Fukaya categories of hyperplane arrangements

TL;DR

This paper establishes a connection between the symplectic topology of hyperplane arrangement complements and their associated Fukaya categories, confirming conjectures linking algebraic and symplectic invariants.

Contribution

It proves that the partially wrapped Fukaya categories of hyperplane arrangement complements are generated by specific Lagrangians and identifies these categories with hypertoric convolution algebras, confirming key conjectures.

Findings

Fukaya categories are generated by Lagrangians from chambers.

Identified Fukaya categories with hypertoric convolution algebras.

Confirmed conjectures relating symplectic and algebraic structures.

Abstract

To a simple polarized hyperplane arrangement (not necessarily cyclic) $\mathbb{V}$, one can associate a stopped Liouville manifold (equivalently, a Liouville sector) $\left(M(\mathbb{V}),\xi\right)$, where $M(\mathbb{V})$ is the complement of finitely many hyperplanes in $\mathbb{C}^d$, obtained as the complexifications of the real hyperplanes in $\mathbb{V}$. The Liouville structure on $M(\mathbb{V})$ comes from a very affine embedding, and the stop $\xi$ is determined by the polarization. In this article, we study the symplectic topology of $\left(M(\mathbb{V}),\xi\right)$. In particular, we prove that their partially wrapped Fukaya categories are generated by Lagrangian submanifolds associated to the bounded and feasible chambers of $\mathbb{V}$. A computation of the Fukaya $A_\infty$-algebra of these Lagrangians then enables us to identity these wrapped Fukaya categories with the $\mathbb{G}_m^d$-equivariant hypertoric convolution algebras $\widetilde{B}(\mathbb{V})$ associated to $\mathbb{V}$. This confirms a conjecture of Lauda-Licata-Manion (arXiv:2009.03981) and provides evidence for the general conjecture of Lekili-Segal (arXiv:2304.10969) on the equivariant Fukaya categories of symplectic manifolds with Hamiltonian torus actions.