Generalized Abelian Turaev-Viro and $\mathrm{U}\!\left(1\right)$ BF Theories

TL;DR

This paper develops a unified framework for Abelian Turaev-Viro invariants and $ ext{U}(1)$ BF theories across multiple dimensions using Deligne-Beilinson cohomology, linking topological invariants with discrete BF theories.

Contribution

It introduces a method to extend $ ext{U}(1)$ BF theory and Turaev-Viro invariants to higher dimensions within an Abelian setting, connecting these invariants to discrete BF theories.

Findings

Extended $ ext{U}(1)$ BF theory to any dimension.

Generalized Turaev-Viro invariants in Abelian framework.

Linked invariants to discrete BF theories.

Abstract

We explain how it is possible to study $\mathrm{U}\!\left(1\right)$ BF theory over a connected closed oriented smooth $3$-manifold in the formalism of path integral thanks to Deligne-Beilinson cohomology. We show how we can straightforwardly extend the definition to families of theories in any dimension. We extend then the definition of the Turaev-Viro invariant of a connected closed oriented smooth $3$-manifold in an Abelian framework to a family of invariants in any dimension. We show that those invariants can be written as discrete BF theories. We explain how the extensions of $\mathrm{U}\!\left(1\right)$ BF theory we defined can be related to the extensions of Turaev-Viro invariant we constructed.