MathPhys Odyssey 2001: Integrable Models and Beyond In Honor of Barry M. McCoy PDF
Preview MathPhys Odyssey 2001: Integrable Models and Beyond In Honor of Barry M. McCoy
Progress in Mathematical Physics Volume 23 Editors-in-Chief Anne Boutet de Monvel, Universite Paris VII Denis Diderot Gerald Kaiser, The Virginia Centerfor Signals andWaves Editorial Board D. Bao, University ofHouston C. Berenstein, University ofMaryland, College Park P. Blanchard, Universitiit Bielefeld A.S. Fokas, Imperial College ofScience, Technology andMedicine C. Tracy, University ofCalifornia, Davis H. van den Berg, Wageningen University MathPhys Odyssey 2001 Integrable Models and Beyond In Honor of Barry M. McCoy Masaki Kashiwara Tetsuji Miwa Editors Springer Science+Business Media, LLC Masaki Kashiwara Tetsuji Miwa Kyoto University Kyoto University RIMS Department of Mathematics Kyoto 606-8502 Kyoto 606-8502 Japan Japan Library of Congress Cataloging.in·Publication Data MathPhys odyssey 2001: integrable models and beyond / Masaki Kashiwara and Tetsuji Miwa, editors. p. cm.-(Progress in mathematical physics ; v. 23) Inc1udes bibliographical references. ISBN 978-1-4612-6605-1 ISBN 978-1-4612-0087-1 (eBook) DOI 10.1007/978-1-4612-0087-1 1. Integral equations. 2. Mathematical physics. 1. Kashiwara, Masaki, 1947-II. Miwa, T. (Tetsuji) III. Series. QC20.7.1S8 M37 2002 S30.1S'S45-dc21 2002018300 AMS SubjectClassifications: 82BXX, 82B20, SITXX, 81T40, 17BXX, 17B69, 81RXX, 81RlO Printed on acid-free paper. © 2002 Springer Science+Business Media New York Originally published by Birkhăuser Boston in 2002 Softcover reprint of the hardcover 1s t edition 2002 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use of general descriptive names, trade names, trademarks, etc., in this publication, even ifthe former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone. ISBN978-1-4612-6605-1 SPIN 10844618 Reformatted from editors' files by 1EXniques, Inc., Cambridge, MA. 987 6 S 4 3 2 1 BarryM. McCoy Photocourtesya/VaughanJones To BarryM. McCoy, ourcolleagueandfriend in honorofhissixtiethbirthday Contents Preface ix Wavevector-DependentSusceptibility inAperiodicPlanarIsingModels H Au-YangandJ.HH Perk 1 CorrelationFunctionsandSusceptibility inthe Z-InvariantIsingModel H Au-YangandJ.HH Perk 23 ARapidity-IndependentParameterintheStar-TriangleRelation R.J. Baxter 49 EvaluationofIntegralsRepresentingCorrelationsin the XXXHeisenbergSpinChain HE. Boosandv.E. Korepin 65 ANoteonQuotientsoftheOnsagerAlgebra E. Date 109 EvaluationParametersandBetheRootsfor theSix-Vertex Model atRootsofUnity K. FabriciusandB.M. McCoy 119 NormalizationFactors,ReflectionAmplitudesand IntegrableSystems V.A. Fateev 145 VertexOperatorAlgebraArisingfrom theMinimal SeriesM(3,p) andMonomialBasis B.L. Feigin, M. JimboandT. Miwa 179 Paths,CrystalsandFermionicFormulae G. Hatayama, A. Kuniba, M. Okada, T. TakagiandZ. Tsuboi 205 viii Contents TheNonlinearSteepestDescentApproach to theAsymptotics oftheSecondPainleveTranscendentintheComplexDomain A.R. Its andA. A. Kapaev 273 GeneralizedUmemuraPolynomialsand theHirota-MiwaEquation A. N. KirillovandM. Taneda 313 CorrelationFunctionsofQuantumIntegrableModels:TheXXZSpinChain 1. M. Maillet 333 OnFormFactorsoftheSU(2) InvariantThirringModel A. NakayashikiandY. Takeyama 357 IntegrableBoundaries andUniversalTBAFunctionalEquations C. H. Otto Chui, C. MercatandP. A. Pearce 391 ConformalFieldTheories, GraphsandQuantumAlgebras V Petkovaand1.-B. Zuber 415 q-SupernomialCoefficients: FromRiggings toRibbons A. Schilling 437 SeparationofVariablesfor Quantum IntegrableModels Relatedto Uq(;[N) F A. Smirnov 455 OnaDistributionFunctionArisinginComputationalBiology C. A. TracyandH. Widom 467 Preface This volumeisdedicated toBarryM. McCoyonthe occasionofhissixtiethbirthday. BarryMcCoy has ledtheresearchonintegrablemodels instatisticalmechanicsand quantumfieldtheoryformorethan30years.Hisbook,cowrittenwithT.T.Wu,TheTwo dimensional/singModel,(HarvardUniversity)containsallthebasicresultsontheIsing modelobtainedbytheearly 1970s.However,McCoy'sjointpaperwithWu,Tracyand Barouch, Spin-spin correlationfunctionsfor the two-dimensional/sing model: Exact results in the scaling region (Physical Review B13, 316-374, 1976), was a giant step beyond the book. A remarkableconnection between the two-point scaling correlation functions andthePainlevetranscendents wasfound. This work madean enormous impacton mathematical physics in the lastquarterof the20thcentury. Itgavethefirstnontrivialexampleofinteractingfield theoryinwhich the Green function can be written explicitly in terms of special functions satisfying nonlineardifferential equations. Later in the 1980s, an extensive listofmassless field theories was added in the innovation of conformal field theory, in which the Green functions arecharacterizedbylineardifferentialequationsofthehypergeometric type. However,evenalmost30yearsaftertheirwork,theIsingmodelisessentiallytheunique casewherealltheGreenfunctionsofmassivefieldtheoryarecharacterizedbynonlinear differentialequations. Sincehisfirstvisitin 1980,BarryMcCoyhasvisitedtheResearchInstituteforMath ematicalSciencesofKyotoUniversityseveraltimes,and influencedresearchactivities in Japan in the field ofintegrable models. The works on isomonodromy deformations and soliton equations by Sato, Jimbo, Date and others, including the present editors, werelargelyinspiredbyhisworkontheIsingmodel.In 1991,hevisitedRIMSforthree months in thesummer, where hedevelopedanew activity which is nowfamous as the fermionic characterformulas. This work connects the representation theory ofinfinite dimensional Lie algebras and the BetheAnsatz in solvable lattice models, and along with Baxter's corner transfer matrix method, has led to a new field in mathematical physicswhich we maycallphysicalcombinatorics. The interaction cannot be in one direction. Most recently, Barry McCoy has been studying the affine Lie algebra content ofthe XXZ model at the root of unity ofthe x Preface parameterq.Theeditorsweremostpleasedtoreceivealetterfromhimstatinghiswish to contribute a paperon this subject to this volume. This might be unusual since the volumeisdedicatedtoMcCoyhimself. However, theadditionofhisjointpapermakes the volume a more completerepresentation ofthe status ofourodyssey in integrable modelsatthetumofthecentury. The editors express their sincere gratitude to the contributors ofthis volume, and toAnn Kostant, the staffofBirkhauser, T£Xniques, and the RIMS secretaries, whose cooperationhasagain beensohelpful inpublication. Kyoto Masaki KashiwaraandTetsujiMiwa November24,2001 Editors
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