N=2 Extremal Black Holes

TL;DR

Ferrara, Kallosh, and Strominger study extremal magnetic BPS black holes in N=2 supergravity with vector multiplets and a general prepotential F(X^Λ). They show these solutions behave as supersymmetric solitons interpolating between asymptotically maximally symmetric vacua and a horizon, deriving first-order BPS equations for the moduli and the metric function U; in a simple basis with q^0=0, they identify a class of solutions in which the spatial conformal factor is given by the vector-multiplet Kähler potential, e^{-U} ∝ e^{K}. The paper provides multiple explicit examples—Calabi–Yau, SU(1,n) white holes, and SO(2,1)×SO(n) families, plus reductions to N=4/2 theories like axion-dilaton dyons and RN black holes—highlighting the deep link between special geometry and spacetime geometry. This framework, grounded in symplectic invariance and Kähler structure, offers a unifying description of a broad class of BPS black holes and white holes in extended supergravity.

Abstract

It is shown that extremal magnetic black hole solutions of N = 2 supergravity coupled to vector multiplets $X^Λ$ with a generic holomorphic prepotential $F(X^Λ)$ can be described as supersymmetric solitons which interpolate between maximally symmetric limiting solutions at spatial infinity and the horizon. A simple exact solution is found for the special case that the ratios of the $X^Λ$ are real, and it is seen that the logarithm of the conformal factor of the spatial metric equals the Kahler potential on the vector multiplet moduli space. Several examples are discussed in detail.