Abstract

In this note we classify the necessary and the sufficient conditions that an index of a superconformal theory in 3 ≤ d ≤ 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 1le mathcal{N}le 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.

Highlights

  • Introduction and summary of resultsSuperconformal theories have been extensively studied in the past 40 years

  • In this note we classify the necessary and the sufficient conditions that an index of a superconformal theory in 3 ≤ d ≤ 6 must obey for the theory to have enhanced supersymmetry

  • We find a set of necessary conditions and a sufficient condition on the d = 4 N = 1 index for the theory to have N > 1 SUSY

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Summary

Introduction and summary of results

Superconformal theories have been extensively studied in the past 40 years. This has partly been because their enhanced symmetries allow for various exact computations to be performed in these theories, even at strong coupling, so that they provide useful windows into strong coupling physics. In 6 dimensions one cannot have relevant or marginal deformations [20, 23] that would preserve superconformal theory; we still list a necessary and sufficient condition on SUSY enhancement through index, as one can obtain N = (1, 0) index for a theory that has N = (2, 0) SUSY from e.g. stringy construction. We first describe the massive deformation method, following [19], and study several of the theories obtained in [25], using the methods we developed for d = 4 indices, to show how different necessary and sufficient conditions on N = 1 → N = 2 enhancement work out in different cases; we prove that several theories found by [25] have only N = 1 SUSY despite having rational central charges.

Index and SUSY enhancement conditions in 4 dimensions
If the coefficient in front of
AMS and SUSY enhancement
Description of the method
SUSY enhancement checking
Index and SUSY enhancement conditions in 3d
Index and SUSY enhancement conditions in 6d
General case
Special cases
If the coefficient in front of t
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