N.J.B. Aza, F. Doresty F.
On Lattice Topological Field Theories with Finite Groups
(3405K, .pdf)
ABSTRACT. The approach of Lattice Topological Field Theories is used to describe
quantities which are independent of manifolds, and the same time Lattice Gauge Theories are important to renormalize continuous theories. Therefore, the natural connection between both theories can be made to understand physical topological theories. In this work, we review the basic concepts of each theory and study gauge theories coupled with matter fields in two-dimensional manifolds. In order to
proceed, we first describe a formalism in two and three dimensions which is based on the idea of Kuperberg of defining a topological invariant in three dimensions using Hopf algebras and Heegaard diagrams. This formalism is useful in the context of our analysis because it allows to easily identify topological limits without
solving the model. Furthermore, we write the gauge model with matter fields choosing the unitary gauge, working with finite groups, in particular with the abelian group $\mathbb{Z}_{n}$ and explaining the $\mathbb{Z}_{2}$ case in detail. We calculate partition functions and Wilson loops for this group in different topological limits. We show
that there were cases in which the results depended on the triangulation although in a trivial way, these cases are called quasi-topological.