Holography as a highly efficient RG flow II: An explicit construction

TL;DR

This paper develops an explicit construction of a highly efficient holographic RG flow that encodes UV data and reproduces Einstein's equations, providing a new framework for non-perturbative QCD-like theories.

Contribution

It introduces a novel coarse-graining method defining operators without elementary fields, generalizes Wilsonian RG, and explicitly connects the flow to classical gravity solutions.

Findings

Reproduces Einstein's equations from the RG flow

Explicitly computes the beta function in a simple example

Shows the flow's endpoint corresponds to non-relativistic equations like Navier-Stokes

Abstract

We complete the reformulation of the holographic correspondence as a \emph{highly efficient 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 generalised by promoting the cut-off 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 criterion is that the endpoint of the scale-evolution of the RG flow can be transformed to a fixed point corresponding to familiar non-relativistic equations with a finite number of parameters, such as incompressible non-relativistic 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. We obtain the explicit coarse-graining which reproduces Einstein's equations. In a simple example, we are also able to 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 non-perturbative frameworks for QCD-like theories.