A few years ago it was shown that the superconformal index of the $\mathcal{N}=4$ supersymmetric $SU(N)$ Yang-Mills theory in the large $N$ limit matches with the entropy of $1/16$-supersymmetric black holes in type IIB string theory on ${\mathrm{AdS}}_5\times S^5$. In some cases, an even more detailed match between the two sides is possible. When the two angular momentum chemical potentials in the index are equal, the superconformal index can be written as a discrete sum of Bethe ansatz solutions, and it was shown that specific terms in this sum are in a one-to-one correspondence to stable black hole solutions, and that the matching can be extended to nonperturbative contributions from wrapped D3-branes. A Bethe ansatz approach to computing the superconformal index exists also when the ratio of the angular momentum chemical potentials is any rational number, but in those cases it involves a sum over a very large number of terms (growing exponentially with $N$). Benini et al. showed that a specific one of these terms matches with the black hole, but the role of the other terms is not clear. In this paper we analyze some of the additional contributions to the index in the Bethe ansatz approach, and we find that their matching to the gravity side is much more complicated than in the case of equal chemical potentials. In particular, we find some contributions that are larger than the one that was found to match the black holes, in which case they must cancel with other large contributions. We give some evidence that cancellations of this type are possible, but we leave a full understanding of how they work to the future.

中文翻译：

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Bethe Ansatz 的不等角动量超共形指数

几年前，研究表明，超共形指数$\mathcal{氮}=4$超对称$SU（氮）$大范围的杨-米尔斯理论$氮$极限与熵匹配$1/16$-IIB型弦理论中的超对称黑洞${\mathrm{广告服务}}_5\times S^5$。在某些情况下，双方之间甚至可能进行更详细的比赛。当指数中的两个角动量化学势相等时，超共形指数可以写成Bethe ansatz解的离散和，并且表明该和中的特定项与稳定黑一一对应孔解，并且匹配可以扩展到来自包裹的 D3 膜的非微扰贡献。当角动量化学势之比为任何有理数时，也存在计算超共形指数的 Bethe ansatz 方法，但在这些情况下，它涉及大量项的总和（随指数增长）$氮$）。贝尼尼等人。表明这些术语中的一个特定术语与黑洞匹配，但其他术语的作用尚不清楚。在本文中，我们分析了 Bethe ansatz 方法中对指数的一些额外贡献，我们发现它们与重力侧的匹配比化学势相等的情况要复杂得多。特别是，我们发现一些贡献大于与黑洞匹配的贡献，在这种情况下，它们必须与其他大贡献抵消。我们提供了一些证据表明这种类型的取消是可能的，但我们对它们如何运作留待未来的充分理解。