Abelian Chern–Simons theory and linking numbers via oscillatory integrals

Abstract

A rigorous mathematical model of Abelian Chern–Simons theory based on the theory of infinite-dimensional oscillatory integrals developed by Albeverio and Ho/egh-Krohn is introduced. A gauge-fixed Chern–Simons path integral is constructed as a Fresnel integral in a certain Hilbert space. Wilson loop variables are defined as Fresnel integrable functions and it is shown in this context that the expectation value of products of Wilson loops with respect to the Chern–Simons path integral is a topological invariant which can be computed in terms of pairwise linking numbers of the loops, as conjectured by Witten. Furthermore, a lattice Chern–Simons action is proposed which converges to the continuum limit.