Recent Titles in This Series

85 Michi o Jimbo and Tetsuji Miwa, Algebrai c analysis of solvable lattice models , 199 5 84 Hug h L. Montgomery, Te n lectures on the interface betwee n analyti c number theor y and

harmonic analysis , 199 4

83 Carlo s E. Kenig, Harmoni c analysi s techniques fo r secon d orde r ellipti c boundary valu e problems, 199 4

82 Susa n Montgomery, Hop f algebras and thei r actions on rings , 199 3 81 Steve n G. Krantz, Geometri c analysi s and functio n spaces , 199 3 80 Vaugha n F. R. Jones, Subfactor s an d knots , 1991 79 Michae l Frazier , Bjorn Jawerth, and Guido Weiss, Littlewood-Pale y theor y and th e study

of function spaces , 199 1

78 Edwar d Formanek, Th e polynomial identitie s and variants of n x n matrices , 199 1

77 Michae l Christ, Lecture s on singular integra l operators , 199 0

76 Klau s Schmidt, Algebrai c ideas in ergodic theory, 199 0

75 F . Thomas Farrell and L. Edwin Jones, Classica l aspherica l manifolds , 199 0

74 Lawrenc e C. Evans, Wea k convergence methods for nonlinear partial differential equations , 1990

73 Walte r A. Strauss, Nonlinea r wave equations, 198 9 72 Pete r Orlik, Introductio n t o arrangements, 198 9 71 Harr y Dym , J contractiv e matri x functions , reproducin g kerne l Hilber t space s an d

interpolation, 198 9 70 Richar d F. Gundy, Som e topics in probability an d analysis , 198 9 69 Fran k D . Grosshans , Gian-Carl o Rota , an d Joe l A . Stein , Invarian t theor y an d

superalgebras, 198 7 68 J . Willia m Helton , Josep h A . Ball , Charle s R . Johnson , an d Joh n N . Palmer ,

Operator theory , analytic functions , matrices , and electrica l engineering , 198 7 67 Haral d Upmeier, Jorda n algebra s i n analysis , operator theory , an d quantu m mechanics ,

1987 66 G . Andrews, ^-Series : Thei r developmen t an d applicatio n i n analysis , numbe r theory ,

combinatorics, physics and computer algebra , 198 6 65 Pau l H . Rabinowitz , Minima x method s i n critica l poin t theor y wit h application s t o

differential equations , 198 6 64 Donal d S. Passman, Grou p rings, crossed product s and Galois theory, 198 6 63 Walte r Rudin, Ne w constructions o f function s holomorphi c i n the uni t bal l of C n, 198 6 62 Bel a Bollobas, Extrema l graph theory with emphasis on probabilistic methods , 198 6 61 Mogen s Flensted-Jensen, Analysi s on non-Riemannia n symmetri c spaces , 198 6 60 Gille s Pisier, Factorizatio n o f linear operators and geometry o f Banac h spaces , 198 6 59 Roge r Howe and Allen Moy, Harish-Chandr a homomorphism s fo r p-adi c groups, 198 5 58 H . Blaine Lawson, Jr., Th e theory of gauge fields in four dimensions , 198 5 57 Jerr y L. Kazdan, Prescribin g the curvature of a Riemannian manifold , 198 5 56 Har i Bercovici, Ciprian Foia$, and Carl Pearcy, Dua l algebras with applications to invariant

subspaces and dilation theory , 198 5 55 Willia m Arveson, Te n lectures on operator algebras , 198 4 54 Willia m Fulton, Introductio n t o intersection theor y i n algebraic geometry, 198 4 53 Wilhel m Klingenberg, Close d geodesies on Riemannia n manifolds , 198 3 52 Tsit-Yue n Lam, Orderings , valuations and quadrati c forms , 198 3 51 Masamich i Takesaki, Structur e of factor s an d automorphism groups , 198 3 50 Jame s Eells and Luc Lemaire, Selecte d topic s i n harmonic maps , 198 3

(See the AMS catalog for earlie r titles )

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Algebraic Analysis of Solvable Lattice Models

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Conference Boar d o f the Mathematica l Science s

C B M S Regional Conference Serie s in Mathematic s

Number 8 5

Algebraic Analysis of Solvable Lattice Models

Michio Jimb o Tetsuji Miw a

Published fo r th e Conference Boar d of th e Mathematica l Science s

by th e American Mathematica l Societ y

Providence, Rhod e Islan d with suppor t fro m th e

National Scienc e Foundatio n

http://dx.doi.org/10.1090/cbms/085

Expository Lecture s from th e NSF-CBMS-Regiona l Conferenc e

held a t Nort h Carolin a Stat e University , Raleigh , Nort h Carolin a June 1-5 , 199 3

Research part ial l y suppor t e d b y Grant- in-Aid fo r Scientifi c Researc h o n

Pr ior i ty Area s 231 , t h e Minis t r y o f Educa t ion , Scienc e and Cul ture , J apan ; an d b y t h e

Nat ional Scienc e Foundat io n G r a n t D M S 921507 5

1991 Mathematics Subject Classification. P r imar y 17B37 , 81R50 , 82B20 , 82B23 .

Library o f Congres s Cataloging- in-Publ icat io n D a t a

Jimbo, M . (Michio ) Algebraic analysi s o f solvabl e lattic e models/Michi o Jimbo , Tetsuj i Miwa .

p. cm . — (Regiona l conferenc e serie s i n mathematics , ISS N 0160-7642 ; no . 85 ) Includes bibliographica l references . ISBN 0-8218-0320- 4 1. Lattic e dynamics . 2 . Statistica l mechanics . 3 . Li e algebras . 4 . Quantu m field theory .

