Symplectic reduction of quasi-morphisms and quasi-states

TL;DR

This paper demonstrates how quasi-morphisms and quasi-states on closed symplectic manifolds can be reduced to hyperplane sections, revealing their behavior under symplectic reduction and linking spectral invariants to the Calabi homomorphism.

Contribution

It establishes the descent of quasi-morphisms and quasi-states under symplectic reduction and characterizes spectral invariant-derived quasi-morphisms as the Calabi homomorphism on certain Hamiltonians.

Findings

Quasi-morphisms descend under symplectic reduction.

Spectral invariant quasi-morphisms equal the Calabi homomorphism on stably displaceable sets.

Provides new insights into the structure of quasi-morphisms and quasi-states in symplectic geometry.

Abstract

We prove that quasi-morphisms and quasi-states on a closed integral symplectic manifold descend under symplectic reduction to symplectic hyperplane sections. Along the way we show that quasi-morphisms that arise from spectral invariants are the Calabi homomorphism when restricted to Hamiltonians supported on stably displaceable sets.