Hypertoric Fukaya categories and categories O
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| Format: | Preprint |
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2024
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| _version_ | 1866917709712719872 |
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| author | Côté, Laurent Gammage, Benjamin Hilburn, Justin |
| author_facet | Côté, Laurent Gammage, Benjamin Hilburn, Justin |
| contents | To a conical symplectic resolution with Hamiltonian torus action, Braden--Proudfoot--Licata--Webster associate a category O, defined using deformation quantization (DQ) modules. It has long been expected, though not stated precisely in the literature, that category O also admits a "Betti-type" realization as the Fukaya--Seidel category of a Lefschetz fibration.
In this paper, we confirm that the category O associated to a toric hyperkähler manifold is equivalent to the partially wrapped Fukaya category of a Liouville manifold stopped by the fiber of a J-holomorphic moment map. The proof involves relating earlier DQ-module computations to a new computation of microlocal perverse sheaves. Leveraging known results on (de Rham) hypertoric category O, we deduce several Floer-theoretic consequences, including formality of simple objects and Koszul duality for the (fully) wrapped Fukaya category; conversely, by applying results about microlocal sheaves, we produce a relative Calabi-Yau structure on category O. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_01379 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Hypertoric Fukaya categories and categories O Côté, Laurent Gammage, Benjamin Hilburn, Justin Symplectic Geometry Representation Theory To a conical symplectic resolution with Hamiltonian torus action, Braden--Proudfoot--Licata--Webster associate a category O, defined using deformation quantization (DQ) modules. It has long been expected, though not stated precisely in the literature, that category O also admits a "Betti-type" realization as the Fukaya--Seidel category of a Lefschetz fibration. In this paper, we confirm that the category O associated to a toric hyperkähler manifold is equivalent to the partially wrapped Fukaya category of a Liouville manifold stopped by the fiber of a J-holomorphic moment map. The proof involves relating earlier DQ-module computations to a new computation of microlocal perverse sheaves. Leveraging known results on (de Rham) hypertoric category O, we deduce several Floer-theoretic consequences, including formality of simple objects and Koszul duality for the (fully) wrapped Fukaya category; conversely, by applying results about microlocal sheaves, we produce a relative Calabi-Yau structure on category O. |
| title | Hypertoric Fukaya categories and categories O |
| topic | Symplectic Geometry Representation Theory |
| url | https://arxiv.org/abs/2406.01379 |