A reciprocity formula from abelian BF and Turaev-Viro theories

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.