Conference Boar d o f the Mathematica l Science s

CBMS Regional Conference Series in Mathematics

Number 9 8

Special Functions , KZ Type Equations , an d Representation Theor y

Alexander Varchenk o

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

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by th e American Mathematica l Societ y

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http://dx.doi.org/10.1090/cbms/098

C B M S Confe renc e o n A r r a n g e m e n t s a n d M a t h e m a t i c a l P h y s i c s

L o u i s i a n a S t a t e U n i v e r s i t y

J a n u a r y 1 5 - 1 9 , 200 2

Par t ia l ly suppor te d b y th e Nat iona l Scienc e Foundat io n

2000 Mathematics Subject Classification. P r i m a r y 14Dxx , 22Exx , 33Cxx ; Secondary 39Axx , 81Rxx , 82Bxx .

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CBMS Conferenc e o n Arrangements an d Mathematica l Physic s (200 2 : Louisiana Stat e University ) Special functions , K Z typ e equations , an d representatio n theor y / Alexande r Varchenko .

p. cm . — (Conferenc e Boar d o f th e Mathematica l Science s regiona l conferenc e serie s i n mathematics, ISS N 0160-764 2 ; no. 98 )

Includes bibliographica l reference s an d index . ISBN 0-8218-2867- 3 (alk . paper ) 1. Functions , Special—Congresses . 2 . Knizhnik-Zamolodchiko v equations—Congresses .

3. Representation s o f groups—Congresses . I . Title . II . Series .

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T O M Y MOTHE R AN D FATHE R

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Contents

Preface VI1

Chapter 1 . Hypergeometri c Solution s o f K Z Equation s 1 1.1. Example s o f Hypergeometri c Integral s 1 1.2. Knizhnik-Zamolodchiko v Equation s 3 1.3. Example s o f Solution s 5 1.4. Solution s of the KZ Equation with Values in Sing M®m [|ra|-2fc ] 6 1.5. Solution s of the KZ Equation with Values in Sing L®m [ \m\-2k] 8 1.6. Th e Classica l Hypergeometri c Serie s 8 1.7. Identitie s fo r Differentia l Form s 1 0 1.8. Hyperplan e Arrangement s 1 1

Chapter 2 . Cycle s of Integrals an d th e Monodrom y o f the K Z Equatio n 1 5

Iml - 2 15 2.1. Cycle s fo r Solution s i n SingM ®

2.2. Th e Quantu m Grou p U q(sl2) 1 8 2.3. Th e Yang-Baxte r Equatio n an d Representation s o f Brai d

Groups 2 0 2.4. Quantu m Singula r Vector s 2 2 2.5. Th e Monodrom y o f KZ Equation s 2 3 2.6. A Remark o n Integratio n Cycle s 2 9

Chapter 3 . Selber g Integral , Determinan t Formulas , an d Dynamica l Equations 3 1

3.1. Th e Selber g Integra l 3 1 3.2. A Connection wit h Finit e Reflectio n Group s 3 2 3.3. A n Exampl e o f a Determinan t Formul a 3 3 3.4. Th e Determinan t Formul a fo r Solution s i n a Weigh t Subspac e 3 6 3.5. Genera l Determinan t Formula s 3 8 3.6. Resonance s 4 0 3.7. Dynamica l Equation s 4 3

Chapter 4 . Critica l Point s o f Maste r Function s an d th e Beth e Ansat z 4 9 4.1. Th e Gaudi n Mode l an d th e Beth e Ansat z 4 9 4.2. Asymptoti c Solution s an d Eigenvector s 5 0 4.3. Quasi-Classica l Asymptotic s o f Solution s t o th e K Z Equatio n 5 3 4.4. Th e Shapovalo v Nor m o f Beth e Vector s 5 6

vi CONTENT S

4.5. Th e Numbe r o f Critical Point s o f a Product o f Powers of Linear Functions 5 8

4.6. Critica l Point s o f $k,n(t, z,m) i f m i , . . . , mn Ar e Natura l Numbers 6 0

4.7. Critica l Point s an d Fuchsia n Equation s wit h Polynomia l Solutions 6 2

4.8. Resonan t Loca l System s 6 7

Chapter 5 . Ellipti c Hypergeometri c Function s 6 9 5.1. Knizhnik-Zamolodchikov-Bernar d Equation s 6 9 5.2. Th e Cas e o f n = 1 and V = L 2p 7 0 5.3. Quasi-Classica l Asymptotic s o f Solution s t o th e KZ B Hea t

