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Lagrangian Floer theory on Woodward's multiplicity-free $U(2)$-manifolds

Yao Xiao

Abstract

In this paper, we study a family of symplectic manifolds introduced by Woodward. These manifolds belong to the broader class of \emph{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.

Lagrangian Floer theory on Woodward's multiplicity-free $U(2)$-manifolds

Abstract

In this paper, we study a family of symplectic manifolds introduced by Woodward. These manifolds belong to the broader class of \emph{multiplicity-free} Hamiltonian -manifolds, a generalization of toric manifolds for non-abelian Hamiltonian group actions. Prominent examples of multiplicity-free spaces include coadjoint orbits of and equipped with multiplicity-free - and -actions, respectively. Although these multiplicity-free -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.
Paper Structure (15 sections, 28 theorems, 130 equations, 7 figures)

This paper contains 15 sections, 28 theorems, 130 equations, 7 figures.

Table of Contents

  1. Introduction
  2. The multiplicity-free U(2)-manifolds
  3. Coadjoint orbits of unitary groups
  4. Construction of Woodward's examples
  5. The split A-infinity algebra
  6. The leading order potential
  7. Critical points of the potential function
  8. Displaceability of Gelfand--Cetlin fibers
  9. Non-displaceability of the central Lagrangian torus
  10. Displaceability of some other fibers
  11. Generation of the Fukaya category
  12. Dimension of HH_*(CF(L_i,L_i))
  13. Surjectivity of OC
  14. Multiplicity-free manifolds
  15. Z/2Z-graded Clifford algebras

Key Result

Theorem 1

There exists a non-displaceable Lagrangian torus in every multiplicity-free $U(2)$-manifold $M$ constructed by Woodward in WoodwardExample.

Figures (7)

  • Figure 1: Left: GC polytope of $M$, Right: $U(2)$-Kirwan polytope of $M$
  • Figure 2: Left: GC polytope of $\mathop{\mathrm{\mathcal{O}}}\nolimits_{\mathop{\mathrm{\lambda}}\nolimits}$, Right: $U(2)$-moment polytope of $\mathop{\mathrm{\mathcal{O}}}\nolimits_{\mathop{\mathrm{\lambda}}\nolimits}$
  • Figure 3: The symplectic cut. Right: Gelfand--Cetlin $T^2$-moment polytope.
  • Figure 4: The pieces $M_+, Y, M_-$.
  • Figure 5: $M_-$ and its $U(2)$-Kirwan polytope
  • ...and 2 more figures

Theorems & Definitions (54)

  • Theorem : Theorem \ref{['thm Existence of a non-displaceable Lagrangian']}
  • Proposition : Proposition \ref{['Proposition Probe-nondisplaceable set']}
  • Theorem
  • Definition 2.1: Multiplicity-free
  • Lemma 2.1: Delzant, WoodwardExample Lemma 3.6
  • Proposition 4.1: Leading order potential of $M$
  • Lemma 4.1
  • proof
  • Lemma 4.2: $H^1(L;\Lambda_0)$ are weak Maurer-Cartan
  • proof
  • ...and 44 more