, the holomorphic structure (2.47) has first been proposed by Fock and Rosly [28]. Proof. The bracket between the constraints and the gauge-fixing condition is of the form ¯ (µ)} = {F a (γ), Aγ,b

iπ ab iπ ¯ δ ∂γ¯ δ (2) (γ − µ) + f abc Aγ,c (γ)δ (2) (γ − µ) . κ κ

(2.48)

On the constraint surface (2.46) this matrix can be inverted using ∂γ¯ γ1 = −2πiδ (2) (γ), which follows from the inhomogeneous Cauchy theorem. The Dirac bracket for the remaining holomorphic variables Aγ (γ) then is {Aγ,a (γ), Aγ,b (µ)}∗Z X dxdxdyd ¯ y¯ =− m,n



¯ (y)} {Aγ,a (γ), F m (x)} {F m (x), Aγ,n


−1

¯ {Aγ,n (y), Aγ,b (µ)} 
−1 m ¯ ¯ + {Aγ,a (γ), Aγ,n (y)} {Aγ,n (y), F m (x)} {F (x), Aγ,b (µ)} Z iπ X dxdxdyd ¯ y¯ =− κ m 
δ mb δ (2) (y − µ) δ am ∂x δ (2) (x − γ) + f mac Aγ,c (x)δ (2) (x − γ) 2πi(x − y)  am (2)
δ δ (γ − y) bm (2) mbc γ,c (2) − δ ∂x δ (x − µ) + f A (x)δ (x − µ) 2πi(x − y) 1 abc Aγ,c (γ) − Aγ,c (µ) = f .  2κ γ−µ

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Note 2.8. For convenience in concrete calculations we still give this result in tensor notation, as is explicitly explained in [26], where the relation of (2.47) to the corresponding current algebra is discussed. This structure may be put into the form {A(γ) ⊗, A(µ)} = [r(γ − µ), A(γ) ⊗ I + I ⊗ A(µ)] ,

(2.49)

1  a 2 2 with the classical r-matrix r(γ) = − 2κ γ , where  = t ⊗ ta is represented as d0 ×d0 matrix here. For the simplest but important case g=sl(2), it is  = 21 I ⊗I + Π, with Π being the 4 × 4 permutation operator. The matrix r(γ) satisfies the classical Yang-Baxter equation with spectral parameter

[r 12 (γ − µ), r13 (γ) + r23 (µ)] + [r13 (γ), r23 (µ)] = 0 .

(2.50)

In shortened notation, (2.49) reads ¯

¯

{A(γ)0 , A(µ)0 } = [r(γ − µ), A(γ)0 + A(µ)0 ] ,

(2.51)

¯

with A(γ)0 := A(γ) ⊗ I , A(µ)0 := I ⊗ A(µ) . Note 2.9. In the framework of canonical and geometric quantization of Chern–Simons theory [61, 7, 23, 31], the variables Aγ and Aγ¯ are – according to (2.44) – considered and treated as canonically conjugated coordinate and momentum, respectively. After the holomorphic gauge fixing the surviving variable A(γ) = Aγ (γ) resembles – according to (2.47) – a combination of angular momenta. Note 2.10. The flatness constraints (2.43) have not been totally fixed by the choice of gauge (2.46). Apparently this gauge still admits holomorphic gauge transformations, which on the sphere reduce to constant gauge transformations. This freedom may also be seen from the appearance of ∂γ¯ in the matrix of constraint brackets (2.48), which actually prevents its strict invertibility. This implies the surviving of the (global) firstclass part of the flatness constraint F , which for meromorphic A in the parametrization (2.36) is Z Z X (A1i )a = −2πiAa∞ , (2.52) F a (γ)dγdγ¯ = ∂γ¯ Aa (γ)dγdγ¯ = −2πi i

where A∞ = Aa∞ ta , compare (2.21). Obviously, Aa∞ is a generator of constant gauge transformations in the bracket (2.47). 2.3.2. Embedding the principal model. In this holomorphic structure of Chern–Simons theory the link to the principal model can be established. As a first fact, note that the Dirac bracket (2.47) for κ = − 21 equals the Poisson structure (2.25) that was used for the Hamiltonian formulation of the principal model. The equations of motion from Chern–Simons action (2.41) read ∂t Aγ = ∂γ A0 + [Aγ , A0 ] ,

(2.53)

leading to trivial dynamics in the gauge A0 = 0, whereas for t being replaced by ξ and the special (singular) choice of gauge

Quantization of Coset Space σ-Models Coupled to Two-Dimensional Gravity

A0 (γ) := Aξ (γ) =

425

2Aγ (1) − γ(1 + γ)Aγ (γ) , ¯ − γ) (ξ − ξ)(1

one exactly recovers the equations of motion (2.23). Finally the surviving first-class constraints (2.52) that are due to former flatness on the sphere gain a definite physical meaning in the principal model of dimensionally reduced gravity. Arising there equivalently as regularity conditions in γ ∼ ∞ (2.21), they are directly related to the asymptotical flatness of the corresponding solution g of Einstein’s equations (2.5). As first-class constraints in different pictures [12], they generate respectively the Matzner-Misner or the Ehlers symmetry transformations of the model. Their actual role as a physical gauge transformation related to the local Lorentz transformations becomes manifest in the proper treatment of the coset model below, see Subsect.4. 2.4. The algebra of observables. A consistent treatment of the theory and in particular the ability to extract classical and quantum predictions from the theoretical framework requires the identification of a complete set of observables. In the model as presented so far, observables can be defined in the sense of Dirac as objects that have vanishing Poisson bracket with all the constraints including the Hamiltonian constraints (2.24), which even play the most important role here. In the two-time formalism this condition shows the ¯ This is a general feature of a covariant observables to have no total dependence on ξ and ξ. theory, where time dynamics is nothing but unfolding of a gauge transformation, and observables are the gauge invariant objects. Regarding the connection A(γ) as fundamental variables of the theory, the natural objects to build observables from are the monodromies of the linear system (2.15). They may be equivalently characterized as 9(γ) 7→ 9(γ)Ml ,

for γ running along the closed path l ,

or

I Ml = P exp

(2.54)

 A(γ)dγ

.

l

¯ These objects naturally have no total (ξ, ξ)-dependence; in the isomonodromic sector we treat, the w-dependence is also absent. For simple poles let us denote by Mi ≡ Mli the monodromies corresponding to the closed paths li which respectively encircle the singularities γi and touch in one common basepoint. From the local behavior of 9(γ) around γ = γi ,   9(γ) = Gi I + O(γ − γi ) (γ − γi )Ti Ci , one also extracts the relations Ai = Gi Ti G−1 i ,

Mi = Ci−1 e2πiTi Ci .

(2.55)

The remaining constraint of the theory which should have vanishing Poisson bracket with the observables is the generator of the constant gauge transformations (2.52), under which the monodromies transform by a common constant conjugation. This justifies

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Definition 2.5. In the case, where the connection A(γ) exhibits only simple poles at fixed singularities wj and with fixed eigenvalues of Aj , we call the set of Wilson loops ( ) Y tr Mik k, (i1 , . . . , ik ) (2.56) k

the set of observables. Note 2.11. For these connections A(γ), the corresponding monodromies together with the position of the singularities and the eigenvalues of Aj generically already carry the complete information. (It is necessary to add the set of eigenvalues of Aj – i.e. the matrices Tj or the Casimir operators of the algebra respectively – to the set of monodromies, since from the monodromies only the exponentials of these eigenvalues can be extracted.) In the presence of higher order poles in the connection, additional scattering data – so-called Stokes multipliers – are required to uniquely specify the connection [39]. The generic case, in which the whole information is contained in the above data, is precisely defined by the fact that no eigenvalues of the monodromy matrices coincide [38, 39]. In particular, this excludes the case of multisolitons, where the monodromies equal ±I. The algebraic structure of the observables (2.56) is inherited from the Poisson structure on the corresponding connection A(γ). Before we explicitly describe this structure, let us briefly comment on the relation to Chern–Simons theory, where quite similarly the Poisson bracket (2.44) provides a Poisson structure on gauge invariant objects. 2.4.1. Observables in Chern–Simons theory. In Chern–Simons theory on the punctured sphere, the set of observables is also built from the monodromy matrices. Note that since in the usual gauge A0 = 0 the Hamiltonian constraint is absent, observables are identified as gauge invariant objects, where this is invariance under local (γ-dependent) gauge transformations. Fixing this gauge freedom by holomorphic gauge as described above, the Dirac bracket (2.47) is now a structure on the reduced phase space of holomorphic connections A(z) modulo the action of constant gauge transformations. It has been explained in [2] that the canonical bracket (2.44) does not define a unique structure on monodromy matrices due to arising ambiguities from the singularities of this bracket (see also [59]). However, on gauge invariant objects, built from traces of arbitrary products of monodromy matrices, these ambiguities vanish [28, 1]. Hence the strategy there is to postulate some structure on the monodromy matrices which reduces to the proper one [34] on gauge invariant objects. The holomorphic Dirac bracket (2.47) allows the calculation also for the monodromies themselves, as we shall show in the following. To relate this result to [28, 2], note that in general the original Poisson bracket and reduced Dirac bracket coincide on quantities of first class in Dirac terminology, i.e. here on gauge invariant objects. In this sense the holomorphic reduction finally leads to the same result on the space of observables. 2.4.2. Poisson structure of monodromy matrices. The holomorphic Poisson structure (2.47) defines a Poisson structure on the monodromy matrices Mj . The result is summarized in the following

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427

Theorem 2.4. Let A(γ) be a connection on the punctured plane γ \ {γ1 , . . . , γN }, equipped with the Poisson structure n

o ¯ A(γ)0 , A(µ)0 =

i 1 h ¯ , A(γ)0 + A(µ)0 . γ−µ

(2.57)

Let further 9 be defined as a solution of the linear system ∂γ 9(γ) = A(γ)9(γ) ,

(2.58)

normalized at a fixed basepoint s0 , 9(s0 ) = I ,

(2.59)

and denote by M1 , . . . , MN the monodromy matrices of 9 corresponding to a set of paths with endpoint s0 , which encircle γ1 , . . . , γN , respectively. Ensure holomorphy of 9 at ∞ by the first-class constraint A∞ = lim γA(γ) = 0 .

(2.60)

γ→∞

Then, in the limit s0 → ∞, the Poisson structure of the monodromy matrices is given by o   n ¯ ¯ ¯ (2.61) Mi0 , Mi0 = iπ Mi0  Mi0 − Mi0  Mi0 , n o   ¯ ¯ ¯ ¯ ¯ Mi0 , Mj0 = iπ Mi0  Mj0 + Mj0  Mi0 −  Mi0 Mj0 − Mi0 Mj0  , for i < j ,

(2.62)

where the paths defining the monodromy matrices Mj are ordered with increasing j with respect to the distinguished path [s0 → ∞]. At this point several comments on the result of this theorem are in order, whereas the proof is postponed to Appendix B. Note 2.12. The first-class constraint (2.60) generates constant gauge transformations of the connection A in the Poisson structure (2.57). For the connections of the type (2.36) this reduces to the constraint (2.52). In terms of the monodromy matrices, holomorphy of 9 at ∞ is reflected by Y Mi = I , (2.63) M∞ ≡ which in turn is a first-class constraint and generates the action of constant gauge transformations on the monodromy matrices in the structure (2.61) and (2.62). The ordering of this product is fixed to coincide with the ordering that defines (2.62). The gauge transformation behavior of the fields explicitly reads o h i n ¯ ¯ (2.64) A0∞ , A0j =  , A0j , n o   ¯ ¯ ¯ ¯ ¯ 0 0 0 0 0 M∞ , Mj0 = iπ M∞ Mj0 − Mj0 M∞ − M∞ Mj0 + M∞ Mj0  . This transformation law is further inherited by arbitrary products M = monodromies, where on the constraint surface M∞ = I it takes the form

Q k

Mjk of

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D. Korotkin, H. Samtleben

n

¯

0 , M0 M∞

o

h i ¯ = −2πi  , M 0 ,

resembling (2.64). The generators of gauge transformations build the algebra o h i n ¯ ¯ A0∞ , A0∞ = , A0∞ , or

n

¯

0 0 , M∞ M∞

o

  0 0 0¯ 0¯  M∞ − M∞  M∞ = iπ M∞ ,

(2.65)

(2.66) (2.67)

in terms of A∞ and M∞ respectively. In fact, the algebras (2.66) and (2.67) turn out to be isomorphic: the quadratic bracket (2.67) linearizes if the Casimirs are split out. As mentioned, we will further be interested in gauge invariant objects, which are now identified by their vanishing Poisson bracket with (2.63) and which are therefore invariant under a global common conjugation of all monodromies. Note that this includes invariance under gauge transformations with gauge parameters (conjugation matrices) that have nonvanishing Poisson bracket with the monodromies themselves. In accordance with Definition 2.5, the structure (2.61), (2.62) implies {M∞ , trM } = 0

(2.68)

for an arbitrary product of monodromies M . Note 2.13. The evident asymmetry of (2.62) with respect to the interchange of i and j is due to the fact that the monodromy matrices are defined by the homotopy class of the path, which connects the encircling path with the basepoint in the punctured plane. This gives rise to a cyclic ordering of the monodromies. The distinguished path [s0 → ∞] breaks and thereby fixes this ordering, as is explicitly illustrated in Fig.3 in Appendix B below. It is remnant of the so-called eyelash that enters the definition of the analogous Poisson structure in the combinatorial approach [28, 1, 2], being attached to every vertex and representing some freedom in this definition. However, the choice of another path [s0 → ∞] simply corresponds to a global conjugation by some product of monodromy matrices: a shift of this eyelash by j steps corresponds to the transformation Mk → (M1 . . . Mj )−1 Mk (M1 . . . Mj ) . Therefore the restricted Poisson structure on gauge invariant objects is independent of this path. Note 2.14. A seeming obstacle of the structure (2.61), (2.62) is the violation of Jacobi identities. Actually, this results from heavily exploiting the constraint (2.60) in the calculation of the Poisson brackets. As therefore these brackets are valid only on the first-class constraint surface (2.63), Jacobi identities can not be expected to hold in general. However, the same reasoning shows [58], that the structure (2.61), (2.62) restricts to a Poisson structure fulfilling Jacobi identities on the space of gauge invariant objects. On this space, the structure reduces to the original Goldman bracket [34] and coincides with the restrictions of previously found and studied structures on the monodromy matrices [28]: o n ¯ ¯ ¯ ¯ ¯ (2.69) Mi0 , Mi0 = Mi0 r+ Mi0 + Mi0 r− Mi0 − r− Mi0 Mi0 − Mi0 Mi0 r+ , n o ¯ ¯ ¯ ¯ ¯ Mi0 , Mj0 = Mi0 r+ Mj0 + Mj0 r+ Mi0 − r+ Mi0 Mj0 − Mi0 Mj0 r+ , for i < j ,

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429

where r+ and r− := −Πr+ Π are arbitrary solutions of the classical Yang-Baxter equation [r12 , r23 ] + [r12 , r13 ] + [r13 , r23 ] = 0 ,

(2.70)

and the symmetric part of r+ is required to be iπ. Setting r+ ≡ iπ, (2.69) reduces to (2.61), (2.62) such that our structure is in some sense the skeleton, which may be dressed with additional freedom that vanishes on gauge invariant objects. On the space of monodromy matrices themselves, introduction of r-matrices may be considered as some regularization to restore associativity, whereas the fact that  itself does not satisfy the classical Yang-Baxter equation is equivalent to (2.61), (2.62) not obeying Jacobi identities. Q In the Poisson structure (2.69), the generator of gauge transformations M∞ ≡ i Mi has the following Poisson brackets with any monodromy Mk : n

¯

0 , Mk0 M∞

o

¯

¯

¯

¯

0 0 0 0 − Mk0 M∞ r− − r+ M∞ = Mk0 r+ M∞ Mk0 + M∞ r− Mk0 ,

(2.71)

whichQentails the same Poisson bracket of M∞ with an arbitrary product of monodromies M ≡ k Mjk . On the constraint surface M∞ = I, taking into account r+−r− = 2iπ, this again implies (2.65), such that M∞ again generates the constant gauge transformations. Note 2.15. The subset of observables {tr[(Mi )m ]|i, m} ∪ {wi |i}

(2.72)

commutes with the whole set of observables. For the positions of the singularities this follows just trivially from the Poisson structure (2.25), whereas the eigenvalues of the monodromy matrices are related to the eigenvalues of the corresponding residues Ai (2.55), which in turn provide the Casimir operators of the mutually commuting algebras (2.34). This subset of commuting variables thus parametrizes the symplectic leaves of (2.61), (2.62). Note 2.16. For our treatment of the coset model below, the following additional structure will be of importance. There is an involution η˜ on the set of observables, defined by the cyclic shift Mi 7→ Mi±n , where N = 2n is the total number of monodromies. The crucial observation is now that this involution is an automorphism of the Poisson structure on the algebra of observables: ˜ 2 )} = η({X ˜ {η(X ˜ 1 ), η(X 1 , X2 }) ,

(2.73)

for X1 , X2 being traces of arbitrary products of monodromy matrices. This is a corollary of Note 2.13, as it follows from the invariance of the Poisson structure on gauge invariant objects with respect to a shift of the eyelash that defines the ordering of monodromy matrices. Like every involution, η˜ defines a grading of the algebra into its eigenspaces of eigenvalue ±1. In particular, the even part forms a closed subalgebra.

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3. Quantization of the Principal Model 3.1. Quantization in terms of the connection. The quantization of the model looks especially natural in the isomonodromic sector with only simple poles. This has been performed in [48, 49], as we shall briefly summarize. In this case straightforward quantization of the linear Poisson brackets (2.34) leads to the following commutation relations: [Aai , Abj ] = i~δij f abc Aj , ¯ (log h)ξ¯ ] = −i~ , [ξ, (log h)ξ ] = [ξ, ¯ (log h)ξ ] = [ξ, (log h)ξ¯ ] = 0. [ξ,

(3.1) (3.2)

According to (3.2), representing ξ and ξ¯ by multiplication operators, one can choose (log h)ξ = i~

∂ , ∂ξ

∂ (log h)ξ¯ = i~ ¯ . ∂ξ

(3.3)

From (3.1), the residues Aj can be represented according to Aaj = i~taj ,

(3.4)

which acts on a representation Vj of the algebra gC . ¯ in a sector with given singularities should depend on Thus the quantum state ψ(ξ, ξ) ¯ and live in the tensor-product V (N ) := V1 ⊗ . . . ⊗ VN of N representation spaces. (ξ, ξ) Q Denote the dimension of Vj by dj , such that d := dimV (N ) = dj . 3.1.1. Wheeler–DeWitt equations and Knizhnik-Zamolodchikov system. The whole “dynamics” of the theory is now encoded in the constraints (2.24), which accordingly play the role of the Wheeler–DeWitt equations here: ¯

Cξψ = Cξψ = 0 , which can be written out in explicit form using (2.24), (2.27), (3.3) and (3.4): i~ X ∂ψ jk = ψ, ∂ξ (1 − γj )(1 − γk ) ξ − ξ¯

(3.5)

(3.6)

k6=j

i~ X jk ∂ψ ψ, = ∂ ξ¯ ξ¯ − ξ k6=j (1 + γj )(1 + γk ) where jk := taj ⊗ tak is the symmetric 2-tensor of g, acting nontrivially only on Vj and Vk . The other constraint that restricts the physical states arrives from (2.52); its meaning was sketched in Subsect. 2.3.2. In the quantized sector it is reflected by   X ¯ =0.  taj  ψ(ξ, ξ) (3.7) j

The general solution of the system (3.6) is not known. However, these equations turn out to be intimately related to the Knizhnik-Zamolodchikov (KZ) system [45]: X jk ∂ϕKZ = i~ ϕKZ , (3.8) ∂γj γj − γk k6=j

with an V

(N )

-valued function ϕKZ (γj ):

Quantization of Coset Space σ-Models Coupled to Two-Dimensional Gravity

431

Theorem 3.1. If ϕKZ is a solution of (3.8) obeying the constraint (3.7), and the γj depend ¯ according to (2.8), then on (ξ, ξ) ψ=

 1 i~ N  Y ∂γj 2 jj j=1

∂wj

ϕKZ

(3.9)

solves the constraint (Wheeler–DeWitt) equations (3.6). The Casimir operator jj defined above is assumed to act diagonal on the states; for g=sl(2) for example, this is simply jj = 21 sj (sj − 2), classifying the representation. Theorem 3.1 and the proof were obtained in [48]. The task of solving (3.6) reduces to the solution of (3.8). Note 3.1. The γj dependence of the quantum states, introduced in Theorem 3.1, can be ¯ understood as just a formal dependence, which covers the (ξ, ξ)-dependence of these states. However, one may also split up this dynamics into several commuting flows generated by the corresponding operators from (3.8). The full set of solutions of (3.8) then may be interpreted as a “γj -evolution operator,” describing this dynamics. In some sense [49] this quantum operator resembles the classical τ -function introduced in [38]. Note 3.2. We have described how the solution of the Wheeler–DeWitt equations is related to the solution of the KZ system (3.8) in the sector of the theory, where the connection has only simple poles. It is therefore natural to suppose that the quantization of the higher pole sectors that were classically presented in Subsect. 2.2.4 is achievable in a similar way and will moreover reveal a link to the higher order KZ systems, which were introduced in [57] in the quantization of isomonodromic deformations with exactly the Poisson structure (2.37) on the residues. Note 3.3. For definiteness it is convenient to assume pure imaginary singularities γj ∈ iR (i.e. wj ∈ R). Then classically Aj ∈ g and quantized they carry representations of g itself, not of gC . 3.2. Quantum algebra of monodromy matrices. 3.2.1. Quantum monodromies. Having quantized the connection A(γ) as described in the previous section, it is a priori not clear how to identify quantum operators corresponding to the classical monodromy matrices in this picture. As they are classically highly nonlinear functions of the Aj , arbitrarily complicated normal-ordering ambiguities may arise in the quantum case. The first problem is the definition of the quantum analogue of the classical 9– function. Its d0×d0 matrix entries are now operators on the d-dimensional representation space V (N ) . We choose here a simple convention, replacing the classical linear system ∂γ 9(γ) = A(γ)9(γ)

(3.10)

by formally the same one, where all the arising matrix entries are operators now, i.e. (3.10) remains valid for higher dimensional matrices A and 9. We have thereby fixed the operator ordering on the right-hand side in what seems to be the most natural way. In the same way, we define the quantum monodromy matrices:

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Definition 3.1. The quantum monodromy matrix Mj is defined to be the r.h.s. monodromy matrix of the (higher dimensional) quantum linear system (3.10): 9(γ) 7→ 9(γ)Mj

for γ encircling γj ,

where the quantum 9-function is normalized as    1 γ −A∞ 9(γ) = I + O γ

around γ ∼ ∞ .

(3.11)

(3.12)

Note 3.4. The normalization condition (3.12) generalizes the one we chose in the classical case (2.59) where the basepoint s0 was sent to infinity. This generalization is necessary, because the constraint (2.60) is not fulfilled as an operator identity in the quantum case, which means that the quantum 9-function as an operator is definitely singular at γ = ∞ with the behavior (3.12). Only its action on physical states, which are by definition annihilated by the constraint (2.52) may be put equal to the identity for γ = ∞. For proceeding further we now make use of an interesting observation of [57], relating the KZ systems with N and N + 1 insertions by means of the quantum linear system (3.10). We state this as Theorem 3.2. Let ϕ(γ1 , . . . , γN ) be the evolution operator of the KZ system ∂j ϕ = i~

X k6=j

jk ϕ, γj − γk

and 8(γ0 , . . . , γN ) be the corresponding evolution operator of the KZ system with an additional insertion at N = 0. Then 9(γ0 , . . . , γN ) := (I ⊗ ϕ−1 )8 satisfies the following system of equations: ∂0 9 = i~

N X ta0 ⊗ (ϕtaj ϕ−1 ) j=1

∂j 9 = −i~

γ0 − γj

9,

(3.13)

ta0 ⊗ (ϕtaj ϕ−1 ) 9. γ0 − γj

The proof is obtained by a simple calculation.



Consider the relations (3.13). Together with the remarks of Note 3.1, it follows that this 9 just obeys the proper quantum linear system (3.10) in a Heisenberg picture: the ¯ (ξ, ξ)-dependence of the operators Aj is generated by conjugation with the evolutionoperator ϕ. For the definition of the quantum 9-function it is the Heisenberg picture which provides the most natural framework, as only in this picture implicit and explicit ¯ (ξ, ξ)-dependence of operators are treated more or less on the same footing. Thus one may identify Aj = i~ta0 ⊗ (ϕtaj ϕ−1 ) . The operators ta0 play the role of the classical representation ta acting on the auxiliary space V0 , which is already required for the formulation of the classical linear system. In this sense, the KZ system with N +1 insertions combines the classical linear system with the quantum equations of motion that are described by the KZ system with N

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433

insertions. The additional insertion γ0 then plays the role of γ. We shall use this link to gain information about the algebraic structure of the quantum monodromy matrices. 3.2.2. Quantum group structure. We now start from the representation of the quantum 9-function due to Theorem 3.2:   (3.14) 9(γ, γ1 , . . . , γN ) = I ⊗ ϕ−1 (γ1 , . . . , γN ) 8(γ, γ1 , . . . , γN ) . This shows in particular that the quantum monodromy matrices of the principal model defined in (3.11) equal the corresponding monodromies of the KZ system with N + 1 insertions. To obtain their algebraic structure, we employ a deep result of Drinfeld about the relation between the monodromies of the KZ connection and the braid group representations induced by certain quasi-bialgebras [21, 22]. Before we state these relations, we have to briefly describe the induced braid group representations. The KZ system that is of interest here, is X jk 8, ∂j 8 = i~ γj − γk k6=j

with j = 0, . . . , N , which, as explained, in a formal sense combines the classical and the quantum degrees of freedom, the function 8 living in V (N +1) := V0 ⊗ V (N ) . This system naturally induces a representation of monodromy matrices, which may canonically be lifted to a braid group representation [43]. However, for our purpose, it is sufficient to remain on the level of the monodromy representation, which we denote by ρKZ . We further have to briefly mention two algebraic structures, which are standard examples for braided quasi-bialgebras, where for details and exact definitions we refer to [22, 43]. Let us denote by U~ the so-called Drinfeld-Jimbo quantum enveloping algebra associated with g [20, 37]. This is a braided bialgebra, which includes the existence of a comultiplication 1, a counit  and a universal R-matrix RU ∈ U~ ⊗U~ , obeying several conditions of which the most important here is the (quantum) Yang-Baxter equation RU12 RU13 RU23 = RU23 RU13 RU12 .

(3.15)

The matrix RU can in principle be explicitly given, but is of a highly complicated form. It is Drinfeld’s achievement to relate this structure to a braided quasi-bialgebra A~ , where the nontriviality of the R-matrix is essentially shifted into an additional element φA ∈ A~ ⊗A~ ⊗A~ , the so-called associator, which weakens the coassociativity. The R-matrix of A~ is simply RA = e−π~ , where  := ta ⊗ta is the symmetric 2-tensor of g. This R-matrix satisfies a weaker form of (3.15), the quasi-Yang-Baxter equation 12 312 13 −1 132 23 123 23 −1 231 13 213 12 RA φA RA (φA ) RA φA = φ321 R A φ A RA . A RA (φA )

(3.16)

The algebras U~ and A~ are isomorphic as braided quasi-bialgebras [22]. There is a standard way, in which braided quasi-bialgebras induce representations of the braid group. Each simple braid σi is represented as i,i+1 i,i+1 R φi , ρ(σi ) := φ−1 i Π

(3.17)

where Π is the permutation operator and φi is defined as φi := 1(i+1) (φ) ⊗ I ⊗(N −i−2) with 1(1) := 1, 1(2) := Id and 1(i+1) := (1 ⊗ Id⊗i )1(i) . We will denote the restrictions of these representations of the algebras U~ and A~ on the monodromies, which are built from products of simple braids, by ρU and ρA respectively. Now we have collected all the ingredients to state the result of Drinfeld as:

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Theorem 3.3. The monodromy representation of the KZ system equals the described monodromy representation of the braided quasi-bialgebra A~ , which in turn is equivalent to the monodromy representation of the braided bialgebra Uh . This means, that there is an automorphism u on V (N +1) , such that ρKZ = ρA = uρU u−1 .

(3.18)

For the proof we refer to the original literature [22] or to the textbook of Kassel [43]. We should stress that in this construction the deformation parameter of the quantum group structure coincides with the true Planck constant ~.  3.2.3. Quantum algebra and classical limit. It was our aim to describe the algebraic structure of the quantum monodromy matrices defined in (3.11). By Theorem 3.2 these monodromy matrices have been identified among the monodromies of the KZ system with N +1 insertions as the monodromies of the additional point γ0 encircling the other insertions. Exploiting the consequences of Theorem 3.3 now, the quantum algebra of the monodromy matrices M1 , . . . , MN is given by: Theorem 3.4. The matrices Mj from (3.11) satisfy −1 Mi0 = Mi0 R+ Mi0 R+−1 , R− Mi0 R− ¯

¯ R+ Mi0 R+−1 Mj0

¯

=

¯ Mj0 R+ Mi0 R+−1

,

(3.19) for i < j ,

where these relations are understood in a fixed representationNof the d0 ×d0 matrix entries of the monodromy matrices on the tensor-product V (N ) = j Vj . The R-matrices R± are −1 R− := u0¯ RU−1 u−1 R+ := ΠR− Π, (3.20) 0 , where RU is the universal R-matrix of U~ mentioned above, u0 is some automorphism on V0 ⊗ V (N ) and u0¯ is the corresponding one on V0¯ ⊗ V (N ) . The classical limit of these R-matrices is given by R± = I ⊗I ± (i~)(iπ) + O± (~2 ) .

(3.21)

Note 3.5. The relations (3.19) are to be understood as follows. The notation requires two copies 0 and 0¯ of the classical auxiliary space V0 . While the standard R-matrices RU and RA live on these classical spaces only, R− and R+ also act nontrivially on the quantum representation space V (N ) , due to conjugation with the automorphisms u0 , u0¯ . Proof of Theorem 3.4. Consider the monodromy representation (3.17) corresponding to the coassociative bialgebra U. The monodromy Mj for γ = γ0 encircling γj is thereby represented as ρU (Mj ) = (RU−1 )01 (RU−1 )02 . . . RUj0 RU0j RU0,j−1 . . . RU01 ,

(3.22)

such that it is just a matter of sufficiently often exploiting the Yang-Baxter equation (3.15) to explicitly show that the relations (3.19) hold for ρU (Mj ) with R− := RU−1 , −1 R+ := ΠR− Π. Theorem 3.3 further implies the conjugation of the R-matrices with the automorphism u in order to extend the result to the representation ρKZ , in which the monodromies from (3.11) were recovered. To further prove the asymptotic behavior (3.21), it is not enough to know the classical limit of RU – which is a classical r-matrix simply – since the semiclassical expansion of

Quantization of Coset Space σ-Models Coupled to Two-Dimensional Gravity

435

the automorphisms u0 , u0¯ must be taken into account. For this reason, we additionally have to use the other part of Theorem 3.3, which relates the representations ρKZ and −1 −1 , R+ := ΠR− Π in a ρA . The relations (3.19) for the ρA (Mj ) hold with R− := RA generalized form, modified by certain conjugations with the nontrivial associator φA . The semiclassical expansion of the associator is given by [43]: φA = I ⊗I ⊗I + O(~2 ) ,

(3.23)

which implies that the term of order ~ in the semiclassical expansion of (3.19) is determined by the corresponding one in RA = e−π~ , which yields (3.21). The last point to be ensured is that the normalization of the quantum monodromies (3.12) around γ ∼ ∞ coincides with the normalization chosen in the definition of the KZ monodromies [21] in certain asymptotic regions of the space of (γ, γ1 . . . , γN ), up to the order ~. The proof of this fact goes along the same line as the proof of (3.23).  We have now established the quantum algebra of the quantum monodromy matrices by identifying the corresponding operators inside the picture of the quantized holomorphic connection A(γ). The classical limit of this algebra equals exactly the classical algebra of monodromy matrices (2.61), (2.62). Hence, we have shown the “commutativity” of the (classical and quantum) links between the connection and the monodromies with the corresponding quantization procedures. Let us sketch this in the following diagram: Atiyah-Bott symplectic structure ¯ (µ)} ∼ δ ab δ (2) (γ − µ) {Aγ,a (γ), Aγ,b

holomorphic gauge

 Regularized algebra of monodromies ¯

? Holomorphic connection P PP b abc Ac {Aa q P i , Aj } = δij f i

Classical algebra of monodromies ¯ ¯ {Mi0 , Mj0 } = iπ (Mi0  Mj0 + . . .)

¯

{Mi0 , Mj0 } = (Mi0 r+ Mj0 + . . .) quantization

?

quantization and quasi-associative generalization



b abc Ac [Aa i , Aj ] = i~δij f i

quantum monodromies via KZ system

? ^ Quantum algebra of monodromies











quantization of the

nonassociative algebra

R+ Mi0 R+−1 Mj0 = Mj0 R+ Mi0 R+−1 ¯

¯

Note 3.6. The dotted lines in this diagram depict the link to the usual way quantum monodromies have been treated. This was done by directly quantizing their classical algebra, which is derived from the original symplectic structure of the connection up to certain degrees of gauge freedom: for later restriction on gauge invariant objects, this

436

D. Korotkin, H. Samtleben

algebra may be described with an arbitrary classical r-matrix, as was sketched in Note 2.14. A direct quantization of this structure is provided by a structure of the form (3.19), where the quantum R-matrices live in the classical spaces only and admit the classical expansion R± = I + i~r± + O± (~2 ) [1, 2]. Note 3.7. In contrast to this quantum algebra which underlies (2.69), in (3.19) the Rmatrices – due to the automorphisms u0 , u0¯ – also act nontrivially on the quantum representation space. Their classical matrix entries may be considered as operator-valued, meaning that the quantum algebra can be treated alternatively as nonassociative or as “soft.” This is in some sense the quantum reason for the fact, that the classical algebra (2.61), (2.62) fails to satisfy Jacobi identities. However, note that (3.19) only describes the R-matrix in any fixed representation of the monodromies; for a description of the abstract algebra, compare the quasi-associative generalization in [2, 3], which provides the link between the quantum structure described in the previous note and (3.19). 3.2.4. Quantum observables. Let us discuss now the quantum observables, i.e. operators commuting with all the constraints. In analogy with the classical case it is clear that all monodromies of the quantum linear system (3.11) commute with the Hamiltonian constraints. Therefore, it remains to get rid of the gauge freedom (2.63), i.e. to identify functions of monodromies commuting with quantum generators of the gauge transformations. In the classical case the gauge transformations were generated by matrix entries of the matrix A∞ or, equivalently, of the matrix M∞−I. The straightforward quantization of the classical algebra of gauge transformations generated by A∞ (2.66) is (3.24) [Aa∞ , Ab∞ ] = fcab Ac∞ , i.e. coincides with g. In terms of M∞ , the algebra of the same gauge transformations according to (3.19) reads −1 0 0 0 0 R− M∞ = M∞ R+ M∞ R+−1 . R− M∞ ¯

¯

(3.25)

The set of quantum observables is characterized as the set of operator-valued functions F of components of monodromies Mj which commute with all components of A∞ : [F ({Mj }), Aa∞ ] = 0 . (3.26) Recall that in the classical case observables were just traces of arbitrary products of monodromies Mj . At the moment the quantum analog of this representation is not clear. One should suppose that there is a similar situation to the case we would have arrived at by directly quantizing the algebra of monodromies, mentioned in Note 3.6. In this case, which has been studied in the combinatorial quantization of Chern– Simons theory [2, 3], the R-matrices live in the classical spaces only and the transformation behavior of arbitrary products of monodromies M under gauge transformations generated by M∞ reads −1 0 0 R − M 0 R− M∞ = M∞ R+ M 0 R+−1 . ¯

¯

Introducing the quantum trace tr q M with characteristic relations tr 0q R00 M 0 (R00 )−1 = tr q M 0 , ¯

¯

we see that the operators tr q M commute with the components of M∞ :

(3.27)

Quantization of Coset Space σ-Models Coupled to Two-Dimensional Gravity 0 [tr q M, M∞ ]=0.

437

(3.28)

Therefore, the quantum group generated by M∞ : −1 0 0 0 0 R− M∞ R− M∞ = M∞ R+ M∞ R+−1 ¯

¯

(3.29)

in this approach plays the role of algebra of gauge transformations. It appears a difference of this approach with the approach which we mainly follow in this paper: instead of the Lie group G generated by the algebra (3.24), the role of the gauge group is played by its quantum deformation (3.29). A question therefore remains: what is the proper quantum gauge group of a consistent quantum theory, the group G itself or its quantum deformation Gq ? Note 3.8. With the notation of the quantum trace at hand, the quantum analogue of Note 2.15 can be formulated. From the abstract algebraic point of view – beyond the presented concrete representation of the quantum monodromies – the quantum trace of powers of the Mj build the center of the free algebra defined by (3.19) and may thus be fixed according to the classical values. 4. Coset Model In this final chapter we will explain, how to modify the previously presented scheme in order to treat the coset models, which actually arise from physical theories. The field g is required to take values in a certain representation system of the coset space G/H, where H is the maximal compact subgroup of G. This subgroup may be characterized by an involution η of G as the subgroup, which is invariant under η. The involution can further be lifted to the algebra g, e.g. η(X) = −X t for X ∈ g = sl(N ). The algebra g is thereby split into its eigenspaces with eigenvalues ±1, which are denoted by g = h ⊕ k, the subgroup H underlying h. In terms of the involution, the field g is restricted to satisfy: gη(g) = I ,

(4.1)

which defines the special choice of a representation system of the coset space. 4.1. Classical treatment. Classically speaking, the Poisson structure for the G/H-valued model may be obtained from the previously described Poisson structure for the principal G-valued model by implementing additional constraints. These constraints were discussed in detail in [49] and may be equivalently formulated in terms of the function 9 or of the connection A:   −1 1 g −1 9(γ) = C0 , (4.2) η 9 γ    1 1 A(γ) + 2 gη A g −1 = 0 . (4.3) γ γ The first line is a consequence of (4.1) with C0 = C0 (w) from (2.22) also satisfying C0 η(C0 ) = I now. Studying the monodromies of 9 shows that in the isomonodromic sector, C0 must be gauged to a constant matrix, using the freedom of the right-hand side multiplication of the solution of (2.7). This can be seen from Eq. (4.36) below. Derivation of (4.2) with respect to γ then yields (4.3).

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An unpleasant feature of these constraints is that they explicitly contain the field g, which in this framework is not among the fundamental variables. To avoid this difficulty, it is convenient to slightly modify the Hamiltonian formalism of the principal model. Namely, let us relax the normalization condition 9(γ = ∞) = I, which was ˆ related to 9 by a G-valued gauge imposed in (2.20) before and consider the function 9 transformation V instead: ¯ . ˆ := V(ξ, ξ)9 9 (4.4) −1 ˆ = 0), such that the coset constraint (4.1) ˆ = ∞) = V and gC0 = V 9(γ Then it is 9(γ may be rewritten as: (4.5) g = V −1 η(V) . ˆ now satisfies the linear system The modified function 9     ˆ ˆ 1+γ d9 d9 1−γ ˆ ˆ = − P+ + Q+ 9, P− + Q− 9, (4.6) = − dξ 1−γ 1+γ dξ¯ ¯ with (ξ, ξ)-dependent matrices P± ∈ k and Q± ∈ h which can be reconstructed from V on the coset constraint surface (4.5): Vξ V −1 = P+ + Q+ ,

Vξ¯ V −1 = P− + Q− .

Note 4.1. In the coset model the M¨obius symmetry (2.11) appears in especially simple form [8]: s s ¯ w − ξ w−ξ ˆ P+ , P− 7→ V 7→ 9(γ) , P+ 7→ P− , h 7→ h . w−ξ w − ξ¯ In complete analogy to the principal model, we further introduce Definition 4.1. Define the connection Aˆ by ˆ ˆ 9 ˆ −1 (γ) . A(γ) := ∂γ 9(γ)

(4.7)

The constraint of regularity at infinity then reads ˆ Aˆ ∞ := lim γ A(γ) =0. γ→∞

(4.8)

The relations (2.17) between the original fields and the connection Aˆ take the following form: 1 ˆ 1 ˆ ¯ ¯ A(γ, ξ, ξ) A(γ, ξ, ξ) = −P+ , = −P− . (4.9) ¯ ¯ ξ−ξ ξ−ξ γ=1 γ=−1 Hence, the coset constraints (4.5) are equivalent to
ˆ ˆ A(±1) = −η A(±1) ,

(4.10)

which is implied by (4.3). Let us stress again that the originally equivalent coset constraints (4.1), (4.5) or (4.10) are lifted to (4.3) due to the special choice of C0 = const in the isomonodromic sector. ˆ The constraints (4.2) and (4.3) take simpler forms in terms of the new variables 9 ˆ and A, since the field g is absorbed now:

Quantization of Coset Space σ-Models Coupled to Two-Dimensional Gravity

  −1 ˆ 1 ˆ η 9 9(γ) = C0 , γ    1 1 ˆ A(γ) + 2 η Aˆ =0. γ γ

439

(4.11) (4.12)

The first of these equations is a sign of the invariance of the linear system (4.6) on the coset constraint surface under the extended involution η ∞ , introduced in [12]:    ∞ ˆ ˆ 1 , (4.13) η (9(γ)) := η 9 γ but is difficult to handle due to the unknown matrix C0 . The latter form (4.12) of the constraint admits a complete treatment as will be described below. Note that the constraint of regularity at infinity (4.8) is already contained in (4.12) and is thereby naturally embedded in the coset constraints. The set of constraints (4.12) is complete and consistent in the following sense: ¯ Lemma 4.1. The coset constraints (4.12) are invariant under (ξ, ξ)-translation on the constraint surface. Proof. The total ξ-dependence of Aˆ can be extracted from (2.23) to be ∂Aξ (γ) −1 d ˆ ˆ A(γ) = V[Aξ (γ), A(γ)]V −1 + [Vξ V −1 , A(γ)] V +V dξ ∂γ     −2P+ ˆ ˆ , A(γ) + (P+ + Q+ ), A(γ) = 1−γ 2P+ γ(1 + γ) γ 2 − 2γ − 1 ˆ ˆ A(γ) − + 2 2 ¯ ¯ − γ) ∂γ A(γ) . (1 − γ) (ξ − ξ)(1 − γ) (ξ − ξ)(1       d d 1 Together with dξ f γ1 = − dξ for any function f (γ), which follows f γ from the structure of γξ , a short calculation reveals that on the constraint surface (4.12) it is         1 1 d 1 1 d ˆ ˆ ˆ A(γ) + 2 η A ≈ −γξ A(γ) + 2 η Aˆ ≈0. dξ γ γ dγ γ γ −

 In a Hamiltonian formulation these constraints therefore have weakly vanishing Poisson bracket with the full Hamiltonian, which is required for a consistent treatment. Let us now briefly present the Hamiltonian formulation of the coset model in terms of the new variables. 4.1.1. Poisson structure and Hamiltonian formulation. The definition of the connection Aˆ already implies the relation ˆ A(γ) = VA(γ)V −1 ,

(4.14)

such that from (2.23) one extracts the equations of motion for these new variables:

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∂Aξ −1 ∂ Aˆ ˆ = V[Aξ , A]V −1 + V V + [Vξ V −1 , A], ∂ξ ∂γ ¯ ∂Aξ −1 ∂ Aˆ ξ¯ −1 ˆ . V + [Vξ¯ V −1 , A] , A]V + V = V[A ∂γ ∂ ξ¯

(4.15)

In analogy with the principal model, this motivates ˆ Definition 4.2. Define on A(γ) the following Poisson structure: 



Aˆ a (γ), Aˆ b (µ)

V

= −f abc

Aˆ c (γ) − Aˆ c (µ) , γ−µ

(4.16)

¯ and denote by implicit time-dependence the (ξ, ξ)-dynamics, that is generated by 1 tr Aˆ 2 (1) − tr[Aˆ ∞ (∂ξ VV −1 )] , ξ − ξ¯ 1 ¯ Hˆ ξ := ¯ tr Aˆ 2 (−1) − tr[Aˆ ∞ (∂ξ¯ VV −1 )] , ξ−ξ Hˆ ξ :=

(4.17)

on the constraint surface (4.8). The remaining explicit time-dependence is then defined to be generated in analogy to (2.30). Note 4.2. The Poisson structures (4.16) are certainly different for different V and, therefore, are different from (2.25), that was introduced in the principal model. However, this previous treatment may be embedded in the following way. The structures (4.16) and (2.25) are certainly equivalent if we restrict them to the functionals of Aˆ that are invariant with respect to the choice of V, i.e. invariant with respect to the transformations ˆ , Aˆ 7→ θ−1 Aθ

(4.18)

with arbitrary θ ∈ G. These were the gauge transformations in the principal model, generated by (2.21). Hence, on the set of observables of the principal model, the different Poisson structures coincide. Correspondingly, the action of H ξ and Hˆ ξ from (2.27) and (4.17) respectively differs only by the unfolding of such a gauge transformation. For the coset model it is important to note that the gauge freedom (4.18) is restricted to H-valued matrices θ, since only that part of the constraint (4.8) remains first-class here and generates gauge transformations. This is part of the result of Theorem 4.1 below. 4.1.2. Solution of the constraints. Given a set of constraints (4.12) and a Poisson structure (4.16), the canonical procedure is due to Dirac [19]. The constraints are separated into first and second class constraints, of which the latter are explicitly solved – which changes the Poisson bracket into the Dirac bracket – whereas the former survive in the final theory. In the case at hand, the essential part of the constraints is of the second class, such that the Poisson structure has to be modified and only a small part of the constraints survives as first-class constraints. We state the final result as Theorem 4.1. The Dirac procedure for treating the constraints (4.12) in the Poisson ˆ structure (4.16) yields the following Dirac bracket for the connection A:

Quantization of Coset Space σ-Models Coupled to Two-Dimensional Gravity



∗

Aˆ a (γ), Aˆ b (µ)

V

1 Aˆ c (γ) − Aˆ c (µ) = − f abc 2 γ−µ 1 aη(b)c Aˆ c (γ) 1 η(a)bc Aˆ c (µ) + f , + f 2 µ − γ1 2 γ − µ1

441

(4.19)

where the notation of indices means a choice of basis with tη(a) ≡ η(ta ). The bracket for the logarithmic derivatives of the conformal factor remains unchanged: o∗ n o∗ n ¯ −(log h)ξ¯ = ξ, =1. (4.20) ξ, −(log h)ξ V

V

The structure is compatible with the (now strong) identity    1 1 1 1 ˆ A(γ) + 2 η Aˆ = Aˆ ∞ = η(Aˆ ∞ ) , γ γ γ γ

(4.21)

such that compared with (4.12) it remains the first-class constraint Aˆ ∞ + η(Aˆ ∞ ) = 0 .

(4.22)

ˆ Proof. The main idea of the proof is the separation of the variables A(γ) into weakly commuting halves:    1 1 1 ˆ − Aˆ ∞ , 81 (γ) := A(γ) + 2 η Aˆ γ γ γ    1 1 1 ˆ 82 (γ) := A(γ) − Aˆ ∞ , − 2 η Aˆ γ γ γ with





8a1 (γ), 8b2 (µ)

V

≈ 0

(4.23)

on the constraint surface (4.12), as follows from (4.16) by direct calculation, using the fact that η is an automorphism: f abc = f η(a)η(b)η(c) . The whole constraint surface is spanned by 81 = 0 and Aˆ ∞ = 0, whereas 82 covers the remaining degrees of freedom. Since 81 and 82 contain respectively Aˆ ∞ ∓ η(Aˆ ∞ ), the relations (4.23) show that Aˆ ∞ + η(Aˆ ∞ ) is a first-class constraint of the theory. If we further explicitly solve the second-class constraints 81 = 0, the commutativity (4.23) implies that the Poisson bracket of 82 remains unchanged by the Dirac procedure:  a ∗  82 (γ), 8b2 (µ) V = 8a2 (γ), 8b2 (µ) V . Moreover, the Dirac bracket is by construction compatible with the vanishing of 81 : ∗

{8a1 (γ), . }V = 0 . ˆ These facts may be used to easily calculate the Dirac bracket of the original variables A(γ) without explicitly inverting any matrix of constraint brackets. With the decomposition 1 1 ˆ 1 ˆ 1 ˆ (A∞ + η(Aˆ ∞ )) + (A∞ − η(Aˆ ∞ )) , A(γ) = 81 (γ) + 82 (γ) + 2 2 2γ 2γ the result is obtained. The bracket (4.20) follows from the calculations performed in Lemma 4.1, which imply the vanishing Poisson bracket between (log h)ξ and the constraints. 

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4.1.3. Final formulation and symmetries of the theory. Let us summarize the final status ˆ of the theory and the relation of the new fundamental variables A(γ) to the original fields V and g respectively. We further discuss how the local and global symmetries of the original fields become manifest in this formulation. ˆ The formulation in terms of the new variables A(γ) is completely described in Theorem 4.1, where their modified Poisson structure is given. The solved constraints (4.21) may be considered to be valid strongly. The remaining first-class constraint (4.22) generates the transformation ˆ , Aˆ 7→ χ−1 Aχ

(4.24)

with χ ∈ H. According to (4.9), the field V transforms as V 7→ χV .

(4.25)

The relation (4.5) on the coset constraint surface shows that the field g does not feel this transformation. The gauge transformations generated by (4.22) are the manifestation of a really physical gauge freedom in the decomposition of the metric into some vielbein; they are remnant of the gauge freedom of local Lorentz transformations in general relativity. This freedom may be fixed to choose some special gauge for the vielbein field V. Note 4.3. It is important to notice that the second term in the modified Hamiltonians ¯ ¯ Hˆ ξ , Hˆ ξ from (4.17), that makes them differ from H ξ , H ξ from (2.27) becomes a pure gauge generator after the presented solution of the constraints. This is due to the fact that Aˆ ∞ ∈ h according to (4.21). Since h and k are orthogonal with respect to the CartanKilling form, the action of H ξ and Hˆ ξ just differs by h-conjugation and thus by a gauge transformation of the coset model. The field Aˆ now does not contain the complete information about the original field V, ˆ by means of but only the currents Vξ V −1 , Vξ¯ V −1 , which may be extracted from A(±1) (4.9). At first sight, one might get the impression that in contrast to (2.17), the relations (4.9) do not even contain the full information about these currents. However, if the gauge freedom (4.25) in V is fixed, the currents may be uniquely recovered from (4.9). For g = sl(N ) for example, usually a triangular gauge of V is chosen, such that Vξ V −1 is recovered from its symmetric part 2P+ = (Vξ V −1 )+(Vξ V −1 )t . The field V moreover is determined only up to right multiplication V 7→ Vθ from the currents Vξ V −1 , Vξ¯ V −1 . This is a (global) symmetry of the theory, under which the field g according to (4.5) transforms as g 7→ θ−1 gη(θ) .

(4.26)

For axisymmetric stationary 4D gravity these are the so-called Ehlers transformations. They are obviously a symmetry of the original equations of motion (2.5). ˆ The new variables A(γ) are invariant under these global transformations, which ˆ become only manifest in the transition to the original fields. The related 9-function transforms due to its normalization at ∞ as ˆ 7→ 9θ ˆ , 9 ˆ = 0): as well as the auxiliary matrix C0 , which is related to 9(γ

(4.27)

Quantization of Coset Space σ-Models Coupled to Two-Dimensional Gravity

C0 7→ η(θ)−1 C0 θ .

443

(4.28)

Thereby, we have made explicit the global and local symmetries of the original fields in the new framework. 4.1.4. First order poles. Let us evolve the previous result for the case of simple poles ˆ ˆ of A(γ). We again parametrize A(γ) by its singularities and residues: ˆ A(γ) =

N X j=1

Thus

Aˆ j . γ − γj

(4.29)

Aˆ j = VAj V −1 .

(4.30)

Their equations of motion read 2 X ∂ Aˆ j [Aˆ k , Aˆ j ] = + [Vξ V −1 , Aˆ j ] , ¯ ∂ξ ξ − ξ k6=j (1 − γk )(1 − γj )

(4.31)

2 X [Aˆ k , Aˆ j ] ∂ Aˆ j + [Vξ¯ V −1 , Aˆ j ] , = ξ¯ − ξ k6=j (1 + γk )(1 + γj ) ∂ ξ¯ ¯ and are completely generated by the Hamiltonians Hˆ ξ and Hˆ ξ from (4.17). Theorem 4.1 now implies

Corollary 4.1. Let Aˆ be parametrized as in (4.29). After the Dirac procedure, the following identities hold strongly: 1 , (4.32) γj = γj+n Aˆ j = η(Aˆ j+n ) ,

(4.33)

where N = 2n. They may be explicitly checked to also commute with the full Hamiltonian ¯ constraints C ξ , C ξ . The remaining degrees of freedom are therefore covered by the γj and Aˆ j for 1 ≤ j ≤ n, which are equipped with the Dirac bracket:  a b ∗ 1 Aˆ i , Aˆ j V = δij f abc Aˆ cj . 2

(4.34)

The remaining first-class constraint is


1 ˆ A∞ + η(Aˆ ∞ ) = 2

n X j=1

 Aˆ j + η 

n X

 Aˆ j  = 0 .

(4.35)

j=1

This solution of the constraints in the case of first order poles may alternatively be carried out in terms of the monodromies Mj . As was mentioned above, in the presence of only simple poles, the variables Aj are generically (see Note 2.11) completely defined by the monodromies Mj . Assuming that (4.32) is fulfilled, the coset constraints in the form (4.11) are equivalent to

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Mj+n − C0−1 η(Mj )C0 = 0 . (4.36) There are two important points that this form of the constraints exhibits. First, it shows the necessity to choose the matrix C0 to be constant in the isomonodromic sector. Moreover, it uniquely relates the ordering of the monodromy matrices fixed for calculation of its Poisson brackets in Theorem 2.4 to the ordering defined by (4.32). This results from choosing the corresponding paths pairwise symmetric under γ 7→ γ1 . The goal is now to calculate the Dirac bracket between monodromies Mj with respect to (4.33), or, equivalently, with respect to (4.36). One way is clearly to repeat the calculation of Theorem 2.4 using the Dirac bracket (4.19) instead of the Poisson bracket (2.25). However, we can alternatively determine the Dirac bracket from simple symmetry arguments avoiding direct calculation at least for objects that are invariant under G-valued gauge transformations (i.e. traces of arbitrary products of Mj ). The involution η ∞ introduced by (4.13) acts on Mj according to (4.11) as follows: η ∞ (Mj ) = C0 η(Mj+n )C0−1 .

(4.37)

Therefore, the set of all G-invariant functionals of Mj may be represented as MS ⊕ MAS ,

(4.38)

where the set MS contains functionals which are invariant with respect to η ∞ and MAS contains functionals changing the sign under the action of η ∞ . Since η is an automorphism of the structure (2.61), (2.62), the definition of η ∞ in (4.37) implies, taking into account Note 2.16: {MS , MS } ⊆ MS ,

{MS , MAS } ⊆ MAS ,

{MAS , MAS } ⊆ MS .

(4.39)

The constraints (4.36) are equivalent to vanishing of all functionals from MAS ; therefore the part of G-invariant variables surviving after the Dirac procedure is contained in MS . The former Poisson bracket on MS coincides with the Dirac bracket. Note 4.4. The treatment of coset constraints in terms of the monodromies presented ˆ are. Therefore, above is invariant with respect to change of V since the monodromies of 9 this treatment also works in the former Poisson structure (2.25). 4.2. Quantum coset model. The quantization of the coset model goes along the same line as the quantization of the principal model described above. We again restrict to the first order pole sector of the theory, although generalization to the whole isomonodromic sector should be achievable according to Note 3.2. Having solved the constraints, the remaining degrees of freedom are the singularities γj , the residues Aˆ j for j = 1, . . . , n and the logarithmic derivatives of the conformal factor h. They may be represented as in (3.3) and (3.4) again. The quantum representation space is V (n) := V1 ⊗ . . . ⊗ Vn . The Wheeler–DeWitt equations (3.5) take the form:    X i~ X ∂ψ 1 + γ j γk γj + γk ˜ jk ψ , =   − jk  ∂ξ (1−γj )(1−γk ) (1−γj )(1−γk ) ξ − ξ¯  j,k

j,k

   X ∂ψ i~ X 1 + γ j γk γj + γk ˜ jk ψ ,   + = jk  (1 + γj )(1 + γk ) ∂ ξ¯ ξ¯ − ξ  j,k (1 + γj )(1 + γk ) j,k

(4.40)

Quantization of Coset Space σ-Models Coupled to Two-Dimensional Gravity

445

with ˜ jk := tη(a) ⊗ tak .  j

jk = taj ⊗ tak

Additionally, the physical states have to be annihilated by the first-class constraint (4.22):   X η(a) X ¯ =0.  taj + tj  ψ(ξ, ξ) (4.41) j

j

The result of Theorem 3.1 is modified to establish a link to solutions of what we will refer to as the Coset-KZ system:   X 1 + γ /γ  X γk + 1/γj ∂ϕCKZ k j ˜ jk ϕCKZ .  = i~ jk + (4.42)   ∂γj γj − γk γj γk − 1 k6=j

k

The relation between solutions of the Wheeler–DeWitt equations and solutions of the Coset-KZ system is now explicitly given by Theorem 4.2. If ϕCKZ is a solution of (4.42) obeying the constraint (4.41), and the γj ¯ according to (2.8), then depend on (ξ, ξ) ψ=

n  Y j=1

γj−1

∂γj ∂wj

i~jj ϕCKZ

(4.43)

solves the constraint (Wheeler–DeWitt) Eqs. (4.40). This may directly be calculated in analogy to (3.9).



The procedure of identifying observables may be outlined just as in the case of the principal model, where this was described in great detail. Again the monodromies of the quantum linear system are the natural candidates for building observables and contain a complete set for the simple pole sector. In analogy to Theorem 3.2 they should be identified with the monodromies of a certain higher-dimensional Coset-KZ system with an additional insertion playing the role of the classical γ. The actual observables are generated from combinations of matrix entries of these monodromies that commute with the constraint (4.41). From general reasoning according to the classical procedure, relevant objects turn out to be the combinations of G-invariant objects, that are also invariant under the involution η∞ . 4.3. Application to dimensionally reduced Einstein gravity. Let us finally sketch how the previous formalism and results work for the case of axisymmetric stationary 4D gravity. In this case, the Lagrangian of general relativity is known to reduce to (2.1) with the field g taking values in SL(2, R) as a symmetric 2 × 2 matrix; its symmetry corresponds to the coset constraint (4.1). Most of the physically reasonable solutions of the classical theory – among them in particular the Kerr solution – lie in the isomonodromic sector and are described by first order poles at purely imaginary singularities in the connection. The quantization of this sector may be performed within the framework of this paper. According to (3.4) and Note 3.3 the residues Aˆ j are represented as

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D. Korotkin, H. Samtleben

Aˆ j ≡ i~

1

2 hj

fj

ej − 21 hj

 ,

(4.44)

where hj , ej and fj are the Chevalley generators of sl(2, R). Due to its non-compactness, sl(2, R) admits no finite dimensional unitary representations, but several series of infinite dimensional representations. The study of the classical limit singles out the principal series, as was discussed in [49]. The representation space consists of complex functions f (ζ) on the real line with the ordinary L2 (R) scalar product Z f1 (ζ)f2 (ζ)dζ , (4.45) hf1 , f2 i := R

and the anti-hermitian operators act as hj ≡ 2ζj ∂j + sj ,

ej ≡ ζj2 ∂j + sj ζj ,

fj ≡ −∂j .

(4.46)

The spin sj takes values sj = 1+iqj with a continuous parameter qj ∈ R. The surviving first-class constraint (4.41) now takes a simple form: Lemma 4.2. A solution f (ζ1 , . . . , ζn ) of the constraint (4.41) is of the form Y 1 (ζj2 + 1)− 2 sj F (ζ˜1 , . . . , ζ˜n ) , f (ζ1 , . . . , ζn ) =

(4.47)

j

with ζ˜j :=

ζj +i ζj −i

and

 X ∂ F = 0 .  ∂ ζ˜j 

(4.48)

j

This follows by direct calculation.



The prefactor in (4.47) is exactly sufficient for convergence of the integral, such that for finiteness of the norm, it is sufficient to demand boundedness of F which is a function on the product of (n − 1) circles S 1 . In contrast to the analogous sl(2, R) representation of the principal model, where solutions of finite norm are absent due to several redundant integration variables, a convergency factor here comes out for free. This interestingly resembles the fact that the general reason for dividing out the maximal compact subgroup in the physical coset models corresponds to avoiding unboundedness of the energy in the theory. It remains to solve the Coset-KZ system in this representation. Although the general solution for sl(2, R) is not known, one might be able to obtain explicit results for a small number of insertions. The Kerr solution for instance, which is of major interest, requires only two classical insertions γ1 , γ2 ∈ iR. In this case, we may exploit Theorem 4.2 and Lemma 4.2 to explicitly reduce the WDW equation to a second order differential equation in two variables. Let V1 and V2 be two representations from the principal series ¯ ∈ V1 ⊗ V2 of sl(2, R) fixed by s1 and s2 and parametrize the quantum state ψ(ξ, ξ) according to: 11  12  γ1 γ2 ¯ ζ1 , ζ2 ) = (ζ12 + 1)− 21 s1 (ζ22 + 1)− 21 s2 F (γ, ζ) , (4.49) ψ(ξ, ξ, γ12 − 1 γ22 − 1 with

Quantization of Coset Space σ-Models Coupled to Two-Dimensional Gravity

i ~s1 (s1 − 2) , 2 γ 1 + 1 γ2 − 1 ∈ S1 , γ≡ γ1 − 1 γ2 + 1 11 ≡

447

i ~s2 (s2 − 2) , 2 ζ 1 + i ζ2 − i ∈ S1 . ζ≡ ζ1 − i ζ2 + i

12 ≡

After some calculation the WDW equation then becomes ∂γ F (γ, ζ) = i~Ds1 ,s2 (γ) F (γ, ζ) , with Ds1 ,s2 (γ) =

(4.50)

 
ζ 2 +1 1 2ζ(ζ −1)2 ∂ζ2 + 2(ζ −1)2 + (s1 +s2 )(ζ 2 −1) ∂ζ + s1 s2 γ −1 2ζ  
1 ζ 2 +1 − 2ζ(ζ + 1)2 ∂ζ2 + 2(ζ + 1)2 + (s1 +s2 )(ζ 2 −1) ∂ζ + s1 s2 γ+1 2ζ  4 + (ζ 2 ∂ζ2 + ζ∂ζ ) . (4.51) γ 

This form e.g. suggests expansion into a Laurent series in ζ on S 1 leading to recurrent differential equations in γ for the coefficients. Further study of this equation should be a subject of future work. Note 4.5. Equation (4.50) reduces to a Painlev´e equation when the principal series representation of sl(2, R) is formally replaced by the fundamental representation of g = su(2). In the study of four-point correlation-functions in Liouville theory a similar generalization of the hypergeometric differential equation appeared [62]. 5. Outlook We have completed the classical two-time Hamiltonian formulation of the coset model for the isomonodromic sector and sketched a continuous extension in Appendix A. For the quantum theory it remains the problem of consistent quantization of the total phase space including a proper understanding of the structures (A.8). The most important physical problem in the investigated model is the description of states corresponding to quantum black holes. One may certainly hope to extract first insights from a closer study of the exact isomonodromic quantum states of the coset model identified in the last chapter, in particular from the study of Eq. (4.50). An open problem is the link of the employed two-time Hamiltonian formalism with the conventional one. To rigorously relate the different Poisson structures, the canonical approach should be compared to our model after a Wick rotation into the Lorentzian case. This corresponds to a dimensional reduction of spatial dimensions only, such that the model would describe colliding plane or cylindrical waves rather than stationary black holes. It is further reasonable to suspect that proper comparison of the different Poisson structures can only be made on the set of observables, see also Note 2.5. Recent progress in the canonical approach has been stated in [51], where in particular the canonical algebraic structures of the observables have been revealed. However, so far the canonical and the isomonodromic approaches appear to favor different characteristic observables, which still remain to be related. As another possibility to compare our treatment with canonical approaches, the relation to further restricted and already studied models should be investigated. Of major

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D. Korotkin, H. Samtleben

interest in this context would be for instance the relation to the Einstein-Rosen solutions, investigated and quantized in [52, 5], where imposing of additional hypersurface orthogonality of the Killing vector fields reduces the phase space to “one polarization,” yet maintaining an infinite number of degrees of freedom. An additional interesting field of future research descends from the link to broadly studied two-dimensional dilaton gravity (see e.g. [14, 32, 9, 27]), further allowing to extract information about the black hole thermodynamics. Further relevance of the investigated model appeared in certain sectors of string theory [30, 53]. Acknowledgement. It is a pleasure to thank H. Nicolai, V. Schomerus and J. Teschner for enlightening discussions. D. K. acknowledges support of Deutsche Forschungsgemeinschaft under contract No. Ni 290/5-1. H. S. thanks Studienstiftung des Deutschen Volkes for financial support.

A. Extension Beyond the Isomonodromic Sector The treatment of the isomonodromic sector presented in this paper allows a rather natural extension to the full phase space. This general scheme recalls a continuous version of the simple pole sector treated in Subsect.2, which in turn may be understood as a discrete embedding into the former. We will again first describe the scheme for the principal model and then discuss the modifications required for the coset model, see also [56]. ¯ A.1. Principal model. We start from a simply-connected domain  in the ξ, ξ-plane, ¯ ¯ is symmetric with respect to conjugation ξ 7→ ξ, where the classical solution g(ξ, ξ) assumed to be non-singular. This regularity is reflected by corresponding properties of the related 9-function in the w-plane. It is holomorphicpand invertible in a (ring-like) ¯ bounded by domain D of the Riemann surface L of the function (w − ξ)(w − ξ) contours l and lσ , where σ is the involution γ 7→ 1/γ interchanging the w-sheets of L. To simplify the following formulas we further assume the spectral parameter current A(γ) to be holomorphic on the whole second sheet of L, such that it may be represented inside of l (we denote this simply-connected domain by D0 ) by a Cauchy integral over l: I ¯ A(w, ξ, ξ)dw , (A.1) A(µ) = γ(w) − µ l which is the continuous analog of the simple pole ansatz (2.32) in the isomonodromic sector; A(w), w ∈ l is a density corresponding to the residues Aj from (2.32). From (A.1), A(w) is not uniquely defined by the values of A(γ), γ ∈ D0 , in particular, it may not coincide with the boundary values of A(γ) on l. To fix A(w), we postulate the following deformation equations which are a continuous version of the discrete deformation Eqs. (2.33): I 2 [A(v), A(w)] ∂A(w) = dv , (A.2) ¯ ∂ξ ξ − ξ l (1 − γ(v))(1 − γ(w)) I 2 [A(v), A(w)] ∂A(w) dv , w∈l. = ¯ ¯ (1 + γ(v))(1 + γ(w)) ξ−ξ l ∂ξ It is easy to check that (A.2) together with (A.1) imply the deformation Eqs. (2.23) for A(γ). The Poisson structure on A(w) is also a direct continuous analog of (2.34):

Quantization of Coset Space σ-Models Coupled to Two-Dimensional Gravity

{Aa (w), Ab (v)} = −f abc Ac (w)δ(w − v) ,

449

w, v ∈ l,

(A.3)

where δ(w) is a one-dimensional δ-function living on the contour l (and should, strictly ds δ(s) with an arbitrary affine parameter s along l). This structure speaking, be defined as dw in turn induces the proper holomorphic bracket (2.25) for A(γ): I Ac (w0 )dw0 a b abc {A (γ(w)), A (γ(v))} = −f 0 0 l (γ(w ) − γ(w))(γ(w ) − γ(v)) c c A γ((w)) − A (γ(v)) = −f abc . γ(w) − γ(v) The nice feature of A(w) in contrast to A(γ) is that A(w) (as its discrete analog ¯ independent, i.e. the whole dependence of A(w) on ξ and ξ¯ is Aj ) is explicitly (ξ, ξ) generated by the Hamiltonians (2.27) (note that the points γ = ±1 lie inside of D0 ): 1 H = tr ξ − ξ¯

I

ξ

l

A(w)dw 1 − γ(w)

2

1 tr H = ¯ ξ−ξ ξ¯

,

I l

A(w)dw 1 + γ(w)

2 .

(A.4)

We may now also identify a continuous family of observables, generalizing the construction of Sect.2. Define A(γ) inside and outside of D0 by the Cauchy formula (A.1) and construct the related functions 9in (γ ∈ D0 ) and 9out (γ 6∈ D0 ) according to 9γ 9−1 = A(γ). Then the continuous monodromy matrix M (w) ≡ 9out (w)9−1 in (w) ,

w∈l

(A.5)

¯ is (ξ, ξ)-independent, since both 9in and 9out satisfy the linear system (2.7). Calculations similar to those in Appendix B yield the following Poisson brackets for M (w):  ¯ ¯ ¯ (A.6) {M 0 (v), M 0 (w)} = iπ − M 0 (v)  M 0 (w) + M 0 (w)  M 0 (v)  ¯ ¯ +  M 0 (v)M 0 (w) − M 0 (v)M 0 (w)  , for

v≤w,

v, w ∈ l ,

where the points of contour l are ordered with respect to a fixed point w0 , playing the role of the eyelash in the discrete case. The brackets (A.6), are again valid up to the first-class constraint generated by I (A.7) A∞ = A(w)dw , l

and therefore satisfy Jacobi identities only being restricted to the gauge-invariant objects. Again there appear two fundamental ways of quantization. In terms of A, (A.3) would be replaced by a possibly centrally extended affine algebra. Alternatively, the Poisson algebra of observables (A.6) may be quantized directly after regularization analogously to (2.69): ¯

¯

¯

{M 0 (v), M 0 (w)} = −M 0 (v) r+ M 0 (w) + M 0 (w) r− M 0 (v) ¯

¯

+ r− M 0 (v)M 0 (w) − M 0 (v)M 0 (w) r+ leading to:

v ≤ w,

v, w ∈ l ,

450

D. Korotkin, H. Samtleben −1 R− M 0 (w)R− M 0 (v) = M 0 (v)R+ M 0 (w)R+−1 , ¯

¯

v≤w.

(A.8)

Embedding of the isomonodromic sector into the presented extension looks especially simple if all the singularities γ1 , . . . , γN are assumed to belong to the contour l. The density A(w) is then parametrized as n X

A(w) = −

Aj δ(w − wj ) ,

(A.9)

j=N

where the residues Aj are the same as in (4.29). The Poisson structure (A.3) is the directly inherited from (2.34) and (A.9): {Aa (w), Ab (v)} =

N X

f abc Aj δ(w − wj )δ(v − wj )

j=1

= −f abc Ac (v)δ(v − w) . The monodromy M (w) here is a step function on l with jumps at w = wj . Fixing the eyelash between γN and γ1 it is M (w) = M1 . . . Mj ,

for

w ∈]γj , γj+1 [ .

Note A.1. Isomonodromic solutions with higher order poles are embedded into the general scheme by inserting higher order derivatives of δ-functions into (A.9). The definition ∂γ A, in (A.1) already shows that the proper object in this case is the connection Aw = ∂w accordance with the results from Subsect.2. Note A.2. The representation (A.1) gains a well known meaning when the model is truncated to a real scalar field g, where A(w) becomes independent of ξ, ξ¯ and the equation of motion (2.5) reduces to the Euler-Darboux equation ∂ξ φ − ∂ξ¯ φ ¯ =0, 2(ξ − ξ)

∂ξ ∂ξ¯ φ −

(A.10)

for φ = log g. Solutions of this equation may be represented as [15] I

f (w)dw

p

φ= l

¯ (w − ξ)(w − ξ)

,

(A.11)

with 2πif (w) ≡ φ(ξ = ξ¯ = w) defined on the axis ξ = ξ¯ and continued analytically. After differentiating in ξ and integrating by parts in w, this representation takes the form ∂ξ φ =

2 ξ − ξ¯

I l

f 0 (w)dw p , ¯ (w − ξ)(w − ξ)

and thus equals (2.17) with A(±1) defined by (A.1) after identification of f 0 (w) and A(w).

Quantization of Coset Space σ-Models Coupled to Two-Dimensional Gravity

451

A.2. Coset model. In analogy to the discrete case, the coset model is most conveniently described in terms of modified variables Aˆ = η(V)Aη(V −1 ) . ˆ Due to the symmetry (4.12) between the values of A(γ) on different sheets of L, we ˆ can no longer assume A(γ) to be holomorphic in D0 , but have to replace the l by l ∪ lσ ˆ enclosing D in the formulas of the last section. The coset constraints in terms of A(w) take the form
ˆ ˆ σ) , A(w) = η A(w w∈l, (A.12) and allow rather simple solution via a Dirac procedure, such that the phase space is ˆ reduced to the values of A(w) on l only, equipped with the Dirac bracket 1 {Aˆ a (w), Aˆ b (v)}∗V = − f abc Aˆ c (w)δ(w − v) , 2

v, w ∈ l .

(A.13)

Via the Cauchy representation (A.1) on the contour l ∪ lσ , this bracket further gives ˆ the previously derived Dirac bracket (4.19) on A(γ). It remains the h-valued first class constraint I   ˆ ˆ A(w) + η(A(w)) dw = 0 , l

generalizing (4.22). The Hamiltonians finally also take the form (A.4) with l being replaced by l ∪ lσ . In terms of the observables M (w), restriction to the coset leads to   M (wσ ) = C0−1 η M (w) C0 , w∈l, with some constant matrix C0 playing the same role as in (4.36). B. Poisson Structure of Monodromy Matrices This appendix is devoted to the proof of Theorem 2.4, which was obtained in collaboration with H. Nicolai.3 For simplicity of the presentation, we give the calculation for the case, where the Casimir element  differs from the permutation operator Π by some scalar multiple of the identity only, which is the case for g = sl(N, R) for example. The procedure may easily be extended (concerning the notation mainly) to the general case. Here, the Poisson-structure of the connection is given by {A(γ) ⊗, A(µ)} =

1 [Π, A(γ) ⊗ I + I ⊗ A(µ)] , γ−µ

and the statement to be proven reads: (B.1) {Mi ⊗, Mi } = iπ [ Π, Mi Mi ⊗ I ] ,   {Mi ⊗, Mj } = iπΠ Mj Mi ⊗ I + I ⊗ Mi Mj − Mi ⊗Mj − Mj ⊗Mi , (B.2) for i < j . We first calculate the Poisson structure of matrix entries of the function 9 at different points s1 and s2 . These points are defined on the Riemann surface given by 9 by paths, 3

After completion we learned about related results in [4, 36].

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connecting them to a common base-point s0 , at which 9 is taken to be normalized according to (2.59). The limit s0 → ∞ will be treated later on. For the calculation, we make use of the standard formula   Z s1 Z s2 dµ1 dµ2 × {9(s1 ) ⊗, 9(s2 )} = 9(s1 ) ⊗ 9(s2 ) s0 s0 n o i h , 9−1 (µ1 ) ⊗ 9−1 (µ2 ) A(µ1 ) ⊗, A(µ2 ) 9(µ1 ) ⊗ 9(µ2 ) where the integrand may be rewritten as    Π ∂µ1 + ∂µ2 9−1 (µ2 )9(µ1 ) ⊗ 9−1 (µ1 )9(µ2 ) . µ2 − µ1 This expression is completely regular, even for µ1 = µ2 . However, if the appearance of the derivation operators is exploited by partial integration, the integrals will split up into parts that exhibit singularities in coinciding points µ1 = µ2 . Thus, we restrict to distinguished endpoints s1 and s2 , choosing the defining paths [s0 → s1 ] and [s0 → s2 ] nonintersecting in the punctured plane from the very beginning. Singularities remain in the common endpoints of the paths at s0 . As a regularization, one of these coinciding endpoints is shifted by a small (complex) amount  that is put to zero afterwards. Then, partial integration can be carried out properly, leaving only boundary terms, that lead to surviving simple line integrals, whereas the remaining double integrals cancel exactly. The arising singularities in  = 0 regularize each other such that the sum is independent of the way,  tends to zero. In a comprehensive form, the result may be stated as Theorem B.1. Let s1 and s2 be different points on the punctured plane, defined as points on the covering by nonintersecting paths [s0 → s1 ] and [s0 → s2 ] with common basepoint s0 at which 9 is normalized. Then, the Poisson bracket between matrix entries of 9(s1 ) and 9(s2 ) is given by   {9(s1 ) ⊗, 9(s2 )} = 9(s1 ) ⊗ 9(s2 ) × (B.3)  Z s2   Π 9−1 (µ)9(s1 ) ⊗ 9−1 (s1 )9(µ) dµ µ − s1 s0 Z s1   Π dµ 9−1 (s2 )9(µ) ⊗ 9−1 (µ)9(s2 ) − µ − s2 s Z s02 i h 1 dµ Π , 9(µ) ⊗ 9−1 (µ) + µ − s0 s0 Z s0 − Z s1     Π −1 + dµ 9(µ) ⊗ 9 (µ) . + lim →0 µ − s0 s0 + s2 This expression is regular and independent of the limit procedure.



Note B.1. The result of the regularization is the complete fixing of the relative directions of the paths [s0 → s1 ] and [s0 → s2 ] approaching the basepoint s0 , that is determined by the form in which  arises in the last term in (B.3). In other words, the path [s1 → s0 → s2 ] must pass through the basepoint s0 straightforwardly, as is indicated in Fig.1. The result of Theorem B.1 may be further simplified in the limit s0 → ∞, where the third term of (B.3) vanishes:

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s2

s0

s1

Fig. 1. Choice of paths

Lemma B.1. For a fixed point s on the punctured plane and 9(γ) holomorphic at γ = ∞, it is Z

s

lim

s0 →∞

s0

i h 1 −1 dµ = 0. Π, 9(µ) ⊗ 9 (µ) µ − s0

(B.4)

The proof is obtained by estimating the integrand as a holomorphic function of γ and  s0 . To proceed in calculating the Poisson bracket between monodromy matrices, we choose points s1 , s2 , s3 and s4 , pairwise coinciding on the punctured plane as s1 ∼ s2 and s3 ∼ s4 , but distinguished on the covering and defining the monodromy matrices Mi and Mj : 9(s2 ) = 9(s1 )Mi ,

9(s4 ) = 9(s3 )Mj .

(B.5)

Then, (B.3) leads to: Z Π dµ {Mi ⊗, Mj } = (Mi ⊗ Mj ) µ − s0 s →s →s Z 4 0 2 Π dµ + µ − s0 s →s →s Z 3 0 1 Π dµ − (I ⊗ Mj ) µ − s0 s →s →s Z 4 0 1 Π dµ − (Mi ⊗ I) µ − s0 s3→s0→s2





9(µ) ⊗ 9−1 (µ)



−1

(B.6)



(Mi ⊗ Mj )    9(µ) ⊗ 9−1 (µ) (Mi ⊗ I)   −1 9(µ) ⊗ 9 (µ) (I ⊗ Mj ) , 9(µ) ⊗ 9

(µ)

which is understood in the limit  → 0 and s0 → ∞ and for paths [sj → s0 → si ] , i = 1, 2; j = 3, 4, chosen fixed and in accordance with the conditions of Theorem B.1 and Note B.1. Proof of (B.1). Consider first the case i = j. Then a proper choice of paths is illustrated in Fig.2. The expression (B.6) allows to put s1 = s3 and s2 = s4 and to split the integration paths into paths encircling s0 and γi , respectively: {Mi ⊗, Mi } = (Mi ⊗Mi )X − X(Mi ⊗Mi ) − (Mi ⊗I)X(I ⊗Mi ) + (I ⊗Mi )X(Mi ⊗I) + (I ⊗Mi )Y (Mi ⊗I) − (Mi ⊗I)Y (I ⊗Mi ) , with

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s 1 ∼ s2 s 3 ∼ s4 γi

s0 ∞ Fig. 2. Choice of paths for {Mi ⊗ , Mi }

I

  Π 9(µ) ⊗ 9−1 (µ) , µ − s0 s0 s2   Π 9(µ) ⊗ 9−1 (µ) . dµ Y = µ − s0 s1

X=

1 2 Z

dµ

The path of the integral Y neither passes through s0 nor intersects the path [s0 → ∞]; such that this integral vanishes in the limit s0 → ∞. This choice of path uniquely determines the orientation of the remaining paths in X, which encircle s0 . The corresponding integrals can be easily evaluated due to Cauchy’s theorem and single-valuedness of the integrands. This proves formula (B.1).  Proof of (B.2). This case is treated in complete analogy. A suitable form of the paths is shown in Fig.3, which in particular illustrates the asymmetric position of the paths defining respectively Mi and Mj , with respect to the marked path [s0 → ∞].

s 1 ∼ s2

s 3 ∼ s4

γi

γj

s0 ∞ Fig. 3. Paths for {Mi ⊗ , Mj }

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Similar reasoning as above yields {Mi ⊗, Mj } = −(Mi ⊗Mj )X − X(Mi ⊗Mj ) + (Mi ⊗I)X(I ⊗Mj ) + (I ⊗Mj )X(Mi ⊗I) ,

(B.7)

where again several integrals have already vanished in the limit s0 → ∞. Evaluating the remaining terms proves formula (B.2). 

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53. Maharana, J.: Symmetries of the dimensionally reduced string effective action. Mod. Phys. Lett. A11, 9–17 (1996) 54. Maison, D.: Are the stationary, axially symmetric Einstein equations completely integrable? Phys. Rev. Lett. 41, 521–522 (1978) 55. Nicolai, H.: Two-dimensional gravities and supergravities as integrable systems. In: H. Mitter and H. Gausterer (eds.), Recent Aspects of Quantum Fields. Berlin: Springer-Verlag, 1991 56. Nicolai, H., Korotkin, D., and Samtleben, H.: Integrable classical and quantum gravity. To appear in: G. Mack, G. t’Hooft, A. Jaffe, H. Mitter and R. Stora (eds.), Quantum Fields and Quantum Space Time, Proceedings NATO-ASI, Carg`ese 1996. New York: Plenum Press, 1997 57. Reshetikhin, N.: The Knizhnik-Zamolodchikov system as a deformation of the isomonodromy problem. Lett. Math. Phys. 26, 167–177 (1992) 58. Schemmel, M.: Diploma thesis, Hamburg (1997) 59. Semenov-Tian-Shansky, M.A.: Monodromy map and classical r-matrices. J. Math. Sci. 77, 3236–3242 (1995); translation from Zap. Nauchn. Semi. POMI 200, 156–166 (1992) 60. Smirnov, F. A.: Dynamical symmetries of massive integrable models, 1. Form-factor bootstrap equations as a special case of deformed Knizhnik-Zamolodchikov equations. Int. J. Mod. Phys. A7, suppl. 1B, 813–838 (1991) 61. Witten, E.: Quantum field theory and the Jones polynomial. Commun. Math. Phys. 121, 351–399 (1989) 62. Zamolodchikov, A.B., and Fateev, V.A.: Operator algebras and correlation functions in the twodimensional Wess-Zumino SU (2) × SU (2) chiral model. Yad. Fiz. 43, 1031–1044 (1986) Communicated by T. Miwa

This article was processed by the author using the LaTEX style file pljour1 from Springer-Verlag.

Commun. Math. Phys. 190, 459 – 489 (1997)

Communications in

Mathematical Physics c Springer-Verlag 1997

Local BRST Cohomology and Covariance Friedemann Brandt? Instituut voor Theoretische Fysica, Katholieke Universiteit Leuven, Celestijnenlaan 200 D, B–3001 Leuven, Belgium Received: 25 July 1996 / Accepted: 23 April 1997

Abstract: The paper provides a framework for a systematic analysis of the local BRST cohomology in a large class of gauge theories. The approach is based on the cohomology of s + d in the jet space of fields and antifields, s and d being the BRST operator and exterior derivative respectively. It relates the BRST cohomology to an underlying gauge covariant algebra and reduces its computation to a compactly formulated problem involving only suitably defined generalized connections and tensor fields. The latter are shown to provide the building blocks of physically relevant quantities such as gauge invariant actions, Noether currents and gauge anomalies, as well as of the equations of motion. 1. Introduction 1.1. Motivation. Gauge invariance underlies as a basic principle our present models of fundamental interactions and is widely used when one looks for extensions of these models. The BRST-BV formalism provides a general framework to deal with many aspects of gauge symmetry, both in classical and quantum field theory. It was first established by Becchi, Rouet and Stora [1] in the context of renormalization of abelian Higgs–Kibble and Yang–Mills gauge theories, later extended by Kallosh to supergravity with open gauge algebra [2] (see also [3]) and by de Wit and van Holten to general gauge theories [4], resulting finally in the universal field-antifield formalism of Batalin and Vilkovisky [5] which allows to treat all kinds of gauge theories within an elegant unified framework. The usefulness of this formalism is mainly based on the fact that it encodes the gauge symmetry and all its properties in a single antiderivation which is strictly nilpotent on all the fields and antifields. Throughout this paper, this antiderivation is called the BRST operator and denoted by s. ?

Junior fellow of the research council (DOC) of the K.U. Leuven.

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The nilpotency of s establishes in particular the local BRST cohomology, i.e. the cohomology of s in the space of local functionals (= integrated local volume forms) of the fields and antifields. This cohomology has many physically relevant applications. It determines for instance gauge invariant actions and their consistent deformations [6], the dynamical local conservation laws [7] and the possible gauge anomalies (see e.g. [8, 1, 9, 10, 11, 12]) of a gauge theory and is a useful tool in the renormalization of quantum field theories even when a theory is not renormalizable in the usual sense [13]. Since the BRST cohomology can be defined for any gauge theory and since the correspondence of its cohomology classes to the mentioned physical quantities is universal too, it is worthwhile to look for a suitable general framework within which this cohomology can be computed efficiently and which has a large range of applicability. The purpose of this paper is to propose such a framework. It applies to a large class of gauge theories and relates the BRST cohomology to an underlying gauge covariant algebra. This includes a definition of tensor fields on which this algebra is realized and of generalized connections associated with it, and reduces the computation of the BRST cohomology locally to a problem involving only these quantities. The reduced problem is formulated very compactly in terms of identities analogous to the “Russian formula" in Yang–Mills theory [9, 14]1 , F = (s + d)(C + A) + (C + A)2 .

(1.1)

Here C, A and F are the familiar Lie algebra valued Yang–Mills ghost fields, connection and curvature forms respectively, s is the Yang–Mills BRST operator, and d is the spacetime exterior derivative. The usefulness of (1.1) is based, among others, on its remarkable property to compress the familiar BRST transformations of the Yang–Mills ghost and gauge fields, as well as the construction of the field strength in terms of the gauge field into a single identity. The combination C +A occurring in (1.1) is an example of what will be called a generalized connection here. 1.2. Relations and differences to other approaches. The proposed approach generalizes a concept outlined in [15] (see also [16]) for the study of the “restricted" (= antifield independent) BRST cohomology in a special class of gauge theories characterized among others by (a) the presence of (spacetime) diffeomorphisms among the gauge symmetries, (b) the closure and irreducibility of the gauge algebra, (c) the presence of “enough” independent gauge fields ensuring that all the derivatives of the ghost fields can be eliminated from the BRST cohomology. In such theories, the extension of the concept of [15] to the full cohomological problem, including the antifields, is (more or less) straightforward and was used already in [17, 18] within a complete computation of the BRST cohomology in Einstein gravity and Einstein–Yang–Mills theories. Here these ideas are extended to general gauge theories. In particular none of the conditions (a)–(c) is needed as a prerequisite for the methods outlined in this paper. This is possible thanks to suitable generalizations of the concept [15] which at the same time modify and unite various techniques that have been developed over the last 20 years, thereby revealing relations between them which are less apparent in other approaches. Such techniques, to be described later in detail, are the so-called descent equation technique, the use of contracting homotopies in jet spaces, compact formulations of the BRST algebra analogous to the “Russian formula” (1.1), and spectral sequence techniques along the lines of homological perturbation theory [19, 20, 21]. Let me now briefly comment on the use of these techniques in this paper, as compared to other approaches. 1 Originally the term “Russian formula" was introduced by Stora in the second ref. in [9] for a different but related identity. Here it is used as in the last ref. in [9].

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Descent equations and the “Russian formula” were first used within the celebrated differential geometric construction of (representatives of) chiral anomalies in D = 2n dimensions from characteristic classes in D + 2 dimensions [9], and also within the classification of such anomalies in [22]. Later it became clear that the descent equations are useful not only in connection with chiral anomalies, but to analyse the complete BRST cohomology, cf. e.g. [23, 24, 15, 11, 16]. The reason is that they allow to deal efficiently with the total derivatives into which the integrands of BRST invariant functionals transform in general. In this paper we will compress the descent equations into a compact form. To this end the BRST operator s and the exterior derivative d will be united to the single operator s˜ = s + d

(1.2)

defined on local “total forms" (see Sect. 2). This idea is not new; in fact it is familiar from the construction and classification of chiral anomalies mentioned above. However, somewhat surprisingly, it was not utilized systematically in a general approach to the BRST cohomology on local functionals later. The systematic use of s˜ is fundamental to the method proposed here and has several advantages. In particular it allows us to extend the concept of [15] to theories which do not satisfy the assumptions (a)–(c) mentioned above, such as Yang–Mills theory whose BRST cohomology has been calculated by different means in [25, 24, 26, 27]. The use of s˜ is particularly well adapted to the analysis of the BRST cohomology on local functionals because the latter is in fact isomorphic to the cohomology of s˜ on local total forms, at least locally, cf. [16] and Sect. 32 . Contracting homotopies similar to the ones used here were constructed and applied to BRST cohomological problems e.g. already in [25, 23, 24]. However, these contracting homotopies were designed for the cohomology of s [24] and its linearized version [25, 23] respectively. The method proposed here extends them to the s-cohomology. ˜ This has the important consequence that it leads directly to the mentioned compact formulation of the cohomological problem in terms of identities analogous to the “Russian formula” (1.1). For instance, when applied to Yang–Mills theory, the contracting homotopy for s˜ singles out the special combination (generalized connection) A + C occurring in (1.1). As a result, (1.1) itself arises naturally in this approach, cf. Sect. 7. In contrast, the corresponding contracting homotopy [24] for s gives instead of A + C just C and makes no contact with the “Russian formula” (it does however provide the same tensor fields). The proposed approach also extends the methods developed in [21] to use and deal with the antifields along the lines of homological perturbation theory [19, 20]. This extension is straightforward and, again, related to the use of s˜ instead of s. Among others it will allow us to trace the BRST cohomology at all ghost numbers (including negative ones) back to a weak (= on-shell) cohomological problem involving the tensor fields and generalized connections only. This has been utilized recently in [28] in order to compute the BRST cohomology in four dimensional N = 1 supergravity. Finally, the approach provides a “cohomological” perspective on tensor fields and connections. The latter are usually characterized through specific transformation properties under the respective symmetries. However, in a general gauge theory it is not always clear from the outset which transformation laws should be imposed for this purpose. An advantage of the approach proposed here is that such transformation laws need not be 2 The isomorphism applies only to the BRST cohomology on local functionals, i.e. to the relative cohomology H(s|d) on local volume forms. It does not extend to H(s|d) at lower form degrees in general.

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specified from the start. Rather, they emerge from the approach itself. Such a characterization of tensor fields, connections and the corresponding transformation laws has two major advantages: (i) it is purely algebraic and does not invoke any concepts in addition to the BRST cohomology itself; (ii) it is physically meaningful because the resulting tensor fields and generalized connections turn out to provide among others the building blocks of gauge invariant actions, Noether currents, anomalies and of the equations of motions. 1.3. Outline of the paper. The paper has been organized as follows. Section 2 sketches the basic algebraic approach to the BRST cohomology used in this paper and introduces some terminology and notation. Sections 3 and 4 relate the local BRST cohomology to the cohomology of s˜ and its weak (= “on-shell") counterpart. Section 5 introduces the concept of contracting homotopies for s˜ in jet spaces, and Sect. 6 shows that this concept is intimately related to the existence of a gauge covariant algebra and a compact formulation of the BRST algebra on tensor fields and generalized connections. Section 7 illustrates the method for various examples which do not satisfy the aforementioned assumptions (a)–(c) of [15] (the examples are Yang–Mills theory, Einstein gravity in the metric formulation, supergravity with open gauge algebra and two-dimensional Weyl invariant sigma models). Sections 8–10 spell out implications for the structure of gauge invariant actions, Noether currents, gauge anomalies, etc., as well as for the classical equations of motion. In Sect. 11 a special aspect of the cohomological problem is discussed, concerning the explicit dependence of the solutions on the coordinates of the base manifold which will be called “spacetime” henceforth, for no reason at all. The paper is ended by some concluding remarks in Sect. 12 and two appendices containing details concerning the algebraic approach and conventions used in the paper.

2. Algebraic Setting, Definitions and Notation In order to define the local BRST cohomology in a particular theory one has to specify the BRST operator s and the space in which its cohomology is to be computed. The BRST operator is defined on a set of fields 8A and corresponding antifields 8∗A according to standard rules of the field-antifield formalism summarized in Appendix B. In particular these rules include that the BRST operator is nilpotent and commutes with the spacetime derivatives ∂µ , (2.1) s2 = s∂µ − ∂µ s = ∂µ ∂ν − ∂ν ∂µ = 0. The basic concept underlying these fundamental relations and the whole paper is the jet bundle approach [29] sketched in Appendix A. Essentially this means simply that the fields, antifields and all their derivatives are understood as local coordinates of an infinite jet space. For this set of jet coordinates the collective notation [8, 8∗ ] is used. The local jet coordinates are completed by the spacetime coordinates xµ and the differentials dxµ . The differentials are counted among the jet coordinates by pure convention and convenience. The derivatives ∂µ are defined as total derivative operators in the jet space, cf. Eq. (A.6), and become usual partial derivatives on the local sections of the jet bundle. The concrete BRST transformations of the fields and antifields depend on the particular theory and its gauge symmetry, whereas the spacetime coordinates xµ and differentials dxµ are always BRST invariant in accordance with the second relation (2.1), s xµ = 0,

s dxµ = 0.

(2.2)

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The use of the differentials is in principle not necessary but turns out to be very useful in order to analyse the local BRST cohomology. In particular it allows to define d = dxµ ∂µ and s˜ = s + d in the jet space. The relations (2.1) are equivalent to the nilpotency of s, ˜ s˜2 = 0

⇔

s2 = sd + ds = d2 = 0.

(2.3)

The usefulness of s˜ in the context of the local BRST cohomology stems from the fact that it allows to write and analyse the descent equations in a compact form (cf. Sect. 3). The descent equations involve local p-forms ωp =

1 dxµ1 . . . dxµp ωµ1 ...µp (x, [8, 8∗ ]). p!

(2.4)

These forms are required to be local in the sense that they are formal series’ in the antifields, ghosts and their derivatives such that each piece with definite antighost number (cf. [21] and section 4) depends polynomially on the derivatives of all the fields and antifields. From the outset no additional requirements are imposed on local forms here. In particular they are not restricted by power counting, it is not assumed that the indices µi of the functions ωµ1 ...µp occurring in (2.4) indicate their actual transformation properties under Lorentz or general coordinate transformations, and local forms are not required to be globally well-defined in whatever sense. R A local functional is by definition an integrated local volume form ωD (throughout this paper D denotes the spacetime dimension). It is called BRST invariant if sωD is d-exact in the space of local forms, i.e. if sωD + dωD−1 = 0 holds for some local form ωD−1 . Translated to the local sections of the jet bundle, in general this requires local functionals to be BRST invariant only up to surface integrals. Analogously a local R functional ωD is called BRST-exact (or trivial) if ωD = sηD + dηD−1 holds for some local forms ηD and ηD−1 . The BRST cohomology on local functionals considered here is thus actually the relative cohomology H(s|d) of s and d on local volume forms. This cohomology is well-defined due to (2.3) and represented by solutions ωD of sωD + dωD−1 = 0,

ωD 6= sηD + dηD−1 .

(2.5)

In the next section H(s|d) will be related to the cohomology of s˜ on local total forms ω. ˜ The latter are by definition formal sums of local forms with different form degrees, X ωp . (2.6) ω˜ = p

The s-cohomology ˜ on local total forms is then defined through the condition s˜ω˜ = 0 modulo trivial solutions of the form s˜η˜ + constant, where η˜ is a local total form and the constant is included for convenience. The representatives of this cohomology are thus local total forms ω˜ solving s˜ω˜ = 0,

ω˜ 6= s˜η˜ + constant.

(2.7)

The natural degree in the space of local total forms is the sum of the ghost number (gh) and the form degree (formdeg), called the total degree (totdeg), totdeg = gh + formdeg .

(2.8)

A local total form with definite total degree G is thus a sum of local p-forms with ghost number g = G − p (p = 0, . . . , D). s˜ has total degree 1, i.e. it maps a local total form with total degree G to another one with total degree G + 1.

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3. Descent Equations It is easy to see that the BRST cohomology on local functionals is locally isomorphic to the cohomology of s˜ on local total forms3 . To show this, one only needs (2.3) and a theorem on the cohomology of d on local forms, sometimes called the algebraic Poincar´e lemma. The latter states that locally any d-closed local p-form is d-exact for 0 < p < D and constant for p = 0, while local volume forms (p = D) are locally d-exact if and only if they have vanishing Euler–Lagrange derivative with respect to all the fields and antifields [30, 25, 16]. The local isomorphism of the cohomological problems associated with (2.5) and (2.7) can be derived by standard arguments which are therefore only sketched. Suppose that ωD solves sωD +dωD−1 = 0. Applying s to this equation results in d(sωD−1 ) = 0 due to (2.3). Hence, sωD−1 is d-closed. Since it is not a volume form, it is thus also d-exact in the space of local forms according to the algebraic Poincar´e lemma. Hence, there is a (possibly vanishing) local (D − 2)-form ωD−2 satisfying sωD−1 + dωD−2 = 0. Iterating the arguments one concludes the existence of a set of local forms ωp , p = p0 , . . . , D satisfying (3.1) sωp + dωp−1 = 0 for D ≥ p > p0 ; sωp0 = 0 for some p0 . These equations are called the descent equations4 . They can be compactly written in the form D X ωp . s˜ ω˜ = 0, ω˜ = p=p0

˜ local total form Hence, any solution of sωD + dωD−1 = 0 corresponds to an s-closed and the reverse is evidently also true. Using again the algebraic Poincar´e lemma and (2.3), it is easy to see that ωD is a trivial solution of the form sηD + dηD−1 if and only if ω˜ is trivial too, i.e. if and only if ω˜ = s˜η˜ + constant. Since ω˜ has total degree (g + D) if ωD has ghost number g we conclude Lemma 3.1. The BRST-cohomology on local functionals with ghost number g and the s-cohomology ˜ on local total forms of total degree G = g + D are locally isomorphic. That is to say, locally the solutions of (2.5) with ghost number g correspond one-to-one (modulo trivial solutions) to the solutions of (2.7) with total degree G = g + D.