5. Mathematica l physics . I . Miwa , T . (Tetsuji ) II . Title . III . Series . QC176.8.L3J56 199 5 530.4'11—dc20 94-2384 0

CIP

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10 9 8 7 6 5 4 3 2 0 3 0 2 0 1 0 0 9 9

Dedicated t o

MIKIO SAT O and

LUDWIG D . FADDEEV

Commemorating th e Fruitfu l Exchang e an d Interactio n Between Thei r School s i n Kyot o an d Leningra d

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Contents

0 Backgroun d o f th e proble m 1 0.1 Statistica l mechanic s 1 0.2 Solvabl e model s 6

1 Th e spi n 1/ 2 XX Z mode l fo r A < - 1 1 1 1.1 Quantu m Hamiltonia n 1 1 1.2 Thre e region s i n A 1 3 1.3 Th e anisotropi c limi t 1 4 1.4 On e poin t functio n (vac|cr f |vac) 1 6

2 Th e six-verte x mode l i n th e anti-ferroelectri c regim e 1 9 2.1 Verte x mode l 1 9 2.2 Groun d state s an d low-temperatur e expansio n 2 1 2.3 Th e correlatio n functio n 2 3 2.4 Transfe r matri x 2 4

3 Solvabilit y an d Symmetr y 2 7 3.1 Commutin g Hamiltonian s 2 7 3.2 Yang-Baxte r equatio n 2 8 3.3 Z-invarian t lattic e 2 9 3.4 Quantu m affin e algebr a U q(sl2) 3 4 3.5 R matri x a s a n intertwine r 3 7 3.6 Dua l module s an d crossin g symmetr y 3 9 3.7 Abelia n an d non-abelia n Symmetrie s 4 2

4 Correlatio n functions—physica l derivatio n 4 5 4.1 Corne r Transfe r Matri x 4 5 4.2 Propertie s o f Vertex Operator s 5 0 4.3 Th e on e poin t functio n 5 7

IX

4.4 Trac e function s an d differenc e equation s 6 0

5 Leve l on e module s an d bosonizatio n 6 3 5.1 Highes t weigh t module s 6 3 5.2 Drinfeld' s generator s 6 4 5.3 Realizatio n o f leve l one module s 6 6 5.4 Principa l vs . homogeneou s picture s 6 9

6 Verte x operator s 7 1 6.1 Th e notio n o f vertex operator s 7 1 6.2 Typ e I vertex operato r 7 2 6.3 Typ e I I verte x operato r 7 6 6.4 Commutatio n relation s 7 6 6.5 Dua l verte x operator s 7 8 6.6 Principa l pictur e 8 0

7 Spac e o f states—mathematica l pictur e 8 3 7.1 Spac e o f states 8 3 7.2 Translatio n an d loca l operator s 8 5 7.3 Transfe r matri x 8 6 7.4 Vacuu m 8 7 7.5 Eigenstate s 8 9

8 Trace s o f verte x operator s 9 3 8.1 Calculatin g th e trac e 9 3 8.2 Resul t 9 8 8.3 Example s 10 2 8.4 Orthogonalit y o f the eigenvector s 10 3

9 Correlatio n function s an d for m factor s 10 9 9.1 Correlatio n function s 10 9 9.2 For m factor s I l l 9.3 Matri x element s 11 3 9.4 Completenes s relatio n 11 5

10 Th e XXX limi t q -• - 1 11 9 10.1 Th e XXX limi t an d th e continuu m limi t 11 9 10.2 Scalin g 12 0 10.3 Critica l value s o f the correlator s 12 1 10.4 For m factor s i n th e limi t 12 3

CONTENTS x i

11 Discussion s 12 7 11.1 Othe r model s 12 7 11.2 Th e q-KZ equatio n 13 0 11.3 Relate d work s 13 2

A Lis t o f formula s 14 1 A.l R matri x 14 1 A.2 U q(si2) 14 2 A.3 Current s an d verte x operator s 14 5 A.4 Propertie s o f Vertex operator s 14 6 A.5 Principa l v s homogeneous picture s 14 8 A.6 Spac e o f states 14 9

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Preface

The ai m o f the presen t volum e i s to giv e a survey o f the recen t developmen t on the interpla y betwee n solvabl e lattice model s in statistical mechanic s an d representation theory of quantum affin e algebras . Th e original papers on this subject wer e publishe d i n th e for m o f a serie s an d th e result s ar e al l scat -tered around . W e thus fel t tha t a systemati c accoun t wa s necessary , whic h develops th e material s fro m scratch , focusin g attentio n o n th e mos t funda -mental case and withou t assumin g prior knowledg e abou t lattic e models no r representation theory .

Schematically, th e basi c problem s o f integrable model s i n field theory o r statistical mechanic s ar e t o diagonaliz e th e give n Hamiltonian , an d t o com -pute the correlation functions . B y correlation functions w e mean a system of functions (^> a(x)), ((/><*(%)<t>(3(y)), ''' obtaine d a s vacuum expectation s o f th e operators i n the theory . I n the contex t o f lattice statistics they ar e function s of th e lattic e site s x,y , • • • ; i n field theor y the y ar e function s o f th e space -time coordinate s o r momenta . I n principl e th e totalit y o f th e correlatio n functions ha s enoug h informatio n t o determin e th e theor y completely .

In a naiv e wa y th e Hamiltonia n i s a n infint e dimensiona l matri x actin g on som e infinit e dimensiona l space . Fo r instance , i n th e lattic e model s th e latter i s typically give n a s a n infinit e tenso r produc t o f 'local ' spaces , e.g .

• • • <g) C2 ® C2 <g > C2 ® • • • .

Obviously suc h a Hamiltonia n canno t b e define d literall y becaus e o f th e difficulty o f divergence . I n fact , a n arbitrar y vecto r i n thi s hug e spac e i s not meaningful ; wha t mak e sens e ar e onl y thos e eigenvector s whic h hav e finite energ y ( = finite eigenvalues) . The y ca n b e though t o f a s constitutin g a non-trivia l space , which w e will refe r t o a s the space of states.

