Institute of Theoretical Physics

Einstein-Hilbert (EH) action can be separated into a bulk and a surface term, with a specific (''holographic'') relationship between the two, so that either can be used to extract information about the other. The surface term can also be interpreted as the entropy of the horizon in a wide class of spacetimes. Since EH action is likely to just the first term in the derivative expansion of an effective theory, it is interesting to ask whether these features continue to hold for more general gravitational actions. We provide a comprehensive analysis of Lagrangians of the form ÿg p L ÿg p Q a bcd R a bcd , in which Q a bcd is a tensor with the symmetries of the curvature tensor, made from metric and curvature tensor and satisfies the condition r c Q a bcd 0, and show that they share these features. The Lanczos-Lovelock Lagrangians are a subset of these in which Q a bcd is a homogeneous function of the curvature tensor. They are all holographic, in a specific sense of the term, and-in all these cases-the surface term can be interpreted as the horizon entropy. The thermodynamics route to gravity, in which the field equations are interpreted as TdS dE pdV, seems to have a greater degree of validity than the field equations of Einstein gravity itself. The results suggest that the holographic feature of EH action could also serve as a new symmetry principle in constraining the semiclassical corrections to Einstein gravity. The implications are discussed.

The AdS/CFT correspondence defines a sector with universal strongly coupled dynamics in the field theory as the dual of pure gravity in AdS described by Einstein's equation with a negative cosmological constant. We explain here, from the field-theoretic viewpoint how the dynamics in this sector gets determined by the expectation value of the energy-momentum tensor alone. We first show that the Boltzmann equation has very special solutions which could be functionally completely determined in terms of the energymomentum tensor alone. We call these solutions conservative solutions. We indicate why conservative solutions should also exist when we refine this kinetic description to go closer to the exact microscopic theory or even move away from the regime of weak coupling so that no kinetic description could be employed. We argue that these conservative solutions form the universal sector dual to pure gravity at strong coupling and large N. Based on this observation, we propose a regularity condition on the energy-momentum tensor so that the dual solution in pure gravity has a smooth future horizon. We also study if irreversibility emerges only at long time scales of observation, unlike the case of the Boltzmann equation.

We study asymptotically slowly varying perturbations of the AdS black brane in Einstein's gravity with a negative cosmological constant. We allow both the induced metric and the Brown-York stress tensor at a given radial cut-off slice to fluctuate. These fluctuations, which determine the radial evolution of the metric, are parametrized in terms of boundary data. We observe that the renormalized energy-momentum tensor at any radial slice takes the standard hydrodynamic form which is relativistically covariant with respect to the induced metric. The RG flow of the fluid takes the form of field redefinitions of the boundary hydrodynamic variables. To show this, up to first order in the derivative expansion, we only need to investigate the radial flow of the boundary data and do not need to impose constraints on them. Imposing the constraints gives unforced nonlinear hydrodynamic equations at any radial slice. Along the way we make a careful study of the choice of counter-terms and hypersurfaces involved in defining the holographic RG flow, while at the same time we do not explicitly set any boundary condition either at the cut-off or at the horizon. We find that η/s does not change along the RG flow, equaling 1/(4π) when the future horizon is regular. We also analyze the flow of the speed of sound and find that it diverges at the horizon.

We show that holographic RG flow can be defined precisely such that it corresponds to emergence of spacetime. We consider the case of pure Einstein's gravity with a negative cosmological constant in the dual hydrodynamic regime. The holographic RG flow is a system of first order differential equations for radial evolution of the energy-momentum tensor and the variables which parametrize it's phenomenological form on hypersurfaces in a foliation. The RG flow can be constructed without explicit knowledge of the bulk metric provided the hypersurface foliation is of a special kind. The bulk metric can be reconstructed once the RG flow equations are solved. We show that the full spacetime can be determined from the RG flow by requiring that the horizon fluid is a fixed point in a certain scaling limit leading to the non-relativistic incompressible Navier-Stokes dynamics. This restricts the nearhorizon forms of all transport coefficients, which are thus determined independently of their asymptotic values and the RG flow can be solved uniquely. We are therefore able to recover the known boundary values of almost all transport coefficients at the first and second orders in the derivative expansion. We conjecture that the complete characterisation of the general holographic RG flow, including the choice of counterterms, might be determined from the hydrodynamic regime.

