The Journal of high energy physics/The journal of high energy physics, Mar 20, 2024
We describe solutions of asymptotically AdS 3 Einstein gravity that are sourced by the insertion ... more We describe solutions of asymptotically AdS 3 Einstein gravity that are sourced by the insertion of operators in the boundary CFT 2 , whose dimension scales with the central charge of the theory. Previously, we found that the geometry corresponding to a black hole two-point function is simply related to an infinite covering of the Euclidean BTZ black hole . However, here we find that the geometry sourced by the presence of a third black hole operator turns out to be a Euclidean wormhole with two asymptotic boundaries. We construct this new geometry as a quotient of empty AdS 3 realized by domes and doors. The doors give access to the infinite covers that are needed to describe the insertion of the operators, while the domes describe the fundamental domains of the quotient on each cover. In particular, despite the standard fact that the Fefferman-Graham expansion is single-sided, the extended bulk geometry contains a wormhole that connects two asymptotic boundaries. We observe that the two-sided wormhole can be made single-sided by cutting off the wormhole and gluing on a "Lorentzian cap". In this way, the geometry gives the holographic description of a three-point function, up to phases. By rewriting the metric in terms of a Liouville field, we compute the on-shell action and find that the result matches with the Heavy-Heavy-Heavy three-point function predicted by the modular bootstrap. Finally, we describe the geometric transition between doors and defects, that is, when one or more dual operators describe a conical defect insertion, rather than a black hole insertion.
We investigate how the complexity=anything observables proposed by can be used to investigate the... more We investigate how the complexity=anything observables proposed by can be used to investigate the interior geometry of AdS black holes. In particular, we illustrate how the flexibility of the complexity=anything approach allows us to systematically probe the geometric properties of black hole singularities. We contrast our results for the AdS Schwarzschild and AdS Reissner-Nordström geometries, i.e., for uncharged and charged black holes, respectively. In the latter case, the holographic complexity observables can only probe the interior up to the inner horizon.
In a recent paper, Duff and Rahmfeld argued that certain massive N R = 1/2 states of the four-dim... more In a recent paper, Duff and Rahmfeld argued that certain massive N R = 1/2 states of the four-dimensional heterotic string correspond to extreme black hole solutions. We provide further, dynamical, evidence for this identification by comparing the scattering of these elementary string states with that of the corresponding extreme black holes, in the limit of low velocities.
New Euclidean solutions, describing the nucleation of a baby universe, are presented for an axion... more New Euclidean solutions, describing the nucleation of a baby universe, are presented for an axion (described by an antisymmetric tensor field strength of rank D -1 in D dimensions) coupled to grav- ity including a cosmological constant. It has long been speculated that the geometry of space-time should fluctuate at distances of the Planck scale. This possibility that quantum gravity allow fluc- tuations in topology has recently been receiving serious consideration by many authors. The work of Gid- dings and Strominger is based on a definite axionic in- stanton. The four-dimensional Euclidean solution in- volves an axion (given as a three-form field strength) cou- pled to Einstein gravity, and describes the nucleation of a baby universe, which carries ofl' a definite axion charge. In the present paper, we briefly report two extensions of this solution. The first is the inclusion of @ cosmological constant. The resulting instanton may be used to de- scribe tunneling between a large de Sitter space and a Planckian Robertson-Walker universe. The second ex- tension is finding analogous solutions for space-time di- mensions other than four. This requires replacing the rank-three antisymmetric tensor field strength by one of rank D -1 in D dimensions. Such higher-rank tensor fields arise in higher-dimensional supergravity theories.
We consider the effects of abelian duality transformations on static, spherically-symmetric, asym... more We consider the effects of abelian duality transformations on static, spherically-symmetric, asymptotically flat string spacetimes in four dimensions, where the dilaton, axion, metric, and gauge fields are allowed to be nonzero. Independent of the α ′ expansion, there is a six-parameter family of such configurations, labelled by the charges characterizing the asymptotic behaviour of the various fields: i.e. their mass, dilaton charge, axion charge, electric charge, magnetic charge, and Taub-NUT parameter. We show that duality, based on time-translation invariance, maps these solutions amongst themselves, with the effect of interchanging two pairs of these six labels, namely: (1) the mass and dilaton charge, and (2) the axion charge and the Taub-NUT parameter. We consider in detail the special case of the purely Schwarzschild black hole, for which the mass of the dual configuration vanishes to leading order in α ′ . Working to next-to-leading order in α ′ for the bosonic and heterotic strings, we find that duality takes a black hole of mass M to a (singular) solution having mass ∼ -1/(α ′ M ). Finally, we argue that two solutions which are related by duality based on a noncompact symmetry are not always physically equivalent.
