2024/10/24 by Er, Arif, Tan, Meng-Chwan
#Algebraic Geometry (math.AG) #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #Geometric Topology (math.GT) #High Energy Physics - Theory (hep-th) #Symplectic Geometry (math.SG)
paper · doi:10.48550/arxiv.2410.18575
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 N=1 gauge theory on a Spin(7)-manifold via its supersymmetric quantum mechanics interpretation. They are associated with G2 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_∞-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_∞-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.