Black holes in N = 2 supergravity theories and harmonic functions

Abstract

We present dyonic BPS static black hole solutions for general d = 4, N = 2 supergravity theories coupled to vector and hypermultiplets. These solutions are generalisations of the spherically symmetric Majumdar-Papapetrou black hole solutions of Einstein-Maxwell gravity and are completely characterised by a set of constrained harmonic functions. In terms of the underlying special geometry, these harmonic functions are identified with the imaginary part of the holomorphic sections defining the special Kähler manifold and the metric is expressed in terms of the symplectic invariant Kähler potential. The relations of the holomorphic sections to the harmonic functions constitute the “ generalised stabilisation equations” for the moduli fields. In addition to asymptotic flatness, the harmonic functions are also constrained by the requirement that the Kähler connection of the underlying Hodge-Kähler manifold has to vanish in order to obtain static solutions. The behaviour of these solutions near the horizon is also explained.