We examine brane induced gravity on codimension-1 branes, a.k.a DGP gravity, as a theory of five-... more We examine brane induced gravity on codimension-1 branes, a.k.a DGP gravity, as a theory of five-dimensional gravity containing a certain class of fourdimensional branes. From this perspective, the model suffers from a number of pathologies which went unnoticed before. By generalizing the 5D geometry from Minkowski to Schwarzschild, we find that when the bulk mass is large enough, the brane hits a pressure singularity at finite radius. Further, on the self-accelerating branch, the five-dimensional energy is unbounded from below, implying that the self-accelerating backgrounds are unstable. Even in an empty Minkowski bulk, standard Euclidean techniques suggest that the spontaneous nucleation of self-accelerating branes is unsuppressed. If so, quantum effects will strongly modify any classical intuition about the theory. We also note that unless considered as Z 2 -orbifold boundaries, self-accelerating branes correspond to 'wormhole' configurations, which introduces the usual problematic issues associated with wormholes. Altogether these pathologies present a serious challenge that any proposed UV completion of the DGP model must overcome.
We discuss holographic models of extremal and non-extremal black holes in contact with a bath in ... more We discuss holographic models of extremal and non-extremal black holes in contact with a bath in d dimensions, based on a brane world model introduced in . The main benefit of our setup is that it allows for a high degree of analytic control as compared to previous work in higher dimensions. We show that the appearance of quantum extremal islands in those models is a consequence of the well-understood phase transition of RT surfaces, and does not make any direct reference to ensemble averaging. For non-extremal black holes the appearance of quantum extremal islands has the right behaviour to avoid the information paradox in any dimension. We further show that for these models the calculation of the full Page curve is possible in any dimension. The calculation reduces to numerically solving two ODEs. In the case of extremal black holes in higher dimensions, we find no quantum extremal islands for a wide range of parameters. In two dimensions, our results agree with [2] at leading order; however a finite UV cutoff introduced by the brane results in subleading corrections. For example, these corrections result in the quantum extremal surfaces moving further outward from the horizon, and shifting the Page transition to a slightly earlier time.
We study the doubly holographic model of in the situation where a black hole in two-dimensional J... more We study the doubly holographic model of in the situation where a black hole in two-dimensional JT gravity theory is coupled to an auxiliary bath system at arbitrary finite temperature. Depending on the initial temperature of the black hole relative to the bath temperature, the black hole can lose mass by emitting Hawking radiation, stay in equilibrium with the bath or gain mass by absorbing thermal radiation from the bath. In all of these scenarios, a unitary Page curve is obtained by applying the usual prescription for holographic entanglement entropy and identifying the quantum extremal surface for the generalized entropy, using both analytical and numeric calculations. As the application of the entanglement wedge reconstruction, we further investigate the reconstruction of the black hole interior from a subsystem containing the Hawking radiation. We examine the roles of the Hawking radiation and also the purification of the thermal bath in this reconstruction.
A holographic dual of a finite-temperature SUN c gauge theory with a small number of flavors N f ... more A holographic dual of a finite-temperature SUN c gauge theory with a small number of flavors N f N c typically contains D-branes in a black hole background. By considering the backreaction of the branes, we demonstrate that, to leading order in N f =N c , the viscosity to entropy ratio in these theories saturates the conjectured universal bound =s 1=4. Given the known results for the entropy density, the contribution of the fundamental matter fund is therefore enhanced at strong 't Hooft coupling ; for example, fund N c N f T 3 in four dimensions. Other transport coefficients are analogously enhanced. These results hold with or without a baryon number chemical potential.
... The conformal factor in the metric (3.10) was chosen to simplify the equations for the remain... more ... The conformal factor in the metric (3.10) was chosen to simplify the equations for the remaining metric components z Rob ZTab + gab(Tee + Y) = 0. (3.12) D2 As boundary conditions we choose , e, q0 0 ... 177B (1986) 341 [23] L. Dixon, J. Harvey, C. Vafa, and E. Witten, Nucl, Phys ...
