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arXiv · 2104.07030

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

Vishnu Jejjala·Yang Lei·Sam van Leuven·Wei Li
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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.

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Vishnu Jejjala, Yang Lei, Sam van Leuven, Wei Li. 2021-04-14. $SL(3,\mathbb{Z})$ Modularity and New Cardy Limits of the $\mathcal{N}=4$ Superconformal Index. https://doi.org/10.1007/jhep11(2021)047

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