We address a number of puzzles relating to the proposed formulae for the degeneracies of dyons in... more We address a number of puzzles relating to the proposed formulae for the degeneracies of dyons in orbifold compactifications of the heterotic string to four dimensions with N = 4 supersymmetry. The partition function for these dyons is given in terms of Siegel modular forms associated with genus-two Riemann surfaces. We point out a subtlety in demonstrating S-duality invariance of the resulting degeneracies and give a prescription that makes the invariance manifest. We show, using M-theory lift of string webs, that the genus-two contribution captures the degeneracy only if a specific irreducibility criterion is satisfied by the charges. Otherwise, in general there can be additional contributions from higher genus Riemann surfaces. We analyze the negative discriminant states predicted by the formula. We show that even though there are no big black holes in supergravity corresponding to these states, there are multi-centered particle-like configurations with subleading entropy in agreement with the microscopic prediction and our prescription for S-duality invariance. The existence of the states is moduli dependent and we exhibit the curves of marginal stability and comment on its relation to S-duality invariance.
We study non-supersymmetric attractors obtained in Type IIA compactifications on Calabi Yau manif... more We study non-supersymmetric attractors obtained in Type IIA compactifications on Calabi Yau manifolds. Determining if an attractor is stable or unstable requires an algebraically complicated analysis in general. We show using group theoretic techniques that this analysis can be considerably simplified and can be reduced to solving a simple example like the STU model. For attractors with D0 -D4 brane charges, determining stability requires expanding the effective potential to quartic order in the massless fields. We obtain the full set of these terms. For attractors with D0 -D6 brane charges, we find that there is a moduli space of solutions and the resulting attractors are stable. Our analysis is restricted to the two derivative action.
HAL (Le Centre pour la Communication Scientifique Directe), May 12, 2010
We explain why multi-centered black hole configurations where at least one of the centers is a la... more We explain why multi-centered black hole configurations where at least one of the centers is a large black hole do not contribute to the indexed degeneracies in theories with N = 4 supersymmetry. This is a consequence of the fact that such configurations, although supersymmetric, belong to long supermultiplets. As a result, there is no entropy enigma in N = 4 theories, unlike in N = 2 theories.
A quarter-BPS dyon in N = 4 super Yang-Mills theory is generically ‘decadent’ in that it is stabl... more A quarter-BPS dyon in N = 4 super Yang-Mills theory is generically ‘decadent’ in that it is stable only in some regions of the moduli space and decays on submanifolds in the moduli space. Using this fact, and from the degeneracy of the system close to the decay, a new derivation for the degeneracy of such dyons is given. The degeneracy obtained from these very simple physical considerations is in precise agreement with the results obtained from index computations in all known cases. Similar considerations apply to dyons in N = 2 gauge theories. The relation between the N = 4 field theory dyons and those counted by the Igusa
The degeneracies of supersymmetric quarter BPS dyons in four dimensions and of spinning black hol... more The degeneracies of supersymmetric quarter BPS dyons in four dimensions and of spinning black holes in five dimensions in a CHL compactification are computed exactly using Borcherds lift. The Hodge anomaly in the construction has a physical interpretation as the contribution of a single M-theory Kaluza-Klein 6-brane in the 4d-5d lift. Using factorization, it is shown that the resulting formula has a natural interpretation as a two-loop partition function of left-moving heterotic string, consistent with the heuristic picture of dyons in the M-theory lift of string webs.
We study non-supersymmetric attractors obtained in Type IIA compactifications on Calabi Yau manif... more We study non-supersymmetric attractors obtained in Type IIA compactifications on Calabi Yau manifolds. Determining if an attractor is stable or unstable requires an algebraically complicated analysis in general. We show using group theoretic techniques that this analysis can be considerably simplified and can be reduced to solving a simple example like the STU model. For attractors with D0 -D4 brane charges, determining stability requires expanding the effective potential to quartic order in the massless fields. We obtain the full set of these terms. For attractors with D0 -D6 brane charges, we find that there is a moduli space of solutions and the resulting attractors are stable. Our analysis is restricted to the two derivative action.
