BPS spectra and 3-manifold invariants

Understand

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 $M_3$.