• Chern-Simons theory
• gauge theory
• three-dimensional torus.
• topology

Chern-Simons theory is a pure gauge theory in three dimensions; this model is interesting not only per se, but also for its applications in Mathematics and in Physics. In the mathematical context, the Chen-Simons theory can be used to compute topological invariants. In Physics, phenomenological models, based on the Chern-Simons lagrangian terms, have been produced to describe the dynamics of quasi-particles with odd statistics in particular solid state systems.

From a theoretical point of view, Chern-Simons theory is interesting for its peculiar characteristics as a topological gauge field theory. In fact, studying this theory, we have the chance to face closely and to manage the main features of the path integral formalism.

Indeed, the Chern-Simons action can be written as a three-form integrated over the whole three-dimensional manifold, for this reasons the theory can be classically defined even if no metric is given, this feature is called general covariance. Even if, at the quantum level, one has to introduce a gauge fixing that explicitly breaks the general covariance, it can be proved that the observables maintain this invariance.

An important expectation value which can be deduced from the theory is the Wilson loop, which corresponds to the trace of the holonomy computed along a closed path in the space. This expectation value is, indeed, a gauge invariant and general covariant quantity. Thus the Wilson loops connect the Chern-Simons theory and the knot theory, because they represent topological invariants associated with knots and links.

Chern-Simons theory is defined starting from the real three-dimensional space and it is extended to all closed, connected and orientable three-dimensional manifolds. This nontrivial extension is made by producing a surgery transformation and its corresponding operator realization.

In this thesis we attempt to define directly the path-integral and the perturbative methods in a particular manifold, i.e. the three-dimensional torus. The above mentioned manifold represents a simple example of a nontrivial manifold. In this framework, we present a general introduction to the Chern-Simons model, we calculate the first orders in the loop expansion of the partition function and the first order of the perturbative expansion of the Wilson loop expectation values of some links.