  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
$\mathcal{N}=2$ and $\mathcal{N}=1$ 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.
