Kleinian Singularities and the Ground Ring of C=1 String Theory

TL;DR

This paper explores the structure of the ground ring in c=1 string theory at special A-D-E points, revealing connections to Kleinian singularities, complex hyperKahler surfaces, and symmetries related to gravitational instantons.

Contribution

It establishes a link between the ground rings of c=1 string theory and Kleinian singularities, highlighting their geometric and symmetry properties at special moduli points.

Findings

Ground rings at A-D-E points define Kleinian singular varieties.

Non-chiral ground rings are U(1) quotients of Kleinian varieties.

Unbroken symmetries are volume-preserving diffeomorphisms.

Abstract

We investigate the nature of the ground ring of c=1 string theory at the special A-D-E 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 A-D-E series of singular varieties introduced by Klein. The non-chiral ground rings relevant to closed-string 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 diffeomorphisms 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.