We investigate background metrics for 2 + 1-dimensional holographic theories where the equilibrium solution behaves as a perfect fluid, and admits thus a thermodynamic description. We introduce stationary perfect-Cotton geometries, where the Cotton-York tensor takes the form of the energy-momentum tensor of a perfect fluid, i.e. they are of Petrov type D t . Fluids in equilibrium in such boundary geometries have non-trivial vorticity. The corresponding bulk can be exactly reconstructed to obtain 3 + 1-dimensional stationary black-hole solutions with no naked singularities for appropriate values of the black-hole mass. It follows that an infinite number of transport coefficients vanish for holographic fluids. Our results imply an intimate relationship between black-hole uniqueness and holographic perfect equilibrium. They also point towards a Cotton/energy-momentum tensor duality constraining the fluid vorticity, as an intriguing boundary manifestation of the bulk mass/nut duality.

We develop holographic prescriptions for obtaining spectral functions in non-equilibrium states and space-time dependent non-equilibrium shifts in the energy and spin of quasi-particle like excitations. We reproduce strongly coupled versions of aspects of non-equilibrium dynamics of Fermi surfaces in Landau's Fermi-liquid theory. We find that the incoming wave boundary condition at the horizon does not suffice to obtain a well-defined perturbative expansion for non-equilibrium observables. Our prescription, based on analysis of regularity at the horizon, allows such a perturbative expansion to be achieved nevertheless and can be precisely formulated in a universal manner independent of the non-equilibrium state, provided the state thermalizes. We also find that the non-equilibrium spectral function furnishes information about the relaxation modes of the system. Along the way, we argue that in a typical non-supersymmetric theory with a gravity dual, there may exist a window of temperature and chemical potential at large N , in which a generic non-equilibrium state can be characterized by just a finitely few operators with low scaling dimensions, even far away from the hydrodynamic limit. Acknowledgments 37 Appendices 38 A. Eddington-Finkelstein vs Schwarzchild coordinates 38 B. The general solution for the non-equilibrium profile of the scalar field 39 C. Vielbeins and spin connections in the hydrodynamically perturbed black-brane metric 39 D. The generalized effective action 41 References 42 I. INTRODUCTION AND AN OUTLINE OF RESULTS

It is known that the solutions of pure classical 5D gravity with AdS 5 asymptotics can describe strongly coupled large N dynamics in a universal sector of 4D conformal gauge theories. We show that when the boundary metric is flat we can uniquely specify the solution by the boundary stress tensor. We also show that in the Fefferman-Graham coordinates all these solutions have an integer Taylor series expansion in the radial coordinate (i.e. no log terms). Specifying an arbitrary stress tensor can lead to two types of pathologies, it can either destroy the asymptotic AdS boundary condition or it can produce naked singularities. We show that when solutions have no net angular momentum, all hydrodynamic stress tensors preserve the asymptotic AdS boundary condition, though they may produce naked singularities. We construct solutions corresponding to arbitrary hydrodynamic stress tensors in Fefferman-Graham coordinates using a derivative expansion. In contrast to Eddington-Finkelstein coordinates here the constraint equations simplify and at each order it is manifestly Lorentz covariant. The regularity analysis, becomes more elaborate, but we can show that there is a unique hydrodynamic stress tensor which gives us solutions free of naked singularities. In the process we write down explicit first order solutions in both Fefferman-Graham and Eddington-Finkelstein coordinates for hydrodynamic stress tensors with arbitrary η/s. Our solutions can describe arbitrary (slowly varying) velocity configurations. We point out some fieldtheoretic implications of our general results.