Equation 7 1 5.4. Ellipti c Hypergeometric Functions Associated with One Marked

Point 7 2 5.5. Integra l Representation s fo r Ellipti c Hypergeometri c Function s 7 3 5.6. Ellipti c Selber g Integral s 7 4 5.7. Transformation s Actin g o n th e Spac e o f Conforma l Block s 7 5 5.8. Relation s Betwee n Thet a Function s 7 6 5.9. Basi c Relation s Betwee n Ellipti c Hypergeometri c Function s 7 7 5.10. Macdonal d Polynomial s an d th e Shif t Operato r 7 9

(k)

5.11. Coefficient s frn,n and Value s o f Macdonal d Polynomial s 8 0 5.12. Modula r Transformation s o f Ellipti c Hypergeometri c

Functions 8 1 5.13. Trac e Function s fo r U q(sl2) 8 2

Chapter 6 . q-Hypergeometri c Solution s o f qKZ Equation s 8 5 6.1. Quantu m Knizhnik-Zamolodchiko v Equation s 8 5 6.2. Quasi-Classica l Asymptotic s o f Solutions an d Eigenvector s 8 8 6.3. A n Exampl e o f Quantizatio n o f Hypergeometri c Function s 8 9 6.4. q-Hypergeometri c Solutions , Genera l Cas e 9 7 6.5. Th e g-Hypergeometri c Pairin g 10 3 6.6. Quantizatio n o f the Kohno-Drinfel d Theore m 10 5

Bibliography 10 9

Index 117

Preface

The las t twent y year s hav e see n a n activ e interactio n betwee n mathe -matics an d physics . Thi s boo k i s devote d t o on e o f th e ne w area s whic h deals with mathematica l structure s relate d t o conforma l field theory an d it s g-deformatons. W e discus s th e interpla y betwee n Knizhnik-Zamolodchiko v type equations , th e Beth e ansat z method , representatio n theory , an d geom -etry o f multi-dimensiona l hypergeometri c integrals .

This boo k aim s t o provid e a n introductio n t o th e are a an d expos e dif -ferent facet s o f th e subject . Th e boo k contain s constructions , discussion s of notions , statement s o f theorems , an d illustratin g examples . Th e exposi -tion i s restricted t o th e simples t cas e o f th e theor y associate d wit h th e Li e algebra 5(2-

The boo k consist s o f six chapters . In Chapte r 1 th e Knizhnik-Zamolodchiko v equatio n i s realize d a s a

Gauss-Manin connection, that i s as the differential equatio n for multi-dimen -sional hypergeometri c integrals .

In Chapte r 2 the space of hypergeometric solution s of a Knizhnik-Zamo-lodchikov equatio n i s identifie d wit h th e multiplicit y spac e o f th e tenso r product o f representations o f the correspondin g quantu m group . Thi s iden -tification allow s on e t o identif y th e monodrom y grou p o f th e Knizhnik -Zamolodchikov equation with the R-matrix representation of the braid group .

In Chapte r 3 determinant formula s fo r hypergeometri c integral s ar e dis-cussed. Th e determinan t formula s allo w one to conclude tha t th e hypergeo -metric solution s give al l solutions t o the Knizhnik-Zamolodchiko v equation . The secon d topi c o f Chapte r 3 i s th e dynamica l equations . Th e dynami -cal equations ar e equation s complementar y t o th e Knizhnik-Zamolodchiko v equations. I n th e simples t exampl e th e equation s respectivel y becom e th e Gauss contiguou s functio n relation s an d th e Eule r hypergeometri c differen -tial equatio n fo r th e classica l hypergeometri c function .

