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

Michael McBreen and Ben Webster

Geometry & Topology 28 (2024) 1005–1063

We consider homological mirror symmetry in the context of hypertoric varieties, showing that an appropriate category of B–branes (that is, coherent sheaves) on an additive hypertoric variety matches a category of A–branes on a Dolbeault hypertoric manifold for the same underlying combinatorial data. For technical reasons, the A–branes we consider are 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 𝔾m–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 𝔾m–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 𝔾m–equivariant grading on coherent sheaves matches a Hodge grading in this category.

mirror symmetry, hypertoric, characteristic $p$, deformation quantization
Mathematical Subject Classification
Primary: 14J33, 47A67
Received: 9 October 2020
Revised: 22 August 2022
Accepted: 24 September 2022
Published: 10 May 2024
Proposed: Mark Gross
Seconded: Richard P Thomas, Dan Abramovich
Michael McBreen
Department of Mathematics
Chinese University of Hong Kong
Hong Kong
Ben Webster
Department of Pure Mathematics
University of Waterloo
Waterloo, ON
Perimeter Institute for Theoretical Physics
Waterloo, ON

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