Integral representations of rank two false theta functions and their modularity properties
Integral representations of rank two false theta functions and their modularity properties
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DOI：
10.1007/s40687-021-00284-1
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发表时间：
2021-01
影响因子：
1.2
通讯作者：
K. Bringmann;Jonas Kaszian;A. Milas;Caner Nazaroglu
中科院分区：
文献类型：
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作者：
K. Bringmann;Jonas Kaszian;A. Milas;Caner Nazaroglu
假theta函数形成了一个具有有趣的模性质和与模拟模形式的联系的函数家族。在本文中，我们采取的第一步，调查高秩假theta函数的模变换，更高的深度模拟模形式的例子。特别是，我们证明了在相当一般的条件下，一个秩二假θ函数确定的迭代，全纯，Eichler型积分。这提供了一个新的方法来检查其模块化的属性，我们将其应用在各种情况下，排名两个假θ函数出现。本文首先考虑型顶点代数的泛仿费米子特征标。这需要一个相当非平凡的分析傅立叶系数的亚纯Jacobi形式的负指数，这是独立的利益。然后讨论了由超共形Schur指标得到的秩为2的伪theta函数的模性。最后，我们分析了Gukov，裴，Putrov和Vafa的某些铅垂图的不变量。沿着的方式，我们的方法澄清了以前的结果深度2量子模块。
False theta functions form a family of functions with intriguing modular properties and connections to mock modular forms. In this paper, we take the first step towards investigating modular transformations of higher rank false theta functions, following the example of higher depth mock modular forms. In particular, we prove that under quite general conditions, a rank two false theta function is determined in terms of iterated, holomorphic, Eichler-type integrals. This provides a new method for examining their modular properties and we apply it in a variety of situations where rank two false theta functions arise. We first consider generic parafermion characters of vertex algebras of typeand. This requires a fairly non-trivial analysis of Fourier coefficients of meromorphic Jacobi forms of negative index, which is of independent interest. Then we discuss modularity of rank two false theta functions coming from superconformal Schur indices. Lastly, we analyze-invariants of Gukov, Pei, Putrov, and Vafa for certain plumbing-graphs. Along the way, our method clarifies previous results on depth two quantum modularity.