The Hesse potential, the c-map and black hole solutions

• Published in: The journal of high energy physics Vol. 2012; no. 7
• Main Authors: Mohaupt, T., Vaughan, O.
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer-Verlag 01.07.2012; Springer Nature B.V
• Subjects: Black holes; Classical and Quantum Gravitation; Elementary Particles; High energy physics; Manifolds (mathematics); Physics; Physics and Astronomy; Quantum Field Theories; Quantum Field Theory; Quantum Physics; Relativity Theory; Rotation; String Theory; Supersymmetry; Symmetry; Black Holes in String Theory; Supergravity Models; Differential and Algebraic Geometry
• ISSN: 1029-8479; 1029-8479
• DOI: 10.1007/JHEP07(2012)163

A bstract We present a new formulation of the local c -map, which makes use of a symplectically covariant real formulation of special Kähler geometry. We obtain an explicit and simple expression for the resulting quaternionic, or, in the case of reduction over time, para-quaternionic Kähler metric in terms of the Hesse potential, which is similar to the expressions for the metrics obtained from the rigid r - and c -map, and from the local r -map. As an application we use the temporal version of the c -map to derive the black hole attractor equations from geometric properties of the scalar manifold, without imposing supersymmetry or spherical symmetry. We observe that for general (non-symmetric) c -map spaces static BPS solutions are related to a canonical family of totally isotropic, totally geodesic submanifolds. Static non-BPS solutions can be obtained by applying a field rotation matrix which is subject to a non-trivial compatibility condition. We show that for a class of prepotentials, which includes the very special (‘cubic’) prepotentials as a subclass, axion-free solutions always admit a non-trivial field rotation matrix.