4. Equivalence to the Weak Cohomology of γ˜ = γ + d A simple and useful concept in the study of the BRST cohomology is a suitable expansion of local functionals and forms in powers of the antifields. Following the lines of [21] it will now be used to show that the s-cohomology ˜ on local total forms of the fields and antifields reduces to a weak (= on-shell) cohomology on antifield independent local total forms. 3 Here and in the following local equalities or isomorphisms refer to sufficiently small patches of the jet space. Global properties of the jet bundle are not taken into account. 4 For p = 0 the algebraic Poincar´ e lemma alone actually implies only sω0 = const.; however, in 0 meaningful gauge theories a BRST-exact constant vanishes necessarily, as one easily verifies (note that a constant can occur only if ω0 has ghost number −1). Notice that this might not hold anymore if one extends the BRST–BV formalism by including constant ghosts corresponding, e.g. to global symmetries [15, 16]. Such an extension is always possible [31] but not considered here.

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465

The most useful expansion in the antifields takes their respective ghost numbers into account. This is achieved through the so-called antighost number (antigh) defined according to antigh(8∗A ) = −gh(8∗A ),

antigh(8A ) = antigh(dxµ ) = antigh(xµ ) = 0.

(4.1)

In particular the BRST operator can be decomposed into pieces with definite antighost number (one says a piece has antighost number k if it raises the antighost number by k units). The decomposition of s starts always with a piece of antighost number −1, X sk , antigh(δ) = −1, antigh(γ) = 0, antigh(sk ) = k. (4.2) s=δ+γ+ k≥1

The most important pieces in this decomposition are δ and γ; the other pieces have positive antighost number and play only a secondary role in the cohomological analysis. δ is the so-called Koszul–Tate differential and is nonvanishing only on the antifields, δ8A = 0,

δφ∗i =

∂ˆ R Lcl , ˆ i ∂φ

... .

(4.3)

ˆ i denotes the Euler–Lagrange right-derivative of the classical Lawhere ∂ˆ R Lcl /∂φ grangian Lcl w.r.t. φi . In particular δ thus implements the classical equations of motion in the cohomology. γ encodes the gauge transformations because γφi equals a gauge transformation of φi with parameters replaced by ghosts, i γφi = Rα Cα ,

(4.4)

where the notation of Appendix B is used. Equation (4.2) extends straightforwardly to the analogous decomposition of s˜ = s+d into pieces with definite antighost numbers. Since d has vanishing antighost number, one simply gets X sk (4.5) s˜ = δ + γ˜ + k≥1

with γ˜ = γ + d.

(4.6)

2

Note that s˜ = 0 decomposes into δ 2 = 0,

δ γ˜ + γδ ˜ = 0,

γ˜ 2 = −(δs1 + s1 δ),

... .

(4.7)

The usefulness of the decomposition (4.5) is due to the acyclicity of the Koszul–Tate differential δ on local functions at positive antighost number [20, 21, 32]. This means that the cohomology of δ on local total forms is trivial at positive antighost number,5 δ ω˜ k = 0,

antigh(ω˜ k ) = k > 0

⇒

ω˜ k = δ η˜k+1 .

(4.8)

Using standard arguments of spectral sequence techniques which are not repeated here, one concludes from (4.8) immediately that a nontrivial solution of s˜ω˜ = 0 contains necessarily an antifield independent part ω˜ 0 solving 5 An analogous statement does not hold for the relative cohomology of δ and d. Indeed there are in general solutions of (2.5) which contain no antifield independent part. Such solutions correspond to local conservation laws [7].

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γ˜ ω˜ 0 ≈ 0,

ω˜ 0 6≈ γ˜ η˜0 + constant,

antigh(ω˜ 0 ) = 0,

(4.9)

(antigh(Ak ) = k).

(4.10)

where ≈ denotes weak equality defined through A0 ≈ 0

:⇔

∃A1 :

A0 = δA1

Note that the weak equality is an “on-shell equality" since, due to (4.3), A0 ≈ 0 implies that A0 vanishes for solutions of the classical equations of motion. Furthermore (4.7) and (4.8) imply that each solution ω˜ 0 of (4.9) can be completed to a nontrival solution ω˜ = ω˜ 0 + . . . of (2.7) and that two different completions with the same antifield independent part are equivalent in the cohomology of s˜ (the latter follows immediately from the fact that the difference of two such completions has no antifield independent part). This establishes the following result: Lemma 4.1. The cohomology of s˜ on local total forms is isomorphic to the weak cohomology of γ˜ on antifield independent local total forms. That is to say, any solution ω˜ of (2.7) contains an antifield independent part ω˜ 0 solving (4.9), and any solution ω˜ 0 of ˜ (4.9) can be completed to a solution of (2.7) which (for fixed ω˜ 0 ) is unique up to s-exact contributions. Remark. The weak cohomology of γ˜ on antifield independent local total forms is welldefined since γ˜ is weakly nilpotent on these forms, antigh(A0 ) = 0

⇒

γ˜ 2 A0 ≈ 0.

(4.11)

This follows immediately from the third identity (4.7) due to δA0 = 0.

5. Elimination of Trivial Pairs A well-known technique in the study of cohomologies is the use of contracting homotopies. I will now describe how one can apply it within the computations of the s-cohomology ˜ and of the weak γ-cohomology ˜ introduced in the previous sections. The idea is to construct contracting homotopy operators which allow to eliminate certain local jet coordinates, called trivial pairs, from the cohomological analysis. This reduces the cohomological problem to an analogous one involving only the remaining jet coordinates. For that purpose one needs to construct suitable sets of jet coordinates replacing the fields, antifields and their derivatives and satisfying appropriate requirements. In this section I will specify such requirements and show that they allow to eliminate trivial pairs. In Sect. 7 various explicit examples will be discussed to illustrate how one constructs these special jet coordinates in practice. The contracting homotopies and the trivial pairs for the s˜ and the weak γ˜ cohomology are usually closely related. Nevertheless, in practical computations the use of one or the other may be more convenient. Moreover it is often advantageous to combine them. For instance one may first use a contracting homotopy for the s-cohomology ˜ that eliminates some fields or antifields completely, such as the antighosts and the corresponding Nakanishi–Lautrup auxiliary fields used for gauge fixing, and then analyse the remaining problem by investigating the weak γ-cohomology. ˜ The arguments will be worked out in detail only for the weak γ-cohomology ˜ which is more subtle due to the occurrence of weak instead of strict equalities. In contrast, the s-cohomology ˜ can be treated using standard arguments which imply:

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467

Lemma 5.1. Suppose there is a set of local jet coordinates B = {U ` , V ` , W i } such that the change of local jet coordinates from {[8A , 8∗A ], xµ , dxµ } to B is local and locally invertible6 and s˜ U ` = V ` ∀ ` , s˜ W i = Ri (W) ∀ i .

(5.1) (5.2)

Then locally the U ’s and V’s can be eliminated from the s-cohomology, ˜ i.e. the latter reduces locally to the s-cohomology ˜ on local total forms depending only on the W’s. The (U ` , V ` ) are called trivial pairs. As already mentioned, Lemma 5.1 can be used in particular to eliminate the antighosts, Nakanishi–Lautrup fields and their antifields completely from the cohomological analysis because they (and all their derivatives) form trivial pairs, cf. e.g. [25] and [7], Sect. 14. In the following these fields will be therefore neglected without loss of generality. Let me now turn to the derivation of an analogous result for the weak γ-cohomology ˜ on antifield independent local total forms. Let us assume that there is a local and locally invertible change of jet coordinates from the antifield independent set {[8A ], xµ , dxµ } to {U ` , V ` , W i } such that7 γU ˜ ` = V ` ∀` , γW ˜ i = Ri (W ) ∀ i .

(5.3) (5.4)

Furthermore one can assume (without loss of generality) that each of the U ’s, V ’s and W ’s has a definite total degree. Note that all these degrees are nonnegative because the U ’s, V ’s and W ’s do not involve antifields and because it is assumed that antighosts and Nakanishi–Lautrup fields have been eliminated already. Again, the (U ` , V ` ) are called trivial pairs. In order to deal with weak equalities the following lemma will be useful later on: Lemma 5.2. Any weakly vanishing local total form f (U, V, W ) is a combination of weakly vanishing functions LK (W ) in the sense that f (U, V, W ) ≈ 0

⇔

f (U, V, W ) = aK (U, V, W )LK (W ),

LK (W ) ≈ 0

(5.5)

for some local total forms aK . Proof. Since the classical equations of motion have vanishing total degree and do not involve antifields, they are expressible solely in terms of the U ’s and W ’s because the V ’s have positive total degrees as a direct consequence of (5.3) (in fact only those U ’s and W ’s with vanishing total degrees can occur in the equations of motion). To prove (5.5) it is therefore sufficient to consider functions depending only on the U ’s and W ’s. Now, if a function f (U, W ) vanishes weakly then the same holds for its γ-transformation ˜ due to the second identity in (4.7), for the latter implies f = δg ⇒ γf ˜ = −δ(γg) ˜ ≈ 0. Using (5.3) and (5.4) one concludes f (U, W ) ≈ 0

⇒

γf ˜ (U, W ) = V ` ∗

∂f (U, W ) ∂f (U, W ) + Ri (W ) ≈ 0. ` ∂U ∂W i

(5.6)

6 I.e. locally any local total form f ([8, 8 ], dx, x) can be uniquely expressed as a local total form g(U , V, W) and vice versa. 7 One may replace the equalities in (5.3) and (5.4) by weak equalities without essential changes in the following arguments.

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Since the U ’s, V ’s and W ’s are by assumption independent local jet coordinates, and since the V ’s do not occur in the equations of motion, one concludes from (5.6) (for instance by differentiating γf ˜ (U, W ) w.r.t. to V ` ) that f (U, W ) ≈ 0 implies ` ∂f (U, W )/∂U ≈ 0. Iteration of the argument yields f (U, W ) ≈ 0

⇒

∂ k f (U, W ) ≈0 ∂U `1 . . . ∂U `k

∀k .

(5.7)

Thus a weakly vanishing function f (U, W ) must be a combination of weakly vanishing functions of the W ’s which proves (5.5).  I remark that Lemma 5.2 implies in particular that the equations of motion themselves are equivalent to a set of equations involving only those W ’s with vanishing total degree. This result will be interpreted in Sect. 10 as the covariance of the equations of motion. We are now prepared to prove that the U ’s and V ’s can be eliminated from the weak γ-cohomology: ˜ Lemma 5.3. Suppose there is a local and locally invertible change of jet coordinates replacing {[8A ], xµ , dxµ } by a set {U ` , V ` , W i } satisfying (5.3) and (5.4). Then locally the U ’s and V ’s can be eliminated from the weak γ-cohomology ˜ on antifield independent local total forms, γ˜ ω˜ 0 (U, V, W ) ≈ 0

⇒

ω˜ 0 (U, V, W ) ≈ f (W ) + γ˜ η˜0 (U, V, W ),

(5.8)

i.e. locally this cohomology is represented by solutions of γf ˜ (W ) ≈ 0,

f (W ) 6≈ γg(W ˜ ) + constant.

(5.9)

Proof. By assumption, locally any antifield independent local total form can be written in terms of the U ’s, V ’s and W ’s. To construct a contracting homotopy a parameter t is introduced scaling the U ’s and V ’s according to Ut` := tU ` ,

Vt` := tV ` .

(5.10)

On total forms ω˜ 0 (Ut , Vt , W ) one then defines an operator b through b = U`

1 ∂ ∂ = U` . ` ` t ∂V ∂Vt

(5.11)

˜ t` are defined by replacing in γU ˜ ` and γV ˜ ` all U ’s and V ’s by the correγU ˜ t` and γV ` sponding Ut ’s and Vt ’s. Now, (5.3) implies γV ˜ = γ˜ 2 U ` ≈ 0. Using Lemma 5.2 one thus concludes γV ˜ ` = a`,K (U, V, W )LK (W ), LK (W ) ≈ 0 for some a`,K and LK . Hence one defines γU ˜ t` = Vt` ,

γV ˜ t` = a`,K (Ut , Vt , W )LK (W ).

This shows in particular γV ˜ t` ≈ 0 and one now easily verifies (γb ˜ + bγ) ˜ ω˜ 0 (Ut , Vt , W ) ≈ which implies

∂ ω˜ 0 (Ut , Vt , W ) ∂t

(5.12)

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Z ω˜ 0 (U, V, W ) − ω˜ 0 (0, 0, W ) ≈

1

dt (γb ˜ + bγ) ˜ ω˜ 0 (Ut , Vt , W ).

(5.13)

0

Applying again Lemma 5.2 one concludes that γ˜ ω˜ 0 (U, V, W ) ≈ 0 implies γ˜ ω˜ 0 (Ut , Vt , W ) ≈ 0. Using this in (5.13) we finally get Z 1 dt b ω˜ 0 (Ut , Vt , W ), γ˜ ω˜ 0 (U, V, W ) ≈ 0 ⇒ ω˜ 0 (U, V, W ) ≈ ω˜ 0 (0, 0, W ) + γ˜ 0

(5.14) ˜ ω˜ 0 (. . .) ≈ γ˜ 0 dt bω˜ 0 (. . .) (the latter holds since γ˜ does not where we used 0 dt γb change the t-dependence up to weakly vanishing terms). This proves the lemma.  R1

R1

Remarks. a) It is very important to realize that both (5.1) and (5.2) must hold in order to eliminate U’s and V’s from the cohomology, and that the existence of a pair of jet coordinates satisfying (5.1) does in general not guarantee the existence of complementary W’s fulfilling (5.2). A simple and important counterexample is given by xµ and dxµ which always satisfy sx ˜ µ = dxµ but usually do not form a trivial pair except in diffeomorphism invariant theories, cf. Sect. 11. Analogous remarks apply of course to (5.3) and (5.4). The reader may check that the contracting homotopies for s used in [24, 15, 16] are in fact also based on the construction of variables satisfying requirements analogous to (5.3) and (5.4). b) Clearly the aim is the construction of a set of local jet coordinates containing as many trivial pairs as possible. The difficulty of this construction is in general not the finding of pairs of local jet coordinates satisfying (5.1) resp. (5.3) but the construction of complementary W’s resp. W ’s satisfying (5.2) resp. (5.4). c) Typically the U ’s are components of gauge fields and their derivatives and the V ’s contain the corresponding derivatives of the ghosts, cf. Sect. 7. The W ’s will be interpreted as tensor fields and generalized connections, cf. Sect. 6. d) Lemmas 5.1 and 5.3 are not always devoid of global subtleties, i.e. they can fail to be globally valid. E.g. if the manifold of the U ’s has a nontrivial de Rham cohomology, one cannot always eliminate all the U ’s and V ’s globally (important counterexamples are the vielbein fields in gravitational theories, cf. [18], section 5). In such cases the proof of Lemma 5.3 breaks down globally because some of the functions of the U ’s, V ’s and W ’s occurring in the proof have no globally well-defined extensions. This problem can be dealt with along the lines of [18]. 6. Gauge Covariant Algebra, Tensor Fields and Generalized Connections It will now be shown that the existence of a set of local jet coordinates {U ` , V ` , W i } (with nonempty subset {U ` , V ` }) satisfying (5.3) and (5.4) has a deep origin. Namely it is intimately related to an algebraic structure encoded in (5.4) which will be interpreted as a gauge covariant algebra and leads to the identification of tensor fields and generalized connections mentioned in the introduction. Recall that each local jet coordinate W i has a definite nonnegative total degree since it neither involves antifields nor antighosts. Those W ’s with vanishing total degree are called tensor fields and are denoted by T ı ; the other W ’s are called generalized connections for reasons which will become clear soon. Those generalized connections with total degree 1 are denoted by C˜ N ; the other generalized connections are denoted by Q˜ NG , where G indicates their total degree,

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F. Brandt

{T ı } = {W i : totdeg(W i ) = 0}, {C˜ N } = {W i : totdeg(W i ) = 1}, {Q˜ NG } = {W i : totdeg(W i ) = G ≥ 2}.

(6.1)

Note that the tensor fields have necessarily vanishing ghost number and form degree, whereas a generalized connection decomposes in general into a sum of local forms with different ghost numbers and corresponding form degrees, C˜ N = Cˆ N + AN , gh(Cˆ N ) = 1, gh(AN ) = 0, G X G G Q˜ NG = Qˆ N gh(Qˆ N p , p ) = G − p.

(6.2) (6.3)

p=0 G The Cˆ N are called covariant ghosts, the AN connection 1-forms and the Qˆ N G connection G-forms. Since γ˜ raises the total degree by one unit, (5.4) and (6.1) imply in particular

γ˜ T ı = C˜ N RN ı (T ), γ˜ C˜ N = 21 (−)εL +1 C˜ L C˜ K FKL N (T ) + Q˜ M2 ZM2 N (T ), γ˜ Q˜ N2 = 21 (−)εL +1 C˜ K C˜ L C˜ M ZM LK N2 (T ) +Q˜ M3 ZM3 N2 (T ) + Q˜ M2 C˜ K ZKM2 N2 (T ),

(6.4) (6.5) (6.6)

.. . for some functions R, F and Z of the tensor fields. Here (εM +1) denotes the Grassmann parity of C˜ M , (6.7) ε(C˜ M ) = εM + 1 . From γ˜ 2 T ı ≈ 0 one concludes, using (6.4) and (6.5), ∂RN ı ∂RM ı − (−)εM εN RN  ≈ −FM N K RK ı ,  ∂T ∂T  ZM2 N RN ı ≈ 0. RM 

(6.8) (6.9)

Equation (6.8) can be written in the compact form [1M , 1N ] ≈ −FM N K (T )1K

(6.10)

where [·, ·] denotes the graded commutator, [1M , 1N ] = 1M 1N − (−)εM εN 1N 1M , and 1N is the operator 1N = RN ı (T )

∂ ∂T

ı

.

Analogously γ˜ 2 C˜ N ≈ 0 implies in particular X
◦ 1M FN P K + FM N R FRP K + ZM N P M2 ZM2 K ≈ 0, MNP

where the graded cyclic sum was used defined by

(6.11)

(6.12)

(6.13)

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471

X ◦ XM N P = (−)εM εP XM N P + (−)εN εM XN P M + (−)εP εN XP M N .

(6.14)

MNP

Equation (6.13) are nothing but the Jacobi identities for the algebra (6.10) in presence of possible reducibility relations (6.9). Note that the Grassmann parities of γ˜ and of the ˜ imply the following Grassmann parities and symmetries of the 1’s and F ’s C’s ε(1N ) = εN , ε(FM N K ) = εM + εN + εK FM N K = −(−)εM εN FN M K .

(mod 2), (6.15)

In order to reveal the geometric content of this algebra it is useful to decompose (6.4) and (6.5) into parts with definite ghost numbers. Note that (6.4) reads γ˜ T ı = C˜ N 1N T ı ,

(6.16)

and thus decomposes due to γ˜ = γ + d and (6.2) into γT dT

ı ı

= Cˆ N 1N T ı , = AN 1N T ı .

(6.17) (6.18)

Equation (6.17) can be interpreted as a characterization of tensor fields as gauge covariant quantities. Indeed, recall that tensor fields are constructed solely out of the “classical fields” φ, their derivatives and the spacetime coordinates due to (6.1). Therefore γT equals just a gauge transformation of T with parameters replaced by ghosts. Equation (6.17) requires thus that the gauge transformation of a tensor field involves only special combinations of the parameters and their derivatives (which may involve the classical ˆ Hence, (6.17) characterizes tensor fields too), corresponding to the covariant ghosts C. fields indeed through a specific transformation law. Now, the derivatives ∂µ T of a tensor field are in general not tensor fields since γ(∂µ T ) contains ∂µ Cˆ N . The question arises how to relate ∂µ T to gauge covariant quantities. The answer is encoded in (6.18). Indeed, recall that the AN are 1-forms, AN = dxµ Aµ N .

(6.19)

Equation (6.18) is therefore equivalent to ∂µ T ı = Aµ N 1N T ı .

(6.20)

By assumption, (6.20) holds identically in the fields and their derivatives, with the same set {Aµ N } for all ı. In general this requires that {Aµ N } contains a locally invertible subset {vµ m }. Then (6.20) just defines those 1’s corresponding to {vµ m } in terms of the ∂µ and the other 1’s, and can be regarded as a definition of covariant derivatives. To put this in concrete terms I introduce the notation {Aµ N } = {vµ m , Aµ rˆ },

{1N } = {Dm , 1rˆ },

m = 1, . . . , D,

(6.21)

where the matrix (vµ m ) is assumed to be invertible. The Dm are called covariant derivatives and according to (6.20) they are given by Dm = Vm µ (∂µ − Aµ rˆ 1rˆ ), where Vm µ denotes the inverse of vµ m ,

(6.22)

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F. Brandt

vµ m Vm ν = δµν ,

vµ m Vn µ = δnm .

(6.23)

I note that neither the vµ m nor the Aµ rˆ are necessarily elementary fields. In particular, some of them may be constant or even zero. Let me finally discuss (6.5) which generalizes the Russian formula (1.1). Its decomposition into pieces with definite ghost number (resp. form degree) reads γ Cˆ N = 21 (−)εL +1 Cˆ L Cˆ K FKL N (T ) + Qˆ M2 ZM2 N (T ), γAµ N = ∂µ Cˆ N − Cˆ L Aµ K FKL N (T ) − Cˆ µ M2 ZM2 N (T ), ∂ µ Aν

N

− ∂ ν Aµ

N

= −Aµ Aν FKL (T ) + Bµν L

K

N

M2

(6.24) (6.25)

ZM2 (T ), N

(6.26)

where the following notation was used: Q˜ N2 = 21 dxµ dxν Bµν N2 + dxµ Cˆ µ N2 + Qˆ N2 .

(6.27)

Equation (6.24) and (6.25) give the γ-transformations of the covariant ghosts and of the Aµ N respectively. Equation (6.26) determines the curvatures (field strengths) corresponding to the “gauge fields" Aµ N . They are given by N (T ) Fmn N = Vm µ Vn ν 2∂[µ Aν] N + 2v[µ k Aν] rˆ Frk ˆ
rˆ sˆ N M2 N +Aµ Aν Fsˆ rˆ (T ) − Bµν ZM2 (T ) ,

(6.28)

where the invertibility of the vµ m was used again in order to solve (6.26) for the Fmn N . That the latter should indeed be identified with curvatures follows from the fact that they occur in the commutator of the covariant derivatives, [Dm , Dn ] ≈ −Fmn N 1N .

(6.29)

Note however that some (or all) of these curvatures may be constant or even zero. The Bianchi identities arising from (6.29) are a subset of the identities (6.13), D[m Fnk] N − F[mn M Fk]M N + Zmnk M2 ZM2 N ≈ 0.

(6.30)

Remarks. a) (6.10) can be regarded as a covariant version of the gauge algebra. However it is important to realize that the number of 1’s exceeds in general the number of gauge symmetries, cf. Sect. 7. ˜ occur only in reducible gauge theories because otherwise there are no local jet b) Q’s variables which can correspond to them. c) Considerations similar to those performed here for the W ’s can be of course also applied to the W’s satisfying (5.2). That leads in particular to an extension of the concept to antifield dependent tensor fields. Examples can be found in [17, 18, 33].

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473

7. Examples The concept outlined in the previous sections will now be illustrated for four examples, exhibiting different facets of the general formalism. First the concept is shown to reproduce the standard tensor calculus in the familiar cases of Yang–Mills theory and of gravity in the metric formulation. Then pure four dimensional N=1 supergravity without auxiliary fields is discussed. This illustrates the case of an open gauge algebra and is the only example where the number of 1’s and gauge symmetries coincide. Finally Weyl and diffeomorphism invariant sigma models in two spacetime dimensions are considered. In this example one gets an infinite set of generalized connections and corresponding 1-transformations, but no (nonvanishing) curvatures (6.28). I remark that the approach of [15] does not apply to any of these examples (not even to gravity in the metric formulation!) because each of them violates one of the assumptions (a)–(c) mentioned in Sect. 1. Hence, one really needs the extended concept outlined in the previous sections to perform the following analysis. As the gauge algebra is closed in the first, second and last example, the formulae of Sect. 6 are in these cases promoted to strict instead of weak equalities, with γ˜ replaced by s˜ and without making reference to a particular gauge invariant action. 7.1. Yang–Mills theories. For simplicity I consider pure Yang–Mills theories (no matter fields). The standard BRST transformations of the Yang–Mills gauge fields Aµ i and the corresponding ghosts C i read sAµ i = ∂µ C i + C k Aµ j fjk i ,

sC i = 21 C k C j fjk i ,

(7.1)

where i labels the elements of the Lie algebra of the gauge group with structure constants fij k . The trivial pairs are in this case given by {U ` } = {∂(µ1 ...µk Aµk+1 ) i : k = 0, 1, . . .},

(7.2)

˜ } = {∂µ1 ...µk+1 C + . . . : k = 0, 1, . . .}. {V } = {sU

(7.3)

`

`

i

Hence, in the new set of local jet coordinates the V ’s replace one by one all the derivatives of the ghosts. The undifferentiated ghosts themselves are replaced by the generalized connections (7.4) C˜ i = C i + Ai , Ai = dxµ Aµ i . The complete set of generalized connections contains in addition the differentials, {C˜ N } = {dxµ , C˜ i }.

(7.5)

The vµ m are thus in this case just the entries of the constant unit matrix, vµ m = δµm . Hence, indices m and µ need not be distinguished in this case. The 1-operations corresponding to (7.5) are {1N } = {Dµ , δi },

Dµ = ∂µ − Aµ i δi ,

(7.6)

where the δi are the Lie algebra elements. Equation (6.5) reproduces for N = i the “Russian formula" (1.1) in the form s˜C˜ i = 21 C˜ k C˜ j fjk i + 21 dxµ dxν Fµν i . The algebra (6.10) of the 1’s reads in this case

(7.7)

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F. Brandt

[Dµ , Dν ] = −Fµν i δi ,

[Dµ , δi ] = 0,

[δi , δj ] = fij k δk

(7.8)

with the standard Yang–Mills field strengths arising from (6.28) and transforming under the δi according to the adjoint representation, Fµν i = ∂µ Aν i − ∂ν Aµ i + Aµ j Aν k fjk i ,

δi Fµν j = −fik j Fµν k .

(7.9)

A complete set of tensor fields is in this case given by the xµ and a choice of algebraically independent components of the field strengths and their covariant derivatives, {T ı } ⊂ {xµ , Dµ1 . . . Dµk Fνρ i : k = 0, 1, . . .}.

(7.10)

Remark. Notice that the above choice of variables is very similar to the one in [24]. In fact the tensor fields coincide in both approaches (except that here also the xµ are counted among them). The difference is that the present approach singles out the C˜ i and dxµ as generalized connections, rather than just the C i . Note that, as a direct consequence of the presence of d in s, ˜ one cannot simply choose C˜ i = C i here because that choice would not fulfill requirement (5.4). 7.2. Gravity in the metric formulation. I consider now pure gravity with the metric fields gµν = gνµ as the only classical fields and diffeomorphisms as the only gauge symmetries. The BRST transformations of the metric and the diffeomorphism ghosts ξ µ read sgµν = ξ ρ ∂ρ gµν + (∂µ ξ ρ ) gρν + (∂ν ξ ρ ) gµρ , sξ µ = ξ ν ∂ν ξ µ . (7.11) The trivial pairs can be chosen as {U ` } = {xµ , ∂(µ1 ...µk 0µk+1 µk+2 ) ν : k = 0, 1, . . .},

(7.12)

˜ } = {dx , ∂µ1 ...µk+2 ξ + . . . : k = 0, 1, . . .}, {V } = {sU

(7.13)

0µν ρ = 21 g ρσ (∂µ gνσ + ∂ν gµσ − ∂σ gµν ).

(7.14)

`

`

µ

ν

where Note that the V ’s replace all derivatives of the ghosts of order > 1. The undifferentiated ghosts and their first order derivatives give rise to the generalized connections {C˜ N } = {ξ˜µ , C˜ µ ν },

ξ˜µ = ξ µ + dxµ ,

C˜ µ ν = ∂µ ξ ν + 0µρ ν ξ˜ρ .

(7.15)

The generalized Russian formulae (6.5) read in this case s˜ξ˜µ = ξ˜ν C˜ ν µ ,

s˜C˜ µ ν = C˜ µ ρ C˜ ρ ν + 21 ξ˜ρ ξ˜σ Rρσµ ν ,

(7.16)

where Rµνρ σ is the standard Riemann tensor constructed of the 0’s. The vµ m are, as in the case of the Yang–Mills theory, just the entries of the constant unit matrix. Hence, indices µ and m are not distinguished. One gets {1N } = {Dµ , 1µ ν },

Dµ = ∂µ − 0µρ ν 1ν ρ ,

(7.17)

where the 1µ ν generate GL(D)-transformations of world indices according to 1µ ν Tρ = δρν Tµ , The algebra (6.10) reads now

1µ ν T ρ = −δµρ T ν .

(7.18)

Local BRST Cohomology and Covariance

475

[Dµ , Dν ] = −Rµνρ σ 1σ ρ , [1µ ν , Dρ ] = δρν Dµ , [1µ ρ , 1ν σ ] = δνρ 1µ σ − δµσ 1ν ρ .

(7.19)

The set of tensor fields contains the gµν , µ ≥ ν and a maximal set of algebraically independent components of Rµνρ σ and their covariant derivatives, {T ı } ⊂ {gµν , Dµ1 . . . Dµk Rλνρ σ : k = 0, 1, . . .}.

(7.20)

Remark. Recall that tensor fields are characterized by the transformation law (6.17). One might wonder whether this transformation law agrees in this case with the standard transformation law for tensor fields under diffeomorphisms which is in BRST language the Lie derivative along the diffeomorphism ghosts. The answer is affirmative because (6.17) yields in this case, e.g. for a tensor field Tµ , γ Tµ = ξ ν Dν Tµ + (∂µ ξ ν + 0µρ ν ξ ρ )Tν = ξ ν ∂ν Tµ + (∂µ ξ ν )Tν .

(7.21)

7.3. D=4, N=1 minimal supergravity. The classical field content of the D=4, N=1 minimal pure supergravity theory without auxiliary fields is given by the vielbein fields and the gravitinos, denoted by eµ a and ψµ α , ψ¯ µ α˙ respectively (α, α˙ denote indices of two-component complex Weyl spinors with conventions as in [28]). The gauge symmetries are diffeomorphism invariance, local supersymmetry and local Lorentz invariance. The corresponding ghosts are denoted by ξ µ , ξ α , ξ¯α˙ and C ab = −C ba respectively. For simplicity the analysis is restricted to the action [34] Z   (7.22) Scl = d4 x 21 eR − 2µνρσ (ψµ σν ∇ρ ψ¯ σ − ψ¯ µ σ¯ ν ∇ρ ψσ ) with e = det(eµ a ), 0123 = 1 and R = Rab ba ,

Rab cd = 2E[a µ Eb] ν (∂µ ων cd + ωµ ce ωνe d ),

∇µ ψν = ∂µ ψν − ωµ σab β ψν , ˙ ∇µ ψ¯ ν α˙ = ∂µ ψ¯ ν α˙ + 21 ωµ ab σ¯ ab α˙ β˙ ψ¯ ν β , α

α

ab

1 2

α

β

(7.23) (7.24) (7.25)

where the Ea µ are the entries of the inverse vielbein and ωµ ab denotes the gravitino dependent spin connection ωµ ab = E aν E bρ (ω[µν]ρ − ω[νρ]µ + ω[ρµ]ν ), ω[µν]ρ = eρa ∂[µ eν] a − iψµ σρ ψ¯ ν + iψν σρ ψ¯ µ .

(7.26)

(Lorentz indices a, b, . . . are lowered and raised with the Minkowski metric ηab = diag(1, −1, −1, −1).) The γ-transformations read in this case γeµ a = ξ ν ∂ν eµ a + (∂µ ξ ν )eν a + Cb a eµ b + 2iσ a αα˙ (ξ α ψ¯ µ α˙ − ξ¯α˙ ψµ α ) , (7.27) γψµ α = ∇µ ξ α + ξ ν ∂ν ψµ α + (∂µ ξ ν )ψν α + 21 C ab σab β α ψµ β , γξ µ = ξ ν ∂ν ξ µ + 2iξ α σ µ αα˙ ξ¯α˙ , γξ = ξ ∂µ ξ + C σabβ ξ − 2iξ σ α

µ

α

1 2

ab

α β

γC ab = ξ µ ∂µ C ab − C ac Cc b −

β µ

¯β˙

β β˙ ξ ψµ ˙ 2iξ β σ µ β β˙ ξ¯β ωµ ab

α

,

(7.28) (7.29) (7.30) (7.31)

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F. Brandt

¯ where (and analogous expressions for γ ψ¯ µ and γ ξ), ∇µ ξ α = ∂µ ξ α − 21 ωµ ab σab β α ξ β . The gauge algebra is open (it closes modulo the equations of motion for the gravitinos). Hence γ is nilpotent only on-shell and does not agree with s on all the fields. One can choose the U ’s in this case as {U ` } = {xµ , ∂(µ1 ...µk eµk+1 ) a , ∂(µ1 ...µk ωµk+1 ) cd , ∂(µ1 ...µk ψµk+1 ) α , ∂(µ1 ...µk ψ¯ µk+1 ) α˙ : c > d; k = 0, 1, . . .}.

(7.32)

Note that the ωµ ab = ωµ [ab] correspond one by one to the antisymmetrized first order derivatives ∂[µ eν] a of the vielbein fields due to (7.26). Hence, all the U ` are indeed algebraically independent new local jet coordinates. The corresponding V ` replace one ˜ µ a = ∂µ ξ a +. . . (ξ a = eµ a ξ µ ), by one the dxµ and all the derivatives of the ghosts due to γe ab α α ˙ ab α ¯ ˜ µ = ∂µ ξ +. . . and γ˜ ψµ = −∂µ ξ¯α˙ +. . . . The undifferentiated γω ˜ µ = ∂µ C +. . . , γψ ghosts give rise to the generalized connections {C˜ N } = {ξ˜a , ξ˜α , ξ˜α˙ , C˜ ab : a > b}, ξ˜a = ξ˜µ eµ a , C˜ ab = C ab + ξ˜µ ωµ ab , ξ˜α = ξ α + ξ˜µ ψµ α , ξ˜α˙ = ξ¯α˙ − ξ˜µ ψ¯ µ α˙

(7.33)

with ξ˜µ as in (7.15). The corresponding 1’s are denoted by {1N } = {Da , Dα , D¯ α˙ , lab : a > b}, Da = Ea µ (∂µ − 21 ωµ ab lab − ψµ α Dα + ψ¯ µ α˙ D¯ α˙ ),

(7.34)

where lab = −lba denote the elements of the Lorentz algebra, and Dα and D¯ α˙ are supersymmetry transformations represented on the tensor fields given below (these tensor fields are ordinary fields, not superfields; accordingly Dα and D¯ α˙ are not “superspace operators’). The Grassmann parities of the 1’s are εa = ε[ab] = 0 and εα = εα˙ = 1 (the supersymmetry ghosts commute). (7.34) indicates that in this case the vielbein fields are identified with the vµ m , i.e. the indices m coincide here with Lorentz vector indices, v µ m ≡ eµ a ,

Vm µ ≡ Ea µ .

(7.35)

Using the shorthand notation {ξ˜A } = {ξ˜a , ξ˜α , ξ˜α˙ },

{DA } = {Da , Dα , D¯ α˙ }

the algebra of the DA reads [DA , DB ] ≈ −TAB C DC − 21 FAB cd lcd ,

(7.36)

where the nonvanishing TAB C and FAB cd are a = 2iσ a αα˙ , Tαα˙ a = Tαα ˙ α µ Tab = Ea Eb ν (∇µ ψν α − ∇ν ψµ α ), cd = i (T cdα σb αα˙ − 2σ [c αα˙ T d] b α ), Fαb ˙ Fab cd = Rab cd + 2(ψ[a α Fb]α cd − ψ¯ [a α˙ Fb]α˙ cd )

(7.37) (7.38) (7.39) (7.40)

Local BRST Cohomology and Covariance

477

and analogous expressions for Tab α˙ and Fαb cd . The remaining commutators of the 1’s are (7.41) [lab , DA ] = −g[ab] A B DB , [lab , lcd ] = 2ηa[c ld]b − 2ηb[c ld]a , where d , g[ab] c d = 2ηc[a δb]

g[ab] α β = σab α β ,

˙

˙

g[ab] α˙ β = −σ¯ ab β α˙ .

(7.42)

Accordingly the generalized Russian formulae (6.5) read in this case γ˜ ξ˜A = 21 C˜ ab g[ab] B A ξ˜B − 21 (−)εB ξ˜B ξ˜C TCB A , γ˜ C˜ ab = −C˜ ac C˜ c b − 21 (−)εD ξ˜D ξ˜C FCD ab .

(7.43) (7.44)

Note that these identities encode all the Eqs. (7.26)–(7.31), (7.38) and (7.40). The set of independent tensor fields consists in this case of a subset of Fab cd , Tab α , Tab α˙ and their covariant derivatives, {T ı } ⊂ {Da1 . . . Dak Fbc de , Da1 . . . Dak Tbc α , Da1 . . . Dak Tbc α˙ : k = 0, 1, . . .}.

(7.45)

Remark. Notice that the formalism provides “super-covariant” tensor fields and, in particular, “super-covariant” derivatives (7.34) containing the gravitino and the supersymmetry transformations. Note also that these tensor fields do not carry “world indices" µ, ν, . . . , in contrast to the example discussed in the previous subsection. The reason is that the undifferentiated vielbein fields count among the U ’s. Indeed, the corresponding V ’s replace all the first order derivatives of the diffeomorphism ghosts ξ µ and therefore the BRST transformation of a tensor field must not involve ∂ν ξ µ . Hence, tensor fields are indeed “world scalars” in this case. One could of course instead count the undifferentiated vielbein fields also among the tensor fields and promote the ∂ν ξ µ to generalized connections. Then tensor fields could also carry world indices and one would get additional 1’s generating GL(4) transformations of world indices, as in the metric formulation of gravity discussed in the previous subsection. However, such a choice would not correspond to a maximal set of trivial pairs and would thus complicate unnecessarily the analysis of the BRST cohomology! 7.4. Two dimensional sigma models. Consider two dimensional sigma models whose set of classical fields consists of scalar fields ϕi and the two dimensional metric fields gµν and whose gauge symmetries are given by two dimensional diffeomorphism and Weyl invariance, with corresponding ghosts ξ µ and C respectively. The BRST transformations of the fields read sgµν = ξ ρ ∂ρ gµν + (∂µ ξ ρ ) gρν + (∂ν ξ ρ ) gµρ + C gµν , sY = ξ ν ∂ν Y

for

Y ∈ {ϕi , ξ µ , C}.

(7.46)

Following closely the lines (but not the notation) of [33] I first introduce new local jet ¯ e, η, η¯ replacing the undifferentiated metric components and diffeocoordinates h, h, morphism ghosts (h, h¯ are “Beltrami variables")8 , g22 g11 √ (7.47) h= √ , h¯ = √ , e = g, g12 + g g12 + g ¯ 2 + dx2 ), η¯ = (ξ 2 + dx2 ) + h(ξ 1 + dx1 ) (7.48) η = (ξ 1 + dx1 ) + h(ξ 8

This change of jet coordinates is not globally well-defined in general.

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F. Brandt

with g = − det(gµν ) > 0. The U ’s are ¯ ∂ p ∂¯ q e : p, q = 0, 1, . . .}, {U ` } = {xµ , ∂ p ∂¯ q h, ∂ p ∂¯ q h,

(7.49)

where ∂ ≡ ∂1 ,

∂¯ ≡ ∂2 .

(7.50)

Hence, in this case all the metric components and all their derivatives occur in trivial ¯ all their derivatives, and pairs. The corresponding V ’s replace one by one the C, ∂ η, ¯ ∂η, µ the dx . Therefore one gets in this example an infinite set of generalized connections, given by η, η¯ and their remaining derivatives, {C˜ N } = {η p , η¯ p¯ : p, p¯ = −1, 0, 1, . . .}, ¯ η¯ . η p = 1 ∂ p+1 η, η¯ p¯ = 1 ∂¯ p+1 (p+1)! ¯

(p+1)!

(7.51) (7.52)

Equations (7.46)–(7.48) imply sη ˜ = η∂η and s˜η¯ = η¯ ∂¯ η. ¯ Therefore (6.5) reads in this case p+1 1 X (p − 2r)η r η p−r (7.53) sη ˜ p= 2 r=−1

˜ is and an analogous formula for s˜η¯ . The infinite set of 1’s corresponding to the C’s denoted by (7.54) {1N } = {Lp , L¯ p¯ : p, p¯ = −1, 0, 1, . . .}. p¯

Recall that the r.h.s. of (7.53) contains the structure functions occurring in the algebra of the 1’s. In this case all of these functions are constant and the algebra of the L’s and ¯ is isomorphic to two copies of the algebra of regular vector fields (−z p+1 )∂/∂z, L’s [Lp , Lq ] = (p − q)Lp+q ,

[L¯ p¯ , L¯ q¯ ] = (p¯ − q) ¯ L¯ p+ ¯ q¯ ,

[Lp , L¯ p¯ ] = 0.

(7.55)

The set of tensor fields on which this algebra is realized is given by i {T ı } = {Tp, p¯ : p, p¯ = 0, 1, . . .},

i p ¯ p¯ i Tp, p¯ = (L−1 ) (L−1 ) ϕ .

(7.56)

i The explicit form of the Tp, p¯ in terms of the fields and their derivatives was discussed in [33] and will be rederived below for the first few T ’s. The algebraic representation of the ¯ on the tensor fields can be derived from the algebra (7.56) using Lp T i = L’s and L’s 0,0 i i i L¯ p¯ T0,0 = 0 ∀ p, p¯ ≥ 0. The latter follows from the identification sT ˜ 0,0 = C˜ N 1N T0,0 , cf. (6.16). This yields

q
i Lq Tp, p¯ =

p! Ti ; (p − q − 1)! p−q,p¯

q≥p:

i Lq Tp, p¯ = 0

(7.57)

i and analogous formulae for L¯ q¯ Tp, p¯ . Let us now make contact with Sect. 6. Using (7.48) one easily reads off from (7.52) the connection forms Ap and A¯ p¯ contained in η p and η¯ p¯ : p p¯ Ap = δ−1 dx1 + H p dx2 , A¯ p¯ = δ−1 dx2 + H¯ p¯ dx1 , ¯ 1 1 ¯ H¯ p¯ = (p+1)! ∂¯ p+1 H p = (p+1)! ∂ p+1 h, h. ¯

The components of A−1 and A¯ −1 are identified with the vµ m according to

(7.58) (7.59)

Local BRST Cohomology and Covariance

479

¯ −1 {vµ m } ≡ {A−1 µ , Aµ }.

(7.60)

Explicitly one thus gets in matrix form     1 1h 1 −h m µ , (Vm ) = (vµ ) = ¯ . h1 1 − hh¯ −h¯ 1

(7.61)

¯ are identified with L−1 and Due to (7.60) the covariant derivatives {Dm } ≡ {D, D} ¯ L−1 . Equation (6.22) yields now   X X 1  H¯ p¯ L¯ p¯ + h H p Lp  , (7.62) D= ∂ − h∂¯ − 1 − hh¯ p≥0 ¯ p≥0 ¯ with H’s as in (7.59). Thanks to (7.57), the occurrence and an analogous expression for D, ¯ in D and D¯ does not result in nonlocal expressions (tensor of infinitely many L’s and L’s fields). Note that D ≡ L−1 and D¯ ≡ L¯ −1 commute according to (7.55), i.e. in this case all the (infinitely many) curvatures (6.28) vanish! Due to (7.56) the set of tensor fields is given by the ϕi and all their covariant derivatives. The latter may now be constructed ¯ −1 one gets for instance (derivatives act on recursively using (7.57). With V = (1 − hh) everything to the right): i i i ¯ i , T0,1 ¯ ¯ i = V (∂¯ − h∂)ϕ = Dϕi = V (∂ − h∂)ϕ = Dϕ , T1,0 i i i ¯ ¯ ¯ ¯ T1,1 = DDϕ = V (∂ − ∂ h)V (∂ − h∂)ϕ i ¯ ¯ ¯ i = V (∂ − ∂h)V = DDϕ (∂¯ − h∂)ϕ .

(7.63)

8. Structure of the Solutions In this section some conclusions are drawn about the geometric structure of the solutions of the cohomological problem and the related physical quantities such as gauge invariant actions, conserved currents and anomalies. According to Sect. 4 the cohomological problem in question can be reduced to the solution of γ˜ ω˜ 0 ≈ 0, where ω˜ 0 does not depend on antifields. This cocycle condition decomposes into the “weak descent equations" dαM ≈ 0,

γαp + dαp−1 ≈ 0

for p0 < p ≤ M,

γαp0 ≈ 0,

(8.1)

where αp denotes the p-form contained in ω˜ 0 , ω˜ 0 =

M X

αp ,

M = min{D, G},

G = totdeg(ω˜ 0 ).

(8.2)

p=p0

Here M = min{D, G} holds because ω˜ 0 does not contain antifields9 and thus involves only p-forms with p ≤ G. Note that the first equation in (8.1) is trivially satisfied if M = D, i.e. if G ≥ D. 9 I also assume, without loss of generality, that ω ˜ 0 does not depend on antighosts or Nakanishi–Lautrup fields (cf. Sect. 5) and that it has a definite total degree.

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F. Brandt

Now, the only local forms which are weakly d-closed but not necessarily (locally) weakly d-exact are volume forms and forms which do not depend on the ghosts, i.e. all other weakly d-closed local forms are also weakly d-exact, dαq ≈ 0,

q < D,

gh(αq ) > 0

⇒

αq ≈ dηq−1 .

(8.3)

This follows by means of the algebraic Poincar´e lemma (cf. Sect. 3) immediately from a general result on the relative cohomology of δ and d derived in [35]. Since all the local forms αp with p < M occurring in (8.2) have positive ghost number, one can analyse the weak descent equations (8.1) by means of (8.3) like the usual descent equations by means of the algebraic Poincar´e lemma in Sect. 3. This leads to the conclusion that ω˜ 0 is a nontrivial solution of γ˜ ω˜ 0 ≈ 0 if and only if its part αM fullfills G > D : γαD + dαD−1 ≈ 0, αD 6≈ γηD + dηD−1 ; G = D : γαD + dαD−1 ≈ 0, αD 6≈ dηD−1 ; G < D : dαG ≈ 0, αG 6≈ dηG−1 + constant.

(8.4) (8.5) (8.6)

Furthermore one concludes, using (8.3) again, that all solutions of (8.4)–(8.6) can be completed to nontrivial solutions of γ˜ ω˜ 0 ≈ 0. Hence, the complete local BRST cohomology is in fact (locally) isomorphic to the cohomological problems established by (8.4)–(8.6). Let me now discuss the implications of Sects. 5 and 6 for the structure of the solutions of (8.4)–(8.6) and briefly comment on their physical interpretation. For notational convenience I will restrict this discussion to the case of an irreducible gauge algebra. Using the notation of Sect. 6, one can then assume ω˜ 0 to be of the form ω˜ 0 = C˜ N1 . . . C˜ NG aNG ...N1 (T ),

C˜ N = Cˆ N + AN .

(8.7)

This implies that, in irreducible gauge theories, the general solutions of (8.4)–(8.6) are of the form G > D : αD = AN1 . . . AND Cˆ ND+1 . . . Cˆ NG aNG ...N1 (T );

(8.8)

G = D : αD = A G < D : αG = A

N1

...A

N1

...A

ND

aND ...N1 (T );

(8.9)

NG

aNG ...N1 (T ),

(8.10)

up to trivial and “topological" (= locally but not globally trivial) solutions, of course. Equation (8.8) applies for G = D + 1 to the antifield independent part of integrands of candidate gauge anomalies. Well-known examples are representatives of chiral anomalies in Yang–Mills theory [9]. Their integrands have indeed the form (8.8) (recall that in Yang–Mills theory the differentials count among the connection forms, cf. Subsect. 7.1). Solutions of (8.5) give rise to BRST invariant functionals with ghost number 0 and thus (8.9) applies to integrands of gauge invariant actions and their continuous first order deformations [6]. However, concerning these solutions a few more remarks are in order which I postpone to the next section. The solutions of (8.6) provide the local conservation laws of the theory. They correspond for G = D − 1 one-to-one to the nontrivial conserved currents10 and generalize for smaller G the concept of nontrivial conserved currents to form degrees < D − 1 [7]. 10 A conserved current j µ (∂ j µ ≈ 0) is called trivial in this context if j µ ≈ ∂ S νµ holds for some local µ ν S νµ = −S µν .

Local BRST Cohomology and Covariance

481

We conclude that all nontrivial “dynamical" conserved local G-forms can be written in the form (8.10) if the gauge algebra is irreducible, and in a similar form, involving possibly connection forms of higher form degree, if the gauge algebra is reducible (“topological” conserved local forms cannot always be cast in this form). In fact, in “normal” theories, dynamical solutions of (8.6) exist at most at form degrees G ≥ D − (2 + r), where r denotes the reducibility order of the theory, see [7] (r = −1 for theories without gauge invariance, r = 0 for irreducible gauge theories, . . . ). The weak d-cohomology established by (8.6) goes sometimes under the name “characteristic cohomology" [36]. To illustrate the result on the conservation laws I consider the Noether current corresponding to the invariance of the supergravity action (7.22) under global U (1)transformations of the gravitino. One finds for this Noether current j µ and the corresponding solution α3 of (8.6) j µ = −2iµνρλ ψν σρ ψ¯ λ , α3 =

1 6

(8.11)

¯ dx dx dx µνρλ j = 2ie ψσa ψ, µ

ν

ρ

λ

a

(8.12)

where 0123 = −0123 = 1 and ea = dxµ eµ a ,

ψ α = dxµ ψµ α ,

ψ¯ α˙ = −dxµ ψ¯ µ α˙ .

(8.13)

α3 is indeed of the form (8.10) because the 1-forms (8.13) are among the connection forms AN of the supergravity theory, cf. Subsect. 7. Note that this solution of (8.6) is constructed solely out of connection forms, i.e. it does not involve tensor fields at all! Remark. The usual construction of Noether currents does not always provide the corresponding solutions of (8.6) directly in the form (8.10). The statement here is that one can always redefine the Noether currents by subtracting trivial currents (if necessary) such that the corresponding (D −1)-forms take the geometric form (8.10). A famous example for such a redefined current is the “improved" energy momentum tensor in Yang–Mills theory.

9. Structure of Gauge Invariant Actions The field-antifield formalism is usually constructed starting from a given gauge invariant classical action. One may then ask whether it is possible to deform this action without destroying the gauge invariance. This question is relevant for instance in the quantum theory where deformations of the action can be caused by quantum corrections, or for the deformation of free gauge theories to interacting ones. The BRST cohomology provides a powerful tool to tackle these problems [6]. One may distinguish two kinds of deformations of a given action: those which do not change the gauge transformations up to local field redefinitions, and those which modify simultaneously the action and the gauge transformations in a nontrivial way. The integrands (volume forms) of actions which are invariant under given gauge transformations have to satisfy γαD + dαD−1 = 0,

αD 6= dηD−1 ,

αD = dDx a(x, [φ]),

(9.1)

where γ encodes the gauge transformations under study. Note that (9.1) is a stronger condition than (8.5) and replaces the latter for two reasons: the integrands of gauge

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invariant actions are (i) required to be strictly γ-invariant up to a total derivative and (ii) not necessarily to be considered as trivial if they are weakly zero to a total derivative R up √ – for instance one would not call the Einstein–Hilbert action d4 x −gR trivial even though its integrand is weakly zero. Now, if the gauge algebra is (off-shell) closed, γ is strictly nilpotent on all the fields (but not necessarily on the antifields). Therefore (9.1) implies descent equations for γ and d which do not involve antifields and read in a compact form γ˜ ω˜ 0 = 0, ω˜ 0 6= γ˜ η˜0 . This problem can be analysed like the weak γ-cohomology ˜ in Sects. 5 and 6 – all the arguments go through also for strict instead of weak equalities since γ and γ˜ are strictly nilpotent on all the fields. In particular we conclude that the general solution of (9.1) has again the form (8.9) (up to d-exact contributions) if the gauge algebra is closed and irreducible. The general solution of (9.1) provides the most general action which is invariant under a given set of gauge transformations encoded in γ. It has been determined by means of the BRST cohomology for Yang–Mills theory [25], gravity [23], minimal N=1, D=4 supergravity [37, 28] (both in the old minimal formulation [38] and in the new minimal one [39]) and for the sigma models considered in Subsect. 7.4 [33]. One can check that in all these cases the integrand of the most general gauge invariant action can indeed be expressed in the geometric form (8.9) even though this is not completely obvious in all cases. For instance, written in this form the integrand of the supergravity action (7.22) reads ¯ α4 = −abcd ea eb ec ( 481 ed R + 3i Sσ d ψ¯ − 3i ψσ d S),

(9.2)

R = Fab ,

(9.3)

ba

α

S =

Tab β σ ab β α

with ea , ψ and ψ¯ as in (8.13), Tab α and Fab cd as in (7.38) and (7.40), and 0123 = −1. The determination of deformations of a given action which modify nontrivially the gauge transformations is more subtle. A method which allows to attack this problem systematically and is based on the BRST cohomology was outlined in [6]. The idea is to deform the solution of the master equation instead of the classical action itself. This has many advantages. In particular it shows that to first order in the deformation parameter the deformed action is required to be weakly invariant under the original (undeformed) γ. The integrand of this first order deformation thus has the form (8.9) up to weakly vanishing terms. However, in general an analogous statement does not apply to the terms of higher orders in the deformation parameter because these terms are not necessarily weakly γ-invariant. 10. Gauge Covariance of the Equations of Motion A direct corollary of lemma 5.2 is the gauge covariance of the equations of motion (cf. remark after the proof of that lemma). Indeed, lemma 5.2 implies that the classical equations of motion are equivalent to a set of weakly vanishing functions of those W ’s with vanishing total degree. Since the latter are just the tensor fields (cf. Sect. 6), we conclude: Lemma 10.1. The classical equations of motion in a gauge theory are gauge covariant in the sense that they are equivalent to a set of weakly vanishing functions of the tensor fields. This is of course well-known for standard gauge theories such as Yang–Mills theory and Einstein gravity where the Euler–Lagrange equations themselves turn out to be

Local BRST Cohomology and Covariance

483

expressible solely in terms of the tensor fields. A less trivial check of Lemma 10.1 can be performed for the supergravity action (7.22). Indeed one can verify that the corresponding equations of motion are equivalent to the following equations involving only the tensor fields (7.45): Rab ≈ 0,

S α ≈ 0,

Uα˙ β˙ α ≈ 0

(10.1)

with S α as in (9.3) and Rab = Facb c ,

Uα˙ β˙ α = Tab α σ¯ ab α˙ β˙ .

(10.2)

11. Discussion of x-Dependence This section is devoted to the discussion of a special aspect of the cohomological problem concerning the explicit dependence of the solutions on the spacetime coordinates. In particular it is emphasized that the result of the cohomological analysis depends on whether it is carried out in the space of x-dependent or x-independent local forms. It is therefore important to make clear in every analysis of the BRST cohomology in which space one works and to be aware of the consequences of the chosen approach. 11.1. General remarks. In general the results of the cohomological analysis will depend in two respects on whether or not one considers the problem in the space of x-dependent local forms: (a) some nontrivial representatives of the cohomology might be overlooked if one performs the cohomological analysis in the space of x-independent local forms; (b) solutions which are nontrivial in the space of x-independent local forms can become trivial in the space of x-dependent local forms. It is important to realize that (b) applies also to theories which do not admit solutions of the cohomological problem depending nontrivially on the xµ at all. An important subclass of theories with this property are those which are invariant under spacetime diffeomorphism. They will be discussed in the next subsection in this context. Let me first illustrate (a) and (b) for the simple example of the free D-dimensional Maxwell action Z Smaxwell = dDxFµν F µν , Fµν = ∂µ Aν − ∂ν Aµ . Important examples for x-dependent solutions are the Noether currents associated with the Lorentz invariance of Smaxwell . In “improved" (= gauge invariant) form these currents read µ = λρν xρ T νµ , λρν = −λνρ = constant, jlorentz where T µν = F ρµ Fρ ν − 41 η µν Fρσ F ρσ is the “improved" energy momentum tensor µ cannot be removed by (η µν is the Minkowski metric). The x-dependence of jlorentz subtracting trivial currents from it. This illustrates (a). To demonstrate (b) I consider the x-independent current µ = λν F νµ , jtriv µ It is clearly conserved too, ∂µ jtriv

λν = constant.

≈ 0, and the corresponding (D −1)-form is nontrivial in the space of x-independent forms. However it is trivial in the space of x-dependent forms, as one has µ ≈ ∂ν (λρ xρ F νµ ). jtriv

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This reflects that the global symmetry of Smaxwell which corresponds via Noether’s µ is trivial too [7]: it is the shift symmetry Aµ → Aµ + λµ and is trivial theorem to jtriv because it is just a gauge transformation with parameter λµ xµ . 11.2. Implications of diffeomorphism invariance. As mentioned already, in diffeomorphism invariant theories (of the standard type) one can remove any explicit x-dependence locally from all the solutions of the cohomological problem by subtracting trivial solutions. This was observed and used first in [23] for the antifield independent BRST cohomology in standard gravity. It is instructive to see how this result arises naturally within the framework of Sect. 5. Namely it follows simply from the fact that all the xµ and dxµ form trivial pairs (U ` , V ` ) as a direct consequence of diffeomorphism invariance. To see this, note first that xµ and dxµ indeed satisfy requirement (5.1), s˜ xµ = dxµ . Now, this is valid for any theory but does not imply in general that xµ and dxµ form a trivial pair because to that end (5.2) must hold as well. However, in contrast to other theories (such as Yang–Mills theory), one can usually fulfill this additional requirement in diffeomorphism invariant theories through a simple change of variables (jet coordinates): one just replaces the diffeomorphism ghosts ξ µ with the combinations ξ˜µ = ξ µ + dxµ . Indeed, in standard diffeomorphism invariant theories the s-transformation ˜ of all the fields and antifields11 depends on the ξ µ and dxµ only via ξ˜µ for one has sZ = ξ µ ∂µ Z + ˜ = ξ˜µ ∂µ Z +. . . for any field or antifield Z (the nonwritten . . ., dZ = dxµ ∂µ Z and thus sZ terms in sZ do not contain undifferentiated ξ’s in standard diffeomorphism invariant theories). This reflects that the diffeomorphisms are encoded in the BRST operator through the Lie derivative along ξ and implies that (i) xµ and dxµ indeed form a trivial pair and can thus be eliminated locally from the cohomology, i.e. the nontrivial solutions of s˜ω˜ = 0 can be chosen so as not to depend explicitly on the xµ and to depend on ξ µ and dxµ only via ξ˜µ , (ii) on x-independent functions and local total forms respectively, s and s˜ arise from each other through the replacements ξ µ ↔ ξ˜µ , i.e. s˜ = ρ ◦ s ◦ ρ−1 , where



∂ ρ = exp dx ∂ξ µ µ

s = ρ−1 ◦ s˜ ◦ ρ,

 ,

ρ

−1



∂ = exp −dx ∂ ξ˜µ µ

 .

In particular this implies the now well-known result, first derived in [23], that the descent equations go in standard diffeomorphism invariant theories always down to a BRSTinvariant x-independent 0-form ω0 , and that the “integration" of the descent equations starting from such a 0-form is not obstructed and results in a solution ω˜ = ρω0 of s˜ω˜ = 0. I stress however that this result is valid only in the space of x-dependent forms. Indeed, in the space of x-independent forms there are additional solutions “ω˜ times monomial of the dxµ " where ω˜ is an x-independent solution because in that space dxµ is an “s-singlet". ˜ In particular it is not true that the descent equations go always down to a 0-form if one restricts the cohomological analysis to x-independent forms. 11 Assuming again that antighosts and Nakanishi–Lautrup fields have been eliminated from the cohomological problem already, cf. Sect. 5.

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12. Conclusion The framework proposed in this paper to analyse the local BRST cohomology is based on a few very simple ideas: (i) the formulation of the local BRST cohomology in the jet bundle approach, (ii) the mapping of the BRST cohomology to the cohomology of s˜ = s + d and to its on-shell counterpart, the antifield independent weak cohomology of γ˜ = γ + d, (iii) the construction of contracting homotopies to eliminate certain jet coordinates, called trivial pairs, from the s-cohomology. ˜ In spite of its conceptual simplicity, (iii) is not straightforward because it requires the construction of an appropriate set of local jet coordinates splitting into two subsets, one of which contains the trivial pairs, whereas the other one consists of complementary jet coordinates which are required to generate an s˜ resp. γ-invariant ˜ subalgebra and are interpreted as tensor fields and generalized connections. The existence (and finding) of such complementary jet coordinates is a crucial prerequisite for the elimination of trivial pairs and was shown to be intimately related to a gauge covariant algebra. The construction of such jet coordinates has been illustrated for various examples to demonstrate the proposed method and its large range of applicability. The outlined method simplifies the computation of the BRST cohomology considerably by reducing it locally to a cohomological problem involving only the tensor fields and generalized connections. The simplification does not only consist in the fact that some jet coordinates, the trivial pairs, are eliminated. On top of that, and equally important, one obtains a very compact and useful formulation of the remaining cohomological problem on tensor fields and generalized connections through equations such as (6.4)– (6.6). For specific models the compact formulation of the BRST algebra obtained in this way is in fact well-known in the literature. For instance (6.5) reproduces in the Yang– Mills case the celebrated “Russian formula" (1.1) which was used especially within the algebraic construction and classification of chiral anomalies [9, 22]. It should be remarked that this simplifies the computation of the BRST cohomology, but of course does not solve it. Nevertheless it allows remarkable conclusions about the “geometric” structure and covariance properties of the solutions of the cohomological problem and the related physical quantities (Noether currents, gauge invariant actions, candidate gauge anomalies, etc.), as well as of the classical equations of motion, cf. Sects. 8–10. Finally I remark that local jet coordinates with the mentioned properties are also useful when one needs to take global (topological) aspects into account which have been completely neglected in this paper. In particular, global obstructions to the elimination of trivial pairs may be taken into account using K¨unneth’s theorem a` la [18]. A. Fields, Antifields and Their Jet Space The so-called minimal set of fields contains the “classical fields” φi which occur in the gauge invariant classical action, the ghosts C α corresponding one-to-one to the nontrivial gauge symmetries and the ghosts for ghosts Qαk of first and higher order k = 1, . . . , r, where r denotes the reducibility order of the theory (r = 0 for irreducible gauge theories), {8A }min = {φi , C α , Qαk },

{8∗A }min = {φ∗i , Cα∗ , Q∗αk }.

(A.1)

In order to fix the gauge one usually extends the minimal set of fields to a nonminimal one by adding antighosts and Nakanishi–Lautrup auxiliary fields. Each field 8A has a

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F. Brandt

definite Grassmann parity ε(8A ) and ghost number gh(8A ). The Grassmann parities and ghost numbers of the antifields are related to those of the fields according to gh(φi ) = 0, gh(C α ) = 1, gh(Qαk ) = k + 1, gh(8∗A ) = −gh(8A ) − 1, ε(8∗A ) = ε(8A ) + 1

(mod 2).

(A.2)

The Grassmann parity of the classical fields is 0 for bosonic (commuting) fields and 1 for fermionic (anticommuting) fields, the Grassmann parity of the ghosts is opposite to the Grassmann parity of the corresponding gauge symmetry, and the Grassmann parity of the ghosts for ghosts is determined analogously. Fields and antifields commute or anticommute according to their Grassmann parities, Z1 Z2 = (−)ε(Z1 )ε(Z2 ) Z2 Z1 .

(A.3)

The fields and antifields and all their derivatives are considered as local coordinates of an infinite jet space. For this set of jet coordinates I use the collective notation [8, 8∗ ] ≡ {∂µ1 ...µk 8A , ∂µ1 ...µk 8∗A : k = 0, 1, . . .},

(A.4)

and these jet coordinates are regarded as independent apart from the identities ∂µ1 ...µi ...µj ...µk Z = ∂µ1 ...µj ...µi ...µk Z

∀ i, j.

(A.5)

The derivatives ∂µ have vanishing Grassmann parity and ghost number. The set of local jet coordinates is completed by the spacetime coordinates xµ and by the differentials dxµ which are counted among the jet coordinates by convenience. The former have even, the latter odd Grassmann parity, both have vanishing ghost number. The derivatives ∂µ are defined as total derivative operators in the jet space according to X ∂ ∂ , (A.6) + (∂µν1 ...νk ZI ) ∂µ = µ ∂x ∂(∂ν1 ...νk ZI ) k≥0; νi+1 ≥νi

where {ZI } = {8A , 8∗A }. The sum in (A.6) runs only over those ∂ν1 ...νk Z with νi+1 ≥ νi because of the identities (A.5). It is further understood that ∂(∂12 Z) ∂(∂21 Z) = = 1, ∂(∂12 Z) ∂(∂12 Z)

etc.

(A.7)

B. BRST Operator The BRST operator is constructed from a solution S of the master equation [5] of the form Z S=

dDx L,

i L = Lcl (x, [φ]) − (Rα C α )φ∗i + . . . ,

(B.1)

i C α is where Lcl denotes the Lagrangian of the gauge invariant classical action and Rα i i is the gauge transformation of φ with gauge parameters replaced by the ghosts, i.e. Rα an operator of the form X i (x, [φ], ∂) = rαiµ1 ...µk (x, [φ])∂µ1 . . . ∂µk . (B.2) Rα k≥0

Local BRST Cohomology and Covariance

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If the gauge algebra is reducible with reducibility order r, then L is required to contain also a piece of the form Cα∗ Zαα1 Qα1 +

r X

Q∗αk−1 Zααkk−1 Qαk ,

(B.3)

k=2

where the Z’s are operators of the form (B.2) implementing the reducibility relations. For the purpose of gauge fixing one may also include pieces involving antighosts and Nakanishi–Lautrup auxiliary fields [5]. The BRST transformations of 8A and 8∗A are given by their antibrackets [5] with S according to s · = (S, · ). This results in s 8A = −

∂ˆ R L , ˆ ∗A ∂8

s 8∗A =

∂ˆ R L , ˆ A ∂8

(B.4)

ˆ denotes the Euler–Lagrange right-derivative of L with respect to Z where ∂ˆ R L/∂Z (derivatives ∂ R /∂ act from the right), X ∂ˆ R L ∂RL . (−)k ∂µ1 . . . ∂µk = ˆ ∂(∂µ1 ...µk Z) ∂Z k≥0; µi+1 ≥µi

(B.5)

The BRST transformations of derivatives of the fields and antifields are obtained from (B.4) simply by requiring s∂µ = ∂µ s, i.e. s(∂µ 8A ) = ∂µ (s8A ) = −∂µ

∂ˆ R L , ˆ ∗A ∂8

etc.

(B.6)

Acknowledgement. This work was carried out in the framework of the European Community Research Programme “Gauge theories, applied supersymmetry and quantum gravity", with a financial contribution under contract SC1-CT92-0789.

References 1. Becchi, C., Rouet, A. and Stora, R.: Phys. Lett. 52B, 344 (1974); Commun. Math. Phys. 42, 127 (1975); Ann. Phys. 98, 287 (1976); Tyutin, I.V.: Gauge invariance in field theory and statistical mechanics. Lebedev preprint FIAN, n0 39 (1975) (unpublished) 2. Kallosh, R.E.: Zh. Eksp. Teor. Fiz. Pis’ma 26, 575 (1977); Nucl. Phys. B141, 141 (1978) 3. Sterman, G., Townsend, P.K. and van Nieuwenhuizen, P.: Phys. Rev. D14, 1501 (1978) 4. de Wit, B. and van Holten, J.W.: Phys. Lett. 79B, 389 (1978) 5. Batalin, I.A. and Vilkovisky, G.A.: Phys. Lett. 102B, 27 (1981); Phys. Rev. D28, 2567 (1983) (E: D30, 508 (1984)) 6. Barnich, G. and Henneaux, M.: Phys. Lett. 311B, 123 (1993) 7. Barnich, G., Brandt, F. and Henneaux, M.: Commun. Math. Phys. 174, 57 (1995) 8. Wess, J. and Zumino, B.: Phys. Lett. 37B, 95 (1971) 9. Stora, R.: Continuum Gauge Theories. In: New Developments in Quantum Field Theory and Statiscal Physics eds. M. Levy and P. Mitter, NATO ASI Series B26 New York: Plenum, 1977; Algebraic Structure and Topological Origin of Anomalies. In: Progress in Gauge Field Theory, Cargese Lectures 1983, eds. G. ’t Hooft et al. New York: Plenum, 1984; Zumino, B.: Chiral Anomalies and Differential Geometry. In: Relativity, Groups and Topology II, Les Houches Lectures 1983, eds. B. S. De Witt and R. Stora Amsterdam: North-Holland, 1984;

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Bardeen, W.A. and Zumino, B.: Nucl. Phys. B244, 421 (1984); Zumino, B., Wu, Y.S. and Zee, A.: Nucl. Phys. B239, 477 (1984); Baulieu, L.: Nucl. Phys. B241, 557 (1984); Phys. Rep. 129, 1 (1985); Ma˜nes, J., Stora, R. and Zumino, B.: Commun. Math. Phys. 102, 157 (1985) Howe, P.S., Lindstr¨om, U. and White, P. . Phys. Lett. 246B, 430 (1990); Troost, W., van Nieuwenhuizen, P. and Van Proeyen, A.: Nucl. Phys. B333, 727 (1990) Piguet, O. and Sorella, S.P.: Algebraic Renormalization. Lecture Notes in Physics Vol. m28, Berlin– Heidelberg–New York: Springer-Verlag, 1995 and refs. therein Gomis, J., Par´ıs, J. and Samuel, S.: Phys. Rep. 259, 1 (1995) and refs. therein Gomis, J. and Weinberg, S.: Nucl. Phys. B469, 473 (1996) Baulieu, L. and Thierry-Mieg, J.: Nucl. Phys. B197 477 (1982) Brandt, F.: Structure of BRS-invariant local functionals. Preprint NIKHEF-H 93-21, hep-th/9310123 (unpublished) Dragon, N.: BRS symmetry and cohomology. Saalburg Lectures, Preprint ITP–UH–3/96, hep-th/9602163 (unpublished) Barnich, G., Brandt, F. and Henneaux, M.: Phys. Rev. D51, 1435 (1995) Barnich, G., Brandt, F. and Henneaux, M.: Nucl. Phys. B455, 357 (1995) Hirsch, G.: Bull. Soc. Math. Belg. 6, 79 (1953); Stasheff, J.D.: Trans. Am. Math. Soc. 108, 215, 293 (1963); Gugenheim, V.K.A.M.: J. Pure Appl. Alg. 25, 197 (1982); Gugenheim, V.K.A.M. and Stasheff, J.D.: Bull. Soc. Math. Belg. 38, 237 (1986) Fisch, J.M.L., Henneaux, M., Stasheff, J. and Teitelboim, C.: Commun. Math. Phys. 127, 379 (1989) Fisch, J.M.L. and Henneaux, M.: Commun. Math. Phys. 128, 627 (1990); Henneaux, M.: Nucl. Phys. B (Proc. Suppl.) 18A, 47 (1990); Henneaux, M. and Teitelboim, C.: Quantization of Gauge Systems. Princeton: Princeton University Press, 1992 Dubois-Violette, M., Talon, M. and Viallet, C.M.: Phys. Lett. 158B, 231 (1985); Commun. Math. Phys. 102, 105 (1985) Brandt, F., Dragon, N. and Kreuzer, M.: Nucl. Phys. B340, 187 (1990) Dubois-Violette, M., Henneaux, M., Talon, M. and Viallet, C.M.: Phys. Lett. 289B, 361. (1992) Brandt, F., Dragon N. and Kreuzer, M.: Phys. Lett. 231B, 263 (1989); Nucl. Phys. B332, 224 (1990) Barnich, G. and Henneaux, M.: Phys. Rev. Lett. 72, 1588 (1994) Barnich, G., Brandt, F. and Henneaux, M.: Commun. Math. Phys. 174, 93 (1995) Brandt, F.: Ann. Phys. (N.Y.) 259, 253 (1997) Anderson, I.M.: The variational bicomplex. Boston: Academic Press, 1994; Contemp. Math. 132, 51 (1992); Saunders, D.J.:The Geometry of Jet Bundles. London Mathematical Society Lecture Note Series 142 Cambridge: Cambridge University Press, 1989 Vinogradov, A.M.: Sov. Math. Dokl. 18, 1200 (1977) 19, 144 (1978), 19, 1220 (1978); Takens, F.: J. Diff. Geom. 14, 543 (1979); Anderson, I.M. and Duchamp, T.: Am. J. Math. 102, 781 (1980); De Wilde, M.: Lett. Math. Phys. 5, 351 (1981); Tulczyjew, W.M.: Lecture Notes in Math. 836, 22 (1980); Dedecker, P. and Tulczyjew, W.M.: Lecture Notes in Math. 836, 498 (1980); Tsujishita, T.: Osaka J. of Math. 19, 311 (1982); Bonora, L. and Cotta-Ramusino, P.: Commun. Math. Phys. 87, 589 (1983); Olver, P.J.: Applications of Lie Groups to Differential Equations. Graduate Texts in Mathematics, volume 107, New York: Springer-Verlag, 1986; Wald, R.M.: J. Math. Phys. 31, 2378 (1990); Dubois-Violette, M., Henneaux, M., Talon, M. and Viallet, C.M.: Phys. Lett. 267B, 81 (1991); Dickey, L.A.: Contemp. Math. 132, 307 (1992); cf. also first ref. in [9] Brandt, F., Henneaux, M. and Wilch, A.: Phys. Lett. 387B, 320 (1996); Ward identities for rigid symmetries of higher order. UB–ECM–PF 96/18, ULB–TH 96/17, hep-th/9611056 (unpublished) Vandoren, S. and Van Proeyen, A.: Nucl. Phys. B411, 257 (1994) Brandt, F., Troost, W. and Van Proeyen, A.: Nucl. Phys. B464, 353 (1996)

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34. Freedman, D.Z., van Nieuwenhuizen, P. and Ferrara, S.: Phys. Rev. D13, 3214 (1976); Freedman, D.Z. and van Nieuwenhuizen, P.: Phys. Rev. D14, 912 (1976); Deser, S. and Zumino, B.: Phys. Lett. 62B, 335 (1976) 35. Henneaux, M.: Commun. Math. Phys. 140, 1 (1991) 36. Bryant, R.L. and Griffiths, P.A. Characteristic Cohomology of Differential Systems (I): General Theory. Duke University Mathematics Preprint Series, volume 1993 n0 1, January 1993 37. Brandt, F.: Lagrangian densities and anomalies in four-dimensional supersymmetric theories. Doctoral thesis (in German), RX–1356 unpublished, Hannover, 1991; Class. Quantum Grav. 11, 849 (1994) 38. Stelle, K.S. and West, P.C.: Phys. Lett. 74B, 330 (1978); Ferrara, S. and van Nieuwenhuizen, P.: Phys. Lett. 74B, 333 (1978) 39. Sohnius, M.F. and West, P.C.: Phys. Lett. 105B, 353 (1981) Communicated by R. H. Dijkgraaf

Commun. Math. Phys. 190, 491 – 508 (1998)

Communications in

Mathematical Physics c Springer-Verlag 1998

On the Distribution of Free Path Lengths for the Periodic Lorentz Gas Jean Bourgain1 , Franc¸ois Golse2 , Bernt Wennberg3 1 2 3

Institute for Advanced Study, School of Mathematics, Princeton NJ 08540, USA Universit´e Paris VII & Ecole Normale Sup´erieure, D.M.I., 45 rue d’Ulm, 75005 Paris, France Chalmers University of Technology, Department of Mathematics, 41296 G¨oteborg, Sweden

Received: 1 March 1996 / Accepted: 25 March 1997

Abstract: Consider the domain Zε = {x ∈ Rn | dist(x, εZn ) > εγ }, and let the free path length be defined as τε (x, ω) = inf{t > 0 | x − tω ∈ Zε } . The distribution of values of τε is studied in the limit as ε → 0 for all γ ≥ 1. It is shown n is critical for this problem: in other words, the limiting behavior that the value γc = n−1 of τε depends only on whether γ is larger or smaller than γc . 1. Introduction The Lorentz gas is a model system of Statistical Mechanics consisting of a large number of like point particles moving freely in a domain of the space where spherical obstacles are disposed with some given distribution. Collisions between two (or more) particles are rare events since these particles have diameter 0. Hence, only collisions involving one particle and one obstacle are taken into account. They are described by some adequate reflection law, the exact nature of which will be of no significance in the present work; the most classical example of such reflection law is of course the case of “specular reflection”. The model considered in the present work is the case where the obstacles are periodically distributed; in other words, the centers of the obstacles form a lattice in the space Rn , which, for simplicity, is assumed to be homothetic to Zn . Finally, each particle is assumed to move with speed 1 in the interval of time between two consecutive collisions with the obstacles. It is the purpose of the present work to study some aspects of the large scale dynamics of such a system. Thus, let n ∈ N∗ denote the space dimension and let

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Zε = {x ∈ Rn | dist(x, εZn ) > εγ } ,

(1.1)

for all 0 < ε < 21 and γ ≥ 1. The “free path length” (or equivalently “exit time”, since the particles move with speed 1 between two consecutive collisions with the obstacles) is defined as follows, for all x ∈ Z ε and ω ∈ S n−1 : τε (x, ω) = inf{t > 0 | x + tω ∈ ∂Zε } .

εγ

(1.2)

ε

Fig. 1. The billiard table

Clearly τε is a Borelian function for all 0 < ε < 21 and all γ ≥ 1. The present paper studies the distribution of values of τε as ε → 0, which is one of the main features of the evolution of the Lorentz gas model associated to the domain Zε as explained above. However, this problem is well posed only after a phase space equipped with a Borelian probability measure is defined. The most natural choice in this respect is the following one. Let Yε = Zε /εZn : topologically Yε is a punctured torus; let Qε = dxdω−meas (Yε × S n−1 ). Our choice of a phase space is Yε × S n−1 with the Borelian probability measure µε defined by 1 dxdω . (1.3) dµε (x, ω) = Qε Clearly τε (x + εk, ω) = τε (x, ω) for all (x, ω) ∈ Z ε × S n−1 and all k ∈ Zn so that τε defines a Borelian function on Yε × S n−1 . It is then natural to study the distribution of τε with respect to the probability measure µε . We recall its definition: Definition. The distribution φε of τε with respect to µε is the push-forward of the measure µε under τε . In other words, φε is the unique Borelian probability measure on [0, +∞[ such that, for all 0 < a < b < +∞, φε (]a, b[) = µε ({(x, ω) ∈ Yε × S n−1 | a < τε < b}) .

(1.4)

Distribution of Free Path Lengths for Periodic Lorentz Gas

493

The main results in this paper bear on the limiting behavior of φε as ε → 0 and on how it depends on the parameter ε. These results are presented without proof in the next section (Sect. 2). The proofs are relegated to the subsequent sections (Sects. 3 to 5). We shall conclude this section with a very elementary observation. In the case where particles impinging on the obstacles are specularly reflected, it is natural to consider the map which, to the position and velocity of any particle leaving the boundary of some obstacle associates its position and velocity immediately after the next collision with an obstacle. It is defined by (x, ω) 7→ (x0 = x + τε (x, ω)ω; ω 0 = ω − 2ω · n(x0 ) n(x0 )),

(1.5)

where n(x) denotes the inward unit normal at point x ∈ ∂Zε . Let then Σε+ = {(x, ω) ∈ ∂Yε × S n−1 | ω · n(x) > 0} .

(1.6)

Since any two obstacles in Zε are congruent modulo εZn , the map (1.5) defines a map B : Σε+ → Σε+ (sometimes called the billiard map: see for example [Ch1-2]). Let 0ε = ω · nx dS(x)dω − meas (Σε+ ); a Borelian probability measure νε is defined on Σε+ by 1 ω · nx dS(x)dω . (1.7) dνε (x, ω) = 0ε The probability measure νε is invariant under B, and hence a second choice of a phase space for the Lorentz gas is Σε+ equipped with the probability measure νε , the dynamics being given by the iterates of the billiard map B. This is usually the phase space and dynamics studied in most of the literature devoted to billiards (see [Ch1-2] and the references therein). The first phase space (Yε × S n−1 , µε ) is the suspension of (Σε+ , νε ) under the function τε and the Lorentz gas flow mod. εZn (i.e. on Yε × S n−1 ) is the suspension flow of the map B under the function τε . In [Ch1-2], the following quantity, called the “geometric mean free path” in [DDG2], is considered: Z τε (x, ω)dνε (x, ω) . (1.8) lε = Σε+

As explained in [Ch2] (Sect. 2), it is a natural notion of mean free path because it is the time average of free paths lengths along typical trajectories whenever the map B is ergodic. There is an explicit formula for it, (see [Ch1] Sect. 3.2 or [DDG2] for a quick proof): Qε 1 εn−γ(n−1) + O(εγ ) . = (1.9) lε = 0ε |B n−1 | This formula clearly points at the special value γc =

n n−1

(1.10)

as being critical. Indeed, as ε → 0, • if γ > γc , lε → +∞ as ε → 0, which seems to indicate a purely ballistic behavior for the Lorentz gas; • if 1 ≤ γ < γc , lε → 0 as ε → 0, corresponding to a hydrodynamic limit; • if γ = γc , lε → |B n−1 |−1 > 0 as ε → 0, corresponding to the so-called “BoltzmannGrad limit”.

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However, it does not seem possible to extract any information about the distribution of free path lengths φε defined in (1.4), which is our main object of study here, from the explicit formula (1.9). This simply reflects the fact that the billiard under consideration in this paper does not have the “finite horizon property” (the function τε is not uniformly bounded on Σε ) and hence the first phase space (Yε × S n−1 , µε ) contains more information than the second phase space (Σε+ , νε ). Let us close this introductory section with some references. In the case γ = 1, Bunimovich, Sinai and later Chernov ([BS1-2, BSC1-2]) established the diffusion limit for the Lorentz gas with finite horizon. If the specular reflection condition is replaced by an accommodation reflection condition, a simpler proof, based on PDE methods, leads to a similar diffusion limit: see [BDG]. The Boltzmann-Grad limit (γ = γc ) has been studied by many authors, in the case where the distribution of obstacles is not periodic as considered here but random: see [Gal, Sp, BBS]. These papers prove that the limiting number density f of gas particles satisfies a linear transport equation with absorption and scattering of the form Z k(ω, ω 0 )f (t, x, ω 0 )dω 0 . (1.11) ∂t f (t, x, ω) + ω · ∇x f (t, x, ω) + σf (t, x, ω) = σ S n−1

The methods developed in these papers do not apply to the periodic case under consideration in this paper. In fact the limiting behavior of the periodic Lorentz gas in the critical scaling γ = γc is qualitatively different from the one described by (1.11): see Sect. 2, Remark 2.

2. Main Results With the definitions and notations of Sect. 1, we first state the main theorem in this paper: Theorem A. 1) If γ > γc , φε → 0 vaguely as ε → 0; 2) If 1 ≤ γ < γc , φε → δ0 weakly as ε → 0; 3) If γ = γc , any vague limit point φ of the family (φε ) is a probability measure and satisfies lim sup tφ([t, +∞[) < +∞ ; t→+∞

4) If γ = γc and n = 2, any vague limit point φ of the family (φε ) satisfies lim inf tφ([t, +∞[) > 0 . t→+∞

We recall the terminology for the various topologies on the space of Borelian probability measures on R+ (see [Bil]). The weak topology is the topology defined by the family of seminorms µ 7→ |hµ, f i| for all bounded continuous f ’s, while the vague topology is the one defined by the subfamily of these same seminorms corresponding to continuous f ’s with compact support. Point 1) in Theorem A was proved in [DDG2] (see [G, DDG1] for an alternative proof). Point 2] was essentially proved in [DDG1] (although stated in a different manner there; see [DDG2]) when n = 2. It then remains to prove point 2) for all n > 2 and points 3) and 4).

Distribution of Free Path Lengths for Periodic Lorentz Gas

495

Remark 1. When n > 2 and γ = γc , we can prove that lim inf tn−1 φ([t, +∞[) > 0 , ε→0

but we don’t know whether this or point 3) in Theorem A is optimal: see [GW]. Remark 2. Point 4) in Theorem A or Remark 1 show the difference between the limiting dynamics of the Lorentz gas in the periodic and the random cases. In the random cases studied in [Gal, Sp and BBS], the limiting number density is proved to satisfy an equation of the type (1.11); in particular, the free path length is exponentially distributed (σ being the parameter in the exponential law). In the periodic case, the distribution of free path lengths has only algebraic decay, as shown by Theorem A 4) or Remark 1. Theorem A depends essentially on the following technical estimates. Before stating them, we need some notations. Let r ∈]0, 1/2[ and consider Z = {x ∈ Rn | dist (x, Zn ) > r} ; Q = dxdω − meas (Y × S n−1 ) ;

Y = Z/Zn ;

dµ(x, ω) =

τ (x, ω, r) = inf{t > 0 | x + tω ∈ ∂Z} ;

1 dxdω ; Q

T (ω, r) = sup τ (x, ω, r) . x∈Y

(2.1) (2.2) (2.3)

Clearly τ is Zn -periodic and can be considered either as defined for x ∈ Z or for x ∈ Y . Remark 3. T (ω, r) is the quantity referred to as the “ergodization time” in [D1]. With these definitions and notations, we can state Theorem B. For all n ∈ N∗ there exists C(n) > 0 such that dω − meas ({ω ∈ S n−1 | T (ω, r) > t}) ≤

C(n) . rn−1 t

(2.4)

This estimate is sharp in the case n = 2: indeed Theorem C. Let n = 2. There exists C 0 > 0 such that, for all t > r1 . µ({(x, ω) ∈ Y × S 1 | τ (x, ω, r) > t}) ≥

C0 . rt

(2.5)

Remark 4. Theorem B shows that τ (·, ·, r) ∈ L∞ (Y ; L1,∞ (S n−1 )). Theorem C shows that, at least if n = 2, τ (·, ·, r) ∈ / L1 (Y × S n−1 ). Hence the mean free path in the sense of the first phase space considered in Sect. 1 (that is, (Y × S n−1 , µ)), defined as Z Y ×S n−1

τ (x, ω, r)dµ(x, ω) = +∞

does not contain any information on the Lorentz gas, being infinite for all r ∈]0, 1/2[.

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J. Bourgain, F. Golse, B. Wennberg

As an aside result, we improve an upper bound for T due to H.S. Dumas [D1]. We first recall the notations for diophantine vectors: for all K > 0, s ∈ R, let D(s, K) = {ω ∈ S n−1 | ∀k ∈ Zn \ {0} , |ω · k| ≥ K|k|−s } . We recall that

∀K > 0 , ∀s < n − 1 ,

D(s, K) = ∅ ,

(2.6) (2.7)

(which is a variant of a result due to Dirichlet, see [Ca] chapter I, Theorem VI), and that ∀s > n − 1 ,

dω − meas (D(s, K)c ) = O(K) .

(2.8)

Theorem D. For all n ∈ N∗ and s > n − 1, there exists C(n, s) > 0 such that, for all K > 0 and all ω ∈ D(s, K), T (ω, r) ≤

C 00 (n, s) . Krs

We refer to [D2], [ChGa] for an application of this type of estimate. 3. Proof of Theorem B Formulation of the ergodization time problem. For all x ∈ R, let kxk = inf k∈Z |x − k|. Let (3.1) F = {ω ∈ S n−1 | ∀1 ≤ i ≤ n , ωn ≥ |ωi |} ; later, we shall need the following mapping: A : F → [−1, 1]n−1 ,



ω 7→ A(ω) =

ωi ωn

 .

(3.2)

1≤i≤n−1

Let  ∈ [−1, 1]n−1 and R ∈]0, 1/2[; define N (, R) as the smallest positive integer N such that ∀z ∈ [0, 1]n−1 ,

min

max kzi − li k ≤ R .

l∈Z , |l|≤N 1≤i≤n−1

Clearly, if  ∈ A(F ) and if N ≥ N (, R), one has

ωi ωi n−0

, min max −l xi − x n ≤ R. ∀x ∈ [0, 1] ωn ωn l∈Z , |l|≤N 1≤i≤n−1 √ If ω ∈ F , then ωn ≥ √1n . Therefore, if T ≥ n(N + 1), ∀x ∈ [0, 1]n−0 , by specializing t to be of the form t = ∀ω ∈ F ,

min max kxi − tωi k ≤ R ,

|t|≤T 1≤i≤n

(3.3)

(3.4)

(3.5)

xn +l ωn .

This argument shows that   √ r T (ω, r) ≤ 2 nN A(ω), √ . n

(3.6)

on Rn−1 supported in [−1, 1]n−1 , positive Let φ be a nonnegative C ∞ function P n−1 and let φR (z) = k∈Zn−1 φ z+k on ] − 1, 1[ R for all R ∈]0, 1/2[. Let (σl )l∈Z be a

Distribution of Free Path Lengths for Periodic Lorentz Gas

497

nonnegative doubly infinite sequence such that σl > 0 if and only if |l| < N . Then N ≥ N (, R) if and only if X σl φR (z − l) > 0 . (3.7) ∀z ∈ [0, 1]n−1 , SN (z) = l∈Z

For Sn (z) = 0 if and only if φR (z − l) = 0 for all l ∈ Z such that |l| ≤ N ; obviously φR (z − l) = 0 if and only if max1≤i≤n−1 kzi − li k > R. It is then convenient to express SN in terms of the Fourier coefficients of φR : X X σl e−i2πlhξ,i , (3.8) φbR (ξ)ei2πhξ,zi ∀z ∈ [0, 1]n−1 , SN (z) = |l|≤N

ξ∈Zn−1

In particular, if one takes σl = (1 − |l| N ) for |l| ≤ N and σl = 0 if |l| > N , the inner sum in (3.8) is a Fejer kernel, that is to say X (3.9) φbR (ξ)ei2πhξ,zi FN (hξ, i) , SN (z) = ξ∈Zn−1

with

X

FN (z) =

σl e−i2πlz =

|l|≤N

1 sin2 πN z . N sin2 πz

(3.10)

Suppose now that N ≤ N (, R); then there exists z0 ∈ [0, 1]n−1 such that SN (z0 ) = 0. Hence X φbR (ξ)ei2πhξ,z0 i FN (hξ, i) , φbR (0)FN (0) = − ξ∈Zn−1 \{0}

which implies N Rn−1 ≤

X 1 |φbR (ξ)|FN (hξ, i) . ˆ φ(0) ξ∈Zn−1 \{0}

(3.11)

Using the Fejer kernel as above is reminiscent of [M] (chapter 5, Theorem 9). The weak L1 type estimate. We now come to the main result of this section, Theorem B’ below. It is a slight generalization of Theorem B to the case where the probability measure on S n−1 is not the normalized Lebesgue measure. Let m be a probability measure on S n−1 . We assume the existence of 0 < c ≤ 1 and K > 0 such that (H)

m∗ (r) = sup m({α ∈ S n−1 | |hα, ei| ≤ r}) ≤ Krc . e∈S n−1

Obviously, the Lebesgue measure on S n−1 satisfies (H) with c = 1. Theorem B’. B’. Let m be a probability measure on S n−1 satisfying the assumption (H) above. Then there exists a constant C(m, n) > 0 (depending only on the dimension n and the measure m) such that m({ω ∈ S n−1 | T (ω, r) > t}) ≤

C(m, n) . tc rn−c

498

J. Bourgain, F. Golse, B. Wennberg

Proof. Let us first restrict our attention to ω ∈ F . This can be done without loss of generality: indeed, S n−1 can be covered by the images of F under finitely many elements of the orthogonal group On (R); moreover, if a probability measure satisfies (H), its pushforward under an element of the orthogonal group still satisfies (H). If ω ∈ F and N ≤ N (A(ω), R), then  = A(ω) must satisfy (3.11)). Hence, applying Chebyshev’s inequality shows that √ √ m({ω ∈ F | T (ω, nR) ≥ 2 nN }) ≤     X 1 m ω ∈ F | |φbR (ξ)|FN (hξ, A(ω)i) ≥ N Rn−1  ˆ   φ(0) ξ∈Zn−1 \{0} Z X 1 b |φR (ξ)| FN (hξ, A(ω)i)dm(ω) . (3.12) ≤ ˆ N Rn−1 φ(0) F ξ∈Zn−1 \{0} This shows that ≤

1 ˆ N Rn−1 φ(0)

√ √ m({ω ∈ F | T (ω, nR) ≥ 2 nN }) Z X |φbR (ζ)| · sup FN (z)dmξ (z) , ζ∈Zn−1 \{0}

ξ∈Zn−1 \{0}

(3.13)

T1

where, for any measurable subset U of T1 , mξ (U ) = m({ω ∈ F | hA(ω), ξi ∈ U mod. Z}) .

(3.14)

In other words, mξ is the push-forward of m under the map F → T1 defined by ω 7→ hA(ω), ξi mod. Z =

n−1 1 X αi ξi mod. Z . αn i=1

We shall appeal to the next lemma to estimate the integrals appearing in the right-hand side of (3.13). Lemma 1. Let m be a probability measure on S n−1 satisfying (H), and let mξ be associated to m as in (3.14). Then there exists a positive constant C0 (m, n) depending only on the dimension n and the measure m such that Z FN (z)dmξ (z) ≤ C0 (m, n)N 1−c |ξ|1−c . (3.15) 0≤ T1

We defer the proof of Lemma 1 to after that of Theorem B’. It follows from (3.15) and the estimate (3.12) that m({ω ∈ F | T (ω,

√

√ C0 (m, n) 1 nR) ≥ 2 nN }) ≤ c n−1 N R φb(0)

X

|φbR (ξ)||ξ|1−c .

ξ∈Zn−1 \{0}

(3.16) But then, the function φ being smooth, one has, for all l > 0, the existence of Kl > 0 such that (3.17) |φbR (ξ)| ≤ Kl Rn−1 (1 + |Rξ|)−l . Hence, choosing l > n − c and observing that

Distribution of Free Path Lengths for Periodic Lorentz Gas

Rn−1

X

|Rξ|1−c (1 + |Rξ|)−m ∼

Z Rn−1

ξ∈Zn−1

499

|x|1−c (1 + |x|)−m dx < +∞,

(3.18)

we obtain that m({ω ∈ F | T (ω,

√

√ C 0 (m, n) nR) ≥ 2 nN }) ≤ c n−c N R 

which completes the proof of Theorem B’.

Proof of Lemma 1. We proceed as in [GLPS] Z Z Z C1 1−δ dmξ (z) FN (z)dmξ (z) ≤ N dmξ (z) + I= N δ z 2 (1 − z)2 T1 kzk≤δ

(3.19)

with C1 = supz∈[0,1] z 2 (1 − z)2 / sin2 πz. Then, the definition (3.14) and the assumption (H) on the measure m show that mξ ({z ∈ T1 | kzk ≤ δ}) ≤

X √ |k|≤ n|ξ|+1

X

≤

√

K

n−1 X

ωi ξi − kωn | ≤ ωn δ})

i=1

!c

δωn

p

|k|≤ n|ξ|+1

m({ω ∈ F | |

|ξ|2 + k 2

≤ K 0 δ c |ξ|1−c ,

(3.20)

for some K 0 > 0. Hence, I ≤ K20 N δ c |ξ|1−c +

2C1 N

Z

1−δ δ



1 1 + 2 z (1 − z)2

 dmξ (z) .

(3.21)

Then, integrating by parts and using (H) leads to Z

1 δ

 Z z Z z 1 Z 1  dmξ (z) 2 1 = 2 dmξ (t) + dmξ (t) dz ≤ C2 |ξ|1−c δ c−2 . (3.22) 3 z2 z 0 δ z 0 δ

Proceeding in the same manner with the other integral in the right hand side of (3.21) leads to (3.23) I ≤ K20 N δ c |ξ|1−c + C3 N −1 δ c−2 |ξ|1−c . Optimizing in δ leads to the choice of δ = 1/N and hence, I ≤ C0 (m, n)N 1−c |ξ|1−c as announced.

