Black hole attractors and the entropy function in four- and five-dimensional N=2 supergravity

Extremal black holes in theories of gravity coupled to abelian gauge fields and neutral scalars, such as those arising in the low-energy description of compactifications of string theory on Calabi-Yau manifolds, exhibit the attractor phenomenon: on the event horizon the scalars settle to values determined by the charges carried by the black hole and independent of the values at infinity. It is so, because on the horizon the energy contained in vector fields acts as an effective potential (the black hole potential), driving the scalars towards its minima. For spherically symmetric black holes in theories where gauge potentials appear in the Lagrangian solely through field strengths, the attractor phenomenon can be alternatively described by a variational principle based on the so-called entropy function, defined as the Legendre transform with respect to electric fields of the Lagrangian density integrated over the horizon. Stationarity conditions for the entropy function then take the form of attractor equations relating the horizon values of the scalars to the black hole charges, while the stationary value itself yields the entropy of the black hole. In this study we examine the relationship between the entropy function and the black hole potential in four-dimensional N=2 supergravity and demonstrate that in the absence of higher-order corrections to the Lagrangian these two notions are equivalent. We also exemplify their practical application by finding a supersymmetric and a non-supersymmetric solution to the attractor equations for a conifold prepotential. Exploiting a connection between four- and five-dimensional black holes we then extend the definition of the entropy function to a class of rotating black holes in five-dimensional N=2 supergravity with cubic prepotentials, to which the original formulation did not apply because of broken spherical symmetry and explicit dependence of the Lagrangian on the gauge potentials in the Chern-Simons term. We also display two types of solutions to the respective attractor equations. The link between four- and five-dimensional black holes allows us further to derive five-dimensional first-order differential flow equations governing the profile of the fields from infinity to the horizon and construct non-supersymmetric solutions in four dimensions by dimensional reduction. Finally, four-dimensional extremal black holes in N=2 supergravity can be also viewed as certain two-dimensional string compactifications with fluxes. Motivated by this fact the recently proposed entropic principle postulates as a probability measure on the space of these string compactifications the exponentiated entropy of the corresponding black holes. Invoking the conifold example we find that the entropic principle would favor compactifications that result in infrared-free gauge theories

Black hole solutions of N=2, d=4 supergravity with a quantum correction, in the H-FGK formalism

We apply the H-FGK formalism to the study of some properties of the general class of black holes in N=2 supergravity in four dimensions that correspond to the harmonic and hyperbolic ansatze and obtain explicit extremal and non-extremal solutions for the t^3 model with and without a quantum correction. Not all solutions of the corrected model (quantum black holes), including in particular a solution with a single q_1 charge, have a regular classical limit.Comment: Latex2e file +Bibtex file, 35 pages, no figure

On anharmonic stabilisation equations for black holes

Abstract: We investigate the stabilisation equations for sufficiently general, yet regular, extremal (supersymmetric and non-supersymmetric) and non-extremal black holes in fourdimensional N = 2 supergravity using both the H-FGK approach and a generalisation of Denef's formalism. By an explicit calculation we demonstrate that the equations necessarily contain an anharmonic part, even in the static, spherically symmetric and asymptotically flat case

Entropy Maximization in the Presence of Higher-Curvature Interactions

Within the context of the entropic principle, we consider the entropy of supersymmetric black holes in N=2 supergravity theories in four dimensions with higher-curvature interactions, and we discuss its maximization at points in moduli space at which an excess of hypermultiplets becomes massless. We find that the gravitational coupling function F^(1) enhances the maximization at these points in moduli space. In principle, this enhancement may be modified by the contribution from higher F^(g)-couplings. We show that this is indeed the case for the resolved conifold by resorting to the non-perturbative expression for the topological free energy.Comment: 22 pages, 8 figures, AMS-LaTe

H-FGK formalism for black-hole solutions of N=2, d=4 and d=5 supergravity

We rewrite the Ferrara-Gibbons-Kallosh (FGK) black-hole effective action of N=2, d=4,5 supergravities coupled to vector multiplets, replacing the metric warp factor and the physical scalars with real variables that transform in the same way as the charges under duality transformations, which simplifies the equations of motion. For a given model, the form of the solution in these variables is the same for all spherically symmetric black holes, regardless of supersymmetry or extremality.Comment: 10 pages; v2: references added, some editing of the text, results unchanged; v3: references added as in the published versio

Entropy function for rotating extremal black holes in very special geometry

We use the relation between extremal black hole solutions in five- and in four-dimensional N=2 supergravity theories with cubic prepotentials to define the entropy function for extremal black holes with one angular momentum in five dimensions. We construct two types of solutions to the associated attractor equations.Comment: 15 pages, minor change

Extremal non-BPS black holes and entropy extremization

At the horizon, a static extremal black hole solution in N=2 supergravity in four dimensions is determined by a set of so-called attractor equations which, in the absence of higher-curvature interactions, can be derived as extremization conditions for the black hole potential or, equivalently, for the entropy function. We contrast both methods by explicitly solving the attractor equations for a one-modulus prepotential associated with the conifold. We find that near the conifold point, the non-supersymmetric solution has a substantially different behavior than the supersymmetric solution. We analyze the stability of the solutions and the extrema of the resulting entropy as a function of the modulus. For the non-BPS solution the region of attractivity and the maximum of the entropy do not coincide with the conifold point.Comment: 19 pages, 4 figures, AMS-LaTeX, reference adde

Brane solutions and integrability: a status report

We review the status of the integrability and solvability of the geodesics equations of motion on symmetric coset spaces that appear as sigma models of supergravity theories when reduced over respectively the timelike and spacelike direction. Such geodesic curves describe respectively timelike and spacelike brane solutions. We emphasize the applications to black holes.Comment: 4 pages, Proceedings of ERE 2010, Granada, 6-10 september 201

Brane solutions and integrability: a status report

We review the status of the integrability and solvability of the geodesics equations of motion on symmetric coset spaces that appear as sigma models of supergravity theories when reduced over respectively the timelike and spacelike direction. Such geodesic curves describe respectively timelike and spacelike brane solutions. We emphasize the applications to black holes.Comment: 4 pages, Proceedings of ERE 2010, Granada, 6-10 september 201

First-order flow equations for extremal black holes in very special geometry

We construct interpolating solutions describing single-center static extremal non-supersymmetric black holes in four-dimensional N=2 supergravity theories with cubic prepotentials. To this end, we derive and solve first-order flow equations for rotating electrically charged extremal black holes in a Taub-NUT geometry in five dimensions. We then use the connection between five- and four-dimensional extremal black holes to obtain four-dimensional flow equations and we give the corresponding solutions.Comment: 21 pages. v2: Summary section adde