Discrete canonical analysis of three dimensional gravity with cosmological constant

TL;DR

This paper explores the canonical analysis of three-dimensional gravity with a cosmological constant, examining how continuum symmetries are affected by discretization and identifying preserved gauge freedoms in the lattice approach.

Contribution

It provides a detailed comparison between continuum and discretized canonical structures, highlighting the preservation of certain gauge symmetries despite explicit symmetry breaking.

Findings

Continuum local symmetries correspond to on-shell space-time diffeomorphisms.

Discretization breaks some symmetries but preserves gauge freedoms related to constant curvature.

The lattice approach maintains gauge invariance through holonomies and Gauss's law.

Abstract

We discuss the interplay between standard canonical analysis and canonical discretization in three-dimensional gravity with cosmological constant. By using the Hamiltonian analysis, we find that the continuum local symmetries of the theory are given by the on-shell space-time diffeomorphisms, which at the action level, corresponds to the Kalb-Ramond transformations. At the time of discretization, although this symmetry is explicitly broken, we prove that the theory still preserves certain gauge freedom generated by a constant curvature relation in terms of holonomies and the Gauss's law in the lattice approach.