BPS spectra and 3-manifold invariants

TL;DR

The paper develops a physical framework to define and categorify new 3-manifold invariants ${\mathcal{H}}_a(M_3)$ by realizing them as the BPS spectrum of the 6d (2,0) theory on $M_3\times D^2$, organized into abelian-flat-connection labeled blocks with associated homological blocks $\widehat{Z}_a(q)$. It shows how these blocks recombine to reproduce Witten–Reshetikhin–Turaev invariants and how they relate to SUSY partition functions (the D^2×S^1 half-index, the S^2×S^1 superconformal index, and the S^2×S^1 topologically twisted index) through factorization formulas, refined gradings, and resurgent structures. The work provides extensive concrete checks across a variety of 3-manifolds (including $S^3$, lens spaces, circle bundles over $\Sigma_g$, and plumbed manifolds) and extends to knots/links via line operators and impurities in $T[M_3]$, linking back to open GW invariants and Langlands duality. Overall, the results establish a robust bridge between high-energy physics constructions (M5-branes, S-duality, and localization) and mathematical invariants, offering new avenues for computable, integrality-rich categorifications of 3-manifold invariants with potential implications for both topology and mathematical physics.

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.

Figures (5)

• Figure 1: The space of BPS states in 3d ${\mathcal{N}}=2$ theory on ${\mathbb R} \times D^2$ with an impurity, relevant to the physical realization of Heegaard Floer homology $HF(M_3)$, monopole Floer homology $HM(M_3)$, as well as categorification of WRT invariants of 3-manifolds with knots.
• Figure 2: Another representation of the background in Figure 1.
• Figure 3: Critical points of $\widetilde{{\mathcal{W}}}$ in 3d ${\mathcal{N}}=2$ theories come in infinite towers because, as in complex Chern-Simons theory, the space of fields is not simply-connected. Moreover, a generic theory has degenerate, non-Morse critical points, which arise, e.g., when the fields at the critical point have a large stabilizer. In localization integrals, such critical points "absorb" the contributions of regular, Morse critical points as trans-series.
• Figure 4: An example of a plumbing graph $\Gamma$ (left) and the corresponding link ${\mathcal{L}}(\Gamma)$ of framed unknots in $S^3$ (right). The associated 3-manifold $M_3(\Gamma)$ can be constructed by performing a Dehn surgery on ${\mathcal{L}}(\Gamma)$.
• Figure 5: 3d Kirby moves that relate plumbing graphs which result in homeomorphic 3-manifolds.

Theorems & Definitions (4)

• Conjecture 2.1
• Conjecture 2.2
• Conjecture 2.3
• Conjecture 4.1