We show that the second order field equations characterizing extremal solutions for spherically symmetric, stationary black holes are in fact implied by a system of first order equations given in terms of a prepotential W . This confirms and generalizes the results in [14]. Moreover we prove that the squared prepotential function shares the same properties of a c-function and that it interpolates between M 2 ADM and M 2 BR , the parameter of the near-horizon Bertotti-Robinson geometry. When the black holes are solutions of extended supergravities we are able to find an explicit expression for the prepotentials, valid at any radial distance from the horizon, which reproduces all the attractors of the four dimensional N > 2 theories. Far from the horizon, however, for N -even, our ansatz poses a constraint on one of the U-duality invariants for the non-BPS solutions with Z = 0. We discuss a possible extension of our considerations to the non extremal case.

We study a large family of metric-affine theories with a projective symmetry, including nonminimally coupled matter fields which respect this invariance. The symmetry is straightforwardly realised by imposing that the connection only enters through the symmetric part of the Ricci tensor, even in the matter sector. We leave the connection completely free (including torsion) and obtain its general solution as the Levi-Civita connection of an auxiliary metric, showing that the torsion only appears as a projective mode. This result justifies the widely used condition of setting vanishing torsion in these theories as a simple gauge choice. We apply our results to some particular cases considered in the literature like the so-called Eddington-inspired-Born-Infeld theories among others. We finally discuss the possibility of imposing a gauge fixing where the connection is metric compatible and comment on the genuine character of the non-metricity in theories where the two metrics are not conformally related.PACS numbers: I. INTRODUCTIONThe remarkable properties of Born-Infeld electromagnetism [1], originally aimed at resolving divergences of point-like charged particles, motivated the search of a similar route to resolve the singularities of General Relativity [2]. Among the different proposals, the so-called Eddington-inspired-Born-Infeld (EiBI) theory [3] has attracted a lot of attention in the last years due to its extraordinary ability to get rid of cosmological and black hole singularities and numerous works have been devoted to constrain the model using different types of observations [4]. Extensions and modifications of that model also lead to interesting results in cosmological and black hole scenarios [5][6][7][8]. The exploration of these theories showed that their natural habitat is the framework of metric-affine geometries and precisely this formulation permitted the mentioned progress (see [9] for a review). The reason for the necessity of considering these theories in the metric-affine approach is the avoidance of ghost-like instabilities that otherwise would be present in the metric formulation of theories with non-linear curvature terms in the action [10]. The metric-affine (sometimes also called Palatini) formulation is characterised by unlocking the affine structure and disentangle it from the metric structure, which amounts to assuming that the geometry is not Riemannian a priori, but of metric-affine type, where the metric and the connection are regarded as fully independent objects. The spirit of this approach is that only the resulting field equations should specify the full geometrical structure of the spacetime and, in particular, the relation between the metric and the connection with the matter fields. In this regard, it must be noted that the EiBI theory has been systematically analysed for a constrained family of connections by assuming a vanishing torsion tensor 1 . Given the * Electronic address: viafonso@df.ufcg.edu.br † Electronic address: cbejarano@iafe.uba.ar ‡ Electronic address: jose.b...

We classify 2-center extremal black hole charge configurations through duality-invariant homogeneous polynomials, which are the generalization of the unique invariant quartic polynomial for single-center black holes based on homogeneous symmetric cubic special Kähler geometries.A crucial role is played by an horizontal SL(p, R) symmetry group, which classifies invariants for p-center black holes. For p = 2, a (spin 2) quintet of quartic invariants emerge. We provide the minimal set of independent invariants for the rank-3 N = 2, d = 4 stu model, and for its lower-rank descendants, namely the rank-2 st 2 and rank-1 t 3 models; these models respectively exhibit seven, six and five independent invariants.We also derive the polynomial relations among these and other duality invariants. In particular, the symplectic product of two charge vectors is not independent from the quartic quintet in the t 3 model, but rather it satisfies a degree-16 relation, corresponding to a quartic equation for the square of the symplectic product itself.

We propose to consider the N = 4, d = 1 supermultiplet with (4, 4, 0) component content as a "root" one. We elaborate a new reduction scheme from the "root" multiplet to supermultiplets with a smaller number of physical bosons. Starting from the most general sigma-model type action for the "root" multiplet, we explicitly demonstrate that the actions for the rest of linear and nonlinear N = 4 supermultiplets can be easily obtained by reduction.Within the proposed reduction scheme there is a natural possibility to introduce Fayet-Iliopoulos terms. In the reduced systems, such terms give rise to potential terms, and in some cases also to terms describing the interaction with a magnetic field.We demonstrate that known N = 4 superconformal actions, together with their possible interactions, appear as results of the reduction from a free action for the "root" supermultiplet. As a byproduct, we also construct an N = 4 supersymmetric action for the linear (3, 4, 1) supermultiplet, containing both an interaction with a Dirac monopole and a harmonic oscillator-type potential, generalized for arbitrary conformally flat metrics.