  We provide a physical definition of new homological invariants $\mathcal{H}_a
(M_3)$ of 3-manifolds (possibly, with knots) labeled by abelian flat
connections. The physical system in question involves a 6d fivebrane theory on
$M_3$ times a 2-disk, $D^2$, whose Hilbert space of BPS states plays the role
of a basic building block in categorification of various partition functions of
3d $\mathcal{N}=2$ theory $T[M_3]$: $D^2\times S^1$ half-index, $S^2\times S^1$
superconformal index, and $S^2\times S^1$ topologically twisted index. The
first partition function is labeled by a choice of boundary condition and
provides a refinement of Chern-Simons (WRT) invariant. A linear combination of
them in the unrefined limit gives the analytically continued WRT invariant of
$M_3$. The last two can be factorized into the product of half-indices. We show
how this works explicitly for many examples, including Lens spaces, circle
fibrations over Riemann surfaces, and plumbed 3-manifolds.