We study supersymmetric and non-supersymmetric extremal black holes obtained in Type IIA string t... more We study supersymmetric and non-supersymmetric extremal black holes obtained in Type IIA string theory compactified on K3 × T 2 , with duality group O(6, 22, Z) × SL(2, Z). In the Cardy limit an internal circle combines with the AdS 2 component in the near horizon geometry to give a BTZ black hole whose entropy is given by the Cardy formula. We study black holes carrying D0 -D4 and D0 -D6 brane charges. We find, both in the supersymmetric and non-supersymmetric cases, that a generic set of charges cannot be brought to the Cardy limit using the duality symmetries. In the non-supersymmetric case, unlike the supersymmetric one, we find that when the charges are large, a small fractional change in them always allows the charges to be taken to the Cardy limit. These results could lead to a microscopic determination of the entropy for extremal non-supersymmetric black holes, including rotating cases like the extreme Kerr black hole in four dimensions.
We address a number of puzzles relating to the proposed formulae for the degeneracies of dyons in... more We address a number of puzzles relating to the proposed formulae for the degeneracies of dyons in orbifold compactifications of the heterotic string to four dimensions with N = 4 supersymmetry. The partition function for these dyons is given in terms of Siegel modular forms associated with genus-two Riemann surfaces. We point out a subtlety in demonstrating S-duality invariance of the resulting degeneracies and give a prescription that makes the invariance manifest. We show, using M-theory lift of string webs, that the genus-two contribution captures the degeneracy only if a specific irreducibility criterion is satisfied by the charges. Otherwise, in general there can be additional contributions from higher genus Riemann surfaces. We analyze the negative discriminant states predicted by the formula. We show that even though there are no big black holes in supergravity corresponding to these states, there are multi-centered particle-like configurations with subleading entropy in agreement with the microscopic prediction and our prescription for S-duality invariance. The existence of the states is moduli dependent and we exhibit the curves of marginal stability and comment on its relation to S-duality invariance.
A quarter-BPS dyon in N = 4 super Yang-Mills theory is generically 'decadent' in that it is stabl... more A quarter-BPS dyon in N = 4 super Yang-Mills theory is generically 'decadent' in that it is stable only in some regions of the moduli space and decays on submanifolds in the moduli space. Using this fact, and from the degeneracy of the system close to the decay, a new derivation for the degeneracy of such dyons is given. The degeneracy obtained from these very simple physical considerations is in precise agreement with the results obtained from index computations in all known cases. Similar considerations apply to dyons in N = 2 gauge theories. The relation between the N = 4 field theory dyons and those counted by the Igusa cusp form in toroidally compactified heterotic string is elucidated.
We generalise the computations of [1] to generate long wavelength, asymptotically locally AdS 5 s... more We generalise the computations of [1] to generate long wavelength, asymptotically locally AdS 5 solutions to the Einstein-dilaton system with a slowly varying boundary dilaton field and a weakly curved boundary metric. Upon demanding regularity , our solutions are dual, under the AdS/CFT correspondence, to arbitrary fluid flows in the boundary theory formulated on a weakly curved manifold with a prescribed slowly varying coupling constant. These solutions turn out to be parametrised by four-velocity and temperature fields that are constrained to obey the boundary covariant Navier Stokes equations with a dilaton dependent forcing term. We explicitly evaluate the stress tensor and Lagrangian as a function of the velocity, temperature, coupling constant and curvature fields, to second order in the derivative expansion and demonstrate the Weyl covariance of these expressions. We also construct the event horizon of the dual solutions to second order in the derivative expansion, and use the area form on this event horizon to construct an entropy current for the dual fluid. As a check of our constructions we expand the exactly known solutions for rotating black holes in global AdS 5 in a boundary derivative expansion and find perfect agreement with all our results upto second order. We also find other simple solutions of the forced fluid mechanics equations and discuss their bulk interpretation. Our results may aid in determining a bulk dual to forced flows exhibiting steady state turbulence. Solutions of Forced Fluid Dynamics 33 7.1 Metric as the Forcing Term 33 7.1.1 Hydrostatic Solution in an arbitrary spacetime 33 7.1.2 Small deviations from hydrostatic equilibrium 34 7.1.3 Flows forced by a time dependent spatial metric 35 7.2 Time Dependent Dilaton and Highly Curved Spacetimes 35 7.2.1 Highly Curved Spacetimes 36 7.3 Spatially varying Dilaton 40 8. Discussion 41 A. Force on the Boundary fluid due to a Varying Dilaton 42 B. AdS Kerr 43 C. Notation 46
We explain why multi-centered black hole configurations where at least one of the centers is a la... more We explain why multi-centered black hole configurations where at least one of the centers is a large black hole do not contribute to the indexed degeneracies in theories with N = 4 supersymmetry. This is a consequence of the fact that such configurations, although supersymmetric, belong to long supermultiplets. As a result, there is no entropy enigma in N = 4 theories, unlike in N = 2 theories.
