$c_M<1$ String Theory as a Constrained Topological Sigma Model

TL;DR

This paper generalizes the connection between $c_M<1$ string theory and constrained topological sigma models, providing a topological framework that reproduces the physical spectrum of the string theory.

Contribution

It extends the construction of $c_M<1$ string theory as a constrained topological sigma model to arbitrary $(p,q)$ minimal models coupled to 2D gravity, establishing a topological representation.

Findings

Mapped energy--momentum tensor and topological charge to string theory counterparts

Recovered the physical spectrum in the absolute cohomology

Provided a topological formulation of the Liouville approach

Abstract

It has been argued by Ishikawa and Kato that by making use of a specific bosonization, $c_M=1$ string theory can be regarded as a constrained topological sigma model. We generalize their construction for any $(p,q)$ minimal model coupled to two dimensional (2d) gravity and show that the energy--momentum tensor and the topological charge of a constrained topological sigma model can be mapped to the energy--momentum tensor and the BRST charge of $c_M<1$ string theory at zero cosmological constant. We systematically study the physical state spectrum of this topological sigma model and recover the spectrum in the absolute cohomology of $c_M<1$ string theory. This procedure provides us a manifestly topological representation of the continuum Liouville formulation of $c_M<1$ string theory.