Quasi-topological quantum field theories and ₂ lattice gauge theories

We consider a two-parameter family of ₂ gauge theories on a lattice discretization $T({\cal M})$ of a three-manifold ${\cal M}$ 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 ⊂ Γ where the partition function and the expectation value 〈W _R(γ) 〉 of the Wilson loop 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 ${\cal M}$. 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.