-Algebras, False Theta Functions and Quantum Modular Forms, I
-Algebras, False Theta Functions and Quantum Modular Forms, I
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-代数、伪 Theta 函数和量子模形式，I
DOI：
10.1093/imrn/rnv033
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发表时间：
2015
影响因子：
1
通讯作者：
A. Milas
中科院分区：
文献类型：
--
作者：
K. Bringmann;A. Milas
本文结合顶点算子代数和共形域理论，研究了某些部分和假θ函数。我们证明了各种结果的修改字符的渐近性的不可约模的某些W-代数的单态型，这使我们特别是要确定其（解析）的量子尺寸。我们的结果是完全一致的，与以前关于这些顶点代数的fusion环。更重要的是，我们证明了单态代数模的不可约特征标的分子部分的量子模性（alaZagier），从而证明了量子模形式自然地出现在许多”无理顶点代数“中.有趣的是，量子模块性在整个有理数集上都是存在的，就像最初的Zagier的例子来自Kontsevich的“奇怪系列”一样。在最后一部分中，我们还讨论了与某些环面纽结的色Jones多项式的尾和（1;p）-单重代数的模的特征标有关的Nahm型q-超度级数。
In this paper, we study certain partial and false theta functions in connection to vertex operator algebras and conformal eld theory. We prove a variety of results concerning the asymptotics of modied characters of irreducible modules of certain W-algebras of singlet type, which allows us in particular to determine their (analytic) quantum dimensions. Our results are fully consistent with the previous conjectures on fusion rings for these vertex algebras. More importantly, we prove quantum modularity ( a la Zagier) of the numerator part of irreducible characters of singlet algebra modules, thus demonstrating that quantum modular forms naturally appear in many \suciently nice" irrational vertex algebras. It is interesting that quantum modularity persists on the whole set of rationals as in the original Zagier’s example coming from Kontsevich’s \strange series". In the last part, slightly independent of all this, we also discuss Nahm-type q-hypergometric series in connection to tails of colored Jones polynomials of certain torus knots and characters of modules for the (1;p)-singlet algebra.