BPS spectra and 3-manifold invariants

Abstract

We provide a physical definition of new homological invariants [Formula: see text] of 3-manifolds (possibly, with knots) labeled by abelian flat connections. The physical system in question involves a 6d fivebrane theory on [Formula: see text] times a 2-disk, [Formula: see text], whose Hilbert space of BPS states plays the role of a basic building block in categorification of various partition functions of 3d [Formula: see text] theory [Formula: see text]: [Formula: see text] half-index, [Formula: see text] superconformal index, and [Formula: see text] 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 [Formula: see text]. 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.