In this brief note we show that the horizon entropy of the largest known class of non-supersymmetric extremal black rings, with up to six parameters, is exactly reproduced for all values of the ring radius using the same conformal field theory of the four-charge four-dimensional black hole. A particularly simple case is a dipole black ring without any conserved charges. The mass gets renormalized, but the first corrections it receives can be easily understood as an interaction potential energy. Finally, we stress that even if the entropy is correctly reproduced, this only implies that one sector of chiral excitations has been identified, but an understanding of excitations in the other sector is still required in order to capture the black ring dynamics.
Using the superpotential approach we generalize Denef's method of deriving and solving first-order equations describing multicenter extremal black holes in four-dimensional N = 2 supergravity to allow non-supersymmetric solutions. We illustrate the general results with an explicit example of the stu model.
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.
String theory is used to compute the microscopic entropy for several examples of black holes in compactifications with $N=2$ supersymmetry. Agreement with the Bekenstein-Hawking entropy and the moduli-independent $N=2$ area formula is found in all cases.
Using the entropy function formalism we compute the entropy of extremal supersymmetric and non-supersymmetric black holes in N=2 supergravity theories in four dimensions with higher derivative corrections. For supersymmetric black holes our results agree with all previous analysis. However in some examples where the four dimensional theory is expected to arise from the dimensional reduction of a five dimensional theory, there is an apparent disagreement between our results for non-supersymmetric black holes and those obtained by using the five dimensional description. This indicates that for t…
We study the dynamics of supertubes in smooth bubbling geometries with three charges and three dipole charges that can describe black holes, black rings and their microstates. We find the supertube Hamiltonian in these backgrounds and show that there exist metastable supertube configurations, that can decay into supersymmetric and non-supersymmetric ones via brane-flux annihilation. We also find stable non-supersymmetric configurations. Both the metastable and the stable non-supersymmetric configuration are expected to describe microstate geometries for non-extremal black holes, and we discuss…
A seven parameter family of five-dimensional black hole solutions depending on mass, two angular momenta, three charges and the asymptotic value of a scalar field is constructed. The entropy is computed as a function of these parameters both from the Bekenstein-Hawking formula and from the degeneracies of the corresponding D-brane states in string theory. The expressions agree at and to leading order away from extremality.
We consider N=2 supergravity in four dimensions with small R^2 curvature corrections. We construct large charge extremal supersymmetric and non-supersymmetric black hole solutions in all space, and analyze their thermodynamic properties.
We consider N=2 supergravity in four dimensions with small R^2 curvature corrections. We construct large charge non-extremal black hole solutions in all space, with either a supersymmetric or a non-supersymmetric extremal limit, and analyze their thermodynamic properties. This generalizes some of the extremal solutions presented in [arXiv:0902.0831]. The indexed entropy of the non-extremal extension of the supersymmetric black hole, has the form of the extremal entropy, with the charges replaced by a function of the charges, the moduli at infinity and the non-extremality parameter. This is the…
We study extremal black hole solutions of the S3 model (obtained by setting S=T=U in the STU model) using group theoretical methods. Upon dimensional reduction over time, the S3 model exhibits the pseudo-Riemannian coset structure G/K with G=G2(2) and K=SO(2,2). We study nilpotent K-orbits of G2(2) corresponding to non-rotating single-center extremal solutions. We find six such distinct K-orbits. Three of these orbits are supersymmetric, one is non-supersymmetric, and two are unphysical. We write general solutions and discuss examples in all four physical orbits. We show that all solutions in …
The large N limit of extremal non-supersymmetric Type-I five-dimensional string black holes is studied from the point of view of D-branes. We find that the agreement between the D-brane and the black-hole picture is due to an asymptotic restoration of supersymmetry in the large $N$ limit in which both pictures are compared. In that limit Type-I string perturbation theory is effectively embedded into a Type-IIB perturbation theory with unbroken supersymmetric charges whose presence guarantees the non-renormalization of mass and entropy as the effective couplings are increased. In this vein, we …
