In this paper, we prove a conjecture by Gukov–Pei–Putrov–Vafa for a wide class of plumbed $3$-manifolds. Their conjecture states that Witten–Reshetikhin–Turaev (WRT) invariants are radial limits of homological blocks, which are $q$-series introduced by them for plumbed $3$-manifolds with negative definite linking matrices. The most difficult point in our proof is to prove the vanishing of weighted Gauss sums that appear in coefficients of negative degree in asymptotic expansions of homological blocks. To deal with it, we develop a new technique for asymptotic expansions, which enables us to compare asymptotic expansions of rational functions and false theta functions related to WRT invariants and homological blocks, respectively. In our technique, our vanishing results follow from holomorphy of such rational functions.

中文翻译：

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管道同调球的 Witten-Reshetikhin-Turaev 不变量和同调块


在本文中，我们证明了 Gukov-Pei-Putrov-Vafa 对一类广泛的 3 美元流形的猜想。他们的猜想指出，Witten–Reshetikhin–Turaev (WRT) 不变量是同调块的径向极限，它们是他们为具有负定连接矩阵的管道 $3$ 流形引入的 $q$ 系列。我们证明中最困难的一点是证明同调块渐进展开中负次数系数中出现的加权高斯和的消失。为了解决这个问题，我们开发了一种渐近展开的新技术，它使我们能够分别比较与 WRT 不变量和同调块相关的有理函数和假 theta 函数的渐近展开。在我们的技术中，我们的消失结果来自于此类有理函数的全纯。