Observations on the Darboux coordinates for rigid special geometry

TL;DR

This paper investigates the structure of Darboux coordinates in rigid special geometry, highlighting the role of a key symplectic matrix and its relation to the metric and K"ahler potential in associated hyperk"ahler manifolds.

Contribution

It clarifies the role of the matrix M in real symplectic coordinates and its connection to the Hessian of a Hamiltonian function, linking special geometry to hyperk"ahler metrics.

Findings

Identifies M as the Hessian of a Hamiltonian function S(P).

Shows M satisfies a symplectic invariance property MΩM=Ω.

Connects the Hamiltonian S(P) to the K"ahler potential of hyperk"ahler manifolds.

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 K\"ahler manifold. When $S(P)=S(U+\bar U)$ is regarded as a "K\"ahler potential'' of a complex manifold with coordinates $U^I=\frac12(P^I+iZ^I)$, then it provides a K\"ahler metric of an hyperk\"ahler manifold which describes the hypermultiplet geometry obtained by c-map from the original n-dimensional special K\"ahler structure.