Advances in Theoretical and Mathematical Physics, 2001
A class of correlation functions of half-BPS composite operators are computed exactly (at finite ... more A class of correlation functions of half-BPS composite operators are computed exactly (at finite N) in the zero coupling limit of N = 4 SYM theory. These have a simple dependence on the four-dimensional spacetime coordinates and are related to correlators in a one-dimensional Matrix Model with complex Matrices obtained by dimensional reduction of N = 4 SYM on a three-sphere. A key technical tool is Frobenius-Schur duality between symmetric and Unitary groups and the results are expressed simply in terms of U (N) group integrals or equivalently in terms of Littlewood-Richardson coefficients. These correlation functions are used to understand the existence/properties of giant gravitons and related solutions in the string theory dual on AdS 5 × S 5. Some of their properties hint at integrability in N = 4 SYM.
We consider the N = 2 gauge theory on N D7-branes wrapping K3, with D3brane probes. In the large ... more We consider the N = 2 gauge theory on N D7-branes wrapping K3, with D3brane probes. In the large N limit, the D7-branes blow up to form an enhancon shell. We probe the region inside and outside the enhancon shell using the D3-branes, and compute the probe metric using the Seiberg-Witten formalism. Supergravity arguments suggest a flat interior up to 1/N corrections, and indeed our results for the D3-brane probes are consistent with that. By including the dynamics of the branes, these results, together with those of hep-th/0204050, demonstrate the robustness of the enhancon mechanism beyond patching together of supergravity solutions with D-brane source junction conditions.
We find new closed string couplings on Dp-branes for the bosonic string. These couplings are quad... more We find new closed string couplings on Dp-branes for the bosonic string. These couplings are quadratic in derivatives and therefore take the form of induced kinetic terms on the brane. For the graviton in particular we find the induced Einstein-Hilbert term as well as terms quadratic in the second fundamental tensor. We comment on tachyon dependences of these brane-localized couplings.
Projector equivalences used in the definition of the K-theory of operator algebras are shown to l... more Projector equivalences used in the definition of the K-theory of operator algebras are shown to lead to generalizations of the solution generating technique for solitons in NC field theories, which has recently been used in the construction of branes from other branes in B-field backgrounds and in the construction of fluxon solutions of gauge theories. The generalizations involve families of static solutions as well as solutions which depend on euclidean time and interpolate between different configurations. We investigate the physics of these generalizations in the brane-construction as well as the fluxon context. These results can be interpreted in the light of recent discussions on the topology of the configuration space of string fields.
The brane world scenario advocated by Arkani-Hamed et al. transmutes the hierarchy problem into e... more The brane world scenario advocated by Arkani-Hamed et al. transmutes the hierarchy problem into explaining why extra dimensions have sizes much larger than the fundamental scale. In this paper we discuss possible solutions to this problem by considering the compactified dimensions to be populated by a large number of branes in a crystal lattice. The experimental consequences of this scenario are described, including the presence of large energy gaps in the spectrum of Kaluza-Klein modes.
As a step toward constructing realistic brane world models in string theory, we consider the inte... more As a step toward constructing realistic brane world models in string theory, we consider the interactions of a pair of non-BPS branes. We construct a dyonic generalization of the non-BPS branes first constructed by Bergman, Gaberdiel and Sen as orbifolds of D-branes on T 4 /Z 2. The force between a dyonic brane and an electric brane is computed and is found to vanish at a nontrivial critical separation. This equilibrium point is unstable. For smaller separations the branes coalesce to form a composite dyonic state, while for larger separations the branes run off to infinity. We suggest generalizations that will lead to potentials with stable local minima.
We study the spectrum of created particles in two-dimensional black hole geometries for a linear,... more We study the spectrum of created particles in two-dimensional black hole geometries for a linear, hermitian scalar field satisfying a Lorentz non-invariant field equation with higher spatial derivative terms that are suppressed by powers of a fundamental momentum scale k 0. The preferred frame is the "free-fall frame" of the black hole. This model is a variation of Unruh's sonic black hole analogy. We find that there are two qualitatively different types of particle production in this model: a thermal Hawking flux generated by "mode conversion" at the black hole horizon, and a non-thermal spectrum generated via scattering off the background into negative free-fall frequency modes. This second process has nothing to do with black holes and does not occur for the ordinary wave equation because such modes do not propagate outside the horizon with positive Killing frequency. The horizon component of the radiation is astonishingly close to a perfect thermal spectrum: for the smoothest metric studied, with Hawking temperature T H ≃ 0.0008k 0 , agreement is of order (T H /k 0) 3 at frequency ω = T H , and agreement to order T H /k 0 persists out to ω/T H ≃ 45 where the thermal number flux is O(10 −20). The flux from scattering dominates at large ω and becomes many orders of magnitude larger than the horizon component for metrics with a "kink", i.e. a region of high curvature localized on a static worldline outside the horizon. This non-thermal flux amounts to roughly 10% of the total luminosity for the kinkier metrics considered. The flux exhibits oscillations as a function of frequency which can be explained by interference between the various contributions to the flux.
