The 5d Superconformal Index at Large $N$ and Black Holes

Abstract

We study the large $N$ limit of the superconformal index of a large class of 5d $\mathcal{N}=1$ superconformal field theories and show it is given by the square of the partition function on the squashed five-sphere. We show this simple relation implies a Cardy formula in 5d, which is valid in an "extended" regime in which fugacities are finite and $N$ is large. For theories with weakly coupled gravity duals we conjecture this large $N$ Cardy formula universally accounts for the microscopic entropy of spinning black holes in AdS${}_6$. We check this explicitly for known black hole solutions in massive type IIA and type IIB string theory, carrying two angular momenta and one electric charge, and predict the entropy of black holes carrying multiple electric charges, yet to be constructed. We also discuss large $N$ properties of the $S_b^3\times {\Sigma }_{\mathfrak{g}}$ partition function, extending previous results to theories with type IIB duals.

arXiv comment28 pages; v2: 30 pages, References added and typos corrected (published version); v3: Typos corrected and references updated

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• Exact QFT duals of AdS black holes
• Phases of five-dimensional supersymmetric gauge theories
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• Explorations of Universality in the Entropy and Hawking Radiation of Non-Extremal Kerr AdS${}_4$ Black Holes