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168
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 324 (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.
Split States, Entropy Enigmas, Holes and Halos
, 2007
"... We investigate degeneracies of BPS states of Dbranes on compact CalabiYau manifolds. We develop a factorization formula for BPS indices using attractor flow trees associated to multicentered black hole bound states. This enables us to study background dependence of the BPS spectrum, to compute e ..."
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Cited by 235 (22 self)
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We investigate degeneracies of BPS states of Dbranes on compact CalabiYau manifolds. We develop a factorization formula for BPS indices using attractor flow trees associated to multicentered black hole bound states. This enables us to study background dependence of the BPS spectrum, to compute explicitly exact indices of various nontrivial Dbrane systems, and to clarify the subtle relation of DonaldsonThomas invariants to BPS indices of stable D6D2D0 states, realized in supergravity as “hole halos. ” We introduce a convergent generating function for D4 indices in the large CY volume limit, and prove it can be written as a modular average of its polar part, generalizing the fareytail expansion of the elliptic genus. We show polar states are “split ” D6antiD6 bound states, and that the partition function factorizes accordingly, leading to a refined version of the OSV conjecture. This differs from the original conjecture in several aspects. In particular we obtain a nontrivial measure factor g −2 top e−K and find factorization requires a cutoff. We show that the main factor determining the cutoff and therefore the error is the existence of “swing states ” — D6 states which exist at large radius but do not form stable D6antiD6 bound states. We point out a likely breakdown of the OSV conjecture at small gtop (in the large background CY volume limit), due to the surprising phenomenon that for sufficiently large background Kähler moduli, a charge ΛΓ supporting single centered black holes of entropy ∼ Λ2S(Γ) also admits twocentered BPS black hole realizations whose entropy grows like Λ3 when Λ → ∞.
N=4 topological strings
 Nucl. Phys. B
, 1995
"... We show how to make a topological string theory starting from an N = 4 superconformal theory. The critical dimension for this theory is ĉ = 2 (c = 6). It is shown that superstrings (in both the RNS and GS formulations) and critical N = 2 strings are special cases of this topological theory. Applicat ..."
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Cited by 225 (23 self)
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We show how to make a topological string theory starting from an N = 4 superconformal theory. The critical dimension for this theory is ĉ = 2 (c = 6). It is shown that superstrings (in both the RNS and GS formulations) and critical N = 2 strings are special cases of this topological theory. Applications for this new topological theory include: 1) Proving the vanishing to all orders of all scattering amplitudes for the selfdual N = 2 string with flat background, with the exception of the threepoint function and the closedstring partition function; 2) Showing that the topological partition function of the N = 2 string on the K3 background may be interpreted as computing the superpotential in harmonic superspace generated upon compactification of type II superstrings from 10 to 6 dimensions; and 3) Providing a new prescription for calculating superstring amplitudes which appears to be free of totalderivative ambiguities. July
Supersymmetry and Attractors
 Phys. Rev. D
, 1996
"... We find a general principle which allows one to compute the area of the horizon of N=2 extremal black holes as an extremum of the central charge. One considers the ADM mass equal to the central charge as a function of electric and magnetic charges and moduli and extremizes this function in the modul ..."
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Cited by 158 (14 self)
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We find a general principle which allows one to compute the area of the horizon of N=2 extremal black holes as an extremum of the central charge. One considers the ADM mass equal to the central charge as a function of electric and magnetic charges and moduli and extremizes this function in the moduli space (a minimum corresponds to a fixed point of attraction). The extremal value of the square of the central charge provides the area of the horizon, which depends only on electric and magnetic charges. The doubling of unbroken supersymmetry at the fixed point of attraction for N=2 black holes near the horizon is derived via conformal flatness of the BertottiRobinsontype geometry. These results provide an explicit model independent expression for the macroscopic BekensteinHawking entropy of N=2 black holes which is manifestly duality invariant. The presence of hypermultiplets in the solution does not affect the area formula. Various examples of the general formula are displayed. We outline the attractor mechanism in N=4,8 supersymmetries and the relation to the N=2 case. The entropyarea formula in five dimensions, recently discussed in the literature, is also seen to be obtained by extremizing the 5d central charge.
Nearhorizon symmetries of extremal black holes
, 2008
"... Recent work has demonstrated an attractor mechanism for extremal rotating black holes subject to the assumption of a nearhorizon SO(2, 1) symmetry. We prove the existence of this symmetry for any extremal black hole with the same number of rotational symmetries as known four and five dimensional so ..."
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Cited by 114 (8 self)
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Recent work has demonstrated an attractor mechanism for extremal rotating black holes subject to the assumption of a nearhorizon SO(2, 1) symmetry. We prove the existence of this symmetry for any extremal black hole with the same number of rotational symmetries as known four and five dimensional solutions (including black rings). The result is valid for a general twoderivative theory of gravity coupled to abelian vectors and uncharged scalars, allowing for a nontrivial scalar potential. We prove that it remains valid in the presence of higherderivative corrections. We show that SO(2, 1)symmetric nearhorizon solutions can be analytically continued to give SU(2)symmetric black hole solutions. For example, the nearhorizon limit of an extremal 5D MyersPerry black hole is related by analytic continuation to a nonextremal cohomogeneity1 MyersPerry solution.
Exact and Asymptotic Degeneracies of Small Black Holes
, 2005
"... We examine the recently proposed relations between black hole entropy and the topological string in the context of type II/heterotic string dual models. We consider the degeneracies of perturbative heterotic BPS states. In several examples with N = 4 and N = 2 supersymmetry, we show that the macrosc ..."
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Cited by 80 (12 self)
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We examine the recently proposed relations between black hole entropy and the topological string in the context of type II/heterotic string dual models. We consider the degeneracies of perturbative heterotic BPS states. In several examples with N = 4 and N = 2 supersymmetry, we show that the macroscopic degeneracy of small black holes agrees to all orders with the microscopic degeneracy, but misses nonperturbative corrections which are computable in the heterotic dual. Using these examples we refine the previous proposals and comment on their domain of validity as well as on the relevance of helicity supertraces.
Deviations from the Area Law for Supersymmetric Black Holes
, 1999
"... We review modifications of the BekensteinHawking area law for black hole entropy in the presence of higherderivative interactions. In fourdimensional N = 2 compactifications of string theory or Mtheory these modifications are crucial for finding agreement between the macroscopic entropy obtained ..."
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Cited by 79 (3 self)
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We review modifications of the BekensteinHawking area law for black hole entropy in the presence of higherderivative interactions. In fourdimensional N = 2 compactifications of string theory or Mtheory these modifications are crucial for finding agreement between the macroscopic entropy obtained from supergravity and the microscopic entropy obtained by counting states in string or Mtheory. Our discussion is based on the effective Wilsonian action, which in the context of N = 2 supersymmetric theories is defined in terms of holomorphic quantities. At the end we briefly indicate how to incorporate nonholomorphic corrections. March 1999 a cardoso@phys.uu.nl b bdewit@phys.uu.nl c mohaupt@hera1.physik.unihalle.de 1 Introduction It is one of the most intriguing properties of black holes in general relativity that one can derive a set of laws, called the laws of black hole mechanics, which are formally equivalent to the laws of thermodynamics [1]. For instance, the first law of ...