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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 $Γ$. We show that there is a region $Γ_0$ of $Γ$ where the partition function and the expectation value $<W_R(γ)>$ of the Wilson loop for a curve $γ$ can be exactly computed. Depending on the point of $Γ_0$, 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 $γ$. However, for a subset of $Γ_0$, $<W_R(γ)>$ depends on the size of $γ$ 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.