In this paper the interrelations between several natural morphisms on the π1 of groups of Hamiltonian diffeomorphisms are investigated. As an application, the equality of the (nonlinear) Maslov index of loops of quantomorphisms of prequantizations of ℂPn and the Calabi–Weinstein invariant is shown, settling affirmatively a conjecture by Givental. We also prove, in the wake of a remark by Woodward, the proportionality of the mixed action-Maslov morphism and the Futaki invariant on loops of Hamiltonian biholomorphisms of Fano Kahler manifolds. Finally, a family of generalized action-Maslov invariants is computed for toric manifolds, on loops coming from the torus action, via barycenters of their moment polytopes, with an application to mass-linear functions recently introduced by McDuff and Tolman. In addition, we reinterpret the quasimorphism of Py on the universal cover of the group of Hamiltonian diffeomorphisms of monotone symplectic manifolds.
A Variational Approach to Givental’s Nonlinear Maslov Index