The "screen mapping" introduced by Susskind to implement 't Hooft's holographic hypothesis is stu... more The "screen mapping" introduced by Susskind to implement 't Hooft's holographic hypothesis is studied. For a single screen time, there are an infinite number of images of a black hole event horizon, almost all of which have smaller area on the screen than the horizon area. This is consistent with the focusing equation because of the existence of focal points. However, the boundary of the past (or future) of the screen obeys the area theorem, and so always gives an expanding map to the screen, as required by the holographic hypothesis. These considerations are illustrated with several axisymmetric static black hole spacetimes.
High frequency dispersion does not alter the low frequency spectrum of Hawking radiation from a s... more High frequency dispersion does not alter the low frequency spectrum of Hawking radiation from a single black hole horizon, whether the dispersion entails subluminal or superluminal group velocities. We show here that in the presence of an inner horizon as well as an outer horizon the superluminal case differs dramatically however. The negative energy partners of Hawking quanta return to the outer horizon and stimulate more Hawking radiation if the field is bosonic or suppress it if the field is fermionic. This process leads to exponential growth or damping of the radiated flux and correlations among the quanta emitted at different times, unlike in the usual Hawking effect. These phenomena may be observable in condensed matter black hole analogs that exhibit "superluminal" dispersion.
Kaluza-Klein theory admits "bubble" configurations, in which the circumference of the fifth dimen... more Kaluza-Klein theory admits "bubble" configurations, in which the circumference of the fifth dimension shrinks to zero on some compact surface. A three parameter family of such bubble initial data at a moment of timesymmetry (some including a magnetic field) has been found by Brill and Horowitz, generalizing the (zero-energy) "Witten bubble" solution. Some of these data have negative total energy. We show here that all the negative energy bubble solutions start out expanding away from the moment of time symmetry, while the positive energy bubbles can start out either expanding or contracting. Thus it is unlikely that the negative energy bubbles would collapse and produce a naked singularity.
We study the Hawking process on lattices falling into static black holes. The motivation is to un... more We study the Hawking process on lattices falling into static black holes. The motivation is to understand how the outgoing modes and Hawking radiation can arise in a setting with a strict short distance cutoff in the free-fall frame. We employ two-dimensional free scalar field theory. For a falling lattice with a discrete time-translation symmetry we use analytical methods to establish that, for Killing frequency ω and surface gravity κ satisfying κ ≪ ω 1/3 ≪ 1 in lattice units, the continuum Hawking spectrum is recovered. The low frequency outgoing modes arise from exotic ingoing modes with large proper wavevectors that "refract" off the horizon. In this model with time translation symmetry the proper lattice spacing goes to zero at spatial infinity. We also consider instead falling lattices whose proper lattice spacing is constant at infinity and therefore grows with time at any finite radius. This violation of time translation symmetry is visible only at wavelengths comparable to the lattice spacing, and it is responsible for transmuting ingoing high Killing frequency modes into low frequency outgoing modes.
We show that the BMN operators in D = 4 N = 4 super Yang Mills theory proposed as duals of string... more We show that the BMN operators in D = 4 N = 4 super Yang Mills theory proposed as duals of stringy oscillators in a plane wave background have a natural quantum group construction in terms of the quantum deformation of the SO(6) R symmetry. We describe in detail how a q-deformed U (2) subalgebra generates BMN operators, with q ∼ e 2iπ J. The standard quantum co-product as well as generalized traces which use q-cyclic operators acting on tensor products of Higgs fields are the ingredients in this construction. They generate the oscillators with the correct (undeformed) permutation symmetries of Fock space oscillators. The quantum group can be viewed as a spectrum generating algebra, and suggests that correlators of BMN operators should have a geometrical meaning in terms of spaces with quantum group symmetry.
