  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.
