Gauge/Gravity Duality & Renormalization

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

of the RG prescription on page55which includes a closed set of operatorsO_a respecting the symmetry of the system

Seff“ ż

d^dx gpxq^a

O

apxq. (3.86)

UV

IR u R^1,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. Keepaverag-ing the “space” in the averagaverag-ing process fixed corresponds to a rescaling of the characteristic length scale. Thus, AdS_d`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. g^a “ g^apxq. Iterating the RG steps

i

_to

iii

_{on page 55}increases the lattice 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 g^apx, uq. The RG flow, as combined operation of infinite many infinitesimal RG steps, of the couplings g^apx, 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

dg^apx, uq

du “Rpg^apx, uq, uq, µBg^apx, µq

Bµ “βpg^apx, 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

i

_to

iii

_{on page}57. Scale invariant theories with Poincaré symmetries give rise to conformal theories which are invariant under the conformal groupSOp2, dqinddimensions. Since at the fixed point the couplings remain invariant under a change in the characteristic length scaleu, the geometry of scale invariant fixed point theories must be scale invariant underx ÝÑ c^´1x anduÝÑc^´1u, which is realized by the AdS-space with metric³⁴

ds²“ L² u²

`´dt<sup>2</sup>`dx²`du²˘

. (3.88)

Note that the isometries of the AdS-spacetime ind`1 dimensionsSOp2, dqis exactly the con-formal group inddimensions. The crucial question arises, which fixed point is described by the AdS-spacetime. Clearly, we need to start with the microscopic theory in the UV in order to de-fine the couplingsg^aof the underlying lattice theory in terms of physical microscopic interaction strengthsJ^a. Thus, we rediscover the holographic dictionary entry that the field living in the bulk geometry of the AdS-spacetime must correspond to the microscopic couplings in the UV,i.e.

g^apxqˇˇˇ

UV“J^apxq “ϕ^apx, uqˇˇˇ

u“0“ϕ^apxqˇˇˇ

BpAdSq. (3.89)

Here we can make the relation between the scaling dimension and the nature of the associated operator clear. The coefficientϕd´∆of the leading term in the boundary expansion of the field scales withd´∆under scaling transformations, which is exactly the eigenvalueyJ of a scaling field close to the fixed point (see (2.123)). The scaling dimension of the operator is related to the scaling of the source field byyJ “d´∆_O. The behavior in the vicinity of the fixed point is thus controlled by the mass of the bulk field. According to Table3.3, for a scalar field with mass mthe corresponding scaling field of the operator is relevant when yJ ą0 which corresponds to ∆ ă dand ´^d2{^4L2 ă m² ă 0. For m “ 0 we find ∆ “ d and thus a marginal operator y_J “0, whereas formą0,∆ądandy_J ă0the operator is irrelevant. Similarly, for all other bulk fields listed in Table 3.3we can do the same analysis. Additionally, from (3.89) follows that the bulk fields in AdS-spacetime must carry the same quantum numbers and charges as the corresponding couplings. Since the effective action at an RG fixed point is a conformal field theory we can extend the dictionary by identifying the coupling with the sources of the effective CFT action (2.156)

Ssource“ ż

d^dx`

ϕ^apxq

O

apxq `A<sup>a</sup><sub>µ</sub>J<sub>a</sub><sup>µ</sup>`g_µνT^µν˘

, (3.90)

which in turn correspond to the associated bulk fields via (3.89) with the same structure,i.e. the effective action of the bulk theory includes scalar fieldsϕpx, uqfor each scalar operator

O

, vector (gauge) fieldsAApx, uqfor each currentJ^µand a spin-two fieldgABfor the canonical energy mo-mentum tensorT^µν, arising due toPolyakov’s Theoremon page65. The existence of a spin-two field is the key to the gauge/gravity correspondence. According to the Weinberg-Witten theo-rem, and precursors [91,199], a Lorentz invariant spin-two field theory describes a topological theory which would not affect the couplings/sources of the QFT side or couple universally due to the equivalence principle which effectively describes gravity. Thus, the AdS-spacetime arises

34Strictly speaking we work in Euclidean signature which corresponds to a statistical system, but we like to replace the computation of physical quantities such as thermodynamic and transport properties by a gravitational computation with the gravity dual. As we will shortly see, we can extend the holographic dictionary to include calculations involving correlation functions by real-time calculations as is made explicit in the holographic fluctuation-dissipation theorem (3.102) discussed in Section3.5.1.

from a gravitational theory described by classical gravity with negative cosmological constant Λ“^dpd´1q{^2L2

SAdS“ 1 2κ²

ż

d^d`1x?

´g

´

R´2Λ`L_matterrϕ, AAs¯

. (3.91)

Note that the energy-momentum tensor of the gravitational bulk theory is given by (3.3) T_AB“ ´ 2

?´g δp?

´gL_matterq

δg^AB “ ´2δL_matter

δg^AB `g_ABL_matter, (3.92) or³⁵

T^AB“ 2

?´g δp?

´gL_matterq δgAB

, (3.93)

and must not be confused with the energy-momentum tensor of the boundary conformal field theory. In summary, the UV fixed point conformal quantum field theory in d dimensions can be viewed as the boundary of a d`1 dimensional gravitational theory described by an AdS-spacetime. The source of the conformal energy-momentum tensor is the boundary value of the spacetime metric and the matter fields in the bulk AdS-spacetime describe the dynamics of the couplings under the RG flow of the quantum field theory operators. The boundary values of these matter fields correspond to the UV fixed point couplings.

