Hypertoric Fukaya categories and categories O

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&q...

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Bibliographic Details
Published in:arXiv.org 2024-07
Main Authors: Côté, Laurent, Gammage, Benjamin, Hilburn, Justin
Format: Article
Language:English
Subjects:
Manifolds
Modules
Sheaves
Toruses
Online Access:Get full text
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Summary: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 hyperk\"ahler 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.
ISSN:2331-8422