Kleinian singularities and the ground ring of c = 1 string theory

Abstract

We investigate the nature of the ground ring of c = 1 string theory at the special ADE points in the c = 1 moduli space associated to discrete subgroups of SU(2). The chiral ground rings at these points are shown to define the ADE series of singular varieties introduced by Klein. The non-chiral ground rings relevant to closed-string theory are 3 real dimensional singular theory are 3 real dimensional singular varieties obtained as U(1) quotients of the kleinian varieties. The unbroken symmetries of the theory at these points are the volume-preserving diffeomophisms of these varieties. The theory of kleinian singularities has a close relation to that of complex hyperKahler surfaces, or gravitational instantons. We speculate on the relevance of these instantons and of self-dual gravity in c = 1 string theory.