At presen t on e know s ver y fe w system s whos e correlatio n function s ca n be described explicitly . Th e representativ e example s ar e 1 ) the Isin g model , and 2 ) conforma l field theory . Th e Isin g mode l i s a two-dimensiona l lattic e

X l l l

XIV PREFACE

model. It s correlation s (o n th e lattic e o r i n th e continuu m limit ) ca n b e characterized b y classica l non-linea r system s suc h a s the Painlev e equation s or soliton equations . Th e conformal field theory deal s with critical , o r mass-less, system s i n th e continuum . Thei r correlatio n function s belon g t o th e linear world , givin g a goo d clas s o f generalize d hype r geometric functions . The succes s i n th e Isin g mode l o r conforma l field theor y i s largel y relate d to th e fac t tha t thei r space s o f state s hav e clea r mathematica l structures : in th e Isin g mode l the y ar e th e fermio n Foc k spaces , an d i n conforma l field theory they are the highest weight representations of infinite-dimensional Li e algebras.

Beyond th e Isin g model , a larg e clas s o f solvabl e lattic e model s hav e been known ; the y ar e buil t o n th e solution s o f th e Yang-Baxte r equation . Our mai n example — th e six-verte x mode l an d it s spin-chai n equivalent , the XX Z model — i s on e o f th e mos t typica l model s o f thi s sort . However , until recentl y th e spac e o f state s an d correlatio n function s hav e no t bee n understood ver y wel l for thes e mor e genera l clas s o f models .

One o f th e ke y insight s t o thi s proble m cam e fro m th e corne r transfe r matrix metho d introduce d b y Baxte r i n 1976 . Th e calculatio n o f th e one -point function s i s reduced t o counting the multiplicitie s of the eigenvalues of the corner transfe r matrix . Amon g others , in the study o f the Hard Hexago n model, i t le d t o a remarkabl e connectio n wit h th e Rogers-Ramanuja n iden -tities. I t wa s the n recognize d that , i n man y interestin g case s includin g th e Hard-Hexagon model , the spectr a o f the corne r transfe r matrice s ca n b e de-scribed i n term s o f th e character s o f affin e Li e algebras . Despit e th e clos e similarity t o certai n structur e i n conforma l field theory , thi s finding ha s re -mained a curiosity for some years. It s combinatorial aspec t was subsequentl y clarified b y th e theor y o f crysta l base s fo r quantu m affin e algebras .

Another ke y emerged throug h th e recen t symmetr y approac h t o massiv e integrable field theories. Bernar d an d others realized that thes e theories pos-sess hidden non-Abelian symmetries by the Yangians. I t was hoped to exploit these symmetries to understand th e integrabilit y i n the massive case, follow -ing the spiri t o f conforma l field theory . I n th e latte r cas e a centra l rol e wa s played b y th e notio n o f verte x operator s an d th e Knizhnik-Zamolodchiko v (KZ) equation s fo r th e correlatio n functions . I t wa s the n foun d tha t thes e structures admi t a remarkabl e deformation : b y Smirnov , wh o showe d tha t the for m factor s h e ha s constructe d ove r th e year s satisf y th e deforme d K Z equations; an d b y Frenke l an d Reshetikhin , wh o studie d th e verte x opera -tors fo r quantu m affin e algebra s an d derive d th e g-deforme d K Z equation s for thei r matri x elements .

PREFACE xv

For lattice models the space on which the corner transfer matri x i s acting can b e viewe d a s 'hal f o f th e spac e o f states . Th e appearanc e o f th e Li e algebra character s suggest s tha t thi s hal f ca n b e identifie d wit h a highes t weight representatio n o f th e quantu m affin e algebra , whic h w e expec t t o govern th e symmtrie s o f th e models . Ou r firs t goa l i n thi s volum e i s t o explain tha t i t i s indee d so . Le t H = V(Ao ) © V^Ai) b e th e direc t su m o f level on e integrabl e representation s o f th e quantu m affin e algebr a U q(sl2). Then th e spac e o f states fo r th e six-verte x mode l ha s the structur e TL ® H*:, the tenso r produc t bein g understoo d i n a certai n complete d sense . Thu s we ar e upgradin g th e dimensio n countin g b y th e character s t o a structura l understanding o f the space of states. Thi s picture will lead to the descriptio n of the correlatio n function s an d th e for m factor s i n terms o f the g-deforme d vertex operators , and , vi a bosonization , t o th e integra l formula s fo r them . This wil l be ou r secon d goal .

Our exposition s ar e organize d a s follows . I n Chapte r 0 w e shal l giv e a brief accoun t o f basic principles in statstical mechanics . W e also touch upo n the histor y o f solvabl e models . Th e firs t thre e Chapter s ar e devote d t o th e standard subject s concerning solvable lattice models in statistical mechanics . Our mai n example s ar e th e spi n 1/ 2 XX Z chai n an d th e six-verte x model . The settin g fo r thes e model s an d thei r mutua l equivalenc e ar e explaine d i n Chapter 1 and Chapte r 2 , respectively . I n Chapte r 3 we discus s th e inte -grability o f the models . Th e rol e of the Yang-Baxte r equatio n an d th e com -muting transfe r matrice s ar e clarified . Th e res t o f th e Chapte r i s devote d to the introductio n o f the quantu m affin e algebr a U q(sl2), an d th e represen -tation theoretica l interpretatio n o f the Yang-Baxte r equation . I n Chapte r 4 we introduc e th e mai n objects , th e corne r transfe r matrice s an d th e verte x operators. B y a physica l argumen t w e then sho w ho w th e correlatio n func -tions ca n b e writte n a s th e trac e o f product s o f th e verte x operators , an d derive differenc e equation s fo r them . Havin g thes e a s physica l motivations , we restart ou r mathematica l discussion s fro m th e nex t Chapters . Chapte r 5 is devoted t o the Frenkel-Jin g bosonizatio n o f the leve l 1 module o f U q(sl2). In Chapte r 6 we derive th e formula s fo r th e verte x operator s usin g bosons . In Chapte r 7 we reformulate th e physica l settin g i n representation theoreti -cal terms, such a s the space of states, vacuum, translation , Hamiltonia n an d its eigenstates . T o derive th e formula s fo r th e correlatio n function s an d th e form factor s w e need t o calculat e th e trac e o f product s o f vertex operators . This computatio n i s carried ou t i n Chapte r 8 , and it s applicatio n i s given i n Chapter 9 . Th e limi t o f the XX X mode l i s briefly discusse d i n Chapte r 10 . We note tha t th e formula s i n Chapter s 8-1 0 ar e presente d her e fo r th e first

XVI PREFACE

time i n such details . Th e las t Chapte r 1 1 is devoted t o the discussio n o f th e other type s o f models , an d relate d works . I n th e Appendi x w e collec t basi c formulas fo r reader' s reference . Th e bibliograph y i s fa r fro m bein g exhaus -tive. W e hav e limite d th e citation s t o onl y thos e whic h ar e directl y relate d to th e discussions .

