Abelian Chern–Simons theory, Stokes’ theorem, and generalized connections

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 the test case for our method. It can also be applied to the non-abelian theory. Results will be reported in a companion article.