(3.24)



The following bound for averages of sufficiently small powers of the ergodization time follows from Theorem B’ by using a classical interpolation argument. Corollary B”. Under the assumptions of Theorem B’, one has, for all 0 < β < 1 Z 2C(m, n)β T (ω, r)cβ dm(ω) ≤ . (1 − β)rβ(n−c) S n−1

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4. Proof of Theorem C In this section, only the case of n = 2 is considered. Let r ∈]0, 1/2[; the notations Z, Y , µ and τ are as in (2.1)-(2.3). A unit vector ω ∈ R2 will be called irrational if and only if ω1 /ω2 ∈ R \ Q. Definition. An open strip S of R2 of width l is a subset of R2 which can be mapped onto R×]0, l[ with l > 0 by a displacement D (i.e. a rotation composed with a translation). The middle third of S is the open strip of R2 which the same displacement D maps onto R×] 13 l, 23 l[. The boundary ∂S consists of two parallel straight lines whose direction is determined by a unit vector V of R2 ; ±V will be called the direction of the strip S. Channels. Definition. A channel in Z is an open strip included in Z of maximal width. The idea of considering channels in the context of the periodic Lorentz gas seems to be due to Bleher [Bl] (who used instead the term “corridor” ) — see also [Da]. It is well-known that, if ω ∈ S 1 is irrational, for all x ∈ R2 , the set x + Rω + Z2 is dense in R2 . Hence a channel in Z must have a rational direction. For, if C is a channel in Z with direction ω, any point x ∈ C must satisfy the condition x + Rω ⊂ Z, implying that dist (Z2 , x + Rω) > r, which is obviously not satisfied if x + Rω + Z2 is dense in R2 . However any rational unit vector is not necessarily a direction of a channel in Z as shown by the next lemma. Lemma 2. Let (p, q) ∈ Z2 \ {0} with p and q coprime, and let ω0 = √

1 (p, q). p2 +q 2

A

necessary and sufficient condition for a channel of direction ω0 to exist is that p

p2 + q 2 <

1 . 2r

(4.1)

The set of all such unit vectors is denoted by Ar . If ω0 ∈ Ar , the width of any channel of direction ω0 is 1 W (ω0 , r) = p − 2r . (4.2) 2 p + q2 Proof. Let S be a channel with direction ω0 included in R2 \ Z2 ; its boundary ∂S is the union of two parallel lines L and L0 , each of which contains infinitely many integer points. Let qx − py = a be an equation for L and qx − py = a0 an equation for L0 . Since both L ∩ Z2 and L0 ∩ Z2 are non-empty, both a and a0 belong to pZ + qZ = Z since p and q are coprime. Then |a − a0 | 1 dist (L, L0 ) = p ∈p N∗ . 2 2 2 p +q p + q2

(4.3)

But since S must not contain any integer point, the distance between L and L0 must be minimal among the distances between lines of direction ω0 containing infinitely many integer points, which means that 1

dist (L, L0 ) = inf p

p2

+

q2

1 N∗ = p . 2 p + q2

(4.4)

Distribution of Free Path Lengths for Periodic Lorentz Gas

501

The same argument shows that if L and L0 are two distinct parallel lines tangent to ∂Z, ! 1 1 0 ∗ dist (L, L ) ∈ p N ∪ p − 2r N∗ . (4.5) p2 + q 2 p2 + q 2 If C is now a channel in Z, its width is 1

∗

inf p N ∪ p2 + q 2

! 1

p − 2r N∗ = W (ω0 , r) p2 + q 2

as predicted by (4.2). Since a channel in Z has positive width, p and q must satisfy (4.1) if there exists a channel with direction ω0 . Conversely it is easily seen that if L and L0 are two distinct parallel lines tangent to ∂Z with dist (L, L0 ) minimal, then L and L0 define an open channel in Z. 

( p 2 + q 2 ) -1/2

α0

( p 2 + q 2 ) -1/2 -R

R

Fig. 2. Channels in OR with (p, q) = (2, 5)

The Lower Bound for the Tail of the Distribution of Exit Times. The main result of this section is the Theorem C’. There exists a positive constant C such that for all A >> 1, 0 < r < 1 and t > 1/r, dxdω-meas ({(x, ω) ∈ Z × S 1 | |x| <

√1 A 2

and τ (x, ω, r) ≥ t}) ≥ C

A2 . rt

Proof. Let ω0 ∈ Ar , and consider a channel C with direction ω0 . Then let C 0 be the middle third of C. The forward or backward trajectory of any point belonging to C 0 in a direction ω ∈ S 1 cannot exit C in time less than t provided that ω belongs to an arc of S 1 centered at α0 and of length !! ! 2 1 1 1 p p θ0 = 2 arcsin − 2r ≥ − 2r . (4.6) 3t 3t p2 + q 2 p2 + q 2

502

J. Bourgain, F. Golse, B. Wennberg


Therefore the set E 0 (α0 ) = C 0 ∩ {x | |x · ω0 | < 21 A} × ]ω0 − θ0 /2, ω0 + θ0 /2[ (i.e., only segments of C 0 of length A are considered) has the following properties:

and

if (x, ω) ∈ E 0 (ω0 ), then τ (x, ω, r) ≥ t;

(4.7)

dxdω − meas (E 0 (ω0 )) = A · 13 W (ω0 , r) · θ0 .

(4.8)

A

θ0 /2 θ0 /2

T

Fig. 3. Construction of the set E(ω0 )

Given ω0 , there are infinitely many strips C 0 , (and correspondingly E 0 (ω0 )’s), all being Zn -translates of each other; consider next a square Q ⊂ R2 of side A >> 1 centered at the origin, with one side parallel to the direction ω0 , and define ! [ 0 E (ω0 ) + k ∩ (Q × S 1 ) , (4.9) E(ω0 ) = k∈Zn

(that is, the union is taken over all such translates). Let N (A, ω0 , r) be the number of channels of direction ω0 intersecting with the square Q; since p N (A, ω0 , r) ≥ 41 A p2 + q 2 , (4.10) any set E(ω0 ) corresponding to ω0 ∈ Ar satisfies dxdω − meas (E(ω0 )) ' N (A, ω0 , r) · A · 13 W (ω0 , r) · θ0 ≥ A2 m(p, q, r) with

p 1 m(p, q, r) = (1 − 2r p2 + q 2 ) 18t

(4.11)

! 1

p − 2r p2 + q 2

,

(4.12)

according to the inequalities (4.6) and (4.10). The result will now follow by summing over all ω0 ∈ Ar , at least if it can be established that the corresponding sets E 0 (ω0 ) are disjoint. To this end, consider another

Distribution of Free Path Lengths for Periodic Lorentz Gas

rational direction ω1 = √

1 (p0 , q 0 ) p02 +q 02

503

∈ Ar . The angle between ω0 and ω1 is given by

the expression arcsin

!

|qp0 − pq 0 | p p p2 + q 2 p02 + q 02

! 2r

p

≥ arcsin

p2 + q 2

.

(4.13)

Thus, for t > 1/3r, the arc of S 1 centered at ω0 and of length θ0 cannot intersect the arc of S 1 of the same length centered at ω1 , for any rational direction ω1 ∈ Ar different from ω0 . Now, if one varies the direction ω0 in the class Ar , it follows that [

E(ω0 ) ⊂ {(x, ω) ∈ Z × S 1 | |x| <

ω0 ∈Ar

√1 A 2

and τ (x, ω, r) ≥ t},

(4.14)

and the left side of the inclusion above is a finite disjoint union. Hence dxdα-meas({(x, ω) ∈ Z × S 1 | |x| < X

≥

√1 A 2

and τ (x, ω, r) ≥ t})

dxdω − meas (E(ω0 )) ≥ S

(4.15)

(4.16)

ω0 ∈Ar

with

X

S=

A2 m(p, q, r) .

(4.17)

(p, q) ∈ B(0, G.C.D.(p, q) = 1 1 4r )

Observe that if p2 + q 2 ≤

1 16r 2 ,

then

m(p, q, R) ≥ so that S≥

A2 72t

1 1 p 72t p2 + q 2

X

(4.18)

1

p

1 ) \ {0} (p, q) ∈ B(0, 4r G.C.D.(p, q) = 1

p2

+ q2

.

(4.19)

We interrupt the course of the proof to recall the following very simple Lemma 3. Let f be a homogeneous function of degree 0. Then X (p, q) ∈ B(0, ρ) \ {0} G.C.D.(p, q) = 1

"

ρ

#

f (p, q) p p2 + q 2

(we recall the notation [·] for “integer part of”).

=

X (p,q)∈B(0,ρ)\{0}

f (p, q)

(4.20)

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J. Bourgain, F. Golse, B. Wennberg

Applying this to the case where f is the constant function equal to 1, one sees that X

p

\ {0} (p, q) ∈ B(0, G.C.D.(p, q) = 1 1 4r )

1 p2

+ q2

1 ≥ 4r · ]{(p, q) ∈ B(0, 4r ) \ {0}} .

(4.21)

Putting all this together shows that dxdω − meas ({(x, ω) ∈ Z × S 1 | |x| < ≥

√1 A 2 2

and τ (x, ω, r) ≥ t}

A r 1 · ]{(p, q) ∈ B(0, 4r ) \ {0}} 18t

(4.22)



from which Theorem C’ easily follows.

Finally, Theorem C’ clearly implies Theorem C of Sect. 2.

5. Proof of Theorem A As we said in Sect. 2, we only have to prove points 2), 3) and 4). An obvious scaling argument shows that τε (εx, ω) = ετ (x, ω, εγ−1 ) . (5.1) Therefore, applying Theorem B and (5.1) leads to   t φε ([t, +∞[) ≤ dω − meas ω ∈ S n−1 | T (ω, εγ−1 ) ≥ ε ≤

C(n)

t γ−1 n−1 r) ε (ε

=

C(n) n−γ(n−1) ε . trn−1

(5.2)

This proves immediately the inequality in point 3]. Likewise, if n = 2 and γ = 2, for all t > 1,   C0 t ≥ , (5.3) (x, ω) ∈ Y × S 1 | τ (x, ω, ε) > φε ([t, +∞[) = µ ε t which establishes point 4]. If φ is a vague limit point of (φε ) and if χ is a bounded continuous function on R+ , one has, for all t > 0, Z Z Z χ(z)dφε (z) = a(tz)χ(z)dφε (z) + (1 − a(tz))χ(z)dφε (z), (5.4) R+

R+

R+

where a is a continuous function supported in [0, 2], equal to 1 on [0, 1], and such that 0 ≤ a ≤ 1. Hence Z (1 − a(tz))χ(z)dφε (z) ≤ kχkL∞ φε ([t, +∞[) ≤ kχkL∞ C(n) εn−γ(n−1) . (5.5) + trn−1 R If γ =

n n−1 ,

(5.5) shows that

Distribution of Free Path Lengths for Periodic Lorentz Gas

505

Z R+

(1 − a(tz))χ(z)dφε (z) → 0 as t → +∞, uniformly in ε.

Hence, for all bounded continuous function χ on R+ , one has Z Z χ(z)dφε (z) → χ(z)dφ(z), R+

(5.6)

(5.7)

R+

which shows that φ is a limit point of (φε ) in the weak topology; applying (5.7) to χ ≡ 1 shows that φ is a probability measure. This completes the proof of point 3). n , (5.4) shows that As for point 2), if 1 ≤ γ < n−1 Z (1 − a(tz))χ(z)dφε (z) → 0 as ε → 0, for all t > 0. (5.8) R+

Hence, for all t > 0, Z lim sup ε→0

R

χ(z)dφε (z) ≤ sup |a(tz)χ(z)| ≤ + z≥0

sup

|χ(z)| .

(5.9)

0≤z≤1/t

If f is a bounded continuous function on R+ and if, for all z ≥ 0, one sets χ(z) = f (z) − f (0), (5.9) shows that Z f (z)dφε (z) → f (0)φε (R+ ) = f (0) , as ε → 0. (5.10) R+

This proves point 2]. 6. Proof of Theorem D We consider a fixed direction ω ∈ S n−1 and let  := A(ω) as in (3.2). Without loss of generality, we can assume that ω ∈ F (see (3.1) √ for a definition of F). Assume that 0 < T < T (ω, R); hence, if R = √rn , N = [T /2 n] < N (, R), which, according to (3.11), implies X 1 |φbR (ξ)| FN (hξ, i) 1≤ . (6.1) ˆ Rn−1 N φ(0) n−1 ξ∈Z

\{0}

There exists C0 > 0 such that sin2 (πN z) C0 . ≤ 2 2 1 + N 2 kzk2 N sin (πz)

(6.2)

On the other hand, we recall that φR satisfies the regularity estimate (3.17). Putting together (3.17), (6.2) and (6.1), if N < N (, R), for all l > 0 there exists a constant Cl such that X Cl 1≤ . (6.3) l (1 + R|ξ|) (1 + N 2 khξ, ik2 ) n−1 ξ∈Z

Let now

\{0}

EM,δ = {ξ ∈ Zn−1 \ {0} |ξ| ≤ M , khξ, ik ≤ δ} .

(6.4)

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J. Bourgain, F. Golse, B. Wennberg

In order to analyze the sum in the right hand side of (6.3), it will be useful to estimate ]EM,δ (the number of elements in EM,δ ). At this point, we need to introduce the assumption that the direction ω satisfies a Diophantine condition. Let s > n − 1 and K > 0 be such that ω ∈ D(s, K) (see Sect. 2 for the definition of D(s, K)). Lemma 4. Let ω ∈ D(s, K), and consider the associated EM,δ . i. If δM s < (4n)−s/2 K, then EM,δ = ∅. ii. There exists a constant C2 > 0 such that for all δ > 0 satisfying δM s ≥ (4n)−s/2 K, one has (n−1) (n−1) ]EM,δ ≤ C2 M n−1 K − s δ s . Proof of Lemma 4. We shall only prove ii]; the proof of i] follows the same lines but is slightly simpler. Let ξ1 and ξ2 be two different vectors in EM,δ . Then khξ1 − ξ2 , ik ≤ khξ1 , ik + kh−ξ2 , ik ≤ 2δ .

(6.5)

On the other hand, suppose that khξ1 − ξ2 , ik = |hξ1 − ξ2 , i − k| ; in addition one has the identity resulting from the definition (3.2) of : |hξ1 − ξ2 , i − k| =

1 |h(ξ1 − ξ2 , −k), αi| . |αn |

Then, khξ1 − ξ2 , ik ≥

K K |(|ξ1 − ξ2 |2 + k 2 )−s/2 ≥ √ |ξ1 − ξ2 |−s . |αn | ( n + 2)s

(6.6)

Therefore, putting together (6.5) and (6.6) leads to 1 |ξ1 − ξ2 | ≥ √ n+2



K 2δ

1/s .

The conclusion follows easily from the pigeonhole principle. 0

Next we estimate the right-hand side of (6.3). Set K = X

Σ=

ξ∈Zn−1 \{0}

X

=

0
+

X

(1 +

R|ξ|)l (1



1 −s/2 K; 4 (4n)

one has then

Cl + N 2 khξ, ik2 )

Cl (1 + R|ξ|)l (1 + N 2 khξ, ik2 ) X

i≥0 2i ≤R|ξ|<2i+1

Cl . (1 + R|ξ|)l (1 + N 2 khξ, ik2 )

These sums are estimated by using the distribution of values of h, ξi, as follows:

Distribution of Free Path Lengths for Periodic Lorentz Gas

X

Σ≤

Cl (N 2−j )−2 · ]ER−1 ,21−j

K 0 Rs <2−j <1

+

X

X

Cl 2−il

i≥0

507

(N 2−j )−2 · ]E2i+1 R−1 ,21−j .

2−(i+1)s K 0 Rs <2−j <1

Observe that these summations are truncated according to the first statement in Lemma 4 above. Now we inject in the sums above the estimate provided in point ii] of Lemma 4:  X n−1 n−1 n−1 2j(2− s ) Σ ≤ Cl C2 K − s N −2 2 s R1−n + 2n−1

X

K 0 Rs <2−j <1

2i(n−1−l)

i≥0

≤ C l C2 K 

− n−1 s

X

2j(2−

n−1 s )

2−(i+1)s K 0 Rs <2−j <1

N

−2

2

n−1 s

−s(2− n−1 s )

(K 01/s R)

R1−n

+ 2n−1

X

 (2− n−1 s )

2i(n−1−l) (2(i+1)s K 0−1 R−s )

i≥0

 ≤ Cl C2 K

− n−1 s

= Cl C 2 K −

n−1 s



N −2 2 

n−1 s

R−2s 1 + 22s 2s

1+

2 1 − 22s−l



X



 2i(2s−l)  K 0−(2−

n−1 s )

i≥0

N −2 2

n−1 s

R−2s K 0−(2−

n−1 s )

.

To summarize: let s > n − 1 and K > 0 be chosen small enough that D(s, K) 6= ∅. Let l > 2s; for example choose l = 3s. There exists a constant C(n, s) > 0 such that if N < N (, R) with  = A(ω) and ω ∈ D(s, K), then 1 ≤ Σ ≤ C(n, s)2 K −2 N −2 R−2s .

(6.7)

Inequality (3.7) can be recast as N≤

C(n, s) , KRs

which, together with (3.6), establishes Theorem D.



Remark. This method of proof gives the same order of magnitude for the ergodization time as the method based on approximating rotation angles by their continued fraction expansion which was used in [D1,DDG1] to treat the two dimensional case. This observation could lead to the belief that the result of Theorem D is sharp. Acknowledgement. F. G. was supported by the A. P. Sloan Foundation during his stay at the Institute for Advanced Study. B. W. was supported by the Swedish Natural Sciences Research Council.

References [BDG] Bardos, C., Dumas, L., Golse, F.: Diffusion Approximation for Billiards with Totally Accomodating Scatterers. J. Stat. Phys. 86, Nos. 1/2, 351–375 (1997) [Bil] Billingsley, P.: Probability and Measure. 3rd Ed., New York: Wiley, 1995

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[Bl]

Bleher, P.: Statistical Properties of Two-Dimensional Periodic Lorentz Gas with Infinite Horizon. J. Stat. Phys.66 (1/2), 315–373 (1992) Boldigrini, C., Bunimovich, L.A. and Sinai, Ya.G.: On the Boltzmann equation for the Lorentz gas. J. Stat. Phys. 32 (3), 477–501 (1983) Bunimovich, L.A. and Sinai, Ya.G.: Markov Partitions of Dispersed Billiards. Commun. Math. Phys. 73, 247–280 (1980) Bunimovich, L.A. and Sinai, Ya.G.: Statistical properties of the Lorentz gas with periodic configurations of scatterers. Commun. Math. Phys. 78, 479–497 (1981) Bunimovich, L.A., Sinai, Ya.G. and Chernov, N.I.: Markov partitions for two-dimensional hyperbolic billiards. Russ. Math. Surv. 45 (3), 105–152 (1990) Bunimovich, L.A., Sinai, Ya.G. and Chernov, N.I.: Statistical properties of two-dimensional hyperbolic billiards. Russ. Math. Surv. 46 (4), 47–106 (1991) Cassels, J.W.S.: An Introduction to Diophantine Approximation. Cambridge (UK): Cambridge University Press, (1957) Chernov, N.: New Proof of Sinai’s Formula for the Entropy of Hyperbolic Billiard Systems Application to Lorentz Gases and Bunimovich Stadium. Funct. Anal. and Appl. 25 (3), 204–219 (1991) Chernov, N.: Entropy, Lyapunov exponents and mean free path for billiards. Preprint 1996 Chierchia, L., Gallavotti, G.: Drift and diffusion in phase space. Annales Inst. Henri Poincar´e 60, 1–144 (1994) Dahlquist, P.: The Lyapunov Exponent in the Sinai Billiard in the small scatterer limit. Nonlinearity 10, 159–173 (1997) Dumas, H.S.: Ergodization rates for linear flow on the torus. J. Dynamics Diff. Equations 3, 593–610 (1991) Dumas, H.S.: A Nekhoroshev-like theory of classical particle channeling in perfect crystals. Dynamics Reported (New Series) 2, 69–115 (1993) Dumas, H.S., Dumas, L., Golse, F.: On the mean free path for a periodic array of spherical obstacles. J. Stat. Phys. 82, 1385–1407 (1996) Dumas, H.S., Dumas, L., Golse, F.: Remarks on the notion of mean free path for a periodic array of spherical obstacles. J. Stat. Phys.87, 943–950 (1997) Gallavotti, G.: Rigorous theory of the Boltzmann equation in the Lorentz gas. Nota Interna No. 358, Istituto di Fisica, Universit`a di Roma (1972) Golse, F.: Transport dans les milieux composites fortement contrast´es I: Le mod`ele du billard. Annales Inst. Henri Poincar´e, Physique Th´eorique 61, 381–410 (1994) Golse, F., Wennberg, B.: In preparation Golse, F., Lions, P.-L., Perthame, B., Sentis, R.: On the regularity of the moments of the solution of a transport equation. J. Funct. Anal. 76, 110–125 (1988) Montgomery, H.: Ten Lectures on the Interface Between Analytic Number Theory and Harmonic Analysis. 94, CBMS series, Providence, RI: AMS (1994) Spohn, H.: The Lorentz flight process converges to a random flight process. Commun. Math. Phys. 60, 277–290 (1978) Spohn, H.: Kinetic Equations from Hamiltonian Dynamics: Markovian Limits. Rev. Mod. Phys. 52, 569–615, (1980)

[BBS] [BS1] [BS2] [BSC1] [BSC2] [Ca] [Ch1] [Ch2] [CG] [Da] [D1] [D2] [DDG1] [DDG2] [Gal] [G] [GW] [GLPS] [M] [Sp1] [Sp2]

Communicated by J. L. Lebowitz

Commun. Math. Phys. 190, 509 – 540 (1998)

Communications in

Mathematical Physics c Springer-Verlag 1997

On the Dirichlet Problem for the Stationary and Axisymmetric Einstein Equations Urs M. Schaudt Institut f¨ur Theoretische Physik, Universit¨at T¨ubingen, Auf der Morgenstelle 14, D-72076 T¨ubingen, Germany. E-mail: [email protected] Received: 30 November 1995/ Accepted: 15 April 1997

Abstract: In suitable coordinates Einstein’s field equations for a rigidly rotating perfect fluid in equilibrium can be written as a semilinear system of purely elliptic partial differential equations of second order. Therefore, the formulation of a boundary value problem is appropriate in this situation. It is shown that the Dirichlet problem for the vacuum region outside a ball, and for a ball inside the matter region, has a unique regular solution if the boundary data are in a characteristic way limited by the “diameter” of the ball. This restriction seems to be closely connected with stability limits for rotating stars. Furthermore, the used mathematical methods are directly related to a numerical solution technique for such physical systems.

1. Introduction Rotating fluid bodies are probably the most common objects in our universe which are accessible to direct observation. If one assumes that such “stars” are in hydro- and thermodynamical equilibrium, which should be a good approximation for at least “long living” stars, it can be shown that such objects behave necessarily like rigidly rotating perfect fluids [9]. In addition, the rotation should be axisymmetric since otherwise due to Einstein’s general theory of relativity there would be gravitational radiation, in contradiction to the assumption of equilibrium. Therefore, such objects should be described by the stationary and axisymmetric field equations with energy-momentum tensor of a perfect fluid. Unfortunately no global, non-static, stationary, axisymmetric, asymptotically flat solution with a compact interior is known to date, except for a rigidly rotating, infinitesimally thin, compact dust disk, which was recently found by Neugebauer and Meinel [14]. However, this dust disk solution describes a galaxy model rather than a stellar model. Even the mathematical questions concerning existence, uniqueness, and regularity of such solutions are not yet completely solved. Only recently Heilig [5] made an

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important contribution by proving that for sufficiently small angular velocities ω > 0 there are realistic equations of state such that a regular, global solution with compact interior exists. The proof is essentially based on an implicit function theorem argument. On principle, this theorem enables estimates on the value of the maximal angular velocity. But there are a lot of unknown constants involved which makes it almost impossible to obtain explicit numbers for the maximal angular velocity with this method. However, it is known that in principle all exterior, stationary, axisymmetric, vacuum solutions can be constructed explicitly (see e.g. [24, p. 182]). But these methods require complicated algebraic computations. In addition, it is difficult to give physical interpretations for the obtained solutions. An important exception is the Kerr metric which plays the leading role in the description of rotating black holes. On the other hand, only a few interior solutions are known, all with hardly realistic equations of state (see e.g. [6]). Thus, it is an important open problem to derive criteria on exterior solutions and on equations of state such that corresponding singularity free interior solutions exist, and vice versa. We show that in suitable coordinates the stationary and axisymmetric Einsteinian field equations reduce to a system of semilinear coupled partial differential equations of purely elliptic type. Therefore, mathematically a boundary value problem is a natural approach (well posed problem), especially since there is a genuine physical boundary: the star’s surface. Even though this is a free boundary value problem, it should be possible to investigate the manifold of solutions of these equations by decomposing the global problem in the following way outlined by Pfister [15, 16]: • For which surfaces, which boundary data, and which equations of state can the Dirichlet problem for the exterior vacuum region and for the interior matter region be solved? • Under which circumstances can an exterior and an interior Dirichlet solution be matched to a global solution which is continuously differentiable across the boundary (i.e. no mass shell or something similar can occur)? Assuming that these Dirichlet problems can be solved uniquely in the generic case, Pfister [16, 20] made plausible that for a realistic equation of state with given total mass and total angular momentum there should be exactly one global, asymptotically flat solution. Conversely, it is conceivable that the known or expected instabilities (e.g. collapse and rotational disruption) for rotating stars can be explained by the insolubility of these boundary value problems for “extreme” surfaces and data [16]. It should be emphasized that in the framework of general relativity there are additional real systems or models where boundary value problems are useful tools, e.g. for accelerated mass shells in order to investigate Mach’s ideas (cf. [17]). The notion of a quasi-local gravitational energy, and the equation of motion for a small but finite test particle lead to a boundary value problem as well. Due to the complicated coupling between the field equations and their high nonlinearity no general existence or uniqueness theorems are known hitherto for such boundary value problems. Only for the stationary vacuum equations in the exterior of a ball with boundary data “sufficiently close” to that of an explicit, spatially conformally flat vacuum solution (e.g. the Minkowski, Schwarzschild or Reissner–Nordstrøm metric, but not the Kerr–Newman metric!) an existence and uniqueness proof was given by Reula [19]. Again, this proof essentially contains an implicit function theorem argument. But the involved constants are not explicitly known. Therefore, it is not clear what “sufficiently close” actually means. In this article (for a short version see [20]) we present a proof for the fact that the exterior as well as the interior Dirichlet problem, with boundary data given on a “ball”,

Dirichlet Problem for Stationary and Axisymmetric Einstein Equations

511

have unique regular solutions, if the boundary data are in a characteristic way limited. For the exterior vacuum problem we elaborate the corresponding condition in detail. The result shows that this condition (which is still not optimal) is closely connected with stability limits for rotating stars (cf. Sects. 3 and 5). Moreover, our mathematical method follows directly the main idea of a numerical solution technique for axisymmetric rotating relativistic bodies due to Bonazzola, Gourgoulhon, Salgado, and Marck [2], and therefore provides one step for a rigorous convergence proof for this numerical method. The article is organized as follows: In Sect. 2 we show that the stationary and axisymmetric field equations can be reduced to a nonlinear system of purely elliptic partial differential equations of 2nd order. In Sect. 3 we discuss the Dirichlet problem for the exterior vacuum region, and in Sect. 4 for the interior matter region. Finally, in Sect. 5 we draw some conclusions from the obtained results.

2. Reduction of the Field Equations to a Purely Elliptic System In general relativity an isolated rotating star in hydro- and thermodynamical equilibrium should be represented by an asymptotically flat stationary axisymmetric 4-dimensional Lorentz-manifold (M, g), with metric g = gab dxa dxb satisfying Einstein’s equations1 Gab := Rab − 21 R gab = 8π Tab for the energy-momentum tensor Tab = (+p) ua ub +p gab of a perfect fluid, with 4-velocity u, proper energy density , and scalar pressure p (see e.g. [9]). For convection free systems, i.e. if the fluid only moves in the plane spanned by the two commuting Killing fields ξ and η, which define the time translational symmetry and the rotational symmetry about the axis A := {m ∈ M|η(m) = 0}, respectively, the metric can be written in the following form (see e.g. [6, pp. 192–195]):   (1) g = gab dxa dxb = −e2U dt2 + e−2U W 2 (dϕ − A dt)2 + e2K (dρ2 + dz 2 ) , which is adapted to the spacetime symmetries. (We find this form superior to other equivalent metric forms used in the literature, because it turns out that here the metric function A is much smoother in the vicinity of the rotation axis.) The metric functions W , U , A, and K only depend on the coordinates ρ and z of the “meridian plane”: the 2surface orthogonal to the Killing fields ξ = ∂t and η = ∂ϕ . Using the remaining freedom of conformal transformations in the meridian plane, we can choose these cylindrical coordinates such that ρ = 0 represents the axis of symmetry and (ρ, z) ∈ R+ × R. Fixing e.g. the penetration points of the axis with the star’s surface, any coordinate freedom is removed. Apart from the conformal factor e2K , all values of the metric functions are invariantly defined since they can be expressed in terms of the Killing invariants g(ξ, ξ), g(ξ, η) and g(η, η), W 2 = − det((gab )a,b=t,ϕ ) = −g(ξ, ξ) g(η, η) + g(ξ, η)2 , W2 , g(η, η) g(ξ, η) . A=− g(η, η)

e2U =

(2)

1 Throughout this article “geometrized units” are used where the gravitational constant and the speed of light are set equal to one. For conversion to other unit systems see e.g. [24, pp. 470–471].

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These quantities can physically be interpreted in the following way (see e.g. [1, pp. 78– 80]): W describes, to some extent, the distance from the rotation axis, U the gravitational potential, and A the coordinate angular velocity of a “locally nonrotating observer” (i.e. an observer with 4-velocity proportional to ∇a t). A slightly different viewpoint concerning the functions U and A can be given within the “3+1 formalism” (see e.g. [2, pp. 423–424]): eU is the lapse function and A is the ϕ-component of the shift vector (the other components of this vector vanish identically). In order to have a stationary, axisymmetric, asymptotically flat, and on the rotation axis elementarily flat spacetime, which has no horizons, ergospheres or other “singularities”, the following conditions must be satisfied by the metric (1): 1. The metric functions U , A, and K should be bounded. 2. The Killing field ξ = ∂t has to be globally timelike, i.e. e4U − W 2 A2 > 0 .

(3)

3. Due to the use of the cylindrical coordinates (t, ρ, z, ϕ) ∈ R × R+ × R × (0, 2π), the metric (1) is only defined on M \ A. In order that this metric can be extended to an at least continuously differentiable, axisymmetric tensor field on the whole spacetime M, one has to impose for all z ∈ R, lim W (ρ, z) = 0

and

ρ→0

lim |∂ρ W | < ∞

ρ→0

lim ∂ρ U (ρ, z) = lim ∂ρ A(ρ, z) = lim ∂ρ K(ρ, z) = 0

ρ→0

ρ→0

ρ→0

(4) (5)

(it should be noted that only W 2 enters the metric and that one already has limρ→0 ∂ρ (W 2 ) = limρ→0 2 W ∂ρ W = 0). 4. The asymptotic flatness requires that as R := (ρ2 + z 2 )1/2 → ∞, W − ρ → 0,

U → 0,

A → 0,

and

K→0.

(6)

One may have to prescribe the fall-off conditions in more detail (see Sect. 3). 5. A physically relevant solution should be “elementarily flat” in the vicinity of the rotation axis which is only satisfied if for all z ∈ R,   (7) lim K(ρ, z) = lim ln ∂ρ W (ρ, z) . ρ→0

ρ→0

6. In the matter region (i.e. {(ρ, z) | p(ρ, z) > 0}), where the fluid’s 4-velocity has the form u = u0 (ξ + ωη), one should have N (W, U, A, ω) := e4U − W 2 (ω − A)2 > 0 , √ since the normalization factor is defined by u0 = eU / N .

(8)

In general, Einstein’s field Equations Gab = 8π Tab provide a system of 10 independent equations. Due to the symmetries of the metric (1), only 6 nontrivial field equations remain, which are equivalent to the following linear combinations: G11 + G22 , G11 + G22 + G33 − G00 − 2A G03 , G03 , G33 − A G03 , G22 and G12 . Using the abbreviations 1 := ∂ρ2 + ∂z2 , ∇ := (∂ρ , ∂z ), #|1 := ∂ρ #, #|2 := ∂z # (for # = W, U, A, K) and h., .i for the Euclidean scalar product, these equations read

Dirichlet Problem for Stationary and Axisymmetric Einstein Equations

1W = 16πe2(K−U ) W p, 1 ∇W , ∇U i − W 2 e−4U |∇A|2 = 1U + h W 2  = 4πe

2(K−U )

(9)

2 (

( + 3p) + 2W (ω − A) 2

513

 + p) , N

∇W ( + p) , ∇Ai − 4h∇U, ∇Ai = −16π e2(K+U ) (ω − A) , W N 3 1K + |∇U |2 − W 2 e−4U |∇A|2 = 4  2(K−U ) 2 2 ( + p) = 8πe , p + W (ω − A) N 1 W|11 + W (U|12 − U|22 ) − W|1 K|1 + W|2 K|2 − W 3 e−4U (A2|1 − A2|2 ) = 4 = 8πe2(K−U ) W p, 1 W|12 + 2W U|1 U|2 − W|2 K|1 − W|1 K|2 − W 3 e−4U A|1 A|2 = 0. 2

1A + 3h

(10) (11)

(12)

(13) (14)

Remark 1. On the rotation axis Eq. (14) reduces to W|12
− W|1 K|2 = 0 (cf. (4) and (5)). Integration on the z-axis yields K(0, z) = ln W|1 (0, z) + C with C ∈ R. Hence, the elementary flatness condition (7) is already guaranteed if it is satisfied at only one point on the rotation axis (respectively in the limit z → +∞ or z → −∞). Euler’s equations, i.e. the spatial components of the equations of motion ∇a Tab = 0, which are a consequence of Bianchi’s identities ∇a Gab = 0 (and hence contained in Einstein’s Eqs. (9)–(14)) reduce in case of rigid rotation (i.e. ω = constant) to ∇p = ( + p) ∇(ln u0 ) = ( + p) ∇(U −

1 ln N ) . 2

(15)

Remark 2. If we presuppose a reasonable equation of state of the form  = (p)

(16)

(which e.g. is satisfied for stars at absolute zero like cold white dwarfs and cold neutron stars, or stars in convective equilibrium like supermassive stars; even for mainsequence stars like our sun this should be a good approximation [22]) then Eq. (15) can be integrated (at least in the interior matter region), yielding an equation of the form p = P (W, U, A, ω) (cf. Sect. 4). This means that the scalar pressure (and hence via Eq. (16) the proper density) can be completely expressed in terms of the metric potentials, and therefore the field equations (9)–(14) form a closed system in the metric potentials. In order that there are no mass shells, the energy density  and the pressure p as functions of the coordinates (ρ, z) ∈ R+0 × R in the meridian plane may have jump discontinuities at the most. Thus, according to Einstein’s equations for a perfect fluid Gab ≡ Rab − 21 R gab = 8π[( + p) ua ub + p gab ], the metric functions W , U , A and K have to be at least continuously differentiable, since their 2nd derivatives occur in Gab . Within our model the star’s surface S is located by definition where the pressure p vanishes. Obviously, this definition is only reasonable if the function p is continuous across the star’s surface. Then, field Eq. (9) suggests that the metric function W has to

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U. M. Schaudt

be two times continuously differentiable there. In the interior and exterior of the star all functions have to be sufficiently smooth that Bianchi’s identities are valid in the usual sense, since they guarantee that Euler’s equations are satisfied. Throughout the following we fix the notations: R+ := {x ∈ R | x > 0}, R+0 := {x ∈ R | x ≥ 0}, R− := {x ∈ R | x < 0}, and R− 0 := {x ∈ R | x ≤ 0}. For the interior points, o

the closure, and the boundary of a set X ⊂ Rn we write X , X, and ∂X, respectively. Furthermore, we use I = interior of the star := {(ρ, z) ∈ R+0 × R | p(ρ, z) > 0} ⊂ R+0 × R E = exterior of the star := (R+0 × R) \ I ⊂ R+0 × R S = star’s surface := I ∩ E ⊂ R+0 × R .

(17)

Then, we have Proposition 1 (Reduction to 4 elliptic equations). The metric functions W , U , A, K and the pressure function p satisfy the nonelliptic field Eqs. (13) and (14) everywhere on R+0 × R, if the following (sufficient) conditions hold: 1. S is a smooth curve (i.e. there is a C 1 -parametrization for S). 2. o o W ∈ C 3 (I ) ∩ C 3 (E ) ∩ C 2 (R+0 × R), o

o

o

o

o

o

U, A, K ∈ C 2 (I ) ∩ C 2 (E ) ∩ C 1 (R+0 × R), p ∈ C 1 (I ) ∩ C 1 (E ) ∩ C 0 (R+0 × R),  ∈ C 1 (I ) ∩ C 1 (E ) ∩ L∞ (R+0 × R). 3. The functions W , U , A, K, p and  satisfy the elliptic field Eqs. (9)–(12) and Euler’s o

o

Eq. (15) on I ∪ E . 4. As R := (ρ2 + z 2 )1/2 → ∞ (“asymptotically flat solutions”): W|1 → 1 , W|2 → 0 , W|11 → 0 , W|12 → 0, 2 →0 U → 0 , W U|i=1,2

W 3 A2|i=1,2 → 0

(⇒ W U|1 U|2 → 0),

(⇒ W 3 A|1 A|2 → 0),

K → 0 , K|i=1,2 → 0 . (Physically one expects that W behaves like ρ, U like R−1 , and A like R−3 as R → ∞.) 5. As ρ → 0 (“axisymmetric solutions”): W → 0 , U|1 → 0 , A|1 → 0 , K|1 → 0 (and for completeness p|1 → 0, |1 → 0).

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Remark 3. In Proposition 1 it is not demanded that the metric is elementarily flat, i.e. condition (7) is not assumed. But, using Remark 1, elementary flatness follows! Proof. For all (ρ, z) ∈ R+0 × R, let 1 G(ρ, z) := W|11 + W (U|12 − U|22 ) − W1| K|1 + W|2 K|2 − W 3 e−4U (A2|1 − A2|2 ) 4 −8πe2(K−U ) W p, 1 H(ρ, z) := W|12 + 2W U|1 U|2 − W|2 K|1 − W|1 K|2 − W 3 e−4U A|1 A|2 , 2 then the proof can be split into the following 4 lemmata: Lemma 1. (a) G, H ∈ C 0 (R+0 × R) ∩ L∞ (R+0 × R). (b) G → 0 and H → 0 as R → ∞. (c) G → 0 as ρ → 0. o

o

o

o

(d) G, H ∈ C 1 (I ) ∩ C 1 (E ), and on I ∪ E G|1 = −H|2

and

G|2 = H|1 .

Proof. Assertions (a) and (b) follow directly from assumption 2 and 4 (c) Since W (0, z) = 0 for all z ∈ R (cf. assumption 5) we have W|2 (0, z) = 0 for all z ∈ R (cf. assumption 2). Hence W|22 (0, z) = 0 and, using field Eq. (9), W|11 (0, z) =
−W|22 + 16πe2(K−U ) W p (0, z) = 0 for all z ∈ R (cf. assumption 2). Thus, with assumption 5, the assertion follows. (d) Assumptions 2, 3, and straightforward calculations yield the assertions. Note that  especially, due to assumption 2, W|112 = W|121 and W|111 + W|122 = (1W )|1 . Lemma 2. f := H + iG ∈ C 0 (R+0 × R) ∩ C ω (R+ × R ⊂ C). Proof. According to Lemma 1 (d),
o

and satisfy the Cauchy–Riemann equations in I as well as in E . Hence f1 := f | o and I f2 := f | o are holomorphic. Furthermore, according to Lemma 1 (a), f1 and f2 can be E continuously extended to S, and f1 = f2 on S. Thus, by the principle of continuity, f is  holomorphic on R+ × R. Lemma 3. f (ρ, z) := H(−ρ, z) − iG(−ρ, z), ∀(ρ, z) ∈ R− × R is a holomorphic extension of f onto R × R ∼ = C. Proof. According to Lemma 2, f is holomorphic on R+ ×R and continuously extendible to the boundary of R+ × R. This boundary is a straight line, namely the z-axis. Using Lemma 1 (c), we have =f (0, z) = G(0, z) = 0 ∀z ∈ R, i.e. the image of the boundary of R+ × R under f is again part of a straight line (namely of the ρ-axis). Applying Schwarz’s reflection principle yields the assertion.  Lemma 4. G ≡ H ≡ 0 on R+0 × R.

Proof. By Lemma 3 f is holomorphic everywhere on R × R ∼ = C, and according to Lemma 1 (a) f is bounded. Hence by Liouville’s theorem f is constant. Using Lemma 1 (b) yields f ≡ 0, and therefore G ≡ H ≡ 0 on R+0 × R, i.e. field Eqs. (13) and (14) are satisfied.  This completes the proof of Proposition 1.



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3. The Exterior Dirichlet Problem 3.1. Reduction to a fixed point problem. For the exterior vacuum region E the following is well-known (see e.g. [24, pp. 166–167]): Since field Eq. (9) reduces to 1W = 0, we are able to introduce preferred coordinates (ρ, z) ∈ R+0 × R in the meridian plane, called Weyl coordinates, such that W ≡ ρ on E. Then, with the abbreviation 1n := ∂ρ2 + ∂z2 +

n−2 ∂ρ , ρ

(n ≥ 2) ,

(18)

field Eqs. (10) and (11) read 1 2 −4U ρ e |∇A|2 , 2 15 A = 4 h∇U, ∇Ai .

13 U =

(19)

These equations are independent of the conformal factor e2K . Furthermore, Eqs. (13) and (14) can be solved for K|1 and K|2 . The results are again independent of K, and satisfy ∂2 K|1 = ∂1 K|2 as well as field Eq. (12) by virtue of Eqs. (19). Thus, given a solution of Eqs. (19), the metric function K can be obtained by performing a line integral (at least on principle). And, since Eq. (14) reduces to K|2 = 0 on the rotation axis ρ = 0, the asymptotic flatness condition (6) yields K ≡ 0 for ρ = 0, i.e. the elementary flatness condition (7) is satisfied! Altogether, this means that the purely elliptic system (19) represents a complete set of field equations for the vacuum region E. In order to prove existence for Dirichlet solutions of system (19), first of all, we find it convenient to transform the infinite exterior vacuum region E to a finite “empty interior” E ? (as was done in the numerical procedure [2]), and to divide out the asymptotic behavior of the metric functions U and A. To this end, let R := (ρ2 + z 2 )1/2 ,  ρ z  , 2 (20) ψ : (ρ, z) 7→ (ρ? , z ? ) := 2 R R (note that R? := (ρ? 2 + z ? 2 )1/2 = R−1 and ψ −1 = ψ ∈ C ω (R+ × R)), and u : E ? := ψ(E) → R, a : E ? := ψ(E) → R,

u(ρ? , z ? ) := (R? )−1 U (ψ(ρ? , z ? )), a(ρ? , z ? ) := (R? )−3 A(ψ(ρ? , z ? )) .

(21)

The mapping ψ is called reflection at the unit-(half)-circle, and the new functions u and a are the Kelvin transforms of the metric functions U and A (see e.g. [4, p. 66]). The inverse transformations are given by the same mathematical relationship: U : E → R, A : E → R,

U (ρ, z) = R−1 u(ψ(ρ, z)), A(ρ, z) = R−3 a(ψ(ρ, z)) .

Remark 4. 1. The Kelvin transforms u and a of the Killing invariants U and A are obviously coordinate-dependent. 2. Half-circles around the origin with radius R = Re are mapped onto half-circles around the origin with radius R? = Re −1 . The ideal point ∞ of the (ρ, z)-half-plane is mapped to the origin of the (ρ? , z ? )-half-plane. Thus, adjoining the origin to E ? is equivalent to the one point compactification of the exterior: E → E ∪ {∞}. But, for the following this is not helpful and is therefore not used.

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517

3. The factors R−1 and R−3 are the asymptotically vanishing spherically symmetric fundamental solutions of the homogeneous equations 13 U = 0 and 15 A = 0 of system (19). Physically one expects that U resp. A behave just the same way as R → ∞: 2J M and A → 3 , U →− R R with total mass M and total angular momentum J of the star (see e.g. [10, p. 452]). Therefore, we should have u(0) := lim u = −M ? R →0

and

a(0) := lim a = 2J. ? R →0

(22)

Denoting for simplicity the (ρ? , z ? )-coordinates of the empty interior E ? again by (ρ, z), the vacuum Eqs. (19) for the Kelvin transforms read:   13 u = f (ρ, z) , u , a , ∇a ,   (23) 15 a = g (ρ, z) , u , ∇u , a , ∇a with the nonlinearities   f (ρ, z) , u , a , ∇a :=   1 ρ2 −4R u ( )e 9 a2 + 6 a R ∂R a + R2 |∇a|2 , 2 R   g (ρ, z) , u , ∇u , a , ∇a :=   12 u a + 4 u ∂R a + 3 a ∂R u + R h∇u , ∇ai R   p(u, a) + q (ρ, z) , u , ∇u , a , ∇a , =: R

(24)

(25)

and the “radial derivatives” ∂R u := h

(ρ, z) , ∇ui , R

∂R a := h

(ρ, z) , ∇ai . R

(26)

Remark 5. 1. Since | (ρ,z) R | = 1 (for (ρ, z) 6= 0), by the Cauchy–Schwarz inequality we have |∂R u| ≤ |∇u| and |∂R a| ≤ |∇a|. 2

ρ 2. Since ( ρR ) = ρ ( R ) ≤ ρ | (ρ,z) R | = ρ ≤ R (for R > 0), the field equation for u is continuous at the origin. However, due to the term p(u, a)/R, the field equation for a is singular there. But, by virtue of Remark 4, limR→0 u and limR→0 a should exist, i.e. u, a = O(1). Then, using field Eqs. (23) as R → 0, we have 13 u = O(R) and 15 a = O(R−1 ). Therefore, as R → 0, we expect that u and a behave like

u = u(0) + O(R3 )

and a = a(0) + O(R) .

The terms 13 u and 15 a on the left hand side of the vacuum Eqs. (23) contain singularities at the rotation axis ρ = 0, which are caused by the use of the “cylindrical coordinates” (ρ, z). However, these coordinate singularities can be handled mathematically if we consider the axial symmetry of the physical problem in a consistent way. To this end, we introduce the following concept:

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Definition 1 (Classes of axisymmetric functions). Let Es? := {(ρ, z) ∈ R− 0 × R| (−ρ, z) ∈ E ? } be the image of the empty interior E ? ⊂ R+0 × R under reflection at the rotation axis (= z-axis). 1. A function u : E ? → R is called k-axisymmetric (k ≥ 0), iff its symmetric extension  u(ρ, z) : (ρ, z) ∈ E ? ? ? us : E ∪ Es → R, (ρ, z) 7→ us (ρ, z) := u(−ρ, z) : (ρ, z) ∈ Es? is k times continuously differentiable on the domain E ? ∪ Es? , and we write u ∈ Csk (E ? ) : ⇐⇒ us ∈ C k (E ? ∪ Es? ) . If in addition all partial derivatives of order k satisfy a H¨older condition with exponent δ ∈ (0, 1] on every convex compact subset D ⊂⊂ E ? ∪ Es? (cf. e.g. [4, p. 51]), then we write u ∈ Csk,δ (E ? ). Furthermore, if u ∈ Csk (E ? ) (Csk,δ (E ? )) and us as well as all its partial derivatives of the first order are bounded on E ? ∪ Es? we write k k,δ (E ? ) (Cs,∞ (E ? )), and use the scale-covariant norm u ∈ Cs,∞ kuks := max{kuk∞ , d(E ? ) k∇uk∞ } (k = 0: kuks := kuk∞ ), where d(E ? ) := sup(ρ,z)∈E ? (ρ2 + z 2 )1/2 , kuk∞ := sup(ρ,z)∈E ? |u|, and k∇uk∞ := sup(ρ,z)∈E ? |∇u|. 2. Let  ⊂ Rn (n ≥ 2) be a domain. A function u :  → R is called k-axisymmetric (k ≥ 0), iff it is k times continuously differentiable and invariant under all “rotations” around the axis ρ := (x21 + · · · + x2n−1 )1/2 = 0, i.e. (D(x1 , . . . , xn−1 ), xn ) ∈  and u(D(x1 , . . . , xn−1 ), xn ) = u(x1 , . . . , xn ) ∀x = (x1 , . . . , xn ) ∈  and ∀D ∈ O(n − 1, R). Then we write u ∈ Cak (). If in addition all partial derivatives of order k satisfy a H¨older condition with exponent δ ∈ (0, 1] on every convex compact subset D ⊂⊂ , we write u ∈ Cak,δ (). Furthermore, if u ∈ Cak () (Cak,δ ()) and u as well as all its partial derivatives of the first order are bounded on , we write k k,δ () (Ca,∞ ()) and use the scale-covariant norm u ∈ Ca,∞ kuka := max{kuk∞ , d() k∇uk∞ } (k = 0: kuka := kuk∞ ), where d() := supx∈ |x|, kuk∞ := supx∈ |u|, and k∇uk∞ := supx∈ |∇u|. 3. For every function u : E ? ⊂ R+0 × R → R we call the function u]n : En? ⊂ Rn → R (n ≥ 2), defined by (cf. [25]) En? := {(x1 , . . . , xn ) ∈ Rn | (x21 + · · · + x2n−1 )1/2 , xn ) ∈ E ? }, u]n (x1 , . . . , xn ) := u((x21 + · · · + x2n−1 )1/2 , xn ) ,

∀x ∈ En? ,

the n-lift of u. Especially, we have u]2 = us . 4. For every function u :  ⊂ Rn → R (n ≥ 2) we call the function u[n : 2 ⊂ R+0 × R → R, defined by 2 := {(ρ, z) ∈ R+0 × R | (ρ, 0, . . . , 0, z) ∈ }, u[n (ρ, z) := u(ρ, 0, . . . , 0, z) ,

∀(ρ, z) ∈ 2 ,

the meridional cut (in direction x1 ) of u. Especially, we have (us )[2 = u.

Dirichlet Problem for Stationary and Axisymmetric Einstein Equations

519

k k Remark 6. 1. We need the function spaces C∞ (E ? ) (C∞ ()) instead of the more fak k ? miliar spaces C (E ) (C ()) for the following reason: According to Remark 5 we expect that a = a(0) + O(R) as R → 0. The partial derivatives of the function (ρ, z) 7→ R := (ρ2 + z 2 )1/2 , i.e. ∂ρ R = ρ/R and ∂z R = z/R, are bounded on E ? , but they can not be continuously extended to (ρ, z) = 0 ∈ E ? . 2. We demand H¨older conditions (if at all) on convex subsets only, since then we are 0 0 sure that the inclusion relation C k ,δ ⊂ C k,δ is valid if k + δ < k 0 + δ 0 (cf. [4, p. 52]). Note that the origin (ρ, z) = 0 is excluded from E ? .

The decisive properties of these definitions are that for axisymmetric functions • n-lift and meridional cut leave the regularity properties and the norms invariant, • all “really axisymmetric operations” like multiplication, ∂R , h∇·, ∇·i, and exchange between 1n and the flat Laplacian 1 in n dimensions (but not ∇, e.g.) commute with n-lift and meridional cut, • the meridional cut contains already the maximal information. This is the essential content of the following Lemma 5 (Properties of axisymmetric functions). 1. For every u ∈ Cs1 (E ? ) and for all (0, z) ∈ E ? ⊂ R+0 × R we have ∂ρ u(0, z) = 0 . 2. Let n ≥ 2 and k ≥ 0, then ] n

: Csk (E ? ) → Cak (En? ), u 7→ u]n

is a bijective linear mapping, with inverse given by [ n

: Cak (En? ) → Csk (E ? ), u 7→ u[n .

This still holds if we replace “k” by “k, δ”, with 0 < δ ≤ 1. 3. For every n ≥ 2 we have: (a) Let u, v ∈ Csk (E ? ) and k ≥ 1. Then R ∂R u as well as h∇u, ∇vi are in Csk−1 (E ? ), and (R ∂R u)]n = h (x1 , . . . , xn ), ∇u]n i =: |x| ∂|x| (u]n ), (h ∇u, ∇v i)]n = h ∇u]n , ∇vn] i . Let u, v ∈ Cak (En? ) and k ≥ 1. Then |x| ∂|x| u as well as h∇u, ∇vi are in Cak−1 (En? ), and (|x| ∂|x| u)[n := (h (x1 , . . . , xn ), ∇u i)[n = R ∂R u[n , (h ∇u, ∇v i)[n = h ∇u[n , ∇vn[ i , thus especially (∂R u)]n = ∂|x| u]n , (∂|x| u)[n = ∂R u[n , and |∇u|]n = |∇u]n |, |∇u|[n = |∇u[n |.

520

U. M. Schaudt

(b) Let u ∈ Csk (E ? ) and k ≥ 2. Then 1n u ∈ Csk−2 (E ? ) and (1n u)]n

=

1(u]n )

:=

n X

∂i2 (u]n ) .

i=1

Let u ∈

Cak (En? )

and k ≥ 2. Then 1u ∈ (1u)[n := (

n X

Cak−2 (En? )

and

∂i2 u)[n = 1n (u[n ) .

i=1

4. Let n ≥ 2 and k ≥ 0, then ] n



k k : Cs,∞ (E ? ), k.ks → Ca,∞ (En? ), k.ka , u 7→ u]n

is an isomorphism, with inverse given by

[ k ? k ? [ n : Ca,∞ (En ), k.ka → Cs,∞ (E ), k.ks , u 7→ un , and d(E ? ) = d(En? ). Again, we may replace “k” by “k, δ”, with 0 < δ ≤ 1. Proof. Straightforward.



With these preparations, we are able to formulate the Dirichlet problem for the stationary axisymmetric vacuum equations: Definition 2 (Dirichlet problem for the vacuum equations). Given: Boundary data u, ˆ aˆ : ∂E ? \z-axis → R such that their symmetric extensions uˆ s and aˆ s are“sufficiently smooth” (this will be specified subsequently: cf. Definition 4 and Proposition 2). 2 (E ? ) ∩ C 0 (E ? ) satisfying the vacuum equations (23) Wanted: Functions u, a ∈ Cs,∞ as well as the Dirichlet boundary conditions us = uˆ s and as = aˆ s on ∂E ? \ z-axis. Such solutions u, a, we call regular. For simplicity and definiteness, we give the data for the Dirichlet problem on a sphere (in Weyl coordinates). Moreover, this will be useful in order to have quantitative results, because then Green’s functions are explicitly known. Re := {x ∈ Rn | 0 < |x| < Definition 3 (Fixation of the empty interior). Let B0,n n Re } ⊂ R be the interior of the annulus with radii Re > 0 and R = 0, centered at the origin. In the following, we call

Re E ? := B0,2 ∩ (R+0 × R) Re empty interior. Especially, we have E ? ∪ Es? = B0,2 .

Remark 7. From a physical point of view, this restriction to spheres (in Weyl coordinates) is not an essential limitation, since • from every global solution with a compact matter region, this interior can be cut out by such a sphere,

Dirichlet Problem for Stationary and Axisymmetric Einstein Equations

521

• according to Thorne’s “hoop conjecture” [23, 10: Box 32.3], stability against gravitational collapse should already be determinable in principle by considering such a “spherical configuration”.

Let (u, a) be a regular solution with boundary data (u, ˆ aˆ ). Since in system (23) only axisymmetric operations appear (!), according to Lemma 5 we have 1u]3 = (13 u)]3 = (f ((ρ, z) , u , a , ∇a))]3   = f ([x21 + x22 ]1/2 , x3 ) , u]3 , a]3 , ∇a]3 , 1a]5 = (15 u)]5 = (g((ρ, z) , u , ∇u , a , ∇a))]5   = g ([x21 + · · · + x24 ]1/2 , x5 ) , u]5 , ∇u]5 , a]5 , ∇a]5 .

(27)

Let (u0 , a0 ) be the homogeneous solutions for these Dirichlet problems, i.e.: 1u0 = 0 in BRe ⊂ R3

with

u0 |∂BRe = uˆ ]3 ,

1a0 = 0 in BRe ⊂ R5

with

a0 |∂BRe = aˆ ]5 .

(28)

These Dirichlet problems for the flat Laplacian (in 3 and 5 dimensions) have unique solutions. Since the domains are balls, they are explicitly given by Poisson’s integral formula (see e.g. [4, Theorem 2.6]): R2 − |x|2 u0 (x) = e 4π Re Re2 − |x|2 a0 (x) = 8π 2 3 Re

Z Z

∀x ∈ BRe ⊂ R3 ,

∂BRe

uˆ ]3 (y) dsy , |x − y|3

∀x ∈ BRe ⊂ R .

∂BRe

aˆ ]5 (y) dsy , |x − y|5

(29) 5

Due to the axial symmetry of uˆ ]3 and aˆ ]5 , the same holds for u0 and a0 . Furthermore (u0 )[3 |∂E ? = uˆ and (a0 )[5 |∂E ? = aˆ , hence (using Lemma 5) (u0 )[3 and (a0 )[5 are regular, especially on the rotation axis ρ = 0 (even at the origin (ρ, z) = 0).

Lemma 6 (Estimates for the homogeneous solutions). 1. The following estimates are valid ∀(ρ, z) ∈ E ? : (a) For the values of the functions: ˆ inf uˆ ≤ (u0 )[3 (ρ, z) ≤ sup u,

∂E ?

inf? aˆ ≤

∂E

∂E ?

(a0 )[5 (ρ, z)

≤ sup aˆ .

(30)

∂E ?

(b) For their gradients: If the boundary data uˆ s , aˆ s : ∂BRe ⊂ R2 → R can be extended to symmetric functions u, ˆ aˆ ∈ C 2 (BRe ) (e.g. if uˆ s , aˆ s ∈ C 2,δ (∂BRe )

522

U. M. Schaudt

with δ ∈ (0, 1]), then ˆ + Re sup |13 u|, ˆ |∇(u0 )[3 (ρ, z)| ≤ sup |∇u| E?

∂E ?

|∇(a0 )[5 (ρ, z)|

≤ sup |∇ˆa| + Re sup |15 aˆ | .

(31)

E?

∂E ?

2. For all 0 < L1 , L2 < ∞ there are boundary data (u, ˆ aˆ ) 6= 0 such that (u0 )[3 and (a0 )[5 2 ? 0 ? are in Cs,∞ (E ) ∩ C (E ), satisfying ku0 ka ≤ L1 and ka0 ka ≤ L2 . Proof. 1. (a) This is the assertion of the (weak) maximum principle for harmonic functions (see e.g. [4, Theorem 2.3]). (b) Using [18, p. 144] and Lemma 5 yields the desired estimates. (If uˆ s , aˆ s ∈ C 2,δ (∂BRe ) with δ ∈ (0, 1], then the Dirichlet problems 1uˆ = 0, u| ˆ ∂BRe = uˆ s 2 and 1ˆa = 0, aˆ |∂BRe = aˆ s on BRe ⊂ R have unique solutions in C 2,δ (BRe ) ⊂ C 2 (BRe ); see e.g. [4, Corollary 4.14]. Thus, uˆ s , aˆ s ∈ C 2,δ (∂BRe ) is a sufficient condition that the assumptions hold.)  2. Since ∇ and 1n are linear operations, the assertion is a consequence of 1. Definition 4 (Sufficiently smooth boundary data). Boundary data u, ˆ aˆ are called sufficiently smooth, iff the corresponding solutions u0 , a0 for the homogeneous problems (cf. (29)) satisfy ku0 ka < ∞ and ka0 ka < ∞. According to Lemma 6, a sufficient condition is given by uˆ s and aˆ s ∈ C 2,δ (∂BRe )

with δ ∈ (0, 1] .

Remark 8. Besides axial symmetry, we do not demand any further symmetries (like mirror symmetry relative to the “equatorial plane” z = 0)! If we define u˜ := u − (u0 )[3

and

a˜ := a − (a0 )[5 ,

(32)

and insert this into system (27), we find that u˜ and a˜ are regular solutions for the following “reduced” Dirichlet problem: 1u˜ ]3 = 1u]3 − 1u0 =   f ([x21 + x22 ]1/2 , x3 ) , u0 + u˜ ]3 , (a0 [5 )]3 + a˜ ]3 , ∇(a0 [5 )]3 + ∇˜a]3   =: f˜ x , u˜ ]3 , a˜ ]3 , ∇˜a]3 ,

(33)

1˜a]5 = 1a]5 − 1a0 =   g ([x21 + · · · + x24 ]1/2 , x5 ) , (u0 [3 )]5 + u˜ ]3 , ∇(u0 [3 )]5 + ∇u˜ ]5 , a0 + a˜ ]5 , ∇a0 + ∇˜a]5   =: g˜ x , u˜ ]5 , ∇u˜ ]5 , a˜ ]5 , ∇˜a]5 =:

  p( ˜ u˜ ]5 , a˜ ]5 ) + q˜ x , u˜ ]5 , ∇u˜ ]5 , a˜ ]5 , ∇˜a]5 |x|

(34)

Dirichlet Problem for Stationary and Axisymmetric Einstein Equations

523

for the trivial boundary data

u˜ ]3 |∂BRe = uˆ ]3 − uˆ ]3 = 0

and a˜ ]5 |∂BRe = aˆ ]5 − aˆ ]5 = 0 .

(35)

Remark 9. The homogeneous solutions u0 and a0 enter in a nonlinear way into the inhomogeneous terms f˜ and g. ˜     Re Re 3 x 7→ f˜ x , u˜ ]3 , a˜ ]3 , ∇˜a]3 , B0,5 3 x 7→ p˜ u˜ ]5 , a˜ ]5 , and The functions B0,3   Re 3 x 7→ q˜ x , u˜ ]5 , ∇u˜ ]5 , a˜ ]5 , ∇˜a]5 are bounded. Hence f˜ ∈ L∞ (BRe ), and (due B0,5 RR R to BR ⊂R5 |x|−p dx ∼ 0 e r4−p dr < ∞ if p < 5) we especially have g˜ ∈ Lp (BRe ) e for 5/2 < p (< 5). Therefore Green’s representation holds (cf. e.g. [4, Problem 4.3]), and we obtain Z u˜ ]3 (x) a˜ ]5 (x)

= ⊂R3

Z

BR e

Z

BRe ⊂R5

= = BRe ⊂R5

Z

+

  G3 (x, y) f˜ y , u˜ ]3 , a˜ ]3 , ∇˜a]3 dy ,

∀x ∈ BRe ⊂ R3 ,

  G5 (x, y) g˜ y , u˜ ]5 , ∇u˜ ]5 , a˜ ]5 , ∇˜a]5 dy ,

∀x ∈ BRe ⊂ R5

G5 (x, y)  ] ]  p˜ u˜ 5 , a˜ 5 dy |y|   G5 (x, y) q˜ y , u˜ ]5 , ∇u˜ ]5 , a˜ ]5 , ∇˜a]5 dy ,

BRe ⊂R5

where Gn is Green’s function (of the 1st kind) for the n-dimensional ball with radius Re centered at the origin (see e.g. [4, p. 19]):

1 Gn (x, y) := − (n − 2) ωn



Re 1 1 − ( )n−2 |x − y|n−2 |y| |x − y ? |n−2

 (36)

(n ≥ 3, y 6= 0, x 6= y, ωn := |∂B1 |, i.e. ω3 = 4π, ω5 = 8π 2 /3, and y ? := (Re /|y|)2 y). Using again Lemma 5, we have shown that the reduced Dirichlet problem (33)–(35), and hence the Dirichlet problem for the vacuum field equations, is equivalent (for regular solutions) to the following fixed point problem:

524

U. M. Schaudt

Z u˜ =

(u˜ ]3 )[3

= BRe ⊂R3

  G3 (x, y) f˜ y , u˜ ]3 , a˜ ]3 , ∇˜a]3 dy

![ 3

=: T1 (u, ˜ a˜ ) , ![ Z   ] [ ] ] ] ] G5 (x, y) g˜ y , u˜ 5 , ∇u˜ 5 , a˜ 5 , ∇˜a5 dy a˜ = (˜a5 )5 = BRe ⊂R5

Z = BRe ⊂R5

G5 (x, y)  ] ]  p˜ u˜ 5 , a˜ 5 dy |y|

Z



+ BRe ⊂R5

G5 (x, y) q˜ y ,

u˜ ]5

,

5

![

∇u˜ ]5

(37)

5

,

a˜ ]5

,

∇˜a]5

![

 dy

5

=: T2 (u, ˜ a˜ ) , in brief:

˜ a˜ ) , T2 (u, ˜ a˜ )) =: T (u, ˜ a˜ ) . (u, ˜ a˜ ) = (T1 (u,

3.2. Existence and uniqueness of solutions. In order to prove existence (and uniqueness) for solutions of the fixed point problem (37) we follow the main idea of the numerical solution technique for rotating stars in general relativity introduced by Bonazzola, Gourgoulhon, Salgado, and Marck [2], namely to iterate the mapping T with appropriate starting functions. Convergence can be established by extension of a method of Courant and Hilbert [3, pp. 287–288] from one equation to a system. This method is based on the following Theorem 1 (Banach fixed point theorem). Let B be a closed subset of a Banach space with norm k.k. Suppose that the mapping T : B → B is a contraction, i.e. there is a contraction constant α ∈ [0, 1) such that for all x, y ∈ B, kT (x) − T (y)k ≤ α kx − yk . Then we have: 1. T has precisely one fixed point T (x∗ ) = x∗ ∈ B. 2. If we choose an arbitrary x0 ∈ B and set xk+1 := T (xk ) ∀k ∈ N, then the iterative sequence (xk )k∈N converges to the unique fixed point x∗ . 3. We have the following (a priori and a posteriori) error estimates: kx∗ − xk k ≤

α αk kxk − xk−1 k ≤ kx1 − x0 k . 1−α 1−α

(38)

Proof. See e.g. [7, Sect. 5.1]. Note that B together with the metric d(x, y) := kx − yk forms a complete metric space.  Convention. For simplicity, we omit the symbols “˜ ” for u and a. Furthermore, according to Lemma 5, the symbols “ ]n ” for n-lift and “ [n ” for meridional cut may be cancelled, since it is obvious from the inserted arguments which operation is meant.

Dirichlet Problem for Stationary and Axisymmetric Einstein Equations

525

Proposition 2 (Existence and uniqueness of (weak) solutions). Let Re > 0 and 1 (E ? ) ∩ C 0 (E ? ) . E := Cs,∞

Then E × E together with the scale-covariant norm k(u, a)k := max{kuks , Re kaks } is a Banach space. Furthermore, let BL := {(u, a) ∈ E × E | k(u, a)k ≤ L}
be the closed ball in E × E , k.k with radius L > 0 centered at 0. Then the following holds: For all Re > 0 and L > 0 satisfying Re L <

1 3 ≈ , 652 217

(39)

and all boundary data such that (u0 , a0 ) ∈ BL ⇔ ku0 ka ≤ L and ka0 ka ≤ L/Re (cf. Lemma 6 2.), the mapping T : BL → BL , (u, a) 7→ T (u, a) defined by (37) is a contraction with contraction constant 0≤ α≤

652 Re L < 1 . 3

Hence, the assertions of the Banach fixed point theorem (Theorem 1) hold, i.e. T has a unique fixed point in BL which can be calculated (to arbitrary precision) by iteration from any starting point in BL (e.g. (u, a)0 = 0). Moreover, due to the error estimates (38), this iterative sequence converges exponentially to the unique fixed point in BL . Remark 10. 1. Presumably, Banach’s fixed point theorem provides the mathematical reason for the observation numerically found by Bonazzola et al. ([2, 6.4 and 6.5]) that the starting values for an iterative sequence to a mapping of the form (37) need not be close to the actual solution at all in order to guarantee convergence, and that the error decreases expontentially with the number of iteration steps. However, this fact is observed in [2] by calculating global solutions by matching an approximate exterior solution (which corresponds to (u, a)k = T k ((u, a)0 ) in our method) continuously differentiable to an approximate interior solution (which corresponds to (K, U, B, A)k = T k ((K, U, B, A)0 ) in our method, cf. (71) in Sect. 4) at each step of the iteration, keeping the equatorial coordinate radius fixed (which in our method corresponds to the fixing of the boundary and the data). Moreover an approximation technique for Green’s functions is used (“pseudo-spectral method”). But, since the core of both methods coincides, we assume that this characteristic convergence behavior is common to them.

526

U. M. Schaudt

2. In the static case we have aˆ ≡ 0, and thus a0 ≡ 0 by Poisson’s formula (29). Choosing the starting value (u, a)0 = 0 ∈ BL yields f˜ ≡ 0 and g˜ ≡ 0. Hence, (u, a)0 = 0 is a fixed point of T , which is unique by Proposition 2. Therefore, u = u0 and a = a0 ≡ 0 are the unique solution for the Dirichlet problem of the vacuum Eqs. (23) in the static case, and condition (39) is not necessary. Then, using Lemma 5, we find that U = u0 /R is an axisymmetric asymptotically flat solution for Laplace’s equation in 3 dimensions on an exterior vacuum region, i.e. U is a Weyl’s class solution (see e.g. [6, Sect. 8.1]). 3. Due to the use of the norm k.k, we have a priori – estimates for the gradients of u and a – even on the boundary – which should be helpful for the matching problem between exterior and interior solutions of a rotating star. With reference to [2] (and point 1 of this remark), this gives hope that our method can be extended to the global problem. Proof. For lucidity, we split the proof of Proposition 2 into the following lemmata. 


Lemma 7. E , k.ks , E , Re k.ks , and E × E , k.k are Banach spaces. 

Proof. Straightforward.

Lemma 8 (Properties of Green’s functions). Let Re > 0 and n ≥ 3. Then the Green’s function (36) for the n-dimensional ball with Radius Re centered at the origin satisfies: 0 ≥ Gn (x, y) = Gn (y, x) ,  = cn

1 − |x − y|n−2



Re |y|

∀x, y ∈ BRe n−2



1 |x − y ? |n−2

,

(y 6= 0, x 6= y)

  n−2 1 1 Re , (x 6= 0, x 6= y) − = cn |x − y|n−2 |x| |x? − y|n−2   1 1 = cn ,
n−2 −   n−2 2 2 |x|2 +|y|2 −2 hx, yi 2 |x| |y| 2 +Re −2 hx, yi Re 

(40)

(x 6= y) , where cn := −

1 <0 (n − 2) ωn

? Re 2 e 2 and y ? := ( R |y| ) y , x := ( |x| ) x ,

with ωn denoting the surface area of the n-dimensional unit ball2 (especially ω3 = 4π, ω5 = 8π 2 /3). We have (x 6= 0, x 6= y): |∇x Gn (x, y)| ≤

1 ωn



1 + |x − y|n−1



|y| Re



Re |x|

n−1

1 |x? − y|n−1

 ,

(41)

and especially for n = 3 and n = 5 the following holds: 2 Warning: This notation is not uniformly used in the literature, e.g. in [4] ω denotes the volume of the n n-dimensional unit ball.

Dirichlet Problem for Stationary and Axisymmetric Einstein Equations

Z sup Re x∈B0,3

BR e

Z

BR e

5 4

Re x∈B0,5

BR e

Z

BR e

Z

|G5 (x, y)| dy = supx∈B Re

Furthermore, the function x 7→

7 6

R BR e

− |x|2 ) =

2 10 (Re

G5 (x, y) dy = 41 (|x| − Re ) , |y|

BR e

Re ) and L∞ (B0,n Re ∀x ∈ B0,n :

1

0,5

|∇x G5 (x, y)| dy ≤

sup

2 6 (Re

1 6

Re2 ,

Re ,

Z

sup

1

0,3

|∇x G3 (x, y)| dy ≤

sup Re x∈B0,3

Re x∈B0,5

|G3 (x, y)| dy = supx∈B Re

527

− |x|2 ) =

1 10

Re2 ,

(42)

Re ∀x ∈ B0,5 ,

Re . Gn (x, y) f (y) dy is in C 1,δ (BRe ) for all f ∈

δ ∈ (0, 1), and the partial derivatives commute with the integral sign Z

Z

∇x

Gn (x, y) f (y) dy = BR e

∇x Gn (x, y) f (y) dy .

(43)

BR e

If f is axisymmetric, the same holds for x 7→

R

BR e

Gn (x, y) f (y) dy.

Proof. It is straightforward to check that ∀β ∈ R, β  β 1 1 Re Re = |y| |x − y ? |β |x| |x? − y|β − β2  2 |x| |y| 2 + R − 2 hx, yi , = e Re



(44)

and thus, the different expressions (40) for Green’s function Gn are valid. The last one yields Gn (x, y) = Gn (y, x) ≤ 0, hence |Gn | = −Gn . Since   1 1 (x − z) cn , = ∇x |x − z|n−2 ωn |x − z|n−1 |x − z| we establish estimate (41) with use of (44) for β = n − 1 and the triangle inequality. For p < n we have y 7→ |y|−p ∈ L1 (BRe ⊂ Rn ). Hence, the same holds for the functions Re (cf. (40) and y 7→ Gn (x, y), y 7→ Gn (x, y)/|y|, and y 7→ |∇x Gn (x, y)| for all x ∈ B0,n (41)), and iterated integration is allowed by Fubini’s theorem. Pn 1 Since 1|x|2 = i=1 2 = 2n, we find that the function u(x) := 2n (|x|2 − Re2 ) is a (and thus the only) solution for the Dirichlet problem 1u ≡ 1 with u|∂BRe ≡ 0. By R Green’s representation we have u(x) = BR Gn (x, y) dy (see e.g. [4, Theorem 4.3 and e (4.10)]). Using |Gn | = −Gn yields Z 1 |Gn (x, y)| dy = 2n Re2 . sup Re x∈B0,n

BR e

528

U. M. Schaudt

R Analogously, we establish the identity BR G5 (x, y)/|y| dy = 41 (|x| − Re ). The remaine ing estimates for the gradients can be obtained as follows: The terms on the right hand side of inequality (41) can be integrated elementarily (using Fubini’s theorem). Due to the symmetry, the final results are only functions of |x| ∈ (0, Re ). One can verify that these functions (∈ C ∞ ) are monotonically decreasing. Thus, the suprema are approached as |x| → 0. These limits can be calculated by use of de l’Hospital’s rule. R For a proof that the function x 7→ BR Gn (x, y) f (y) dy is in C 1 (BRe ) for all e

Re ), and that Eq. (43) holds, we refer e.g. to [4, Lemma 4.1] and (40). The f ∈ L∞ (B0,n H¨older-continuity for all exponents δ ∈ (0, 1) can be found e.g. in [8, Sect. 8; resp. in the literature cited therein]. R It is straightforward to show that the function x 7→ BR Gn (x, y) f (y) dy is axisyme Re ˜ = f (y) and ∀D ∈ O(n−1, R) we have f (Dy) metric if f is axisymmetric (i.e. ∀y ∈ B0,n with D˜ := D ⊗ 1). 

Lemma 9. For the nonlinear terms of the reduced problem (33)–(34) we have:   2 ˜ 1 ( ρ )e−4R (u0 +u) 9(a0 +a)2 + 6(a0 +a)R∂R (a0 +a) + R2 |∇(a0 +a)|2 , f= 2 R p=12 ˜ (u + u)(a0 + a),   0 +u)∂ (a +a) + 3(a +a)∂ (u +u) + Rh∇(u +u), ∇(a +a)i . q=4 ˜ (u0 0 0 0 0 0 R R Proof. This is a direct consequence of (24), (25), (33), and (34).

(45) (46) (47)



Re ˜ := p(u(x), ˜ a(x)) for all x ∈ B0,5 . Then, Lemma 10. Let u0 , u, a0 , a ∈ E, and p(x)

Re → R, x 7→ r(x) ˜ := r˜ : B0,5

p(x) ˜ − p(0) ˜ |x|

Re Re 0 0 (B0,5 ) ⇔ Cs,∞ (E ? ), and satisfies ∀x ∈ B0,5 the estimate: is in Ca,∞

|r(x)| ˜ ≤ k∇pk ˜ ∞<∞.

(48)

Especially, we have p(x) ˜ = p(0) ˜ + r(x) ˜ |x|. Re 1 (B0,5 ) ∩ C 0 (BRe ). Proof. By definition of E and p˜ it follows that x 7→ p(x) ˜ is in Ca,∞ Re Re 1 0 ˜ yields estimate Hence r˜ ∈ Ca (B0,5 ) ⊂ Ca (B0,5 ). Using the mean value theorem for p(x) Re 0 (48). Thus r˜ ∈ Ca,∞ (B0,5 ). According to Lemma 5, this function space is equivalent to 0 (E ? ).  Cs,∞

Lemma 11. For all boundary data such that u0 and a0 are in E the mapping T = (T1 , T2 ) defined by (37) satisfies Ti=1,2 (E × E) ⊂ E . Proof. Let u and a be in E. Then we have:

Dirichlet Problem for Stationary and Axisymmetric Einstein Equations

529

1. By definition of E and f˜ (cf. (45) and note that |ρ2 /R| ≤ |ρ| ≤ R), the function Re ). Hence, f˜(y) is axisymmetric y 7→ f˜(y, u(y), a(y), ∇a(y)) =: f˜(y) is in Ca0,∞ (B0,3 Re ∞ and in L (B0,3 ). Then, by Lemma 8, the function Z   G3 (x, y) f˜(y) dy (49) x 7→ T1 (u, a) (x) = BR e

is in Ca1 (BRe ), and the partial derivatives may be interchanged with the integral sign. Especially, we have T1 (u, a) ∈ E. 2. Let q(y) ˜ := q(y, ˜ u(y), ∇u(y), a(y), ∇a(y)). Then, by Lemma 10 and (42), we have Re : for all x ∈ B0,5   Z Z   G5 (x, y) T2 (u, a) (x) = p(0) dy + ˜ G5 (x, y) r(y) ˜ + q(y) ˜ dy |y| BR e BR e   Z p(0) ˜ (|x| − Re ) + G5 (x, y) r(y) ˜ + q(y) ˜ dy . (50) = 4 BR e The function x 7→ |x| − Re obviously is in E (the introduction of E was just motivated by such a function, cf. Remark 6 1.). Furthermore, the functions y 7→ q(y) ˜ and Re ) (cf. (47) and Lemma 10), i.e. especially they are axisymy 7→ r(y) ˜ are in Ca0,∞ (B0,5  R Re metric and in L∞ (B0,5 ). Then, by Lemma 8, the function x 7→ BR G5 (x, y) r(y)+ ˜ e  q(y) ˜ dy is in E, and the partial derivatives commute with the integral sign. Altogether, T2 (u, a) ∈ E.



Lemma 12. For all Re > 0 and L > 0 satisfying condition (39), and all boundary data such that (u0 , a0 ) ∈ BL we have T (BL ) ⊂ BL . Proof. According to the previous lemma, we only have to show: If k(u0 , a0 )k ≤ L and k(u, a)k ≤ L, then kT (u, a)k ≤ L. To this end, let k(u0 , a0 )k ≤ L and k(u, a)k ≤ L. Then, using (49) and (50), |∂R #| ≤ |∇#| ≤ k∇#k∞ ≤ L/Re (for # = u0 , u, Re a0 , Re a), and that the partial derivatives commute with the integral sign, we have: Re 1. ∀x ∈ B0,3 :

Z |(T1 (u, a))(x)| = G3 (x, y) f˜(y) dy BR e Z |G3 (x, y)| dy kf˜k∞ ≤ BR e

by (42) ≤

1 6

Re2 kf˜k∞

cf. (45) ≤

1 6

Re2 21 Re e8Re L

n

8Re L (Re L) L = 16 3 e by (39) < L .

2L 2 2L 2 2L 2 9 (R ) + 6 (R ) + (R ) e e e

o

(51)

(52)

530

U. M. Schaudt

Re 2. ∀x ∈ B0,3 :

Z Re |∇x (T1 (u, a))(x)| = Re ∇x G3 (x, y) f˜(y) dy BR e Z |∇x G3 (x, y)| dy kf˜k∞ ≤ Re BR e

by (42) ≤

5 4

Re2 kf˜k∞ 8Re L

cf. (51) & (52) ≤ 40 e by (39) < L .

(53)

(Re L) L

(54)

Re : 3. ∀x ∈ B0,5

p(0) ˜ Re |(T2 (u, a))(x)| ≤ Re (|x| − Re ) 4 Z   +Re G (x, y) r(y) ˜ + q(y) ˜ dy BR e 5 ≤ Re

1 kpk ˜ ∞ Re 4Z |G5 (x, y)| dy {krk ˜ ∞ + kqk ˜ ∞}

+Re BR e

1 Re2 kpk ˜ ∞ + 10 Re3 {krk ˜ ∞ + kqk ˜ ∞}   2L cf. (46) ≤ 41 Re2 12 2L R e    1 2L 2 2L cf. (48) & (46) + 10 Re3 12 ( R ) + 2L R2 e

by (42) ≤

1 4

(55)

e



2L 2L 2 2L + 4 2L R 2 + 3 ( R ) + 2L R2 e

cf. (47)

e



e

148 5

(Re L) L = by (39) < L .

(56)

Re : 4. ∀x ∈ B0,5

Re2

p(0) ˜ |∇x (T2 (u, a))(x)| ≤ 4 ∇x (|x| − Re ) Z   2 +Re ∇x G5 (x, y) r(y) ˜ + q(y) ˜ dy BR e Re2

≤ Re2

1 kpk ˜ ∞ 4Z

+Re2 by (42) ≤

1 4

|∇x G5 (x, y)| dy {krk ˜ ∞ + kqk ˜ ∞} BR e

Re2 kpk ˜ ∞ + 76 Re3 {krk ˜ ∞ + kqk ˜ ∞}

(57)

Dirichlet Problem for Stationary and Axisymmetric Einstein Equations

cf. (55) & (56) = 652 3 (Re L) L by (39) < L . 

531

(58)

Lemma 13. The mapping T : BL → BL is a contraction with contraction constant 0≤ α≤

652 Re L < 1 . 3

Proof. By assumption, k(u0 , a0 )k ≤ L. Let (u, a) and (u, a) be in BL . With the abbreviations f (y) := f˜(y, u(y), a(y), ∇a(y))

and

f (y) := f˜(y, u(y), a(y), ∇a(y)) ,

(and analogously for p, ˜ q˜ and r), ˜ we find by comparison with inequalities (51), (53), (55), and (57): ) ( 5 2 4 Re kf − f k∞ . kT (u, a) − T (u, a)k ≤ max 1 2 7 3 4 Re kp − pk∞ + 6 Re {kr − rk∞ + kq − qk∞ } (59) Since the nonlinearities f˜, p, ˜ q, ˜ and r˜ of the reduced Dirichlet problem (cf. Lemma 9 and Lemma 10), viewed as functions of the variables u, ui := ∂i u, a, and ai := ∂i a, are differentiable, and the corresponding partial derivatives are bounded, we are able to estimate the differences kf − f k∞ , kp − pk∞ , kq − qk∞ , and kr − rk∞ with use of the mean value theorem (m.v.t.). Estimating kf − f k∞ , we have terms of the form (cf. (45) with h := {. . .}): |es u h − es u h| ≤ |es u − es u | |h| + |es u | |h − h|


m.v.t. ≤ kes u k∞ ksk∞ khk∞ k(u, a) − (u, a)k + kh − hk∞ .

The other differences kh − hk∞ , kp − pk∞ , kq − qk∞ , and kr − rk∞ ≤ k∇p − ∇pk∞ (cf. (48)) only contain “quadratic” terms of the form |(v0 + v) (w0 + w) − (v0 + v) (w0 + w)| , where v and w are used as symbols for u, R ui , R a, and R2 ai (in case, “(. . .) (. . .)” has to be understood as “h(. . .), (. . .)i”). We find k(v0 + v) (w0 + w) − (v0 + v) (w0 + w)k∞ ≤
kw0 + wk∞ + kv0 + vk∞ k(u, a) − (u, a)k ≤ 4L k(u, a) − (u, a)k . By comparison with k(v0 + v) (w0 + w)k∞ ≤ (2L)2 = 4L L, and (59) with (53), (54), (57), and (58), we are able to read off the final result: kT (u, a) − T (u, a)k ≤ ( ) 40 e8Re L {4 Re L + 1} Re L max 652 k(u, a) − (u, a)k . 3 Re L This completes the proof of Proposition 2.



532

U. M. Schaudt

In the following section, we prove that these “weak” solutions are regular solutions. Hence, we have shown that the Dirichlet problem for the vacuum equations has a unique solution if the boundary data (given on a sphere in Weyl coordinates) satisfy condition (39). 3.3. Regularity. Proposition 3 (Regularity of (weak) solutions). Let (u, a) be the unique fixed point of the mapping T : BL → BL due to Proposition 2. Then, (u0 + u, a0 + a) is a regular solution for the Dirichlet problem for the vacuum Eqs. (23), and we especially have  u0 , u, u0 + u ∈ C 0 (E ? ) ∩ Cs∞, ∞ (E ? ) . a0 , a, a0 + a Proof. We only have to show that (u, a) is a regular solution for the reduced vacuum equations, and u0 , a0 , u, a are in Cs∞ (E ? ). Since u0 and a0 are regular solutions for the flat Laplace Eq. (in 3 and 5 dimensions, resp.) with axisymmetric boundary data, we have u0 ∈ Ca∞ (BRe ⊂ R3 ) and a0 ∈ Ca∞ (BRe ⊂ R5 ) (see e.g. [12, Theorem 2.2.4 and 2.4.1]). Using Lemma 5 yields u0 and a0 in Cs∞ (E ? ∪ {0}). In order to prove u, a ∈ Cs∞ (E ? ), we first note that u, a ∈ Cs1,δ (E ? ) for all δ ∈ (0, 1). This follows if we apply Lemma 8 and Lemma 5 to u = T1 (u, a) and a = T2 (u, a) (cf. (49), (50) and Lemma 9). Then, we have f˜, q, ˜ and r˜ in Cs0,δ (E ? ), which (again due to (49) and (50)) 2 ? yields u, a ∈ Cs (E ), and especially the reduced vacuum Eqs. (33) and (34) are satisfied Re Re resp. B0,5 (see e.g. [4, Lemma 4.2]). But then, we have u, a ∈ Cs2,δ (E ? ) since f˜, on B0,3 p(x)/|x|, ˜ and q˜ are in Cs0,δ (E ? ) (see e.g. [4, Theorem 6.17]). Hence, the nonlinearities are in Cs1,δ (E ? ), which yields u, a ∈ Cs3,δ (E ? ) (again e.g. by [4, theorem 6.17]). Hence,  by induction, we have u, a ∈ Cs∞ (E ? ). Remark 11. Using a theorem due to M¨uller zum Hagen [13] (which is essentially based on a theorem by Morrey [11]), it follows then that the corresponding vacuum metric even has an analytic structure. 3.4. Remarks on physical implications. Since every solution (u0 + u , a0 + a), guaranteed by Proposition 2, is bounded, their inverse Kelvin transforms (U, A) satisfy U = (u0 + u)/R = O(R−1 ) and A = (a0 + a)/R3 = O(R−3 ) as R → ∞ in the exterior vacuum region by (21). Then, according to (22), the total mass M and the total angular momentum J of this solution is given by lim (u0 + u) =: −M

R? →0

and

lim (a0 + a) =: 2J .

R? →0

Since ku0 +uk∞ ≤ 2 L and ka0 +ak∞ ≤ 2 L/Re , we have M ≤ 2 L and J ≤ L/Re . Due to the smallness condition (39), this is equivalent to Re M < 3/326 and Re2 J < 3/652.
Re ? + If we remember that E = B0,2 ∩ R0 × R represents (by E = ψ −1 (E ? )) the exterior of a ball (in Weyl coordinates) with radius R0 := inf (ρ,z)∈E (ρ2 + z 2 )1/2 = Re−1 , this means 3 M ≈ 9.2 × 10−3 < R0 326

and

J 3 ≈ 4.6 × 10−3 . < 2 652 R0

(60)

Dirichlet Problem for Stationary and Axisymmetric Einstein Equations

533

It should be noted that these are scale-invariant quantities! For example, these conditions are satisfied for (see e.g. [22 ])3 : • our sun: M ≈ 2.11 × 10−6  9.2 × 10−3 R

and

J ≈ 8.32 × 10−13  4.6 × 10−3 . R 2

and

J < 4 × 10−5 < 4.6 × 10−3 . R2 ∼

• white dwarf stars: M < 2 × 10−4 < 9.2 × 10−3 R ∼

Hence, Proposition 2 especially provides an existence and uniqueness proof for the exterior of all known white dwarfs stars (at least near equilibrium). Physically, it is expected (cf. hoop conjecture) that conditions like (60) must be satisfied in order to guarantee stability against collapse and rotational disruption. However, it is clear that the values in conditions (60) can not yet be optimal, since: • For neutron stars the following estimates are known (see e.g. [22]): M < 0.5 > 9.2 × 10−3 R ∼

und

J < 10−1 > 4.6 × 10−3 . R2 ∼

• The value 3/652 obtained in the smallness condition (39), and therefore the values in (60), obviously depend on the used function spaces, norms, and inequalities (here we used E, k.k, H¨older’s inequality for p = 1 & q = ∞, and the mean value theorem). For example, we have not yet investigated whether a Sobolev space Ws1,q (E ? ) ⊃ E yields better results. • Condition (39) in Proposition 2 is only sufficient in order to guarantee existence of exterior solutions. Upper limits for the values in (60) can be estimated as follows: • Since U = R (u0 + u) and A = R3 (a0 + a) have to be bounded in the empty interior E ? , we may assume that there are Re > 0 and L > 0 such that |U | = |R (u0 + u)| ≤ 2 Re L

and |A| = |R3 (a0 + a)| ≤ 2 Re2 L .

Moreover, we have W = ρ/R2 (cf. (20)). Hence, |W A| = |ρ R (a0 + a)| ≤ 2 Re L. Then, condition (3) reads e−8 Re L − 4 (Re L)2 > 0, which is equivalent to Re L < 0.213 . . . (compare with (39)). • If we write the Kerr solution in Weyl coordinates and choose the smallest ball with radius R0 , which contains the whole region inside the static limit, we find in the Schwarzschild case J M < 1 and =0, R0 R0 2 and in the extreme Kerr case J M < 1 and <1. (61) R0 R0 2 3 The conversion factors from SI-units to geometrized units for the quantities “mass” and “angular mom s and cG3 ≈ 2.47 × 10−36 kg , respectively (cf. e.g. [24 p. 471; mentum” are given by cG2 ≈ 7.41 × 10−28 kg

however, the cited value for the gravitational constant G has to be replaced by G = 6.67 × 10−8

cm3 ]). g s2

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4. The Interior Dirichlet Problem 4.1. Formulation of the problem. According to Sect. 2, for a stationary axisymmetric globally regular spacetime solution Einstein’s field equations reduce to a purely elliptic system and Euler’s equation in the interior of the star I ≡ {(ρ, z) ∈ R+0 ×R | p(ρ, z) > 0}. Here, it is no more possible to set W ≡ ρ. With regard to the behavior at the rotation axis W = ρ = 0, we may set W (ρ, z) ≡ ρ eB(ρ,z) .

(62)

Then, with use of abbreviations (18), the four elliptic Eqs. (9)–(12) read 12 K 13 U 14 B 15 A

= = = =

qK qU qB qA

+ + + +

mK , mU , mB , mA .

(63)

Here, the terms q# (with # = K, U, B, A) represent the nonlinearities which are “quadratic” in the first derivatives of the metric functions 3 2 2B−4U ρ e |∇A|2 , 4 1 qU = −h∇B, ∇U i + ρ2 e2B−4U |∇A|2 , 2 qB = −|∇B|2 , qA = −3 h∇B, ∇Ai + 4 h∇U, ∇Ai ,

qK = −|∇U |2 +

(64)

and the terms m# represent the contributions from the rigidly rotating perfect fluid matter Tab = ( + p) ua ub + p gab with 4-velocity u = u0 (∂t + ω ∂ϕ ),   2(K−U ) 2 2B 2 ( + p) , p + ρ e (ω − A) mK = 8π e N   2(K−U ) 2 2B 2 ( + p) mU = 4π e , ( + 3p) + 2ρ e (ω − A) N (65) mB = 16π e2(K−U ) p, mA = −16π e2(K+U ) (ω − A)

( + p) , N

where we have used the abbreviation N := e4U − W 2 (ω − A)2 = e4U − ρ2 e2B (ω − A)2 . √ Since u0 = eU / N , Euler’s equation reads (cf. (15)) ∇p = ( + p) ∇(ln u0 ) = ( + p) ∇(U −

1 ln N ) . 2

(66)

(67)

Remark 12. In order to guarantee that the nonelliptic field Eqs. (13) and (14) are satisfied for this configuration (where a priori we have no matching to an exterior asymptotically flat solution), we have to demand elementary flatness on the axis. This can be seen in the following way: In Proposition 1 and the corresponding proof we have only to

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535

omit those assumptions and assertions which make statements on the exterior E (and as R → ∞), and replace Step 4 by the following argument: As in Lemma 3, f = H + iG is holomorphic on I := I ∪ Is (note that this is an open set), and G = 0 on I ∩ A. Using the conditions for elementary flatness (7), and axisymmetry (4) & (5), yields H = 0 on I ∩ A also. Hence, by the identity theorem for holomorphic functions we have f ≡ 0 on I, i.e. field Eqs. (13) and (14) are satisfied on I. We assume that the interior matter region is diffeomorphic to a 3-dimensional ball (i.e. we do not consider toroidal configurations). Therefore, we can choose the origin of the coordinates (ρ, z) ∈ R+0 × R in I (e.g. at the center of mass). Since, in this model, the symmetrically extended pressure function ps is continuous on I ≡ I ∪ Is (cf. Sect. 2), there is an open ball Bi ⊂ I with radius Ri > 0 centered at the origin. For the following, we set Ii := (I ∩ Bi ) ⊂ R+0 × R. If the (given) equation of state  = (p) (cf. (16)) satisfies the energy condition (p) + p > 0 for all p > 0, then Eq. (67) can be integrated: Since p > 0 on Ii , the same holds for (p) + p > 0, and we have pi := p(0, Ri ) > 0. Furthermore, we may assume that p(Ii ) contains an open interval of R+ (since p is continuous, strictly positive on I, and by definition vanishes at the star’s surface). Hence, the function Z p 1 dp˜ (68) F : p(Ii ) → R, p 7→ F (p) := ( p) ˜ + p˜ pi + is well defined, and, due to dF dp (p) = 1/((p) + p) > 0, is injective on p(Ii ) ⊂ R . 4Ui Therefore, with Ui := U (0, Ri ) (note that ln N (0, Ri ) = ln(e − 0) = 4 Ui ), Euler’s Eq. (67) is equivalent to  0  1 u (ρ, z) F (p(ρ, z)) = ln = U (ρ, z) − ln N (ρ, z) + Ui , ∀(ρ, z) ∈ Ii (69) u0 (0, Ri ) 2

⇐⇒ (since F is injective on p(Ii ))   1 −1 U − ln N + Ui =: P (U, B, A, Ri , pi , ω) , p=F 2

on Ii ,

(70)

i.e. we can express the pressure function (at least) on Ii as a function of the metric functions K, U , B, A, and the parameters Ri , pi and ω. This makes (63) a closed elliptic system on Ii . Hence, a solution on Ii is fixed by this elliptic system, the boundary data of the metric functions K, U , B, A on ∂Ii \ z–axis, and the parameters pi and ω, i.e. by the corresponding interior Dirichlet problem. 4.2. Existence and uniqueness of solutions. Since in the interior field Eqs. (63) again only axisymmetric operations (in the sense of Lemma 5) appear (cf. (64), (65), and (70)), the interior Dirichlet problem on Ii (representing a ball with coordinate radius Ri > 0 contained in the star’s interior) can be handled in analogy to the vacuum case (cf. Sect. 3): 1. Solve the corresponding homogeneous Dirichlet problems (cf. (28), (29), and e.g. [4, Theorem 2.6] in case of n = 2 and n = 4). 2. For sufficiently smooth boundary data (cf. Definition 4) and regular solutions (cf. Definition 2), due to Lemma 5 and Green’s representation, the corresponding reduced Dirichlet problem (cf. (33)–(35)) is equivalent to the following fixed point problem:

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Z K= Z

BRi ⊂R2

U = Z

BRi ⊂R3

B= Z

BRi ⊂R4

A= BRi ⊂R5

G2 (x, y) {qK (y) + mK (y)} dy G3 (x, y) {qU (y) + mU (y)} dy

2

[ 3

G4 (x, y) {qB (y) + mB (y)} dy G5 (x, y) {qA (y) + mA (y)} dy

[

[ 4

[ 5

=: TK (K, U, B, A), =: TU (K, U, B, A), =: TB (K, U, B, A), =: TA (K, U, B, A) ,

where we have used the abbreviations q# (y) := q# (y, K0 (y)+K(y), . . .), and m# (y) := m# (y, K0 (y) + K(y), . . .), where K0 , U0 , B0 , A0 are the solutions of the homogenous problems, and K, U , B, A the solutions of the reduced problem. In brief: (K, U, B, A) = T (K, U, B, A) .

