Chern-Simons field theory and invariants of 3-manifolds

Abstract

The Chern-Simons (CS) theory in three dimensions with a compact gauge group G is studied. Starting from the BRST quantization of the theory defined in R^3, the values of gauge invariants observables are computed in any closed and orientable three manifold M by constructing a surgery operator associated with a Dehn's surgery presentation of M. We have given the general rules to find the reduced tensor algebra whose elements give the physical inequivalent quantum numbers that characterize uniquely the CS observables. Some general properties of the reduced tensor algebra and of the surgery operator are studied. Many examples of the general construction are given in the case of G=SU(2) and G=SU(3), in particular the expectation value of a Wilson line associated with the unknot and the Hopf link in R^3(S^3) is computed for any irreducible representation of the gauge group. The relation of the topological invariant I(M), defined as a suitable modified partition function of the CS theory in M, and the fundamental group of M is studied. Finally, the relationship between CS theory and WZWN model in two dimensions is exploited to derive from a full three-dimensional point of view some classical results in conformal field theory.