Moduli space of symplectic connections of Ricci type on $T^{2n}$; a formal approach

Abstract

We consider analytic curves ${\nabla }^t$ of symplectic connections of Ricci type on the torus $T^{2n}$ with ${\nabla }^0$ the standard connection. We show, by a recursion argument, that if ${\nabla }^t$ is a formal curve of such connections then there exists a formal curve of symplectomorphisms ${\psi }_t$ such that ${\psi }_t\cdot {\nabla }^t$ is a formal curve of flat invariant symplectic connections and so ${\nabla }^t$ 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 ${\nabla }^0$ are also flat. The group $G$ of symplectomorphisms of the torus $(T^{2n},\omega )$ 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.

LaTeX, 19 pages. Removed claim to handle all symplectic structures on the torus