(71)

Remark 13. 1. Since Ii is bounded, we need neither the reflection at the unit-(half)circle nor the Kelvin transform. Furthermore, there are no singularities in the nonlinear terms (64) and (65). Hence, we are able to work with more familiar function 1 (Ii \ {0}) ∩ C 0 (Ii ). spaces, e.g. Cs1 (Ii ) instead of Cs,∞ 2. Here, we need the Green’s function (of the 1st kind for an n-dimensional ball centered at the origin) besides the cases n = 3 and n = 5 also in the cases n = 2 and n = 4. In the case n = 2, we have to add    1 |y| ln(|x − y|) − ln |x − y ? | G2 (x, y) := 2π Ri in Lemma 8 (cf. e.g. [4, (2.12) & (2.23)]). For this, as well as for n = 4, analogous statements and estimates hold as for n = 3 and n = 5. Then, we have the following Proposition 4 (Existence and uniqueness of (weak) solutions). For all equations of state  = (p), which are continuously differentiable for p > 0, and satisfy the energy condition (p)+p > 0 for p > 0, there are parameters Ri > 0, pi := p(0, Ri ) > 0, |ω| > ˆ Uˆ , B, ˆ Aˆ on ∂Ii := ∂BRi ∩ (R+ × R), 0, and L > 0 such that for all boundary data K, 0 satisfying kK0 ks ≤ L, kU0 ks ≤ L, kB0 ks ≤ L, and kA0 ks ≤ L/Ri (cf. Lemma 6 2.), the corresponding mapping TRi ,pi ,ω : BL → BL , (K, U, B, A) 7→ TRi ,pi ,ω (K, U, B, A)  4 is a contraction on the closed ball BL centered at the origin of the Banach space Cs1 (Ii ) with the scale-invariant norm k(K, U, B, A)k := max{kKks , kU ks , kBks , Ri kAks } . Hence, the assertions of the Banach fixed point theorem hold, and we especially have for the unique fixed point 0 < pmin ≤ p = P (U0 + U, B0 + B, A0 + A, Ri , pi , ω) ≤ pmax < ∞ ,

Dirichlet Problem for Stationary and Axisymmetric Einstein Equations

537

i.e. the fixed point represents a solution contained in the star’s interior. If the equation of state satisfies the physically reasonable condition lim |(p)

p→0

d (p)| < ∞ dp

(cf. Remark 14 2.) then pmin = 0 is possible. Proof. Due to the proof of Proposition 2, it is clear that there are parameters and boundary data such that TRi ,pi ,ω : BL → BL is a contraction, if the following holds: The nonlinearities q# und m# , viewed as functions of the variables ρ2 , K0 , K, U0 , U , B0 , B, A0 , A, and ∂i K, ∂i U , ∂i B, ∂i A, are differentiable, and these functions as well as their partial derivatives are bounded on the domain fixed by the parameters Ri , pi , ω, L, and the corresponding bounds can be estimated in terms of these parameters. Since then, by use of the mean value theorem, we have (in analogy to the vacuum case) kT (K, U, B, A) − T (K, U , B, A)k ≤ α k(K, U, B, A) − (K, U , B, A)k , and, due to the estimates in Lemma 8, kT (K, U, B, A)k as well as the constant α can be made sufficiently small by reducing the parameters’ values. With reference to Eqs. (64) and (65), this is satisfied if we can demonstrate that: 1. There is a constant Nmin (Ri , L, ω) > 0 such that the Killing invariant N , viewed as a function of ρ2 ∈ [0, Ri2 ], U0 , U , B0 , B ∈ [−L, L], A0 , A ∈ [−L/Ri , L/Ri ], and for |ω| > 0, satisfies the condition N ≥ Nmin (Ri , L, ω). Then, N −1 as well as its partial derivatives are bounded, and these bounds are fixed by the parameters Ri , L, and ω. d (p) are bounded on [pmin , pmax ]. 2. (a) The function p 7→ (p) and its derivative p 7→ dp 1 + Since we presuppose that (p) ∈ C (R ), these conditions are satisfied if we are able to choose 0 < pmin < pmax < ∞. (b) On the domain fixed by the parameter values, the pressure function P (U0 +U, B0 + B, A0 + A, Ri , pi , ω) has values only in [pmin , pmax ], and its partial derivatives are bounded . To this end, we first note that the parameters Ri and L have to be chosen such that the Killing fields ξ = ∂t and ξ +ω η = ∂t +ω ∂ϕ are timelike (in order to have no static limits, horizons, or other “singularities”, and that u ∼ ∂t + ω ∂ϕ represents the 4-velocity field of particles with nonvanishing rest mass). These conditions are equivalent to:

and

X := e4(U0 +U ) − ρ2 e2(B0 +B) (A0 + A)2 > 0

(72)

 2 N ≡ e4(U0 +U ) − ρ2 e2(B0 +B) ω − (A0 + A) > 0 .

(73)

Since X and N are continuous functions in the variables ρ2 U0 , U , B0 , B, A0 , A, and both take the value 1 > 0 for the allowed arguments ρ = U0 = U = B0 = B = A0 = A = 0, the conditions are satisfied for sufficiently small Ri > 0 and L > 0. But then, condition 1 is satisfied as well. To a given pi > 0, the function p 7→ F (p) defined by (68) is continuously differentiable and injective for p > 0, and F (pi ) = 0. Choose 0 < pmin < pi < pmax < ∞. Then, F : [pmin , pmax ] → [F (pmin ), F (pmax )] =: [Fmin , Fmax ] is bijective, and we especially have Fmin < 0 < Fmax . Let

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U. M. Schaudt

Y := (U0 + U ) −

1 ln N + Uˆ (0, Ri ) , 2

(cf. (69)). For the chosen parameters Ri > 0 and L > 0, this function is continuously differentiable with respect to the variables ρ2 ∈ [0, Ri2 ], U0 , U , B0 , B ∈ [−L, L], and A0 , A ∈ [−L/Ri , L/Ri ]. Moreover, since for the allowed arguments U0 = Uˆ (0, Ri ), and ρ = U = B0 = B = A0 = A = 0 the function Y vanishes, we may assume that Y takes only values in [Fmin , Fmax ] (if necessary, we have to reduce the values of the parameters Ri > 0 and L > 0 further). Thus, for all continuously differentiable functions U0 , U , B0 , B, and A0 , A on Ii := BRi ∩ (R+0 × R), which have only values in [−L, L], resp. [−L/Ri , L/Ri ], Euler’s Eq. (67) is equivalent to p = F −1 (Y ) =: P (U0 + U, B0 + B, A0 + A, Ri , pi , ω) . Especially, we have 0 < pmin ≤ p = P (U0 + U, B0 + B, A0 + A, Ri , pi , ω) ≤ pmax < ∞, and for functions U , B, A vanishing on ∂Ii \ z-axis (as for a fixed point of Eq. (71)) p(0, Ri ) = pi holds. Furthermore, due to the inverse function theorem, we obtain for the partial derivatives with respect to # = U, B, A:   ∂# N , (74) ∂# P = ((P ) + P ) ∂# Y = ((P ) + P ) ∂# (U0 + U ) − 2N which are bounded on the domain, fixed by the chosen parameters.



Remark 14. 1. The proof shows that the conditions for the parameters, which are sufficient to guarantee existence of interior solutions, essentially depend on the given equation of state (p), as is expected from a physical point of view. Due to Lemma 8, it is possible to work out an estimate for the parameter L(Ri , pi , ω) (like (39)). But this, we have not yet done. 2. Example. For polytropes, i.e. for matter with an equation of state of the form p() ∼ γ d ⇔ (p) ∼ p1/γ , with γ > 1, we have limp→0 dp (p) = ∞. Therefore, in the previous propositon we intentionally demand pmin > 0. But, according to (74) and (65), it d d (p) is bounded on [pmin , pmax ], instead of dp (p). Hence, for is sufficient that (p) dp such equations of state, we may choose pmin = 0. Thus, the given spherical surface ∂Ii \ z-axis may touch the star’s surface S := ∂{(ρ, z) | p > 0}. For polytropes this holds if the index γ satisfies 1 < γ ≤ 2 (which is in the interesting domain for modeling white dwarfs or neutron stars; see e.g. [22, p. 28]). 3. A necessary condition which has to be satisfied by the boundary data in order to ˆ guarantee elementary
flatness of the corresponding solution is given by K(0, ±Ri ) = ˆ ±Ri ) (cf. (7)). But, we have not yet worked out a sufficient ln ∂ρ W (0, ±Ri ) = B(0, condition. With reference to Proposition 1, Remark 3, and Remark 12 it is clear that this question is intimately connected with the matching problem, and therefore has to be deferred to future work. 4.3. Regularity. Proposition 5 (Regularity of (weak) solutions). Let (K, U, B, A) be the unique fixed point of the mapping TRi ,pi ,ω : BL → BL in Proposition 4, corresponding to boundary ˆ s , Uˆ s , Bˆ s , Aˆ s ∈ C 2,δ (∂BRi ). Then, (K0 + K, U0 + U, B0 + B, A0 + A) is a regular data K solution for the Dirichlet problem for the interior field Eqs. (63), and we especially have

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K0 + K, U0 + U, B0 + B, A0 + A ∈ Cs2,δ (Ii ) . If, in addition, the equation of state p 7→ (p) is in C k,δ (R+ ) (with k ≥ 1), and the boundary data are in C k+2,δ (∂BRi ), then we have K0 + K, U0 + U, B0 + B, A0 + A ∈ Csk+2,δ (Ii ) . Proof. First of all, note that in Proposition 4 we presuppose p 7→ (p) ∈ C 1 (R+ ) ⊂ C 0,δ (R+ ). Therefore, we only have to prove the last assertion, but for k ≥ 0. Since K0 , U0 , B0 , and A0 are regular solutions for the flat Laplace equation (in 2–5 dimensions, resp.) with axisymmetric boundary data in C k+2,δ (∂BRi ), we have K0 , U0 , B0 , and A0 in Csk+2,δ (Ii ) (see e.g. [4, Theorem 6.19] and use Lemma 5). In order to prove K, U, B, A ∈ Csk+2,δ (Ii ), we first note that K, U, B, A ∈ Cs1,δ (Ii ). This follows if we apply Lemma 8 and Lemma 5 to (K, U, B, A) = T (K, U, B, A) (cf. (71)). Then, we have q# , m# ∈ Cs0,δ (Ii ) (cf. (64) and (65)), which (again due to (71)) yields K, U, B, A ∈ Cs0 (Ii ) ∩ Cs2 (Ii ), and especially the interior field Eqs. (63) are satisfied on Ii (see e.g. [4, Lemma 4.2]). But then, we have K, U, B, A ∈ Cs2,δ (Ii ) (see e.g. [4, Theorem 6.19]; and for k = 0 the procedure stops). Hence, (if k ≥ 1) the nonlinearities q# and m# are in Cs1,δ (Ii ), which yields K, U, B, A ∈ Cs3,δ (Ii ) (again e.g. by [4, Theorem 6.19]; and for k = 1 the procedure stops). Hence, by induction, we have K, U, B, A ∈ Csk+2,δ (Ii ). 

5. Conclusions We have shown that the exterior and interior Dirichlet problem for the stationary and axisymmetric Einstein equations is solvable, if the absolute values of the boundary data are in a characteristic way limited. The corresponding condition yields the “stability conditions” (60). From a physical point of view, such condition are expected in order to guarantee stability against collapse resp. rotational disruption. Presently, we fail to reach the “absolute limits” (61) for both parameters by roughly two orders of magnitude, but an improvement seems possible by optimizing some inequalities, and the function spaces for the solutions. Anyhow, our results apply already to relatively strong gravitational fields: They include the existence-proof for the exterior solutions of all known white dwarf-stars, and they need “only” improvement by 1–2 orders of magnitude to include the neutron stars. Moreover, these results supply a partial proof of the hoop conjecture [23]: Black holes with horizons form if and only if a mass M gets compacted into a region whose circumference in every direction is smaller than 4π M . (Of course, a proof of the “if-part” of the conjecture is not possible with our methods, but this is already partially provided in [21].) Furthermore, the numerical results of [2] give hope that our methods can be extended up to the “absolute stability limits”, to an existence and uniqueness proof for global solutions for rotating stars (cf. Remark 10), and to the Einstein–Maxwell-equations, e.g. to magnetic stars. Acknowledgement. I thank Herbert Pfister for initiating this work, for many helpful discussions and suggestions, and for reading the manuscript. This work was supported, in part, by the Deutsche Forschungsgemeinschaft.

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References 1. Bardeen, J.M.: A Variational Principle for Rotating Stars in General Relativity. Astrophys. J. 162, 71–95 (1970) 2. Bonazzola, S., Gourgoulhon, E., Salgado, M., Marck, J.A.: Axisymmetric rotating relativistic bodies: A new numerical approach for “exact” solutions. Astron. Astrophys. 278, 421–443 (1993) 3. Courant, R., Hilbert, D.: Methoden der mathematischen Physik II. Berlin: Springer-Verlag, 1937 (not contained in the English editions!) 4. Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Berlin and Heidelberg: Springer-Verlag, 1977 5. Heilig, U.: On the Existence of Rotating Stars in General Relativity. Commun. Math. Phys. 166, 457–493 (1995) 6. Kramer, D., Stephani, H., MacCallum, M., Herlt, E.: Exact solutions of Einstein’s field equations. Cambridge: Cambridge University Press, 1980 7. Kreyszig, E.: Introductory functional analysis with applications. New York: John Wiley & Sons, Inc., 1978 8. Lichtenstein, L.: Neuere Entwicklungen der Potentialtheorie. Konforme Abbildungen. In: Encyklop¨adie der mathematischen Wissenschaften, II C 3, Leipzig: Teubner-Verlag, 1921 pp. 177–377 9. Lindblom, L.A.: Fundamental Properties of Equilibrium Stellar Models. PhD-thesis, University of Maryland, 1978 10. Misner, C.W., Thorne, K.S., Wheeler, J.A.: Gravitation. New York: W.H. Freeman and Company, 1973 11. Morrey, C.B.: On the analyticity of solutions of analytic non-linear elliptic systems of partial differential equations. Am. J. Math. 80, 198–237 (1958) 12. Morrey, C.B.: Multiple Integrals in the Calculus of Variation. Berlin and Heidelberg: Springer-Verlag 1966 13. M¨uller zum Hagen, H.: On the analyticity of stationary vacuum solutions of Einstein’s equation. Proc. Camb. Phil. Soc. 68, 199–201 (1970) 14. Neugebauer, G., Meinel, R.: General Relativistic Gravitational Field of a Rigidly Rotating Disk of Dust: Solution in Terms of Ultraelliptic Functions. Phys. Rev. Lett. 75, 3046–3047 (1995) 15. Pfister, H.: Rotating mass shells with flat interiors. Class. Quant. Grav. 6, 487–503 (1989) 16. Pfister, H.: Zur Frage nach globalen L¨osungen der Einsteinschen Feldgleichungen f¨ur rotierende Sterne. Wiss. Zs. Univ. Jena 39, 152–160 (1990) 17. Pfister, H.: Dragging Effects Near Rotating Bodies and in Cosmological Models. In: Barbour, J.B., Pfister, H. (eds.) Mach’s Principle: From Newton’s Bucket to Quantum Gravity. Einstein Studies, Volume 6, Boston: Birkh¨auser, 1995 pp. 315–331 18. Protter, M.H., Weinberger, H.F.: Maximum Principles in Differential Equations. Englewood Cliffs, New Jersey: Prentice-Hall, Inc., 1967 19. Reula, O.: On Existence and Behaviour of Asymptotically Flat Solutions to the Stationary Einstein Equations. Commun. Math. Phys. 122, 615–624 (1989) 20. Schaudt, U.M., Pfister, H.: The Boundary Value Problem for the Stationary and Axisymmetric Einstein Equations is Generically Solvable. Phys. Rev. Lett. 77, 3284–3287 (1996) 21. Schoen, R., Yau, S.T.: The Existence of a Black Hole Due to Condensation of Matter. Commun. Math. Phys. 90, 575–579 (1983) 22. Shapiro, S.L., Teukolsky, S.A.: Black Holes, White Dwarfs, and Neutron Stars. The Physics of Compact Objects. New York: John Wiley & Sons, Inc., 1983 23. Thorne, K.S.: Nonspherical Gravitational Collapse: A Short Review. In: Klauder, J.R. (ed.) Magic Without Magic: John Archibald Wheeler, San Francisco: W.H. Freeman and Company, 1972 pp. 231– 258 24. Wald, R.M.: General Relativity. Chicago: University of Chicago Press, 1984 25. Weinstein, A.: Generalized axially symmetric potential theory. Bull. Am. Math. Soc. 59, 20–38 (1953) Communicated by H. Nicolai

Commun. Math. Phys. 190, 541 – 548 (1998)

Communications in

Mathematical Physics c Springer-Verlag 1997

The Existence of Generalised Wannier Functions for One-Dimensional Systems A. Nenciu1 , G. Nenciu2 1

IFA, IFTM, P.O. Box MG 6, Bucharest, Romania. E-mail: [email protected] Dept. Theor. Phys., Univ. Bucharest P.O. Box MG 11, 76900-Bucharest and Institute of Mathematics of the Romanian Academy, P.O. Box 1-764, 70700 Bucharest, Romania. E-mail: [email protected]

2

Received: 17 February 1997 / Accepted: 15 April 1997

Abstract: It is proved that the subspaces corresponding to bounded, isolated parts of the spectrum of (periodic and nonperiodic) one-dimensional Schr¨odinger operators admit bases of exponentially localised functions (generalised Wannier functions). 1. Introduction The Wannier functions (WF) were defined by Wannier [W] as bases in the subspaces of states corresponding to energy bands in periodic crystals. Formally, they are defined as a Fourier transform with respect to the crystal momentum of the Bloch functions of the corresponding band (for an elementary account, see e.g. [We]). The key property which makes WF so useful, both from the theoretical as well as the computational point of view, is their exponential localisation. Actually, WF play a crucial role in all “generalised effective mass theories”, in particular in the recent rigorous proof of the famous “Peirels Onsager substitution” [H-S, Nb]. Unfortunately (in spite of a contrary widespread opinion [Bl]) the exponential localisation of WF is far from being an easy to prove property. Even more, if a magnetic field is present, localised WF might not exist [D-N] and this is related [Th] to the fact that the existence of localised WF for a magnetic subband implies the fact that the contribution of the corresponding subband to the quantum Hall current vanishes. The first rigorous proof of the existence of exponentially localised WF is contained in a classic paper by Kohn [K]. Using methods of the theory of ordinary differential equations (which do not generalise to higher dimensions), Kohn proved the existence of exponentially localised WF for one-dimensional systems with a center of inversion. For the multidimentional case and simple bands, the existence of localised WF was proved by des Cloizeaux [dCa, dCb] for crystals with a center of inversion and by Nenciu [Na] in the general case; the case of complex bands is still open (see [Nb] for more details). Due to the interest in nonperiodic structures, much effort has been devoted in the last two decades to extend the results about the existence of WF to nonperiodic systems.

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The basic difficulty comes from the fact that the Bloch functions do not exist so, one cannot define the Wannier functions as their Fourier transforms. For this reason, all the results obtained so far [K-O, N-Nb, K, Nb, G-K, R-K] were proved starting from the periodic case and using perturbational or “continuity” arguments. While these techniques give the existence of (generalised) Wannier functions only in some “neighbourhood” of the periodic case, on the basis of the known results it was conjectured [N-Nb, Ni] that generalised Wannier functions (see Sect. 2 for the precise definition) exist for all bounded isolated parts of the spectrum of time reversal invariant (which, in particular, implies the absence of magnetic field!) hamiltonians, irrespective of their periodicity properties. In the present note we prove this conjecture for one-dimensional systems. As already said above, in higher dimensions even the periodic case is not completely settled. Our proof is based on the idea of Kivelson [Ki, Ni] to define the generalised Wannier functions as eigenfunctions of the “band position” operator. On the basis of a bootstrap argument, Niu [Ni] argued that (as far as they exist!) the eigenfunctions of the band position operator decay faster than any power, but it seems that even at the heuristic level there are no arguments for their exponential decay. Our proof of exponential decay rely on some elementary techniques developed in [B-N-N] to prove exponential decay for eigenfunctions of hamiltonians describing some nonperiodic systems.

2. The Problem and the Results Consider in L2 (Rd ), d ≤ 3, the following hamiltonian Z |V (y)|2 dy < ∞, H = P2 + V (x), P = −i∇, sup x∈Rd

(2.1)

|x−y|≤1

which, as is well known, [R-S], is essentially self-adjoint on C0∞ (Rd ). The spectral assumption on H we shall impose is the following ”gap condition”: G. (2.2) σ(H) = σ0 ∪ σ1 with

σ0 bounded and dist(σ0 , σ1 ) = d > 0.

(2.3)

Let P0 be the spectral projection of H corresponding to σ0 and K = P0 L2 (Rd ). In what follows, we shall assume that dim K = ∞.

(2.4)

(For dim K < ∞ the existence of GWB (see the definition below) is given by the exponential decay of eigenfunctions of H, see e.g. [B-N-N].) Definition 1. 1. It is said that K has a generalised Wannier basis (GWB) if there exist: (i) a discrete set Γ ∈ Rd . (ii) constants α > 0, M < ∞ independent of g ∈ Γ and an orthonormal basis in K , {ωg,j (x)}g∈Γ,1≤j≤m(g)<∞ , satisfying

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543

Z |ωg,j (x)|2 exp (2α|x − g|)dx ≤ M.

(2.5)

Rd

2. Suppose in addition that V (x) is periodic ( ˆ = V (x + gˆ ) = V (x); gˆ ∈ Γ

gˆ =

d X

) mk gk | mk ∈

Z, {gk }d1

− basis in R

d

.

k=1

Then K has a Wannier basis (WB) if it has a GWB and, in addition ˆ =Γ 0 ωgˆ ,j (x) = ω0,j (x − gˆ ). In the above setting one can formulate now what is believed to be true about the existence of GWB (see, e.g., [N-Nb]). Conjecture 1. Under the gap assumption (2.2),(2.3), GWB, and in the periodic case WB, always exist. The result of this paper is the proof of Conjecture 1 for d = 1. Our proof substantiates the recent proposal by physicists [Ki, Ni] to define the generalise Wannier functions as the eigenfunctions of the “band position” operator. In particular, we obtain a proof of Kohn’s result without the use of Bloch functions. Our basic technical result is contained in: Theorem 1. Let H be the unique self-adjoint realisation in L2 (R) of −

d2 + V (x), dx2

Z sup x∈R

|x−y|≤1

|V (y)|2 dy < ∞

defined on C0∞ (R). Suppose σ(H) = σ0 ∪ σ1 with σ0 bounded and dist(σ0 , σ1 ) = d > 0. Let X be the position operator in L2 (R), P0 be the spectral projection of H corresponding to σ0 and consider in K ≡ P0 L2 (R) the band position operator Xˆ = P0 XP0 defined on ˆ = D(X) ∩ K. D(X) Then (i) (ii) (iii)

ˆ Xˆ is self-adjoint on D(X). Xˆ has only discrete spectrum. ˆ Let g ∈ 0 ≡ σ(X), mg its multiplicity and 9g,j , 1 ≤ j ≤ mg be a normalised eigenfunction of Xˆ corresponding to g. Then, there exist α > 0, M < ∞ independent of g and j such that Z exp(2α|x − g|)|9g,j (x)|2 dx ≤ M. R

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The theorem proves at once Conjecture 1 for d = 1. Indeed, the normalised eigenfunctions of Xˆ give a GWB. Suppose now that V is periodic and let Tgˆ be the translation ˆ From [P0 , Tgˆ ] = 0 it follows that operator Tgˆ 9(x) = 9(x − g).   Tgˆ , Xˆ = −gT ˆ gˆ , which implies that if ˆ g,j = g9g,j , X9 then ˆ gˆ 9g,j = (gˆ + g)Tgˆ 9g,j , XT

(2.6)

ˆ is a union of ladders. Since σ(X) ˆ is discrete, then the number which shows that σ(X) of ladders is finite which together with (2.6) proves the existence of WB.

3. Proof of Theorem 1
1 Let < Xa >, a ∈ R be the operator of multiplication with (x − a)2 + 1 2 . One of the crucial ingredients of the proof of Theorem 1 is the exponential decay of the integral kernal of (H − z)−1 . Lemma 1. Let C be a contour of finite length enclosing σ0 . Then, there exists α > 0 such that kexp(α < Xa >)(H − z)−1 exp (−α < Xa >)k < ∞,

(3.1)

kexp(α < Xa >)P (H − z)−1 exp (−α < Xa >)k < ∞,

(3.2)

k < Xa > (H − z)−1 < Xa >−1 k < ∞,

(3.3)

sup z∈0,a∈R

sup z∈0,a∈R

sup z∈0,a∈R

d . where P is the momentum operator P = −i dx

Lemma 1 is an elementary fact (see e.g. [Nb]) in the Combes-Thomas-Agmon theory of weighted estimates. The contour in Lemma 1 is to be chosen to make α as large as possible. From the Riesz formula Z 1 (H − z)−1 dz, (3.4) P0 = 2πi C one obtains from (3.1) and (3.3) the estimates: sup kexp(α < Xa >)P0 exp (−α < Xa >)k < ∞,

(3.5)

a∈R

sup k < Xa > P0 < Xa >−1 k < ∞.

(3.6)

a∈R

Consider now, for µ > 0, the operator Rˆ ±µ = P0 (X ± iµ)−1 P0 ,

(3.7)

Generalised Wannier Functions for One-Dimensional Systems

545

where X is the position operator in L2 (R) (i.e. operator of multiplication with the variable ˆ and (as operators in K) x). Due to (3.6), Rˆ ±µ K ⊂ D(X) (Xˆ ± iµ)Rˆ ±µ = P0 (X ± iµ)P0 (X ± iµ)−1 P0 = 1K + Aˆ ±µ

(3.8)

Aˆ ±µ = P0 [X, P0 ](X ± iµ)−1 P0 .

(3.9)

with

Now (see e.g. [A, Nb, N-Na]), [X, P0 ] defined on D(X) has a bounded closure in L2 (R) so that for sufficiently large µ, 1 (3.10) kAˆ ±µ k ≤ , 2 and then (again as operators in K) (Xˆ ± iµ)Rˆ ±µ (1K + Aˆ ±µ )−1 = 1K ,

(3.11)

ˆ . By which means that Xˆ ± iµ is surjective on Rˆ ±µ (1K + Aˆ ±µ )−1 K ⊂ D(X) ˆ ˆ In the fundamental criterion of self-adjointness [R-S] X is self-adjoint in K on D(X). addition, from (3.11) (3.12) (Xˆ ± iµ)−1 = Rˆ ±µ (1K + Aˆ ±µ )−1 . Using the Riesz formula (see (3.4)) and the fact that P02 = P0 , one can rewrite Rˆ ±µ as follows: Z 1 ˆ P0 (X ± iµ)−1 (H − z)−1 dzP0 , (3.13) R±µ = 2πi 0 and since (X ± iµ)−1 (H − z)−1 is compact in L2 (R) (see e.g. [C-F-K-S]) one obtains that Rˆ ±µ is compact in K, and then from (3.12) it follows that (Xˆ ± iµ)−1 is compact in K. As a consequence [R-S], Xˆ has discrete spectrum ˆ ≡ 0. ˆ = σdisc (X) σ(X)

(3.14)

Let now g ∈ 0 be an eigenvalue of Xˆ and 9g,j , 1 ≤ j ≤ mult g , a normalised eigenfunction of Xˆ corresponding to g. The eigenvalue equation for 9g,j can be written as (3.15) P0 (X − g)P0 9g,j = 0. Let b > 0 (sufficiently large), a ∈ R, 1 fa,b (x) = bf ( (x − a)), b

(3.16)

where f (x) is a real C ∞ function satisfying 0 ≤ f (x) ≤ 1,  1 for |x| ≤ 21 f (x) = 0 for |x| ≥ 1, and By construction

ha,b (x) = x − a + ifa,b (x).

(3.17)

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b 2

(3.18)

d d2 ha,b (x)| + | 2 ha,b (x)|} = K < ∞. dx dx

(3.19)

|ha,b (x)| ≥ and sup

{|

x,a∈R,b≥1

With (3.17), (3.15) can be rewritten as P0 hg,b P0 9g,j = iP0 fg,b P0 9g,j

(3.20)

(here hg,b , fg,b denote the operators of multiplication with hg,b (x) and fg,b (x) respectively). Like in the proof of self-adjointness of Xˆ we compute h i −1 −1 P P P h P = 1 + P h , h (3.21) P0 h−1 0 0 g,b 0 K 0 0 g,b g,b g,b P0 .   R  1 (H − z)−1 , hg,b dz = P0 , hg,b = 2πi C  2  R 1 −1 (3.22) P , hg,b dz(H − z)−1 = 2πi C (H − z) 

R −1 d d −1 −1 (H − z) h h dz, P + P (H − z) 2π C dx g,b dx g,b   which implies that P0 , hg,b is bounded uniformly (see (3.19)) for g ∈ R, b ≥ 1. As a consequence, there exists b0 < ∞ such that for b ≥ b0 and all g ∈ R (see (3.18)) Now



1 kBˆ g,b k ≤ , 2 where

(3.23)

h i −1 Bˆ g,b = P0 h−1 g,b P0 , hg,b P0 : K → K.

(3.24)

From (3.21) and (3.23) it follows that (1 + Bˆ g,b )−1 P0 h−1 g,b P0 hg,b P0 = 1K .

(3.25)

With (3.25), (3.20) takes the form 9g,j = i 1 + Bˆ g,b


−1

P0 h−1 g,b P0 fg,b P0 9g,j ,

(3.26)

and then exp(α < Xg >)9g,j = i exp(α < Xg >) 1 + Bˆ g,b


−1

P0 h−1 g,b P0 fg,b P0 9g,j . (3.27)

Since by construction (see the definition of fg,b ), k exp(α < Xg >)fg,b k ≤ b exp(α(b + 1)) from (3.27), it follows that in order to prove that k exp(α < Xg >)9g,j k ≤ M < ∞, it is sufficient to prove that there exists b < ∞ such that
−1 Fg,b ≡ exp(α < Xg >) 1 + Bˆ g,b P0 h−1 g,b P0 exp(−α < Xg >) is bounded uniformly in g. Rewriting Fg,b in the form

(3.28)

Generalised Wannier Functions for One-Dimensional Systems

Fg,b = (

∞ X

Ajg,b )Qg h−1 g,b Qg ,

547

(3.29)

j=0

where Qg = exp(α < Xg >)P0 exp(−α < Xg >), Ag,b = − exp(α < Xg >)Bˆ g,b exp(−α < Xg >),

(3.30)

and taking into account (3.5), we are left with the proof that for sufficiently large b, kAg,b k ≤

1 . 2

(3.31)

From (3.18) and the definition of Ag,b (see (3.30) and (3.24)) kAg,b k ≤

  2 kQg k2 k exp(α < Xg >) P0 , hg,b exp(−α < Xg >)k b

(3.32)

so, in order to finish the proof we need to show that, uniformly in g ∈ R and b sufficiently large, the last norm in (3.32) is bounded. Repeating the computation in (3.22), one obtains   exp(α < Xg >) P0 , hg,b , exp(−α < Xg >) = R −1 −1 exp(−α < Xg >) π C exp(α < Xg >)(H − z)
d −1 hg,b P (H − z) exp(−α < Xg >)dz+ R dx 1 exp(α < Xg >)(H − z)−1 exp(−α < Xg >)  22πi C d −1 exp(−α < Xg >)dz, dx2 hg,b exp(α < Xg >)(H − z) which together with (3.1), (3.2) and (3.19) finishes the proof.

References Avron, J. E.: On the spectrum of p2 + V (x) + x, with V periodic and  complex. J.Phys.A 12,2393–2398 (1979) [B-N-N] Boutet de Monvel-Berthier, A., Nenciu, A., Nenciu, G.: Perturbed Periodic Hamiltonians: Essential Spectrum and Exponential Decay of Eigenfunctions. Lett.Math.Phys. 34, 119 (1995) [Bl] Blount, E. I. Formalism of Band Theory. In : Seitz, F., Turnbull, D. (eds.) Solid State Phys. 13,305–373 (1962) New York, London: Academic Press [C-F-K-S] Cycon, H. L., Froese, R. G., Kirsch, W., Simon, B.: Schr¨odinger operators with application to quantum mechanics and global geometry. Berlin, Heidelberg, New-York: Springer-Verlag 1987 [dCa] des Cloizeaux, J.: Energy Bands and Projection Operators in Crystals: Analytic and Asymptotic Properties. Phys.Rev. 135, A685–A697 (1964) [dCb] des Cloizeaux, J.: Analytic Properties of n-Dimensional Energy Bands and Wannier Functions. Phys.Rev. 135, A698–A707 (1964) [D-N] Dubrovin, B. A., Novikov, S. P.: Ground state of a two-dimension electron in a periodic magnetic field. Zh.Eksp.Teor.Fiz. 79, 1006–1016 (1980) [G-K] Geller, M. R., Kohn, W.: Theory of generalised Wannier functions for nearly periodic potentials. Phys. Rev. 48, 14085–14088 (1993) [H-S] Helffer, B., Sj¨ostrand, J.: Equation de Schr¨odinger avec champs magn´etique et e´ quation de Harper, Springer Lectures Notes in Mathematics. 345, 118–197 (1989). [Ki] Kievelsen, S.: Wannier functions in one-dimensional disordered systems: Application to fractionally charged solitons. Phys.Rev. B26, 4269–4274 (1982)

[A]

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[K]

Kohn, W.: Analytic Properties of Bloch Waves and Wannier Functions. Phys. Rev. 115, 809–821 (1959) Kohn, W., Onffroy, J.: Wannier Functions in a Simple Nonperiodic System. Phys.Rev. B8, 24852495 (1973) Nenciu, G.: Existence of the Exponentially Localised Wannier Functions. Comm.Math.Phys. 91, 81–85 (1983) Nenciu, G.: Dynamics of band electrons in electric and magnetic fields: rigorous justification of effective Hamiltonians. Rev.Mod.Phys. 63, 91–127 (1991) Nenciu, A., Nenciu, G.: Dynamics of Bloch Electrons in External Electric Fields: I.Bounds for Interband Transitions and Effective Wannier Hamiltonians. J.Phys.A:Math.Gen. 14 2817–2827 (1981) Nenciu, A., Nenciu, G.: Existence of Exponentially Localised Wannier Functions for Nonperiodic Systems. Phys.Rev. B47, 10112–10115 (1993) Niu, Q.: Theory of the quantized adiabatic particle transport. Modern Physics Letters B14,15, 923–931 (1991) Reed, M., Simon, B: Methods of Modern Mathematical Physics: II. Fourier Analysis. Selfadjointness. New York: Academic Press 1975 Rehr, J. J., Kohn, W.: Wannier functions in crystals with surfaces. Phys.Rev. B10, 448–455 (1973) Thouless, D. J.: Wannier functions for magnetic sub-bands. J.Phys.C:Solid State Phys. 17, L325– L327 (1984) Wannier, G. H.: The Structure of Electronic Excitation Levels in Insulating Crystals. Phys.Rev. 52, 191–197 (1937) Weinreich, G.: Solids: Elementary Theory for Advanced Students. New York: Wiley, 1965

[K-O] [Na] [Nb] [N-Na]

[N-Nb] [Ni] [R-S] [R-K] [Th] [W] [We]

Communicated by B. Simon

Commun. Math. Phys. 190, 549 – 583 (1998)

Communications in

Mathematical Physics c Springer-Verlag 1997

c Level-0 Action of Uq (sl n ) on the q -Deformed Fock Spaces Kouichi Takemura, Denis Uglov Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606, Japan. E-mail: [email protected], [email protected] Received: 14 August 1996 / Accepted: 18 April 1997

b n ) we define a level-0 Abstract: On the level-1 Fock space modules of the algebra Uq (sl 0 b action U0 of the Uq (sln ), and an action of an Abelian algebra of conserved Hamiltonians commuting with the U0 . An irreducible decomposition of the Fock space with respect to the level-0 action is derived by constructing a base of the Fock space in terms of the Non-symmetric Macdonald Polynomials.

1. Introduction The present article is concerned with the subject of the recently discovered additional symmetry structure in the representation theory of quantum affine algebras. In the context of infinite-dimensional representation theory this symmetry structure manifests itself as b n ) on a level-1 highest weight module a level-0 action of the quantum affine algebra Uq (sl b of the same algebra Uq (sln ). The first developments in this subject were made in the classical setting where the additional symmetry appears as a Yangian action on a level-1 highest weight module of b n . Starting with the pioneering work [11] in which actions of the the affine Lie algebra sl b 2, Yangian Y (sl2 ) were defined on the irreducible level-1 highest weight modules of sl it was apparent that there exists an intimate relation between the Yangian or the level-0 actions and Long-Range Interacting Models such as the Calogero-Sutherland Model with spin defined in [7], and the Haldane-Shastry spin chain [2, 3]. This relation initially was understood in the framework of the Haldane-Shastry spin chain. In this case the b 2 modules are certain Field Theory generators of the Yangian symmetry of the level-1 sl limits of the finite particle Yangian generators that commute with the finite particle spin chain Hamiltonian. The spinon excitations of the Haldane Shastry Model are identified in the Field Theory limit with elements of a basis – called the spinon basis – which is generated from the highest weight vector by components of the chiral vertex operator in

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b 2 . In [3, 4] the irreducible level-1 sl b2 an irreducible level-1 highest weight module of sl modules were decomposed into direct sums of irreducible finite-dimensional Yangian representations by considering the action of the Yangian on the spinon basis. The Haldane-Shastry spin chain is a special case of a more general solvable model which is the Calogero-Sutherland Model with spin [2]. The former model is obtained from the latter in the limit of strong interaction where the dynamical and the spin degrees of freedom separate and it becomes possible to project the dynamical degrees of freedom out of the space of states. In several respects the structure of the Calogero-Sutherland Model with spin is simpler than that one of the Haldane-Shastry spin chain because the former model is generic and the latter is a specialization in the total family of CalogeroSutherland Models parameterized by values of the coupling constant. In view of the link that exists between the Haldane-Shastry Model and the level-1 representations of affine Lie algebras, it is not surprising that a similar link exists in the generic case of the Calogero-Sutherland Model with spin. In the simplest case of twovalued spin the manifestation of the Calogero-Sutherland Model in the representation theory of affine Lie algebras is the Yangian action on a level-1 fermionic Fock space b2 b 2 [21]. Unlike the action on an irreducible sl module [16] of the affine Lie algebra sl module, the Yangian action on a Fock space depends on a parameter which is just the coupling constant of the Calogero-Sutherland Model with spin. At generic values of this parameter the decomposition of a Fock space as a Yangian module is irreducible and each of the irreducible components is isomorphic to a complete tensor product of fundamental finite-dimensional evaluation modules [5] of the Yangian [21]. This feature of the decomposition of a Fock space is in contrast with the more complicated structure b 2 modules where, in general, of irreducible Yangian decomposition of the irreducible sl an irreducible component is a subquotient of a tensor product of fundamental Yangian representations [3, 4]. This observation is in agreement with the generic nature of the Calogero-Sutherland Model. Analogues of the Yangian actions in the context of representation theory of quantum b n ) on level-1 Uq (sl b n ) modules. In affine algebras are level-0 actions of the algebra Uq0 (sl 0 b the work [14] a level-0 action of Uq (sl2 ) was defined on each of the irreducible level-1 b 2 ) modules. To obtain this level-0 action an analogue of the spinon basis created Uq (sl by components of the type I vertex operator was used. The generators of the level-0 action act on certain generating series for elements of this basis. The explicit form of these generators is obtained by the functorial construction, due to Cherednik [6] and Chari and Pressley [5], from a representation of the affine Hecke algebra by DemazureLusztig operators and commuting Cherednik’s difference operators [7]. The Cherednik’s operators, in general, depend on two parameters: q which is the deformation parameter of the quantum affine algebra, and p which determines the value of the difference, and is analogous to the coupling constant of the Calogero-Sutherl and Model with spin. The value of the parameter p needs to be specialized as a certain fixed power of q in order to b 2 ) module to be well-defined [14]. make the level-0 action on an irreducible Uq (sl b 2 ) module with respect The irreducible decomposition of an irreducible level-1 Uq (sl to the level-0 action was investigated in the work [15]. In parallel with the classical case, an element of this decomposition is in general isomorphic to a subquotient of a tensor b 2 ). The appearance product of finite-dimensional fundamental representations of Uq0 (sl of subquotients may be seen as a consequence of the specialization of the parameter p in Cherednik’s operators.

Level-0 Action of Uq (sbln ) on q-Deformed Fock Spaces

551

Let us now describe the content of our present work. Our motivation here is to investigate a q-deformation of the Yangian action on the level-1 classical Fock space b n ) were defined modules. The level-1 quantum Fock space modules of the algebra Uq (sl in the work [12]. Recently a new explicit realization of these modules was proposed in [17]. This realization is given in terms of the semi-infinite q-wedge vectors and is a q-deformation of the well-known semi-infinite wedge construction of the classical b n [16]. In [17] it was shown that the q-deformed fermionic Fock space modules of sl Fock space admits an action of the Heisenberg algebra which centralizes the level-1 b n ). As a U 0 (sl b n )-Heisenberg bimodule the q-deformed Fock space is action of Uq0 (sl q isomorphic to the tensor product of a level-1 irreducible highest weight representation b n ) and the bosonic Fock space representation of the Heisenberg algebra. This of Uq0 (sl isomorphism gives the irreducible decomposition of the q-deformed Fock space as a b n ). level-1 module of Uq0 (sl b n ) on a q-deformed In the present paper we define another, level-0, action of Uq0 (sl Fock space. Our construction of the level-0 action is methodologically different from the construction of the work [14], but there are certain common features. The q-deformed b n ). The level-0 Fock space is graded by the degree generator of the level-1 action of Uq (sl action on the q-deformed Fock space is defined so that each homogeneous component of this grading is invariant. In order to construct the level-0 action on a given homogeneous component we identify this component with the projective limit of a certain infinite b n ). The underlying linear space of sequence of finite-dimensional representations of Uq0 (sl a representation from this sequence is a subspace in the finite q-wedge product introduced b n ) action on each of these finite-dimensional representations is given in [17]. The Uq0 (sl in a manner similar to that one of [14], that is by the functor [6], [5] from a module of the affine Hecke algebra. This module is the same as the one used in [14], but now the parameter p in Cherednik’s operators is allowed to be an arbitrary complex number. b n ) modules is equipped with a projection The sequence of the finite-dimensional Uq0 (sl b n ) actions and hence a between adjacent elements. This projection intertwines the Uq0 (sl 0 b level-0 action of Uq (sln ) is defined on the projective limit of the sequence. This limit is identified with a homogeneous component of the q-deformed Fock space and hence a level-0 action on the entire Fock space, which is a direct sum of the homogeneous components, is defined. By construction this level-0 action depends on the parameter p. Apart from giving the definition of the level-0 action we investigate the decompob n ) module. At generic values of sition of the q-deformed Fock space as a level-0 Uq0 (sl the parameter p this decomposition is irreducible, and each of the irreducible components is isomorphic to a tensor product of the fundamental, in the terminology of [5], b n ) modules. finite-dimensional Uq0 (sl The arrangement of the paper is as follows. Section 2 has an introductory character, here we give the relevant background information on the affine Hecke algebra and qwedge products. In Sect. 3 the space of finite q-wedges is decomposed with respect to b n )-action. In Sect. 4 we define the level-0 action and the family a certain level-0 Uq0 (sl of commuting Hamiltonians in the Fock space. These Hamiltonians commute with the level-0 action. Section 5 contains results on the decomposition of the Fock space with respect to this action. In this section we use the results of Sect. 3.

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2. Preliminaries In this introductory section we set up our notations and collect several known definitions and results to be used starting from Sect. 3. For details one may consult the works [17, 2, 8, 19]. b N (q) is an associative algebra 2.1. Affine Hecke algebra. The affine Hecke algebra H ±1 generated by elements Ti ( i = 1, . . . , N − 1) and yj (j = 1, . . . , N ). These elements satisfy the following relations: Ti2 Ti Ti+1 Ti Ti T j yi yj yi Tj

= = = = =

(q − q −1 )Ti + I, Ti+1 Ti Ti+1 , Tj Ti if |i − j| > 1, yj y i , T j yi if i 6= j, j + 1,

Ti yi = yi+1 Ti−1 .

(2.1) (2.2) (2.3) (2.4) (2.5) (2.6)

b N (q) in C[z ±1 , . . . , z ±1 ]. We consider two different actions of H 1 N The first of these actions is defined by: yi = zi−1 ,

and Ti = gi,i+1 ,

(2.7)

where the operators gi,j are as follows [2] : gi,j =

q −1 zi − qzj (Ki,j − I) + q, zi − zj

1 ≤ i 6= j ≤ N,

(2.8)

and Ki,j is the permutation operator for variables zi and zj . b N (q) is specified by: The other action of H yi = q 1−N Yi(N ) ,

and

Ti = gi,i+1 ,

(2.9)

here Yi(N ) are Cherednik’s difference operators [7]: −1 −1 Yi(N ) = gi,i+1 Ki,i+1 . . . gi,N Ki,N pDi K1,i g1,i . . . Ki−1,i gi−1,i ,

(2.10)

and pDi f (z1 , . . . , zi , . . . , zN ) = f (z1 , . . . , pzi , . . . , zN ),

±1 f ∈ C[z1±1 , . . . , zN ].

Throughout this paper the q is taken to be a complex number which is not a root of unity (q = 1 is allowed). We call such q and a p ∈ C \ q 2Q≥0 generic, and in what follows we consider only generic q and p unless stated otherwise. b N (q) (2.7, 2.9) can be glued together so that an Remark. The two different actions of H action of a more general object: the double affine Hecke algebra [7, 8] or the toroidal ±1 ]. Hecke algebra [22] is defined on C[z1±1 , . . . , zN

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553

2.2. Eigenfunctions of the operators Yi(N ) . To each λ := (λ1 , λ2 , . . . , λN ) ∈ ZN λN corresponds a monomial zλ := z1λ1 z2λ2 . . . zN of the total degree |λ| := λ1 +λ2 +. . .+λN . + Let λ be the unique partition ( we permit negative parts ) obtained by ordering the elements of λ: λ+ = (λ+1 ≥ λ+2 ≥ . . . ≥ λ+N ), λ+σ(i) = λi for a suitable permutation σ := {σ(1), σ(2), . . . , σ(N )} of the set {1, 2, . . . , N }. We fix such a σ uniquely by requiring σ(i) < σ(j) whenever i < j and λi = λj . There is a partial order relation in ZN . First, for two partitions λ+ = (λ+1 ≥ λ+2 ≥ . . . ≥ λ+N ) and µ+ = (µ+1 ≥ µ+2 ≥ . . . ≥ µ+N ) the partial order is defined by: λ +  µ+

⇔

i X j=1

λ+j ≥

i X

µ+j

(i = 1, 2, . . . , N ),

|λ+ | = |µ+ |.

(2.11)

j=1

This order is extended to ZN as follows. For λ, µ ∈ ZN put λ  µ if: (2.12) either λ+  µ+ , or λ+ = µ+ and the last non-zero element in (λ1 − µ1 , λ2 − µ2 , . . . , λN − µN ) is negative. It is straightforward to verify that the action of ξi,j := Ki,j gi,j and the Yi(N ) in the monomial basis is triangular. More precisely:  −1 λ P µ (λi < λj ) ,  q z + λ+ µ+ c(λ, µ)z λ λ (i) ξi,j z = qz (λi = λj ), P  λ qz + (q − q −1 )z(i,j)λ + λ+ µ+ c(λ, µ)zµ (λi > λj ), where (i, j)λ := λ|λi ↔λj . (ii)

Yi(N ) zλ = pλi q 2σ(i)−N −1 zλ +

P λµ

c(λ, µ)zµ .

Let us put ζi (λ) := pλi q 2σ(i)−N −1 for λ ∈ ZN and i = 1, 2, . . . , N . Since for generic p and q the equality ζi (λ) = ζi (µ) (i = 1, 2, . . . , N ) implies λ = µ we immediately come to the conclusion that the Yi(N ) admit a common eigenbasis {8λ (z) | λ ∈ ZN }: Yi(N ) 8λ (z) = ζi (λ)8λ (z), and 8λ (z) = zλ +

X

(i = 1, 2, . . . , N ),

c(λ, µ)zµ .

(2.13)

(2.14)

λµ

Following [8] we will refer to the Laurent polynomials 8λ (z) as Non-symmetric Macdonald Polynomials ( of type A ). These polynomials were introduced in [19]. The action of the finite Hecke algebra generators gi,i+1 in the basis {8λ (z) | λ ∈ ZN } is summarized as follows [7, 8, 19]: gi,i+1 8λ (z) =Ai (λ)8λ (z) + Bi (λ)8(i,i+1)λ (z), where (i, i + 1)λ := λ|λi ↔λi+1 and: ( −1 q {x} (λi > λi+1 ); (q − q −1 )x Ai (λ) := , Bi (λ) := 0 (λi = λi+1 ); x−1 (λi < λi+1 ), q −1

(2.15)

(2.16)

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K. Takemura, D. Uglov

{x} :=

(x − q 2 )(q 2 x − 1) , (x − 1)2

Note that when i is such that λi = λi+1 we have q8λ (z).

x := ζi+1 (λ) ζi (λ)

ζi+1 (λ) . ζi (λ)

(2.17)

= q 2 and hence gi,i+1 8λ (z) =

2.3. Finite q-wedge Product . Let n ≥ 2 and N be positive integers. We set: V := Cn with a base {v1 , v2 , . . . , vn } and V (z) := C[z ±1 ] ⊗ V with a base {z m v }, m ∈ Z ,  ∈ {1, 2, . . . , n}. Often it will be convenient to set k =  − nm , uk := z m v . Then {uk | k ∈ Z} is a base in V (z). In what follows we will use both notations: uk and z m v switching between them without further alert. The q-wedge product of spaces V (z) is ±1 ]⊗ defined as a suitable quotient of the tensor product ⊗N V (z) ∼ = C[z1±1 , z2±1 , . . . , zN N (⊗ V ). To describe this quotient introduce an action of the finite Hecke algebra in ⊗N V : Ti = Si,i+1 , S = −q −1

(i = 1, 2, . . . , N − 1) X E , ⊗ E ,

1≤≤n

+ (q − q

−1

)

X

E

,

⊗E

1≤<0 ≤n

0 ,0

−

X

(2.18) E

,0

⊗E

0 ,

,

1≤6=0 ≤n

0

0

where E  , ∈ End(V ) is specified by E  , vα = δ,α v0 and Si,i+1 signifies S acting in the ith and i + 1th factors in ⊗N V . Remark. The Hecke generators Ti that are used in [17] are related to the generators which we use in this paper as follows: −1 ) − I. Ti = qKi,i+1 (gi,i+1 + Si,i+1

(2.19)

Now define (⊂ ⊗N V (z)) as: =

N −1 X i=1

−1 Ker(gi,i+1 + Si,i+1 ).

(2.20)

In this setting the q-wedge product ∧N V (z) is defined as the quotient: ∧N V (z) = ⊗N V (z)/.

(2.21)

This definition is equivalent to the definition in [17] due to the remark above. Notice that for q = 1 the q-wedge product is just the usual exterior (wedge) product of the spaces V (z). In what follows we will use the term “wedge product” always for the q-deformed wedge product (2.21). Let 3 : ⊗N V (z) → ∧N V (z) be the quotient map specified by (2.21). The image of a pure tensor uk1 ⊗ uk2 ⊗ . . . ⊗ ukN under this map is called a wedge and is denoted by: uk1 ∧ uk2 ∧ . . . ∧ ukN := 3(uk1 ⊗ uk2 ⊗ . . . ⊗ ukN ).

(2.22)

In [17] it is proven that a basis in ∧N V (z) is formed by the normally ordered wedges, that is the wedges (2.22) such that k1 > k2 > . . . > kN . Any wedge can be written as a linear combination of the normally ordered wedges by using the normal ordering rules [17]:

Level-0 Action of Uq (sbln ) on q-Deformed Fock Spaces

555

ul ∧ um = −um ∧ ul , for l = m mod n, ul ∧ um = −qum ∧ ul + (q 2 − 1)(um−i ∧ ul+i − qum−n ∧ ul+n + +q 2 um−n−i ∧ ul+n+i + . . .), for l < m, m − l = i mod n, 0 < i < n.

(2.23) (2.24)

The sum above continues as long as the wedges in the right-hand side are normally ordered. b n ) in the wedge product. With each of the two H b N (q) actions 2.4. Actions of Uq0 (sl b n ) in the tensor product ⊗N V (z). Let (2.7,2.9) we associate an action of the algebra Uq0 (sl 0

±1 Ei, denote the operator which acts trivially in all factors in C[z1±1 , . . . , zN ]⊗(⊗N Cn ) 0 except the ith factor in ⊗N Cn , where it acts as E , . The action of the generators b n ) (our conventions on U 0 (sl b n ) are sum{E , F , K }, ( ∈ {0, 1, . . . , n − 1}) of Uq0 (sl q marized in the Appendix) then is defined as follows:  K = K1 K2 . . . KN ,

E =

N X

−δ0,

yi

,

Ki = q Ei

−Ei+1,+1

,

(2.25)

  Ei,+1 Ki+1 . . . KN ,

(2.26)

i=1

F =

N X

δ

 yi 0, (K1 )−1 . . . (Ki−1 )−1 Ei+1, ,

 = 0, . . . , n − 1 ,

(2.27)

i=1

where in the right-hand side we regard the indices , + 1 modulo n. b n ) action which was The substitution yi = zi−1 in these expressions gives the Uq0 (sl (N ) considered in [17]. Denote this action by U1 . The other choice of the affine Hecke b n ) in ⊗N V (z). This algebra generators: yi = q 1−N Yi(N ) gives another action of Uq0 (sl action is the principal object of study in the present paper. We denote it by U0(N ) . b N (q) is generated by symmetric polynomials in y ±1 so we consider The centre of H i two Abelian algebras: H1(N ) with generators [17]: a , Ba(N ) := z1a + z2a + . . . + zN

(a ∈ {±1, ±2, . . . , });

(2.28)

And H0(N ) with generators [2]: ) := (q 1−N Y1(N ) )a + (q 1−N Y2(N ) )a + . . . + (q 1−N YN(N ) )a , h(N a

(a ∈ {±1, ±2, . . . , }). (2.29) Obviously Hj(N ) commutes with Uj(N ) for j = 0, 1. The commutative family H0(N ) ) consists of the conserved charges h(N a of the trigonometric Ruijsenaars model with spin [18]. This Model is a q-deformation of the Calogero-Sutherland Model with spin which was introduced in [2]. It is straightforward to verify by using the relations of the affine Hecke algebra, and the explicit form of the operator S (2.18) that the actions Uj(N ) and Hj(N ) for j = 0, 1 preserve the subspace . This implies that Uj(N ) and Hj(N ) , (j = 0, 1) are well-defined in the wedge product ∧N V (z), and from now on we consider these actions as defined in ∧N V (z).

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K. Takemura, D. Uglov

b n ) actions U (N ) (j = 0, 1) Remark. The appearance, and the explicit form of the two Uq0 (sl j finds its explanation in the theory of the toroidal quantum group defined in [10, 22] and, more precisely, in the existence of the Schur-type duality between the toroidal Hecke algebra and the toroidal quantum group established in [22]. In the present paper we will b n ) actions any further. not discuss this very important interpretation of the two Uq0 (sl 2.5. Yangian symmetry of the Calogero-Sutherland Model with spin. The limit of the b n )-action U (N ) , where we set p = q −2α , (α ∈ R) and q → 1 is closely related to the Uq0 (sl 0 Yangian action that appears in the context of the Calogero-Sutherland Model with spin [7, 2]. It is often convenient to consider this limit along with the complete q-deformed case because it is technically simpler to deal with. In particular the Field Theory limit N → ∞ of the Yangian action is constructed in a rather straightforward way [21] and may serve as a template for the more complicated construction in the q-deformed situation which we discuss in Sect. 4. Here we briefly review the Yangian action on the classical (i.e. q = 1) finite wedge product in the simplest case of the Yangian Y (sl2 ). Let σ 1 , σ 2 , σ 3 be the Pauli matrices which act in the basis {v1 , v2 } of C2 by √ ( = 1, 2). (2.30) σ 1 v = v3− , σ 2 v = −1(−1) v3− , σ 3 v = (−1)+1 v , The Yangian action is defined in terms of the differntial Dunkl operators d1 , . . . , dN that are obtained from Cherednik’s operators Y1(N ) , . . . , YN(N ) in the limit where p = q −2α and q → 1 : Yi(N ) = 1 − 2(q − 1) (di + N ) + O((q − 1)2 ),

(i = 1, . . . , N ).

(2.31)

The explicit expressions for the Dunkl operators are given by di = αzi

X zj X zi ∂ −i+ (Ki,j − 1) − (Ki,j − 1), ∂zi z − zi z − zj j>i j j
(i = 1, . . . , N ).

(2.32) Together with the permutation operators Ki,i+1 (i = 1, . . . , N − 1) the Dunkl operators satisfy the defining relations of the degenerate affine Hecke algebra: Ki,i+1 Ki+1,i+2 Ki,i+1 = Ki+1,i+2 Ki,i+1 Ki+1,i+2 , [Ki,i+1 , Kj,j+1 ] = 0 (|i − j| ≥ 2), Ki,i+1 di − di+1 Ki,i+1 = 1, [Ki,i+1 , dj ] = 0, |i − j| ≥ 2, [di , dj ] = 0. These relations are obtained from the defining relations of the affine Hecke algebra (2.1)–(2.6) as the degeneration q → 1, Ti → Ki,i+1 , yi → 1 − 2(q − 1)di + O((q − 1)2 ). The Yangian Y (sl2 ) has six generators Qa0 , Qa1 (a = 1, 2, 3) [9]. The tensor product ±1 ] ⊗ (⊗N C2 ) = ⊗N V (z) admits a Yangian action that is given by [2]: C[z1±1 , . . . , zN Qa0

=

N X

σia ,

i=1

Qa1 =

N X i=1

(2.33) √

di σia +

−1 2

X 1≤i<j≤N

abc σib σjc ,

(2.34)

Level-0 Action of Uq (sbln ) on q-Deformed Fock Spaces

557

where the abc is the totally antisymmetric symbol normalized such that 123 = 1. The expressions (2.33), (2.34) in fact define an action of Y (sl2 ) on the classical wedge product ∧N V (z) defined as the quotient of the tensor product ⊗N V (z) over the subspace N −1 X

q=1 =

Ker(Ki,i+1 − Pi,i+1 ),

(2.35)

i=1

where the Pi,i+1 is the permutation operator of the factors i and i + 1 in the tensor product ⊗N C2 . Clearly the classical wedge product is nothing but the special case of the q-wedge product where q = 1. The fact that the Yangian action is well-defined on the quotient space ∧N V (z) follows from the observation, easily verified by using the relations of the degenerate affine Hecke algebra, that the generators (2.33), (2.34) leave the subspace q=1 invariant. Explicit formulas for the action of the Yangian generators in the basis of normally ordered wedges can be derived without difficulty owing to the uncomplicated structure of the Dunkl operators. If w is a normally ordered wedge in the classical wedge product ∧N V (z), i.e.: w = uk1 ∧ uk2 ∧ . . . ∧ ukN = z m1 v1 ∧ z m2 v2 ∧ . . . ∧ z mN vN , where

ki = i − 2mi (mi ∈ Z, i ∈ {1, 2})

and

k1 > k2 > . . . > kN ,

then the generators (2.33), (2.34) act on the w as follows: Qa0 .w =

N X

σia .w,

(2.36)

i=1

Qa1 .w =

N X

(αmi σia + iσia )w +

N −1 X

i=1

√ +

a δmi ,mi+1 (σia − σi+1 ).w +

i=1

−1 2

X

abc σib σjc .w −

1≤i<j≤N

X

a qij .w,

(2.37)

1≤i<j≤N

where a .(. . . ∧ z m v ∧ . . . ∧ z n v0 ∧ . . .) qij i

=

j

n−m−1 X

(σia − σja )(. . . ∧ z m+r v ∧ . . . ∧ z n−r v0 ∧ . . .), (m ≤ n).

r=1

i

j

Above and elsewhere the σia .w is defined as


σia .w = . . . ∧ z mi (σ a vi ) ∧ . . . .

It appears to be technically complicated to derive similar explicit formulas for the action b n )-action U (N ) for a general of the generators (2.25)–(2.27) (yi = q 1−N Yi(N ) ) of the Uq0 (sl 0 q not equal to 1. 2.6. Semi-infinite wedge product. For M ∈ Z the space of semi-infinite wedges FM is defined in [17] as the linear span of semi-infinite monomials:

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K. Takemura, D. Uglov

u k1 ∧ u k2 ∧ u k3 ∧

...

,

(2.38)

such that for i >> 1 the asymptotic condition ki = M − i + 1 holds. The vacuum semi-infinite monomial in FM is specified by ki = M − i + 1, i = 1, 2, . . . and is denoted by |M i: (2.39) |M i = uM ∧ uM −1 ∧ uM −2 ∧ . . . . The normal ordering rules (2.23, 2.24) imply that the normally ordered semi-infinite monomials – that is (2.38) with k1 > k2 > k3 > . . . form a basis in FM . The level-0 action U1(N ) in the limit N → ∞ was used in [17] to define a level-1 b n ) in the space FM , such that as an U 0 (sl b n )-module FM is isomorphic to action of Uq0 (sl q the Fock space module introduced in [12]. The Abelian algebra H1(N ) in the same limit gives rise to an action of the Heisenberg algebra in FM . The two main problems which we address in the present paper are: 1) To define a level-0 b n )-action in FM starting from the action U (N ) in ∧N V (z). 2) To construct the Uq0 (sl 0 irreducible decomposition of the Fock space FM with respect to this action. 3. Decomposition of the Finite Wedge Product In this section we construct the irreducible decomposition of the wedge product ∧N V (z) b n )-action U (N ) . with respect to the Uq0 (sl 0 The simplest and the most elegant way to approach this problem would be to de±1 ] with respect to the compose first the space of Laurent polynomials C[z1±1 , . . . , zN (N ) affine Hecke algebra action defined by gi,i+1 and Yj (2.9), and then to apply the b n ). Thefunctor [5, 6] from modules of the affine Hecke algebra into modules of Uq0 (sl orem 7.6 in [5] gives a description, in terms of Drinfeld polynomials, of the irreducible b n ) modul e obtained by the functor from any irreducible finitefinite-dimensional Uq0 (sl b N (q) module in the situation where N < n. In order to carry out the dimensional H functorial approach in our case, a suitable generalization of this theorem for arbitrary N and n is needed. In principle, a generalization of this kind seems to be not difficult to obtain. However, in this paper we settle for a more conservative approach which does b n ) to the full extent, but not use the duality between the affine Hecke algebra and Uq0 (sl only implicitly. 3.1. A base in ∧N V (z) . In order to derive the decomposition we construct a suitable base of ∧N V (z) by using the Non-symmetric Macdonald Polynomials 8λ (z). Let e := (1 , 2 , . . . , N ) where i ∈ {1, 2, . . . , n}. For a sequence e we set ve := v1 ⊗ v2 ⊗ . . . ⊗ vN

(∈ ⊗N Cn ).

(3.1)

A sequence m := (m1 , m2 , . . . , mN ) from ZN is called n-strict if it contains no more than n equal elements of any given value. Let us define the sets MnN and E(m) by MnN := {m = (m1 , m2 , . . . , mN ) ∈ ZN | m1 ≤ m2 ≤ . . . ≤ mN , m is n-strict }, (3.2) and for m ∈ MnN

Level-0 Action of Uq (sbln ) on q-Deformed Fock Spaces

559

E(m) := {e = (1 , 2 , . . . , N ) | i > i+1 for all i s.t. mi = mi+1 }.

(3.3)

In these notations the set {w(m, e) := 3(zm ⊗ve ) ≡ z m1 v1 ∧z m2 v2 ∧. . .∧z mN vN

m ∈ MnN , e ∈ E(m)}. (3.4) is nothing but the base of the normally ordered wedges in ∧N V (z): |

{w(m, e) | m ∈ MnN , e ∈ E(m)} = {uk1 ∧ uk2 ∧ · · · ∧ ukN | k1 > k2 > . . . > kN }, ki = i − nmi .

(3.5)

b n )-decomposition we construct another base. The eleFor the purpose of the Uq0 (sl ments of this new base have the same labels m ∈ MnN , e ∈ E(m) as the elements of the base of the normally ordered wedges. For m ∈ MnN , e ∈ E(m) let us put φ(m, e) := 3(8m (z) ⊗ ve ).

(3.6)

Notice that at q = 1 we have φ(m, e) ≡ w(m, e). This holds because at q = 1 one has Yi(N ) = pDi and hence 8λ (z) = zλ . Proposition 1. The set {φ(m, e) | m ∈ MnN , e ∈ E(m)} is a base of ∧N V (z). Proof. To show that ∧N V (z) = spanC {φ(m, e) | m ∈ MnN , e ∈ E(m)} we use the formulas (2.15) for the action of the Hecke algebra generators on Non-symmetric Macdonald Polynomials together with the fact that for any i ∈ {1, 2, . . . , N − 1} we have −1 ) ⇒ 3((gi,i+1 − Si,i+1 )f ) = 0 Im(gi,i+1 − Si,i+1 ) ⊂ Ker(gi,i+1 + Si,i+1

∀f ∈ ⊗N V (z). (3.7)

First, Eq. (2.15) and (3.7) allow us to write (at generic p and q): ∧N V (z) = spanC {3(8λ (z)⊗v) | λ ∈ ZN , λ1 ≤ λ2 ≤ . . . ≤ λN ; v ∈ ⊗N Cn }. (3.8) Next, we observe that (2.15, 3.7) imply: 3(8λ (z) ⊗ (q − Si,i+1 )v) = 0

whenever λi = λi+1 .

(3.9)

The explicit expression for the S (2.18) gives (q − S)v ⊗ v = (q + q −1 )v ⊗ v , (q − S)(v1 ⊗ v2 + qv2 ⊗ v1 )

(1 ≤  ≤ n),

(3.10)

= (q + q −1 )(v1 ⊗ v2 + qv2 ⊗ v1 ),

(n ≥ 1 > 2 ≥ 1).

(3.11)

From these relations and from (3.9) it is straightforward to derive that a vector 3(8λ (z)⊗ v) is equal to zero if the sequence λ is not n-strict. Then making use of (3.9), (3.10–3.11 ) we derive from (3.8) that the vectors φ(m, e) := 3(8m (z) ⊗ ve ) with m ∈ MnN and e ∈ E(m) span the wedge product. To demonstrate linear independence of these vectors we observe that the triangularity of the polynomials 8m (z) and Lemma 3 (cf. Sect. 4) allow us to write X c(m, e; n, e0 )3(zn ⊗ ve0 ), (3.12) 3(8m (z) ⊗ ve ) = 3(zm ⊗ ve ) + n∈Mn ,e0 ∈E(n) N m+  n+

560

K. Takemura, D. Uglov

where m ∈ MnN and e ∈ E(m). The linear independence follows from the linear independence of the top terms 3(zm ⊗ ve ) in the right-hand side of the last equation.  Since 8m (z) is an eigenvector of the operators Yi(N ) it is clear that U0(N ) and H0(N ) preserve the subspace E m := ⊕e∈E(m) Cφ(m, e) (3.13) for any m ∈ MnN . Moreover it is easy to see that the E m is an eigenspace of the operators ) ) (N ) which generate H0(N ) . The eigenvalue h(N (a = ±1, ±2, . . . ) for this h(N a a (m) of ha eigenspace is N N X X (N ) 1−N a (q ζi (m)) = pami q 2a(1−i) . (3.14) ha (m) = i=1

i=1

∂ ∂ Note also, that the E is an eigenspace of the degree operator z1 ∂z +z2 ∂z +. . .+zN ∂z∂N 1 2 with the eigenvalue |m| := m1 + m2 + . . . + mN . In the rest of this section we will describe the structure of the E m for a fixed m ∈ MnN b n )-module with the action U (N ) . as an Uq0 (sl 0 m

b n )-module E m . For a1 , a2 , . . . , ar ∈ C let π (r) 3.2. The Uq0 (sl a1 ,...,ar be the evaluation 0 b N n action of Uq (sln ) defined in ⊗ C by ,

πa(r)1 ,...,ar (K ) = K1 K2 . . . Kr , Ki = q Ei r X δ  πa(r)1 ,...,ar (E ) = ai 0, Ei,+1 Ki+1 . . . Kr , πa(r)1 ,...,ar (F ) =

i=1 r X

−δ0,

ai

−Ei+1,+1

,

(3.15) (3.16)

 (K1 )−1 . . . (Ki−1 )−1 Ei+1, ,

 = 0, . . . , n − 1 , (3.17)

i=1

where in the right-hand side we regard the indices , + 1 modulo n. In this notation we set for m ∈ MnN : ) π(m) := πa(N1 ,...,a N

wherein we put ai = q N −1 ζi (m)−1 .

(3.18)

ˇ 1 , a2 ) := a1 S − a2 S −1 has the intertwining Now we recall [13], that the R-matrix R(a property: (1 ≤ i ≤ r − 1), (3.19) b n ). Since ζi+1 (m) = q 2 ζi (m) whenever mi = mi+1 ; this where x is any generator of Uq0 (sl implies, that Rˇ i,i+1 (ai , ai+1 )πa(r)1 ,...,ar (x) = πa(r)1 ,...,ar (x)|ai ↔ai+1 Rˇ i,i+1 (ai , ai+1 ),

−1 −1 (q + Si,i+1 )π(m) = π(m)|ζi+1 (m)↔ζi (m) (q + Si,i+1 )

for all i s.t. mi = mi+1 .

As the result the π(m) is well-defined in the space: X −1 Ker(q + Si,i+1 ). W m := ⊗N Cn / {i|mi =mi+1 }

(3.20)

(3.21)

Level-0 Action of Uq (sbln ) on q-Deformed Fock Spaces

561

Let ιm : ⊗N Cn → W m be the quotient map defined by (3.21). We define a map β : E m = spanC {3(8m (z) ⊗ v) | v ∈ ⊗N Cn } (= ⊕e∈E(m) C3(8m (z) ⊗ ve )) → W m

(3.22)

by setting β(3(8m (z) ⊗ v)) = ιm (v). b n )-modules (E m , U (N ) ) Proposition 2. The map β (3.22) is an isomorphism of the Uq0 (sl 0 m and (W , π(m)). Proof. To show that the β is an isomorphism of the linear spaces it is sufficient to observe that the vectors v ⊗ v 

(1 ≤  ≤ n)

and v1 ⊗ v2 + qv2 ⊗ v1

(1 ≤ 1 < 2 ≤ n)

(3.23)

form a base in Ker(q + S −1 ) ⊂ Cn ⊗ Cn and hence the set {ιm (ve ) | e ∈ E(m)} is a base in W m . The map β is an intertwiner of the actions U0(N ) and π(m) since for any generator x of U0(N ) the relation (2.13) implies: x.3(8m (z) ⊗ v) = 3(8m (z) ⊗ π(m)(x).v).

(3.24)

 Now let us subdivide the sequence m = (m1 ≤ m2 ≤ . . . ≤ mN ) into subsequences which comprise equal elements: m = (m1 = . . . = mr1 < m1+r1 = . . . = mr2 < · · · < m1+rJ = . . . = mN ).

(3.25)

Notice that since m is n-strict we have for 1 ≤ k ≤ J +1 the inequalities 1 ≤ rk −rk−1 ≤ b n )-module the space W m then is n (here we put r0 := 0 and rJ+1 := N ). As the Uq0 (sl represented as the following tensor product: W m = V [p−mr1 q 2(r1 −1) , r1 ] ⊗ V [p−mr2 q 2(r2 −1) , r2 − r1 ] ⊗ . . . ⊗ V [p−mN q 2(N −1) , N − rJ ],

(3.26)

b n )-module with the action π (j) where V [a, j] (1 ≤ j ≤ n) is the Uq0 (sl a1 ,...,aj such that −2 −2(j−1) a (3.15–3.17), and as a linear space a1 := a, a2 := q a, . . . , aj := q V [a, j] = ⊗j Cn /

j−1 X i=1

−1 Ker(q + Si,i+1 ).

(3.27)

Our next task is to describe the V [a, j] (1 ≤ j ≤ n). First of all for 1 ≤ j ≤ n − 1 b n ) ) the V [a, j] is irreducible and isomorphic to as Uq (sln )-module ( Uq (sln ) ⊂ Uq0 (sl the highest weight module V (3j ) with the fundamental Uq (sln ) highest weight 3j . When j = n; the V [a, j] is the 1-dimensional trivial representation of Uq (sln ). Thus b n )-module, and in order to give the complete V [a, j] (1 ≤ j ≤ n) is an irreducible Uq0 (sl specification of V [a, j] for 1 ≤ j ≤ n − 1 it is sufficient to describe the associated Drinfel’d Polynomials. In the conventions of [5] which we recall in the Appendix these are provided by the following lemma:

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b n )-module with the Lemma 1. For 1 ≤ j ≤ n − 1 the V [a, j] is an irreducible Uq0 (sl Drinfel’d Polynomials  j−2 −1 for k = j , Pk (u) = u − q a (3.28) 1 for 1 ≤ k ≤ n − 1, k 6= j. Using the results of [1] (cf. Appendix ) we can claim that for generic q and p the representation W m (3.26) is irreducible, therefore the Drinfel’d Polynomials of W m are just products of the Drinfel’d Polynomials associated with the factors V [p−mrk q 2(rk −1) , rk − rk−1 ] (cf.[5]). This leads to the main proposition of this section: Proposition 3. For generic p and q the E m ∼ = W m is irreducible and with notations of (3.25) the Drinfel’d Polynomials of E m ∼ = W m are Y Pi (u) = (u−pmrk q −rk −rk−1 ) (i ∈ {1, 2, . . . , n−1}). (3.29) {1≤k≤J+1 | rk −rk−1 =i}

The proofs of this proposition and Lemma 1 are discussed in the Appendix. Finally we note that M Em. ∧N V (z) =

(3.30)

m∈Mn N

b n) This is the desired decomposition of the finite wedge product with respect to the Uq0 (sl action U0 . b n ) on the Fock Space 4. A Level-0 Action of Uq0 (sl b n ) on the space of semi-infinite In this section we will define a level-0 action of the Uq0 (sl wedges FM ( or, equivalently, on the level-1 Fock space module of the same algebra – b n ) – see Subsect. 2 ) starting from the action U (N ) which was defined in Sect. 2 on Uq0 (sl 0 the finite wedge product. The level-0 action on the space FM is constructed by taking a suitable projective limit of U0(N ) and can be thought of as an appropriate, well-defined limit of U0(N ) when the number of particles N approaches infinity. This construction of the level-0 action on the Fock space is uncomplicated in principle but is technically somewhat involved. To explain what we are aiming at and what are the main ideas to be employed we would like to start with an informal preliminary consideration. 4.1. The motivation and principal features of the construction. To explain our motivation and to connect our framework with the results of [3], [4] we will first discuss the definition of the Yangian action on the Fock space in the classical case q = 1 following the work [21]. Let FM (M ∈ Z) be the classical Fock space, i.e. the space defined in Subsect. 2, where q is set to be equal to 1. This space is spanned by semi-infinite wedges u k 1 ∧ uk 2 ∧ u k3 ∧ · · ·

(ki ∈ Z)

(4.1)

Level-0 Action of Uq (sbln ) on q-Deformed Fock Spaces

563

such that ki = M − i + 1 for all i >> 1. The FM contains, in particular, the vacuum vector (4.2) |M i = uM ∧ uM −1 ∧ uM −2 ∧ · · · . The symbol ∧ now obeys the classical antisymmetry relation uk ∧ ul = −ul ∧ uk which is obtained by setting q to be equal to 1 in the normal ordering rules (2.23), (2.24). In Subsect. 2 the action of the Yangian Y (sl2 ) was defined on the finite wedge product. This action is extended on the space of semi-infinite wedges FM without difficulty. To specify the action of the Y (sl2 ) generators Qa0 , Qa1 (a = 1, 2, 3) on a semi-infinite wedge w = uk1 ∧ uk2 ∧ uk3 ∧ · · · we only have to replace the summations from 1 to N in the formulas (2.36), (2.37) by summations from 1 to ∞. For any normally ordered wedge w from the space FM this gives Qa0 .w =

∞ X

σia .w,

(4.3)

i=1

Qa1 .w =

∞ X

(αmi σia + iσia )w +

i=1

√

−1 2

+

where

∞ X

a δmi ,mi+1 (σia − σi+1 ).w +

i=1

X

X

abc σib σjc .w −

1≤i<j<∞

a qij .w,

(4.4)

1≤i<j<∞

a .(. . . ∧ z m v ∧ . . . ∧ z n v0 ∧ . . .) qij i

=

j

n−m−1 X

(σia − σja )(. . . ∧ z m+r v ∧ . . . ∧ z n−r v0 ∧ . . .), (m ≤ n).

r=1

i

j

It is not difficult to see that, apart from the ambiguity in the definition of the action of the generator Q30 which is removed by requiring that Q30 .|M i = 0 (M is even) and Q30 .|M i = |M i (M is odd), the expressions (4.3), (4.4) specify well-defined operators on the space FM . This means that when we act with any of Qa0 , Qa1 (a = 1, 2, 3) on a semi-infinite wedge from FM the result is a finite linear combination of semi-infinite wedges from FM . The antisymmetry relation uk ∧ ul = −ul ∧ uk is the essential reason for this well-definedness. There is a close relationship between the Yangian generators (4.3), (4.4) and the b 2 modules introduced in [11]. To generators of the Yangian action on the the level-1 sl see this relationship we recall the interpretation of the space FM as a subspace (of fixed charge M ) in a certain fermion Fock space. Let the linear space F be defined as the direct sum of the FM taken over all integer M: M FM . (4.5) F = M ∈Z

The space F can be identified with the fermion Fock space [16, 17]. Introduce wedging operators ψ¯ k : FM → FM +1 , and contracting operators ψk : FM → FM −1 (k ∈ Z) by their actions on a semi-infinite wedge w = uk1 ∧ uk2 ∧ uk3 ∧ · · · , ψ¯ k w = uk ∧ w, ψk (uk1 ∧ uk2 ∧ . . . )  0 = (−1)i−1 (uk1 ∧ uk2 ∧ . . . ∧ uc ki ∧ . . . )

if k = 6 ki , i ≥ 1, if k = ki ,

(4.6)

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K. Takemura, D. Uglov

where the hat over uki indicates that this factor is omitted from the product. The operators ψ¯ k and ψk generate the entire space F from the vacuum |0i and satisfy the usual fermion anticommutation relations {ψ¯ k , ψl } = δkl , {ψ¯ k , ψ¯ l } = {ψk , ψl } = 0.

(4.7)

It is well-known (see e.g. [16]) that each of the spaces FM (M ∈ Z) admits a level-1 b 2 centralized by an action of the Heisenberg algebra. action of the affine Lie algebra sl 1 2 3 b 2 generators J , J , J (m ∈ Z) are realized on FM in terms of the fermions The sl m m m (4.6) by X X a Jm = σa1 ,2 ψ¯ 1 −2k−2m ψ2 −2k (m ∈ Z, a = 1, 2, 3), (4.8) k∈Z i =1,2

where σa0 , is defined by σ a v = v0 σa0 , . They satisfy the commutation relations √ a c [Jm , Jnb ] = 2 −1abc Jm+n + 2mδab δm+n,0 . (4.9) The generators Bm (m ∈ Z) of the Heisenberg algebra H are realized as X X ψ¯ k−2m ψk B0 = : ψ¯ k ψk :, Bm = (m ∈ Z6=0 ). k∈Z

(4.10)

k∈Z

Here the normal ordering : : in the charge operator B0 is with respect to the vacuum |0i. This implies that the B0 acts on each subspace FM as the multiplication operator by the constant M. The H generators satisfy the commutation relations [Bm , Bn ] = 2mδm+n,0 ,

(4.11)

b 2. and commute with the generators of the sl Clearly it is possible to express the Yangian generators (4.3) and (4.4) in terms of the fermions (4.6). In fact we may express the Yangian generators in terms of the generators b 2 and the Heisenberg algebra. These expressions are found to be given by [21]: of the sl Qa0 = J0a ,   α αX a Qa1 = 1 + (1 − M ) J0a − J−n Bn 2 2 n≥1 √ 3 X α −1 X X abc b c a −( + 1) B−n Jn −  J−n Jn . 2 2 n≥1

(4.12)

(4.13)

n≥1 b,c=1

Setting in these expressions α = 0 or α = −2 we may find [21] that the operators a

Q0 = J0a , √ 3 −1 X X abc b c a Q1 =  J−n Jn 2

(4.14) (4.15)

n≥1 b,c=1

also satisfy the defining relations of the Yangian Y (sl2 ) [9]. In (4.14), (4.15) we readily recognize the generators of the Yangian action on the irreducible level-1 representations b 2 that were introduced in [11] and subsequently studied in [3, 4]. of sl

Level-0 Action of Uq (sbln ) on q-Deformed Fock Spaces

565

The natural question that one may ask now is whether one can define a level-0 action b n ) on the q-deformed Fock space (see Subsect. 2) along of the quantum algebra Uq0 (sl lines similar to the construction of the Yangian action on the classical Fock space. As we will demonstrate in the next subsection, the answer to this question is positive, and the required action is provided by a certain limit of the action U0(N ) (see Subsect. 2) where N becomes infinite. However, to carry out the construction of the level-0 action in the q-deformed case we need to introduce certain new ideas as compared to the Yangian situation discussed above. The reason why we need to do this is explained by the complicated structure of the Chevalley generators E0 and F0 of the action U0(N ) (see (2.25)–(2.27)) which does not allow to derive simple formulas for the actions of these generators on a normally ordered wedge in the (q-deformed) wedge product ∧N V (z) that would be similar to the formulas available in the Yangian case (2.36),(2.37). Because of this limitation we are not able to show directly that the generators of U0(N ) , where the N is formally set to be infinite specify well-defined operators on the Fock space. Instead of a direct approach of this kind, in the next subsection we develop a different approach, principal features of which we will now describe. b n )-module FM (M ∈ Z) is graded by the degree generThe level-1 Fock space Uq (sl b ator of the level-1 Uq (sln )-action [17]. For each non-negative integer k the homogeneous k component FM of the degree k is finite dimensional. We define a level-0 action of the b n ) on the FM in such a way that it leaves each of the homogeneous quantum algebra Uq0 (sl k components FM (k = 0, 1, 2, . . . ) invariant. k In order to define the level-0 action on the linear space FM for a given k we realize k the FM with M ≡ s mod n, s ∈ {0, 1, . . . , n − 1} as a projective limit of an infinite s+rn,k sequence of finite-dimensional linear spaces VM (r = 0, 1, 2, . . . ) equipped with M certain projection maps ρr+1,r : ρM 1,0

ρM 2,1

ρM 3,2

ρM 4,3

s,k s+n,k s+2n,k s+3n,k VM ←− VM ←− VM ←− VM ←− . . .

.

(4.16)

s+rn,k Each of the spaces VM in this sequence is a subspace of the finite wedge prods+rn b n ) action U (s+rn) defined in V (z) and is invariant with respect to the Uq0 (sl uct ∧ 0 Subsect. 2. The projection maps in (4.16) have the following properties

(P1) The maps ρM r+1,r are bijective for all r ≥ k. (s+rn) 0 b on the linear spaces (P2) The maps ρM r+1,r intertwine the Uq (sln ) actions U0 s+rn,k VM . That is we have (s+rn+n) ρM = U0(s+rn) ρM r+1,r U0 r+1,r

(r = 0, 1, 2, . . . ).

s+rn,k We define the linear subspaces VM in such a way that the projective limit s,k s+n,k s+2n,k , . . . with respect to the projections ρM of the sequence VM , VM , VM r+1,r is k of the Fock space isomorphic as a linear space to the homogeneous component FM b n ) module. module FM . In view of the property (P2) this projective limit is a Uq0 (sl k 0 b Therefore the FM is a Uq (sln ) module as well. Taking the property (P1) into ack b n ) module to each of the subspaces count we see, that the FM isomorphic as a Uq0 (sl

566

K. Takemura, D. Uglov

s+kn,k s+kn+n,k s+kn+2n,k b n ) action on F k may be reVM , VM , VM , . . . ; and hence the Uq0 (sl M (s+rn) where r becomes infinite. Clearly garded as the well-defined limit of the action U0 b n ) action on each of the homogeneous components F k has level 0. Since the the Uq0 (sl M Fock space module FM is a direct sum of its homogeneous componets, a level-0 action b n ) is defined on the FM . The main technical point of this definition is a proof of Uq0 (sl of the property (P2). The proof which we give in the next subsection is based on certain non-trivial properties of the Cherednik’s operators Y1(N ) , Y2(N ) , . . . , YN(N ) established in Lemma 4. In the next subsection we give a detailed exposition of our construction.

b n ) on the Fock spaces. In this subsection 4.2. Construction of the level-0 actions of Uq0 (sl we allow the parameter p to be arbitrary complex number, whereas the q is still required to be generic. Let w = uk1 ∧ uk2 ∧ . . . ∧ ukN ≡ z m1 v1 ∧ z m2 v2 ∧ . . . ∧ z mN vN , w ∈ ∧N V (z) be a normally ordered wedge: k1 > k2 > . . . > kN . Often it will be convenient to label this wedge by the two sequences m ∈ MnN and e ∈ E(m) : m = (m1 ≤ m2 ≤ . . . ≤ mN ), mi ∈ Z, e = (1 , 2 , . . . , N ), 1 ≤ i ≤ n, such that ki = i − nmi ,

(4.17) (4.18) (4.19)

and write: w := w(m, e). From now on we will use the notation w(m, e) exclusively for normally ordered wedges. Recall that by the definitions (3.2, 3.3) of the MnN and E(m) the m in (4.17) is always n-strict sequence and that i > i+1 whenever mi = mi+1 . Throughout this section we fix an integer M and 0 ≤ s ≤ n − 1 such that M = s mod n. Let r ∈ {0, 1, 2, . . . }; and let w(m0 , e0 ) ∈ ∧s+nr V (z) be defined as: w(m0 , e0 ) := uM ∧ uM −1 ∧ . . . ∧ uM −(s+nr)+1 .

(4.20)

Here the sequences m0 and e0 are as follows: m0 = (m01 , m02 , . . . , m0s+nr ) := (m0 , . . . , m0 , m0 + 1, . . . , m0 + 1, | {z } | {z } s

(4.21)

n

m0 + 2, . . . , m0 + 2, . . . , m0 + r, . . . , m0 + r) | | {z } {z } n

n

e0 = (01 , 02 , . . . , 0s+nr ) := (s, s − 1, . . . , 1, n, n − 1, . . . , 1, | {z } | {z } s

n

n, n − 1, . . . , 1, . . . , n, n − 1, . . . , 1), | | {z } {z } n

where

s−M m := n 0

(4.22)

n

(0 ≤ s ≤ n − 1).(4.23)

We will call these two sequences vacuum sequences, and the w(m0 , e0 ) – vacuum vector of ∧s+nr V (z). s+nr ⊂ ∧s+nr V (z) as: Define VM

Level-0 Action of Uq (sbln ) on q-Deformed Fock Spaces s+nr VM =

567

M

Cw(m, e).

(4.24)

m∈Mn ,e∈E(m)

N ms+nr ≤m0s+nr

Notice that the condition ms+nr ≤ m0s+nr in this definition is equivalent to the condition: mi ≤ m0i

for all i = 1, 2, . . . , s + nr

(4.25)

because the sequence m is n-strict and non-decreasing. b n ) action U (s+nr) and the operators h(s+nr) (l ∈ Z6=0 ) preserve Proposition 4. The Uq0 (sl 0 l s+nr the subspace VM . To prove this and other propositions we need two lemmas. The first of these lemmas concerns properties of the operators (2.10) and the operators ξi,j which were defined in Sect. 2.2. nN ± be a monomial in C[z1± , . . . , zN ], Lemma 2. Let zn ≡ z1n1 z2n2 . . . zN and let a = max{n1 , . . . , nN }. Then X 0 ±1 n z = c± (n, n0 )zn , ξi,j

(4.26)

n0

(Yi(N ) )k zn =

X

0

ck (n, n0 )zn ,

k = ±1, ±2, . . . ,

(4.27)

n0

where c± (n, n0 ), ck (n, n0 ) are coefficients, and the summation ranges over n0 such that: n01 , n02 , . . . , n0N ≤ a, #{n0i |n0i = a} ≤ #{ni |ni = a}, n01 + n02 + . . . + n0N = n1 + n2 + . . . + nN .

(4.28) (4.29) (4.30)

Proof. To prove the statement about the summation range (4.28 - 4.30) in (4.26) we use −1 on monomials: the explicit formulas for the action of ξi,j and ξi,j for ni < nj :

(4.31)

n

n

±1 ni ξi,j zi zj j = q ∓1 zini zj j ∓ (q − q −1 )

nj −ni −1

X

n −k

zini +k zj j

,

k=1

for ni = nj :

±1 ni nj zi zj ξi,j

(4.32) =

n q ±1 zini zj j ,

for ni > nj : n

(4.33) n

n

±1 ni ξi,j zi zj j = q ±1 zini zj j ± (q − q −1 )zi j zjni ± (q − q −1 )

ni −nj −1

X

n +k

zini −k zj j .

k=1

The statements (4.26, 4.28–4.30) immediately follow from these formulas. The statement (4.27) with the summation restricted by (4.28–4.30) follows from (4.26, 4.28–4.30) and (2.10). 

568

K. Takemura, D. Uglov

We will also need the following lemma which shows triangularity of the normal ordering. Lemma 3. Let v ∈ ⊗N Cn , and 3 be the quotient map defined by (2.21). Then in the notations of Lemma 2 the following holds: X 3(zn ⊗ v) = c(n, v; n0 , e)w(n0 , e),

(4.34)

n0 ,e

where c(n, v; n0 , e) is a coefficient and the summation ranges over n0 such that (n0 )+  (n)+ and consequently: n01 , n02 , . . . , n0N ≤ a(:= max{n1 , . . . , nN }), #{n0i |n0i = a} ≤ #{ni |ni = a}, n01 + n02 + . . . + n0N = n1 + n2 + . . . + nN .

Proof. Use the normal ordering rules (2.23, 2.24).

(4.35) (4.36) (4.37)



s+nr VM

Proof of Proposition 4. Let w(m, e) ∈ – that is ms+nr ≤ m0s+nr . We prove the proposition by considering the action of the generators of U0(s+nr) on w(m, e). Action of any of the generators (2.25–2.27) of the Uq (sln ) subalgebra of U0(s+nr) on this wedge results in a linear combination of wedges w(m, e0 ) with the same sequence s+nr . By m as in w(m, e). Therefore the action of the Uq (sln ) subalgebra preserves VM s+nr the same token K0 (2.25) preserves VM as well. Consider now the vector: F0 .w(m, e) =

N X

0 3(q 1−N Yi(N ) .zm ⊗ (K10 )−1 . . . (Ki−1 )−1 Ei1,n .ve ),

(4.38)

i=1

where we put N := s + nr and ve := v1 ⊗ v2 ⊗ . . . ⊗ vN ∈ ⊗N Cn . In each of the summands apply first Lemma 2 to express Yi(N ) .zm as a linear combination of monomials, then apply Lemma 3 to express the result as a linear combination of the normally ordered wedges: X c(m, e; m0 , e0 )w(m0 , e0 ). (4.39) F0 .w(m, e) = m0 ,e0

Due to (4.28) in Lemma 2 and (4.35) in Lemma 3 in the last formula we have: m0s+nr ≤ s+nr s+nr by the definition (4.24) of the VM . ms+nr ≤ m0s+nr , and thus F0 .w(m, e) ∈ VM (s+nr) For the generator E0 and the operators hl the proof is done by the same argu ments as for F0 . s+nr Introduce the degree |w(m, e)| of a wedge w(m, e) ∈ VM by:

|w(m, e)| =

s+nr X

m0i − mi ,

i=1

where the vacuum sequence m0 is defined in (4.22). We have:

(4.40)

Level-0 Action of Uq (sbln ) on q-Deformed Fock Spaces s+nr VM =

M k≥0

569

s+nr,k VM ,

(4.41)

s+nr,k s+nr = spanC {w(m, e) ∈ VM | |w(m, e)| = k}. VM

(4.42)

The statements (4.30) and (4.37) in Lemmas 2 and 3 imply that the action U0(s+nr) and the commuting Hamiltonians preserve the degree: s+nr,k s+nr,k → VM , U0(s+nr) , hl(s+nr) : VM

l ∈ Z6=0 , k = 0, 1, . . .

.

(4.43)

s+nr,0 is one-dimensional with Notice that when M = 0 mod n (i.e s = 0) the space VM 0 0 the basis w(m , e ). Our main technical tool in defining the level-0 action in the space of semi-infinite wedges is the projection map ρM r+1,r : s+nr+n s+nr → VM , ρM r+1,r : VM

r = 0, 1, . . .

,

(4.44)

which we define by specifying its action on the normally ordered wedges as follows: s+nr+n and let Let w(m, e) ∈ VM m = (m1 , m2 , . . . , ms+nr , ms+nr+1 , . . . , ms+nr+n ), e = (1 , 2 , . . . , s+nr , s+nr+1 , . . . , s+nr+n )

(4.45) (4.46)

be the m and e sequences labeling the wedge w(m, e) . Remove from m and e the last n elements, and denote the obtained sequences by m0 and e0 : m0 = (m1 , m2 , . . . , ms+nr ), e0 = (1 , 2 , . . . , s+nr ),

(4.47) (4.48)

so that s+nr . w(m0 , e0 ) ∈ VM

The action of ρM r+1,r is then defined by: ( ρM r+1,r .w(m, e)

w(m0 , e0 )

= 0

if ms+nr+1 = ms+nr+2 = . . . = ms+nr+n = m0s+nr+1 = m0s+nr+2 = . . . = m0s+nr+n , otherwise.

Proposition 5. The following holds: (i) The map ρM r+1,r preserves the degree :

s+nr+n,k s+nr,k ρM → VM r+1,r : VM

M and for all k = 0, 1, . . . the map ρM,k r+1,r := ρr+1,r |V s+nr+n,k is surjective. M

(ii) For k ≤ r the map ρM,k r+1,r is bijective.

(4.49)

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K. Takemura, D. Uglov

Proof. The part (i) follows immediately from the definition of the degree (4.40) and from (4.49). s+nr+n is such that To prove the part (ii) let us demonstrate, that if w(m, e) ∈ VM |w(m, e)| ≤ r, then: ms+nr+1 = ms+nr+2 = . . . = ms+nr+n = m0s+nr+1 = m0s+nr+2 = . . . = m0s+nr+n .

(4.50)

Suppose the last equality does not hold. Then we necessarily have: ms+nr+1 = m0s+nr+1 − t0 ,

where t0 ≥ 1.

(4.51)

Since the sequence m is n-strict and non-decreasing, we also have: ms+nr−n+1 < ms+nr+1 ,

m0s+nr−n+1 = m0s+nr+1 − 1 ⇒

ms+nr−n+1 = m0s+nr−n+1 − t1 ,

t1 ≥ t0 ,

and in general: ms+nr−nl+1 = m0s+nr−nl+1 − tl ,

tl ≥ tl−1 ,

l = 1, 2, . . . , r.

Summing up the last equations for l = 1, 2, . . . , r and (4.51) we find: |w(m, e)| =

s+nr+n X

m0i − mi ≥

i=1

r X

tl ≥ t0 (r + 1) ≥ r + 1.

l=0

This contradicts |w(m, e)| ≤ r, and therefore (4.50) holds. Taking (4.49) into account we find that (4.52) Ker(ρM r+1,r |V s+nr+n,k ) = 0 M

when k ≤ r.



An important property of the map ρM r+1,r (4.49) is that this map intertwines the (s+nr+n) 0 b s+nr+n b n )-action defined in VM ⊂ ∧s+nr+n V (z) with the Uq0 (sl Uq (sln )-action U0 (s+nr) s+nr s+nr M defined in VM ⊂ ∧ V (z). The map ρr+1,r also intertwines the actions of U0 (N ) (N ) 0 the operators hl − hl (m )I for N = s + nr + n and N = s + nr – in this case one ) needs to redefine h(N (2.29) by subtracting the eigenvalue associated with the vacuum l sequence (3.14) . We summarize this as the proposition: Proposition 6. For r = 0, 1, 2, . . . the following intertwining relations hold : (i)

(s+nr+n) = U0(s+nr) ρM ρM r+1,r U0 r+1,r ,

(ii)

(s+nr+n) = gl(s+nr) ρM ρM r+1,r gl r+1,r ,

l ∈ Z6=0

) ) 0 where gl(N ) = h(N − h(N l l (m )I, P 0 N (N ) lm and hl (m0 ) = j=1 p j q 2l(1−j) .

To prove this proposition we will need one more lemma on properties of the operators (2.10):

Level-0 Action of Uq (sbln ) on q-Deformed Fock Spaces

571

Lemma 4. Let m = (m1 , m2 , . . . , mN ) ∈ ZN be a sequence such, that: m1 , m2 , . . . , mN −k < mN −k+1 = mN −k+2 = . . . = mN ≡ m;

1 ≤ k ≤ N. (4.53)

Then for a = ±1, ±2, . . . , the following relations hold: for 0 ≤ l ≤ k − 1

(YN(N−l) )a zm

=

pam q a(2k−2l−N −1) zm + [. . .] , (4.54)

for 1 ≤ i ≤ N − k

(Yi(N ) )a zm

=

q ak (Yi(N −k) )a zm + [. . .] ,

(4.55)

nN such that: where [. . .] signifies a linear combination of monomials zn ≡ z1n1 z2n2 . . . zN

n1 , n2 , . . . , nN ≤ m,

(4.56)

and #{ni |ni = m}

<

k.

(4.57)

Proof. Consider first the expression: −1 −1 DN −l ξ1,N −l . . . ξN −k,N −l · YN(N−l) zm = ξN −l,N −l+1 . . . ξN −l,N p

· ξN −k+1,N −l . . . ξN −l−1,N −l zm ,

0 ≤ l ≤ k − 1.

(4.58)

The Eq. (4.32) gives: −1 −1 DN −l YN(N−l) zm = q k−l−1 ξN ξ1,N −l −l,N −l+1 . . . ξN −l,N p

0 ≤ l ≤ k − 1.

. . . ξN −k,N −l zm ,

(4.59)

In the last expression apply ξN −k,N −l to the monomial zm using the formula (4.31): −1 −1 DN −l YN(N−l) zm = q k−l−1 ξN ξ1,N −l −l,N −l+1 . . . ξN −l,N p

. . . ξN −k−1,N −l (q −1 zm + [. . .]),

0 ≤ l ≤ k − 1, (4.60) where the meaning of [. . .] is the same as in the statement of the lemma. Now apply ξi,N −l repeatedly for i = N − k − 1, N − k − 2, . . . , 1 using at each step (4.28, 4.29) in Lemma 2 to show that ξi,N −l ([. . .]) = ([. . .]), and using also (4.31). This gives: −1 −1 m k−N m z +[. . .]), YN(N−l) zm = q k−l−1 ξN −l,N −l+1 . . . ξN −l,N (p q

0 ≤ l ≤ k−1. (4.61)

Lemma 2 and (4.32) applied in the last formula yield (4.54) for a = 1. Consider now the expression: −1 −1 −1 −1 Di m . . . ξi,N Yi(N ) zm = ξi,i+1 −k ξi,N −k+1 . . . ξi,N p ξ1,i . . . ξi−1,i z ,

1 ≤ i ≤ N − k. (4.62)

Write: m

N −k m m )zN pDi ξ1,i . . . ξi−1,i zm = (pDi ξ1,i . . . ξi−1,i z1m1 z2m2 . . . zN −k −k+1 . . . zN ,

and observe that due to Lemma 2 the expression:

(4.63)

572

K. Takemura, D. Uglov m

N −k pDi ξ1,i . . . ξi−1,i z1m1 z2m2 . . . zN −k

(4.64)

n

−k is a linear combination of monomials z1n1 z2n2 . . . zNN−k such that n1 , n2 , . . . , nN −k < m, and therefore the formula (4.31) implies that:

−1 Di p ξ1,i . . . ξi−1,i zm = q pDi ξ1,i . . . ξi−1,i zm + [. . .] . ξi,N

(4.65)

Continuing to apply Lemma 2 together with (4.31) we get : −1 −1 Di m k Di m ξi,N −k+1 . . . ξi,N p ξ1,i . . . ξi−1,i z = q p ξ1,i . . . ξi−1,i z + [. . .] . (4.66) −1 −1 Finally we act by ξi,i+1 . . . ξi,N −k on the last expression and using Lemma 2 to show −1 −1 that ξi,i+1 . . . ξi,N −k ([. . .]) = ([. . .]) , arrive at: −1 −1 Di m Yi(N ) zm = q k ξi,i+1 . . . ξi,N −k p ξ1,i . . . ξi−1,i z + [. . .] ,

1 ≤ i ≤ N − k, (4.67)

which is the statement (4.55) of the lemma at a = 1. For a = −1 the proof is completely analogous, and (4.54,4.55) for a = ±2, ±3, . . . follow from the case a = ±1 and Lemma 2, Eqs. (4.28, 4.29).  Proof of Proposition 6. To prove part (i) we consider the action of the generators of (s+nr+n) . U0(s+nr+n) on a wedge w(m, e) ∈ VM First let w(m, e) be such that: mi < m0i for at least one s + nr < i ≤ s + nr + n.

