Fukaya categories of hyperplane arrangements

Lee, Sukjoo; Li, Yin; Liu, Si-Yang; Mak, Cheuk Yu

doi:10.2140/gt.2025.29.4841

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Fukaya categories of hyperplane arrangements

Sukjoo Lee, Yin Li, Si-Yang Liu and Cheuk Yu Mak

Geometry & Topology 29 (2025) 4841–4909

DOI: 10.2140/gt.2025.29.4841

Abstract

To a simple polarized hyperplane arrangement (not necessarily cyclic) $𝕍$ , one can associate a stopped Liouville manifold (equivalently, a Liouville sector) $(M (𝕍), ξ)$ , where $M (𝕍)$ is the complement of finitely many hyperplanes in $ℂ^{d}$ , obtained as the complexifications of the real hyperplanes in $𝕍$ . The Liouville structure on $M (𝕍)$ comes from a very affine embedding, and the stop $ξ$ is determined by the polarization. In this article, we study the symplectic topology of $(M (𝕍), ξ)$ . In particular, we prove that their partially wrapped Fukaya categories are generated by Lagrangian submanifolds associated to the bounded and feasible chambers of $𝕍$ . A computation of the Fukaya $A_{\infty}$ -algebra of these Lagrangians then enables us to identify the partially wrapped Fukaya categories $𝒲 (M (𝕍), ξ)$ with the $𝔾_{m}^{d}$ -equivariant hypertoric convolution algebras $\tilde{B} (𝕍)$ associated to $𝕍$ . This confirms a conjecture of Lauda, Licata and Manion (2024) and provides evidence for the general conjecture of Lekili and Segal (2023) on the equivariant Fukaya categories of symplectic manifolds with Hamiltonian torus actions.

Keywords

symplectic geometry, Fukaya categories, hyperplane arrangements, hypertoric varieties, representation theory

Mathematical Subject Classification

Primary: 53D40

References

Publication

Received: 17 July 2024

Revised: 5 February 2025

Accepted: 16 April 2025

Published: 31 December 2025

Proposed: Mohammed Abouzaid

Seconded: Leonid Polterovich, Ciprian Manolescu

Authors

Sukjoo Lee
Center for Geometry and Physics Institute for Basic Science (IBS) Pohang South Korea

Yin Li
Department of Mathematics Uppsala University Uppsala Sweden

Si-Yang Liu
Department of Mathematics Dornsife College of Letters, Arts and Sciences University of Southern California Los Angeles, CA United States	Hausdorff Center for Mathematics University of Bonn Bonn Germany

Cheuk Yu Mak
School of Mathematical and Physical Sciences University of Sheffield Sheffield United Kingdom

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Volume 29, issue 9 (2025)