3.4.2. Finite temperature & density deformations

In order to understand the global RG flow diagram and critical phenomena, we need to deform the fixed point CFT by relevant deformations allowing us to flow to other fixed points. Gener-ically, the β-functions encoding the global RG flow are not accessible in complicated strongly coupled or strongly correlated systems. A holographic realization of fixed point deformation is realized by deforming the spacetime geometry in such a way that we recover the AdS-spacetime asymptotically. This amounts to a theory with a UV fixed point and a non-trivial IR behavior.

Such a scenario is known from almost all condensed matter theories, where the short-range mi-croscopic theory is known, but the long-range/low-energy behavior emerges non-trivially from the microscopic degrees of freedom. There are many possibilities for non-trivial IR geometries, but the holographic principle provides us already with the most simple ones. In order to define a field theory with thermodynamic properties as temperature, entropy and a free energy, we may consider a black hole geometry with horizon at u“ uH which approaches AdS-spacetime asymptotically foruÑ0. Two well-known AdS-black hole solutions to Einstein’s equations with negative cosmological constant are given by the AdS-Schwarzschild and AdS-Reissner-Nordström black hole, extensively discussed in Chapter4and applied to render a holographic dual of su-perconductors and charged superfluids. Therefore, our holographic dictionary can be extended by including thermal field theories with finite temperature, set by the Hawking temperature (3.7) TH, finite entropy, defined by the black hole horizon or the Bekenstein-Hawking entropy SBH (3.8) and a finite chemical potential related to the charge of the Reissner-Nordström black

35The minus sign arises from

δ´ g^ABgBC

¯

“0 ñ δg^AB“ ´g^ACg^BDδgCD

and

g_ACg_BDT^AB“T_CD“ 2

?´gg^ACg^BDδg^CD δgAB

looooooooomooooooooon

“´1

δp?

´gL_matterq δg^CD .

Boundary field theory inddimensions Bulk gravity ind`1dimensions

Global current

J^µpxq é A_Apx, uq

Gauge field Aµpxq é AApx, uqˇˇ

BpAdSq

xJ^µpxq y é ΠrAs^Apx, uqˇˇ

BpAdSq

Energy-momentum tensor

T^µνpxq é g_abpx, uq

spacetime metric g^µνpxq é g_ABpx, uqˇˇ

BpAdSq

xT^µνpxq y é Πrgs^ABpx, uqˇˇ

BpAdSq

Entropy S é SBH

Bekenstein-Hawking entropy

Free energy F é IGravity Euclidean on-shell action

Temperature T é TH Hawking Temperature

Chemical potential µ é QBH Charge of black hole

Table 3.5. From the holographic RG flow viewpoint the couplings of the strongly coupled QFT correspond to the fields with the same symmetries, quantum numbers and tensorial structure in the gravitational theory. The UV fixed point couplings are the sources of the fixed point CFT operators that correspond to the boundary values of the fields in asymptotic AdS spacetime.

hole. According to Table2.1, once we have a thermal field theory, the thermodynamic potentials such as the free energy are determined by the logarithm of the partition function. In the case of gauge/gravity dualities we may employ the saddle-point approximation to the gravitational theory for strongly coupled field theories and thus the thermodynamic potentials reduce to the regularized Euclidean on-shell action. The extended holographic dictionary is listed in Table3.5.

The non-trivial geometry arises from a matter Lagrangian³⁶L_matterdesigned in such a way that the boundary values of the matter fields correspond to the sources of the strongly coupled QFT we want to describe holographically. In general, the so-called backreaction of the matter fields onto the simple AdS geometry generates the non-trivial IR geometries which arise as consistent solutions of Einstein’s equations with negative cosmological constant, c.f. Figure 3.13. Apart from the black hole solutions there are scaling solutions for non-trivial IR fixed points and there are other geometries like hard-wall solutions with a hard cut-off in the geometry introducing a mass-gap, or solitonic solutions connecting two AdS-spaces with different radii. There are also more exotic black hole solutions, such as the dionic black hole, including sources for magnetic fields. The main task to apply holography to physical systems is to identify the correct gravita-tional dual encoding the properties of the system and consistently solve the coupled equations of motion with two constraints. The first constraint arises from regularity in the IR,i.e. infalling boundary conditions at the black-hole horizon, that fixes one of the two solutions of the bulk equations of motion. The second constraint is that the geometry must be asymptotically AdS at the boundary,i.e. approach the UV fixed point CFT. In the next section, we will describe how

36In the following, we typically denote every Lagrangian including non-gravitational degrees of freedom as matter Lagrangian, including gauge fields or other massless fields.

Figure 3.13.

The IR geometry is deformed by a relevant opera-tor which corresponds to a AdS-black-hole geom-etry, where the boundary is still asymptotic AdS-space corresponding to a UV fixed point CFT. More accurately, the black hole is a spatially infinite black brane extending across the flat spatial direction of the field theory. The black brane horizon sets the temperature of the deformed UV fixed point CFT, but the matter content on the gravity side is intro-duced at zero temperature. As explained in Section 3.5, fluctuations about the background solution to the full Einstein equations are related to dissipative effects described by the infalling bulk field fluctua-tions. For charged black branes, the electric flux em-anating from the black brane horizon sets the charge density on the boundary field theory.

u ddimensional CFT

onR^1,d´1

d´1dimensional black brane with

temperatureT_H

to retrieve physical properties in terms of response functions by applying linear response theory from Section2.2to our holographic setup.

Im Dokument Gauge/Gravity duality (Seite 124-129)