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

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.

Dyonic black holes at arbitrary locations

“…Freudenthal duality equations, which give the components of the Freudenthal dual 19Ĩ M (I) in terms of the functions I M [64]; these relations completely characterize the model of N = 2, d = 4 supergravity, but they may be not unique [65,66]. Equivalently, theĨ M (I) can be derived from a homogeneous function of degree 2 called the Hesse potential, W (I), as [29,67,69]…”

Rotating black holes in 4d gauged supergravity

“…We also discuss an alternative but complementary approach to constructing solutions which is based on dimensional reduction and the real formulation of special geometry, as developed in [38]. Within this formalism, the problem of constructing stationary solutions of 4D, N = 2 Fayet-Iliopoulos gauged supergravity reduces to solving a particular threedimensional Euclidean non-linear sigma model (with potential).…”

“…The simplicity of the geometry (2.1), and the fact that it is particularly suited for a formalism based on timelike dimensional reduction like the one used in [38], should help constructing new nonextremal rotating black holes in matter-coupled gauged supergravity with an arbitrary number of vector multiplets and general prepotentials. Unfortunately even if the universal structure should remain the same, the equations of motion of gauged supergravity depend crucially on the given model and cannot be solved in complete generality, therefore we first restrict ourselves to considering the simplest interesting prepotentials with a single vector multiplet.…”

“…In the ungauged case, the resulting three-dimensional Lagrangian, which takes the form of a non-linear sigma model, was constructed in [38] (see eq. (29) ∂q a ∂q b log(−2H).…”

Nonextremal black holes in gauged supergravity and the real formulation of special geometry

“…In [1] a new formulation of dimensional reduction of N = 2, D = 4 ungauged supergravity (the c-map) was presented, which made extensive use of a real, symplectically covariant, formulation of special geometry. A crucial step in the procedure was to absorb a metric degree of freedom into the moduli fields in order to lift a hypersurface constraint, an idea that was first developed in [18] for D = 5.…”