Finite group discretization of Yang-Mills and Einstein actions

TL;DR

This paper introduces discrete formulations of Yang-Mills and Einstein actions for finite groups, maintaining gauge and coordinate invariance, and recovers known and new discretized actions including a gravity model.

Contribution

It proposes a novel finite group discretization framework for Yang-Mills and Einstein actions, extending gauge and gravity theories to discrete settings.

Findings

Recover the Wilson action for Yang-Mills theories

Introduce a new discretized action for gravity

Demonstrate invariance under gauge and coordinate transformations

Abstract

Discrete versions of the Yang-Mills and Einstein actions are proposed for any finite group. These actions are invariant respectively under local gauge transformations, and under the analogues of Lorentz and general coordinate transformations. The case Z_n \times Z_n \times...\times Z_n is treated in some detail, recovering the Wilson action for Yang-Mills theories, and a new discretized action for gravity.