One of the main challenges in 3d-3d correspondence is that no existent approach offers a complete description of 3d $N=2$ SCFT $T[M_3]$ --- or, rather, a "collection of SCFTs" as we refer to it in the paper --- for all types of 3-manifolds that include, for example, a 3-torus, Brieskorn spheres, and hyperbolic surgeries on knots. The goal of this paper is to overcome this challenge by a more systematic study of 3d-3d correspondence that, first of all, does not rely heavily on any geometric structure on $M_3$ and, secondly, is not limited to a particular supersymmetric partition function of $T[M_3]$. In particular, we propose to describe such "collection of SCFTs" in terms of 3d $N=2$ gauge theories with "non-linear matter'' fields valued in complex group manifolds. As a result, we are able to recover familiar 3-manifold invariants, such as Turaev torsion and WRT invariants, from twisted indices and half-indices of $T[M_3]$, and propose new tools to compute more recent $q$-series invariants $\hat Z (M_3)$ in the case of manifolds with $b_1 > 0$. Although we use genus-1 mapping tori as our "case study," many results and techniques readily apply to more general 3-manifolds, as we illustrate throughout the paper.

中文翻译：

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映射 tori 的 3d-3d 对应关系

3d-3d 对应的主要挑战之一是，没有任何现有的方法提供 3d $N=2$ SCFT $T[M_3]$ 的完整描述——或者更确切地说，我们所指的“SCFT 集合”它在论文中 --- 适用于所有类型的 3 流形，例如，3 环面、Brieskorn 球体和结上的双曲线手术。本文的目标是通过更系统地研究 3d-3d 对应来克服这一挑战，首先，不严重依赖 $M_3$ 上的任何几何结构，其次，不限于特定的超对称分区$T[M_3]$ 的函数。特别是，我们建议用 3d $N=2$ 规范理论来描述这种“SCFT 的集合”，其中“非线性物质”场在复杂群流形中具有价值。因此，我们能够从 $T[M_3]$ 的扭曲指数和半指数中恢复熟悉的 3-流形不变量，例如 Turaev 扭转和 WRT 不变量，并提出新的工具来计算最近的 $q$ 系列不变量 $\在 $b_1 > 0$ 的流形的情况下，帽子 Z (M_3)$。尽管我们使用 genus-1 mapping tori 作为我们的“案例研究”，但正如我们在整篇论文中所说明的那样，许多结果和技术很容易应用于更一般的 3-流形。