The ultraviolet limit and sum rule for the shear correlator in hot Yang-Mills theory

We determine a next-to-leading order result for the correlator of the shear stress operator in high-temperature Yang-Mills theory. The computation is performed via an ultraviolet expansion, valid in the limit of small distances or large momenta, and the result is used for writing operator product expansions for the Euclidean momentum and coordinate space correlators as well as for the Minkowskian spectral density. In addition, our results enable us to confirm and refine a shear sum rule originally derived by Romatschke, Son and Meyer.Comment: 16 pages, 2 figures. v2: small clarifications, one reference added, published versio

Shear Modes, Criticality and Extremal Black Holes

We consider a (2+1)-dimensional field theory, assumed to be holographically dual to the extremal Reissner-Nordstrom AdS(4) black hole background, and calculate the retarded correlators of charge (vector) current and energy-momentum (tensor) operators at finite momentum and frequency. We show that, similar to what was observed previously for the correlators of scalar and spinor operators, these correlators exhibit emergent scaling behavior at low frequency. We numerically compute the electromagnetic and gravitational quasinormal frequencies (in the shear channel) of the extremal Reissner-Nordstrom AdS(4) black hole corresponding to the spectrum of poles in the retarded correlators. The picture that emerges is quite simple: there is a branch cut along the negative imaginary frequency axis, and a series of isolated poles corresponding to damped excitations. All of these poles are always in the lower half complex frequency plane, indicating stability. We show that this analytic structure can be understood as the proper limit of finite temperature results as T is taken to zero holding the chemical potential fixed.Comment: 28 pages, 7 figures, added reference

Wilsonian Approach to Fluid/Gravity Duality

The problem of gravitational fluctuations confined inside a finite cutoff at radius $r=r_c$ outside the horizon in a general class of black hole geometries is considered. Consistent boundary conditions at both the cutoff surface and the horizon are found and the resulting modes analyzed. For general cutoff $r_c$ the dispersion relation is shown at long wavelengths to be that of a linearized Navier-Stokes fluid living on the cutoff surface. A cutoff-dependent line-integral formula for the diffusion constant $D(r_c)$ is derived. The dependence on $r_c$ is interpreted as renormalization group (RG) flow in the fluid. Taking the cutoff to infinity in an asymptotically AdS context, the formula for $D(\infty)$ reproduces as a special case well-known results derived using AdS/CFT. Taking the cutoff to the horizon, the effective speed of sound goes to infinity, the fluid becomes incompressible and the Navier-Stokes dispersion relation becomes exact. The resulting universal formula for the diffusion constant $D(horizon)$ reproduces old results from the membrane paradigm. Hence the old membrane paradigm results and new AdS/CFT results are related by RG flow. RG flow-invariance of the viscosity to entropy ratio $\eta /s$ is shown to follow from the first law of thermodynamics together with isentropy of radial evolution in classical gravity. The ratio is expected to run when quantum gravitational corrections are included.Comment: 34 pages, harvmac, clarified boundary conditio

Higher spin fermions in the BTZ black hole

Recently it has been shown that the wave equations of bosonic higher spin fields in the BTZ background can be solved exactly. In this work we extend this analysis to fermionic higher spin fields. We solve the wave equations for arbitrary half-integer spin fields in the BTZ black hole background and obtain exact expressions for their quasinormal modes. These quasinormal modes are shown to agree precisely with the poles of the corresponding two point function in the dual conformal field theory as predicted by the AdS/CFT correspondence. We also obtain an expression for the 1-loop determinant in terms of the quasinormal modes and show it agrees with that obtained by integrating the heat kernel found by group theoretic methods.Comment: 29 page

Mixed RG Flows and Hydrodynamics at Finite Holographic Screen

We consider quark-gluon plasma with chemical potential and study renormalization group flows of transport coefficients in the framework of gauge/gravity duality. We first study them using the flow equations and compare the results with hydrodynamic results by calculating the Green functions on the arbitrary slice. Two results match exactly. Transport coefficients at arbitrary scale is ontained by calculating hydrodynamics Green functions. When either momentum or charge vanishes, transport coefficients decouple from each other.Comment: 22 pages, 6 figure