We consider the N = 4 SYM theory in flat 3 + 1 dimensional spacetime with a time dependent coupli... more We consider the N = 4 SYM theory in flat 3 + 1 dimensional spacetime with a time dependent coupling constant which vanishes at t = 0, like g 2 Y M = t p . We find that the response of the gauge theory is singular. The energy diverges at t = 0, for a generic state in the gauge theory. In addition if p > 1 the phase of the wave function has a wildly oscillating behavior, which does not allow it to be continued past t = 0. The gravity dual in some cases is known to be a time dependent cosmology which exhibits a space-like singularity at t = 0. Our results imply that the singularity is a genuine sickness and does not admit a meaningful continuation. When the coupling remains non-zero and becomes small at t = 0, the curvature in the bulk becomes of order the string scale. The gauge theory now admits a time evolution beyond this point. In this case, a finite amount of energy is produced which possibly thermalizes and leads to a black hole in the bulk.
Abstract: The product of the areas of the event horizon and the Cauchy horizon of a non-extremal ... more Abstract: The product of the areas of the event horizon and the Cauchy horizon of a non-extremal black hole equals the square of the area of the horizon of the black hole obtained from taking the smooth extremal limit. We establish this result for a large class of black holes using the second order equations of motion, black hole thermodynamics, and the attractor mechanism for extremal black holes. This happens even though the area of each horizon generically depends on the moduli, which are asymptotic values of scalar fields. The conformal field theory dual to the BTZ black hole facilitates a microscopic interpretation of the result. In addition, we demonstrate that certain quantities which vanish in the extremal case are zero when integrated over the region between the two horizons. We corroborate these conclusions through an analysis of known solutions. ar X iv
In this note, we model the Bekenstein-Hawking entropy of a four dimensional extremal black hole i... more In this note, we model the Bekenstein-Hawking entropy of a four dimensional extremal black hole in terms of classifying particles moving in its near horizon AdS 2 geometry. We use the framework of SLE curves in AdS 2 to classify these particle trajectories in terms of their boundary conditions. These turn out to be related to singular vectors in two-dimensional conformal quantum gravity theory in AdS 2 and the dynamics of these particles are governed by the Hamiltonians of the integrable Calogero-like models, for these boundary conditions. We use this classification to count the leading order Bekenstein-Hawking entropy of the black hole and arrive at a first principle microscopic computation of black hole degeneracy.
We analyze zero mode counting problems for Dirac operators that find their origin in string theor... more We analyze zero mode counting problems for Dirac operators that find their origin in string theory backgrounds. A first class of quantum mechanical models for which we compute the number of ground states arises from a string winding an isometric direction in a geometry, taking into account its energy due to tension. Alternatively, the models arise from deforming marginal bound states of a string winding a circle, and moving in an orthogonal geometry. After deformation, the number of bound states is again counted by the zero modes of a Dirac operator. We count these bound states in even dimensional asymptotically linear dilaton backgrounds as well as in Euclidean Taub-NUT. We show multiple pole behavior in the fugacities keeping track of a U(1) charge. We also discuss a second class of counting problems that arises when these backgrounds are deformed via the application of a heterotic duality transformation. We discuss applications of our results to Appell-Lerch sums and the counting...
