A reciprocity formula from abelian BF and Turaev-Viro theories

TL;DR

This paper demonstrates that using Deligne-Beilinson cohomology in $U(1)$ BF theory on closed 3-manifolds produces a discrete $ ext{Z}_N$ BF theory with a partition function matching an abelian Turaev-Viro invariant, leading to a reciprocity formula.

Contribution

It introduces a novel connection between $U(1)$ and $ ext{Z}_N$ BF theories via Deligne-Beilinson cohomology and derives a reciprocity formula comparing their expectation values.

Findings

Partition function of discrete $ ext{Z}_N$ BF theory equals abelian Turaev-Viro invariant.

Derived a reciprocity formula relating $U(1)$ and $ ext{Z}_N$ holonomies.

Established a new link between cohomology, topological invariants, and gauge theories.

Abstract

In this article we show that the use of Deligne-Beilinson cohomology in the context of the $U(1)$ BF theory on a closed 3-manifold $M$ yields a discrete $\Z_N$ BF theory whose partition function is an abelian TV invariant of $M$. By comparing the expectation values of the $U(1)$ and $\mathbb{Z}_N$ holonomies in both BF theories we obtain a reciprocity formula.