Quasi-states, quasi-morphisms, and the moment map

Abstract

We prove that symplectic quasi-states and quasi-morphisms on a symplectic manifold descend under symplectic reduction on a superheavy level set of a Hamiltonian torus action. Using a construction due to Abreu and Macarini, in each dimension at least four we produce a closed symplectic toric manifold with infinite dimensional spaces of symplectic quasi-states and quasi-morphisms, and a one-parameter family of non-displaceable Lagrangian tori. By using McDuff's method of probes, we also show how Ostrover and Tyomkin's method for finding distinct spectral quasi-states in symplectic toric Fano manifolds can also be used to find different superheavy toric fibers.

22 pages, 7 figures; v3: minor corrections, added remarks, and altered numbering scheme to match published version. To appear in International Mathematics Research Notices

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Cited by in corpus (11)

• Quasi-morphisms on contactomorphism groups and contact rigidity
• Spectral invariants with bulk, quasimorphisms and Lagrangian Floer theory
• Vanishing of Rabinowitz Floer homology on negative line bundles
• Givental's non-linear Maslov index on lens spaces
• Low-area Floer theory and non-displaceability
• Displacing Lagrangian toric fibers by extended probes
• Remarks on Lagrangian intersections in toric manifolds
• $A_n$-type surface singularity and nondisplaceable Lagrangian tori
• Existence of pseudo-heavy fibers of moment maps
• On boundedness of characteristic class via quasi-morphism
• Gauged Floer homology and spectral invariants