标题： 拓扑5d $\mathcal {N} = 2$ 规范理论：新型弗洛尔同调，它们的对偶性，以及三维流形的 $A_\infty$-范畴 显示英文标题

标题： Topological 5d $\mathcal {N} = 2$ Gauge Theory: Novel Floer Homologies, their Dualities, and an $A_\infty$-category of Three-Manifolds

摘要： 我们展示如何通过某种拓扑扭曲的5d${\cal N}=2$规范理论的超对称量子力学解释，从该理论的物理中定义四维、三维和二维流形的新规范理论Floer同调。它们分别与四维、三维和二维流形上的Vafa-Witten、Hitchin和$G_{\mathbb{C}}$-BF构型相关联。我们还展示了如何定义Hitchin空间的新辛Floer同调，这将使我们能够推导出新的Atiyah-Floer对应关系，将我们的规范理论Floer同调与Higgs丛的辛交点Floer同调联系起来。此外，拓扑不变性和5d“S对偶性”暗示了这些新Floer同调及其环/环面群推广之间的关系网和朗兰兹对偶性。最后，通过将5d理论解释为二维规范Landau-Ginzburg模型，我们从它所定义的孤子弦理论和5d分划函数中推导出一种Fukaya-Seidel类型的$A_\infty$-范畴，该范畴描述了三维流形上的Hitchin构型——从而对上述三维流形的Floer同调进行了范畴化——以及其新的Atiyah-Floer类型对应关系。因此，我们的工作提供了Haydys[1]、Abouzaid-Manolescu[2]和Bousseau[3]等数学猜想的纯物理证明和推广，以及其他内容。 显示英文摘要

摘要： We show how one can define novel gauge-theoretic Floer homologies of four, three and two-manifolds from the physics of a certain topologically-twisted 5d ${\cal N}=2$ gauge theory via its supersymmetric quantum mechanics interpretation. They are associated with Vafa-Witten, Hitchin, and $G_{\mathbb{C}}$-BF configurations on the four, three, and two-manifolds, respectively. We also show how one can define novel symplectic Floer homologies of Hitchin spaces, which in turn will allow us to derive novel Atiyah-Floer correspondences that relate our gauge-theoretic Floer homologies to symplectic intersection Floer homologies of Higgs bundles. Furthermore, topological invariance and 5d "S-duality" suggest a web of relations and a Langlands duality amongst these novel Floer homologies and their loop/toroidal group generalizations. Last but not least, via a 2d gauged Landau-Ginzburg model interpretation of the 5d theory, we derive, from the soliton string theory that it defines and the 5d partition function, a Fukaya-Seidel type $A_\infty$-category of Hitchin configurations on three-manifolds -- thereby categorifying the aforementioned Floer homology of three-manifolds -- and its novel Atiyah-Floer type correspondence. Our work therefore furnishes purely physical proofs and generalizations of the mathematical conjectures by Haydys [1], Abouzaid-Manolescu [2], and Bousseau [3], and more.