Structure of Topological Lattice Field Theories in Three Dimensions

TL;DR

This paper classifies three-dimensional topological lattice field theories by linking invariance under lattice moves to Hopf algebras, connecting to known theories like lattice gauge theory and Ponzano-Regge.

Contribution

It introduces a general framework for topological lattice theories in 3D, establishing a correspondence with Hopf algebras and providing explicit examples.

Findings

Invariant solutions correspond to Hopf algebras satisfying specific constraints.

The Hopf algebra based on the group ring $ ext{C}[G]$ reproduces lattice gauge theory at zero coupling.

The same Hopf algebra yields the Ponzano-Regge model for $G=SU(2)$.

Abstract

We construct and classify topological lattice field theories in three dimensions. After defining a general class of local lattice field theories, we impose invariance under arbitrary topology-preserving deformations of the underlying lattice, which are generated by two new local lattice moves. Invariant solutions are in one--to--one correspondence with Hopf algebras satisfying a certain constraint. As an example, we study in detail the topological lattice field theory corresponding to the Hopf algebra based on the group ring $\C[G]$, and show that it is equivalent to lattice gauge theory at zero coupling, and to the Ponzano--Regge theory for $G=$SU(2).