Breaking and restoring of diffeomorphism symmetry in discrete gravity

TL;DR

This work analyzes the fate of diffeomorphism invariance in discrete gravity, arguing that discretization generally breaks this symmetry but can be countered by constructing perfect actions that reproduce continuum gauge invariance. Using one-dimensional reparametrization-invariant models as tractable proxies, it shows how discretization introduces gauge-breaking effects and how a Wilsonian blocking-from-the-continuum approach can yield exact, symmetry-preserving discrete actions. The authors apply these ideas to 3D Regge gravity with a cosmological constant, demonstrating that a perfect action capturing the continuum dynamics and the vertex-displacement gauge exists when appropriately formulated, while in higher dimensions the situation is more complex but perturbative and area-based formulations offer paths toward symmetry restoration. They further discuss the canonical implications, showing that exact symmetries lead to first-class constraints, while approximate symmetries induce pseudo constraints; understanding these structures is essential for connecting discrete quantum gravity to continuum GR and guiding quantization strategies.

Abstract

We discuss the fate of diffeomorphism symmetry in discrete gravity. Diffeomorphism symmetry is typically broken by the discretization. This has repercussions for the observable content and the canonical formulation of the theory. It might however be possible to construct discrete actions, so--called perfect actions, with exact symmetries and we will review first steps towards this end.