Field-operator map

In this paragraph we will discuss the precise mapping of the classical fields in type IIB supergrav-ity on AdS5ˆS⁵onto gauge invariant operators of theN “4SUpNcqsupersymmetric Yang-Mills theory. As explained in Section3.2the basic claim of the AdS/CFT correspondence and gauge/-gravity duality in general is the reorganization of ad-dimensional CFT into ad`1-dimensional (quantum) theory of gravity. Independent of the internal distribution of the degrees of freedom the partition functions of both theories must agree in order to encode the equivalent physical information in both theories. Therefore, the first entry in the holographic dictionary is given by [76]

Z_QFT^pdqrJs “Z_QG^pd`1qrϕsˇˇˇ

ϕ_BpAdSq“J, (3.59)

where the operators of the field theory are sourced byJ which are determined by the boundary values of the quantum gravity fields. Operationally, the quantum gravity partition function is not known, so we have to take the classical limit on the gravity side. For a well defined classi-cal gravity the curvature must be small compared to the Planck lengthℓP in order to suppress quantum corrections and additionally small compared to the string lengthℓssuppressing stringy corrections, thus

ℓP

L !1, ℓs

L !1. (3.60)

This is exactly satisfied by the Maldacena limit and the conditions of the weak form of the AdS/CFT correspondence (3.54). So the second entry in the holographic dictionary relates the local degrees of freedom N_c and the ’t Hooft coupling λto the string length (following from (3.49) withp“3) and the Planck length following from theHolographic Principleon page74

N_c²„ ˆL

ℓP

˙d´1

, λ„

ˆL ℓs

˙d

. (3.61)

In the largeNcand strong coupling limit, we can find weakly curved classical supergravity in the saddle-point approximation which reduces (3.59) to

A

e^{´şddx JpxqOpxq}E

QFT“Z_QFT^pdqrJs “Z_SUGRA^pd`1qrϕsˇˇˇ

ϕBpAdSq“J« e^´ISUGRAˇˇˇ

ϕBpAdSq“J. (3.62) Note that in the saddle-point approximationISUGRAis the regularized Euclidean on-shell action evaluated on the classical solution to the saddle-point equations (2.18) (see InfoboxSaddle Point Approximationin Section2.1.1for more details). From the relation between quantum statistical partition functions and Euclidean QFT we can deduce that thermodynamic quantities scale with N_c², and λ amounts to a scale setting the anomalous dimensions. Thus (3.62) constitutes a

QuantumSUpNcq Gauge Theory

N_cÑ 8 Gauge Theory

String Theory Quantum Gravity

Classical (Super-) Gravity conjectured duality

saddlepoint approximation

holographic dictionary weak/strong duality

treelevellimit gsÑ0 pointparticle limitα1Ñ0 large

N^cgaugetheories ind closedstring theory ind`1

Figure 3.11. Conceptual overview of the generalized gauge/gravity duality. Strongly coupled largeN_cgauge theory is effectively described by classical gravity. The Weinberg-Witten theorem [91] is evaded by the holographic principle allowing gravity to emerge from a gauge theory in a lower spacetime dimensions as the gravitational theory.

typical weak/strong duality, where a strongly coupled non-perturbative QFT is mapped onto a weakly curved gravity. In this sense the quasi particle description of perturbative QFT is replaced by a geometrical description in terms of gravity when flowing to strong coupling. Correlation functions of the strongly coupled QFT inddimensions are calculated by functional derivatives of the regularized on-shell actionISUGRAwith the boundary conditionϕ_BpAdSd`1q“ J “0. The main connection drawn from the holographic dictionary is summarized in Figure3.11.

Let us derive the explicit relation between the fields in type IIB supergravity on AdS<sub>d`</sub>1. In order to make contact with the original AdS5ˆS<sup>5</sup> supergravity, let us do the calculation in the simplest case of a scalar field in the AdSd`1ˆS^9´d background where we generalize it to arbitrary dimensionality of the AdS-space. In this case we can look at the effective saddle-point action with the most relevant terms. As we have learned in Section3.1.4the Euclidean action for a scalar field reads

Seff“ ´ 1 2κ0

ż

d¹⁰x?