We show that the spinning magnetic one-brane in minimal five-dimensional supergravity admits a decoupling limit that interpolates smoothly between a self-dual null orbifold of AdS_3 \times S^2 and the near-horizon limit of the extremal Kerr black hole times a circle. We use this interpolating solution to understand the field theory dual to spinning M5 branes as a deformation of the Discrete Light Cone Quantized (DLCQ) Maldacena-Stominger-Witten (MSW) CFT. In particular, the conformal weights of the operators dual to the deformation around AdS_3 \times S^2 are calculated. We present pieces of e…
We study the entropy of non-supersymmetric extremal black holes which exhibit attractor mechanism by making use of the entropy function. This method, being simple, can be used to calculate corrections to the entropy due to higher order corrections to the action. In particular we apply this method for five dimensional non-supersymmetric extremal black hole which carries two magnetic charges and find the R^2 corrections to the entropy. Using the behavior of the action evaluated for the extremal black hole near the horizon, we also present a simple expression for C-function corrected by higher or…
We investigate extremal Reissner-Nordstrom-AdS black holes in four-dimensional N=2 abelian gauged supergravity. We find a new attractor equation which is not reduced to the one in the asymptotically flat spacetime. Focusing on so-called the T^3-model with a single neutral vector multiplet, we obtain non-supersymmetric extremal Reissner-Nordstrom-AdS black hole solutions with regular event horizon in the D0-D4 and the D2-D6 charge configurations. The negative cosmological constant emerges even without the Fayet-Iliopoulos parameters. Furthermore, we also argue the formal description of the non-…
We derive a general form of first-order flow equations for extremal and non-extremal, static, spherically symmetric black holes in theories with massless scalars and vectors coupled to gravity. By rewriting the action as a sum of squares a la Bogomol'nyi, we identify the function governing the first-order gradient flow, the `generalised superpotential', which reduces to the `fake superpotential' for non-supersymmetric extremal black holes and to the central charge for supersymmetric black holes. For theories whose scalar manifold is a symmetric space after a timelike dimensional reduction, we …
In this article we study a family of four-dimensional, $\mathcal{N}=2$ supergravity theories that interpolates between all the single dilaton truncations of the $\mathrm{SO}(8)$ gauged $\mathcal{N}=8$ supergravity. In this infinitely many theories characterized by two real numbers -- the interpolation parameter and the dyonic "angle" of the gauging -- we construct non-extremal electrically or magnetically charged black hole solutions and their supersymmetric limits. All the supersymmetric black holes have non-singular horizons with spherical, hyperbolic or planar topology. Some of these supers…
This is a short review summarizing the current status of the comparison between microscopic and macroscopic entropy of extremal BPS black holes in string theory.
We study non-supersymmetric extremal black hole excitations of 4d N=2 supersymmetric string vacua arising from compactification on Calabi-Yau threefolds. The values of the (vector multiplet) moduli at the black hole horizon are governed by the attractor mechanism. This raises natural questions, such as "what is the distribution of attractor points on moduli space?" and "how many attractor black holes are there with horizon area up to a certain size?" We employ tools developed by Denef and Douglas to answer these questions.
We study extremal cohomogeneity one five-dimensional asymptotically flat black holes of minimal supergravity in terms of the geodesics generated by nilpotent elements of the Lie algebra g2(2) on the coset manifold G2(2)/SO(2,2). There are two branches of regular extremal black holes with these properties: (i) the supersymmetric BMPV branch, and (ii) the non-supersymmetric extremal branch. We show that both of these branches are reproduced by nilpotent SO(2,2)-orbits. Furthermore, we show that the partial ordering of nilpotent orbits of G2(2) is in one-to-one correspondence with the phase diagr…
We study the low-energy effective theory in N=2 super Yang-Mills theories by microscopic and exact approaches. We calculate the one-instanton correction to the prepotential for any simple Lie group from the microscopic approach. We also study the Picard-Fuchs equations and their solutions in the semi- classical regime for classical gauge groups with rank r \leq 3. We find that for gauge groups G=A_r, B_r, C_r (r \leq 3) the microscopic results agree with those from the exact solutions.
