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arXiv 2104.07030hep-th
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$\mathcal{N}=4$超共形指数的$SL(3,\mathbb{Z})$模性和新Cardy极限

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

Vishnu Jejjala, Yang Lei, Sam van Leuven, Wei Li

备注：
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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摘要：

$1/16$ -th BPS AdS $_5$黑洞的熵可以用$\mathcal{N}=4$ super-Yang-Mills理论的超共形指数进行微观解释。计算这一点的一种方法是通过[1]中使用椭圆$Γ$函数的“$S$ -变换”获得的指数公式的Cardy式极限。在本文中，我们得到了更一般的椭圆$Γ$函数的$SL(3,\mathbb{Z})$模性质。然后利用这些性质得到了$\mathcal{N}=4$超共形指数的广义Cardy极限的三个整数参数族.从这些限制，我们得到的熵公式，有一个类似的形式，原来的AdS $_5$黑洞，到一个整体重新缩放的熵。我们从场论和引力两方面对此进行解释。最后，我们评论如何我们的工作建议一个概括的法里尾四维。

英文摘要：

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 $Γ$ function. In this paper, we derive more general $SL(3,\mathbb{Z})$ modular properties of the elliptic $Γ$ 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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