Topological 8d $\mathcal{N}=1$ Gauge Theory: Novel Floer Homologies, and $A_\infty$-categories of Six, Five, and Four-Manifolds

TL;DR

This work constructs a comprehensive physical program that derives and connects multiple Floer homologies for manifolds of dimension seven and below from a topologically twisted 8d $N=1$ Spin$(7)$ gauge theory. By performing successive Kaluza–Klein reductions along circles and Calabi–Yau factors, the authors realize gauge-theoretic and holomorphic Floer homologies (e.g., Spin$(7)$-instanton, holomorphic $G_2$ monopole, holomorphic DT, HW, VW) as supersymmetric quantum mechanics on moduli spaces and relate them to symplectic and hyperkähler Floer theories via Atiyah–Floer-type dualities. They also formulate Fukaya–Seidel type $A_ullet$-categories categorifying these Floer theories across six-, five-, and four-manifolds, establishing at least physical proofs or generalizations of conjectures by Haydys, Donaldson–Thomas, Cherkis, Salamon, and Bousseau, and propose a web of topological invariances connecting these structures. The results consolidate a unifying framework that translates gauge-theoretic data into categorified invariants and open-string/branes language, with potential implications for higher-categorical structures and future 2-category generalizations via Fueter-type theories. Overall, the paper provides a physically driven, interconnected panorama of Floer homologies and $A_$-categories across multiple dimensions, anchored in Spin$(7)$ geometry and its KK reductions, and offers concrete evidence for longstanding mathematical conjectures within a physical setting.

Abstract

This work is a continuation of the program initiated in [arXiv:2311.18302]. We show how one can define novel gauge-theoretic (holomorphic) Floer homologies of seven, six, and five-manifolds, from the physics of a topologically-twisted 8d $\mathcal{N}=1$ gauge theory on a Spin$(7)$-manifold via its supersymmetric quantum mechanics interpretation. They are associated with $G_2$ instanton, Donaldson-Thomas, and Haydys-Witten configurations on the seven, six, and five-manifolds, respectively. We also show how one can define hyperkähler Floer homologies specified by hypercontact three-manifolds, and symplectic Floer homologies of instanton moduli spaces. In turn, this will allow us to derive Atiyah-Floer type dualities between the various gauge-theoretic Floer homologies and symplectic intersection Floer homologies of instanton moduli spaces. Via a 2d gauged Landau-Ginzburg model interpretation of the 8d theory, one can derive novel Fukaya-Seidel type $A_\infty$-categories that categorify Donaldson-Thomas, Haydys-Witten, and Vafa-Witten configurations on six, five, and four-manifolds, respectively -- thereby categorifying the aforementioned Floer homologies of six and five-manifolds, and the Floer homology of four-manifolds from [arXiv:2311.18302] -- where an Atiyah-Floer type correspondence for the Donaldson-Thomas case can be established. Last but not least, topological invariance of the theory suggests a relation amongst these Floer homologies and Fukaya-Seidel type $A_\infty$-categories for certain Spin$(7)$-manifolds. Our work therefore furnishes purely physical proofs and generalizations of the conjectures by Donaldson-Thomas [2], Donaldson-Segal [3], Cherkis [4], Hohloch-Noetzel-Salamon [5], Salamon [6], Haydys [7], and Bousseau [8], and more.

Figures (4)

• Figure 1: Tyurin degeneration of $CY_3$ and Spin$(7)$-manifolds.
• Figure 2: Union of 2d A-models along their common boundary $\mathscr{R}$.
• Figure 3: BPS worldsheet with solitons $\gamma^{IJ}_\pm(t, \theta, \mathfrak{A}_6)$ and boundaries corresponding to $\mathcal{E}^I_{\text{DT}}(\theta)$ and $\mathcal{E}^J_{\text{DT}}(\theta)$.
• Figure 4: Tree-level scattering BPS worldsheet of incoming ($-$) and outgoing ($+$) LG $\mathfrak{A}_6^\theta$-soliton strings.