Fukaya categories of hyperplane arrangements

Abstract

To a simple polarized hyperplane arrangement (not necessarily cyclic) $\mathbb{V}$, one can associate a stopped Liouville manifold (equivalently, a Liouville sector) $\left(M(\mathbb{V}),\hat{I}¾\right)$, 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 $\hat{I}¾$ is determined by the polarization. In this article, we study the symplectic topology of $\left(M(\mathbb{V}),\hat{I}¾\right)$. 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 $\tilde{B}(\mathbb{V})$ 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.

v3: Accepted version. Extended introductions and expositions, several typos fixed. 65 pages