We study general black-hole solutions of the low-energy string effective action in arbitrary dimensions using a general metric that can describe them all in a unified way both in the extreme and non-extreme cases. We calculate the mass, temperature and entropy and study which relations amongst the charges and the mass lead to extremality. We find that the temperature always vanishes in the extreme limit and we find that, for a set of n charges (no further reducible by duality) there are 2^{(n-1)} combinations of the charges that imply extremality. Not all of these combinations can be central c…
We present a method which allows to deform extremal black hole solutions into non-extremal solutions, for a large class of supersymmetric and non-supersymmetric Einstein-Vector-Scalar type theories. The deformation is shown to be largely independent of the details of the matter sector. While the line element is dressed with an additional harmonic function, the attractor equations for the scalars remain unmodified in suitable coordinates, and the values of the scalar fields on the outer and inner horizon are obtained from their fixed point values by making specific substitutions for the charges…
A five-dimensional dyonic black hole in Type-I theory is considered that is extremal but non-supersymmetric. It is shown that the Bekenstein-Hawking entropy of this black hole counts precisely the microstates of a D-brane configuration with the same charges and mass, even though there is no apparent supersymmetric nonrenormalization theorem for the mass. A similar result is known for the entropy at the stretched horizon of electrically charged, extremal, but non-supersymmetric black holes in heterotic string theory. It is argued that classical nonrenormalization of the mass may partially expla…
Families of exact $(0,2)$ supersymmetric conformal field theories of magnetically and electrically charged extremal 4D black hole solutions of heterotic string theory are presented. They are constructed using a $(0,1)$ supersymmetric $SL(2,R)\times SU(2)$ WZW model where anomalously embedded $U(1)\times U(1)$ subgroups are gauged. Crucial cancelations of the $U(1)$ anomalies coming from the supersymmetric fermions, the current algebra fermions and the gauging ensure that there is a consistency of these models at the quantum level. Various 2D models, which may be considered as building blocks f…
We present a c-function for spherically symmetric, static and asymptotically flat solutions in theories of four-dimensional gravity coupled to gauge fields and moduli. The c-function is valid for both extremal and non-extremal black holes. It monotonically decreases from infinity and in the static region acquires its minimum value at the horizon, where it equals the entropy of the black hole. Higher dimensional cases, involving $p$-form gauge fields, and other generalisations are also discussed.
The extremal Reissner-Nordstrom black hole embedded in a Melvin-like magnetic universe is studied in the framework of the Kerr/CFT correspondence. The near horizon geometry can be written as a warped and twisted product of $AdS_2 \times S^2$, also in presence of an axial external magnetic field that deforms the black hole. The central charge of the Virasoro algebra can be extracted from the asymptotic symmetries. Using the Cardy formula for the microscopic statistical entropy of the dual two-dimensional CFT, the Bekenstein-Hawking entropy, for this charged and magnetised black hole, is reprodu…
We study five-dimensional supersymmetric field theories with one-dimensional Coulomb branch. We extend a previous analysis which led to non-trivial fixed points with $E_n$ symmetry ($E_8$, $E_7$, $E_6$, $E_5=Spin(10)$, $E_4=SU(5)$, $E_3=SU(3)\times SU(2)$, $E_2=SU(2)\times U(1)$ and $E_1=SU(2)$) by finding two new theories: $\tilde E_1$ with $U(1)$ symmetry and $E_0$ with no symmetry. The latter is a non-trivial theory with no relevant operators preserving the super-Poincaré symmetry. In terms of string theory these new field theories enable us to describe compactifications of the type I' theo…
We present the main features of the physics of extremal black holes embedded in supersymmetric theories of gravitation, with a detailed analysis of the attractor mechanism for BPS and non-BPS black-hole solutions in four dimensions.
