Cohomology of symplectomorphism groups and critical values of hamiltonians

Abstract

Let (P, co) be a compact, simply connected symplectic manifold, Sym (P, co) its group of automorphisms. The period group F(P, co)c R is defined to be the image of [co] x H2 (P; Z) under the integration pairing H 2 (P; R) x H2 (P; Z) ~ R. Since [co] 4:0, the quotient G(P, co)=R/F(P, co) is either a circle or the quotient of a circle by a countable dense subgroup. The purpose of this paper is to define and study a natural homomorphism A from 7h (Sym (P, co)) to G (P, co). Simply stated, A measures the average over P (with respect to the symplectic measure) of the action integral around the trajectories of a loop of symplectomorphisms. In terms of hamiltonians, we give a formula for A which becomes especially simple when the loop is a 1-parameter subgroup, in which case we can make some concrete computations. These are related to the work of Guillemin-Sternberg [9], and Duistermaat-Heckmann [7] on momentum mappings. We also give an interpretation of A in terms of lifting symplectomorphisms from P to a prequantization, i.e. a G(P, co) bundle over P with a connection whose curvature form is co. Here, we extend the theory from the usual case where G(P, co) is a circle by using the concept of diffeological structure introduced by Souriau [16]. In a sequel to this paper [19], taking as a starting point the observation that the graphs of symplectomorphisms are lagrangian submanifolds in a product manifold, we shall study the possible extension of A to an invariant on loops in the space of lagrangian submanifolds in a symplectic manifold. It turns out that this invariant should be defined as the holonomy of a connection whose definition involves choosing a probability measure on each lagrangian submanifold. The connection is flat only for certain ways of choosing these measures, and the curvature in the general case is closely related to the "geometric phase" introduced by Berry [4], which is a subject of current interest among theoretical and experimental physicists as well as mathematicians [1, 6, I0, 11, 13, 173.