BPS spectra and 3-manifold invariants

We provide a physical definition of new homological invariants 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 N = 2 theory T[M-3]: D-2 x S-1 half-index, S-2 x S-1 superconformal index, and S-2 x 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.