Hydrodynamics of fundamental matter

First and second order transport coefficients are calculated for the strongly coupled N=4 SYM plasma coupled to massless fundamental matter in the Veneziano limit. The results, including among others the value of the bulk viscosity and some relaxation times, are presented at next-to-leading order in the flavor contribution. The bulk viscosity is found to saturate Buchel's bound. This result is also captured by an effective single-scalar five-dimensional holographic dual in the Chamblin-Reall class and it is suggested to hold, in the limit of small deformations, for generic plasmas with gravity duals, whenever the leading conformality breaking effects are driven by marginally (ir)relevant operators. This proposal is then extended to other relations for hydrodynamic coefficients, which are conjectured to be universal for every non-conformal plasma with a dual Chamblin-Reall-like description. Our analysis extends to any strongly coupled gauge theory describing the low energy dynamics of Nc>>1 D3-branes at the tip of a generic Calabi-Yau cone. The fundamental fields are added by means of 1<<Nf<<Nc homogeneously smeared D7-branes.Comment: 24 pages. V2: Important improvements in the discussion of the results in section 1. References adde

Shear sum rules at finite chemical potential

We derive sum rules which constrain the spectral density corresponding to the retarded propagator of the T_{xy} component of the stress tensor for three gravitational duals. The shear sum rule is obtained for the gravitational dual of the N=4 Yang-Mills, theory of the M2-branes and M5-branes all at finite chemical potential. We show that at finite chemical potential there are additional terms in the sum rule which involve the chemical potential. These modifications are shown to be due to the presence of scalars in the operator product expansion of the stress tensor which have non-trivial vacuum expectation values at finite chemical potential.Comment: The proof for the absence of branch cuts is corrected.Results unchange

Holographic GB gravity in arbitrary dimensions

We study the properties of the holographic CFT dual to Gauss-Bonnet gravity in general $D \ge 5$ dimensions. We establish the AdS/CFT dictionary and in particular relate the couplings of the gravitational theory to the universal couplings arising in correlators of the stress tensor of the dual CFT. This allows us to examine constraints on the gravitational couplings by demanding consistency of the CFT. In particular, one can demand positive energy fluxes in scattering processes or the causal propagation of fluctuations. We also examine the holographic hydrodynamics, commenting on the shear viscosity as well as the relaxation time. The latter allows us to consider causality constraints arising from the second-order truncated theory of hydrodynamics.Comment: 48 pages, 9 figures. v2: New discussion on free fields in subsection 3.3 and new appendix B on conformal tensor fields. Added comments on the relation between the central charge appearing in the two-point function and the "central charge" characterizing the entropy density in the discussion. References adde

Hydrodynamics of R-charged D1-branes

We study the hydrodynamic properties of strongly coupled $SU(N)$ Yang-Mills theory of the D1-brane at finite temperature and at a non-zero density of R-charge in the framework of gauge/gravity duality. The gravity dual description involves a charged black hole solution of an Einstein-Maxwell-dilaton system in 3 dimensions which is obtained by a consistent truncation of the spinning D1-brane in 10 dimensions. We evaluate thermal and electrical conductivity as well as the bulk viscosity as a function of the chemical potential conjugate to the R-charges of the D1-brane. We show that the ratio of bulk viscosity to entropy density is independent of the chemical potential and is equal to $1/4\pi$. The thermal conductivity and bulk viscosity obey a relationship similar to the Wiedemann-Franz law. We show that at the boundary of thermodynamic stability, the charge diffusion mode becomes unstable and the transport coefficients exhibit critical behaviour. Our method for evaluating the transport coefficients relies on expressing the second order differential equations in terms of a first order equation which dictates the radial evolution of the transport coefficient. The radial evolution equations can be solved exactly for the transport coefficients of our interest. We observe that transport coefficients of the D1-brane theory are related to that of the M2-brane by an overall proportionality constant which sets the dimensions.Comment: 57 pages, 12 figure

Causality and the AdS Dirichlet problem

The (planar) AdS Dirichlet problem has previously been shown to exhibit superluminal hydrodynamic sound modes. This problem is defined by bulk gravitational dynamics with Dirichlet boundary conditions imposed on a rigid timelike cut-off surface. We undertake a careful examination of this set-up and argue that, in most cases, the propagation of information between points on the Dirichlet hypersurface is nevertheless causal with respect to the induced light cones. In particular, the high-frequency dynamics is causal in this sense. There are however two exceptions and both involve boundary gravitons whose propagation is not constrained by the Einstein equations. These occur in i) AdS$_3$, where the boundary gravitons generally do not respect the induced light cones on the boundary, and ii) Rindler space, where they are related to the infinite speed of sound in incompressible fluids. We discuss implications for the fluid/gravity correspondence with rigid Dirichlet boundaries and for the black hole membrane paradigm.Comment: 29 pages, 5 figures. v2: added refs. v3: minor clarification