• abelian Chern-Simons
• Chern-Simons
• gauge theory
• Gauss sum.
• homology
• homology spheres
• knot theory
• knots
• non-Abelian Chern-Simons
• Reshetikhin
• surgery
• surgery invariant
• surgery rules
• three-manifolds
• topological field theory
• Turaev
• Wilson loop

RIASSUNTO DELLA TESI DI LAUREA SPECIALISTICA

Titolo: Some computations in Chern-Simons quantum field theory.

In this work, the main properties of the quantum Chern-Simons theory- which is a topological quantum field theory in three dimensions- are exposed, and the computation of a certain class of observables is produced.

In the first part of the thesis, several concepts of three-dimensional topology and of knot theory are introduced. Thus, the Abelian Chern-Simons action functional is considered, and the symmetry properties of the theory are discussed. The gauge-fixing procedure together with the construction of the perturbative expansion are reported; moreover, a definition of the observables associated with oriented framed and coloured links is given. The solution of the theory in R^3 is produced and the rules for the computation of the observables in a generic closed oriented 3-manifold are derived. These rules are applied to compute a set of link observables which are defined on lens spaces.

We give a brief description of the homology groups for closed manifolds; then we concentrate on 3-manifolds which are homology spheres and we give a convenient surgery presentation of these manifolds. We define a combinatorial invariant for closed 3-manifolds which corresponds to the normalized partition function of the Abelian Chern-Simons theory. This invariant is computed for various examples of 3-manifolds and it is shown that it is trivial for homology spheres. It is shown that such an invariant does not depend only on the homology of the manifold.