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269
Liouville Correlation Functions from Fourdimensional Gauge Theories
 SIMONS CENTER FOR GEOMETRY AND PHYSICS, STONY BROOK UNIVERSITY
, 2009
"... We conjecture an expression for the Liouville theory conformal blocks and correlation functions on a Riemann surface of genus g and n punctures as the Nekrasov partition function of a certain class of N = 2 SCFTs recently defined by one of the authors. We conduct extensive tests of the conjecture ..."
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Cited by 393 (22 self)
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We conjecture an expression for the Liouville theory conformal blocks and correlation functions on a Riemann surface of genus g and n punctures as the Nekrasov partition function of a certain class of N = 2 SCFTs recently defined by one of the authors. We conduct extensive tests of the conjecture at genus 0, 1.
Localization of gauge theory on a foursphere and supersymmetric Wilson loops
, 2007
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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 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.
Supersymmetric Branes in . . .
, 2004
"... The problem of finding supersymmetric brane configurations in the nearhorizon attractor geometry of a CalabiYau black hole with magneticelectric charges (pI, qI) is considered. HalfBPS configurations, which are static for some choice of global AdS2 coordinate, are found for wrapped brane configu ..."
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Cited by 123 (10 self)
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The problem of finding supersymmetric brane configurations in the nearhorizon attractor geometry of a CalabiYau black hole with magneticelectric charges (pI, qI) is considered. HalfBPS configurations, which are static for some choice of global AdS2 coordinate, are found for wrapped brane configurations with essentially any fourdimensional charges (uI, vI). HalfBPS multibrane configurations can also be found for any collection of wrapped branes provided they all have the same sign for the symplectic inner product p I vI − u I qI of their charges with the black hole charges. This contrasts with the Minkowski problem for which a mutually preserved supersymmetry requires alignment of all the charge vectors. The radial position of the branes in global AdS2 is determined by
Microscopic black hole entropy in theories with higher derivatives
"... We discuss higher derivative corrections to black hole entropy in theories that allow a near horizon AdS3 × X geometry. In arbitrary theories with diffeomorphism invariance we show how to obtain the spacetime central charge in a simple way. Black hole entropy then follows from the Euclidean partitio ..."
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Cited by 116 (15 self)
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We discuss higher derivative corrections to black hole entropy in theories that allow a near horizon AdS3 × X geometry. In arbitrary theories with diffeomorphism invariance we show how to obtain the spacetime central charge in a simple way. Black hole entropy then follows from the Euclidean partition function, and we show that this gives agreement with Wald’s formula. In string theory there are certain diffeomorphism anomalies that we exploit. We thereby reproduce some recent computations of corrected entropy formulas, and extend them to the nonextremal, nonsupersymetric context. Examples include black holes in Mtheory on K3×T 2, whose entropy reproduces that of the perturbative heterotic string with both right and left movers excited and angular momentum included. Our anomaly based approach also sheds light on why exact results have been obtained in four dimensions while ignoring R 4 type corrections.
Holographic gravitational anomalies
 JHEP
, 2006
"... In the AdS/CFT correspondence one encounters theories that are not invariant under diffeomorphisms. In the boundary theory this is a gravitational anomaly, and can arise in 4k + 2 dimensions. In the bulk, there can be gravitational ChernSimons terms which vary by a total derivative. We work out the ..."
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Cited by 102 (7 self)
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In the AdS/CFT correspondence one encounters theories that are not invariant under diffeomorphisms. In the boundary theory this is a gravitational anomaly, and can arise in 4k + 2 dimensions. In the bulk, there can be gravitational ChernSimons terms which vary by a total derivative. We work out the holographic stress tensor for such theories, and demonstrate agreement between the bulk and boundary. Anomalies lead to novel effects, such as a nonzero angular momentum for global AdS3. In string theory such ChernSimons terms are known with exact coefficients. The resulting anomalies, combined with symmetries, imply corrections to the BekensteinHawking entropy of black holes that agree exactly with the microscopic counting.
Black holes, qdeformed 2d YangMills, and nonperturbative topological strings
, 2004
"... We count the number of bound states of BPS black holes on local CalabiYau threefolds involving a Riemann surface of genus g. We show that the corresponding gauge theory on the brane reduces to a qdeformed YangMills theory on the Riemann surface. Following the recent connection between the black h ..."
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Cited by 99 (11 self)
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We count the number of bound states of BPS black holes on local CalabiYau threefolds involving a Riemann surface of genus g. We show that the corresponding gauge theory on the brane reduces to a qdeformed YangMills theory on the Riemann surface. Following the recent connection between the black hole entropy and the topological string partition function, we find that for a large black hole charge N, up to corrections of O(e−N), ZBH is given as a sum of a square of chiral blocks, each of which corresponds to a specific Dbrane amplitude. The leading chiral block, the vacuum block, corresponds to the closed topological string amplitudes. The subleading chiral blocks involve topological string amplitudes with Dbrane insertions at (2g − 2) points on the Riemann surface analogous to the Ω points in the large N 2d YangMills theory. The finite N amplitude provides a nonperturbative definition of topological strings in these backgrounds. This also leads to a novel nonperturbative formulation of c = 1 noncritical string at the selfdual radius.