Fivebranes and 3-manifold homology

TL;DR

This work unifies 3-manifold homology with fivebrane constructions by formulating a universal framework where ${\mathcal{H}}_N^{*,*}(M_3)$ arises as $Q$-cohomology of partially twisted 3d ${\mathcal{N}}=2$ theories $T[M_3]$, connected to the categorification of Chern–Simons invariants and to Seiberg–Witten/Heegaard Floer theories in the $N=0$ case. By developing a 2d A-model perspective and exploiting the 3d/3d correspondence, the authors derive modular, Verlinde-like structures, and explicit calculations for lens spaces and plumbed 3-manifolds, including Turaev torsion and $HF^+(M_3)$. They extend the framework to higher ranks, revealing connections to mock modular forms, Eichler integrals, and homological blocks, thereby offering a path to 3-manifold analogues of Khovanov–Rozansky homology and their BPS interpretations. The paper also links these invariants to 4- and 5-dimensional theories via Rozansky–Witten refinements and large-$N$ dualities, suggesting deep ties between topological quantum field theory, algebraic geometry, and condensed-mmatter-like BPS spectra that may yield computational tools and new invariants for 3-manifolds.

Abstract

Motivated by physical constructions of homological knot invariants, we study their analogs for closed 3-manifolds. We show that fivebrane compactifications provide a universal description of various old and new homological invariants of 3-manifolds. In terms of 3d/3d correspondence, such invariants are given by the Q-cohomology of the Hilbert space of partially topologically twisted 3d N=2 theory T[M_3] on a Riemann surface with defects. We demonstrate this by concrete and explicit calculations in the case of monopole/Heegaard Floer homology and a 3-manifold analog of Khovanov-Rozansky link homology. The latter gives a categorification of Chern-Simons partition function. Some of the new key elements include the explicit form of the S-transform and a novel connection between categorification and a previously mysterious role of Eichler integrals in Chern-Simons theory.

Figures (12)

• Figure 1: The landscape of knot homologies shows peculiar behavior at $N=-1, 0$ and $1$.
• Figure 2: Introducing a line operator labelled by $h\in H$ along a non-trivial cycle of the solid torus creates a state in ${\mathcal{H}}_{T^A[M_3]}(S^1) = {\mathbb C}[H]$.
• Figure 3: The brane realization of ${\mathcal{N}}=4$ 3d SQED in type IIB string theory.
• Figure 4: The space-time of $T[M_3]$: $\Sigma_\text{SW}\times S^1$ where $\Sigma_\text{SW}$ is the Seiberg-Witten curve.
• Figure 5: a) Type IIA brane construction of 4d ${\mathcal{N}}=2$ SQED with zero FI-parameter. b) Equivalent description achieved by pulling $D6$ brane to $x^7=-\infty$. c) The curve appearing in M-theory lift.