Fukaya Categories of Hyperplane Arrangements

To a simple polarized hyperplane arrangement (not necessarily cyclic) V, one can associate a stopped Liouville manifold (equivalently, a Liouville sector) (M (V), ), where M (V) is the complement of finitely many hyperplanes in Cᵈ, 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 is determined by the polarization. In this article, we study the symplectic topology of (M (V), ). 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_-algebra of these Lagrangians then enables us to identity these wrapped Fukaya categories with the Gₘᵈ-equivariant hypertoric convolution algebras B (V) associated to V. This confirms a conjecture of Lauda-Licata-Manion and provides evidence for the general conjecture of Lekili-Segal on the equivariant Fukaya categories of symplectic manifolds with Hamiltonian torus actions.