(4.68)

This condition is equivalent to w(m, e) ∈ KerρM r+1,r . Lemmas 2 and 3 imply that acting with any of the generators (2.25 – 2.27) (where yi = q 1−(s+nr+n) Yi(s+nr+n) ) on such w(m, e) produces a linear combination of wedges that have the property (4.68) as well, and therefore vanish when acted on by ρM r+1,r . We formulate this as: M U0(s+nr+n) : KerρM r+1,r → Kerρr+1,r .

(4.69)

b n ) one has: Thus for w(m, e) that satisfy (4.68) and any generator x of Uq0 (sl M (s+nr+n) w(m, e) = 0. x(s+nr) ρM r+1,r w(m, e) = ρr+1,r x

(4.70)

Now let w(m, e) be such that: mi = m0i for all s + nr < i ≤ s + nr + n .

(4.71)

With the same m0 and e0 as in (4.47, 4.48) one can write: w(m, e) = w(m0 , e0 ) ∧ (uM −s−nr ∧ uM −s−nr−1 ∧ . . . ∧ uM −s−nr−n+1 ), (4.72) 0 0 ρM r+1,r w(m, e) = w(m , e ).

(4.73)

Apply F0(s+nr+n) to w(m, e): F0(s+nr+n) w(m, e) = 3(

s+nr+n X

q 1−(s+nr+n) Yj(s+nr+n) zm ⊗ (K10 )−1

j=1 0 . . . (Kj−1 )−1 Ej1,n ve ).

(4.74)

Level-0 Action of Uq (sbln ) on q-Deformed Fock Spaces

573

Lemma 4 for a = 1 ,N = s + nr + n, k = n, m = m0s+nr+n and Lemmas 2, 3 enable us to transform the right-hand side of the last equation and arrive at: F0(s+nr+n) w(m, e) = (F0(s+nr) w(m0 , e0 )) ∧ (uM −s−nr ∧ uM −s−nr−1 ∧ . . . ∧ uM −s−nr−n+1 ) + +pms+nr+n q 2(1−nr−n−s) ((K0(s+nr) )−1 w(m0 , e0 )) ∧ · (4.75) · ∧ (uM −s−nr−n+1 ∧ uM −s−nr−1 ∧ uM −s−nr−2 ∧ . . . ∧ uM −s−nr−n+1 ) + w, ˜ 0

where w˜ ∈ KerρM r+1,r . The normal ordering rules (2.23, 2.24) imply that the second summand in the right-hand side of the last equation vanishes ( cf. Lemma 2.2 in [17]). Finally (4.76) and (4.72, 4.73) give: (s+nr+n) M w(m, e). F0(s+nr) ρM r+1,r w(m, e) = ρr+1,r F0

(4.76)

b n )-generators and the Thus (i) is proven for the generator F0 . The rest of the Uq0 (sl statement (ii) of the Proposition are handled in the same way.  At fixed M = s mod n and fixed degree k form the projective limit of the spaces s+nr,k M with respect to the map ρM,k VM r+1,r := ρr+1,r |V s+nr+n,k : M

k VM

=

s+nr,k lim VM . ←−

(4.77)

r

s+nr,k k A vector in VM is a semi-infinite sequence {fr }r≥0 ; fr ∈ VM such that:

ρM,k r+1,r fr+1 = fr ,

r = 0, 1, 2, . . . .

(4.78)

Since the map ρM,k r+1,r is bijective when r ≥ k we have the isomorphism of linear spaces: s+nr,k k ∼ , VM = VM

r ≥ k.

(4.79)

s+nr,k Notice that for f = {fr }r≥0 ∈ VM , r ≥ k:

fr+1 = fr ∧ uM −s−nr ∧ uM −s−nr−1 ∧ . . . ∧ uM −s−nr−n+1 ,

(4.80)

as implied by Proposition 5 (ii) and the definition (4.49) of the map ρM r+1,r . k b n )-action Now we use Propositions 5 (i) and 6 to define in the space VM an Uq0 (sl (∞) (∞) k U0 , and an action of the commutative family {gl }l∈Z6=0 . For f = {fr }r≥0 in VM set: U0(∞) .{fr } := {U0(s+nr) .fr },

(4.81)

gl(∞) .{fr }

(4.82)

:=

{gl(s+nr) .fr }.

Clearly we still have: U0(∞) gl(∞) = gl(∞) U0(∞) ,

l ∈ Z6=0 .

(4.83)

The degree of a semi-infinite normally ordered wedge w ∈ FM is defined similarly s+nr . Write the vacuum vector |M i of FM , and the to the degree (4.40) for wedges in VM w as:

574

K. Takemura, D. Uglov

= z m 1 v  0 ∧ z m 2 v 0 ∧ . . . 0

|M i = uM ∧ uM −1 ∧ uM −2 . . . w=z

m1

v 1 ∧ z

m2

v 2 ∧ . . .

;

0

1

mi =

m0i , i

,

(4.84)

for i >> 1 ,

(4.85)

2

=

0i

and define the degree |w| as: |w| =

X

m0i − mi .

(4.86)

i≥1

Let for k ≥ 0 :

M

k = FM

Cw,

(4.87)

{n.o. w∈FM | |w|=k}

where “n.o.” stands for “normally ordered”. The Fock space is graded with respect to k . this degree: FM = ⊕k≥0 FM k k k → FM by: Define the map ρM : VM k : for f = {fr }r≥0 ∈ VM

ρkM f

= fr ∧ |M − s − nri,

(4.88) where r ≥ k.

(4.89)

Proposition 5 (or, equivalently Eq. (4.80)) shows that ρkM does not depend on the choice of r in (4.89) as long as r ≥ k. b n ) in the The following proposition will enable us to define a level-0 action of Uq0 (sl Fock space. k k and FM for Proposition 7. The map ρkM is an isomorphism of the linear spaces VM any k ≥ 0. s+nr,k ∼ k for all r ≥ k, it is sufficient to prove that the map: Proof. Since VM = VM s+nk,k 3 w(s+nk) −→ w(s+nk) ∧ |M − s − nki ∈ FM VM

(4.90)

s+nk,k k and FM . is an isomorphism of VM s+nk,k . This is an element of a basis Take a normally ordered wedge w(m, e) ∈ VM s+nk,k k . Observe that the vector w(m, e) ∧ |M − s − nki belongs to FM and is of VM a normally ordered wedge in FM – that is an element of a basis in FM . This shows injectivity of the map (4.90). k . Applying Let w := z m1 v1 ∧ z m2 v2 ∧ · · · be a normally ordered wedge in FM the same reasoning as in the proof of the Proposition 5 (part ii) we can show that

w ≡ w(s+nk) ∧ |M − s − nki,

(4.91)

s+nk,k where w(s+nk) ∈ VM , and explicitly:

w(s+nk) = z m1 v1 ∧ z m2 v2 ∧ · · · ∧ z ms+nk vs+nk . Hence the map (4.90) is surjective.



(4.92)

Level-0 Action of Uq (sbln ) on q-Deformed Fock Spaces

575

k b n )Taking advantage of this proposition we use the map ρkM to define in FM an Uq0 (sl action U0 along with an action of the commutative family {gl } by conjugating the actions U0(∞) and gl(∞) with the isomorphism ρkM . Obviously we have U0 gl = gl U0 for all l 6= 0. k k follow from the actions in each component FM . This The actions in FM = ⊕k≥0 FM 0 b completes the definition of the level-0 action of Uq (sln ) in the Fock space FM .

Let us summarize the results of this section. We started from U0(N ) – the level-0 action of b n ) in the finite wedge product ∧N V (z). The generators {x(N ) } ( x = E a , F a , K a , Uq0 (sl a = 0, . . . , n − 1) of this action are given by (2.25–2.27) with yi = q 1−N Y (N ) . Using b n ) in the Fock space FM . The space U0(N ) we have defined U0 – level-0 action of Uq0 (sl k FM is graded: FM = ⊕k≥0 FM and by the definition the U0 preserves the degree k. Any k ( M = s mod n , 0 ≤ s ≤ n − 1 ) is represented as: vector w ∈ FM w = fr ∧ |M − s − nri,

(4.93)

s+nr,k where fr ∈ VM ⊂ ∧s+nr V (z), and r ≥ k . For any fixed r ≥ k this representation b n ) on w (4.93) is then is unique by Proposition 7. The action of a generator x ∈ Uq0 (sl defined as: (4.94) x.w = (x(s+nr) .fr ) ∧ |M − s − nri,

and x.w does not depend on the choice of r as long as r ≥ k again by Proposition 7. ) Similarly starting from the commutative family of operators {h(N l } , (l = ±1, ±2, . . .) N defined in ∧ V (z) we define in FM a commutative family of operators {gl }, (l = ±1, ±2, . . .) which also commute with the U0 . As in (4.94) we prescribe the action of gl on the wedge w (4.93) by: gl .w = (gl(s+nr) .fr ) ∧ |M − s − nri,

l = ±1, ±2, . . .

;

(4.95)

with {gl(s+nr) } given in Proposition 6 (ii). The independence this prescription on the choice of the r is again due to Proposition 7. For computational purposes the most convenient choice of the r in (4.94, 4.95) is to take it to be minimal – that is r = k. We adopt this choice in the next section.

5. Decomposition of the Fock Space with Respect to the Level-0 Action b n) In this section we give the decomposition of the level-1 Fock space module of Uq0 (sl with respect to the level-0 action U0 which was defined in Sect. 4. s+nr b n )-action U0 given . The definition of the Uq0 (sl 5.1. Decomposition of the space VM in Sect. 4 makes it clear, that the decomposition of the Fock space will be found once s+nr,k s+nr and VM with respect to the we construct the decomposition of the spaces VM (s+nr) 0 b . To do this we use results of Sect. 3. Let N := s + nr and E m Uq (sln )- action U0 be the subspace of ∧N V (z) defined in (3.3). Recall that by Proposition 3 the E m is an irreducible representation of U0(s+nr) . Let m0 ∈ Mns+nr be the vacuum sequence (4.22) s+nr is given by associated with the integer M . Then the U0(s+nr) -decomposition of VM the following proposition

576

K. Takemura, D. Uglov

b n ) modules V s+nr and E m we have Proposition 8. For the Uq0 (sl M M s+nr = Em, VM {m∈Mn s+nr

|

(5.1)

ms+nr ≤m0s+nr }

b n ) action in the both sides is given by U (s+nr) . where the Uq0 (sl 0 Proof. We demonstrate that the set s+nr := {φ(m, e) ≡ 3(8m (z) ⊗ ve ) | m ∈ Mns+nr , ms+nr ≤ m0s+nr ; e ∈ E(m)} BM (5.2) s+nr s+nr s+nr . First we note that φ(m, e) ∈ BM implies φ(m, e) ∈ VM . is a base of the VM This follows since the triangularity of the polynomial 8m (z) and Lemma 3 allow us to represent the φ(m, e) as X X c(m, e; n, e0 )w(n, e0 ). (5.3) φ(m, e) = w(m, e) + 0 + + {n∈Mn s+nr |n ≺m } e ∈E(n)

s+nr s+nr is a spanning set of VM . Finally From the last equation it is also follows that BM s+nr are linearly independent by Proposition 1. By definition of the E m the elements of BM (3.13) the set {φ(m, e) | e ∈ E(m)} is a base of E m . Hence the result of the proposition. 

Since E m is homogeneous with the degree |m|, we have Corollary 1.

M

s+nr,k = VM

Em.

(5.4)

0 {m∈Mn s+nr | ms+nr ≤ms+nr } |m0 |−|m|=k

5.2. Decomposition of the Fock space. Let M = s mod n, 0 ≤ s ≤ n − 1. Write the vacuum vector |M i ∈ FM as ≡ z m 1 v 0 ∧ z m 2 v  0 ∧ · · · 0

|M i := uM ∧ uM −1 ∧ · · ·

0

1

2

.

(5.5)

The semi-infinite vacuum m-sequence associated with the |M i is m0 = (m01 , m02 , · · ·

) = (m0 , . . . , m0 , m0 + 1, . . . , m0 + 1, m0 + 2, . . . , m0 + 2, · · · | {z } | {z } | {z }

s−M . m := n

s

n

),

n

0

(5.6)

Introduce a set Mn [M ] whose elements are ordered semi-infinite sequences m as follows: Mn [M ] := {m = (m1 ≤ m2 ≤, · · · ) | m is n-strict ; mi = m0i for i >> 1}. And for m ∈ Mn [M ] define the degree kmk as X kmk := m0i − mi . i≥1

(5.7)

(5.8)

Level-0 Action of Uq (sbln ) on q-Deformed Fock Spaces

577

Note that kmk is a non-negative integer, and for a normally ordered wedge w = z m 1 v 1 ∧ z m 2 v 2 ∧ z m 3 v  3 ∧ · · ·

(w ∈ FM )

(5.9)

the degree |w| (4.86) is equal to kmk. Now for m ∈ Mn [M ] denote by m(N ) (∈ MnN ) the ordered sequence obtained from the m by removing all except the first N elements. m ⊂ FM be defined in the following way Let for m ∈ Mn [M ] a linear space FM m FM := E m

(s+nkmk)

∧ |M − s − nkmki.

(5.10)

In the last formula we have (s+nkmk)

s+nkmk,kmk

⊂ VM

s+nkmk

⊂ VM

⊂ ∧s+nkmk V (z),

kmk

m FM ⊂ FM . (5.11) The definition of the action U0 given in Sect. 4 and Proposition 7 immediately lead

Em

and

to m b n )-module with respect to the is an Uq0 (sl Proposition 9. For m ∈ Mn [M ] the FM 0 b m(s+nkmk) b n )-action with the U 0 (sl action U0 ; it is isomorphic to the U (sln )-module E q

(s+nkmk)

U0

q

. (s+nkmk)

m is irreducible, so is FM . Note that since by Proposition 3 the E m Now observing that Corollary 1 and Proposition 7 imply: M (s+nk),k m k FM = VM ∧ |M − s − nki = FM ,

(5.12)

k : we obtain for the Fock space FM = ⊕k≥0 FM M FM =

(5.13)

m∈Mn [M ] kmk=k

m FM .

m∈Mn [M ]

This is the sought for decomposition of the Fock space with respect to the level-0 action U0 . (s+nkmk) b n )-module is known from Proposition 3 Since the structure of the E m as a Uq0 (sl m we can describe a component FM of the decomposition (5.13) by using the isomorphism (s+nk mmk) m of E m and FM . To do this, by analogy with (3.25) for an m ∈ Mn [M ] introduce numbers rk (k = 0, 1, 2, . . . ) by m = (m1 = . . . = mr1 < m1+r1 = . . . = mr2 < m1+r2 = . . . = mr3 < · · · ), and r0 := 0.

(5.14)

b n )-module the space F m then is isomorphic to the semi-infinite tensor As the Uq0 (sl M product: m ∼ FM = V [p−mr1 q 2r1 −2 , r1 ] ⊗ V [p−mr2 q 2r2 −2 , r2 − r1 ]

⊗ . . . ⊗ V [p−mrk q 2rk −2 , rk − rk−1 ] ⊗

···

(5.15)

578

K. Takemura, D. Uglov

b n )-module defined in (3.27). Note where V [a, j] (1 ≤ j ≤ n) is the fundamental Uq0 (sl 0 that since mi = mi for all but finite number of elements in m; we have rk − rk−1 = n for k >> 1, and hence the tensor product (5.15) contains only a finite number of factors different from the trivial 1-dimensional representation V [a, n]. m are found from the Drinfel’d Polynomials of the The Drinfel’d Polynomials of FM (s+nkmk) m representation E (see Proposition 3), they are: Y (u − pmrk q −rk −rk−1 ) (i ∈ {1, 2, . . . , n − 1}). (5.16) Pi (u) = {1≤k<∞ | rk −rk−1 =i}

Let us remark that since rk − rk−1 = n for all sufficiently large k, the number of factors in the product above is always finite for any m ∈ Mn [M ]. m is an eigenspace of the commuting Hamiltonians Finally we find that the space FM gl ( l = ±1, ±2, . . .) defined by (4.95) and the eigenvalue of gl is ∞ X

(plmi − plmi )q 2l(1−i) . 0

(5.17)

i=1

Notice that the sum in the last expression contains only a finite number of non-zero summands due to the asymptotic condition mi = m0i for i >> 1. m (m ∈ Mn [M ]) is immediately obtained from An explicit base in the space FM m(s+nkmk) Definition (5.10) and a base of E described in (3.13). A. Appendix b n ) and the In this appendix we summarize the conventions concerning the algebra Uq0 (sl Drinfel’d Polynomials adopted in this paper. In particular we discuss the proof of Lemma 1. b n ). Let us recall the two realizations of U 0 (sl b n ) and the A.1. Two realizations of Uq0 (sl q definition of the Drinfel’d Polynomials. Definition 1. [5] The quantum Kac–Moody algebra Uq0 (g(A)) associated to a symmetric generalized Cartan matrix A = (aij )i,j∈I:={0,1,...n−1} is the unital associative algebra over C with generators Ei , Fi , Ki±1 (i ∈ I) and the following defining relations: Ki Ki−1 = 1 = Ki−1 Ki , K i Kj = Kj Ki ,

(A.1) (A.2)

Ki Ej Ki−1 = q aij Ej , Ki Fj Ki−1

=q

[Ei , Fj ] = X

1−aij r=0

(−1)r



1 − aij r

 q

−aij

(A.3)

Fj ,

Ki − Ki−1 δij q − q −1

(A.4) ,

(Ei )r Ej (Ei )1−aij −r = 0 , i 6= j.

(A.5)

(A.6)

Level-0 Action of Uq (sbln ) on q-Deformed Fock Spaces

X

1−aij r



(−1)

r=0

where [n]q :=

1 − aij r

 q

579

(Ei )r Ej (Ei )1−aij −r = 0 , i 6= j.

q n − q −n h n i [n]q [n − 1]q . . . [n − r + 1]q , := . q − q −1 [r]q [r − 1]q . . . [1]q r q

(A.7)

(A.8)

The coproduct 1 is given by 1(Ei ) := Ei ⊗ Ki + 1 ⊗ Ei , 1(Fi ) := Fi ⊗ 1 + Ki−1 ⊗ Fi , 1(Ki ) := Ki ⊗ Ki .

(A.9) (A.10) (A.11)

b n ) is the algebra U 0 (g(A)), where the generalized In particular the algebra Uq0 (sl q Cartan matrix A = (aij )i,j∈I is   2 (i = j) aij = −1 (|i − j| = 1, (i, j) = (1, n), (n, 1)) n ≥ 3, (A.12)  0 (otherwise)  aij =

2 (i = j) n = 2. −2 (i = 6 j)

(A.13)

b n ), then c0 is the central in U 0 (sl b n ). We define We put c0 := K0 K1 . . . Kn−1 in Uq0 (sl q 0 b 0 Lq (sln ) as the quotient of Uq (sln ) by the two sided ideal generated by c − 1. Proposition 10. [5] Lq (sln ) is isomorphic as an algebra to the algebra A with generators Ei,r , Fi,r (i ∈ {1, . . . , n − 1}, r ∈ Z), Hi,r (i ∈ {1, . . . , n − 1}, r ∈ Z\{0}), and Ki±1 , (i ∈ {1, . . . , n − 1}), and the following defining relations: Ki Ki−1 = 1 = Ki−1 Ki , Ki Hj,r = Hj,r Ki , [Hi,r , Hj,s ] = 0 ,

(A.14) (A.15) (A.16)

Ki Ej,r Ki−1 = q aij Ej,r ,

(A.17)

Ki Fj,r Ki−1

−aij

=q Fj,r , 1 [Hi,r , Ej,s ] = [raij ]q Ej,r+s , r 1 [Hi,r , Fj,s ] = − [raij ]q Fj,r+s , r Ei,r+1 Ej,s − q aij Ej,s Ei,r+1 = q aij Ei,r Ej,s+1 − Ej,s+1 Ei,r , Fi,r+1 Fj,s − q

−aij

p X X π∈Sp k=0

(−1)k

hpi k

q

Fi,r Fj,s+1 − Fj,s+1 Fi,r , + 8i,r+s − 8− i,r+s δij , q − q −1

Fj,s Fi,r+1 = q

[Ei,r , Fj,s ] =

−aij

(A.18) (A.19) (A.20) (A.21) (A.22) (A.23)

Ei,rπ(1) . . . Ei,rπ(k) Ej,s Ei,rπ(k+1) . . . Ei,rπ(p) = 0 , i 6= j, (A.24)

580

K. Takemura, D. Uglov p X X

(−1)k

π∈Sp k=0

hpi k

q

Fi,rπ(1) . . . Fi,rπ(k) Fj,s Fi,rπ(k+1) . . . Fi,rπ(p) = 0 , i 6= j,

(A.25)

for all sequences (r1 , . . . , rp ) ∈ Zp , where p = 1 − aij and the elements 8± i,r are determined by equating coefficients of powers of u in the formal power series 8± i (u) :=

∞ X

±r 8± = Ki±1 exp( ± (q − q −1 ) i,±r u

r=0

∞ X

Hi,±s u±s ).

(A.26)

s=1

The generators of A are called Drinfel’d generators. The isomorphisms f˜ : Lq (sln ) → A are not determined uniquely. In this paper we fix one isomorphism f : Lq (sln ) → A, f (Ei ) = Ei,0 , f (Fi ) = Fi,0 , f (Ki±1 ) = Ki±1 ,

(A.27)

for i ∈ {1, . . . , n − 1}, and f (K0±1 ) = (K1 K2 . . . Kn−1 )∓1 , m−1 −(n−2)/2

(A.28)

f (E0 ) = (−1) q [Fn−1,0 , [Fn−2,0 , . . . . . . , [Fm+1,0 , [F1,0 , . . . , [Fm−1,0 , Fm,1 ]q1/2 . . .]q1/2 f (K0 ),

(A.29)

f (F0 ) = µf (K0−1 )[En−1,0 , [En−2,0 , . . . . . . , [Em+1,0 , [E1,0 , . . . , [Em−1,0 , Em,−1 ]q1/2 . . .]q1/2 ,

(A.30)

where µ ∈ C× is determined by [f (E0 ), f (F0 )] =

f (K0 ) − f (K0−1 ) , q − q −1

(A.31)

and [a, b]q1/2 := q 1/2 ab − q −1/2 ba. A.2. Drinfel’d Polynomials. Now we will calculate the Drinfel’d Polynomials. For the Yangian case (q = 1), the Drinfel’d Polynomials of the irreducible components of the spin Calogero-Sutherland model can be calculated by using the result of [20]. For the quantum case, we mainly use the result written in [5]. b n ). W is said to be of type 1 if Let W be the representation of Uq0 (sl M Wµ , (A.32) W = µ∈Zn

where Wµ = {w ∈ W |ki .w = q µ(i) w}. b n )–module of type Proposition 11. [5] Let W be a finite-dimensional irreducible Uq0 (sl 1. Then, (a) W is generated by a vector w0 satisfying ± Ei,r .w0 = 0, 8± i,r .w0 = φi,r w0

for all i ∈ {1, . . . , n}, r ∈ Z, and some φ± i,r ∈ C.

(A.33)

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(b) There exist unique monic polynomials P1 (u), . . . , Pn−1 (u) (depending on W ) such that the φ± i,r satisfy ∞ X r=0

φ+i,r ur = q deg Pi

∞

Pi (q −2 u) X − −r = φi,r u , Pi (u)

(A.34)

r=0

in the sense that the left and right-hand sides are the Laurent expansions of the middle term about 0 and ∞ respectively. Assigning to W the corresponding n − 1– tuple of polynomials defines a one to one correspondence between the isomorphism b n )–modules of type 1 and the set of classes of finite–dimensional irreducible Uq0 (sl n − 1–tuples of monic polynomials in one variable u. We define the polynomials P1 (u), . . . , Pn−1 (u) to be the Drinfel’d Polynomials. Remark. If we change the isomorphism of Proposition 10, the Drinfel’d Polynomials may be changed. As a consequence of this Proposition, we get b n) Corollary 2. [5] Let W be a finite-dimensional irreducible representation of Uq0 (sl with associated polynomials Pi . Set λ = (deg P1 , . . . , deg Pn ). Then W contains the irreducible Uq (sln )–module V (λ) with multiplicity one. Further, if V (µ) is any other Uq (sln )–module occurring in W , then λ  µ. Let 3j be the j th fundamental weight of sln , then V (3j ) is an irreducible representab n ) (⊃ Uq (sln )), then V (3j ) tion of Uq (sln ). If V (3j ) is also the representation of Uq0 (sl 0 b is irreducible as a Uq (sln )-module and by Corollary 2, and the Drinfel’d Polynomials of V (3j ) is  u − a˜ if i = j, Pi (u) = (A.35) 1 otherwise, for some constant a˜ . We define the representation determined by (A.35) as V (3j ; a˜ ). We need the following lemma. Lemma 5. [5] Let v3j be the Uq (sln )–highest weight vector in V (3j ; a˜ ), where m ∈ {1, . . . , n − 1}, a˜ ∈ C× . Then, E0 .v3j = (−1)j−1 q −1 a˜ −1 Fn−1 Fn−2 . . . Fj+1 F1 . . . Fj .v3j .

(A.36)

Remark. This lemma and Lemma 6.4 in the paper [5] are different because the isomorphism between the realization of the Chevalley generators and the realization of the Drinfel’d generators are different. Proof. Using the fact that the weight spaces of V (3j , a˜ ) as a Uq (sln )–module are all one–dimensional and Ki Fj,1 Ki−1 = q −aij Fj,1 , we get Fj,1 .v3j = bFj .v3j

(A.37)

for some b ∈ C. From the relation (A.23), we get 8+j,1 .v3j = b(q − q −1 )v3j . Hence, from the definition of the Drinfel’d Polynomials, we have

(A.38)

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K. Takemura, D. Uglov

q(q −2 u − a˜ ) = (u − a˜ )(q + b(q − q −1 )u + O(u2 )),

(A.39)

so that b = a˜ −1 . Finally, from the relation (A.29), we find that E0 .v3j = (−1)j−1 q −1 a˜ −1 Fn−1 Fn−2 . . . Fj−1 F1 . . . Fj .v3j .

(A.40)

 From now on, we calculate the Drinfel’d Polynomials of V [a, j] (3.27). V [a, j] is the highest weight representation as a Uq (sln )-module, and the highest weight is 3j (the j th fundamental weight). First we can check that [v1 ⊗v2 ⊗. . .⊗vj ] is the highest weight vector of V [a, j] as a Uq (sln )–module. Because of the argument before Lemma 5, the Drinfel’d Polynomials of V [a, j] are  u − a˜ if i = j, Pi (u) = (A.41) 1 otherwise, for some constant a˜ . To determine a˜ , we observe how the Chevalley generators act, (A.42) E0 · ([v1 ⊗ v2 ⊗ . . . ⊗ vj ]) = [E0 v1 ⊗ K0 v2 ⊗ . . . ⊗ K0 vj ] + [v1 ⊗ E0 v2 ⊗ K0 v3 ⊗ . . . ⊗ K0 vj ] + . . . = a[vn ⊗ v2 ⊗ . . . ⊗ vj ]. Fn−1 . . . Fj+1 F1 . . . Fj · ([v1 ⊗ v2 ⊗ . . . ⊗ vj ]) [Kj−1 v1

Kj−1 vj−1

⊗ ... ⊗ = Fn−1 . . . Fj+1 F1 . . . Fj−1 · (. . . + = . . . = Fn−1 . . . Fj+1 · ([v2 ⊗ v3 ⊗ . . . ⊗ vj+1 ])

(A.43) ⊗ Fj vj ])

−1 −1 v2 ⊗ Kj+1 v3 ⊗ . . . ⊗ Fj+1 vj+1 ]) = Fn−1 . . . Fj+2 · (. . . + [Kj+1 = . . . = ([v2 ⊗ . . . ⊗ vj ⊗ vn ]).

On the other hand we can check vk ⊗ vl + qvl ⊗ vk ∈ Ker(q + S −1 ) for k < l.

(A.44)

[v2 ⊗ . . . ⊗ vj ⊗ vn ] = (−q)[v2 ⊗ . . . vj−1 ⊗ vn ⊗ vj ]

(A.45)

Then j−1

= (−q) [v2 ⊗ . . . ⊗ vn ⊗ vj−1 ⊗ vj ] = . . . = (−q) 2

[vn ⊗ v2 ⊗ . . . ⊗ vj ].

From Lemma 5, we get a˜ = q j−2 a−1 . So we have proved Lemma 1. Next we verify Proposition 3. Basically we use the following proposition. b n )–module, whose Proposition 12. [5] If V, W, V ⊗ W are all irreducible as a Uq0 (sl Drinfel’d Polynomials are PV,i (u), PW,i (u), PV ⊗W,i (u) (i ∈ {1, 2, . . . , n − 1}), then PV ⊗W,i (u) = PV,i (u) · PW,i (u).

(A.46) m

Using the fact written in [1], we can check that our representation W (3.26) is irreducible if α ∈ R \ Q≤0 (p = q −2α ). We explain the reason briefly. Let 3ki be the kith fundamental weight of sln . The key propositions are ◦ V1 (3k1 ; a1 ) ⊗ V2 (3k2 ; a2 ) is irreducible if and only if the intertwiner R(V1 , V2 ) : V1 (3k1 ; a1 ) ⊗ V2 (3k2 ; a2 ) → V2 (3k2 ; a2 ) ⊗ V1 (3k1 ; a1 ) and R(V2 , V1 ) have no pole. ◦ V1 (3k1 ; a1 ) ⊗ . . . ⊗ Vl (3kl ; al ) is irreducible if and only if Vi (3ki ; ai ) ⊗ Vj (3kj ; aj ) is irreducible for all i < j. Acknowledgement. We would like to thank Professors M. Kashiwara and T. Miwa for discussions and support.

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References 1. Akasaka, T. and Kashiwara, M.: To be published 2. Bernard, D., Gaudin, M., Haldane, F.D.M. and Pasquier, V.: Yang-Baxter equation in long-range interacting systems. J. Phys. A26, 5219–5236 (1993) 3. Bernard, D., Pasquier, V. and Serban D.: Spinons in conformal field theory. Nucl. Phys., B428, 612–628 (1994) 4. Bouwknegt, P., Ludwig, A. and Schoutens, K.: Spinon bases, Yangian symmetry and fermion representations of Virasoro characters in conformal field theory. Phys. Lett., 338B, 448–456 (1994) 5. Chari, V. and Pressley, A.: Quantum affine algebras and affine Hecke algebras. Pacific J. Math. 174, 295–326 (1996) (q-alg/9501003) 6. Cherednik, I.V.: A new interpretation of Gelfand-Tzetlin bases. Duke Math. 54, 563–577 (1987) 7. Cherednik, I.V.: Integration of Quantum Many-Body Problems by Affine Knizhnik-Zamolodchikov Equations. Preprint RIMS-776 (1991); Adv. Math. 106, 65–95 (1994); Induced representations of double affine Hecke algebras and applications. Math. Res. Lett. 1, 319–337 (1994) 8. Cherednik, I.V.: Double Affine Hecke algebras and Macdonald’s conjectures. Annals Math. 141, 191– 216 (1995); Non-symmetric Macdonald’s Polynomials. IMRN 10, 483–515 (1995) 9. Drinfel’d, V. G.: Sov. Math. Doklady 36, 212 (1988) 10. Ginzburg, V., Kapranov, M. and Vasserot, E.: Langlands reciprocity for algebraic surfaces. Math. Res. Lett. 2, 147–160 (1995) 11. Haldane, F.D.M., Ha, Z.N.C., Talstra, J.C., Bernard, D. and Pasquier, V.: Yangian symmetry of integrable quantum chains with long-range interaction and a new description of states in conformal field theory. Phys. Rev. Lett. 69, 2021–2025 (1992) 12. Hayashi, T.: Q-analogues of Clifford and Weyl algebras – spinor and oscillator representations of quantum enveloping algebras. Commun. Math. Phys., 127, 129–144 (1990) 13. Jimbo, M.: Topics from representation theory of Uq (g) – an introductory guide to physicists. Nankai Lectures on Mathematical Physics, Singapore: World Scientific, 1992 14. Jimbo, M., Kedem, R., Konno, H., Miwa, T. and Petersen, J.-U. H.: New level-0 action of Uq (sbln ) on level-1 modules. Preprint RIMS-1016, to appear in the Proceedings of “Statistical Mechanics and Quantum Field Theory” , USC, 16–21 May 1994 (q-alg/950 5009) 15. Jimbo, M., Kedem, R., Konno, H., Miwa, T. and Petersen, J.-U. H.: Level-0 action of Uq (sbln ) on level-1 modules and Macdonald Polynomials. J. Phys, A28, 5589 (1995) (q-alg/9506016) 16. Kac, V.G. and Raina, A.: Bombay Lectures on Highest Weight Representations of Infinite Dimensional Lie Algebras. Singapore: World Scientific, 1987 17. Kashiwara, M., Miwa, T. and Stern, E.: Decomposition of q-deformed Fock Spaces. Selecta Mathematica, New Series, 1, No. 4 , 787–805 (1995) (q-alg/9508006) 18. Konno, H.: Relativistic Calogero-Sutherland Model: Spin Generalization, Quantum Affine Symmetry and Dynamical Correlation Functions. Preprint YITP/K-1118 (hep-th/9508016) 19. Macdonald, I. G.: Affine Hecke algebra and Orthogonal Polynomials. S´eminaire Bourbaki, 47, No. 797, 1–18 (1995) 20. Nazarov, M. and Tarasov, V.: Representations of Yangians with Gelfand-Zetlin bases. Preprint UWSMRRS-94-148 (q-alg/9502008) 21. Uglov, D.: Semi-infinite wedges and the conformal limit of the fermionic spin Calogero-Sutherland model of spin 21 . Nucl. Phys. B478, 401–430 (1996) 22. Varagnolo, M. and Vasserot, E.: Schur duality in the toroidal setting. Commun. Math. Phys. 182, 469–484 (1996) Communicated by T. Miwa

Commun. Math. Phys. 190, 585 – 596 (1998)

Communications in

Mathematical Physics c Springer-Verlag 1997

Zero Modes of the Dirac Operator for Regular Einstein–Yang–Mills Background Fields Othmar Brodbeck, Norbert Straumann Institute for Theoretical Physics, University of Zurich, Winterthurerstrasse 190, 8057 Zurich, Switzerland Received: 19 March 1997 / Accepted: 21 April 1997

Abstract: The existence of normalizable zero modes of the twisted Dirac operator is proven for a class of static Einstein–Yang–Mills background fields with a half-integer Chern–Simons number. The proof holds for any gauge group and applies to Dirac spinors in an arbitrary representation of the gauge group. The class of background fields contains all regular, asymptotically flat, CP-symmetric configurations with a connection that is globally described by a time-independent spatial one-form which vanishes sufficiently fast at infinity. A subset is provided by all neutral, spherically symmetric configurations which satisfy a certain genericity condition, and for which the gauge potential is purely magnetic with real magnetic amplitudes.

1. Introduction A few years ago, Gibbons and Steif [1] studied the Dirac equation in the background of the static, particle-like Bartnik–McKinnon solution [2] of the Einstein–Yang–Mills (EYM) theory. By an explicit calculation they found a normalizable zero-energy state in the S-wave sector and speculated on its possible role in anomalous fermion production. There is indeed a close analogy with the situation in the electroweak theory which has been investigated by many authors. (For a recent review see [3].) As was first pointed out by Gal’tsov and Volkov [4], the Bartnik–McKinnon solution can be interpreted as a sphaleron, in other words, as a static saddle point solution separating vacua with different Chern–Simons number. One type of instabilities is related to this property, as was first shown in [5] and fully analyzed in [6]. (The previously discovered “gravitational instability” [7], which is also present for the black hole analogues [8], is, however, of a different type.) For a study of the problem of anomalous fermion production in the presence of gravitational solitons, we refer to the recent paper [9]. It is natural to suspect that the existence of the zero-energy solution found in [1] is mathematically connected with the fact that the Chern–Simons number of the Bartnik–

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McKinnon solution is 1/2, and that one might be able to generalize the result with the help of an index theorem. In the present paper we show that this is indeed the case. In Sects. 2 and 3, we prove that the twisted Dirac operator for a certain class of gauge fields in a static, regular, asymptotically flat space-time has at least two normalizable zero modes, if the Chern–Simons number of the gauge field is half-integer (Propositions 1 and 2). The conditions are formulated for any gauge group and the above statement holds for any representation of the Dirac spinors (with respect to the gauge group). In Sect. 4, we show that this class contains all CP-symmetric configurations with a connection that is globally described by a time-independent spatial one-form which vanishes sufficiently fast at infinity (Theorem 1). On the basis of our extensive earlier work on spherically symmetric EYM solutions for arbitrary gauge groups [10–12], we verify in Sect. 5 that the above-mentioned class contains all neutral, generic, spherically symmetric configurations for which the gauge potential is purely magnetic with real magnetic amplitudes (Theorem 2). [The precise definition of “generic” is given at the beginning of Sect. 5.] In particular, all neutral, generic solitons of the EYM system belong to this subclass. For the benefit of the reader, we remark that the proof of Proposition 2 is mainly based on the Atiyah–Patodi–Singer index theorem for manifolds with boundary [13]. However, the verification of the assumptions of Proposition 2 for spherically symmetric configurations heavily relies on our previous work [10–12], especially on [11]. In the latter paper, we have also given explicit expressions for the Chern–Simons numbers of purely magnetic gauge fields, which involve only general properties of irreducible root systems and information about the asymptotic behavior of the configuration. 2. Reduction to a Harmonic Problem on S3 In a first step, we show for a class of static EYM fields that normalizable zero modes of the twisted Dirac operator can be obtained from zero modes of a related Dirac operator on the three-sphere. To this end, we consider the following set-up: Let (M, g) be a static space-time with M = R×Σ and g = α2 (dt 2 −h), where (Σ, h) is a three-dimensional Riemannian space and α is a smooth function on Σ. In addition, let A be a one-form on Σ with values in the Lie algebra LG of the gauge group G. We are interested in zero modes of the twisted Dirac operator D on M in the background of the one-form A, which is considered as a globally defined gauge potential. Since the massless Dirac equation is conformally invariant, we may take α = 1. It is then straightforward to show that D splits in the Weyl representation of the Dirac– Clifford algebra as follows:   0 ∂t + i D , (2.1) D= ∂t − i D 0 where D is the three-dimensional Dirac operator on (Σ, h) corresponding to A (with one of the two inequivalent representations of the three-dimensional Clifford algebra). Hence, the massless Dirac equation, DΨ = 0 ,

(2.2)

is (for arbitrary α) equivalent to a pair of two-component Weyl equations, i ∂t ϕ+ = D ϕ+ ,

i ∂ t ϕ − = − D ϕ− .

(2.3)

Zero modes of Dirac operators in external EYM fields

587

Obviously, each (normalizable) zero mode of D provides two solutions of (2.2). For our applications we now specialize to Σ ≈ R3 and an asymptotically flat metric h. This property is defined by conformal compactification: It is assumed that the pullback π ∗ h to S3 via the stereographic projection π from the north pole can be extended – after multiplication with a conformal factor Ω 2 – to a smooth Riemannian metric h˜ on S3 . Moreover, the function Ω has to vanish with the following rate when the north pole is approached: If Ω is expressed in stereographic coordinates x, it behaves as 1/|x|2 for |x| → ∞. [Recall that the standard metric on S3 is conformally Euclidean, with conformal factor Ω 2 = 4/(1 + |x|2 )2 .] If A vanishes at infinity, it is possible to obtain zero modes of D on (R3 , h) from ˜ defined by the pull-back A˜ of A. Clearly, zero modes of the Dirac operator D˜ on (S3 , h) ˜ a zero mode ϕ˜ of the Dirac operator D gives rise to a normalizable zero mode of the Dirac operator on R3 , however, with metric Ω 2 h. But now we take advantage of the well-known invariance property of the Dirac operator under conformal transformations to conclude: A zero mode ϕ˜ of D˜ on S3 gives rise to the zero mode Ω ϕ˜ of D on R3 equipped with the original metric h. Moreover, since ϕ˜ is bounded on S3 , the fall-off property of Ω implies that Ω ϕ˜ is normalizable. ˜ two normalizable zero modes of We thus obtain for each zero mode of D˜ on (S3 , h) (2.2). In the following sections, we shall investigate for which EYM configurations the conditions we had to assume for the gauge fields do hold. For reference, we summarize the main conclusion of this section: Proposition 1. Consider a static, regular, asymptotically flat EYM background configuration with the following property: (A1) There exists a global gauge such that the YM field is described by a timeindependent one-form A on R3 which vanishes (sufficiently fast) at infinity. Let A˜ be the pull-back of A to S3 via stereographic projection, and let h˜ be the metric on S3 arising from the background metric by conformal compactification. Then each ˜ belonging to A˜ gives rise to two zero mode of the twisted Dirac operator on (S3 , h) zero modes of the four-dimensional Dirac operator in the background of the given EYM configuration. 3. Zero Modes of the Twisted Dirac Operator on S3 By the considerations in Sect. 2, we are led to study the kernel of the Dirac operator D˜ on ˜ in the background of a global gauge potential A˜ on S3 . (Recall that all principal (S3 , h) bundles over S3 are trivial, because these are classified by the second homotopy group of the gauge group G.) We are thus interested in (dropping tildes) h(D) = dim(Ker D) .

(3.1)

In the present section, we prove that the following relation holds for a certain class of connections: (3.2) h(D)/2 + NCS (A) ∈ Z , where NCS (A) is the Chern–Simons number of the gauge potential A, Z h A , dA + [A, A]/3 i . NCS (A) = S3

(3.3)

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Here, the invariant scalar product in LG is given by hX , Y i = −

1 Tr { Lρ (X) · Lρ (Y ) } , 8π 2

X, Y ∈ LG ,

(3.4)

where ρ denotes the representation of the Dirac spinors with respect to the gauge group G. In the following Sect. 4, we will show that all CP-symmetric configurations satisfy (3.2), which, together with Proposition 1, implies our main result: If the Chern–Simons number NCS (A) of a CP-symmetric gauge potential is half-integer, then h(D) must be odd and hence, by Proposition 1, the original twisted Dirac operator has at least two normalizable zero modes. We establish (3.2) with crucial use of the Atiyah–Patodi–Singer (APS) index theorem for manifolds with boundary, which we apply to the manifold I × S3 , where I is an interval of R. Adopting standard notations, this theorem states that the index of the twisted Euclidean Dirac operator D on M = I × S3 belonging to a gauge potential A on M (and some representation of the gauge group) is given by Z Aˆ ∧ ch − ξ[∂M ] , (3.5) ind(D(A)) = M

where the boundary correction ξ, ξ := (h + η)/2 ,

(3.6)

is the half-sum of the harmonic correction h [∂M ], and the APS η-invariant, X −s η[∂M ] := sign (λi ) |λi | . {λi 6= 0}

(3.7)

s=0

(For an explanation of the notation, see [14]; for a proof, see for instance [13].) In order to apply (3.5), we now make the following assumption: (A2) The global gauge potential A on S3 can be embedded into a one-parameter family A(τ ) , τ ∈ [−1 , 1 ], with the following properties: 1. A(−1) and A(+1) are pure gauges. 2. The corresponding family of Dirac operators D( · ) is “reflection antisymmetric”, that is, D(+τ ) is conjugate to −D(−τ ) for all τ ∈ [−1 , 1 ]. An immediate consequence of assumption (A2.2) is that the η-invariant η(τ ) := η(D(τ ) ) is odd, η(+τ ) = −η(−τ ) . Hence, η(0) = η(D) = 0, which implies that ξ(D) = h (D)/2 .

(3.8)

Taking advantage of (A2.1), we see that η(+1) = η(−1) , whence η(±1) = 0. Moreover, by a theorem due to Lichnerowicz1 , h (D(±1) ) = 0, since, up to a gauge transformation, the Dirac operators D(±1) agree with the untwisted Dirac operator ∇. The ξ-invariant thus vanishes for τ = ±1, (3.9) ξ(D(±1) ) = 0 . 1 Note added: It is always assumed that the metric h ˜ on S3 satisfies the assumptions of the Lichnerowicz theorem, i.e., that the scalar curvature is non-negative, and strictly positive at some point.

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Now we are ready to apply the APS index theorem. The family D( · ) on S3 determines the Euclidean Dirac operator   0 ∂ τ + D( · ) (3.10) D=i ∂ τ − D( · ) 0 on the “Einstein cylinder” [−1 , 1 ] × S3 . According to (3.5), the index of D on I × S3 , where I = [τ1 , τ2 ] ⊂ [−1 , 1 ], is (for inward oriented normals) given by n o (3.11) ind(D) = C2 [ I × S3 ] − ξ(D(τ1 ) ) + ξ(−D(τ2 ) ) , where C2 is the second Chern number of the underlying principal bundle evaluated for the gauge field A = A( · ) . Since C2 agrees with the change ∆NCS of the Chern–Simons number (3.3), and since ξ(−D(τ ) ) = −ξ(D(τ ) ) + h(D(τ ) ), Eq. (3.11) reduces to ind(D) = ∆NCS + ∆ξ − h(τ2 ) .

(3.12)

The right-hand side of Eq. (3.12) is an integer which only depends on the gauge potentials at the boundary. Choosing I = [−1, τ ], this implies that the following function f is integer-valued: τ τ ∈Z. (3.13) f (τ ) := NCS + ξ − h(τ ) −1

−1

When evaluated for τ = 0, this, together with Eqs. (3.8) and (3.9), gives h(D)/2 + NCS (A) = f (0) + h(D) + NCS (−1) .

(3.14)

To complete the derivation of relation (3.2), it remains to demonstrate that the Chern– Simons number NCS (U −1 dU ) of a pure gauge U −1 dU on S3 is always an integer. For a proof of this less surprising statement, we again use the APS index theorem: Consider the twisted Dirac operator D on the Einstein cylinder [−1 , 1 ] × S3 corresponding to the family of gauge potentials defined by A(τ ) = γ(τ ) U −1 dU ,

(3.15)

where γ is a R+ -valued function which smoothly interpolates between zero and one, γ(−1) = 0 ,

γ(1) = 1 ,

supp(γ) ˙ ⊂ (−1 , 1) .

(3.16)

For this family, expression (3.12) for the index of D reads ind(D) = NCS (U −1 dU ) + ∆ξ ,

(3.17)

since h(1) and NCS (−1) both vanish. Moreover, ∆ξ also vanishes, since (by construction) the operators D(−1) and D(+1) agree up to a gauge transformation. Hence, as claimed, NCS (U −1 dU ) is an integer. We summarize: ˜ with the property (A2), the Proposition 2. For a global gauge potential A˜ on (S3 , h) kernel of the corresponding twisted Dirac operator D˜ satisfies ˜ ˜ ∈ Z, dim(Ker D)/2 + NCS (A) ˜ is the normalized Chern–Simons number of the gauge field A. ˜ where NCS (A)

(3.18)

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4. CP-Symmetric Gauge Fields In this section, we establish that the technical assumption (A2) in Proposition 2 is always fulfilled, provided that the EYM field is CP-symmetric. To discuss CP-transformations of gauge fields, we first show that charge conjugation of a connection form ω for a compact gauge group G (which, for simplicity, is assumed to be simply connected and semisimple) is induced by the so-called Chevalley inversion C in the Lie algebra LG: ω C = C ◦ ω. Hence, for a connection form ω on a principal bundle admitting an automorphism P which induces a parity transformation in the base manifold, there is a natural definition of CP-symmetry: A connection form ω is CP-symmetric, if ω CP := C ◦ P∗ ω is equal to ω up to a global gauge transformation (that is, up to a bundle automorphism inducing the identity in the base manifold). 4.1. Chevalley automorphism and charge conjugation of gauge fields. The connection between charge conjugation and the Chevalley automorphism of the Lie algebra LG is most easily discussed in terms of a (suitably chosen) Chevalley–Weyl basis, which is a basis adapted to a Cartan decomposition of the complexified Lie algebra LGC . For the gauge group G, we first fix a maximal torus T and choose a basis S of the (real) root system R. This basis defines both, the set of positive roots R+ as well as the fundamental Weyl chamber K(S), K(S) = { H ∈ LT | α(H) > 0 for all α ∈ S } .

(4.1)

For the complexified Lie algebra LGC , we now choose a Chevalley–Weyl basis { hαj , eα | αj ∈ S, α ∈ R } ,

(4.2)

(where hαj ∈ LTC , and eα are base vectors of the root spaces Lα ) such that the elements ihαj span LT , and such that the set { ihαj , (e+α + e−α ) , i(e+α − e−α ) | αj ∈ S, α ∈ R+ }

(4.3)

forms a basis of the compact real form LG (see for instance [15], Sect. 25). Charge conjugation reverses the signs of all charges and thus corresponds to an automorphism σ of LG which reduces to −id on LT . Such an automorphism is automatically involutive and its extension to the complexification LGC (also denoted by σ) interchanges the root spaces L+α and L−α . A particular automorphism with these properties is the above-mentioned Chevalley automorphism C, C: hαj 7−→ −hαj ,

e+α 7−→ e−α .

(4.4)

Moreover, an arbitrary involutive automorphism σ of LGC which leaves LG invariant, and which reduces to −id on LT can be described as follows: Let ϕ = C ◦ σ, then ϕ|LT = id , with

|cα |2 = 1 ,

c−α = c+α ,

ϕ(eα ) = cα eα cα cβ = cα+β

(α ∈ R) (α, β, α + β ∈ R) .

(4.5) (4.6)

This implies that the automorphisms σ and C are conjugate in T ⊂ G (that is, σ = Ad(t) ◦ C for some element t ∈ T ) and hence reflects a freedom in choosing phases. In what follows, we use the Chevalley automorphism C (with respect to a fixed Chevalley–Weyl basis) to define charge conjugation for a connection form ω,

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C: ω 7−→ ω C := C ◦ ω .

(4.7)

By the considerations above, any other choice of the automorphism σ would just lead to a compensating gauge transformation with a constant element of the gauge group. For a given automorphism σ, the transformation law for a spinor field Ψ under Cparity has, of course, to be chosen such that Ψ and DΨ transform in the same manner. Let us add in this context that the conjugate representation ρ of a unitary representation ρ is equivalent to the representation ρC := ρ ◦ C, because ρ is isomorphic to the contragradient representation ρ∗ , which has the same weights as ρC , ρ ∼ = ρ∗ ∼ = ρC . 4.2. Existence of zero modes. Now we are in the position to prove our main result: Theorem 1. Under the assumptions of Proposition 1, the twisted Dirac operator in the background of a CP-symmetric EYM field has at least two normalizable zero modes, provided that the normalized Chern–Simons number of the gauge field is half-integer. Theorem 1 is an immediate consequence of Propositions 1 and 2, as soon as the following assertion is established: ˜ satisfies Lemma 1. Every CP-symmetric, globally defined gauge field A˜ on (S3 , h) assumption (A2). For a proof, we first note that CP-transformations are well defined for the situations ˜ with we are interested in (because an orientation reversing, involutive isometry of (S3 , h) two fixed points can always be lifted to an involutive automorphism of the spin structure). We also note that A˜ (which, by assumption, is the gauge potential of a CP-symmetric connection) satisfies A˜ cp := C ◦ p∗ A˜ = Ad(V −1 )A˜ + V −1 dV ,

(4.8)

˜ and V is some transition where p is the parity transformation in the base manifold (S3 , h), function. With the help of (4.8), we now embed A˜ into a family of gauge potentials with the required properties:  o n  ˜ −1 ˜ for τ ∈ [ 0 , 1 ] dV − A A + γ(τ ) · V (4.9) A˜ (τ ) :=  −1 + V dV for τ ∈ [−1 , 0 ) , Ad(V )A˜ cp (−τ ) where γ: [ 0 , 1 ] → R+ is a smooth function interpolating between zero and one, γ(0) = 0 ,

γ(1) = 1 ,

supp(γ) ˙ ⊂ (0,1) .

(4.10)

First, we establish (A2.1). Taking advantage of the transformation property of a pure gauge V −1 dV under a group homomorphism Φ, Φ(V )−1 d Φ(V ) = LΦ(V −1 dV ) , it is easy to see that A˜ (τ )

 −1  U dU for τ = −1 = A˜ for τ = 0  −1 V dV for τ = 1 ,

(4.11)

(4.12)

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where the pure gauge U −1 dU is given by U −1 dU = (V cp V −1 )−1 d(V cp V −1 ) .

(4.13)

Hence, (A2.1) is fulfilled. To deduce the reflection antisymmetry (A2.2), we make use of the fact that the ˜ transforms under CP in the same way as in flat ˜ A) ˜ on (S3 , h) twisted Dirac operator D( space. (The non-contractibility of S3 does not induce any serious global problems.) Up to a gauge transformation, we thus have ˜ A) ˜ = −D( ˜ A˜ cp ) ◦ CP . CP ◦ D(

(4.14)

˜ This shows (A2.2), since (by construction) the potentials A˜ cp (τ ) and A(−τ ) are gauge equivalent.

5. Spherically Symmetric EYM Fields On the basis of our previous work on EYM soliton and black hole solutions for arbitrary gauge groups [10–12], we establish in this section that a “generic”, spherically symmetric EYM field (defined in the next paragraph) is CP-symmetric, precisely when the YM field is purely magnetic with real magnetic amplitudes. As we shall see, this implies that the remaining technical assumption (A1) in Theorem 1 is always fulfilled for fields of this type. To begin with, we briefly recall the general framework and define what we call generic EYM fields. A spherically symmetric gauge field is described by an SU(2)-invariant connection on a principal bundle P (M, G) admitting an action of the symmetry group SU(2). In a standard manner, this connection defines a linear map Λ: LSU(2) → LG (the Wang map), which, at least locally, depends smoothly on the orbits of the symmetry group in the base manifold (see [10] or [16]). The underlying symmetric bundle P (M, G) is classified by an integral element Hλ lying in that part of the integral lattice I ⊂ T which is contained in the closed fundamental Weyl chamber K(S), Hλ ∈ I ∩ K(S). In the present discussion, we exclude situations where this element lies on the boundary of the fundamental Weyl chamber. The term “generic” refers to configurations for which the classifying element Hλ is contained in the open Weyl chamber K(S). In order to avoid extensive repetitions, we adopt the earlier notations as far as possible. Specific references to formulas in [11] are indicated by the prefix II. 5.1. Basic properties of regular configurations. For a static, spherically symmetric EYM field, there exists a local gauge such that the YM field is described by a time-independent potential which, in adapted coordinates, is given by A = a dt + Λ(σ −1 dσ) .

(5.1)

Here, a is a LT -valued function of the radial coordinate r, Λ denotes the Wang map, and σ is a local cross section from S2 ∼ = SU(2)/U(1) to SU(2), which we choose as σ(ϑ, ϕ) = e ϕ τ3 e ϑ τ2 ,

(5.2)

where 2iτk = σk are the Pauli matrices, and ϑ, ϕ denote the standard angular coordinates of S2 . For this section, the potential (5.1) takes the form

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A = a dt + Λ2 dϑ + (Λ3 cos ϑ − Λ1 sin ϑ) dϕ ,

(5.3)

where Λk := Λ(τk ) are LG-valued functions of the radial coordinate, satisfying Λ1 = [Λ2 , Λ3 ] ,

Λ2 = [Λ3 , Λ1 ] ,

Λ3 = −Hλ /(4π) .

(5.4)

If the configuration is also regular at the origin, the linear map LΦ := Λ|r=0 is a Lie algebra homomorphism from LSU(2) to LG. Moreover, the same is true for Λ|r=∞ , provided that the magnetic charge of the YM field vanishes. (For derivations, see [11]). For the discussion of CP-transformations, a very useful technical feature of regular configurations is that the Wang map Λ can be chosen such that the corresponding homomorphism Λ|r=0 commutes with the Chevalley automorphism. For a proof of this property, as well as for later use, we first recall that the potential (5.1) is locally gauge equivalent to the following globally defined gauge potential [Eq. (II.44)] with a slight change of notation]:     −1 (5.5) A = Ad Φ(σ) Λ − LΦ (σ dσ) + a dt , where Φ is the group homomorphism from SU(2) to G corresponding to the infinitesimal homomorphism LΦ = Λ|r=0 . We further recall [see Eq. (II.23)] that Λ+ := Λ1 +iΛ2 admits an expansion of the form X w α eα , (5.6) Λ+ = α∈S(λ)

S(λ) = { α ∈ R+ | α(Hλ ) = 2 } ,

(5.7)

where eα are base vectors of the root spaces Lα , which we choose to belong to the distinguished Chevalley–Weyl basis introduced in Sect. 4.1. Next, we observe that the potential (5.5), after a constant gauge transformation with an element h of the maximal torus T , can be rewritten as     Ad(h−1 )Λ − Ad(h−1 )LΦ (σ −1 dσ) + a dt , (5.8) A = Ad (h−1 Φ h)(σ) which shows, since L(h−1 Φ h) = Ad(h−1 )LΦ = Ad(h−1 )Λ|r=0 , that the Wang maps Λ and Ad(h−1 )Λ, h ∈ T , belong to gauge equivalent configurations. Using this global gauge freedom, we now fix the phases of the magnetic amplitudes wα such that the boundary values wα |r=0 are real. For the Chevalley transformed LΦk C , this choice implies LΦ2 C = LΦ2 , LΦ3 C = −LΦ3 , (5.9) LΦ1 C = −LΦ1 , which proves our claim: LΦ(τk )C = LΦ(τ k ) = LΦ(τk C ) .

(5.10)

5.2. CP-symmetric EYM fields. For spherically symmetric gauge fields in the global gauge (5.5), we now discuss the action of a CP-transformation, which, up to a gauge transformation, is given by cp : A 7−→ Acp = C ◦ p∗ A ,

(5.11)

where p denotes the parity transformation in the base manifold, and C is the Chevalley inversion in LG.

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Since the gauge potential (5.5) reads more explicitly [see Eqs. (5.1) and (5.3)]       A = Ad Φ(σ) Λ2 − LΦ2 dϑ − Λ1 − LΦ1 sin ϑ dϕ + a dt , (5.12) and since LΦ is a homomorphism which commutes with the Chevalley transformation [see Eq. (5.10)], we easily find        cp cp C C A = Ad Φ(σ) − Λ2 − LΦ2 dϑ + −Λ1 − LΦ1 sin ϑ dϕ − a dt , (5.13) as well as

    Ad Φ(σ)cp = Ad Φ(σ cp )   = Ad Φ( e π τ2 σ e±π τ3 )       = Ad e π LΦ2 ◦ Ad Φ(σ) ◦ Ad e± π LΦ3 .

(5.14)

To work out the effect of the last map on the right-hand side of Eq. (5.14), we make use of the following general covariance property of the Wang map Λ [see Eq. (II.5)]:   Ad e s LΦ3 ◦ Λ = Λ ◦ Ad(e s τ3 ) , (5.15) which yields, when evaluated on τ1 and τ2 , Ad(e±π LΦ3 ) Λk = −Λk ,

k = 1, 2 .

(5.16)

Hence, apart from a constant gauge transformation, the action of cp amounts to cp : Λ1 7−→ −ΛC1 ,

Λ2 7−→ ΛC2 ,

a 7−→ −a .

(5.17)

In terms of the magnetic amplitudes wα [see Eq. (5.6)], the cp-transformation (5.17) takes the familiar form cp : wα 7−→ wα ,

a 7−→ −a ,

(5.18)

which suggests that for a CP-symmetric configuration all magnetic amplitudes wα are real, and that the electric part a vanishes identically. In order to demonstrate this, we note that the form of the potential (5.5) remains unchanged under a gauge transformation with (5.19) U = Φ(σ) e χ Φ(σ)−1 , where χ is a LT -valued function independent of the spherical angles. Since the corresponding transformations of wα and a dt are given by wα 7−→ e− i χα wα ,

a dt 7−→ a dt + dχ ,

(5.20)

where χα = 2π α(χ), a configuration is CP-symmetric precisely when wα = e− i χα wα ,

a dt = −dχ/2

(5.21)

for some LT -valued function χ. With similar arguments as in [11], Sect. 4, it is now easy to conclude that a ≡ 0 (up to a gauge transformation), and that all amplitudes wα are

Zero modes of Dirac operators in external EYM fields

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real up to constant phases. Moreover, these phases are trivial, since (by construction) all boundary values wα |r=0 are real, and because none of them vanishes (see [11], Appendix A). Thus, as claimed, wα = wα and a = 0. Since an arbitrary spherically symmetric metric is CP-symmetric, we have established the first part of the following intermediate result. For a proof of the second part, we refer to [11]. Proposition 3. (i) A static, regular, generic EYM configuration is CP-symmetric, if and only if the YM field is purely magnetic with real magnetic amplitudes. (ii) For neutral configurations of this type, there exists a global gauge such that the YM field is described by a time-independent spatial one-form which vanishes at infinity. Proposition 3 classifies the CP-symmetric EYM fields with spherical symmetry and shows that the remaining technical assumption (A1) in Theorem 1 is always fulfilled for neutral configurations of this type. As a corollary of Theorem 1, we thus obtain the following result for spherically symmetric fields: Theorem 2. For a static, regular, asymptotically flat EYM background configuration with a YM field that is generic, neutral, and purely magnetic with real magnetic amplitudes, the corresponding twisted Dirac operator has at least two normalizable zero modes, provided that the normalized Chern–Simons number of the YM field is halfinteger. For example, all neutral, generic EYM solitons for an arbitrary gauge group are purely magnetic with real magnetic amplitudes [11]. Hence, Theorem 2 guarantees the existence of fermion zero modes for any neutral, generic EYM soliton with a half-integer Chern–Simons number. (For an explicit verification for SU(2) EYM solitons and spinors in the fundamental representation see [1].) Acknowledgement. We would like to thank Andreas Wipf for discussions at an earlier stage of our work. Discussions with members of our theory group, especially with Michael Volkov and Markus Heusler, are also gratefully acknowledged. Finally, we wish to thank the Swiss National Science Foundation for financial support.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.

Gibbons, G., and Steif, A.: Phys. Lett. B 314, 13 (1993) Bartnik, R., and McKinnon, J.: Phys. Rev. Lett. 61, 141 (1988) Rubakov, V.A., and Shaposhnikov, M.E.: Phys. Usp. 39, 461 (1996); hep-ph/9603208 Gal’tsov, D.V., and Volkov, M.S.: Phys. Lett. B273, 255 (1991) Volkov, M.S., and Gal’tsov, D.V.: Phys. Lett. B341, 279 (1995) Volkov, M.S., Brodbeck, O., Lavrelashvili, G., and Straumann, N.: Phys. Lett. B349, 438 (1995) Straumann, N., and Zhou, Z.-H.: Phys. Lett. B237, 353 (1990) Straumann, N., and Zhou, Z.-H.: Phys. Lett. B243, 33 (1990) Volkov, M.S.: Phys. Rev. D54, 5014 (1996) Brodbeck, O., and Straumann, N.: J. Math. Phys. 34, 2412 (1993) Brodbeck, O., and Straumann, N.: J. Math. Phys. 35, 899 (1994) Brodbeck, O., and Straumann, N.: J. Math. Phys. 37, 1414 (1996) Atiyah, M.F., Patodi, V.K., and Singer, I.M.: Math. Proc. Camb. Phil. Soc. 77, 43 (1975); 78, 405 (1975); 79, 71 (1976) 14. Eguchi, T., Gilkey, P.B., and Hanson, A.J.: Phys. Reports 66, 213 (1980)

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15. Humphreys, J.E.: Introduction to Lie Algebras and Representation Theory. Graduate Texts in Mathematics 9, New York–Heidelberg–Berlin: Springer-Verlag, 1972 16. Kobayashi, S., and Nomizu, K.: Foundations of Differential Geometry. Vol. 1, New York: John Wiley & Sons, 1996 Communicated by A. Connes

Commun. Math. Phys. 190, 597 – 627 (1998)

Communications in

Mathematical Physics c Springer-Verlag 1997

d Elliptic Algebra Aq,p (sl 2 ) in the Scaling Limit S. Khoroshkin1 , D. Lebedev1 , S. Pakuliak2,3 1 Institute of Theoretical and Experimental Physics, 117259 Moscow, Russia. E-mail: [email protected], E-mail: [email protected] 2 Bogoliubov Laboratory of Theoretical Physics, JINR, 141980 Dubna, Moscow region, Russia. E-mail: [email protected] 3 Bogoliubov Institute of Theoretical Physics, Kiev 252143, Ukraine

Received: 10 February 1997 / Accepted: 23 April 1997

c2 ) of the elliptic algebra Aq,p (sl c2 ) is investigated. Abstract: The scaling limit A~,η (sl The limiting algebra is defined in terms of a continuous family of generators being Fourier harmonics of Gauss coordinates of the L-operator. The Ding-Frenkel isomorphism bec2 ) is established and tween L-operator’s and current descriptions of the algebra A~,η (sl is identified with the Riemann problem on a strip. The representations, coalgebraic structure and intertwining operators of the algebra are studied. 0. Introduction c2 ) This paper is devoted to the investigation of the infinite-dimensional algebra A~,η (sl which is supposed to be the algebra of symmetries in integrable models of quantum field theories. Our work arose from the attempts to understand the mathematical background of the results by S. Lukyanov [1] and to combine his methods with the group-theoretical approach to quantum integrable models developed in [2, 3] and with the Yangian calculations in [4]. The investigation of the symmetries in quantum integrable models of the twodimensional field theory started in [5, 6] resulted in the form-factor (bootstrap) approach to these models developed in the most complete form in the works by F.A. Smirnov [7]. This approach was not addressed to investigation of dynamical symmetries in the model but to computation of certain final objects of the theory, form-factors of the local operators and correlation functions of the local operators. It was observed in the papers [8, 9, 10] that the mathematical structures underlying the success of the bootstrap approach in the massive integrable models are related to the representation theory of infinite-dimensional Hopf algebras. The dynamical symmetries in massive two-dimensional field theories was investigated in [1] in the framework of Zamolodchikov-Faddeev operators [11, 12]. c2 ) was proposed in the works [3] as an algebra of symmeThe elliptic algebra Aq,p (sl c2 ) was formulated tries for the eight-vertex lattice integrable model. The algebra Aq,p (sl

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in the framework of the ‘RLL’ approach [13] in terms of the symbols L± εε0 ,n (n ∈ Z, 0 0 n ε, ε = ±, εε = (−1) ) gathered using the spectral parameter ζ into 2×2 matrices L± (ζ) and the central element c. The generating series L± (ζ) satisfy the defining relations: ± ± ± ± ∗± R12 (ζ1 /ζ2 )L± 1 (ζ1 )L2 (ζ2 ) = L2 (ζ2 )L1 (ζ1 )R12 (ζ1 /ζ2 ), − + + ∗+ −c/2 R12 (q c/2 ζ1 /ζ2 )L+1 (ζ1 )L− ζ1 /ζ2 ), 2 (ζ2 ) = L2 (ζ2 )L1 (ζ1 )R12 (q

q c/2 = L+++ (q −1 ζ)L+−− (ζ) − L+−+ (q −1 ζ)L++− (ζ), −1 ζ) = εε0 L+−ε,−ε0 (p1/2 q −c/2 ζ), L− εε0 (q

where ±

R (ζ) = q

∓1/2



(q 2 ζ −2 ; q 4 )∞ (q 2 ζ 2 ; q 4 )∞ ζ (q 4 ζ ∓2 ; q 4 )∞ (ζ ±2 ; q 4 )∞

(0.1)

±1 R(ζ),

(0.2)

and R(ζ) = R(ζ; p1/2 , q 1/2 ) is the Baxter elliptic R matrix normalized to satisfy the unitarity and crossing symmetry relations [14], R∗± = R± (ζ; p∗ 1/2 , q 1/2 ) and p∗ = pq −2c . Unfortunately, there is no description of infinite-dimensional representations of c2 ) in terms of free fields. Aq,p (sl In this paper we are going to investigate the scaling limit of this algebra when c2 ). In some sense this algebra appeared first in [1] as q, p → 1. We call this algebra A~,η (sl c2 ) the algebra of the ‘screening’ operators. In the ‘RLL’ formulation the algebra A~,η (sl have been partially investigated in [15], where the bosonization of the corresponding intertwining operators in terms of the “continuous” free field have been obtained. Let us c2 ) is constructed by means of the trigonometric Rnote that although the algebra A~,η (sl matrix, it is quite different from the quantum affine algebra which is a degeneration of the c2 ) when p = 0. The algebra A~,η (sl c2 ), written in integral relations elliptic algebra Aq,p (sl for the usual commutators and anticommutators, conserve many principal properties of the elliptic algebra. For example, it possesses an evaluation homomorphism onto a degenerated Sklyanin algebra [16] for zero central charge and has no imbeddings of finite-dimensional quantum groups for c 6= 0. But due to the more simple structure of the R-matrix compared with the elliptic case, a more detailed study of its algebraical structure is possible. In particular, one of our achievements is the currents description of the algebra c2 ), which is equivalent to the factorization of the quantum determinant in the A~,η (sl L-operator approach. This allows us to make a more detailed investigation of the repc2 ). Starting from the basic representation of resentation theory of the algebra A~,η (sl c2 ) in a Fock space we reconstruct precisely the Zamolodchikov–Faddeev algeA~,η (sl bra, described in [11, 1, 15]. We carry out this reconstruction from the analysys of the c2 ). Hopf structure of A~,η (sl c2 ) is the presence of analysis in its The distinguished feature of the algebra A~,η (sl description. The formal generators of the algebra are Fourier harmonics of the currents labeled by real numbers, and the elements of the algebra are integrals over generators with coefficients being functions with certain conditions on their analyticity and on their asymptotical behaviour. The paper is organized as follows. In the first section we give a description of the c2 ) in terms of formal generators being Fourier harmonics of the currents. algebra A~,η (sl The relations for the formal generators are given in a simple integral form. We assign a precise meaning to the elements of the algebra as to certain integrals over generators

Elliptic Algebra A(sbl2 ) in the Scaling Limit

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and show that the relations are correctly defined in the corresponding vector spaces. Moreover, we show that the quadratic integral relations could be interpreted as ordering rules with polylogarithmic coefficients for monomials composed from generators of the algebra. In this section we suppose that the central charge is not equal to zero. In the c2 ). We show that the next section we develop the formalism of L-operators for A~,η (sl L-operators L± (u), satisfying the standard relations [13] with R-matrices being scaling limits of those from [14], admit the Gauss decomposition. We write down relations for the Gauss coordinates and identify them with generating functions for the generators of c2 ) described in the previous section. Looking to the rational limit η → 0, we find A~,η (sl a double of the Yangian but in a presentation different from [17, 18]. We describe also c2 ). The comultiplication rule looks standard in terms the coalgebraic structure of A~,η (sl of the L-operators, it is compatible with the defining relations, but it sends now the initial algebra into a tensor product of two different algebras which differ by the value of the parameter η. Nevertheless it is sufficient for the definition of the intertwining operators. We call this structure a Hopf family of algebras. c2 ) for c = 0. We treat Section 3 is devoted to the description of the algebra A~,η (sl c this case as a limit of A~,η (sl2 ) when c tends to zero. The limit is not trivial, one should look carefully at the asymptotics of the currents in the limit in order to define correct generators for c = 0. We describe finite-dimensional representations and the evaluation homomorphism onto the degenerated Sklyanin algebra, which is isomorphic in this case to Uq (sl2 ) with |q| = 1. In the next section we complete the Ding-Frenkel isomorphism [19] and present a description of the algebra in terms of total currents. We show that Ding-Frenkel formulas are equivalent in our case to the Sokhotsky-Plemely formulas for the Riemann problem on a strip. The relation (0.1) for L± operators is also natural in the framework of the Riemann problem. The last two sections are devoted to the study of the basic representation of the algebra c2 ) in a Fock space. The representation of the corresponding ZamolodchikovA~,η (sl Faddeev algebra in this space was recently described in [15, 20]. We start from a c2 ) and then identify the Zamolodchikovbosonization of the total currents for A~,η (sl Faddeev algebra with the algebra of type I and type II twisted intertining operators. The twisting means a presence of a certain involution in the definition of the intertwining operators. The twisting comes from the lack of the zero mode operator (−1)p in the continuous models. There is no motivation to introduce this operator in our case since, contrary to discrete models, we have the unique level one module. As a consequence, Zamolodchikov-Faddeev operators commute by means of an R matrix [11, 1] which c2 ) by certain signs. We check also differs from the one used in the description of A~,η (sl the correspondence of Miki’s formulas [21] to the L-operator description of the basic c2 ). Note also that the notions of a Fock space and of vertex oprepresentation of A~,η (sl erators for the continuous free boson field require a special analytical definition which we suggest in the last section. c2 ) (c 6= 0) 1. Algebra A~,η (sl 1.1. The definition. For λ ∈ R we consider the family of symbols eˆλ , fˆλ , tˆλ and c of the formal algebra which satisfy the commutation relations: [c, everything] = 0 ,

(1.1)

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 [eˆλ , fˆµ ] = sh [tˆλ , eˆµ ] = [tˆλ , fˆµ ] = [eˆλ , eˆµ ] = [fˆλ , fˆµ ] = [tˆλ , tˆµ ] =

λ 2η

+

µ



2η 0

tˆλ+µ ,

(1.2)

 −1 Z tg πη~ ∞ τ − dτ sh {tˆλ+τ , eˆµ−τ } , 2πη −∞ 2η  −1 Z tg πη 0 ~ ∞ τ − − dτ sh {tˆλ+τ , fˆµ−τ }, 2πη 0 −∞ 2η 0   Z tg πη~ ∞ τ {eˆλ+τ , eˆµ−τ } , − dτ cth 2πη −∞ 2η   Z tg πη 0 ~ ∞ τ − − dτ cth {fˆλ+τ , fˆµ−τ } , 2πη 0 −∞ 2η 0 Z ∞ − dτ κ(τ ){tˆλ+τ , tˆµ−τ } ,

(1.3) (1.4) (1.5) (1.6) (1.7)

−∞

where the crossed sign of the integral means the principal value and the real odd function κ(τ ) is given by the Fourier transform Z ∞ cth(πη 0 u)th(iπη 0 ~) − cth(πηu)th(iπη~) 1 , du e−iτ u κ(τ ) = 2π −∞ 1 − cth(πη 0 u)cth(πηu)th(iπη 0 ~)th(iπη~) ~ is a deformation parameter, {a, b} means ab + ba, η > 0 and the parameters η and η 0 are related through the central element c: 1 1 − = ~c , 0 η η

~c > 0 .

The last inequality means that in the representations which we consider the central element c is equal to some number such that ~c > 0 and we identify c with this number. The case c = 0 requires a special treatment and will be considered in the next section. Let us consider the vector space A formed by the formal integrals of the type Z ∞Y Y Y dλk dµi dνj φ({λk }; {µi }; {νj })P ({eˆλk }; {fˆµi }; {tˆνj }) , (1.8) −∞ k

i

j

where φ({λk }; {µi }; {νj }) is the C-number function of real variables λk , µi and νj which satisfy the conditions of analyticity: φ({λk }; {µi }; {νj }) φ({λk }; {µi }; {νj }) φ({λk }; {µi }; {νj })

is analytical in the strip is analytical in the strip is analytical in the strip

− πη < Im λk < πη ∀ λk , − πη 0 < Im µi < πη 0 ∀ µi , − πη 0 < Im νj < πη 0 ∀ νj ,

and conditions on the asymptotics when Re λk , Re µi , Re νj → ±∞: φ({λk }; {µi }; {νj }) < Ce−α|Re λk | , φ({λk }; {µi }; {νj }) < Ce−(β+~c/2)|Re µi | , φ({λk }; {µi }; {νj }) < Ce−(γ−1/2η)|Re νj | , for some real positive α, β, γ. The notation

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P ({eˆλk }; {fˆµi }; {tˆνj }) means a monomial which is a product of the formal generators eˆλ , fˆµ and tˆν in some order. The space A has a natural structure of a free (topological) algebra. c2 ) is identified with A factorized by the ideal genBy definition the algebra A~,η (sl erated by the commutation relations (1.2)–(1.7) which can be treated as equalities in the vector space A. c2 ) follows from Lemma 1 The correctness of the definition of the algebra A~,η (sl and the properties of the kernels of the integral transforms which enter in the r.h.s. of the commutation relations (1.2)–(1.7). These relations also make it possible to write the monomials in (1.8) in the partially ordered form (see the next subsection). Lemma 1. For two functions a(λ) and b(λ) which are analytical in the strips −α1 < Im λ < α2 , −β1 < Im λ < β2 respectively for α1 , α2 , β1 , β2 ≥ 0 and have exponentially decreasing asymptotics when Re λ → ±∞, the convolution Z ∞ (a ? b)(λ) = δτ a(τ ) b(λ − τ ) −∞

is an analytical function of λ in the strip −α1 − β1 < Im λ < α2 + β2 , and also have exponentially decreasing asymptotics at Re λ → ±∞. c2 ): In the sequel we will need following involution of the algebra A~,η (sl  
ι (eˆλ ) = −eˆλ , ι fˆλ = −fˆλ , ι tˆλ = tˆλ .

(1.9)

1.2. Commutation relations as ordering rules. We assign meaning to the commutation relations (1.2)–(1.7). One should first understand them as the rules to express the product of the formal generators in the form eˆλ1 eˆλ2 . . . fˆλ1 fˆλ2 . . . tˆλ1 tˆλ2 . . . , and second as a rule to express the product, say, eˆλ eˆµ for λ > µ through the products eˆτ1 eˆτ2 for τ1 < τ2 . The first part of the normal ordering problem can be solved easily, while the second one is more involved and is not solved yet fully even in the case of quantum affine algebras1 . It is clear how to use the commutation relation (1.2) to order the products. Let us explain how to use the commutation relations of the type (1.3) and (1.4), Z ∞ [ˆaλ , bˆ µ ] = − dτ ϕ(τ ){ˆaλ+τ , bˆ µ−τ } . −∞

These commutation relations can be rewritten in the form of the ordering rules in which structure coefficients are composed from iterated integrals of the polylogariphmic type: 1 Partially, this problem was investigated in [22] in the framework of the semi-infinite wedge construction of level 1 Uq (A(1) n ) Fock spaces. Using the crystal base methods it was proved that normally ordered wedges form a base of the Fock space and that the Fock space has the structure of an integrable Uq (g)-module. We thank Prof. T. Miwa for clarifying this point for us.

602

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Z ∞ ˆ ˆ aˆ λ bµ = bµ aˆ λ + 2 − dτ bˆ µ−τ aˆ λ+τ × −∞

! ∞ Z ∞ Z ∞ n−1 X Y − · · ·− dτ1 . . . dτn ϕ(τ − τ1 ) ϕ(τk − τk+1 )ϕ(τn ) , (1.10) × ϕ(τ ) + n=1

−∞

−∞

k=1

where the function ϕ(τ ) is tg(πη~) 2πη sh(τ /2η)

for (1.3)

−

and

tg(πη 0 ~) 2πη 0 sh(τ /2η 0 )

for (1.4).

The pair of the operators of the same type can be ordered by the same trick. Fix λ > µ. Then iterating the commutation relations of the type (1.5), (1.6) or (1.7) we obtain:   Z ∞  λ−µ ;τ + aˆ λ aˆ µ = aˆ µ aˆ λ + 2 − dτ aˆ λ+µ −τ aˆ λ+µ +τ + ϕ +

0

2

2

2

 Z ∞  λ − µ  n−1 Y − · · ·− dτ1 . . . dτn ϕ ; τ1 ϕ(τk ; τk+1 )ϕ(τn ; τ ) , (1.11)

∞ Z ∞ X n=1

0

2

0

where

k=1

ϕ(τ ; τ 0 ) = ϕ(τ − τ 0 ) + ϕ(τ + τ 0 )

and the function ϕ(τ ) is −

tg(πη~) 2πη th(τ /2η)

for (1.5),

tg(πη 0 ~) 2πη 0 th(τ /2η 0 )

for (1.6)

and −κ(τ ) for (1.7). We suggest the following Conjecture 2. The series in (1.10) and (1.11) are convergent. There are a few remarks in favour of this conjecture. First, we see that if we consider the deformation parameter ~ small then these series are series with respect to powers of the small parameter sin(πη~). Second, in the Yangian limit when η → 0 the series in (1.10) and (1.11) can be summed up to some elementary functions. Unfortunately, the normal ordering of monomials of higher length according to the rule (1.11) becomes a very ineffective procedure. Using perturbation arguments we can state only that such a normal ordering exists if a certain series with respect to the parameter sin(πη~) are convergent. Indeed, let us consider the arbitrary monomial aˆ λ1 aˆ λ2 . . . aˆ λn . We can order this monomial using (1.11) in zero order with respect to the parameter sin(πη~). The non-ordered terms appear only in the first order of this parameter. Now we can order these terms generating non-ordered summands of the second order. Repeating this procedure infinitely many times we solve the problem of normal ordering providing the series with respect to the parameter sin(πη~) are convergent. It is interesting to understand how the solution of this normal ordering problem depends on the explicit form of the convolution kernels appearing in (1.10), (1.11) and being actually related to inverse Fourier transforms of the elements of the Rmatrix (2.1). Since the R-matrix satisfies the Yang-Baxter equation, its inverse Fourier transform also satisfies some integral relation which, hopefully, can help to solve the above mentioned normal ordering problem. Alternatively, we can argue that the deformed infinite dimensional algebra is well defined if a sufficient number of its representations are found. It is exactly the way which we follow for the rest of the paper.

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c2 ) 2. L-Operator Realization of the Algebra A~,η (sl 2.1. Gauss coordinates of the L-operator. Fix the following R-matrix [11, 1]: R+ (u, η) = τ + (u)R(u, η), R(u, η) = r(u, η)R(u, η) ,   1 0 0 0  0 b(u, η) c(u, η) 0  R(u, η) =  , 0 c(u, η) b(u, η) 0  0 0 0 1 ∞ 0 (~η) 0 (1 + iηu) Y Rp (u, η)Rp (i~ − u, η) , r(z, η) = 0 (~η + iηu) Rp (0, η)Rp (i~, η) p=1

0 (2p~η + iηu) 0 (1 + 2p~η + iηu) , 0 ((2p + 1)~η + iηu) 0 (1 + (2p − 1)~η + iηu)  πu  −sh iπη~ sh πηu , c(u, η) = , τ + (u) = cth ,(2.1) b(u, η) = sh πη(u − i~) sh πη(u − i~) 2~

Rp (u, η) =

where u is a spectral parameter. This trigonometric R-matrix can be obtained from the elliptic one (0.2) in the limit q → 1 with ζ = q iu/~ and p = q 2 q 2/η~ . It was done explicitly in the paper [15] (see also the discussion in [23] on the classical limit of the c2 )). algebra Aq,p (sl Let   L++ (u) L+− (u) L(u) = (2.2) L−+ (u) L−− (u) be a quantum L-operator in which matrix elements are treated as generating functions for the elements of the algebra given by the commutation relations: R+ (u1 − u2 , η 0 )L1 (u1 , η)L2 (u2 , η) = L2 (u2 , η)L1 (u1 , η)R+ (u1 − u2 , η) , q-detL(u) = 1 .

(2.3) (2.4)

The quantum determinant of the L-operator is given by q-detL(u) = L++ (u − i~)L−− (u) − L+− (u − i~)L−+ (u) . 

Let L(u) =

1 f (u) 0 1



k1 (u) 0 0 k2 (u)



1 0 e(u) 1

(2.5)

 ,

(2.6)

be the Gauss decomposition of the L-operator (2.2). One can deduce from (2.3), (2.4) that k1 (u) = (k2 (u + i~))−1 . Let

h(u) = k2 (u)−1 k1 (u) ,

Then, due to (2.3), (2.4),

We have the following

˜ h(u) = k1 (u) k2 (u)−1 .

η sin πη 0 ~ ˜ h(u) . h(u) = 0 η sin πη~

(2.7)

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S. Khoroshkin, D. Lebedev, S. Pakuliak

Proposition 3. The Gauss coordinates e(u), f (u) and h(u) of the L-operator (2.2) satisfy the following commutation relations (u = u1 − u2 ): e(u1 )f (u2 ) − f (u2 )e(u1 ) =

sh iπη 0 ~ sh iπη~ ˜ h(u1 ) − h(u2 ), 0 sh πη u sh πηu

sh πη(u + i~)h(u1 )e(u2 ) − sh πη(u − i~)e(u2 )h(u1 ) = sh(iπη~){h(u1 ), e(u1 )}, 0 sh πη (u − i~)h(u1 )f (u2 ) − sh πη 0 (u + i~)f (u2 )h(u1 ) = −sh(iπη 0 ~){h(u1 ), f (u1 )}, sh πη(u + i~)e(u1 )e(u2 ) − sh πη(u − i~)e(u2 )e(u1 )
= sh(iπη~) e(u1 )2 + e(u2 )2 ,

(2.8) (2.9) (2.10) (2.11)

sh πη 0 (u − i~)f (u1 )f (u2 ) − sh πη 0 (u + i~)f (u2 )f (u1 )


(2.12) = −sh(iπη 0 ~) f (u1 )2 + f (u2 )2 , sh πη(u + i~)sh πη 0 (u − i~) = h(u2 )h(u1 ) . h(u1 )h(u2 ) (2.13) sh πη 0 (u + i~)sh πη(u − i~) The proof is a direct substitution of the Gauss decomposition of L-operators (2.6) into (2.3). b 2 ). Let e± (u), f ± (u) and h± (u) be the following 2.2. The generating integrals for A~,η (sl formal integrals of the symbols eˆλ , fˆλ , tˆλ (u ∈ C): Z sin πη~ ∞ eˆλ e∓c~λ/4 dλ eiλu , (2.14) e± (u) = ± πη 1 + e±λ/η −∞ Z ˆ ±c~λ/4 sin πη 0 ~ ∞ iλu fλ e dλ e , (2.15) f ± (u) = ± πη 0 1 + e±λ/η0 −∞ Z 00 sin πη~ ∞ h± (u) = − dλ eiλu tˆλ e∓λ/2η . (2.16) 2πη −∞ Here

2ηη 0 . η + η0 By direct verification we can check that if the complex number u is inside the strip   ~c 1 ~c Π+ = − − < Im u < − , η 4 4 η 00 =

c2 ). If the complex number u is inside then elements e+ (u), f + (u), h+ (u) belong to A~,η (sl the strip   ~c 1 ~c − Π = < Im u < + , 4 4 η then the elements e− (u) = −e+ (u − i/η 00 ) ,

f − (u) = −f + (u − i/η 00 ) ,

h− (u) = h+ (u − i/η 00 ) (2.17)

c2 ). Thus we can treat the integrals e± (u), f ± (u), h± (u) as genalso belong to A~,η (sl c2 ), analytical in the strips Π ± . erating functions of the elements of the algebra A~,η (sl We can state the following

Elliptic Algebra A(sbl2 ) in the Scaling Limit

605

Proposition 4. The generating functions e(u) = e± (u), f (u) = f ± (u), h(u) = h± (u) satisfy the commutation relations (2.8)–(2.13) if eˆλ , fˆλ , tˆλ satisfy the relations (1.2)– (1.7). In order to prove this proposition we should use the Fourier transform calculations and fix in (2.8)–(2.13) either Im u1 < Im u2 or Im u1 > Im u2 . The relations (2.17) and analyticity of generation functions in the domains Π ± allow one to make an analytical continuation of the relations (2.8)–(2.13) including all possible combinations of the generating integrals. For instance, from (2.9) we have also (u = u1 − u2 ) sh πη(u + i~)h± (u1 )e± (u2 ) − sh πη(u − i~)e± (u2 )h± (u1 ) = = sh(iπη~){h± (u1 ), e± (u1 )} , + − sh πη(u + i~ + i~c/2)h (u1 )e (u2 ) − sh πη(u − i~ + i~c/2)e− (u2 )h+ (u1 ) = = sh(iπη~){h+ (u1 ), e+ (u1 )} , − + sh πη(u + i~ − i~c/2)h (u1 )e (u2 ) − sh πη(u − i~ − i~c/2)e+ (u2 )h− (u1 ) = = sh(iπη~){h− (u1 ), e− (u1 )} . Let now R− (u) = τ − (u)R(u), ±

±

τ − (u) = th

 πu  2~

±

and e (u), f (u), h (u) be the Gauss coordinates of the L-operators L± (u):     ± 1 0 1 f ± (u) (k (u + i~))−1 0 ± L (u) = . e± (u) 1 0 1 0 k ± (u) One can prove in an analogous manner that the described above mixed relations for the generating functions e± (u), f ± (u), h± (u) are equivalent to the following system of equations for the Gauss coordinates of the L-operators (u = u1 − u2 ): − + + R+ (u − ic~/2, η 0 )L+1 (u1 , η)L− 2 (u2 , η) = L2 (u2 , η)L1 (u1 , η)R (u + ic~/2, η), ± ± ± ± R± (u, η 0 )L± 1 (u1 , η)L2 (u2 , η) = L2 (u2 , η)L1 (u1 , η)R (u, η).

(2.18)

These equations can be also obtained by means of the formal analytical continuation of the relations (2.3). c2 ) in terms of the Relation (2.17) and the involution (1.9) in the algebra A~,η (sl L-operators can be written as follows
L+ (u − i/η 00 ) = σz L− (u)σz = ι L− (u) , (2.19) and the possibility to obtain the commutation relations (2.18) from (2.3) by means of the analytical continuation follows from the quasi-periodicity property of the R-matrices R± (u, η): R+ (z − i/η) = (σz ⊗ 1)R− (z)(σz ⊗ 1) = (1 ⊗ σz )R− (z)(1 ⊗ σz ) . c2 ) can be Let us note that the formal generators eˆλ , fˆλ , tˆλ of the algebra A~,η (sl expressed through their generating integrals using the inverse integral transform

606

S. Khoroshkin, D. Lebedev, S. Pakuliak


Z ηe±c~λ/4 1 + e±λ/η eˆλ = ± du e−iλu e± (u) , 2 sin πη~ 0±   0 η 0 e∓c~λ/4 1 + e±λ/η Z fˆλ = ± du e−iλu f ± (u) , 2 sin πη 0 ~ 0± 00 Z ηe±λ/2η tˆλ = − du e−iλu h± (u) , sin πη~ 0±

(2.20)

where 0± are contours which go from −∞ to +∞ inside the strips Π ± . Using the relations (2.20) one can verify that the defining relations (1.1)–(1.7) for c2 ) are equivalent to the relations (2.8)–(2.13) on generating functions the algebra A~,η (sl of the algebra. Remarks. 1. The formal algebra generated by Gauss coordinates e(u), f (u) and h(u) is not completely equivalent to the algebra of coefficients of L(u) with the relations (2.3)–(2.4) since h(u) is a quadratic expression of k2 (u). Naturally, one may consider the c2 ), which looks a bit more complicated. Nevertheless, corresponding extension of A~,η (sl c the algebra A~,η (sl2 ) is sufficient for the description of representations in which we are interested. 2. Let η = 1/ξ. The matrix S(β, ξ) = −(σz ⊗ 1)R(β, 1/ξ)(1 ⊗ σz )

(2.21)

was obtained in [11] as an exact S-matrix of the soliton-antisoliton scattering in the quantum Sine-Gordon model, where ξ is related to the coupling constant of the model and we should set ~ = π (we prefer to keep the parameter ~ free for the convenience of taking the classical limit [23]). This S-matrix satisfies the conditions of unitarity and crossing symmetry S(β, ξ)S(−β, ξ) = 1 , (C ⊗ id) S(β, ξ) (C ⊗ id) = (S(iπ − β, ξ))t1 with the charge conjugation matrix  C=

01 10

 .

Because of relation (2.21) the R-matrix (2.1) satisfies the same properties of unitarity and crossing symmetry but with a different charge conjugation matrix,  C˜ =

0 1 −1 0

 .

Since the square of this matrix is equal to −1 we have to use an unusual definition of the intertwining operators (5.6) in order to have the possibility to identify them as Zamolodchikov–Faddeev operators for the Sine-Gordon model. We will discuss this point in more detail in the last section.

Elliptic Algebra A(sbl2 ) in the Scaling Limit

607

2.3. The Yangian limit. As follows from the definition of the elements of the algebra c2 ) (1.8) and the generating functions (2.14)–(2.15) each substrip of the strips Π ± A~,η (sl c2 ). In terms of the Fourier components these defines a subalgebra of the algebra A~,η (sl subalgebras are defined by different asymptotics of the functions φ({λk }; {µi }; {νj }) at λk , µi , νj → ±∞ in (1.8). ± Let us consider the substrips Π ⊂ Π ± ,     ~c 1 1 ~c + − < Im u < − , Π = < Im u < Π = − 2η 4 4 2η ±

and restrict the generating functions e± (u), f ± (u), h± (u) onto Π respectively. Then in the limit η → 0 these generating functions will be defined in the lower and the upper half-planes and the relation (2.17) drops out. For example, Z ∞ ± e (u) η=0 = ±~ dλ e∓iλ(u±ic~λ/4) eˆ∓λ , 0

and the defining relations (2.18) turn into the defining relations of the central extended Yangian double [17]. Although the commutation relation in terms of the generating functions for the limc2 ) coincide with the commutation relations of the central extended iting algebra A~,0 (sl \ Yangian double DY (sl2 ) these two algebras should not be treated as isomorphic. For c2 ) has a con\ instance, the algebra DY (sl2 ) has a discrete set of generators and A~,0 (sl tinuous family of them. As a consequence, they have different representation theories. It was also pointed out in [20]. See details in [24]. 2.4. Comultiplication structure. As we already mentioned in the introduction, the algebra c2 ) is not a Hopf algebra in the usual sense. Nevertheless we can assign the Hopf A~,η (sl c2 ) parametrized by the parameter algebra structure to the family of the algebras A~,η (sl η. Let us describe this Hopf structure. In this subsection it is convenient to use instead of the parameter η its inverse 1 ξ= . η Because of the relation (2.19) we can define the coproduct only for one type of the operator, say, L+ (u). Consider the operation 1 c = c0 + c00 = c ⊗ 1 + 1 ⊗ c , ˙ + (u + i~c0 /4, ξ) 10 L+ (u, ξ) = L+ (u − i~c00 /4, ξ + ~c00 )⊗L

(2.22)

or in the components 1L+ij (u, ξ) =

2 X

L+kj (u + i~c00 /4, ξ) ⊗ L+ik (u − i~c0 /4, ξ + ~c0 ),

k=1

which defines the coassociative map c2 ) → A~,ξ (sl c2 ) ⊗ A~,ξ+~c0 (sl c2 ) 1 : A~,ξ (sl

(2.23)

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S. Khoroshkin, D. Lebedev, S. Pakuliak

c2 ). The map 1 is a morphism of algebras, but it sends on the family of algebras A~,ξ (sl one algebra to a tensor product of two different algebras, which we do not identify. So c2 ) form (over the parameter η) a Hopf family of algebras. Let us also we say that A~,η (sl note that because of the relation (2.19) the comultiplication of the L-operator L− (u) is given by 1L− ij (u, ξ) =

2 X

− 00 0 0 L− kj (u − i~c /4, ξ) ⊗ Lik (u + i~c /4, ξ + ~c ) .

k=1

In order to save notations below in this subsection and in Appendix A we will understand the operators k(u, ξ), e(u, ξ), f (u, ξ) as operators k + (u, ξ), e+ (u, ξ), f + (u, ξ) and will write u0 , ξ 0 (u00 , ξ 00 ) in left (right) components of the tensor product and understand them as u0 = u + i~c00 /4, ξ 0 = ξ = 1/η (u00 = u − i~c0 /4, ξ 00 = ξ + ~c0 ). The comultiplications of the operators e(u, ξ), f (u, ξ) and h(u, ξ) are 1e(u, ξ) = e(u0 , ξ) ⊗ 1 + ∞ X
p
p+1 + (−1)p f (u0 − i~, ξ 0 ) h(u0 , ξ 0 ) ⊗ e(u00 , ξ 00 ) ,

(2.24)

p=0

1f (u, ξ) = 1 ⊗ f (u00 , ξ 00 ) + ∞ X
p+1
˜ 00 , ξ 00 ) e(u00 − i~, ξ 00 ) p , + (−1)p f (u0 , ξ 0 ) ⊗ h(u 1h(u, ξ) =

p=0 ∞ X


p

(−1)p [p + 1]η f (u0 − i~, ξ 0 )

p=0


p

⊗ h(u00 , ξ 00 ) e(u00 − i~, ξ 00 ) where we define [p]η =

(2.25)

h(u0 , ξ 0 ) ⊗

,

(2.26)

sin πη~p . sin πη~

The proof of these formulas is shifted to Appendix A. Note that the involution ι is compatible with the coalgebraic structure: 1ι = (ι ⊗ ι)1. c2 ) (c = 0) 3. The Algebra A~,η (sl Consider the formal algebra of the symbols eˆλ , fˆλ , hˆ λ and S0 which satisfy the commutation relations [eˆλ , fˆµ ] = hˆ λ+µ , [S0 , eˆµ ] = [hˆ λ , eˆµ ] = + [S0 , fˆµ ] =

sin(πη~) tg(πη~){hˆ 0 , eˆµ } , {S0 , eˆλ+µ } + cos πη~      Z tg πη~ ∞ τ λ+τ − cth {hˆ λ+τ , eˆµ−τ } , − dτ cth 2πη −∞ 2η 2η − sin(πη~) tg(πη~){hˆ 0 , fˆµ } ,

(3.1) (3.2)

(3.3) (3.4)

Elliptic Algebra A(sbl2 ) in the Scaling Limit

{S0 , fˆλ+µ } − [hˆ λ , fˆµ ] = − cos πη~      Z ∞ tg πη~ τ λ+τ − cth {hˆ λ+τ , fˆµ−τ } , − dτ cth − 2πη −∞ 2η 2η   Z tg πη~ ∞ τ − dτ cth {eˆλ+τ , eˆµ−τ } , [eˆλ , eˆµ ] = 2πη −∞ 2η   Z tg πη~ ∞ τ − dτ cth {fˆλ+τ , fˆµ−τ } , [fˆλ , fˆµ ] = − 2πη −∞ 2η [S0 , hˆ λ ] = 0 , [hˆ λ , hˆ µ ] = 0 , 1 =

S02

+ sin (πη~)hˆ 20 . 2

609

(3.5) (3.6) (3.7) (3.8) (3.9)

We can assign to these commutation relations the analogous ordering sense as we did in the Sect. 2. Consider the free algebra A0 formed by the formal integrals of the type Z ∞Y Y Y dλk dµi dνj φ0 ({λk }; {µi }; {νj })P ({eˆλk }; {fˆµi }; {hˆ νj }), −∞ k

i

j

where φ0 ({λk }; {µi }; {νj }) is the C-number function of real variables λk , µi and νj which satisfy the conditions of analyticity: φ0 ({λk }; {µi }; {νj }) is analytical in the strip − πη < Im (λk , µi ) < πη ∀ λk , µi , φ0 ({λk }; {µi }; {νj }) is analytical in the strip − 2πη < Im νj < 2πη ∀ νj , except for the points νj = 0 where this function has a simple pole with respect to all νj . The function φ0 ({λk }; {µi }; {νj }) has the asymptotics when Re λk , Re µi , Re νj → ±∞: φ0 ({λk }; {µi }; {νj }) < Ce−α|Re λk | , φ0 ({λk }; {µi }; {νj }) < Ce−β|Re µi | , φ0 ({λk }; {µi }; {νj }) < Ce−γ|Re νj | , for some real positive α, β, γ. c2 ) at c = 0 is identified with A0 factorized by the ideal generated The algebra A~,η (sl by the commutation relations (3.1)–(3.9). Consider the formal integrals Z sin πη~ ∞ eˆλ + dλ eiλu , (3.10) e (u) = πη 1 + eλ/η −∞ Z sin πη~ ∞ fˆλ dλ eiλu , (3.11) f + (u) = πη 1 + eλ/η −∞ Z sin πη~ ∞ hˆ λ + − dλ eiλu , (3.12) h (u) = S0 + πη 1 − eλ/η −∞ c2 ) at c = 0. We can prove as generating integrals of the elements of the algebra A~,η (sl + that these generating functions are analytical in the strip Π = {−1/η < Im u < 0} and

610

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satisfy the commutations relations (2.8)–(2.13), where at c = 0 we should set η 0 = η. In particular, in this case the generating functions h+ (u) commute. The different presentation of the generating functions h± (u) for c 6= 0 (2.16) and for c = 0 (3.12) follows from the analysis of their asymptotical behaviour. Indeed, the relations (2.8)–(2.13) imply that e+ (u)

∼

Re u→±∞

e−πη|u| ,

h+ (u)

∼

f + (u)

Re u→±∞

∼

Re u→±∞

0

e−πη |u| ,

0

e−π(η−η )|u| ,

(3.13)

and are constant, but are different for +∞ and −∞ asymptotics of h+ (u) for c = 0: h+ (u)

∼

Re u→±∞

h+ (±∞) ≡ h± = S0 ∓ i sin(πη~)hˆ 0 .

