Bombay Lectures On Highest Weight Representations Of Infinite Dimensional Lie Algebras (2nd Edition)

by Ashok K Raina, Victor G Kac, Natasha Rozhkovskaya

Overview

The first edition of this book is a collection of a series of lectures given by Professor Victor Kac at the TIFR, Mumbai, India in December 1985 and January 1986. These lectures focus on the idea of a highest weight representation, which goes through four different incarnations.The first is the canonical commutation relations of the infinite dimensional Heisenberg Algebra (= oscillator algebra). The second is the highest weight representations of the Lie algebra gℓ∞ of infinite matrices, along with their applications to the theory of soliton equations, discovered by Sato and Date, Jimbo, Kashiwara and Miwa. The third is the unitary highest weight representations of the current (= affine Kac-Moody) algebras. These Lie algebras appear in the lectures in connection to the Sugawara construction, which is the main tool in the study of the fourth incarnation of the main idea, the theory of the highest weight representations of the Virasoro algebra. In particular, the book provides a complete proof of the Kac determinant formula, the key result in representation theory of the Virasoro algebra.The second edition of this book incorporates, as its first part, the largely unchanged text of the first edition, while its second part is the collection of lectures on vertex algebras, delivered by Professor Kac at the TIFR in January 2003. The basic idea of these lectures was to demonstrate how the key notions of the theory of vertex algebras — such as quantum fields, their normal ordered product and lambda-bracket, energy-momentum field and conformal weight, untwisted and twisted representations — simplify and clarify the constructions of the first edition of the book.This book should be very useful for both mathematicians and physicists. To mathematicians, it illustrates the interaction of the key ideas of the representation theory of infinite dimensional Lie algebras and of the theory of vertex algebras; and to physicists, these theories are turning into an important component of such domains of theoretical physics as soliton theory, conformal field theory, the theory of two-dimensional statistical models, and string theory.

Product Details

ISBN-13:	9789814522182
Publisher:	World Scientific Publishing Company, Incorporated
Publication date:	08/28/2013
Series:	Advanced Series In Mathematical Physics , #29
Edition description:	2nd Revised ed.
Pages:	252
Product dimensions:	6.00(w) x 9.00(h) x 0.63(d)

Table of Contents

Preface v

Preface to the second edition vii

Lecture 1 1

1.1 The Lie algebra ∂ of complex vector fields on the circle 1

1.2 Representations V_α,β of ∂ 4

1.3 Central extensions of ∂: the Virasoro algebra 7

Lecture 2 11

2.1 Definition of positive-energy representations of Vir 11

2.2 Oscillator algebra A 12

2.3 Oscillator representations of Vir 15

Lecture 3 19

3.1 Complete reducibility of the oscillator representations of Vir 19

3.2 Highest weight representations of Vir 21

3.3 Verma representations M(c, h) and irreducible highest weight representations V(c, h) of Vir 23

3.4 More (unitary) oscillator representations of Vir 26

Lecture 4 31

4.1 Lie algebras of infinite matrices 31

4.2 Infinite wedge space F and the Dirac positron theory 33

4.3 Representations of GL_∞ and gl_∞ in F. Unitarity of highest weight representations of gl_∞ 36

4.4 Representation of a_∞ in F 40

4.5 Representations of Vir in F 42

Lecture 5 45

5.1 Boson-fermion correspondence 45

5.2 Wedging and contracting operators 47

5.3 Vertex operators. The first part of the boson-fermion correspondence 49

5.4 Vertex operator representations of gl_∞ and a_∞ 52

Lecture 6 55

6.1 Schur polynomials 55

6.2 The second part of the boson-fermion correspondence 57

6.3 An application: structure of the Virasoro representations for c = 1 60

Lecture 7 35

7.1 Orbit of the vacuum vector under GL_∞ 65

7.2 Denning equations for Ω in F⁽⁰⁾ 65

7.3 Differential equations for Ω in C[x¹,x²,…] 68

7.4 Hirota's bilinear equations 69

7.5 The KP hierarchy 71

7.6 N-soliton solutions 73

Lecture 8 77

8.1 Degenerate representations and the determinant det_n (c, h) of the contravariant form 77

8.2 The determinant det_n (c, h) as a polynomial in h 79

8.3 The Kac determinant formula 82

8.4 Some consequences of the determinant formula for unitarity and degeneracy 84

Lecture 9 89

9.1 Representations of loop algebras in α_∞ 89

9.2 Representations of gl_n in F^(m) 92

9.3 The invariant bilinear form on gl_n. The action of GL_n on gl_n 93

9.4 Reduction from a_∞ to sl_n and the unitarity of highest weight representations of sl_n 96

Lecture 10 101

10.1 Nonabelian generalization of Virasoro operators: the Sugawara construction 101

10.2 The Goddard-Kent-Olive construction 109

Lecture 11 113

11.1 sl₂ and its Weyl group 113

11.2 The Weyl-Kac character formula and Jacobi-Riemann-theta functions 115

11.3 A character identity 120

Lecture 12 123

12.1 Preliminaries on sl₂ 123

12.2 A tensor product decomposition of some representations OF sl₂ 124

12.3 Construction and unitarity of the discrete series representations of Vir 127

12.4 Completion of the proof of the Kac determinant formula 131

12.5 On non-unitarity in the region 0 ≤ c < 1, h ≥ 0 132

Lecture 13 135

13.1 Formal distributions 135

13.2 Local pairs of formal distributions 141

13.3 Formal Fourier transform 143

13.4 Lambda-bracket of local formal distributions 144

Lecture 14 151

14.1 Completion of U, restricted representations and quantum fields 151

14.2 Normal ordered product 157

Lecture 15 163

15.1 Non-commutative Wick formula 163

15.2 Virasoro formal distribution for free boson 168

15.3 Virasoro formal distribution for neutral free fermions 170

15.4 Virasoro formal distribution for charged free fermions 171

Lecture 16 175

16.1 Conformal weights 175

16.2 Sugawara construction 177

16.3 Bosonization of charged free fermions 179

16.4 Irreducibility theorem for the charge decomposition 182

16.5 An application: the Jacobi triple product identity 186

16.6 Restricted representations of free fermions 187

Lecture 17 191

17.1 Definition of a vertex algebra 1 191

17.2 Existence Theorem 195

17.3 Examples of vertex algebras 198

17.4 Uniqueness Theorem and n-th product identity 201

17.5 Some constructions 202

17.6 Energy-momentum fields 204

17.7 Poisson like definition of a vertex algebra 206

17.8 Borcherds identity 208

Lecture 18 211

18.1 Definition of a representation of a vertex algebra 211

18.2 Representations of the universal vertex algebras 213

18.3 On representations of simple vertex algebras 214

18.4 On representations of simple affine vertex algebras 215

18.5 The Zhu algebra method 218

18.6 Twisted representations 223

References 229

Index 235