We demonstrate that the Navier-Stokes equation can be covariantized under the full infinite dimensional Galilean Conformal Algebra (GCA), such that it reduces to the usual Navier-Stokes equation in an inertial frame. The covariantization is possible only for incompressible flows, i.e when the divergence of the velocity field vanishes. Using the continuity equation, we can fix the transformation of pressure and density under GCA uniquely. We also find that when all chemical potentials vanish, c s , which denotes the speed of sound in an inertial frame comoving with the flow, must either be a fundamental constant or given in terms of microscopic parameters. We will discuss how both could be possible. In absence of chemical potentials, we also find that the covariance under GCA implies that either the viscosity should vanish or the microscopic theory should have a length scale or a time scale or both. We further find that the higher derivative corrections to the Navier-Stokes equation, can be covariantized, only if they are restricted to certain possible combinations in the inertial frame. We explicitly evaluate all possible three derivative corrections. Finally, we argue that our analysis hints that the parent relativistic theory with relativistic conformal symmetry needs to be deformed before the contraction is taken to produce a sensible GCA invariant dynamical limit.

We continue to investigate planar four point worldsheet correlators of string theories which are conjectured to be duals of free gauge theories. We focus on the extremal correlators Tr(Z J 1 (x))Tr(Z J 2 (y))Tr(Z J 3 (z))Tr(Z J (0))) of N = 4 SYM theory, and construct the corresponding worldsheet correlators in the limit when the J i ≫ 1. The worldsheet correlator gets contributions, in this limit, from a whole family of Feynman graphs. We find that it is supported on a curve in the moduli space parametrised by the worldsheet crossratio. In a further limit of the spacetime correlators we find this curve to be the unit circle. In this case, we also check that the entire worldsheet correlator displays the appropriate crossing symmetry. The nonrenormalization of the extremal correlators in the 't Hooft coupling offers a potential window for a comparison of these results with those from strong coupling. * On lien from Harish-Chandra Research Institute, Allahabad. 2 1 3 4 1 2 3 4

We investigate how the holographic correspondence can be reformulated as a generalisation of Wilsonian RG flow in a strongly interacting large N quantum field theory. We firstly define a highly efficient RG flow as one in which the Ward identities related to local conservation of energy, momentum and charges preserve the same form at each scale -to achieve this it is necessary to redefine the background metric and external sources at each scale as functionals of the effective single trace operators. These redefinitions also absorb the contributions of the multi-trace operators to these effective Ward identities. Thus the background metric and external sources become effectively dynamical reproducing the dual classical gravity equations in one higher dimension. Here, we focus on reconstructing the pure gravity sector as a highly efficient RG flow of the energy-momentum tensor operator, leaving the explicit constructive field theory approach for generating such RG flows to the second part of the work. We show that special symmetries of the highly efficient RG flows carry information through which we can decode the gauge fixing of bulk diffeomorphisms in the corresponding gravity equations. We also show that the highly efficient RG flow which reproduces a given classical gravity theory in a given gauge is unique provided the endpoint can be transformed to a non-relativistic fixed point with a finite number of parameters under a universal rescaling.

We demonstrate that the Navier-Stokes equation can be covariantized under the full infinite dimensional Galilean Conformal Algebra (GCA), such that it reduces to the usual Navier-Stokes equation in an inertial frame. The covariantization is possible only for incompressible flows, i.e when the divergence of the velocity field vanishes. Using the continuity equation, we can fix the transformation of pressure and density under GCA uniquely. We also find that when all chemical potentials vanish, $c_{s}$, which denotes the speed of sound in an inertial frame comoving with the flow, must either be a fundamental constant or given in terms of microscopic parameters. We will discuss how both could be possible. In absence of chemical potentials, we also find that the covariance under GCA implies that either the viscosity should vanish or the microscopic theory should have a length scale or a time scale or both. We further find that the higher derivative corrections to the Navier-Stokes equation, can be covariantized, only if they are restricted to certain possible combinations in the inertial frame. We explicitly evaluate all possible three derivative corrections. Finally, we argue that our analysis hints that the parent relativistic theory with relativistic conformal symmetry needs to be deformed before the contraction is taken to produce a sensible GCA invariant dynamical limit.

