We study the large $N$ limit of the superconformal index of a large class of 5d $\mathcal{N}=1$ gauge 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 $S^{3}_{b}\times \Sigma_{\mathfrak{g}}$ partition function, extending previous results to theories with type IIB duals.

中文翻译：

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大 N 和黑洞的 5d 超共形指数

我们研究了一大类 5d $\mathcal{N}=1$ 规范理论的超共形指数的大 $N$ 极限，并表明它是由压扁的五球面上的配分函数的平方给出的。我们展示了这个简单的关系意味着 5d 中的 Cardy 公式，它在逸度有限且 $N$ 很大的“扩展”状态中有效。对于弱耦合引力对偶理论，我们推测这个大 $N$ Cardy 公式普遍解释了 AdS$_{6}$ 中旋转黑洞的微观熵。我们在大质量 IIA 型和 IIB 型弦理论中明确检查已知黑洞解，携带两个角动量和一个电荷，并预测携带多个电荷的黑洞的熵，但尚未构建。