In the 80s, Atiyah, Donaldson and Witten initiated a program consisting in researching some relations between gauge theories and the topology of low dimensional manifolds. In this framework, Witten related the Jones polynomial of a knot to the expectation value of this knot in Chern-Simons theory, providing thus an interesting application of quantum field theory tools to the computation of topological invariants.
In this talk, we will be interested in the abelian BF theory defined on a 3-manifold M. We will show how Deligne-Beilinson cohomology, that classifies U(1) connections, makes it possible to extract relevant topological quantities from the abelian BF partition function and expectation value of observables. We will see also how the relations between the abelian BF theory and the abelian Reshetikhin-Turaev and Turaev-Viro theories, which provide usually invariants of 3-manifolds, lead to a ``surgery formula'' that transforms the computation of expectation value of observables in M into the computation of expectation value of observables in the 3-sphere.