Observations on the Darboux coordinates for rigid special geometry

Abstract

We exploit some relations which exist when (rigid) special geometry is formulated in real symplectic special coordinates $P^I=(p^Λ,q_Λ), I=1,...,2n$. The central role of the real $2n\times 2n$ matrix $M(\Re \mathcal{F},\Im \mathcal{F})$, where $\mathcal{F} = \partial_Λ\partial_ΣF$ and $F$ is the holomorphic prepotential, is elucidated in the real formalism. The property $MΩM=Ω$ with $Ω$ being the invariant symplectic form is used to prove several identities in the Darboux formulation. In this setting the matrix $M$ coincides with the (negative of the) Hessian matrix $H(S)=\frac{\partial^2 S}{\partial P^I\partial P^J}$ of a certain hamiltonian real function $S(P)$, which also provides the metric of the special Kähler manifold. When $S(P)=S(U+\bar U)$ is regarded as a "Kähler potential'' of a complex manifold with coordinates $U^I=\frac12(P^I+iZ^I)$, then it provides a Kähler metric of an hyperkähler manifold which describes the hypermultiplet geometry obtained by c-map from the original n-dimensional special Kähler structure.