Homological Mirror Symmetry for Hypertoric Varieties I

Understand

We consider homological mirror symmetry in the context of hypertoric varieties, showing that appropriate categories of B-branes (that is, coherent sheaves) on an additive hypertoric variety match a category of A-branes on a Dolbeault hypertoric manifold for the same underlying combinatorial data.

• For technical reasons, the category of A-branes we consider is the modules over a deformation quantization (that is, DQ-modules).
• We consider objects in this category equipped with an analogue of a Hodge structure, which corresponds to a $\mathbb{G}_m$-action on the dual side of the mirror symmetry.
• This result is based on hands-on calculations in both categories.