Infinite Dimensional Algebras and Quantum Integrable Systems (Progress in Mathematics)

Progress in Mathematics Volume 237 Series Editors H. Bass X Oesterle A. Weinstein Petr P. Kulish Nenad Manojlovic He...

Author: Petr P. Kulish | Nenad Manojlovic | Henning Samtleben

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Progress in Mathematics Volume 237

Series Editors H. Bass X Oesterle A. Weinstein

Petr P. Kulish Nenad Manojlovic Henning Samtleben Editors

Infinite Dimensional Algebras and Quantum Integrable Systems

Birkhauser Verlag Basel • Boston • Berlin

Authors'. Petr P. Kulish St. Petersburg Department of Steklov Mathematical Institute Rassian Academy of Sciences Fontaka27 191011 St. Petersburg Russia e-mail: [email protected]

Nenad Manojlovich Departamcnto de Matematica Faculdade Ue Cicncias e Tecnologia Universidade do Algarve Campus de Gambelas 8005-139 Faro Portugal e-mail: [email protected]

Henning Sanitleben Ilnd Institute for Theoretical Physics University of Hamburg Lumper Chaussee 149 22761 Hamburg Germany e-mail: henning s;imUeben(«)desy.de

2000 Mathematics Subject Classification 14H15, I4H70, 17B37, 17B55, 17B67, 17B68, 17B69, 17B80, 17B81, 20GI0, 32V60, 32G15, 32G34, 33C70, 33C80, 35Q58, 37J35, 37K10, 46E20, 58B20, 5RFO7, 81R10, 81R50, 81U15, 8IU20, 81T10, 81T40, 82B20, 82B23

A CIP catalogue record for this book is available from the Library of Congress, Washington D.C., USA Bibliographic information published b; Hie Deutsche Ribliothck Die Deutsche Bihliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at .

ISBN 3-7643-7215-X Birkhauser Verlag, Basel - Boston - Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use whatsoever, permission from the copyright owner must be obtained. © 2005 Birkhauscr Verlag, P.O. Box 133, CH-4010 Basel, Switzerland Part of Springer Science (-Business Media Printed on acid-free paper produced of chlorine-free pulp, TCF « Printed in Germany ISBN-10:3-7643-7215-X ISBN-13:978-3-7643-7215-fi 987654321

www.birkhauser.cb

Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

vii

E. Frenkel Gaudin Model and Opers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

O.A. Castro-Alvaredo and A. Fring Integrable Models with Unstable Particles . . . . . . . . . . . . . . . . . . . . . . . . . . .

59

V.G. Kac and M. Wakimoto Quantum Reduction in the Twisted Case . . . . . . . . . . . . . . . . . . . . . . . . . . . .

89

A. Gerasimov, S. Kharchev and D. Lebedev Representation Theory and Quantum Integrability . . . . . . . . . . . . . . . . . . . 133 H.E. Boos, V.E. Korepin and F.A. Smirnov Connecting Lattice and Relativistic Models via Conformal Field Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

157

Kanehisa Takasaki Elliptic Spectral Parameter and Infinite-Dimensional Grassmann Variety . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 Takashi Takebe Trigonometric Degeneration and Orbifold Wess-Zumino-Witten Model. II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 L.A. Takhtajan and Lee-Peng Teo Weil-Petersson Geometry of the Universal Teichm¨ uller Space . . . . . . . . . 225 V. Tarasov Duality for Knizhinik-Zamolodchikov and Dynamical Equations, and Hypergeometric Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235

Preface The workshop “Infinite dimensional algebras and quantum integrable systems” was held in July 2003 at the University of Algarve, Faro, Portugal, as a satellite workshop of the XIV International Congress on Mathematical Physics. Recent developments in the theory of infinite dimensional algebras and their applications to quantum integrable systems were reviewed in invited lectures and a number of contributions from the participants. This volume presents the invited lectures of the workshop. V. Kac and M. Wakimoto describe the representation theory of twisted vertex algebras obtained by quantum Hamiltonian reduction from affine superalgebras. They present a unified representation theory of twisted superconformal algebras. In particular this leads to unified free field realizations and determinant formulas. Examples include the Ramond type sectors and twisted sectors of the N = 1, 2, 3, 4 and the big N = 4 superconformal algebras. E. Frenkel reviews relations between the Gaudin model and opers. He introduces the Gaudin algebra to a Lie algebra g as a commutative subalgebra of U (g)⊗N that contains in particular the Hamiltonians of the Gaudin model. The spectrum of this algebra can be identified with the space of opers associated to the Langlands dual Lie group L G to g. Eventually, that allows to relate solutions of the Bethe Ansatz equations to Miura opers and further to the flag varieties associated to L G. L. Takhtajan and Lee-Peng Teo give a brief summary of recent work on geometrical structures on the universal Teichm¨ uller space T (1). They define a Weil-Petersson metric on T (1) by Hilbert space inner products on tangent spaces, compute its Riemann curvature tensor, and show that T (1) is a K¨ ahler-Einstein manifold with negative constant Ricci curvature. Several lectures are devoted to the applications to quantum integrable models, conformal field theory, and in particular the Knizhnik-Zamolodchikov equations. A. Gerasimov, S. Kharchev and D. Lebedev describe various constructions in the representation theory of classical and quantum groups that are inspired by the Quantum Inverse Scattering Method. Using the separation of variable method in the modern group-theoretical framework, they review recent results on the analytic continuation of Gelfand-Zetlin theory to infinite-dimensional representations of U (gln ) and present the generalization to the quantum groups Uq (gln ). They further demonstrate the applications to quantum integrable systems of Toda type.

viii

Preface

H. Boos, V. Korepin and F. Smirnov present new results on correlation functions of the quantum group invariant XXZ-model. These results are based on the relation previously found by Jimbo and Miwa between XXZ correlators and solutions of the q-deformed Knizhnik-Zamolodchikov equations on level −4. These solutions are further related to level 0 solutions; the new formulae suggest the decomposition of general matrix elements with respect to states of the infrared CFT. Takashi Takabe in his lecture discusses the trigonometric Wess-ZuminoWitten (WZW) model. Based on the result that the trigonometric WZW model is factorized into the orbifold WZW models, he shows that it arises as degeneration of the twisted WZW model on elliptic curves. This is natural as the elliptic r-matrix describing the elliptic Knizhnik-Zamolodchikov equations likewise degenerates to the trigonometric r-matrix. The rigorous proof requires careful algebro-geometric arguments. V. Tarasov reviews the generalization of the Knizhnik-Zamolodchikov equations to the system of so-called differential dynamical equations. Both systems have a complete set of hypergeometric solutions. It is shown how the known (glk , gln ) dualities between the two systems of differential equations lead to nontrivial relations between hypergeometric integrals of different dimensions. Extensions to trigonometric and difference versions of the Knizhnik-Zamolodchikov and dynamical equations are briefly discussed. Recent progress in the theory of classical integrable systems is reported by Kanehisa Takasaki. He analyzes new classes of integrable partial differential equations admitting a zero-curvature representation on algebraic curves of arbitrary genus. He first reviews how conventional soliton equations are treated in the Grassmannian perspective, considering as example the nonlinear Schr¨ odinger hierarchy in great detail. Subsequently, recent results on the elliptic analogues of these systems are presented. Finally, O. Castro-Alvaredo and A. Fring present a lecture on two-dimensional quantum field theories with unstable particles. They review the main facts on analytic scattering theory of factorizable integrable models before presenting a new bootstrap principle that allows to include unstable particles in the spectrum. They describe the underlying Lie algebraic structure and the construction of an S-matrix like object characterizing the scattering between unstable particles. We gratefully acknowledge the financial support provided by the Centre for Mathematics and its Applications (CEMAT) of the Instituto Superior Técnico, the Luso-American Foundation and the Portuguese Foundation for Science and Technology, project POCTI/33858/MAT/2000. We wish to express our gratitude to José Ferreira Pereira Ferraz, Vice-Rector of the University of Algarve, and Ant´ onio Ferreira dos Santos, CEMAT and Department of Mathematics of Instituto Superior Técnico, for their support. Finally, we would like to thank all the participants for creating an excellent atmosphere of the workshop, and especially the contributors of this volume for writing a wonderful set of lecture notes. P.P. Kulish, N. Manojlović, H. Samtleben

Progress in Mathematics, Vol. 237, 1–58 c 2005 Birkh¨ auser Verlag Basel/Switzerland

Gaudin Model and Opers Edward Frenkel Abstract. This is a review of our previous works [FFR, F1, F3] (some of them joint with B. Feigin and N. Reshetikhin) on the Gaudin model and opers. We define a commutative subalgebra in the tensor power of the universal enveloping algebra of a simple Lie algebra g. This algebra includes the Hamiltonians of the Gaudin model, hence we call it the Gaudin algebra. It is constructed as a quotient of the center of the completed enveloping algebra of the affine KacMoody algebra g at the critical level. We identify the spectrum of the Gaudin algebra with the space of opers associated to the Langlands dual Lie algebra L g on the projective line with regular singularities at the marked points. Next, we recall the construction of the eigenvectors of the Gaudin algebra using the Wakimoto modules over g of critical level. The Wakimoto modules are naturally parameterized by Miura opers (or, equivalently, Cartan connections), and the action of the center on them is given by the Miura transformation. This allows us to relate solutions of the Bethe Ansatz equations to Miura opers and ultimately to the flag varieties associated to the Langlands dual Lie algebra L g. Mathematics Subject Classification (2000). 17B67 and 82B23. Keywords. Gaudin model, Bethe Ansatz, oper, Miura transformation, Wakimoto module.

Introduction Let g be a finite-dimensional simple Lie algebra over C and U (g) its universal enveloping algebra. Choose a basis {Ja }, a = 1, . . . , d, of g, and let {J a } the dual basis with respect to a non-degenerate invariant bilinear form on g. Let z1 , . . . , zN be a collection of distinct complex numbers. The Gaudin Hamiltonians are the following elements of the algebra U (g)⊗N : Ξi =

d (i) Ja J a(j) j=i a=1

zi − zj

,

i = 1, . . . , N,

Partially supported by grants from the NSF and DARPA.

(0.1)

2

E. Frenkel

where for A ∈ g we denote by A(i) the element of U (g)⊗N which is the tensor product of A in the ith factor and 1 in all other factors. One checks easily that these elements commute with each other and are invariant with respect to the diagonal action of g on U (g)⊗N . For any collection M1 , . . . , MN of g-modules, the Gaudin Hamiltonians give N rise to commuting linear operators on i=1 Mi . We are interested in the diagonalization of these operators. More specifically, we will consider the following two cases: when all of the Mi ’s are Verma modules and when they are finite-dimensional irreducible modules. It is natural to ask: are there any other elements in U (g)⊗N which commute with the Gaudin Hamiltonians? Clearly, the N -fold tensor product Z(g)⊗N of the center Z(g) of U (g) is the center of U (g)⊗N , and its elements obviously commute with the Ξi ’s. As shown in [FFR], if g has rank grater than one, then in addition to the Gaudin Hamiltonians and the central elements there are other elements in U (g)⊗N of orders higher than two which commute with the Gaudin operators and with each other (but explicit formulas for them are much more complicated and unknown in general). Adjoining these “higher Gaudin Hamiltonians” to the Ξi ’s together with the center Z(g)⊗N , we obtain a large commutative subalgebra of U (g)⊗N . We will call it the Gaudin algebra and denote it by Z(zi ) (g). The construction of Z(zi ) (g) will be recalled in Section 2. The key point is the realization of U (g)⊗N as the space of coinvariants of induced modules over the affine Kac-Moody algebra g on the projective line. Using this realization, we obtain a surjective map from the center of the completed universal enveloping algebra of g at the critical level onto Z(zi ) (g). The next natural question is what is the spectrum of Z(zi ) (g), i.e., the set of all maximal ideals of Z(zi ) (g), or equivalently, algebra homomorphisms Z(zi ) (g) → C. Knowing the answer is important, because then we will know how to think about the common eigenvalues of the higher Gaudin operators on the tensor products N i=1 Mi of g-modules. These common eigenvalues correspond to points of the spectrum of Z(zi ) (g). The answer comes from the description of the center of the completed universal enveloping algebra of g at the critical level. In [FF2, F2] it is shown that the spectrum of this center (more precisely, the center of the corresponding vertex algebra) is canonically identified with the space of the so-called L G-opers, where L G is the Langlands dual Lie group to g (of adjoint type), on the formal disc. This result leads us to the following description of the spectrum of the algebra Z(zi ) (g) of higher Gaudin Hamiltonians: it is the space of L G-opers on P1 with regular singularities at the points z1 , . . . , zN and ∞. We obtain this description from some basic facts about the spaces of coinvariants from [FB]. Recall that the space of coinvariants HV (X; (xi ); (Mi )) is defined in [FB] for any (quasi-conformal) vertex algebra V , a smooth projective curve X, a collection x1 , . . . , xn of distinct points of X and a collection of V -modules M1 , . . . , Mn attached to those points. Suppose that V is a commutative vertex

Gaudin Model and Opers

3

algebra, and so in particular it is a commutative algebra. Suppose that the spectrum of V is the space S(D) of certain geometric objects, such as L G-opers, on the disc D = Spec C[[t]]. Then a V -module is the same as a smooth module over the complete topological algebra U (V ) of functions on S(D× ), which is the space of our objects (such as L G-opers) on the punctured disc D× = Spec C((t)). Suppose in addition that each V -module Mi is the space of functions on a subspace Si of S(D× ) (with its natural Fun S(D× )-module structure). Then the space of coinvariants HV (X; (xi ); (Mi )) is naturally a commutative algebra, and its spectrum is the space of our objects (such as L G-opers) on X\{x1 , . . . , xn } whose restriction to the punctured disc Dx×i around xi belongs to Si ⊂ S(Dx×i ), i = 1, . . . , n. For example, if g = sl2 , then L G = P GL2 , and P GL2 -opers are the same as second order differential operators ∂t2 − q(t) acting from sections of the line bundle Ω−1/2 to sections of Ω3/2 . A P GL2 -oper on P1 with regular singularities at z1 , . . . , zN and ∞ may be written as the Fuchsian differential operator of second order with regular singularities at z1 , . . . , zN , N N ci µi ∂t2 − − , 2 i=1 (t − zi ) i=1 t − zi N satisfying the condition i=1 µi = 0 that insures that it also has regular singularity at ∞. Defining such an operator is the same as giving a collection of numbers N ci , µi , i = 1, . . . , N , such that i=1 µi = 0. The set N µi = 0 , ci , µi , i = 1, . . . , N i=1

is then the spectrum of the Gaudin algebra Z(zi ) (g), which in this case is the (i) polynomial algebra generated by the Casimir operators Ci = 12 a Ja J a(i) , i = 1, . . . , N , and the Gaudin Hamiltonians Ξi , subject to the relation N i=1 Ξi = 0. In other words, the numbers ci record the eigenvalues of the Ci ’s, while the numbers µi record the eigenvalues of the Ξi ’s. For a general simple Lie algebra g, the Gaudin algebra ZN (g) has many more generators, and its spectrum does not have such a nice system of coordinates as the ci ’s and the µi ’s in the above example. Therefore the description of the spectrum as a space of L G-opers is very useful. In particular, we obtain that common eigenvalues of the higher Gaudin Hamiltonians are encoded by L G-opers on P1 , with regular singularities at prescribed points. These L G-opers appear as generalizations of the above second order Fuchsian operators. Next, we ask which points in the spectrum of Z(zi ) (g) might occur as the N common eigenvalues on particular tensor products i=1 Mi . We answer this question first in the case when each Mi admits a central character: namely, it turns out that the central character of Mi fixes the residue of the L G-oper at the point zi . We then show that if all g-modules Mi are finite-dimensional and irreducible, then the L G-opers encoding possible eigenvalues of the higher Gaudin Hamiltonians in N i=1 Mi necessarily have trivial monodromy representation.

4

E. Frenkel

We conjecture that there is a bijection between the eigenvalues of the Gaudin N Hamiltonians on i=1 Mi , where the Mi ’s are irreducible finite-dimensional gmodules, and L G-opers on P1 with prescribed singularities at z1 , . . . , zN , ∞ and trivial monodromy. Thus, we obtain a correspondence between two seemingly unrelated objects: the eigenvalues of the generalized Gaudin Hamiltonians and the L G-opers on P1 . The connection between the eigenvalues of the Gaudin operators and differential operators of some sort has been observed previously, but it was not until [FFR, F1] that this phenomenon was explained conceptually. We present a more geometric description the L G-opers without monodromy (which occur as the eigenvalues of the Gaudin Hamiltonians) as isomorphism classes of holomorphic maps from P1 to L G/L B, the flag manifold of L G, satisfying a certain transversality condition. For example, if L G = P GL2 , they may be described as holomorphic maps P1 → P1 whose derivative vanishes to prescribed orders at the marked points z1 , . . . , zN and ∞, and does not vanish anywhere else (these orders correspond to the highest weights of the finite-dimensional representations inserted at those points). If the L G-oper is non-degenerate (in the sense explained in Section 5.2), then we can associate to it an eigenvector of the Gaudin Hamiltonians called a Bethe vector. The procedure to construct eigenvectors of the Gaudin Hamiltonians that produces these vectors is known as the Bethe Ansatz. In [FFR] we explained that this procedure can also be understood in the framework of coinvariants of g-modules of critical level. We need to use a particular class of g-modules, called the Wakimoto modules. Let us recall that the Wakimoto modules of critical level are naturally parameterized by objects closely related to opers, which we call Miura opers. They may also be described more explicitly as certain connections on a particular L Hbundle Ω−ρ on the punctured disc, where L H is the Cartan subgroup of L G. The center acts on the Wakimoto module corresponding to a Cartan connection by the Miura transformation of this connection (see [F2]). The idea of [FFR] was to use the spaces of coinvariants of the tensor product of the Wakimoto modules to construct eigenvectors of the generalized Gaudin Hamiltonians. We found in [FFR] that the eigenvalues of the Gaudin Hamiltonians on these vectors are encoded by the L G-opers which are obtained by applying the Miura transformation to certain Cartan connections on P1 . More precisely, the Bethe vector depends on an m-tuple of complex numbers wj , where j = 1, . . . , m, with an extra datum attached to each of them, ij ∈ I, where I is the set of nodes of the Dynkin diagram of g (or equivalently, the set of simple roots of g). These numbers have to be distinct from the zi ’s and satisfy the following system of Bethe Ansatz equations: N λi , α ˇ ij αis , α ˇ ij − = 0, i=1 wj − zi s=j wj − ws

j = 1, . . . , m,

(0.2)


5

where λi denotes the highest weight of the finite-dimensional g-module Mi = Vλi , i = 1, . . . , N . We can compute explicitly the L G-oper encoding the eigenvalues of the generalized Gaudin Hamiltonians on this vector. As shown in [FFR], this L G-oper is obtained by applying the Miura transformation of the connection N m αij λi − (0.3) ∂t + i=1 t − zi j=1 t − wj on the L H-bundle Ω−ρ on P1 . This L G-oper automatically has trivial monodromy. The Bethe vector corresponding to a solution of the system (0.2) is a highest N weight vector in i=1 Vλi of weight N m µ= λi − αij , i=1

j=1

so it can only be non-zero if µ is a dominant integral weight of g. But it is still interesting to describe the set of all solutions of the Bethe Ansatz equations (0.2), even for non-dominant weights µ. While the eigenvalues of the Gaudin Hamiltonians are parameterized by L Gopers, it turns out that the solutions of the Bethe Ansatz equations are parameterized by the (non-degenerate) Miura L G-opers. As mentioned above, those may in turn be related to very simple objects, namely, connections on an L H-bundle Ω−ρ of the kind given above in formula (0.3). A L G-oper on a curve X is by definition a triple (F, ∇, FL B ), where F is L a G-bundle on X, ∇ is a connection on F and FL B is a reduction of F to a Borel subgroup L B of L G, which satisfies a certain transversality condition with ∇. A Miura L G-oper is by definition a quadruple (F, ∇, FL B , FL B ) where FL B is another L B-reduction of F, which is preserved by ∇. The space of Miura opers on a curve X whose underlying oper has regular singularities and trivial monodromy representation (so that F is isomorphic to the trivial bundle) is isomorphic to the flag manifold L G/L B of L G. Indeed, in order to define the L B-reduction FL B of such F everywhere, it is sufficient to define it at one point x ∈ X and then use the connection to “spread” it around. But choosing a L B-reduction at one point means choosing an element of the twist of L G/L B by Fx , and so we see that the space of all reductions is isomorphic to the flag manifold of L G. The relative position of the two reductions FL B and FL B at each point of X is measured by an element w of the Weyl group W of G. The two reductions are in generic relative position (corresponding to w = 1) almost everywhere on X. The mildest possible non-generic relative positions correspond to the simple reflections si from W . We call a Miura oper on P1 with marked points z1 , . . . , zN non are in generic position at z1 , . . . , zN , degenerate if the two reductions FB and FB 1 and elsewhere on P their relative position is either generic or corresponds to a simple reflection. We denote the points where the relative position is not generic by wj , j = 1, . . . , m; each point wj comes together with a simple reflection sij , or equivalently a simple root αij attached to it.

6

E. Frenkel

It is then easy to see that this collection satisfies the equations (0.2), and conversely any solution of (0.2) corresponds to a non-degenerate Miura oper (or to an L H-connection (0.3)). Thus, we obtain that there is a bijection between the set of solutions of (0.2) (for all possible collections {i1 , . . . , im }) and the set of non-degenerate Miura L G-opers such that the underlying L G-opers have prescribed residues at the points z1 , . . . , zN , ∞ and trivial monodromy. Now let us fix λ1 , . . . , λN and µ. Then every L G-oper τ on P1 with regular singularities at z1 , . . . , zN and ∞ and with prescribed residues corresponding to λ1 , . . . , λN and µ and trivial monodromy admits a horizontal L B-reduction FL B satisfying the conditions of a non-degenerate Miura oper. Since these are open conditions, we find that for such z1 , . . . , zN the non-degenerate Miura oper structures on a particular L G-oper τ on P1 form an open dense subset in the set of all Miura oper structures on τ . But the set of all Miura structures on a given L G-oper τ is isomorphic to the flag manifold L G/L B. Therefore we find that the set of non-degenerate Miura oper structures on τ is an open dense subset of L G/L B! Recall that the set of all solutions of the Bethe Ansatz equations (0.2) is the union of the sets of non-degenerate Miura oper structures on all L G-opers with trivial monodromy. Hence it is naturally a disjoint union of subsets, parameterized by these L G-opers. We have now identified each of these sets with an open and dense subset of the flag manifold L G/L B. Let us summarize our results: • the eigenvalues of the Hamiltonians of the Gaudin model associated to a simple Lie algebra g on the tensor product of finite-dimensional representations are encoded by L G-opers on P1 , where L G is the Langlands dual group of G, which have regular singularities at the marked points z1 , . . . , zN , ∞ and trivial monodromy; • if such an oper τ is non-degenerate, then we can associte to it a solution of the Bethe Ansatz equations (0.2), which gives rise to the Bethe eigenvector of dominant integral weight whose eigenvalues are encoded by τ ; • there is a one-to-one correspondence between the set of all solutions of the Bethe Ansatz equations (0.2) and the set of non-degenerate Miura opers corresponding to a fixed L G-oper; • the set of non-degenerate Miura opers corresponding to the same underlying L G-oper is an open dense subset of the flag manifold L G/L B of the Langlands dual group, and therefore the set of all solutions of the Bethe Ansatz equations (0.2) is the union of certain open dense subsets of the flag manifold of the Langlands dual group, one for each L G-oper. Finally, to a degenerate L G-oper we can also attach, at least in some cases, an eigenvector of the Gaudin hamiltonians by generalizing the Bethe Ansatz procedure, as explained in Section 5.5. The paper is organized as follows. In Section 1 we introduce opers and discuss their basic properties. We define opers with regular singularities and their residues.


7

In Section 2, following [FFR], we define the Gaudin algebra using the coinvariants of the affine Kac-Moody algebra g of critical level. We recall the results of [FF2, F2] on the isomorphism of the center of the completed universal enveloping algebra of g at the critical level and the algebra of functions on the space of L G-opers on the punctured disc. Using these results and general facts about the spaces of coinvariants from [FB], we describe the spectrum of the Gaudin algebra. In Section 3 we introduce Miura opers, Cartan connections and the Miura transformation and describe their properties, following [F2, F3]. We use these results in the next section, Section 4, to describe the Bethe Ansatz, a construction of eigenvectors of the Gaudin algebra. We introduce the Wakimoto modules of critical level, following [FF1, F2]. The Wakimoto modules are naturally parameterized by the Cartan connections on the punctured disc introduced in Section 3. The action of the center on the Wakimoto modules is given by the Miura transformation. We construct the Bethe vectors, following [FFR], using the coinvariants of the Wakimoto modules. We show that the Bethe Ansatz equations which ensure that this vector is an eigenvector of the Gaudin Hamiltonians coincide with the requirement that the Miura transformation of the Cartan connection on P1 encoding the Wakimoto modules has no singularities at the points w1 , . . . , wm . Finally, in Section 5 we consider the Gaudin model in the case when all modules Mi finite-dimensional modules. We describe the precise connection between the spectrum of the Gaudin algebra on the tensor product of finite-dimensional modules and the set of L G-opers with prescribed singularities at z1 , . . . , zN , ∞ and trivial monodromy.

1. Opers 1.1. Definition of opers Let G be a simple algebraic group of adjoint type, B a Borel subgroup and N = [B, B] its unipotent radical, with the corresponding Lie algebras n ⊂ b ⊂ g. The quotient H = B/N is a torus. Choose a splitting H → B of the homomorphism B → H and the corresponding splitting h → b at the level of Lie algebras. Then we will have a Cartan decomposition g = n− ⊕ h ⊕ n. We will choose generators {ei }, i = 1, . . . , , of n and generators {fi }, i = 1, . . . , of n− corresponding to simple roots, and denote by ρˇ ∈ h the sum of the fundamental coweights of g. ρ, fi ] = −1. Then we will have the following relations: [ˇ ρ, ei ] = 1, [ˇ A G-oper on a smooth curve X (or a disc D Spec C[[t]] or a punctured disc D× = Spec C((t))) is by definition a triple (F, ∇, FB ), where F is a principal G-bundle F on X, ∇ is a connection on F and FB is a B-reduction of F such that locally on X (with respect to a local coordinate t and a local trivialization of FB ) the connection has the form ∇ = ∂t +

i=1

ψi (t)fi + v(t),

(1.1)

8

E. Frenkel

where each ψi (t) is a nowhere vanishing function, and v(t) is a b-valued function. The space of G-opers on X is denoted by OpG (X). This definition is due to A. Beilinson and V. Drinfeld [BD1] (in the case when X is the punctured disc opers were first introduced in [DS]). In particular, if U = Spec R is an affine curve with the ring of functions R and t is a global coordinate on U (for example, if U = Spec C[[t]]), then OpG (U ) is isomorphic to the quotient of the space of operators of the form ∇ = ∂t +

v(t) ∈ b(R),

fi + v(t),

(1.2)

i=1

by the action of the group N (R) (we use the action of H(R) to make all functions ψi (t) equal to 1). Recall that the gauge transformation of an operator ∂t + A(t), where A(t) ∈ g(R) by g(t) ∈ G(R) is given by the formula g · (∂t + A(t)) = ∂t + gA(t)g −1 − ∂t g · g −1 . The operator ad ρˇ defines the principal gradation on b, with respect to which we have a direct sum decomposition b = i≥0 bi . Set p−1 =

fi .

i=1

Let p1 be the unique element of degree 1 in n, such that {p−1 , 2ρˇ, p1 } is an sl2 triple. Let Vcan = ⊕i∈E Vcan,i be the space of ad p1 -invariants in n. Then p1 spans Vcan,1 . Choose a linear generator pj of Vcan,dj (if the multiplicity of dj is greater (1) than one, which happens only in the case g = D2n , dj = 2n, then we choose linearly independent vectors in Vcan,dj ). The following result is due to Drinfeld and Sokolov [DS] (the proof is reproduced in Lemma 2.1 of [F3]). G (Spec R) is free, and each gauge Lemma 1.1. The gauge action of N (R) on Op equivalence class contains a unique operator of the form ∇ = ∂t +p−1 +v(t), where v(t) ∈ Vcan (R), so that we can write v(t) =

vj (t) · pj .

(1.3)

j=1

1.2. P GL2 -opers For g = sl2 , G = P GL2 we obtain an identification of the space of P GL2 -opers with the space of operators of the form

0 v(t) ∂t + . 1 0 If we make a change of variables t = ϕ(s), then the corresponding connection operator will become

0 ϕ (s)v(ϕ(s)) . ∂s + ϕ (s) 0


9

Applying the B-valued gauge transformation with

(s) 0 (ϕ (s))1/2 1 12 ϕϕ (s) , 0 (ϕ (s))1/2 0 1 we obtain the operator

∂s +

0 1

v(ϕ(s))ϕ (s)2 − 12 {ϕ, s} 0

,

where

2 ϕ 3 ϕ {ϕ, s} = − ϕ 2 ϕ is the Schwarzian derivative. Thus, under the change of variables t = ϕ(s) we have 1 v(t) → v(ϕ(s))ϕ (s)2 − {ϕ, s}. 2 this coincides with the transformation properties of the second order differential operators ∂t2 −v(t) acting from sections of Ω−1/2 to sections of Ω3/2 , where Ω is the canonical line bundle on X. Such operators are known as projective connections on X (see, e.g., [FB], Sect. 9.2), and so P GL2 -opers are the same as projective connections. For a general g, the first coefficient function v1 (t) in (1.3) transforms as a projective connection, and the coefficient vi (t) with i > 1 transforms as a (di + 1)differential on X. Thus, we obtain an isomorphism OpG (X) Proj(X) ×

Γ(X, Ω(di +1) ).

(1.4)

i=2

1.3. Opers with regular singularities Let x be a point of a smooth curve X and Dx = Spec Ox , Dx× = Spec Kx , where Ox is the completion of the local ring of x and Kx is the field of fractions of Ox . Choose a formal coordinate t at x, so that Ox C[[t]] and Kx = C((t)). Recall that the space OpG (Dx ) (resp., OpG (Dx× )) of G-opers on Dx (resp., Dx× ) is the quotient of the space of operators of the form (1.1) where ψi (t) and v(t) take values in Ox (resp., in Kx ) by the action of B(Ox ) (resp., B(Kx )). A G-oper on Dx with regular singularity at x is by definition (see [BD1], Sect. 3.8.8) a B(Ox )-conjugacy class of operators of the form

−1 ψi (t)fi + v(t) , (1.5) ∇ = ∂t + t i=1

where ψi (t) ∈ Ox , ψi (0) = 0, and v(t) ∈ b(Ox ). Equivalently, it is an N (Ox )equivalence class of operators ∇ = ∂t +

1 (p−1 + v(t)) , t

v(t) ∈ b(Ox ).

(1.6)

10

E. Frenkel

Denote by OpRS G (Dx ) the space of opers on Dx with regular singularity. It is easy × to see that the natural map OpRS G (Dx ) → OpG (Dx ) is injective. Therefore an oper with regular singularity may be viewed as an oper on the punctured disc. But to an oper with regular singularity one can unambiguously attach a point in g/G := Spec C[g]G C[h]W =: h/W, its residue, which in our case is equal to p−1 + v(0). ˇ ∈ h, we denote by OpRS (Dx ) ˇ the subvariety of OpRS (Dx ) which Given λ G G λ ˇ − ρˇ) ∈ h/W , where is the consists of those opers that have residue (−λ projection h → h/W . In particular, the residue of a regular oper ∂t +p−1 +v(t), where v(t) ∈ b(Ox ), is equal to (−ρˇ) (see [BD1]). Indeed, a regular oper may be brought to the form (1.6) by using the gauge transformation with ρˇ(t) ∈ B(Kx ), after which it takes the form 1 p−1 − ρˇ + t · ρˇ(t)(v(t))ˇ ∂t + ρ(t)−1 . t If v(t) is regular, then so is ρˇ(t)(v(t))ˇ ρ (t)−1 . Therefore the residue of this oper in h/W is equal to (−ρˇ), and so OpG (Dx ) = OpRS G (Dx )0 . ˇ with a complex number. Then For G = P GL2 we identify h with C and so λ RS one finds that OpP GL2 (Dx )λˇ is the space of second order operators of the form ∂t2 −

ˇ λ ˇ + 2)/4 λ( − vn tn . t2

(1.7)

n≥−1

ˇ is a dominant integral coweight of g. Following Drinfeld, Now suppose that λ introduce the variety OpG (Dx )λˇ as the quotient of the space of operators of the form ψi (t)fi + v(t), (1.8) ∇ = ∂t + i=1

where

ˇ

ψi (t) = tαi ,λ (κi + t(. . .)) ∈ Ox , κi = 0 and v(t) ∈ b(Ox ), by the gauge action of B(Ox ). Equivalently, OpG (Dx )λˇ is the quotient of the space of operators of the form ∇ = ∂t +

ˇ

tαi ,λ fi + v(t),

(1.9)

i=1

where v(t) ∈ b(Ox ), by the gauge action of N (Ox ). Considering the N (Kx )-class of such an operator, we obtain an oper on Dx× . Thus, we have a map OpG (Dx )λˇ → OpG (Dx× ). Lemma 1.2 ([F3], Lemma 2.4). The map OpG (Dx )λˇ → OpG (Dx× ) is injective and its image is contained in the subvariety OpRS ˇ . Moreover, the points of G (Dx )λ ˇ which OpG (Dx )λˇ are precisely those G-opers with regular singularity and residue λ have no monodromy around x.


11

The space OpP GL2 (Dx )λˇ is the subspace of codimension one in OpP GL2 (Dx )RS ˇ . In λ terms of the coefficients vn , n ≥ −1, appearing in formula (1.7) the corresponding equation has the form Pλ (vn ) = 0, where Pλ is a polynomial of degree λ + 1, where 2 − v0 , etc. In general, we set deg vn = n + 2. For example, P0 = v−1 , P2 = 2v−1 RS the subspace OpG (Dx )λˇ ⊂ OpG (Dx )λˇ is defined by dim N polynomial equations, where N is the unipotent subgroup of G.