´gpR`4∇aΦ∇<sup>a</sup>Φq. (3.63) The AdS<sub>d`</sub>1ˆS^9´d metric is given by (3.51) in the near horizon limitr Ñ0. It is convenient to write the metric in more suitable coordinatesu “ ^L2{^r which are local coordinates on the Poincaré patch of global AdS-space (for details seee.g. [125])

ds²“L² u²

`´dt<sup>2</sup>`dx²`du²˘

`L²dΩ²₅. (3.64)

Thed “D`1 spatial coordinates will be denoted byx“ pt,xq, the radial coordinate of AdS-space by uand the coordinates on the S^9´d by y. First, we remove the extra dimensions of theS^9´dby dimensional reduction where we expand the fields in spherical harmonicsYpyq. As a simple example, we may consider a one dimensional analog: the compactification yields a

“Kaluza-Klein tower” following from the periodicity of compact spaces²⁸ Φpx, u, yq “ ÿ

nPZ

ϕ_npx, uqe^iny{L ÝÑ

ˆ

∇_A∇^A´n² L²

˙

ϕ_npx, uq “0. (3.65) In the case of spherical harmonicsYpyqthe ten-dimensional Klein-Gordon equation arising from (3.63) acquires an additional mass term

Φpx, u,yq “ ÿ8

∆“0

ϕ^∆px, uqY∆pyq ÝÑ p´∇²`m²_∆qϕ^∆px, uq “0, (3.66) where the mass of the scalar field is given by

m²_∆L²“∆p∆´dq. (3.67)

Note that the dimensional reduction of the action (3.63) yields the effective action of a scalar in AdSd`1space with the most relevant terms included being a Gaussian theory as explained in Section2.3.6and (2.129)²⁹

S_effrϕs “ 1 2 ż

d^d`1x?

´g`

g^AB∇Aϕ^∆∇Bϕ^∆`m²_∆ϕ^∆˘

. (3.68)

In the following we will suppress the index∆onϕandm. The Laplacian, or more precisely the Laplace-Beltrami operator,∇²for scalar fields can be reduced to

∇A∇^Aϕpx, uq “ 1

?´gBA

`?´gg^ABBBϕpx, uq˘

“ 1 L²

`u<sup>2</sup>Bu<sup>2</sup>´ pd´1quBu`u²BµB^µ˘

ϕpx, uq. (3.69) The equation of motion (AdSd`1 Klein-Gordon equation) (3.66) in momentum space with the Fourier transform of the scalar field, imposed by Lorentz invariance,

ϕpx, uq “ ż d^dk

p2πq^dϕpk, uqe^ikµxµ, (3.70) replacingBµB^µÝÑ ´k_µk^µis given by

„

B²u´d´1 u Bu´

ˆ

kµk^µ`m²L² u²

˙

ϕpk, uq “0. (3.71)

According to [187] (10.13.4), the solution to (3.71) can be written in terms of modified Bessel functions of the firstKνpkuqand second kindIνpkuq

ϕpk, uq “u^d{2rϕregK_νpkuq `ϕirregI_νpkuqs, (3.72)

28The AdS directions are labeled by capitalized Latin indicesA, B, . . .and theS^9´ddirection are indexed by lower case Latin charactersa, b, . . ..

29There are two possibilities to write the effective action. Firstly, we can write the action with covariant derivatives allowing us to simply deduce the equations of motion due to the metric property∇g “ 0, so the integration by parts can be conducted easily. Note that the covariant Laplacian of a scalar field involves connection terms arising from∇ApB^Aϕq, where the gradient of the scalar field needs to be treated as a vector. This gives rise to connection terms in the fully expanded Laplacian. Secondly, we know that the covariant derivative of a scalar field reduce to partial derivatives, so we can write the effective action in terms of partial derivatives. Here the integration by parts introduces derivatives of the metric since the partial derivativedoes notcommute with the metric. Of course, both approaches will yield the same equations of motion.

whereν “∆´^d{²andk“a

k_µk^µ. Note that∆denotes the larger solution of (3.67),i.e.