We construct rotating extremal black hole and attractor solutions in gravity theories by solving ... more We construct rotating extremal black hole and attractor solutions in gravity theories by solving a Riemann-Hilbert problem associated with the Breitenlohner-Maison linear system. By employing a vectorial Riemann-Hilbert factorization method we explicitly factorize the corresponding monodromy matrices, which have second order poles in the spectral parameter. In the underrotating case we identify elements of the Geroch group which implement Harrison-type transformations which map the attractor geometries to interpolating rotating black hole solutions. The factorization method we use yields an explicit solution to the linear system, from which we do not only obtain the spacetime solution, but also an explicit expression for the master potential encoding the potentials of the infinitely many conserved currents which make this sector of gravity integrable.
The product of the areas of the event horizon and the Cauchy horizon of a non-extremal black hole... more The product of the areas of the event horizon and the Cauchy horizon of a non-extremal black hole equals the square of the area of the horizon of the black hole obtained from taking the smooth extremal limit. We establish this result for a large class of black holes using the second order equations of motion, black hole thermodynamics, and the attractor mechanism for extremal black holes. This happens even though the area of each horizon generically depends on the moduli, which are asymptotic values of scalar fields. The conformal field theory dual to the BTZ black hole facilitates a microscopic interpretation of the result. In addition, we demonstrate that certain quantities which vanish in the extremal case are zero when integrated over the region between the two horizons. We corroborate these conclusions through an analysis of known solutions.
CONTENTS ifcation of Type-II string on K 3 × T 2. This leads to a four-dimensional theory with N ... more CONTENTS ifcation of Type-II string on K 3 × T 2. This leads to a four-dimensional theory with N = 4 supersymmetry and 22 vector multiplets. Our objective will be to understand the entropy of half-BPS and quarter-BPS black holes in this theory both from the thermodynamic and statistical view points. A lot is known about generalization of these results to other compactifications. For a review of these generalization and of some of the material covered here see [1, 2]. There has also been more progress both in defining the quantum entropy using AdS/CF T correspondence and in computing it using localization. For a review see [3]. We will not discuss these more recent topics here to keep the discussion simple and more accessible. The organization is as follows. We review aspects of classical and semiclassical black holes in chapters §1 and §2, and elements of string theory in chapter §3. The microscopic counting is then described in chapters §4 and §5 and the comparison with macroscopic entropy is discussed in §6. Relevant mathematical background is covered in §7. These lecture notes are aimed at beginning graduate students but assume some basic background in General Theory of Relativity, Quantum Field Theory, and String Theory. A good introductory textbook on general relativity from a modern perspective see [4]. For a more detailed treatment see [5] which has become a standard reference among relativists, and [6] which remains a classic for various aspects of general relativity. For quantum field theory in curved spacetime see [7]. For relevant aspects of string theory see [8, 9, 10, 11].
We analyze zero mode counting problems for Dirac operators that find their origin in string theor... more We analyze zero mode counting problems for Dirac operators that find their origin in string theory backgrounds. A first class of quantum mechanical models for which we compute the number of ground states arises from a string winding an isometric direction in a geometry, taking into account its energy due to tension. Alternatively, the models arise from deforming marginal bound states of a string winding a circle, and moving in an orthogonal geometry. After deformation, the number of bound states is again counted by the zero modes of a Dirac operator. We count these bound states in even dimensional asymptotically linear dilaton backgrounds as well as in Euclidean Taub-NUT. We show multiple pole behavior in the fugacities keeping track of a U(1) charge. We also discuss a second class of counting problems that arises when these backgrounds are deformed via the application of a heterotic duality transformation. We discuss applications of our results to Appell-Lerch sums and the counting of domain wall bound states.
