Homological Mirror Symmetry for Hypertoric Varieties I
Michael McBreen, Ben Webster
TL;DR
This work establishes a homological mirror symmetry for hypertoric varieties by relating coherent sheaves on the additive hypertoric variety to deformation quantization modules on the Dolbeault mirror, with a conical G_m-action tracked via a graded Hodge-type structure. The authors construct a tilting generator on the B-side, compute its endomorphism algebra H^λ, and identify its Koszul dual H^! with the Ext-algebra of simple A-branes on the mirror, enabling a derived equivalence between D^b(Coh(𝔐))_o and the derived category of DQ-modules on 𝔇. The characteristic p localization plays a central role in building tilting generators and transporting them to characteristic zero, while microlocal mixed Hodge structures provide a refined, graded HMS framework that matches the G_m-grading on the A-side. The results yield exotic t-structures on coherent sheaves and a Koszul dual narrative tying the A- and B-model sides through toric and Coulomb-branch data, with extensions to a graded category of mixed Hodge DQ-modules and a pathway to future generalizations via Coulomb branches.
Abstract
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. We analyze coherent sheaves by constructing a tilting generator, using the characteristic $p$ approach of Kaledin; the result is a sum of line bundles, which can be described using a simple combinatorial rule. The endomorphism algebra $H$ of this tilting generator has a simple quadratic presentation in the grading induced by $\mathbb{G}_m$-equivariance. In fact, we can confirm it is Koszul, and compute its Koszul dual $H^!$. We then show that this same algebra appears as an Ext-algebra of simple A-branes in a Dolbeault hypertoric manifold. The $\mathbb{G}_m$-equivariant grading on coherent sheaves matches a Hodge grading in this category.