• topological quantum field theory
• Reshetikhin-Turaev invariant
• path integral
• Deligne-Beilinson cohomology
• Chern-Simons theory

In order to quantize the U(1) Chern-Simons field theory via path-integration an action is required, and the usual definition as the integral of the Lagrangian density fails as this is only defined locally. To this end, the Deligne-Beilinson (DB) cohomology groups are introduced as a tool to classify and manipulate the connection over the base M. In this context a product is defined, as well as an integration over singular cycles with values in the real numbers modulo integers. If A is a configuration of the gauge field, the associated action is then given by integration over M of the product of A (seen as a DB class) with itself.

The observables of the quantum version of the theory are (generated by) the holonomies along framed coloured links in the base, and once the action is defined it is possible to introduce a path integral and a partition function as sums performed on a suitable DB cohomology group. Due to the structure of this group, the computation of the integrals is reduced to a sum over the first homology group of M of integrals over a simpler space.

The actual computation leads then to a topological invariant for the manifold, this being relater to the Witten-Reshetikhin-Turaev surgery invariant for M.