∆_˘“ d 2 ˘1

2

ad²`4m<sup>2</sup>L<sup>2</sup> ÝÑ ∆<sub>`</sub>`∆<sub>´</sub>“d ñ ∆<sub>`</sub>”∆, ∆_´ “d´∆. (3.73) Expanding the solution in the deep interior of the AdS-spaceuÑ 8, we see that the modified Bessel functions are exponentially increasingIνpkuq „ e^ku or decayingKνpkuq „ e^´ku. Impos-ing regularity requires the irregular coefficient to vanish,ϕirreg “ 0³⁰. Expanding the solution (3.72) at the AdS boundaryuÑ0yields

ϕpk, uq «ϕ_d´∆pkqu^d´∆`ϕ∆pkqu^∆. (3.74) Considering the equivalence of the partition functions (3.59), we immediately see that∆sets the scaling dimensions of the operator

O

sourced by the leading termϕ_d´∆pkq(for more details see the discussion in Section3.4). Here some clarifications are in order. First, the scaling dimen-sion must be real which requires the mass of the scalar to be larger than´^d2{^4L2, the so-called Breitenlohner-Freedman bound [188]. Due to the hyperbolic form of the AdS-space, which is a maximally symmetric space with negative curvature, some peculiar properties arisee.g. there is a boundary at infinity which can be reached in finite time. In fact, AdS-space acts as an isotropic and homogeneous harmonic box, so a photon sent out inside AdS will run to the boundary and return in finite time. This harmonic box-like structure allows stable fields with negative m² terms. The instability expected from flat space is removed by the harmonic potential generated by the AdS-space³¹. Moreover, form²L² ą 1´^d2{⁴ we can characterize the two independent solutions by their asymptotic behavior. Defining the norm on a constant time slice of AdS-space

pϕ₁, ϕ₂q “ ż

d^d´1xdz?

´gg^ttpϕ^˚₁Btϕ₂´ϕ^˚₂Btϕ₁q, (3.75) we see that

ϕpk, uq „

$&

%

u^d´∆ non-normalizable u^∆ normalizable.

(3.76) In order to have a well-posed variational problem of the action in (3.68), we need to fix the non-normalizable modeϕ_d´∆by boundary conditions atuÑ0which can be identified with the source for the dual operator with proper wave function renormalization³²

Jpkq ”ϕ_d´∆pkq “lim

uÑ0u^∆´dϕpk, uq, (3.77)

30In Minkowski signature this amounts to an outgoing mode, where the in-going mode is normalized byϕ_reg. When considering finite temperature, we will deal with a black hole horizon in the deep interior of the modified AdS-space, so physically only the in-falling boundary condition will yield the correct causal structure for the Green functions, see Section2.2.2.

31Furthermore, black holes in AdS-space are thermodynamically stable objects,i.e. their specific heat is positive. Thus, adding energy/matter to the black hole increases its temperature unlike the flat space black holes and therefore AdS black holes can come to an equilibrium state. Heuristically, this follows again from the harmonic box property, where any radiation (or even gravity) cannot leave the AdS-space, so the evaporation radiation will fall back into the black hole.

32For brevity we have omitted a proper discussion on holographic renormalization. It follows the same logic of pertur-bative QFT renormalization in removing divergent terms. The QFT correlation functions computed in a holographic context suffer from UV divergences which can be regularized by a near boundary cut-off. Due to the IR/UV connection of holographic theories (see Figure3.1), the UV divergences of the field theory are related to IR divergences of the gravitational theory [189]. Heuristically, this IR divergence arises from the infinite spacetime volume of AdS-space.

