Quasi-Topological Field Theories in Two Dimensions as Soluble Models

We study a class of lattice field theories in two dimensions that includes gauge theories. Given a two dimensional orientable surface of genus g, the partition function Z is defined for a triangulation consisting of n triangles of area ǫ. The reason these models are called quasi-topological is that Z depends on g, n and ǫ but not on the details of the triangulation. They are also soluble in the sense that the computation of their partition functions can be reduced to a soluble one dimensional problem. We show that the continuum limit is well defined if the model approaches a topological field theory in the zero area limit, i.e., ǫ → 0 with finite n. We also show that the universality classes of such quasi-topological lattice field theories can be easily classified. Yang-Mills and generalized Yang-Mills theories appear as particular examples of such continuum limits.