We propose entanglement entropy as a probe of the architecture of spacetime in quantum gravity. W... more We propose entanglement entropy as a probe of the architecture of spacetime in quantum gravity. We argue that the leading contribution to this entropy satisfies an area law for any sufficiently large region in a smooth spacetime, which, in fact, is given by the Bekenstein-Hawking formula. This conjecture is supported by various lines of evidence from perturbative quantum gravity, simplified models of induced gravity and loop quantum gravity, as well as the AdS/CFT correspondence.
We examine the idea that in quantum gravity, the entanglement entropy of a general region should ... more We examine the idea that in quantum gravity, the entanglement entropy of a general region should be finite and the leading contribution is given by the Bekenstein-Hawking area law. Using holographic entanglement entropy calculations, we show that this idea is realized in the Randall-Sundrum II braneworld for sufficiently large regions in smoothly curved backgrounds. Extending the induced gravity action on the brane to include the curvature-squared interactions, we show that the Wald entropy closely matches the expression describing the entanglement entropy. The difference is that for a general region, the latter includes terms involving the extrinsic curvature of the entangling surface, which do not appear in the Wald entropy. We also consider various limitations on the validity of these results.
There appears to be no natural explanation for the cosmological constant's small size within the ... more There appears to be no natural explanation for the cosmological constant's small size within the framework of local relativistic field theories. We argue that the recentlydiscussed framework for which the observable universe is identified with a p-brane embedded within a higher-dimensional 'bulk' spacetime, has special properties that may help circumvent the obstacles to this understanding. This possibility arises partly due to several unique features of the brane proposal. These are: (1) the potential such models introduce for partially breaking supersymmetry, (2) the possibility of having low-energy degrees of freedom which are not observable to us because they are physically located on a different brane, (3) the fundamental scale may be much smaller than the Planck scale. Furthermore, although the resulting cosmological constant in the scenarios we outline need not be exactly zero, it may be suppressed relative to the mass splittings of supermultiplets by weak coupling constants of gravitational strength, in accord with cosmological observations.
The existence of a new fundamental scale may lead to modified dispersion relations for particles ... more The existence of a new fundamental scale may lead to modified dispersion relations for particles at high energies. Such modifications seem to be realized with the Planck scale in certain descriptions of quantum gravity. We apply effective field theory to this problem and identify dimension 5 operators that would lead to cubic modifications of dispersion relations for Standard Model particles. We also discuss other issues related to this approach including various experimental bounds on the strength of these interactions. Further we sketch a scenario where mixing of these operators with dimensions 3 and 4 due to quantum effects is minimal.
This is a short summary of my lectures given at the Fourth Mexican School on Gravitation and Math... more This is a short summary of my lectures given at the Fourth Mexican School on Gravitation and Mathematical Physics. These lectures gave a brief introduction to black holes in string theory, in which I primarily focussed on describing some of the recent calculations of black hole entropy using the statistical mechanics of D-brane states. The following overview will also provide the interested students with an introduction to the relevant literature. * The surface gravity may be thought of as the redshifted acceleration of a fiducial observer moving just outside the horizon .
We discuss the low-energy dynamics of generalized extremal higher membrane black hole solutions o... more We discuss the low-energy dynamics of generalized extremal higher membrane black hole solutions of string theory and higher membrane theories following Manton's prescription for multi-soliton solutions. A flat metric is found for those solutions which possess κ-symmetry on the worldvolume. McGill/95-25
In theories of gravity with a positive cosmological constant, we consider product solutions with ... more In theories of gravity with a positive cosmological constant, we consider product solutions with flux, of the form (A)dS p × S q . Most solutions are shown to be perturbatively unstable, including all uncharged dS p × S q spacetimes. For dimensions greater than four, the stable class includes universes whose entropy exceeds that of de Sitter space, in violation of the conjectured "N-bound". Hence, if quantum gravity theories with finite-dimensional Hilbert space exist, the specification of a positive cosmological constant will not suffice to characterize the class of spacetimes they describe.
Recently certain nonlocal operators were proposed to provide quark masses for the holographic mod... more Recently certain nonlocal operators were proposed to provide quark masses for the holographic model of QCD developed by Sakai and Sugimoto. The properties of these operators at strong coupling are examined in detail using holographic techniques. We find the renormalization procedure for these operators is modified by the running of the fivedimensional gauge coupling. We explicitly evaluate the chiral condensate characterized by these operators.
In this addendum we modify the acknowledgements to include the support provided by MURI grant W91... more In this addendum we modify the acknowledgements to include the support provided by MURI grant W911NF-14-1-0003 from ARO.