2. The Gaudin model Let g be a simple Lie algebra. The Langlands dual Lie algebra L g is by definition the Lie algebra whose Cartan matrix is the transpose of that of g. We will identify the set of roots of g with the set of coroots of L g and the set of weights of g with the set of coweights of L g. The results on opers from the previous sections will be applied here to the Lie algebra L g. Thus, in particular, L G will denote the adjoint group of L g. 2.1. The definition of the Gaudin model Here we recall the definition of the Gaudin model and the realization of the Gaudin Hamiltonians in terms of the spaces of conformal blocks for affine Kac-Moody algebras of critical level. We follow closely the paper [FFR]. Choose a non-degenerate invariant inner product κ0 on g. Let {Ja }, a = 1, . . . , d, be a basis of g and {J a } the dual basis with respect to κ0 . Denote by ∆ the quadratic Casimir operator from the center of U (g): 1 Ja J a . 2 a=1 d

∆=

The Gaudin Hamiltonians are the elements Ξi =

d (i) Ja J a(j) j=i a=1

zi − zj

,

i = 1, . . . , N,

(2.1)

of the algebra U (g)⊗N . Note that they commute with the diagonal action of g on U (g)⊗N and that N Ξi = 0. i=1

2.2. Gaudin model and coinvariants Let gκc be the affine Kac-Moody algebra corresponding to g. It is the extension of the Lie algebra g ⊗ C((t)) by the one-dimensional center CK. The commutation relations in gκc read [A ⊗ f (t), B ⊗ g(t)] = [A, B] ⊗ f g − κc (A, B) Rest=0 f dg · K,

(2.2)

12

E. Frenkel

where κc is the critical invariant inner product on g defined by the formula 1 κc (A, B) = − Trg ad A ad B. 2 Note that κc = −h∨ κ0 , where κ0 is the inner product normalized as in [K] and h∨ is the dual Coxeter number. gκc . Let M be a gDenote by g+ the Lie subalgebra g ⊗ C[[t]] ⊕ CK of module. We extend the action of g on M to g ⊗ C[[t]] by using the evaluation at zero homomorphism g ⊗ C[[t]] → g and to g+ by making K act as the identity. Denote by M the corresponding induced gκc -module M = U ( gκc ) ⊗ M. U( g+ )

By construction, K acts as the identity on this module. We call such modules the gκc -modules of critical level. For example, for λ ∈ h∗ let Cλ be the one-dimensional b-module on which h acts by the character λ : h → C and n acts by 0. Let Mλ be the Verma module over g of highest weight λ, Mλ = U (g)U (b)Cλ . ⊗

The corresponding induced module Mλ is the Verma module over gκc with highest weight λ. For a dominant integral weight λ ∈ h∗ denote by Vλ the irreducible finitedimensional g-module of highest weight λ. The corresponding induced module Vλ is called the Weyl module over gκc with highest weight λ. Consider the projective line P1 with a global coordinate t and N distinct finite points z1 , . . . , zN ∈ P1 . In the neighborhood of each point zi we have the local coordinate t − zi and in the neighborhood of the point ∞ we have the local g(zi ) = g ⊗ C((t − zi )) and g(∞) = g ⊗ C((t−1 )). Let gN be coordinate t−1 . Set N the extension of the Lie algebra i=1 g(zi ) ⊕ g(∞) by a one-dimensional center g(∞) coincides with the above CK whose restriction to each summand g(zi ) or central extension. Suppose we are given a collection M1 , . . . , MN and M∞ of g-modules. Then the Lie algebra gN naturally acts on the tensor product N i=1 Mi ⊗ M∞ (in particular, K acts as the identity). Let g(zi ) = gz1 ,...,zN be the Lie algebra of g-valued regular functions on P1 \{z1 , . . . , zN , ∞} (i.e., rational functions on P1 , which may have poles only at the points z1 , . . . , zN and ∞). Clearly, such a function can be expanded into a Laurent power series in the corresponding local coordinates at each point zi and at ∞. Thus, we obtain an embedding g(zi ) →

N i=1

g(∞). g(zi ) ⊕


13

It follows from the residue theorem and formula (2.2) that the restriction of the central extension to the image of this embedding is trivial. Hence this embedding lifts to the embedding g(zi ) → gN . N Denote by H(M1 , . . . , MN , M∞ ) the space of coinvariants of i=1 Mi ⊗ M∞ with respect to the action of the Lie algebra g(zi ) . By construction, we have a canonical embedding of a g-module M into the induced gκc -module M: x ∈ M → 1 ⊗ x ∈ M, which commutes with the action of g on both spaces (where g is embedded into gκc as the constant subalgebra). Thus we have an embedding N

Mi ⊗ M∞ →

i=1

N

Mi ⊗ M∞ .

i=1

The following result is proved in the same way as Lemma 1 in [FFR]. Lemma 2.1. The composition of this embedding and the projection N

Mi ⊗ M∞ H(M1 , . . . , MN , M∞ )

i=1

gives rise to an isomorphism N H(M1 , . . . , MN , M∞ ) ( Mi ⊗ M∞ )/gdiag . i=1

Let V0 be the induced gκc -module of critical level, which corresponds to the one-dimensional trivial g-module V0 ; it is called the vacuum module. Denote by v0 the generating vector of V0 . We assign the vacuum module to a point u ∈ P1 which is different from z1 , . . . , zN , ∞. Denote by H(M1 , . . . , MN , M∞ , C) the space N of g(zi ),u -invariant functionals on i=1 Mi ⊗ M∞ ⊗ V0 with respect to the Lie algebra g(zi ),u . Lemma 2.1 tells us that we have a canonical isomorphism H(M1 , . . . , MN , M∞ , C) H(M1 , . . . , MN , M∞ ). Now observe that by functoriality any endomorphism X ∈ Endgκc V0 gives rise to an endomorphism of the space of coinvariants H(M1 , . . . , MN , M∞ , C), and hence of H(M1 , . . . , MN , M∞ ). Thus, we obtain a homomorphism of algebras Endgκc V0 → EndC H(M1 , . . . , MN , M∞ ). Let us compute this homomorphism explicitly.

14

E. Frenkel First of all, we identify the algebra Endgκc V0 with the space g[[t]]

z( g ) = V0

of g[[t]]-invariant vectors in V0 . Indeed, a g[[t]]-invariant vector v gives rise to an endomorphism of V0 commuting with the action of gκc which sends the generating gκc -endomorphism of V0 is uniquely determined by vector v0 to v. Conversely, any the image of v0 which necessarily belongs to z( g). Thus, we obtain an isomorphism z( g) Endgκc (V0 ) which gives z( g) an algebra structure. The opposite algebra structure on z( g) coincides with the algebra structure induced by the identification of V0 with the algebra U (g ⊗ t−1 C[t−1 ]). But we will see in the next section that the algebra z( g) is commutative and so the two algebra structures on it coincide. Now let v ∈ z( g) ⊂ V0 . For any N Mi ⊗ M∞ )/gdiag H(M1 , . . . , MN , M∞ ) x∈( i=1

take a lifting x to

N

i=1

Mi ⊗ M∞ . By Lemma 2.1, the projection of the vector x ⊗v ∈

N

Mi ⊗ M∞ ⊗ V0

i=1

onto

N H(M1 , . . . , MN , M∞ , C) ( Mi ⊗ M∞ )/gdiag i=1

is equal to the projection of a vector of the form (Ψu (v) · x ) ⊗ v0 , where N Ψu (v) · x ∈( Mi ⊗ M∞ )/gdiag . i=1

For A ∈ g and n ∈ Z, denote by An the element A ⊗ tn ∈ gκc . Then V0 U (g ⊗ t−1 C[t−1 ])v0 has a basis of lexicographically ordered monomials of the form m Jna11 . . . Jnam v0 with ni < 0. Let us set Jan (u) = −

N i=1

J a(i) . (zi − u)n

Define an anti-homomorphism Φu : U (g ⊗ t−1 C[t−1 ]) → U (g)⊗N ⊗ C[(u − zi )−1 ]i=1,...,N by the formula m v0 ) = Janm (u)Janm−1 (u) . . . Jan11 (u). Φu (Jna11 . . . Jnam m m−1

(2.3)

According to the computation presented in the proof of Proposition 1 of [FFR], we have Ψu (v) · x = Φu (v) · x .


15

In general, Φv (u) does not commute with the diagonal action of g, and so depends on the choice of the lifting x . But if v ∈ z( g) ⊂ V0 , then Φv (u) Ψv (u) · x commutes with the diagonal action of g and hence gives rise to a well-defined N endomorphism ( i=1 Mi ⊗ M∞ )/gdiag . Thus, we restrict Φu to z( g). This gives us a homomorphism of algebras G z( g) → U (g)⊗N ⊗ C[(u − zi )−1 ]i=1,...,N , which we also denote by Φu . For example, consider the Segal-Sugawara vector in V0 : 1 a Ja,−1 J−1 v0 . 2 a=1 d

S=

(2.4)

One shows (see, e.g., [FB]) that this vector belongs to z( g). Consider the corresponding element Φu (S). 1 Denote by ∆ the Casimir operator Ja J a from U (g). 2 a Proposition 2.2 ([FFR], Proposition 1). We have Φu (S) =

N i=1

∆(i) Ξi + , u − zi i=1 (u − zi )2 N

where the Ξi ’s are the Gaudin operators (2.1). We wish to study the algebra generated by the image of the map Φu . Proposition 2.3 ([FFR], Proposition 2). For any Z1 , Z2 ∈ z( g) and any points u1 , u2 ∈ P1 \{z1, . . . , zN , ∞} the linear operators ΨZ1 (u1 ) and ΨZ2 (u2 ) commute. Let Z(zi ) (g) be the span in U (g)⊗N of the coefficients in front of the monomiN g). Since Φu is an algebra homomorals i=1 (u − zi )ni of the series Φu (v), v ∈ z( G phism, we find that Z(zi ) (g) is a subalgebra of U (g)⊗N , which is commutative by Proposition 2.3. We call it the Gaudin algebra associated to g and the collection z1 , . . . , zN , and its elements the generalized Gaudin Hamiltonians. 2.3. The center of V0 and L G-opers In order to describe the Gaudin algebra Z(zi ) (g) and its spectrum we need to recall the description of z( g). According to [FF2, F2], z( g) is identified with the algebra Fun OpL G (D) of (regular) functions on the space OpL G (D) of L G-opers on the disc D = Spec C[[t]], where L G is the Langlands dual group to G. Since we have assumed that G is simply-connected, L G may be defined as the adjoint group of the Lie algebra L g whose Cartan matrix is the transpose of that of g. This isomorphism satisfies various properties, one of which we will now recall. Let Der O = C[[t]]∂t be the Lie algebra of continuous derivations of the topological algebra O = C[[t]]. The action of its Lie subalgebra Der0 O = tC[[t]]∂t on O exponentiates to an action of the group Aut O of formal changes of variables.

16

E. Frenkel

Both Der O and Aut O naturally act on V0 in a compatible way, and these actions preserve z( g). They also act on the space OpL G (D). Denote by Fun OpL G (D) the algebra of regular functions on OpL G (D). In view of Lemma 1.1, it is isomorphic to the algebra of functions on the space of -tuples (v1 (t), . . . , v (t)) of formal Taylor series, i.e., the space C[[t]] . If we write vi (t) = n≥0 vi,n tn , then we obtain Fun OpL G (D) C[vi,n ]i∈I,n≥0 .

(2.5)

Note that the vector field −t∂t acts naturally on OpL G (D) and defines a Z-grading on Fun OpL G (D) such that deg vi,n = di + n + 1. The vector field −∂t acts as a derivation such that −∂t · vi,n = −(di + n + 1)vi,n+1 . Theorem 2.4 ([FF2, F2]). There is a canonical isomorphism z( g) Fun OpL G (D) of algebras which is compatible with the action of Der O and Aut O. We use this result to describe the twist of z( g) by the Aut O-torsor Autx of formal coordinates at a smooth point x of an algebraic curve X, z( g)x = Autx × z( g) Aut O

(see Ch. 6 of [FB] for more details). It follows from the definition that the corresponding twist of Fun OpL G (D) by Autx is nothing but Fun OpL G (Dx ), where Dx is the disc around x. Therefore we obtain from Theorem 2.4 an isomorphism z( g)x Fun OpL G (Dx ).

(2.6)

The module V0 has a natural Z-grading defined by the formulas deg v0 = 0, deg Jna = −n, and it carries a translation operator T defined by the formulas a T v0 = 0, [T, Jna ] = −nJn−1 . Theorem 2.4 and the isomorphism (2.5) imply that g[[t]] of degrees di + 1, i ∈ I, such that there exist non-zero vectors Si ∈ V0 z( g) = C[T n Si ]i∈I,n≥0 v0 . Then under the isomorphism of Theorem 2.4 we have Si → vi,0 , the Z-gradings on both algebras get identified and the action of T on z( g) becomes the action of −∂t on Fun OpL G (D). Note that the vector S1 is nothing but the vector (2.4), up to a non-zero scalar. Recall from [FB] that V0 is a vertex algebra, and z( g) is its commutative vertex subalgebra; in fact, it is the center of V0 . We will also need the center of the completed universal enveloping algebra of g of critical level. This algebra is defined as follows. Let Uκc ( g) be the quotient of the universal enveloping algebra U ( gκc ) of g κc κc ( by the ideal generated by (K − 1). Define its completion U g) as follows: κc ( U g) = lim Uκc ( g)/Uκc ( g) · (g ⊗ tN C[[t]]). ←−


17

κc ( It is clear that U g) is a topological algebra which acts on all smooth g κc gκc -module such that any vector module. By definition, a smooth gκc -module is a is annihilated by g ⊗ tN C[[t]] for sufficiently large N , and K acts as the identity. κc ( g). Let Z( g) be the center of U × Denote by Fun OpL G (D ) the algebra of regular functions on the space OpL G (D× ) of L G-opers on the punctured disc D× = Spec C((t)). In view of Lemma 1.1, it is isomorphic to the algebra of functions on the space of -tuples (v1 (t), . . . , v (t)) Laurent series, i.e., the ind-affine space C((t)) . If we of formal n write vi (t) = n∈Z vi,n t , then we obtain that Fun OpL G (D) is isomorphic to the completion of the polynomial algebra C[vi,n ]i∈I,n∈Z with respect to the topology in which the basis of open neighborhoods of zero is formed by the ideals generated by vi,n , i ∈ I, n ≤ N , for N ≤ 0. Theorem 2.5 ([F2]). There is a canonical isomorphism Z( g) Fun OpL G (D× ) of complete topological algebras which is compatible with the action of Der O and Aut O. If M is a smooth gκc -module, then the action of Z( g) on M gives rise to a homomorphism Z( g) → Endgκc M. For example, if M = V0 , then using Theorems 2.4 and 2.5 we identify this homomorphism with the surjection Fun OpL G (D× ) Fun OpL G (D) induced by the natural embedding OpL G (D) → OpL G (D× ). Recall that the Harish-Chandra homomorphism identifies the center Z(g) of U (g) with the algebra (Fun h∗ )W of polynomials on h∗ which are invariant with respect to the action of the Weyl group W . Therefore a character Z(g) → C is the same as a point in Spec(Fun h∗ )W which is the quotient h∗ /W . For λ ∈ h∗ we denote by (λ) its projection onto h∗ /W . In particular, Z(g) acts on Mλ and Vλ via its character ϕ(λ + ρ). We also denote by Iλ the maximal ideal of Z(g) equal to the kernel of the homomorphism Z(g) → C corresponding to the character ϕ(λ + ρ). In what follows we will use the canonical identification between h∗ and the Cartan subalgebra L h of the Langlands dual Lie algebra L g. Recall that in SecL tion 1.3 we defined the space OpRS G-opers on D× with regular singularity L G (D) of RS and its subspace OpL G (D)λ of opers with residue (−λ − ρ). We also defined the subspace × OpL G (D)λˇ ⊂ OpRS L G (D)λ ˇ ⊂ OpL G (D ) of those L G-opers which have trivial monodromy. Here we identify the coweights of the group L G with the weights of G.

18

E. Frenkel

The following result is obtained by combining Theorem 12.4, Lemma 9.4 and Proposition 12.8 of [F2]. Theorem 2.6. (1) Let U be the gκc -module induced from the g[[t]] ⊕ CK-module U (g). Then the homomorphism Z( g) → Endgκc U factors as Z( g) Fun OpL G (D× ) Fun OpRS L G (D) → End gκc U. (2) Let M be a g-module on which the center Z(g) acts via its character (λ + g) → ρ), and let M be the induced gκc -module. Then the homomorphism Z( Endgκc M factors as follows Z( g) Fun OpL G (D× ) Fun OpRS L G (D)λ → End gκc M. Moreover, if M = Mλ , then the last map is an isomorphism Endgκc M Fun OpRS L G (D)λ . (3) For an integral dominant weight λ ∈ h∗ the homomorphism Fun OpL G (D× ) → Endgκc Vλ factors as Fun OpL G (D× ) → Fun OpL G (D)λ → Endgκc Vλ , and the last map is an isomorphism Endgκc Vλ Fun OpL G (D)λ . 2.4. Example Let us consider the case g = sl2 in more detail. Introduce the Segal-Sugawara operators Sn , n ∈ Z, by the formula 1 a Sn z −n−2 = :J (z)Ja (z): , S(z) = 2 a n∈Z

where the normal ordering is defined as in [FB]. Then the center Z(sl2 ) is the completion C[Sn ]∼ n∈Z of the polynomial algebra C[Sn ]n∈Z with respect to the topology in which the basis of open neighborhoods of zero is formed by the ideals of Sn , n > N , for N ≥ 0. We have the following diagram of (vertical) isomorphisms and (horizontal) surjections 2) Z(sl ⏐ ⏐

−−−−→ Endgκc U −−−−→ Endgκc Mλ −−−−→ ⏐ ⏐ ⏐ ⏐

Endgκc Vλ ⏐ ⏐

C[Sn ]∼ n∈Z −−−−→ C[Sn ]n≤0 −−−−→ C[Sn ]n≤0 /Jλ −−−−→ C[Sn ]n≤0 /Jλ

where Jλ is the ideal generated by (S0 − 14 λ(λ + 2)) and Jλ is the ideal generated by Iλ and the polynomial Pλ introduced at the end of Section 1.3.


19

× The space of P GL2 -opers on D nis identified with the space of projective 2 connections of the form ∂t − n∈Z vn t . The isomorphism of Theorem 2.5 sends Sn to v−n−2 . The relevant spaces of P GL2 -opers with regular singularities were described at the end of Section 1.3, and these descriptions agree with the above diagram and Theorem 2.6.

2.5. The Gaudin algebra Now we are ready to identify the Gaudin algebra Z(zi ) (g) with the algebra of functions on a certain space of opers on P1 . 1 L G-opers on P1 with regular singularities Let OpRS L G (P )(zi ),∞ be the space of at z1 , . . . , zN and ∞. For an arbitrary collection of weights λ1 , . . . , λN and λ∞ , let 1 OpRS L G (P )(zi ),∞;(λi ),λ∞

be its subspace of those opers whose residue at the point zi (resp., ∞) is equal to (−λi − ρ), i = 1, . . . , N (resp., (−λ∞ − ρ)). Finally, if all of the weights λ1 , . . . , λN , λ∞ are dominant integral, we introduce a subset 1 OpL G (P1 )(zi ),∞;(λi ),λ∞ ⊂ OpRS L G (P )(zi ),∞;(λi ),λ∞

which consists of those L G-opers which have trivial monodromy representation. On the other hand, for each collection of points z1 , . . . , zN on P1 \∞ we have the Gaudin algebra G G Z(zi ) (g) ⊂ U (g)⊗N U (g)⊗(N +1) /gdiag , where the second isomorphism is obtained by identifying U (g)⊗(N +1) /gdiag with U (g)⊗N ⊗ 1. We have a homomorphism ci : Z(g) → U (g) → U (g)⊗(N +1) corresponding to the ith factor, for all i = 1, . . . , N , and a homomorphism c∞ : Z(g) → U (g)⊗(N +1) corresponding to the (N + 1)st factor. It is easy to see that the images of ci , i = 1, . . . , N , and c∞ belong to Z(zi ) (g). For a collection of weights λ1 , . . . , λN and λ∞ , let I(λi ),λ∞ be the ideal of Z(zi ) (g) generated by ci (Iλi ), i = 1, . . . , N , and c∞ (Iλ∞ ). Let Z(zi ),∞;(λi ),λ∞ be the quotient of Z(zi ) (g) by I(λi ),λ∞ . The algebra Z(zi ),∞;(λi ),λ∞ (g) acts on the space of g-coinvariants in N

Mi ⊗ M∞ ,

i=1

where Mi is a g-module with central character (λi + ρ), i = 1, . . . , N , and M∞ is a g-module with central character (λ∞ + ρ). In particular, if all the weights λ1 , . . . , λN , λ∞ are dominant integral, then we can take as the Mi ’s the finitedimensional irreducible modules Vλi for i = 1, . . . , N , and as M∞ the module Vλ∞ . The corresponding space of g-coinvariants is isomorphic to the space

G N Vλi ⊗ Vλ∞ i=1

20

E. Frenkel

of G-invariants in

N

Vλi ⊗ Vλ∞ . Let Z(zi ),∞;(λi ),λ∞ (g) be the image of the G N in End V ⊗ V . λ λ i ∞ i=1

i=1

algebra Z(zi ),∞;(λi ),λ∞

We have the following result. Theorem 2.7. (1) The algebra Z(zi ) (g) is isomorphic to the algebra of functions on the space 1 OpRS L G (P )(zi ),∞ . (2) The algebra Z(zi ),∞;(λi ),λ∞ (g) is isomorphic to the algebra of functions on the 1 space OpRS L G (P )(zi ),∞;(λi ),λ∞ . (3) For a collection of dominant integral weights λ1 , . . . , λN , λ∞ , there is a surjective homomorphism from the algebra of functions OpL G (P1 )(zi ),∞;(λi ),λ∞ to the algebra Z(zi ),∞;(λi ),λ∞ (g). Proof. In [FB] we defined, for any quasi-conformal vertex algebra V , a smooth projective curve X, a set of points x1 , . . . , xN ∈ X and a collection of V -modules M1 , . . . , MN , the space of coinvariants HV (X, (xi ), (Mi )), which is the quotient N of i=1 Mi by the action of a certain Lie algebra. This construction (which is recalled in the proof of Theorem 4.7 in [F3]) is functorial in the following sense. Suppose that we are given a homomorphism W → V of vertex algebras (so that each Mi becomes a V -module), a collection R1 , . . . , RN of W -modules and a collection of homomorphisms of W -modules Mi → Ri for all i = 1, . . . , N . Then the N N corresponding map i=1 Ri → i=1 Mi gives rise to a map of the corresponding spaces of coinvariants HW (X, (xi ), (Ri )) → HV (X, (xi ), (Mi )). Suppose now that W is the center of V (see [FB]). Then the action of W (W ) → EndC M , where on any V -module M factors through a homomorphism U U (W ) is the enveloping algebra of W (see [FB], Sect. 4.3). Let W (M ) be the image of this homomorphism. Then HW (X, (xi ), (W (Mi ))) is an algebra, and we obtain a natural homomorphism of algebras HW (X, (xi ), (W (Mi ))) → EndC HV (X, (xi ), (Mi )).

(2.7)

If V = V0 , then the center of V is precisely the subspace z( g) of g[[t]]-invariant vectors in V0 (see [FB]). In particular, z( g) is a commutative vertex subalgebra of V0 . A module over the vertex algebra z( g) is the same as a module over the κc ( topological algebra U (z( g)) which is nothing but the center Z( g) of U g) (see [F2], Sect. 11). The action of Z( g) on any gκc -module M factors through the homomorphism Z( g) → Endgκc M . Let Z(M ) denote the image of this homomorphism. Recall that we have identified Z( g) with Fun OpL G (D× ) in Theorem 2.5. For each gκc -module M , the algebra Z(M ) is a quotient of Fun OpL G (D× ), and × hence Spec Z(M ) is a subscheme in OpL G (D× ) which we denote by OpM L G (D ).


21

The space of coinvariants Hz(g) (X, (xi ), Z(Mi )) is computed in the same way as in Theorem 4.7 of [F3]: Hz(g) (X, (xi ), Z(Mi )) Fun OpL G (X, (xi ), (Mi ))

(2.8)

L

where OpL G (X, (xi ), (Mi )) is the space of G-opers on X which are regular on X\{x1 , . . . , xN } and such that their restriction to Dx× belongs to OpLMGi (Dx×i ) for all i = 1, . . . , N . Let u be an additional point of X, different from x1 , . . . , xN , and let us insert z( g) Z(V0 ) at this point. Then by Theorem 10.3.1 of [FB] we have an isomorphism Hz(g) (X, (xi ), (Z(Mi ))) Hz(g) (X; (zi ), u; (Z(Mi )), Z(V0 )). Hence we obtain a homomorphism z( g)u Z(V0 )u → Hz(g) (X, (xi ), (Z(Mi ))).

(2.9)

The corresponding homomorphism Fun OpL G (Du ) → Fun OpL G (X, (xi ), (Mi )) (see formula (2.6)) is induced by the embedding OpL G (X, (xi ), (Mi )) → OpL G (Du ) obtained by restricting an oper to Du . We apply this construction in the case when the curve X is P1 , the points are z1 , . . . , zN and ∞, and the modules are gκc -modules M1 , . . . , MN and M∞ . It is proved in [FB] (see Theorem 9.3.3 and Remark 9.3.10) that the corresponding N space of coinvariants is the space of g(zi ) -coinvariants of i=1 Mi ⊗ M∞ , which is the space H(M1 , . . . , MN , M∞ ) that we have computed in Lemma 2.1. The homomorphism (2.7) specializes to a homomorphism Hz(g) (P1 ; (zi ), ∞; (Z(Mi )), Z(M∞ )) → EndC H(M1 , . . . , MN , M∞ ).

(2.10)

Observe that by its very definition the homomorphism g)u → EndC H(M1 , . . . , MN , M∞ ) Φu : z( constructed in Section 2.2 factors through the homomorphisms (2.10) and (2.9). Hence the image of Φu is a quotient of the algebra Hz(g) (P1 ; (zi ), ∞; (Z(Mi )), Z(M∞ )) Fun OpL G (P1 ; (zi ), ∞; (Z(Mi )), Z(M∞ )), according to the isomorphism (2.8). Let us specialize this result to our setting. First we suppose that all Mi ’s and M∞ are equal to U (g). By Theorem 2.6,(1), we have OpL G (D× ) = OpRS L G (D). U(g)

Therefore we find that 1 Hz(g) (P1 ; (zi ), ∞; (U (g)), U (g)) Fun OpRS L G (P )(zi ),∞ .

(2.11)

22

E. Frenkel Next, by Theorem 2.6,(2), we have U(g)/Iλ

OpL G

(D× ) = OpRS L G (D)λ .

Therefore 1 Hz(g) (P1 ; (zi ), ∞; (U (g)/Iλi ), U (g)/Iλ∞ ) Fun OpRS L G (P )(zi ),∞;(λi ),λ∞ .

(2.12)

Finally, by Theorem 2.6,(3), we have OpLVλG (D× ) = OpL G (D)λ . Therefore Hz(g) (P1 ; (zi ), ∞; (Vλi ), Vλ∞ ) Fun OpL G (P1 )(zi ),∞;(λi ),λ∞ .

(2.13)

Moreover, in all three cases the homomorphism Φu is just the natural homomorphism from Fun OpL G (Du ) to the above algebras of functions that is induced by the restrictions of the corresponding opers to the disc Du . Now consider the homomorphism (2.10) in the case of the space of coinvariants given by the left-hand side of formula (2.11), 1 ⊗(N +1) /gdiag EndC U (g)⊗N , Fun OpRS L G (P )(zi ),∞ → EndC U (g)

where the second isomorphism we use the identification of U (g)⊗(N +1) /gdiag with U (g)⊗N corresponding to the first N factors. It follows from the explicit computation of this map given above that its image belongs to G U (g)⊗N ⊂ EndC U (g)⊗N , and so we have a homomorphism 1 ⊗N G Fun OpRS . L G (P )(zi ),∞ → U (g)

(2.14)

By definition, the Gaudin algebra Z(zi ) (g) is the image of this homomorphism. Likewise, we obtain from formula (2.12) that the homomorphism (2.10) gives rise to a homomorphism G 1 Fun OpRS U (g)⊗(N +1) /gdiag /I(λi ),λ∞ , (2.15) L G (P )(zi ),∞;(λi ),λ∞ → whose image is the algebra Z(zi ),∞;(λi ),λ∞ (g). Finally, formula (2.13) gives us a homomorphism N

1 Vλi ⊗ Vλ∞ , Fun OpL G (P )(zi ),∞;(λi ),λ∞ → EndC i=1

whose image is the algebra Z(zi ),∞;(λi ),λ∞ (g). This proves part (3) of the theorem. To prove parts (1) and (2), it remains to show that the homomorphisms (2.14) and (2.15) are injective. It is sufficient to prove that the latter is injective. To see that, we pass to the associate graded spaces on both sides with respect to natural filtrations which we now describe.


23

According to the identification given in formula (1.4), the algebra of functions 1 on OpRS L G (P )(zi ),∞;(λi ),λ∞ is filtered, and the corresponding associated graded algebra is the algebra of functions on the vector space RS C(z = i ),∞

Γ(P1 , Ω⊗(di +1) (−di z1 − . . . − di zN − di ∞)),

i=1

where Ω is the canonical line bundle on P1 . The algebra G U (g)⊗(N +1) /gdiag /I(λi ),λ∞ carries a PBW filtration, and the associated graded is the algebra of functions on the space µ−1 ((T ∗ G/B)N +1 )/G, where µ : (T ∗ G/B)N +1 → g∗ is the moment map corresponding to the diagonal action of G on (T ∗ G/B)N +1 . The two filtrations are compatible according to [F2]. The corresponding homomorphism of the associate graded algebras RS → Fun µ−1 ((T ∗ G/B)N +1 )/G Fun C(z i ),∞

(2.16)

is induced by a map RS , h(zi ),∞ : µ−1 ((T ∗ G/B)N +1 )/G → C(z i ),∞

that we now describe. Let us identify the tangent space to a point gB ⊂ G/B with (g/gbg −1)∗ −1 gng . Then a point in µ−1 ((T ∗ G/B)N +1 )/G consists of an (N +1)-tuple of points gi B of G/B and an (N + 1)-tuple of vectors (ηi ), where ηi ∈ gi ngi−1 ⊂ g such that N +1 i=1 ηi = 0, considered up to simultaneous conjugation by G. We attach to it the g-valued one-form η=

N i=1

ηi dt t − zi

on P with poles at z1 , . . . , zN , ∞. Let P1 , . . . , P be generators of the algebra of G-invariant polynomials on g of degrees di + 1. Then 1

RS . h(zi ),∞ ((gi ), (ηi )) = (Pi (η))i=1 ∈ C(z i ),∞

The space µ−1 ((T ∗ G/B)N +1 )/G is identified with the moduli space of Higgs fields on the trivial G-bundle with parabolic structures at z1 , . . . , zN , ∞, and the map h(zi ),∞ is nothing but the Hitchin map (see [ER]). The Hitchin map is known to be proper, so in particular it is surjective (see, e.g., [M]). Therefore the corresponding homomorphism (2.16) of algebras of functions is injective. This implies that the homomorphism (2.15) is also injective and completes the proof of the theorem. This theorem has an important application to the question of simultaneous diagonalization of generalized Gaudin Hamiltonians, or equivalently, of the comN mutative algebra Z(zi ) (g), on the tensor product i=1 Mi of g-modules. Indeed, the joint eigenvalues of the generalized Gaudin Hamiltonians on any eigenvector

24

E. Frenkel

in N i=1 Mi correspond to a point in the spectrum of the algebra Z(zi ) (g), which, 1 according to Theorem 2.7,(1), is a point of the space OpRS L G (P )(zi ),∞ . If we assume in addition that each of the modules Mi admits a central character (λi + ρ) (for instance, if Mi is the Verma module Mλi ) and we are looking for eigenvectors in the component of N i=1 Mi corresponding to the central character (−λ∞ − ρ) with respect to the diagonal action of g, then the joint eigenvalues define a point in the spectrum of the algebra Z(zi ),∞;(λi ),λ∞ (g), i.e., a point of 1 OpRS L G (P )(zi ),∞;(λi ),λ∞ . Finally, for a collection of dominant integral weights λ1 , . . . , λN , λ∞ , the G joint eigenvalues of the generalized Gaudin Hamiltonians on ( N i=1 Vλi ⊗ Vλ∞ ) is a point in the spectrum of the algebra Z(zi ),∞;(λi ),λ∞ (g), which is a point of OpL G (P1 )(zi ),∞;(λi ),λ∞ . A natural question is whether, conversely, one can attach to a L G-oper on 1 P with regular singularities at z1 , . . . , zN , ∞ (and satisfying additional conditions N as above) an eigenvector in i=1 Mi with such eigenvalues. It turns out that for general modules this is not true, but if these modules are finite-dimensional, then we conjecture that it is true. In order to construct the eigenvectors we use the procedure called Bethe Ansatz. As shown in [FFR], this procedure may be cast in the framework of coinvariants that we have discussed in this section, using the Wakimoto modules over gκc . We will explain that in Section 4 and Section 5.5. But first we need to introduce Miura opers and Cartan connections.

3. Miura opers and Cartan connections By definition (see [F2], Sect. 10.3), a Miura G-oper on X (which is a smooth curve ), where (F, ∇, FB ) is a G-oper on X and or a disc) is a quadruple (F, ∇, FB , FB FB is another B-reduction of F which is preserved by ∇. We denote the space of Miura G-opers on X by MOpG (X). 3.1. Miura opers and flag manifolds A B-reduction of F which is preserved by the connection ∇ is uniquely determined by a B-reduction of the fiber Fx of F at any point x ∈ X (in the case when U = D, x has to be the origin 0 ∈ D). The set of such reductions is the Fx -twist (G/B)Fx = Fx × G/B = FB,x × G/B = (G/B)FB,x G

(3.1)

B

of the flag manifold G/B. If X is a curve or a disc and the oper connection has a regular singularity and trivial monodromy representation, then this connection gives us a global (algebraic) trivialization of the bundle F. Then any B-reduction of the fiber Fx gives rise to a global (algebraic) B-reduction of F. Thus, we obtain: Lemma 3.1. Suppose that we are given an oper τ on a curve X (or on the disc) such that the oper connection has a regular singularity and trivial monodromy.


25

Then for each x ∈ X there is a canonical isomorphism between the space of Miura opers with the underlying oper τ and the twist (G/B)FB,x . Recall that the B-orbits in G/B, known as the Schubert cells, are parameterized by the Weyl group W of G. Let w0 be the longest element of the Weyl group of G. Denote the orbit Bw−1 w0 B ⊂ G/B by Sw (so that S1 is the open orbit). We obtain from the second description of (G/B)Fx given in formula (3.1) that (G/B)Fx decomposes into a union of locally closed subvarieties Sw,FB,x , which are the FB,x -twists of the Schubert cells Sw . The B-reduction FB,x defines a point are in relative in (G/B)FB,x . We will say that the B-reductions FB,x and FB,x position w with if FB,x belongs to Sw,FB,x . In particular, if it belongs to the open orbit S1,FB,x , we will say that FB,x and FB,x are in generic position. A Miura G-oper is called generic at the point x ∈ X if the B-reductions FB,x and FB,x of Fx are in generic position. In other words, FB,x belongs to the stratum OpG (X) × S1,FB,x ⊂ MOpG (X). Being generic is an open condition. Therefore if a Miura oper is generic at x ∈ X, then there exists an open neighborhood U of x such that it is also generic at all other points of U . We denote the space of generic Miura opers on U by MOpG (U )gen . Lemma 3.2. Suppose we are given a Miura oper on the disc Dx around a point x ∈ X. Then its restriction to the punctured disc Dx× is generic. Proof. Since being generic is an open condition, we obtain that if a Miura oper is generic at x, it is also generic on the entire Dx . Hence we only need to consider the situation where the Miura oper is not generic at x, i.e., the two reductions are in relative position w = 1. Let us trivialize the B-bundle FB , FB,x and FB,x and hence the G-bundle FG over Dx . Then ∇ gives us a connection on the trivial G-bundle which we can bring to the canonical form ∇ = ∂t + p−1 +

vj (t) · pj

j=1

(see Lemma 1.1). It induces a connection on the trivial G/B-bundle. We are given a point gB in the fiber of the latter bundle which lies in the orbit Sw = Bw0 wB, where w = 1. Consider the horizontal section whose value at x is gB, viewed as a map Dx → G/B. We need to show that the image of this map lies in the open B-orbit S1 = Bw0 B over Dx× , i.e., it does not lie in the orbit Sy for any y = 1. Suppose that this is not so, and the image of the horizontal section actually lies in the orbit Sy for some y = 1. Since all B-orbits are H-invariant, we obtain that the same would be true for the horizontal section with respect to the connection ∇ = h∇h−1 for any constant element of H. Choosing h = ρˇ(a) for a ∈ C× , we can bring the connection to the form ∂t + a−1 p−1 +

j=1

adj vj (t) · pj .

26

E. Frenkel

Changing the variable t to s = a−1 t, we obtain the connection ∂s + p−1 +

adj +1 vj (t),

j=1

so choosing small a we can make the functions vj (t) arbitrarily small. Therefore without loss of generality we can consider the case when our connection operator is ∇ = ∂t + p−1 . In this case our assumption that the horizontal section lies in Sy , y = 1, means that the vector field ξp−1 corresponding to the infinitesimal action of p−1 on G/B is tangent to an orbit Sy , y = 1, in the neighborhood of some point gB of Sw ⊂ G/B, w = 1. But then, again because of the H-invariance of the B-orbits, the vector field ξhp−1 h−1 is also tangent to this orbit for any h ∈ H. For any i = 1 . . . , , there exists a one-parameter subgroup h , ∈ C× in H, such that lim p−1 −1 = fi . Hence we obtain that each of the vector fields ξfi , i = 1 . . . , , (i)

→0

is tangent to the orbit Sy , y = 1, in the neighborhood of gB ∈ Sw , w = 1. But then all commutators of these vectors fields are also tangent to this orbit. Hence we obtain that all vector fields of the form ξp , p ∈ n− , are tangent to Sy in the neighborhood of gB ∈ Sw . Consider any point of G/B that does not belong to the open dense orbit S1 . Then the quotient of the tangent space to this point by the tangent space to the B-orbit passing through this point is non-zero and the vector fields from the Lie algebra n− map surjectively onto this quotient. Therefore they cannot be tangent to the orbit Sy , y = 1, in a neighborhood of gB. Therefore our Miura oper is generic on Dx× .

This lemma shows that any Miura oper on any smooth curve X is generic over an open dense subset. 3.2. Cartan connections Introduce the H-bundle Ωρˇ on X which is uniquely determined by the following property: for any character λ : H → C× , the line bundle Ωρˇ × λ associated to the H

ˇ corresponding one-dimensional representation of H is Ωλ,ρ . ρˇ Explicitly, connections on Ω may be described as follows. If we choose a local coordinate t on X, then we trivialize Ωρˇ and represent the connection as an operator ∂t + u(t), where u(t) is an h-valued function on X. If s is another coordinate such that t = ϕ(s), then this connection will be represented by the operator ϕ (s) . (3.2) ∂s + ϕ (s)u(ϕ(s)) − ρˇ · ϕ (s) Let Conn(Ωρˇ)X be the space of connections on the H-bundle Ωρˇ on X. When no confusion can arise, we will simply write ConnX . We define a map

bX : ConnX → MOpG (X)gen .


