A Hamiltonian Formulation of Topological Gravity

TL;DR

This paper develops an exact Hamiltonian lattice formulation of topological gravity, emphasizing gauge symmetries that eliminate local degrees of freedom and proposing methods to extend this to curved space-times.

Contribution

It introduces a Hamiltonian lattice model for topological gravity with novel gauge symmetries and suggests approaches for formulating lattice theories of curved space-times.

Findings

Gauge symmetries eliminate local degrees of freedom.

In a specific gauge, the theory describes flat space-times.

Proposes methods for extending to curved space-times.

Abstract

Topological gravity is the reduction of Einstein's theory to spacetimes with vanishing curvature, but with global degrees of freedom related to the topology of the universe. We present an exact Hamiltonian lattice theory for topological gravity, which admits translations of the lattice sites as a gauge symmetry. There are additional symmetries, not present in Einstein's theory, which kill the local degrees of freedom. We show that these symmetries can be fixed by choosing a gauge where the torsion is equal to zero. In this gauge, the theory describes flat space-times. We propose two methods to advance towards the holy grail of lattice gravity: A Hamiltonian lattice theory for curved space-times, with first-class translation constraints.