In this note we study four dimensional theories with N = 3 superconformal symmetry, that do not also have N = 4 supersymmetry. No examples of such theories are known, but their existence is also not ruled out. We analyze several properties that such theories must have. We show that their conformal anomalies obey a = c. Using the N = 3 superconformal algebra, we show that they do not have any exactly marginal deformations preserving N = 3 supersymmetry, or global symmetries (except for their R-symmetries). Finally, we analyze the possible dimensions of chiral operators labeling their moduli space.

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 N = 1 and N > 1 equivalence classes. Using this correspondence, 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. We also find a necessary and sufficient condition on a d = 4 N > 1 index to correspond to a theory with 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 N = 1 SUSY despite having rational central charges. In d = 3 we classify the equivalence classes, and build a correspondence between N = 2 and N > 2 equivalence classes. Using this correspondence, we classify all necessary or sufficient conditions on an 1 ≤ N ≤ 3 superconformal index in d = 3 to correspond to a theory with higher SUSY, and find a necessary and sufficient condition on an N = 4 index to correspond to an N > 4 theory. Finally, in d = 6 we find a necessary and sufficient condition for an N = 1 index to correspond to an N = 2 theory.

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In this paper, we study the 6d Little String Theory (LST) (the decoupled theory on the worldvolume of N NS5-branes) on curved manifolds, by using its holographic duality to Type II string theory in asymptotically linear dilaton backgrounds. We focus on backgrounds with a large number of Killing vectors (namely, products of maximally symmetric spaces), without requiring supersymmetry (we do not turn on any background fields except the metric). LST is non-local so it is not obvious which spaces it can be defined on; we show that holography implies that the theory cannot be put on negatively curved spaces, but only on spaces with zero or positive curvature. For example, one cannot put LST on a product of an anti-de Sitter space times another space, without turning on extra background fields. On spaces with positive curvature, such as S 6 , R 2 × S 4 , S 3 × S 3 , etc., we typically find (for large N ) dual holographic backgrounds which are weakly coupled and weakly curved everywhere, so that they can be well-described by Type II supergravity. In some cases more than one smooth solution exists for LST on the same space, and they all contribute to the partition function. We also study the thermodynamical properties of LST compactified on spheres, finding the leading correction to the Hagedorn behavior of the spectrum, which is different on curved space than on flat space. We discuss the holographic renormalization procedure, which must be implemented in order to get a finite free energy for the LST; we do not know how to implement it for general spaces, but we can (and we do) implement it for the theory compactified on S 4 .

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