Anti-de Sitter/conformal Field Theory Research Articles

Black holes, AdS, and CFTs

This brief conference proceeding attempts to explain the implications of the anti-de Sitter/conformal field theory (AdS/CFT) correspondence for black hole entropy in a language accessible to relativists and other non-string theorists. The main conclusion is that the Bekenstein–Hawking entropy S BH is the density of states associated with certain superselections sectors, defined by what may be called the algebra of boundary observables. Interestingly while there is a valid context in which this result can be restated as “S BH counts all states inside the black hole,” there may also be another in which it may be restated as “S BH does not count all states inside the black hole, but only those that are distinguishable from the outside.” The arguments and conclusions represent the author’s translation of the community’s collective wisdom, combined with a few recent results.

Characteristic length of an AdS/CFT superconductor

We investigate in more detail the holographic model of a superconductor recently found by Hartnoll, Herzog, and Horowitz [Phys. Rev. Lett. 101, 031601 (2008)], which is constructed from a condensate of a charged scalar field in AdS{sub 4}-Schwarzschild background. By analytically studying the perturbation of the gravitational system near the critical temperature T{sub c}, we obtain the superconducting coherence length proportional to 1/{radical}(1-T/T{sub c}) via AdS/CFT (anti-de Sitter/conformal field theory) correspondence. By adding a small external homogeneous magnetic field to the system, we find that a stationary diamagnetic current proportional to the square of the order parameter is induced by the magnetic field. These results agree with Ginzburg-Landau theory and strongly support the idea that a superconductor can be described by a charged scalar field on a black hole via AdS/CFT duality.

Low-lying quasinormal modes of topological AdS black holes and hydrodynamics

We analytically calculate the low-lying gravitational quasinormal modes of a topological AdS black hole of arbitrary dimension in the high temperature limit. We show that they are in agreement with corresponding results from the hydrodynamics of the gauge theory plasma on the boundary, as required by the AdS/CFT (anti--de Sitter/conformal field theories) correspondence. For some of these modes, we obtain a lifetime which is comparable to or longer than the longest lifetime of perturbations of spherical black holes. These modes might play a significant role in the late-time behavior of the gauge theory plasma.

Near-zone Navier-Stokes analysis of heavy quark jet quenching in anN=4super-Yang-Mills plasma

The near zone energy-momentum tensor of a supersonic heavy quark jet moving through a strongly coupled N=4 super Yang-Mills (SYM) plasma is analyzed in terms of first-order Navier-Stokes hydrodynamics. It is shown that the hydrodynamic description of the near quark region worsens with increasing quark velocities. For realistic quark velocities, v=0.99, the nonhydrodynamic region is located at a narrow band surrounding the quark with a width of approximately 3/{pi}T in the direction parallel to the quark's motion and with a length of roughly 10/{pi}T in the perpendicular direction. Our results can be interpreted as an indication of the presence of coherent Yang-Mills fields where deviation from hydrodynamics is at its maximum. In the region where hydrodynamics does provide a good description of the system's dynamics, the flow velocity is so small that all the nonlinear terms can be dropped. Our results, which are compatible with the thermalization timescales extracted from elliptic flow measurements, suggest that if anti-de-Sitter/conformal field theory (AdS/CFT) provides a good description of the BNL Relativistic Heavy Ion Collider system, the bulk of the quenched jet energy has more than enough time to locally thermalize and become encoded in the collective flow. The resulting flow pattern close tomore » the quark, however, is shown to be considerably different from the superposition of Mach cones and diffusion wakes observed at large distances.« less

Radiation from an accelerated quark via AdS/CFT correspondence

In this paper we investigate radiation by an accelerated quark in a strongly coupled gauge theory via anti―de Sitter/conformal field theory (AdS/CFT) correspondence. According to the AdS/CFT dictionary, we can read off the energy density or energy flux of the radiation from the asymptotic gravitational field in AdS bulk sourced by an accelerated string trailing behind the quark. In the case of an oscillating quark with frequency Ω, we show that the time averaged energy density is asymptotically isotropic, and it falls off as (g 2 YM N) 1/2 Ω 4 /R 2 with distance R from the source. In a toy model of a scattered quark by an external field, we simply estimate the Poynting vector by the bremsstrahlung radiation and show that the energy flux is anisotropic outgoing radiation. Based on these investigations, we discuss the properties of strongly coupled gauge theory radiation in comparison with electromagnetic radiation.

