一个现代的Fareytail
A Modern Fareytail
摘要：
我们重新讨论了在AdS_3/CFT_2对应和OSV猜想的讨论中使用的椭圆亏格的“fareytail展开”。我们展示了如何编写这样的扩展，而不使用有问题的“fareytail变换。特别地，我们展示了如何将非正权重的一般向量值模形式写为SL（2，Z）陪集上的收敛和。这个和建议一个新的正则化的重力路径积分在AdS_3，解决了与“fareytail变换”，并导致几个新的见解。我们讨论的极性系数的负权重模块化形式所产生的模块不变性的限制，显示这些是如何相关的傅立叶系数的正权重尖点形式。此外，我们还讨论了全纯异常的外观的背景下的fareytail。
英文摘要：
We revisit the "fareytail expansions" of elliptic genera which have been used in discussions of the AdS_3/CFT_2 correspondence and the OSV conjecture. We show how to write such expansions without the use of the problematic "fareytail transform." In particular, we show how to write a general vector-valued modular form of non-positive weight as a convergent sum over cosets of SL(2,Z). This sum suggests a new regularization of the gravity path integral in AdS_3, resolves the puzzles associated with the "fareytail transform," and leads to several new insights. We discuss constraints on the polar coefficients of negative weight modular forms arising from modular invariance, showing how these are related to Fourier coefficients of positive weight cusp forms. In addition, we discuss the appearance of holomorphic anomalies in the context of the fareytail.
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