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

The 3d A A -model is a two-dimensional approach to the computation of supersymmetric observables of three-dimensional \mathcal{N}=2 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 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 G = G ̃ / Γ , which is obtained from the \widetilde G G ̃ theory by gauging a discrete one-form symmetry. In this paper, we discuss in detail the case of pure \mathcal{N}=2 𝒩 = 2 Chern-Simons theories (without matter) for simple groups G G . When G=\widetilde G G = 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 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 G k can be obtained from \widetilde G_k G ̃ k by the condensation of abelian anyons which are bosonic. Along the way, we revisit the 3d A A -model formalism by emphasising its 2d TQFT underpinning.