27

Suppose we are given a connection ∇ on the H-bundle Ωρˇ on D. We associate to it a generic Miura oper as follows. Let us choose a splitting H → B of the homomorphism B → H and set F = Ωρˇ × G, FB = Ωρˇ × B, where we consider H

H

the adjoint action of H on G and on B obtained through the above splitting. The choice of the splitting also gives us the opposite Borel subgroup B− , which is the unique Borel subgroup in generic position with B containing H. Let again w0 be the longest element of the Weyl group of g. Then w0 B is a B-torsor equipped with a left action of H, so we define the B-subbundle FB of F as Ωρˇ × w0 B. H

Observe that the space of connections on F is isomorphic to the direct product ˇ ConnX × Γ(X, Ωα(ρ)+1 ). α∈∆

Its subspace corresponding to negative simple roots is isomorphic to the tensor product of i=1 g−αi and Fun X. Having chosen a basis element fi of g−αi for each i = 1, . . . , , we now construct an element p−1 = i=1 fi ⊗ 1 of this space. Now we set ∇ = ∇+p−1 . By construction, ∇ has the correct relative position with . Therefore the quadruple the B-reduction FB and preserves the B-reduction FB (F, ∇, FB , FB ) is a generic Miura oper on X. We define the morphism bX by ). setting bX (∇) = (F, ∇, FB , FB This map is independent of the choice of a splitting H → B and of the generators fi , i = 1, . . . , . Proposition 3.3 ([F2],Proposition 10.4). The map bX is an isomorphism of algebraic varieties ConnX → MOpG (X)gen . Thus, generic Miura opers are the same as Cartan connections, which are much simpler objects than opers. The composition bX of bX and the forgetful map MOpG (X)gen → OpG (X) is called the Miura transformation. For example, in the case of g = sl2 , we have a connection ∇ = ∂t − u(t) on t the line bundle Ω1/2 (equivalently, a connection ∇ = ∂t + u(t) on Ω−1/2 ), and the Miura transformation assigns to this connection the P GL2 -oper

−u(t) 0 ∂t + . 1 u(t) The oper B reduction FB corresponds to the upper triangular matrices, and the corresponds to the lower triangular matrices. The correMiura B-reduction FB sponding projective connection is ∂t2 − v(t) = (∂t − u(t))(∂t + u(t)), i.e., u(t) → v(t) = u(t)2 − u (t).

28

E. Frenkel

3.3. Singularities of Cartan connections Consider the Miura transformation bDx× in the case of the punctured disc Dx× , bDx× : Conn(Ωρˇ)Dx× → OpG (Dx× ). ρˇ × be the space of all connections on the H-bundle Let Conn(Ωρˇ)RS Dx ⊂ Conn(Ω )Dx ρˇ Ω on Dx with regular singularity, i.e., those for which the connection operator has the form ˇ λ ∇ = ∂t + + u n tn . t n≥0

We define a map resh : Conn(Ωρˇ)RS D →h ˇ assigning to such a connection its residue λ. It follows from the definition of the Miura transformation bDx× that its restricRS ρˇ × takes values in Op tion to Conn(Ωρˇ)RS G (Dx ). Hence we obtain Dx ⊂ Conn(Ω )Dx a morphism RS

RS bx : Conn(Ωρˇ)RS Dx → OpG (Dx ).

Explicitly, after choosing a coordinate t on Dx , we can write ∇ as ∂t + t−1 u(t), where u(t) ∈ h[[t]]. Its residue is u(0). Then the corresponding oper with regular singularity is by definition the N ((t))-equivalence class of the operator ∇ = ∂t + p−1 + t−1 u(t), which is the same as the N [[t]]-equivalence class of the operator ρˇ(t)∇ˇ ρ(t)−1 = ∂t + t−1 (p−1 − ρˇ + u(t)), so it is indeed an oper with regular singularity. Therefore it follows from the definition that we have a commutative diagram b

RS

x RS Conn(Ωρˇ)RS Dx −−−−→ OpG (Dx ) ⏐ ⏐ ⏐res resh ⏐

h

−−−−→

(3.3)

h/W

ˇ → λ ˇ − ρˇ and the where the lower horizontal map is the composition of the map λ projection : h → h/W . RS

Now let Connreg ˇ ⊂ ˇ be the preimage under bx of the subspace OpG (Dx )λ Dx ,λ RS × ˇ OpG (Dx ) of G-opers on Dx with regular singularity, residue (−λ − ρˇ) and trivial monodromy. By the commutativity of the above diagram, a connection in ˇ Connreg ˇ) + ρˇ for some element ˇ necessarily has residue of the form −w(λ + ρ Dx ,λ reg w of the Weyl group of G, so that ConnD ,λˇ is the disjoint union of its subsets x ˇ + ρˇ) + ρˇ. Connreg consisting of connections with residue −w(λ ˇ Dx ,λ,w


29

RS

The restriction of bx to Connreg is a map ˇ D ,λ,w x

bλ,w : ˇ

Connreg ˇ Dx ,λ,w

→ OpG (Dx )λˇ .

Let us recall that by construction of the Miura transformation b, each oper on Dx× which lies in the image of the map b (hence in particular, in the image of bλ,w ˇ ) carries a canonical horizontal B-reduction FB = Ωρˇ × w0 B H

(i.e., it carries a canonical structure of Miura oper on Dx× ). But if this oper is in the image of bλ,w ˇ , i.e., belongs to OpG (Dx )λ ˇ , then the oper B-reduction FB (and hence the oper bundle F) has a canonical extension to a B-bundle on the entire disc Dx , namely, one for which the oper connection has the form (1.8). Therefore the B-reduction FB = Ωρˇ × w0 B may also be extended to Dx . H

to a map Therefore we can lift bλ,w ˇ bλ,w : Connreg → MOpG (Dx )λˇ . ˇ ˇ D ,λ,w x

⊂ MOpG (Dx )λˇ be the subvariety of those Miura opers of Let MOpG (Dx )λ,w ˇ ˇ coweight λ which have relative position w at x. Then MOpG (Dx )λ,w OpG (Dx )λˇ × Sw,FB,x . ˇ

The following result is due to D. Gaitsgory and myself [FG] (see [F3], Proposition 2.9). Proposition 3.4. For each w ∈ W the morphism bλ,w is an isomorphism between ˇ the varieties Connreg and MOp (D ) . ˇ x λ,w G ˇ D ,λ,w x

Proof. First we observe that at the level of points the map defined by bλ,w ˇ ,w ∈ W, , w ∈ W , to MOp (D ) , is a bijection. Indeed, by from the union of Connreg ˇ x G ˇ λ Dx ,λ,w Proposition 3.3 we have a map taking a Miura oper from MOpG (Dx )λˇ , considered as a Miura oper on the punctured disc Dx× , to a connection ∇ on the H-bundle Ωρˇ over Dx× . We have shown above that ∇ has regular singularity at x and that its ˇ + ρˇ)+ ρˇ, w ∈ W . Thus, we obtain a map from the set of residue is of the form −w(λ points of MOpG (Dx )λˇ to the union of Connreg ˇ , w ∈ W , and by Proposition 3.3 Dx ,λ,w it is a bijection. It remains to show that if the Miura oper belongs to MOpG (Dx )λ,w ˇ , then ˇ the corresponding connection has residue precisely −w(λ + ρˇ) + ρˇ. ˇ Let us choose a ) of coweight λ. Thus, we are given a G-oper (F, ∇, FB , FB trivialization of the B-bundle FB . Then the connection operator reads ∇ = ∂t +

i=1

ˇ

tαi ,λ fi + v(t),

v(t) ∈ b[[t]].

(3.4)

30

E. Frenkel

Suppose that the horizontal B-reduction FB of our Miura oper has relative position w with FB at x (see Section 3 for the definition of relative position). We need to ˇ + ρˇ) + ρˇ. show that the corresponding connection on FH Ωρˇ has residue −w(λ This is equivalent to the following statement. Let Φ(t) be the G-valued solution of the equation

ˇ αi ,λ t fi + v(t) Φ(t) = 0, (3.5) ∂t + i=1

such that Φ(0) = 1. Since the connection operator is regular at t = 0, this solution exists and is unique. Then Φ(t)w−1 w0 is the unique solution of the equation (3.5) whose value at t = 0 is equal to w−1 w0 . By Lemma 3.2, we have Φ(t)w−1 w0 = Xw (t)Yw (t)Zw (t)w0 , where Xw (t) ∈ N ((t)),

Yw (t) ∈ H((t)), Zw (t) ∈ N− ((t)). ˇw (t)Yw (t), where µ ˇw is a coweight and Yw (t) ∈ H[[t]]. We can write Yw (t) = µ Since the connection ∇ preserves Φ(t)w0 b+ w0 Φ(t)−1 = Φ(t)b− Φ(t)−1 , the connection X(t)−1 w ∇Xw (t) preserves Yw (t)Zw (t)b− Zw (t)−1 Yw (t)−1 = b− , and therefore has the form ∂t +

ˇ

tαi ,λ fi −

i=1

µ ˇw + u(t), t

u(t) ∈ h[[t]].

ˇ By conjugating it with λ(t) we obtain a connection ˇ+µ λ ˇw + u(t), t Therefore we need to show that ∂t + p−1 −

u(t) ∈ h[[t]].

ˇ + ρˇ) − (λ ˇ + ρˇ). µ ˇ w = w(λ

(3.6)

To see that, let us apply the identity Φ(t)w−1 = Xw (t)Yw (t)Zw (t) to a nonzero vector vw0 (ν) of weight w0 (ν) in a finite-dimensional irreducible g-module Vν of highest weight ν (so that vw0 (ν) is a lowest weight vector and hence is unique up to scalar). The right-hand side will then be equal to a P (t)vw0 (ν) plus the sum of terms of weights greater than w0 (ν), where P (t) = ctw0 (ν),ˇµw , c = 0, plus the sum of terms of higher degree in t. Applying the left-hand side to vw0 (ν) , we obtain Φ(t)vw−1 w0 (ν) , where vw−1 w0 (ν) ∈ Vν is a non-zero vector of weight w−1 w0 (ν) which is also unique up to a scalar.


31

Thus, we need to show that the coefficient with which vw0 (ν) enters the expression Φ(t)vw−1 w0 (ν) is a polynomial in t whose lowest degree is equal to ˇ + ρˇ) − (λ ˇ + ρˇ), w0 (ν), w(λ because if this is so for all dominant integral weights ν, then we obtain the desired equality (3.6). But this formula is easy to establish. Indeed, from the form (3.4) of the oper connection ∇ it follows that we can obtain a vector proportional to vw0 ˇ by applying the operators α ,1λ+1 tαi ,λ+1 fi , i = 1, . . . , , to vw−1 w0 (ν) in some ˇ i order. The linear combination of these monomials appearing in the solution is the term of the lowest degree in t with which vw0 (ν) enters Φ(t)vw−1 w0 (ν) . It follows from Lemma 3.2 that it is non-zero. The corresponding power of t is nothing but ˇ + ρˇ)-degrees of the vectors vw−1 w and vw , i.e., the difference between the (λ 0 0 ˇ + ρˇ − w0 (ν), λ ˇ + ρˇ = w0 (ν), w(λ ˇ + ρˇ) − (λ ˇ + ρˇ), w−1 w0 (ν), λ

as desired. This completes the proof.

of connections on the HThus, we have identified the space Connreg ˇ Dx ,λ,w ρˇ × bundle Ω on Dx of the form with the space of Miura opers on Dx× such that the underlying oper belongs to OpG (Dx )λˇ and the corresponding B-reductions FB and FB have relative position w at x. The condition that the image under bλ,w of a connection of the form ˇ ˇ + ρˇ) − ρˇ w(λ + u(t), u(t) ∈ h[[t]], t is an oper without monodromy imposes polynomial equations on the coefficients of ˇ = 0 and w = si , the ith simple the series u(t). Consider the simplest case when λ ˇ + ρˇ) + ρˇ = α reflection. Then −w(λ ˇ i , so we write this connection as ∂t −

∇ = ∂t +

α ˇi + u(t), t

u(t) ∈ h[[t]].

(3.7)

Lemma 3.5 ([F3],Lemma 2.10). A connection of the form (3.7) belongs to Connreg Dx ,si (i.e., the corresponding G-oper is regular at x) if and only if αi , u(0) = 0.

4. Wakimoto modules and Bethe Ansatz In this section we explain how to construct Bethe eigenvectors of the generalized Gaudin Hamiltonians. For that we utilize the Wakimoto modules over gκc which are parameterized by Cartan connections on the punctured disc. As the result, the eigenvectors will be parameterized by the Cartan connections on P1 with regular singularities at z1 , . . . , zN , ∞ and some additional points w1 , . . . , wm with residues λ1 , . . . , λN , λ∞ and −αi1 , . . . , −αim and whose Miura transformation is an oper that has no singularities at w1 , . . . , wm .

32

E. Frenkel

4.1. Definition of Wakimoto modules We recall some of the results of [FF1, F2] on the construction of the Wakimoto realization. Let Ag be the Weyl algebra with generators aα,n , a∗α,n , α ∈ ∆+ , n ∈ Z, and relations [aα,n , a∗β,m ] = δα,β δn,−m ,

[aα,n , aβ,m ] = [a∗α,n , a∗β,m ] = 0.

(4.1)

Introduce the generating functions aα (z) =

aα,n z −n−1 ,

(4.2)

a∗α,n z −n .

(4.3)

n∈Z

a∗α (z) =

n∈Z

Let Mg be the Fock representation of Ag generated by a vector |0 such that aα,n |0 = 0,

n ≥ 0;

a∗α,n |0 = 0,

n > 0.

It carries a vertex algebra structure (see [F2]). Let π0 be the commutative algebra C[bi,n ]i=1,...,;n 0. uH H

α1

u α2

Taking also in this case for simplicity ν = 1/2, we find the following bootstrap satisfied S˜l(12) (θ) = S˜l2 (θ − σ12 /2 + iπ/4)S˜l1 (θ + σ12 /2 − iπ/4), (2.36) which yields the S-matrix ⎛

⎞ −(σ12 /2, 1) −1 (σ12 , 2) −1 −(σ21 /2, 3) ⎠ . S˜SU(3) (θ, σ21 ) = ⎝ −(σ21 , 2) −(σ21 /2, 3) −(σ12 /2, 1) −1

(2.37)

The S-matrix (2.37) allows for the processes 2 + 1 → (12),

1 + (12) → 2,

(12) + 2 → 1,

(2.38)

Integrable Models with Unstable Particles

69

instead of (2.33). Now the fusing angles are read off as iπ 3iπ σ12 3iπ σ12 (12) 2 1 γ21 = − + σ12 , γ1(12) − , γ2(12) − (2.39) =− =− 2 4 2 4 2 and also satisfy (2.26). The masses and decay width are obtained again from (2.12) and (2.13) with σ12 → σ21 . As a whole, we can think of this theory simply as being obtained from the Z2 -Dynkin diagram automorphism which exchanges the roles of the particles 1 and 2. However, since parity invariance is now broken this is not a symmetry any more and the two theories are different. In the asymptotic limit σ12 → ∞, we obtain once again a simple version of the decoupling rule (3.3) and the theory decouples into two SU (2)2 -models. The next example, SU (4)2 -HSG, is more intriguing as it leads to the prediction a new unstable particle. Proceeding in the way as before we construct the ˜ for details see [19]. We found there the processes corresponding amplitudes S, 1 + 2 → (12), 3 + 2 → (23),

(12) + 1 → (23) + 3 →

2, 2,

2 + (12) → 2 + (23) →

1, 3,

(2.40)

which simply correspond to two copies of SU (3)2 -HSG. It is interesting to note that the amplitudes S˜(12)3 and S˜(23)1 contain poles at 3iπ 3iπ σ21 − 2σ23 σ23 − 2σ21 (123) − and γ(23)1 = − , 2 4 2 4 which yield the possible processes (123)

γ(12)3 =

(12) + 3 → (123), (23) + 1 → (123),

(123) + (12) → (123) + (23) →

3, 1,

3 + (123) → (12), 1 + (123) → (23).

(2.41)

(2.42)

An interesting prediction results from the consideration of the first two processes in (2.42). Making in the first process the particle (12) and in the second the particle (23) stable, by σ2 → σ1 and by σ2 → σ3 , respectively, both predict the mass of the particle (123) as (2.43) m(123) ∼ me|σ13 |/2 . This value is precisely the one we expect from the approximation in the BreitWigner formula (2.15). Note that in one case we obtain σ13 and in the other σ31 as a resonance parameter. The difference results from the fact that according to the processes (2.42), the particle (123) is either formed as (1 + 2) + 3 or 3 + (2 + 1). Thus the different parity shows up in this process, but this has no effect on the values for the mass. In [19] we presented more examples and remarkably we found consistency in each case. We take the closure of the bootstrap equations as a non-trivial confirmation for our proposal.

3. Lie algebraic structure for theories with unstable particles There exist some concrete Lagrangian formulations for integrable theories with unstable particles, such as the aforementioned HSG-models (2.27). Inspired by the

70

O.A. Castro-Alvaredo and A. Fring

structure of these models, we present here a slightly more general Lie algebraic picture. We keep the discussion here abstract and supply below concrete examples. ˜ (possibly with For our formulation we need an arbitrary simply laced Lie algebra g ˜ ˜ a subalgebra h) with rank together with its associated Dynkin diagram (see for instance [33]). To each node we attach a simply laced Lie algebra gi with rank i and to each link between the nodes i and j a resonance parameter σij = σi − σj , as depicted in the following ˜g/ ˜h-coset Dynkin diagram σij σjk uH u ... u H u. . . u H H g1 g˜ gk gi gj

σlm σmn uH ... u H u. . . H H gn gl gm

Besides the usual rules for Dynkin diagrams, we adopt here the convention that we add an arrow to the link, which manifests the parity breaking and allows to identify the signs of the resonance parameters. An arrow pointing from the node i to j simply indicates that σij > 0. Since we are dealing exclusively with simply laced Lie algebras, this should not lead to confusion. To each simple root of the algebras gi , we associate now a stable particle and to each positive non-simple ˜ an unstable particle, such that root of g # of stable particles =

˜

i ,

# of unstable particles =

i=1

˜ − 2) ˜(h . 2

(3.1)

From the discussion above, we expect that the σ’s will be associated to unstable particles, but we note that the # of resonance parameters =

˜ ˜ − 1) ( 2

(3.2)

˜ = ˜ + 1, e.g., for g ˜ = only agrees with the amount of unstable particles for h ˜ SU ( + 1). Since the resonance parameters govern the mass of the unstable particles, this discrepancy is interpreted as an unavoidable mass degeneracy. ˜ k -homogeneous sine-Gordon Concrete examples for this formulations are the g ˜ to be simply laced and g1 = · · · = models [20, 21], for which one can choose g ˜ to be non-simply laced g˜ = SU (k). This is generalized [34] when taking instead g ˜ . The choice g1 = · · · = and gi = SU (2k/α2i ), with αi being the simple roots of g g˜ = g with g being any arbitrary simply laced Lie algebra gives the g|˜ g-theories [35]. An example for a theory associated to a coset is the roaming sinh-Gordon model [36], which can be thought of as ˜g/ ˜h ≡ limk→∞ SU (k + 1)/SU (k) with g1 = · · · = g˜ = SU (2). It is clear that the examples presented here do not exhaust yet all possible combinations and the structure mentioned above allows for more combinations of algebras, which are not yet explored. One is also not limited to Dynkin diagrams and may consider more general graphs which have multiple links, i.e., resonance parameters, between various nodes. Examples for such theories were proposed and studied in [37].


71

3.1. Decoupling rule Of special interest is to investigate the behavior of previously defined systems when certain resonance parameters σ become very large or tend to infinity. The physical motivation for that is to describe a renormalization group (RG) flow, which we shall discuss in more detail below. Here we present first the mathematical set-up. Decoupling rule: Call the overall Dynkin diagram C and denote the associated ˜ C and gC , respectively. Let σij be some resonance Lie group and Lie algebra by G parameter related to the link between the nodes i and j. To each node i attach a simply laced Lie algebra gi . Produce a reduced diagram Cji containing the node j by cutting the link adjacent to it in the direction i. Likewise produce a reduced diagram Cij containing the node i by cutting the link adjacent to it in the direction ˜ C -theory decouples according to the rule j. Then the G ˜C = G ˜ (C−C ) ⊗ G ˜ (C−C ) /G ˜ (C−C −C ) . lim G (3.3) ij

σij →∞

ji

ij

ji

We depict this rule also graphically in terms of Dynkin diagrams: σij → ∞ C ... u e ... e u. . . ⇒ gi gj ... u gi

C − Cji e ...

e ⊗

e

C − Cij ... e

u. . . gj

C − Cij − Cji e e ...

According to the GKO-coset construction [32], this means that the Virasoro central charge flows as cg˜C → cg˜C−Cij + cg˜C−Cji − cg˜C−Cij −Cji . (3.4) The rule may be applied consecutively to each disconnected subgraph produced according to the decoupling rule (3.3). Note that this rule describes a decoupling and not a fusing, as it only predicts the flow in one direction and the limit is not reversible. From a physical point of view this is natural as analogously the RG flow is also irreversible. The rule (3.3) generalizes a rule proposed in [26], which was based on the assumption that unstable particles are associated exclusively to positive roots of height two. More familiar in the mathematical literature is a decoupling rule found by ˜ from a given algebra Dynkin [38] for the construction of semi-simple2 subalgebras h ˜ . For the more general diagrams which can be related to the g ˜ k -HSG models the g generalized rule can be found in [39]. These rules are all based on removing some of the nodes rather than links. For our physical situation at hand this corresponds to sending the masses of all stable particles which are associated to the algebra of a particular node to infinity. As in the decoupling rule (3.3) the number of stable 2 The

subalgebras constructed in this way are not necessarily maximal and regular. A guarantee for obtaining those, except in six special cases, is only given when one manipulates adequately the extended Dynkin diagram.

72


particles remains preserved, it is evident that the two rules are inequivalent. Letting for instance the mass scale in gj go to infinity, the generalized (in the sense that gj can be different from A ) rule of Kuniba is simply depicted as ... u gi

C u gj

u. . . gk

mj → ∞

⇒

C − Cji ... u gi

⊗

C − Cjk u. . . gk

Clearly this cannot be produced with (3.3). 3.2. A simple example: The SU (4)2 -HSG model ˜ We illustrate the working of the rule (3.3) with a simple example. We take g to be SU (4), attach to each node simply an SU (2) algebra and to the links the resonance parameters σ12 , σ13 , σ23 . This corresponds to the SU (4)2 -HSG model. For the ordering σ13 > σ12 > σ23 the rule (3.3) predicts the following flow u

u u g = SU (4)2 ˜ c=2 α1 α2 α3 → σ13 u u ⊗ u u u g = SU (3)⊗2 ˜ 2 /SU (2)2 c = 1.9 α2 α1 α2 α3 α2 → σ12 u ⊗ u u g = SU (3)2 ⊗ SU (2)2 c = 1.7 ˜ α1 α3 α2 → σ23 u ⊗ u ⊗ u g = SU (2)⊗3 ˜ c = 1.5 2 α1 α2 α3 The central charges are obtained from (2.28) using (3.4). Choosing instead the ordering σ23 > σ13 > σ12 , we compute u u u g = SU (4)2 ˜ c=2 α1 α2 α3 u u ⊗ u → σ23 g = SU (3)2 ⊗ SU (2)2 c = 1.7 ˜ α1 α2 α3 → σ13 is already decoupled → σ12

u ⊗ u ⊗ u g = SU (2)⊗3 ˜ c = 1.5 2 α2 α3 α1 It is important to note the non-commutative nature of the limiting procedures. For more complicated algebras it is essential to keep track of the labels on the nodes, since only in this way one can decide whether they cancel against the subgroup diagrams or not. 3.3. A non-trivial example: The (E6 )2 -HSG model As by now we do not have a rigorous proof of the decoupling rule (3.3), we take the support for its validity from the working of various examples. We will check below the analytic predictions of the rule against some alternative method. As the


73

previous example was a simple pedagogical one, we will consider next a non-trivial one leading to an intricate prediction for the RG-flow. The confirmative double check below can hardy be accidental and we take that as very strong support for the validity of (3.3). We consider now the (E6 )2 -HSG model. In this case we have ˜ = 6, ˜h = 12 such that we have 6 stable particles, 30 unstable particles and 15 resonance parameters. From the 5! possible orderings for the resonance parameters we present here only two concrete ones, which will predict different types of flows and mass degeneracies. Note that this degeneracy is not the unavoidable one resulting from the difference between the number of resonance parameters and non-simple positive roots that is 30 − 15, as discussed for (3.1) and (3.2). The degeneracies discussed here are a consequence of the particular choices of the resonance parameters. The conventions for the labelling of our particles are indicated in the following Dynkin diagram: u α2 u α3

u α1

u α4

u α5

u α6

We choose first the ordering and values for resonance parameters as σ13 = 100 > σ34 = 80 > σ45 = 60 > σ56 = 40 > σ24 = 20 .

(3.5)

According to the decoupling rule (3.3), we predict therefore the flow: 36 7

E6 ⊗2

→ σ16 = 280

SO(10)

→ σ15 = 240

SO(10) ⊗ SU (5)/SU (4)

→ σ14 , σ36 = 180

SO(8) ⊗ SU (5) ⊗ SU (3)/SU (4) ⊗ SU (2)

→ σ12 = 160

is already decoupled

→ σ35 = 140

SU (5) ⊗ SU (4) ⊗ SU (3)/SU (3) ⊗ SU (2)

→ σ26 = 120 → σ13 , σ46 = 100

/SO(8)

5

⊗3

⊗ SU (3)/SU (3)

⊗2

⊗ SU (3) ⊗ SU (2)/SU (3) ⊗ SU (2)

SU (4) SU (4)

⊗3

⊗2

⊗2

⊗ SU (2)

→ σ25 , σ34 = 80

SU (3)

→ σ32 , σ45 = 60

SU (3)⊗2 ⊗ SU (2)⊗2

→ σ56 = 40 → σ24 = 20

⊗4

SU (3) ⊗ SU (2) SU (2)⊗6

∼ 5. 14

⊗ SU (2) ⊗2

/SU (2)

34 7 ∼ 4. 86 319 70 ∼ 4. 56 61 14

∼ 4. 36

4.3 4 3.6 3.4 3.2 3

Note that eight particles are pairwise degenerate and we therefore expect to find 15−8/2 = 11 plateaux in the flow. The first step which corresponds to one of these degeneracies occurs for instance at σ14 = σ36 and we have to apply the decoupling rule twice at this point before we get a new fixed point theory.

74

O.A. Castro-Alvaredo and A. Fring Next we arrange the couplings as σ45 = 100 > σ34 = 80 > σ13 = 60 > σ56 = 40 > σ24 = 20 .

(3.6)

and compute from (3.3) the flow → σ16 → σ15 → σ36 → σ35 → σ26 → σ14 → σ12 → σ45 → σ34 → σ13 → σ56 → σ24

= 280 = 240 = 220 = 180 = 160 = σ46 = 140 = σ25 = 120 = 100 = 80 = σ32 = 60 = 40 = 20

E6 SO(10)⊗2 /SO(8) SO(10) ⊗ SU (5)/SU (4) SU (5)⊗2 ⊗ SO(8)/SU (4)⊗2 SU (5)⊗2 /SU (3) SU (4)⊗2 ⊗ SU (5)/SU (3)⊗2 SU (4)⊗2 ⊗ SU (3)⊗2 /SU (2)⊗2 ⊗ SU (3) SU (4) ⊗ SU (3)⊗3 /SU (2)⊗3 SU (4) ⊗ SU (3)⊗2 /SU (2) SU (3)⊗3 SU (3)⊗2 ⊗ SU (2)⊗2 SU (3) ⊗ SU (2)⊗4 SU (2)⊗6

36 7

∼ 5. 14

5 34 7 ∼ 4. 86 33 7 ∼ 4. 71 158 35 ∼ 4. 51 156 35 ∼ 4. 46

4.2 4.1 3.9 3.6 3.4 3.2 3

In this case we have only six particles pairwise degenerate and we expect to find 15 − 6/2 = 12 plateaux. In the next section we find that the predictions made here are confirmed even for this involved case.

4. How to detect unstable particles? In Section 2 we described several arguments which predict the spectrum of unstable particles and now we will present some methods which allow to test these predictions. In particular with regard to the bootstrap proposal this will be important, as it is not yet rigorously supported. Computing renormalization group (RG) flows will allow to detect the unstable particles. Roughly speaking, the central idea of an RG analysis is to probe different energy scales of a theory. We can flow from an energy regime so large that the unstable particle can energetically not exist to one in which it is formed. As a consequence, the particle content of the theory will be altered, which is visible in form of a typical staircase pattern of the RG scaling function. There are various ways to compute such scaling functions, such as the evaluation of the c-theorem [40] or an analysis by means of the thermodynamic Bethe ansatz (TBA) [41]. In the first case we have to evaluate the expression ∞ 3 dr r3 Θ(r)Θ(0) . (4.1) c(r0 ) = 2 r0

The main difficulty in this approach is to evaluate the two-point correlation function Θ(r)Θ(0) for the trace of the energy-momentum tensor Θ depending on the radial distance r. Most effectively, one can do this by expanding it in terms of


75

form factors, for a general recent introduction see, e.g., [42] and references therein. It is well known that for many, even quite non-trivial, theories such form factor expansions converge extremely fast, see [27] for the computation of (4.1) for the SU (3)2 -HSG model. Here we will concentrate more on the TBA, which is simpler to handle in most cases. As a prerequisite, one assumes to know all scattering matrices Sij (θ) for the stable particles of the type i,j with masses mi , mj . Besides this dynamical interaction one also makes an assumption on the statistical interaction between the particles, which are chosen here to be of fermionic type. The TBA consists now of compactifying the space of this 1 + 1-dimensional relativistic model into a circle of finite circumference R, such that all energies become discrete and functions of R. The function similar to (4.1), which scales now these energies takes on the form ceff (r) =

∞ 3r m dθ cosh θ ln(1 + e−εi (θ,r) ) . i π2 i

(4.2)

−∞

One identifies the circumference R with the inverse temperature T and introduces the scaling parameter r = m/T , with m being an overall mass scale. The εi (θ, r) are the so-called the pseudo-energies which can be obtained as solutions of the thermodynamic Bethe ansatz equations [ϕij ∗ ln(1 + e−εj )](θ, r) . (4.3) rmi cosh θ = εi (θ, r) + j

Here the ∗ denotes the convolution of two functions (f ∗ g) (θ) := 1/(2π) dθ f (θ− θ )g(θ ) and the S (for the stable particles only!) enter via their logarithmic derivatives ϕij (θ) = −id ln Sij (θ)/dθ. The main difficulty in this approach is to solve (4.3), which are coupled non-linear integral equations and therefore not solvable in a systematic analytical way. Now it is clear, that the two functions (4.1) and (4.2) cannot be the same, but nevertheless they contain the same qualitative information. The functions will flow through various fixed points, at which the theory become effectively conformal field theories and the normalizations are chosen in such a way that the values of both functions coincide with the corresponding Virasoro central charges. When the theory is not unitary, (4.2) has to be corrected by an additive term to achieve this. Computing then a flow from the infrared to the ultraviolet, one passes now various CFT plateaux, where the changes are associated to the formation of unstable particles with mass (2.15). The challenge is of course to predict the positions, that is, the height and the on-set of the plateaux, as a function of the scaling parameter. The on-set is related to the energy scale of the unstable particles and thus simply determined by the formula (2.15). To predict the height is less trivial and the proposal made in [19] is that the decoupling rule (3.3) achieves this. It is important to note here that σ → ∞ in (3.3), which means in the RG context σ all other resonance parameters. In the following picture we present the numerical

76


computation for the (E6 )2 -HSG model, which precisely confirms our analytical predictions made by the decoupling rule in Section 3.3

Having confirmed the predictions of our decoupling rule with a TBA-analysis, let us now discuss how the results of this analysis are compatible with our bootstrap proposal with a simple example: We consider the processes (2.40), (2.42). In order to be able to interpret the BW-formula for the production of the particle (123) from (12) + 3 or (23) + 1 one has to “make” (12) and (23) stable, which is achieved when σ12 or σ23 is zero. One has to do that as otherwise the BW cannot be applied, it only makes sense for stable particles. The first not obvious result here is that the resulting mass for (123) turns out to be the same from both cases (2.42) (and in all other examples!!!). Looking at the outcome of the TBA calculation (see [19] for the numerics on this case) one finds precisely the value (2.43) reproduced by the TBA at the onset ln(r/2) ∼ −σ13 /2 = −(σ12 + σ23 )/2. Now apparently in the TBA analysis σ12 or σ23 are not zero, which seems to contradict the previous assumptions in the bootstrap analysis. To understand this, one should keep in mind the meaning of the steps in the TBA. The formation of the particle (123) takes place when its mass becomes greater than the energy scale of the RG-flow, i.e., when m exp(σ13 /2) > 2m/r. Let us chose for instance σ12 = 30, σ23 = 60, then exp(σ13 /2) ∼ exp(45) ∼ 3.49 × 1019 . To resolve the apparent contradiction, it is now important to note that the other unstable particles are formed several orders of magnitude below at exp(30) ∼ 1.06 × 1013 and exp(15) ∼ 3.72 × 106. This means in comparison to the formation energy scale of particle (123) the parameters σ12 , σ23 can be regarded as approximately zero, which is in agreement with the assumption in the bootstrap analysis! This is just the same picture as put forward in the decoupling rule: In the formulation we say σ13 → ∞, but inside the TBA analysis this is a milder statement and just means σ13 σ12 , σ23 . Further quite non-obvious confirmation


77

comes from the results when choosing the parameters differently, i.e., in the example discussed here σ12 → −σ12 . The two pictures completely coincide. With regard to previous studies, it is very important to note that the occurrence of the step at exp(σ13 /2) ∼ 2/r had no explanation at all before. Only the onsets at exp(σ23 /2) ∼ 2/r and exp(σ12 /2) ∼ 2/r could be explained as they correspond to the formation of unstable particles from two stable ones. The additional step (for other algebras there are far more) was a mystery pointed out first in [29]. In [19] we provided for the first time an explanation for this feature: We predict its height and on-set, thus explaining also why it is absent when the resonance parameters are chosen differently. For all other examples studied (not even all have been presented in this proceeding, see [19] for more) this picture is completely consistent.

5. Theories with an infinite amount of unstable particles We address now the question of how to enlarge a given finite particle spectrum of a theory to an infinite one. In general the bootstrap (2.10), which is the central construction principle for the S-matrix, is assumed to close after a finite number of steps, which means it involves a finite number of particles. However, from a physical as well as from a mathematical point of view, it appears to be natural to extend the construction in such a way that it would involve an infinite number of particles. The physical motivation for this are string theories, which admit an infinite particle spectrum. Mathematically the infinite bootstrap would be an analogy to infinite-dimensional groups, in the sense that two entries of the Smatrix are combined into a third, which is again a member of the same infinite set. It appears to us that it is impossible to construct an infinite bootstrap system involving asymptotic states, although we do not know a rigorous proof of such a no-go theorem. Instead, we will demonstrate that it is possible to introduce an infinite number of unstable particles into the spectrum. 5.1. q-deformed gamma functions and Jacobian elliptic functions In general, the S-matrix amplitudes consist of (in)finite products of hyperbolic or/and gamma functions. Here we will argue, that to enlarge the spectrum to an infinite number one should replace these functions by q-deformed quantities or elliptic functions. Let us first recall some mathematical facts in this section. We start with some properties of q-deformed quantities, which have turned out to be very useful objects as they allow for instance to carry out elegantly (semi)-classical limits when the deformation parameter is associated to Planck’s constant. Here we define a deformation parameter q and its Jacobian imaginary transformed version, i.e., τ → −1/τ , as q = exp(iπτ ),

qˆ = exp(−iπ/τ ),

τ = iK1− /K .

(5.1)

We introduced here the quarter periods K of the Jacobian elliptic functions depending on the parameter ∈ [0, 1], defined in the usual way through the complete

78


elliptic integrals K = lim

→0,ˆ q→1

K =

π/2 0

lim

→1,q→1

(1 − sin2 θ)−1/2 dθ . Then

K1− = π/2,

lim

→0,ˆ q →1

K1− =

lim

→1,q→1

K → ∞ .

(5.2) It will turn out below that quantities in qˆ will be most relevant for our purposes and therefore we state several identities directly in qˆ, rather than q, even when they hold for generic values. The most basic q-deformed objects one defines are q-deformed integers (numbers), for which we take the convention qˆn − qˆ−n . qˆ − qˆ−1

[n]qˆ :=

(5.3)

They have the obvious properties lim [n]qˆ = n, [n + mτ ]qˆ 1 for m, m = 0 lim = n/n for m = m = 0 →0 [n + m τ ]qˆ

(5.4)

→0

.

