Studying superconformal symmetry enhancement through indices

Abstract

In this note we classify the necessary and the sufficient conditions that an index of a superconformal theory in $3\le d\le 6$ must obey for the theory to have enhanced supersymmetry. We do that by noting that the index distinguishes a superconformal multiplet contribution to the index only up to a certain equivalence class it lies in. We classify the equivalence classes in $d=4$ and build a correspondence between $\mathcal{N}=1$ and $\mathcal{N}>1$ equivalence classes. Using this correspondence, we find a set of necessary conditions and a sufficient condition on the $d=4$ $\mathcal{N}=1$ index for the theory to have $\mathcal{N}>1$ SUSY. We also find a necessary and sufficient condition on a $d=4$ $\mathcal{N}>1$ index to correspond to a theory with $\mathcal{N}>2$. We then use our results to study some of the $d=4$ theories described by Agarwal, Maruyoshi and Song, and find that the theories in question have only $\mathcal{N}=1$ SUSY despite having rational central charges. In $d=3$ we classify the equivalence classes, and build a correspondence between $\mathcal{N}=2$ and $\mathcal{N}>2$ equivalence classes. Using this correspondence, we classify all necessary or sufficient conditions on an $\mathcal{N}=1-3$ superconformal index in $d=3$ to correspond to a theory with higher SUSY, and find a necessary and sufficient condition on an $\mathcal{N}=4$ index to correspond to an $\mathcal{N}>4$ theory. Finally, in $d=6$ we find a necessary and sufficient condition for an $\mathcal{N}=1$ index to correspond to an $\mathcal{N}=2$ theory.

40 pages, v3: improved typography, added a remark on using the analysis for the theories with many sectors

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Cited by in corpus (26)

• Mixed Anomalies, Two-groups, Non-Invertible Symmetries, and 3d Superconformal Indices
• Three-dimensional gauge theories with supersymmetry enhancement
• RG flows with supersymmetry enhancement and geometric engineering
• A tale of exceptional 3d dualities
• Non-unitary TQFTs from 3D $\mathcal{N}=4$ rank 0 SCFTs
• Zero-form and one-form symmetries of the ABJ and related theories
• Landscape of Simple Superconformal Field Theories in 4d
• Supersymmetric Indices of 3d S-fold SCFTs
• 4d $\mathcal{N}=3$ indices via discrete gauging
• SUSY enhancement from T-branes
• An N=1 Lagrangian for an N=3 SCFT
• Four-dimensional Lens Space Index from Two-dimensional Chiral Algebra
• Infrared enhancement of supersymmetry in four dimensions
• The $SL(2,\mathbb{Z})$ dualization algorithm at work
• 3d TQFTs from Argyres-Douglas theories
• Marginal operators and supersymmetry enhancement in 3d $S$-fold SCFTs
• The Coulomb and Higgs Branches of $\mathcal{N}=1$ Theories of Class ${\mathcal{S}}_k$
• Notes on Integral Identities for 3d Supersymmetric Dualities
• Nilpotent Networks and 4D RG Flows
• 3D Dualities and Supersymmetry Enhancement from Domain Walls
• 3d SUSY enhancement and non-semisimple TQFTs from four dimensions
• A Family of Vertex Algebras from Argyres-Douglas Theory
• Generalized symmetry constraints on deformed 4d (S)CFTs
• Vacua, Symmetries, and Higgsing of Chern-Simons Matter Theories
• A landscape of 4d N=1 SCFTs with a=c
• Geometric Approaches to Quantum Fields and Strings at Strong Couplings