[Paper Review] An open quantum Kirwan map

Chris Woodward, Guangbo Xu|arXiv (Cornell University)|Jun 18, 2018

Geometric and Algebraic Topology49 references3 citations

TL;DR

This paper constructs an open quantum Kirwan map—a morphism between the equivariant Fukaya algebra of a Lagrangian brane in the zero level set of a moment map and the Fukaya algebra of its quotient brane—proving it intertwines disk potentials and induces a map on Maurer–Cartan solution spaces. The construction verifies a conjecture on coordinate changes in toric mirror symmetry and provides a new proof of the open mirror theorem for semi-Fano toric manifolds.

ABSTRACT

We construct a morphism from the equivariant Fukaya algebra of a Lagrangian brane in the zero level set of a moment map of a Hamiltonian action to the Fukaya algebra of the quotient brane. This morphism induces a map between Maurer-Cartan solution spaces, and intertwines the disk potentials. As an application, we show under some technical hypotheses that weak unobstructedness of an invariant Lagrangian brane implies weak unobstructedness of its quotient. For semi-Fano toric manifolds we give a different proof of the open mirror theorem of Chan-Lau-Leung-Tseng by showing that the potential of a Lagrangian toric orbit in a toric manifold is related to the Givental-Hori-Vafa potential by a change of variable. We also reprove the results of Fukaya-Oh-Ohta-Ono on weak unobstructedness of these toric orbits. In the case of polygon spaces we show the existence of weakly unobstructed and Floer nontrivial products of spheres.

Motivation & Objective

• To establish a morphism between the equivariant Fukaya algebra of a Lagrangian in the zero level set of a moment map and the Fukaya algebra of its quotient under a Hamiltonian group action.
• To show this morphism intertwines disk potentials and maps solutions of the Maurer–Cartan equations between the two algebras.
• To prove that weak unobstructedness of an invariant Lagrangian implies weak unobstructedness of its quotient under suitable technical conditions.
• To reprove the open mirror theorem for semi-Fano toric manifolds using the open quantum Kirwan map and relate the Givental–Hori–Vafa potential to the Lagrangian Floer potential via a change of variables.
• To extend the framework to polygon spaces and demonstrate the existence of weakly unobstructed, Floer-nontrivial products of spheres.

Proposed method

• Constructs a morphism—called the open quantum Kirwan map—using moduli spaces of treed vortices with boundary conditions on Lagrangian submanifolds.
• Employs perturbation data and coherent perturbation systems to achieve regularity and orientability of the moduli spaces of treed vortices.
• Applies transversality and compactness arguments to control the structure of one-dimensional moduli spaces and handle boundary components.
• Uses weighted treed disks and a scaling limit to relate the equivariant Fukaya algebra on the original space to the quotient algebra on the symplectic quotient.
• Implements a local model for affine vortices and analyzes their contribution to the potential function via index and dimension counting.
• Applies the construction to toric manifolds by realizing them as GIT quotients and computing the potential function using the Givental–Hori–Vafa potential with a change of variables.

Experimental results

Research questions

• RQ1 How can one relate the equivariant Fukaya algebra of a Lagrangian in the zero level set of a moment map to the Fukaya algebra of its quotient under a Hamiltonian group action?
• RQ2 What is the precise relationship between the disk potential in the equivariant setting and the potential in the quotient setting?
• RQ3 Does weak unobstructedness of an invariant Lagrangian imply weak unobstructedness of its quotient under the group action?
• RQ4 Can the open mirror theorem for semi-Fano toric manifolds be re-proven using a geometric construction of the quantum Kirwan map?
• RQ5 What is the role of affine vortices in the coordinate change between the Givental–Hori–Vafa potential and the Lagrangian Floer potential?

Key findings

• The open quantum Kirwan map is constructed as a well-defined morphism of $A_∞$-algebras that intertwines the disk potentials of the original and quotient Lagrangians.
• The map induces a well-defined, injective map between the solution spaces of the Maurer–Cartan equations of the two algebras.
• Under technical hypotheses, weak unobstructedness of an invariant Lagrangian implies weak unobstructedness of its quotient, extending results of Fukaya–Oh–Ohta–Ono.
• For semi-Fano toric manifolds, the open mirror theorem of Chan–Lau–Leung–Tseng is reproven by showing the Lagrangian Floer potential is related to the Givental–Hori–Vafa potential via a change of variables.
• The construction verifies the conjecture that the coordinate change in mirror symmetry arises from counts of affine vortices over the complex plane and upper half-plane.
• In the case of polygon spaces, the existence of weakly unobstructed Lagrangians with nontrivial Floer cohomology is established, demonstrating the map's applicability beyond toric geometry.