A Hamiltonian Formulation of Topological Gravity

TL;DR

This work develops a Hamiltonian lattice formulation of topological gravity in 3+1 dimensions, where flat spacetimes emerge from a geometrical, torsion-free gauge and all local degrees of freedom are gauge artifacts. By discretizing with $SO(3,1)$ transport matrices and area bivectors, the authors derive a set of first-class constraints that enforce cell closure and vanishing curvature, and identify translation generators that mirror diffeomorphisms in the continuum. A geometrical gauge further expresses bivectors in terms of edge-vectors and imposes conditions that guarantee flat geometricity, leaving only the vertex-translation constraints as the physical symmetry, and yielding zero local DOF for typical topologies via $d=6\chi$. The paper outlines a path toward curved-space lattice gravity by seeking first-class extensions of the translation constraints (potentially via BRST methods) or by adopting a teleparallel viewpoint, suggesting that a consistent lattice theory of gravity with curvature may be achievable.

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.