By incorporating higher-form symmetries, we propose a refined definition of the theories obtained by compactification of the 6d (2 , 0) theory on a three-manifold M 3 . This generalization is applicable to both the 3d N $$ \mathcal{N} $$ = 2 and N $$ \mathcal{N} $$ = 1 supersymmetric reductions. An observable that is sensitive to the higher-form symmetries is the Witten index, which can be computed by counting solutions to a set of Bethe equations that are determined by M 3 . This is carried out in detail for M 3 a Seifert manifold, where we compute a refined version of the Witten index. In the context of the 3d-3d correspondence, we complement this analysis in the dual topological theory, and determine the refined counting of flat connections on M 3 , which matches the Witten index computation that takes the higher-form symmetries into account.

中文翻译：

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更高形式的对称性、Bethe vacua 和 3d-3d 对应关系

通过结合更高形式的对称性，我们提出了通过对三流形 M 3 上的 6d (2 , 0) 理论进行紧化而获得的理论的精确定义。这种概括适用于 3d N $$ \mathcal{N} $$ = 2 和 N $$ \mathcal{N} $$ = 1 超对称归约。对更高形式的对称性敏感的可观察量是 Witten 指数，它可以通过计算由 M 3 确定的一组 Bethe 方程的解来计算。这在 M 3 a Seifert 流形中详细执行，我们在其中计算 Witten 指数的改进版本。在 3d-3d 对应的背景下，我们在对偶拓扑理论中补充了这一分析，并确定了 M 3 上平面连接的精细计数，这与考虑更高形式对称性的 Witten 指数计算相匹配。