(5.5)

Next we define a q-deformed version of Euler’s gamma function ∞ [1 + n]xqˆ [n]qˆ . [x + n]qˆ[n]xqˆ n=1

Γqˆ(x + 1) :=

(5.6)

The crucial property of the function Γqˆ, which coins also its name, is lim Γqˆ(x + 1) = lim Γqˆ(x + 1) =

→0

qˆ→1

∞

n n+x n=1

1+n n

x = Γ(x + 1) .

(5.7)

We can relate deformations in q and qˆ through 2

Γq (−y/τ )Γq (1 + y/τ ) qˆ(x+τ /2−1/2) Γqˆ(y)Γqˆ(1 − y) = . Γq (−x/τ )Γq (1 + x/τ ) qˆ(y+τ /2−1/2)2 Γqˆ(x)Γqˆ(1 − x)

(5.8)

Frequently we have to shift the argument by integer values Γqˆ(x + 1) = qˆx−1 [x]qˆΓqˆ(x) .

(5.9)

Relation (5.9) can be obtained directly from (5.6). As a consequence of this we also have Γqˆ(x + m) =

Γqˆ(x)

m−1

qˆx+l−1 [x + l]qˆ

m∈Z

(5.10)

l=0

Γqˆ(x)

=

Γqˆ(x − m)

m−1 l=0

qˆx−l−2 [x − l − 1]qˆ

m∈Z.

(5.11)


79

Whereas (5.9)–(5.10) hold for generic q, the following identities are only valid for qˆ Γqˆ(1/2 − τ /2)Γqˆ(1/2 + τ /2) = 1/4 Γqˆ(1/2)2 Γqˆ(x) Γqˆ(x + 2τ ) = Γqˆ(y + 2τ ) Γqˆ(y) p p Γqˆ(xi )Γqˆ(xi ± τ /2) Γqˆ2 (xi ) = Γ (y )Γqˆ(yi ± τ /2) Γ 2 (yi ) i=1 qˆ i i=1 qˆ

lim

qˆ→1

1

p

Γqˆ(xi ± τ /2) = 1 if Γqˆ(yi ± τ /2)

i=1

p

(5.12) (5.13) if

p

xi =

i=1

xi =

i=1

p

p i=1

yi

yi (5.14) (5.15)

i=1

x τ x τ (5.16) ∓ )Γqˆ(1 − ± ) = π for x = 0 2K 2 2K 2 Most of these properties can be checked directly by means of the defining relation (5.6). The singularity structure will be important for the physical applications. It follows from (5.6) that the Γqˆ-function has no zeros, but poles lim

qˆ→1 1/4

Γqˆ(

lim

nm =mτ −n θ→θΓ,p

Γqˆ(θ + 1) → ∞

for m ∈ Z, n ∈ N .

(5.17)

Next we define tanh(θ − iπx + σ)/2 , tanh(θ + iπx + σ)/2

∞

sc θ− dn θ+ sc θ+ dn θ− n=−∞ (5.18) with x ∈ Q, σ ∈ R and θ± = (θ ± iπx + σ)iK /π. We employed here the Jacobian elliptic functions for which we use the common notation pq(z) with p, q ∈ {s,c,d,n} (see, e.g., [43] for standard properties). We derive important relations between the q-deformed gamma functions and the Jacobian elliptic sn-function {x}σθ :=

sn(x)

=

x τ 1 Γqˆ( 2K ∓ 2 )Γqˆ(1 − 1 x Γqˆ( 2K )Γqˆ(1 − 4 1

=

{x}σθ, :=

q4

− 2Kix

1−

Γq ( 12 +

1

i 4

Γq (1

{x}σθ−n log q =

x τ 2K ± 2 ) , x 2K )

ix 1 ix 2K1− )Γq ( 2 − 2K1− ) − 2Kix1− )Γq ( 2Kix1− )

(5.19)

.

(5.20)

These relations can be used to obtain some of the above-mentioned expressions. For instance, recalling that sn(K ) = 1, we obtain (5.12). With (5.6) we recover from this the well-known identity sn(iK1− /2) = i/1/4 . The trigonometric limits π = sin(x) (5.21) lim sn(x) = lim sn(x) = →0 qˆ→1 Γ( πx )Γ(1 − πx ) lim sn(x)

→1

=

lim sn(x) =

q→1

1 ix 1 Γ( 12 + ix π )Γ( 2 − π ) = tanh(x). ix i Γ(1 − ix π )Γ( π )

(5.22)

can be read off directly recalling (5.2), (5.7) and presuming that (5.16) holds. We recall the zeros and poles of the Jacobian elliptic sn(θ)-function, which in our

80


conventions are located at zeros:

lm θsn,0 = 2lK + i2mK1−

l, m ∈ Z

(5.23)

poles:

lm θsn,p

l, m ∈ Z .

(5.24)

= 2lK + i(2m + 1)K1−

We have now assembled the main properties of the q-deformed functions which we shall use below. 5.2. Generalizing diagonal S-matrices Here we follow [37] and propose a quite simple principle which introduces an infinite number of unstable particles into the spectrum. We note first, that in general many scattering matrices factorize in the following form min CDD Sab (θ) = Sab (θ)Sab (θ) .

(5.25)

min Here Sab (θ) denotes the so-called minimal S-matrix which satisfies the consistency relations i)–vii) of Section 2. The CDD-factor [18], only satisfies i)–vi) and has its poles in the sheet −π ≤ Im θ ≤ 0, which is the “physical one” for resonance states. We note now that the minimal part is of the general form Sab (θ) = {x}σθ , (5.26) x∈A

with A being a finite set specific to each theory. Then we may define a new Smatrix Sâb (θ) = {x}σθ {x}σθ, (5.27) x∈A

and note that the additional factor in (5.27) is just of CDD-type. Therefore (5.27) constitutes a solution to the consistency relations i)–vii) of Section 2, and thus a strong candidate for a scattering matrix of a proper quantum field theory. Note that whereas (5.26) was a finite product of hyperbolic functions, the new proposal (5.27) contains, according to the identity (5.18) in addition elliptic functions, which lead to the desired spectrum of infinitely many unstable particles according to the principles outlined in Section 2. 5.3. Non-diagonal S-matrices We discuss now the elliptic sine-Gordon model, which may be related to the continuum limit of the eight-vertex model. The (anti)-soliton sector was studied many years ago in [44]. In [45] we demonstrated that it is possible to associate a consistent breather sector to this model. Let us recall the argument by recalling the Zamolodchikov algebra for the soliton sector Z(θ1 )Z(θ2 ) = ¯ 2) = Z(θ1 )Z(θ

¯ 2 )Z(θ ¯ 1) , a(θ12 )Z(θ2 )Z(θ1 ) + d(θ12 )Z(θ ¯ 2 )Z(θ1 ) + c(θ12 )Z(θ2 )Z(θ ¯ 1) . b(θ12 )Z(θ

(5.28) (5.29)

In comparison with the more extensively studied sine-Gordon model the difference is the occurrence of the amplitude d in (5.28), i.e., the possibility that two solitons


81

change into two anti-solitons and vice versa. Invoking the consistency equations i)–v) one finds [44, 45] ∞ ˆ 1+2k λ] 2 Γqˆ2 [−θˆ − 1+2k 2 λ]Γqˆ [1 − θ − 2 a(θ) = Φ(θ) (5.30) 1+2k 1+2k ˆ ˆ 2 2 Γ [ θ − λ]Γ [1 + θ − qˆ qˆ 2 2 λ] k=0

Γqˆ2 [θˆ − (k + 1)λ]Γqˆ2 [1 + θˆ − kλ] × Γqˆ2 [−θˆ − (k + 1)λ]Γqˆ2 [1 − θˆ − kλ] b(θ) c(θ) d(θ) Φ(θ)

sn(iθ/ν) a(θ), sn(iθ/ν + π/ν) sn(π/ν) = a(θ), sn(iθ/ν + π/ν) √ = − sn(iθ/ν) sn(π/ν)a(θ), Γqˆ[1 + τ2 ]Γqˆ[− τ2 ]Γqˆ[1 − θˆ + λ2 + τ2 ]Γqˆ[θˆ − λ2 − τ2 ] = . Γqˆ[1 + θˆ + τ2 ]Γqˆ[−θˆ − τ2 ]Γqˆ[1 + λ2 + τ2 ]Γqˆ[− λ2 − τ2 ] = −

(5.31) (5.32) (5.33) (5.34)

Here we used λ = −π/K ν, θˆ = iθ/2Kν with ν ∈ R being the coupling constant of the model.With regard to property vii), it is clear that it is important to analyze the singularity structure of the amplitudes (5.30)–(5.33) to judge whether there exists a breather sector. For this we appeal to the relations (5.17), (5.23) and (5.24) and find the following pole structure inside the physical sheet = 2mνK1− + i2nνK, θanm 1 ,p = 2mνK1− + i(π − 2lνK ), θblm 1 ,p lm θc1 ,p = 2mνK1− + i2lνK , = (2m + 1)νK1− + i2lνK, θdlm 1 ,p

θanm = (2m + 1)νK1− + i(π − 2nνK ), 2 ,p θblm = (2m + 1)νK1− + i2lνK, 2 ,p lm θc2 ,p = 2mνK1− + i(π − 2lνK), θdlm = (2m + 1)νK1− + i(π − 2nνK ). 2 ,p

We took l, m ∈ Z, n ∈ N and associated always two sets of poles θanm and θanm 1 ,p 2 ,p to a(θ), θbnm and θbnm to b(θ) etc. One readily sees from this that if one restricts 1 ,p 2 ,p the parameter ν ≥ π/2K all poles move out of the physical sheet into the nonphysical one, where they can be interpreted in principle as unstable particles. This was already stated in [44], where the choice ν ≥ π/2K was made in order to avoid the occurrence of non-physical states. This is clear from our discussion of property vii) in Section 2, as we would have poles in the physical sheet beyond the imaginary axis, which when interpreted with the Breit-Wigner formula leave the choice that either mc¯ < 0 or Γc¯ < 0, i.e., we either violate causality or we have Tachyons. The restriction on the parameters makes the model somewhat unattractive as this limitation eliminates the analogue of the entire breather sector which is present in the sine-Gordon model, such that also in the trigonometric limit one only obtains the soliton-antisoliton sector of that model, instead of a theory with a richer particle content. For this reason we relax here the restriction on ν and note that the poles = θcn0 θbn0 1 ,p 2 ,p

for 0 < n < nmax = [π/2νK ], n ∈ N

(5.35)

82


are located on the imaginary axis inside the physical sheet and are therefore candidates for the analogue of the nth-breather bound states in the sine-Gordon model. We indicate here the integer part of x by [x]. In other words, there are at most nmax − 1 breathers for fixed ν and . The price one pays for the occurrence of these new particles in the elliptic sine-Gordon model is that one unavoidably also introduces additional Tachyons into the model as the poles always emerge in “strings”. It remains to be established whether the poles (5.35) may really be associated to a breather type behavior. Let us now see if the poles on the imaginary axis inside the physical sheet can be associated consistently with breathers. We proceed similarly as for the sine-Gordon model [46], even though in the latter approach the following ansatz is inspired by the classical theory and here we do not have a classical counterpart. We define the auxiliary state ! 1 ¯ 1 )Z(θ2 ) . ¯ 2 ) + (−1)n Z(θ (5.36) Zn (θ1 , θ2 ) := √ Z(θ1 )Z(θ 2 This state has properties of the classical sine-Gordon breather being chargeless and having parity (−1)n . Choosing thereafter the rapidities such that the state (5.36) is on-shell, we can speak of a breather bound state lim

(p1 +p2 )2 →m2bn

Zn (θ1 , θ2 ) ≡

lim

bn θ12 →θ+θ12

Zn (θ1 , θ2 ) = Zn (θ) .

(5.37)

bn is the fusing angle related to the poles in the soliton-antisoliton scattering Here θ12 amplitudes. We compute now with the help of (5.28) and (5.29) the exchange relation (5.38) Zn (θ1 )Z(θ2 ) = Sbn s (θ12 )Z(θ2 )Zn (θ1 ) , where

π sn( iθ iθ π 2π 2 ν − 2ν + nK ) Sbn s (θ) = a (5.39) sn + + nK [sn − 1]¯ π ν ν 2ν sn( iθ ν + 2ν + nK )

and Γqˆ2 [1 + θˆ + λ4 − n2 ]Γqˆ2 [−θˆ − λ4 − n2 ]Γqˆ2 [−θˆ + λ4 + n2 ]Γqˆ2 [θˆ + λ4 − n2 ] Γqˆ2 [1 − θˆ + λ4 − n2 ]Γqˆ2 [θˆ − λ4 − n2 ]Γqˆ2 [θˆ + λ4 + n2 ]Γqˆ2 [−θˆ + λ4 − n2 ] n−1 [θˆ − n2 + λ4 − kλ + l]2qˆ2 [−θˆ + n2 − λ4 − kλ − l]2qˆ2 ×Φ13 Φ23 [−θˆ − n2 + λ4 − kλ + l]2qˆ2 [θˆ + n2 − λ4 − kλ − l]2qˆ2 l=1

a ¯=

× ×

∞ ˆ [θ − n2 + λ4 − kλ]qˆ2 [−θˆ + ˆ n λ 2 ˆ k=0 [−θ − 2 + 4 − kλ]qˆ [θ +

n 2 n 2

−

∞

n 2 n 2

−

[θˆ + n2 + λ4 − kλ]qˆ2 [−θˆ − ˆ n λ 2 ˆ k=0 [−θ + 2 + 4 − kλ]qˆ [θ −

− −

λ 4 λ 4

− kλ]qˆ2

λ 4 λ 4

− kλ]qˆ2

− kλ]qˆ2 − kλ]qˆ2

.

We abbreviate Φij = Φ(θij ) with θij being the difference of the on-shell rapidities. What is remarkable here and cannot be anticipated a priori, is that all off-diagonal


83

terms vanish, thus as (5.38) expresses in the soliton breather scattering there is no backscattering. Similarly, but more lengthy, we compute the scattering amplitude between the nth-breather and mth-breather Zn (θ1 )Zm (θ2 ) = Sbn bm (θ12 )Zm (θ2 )Zn (θ1 ) where

"

(5.40)

# iθ π + (n + m)K + Sbn bm (θ) = 1 − sn sn (5.41) ν ν ν

# " iθ sn(iθ/ν − π/ν + (n + m)K ) π + (n + m)K a ˜ × 1 − sn2 sn2 ν ν sn(iθ/ν + π/ν + (n + m)K ) and n −m n ˆ λ ˆ λ 2 Γqˆ2 (1 + m 2 + 2 + θ + 2 ) Γqˆ ( 2 − 2 − θ − 2 ) a ˜ = Φ13 Φ14 Φ23 Φ24 n −m n ˆ λ ˆ λ 2 Γqˆ2 (1 + m 2 + 2 − θ + 2 )Γqˆ ( 2 − 2 + θ − 2 ) ×

∞ n−1 [m + 2 k=1 l=1

× × ×

2π

[m 2 +

∞ n−1 [m + 2 m k=0 l=1 [ 2

+

∞ m−1 [m + 2 m k=1 l=0 [ 2 ∞ m−1 [m 2 m [ 2 k=0 l=0

+ + +

2

n 2 n 2

− l − θˆ − k λ + λ]qˆ2 [ −m 2 − −m ˆ − l + θ − k λ + λ]qˆ2 [ 2 −

n 2 n 2

− l + θˆ − − l − θˆ −

n 2 n 2

+ l + θˆ − k λ]qˆ2 + l − θˆ − k λ]qˆ2

−

n 2 n 2

+ l − θˆ − + l + θˆ −

n 2 n 2

− l − θˆ − k λ + λ]qˆ2 [− m 2 − m ˆ − l + θ − k λ + λ]qˆ2 [− 2 −

n 2 n 2

+ l + θˆ − k λ]qˆ2 + l − θˆ − k λ]qˆ2

n 2 n 2

− l + θˆ − − l − θˆ −

n 2 n 2

+ l − θˆ − + l + θˆ −

λ 2 λ 2

λ 2 λ 2

− k λ]qˆ2 [− m 2 − −

k λ]qˆ2 [− m 2

− k λ]qˆ2 [− m 2 − −

k λ]qˆ2 [− m 2

−

λ 2 λ 2

λ 2 λ 2

− k λ]qˆ2

(5.42)

− k λ]qˆ2

− k λ]qˆ2 − k λ]qˆ2

.

The latter expression (5.42) is tailored to make contact to the expressions in the literature corresponding to the trigonometric limit. Also for this amplitude the backscattering is zero. The matrix Sbn bm (θ) also exhibits several types of poles: a) simple and double poles inside the physical sheet beyond the imaginary axis, b) double poles located on the imaginary axis, c) simple poles in the non-physical sheet and d) one simple pole on the imaginary axis inside the physical sheet at θ = θb = iν(n+m)K which is related to the fusing process of two breathers bn + bm → bn+m . To be really sure that this pole admits such an interpretation, we have to establish according to (2.11) that the imaginary part of the residue is strictly positive, i.e., −i lim (θ − θb )Sbn bm (θ) > 0 . θ→θb

(5.43)

The explicit computation shows that this is indeed the case, see [45]. Furthermore, it is very interesting to check if also (2.10) is satisfied for the fusing process bn + bm → bn+m . For consistency, all amplitudes have to satisfy the bootstrap equations Slbn+m (θ) = Slbn (θ + iνmK )Slbm (θ − iνnK ) ,

(5.44)

84


for l ∈ {bk , s, s¯} ; k, m + n < nmax . Indeed, we verify with some algebra that (5.44) holds for the above amplitudes (5.39) and (5.41). Finally, we carry out various limits. Our formulation in terms of q-deformed quantities and elliptic functions is useful to make this task fairly easy. We state our results here only schematically and refer the reader for details to [45]. We find elliptic sine-Gordon

1/ν→2nK /π+2imK1− /π

− − − − − − − −→

| | →0 | ↓ sine-Gordon

m = 0, → 0 ↓ free theory ↑ 1/ν → i∞

− − − − − − − −→ 1/ν→n

(1)

elliptic Dn+1 -ATFT | | m = 0, → 0 | ↓ (1)

minimal Dn+1 -ATFT

Thus we can view the elliptic sine-Gordon model as a master theory for several other models. In the limit → 0 we recover now all sectors, including the breathers, of the sine-Gordon model. The diagonal limit 1/ν → 2nK /π + 2imK1− /π is interesting as it yields a new type of theory, which we refer to as elliptic SO(2n + (1) 2) ≡ Dn+1 -affine Toda field theory (ATFT). To coin this name for these theories seems natural as in the trigonometric limit we obtain from it the ordinary minimal (1) Dn+1 -ATFT.

6. Conclusions We reviewed the general analytical scattering theory related to integrable quantum field theories in 1+1 space-time dimensions. We made a proposal for a construction principle of an S-matrix like object which describes the scattering between two unstable particles or an unstable particle and a stable one. We tested this proposal with various examples and found a remarkable agreement with the outcome of the thermodynamic Bethe ansatz in what concerns the particle content and the RG flow of the theories. We described the general Lie algebraic structure of theories with unstable particles and propose a decoupling rule which predicts the RG flow when some of the parameters in the theory become very large. Alternatively, we tested these analytical prediction with the TBA. Finally, we discussed how one can construct theories with and without backscattering which contain an infinite number of unstable particles. Acknowledgment We are grateful to the Deutsche Forschungsgemeinschaft (Sfb288), for financial support. This work is supported by the EU network EUCLID, Integrable models and applications: from strings to condensed matter, HPRN-CT-2002-00325.


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Quantum Reduction in the Twisted Case Victor G. Kac and Minoru Wakimoto Mathematics Subject Classification (2000). 17B55, 17B67, 17B68, 17B69, 81R10. Keywords. Twisted vertex algebra, quantum Hamiltonian reduction, superconformal algebra, Euler-Poincaré character, free-field realization, determinant formula.

0. Introduction This paper is a continuation of the papers [KRW] and [KW] on structure and representation theory of vertex algebras Wk (g, x) obtained by quantum Hamiltonian reduction from the affine superalgebra g. The datum one begins with is a quadruple (g, x, f, k), where g is a simple finite-dimensional Lie superalgebra with a non-zero invariant even supersymmetric bilinear form ( . | . ), x is an element of g such that ad x is diagonalizable with eigenvalues in 12 Z, f is an even element of g such that [x, f ] = −f and the eigenvalues of ad x on the centralizer gf of f in g are non-positive, and k ∈ C. Recall that a pair {x, f } satisfying the above properties can be obtained from an s2 -triple {e, x, f }, so that [x, e] = e, [x, f ] = −f , [e, f ] = x. We associate to the quadruple (g, x, f, k) a homology complex C(g, x, k) = (Vk (g) ⊗ Fch ⊗ Fne , d0 ), where Vk (g) is the universal affine vertex algebra of level k associated to the affine superalgebra g, Fch is the vertex algebra of free charged superfermions based on g+ + g∗+ with reversed parity, Fne is the vertex algebra of free neutral superfermions based on g1/2 , and d0 is an explicitly constructed odd derivation of the vertex algebra C(g, x, k) whose square is 0. Here g+ (resp. g1/2 ) denotes the sum of eigenspaces of ad x with positive eigenvalues (resp. with eigenvalue 1/2), and we drop f from the notation since its different choices are conjugate. The vertex algebra Wk (g, x) is the homology of the complex (C(g, x, k), d0 ). The first author was supported in part by NSF grant DMS-0201017. The second author was supported by Grant-in-aid 13440012 for scientific research Japan.

90

V.G. Kac and M. Wakimoto

In the present paper we begin with a diagonalizable automorphism σ of g with modulus 1 eigenvalues, which leaves invariant the bilinear form (. | . ) and keeps the elements x and f fixed. The automorphism σ gives rise to the twisted affine superalgebra gtw and the corresponding twisted vertex algebra Vk (g, σ), and tw tw to the twisted vertex algebras Fch and Fne , and we consider the twisted complex tw tw (C(g, σ, x, k) = Vk (g, σ) ⊗ Fch ⊗ Fne , dtw 0 ).

Its homology is the twisted vertex algebra Wk (g, σ, x), which is the main object of our study. In the case when σ = 1 we recover the “Neveu-Schwarz sector” Wk (g, x) studied in our previous papers [KRW] and [KW] (and earlier in [FF, FKW, BT, FB, ST], and many other works, see [BS].) In the case when σ = σR , where σR |g¯0 = 1 and σR |g¯1 = −1, we obtain the “Ramond sector”. This terminology comes from the fact that taking the smallest simple Lie superalgebra g = osp(1|2) and the only possible choice of x, we obtain as Wk (g, x) the vertex algebra associated to the usual Neveu-Schwarz algebra, and as Wk (g, σR , x) the twisted vertex algebra associated to the usual Ramond algebra. Likewise, taking g = s(2|1), s(2|2))/CI, osp(3|2) or D(2, 1; a), and x the suitable multiple of the highest root θ of one of the simple components of g¯0 , all possible choices of σ produce all possible twists of the N = 2, N = 4, N = 3 and big N = 4 superconformal algebras. This leads us to a unified representation theory of all twisted superconformal algebras, in particular to unified free field realizations and determinant formulas. As in [FKW] and [KRW], we construct also a functor M → H(M ) from the category of restricted gtw -modules to the category of Z-graded Wk (g, σ, x)-modules and compute the Euler-Poincaré character of H(M ) in terms of the character of M . In a forthcoming paper [KW4] we shall develop a theory of characters of Wk (g, σ, x) using this functor.

1. An overview of twisted formal distributions Let R be a Lie conformal superalgebra. Recall that this is a Z/2Z-graded C[∂]module, endowed with jth products denoted by a(j) b, j ∈ Z+ , satisfying certain axioms [K4]. One associates to R a Lie superalgebra Lie (R) = R [t, t−1 ]/ Image (∂ ⊗ 1 + 1 ⊗ ∂t ) ,

(1.1) −1

endowed with the following bracket, where a(µ) stands for a ⊗ t ∈ R [t, t ] = R ⊗ C [t, t−1 ]: µ [a(µ) , b(ν) ] = (1.2) (a(j) b)(µ+ν−j) . j µ

j∈Z+

Introducing formal distributions a(z) = a(µ) z −µ−1 , µ∈Z

a ∈ R,

(1.3)

Quantum Reduction in the Twisted Case one rewrites (1.2) as [a(z) , b(w)] =

j (a(j) b)(w)∂w δ(z − w)/j! .

91

(1.4)

j∈Z+

One also has: (∂a)(z) = ∂z a(z) .

(1.5)

(The fact that Lie R is a Lie superalgebra and the distributions {a(z)}a∈R form a local system is encoded in the axioms of R.) Let now σ be a diagonalizable automorphism of R. We shall always assume for simplicity that all eigenvalues of σ have modulus 1. We have: R= Rµ¯ , where Rµ¯ = {a ∈ R|σ(a) = e2πi¯µ a} . (1.6) µ ¯ ∈R/Z

Here and further µ ¯ denotes the coset µ + Z of µ ∈ R. We associate to the pair (R, σ) the σ-twisted Lie superalgebra Lie (R, σ) = (Rµ¯ ⊗ tµ )/ Image (∂ ⊗ 1 + 1 ⊗ ∂t ) , (1.7) µ∈R

endowed with bracket (1.2) (except that now µ and ν are not necessarily integers). Denoting by a(µ) the image of a ⊗ tµ in Lie (R, σ), and introducing the twisted formal distributions atw (z) = a(µ) z −µ−1 , a ∈ Rµ¯ , (1.8) µ∈¯ µ

we get the twisted analogue of (1.4): j (a(j) b)tw (w)∂w δµ¯ (z − w)/j! , [atw (z) , btw (w)] =

(1.9)

j∈Z+

where δµ¯ (z − w) = z −1

w µ µ∈¯ µ

z

is the twisted formal δ-function. Assuming that the C[∂]-module R is generated by a finite set Q, introduce a descending filtration of the Lie superalgebra L = Lie (R, σ) by subspaces Fj L = com {a(µ) |a ∈ Q, µ j}, and define a completion of its universal enveloping U (L) algebra U (L), which consists of all series i ui such that for each N ∈ R all but finitely many of the ui ’s lie in U (L)(FN L). The automorphism σ of R induces one of L and of U (L)com in the obvious way, which we again denote by σ. A U (L)com -valued twisted formal distribution is an expression of the form a(z) = a(µ) z −µ−1 , µ∈¯ µ

where µ ¯ = µ + Z, σ(a(µ) ) = e a(µ) and a(µ) ∈ U (L)com satisfy the property that for each N ∈ R, a(µ) ∈ U (L)(FN L) for µ 0 and all the a(µ) have the same 2πi¯ µ

92


parity, denoted by p(a) ∈ Z/2Z. It is clear that the derivative ∂z a(z) of a twisted formal distribution a(z) is also a twisted formal distribution. In order to define a normally ordered product of twisted formal distributions a(z) and b(z), we need to define a splitting a(z) = a(z)+ + a(z)− into creation and annihilation parts a(z)+ and a(z)− . For that we choose sa in the coset µ ¯ and let a(z)+ =

a(µ) z −µ−1 , a(z)− =

µ<sa

a(µ) z −µ−1 .

(1.10)

µsa

If a(z) is a non-twisted formal distribution, i.e., µ ¯ = Z, then one may choose sa = 0, so that ∂z (a(z)± ) = (∂z a(z))± , but for twisted formal distributions such a choice is impossible. After making a choice of sa , one defines the normally ordered product of twisted formal distributions in the usual way: : a(z)b(z) := a(z)+ b(z) + (−1)p(a)p(b) b(z)a(z)− . It is easy to see that this is again a U (L)com -valued formal distribution. As usual, one defines the normally ordered product of more than two formal distributions from right to left, e.g. : abc :=: a : bc ::. Denote by V (R) the subspace of U (Lie R)com consisting of all normally ordered products of formal distributions (1.3) and 1. This is one of the constructions of the universal enveloping vertex algebra of the Lie conformal algebra R [KRW] (cf. [K4], [GMS], [BK]). The infinitesimal translation operator ∂ of V (R) is defined by (1.5). The jth product a(j) b on V (R) is defined by (1.4) for j ∈ Z+ , and by a(−j−1) b =: (∂ j a)b : /j! for j ∈ Z+ . The automorphism σ of R induces an automorphism of V (R), and we have its eigenspace decomposition: V (R) =

V µ¯ (R) .

µ∈R/Z ¯

Likewise, denote by V (R, σ) the subspace of U (Lie (R, σ))com [[z, z −1 ]] consisting of all normally ordered products of twisted formal distributions (1.8) and 1. This is called a σ-twist of the vertex algebra V (R). (It is independent of the choices of sa used in the definition of normally ordered products.) The subspace V (R, σ) is σ-invariant, so that we have the decomposition into its eigenspaces: V (R, σ) =

V µ¯ (R, σ) .

µ∈R/Z ¯

The following result is well known. Proposition 1.1. The map a(z) → atw (z) (a ∈ Rµ¯ , µ ¯ ∈ R/Z) extends uniquely to a σ-eigenspace preserving vector space isomorphism V (R) → V (R, σ), a(z) →

Quantum Reduction in the Twisted Case

93

atw (z), satisfying the following properties (a ∈ V µ¯ (R), b ∈ V (R)): 1tw

=

tw

(∂a) (z) = tw

tw

[a (z), b (w)]

=

1,

(1.11) tw

∂z a (z) , j (a(j) b)tw (w)∂w δµ¯ (z − w)/j! ,

(1.12) (1.13)

j∈Z+ tw

tw

: a (z)b (z) : =

sa (a(j−1) b)tw (z)z −j . j

(1.14)

j∈Z+

A module M over the filtered Lie superalgebra L = Lie (R, σ) is called restricted if any vector of M is annihilated by some Fj L. Such an L-module can be uniquely extended to a module over the associative algebra U (L)com . Restricting this module to V (R, σ), we obtain, in view of Proposition 1.1, what is called a σ-twisted module M over the vertex algebra V (R). In the examples of Lie conformal superalgebras R that follow we use the λ j bracket [aλ b] = j∈Z+ λj! a(j) b. Due to sesquilinearity ([∂aλ b] = −λ[aλ b], [aλ ∂b] = (∂ + λ)[aλ b]), the λ-brackets of generators of the C[∂]-module R determine the λ-bracket on R. Recall also that an element K of R is called central if [Kλ R] = 0 = [Rλ K]. Example 1.1. (twisted currents and Sugawara construction). Let g be a simple finite-dimensional Lie superalgebra with a non-degenerate supersymmetric invariant bilinear form ( . | . ). The associated Lie conformal superalgebra is Curg = (C[∂] ⊗ g) ⊕ CK , where K is a central element and [aλ b] = [a, b] + λ(a|b)K ,

a, b ∈ 1 ⊗ g ≡ g .

Given a complex number k, denote by Vk (g) the quotient of the universal enveloping vertex algebra V (Curg) by the ideal generated by K − k. This is called the universal affine vertex algebra of level k. Let σ be a diagonalizable automorphism of the Lie superalgebra g, keeping the bilinear form ( . | . ) invariant. It extends to an automorphism of Curg, also denoted by σ, by letting σ(P (∂) ⊗ a) = P (∂) ⊗ σ(a), σ(K) = K. Let g = ⊕µ¯∈R/Z gµ¯ , where gµ¯ = {a ∈ g| σ(a) = e2πi¯µ a}, be the eigenspace decomposition of g for σ. Then the corresponding σ-twisted Lie superalgebra Lie (Curg, σ) is a twisted Kac-Moody affinization (gµ¯ ⊗ tµ ) ⊕ CK , gtw = µ∈R

with the bracket (a ∈ g , b ∈ g ): µ ¯

ν ¯

[atµ , btν ] = [a, b]tµ+ν + µ(a|b)δµ,−ν K ,

[K, gtw ] = 0 .

94


The formal distributions atw (z) =

(atµ )z −µ−1 ,

a ∈ gµ¯ ,

µ∈¯ µ

are called twisted currents. They generate (by taking derivatives and normally ordered products) the σ-twist V (Curg, σ) of the vertex algebra V (Curg). As in the non-twisted case, denote by Vk (g, σ) the quotient by the ideal generated by K − k; this is the σ-twist of the vertex algebra Vk (g). Choosing dual bases {ai } and {ai } of g, compatible with the eigenspace decomposition for σ, so that (ai |aj ) = δij , define the twisted Sugawara field in Vk (g, σ) (assuming that k + h∨ = 0): 1 (−1)p(ai ) : ai ai :tw (z) . Lg,tw (z) = 2(k + h∨ ) i Writing Lg,tw (z) = n∈Z Lg,tw z −n−2 , and using the non-twisted Sugawara conn struction and formula (1.13), we obtain that the Lg,tw satisfy the relations of the n Virasoro algebra with central charge c(k) = k sdim g/(k + h∨ ). Using formula (1.14), we can rewrite Lg,tw (z) in terms of twisted currents and numbers si = sai (see (1.10)): 1 g,tw i,tw (z) = (−1)p(ai ) : atw (z) : (1.15) L i (z)a ∨ 2(k + h ) i

s i − (−1)p(ai ) si [ai , ai ]tw (z)z −1 − k (−1)p(ai ) z −2 . 2 i i Example 1.2. (twisted neutral free superfermions). Let A be a finite-dimensional vector superspace with a non-degenerate skew-supersymmetric bilinear form . , . . The associated Clifford Lie conformal superalgebra is C(A) = (C[∂] ⊗ A) ⊕ CKA , where KA is a central element and [aλ b] = a, bKA . Denote by F (A) the quotient of the universal enveloping vertex algebra of C(A) by the ideal generated by KA − 1. Let σ be a diagonalizable automorphism of the space A, keeping the bilinear form . , . invariant. As above, it extends to an automorphism σ of the Lie conformal superalgebra C(A). Let A = ⊕µ¯∈R/Z Aµ¯ be the eigenspace decomposition for σ. Then the corresponding σ-twisted Lie superalgebra Lie (C(A), σ) is a twisted Clifford affinization tw = ⊕µ∈R (Aµ¯ ⊗ tµ ) ⊕ CKA A with the bracket tw ] = 0 . [atµ , btν ] = a, bδµ,−ν−1 KA , [KA , A


95

tw )com /(KA −1). The formal distributions We shall work in the Clifford algebra U (A Φtw (z) = (Φtµ )z −µ−1 , Φ ∈ Aµ¯ , µ∈¯ µ

are called twisted neutral free superfermions. They generate the σ-twist F (A, σ) of the vertex algebra F (A). Choosing dual bases {Φi } and {Φi } of A, compatible with the eigenspace decomposition for σ, we let Lne ,tw (z) =

1 (−1)p(Φi ) : Φi ∂Φi :tw (z) . 2 i

,tw −n−2 Writing Lne ,tw (z) = n∈Z Lne z , we obtain a Virasoro algebra with central n 1 charge c = − 2 sdim A. As in the previous example, using formula (1.14), we obtain: Lne ,tw (z) =

1 i,tw (−1)p(Φi ) : Φtw (z) : i (z)∂Φ 2 i

1 p(Φi ) si (−1) − z −2 . 2 i 2

(1.16)

Example 1.3. (twisted charged free superfermions). In notation of Example 1.2, assume that A = A+ ⊕ A− , where both A+ and A− are isotropic and σ-invariant subspaces. Choose a basis ϕi of A+ , compatible with the eigenspace decomposition of A+ for σ, and its dual basis ϕ∗i of A− , so that ϕi , ϕ∗j = δij , and define charge by charge(ϕi ) = 1 ;

charge(ϕ∗i ) = −1 .

(1.17)

∗tw The formal distributions ϕtw (z) are called twisted charged free i (z) and ϕi superfermions. Relation (1.17) gives rise to the charge decomposition:

F (A , σ) = ⊕m∈Z Fm (A , σ) .

(1.18)

For a collection of complex numbers (mj ) ∈ Cdim A+ we can define a Virasoro formal distribution mi : ϕ∗i ∂ϕi :tw (z) + (1 − mi ) : ∂ϕ∗i ϕi :tw (z) Lch,tw (z) = − i

with central charge

i

p(ϕi ) (12m2i −12mi +2). i (−1)

Lch,tw (z) = −

i

Using formula (1.14), we obtain

mi : ϕ∗tw (z)∂ϕtw i i (z) :

∗tw tw p(ϕi ) si + (1 − mi ) : ∂ϕi (z)ϕi (z) : + (−1) z −2 . 2 i i

(1.19)

96


2. The twisted complex Let g be a simple finite-dimensional Lie superalgebra with a non-degenerate even supersymmetric invariant bilinear form ( . | . ). Fix an even element x of g such that ad x is diagonalizable with half-integer eigenvalues, and let g = ⊕j∈ 12 Z gj

(2.1)

be the eigenspace decomposition. Let g+ = ⊕j>0 gj , g− = ⊕j0

(nj (σ)− ⊗ t−j +

µ∈R j+µ −mi } if ui ∈ n(σ)+ , (3.2) si = −mi if ui ∈ n(σ)+ . µ i It is easy to see that for a dual basis element ui ∈ g−¯ −mi we have for s = sui :

si = 1 − si for all i ∈ S .

