A 4d N=1 Cardy Formula

Abstract

We study the asymptotic behavior of the (modified) superconformal index for 4d $\mathcal{N}=1$ gauge theory. By considering complexified chemical potential, we find that the `high-temperature limit' of the index can be written in terms of the conformal anomalies $3c-2a$. We also find macroscopic entropy from our asymptotic free energy when the Hofman-Maldacena bound $1/2<a/c<3/2$ for the interacting SCFT is satisfied. We study $\mathcal{N}=1$ theories that are dual to AdS${}_5\times Y^{p,p}$ and find that the Cardy limit of our index accounts for the Bekenstein-Hawking entropy of large black holes.

31 pages, 51 figures, v2: minor corrections, reference added

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• Classification of large N superconformal gauge theories with a dense spectrum
• Quantum Phases of $4d$ $SU(N)$ $\mathcal{N}=4$ SYM
• Microscopic Entropy of AdS${}_3$ Black Holes Revisited
• Twisted 6d $(2,0)$ SCFTs on a Circle
• The growth of the $\frac{1}{16}$-BPS index in 4d $\mathcal{N}=4$ SYM
• The giant graviton expansion in $Ad{S}_5\times S^5$
• From multi-gravitons to Black holes: The role of complex saddles
• Cardy Limits of 6d Superconformal Theories
• Anomaly Matching Across Dimensions and Supersymmetric Cardy Formulae
• Supersymmetric Grey Galaxies, Dual Dressed Black Holes and the Superconformal Index
• The large-$N$ limit of 4d superconformal indices for general BPS charges
• The superconformal index of $\mathcal{N}=4$ $USp(2N_c)$ and $SO(N_c)$ SYM as a matrix integral
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