The 3d $A$-model and generalised symmetries, Part I: bosonic Chern-Simons theories

TL;DR

This paper explores the 3d $A$-model for supersymmetric observables in 3D $ ext{N}=2$ gauge theories, focusing on pure Chern-Simons theories with general gauge groups, and establishes precise links between supersymmetric and topological quantum field theory approaches.

Contribution

It extends the 3d $A$-model framework to non-simply-connected gauge groups and clarifies the relation between supersymmetric computations and topological surgery in Chern-Simons theories.

Findings

Exact matching between supersymmetric and TQFT approaches for simply-connected groups.

Extension to gauge groups obtained via anyon condensation.

Revisiting the 2d TQFT underpinning of the 3d $A$-model.

Abstract

The 3d $A$-model is a two-dimensional approach to the computation of supersymmetric observables of three-dimensional $\mathcal{N}=2$ supersymmetric gauge theories. In principle, it allows us to compute half-BPS partition functions on any compact Seifert three-manifold (as well as of expectation values of half-BPS lines thereon), but previous results focussed on the case where the gauge group $\widetilde G$ is a product of simply-connected and/or unitary gauge groups. We are interested in the more general case of a compact gauge group $G=\widetilde G/\Gamma$, which is obtained from the $\widetilde G$ theory by gauging a discrete one-form symmetry. In this paper, we discuss in detail the case of pure $\mathcal{N}=2$ Chern-Simons theories (without matter) for simple groups $G$. When $G=\widetilde G$ is simply-connected, we demonstrate the exact matching between the supersymmetric approach in terms of Seifert fibering operators and the 3d TQFT approach based on topological surgery in the infrared Chern-Simons theory $\widetilde G_k$, including through the identification of subtle counterterms that relate the two approaches. We then extend this discussion to the case where the Chern-Simons theory $G_k$ can be obtained from $\widetilde G_k$ by the condensation of abelian anyons which are bosonic. Along the way, we revisit the 3d $A$-model formalism by emphasising its 2d TQFT underpinning.