(3.3)

tw and A tw as follows: We extend this definition of annihilation operators to A ne ch sΦi = si (i ∈ S1/2 ) , sϕi = si , sϕ∗i = 1 − si (i ∈ S+ ) .

(3.4)

It is easy to see that we have sΦi = ∓1/2 if Φi ∈ n1/2 (σ)± , |sΦi | < 1/2 otherwise. sΦi + sΦi = δi,i0 , where Φi0 , Φi0 ne = 0. We write the generating fields in the form: ui,n z −n−1 , Φi (z) = Φi,n z −n−1/2 , ui (z) = n∈¯ si

ϕi (z) =

(3.5) (3.6)

n∈¯ si +1/2

ϕi,n z −n−1 , ϕ∗i (z) =

n∈¯ si

ϕ∗i,n z −n .

n∈−¯ si

tw and A tw has a unique irreducible modEach of the Clifford affinizations A ne ch tw tw ule, denoted by Fne and Fch , respectively, admitting a non-zero vector |0ne and |0ch , respectively, killed by all annihilation operators: Φi,n |0ne = 0 for n si + 1/2 ,

(3.7)

ϕ∗i,n |0

= 0 for n 1 − si . (3.8) ϕi,n |0ch = 0 for n si , Since these modules are restricted, they extend to the modules over F (Ane , σ) and F (Ach ,σ) (= twisted modules over the vertex algebras F (Ane ,1) and F (Ach ,1)), respectively), hence tw tw F tw = Fne ⊗ Fch is a module over F (g, σ, x) (= twisted module over the tensor product of these vector algebras, F (g, 1, x)). We let |0 = |0ne ⊗ |0ch ∈ F tw . Thus, given a restricted gtw -module M with K = kI, we extend it to a module over Vk (g, σ) (= twisted module over the vertex algebra Vk (g, 1)), then M ⊗ F tw becomes a module over C(g, σ, x, k) (= twisted module over the vertex algebra C(g, x, k)). Passing to the homology of the complex C tw (M ) = (M ⊗ F tw , dtw 0 ), we obtain a Wk (g, σ, x)-module (= twisted Wk (g, x)-module) H tw (M ). One has

102


the charge decomposition of C tw (M ) induced by that of F (g, σ, x) by setting the charge of M to be zero. This induces a decomposition as Wk (g, σ, x)-modules: H tw (M ) = j∈Z Hjtw (M ). Let ∆σ ⊂ hσ∗ be the set of non-zero roots of g with respect to hσ , counted with their multiplicities. We may identify ∆σ with a subset of S , which indexes root vectors attached to non-zero roots. (Then the remaining elements of S index a basis of h mod hσ .) Given one of the above basis root vectors eα , attached to α ∈ ∆σ , we let sα = seα . One should keep in mind that the sα corresponding to root vectors with the same α may be different (in the case hσ = h). ⊂ gtw Recall that the set of roots ∆ h∗ of the twisted affine Lie superalgebra re im is ∆ = ∆ ∪ ∆ , where: re = {α + (m + sα )δ| m ∈ Z , α ∈ ∆σ } , ∆ im = {mδ| m ∈ E0 \{0}} , ∆ where E0 = {µ ∈ R| e2πiµ is an eigenvalue of σ on h}, and the roots are considered + = ∆ re ∪ ∆ im of positive roots with their multiplicities. Then we have a subset ∆ + + corresponding to in ∆, n+ (see (2.8)), where σ re im ∆ + = {α + (m + sα )δ| m ∈ Z+ , α ∈ ∆ } , ∆+ = {mδ| m ∈ E0 , m > 0} .

re Introduce the following subset of ∆ +: re = {α + (m + sα )δ| α ∈ ∆σ , α(x) 0 , m ∈ Z+ } . ∆ ++ Proposition 3.1. (a) If M is a restricted gtw -module and v ∈ M is a singular vector, i.e., ntw + v = 0, then dtw 0 (v ⊗ |0) = 0 . (b) If M is a Verma module over gtw with the highest weight vector vΛ and v ∈ M − nα, where α ∈ ∆ re , then the is a singular vector with highest weight Λ ++ homology class of v ⊗ |0 in H0tw (M ) is non-zero. Proof. We have: dtw 0 = A + B + C + D, where (−1)p(ui ) ui,p ϕ∗i,q , A = µi i∈S+ p∈¯ q∈−¯ µi p+q=0

B

C

=

=

−

1 2

i,j,k∈S+

p∈¯ µk q∈−¯ µi r∈−¯ µj p+q+r=0

i∈S+

(−1)p(ui )p(uk ) ckij ϕk,p ϕ∗i,q ϕ∗j,r ,

(f |ui )ϕ∗i,1 , D =

i∈S1/2

p∈−¯ µi q∈¯ µi +1/2 p+q=0

ϕ∗i,p Φi,q+1/2 .


103

A summand of A does not annihilate v ⊗ |0 only if p si − 1, q si , hence there are no such summands since p + q = 0. A summand of B does not annihilate v ⊗ |0 only if p sk − 1, q −si , r sj , which happens only if p + q + r sk − si − sj − 1 −1, since sk si + sj when ckij = 0. Hence there are no such summands. If (f |ui ) = 0, then (f t|ui t−1 ) = 0, and since f t ∈ n+ , we obtain that ui t−1 ∈ ∗ n− and therefore si 0, by definition of si . Hence ϕi,1 is an annihilation operator (see (3.8)) and C(v ⊗ |0) = 0. Finally, if a summand of D does not annihilate the v ⊗ |0, then p si and q + 1/2 si − 1/2 and therefore p + q = −1, which is impossible since p + q = 0. This proves (a). The proof of (b) is the same as in the non-twisted case, see [KW], Lemma 7.3. Next, we study the formal distribution Ltw (z) of Wk (g, σ, x). Using formulas (1.15), (1.16) and (1.19) first three summands, we obtain an explicit ex for the −n−2 z . Note that the Ltw pression for Ltw (z) = n∈Z Ltw n n form a Virasoro algebra with the same central charge as in the non-twisted case. Examples 1.1, 1.2 and 1.3 give the following important formulas. Proposition 3.2. Introducing the constants

k p(α) sα (−1) sg = − , 2(k + h∨ ) 2

(3.9)

α∈S

=

sne

sα 1 1 (σ) − (−1)p(α) , 8 2 2

(3.10)

α∈S1/2

sch

=

p(α)

(−1)

α∈S+

we have Lg,tw 0 ,tw Lne |0ne 0

Ltw 0

=

1 2(k + h∨ )

sα + m α sα , 2

hi h −

i

= (sne + ann )|0ne ,

i

(3.11)

p(α)

(−1)

i∈S ch,tw L0 |0ch

sα α

+ sg + ann ;

= (sch + ann )|0ch ;

,tw = Lg,tw + Lne + Lch,tw − x, 0 0 0

where ann (resp. ann |0) denotes the sum of terms which annihilate any singular vector in a gtw -module M of level k (resp. annihilate the vacuum vector), and {hi } and {hi } are dual bases of hσ . Proof. We have: α∈S (−1)p(α) sα [uα , uα ] = i ai hi , where ai ∈ C. Hence ai = (−1)p(α) sα ([uα , uα ]| hi ) = (−1)p(α) sα α(hi ). α∈S

α∈S

104 Hence


(−1)p(α) sα [uα , uα ] =

α∈S

(−1)p(α) sα α.

α∈S

The rest of the calculation is straightforward.

Corollary 3.1. Let v be a singular vector of a gtw -module M of level k such that σ σ∗ av = Λ(a)v, a ∈ h , for some Λ ∈ h . Then Ltw 0 (v ⊗ |0) = h v ⊗ |0, where 1 ((Λ|Λ) − (−1)p(α) sα (Λ|α)) − Λ(x) + sg + sne + sch . h= ∨ 2(k + h ) α∈S

1

Corollary 3.2. Let γ = 2 α∈S (−1)p(α) α ∈ hσ∗ , and let ρtw be the Weyl ρtw |αi ) = 12 (αi |αi ) for all simple roots αi of gtw ). vector for gtw (i.e., ( Then ρtw |hσ = −γ . Proof. By Proposition 3.2 we have in any highest weight gtw -module of level k with highest weight Λ ∈ hσ∗ : vΛ = Lg,tw 0

1 (((Λ|Λ) − 2(Λ|γ )) + c1 ) vΛ , 2(k + h∨ )

(3.12)

where c1 ∈ C is independent of Λ. On the other hand, the operator Lg,tw +D 0 commutes with gtw , hence equals c2 Ωtw + c3 , where c2 , c3 ∈ C are independent of Λ and Ωtw is the Casimir operator of gtw . But Ωtw vΛ = (Λ|Λ + 2 ρtw )vΛ (see [K3]) and DvΛ = 0, hence, comparing with (3.12) we obtain for any Λ ∈ hσ∗ : 1 ((Λ|Λ) − 2(Λ|γ )) + c1 = c2 (Λ|Λ) + 2( ρtw |Λ) + c3 . ∨ 2(k + h ) Comparing quadratic terms in Λ we obtain c2 = (2k + 2h∨ )−1 . Comparing linear terms in Λ, we get ρtw |hσ = −γ . Recall that the conformal weight 1 formal distributions of the vertex algebra Wk (g, x) are [KW]: 1 (−1)p(ui ) cij (v) : Φi (z)Φj (z) : (v ∈ gf0 ). J {v} (z) = J (v) (z) − 2 i,j∈S1/2

Hence, by Equation (1.14), the corresponding twisted formal distributions of Wk (g, σ, x) can be explicitly expressed via twisted currents and twisted ghosts. In the sequel we shall need the following formula, in the case when a ∈ hσf : 1 {a}tw =a− (−1)p(ui ) si cii (a) + (−1)p(ui ) si cii (a) + ann , (3.13) J0 2 i∈S+

i∈S1/2

where ann denotes an operator which annihilates any vector of the form v ⊗ |0 ∈ M ⊗ F tw . Formula (3.13) implies the following corollary.


105 {H},tw

Corollary 3.3. Under the conditions of Corollary 3.1, the eigenvalue of J0 (H ∈ hσf ) on the vector v ⊗ |0 is equal to 1 Λ(H) + (−1)p(α) sα α(H) − (−1)p(α) sα α(H) . 2 α∈S1/2

α∈S+

As in [KRW], define the Euler-Poincaré character of H tw (M ) by the following formula, where h ∈ hσf and τ ∈ C, Im τ > 0: {h} tw (−1)j tr Htw e2πiτ L0 e2πiJ0 . chH tw (M) (τ, h) = j (M) j∈Z

The same argument as in [KRW] gives an explicit formula in terms of the character g,tw

chM (τ, z) = tr M e2πi(z+τ L0

)

, z ∈ hσ ,

of the gtw -module M : chH tw (M) (τ, h) = e2πiτ (sne +sch )

−α p(α) ˜ mult α × chM (τ, −τ x + h). (1 − p˜(α)e )

(3.14)

+ α∈∆ α(x)=0,−1/2

Here and further, in order to simplify notation, we let p˜(α) = (−1)p(α) , and for α∈ g∗ , we define α(τ, z) = 2πiα(z − τ D). The conditions of non-vanishing of chH tw (M) are similar to those in the nontwisted case [KRW]. Namely, the same argument as in [KRW], Theorem 3.2, gives the following result. Proposition 3.3. Let M be a restricted gtw -module of level k = −h∨ and assume that chM (τ, h) extends to a meromorphic function on the upper half space Im τ > 0, h ∈ hσ , with at most simple poles at the hyperplanes α = 0, where α are real even roots. Then chH tw (M) (τ, h) is not identically zero iff the gtw -module M is not locally nilpotent with respect to all root spaces g−α , such that α are positive even real roots satisfying the following three properties: (i) α(D + x) = 0 ,

(ii) α|(hσ )f = 0 ,

(iii) α(x) = 0 , −1/2 .

We shall use formula (3.14) and [KW2, KW3] to compute the characters of Wk (g, σ, x)-modules in a subsequent paper [KW4] (cf. [FKW, KRW]). Remark 3.1. A slightly more explicit form of (3.14) is as follows: chH tw (M) (τ, h) = e2πiτ (sne +sch ) ∞ p(α) ˜ mult α (1 − p˜(α)e−(n−sα )δ+α )(1 − p˜(α)e−(n−1+sα )δ−α ) × chM α∈S n=1 α(x)>0

×

∞

˜ mult α (1 − p˜(α)e−(n−sα )δ+α )−p(α) (τ, −τ x + h) .

α∈S n=1 α(x)=1/2

106


¯ f Let a ∈ (gµ−j ) , j 0, and let J {a},tw (z) be the corresponding formal distribution of Wk (g, σ, x) (see Theorem 2.1). As in the non-twisted case [KW], its conformal weight with respect to Ltw (z) equals ∆a = j + 1. We therefore write Jn{a},tw z −n−∆a . (3.15) J {a},tw (z) = n∈¯ µ−∆a

Recall the isomorphism as gf ∼ = g0 + g1/2 given by [KW], (1.12). We shall identify g0 (and its subspace h ) with a σ-invariant subspace of gf , using this isomorphism. A Wk (g, σ, x)-module M is called a highest weight module with highest weight λ ∈ (hσ )∗ if there exists a non-zero vector vλ ∈ M such that: gf0 -modules σ

polynomials in the operators Jn{a},tw applied to vλ span M , {a},tw J0 vλ

= λ(a)vλ if a ∈ h , σ

{a},tw vλ = 0 if m > 0 or m = 0 and a ∈ n0 (σ)+ . Jm

(3.16) (3.17) (3.18)

The Verma module is defined in the same way as in [KW], and we have the following twisted analogue of Theorem 6.3 from [KW]. Theorem 3.1. If P is a Verma module over the Lie superalgebra gtw, then H tw (P ) = tw H0 (P ), and it is a Verma module over Wk (g, σ, x).

4. Modules over Wk (g, σ, θ/2), the free field realizations and determinant formulas Of particular interest are the vertex algebras Wk (g, θ/2) associated to a minimal gradation of g [KRW, KW] (cf. [FL]). In this case g is one of the simple Lie superalgebras s(m|n)/δm,n CI, osp(m|n) (= spo(n|m)), D(2, 1 ; a), F (4), G(3) or one of the five exceptional Lie algebras, θ is the highest root of one of the simple components of the even part of g, the bilinear form ( . | . ) is normalized by the condition (θ|θ) = 2, and x = θ/2. The corresponding 12 Z-gradation (2.1) looks as follows: g = Cf + g−1/2 + g0 + g1/2 + Ce , (4.1) where {e, x, f } form an s2 triple , and gf0 = {a ∈ g0 |(x|a) = 0} , Then

gf = Cf + g−1/2 + gf0 .

−σ σ g(σ) = Cf + g−σ −1/2 + g0 + g1/2 + Ce

is a minimal gradation of g(σ). Since gσ0 = (gσ0 )f + Cx, it follows that there exists an element h0 ∈ hσf of the Lie superalgebra g(σ) such that the eigenvalues of ad h0 are real, h0 is a regular element of gσ0 , and the 0th eigenspace of ad h0 on −σ g−σ 1/2 (resp. g−1/2 ) is Ceθ/2 (resp. Ce−θ/2 ) if eθ/2 is a root vector of g(σ). (Here θ/2 stands for the restriction of θ/2 to hσ .) Letting n+ (σ) (resp. n− (σ)) be the span of all eigenvectors of ad h0 with positive (resp. negative) eigenvalues and the vectors


107

f = e−θ and e−θ/2 (resp. e = eθ and e = eθ/2 ), we obtain the decomposition (2.4), satisfying the properties (i)–(iv). Note also that in the decomposition (2.5), h1/2 (σ) (resp. n1/2 (σ) ) is the span of all eigenvectors of ad h0 with positive (resp. negative) eigenvalues, and g1/2 (σ)0 = Ceθ/2 ∈ g(σ). Thus, (σ) = 0 iff θ/2 is a root of g with respect to hσ and σ(eθ/2 ) = −eθ/2 . Example 4.1. For the minimal gradation the numbers sα (α ∈ ∆ ⊂ h∗ ) are as follows (cf. (3.2)): (a) If σ = 1, then sα = 0 (resp. 1) for α ∈ ∆+ (resp. −α ∈ ∆+ ). (b) If σ|gj = (−1)2j , then sα = 0 (resp. 1) if α(x) = 0 and α(h0 ) > 0 (resp. α(h0 ) < 0), sα = −1/2 (resp. 12 ) if α(x) = 12 and α(h0 ) > 0 (resp. 0), sθ = 0 and sα + s−α = 1, α ∈ ∆. Recall that the (Virasoro) central charge of Wk (g, θ/2) is [KW]: k sdim g − 6k + h∨ − 4 , c(k) = k + h∨ and it is, of course, the same for the twisted vertex algebras Wk (g, σ, θ/2). Introduce the following vectors in hσ∗ : 1 1 p˜(α)sα α , γ1/2 = p˜(α)sα α . γ = 2 2 α∈S

α∈S1/2

Corollaries 3.1 and 3.3 give the following result, which will be used in the calculation of the determinant formula. Proposition 4.1. Let M be a restricted gtw -module of level k. Let v ∈ M be a singular vector of M with weight Λ ∈ hσ∗ . Then we have in the case of Wk (g, σ , θ/2): (a) The eigenvalue of Ltw 0 on v ⊗ |0 is equal to 1 ((Λ|Λ) − 2(Λ|γ )) − Λ(x) + sg + sgh , h= 2(k + h∨ ) where k 1 sg = − p ˜ (α)s (s − 1) , s = p˜(α)s2α . α α gh 4(k + h∨ ) 4 α∈S

(b) The eigenvalue of J

{H},tw

α∈S1/2

for H ∈ h

σf

on v ⊗ |0 is equal to

(Λ − γ1/2 )(H) . Proof. Letting sgh := sch + sne , we have (see (3.10) and (3.11)): sgh = 1 ˜(α)(s2α + sα ). Since α ∈ S1/2 iff θ − α ∈ S1/2 , we obtain that α∈S1/2 p 4 1 p˜(α)sα = − (σ) . 2

1 8 (σ)

+

(4.2)

α∈S1/2

It is because sα + sθ−α = δα,θ/2 , which holds due to (3.6). This proves the formula for sgh . The rest is straightforward.

108


Proposition 4.2. For the 12 Z-gradation of g defined by ad x one has: (a) 2γ (x) = 1 − h∨ − 12 (σ). (b) γ = 2γ1/2 + γ0 − 12 (h∨ − 1)θ, where γ0 =

1 2

= 12 (γ − γ0 ). (c) γ1/2

Proof. We have: 1 1 2γ (x) = p˜(α)sα − 2 2 α∈S1/2

p˜(α)sα − 1 =

α∈S−1/2

α∈S α(x)=0

p˜(α)sα α.

p˜(α)sα −

α∈S1/2

1 sdim g1/2 − 1. 2

∨

Since sdim g1/2 = 2h −4 (see [KW], (5.6)), formula (4.2) completes the proof of (a). Similar calculations establish (b), and (c) is immediate by (b). In [KW], Theorem 5.2, we gave a realization of the vertex algebra Wk (g, θ/2) as a subalgebra of Vαk (g0 )⊗F (Ane ), where g0 is the 0th grading component in (2.1) and αk is the “shifted” 2-cocycle: αk (atm ,btn ) = ((k+h∨ )(a|b)− 12 κg0 (a,b))mδm,−n , where κg0 is the Killing form on g0 . The twisted version of this result is derived from [KW], Theorem 5.2, by making use of (1.14), Theorem 2.1, and the following identity for formal distributions a, b, c such that [aλ b] = a, b ∈ C, [aλ c] = a, c ∈ C, [bλ c] = b, c ∈ C: (4.3) : abc :tw (z) = : atw (z)btw (z)ctw (z) : −z −1 sb b, catw (z) tw p(a)p(b) tw +sa a, bc (z) + sa (−1) a, cb (z) . As in [KW], we keep the notation J {a},tw if a ∈ gf0 , but let G{v},tw = J {v},tw if v ∈ g−1/2 . Due to Theorem 2.1, the formal distributions J {a},tw , G{v},tw and Ltw strongly and freely generate the twisted vertex algebra Wk (g, σ, θ/2). Theorem 4.1. The following formulas define an injective vertex algebra homomorphism of Wk (g, σ, θ/2) to Vαk (g0 , σ) ⊗ F (Ane , σ): (−1)p(a) : Φα,tw (z)Φtw [uα ,a] (z) : 2 α∈S1/2 (−1)p(a) sΦα Φα , Φ[uα ,a] ne z −1 (a ∈ gf0 ) , − 2 α∈S1/2 G{v},tw (z) → : [v, uα ]tw (z)Φα,tw (z) : −(k + 1) (v|uα )∂Φα,tw (z)

J {a},tw (z) → atw (z) +

α∈S1/2 p(v)

(−1) − 3

(−1)p(v) + 3 +

α,β∈S1/2

α∈S1/2

: Φα,tw (z)Φβ,tw (z)Φtw [uβ ,[uα ,v]] (z) : sΦβ Φβ , Φ[uβ ,[uα ,v]] ne Φα,tw (z)

α,β∈S1/2 p(α)p(β) (−1) sΦα Φα , Φ[uβ ,[uα ,v]] ne Φβ,tw (z)

−1 + sΦα Φα , Φβ ne Φtw (v ∈ g−1/2 ) , [uβ ,[uα ,v]] (z) z


109

1 k+1 α,tw (−1)p(α) : utw (z) : + ∂x(z) α (z)u ∨ 2(k + h ) k + h∨ α∈S0 1 1 α,tw + (−1)p(α) : Φtw (z) : − (−1)p(α) sα α(z)z −1 α (z)∂Φ ∨ 2 2(k + h ) α∈S1/2 α∈S0

1 1 p(α) sα α p(α) sα + (−1) (−1) z −2 . κg0 (uα , u ) − 4(k + h∨ ) 2 2 2 Ltw (z) →

α∈S+ ∪S0

α∈S0

f ∨ (For gf0 simple, κg0 (uα , uα ) = 2h∨ 0 ,where h0 is the dual Coxeter number of g0 with respect to ( . | . ).)

In the case of Wk (g, σ, θ/2), Proposition 3.1 gives the following result. Proposition 4.3. Let M be a gtw -module satisfying the conditions of Proposition 3.1. Then the Euler-Poincaré character of the Wk (g, σ, θ/2)-module H tw (M ) is not identically zero iff eθ t−1 is not locally nilpotent on M . Now we turn to the determinant formula for the Verma modules over Wk (g, σ, θ/2). To simplify notation, we let g = gf0 (resp. h = (hσ )f ), the centralizer of f in g0 (resp. in hσ ). Let λ → λ denote the restriction map hσ → h . Let S0 (resp. S−1/2 ) = {α ∈ S | α(x) = 0 (resp. α(x) = −1/2)}, and let ∆W,σ = {α | α ∈ S0 ∪ S−1/2 } ⊂ h∗ , the multiplicity of α being the multiplicity of α ∈ S . Note that ∆W,σ may contain 0 (this happens iff θ/2 ∈ ∆σ ). Define the set of roots ∆W,σ of Wk (g, σ, θ/2) as a subset of the dual of its Cartan algebra hW,σ = h CLtw 0 , defined as follows. We embed h∗ in h∗W,σ by letting α ∈ h∗ to be zero on Ltw 0 , and define δ ∈ h∗W,σ by δ |h = 0 ,

δ (Ltw 0 ) = −1 .

im Then ∆W,σ = ∆re W,σ ∪ ∆W,σ , where

∆re W,σ

= {(n + sα + α(x))δ + α | α ∈ ∆W,σ ,

∆im W,σ

= {nδ | n ∈ E0 ,

n ∈ Z} ,

n = 0} ,

where the multiplicity of a root (n + sα + α(x))δ + α is equal to the multiplicity of α ∈ ∆W,σ with given sα , and the multiplicity of nδ is equal to the multiplicity of the root nδ of gtw . Note that 0 is a root in ∆re W,σ of multiplicity (σ)( 1). We denote by ∆+ the subset of positive roots, which consists of the subset W,σ im ,+ im ∆W,σ of elements of ∆W,σ for which n > 0, and the subset ∆re,+ W,σ of elements of for which n ∈ Z . ∆re + W,σ

110


Define the corresponding partition function PW,σ (η) on h∗W,σ as the number of ways η can be represented in the form (counting root multiplicities): η= kα α , where kα ∈ Z+ and kα 1 if α is odd. α∈∆+ W,σ Remark 4.1. Denote by PW,σ (η) the partition function for the set ∆+ W,σ \{0}. Of course, PW,σ (η) = PW,σ (η) if (σ) = 0, but PW,σ (η) = 12 PW,σ (η) if (σ) = 1 and η = 0.

The definition (3.16)–(3.18) of a highest weight module M over the vertex algebra Wk (g, σ, θ/2) can be made a bit more explicit: the highest weight λ is an element of h∗W , and condition (3.17) can be replaced by {H}

J0

vλ = λ (H)vλ , H ∈ h , and Ltw 0 vλ = hvλ ,

(4.4)

where λ denotes the restriction of λ to h and h is the minimal eigenvalue of Ltw 0 on M . We have the weight space decomposition of M : {H} tw M= Mµ , Mµ = {v ∈ M |J0 v = µ (H)v , H ∈ h , Ltw 0 v = µ(L0 )v} .

µ∈h∗ W,σ

The Verma module M (λ) over Wk (g, σ, x) is a highest weight module for which dim M (λ)µ = PW,σ (λ − µ) . In the case when (σ)(= dim g1/2 (σ)0 ) = 1 choose a non-zero vector e ∈ g1/2 (σ)0 and let f = [f, e ]. Rescaling e , if necessary, we may assume that [f , f ] = f . The vector f is a weight vector for h in g−1/2 with weight zero. Due to Theorem 2.1, we have the corresponding formal distribution in Wk (g, σ, θ/2): Gn z −n−3/2 . (4.5) G(z) := (−k − h∨ )−1/2 G{f },tw (z) = n∈Z

We have the following description of the highest weight subspace of M (λ): Cvλ if (σ) = 0 , M (λ)λ = Cvλ + CG0 vλ if (σ) = 1 . We shall need an explicit formula for the eigenvalue of [G0 , G0 ] on vλ ∈ M (λ), which we shall denote by ϕ0 (k, h, λ ). In order to compute the function ϕ0 , recall that Theorem 5.1(e) from [KW] provided an explicit expression for [G{u} λ G{v} ], 2 u, v ∈ g−1/2 , in Wk (g, θ/2). Unfortunately, the coefficient of λ6 in this expression is correct only when g = gf0 is simple. Here is a correct expression for this coefficient, which we shall denote by γk : βk (uα , uα ) γk (u, v) = −(k + h∨ )g(u, v)c(k) + g(u, v) +2

j∈S1/2

α∈S

βk ([u, uj ] , [uj , v] ) ,


111

where βk (a, b) = (k + 12 h∨ )(a|b) − 14 κg0 (a, b), a, b ∈ g0 , g(u, v) ∈ C is defined by [u, v] = g(u, v)f , a stands for the orthogonal projection of a ∈ g0 on g , and S indexes a basis of g . If g is simple or, more generally, if κg0 (a, b) = 2h∨ 0 (a|b), a, b ∈ g , we have a much simpler formula: 1 ) (sdim g + sdim g ) . γk (u, v) = −g(u, v) k + h∨ c(k) − k + (h∨ − h∨ 0 1/2 0 2 In the case when u = v = f , so that g(u, v) = 1, we obtain from [KW], Theorem 5.1(e): [G{f

}

{f } ] λG

+

1 2

= −(k + h∨ )L : J {h } J {hi } : + i

: J {u } J {uα } :) + α

α∈S0

i

λ2 γk (f , f ) . 6

i Here {h i } and {h } are dual bases of h , and S0 is a basis of the kernel of the map ad f : α∈S Cuα → g−1/2 . 0 This formula is used to obtain: 1 − γ |2 (4.6) |λ + γ1/2 ϕ0 (k, h, λ ) = h − ∨ 2(k + h )

sα − |γ1/2 − γ |2 − (−1)p(α) βk (uα , uα ) 2 α∈S0

−

1 24(k + h∨ )

i

βk (hi , hi ) +

c(k) . (−1)p(α) βk (uα , uα ) − 24

α∈S0

Note that in the case when κg0 (a, b) = 2h∨ 0 (a|b), this using βk (uα , uα ) = βk (hi , hi ) = k + 12 (h∨ − h∨ 0 ). Then

formula can be simplified, (4.6) becomes:

1 |λ + γ1/2 − γ |2 (4.7) 2(k + h∨ )

sα h∨ + h∨ 1 1 0 − |γ1/2 − γ |2 + (−1)p(α) − (k + )2 2 2 2 2

ϕ0 (k, h, λ ) = h −

α∈S0

h∨ (h∨ − 1) 1 h∨ − h∨ 0 + − (sdim g0 + sdim g1/2 ) + 3 8 24 1 h∨ p(α) sα . (−1) + + 2 2 8 α∈S0

In order to define the contravariant bilinear form on a Verma module M (λ) over Wk (g, σ, x), we use the anti-involution ω of g introduced in [KW]; we shall assume that it commutes with σ. As in [KW], we have the following anti-involution {a},tw of the associative algebra A generated by coefficients Jn of formal distributions {a},tw (z), where a ∈ g ⊕ g−1/2 (see (3.14)), and the Ltw : J n {ω(a)},tw

tw {a},tw ) = J−n ω(Ltw n ) = L−n , ω(Jn

.

112


The contravariant bilinear form B( . , . ) on a Verma module M (λ) over Wk (g, σ, x) with highest weight vector vλ is defined in the usual way: B(avλ , bvλ ) = vλ∗ , ω(a)bvλ ,

a, b ∈ A ,

where vλ∗ is the linear function on M (λ), equal to 1 on vλ and 0 on G0 vλ and all weight spaces M (λ)µ , µ = λ. This is a supersymmetric bilinear form, which is contravariant, i.e., B(au, v) = B(u, ω(a)v), u, v ∈ M (λ), a ∈ A, and B(vλ , vλ ) = 1, and these properties determine B( . , . ) uniquely. Different weight spaces are orthogonal with respect to this form and its kernel is the maximal submodule of M (λ). Denote by detη (k, h, λ ) the determinant of the bilinear form B( . , . ) restricted to the weight space M (λ)λ−η , η ∈ h∗W,σ . This is a function in k, h and λ (see (4.4)) and it depends on the choice of a basis of M (λ)λ−η only up to a constant factor. → h∗ , defined by Consider the map π : ∆ W,σ π(α + mδ) = α + (m + α(x))δ , π(mδ) = mδ . It is easy to see that, counting root multiplicities, π induces a bijective map: ∼

re ∪ ∆ im ) → ∆ + \{0}. π(∆ ++ + W,σ tw Denote by mult 0 mδ the multiplicity of the root mδ in g (⊂ gtw ).

Theorem 4.2. Up to a non-zero constant factor, the determinant detη (k, h, λ ) is given by the following formula: (k + h∨ )(mult 0 mδ)PW,σ (η−mnδ ) ϕ0 (k, h, λ )(σ)PW,σ (η)

×

m∈E0+ n∈N ˜ ϕα,n (k, h, λ )p(α)

n+1

PW,σ (η−nπ(α))

,

re α∈∆ ++ n∈N

where the factor ϕ0 (occurring only when (σ) = 1) is given by (4.6), and all the remaining factors are as follows (α ∈ ∆σ , m ∈ E0 ): n ϕmδ+α,n =m(k + h∨ ) + (λ + γ1/2 − γ | α) − |α|2 if α(x)= 0, 2 ϕmδ+α,n = 2 1 1 n 2 |α| − (m + )(k + h∨ ) − (λ + γ1/2 − γ |α) h− ∨ k+h 2 2 1 1 1 1 2 + |λ + γ1/2 − γ | − (k + 1 − (σ))2 − |γ |2 − sg − sgh 2 4 2 2

(4.8) (4.9)


113

if α(x) = 12 , 2 1 n − (m + 1)(k + h∨ ) 4(k + h∨ ) 1 +2|λ + γ1/2 − γ |2 − (k + 1 − (σ))2 − 2|γ |2 − sg − sgh . 2

ϕmδ+θ,n = h −

(4.10)

(Formulas for sg and sgh are given in Proposition 4.1(a).) Proof. The proof follows the traditional lines, as in [KW]. First, let M be a Verma = Λ + kD, where Λ ∈ hσ∗ . Then for each module over gtw with highest weight Λ + and a positive integer n such that α ∈∆ + ρtw | 2(Λ α) = n( α| α)

(4.11)

− n under certain conditions (stated in Lemma 7.1 of [KW]), Λ α is a singular weight, of multiplicity at least mult α, of M . This follows from the determinant formula for gtw in [K2] (as corrected in Remark 7.1 of [KW]). tw Let h∨ = ( ρtw |δ) be the dual Coxeter number of gtw . We have: h∨

tw

= h∨ .

(4.12)

∨ (see [K3], Exercise 12.20). Indeed, the Ltw n can be constructed for all k = −h But the central charge of Lg,tw is independent of σ and has singularity only at k = −h∨ . This implies (4.12). Hence for α = α + mδ, where α ∈ ∆σ , m ∈ E0 , (4.11) can be rewritten, using also Corollary 3.2, as follows: tw

2(Λ|α) − 2(γ |α) + 2m(k + h∨ ) = n(α|α) .

(4.13)

We decompose α ∈ hσ (= hσ∗ ) with respect to the orthogonal direct sum decomposition hσ = Cx + h : α = 2α(x)x + α .

(4.14)

Next, by Theorem 3.1, the Wk (g, σ, θ/2)-module H tw (M ) is a Verma module, and its highest weight is λ = hδ + λ , where h is given by Proposition 4.1(a), and (see Proposition 4.1(b)): . λ = Λ − γ1/2

(4.15)

− n By Proposition 3.1, each singular weight Λ α of M satisfying (4.11) and re ++ , gives rise to a singular weight of H tw (M ) (which is a Verma such that α ∈∆ module over Wk (g, σ, θ/2) with highest weight λ). This gives rise to a factor of detη . We now rewrite (4.13) in terms of k, h and λ . In the case α(x) = 0, substituting (4.15) in (4.13), we obtain (4.8). In the case α(x) = 0, we substitute Λ = 2Λ(x)x + λ + γ1/2 (obtained from (4.14) and

114


(4.15)) in the formula for h given

1 h= (Λ(x) − γ (x) − k + h∨

by Proposition 4.1(a) to obtain:

1 1 (k + h∨ ))2 + |λ + γ1/2 − γ |2 (4.16) 2 2 1 1 − (k + h∨ + 2γ (x))2 − |γ |2 + sg + sgh . 4 2

Substituting (4.14) and (4.15) in (4.13), we obtain: n 2α(x)Λ(x) = |α|2 − (λ + γ1/2 − γ |α) − m(k + h∨ ) . 2 Finally, substituting the obtained expression for Λ(x) in (4.16) and using Proposition 4.2(a), we get (4.9) and (4.10). The rest of the proof is the same as in [KW]. Remark 4.2. re is such that (α|α) = 0, the condition (4.1) becomes (a) If α + mδ ∈ ∆ ++ tw (Λ + ρ |α + mδ) = 0. Hence in this case the function ϕα+mδ,n (k, h, λ ) is independent of n. Since α + mδ is an odd root, we therefore can simplify the corresponding factor in the formula for detη (cf. [KW], Remark 7.2): p(α) n+1 PW,σ ; π(α+mδ) (η−π(α+mδ)) ˜ P (η−nπ(α+mδ)) ϕα+mδ,n W,σ = ϕα+mδ,1 . n∈N

Here PW,σ ; α stands for the partition function of the set ∆+ α} (i.e., we W,σ \{ reduce by 1 the multiplicity of α ). re re (b) If α + mδ ∈ ∆ ++ is such that 2(α + mδ) ∈ ∆++ , and n ∈ N, then condition (4.11) for the pair {2(α + mδ), n} is the same as that for the pair {α + mδ, 2n}, hence in this case we have ϕα+mδ,2n = ϕ2(α+mδ),n , and the corresponding factors in detη cancel as in [KW], Remark 7.2. (c) In all examples we have: ϕ0 = ϕ(−δ+θ)/2,0 , but we do not know how to prove this in general. We conjecture that this is always the case, i.e.,

1 1 1 2 2 2 |λ + γ1/2 − γ | − (k + ) − |γ | − sg − sgh . ϕ0 = h − 2(k + h∨ ) 2 2

5. Examples 5.1. Ramond N = 1 algebra Recall that the Neveu-Schwarz vertex algebra is Wk (spo(2|1), θ/2) [KW]. It corresponds to the minimal gradation of g = spo(2|1), which looks as follows: g = Ce−θ ⊕ Ce−θ/2 ⊕ Cx ⊕ Ceθ/2 ⊕ Ceθ , e−θ/2 = 12 (E31 − E23 ), x = 12 (E11 − E22 ), eθ/2 = E13 + E32 , where e−θ = eθ = 2E12 , h = Cx, and θ ∈ h∗ is defined by θ(x) = 1. Then ∆+ = {θ/2, θ}. Choose 1 2 E21 ,


115

the invariant bilinear form (a|b) = strab. Then h∨ = 3/2 and (eθ/2 |e−θ/2 ) = (eθ |e−θ ) = 1, (x|x) = 1/2. We have x = θ/2 under the identification of h with h∗ . We take f = e−θ . The only non-trivial automorphism σ that fixes f and x, also fixes e = eθ /2 and σ(e±θ/2 ) = −e±θ/2 . Then we have: (σ) = 1 , sθ/2 = 1/2 , s−θ/2 = 1/2 , sθ = 0 , s−θ = 1 . Hence we have: γ = −θ/2, γ1/2 = −θ/8, sg = −k/4(2k + 3),sgh = −1/16. In this case we have one twisted neutral free fermion Φtw (z) = n∈Z Φn z −n−1/2 , where [Φm , Φn ] = δm,−n and Φtw (z)− = n>0 Φn z −n−1/2 . The twisted vertex algebra Wk (g, σ, θ/2) is strongly generated by the Virasoro field −n−2 Ltw (z) = Ltw n z n∈Z

and the odd Ramond field Gtw (z) =

−n−3/2 Gtw , n z

n∈Z

so that the Ln and Gn satisfy the relations of the Ramond (N = 1) superalgebra [R] with central charge c(k) = 3/2 − 12γ 2 , where γ 2 = (k + 1)2 /(2k + 3) . In particular, we have: tw tw [Gtw 0 , G0 ] = 2L0 − c(k)/12 .

