SYZ Mirror Symmetry for Hypertoric Varieties

Abstract

We construct a Lagrangian torus fibration on a smooth hypertoric variety and a corresponding SYZ mirror variety using T-duality and generating functions of open Gromov-Witten invariants. The variety is singular in general. We construct a resolution of the variety using the wall and chamber structure on the base of the SYZ fibration.

Homological mirror symmetry for hypertoric varieties, I: Conic equivariant sheaves

“…As a result of our use of DQ-modules as a substitute for the Fukaya category, this paper contains little about Lagrangian branes, pseudoholomorphic disks and other staples of symplectic geometry. The reader may wish to compare with the interesting recent preprint by Lau and Zheng [27], which appeared a few days before this paper and treats the problem of nonequivariant mirror symmetry for hypertoric varieties from the perspective of SYZ fibrations.…”