MEDIOS NO GUIADOS ]

Overview

• The general gauge/gravity duality is intimately connected to the RG flow of field theo- ries. In fact the characteristic RG length/energy scale gives rise to an emergent dimen- sionÝÑRG flow is geometrized.

• Strongly correlated effects at finite temperature and density are geometrized by certain IR geometries that correspond to deformations by relevant scaling fieldsÝÑglobal RG flow of strongly coupled theories.

Now let us switch gears and motivate the gauge/gravity duality from a very different angle and reconstruct the basic ingredients of the dictionary in a bottom-up approach, i.e. without explicit use of string theory or supersymmetry. As already mentioned at the end of the previous section, the original discovery of holography and in particular the AdS/CFT correspondence is build on supersymmetric string theories and requires conformal invariance, the more general holographic dualities allow for relaxation of these constraints. Geared towards condensed matter applications, the renormalization flow viewpoint developed in Section2.3.5on holography has been proven very powerful in understanding strongly correlated systems at finite temperature and density. In a wider sense, the RG flow viewpoint of holography resembles the theme of emergent phenomena found in various intricate many-body systems.

3.4.1. Emergent holography

Let us start with a generic effective action (2.116) generated by a decimation process as ex- plained in step

ii

respecting the symmetry of the system

Seff“ ż

ddx gpxqa

_O

UV IR u R1,d´1 AdSd`1 lattice spacinga lattice spacingca

Figure 3.12. The RG flow prescription outlined in Section2.3.5as a local operation of an en- ergy scale ucan be geometrized by including the energy scale as an additional direction. Under each RG step the degrees of freedom are reduced by an averag- ing process. Keeping the “space” in the averaging process fixed corresponds to a rescaling of the characteristic length scale. Thus, AdSd`1 can be understood as an emergent space where each radial slice can be viewed as a step in the coarse- graining of the degrees of freedom. The running couplings in the UV are then identified according to the holographic dictionary with the boundary value of a field in AdS space. The RG flow of the boundary quantum field theory to the IR is determined by the equations of motion of the gravitational dual.

Note that we are looking at a statistical theory which can be mapped onto a Euclidean field theory, where we made the discretization of the effective action explicit which is encoded in the spatial positionsxof the operators on a lattice. Generically, the couplings can vary between each lattice site, i.e. ga _{“ g}a_{pxq. Iterating the RG steps}

_i

_iii

spacing where the degrees of freedom on the larger lattice represent a well-defined average of the degrees of freedom of the original lattice. The couplings are adjusted in such a way to preserve the physical content of the low-energy excitations above the ground state. Therefore, we can view the couplings as functions that depend on the lattice/spatial position and on the characteristic length scale uprobing the system ga_{px, uq. The RG flow, as combined operation}

of infinite many infinitesimal RG steps, of the couplings ga_{px, uq is encoded in the β-function}

(2.120) or (2.143) which is local in the characteristic length scale, or alternatively, in the energy scaleµ_„u´1 dga_{px, uq} du “Rpg a_{px, uq, uq, µ}Bgapx, µq Bµ “βpg a_{px, µq, µq, (3.87)}

Thus, from the locality of the RG flow emerges another dimension, so we can view the couplings as fluctuating fields ind`1dimensional space described by the originalddimensional spatial directionxand an additional direction, the characteristic length or energy scale of the RG flow

u. In this sense the RG flow is geometrized, whereuÑ0denotes the flow to the UV anduÑ 8

the flow to the IR. The geometry of the system must encode the scaling transformation (2.115)

x ÝÑc´1x. Heuristically, the decimation process is then carried over to a “shrinking” process of the (lattice) space. Pictorially, this is shown in Figure 3.12. The nature of the underlying geometry can be uncovered by taking the Wilsonian view of the RG method. Here we start with

all known scale invariant theories describing the fixed points in the global RG flow diagram (see Figure2.4) and try to connect them by global RG flows as outline in the Wilsonian approach