Hypertoric Fukaya categories and categories O

TL;DR

This paper establishes an equivalence between a category O from conical symplectic resolutions and a Fukaya category, providing new insights into their structure and relationships through microlocal sheaves and Floer theory.

Contribution

It proves that the category O for toric hyperkähler manifolds is equivalent to a partially wrapped Fukaya category, connecting deformation quantization and symplectic geometry.

Findings

Equivalence between category O and Fukaya categories for toric hyperkähler manifolds.

Formality of simple objects in the Fukaya category.

Koszul duality and Calabi-Yau structures in the context of these categories.

Abstract

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.