Homological Mirror Symmetry for Hypertoric Varieties I

• Published in: arXiv.org 2021-10
• Main Authors: McBreen, Michael, Webster, Ben
• Format: Article
• Language: English
• Subjects: Algebra; Branes; Coherence; Combinatorial analysis; Deformation; Grading; Homology; Manifolds; Modules; Sheaves; Symmetry
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.1804.10646

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.