(5.1)

The free field realization, provided by Theorem 4.1, of this algebra is given bn z −n−1 , where [bm , bn ] = mδm,−n and in terms of a free boson b(z) = n∈Z b(z)− = n0 bn z −n−1 , and the twisted fermion Φtw (z). We have: 1 1 1 : b(z)2 : +γ∂b(z) − : Φtw (z)∂Φtw (z) : − z −2 , 2 2 16 √ 1 Gtw (z) = √ : Φtw (z)b(z) : + 2γ∂Φtw (z) . 2 In order to compute the determinant formula for the Ramond algebra we need the σ-twisted affinization gtw = m∈Z g¯0 tm + m∈1/2+Z g¯1 tm + CK + CD, re where g¯0 = Ceθ + Cx + Ce−θ , g¯1 = Ceθ/2 + Ce−θ/2 . The set ∆ ++ is a union of two subsets: 1 {mδ + θ/2| m ∈ + Z+ } and {mδ + θ| m ∈ Z+ } . 2 From (4.10) and Remark 4.2(b) we obtain that ϕmδ+θ/2,n (k, h) = h − hR n,2m+1 (k) (k), where and ϕmδ+θ,n = h − hR 2n,m+1

2 n 3 1 1 hR −m k+ . (5.2) − (k + 1)2 + n,m (k) = 2 2 16 4(k + 32 ) Ltw (z) =

It follows from (5.1) that the extra factor is equal to h − c(k)/24.

116


The set of positive even (resp. odd) roots for Wk (spo(2|1), σ, θ/2) is Nδ (resp. Z+ δ ). Hence PW,σ (η) = pR (η), where pR (η) is defined by the generat ∞ 1+qn ing series η∈Z+ pR (η)q η = n=1 1−q n . Hence, by Theorem 4.2, we obtain the following determinant formula for the Ramond algebra, where η ∈ N (cf. [KW]): c(k) pR (η) 2pR (η− 12 mn) ) (h − hR . detη (k, h) = (h − n,m (k)) 24 m,n∈N m+n odd

5.2. N = 2 Ramond type sector Recall that the N = 2 vertex algebra is Wk (s(2|1), θ/2). In this section the Lie superalgebra g = s(2|1) consists of supertraceless matrices in the superspace C2|1 , whose even part is C1 + C3 and odd part is C2 , where C1 , 2 , 3 is the standard basis. We shall work in the following basis of g: e1 = E12 , e2 = E23 , −[e1 , e2 ] , f1 = E21 , f2 = −E32 , [f1 , f2 ] , h1 = E11 + E22 , h2 = −E22 − E33 . The elements ei , fi , hi (i = 1, 2) are the Chevalley generators of g and h = Ch1 + Ch2 . The elements ei , fi (i = 1, 2) are all odd elements of g, both simple roots αi (i = 1, 2), attached to ei , are odd, and ∆+ = {α1 , α2 , θ = α1 + α2 }. Since g¯0 = C[e1 , e2 ] + C[f1 , f2 ] + h g2 , there is only one, up to conjugacy, nilpotent element f = [f1 , f2 ], which embeds in the following s2 -triple: {e = − 21 [e1 , e2 ] , x = 1 2 (h1 + h2 ) , f }. The minimal gradation of g, defined by ad x, looks as follows: g = Cf ⊕ (Cf1 + Cf2 ) ⊕ h ⊕ (Ce1 + Ce2 ) ⊕ Ce, . The invariant bilinear form on g is (a|b) = strab, and h∨ = 1. First consider the Ramond type automorphisms σa (−1/2 < a 1/2), defined by σa (e1 ) = e2πia e1 , σa (f1 ) = e−2πia f1 , σa (e2 ) = e−2πia e2 , σa (f2 ) = e2πia f2 . Then g(σa ) = g if a = 1/2 (resp. = g¯0 if a < 1/2), and we choose n(σa )+ = Ce2 + Cf1 + Cf , n(σa )− = Cf2 + Ce1 + Ce (resp. Cf and Ce) , so that in all cases (σa ) = 0, and sα1 = a , sα2 = −a , sθ = 0 , s−α1 = 1 − a , s−α2 = 1 + a , s−θ = 1 . In this case we have two twisted neutral free fermions Φ1n z −n−1/2 , Φtw Φtw 1 (z) = 2 (z) = n∈1/2+a+Z

Φ2n z −n−1/2 ,

n∈1/2−a+Z

where [Φim , Φjn ] = (δij − 1) δm,−n , Φtw 1 (z)− =

Φ1n z −n−1/2 , Φtw 2 (z)−

n∈1/2+a+Z+

=

n∈1/2−a+Z+

Φ2n z −n−1/2 .


117

The twisted vertex Wk (g, σa , θ/2) is strongly generated by the Vira algebra−n−2 tw tw −n−1 z , the current J (z) = J and soro field Ltw (z) = n∈Z Ltw n∈Z n z n ±,tw ±,tw −n−3/2 (z) = n∈1/2∓a+Z Gn z so that Ln , Jn (n ∈ Z) and two odd fields G G± (n ∈ 1/2 ∓ a + Z) satisfy the relations of N = 2 Ramond type superconformal n algebra with central charge c(k) = −3(2k + 1). The free field realization, provided by Theorem 4.1, of this algebra is given in terms of free bosons hi (z) = n∈Z hin z −n−1 (i = 1, 2), where [him , hjn ] = (k + 1)m(1 − δij )δm,−n , and the twisted neutral free fermions Φtw i (z)(i = 1, 2): 1 1 tw : h1 (z)h2 (z) : + (: Φtw 1 (z)∂Φ2 (z) : k+1 2

Ltw (z) =

tw + : Φtw 2 (z)∂Φ1 (z) : +∂(h1 (z) + h2 (z))) +

J tw (z) =

a2 −2 z , 2

tw −1 h1 (z) − h2 (z)+ : Φtw , 1 (z)Φ2 (z) : +az

G+,tw (z) =

tw (−k − 1)−1/2 (: Φtw 2 (z)h1 (z) : +(k + 1)∂Φ2 (z))

G−,tw (z) =

tw (−k − 1)−1/2 (: Φtw 1 (z)h2 (z) : +(k + 1)∂Φ1 (z)) .

+ = ∆ re ∪ ∆ im of positive roots of re = {(m + The set ∆ gtw is as follows: ∆ + + + a)δ + α1 , (m − a + 1)δ − α1 , (m − a)δ + α2 , (m + a + 1)δ − α2 , mδ + θ , (m + im 1)δ − θ| m ∈ Z+ }, where all roots have multiplicity 1, and ∆ + = {mδ| m ∈ N}, all having multiplicity 2. Next, γ = −aH, γ1/2 = −aH/2, where H = h1 − h2 , re is as follows: sg = ka2 /(k + 1), sgh = −a2 /2, and the set of roots ∆ ++ {mδ + θ| m ∈ Z+ } ∪ {mδ + α1 | m ∈ a + Z+ } ∪ {mδ + α2 | m ∈ −a + Z+ } . ∗ We have: hW,σ = CH + CLtw 0 , where H = h1 − h2 . Define α ∈ hW,σ by α(H) = 1, re ,+ α(Ltw 0 ) = 0. Then ∆W,σ = {mδ − α| m ∈ 1/2 + a + Z+ } ∪ {mδ + α| m ∈ im ,+ 1/2 − a + Z+ }, all of multiplicity 1, and ∆W,σ = {mδ | m ∈ N} of multiplicity 2. Let PNa =2 (η) , η ∈ h∗W,σ , be the corresponding partition function. Let h and j be tw the eigenvalues of Ltw 0 and of J0 respectively on the highest weight vector vλ . By Theorem 4.2 and Remark 4.2(a), we obtain the following formula for detη (k, h, j), conjectured in [BFK] (cf. [KM]):

PNa =2 (η−mnδ ) (k + 1)(h − hm,n (k, j)) m,n∈N

×

ϕm,− (k, h, j)PN =2;mδ −α (η−(mδ −α)) a

m∈1/2+a+Z+

×

m∈1/2−a+Z+

a

ϕm,+ (k, h, j)PN =2;mδ +α (η−(mδ +α)) ,

118


where

a2 1 (n − m(k + 1))2 − (j − a)2 − (k + 1)2 + , 4(k + 1) 2 a2 1 . ϕm,± (k, h, j) = h − (m2 + m)(k + 1) ∓ (m + )(j − a) − 2 2 hm,n (k, j) =

5.3. N = 2 twisted sector In this subsection we consider the involution σ = σtw of g = s(2|1) defined by: σtw (e1 ) = e2 , σtw (f1 ) = f2 , σtw (h1 ) = h2 . √ √ √ Let e√ = (e1 + e2 )/ 2 , e(2) = (e1 − e2 )/ 2 , f (1) = (f1 + f2 )/ 2 , f (2) = (f1 − f2 )/ 2 , H = h1 − h2 . Then (1)

g(σtw ) = Cf ⊕ Cf (2) ⊕ Cx ⊕ Ce(2) ⊕ Ce , and the only possible choice for n(σtw )± is as follows: n(σtw )+ = Cf (2) + Cf , n(σtw )− = Ce(2) + Ce . Note that n1/2 (σ)+ = n1/2 (σ)− = 0 and g1/2 (σ)0 = Ce(2) (see (2.5)), so that (σtw ) = 1. Note also that hσ = Cx, so that the set ∆σ (⊂ hσ∗ ) of non-zero roots of hσ in g is ∆σ = {±θ , ±θ/2}, where θ(x) = 1, the roots ±θ (resp. ±θ/2) being of multiplicity 1 (resp. 2). Thus the sa are as follows: sH = se(2) = sf (2) = 1/2 , se(1) = se = 0 , sf (1) = sf = 1 . (Note that here sa depends not only on the root, but also on the root vector.) The free field realization of the twisted vertex algebra Wk (g, σtw , θ/2), provided by Theorem 4.1, is given in terms of free neutral fermions −n−1/2 −n−1/2 Φ(1) (z) = Φ(1) , Φ(2)tw (z) = Φ(2) , n z n z n∈Z

n∈1/2+Z

where (j) j [Φ(i) m , Φn ] = (−1) δij δm,−n , −n−1/2 Φ(1) (z)− = Φ(1) , n z

Φ(2)tw (z)− =

n>0

and free commuting bosons x(z) = xn z −n−1 , n∈Z

−n−1/2 Φ(2) , n z

n>0

H tw (z) =

Hn z −n−1 ,

n∈1/2+Z

where 1 (k + 1)mδm,−n , 2 [Hm , Hn ] = −2(k + 1)mδm,−n , x(z)− = xn z −n−1 [xm , xn ] =

n0

Quantum Reduction in the Twisted Case and H tw (z)− =

119

Hn z −n−1 :

1 1 1 (1) 2 tw 2 : Φ (z)∂Φ(1) (z) : : x(z) : − : H (z) : + k+1 4 2 − : Φ(2),tw (z)∂Φ(2),tw (z) : + ∂x(z) ,

n>0

Ltw (z) =

J tw (z) = H tw (z)− : Φ(1) (z)Φ(2),tw (z) : ,

1 G(1),tw (z) = (−k − 1)−1/2 : Φ(1) (z)x(z) : − : Φ(2),tw (z)H tw (z) : 2 (1) +(k + 1)∂Φ (z) , (2),tw

G

1 (z) = (−k − 1) : Φ(2),tw (z)x(z) : + : Φ(1) (z)H tw (z) : 2 −(k + 1)∂Φ(2),tw (z) , −1/2

where G(1),tw =

√1 (G+,tw 2

+ G−,tw ), G(2),tw =

√1 (G+,tw − G−,tw ). 2 re im ∆+ ∪ ∆ + of positive

+ = roots of gtw Furthermore, in this case the set ∆ re is as follows: ∆ + = {mδ + θ/2, (m + 1)δ − θ/2, mδ + θ, (m + 1)δ − θ| m ∈ 1 im Z+ } ∪ {mδ ± θ/2| m ∈ 12 + Z+ }, ∆ + = {mδ| m ∈ 2 N}, all having multiplicity 1. Note also that the roots mδ ± θ/2 are odd and all the other roots are even. We have: h = 0, sg = −k/16(k + 1), sgh = −1/16, and 1 re ∆ ++ = {mδ + θ| m ∈ Z+ } ∪ {mδ + θ/2| m ∈ Z+ } , 2 all of multiplicity 1. From (4.10) and Remark 4.2(b) we obtain that ϕmδ+θ/2,n (k, h) = h − htw n,2m+1 (k) where htw n,m (k)

and ϕmδ+θ,n (k, h) = h − htw 2n,m+1 (k),

2 1 2 − m(k + 1) − (k + 1) + . 2 8

n

1 = 4(k + 1) (2)

(2)

(5.3)

It is easy to compute that [G0 , G0 ] = −(Ltw 0 − c(k)/24), hence the extra factor equals ϕ(θ−δ)/2,0 (h, k) = h − c(k)/24 = h + (2k + 1)/8. The set of positive even (resp. odd) roots for Wk (s(2|1), σtw , θ/2) is 12 Nδ (resp. 12 Z+ δ ), all of multiplicity 1. Hence PW,σ (η) = ptw (η), where ptw (η) is tw defined by the generating series η∈ 12 Z+

ptw (η)q η =

∞ 1 + q n/2 . 1 − q n/2 n=1

120


Hence by Theorem 4.2, we obtain the following determinant formula for the N = 2 twisted superconformal algebra, conjectured in [BFK]: 2k + 1 ptw (η) 2ptw (η− 12 mn) detη (k, h) = (h + ) (h − htw . n,m (k)) 8 m,n∈N n odd

5.4. N = 4 Ramond type sector Recall that the N = 4 vertex algebra is Wk (g, θ/2), where g = s(2|2)/CI. We shall use the same basis of g and keep the same notation as in [KW], Section 8.4. In particular, the simple roots are α1 , α2 , α3 , where α1 and α3 are odd and α2 is even, all the non-zero scalar products between them being (α1 |α2 ) = (α2 |α3 ) = 1, (α2 |α2 ) = −2. The dual Coxeter number h∨ = 0. Consider the Ramond type automorphisms σ = σa,b of g, where −1/2 < a, b 1/2, defined by σ(e1 ) = e2πia e1 , σ(e2 ) = e−2πi(a+b) e2 , σ(e3 ) = e2πib e3 , σ(hi ) = hi . Note that (σa,b ) = 0. We consider first the case when a + b > 0. Then we have the following possibilities for n(σ)± : (i) (ii) (iii) (iv)

a, b = 1/2: n(σ)− = Ce, where e = e123 , n(σ)+ = Cf , where f = f123 ; a = 1/2, b = 1/2: n(σ)− = Ce + Ce1 + Cf23 , n(σ)+ = Cf + Ce23 ; +Cf1 ; a = 1/2, b = 1/2: n(σ)− = Ce + Ce3 + Cf12 , n(σ)+ = Cf + Ce12 + Cf3 ; a = b = 1/2: n(σ)− = span {e, e1 , e3 , f12 , f23 , f2 }, n(σ)+ = span {f, e12 , e23 , e2 , f1 , f3 }. In these four cases the sα are as follows: sα1 = a, sα2 = 1 − a − b, sα3 = b, sα1 +α2 = −b, sα2 +α3 = −a, sθ = 0 ,

where θ = α1 + α2 + α3 , and, as usual, s−α = 1 − sα . Consequently, we have: re ∆ ++

=

{(m + a)δ + α1 , (m + b)δ + α3 , (m − b)δ + α1 + α2 , (m − a)δ + α2 + α3 , (m + 1 − a − b)δ + α2 , (m + a + b)δ − α2 , mδ + θ| m ∈ Z+ } .

Next γ = 12 α2 , γ1/2 =

a+b 2 α2 ,

sg + sgh = −ab + (a + b)/2.

∗ We have: h = Ch2 , hence hW,σ = Ch2 +CLtw 0 . Define α ∈ hW,σ by α(h2 ) = 2, tw α(L0 ) = 0. Then

∆+ W,σ = {(m +

1 1 1 α α α + a)δ − , (m + + b)δ − , (m + − b)δ + , 2 2 2 2 2 2

1 α (m+ −a)δ + , (m+1 − a − b)δ +α, (m+a+b)δ −α, (m+1)δ | m ∈ Z+ } 2 2 is the set of positive roots of W (g, σa,b , θ/2), all having multiplicity 1, except for mδ which have multiplicity 2. We have: α1 = α3 = −α/2, α2 = α. Let PNa,b=4 (η) be the corresponding partition function. Let h and j be the {h2 },tw eigenvalues of Ltw on vλ , so that λ = 2j α. Formulas (4.8)–(4.10) give 0 and J0


121

the following factors of the determinant (we introduce a simplifying notation consistent with the map π): ϕm,n := kϕ(m−1)δ+θ,n , ϕm,−α/2 := ϕmδ+α1 ,1 = ϕmδ+α3 ,1 , ϕm,α/2 := ϕmδ+α1 +α2 ,1 = ϕmδ+α2 +α3 ,1 , ϕm,n,±α = ϕmδ±α2 ,n , where: ϕm,n (k, h, j) = 4kh−(n−mk)2+(a+b+j −1)2+k(k+1)+k(2a−1)(2b−1), k+1 1 2 1 (a + b + j − 1) + ϕm,±α/2 (k, h, j) = h − m + k± m+ 2 2 4 1 1 + a− b− , 2 2 ϕm,n,±α (k, j) = mk ∓ (a + b + j − 1) + n . By Theorem 4.2 and Remark 4.2(a), we obtain the following formula for detη (k, h, j) in the case a + b > 0: a,b ϕm,n (k, h, j)PN =4 (η−mnδ ) m,n∈N

×

a,b PN (η−(mδ −α/2)) =4;mδ −α/2

ϕm,−α/2 (k, h, j)

m∈ 12 +{a,b}+Z+

×

a,b PN (η−(mδ +α/2)) =4;mδ +α/2

ϕm,α/2 (k, h, j)

m∈ 12 −{a,b}+Z+

×

a,b

ϕm,n,−α (k, j)PN =4 (η−n(mδ −α))

m∈a+b+Z+ n∈N

×

a,b

ϕm,n,α (k, j)PN =4 (η−n(mδ +α)) .

m∈−a−b+N n∈N

The case a + b 0 is treated in the same fashion. The sα ’s in this case are the same as in the case a + b > 0, except for sα2 = −a − b. After the calculation, it turns out that the determinant formula in this case can be obtained from the above determinant formula by replacing a by a + 1 and b by b + 1 in all factors and by changing the range of m in the last two factors by exchanging Z+ and N. Some cases of this determinant formula were conjectured in [KR]. We shall omit the free field realization of the Ramond type sector of N = 4 and other remaining superconformal algebras as being quite long. On the other hand, as in the simplest cases of N = 1 and 2, they are straightforward applications of Theorem 4.1. 5.5. N = 3 Ramond type sector Recall that the N = 3 vertex algebra is Wk (g, θ/2), where g = spo(2|3). (To get the “linear” N = 3 superconformal algebra one needs to tensor the above vertex algebra with one free fermion, and the results of this section can easily be extended to the latter case as in [KW].) We shall keep the notation of [KW], Section 8.5. In particular, the simple roots are α1 and α2 , where α1 is odd and α2 is even, the scalar products between them being (α1 |α1 ) = 0, (α1 |α2 ) = 1/2, (α2 |α2 ) = −1/2.

122


Since θ = 2α1 + 2α2 , we have: α2 = −α1 = α2 . Recall also that S0 = {±α2 }, S1/2 = {α1 , θ/2, α1 + 2α2 }, S1 = {θ}. The dual Coxeter number h∨ = 1/2. Consider the Ramond type automorphisms σ = σa,b of g defined by σ(e10 ) = 2πia e e10 , σ(e01 ) = e2πib e01 , σ|h = 1, where a, b ∈ R are such that a + b ∈ 12 Z. We consider the following three cases: I (resp. II): a = −b, −1/2 < a 0 (resp. a = −b , 0 < a 1/2) , III: a + b = 1/2, −1/2 < a 1/2 . Note that (σ) = 0 in cases I and II, and (σ) = 1 in case III, when g1/2 (σ)0 = Ce11 in (2.5). We have the following possibilities for n(σ)± : I, II , a = 0, 1/2 : n(σ)− = Ce22 , n(σ)+ = Cf22 , I, II, a = 0 : n(σ)− = Ce22 + Cf01 , n(σ)+ = Cf22 + Ce01 , I, II, a = 1/2 : n(σ)− = span {e22 , e10 , f12 }, n(σ)+ = span {f22 , e12 , f10 }, III, a = 1/2 : n(σ)− = Ce22 + Ce11 , n(σ)+ = Cf22 + Cf11 , III, a = 1/2 : n(σ)− = span {e22 , e11 , f12 , f01 , e10 }, n(σ)+ = span {f22 , f11 , e12 , e01 , f10 }. In these cases the sα are as follows (up to the relation (3.3)): I. : sα1 = a, sα2 = −a, sθ/2 = 0, sα1 +2α2 = −a, sθ = 0; II. : the same as in I, except for sα2 = 1 − a; III. : sα1 = a, sα2 = 1/2 − a, sθ/2 = 1/2, sα1 +2α2 = −a, sθ = 0. One finds that in these three cases: a I. : γ = (a − 12 )α2 , γ1/2 = aα2 , sg + sgh = a(1−a) 4k+2 + 2 ; a = aα2 , sg + sgh = − a(a+1) II. : γ = (a + 12 )α2 , γ1/2 4k+2 + 2 ; 2

a III. : γ = aα2 , γ1/2 = aα2 , sg + sgh = − 4k+2 −

1 16 .

Consequently we have in these cases (m ∈ Z+ ): re = {(m− a)δ + α2 , (m+ 1 + a)δ − α2 , (m+ a)δ + α1 , (m− a)δ + α1 + 2α2 , I. : ∆ ++ mδ + θ/2, mδ + θ}, re = {(m+ 1 − a)δ + α2 , (m+ a)δ − α2 , (m+ a)δ + α1 , (m− a)δ + α1 + 2α2 , II. : ∆ ++ mδ + θ/2, mδ + θ}, re III. : ∆ ++ = {(m + 1/2 − a)δ + α2 , (m + 1/2 + a)δ − α2 , (m + a)δ + α1 , (m − a)δ + α1 + 2α2 , (m + 1/2)δ + θ/2, mδ + θ}. ∗ We have: h = Cα2 , hence hW,σ = Cα2 +CLtw 0 . Define α ∈ hW,σ by α = α2 |h , tw α(L0 ) = 0. Then we have in the three cases (m ∈ Z+ ): 1 I. : ∆+ W,σ = {(m−a)δ +α , (m+1+a)δ −α, (m+1/2+a)δ −α, (m+ 2 −a)δ +α, (m + 1/2)δ , (m + 1)δ }; II. : ∆+ W,σ = {(m+1−a)δ +α, (m+a)δ −α, (m+1/2+a)δ −α, (m+1/2−a)δ +α, (m + 1/2)δ , (m + 1)δ }; III. : ∆+ W,σ = {(m + 1/2 − a)δ + α, (m + 1/2 + a)δ − α, (m + 1)δ }. The multiplicities of these positive roots of W (g, σ, θ/2) are 1, except for the following cases: mult (m + 1)δ = 2 in cases I and II, mult (m + 1/2 ∓ a)δ ± α = 2


123

and mult (m + 1)δ = 3 in case III (m ∈ Z+ ). Note, however, that in case III we have, in fact, one even root and one odd root equal (m+1/2∓a)δ ±α, each having multiplicity 1, and an even (resp. odd) root (m + 1)δ of multiplicity 2 (resp. 1). We have: α2 = −α1 = α. Note that 0 is a (odd) root of ∆+ W,σ only in case III. Let PNa,b=3 (η) be the corresponding partition function. Let h and j be the {−4α2 },tw on vλ , so that λ = 2j α. respective eigenvalues of Ltw 0 and J0 Introduce the following notations for the factors of the determinant: ϕm,n = ϕ(m−1)δ+θ,n , ϕm,α = ϕmδ+α1 +2α2 ,1 , ϕm,−α = ϕmδ+α1 ,1 , ϕm,n,±α = ϕmδ±α2 ,n . Formulas (4.8)–(4.10) give the following expressions in case I: n 2 (j + 1)2 1 3 a 1 1 + (k + ) + , ϕm,n (k, h, j) = h − − m(k + ) − 4k + 2 2 2 4 4 2 2 1 1 1 3 a 1 2 1 ϕm,±α (k, h, j) = h − (m + ) (k + ) ± (m + )(j + 1) + (k + ) + , 2 2 2 2 4 2 2 1 n j+1 ϕm,n,±α (k, j) = m(k + ) + ∓ . 2 4 4 By Theorem 4.2 and Remarks 4.2(a) and (b) we obtain the following formula for detη (k, h, j) in case I (a special case of this formula was conjectured in [KMR] and partially proved in [M]): a 1 a 1 (k + )PN =3 (η−mnδ ) ϕm,n (k, h, j)PN =3 (η− 2 mnδ ) 2 m,n∈N

× ×

m,n∈N m+n even

a

m∈∓a+ 12 +Z+ a

ϕm,n,α (k, j)PN =3 (η−n(mδ +α))

m∈−a+Z+ n∈N

ϕm,±α (k, h, j)PN =3;mδ ±α (η−(mδ ±α))

ϕm,n,−α (k, j)PN =3 (η−n(mδ −α)) . a

m∈a+N n∈N

In case II the determinant formula is similar. It can be obtained from the above formula by replacing j + 1 by j − 1 and a by −a in all factors and by changing the range of m in the last two factors by exchanging Z+ and N. In case III we have: 3 1 n 2 j 2 k 1 + + , m k+ − ϕm,n (k, h, j) = h − − 4k + 2 2 2 4 4 16 1 2 1 k 3 1 j ϕm,±α (k, h, j) = h − m + ± m+ + + , k+ 2 2 2 2 4 16 1 n j ϕm,n,±α (k, j) = m k + + ∓ . 2 4 4

124

V.G. Kac and M. Wakimoto 2

j 1 The extra factor is ϕ0 = h + 16 (4k + 3 + k+1/2 ), which is computed, using formula (4.7) (in this case h∨ = −1/2). 0 By Theorem 4.2 and Remarks 4.1 and 4.2(a) and (b) we obtain the following formula for detη (k, h, j) in case III: a a 1 m,n∈N PN =3 (η−mnδ )+ m∈N PN =3;mδ (η−mδ ) k+ 2 a a j 2 PN =3 (η) 1 1 4k + 3 + × h+ ϕm,n (k, h, j)PN =3 (η− 2 mnδ ) 16 k + 1/2

×

m,n∈N m+n odd

ϕm,±α (k, h, j)PN =3;mδ ±α (η−(mδ ±α)) a

m∈ 12 ∓a+Z+

×

ϕm,n,±α (k, j)PN =3 (η−n(mδ ±α)) . a

m∈ 12 ∓a+Z+ n∈N

5.6. Big N = 4 Ramond type sector Recall that the big N = 4 vertex algebra is Wk (g, θ/2), where g = D(2,1; a). (To get the “linear” N= 4 superconformal algebra ([KL],[S],[STP]) one needs to tensor the above vertex algebra with four free fermions and one free boson [GS], and the results of this and the next section can easily be extended to the latter case as in [KW].) We shall keep the notation of [KW], Section 8.6. In particular, the simple roots are α1 , α2 , α3 , where α1 and α3 are even, and α2 is odd, the non-zero scalar products between them being (a = 0, −1): 1 , a+1 2 , (α1 |α1 ) = − a+1

a , a+1 2a (α3 |α3 ) = − . a+1

(α1 |α2 ) =

(α2 |α3 ) =

We shall slightly simplify notation of [KW] by letting e1 = e100 ,

e2 = e010 ,

e3 = e001 ,

f1 = f100 ,

f2 = f010 ,

f3 = f001 ,

e = e121 ,

f = f121 .

In this subsection we consider the Ramond type automorphisms σ = σµ,ν of g defined by σ(e1 ) = e2πiµ e1 , σ(e2 ) = e−πi(µ+ν) e2 , σ(e3 ) = e2πiν e3 , σ|h = 1, where µ, ν ∈ R are such that −1 µ ± ν < 1. We consider separately the following four cases: (++) : µ, ν 0; (−+) : µ < 0, ν 0; (+−) : µ 0, ν < 0; (−−) : µ, ν < 0. In all cases, (σ) = 0 and hσ = h. Since θ = α1 + 2α2 + α3 , we have: h = Cα + Cα , where α := α1 |h = α1 , α := α3 |h = α3 , and α2 = −(α + α )/2. Recall also that S0 = {±α1 , ±α3 }, S1/2 = {α2 , α1 + α2 , α2 + α3 , α1 + α2 + α3 }, S1 = {θ}. The dual Coxeter number h∨ = 0.


125

We have the following possibilities for n(σ)± : (i) µ = −1, ν = 0: n(σ)− = span {e, e2 , e011 , f1 , f3 , f111 , f110 }, n(σ)+ = span {f, f2 , f011 , e1 , e3 , e111 , e110 }; (ii) µ + ν = −1, ν = 0: n(σ)− = span {e, e2 , f111 }, n(σ)+ = span {f, e111 , f2 }; (iii) µ − ν = −1, ν = 0: n(σ)− = span{e, e011,f110 }, n(σ)+ = span{f, e110 ,f011}; (iv) µ = ν = 0: n(σ)− = span {e, f1 , f3 }, n(σ)+ = span {f, e1 , e3 }; (v) µ = 0, ν = 0: n(σ)− = Ce + Cf1 , n(σ)+ = Cf + Ce1 ; (vi) µ = 0, −1, ν = 0: n(σ)− = Ce + Cf3 , n(σ)+ = Cf + Ce3 ; (vii) in all other cases: n(σ)− = Ce, n(σ)+ = Cf . The sα are as follows: sθ = 0,

µ+ν µ+ν , sα1 +α2 +α3 = , 2 2 µ−ν µ−ν , sα2 +α3 = − = 2 2

sα2 = −

sα1 +α2

in all cases; the remaining sα (up to the relation (3.3)) are: sα1 = µ in cases (++) and (+−), sα1 = 1 + µ in cases (−+) and (−−); sα3 = ν in cases (++) and (−+), sα3 = 1 + ν in cases (+−) and (−−). 3 Using these data one finds that γ1/2 = − µα1 +να in all cases and that in 2

α1 + α3 , the four cases (, ), where each and 2 1 is + or − one has: γ = − 1 2 2 sg + sgh = − 4 (µ − 1) + (ν − 1) + 2 . Furthermore, let µ+ν µ+ν (1/2) δ + α2 , m + δ + α1 + α2 + α3 , ∆ ++ = { m − 2 2 µ−ν µ−ν m+ δ + α1 + α2 , m − δ + α2 + α3 |m ∈ Z+ } , 2 2

(0) in the four cases as follows (m ∈ Z+ ): and define ∆ ++ (++) : {(m + µ)δ + α1 , (m + ν)δ + α3 , (m + 1 − µ)δ − α1 , (m + 1 − ν)δ − α3 } , (+−) : {(m + µ)δ + α1 , (m + 1 + ν)δ + α3 , (m + 1 − µ)δ − α1 , (m − ν)δ − α3 }, (−+) : {(m + 1 + µ)δ + α1 , (m + ν)δ + α3 , (m − µ)δ − α1 , (m + 1 − ν)δ − α3 }, (−−) : {(m + 1 + µ)δ + α1 , (m + 1 + ν)δ + α3 , (m − µ)δ − α1 , (m − ν)δ − α3 } . (0) (1/2) re Then ∆ ++ = ∆++ ∪ ∆++ ∪ {mδ + θ| m ∈ Z+ }.

126

V.G. Kac and M. Wakimoto Next, hW,σ = h ⊕ CLtw 0 ,

+(1/2)

and ∆+,re W,σ = ∆W,σ

∪ ∆W,σ ⊂ h∗W,σ , +(0)

where 1 µ + ν α + α 1 µ + ν α + α δ − , m+ + δ + , ={ m+ − 2 2 2 2 2 2 1 µ − ν α − α 1 µ − ν α − α m+ + δ + , m+ − δ − | m ∈ Z+ } , 2 2 2 2 2 2 +(1/2)

∆W,σ

+(0)

and ∆W,σ in the four cases is as follows (m ∈ Z+ ): (++) : {(m + µ)δ + α, (m + ν)δ + α , (m + 1 − µ)δ − α, (m + 1 − ν)δ − α } , (+−) : {(m + µ)δ + α, (m + 1 + ν)δ + α , (m + 1 − µ)δ − α, (m − ν)δ − α }, (−+) : {(m + 1 + µ)δ + α, (m + ν)δ + α , (m − µ)δ − α, (m + 1 − ν)δ − α }, (−−) : {(m + 1 + µ)δ + α, (m + 1 + ν)δ + α , (m − µ)δ − α, (m − ν)δ − α } . The multiplicities of all these roots of W (g, σ, θ/2) are 1. There are, in addition, roots mδ (m ∈ N), all of multiplicity 3. Let PNµ,ν =4 (η) be the corresponding partition function. Let h, j and j be the 1 tw tw tw respective eigenvalues of L0 , J0 and J0 on vλ , so that λ = 2 (jα + j α ). Formulas (4.8)–(4.10) give the following expressions for the factors of the determinant in the (++) case: ϕ(m−1)δ+θ,n

=

h−

a(j + 1 − ν)2 1 (j + 1 − µ)2 (n − mk)2 + + 4k 4k(a + 1) 4k(a + 1)

k (µ − 1)2 + (ν − 1)2 + ; 4 4 2 j+1−µ 1 1 j + 1 − ν (β|α1 ) + (β|α3 ) m+ k+ h− k 2 2 2 +

ϕmδ+β,1

=

+

a(j + 1 − ν)2 k (µ − 1)2 + (ν − 1)2 (j + 1 − µ)2 + + + 4k(a + 1) 4k(a + 1) 4 4

if β ∈ S1/2 ; ϕmδ+β,n

=

j+1−µ j + 1 − ν n(β|β) (β|α1 ) + (β|α3 ) − 2 2 2 if β ∈ S0 .

mk +


127

The factors in the remaining three cases are obtained from the above formulas by a shift of µ and ν as follows: (−+) : µ → µ + 2 , ν → ν ; (+−) : µ → µ , ν → ν + 2 ; (−−) : µ → µ + 2 , ν → ν + 2 . By Theorem 4.2 and Remark 4.2(a) we obtain the following formula for detη (k, h, j, j ): µ,ν (k 2 ϕ(m−1)δ+θ,n (h, k, j, j ))PN =4 (η−mnδ ) m,n∈N

×

P µ,ν

ϕmδ+β,1 (k, h, j, j )

N =4;(m+1/2)δ +β

(η−(m+1/2)δ −β )

(1/2)

mδ+β∈∆ ++

×

µ,ν

ϕmδ+β,n (k, h, j, j )PN =4 (η−n(mδ +β

))

.

