Mixed Anomalies, Two-groups, Non-Invertible Symmetries, and 3d Superconformal Indices

TL;DR

This work develops and deploys a 3d ${\cal N} \ge 3$ index-based method to detect mixed anomalies among discrete zero-form and one-form symmetries in diverse gauge theories, linking monopole sectors with fractional flux to obstructions in gauging symmetries. By applying it to $U(1)_k$ CS-matter models, $T(\mathrm{SU}(N))$, ${\rm SO}(2N)_{2k}$ with vectors, and ABJ-type orthosymplectic theories, the authors uncover a rich pattern of global-form constraints, two-group structures, and non-invertible symmetries that arise when gauging discrete symmetries. They demonstrate that the superconformal index encodes these anomalies and their consequences for dualities and IR global symmetries, providing concrete evidence for two-group couplings such as $\delta B_2^{\mathcal M} = B_1^{\cal C} \cup w_2^f$ or its variants, and identifying cases where non-invertible defects emerge via Kaidi-like constructions. The results illuminate how gauging discrete symmetries reshapes the spectrum and symmetry content, with implications for class ${\cal S}$ mirrors and potential holographic checks, and they outline open questions for odd levels and broader supersymmetric settings. Overall, the work establishes index-based diagnostics as a powerful tool to map the landscape of generalized global symmetries in 3d SCFTs and their IR duals.

Abstract

Mixed anomalies, higher form symmetries, two-group symmetries and non-invertible symmetries have proved to be useful in providing non-trivial constraints on the dynamics of quantum field theories. We study mixed anomalies involving discrete zero-form global symmetries, and possibly a one-form symmetry, in 3d $\mathcal{N} \geq 3$ gauge theories using the superconformal index. The effectiveness of this method is demonstrated via several classes of theories, including Chern-Simons-matter theories, such as the $\mathrm{U}(1)_k$ gauge theory with hypermultiplets of diverse charges, the $T(\mathrm{SU}(N))$ theory of Gaiotto-Witten, the theories with $\mathfrak{so}(2N)_{2k}$ gauge algebra and hypermultiplets in the vector representation, and variants of the Aharony-Bergman-Jafferis (ABJ) theory with the orthosymplectic gauge algebra. Gauging appropriate global symmetries of some of these models, we obtain various interesting theories with non-invertible symmetries or two-group structures.