$SL(3,\mathbb{Z})$ Modularity and New Cardy Limits of the $\mathcal{N}=4$ Superconformal Index

Abstract

The entropy of $1/16$-th BPS AdS${}_5$ black holes can be microscopically accounted for by the superconformal index of the $\mathcal{N}=4$ super-Yang-Mills theory. One way to compute this is through a Cardy-like limit of a formula for the index obtained in [1] using the "$S$-transformation" of the elliptic $\Gamma$ function. In this paper, we derive more general $SL(3,\mathbb{Z})$ modular properties of the elliptic $\Gamma$ function. We then use these properties to obtain a three integer parameter family of generalized Cardy-like limits of the $\mathcal{N}=4$ superconformal index. From these limits, we obtain entropy formulae that have a similar form as that of the original AdS${}_5$ black hole, up to an overall rescaling of the entropy. We interpret this both on the field theory and the gravitational side. Finally, we comment on how our work suggests a generalization of the Farey tail to four dimensions.

63 pages + appendices. v2: refs added, various corrections and improvements, discussion section rewritten. v3: corrected defn normalized partition function, added tau=sigma limit modular property in section 3.5 + its physical interpretation in section 5.2. (published version)

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Cited by in corpus (11)

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• The Panorama of Spin Matrix Theory
• Quantum Phases of $4d$ $SU(N)$ $\mathcal{N}=4$ SYM
• Supersymmetric Grey Galaxies, Dual Dressed Black Holes and the Superconformal Index
• The large-$N$ limit of 4d superconformal indices for general BPS charges
• Root of unity asymptotics for Schur indices of 4d Lagrangian theories
• S-duality in the Cardy-like limit of the superconformal index
• Off-shell phase diagram of BPS black holes in AdS${}_5$