Mixed anomalies, two-groups, non-invertible symmetries, and 3d superconformal indices

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 $$ \mathcal{N} $$ N ≥ 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 U(1)k gauge theory with hypermultiplets of diverse charges, the T (SU(N)) theory of Gaiotto-Witten, the theories with 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.

Trinions for the $3d$ compactification of the $5d$ rank 1 $E_{N_f+1}$ SCFTs

“…In the 5d/3d setup, even though there are no anomalies for continues symmetries, there are anomalies involving discrete symmetries. Specifically, certain anomalies of 5d SCFTs are known [55,56,[58][59][60][61] and we can try to match their dimensional reduction with those computed directly in the 3d models, along the lines of the field theory analysis of [61][62][63][64][65][66][67]. 7 This would provide an additional check of the proposals both of this paper and of [38,39], and possibly have new predictions for the anomalies of certain 5d and 3d theories.…”

3d $\mathcal{N}=4$ mirror symmetry with 1-form symmetry

“…The resulting theories are straightforwardly derived. Gauging a Z q ⊂ U(1) t for SQED with N charge 1 hypermultiplets leads to SQED with N charge q hypermultiplets, see also [12]. Similarly, gauging a Z q ⊂ U(1) f of the abelian mirror quiver leads to an abelian quiver with a Z q 1-form symmetry.…”

“…In summary, both results confirm the expectation and provide the explicit parameter map. As a remark, the superconformal index is equally well suited to probe such dualities; see for -4 -instance [12] for SQED with charge q = 2 hypermultiplets. Since either Higgs or Coulomb branch operators are unaffected by gauging a Z t/f q , the Hilbert series is a more convenient tool.…”

“…Specifically for 3d theories, discrete 1-form symmetries can be generated by gauging discrete 0-form symmetries. The structure of generalised symmetries in 3d supersymmetric theories has been the focus of recent research, including [3][4][5][6][7][8][9][10][11][12][13][14] and references therein 1 . Given the vast catalogue of 3d mirror pairs with trivial 1-form symmetry, one might wonder what mirror symmetry implies for 3d theories with 1-form symmetry.…”

Generalized Symmetries and Anomalies of 3d N=4 SCFTs

“…These configurations survive in the IR SCFT and the charges of such non-genuine local operators under IR Coulomb symmetry determine the precise form of the mixed 't Hooft anomaly between the Coulomb 0-form symmetry and the 1-form symmetry of the IR 3d N = 4 SCFT. While this work was being written, we received [13] which also used the analysis of [12] to obtain some of the results appearing in sections 3 and 6 of this paper.…”