Cardy Formulae for SUSY Theories in d=4 and d=6

Abstract

We consider supersymmetric theories on a space with compact space-like slices. One can count BPS representations weighted by (-1)^F, or, equivalently, study supersymmetric partition functions by compactifying the time direction. A special case of this general construction corresponds to the counting of short representations of the superconformal group. We show that in four-dimensional N=1 theories the "high temperature" asymptotics of such counting problems is fixed by the anomalies of the theory. Notably, the combination a-c of the trace anomalies plays a crucial role. We also propose similar formulae for six-dimensional (1,0) theories.

33 pages; added references

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• Generalized Indices for $\mathcal{N}=1$ Theories in Four-Dimensions
• Comments on the Casimir energy in supersymmetric field theories
• Supersymmetric counterterms from new minimal supergravity
• Central charges from the $\mathcal{N}=1$ superconformal index
• Needles in a haystack: An algorithmic approach to the classification of 4d $\mathcal{N}=2$ SCFTs
• Casimir energy of confining large $N$ gauge theories
• 4d/2d $\to$ 3d/1d: A song of protected operator algebras
• Lectures on chiral algebras of $\mathcal{N}⩾2$ superconformal field theories
• Supersymmetric zeta functions and determinants
• Explorations of Universality in the Entropy and Hawking Radiation of Non-Extremal Kerr AdS${}_4$ Black Holes
• Running of the Number of Degrees of Freedom in Quantum Conformal Gravity
• Holographic spectral functions for Sasaki-Einstein 5-manifolds