SYM, with U (N) gauge group, is shown to satisfy finite factorization equations reminiscent of to... more SYM, with U (N) gauge group, is shown to satisfy finite factorization equations reminiscent of topological gauge theories. The finite factorization equations can be generalized, beyond the extremal case, to a class of correlators involving observables with a simple pattern of SO(6) charges. The simple group theoretic form of the correlators allows equalities between ratios of correlators in N = 4 SYM and Wilson loops in Chern-Simons theories at k = ∞, correlators of appropriate observables in topological G/G models and Wilson loops in two-dimensional Yang-Mills theories. The correlators also obey sum rules which can be generalized to off-extremal correlators. The simplest sum rules can be viewed as large k limits of the Verlinde formula using the Chern-Simons correspondence. For special classes of correlators, the saturation of the factorization equations by a small subset of the operators in the large N theory is related to the emergence of semiclassical objects like KK modes and giant gravitons in the dual ADS × S background. We comment on an intriguing symmetry between KK modes and giant gravitons.
We compute the moduli space metric of SU (N) Yang-Mills theory with N = 2 supersymmetry in the vi... more We compute the moduli space metric of SU (N) Yang-Mills theory with N = 2 supersymmetry in the vicinity of the point where the classical moduli vanish. This gauge theory may be realized as a set of N D7-branes wrapping a K3 surface, near the enhancon locus. The moduli space metric determines the low-energy worldvolume dynamics of the D7 branes near this point, including stringy corrections. Non-abelian gauge symmetry is not restored on the worldvolume at the enhancon point, but rather the gauge group remains U (1) N−1 and light electric and magnetically charged particles coexist. We also study the moduli space metric for a single probe brane in the background of N − 1 branes near the enhancon point. We find quantum corrections to the supergravity probe metric that are not suppressed at large separations, but are down by 1/N factors, due to the response of the N − 1 enhancon branes to the probe. A singularity appears before the probe reaches the enhancon point where a dyon becomes massless. We compute the masses of W-bosons and monopoles in a large N limit near this critical point.
An initial step is taken in investigating the duality between the near horizon region of a four d... more An initial step is taken in investigating the duality between the near horizon region of a four dimensional extremal Reissner-Nordström black hole and the n-particle, N = 4 Calogero model as conjectured by Gibbons and Townsend. Specifically we compute the mass spectrum of d = 4, N = 8 supergravity about the Bertotti-Robinson solution and find the corresponding set of conformal dimensions of states in the dual conformal quantum mechanics. We find that the dual states fill irreducible representations of the supergroup SU (1, 1|2), and furthermore transform under various irreducible representations of the group SU (2) × SU (6) spontaneously broken down from the E 7(7) duality group of N = 8 supergravity.
Advances in Theoretical and Mathematical Physics, 2001
A class of correlation functions of half-BPS composite operators are computed exactly (at finite ... more A class of correlation functions of half-BPS composite operators are computed exactly (at finite N) in the zero coupling limit of N = 4 SYM theory. These have a simple dependence on the four-dimensional spacetime coordinates and are related to correlators in a one-dimensional Matrix Model with complex Matrices obtained by dimensional reduction of N = 4 SYM on a three-sphere. A key technical tool is Frobenius-Schur duality between symmetric and Unitary groups and the results are expressed simply in terms of U (N) group integrals or equivalently in terms of Littlewood-Richardson coefficients. These correlation functions are used to understand the existence/properties of giant gravitons and related solutions in the string theory dual on AdS 5 × S 5. Some of their properties hint at integrability in N = 4 SYM.
We consider the N = 2 gauge theory on N D7-branes wrapping K3, with D3brane probes. In the large ... more We consider the N = 2 gauge theory on N D7-branes wrapping K3, with D3brane probes. In the large N limit, the D7-branes blow up to form an enhancon shell. We probe the region inside and outside the enhancon shell using the D3-branes, and compute the probe metric using the Seiberg-Witten formalism. Supergravity arguments suggest a flat interior up to 1/N corrections, and indeed our results for the D3-brane probes are consistent with that. By including the dynamics of the branes, these results, together with those of hep-th/0204050, demonstrate the robustness of the enhancon mechanism beyond patching together of supergravity solutions with D-brane source junction conditions.
We find new closed string couplings on Dp-branes for the bosonic string. These couplings are quad... more We find new closed string couplings on Dp-branes for the bosonic string. These couplings are quadratic in derivatives and therefore take the form of induced kinetic terms on the brane. For the graviton in particular we find the induced Einstein-Hilbert term as well as terms quadratic in the second fundamental tensor. We comment on tachyon dependences of these brane-localized couplings.
Projector equivalences used in the definition of the K-theory of operator algebras are shown to l... more Projector equivalences used in the definition of the K-theory of operator algebras are shown to lead to generalizations of the solution generating technique for solitons in NC field theories, which has recently been used in the construction of branes from other branes in B-field backgrounds and in the construction of fluxon solutions of gauge theories. The generalizations involve families of static solutions as well as solutions which depend on euclidean time and interpolate between different configurations. We investigate the physics of these generalizations in the brane-construction as well as the fluxon context. These results can be interpreted in the light of recent discussions on the topology of the configuration space of string fields.