We woul d lik e t o than k ou r colleague s B.Davies , O.Foda , M.Idzumi , K.Iohara, K.Miki , T.Nakashima , A.Nakayashiki , Y.Oht a an d T.Tokihir o fo r the collaboratio n i n these works . Specia l thank s ar e du e t o F.Smirno v fro m whom w e learne d a grea t dea l durin g hi s sta y i n Kyoto . Th e presen t vol -ume i s a n outgrowt h o f a serie s o f lecture s delivere d b y on e o f u s (T.M. ) at th e regiona l conferenc e a t Nort h Carolin a Stat e University , Jun e 1993 . He wishe s t o than k K.Misr a fo r th e organizatio n o f th e conferenc e an d th e hospitality. W e ar e indebte d t o S.-J . Kan g an d N . Kawakam i wh o rea d th e manuscript an d offere d invaluabl e comments . Thi s work i s partly supporte d by Grant-in-Ai d fo r Scientifi c Researc h o n Priorit y Area s 231 , the Ministr y of Education , Scienc e an d Culture , Japan .

Michio Jimb o an d Tetsuj i Miw a Kyoto, Japan April 199 4

Bibliography

[1] A . Abada , A . H . Bougourzi , an d El . Gradechi . Deformatio n o f th e Wakimoto construction . Mod. Phys. Lett. A, 8:715-724 , 1993 .

[2] E . Abe . Hopf Algebra. Cambridg e Universit y Press , Cambridge , 1977 .

[3] G . E . Andrews , R . J . Baxter , an d P . J . Forrester . Eight-verte x SOS model an d generalize d Rogers-Ramanujan-typ e identities . J. Stat. Phys., 35:193-266 , 1984 .

[4] Y . Asai , M . Jimbo , T . Miwa , an d Y . Pugai . Bosonizatio n o f verte x

operators fo r th e A^ fac e model . J. Phys. A, 29:6595-6616 , 1996 .

[5] H . Au-Yang , B . M . McCoy , J . H . H . Perk , S . Tang , an d M . L . Yan . Commuting transfe r matrice s i n th e chira l Pott s models : solution s o f star-triangle equation s wit h genus > 1 . Phys. Lett. A, 123:219-223 , 1987.

[6] H . Awata, H. Kubo, S . Odake, and J . Shiraishi . Quantu m WN algebra s and Macdonal d polynomials . Commun. Math. Phys., 179:401-416 , 1996.

[7] H . Awata , S . Odake , an d J . Shiraishi . Fre e boso n representatio n o f Uq(sls). Lett. Math. Phys., 30:207-216 , 1994 .

[8] H . Awata , S . Odake , an d J . Shiraishi . Fre e boso n representatio n o f Uq(slN). Commun. Math. Phys., 162:61-83 , 1994 .

[9] O . Babelon , H . d e Vega , an d C . M . Viallet . Analysi s o f th e Bethe -Ansatz equation s o f the XXZ model . Nucl. Phys. B, 220:13-34 , 1983.

[10] R . J . Baxter . One-dimensiona l anisotropi c Heisenber g chain . Annals of Phys., 70:323-337 , 1972 .

151

152 BIBLIOGRAPHY

[11] R . J . Baxter . Partitio n functio n o f th e eight-verte x lattic e model . Annals of Phys., 70:193-228 , 1972 .

[12] R . J . Baxter . Eigh t verte x mode l i n lattic e statistic s an d one -dimensional anisotropi c heisenber g chain , ii . Equivalence t o a general -ized ice-typ e lattic e model . Annals Phys., 76:1-24 , 1973 .

[13] R . J . Baxter . Spontaneou s staggere d polarizatio n o f th e F model . J . Stat. Phys., 9:145-182 , 1973 .

[14] R . J . Baxter . Corne r transfe r matrice s o f the eigh t verte x model . I . J . Stat. Phys., 15:485-503 , 1976 .

[15] R . J . Baxter . Corne r transfe r matrice s o f th e eigh t verte x model . II . J. Stat. Phys., 17:1-14 , 1977 .

[16] R . J . Baxter . Solvabl e eight-verte x mode l o n a n arbitrar y plana r lat -tice. Phil. Trans. Royal Soc. London, 289:315-346 , 1978 .

[17] R . J . Baxter . Variationa l approximation s fo r squar e lattic e model s i n statistical mechanics . J. Stat. Phys., 19:461-478 , 1978 .

[18] R . J . Baxter. Exactly Solved Models in Statistical Mechanics. Academi c Press, London , 1982 .

[19] R . J . Baxter , J . H . H. Perk, an d H . Au-Yang. Ne w solutons of the star -triangle relation s fo r th e chira l Pott s model . Phys. Lett. A, 128:138 -142, 1988 .

[20] A . A . Belavin , A . M . Polyakov , an d A . B . Zamolodchikov . Infinit e conformal symmetr y i n tw o dimensiona l quantu m fiel d theory . Nucl. Phys. B, 241:333-380 , 1984 .

[21] D . Bernard . Verte x operato r representation s o f th e quantu m affin e

algebra U q{B^). Lett. Math. Phys., 17:239-245 , 1989 .

[22] Bougourz i an d Weston . N-poin t correlatio n function s o f th e spin- 1 XXZ model , preprin t 1896 , CRM , Univ . d e Montreal , 1993 .

[23] V . Brazhniko v an d S . Lukyanov . Angula r quantizatio n an d form -factors i n massive integr able models. Nucl. Phys. B, 512:616-636, 1998.