Flat space holography

The implementation of holography in gravitational physics has its most concrete realization in the context of the anti-de Sitter/conformal field theory (AdS/CFT) correspondence conjecture, an implication of which is that counterterms from the boundary CFT can be understood as surface terms that render the variational principle finite and well-defined for the gravity theory in the bulk. I discuss recent developments that show how such gravitational counterterms can be deployed for asymptotically flat spacetimes in any spacetime dimension d ≥ 4. These actions yield conserved quantities at spacelike infinity that agree with the usual Arnowitt–Deser–Misner results but are more general. This approach removes the need for ill-defined background subtraction methods and suggests the possibility of obtaining a dual field theory to gravity theories in asymptotically flat spacetimes.PACS Nos.: 04.20.Ha, 04.60.–m, 11.25.Tq

Viscosity Information from Relativistic Nuclear Collisions: How Perfect is the Fluid Observed at RHIC?

Relativistic viscous hydrodynamic fits to Brookhaven Relativistic Heavy Ion Collider data on the centrality dependence of multiplicity, transverse, and elliptic flow for square root s = 200 GeV Au+Au collisions are presented. For standard (Glauber-type) initial conditions, while data on the integrated elliptic flow coefficient v(2) are consistent with a ratio of viscosity over entropy density up to eta/s approximately 0.16, data on minimum bias v(2) seem to favor a much smaller viscosity over entropy ratio, below the bound from the anti-de Sitter conformal field theory conjecture. Some caveats on this result are discussed.

Reconstructing1/2BPS space-time metrics from matrix models and spin chains

Using the anti-de Sitter/conformal field theories (AdS/CFT) correspondence, we address the question of how to measure complicated space-time metrics using gauge theory probes. In particular, we consider the case of the 1/2 Bogomol'nyi-Prasad-Sommerfield geometries of type IIB supergravity. These geometries are classified by certain droplets in a two-dimensional spacelike hypersurface. We show how to reconstruct the full metric inside these droplets using the one-loop N=4 super Yang-Mills theory dilatation operator. This is done by considering long operators in the SU(2) sector, which are dual to fast rotating strings on the droplets. We develop new powerful techniques for large N complex matrix models that allow us to construct the Hamiltonian for these strings. We find that the Hamiltonian can be mapped to a dynamical spin chain. That is, the length of the chain is not fixed. Moreover, all of these spin chains can be explicitly constructed using an interesting algebra which is derived from the matrix model. Our techniques work for general droplet configurations. As an example, we study a single elliptical droplet and the hypotrochoid.

Quantum critical transport, duality, and M theory

We consider charge transport properties of $2+1$ dimensional conformal field theories at nonzero temperature. For theories with only Abelian U(1) charges, we describe the action of particle-vortex duality on the hydrodynamic-to-collisionless crossover function: this leads to powerful functional constraints for self-dual theories. For $\mathcal{N}=8$ supersymmetric, $\mathrm{SU}(N)$ Yang-Mills theory at the conformal fixed point, exact hydrodynamic-to-collisionless crossover functions of the SO(8) R-currents can be obtained in the large $N$ limit by applying the anti-de Sitter/conformal field theory (AdS/CFT) correspondence to M theory. In the gravity theory, fluctuating currents are mapped to fluctuating gauge fields in the background of a black hole in $3+1$ dimensional anti-de Sitter space. The electromagnetic self-duality of the $3+1$ dimensional theory implies that the correlators of the R-currents obey a functional constraint similar to that found from particle-vortex duality in $2+1$ dimensional Abelian theories. Thus the $2+1$ dimensional, superconformal Yang Mills theory obeys a ``holographic self-duality'' in the large $N$ limit, and perhaps more generally.

