A Model of Three-Dimensional Lattice Gravity

TL;DR

This paper introduces a model for 3D lattice gravity using topological invariants, connecting quantum gravity with lattice gauge theories and topological quantum field theories like Turaev-Viro and Dijkgraaf-Witten.

Contribution

It proposes a unified model generating 3D simplicial complexes weighted by topological invariants, linking quantum gravity to lattice gauge theories and topological invariants.

Findings

When gauge group is SU_q(2), the model reproduces the Turaev-Viro invariant.

For finite abelian groups, the invariant relates to the rank of the first cohomology group.

The model encompasses a topological expansion based on Betti numbers.

Abstract

A model is proposed which generates all oriented $3d$ simplicial complexes weighted with an invariant associated with a topological lattice gauge theory. When the gauge group is $SU_q(2)$, $q^n=1,$ it is the Turaev-Viro invariant and the model may be regarded as a non-perturbative definition of $3d$ simplicial quantum gravity. If one takes a finite abelian group $G$, the corresponding invariant gives the rank of the first cohomology group of a complex \nolinebreak $C$: $I_G(C) = rank(H^1(C,G))$, which means a topological expansion in the Betti number $b^1$. In general, it is a theory of the Dijkgraaf-Witten type, $i.e.$ determined completely by the fundamental group of a manifold.