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

Abstract

We relate the Fukaya category of the symmetric power of a genus zero surface to deformed category \(\mathcal {O}\) of a cyclic hypertoric variety by establishing an isomorphism between algebras defined by Ozsváth–Szabó in Heegaard–Floer theory and Braden–Licata–Proudfoot–Webster in hypertoric geometry. The proof extends work of Karp–Williams 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 \(\mathfrak {gl}(1|1)\) , and generalize our isomorphism to give a conjectural algebraic description of the Fukaya category of a complexified hyperplane complement.