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.

arXiv comment40 pages, v3: improved typography, added a remark on using the analysis for the theories with many sectors

References in corpus (36)

• An Index for 4 dimensional Super Conformal Theories
• New Phenomena in SU(3) Supersymmetric Gauge Theory
• The Exact Superconformal R-Symmetry Maximizes a
• Counting chiral primaries in N=1 d=4 superconformal field theories
• Applications of the Superconformal Index for Protected Operators and q-Hypergeometric Identities to N=1 Dual Theories
• On Short and Semi-Short Representations for Four Dimensional Superconformal Symmetry
• 3d dualities from 4d dualities
• Indices for Superconformal Field Theories in 3,5 and 6 Dimensions
• Central charges of N=2 superconformal field theories in four dimensions
• Exactly Marginal Deformations and Global Symmetries
• Deformations of Superconformal Theories
• 4d Index to 3d Index and 2d TQFT
• N=1 Deformations and RG Flows of N=2 SCFTs
• Enhancement of Supersymmetry via Renormalization Group Flow and the Superconformal Index
• Central Charges and $U(1{)}_R$ Symmetries in $\mathcal{N}=1$ Super Yang-Mills
• N=1 Deformations and RG Flows of N=2 SCFTs, Part II: Non-principal deformations
• Lagrangians for generalized Argyres-Douglas theories
• Tinkertoys for the Twisted D-Series
• Supersymmetric gauge theories with decoupled operators and chiral ring stability
• Calculating the Superconformal Index and Seiberg Duality
• Counting BPS operators in N=4 SYM
• The superconformal index of N=1 class S fixed points
• A "Lagrangian" for a non-Lagrangian theory
• New N=1 Dualities
• N=1 Lagrangians for generalized Argyres-Douglas theories
• Abelianization and Sequential Confinement in $2+1$ dimensions
• N=4 Superconformal Characters and Partition Functions
• Constraints on Chiral Operators in N=2 SCFTs
• The Veneziano Limit of N=2 Superconformal QCD: Towards the String Dual of N=2 SU(N_c) SYM with N_f =2 N_c
• Supersymmetric AdS_7 backgrounds in half-maximal supergravity and marginal operators of (1,0) SCFTs
• New and old N=8 superconformal field theories in three dimensions
• On Superconformal Characters and Partition Functions in Three Dimensions
• Central charges from the $\mathcal{N}=1$ superconformal index
• Tinkertoys for the Twisted $E_6$ Theory
• Symmetry enhancements via 5d instantons, qW-algebrae and (1,0) superconformal index
• A note on $\mathcal{N}\ge 6$ Superconformal Quantum Field Theories in three dimensions

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• The Coulomb and Higgs Branches of $\mathcal{N}=1$ Theories of Class ${\mathcal{S}}_k$
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