Abelian Chern-Simons theory, Stokes' theorem, and generalized connections

TL;DR

This paper applies generalized connections and Stokes' theorem to abelian Chern-Simons theory, deriving holonomy expectation values and framing effects, serving as a test case for broader non-abelian applications.

Contribution

It introduces a novel method using generalized connections and Stokes' theorem to compute expectation values in abelian Chern-Simons theory, with potential extension to non-abelian cases.

Findings

Derived expectation values of holonomies in U(1) Chern-Simons theory.

Identified the role of framing in the construction of expectation values.

Established a functional framework over gauge invariant cylindrical functions.

Abstract

Generalized connections and their calculus have been developed in the context of quantum gravity. Here we apply them to abelian Chern-Simons theory. We derive the expectation values of holonomies in U(1) Chern-Simons theory using Stokes' theorem, flux operators and generalized connections. A framing of the holonomy loops arises in our construction, and we show how, by choosing natural framings, the resulting expectation values nevertheless define a functional over gauge invariant cylindrical functions. The abelian theory considered in the present article is test case for our method. It can also be applied to the non-abelian theory. Results for that case will be reported elsewhere.