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.