Moduli space of symplectic connections of Ricci type on T 2 n ; a formal approach

We consider analytic curves ∇ t of symplectic connections of Ricci type on the torus T 2n with ∇ 0 the standard connection. We show, by a recursion argument, that if ∇ t is a formal curve of such connections then there exists a formal curve of symplectomorphisms ψ t such that ψ t · ∇ t is a formal curve of flat T 2n-invariant symplectic connections and so ∇ t is flat for all t. Applying this result to the Taylor series of the analytic curve, it means that analytic curves of symplectic connections of Ricci type starting at ∇ 0 are also flat. The group G of symplectomorphisms of the torus (T 2n , ω) acts on the space E of symplectic connections which are of Ricci type. As a preliminary to studying the moduli space E /G we study the moduli of formal curves of connections under the action of formal curves of symplectomorphisms. Research of the first three authors supported by an ARC of the communauté française de Belgique.