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Black Hole Entropy Function, Attractors and Precision Counting of Microstates
, 2007
"... In these lecture notes we describe recent progress in our understanding of attractor mechanism and entropy of extremal black holes based on the entropy function formalism. We also describe precise computation of the microscopic degeneracy of a class of quarter BPS dyons in N = 4 supersymmetric strin ..."
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Cited by 326 (28 self)
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In these lecture notes we describe recent progress in our understanding of attractor mechanism and entropy of extremal black holes based on the entropy function formalism. We also describe precise computation of the microscopic degeneracy of a class of quarter BPS dyons in N = 4 supersymmetric string theories, and compare the statistical entropy of these dyons, expanded in inverse powers of electric and magnetic charges, with a similar expansion of the corresponding black hole entropy. This comparison is extended to include the contribution to the entropy from multicentered black holes as well.
Rotating Attractors
, 2006
"... We prove that, in a general higher derivative theory of gravity coupled to abelian gauge fields and neutral scalar fields, the entropy and the near horizon background of a rotating extremal black hole is obtained by extremizing an entropy function which depends only on the parameters labeling the ne ..."
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Cited by 64 (19 self)
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We prove that, in a general higher derivative theory of gravity coupled to abelian gauge fields and neutral scalar fields, the entropy and the near horizon background of a rotating extremal black hole is obtained by extremizing an entropy function which depends only on the parameters labeling the near horizon background and the electric and magnetic charges and angular momentum carried by the black hole. If the entropy function has a unique extremum then this extremum must be independent of the asymptotic values of the moduli scalar fields and the solution exhibits attractor behaviour. If the entropy function has flat directions then the near horizon background is not uniquely determined by the extremization equations and could depend on the asymptotic data on the moduli fields, but the value of the entropy is still independent of this asymptotic data. We illustrate these results in the context of two derivative theories of gravity in several examples. These include Kerr black hole, KerrNewman black hole, black holes in KaluzaKlein theory, and
Moduli and (un)attractor black hole thermodynamics
, 2007
"... We investigate fourdimensional spherically symmetric black hole solutions in gravity theories with massless, neutral scalars nonminimally coupled to gauge fields. In the nonextremal case, we explicitly show that, under the variation of the moduli, the scalar charges appear in the first law of bla ..."
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Cited by 59 (9 self)
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We investigate fourdimensional spherically symmetric black hole solutions in gravity theories with massless, neutral scalars nonminimally coupled to gauge fields. In the nonextremal case, we explicitly show that, under the variation of the moduli, the scalar charges appear in the first law of black hole thermodynamics. In the extremal limit, the near horizon geometry is AdS2 ×S 2 and the entropy does not depend on the values of moduli at infinity. We discuss the attractor behaviour by using Sen’s entropy function formalism as well as the effective potential approach and their relation with the results previously obtained through special geometry method. We also argue that the attractor mechanism is at the basis of the matching between the microscopic and macroscopic entropies for the extremal nonBPS KaluzaKlein black hole.
New attractor, entropy function and black hole partition function
 JHEP
"... By making use of the entropy function formalism we study the generalized attractor equations in the four dimensional N = 2 supergravity in the presence of higher order corrections. This result might be used to understand a possible ensemble one could associate to an extremal black hole. Using the ge ..."
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Cited by 43 (2 self)
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By making use of the entropy function formalism we study the generalized attractor equations in the four dimensional N = 2 supergravity in the presence of higher order corrections. This result might be used to understand a possible ensemble one could associate to an extremal black hole. Using the generality and simplicity of this formalism we establish a duality between a four gravitational theory on AdS2 × S 2 background and the extremal black hole of the theory whose near horizon geometry is fixed by the AdS2 background. In this sense the attractor mechanism plays the role of decoupling limit in the context of AdS/CFT correspondence.
4d/5d Correspondence for the Black Hole Potential and its Critical
"... 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 ..."
