Gravitational action for a massive Majorana fermion in 2d quantum gravity

TL;DR

This paper calculates the gravitational action for a massive Majorana fermion in 2D quantum gravity, revealing connections to Liouville and Mabuchi actions and highlighting differences from the DDK ansatz.

Contribution

It provides the first detailed computation of the gravitational action for a massive Majorana fermion in 2D gravity, extending previous scalar field results.

Findings

Small-mass expansion yields Liouville action at zeroth order.

First-order correction identified as Mabuchi action.

Result differs from the DDK ansatz for conformal deformations.

Abstract

We compute the gravitational action of a free massive Majorana fermion coupled to two-dimensional gravity on compact Riemann surfaces of arbitrary genus. The structure is similar to the case of the massive scalar. The small-mass expansion of the gravitational yields the Liouville action at zeroth order, and we can identify the Mabuchi action at first order. While the massive Majorana action is a conformal deformation of the massless Majorana CFT, we find an action different from the one given by the David-Distler-Kawai (DDK) ansatz.