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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.

Submission history

From: Meng-Chwan Tan [view email]
[v1] Thu, 24 Oct 2024 09:29:21 UTC (78 KB)
[v2] Thu, 5 Dec 2024 03:05:02 UTC (78 KB)
[v3] Thu, 16 Jan 2025 09:34:49 UTC (79 KB)
[v4] Thu, 8 May 2025 04:07:50 UTC (80 KB)
[v5] Thu, 25 Jun 2026 07:45:40 UTC (73 KB)