  In this article we construct the Reshetikhin-Turaev invariant associated with
the Drinfeld Center of the spherical category arising from the $U(1)$ BF theory
on a closed $3$-manifold $M$. This invariant is shown to coincide with the
Turaev-Viro invariant of $M$ thus providing an example of the Turaev-Virelizier
theorem. Finally we exhibit some surgery formulas for the abelian Turaev-Viro
invariant which are very similar to the surgery formulas of the abelian
Reshetikhin-Turaev invariant obtained in the $U(1)$ Chern-Simons context.
