INFINITE DIMENSIONAL LIE ALGEBRAS

Transcription

1 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

2 CONTENTS Preface v Preface to the second edition vii Lecture The Lie algebra 0 of complex vector fields on the circle Representations Va$ of D Central extensions of t>: the Virasoro algebra 7 Lecture Definition of positive-energy representations of Vir Oscillator algebra sf Oscillator representations of Vir 15 Lecture Complete reducibility of the oscillator representations of Vir Highest weight representations of Vir Verma representations M(c, h) and irreducible highest weight representations V(c,h) of Vir More (unitary) oscillator representations of Vir 26 Lecture Lie algebras of infinite matrices Infinite wedge space F and the Dirac positron theory 33 ix

3 X 4.3. Representations of GL^ and gz^ in F. Unitarity of highest weight representations of gz^ Representation of in F Representations of Vir in F 42 Lecture Boson-fermion correspondence Wedging and contracting operators Vertex operators. The first part of the boson-fermion correspondence Vertex operator representations of gt^ and Ooo 52 Lecture Schur polynomials The second part of the boson-fermion correspondence An application: structure of the Virasoro representations for c = 1 60 Lecture Orbit of the vacuum vector under GL^ Defining equations for in F^ Differential equations for ft in C[xi, X2, } Hirota's bilinear equations The KP hierarchy V-soliton solutions 73 Lecture Degenerate representations and the determinant detn(c, h) of the contravariant form The determinant det (c,h) as a polynomial in h The Kac determinant formula Some consequences of the determinant formula for unitarity and degeneracy 84 Lecture Representations of loop algebras in 5oo Representations of g n in F^ 92

4 xi 9.3. The invariant bilinear form on g n. The action of GLn on gln Reduction from to s n and the unitarity of highest weight representations of s n 96 Lecture Nonabelian generalization of Virasoro operators: the Sugawara construction The Goddard-Kent-Olive construction 109 Lecture s '2 and its Weyl group The Weyl-Kac character formula and Jacobi-Riemann theta functions A character identity 120 Lecture Preliminaries on sd A tensor product decomposition of some representations of s\ Construction and unitarity of the discrete series representations of Vir Completion of the proof of the Kac determinant formula On non-unitarity in the region 0<c<l,/i^0 132 Lecture Formal distributions Local pairs of formal distributions Formal Fourier transform Lambda-bracket of local formal distributions 144 Lecture Completion of U, restricted representations and quantum fields Normal ordered product 157

5 xii Lecture Non-commutative Wick formula Virasoro formal distribution for free boson Virasoro formal distribution for neutral free fermions Virasoro formal distribution for charged free fermions 171 Lecture Conformal weights Sugawara construction Bosonization of charged free fermions Irreducibility theorem for the charge decomposition An application: the Jacobi triple product identity Restricted representations of free fermions 187 Lecture Definition of a vertex algebra Existence Theorem Examples of vertex algebras Uniqueness Theorem and n-th product identity Some constructions Energy-momentum fields Poisson like definition of a vertex algebra Borcherds identity 208 Lecture Definition of a representation of a vertex algebra Representations of the universal vertex algebras On representations of simple vertex algebras On representations of simple affine vertex algebras The Zhu algebra method Twisted representations 223 References 229 Index 235