The Journal of high energy physics/The journal of high energy physics, Mar 20, 2024
We describe solutions of asymptotically AdS 3 Einstein gravity that are sourced by the insertion ... more We describe solutions of asymptotically AdS 3 Einstein gravity that are sourced by the insertion of operators in the boundary CFT 2 , whose dimension scales with the central charge of the theory. Previously, we found that the geometry corresponding to a black hole two-point function is simply related to an infinite covering of the Euclidean BTZ black hole . However, here we find that the geometry sourced by the presence of a third black hole operator turns out to be a Euclidean wormhole with two asymptotic boundaries. We construct this new geometry as a quotient of empty AdS 3 realized by domes and doors. The doors give access to the infinite covers that are needed to describe the insertion of the operators, while the domes describe the fundamental domains of the quotient on each cover. In particular, despite the standard fact that the Fefferman-Graham expansion is single-sided, the extended bulk geometry contains a wormhole that connects two asymptotic boundaries. We observe that the two-sided wormhole can be made single-sided by cutting off the wormhole and gluing on a "Lorentzian cap". In this way, the geometry gives the holographic description of a three-point function, up to phases. By rewriting the metric in terms of a Liouville field, we compute the on-shell action and find that the result matches with the Heavy-Heavy-Heavy three-point function predicted by the modular bootstrap. Finally, we describe the geometric transition between doors and defects, that is, when one or more dual operators describe a conical defect insertion, rather than a black hole insertion.
We investigate how the complexity=anything observables proposed by can be used to investigate the... more We investigate how the complexity=anything observables proposed by can be used to investigate the interior geometry of AdS black holes. In particular, we illustrate how the flexibility of the complexity=anything approach allows us to systematically probe the geometric properties of black hole singularities. We contrast our results for the AdS Schwarzschild and AdS Reissner-Nordström geometries, i.e., for uncharged and charged black holes, respectively. In the latter case, the holographic complexity observables can only probe the interior up to the inner horizon.
In a recent paper, Duff and Rahmfeld argued that certain massive N R = 1/2 states of the four-dim... more In a recent paper, Duff and Rahmfeld argued that certain massive N R = 1/2 states of the four-dimensional heterotic string correspond to extreme black hole solutions. We provide further, dynamical, evidence for this identification by comparing the scattering of these elementary string states with that of the corresponding extreme black holes, in the limit of low velocities.
New Euclidean solutions, describing the nucleation of a baby universe, are presented for an axion... more New Euclidean solutions, describing the nucleation of a baby universe, are presented for an axion (described by an antisymmetric tensor field strength of rank D -1 in D dimensions) coupled to grav- ity including a cosmological constant. It has long been speculated that the geometry of space-time should fluctuate at distances of the Planck scale. This possibility that quantum gravity allow fluc- tuations in topology has recently been receiving serious consideration by many authors. The work of Gid- dings and Strominger is based on a definite axionic in- stanton. The four-dimensional Euclidean solution in- volves an axion (given as a three-form field strength) cou- pled to Einstein gravity, and describes the nucleation of a baby universe, which carries ofl' a definite axion charge. In the present paper, we briefly report two extensions of this solution. The first is the inclusion of @ cosmological constant. The resulting instanton may be used to de- scribe tunneling between a large de Sitter space and a Planckian Robertson-Walker universe. The second ex- tension is finding analogous solutions for space-time di- mensions other than four. This requires replacing the rank-three antisymmetric tensor field strength by one of rank D -1 in D dimensions. Such higher-rank tensor fields arise in higher-dimensional supergravity theories.
We consider the effects of abelian duality transformations on static, spherically-symmetric, asym... more We consider the effects of abelian duality transformations on static, spherically-symmetric, asymptotically flat string spacetimes in four dimensions, where the dilaton, axion, metric, and gauge fields are allowed to be nonzero. Independent of the α ′ expansion, there is a six-parameter family of such configurations, labelled by the charges characterizing the asymptotic behaviour of the various fields: i.e. their mass, dilaton charge, axion charge, electric charge, magnetic charge, and Taub-NUT parameter. We show that duality, based on time-translation invariance, maps these solutions amongst themselves, with the effect of interchanging two pairs of these six labels, namely: (1) the mass and dilaton charge, and (2) the axion charge and the Taub-NUT parameter. We consider in detail the special case of the purely Schwarzschild black hole, for which the mass of the dual configuration vanishes to leading order in α ′ . Working to next-to-leading order in α ′ for the bosonic and heterotic strings, we find that duality takes a black hole of mass M to a (singular) solution having mass ∼ -1/(α ′ M ). Finally, we argue that two solutions which are related by duality based on a noncompact symmetry are not always physically equivalent.