The brane world scenario advocated by Arkani-Hamed et al. transmutes the hierarchy problem into e... more The brane world scenario advocated by Arkani-Hamed et al. transmutes the hierarchy problem into explaining why extra dimensions have sizes much larger than the fundamental scale. In this paper we discuss possible solutions to this problem by considering the compactified dimensions to be populated by a large number of branes in a crystal lattice. The experimental consequences of this scenario are described, including the presence of large energy gaps in the spectrum of Kaluza-Klein modes.
As a step toward constructing realistic brane world models in string theory, we consider the inte... more As a step toward constructing realistic brane world models in string theory, we consider the interactions of a pair of non-BPS branes. We construct a dyonic generalization of the non-BPS branes first constructed by Bergman, Gaberdiel and Sen as orbifolds of D-branes on T 4 /Z 2. The force between a dyonic brane and an electric brane is computed and is found to vanish at a nontrivial critical separation. This equilibrium point is unstable. For smaller separations the branes coalesce to form a composite dyonic state, while for larger separations the branes run off to infinity. We suggest generalizations that will lead to potentials with stable local minima.
We study the spectrum of created particles in two-dimensional black hole geometries for a linear,... more We study the spectrum of created particles in two-dimensional black hole geometries for a linear, hermitian scalar field satisfying a Lorentz non-invariant field equation with higher spatial derivative terms that are suppressed by powers of a fundamental momentum scale k 0. The preferred frame is the "free-fall frame" of the black hole. This model is a variation of Unruh's sonic black hole analogy. We find that there are two qualitatively different types of particle production in this model: a thermal Hawking flux generated by "mode conversion" at the black hole horizon, and a non-thermal spectrum generated via scattering off the background into negative free-fall frequency modes. This second process has nothing to do with black holes and does not occur for the ordinary wave equation because such modes do not propagate outside the horizon with positive Killing frequency. The horizon component of the radiation is astonishingly close to a perfect thermal spectrum: for the smoothest metric studied, with Hawking temperature T H ≃ 0.0008k 0 , agreement is of order (T H /k 0) 3 at frequency ω = T H , and agreement to order T H /k 0 persists out to ω/T H ≃ 45 where the thermal number flux is O(10 −20). The flux from scattering dominates at large ω and becomes many orders of magnitude larger than the horizon component for metrics with a "kink", i.e. a region of high curvature localized on a static worldline outside the horizon. This non-thermal flux amounts to roughly 10% of the total luminosity for the kinkier metrics considered. The flux exhibits oscillations as a function of frequency which can be explained by interference between the various contributions to the flux.
The "screen mapping" introduced by Susskind to implement 't Hooft's holographic hypothesis is stu... more The "screen mapping" introduced by Susskind to implement 't Hooft's holographic hypothesis is studied. For a single screen time, there are an infinite number of images of a black hole event horizon, almost all of which have smaller area on the screen than the horizon area. This is consistent with the focusing equation because of the existence of focal points. However, the boundary of the past (or future) of the screen obeys the area theorem, and so always gives an expanding map to the screen, as required by the holographic hypothesis. These considerations are illustrated with several axisymmetric static black hole spacetimes.
High frequency dispersion does not alter the low frequency spectrum of Hawking radiation from a s... more High frequency dispersion does not alter the low frequency spectrum of Hawking radiation from a single black hole horizon, whether the dispersion entails subluminal or superluminal group velocities. We show here that in the presence of an inner horizon as well as an outer horizon the superluminal case differs dramatically however. The negative energy partners of Hawking quanta return to the outer horizon and stimulate more Hawking radiation if the field is bosonic or suppress it if the field is fermionic. This process leads to exponential growth or damping of the radiated flux and correlations among the quanta emitted at different times, unlike in the usual Hawking effect. These phenomena may be observable in condensed matter black hole analogs that exhibit "superluminal" dispersion.
Kaluza-Klein theory admits "bubble" configurations, in which the circumference of the fifth dimen... more Kaluza-Klein theory admits "bubble" configurations, in which the circumference of the fifth dimension shrinks to zero on some compact surface. A three parameter family of such bubble initial data at a moment of timesymmetry (some including a magnetic field) has been found by Brill and Horowitz, generalizing the (zero-energy) "Witten bubble" solution. Some of these data have negative total energy. We show here that all the negative energy bubble solutions start out expanding away from the moment of time symmetry, while the positive energy bubbles can start out either expanding or contracting. Thus it is unlikely that the negative energy bubbles would collapse and produce a naked singularity.
