Chern-Simons theory, Stokes' Theorem, and the Duflo map

TL;DR

This paper introduces a novel derivation of holonomy expectation values in Chern-Simons theory using Stokes' Theorem and the Duflo map, connecting functional derivatives and ordering choices in path integrals.

Contribution

It demonstrates that the Duflo isomorphism provides the correct ordering in the derivation of holonomy expectation values in Chern-Simons theory.

Findings

Duflo map yields correct ordering for simple cases

Expectation values for unknotted, linked loops in SU(2) and SU(3) computed

Method potentially applicable to more complex loop configurations

Abstract

We consider a novel derivation of the expectation values of holonomies in Chern-Simons theory, based on Stokes' Theorem and the functional properties of the Chern-Simons action. It involves replacing the connection by certain functional derivatives under the path integral integral. It turns out that ordering choices have to be made in the process, and we demonstrate that, quite surprisingly, the Duflo isomorphism gives the right ordering, at least in the simple cases that we consider. In this way, we determine the expectation values of unknotted, but possibly linked, holonomy loops for SU(2) and SU(3), and sketch how the method may be applied to more complicated cases. Our manipulations of the path integral are formal but well motivated by a rigorous calculus of integration on spaces of generalized connections which has been developed in the context of loop quantum gravity.