Chern-Simons theory on L(p,q) lens spaces and Localization

TL;DR

This work extends exact results for Chern-Simons theory obtained by localization from the three-sphere to lens spaces $L(p,q)$ by applying the Pestun-Kapustin localization framework to $ ext{N}=2$ Chern-Simons theory with no matter. It derives explicit matrix-model representations for the partition function and Wilson loop expectations, incorporating sums over discrete flat connections and modified one-loop determinants due to lens-space geometry. For $L(0,1)=S^2 imes S^1$ the results reproduce the known trivial outcome, while for general $L(p,q)$ the formalism yields concrete expressions that match previously known results (e.g., for $q=-1$) up to anticipated overall phases. The analysis highlights how flat connections and spectrum data encode the topological information of the manifold and suggests routes toward systematic localization on broader classes of three-manifolds via surgery and symmetry structure.

Abstract

Using localization technique, we calculate the partition function and the expectation value of Wilson loop operator in Chern-Simons theory on general lens spaces L(p,q)(including S2XS1). Our results are consistent with known results.