Dirac quasinormal frequencies of Schwarzschild-anti-de Sitter and Reissner-Nordstrom-anti-de Sitter black holes

The quasinormal modes (QNMs) of Dirac field perturbations of a Reissner-Nordstr\"om black hole in an asymptotically Anti-de Sitter spacetime are investigated. We find that both the real and imaginary parts of the fundamental quasinormal frequencies for large black holes are the linear functions of the Hawking temperature, and the slope of the lines for the real parts decreases while that for the magnitude of the imaginary parts increases as the black hole charge increases. According to the Anti-de Sitter/Conformal Field Theory (AdS/CFT) correspondence, the fact shows that different charge presents different time scale in three-dimensional CFT. Another interesting result is that the quasinormal frequencies become evenly spaced for high overtone number, and in the spacing expressions the real part decreases while the magnitude of the imaginary part increases as the charge increases. We also study the relation between quasinormal frequencies and angular quantum number and find that the real part increases while the magnitude of the imaginary part decreases as the angular quantum number increases.

Wilson loops and anomalous dimensions in cascading theories

We use lightlike Wilson loops and the AdS Conformal Field Theory correspondence to compute the anomalous dimensions of twist-2 operators in the cascading [Klebanov-Strassler (KS)] theory. The computation amounts to find a minimal surface in the uv region of the KS background, which is described by the Klebanov-Tseytlin solution. The result is similar to the one for $\mathrm{SU}(N)\mathcal{N}=4$ Super Yang-Mills theory but with N replaced by an effective, scale-dependent N. We performed also a calculation using a rotating string and find agreement. In fact, we use a double Wick rotated version of the rotating string solution which is a Euclidean world-sheet ending on a lightlike line in the boundary. It gives the same result as the rotating string for the $\mathcal{N}=4$ case, but it is more appropriate than the rotating string in the $\mathcal{N}=1$ case.

A uniqueness theorem for the anti-de Sitter soliton.

The stability of physical systems depends on the existence of a state of least energy. In gravity, this is guaranteed by the positive energy theorem. For topological reasons, this fails for nonsupersymmetric Kaluza-Klein compactifications, which can decay to arbitrarily negative energy. For related reasons, this also fails for the anti-de Sitter (AdS) soliton, a globally static, asymptotically toroidal Lambda<0 spacetime with negative mass. Nonetheless, arguing from the AdS conformal field theory (AdS/CFT) correspondence, Horowitz and Myers proposed a new positive energy conjecture, which asserts that the AdS soliton is the unique state of least energy in its asymptotic class. We give a new structure theorem for static Lambda<0 spacetimes and use it to prove uniqueness of the AdS soliton. Our results offer significant support for the new positive energy conjecture and add to the body of rigorous results inspired by the AdS/CFT correspondence.

Generalized AdS/CFT dualities and unitary realizations ofspace-time symmetries of M-theory

I review the AdS/CFT (anti-de Sitter/conformal field theory) dualitieswithin the framework of the general theory of unitary lowest-weight(ULWR) (positive-energy) representations of non-compact spacetimegroups and supergroups. The ULWRs have the remarkable property thatthey can be constructed by tensoring some fundamental ULWRs (singletonsor doubletons). The conformally invariant theory in d dimensions towhich M/superstring theory over AdS(d + 1)×Sp is dual to is asingleton or doubleton field theory. One can work either in a manifestlyunitary compact `particle' basis (Wigner picture)or manifestly covariant non-compact `coherent state' basis (Diracpicture)of the ULWRs of conformal (super)groups. These coherent states arelabelled by spacetime coordinates and correspond to covariant fields withdefinite conformal dimensions. On the other hand, the supercoherentstates of the ULWRs of superconformal algebras correspond tosuperfields. These results extend to higher-dimensional generalizedspacetimes (superspaces) defined by Jordan (super)algebras and Jordan(super)triple systems. In particular, they extend to the ULWRs of theM-theory symmetry superalgebra OSp(1/32,R).

