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

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Abstract

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.

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Authors and Affiliations

  1. Department of Mathematical Sciences, University of Liverpool, Peach Street, Liverpool, L69 7ZL, U.K.

    T. Mohaupt & O. Vaughan

  2. Department of Mathematics and Center for Mathematical Physics, University of Hamburg, Bundesstraße 55, D-20146, Hamburg, Germany

    O. Vaughan

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  1. T. Mohaupt
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Mohaupt, T., Vaughan, O. The Hesse potential, the c-map and black hole solutions. J. High Energ. Phys. 2012, 163 (2012). https://doi.org/10.1007/JHEP07(2012)163

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