Localization for Wilson Loops in Chern–Simons Theory

Abstract

We reconsider Chern–Simons gauge theory on a Seifert manifold $M$, which is the total space of a non-trivial circle bundle over a Riemann surface $\Sigma$, possibly with orbifold points. As shown in previous work with Witten, the path integral technique of non-abelian localization can be used to express the partition function of Chern–Simons theory in terms of the equivariant cohomology of the moduli space of flat connections on $M$. Here we extend this result to apply to the expectation values of Wilson loop operators that wrap the circle fibers of $M$ over $\Sigma$. Under localization, such a Wilson loop operator reduces naturally to the Chern character of an associated universal bundle over the moduli space. Along the way, we demonstrate that the stationary-phase approximation to the Wilson loop path integral is exact for torus knots in $S^3$, an observation made empirically by Lawrence and Rozansky prior to this work.