We address a number of puzzles relating to the proposed formulae for the degeneracies of dyons in... more We address a number of puzzles relating to the proposed formulae for the degeneracies of dyons in orbifold compactifications of the heterotic string to four dimensions with N = 4 supersymmetry. The partition function for these dyons is given in terms of Siegel modular forms associated with genus-two Riemann surfaces. We point out a subtlety in demonstrating S-duality invariance of the resulting degeneracies and give a prescription that makes the invariance manifest. We show, using M-theory lift of string webs, that the genus-two contribution captures the degeneracy only if a specific irreducibility criterion is satisfied by the charges. Otherwise, in general there can be additional contributions from higher genus Riemann surfaces. We analyze the negative discriminant states predicted by the formula. We show that even though there are no big black holes in supergravity corresponding to these states, there are multi-centered particle-like configurations with subleading entropy in agreement with the microscopic prediction and our prescription for S-duality invariance. The existence of the states is moduli dependent and we exhibit the curves of marginal stability and comment on its relation to S-duality invariance.
We study non-supersymmetric attractors obtained in Type IIA compactifications on Calabi Yau manif... more We study non-supersymmetric attractors obtained in Type IIA compactifications on Calabi Yau manifolds. Determining if an attractor is stable or unstable requires an algebraically complicated analysis in general. We show using group theoretic techniques that this analysis can be considerably simplified and can be reduced to solving a simple example like the STU model. For attractors with D0 -D4 brane charges, determining stability requires expanding the effective potential to quartic order in the massless fields. We obtain the full set of these terms. For attractors with D0 -D6 brane charges, we find that there is a moduli space of solutions and the resulting attractors are stable. Our analysis is restricted to the two derivative action.
HAL (Le Centre pour la Communication Scientifique Directe), May 12, 2010
We explain why multi-centered black hole configurations where at least one of the centers is a la... more We explain why multi-centered black hole configurations where at least one of the centers is a large black hole do not contribute to the indexed degeneracies in theories with N = 4 supersymmetry. This is a consequence of the fact that such configurations, although supersymmetric, belong to long supermultiplets. As a result, there is no entropy enigma in N = 4 theories, unlike in N = 2 theories.
A quarter-BPS dyon in N = 4 super Yang-Mills theory is generically ‘decadent’ in that it is stabl... more A quarter-BPS dyon in N = 4 super Yang-Mills theory is generically ‘decadent’ in that it is stable only in some regions of the moduli space and decays on submanifolds in the moduli space. Using this fact, and from the degeneracy of the system close to the decay, a new derivation for the degeneracy of such dyons is given. The degeneracy obtained from these very simple physical considerations is in precise agreement with the results obtained from index computations in all known cases. Similar considerations apply to dyons in N = 2 gauge theories. The relation between the N = 4 field theory dyons and those counted by the Igusa
The degeneracies of supersymmetric quarter BPS dyons in four dimensions and of spinning black hol... more The degeneracies of supersymmetric quarter BPS dyons in four dimensions and of spinning black holes in five dimensions in a CHL compactification are computed exactly using Borcherds lift. The Hodge anomaly in the construction has a physical interpretation as the contribution of a single M-theory Kaluza-Klein 6-brane in the 4d-5d lift. Using factorization, it is shown that the resulting formula has a natural interpretation as a two-loop partition function of left-moving heterotic string, consistent with the heuristic picture of dyons in the M-theory lift of string webs.
We study non-supersymmetric attractors obtained in Type IIA compactifications on Calabi Yau manif... more We study non-supersymmetric attractors obtained in Type IIA compactifications on Calabi Yau manifolds. Determining if an attractor is stable or unstable requires an algebraically complicated analysis in general. We show using group theoretic techniques that this analysis can be considerably simplified and can be reduced to solving a simple example like the STU model. For attractors with D0 -D4 brane charges, determining stability requires expanding the effective potential to quartic order in the massless fields. We obtain the full set of these terms. For attractors with D0 -D6 brane charges, we find that there is a moduli space of solutions and the resulting attractors are stable. Our analysis is restricted to the two derivative action.
