**CBMS Regional Conference Series in Mathematics**

Volume: 98;
2003;
118 pp;
Softcover

MSC: Primary 14; 22; 33;
Secondary 39; 81; 82

**Print ISBN: 978-0-8218-2867-0
Product Code: CBMS/98**

List Price: $44.00

Individual Price: $35.20

**Electronic ISBN: 978-1-4704-2458-9
Product Code: CBMS/98.E**

List Price: $41.00

Individual Price: $32.80

#### Supplemental Materials

# Special Functions, KZ Type Equations, and Representation Theory

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*Alexander Varchenko*

A co-publication of the AMS and CBMS

The last twenty years have seen an active interaction between
mathematics and physics. This book is devoted to one of the new areas which
deals with mathematical structures related to conformal field theory and its
\(q\)-deformations. In the book, the author discusses the interplay
between Knizhnik–Zamolodchikov type equations, the Bethe ansatz method,
representation theory, and geometry of multi-dimensional hypergeometric
functions.

This book aims to provide an introduction to the area and expose different
facets of the subject. It contains constructions, discussions of notions,
statements of main results, and illustrative examples. The exposition is
restricted to the simplest case of the theory associated with the Lie algebra
\(\mathfrak{sl}_2\).

This book is intended for researchers and graduate students in mathematics
and in mathematical physics, in particular to those interested in applications of
special functions.

#### Readership

Graduate students and research mathematicians interested in mathematical physics, in particular to those interested in application of special functions.

#### Reviews & Endorsements

The book is a result of many years work and reading of lectures and, on my opinion, at this moment is the best exposition of the theory of KZ equations in which different facets and their connections are considered. It would be desirable to construct so complete the theory for the KZ equations associated to the other root systems.

-- Zentralblatt MATH

#### Table of Contents

# Table of Contents

## Special Functions, KZ Type Equations, and Representation Theory

- Cover Cover11 free
- Title i2 free
- Copyright ii3 free
- Contents v6 free
- Preface vii8 free
- Chapter 1. Hypergeometric Solutions of KZ Equations 110 free
- 1.1. Examples of Hypergeometric Integrals 110
- 1.2. Knizhnik-Zamolodchikov Equations 312
- 1.3. Examples of Solutions 514
- 1.4. Solutions of the KZ Equation with Values in Sing M[sup(⊗m)] [|m| … 2k] 615
- 1.5. Solutions of the KZ Equation with Values in Sing L[sup(⊗m)] [|m| … 2k] 817
- 1.6. The Classical Hypergeometric Series 817
- 1.7. Identities for Differential Forms 1019
- 1.8. Hyperplane Arrangements 1120

- Chapter 2. Cycles of Integrals and the Monodromy of the KZ Equation 1524
- Chapter 3. Selberg Integral, Determinant Formulas, and Dynamical Equations 3140
- Chapter 4. Critical Points of Master Functions and the Bethe Ansatz 4958
- 4.1. The Gaudin Model and the Bethe Ansatz 4958
- 4.2. Asymptotic Solutions and Eigenvectors 5059
- 4.3. Quasi-Classical Asymptotics of Solutions to the KZ Equation 5362
- 4.4. The Shapovalov Norm of Bethe Vectors 5665
- 4.5. The Number of Critical Points of a Product of Powers of Linear Functions 5867
- 4.6. Critical Points of Φ[sub(k,n)](t,z,m) if m[sub(1)],ƒ,m[sub(n)] Are Natural Numbers 6069
- 4.7. Critical Points and Fuchsian Equations with Polynomial Solutions 6271
- 4.8. Resonant Local Systems 6776

- Chapter 5. Elliptic Hypergeometric Functions 6978
- 5.1. Knizhnik-Zamolodchikov- Bernard Equations 6978
- 5.2. The Case of n = 1 and V = L[sub(2p)] 7079
- 5.3. Quasi- Classical Asymptotics of Solutions to the KZB Heat Equation 7180
- 5.4. Elliptic Hypergeometric Functions Associated with One Marked Point 7281
- 5.5. Integral Representations for Elliptic Hypergeometric Functions 7382
- 5.6. Elliptic Selberg Integrals 7483
- 5.7. Transformations Acting on the Space of Conformal Blocks 7584
- 5.8. Relations Between Theta Functions 7685
- 5.9. Basic Relations Between Elliptic Hypergeometric Functions 7786
- 5.10. Macdonald Polynomials and the Shift Operator 7988
- 5.11. Coefficients f[sup((k))[sub(m,n)] and Values of Macdonald Polynomials 8089
- 5.12. Modular Transformations of Elliptic Hypergeometric Functions 8190
- 5.13. Trace Functions for U[sub(q)](sl[sub(2)]) 8291

- Chapter 6. q-Hypergeometric Solutions of qKZ Equations 8594
- 6.1. Quantum Knizhnik-Zamolodchikov Equations 8594
- 6.2. Quasi-Classical Asymptotics of Solutions and Eigenvectors 8897
- 6.3. An Example of Quantization of Hypergeometric Functions 8998
- 6.4. q-Hypergeometric Solutions, General Case 97106
- 6.5. The g-Hypergeometric Pairing 103112
- 6.6. Quantization of the Kohno-Drinfeld Theorem 105114

- Bibliography 109118
- Index 117126
- Back Cover Back Cover1130