Chern–Simons field theories with non-semisimple gauge group of symmetry

Abstract

The subject of this work is a class of Chern–Simons field theories with non-semisimple gauge group, which may well be considered as the most straightforward generalization of an Abelian Chern–Simons field theory. As a matter of fact, these theories, which are characterized by a non-semisimple group of gauge symmetry, have cubic interactions like those of non-Abelian Chern–Simons field theories, but are free from radiative corrections. Moreover, at the tree level in the perturbative expansion, there are only two connected tree diagrams, corresponding to the propagator and to the three vertex originating from the cubic interaction terms. For such theories it is derived here a set of BRST invariant observables, which lead to metric independent amplitudes. The vacuum expectation values of these observables can be computed exactly. From their expressions it is possible to isolate the Gauss linking number and an invariant of the Milnor type, which describes the topological relations among three or more closed curves.