Emergence of Gravity and RG Flow
01 Jan 2017-Vol. 187, pp 283-302
TL;DR: In this paper, Padmanabhan et al. reformulated Einstein's equations in AdS as a non-perturbative RG flow that further leads to a new approach towards constructing strongly interacting QFTs.
Abstract: This is a tribute to Padmanabhan’s works on the holographic principle which have consistently enunciated the profound philosophy that the classical equations of gravity themselves hold the key to understanding their holographic origin. I discuss how this can be realised by reformulating Einstein’s equations in AdS as a non-perturbative RG flow that further leads to a new approach towards constructing strongly interacting QFTs. For a concrete demonstration, I focus on the hydrodynamic limit in which case this RG flow connects the AdS/CFT correspondence with the membrane paradigm.
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TL;DR: In this paper, the authors present a collection of notes based on lectures given at IIT Madras in September 2019 and at IFT Madrid in November 2019 on applied holography and especially the analytic and numerical techniques involved.
Abstract: This is a collection of notes based on lectures given at IIT Madras in September 2019 and at IFT Madrid in November 2019. It is supposed to be a concise (and therefore not comprehensive) and pragmatic course on applied holography and especially the (basic) analytic and numerical techniques involved. The lectures are not focused on the large theoretical and fundamental background which can be found already in several places in the literature, but rather on concrete applications of Bottom-Up AdS-CFT to Hydrodynamics, QCD and Condensed Matter. The idea is to accompany the reader step by step through the various benchmark examples with a classmate attitude, providing details of the computations and open-source numerical codes in Mathematica, and sharing simple tricks and warnings collected during my research experience. At the end of this path, the reader will be in possess of all the fundamental skills and tools to learn by himself/herself more advanced techniques and to produce independent and novel research on the topic.
81 citations
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TL;DR: In this article, it was shown that the Boltzmann equation has very special solutions which could be functionally completely determined in terms of the energy-momentum tensor alone, and that these conservative solutions form the universal sector dual to pure gravity at strong coupling and large N. They also showed that 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.
Abstract: 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 energy-momentum 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.
18 citations
TL;DR: In this paper, the holographic correspondence can be reformulated as a generalisation of the Wilsonian RG flow in a strongly interacting large $N$ quantum field theory, where the background metric and external sources become effectively dynamical reproducing the dual classical gravity equations in one higher dimension.
Abstract: 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 \textit{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 \textit{unique} provided the endpoint can be transformed to a non-relativistic fixed point with a finite number of parameters under a universal rescaling. The results obtained here are used in the second part of this work, where we do an explicit field-theoretic construction of the RG flow, and obtain the dual classical gravity theory.
18 citations
TL;DR: In this article, the authors construct asymptotically AdS solutions of Einstein's gravity dual to such processes at strong coupling, perturbatively in the amplitude expansion, where the expansion parameter is the ratio of the amplitude of the non-hydrodynamic shear-stress tensor to the pressure.
Abstract: Homogeneous relaxation is a ubiquitous phenomenon in semiclassical kinetic theories where the quasiparticles are distributed uniformly in space, and the equilibration involves only their velocity distribution. For such solutions, the hydrodynamic variables remain constant. We construct asymptotically AdS solutions of Einstein's gravity dual to such processes at strong coupling, perturbatively in the amplitude expansion, where the expansion parameter is the ratio of the amplitude of the non-hydrodynamic shear-stress tensor to the pressure. At each order, we sum over all time derivatives through exact recursion relations. We argue that the metric has a regular future horizon, order by order in the amplitude expansion, provided the shear-stress tensor follows an equation of motion. At the linear order, this equation of motion implies that the metric perturbations are composed of zero wavelength quasinormal modes. Our method allows us to calculate the non-linear corrections to this equation perturbatively in the amplitude expansion. We thus derive a special case of our previous conjecture on the regularity condition on the boundary stress tensor that endows the bulk metric with a regular future horizon, and also refine it further. We also propose a new outlook for heavy-ion phenomenology at RHIC and ALICE.
18 citations
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TL;DR: In this paper, a semi-holographic model for heavy-ion collisions is proposed, which includes self-consistent couplings between the Yang-Mills fields of the Color Glass Condensate framework and an infrared AdS/CFT sector, such as to guarantee the existence of a conserved energy-momentum tensor for the combined system.
Abstract: We present an extended version of a recently proposed semi-holographic model for heavy-ion collisions, which includes self-consistent couplings between the Yang-Mills fields of the Color Glass Condensate framework and an infrared AdS/CFT sector, such as to guarantee the existence of a conserved energy-momentum tensor for the combined system that is local in space and time, which we also construct explicitly. Moreover, we include a coupling of the topological charge density in the glasma to the same of the holographic infrared CFT. The semi-holographic approach makes it possible to combine CGC initial conditions and weak-coupling glasma field equations with a simultaneous evolution of a strongly coupled infrared sector describing the soft gluons radiated by hard partons. As a first numerical test of the semi-holographic model we study the dynamics of fluctuating homogeneous color-spin-locked Yang-Mills fields when coupled to a homogeneous and isotropic energy-momentum tensor of the holographic IR-CFT, and we find rapid convergence of the iterative numerical procedure suggested earlier.
11 citations
TL;DR: In this paper, a renormalization group (RG) flow is proposed to reproduce the UV data in the field theory in the strong coupling and large-$N$ limit, which can also be used to define operators at any given scale in terms of appropriate coarse-grained collective variables.
Abstract: We complete the reformulation of the holographic correspondence as a highly efficient renormalization group (RG) flow that can also determine the UV data in the field theory in the strong-coupling and large-$N$ limit. We introduce a special way to define operators at any given scale in terms of appropriate coarse-grained collective variables, without requiring the use of the elementary fields. The Wilsonian construction is generalized by promoting the cutoff to a functional of these collective variables. We impose three criteria to determine the coarse-graining. The first criterion is that the effective Ward identities for local conservation of energy, momentum, etc. should preserve their standard forms, but in new scale-dependent background metric and sources which are functionals of the effective single-trace operators. The second criterion is that the scale-evolution equations of the operators in the actual background metric should be state-independent, implying that the collective variables should not explicitly appear in them. The final required criterion is that the end point of the scale-evolution of the RG flow can be transformed to a fixed point corresponding to familiar nonrelativistic equations with a finite number of parameters, such as incompressible nonrelativistic Navier-Stokes, under a certain universal rescaling of the scale and of the time coordinate. Using previous work, we explicitly show that in the hydrodynamic limit each such highly efficient RG flow reproduces a unique classical gravity theory with precise UV data that satisfy our IR criterion and also lead to regular horizons in the dual geometries. We obtain the explicit coarse-graining which reproduces Einstein's equations. In a simple example, we are also able to construct a low-energy effective action and compute the beta function. Finally, we show how our construction can be interpolated with the traditional Wilsonian RG flow at a suitable scale and can be used to develop new nonperturbative frameworks for QCD-like theories.
8 citations