[24] V . Char i an d A . Pressley . Quantu m affin e algebras . Commun. Math. Phys., 142:261-283 , 1991.

BIBLIOGRAPHY 153

[25] E . Date , M . Jimbo , A . Kuniba , T . Miwa , an d M . Okado . Exactl y solvable SOS model s :I . loca l heigh t probabilitie s an d thet a functio n identities. Nucl. Phys. B, 290[FS20]:231-273 , 1987 .

[26] E . Date , M . Jimbo , A . Kuniba , T . Miwa , an d M . Okado . Exactl y solvable SOS model s :II . proo f o f th e star-triangl e relatio n an d com -binatorial identities . Adv. Stud, in Pure Math., 16:17-122 , 1988 .

[27] E . Date , M . Jimbo , A . Kuniba , T . Miwa , an d M . Okado . On e di -mensional configuratio n sum s i n verte x model s an d affin e Li e algebr a characters. Lett. Math. Phys., 17:69-77 , 1989 .

[28] E . Date, M. Jimbo, and M. Okado. Crysta l base and q vertex operators . Commun. Math. Phys., 155:47-69 , 1993 .

[29] E . Dat e an d M . Okado . Calculatio n o f excitatio n spectr a o f th e spi n model relate d wit h th e vecto r representatio n o f th e quantize d affin e algebra o f type A&. Int. J. Mod. Phys. A, 9:399-417 , 1994 .

[30] B . Davies , O . Foda , M . Jimbo , T . Miwa , an d A . Nakayashiki . Diag -onalization o f th e XXZ Hamiltonia n b y verte x operators . Commun. Math. Phys., 151:89-153 , 1993.

[31] V . G. Drinfeld . Quantu m groups . I n Proceedings of the International Congress of Mathematicians, page s 798-820 , Berkeley , 1987 .

[32] P . I . Etingof . Differenc e equation s wit h ellipti c coefficient s an d quan -tum affin e algebras , preprint , Yal e Univ. , 1993 .

[33] P . I . Etingof . Representation s o f affin e Li e algebras , ellipti c r-matri x systems, an d specia l functions . Commun. Math. Phys., 159:471-502 , 1994.

[34] P . I. Etingof an d Jr . A . Kirillov. Representation s o f affine Li e algebras, parabolic differentia l equation s an d Lam e functions . Duke Math. J., 74:585-614, 1994 .

[35] L . D . Faddeev . Quantu m invers e scatterin g method . Sov. Sci. Rev. Math. Phys., Cl:107-160 , 1981 .

[36] L . D . Faddeev an d L . A . Takhtajan . Spectru m an d scatterin g o f exci-tations i n th e one-dimensiona l isotropi c Heisenber g model . J. Soviet. Math., 24:241-267 , 1984 .

154 BIBLIOGRAPHY

[37] V . Fateev , D . Fradkin , S . Lukyanov , A . Zamolodchikov , an d Al . Zamolodchikov. Expectatio n value s o f descenden t fields i n th e sine -Gordon model . Nucl Phys. B, 540:587-609 , 1999 .

[38] V . Fateev , S . Lukyanov , A . Zamolodchikov , an d Al . Zamolodchikov . Expectation value s o f boundar y fields i n th e boundar y sine-Gordo n model. Phys. Lett. B, 406:83-88 , 1997 .

[39] V . Fateev , S . Lukyanov , A . Zamolodchikov , an d Al . Zamolodchikov . Expectation value s o f loca l fields i n Bullough-Dod d mode l an d inte -grable perturbed conforma l field theories . Nucl. Phys. B, 516:652-674 , 1998.

[40] B . Feigin , M . Jimbo , T . Miwa , A . Odesskii , an d Ya . Pugai . Algebr a of screenin g operator s fo r th e deforme d W n algebra . Commun. Math. Phys., 191:501-541 , 1998.

[41] B . L . Feigi n an d E . V . Frenkel . Quantu m W-algebra s an d ellipti c algebras. Commun. Math. Phys., 178:653-678 , 1996 .

[42] G. Felder . Ellipti c quantu m groups . Proc. ICMP Paris 1994, page s 211-218, 1995 .

[43] G. Felde r an d A . Varchenko . Algebrai c Beth e ansat z fo r th e ellipti c quantum grou p E T^(sl2). Nucl. Phys. B, 480:485-503 , 1996 .

[44] G. Felder and A. Varchenko. On representations of the elliptic quantum group E TiV(sl2). Commun. Math. Phys., 181:741-761 , 1996.

[45] O . Foda , K . Iohara , M . Jimbo , R . Kedem , T . Miwa , an d H . Yan . A n elliptic quantu m algebr a fo r sl 2. Lett. Math. Phys., 32:259-268 , 1994 .

[46] O . Foda , K . Iohara , M . Jimbo , R . Kedem , T . Miwa , an d H . Yan . Notes on highest weigh t modules of the elliptic algebra Aq iP(sl2). Prog. Theor. Phys. Suppl, 118:1-34 , 1995 .

[47] O . Foda , M . Jimbo , T . Miwa , K . Miki , an d A . Nakayashiki . Verte x operators i n solvabl e lattic e models . J. Math. Phys., 35:13-46 , 1994 .

[48] I . B . Frenke l an d N . Jing . Verte x representation s o f quantu m affin e algebras. Proc. Natl. Acad. Sci. USA, 85:9373-9377 , 1988 .

BIBLIOGRAPHY 155

[49] I . B . Frenke l an d N . Yu . Reshetikhin . Quantu m affin e algebra s an d holonomic differenc e equations . Commun. Math. Phys., 146:1-60 , 1992.

[50] Y . Hara , M . Jimbo , H . Konno , S . Odake , an d J . Shiraishi . Fre e fiel d approach t o th e dilut e AL models . Jour. Math. Phys., 40:3791-3826 , 1999.

[51] J . Hong , S.-J . Kan g ad n T . Miwa , an d R . Weston . Mixin g o f groun d states i n vertex models . J. Phys. A, 31:L515-L525 , 1998 .

