We consider a formulation of local special geometry in terms of Darboux special coordinates PI= ( pi, qi), I = 1, ..., 2 n. A general formula for the metric is obtained which is manifestly Sp ( 2 n, R ) covariant. Unlike the rigid case the metric is not given by the Hessian of the real function S ( P ) which is the Legendre transform of the imaginary part of the holomorphic prepotential. Rather it is given by an expression that contains S, its Hessian and the conjugate momenta SI= frac(∂ S, ∂ PI). Only in the one-dimensional case ( n = 1) is the real (two-dimensional) metric proportional to the Hessian with an appropriate conformal factor. © 2006 Elsevier B.V. All rights reserved.
Ferrara, S., & Maciá, Ó. (2006). Real symplectic formulation of local special geometry. Physics Letters, Section B: Nuclear, Elementary Particle and High-Energy Physics, 637(1–2), 102–106. https://doi.org/10.1016/j.physletb.2006.04.010