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

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.