[52] J . Hong , S.-J . Kang , T . Miwa , an d R . Weston . Verte x model s wit h alternating spins , math/9811175, 1998 .

[53] M . Idzumi . Leve l tw o irreducibl e representation s o f U q(sl2), verte x operators, an d thei r correlations . Int. J. Mod. Phys. A, 9:4449-4484 , 1994.

[54] M . Idzumi, K . Iohara, M . Jimbo, T . Miwa , T. Nakashima , an d T . Tok -ihiro. Quantu m affin e symmetr y i n vertex models . Int. J. Mod. Phys. A, 8:1479-1511 , 1993.

[55] I.S.Gradshtey n an d I.M.Ryzhik . Tables of integrals, series, and prod-ucts. Academi c Press , Ne w York , 1981 .

[56] M . Jimbo . Topics from representations of U q(g)—an introductory guide to physicists, page s 1-61 . Nanka i Lecture s o n Mathematica l Physics. Wolr d Scientific , Singapore , 1992 .

[57] M . Jimbo, R. Kedem, T . Kojima , H . Konno, and T . Miwa . XXZ chai n with a boundary . Nucl. Phys. B[FS], 441:437-470, 1995 .

[58] M . Jimbo , R . Kedem , H . Konno , T . Miwa , an d R . Weston . Differenc e equations in spin chains with a boundary. Nucl. Phys. B[FS], 448:429-456, 1995 .

[59] M . Jimbo, T . Kojima , T . Miwa , and Y.-H. Quano . Smirnov' s integral s and quantu m Knizhnik-Zamolodchiko v equatio n o f leve l 0 . J. Phys. A, 27:3267-3283 , 1994 .

[60] M . Jimbo, H. Konno, and T . Miwa. Massles s XXZ mode l and degener -ation o f the ellipti c algebr a A qiP(sl2)- I n Proceedings of the conference "Deformation Theory and Symplectic Geometry", eds. D. Sternheimer,

156 BIBLIOGRAPHY

J. Rownsley, and S. Gutt. Math. Phys. Studies. , volum e 20, pages 117— 138, Kluwer , 1997 .

[61] M . Jimbo , H . Konno , S . Odake , an d J . Shiraishi . Quasi-Hop f twistor s for ellipti c quantu m groups , q-alg/9712029 To appear in Transforma-tion Groups., 1997 .

[62] M . Jimbo , H . Konno , S . Odake , an d J . Shiraishi . Ellipti c algebr a Uq,p(sl2)' Drinfel d current s an d verte x operators . Commun. Math. Phys., 199:605-647 , 1999 .

[63] M . Jimbo , A . Kuniba , T . Miwa , an d M.Okado . Th e A& fac e models . Commun. Math. Phys., 119:543-565 , 1988 .

[64] M . Jimbo , M . Lashkevich , T . Miwa, an d Ya . Puga L Lukyanov' s screening operator s fo r th e deforme d Virasor o algebra . Phys. Lett. A, 229:285-292, 1997 .

[65] M . Jimbo , K . Miki , T . Miwa , an d A . Nakayashiki . Correlatio n func -tions o f th e XXZ mode l fo r A < - 1 . Phys. Lett. A, 168:256-263 , 1992.

[66] M . Jimb o an d T . Miwa . qKZ equatio n wit h \q\ = 1 and correlatio n functions o f the XXZ mode l in the gapless regime. J. Phys. A, 29:2923-2958, 1996 .

[67] M . Jimbo , T . Miwa , an d A . Nakayashiki . Differenc e equation s fo r th e correlation function s o f th e eight-verte x model . J. Phys. A, 26:2199 -2209, 1993 .

[68] M . Jimbo , T . Miwa , an d Y . Ohta . Structur e o f th e spac e o f state s i n RSOS models . Int. J. Mod. Phys. A, 8:1457-1477 , 1993 .

[69] M . Jimbo , T . Miwa , an d M . Okado . Solvabl e lattic e model s wit h broken Z n symmetr y an d Hecke' s indefinit e modula r forms . Nucl. Phys. B, 27 5 [FS17 ] :517-545, 1986 .

[70] M . Jimbo, T. Miwa, and M. Okado. Local state probabilities of solvable

lattice models : a n A^\ family . Nucl. Phys. B, 30 0 [FS22]:74-108 , 1988.

BIBLIOGRAPHY 157

[71] M . Jimbo , T . Miwa , an d M . Okado . Solvabl e lattic e model s relate d to the vecto r representatio n o f classical simple Lie algebras. Commun. Math. Phys., 116:507-525 , 1988 .

[72] M . Jimbo , T . Miwa, an d M . Okado . Solvabl e lattic e model s relate d to the vecto r representatio n o f classical simple Lie algebras. Commun. Math. Phys., 116:507-525 , 1988 .

[73] J . D. Johnson, S . Krinsky, and B. M. McCoy. Vertical-arrow correlatio n length in the eight-vertex model and the low-lying excitations of the X-Y-Z Hamiltonian . Phys. Rev. A, 8:2526-2547 , 1973 .

[74] V . G . Kac . Infinite Dimensional Lie Algebras. Cambridg e Universit y Press, Cambridge , thir d edition , 1990 .

[75] S.-J . Kang , M . Kashiwara , K . Misra , T . Miwa , T . Nakashima , an d A. Nakayashiki . Affin e crystal s an d verte x models . Int. J. Mod. Phys. A, 7(Suppl.lA):449-484 , 1992 .

[76] M . Kashiwar a an d T . Miwa . A clas s o f ellipti c solution s t o th e star -triangle relation . Nucl. Phys. B, 275:121-134 , 1986 .

[77] A . Kato , Y.-H . Quano , an d J . Shiraishi . Fre e boso n representatio n o f g-vertex operator s an d thei r correlatio n functions . Commun. Math. Phys., 157:119-137 , 1993 .

[78] S . Khoroshkin , D . Lebedev , an d S . Pakuliak . Intertwinin g operator s for the central extension of the Yangian double. Phys. Lett. A, 222:381 -392, 1996 .

