N=4双子的Farey故事
A Farey tale for N=4 dyons
摘要：
我们研究指数抑制的贡献极端黑洞的退化。在Sen的量子熵函数框架下，我们研究了具有中间AdS 3区域的极端黑洞，确定了一个无限族的半经典AdS 2几何，它可以贡献exp（S_0/c）阶的效应，其中S_0是Bekenstein-Hawking-Wald熵，c是大于1的整数.这些解提升到SL（2，Z）族BTZ黑洞的极值极限，类似于“黑洞Farey尾”。我们测试这种理解在N=4弦真空，确切的双子退化是已知的西格尔模形式的傅立叶系数。我们把Siegel上半平面的极点和与Farey尾展开联系起来，导出了双子配分函数的“Farey尾”展开式。在数学上，这提供了从希尔伯特模形式到在对角因子处具有极点的西格尔模形式的（形式上的）提升。
英文摘要：
We study exponentially suppressed contributions to the degeneracies of extremal black holes. Within Sen's quantum entropy function framework and focusing on extremal black holes with an intermediate AdS3 region, we identify an infinite family of semi-classical AdS2 geometries which can contribute effects of order exp(S_0/c), where S_0 is the Bekenstein-Hawking-Wald entropy and c is an integer greater than one. These solutions lift to the extremal limit of the SL(2,Z) family of BTZ black holes familiar from the "black hole Farey tail". We test this understanding in N=4 string vacua, where exact dyon degeneracies are known to be given by Fourier coefficients of Siegel modular forms. We relate the sum over poles in the Siegel upper half plane to the Farey tail expansion, and derive a "Farey tale" expansion for the dyon partition function. Mathematically, this provides a (formal) lift from Hilbert modular forms to Siegel modular forms with a pole at the diagonal divisor.
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