2D quantum gravity on compact Riemann surfaces with non-conformal matter

TL;DR

This paper investigates the gravitational action in 2D quantum gravity coupled with non-conformal massive matter on compact Riemann surfaces, providing a finite expression valid for any mass and connecting to known actions in the massless limit.

Contribution

It introduces a finite, well-defined gravitational action for non-conformal matter on Riemann surfaces, extending previous conformal results to massive cases with a systematic expansion.

Findings

Recover the Liouville action in the massless limit

Derive Mabuchi and Aubin-Yau actions at first order in mass

Present an infinite series of geometric contributions for finite mass

Abstract

We study the gravitational action induced by coupling two-dimensional non-conformal, massive matter to gravity on a compact Riemann surface. We express this gravitational action in terms of finite and well-defined quantities for any value of the mass. A small-mass expansion gives back the Liouville action in the massless limit, the Mabuchi and Aubin-Yau actions to first order, as well as an infinite series of higher-order contributions written in terms of purely geometric quantities.