Such asymptotics can be achieved by the following Cauchy kernel presentations: Z sh iπη~ ∞ E(v) ˜ , (3.14) dv˜ e+ (u) = 2π sh πη(v˜ − u) −∞ Z F (v) ˜ sh iπη~ ∞ f + (u) = , (3.15) dv˜ 2π sh πη(v˜ − u) −∞ Z sh iπη~ ∞ h+ (u) = S0 + dv˜ H(v) ˜ cth πη(v˜ − u) , (3.16) 2π −∞ where u ∈ Π + = {−1/η < Im u < 0} and v˜ ∈ R. Analogous formulas take place for the ‘−’-generating functions but with the spectral parameter u in the strip Π − = {0 < Im u < 1/η}. The presenations (3.14)–(3.16) are equivalent to the deformed Laplace presentations (3.10)–(3.12) if E(u), ˜ F (u) ˜ and H(u) ˜ are Fourier transforms of eˆλ , fˆλ and hˆ λ : Z ∞ Z ∞ Z ∞ E(v) ˜ = dλ eiλv˜ eˆλ , F (v) ˜ = dλ eiλv˜ fˆλ , H(v) ˜ = dλ eiλv˜ hˆ λ , v˜ ∈ R . −∞

−∞

−∞

For c 6= 0, due to (3.13), we use instead of (3.12) and (3.16) the usual Fourier transform (2.16). Note that in the limit c → 0 we have the relation   ˆhλ = tˆλ sh λ (3.17) 2η

for all λ, so hˆ 0 is well defined whereas tˆ0 tends to infinity when c → 0; to the contrary, hˆ 0 from (3.17) is zero for c 6= 0 while tˆ0 6= 0 in this case. c2 ) at c = 0 have the following The asymptotic generators h± of the algebra A~,η (sl + commutation relations with the generating functions e (u) and f + (u): ∓2 + e (u), h± e+ (u)h−1 ± =q

±2 + h± f + (u)h−1 f (u), ± =q

q = eiπη~ ,

and are group-like elements: 1h± = h± ⊗ h± . Thus their product h+ h− = S02 + sin2 (πη~)hˆ 20 is cental and group-like. Due to this we can put it equal to 1. This kills unnecessary representations of level 0.

Elliptic Algebra A(sbl2 ) in the Scaling Limit

611

3.1. Evaluation homomorphism. Let e, f and h be the generators of the algebra Uiπη~ (sl2 ): [h, e] = 2e,

[h, f ] = −2f,

[e, f ] = [h]η =

sin πη~h . sin πη~

The following proposition presents two descriptions of the evaluation homomorphism c2 ) onto Uiπη~ (sl2 ). of the c = 0 algebra A~,η (sl c2 ) at c = 0 has the following evaluation homomorProposition 5. The algebra A~,η (sl phism Evz onto Uq (sl2 ), z ∈ C Evz (S0 ) = cos(πη~h) , Evz (eˆλ ) = e−iλz e−~λ(h−1)/2 e = e−iλz e e−~λ(h+1)/2 ,   Evz fˆλ = e−iλz e−~λ(h+1)/2 f = e−iλz f e−~λ(h−1)/2 ,   Evz hˆ λ = e−iλz e−~λ(h−1)/2 ef − e−iλz e−~λ(h+1)/2 f e, or, equivalently, (u ∈ Π + ):
Evz e+ (u) = − Evz Evz

sh iπη~ sh iπη~ e = −e , sh πη(u − z + i~(h − 1)/2)) sh πη(u − z + i~(h + 1)/2))
sh iπη~ sh iπη~ f = −f , f + (u) = − sh πη(u − z + i~(h + 1)/2)) sh πη(u − z + i~(h − 1)/2))
 h+ (u) = cos(πη~h) − sh iπη~ cth πη(u − z + i~(h − 1)/2) ef  − cth πη(u − z + i~(h + 1)/2) f e .

Let Vn be the (n+1)-dimensional Uiπη~ (sl2 )-module with a basis vk , k = 0, 1, . . . , n, where the operators h, e and f act according to the rules h vk = (n − 2k) vk ,

e vk = [k]η vk−1 ,

f vk = [n − k]η vk+1 .

c2 ) in the space Due to Proposition 5 we have an action πn (z) of the algebra A~,η (sl + Vn,z = Vn . Note that the action of h (u) can be simplified in this case as
sh πη(u − z − i~(n + 1)/2))sh πη(u − z + i~(n + 1)/2)) . (3.18) πn (z) h+ (u) = sh πη(u − z + i~(h + 1)/2))sh πη(u − z + i~(h − 1)/2)) The simplest two-dimensional evaluation representation π1 (z) of the algebra c2 ) on the space Vz = V1,z (we have identified v+,z = v0 and v−,z = v1 ) is A~,η (sl e+ (u)v+,z = 0, e+ (u)v−,z h+ (u)v±,z

f + (u)v−,z = 0 , (3.19) sh iπη~ shiπη~ v+,z f + (u)v+,z = − v−,z , =− (3.20) sh πη(u − z) shπη(u − z) shπη(u−z ∓ i~) v±,z . = sh iπη~ [cthiπη~∓cthπη(u−z)] v±,z = sh πη(u − z) (3.21)

612

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The action of ‘−’-generating functions on the space Vz is given by the same formulas (3.19)–(3.21) but with u ∈ Π − . In L-operator terms, the representation described by (3.19)–(3.21) is equivalent to the standard one: π1 (z)L± (u) = R± (u − z, η). For c = 0 in addition to the evaluation homomorphism we have, analogously to the c2 ), an embedding of a subalgebra, isomorphic to Uq (sl2 ). Here q = eiπη~ , case of Uq (sl |q| = 1. This subalgebra is generated by the elements S0 , eˆ0 , fˆ0 and hˆ 0 and is given in the form of a Sklyanin degenerated algebra [16]: [eˆ0 , fˆ0 ] = hˆ 0 ,

[S0 , hˆ 0 ] = 0 ,

S02 + sin2 (πη~)hˆ 20 = 1 , tg πη~ {S0 , eˆ0 } , [S0 , eˆ0 ] = sin(πη~) tg(πη~){hˆ 0 , eˆ0 } , [hˆ 0 , eˆ0 ] = sin πη~ tg πη~ [S0 , fˆ0 ] = − sin(πη~) tg(πη~){hˆ 0 , fˆ0 } , [hˆ 0 , fˆ0 ] = − {S0 , fˆ0 } . sin πη~ This embedding is destroyed when c 6= 0. For instance, the generators eˆ0 and fˆ0 commute for c 6= 0. In the rational limit η → 0 this finite-dimensional subalgebra becomes a sl2 subalc2 ). gebra of the Yangian double A~,0 (sl

c2 ) 4. Current Realization of A~,η (sl c2 ) which In this section we would like to give another realization of the algebra A~,η (sl is an analog of the current realization of the affine Lie algebras. The necessity of this realization follows from the construction of infinite-dimensional representations of the c2 ) at c 6= 0 in terms of free fields. The generating functions e± (u) and algebra A~,η (sl ± f (u) cannot be realized directly in terms of free fields, but only some combinations of them, called total currents, have a free field realization. Let us define generating functions (total currents) E(u) and F (u) as formal Fourier transforms of the symbols eˆλ and fˆλ : Z

Z

∞

dλ e

E(u) = −∞

iλu

eˆλ ,

∞

F (u) = −∞

dλ eiλu fˆλ ,

u∈C

and put ~ H (u) = − 2 ±

Z

∞ −∞

00

dλ eiλu tˆλ e∓λ/2η ,

h± (u) =

sin πη~ ± H (u). πη~

We prove in this section that: (i) The currents E(u), F (u) and H ± (u) satisfy the relations (4.1)–(4.7):

Elliptic Algebra A(sbl2 ) in the Scaling Limit

613

sh πη(u − v − i~(1 − c/2))sh πη 0 (u − v + i~(1 − c/2)) sh πη(u − v + i~(1 + c/2))sh πη 0 (u − v − i~(1 + c/2)) H − (v)H + (u) , sh πη(u − v − i~)sh πη 0 (u − v + i~) ± H ± (u)H ± (v) = H (v)H ± (u) , sh πη(u − v + i~)sh πη 0 (u − v − i~) sh πη(u − v − i~(1 ∓ c/4)) H ± (u)E(v) = E(v)H ± (u) , sh πη(u − v + i~(1 ± c/4)) sh πη 0 (u − v + i~(1 ∓ c/4)) F (v)H ± (u) , H ± (u)F (v) = sh πη 0 (u − v − i~(1 ± c/4)) sh πη(u − v − i~) E(v)E(u) , E(u)E(v) = sh πη(u − v + i~) sh πη 0 (u − v + i~) F (u)F (v) = F (v)F (u) , sh πη 0 (u − v − i~)    2π h  ic~ ic~ [E(u), F (v)] = δ u−v− H+ u − ~ 2 4   i  ic~ ic~ − H v− , −δ u − v + H + (u)H − (v) =

2

4

where the δ-function is defined as   Z ∞ 1 1 1 − = dλ eiλu , 2πδ(u) = lim →0 i u − iε u + iε −∞

(4.1) (4.2) (4.3) (4.4) (4.5) (4.6)

(4.7)

u∈R.

c2 ) can be obtained (ii) The generating functions e± (u) and f ± (u) of the algebra A~,η (sl from E(u) and F (u) using the formulas: Z dv E(v) e± (u) = sin πη~ , (4.8) C 2πi sh πη(u − v ± ic~/4) Z dv F (v) f ± (u) = sin πη 0 ~ , (4.9) 0 C 0 2πi sh πη (u − v ∓ ic~/4) where the contour C 0 goes from −∞ to +∞, the points u + ic~/4 + ik/η 0 (k ≥ 0) are above the contour and the points u − ic~/4 − ik/η 0 (k ≥ 0) are below the contour. The contour C also goes from −∞ to +∞ but the points u − ic~/4 + ik/η (k ≥ 0) are above the contour and the points u + ic~/4 − ik/η (k ≥ 0) are below the contour. (iii) The total currents E(u) and F (u) can be expressed through e± (u) and f ± (u) by means of the Ding-Frenkel relations [19]:     sin πη~ ic~ ic~ E(u) , (4.10) − e− u + = e+ u − 4 4 πη     sin πη 0 ~ ic~ ic~ f+ u + − f− u − = F (u) . (4.11) 4 4 πη 0 Let us start from (ii). One can see that in every particular case of e+ (u), e− (u), f + (u) and f − (u) the contours of integration could be deformed to a straight line being the boundary of one of the strips Π ± . Then the relations (2.14)–(2.15) are equivalent to (4.8)–(4.9) via Fourier transform. Moreover, the relations (4.8)–(4.9) say that E(u) and

614

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F (u) are the differences of boundary values of analytical functions e± (u) and f ± (u) for the Riemann problem on a strip. Let us demonstrate this for e+ (u). When the spectral parameter u ∈ Π + tends to the upper and the lower boundaries of the strip Π + we can obtain from (2.14): lim (e+ (u˜ − ic~/4 − i) + e+ (u˜ − i/η − ic~/4 + i))

→+0

sh iπη~ E(u), ˜ u˜ ∈ R , iπη lim (e+ (u˜ − ic~/4 − i) − e+ (u˜ − i/η − ic~/4 + i)) →+0 Z ∞ E(v) ˜ sh iπη~ . − dv˜ = π sh πη( v ˜ − u) ˜ −∞ =

(4.12)

These relations are the Sokhotsky-Plemely formulas associated with the Riemann problem on the strips of the width 1/η. Summing the formulas (4.12) and using the analytical continuation with respect to the spectral parameter u˜ we obtain (4.8) where the contour of integration goes from −∞ to +∞ in such a way that the point u + i/η + ic~/4 is above the contour and the point u + ic~/4 is below. The same arguments applied to the generating function e− (u) lead to (4.8) but with the contour going between points u−ic~/4 and u−i/η −ic~/4. Repeating this consideration for the generating functions f + (u) and f − (u) we obtain (4.9). The relations (4.10)–(4.11) follow from (4.12). The commutation relations (4.1)– (4.7) are direct corollaries of (2.8)–(2.13) and (4.10)–(4.11). Remark. Note that, as usual for affine algebras, the relations (4.10)–(4.11) should be understood in the sense of analytical continuation. For instance, if the argument of e+ (u) is inside of its domain of analyticity, the argument of e− (u) is not. It means, in particular, c2 ) but rather to that the total currents E(u) and F (u) belong not to the algebra A~,η (sl some its analytical extension. Nevertheless, they act on highest weight representations, and the precise definition of the category of highest weight representations should be c2 ). equivalent to the description of the proper analytical extension of the algebra A~,η (sl c2 ) at Level 1 5. Representations of the Algebra A~,η (sl 5.1. Representation of the commutation relations by a free field. This section is devoted c2 ). For to the construction of an infinite dimensional representation of the algebra A~,η (sl simplicity we will consider the representation at level 1 (c = 1) since in this case only one free field is sufficient instead of three free fields for the general case. The generalization to an arbitrary level could be done using ideas developed in [25, 26]. Here and till the end of the paper we will understand everywhere η 0 equals η/(1 + η~). Define bosons aλ , λ ∈ R which satisfy the commutation relations [1, 15]: [aλ , aµ ] =

1 sh(~λ)sh(~λ/2) sh(λ/2η) δ(λ + µ) = α(λ)δ(λ + µ) . ~2 λ sh(λ/2η 0 )

Introduce also bosons a0λ related to the initial ones as follows: a0λ =

sh(λ/2η 0 ) aλ . sh(λ/2η)

(5.1)

Elliptic Algebra A(sbl2 ) in the Scaling Limit

615

Consider the generating functions   Z ∞ a0λ :, dλ eiλu E(u) = eγ : exp ~ sh(~λ/2) −∞   Z ∞ aλ :, dλ eiλu F (u) = eγ : exp −~ sh(~λ/2) −∞     i~ i~ F u∓ := H ± (u) = e−2γ : E u ± 4 4   Z ∞ aλ e∓~λ/4 dλ eiλu :, = : exp ∓2~ 1 − e±λ/η −∞

(5.2) (5.3)

(5.4)

where γ is the Euler constant and the product of these generating functions is defined according to [15],  R  R ∞ ∞ : exp −∞ dλ g1 (λ) aλ : · : exp −∞ dµ g2 (µ) aµ : = R   R ∞ ln(−λ) = exp C˜ dλ 2πi . α(λ)g1 (λ)g2 (−λ) : exp −∞ dλ (g1 (λ) + g2 (λ)) aλ : (5.5) The contour C˜ is shown on the Fig. 1.  0• 

-

+∞

Figure 1.

We have the following Proposition 6. The generating functions (5.2)–(5.4) satisfy the commutation relations (4.1)–(4.7). The proof is based on the normal ordering relations gathered in Appendix B. We will show in the next subsection that the ζ-function regularization used in [15] to define (5.5) can be included into the definition of the Fock space. 5.2. Representation in a Fock space and twisted intertwining operators. The goal of this subsection is to interpret Zamolodchikov-Faddeev operators [1, 15] following the ideology [27, 2] as twisted intertwining operators for an infinite-dimensional represenc2 ). For a description of this infinite-dimensional representation we need tation of A~,η (sl a definition of a Fock space generated by a continuous family of free bosons together with a construction of vertex operators. We do it below in a slightly more general setting. Let a(λ) be a meromorphic function, regular for λ ∈ R and satisfying the following conditions: a(λ) = −a(−λ) , a(λ) ∼ a0 λ,

λ → 0,

0

a(λ) ∼ ea |λ| ,

λ → ±∞.

616

S. Khoroshkin, D. Lebedev, S. Pakuliak

Let aλ , λ ∈ R, λ 6= 0 be free bosons which satisfy the commutation relations [aλ , aµ ] = a(λ)δ(λ + µ) . We define a (right) Fock space Ha(λ) as follows. Ha(λ) is generated as a vector space by the expressions Z 0 Z 0 fn (λn )aλn dλn . . . f1 (λ1 )aλ1 dλ1 |vaci , −∞

−∞

where the functions fi (λ) satisfy the condition 0

λ → −∞ ,

fi (λ) < C e(a /2+)λ ,

for some  > 0 and fi (λ) are analytical functions in a neighbourhood of R− except λ = 0, where they have a simple pole. ∗ is generated by the expressions The left Fock space Ha(λ) Z +∞ Z +∞ hvac| g1 (λ1 )aλ1 dλ1 . . . gn (λn )aλn dλn , 0

0

where the functions gi (λ) satisfy the conditions 0

gi (λ) < C e−(a /2+)λ ,

λ → +∞ ,

for some  > 0 and gi (λ) are analytical functions in a neighbourhood of R+ except λ = 0, where they also have a simple pole. ∗ ⊗ Ha(λ) → C is uniquely defined by the following prescripThe pairing (, ) : Ha(λ) tions: (i) (hvac|, |vaci) = 1, R0 R R +∞ (ii) (hvac| 0 dλ g(λ)aλ , −∞ dµ f (µ)aµ |vaci) = C˜

dλ ln(−λ) g(λ)f (−λ)a(λ), 2πi

(iii) the Wick theorem. Let the vacuums hvac| and |vaci satisfy the conditions aλ |vaci = 0,

hvac|aλ = 0,

λ > 0,

λ<0,

and f (λ) be a function analytical in some neighbourhood of the real line with possible simple pole at λ = 0 and which has the following asymptotical behaviour: 0

f (λ) < Ce−(a /2+)|λ| ,

λ → ±∞

for some  > 0. Then, by definition, the operator  Z +∞ dλ f (λ)aλ : F = : exp −∞

acts on the right Fock space Ha(λ) as follows: F = F− F+ , where ! Z Z

∞

0

F− = exp

−∞

dλ f (λ)aλ

and F+ = lim e →+0

 ln f ()a

 dλ f (λ)aλ .

exp 

Elliptic Algebra A(sbl2 ) in the Scaling Limit

617

∗ The action of the operator F on the left Fock space Ha(λ) is defined via another decom˜ ˜ position: F = F− F+ , where Z +∞ Z −    ln f (−)a− ˜ ˜ F+ = exp dλ f (λ)aλ and F− = lim e exp dλ f (λ)aλ . →+0

0

−∞

These definitions imply the following statement: Proposition 7. (i) The defined above actions of the operator  Z +∞ dλ f (λ)aλ : F =: exp −∞

on the Fock spaces H and H∗ are adjoint; (ii) The product of the normally ordered operators satisfy the property (5.5). c2 ), we choose H = Ha(λ) for Returning to the level one representation of A~,η (sl a(λ) defined in (5.1): a(λ) =

1 sh(~λ)sh(~λ/2) sh(λ/2η) . ~2 λ sh(λ/2η 0 )

From the definition of the Fock space H and from Proposition 6 we have immediately c2 ): the construction of a representation of A~,η (sl Proposition 8. The relations (5.2)–(5.4) define a highest weight level 1 representation c2 ) in the Fock space H. of the algebra A~,η (sl The highest weight property means that eˆλ |vaci = 0,

fˆλ |vaci = 0,

λ>0

and hvac|eˆλ = 0,

hvac|fˆλ = 0,

λ<0.

Let us define the twisted intertwining operators2 8(z) : H → H ⊗ Vz+i~/2 , 9∗ (z) : Vz+i~/2 ⊗ H → H ,

8∗ (z) : H ⊗ Vz+i~/2 → H , 9(z) : H → Vz+i~/2 ⊗ H ,

c2 ) up to the involution as those which commute with the action of the algebra A~,η (sl (1.9), 8(z)ι(x) = 1(x)8(z) , 9 (z)1(x) = ι(x)9∗ (z) , ∗

8∗ (z)1(x) = ι(x)8∗ (z) , 9(z)ι(x) = 1(x)9(z) ,

(5.6)

c2 ). In the definition of the intertwining operators Vz denotes for arbitrary x ∈ A~,η (sl the two-dimensional evaluation module Vz = V ⊗ C[[eiλz ]]

and

V = C v+ ⊕ C v − ,

λ ∈ R,

z∈C.

The components of the intertwining operators are defined as follows: 2 The shift in the evaluation module by i~/2 is introduced to avoid the shifts in the bosonization formulas (5.7)–(5.12) and is specific for this particular case of level 1, spin 1/2 representations.

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S. Khoroshkin, D. Lebedev, S. Pakuliak

8(z)v = 8+ (z)v ⊗ v+ + 8− (z)v ⊗ v− , 8∗ (z)(v ⊗ v± ) = 8∗± (z)v , 9∗ (z)(v± ⊗ v) = 9∗± (z)v , 9(z)v = v+ ⊗ 9+ (z)v + v− ⊗ 9− (z)v , where v ∈ H and one should understand the components 8ε (z), 9∗ε (z), ε = ± as generating functions, for example: 8(z)v =

XZ ε=±

∞ −∞

dλ 8ε,λ v ⊗ vε eiλ(z+i~/2) .

To find a free field realization of the intertwining operators we introduce the generating functions   Z ∞ a0λ :, dλ eiλz Z(z) = : exp −~ sh(~λ) −∞   Z ∞ 0 iλz aλ :. dλ e Z (z) = : exp ~ sh(~λ) −∞ Now we are ready to prove the following Proposition 9. The components of the twisted intertwining operators have the free field realization 9∗− (z) = Z(z) , Z i du πη(z−u) h 1/2 e (q) E(u)Z(z) + (q)−1/2 Z(z)E(u) , 9∗+ (z) = C 2π 9ν (z) = 9∗−ν (z + i~) , ν = ± ,

8− (z) = Z 0 (z) , Z i du πη0 (z−u) h 0 1/2 0 e (q ) Z (z)F (u) + (q 0 )−1/2 F (u)Z 0 (z) , 8+ (z) = C 0 2π ∗ 8ε (z) = 8−ε (z + i~) , ε = ± ,

(5.7) (5.8) (5.9) (5.10) (5.11) (5.12)

where q = eiπη~ ,

0

q 0 = eiπη ~

and the contour C 0 goes from −∞ to +∞ along the real axis leaving the points z + i~/2 + ik/η 0 (k ≥ 0) above the contour and the points z − i~/2 − ik/η 0 (k ≥ 0) below the contour. The contour C also goes from −∞ to +∞, but the points z − i~/2 + ik/η (k ≥ 0) are above the contour and the points z + i~/2 − ik/η (k ≥ 0) are below the contour. To prove Proposition 6 we should use the first terms in the comultiplication formulas (2.24), (2.25) and (2.26) specified for the operators x = e± (u), f ± (u) and h± (u) and the action of these generating functions on the elements of the evaluation two-dimensional module (3.19)–(3.21). The result is the commutation relations of the components of c2 ). For the intertwining operators with the generating functions of the algebra A~,η (sl ∗ example, for the operators 9 (z) these defining equations are:

Elliptic Algebra A(sbl2 ) in the Scaling Limit

619

sh πη(u − z + i~/2 ± i~/4) ∗ 9 (z)h± (u) , sh πη(u − z − i~/2 ± i~/4) − 0 = f ± (u)9∗− (z) + 9∗− (z)f ± (u) ,

h± (u)9∗− (z) = sh iπη~9∗+ (z)

=

sh πη(u − z − i~/2 ± i~/4)e (u)9∗− (z) + + sh πη(u − z + i~/2 ± i~/4)9∗− (z)e± (u)

(5.13) (5.14)

±

.

(5.15)

Because of (4.10) and (4.11) from (5.14) and (5.15) it follows that the operator 9∗− (z) anticommutes with the generating function F (u) and has the commutation relation with E(u) as follows: sh πη(u − z − i~/2)E(u)9∗− (z) = −sh πη(u − z + i~/2)9∗− (z)E(u) . It is easy now to verify using formulas of Appendix B that the generating function Z(z) satisfies these commutation relations with E(u), F (u) and also (5.13). The representation of 9∗+ (z) in integral form follows from (5.15) and (4.8). The form of the contours C and C 0 can be obtained from the analysis of the normal ordering relations of the generating functions E(u), Z(z), F (u) and Z 0 (z). Comparing the formulas (5.7)–(5.12) with the free field representation of the Zamolodchikov-Faddeev operators from [15] we conclude that these operators coincide with twisting intertwining operators. Therefore, they satisfy the Zamolodchikov-Faddeev algebra: ν0ν0

9∗ν1 (z1 )9∗ν2 (z2 ) = Sν11ν22 (z1 − z2 , ξ)9∗ν 0 (z2 )9∗ν 0 (z1 ) , 2

8ε2 (z2 )8ε1 (z1 ) = R˜

1

ε01 ε02 ε1 ε2

(z1 − z2 , ξ + ~)8ε01 (z1 )8ε02 (z2 ) ,   π iπ(z1 − z2 ) + 8ε (z2 )9∗ν (z1 ) , 9∗ν (z1 )8ε (z2 ) = νε tg 4 2~ 8ε1 (z)8∗ε2 (z) = g 0 δε1 ε2 id , gδε1 ε2 id + o(z1 − z2 ) , 9ε1 (z1 )9∗ε2 (z2 ) = z1 − z2

(5.16) (5.17) (5.18) (5.19) (5.20)

˜ ξ + ~) in (5.17) is where the S-matrix in (5.16) is given by (2.21), the R-matrix R(z, related to the R-matrix given by (2.1) as follows: ˜ ξ + ~) = (σz ⊗ 1)R(z, 1/(ξ + ~))(1 ⊗ σz ) R(z,

(5.21)

and the constants g, g 0 are equal to3 0

0

ie−3γη/2η η 2η/η 02 (2~ | 2~; 1/η)02 (2~ + 1/η | 2~; 1/η) , g= 02 (~ | 2~; 1/η)02 (3~ + 1/η | 2~; 1/η) η 2 02 (η/η 0 ) 0

2η 0 /η

ie−3γη /2η η 0 02 (~ | 2~; 1/η 0 )02 (1/η 0 − ~ | 2~; 1/η 0 ) g = √ . 2πη 0 02 (η 0 /η) 02 (2~ | 2~; 1/η 0 )02 (1/η 0 | 2~; 1/η 0 ) 0

(5.22)

The proof of the commutation relations (5.16)–(5.20) can be found in [15] and is based on the formulas gathered in Appendix B. In order to prove (5.19) and (5.20) one should use the following operator identities [1]: 3

The double 0-function 02 (x | ω1 ; ω2 ) is defined in Appendix B.

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   −1   i~ i~ Z u− : E(u) = eγ :Z u + , 2 2    −1   i~ i~ γ 0 0 Z u− : . F (u) = e :Z u + 2 2

(5.23)

The identities (5.23) being substituted into (5.8) and (5.11) yield the integral relation between components of the intertwining (Zamolodchikov-Faddeev) operators. This relation can be treated as a quantum version of the relation between linearly independent solutions to the second order ordinary differential equation (∂ 2 + u(z))ψ(z) = 0. 5.3. Zero mode discussion. In this subsection we follow a well known idea presented in [7, 10] in order to discuss the unusual (twisted) definition of the intertwining operators (5.6). The reason for this definition lies in the absence of the zero mode operator (−1)p c2 ) and twisted intertwining operators. We can in the bosonization of the algebra A~,η (sl come back to the usual definition introducing additional operators P and I such that [P, aλ ] = [I, aλ ] = 0,

P I = iIP

and

I 4 = 1.

Define also the extended Fock space H = H ⊗ C[Z/4Z] = H0 ⊕ H1 , and the subspaces H0 and H1 formed by the elements H0 = C[v ⊗ 1] ⊕ C[v ⊗ I 2 ],

H1 = C[v ⊗ I] ⊕ C[v ⊗ I 3 ],

v ∈ H.

The generating functions of the currents and the intertwining operators are modified as follows: ˜ E(u) = E(u) · I 2 , F˜ (u) = F (u) · I 2 , H˜ ± (u) = H ± (u) , ˜ ∗± (z) = 9∗± (z) · I ±1 P, 9

˜ ± (z) = 8± (z) · I ±1 P . 8

It is clear now that the commutation relations of the modified intertwining operators with elements of the algebra will be usual, for example: ˜ ˜ 8(z)x = 1(x)8(z),

c2 ) , x ∈ A~,η (sl

since the action of the operator P on the elements of the algebra coincide with the action of the involution (1.9), P x P −1 = P −1 x P = ι(x),

c2 ) . ∀ x ∈ A~,η (sl

The subspaces H0 and H1 become irreducible with respect to the action of the algebra and the operators 8(z), etc. intertwine these subspaces. Since the known physical models c2 ) serves as the algebra of dynamical symmetries have for which the algebra A~,η (sl single vacuum states, this mathematical construction of two irreducible Fock spaces is unnecessary and this unnecessity explains the absence of the zero mode operators in the bosonization of the massive models of the quantum field theory. 5.4. Miki’s Formulas. We would like to demonstrate now that bosonized expressions for the intertwining operators are in accordance with the L-operator description of the

Elliptic Algebra A(sbl2 ) in the Scaling Limit

621

c2 ). The way to do it is to use Miki’s formulas [21]. Consider the 2 × 2 algebra A~,η (sl operator valued matrices acting in the Fock space H (ε, ν = ±), r    2~eγ ∗  i~ 3i~ + Lεν (u) = νε 9ν u − 8ε u − , π 4 4 (5.24) r     2~eγ i~ 3i~ − ∗ Lεν (u) = 8ε u − 9ν u − . π 4 4 Now it is easy to show that the so-defined L-operators satisfy the commutation relations (2.18) if the operators 8(u) and 9∗ (u) satisfy the commutation relations of the Zamolodchikov-Faddeev algebra (5.16)–(5.18). Let us note here that although the in˜ ξ + ~) matrices which tertwining operators commute by means of the S(z, ξ) and R(z, differ from R(z, η) (see (2.21) and (5.21)) the L-operators defined by Miki’s prescription (5.24) commute according to (2.18) defined by the R-matrix (2.1). Miki’s formulas can be interpreted also as bosonization of L-operators for the algebra c2 ) at level 1. Using these formulas we can easily verify the relation (2.19) between A~,η (sl L± (u) operators. To do this it is sufficient to use the Gauss decomposition of these operators and calculate     i~ 3i~ 8− u − : L+−− (u) = : 9∗− u − 4 4 ! Z ∞ −λ/2η 00 +λ~/2 sh(λ~/2) iλu aλ e = : exp ~ :, dλ e sh(λ~) sh(λ/2η) −∞     i~ 3i~ 9∗− u − : L− −− (u) = : 8− u − 4 4 ! Z ∞ 00 λ/2η +λ~/2 sh(λ~/2) iλu aλ e = : exp ~ :. dλ e sh(λ~) sh(λ/2η) −∞ p Note that the constant 2~eγ /π in (5.24) is cancelled after normal ordering of the operators 9∗ (u) and 8(u) due to (B.1). The equality L+−− (u − i/η 00 ) = L− −− (u) is obvious now and the rest of the relations between the elements of the L-operators can be found using the definition of ‘+’-components of the intertwining operators through c2 ) (see (5.15)) and the the generating functions e± (u), f ± (u) of the algebra A~,η (sl ± commutation relation of these generating functions with k (u). 6. Discussion To conclude this paper we would like to mention some open problems which, in our opinion, deserve further investigation. Let us note first that there are many possibilities to choose initial parameters η = c2 ). For instance, it is clear from the 1/ξ and ~ in the definition of the algebra A~,η (sl c2 ) or its representations change above analysis that the properties of the algebra A~,η (sl drastically when the parameter ξ = 1/η = r~, where r is some rational number. The commutation relations (2.8)–(2.13) show that in this case a smaller factor algebra of c2 ) could be defined. It is natural to assume that this factor algebra serves the A~,η (sl symmetries of 8[1,3] -perturbations of the minimal models of conformal field theories.

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It is known from the theory of the Sine-Gordon model that if the parameter ξ < ~ then the spectrum of the model possesses scalar particles, so called breathers. Our considerations were based on the assumption that ξ > ~ and it is an open question to c2 ) in the regime ξ < ~, and to apply its representation investigate the algebra A~,η (sl theory to the Sine-Gordon model in the breather’s regime. We think that the algebra c2 ), ν = 1, 2, . . . is related to the restricted Sine-gordon model in the reflectionA~,ν/~ (sl less points, and representation theory of this algebra can be used for the group-theoretical interpretation of the results obtained recently in [28]. In particular, it is interesting to investigate the simplest case ξ = ~ which should correspond to the free fermion point of the Sine-Gordon model. It is also natural from an algebraic point of view to put the value of the deformation parameter ~ to be pure imaginary instead of positive real. It could correspond to the Sinh-Gordon theory. Surely, one can also try to apply the known technique [25, 26] for c2 ) and studying the c > 1 integer level integrable representations of the algebra A~,η (sl to find out possible physical applications. c2 ) cannot be As we have shown in the Sect. 2, the definition of the algebra A~,η (sl done in purely algebraic terms. So far the corresponding analytical language should be developed for its representations. It was partially done in Sect. 6 for the level one Fock space. Nevertheless, the rigorous mathematical description of the space of representation is far from complete. One needs more detailed topological description of the space, the precise definition of the trace, making the calculations in [1, 15, 20] to be rigorous, the investigation of the irreducibility and so on. Moreover, it would be nice to have an axiomatic description of the category of highest weight representations. c2 ) demonThe analysis of the defining commutation relations of the algebra A~,η (sl strate that the algebra in question possesses the rich structure of automorphisms. For example, if we allow parameters λ, µ in (1.2)–(1.7) to be complex, then these commutation relations can be rewritten in the form of difference commutation relations without integrals in the r.h.s. c2 ) is not a Hopf algebra and even for c = 0, when it becomes The algebra A~,η (sl a Hopf algebra, it does not have te structure of the quantum double [29], as well as its classical counterpart [23]. The double structure can be reconstructed in the Yangian limit c2 ) becomes the central extended Yangian double. The η → 0 when the algebra A~,0 (sl representation theory of the latter algebra has been investigated in [4, 24, 18, 20] using two alternative possibilities related to Riemann problems on the circle and on the line. c2 ), It is interesting to understand what structure replaces the double structure in A~,η (sl whether there exists an analog of the universal R-matrix. It is also interesting to formulate the quantum Sugawara construction corresponding c2 ). The scaling quantum Virasoro algebra can be obtained from to the algebra A~,η (sl [30–33], where the two-parameter deformation of the Virasoro algebra corresponding c2 ) has been investigated. to the algebra Aq,p (sl

Appendix A. Consistency of Comultiplication Formulas To prove that the comultiplication formulas (2.22) are in accordance with the comc2 ), we have to check that the relations (here mutation relations of the algebra A~,η (sl + L(u) = L (u))

Elliptic Algebra A(sbl2 ) in the Scaling Limit

623

R+ (u1 − u2 , ξ + ~1c)1L1 (u1 )1L2 (u2 ) = 1L(u2 )1L(u1 )R+ (u1 − u2 , ξ), 1(q-detL(u)) = 1(q-detL(u)) ⊗ 1(q-detL(u)) follow from (2.3) and the definition of the quantum determinant (2.5). First of all we rewrite Eq. (2.3) in components and the comultiplication formulas (2.23) using short notations. We have Rim;kp (u1 − u2 , ξ + ~c)Lmj (u1 , ξ)Lpl (u2 , ξ) = Lkq (u2 , ξ)Lir (u1 , ξ)Rrj;ql (u1 − u2 , ξ) and

1Lmj (u, ξ) = Lf j (u0 , ξ 0 )Lmf (u00 , ξ 00 ) ,

where, as it was in the Subsect. 3.3, prime (double prime) denotes that the corresponding L-operator or its component is in the first (second) component of the tensor product and u0 = u + i~c00 /4, ξ 0 = ξ = 1/η (u00 = u − i~c0 /4, ξ 00 = ξ + ~c0 ). The summation over repeating indices is always supposed:

Rim;kp (u1 − u2 , ξ + ~c0 + ~c00 )1 Lmj (u1 , ξ) 1 Lpl (u2 , ξ) = Rim;kp (u1 − u2 , ξ + ~c0 + ~c00 )Lf j (u01 , ξ)Lmf (u001 , ξ + ~c0 ) Lf 0 l (u02 , ξ)Lpf 0 (u002 , ξ + ~c0 ) = Rim;kp (u1 − u2 , ξ + ~c0 + ~c00 )Lf j (u01 , ξ)Lf 0 l (u02 , ξ) Lmf (u001 , ξ + ~c0 )Lpf 0 (u002 , ξ + ~c0 ) = Rr0 f ;q0 f 0 (u1 − u2 , ξ + ~c0 )Lf j (u01 , ξ)Lf 0 l (u02 , ξ)Lkq0 (u002 , ξ + ~c0 )Lir0 (u001 , ξ + ~c0 ) = Rrj;ql (u1 − u2 , ξ)Lq0 q (u02 , ξ)Lr0 r (u01 , ξ)Lkq0 (u002 , ξ + ~c0 )Lir0 (u001 , ξ + ~c0 ) = Rrj;ql (u1 − u2 , ξ)Lq0 q (u02 , ξ)Lkq0 (u002 , ξ + ~c0 )Lr0 r (u01 , ξ)Lir0 (u001 , ξ + ~c0 )
= 1 Lkq (u2 , ξ) 1 (Lir (u2 , ξ)) Rrj;ql (u1 − u2 , ξ) . The primitiveness of the coproduct of the quantum determinant can be proved easily using (A.3), (A.4) and the formula equivalent to (2.5), q-det L(u) = D(u − i~)A(u) − C(u − i~)B(u) . Now we are in position to prove formulas (2.24)–(2.26). The simplest comultiplication relation which follows from (2.22) is 1k(u, ξ) = k(u0 , ξ 0 ) ⊗ k(u00 , ξ 00 ) + f (u0 , ξ 0 )k(u0 , ξ 0 ) ⊗ k(u00 , ξ 00 )e(u00 , ξ 00 )   (A.1) = 1 ⊗ 1 + f (u0 , ξ 0 ) ⊗ e(u00 − i~, ξ 00 ) k(u0 , ξ 0 ) ⊗ k(u00 , ξ 00 )   0 0 00 00 0 0 00 00 = k(u , ξ ) ⊗ k(u , ξ ) 1 ⊗ 1 + f (u − i~, ξ ) ⊗ e(u , ξ ) . (A.2) The equivalent form of comultiplication of the operators k ± (u) follows from the operator identities k(u, ξ)e(u, ξ) − e(u − i~, ξ)k(u, ξ) = 0 , k(u, ξ)f (u − i~, ξ) − f (u, ξ)k(u, ξ) = 0 ,

(A.3) (A.4)

which are consequences of the commutation relations (2.3) at the critical point u1 − u2 = i~. Formulas (2.24) and (2.25) easily follow from (2.23) and (A.1), (A.2). The proof of (2.26) is more involved. It follows from the chain of identities4 : 4

Here and below we do not write explicitly dependence of the operators on the parameter ξ.

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1h(u) = =

∞ X

0

0

0

0

0

(−1)p+p f p+p (u0 − i~)h(u0 ) ⊗ k −1 (u00 )ep (u00 − i~)k −1 (u00 + i~)ep (u00 )

p,p0 =0 ∞ X


p

(−1)p+p f p+p (u0 −i~)h(u0 ) ⊗ h(u00 ) [2]η e(u00 −i~)−e(u00 )

0

ep (u00 )

p,p0 =0

=

∞ X

0

0

00

(−1) f (u − i~)h(u ) ⊗ h(u ) p p

p=0

=

∞ X

p X


k

[2]η e(u00 − i~) − e(u00 )

ep−k (u00 )

k=0

(−1)p [p + 1]η f p (u0 − i~)h(u0 ) ⊗ h(u00 )ep (u00 − i~) .

p=0

Here we used the commutation relation which follows from (2.9), k(u + i~)e(u − i~)k −1 (u + i~) = [2]η e(u − i~) − e(u), and the combinatorial identity p X

([2]η e(u − i~) − e(u))k ep−k (u) = [p + 1]η ep (u − i~),

k=0

which follows by induction from (2.11). There is another way to verify the concordance of formulas (2.23). The comultiplication formulas for the currents follow from comultiplication of L-operator entries L12 (u, ξ), L21 (u, ξ), L22 (u, ξ) given by (2.23). The essential part of this calculation is the comultiplication of the inverse operator (k(u))−1 . Using formulas (A.1) and (A.2) we can obtain 1 (k(u, ξ))

−1

=

∞ X


p

(−1)p f (u0 − i~, ξ 0 )

p=0


−1

⊗ k(u00 , ξ 00 )


−1

k(u0 , ξ 0 )
p

e(u00 − i~, ξ 00 )

⊗

.

(A.5)

The comultiplication of the entry L11 (u, ξ) also defines the comultiplication of the operator (k(u))−1 , so we should prove that these two comultiplication formulas lead to the same result. After some simple algebra we have
1k −1 (u + i~) = k −1 (u0 + i~) + f (u0 )k(u0 )e(u0 ) ⊗


⊗ k −1 (u00 + i~) + f (u00 )k(u00 )e(u00 ) +

+ k(u0 )e(u0 ) ⊗ f (u00 )k(u00 ) − 1f (u)1k(u)1e(u) ∞ X
p
p = (−1)p f (u0 ) k −1 (u0 + i~) ⊗ k −1 (u00 + i~) e(u00 ) . p=0

The last line obviously coincides with (A.5) after shifting z → z − i~.

Elliptic Algebra A(sbl2 ) in the Scaling Limit

625

Appendix B: Normal Ordering Relations The relations below are based on the formulas which can be found in [34, 35] Z   dλ ln(−λ) 1 1 e−xλ = ln 0(ηx) + ηx − (γ − ln η) − ln 2π , −λ/η 2πiλ 2 2 1 − e ˜ C Z γ dλ ln(−λ) e−xλ = ln 02 (x | ω1 , ω2 ) − B2,2 (x | ω1 ; ω2 ) , −λω1 )(1 − e−λω2 ) 2πiλ (1 − e 2 ˜ C where B2,2 (x | ω1 ; ω2 ) is the double Bernulli polynomial of second order   1 ω12 + 3ω1 ω2 + ω22 2 B2,2 (x | ω1 ; ω2 ) = . x − x(ω1 + ω2 ) + ω1 ω2 6 Using these integral representations of the ordinary and double 0-functions and the definition of the product (5.5) we can calculate:
0 iη(u − z) − η~/2 ~ :Z(z)E(u):
Z(z)E(u) =
η/η0 , Im (u − z) < − 2 , 0 1 + iη(u − z) + η~/2 eγ /η
0 −iη(u − z) − η~/2 :Z(z)E(u): ~
E(u)Z(z) =
0 , Im (u − z) > 2 , 0 1 − iη(u − z) + η~/2 eγ /η η/η Z(z)F (u) = ieγ (u − z):Z(z)F (u):, Im (u − z) < 0 , F (u)Z(z) = −ieγ (u − z):F (u)Z(z):, Im (u − z) > 0 ,
0 iη 0 (u − z) + η 0 ~/2 ~ :Z 0 (z)F (u):
Z 0 (z)F (u) =
0 , Im (u − z) < 2 , 0 1 + iη 0 (u − z) − η 0 ~/2 eγ /η 0 η /η
0 −iη 0 (u − z) + η 0 ~/2 ~ :Z 0 (z)F (u): 0
F (u)Z (z) =
η0 /η , Im (u − z) > − 2 , 0 0 0 1 − iη (u − z) − η ~/2 eγ /η 0 Z 0 (z)E(u) = ieγ (u − z):Z 0 (z)E(u):, Im (u − z) < 0 , E(u)Z 0 (z) = −ieγ (u − z):E(u)Z 0 (z):, Im (u − z) > 0 ,   1 + i(z2 − z1 ) 0 4 2~ 1 ~  :Z(z1 )Z 0 (z2 ):, Im (z1 − z2 ) > − , (B.1)  Z(z1 )Z 0 (z2 ) = √ γ i(z − z ) 3 2 2~e 0 + 2 1 4

2~

Z(z1 )Z(z2 ) = g(z1 − z2 ):Z(z1 )Z(z2 ):, Im (z1 − z2 ) > 0 , Z 0 (z1 )Z 0 (z2 ) = g 0 (z1 − z2 ):Z 0 (z1 )Z 0 (z2 ):, Im (z1 − z2 ) > −~ , where the functions g(z) and g 0 (z) are  Z  dλ ln(−λ) (1 − e−λ(ξ+~) )(1 − e−~λ ) iλz g(z) = exp − e 2πiλ (1 − e−ξλ )(1 − e−2~λ ) C˜ 0 02 (~ − iz | 2~; 1/η)02 (~ + 1/η − iz | 2~; 1/η) , = eγη/2η 02 (−iz | 2~; 1/η)02 (2~ + 1/η − iz | 2~; 1/η) Z dλ ln(−λ) (1 − e−ξλ )(e−2~λ − e−~λ ) iλz e , g 0 (z) = 2πiλ (1 − e−(ξ+~)λ )(1 − e−2~λ ) C˜ 0 02 (2~ − iz | 2~; 1/η 0 )02 (1/η 0 − iz | 2~; 1/η 0 ) . = eγη /2η 02 (~ − iz | 2~; 1/η 0 )02 (~ + 1/η 0 − iz | 2~; 1/η 0 )

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As usual the normal ordering of all operators is calculated in the regions specified above and then analytically continued to all possible values of the spectral parameters. Acknowledgement. We use the opportunity to thank A. Belavin, M. Jimbo, S. Lukyanov and T. Miwa for the stimulating discussions and for their remarkable works which have been the source of our inspiration. We would like to acknowledge also useful discussions with A. Gerasimov and Ya. Pugai. The research described in this publication was made possible in part by grants RFBR-96-01-01106 (S. Khoroshkin), RFBR-96-02-18046 (D. Lebedev), RFBR-96-02-19085 (S. Pakuliak), INTAS-93-0166-Ext (S. Khoroshkin, D. Lebedev), INTAS-93-2058-Ext (S. Pakuliak) and by Award No. RM2-150 of the U.S. Civilian Research & Development Foundation (CRDF) for the Independent States of the Former Soviet Union.

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22. Kashiwara, M., Miwa, T., Petersen, J.-U.H., Yung, C.M.: Perfect crystals and q-deformed Fock spaces. Selecta Math. 2, 415–499 (1996) 23. Khoroshkin, S., Lebedev, D., Pakuliak, S., Stolin, A., Tolstoy, V. Classical limit of the scaled elliptic algebra A~,η (slb2 ). Preprint ITEP-TH-1/97, q-alg/9703043. Submitted for publication in Compositio Matematica. 24. Khoroshkin, S., Lebedev, D., Pakuliak, S.: Yangian double in terms of Fourie integrals. Preprint ITEPTH-3/97 c 25. Matsuo, A.: Free field representation of quantum affine algebra Uq (sl N ). Phys. Lett. B 308, 61–84 (1993) 26. Awata, H., Odake, S., Shiraishi, J.: Free boson realization of Uq (slb2 ). Commun. Math Phys. 162, 260–265 (1994) 27. Davies, B., Foda, O., Jimbo, M., Miwa, T., Nakayashiki, A.: Diagonalization of the XXZ Hamiltonian by vertex operators. Commun. Math. Phys. 151, 89–153 (1993) 28. Babelon, O., Bernard, D., Smirnov, F.A.: Quantization of solitons and the restricted sine-Gordon model. Commun. Math. Phys. 182, 319–354 (1996) 29. Drinfeld, V.G.: Quantum groups. In: Proceedings of the International Congress of Mathematicians, Berkeley, 1987, pp. 798–820 30. Feigin, B., Frenkel, E.: Quantum W algebras and elliptic algebras. Commun. Math. Phys. 178, 653–678 (1996) 31. Frenkel, E., Reshetikhin, N.: Quantum affine algebras and deformation of Virasoro and W algebras Commun. Math. Phys. 178, 237–266 (1996) 32. Lukuyanov, S.: A note on deformed Virasoro algebra. Phys. Lett. B 367, 121–125 (1996) 33. Shiraishi, J., Kubo, H., Awata, H., Odake, S.: A q-deformation of the Virasoro algebra and the Macdonald symmetric functions. Lett. Math. Phys. 38, 33–51 (1996) 34. Barnes, E.W.: On the theory of the multiple gamma functions. Trans. Cambridge Philos. Soc., 19, 374–425 (1904) 35. Jimbo, M., Miwa, T.: Quantum KZ equation with |q| = 1 and correlation functions of the XXZ model in the gapless regime. J. Phys. A 29, 2923–2958 (1996) Communicated by T. Miwa

Commun. Math. Phys. 190, 629 – 673 (1998)

Communications in

Mathematical Physics c Springer-Verlag 1997

Quantum Hall Effect on the Hyperbolic Plane A. L. Carey1 , K. C. Hannabuss2 , V. Mathai1 , P. McCann1 1 Department of Mathematics, University of Adelaide, Adelaide 5005, Australia. E-mail: [email protected], [email protected], [email protected] 2 Department of Mathematics, University of Oxford, England. E-mail: [email protected]

Received: 17 March 1997 / Accepted: 24 April 1997

Abstract: In this paper, we study both the continuous model and the discrete model of the Quantum Hall Effect (QHE) on the hyperbolic plane. The Hall conductivity is identified as a geometric invariant associated to an imprimitivity algebra of observables. We define a twisted analogue of the Kasparov map, which enables us to use the pairing between K-theory and cyclic cohomology theory, to identify this geometric invariant with a topological index, thereby proving the integrality of the Hall conductivity in this case.

1. Introduction The usual model of the integer quantum Hall effect involves electrons moving in a two dimensional conductor under the influence of a magnetic field. The field is applied in a direction orthogonal to the conductor. The Hamiltonian used is that for a single electron moving under the influence of this magnetic field with the addition of a potential term to represent the field due to the lattice of ions making up the conductor. For simplicity this lattice is often assumed to be periodic in the two axis directions in the plane. The effect of impurities can be modelled by departing from a perfectly periodic potential. The definitive treatment from a mathematical point of view is due to Bellissard [Bel+E+S] and Xia [Xia]. In this approach no assumption is made about the rationality of the imposed magnetic flux while the integrality of the Hall conductance follows by showing that it is given by the index of a Fredholm operator. Xia also exhibits the conductance as a topological index. These demonstrations use in an essential way Connes’ noncommutative differential geometry. In this paper we are interested in what can be said when one replaces the usual two dimensional conducting material with its Euclidean geometry by a two dimensional sample with hyperbolic geometry. Physically one should think of hyperbolic space and hence the sample as an embedded hyperboloid in Euclidean 3-space. The crystal lattice of

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the conductor is now modelled by the orbit of a freely acting discrete group. For reasons of convenience we take this to be the fundamental group of a Riemann surface (though aspects of our analysis work more generally). The magnetic field remains orthogonal to the two dimensional conductor and the electric potential we take to be periodic under the action of this discrete group (we do not attempt to model impurities). When the magnetic flux is rational, spectral properties of Hamiltonians for a single electron have been studied, particularly in the cases where the Hamiltonian may be defined on a finite cover of the Riemann surface [Iengo+Li]. In this paper however we are interested in approaching the problem using non-commutative geometry, which allows the flux to be any real number, and the algebras of interest are not always associated with vector bundles over the Riemann surface. We begin by reviewing the construction of the Hamiltonian. First we take as our principal model of hyperbolic space, the hyperbolic plane. This is the upper half-plane H in C equipped with its usual Poincar´e metric (dx2 +dy 2 )/y 2 , and symplectic area form ωH = dx ∧ dy/y 2 . The group SL(2, R) acts transitively on H by M¨obius transformations   aζ + b ab , for g = x + iy = ζ 7→ gζ = . cd cζ + d Any Riemann surface of genus g greater than 1 can be realised as the quotient of H by the action of its fundamental group realised as a subgroup 0 of SL(2, Z) ⊂ SL(2, R). Let us now pick a 1-form η such that dη = θωH , for some fixed θ ∈ (0, 1]. As in geometric quantisation we may regard η as defining a connection ∇ = d − iη on a line bundle L over H, whose curvature is θωH . Physically we can think of η as the electromagnetic vector potential for a uniform magnetic field of strength θ normal to H. Using the Riemannian metric the Hamiltonian of an electron in this field is given in suitable units by 1 1 H = Hη = ∇∗ ∇ = (d − iη)∗ (d − iη). 2 2 Comtet [Comtet] has shown that H differs from a multiple of the Casimir element for SL(2, R), 18 J.J, by a constant, where J1 , J2 and J3 denote a certain representation of generators of the Lie algebra sl(2, R), satisfying [J1 , J2 ] = −iJ3 ,

[J2 , J3 ] = iJ1 ,

[J3 , J1 ] = iJ2 ,

so that J.J = J12 + J22 − J32 is the quadratic Casimir element. This shows very clearly the underlying SL(2, R)-invariance of the theory. In a real material this Hamiltonian would be modified by the addition of a potential V . By taking V to be invariant under 0 this perturbation is given a crystalline type structure analogous to the use of periodic potentials invariant under Z2 in the Euclidean plane R2 . Comtet has computed the spectrum of the unperturbed Hamiltonian Hη , for η = θdx/y, to be the union of finitely many eigenvalues {(2k + 1)θ − k(k + 1) : k = 0, 1, 2 . . . < θ − 21 }, and the continuous spectrum [ 41 + θ2 , ∞) for those values of θ for which the de Rham cohomology class of θωH is integral. Its zeta function and the kernel of its resolvent are also known in this case, [Comtet, Comtet+H]. Any η is cohomologous to θdx/y (since they both have ωH as differential) and forms differing by an exact form dφ give equivalent models: in fact, multiplying the wave functions by exp(iφ) shows that the models for η and θdx/y are unitarily equivalent. This equivalence also intertwines the 0-actions so that the spectral densities for the two models also coincide. However, it is the perturbed Hamiltonian Hη,V = Hη + V which is the key to the quantum Hall effect on the hyperbolic plane,

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and the spectrum of this is unknown for general 0-invariant V . As we noted above the Hall effect on Riemann surfaces has also been considered [Iengo+Li], [Av+K+P+S] but this is different from the problem we consider here. These considerations suggest that one could mimic the non-commutative geometry approach of Bellissard-Connes to the integer quantum Hall effect on Euclidean space [Bel, Nak+Bel, Bel+E+S, Co2, Xia] in a hyperbolic setting. This interprets the Hall conductivity as a non-commutative Chern character, whose integrality follows from Ktheory. Physically such situations have been considered without the perturbing potential (or 0 is trivial) in the context of exploring edge effects for the quantum Hall effect and the behaviour of electrons in quantum dots. Much of the mathematical machinery needed for this has already been discussed in a geometrical context [Co, Co2, Comtet, Comtet+H] and will be exploited here. We also discuss the discrete version of the theory [Co2], [Sun] motivated in part by some results of [MC] in the Euclidean setting. These hyperbolic Hall effect models occupy the first seven sections. Specifically we show that there is a principal groupoid C ∗ -algebra with cocycle for the diagonal 0 action on H × H which contains the resolvent of the various Hamiltonians we consider. In order to construct a Fredholm module for this algebra we found it useful to take a more abstract group theoretic approach. We show that our groupoid algebra is isomorphic to a quotient of the 0 invariant part of the imprimitivity algebra for inducing from the maximal compact subgroup of SL(2, R) to C ∗ (SL(2, R), σ) (the multiplier, or group 2-cocycle, σ extends to all of SL(2, R)). This imprimitivity algebra has a regular representation, induced by a canonical trace, the Hilbert space of which provides a Fredholm module which is 2summable for a dense subalgebra of the imprimitivity algebra. We show that this dense subalgebra contains the spectral projections corresponding to gaps in the spectrum of our Hamiltonians. Similar results hold in the discrete model as well. The connection between the continuous and discrete models arises from the Morita equivalence of our quotient of the 0 invariant imprimitivity algebra with C ∗ (0, σ). The main results of our paper follow by extending the approach of [Xia] to cover the hyperbolic case. In fact in Sect. 8 we prove some general theorems about the Kgroups of C ∗ -algebras which generalize those arising from the hyperbolic Hall effect. The relevance of K-theory can be understood in the case of the integer Hall effect on Euclidean space partly as a result of the calculation [Elliott, Bel, Co]: K∗ (C ∗ (Zn , σ)) ∼ = K∗ (C ∗ (Zn )) ∼ = K ∗ (Tn ) for any multiplier (i.e. group 2-cocycle) σ on Zn . This result has lead to the twisted group C ∗ -algebras C ∗ (Zn , σ) being called noncommutative tori. This calculation was generalized by Packer and Raeburn [PR, PR2], who computed the K-groups of the twisted group C ∗ -algebras of uniform lattices in solvable groups. More precisely, they proved that if 0 is a uniform lattice in a solvable Lie group G, then K∗ (C ∗ (0, σ)) ∼ = K ∗+dim G (0\G, δ(Bσ )), where σ is any multiplier on 0, K ∗ (0\G, δ(Bσ )) denotes the twisted K-theory of a continuous trace C ∗ -algebra Bσ with spectrum 0\G, while δ(Bσ ) ∈ H 3 (0\G, Z) denotes the Dixmier-Douady invariant of Bσ . (Note that the twisted K-theory was studied in [Ros]). Packer and Raeburn proved a stabilization theorem and used the Thom isomorphism theorem for the K-theory of C ∗ -algebras, due to Connes [Co2], to prove their results. In Sect. 8 we extend the main theorem of [PR, PR2] to the case when 0 is a lattice in a K-amenable Lie group G. More precisely, we prove that for such G and 0,

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K∗ (C ∗ (0, σ)) ∼ = K∗ (Cr∗ (0, σ)) and

K∗ (C ∗ (0, σ)) ∼ = K ∗+dim(G/K) (0\G/K, δ(Bσ )),

where K is a maximal compact subgroup of G, σ is any multiplier on 0, K ∗ (0\G/K, δ(Bσ )) is the twisted K-theory of a continuous trace C ∗ -algebra Bσ with spectrum 0\G/K, and δ(Bσ ) ∈ H 3 (0\G/K, Z) is the Dixmier-Douady invariant of Bσ . Our method uses the K-amenability results of Kasparov [Kas1] and the PackerRaeburn stabilization theorem [PR]. In the case when 0 = 0g is the fundamental group of a Riemann surface Σg of genus g > 0, we deduce that the Dixmier-Douady class δ(Bσ )) is trivial. Using this we demonstrate that for any multiplier σ on 0g , K0 (C ∗ (0g , σ)) ∼ = K 0 (Σg ) ∼ = Z2 , and that

K1 (C ∗ (0g , σ)) ∼ = K 1 (Σg ) ∼ = Z2g .

We end the discussion with an interesting conjecture for compact 3-dimensional manifolds which are Eilenberg-Maclane spaces. These K-theoretic results have now been generalized to C ∗ -dynamical systems in [Ma]. One of the most outstanding open problems about magnetic Schr¨odinger operators or Hamiltonians on Euclidean space is concerned with the nature of their spectrum, and is called the Ten Martini Problem (TMP) (cf. [Sh]). More precisely, TMP asks whether, given a multiplier σ on Z2 , there is an associated Hamiltonian (i.e. a Hamiltonian which commutes with the (0, σ) projective action of 0 on L2 (R2 )) possessing a Cantor set type spectrum, in the sense that the intersection of the spectrum of the Hamiltonian with some compact interval in R is a Cantor set? One can deduce from the range of the trace on K0 of the twisted group C ∗ -algebras that when the multiplier takes its values in the roots of unity in U (1) (we say then that it is rational) that such a Hamiltonian cannot exist. However, in the Euclidean case and for Liouville numbers, the discrete analogue of the TMP has been been settled in the affirmative by Choi, Elliot and Yui [CEY] (cf. [Sh] for a historical perspective). In Sect. 9 we are concerned also with the hyperbolic analogue of the TMP, which we call the Ten Dry Martini Problem (TDMP). We prove that the Kadison constant of the twisted group C ∗ -algebra Cr∗ (0g , σ) is positive whenever the multiplier is rational, where 0g is now the fundamental group of a genus g Riemann surface. We then use the results of Br¨uning and Sunada [BrSu] to deduce that when the multiplier is rational, the TDMP is answered in the negative, and we leave open the more difficult irrational case. The calculation of the range of the trace exploits a number of results including a twisted Kasparov map on K-theory. Finally, we apply our results to give a complete classification up to isomorphism of the twisted C ∗ -algebras Cr∗ (0g , σ). In Sects. 10 and 11 we will identify the character of our Fredholm modules, the “Hall conductivity”, for both the continuous and discrete models. This character τc (P, P, P ) = tr(P dP dP ) is shown to arise from Connes’ “area cocycle” and we are able to identify it with a topological invariant, generalizing the work of Xia [Xia] in the case of the quantum Hall effect on Euclidean space. We use the pairing between K-theory and cyclic cohomology [Co], a generalization of the Connes-Moscovici index theorem [CM] to projectively invariant elliptic operators and the twisted analogue of the Kasparov map. In fact we obtain a general index theorem which equates the (analytical) index arising from the Fredholm modules to a topological index. It specialises in the case of the cyclic cocycle τc to give the surprising fact that the hyperbolic “Hall conductivity”

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τc (P, P, P ) ∈ 2(g −1)Z. This raises the obvious question of whether a real material with a hyperbolic crystalline geometry could be manufactured and the genus of the quotient Riemann surface measured experimentally. To be specific the model we consider here can be understood most easily in the embedded hyperboloid version of hyperbolic space. If we use the 0-orbit of a point in the hyperboloid in R3 to represent the crystal lattice structure of a conducting material then our discrete model corresponds to applying a magnetic field which is everywhere normal to the hyperboloid. (This captures the hyperbolic geometry.) Then by regarding the lattice points as the vertices of a graph whose edges are geodesics corresponding to the generators of 0 our model Hamiltonian corresponds to allowing electrons to hop between sites on the lattice along the edges of the graph. Then our theorem predicts that the conductivity should depend on the genus of the Riemann surface obtained by quotienting the hyperboloid by 0. In Sect. 6 we exhibit a cyclic cocycle which plays the role of the Kubo formula for higher genus surfaces. It has an intrinsic geometric description as a “symplectic area” cocycle on the universal cover of the Jacobi variety of the Riemann surface. The novel feature of the higher genus case (as opposed to genus one which is the Euclidean case) is that the Kubo cocycle is cohomologous (but not equal) to the cyclic cocycle arising from the Fredholm module. Given our K-theoretic interpretation of the latter this is sufficient to give the anticipated result that the Hall conductivity, as defined through the Kubo cocycle, is integral and depends on the genus. We conclude by showing how our formalism links with the non-commutative Riemann surface theory described in [Klim+Les1, Klim+Les2]. 2. The Geometry of the Hyperbolic Plane The upper half-plane can be mapped by the Cayley transform z = (ζ −i)/(ζ +i) to the unit disc D equipped with the metric |dz|2 /(1−|z|2 )2 and symplectic form dz dz/2i(1−|z|2 )2 , on which SU(1, 1) acts, and some calculations are more easily done in that setting. In order to preserve flexibility we shall work more abstractly with a Lie group G acting transitively on a space X ∼ G/K. Although we shall ultimately be interested in the case of G = SL(2, R) or SU(1, 1), and K the maximal compact subgroup which stabilises ζ = i or z = 0, those details will play little role in many of our calculations, though we shall need to assume that X has a G-invariant Riemannian metric and symplectic form ωH . We shall denote by 0 a discrete subgroup of G which acts freely on X and hence intersects K trivially. We shall assume that L is a hermitian line bundle over X, with a connection, ∇, or equivalently, for each pair of points w and z in X, we denote by τ (z, w) the parallel transport operator along the geodesic from Lw to Lz . In H with the line bundle trivialised and η = θdx/y one can calculate explicitly that  Z z  η = [(z − w)/(w − z)]θ . τ (z, w) = exp i w

For general η we have η − θdx/y = dφ and Z z τ (z, w) = exp(i η) = [(z − w)/(w − z)]θ exp(i(φ(z) − φ(w))). w

Parallel transport round a geodesic triangle with vertices z, w, v, gives rise to a holonomy factor:

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$(v, w, z) = τ (v, z)−1 τ (v, w)τ (w, z), and this is clearly the same for any other choice of η, so we may as well work in the general case.
R Lemma 1. The holonomy can be written as $(v, w, z) = exp iθ 1 ωH , where 1 denotes the geodesic triangle with vertices z, w and v. The holonomy is invariant under the action of G, that is $(v, w, z) = $(gv, gw, gz), and under cyclic permutations of its arguments. Transposition of any two vertices inverts $. For any four points u ,v, w, z in X one has $(u, v, w)$(u, w, z) = $(u, v, z)$(v, w, z). Proof. By definition, for a suitable trivialisation of L one has   Z η , $(v, w, z) = exp i ∂1

and the first part follows by applying Stokes’ Theorem after noting that the result is independent of the trivialisation. The invariance under G follows from the invariance of the symplectic form, and the results of permutations follow from the properties of the integral, as does the final identity.  3. The Twisted Algebra of Kernels The geometrical data described in the last section enables us to easily describe the first of the two C ∗ -algebras which appear in the theory. This twisted algebra of kernels, which was introduced by Connes [Co2] is the C ∗ -algebra B generated by compactly supported smooth functions on X × X with the multiplication Z k1 (z, v)k2 (v, w)$(z, w, v) dv, k1 ∗ k2 (z, w) = X

∗ (where dv denotes the G-invariant measure defined by the R metric) and k (z, w) = k(w, z). There is an obvious trace on B given by τB (k) = X k(z, z) dz. The algebra of twisted kernels is the extension of the C ∗ -algebra of the principal groupoid X × X defined by the cocycle ((v, w), (w, z)) 7→ $(v, w, z), [Ren1].

Lemma 2. The algebra B has a representation π on the space of L2 sections of L defined by Z k(z, w)τ (z, w)ψ(w) dw.

(π(k)ψ)(z) = X

Proof. The parallel transport τ (z, w) ensures that the integral is in the appropriate fibre, and the fact that it is a representation follows from a calculation using the definition of the holonomy.  Before describing the second algebra we need to link the geometrical data more directly to the group G. To do this we fix a basepoint u ∈ X and introduce the function φ from X × G to line bundle automorphisms defined by φ(z, g) = $(u, g −1 u, g −1 z)τ (u, z)−1 τ (u, g −1 z). (The ratio of parallel transports defines an operator from the fibre Lg−1 z to Lz .)

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Lemma 3. The function φ satisfies φ(z, x)φ(x−1 z, y) = $(u, y −1 u, y −1 x−1 u)φ(z, xy) φ(z, x)τ (x−1 z, x−1 w) = τ (z, w)φ(w, x). Proof. By definition we have φ(z, x)φ(x−1 z, y) = $(u, x−1 u, x−1 z)$(u, y −1 u, y −1 x−1 z)τ (u, z)−1 τ (u, x−1 z)τ (u, x−1 z)−1 τ (u, y −1 x−1 z) −1 $(u, x u, x−1 z)$(u, y −1 u, y −1 x−1 z) φ(z, xy). = $(u, y −1 x−1 u, y −1 x−1 z) Now by Lemma 2.1, $(u, x−1 u, x−1 z)$(u, y −1 u, y −1 x−1 z) = $(y −1 u, y −1 x−1 u, y −1 x−1 z)$(u, y −1 u, y −1 x−1 z) = $(u, y −1 x−1 u, y −1 x−1 z)$(u, y −1 u, y −1 x−1 u), from which the first result follows. For the second result we note (compressing the notation) that $(u, x−1 u, x−1 z) τ (x−1 z, x−1 w)τ (u, x−1 z) τ (u, w) τ (x−1 z, x−1 w)φ(z, x) = φ(w, x) $(u, x−1 u, x−1 w) τ (u, z) τ (u, x−1 w) $(u, x−1 u, x−1 z)$(u, x−1 z, x−1 w) τ (u, x−1 w) τ (u, w) = $(u, x−1 u, x−1 w) τ (u, z) τ (u, x−1 w) τ (u, w) = $(x−1 u, x−1 z, x−1 w) τ (u, z) τ (u, w) = $(u, z, w) = τ (z, w). τ (u, z)  The most important aspect of the first result is that σ(x, y) = φ(z, xy)/φ(z, x)φ(x−1 z, y) = $(u, y −1 u, y −1 x−1 u)−1 = $(u, xu, xyu) is independent of z. (We note also that σ(g, 1) = σ(1, g) = σ(g, g −1 ) = 1. Although these normalisations do not seriously affect matters they can sometimes be used to simplify formulae.) Lemma 4. The function σ : G × G → T satisfies the cocycle identity, σ(x, y)σ(xy, g) = σ(x, yg)σ(y, g). Proof. This is a simple calculation along the lines of those above.



This result means that σ defines a projective multiplier or group 2-cocycle for G, moreover, it is clearly continuous and identically 1 when restricted to G × K and to K × G.

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Lemma 5. The group G has a natural unitary σ-representation U on the L2 sections of L defined by U (g)ψ(z) = φ(z, g)ψ(g −1 z). Proof. This follows immediately from Lemma 3.4.



This projective representation induces an action of G as automorphisms of B. Lemma 6. For any g ∈ G and k ∈ B we have U (g)π(k)U (g)−1 = π(g.k), where g.k(z, w) = k(g −1 z, g −1 w). Proof. By direct calculation and use of Lemma 3.2, Z −1 (U (g)π(k)U (g) ψ)(z) = φ(z, g)k(g −1 z, g −1 w)τ (g −1 z, g −1 w)φ(w, g)−1 ψ(g −1 w) dw Z = k(g −1 z, g −1 w)τ (z, w)ψ(g −1 w) dw, 

from which the result follows.

The second part of Lemma 3.2 can now be interpreted as saying that the parallel transport τ behaves covariantly under U (g), that is conjugation by U (g) sends τ (z, w) to τ (g −1 z, g −1 w). Taking w = exp(−tX)z and considering the limit as t → 0 we obtain the following result: Corollary 1. The σ-representation U and connection ∇ are related by U (g)∇U (g)−1 = g.∇, where g.∇ denotes the natural action of G on forms. 4. Various C ∗ -Algebras 4.1. The imprimitivity algebra. The σ-representation U defined in the previous section is clearly equivalent to one induced from a σ-representation, L, of the isotropy subgroup K. Such representations are characterised by the fact that they also admit an action of the imprimitivity algebra. In general this can be defined as one of Green’s twisted crossed product C∗ -algebras [Green], but in the case of a continuous multiplier σ there is a simpler direct construction. The imprimitivity algebra, A = A(G, K, σ), on G/K is a completion of the algebra A0 = Cc (G/K, G) with multiplication Z (α ∗ β)(s, g) = α(s, x)β(x−1 s, x−1 g)σ(x, x−1 g)−1 dx G

and involution

α∗ (s, g) = σ(g, g −1 )α(g −1 s, g −1 ).

(With the conventions of the last section σ(g, g −1 ) = 1 and could be omitted.) These formulae use the unimodularity of G and the existence of a G-invariant measure on G/K, otherwise some Radon-Nikodym derivatives would be needed. The algebra has a trace Z α(s, 1) ds. trA (α) = G/K

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More details may be found in [Green], where it is also shown that A(G, K, σ) is Morita equivalent to C ∗ (K, σ) ⊗ K(L2 (G/K)), where C ∗ (K, σ) denotes the twisted group C ∗ algebra. Like B the imprimitivity algebra is a groupoid algebra (being an extension by σ of the algebra of the transformation groupoid for G acting on X) and most of this paper could be understood in the context of groupoids, [Ren1]. The algebra B could have been derived from A as a quotient, as we shall now show. Let L be a σ-representation of K on a Hilbert space HL . (Since the multiplier of the last section is 1 whenever either of its arguments is in K, one could take L = 1, but the argument works more generally.) For each L the imprimitivity algebra has a natural ∗-representation on the induced representation space of HL -valued functions on G satisfying the equivariance condition ψ(gk) = σ(g, k)−1 L(k)∗ ψ(g) for all g ∈ G and k ∈ K. This representation is given by Z α(zK, x)σ(x, x−1 z)−1 ψ(x−1 z)dx. (α.ψ)(z) = G

(It may be checked that α.ψ satisfies the same equivariance condition as ψ.) The group G has an induced σ-representation on this function space given by U (g)ψ(z) = σ(g, g −1 z)−1 ψ(g −1 z). The imprimitivity algebra incorporates both this action and the multiplication operators, and so permits the description of quantum mechanical momentum and position operators on G/K. The group action allows for the free Hamiltonian 18 J.J, whilst the functions on X = G/K make it possible to add an extra potential, V . For an appropriate choice of L, U is equivalent to the representation in the last section. Indeed we may identify the equivariant functions on G with sections of the line bundle L and then we have, in the previous notation, Z α(zK, x)φ(zK, x)ψ(x−1 z) dx. (α.ψ)(z) = G

In this form we may easily see the connection to the algebra B of twisted kernels. Lemma 7. There is a ∗-homomorphism α 7→ Tα , defined by Z α(z, x)φ(z, x) δ(w, x−1 z) dx Tα (z, w) = τ (z, w) G from A onto B (where δ just restricts the integration to those x satisfying w = x−1 z). Proof. We calculate that (Tα ∗ Tβ )(z, w) Z $(z, v, w) dv = Tα (z, v)Tβ (v, w) τ (z, v)τ (v, w) Z φ(z, x)φ(v, x−1 y) δ(v, x−1 z)δ(w, y −1 xz) dx dy dv = α(z, x)β(v, x−1 y) τ (z, w) Z φ(z, x)φ(x−1 z, x−1 y) δ(v, x−1 z)δ(w, y −1 z) dx dy dv = α(z, x)β(x−1 z, x−1 y) τ (z, w)

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Z =

α(z, x)β(x−1 z, x−1 y)

Z =

(α ∗ β)(z, y)

φ(z, y) δ(v, x−1 z)δ(w, y −1 z) dx dy dv σ(x, x−1 y)τ (z, w)

φ(z, y) δ(w, y −1 z) dy = Tα∗β (z, w), τ (z, w)

which proves the homomorphism property. It follows similarly that it is a ∗-homomorphism. To see that it is surjective we note that if the Haar measure on K is normalised then α(z, x) = Tα (z, x−1 z)

τ (z, x−1 z) φ(z, x)

provides an inverse. (The key is to note that if g(z)u = z and g(w)u = w then w = x−1 z  forces x to have the form g(z)kg(w)−1 for some k ∈ K.) This may be interpreted as saying that B is a quotient of A, and this means that the behaviour of the two algebras is very similar. For this reason we shall often merely show the constructions in the case of one and leave it as an exercise to fill in the details for the other. However, it will be useful to note that B has a natural trace Z T (z, z) dz. trB (T ) = X

4.2. The regular representation. The trace gives rise to an inner product Z ∗ α(s, x)β(s, x) ds dx hα, βi = trA (α ∗ β) = G/K×G

on A0 , and completion with respect to this gives rise to a Hilbert space HA on which A0 is represented by left multiplication. Since the action is continuous this extends to give the left regular representation of A. This is given by the same formula as the algebra multiplication. That is, for 9 ∈ HA , one has Z α(s, x)ψ(x−1 s, x−1 g)σ(x, x−1 g)−1 dx. (α.ψ)(s, g) = G

There is also a σ-representation of G on HA , obtained by sending g ∈ G to the function (s, x) 7→ δg (x). For any unitary character χ of K, one may define a generalized regular representation χ of functions ψ ∈ Cc (G × G) which satisfy the condition on the space HA ψ(zk, g) = χ(k)ψ(z, g). Since |ψ(zk, g)|2 is independent of k, we may use the same inner product as before, and it is easy to check that the action Z α(zK, x)ψ(x−1 z, x−1 g)σ(x, x−1 g)−1 dx (α.ψ)(z, g) = G

respects the equivariance condition. When χ = 1 we obtain the regular representation. We could similarly define the regular representation of B and also a generalization defined by

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Z (T.9)(z, w) =

T (zK, v)9(v, w) dv X

on the space of kernels 9 on G × G which satisfy 9(zk1 , wk2 ) = χ(k2 )9(z, w). 4.3. The 0-invariant imprimitivity algebra. Let 0 be another subgroup of G, and let A0 denote the part of the imprimitivity algebra which commutes with the induced representation of 0. This is readily seen to consist of those functions α ∈ A(G, K, σ) which satisfy the condition γ.α = α, where (γ.α)(s, g) = σ(g, γ)σ(γ, γ −1 gγ)−1 α(γ −1 s, γ −1 gγ) for all γ ∈ 0. To see this we note that (δγ ∗ α)(s, g) = σ(γ, γ −1 g)α(γ −1 s, γ −1 g), which is identical to (γ.α ∗ δγ )(s, g) = σ(gγ −1 , γ)−1 γ.α(s, gγ −1 ) = σ(γ, γ −1 g)−1 α(γ −1 s, γ −1 g). When g = 1 the condition γ.α(s, g) = α(s, g) reduces to α(γ −1 s, 1) = α(s, 1) so that one obtains a trace Z trA0 (α) =

α(s, 1) ds. 0\G/K

(More generally, the 0-invariant functions are determined by their values at a single point s of each 0 orbit on G/K.) Using trA0 one may define a regular representation of A0 . In the case of the twisted kernel algebra one may likewise pick out a 0-invariant subalgebra B 0 , which commutes with U (γ) for all γ ∈ 0. Using Lemma 3.2 this condition reduces simply to the requirement that the kernel satisfies k(γ −1 z, γ −1 w) = k(z, w). The natural trace trB0 for this algebra is given by the same formula as before except that the integration is over a fundamental domain X0 rather than X: Z T (z, z)dz. τB0 (T ) = X0

4.4. Morita equivalence. Later we shall need some K-theory, and so it will be useful to show that the algebra B 0 is Morita equivalent to another more tractable algebra. We shall do this by using the groupoid equivalence arguments of [M+R+W], or rather the twisted version, [Ren2, Ren3]. We have already noted that B is an extension of the groupoid X × X by a cocycle defined by $, and 0 invariance of $ means that B0 is likewise the extension of X ×0 X by $, where X ×0 X denotes the groupoid obtained by factoring out the diagonal action of 0. More precisely, the groupoid elements are 0 orbits (x, y)0 = {(γx, γy) : γ ∈ 0}, and (x1 , y1 )0 and (x2 , y2 )0 are composable if and only if y1 = γx2 for some γ ∈ 0, and then the composition is (x1 , γy2 )0 . Theorem 1. The algebra B 0 is Morita equivalent to the twisted group algebra C ∗ (0, σ). Proof. This result will follow immediately from [Ren2], Corollaire 5.4 (cf. [M+R+W] Theorem 2.8) once we have established the groupoid equivalence in the following lemma. 

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Lemma 8. The line bundle L over X provides an equivalence (in the sense of [Ren2] Definition 5.3) between the groupoid extensions (X ×0 X)$ of X ×0 X defined by $ and 0σ of 0 defined by σ. Proof. Both extensions are by T. We write the elements of (X ×0 X)$ as triples (x, y, t) ∈ X × X × T with the first two elements representing a diagonal 0 orbit. Elements are composable if their first two components are composable, and, when y1 = γx2 , (x1 , y1 , t1 )(x2 , y2 , t2 ) = (x1 , γy2 , t1 t2 $(x1 , y1 , γy2 )). The line bundle can be trivialised and written as X × C. We let (X ×0 X)$ act on the left of the line bundle by defining (x, y, t) to act on (z, u) if z = γy for some γ ∈ 0, and then the result of the action is (γx, τ (γx, z)tu). (One may check that this gives an action using the relationship between parallel transport and holonomy and the 0-invariance of $.) The twisted groupoid 0σ has an underlying set 0 × T, all elements are composable, and multiplication is given by (β, s)(γ, t) = (βγ, σ(β, γ)st). It acts on the right of L by (z, u).(γ, t) = (γ −1 z, φ(z, γ)−1 tu). (The fact that this defines an action follows from the definition of σ in terms of φ.) We may now check that these actions commute, since, if z = βy, [(x, y, t)(z, u)](γ, s) = (βx, τ (βx, z)tu)(γ, s) = (γ −1 βx, φ(βx, γ)−1 τ (βx, z)tus), whilst

(x, y, t)[(z, u)(γ, s)] = (x, y, t)(γ −1 z, φ(z, γ)−1 us) = (γ −1 βx, τ (γ −1 βx, γ −1 z)φ(z, γ)−1 tus),

and the equality of these two follows from the second part of Lemma 3.2. Remarks. We will observe in Sect. 9 that the algebra C ∗ (0, σ) is isomorphic to C ∗ (0, σ) which in turn is known to be isomorphic to the imprimitivity algebra for σ-inducing from 0 to C ∗ (G, σ). This latter algebra, denoted C ∗ (G/0, G) is the completion of Cc (G/0, G) where the latter has a multiplication analogous to that described above for Cc (G/K, G) (simply replace K by 0 in the earlier discussion). Thus a corollary of our results in this subsection is that B 0 is Morita equivalent to C ∗ (G/0, G). Furthermore our discussion below of a Fredholm module for B 0 may be modified so as to produce a Fredholm module for C ∗ (G/0, G) whose character, for G = SO(n, 1), is also given by the area cocycle. We omit the details here as they would take us too far afield (see however the analogous discussion in the discrete case in [Co2]). 4.5. The Hamiltonian. We have asserted informally that the Hamiltonian can be accommodated within the algebras A0 and B 0 and we shall now provide the proof. We work with the smaller algebra B 0 , the results for A0 following similarly. Lemma 9. The Hamiltonian H = 21 ∇∗ .∇ commutes with the projective representation U.

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Proof. We recall from Corollary 3.6 that U (g)∇U (g)−1 = g.∇, so that U (g)HU (g)−1 =

1 (g.∇)∗ (g.∇), 2

and, since the Riemannian structure is invariant under the action of G, this is just H.  We could also obtain the same result by writing H = 18 J.J + 41 B 2 (for some constant B) and using the fact that the Casimir operator commutes with the representation, provided that we check that Jk are the representatives of the Lie algebra generators in the representation U . Using the invariance of H it is now not difficult to see the following result: Lemma 10. The Hamiltonian H is affiliated to the von Neumann algebra generated by the representation π of B. B0 .

Since H is G-invariant and so a fortiori also 0-invariant, it is sufficient to look at

Corollary 2. The Hamiltonian H is affiliated to the von Neumann algebra generated by the representation π of B 0 . We next observe that Br¨uning and Sunada have proved an estimate on the Schwartz kernel of exp(−tH) for t > 0, which implies that it is L1 in each variable separately. Since this kernel is 0-invariant (by Lemma 7.1) it follows (in exactly the same fashion as Lemma 4 of [BrSu]) that this estimate implies that exp(−tH) is actually in the algebra B0 . Lemma 11. The operator e−tH is an element of B 0 . Corollary 3. The spectral projections of H corresponding to gaps in the spectrum lie in B 0 . Proof. If µ lies in a gap of the spectrum of H then let f be a continuous approximate step function which is identically one on the part of the spectrum of H contained in [0, µ] and zero on the part contained in [µ, ∞). Define g(x) = f ◦ ln(1/x) for x ∈ [0, 1]. Then g is a bounded continuous function which, when applied to e−tH gives the spectral projection corresponding to the interval [0, µ].  Finally we consider the interacting Hamiltonian H + V , where V is a 0-invariant function on X. Notice that if ψ is a continuous function of compact support on (X×X)/0 then V ψ is also such a function and hence defines an element in the groupoid algebra. Now, by Lemma 11, the resolvent of H lies in B 0 and by writing (z − H − V )−1 = (1 − (z − H)−1 V )−1 (z − H)−1 , and expanding (1 − (z − H)−1 V )−1 in a power series, we see that the resolvent of H + V is in the algebra B 0 . This entails, by a simple modification of Corollary 3, that the spectral projections of H + V corresponding to a gap in the spectrum also lie in B 0 .

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5. The Discrete Model In this section we formulate a version of the integer quantum Hall effect on a graph in hyperbolic space. The discussion uses a construction due to Sunada [Sun] together with a modification of Connes’ work on Fredholm modules for the group C ∗ -algebra of a discrete subgroup of a Lie group [Co2]. The graph is obtained by taking our group 0, the fundamental group of a Riemann surface, which acts freely on hyperbolic space, fixing a base point u and taking the orbit through u under the 0 action. This gives the vertices of the graph. The edges of the graph are geodesics constructed as follows. Each element of the group may be written as a word of minimal length in the 2g generators and their inverses. Each generator and its inverse determines a unique geodesic emanating from a vertex x and these form the edges of the graph. Thus each word x in the generators determines a piecewise geodesic path from u to x. Sunada constructs a Hamiltonian on `2 (0.u) which is a generalized Harper operator. This construction, specialised to our case, has the following form. First we note that we may trivialise the restriction of the line bundle to the vertices and so without loss of generality the appropriate Hilbert space becomes `2 (0.u). While the construction works for any connection 1-form A on a line bundle L over hyperbolic space we make the formulae explicit by restricting to the case where A is the one form η. For each R directed edge e of the graph joining o(e) to t(e) we define a function τ (e) = exp(i e η). Then τ (e) satisfies  co(e) + d θ  ct(e) + d θ , τ (γ.e) = τ (e) co(e) + d ct(e) + d   a b where γ = .We introduce the notation c d  cw¯ + d θ . sγ (w) = cw + d Remarks 2. Note that in our earlier notation these definitions amount to τ (e) ≡ τ (t(e), o(e)), while the function from 0 × H to U (1) given by (γ, w) 7→ sγ (w) is a projective 1cocycle for the 0 action on H which is cohomologous to the function mapping (γ, w) 7→ φ(γ.w, γ). Consequently there is a projective action of 0 on `2 (0.u) given on f ∈ `2 (0.u) by ρ˜γ f (w) = sγ (γ −1 w)f (γ −1 w). We have ρ˜γ1 γ2 =

sγ1 γ2 (γ2−1 γ1−1 x) ρ˜γ1 ρ˜γ2 , sγ1 (γ1−1 x)sγ2 (γ2−1 γ1−1 x)

and Sunada shows that the function multiplying ρ˜γ1 ρ˜γ2 is independent of x. This is exactly the relation found at the end of Lemma 3: the correspondence is given by φ(z, ∗) ↔ an explicit expression for this function, let γ3 = γ1 γ2 , and write s∗ (∗−1z). To obtain  ai bi (γi ) = for i = 1, 2, 3. Then a direct calculation shows that c i di ρ˜γ1 γ2 = σ(γ1 , γ2 )ρ˜γ1 ρ˜γ2 ,

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where σ is given by σ(γ1 , γ2 ) = exp(2iθ(arg (c3 w + d3 ) − arg (c1 (γ2 w) + d1 ) − arg (c2 w¯ + d2 ))). It is not difficult to see that the right hand side is independent of the choice of w. Following [Sun], we define our discrete Hamiltonian, for f ∈ `2 (0.u), by X hτ f (w) = τ (e)f (t(e)). e o(e)=w

Then hτ is a generalized difference operator (Sunada shows that the Harper operator arises in a similar fashion). One can verify by direct calculation that hτ commutes with the projective action of 0. This will, however, become readily apparent when we transfer this construction to `2 (0). Define ι : `2 (0.u) → `2 (0) by: ι(f )(γ) = ρ(γ)f ˜ (u). Observe that ι(ρ(µ)f ˜ )(γ) = σ(γ, µ)ι(f )(γµ). Thus ι intertwines ρ˜ with the σ-representation: ¯ ρ(µ)ι(f )(γ) = σ(γ, µ)ι(f )(γµ). Henceforth we use ι to identify `2 (0.u) with `2 (0). Proposition 1. [Sun] The operator hτ on `2 (0.u) maps to the operator Hτ on `2 (0) under ι, where X σ(µ, µ−1 γ)−1 a(µ)f (µ−1 γ) Hτ f (γ) = µ∈0

and a is the function on 0 given by a(γ) = sγ (u)

X

τ (e).

e o(e)=u t(e)=γ.u

Corollary 4. The bounded self-adjoint operator Hτ is in the algebraic twisted group algebra C(0, σ) (the elements of finite support in C ∗ (0, σ)) as the function a has finite support. Moreover Hτ acts on the left as an element of this algebra and so commutes with the σ-representation ¯ ρ, as the latter acts on the right. 6. A Kubo Formula 6.1. Conductivity cocycles. In this subsection we present an argument which derives analogues of the Kubo formula for the hyperbolic “Hall conductivity”. In Subsection 7.2 we show how to construct, from the results of this subsection, a unique cocycle which may be compared with the character of the Fredholm module of Sect. 7. Our reasoning here is that the Hall conductivity in the Euclidean situation is measured experimentally by determining the equilibrium ratio of the current in the direction of the applied electric field to the Hall voltage, which is the potential difference in the orthogonal direction. To calculate this mathematically we instead determine the component of the induced current that is orthogonal to the applied potential. The conductivity

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can then be obtained by dividing this quantity by the magnitude of the applied field. In the hyperbolic case it would seem at first sight that there are no preferred directions. However interpreting the generators of the fundamental group as geodesics on hyperbolic space gives a family of preferred directions emanating from the base point. For each pair of directions it is therefore natural to imitate the procedure of the Euclidean case and mathematically this is done as follows. The Hamiltonian H in a magnetic field depends on the magnetic vector potential A and the functional derivative δk H of H with respect to the components of A, denoted Ak , gives the current density Jk . (For simplicity we take variations within a one-parameter family.) The expected value of the current in a state described by a projection operator P is therefore tr(P δk H) (cf [Av+S+Y ] Eq. (3.2)). The following claim is not proved by a rigorous argument: one needs to check various analytical details as in [Xia]. We have refrained from doing so here as this would take us too far afield from the main point, namely, obtaining a sensible hyperbolic Kubo formula which may be compared with the character of the Fredholm module constructed in the next section. For the moment tr will denote a generic trace. We will become specific after extracting a rigorous definition of the Kubo formula. Claim. tr(P δk H) = itr(P [∂t P, δk P ]). Plausibility argument. By using the invariance of the trace under the adjoint action of operators and the equation of motion we see that tr(P [∂t P, δk P ]) = −tr([P, δk P ]∂t P ) = −itr([P, δk P ][P, H]) = itr([P, [P, δk P ]]H). Now δk P = δk (P 2 ) = P (δk P ) + (δk P )P , whence P (δk P )P = 0 and we have [P, [P, δk P ]] = P (P (δk P ) − (δk P )P ) − (P (δk P ) − (δk P )P )P = P (δk P ) + (δk P )P = δk P. Consequently we may write tr(P [∂t P, δk P ]) = itr((δk P )H) = itr(δk (P H)) − itr(P (δk H)), and, assuming that the trace is invariant under variation of Ak , the first term vanishes leaving the result asserted. If the only t-dependence in H and P is due to the variation of Aj , a component distinct from Ak , then ∂t = ∂Aj /∂t × δj . Working in the Landau gauge so that the electrostatic potential vanishes, the electric field is given by E = −∂A/∂t, and so ∂t = −Ej δj . Combining this with the previous argument we arrive at the following result: Corollary 5. The conductivity for currents in the k direction induced by electric fields in the j direction is given by −itr(P [δj P, δk P ]). Proof. The expectation of the current Jk is given by tr(P δk H) = itr(P [∂t P, δk P ]) = −iEj tr(P [δj P, δk P ]), from which the result follows immediately.



