Real symplectic formulation of local special geometry

TL;DR

This paper presents a symplectic covariant formulation of local special geometry using Darboux coordinates, providing a new explicit metric expression that differs from the rigid case and generalizes previous results.

Contribution

It introduces a general formula for the metric in local special geometry that is manifestly symplectic covariant and extends the rigid case formulation.

Findings

Derived a general symplectic covariant metric formula

Showed the metric differs from the Hessian of the Legendre transform in general

Identified the special case where the metric is proportional to the Hessian

Abstract

We consider a formulation of local special geometry in terms of Darboux special coordinates $P^I=(p^i,q_i)$, $I=1,...,2n$. A general formula for the metric is obtained which is manifestly $\mathbf{Sp}(2n,\mathbb{R})$ 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 $S_I=\frac{\partial S}{\partial P^I}$. Only in the one-dimensional case ($n=1$) is the real (two-dimensional) metric proportional to the Hessian with an appropriate conformal factor.