q-deformed lattice gauge theory and 3-manifold invariants

TL;DR

This paper introduces a q-deformed lattice gauge theory framework that produces 3-manifold invariants at roots of unity, generalizing Turaev-Viro invariants and linking them to fundamental group actions and link invariants.

Contribution

It develops a new q-deformed lattice gauge theory approach to define and analyze 3-manifold invariants, extending existing topological invariants like Turaev-Viro.

Findings

Defines a q-deformed lattice gauge theory at roots of unity.

Establishes a generalized Turaev-Viro invariant for cell complexes.

Connects the invariant to fundamental group actions and link invariants.

Abstract

The notion of $q$-deformed lattice gauge theory is introduced. If the deformation parameter is a root of unity, the weak coupling limit of a 3-$d$ partition function gives a topological invariant for a corresponding 3-manifold. It enables us to define the generalized Turaev-Viro invariant for cell complexes. It is shown that this invariant is determined by an action of a fundamental group on a universal covering of a complex. A connection with invariants of framed links in a manifold is also explored. A model giving a generating function of all simplicial complexes weighted with the invariant is investigated.