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77
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 → ∞.
Wall crossing in local Calabi Yau manifolds
, 2008
"... We study the BPS states of a D6brane wrapping the conifold and bound to collections of D2 and D0 branes. We find that in addition to the complexified Kähler parameter of the rigid P 1 it is necessary to introduce an extra real parameter to describe BPS partition functions and marginal stability wa ..."
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Cited by 46 (3 self)
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We study the BPS states of a D6brane wrapping the conifold and bound to collections of D2 and D0 branes. We find that in addition to the complexified Kähler parameter of the rigid P 1 it is necessary to introduce an extra real parameter to describe BPS partition functions and marginal stability walls. The supergravity approach to BPS statecounting gives a simple derivation of results of Szendrői concerning DonaldsonThomas theory on the noncommutative conifold. This example also illustrates some interesting limitations on the supergravity approach to BPS statecounting and wallcrossing.
Four Dimensional Black Hole Microstates: From Dbranes to Spacetime Foam,” arXiv:hepth/0606118
"... We propose that every supersymmetric four dimensional black hole of finite area can be split up into microstates made up of primitive halfBPS “atoms”. The mutual nonlocality of the charges of these “atoms ” binds the state together. In support of this proposal, we display a class of smooth, horizo ..."
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Cited by 44 (6 self)
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We propose that every supersymmetric four dimensional black hole of finite area can be split up into microstates made up of primitive halfBPS “atoms”. The mutual nonlocality of the charges of these “atoms ” binds the state together. In support of this proposal, we display a class of smooth, horizonfree, four dimensional supergravity solutions carrying the charges of black holes, with multiple centers each carrying the charge of a halfBPS state. At vanishing string coupling the solutions collapse to a bound system of intersecting Dbranes. At weak coupling the system expands into the noncompact directions forming a topologically complex geometry. At strong coupling, a new dimension opens up, and the solutions form a “foam ” of spheres threaded by flux in Mtheory. We propose that this transverse growth of the underlying bound state of constitutent branes is responsible for the emergence of black hole horizons for coarsegrained observables. As such, it suggests the link between the Dbrane and “spacetime foam ” approaches to black hole entropy.
Quantum Entropy Function from AdS(2)/CFT(1) Correspondence
, 2008
"... We review and extend recent attempts to find a precise relation between extremal black hole entropy and degeneracy of microstates using AdS2/CFT1 correspondence. Our analysis leads to a specific relation between degeneracy of black hole microstates and an appropriately defined partition function of ..."
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Cited by 34 (9 self)
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We review and extend recent attempts to find a precise relation between extremal black hole entropy and degeneracy of microstates using AdS2/CFT1 correspondence. Our analysis leads to a specific relation between degeneracy of black hole microstates and an appropriately defined partition function of string theory on the near horizon geometry, – named the quantum entropy function. In the classical limit this reduces to the usual relation between statistical
Bubbling supertubes and foaming black holes
 hepth/0505166. 32 G. Mandal, “Fermions from halfBPS supergravity,” hepth/0502104
"... We construct smooth BPS threecharge geometries that resolve the zeroentropy singularity of the U(1)×U(1) invariant black ring. This singularity is resolved by a geometric transition that results in geometries without any branes sources or singularities but with nontrivial topology. These geometri ..."
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Cited by 30 (5 self)
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We construct smooth BPS threecharge geometries that resolve the zeroentropy singularity of the U(1)×U(1) invariant black ring. This singularity is resolved by a geometric transition that results in geometries without any branes sources or singularities but with nontrivial topology. These geometries are both ground states of the black ring, and nontrivial microstates of the D1D5P system. We also find the form of the geometries that result from the geometric transition of N zeroentropy black rings, and argue that, in general, such geometries give a very large number of smooth boundstate threecharge solutions, parameterized by 6N functions. The generic microstate solution is specified by a fourdimensional hyperKähler geometry of a certain signature, and contains a “foam ” of nontrivial twospheres. We conjecture that these geometries will account for a significant part of the entropy of the D1D5P black hole, and that Mathur’s conjecture might reduce to counting certain hyperKähler manifolds. May
Two centered black holes and N=4 dyon spectrum
"... The exact spectrum of dyons in a class of N=4 supersymmetric string theories is known to change discontinuously across walls of marginal stability. We show that the change in the degeneracy across the walls of marginal stability can be accounted for precisely by the entropy ..."
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Cited by 30 (16 self)
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The exact spectrum of dyons in a class of N=4 supersymmetric string theories is known to change discontinuously across walls of marginal stability. We show that the change in the degeneracy across the walls of marginal stability can be accounted for precisely by the entropy
Wall Crossing from Boltzmann Black Hole Halos
, 2011
"... A key question in the study of N = 2 supersymmetric string or field theories is to understand the decay of BPS bound states across walls of marginal stability in the space of parameters or vacua. By representing the potentially unstable bound states as multicentered black hole solutions in N = 2 s ..."
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Cited by 30 (8 self)
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A key question in the study of N = 2 supersymmetric string or field theories is to understand the decay of BPS bound states across walls of marginal stability in the space of parameters or vacua. By representing the potentially unstable bound states as multicentered black hole solutions in N = 2 supergravity, we provide two fully general and explicit formulæ for the change in the (refined) index across the wall. The first, “Higgs branch” formula relies on Reineke’s results for invariants of quivers without oriented loops, specialized to the Abelian case. The second, “Coulomb branch ” formula results from evaluating the symplectic volume of the classical phase space of multicentered solutions by localization. We provide extensive evidence that these new formulæ agree with each other and with the mathematical results of Kontsevich and Soibelman (KS) and Joyce and Song (JS). The main physical insight behind our results is that the BoseFermi statistics of individual black holes participating in the bound state can be traded for MaxwellBoltzmann statistics, provided the (integer) index Ω(γ) of the internal degrees of freedom carried by each black hole is replaced by an effective (rational) index Ω̄(γ) = mγ Ω(γ/m)/m 2. A similar map also exists for the refined index. This observation provides a physical rationale for the appearance of the rational DonaldsonThomas invariant Ω̄(γ) in the works of KS and JS. The simplicity of the wallcrossing formula for rational invariants allows us to generalize the “semiprimitive wallcrossing formula ” to arbitrary decays of the type γ →Mγ1 +Nγ2 with M = 2, 3.