Black holes in N=2 supergravity theories and harmonic functions

TL;DR

The paper derives general dyonic BPS static black hole solutions in four-dimensional, ungauged N=2 supergravity with vector and hypermultiplets, showing these solutions are completely characterized by constrained harmonic functions tied to the imaginary parts of covariantly holomorphic sections of the underlying special Kähler geometry. The metric is governed by the symplectic-invariant Kähler potential via $ds^2=e^K dt^2-e^{-K}d\vec{x}^2$, and the moduli satisfy generalized stabilisation equations that relate spatially varying scalars to electric and magnetic charges; staticity requires the Kähler connection to vanish, and near the horizon the geometry becomes Bertotti–Robinson with horizon data fixed by the central charge, yielding $S=\pi|Z_h|^2$ and attractor behavior. As a check, the Reissner–Nordström solution is recovered in the pure Einstein–Maxwell limit, illustrating how the harmonic-function framework encapsulates horizon physics across N=2 theories. The results illuminate how the central charge, moduli stabilisation, and Kähler structure control black hole microphysics and horizon data, and set the stage for extending to stationary solutions where the Kähler connection nontrivially contributes.

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.