Remarks on invariants of hamiltonian loops

TL;DR

This paper explores invariants of Hamiltonian loops, establishing key equalities and proportionalities among them, and computes generalized invariants for toric manifolds, advancing understanding in symplectic and complex geometry.

Contribution

It proves the equality of the Maslov index and Calabi-Weinstein invariant, and the proportionality of the action-Maslov morphism with the Futaki invariant, plus computes new invariants for toric manifolds.

Findings

Equality of Maslov index and Calabi-Weinstein invariant for loops of quantomorphisms.

Proportionality of the mixed action-Maslov morphism and Futaki invariant.

Explicit computation of generalized action-Maslov invariants for toric manifolds.

Abstract

In this note the interrelations between several natural morphisms on the $\pi_1$ of groups of Hamiltonian diffeomorphisms are investigated. As an application, the equality of the (non-linear) Maslov index of loops of quantomorphisms of prequantizations of $\C P^n$ and the Calabi-Weinstein invariant is shown, settling affirmatively a conjecture by A. Givental. We also prove the proportionality of the mixed action-Maslov morphism and the Futaki invariant on loops of Hamiltonian biholomorphisms of Fano Kahler manifolds, as suggested by C. Woodward. Finally, a family of generalized action-Maslov invariants is computed for toric manifolds via barycenters of their moment polytopes, with an application to mass-linear functions recently introduced by D. McDuff and S. Tolman.