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 $P^I=(p^{\Lambda },q_{\Lambda }),I=1,...,2n$. The central role of the real $2n\times 2n$ matrix $M(\Re \mathcal{F},\Im \mathcal{F})$, where $\mathcal{F}={\partial }_{\Lambda }{\partial }_{\Sigma }F$ and $F$ is the holomorphic prepotential, is elucidated in the real formalism. The property $M\Omega M=\Omega$ with $\Omega$ 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 $U^I=\frac{1}{2}(P^I+i{Z}^I)$, 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.

15 pages; final version to appear on JHEP. Misprints corrected, references added

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• Holomorphic Anomaly in Gauge Theories and Matrix Models
• Stringy Black Holes and the Geometry of Entanglement
• Twistors and Black Holes
• Quantum Attractor Flows
• Dualities Near the Horizon
• Special Geometry, Hessian Structures and Applications
• BPS dyons and Hesse flow
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