Holography as a highly efficient renormalization group flow. II. An explicit construction

Abstract

We complete the reformulation of the holographic correspondence as a 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 required 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 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 non-perturbative frameworks for QCD-like theories.

Semi-holography for heavy ion collisions: self-consistency and first numerical tests

“…In the future, we may also hope to determine the (so far free) hard-soft coupling constants α, β, and γ in terms of α s (Q s ) to a certain degree of approximation, by using renormalisation group techniques as discussed in [23,25]. In this case, the semi-holographic model could eventually lead to sharp predictions.…”

“…Going along the radial coordinate both the sources (including the background metric) and the single trace operators of the CFT evolve in the dual geometries. It was shown in [23,25] that it can be recast in a precise form of RG flow in which the Ward identities of single trace operators giving local conservation laws preserve their form along the scale evolution, so that the sources become functionals of the ultraviolet degrees of freedom. Although this RG flow occurs in a fixed background metric, the sources emerge as scale-dependent effective dynamical objects whose evolution is described by a dual diffeomorphism-invariant classical gravity theory in one higher dimension in the large N limit.…”

“…Following [22], in this section we construct an action where the description of a semi-hard gluon (UV) sector and that of a strongly-interacting soft gluon (IR) sector can be coupled self-consistently. In this paper, we will treat the semi-holographic model phenomenologically including minimalistic hard-soft couplings only, without attempting a derivation from first principles; a rigorous framework for a possible derivation has been recently proposed in [23]. The assumptions that provide the basis of the semi-holographic model for heavy-ion collisions in [22] are as follows:…”

Understanding the holographic principle via RG flow

“…As an illustration, let us see how we construct highly efficient RG flows in the hydrodynamic limit. 11 Once again, let us revert back to the sector of states where t µ ν (Λ) is the only single-trace operator with a non-vanishing expectation value for the sake of simplicity. The expectation value of t µ ν (Λ) is parametrised by the (collective) hydrodynamic variables u µ (Λ) and T (Λ) which thus define the quantum operator.…”

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

“…Above we have used the CFT Ward identities ∂ µ t µ ν ∞ = 0 and Tr t ∞ = 0 to constrain (4) up to redundancies. In the second part of our work [8], we will also extend our construction partly by including another scalar single-trace operator.…”