2026/03/16 by Yao Xiao · 1 voice
Mathematics · #math.SG
arxiv published 2026/03/16 · arxiv updated 2026/03/17
In this paper, we study a family of symplectic manifolds introduced by Woodward. These manifolds belong to the broader class of multiplicity-free Hamiltonian G-manifolds, a generalization of toric manifolds for non-abelian Hamiltonian group actions. Prominent examples of multiplicity-free spaces include coadjoint orbits of U(n) and SO(n) equipped with multiplicity-free U(n-1)- and SO(n-1)-actions, respectively. Although these multiplicity-free U(2)-manifolds are not toric, we may study a family of Lagrangian tori by performing a symplectic cut that allows us to apply the toric Lagrangian Floer theory. In particular, we employ Venugopalan--Woodward's study of pseudoholomorphic curves under symplectic cuts to obtain the leading order potential. This allows us to identify a number of critical points of the potential function which correspond to a non-displaceable Lagrangian submanifold. Moreover, we adapt McDuff's probe method to show that the majority of the other Lagrangian submanifolds in these spaces are displaceable. Finally, we prove that the open-closed map for the Fukaya subcategory generated by these branes is an isomorphism. It follows that they satisfy Abouzaid--Fukaya--Oh--Ohta--Ono's generation criterion.