[79] S . Khoroshkin , D . Lebedev , an d S . Pakuliak . Trace s o f intertwinin g operators fo r th e Yangia n double . Lett. Math. Phys., 41:31-47 , 1997 .

[80] S . Khoroshkin, D . Lebedev, an d S . Pakuliak. Zamolodchikov-Faddee v algebras for the Yangian double of level 1 . Theor. Math. Phys., 110:25 -45, 1997 .

[81] S . Khoroshkin, D . Lebedev, an d S . Pakuliak. Ellipti c algebra Aq^sfa) in th e scalin g limit . Commun. Math. Phys., 190:599-627 , 1998 .

[82] S . M . Khoroshkin . Centra l extensio n o f th e Yangia n double . Alge bre non-commutative, groupes quantiques et invariants, (Reims, 1995) 119-135, Smin. Congr. 2, Soc. Math. France, Paris, 1997 .

158 BIBLIOGRAPHY

[83] N . Kitanine , J . M . Maillet , an d V . Terras . For m factor s o f th e XXZ Heisenberg spin-1/ 2 finite chain , math-ph/9807020, 1998 .

[84] N . Kitanine, J . M . Maillet, an d V . Terras. Correlatio n function s o f the XXZ Heisenber g spin-1/2 chain in a magnetic field, math-ph/9901 019, 1999.

[85] T . Kojima and Y.-H. Quano. Quantum Knizhnik-Zamolodchikov equa -

tion for U q (sO) an d integra l formula. J. Phys. A, 27:6807-6826 , 1994 .

[86] H . Konno . Fre e field representation o f quantum affin e algebr a U q(sl2) and for m factor s i n higher spi n XXZ model . Nucl Phys. B, 432:457 -486, 1994 .

[87] H . Konno . A n ellipti c algebr a Uq^sfo) an d th e fusio n RSO S models . Commun. Math. Phys., 195:373-403 , 1998.

[88] V . E. Korepin, N . M. Bogoliubov, an d A. G. Izergin. Quantum Inverse Scattering Method and Correlation Functions. Cambridg e Universit y Press, Cambridge , 1993 .

[89] V.E . Korepin, A.G. Izergin, F.H.L. Essler, and D.B. Uglov. Correlatio n function o f the spin-1/2 XXX antiferromagnet. Phys. Lett. A, 190:182 -184, 1994 .

[90] Y . Koyama . Staggere d polarizatio n o f verte x model s wit h U q(sl(n)) symmetry. Commun. Math. Phys., 164:277-291 , 1994.

[91] M . Lashkevic h an d Ya . Pugai . Fre e field constructio n fo r correlatio n functions o f th e eight-verte x model . Nucl. Phys. B[FS], 516:623-651, 1998.

[92] S . Lukyanov . Correlator s o f th e Jos t function s i n th e sine-Gordo n model. Phys. Lett. B, 325:409-417 , 1994 .

[93] S . Lukyanov . Fre e field representatio n fo r massiv e integrabl e models . Commun. Math. Phys., 167:183-226 , 1995 .

[94] S . Lukyanov . Form-factor s o f exponentia l fields i n th e affin e ^ v - i Toda model . Phys. Lett. B, 408:192-200 , 1997 .

[95] S . Lukyanov . Form-factor s o f exponentia l fields i n th e sine-Gordo n model. Mod. Phys. Lett. A, 12:2543-2550 , 1997 .

BIBLIOGRAPHY 15 9

[96

[97

[98

[99

[ioo;

[101

[102;

[103

[104

[105;

[106

[107

[108

S. Lukyano v an d Ya . Pugai . Multipoin t loca l heigh t probabilitie s i n the integrabl e RSO S model . Nucl. Phys. BfFSJ, 473:631-658, 1996 .

S. Lukyanov an d A . Zamolodchikov . Exac t expectatio n value s of loca l fields i n quantu m sine-Gordo n model . Nucl. Phys. B, 493:571-587 , 1997.

J. R . Reye s Martinez . Correlatio n function s fo r th e Z-invarian t isin g model. Phys. Lett A, 227:203-208 , 1997 .

J. R . Reye s Martinez . Multi-spi n correlatio n function s fo r th e Z-invariant isin g model . Physica A, 256:463-484 , 1998 .

A. Matsuo. Fre e field representatio n o f quantum affin e algebr a U q{sl2). preprint, Nagoy a Univ. , 1992 .

A. Matsuo . Jackso n integral s o f Jordan-Pochhamme r typ e an d quan -tum Knizhnik-Zamolodchiko v equations . Commun. Math. Phys., 151:263-273, 1993 .

A. Matsuo . Quantu m algebr a structur e o f certai n Jackso n integrals . Commun. Math. Phys., 157:479-498 , 1993.

A. Matsuo. A g-deformation o f Wakimoto modules , primary field s an d screening operators . Commun. Math. Phys., 160:33-48 , 1994 .

B. M. McCoy. The connection between statistical mechanics and quan -tum fiel d theory , preprint , hetp-t h 9403084 , ITP-SB-94-07 , 1994 . To appea r i n "Fiel d Theor y an d Statistica l Mechanics" , Proc.7-t h Physics Summe r School , Australia n Nationa l University , 1994 . ed . V.V.Bazhanov C.J.Burden , Worl d Scientific , Singapore , 1995 .

B. M . McCoy , J . H . H . Perk , S . Tang , an d C . H . Sah . Commutin g transfer matrice s fo r th e four-stat e self-dua l chira l Pott s mode l wit h a genus-three uniformizin g Ferma t curve . Phys. Lett A, 125:9-14 , 1987 .

K. Miki. Creation/annihilation operator s and form factors of the XXZ model. Phys. Lett. A, 186:217-224 , 1994 .

T. Miw a an d R . Weston . Boundar y AB F models . Nucl. Phys. B[PM], 486:517-545, 1997 .

A. Nakayashiki. Som e integral formulas t o the solutions of the sh dK Z equations wit h leve l - 4 . Int. J. Mod. Phys. A, 9:5673-5687 , 1994 .