We study the Hawking process on lattices falling into static black holes. The motivation is to un... more We study the Hawking process on lattices falling into static black holes. The motivation is to understand how the outgoing modes and Hawking radiation can arise in a setting with a strict short distance cutoff in the free-fall frame. We employ two-dimensional free scalar field theory. For a falling lattice with a discrete time-translation symmetry we use analytical methods to establish that, for Killing frequency ω and surface gravity κ satisfying κ ≪ ω 1/3 ≪ 1 in lattice units, the continuum Hawking spectrum is recovered. The low frequency outgoing modes arise from exotic ingoing modes with large proper wavevectors that "refract" off the horizon. In this model with time translation symmetry the proper lattice spacing goes to zero at spatial infinity. We also consider instead falling lattices whose proper lattice spacing is constant at infinity and therefore grows with time at any finite radius. This violation of time translation symmetry is visible only at wavelengths comparable to the lattice spacing, and it is responsible for transmuting ingoing high Killing frequency modes into low frequency outgoing modes.
We show that the BMN operators in D = 4 N = 4 super Yang Mills theory proposed as duals of string... more We show that the BMN operators in D = 4 N = 4 super Yang Mills theory proposed as duals of stringy oscillators in a plane wave background have a natural quantum group construction in terms of the quantum deformation of the SO(6) R symmetry. We describe in detail how a q-deformed U (2) subalgebra generates BMN operators, with q ∼ e 2iπ J. The standard quantum co-product as well as generalized traces which use q-cyclic operators acting on tensor products of Higgs fields are the ingredients in this construction. They generate the oscillators with the correct (undeformed) permutation symmetries of Fock space oscillators. The quantum group can be viewed as a spectrum generating algebra, and suggests that correlators of BMN operators should have a geometrical meaning in terms of spaces with quantum group symmetry.
SYM, with U (N) gauge group, is shown to satisfy finite factorization equations reminiscent of to... more SYM, with U (N) gauge group, is shown to satisfy finite factorization equations reminiscent of topological gauge theories. The finite factorization equations can be generalized, beyond the extremal case, to a class of correlators involving observables with a simple pattern of SO(6) charges. The simple group theoretic form of the correlators allows equalities between ratios of correlators in N = 4 SYM and Wilson loops in Chern-Simons theories at k = ∞, correlators of appropriate observables in topological G/G models and Wilson loops in two-dimensional Yang-Mills theories. The correlators also obey sum rules which can be generalized to off-extremal correlators. The simplest sum rules can be viewed as large k limits of the Verlinde formula using the Chern-Simons correspondence. For special classes of correlators, the saturation of the factorization equations by a small subset of the operators in the large N theory is related to the emergence of semiclassical objects like KK modes and giant gravitons in the dual ADS × S background. We comment on an intriguing symmetry between KK modes and giant gravitons.
We compute the moduli space metric of SU (N) Yang-Mills theory with N = 2 supersymmetry in the vi... more We compute the moduli space metric of SU (N) Yang-Mills theory with N = 2 supersymmetry in the vicinity of the point where the classical moduli vanish. This gauge theory may be realized as a set of N D7-branes wrapping a K3 surface, near the enhancon locus. The moduli space metric determines the low-energy worldvolume dynamics of the D7 branes near this point, including stringy corrections. Non-abelian gauge symmetry is not restored on the worldvolume at the enhancon point, but rather the gauge group remains U (1) N−1 and light electric and magnetically charged particles coexist. We also study the moduli space metric for a single probe brane in the background of N − 1 branes near the enhancon point. We find quantum corrections to the supergravity probe metric that are not suppressed at large separations, but are down by 1/N factors, due to the response of the N − 1 enhancon branes to the probe. A singularity appears before the probe reaches the enhancon point where a dyon becomes massless. We compute the masses of W-bosons and monopoles in a large N limit near this critical point.
An initial step is taken in investigating the duality between the near horizon region of a four d... more An initial step is taken in investigating the duality between the near horizon region of a four dimensional extremal Reissner-Nordström black hole and the n-particle, N = 4 Calogero model as conjectured by Gibbons and Townsend. Specifically we compute the mass spectrum of d = 4, N = 8 supergravity about the Bertotti-Robinson solution and find the corresponding set of conformal dimensions of states in the dual conformal quantum mechanics. We find that the dual states fill irreducible representations of the supergroup SU (1, 1|2), and furthermore transform under various irreducible representations of the group SU (2) × SU (6) spontaneously broken down from the E 7(7) duality group of N = 8 supergravity.
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Papers by Steven Corley