In this paper, we study new invariants $$\widehat{Z}_{{{\varvec{a}}}}(q)$$ Z ^ a ( q ) attached to plumbed 3-manifolds that were introduced by Gukov, Pei, Putrov, and Vafa. These remarkable q -series at radial limits conjecturally compute WRT invariants of the corresponding plumbed 3-manifold. Here, we investigate the series $$\widehat{Z}_{0}(q)$$ Z ^ 0 ( q ) for unimodular plumbing H -graphs with six vertices. We prove that for every positive definite unimodular plumbing matrix, $$\widehat{Z}_{0}(q)$$ Z ^ 0 ( q ) is a depth two quantum modular form on $${\mathbb {Q}}$$ Q .

中文翻译：

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更高深度的量子模块化形式和管道 3 流形

在本文中，我们研究了新的不变量 $$\widehat{Z}_{{{\varvec{a}}}}(q)$$ Z ^ a ( q ) 附加到由 Gukov 引入的管道 3-流形，贝、普特罗夫和瓦法。这些在径向极限处的非凡 q 系列推测性地计算了相应管道 3 流形的 WRT 不变量。在这里，我们研究了具有六个顶点的单模管道 H 图形的系列 $$\widehat{Z}_{0}(q)$$ Z ^ 0 ( q )。我们证明对于每个正定单模管道矩阵，$$\widehat{Z}_{0}(q)$$ Z ^ 0 ( q ) 是 $${\mathbb {Q}} 上的深度二量子模形式$$ Q 。