We derive the Bekenstein-Hawking entropy formula for four and five dimensional non-supersymmetric black holes (which include the Schwarzchild ones) by counting microscopic states. This is achieved by first showing that these black holes are U-dual to the three-dimensional black hole of Banados-Teitelboim-Zanelli and then counting microscopic states of the latter following Carlip's approach. Higher than five dimensional black holes are also considered. We discuss the connection of our approach to the D-brane picture.
Computations in general relativity have revealed an interesting phase diagram for the black hole - black string phase transition, with three different black objects present for a range of mass values. We can add charges to this system by `boosting' plus dualities; this makes only kinematic changes in the gravity computation but has the virtue of bringing the system into the near-extremal domain where a microscopic model can be conjectured. When the compactification radius is very large or very small then we get the microscopic models of 4+1 dimensional near-extremal holes and 3+1 dimensional n…
We show that there exist supersymmetric Lifshitz vacua in off-shell Einstein-Weyl supergravity, in addition to the BPS AdS_4 vacuum. The Lifshitz exponents are determined by the product of the cosmological constant and the coupling of the Weyl-squared term. We then obtain the equations of the supersymmetric solutions that are asymptotic to the AdS or Lifshitz vacua. We obtain many examples of exact solutions as well as numerical ones. We find examples of extremal AdS black branes whose near-horizon geometry is AdS_2\times T^2. We also find an extremal Lifshitz black hole with z=-2, whose horiz…
We construct non-extremal as well as extremal black string solutions in minimal five-dimensional supergravity coupled to vector multiplets using dimensional reduction to three Euclidean dimensions. Our method does not assume that the scalar manifold is a symmetric space, and applies as well to a class of non-supersymmetric theories governed by a generalization of special real geometry. We find that five-dimensional black string solutions correspond to geodesics in a specific totally geodesic para-Kähler submanifold of the scalar manifold of the dimensionally reduced theory, and identify the su…
As a test for the non-supersymmetric attractor mechanism, we consider extremal Reissner-Nordström-(anti-)de Sitter black holes. Based on the simple observation that the near-horizon geometry of a generic extremal black hole contains two-dimensional anti-de Sitter factor even in the presence of the positive cosmological constant, we apply Ashoke Sen's entropy function method to compute the entropy of these black holes. We find the results which exactly agree with the Bekenstein-Hawking entropy. We also obtain the constant higher-order correction to the entropy due to the Gauss-Bonnet term.
We construct the general solution for non-extremal charged rotating black holes in five-dimensional minimal gauged supergravity. They are characterised by four non-trivial parameters, namely the mass, the charge, and the two independent rotation parameters. The metrics in general describe regular rotating black holes, providing the parameters lie in appropriate ranges so that naked singularities and closed timelike curves (CTC's) are avoided. We calculate the conserved energy, angular momenta and charge for the solutions, and show how supersymmetric solutions arise in a BPS limit. These have n…
In this article we study correspondence between the microscopic spectrum and macroscopic properties of a class of extremal and non-extremal black branes and outline an origin of the interactions among various microstates of a given black brane configuration from the perspective of an intrinsic Riemannian geometry arising from the coarse graining entropy over a large number of microstates. We have analyzed the state-space geometry in the case of various extremal and non-extremal black branes arising from the string theories, multi-centered black brane configurations, small black holes with frac…
We present a holographic derivation of the entropy of supersymmetric asymptotically AdS$_5$ black holes. We define a BPS limit of black hole thermodynamics by first focussing on a supersymmetric family of complexified solutions and then reaching extremality. We show that in this limit the black hole entropy is the Legendre transform of the on-shell gravitational action with respect to three chemical potentials subject to a constraint. This constraint follows from supersymmetry and regularity in the Euclidean bulk geometry. Further, we calculate, using localization, the exact partition function…