160 BIBLIOGRAPHY

[109] A . Nakayashiki . Fusio n o f th e ^-verte x operator s an d it s applicatio n to solvabl e verte x models . Commun. Math. Phys., 177:27-62 , 1996 .

[110] A . Nakayashiki , S . Pakuliak , an d V . Tarasov . O n solution s o f the K Z and gK Z equation s a t leve l zero , q-alg/9712002, 1997 .

[Ill] C . Fr0nsdal . Generalizatio n an d exac t deformation s o f quantu m groups. Publ RIMS, Kyoto Univ., 33:91-149 , 1997 .

[112] C . Fr0nsdal . Quasi-Hop f deformation s o f quantum groups . Lett. Math. Phys., 40:117-134 , 1997 .

[113] C . Fr0nsda l an d A . Galindo . 8-verte x correlatio n function s an d twis t covariance o f q-KZ equation . Rev. Math. Phys., 10:1027-1059 , 1998 .

[114] L . Onsager . Crysta l statistics.I . A two-dimensiona l mode l wit h a n order-disorder transition . Phys. Rev., 65:117-149 , 1944 .

[115] Y.-H . Quano . Spontaneou s staggere d polarizatio n o f th e ZN Baxte r model. Mod. Phys. Lett. A, 8:3363-3375 , 1993 .

[116] N . Yu. Reshetikhin . 5-matrice s i n integrabl e model s o f isotropic mag -netic chain s : I . J. Phys. A, 24:3299-3309 , 1991.

[117] N . Yu. Reshetikhin. Jackson-typ e integrals , Bethe vectors, and solutio n to a difference analo g o f Knizhnik-Zamolodchikov system . Lett. Math. Phys., 26:153-165 , 1992 .

[118] M . Sato , T . Miwa , an d M . Jimbo . Holonomi c quantu m fields.Ill,IV. Publ. RIMS, Kyoto Univ., 15:577-629 , 871-972 , 1979 .

[119] J . Shiraishi . Fre e boso n representatio n o f U q(sl2)' Phys. Lett. A, 171:243-248, 1992 .

[120] J . Shiraishi , H . Kubo , H . Awata , an d S . Odake . A quantum deforma -tion o f the Virasor o algebr a an d th e Macdonal d symmetri c functions . Lett. Math. Phys., 38:33-51 , 1996 .

[121] E . K . Sklyanin . Boundar y condition s fo r integrabl e quantu m systems . J. Phys. A, 21:2375-2389 , 1988 .

[122] E . K. Sklyanin , L . A. Takhtajan, an d L . D. Faddeev. Quantu m invers e problem. I . Theor. Math. Phys., 40:688-706 , 1979 .

BIBLIOGRAPHY 16 1

[123] E . K. Sklyanin , L . A. Takhtajan, an d L . D. Faddeev. Quantu m invers e scattering proble m method . Zeit. fur Phys., 40:194-220 , 1979 .

[124] F . A. Smirnov. Form Factors in Completely Integrable Models of Quan-tum Field Theory. Worl d Scientific , Singapore , 1992 .

[125] F . A . Smirnov . Remark s o n deforme d an d undeorme d Knizhnik -Zamolodchikov equations , preprint RIMS, 860 , 1992 .

[126] F . A. Smirnov. For m factors, deforme d Knizhnik-Zamolodchiko v equa -tions an d finite-gap integration . Commun. Math. Phys., 155:459-487 , 1993.

[127] F . A . Smirnov . Wha t ar e we quantizing i n integrable field theory? St Petersburg Math. J. , 6:417-428 , 1995 .

[128] K . Sogo . Groun d stat e an d low-lyin g excitation s i n th e Heisenber g XXZ chai n o f arbitrar y spins . Phys. Lett. A, 104:51-54 , 1984 .

[129] K . Sogo and M. Wadati. Boos t operator an d its application to quantu m Gelfand-Levitan equatio n fo r Heisenberg-Isin g chai n wit h spi n one -half. Progr.Theoret.Phys ., 69:431-450 , 1983 .

[130] M . E . Sweedler . Hopf Algebras. Benjamin , 1969 .

[131] V . Taraso v an d A . Varchenko . Jackso n integra l representation s fo r solutions o f the quantize d Knizhnik-Zamolodchiko v equation . St Pe-tersburg Math. J. , 6:275-313 , 1995 .

[132] M . G . Tetel'man . Loren z grou p fo r two-dimensiona l integrabl e lattic e systems. Zh. Eksp.Theor.Fiz., 82:528-535 , 1982 .

[133] H . B . Thacker . Corne r transfe r matrice s an d Loren z invarianc e o n a lattice. Physica D, 18:348-359 , 1986 .

[134] A . Tsuchiya an d Y . Kanie . Verte x operator s i n conforma l field theor y on P 1 an d monodrom y representation s o f brai d group . Adv. Stud, in Pure Math., 16:297-372 , 1988 .

[135] A . Varchenko . Quantize d Knizhnik-Zamolodchiko v equations , quantum Yang-Baxte r equation , an d differenc e equation s fo r q-hypergoemtric functions , preprint , Univ . o f Nort h Carolina , 1993 .

162 BIBLIOGRAPHY

[136] T . T . Wu , B . M . McCoy , C . A . Tracy , an d E . Barouch . Spin-spi n correlation functions fo r the two-dimensional Ising modehExact result s in the scalin g region . Phys. Rev. B, 13:316-374 , 1976 .

[137] C . N. Yang. The spontaneous magnetization of a two-dimensional Isin g model. Phys. Rev., 85:808-816 , 1952 .

[138] C.N . Yan g and C.P . Yang . One-diemnsiona l chai n o f anisotropic spin -spin interactions . I . Proo f o f Bethe' s hypothesi s fo r groun d stat e i n finite system . Phys. Rev., 150:321-327 , 1966 .

[139] C.N . Yan g an d C.P . Yang . One-diemnsiona l chai n o f anisotropic spin -spin interactions . II . Propertie s o f the groun d stat e energ y pe r lattic e site fo r a n infinit e system . Phys. Rev., 150:327-339 , 1966 .