Mabuchi spectrum from the minisuperspace

TL;DR

This paper introduces a minisuperspace model for the Mabuchi functional in two-dimensional gravity, showing it shares the same spectrum and eigenfunctions as Liouville theory in a simplified zero-mode approximation.

Contribution

It proposes a minisuperspace action for the Mabuchi functional that reproduces key features and demonstrates its spectral equivalence with Liouville theory.

Findings

The Hamiltonian of the Mabuchi minisuperspace model matches the Liouville Hamiltonian.

Both theories share the same spectrum and eigenfunctions in this approximation.

The model provides insights into the zero-mode dynamics of the Mabuchi functional.

Abstract

It was recently shown that other functionals contribute to the effective action for the Liouville field when considering massive matter coupled to two-dimensional gravity in the conformal gauge. The most important of these new contributions corresponds to the Mabuchi functional. We propose a minisuperspace action that reproduces the main features of the Mabuchi action in order to describe the dynamics of the zero-mode. We show that the associated Hamiltonian coincides with the (quantum mechanical) Liouville Hamiltonian. As a consequence the Liouville theory and our model of the Mabuchi theory both share the same spectrum, eigenfunctions and - in this approximation - correlation functions.