Infinite Dimensional Algebras and Quantum Integrable Systems

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 Nenad Manojlovich St. Petersburg Department of Departamcnto de Matematica Steklov Mathematical Institute Faculdade Ue Cicncias e Tecnologia Rassian Academy of Sciences Universidade do Algarve Fontaka27 Campus de Gambelas 191011 St. Petersburg 8005-139 Faro Russia Portugal e-mail: [email protected] 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 <http://dnb.ddb.de>. 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 illustra- tions, 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 Teichmüller 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 in- troduces 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 Teichmüller 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 Kähler-Einstein manifold with negative constant Ricci curvature. Several lectures are devoted to the applications to quantum integrable mod- els, conformal field theory, and in particular the Knizhnik-Zamolodchikov equa- tions. 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 ana- lytic 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 func- tions of the quantum group invariant XXZ-model. These results are based on the relation previously found by Jimbo and Miwa between XXZ correlators and so- lutions of the q-deformed Knizhnik-Zamolodchikov equations on level −4. These solutions are further related to level 0 solutions; the new formulae suggest the de- composition of general matrix elements with respect to states of the infrared CFT. Takashi Takabe in his lecture discusses the trigonometric Wess-Zumino- Witten (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 equa- tions 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 re- lations between hypergeometric integrals of different dimensions. Extensions to trigonometric and difference versions of the Knizhnik-Zamolodchikov and dynam- ical 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 equa- tions admitting a zero-curvature representation on algebraic curves of arbitrary genus. He first reviews how conventional soliton equations are treated in the Grass- mannian perspective, considering as example the nonlinear Schrödinger hierarchy in great detail. Subsequently, recent results on the elliptic analogues of these sys- tems are presented. Finally, O. Castro-Alvaredo and A. Fring present a lecture on two-dimensi- onal 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 António Ferreira dos Santos, CEMAT and Department of Mathematics of Insti- tuto 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