Quasi-Topological Field Theories in Two Dimensions as Soluble Models

TL;DR

This paper introduces a class of two-dimensional lattice field theories called quasi-topological models, which are solvable and depend only on global surface properties, with continuum limits connecting to topological field theories.

Contribution

It defines quasi-topological lattice models in 2D, shows their solvability, and classifies their universality classes, linking them to topological and Yang-Mills theories.

Findings

Partition functions depend only on genus, number of triangles, and area.

Continuum limit yields topological field theories.

Yang-Mills theories are special cases within this framework.

Abstract

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 $\epsilon$. The reason these models are called quasi-topological is that $Z$ depends on $g$, $n$ and $\epsilon$ 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., $\epsilon \to 0$ 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.