The holographic renormalization proceeds as follows. First, we integrate only to a shell near the boundaryuÑǫand isolate the divergent terms underǫÑ0. Then, we introduce appropriate counterterms respecting all symmetries to remove the divergences. In the scalar field case discussed in the main text, the renormalization was easy to obtain, but in generic situations a true holographic renormalization must be performed. An important counterterm of the

Supergravity field Mass of field Scaling of dual operator

Scalar m²L²“∆p∆´dq ∆_˘“ ¹₂`

d˘?

d²`4m²L²˘

massless spin two m²L²“0 ∆“d

p-form m²L²“ p∆´pqp∆`p´dq ∆_˘“ ¹₂´ d˘a

pd´2pq²`4m²L²¯

spinors |m|L“∆´^d2 ∆“ ¹2pd`2|m|Lq

Table 3.3. Considering different types of supergravity fields, one can derive relations between the mass of the supergravity field ind`1 dimensions and the scaling of the dual sourced operator in theddimensional supersymmetric Yang-Mills CFT. Many results are obtained by various groups, for references see [191].

adding another piece to our dictionary. In the case of ´^d2{⁴ ă m²L² ă 1´^d2{⁴ we can have two different quantizations [77] where we find operators with scaling dimension∆andd´∆, respectively. The field theory with operator scalingd´∆ is unstable against relevant double-trace deformations and thus will flow to the theory with operator scaling∆. Exactly at the BF bound, the two scaling dimensions are identical, which usually gives rise to additional log-like terms in the boundary expansion. To conclude, the scalar field solution is completely fixed by the regularity condition deep in the AdS-space, which connects the coefficient of the normalizable solution to the non-normalizable solution as computed by setting ϕirreg “ 0 in the boundary expansion of (3.72)

ϕ∆“2^d{2´∆Γp^d₂ ´∆q

Γp∆´^d₂qk^2∆´dϕd´∆. (3.78) This calculation can be generalized to other types of fields on the gravity side in particular for vector fields (p-form fields), spinors and massless spin two fields. The relation between the field’s masses and the scaling of the dual operator sourced by the supergravity field are listed in Table 3.3. Finally, let us determine the dictionary entries for correlation functions, in particular the vacuum expectation values and the Green functions related to the scalar field sourced operator.

Applying the general formula for correlation functions from generating functionals (2.31) to the partition function of supergravity in the saddle-point approximation (3.62) yields

xx

O

px1q ¨ ¨ ¨

O

px_nq yy “ p´1q^n`1 δⁿISUGRA

δJpx1q ¨ ¨ ¨δJpx_nq ˇˇ ˇˇ

J“ϕ_BpAdSq

. (3.79)

The expectation value can be calculated from a single functional derivative with respect to the boundary value of the field. From Hamilton-Jacobi theory we know that the variation of the action with respect to the boundary value of a generalized coordinate yields the canonical con-jugate momentum at the boundary. Treating the radial coordinate as “time” and applying the

Einstein action leading to AdS spacetime is the Gibbons-Hawking-York term given by

S“ 1 κ²

ż

BM

d^dxa

|γ|K

where γdenotes the determinant of the induces metricγ^µν on the near boundary shell andK is the extrinsic curvature given by

K“γ^µν∇µnν

withnνbeing the outward pointing normal vector to the tangent surface of the near boundary shell. More details can be found in [190].

Leibniz integral rule or for higher dimensions the Reynold’s transport theorem, the functional derivative at the boundary reduces to a normal gradient to the boundary (with normal vector pointing in the interior of AdS-space)

Πpk, uq “δISUGRArϕs δϕ_BpAdSq “ ´?

´gg^uuBuϕpk, uq (3.80)

ñ x

O

pkq y “ δISUGRArϕs δϕ_d´∆pkq “lim

uÑ0u^d´∆Πpk, uq. (3.81) Note that the factoru^d´∆ arises from the wavefunction renormalization of the source from the boundary value in (3.77) and the “functional chain rule”. Inserting the boundary expansions (3.74) and the definition of the canonical conjugate momentum yields

x

O

pkq y “ 2∆´d

L ϕ∆pkq, (3.82)

so the vacuum expectation value of an operator is given by the coefficient of the subleading term in the boundary expansion ϕ∆, characterizing the normalizable solution according to (3.76).

