Topological anomalies for Seifert 3-manifolds

TL;DR

This paper explores the coupling of 3d topological gauge theories to background topological gravity on Seifert manifolds, deriving conditions for supersymmetry, analyzing anomalies, and computing partition functions on squashed spheres.

Contribution

It introduces a cohomological framework for understanding supersymmetric 3d gauge theories on Seifert manifolds and explicitly computes the partition function dependence on moduli without functional determinants.

Findings

Seifert condition for supersymmetry is derived from topological gravity BRST transformations.

Chern-Simons framing anomaly is shown to be BRST trivial and its Wess-Zumino functional computed.

Partition function dependence on Seifert moduli is obtained via anomalous Ward identities.

Abstract

We study globally supersymmetric 3d gauge theories on curved manifolds by describing the coupling of 3d topological gauge theories, with both Yang-Mills and Chern-Simons terms in the action, to background topological gravity. In our approach the Seifert condition for manifolds supporting global supersymmetry is elegantly deduced from the topological gravity BRST transformations. A cohomological characterization of the geometrical moduli which affect the partition function is obtained. In the Seifert context Chern-Simons topological (framing) anomaly is BRST trivial. We compute explicitly the corresponding local Wess-Zumino functional. As an application, we obtain the dependence on the Seifert moduli of the partition function of 3d supersymmetric gauge theory on the squashed sphere by solving the anomalous topological Ward identities, in a regularization independent way and without the need of evaluating any functional determinant.