Classical Gravity Coupled to Liouville Theory

TL;DR

This paper studies a two-dimensional gravity model coupled with Liouville theory and scalar fields, showing it can be viewed as a critical string in a higher-dimensional target space, and analyzes its renormalizability depending on the potential.

Contribution

It demonstrates the interpretation of the coupled gravity-Liouville system as a critical string and analyzes the conditions for its renormalizability with different potentials.

Findings

The combined system acts as a critical string in a target space of dimension c+2.

Renormalizability depends on the form of the potential, with Liouville-like potentials being renormalizable.

The model can be treated as a non-linear sigma model, allowing for perturbative analysis.

Abstract

We consider the two dimensional Jackiw-Teitelboim model of gravity. We first couple the model to the Liouville action and $c$ scalar fields and show, treating the combined system as a non linear sigma model, that the resulting theory can be interpreted as a critical string moving in a target space of dimension $D=c+2$. We then analyse perturbatively a generalised model containing a kinetic term and an arbitrary potential for the auxiliary field. We use the background field method and work with covariant gauges. We show that the renormalisability of the theory depends on the form of the potential. For a general potential, the theory can be renormalised as a non linear sigma model. In the particular case of a Liouville-like potential, the theory is renormalisable in the usual sense.