Table Of ContentProgress 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