We examine brane induced gravity on codimension-1 branes, a.k.a DGP gravity, as a theory of five-... more We examine brane induced gravity on codimension-1 branes, a.k.a DGP gravity, as a theory of five-dimensional gravity containing a certain class of fourdimensional branes. From this perspective, the model suffers from a number of pathologies which went unnoticed before. By generalizing the 5D geometry from Minkowski to Schwarzschild, we find that when the bulk mass is large enough, the brane hits a pressure singularity at finite radius. Further, on the self-accelerating branch, the five-dimensional energy is unbounded from below, implying that the self-accelerating backgrounds are unstable. Even in an empty Minkowski bulk, standard Euclidean techniques suggest that the spontaneous nucleation of self-accelerating branes is unsuppressed. If so, quantum effects will strongly modify any classical intuition about the theory. We also note that unless considered as Z 2 -orbifold boundaries, self-accelerating branes correspond to 'wormhole' configurations, which introduces the usual problematic issues associated with wormholes. Altogether these pathologies present a serious challenge that any proposed UV completion of the DGP model must overcome.
We discuss holographic models of extremal and non-extremal black holes in contact with a bath in ... more We discuss holographic models of extremal and non-extremal black holes in contact with a bath in d dimensions, based on a brane world model introduced in . The main benefit of our setup is that it allows for a high degree of analytic control as compared to previous work in higher dimensions. We show that the appearance of quantum extremal islands in those models is a consequence of the well-understood phase transition of RT surfaces, and does not make any direct reference to ensemble averaging. For non-extremal black holes the appearance of quantum extremal islands has the right behaviour to avoid the information paradox in any dimension. We further show that for these models the calculation of the full Page curve is possible in any dimension. The calculation reduces to numerically solving two ODEs. In the case of extremal black holes in higher dimensions, we find no quantum extremal islands for a wide range of parameters. In two dimensions, our results agree with [2] at leading order; however a finite UV cutoff introduced by the brane results in subleading corrections. For example, these corrections result in the quantum extremal surfaces moving further outward from the horizon, and shifting the Page transition to a slightly earlier time.
We study the doubly holographic model of in the situation where a black hole in two-dimensional J... more We study the doubly holographic model of in the situation where a black hole in two-dimensional JT gravity theory is coupled to an auxiliary bath system at arbitrary finite temperature. Depending on the initial temperature of the black hole relative to the bath temperature, the black hole can lose mass by emitting Hawking radiation, stay in equilibrium with the bath or gain mass by absorbing thermal radiation from the bath. In all of these scenarios, a unitary Page curve is obtained by applying the usual prescription for holographic entanglement entropy and identifying the quantum extremal surface for the generalized entropy, using both analytical and numeric calculations. As the application of the entanglement wedge reconstruction, we further investigate the reconstruction of the black hole interior from a subsystem containing the Hawking radiation. We examine the roles of the Hawking radiation and also the purification of the thermal bath in this reconstruction.
A holographic dual of a finite-temperature SUN c gauge theory with a small number of flavors N f ... more A holographic dual of a finite-temperature SUN c gauge theory with a small number of flavors N f N c typically contains D-branes in a black hole background. By considering the backreaction of the branes, we demonstrate that, to leading order in N f =N c , the viscosity to entropy ratio in these theories saturates the conjectured universal bound =s 1=4. Given the known results for the entropy density, the contribution of the fundamental matter fund is therefore enhanced at strong 't Hooft coupling ; for example, fund N c N f T 3 in four dimensions. Other transport coefficients are analogously enhanced. These results hold with or without a baryon number chemical potential.
... The conformal factor in the metric (3.10) was chosen to simplify the equations for the remain... more ... The conformal factor in the metric (3.10) was chosen to simplify the equations for the remaining metric components z Rob ZTab + gab(Tee + Y) = 0. (3.12) D2 As boundary conditions we choose , e, q0 0 ... 177B (1986) 341 [23] L. Dixon, J. Harvey, C. Vafa, and E. Witten, Nucl, Phys ...