Several distinct mechanisms of confinement and dynamical symmetry breaking (DSB) are identified, in a class of supersymmetric $SU(n_c)$, $USp(2n_c)$ and $SO(n_c)$ gauge theories. In some of the vacua, the magnetic monopoles carrying nontrivial flavor quantum numbers condense, causing confinement and symmetry breaking simultaneously. In more general classes of vacua, however, the effective low-energy degrees of freedom are found to be constituents of the monopoles - dual (magnetic) quarks. These magnetic quarks condense and give rise to confinement and DSB. We find two more important classes of…
We search for stable bound states of non-extremal rotating three-charge black holes in five dimensions (Cvetic-Youm black holes) and supertubes. We do this by studying the potential of supertube probes in the non-extremal black hole background and find that generically the marginally bound state of the supersymmetric limit becomes metastable and disappears with non-extremality (higher temperature). However near extremality there is a range of parameters allowing for stable bound states, which have lower energy than the supertube-black hole merger. Angular momentum is crucial for this effect. W…
We explicitly compute the entropy of an extremal dyonic black hole in heterotic string theory compactified on T^6 or K3\times T^2 by taking into account all the tree level four derivative corrections to the low energy effective action. For supersymmetric black holes the result agrees with the answer obtained earlier 1) by including only the Gauss-Bonnet corrections to the effective action 2) by including all terms related to the curvature squared terms via space-time supersymmetry transformation, and 3) by using general arguments based on the assumption of AdS_3 near horizon geometry and space…
We have learnt that the Bekenstein-Hawking entropy of supersymmetric black holes in five-dimensional minimal supergravity can be reproduced from microstate counting via the AdS/CFT and Kerr/CFT correspondence. In this note, we demostrate the coincidence of both formalisms for extremal and near-extremal black holes in five-dimensional minimal supergravity. Connection to AdS_2 gravity is also mentioned.
We investigate unattractor behavior in non-extremal black holes in Einstein-Born-Infeld-Dilaton theory of gravity in four-dimensional asymptotically flat spacetime. We obtain solutions which are non-singular near the horizon and dependent on the value of the dilaton field at the infinity, using perturbation method. It is shown that the value of the scalar field at the horizon is determined by its asymptotic value and the charges carried by the black hole. And we also find it is not true in general that the dilaton value at the horizon is a monotonically increasing function of the first coeffic…
We classify all pseudo-supersymmetric extremal near-horizon geometries in minimal five-dimensional de-Sitter supergravity. It is shown that the only such near-horizon geometry is the near-horizon geometry of the de-Sitter BMPV solution, and hence there are no regular extremal pseudo-supersymmetric asymptotically de-Sitter black rings.
Extremal black holes in general dimensions are well known to contain an AdS$_2$ factor in their near-horizon geometries. If the extremal limit is taken in conjunction with a specific vanishing horizon limit, the so-called Extremal Vanishing Horizon (EVH) limit, the AdS$_2$ factor lifts to a locally AdS$_3$ factor with a pinching angular direction. In this paper, we study the EVH limit of asymptotically AdS black holes which preserve some supersymmetry. The primary example we consider is the 1/16$^{\rm th}$ BPS asymptotically AdS$_5$ black hole, whose EVH limit has an AdS$_3$ factor in its near…
Logarithmic corrections to the entropy of extremal black holes have been successfully used to accurately match degeneracies from microscopic constructions to calculations of the gravitational path integral. In this paper, we revisit the problem of deriving such corrections for the case of extremal black holes, either non-supersymmetric or supersymmetric, and for near-extremal black holes. The zero-modes that are present at extremality are crucial, since their path integral cannot be treated quadratically and needs to be regulated. We show how the regulated result can be obtained by taking the …
We propose a generic recipe for deforming extremal black holes into non-extremal black holes and we use it to find and study the non-extremal black-hole solutions of several N=2,d=4 supergravity models (SL(2,R)/U(1), CPn and STU with four charges). In all the cases considered, the non-extremal family of solutions smoothly interpolates between all the different extremal limits, supersymmetric and not supersymmetric. This fact can be used to find explicitly extremal non-supersymmetric solutions in the cases in which the attractor mechanism does not completely fix the values of the scalars on the…
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 a…
Using a simple hypothesis about the degrees of freedom of intersecting branes we find a microscopic counting argument that reproduces the entropy of a class of BPS black holes of type IIA string theory on general Calabi Yau three folds.