The two-point correlation function is now easily obtained. Following the “traditional” way, we would take another derivative with respect to the source of the regularized Euclidean on-shell actionISUGRA, but we can do much better: we already know that the two boundary solutions are connected by the regularity condition in the deep AdS-space interior u Ñ 8as shown in (3.78). Therefore, we can take directly the derivative with respect toJ “ϕ_d´∆of the conjugate momentumΠ

xx

O

pkq

O

p´kq yy “ δ²ISUGRA

δϕ²_d´∆ “ δΠ δϕd´∆

33

» 2∆´d L

Bϕ_∆ Bϕd´∆

. (3.83)

Inserting the relation betweenϕ∆andϕd´∆yields xx

O

pkq

O

p´kq yy “ 2∆´d

L 2^d{2´∆Γp^d₂´∆q

Γp∆´^d2qk^2∆´d, (3.84) and transforming (3.84) to real space reads

xx

O

px1q

O

px2q yy “ 2∆´d L

Γp∆q π^d{2Γp∆´^d2q

1

|x1´x2|^2∆. (3.85) which is exactly the conformal correlator of a quasi primary field with scaling∆ as shown in (2.154). In more complicated cases we cannot find an analytic solution to the classical equation of motion in the bulk AdS-space. Therefore, we must resort to matched asymptotic expansions or even numerical methods to determine the relation betweenϕ∆andϕd´∆. This is extensively shown in all applications in Chapter5and4. For fermions there are more subtleties. Let us just hint at the fact that the Dirac equation for spinors is already a first order differential equation, so imposing boundary conditions in the AdS interior and the AdS boundary would overdetermine the system of differential equations. Therefore half of the spinor components are related to the other half, usually in a decomposition with respect to a projector constructed from the radial Dirac matrix [192]. The dictionary discussed so far is summarized in Table3.4. Further entries will be added when systems with deformed IR geometry will be discussed.

33Proper renormalization of all quantities involved is already taken into account,i.e. working directly with the coeffi-cients of the boundary expansion includes all limit-taking processes.

Boundary field theory inddimensions Bulk gravity ind`1dimensions

#(degrees of freedom) N_c² é ^L_κ^d´21 “´

L ℓP

¯d´1

AdS radius/curvaturerℓps

Interaction strength λ é ´

L ℓs

¯d

AdS radius/curvaturerℓss QFT partition function Z_QFT^pdqrJs é Z_QG^pd`1qrϕs Quantum gravity partition function Strongly coupled QFT Z_QFT^pdqrJsˇˇˇ

N,λ"1 é e^´Ipd`1q Classical Gravity

Operator

O

pxq é ϕpx, uq Field

Scaling dimensions ∆_O é m mass carried by the field

Source of operator Jpxq é ϕpx, uqˇˇ

BpAdSq Field at boundary Vacuum

x

O

y é Πpx, uqˇˇ

BpAdSq

Conjugate momentum

expectation value at boundary

Quantum numbers n_O é n Properties of fields

Global symmetry Gglobal é Glocal Local Symmetry

Global charge of operator Q_O é q gauge charge

carried by the field Table 3.4. The holographic dictionary as explained in the text so far. In general, a gravity

dual to a certain field theory is found by incorporating all the symmetry properties, dimensionality and the nature of the interesting operators. However, the implemen-tation of the field theory properties might not be simple, so in many cases one has to work with a universal QFT, capturing the relevant properties (the so-called bottom-up approach). The dictionary for additional symmetries follows from the fact that gauge symmetries include so-called large gauge transformations which are reduced to global symmetries at the spacetime boundary. The quantum numbers of the field theory (e.g. angular momentum/spin of the operator) are encoded in certain prop-erties of the fields. This also includes certain constraints on the metric (which is viewed as yet another field on the gravity side), so conserved quantum numbers from global symmetries may appear as additional dimensions in the gravity dual (e.g. the Schrödinger metric [193]).

From the holographic principle we already know that black holes can be described as charged objects with temperature and entropy. Thus, AdS-black hole geometries will give rise to finite density and temperature field theories as explained in the RG context in Section3.4.

Im Dokument Gauge/Gravity duality (Seite 117-123)