We propose entanglement entropy as a probe of the architecture of spacetime in quantum gravity. W... more We propose entanglement entropy as a probe of the architecture of spacetime in quantum gravity. We argue that the leading contribution to this entropy satisfies an area law for any sufficiently large region in a smooth spacetime, which, in fact, is given by the Bekenstein-Hawking formula. This conjecture is supported by various lines of evidence from perturbative quantum gravity, simplified models of induced gravity and loop quantum gravity, as well as the AdS/CFT correspondence.
We examine the idea that in quantum gravity, the entanglement entropy of a general region should ... more We examine the idea that in quantum gravity, the entanglement entropy of a general region should be finite and the leading contribution is given by the Bekenstein-Hawking area law. Using holographic entanglement entropy calculations, we show that this idea is realized in the Randall-Sundrum II braneworld for sufficiently large regions in smoothly curved backgrounds. Extending the induced gravity action on the brane to include the curvature-squared interactions, we show that the Wald entropy closely matches the expression describing the entanglement entropy. The difference is that for a general region, the latter includes terms involving the extrinsic curvature of the entangling surface, which do not appear in the Wald entropy. We also consider various limitations on the validity of these results.
There appears to be no natural explanation for the cosmological constant's small size within the ... more There appears to be no natural explanation for the cosmological constant's small size within the framework of local relativistic field theories. We argue that the recentlydiscussed framework for which the observable universe is identified with a p-brane embedded within a higher-dimensional 'bulk' spacetime, has special properties that may help circumvent the obstacles to this understanding. This possibility arises partly due to several unique features of the brane proposal. These are: (1) the potential such models introduce for partially breaking supersymmetry, (2) the possibility of having low-energy degrees of freedom which are not observable to us because they are physically located on a different brane, (3) the fundamental scale may be much smaller than the Planck scale. Furthermore, although the resulting cosmological constant in the scenarios we outline need not be exactly zero, it may be suppressed relative to the mass splittings of supermultiplets by weak coupling constants of gravitational strength, in accord with cosmological observations.
The existence of a new fundamental scale may lead to modified dispersion relations for particles ... more The existence of a new fundamental scale may lead to modified dispersion relations for particles at high energies. Such modifications seem to be realized with the Planck scale in certain descriptions of quantum gravity. We apply effective field theory to this problem and identify dimension 5 operators that would lead to cubic modifications of dispersion relations for Standard Model particles. We also discuss other issues related to this approach including various experimental bounds on the strength of these interactions. Further we sketch a scenario where mixing of these operators with dimensions 3 and 4 due to quantum effects is minimal.
This is a short summary of my lectures given at the Fourth Mexican School on Gravitation and Math... more This is a short summary of my lectures given at the Fourth Mexican School on Gravitation and Mathematical Physics. These lectures gave a brief introduction to black holes in string theory, in which I primarily focussed on describing some of the recent calculations of black hole entropy using the statistical mechanics of D-brane states. The following overview will also provide the interested students with an introduction to the relevant literature. * The surface gravity may be thought of as the redshifted acceleration of a fiducial observer moving just outside the horizon .
We discuss the low-energy dynamics of generalized extremal higher membrane black hole solutions o... more We discuss the low-energy dynamics of generalized extremal higher membrane black hole solutions of string theory and higher membrane theories following Manton's prescription for multi-soliton solutions. A flat metric is found for those solutions which possess κ-symmetry on the worldvolume. McGill/95-25
In theories of gravity with a positive cosmological constant, we consider product solutions with ... more In theories of gravity with a positive cosmological constant, we consider product solutions with flux, of the form (A)dS p × S q . Most solutions are shown to be perturbatively unstable, including all uncharged dS p × S q spacetimes. For dimensions greater than four, the stable class includes universes whose entropy exceeds that of de Sitter space, in violation of the conjectured "N-bound". Hence, if quantum gravity theories with finite-dimensional Hilbert space exist, the specification of a positive cosmological constant will not suffice to characterize the class of spacetimes they describe.
Recently certain nonlocal operators were proposed to provide quark masses for the holographic mod... more Recently certain nonlocal operators were proposed to provide quark masses for the holographic model of QCD developed by Sakai and Sugimoto. The properties of these operators at strong coupling are examined in detail using holographic techniques. We find the renormalization procedure for these operators is modified by the running of the fivedimensional gauge coupling. We explicitly evaluate the chiral condensate characterized by these operators.
In this addendum we modify the acknowledgements to include the support provided by MURI grant W91... more In this addendum we modify the acknowledgements to include the support provided by MURI grant W911NF-14-1-0003 from ARO.
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Papers by Robert Myers