# 2D gravitational Mabuchi action on Riemann surfaces with boundaries

## TL;DR

This paper investigates the gravitational action for 2D non-conformal massive matter on Riemann surfaces with boundaries, revealing boundary-dependent corrections and generalizing the Mabuchi action.

## Contribution

It introduces a first-order mass correction to the Mabuchi action on surfaces with boundaries, extending previous boundaryless formulations.

## Key findings

- Mass correction reduces to Liouville action in the massless limit
- Gravitational Lagrangian depends on boundary distances and modular parameters
- Explicit formulas derived for the cylinder case

## Abstract

We study the gravitational action induced by coupling two-dimensional non-conformal, massive matter to gravity on a Riemann surface with boundaries. A small-mass expansion gives back the Liouville action in the massless limit, while the first-order mass correction allows us to identify what should be the appropriate generalization of the Mabuchi action on a Riemann surface with boundaries. We provide a detailed study for the example of the cylinder. Contrary to the case of manifolds without boundary, we find that the gravitational Lagrangian explicitly depends on the space-point, via the geodesic distances to the boundaries, as well as on the modular parameter of the cylinder, through an elliptic theta-function.