Kac-Moody and Virasoro algebras

Abstract

These are the notes for a Part III course given in the University of Cambridge in autumn 1998. They contain an exposition of the representation theory of the Lie algebras of compact matrix groups, affine Kac-Moody algebras and the Virasoro algebra from a unitary point of view. The treatment uses many of the methods of conformal field theory, in particular the Goddard-Kent-Olive construction and the Kazami-Suzuki supercharge operator, a generalisation of the Dirac operator. The proof of the Weyl character formula is taken from unpublished notes of Peter Goddard. The supersymmetric proof of the Kac character formula for affine Kac-Moody algebras was also found independently at roughly the same time by Greg Landweber in his Harvard Ph.D. dissertation. One of the main novelties of this approach is a very rapid proof of the Feigin-Fuchs character formula for the discrete series representations of the Virasoro algebra. It relies only on the first part of the Friedan-Qiu-Shenker unitarity theorem placing restrictions on the parameter h. Their result, which only uses elementary properties of plane curves, is presented in detail in these notes along with the somewhat harder result for c. A substantially shorter treatment is possible if the final goal is just the Feigin-Fuchs formula, since in that case only the representation theory of affine sl(2) is needed (cf the MSRI summer course given in 2000, http://www.msri.org/publications/ln/msri/2000/opal/wassermann/). The same method has been applied by S. Palcoux to the Neveu-Schwarz algebra and also works for the N=2 superconformal algebra.