In Chapte r 4 the relation s betwee n th e Beth e ansat z metho d an d quasi -classical asymptotic s o f hypergeometric function s ar e discussed .

Chapter 5 is devoted t o elliptic hypergeometric functions , thei r modula r properties, an d relation s t o Macdonald polynomial s an d trace s o f intertwin -ing operators .

In Chapte r 6 w e explai n ho w th e geometri c theor y o f th e Knizhnik -Zamolodchikov differentia l equatio n ha s to b e quantized t o giv e solutions t o the quantu m Knizhnik-Zamolodchiko v differenc e equation .

vii

V l l l PREFACE

The boo k gre w ou t o f graduat e course s a t UN C i n th e fal l o f 2001 , a t MIT i n the sprin g of 2002, and ten lecture s given by the autho r a t th e NSF-CBMS Conferenc e "Arrangement s an d Mathematic s Physics" , a t Louisian a State University , Bato n Rouge , durin g th e wee k o f Januar y 11-15 , 2002.

I a m gratefu l t o student s i n m y courses : A . Boysal , M . Graha , S . Lau , I. Mencattini , A . Oblomkov , V . Ostrik , L . Stevens , J . Scott , an d t o m y colleagues: I . Cherednik , D . Cohen , J . Damon , P . Eberlein , P . Etingof , G . Felder, Y . Markov , E . Mukhin , P . Orlik , I . Scherbak , V . Tarasov , H . Tera o for man y valuabl e discussions . M y specia l thank s ar e t o Danie l Cohe n wh o organized th e conferenc e i n Bato n Rouge .

Alexander Varchenk o Chapel Hill , Marc h o f 200 3

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114 BIBLIOGRAPH Y

[TV2] V . Taraso v an d A . Varchenko , Geometry of q-Hypergeometric Functions, Quan-tum Affine Algebras and Elliptic Quantum Groups, Asterisqu e 246 (1997) , 1-135 , math.QA/9703044.

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Index

p-cycles, 9 1 p-homology group , 9 1 p-master function , 10 3

amplitude, 5 3 asymptotic solution , 51 , 88 asymptotic solution in an asymptotic zone,

106

base point s o f a vecto r spac e o f polyno -mials, 66

Bethe ansat z equations , 50 , 5 5 Bethe ansat z method , 49 , 50 , 55 , 72 Bethe ansat z vector , 50 , 5 5 braid group , 2 0

conformal blocks , 41 , 72 critical valu e o f f a o n a polygon , 3 7

difference equation s of the discrete Gauss -Manin connection , 9 3

differential equatio n associate d with a crit -ical point , 6 4

discrete Gauss-Mani n connection , 9 2 discriminantal arrangement , 2 6 dynamical equations , 43 , 46

elliptic Calogero-Mose r system , 7 0 elliptic hypergeometri c functions , 69 , 7 2 elliptic maste r function , 7 0 elliptic Selber g integrals , 7 5 exponents o f a regula r singula r point , 6 3

flat section s of the discret e Gauss-Mani n connection, 9 3

Fuchsian differentia l equation , 6 3 fundamental spac e o f a critica l point , 6 5

Gauss-Manin connection , 2 4

Hamiltonians o f th e Gaudi n model , 4 9 Hamiltonians o f th e XX X model , 8 8

1

hypergeometric functions associate d wit h a maste r function , 7

hypergeometric pairing , 10 4 hypergeometric solutions to KZ equation ,

7 hypergeometric solution s t o trigonomet -

ric K Z equation , 4 5

indicial equation , 6 3 integrals o f the steepes t descen t method ,

53

Jacobi polynomial , 6 4

Knizhnik-Zamolodchikov equation , 3

Lame equation , 7 2 Lame function , 6 4

Macdonald polynomials , 7 9 master function , 6 , 5 0 modular group , 7 5

non-degenerate spac e o f polynomials , 6 6

phase, 5 3 pure brai d group , 2 0

q-hypergeometric functions , 10 4 qKZ equation , 8 5 qKZ operators , 8 7

rational R-matrix , 8 6 real discriminanta l arrangement , 2 9 regular singula r point , 6 2

scattering matrix , 10 6 Shapovalov form , 5 1 shift operator , 7 9

transition matrix , 10 6 trigonometric classica l r-matrix , 4 4 trigonometric K Z equation , 4 4

