Gauge Invariance in Simplicial Gravity

TL;DR

This paper demonstrates that local gauge invariance in simplicial lattice gravity accurately reflects continuum diffeomorphism invariance, with explicit analysis of gauge properties, zero modes, and scalar field coupling, primarily in two dimensions.

Contribution

It explicitly establishes the exact local gauge invariance of the simplicial gravity action and relates discrete variables to continuum metrics, extending to scalar fields and conformal gauges.

Findings

Discrete gauge invariance matches continuum diffeomorphisms

No Faddeev-Popov determinant needed unless gauge fixing is applied

Results primarily validated in two-dimensional models

Abstract

The issue of local gauge invariance in the simplicial lattice formulation of gravity is examined. We exhibit explicitly, both in the weak field expansion about flat space, and subsequently for arbitrarily triangulated background manifolds, the exact local gauge invariance of the gravitational action, which includes in general both cosmological constant and curvature squared terms. We show that the local invariance of the discrete action and the ensuing zero modes correspond precisely to the diffeomorphism invariance in the continuum, by carefully relating the fundamental variables in the discrete theory (the edge lengths) to the induced metric components in the continuum. We discuss mostly the two dimensional case, but argue that our results have general validity. The previous analysis is then extended to the coupling with a scalar field, and the invariance properties of the scalar field action under lattice diffeomorphisms are exhibited. The construction of the lattice conformal gauge is then described, as well as the separation of lattice metric perturbations into orthogonal conformal and diffeomorphism part. The local gauge invariance properties of the lattice action show that no Fadeev-Popov determinant is required in the gravitational measure, unless lattice perturbation theory is performed with a gauge-fixed action, such as the one arising in the lattice analog of the conformal or harmonic gauges.