Black hole entropy functions and attractor equations

The entropy and the attractor equations for static extremal black hole solutions follow from a variational principle based on an entropy function. In the general case such an entropy function can be derived from the reduced action evaluated in a near-horizon geometry. BPS black holes constitute special solutions of this variational principle, but they can also be derived directly from a different entropy function based on supersymmetry enhancement at the horizon. Both functions are consistent with electric/magnetic duality and for BPS black holes their corresponding OSV-type integrals give identical results at the semi-classical level. We clarify the relation between the two entropy functions and the corresponding attractor equations for N=2 supergravity theories with higher-derivative couplings in four space-time dimensions. We discuss how non-holomorphic corrections will modify these entropy functions.Comment: 21 pages,LaTeX,minor change

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

The world-sheet corrections to dyons in the Heterotic theory

All the linear alpha-prime corrections, however excluding the gravitational Chern-Simons correction, are studied in the toroidally compactified critical Heterotic string theory. These corrections are computed to the entropy for a BPS static spherical four dimensional dyonic black hole which represents a wrapped fundamental string carrying arbitrary winding and momentum charges along one cycle in the presence of KK-monopole and H-monopole charges associated to another cycle. It is verified that after the inclusion of the gravitational Chern-Simons corrections [hep-th/0608182], all the linear alpha-prime corrections to the entropy for the supersymmetric dyon can be reproduced by the inclusion of only the Gauss-Bonnet Lagrangian to the supergravity approximation of the induced Lagrangian.Comment: JHEP style, 17 Pages; v2: a typo corrected ; v3: The coupling of the gravitational Chern-Simons terms to the three form field strength taken into account. The conclusion correcte

How Does a Fundamental String Stretch its Horizon?

It has recently been shown that if we take into account a class of higher derivative corrections to the effective action of heterotic string theory, the entropy of the black hole solution representing elementary string states correctly reproduces the statistical entropy computed from the degeneracy of elementary string states. So far the form of the solution has been analyzed at distance scales large and small compared to the string scale. We analyze the solution that interpolates between these two limits and point out a subtlety in constructing such a solution due to the presence of higher derivative terms in the effective action. We also study the T-duality transformation rules to relate the moduli fields of the effective field theory to the physical compactification radius in the presence of higher derivative corrections and use these results to find the physical radius of compactification near the horizon of the black hole. The radius approaches a finite value even though the corresponding modulus field vanishes. Finally we discuss the non-leading contribution to the black hole entropy due to space-time quantum corrections to the effective action and the ambiguity involved in comparing this result to the statistical entropy.Comment: LaTeX file, 38 pages; v2: minor changes and added reference

Entropy Function for Heterotic Black Holes

We use the entropy function formalism to study the effect of the Gauss-Bonnet term on the entropy of spherically symmetric extremal black holes in heterotic string theory in four dimensions. Surprisingly the resulting entropy and the near horizon metric, gauge field strengths and the axion-dilaton field are identical to those obtained by Cardoso et. al. for a supersymmetric version of the theory that contains Weyl tensor squared term instead of the Gauss-Bonnet term. We also study the effect of holomorphic anomaly on the entropy using our formalism. Again the resulting attractor equations for the axion-dilaton field and the black hole entropy agree with the corresponding equations for the supersymmetric version of the theory. These results suggest that there might be a simpler description of supergravity with curvature squared terms in which we supersymmetrize the Gauss-Bonnet term instead of the Weyl tensor squared term.Comment: LaTeX file, 23 pages; v2: references added; v3: minor addition; v4: 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

Euclidean N=2 Supergravity

Euclidean special geometry has recently been investigated in the context of Euclidean supersymmetric theories with vector multiplets. In the rigid case, the scalar manifold is described by affine special para-Kahler geometry while the target geometries of Euclidean vector multiplets coupled to supergravity are given by projective special para-Kahler manifolds. In this letter, we derive the Killing spinor equations of Euclidean N=2 supergravity theories coupled to vector multiplets. These equations provide the starting point for finding general supersymmetric instanton solutions.Comment: 12 pages, latex. Minor sign corrections in section

Critical points of the Black-Hole potential for homogeneous special geometries

We extend the analysis of N=2 extremal Black-Hole attractor equations to the case of special geometries based on homogeneous coset spaces. For non-BPS critical points (with non vanishing central charge) the (Bekenstein-Hawking) entropy formula is the same as for symmetric spaces, namely four times the square of the central charge evaluated at the critical point. For non homogeneous geometries the deviation from this formula is given in terms of geometrical data of special geometry in presence of a background symplectic charge vector.Comment: 17 pages, LaTeX fil

Black Holes, Elementary Strings and Holomorphic Anomaly

In a previous paper we had proposed a specific route to relating the entropy of two charge black holes to the degeneracy of elementary string states in N=4 supersymmetric heterotic string theory in four dimensions. For toroidal compactification this proposal works correctly to all orders in a power series expansion in inverse charges provided we take into account the corrections to the black hole entropy formula due to holomorphic anomaly. In this paper we demonstrate that similar agreement holds also for other N=4 supersymmetric heterotic string compactifications.Comment: LaTeX file, 28 pages, reference added, minor changes in appendix

Exact solutions for supersymmetric stationary black hole composites

Four dimensional N=2 supergravity has regular, stationary, asymptotically flat BPS solutions with intrinsic angular momentum, describing bound states of separate extremal black holes with mutually nonlocal charges. Though the existence and some properties of these solutions were established some time ago, fully explicit analytic solutions were lacking thus far. In this note, we fill this gap. We show in general that explicit solutions can be constructed whenever an explicit formula is known in the theory at hand for the Bekenstein-Hawking entropy of a single black hole as a function of its charges, and illustrate this with some simple examples. We also give an example of moduli-dependent black hole entropy.Comment: 13 pages, 1 figur