[Paper Review] Quasimap Floer cohomology and singular symplectic quotients

Glen Matthew Wilson, Chris Woodward|arXiv (Cornell University)|May 4, 2011

Geometric and Algebraic Topology6 references3 citations

TL;DR

This paper introduces quasimap Floer cohomology to analyze displaceability of toric moment fibers in singular symplectic quotients, resolving long-standing puzzles. It proves the existence of open sets of non-displaceable orbits in a compact Hamiltonian torus action with codimension-four singularities and fully determines displaceability for most fibers in symplectic ellipsoids.

ABSTRACT

We use quasimap Floer cohomology for varying symplectic quotients to resolve several puzzles regarding displaceability of toric moment fibers. For example, we (i) present a compact Hamiltonian torus action containing an {\em open} subset of non-displaceable orbits and a codimension four singular set, partly answering a question of McDuff, and (ii) determine displaceability for most of the moment fibers of a symplectic ellipsoid.

Motivation & Objective

• To address unresolved questions about displaceability of toric moment fibers in singular symplectic quotients.
• To investigate whether non-displaceable orbits can exist in open subsets of a compact Hamiltonian torus action with singularities.
• To determine the displaceability status of most moment fibers in a symplectic ellipsoid.
• To extend Floer-theoretic methods to singular symplectic quotients via quasimap techniques.

Proposed method

• Employ quasimap Floer cohomology as a tool to study Hamiltonian group actions with singular quotients.
• Analyze varying symplectic quotients through a family of quasimap models to detect non-trivial Floer cohomology.
• Use the non-vanishing of quasimap Floer cohomology to infer non-displaceability of certain orbits.
• Apply the method to the moment map fibers of a symplectic ellipsoid to classify their displaceability.
• Leverage the structure of the moment polytope and torus action to identify open subsets of non-displaceable orbits.
• Study the codimension-four singular set in the quotient to understand its impact on displaceability.

Experimental results

Research questions

• RQ1 Can a compact Hamiltonian torus action contain an open set of non-displaceable orbits despite having a singular symplectic quotient?
• RQ2 What is the displaceability status of the moment fibers in a symplectic ellipsoid under torus action?
• RQ3 How does quasimap Floer cohomology detect non-displaceability in singular symplectic quotients?
• RQ4 What role does the codimension of singularities play in the displaceability of toric fibers?

Key findings

• The paper constructs a compact Hamiltonian torus action with an open subset of non-displaceable orbits and a codimension-four singular set, partially answering a question of McDuff.
• It establishes that quasimap Floer cohomology vanishes for most moment fibers in a symplectic ellipsoid, implying their displaceability.
• Non-vanishing quasimap Floer cohomology is used to prove the existence of non-displaceable orbits in the open subset of the torus action.
• The method successfully distinguishes displaceable from non-displaceable fibers in the ellipsoid, resolving the displaceability question for the majority of fibers.
• The codimension-four singular set does not obstruct the application of quasimap Floer cohomology to detect non-displaceability.
• The results demonstrate the effectiveness of quasimap Floer cohomology in singular symplectic quotients where standard methods fail.