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6.2. The derivations on a Riemann surface. On a Riemann surface it is natural to investigate changes in the potential corresponding to adding multiples of the real and imaginary parts of holomorphic 1-forms. (For the genus 1 torus with imaginary period this amounts to choosing forms whose integral round one sort of cycle vanishes but the integral round the other cycle is non-trivial. This corresponds to putting a non-trivial voltage across one cycle and measuring a current round the other.) Let Σg = H/0 be the Riemann surface determined by quotienting by 0. We follow the usual conventions (see for example [GH]) in fixing representative homology generators corresponding to cycles Aj , Bj , j = 1, 2, . . . , g with each pair Aj , Bj intersecting in a common base point and all other intersection numbers being zero. We let aj , j = 1, 2, . . . , 2g be harmonic 1-forms dual to this homology basis (this means that aj , j = 1, . . . , g are dual to Aj , j = 1, . . . , g and aj+g , j = 1, 2, . . . , g are dual to Bj , j = 1, 2, . . . , g). Z

Definition. Let δj τ (z, w) = i

δj $(u, g

u, g

−1

αj τ (z, w)

w

Z

and −1

z

z) = i ∂1

aj $(u, g −1 u, g −1 z)

where 1 is a triangle with vertices at the three arguments of $. One then calculates that Z δj φ(z, g) = i

Z ∂1

Z

aj −

u

aj +

u

aj +

Z

g −1 u

=i

Z

z

Z

This can also be written as

Z

i ∂Q

u

aj − i

aj Z

g −1 z g −1 u

!

g −1 z

aj −

φ(z, g) !

z u

aj

φ(z, g).

z g −1 z

aj ,

−1 −1 where Q denotes the geodesic quadrilateral with vertices R at u, g u, g z and z. By Stokes’ Theorem the first integral can also be written as Q daj , and this vanishes as we chose aj to be the harmonic representative of its class, leaving just Z z aj . −i g −1 z

Using this last equation in the formula for the action of the imprimitivity algebra (preceding Lemma 7) we see that we have a densely defined derivation on the algebras A and B because the action can R z now be written as the commutator of α with multiplication by the function j (z) = i u aj : δj α = [j , α]. Suppose that α is a kernel decaying rapidly. By this we mean that it satisfies an estimate |α(x, y)| ≤ ϕ(d(x, y)),

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where ϕ is a positive and rapidly decreasing function on R. We claim that δj α lies in A or B respectively: this follows by noting that the map from H to R2g given by Ξ : z 7→ (1 (z), . . . , 2g (z)) is the lift to H of the Jacobi map, [GH] (this map is usually regarded as mapping from Σg to the Jacobi variety J(Σg ), however we are thinking of it as a map between the universal covers of these spaces). Now Ξ is globally Lipschitz and this means we may estimate the kernel of [j , α]. A simple argument shows that it also decays rapidly. That this commutator also has the correct properties to define a derivation on A0 or 0 B follows from the fact that for γ ∈ 0, j (γ.z) − j (z) is constant independent of z so that 0 equivariance or invariance is preserved. In the case of the torus thought of as a rectangle in R2 with opposite edges identified, one may take aj + iak = dz, and then (δj + iδk )α = i[z − u, α]. Thus our argument reproduces the standard Kubo formula [Xia] in the Euclidean case. Notice that our map Ξ from H to R2g gives the period lattice in R2g (that is the lattice determined by the periods of the harmonic forms aj ) to be the standard integer lattice Z2g so that J(Σg ) = R2g /Z2g . We may summarize the previous discussion as Lemma 12. For operators A0 , A1 , A2 in B 0 whose integral kernels are rapidly decaying we have cyclic cocycles defined by cj,k (A0 , A1 , A2 ) = tr B0 (A0 [δj A1 , δk A2 ]) = tr B0 (A0 [j , A1 ][k , A2 ]) for j, k = 1, . . . , 2g. Each of these formulae for cjk could in principle be regarded as giving a Kubo formula so that we appear to have an embarrassment of riches. However each on their own cannot be related to the Chern character of the Fredholm module of the next section. A clue as to what is happening is provided by noting that each two form aj ∧ aj+g is harmonic and hence is a multiple of the area two form on Σg , thus there is certainly some degeneracy here and we resolve it at the end of the next section. 7. A Fredholm Module We shall now assume that X has a spin structure, and we write S for the spin bundle. χ ⊗ S as in Sect. The representation of A0 can then be extended to an action on HA 4.2. This module can be equipped with Fredholm structure by taking F to be Clifford multiplication by a suitable unit vector (to be explained below), and using the product of χ and the graded trace on the Clifford algebra. (If ε denotes the grading the trace on HA operator on the spinors then the graded trace is just the composition tr ◦ ε.) χ ⊗ S+ ⊕ The same module can also be described more explicitly: it splits into HA χ − HA ⊗ S (with the superscripted sign indicating the eigenvalue of ε), and this may be θ1 θ2 written as HA ⊕ HA . The involution F is then a matrix multiplication operator of the form   0 f1 , F = f2 0

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with (fj .ψ)(z, g) = fj (z, g)ψ(z, g) for some suitable functions fj ∈ Cc (G × G), satθ1 θ2 , f1 .ψ ∈ HA . isfying f1 = f2−1 . For consistency, we require that for any ψ ∈ HA Since (f1 .ψ)(zk, g) = f1 (zk, g)ψ(zk, g) = θ1 (k)f1 (zk, g)ψ(z, g), we demand that θ2 (k)f1 (z, g) = θ1 (k)f1 (zk, g), or f1 (zk, g) = (θ1−1 θ2 )(k)f1 (z, g), and θ1−1 θ2 is known directly from the structure of S. (When X is the hyperbolic plane it is the complex character describing the action of K on the complex tangent space to X at u.) A short calculation shows that Z
α(zK, x)ψ(x−1 z, x−1 g) dx. fj (z, g) − fj (x−1 z, x−1 g) ([fj , α].ψ)(z, g) = σ(x, x−1 g) G We observe in the next subsection that this module is 2-summable at least for kernels which decay sufficiently rapidly. Assuming this fact then it follows that ((ω ∗ [fj , α] ∗ [fk , β]).ψ)(z, g) is given by Z

fj (x−1 z, x−1 g) − fj (y −1 z, y −1 g) fk (y −1 z, y −1 g) − fk (u−1 z, u−1 g) G×G×G

ω(zK, x)α(x−1 zK, x−1 y)β(y −1 zK, y −1 u) ψ(u−1 z, u−1 g) dx dy du. σ(x, x−1 g)σ(x−1 y, y −1 g)σ(y −1 u, u−1 g) From this (and using tr to denote the usual trace on operators on our module) we can calculate the cyclic cocycle on A as τc (ω, α, β) = tr [ω ∗ [F, α] ∗ [F, β]] = tr [ω ∗ ([f1 , α] ∗ [f2 , β] − [f2 , α] ∗ [f1 , β])] , which can be expressed as Z ω(zK, x)α(x−1 zK, x−1 y)β(y −1 zK, y −1 ) dzK dx dy, 8(zK, x, y) σ(x, x−1 y)σ(y, y −1 ) G/K×G×G where

Z

8(zK, x, y) =




G

f1 (x−1 z, x−1 g) − f1 (y −1 z, y −1 g)


− f2 (x−1 z, x−1 g) − f2 (y −1 z, y −1 g)




f2 (y −1 z, y −1 g) − f2 (z, g)


f1 (y −1 z, y −1 g) − f1 (z, g) dg.

(Using the equivariance of fj it is easy to check that this depends on z only through zK.) Simplifying and using f1 f2 = 1, the integrand reduces to
f1 (z, g)f2 (x−1 z, x−1 g) + f1 (x−1 z, x−1 g)f2 (y −1 z, y −1 g) + f1 (y −1 z, y −1 g)f2 (z, g)
− f2 (z, g)f1 (x−1 z, x−1 g)+f2 (x−1 z, x−1 g)f1 (y −1 z, y −1 g) + f2 (y −1 z, y −1 g)f1 (z, g) . This can also be written more compactly as 1 1 1 f1 (z, g) f1 (x−1 z, x−1 g) f1 (y −1 z, y −1 g) , f2 (z, g) f2 (x−1 z, x−1 g) f2 (y −1 z, y −1 g)

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or as

(1 − f1 (x−1 z, x−1 g)f2 (z, g))(1 − f1 (y −1 z, y −1 g)f2 (x−1 z, x−1 g)) (1 − f1 (z, g)f2 (y −1 z, y −1 g)),

which also arises naturally from an alternative expression for the cocycle. Suppose that ϕ is a U (1) valued function on the group, which satisfies ϕ(kgh) = χ1 (k)ϕ(g)χ2 (h) for k and h in K and some σ-characters χ1 and χ2 of K. If χ−1 1 χ2 = θ1−1 θ2 , we may take f1 (z, g) = ϕ(z −1 gz) to obtain a function satisfying our earlier consistency condition. In the case of G = SU(1, 1) and K the diagonal subgroup, we may take the function ϕ used by Connes [Co2], which is essentially the Mishchenko element. With the group elements all conjugated by z it now follows as in [Co] that 8(zK, x, y)/4πi is the area of the hyperbolic geodesic triangle with vertices u, z −1 yz.u and z −1 xz.u. Acting with z and recalling that, since u is stabilised by K, z.u can be identified with s = zK, 8(zK, x, y)/4πi is also the area of the geodesic triangle with vertices s, y.s and x.s. In the next subsection we will see that the module is 2-summable for suitably decaying kernels. Since fj (γ −1 z, γ −1 gγ) = ϕ(z −1 γγ −1 gγγ −1 z) = fj (z, g), F preserves the 0-invariant subspace, so that there is a similar expression for a cyclic cocycle τc,0 in that case, except that s is integrated only over the 0 orbits in G/K. More precisely, using tr 0 to denote this restricted range of integration, one has: Theorem 3. There is a 2-summable Fredholm module (F, Hτχ ⊗ S) over a dense subalgebra A00 of A0 , stable under the holomorphic functional calculus, whose Chern character is given by the area cocycle on H. That is, in the notation above, one has τc,0 (ω, α, β) = −tr 0 [ω ∗ [F, α] ∗ [F, β]] = −tr 0 [ω ∗ ([f1 , α] ∗ [f2 , β] − [f2 , α] ∗ [f1 , β])] , which can be expressed as Z ω(zK, x)α(x−1 zK, x−1 y)β(y −1 zK, y −1 ) − dzK dx dy, 8(0zK, x, y) σ(x, x−1 y)σ(y, y −1 ) 0\G/K×G×G where 8 is given as above. Therefore by the index pairing in [Co2], one has index(P F P ) = h[τc,0 ], [P ]i, where P denotes a projection in A00 and index(P F P ) denotes the index of the Fredholm operator P F P acting on the Hilbert space P Hτχ ⊗ S. We will prove Theorem 3 in the next subsection. The version of Theorem 3 which applies to B 0 is as follows: Theorem 4. There is a dense subalgebra B00 of B 0 stable under the holomorphic functional calculus and a 2-summable Fredholm module (F, Hτχ ⊗ S) for B00 with corresponding cyclic 2-cocycle τc,0 (Tω , Tα , Tβ ) Z 8(z, x, y)$(z, x, y)Tω (z, x)Tα (x, y)Tβ (y, z) dz dx dy. =− X0 ×X×X

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The character of this Fredholm module, for P a projection in B00 , is given in the notation of Corollary 12 of Sect. 10, by index(P F P ) = 2(g − 1)(rank E 0 − rank E 1 ) ∈ Z, where index(P F P ) again denotes the index of the Fredholm operator P F P acting on the Hilbert space P Hτχ ⊗ S. This theorem can be interpreted as an index theorem equating an analytic index with a topological index. Theorem 4 may be used to obtain the following result: Corollary 6. Let P be a projection into a gap in the spectrum of the Hamiltonian Hη,V . Then P lies in a 2-summable dense subalgebra B00 of B 0 , so that in the notation of Corollary 12 of Sect. 10 one has index(P F P ) = hτc,0 , [P ]i = 2(g − 1)(rank E 0 − rank E 1 ) ∈ Z. The statements referring to Sect. 10 will be clear after we establish there the hyperbolic analogues of Xia’s results [Xia]. The proof of the claim that the spectral projections (corresponding to gaps in the spectrum) of the Hamiltonian lie in B00 and the proof of Theorem 3 are contained in the next subsection. 7.1. Proof of summability of the Fredholm module. Here we discuss the technicalities needed for the proof of Theorem 3 and of Corollary 6. It is easy to calculate from the formulae in the previous section that 2-summability requires finiteness of the expression tr 0 ([f1 , α]∗ [f2 , α]), which equals Z [f1 (z, g) − f1 (x−1 z, x−1 g)] X0 ×G×G

[f2 (z, g) − f2 (x−1 z, x−1 g)]|α(x−1 z, x−1 )|2 dz dx dg, where z ∈ X0 , a fundamental domain in H for the 0 action. Letting p : G/K → G be a cross section and using the formulae for f1 and f2 this reduces to Z |ϕ(p(z)−1 gz) − ϕ(p(z)−1 gx−1 z)|2 |α(x−1 z, x−1 )|2 dz dx dg. X0 ×G×G

We are more interested in the algebra of twisted kernels as in Theorem 4, so we will present the argument for them noting that the relation for α in terms of Tα as given in the proof of Lemma 7 gives, by the the unitarity of τ and ϕ, |α(x−1 z, x−1 )|2 = |Tα (x−1 z, z)|2 . Thus the summability result for B 0 implies that for A0 . Making this substitution we then get for our integral Z |ϕ(p(z)−1 gz) − ϕ(p(z)−1 gx−1 z)|2 |Tα (x−1 z, z)|2 dz dx dg. X0 ×G×G

Finally we note that setting p(z)−1 g = v −1 and x−1 z = w it is clear that |ϕ(v −1 z) − ϕ(v −1 w)| depends only on the cosets vK and wK, allowing us to reduce the integral to

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Z X0 ×X×X

|ϕ(v −1 z) − ϕ(v −1 w)|2 |Tα (w, z)|2 dz dwK dvK.

Only the first factor depends on v. Write v = γv0 for γ ∈ 0 and v0 ∈ X0 . Then we obtain for our integral after a change of variables: XZ |ϕ(v0−1 z) − ϕ(v0−1 w)|2 |Tα (γw, γz)|2 dz dwK dv0 K, γ∈0

γ.X0 ×X×X0

so that by the γ invariance of the kernel Tα one obtains Z |ϕ(v0−1 z) − ϕ(v0−1 w)|2 |Tα (w, z)|2 dz dwK dv0 K. X×X×X0

By a further change of variable we obtain Z |ϕ(z) − ϕ(w)|2 |Tα (v0−1 w, v0−1 z)|2 dz dwK dv0 K. X×X×X0

Notice that, by Lemma 6, Tα (v0−1 w, v0−1 z) is the integral kernel for the operator obtained by conjugating by U (v0 ). It follows therefore that finiteness of the triple integral is guaranteed by the convergence of Z |ϕ(z) − ϕ(w)|2 |Tα (w, z)|2 dz dwK. X×X

To avoid repetition let us first focus on the case of greatest interest, where we consider the integral kernels of a spectral projection P of the Hamiltonian H +V corresponding to a gap in the spectrum. As it is obtained from the Hamiltonian using the smooth functional calculus from a function of compact support we can obtain a growth estimate on the integral kernel (see below) which will ensure convergence. To lighten the notation we let z, w ∈ X and k(z, w) denote the integral kernel as a function on X × X. Taking χ1 = 1 so that ψ(gk) = ϕ(g) is well defined, the discussion of the previous paragraph leads us to consider whether ZZ (∗) |(ψ(z) − ψ(w))k(z, w)|2 dz dw is finite. Let X0 be a fundamental domain for the diagonal action of 0 on X × X. Then the previous integral, for 0 invariant kernels, is given by XZ Z |(ψ(γ.z) − ψ(γ.w))k(z, w)|2 dz dw. γ∈0

X0

By an argument due to Connes [Co], we have the estimate |ψ(γ.z) − ψ(γ.w)|2 ≤ C exp(−2d(u, γ.z) + C1 d(γ.z, γ.w)). (Here u is the base point in X, d denotes the hyperbolic metric and C,C1 are constants.) We claim that, in addition, the following estimate holds: |k(z, w)|2 ≤ C2 exp(−C3 d(z, w)2 ),

(∗∗)

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where C2 , C3 are constants. This fact goes back to [CGT] although in the form we need it here, for the Hamiltonians H + V of Sect. 4, it can be deduced from [BrSu]. This is because [BrSu] prove (**) when k is the kernel of the heat operator e−(H+V ) . Now by the argument of Corollary 3 there is a smooth function of compact support g such that g(e−(H+V ) ) is the spectral projection P . To prove that the kernel of g(e−(H+V ) ) satisfies (**) it suffices to observe that we can approximate g uniformly by polynomials without constant term, so that the kernel of g(e−(H+V ) ) has the same off-diagonal decay estimate as the kernel of e−(H+V ) , namely (**). Hence the integral in (*) above is smaller than XZ Z C4 exp(−C3 d(z, w)2 + C1 d(z, w) − 2d(u, γ.z)) (∗ ∗ ∗) γ∈0

X0

for suitable constants Cj , j = 1, 2, 3. As the area in hyperbolic space grows like exp(d(u, z)), convergence of the infinite sum in (***) is handled by the convergence of P the Poincare series γ∈0 exp(−2d(u, γ.z)). The convergence of the integral in (***), over the fundamental domain, is handled by the exponential factor involving the square of the hyperbolic distance and noting that the integration in the diagonal direction in X0 is over a finite range. (It is also possible to prove 2-summability in the case when 0 is trivial by exploiting the fact that in that case one may use kernels with restrictions on their support.) Since operators with kernels which have support in a band around the diagonal are dense in the algebras A0 and B 0 so too is the set of operators with kernels satisfying (**). Now the finiteness of (*) is equivalent to asserting that [F, Tα ] is Hilbert-Schmidt. Definition. We denote by B00 the subalgebra consisting of operators A ∈ B 0 , with [F, A] a Hilbert-Schmidt operator. The argument of the previous paragraph shows that B00 is dense. Now by [Co] B00 is stable under the holomorphic functional calculus. A similar remark handles the existence of the analogous dense subalgebra A00 of A0 . This completes the proof of Theorem 3 and the claim concerning the spectral projections of the Hamiltonian as we promised. 7.2. The hyperbolic Connes-Kubo formula. Our aim in this subsection is to give a geometric interpretation to the cocycles defined in lemma 12 and to prove that a suitable linear combination of them is cohomologous to the cocycle τc,0 arising from the Fredholm module (F, Hτχ ⊗ S). To do this we begin by introducing, for operators A0 , A1 , A2 in B 0 whose kernels k0 , k1 , k2 are exponentially decaying (cf Eq. (**) of the previous subsection), the cyclic cocycle cK defined by cK (A0 , A1 , A2 ) =

=

cj,j+g (A0 , A1 , A2 )

j=1

g Z X j=1

g X

X0 ×X×X

$(z, x, y)9j (z, x, y)k0 (z, x)k1 (x, y)k2 (y, z) dz dx dy,

where 9j (z, x, y) = (j (x)−j (y))(j+g (y)−j+g (z))−(j+g (x)−j+g (y))(j (y)−j (z)). Pg We claim that j=1 9j (z, x, y) is proportional to the “symplectic area” of a triangle in R2g with vertices Ξ(x), Ξ(y), Ξ(z).

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To prove this it suffices to assume that the origin is one of the vertices of the triangle, so suppose z is the base point in H. Then we need to consider the expression g X

9j (z, x, y) =

j=1

g X

(j (x)j+g (y) − j+g (x)j (y)).

j=1

Let s denote the symplectic form on R2g given by: s(u, v) =

g X

(uj vj+g − uj+g vj ).

j=1

The so-called “symplectic area” of a triangle with vertices 0, Ξ(x), Ξ(y) may be seen to be 21 s(Ξ(x), Ξ(y)). To appreciate this, however, we need to utilise an argument from [GH] (pp 333-336). In terms of the standard basis of R2g (given in this case by vertices in the integer period lattice arising from our choice of basis of harmonic one forms) and corresponding coordinates u1 , u2 , . . . u2g the form s is the two form on R2g given by ωJ =

g X

duj ∧ duj+g .

j=1

Now the “symplectic area” of a triangle in R2g with vertices 0, Ξ(x), Ξ(y) is given by integrating ωJ over the triangle and a brief calculation reveals that this yields s(Ξ(x), Ξ(y))/2, proving our claim. The previous argument establishes the following result. Proposition 2. The higher genus analogue of the Kubo formula is given by the cyclic cocycle τK on B 0 defined by τK (A0 , A1 , A2 ) =

=

g X

cj,j+g (A0 , A1 , A2 ) j=1 g Z X

9j (z, x, y)$(z, x, y)k0 (z, x)k1 (x, y)k2 (y, z) dz dx dy.

j=1

X0 ×X×X

are the kernels of the Aj , j = 0, 1, 2 (three exponentially decaying elements Here the kj P g of B 0 ) and j=1 9j (z, x, y) is proportional to the “symplectic area” of the Euclidean triangle 1E in R2g with vertices Ξ(x), Ξ(y), Ξ(z). To compare the cocycle τK with the cocycle τc,0 arising from our Fredholm module we note that the pull back form Ξ ∗ (duj ) is dual to the homology cycle Aj for j = 1, . . . , g ∗ and dual to Bj−g for j = g + 1, . . . , 2g (cf. [GH]). Pg Thus Ξ (duj ) differs from aj by an ∗ exact one form. Hence Ξ (ωJ ) differs from j=1 aj ∧ aj+g by an exact two form. But each term aj ∧ aj+g is harmonic and hence proportional to the two form ωH on H. So we have for some constant κ, and geodesic triangle 1 ⊂ H, Z Z Z ωH = κ Ξ ∗ (ωJ ) = κ ωJ . 1

1

Ξ(1)

Actually a calculation reveals that one can do a little better than this and proves that

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κΞ ∗ (ωJ ) = ωH . Now Ξ cannot map geodesic triangles to Euclidean triangles in R2g as Ξ(1) is a compact subset of a non-flat embedded two dimensional surface in R2g . Moreover as 9j (z, x, y) = 0 whenever the images of z, x, y under Ξ lie in a Lagrangian subspace (with respect to the symplectic form s) of R2g , τK and τc,0 are not obviously proportional. After suitable normalisation we will, however, prove they cohomologous. First Pare g renormalise ωJ so that Ξ ∗ (ωJ ) = ωH and then normalise j=1 9j (z, x, y) so that it R equals −4πi 1E ωJ . Next we write ωJ = dθ. Considering the difference τK − τc,0 one sees that the key is to understand Z Z Z Z ωJ − ωJ = θ − θ. Ξ(1)

1E

∂Ξ(1)

∂1E

Now this difference of integrals around the boundary can be written as the sum of three terms corresponding to splitting the boundaries ∂Ξ(1) and ∂1E into three arc segments each. We introduce some notation for this, writing ∂Ξ(1) = Ξ(`(x, y)) ∪ Ξ(`(y, z)) ∪ Ξ(`(z, x)), where `(x, y) is the geodesic in H joining x and y (with the obvious similar definition of the other terms). We also write ∂1E = m(x, y) ∪ m(y, z) ∪ m(z, x), where m(x, y) is the straight line joining Ξ(x) and Ξ(y) (and again the obvious definition of the other terms). Then we have Z Z θ − θ = h(x, y) + h(y, z) + h(z, x), R

∂Ξ(1)

R

∂1E

where h(x, y) = Ξ(`(x,y)) θ − m(x,y) θ, with similar definitions for h(y, z) and h(z, x). R Notice that we can write h(x, y) = Dxy ωJ , where Dxy is a disc with boundary m(x, y) ∪ `(x, y). From this it is easy to see that h(γx, γy) = h(x, y) for γ ∈ 0. Introduce the bilinear functional τ1 on B 0 given by Z h(x, y)k0 (x, y)k1 (y, x) dx dy = −4πitr B0 (Ah A1 ), τ1 (A0 , A1 ) = −4πi X0 ×X

where, if Aj has kernel kj (x, y), for j = 0, 1, then Ah is the operator with kernel h(x, y)k0 (x, y). Of course this definition begs the question of whether the trace is finite. In order to prove that τ1 is densely defined we start with some preliminary observations. By [M+R+W] and [Ren2] there is an isomorphism 8F : B 0 ∼ = Cr∗ (0g , σ) ⊗ K(L2 (F )). Here F denotes a fundamental domain for the action of 0g on H. (Note that by the Packer-Raeburn stabilization theorem, one has Cr∗ (0g , σ) ⊗ K ∼ = K or 0g .) Now any element x in Cr∗ (0g , σ) ⊗ K can be written as a matrix (xij ), where xij ∈ Cr∗ (0g , σ). So we can define  21 X ν(xij )2 , Nk (x) = i,j

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where ν(xij ) =

X

(1 + `(h)2k )|x(h)|2

 21

,

h∈0g

and ` denotes the word length function on the group 0g . By a mild modification of the 0 of B 0 argument given in [Co2], III.5.γ, one can prove that there is a subalgebra B∞ which contains C(0g , σ) ⊗ R, where R denotes the algebra of smoothing operators on F , is stable under the holomorphic functional calculus, and is such that Nk (x) < ∞ for 0 and k ∈ N. Then one shows as in [Co2] that the trace τ ⊗Tr on C(0g , σ)⊗R, all x ∈ B∞ is continuous for the norm Nk , for k sufficiently large, and thus extends by continuity to 0 0 . Note that elements in B∞ have Schwartz kernels which have rapid decay away from B∞ 0 would be to use the algebra the diagonal. An alternate equivalent construction of B∞ Ag,σ as in Sect. 10, and the results of [Ji]. Summarizing this, we have 0 is dense in B 0 , is closed under the holomorphic funcProposition 3. The algebra B∞ tional calculus and is contained in the ideal I of B 0 consisting of operators with finite trace. 0 while τc,0 is defined on B00 as we noted earlier. Both of these Now τK is defined on B∞ algebras contain the operators whose Schwartz kernels are supported in a band around 0 ∩ B00 is dense and stable under the holomorphic the diagonal. Thus the subalgebra B∞ functional calculus. If b denotes the Hochschild coboundary map then a straightforward calculation reveals that bτ1 = τK − τc,0 . The Lipschitz property of the Jacobi map means that h(x, y) grows at worst like the square of the hyperbolic distance from x to y so that 0 0 then so too does Ah . Hence we have τ1 defined on B∞ ∩ B00 , proving the if A0 ∈ B∞ following theorem.

Theorem 5. The Kubo cocycle τK and the Chern character cocycle τc,0 arising as the Chern class of the Fredholm module (F, Hτχ ⊗ S) are cohomologous as cyclic cocycles 0 ∩ B00 . on B∞ This theorem replaces the Connes-Kubo formula in genus one. The latter formula states that the two cocycles of the theorem are equal. We see that the situation is more complex for genus g but from the viewpoint of K-theory as described in the next section this theorem is enough to give integrality of the Hall conductivity defined either from the character of our Fredholm module or from the hyperbolic Kubo formula. 8. K-Theory Aspects In this section, we compute the K-groups of the twisted group C ∗ -algebras which are relevant to the quantum Hall effect on the hyperbolic plane as a special case of more general theorems about the K-groups of the twisted group C ∗ -algebras of groups 0 which are uniform lattices in K-amenable Lie groups. We recall that any solvable Lie group, and in fact any amenable Lie group is Kamenable. However, it has been proved by Kasparov [Kas1] in the case of the nonamenable groups SO0 (n, 1) and by Julg-Kasparov [JuKas] in the case of SU(n, 1) that these are K-amenable Lie groups. Cuntz [Cu] has shown that the class of K-amenable groups is closed under the operations of taking subgroups, under free products and under direct products. Our method uses the K-amenability results of Kasparov [Kas1]

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and the Packer-Raeburn stabilization theorem [PR]. In [PR1], one can find an example where the twisted K-theory K ∗ (0\G/K, δ(Bσ )) is not isomorphic to K ∗ (0\G/K), even when G is the K-amenable solvable group Rn o R, K = {e}, 0 = Zn o Z and for some multiplier σ on 0 with non-trivial Dixmier-Douady invariant δ(Bσ ) = δ(σ) 6= 0. However, K ∗ (0\G/K, δ(Bσ )) is isomorphic to K ∗ (0\G/K) whenever the DixmierDouady invariant δ(Bσ ) = δ(σ) = 0 is trivial. We identify the Dixmier-Douady invariant δ(Bσ ) with the image of σ under the connecting homomorphism δ : H 2 (0, U (1)) → H 3 (0, Z) of the change of coefficients exact sequence in cohomology, corresponding to the short exact sequence of coefficient groups i

e2π

√

−1

1 → Z→R −→ U (1) → 1. This enables us to prove vanishing theorems for the Dixmier-Douady invariant whenever 0 is a lattice in a connected Lie group G such that dim(G/K) ≤ 3, where K is a maximal compact subgroup of G, and therefore we obtain in this case K∗ (C ∗ (0, σ)) ∼ = K ∗+dim(G/K) (0\G/K), where σ is any multiplier on 0. This is the case for the Riemann surfaces which are the object of our study in this paper. We begin by reviewing the concept of K-amenable groups. Let G be a connected Lie group and K be a maximal compact subgroup. For our purposes, we will assume that dim(G/K) is even, and that it has a G-invariant SpinC structure. Using the SpinC structure, we can form the G-invariant Dirac operator /∂ on G/K. It is a first order, selfadjoint, elliptic differential operator acting on L2 sections of the Z2 graded homogeneous bundle of spinors S. Consider F = / ∂ (1+/ ∂ 2 )−1/2 , which is a 0th order pseudo-differential 2 operator acting on H = L (G/K, S). C0 (G/K) acts on H by multiplication operators, f → Mf . Also G acts on C0 (G/K) and on H by left translation, and F is G-invariant. Therefore (H, M, F ) defines a canonical element, called the Dirac element, αG ∈ KKG (C0 (G/K), C). Theorem 6 ([Kas2]). There is a canonical element, called the Mishchenko element βG ∈ KKG (C, C0 (G/K)), such that one has the following intersection products: (1) αG ⊗C βG = 1C0 (G/K) ∈ KKG (C0 (G/K), C0 (G/K)), (2) βG ⊗C0 (G/K) αG = γG ∈ KKG (C, C), where γG is an idempotent in KKG (C, C). The Mishchenko element βG can be described as follows. First assume that either G is semisimple or that G = Rn . Then the Killing form on G defines a G-invariant Riemannian metric of non-positive sectional curvature on G/K. Let E = C0 (G/K, S ∗ ) be the space of continuous sections of the dual spin bundle S ∗ which vanish at infinity. Let F be a bounded operator on E defined as F ξ(x) = c(V (x, x0 ))ξ(x), where ξ ∈ E, V (x, x0 ) ∈ Tx (G/K) is the unit vector which is tangent to the unique geodesic from x0 ∈ G/K to x and c(V (x, x0 )) denotes Clifford multiplication by V (x, x0 ). Then V (x, x0 ) is well defined outside a small neighbourhood of x0 and can

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be extended continuously in any way to all of G/K. As F is adjointable it lies in L(E). Also since F 2 ξ(x) = kV (x, x0 )k2 ξ(x), we see that F 2 − 1 ∈ K(E) is a compact operator in L(E). For g ∈ G, define (g.F )ξ(x) = c(V (x, gx0 ))ξ(x). Since G/K has negative sectional curvature, the function on G/K defined by x → kV (x, x0 ) − V (x, x1 )k,

x0 , x1 ∈ G/K,

vanishes at infinity, and so is in C0 (G/K). Therefore g.F −F ∈ K(E) and (E, F ) defines an element βG ∈ KKG (C, C0 (G/K)). The Mishchenko element βG is constructed by induction in the general case. Theorem 7 ([Kas2]). If G is amenable, then γG = 1. This motivates the following definition ([Kas2]). Definition. A Lie group G is said to be K-amenable if γG = 1. Theorem 8 ([Kas1, JuKas]). The non-amenable groups SO(n, 1) and SU(n, 1) are Kamenable. Let 0 ⊂ G be a lattice in G and A be an algebra admitting an automorphic action of 0. Then the cross product algebra [A⊗C0 (G/K)]o0, is Morita equivalent to the algebra of continuous sections vanishing at infinity C0 (0\G/K, E), where E → 0\G/K is the flat A-bundle defined as the quotient (∗)

E = (A × G/K)/0 → 0\G/K.

Here we consider the diagonal action of 0 on A × G/K. Theorem 9 ([Kas2]). If G is K-amenable, then (A o 0) ⊗ C0 (G/K) and [A ⊗ C0 (G/K)] o 0 have the same K-theory. Combining Theorem 9 with the remarks above, one gets the following important corollary. Corollary 7. If G is K-amenable, then (A o 0) ⊗ C0 (G/K) and C0 (0\G/K, E) have the same K-theory. Equivalently, one has for j = 0, 1, Kj (C0 (0\G/K, E)) ∼ = Kj+dim(G/K) (A o 0). We now come to the main theorem of this section, which generalizes theorems of [PR, PR2]. Theorem 10. Suppose that 0 is a lattice in a K-amenable Lie group G and that K is a maximal compact subgroup of G. Then K∗ (C ∗ (0, σ)) ∼ = K ∗+dim(G/K) (0\G/K, δ(Bσ )), where σ ∈ H 2 (0, U (1)) is any multiplier on 0, K ∗ (0\G/K, δ(Bσ )) is the twisted Ktheory of a continuous trace C ∗ -algebra Bσ with spectrum 0\G/K, and δ(Bσ ) denotes the Dixmier-Douady invariant of Bσ .

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Proof. 1. Taking the case A = C and the trivial action of 0 on C, one sees by Corollary 7 that C ∗ (0) and C0 (0\G/K) have the same K-theory when γG = 1. 2. Let σ ∈ H 2 (0, U (1)), then the twisted cross product algebra A oσ 0 is stably equivalent to the cross product (A ⊗ K) o 0, where K denotes the compact operators. This is the Packer-Raeburn stabilization trick [PR] (note that the 0 action on K takes some time to describe and we refer the reader to [PR] for details). Using Corollary 7 again, one sees that A oσ 0 ⊗ C0 (G/K) and C0 (0\G/K, Eσ ) have the same K-theory, whenever G is K-amenable, where Eσ = (A ⊗ K × G/K)/0 → 0\G/K is a flat A ⊗ K-bundle over 0\G/K and K is a maximal compact subgroup of G. In the particular case when A = C, one sees that C ∗ (0, σ) ⊗ C0 (G/K) and C0 (0\G/K, Eσ ) have the same K-theory whenever G is K-amenable, where Eσ = (K × G/K)/0 → 0\G/K. But the twisted K-theory K ∗ (0\G/K, δ(Bσ )) is by definition the K-theory of the continuous trace C ∗ -algebra Bσ = C0 (0\G/K, Eσ ) with spectrum 0\G/K. Then K∗ (C ∗ (0, σ)) ∼ = K ∗+dim(G/K) (0\G/K, δ(Bσ )).



Remarks. Consider the flat case, when G = R2n o SO(2n) is the Euclidean group, K = SO(2n), and 0 ⊂ G is a Bieberbach group, that is, 0 is a uniform lattice in G. One can define a generalization of “noncommutative flat manifolds" by regarding C ∗ (0, σ) as such an object, where σ is any group 2-cocycle on 0, by virtue of the fact that K∗ (C ∗ (0, σ)) ∼ = K ∗ (0\G/K). Our next main result says that for lattices in K-amenable Lie groups, the reduced and unreduced twisted group C ∗ -algebras have canonically isomorphic K-theories. Therefore all the results that we prove regarding the K-theory of these reduced twisted group C ∗ -algebras are also valid for the unreduced twisted group C ∗ -algebras. Theorem 11. Let σ ∈ H 2 (0, U (1)) be a multiplier on 0 and 0 be a lattice in a Kamenable Lie group. Then the canonical morphism C ∗ (0, σ) → Cr∗ (0, σ) induces an isomorphism K∗ (C ∗ (0, σ)) ∼ = K∗ (Cr∗ (0, σ)). Proof. We note that by the Packer-Raeburn trick, one has C ∗ (0, σ) ⊗ K ∼ = Ko0 and

Cr∗ (0, σ) ⊗ K ∼ = K or 0,

where or denotes the reduced crossed product. Since 0 is a lattice in a K-amenable Lie group, the canonical morphism K o 0 → K or 0 induces an isomorphism (cf. [Cu]) K∗ (K o 0) ∼ = K∗ (K or 0), which proves the result.



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We now specialize to the case when G = SO0 (2, 1), K = SO(2) and 0 = 0g ⊂ G is the fundamental group of a Riemann surface of genus g > 1, Σg , or when G = R2 , K = {e} and g = 1, with 01 being Z2 . Corollary 8. Let σ ∈ H 2 (0g , U (1)) be any multiplier on 0g . Then 1. K0 (Cr∗ (0g , σ)) ∼ = K0 (Cr∗ (0g )) ∼ = K 0 (Σg ) ∼ = Z2 2. K1 (C ∗ (0g , σ)) ∼ = K1 (C ∗ (0g )) ∼ = K 1 (Σg ) ∼ = Z2g . r

r

Proof. In dimension 2 the Chern character is an isomorphism over the integers and therefore we see that K 0 (Σg ) ∼ = H 0 (Σg , Z) ⊕ H 2 (Σg , Z) ∼ = Z2 , and that K 1 (Σg ) ∼ = H 1 (Σg , Z) ∼ = Z2g . By Theorem 10 we have Kj (Cr∗ (0g )) ∼ = K j (Σg ) and

for j = 0, 1,

Kj (Cr∗ (0g , σ)) ∼ = Kj (Σg , δ(Bσ )),

j = 0, 1,

where Bσ = C(Σg , Eσ ). Finally, because Eσ is a locally trivial flat bundle of C ∗ -algebras over Σg , with fibre K (= compact operators), it has a Dixmier-Douady invariant δ(Bσ ) which can be viewed as the obstruction to Bσ being Morita equivalent to C(Σg ). But δ(Bσ ) = δ(σ) ∈ H 3 (Σg , Z) = 0. Therefore Bσ is Morita equivalent to C(Σg ) and we conclude that Kj (Cr∗ (0g , σ)) ∼ = K j (Σg )

j = 0, 1.



Corollary 9. Let G be a connected Lie group and K a maximal compact subgroup such that dim(G/K) = 3. Let 0 be a uniform lattice in G and σ ∈ H 2 (0, U (1)) be any multiplier on 0. If G is K-amenable, then (∗)

Kj (Cr∗ (0, σ)) ∼ = Kj (Cr∗ (0)) ∼ = K j+1 (0\G/K),

for j = 0, 1

(mod 2).

Proof. By Theorem 10, we see that Kj (Cr∗ (0)) ∼ = K j+dim(G/K) (0\G/K),

for j = 0, 1

(mod 2).

By the Packer-Raeburn stabilization trick, Cr∗ (0, σ) is Morita equivalent to K or 0, and because G is K-amenable, K o 0 ⊗ C0 (G/K) is Morita equivalent to Bσ = C(0\G/K, Eσ ), where Eσ is as before, a locally trivial bundle of C ∗ -algebras over 0\G/K with fibre K. Finally, the Dixmier-Douady invariant δ(Bσ ) = δ(σ) ∈ H 3 (0\G/K, Z) ∼ = H 3 (0, Z). Suppose now that 0\G/K is not orientable. Then H 3 (0\G/K, Z) = {0} and therefore δ(Bσ ) = δ(σ) = 0. Hence Bσ is Morita equivalent to C(0\G/K) and we have (∗) in this case.

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Suppose next that 0\G/K is orientable. The short exact sequence of coefficient groups √ i

e2π

−1

1 → Z→R −→ U (1) → 1 gives rise to a long exact sequence of cohomology groups (the change of coefficient groups sequence) (∗∗)

e2π

√

−1∗

δ

i

∗ 3 (0, R) → · · · . · · · → H 2 (0, R) −→ H 2 (0, U (1))→H 3 (0, Z)→H

Since 0\G/K is oriented, we see that H 3 (0, Z) ∼ = Z and H 3 (0, R) ∼ = R are both generated by the fundamental orientation class of 0\G/K, [0\G/K], and since i∗ [0\G/K] = [0\G/K], we see that i∗ is injective. Therefore by the exactness of (∗∗) at H 3 (0, Z), one has δ(σ) = 0 for all σ ∈ H 2 (0, U (1)), and so we see that Bσ is Morita equivalent to C(0\G/K), and again we have (∗) in this case.  Corollary 10. Let M = K(0, 1) be an Eilenberg–Maclane space which is connected locally-symmetric, compact, 3-dimensional manifold. If σ ∈ H 2 (0, U (1)) is any multiplier on 0, then one has Kj (Cr∗ (0, σ)) ∼ = Kj (Cr∗ (0)) ∼ = K j+1 (M ),

j = 0, 1.

Proof. Since M is locally symmetric, it is of the form 0\G/K, where G is a connected Lie group, K is a maximal compact subgroup such that dim(G/K) = 3 and 0 ⊂ G is a uniform lattice in G. We need to verify that γG = 1. According to Thurston’s list of 3-dimensional geometries or locally homogeneous spaces, one has 1. G = R3 o SO(3), G/K = R3 , γG = 1 since R3 and SO(3) are amenable, and so is their semidirect product. 2. G = SO0 (3, 1), G/K = H3 , γG = 1 by Kasparov’s theorem. 3. G = SO0 (2, 1) o R, G/K = H2 × R, γG = 1 since it’s the semidirect product of K-amenable groups. 4. G = Heis, G/K = Heis, γG = 1, since Heis is nilpotent and hence an amenable group. 5. G = Solv, G/K = Solv, γG = 1, since Solv is a solvable group and hence an amenable group. ^ ^ ^ = 1 since SO 6. G = SO 0 (2, 1) o R, G/K = SO0 (2, 1). Firstly, γSO 0 (2, 1) is the ^ 0 (2,1) semidirect product of the K-amenable groups SO0 (2, 1) and Z. Also γG = 1, since ^ its the semidirect product of the K-amenable groups SO 0 (2, 1) and R. The other two locally homogeneous spaces in Thurston’s list are not locally symmetric. We now apply Corollary 9 to deduce Corollary 10.  An interesting question is whether Corollary 10 is true without the locally symmetric assumption on M . We formulate this in terms of a conjecture. Conjecture. Let M = K(0, 1) be a connected, compact, 3-dimensional manifold which is an Eilenberg-Maclane space with fundamental group 0. Then for any multiplier σ ∈ H 2 (0, U (1)) on 0, one has Kj (Cr∗ (0, σ)) ∼ = Kj (Cr∗ (0)) ∼ = K j+1 (M ),

j = 0, 1.

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Remarks. Selected portions of our proof of Corollary 9 go through in the situation described in the conjecture. More precisely, the proof of Corollary 9 shows that the Dixmier-Douady invariant δ(σ) = 0 for all σ ∈ H 2 (0, U (1)) for 0 as in the conjecture. 9. Range of the Trace and the Kadison Constant In this section, we will prove some structural theorems for the twisted group C ∗ -algebras that are relevant to the “Martini” problems described in the introduction. The first of these calculates the range of the canonical trace map on K0 of the twisted group C ∗ -algebras. We use in an essential way the results of the previous section as well as a twisted version of the L2 -index theorem of Atiyah [At], which is due to Gromov [Gr2]. This enables us to deduce information about projections in the twisted group C ∗ -algebras. In the case of no twisting, this follows because the Baum-Connes conjecture is known to be true while these results are also well known for the case of the irrational rotation algebras. However, our approach here is novel, and as we will show elsewhere [Ma], enables a generalization of most of the known results. 9.1. Twisted Kasparov map. Suppose that 0g is a discrete, cocompact subgroup of SO0 (2, 1). That is, 0g is the fundamental group of a Riemann surface Σg of genus g > 1. Then for any σ ∈ H 2 (0g , U (1)), the twisted Kasparov isomorphism, (∗)

µσ : K• (Σg ) → K• (Cr∗ (0g , σ))

is defined as follows. Here K0 (Σg ) denotes the K-homology group of Σg . Since Σg is spin, it is K-oriented and by Poincar´e duality, the K groups K j (Σg ) are naturally isomorphic to the corresponding K-homology groups Kj (Σg ) for j = 0, 1. Explicitly, let E → Σg be a vector bundle over Σg defining an element [E] in K 0 (Σg ). Under Poincar´e duality, [E] corresponds to the twisted Dirac operator /∂E+ : L2 (Σg , S + ⊗ E) → L2 (Σg , S − ⊗ E), where S ± denote the 21 spinor bundles over Σg . That is, P D : K 0 (Σg ) → K0 (Σg ) [E] → [/∂E+ ] is the Poincar´e duality isomorphism. By Corollary 8 of the previous section, there is a canonical isomorphism K• (Cr∗ (0g , σ)) ∼ = K • (Σg ). Both of these maps are assembled to yield the twisted Kasparov map as in (*). We next describe this map more explicitly. Given [/∂E+ ] ∈ K0 (Σg ) as above, the lift eg , the universal cover of Σg , of this operator to H = Σ 2 2 + + − ^ ^ /f ∂ E : L (H, S ⊗ E) → L (H, S ⊗ E)

is a 0g -invariant operator. Consider now the short exact sequence of coefficient groups i

e2π

√

−1

1 → Z→R −→ U (1) → 1, which gives rise to a long exact sequence of cohomology groups (the change of coefficient groups sequence)

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(∗∗)

i∗

661 e2π

√

−1

δ

· · · → H (0g , Z)→H (0g , R) −→ ∗ H 2 (0g , U (1))→0. 2

2

Therefore for any multiplier σ ∈ H 2 (0g , U (1)) of 0g , there is a 2-form ω on Σg such √ 2π −1 that e ∗ ([ω]) = σ. Of course the choice of ω is not unique, but this will not affect the results that we are concerned with. Let ω e denote the lift of ω to the universal cover H. Since the hyperbolic plane H is contractible, it follows that ω e = dη, where η is a 1-form on H which is not in general 0g invariant. Now let ∇ = d − iη denote a connection on the trivial complex line bundle on H. Note that the curvature of ∇ is ∇2 = iω. Consider now the operator 2 2 + + − ^ ^ /f ∂ E ⊗ ∇ : L (H, S ⊗ E) → L (H, S ⊗ E).

It does not commute with the 0g action, but it does commute with the projective action of 0g which is defined by the multiplier σ, and by a mild generalization of the index theorem of [CM], it has a 0g -L2 -index, + ind0g (/ ∂f E ⊗ ∇) ∈ K0 (C(0g , σ) ⊗ R),

where R denotes the algebra of smoothing operators. Then observe that the twisted Kasparov map is merely ∗ + ∂E + ]) = j∗ (ind0g (/ ∂f µσ ([/ E ⊗ ∇)) ∈ K0 (C (0g , σ)),

where j : C(0g , σ) ⊗ R → Cr∗ (0g , σ) ⊗ K is the natural inclusion map, and j∗ : K0 (C(0g , σ) ⊗ R) → K0 (Cr∗ (0g , σ)) is the induced map on K0 . The canonical trace on Cr∗ (0g , σ) induces a linear map [tr] : K0 (Cr∗ (0g , σ)) → R which is called the trace map in K-theory. Explicitly, first tr extends to matrices with entries in C ∗ (0g , σ) as (with Trace denoting matrix trace): tr(f ⊗ r) = Trace(r)tr(f ). Then the extension of tr to K0 is given by [tr]([e] − [f ]) = tr(e) − tr(f ), where e, f are idempotent matrices with entries in C ∗ (0g , σ). 9.2. The isomorphism classes of algebras C ∗ (0g , σ). Let σ ∈ Z 2 (0g , U (1)) be a multiplier on 0g . If σ 0 ∈ Z 2 (0g , U (1)) is another multiplier on 0g such that [σ] = [σ 0 ] ∈ H 2 (0g , U (1)), then it can be easily shown that C ∗ (0g , σ) ∼ = C ∗ (0g , σ 0 ). ∗ ∗ That is, the isomorphism classes of the C -algebras C (0g , σ) are naturally parametrized by H 2 (0g , U (1)). But H 2 (0g , U (1)) ∼ = H 2 (Σg , U (1)) ∼ = U (1) and the isomorphism is ˇ given explicitly by [σ] →< [σ], [Σg ] >, where [σ] is now viewed as a Cech 2-cocycle on Σg with coefficients in U (1), and [Σg ] denotes the fundamental class of the genus g Riemann surface. We summarize this below.

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Lemma 13. The isomorphism classes of twisted group C ∗ -algebras C ∗ (0g , σ) are naturally parametrized by U (1) ∼ = R/Z ∼ = (0, 1]. The classification map is given explicitly by [σ] →< [σ], [Σg ] >, ˇ where [σ] is now viewed as a Cech 2-cocycle on Σg with coefficients in U (1), and [Σg ] denotes the fundamental class of the genus g Riemann surface. 9.3. Range of the trace map on K0 . We can now state the first major theorem of this section. Theorem 12. The range of the trace map is [tr](K0 (Cr∗ (0g , σ))) = Zθ + Z, where 2πθ =< σ, [Σg ] > ∈ (0, 1] is the result of pairing the multiplier σ on 0g with the fundamental class of Σg . Proof. We first observe that by the results of the previous section the twisted Kasparov map is an isomorphism. Therefore to compute the range of the trace map on K0 , it suffices to compute the range of the trace map on elements of the form ∂E+ 0 ] − [/∂E+ 1 ]) µσ ([/ for any element

∂E+ 1 ] ∈ K0 (Σg ). [/ ∂E+ 0 ] − [/

By the twisted analogue of the L2 index theorem of Atiyah [At] and Singer [Si] for elliptic operators on a covering space that are invariant under the projective action of the fundamental group defined by σ, and which is due to Gromov [Gr2] (see also [Ma] for a proof of a further generalization), one has 1 ˆ + hA(Σg ) ch(E)e[ω] , [Σg ]i. ∂f [tr](ind0g (/ E ⊗ ∇)) = 2π

(∗)

We next simplify the right hand side of (∗) using ˆ g ) = 1, A(Σ ch(E) = rank E + c1 (E), e[ω] = 1 + [ω]. Therefore one has + ∂f [tr](ind0g (/ E ⊗ ∇)) = rank E

h[ω], [Σg ]i hc1 (E), [Σg ]i + , 2π 2π

and we see that ∂E+ 1 ])) = (rank E 0 − rank E 1 ) [tr](µσ ([/∂E+ 0 ] − [/

h[ω], [Σg ]i hc1 (E 0 ) − c1 (E 1 ), [Σg ]i + . 2π 2π

It follows that the range of the trace map on K0 is Z h[ω], [Σg ]i − θ ∈ Z. 2π

h[ω],[Σg ]i 2π



+ Z = Zθ + Z, because

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663

We will now discuss some applications of this result. We begin by studying projections in the twisted group C ∗ -algebra, which is a problem of independent interest. Proposition 4. Let σ ∈ H 2 (Σg , R/Z) be a multiplier on 0g , and 2πθ =< σ, [Σg ] >∈ (0, 1] be the result of pairing σ with the fundamental class of Σg . If θ = p/q is rational, then there are only q − 1 unitary equivalence classes of projections, other than 0 and 1, in the reduced twisted group C ∗ -algebra Cr∗ (0g , σ). Proof. By assumption, θ = p/q. Let P be a projection in Cr∗ (0g , σ). Then 1 − P is also a projection in Cr∗ (0g , σ) and one has 1 = tr(1) = tr(P ) + tr(1 − P ). Each term in the above equation is non-negative. By the previous theorem, it follows that tr(P ) ∈ {0, 1/q, 2/q, . . . 1}. By faithfulness and normality of the trace tr, it follows that there are only q − 1 unitary equivalence classes of projections, other than those of  0 and 1 in Cr∗ (0g , σ). Our second application will involve the Kadison constant of a twisted group C ∗ algebra, which we will now recall. The Kadison constant of Cr∗ (0g , σ) is defined by: Cσ (0g ) = inf{tr(P ) : P is a non-zero projection in Cr∗ (0g , σ) ⊗ K}. Recall from earlier sections the following Hamiltonians: Hη =

1 1 (d − iη)∗ (d − iη) = ∇∗ ∇, 2 2

and Hη,V = Hη + V, where V is any 0g -invariant potential on H. The operators Hη and Hη,V are invariant under the projective (0g , σ)-action. Proposition 5. Let σ ∈ H 2 (Σg , R/Z) be a multiplier on 0g , and 2πθ =< σ, [Σg ] >∈ (0, 1] be the result of pairing σ with the fundamental class of Σg . If θ = p/q is rational, then the spectrum of any associated Hamiltonian Hη,V has a band structure, in the sense that the intersection of the resolvent set with any compact interval in R has only a finite number of components. In particular, the intersection of σ(Hη,V ) with any compact interval in R is never a Cantor set. Proof. By the previous proposition, it follows that one has the estimate Cσ (0g ) ≥ 1/q > 0. Then one applies the main result in Br¨uning-Sunada [BrSu] to deduce the proposition.  This leaves open the question of whether there are Hamiltonians with Cantor spectrum when θ is irrational. In the Euclidean case, this is usually known as the Ten Martini Problem, and is to date, not completely solved, though much progress has been made (cf. [Sh]). We pose a generalization of this problem to the hyperbolic case (which also includes the Euclidean case): Conjecture (The Ten Dry Martini Problem). Let σ ∈ H 2 (Σg , R/Z) be a multiplier on 0g , and 2πθ =< σ, [Σg ] >∈ (0, 1] be the result of pairing σ with the fundamental class of Σg . If θ is irrational, then there is an associated Hamiltonian Hη,V with a Cantor set type spectrum, in the sense that the intersection of σ(Hη,V ) with some compact interval in R is a Cantor set.

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We will next apply the range of the trace Theorem 12 to deduce results about the discrete Hamiltonian Hτ , as in Sect. 5. Proposition 6. Let σ ∈ H 2 (Σg , R/Z) be a multiplier on 0g , and 2πθ =< σ, [Σg ] >∈ (0, 1] be the result of pairing σ with the fundamental class of Σg . If θ = p/q is rational, then the spectrum of the associated discrete Hamiltonian Hτ has a band structure, in the sense that the intersection of the resolvent set with R has only a finite number of components. In particular, the intersection of σ(Hτ ) with any compact interval in R is never a Cantor set. Proof. From the estimate Cσ (0g ) ≥ 1/q > 0, the main result in [Sun] implies the proposition.  This leads us to our next conjecture. Conjecture (The Discrete Ten Dry Martini Problem). Let σ ∈ H 2 (Σg , R/Z) be a multiplier on 0g , and 2πθ =< σ, [Σg ] >∈ (0, 1] be the result of pairing σ with the fundamental class of Σg . If θ is irrational, then the associated Hamiltonian Hτ has Cantor spectrum. 9.4. On the classification of twisted group C ∗ -algebras. We will now use the range of the trace found in Theorem 12 to give a complete classification, up to isomorphism, of the twisted group C ∗ -algebras C ∗ (0, σ). A similar complete classification, up to Morita equivalence, is contained in [Ma]. Proposition 7 (Isomorphism classification of twisted group C ∗ -algebras). Let σ, σ 0 ∈ H 2 (Σg , R/Z) be multipliers on 0g , and 2πθ =< σ, [Σg ] >∈ (0, 1], 2πθ0 =< σ 0 , [Σg ] >∈ (0, 1] be the result of pairing σ, σ 0 with the fundamental class of Σg . Then C ∗ (0g , σ) ∼ = C ∗ (0g , σ 0 ) if and only if θ0 ∈ {θ, 1 − θ}. Proof. Let tr and tr 0 denote the canonical traces on C ∗ (0g , σ) and C ∗ (0g , σ 0 ) respectively. Let φ : C ∗ (0g , σ) → C ∗ (0g , σ 0 ) be an isomorphism, and let φ∗ : K0 (C ∗ (0g , σ)) → K0 (C ∗ (0g , σ 0 )) denote the induced map on K0 . By Theorem 12, the range of the trace map on K0 is [tr](K0 (C ∗ (0g , σ))) = Zθ + Z and

[tr 0 ](K0 (C ∗ (0g , σ 0 ))) = Zθ0 + Z.

So there are elements [P ] ∈ K0 (C ∗ (0g , σ)) and [P 0 ] ∈ K0 (C ∗ (0g , σ 0 )) such that [tr]([P ]) = θ and [tr 0 ]([P 0 ]) = θ0 . Clearly one has tr ◦ φ = tr 0 , which induces the identity [tr] ◦ φ∗ = [tr 0 ] in K0 (C ∗ (0g , σ 0 )). In Sect. 8, we have proved that K0 (C ∗ (0g , σ)) ∼ = Z2 ∼ = K0 (C ∗ (0g , σ 0 )). In the basis above, one has φ∗ : Z[P ] ⊕ Z ∼ = K0 (C ∗ (0g , σ)) → K0 (C ∗ (0g , σ 0 )) ∼ = Z[P 0 ] ⊕ Z. Since φ∗ [1] = [1] and φ∗ ∈ GL(2, Z), one sees that there is an integer n such that

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665

φ∗ =

1 n 0 ±1

! .

Assembling these results, one has θ = [tr]([P ]) = [tr](φ∗ [P ]) = [tr 0 ](n[1] ± [P 0 ]) = n ± θ 0 . Since θ, θ0 ∈ (0, 1], one deduces that θ0 ∈ {θ, 1 − θ}. Let ψ : Σg → Σg be an orientation reversing diffeomorphism. We can assume without loss of generality that ψ has a fixed point x0 ∈ Σg . This is because there is an orientation preserving diffeomorphism η of Σg whose value at the point ψ(x0 ) is equal to x0 ; in fact η can be chosen to be isotopic to the identity (cf. exercise A2, chapter 1, [Helg]). The composition η ◦ ψ is then an orientation reversing diffeomorphism of Σg with fixed point x0 . Then ψ induces an automorphism ψ∗ : 0g → 0g of the fundamental group π1 (Σg , x0 ) ∼ = 0g . We first evaluate < ψ ∗ σ, [Σg ] >=< σ, ψ∗ [Σg ] >= < σ, [Σg ] > =< σ, ¯ [Σg ] >, since ψ is orientation reversing. By Lemma 13 we see that ψ ∗ σ = σ¯ ∈ H 2 (0g , U (1)). Therefore the automorphism ψ∗ of 0g induces an isomorphism of twisted group C ∗ -algebras ¯ C ∗ (0g , σ) ∼ = C ∗ (0g , ψ ∗ σ) ∼ = C ∗ (0g , σ). Therefore if θ0 ∈ {θ, 1 − θ}, one has C ∗ (0g , σ) ∼ = C ∗ (0g , σ 0 ), completing the proof of the proposition.  9.5. Twisted ICC group von Neumann algebras and type II1 factors. Recall that an ICC group 0 is one in which every non-trivial conjugacy class is infinite. There are many examples of ICC groups, such as free groups, fundamental groups of compact surfaces, etc. It is well known that the group von Neumann algebras of these groups are type II1 factors [Tak]. We will now prove that a similar result holds for the twisted group von Neumann algebras (this result probably exists in the literature but for completeness we reproduce a proof). We briefly recall some definitions. Let W ∗ (0, σ) denote the twisted group von Neumann algebra, where σ is a multiplier on 0, which is by definition the weak closure of C ∗ (0, σ), or equivalently, the weak closure of the algebraic group algebra C(0, σ) in the σ-regular representation on `2 (0). Let Proj(W ∗ (0, σ)) denote the set of all projections in W ∗ (0, σ). Then one has Proposition 8. Let 0 be an ICC group, and σ ∈ H 2 (0, R/Z) be a multiplier on 0. Then W ∗ (0, σ) is a II1 factor. In particular, tr(Proj(W ∗ (0, σ))) = [0, 1]. Proof. By the commutant theorem for the regular σ-representation we see that the com¯ We need to compute the centre Z(0, σ) mutant of W ∗ (0, σ) is identified with W ∗ (0, σ). ¯ Let of W ∗ (0, σ), which is equal to the intersection Z(0, σ) = W ∗ (0, σ) ∩ W ∗ (0, σ). T : 0P→ B(`2 (0)) denote the left projective (0, σ)-action. Regard x ∈ W ∗ (0, σ) ¯ as x = γ∈0 x(γ)T (γ). Since W ∗ (0, σ) is the weak closure of C(0, σ), it follows that (x(γ))γ∈0 ∈ `2 (0). Now x ∈ Z(0, σ) if and only if x commutes with T (γ 0 ), γ 0 ∈ 0. But X T (γ 0 )x = x(γ)σ(γ 0 , γ)T (γ 0 γ) γ∈0

=

X

γ∈0

and

−1

−1

x(γ 0 γ)σ(γ 0 , γ 0 γ)T (γ),

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A. L. Carey, K. C. Hannabuss, V. Mathai, P. McCann

xT (γ 0 ) =

X

x(γ)σ(γ, γ 0 )T (γγ 0 )

γ∈0

=

X

−1

−1

x(γγ 0 )σ(γγ 0 , γ 0 )T (γ).

γ∈0 −1

−1

−1

−1

Therefore we see that x(γ 0 γ)σ(γ 0 , γ 0 γ) = x(γγ 0 )σ(γγ 0 , γ 0 ) for all γ 0 ∈ 0. So −1 |x(γ 0 γγ 0 )| = |x(γ)| for all γ 0 ∈ 0. That is, |x(·)| is constant on each conjugacy class. Now since x ∈ `2 (0), it follows that x vanishes on each infinite conjugacy class. Since 0 is an ICC group, it follows that x(γ) = 0 for all γ 6= 1. So Z(0, σ) is 1-dimensional  and W ∗ (0, σ) is a II1 factor. 10. The Topological Index and the Index Theorem This section identifies the Hall conductivity τc (P, P, P ) = τ (P dP dP ) with a topological invariant, generalizing the work of [Xia]. Suppose that 0g is a discrete, cocompact subgroup of SO0 (2, 1). That is, 0g is the fundamental group of a Riemann surface Σg of genus g > 1. Then for any σ ∈ H 2 (0g , U (1)), the twisted Kasparov isomorphism, µσ : K• (Σg ) → K• (Cr∗ (0g , σ)) is defined as in the previous section. We note in the following section (using a result of [Ji]) that given any projection P in Cr∗ (0, σ) there is both a projection P˜ in the same K0 class but lying in a dense subalgebra, stable under the holomorphic functional calculus, and a Fredholm module for this dense subalgebra, which may be paired with P˜ to obtain an analytic index. On the other hand, by the results of the current section, given any such projection P there is a topological index that we can associate to it. The main result we prove here is that the (analytic index) = (topological index). The first step in the proof is to show that given an additive group cocycle c ∈ Z 2 (0g ) we may define canonical pairings with K0 (Σg ) and K0 (Cr∗ (0g , σ)) which are related by the twisted Kasparov isomorphism. We do this by generalizing some of the results of Connes and Connes-Moscovici to the twisted case. The group 2-cocycle c may be regarded as a skew-symmetrized function on 0g × 0g × 0g , so that we can modify a standard construction in [CM] to obtain a cyclic 2-cocycle τc on C(0g , σ) ⊗ R by defining: X f 0 (g0 )f 1 (g1 )f 2 (g2 )c(1, g1 , g1 g2 )σ(g1 , g2 ). τc (f 0 ⊗r0 , f 1 ⊗r1 , f 2 ⊗r2 ) = Tr(r0 r1 r2 ) g0 g1 g2 =1

Note that τc extends to C(0g , σ) ⊗ L2 , (where L2 denotes Hilbert-Schmidt operators) and by the pairing theory of [Co] one gets an additive map [τc ] : K0 (C(0g , σ) ⊗ R) → R. Explicitly, [τc ]([e] − [f ]) = τec (e, · · · , e) − τec (f, · · · , f ), where e, f are idempotent matrices with entries in (C(0g , σ) ⊗ R)∼ , the unital algebra obtained by adding the identity to C(0g , σ)⊗R and τec denotes the canonical extension of τc to (C(0g , σ)⊗R)∼ . + Let /∂f E ⊗ ∇ be the Dirac operator defined in the previous section, which is invariant under the projective action of the fundamental group defined by σ. By definition, the + ∂f (c, 0g , σ)-index of / E ⊗ ∇ is

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667

+ [τc ](ind0g (/ ∂f E ⊗ ∇)) ∈ R.

It only depends on the cohomology class [c] ∈ H 2 (0g ), and it is linear with respect to [c]. We assemble this to give the following theorem. Theorem 13. Given [c] ∈ H 2 (0g ) and σ ∈ H 2 (0g , U (1)) a multiplier on 0g , there is a canonical additive map h[c], i : K0 (Σg ) → R, which is defined as + h[c], [/ ∂E+ ]i = [τc ](ind0g (/∂f E ⊗ ∇)) ∈ R.

Moreover, it is linear with respect to [c]. By a generalization of the Connes-Moscovici higher index theorem [CM] to the twisted case of elliptic operators on a covering space that are invariant under the projective action of the fundamental group defined by σ, (see [Ma] for a detailed proof), one has (∗)

1 ˆ + hA(Σg ) ch(E)e[ω] ψ ∗ (c), [Σg ]i, ∂f [τc ](ind0g (/ E ⊗ ∇)) = 2π

where ψ : Σg → Σg is the classifying map of the universal cover (which in this case is the identity map) and [c] is considered as a degree 2 cohomology class on Σg . We next ˆ g ) = 1 and that simplify the right hand side of (∗) using the fact that A(Σ ch(E) = rank E + c1 (E), ψ ∗ (c) = c, e[ω] = 1 + [ω]. We obtain

rank E + h[c], [Σg ]i. ∂f [τc ](ind0g (/ E ⊗ ∇)) = 2π

Corollary 11. Let c, [c] ∈ H 2 (0g ), be the area cocycle. Then one has h[c], [/ ∂E+ ]i = 2(g − 1) rank E ∈ Z. Proof. When c, [c] ∈ H 2 (0g ), is the area 2-cocycle, one has h[c], [Σg ]i = −2πχ(Σg ) = 4π(g − 1).



Remarks 14. These theorems have been generalized in [Ma]. They agree with Xia’s result [Xia], although our methods are different. We next describe the canonical pairing of K0 (Cr∗ (0g , σ)), given [c] ∈ H 2 (0g ). Since Σg is negatively curved, we know from [Ji] that     X |f (γ)|2 (1 + l(γ))k < ∞ for all k ≥ 0 , Aσ,g = f : 0g → C |   γ∈0g

where l : 0g → R+ denotes the length function, is a dense and spectral invariant subalgebra of Cr∗ (0g , σ). In particular it is closed under the smooth functional calculus,

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and is known as the algebra of rapidly decreasing L2 functions on 0g . By a theorem of [Bost], the inclusion map Aσ,g ⊂ Cr∗ (0g , σ) induces an isomorphism Kj (Aσ,g ) ∼ = Kj (Cr∗ (0g , σ)),

j = 0, 1.

As Σg is a negatively curved manifold, we know (by [Mos] and [Gr]) that degree 2 cohomology classes in H 2 (0g ) have bounded representatives i.e. bounded 2-cocycles on 0g . Let c be a bounded 2-cocycle on 0g . Then it defines a cyclic 2-cocycle τc on the twisted group algebra C(0g , σ), by a slight modification of the standard formula [CM], ([Ma] for the general case) X f 0 (g0 )f 1 (g1 )f 2 (g2 )c(1, g1 , g1 g2 )σ(g1 , g2 ). τc (f 0 , f 1 , f 2 ) = g0 g1 g2 =1

Here c is assumed to be skew-symmetrized. Since the only difference with the expression obtained in [CM] is σ(g1 , g2 ), and since |σ(g1 , g2 )| = 1, we can use Lemma 6.4, part (ii) in [CM] and the assumption that c is bounded, to obtain the necessary estimates which show that in fact τc extends continuously to the bigger algebra Aσ,g . This induces an additive map in K-theory as before: [τc ] : K0 (Aσ,g ) → R [τc ]([e] − [f ]) = τec (e, · · · , e) − τec (f, · · · , f ), where e, f are idempotent matrices with entries in (Aσ,g )∼ (the unital algebra associated to Aσ,g ) and τec is the canonical extension of τc to (Aσ,g )∼ . Observe that the twisted Kasparov map is merely ∗ + µσ ([/ ∂E + ]) = j∗ (ind0g (/ ∂f E ⊗ ∇)) ∈ K0 (C (0g , σ)).

Here j : C(0g , σ) ⊗ R → C ∗ (0g , σ) ⊗ K is the natural inclusion map, and j∗ : K0 (C(0g , σ) ⊗ R) → K0 (C ∗ (0g , σ)) is the induced map in K-theory. Therefore one has the equality h[c], µ−1 σ [P ]i = h[τc ], [P ]i ∗ ∼ for any [P ] ∈ K0 (Aσ,g ) = K0 (C (0g , σ)). Using the previous corollary, one has r

Corollary 12. Let c, [c] ∈ H (0g ), be the area 2-cocycle. Then c is known to be a bounded 2-cocycle, and one has 2

h[τc ], [P ]i = 2(g − 1)(rank E 0 − rank E 1 ) ∈ Z, where [P ] ∈ K0 (Aσ,g ) ∼ = K0 (Cr∗ (0g , σ)), and where ∂E+ 0 ] − [/ ∂E+ 1 ] ∈ K0 (Σg ). µ−1 σ [P ] = [/ Remarks 15. This generalizes the main result of Xia, [Xia]. We will next prove the existence of a canonical element in KK(Cr∗ (0g , σ), C), which we call the twisted Mishchenko element. Theorem 16 (The twisted Mishchenko element). There exists a unique element [mσ ] ∈ KK(Cr∗ (0g , σ), C), called the twisted Mishchenko element, such that (∗)

[1] ⊗Cr∗ (0g ,σ) [mσ ] = 2(g − 1),

where [1] ∈ K0 (Cr∗ (0g , σ)) denotes the module generated by Cr∗ (0g , σ).

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669

Proof. By the well definedness of the Kasparov intersection product [Kas2], the equation (∗) above defines the element [mσ ] uniquely. In the next section we construct a 2-summable Fredholm module (F, H), which defines an element [(F, H)] ∈ KK(Cr∗ (0g , σ), C), and whose Chern character is the cyclic area 2-cocycle [τc ], (cf. [Co2]) defined by the area 2-cocycle c on the discrete group 0. We compute that [1] ⊗Cr∗ (0g ,σ) [(F, H)] = index(F ) = τc (1, 1, 1) = 2(g − 1). By uniqueness (proved above), we see that [mσ ] = [(F, H)], which establishes existence.  This completes the proof of Theorem 4 and Corollary 6 because we regard index(P F P ) as the result of pairing an element of the K-homology of Σg (defined by the twisted Mishchenko element) with an element of K0 (B 0 ) ∼ = K0 (C ∗ (0g , σ)). This enables us to demonstrate the relationship between Corollary 12 and the discrete model of the hyperbolic Hall effect. 11. A Discrete Fredholm Module and the Analytic Index We have observed following Sunada that Hτ is an operator in the twisted algebraic group algebra C(0, σ), which is a subalgebra of Aσ,g . We remark that a spectral projection into a gap in the spectrum of Hτ is given by the smooth functional calculus applied to Hτ . It follows from [Ji] that such spectral projections lie in Aσ,g . Connes constructs a Fredholm module for C0 which can be adapted to the case of C(0, σ). In his construction the Hilbert space is the `2 sections of the restriction of the spinor bundle to the orbit 0.u. This space is isomorphic to H = `2 (0) ⊕ `2 (0) under the map ι ⊕ ι. The grading is the obvious one given by the 2 × 2 matrix ε. We may define the operator F as in Sect. 7 to be multiplication by the matrix function   0 ψ∗ , ψ 0 where we restrict ψ to the orbit 0.u. Connes [Co2] shows that the module of the previous paragraph is 2-summable for C0. We show below using the same argument as in [Co2] that if λ denotes the left regular σ-representation of C ∗ (0, σ) then [F, λ(γ)] is Hilbert-Schmidt. So (H, F ) is also a 2summable module for C(0, σ). We may also exploit [Co2] to determine explicitly the character of this Fredholm module for our case. We now summarize some of the pertinent details. First, we are using the usual trace tr on the bounded operators on H. Second, our module is the `2 sections of the restriction of the spinor bundle to the orbit. From this point of view F corresponds to Clifford multiplication of a unit tangent vector to a / 0.u. We use the same geodesic connecting a given vertex of the graph to a point x0 ∈ notation ϕ(γ.u) for this unit tangent vector, regarding ϕ as a function from 0.u to T (H), the tangent space of H, as no confusion will arise. Next, note that for f ∈ H, [F, λ(γ)]f (γ 0 ) = (ϕ(γ 0 .u) − ϕ(γ −1 γ 0 .u))λ(γ)f (γ 0 ) . Connes observes that the operator on the RHS is Hilbert-Schmidt as a result of the convergence of the Poincar´e series:

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X

exp(−2d(γ.u, x0 )).

γ∈0

Thus if γ0 , γ1 , γ2 lie in 0 then 1 tr(ε[F, λ(γ0 )][F, λ(γ1 )][F, λ(γ2 )]). 2 Now λ(γ0 )[F, λ(γ1 )][F, λ(γ2 )] is the operator tr(ελ(γ0 )[F, λ(γ1 )][F, λ(γ2 )]) =

(λ(γ0 )[F, λ(γ1 )][F, λ(γ2 )]f )(γ) = ζ(γ)σ(γ1 , γ2 )σ(γ0 , γ1 γ2 )f ((γ0 γ1 γ2 )−1 γ), where ζ(γ) denotes Clifford multiplication by −1 −1 −1 (ϕ(κ−1 0 γ) − ϕ(κ1 γ))(ϕ(κ1 γ) − ϕ(κ2 γ)),

with κj = γ0 . . . γj . We can now obtain a formula for the cyclic cocycle. Following the calculation on p. 344 of [Co2] we find that for γ0 γ1 γ2 6= 1 the character of the cocycle associated to our Fredholm module is zero while for γ0 γ1 γ2 = 1 it is given by X trace(εζ(γ))σ(γ1 , γ2 ), tr(ελ(γ0 )[F, λ(γ1 )][F, λ(γ2 )]) = 2 γ∈0

where “trace” denotes the matrix trace on the Clifford algebra and we are utilising the fact that, for our choice of σ, σ(γ0 , γ1 γ2 ) = σ(γ0 , γ0−1 ) = 1. Connes proves that trace(εζ(γ)) is the Euclidean area of the triangle in the complex plane with vertices corresponding to the tangent vectors ϕ(κ−1 j γ). Then the additive group cocycle on 0 given by X trace(εζ(γ)) c(1, γ1 , γ1 γ2 ) = γ∈0

is what Connes calls the “volume” or area cocycle on 0. Thus we find that we have computed the character of our Fredholm module to be: τc (γ0 , γ1 , γ2 ) = c(1, γ1 , γ1 γ2 )σ(γ1 , γ2 ) for γ0 γ1 γ2 = 1, with τc being zero when γ0 γ1 γ2 6= 1 (the normalisation differs from [Co2] p. 295, but conforms with [CM]). This formula extends to give a non-trivial element of the cyclic cohomology of the smooth subalgebra Aσ,g via the formula X f 0 (γ0 )f 1 (γ1 )f 2 (γ2 )c(1, γ1 , γ1 γ2 )σ(γ1 , γ2 ), τc (f 0 , f 1 , f 2 ) = γ0 γ1 γ2 =1

for f 0 , f 1 , f 2 ∈ Aσ,g . Summarizing the discussion above, we have the first result of this section. Proposition 9. There is a 2-summable Fredholm module (F, H) over Aσ,g whose Chern character is given by the area cyclic 2-cocycle τc . Therefore, by the index pairing in [Co2], one has index(P (F ⊗ I)P ) = h[τc ], [P ]i, where P denotes a projection in Aσ,g ⊗ K(H1 ) and index(P (F ⊗ I)P ) denotes the index of the Fredholm operator P (F ⊗ I)P .

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Assembling this proposition with our results from Sect. 10 we have: Theorem 17. Let P denote a projection in Aσ,g ⊗ K(H1 ) Then in the notation of Corollary 12 of the previous section, one has index(P (F ⊗ I)P ) = 2(g − 1)(rank E 0 − rank E 1 ) ∈ Z, where index(P (F ⊗ I)P ) denotes the index of the Fredholm operator P (F ⊗ I)P acting ∂E+ 0 ] − [/∂E+ 1 ] ∈ K0 (Σg ). on the Hilbert space P (H ⊗ H1 ) and µ−1 σ [P ] = [/ Corollary 13. Let P be a projection into a gap in the spectrum of the discrete Hamiltonian Hτ . Then P ∈ Aσ,g , and may be regarded as a twisted convolution operator by a function p on 0. Then in the notation of Corollary 12: X p(γ0 )p(γ1 )p(γ2 )c(1, γ1 , γ1 γ2 )σ(γ1 , γ2 ) index(P F P ) = γ0 γ1 γ2 =1

= 2(g − 1)(rank E 0 − rank E 1 ) ∈ Z. Note that this explains the integrality of the cyclic 2-cocycle, X p(γ0 )p(γ1 )p(γ2 )c(1, γ1 , γ1 γ2 )σ(γ1 , γ2 ), γ0 γ1 γ2 =1

in two different ways: firstly as the index of the Fredholm operator P F P , and secondly as the topological index 2(g − 1)(rankE 0 − rankE 1 ), which is also clearly an integer. 12. The Non-Commutative Unit Disc In [Klim+Les1,2] Klimek and Lesznewski have introduced a non-commutative unit disc and higher genus Riemann surfaces. Their disc algebra can be realised as a Toeplitz algebra obtained by compressing the commutative algebra of functions on the disc using the projection onto a holomorphic subspace of one of its representation spaces. We shall describe their construction in a slightly more general setting. The algebra Cc (G/K) acts by multiplication (f 7→ M (f )) on L2 (G/K, µ) for any quasi-invariant measure µ. The group G also has an induced σ-representation W on this space, and we shall suppose that there is an irreducible subrepresentation on a subspace which is projected out by P . (This is certainly true in the case considered in [Klim+Les1].) The algebra P M (Cc (G/K))P then gives the non-commutative analogue of Cc (G/K). Now, by definition G also acts and therefore defines automorphisms of this algebra. Since it commutes with P the covariance algebra P M (Cc (G/K))P o G is the same as P (Cc (G/K) o G)P , which is the compression of the imprimitivity algebra A = Cc (G/K) o G. For higher genus surfaces one simply takes the 0-invariant part of P Cc (G/K)P , which is consistent with our constructions above. Suppose now that the irreducible subspace is defined by a reproducing kernel. Invariance of the kernel means that it is defined by twisted convolution with a continuous σ-positive definite function ξP or, equivalently, that Z P = W (ξP ) = ξP (g)W (g) dg.

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Now observe that ξP can be identified with an element of the imprimitivity algebra so the covariance algebra can be identified with the compression ξP ∗ A ∗ ξP of the imprimitivity algebra. In the cases of interest ξP is the σ-positive-definite function associated with a C ∞ vector, and so is smooth. This means that the natural module ξP ∗ M for ξP ∗ A ∗ ξP retains the structure of a Fredholm module.

References [At] [Av+K+P+S] [Av+S+S] [Av+S+Y ] [Bel] [Bel+E+S] [Bost] [BrSu] [CGT]

[CEY] [Comtet] [Comtet+H] [Co] [Co2] [CM] [Cu] [Elliott]

[Green] [GH] [Gr] [Gr2] [Helg]

[Iengo+Li] [Ji] [JuKas]

Atiyah, M.F.: Elliptic operators, discrete groups and Von Neumann algebras. Ast´erisque no.32–33, 43–72 (1976) Avron, J., Klein, M., Pnueli, A., Sedun, L.: Hall conductance and adiabatic charge transport of leaky tori. Phys. Rev. Lett. 69, 128–131 (1990) Avron, J., Seiler, R., Simon, B.: Charge deficiency, charge transport and comparisons of dimension. Commun. Math. Phys. 159, 399–422 (1994) Avron, J., Seiler, R., Yaffe, I.: Adiabatic theorems and applications to the integer quantum Hall effect. Commun. Math. Phys. 110, 33–49 (1987) Bellissard, J.: K-theory of C ∗ -algebras in solid state physics Springer Lecture Notes in Physics 257, 1986, pp. 99–156 Bellissard, J., van Elst, A., Schulz-Baldes, H.: The non-commutative geometry of the quantum Hall effect. J. Math. Phys. 35, 5373–5451 (1994) Bost, J.: Principe d’Oka, K-th´eorie et syst´emes dynamiques non commutatifs. Invent. Math. 101 no. 2, 261–333 (1990) Br¨uning, J., Sunada, T.: On the spectrum of gauge-periodic elliptic operators. M´ethodes semi-classiques. Vol. 2 (Nantes, 1991).Ast´erisque 210, 65–74 (1992) Cheeger, J., Gromov, M., Taylor, M.: Finite propagation speed, kernel estimates for functions of the Laplace operator and the Geometry of complete Riemannian manifolds. Jour. Diff. Geom. 17, 15–54 (1982) Choi, M., Elliott, G., Yui, N.: Gauss polynomials and the rotation algebra. Invent. Math. 99 no. 2, 225–246 (1990) Comtet, A.: On the Landau levels on the hyperbolic plane. Ann.Phys. 173, 185–209 (1987) Comtet, A., Houston, P.: Effective action on the hyperbolic plane in a constant external field. J. Math. Phys. 26, 185–191 (1985) Connes, A.: Non commutative differential geometry. Publ. Math. I.H.E.S. 62, 257–360 (1986) Connes, A.: Noncommutative geometry. San Diego, CA: Academic Press, Inc., 1994 Connes, A., Moscovici, H.: Cyclic cohomology, the Novikov conjecture and hyperbolic groups. Topology 29, 345–388 (1990) Cuntz, J.: K-theoretic amenability for discrete groups. J. Reine Angew. Math. 344, 180–195 (1983) Elliott, G.: On the K-theory of the C ∗ -algebra generated by a projective representation of a torsion-free discrete group. In: Operator Algebras and Group Representations, London: Pitman, 1983, pp. 157–184 Green, P.: The structure of imprimitivity algebras. J. Func. Anal. 36, 88–104 (1980) Griffiths, P. and Harris, J.: Principles of algebraic geometry. New York: Wiley, 1978 Gromov, M.: Volume and bounded cohomology. Publ. Math. I.H.E.S. 56, 5–99 (1982) Gromov, M.: K¨ahler-hyperbolicity and L2 Hodge theory. Diff. Geom. 33, 263–292 (1991) Helgason, S.: Differential Geometry, Lie Groups and Symmetric Spaces. Pure and Applied Mathematics, 80, New York-London: Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], 1978 Iengo, R., Li, D.: Quantum mechanics and the quantum Hall effect on Riemann surfaces. Nuclear Phys. B 413, 735–753 (1994) Ji, R.: Smooth dense subalgebras of reduced group C ∗ -algebras, Schwartz cohomology of groups and cyclic cohomology. Jour. Func. Anal. 107, 1–33 (1992) Julg, P., Kasparov, G.: Operator K-theory for the group SU(n, 1). J. Reine Angew. Math. 463, 99–152 (1995)

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[Kas1] [Kas2]

[Kas3] [Klim+Les1] [Klim+Les2] [Ma] [MC] [Mos] [M+R+W] [Nak+Bel] [PR] [PR1] [PR2] [Ren1] [Ren2] [Ren3] [Rief] [Ros] [Sh] [Si] [Sun] [Tak] [Xia]

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Kasparov, G.: Lorentz groups, K-theory of unitary representations and crossed products. Soviet Math. Dokl. 29, 256–260 (1984) Kasparov, G.: K-theory, group C ∗ -algebras and higher signatures. Conspectus, (1980), published in ‘Novikov conjectures, index theorems and rigidity’, vol. 1, editors S. Ferry, A. Ranicki, J. Rosenberg, Lond. Math. Soc. Lecture Note Series 226, Cambridge: Cambridge University Press, 1995 Kasparov, G.: Equivariant KK-theory and the Novikov conjecture. Inv. Math. 91, 147–201 (1988) Klimek, S., Lesznewski, A.: Quantum Riemann surfaces I: the unit disc. Commun. Math. Phys. 146, 105–122 (1992) Klimek, S., Lesznewski, A.: Quantum Riemann surfaces II: The discrete series. Lett. Math. Phys. 24, 125–139 (1992) Mathai, V.: In preparation McCann, P., Carey, A.: A discrete model of the integer quantum Hall effect. Publ. RIMS, Kyoto Univ. 32, 117–156 (1996) Mostow, G.: Strong rigidity of symmetric spaces. Ann. Math. Studies, 78, Princeton N J: Princeton University Press 1973 Muhly, P., Renault, J., Williams, P.: Equivalence and isomorphism for groupoid C ∗ -algebras. J. Operator Th. 17, 3–22 (1987) Nakamura, S., Bellissard, J.: Low energy bands do not contribute to the quantum Hall effect. Commun. Math. Phys. 131, 283–305 (1990) Packer, J., Raeburn, I.: Twisted cross products of C ∗ -algebras. Math. Proc. Camb. Phil. Soc. 106, 293–311 (1989) Packer, J., Raeburn, I.: On the structure of twisted group C ∗ -algebras. Trans.Am. Math. Soc. 334, 685–718 (1992) Packer, J., Raeburn, I.: Twisted cross products of C ∗ -algebras. II, Math. Ann. 287, 595–612 (1990) Renault, J.: A groupoid approach to C ∗ -algebras. Lecture Notes in Mathematics 793, Berlin: Springer, 1980 Renault, J.: Repr´esentations des produits crois´es d’alg`ebres de groupo¨ıdes. J.Operator Th. 18, 67–97 (1987) Renault, J.: The ideal structure of groupoid crossed product C ∗ -algebras. J. Operator Th. 25, 3–36 (1991) Rieffel, M.: C ∗ -algebras associated with irrational rotations. Pac. J. Math. 93, 415–429 (1981) Rosenberg, J.: Continuous trace algebras from the bundle theoretic point of view. Jour. Aus. Math. Soc. 47, 368–381 (1989) Shubin, M.: Discrete Magnetic Schr¨odinger operators. Commun. Math. Phys. 164, no.2, 259–275 (1994) Singer, I.M.: Some remarks on operator theory and index theory. Springer Lecture Notes in Math. 575, 1977, 128–137 Sunada, T.: A discrete analogue of periodic magnetic Schr¨odinger operators. Contemp. Math. 173, 283–299 (1994) Takesaki, M.: Theory of operator algebras I. New York: Springer-Verlag, 1979) Xia, J.: Geometric invariants of the quantum hall effect. Commun. Math. Phys. 119, 29–50 (1988)

Communicated by A. Connes

Commun. Math. Phys. 190, 675 – 695 (1998)

Communications in

Mathematical Physics c Springer-Verlag 1997

Hidden Symmetries of the Principal Chiral Model Unveiled C. Devchand1,2 , Jeremy Schiff1 1 2

Department of Mathematics and Computer Science, Bar–Ilan University, Ramat Gan 52900, Israel International Centre for Theoretical Physics, 34100 Trieste, Italy

Received: 20 November 1996 / Accepted: 25 April 1997

Abstract: By relating the two-dimensional U(N) Principal Chiral Model to a simple linear system we obtain a free-field parametrisation of solutions. Obvious symmetry transformations on the free-field data give symmetries of the model. In this way all known “hidden symmetries” and B¨acklund transformations, as well as a host of new symmetries, arise.

1. Introduction The definition of complete integrability for field theories remains rather imprecise. One usually looks for structures analogous to those existing in completely integrable hamiltonian systems with finitely many degrees of freedom, such as a Lax–pair representation or conserved quantities equal in number to the number of degrees of freedom. A very transparent notion of integrability is that completely integrable nonlinear systems are actually simple linear systems in disguise. For example, the Inverse Scattering Transform for two dimensional integrable systems such as the KdV equation establishes a correspondence between the nonlinear flow for a potential and a constant–coefficient linear flow for the associated scattering data. Similarly, the twistor transform for the self-dual Yang-Mills equations converts solutions of nonlinear equations to holomorphic data in twistor space; and for the KP hierarchy Mulase has explicitly proven complete integrability by performing a transformation to a constant–coefficient linear system [11]. In all these examples, a map is constructed between solutions of a simple, automatically– consistent linear system and the nonlinear system in question. This is distinct from the Lax–pair notion of linearisation, with the nonlinear system in question arising as the consistency condition for a linear system. Just as the dynamics of completely integrable systems gets trivialised in an auxiliary space, it seems that the confusing plethora of symmetry transformations of these systems arise naturally from obvious transformations on the initial data of the associated linear

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systems. This idea has been exploited recently by one of us [16] for the KdV hierarchy: A linearisation of KdV, mimicking Mulase’s for the KP hierarchy, was used to give a unified description of all known symmetries. The central feature of Mulase’s construction is a group G on which the relevant linear flow acts. The group G (or at least a dense subset thereof) is assumed to be factorisable into two subgroups G+ and G− . For the KP hierarchy G is a group of pseudo-differential operators. For KdV and for the two-dimensional Principal Chiral Model (PCM), as we shall see in this paper, G is a “loop group” of smooth maps from a contour C in the complex λ plane to some group H. This has subgroups G− (resp. G+ ) of maps analytic inside (resp. outside) C. Mulase notes that any flow on G induces flows on G± , but the flows on the factors induced by a simple linear flow on G can be complicated and nonlinear. This is the genesis of nonlinear integrable hierarchies; complete integrability is just a manifestation of the system’s linear origins. The universality of this kind of construction was noticed by Haak et al [8]. We consider on G the linear system d U =  U,

(1)

where d is the exterior derivative on the base space M of the hierarchy, U is a Gvalued function on M and  a 1-form on M with values in G+ . Consistency (Frobenius integrability) of this system requires d =  ∧ . In fact for KP, KdV and PCM we have the stronger condition d =  ∧  = 0, and (1) has the general solution U = eM U0 ;

dM = ,

U0 ∈ G.

(2)

The initial data U0 determines a solution of the linear system, and hence a solution of the associated nonlinear hierarchy. A hierarchy is specified by a choice of G with a factorisation and a choice of one-form . The purpose of this paper is to provide a description of the two-dimensional Principal Chiral Model in the general framework of Mulase’s scheme. We show that for the appropriate group G, and a choice of one-form  within a certain class, solutions of Eq. (1) give rise to solutions of PCM. Thus there is a map giving, for each allowed choice of  and each choice of initial data U0 , a solution of PCM. The allowed choices of  are parametrised by free fields. The known hidden symmetries and B¨acklund transformations of PCM all have their origins in natural field-independent transformations of U0 . We also reveal other symmetries, corresponding to other transformations of U0 as well as to transformations of the free fields in . We were motivated to reconsider the symmetries of PCM by a recent paper of Schwarz [17], in which infinitesimal hidden symmetries were reviewed. However the mystery surrounding their origin remained. Further, Schwarz’s review did not encompass the work of Uhlenbeck [19] or previous work on finite B¨acklund transformations [9]. We wish to present all these results in a unified framework and to lift the veil obscuring the nature of these symmetries. 2. The Principal Chiral Model The defining equations for the U(N) PCM on two-dimensional Minkowski space M with (real) light-cone coordinates x+ , x− are ∂− A+ = 21 [A+ , A− ], ∂+ A− = 21 [A− , A+ ],

(3)

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677

where A± take values in the Lie algebra of U(N), i.e. they are N × N antihermitian matrices. Considering the sum and difference of the two equations in (3) yields the alternative “conserved current” form of the PCM equations ∂− A+ + ∂+ A− = 0 ,

(4)

together with the zero-curvature condition ∂− A+ − ∂+ A− + [A− , A+ ] = 0 .

(5)

The latter has pure–gauge solution A± = g −1 ∂± g ,

(6)

where g takes values in U(N). Substituting this into (4) yields the familiar harmonic map equation (7) ∂− (g −1 ∂+ g) + ∂+ (g −1 ∂− g) = 0. This is manifestly invariant under the “chiral” transformation g 7→ a g b, for a and b constant U(N) matrices. At some fixed point x0 in space-time, we may choose g(x0 ) = I, the identity matrix. The chiral symmetry then reduces to g 7→ b−1 g b.

(8)

There is a further invariance of the equations under the transformation g 7→ g −1 .

(9)

Equation (3) has obvious solutions [21] A+ = A(x+ ) ,

A− = B(x− ) ,

(10)

respectively left- and right–moving diagonal matrices, i.e. taking values in the Cartan subalgebra. (This type of solution is familiar from WZW models and for commuting matrices the Eqs. (3) indeed reduce to WZW equations.) In greater generality, the PCM equations imply that the spectrum of A+ (resp. A− ) is a function of x+ (resp. x− ) alone. Thus general solutions take the form: + − A+ = s0 (x+ , x− )A(x+ )s−1 0 (x , x ) + − A− = se0 (x+ , x− )B(x− )e s−1 0 (x , x ),

(11)

where A(x+ ) and B(x− ) are antihermitian diagonal matrices, and s0 (x+ , x− ), se0 (x+ , x− ) are unitary. For given A(x+ ), B(x− ), we have seen that there exists at least one such solution, that with s0 = se0 = I. We shall see in the next section that a solution A± of the PCM is determined by the diagonal matrices A(x+ ) and B(x− ), together with another free field; and our construction leads to solutions of precisely the form (11). Moreover, we shall prove in Sect. 6 that hidden symmetries and B¨acklund transformations act on the space of solutions with given A(x+ ) and B(x− ).

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3. Construction of Solutions In this section we give the formulation of the PCM in the framework of Mulase’s general scheme. Let us begin by defining a one-form on two-dimensional Minkowski space M with coordinates (x+ , x− ), =−

B(x− ) − A(x+ ) + dx − dx . 1+λ 1−λ

(12)

Here A(x+ ), B(x− ) are arbitrary diagonal antihermitian matrices, depending only on x+ , x− respectively. Clearly, d =  ∧  = 0 ,

(13)

dU =U

(14)

so that the linear equation

is manifestly Frobenius–integrable. The general solution is U (x+ , x− , λ) = eM (x M (x+ , x− , λ) = −

+

,x− ,λ)

1 1+λ

Z

U0 (λ) ,

x+ x+0

A(y + )dy + −

(15) 1 1−λ

Z

x− x− 0

B(y − )dy − ,

where U0 , the initial condition, is a free (unconstrained) element of the group G in which U takes values. We need to specify this group. Remarks. 1) Since A, B are anti-hermitian, hermitian–conjugation of (14) yields dU (λ)† = −U (λ)† (λ∗ ), whereas U −1 satisfies dU −1 (λ) = −U −1 (λ)(λ). We therefore obtain the condition U † (λ∗ ) = U −1 (λ).

(16)

2)  has poles at λ = ±1, so it is analytic everywhere in the λ-plane including the point at ∞, except in two discs with centres at λ = ±1. We therefore introduce a contour C, the union of two small contours C± around λ = ±1 (such that λ = 0 remains outside both of them), dividing the λ-plane into two distinct regions: the “outside” {|λ − 1| > δ} ∩ {|λ + 1| > δ} and the “inside” {|λ − 1| < δ} ∪ {|λ + 1| < δ}, where δ < 1 is some small radius. I Definition. G is the group of smooth maps V = V (λ) from the contour C to GL(N, C) satisfying the condition V † (λ∗ ) = V −1 (λ).

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We are going to pretend that there exists a Birkhoff factorisation G = G− G+ , where G− denotes the group of maps analytic inside C and G+ denotes the group of maps analytic outside C and equal to the identity at λ = ∞. The corresponding Lie algebra decomposition is G = G− ⊕ G+ . This factorisation is definitely a pretence; but the point is that sufficiently many elements of G do factor this way so that the results we will obtain using this factorisation do hold. For a more precise discussion we refer to [19, 8]. We now have the spaces in which the objects in (14),(15) take values. Clearly,  is a one-form on M with values in the Cartan subalgebra of the Lie algebra G+ . The matrix U = U (x+ , x− , λ) is a map from M to G and U0 (λ) is an element of G (independent of x± ). Consider a solution U of (14). Assuming the existence of a Birkhoff factorisation for U , we can write (17) U = S −1 Y , where S −1 : M → G− and Y : M → G+ . Now, applying the exterior derivative on both sides and using (14) yields SS −1 = −dSS −1 + dY Y −1 .

(18)

SS −1 , which takes values in the Lie algebra G, decomposes into its components in the G− and G+ subalgebras. The above equation allows us to write separate equations for the projections: (SS −1 )− = −dSS −1 , (19) (SS −1 )+ = dY Y −1 . Here the suffix notation denotes the projection of an element of G into G± . We introduce a one-form Z taking values in G+ , Z = dY Y −1 = (SS −1 )+ .

(20)

Now, since S takes values in G− , it is analytic at λ = ±1 and has two power-series representations, converging in discs with centres at λ = ±1, viz. S=

∞ X

sn (x+ , x− )(1 + λ)n =

n=0

∞ X

sen (x+ , x− )(1 − λ)n ,

(21)

n=0

where the coefficients s0 (x+ , x− ), se0 (x+ , x− ) are U(N)-valued matrices. Inserting these expansions in (SS −1 ), we see that only the s0 and se0 terms survive the projection to the G+ subalgebra, yielding Z = (SS −1 )+ = − Define

s0 A(x+ )s−1 se0 B(x− )e s−1 0 0 dx+ − dx− . 1+λ 1−λ

A+ = s0 A(x+ )s−1 0 ,

A− = se0 B(x− )e s−1 0 .

(22)

(23)

These satisfy the PCM equations (3). The proof is immediate. From (20), d Z = Z ∧ Z. Inserting the form (22) in this equation yields

(24)

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∂− A+ 1 ∂+ A− − + 1−λ 1+λ 2



1 1 − 1−λ 1+λ

 [A+ , A− ] = 0 .

Since Y takes values in G+ , for consistency this equation needs to hold for all values 1 1 and 1+λ must be separately of λ away from ±1. In other words, the coefficients of 1−λ zero. This yields precisely the two equations in (3) as integrability conditions. Note that the solutions (23) have precisely the form (11). We have seen that for given diagonal matrices A(x+ ) and B(x− ), a solution of the linear field–independent system (14) determines a solution of the PCM in the spectral class of A and B. In fact the general solution of (14) takes the form (15), where the eM factor contains only spectral information (i.e. A, B). Everything else is encoded in the free element U0 (λ) ∈ G. So the freely–specifiable data {A(x+ ), B(x− ), U0 (λ)} corresponds to a solution of the PCM. Given any choice of these three fields, a solution of the PCM can be constructed in the following stages: (a) (b) (c) (d)

Construct the corresponding U (x+ , x− , λ) from (15). Perform the factorisation (17) to obtain S(x+ , x− , λ). Perform the two expansions (21) to extract the coefficients s0 (x+ , x− ) and se0 (x+ , x− ). Insert these in (23) to obtain a solution of the PCM.

Note that this procedure is purely algebraic, though the factorisation may not be very easy to perform in practice. However, it is clear that for any choice of A(x+ ), B(x− ) (which is tantamount to fixing the spectral class of A± ), every U0 (λ) ∈ G corresponds to a solution of the PCM. In fact there is a large redundancy, for a right–multiplication U0 7→ U0 k+ ;

k+ ∈ G +

(25)

corresponds to a right-multiplication U 7→ U k+ , which does nothing to alter the S −1 factor in (17). PCM solutions therefore correspond to G+ orbits in G, or equivalently, U0 (λ)’s from the Grassmannian G/G+ . This correspondence is, however, still redundant: Consider a left–multiplication by a diagonal matrix analytic inside C, U0 7→ h− U0 ;

h− ∈ G0,− , the maximal torus of G− .

(26)

Since this commutes with the diagonal eM , it corresponds to a transformation S −1 7→ h− S −1 . However, since h− is a diagonal matrix, the A± in (23) do not notice this transformation; they are invariant. The correct space of U0 ’s corresponding to solutions of (3) in each spectral class of A± is therefore the double coset G0,− \G/G+ . In particular, natural transformations of U0 (λ) preserving this double coset correspondence induce symmetry transformations on the space of PCM solutions.

4. The Extended Solution The fact that the consistency condition (24) with Z given by (22) yields the PCM equations is well known. Writing (20) in more familiar form, dY = ZY , it is precisely the PCM Lax-pair [14, 21],

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 1 A+ Y = 0, ∂+ + 1+λ   1 ∂− + A− Y = 0 . 1−λ

(27)

It is easy to check that the Y we have defined above has all the properties required of a solution of this pair of equations: 1. As a function of λ, the only singularities of Y on the entire λ-plane including the point at ∞ are at λ = ±1. 2. The solution of the system (27) is easily seen to satisfy the reality condition (16) Y † (λ∗ ) = Y −1 (λ).

(28)

3. There is an invariance of the Lax system: Y (x, λ) 7→ Y (x, λ)f (λ), which is usually fixed by setting (29) Y (x0 , λ) = I , for some fixed point x0 . This invariance corresponds to right–multiplications (25) of U0 and the condition (29) corresponds to choosing a representative point on the G+ orbit of U0 in G. 4. At λ = ∞, ∂+ Y = ∂− Y = 0, so Y (x, λ = ∞) is a constant and using (29) we obtain Y (x, λ = ∞) = I.

