One-loop effective potential in 2D dilaton gravity on hyperbolic plane

TL;DR

This paper calculates the one-loop effective potential in 2D dilaton gravity on hyperbolic and topologically non-trivial planes, revealing dependence on the reference metric except for special scalar potentials like Liouville.

Contribution

It provides the first explicit calculation of the one-loop effective potential in 2D dilaton gravity on hyperbolic spaces and analyzes its dependence on the reference metric.

Findings

Effective potential depends on the reference metric for general scalar potentials.

The theory becomes ultraviolet finite for the Liouville potential.

Discussion of the effective equations and interpretation of metric dependence.

Abstract

The one-loop effective potential in $2D$ dilaton gravity in conformal gauge on the topologically non-trivial plane $\reals \times S^1$ and on the hyperbolic plane $H^2/\Gamma$ is calculated. For arbitrary choice of the tree scalar potential it is shown, that the one-loop effective potential explicitly depends on the reference metric (through the dependence on the radius of the torus or the radius of $H^2/\Gamma$). This phenomenon is absent only for some special choice of the tree scalar potential corresponding to the Liouville potential and leading to one-loop ultraviolet finite theory. The effective equations are discussed and some interpretation of the reference metric dependence of the effective potential is made.