Remarks on Wilson Loops and Seifert Loops in Chern-Simons Theory

Abstract

As noted long ago by Atiyah and Bott, the classical Yang-Mills action on a Riemann surface admits a beautiful symplectic interpretation as the norm-square of a moment map associated to the Hamiltonian action by gauge transformations on the affine space of connections. Here I explain how certain Wilson loop observables in Chern-Simons gauge theory on a Seifert three-manifold can be given an analogous symplectic description. Among other results, this symplectic description implies that the stationary-phase approximation to the Wilson loop path integral is exact for torus knots, an empirical observation made previously by Lawrence and Rozansky. This article reviews selected material from the larger work "Localization for Wilson Loops in Chern-Simons Theory," arXiv:0911.2687.

17 pages, to appear in the proceedings of the conference "Chern-Simons Gauge Theory: 20 Years After"