From hypertoric geometry to bordered Floer homology via the m=1 amplituhedron

TL;DR

This paper conjectures an algebraic description of the Fukaya category for hyperplane complements, proves it for cyclic arrangements, and connects it to bordered Floer homology and the m=1 amplituhedron, advancing understanding of symplectic and algebraic geometry.

Contribution

It provides a conjectural algebraic framework for the Fukaya category of hyperplane complements and proves it for cyclic arrangements, linking to bordered Floer homology and amplituhedron combinatorics.

Findings

Proved the conjecture for cyclic arrangements using isomorphisms with bordered Floer algebras.

Extended work on sign variation and amplituhedron combinatorics to the algebraic setting.

Constructed categorical actions of gl(1|1) from cyclic arrangement algebras.

Abstract

We give a conjectural algebraic description of the Fukaya category of a complexified hyperplane complement, using the algebras defined in arXiv:0905.1335 from the equivariant cohomology of toric varieties. We prove this conjecture for cyclic arrangements by showing that these algebras are isomorphic to algebras appearing in work of Ozsvath-Szabo arXiv:1603.06559 in bordered Heegaard Floer homology arXiv:0810.0687. The proof of our conjecture in the cyclic case extends work of Karp-Williams arXiv:1608.08288 on sign variation and the combinatorics of the m=1 amplituhedron. We then use the algebras associated to cyclic arrangements to construct categorical actions of gl(1|1).