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231
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 multi-centered black holes as well.
Curve counting via stable pairs in the derived category
, 2009
"... For a nonsingular projective 3-fold X, we define integer invariants virtually enumerating pairs (C,D) where C ⊂ X is an embedded curve and D ⊂ C is a divisor. A virtual class is constructed on the associated moduli space by viewing a pair as an object in the derived category of X. The resulting in ..."
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Cited by 114 (21 self)
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For a nonsingular projective 3-fold X, we define integer invariants virtually enumerating pairs (C,D) where C ⊂ X is an embedded curve and D ⊂ C is a divisor. A virtual class is constructed on the associated moduli space by viewing a pair as an object in the derived category of X. The resulting invariants are conjecturally equivalent, after universal transformations, to both the Gromov-Witten and DT theories of X. For Calabi-Yau 3-folds, the latter equivalence should be viewed as a wall-crossing formula in the derived category. Several calculations of the new invariants are carried out. In the Fano case, the local contributions of nonsingular embedded curves are found. In the local toric Calabi-Yau case, a completely new form of the topological vertex is described. The virtual enumeration of pairs is closely related to the geometry underlying the BPS state counts of Gopakumar and Vafa. We
Non-commutative Donaldson-Thomas theory and the conifold
, 2008
"... Given a quiver algebra A with relations defined by a superpotential, this paper defines a set of invariants of A counting framed cyclic A-modules, analogous to rank-1 Donaldson–Thomas invariants of Calabi–Yau threefolds. For the special case when A is the non-commutative crepant resolution of the th ..."
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Cited by 63 (0 self)
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Given a quiver algebra A with relations defined by a superpotential, this paper defines a set of invariants of A counting framed cyclic A-modules, analogous to rank-1 Donaldson–Thomas invariants of Calabi–Yau threefolds. For the special case when A is the non-commutative crepant resolution of the threefold ordinary double point, it is proved using torus localization that the invariants count certain pyramid-shaped partition-like configurations, or equivalently infinite dimer configurations in the square dimer model with a fixed boundary condition. The resulting partition function admits an infinite product expansion, which factorizes into the rank-1 Donaldson–Thomas partition functions of the commutative crepant resolution of the singularity and its flop. The different partition functions are speculatively interpreted as counting stable objects in the derived category of A-modules under different stability conditions; their relationship should then be an instance of wall crossing in the space of stability conditions on this triangulated category.
Near Extremal Black Hole Entropy as Entanglement Entropy via AdS2/CFT1
, 2007
"... We point out that the entropy of (near) extremal black holes can be interpreted as the entanglement entropy of dual conformal quantum mechanics via AdS2/CFT1. As an explicit example, we study near extremal BTZ black holes and derive this claim from AdS3/CFT2. We also analytically compute the entangl ..."
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Cited by 48 (8 self)
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We point out that the entropy of (near) extremal black holes can be interpreted as the entanglement entropy of dual conformal quantum mechanics via AdS2/CFT1. As an explicit example, we study near extremal BTZ black holes and derive this claim from AdS3/CFT2. We also analytically compute the entanglement entropy in the two dimensional CFT of a free Dirac fermion compactified on a circle at finite temperature. From this result, we clarify the relation between the thermal entropy and entanglement entropy, which is essential for the entanglement interpretation of black hole entropy.
Wall crossing in local Calabi Yau manifolds
, 2008
"... We study the BPS states of a D6-brane 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 D6-brane 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 Donaldson-Thomas theory on the noncommutative conifold. This example also illustrates some interesting limitations on the supergravity approach to BPS state-counting and wall-crossing.
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
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 multi-centered 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 multi-centered 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 multi-centered 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 Bose-Fermi statistics of individual black holes participating in the bound state can be traded for Maxwell-Boltzmann 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) = m|γ 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 Donaldson-Thomas invariant Ω̄(γ) in the works of KS and JS. The simplicity of the wall-crossing formula for rational invariants allows us to generalize the “semi-primitive wall-crossing formula ” to arbitrary decays of the type γ →Mγ1 +Nγ2 with M = 2, 3.