We study supersymmetric and non-supersymmetric extremal black holes obtained in Type IIA string t... more We study supersymmetric and non-supersymmetric extremal black holes obtained in Type IIA string theory compactified on K3 × T 2 , with duality group O(6, 22, Z) × SL(2, Z). In the Cardy limit an internal circle combines with the AdS 2 component in the near horizon geometry to give a BTZ black hole whose entropy is given by the Cardy formula. We study black holes carrying D0 -D4 and D0 -D6 brane charges. We find, both in the supersymmetric and non-supersymmetric cases, that a generic set of charges cannot be brought to the Cardy limit using the duality symmetries. In the non-supersymmetric case, unlike the supersymmetric one, we find that when the charges are large, a small fractional change in them always allows the charges to be taken to the Cardy limit. These results could lead to a microscopic determination of the entropy for extremal non-supersymmetric black holes, including rotating cases like the extreme Kerr black hole in four dimensions.
We address a number of puzzles relating to the proposed formulae for the degeneracies of dyons in... more We address a number of puzzles relating to the proposed formulae for the degeneracies of dyons in orbifold compactifications of the heterotic string to four dimensions with N = 4 supersymmetry. The partition function for these dyons is given in terms of Siegel modular forms associated with genus-two Riemann surfaces. We point out a subtlety in demonstrating S-duality invariance of the resulting degeneracies and give a prescription that makes the invariance manifest. We show, using M-theory lift of string webs, that the genus-two contribution captures the degeneracy only if a specific irreducibility criterion is satisfied by the charges. Otherwise, in general there can be additional contributions from higher genus Riemann surfaces. We analyze the negative discriminant states predicted by the formula. We show that even though there are no big black holes in supergravity corresponding to these states, there are multi-centered particle-like configurations with subleading entropy in agreement with the microscopic prediction and our prescription for S-duality invariance. The existence of the states is moduli dependent and we exhibit the curves of marginal stability and comment on its relation to S-duality invariance.
A quarter-BPS dyon in N = 4 super Yang-Mills theory is generically 'decadent' in that it is stabl... more A quarter-BPS dyon in N = 4 super Yang-Mills theory is generically 'decadent' in that it is stable only in some regions of the moduli space and decays on submanifolds in the moduli space. Using this fact, and from the degeneracy of the system close to the decay, a new derivation for the degeneracy of such dyons is given. The degeneracy obtained from these very simple physical considerations is in precise agreement with the results obtained from index computations in all known cases. Similar considerations apply to dyons in N = 2 gauge theories. The relation between the N = 4 field theory dyons and those counted by the Igusa cusp form in toroidally compactified heterotic string is elucidated.
We generalise the computations of [1] to generate long wavelength, asymptotically locally AdS 5 s... more We generalise the computations of [1] to generate long wavelength, asymptotically locally AdS 5 solutions to the Einstein-dilaton system with a slowly varying boundary dilaton field and a weakly curved boundary metric. Upon demanding regularity , our solutions are dual, under the AdS/CFT correspondence, to arbitrary fluid flows in the boundary theory formulated on a weakly curved manifold with a prescribed slowly varying coupling constant. These solutions turn out to be parametrised by four-velocity and temperature fields that are constrained to obey the boundary covariant Navier Stokes equations with a dilaton dependent forcing term. We explicitly evaluate the stress tensor and Lagrangian as a function of the velocity, temperature, coupling constant and curvature fields, to second order in the derivative expansion and demonstrate the Weyl covariance of these expressions. We also construct the event horizon of the dual solutions to second order in the derivative expansion, and use the area form on this event horizon to construct an entropy current for the dual fluid. As a check of our constructions we expand the exactly known solutions for rotating black holes in global AdS 5 in a boundary derivative expansion and find perfect agreement with all our results upto second order. We also find other simple solutions of the forced fluid mechanics equations and discuss their bulk interpretation. Our results may aid in determining a bulk dual to forced flows exhibiting steady state turbulence. Solutions of Forced Fluid Dynamics 33 7.1 Metric as the Forcing Term 33 7.1.1 Hydrostatic Solution in an arbitrary spacetime 33 7.1.2 Small deviations from hydrostatic equilibrium 34 7.1.3 Flows forced by a time dependent spatial metric 35 7.2 Time Dependent Dilaton and Highly Curved Spacetimes 35 7.2.1 Highly Curved Spacetimes 36 7.3 Spatially varying Dilaton 40 8. Discussion 41 A. Force on the Boundary fluid due to a Varying Dilaton 42 B. AdS Kerr 43 C. Notation 46
We explain why multi-centered black hole configurations where at least one of the centers is a la... more We explain why multi-centered black hole configurations where at least one of the centers is a large black hole do not contribute to the indexed degeneracies in theories with N = 4 supersymmetry. This is a consequence of the fact that such configurations, although supersymmetric, belong to long supermultiplets. As a result, there is no entropy enigma in N = 4 theories, unlike in N = 2 theories.
