Topological gravity on the lattice

TL;DR

This paper proposes a lattice formulation of a twisted super Yang-Mills theory in three dimensions that shares features with Chern-Simons gravity, offering a potential non-perturbative approach to 3D quantum gravity.

Contribution

It introduces a supersymmetric, gauge-invariant lattice construction of twisted super Yang-Mills theory related to 3D gravity, enabling non-perturbative studies.

Findings

Identifies classical vacua corresponding to flat complexified gauge connections.

Establishes a set of topological observables linked to Wilson loops.

Proposes a lattice model as a non-perturbative definition of 3D gravity.

Abstract

In this paper we show that a particular twist of $\mathcal{N}=4$ super Yang-Mills in three dimensions with gauge group SU(2) possesses a set of classical vacua corresponding to the space of flat connections of the {\it complexified} gauge group $SL(2,C)$. The theory also contains a set of topological observables corresponding to Wilson loops wrapping non-trivial cycles of the base manifold. This moduli space and set of topological observables is shared with the Chern Simons formulation of three dimensional gravity and we hence conjecture that the Yang-Mills theory gives an equivalent description of the gravitational theory. Unlike the Chern Simons formulation the twisted Yang-Mills theory possesses a supersymmetric and gauge invariant lattice construction which then provides a possible non-perturbative definition of three dimensional gravity.