118 INDEX

trigonometric K Z operators , 4 4 trigonometric R-matrix , 10 1

Weierstrass p-function , 7 0

Yangian, 8 5

Titles i n Thi s Serie s

98 Alexande r Varchenko , Specia l functions , K Z typ e equations , an d representatio n theory ,

2003

97 Bern d Sturmfels , Solvin g system s o f polynomia l equations , 200 2

96 Nik y Kamran , Selecte d topic s i n th e geometrica l stud y o f differentia l equations , 200 2

95 Benjami n Weiss , Singl e orbi t dynamics , 200 0

94 Davi d J . Saltman , Lecture s o n divisio n algebras , 199 9

93 Gor o Shimura , Eule r product s an d Eisenstei n series , 199 7

92 Fa n R . K . Chung , Spectra l grap h theory , 199 7

91 J . P . Ma y e t al . , Equivarian t homotop y an d cohomolog y theory , dedicate d t o th e

memory o f Rober t J . Piacenza , 199 6

90 Joh n Roe , Inde x theory , coars e geometry , an d topolog y o f manifolds , 199 6

89 Cliffor d Henr y Taubes , Metrics , connection s an d gluin g theorems , 199 6

88 Crai g Huneke , Tigh t closur e an d it s applications , 199 6

87 Joh n Eri k Fornaess , Dynamic s i n severa l comple x variables , 199 6

86 Sori n Popa , Classificatio n o f subfactor s an d thei r endomorphisms , 199 5

85 Michi o J imb o an d Tetsuj i Miwa , Algebrai c analysi s o f solvabl e lattic e models , 199 4

84 Hug h L . Montgomery , Te n lecture s o n th e interfac e betwee n analyti c numbe r theor y an d

harmonic analysis , 199 4

83 Carlo s E . Kenig , Harmoni c analysi s technique s fo r secon d orde r ellipti c boundar y valu e

problems, 199 4

82 Susa n Montgomery , Hop f algebra s an d thei r action s o n rings , 199 3

81 Steve n G . Krantz , Geometri c analysi s an d functio n spaces , 199 3

80 Vaugha n F . R . Jones , Subfactor s an d knots , 199 1

79 Michae l Frazier , Bjor n Jawerth , an d Guid o Weiss , Littlewood-Pale y theor y an d th e

study o f functio n spaces , 199 1

78 Edwar d Formanek , Th e polynomia l identitie s an d variant s o f n x n matrices , 199 1

77 Michae l Christ , Lecture s o n singula r integra l operators , 199 0

76 Klau s Schmidt , Algebrai c idea s i n ergodi c theory , 199 0

75 F . Thoma s Farrel l an d L . Edwi n Jones , Classica l aspherica l manifolds , 199 0

74 Lawrenc e C . Evans , Wea k convergenc e method s fo r nonlinea r partia l differentia l

equations, 199 0

73 Walte r A . Strauss , Nonlinea r wav e equations , 198 9

72 Pete r Orlik , Introductio n t o arrangements , 198 9

71 Harr y D y m , 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 topic s i n probabilit y 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 . Wil l ia m Helton , Josep h A . Ball , Charle s R . Johnson , an d Joh n N . Palmer ,

Operator theory , analyti c functions , matrices , an d electrica l engineering , 198 7

67 Haral d Upmeier , Jorda n algebra s i n analysis , operato r theory , an d quantu m mechanics ,

1987

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

combinatorics, physic s an d compute r 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 , crosse d product s an d Galoi s theory , 198 6

