Hikami关于统一WRT不变量与假theta函数的观察
Hikami's observations on unified WRT invariants and false theta functions
AI总结：
本文利用Bailey引理和假theta函数理论，解释了Poincaré同调球面统一WRT不变量在单位根处不连续时出现1/2因子的原因，并推广了Hikami的观察。
摘要：
本文的研究对象是源自Habiro关于Witten-Reshetikhin-Turaev不变量工作的一族$q$-级数。这些$q$-级数通常仅在$q$为单位根时才有意义，但在某些情况下，它也能确定开单位圆盘上的全纯函数。Habiro针对Poincaré同调球面的统一WRT不变量$H(q)$便是这样一个例子。2007年，Hikami观察到其在单位根处的不连续性。更确切地说，在单位根处$H(ζ)$的值，等于$q$在单位圆盘内沿径向趋于$ζ$时$H(q)$极限值的$1/2$。本文中，我们利用Bailey引理与假theta函数理论解释了该$1/2$因子的出现，并推广了Hikami的观察。
英文摘要：
The object of this article is a family of $q$-series originating from Habiro's work on the Witten-Reshetikhin-Turaev invariants. The $q$-series usually make sense only when $q$ is a root of unity, but for some instances, it also determines a holomorphic function on the open unit disc. Such an example is Habiro's unified WRT invariant $H(q)$ for the Poincaré homology sphere. In 2007, Hikami observed its discontinuity at roots of unity. More precisely, the value of $H(ζ)$ at a root of unity is $1/2$ times the limit value of $H(q)$ as $q$ tends towards $ζ$ radially within the unit disc. In this article, we explain the appearance of the $1/2$-factor and generalize Hikami's observations by using Bailey's lemma and the theory of false theta functions.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
