Black Holes and Attractors in Supergravity

We discuss some of the basic features of extremal black holes in four-dimensional extended supergravities. Firstly, all regular solutions display an attractor behavior for the scalar field evolution towards the black hole horizon. Secondly, they can be obtained by solving first order flow equations even when they are not supersymmetric, provided one identifies a suitable superpotential W which also gives the black hole entropy at the horizon and its ADM mass at spatial infinity. We focus on N=8 supergravity and we review the basic role played by U-duality of the underlying supergravity in determining the attractors, their entropies, their masses and in classifying both regular and singular extremal black holes.Comment: Contribution to the Proceedings of the Conference in Honor of Murray Gell-Mann's 80th Birthday, Singapore, 24th-26th February 2010 (talk delivered by S. Ferrara); 15 page

General matter coupled N=2, D=5 gauged supergravity

We give the full lagrangean and supersymmetry transformation rules for D=5, N=2 supergravity interacting with an arbitrary number of vector, tensor and hyper-multiplets, with gauging of the R-symmetry group SU(2)_R as well as a subgroup K of the isometries of the scalar manifold. Among the many possible applications, this theory provides the setting where a supersymmetric brane-world scenario could occur. We comment on the presence of AdS vacua and BPS solutions that would be relevant towards a supersymmetric smooth realization of the Randall-Sundrum "alternative to compactification". We also add some remarks on the connection between this most general 5D fully coupled supergravity model and type IIB theory on the T^{11} manifold.Comment: 29+1 pages, LaTeX. References added, minor corrections, final version for Nucl. Phys.

4d/5d Correspondence for the Black Hole Potential and its Critical Points

We express the d=4, N=2 black hole effective potential for cubic holomorphic F functions and generic dyonic charges in terms of d=5 real special geometry data. The 4d critical points are computed from the 5d ones, and their relation is elucidated. For symmetric spaces, we identify the BPS and non-BPS classes of attractors and the respective entropies. These are related by simple formulae, interpolating between four and five dimensions, depending on the volume modulus and on the 4d magnetic (or electric) charges, and holding true also for generic field configurations and for non-symmetric cubic geometries.Comment: 1+24 pages; v2: references added, minor improvements; v3: further minor improvements and clarification

General Matter Coupled N=2 Supergravity

The general form of N=2 supergravity coupled to an arbitrary number of vector multiplets and hypermultiplets, with a generic gauging of the scalar manifold isometries is given. This extends the results already available in the literature in that we use a coordinate independent and manifestly symplectic covariant formalism which allows to cover theories difficult to formulate within superspace or tensor calculus approach. We provide the complete lagrangian and supersymmetry variations with all fermionic terms, and the form of the scalar potential for arbitrary quaternionic manifolds and special geometry, not necessarily in special coordinates. Our results can be used to explore properties of theories admitting $N=2$ supergravity as low energy limit.Comment: LaTex, 20 pg

On the Scalar Manifold of Exceptional Supergravity

We construct two parametrizations of the non compact exceptional Lie group G=E7(-25), based on a fibration which has the maximal compact subgroup K=(E6 x U(1))/Z_3 as a fiber. It is well known that G plays an important role in the N=2 d=4 magic exceptional supergravity, where it describes the U-duality of the theory and where the symmetric space M=G/K gives the vector multiplets' scalar manifold. First, by making use of the exponential map, we compute a realization of G/K, that is based on the E6 invariant d-tensor, and hence exhibits the maximal possible manifest [(E6 x U(1))/Z_3]-covariance. This provides a basis for the corresponding supergravity theory, which is the analogue of the Calabi-Vesentini coordinates. Then we study the Iwasawa decomposition. Its main feature is that it is SO(8)-covariant and therefore it highlights the role of triality. Along the way we analyze the relevant chain of maximal embeddings which leads to SO(8). It is worth noticing that being based on the properties of a "mixed" Freudenthal-Tits magic square, the whole procedure can be generalized to a broader class of groups of type E7.Comment: Talk given at the XVII European Workshop on String Theory, held at the University of Padua, September 5-9, 201

Small Black Hole Constituents and Horizontal Symmetry

By exploiting the role of the horizontal symmetry SL(2,R), we extend the analysis and classification of two-centered extremal black hole charge configurations to the case of "small" single-centered constituents. These latter are seen to decrease the number of independent horizontal-invariant polynomials from four to one, depending on the rank of the charge orbit supporting each of the two centers. Within U-duality groups of type E7, both reducible and irreducible symmetric supergravity models in four space-time dimensions are considered, thus encompassing N = 2 and N = 8 theories.Comment: 1+24 pages, 3 Table

Supergravity Predictions on Conformal Field Theories

We give an update on recent results about the matching between CFT operators and KK states in the AdS/CFT correspondence, and add some new comments on the realization of the baryonic symmetries from the supergravity point of view.Comment: 8 pages, uses JHEP.cls, Contribution to the proceedings of the TMR Conference on Quantum Aspects of Gauge Theories, Supersymmetry and Unification, Paris, 1-7 September 199

Small N=2 Extremal Black Holes in Special Geometry

We provide an intrinsic classification of the large and small orbits for N=2, 4D extremal black holes on symmetric spaces which does not depend on the duality frame used for the charges or on the special coordinates. A coordinate independent formula for the fake superpotential W, which (at infinity) represents the black hole ADM mass, is given explicitly in terms of invariants of the N=2 special geometry.Comment: 14 page

M Theory on the Stiefel manifold and 3d Conformal Field Theories

We compute the mass and multiplet spectrum of M theory compactified on the product of AdS(4) spacetime by the Stiefel manifold V(5,2)=SO(5)/SO(3), and we use this information to deduce via the AdS/CFT map the primary operator content of the boundary N=2 conformal field theory. We make an attempt for a candidate supersymmetric gauge theory that, at strong coupling, should be related to parallel M2-branes on the singular point of the non-compact Calabi-Yau four-fold $\sum_{a=1}^5 z_a^2 = 0$, describing the cone on V(5,2).Comment: Latex, 28 page

A Simple Derivation of Supersymmetric Extremal Black Hole Attractors

We present a simple and yet rigorous derivation of the flow equations for the supersymmetric black-hole solutions of all 4-dimensional supergravities based on the recently found general form of all those solutions.Comment: 9 pages, latex2e file, no figures. Several references adde