Calabi quasimorphism and quantum homology

TL;DR

This paper constructs a non-trivial homogeneous quasimorphism on the group of area-preserving diffeomorphisms of the 2-sphere, extending to certain symplectic manifolds, linking quantum homology and symplectic topology.

Contribution

It introduces a new quasimorphism related to the Calabi invariant, applicable to the 2-sphere and certain monotone symplectic manifolds with semi-simple quantum homology.

Findings

Existence of a non-trivial homogeneous quasimorphism on the 2-sphere diffeomorphism group.

Extension of the quasimorphism to monotone symplectic manifolds with semi-simple quantum homology.

Connection established between quantum homology algebra and symplectic diffeomorphism groups.

Abstract

We prove that the group of area-preserving diffeomorphisms of the 2-sphere admits a non-trivial homogeneous quasimorphism to the real numbers with the following property. Its value on any diffeomorphism supported in a sufficiently small open subset of the sphere equals to the Calabi invariant of the diffeomorphism. This result extends to more general symplectic manifolds: If the symplectic manifold is monotone and its quantum homology algebra is semi-simple we construct a similar quasimorphism on the universal cover of the group of Hamiltonian diffeomorphisms.