Abelian BF theory

This dissertation considers the abelian BF theory on a closed manifold of dimension 3. It is formulated in terms of gauge classes which turn out to be Deligne-Beilinson cohomology classes. This formulation offers the possibility to extract the mathematically relevant quantities from formal functional integrals. The partition function and the expectation value of observables are therefore computed in this sense. They are proved to be connected, on the one hand, with their analogue obtained in the same manner from the abelian Chern-Simons theory, and, on the other hand, with their analogue defined in the framework of the abelian Reshetikhin-Turaev and Turaev-Viro theories. Two extensions of this study are also discussed. Firstly, a graphical approach is proposed to compute the SU(N) classical Chern- Simons invariant. Secondly, a geometric interpretation of the gauge fixing procedure is presented for the abelian Chern-Simons theory in R^4l+3. (author)