[Paper Review] Remarks on invariants of hamiltonian loops

Egor Shelukhin|ArXiv.org|May 11, 2009

Geometry and complex manifolds22 references3 citations

TL;DR

This paper establishes the proportionality between the nonlinear Maslov index and the Calabi-Weinstein invariant for Hamiltonian loops on complex projective space $\mathbb{C}P^n$, confirming a conjecture by Givental. It further proves proportionality between the mixed action-Maslov invariant and the Futaki invariant on Fano Kähler manifolds, and computes generalized action-Maslov invariants for toric manifolds via barycenters of moment polytopes, with applications to mass-linear functions.

ABSTRACT

In this note the interrelations between several natural morphisms on the $π_1$ 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.

Motivation & Objective

• To resolve Givental's conjecture on the equality of the nonlinear Maslov index and the Calabi-Weinstein invariant for $\mathbb{C}P^n$.
• To verify a suggestion by C. Woodward on the proportionality between the mixed action-Maslov invariant and the Futaki invariant on Fano Kähler manifolds.
• To compute generalized action-Maslov invariants for toric manifolds using barycenters of moment polytopes.
• To apply these results to the theory of mass-linear functions introduced by McDuff and Tolman.
• To explore topological and geometric implications of these invariants in Hamiltonian and quantomorphism groups.

Proposed method

• The paper uses the Seidel representation and Floer theory to analyze the topology of $\pi_1$ of Hamiltonian diffeomorphism groups.
• It applies the definition of the mixed action-Maslov homomorphism $I$ via symplectic linearization and Maslov index computation along disks filling loops.
• For toric manifolds, the generalized action-Maslov invariants are computed using the barycenter of the moment polytope and integration over invariant submanifolds.
• The Calabi-Weinstein homomorphism is evaluated via integration of contact Hamiltonians over the base manifold.
• The proof of proportionality between invariants relies on comparing the coupling class and Poincaré duality in equivariant cohomology.
• It uses the moment map and equivariant geometry to relate the invariants to integrals over faces of the moment polytope.

Experimental results

Research questions

• RQ1 Does the nonlinear Maslov index on loops of quantomorphisms of $\mathbb{C}P^n$ equal the Calabi-Weinstein invariant, as conjectured by Givental?
• RQ2 Is the mixed action-Maslov invariant on Fano Kähler manifolds proportional to the Futaki invariant, as suggested by Woodward?
• RQ3 Can generalized action-Maslov invariants on toric manifolds be computed via barycenters of their moment polytopes?
• RQ4 Does vanishing of all $I_{\alpha}(\gamma)$ for loops in the toric action imply contractibility of $\gamma$?
• RQ5 Can the mixed action-Maslov invariant be extended to a quasimorphism on the universal cover of $Ham$ in non-monotone Kähler manifolds?

Key findings

• The nonlinear Maslov index and the Calabi-Weinstein invariant are equal for loops of quantomorphisms of $\mathbb{C}P^n$, confirming Givental's conjecture.
• The mixed action-Maslov invariant is proportional to the Futaki invariant on Fano Kähler manifolds, verifying a suggestion by C. Woodward.
• Generalized action-Maslov invariants on toric manifolds are computed as $-(n+1-L)!(B_{n-L} - B_n)$, where $B_k$ are barycenters of moment polytope faces.
• The computation shows that the invariant depends linearly on the barycenter of the moment polytope's codimension-$L$ faces.
• The results provide a geometric interpretation of mass-linear functions in terms of barycenters and invariants of Hamiltonian loops.
• The paper establishes that for toric Fano manifolds, the vanishing of the mixed action-Maslov invariant is equivalent to the existence of a Kähler-Einstein metric.