Quasi-Topological Quantum Field Theories and $Z_2$ Lattice Gauge Theories

TL;DR

This paper explores a family of $Z_2$ lattice gauge theories on 3-manifolds, identifying regions where they behave as topological or quasi-topological theories, and analyzing their partition functions and Wilson loop expectations.

Contribution

It characterizes the parameter space regions where $Z_2$ gauge theories are exactly solvable and exhibit topological or quasi-topological behavior, linking them to topological field theories.

Findings

Partition function is a topological invariant up to scaling.

Wilson loop expectation can be independent of topology or follow an area law.

Behavior depends on the parameter region and temperature limit.

Abstract

We consider a two parameter family of $Z_2$ gauge theories on a lattice discretization $T(M)$ of a 3-manifold $M$ and its relation to topological field theories. Familiar models such as the spin-gauge model are curves on a parameter space $\Gamma$. We show that there is a region $\Gamma_0$ of $\Gamma$ where the partition function and the expectation value $<W_R(\gamma)>$ of the Wilson loop for a curve $\gamma$ can be exactly computed. Depending on the point of $\Gamma_0$, the model behaves as topological or quasi-topological. The partition function is, up to a scaling factor, a topological number of $M$. The Wilson loop on the other hand, does not depend on the topology of $\gamma$. However, for a subset of $\Gamma_0$, $<W_R(\gamma)>$ depends on the size of $\gamma$ and follows a discrete version of an area law. At the zero temperature limit, the spin-gauge model approaches the topological and the quasi-topological regions depending on the sign of the coupling constant.