(0) mδ+β∈∆ ++ n∈N

5.7. Big N = 4 twisted sector In this subsection we consider the involutions σ = σtw,b of g = D(2, 1; 1) = osp(4, 2) defined by: σ(e1 ) = e3 , σ(e2 ) = e−πib e2 , σ(e3 ) = e2πib e1 , σ(f1 ) = f3 , σ(f2 ) = eπib f2 , σ(f3 ) = e−2πib f1 , where b ∈ R, −1 b < 1. Introduce the following elements of g : e(1) =

√1 (e1 + 2 e−πib e3 ), f (1) = √12 (f1 + eπib f3 ), e(3) = √12 (e1 − e−πib e3 ), f (3) = √12 (f1 − eπib f3 ), e(110) = √12 (e110 + e−πib e011 ), f (110) = √12 (f110 + eπib f011 ), e(011) = √12 (e110 − e−πib e011 ), f (011) = √12 (f110 − eπib f011 ). We have the following eigenspace decomµ 2πiµ

position of g with respect to σ (here, as before, g = {a ∈ g|σ(a) = e g = g0 + g1/2 + gb/2 + g−b/2 + g(1+b)/2 + g(1−b)/2 , where

a}):

g0 = span{e(011) , f (011) , e, f, α2 , α1 + α3 }, gb/2 = span{e(1) , e111 , f2 } , g−b/2 = span{e2 , f (1) , f111 }, g(1+b)/2 = Ce(3) , g−(1+b)/2 = Cf (3) , g1/2 = span{e(110) , f (110) , α1 − α3 } . gtw are described in terms of α ˜ i = αi |hσ Then hσ = Cθ+C(α1 +α3 ) and the roots of (i = 1, 2) and δ, the non-zero inner products between them being (˜ α1 |˜ α1 ) = −1/2, (α˜1 |˜ α2 ) = 1/2. The union of the above bases of the eigenspaces of σ is a basis of g, compatible with the 12 Z-gradation and the root space decomposition with respect to hσ , which we denoted by S. Furthermore, h = Cα, where α = α1 = −α2 .

128


We have: (σ) = 1, g1/2 (σ)0 = Ce(110) , and the following possibilities for n(σ)± : (i) b ∈ 2Z : n(σ)− = span {e, e(110) , f (1) }, n(σ)+ = span {f, f (110) , e(1) }; (ii) b ∈ 2Z + 1 : n(σ)− = span {e, e(110) , e2 , f111 , f (3) }, n(σ)+ = span {f, f (110) , f2 , e111 , e(3) }; (iii) b ∈ Z : n(σ)− = Ce + Ce(110) , n(σ)+ = Cf + Cf (110) . We consider separately the following two cases: (+): 0 b < 1 ; (−): −1 b 0, t ∈ C) characterized by the following properties were introduced:

Trigonometric Degeneration and Orbifold WZW

211

• Additive quasi-periodicity: wab (τ ; t + 1) = εa wab (τ ; t),

wab (τ ; t + τ ) = εb wab (τ ; t);

• As a function of t ∈ C, wab (τ ; t) has a simple pole with residue 1 at Z + Zτ . add Let us denote this function by wab (τ ; t). (The superscript “add” stands for “admul ditive”.) Let us rewrite it to a multiplicatively quasi-periodic function wab (q; u) as follows: mul (q; u) := wab

2πiua uN − 1

(q N −a εb u−N ; q N )∞ (q a ε−b uN ; q N )∞ , (28) (q N −a εb ; q N )∞ (q N u−n ; q N )∞ (q a ε−b ; q N )∞ (q N uN ; q N )∞ n where (x; q)∞ = ∞ n=0 (1 − xq ) is the standard infinite product symbol. The funcmul add mul add (q; u) is related to wab by wab (q; u) = wab (N log u/2πi, N log q/2πi) tion wab add when q = 0, that is, we replaced the arguments of wab by e2πiz = uN , e2πiτ = q N mul and used the product formula for the theta function. When q = 0, wab becomes a rational function of u: 7 (a = 0), 2πiua (uN − 1)−1 , mul (29) wab (0; u) = 2πi(1 − εb )−1 (uN − εb )(uN − 1)−1 , (a = 0). ×

mul add is that it inherits the quasi-periodicity of wab : The important property of wab mul mul (q; εu) = εa wab (q; u), wab

mul mul wab (q; qu) = εb wab (q; u).

(30)

˜ in terms of wmul : We define the function wab,i (P ) (i = 1, . . . , L) on X ab for (q, xk , yk )k ∈ Uk (cf. (14)), q = 0,

mul wab,i (((q, xk , yk )k , s)) := wab (q; q k−k xk /xk ).

(31)

Here s = (q; Q1 , . . . , QL ) ∈ S and we fix an index k to express Qi as a point (q, xk , yk )k in Uk . (The function wab,i is determined up to this choice.) This function is extended to the points with q = 0. (For example, wab,i ((0, x0 , y0 )0 ) is the rational function (29) of x = x0 /x0 when Qi is represented as a point (0; x0 , y0 )0 in U0 .) The main properties of this function are 2 • Quasi-periodicity with respect to CN -action: wab,i ((1, 0) · P ) = εa wab,i (P ),

wab,i ((0, 1) · P ) = εb wab,i (P ),

(32)

2 action. • All poles are simple and located at Qi modulo CN It is easy to see that any section of gX,out is a linear combination of Jab ⊗ wab,i (P )’s and their derivatives along the fibre.

ˆD Lemma 3.1. g gD S = gX,out ⊕ ˆ S,+ . ˆD Proof. The singular part of an element of g S can be expressed by a linear combination of Jab ⊗wab,i (P ) and derivatives in a unique way. Subtracting such combination which belongs to gX,out , we end up with a regular element of ˆgD S,+ .

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Takashi Takebe

4. Sheaves of conformal coinvariants and conformal blocks In this section we introduce the sheaf CC of conformal coinvariants and the sheaf CB of conformal blocks and show their basic properties. Definitions of CC and CB are literally the same as those for the non-singular case, Definition 3.3 of [KT]. ˆ Definition 4.1. For any ˆ gD S -module M of level k (i.e., k acts as k · id), we define the sheaf CC(M) of conformal coinvariants and the sheaf CB(M) of conformal blocks by CC(M) := M/gX,outM,

(33)

CB(M) := Hom OS (CC(M), OS ).

(34)

ˆD We can regard CC(·) as a covariant right exact functor from the category of g Smodules to that of OS -modules and similarly CB(·) as a contravariant left exact functor. The goal of this paper is to prove that CC and CB are locally free. Since CB is the dual of CC, we mainly discuss about CC and briefly mention on CB when it is necessary. We assume that the ˆ gD S -module M are of the following type: M = OS ⊗

L i=1

Mi =

L (OS ⊗ Mi ),

(35)

i=1 ˆ

g-Weyl module M (Vi ) := IndgQi ⊕Ck Vi of level k where each Mi is a quotient of a ˆ Qi

ˆ+ g

for a finite dimensional irreducible g-module Vi . (See §6 of [T] or §2.4 of [KL].) To endow M in (35) with the ˆ gD S -module structure, we need to fix the coordinate of E and the trivialisation of gtw X , which is irrelevant to the statements of theorems below. In the concrete computations, we use the coordinates and the trivialisation obtained naturally from the construction in §3. Proposition 4.2. CC(M) is a coherent OS -sheaf. When all Mi ’s are Weyl modules, CC(M) is (and hence CB(M) is) locally free. Proof. Lemma 3.1 makes it possible to apply the same argument as the proof for the non-singular case, Corollary 3.5 of [KT]. In fact, if each Mi in (35) is a Weyl module M (Vi ), M is expressed as ∼

gD (V ⊗ OS ) ← US (gX,out ) ⊗OS (V ⊗ OS ), M = US (ˆ gD S ) ⊗US (ˆ S,+ )

(36)

by the Poincaré-Birkhoff-Witt theorem (V = ⊗L i=1 Vi ). Here US (·) denotes the universal OS -enveloping algebra. Modding out by gX,out M, we have ∼

CC(M) = M/gX,outM ← V ⊗ OS , which means that CC(M) is locally free, in particular, coherent.

(37)


213

If Mi ’s are quotients of Weyl modules, M is a quotient of M (Vi ) ⊗ OS . Hence by the right exactness of the functor CC, CC(M) is a quotient of a coherent sheaf, and therefore coherent. In §6 we prove the locally freeness of CC(M) for integrable Mi ’s, examining the behavior of CC(M) at the boundary of the moduli space (S0 = {q = 0} ⊂ S) carefully.

5. Sheaf version of trigonometric and orbifold WZW model Everything in previous two sections can be restricted on S0 , namely on the configuration space of points on a singular rational curve with one ordinary double point. (As we have mentioned, the restriction of the functions wab,i needs special care.) Hence we can define the corresponding sheaves CC(M) and CB(M) which we denote by CC trig (M) and CBtrig (M). The subscript “trig” is put here because, as we shall see below, there are connections on them expressed in terms of the trigonometric r-matrix. In the proof of Theorem 6.1 we shall use the sheaf of conformal coinvariants of the orbifold WZW model, CC orb (M0 ⊗M⊗M∞ ) on S0 , where M∗ = M∗ ⊗OS0 for a ˆ g(∗) -module M∗ (∗ = 0, ∞). We shall recall the definition of the twisted affine algebras ˆ g(0) and ˆ g(∞) and the details of CCorb soon later. Here we only say that CC orb is defined exactly in the same way as CC trig (M) if we replace the degenerate elliptic curve (the fibre of E at q = 0) by the orbifold P1 /CN . Note that S0 can be regarded as the configuration space of points on P1 /CN . (See §3 of [T].) g(∗) -Verma module. Then Proposition 4.2 Let M be as in (35) and M∗ be a ˆ holds for CC trig , CBtrig , CC orb and CBorb as well. In fact, we can prove locally freeness under this assumption. Proposition 5.1. (i) The sheaf CC trig (M) and the sheaf CBtrig (M) are locally free OS0 -sheaves. (ii) The sheaf CC orb (M0 ⊗ M ⊗ M∞ ) and the sheaf CBorb (M0 ⊗ M ⊗ M∞ ) are locally free OS0 -sheaves. Proof. (i) The proof of the locally freeness for the non-singular case, Corollary 5.3 of [KT] is true also in this case: CC trig (M) is coherent as shown at the end of §4 and there is a connection and D-module structure on it, which implies that it is locally free OS0 -sheaf. The only difference is that we do not change the curve itself (the modulus q is fixed to 0) in the present case, and hence there is nothing corresponding to the connection in the direction of ∂/∂τ in [KT]. The connection in the direction of ∂/∂zi (zi is the coordinate of Qi ) is ∇i = ∂/∂zi − ρi (T [−1]) (cf. (5.14) of [KT]) as is well known, where ρi is the representation of the Virasoro algebra on Mi constructed via the Sugawara construction and T [−1] is one of the Virasoro generator, usually denoted by L−1 . (ii) We might proceed as the proof of (i) from the beginning but the short cut is to use the result of (i). The coherence of CC orb (M0 ⊗ M ⊗ M∞ ) having

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Takashi Takebe

been proved, we have only to check that the above connection operators ∇i (i = 1, . . . , L) on CC trig (M) also define the flat connection on CC orb . What we need to check is orb • [∇i , gorb out ] ⊂ gout , • [∇i , ∇j ] = 0 on CC orb ,

which is proved in the same way as in the case of the ordinary WZW model, e.g., Lemma 4 of [FFR]. The connection on CC trig (M) mentioned in the proof of (i) is obtained by the degeneration q → 0 of the elliptic Knizhnik-Zamolodchikov connection in [E] and [KT]. They are expressed as the first order differential operators on V ⊗ OS0 in terms of the trigonometric r-matrix. In fact, by tracing the argument which leads to the explicit form (Theorem 5.9 in [KT]) of the connection, we have only to replace the functions wab (zj − zi ) there with wab,i (Qj ) which is expressed by the rational function of the form (29) on S0 . Hence the KZ equation for the WZW model on the degenerate elliptic curve is the trigonometric KZ equation. Lemma 5.2. (i) The fibre of CC trig (M) at s ∈ S0 , CC trig (M)|s , is isomorphic to CCtrig (M ), the space of conformal coinvariants of the trigonometric WZW model for the geometric data corresponding to s. (ii) The fibre of CC orb (M) at s, CC orb (M)|s , is isomorphic to CCorb (M ), the space of conformal coinvariants of the orbifold WZW model for the geometric data corresponding to s. See Definition 3.2 of [T] for the definition of CCtrig and CCorb . Proof. We can modify the proof of the corresponding statement for the nonsingular case, Corollary 5.4 of [KT]. For example, for the case (i), the isomorphism trig gX,out |s ∼ = gout is a consequence of the existence of the sections wab,i and their derivatives (cf. the end of §3) which span both gX,out |s and gtrig out . The rest of the proof can be translated to the present case without change. The proof of (ii) is similar. Combining Proposition 5.1, Lemma 5.2 and Theorem 5.1 of [T], we have the following isomorphism: ∼ (0) (∞) ι : CC trig (M) → CC orb (Mλ˜ ⊗ M ⊗ Mλ˜ ), (38) λ∈wt(V )

L where V = i=1 Vi is the g-module generating M = i=1 Mi (recall Mi is a quotient of Weyl module M (Vi )), wt(V ) is the set of its weights, L

˜ := −λ ◦ (1 − Ad β −1 )−1 = λ ◦ (1 − Ad β)−1 ◦ Ad β, λ ˜ := −λ ◦ (1 − Ad β)−1 , λ

(39)


215

(∗) (∗) (∗) ˆ(∗) with the highest weight µ Mµ = Mµ ⊗ OS0 for a Verma module Mµ of g (cf. Definition 4.1 (i) of [T]). In §6 the modules Mi are assumed to be integrable highest weight modules. (cf. Chapter 10 of [K].) In this case the above result can be refined. For this purpose we recall the details of the orbifold WZW model defined in §3 of [T]. Let us denote the standard coordinate of P1 (C) by t. The cyclic group CN acts as t → εa t (a ∈ CN ) and the quotient Eorb = P1 /CN is an ordinary orbifold. The definition of the space of conformal coinvariants/blocks of the orbifold WZW model on Eorb is almost the same as that on elliptic curves, Definition 2.1, except that we also insert modules to the singular points 0 and ∞. The Lie algebra gout in (13) is replaced by gorb out which consists of g-valued meromorphic functions f (t) on P1 such that: (1) poles belong to {0, Q1 , . . . , QL , ∞}; (2) f (εt) = Ad γ(f (t)). Accordingly, the module inserted at 0 is the ˆg(0) -module and the module inserted at ∞ is the ˆ g(∞) -module, where ˆ CJa,b ⊗ ta+mN ⊕ Ck, (40) gˆ(0) = a,b=0,...,N −1 m∈Z (a,b)=(0,0)

g(∞) =

CJa,b ⊗ ta+mN ⊕ Ckˆ

(41)

a,b=0,...,N −1 m∈Z (a,b)=(0,0)

=

ˆ CJa,b ⊗ s−a+mN ⊕ Ck.

(s = t−1 )

a,b=0,...,N −1 m∈Z (a,b)=(0,0)

The cocycles which defines the central extension of g(0) and g(∞) are: ca,0 (A, B) :=

1 Rest=0 (dA|B), N

ca,∞ (A, B) :=

1 Ress=0 (dA|B), N

(42)

for A, B ∈ g(0) and A, B ∈ g(∞) respectively. (g(∗) is the loop algebra part of ˆg(∗) .) As Etingof showed (Lemma 1.1 of [E]), ˆ g(0) and ˆg(∞) are isomorphic to the ∼ (1) ordinary affine Lie algebra ˆ g of AN −1 type. Explicitly the isomorphism φ0 : ˆg(0) → ˆ g is defined by φ0 (Ei,i+1 ⊗ t) = ei ,

φ0 (Ei+1,i ⊗ t−1 ) = fi ,

φ0 (EN,1 ⊗ t) = e0 ,

φ0 (E1,N ⊗ t−1 ) = f0 ,

kˆ φ0 (Hi,i+1 ⊗ 1) = α∨ , i − N

(43)

ˆ = k, ˆ φ0 (k)

for i = 1, . . . , N − 1, where Eij is the matrix unit, Hij = Eii − Ejj , ei , fi (i = 0, . . . , N − 1) are the Chevalley generators of ˆg and α∨ i (i = 0, . . . , N − 1) are coroots of ˆ g. (cf. §6.2 and §7.4 of [K].) Since the positive powers of s (cf. (41)) kill (∞) the highest weight vector of the Verma module Mµ inserted at ∞ (cf. Definition (∞) 4.1 (i) of [T]), we identify ˆ g and ˆ g through an isomorphism which is essentially

216

Takashi Takebe

a composition of φ0 and the Chevalley involution: φ∞ (Ei,i+1 ⊗ s−1 ) = −fi , φ∞ (EN,1 ⊗ s

−1

) = −f0 ,

φ∞ (Ei+1,i ⊗ s) = −ei , φ∞ (E1,N ⊗ s) = −e0 ,

kˆ φ∞ (Hi,i+1 ⊗ 1) = −α∨ , i + N

(44)

ˆ = k, ˆ φ∞ (k)

for i = 1, . . . , N − 1. Identified through φ0 and φ∞ , the Verma modules of ˆg(0) and ˆ g(∞) are Verma modules of ˆ g in ordinary sense. In this section, ei and fi denote the Chevalley generators of ˆg identified with the elements in ˆ g(0) or ˆ g(∞) by means of φ0 or φ∞ . Proposition 5.3. Assume that all Mi (i = 1, . . . , L) are integrable highest weight (∗) modules and that Mµ∗ (∗ = 0, ∞) is a Verma modules of ˆg(∗) . (0) (∞) Then CCorb (Mµ0 ⊗ M ⊗ Mµ∞ ) is 0 unless µ0 and µ∞ are dominant integral (0) weights of ˆ g identified with ˆ g and ˆ g(∞) . If µ0 and µ∞ are dominant integral weights, (∞) CCorb (Mµ(0) ⊗ M ⊗ Mµ(∞) )∼ (45) = CCorb (L(0) µ0 ⊗ M ⊗ Lµ∞ ), 0 ∞ (∗)

(∗)

where Lµ∗ (∗ = 0, ∞) is the irreducible quotient of Mµ∗ . Remark 5.4. In physics context, this proposition is a consequence of the propagation of the null field. See §4 of [Z]. The author thanks Yasuhiko Yamada for this comment. Remark 5.5. Proposition 5.3 is in sharp contrast to the Weyl module case. See §6 of [T]. Proof. The following lemma shall be proved later. (∗)

Lemma 5.6. Let N∗ (∗ = 0, ∞) be a quotient of the Verma module Mµ∗ . Suppose vκ ∈ N0 is a singular vector of weight κ which is not a dominant integral weight. (The weight κ may possibly be the highest weight µ0 .) Then for any v ∈ M and v∞ ∈ N∞ , (46) vκ ⊗ v ⊗ v∞ ≡ 0 mod gorb out (N0 ⊗ M ⊗ N∞ ). The same is true for a singular vector vκ ∈ N∞ . The first statement of Proposition 5.3 is a consequence of Lemma 5.6. For example, assume µ0 is not a dominant integral weight. Let us show X1 [−n1 ] · · · Xl [−nl ]|µ0 ⊗ v ⊗ v∞ ≡ 0 mod gorb out ,

(47)

(∞) g(0) (Xi ∈ g, ni > 0), v ∈ M , v∞ ∈ Mµ∞ and mod gorb where Xi [−ni ] ∈ ˆ out denotes (0) (∞) mod gorb (M ⊗ M ⊗ M ). (This abbreviation shall be used throughout this µ µ 0 ∞ out −n1 n paper.) Let f1 (t) be an element of gorb such that f (t) ∼ X ⊗ t + O(t ) for 1 1 out sufficiently large n. (Such f1 exists due to the Riemann-Roch theorem. It is not


217

difficult to construct such a function concretely.1 ) Then we may replace X1 [−n1 ] by ρ0 (f1 (t)): X1 [−n1 ] · · · Xl [−nl ]|µ0 ⊗ v ⊗ v∞ =ρ0 (f1 (t))X2 [−n2 ] · · · Xl [−nl ]|µ0 ⊗ v ⊗ v∞ ≡ − X2 [−n2 ] · · · Xl [−nl ]|µ0 ⊗

L ρi (f1 (t))v ⊗ v∞ + v ⊗ ρ∞ (f1 (t))v∞ ⊗

(48) mod gorb out .

i=1

By induction on l, the problem is reduced to showing |µ0 ⊗ v ⊗ v∞ ≡ 0

mod gorb out ,

(49)

which immediately follows from Lemma 5.6. To prove the second statement of Proposition 5.3, assume that µ0 and µ∞ (∗) are dominant integral weight. Then the irreducible quotients of Mµ∗ are expressed as N −1 µ ,α∨ +1 (∗) = M / U (n− )fi ∗ i |µ∗ . (50) L(∗) µ∗ µ∗ i=0

See (10.4.6) of [K]. Therefore to prove (45), it is enough to show µ0 ,α∨ i +1

U (n− )fi

|µ0 ⊗ M ⊗ Mµ(∞) ≡0 ∞

mod gorb out ,

(51) (∗)

and a similar statement with the indices “0” and “∞” for µ∗ , Mµ∗ etc. interchanged. They are proved as above, namely by the arguments like (48) and (49), µ ,α∨ +1 because the weight of the singular vector fi ∗ i |µ∗ is not a dominant integral weight. Proof of Lemma 5.6. Since κ is not a dominant integral weights, there is an index i (0 i N − 1) such that κ(α∨ i ) is not a non-negative integer. By an easy calculation, we have eni fin vκ = cvκ ,

c = n!

n (κ(α∨ i ) − l + 1),

(52)

l=1

for any n ∈ N. Note that the constant c never vanishes. −1 n Let e(t) be an element of gorb out such that: (1) e(t) ∼ φ0 (ei ) + O(t ) for sufficiently large n; (2) ρ∞ (e(t))v∞ = 0 (i.e., e(t) has a zero of large order at t = ∞). Such an element can be constructed in the form X ⊗ F (t), where X = Ei,i+1 (i = 1, . . . , N − 1) or X = EN,1 (i = 0) and F (t) is a rational function. Hence we 1A

useful technique: for any g-valued function f (t) with poles in {0, Q1 , . . . , QL , ∞}, f (t) + Ad γ(f (ε−1 t)) + (Ad γ)2 (f (ε−2 t)) + · · · + (Ad γ)N−1 (f (ε−N+1 t)) ∈ gorb out .

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Takashi Takebe

can rewrite vκ ⊗ v ⊗ v∞ modulo gorb out as follows (cf. p.479 of [TUY]): vκ ⊗ v ⊗ v∞ = c−1 ρ0 (eni fin )vκ ⊗ v ⊗ v∞ = c−1 ρ0 (e(t)n fin )vκ ⊗ v ⊗ v∞ ≡ (−1)n c−1

(53) L

n1 +···+nL

n! ρ0 (fin )vκ ⊗ ρj (e(t))nj v ⊗ v∞ . n ! · · · n ! 1 L =n j=1

Recall that ρj (e(t)) = ρj (X ⊗ F (t)) is locally nilpotent on Mj (Corollary 1.4.6 of [TUY]). Thus the right-hand side of (53) is 0 for large n, which completes the proof of the lemma. Because of the difference of the sign in (43) and (44) and the fact that Ad β ˜ is a dominant integral weight for ˆg(0) , λ ˜ is a is a Dynkin automorphism, if λ dominant integral weight for ˆ g(∞) and vice versa. (See (39).) Corollary 5.7. Under the same assumption as Proposition 5.3 we have (0) (∞) CCtrig (M ) ∼ CCorb (Lλ˜ ⊗ M ⊗ Lλ˜ ), =

(54)

˜ λ∈wt(V ),λ:dom. int.

where “dom. int.” means “dominant integral weight”. Similarly the decomposition (38) becomes ∼ (0) (∞) ι : CC trig (M) → CC orb (Lλ˜ ⊗ M ⊗ Lλ˜ ), (55) ˜ λ∈wt(V ),λ:dom. int.

(∗)

(∗)

where Lµ = Lµ ⊗ OS0 (∗ = 0, ∞).

6. Locally freeness The main theorem of this paper is proved in this section. We show that CC is locally free at the discriminant locus, S0 = {q = 0} ⊂ S, provided that all modules inserted are integrable highest weight modules. Thus, combining the result in [KT], we have locally freeness of the sheaf of conformal coinvariants and consequently locally freeness of its dual, the sheaf of conformal blocks. Corresponding statement for the Weyl modules has been proved in Proposition 4.2. The arguments in this section is parallel to those in §7.3 of [TK]. We assume the condition (35) for M. Moreover we assume that all Mi ’s are integrable highest weight modules. In particular, the level k is a non-negative integer. Theorem 6.1. The sheaf CC(M) and hence the sheaf CB(M) are locally free OS sheaves. The rest of the paper is devoted to the proof of this theorem. The main strategy of the proof is the same as that of [TK]. See also [SU], [NT] and [U]:


219

• Since CC(M) is coherent, it is sufficient to prove that each stalk CC(M)s (s ∈ S) is a free OS,s -module. • We proved the locally freeness of CC(M) on the non-singular part S S0 in [KT]. Thus we have only to prove the case s ∈ S0 . • When s ∈ S0 , we prove that the stalk of the completion of CC(M) at s is isomorphic to CC trig (M)s [[q]]. • CC(M)s is a free OS,s -module because of Proposition 5.1 and the faithfully flatness of the completion functor. We define completion of the sheaf CC(M) along the divisor S0 of S by taking ˆS/S be the completion completion of each ingredient of the definition (33). Let O 0 of OS along S0 : ˆS/S := proj lim OS /mnS . O (56) 0 0 n→∞

(mS0 is the defining ideal of S0 .) As an OS0 -module, it is isomorphic to the ring ˆS/S ∼ OS0 [[q]]. The completion of M = M ⊗ OS is of formal power series: O 0 = obviously ˆS/S ∼ 8 := M ⊗OS O (57) M MS0 [[q]], 0 = 8 naturally as where MS0 := M ⊗ OS0 . The OS -Lie algebra gX,out acts on M follows: a germ f (P ) of gX,out at (q = 0; Q1 , . . . , QL ) ∈ S0 is expanded at (q = ˜ in terms of the coordinates (q; xk , yk )k as 0; (0, 0, 0)k ; Q1 , . . . , QL ) ∈ X ∞

f (P ) =

=

=

m,n=0 ∞

n fk,m,n (s)xm k yk

fk,m,n (s)xm−n qn = k

m,n=0 ∞

∞

n=0

m=0

fk,n,x (s, xk )q n =

∞

fk,m,n (s)ykn−m q m

(58)

m,n=0

fk,m,y (s, yk )q m ,

where fk,m,n (s) ∈ g ⊗ OS0 (s ∈ S0 ) and fk,n,x (s, xk ) =

∞

fk,m,n (s)xm−n , k

fk,m,y (s, yk ) =

m=0

∞

fk,m,n (s)ykn−m . (59)

n=0

Note that the periodicity condition (27) implies Ad γ(fk,m,n (s)) = εm−n fk,m,n (s),

Ad β(fk,m,n (s)) = fk+1,m,n (s),

(60)

and hence fk,n,x (s, xk ) and fk,m,y (s, yk ) are meromorphic (rational) function on P1 with quasi-periodicity fk,n,x (s, εxk ) = Ad γ(fk,n,x (s, xk )), fk,m,y (s, ε−1 yk ) = Ad γ(fk,m,y (s, yk )), fk+1,n,x (s, xk ) = Ad β(fk,n,x (s, xk )), fk+1,m,y (s, yk ) = Ad β(fk,m,y (s, yk )),

(61)

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Takashi Takebe

the poles of which are in the divisor ˜ 0 := ∗D

L

∗[εj Qi ] + ∗[0] + ∗[∞].

(62)

i=1 j∈Z/N Z

Therefore {fk,n,x (s, xk )}k∈Z/N Z and {fk,n,y (s, yk )}k∈Z/N Z , namely the nth coefficients of the expansion ∞ f (P ) = fn (P )q n , (63) n=0

define a section fn (P ) of gtw X |π −1

X/S

(S0 )

with poles at Q1 , . . . , QL , 0, ∞. See (23) and

8 (v ∈ M, g ∈ O ˆS/S ) is defined by (27). The action of f (P ) on v ⊗ g ∈ M 0 f (P ) · (v ⊗ g) :=

L ∞

ρi (fn (P ))v ⊗ q n g.

(64)

i=1 n=0

Here ρi denotes the usual action of the Laurent expansion of fn (P ) at Qi on Mi . 8 with respect to this action is the completion of The space of coinvariants of M CC(M): 8 8 X,out (M). 8 CC(M) := M/g (65) Lemma 6.2.

∼ 8 ˆS/S . CC(M) = CC(M) ⊗OS O 0

(66)

Proof. By definition, we have an exact sequence gX,out ⊗ M → M → CC(M) → 0.

(67)

ˆS/S we obtain an exact sequence Tensoring O 0 8 → CC(M) ⊗ O ˆS/S → 0. (gX,out ⊗ M)[[q]] → M 0

(68)

8 is It is sufficient to show that the image of the map (gX,out ⊗ M)[[q]] → M 8 defined by the action (64). This is almost trivial since f (P ) · v for gX,out (M) f (P ) ∈ gX,out and v ∈ M is expressed as f (P ) · v =

L ∞

q n ρi (fn (P ))v,

i=1 n=0

because of the expansion (63).

The next step is to make a completion of the isomorphism (55). For this purpose we need a lemma on Verma modules of ˆg(0) and ˆg(∞) . Note that the (∗) g(∗) (∗ = 0, ∞) is graded by the degree: Verma module Mµ of ˆ Mµ(∗) = Mµ(∗) (d), Mµ(∗) (d) := CX1 [−n1 ] · · · Xl [−nl ]|µ, (69) d≥0

n1 +···+nl =d (∗)

where |µ’s are the highest weight vectors of Mµ ni ∈ N).

ˆ(∗) (Xi ∈ g, and Xi [−ni ] ∈ g


221

Lemma 6.3. (0) (i) For any µ ∈ h∗ there exists a pairing between the Verma modules Mµ◦Ad β (∞)

and M−µ : (0)

(∞)

× M−µ : (u, v) → u, v ∈ C, M 9 µ◦Ad β : which satisfies |µ ◦ Ad β, | − µ = 1 and

(70)

X[n]u, v + u, Ad β(X)[−n]v = 0, (0)

(71)

(∞)

for any u ∈ Mµ◦Ad β , v ∈ M−µ , X ∈ g and n ∈ Z. ; < (0) (∞) (ii) Mµ◦Ad β (n), M−µ (n ) = 0 if n = n . (0)

(∞)

(iii) The radical R(0) = {u ∈ Mµ◦Ad β | u, v = 0 for all v ∈ M−µ } is the largest (0)

(∞)

proper submodule of Mµ◦Ad β . Similarly the radical in M−µ is the largest proper submodule. Hence the pairing descends to a non-degenerate pairing (0) (∞) between the irreducible quotients Lµ◦Ad β and L−µ . Proof. Let ν be the anti-isomorphism ν : Uˆ g(0) & X[n] = X ⊗ tn → −X ⊗ tn = −X[−n] ∈ U ˆg(∞) , (0)

ˆ = k. ˆ (72) ν(k)

(∞)

This induces a linear isomorphism νβ : Mµ◦Ad β → HomC (M−µ , C) defined by νβ (x|µ ◦ Ad β) = −µ|ν(Ad β(x)),

(73)

(∞) ˆ(∞) -module HomC (M−µ where −µ| is the generating vector of the right g , C), normalised by −µ | −µ = 1. We define the pairing , by

v, v := νβ (v)v .

(74)

Straightforward computation shows that for x ∈ U ˆg(0) we have xv, v = v, ν(Ad β(x))v ,

(75)

which means (71) for x = X[n]. (ii) follows from the construction. (iii) is proved in the same way as Proposition 3.26 of [Wa]. Let {eλ,d,i } be a basis of respect to , .

(0) Lλ˜ (d)

and {eiλ,d } be its dual basis of

(∞) Lλ˜ (d)

Proposition 6.4. There exists an isomorphism 8 CC orb (Lλ˜ ⊗ M ⊗ Lλ˜ )[[q]], ˆι : CC(M) →

with

(76)

˜ λ∈wt(V ),λ:dom. int.

ˆS/S -modules defined by of O 0 ˆι([v]) :=

=∞ λ

for v ∈ M.

d=0

i

> eλ,d,i ⊗ v ⊗

eiλ,d

qd ,

(77)

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Takashi Takebe

Proof. First we prove the well-definedness of (77), for which it is enough to show 8 the well-definedness of its component ιλ : CC(M) → CC orb (Mλ˜ ⊗ M ⊗ Mλ˜ )[[q]], namely, ∞ eλ,d,i ⊗ f (P ) · v ⊗ eiλ,d q d ∈ gorb (78) ˜ ⊗ M ⊗ Mλ ˜ ) out (Mλ d=0

i

for f (P ) ∈ gX,out and v ∈ M. Since ∞

i f (P ) · eλ,d,i ⊗ v ⊗ eλ,d q d ∈ gorb ˜ ⊗ M ⊗ Mλ ˜ ), out (Mλ d=0

(79)

i

we have only to show that the left-hand side of (78) is equal to the left-hand side of (79), which is equivalent to an equation in Lλ˜ ⊗ Lλ˜ : ∞ ρ0 (f (P )) · eλ,d,i ⊗ eiλ,d + eλ,d,i ⊗ ρ∞ (f (P )) · eiλ,d q d = 0. d=0

(80)

i

Recall that the germ of f (P ) at 0 and the germ at ∞ is related by f (P )∞ = Ad β(f (P )0 ). See (27) of this paper or (18) of [T]. Hence using the expansion (58) and the invariance (71) of the pairing, we can show (80) in the same way as the proof of Claim 3 in the proof of Theorem 6.2.1 in [TUY]. Thus ˆι is well defined. Obviously the q = 0 part of ˆι is the isomorphism ι, (55). Therefore by termwise approximation (in analytic language) or, in other words, by Nakayama’s lemma (in algebraic language), ˆι is shown to be an isomorphism. With these preparations, the proof of the locally freeness of CC(M) goes as follows. As is mentioned after the statement of Theorem 6.1, it is enough to prove 8 that the stalk CC(M)s at s ∈ S0 is a free OS,s -module. Since CC(M) s is isomorphic ˆ to CC(M)s ⊗OS,s ⊗OS/S0 ,s (Lemma 6.2) and to ˜ ⊗ M ⊗ Lλ ˜ )s [[q]] λ CC orb (Lλ (Proposition 6.4), we have an isomorphism ˆS/S ,s ∼ CC orb (Lλ˜ ⊗ M ⊗ Lλ˜ )s [[q]]. (81) CC(M)s ⊗OS,s O = 0 λ

ˆS/S ,s -module (Proposition 5.1 (ii) and The right-hand side of (81) being a free O 0 ˆS/S ,s over OS,s implies that CC(M)s is a Proposition 5.3), faithfully flatness of O 0 free OS,s -module. Thus Theorem 6.1 is proved.

7. Concluding comments We have proved locally freeness of CC(M) in two cases; Weyl module case (Proposition 4.2) and integrable highest weight module case (Theorem 6.1). A few comments are in order: • In the Weyl module case, CC(M) ∼ = V ⊗ OS as shown in the proof of Proposition 4.2 and the rank of CC(M) is dim V .


223

• In the integrable highest module case, the rank is computed by further degenerating the orbifold. Degeneration of the type Qi → Qj should be considered in the same way as in [TUY] or [NT]. The final results of the degeneration is a combination of the three-punctured orbifold P1 /CN . In principle a Verlinde-type formula would be obtained in this way. • In [KT] we have shown that CC(M) has a flat connection. It has a regular singularity along S0 = {q = 0}, which is easily deduced from the explicit form of the connection, Theorem 5.9 of [KT]. Hence there is a one-to-one correspondence between flat sections around S0 and its restriction to S0 or, in other words, the “initial value” at S0 because of the locally flatness. Acknowledgments The author expresses his gratitude to Akihiro Tsuchiya who explained details of [TUY] and [NT], Toshiro Kuwabara who showed the manuscript of [TK] (the best guide to [TUY] for P1 case) before publishing, Michio Jimbo, Gen Kuroki, Tetsuji Miwa, Hiroyuki Ochiai, Kiyoshi Ohba, Nobuyoshi Takahashi, Tomohide Terasoma and Yasuhiko Yamada for discussion and comments. The atmosphere and environment of Institute for Theoretical and Experimental Physics (Moscow, Russia) and the conference “Infinite-Dimensional Algebras and Integrable Systems” (Faro, Portugal) were very important. The author thanks their hospitality.

References [BD]

A.A. Belavin, V.G. Drinfeld, Solutions of the classical Yang-Baxter equations for simple Lie algebras. Funkts. Anal. i ego Prilozh. 16-3, 1–29 (1982) (in Russian); Funct. Anal. Appl. 16, 159–180 (1982) (English transl.)

[E]

P.I. Etingof, Representations of affine Lie algebras, elliptic r-matrix systems, and special functions. Comm. Math. Phys. 159, 471–502 (1994).

[FFR] B. Feigin, E. Frenkel, N. Reshetikhin, Gaudin model, Bethe Ansatz and critical level. Commun. Math. Phys. 166, 27–62 (1994) [K]

V.G. Kac, Infinite-dimensional Lie algebras, 3rd Edition, Cambridge University Press 1990.

