BPS spectra and 3-manifold invariants

We provide a physical definition of new homological invariants H_a(M₃) of 3-manifolds (possibly, with knots) labeled by abelian flat connections. The physical system in question involves a 6d fivebrane theory on M₃ times a 2-disk, D², 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[M₃]: D²×S¹ half-index, S²×S¹ superconformal index, and S²×S¹ 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 M₃. 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.