We parameterize all classical probe brane configurations that preserve 4 supersymmetries in (a) the extremal D1-D5 geometry, (b) the extremal D1-D5-P geometry, (c) the smooth D1-D5 solutions proposed by Lunin and Mathur and (d) global $AdS_3 \times S_3 \times T^4/K3$. These configurations consist of D1 branes, D5 branes and bound states of D5 and D1 branes with the property that a particular Killing vector is tangent to the brane worldvolume at each point. We show that the supersymmetric sector of the D5 brane worldvolume theory may be analyzed in an effective 1+1 dimensional framework that pl…
We investigate the stabilisation equations for sufficiently general, yet regular, extremal (supersymmetric and non-supersymmetric) and non-extremal black holes in four-dimensional 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.
We study the low-energy effective theory of N=2 supersymmetric Yang-Mills theory with the exceptional gauge group $G_{2}$. We obtain the Picard-Fuchs equations for the $G_{2}$ spectral curve and compute multi-instanton contribution to the prepotential. We find that the spectral curve is consistent with the microscopic supersymmetric instanton calculus. It is also shown that $G_{2}$ hyperelliptic curve does not reproduce the microscopic result.
We discuss the microscopic states of the extremal BTZ black holes. Degeneracy of the primary states corresponding to the extremal BTZ black holes in the boundary N=(4,4) SCFT is obtained by utilizing the elliptic genus and the unitary representation theory of N=4 SCA. The degeneracy is consistent with the Bekenstein-Hawking entropy.
We study the low-energy effective theory in N=2 SU(Nc) supersymmetric QCD with Nf =< 2Nc fundamental hypermultiplets in the Coulomb branch by microscopic and exact approaches. We calculate the one-instanton correction to the modulus u=< 1/2 Tr A^2 > from microscopic instanton calculation. We also study the one-instanton corrections from the exact solutions for Nc=3 with massless hypermultiplets. They agree with each other except for Nf=2Nc-2 and 2Nc cases. These differences come from possible ambiguities in the constructions of the exact solutions or the definitions of the operators in the mic…
We consider the near-horizon geometry of supersymmetric extremal black holes in un-gauaged and gauged 5-dimensional supergravity, coupled to abelian vector multiplets. By analyzing the global properties of the Killing spinors, we prove that the near-horizon geometries undergo a supersymmetry enhancement. This follows from a set of generalized Lichnerowicz-type theorems we establish, together with an index theory argument. As a consequence, these solutions always admit a $\mathfrak{sl}(2,\mathbb{R})$ symmetry group.
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.
We discuss string theory alpha' corrections to charged near-extremal black 3-branes/black holes in type IIB supergravity. We find that supersymmetric global AdS_5 x S^5 geometry is not corrected to leading order in alpha', while charged or non-extremal black 3-branes/black hole geometries receive alpha' corrections. Following gauge theory-string theory correspondence the thermodynamics of these geometries is mapped to the thermodynamics of large-n_c N=4 supersymmetric Yang-Mills theory at finite (large) 't Hooft coupling with the U(1)_R-charge chemical potential. We use holographic renormaliza…
We consider issues of moduli stabilization and "area codes" for type II flux compactifications, and the "Inverse Problem" and "Fake Superpotentials" for extremal (non)supersymmetric black holes in type II compactifications on (orientifold of) a compact two-parameter Calabi-Yau expressed as a degree-18 hypersurface in WCP^4[1,1,1,6,9] which has multiple singular loci in its moduli space. We argue the existence of extended "area codes" [1] wherein for the same set of large NS-NS and RR fluxes, one can stabilize all the complex structure moduli and the axion-dilaton modulus (to different sets of …
We investigate the hypothesis that the higher-derivative corrections always make extremal non-supersymmetric black holes lighter than the classical bound and self-repulsive. This hypothesis was recently formulated in the context of the so-called swampland program. One of our examples involves an extremal heterotic black hole in four dimensions. We also calculate the effect of general four-derivative terms in Maxwell-Einstein theories in D dimensions. The results are consistent with the conjecture.