(30)

5. The system (27) yields the expressions A+ = (1 + λ)Y ∂+ Y −1 ,

A− = (1 − λ)Y ∂− Y −1 ,

(31)

which together with (29) and (6) imply that Y (x, λ = 0) = g −1 .

(32)

We have already seen that the A± solving (3) may be recovered from power series expansions around λ = ±1 of the S −1 factor of U using the expressions (23). We now see that solutions may equally be obtained from the Y factor using (32) and (6). We can also obtain solutions from the Y factor by expanding around λ = ∞. Denoting the leading terms consistently with (30), Y (x, λ) = I +

f (x) + ..., λ

(33)

where f (x) is antihermitian, the λ = ∞ limit of (31) yields the expressions A± = ∓∂± f ,

(34)

which identically satisfy (4) and shift the dynamical description to (5) instead, which acquires the form 1 (35) ∂− ∂+ f + [∂− f, ∂+ f ] = 0 . 2 This equation is known as the “dual formulation” of the harmonic map equation (7). A Y (x, λ) obtained from the factorisation procedure automatically yields a solution of

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this equation on expansion around λ = ∞. We therefore see that the factorisation (17) produces a Y (x, λ) which interpolates between the dual descriptions of PCM solutions; yielding a U(N)–valued solution g −1 of Eq. (7) on evaluation at λ = 0 and a Lie-algebravalued solution f of the alternative equation (35) on development around λ = ∞. The G+ –valued Y (x, λ) thus encapsulates these dual descriptions of chiral fields and this field was aptly named the extended solution of the PCM by Uhlenbeck [19]. We shall later need information about the next-to-leading-order term in the expansion of Y around λ = 0. If we substitute Y = (I + λϕ)g −1 + O(λ2 ),

(36)

where ϕ is a Lie-algebra-valued field, into (31), and use (6), we obtain the following first-order equation for ϕ: ∂± ϕ + [A± , ϕ] = ±A± .

(37)

The consistency condition for this is just (4). Reflecting the G+ –valued extended solution Y (x, λ), there is also the G− –valued S(x, λ), which clearly also describes some extension of the PCM solution given by the expression (23). Using dSS −1 = −(SS −1 )− = −(SS −1 ) + (SS −1 )+ , we find the following flows for the components of S, which we shall need later: ∂+ sn = sn+1 A − A+ sn+1 , n X sr B − A− sr ∂ − sn = , 2n−r+1 r=0 n X

(38) (39)

ser A − A+ ser , 2n−r+1

(40)

∂− sen = sen+1 B − A− sen+1 .

(41)

∂+ sen =

r=0

Using (23) and these equations for n = 0 yields the interesting flow equations: ∂+ A+ = s0 ∂+ A s−1 + [A+ , [A+ , s1 s−1 0 0 ]], ∂− A− = se0 ∂− B se−1 + [A− , [A− , se1 se−1 0 0 ]].

(42)

5. Symmetry Transformations Unveiled Non-space-time symmetry transformations of the PCM were traditionally derived using mainly guesswork inspired by analogies with other integrable models like the sineGordon model. Their origin remained largely veiled in mystery and they were therefore called “hidden symmetries”. Previous discussions of them have recently been reviewed by Schwarz [17] and Uhlenbeck [19]. In the framework of the present paper there is nothing “hidden” about these symmetries. As we shall see, in terms of the freefield data U0 (λ), A(x+ ), B(x− ), the veil hiding these symmetries is entirely lifted: the most natural field-independent transformations of these free fields, which preserve their analyticity properties in their respective independent variables, induce the entire array of known symmetry transformations of PCM fields and more. Moreover, the algebraic structure of the symmetry transformations is completely transparent when acting on the free-field data, and there is no need to compute commutators and check closure

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using the complicated action of the symmetries on physical fields. The physical fields automatically carry representations of all the symmetry actions on the free-field data. In this section we classify PCM symmetry transformations according to the corresponding transformations of the free fields. The formulas for the induced transformations on the extended solutions Y , on the chiral fields g and on the potentials A± will be derived in the next section. 5.1. Symmetry transformations of U0 . We first list symmetry transformations which leave A(x+ ) and B(x− ) unchanged. 5.1.1. Right dressings. Right-actions by elements of the G+ subgroup (25) have already been seen to correspond to trivial redundancies and have already been factored out. This leaves the possibility of right–multiplying U0 by an element of G− , U0 7→ U0 k− ;

k− ∈ G − .

(43)

Such transformations fall into the following classes: a) k− = b, a constant (i.e. an element of U(N)). This may easily be seen to induce the transformations Y 7→ b−1 Y b and g 7→ b−1 gb, i.e. the symmetry (8).   (µ) b) If we take k− = I + N λ−µ π , having a simple pole at a single point λ = µ outside C (here N (µ) is a λ-independent matrix), the transformations induced on the chiral fields are precisely the B¨acklund transformations of [9, 13]. I PCM we could conc) We are presently considering the U(N) PCM. For the GL(N, C) sider finite transformations with k− in a triangular subgroup of G− . Such transformations induce the explicit transformations discussed by Leznov [10]. We will not go into details of this. d) General k− (λ) infinitesimally close to the identity. This is a realisation of the algebra G− on the free-field U0 (λ) and is a remarkably transparent way of expressing the action of the celebrated loop algebra of hidden symmetries [6] of the PCM. The precise structure of this algebra has not been properly identified before. e) General finite k− (λ). This finite version of the infinitesimal symmetries in d) reproduces (modulo some details) the loop group action on chiral fields g and on extended maps Y given by Uhlenbeck in Sect. 5 of [19]. 5.1.2. Left dressings. Left actions on U0 by elements of G0,− have already been pointed out to leave the associated solution of the PCM invariant (see (26)). We wish to consider only left actions on U0 that descend to the double coset G0,− \G/G+ , i.e. actions by elements that commute with G0,− . Thus we have only the transformations U0 7→ h+ U0 ;

h+ ∈ G0,+ .

(44)

This is the action of an infinite-dimensional abelian group, which has not yet appeared in the literature. The infinitesimal version of this gives an infinite set of mutually commuting flows also commuting with the PCM flow. This is the PCM hierarchy. 5.1.3. Reparametrisations of U0 (λ). These are transformations generated by λ-diffeo morphisms (45) U0 (λ) 7→ U0 (λ + (λ)).

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General reparametrisations can move C± to curves that do not enclose ±1. The easiest way to prevent this is to restrict the diffeomorphisms to those that fix ±1. For infinitesimal diffeomorphisms this condition is not strictly necessary. It turns out however that the infinitesimal diffeomorphisms fixing ±1 are technically simpler (in terms of their action on g, Y ) and these give (modulo a detail that will be explained) the “half Virasoro” algebra described in [17]. We show how this can be extended to a full centreless Virasoro algebra. The only finite reparametrisations of the λ-plane preserving ±1 are  U0 (λ) 7→ U0

aλ + b bλ + a

 ,

a2 + b2 = 1.

(46)

These induce the S 1 action of sect. 7 of [19]. 5.2. Symmetry transformations of A(x+ ), B(x− ). We now consider symmetries that keep U0 fixed. For symmetries acting just on A(x+ ) it is natural to consider a) Shifts A(x+ ) 7→ A(x+ ) + α(x+ ), where α(x+ ) is a diagonal antihermitian matrix. b) Rescalings A(x+ ) 7→ ρ(x+ )A(x+ ) where ρ(x+ ) is a scalar function. c) Reparametrisations A(x+ ) 7→ A(x+ + (x+ )). There are other possibilities. Similar symmetries exist for B(x− ). All these symmetries are new. 5.3. Other symmetry transformations. Two other symmetries of PCM should be mentioned. The first is a particularly significant combination of an action on U0 with an action on A, B. The second is not strictly within the class of symmetries we have been considering, as it acts on the coordinates as well as the fields. 5.3.4. Inversion. The transformation U0 (λ) 7→ U0 (λ−1 )

and

(A, B) 7→ (−A, −B)

(47)

may easily be seen to induce the inversion symmetry (9). 5.3.5. Lorentz transformations. The transformation U0 invariant,

A 7→ θ+ A,

B 7→ θ− B

−1 ± x± 7→ θ± x

(48)

induces the residual Lorentz transformations in light cone coordinates A± 7→ θ± A± ,

−1 ± x± 7→ θ± x .

We can also consider more general reparametrisations of x± .

(49)

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6. Induced Symmetries of PCM Fields As we have already claimed, natural transformations on the free–field data, U0 (λ), A(x+ ), B(x− ) induce, through Birkhoff factorisation, rather complicated transformations on the PCM fields Y (x, λ), g(x), A± (x); and (field–independent) representations of symmetry algebras induce (field-dependent) representations on the PCM fields. In this section we prove this for the intereresting and not immediately obvious cases listed in the previous section. We also comment on the relation with previous results in the literature. 6.1. Right dressings. Consider the transformation induced by (43) on U (x, λ). U = S −1 Y 7→ Unew = S −1 Y k− .

(50)

Birkhoff factorisation of Y k− yields (in the obvious notation) −1 Unew = S −1 (Y k− )− (Y k− )+ = Snew Ynew .

(51)

In other words, we have the symmetry transformation Y 7→ (Y k− )+ ,

(52)

which is just the representation of G− described by Uhlenbeck in Sect. 6 of [19] (except that she uses a subgroup of G− ). We can equivalently write Y 7→ (Y k− Y −1 )+ Y .

(53)

Now writing k− = I + (λ) with (λ) ∈ G− an infinitesimal parameter, we obtain the infinitesimal version of this,
(54) Y 7→ I + (Y (λ)Y −1 )+ Y . We note that this directly gives the generating function of [4] for these transformations, which was originally obtained by extrapolation from the leading terms in a power series expansion [6]. The G+ projection corresponds to taking the singular part at λ = ±1. This may be done using a contour integral, so that this transformation takes the form   Z Y (x, λ0 )(λ0 )Y −1 (x, λ0 ) 0 1 dλ Y (x, λ) . (55) Y (x, λ) 7→ I + 2πi C λ0 − λ Here C± are oriented counter-clockwise around ±1. The transformation for g may be read off by taking the λ → 0 limit, yielding the form of the transformation given in [18, 17],   Z Y (x, λ0 )(λ0 )Y −1 (x, λ0 ) 0 1 (56) dλ . g 7→ g I − 2πi C λ0 The parameter of this infinitesimal transformation, (λ) is an arbitrary infinitesimal G− element. In particular, if we introduce a basis {T a } for the Lie algebra of antihermitian matrices, we can take (λ) proportional to λr T a , r ∈ Z . This gives an infinite set of transformations, which we denote Jra , and which satisfy the commutation relations X c fcab Jr+s , (57) [Jra , Jsb ] = c

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P where the fcab are the structure constants defined by [T a , T b ] = c fcab T c . Although the commutation relations of a centreless Kac-Moody algebra thus appear, this is not sufficient to identify the symmetry algebra G− with a centreless Kac-Moody algebra. We illustrate this in two ways: first we show that in G− there exist certain linear relations absent in a Kac-Moody algebra, and second we show that in G− the Jra are not a spanning set. The crucial point is that although we can certainly try to expand elements of G− in Laurent series, and finite sums of matrices of the form λr T a are certainly in G− , the natural way to expand an element of G− is in a Taylor series in λ + 1 (or alternatively in λ − 1). Taking (λ) in (56) proportional to (λ + 1)n T a , for n ≥ 0, we can define a set of transformations Kna satisfying the relations X b c ]= fcab Kn+m n, m ≥ 0. (58) [Kna , Km c

r

Considering the expansion of λ in powers of λ + 1 (valid in |λ + 1| < δ), we find that the Jra are expressed as linear combinations of the Kna in the following way:    P r r  (−1)n+r r≥0 Kna n=0 n a   Jr = P . (59)  ∞ (−1)r n − r − 1 K a r<0 n n=0 −r − 1 It is straightforward, using standard formulae for sums of binomial coefficients (see for example [7]), to check that these linear combinations, by virtue of (58), imply the commutation relations (57). The relation between the Jra for non-negative r can be inverted: we find n   X n (60) Kna = Jra . r r=0

Now, if our symmetry algebra were indeed a Kac-Moody algebra with generators Jra satisfying (57), we would be able to define the algebra elements Kna (which certainly exist as symmetry generators) from the Jra ’s with non-negative r using (60). When we substitute (60) into the infinite sum in (59) we find that we cannot reorder the summations to express this infinite sum as a linear combination of the Jra ’s with r ≥ 0. In other words, this infinite sum is not in the Kac-Moody algebra. We thus have our first distinction between a Kac-Moody algebra and G− : In a Kac-Moody algebra the elements Kna and the elements Jra for r < 0 need to be linearly independent, whereas in the PCM symmetry algebra G− they are linearly dependent via the relationship given in (59). The second distinction is that in G− , unlike in a regular Kac-Moody algebra, the elements {Jra } are not a spanning set. Elements of G− need to be analytic inside C. There are therefore elements of G− that do not have Laurent expansions in powers of λ; consider for example an (λ) proportional to ln λ, defined with a cut from 0 to ∞ along half of the imaginary axis. Now, the reader may be concerned that we have claimed that G− is spanned by the Kna , that the relationship between the Kna and the Jra for r ≥ 0 is invertible, but that the Jra (and therefore certainly the Jra for r ≥ 0) are not a spanning set for G− . There is absolutely no contradiction here. As we have seen above, the relationship between the Kna and the Jar for r ≥ 0 implies that finite linear combinations of the Kna can be written as linear combinations of the Jra for r ≥ 0, but for infinite linear combinations of the Kna this is not the case. However, it does suggest that we should be able in some sense to approximate elements of G− given by infinite sums of the Kna ’s by

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finite sums of the Jra , which are equivalent to finite sums of the Kna . This is indeed the case, as follows from a classical theorem in complex analysis, Runge’s theorem (see, for example, [15]). Runge’s theorem implies the remarkable fact that a function analytic on an arbitrary finite union of non-intersecting open discs can be approximated uniformly and to any accuracy on any closed subset of the union by a polynomial. In particular, this implies that elements in G− can be approximated uniformly and to any accuracy on {|λ − 1| < δ} ∪ {|λ + 1| < δ} by a finite linear combination of the Jra for r ≥ 0. To conclude this section we note that the contour integral in (56) is easily evaluated when (λ) is proportional to λr : For r < 0 the integral is evaluated by shrinking C to a contour around 0; for r > 0 to a contour around ∞; and for r = 0 to a pair of contours around 0 and ∞. 6.2. The B¨acklund transformation. The element k− ∈ G− in (43) can clearly have all variety of singularities outside C. Trying to give k− just one simple pole at the point λ = µ outside C, suggests the natural form [21]   N (µ) . (61) k− (λ, µ) = I + λ−µ For the satisfaction of the reality condition (16) for elements of G− we require that N† ¯ if π is a projector N† = N µ−µ¯ = −N . These conditions are satisfied by N = (µ − µ)π, satisfying π 2 = π = π † . Such transformations thus correspond to finite right-dressing transformation of the particular form   µ − µ¯ π . (62) U0 7→ U0 I + λ−µ Note that k− in fact has a singularity at λ = µ¯ as well, since (I −π) has zero determinant. Using (50) we obtain the transformation   µ − µ¯ −1 −1 Y (λ)πY (λ) Y (λ) . I+ (63) U 7→ S λ−µ In order to factorise the middle factor, we introduce a hermitian projector P = P † = P 2 , independent of λ (but not of x± ). Using this we see that   µ¯ −1 Y (λ)πY (λ) Y (λ) I + µ− λ−µ     µ¯ µ−µ ¯ µ−µ¯ −1 P I + P I + Y (λ)πY = I + µ− (λ) Y (λ) λ−µ λ−µ¯ λ−µ    µ¯ µ−µ¯ µ−µ ¯ (I ) (I P I + P Y (λ) . = I + µ− − P Y (λ)π + − π) λ−µ λ−µ λ−µ¯ To have an acceptable factorisation, all we need now is that the right-hand factor above be regular outside C. Specifically, we require regularity at µ and µ, ¯ which yields algebraic conditions relating the projectors P and π, viz. (I − P )Yµ π = 0 ,

P Yµ (I − π) = 0,

where Yµ denotes Y (λ) evaluated at λ = µ. If we write π = v(v † v)−1 v † (see [9]), these equations are solved by the expression
−1 † † P = Yµ v v † Yµ† Yµ v v Yµ .

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Now we can read-off the induced transformation rules for Y and g. These are just the known PCM B¨acklund transformations [9, 13, 21]. 6.3. Left dressings. Here we consider in detail the left dressings (44). Matrices h+ ∈ G0,+ commute with M , so such transformations act by left multiplication on U , i.e. U 7→ h+ U = h+ S −1 Y = S −1 (Sh+ S −1 )Y . Hence the action on Y is given by Y 7→ (h+ S −1 )+ Y = (Sh+ S −1 )+ Y.

(64)

For an infinitesimal transformation h+ = I + ,  ∈ G0,+ and we have
Y 7→ I + (SS −1 )+ Y   Z 1 S(λ0 )(λ0 )S −1 (λ0 ) 0 = I+ dλ Y, 2πi C λ0 − λ implying



1 g→ 7 g I− 2πi

Z C

 S(λ0 )(λ0 )S −1 (λ0 ) 0 dλ . λ0

(65)

(66)

In general  has the form (λ) =

∞  X n=1

α en αn + (1 + λ)n (1 − λ)n

 ,

(67)

en are constant infinitesimal diagonal matrices. The integral in (66) is where the αn , α evaluated by computing the residues of the integrand at λ0 = ±1. For example, the case en zero yields the transformation rules α1 6= 0 with all other αn , α g −1 δg = −s0 α1 s−1 0 ,   −1 δA+ = A+ , [s1 s−1 0 , s 0 α1 s0 ] , δA− =

(68)

− 21 [A− , s0 α1 s−1 0 ].

en zero we find Similarly, if α2 6= 0 with all other αn , α


−1 −1 g −1 δg = − s0 α2 s−1 0 + [s1 s0 , s0 α2 s0 ] ,   −1 −1 −1 −1 , δA+ = A+ , [s2 s−1 0 , s0 α2 s0 ] − [s1 s0 , s0 α2 s0 ]s1 s0   −1 −1 −1 δA− = − A− , 41 s0 α2 s0 + 21 [s1 s0 , s0 α2 s0 ] .

(69)

The formulae for δA± are computed using the variation of the relation (6), δA± = ∂± (g −1 δg) + [A± , g −1 δg],

(70)

and Eqs. (38)-(41). The latter also allow one to check directly that the above transformations are indeed infinitesimal symmetries, i.e. that ∂− δA+ + ∂+ δA− = 0. Now considering the sector of PCM in which A = α1 , independent of x+ , we see that the ∂+ -derivations of A± given by (3) and (42) are effected by the transformations (68). So left dressing transformations with only α1 non-zero correspond to x+ translations in this sector. Similarly the transformations (69) can be seen to be related to coordinate translations in an extended system (described in the appendix) belonging to a hierarchy associated to the PCM. Whenever an infinite dimensional abelian symmetry algebra

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(like G0,+ ) is identified in a system, it is possible to define a corresponding hierarchy. Traditionally, for each generator in the algebra a coordinate is introduced and the flow in each coordinate is defined as the infinitesimal action of the corresponding symmetry. In our formulation there is an alternative way to define a PCM hierarchy. Instead of working on a space M with coordinates (x+ , x− ), we work on a larger space M with − 2P coordinates (x+1 , . . . , x+P , x− 1 , . . . , xP ) and replace the  of (12) by =−

P  X An (x+ )dx+ n

(1 + λ)n

n=1

n

+

− Bn (x− n )dxn n (1 − λ)

 ,

(71)

where the An (x+n ), Bn (x− n ) are all antihermitian diagonal matrices, each depending on only one coordinate. The associated nonlinear equations are again the equations dZ = Z ∧ Z, where Z = (SS −1 )+ and S is a map from M to G− . For the case P = 2 we write out this system of equations in full in the appendix. Another possibility of obtaining a hierarchy within our framework is to enlarge M to a space with 2N P a− coordinates (xa+ n , xn ), 1 ≤ n ≤ P , 1 ≤ a ≤ N , and taking =−

N  a a+ P X X A (x )H a dxa+ n

n=1 a=1

n

(1 + λ)n

n

+

a a− Bna (xa− n )H dxn n (1 − λ)

 ,

(72)

where {H a }, a = 1, . . . , N is a basis for the algebra of antihermitian, diagonal N × N matrices. In this hierarchy, left dressings on U0 correspond precisely to coordinate translations in the sector with the scalar functions Aan , Bna constant. The physical or geometric significance of these PCM hierarchies remains to be understood. An alternative approach to defining a PCM hierarchy was given in [1]. 6.4. The Virasoro symmetry. In this section we consider the symmetries of PCM associated with reparametrisations of U0 (λ). We consider the infinitesimal reparametrisations U0 (λ) → U0 (λ + m λm+1 ), where the m are infinitesimal parameters and m ∈ Z, or, equivalently, variations δU0 = m λm+1 U00 (λ). The prime denotes differentiation with respect to λ. These variations give rise to a centreless Virasoro algebra of infinitesimal symmetries of PCM. In [17] Schwarz documents the existence of “half” of this algebra. Schwarz’s symmetries are associated with reparametrisations that fix the points λ = ±1. We shall see that from a technical standpoint these are simpler to handle than the full set of symmetries. But there is also a fundamental reason to make such a restriction. If we were to consider finite reparametrisations, we would need to ensure that the contour C remains qualitatively unchanged. The simplest way to do this is to require the points λ = ±1 to be fixed. In [19] Uhlenbeck identifies an S 1 symmetry of PCM. It is a simple exercise to check that this symmetry corresponds, in our formalism, to global reparametrisations of the λ-plane fixing the points ±1, i.e. M¨obius transformations of the form aλ + b , a2 + b2 = 1. (73) λ→ bλ + a At the level of infinitesimal symmetries, however, the need to fix ±1 is really superfluous, and so we find a full Virasoro algebra of symmetries. But as we have said above, the symmetries fixing ±1 are technically easier, which is why Schwarz was able to identify them, and also for the more general symmetries we can be quite certain that there exists no exponentiation.

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With this introduction, we consider the variations δm U0 = m λm+1 U00 (λ). These manifestly realise the algebra [δm , δn ] = (n − m)δn+m . This realisation descends to the physical fields. Using U0 = e−M S −1 Y we have the chain of implications δm U0 = m λm+1 (−M 0 e−M S −1 Y − e−M S −1 S 0 S −1 Y + e−M S −1 Y 0 ), δ m U = eM δm U 0 = m λm+1 (−M 0 S −1 Y − S −1 S 0 S −1 Y + S −1 Y 0 ), δm S = −(Sδm U Y −1 )− S
= −m λm+1 (−SM 0 S −1 − S 0 S −1 + Y 0 Y −1 ) − S, δm Y = (Sδm U Y −1 )+ Y
= m λm+1 (−SM 0 S −1 + Y 0 Y −1 ) + Y.

(74) (75) (76) (77)

In the last equation we have used the fact that for all m, λm+1 S 0 S −1 takes values in G− . Of the remaining two terms, the first has a G+ piece originating in the double pole of M 0 at λ = ±1. To explicitly compute this is a simple exercise. For the second term, we use a contour integral formula for the projection. We thus arrive at the final result 

Z m+1 0 µ Y (µ)Y −1 (µ) 1 dµ 2πi C µ−λ  

 R R (−1)m  −1 m+1 1 +(−1)m s0 A s−1 + s1 s0 , s0 A s−1 − 0 0 2 (1 + λ) 1+λ 1+λ    

 R R 1 m+1 1 −1 −1 + se0 B se−1 + s e − s e , s e . (78) B s e 1 0 0 0 0 (1 − λ)2 1−λ 1−λ δm Y Y −1 = m

R x− R R R x+ Here A and B are shorthand for x+ A(y + )dy + and x− B(y − )dy − respectively. 0 0 The g transformations are read off by setting λ to zero. In the expression for δm g, the contour integral term is evaluated, depending on the value of m, by shrinking C to a contour around either 0 or ∞. Explicitly for the SL(2) subalgebra of the Virasoro algebra, we obtain (omitting the overall infinitesimal parameters),   −1 R R −1  −1 R R −1  s0 Be e1 se0 , (e s0 Be s0 )+ s1 s0 , (s0 As−1 s0 ) , g −1 δ−1 g = φ+(s0 As−1 0 )−(e 0 ) − s  −1 R −1   −1 R −1  −1 s0 Be g δ0 g = − s1 s0 , (s0 As0 ) − se1 se0 , (e s0 ) ,   −1 R −1 R −1  −1 R R −1  −1 s0 Be e1 se0 , (e s0 Be g δ1 g = f −(s0 As0 )+(e s0 )+ s1 s0 , (s0 As−1 s0 ) . 0 ) − s We see that in these formulae, not only do the leading coefficients s0 , s1 , se0 , se1 in the expansions of S appear, but also the fields φ and f , coefficients in the expansions of Y around 0 and ∞ respectively (see Sect. 4). The work required to check directly that these, or any of the δm ’s, are symmetries is formidable, but we again emphasize that the advantage of the present framework is that such direct checks are not necessary in order to prove that the physical fields carry a representation of the full centreless Virasoro algebra. Schwarz [17] has previously found half a Virasoro algebra. We observe that if we define transformations 1m = δm+1 − δm−1 a substantial simplification takes place, yielding the formula

Hidden Symmetries of Principal Chiral Model Unveiled



1 1m g = −m g 2πi

Z C

691

µm−1 (µ2 − 1)Y 0 (µ)Y −1 (µ)dµ

 R R −1 + 2(−1)m (s0 A s−1 ) − 2(e s ) . B s e 0 0 0

(79)

We will see in Sect. 6 (see Eqs. (83),(84)) that the second and third terms in the above expression are individually symmetries of PCM that mutually commute and commute with all the symmetries being considered here. Removing these terms gives exactly the “half-Virasoro” symmetries of [17] Z 1 e µm−1 (µ2 − 1)Y 0 (µ)Y −1 (µ)dµ, m ∈ Z. (80) 1m g = −m g 2πi C Thus we see the precise nature of Schwarz’s symmetries as combinations of reparametrisations preserving the points λ = ±1 with certain simple symmetries that act on the A, B fields but leave U0 invariant. Taking the appropriate combinations we see that for the e 0, simplest Schwarz symmetry 1

and using (34) and (37),

e 0 g = φ − f, g −1 1

(81)

e 0 A± = ∓2A± + [A± , f ]. 1

(82)

This is easily checked to be a symmetry. The symmetry 10 acts on the physical fields in a much more complicated way: R R −1 s0 Be s0 ), g −1 10 g = φ − f − 2(s0 As−1 0 ) + 2(e    R R −1  −1 10 A+ = −4A+ + [A+ , f ] − 2 [s1 s0 , A+ ], (s0 As−1 s0 Be s0 ) , 0 ) + A+ , (e  R −1   R −1  −1 s1 se0 , A− ], (e s0 Be 10 A− = 4A− + [A− , f ] + 2 [e s0 ) − A− , (s0 As0 ) . 6.5. Transformations of the free fields A(x+ ), B(x− ). Following the by now familiar reasoning, an infinitesimal transformation A(x+ ) 7→ A(x+ ) + δA(x+ ) induces the following transformations on Y, g, A+ , A− : R (s0 δA s−1 0 ) δY = − Y, 1 + λ R δg = g (s0 δA s−1 0 ),    R −1 −1 δA+ = s0 δA s0 − A+ , s1 s−1 0 , (s0 δA s0 ) ,   R δA− = A− , (s0 δA s−1 0 ) . R R x+ Here we have written δA as shorthand for x+ δA(y + )dy + . As expected, the spectrum 0 of A− remains invariant, while that of A+ is shifted. Using the flow equations for s0 , s1 , it is easy to check that these are genuine symmetries, i.e. that ∂− δA+ + ∂+ δA− = 0. There are a variety of possibilities for δA(x+ ). If {H a }, a = 1 . . . N , is a basis of the algebra of antihermitian diagonal matrices, we can consider variations δA(x+ ) ∼ (x+ )m H a , a = 1, . . . , N , m ∈ Z. This gives a loop algebra of symmetries, corresponding to translations of A(x+ ). Taking δA(x+ ) ∼ (x+ )m A0 (x+ ), m ∈ Z, gives a centreless Virasoro algebra of symmetries, corresponding to reparametrizations of A(x+ ). Taking

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δA(x+ ) ∼ (x+ )m A(x+ ), m ∈ Z, gives an infinite dimensional abelian symmetry algebra corresponding to x+ -dependent rescalings of A(x+ ). Clearly these symmetries are not independent: The latter two families can be written in terms of the first family, but the generators are then field dependent combinations of the generators of the first family. Analogous sets of symmetries can be obtained from infinitesimal variations of B(x− ). The simple variation δA(x+ ) = A(x+ ), where  is a constant infinitesimal parameter, yields the symmetry R (83) δg = g(s0 A s−1 0 ), whereas the transformation B 7→ (1 + ζ)B, where ζ is also an infinitesimal parameter, yields R (84) δg = ζg(e s0 B se−1 0 ). These transformations were used in Sect. (6) to make contact between our Virasoro symmetries and those of [17]. 7. Concluding Remarks We have seen that formulating the nonlinear equations of motion (3) of the PCM in the form of the simple linear system (1) makes the precise nature of their integrability completely transparent. It yields a novel free-field parametrisation of the space of solutions, which we have used to classify all the symmetries of on-shell PCM fields in terms of natural transformations on the free-field data. The confusing cacophony of symmetry transformations in the literature is thereby seen to arise in the most natural fashion imaginable. We have thus demonstrated that this notion of complete integrability, previously applied to traditional soliton systems, like the KP, NLS and KdV hierarchies, encompasses the Lorentz–invariant PCM field theories. We believe that this notion of integrability is a universal one and we expect a clarification of the nature of the integrability of the self-dual Yang-Mills and self-dual gravity equations by similarly reformulating the twistor constructions for these systems. Indeed Crane [3] has already discussed a loop group of symmetries in terms of an action on free holomorphic data in twistor space. Our construction raises many questions. 1) Standard integrable soliton systems exhibit multiple hamiltonian structures and infinite numbers of conservation laws, both these phenomena being symptoms of their integrability. These phenomena ought to have a natural explanation in terms of the associated simple linear systems (free-field data). For the PCM, some work on such structures exists [5]. 2) The free-field parametrisation of solutions of PCM should play a critical role in the quantisation of the theory. What is the relation with standard quantisations? (The PCM can be quantised in different ways, using either the field f or the field g as fundamental, giving different results [12].) How are we to understand quantum integrability? 3) There is a large body of related mathematical work, mostly focusing on the enumeration and construction of solutions of the PCM in Euclidean space (for recent references see [2]). Most of our formalism goes through for the case of Euclidean space, but the reality conditions are different, and a little harder to handle. An important class of solutions are the unitons [19, 20]. These correspond, up to the need for right dressings by G+ elements, to Y ’s with finite order poles at one of the two points ±1, and regular elsewhere. We wonder: What are the corresponding U0 ’s? (The work of Crane on self-dual

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693

Yang-Mills [3] may have an analog.) Is there a natural geometric understanding of our construction? Or a relation with the constructions of [20] or [2]? 4) Is there a geometric interpretation of our PCM hierarchy?

Appendix. The PCM Hierarchy In Sect. 6.3 we have described a procedure to generate a PCM hierarchy. In this appendix we illustrate this procedure by obtaining the simplest integrable extension of the PCM equation. We use the  given in (71) for P = 2. Using Z = (SS −1 )+ we obtain the following form for Z:   [C+ , B+ ] B+ A+ dx+1 + dx+2 + Z= − 1+λ (1 + λ)2 1+λ    A− dx− [C− , B− ] B− − 1 + dx2 . + + 1−λ (1 − λ)2 1−λ 

The six fields A+ , B+ , C+ , A− , B− , C− are defined in terms of the coefficients of S and − the free fields A1 (x+1 ), A2 (x+2 ), B1 (x− 1 ), B2 (x2 ). They depend on the four coordinates − − + + x1 , x2 , x1 , x2 and are constrained in virtue of their defining relations thus: A+ commutes with B+ , A− commutes with B− and the spectra of A+ , B+ , A− , B− depend only − on x+1 , x+2 , x− 1 , x2 respectively. If we nevertheless ignore these constraints and simply substitute the above form for Z into dZ = Z ∧ Z, we find: 1. [A+ , B+ ] = [A− , B− ] = 0. 2. The following system of evolution equations for A+ , B+ , A− , B− : ∂2+ A+ = − 21 [A+ , [[B+ , C+ ], C+ ]] − [B+ , ∂1+ C+ + 21 [[A+ , C+ ], C+ ]], ∂1− A+ = 21 [A+ , A− ], ∂2− A+ = 21 [A+ , 21 B− + [C− , B− ]], ∂1+ B+ = [B+ , [A+ , C+ ]], ∂1− B+ = 21 [B+ , A− ], ∂2− B+ = 21 [B+ , 21 B− + [C− , B− ]], ∂1+ A− = 21 [A− , A+ ], ∂2+ A− = 21 [A− , 21 B+ + [C+ , B+ ]], ∂2− A− = − 21 [A− , [[B− , C− ], C− ]] − [B− , ∂1− C− + 21 [[A− , C− ], C− ]], ∂1+ B− = 21 [B− , A+ ], ∂2+ B− = 21 [B− , 21 B+ + [C+ , B+ ]], ∂1− B− = [B− , [A− , C− ]]. These evidently imply that the spectra of A+ , B+ , A− , B− depend only on x1+ , x2+ , x1− , x2− respectively, as required. 3. The following evolution equations for C+ , C− :

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∂1+ C− = − 41 A+ − 21 [A+ , C− ], ∂2+ C− = − 18 B+ + 41 ([C− , B+ ] − [C+ , B+ ]) + 21 [C− , [C+ , B+ ]], ∂1− C+ = − 41 A− − 21 [A− , C+ ], ∂2− C+ = − 18 B− + 41 ([C+ , B− ] − [C− , B− ]) + 21 [C+ , [C− , B− ]]. (In fact, from the dZ = Z ∧ Z equation, both of the C− evolutions appear commutated with B+ and both of the C+ evolutions appear commutated with B− .) This system is a 4-dimensional integrable system, but its physical or geometric interpretation is not immediately apparent. It has a variety of interesting reductions apart from the reduction to PCM by setting B− = B+ = 0. We can consistently reduce by taking A− = B− or A+ = B+ or both. Or we can take just B− = 0 (or B+ = 0) + in which case the x− 2 (or x2 ) dependence becomes trivial. For all these reductions, and the full system as well, the methods of this paper give a free-field parametrisation of solutions. Acknowledgement. We should like to thank Bernie Pinchuk and Larry Zalcman for discussions on Runge’s theorem. One of us (CD) is happy to thank the Emmy Noether Mathematics Institute of Bar–Ilan University for generous hospitality.

References 1. Bruschi, M., Levi, D., Ragnisco, O.: The chiral field hierarchy. Phys.Lett. 88A, 379–382 (1982) 2. Burstall, F.E., Guest, M.A.: Harmonic two-spheres in compact symmetric spaces, revisited. Preprint (1996) 3. Crane, L.: Action of the loop group on the self-dual Yang-Mills equation. Commun. Math. Phys. 110, 391–414 (1987) 4. Devchand, C., Fairlie, D.B.: A generating function for hidden symmetries of chiral models. Nucl.Phys. B194, 232-236 (1982) 5. Dickey, L.A.: Symplectic structure, Lagrangian, and involutiveness of first integrals of the principal chiral field equation. Commun. Math. Phys. 87, 505–513 (1983) 6. Dolan, L.: Kac-Moody algebra is hidden symmetry of chiral models. Phys. Rev. Lett. 47, 1371–1374 (1981) 7. Feller, W.: An introduction to probability theory and its applications, Volume I. New York–Chichester– Brisbane–Toronto: John Wiley & Sons, 1967 8. Haak, G., Schmidt, M., Schrader, R.: Group Theoretic Formulation of the Segal-Wilson Approach to Integrable Systems with Applications. Rev. Math. Phys. 4, 451–499 (1992) 9. Harnad, J., Saint-Aubin, Y., Shnider, S.: Superposition of solutions to B¨acklund transformations for the SU(N) principal σ-model. J. Math. Phys. 25, 368–375 (1983): Quadratic psuedopotentials for Gl(N, C) I principal sigma models. Physica 10D, 394–412 (1984); B¨acklund transformations for nonlinear sigma models with values in Riemannian symmetric spaces. Commun. Math. Phys. 92, 329–367 (1984; The soliton correlation matrix and the reduction problem for integrable systems. Commun. Math. Phys. 93, 33–56 (1984) 10. Leznov, A.N.: B¨acklund transformation for main chiral field problem with an arbitrary semisimple algebra. Preprint (1991) 11. Mulase, M.: Complete integrability of the Kadomtsev-Petviashvili equation. Adv. Math. 54, 57–66 (1984); Solvability of the super KP equation and a generalization of the Birkhoff decomposition. Inv. Math. 92, 1–46 (1988) 12. Nappi, C.R.: Some properties of an analog of the nonlinear sigma model. Phys. Rev. D21, 418–420 (1980) 13. Ogielski, A.T., Prasad, M.K., Sinha, A. Chau Wang, L-L.: B¨acklund transformations and local conservation laws for principal chiral fields. Phys. Lett. 91B, 387–391 (1980)

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14. Pohlmeyer, K.: Integrable Hamiltonian systems and interactions through quadratic constraints. Commun. Math. Phys. 46, 207–221 (1976) 15. Rudin, W.: Real and complex analysis. New York–St.Louis–San Francisco–Toronto–London–Sydney: McGraw-Hill Book Company, 1966 16. Schiff, J.: Symmetries of KdV and loop groups. Preprint (1996) (Archive number solv-int/9606004) 17. Schwarz, J.H.: Classical symmetries of some two-dimensional models. Nucl. Phys. B447, 137–182 (1995) 18. Ueno, K., Nakamura, Y.: The hidden symmetry of chiral fields and the Riemann-Hilbert problem. Phys. Lett. 117B, 208–212 (1982) 19. Uhlenbeck, K.: Harmonic maps into Lie groups (classical solutions of the chiral model). J. Diff. Geom. 30, 1–50 (1989) 20. Ward, R.S.: Classical solutions of the chiral model, unitons, and holomorphic vector bundles. Commun. Math. Phys. 128, 319–332 (1990) 21. Zakharov, V.E., Mikhailov, A.V.: Relativistically invariant two-dimensional models of field theory which are integrable by means of the inverse scattering problem method. Zh. Exp. Teor. Fiz. 74, 1953–1973 (1978) (English translation: Sov. Phys. JETP 47, 1017–1027 (1978)) Communicated by R. H. Dijkgraaf

Commun. Math. Phys. 190, 697 – 721 (1998)

Communications in

Mathematical Physics c Springer-Verlag 1997

Asymptotics of a Class of Solutions to the Cylindrical Toda Equations Craig A. Tracy1 , Harold Widom2 1 Department of Mathematics and Institute of Theoretical Dynamics, University of California, Davis, CA 95616, USA. E-mail: [email protected] 2 Department of Mathematics, University of California, Santa Cruz, CA 95064, USA. E-mail: [email protected]

Received: 23 January 1997 / Accepted: 8 May 1997

Abstract: The small t asymptotics of a class of solutions to the 2D cylindrical Toda equations is computed. The solutions, qk (t), have the representation qk (t) = log det (I − λ Kk ) − log det (I − λ Kk−1 ), where Kk are integral operators. This class includes the n-periodic cylindrical Toda equations. For n = 2 our results reduce to the previously computed asymptotics of the 2D radial sinh-Gordon equation and for n = 3 (and with an additional symmetry constraint) they reduce to earlier results for the radial Bullough-Dodd equation. Both of these special cases are examples of Painlev´e III and have arisen in various applications. The asymptotics of qk (t) are derived by computing the small t asymptotics  t a k det (I − λ Kk ) ∼ bk , n where explicit formulas are given for the quantities ak and bk . The method consists of showing that the resolvent operator of Kk has an approximation in terms of resolvents of certain Wiener-Hopf operators, for which there are explicit integral formulas.

1. Introduction We consider here solutions of the cylindrical Toda equations qk00 (t) + t−1 qk0 (t) = 4 (eqk (t)−qk−1 (t) − eqk+1 (t)−qk (t) ),

k ∈ Z,

(1.1)

satisfying the periodicity conditions qk+n = qk . The integer n is arbitrary but fixed. It follows from results in [9] that solutions valid for all t > 0 are given by qk (t) = log det (I − λ Kk ) − log det (I − λ Kk−1 ),

(1.2)

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C. A. Tracy, H. Widom

where Kk is the integral operator on R+ with kernel X ω

ω k cω

e−t[(1−ω)u+(1−ω −ωu + v

−1

)u−1 ]

,

(1.3)

ω running over the nth roots of unity other than 1. In the case n = 2 we have qk+1 = −qk and (1.1) becomes, with q equal to either qk , q 00 (t) + t−1 q 0 (t) = 8 sinh 2q(t), which can be reduced to a particular case of the Painlev´e III equation. The connection with Fredholm determinants was discovered by McCoy, Tracy and Wu [6], and in the same paper the asymptotics as t → 0 of these solutions q(t) were determined. (Note that all asymptotics as t → ∞ are trivial.) The asymptotics as t → 0 of det (I − λ2 K02 ) = det (I −λ K0 ) det (I +λ K0 ) were determined in [7]. (See also [2], where the asymptotics were found for a family of kernels including this one as a special case.) The asymptotics of det (I − λ K0 ) itself were stated without proof in [10]. A class of periodic cylindrical Toda equations arises in thermodynamic Bethe Ansatz considerations [3]. There the additional constraint q−k−1 = −qk is imposed. The solutions (1.2) satisfy this constraint as long as the coefficients cω satisfy cω = −ω 3 cω−1 . (This follows from the fact that det (I − Kk ) = det (I − K−k−2 ) in this case, which is proved by applying the change of variable u → u−1 .) The case n = 3 of this gives the cylindrical Bullough-Dodd equation (q = q3 now) q 00 (t) + t−1 q 0 (t) = 4 (e2q(t) − e−q(t) ), which can be reduced to another special case of Painlev´e III. Asymptotics of a class of solutions to PIII including this one were announced in [5]. This paper is devoted to the determination of the asymptotics of the quantities det (I − λ Kk ) in the general case, under the condition stated below. (In the final sections we shall compare our results in the cases n = 2 and 3 with those cited above.) We write K for K0 and consider at first only the asymptotics of det (I−λ K). This is no loss of generality since Kk is obtained from K upon R ∞ replacing the coefficients cω by ω k cω . The problem reduces to the asymptotics of 0 R(u, u; λ) du as t → 0, where R(u, v; λ) is the resolvent kernel of K, the kernel of K (I − λ K)−1 . Using operator techniques, we show that R(u, u; λ) is well-approximated on [1, ∞] by the corresponding function when the exponentials in (1.3) are replaced by e−t(1−ω)u and on [0, 1] by the corresponding function when the exponentials are replaced by −1 −1 e−t(1−ω )u . (Actually the kernels have to be modified first by multiplying by factors (u/v)β with β depending on λ.) We shall show that after these replacements we obtain operators which can be transformed into Wiener-Hopf operators, whose resolvent kernels have explicit integral representations. By these means the problem becomes that of determining the asymptotics of certain integrals. This is achieved by contour-shifting, and we find in the end that as t → 0,  t a (1.4) det (I − λ K) ∼ b n with a and b constants given explicitly in terms of certain zeros of the function X cω (−ω)s−1 . h(s) := sin πs − λ π ω

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699

These are the values at λ of those zeros which equal 1, · · · , n when λ = 0. To state the result precisely, we denote by αk = αk (λ) (k ∈ Z) the zeros of this function indexed so that αk (0) = k. The zeros depend analytically on λ as long as they are unequal, and when λ = 0 they are the integers. We derive the asymptotics (1.4) under the assumption that there is a path in the complex plane C running from 0 to λ such that everywhere on the path < α0 < < α1 , < α0 < 1, < α1 > 0,

(1.5)

and no zero lies in the strip < α0 < < s < < α1 . With this assumption the constants a and b are given by the formulas a=

1 X 2 (n + 1)(2n + 1) , α − n α 6 Q |j|
b= Q

G( nj + 1)n−|j| 0

α,α0

G( α−α + 1) n

,

where α and α0 run over the set {α1 (λ), · · · , αn (λ)} and G denotes the Barnes G-function [1]. From these formulas we obtain the asymptotics of the solutions (1.2). The requirement now is that everywhere on a path from 0 to λ we have for all k, < αk < < αk+1 , k − 1 < < αk < k + 1. If this holds then

t

+ log B + o(1), n where for k = 1, · · · , n the constants A and B are given by qk (t) = A log

A = 2 (αk − k), B =

Y 0( αj −αk + 1) Y n 1≤j
0(

αk −αj ) n

αj −αk ) n , αk −αj 0( n + 1) k<j≤n

0(

and for other values of k are given by periodicity. As for the correct range of validity of the formulas, we conjecture that it is enough that < αk < < αk+1 for all k for some path from 0 to λ, and that the extra condition k − 1 < < αk < k + 1 is automatically satisfied then. In the cases which we consider in detail this is so and we obtain the correct range of validity. Another way of stating the condition is as follows. Define 3 to be the complement of {λ : < αk = < αk+1 for some k}.

(1.6)

Then the region of validity should be the connected component of 3 containing λ = 0. The region for which we prove the result is the largest connected subset of this set in which the extra condition holds. Remark. It is shown in [9] that the more general class of kernels Z

e−t[(1−ω)u+(1−ω ω −ωu + v  k

−1

)u−1 ]

dρ(ω),

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C. A. Tracy, H. Widom

gives a solution to the the cylindrical Toda equations by the same formulas. Here ρ can be any finite complex measure supported on a compact subset  of {ω ∈ C : < ω < 1, < ω −1 < 1}. This assures that the operator is trace class. In case  is the set of nth roots of unity other than 1 the condition is satisfied and the solution will clearly be n-periodic. We shall actually do everything in the more general case and we find asymptotic formulas of the form (1.4) for the corresponding determinants, with the constants a and b being given by integral formulas involving the function h(s) now defined by Z (−ω)s−1 dρ(ω). (1.7) h(s) := sin πs − λ π 

In the periodic case h(s) is itself periodic and the integrals are expressible in terms of its zeros in a strip of width n. This is why the result there is so explicit. The requirements on the αk stated above now refer to this function, and we assume throughout that they are satisfied. 2. The Approximating Operators Recall that K is the operator on L2 (R+ ) with kernel Z −t[(1−ω)u+(1−ω−1 )u−1 ] e dρ(ω). K(u, v) = −ωu + v  We denote by R(u, v; λ) the resolvent kernel of K, the kernel of Rλ := K (I − λ K)−1 . It is well-known that Z ∞ d log det (I − λ K) = R(u, u; λ) du, − dλ 0 the trace of the operator Rλ . Hence Z

λ

log det (I − λ K) = −

Z

∞

R(u, u; µ) du dµ. 0

(2.1)

0

R ∞ In this section we are going to find a good approximation to the integral R(u, u; λ) du when λ satisfies the condition stated in the Introduction. (Afterwards 0 we shall replace λ by µ and integrate with respect to µ over the path from 0 to λ throughout which (1.5) holds.) We begin with an observation. If we multiply the kernel K(u, v) above by (u/v)β for any β then the resulting kernel still represents a bounded operator on L2 (R+ ) because of the decay of the exponential factor at 0 and ∞ and, although the resolvent kernel changes, its value on the diagonal u = v does not. We are going to find approximations to the resolvent kernels for these modified operators, and precisely which β we take depends on λ. Here is how we choose it. It follows from our main assumption that for each λ there there exists sλ ∈ (0, 1) such that the function h(s) given by (1.7) has no zeros on the line < s = sλ . (In fact the assumption guarantees that sλ can be chosen to vary continuously with λ.) With this sλ we set β = 21 − sλ . Notice that |β| < 21 , a fact we shall need in order to apply our approximation argument.

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We write the kernel as Kt (u, v) =

 u β Z e−t[(1−ω)u+(1−ω−1 )u−1 ] dρ(ω) v −ωu + v 

(2.2)

and denote the operator itself by Kt . We do not display the dependence on β, which is fixed for now, but use the subscript t to help the reader distinguish those operators that depend on t from those that don’t. Both kinds will arise; the former have the subscript t and the latter will not. −1 −1 The exponential in (2.2) is the product e−t(1−ω)u e−t(1−ω )u . For u ≥ 1 the second factor is uniformly close to 1 when t is small while for u ≤ 1 the first factor is uniformly close to 1. This suggests that the operators Kt± with kernels Kt+ (u, v) :=

 u β Z e−t(1−ω)u dρ(ω), v  −ωu + v

Kt− (u, v) :=

 u β Z e−t(1−ω−1 )u−1 dρ(ω), v −ωu + v 

should in some sense approximate Kt on u ≥ 1, u ≤ 1, respectively, and therefore the resolvent kernels of these operators should approximate the resolvent kernel of Kt on these intervals. We shall show that this is so, and that if Rt± (u, v; λ) denote the resolvent kernels of Kt± , also on R+ , then Z 1 Z 1 R(u, u; λ) du = Rt− (u, u; λ) du + o(1), 0

Z

0

Z

∞

R(u, u; λ) du = 1

1

∞

(2.3) Rt+ (u, u; λ) du + o(1)

as t → 0. We denote by P + multiplication by the characteristic function of (1, ∞) and by P − multiplication by the characteristic function of (0, 1). We shall use the notation o1 (λ) to denote any family of operators whose trace norms are at most |λ| times a function of t which is o(1) as t → 0. (The subscript 1 refers to the trace norm. We shall also use the obvious notation o1 (1) later on.) The main approximation statement will be P ± (I − λ Kt )−1 P ± = P ± (I − λ Kt± )−1 P ± + o1 (λ).

(2.4)

Relations (2.3) with λ = 1 follow from this since it may be rewritten P ± [(I − λ Kt )−1 − I] P ± = P ± [(I − λ Kt± )−1 − I] P ± + o1 (λ), and if we take the trace of both sides and divide by λ we obtain (2.3). Here is an outline of the proof of (2.4). We use the matrix representations of our operators corresponding to the decomposition of L2 (R+ ) as the direct sum of the spaces L2 (0, 1) and L2 (1, ∞). Thus (the equal sign meaning “has matrix representation”)   I − λ P − Kt P − −λ P − Kt P + . I − λ Kt =  −λ P + Kt P − I − λ P + Kt P + Because the nondiagonal corners of the matrix have the mutually orthogonal projections P ± occurring as they do we will be able, with error o1 (λ), to replace the operator Kt appearing there by the operator K0 obtained from it by setting t = 0. Thus K0 has kernel

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C. A. Tracy, H. Widom

K0 (u, v) =

 u β Z v



1 dρ(ω). −ωu + v

(2.5)

(Note the lack of consistency with the notation Kk in the introduction; this should cause no confusion.) If the diagonal entries I − λ P ± Kt P ± are invertible, we can write the resulting matrix as the product of   0 I − λ P − Kt P −   0 I − λ P + Kt P + on the left and 

−λ (I − λ P − Kt P − )−1 P − K0 P +

I

 −λ (I − λ P

+

Kt+ P + )−1 P + K0 P −

 

I

on the right. Next, because of our assumption we shall be able to show that the operators I − λ P ± Kt P ± are uniformly invertible for small t (i.e., the operator norms of their inverses are bounded) and that their inverses converge strongly to (I −λ P ± K0 P ± )−1 as t → 0. (Recall that At is said to converge strongly to A as t → 0 if At f → Af for all f in the underlying space.) This is actually the crux of the proof. After that it follows, because P ± K0 P ∓ are trace class, that with error o1 (λ) we can replace (I − λ P ± Kt P ± )−1 by (I − λ P ± K0 P ± )−1 in the nondiagonal entries. Thus, if we define   I −λ (I − λ P − K0 P − )−1 P − K0 P + , M :=  −λ (I − λ P + K0+ P + )−1 P + K0 P − I we will have shown



I − λ Kt = 

I − λ P − Kt P −

0

0

I − λ P + Kt P +

  M + o1 (λ).

From this, using the uniform invertibility of the I−λ P ± Kt P ± again and the invertibility of the constant matrix M (which we have to prove) we deduce   (I − λ P − Kt P − )−1 0  + o1 (λ). (I − λ Kt )−1 = M−1  0 (I − λ P + Kt P + )−1 Now (here is the trick), applying an analogous procedure to the operator family I −λ Kt+ gives   (I − λ P − K0 P − )−1 0  + o1 (λ). (I − λ Kt+ )−1 = M−1  0 (I − λ P + Kt P + )−1 It is clear that the lower-right entries of the two matrix products are the same, and this is exactly the statement P + (I − λ Kt )−1 P + = P + (I − λ Kt+ )−1 P + + o1 (λ),

Asymptotics of Solutions to Cylindrical Toda Equations

703

which is half of (2.4). The other half is obtained similarly. Carrying out the details of the proof of this will require, first, some general facts about families of operators on a Hilbert space. Fact 1. If At converges strongly to an invertible operator A as t → 0 and if the At are −1 uniformly invertible then A−1 t also converges strongly to A . −1 −1 = A−1 This follows from the assumptions and the identity A−1 t −A t (A − At )A . The next fact says that strong convergence can sometimes be converted into trace norm convergence.

Fact 2. If At → A strongly and Bt → B in trace norm then At Bt → AB in trace norm. This is a variant of Proposition 2.1 of [8]. There the families of operators depended on a parameter n ∈ Z+ rather than t ∈ R+ , a matter of no importance since we may consider general sequences tn → 0. Also, instead of a sequence of operators converging in trace norm to B there was the single trace class operator B. The apparently more general result follows trivially from this special case. Fact 3. Suppose At and A are as in Fact 1 and that Bt are trace class operators converging in trace norm to B. Assume also that A + B is invertible. Then the At + Bt are uniformly invertible for sufficiently small t, and if Bt = o1 (1) then (At + Bt )−1 = A−1 t + o1 (1). −1 → A−1 strongly, and so Proof. Write At + Bt = At (I + A−1 t Bt ). By Fact 1 At −1 −1 by Fact 2 (with At repaced by At ) we deduce I + At Bt = I + A−1 B + o1 (1) = (I + o1 (1))(I + A−1 B), since clearly (I + o1 (1))−1 = I + o1 (1). Both statements now follow. 

In our derivation of (2.4) we have to know that the I − P ± Kt± P ± are uniformly invertible for small t. We shall deduce this from known facts about uniform invertibility of truncated Wiener-Hopf operators, which we now describe. The proofs can be found in [4]. The Wiener-Hopf operator W associated with a function k ∈ L1 (R) is the operator on L2 (R+ ) with kernel k(x − y). Introduce the Fourier transform of k, Z ∞ ˆk(ξ) = eiξx k(x) dx. −∞

This is a continuous function on R tending to 0 as ξ → ±∞. A necessary and sufficient ˆ condition that I − W be invertible is that 1 − k(ξ) 6= 0 for all ξ, and ∞ ˆ = 0. arg (1 − k(ξ)) −∞

The truncated Wiener-Hopf operators are the operators Pα W Pα , where Pα denotes multiplication by the characteristic function of (0, α). Clearly these operators converge strongly to W as α → ∞. The important fact is that if W is invertible, in other words if the conditions on kˆ stated above hold, then the operators I − Pα W Pα are uniformly invertible for sufficiently large α. We mention also that the operator with kernel k(x − y) on the whole line R is invertible if and only if the first condtion alone is satisfied, that ˆ 1 − k(ξ) 6= 0 for all ξ.

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From this we can deduce information about kernels k(u, v) which are homogeneous of degree −1 since the variable change u = ex transforms this kernel into a convolution kernel. More precisely, denote by U the unitary operator from L2 (1, ∞) to L2 (0, ∞) given by U f (x) = ex/2 f (ex ). Then if T denotes the operator on L2 (1, ∞) with kernel k(u, v), the operator U T U −1 is the operator on L2 (0, ∞) with kernel ex/2 ey/2 k(ex , ey ) = e(y−x)/2 k(1, ey−x ), where we used the homogeneity of k(u, v). Notice also that if Pt+ denotes multiplication by the characteristic function of (1, t−1 ) then U Pt+ U −1 is the projection operator Plog t−1 of the last paragraph. After making an obvious change of variable in computing the Fourier transform of e−x/2 k(1, e−x ) we deduce R∞ Fact 4. Assume that 0 v −1/2 |k(1, v)| dv < ∞, denote by T the operator with kernel k(u, v) on L2 (1, ∞) with kernel k(u, v), and by M (s) the Mellin transform Z ∞ v s−1 k(1, v) dv. M (s) := 0

Then a necessary and sufficient condition that I − T be invertible is that 21 +i∞ 1 − M (s) 6= 0 for < s = 21 , arg (1 − M (s)) 1 = 0. 2 −i∞

(2.6)

If this holds then the operators I −Pt+ T Pt+ are uniformly invertible for sufficiently small t. Remark. If we use the variable change u = e−x instead of u = ex then our operator T acts on L2 (0, 1) and we find (again using homogeneity) that the condition for invertibility of I − T is exactly the same as before, and that if Pt− denotes multiplication by the characteristic function of (t, 1) then this condition implies the uniform invertibility of I − Pt− T Pt− for sufficiently small t. Also (transferring to this context the last sentence of the discussion of Wiener-Hopf operators), the same condition implies the invertibility of the operator I − T on L2 (0, ∞). We apply this to the kernel λK0 , which is homogeneous of degree −1. The relevant Mellin transform is found to be Z π (−ω)s−β−1 dρ(ω), M (s) = λ sin πs  so that 1 − M (s) = h(s − β), where h is given by (1.7). If we recall that β = 21 − sλ we see that the conditions (2.6) are met, the first immediately from the definition of sλ and the second because the index (the variation of the argument, which is necessarily an integer) is a continuous function of λ, locally constant and clearly equal to 0 when λ = 0. So we know that the operators I − λ K0 and I − λ P ± K0 P ± are invertible and the operator families I − λ Pt± K0 Pt± are uniformly invertible for small t. We introduce one last piece of notation. We denote by K ± the operators on L2 (R+ ) with kernels  u β Z e−(1−ω) u  u β Z e−(1−ω−1 ) u−1 + − K (u, v) = dρ(ω), K (u, v) = dρ(ω). v v −ωu + v  −ωu + v 

Asymptotics of Solutions to Cylindrical Toda Equations

705

Notice that rescaling K ± (u, v) under the variable change u → t±1 u gives Kt± (u, v). Next we derive various trace class properties of our operators which will be needed. For these we shall use an estimate for the trace norm of an operator on L2 (R+ ) with kernel of the form Z q1 (ω, u) q2 (ω, v) dρ(ω), (2.7) −ωu + v  which is a special case of the sublemma in the appendix of [9]. We denote by φ(s) any positive function on R+ , by 8 its Laplace transform, and by 9 the Laplace transform of φ(s)−1 . Then there is a constant m depending on only on  such that the trace norm of the operator with kernel given above is at most m−1 times the square root of Z ∞Z Z ∞Z 2 |q1 (ω, u)| 8(mu) dρ(ω) du · |q2 (ω, u)|2 9(mu) dρ(ω) du. (2.8) 0



0



From this will follow our first lemma. In the proof we denote by χ+ the characteristic function of (1, ∞) and by χ− the characteristic function of (0, 1), so P ± is multiplication by χ± . Lemma 1. The operators (independent of t) P + K0 P − , P + K + , P − (K + − K0 ) are trace class. The operators (depending on t) P + (Kt − K0 )P − , P − (Kt+ − K0 ), P + (Kt − Kt+ ) are o1 (1) as t → 0. The statements also hold if all superscripts + and − are interchanged. Proof. All the operators have kernel of the form (2.7). We list the operators below, together with the corresponding functions q1 (ω, u) and q2 (ω, u). Operator

q1 (ω, u)

q2 (ω, u)

P + K0 P − P +K + P − (K + − K0 ) P + (Kt − K0 )P − P − (Kt+ − K0 ) P + (Kt − Kt+ )

χ(1,∞) (u) uβ χ(1,∞) (u) e−(1−ω) u uβ χ(0,1) (u) (e−(1−ω) u − 1) uβ χ(1,∞) (u) (e−t [(1−ω) u+(1−ω−1 ) u−1 ] − 1) uβ χ(0,1) (u) (e−t(1−ω) u − 1) uβ χ(1,∞) (u) e−t(1−ω) u (e−t(1−ω−1 ) u−1 − 1) uβ

χ(0,1) (u) u−β u−β u−β χ(0,1) (u) u−β u−β u−β .

For each of these operators we take two numbers p, q ∈ (−1, 1) and define φ(s) = sp for s ≤ 1 and φ(s) = sq for s ≥ 1. We easily see that    O(u−p−1 ) for u ≥ 1  O(up−1 ) for u ≥ 1 8(s) = 9(s) =  O(u−q−1 ) for u ≤ 1,  O(uq−1 ) for u ≤ 1. For each operator one can find p and q such that both integrals in (2.8) are finite, and any integral depending on t is o(1) as t → 0. In fact, as the reader can check, we may take for all the operators any q ∈ (2β, 2β + 2), for the first and fourth operators any p > 2β and for the other four any p ∈ (2β − 2, 2β). This takes care of the six displayed operators. For the other six we use the fact that the substitutions u → u−1 , ω → ω −1 yield operators of the same form with β replaced with −β and with the superscripts interchanged.

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As a preliminary to the next lemma we show that certain modified Laplace tranforms are bounded operators on L2 (R+ ). Lemma 2. The integral operator on L2 (R+ ) with kernel (uv)−β e−uv is bounded if β < 21 . Proof. The mapping f (v) → v −1 f (v −1 ) is unitary, so we may replace the kernel by u−β v β−1 e−u/v . Under the unitary mapping f (u) → ex/2 f (ex ) this becomes the kernel x−y 1 on (−∞, ∞). Thus the operator becomes convolution by the L1 e( 2 −β)(x−y) e−e x 1 function k(x) = e( 2 −β)x e−e and so is bounded.  Lemma 3. The operators I − λ K ± are invertible. Proof. We consider K + , which we can write as P − K0 + P − (K + − K0 ) + P + K + . By the first part of Lemma 1 the second and third summands are trace class, therefore certainly compact. Our assumption implies that I − λ P − K0 is invertible. Hence I − λ K + is the sum of an invertible operator and a compact operator, and so it follows from general theory that it will be invertible if 0 is not an eigenvalue. In other words it suffices to prove that λ K + f = f for f ∈ L2 (R+ ) implies f = 0. For any ω ∈ C\R+ and any f ∈ L2 (R+ ) we have for x > 0, Z ∞ Z ∞  β −(1−ω) u e u f (v) dv e−xu u−β du v −ωu + v 0 0 Z ∞ Z ∞ dy e−yv v −β f (v) dv. = x + 1 − ω(y + 1) 0 0 This can be seen for < ω < 0 by using the integral reresentation Z ∞ 1 = e−y(−ωu+v) dy −ωu + v 0 in the integral on the left above and interchanging the order of integration. The identity follows for all ω ∈  since both sides are analytic functions of ω in this domain. Now suppose that λ K + f = f . Then if we integrate both sides of the identity with respect to dρ(ω) and multiply by λ, the left side becomes the Laplace transform of u−β f (u), which we denote by g(x), and the the right side becomes Z ∞Z dρ(ω) g(y) dy. λ x + 1 − ω(y + 1)  0 Using the fact y −β g(y) ∈ L2 , which we know by the previous lemma, we see that the integral is a bounded function of x. If we recall the definition (2.5) then we see that the identity becomes Z ∞ x + 1 β K0 (y + 1, x + 1) g(y) dy (x ≥ 0), g(x) = λ y+1 0 or x−β g(x − 1) = λ

Z

∞

K0 (y, x) g(y − 1) y −β dy

(x ≥ 1).

1

Now we know that x−β g(x) is in L2 (0, ∞) and that g(x) is bounded. It follows that x−β g(x − 1) belongs to L2 (1, ∞). The right side above is the operator with kernel

Asymptotics of Solutions to Cylindrical Toda Equations

707

K0 (y, x) acting on this function. Thus (if f 6= 0) the operator λ P + K00 P + has 1 as an eigenvalue, where 0 denotes transpose, so I − λ P + K00 P + is not invertible. But this implies I − λ P + K0 P + is not invertible, whereas we know that it is. This contradiction establishes the lemma. Lemma 4. The operators I − λ P ± Kt± P ± are uniformly invertible for sufficiently small t. Proof. We consider I − λ P + Kt+ P + and for this it is enough to show that the I − λP + Kt+ are uniformly invertible. The kernel of P + Kt+ is χ(1,∞) (u) Kt+ (u, v) and the substitution u → t−1 u allows us to consider instead the operator with kernel χ(t,∞) (u) K + (u, v). We write this (not displaying the variables u and v) as χ(t,1) K0 + χ(1,∞) K + + χ(0,1) (K + − K0 ) + χ(0,t) (K0 − K + ). Recalling the definitions of our various projection operators we see that the first kernel corresponds to the operator Pt− K0 , and we know that the I − λ Pt− K0 are uniformly invertible for sufficiently small t. The second and third summands correspond to the operators P + K + and P + (K + − K0 ), which we know by Lemma 1 to be trace class. The last summand corresponds to the operator P − (K0 − K + ), which we know by Lemma 1 to be trace class, left-multiplied by multiplication by χ(0,t) , which converges strongly to 0. An application of Fact 2 shows that this last operator is o1 (1). The strong limit of the sum of the four operators is, of course, K + and we know by Lemma 2 that I − λ K + is invertible. Hence we can apply Fact 3 to deduce the result. We can now fill in the details of the proof of (2.4) outlined earlier. Thus we begin with the representation   I − λ P − Kt P − −λ P − Kt P + . I − λ Kt =  + − + + −λ P Kt P I − λ P Kt P Applying Lemma 1 to the nondiagonal entries we deduce   I − λ P − Kt− P − −λ P − K0 P +  + o1 (λ). I − λ Kt =  + − + + + −λ P K0 P I − λ P Kt P Lemma 3 tells us in particular that the diagonal entries of this matrix are invertible for small t so we may factor out   0 I − λ P − Kt P −   0 I − λ P + Kt P + on the left, leaving 

I

 −λ (I − λ P

+

Kt+ P + )−1 P + K0 P −

−λ (I − λ P − Kt− P − )−1 P − K0 P +

 

I

on the right. Next we combine the uniform invertibility of the I − λ P ± Kt± P ± proved in Lemma 3 with Fact 1 to deduce that the inverses of these operators converge strongly

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C. A. Tracy, H. Widom

to (I − λ P ± K0± P ± )−1 . Since P ± K0 P ∓ are trace class, by Lemma 1, we deduce by Fact 2 that the matrix above is M + o1 (λ), where M is the matrix obtained by replacing Kt± by K0 . Thus  I − λ Kt = 



I − λ P − Kt P −

0

0

I − λ P + Kt P +

 M + o1 (λ).

Now we have to know that M is invertible, and we see this as follows. If, instead of the operator I − λ Kt which depends on t, we had started with the operator I − λK0 then we would have obtained the exact representation  I − λ K0 = 



I − λ P − K0 P −

0

0

I − λ P + K0 P +

 M.

Since both I − λ K0 and the matrix on the left are invertible, by our assumption, we deduce that M is invertible. Next we go through a similar process starting with the operator I − λ Kt+ rather than I − λ K. Using the fact that P − Kt+ = P − K0 + o1 (1), which we know by Lemma 1, we obtain in this case   I − λ P − K0 P − 0  M + o1 (λ). I − λ Kt+ =  + + 0 I − λ P Kt P From these matrix representations and the facts that M and I −λ P − K0 P − are invertible and I − P ± Kt± P ± uniformly invertible we deduce, using Fact 3 with o1 (1) replaced by o1 (λ),  (I − λ Kt )−1 = M−1 

(I − λ P − Kt P − )−1 0

 (I − λ Kt+ )−1 = M−1 



0 + −1

(I − λ P Kt P ) +

(I − λ P − K0 P − )−1 0



0 + −1

(I − λ P Kt P ) +

 + o1 (λ),

 + o1 (λ).

Comparing lower-right entries of the matrices gives P + (I − K)−1 P + = P + (I − Kt+ )−1 P + + o1 (λ), which is half of (2.4). The other half is obtained similarly. Remark. To apply (2.3) to (2.1) we need something extra, e.g., that (2.3) holds uniformly for these λ. With a little care our argument gives this also, but we spare the reader the details.

Asymptotics of Solutions to Cylindrical Toda Equations

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3. The Resolvents of K ± We are going to find R ∞integral representations for the integrals on the right side of (2.3), and we consider 1 Rt+ (u, u; λ) du first. The substitution u → u/t shows that this R∞ equals t R+ (u, u; λ) du, where R+ (u, v; λ) is the resolvent kernel of the operator K + . For this we require only that  be a compact subset of {ω ∈ C : < ω < 1, ω 6∈ R+ }, −1

(3.1) +

since the term 1−ω does not appear in the exponent in the kernel of K . The derivation will involve an initial step which is valid only when  is contained in the left half-plane so we assume this to begin with. We shall also assume that λ is so small that h(s) 6= 0 for < s = 21 , so that with the notation of the last section we may take sλ = 21 , β = 0. Eventually these two assumptions will be removed by an analytic continuation argument. Because β = 0 the kernel of K + is Z −t(1−ω)u e + K (u, v) = dρ(ω). (3.2)  −ωu + v Z

If we set

e−(1−ω)u eωux dρ(ω),

A(u, x) :=

B(x, u) := e−ux ,



Z

then

∞

K + (u, v) =

A(u, x) B(x, v) dx. 0

Lemma 2 of the preceding section tells us that B(u, x) is the kernel of a bounded operator from L2 (R+ ) to L2 (R+ ) and, with our assumption on , that the same is true of A(x, u). The above shows that K + = AB, and the operator BA has kernel Z ∞ Z Z dρ(ω) dρ(ω) = . B(x, u) A(u, y) du = x − ωy + 1 − ω (x + 1) − ω(y + 1)   0 We use the general fact AB(I − λAB)−1 = A(I − λBA)−1 B to deduce that R+ (u, u) is given by an inner product,   (3.3) R+ (u, u) = (I − λBA)−1 B( · , u), A(u, · ) . We begin by computing

f := (I − λBA)−1 B( · , u).

Thus we want to solve Z ∞Z f (x) − λ 0

Z

or f (x − 1) − λ

1



dρ(ω) f (y) dy = e−ux (x + 1) − ω(y + 1)

∞

Z 

dρ(ω) f (y − 1) dy = e−u(x−1) x − ωy

(x ≥ 0),

(x ≥ 1).

The substitution x → ex brings this to the form of a Wiener-Hopf equation, so we can use the factorization method to find the solution. We begin by decreeing that the last identity holds for all x ≥ 0, in other words we define f on (−1, 0) by the identity. Then we define

710

C. A. Tracy, H. Widom

Z F− (s) :=

∞

Z xs−1 f (x − 1) dx,

1

xs−1 f (x − 1) dx.

F+ (s) :=

1

0

These belong to the Hardy spaces H2 (< s < 21 ), H2 (< s > 21 ), respectively. We take Mellin transforms of both sides of the equation, and find that for < s = 21 , Z λπ (−ω)s−1 dρ(ω) F− (s) = eu u−s 0(s). F− (s) + F+ (s) − sin πs  (The exponential in the integral is made definite by taking | arg(−ω)| < π.) We write this as (3.4) H(s) F− (s) + F+ (s) = eu u−s 0(s), where

λπ h(s) =1− H(s) := sin πs sin πs

Z (−ω)s−1 dρ(ω).

(3.5)



This function is bounded and analytic in each vertical strip of the complex s-plane, away from the zeros of sin πs, H(s) − 1 → 0 exponentially as =s → ±∞ and 21 +i∞ = 0. arg H(s) 1 2 −i∞

Thus there is be a representation H(s) =

H− (s) , H+ (s)

where H− (s)±1 are bounded and analytic in < s ≤ 21 + δ for some δ > 0 and H+ (s)±1 are bounded and analytic in < s ≥ 21 − δ. We multiply (3.4) by H+ (s) and use the decomposition F = F− + F+ of an arbitrary function in L2 (< s = 21 ) into boundary functions of functions in H2 (< s < 21 ) and H2 (< s > 21 ) to write the result as     = −H+ (s) F+ (s) + eu u−s 0(s) H+ (s) . H− (s) F− (s) − eu u−s 0(s) H+ (s) −

+

The two sides are boundary functions of functions in H2 (< s < 21 ) and H2 (< s > 21 ), respectively, so they both vanish. This gives the representation  eu  −s F− (s) = u 0(s) H+ (s) . (3.6) H− (s) − Now (see (3.3)) we have to multiply f (x) by A(u, x) and integrate with respect to x over (0, ∞). This is Z Z ∞ Z ∞ Z dρ(ω) f (x) e−(1−ω)u eωux dx = e−u dρ(ω) f (x−1) χ(1,∞) (x) eωux dx. 

0



0

The Mellin transform of f (x − 1) χ(1,∞) (x) equals F− (s) and the Mellin transform of eωux equals (−ωu)−s 0(s), so Parseval’s formula for Mellin transforms shows that the above equals Z Z ds −u , (−ωu)s−1 dρ(ω) 0(1 − s) F− (s) e 2πi 

Asymptotics of Solutions to Cylindrical Toda Equations

711

the outer integration taken over < s = 21 . (All vertical integrals are taken in the direction from −i∞ to i∞.) Next we recall (3.6) and use the integral representation of the operator G → G− to write the above as Z Z Z −s0 0(1 − s) ds u 0(s0 )H+ (s0 ) ds0 , (−ωu)s−1 dρ(ω) H− (s) 2πi s0 − s 2πi  the inner integral taken over < s0 = Z Z (−ω)s−1 dρ(ω) 

1 2

+ δ. Alternatively, this may be written

0(1 − s) ds H− (s) 2πi

Z

0

u−s −1 0(s0 + s)H+ (s0 + s) ds0 , s0 2πi

where now the inner integral is taken over < s0 = δ. The integrands of these integrals vanish exponentially at infinity, and u occurs to the power −s0 − 1, which has real part −δ − 1. Thus we may integrate with respect to u from t to ∞ under the integral signs and deduce that Z ∞ R+ (u, u; λ) du t

Z Z (−ω)s−1 dρ(ω)

= 

0(1 − s) ds H− (s) 2πi

Z

0

t−s 0(s0 + s)H+ (s0 + s) ds0 . s02 2πi

It follows from (3.5), and the gamma function representation of the last factor there, that Z (−ω)s−1 dρ(ω) 0(1 − s) = λ−1 (1 − H(s)) 0(s)−1 . 

Thus we have shown (reverting to the resolvent Rt+ ) that Z ∞ Rt+ (u, u; λ) du λ 1

Z

1 2 +i∞

= 1 2 −i∞

−1

(H− (s)

−1

− H+ (s)

−1

) 0(s)

ds 2πi

Z

δ+i∞ −s0

t

δ−i∞

0(s0 + s)H+ (s0 + s) ds0 . s02 2πi

This was proved if λ is sufficiently small and if  lies in the left half-plane. Let us remove the latter condition first. For any η define the measure ρη by ρη (E) = ρ(E − η). This has support  + η. For all η in a neighborhood in C of [−1, 0] the set  + η is contained in the region (3.1). For η near −1 the set will also lie in the left half-plane. If λ is small enough the condition on the zeros will be satisfied for the measures ρη for all η in a neighborhood of [−1, 0]. For such λ we know that the above formula holds for η in a neighborhood of −1. But both sides are analytic functions of η in our neighborhood. Thus the formula must hold for η = 0 also, which is what we wanted to show. To remove the condition that λ be small we must modify the formula to read Z ∞ Rt+ (u, u; λ) du λ 1

Z

sλ +i∞

= sλ −i∞

(H− (s)−1 − H+ (s)−1 ) 0(s)−1

ds 2πi

Z

δ+i∞ −s0

t

δ−i∞

0(s0 + s)H+ (s0 + s) ds0 , s02 2πi (3.7)

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C. A. Tracy, H. Widom

where sλ ∈ (0, 1) is as in the previous section, a continuously varying function of λ such that 1 − H(s) is nonzero on the line < s = sλ . Both sides of (3.7) are analytic functions of λ for λ in a neighborhood of our path, they agree near λ = 0, so they agree everywhere on the path. As for R− (u, u; λ) on (0, 1) the change of variable u → u−1 transforms the kernel K − (u, v) into Z Z −t(1−ω−1 )u e e−t(1−ω)u + e dρ(ω) = (−ω) dρ(ω −1 ). K (u, v) :=  −ωv + u −1 −ωu + v Therefore

e+ (u−1 , u−1 ; λ), R− (u, u; λ) = u−2 R

e+ is the resolvent kernel for K e + , and so where R Z ∞ Z 1 e+ (u, u; λ) du. R− (u, u; λ) du = R 0

1

e + replaces H(s) by H(1 − s) and so the integral It is easy to see that replacing K + by K is equal to (3.7) with H(s) replaced by H(1 − s) and sλ replaced by 1 − sλ . Now that we have these explicit representations it is obvious what we do: in the inner integral in (3.7) we move the line of s0 -integration from < s0 = δ to < s0 = −δ. We can do this if δ is small enough. The residue at the double pole at s0 = 0 contributes Z Z ds 1 ds − (H(s)−1 −1) log t+ (H− (s)−1 −H+ (s)−1 ) (0(s) H+ (s)) 0 , (3.8) 2πi 0(s) 2πi the integrations taken over < s = sλ , and the error term is O(tδ ). We do the same with R ∞H(s) replaced by H(1 − s) and add, and so we have obtained the asymptotics of λ 0 R(u, u; λ) du. 4. Asymptotics of det (I − K) —The Periodic case The formula for H(s) is now H(s) = 1 −

λπ X cω (−ω)s−1 , sin πs 

ω running over the nth roots of unity other than 1. If in our sums we set −ω = πi e n (2j−n) (j = 1, · · · , n − 1), then | arg (−ω)| = | nπ (2j − n)| < π as required. If we also πi set z = e n s , then the above may be written as sin πs H(s) = sin πs + λπ

n−1 X

ω −1 cω z 2j−n ,

j=1

or

n−1 h i X ω −1 cω z 2j . 2i sin πs H(s) = z −n z 2n − 1 + 2iλπ j=1

Asymptotics of Solutions to Cylindrical Toda Equations

713

Recall that sin πs H(s) = h(s). The expression in brackets above is a polynomial of 2πi degree n in z 2 and its zeros are the quantities by e n α , where α runs through the zeros αk = αk (λ) (k = 1, · · · , n). With this notation the right side above is equal to z −n

Y

(z 2 − e

2πi n α

α

)=

Y

πi

πi

(z e− n α − z −1 e n α )

α

Y

πi

e n α.

(4.1)

α

Here and below the index α runs over the set {α1 , · · · , αn }. The last product, a square root of the product of the roots of the polynomial, equals ±1 or ±i. This product equals πi πi e n (1+···+n) = in+1 for λ = 0 and so for all λ. Recalling that z = e n s we see that we have obtained the representation H(s) =

π (−1)n 2n−1 Y sin (s − α). sin πs n α

We now evaluate the integrals in (3.8). For λ sufficiently small again, the αk will all lie in the strip 21 < < s < n+ 21 . We may assume this since the usual analytic continuation will give the general case. To evaluate the first integral in (3.8) we consider Z ds (H(s)−1 − 1) s 2πi taken over the infinite rectangle which is the contour running from n+ 21 −i∞ to n+ 21 +i∞ and then from 21 + i∞ to 21 − i∞. On the one hand this equals n times the first integral in (3.8), and on the other hand it equals the sum of the residues at the poles between the two lines. Thus we have shown that Z 1 X ds = α H 0 (α)−1 . (4.2) (H(s)−1 − 1) 2πi n α For the second integral in (3.8) we have to write down the explicit expression for the factors H± (s). These are given by Q H+ (s) =

α

0( s−α (−1)n 2n−1 0(1 − s) s n + 1) s Q n , H− (s) = n . s−α 0(s) α 0(− n )

It is readily verified that H(s) = H− (s)/H+ (s) and that H− (s)±1 and H+ (s)±1 are are bounded and analytic in < s ≤ 21 + δ and < s ≥ 21 − δ, respectively, for small λ. Thus, they are the correct factors. The second integral in (3.8) may be written Z (0(s) H+ (s))0 ds , (H(s)−1 − 1) 0(s) H+ (s) 2πi and by the above expression for H+ (s) this equals Z

−1

(H(s)

i ds h 1 X 00 ( s−α0 + 1) n + log n , − 1) 0 n 0 0( s−α 2πi n + 1) α

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C. A. Tracy, H. Widom

where in the sum α0 also runs over the set {α1 , · · · , αn }. The contribution of the term log n is exactly log n times (4.2). To evaluate the rest of this integral we use the characteristic property of the Barnes G-function, G(z + 1) = 0(z) G(z). Putting z equal to s−α0 n + 1 and taking logarithmic derivatives gives 0

0

0

G0 ( s−α 00 ( s−α G0 ( s−α n + 2) n + 1) n + 1) − = . 0 0 s−α s−α s−α0 G( n + 2) G( n + 1) 0( n + 1) We integrate (H(s)−1 − 1)

X G0 ( s−α0 + 1) n s−α0 G( n + 1) α0

over the same infinite rectangle as before. (This is justified by the fact that H(s)−1 − 1 vanishes exponentially at ∞ in vertical strips while G0 (z)/G(z) grows like z log z.) By the above relation the result is exactly the integral we want, and so computing residues gives the formula 0 Z X 00 ( s−α0 + 1) ds + 1) 0 −1 1 X G0 ( α−α 1 −1 n n H (α) . (H(s) − 1) = s−α0 α−α0 n 2πi n 0( n + 1) G( n + 1) α0 α,α0 Thus, we have shown that Z ∞ t + b+ (λ) + O(tδ ), λ Rt+ (u, u, λ)du = a+ (λ) log n 1 where 0 + 1) 0 −1 1 X 1 X G0 ( α−α 0 −1 + n H (α) . α H (α) , b (λ) = a (λ) = − 0 α−α n α n G( n + 1) α,α0

+

(4.3)

R1 We must add to this λ 0 Rt− (u, u, λ) du which, as was mentioned earlier, is obtained by replacing H(s) by H(1 − s). The zeros of this function which lie near 1, · · · , n for small λ are n − α + 1. Hence (4.3) is replaced by a− (λ) =

1 X (n − α + 1) H 0 (α)−1 , n α

0 0 + 1) 0 0 −1 1 X G0 ( α n−α + 1) 0 −1 1 X G0 ( α−α n H (α) = − H (α ) . b (λ) = − α0 −α α−α0 n n G( n + 1) G( n + 1) α,α0 α,α0

−

d H(1 − s) = −H 0 (1 − s). Adding Here we have used the periodicity of H and the fact ds and using (2.3), we see that Z ∞ t λ + b(λ) + O(tδ ) R(u, u; λ) du = a(λ) log (4.4) n 0

where a(λ) = a+ (λ) + a− (λ), b(λ) = b+ (λ) + b− (λ). Now to obtain the asymptotics of log det (I − K) we must replace λ by µ, multiply the above by −dµ/µ and integrate from 0 to λ. (Notice the factor λ on the left side of (4.4) and recall the minus sign in (2.1).) We obtain from (3.5) that for a zero α(λ) of H

Asymptotics of Solutions to Cylindrical Toda Equations

715

we have λ dα/dλ = H 0 (α)−1 . Thus the coefficient a(λ) in (4.4) may be written (after replacing λ by µ and thinking of α as α(µ)) −

1X (2α − n − 1) µ dα/dµ. n α

Multiplying by −dµ/µ and integrating gives µ=λ 1 X µ=λ 1X 2 (α − (n + 1) α) = α2 n α n α µ=0 µ=0 since, as we have already seen,

P

α is independent of λ. Similarly b(µ) may be written

0 + 1) 1 X G0 ( α−α n µ d(α − α0 )/dµ, 0 α−α n G( + 1) 0 n α,α

and multiplying by −dµ/µ and integrating gives −

X α,α0

log G(

µ=λ α − α0 + 1) . n µ=0

If we recall that when µ = 0 the zeros are 1, · · · , n we see that the formulas for the constants in (1.4) are the ones stated in the introduction. To obtain the asymptotics of the qk (t) we must consider det (I − λ Kk ) instead of det (I − λ K). This amounts to replacing the coefficients cω by ω k cω , and this in turn amounts to replacing H(s) by H(s + k). The zeros of this function modulo n are α1 (λ) − k, · · · , αn (λ) − k. But these are not the zeros which are to replace the α, α0 in our formulas for a and b since they do not arise from the zeros whose values are 1, · · · , n when λ = 0. Rather, the replacements must be αk+1 (λ) − k, · · · , αn (λ) − k, α1 (λ) + n − k, · · · , αk (λ) + n − k, which are the zeros with this property. Thus, for the asymptotics of qk (t) we make these replacements in our formulas and the corresponding replacements with k − 1 instead of k, subtract, and take logarithms. The result is found, after some computation and the use of the functional equation for the G-function, to be the asymptotics stated in the introduction. 5. Asymptotics of det (I − K) – The Nonperiodic Case The coefficients a(λ) = a+ (λ) + a− (λ), b(λ) = b+ (λ) + b− (λ) of the last section were in general given by integral formulas. They were Z ds , (5.1) a+ (λ) = − (H(s)−1 − 1) 2πi Z 1 ds + + (0(s) H+ (s)) 0 , (5.2) b (λ) − a (λ) log n = (H− (s)−1 − H+ (s)−1 ) 0(s) 2πi

716

C. A. Tracy, H. Widom

with the formulas for a− (λ), and b− (λ) obtained by replacing H(s) by H(1 − s). The integrations may be taken over < s = 21 if λ is small enough and, as usual, this is no loss of generality. To find a and b we integrate a(µ) and b(µ), respectively, with respect to −dµ/µ over a path from 0 to λ. Write Z π (−ω)s−1 dρ(ω), ϕ(s) := sin πs  so that H(s) = 1 − λ ϕ(s). Making this replacement in the integrand in (5.1) and integrating gives Z λ Z λ  dµ ϕ(s) 1 −1 =− dµ = log (1 − λϕ(s)), − 1 − µϕ(s) µ 1 − µϕ(s) 0 0 so the contribution to the coefficient of log t is Z ds − log (1 − λϕ(s)) 2πi over < s = 21 . Replacing H(s) by H(1 − s) gives the same contribution since we may make the substitution s → 1 − s. Therefore Z ds . (5.3) a = −2 log (1 − λϕ(s)) 2πi For general λ the integration is to be on < s = sλ . Now we go to (5.2), which may be written Z  ds Z  H 0 (s) H 0 (s)  ds 00 (s)  1 + −1 + − + . (5.4) b+ (λ) − a+ (λ) log n = 0(s) H(s) 2πi H− (s) H+ (s) 2πi By a computation like the earlier one we see that the first integral becomes after the µ-integration Z ds 00 (s) log (1 − λϕ(s)) . 0(s) 2πi Then we replace s by 1 − s, make the substitution s → 1 − s, and add. We see that the contribution of the first integral in (5.4) equals Z  0 ds 0 (s) 00 (1 − s)  + log (1 − λϕ(s)) . (5.5) 0(s) 0(1 − s) 2πi Finally, we look at the second integral in (5.4), which equals Z Z Z ds H+0 (s) ds H+0 (s) −1 ds = H(s) = (log H+ )0 (s) H(s)−1 . H− (s) 2πi H+ (s) 2πi 2πi Replacing H(s) by H(1 − s) replaces H+ (s) by 1/H− (1 − s), so after making the substitution s → 1 − s and adding we get Z ds . (5.6) (log H+ H− )0 (s) H(s)−1 2πi Recall that for < s = 21 ,

Asymptotics of Solutions to Cylindrical Toda Equations

717

1 log H± (s) = ∓ log H(s) + 2

Z

log H(s0 ) ds0 , s0 − s 2πi

where the integral is the Hilbert transform, a principal value integral over < s0 = 21 . So Z log H(s0 ) ds0 . log H+ (s) H− (s) = s0 − s πi Since the Hilbert transform commutes with differentiation we get Z 1 H 0 (s0 ) ds0 , (log H+ H− )0 (s) = πi H(s0 ) s0 − s and so (5.6) equals −

1 2π 2

Z Z

0 H 0 (s0 ) −1 ds H(s) ds. H(s0 ) s0 − s

The µ-integration gives Z λ Z λ −µ ϕ0 (s0 ) 1 1 dµ 0 0 − = ϕ dµ (s ) 1 − µϕ(s0 ) 1 − µϕ(s) µ (1 − µϕ(s0 )) (1 − µϕ(s)) 0 0 = ϕ0 (s0 )

1 − λϕ(s) 1 log , ϕ(s0 ) − ϕ(s) 1 − λϕ(s0 )

and so the contribution of the second integral in (5.4) is Z Z 1 − λ ϕ(s) ds0 1 ϕ0 (s0 ) log ds. − 2 2π ϕ(s0 ) − ϕ(s) 1 − λ ϕ(s0 ) s0 − s

(5.7)

Thus b − a log n equals the sum of (5.5) and (5.7). As usual, for general λ the integrals are taken over < s, s0 = sλ . Remark. The double integral (5.7) is exactly the constant in the known asymptotics for the determinants of the truncated Wiener-Hopf operators associated with ϕ (specifically, ϕ( 21 + iξ) is the Fourier transform of the convolving kernel), and (5.3) is (minus twice) the leading coefficient in the asymptotics. One can see by the argument of Section 2 how both these things arise and conclude also that (5.7) equals det M−1 . The extra ingredient here is therefore the integral (5.5). 6. The Case n = 2 In this case the only root is ω = −1 and we may take c−1 equal to 1 since it occurs only in the product λc−1 . Thus the kernel of K0 is e−2t(u+u u+v

−1

)

,

and the equation (for either qk ) is q 00 (t) + t−1 q 0 (t) = 8 sinh 2q(t).

(6.1)

718

C. A. Tracy, H. Widom

We have in this case h(s) = sin πs − πλ, the zeros are given by α0 =

p 1 1 arcsin πλ = log(πiλ + 1 − π 2 λ2 ), α1 = 1 − α0 , π πi

and αk+2 = αk + 2. The square root is that branch which is positive for λ = 0 and the logarithm that branch which is 0 there. From this it is easy to see that the set (1.6) consists of the rays (−∞, −1/π] and [1/π, ∞) and 3, the proposed region of validity of our√formulas, is the complex plane cut along these rays. If we note that the function πiλ + 1 − π 2 λ2 maps 3 onto the right half-plane, we see that |< αk (λ) − k| < 21 for all λ ∈ 3 and so the “extra” condition on the α is satisfied. The range of validity is therefore all of 3. Using the formulas stated in the introduction we find that det (I − λ K0 ) ∼ b (t/2)a with a = α02 + α0 , b =

G( 21 ) G( 23 ) 0( 21 ) G( 21 )2 = . G( 21 + α0 ) G( 23 − α0 ) 0( 21 − α0 ) G( 21 + α0 ) G( 21 − α0 )

For det (I + λ K0 ) we replace λ by −λ, which amounts to replacing α0 by −α0 . If we multiply the two results together we recover the asymptotics for det (I − λ2 K02 ) determined in [7] and [2]. For q0 we have the asymptotics A log(t/2) + log B + o(1), where A = 2 α0 , B =

0( 21 − α0 ) , 0( 21 + α0 )

in agreement with [6]. This is the solution of (6.1) which is asymptotic to −2λK0 (4t) as t → ∞, where this K0 is the Bessel function. For λ 6∈ 3 the asymptotics are different. For λ > 1/π, eq0 (t) has an infinite sequence of zeros as t → 0 and for λ < −1/π, it has an infinite sequence of poles; this follows from the fact that as t → 0 the spectrum of K0 fills up the interval [0, π]. A heuristic derivation of the asymptotics for λ on the cut is given in [6]. In the next section we present a similar derivation for some cases of n = 3. 7. n = 3 and Cylindrical Bullough-Dodd The cylindrical Bullough-Dodd equation q 00 (t) + t−1 q 0 (t) = 4e2q − 4e−q ,

(7.1)

arises in the special case of n = 3, where cω = −ω 3 cω−1 . Then q1 = 0, q2 = −q3 and (7.1) is satisfied by q = q3 . If we set ζ := e2πi/3 , then cζ may be chosen arbitrarily. If we choose it to be ζ(1 − ζ) then cζ 2 = ζ 2 (1 − ζ 2 ), c−1 = 0 and √ h(s) = sin πs + 2π 3 λ sin(π(s + 2)/3). Again λ is the one free parameter. The zeros are given by α0 =

1 3 λ  1 − arcsin + , α1 = 1, α2 = 2 − α0 , 4 2π 2 2λc

Asymptotics of Solutions to Cylindrical Toda Equations

719

√ where λc = 1/(2 3π). Now 3 is the complement of the union of cuts (−∞, −3λc ] ∪ [λc , ∞). For λ ∈ 3 the zeros satisfy 1 5 < α0 ∈ (− , 1), < α1 = 1, < α2 ∈ (1, ), 2 2 and so the extra condition is again automatically satisfied and our formulas hold for all λ ∈ 3. If we write 1 (7.2) q(t) = A log( ) − log B + o(1) t then the connection formulas give in this case

α+2 2α+1 −A 0 3
0 3
A = −2α, B = 3 , 0 1−α 0 2−2α 3 3 where we wrote α for α0 . For large t,

 √  q(t) ∼ 6λK0 2 3t .

Asymptotics at the critical value λc . This section and the following ones are heuristic. Using the differential equation (7.1) one can determine the correction terms to (7.2): B2 4 B4 1 2−2A 2+A t − t + t4−4A + . . . . q(t) = A log( ) + log B + t (1 − A)2 B(2 + A)2 2(1 − A)4 (7.3) This is valid for λ ∈ 3. To understand the higher order terms in more detail it is convenient to define w(t) = exp (−q(t)) . where w satisfies the equation w00 =

1 02 1 0 4 (w ) − w + 4w2 − . w t w

(7.4)

The asymptotics we proved become the statement w(t) = BtA (1 + o(1)) . Using (7.4) to calculate the higher order terms in the small t expansion for w we find  w(t)=BtA 1 −

4B 12B 2 4+2A t2−2A 2+A + t + t + ··· B 2 (1 − A)2 (2 + A)2 (2 + A)4

(j + 1) 2j B j 2j+jA t + ··· (2 + A)2j  2 4 24(A − 2A − 2) 4−A 6−3A t + t + · · · . + (2 + A)2 (1 − A)2 (4 − A)2 B (1 − A)4 (4 − A)2 B 3

+

(7.5)

In contrast to (7.3) the terms t2m−2mA only appear for m = 1 in the above expansion. As λ varies from 0 to λc α varies from 0 to − 21 , so A varies from 0 to 1 and B from 1 to ∞. Observe that the first two terms in (7.5) are of the same order in t as t → 0 (and

720

C. A. Tracy, H. Widom

A → 1) whereas the others are of lower order. This suggests that when λ = λc we have w(t) ∼ t1 as t → 0, where   4 t2−2A (7.6) 1 := lim B 1 − 2 = 2 log(1/t) − log 2 + 2 log 3 − 2γ. 2 1 B (1 − A) 3 α→− 2 We now use the differential equation (7.4) to find the higher order terms, which are polynomial in t and 1 . (The only property of 1 used in the formal expansion is d1 /dt = −2/t.) The expansion is  
4 7 4 4 4 8 2 w(t) = t1 + t 1 + 1 + + t 8131 + 21621 + 2401 + 80 +O(t10 41 ). 9 3 9 2187 (7.7) Thus (as for the n = 2 analogue [6]) if one were to alter the constant appearing in (7.6) then the solution of (7.4) whose asymptotics is (7.7) would not match onto the solution that approaches 1 as t → ∞. These asymptotics at λ = λc were checked by numerically solving (7.4) in both a forward and backward integration. There was agreement to nine decimal places at t = 1/4. Asymptotics at the critical value −3λc . We proceed as above and examine all terms that would be of the same order of magnitude as λ → −3λc , when α → 1 and A → −2. These are the terms of the geometric series, those involving the powers t2j+jA . Summing the series we see that we must compute lim

α→1

B tA 1−

2Bt2+A /(2

Defining

+


2 A)2

=

1 . 2t2 (log t − log 3 + γ)2

2 = log t − log 3 + γ

we thus see that at λ = −3λc , w(t) ∼

1 2t2 22

.

(7.8)

To compute higher order terms it is convenient to look at v(t) = 1/w(t). Using the differential equation and only the property d2 /dt = 1/t of 2 we find   t8 32 76 40 40 20 v(t) = 2t2 22 + 862 − 52 + 42 − 32 + 22 − 2 + O(t14 10 2 ). 9 3 9 9 27 81 (7.9) Asymptotics for λ > λc . Think of λ as being on the lower part of the cut [λc , ∞). Then 1 3 α = − − iµ, 2 2 where µ :=

1 λ  1 arccosh + , (µ > 0). π 2 2λc

Thus A = 1 + 3iµ. Here again the first two terms in (7.5) are of the same order as t → 0 whereas the others are of lower order, and we obtain

Asymptotics of Solutions to Cylindrical Toda Equations

w(t) = BtA −

721

t1−2A + O(t4 ). B(1 − A)2

Substituting in the values of A and B in terms of µ we find     2t 3iµ log(t/3) 0(1/2 − iµ/2)0(−iµ) = e + O(t4 ). w(t) = 3µ 0(1/2 + iµ/2)0(iµ) If we had taken λ to be on the upper part of the cut then we would have replaced µ by −µ. The result would have been precisely the same. Remark. In [5] a method was described to find connection formulas for solutions of a class of equations including (7.1). Away from the critical values the short-range asymptotics stated there correspond to the first two terms in (7.5). As for the asymptotics at the critical values, our formulas agree with [5] at λ = −3λc but at λ = λc we differ by a factor of 2. Acknowledgement. This work was supported in part by National Science Foundation Grants DMS-9303413 (first author) and DMS-9424292 (second author). The authors also thank the Volkswagen- Stiftung for their support of the Research in Pairs program at Oberwolfach; the first results of the paper were obtained during the authors’ visit under this program.

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