Topological Lattice Gravity Using Self-Dual Variables

TL;DR

This paper develops a lattice model for topological gravity using self-dual variables, connecting it to Ashtekar's formalism and discussing potential extensions and quantization strategies.

Contribution

It introduces a lattice formulation of topological gravity with self-dual variables, aligning symmetry generators with Ashtekar's constraints and removing extra symmetries of $B ext{-}F$ theory.

Findings

Symmetry generators resemble Ashtekar's constraints.

Constraints close only after initial flat lattice conditions.

Framework suggests pathways for non-flat and quantum extensions.

Abstract

Topological gravity is the reduction of general relativity to flat space-times. A lattice model describing topological gravity is developed starting from a Hamiltonian lattice version of $B\w F$ theory. The extra symmetries not present in gravity that kill the local degrees of freedom in $B\wedge F$ theory are removed. The remaining symmetries preserve the geometrical character of the lattice. Using self-dual variables, the conditions that guarantee the geometricity of the lattice become reality conditions. The local part of the remaining symmetry generators, that respect the geometricity-reality conditions, has the form of Ashtekar's constraints for GR. Only after constraining the initial data to flat lattices and considering the non-local (plus local) part of the constraints does the algebra of the symmetry generators close. A strategy to extend the model for non-flat connections and quantization are discussed.