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

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 $\mathrm{\Gamma }$. We show that there is a region ${\mathrm{\Gamma }}_0$ of $\mathrm{\Gamma }$ 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 ${\mathrm{\Gamma }}_0$, 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 ${\mathrm{\Gamma }}_0$, $<W_R(\gamma )>$ 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.