We present new results towards the construction of the most general black hole solutions in four-dimensional Fayet-Iliopoulos gauged supergravities. In these theories black holes can be asymptotically AdS and have arbitrary mass, angular momentum, electric and magnetic charges and NUT charge. Furthermore, a wide range of horizon topologies is allowed (compact and noncompact) and the complex scalar fields have a nontrivial radial and angular profile. We construct a large class of solutions in the simplest single scalar model with prepotential F = −iX 0 X 1 and discuss their thermodynamics. Moreover, various approaches and calculational tools for facing this problem with more general prepotentials are presented.

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.

We find a class of five-dimensional Einstein-Maxwell type Lagrangians which contains the bosonic Lagrangians of vector multiplets as a subclass, and preserves some features of supersymmetry, namely the existence of multi-centered black hole solutions and of attractor equations. Solutions can be expressed in terms of harmonic functions through a set of algebraic equations. The geometry underlying these Lagrangians is characterised by the existence of a Hesse potential and generalizes the very special real geometry of vector multiplets. Our construction proceeds by first obtaining instanton solutions for a class of fourdimensional Euclidean sigma models, which includes those occuring for four-dimensional Euclidean N = 2 vector multiplets as a subclass. For solutions taking values in a completely isotropic submanifold of the target space, we show that the solution can be expressed in terms of harmonic functions if an integrability condition is met. This condition can either be solved by imposing that the solution depends on a single coordinate, or by imposing that the target space is a para-Kähler manifold which can be obtained from a real Hessian manifold by a generalized r-map. In the latter case one obtains multi-centered solutions. Moreover, if the integrability condition is met, the second order equations of motion can always be reduced to first order equations, which become gradient flow equations if the solution is further required to depend on one coordinate only. The dualization of axions into tensor fields and the lifting of four-dimensional instantons to five-dimensional solitons are used to motivate the addition of a boundary term to the action, which accounts for the instanton action. If the sigma model is coupled to gravity, and if the Hesse potential is of a suitable form which we specify, then the four-dimensional Euclidean Lagrangian can be lifted consistently to a five-dimensional Einstein-Maxwell type Lagrangian. Instanton solutions lift to extremal black hole solutions, and the instanton action equals the ADM mass.

Abstract:We consider timelike and spacelike reductions of 4D, N = 2 Minkowskian and Euclidean vector multiplets coupled to supergravity and the maps induced on the scalar geometry. In particular, we investigate (i) the (standard) spatial c-map, (ii) the temporal c-map, which corresponds to the reduction of the Minkowskian theory over time, and (iii) the Euclidean c-map, which corresponds to the reduction of the Euclidean theory over space. In the last two cases we prove that the target manifold is para-quaternionic Kähler.In cases (i) and (ii) we construct two integrable complex structures on the target manifold, one of which belongs to the quaternionic and para-quaternionic structure, respectively. In case (iii) we construct two integrable para-complex structures, one of which belongs to the para-quaternionic structure.In addition we provide a new global construction of the spatial, temporal and Euclidean c-maps, and separately consider a description of the target manifold as a fibre bundle over a projective special Kähler or para-Kähler base.