Topological invariants of connections on symplectic manifolds

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Let M 2n be a 2n-dimensional symplectic manifold, let E --~ M be a vector bundle whose structure group is a connected semisimple Lie group G, let V M be a symplectic connection on M , and let V E be a connectipn on E . ~ This paper is devoted to the problem of finding all closed differential forms on M that can be written in local .Darboux coordinates as polynomials in finite-order derivatives of the coefficients of the connections ~7E and V M on condition that the cohomology class of the manifold M defined by such a form is preserved under the deformations of the connections. It is required that this differential form could be well defined on M . This is possible only for the case in which the dependence of this form on the connection coefficients is preserved under the transformations of Darboux coordinates. These forms are said to be invariant. A similar problem for Riemannian manifolds was solved by Abramov [1] (also see Gilkey [2]). For a more detailed statement of this problem and its solution see Atiyah, Bott, and Patody [3]. By analogy with the case of Riemannian manifolds, every invariant form on M is a polynomial in Pontryagin classes of the manifold M , characteristic classes of the bundle E , and the sympleetic form w (up to the so-called trivial forrXs whose eohomology classes are always trivial). The author thanks B. V. Fedosov for the statement of the problem and constant at tention and B. L. Feigin for valuable advice.