Super Liouville action for Regge surfaces

TL;DR

This paper calculates the super Liouville action for Regge surfaces on different topologies, revealing how invariance principles determine the action up to certain topological and modular invariants.

Contribution

It extends the computation of super Liouville action to Regge surfaces, incorporating superconformal and supermodular invariance for various topologies.

Findings

Action fixed up to topological invariant for sphere and even spin torus

Normalization derived from perturbation theory

Additional fermionic zero mode contributions for odd spin torus

Abstract

We compute the super Liouville action for a two dimensional Regge surface by exploiting the invariance of the theory under the superconformal group for sphere topology and under the supermodular group for torus topology. For sphere topology and torus topology with even spin structures, the action is completely fixed up to a term which in the continuum limit goes over to a topological invariant, while the overall normalization of the action can be taken from perturbation theory. For the odd spin structure on the torus, due to the presence of the fermionic supermodulus, the action is fixed up to a modular invariant quadratic polynomial in the fermionic zero modes.