  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.
