SHANGHAI TAIPEI Bombay Lectures on HIGHEST WEIGHT REPRESENTATIONS of INFINITE DIMENSIONAL LIE ALGEBRAS Second Edition Victor G. Kac Massachusetts Institute of Technology, USA Ashok K. Raina Tata Institute of Fundamental Research, India Natasha Rozhkovskaya Kansas State University, USA ^ NEW JERSEY LONDON SINGAPORE BEIJING World Scientific HONG KONG CHENNAI
CONTENTS Preface v Preface to the second edition vii Lecture 1 1 1.1. The Lie algebra 0 of complex vector fields on the circle 1 1.2. Representations Va$ of D 4 1.3. Central extensions of t>: the Virasoro algebra 7 Lecture 2 11 2.1. Definition of positive-energy representations of Vir 11 2.2. Oscillator algebra sf 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 ix
X 4.3. Representations of GL^ and gz^ in F. Unitarity of highest weight representations of gz^ 36 4.4. Representation of 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 gt^ and Ooo 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 65 7.1. Orbit of the vacuum vector under GL^ 65 7.2. Defining equations for in F^ 65 7.3. Differential equations for ft in C[xi, X2, } 68 7.4. Hirota's bilinear equations 69 7.5. The KP hierarchy 71 7.6. 7V-soliton solutions 73 Lecture 8 77 8.1. Degenerate representations and the determinant detn(c, h) of the contravariant form 77 8.2. The determinant det (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 5oo 89 9.2. Representations of g n in F^ 92
xi 9.3. The invariant bilinear form on g n. The action of GLn on gln 93 9.4. Reduction from to s n and the unitarity of highest weight representations of s 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. s '2 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 sd.2 123 12.2. A tensor product decomposition of some representations of s\ 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<l,/i^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
xii 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 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