[KL]

D. Kazhdan, G. Lusztig, Tensor structures arising from affine Lie algebras. I, II, J. Amer. Math. Soc. 6, 905–948, 949–1011 (1993).

[KT]

G. Kuroki, T. Takebe, Twisted Wess-Zumino-Witten models on elliptic curves. Comm. Math. Phys. 190, 1–56 (1997).

[NT]

K. Nagatomo, A. Tsuchiya, Conformal field theories associated to regular chiral vertex operator algebras I: theories over the projective line, math.QA/0206223.

[SU]

Y. Shimizu, K. Ueno, Moduli theory III, (Iwanami, Tokyo, 1999) Gendai Suugaku no Tenkai series (in Japanese); Advances in moduli theory, Translations of Mathematical Monographs, 206, Iwanami Series in Modern Mathematics, American Mathematical Society, Providence, U.S.A. (2002) (English translation)

224 [T]

Takashi Takebe

T. Takebe, Trigonometric Degeneration and Orbifold Wess-Zumino-Witten Model. I In the Proceedings of the 6th International workshop on Conformal and Integrable models, Chernogolovka, Sep. 2002, International Journal of Modern Physics, A, 19, Supplement, 418–435 (2004) [TK] A. Tsuchiya, T. Kuwabara, Introduction to Conformal Field Theory, to appear as MSJ Suugaku Memoir of Mathematical Society of Japan. [TUY] A. Tsuchiya, K. Ueno, Y. Yamada, 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). [U] K. Ueno, 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 Math. Soc. Lecture Note Ser. 208, (Cambridge Univ. Press, Cambridge, 1995) pp. 283–345, [Wa] M. Wakimoto, Infinite-dimensional Lie algebras, (Iwanami, Tokyo, 1999) Gendai Suugaku no Tenkai series (in Japanese); Translations of Mathematical Monographs, 195, Iwanami Series in Modern Mathematics, American Mathematical Society, Providence, U.S.A. (2001) (English translation by K. Iohara) [Wo] S. Wolpert, On the homology of the moduli space of stable curves. Ann. of Math. 118, 491–523 (1983). [Z] A.B. Zamolodchikov, Exact solutions of conformal field theory in two dimensions and critical phenomena. Rev. Math. Phys. 1 197–234 (1989). (Translated from the Russian by Y. Kanie.) Takashi Takebe Department of Mathematics Ochanomizu University Otsuka 2-1-1, Bunkyo-ku Tokyo, 112-8610, Japan e-mail: [email protected]


Weil-Petersson Geometry of the Universal Teichm¨ uller Space Leon A. Takhtajan and Lee-Peng Teo Mathematics Subject Classification (2000). 32F60 (Primary) 32G15, 46E20, 58B20 (Secondary). Keywords. Universal Teichm¨ uller space, Bers embedding, Hilbert manifold, Velling-Kirillov metric, Weil-Petersson metric, Riemann curvature tensor.

1. Introduction The universal Teichm¨ uller space T (1) is the simplest Teichm¨ uller space that bridges spaces of univalent functions and general Teichm¨ uller spaces. It was introduced by Bers [Ber65, Ber72, Ber73] and it is an infinite-dimensional complex Banach manifold. The universal Teichm¨ uller space T (1) contains Teichm¨ uller spaces of Riemann surfaces as complex submanifolds. The universal Teichm¨ uller space T (1) plays an important role in one of the approaches to non-perturbative bosonic closed string field theory based on K¨ ahler geometry. Namely, in the “old approach” to string field theory as the K¨ ahler geometry of the loop space [BR87a, BR87b], the loop space L(Rd ) is the configuration space for the closed strings, L(Rd ) = Rd × Ω(Rd ). The space Ω(Rd ) of based loops has a natural structure of an infinite-dimensional K¨ ahler manifold. The space of all complex structures of Ω(Rd ) is M = S 1 \ Diff + (S 1 ). The space M parameterizes vacuum states for Faddeev-Popov ghosts in the string field theory. The “flag manifolds” M and N = M¨ ob(S 1 )\ Diff + (S 1 ) Talk given by the first author at the workshop “Infinite-Dimensional Algebras and Quantum Integrable Systems” in Faro, Portugal, July 21–25, 2003. Detailed exposition and proofs can be found in [TT03].

226

L.A. Takhtajan and Lee-Peng Teo

are infinite-dimensional complex Fréchet manifolds carrying a natural K¨ ahler metrics [BR87a, BR87b, Kir87, KY87]. These manifolds also have an interpretation as coadjoint orbits of the Bott-Virasoro group, and the corresponding K¨ ahler forms coincide with Kirillov-Kostant symplectic forms [Kir87, KY87]. Ricci tensor for M is related to the problem of constructing reparametrization-invariant vacuum for ghosts. The natural inclusion N → T (1) is holomorphic (N is a leaf of a holomorphic foliation of T (1)), and the Kirillov-Kostant symplectic form at the origin of N is a pull-back of a certain symplectic form on the subspace of the tangent space to T (1) at the origin [NV90] (an avatar of the Weil-Petersson structure on T (1)).

2. Basic facts 2.1. Definitions Let D = {z ∈ C : |z| < 1}, D∗ = {z ∈ C : |z| > 1}. The complex Banach spaces L∞ (D∗ ) and L∞ (D) are the spaces of bounded Beltrami differentials on D∗ and D respectively. Let L∞ (D∗ )1 be the unit ball in L∞ (D∗ ). Two classical models of Bers’ universal Teichm¨ uller space T (1) are the following. Model A. Extend every µ ∈ L∞ (D∗ )1 to D by the reflection

2 1 z µ(z) = µ , z ∈ D, z¯ z¯2 and consider the unique quasiconformal mapping wµ : C → C, which fixes −1, −i and 1, and satisfies the Beltrami equation ∂wµ ∂wµ =µ . ∂ z¯ ∂z The mapping wµ satisfies 1 = wµ wµ (z)

1 z¯

and fixes the domains D, D∗ , and the unit circle S 1 . For µ, ν ∈ L∞ (D∗ )1 set µ ∼ ν if wµ |S 1 = wν |S 1 . The universal Teichm¨ uller space T (1) is defined as the set of equivalence classes of the mappings wµ , T (1) = L∞ (D∗ )1 / ∼ .

Weil-Petersson Geometry

227

Model B. Extend every µ ∈ L∞ (D∗ )1 to be zero outside D∗ and consider the unique solution wµ of the Beltrami equation ∂wµ ∂wµ =µ , ∂ z¯ ∂z satisfying f (0) = 0, f (0) = 1 and f (0) = 0, where f = wµ |D is holomorphic on D. For µ, ν ∈ L∞ (D∗ )1 set µ ∼ ν if wµ |D = wν |D . The universal Teichm¨ uller space is defined as the set of equivalence classes of the mappings wµ , T (1) = L∞ (D∗ )1 / ∼ . Since wµ |S 1 = wν |S 1 if and only if wµ |D = wν |D , the two definitions of the universal Teichm¨ uller space are equivalent. The set T (1) is a topological space with the quotient topology induced from L∞ (D∗ )1 . 2.2. Properties of T (1) 1. The universal Teichm¨ uller space T (1) has a unique structure of a complex Banach manifold such that the projection map Φ : L∞ (D∗ )1 → T (1) is a holomorphic submersion. 2. The holomorphic tangent space T0 T (1) at the origin is identified with the Banach space Ω−1,1 (D∗ ) of harmonic Beltrami differentials, Ω−1,1 (D∗ ) = {µ ∈ L∞ (D∗ ) : µ(z) = (1 − |z|2 )2 φ(z), φ ∈ A∞ (D∗ )}, where A∞ (D∗ ) = {φ holomorphic on D∗ : φ ∞ = sup (1 − |z|2 )2 φ(z) < ∞}. z∈D∗

3. The universal Teichm¨ uller space T (1) is a group (not a topological group!) under the composition of the quasiconformal mappings. The group law on L∞ (D∗ )1 λ = ν ∗ µ−1 is defined through wλ = wν ◦ wµ−1 and projects to T (1). Explicitly,

ν − µ (wµ )z λ= ◦ wµ−1 . 1−µ ¯ ν (w µ )z¯ For every µ ∈ L∞ (D∗ )1 the right translations R[µ] : T (1) −→ T (1),

[λ] −→ [λ ∗ µ],

where [λ] = Φ(λ) ∈ T (1), are biholomorphic automorphisms of T (1). The left translations, in general, are not even continuous mappings.

228


4. The group T (1) is isomorphic to the subgroup of the group Homeoqs (S 1 ) of quasisymmetric homeomorphisms of S 1 fixing −1, −i and 1. By definition, γ ∈ Homeoqs (S 1 ) if it is orientation preserving and satisfies γ ei(θ+t) − γ eiθ 1 ≤ M ≤ γ (eiθ ) − γ ei(θ−t) M for all θ and all |t| ≤ π/2 with some constant M > 0. Remark 1. The closure of N in T (1) is the subgroup of symmetric homeomorphisms in M¨ ob(S 1 )\Homeoqs (S 1 ) satisfying the above inequality with M replaced by 1 + o(t) as t → 0. 2.3. Bers embedding and the complex structure of T (1) 2 2 Let A∞ (D) = φ holomorphic on D : φ ∞ = sup (1 − |z| ) φ(z) < ∞ . z∈D

and let S(f ) be the Schwarzian derivative, fzzz 3 S(f ) = − fz 2

fzz fz

2 .

For every µ ∈ L∞ (D∗ )1 the holomorphic function S(wµ )|D ∈ A∞ (D) and, by Kraus-Nehari inequality, lies in the ball of radius 6. The Bers embedding β : T (1) → A∞ (D) is defined by β([µ]) = S(wµ |D ), and is a holomorphic map of complex Banach manifolds. Define the mapping Λ : A∞ (D) → Ω−1,1 (D∗ ) by

1 1 1 2 2 Λ(φ)(z) = − (1 − |z| ) φ . 2 z¯ z¯4 By Ahlfors-Weill theorem, the mapping Λ is inverse to the Bers embedding β over the ball of radius 2 in A∞ (D). The complex structure of T (1) is explicitly described as follows. For every µ ∈ L∞ (D∗ )1 let Uµ ⊂ T (1) be the image of the ball of radius 2 in A∞ (D) under −1 the map h−1 µ = R[µ] ◦ Λ. The inverse map hµ = β ◦ R[µ] : Uµ → A∞ (D) and the ? ? maps hµν = hµ ◦ h−1 hν (Uν ) → hµ (Uµ ) hν (Uν ) are biholomorphic (as ν : hµ (Uµ ) @ functions in the Banach space A∞ (D)). The open covering T (1) = µ∈L∞ (D∗ )1 Uµ with coordinate maps hµ and transition maps hµν defines a complex-analytic atlas on T (1) modelled on the Banach space A∞ (D). The canonical projection Φ : L∞ (D∗ )1 → T (1) is a holomorphic submersion and the Bers embedding β : T (1) → A∞ (D) is a biholomorphic map with respect to this complex structure. Complex coordinates on T (1) defined by the coordinate charts (Uµ , hµ ) are called Bers coordinates.


229

2.4. The universal Teichm¨ uller curve The universal Teichm¨ uller curve T (1) is a complex fiber space over T (1) with a holomorphic projection map π : T (1) → T (1). ˆ = C ∪ {∞} with the The fiber over each point [µ] is the quasi-disk wµ (D∗ ) ⊂ C ˆ complex structure induced from C and T (1) = {([µ], z) : [µ] ∈ T (1), z ∈ wµ (D∗ )} . The fibration π : T (1) −→ T (1) has a natural holomorphic section given by T (1) & [µ] → ([µ], ∞) ∈ T (1) which defines the embedding T (1) → T (1). The universal Teichm¨ uller curve is a complex Banach manifold modelled on A∞ (D) ⊕ C. 2.5. Velling-Kirillov metric on T (1) The Velling-Kirillov metric at the origin of T (1) is defined by υ

2 V K=

∞

n|cn |2 ,

where

υ=

n=1

n=0

cn einθ

∂ ∈ T0 S 1 \Homeoqs (S 1 ) ∂θ

– the tangent space at the origin of a real Banach manifold S 1 \Homeoqs (S 1 ). (The series in the definition of υ 2V K is always convergent.) At other points the Velling-Kirillov metric is defined by the right translations. The Velling-Kirillov metric on T (1) is K¨ ahler with symplectic form ωV K . Remark 2. For the space S 1 \ Diff + (S 1 ) this metric was introduced by Kirillov [Kir87] and has been studied by Kirillov-Yuriev [KY87]. Velling [Vel] introduced a Hermitian metric for T (1) using geometric theory of functions, and in [Teo02] the second author extended Kirillov’s metric to T (1) and proved that it coincides with the metric introduced by Velling. The Velling-Kirillov metric is the unique K¨ ahler metric on T (1) invariant under the right translations [Kir87, Teo02].

3. Weil-Petersson metric on T (1) As a Banach manifold, the universal Teichm¨ uller space does not carry a natural Hermitian metric. However, it is possible (see [TT03] for detailed construction and proofs) to introduce a new Hilbert manifold structure on T (1) such that it has a natural Hermitian metric. Namely, define the Hilbert space of harmonic Beltrami differentials on D∗ by A ¯ φ holomorphic on D∗ : H −1,1 (D∗ ) = µ = ρ−1 φ, B µ 22 = |µ|2 ρ(z)d2 z < ∞ , D∗

230


where

4 (1 − |z|2 )2 is the density of the hyperbolic metric on D∗ . The natural inclusion map H −1,1 (D∗ ) → Ω−1,1 (D∗ ) is bounded, and it can be shown that the family D, defined by T (1) & [µ] → D0 R[µ] H −1,1 (D∗ ) ⊂ T[µ] T (1), ρ(z) =

is an integrable distribution on T (1). Integral manifolds of the distribution D are Hilbert manifolds modelled on the Hilbert space H −1,1 (D∗ ). Thus the universal Teichm¨ uller space T (1) carries a new structure of a Hilbert manifold. Similarly to the Banach manifold structure, the Hilbert manifold structure can be also described by a complex-analytic atlas. Let T0 (1) be the component of origin of the Hilbert manifold T (1), M¨ ob(S 1 )\ Diff + (S 1 ) ⊂ T0 (1). As a Hilbert manifold, the universal Teichm¨ uller space T (1) has a natural Hermitian metric, defined by the Hilbert space inner product on tangent spaces. Thus the Weil-Petersson metric is a right-invariant metric on T (1), defined at the origin of T (1) by µ¯ ν ρ(z)d2 z, µ, ν ∈ H −1,1 (D∗ ) = T0 T (1). gµ¯ν = µ, ν = D∗

If υ=

cn einθ

n=−1,0,1

∂ ∈ T0 M¨ ob(S 1 )\Homeoqs (S 1 ) ∂θ

– the tangent space to a real Hilbert manifold M¨ ob(S 1 )\Homeoqs (S 1 ) at the origin – then υ

2 WP =

∞

(n3 − n)|cn |2 ,

n=2

The Weil-Petersson metric on T (1) is K¨ ahler with symplectic form ωW P .

4. Riemann tensor of the Weil-Petersson metric

−1 Let G = 12 ∆0 + 12 be (the one-half of) the resolvent kernel of the LaplaceBeltrami operator of the hyperbolic metric on D∗ (acting on functions) at λ = 12 . Explicitly G(z, w) =

2u + 1 u+1 1 log − , 2π u π

where

u(z, w) =

Set

G(z, w)f (w)ρ(w)d2 w.

G(f )(z) = D∗

|z − w|2 . (1 − |z|2 )(1 − |w|2 )


231

Theorem A. (i) The Weil-Petersson metric is a K¨ ahler metric on a Hilbert manifold T (1), and the Bers coordinates are geodesic coordinates at the origin of T (1). (ii) Let µα , µβ , µγ , µδ ∈ H −1,1 (D∗ ) T0 T (1) be orthonormal tangent vectors. Then the Riemann tensor at the origin of T(1) is given by Rαβγ ¯ δ¯ = −

∂ 2 gαβ¯ = −G(µα µ ¯δ ), µβ µ ¯γ − µα µ ¯β , G(¯ µγ µδ ). ∂tγ ∂ t¯δ

ahler-Einstein with the negative definite Ricci (iii) The Hilbert manifold T0 (1) is K¨ tensor, 13 ωW P . RicW P = − 12π

5. Characteristic forms of T (1) Let V = Tv T (1) be the vertical tangent bundle of the fibration π : T (1) → T (1). µ

∗

The hyperbolic metric on w (D ) defines a Hermitian metric on V , defining the first Chern form c1 (V ) – a (1, 1)-form on T (1). Mumford-Morita-Miller characteristic forms (“κ-forms”) are (n, n)-forms on the Hilbert manifold T (1), defined by κn = (−1)n+1 π∗ c1 (V )n+1 , where π∗ : Ω∗ (T (1)) → Ω∗−2 (T (1)) is the operation of “integration over the fibers” of π : T (1) → T (1), considered as a fibration of Hilbert manifolds. Theorem B. (i) On T (1), considered as a Banach manifold, c1 (V ) = −

2 ωV K . π

(ii) On T (1), considered as a Hilbert manifold, κ1 =

1 ωW P . π2

(iii) The characteristic forms κn are right-invariant on the Hilbert manifold T (1) and for µ1 , . . . , µn , ν1 , . . . , νn ∈ H −1,1 (D∗ ) T0 T (1), κn (µ1 , . . . , µn , ν¯1 , . . . , ν¯n ) in (n + 1)! sgn(σ) G µ1 ν¯σ(1) . . . G µn ν¯σ(n) ρ(z)d2 z. = n+1 (2π) σ∈Sn

D∗

232


6. Applications The Weil-Petersson properties of the universal Teichm¨ uller space T (1) are “universal” in the sense that all curvature properties of finite-dimensional Teichm¨ uller spaces can be deduced from them. In particular, Wolpert explicit formulas [Wol86] follow from Theorems A and B by using an “averaging procedure”, based on a uniform distribution of lattice points of a cofinite Fuchsian group in the hyperbolic plane (see [TT03] for details). The Kähler potential for the Weil-Petersson metric on the universal Teichm¨ uller space T (1) – “the universal Liouville action” – is constructed in [TT04]. Acknowledgments The first author is grateful to the organizers of the workshop “Infinite-Dimensional Algebras and Quantum Integrable Systems” in Faro, Portugal, July 21-25, 2003, for their kind hospitality.

References [Ber65] L Bers, Automorphic forms and general Teichm¨ uller spaces, Proc. Conf. Complex Analysis (Minneapolis, 1964), Springer, Berlin, 1965, pp. 109–113. [Ber72] Lipman Bers, Uniformization, moduli and Kleinian groups, Bull. London. Math. Soc. 4 (1972), 257–300. , Fiber spaces over Teichm¨ uller spaces, Acta. Math. 130 (1973), 89–126. [Ber73] [BR87a] M.J. Bowick and S.G. Rajeev, The holomorphic geometry of closed bosonic string theory and Diff S 1 /S 1 , Nuclear Phys. B 293 (1987), no. 2, 348–384. [BR87b] , String theory as the K¨ ahler geometry of loop space, Phys. Rev. Lett. 58 (1987), no. 6, 535–538. [Kir87] A.A. Kirillov, K¨ ahler structure on the K-orbits of a group of diffeomorphisms of the circle, Funktsional. Anal. i Prilozhen. 21 (1987), no. 2, 42–45. [KY87] A.A. Kirillov and D.V. Yur ev, K¨ ahler geometry of the infinite-dimensional homogeneous space M = diff + (S 1 )/rot(S 1 ), Funktsional. Anal. i Prilozhen. 21 (1987), no. 4, 35–46. [NV90] Subhashis Nag and Alberto Verjovsky, diff(S 1 ) and the Teichm¨ uller spaces, Comm. Math. Phys. 130 (1990), no. 1, 123–138. [Teo02] Lee-Peng Teo, Velling-Kirillov metric on the universal Teichm¨ uller curve, J. Analyse Math. 93 (2004), 271–308. [TT03] Leon A. Takhtajan and Lee-Peng Teo, Weil-Petersson metric on the universal Teichm¨ uller space I: Curvature properties and Chern forms, Preprint arXiv: math.CV/0312172 (2003). [TT04] Leon A. Takhtajan and Lee-Peng Teo, Weil-Petersson metric on the universal Teichm¨ uller space II: K¨ ahler potential and period mapping, Preprint arXiv: math.CV/0406408. [Vel] John A. Velling, A projectively natural metric on Teichm¨ uller’s spaces, unpublished manuscript.


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[Wol86] Scott A. Wolpert, Chern forms and the Riemann tensor for the moduli space of curves, Invent. Math. 85 (1986), no. 1, 119–145. Leon A. Takhtajan Department of Mathematics SUNY at Stony Brook Stony Brook NY 11794-3651, USA e-mail: [email protected] Lee-Peng Teo Department of Applied Mathematics National Chiao Tung University 1001, Ta-Hsueh Road Hsinchu City, 30050 Taiwan, R.O.C. e-mail: [email protected]


Duality for Knizhinik-Zamolodchikov and Dynamical Equations, and Hypergeometric Integrals V. Tarasov Mathematics Subject Classification (2000). 17B37, 17B80, 33C70, 33C80, 81R10. Keywords. Knizhnik-Zamolodchikov equations, dynamical equations, (glk , gl n ) duality, hypergeometric integrals.

1. Introduction The Knizhnik-Zamolodchikov (KZ ) equations is a holonomic system of differential equations for correlation functions in conformal field theory on the sphere [KZ]. The KZ equations play an important role in representation theory of affine Lie algebras and quantum groups, see for example [EFK]. There are rational, trigonometric and elliptic versions of KZ equations, depending on what kind of coefficient functions the equations have. In this paper we will consider only the rational and trigonometric versions of the KZ equations. The rational KZ equations associated with a reductive Lie algebra g is a system of equations for a function u(z1 , . . . , zn ) of complex variables z1 , . . . , zn , which takes values in a tensor product V1 ⊗ · · ·⊗ Vn of g-modules V1 , . . . , Vn . The equations depend on a complex parameter κ , and their coefficients are expressed in terms of the symmetric tensor Ω ∈ U (g) ⊗ U (g) corresponding to a nondegenerate invariant bilinear form on g . For example, if g = sl 2 and e, f, h are its standard generators such that [e , f ] = h , then Ω = e ⊗ f + f ⊗ e + h ⊗ h/2 . The rational KZ equations are n Ω(ij) ∂u κ = u, i = 1, . . . , n , (1.1) ∂zi z − zj j=1 i j=i

where Ω(ij) ∈ End (V1 ⊗ · · · ⊗ Vn ) is the operator acting as Ω on Vi ⊗ Vj and as the identity on all other tensor factors; for instance, Ω(12) (v1 ⊗ · · · ⊗ vn ) = Ω (v1 ⊗ v2 ) ⊗ v3 ⊗ · · · ⊗ vn .

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All over the paper we will assume that κ is not a rational number. Properties of solutions of the KZ equations depend much on whether κ is rational or not. Equations (1.1) can be generalized to a holonomic system of differential equations depending on an element λ ∈ g : κ

n ∂u Ω(ij) = λ(i) u + u, ∂zi z − zj j=1 i

i = 1, . . . , n .

(1.2)

j=i

Here λ ∈ End (V1 ⊗ · · · ⊗ Vn ) acts as λ on Vi and as the identity on all other tensor factors: λ(i) (v1 ⊗ · · · ⊗ vn ) = v1 ⊗ · · · ⊗ λvi ⊗ · · · ⊗ vn . System (1.2) is also called the rational KZ equations. Further on we will assume that λ is a semisimple regular element of g . Let h ⊂ g be the Cartan subalgebra containing λ , and let eα ∈ g be a root vector corresponding to a root α ∈ h∗ . We normalize the root vectors by (eα , e−α ) = 1 , where ( , ) is the bilinear form on g corresponding to the tensor Ω . In [FMTV] system (1.2) was extended to a larger system of holonomic differential equations for a function u(z1 , . . . , zn ; λ) on C n ⊕ h . In addition to equations (1.2) the extended system includes the following equations with respect to λ : (i)

(µ, α) eα e−α u , µ ∈ h, (1.3) 2 (λ , α) α i=1 where Dµ is the directional derivative: Dµ u(λ) = ∂t u(λ + tµ) t=0 . Equations (1.3) are called the rational dynamical differential (DD) equations. A special case of equations (1.3), when n = 1 and z1 = 0 , was discovered for a completely different reason. Around 1995 studying hyperplanes arrangements De Concini and Procesi introduced in an unpublished work a connection on the set of regular elements of the Cartan subalgebra h . The equations for horizontal sections of the De Concini -Procesi connection coincide with the rational DD equations. The same connection also appeared later in [TL]. De Concini and Procesi conjectured that the monodromy of their connection is described in terms of the quantum Weyl group of type g . For g = sl n this conjecture was proved in [TL]. If all g-modules V1 , . . . , Vn are highest weight modules, solutions of the KZ equations (1.1) can be written in terms of multidimensional hypergeometric integrals [SV], [V]. The construction of hypergeometric solutions can be generalized in a straightforward way to the case of KZ equations (1.2), see [FMTV]. Moreover, it is shown in [FMTV] that the hypergeometric solutions of the KZ equations obey the DD equations (1.3) as well. Generically, hypergeometric solutions of the KZ and DD equations are complete, that is, they form a basis of solutions of those systems of differential equations. An amusing fact about the hypergeometric solutions is that though systems (1.2) and (1.3) have rather similar look, and the variables z1 , . . . , zn and λ seem to play nearly interchangeable roles, the formulae for the hypergeometric solutions of the KZ and DD equations involve z1 , . . . , zn and λ in a highly nonsymmetric way. κ Dµ u =

n

zi µ(i) u +

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237

While the variables z1 , . . . , zn determine singularities of integrands of the hypergeometric integrals and enter there in a rather complicated manner, λ appears in the integrands only in a very simple way via the exponential of a linear form. Such asymmetry suggests the following idea. Suppose that a certain holonomic system of differential equations can be viewed both as a special case of system (1.2) and as a special case of system (1.3), maybe not for the same Lie algebra g . Then one can get two types of integral formulae for solutions of that system, and solutions of one kind should be linear combinations of solutions of the other kind. Thus, this can lead to nontrivial relations between hypergeometric integrals of different dimensions. It turns out that the mentioned idea indeed can be realized in the framework of the (glk , gl n ) duality. This duality plays an important role in the representation theory and the classical invariant theory, see [Zh1], [Ho]. It was observed in [TL] that under the (glk , gl n ) duality the KZ equations (1.1) for the Lie algebra sl k correspond to the DD equations (1.3) (with n replaced by k and all z ’s being equal to zero) for the Lie algebra sl n . This fact was used in [TL] to compute the monodromy of the De Concini –Procesi connection in terms of the quantum Weyl group action. Systems (1.2) and (1.3) are counterparts of each other under the (glk , gl n ) duality in general as well, see [TV4]. Employing this claim for k = n = 2 , after all one arrives to identities for hypergeometric integrals of different dimensions [TV6]. One can expect that there are similar identities for hypergeometric integrals for an arbitrary pair k, n . There are various generalizations of the KZ equations. The function Ω/z , describing the coefficients of the KZ equations, is the simplest example of a classical r-matrix – a solution of the classical Yang-Baxter equation. Starting from any classical r-matrix with a spectral parameter one can write down a holonomic system of differential equations, see [Ch2]. The obtained system is called the KZ equations associated with the given r-matrix. For example, the standard trigonometric r-matrix is 1 Ω + ξa ⊗ ξa + eα ⊗ e−α , r(z) = z−1 2 a α>0 where { ξa } is an orthonormal basis of the Cartan subalgebra, and the second sum is taken over all positive roots α , cf. (3.1) for the Lie algebra glk . The trigonometric r-matrix satisfies the classical Yang-Baxter equation ! ! r12 (z/w), r13 (z) + r23 (w) + r13 (z), r23 (w) = 0 . The corresponding KZ equations are n ∂u (i) κ zi = λ u+ r(ij) (zi /zj ) u , ∂zi j=1 j=i

i = 1, . . . , n ,

(1.4)

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where λ is an element of the Cartan subalgebra. They are called the trigonometric KZ equations associated with the Lie algebra g . System (1.2) can be considered as a limiting case of system (1.4) by the following procedure: one replaces the variables z1 , . . . , zn by e εz1, . . . , e εzn and λ by λ/ε , and then sends ε to 0 . The difference analogue of the KZ equations – the quantized Knizhnik-Zamolodchikov (qKZ ) equations – were introduced in [FR]. Coefficients of the qKZ equations are given in terms of quantum R-matrices – solutions of the quantum Yang-Baxter equation: R12 (z − w) R13 (z) R23 (w) = R23 (w) R13 (z) R12 (z − w) . There are rational, trigonometric and elliptic versions of KZ equations, the corresponding R-matrices coming from the representation theory of Yangians, quantum affine algebra algebras and elliptic quantum groups, respectively. The rational qKZ equations associated with the Lie algebra g is a holonomic system of difference equations for a function u(z1 , . . . , zn ) with values in a tensor product V1 ⊗ · · · ⊗ Vn of modules over the Yangian Y (g) : −1 (1.5) u(z1 , . . . , zi + κ, . . . , zn ) = R1i (z1 − zi − κ) . . . Ri−1,i (zi−1 − zi − κ) × (e µ )(i) Rin (zi − zn ) . . . Ri,i+1 (zi − zi+1 ) u(z1 , . . . , zn ) , i = 1, . . . , n . Here µ is an element of the Cartan subalgebra and Rij (z) is the R-matrix for the tensor product Vi ⊗ Vj of the Yangian modules. There are also several generalizations of the rational differential dynamical equations. The difference analogue of the DD equations – the rational difference dynamical (qDD) equations – was suggested in [TV3]. The idea was to extend the trigonometric KZ equations (1.4) by equations with respect to λ similarly to the way in which system (1.3) extends the rational KZ equations (1.2), and to obtain a holonomic system of differential -difference equations for a function u(z1 , . . . , zn ; λ) on C n ⊕ h . The rational qDD equations have the form u(z1 , . . . , zn ; λ + κω) = Yω (z1 , . . . , zn ; λ) u(z1 , . . . , zn ; λ)

(1.6)

where ω is an integral weight of g , and the operators Yω are written in terms of the extremal cocycle on the Weyl group of g . The extremal cocycles and their special values, the extremal projectors, are important objects in the representation theory of Lie algebras and Lie groups, see [AST], [Zh2], [Zh3], [ST]. The ideas used in [TV3] were further developed in [EV] where a new concept of the dynamical Weyl group was introduced, and the trigonometric version of the difference dynamical equations was suggested. There is also the trigonometric version of the differential dynamical equations, which, in principle, can be obtained by degenerating the trigonometric difference dynamical equations. The explicit form of the trigonometric differential dynamical equations for the Lie algebras glk and sl k was obtained in [TV4] by extending the rational qKZ equations (1.5) by equations with respect to µ in such a way that

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the result is a holonomic system of difference -differential equations for a function u(z1 , . . . , zn ; µ) on C n ⊕ h . The (glk , gl n ) duality naturally applies to the trigonometric and difference versions of the KZ and dynamical equations. Under the duality, the trigonometric KZ equations (1.4) for the Lie algebra glk correspond to the trigonometric differential dynamical equations for the Lie algebra gl n , and vice versa. At the same time the rational qKZ equations for glk are counterparts of the rational qDD equations for gl n . To relate the trigonometric qKZ and qDD equations, one has to employ the q-analogue of the (glk , gl n ) duality: the Uq (glk ) , Uq (gl n ) duality described in [B], [TL]. Hypergeometric solutions of the trigonometric KZ equations (1.4) can be written almost in the same manner as those of the rational KZ equations (1.1), see [Ch1], [MV]. Conjecturally, the hypergeometric solutions of the trigonometric KZ equations obey the corresponding rational qDD equations. For the Lie algebra sl k this claim was proved in [MV]. On the other hand, solutions of the rational qKZ equations can be written in terms of suitable q-hypergeometric Jackson integrals [TV1], or q-hypergeometric integrals of Mellin-Barnes type [TV2]. Thus, using the (glk , gl n ) duality, one can obtain solutions of a certain system of differential -difference equations both in terms of ordinary hypergeometric integrals and q-hypergeometric integrals of Mellin-Barnes type, and establish nontrivial relations between those integrals. For k = n = 2 this has been done in [TV7]. The obtained relations are multidimensional analogues of the equality of two integral representations for the Gauss hypergeometric function 2 F1 : 1 Γ(γ) uα−1 (1 − u)γ−α−1 (1 − uz)−β du 2 F1 (α, β ; γ ; z) = Γ(α) Γ(γ − α) =

Γ(γ) 1 2πi Γ(α) Γ(β)

0 +i∞−ε

(−z)s −i∞−ε

Γ(−s) Γ(s + α) Γ(s + β) ds . Γ(s + γ)

As it was pointed out by J.Harnad, the duality between the KZ and DD equations in the rational differential case is essentially the “quantum” version of the duality for isomonodromic deformation systems [H1]. The relation of the differential KZ equations and the isomonodromic deformation systems is described in [R], [H2]. From this point of view the rational qDD equations can be considered as “quantum” analogues of the Schlesinger transformations, though the correspondence is not quite straightforward. The paper is organized as follows. After introducing basic notation we subsequently describe the differential KZ and DD equations, and the rational difference qKZ and qDD equations, for the Lie algebra glk . This is done in Sections 2 – 5. Then we consider the (glk , gl n ) duality in application to the KZ and dynamical equations. In the last two sections we describe the hypergeometric solutions of the equations, and use the duality relations to establish identities for hypergeometric and q-hypergeometric integrals of different dimensions.

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2. Basic notation Let n be a nonnegative integer. A partition λ = (λ1 , λ2 , . . . ) with at most k parts is an infinite nonincreasing sequence of nonnegative integers such that λk+1 = 0 . Denote by Pk the set of partitions with at most k parts and by P the set of all partitions. We often make use of the embedding Pk → C k given by truncating the zero tail of a partition: (λ1 , . . . , λk , 0 , 0 , . . . ) → (λ1 , . . . , λk ) . Since obviously Pm ⊂ Pk for m k , in fact, one has a collection of embeddings Pm → C k for any m k . What particular embedding is used will be clear from the context. Let eab , a, b = 1, . . . , k , be the standard basis of the Lie algebra glk : [eab , ecd ] = δbc ead − δad ecb . We take the Cartan subalgebra h ⊂ glk spanned by e11 , . . . , ekk , and the nilpotent subalgebras n+ and n− spanned by the elements eab for a < b and a > b , respectively. One has the standard Gauss decomposition glk = n+ ⊕ h ⊕ n− . Let ε1 , . . . , εk be the basis of h∗ dual to e11 , . . . , ekk : εa , ebb = δab . We identify h∗ with C k mapping λ1 ε1 + · · · + λk εk to (λ1 , . . . , λk ) . The root vectors of glk are eab for a = b , the corresponding root being equal to αab = εa − εb . The roots αab for a < b are positive. We choose the standard invariant bilinear form ( , ) on glk : (eab , ecd ) = δad δbc . It defines an isomorphism h → h∗ . The induced bilinear form on h∗ is (εa , εb ) = δab . For a glk -module W and a weight λ ∈ h∗ let W [λ] be the weight subspace of W of weight λ . For any λ ∈ Pk we denote by Vλ the irreducible glk -module with highest weight λ . By abuse of notation, for any l ∈ Z0 we write Vl instead of V(l,0,...,0) . Thus, V0 = C is the trivial glk -module, V1 = C k with the natural action of glk , and Vl is the l-th symmetric power of V1 . Define a glk -action on the polynomial ring C[x1 , . . . , xk ] by differential operators: eab → xa ∂b , where ∂b = ∂/∂xb , and denote the obtained glk -module by V . Then ∞ V= Vl , (2.1) l=0

the submodule Vl being spanned by homogeneous polynomials of degree l . The highest weight vector of the submodule Vl is x1l .

3. Knizhnik-Zamolodchikov and differential dynamical equations ⊗n For any g ∈ U (glk ) set g (i) = id ⊗ · · · ⊗ g ⊗ · · · ⊗ id ∈ U (glk ) . We consider ⊗n ith ⊗n U (glk ) as a subalgebra of U (glk ) , the embedding U (glk ) → U (glk ) being given by the n-fold coproduct, that is, x → x(1) + · · · + x(n) for any x ∈ glk .

Duality for Knizhinik-Zamolodchikov Equations Let Ω =

k

241

eab ⊗ eba be the Casimir tensor, and let

a,b=1

Ω+ =

Ω− =

k 1 eaa ⊗ eaa + 2 a=1 k 1 eaa ⊗ eaa + 2 a=1

eab ⊗ eba ,

1a

Infinite Dimensional Algebras and Quantum Integrable Systems (Progress in Mathematics)

Infinite Dimensional Algebras and Quantum Integrable Systems (Progress in Mathematics)

Infinite Dimensional Algebras and Quantum Integrable Systems