Hypertoric 2-categories O and symplectic duality

TL;DR

This paper constructs 2-categories related to hypertoric varieties and demonstrates their role in categorifying Koszul duality, providing insights into symplectic duality and 3d mirror symmetry.

Contribution

It introduces 2-categories of sheaves on hypertoric skeletons that lift projectives and simples, establishing equivalences that categorify Koszul duality.

Findings

Established 2-categories categorifying Koszul duality

Connected 2-categories to symplectic duality and mirror symmetry

Provided a prototype framework for symplectic duality studies

Abstract

We define 2-categories of microlocal perverse (resp. coherent) sheaves of categories on the skeleton of a hypertoric variety and show that the generators of these 2-categories lift the projectives (resp. simples) in hypertoric category $\mathcal{O}$. We then establish equivalences of 2-categories categorifying the Koszul duality between Gale dual hypertoric categories $\mathcal{O}$. These constructions give a prototype for understanding symplectic duality via the fully extended 3d mirror symmetry conjecture.