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” 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 .