# Functional integration on Regge geometries

## TL;DR

This paper derives exact expressions for the Liouville action and integration measure in Regge gravity, extending the approach to higher dimensions with geometrically invariant measures.

## Contribution

It provides the first exact discretized Liouville action and invariant measure for Regge geometries in multiple dimensions, generalizing previous continuum results.

## Key findings

- Exact Liouville action in 2D Regge gravity
- Explicit invariant measure for conformal factors
- Extension of the approach to higher dimensions

## Abstract

We adopt the standard definition of diffeomorphism for Regge gravity in D=2 and give an exact expression of the Liouville action in the discretized case. We also give the exact form of the integration measure for the conformal factor. In D>2 we extend the approach to any family of geometries described by a finite number of parameters. The ensuing measure is a geometric invariant and it is also invariant in form under an arbitrary change of parameters.