Fukaya categories of hyperplane arrangements
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866914235299135488 |
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| author | Lee, Sukjoo Li, Yin Liu, Si-Yang Mak, Cheuk Yu |
| author_facet | Lee, Sukjoo Li, Yin Liu, Si-Yang Mak, Cheuk Yu |
| contents | To a simple polarized hyperplane arrangement (not necessarily cyclic) $\mathbb{V}$, one can associate a stopped Liouville manifold (equivalently, a Liouville sector) $\left(M(\mathbb{V}),ξ\right)$, where $M(\mathbb{V})$ is the complement of finitely many hyperplanes in $\mathbb{C}^d$, obtained as the complexifications of the real hyperplanes in $\mathbb{V}$. The Liouville structure on $M(\mathbb{V})$ comes from a very affine embedding, and the stop $ξ$ is determined by the polarization. In this article, we study the symplectic topology of $\left(M(\mathbb{V}),ξ\right)$. In particular, we prove that their partially wrapped Fukaya categories are generated by Lagrangian submanifolds associated to the bounded and feasible chambers of $\mathbb{V}$. A computation of the Fukaya $A_\infty$-algebra of these Lagrangians then enables us to identity these wrapped Fukaya categories with the $\mathbb{G}_m^d$-equivariant hypertoric convolution algebras $\widetilde{B}(\mathbb{V})$ associated to $\mathbb{V}$. This confirms a conjecture of Lauda-Licata-Manion (arXiv:2009.03981) and provides evidence for the general conjecture of Lekili-Segal (arXiv:2304.10969) on the equivariant Fukaya categories of symplectic manifolds with Hamiltonian torus actions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_05856 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Fukaya categories of hyperplane arrangements Lee, Sukjoo Li, Yin Liu, Si-Yang Mak, Cheuk Yu Symplectic Geometry Representation Theory 53D40(Primary), 14M25(Secondary) To a simple polarized hyperplane arrangement (not necessarily cyclic) $\mathbb{V}$, one can associate a stopped Liouville manifold (equivalently, a Liouville sector) $\left(M(\mathbb{V}),ξ\right)$, where $M(\mathbb{V})$ is the complement of finitely many hyperplanes in $\mathbb{C}^d$, obtained as the complexifications of the real hyperplanes in $\mathbb{V}$. The Liouville structure on $M(\mathbb{V})$ comes from a very affine embedding, and the stop $ξ$ is determined by the polarization. In this article, we study the symplectic topology of $\left(M(\mathbb{V}),ξ\right)$. In particular, we prove that their partially wrapped Fukaya categories are generated by Lagrangian submanifolds associated to the bounded and feasible chambers of $\mathbb{V}$. A computation of the Fukaya $A_\infty$-algebra of these Lagrangians then enables us to identity these wrapped Fukaya categories with the $\mathbb{G}_m^d$-equivariant hypertoric convolution algebras $\widetilde{B}(\mathbb{V})$ associated to $\mathbb{V}$. This confirms a conjecture of Lauda-Licata-Manion (arXiv:2009.03981) and provides evidence for the general conjecture of Lekili-Segal (arXiv:2304.10969) on the equivariant Fukaya categories of symplectic manifolds with Hamiltonian torus actions. |
| title | Fukaya categories of hyperplane arrangements |
| topic | Symplectic Geometry Representation Theory 53D40(Primary), 14M25(Secondary) |
| url | https://arxiv.org/abs/2405.05856 |