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

TL;DR

This paper demonstrates how symplectic quasi-states and quasi-morphisms behave under symplectic reduction, constructs examples of manifolds with infinite-dimensional spaces of these objects, and explores methods to identify superheavy Lagrangian fibers.

Contribution

It proves the descent of symplectic quasi-states and quasi-morphisms under reduction, constructs new examples with infinite-dimensional spaces, and applies methods to identify superheavy fibers in toric manifolds.

Findings

Symplectic quasi-states descend under reduction on superheavy level sets.

Constructed toric manifolds with infinite-dimensional spaces of quasi-states.

Identified superheavy Lagrangian tori using McDuff's probes and spectral methods.

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.