63 Walte r Rudin , Ne w construction s o f function s holomorphi c i n th e uni t bal l o f C n , 198 6

TITLES I N THI S SERIE S

62 Bel a Bollobas , Extrema l grap h theor y wit h emphasi s o n probabilisti c methods , 198 6

61 Mogen s Flensted-Jensen , Analysi s o n non-Riemannia n symmetri c spaces , 198 6

60 Gille s Pisier , Factorizatio n o f linea r operator s an d geometr y o f Banac h spaces , 198 6

59 Roge r How e an d Alle n Moy , Harish-Chandr a homomorphism s fo r p-adi c groups , 198 5

58 H . Blain e Lawson , Jr. , Th e theor y o f gaug e fields i n fou r dimensions , 198 5

57 Jerr y L . Kazdan , Prescribin g th e curvatur e o f a Riemannia n manifold , 198 5

56 Har i Bercovici , Cipria n Foia§ , an d Car l Pearcy , Dua l algebra s wit h application s t o

invariant subspace s an d dilatio n theory , 198 5

55 Wil l ia m Arveson , Te n lecture s o n operato r algebras , 198 4

54 Wil l ia m Fulton , Introductio n t o intersectio n theor y i n algebrai c geometry , 198 4

53 Wi lhe l m Klingenberg , Close d geodesie s o n Riemannia n manifolds , 198 3

52 Ts i t -Yue n Lam , Orderings , valuation s an d quadrati c forms , 198 3

51 Masamich i Takesaki , Structur e o f factor s an d automorphis m groups , 198 3

50 Jame s Eell s an d Lu c Lemaire , Selecte d topic s i n harmoni c maps , 198 3

49 Joh n M . Franks , Homolog y an d dynamica l systems , 198 2

48 W . Stephe n Wilson , Brown-Peterso n homology : a n introductio n an d sampler , 198 2

47 Jac k K . Hale , Topic s i n dynami c bifurcatio n theory , 198 1

46 Edwar d G . Effros , Dimension s an d C*-algebras , 198 1

45 Ronal d L . Graham , Rudiment s o f Ramse y theory , 198 1

44 Phil l i p A . Griffiths , A n introductio n t o th e theor y o f specia l divisor s o n algebrai c curves ,

1980

43 Wil l ia m Jaco , Lecture s o n three-manifol d topology , 198 0

42 Jea n Dieudonne , Specia l function s an d linea r representation s o f Li e groups , 198 0

41 D . J . N e w m a n , Approximatio n wit h rationa l functions , 197 9

40 Jea n Mawhin , Topologica l degre e method s i n nonlinea r boundar y valu e problems , 197 9

39 Georg e Lusztig , Representation s o f finit e Chevalle y groups , 197 8

38 Charle s Conley , Isolate d invarian t set s an d th e Mors e index , 197 8

37 Masayosh i Nagata , Polynomia l ring s an d affin e spaces , 197 8

36 Car l M . Pearcy , Som e recen t development s i n operato r theory , 197 8

35 R . Bowen , O n Axio m A diffeomorphisms , 197 8

34 L . Auslander , Lectur e note s o n nil-thet a functions , 197 7

33 G . Glauberman , Factorization s i n loca l subgroup s o f finite groups , 197 7

32 W . M . Schmidt , Smal l fractiona l part s o f polynomials , 197 7

31 R . R . Coifma n an d G . Weiss , Transferenc e method s i n analysis , 197 7

30 A . Pelczyriski , Banac h space s o f analyti c function s an d absolutel y summin g operators ,

1977

29 A . Weinste in , Lecture s o n symplecti c manifolds , 197 7

28 T . A . Chapman , Lecture s o n Hilber t cub e manifolds , 197 6

27 H . Blain e Lawson , Jr. , Th e quantitativ e theor y o f foliations , 197 7

26 I . Reiner , Clas s group s an d Picar d group s o f grou p ring s an d orders , 197 6

25 K . W . Gruenberg , Relatio n module s o f finite groups , 197 6

24 M . Hochster , Topic s i n th e homologica l theor y o f module s ove r commutativ e rings , 197 5

23 M . E . Rudin , Lecture s o n se t theoreti c topology , 197 5

For a complet e lis t o f t i t le s i n thi s series , visi t th e AMS Bookstor e a t www.ams.org/bookstore/ .