We consider the N = 4 SYM theory in flat 3 + 1 dimensional spacetime with a time dependent coupli... more We consider the N = 4 SYM theory in flat 3 + 1 dimensional spacetime with a time dependent coupling constant which vanishes at t = 0, like g 2 Y M = t p . We find that the response of the gauge theory is singular. The energy diverges at t = 0, for a generic state in the gauge theory. In addition if p > 1 the phase of the wave function has a wildly oscillating behavior, which does not allow it to be continued past t = 0. The gravity dual in some cases is known to be a time dependent cosmology which exhibits a space-like singularity at t = 0. Our results imply that the singularity is a genuine sickness and does not admit a meaningful continuation. When the coupling remains non-zero and becomes small at t = 0, the curvature in the bulk becomes of order the string scale. The gauge theory now admits a time evolution beyond this point. In this case, a finite amount of energy is produced which possibly thermalizes and leads to a black hole in the bulk.
Abstract: The product of the areas of the event horizon and the Cauchy horizon of a non-extremal ... more Abstract: The product of the areas of the event horizon and the Cauchy horizon of a non-extremal black hole equals the square of the area of the horizon of the black hole obtained from taking the smooth extremal limit. We establish this result for a large class of black holes using the second order equations of motion, black hole thermodynamics, and the attractor mechanism for extremal black holes. This happens even though the area of each horizon generically depends on the moduli, which are asymptotic values of scalar fields. The conformal field theory dual to the BTZ black hole facilitates a microscopic interpretation of the result. In addition, we demonstrate that certain quantities which vanish in the extremal case are zero when integrated over the region between the two horizons. We corroborate these conclusions through an analysis of known solutions. ar X iv
In this note, we model the Bekenstein-Hawking entropy of a four dimensional extremal black hole i... more In this note, we model the Bekenstein-Hawking entropy of a four dimensional extremal black hole in terms of classifying particles moving in its near horizon AdS 2 geometry. We use the framework of SLE curves in AdS 2 to classify these particle trajectories in terms of their boundary conditions. These turn out to be related to singular vectors in two-dimensional conformal quantum gravity theory in AdS 2 and the dynamics of these particles are governed by the Hamiltonians of the integrable Calogero-like models, for these boundary conditions. We use this classification to count the leading order Bekenstein-Hawking entropy of the black hole and arrive at a first principle microscopic computation of black hole degeneracy.
We analyze zero mode counting problems for Dirac operators that find their origin in string theor... more We analyze zero mode counting problems for Dirac operators that find their origin in string theory backgrounds. A first class of quantum mechanical models for which we compute the number of ground states arises from a string winding an isometric direction in a geometry, taking into account its energy due to tension. Alternatively, the models arise from deforming marginal bound states of a string winding a circle, and moving in an orthogonal geometry. After deformation, the number of bound states is again counted by the zero modes of a Dirac operator. We count these bound states in even dimensional asymptotically linear dilaton backgrounds as well as in Euclidean Taub-NUT. We show multiple pole behavior in the fugacities keeping track of a U(1) charge. We also discuss a second class of counting problems that arises when these backgrounds are deformed via the application of a heterotic duality transformation. We discuss applications of our results to Appell-Lerch sums and the counting...
