Topological 5d $\mathcal {N} = 2$ Gauge Theory: Novel Floer Homologies, their Dualities, and an $A_\infty$-category of Three-Manifolds
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| Format: | Preprint |
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2023
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| _version_ | 1866918150103105536 |
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| author | Er, Arif Ong, Zhi-Cong Tan, Meng-Chwan |
| author_facet | Er, Arif Ong, Zhi-Cong Tan, Meng-Chwan |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2311_18302 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Topological 5d $\mathcal {N} = 2$ Gauge Theory: Novel Floer Homologies, their Dualities, and an $A_\infty$-category of Three-Manifolds Er, Arif Ong, Zhi-Cong Tan, Meng-Chwan High Energy Physics - Theory Algebraic Geometry Differential Geometry Geometric Topology Symplectic Geometry 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. |
| title | Topological 5d $\mathcal {N} = 2$ Gauge Theory: Novel Floer Homologies, their Dualities, and an $A_\infty$-category of Three-Manifolds |
| topic | High Energy Physics - Theory Algebraic Geometry Differential Geometry Geometric Topology Symplectic Geometry |
| url | https://arxiv.org/abs/2311.18302 |