Real symplectic formulation of local special geometry

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.

10 pages, corrected typos

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Cited by in corpus (11)

• The Hesse potential, the c-map and black hole solutions
• The Freudenthal gauge symmetry of the black holes of N=2,d=4 supergravity
• Special Geometry, Hessian Structures and Applications
• Non-extremal black hole solutions from the c-map
• Black holes and general Freudenthal transformations
• Non-holomorphic deformations of special geometry and their applications
• Developments in special geometry
• BPS dyons and Hesse flow
• On the sigma-model of deformed special geometry
• Special geometry for arbitrary signatures
• Special Geometry of Euclidean Supersymmetry IV: the local c-map