# BPS spectra and 3-manifold invariants

## TL;DR

This paper introduces new homological invariants of 3-manifolds derived from a 6d fivebrane theory, linking them to categorifications of known 3d partition functions and providing new insights into 3-manifold invariants.

## Contribution

It provides a physical framework for defining homological invariants of 3-manifolds using 6d fivebrane theories, connecting them to categorifications of partition functions and extending the understanding of WRT invariants.

## Key findings

- Defined homological invariants $\\mathcal{H}_a (M_3)$ for 3-manifolds.
- Connected these invariants to categorifications of 3d partition functions.
- Explicitly demonstrated the factorization of partition functions for various examples.

## Abstract

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.