Two Dimensional String Theory and the Topological Torus

TL;DR

This paper explores the symmetries and algebraic structures of topological string theory on a two-dimensional torus, revealing an infinite symmetry algebra and connections to string theory with specific central charges.

Contribution

It provides a detailed analysis of the symmetry algebra and ground ring structure in topological string theory on a torus, including extensions to supermanifolds and implications for $c=1$ string theory.

Findings

Infinite symmetry algebra from point-like observables

Topological ground ring forms a spacetime manifold

Connections to $c=1$ string theory with topological phases

Abstract

We analyze topological string theory on a two dimensional torus, focusing on symmetries in the matter sector. Even before coupling to gravity, the topological torus has an infinite number of point-like physical observables, which give rise via the BRST descent equations to an infinite symmetry algebra of the model. The point-like observables of ghost number zero form a topological ground ring, whose generators span a spacetime manifold; the symmetry algebra represents all (ground ring valued) diffeomorphisms of the spacetime. At nonzero ghost numbers, the topological ground ring is extended to a superring, the spacetime manifold becomes a supermanifold, and the symmetry algebra preserves a symplectic form on it. In a decompactified limit of cylindrical target topology, we find a nilpotent charge which behaves like a spacetime topological BRST operator. After coupling to topological gravity, this model might represent a topological phase of $c=1$ string theory. We also point out some analogies to two dimensional superstrings with the chiral GSO projection, and to string theory with $c=-2$.