Cohomological localization of Chern-Simons theory

TL;DR

This work addresses the problem of obtaining exact path-integral results for Chern-Simons theory on a broad class of 3-manifolds by extending a supersymmetric localization framework from $S^3$ to Seifert manifolds.The main approach is to perform a topological twist of the $\\mathcal{N}=2$ Chern-Simons theory using a contact structure and a Reeb vector, recasting the theory in a cohomological form with a symmetry $Q$ that squares to a combination of a Lie derivative and a gauge transformation, enabling localization.Key contributions include the explicit localization locus ($F=0$, $d_A\sigma=0$, $\sigma+D=0$), a detailed gauge-fixing and cohomological reformulation, a closed-form partition function for Seifert homology spheres (including a level shift $k\rightarrow k+\check{c}_{\mathfrak{g}}$) and exact Wilson-loop results wrapping the Seifert fiber, all in agreement with prior methods such as non-abelian localization and knot invariants calculations.The results illuminate a deep link between supersymmetric localization and Cohomological TFTs, provide a robust framework for exact CS calculations on Seifert manifolds, and suggest avenues for extending to more general flat connections and background configurations.

Abstract

We generalize the framework introduced by Kapustin et al. for doing path integral localization in Chern-Simons theory to work on any Seifert manifold. This is done by topologically twisting the supersymmetric theory considered by Kapustin et al., after which the theory takes a cohomological form. We also consider Wilson loops which wrap the fiber directions and compute their expectation values. We discuss the relation with other approaches to exact path integral calculations in Chern-Simons theory.