We derive and analyse the full set of equations of motion for non-extreme static black holes (including examples with the spatial curvatures k = −1 and k = 0) in D=5 N =2 gauged supergravity by employing the techniques of "very special geometry". These solutions turn out to differ from those in the ungauged supergravity only in the non-extremality function, which has an additional term (proportional to the gauge coupling g), responsible for the appearance of naked singularities in the BPS-saturated limit. We derive an explicit solution for the ST U model of gauged supergravity which is incidentally also a solution of D=5 N =4 and N =8 gauged supergravity. This solution is specified by three charges, the asymptotic negative cosmological constant (minimum of the potential) and a non-extremality parameter. While its BPS-saturated limit has a naked singularity, we find a lower bound on the non-extremality parameter (or equivalently on the ADM mass) for which the non-extreme solutions are regular. When this bound is saturated the extreme (non-supersymmetric) solution has zero Hawking temperature and finite entropy. Analogous qualitative features are expected to emerge for black hole solutions in D = 4 gauged supergravity as well. a behrndt@physik.hu-berlin.de b cvetic@cvetic.hep.upenn.edu c ws00@aub.edu.lb Within this more general setting we address such static black holes, with k = ±1, 0. After briefly reviewing D=5 N = 2 gauged supergravity theory in Section 2 we derive d BPS-saturated topological black holes in gauged supergravity, also with naked singularities, were obtained in [5,6].

BPS black hole solutions of U (1) gauged five-dimensional supergravity are obtained by solving the Killing spinor equations. These extremal static black holes live in an asymptotic AdS 5 space time. Unlike black holes in asymptotic flat space time none of them possess a regular horizon. We also calculate the influence, of a particular class of these solutions, on the Wilson loops calculation.

We discuss general bosonic stationary configurations of N = 2, D = 4 supergravity coupled to vector multiplets. The requirement of unbroken supersymmetries imposes constraints on the holomorphic symplectic section of the underlying special Kähler manifold. The corresponding solutions of the field equations are completely determined by a set of harmonic functions. As examples we discuss rotating black holes, Taub-NUT and Eguchi-Hanson like instantons for the ST U model. In addition, we discuss, in the static limit, worldsheet instanton corrections to the ST U black hole solution, in the neighbourhood of a vanishing 4-cycle of the Calabi-Yau manifold. Our procedure is quite general and includes all known black hole solutions that can be embedded into N = 2 supergravity. a behrndt@qft2.physik.hu-berlin.de b

We discuss general bosonic stationary configurations of N = 2, D = 4 supergravity coupled to vector multiplets. The requirement of unbroken supersymmetries imposes constraints on the holomorphic symplectic section of the underlying special Kähler manifold. The corresponding solutions of the field equations are completely determined by a set of harmonic functions. As examples we discuss rotating black holes, Taub-NUT and Eguchi-Hanson like instantons for the ST U model. In addition, we discuss, in the static limit, worldsheet instanton corrections to the ST U black hole solution, in the neighbourhood of a vanishing 4-cycle of the Calabi-Yau manifold. Our procedure is quite general and includes all known black hole solutions that can be embedded into N = 2 supergravity. a behrndt@qft2.physik.hu-berlin.de b

Based on special geometry, we consider corrections to N = 2 extremal blackhole solutions and their entropies originating from higher-order derivative terms in N = 2 supergravity. These corrections are described by a holomorphic function, and the higher-order black-hole solutions can be expressed in terms of symplectic Sp(2n+2) vectors. We apply the formalism to N = 2 type-IIA Calabi-Yau string compactifications and compare our results to recent related results in the literature.Dedicated to the memory of Constance Caris a behrndt@qft2.physik.hu-berlin.de b cardoso@fys.ruu.nl c bdewit@fys.ruu.nl d