We construct rotating extremal black hole and attractor solutions in gravity theories by solving ... more We construct rotating extremal black hole and attractor solutions in gravity theories by solving a Riemann-Hilbert problem associated with the Breitenlohner-Maison linear system. By employing a vectorial Riemann-Hilbert factorization method we explicitly factorize the corresponding monodromy matrices, which have second order poles in the spectral parameter. In the underrotating case we identify elements of the Geroch group which implement Harrison-type transformations which map the attractor geometries to interpolating rotating black hole solutions. The factorization method we use yields an explicit solution to the linear system, from which we do not only obtain the spacetime solution, but also an explicit expression for the master potential encoding the potentials of the infinitely many conserved currents which make this sector of gravity integrable.
The product of the areas of the event horizon and the Cauchy horizon of a non-extremal black hole... more The product of the areas of the event horizon and the Cauchy horizon of a non-extremal black hole equals the square of the area of the horizon of the black hole obtained from taking the smooth extremal limit. We establish this result for a large class of black holes using the second order equations of motion, black hole thermodynamics, and the attractor mechanism for extremal black holes. This happens even though the area of each horizon generically depends on the moduli, which are asymptotic values of scalar fields. The conformal field theory dual to the BTZ black hole facilitates a microscopic interpretation of the result. In addition, we demonstrate that certain quantities which vanish in the extremal case are zero when integrated over the region between the two horizons. We corroborate these conclusions through an analysis of known solutions.
CONTENTS ifcation of Type-II string on K 3 × T 2. This leads to a four-dimensional theory with N ... more CONTENTS ifcation of Type-II string on K 3 × T 2. This leads to a four-dimensional theory with N = 4 supersymmetry and 22 vector multiplets. Our objective will be to understand the entropy of half-BPS and quarter-BPS black holes in this theory both from the thermodynamic and statistical view points. A lot is known about generalization of these results to other compactifications. For a review of these generalization and of some of the material covered here see [1, 2]. There has also been more progress both in defining the quantum entropy using AdS/CF T correspondence and in computing it using localization. For a review see [3]. We will not discuss these more recent topics here to keep the discussion simple and more accessible. The organization is as follows. We review aspects of classical and semiclassical black holes in chapters §1 and §2, and elements of string theory in chapter §3. The microscopic counting is then described in chapters §4 and §5 and the comparison with macroscopic entropy is discussed in §6. Relevant mathematical background is covered in §7. These lecture notes are aimed at beginning graduate students but assume some basic background in General Theory of Relativity, Quantum Field Theory, and String Theory. A good introductory textbook on general relativity from a modern perspective see [4]. For a more detailed treatment see [5] which has become a standard reference among relativists, and [6] which remains a classic for various aspects of general relativity. For quantum field theory in curved spacetime see [7]. For relevant aspects of string theory see [8, 9, 10, 11].
We analyze zero mode counting problems for Dirac operators that find their origin in string theor... more We analyze zero mode counting problems for Dirac operators that find their origin in string theory backgrounds. A first class of quantum mechanical models for which we compute the number of ground states arises from a string winding an isometric direction in a geometry, taking into account its energy due to tension. Alternatively, the models arise from deforming marginal bound states of a string winding a circle, and moving in an orthogonal geometry. After deformation, the number of bound states is again counted by the zero modes of a Dirac operator. We count these bound states in even dimensional asymptotically linear dilaton backgrounds as well as in Euclidean Taub-NUT. We show multiple pole behavior in the fugacities keeping track of a U(1) charge. We also discuss a second class of counting problems that arises when these backgrounds are deformed via the application of a heterotic duality transformation. We discuss applications of our results to Appell-Lerch sums and the counting of domain wall bound states.
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