A Model of Three-Dimensional Lattice Gravity

TL;DR

This paper constructs a 3D lattice gravity framework by summing over oriented 3D simplicial complexes weighted with topological lattice gauge theory invariants. It presents a general construction where finite groups yield invariants tied to $H^1$ and, for abelian groups, a Betti-number expansion, while a $q$-deformed $SU(2)$ at roots of unity connects to the Turaev–Viro invariant, offering a non-perturbative definition of 3D quantum gravity. By showing topological invariance under 3D moves and detailing the algebraic structure of the invariants, the work provides a controllable, universal approach to 3D quantum gravity with potential for continuum limits and new manifold invariants. The framework unifies Dijkgraaf–Witten, Ponzano–Regge, and TV-type constructions and suggests rich future directions in topological lattice gravity and quantum-group regularizations.

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.