The AdS/CFT correspondence defines a sector with universal strongly coupled dynamics in the field theory as the dual of pure gravity in AdS described by Einstein's equation with a negative cosmological constant. We explain here, from the field-theoretic viewpoint how the dynamics in this sector gets determined by the expectation value of the energy-momentum tensor \emph{alone}. We first show that the Boltzmann equation has very special solutions which could be \textit{functionally} completely determined in terms of the energy-momentum tensor alone. We call these solutions \textit{conservative solutions}. We indicate why conservative solutions should also exist when we refine this kinetic description to go closer to the exact microscopic theory or even move away from the regime of weak coupling so that no kinetic description could be employed. We argue that these \textit{conservative solutions} form the universal sector dual to pure gravity at strong coupling and large $N$. Based on this observation, we propose a \textit{regularity condition} on the energy-momentum tensor so that the dual solution in pure gravity has a smooth future horizon. We also study if irreversibility emerges only at long time scales of observation, unlike the case of the Boltzmann equation.

We extend our previous study of a toy model for coupling classical Yang-Mills equations for describing overoccupied gluons at the saturation scale with a strongly coupled infrared sector modeled by AdS/CFT. Including propagating modes in the bulk we find that the Yang-Mills sector loses its initial energy to a growing black hole in the gravity dual such that there is a conserved energy-momentum tensor for the total system while entropy grows monotonically. This involves a numerical AdS simulation with a backreacted boundary source far from equilibrium.

Semiholographic models of non-Fermi liquids have been shown to have generically stable generalized quasiparticles on the Fermi surface. Although these excitations are broad and exhibit particle-hole asymmetry, they were argued to be stable from interactions at the Fermi surface. In this work, we use this observation to compute the density response and collective behaviour in these systems. Compared to the Fermi liquid case, we find that the boundaries of the particle-hole continuum are blurred by incoherent contributions. However, there is a region inside this continuum, that we call inner core, within which salient features of the Fermi liquid case are preserved. A particularly striking prediction of our work is that these systems support a plasmonic collective excitation which is well-defined at large momenta, has an approximately linear dispersion relation and is located in the low-energy tail of the particle-hole continuum. Furthermore, the dynamic screening potential shows deep attractive regions as a function of the distance at higher frequencies which might lead to long-lived pair formation depending on the behavior of the pair susceptibility. We also find that Friedel oscillations are present in these systems but are highly suppressed.

The nonequilibrium evolution in a boost-invariant Bjorken flow of a hybrid viscous fluid model containing two interacting components with different viscosities, such that they represent strongly and weakly self-coupled sectors, is shown to be characterized by a hydrodynamic attractor which has an early-time behavior that is reminiscent of the so-called bottom-up thermalization scenario in heavy-ion collisions. The hydrodynamization times for the two sectors can differ strongly, with details depending on the curve realized on the two-dimensional attractor surface, which might account for different scenarios for small and large systems in nuclear collisions. The total system behaves as a single viscous fluid with a dynamically determined effective shear viscosity.

We study the quasinormal modes and non-linear dynamics of a simplified model of semi-holography, which consistently integrates mutually interacting perturbative and strongly coupled holographic degrees of freedom such that the full system has a total conserved energy. We show that the thermalization of the full system can be parametrically slow when the mutual coupling is weak. For typical homogeneous initial states, we find that initially energy is transferred from the black brane to the perturbative sector, later giving way to complete transfer of energy to the black brane at a slow and constant rate, while the entropy grows monotonically for all time. Larger mutual coupling between the two sectors leads to larger extraction of energy from the black brane by the boundary perturbative system, but also quicker irreversible transfer of energy back to the black brane. The quasinormal modes replicate features of a dissipative system with a softly broken symmetry including the so-called...