Homological mirror symmetry for hypertoric varieties II (with an Appendix written jointly with Laurent Côté and Justin Hilburn)
Benjamin Gammage, Michael McBreen, Ben Webster
TL;DR
The paper establishes a homological mirror symmetry between Fukaya categories of multiplicative hypertoric varieties and coherent sheaves on their mirrors, realized as noncommutative crepant resolutions. It develops a detailed A-model skeleton-based computation leveraging microlocal perverse sheaves, and a B-model noncommutative-algebraic description built from tilting bundles, with precise quiver presentations. The HMS statement is elevated to a perverse-schober equivalence, encoding monodromy across real-wall stratifications and yielding a rich, wall-crossing-aware picture of the mirror correspondence. Collectively, the work provides a blueprint for HMS and SYZ-type dualities for pairs of K-theoretic Coulomb branches and their monodromies in a hyperkähler setting.
Abstract
In this paper, we prove a homological mirror symmetry equivalence for pairs of multiplicative hypertoric varieties, and we calculate monodromy autoequivalences of these categories by promoting our result to an equivalence of perverse schobers. We prove our equivalence by matching holomorphic Lagrangian skeleta, on the A-model side, with non-commutative resolutions on the B-model side. The hyperkähler geometry of these spaces provides each category with a natural t-structure, which helps clarify SYZ duality in a hyperkähler context. Our results are a prototype for mirror symmetry statements relating pairs of K-theoretic Coulomb branches.