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

TL;DR

The paper develops a systematic framework to study generalized 3d $\mathcal N=2$ gauge theories using the 3d $A$-model, focusing on one-form symmetry gauging and pure Chern–Simons dynamics for simple groups. It constructs Seifert fibering operators for $G_K=(\widetilde G/\Gamma)_K$ theories and proves that, for bosonic gaugings, the SUSY partition functions on Seifert manifolds match the 3d TQFT results up to a framing counterterm proportional to the WZW central charge, confirming a deep link between the 2d $A$-model and infrared TQFT data. The work develops an explicit anyon-condensation (condensing abelian bosons) perspective, showing how the torus Hilbert space and modular data descend to the gauged theory via orbit sums and orbifold-flux sums, with detailed treatment for SU$(N)$ cases. It further extends the analysis to supersymmetric Chern–Simons theories with gauged one-form symmetry, defining the GK Seifert fibering operator as an orbit-averaged object $\mathcal{G}^G_{q,p}(\hat{\omega})$ and providing concrete partition-function formulas on Seifert manifolds, lens spaces, and torus bundles, validated against known TQFT results. The results illuminate the interplay between 2d $\,A$-model topology, 3d CS TQFT, and generalized symmetries, and lay groundwork for future exploration of spin-TQFTs and non-invertible symmetries.

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 $G=\widetilde G/Γ$, 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.