Integrating out geometry: holographic Wilsonian RG and the membrane paradigm

Abstract

We formulate a holographic Wilsonian renormalization group flow for strongly coupled systems with a gravity dual, motivated by the need to extract efficiently low energy behavior of such systems. Starting with field theories defined on a cut-off surface in a bulk spacetime, we propose that integrating out high energy modes in the field theory should correspond to integrating out a part of the bulk geometry. We describe how to carry out this procedure in practice in the classical gravity approximation using examples of scalar and vector fields. By integrating out bulk degrees of freedom all the way to a black hole horizon, this formulation defines a refined version of the black hole membrane paradigm. Furthermore, it also provides a derivation of the semi-holographic description of low energy physics. Contents

Charge transport by holographic Fermi surfaces

“…To understand why, note that in our gravity treatment the role played by the gapless boson in the above example is instead filled by the AdS 2 region. From a field theoretical point of view, our system can be described by a low energy effective action [14,71,72] in which fermionic excitations É around a free fermion Fermi surface hybridize with those of a strongly coupled sector, which can be considered as the field theory dual of the AdS 2 region and was referred to as a semilocal quantum liquid (SLQL) in [73]. See Fig.…”

“…The system can be described by a low energy effective action where fermionic excitations É around a free fermion Fermi surface hybridize with those in a strongly coupled sector (labeled as SLQL in the figure) described on the gravity side by the AdS 2 region [14,71,72] (see [18] for a more extensive review).…”

Effective actions for anomalous hydrodynamics

“…Vis a vis, the cross-contour term in (7.1) as remarked in section 1 one can give a heuristic argument by tracking the anomaly inflow between the two contours via the Hall insulator. We believe that this picture can be developed further using holographic embedding of anomalous fluid dynamics and in fact one should be able to derive (7.1) using the techniques described in [9,54].…”

“…The hydrodynamic shadow gauge field can roughly be viewed as living on the horizon of the black hole (it secretly is a proxy for the horizon gauge field as we will describe later). The individual terms on the left and right given by the transgression form in (1.1) can be viewed as the effective action obtained by working with the Goldstone mode which is the Wilson line interpolating between the boundary and the horizon as described in [45] (see also [54] for a derivation from a holographic renormalization group perspective). While this individual decoupled left-right construction suffices for non-anomalous pieces of the transport, to correctly incorporate the failure of gauge invariance of the effective due to the anomaly, we need to demand a particular gluing condition across the horizon.…”

Holographic RG flow of the shear viscosity to entropy density ratio in strongly coupled anisotropic plasma

“…This shows that there is conformal anomaly in the anisotropic N = 4 plasma due to the anisotropy. Hence, the Callan-Symanzik RG flow equation for the two-point function, consequently, the RG flow of some components of the shear viscosity tensor η b viscosity η i z i z of the anisotropic N = 4 plasma, using the equation of motion for the shear mode gravitational fluctuations [13], the holographic Wilsonian RG method [23][24][25]28], and Kubo's formula, and give analytical solutions up to first order in the anisotropy parameter a. From the solution of the RG flow equation, we find that at the boundary (UV) η i z i z ( = 0) is equivalent to η z i z i ( = 0) which is consistent with the fact that the one index up and one index down energy-momentum tensor operator at the boundary is symmetric, and the shear viscosity tensor has only two independent components: η j i j i and η z i z i [18].…”