Sphalerons, merons, and unstable branes in AdS space

We construct unstable classical solutions of Yang-Mills theories and their dual unstable states of type IIB on anti--de Sitter $({\mathrm{AdS}}_{5})$ space. An example is the unstable D0-brane of type IIB located at the center of AdS space. This has a field theory dual which is a sphaleron in gauge theories on ${S}^{3}\ifmmode\times\else\texttimes\fi{}R.$ We argue that the two are dual because both are sphalerons associated with the topology of the instanton or D-instanton. This agreement provides a non-supersymmetric test of the AdS-conformal field theory duality. As an illustration, many aspects of Sen's hypothesis regarding the unstable branes can be seen easily in the weakly coupled dual field theory description. In Euclidean AdS space the D0-branes are dual to gauge theory merons. This implies that the two ends of a D0-brane world-line carry half the charge of a D-instanton. Other examples involve unstable strings in the Coulomb phase.

Remarks on the quantum modes of the scalar field onAdSd+1spacetime

The problem of the quantum modes of the scalar free field on anti--de Sitter (AdS) backgrounds with an arbitrary number of space dimensions is considered. It is shown that this problem can be solved by using the same quantum numbers as those of the nonrelativistic oscillator and two parameters which give the energy quanta and, respectively, the ground-state energy. This last one is known to be just the conformal dimension of the boundary field theory of the AdS-conformal field theory (CFT) conjecture. The obtained normalized energy eigenfunctions represent a new result which corrects that of Burgess and L\"utken.

Scattering in anti–de Sitter space and operator product expansion

We develop a formalism to evaluate generic scalar exchange diagrams in anti--de Sitter $(\mathrm{AdS}{)}_{d+1}$ relevant for the calculation of four-point functions in AdS conformal field theory (CFT) correspondence. The result may be written as an infinite power series of functions of cross ratios. Logarithmic singularities appear in all orders whenever the dimensions of involved operators satisfy certain relations. We show that the ${\mathrm{AdS}}_{d+1}$ amplitude can be written in a form recognizable as the conformal partial wave expansion of a four-point function in ${\mathrm{CFT}}_{d}$ and identify the spectrum of intermediate operators. We find that, in addition to the contribution of the scalar operator associated with the exchanged field in the ${\mathrm{AdS}}_{d+1}$ diagram, there are also contributions of some other operators which may possibly be identified with two-particle bound states in ${\mathrm{AdS}}_{d+1}.$ The ${\mathrm{CFT}}_{d}$ interpretation also provides a useful way to ``regularize'' the logarithms appearing in ${\mathrm{AdS}}_{d+1}$ amplitude.

AdS-CFT correspondence and the information paradox

The information paradox in the quantum evolution of black holes is studied within the framework of the anti--de Sitter--conformal field theory (AdS-CFT) correspondence. The unitarity of the CFT strongly suggests that all information about an initial state that forms a black hole is returned in the Hawking radiation. The CFT dynamics implies an information retention time of the order of the black-hole lifetime. This fact determines many qualitative properties of the nonlocal effects that must show up in a semiclassical effective theory in the bulk. We argue that no violations of causality are apparent to local observers, but the semiclassical theory in the bulk duplicates degrees of freedom inside and outside the event horizon. Nonlocal quantum effects are required to eliminate this redundancy. This leads to a breakdown of the usual classical-quantum correspondence principle in Lorentzian black-hole spacetimes.

Black hole absorption cross-sections and the anti-de Sitter–conformal field theory correspondence

Recent work has uncovered a correspondence between theories in anti-de Sitter space, and those on its boundary. This has important implications for black holes in string theory which have near-horizon AdS geometries. Using the effective coupling to the boundary conformal field theory, I compute the low-energy, s-wave absorption cross-sections for a minimally coupled scalar in the near-extremal four- and five-dimensional black holes. The results agree precisely with semi-classical gravity calculations. Agreement for fixed scalars, and for the BTZ black hole, is also found.