Modular Invariance and Infinite-Dimensional Algebras

Abstract

This lecture is concerned with certain infinite-dimensional algebras, namely affine Kac-Moody algebras and the Virasoro algebra, their representations and the modular transformation properties of the characters of these representations. These algebras and their properties are of central importance in the study of conformally invariant two-dimensional systems, and hence in analysing the critical behaviour of suitable two-dimensional statistical systems, and in string theories where they are of relevance because of the two-dimensional nature of the string world-sheet. For a recent review of these algebras in relation to their applications in physics see [1]. Our objective is to show how the properties of representations of the Virasoro algebra can be understood in terms of the corresponding properties of affine Kac-Moody representations [2–4].