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Cited by 42 (23 self)
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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 nonBPS classes of attractors and the respective entropies. These always derive from simple interpolating formulæ between four and five dimensions, depending on the volume Recently there has been an increasing amount of work on extremal charged black holes in an environment of scalar background fields, as they naturally arise in modern theories of gravity: superstrings, Mtheory, and their lowenergy description through supergravity. In particular, the Attractor
R 4 corrections to D1D5p black hole entropy from entropy function formalism
"... We show that in IIB string theory and for D1D5p black holes in ten dimensions the method of entropy function works. Despite the more complicated Wald formula for the entropy of D1D5p black holes in ten dimensions, their entropy is given by entropy function at its extremum point. We use this method f ..."
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Cited by 24 (1 self)
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We show that in IIB string theory and for D1D5p black holes in ten dimensions the method of entropy function works. Despite the more complicated Wald formula for the entropy of D1D5p black holes in ten dimensions, their entropy is given by entropy function at its extremum point. We use this method for computing the entropy of the system both at the level of Black holes in string theory can be constructed out of various Dbrane configurations. This is a well known result that the entropy of these black holes can be computed from the Wald formula [1], [2]. Further more it has been shown that this entropy is in exact agreement with counting the degeneracy of the microstates of
Extremal Black Hole and Flux Vacua Attractors
, 2007
"... These lectures provide a pedagogical, introductory review of the socalled Attractor Mechanism (AM) at work in two different 4dimensional frameworks: extremal black holes in N = 2 supergravity and N = 1 flux compactifications. In the first case, AM determines the stabilization of scalars at the bla ..."
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Cited by 23 (15 self)
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These lectures provide a pedagogical, introductory review of the socalled Attractor Mechanism (AM) at work in two different 4dimensional frameworks: extremal black holes in N = 2 supergravity and N = 1 flux compactifications. In the first case, AM determines the stabilization of scalars at the black hole event horizon purely in terms of the electric and magnetic charges, whereas in the second context the AM is responsible for the stabilization of the universal axiondilaton and of the (complex structure) moduli purely in terms of the RR and NSNS fluxes. Two equivalent approaches to AM, namely the socalled “criticality conditions ” and “New Attractor ” ones, are analyzed in detail in both frameworks, whose analogies and differences are discussed. Also a stringy analysis of both frameworks (relying on Hodgedecomposition techniques) is performed, respectively considering 2 CY3×T Type IIB compactified on CY3 and its orientifolded version, associated with. Finally, recent Z2 results on the Uduality orbits and moduli spaces of nonBPS extremal black hole attractors in
Nonsupersymmetric Attractors in BornInfeld Black Holes with A Cosmological Constant
, 2007
"... We investigate the attractor mechanism for spherically symmetric extremal black holes in EinsteinBornInfelddilaton theory of gravity in fourdimensions, in the presence of a cosmological constant. We look for solutions analytic near the horizon by using perturbation method. It is shown that the v ..."
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Cited by 18 (0 self)
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We investigate the attractor mechanism for spherically symmetric extremal black holes in EinsteinBornInfelddilaton theory of gravity in fourdimensions, in the presence of a cosmological constant. We look for solutions analytic near the horizon by using perturbation method. It is shown that the values of the scalar fields at the horizon are only dependent on the charges carried by the black hole and are irrelevant in the asymptotic values. This analysis supports the validity of nonsupersymmetric attractors in the presence of higher derivative interactions in the gauge fields part and in nonasymptotically flat spacetime.
On R 2 corrections for 5D black holes
 JHEP 0708
, 2007
"... We study higher order corrections to extremal black holes/black string in five dimensions. These higher order corrections are due to supersymmetric completion of R 2 term in five dimensions. By making use of the results we extend the notion of very special geometry when higher derivative terms are a ..."
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Cited by 15 (0 self)
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We study higher order corrections to extremal black holes/black string in five dimensions. These higher order corrections are due to supersymmetric completion of R 2 term in five dimensions. By making use of the results we extend the notion of very special geometry when higher derivative terms are also taken into account. This can be used to make a connection between total bundle space of near horizon wrapped M2’s and wrapped M5’s in the presence of higher order corrections. We also show how the corrected geometry removes the singularity of a small black hole. 1