Holography as a highly efficient renormalization group flow. I. Rephrasing gravity

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 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.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.

Understanding the holographic principle via RG flow

“…A precise general statement of the holographic principle is still elusive although we do have a very concrete realisation in the form of AdS/CFT correspondence of string theory in which certain supergravity theories with stringy corrections in antide Sitter (AdS) space have been shown to have dual descriptions given by specific types of conformal Yang-Mills theories without gravity living at the boundary. [5][6][7] Recently, a specific approach towards understanding of the holographic principle (specially [8][9][10][11] ) has been developed which has been directly influenced by the broad philosophy that the classical gravity equations themselves hold the key to unravelling gravity's holographic origin. In the context of AdS/CFT correspondence, which is the most concrete example of holography, this question can be formulated in a precise way.…”

Holographic renormalisation group flows and renormalisation from a Wilsonian perspective

“…(On the other hand, we keep hats explicit whenever they appear; for instance, φ and Π refer to the the original definitions of the 5D fields and momenta as functions of the dimensionful coordinates.) The solution to (3.11), analogous to (3.4), reads in path-integral notation 15) whereφ l (x) = ϕ(x/l) and l 0 is a dummy length introduced for dimensional reasons, due to our choice of 5D fields. The boundary condition imposed by the redefined S B (g) reads…”

“…15 In this case the relevant directions are given by the basic perturbations T b , and can be associated to the scalar fields φ b with negative squared mass (satisfying the Breitenlohner-Freedman bound [34,35]), which we will call relevant fields. The remaining relevant direction is the constant term in S B .…”

“…level through this framework (in the line, for instance, of [10][11][12][13][14][15]). A number of attempts have been made to formulate holographic the RG in Wilsonian terms, but making this map precise has proved challenging.…”

“…The remaining relevant direction is the constant term in S B . The bare manifold B in holographic renormalisation is defined by singular boundary actions that impose Dirichlet boundary 15 In [33], this method is generalised to include the alternate quantisation and multi-trace deformations.…”