Chern-Simons Field Theories with Non-semisimple Gauge Group of Symmetry

TL;DR

This paper explores a class of Chern-Simons theories with non-semisimple gauge groups, highlighting their unique properties such as absence of radiative corrections and exact computability of topological invariants.

Contribution

It introduces a new class of gauge theories with non-semisimple groups, deriving BRST invariant observables and exact topological invariants including linking numbers.

Findings

Exact computation of metric-independent amplitudes

Derivation of BRST invariant observables

Identification of topological invariants like linking numbers

Abstract

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.