Topological phase, spin Chern-Simons theory and level rank duality on lens space

TL;DR

This paper introduces a supersymmetry-preserving regularization method to compute topological phase factors in Chern-Simons theory on lens spaces, confirming level-rank dualities and matching partition functions.

Contribution

Develops a supersymmetric regularization technique to compute phase factors in Chern-Simons theory, validating dualities and enhancing understanding of topological phases on lens spaces.

Findings

Confirmed level-rank dualities for all ranks and levels

Computed phase factors matching dual partition functions

Demonstrated the regularization preserves supersymmetry

Abstract

We study a method to compute a topological phase factor of partition function for pure Chern-Simons theory incorporating the supersymmetric localization. We develop a regularization preserving supersymmetry and the topological phase appears as a result of the supersymmetric regularization. Applying this method to pure Chern-Simons theory on lens space we compute the background dependent phase factor coming from the Chern-Simons term. We confirm that the partition function computed in this method enjoys a couple of level rank dualities including the one recently proposed in arXiv:1607.07457 for all ranks and levels within our numerical calculation. We also present a phase factor with which the lens space partition function exhibits the perfect match between any level rank dual pair including the total phase.