Gravitational Actions in Two Dimensions and the Mabuchi Functional

TL;DR

This paper demonstrates how the Mabuchi energy functional influences two-dimensional gravitational actions, leading to new quantum gravity models that extend traditional Liouville theory by coupling with massive scalar fields.

Contribution

It introduces a novel connection between the Mabuchi functional and 2D quantum gravity, expanding the theoretical framework beyond standard Liouville models.

Findings

Mabuchi energy contributes to effective 2D gravitational actions.

New quantum gravity theories generalize Liouville models.

Coupling with massive scalar fields modifies gravitational dynamics.

Abstract

The Mabuchi energy is an interesting geometric functional on the space of K\"ahler metrics that plays a crucial r\^ole in the study of the geometry of K\"ahler manifolds. We show that this functional, as well as other related geometric actions, contribute to the effective gravitational action when a massive scalar field is coupled to gravity in two dimensions in a small mass expansion. This yields new theories of two-dimensional quantum gravity generalizing the standard Liouville models.