Hypertoric 2-categories O and symplectic duality

TL;DR

This work develops a 2-categorical framework for hypertoric category ${\mathcal O}$ by building a pair of fully extended 2-categories on dual geometric sides: the A-side of microlocal perverse schobers on a Lagrangian skeleton and the B-side of microlocal coherent schobers. It proves a 2-categorical Koszul duality between Gale dual hypertoric data, connecting to fully extended 3d mirror symmetry. By relating Betti (microlocal perverse) and de Rham (deformation-quantized) realizations, and employing periodic cyclic homology for decategorification, the authors recover the classical ${\mathcal O}$ and its Koszul duality while revealing richer 2-categorical structures that encode symplectic duality. The results furnish a concrete prototype for understanding symplectic duality and Fueter-type theories, with explicit generators and endomorphism algebras tied to hypertoric skeleta.

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.

Theorems & Definitions (141)

• Conjecture
• Conjecture : Fully extended 3d mirror symmetry
• Theorem 1
• Example 2
• Definition 3
• Definition 4
• Remark 5
• Definition 6: \ref{['defn:betti-2cat-o']}
• Remark 7
• Definition 8: \ref{['defn:cohcats-microlocal']}