2.1 The original AdS/CFT correspondence
2.1.4 An AdS/CFT dictionary
So far we recognized that the gauge/gravity duality allows for the reformulation of some problem defined in a gauge theory in terms of a gravity theory. In order to obtain quantitative answers, it is necessary to identify the corresponding quantities in both theories. The supergravity theory is formulated in terms of classical fields on a ten dimensional background, while theN = 4SYM theory describes the dynamics of operators acting on quantum states in four spacetime dimensions. The relations between the parameters of the theories were introduced with the correspondence on the preceding page. For the coupling constants gYM, gs and λ, as well as the AdS radius R, the string tensionα0and the number of colorsN, they are
R4= 4πgsN α02, 2πgs=g2YM, λ=g2YMN. (2.30)
Observables, however, are expressed in terms of correlation functions of gauge invariant operators of the quantum field theory. It is possible to translate correlation functions of the field theory to expressions in terms of supergravity fields. A precise prescription of how to accomplish this was given in two seminal papers from 1998 by Edward Witten[13], and Gubser, Klebanov, Polyakov[14]. As a result it is possible to establish a complete dictionary, which translates quantities from on side of the correspondence to the other.
Since the domain on which the field theory is defined can be identified with the boundary ofAdS5 space, one can imagine supergravity fieldsφin AdS5to interact with some conformally invariant operatorOon the boundary.
We denote the boundary value of the supergravity field byφ0 = lim∂AdS5φ.
A coupling would look like Sint=
Z
∂AdS5
d4x φ0(~x)O(~x). (2.31)
In this sense the boundary valueφ0 of the supergravity field acts as the source of the operatorOin the field theory. Such an interaction term appears in the generating functional for correlation functions, which we write schematically as
* exp
Z
∂AdS5
φ0O +
CFT
. (2.32)
Witten’s proposal was to identify the generating functional for correlation functions of operatorsOwith the partition functionZsugraof the supergravity theory, which is given by
Zsugra[φ0] = exp −Ssugra[φ]
φ=φ0
, (2.33)
where Ssugra is the supergravity action. So the ansatz for the generating functional of correlation functions of operators of the field theory can be written as
* exp
Z
∂AdS5
φ0O +
CFT
= exp −Ssugra[φ]
φ=φ0
. (2.34)
Correlation functions forOcan then be obtained in the usual way by evaluating the functional derivative of the generating functional with respect to the source φ0of the operator. Explicit calculations will be performed in later chapters.
As a general example, some two point function would be obtained by solving the supergravity equations of motion, plugging these solutions into the action Ssugra, then expressing the result in terms of solutionφ0on the boundary, and eventually evaluating
hO(x)O(y)i= δ δφ0(x)
δ
δφ0(y) exp −Ssugra[φ]
φ0=0
. (2.35)
The remaining question is which operators are dual to which fields. The fact that there exists such a dictionary relies heavily on the symmetries of the two related theories. The symmetry of a theory reflects the transformation behavior of the field content and by the Noether theorem accounts for the conserved quantities (charges). We can expect that two equivalent theories share the same amount of degrees of freedom, which must be reflected in their symmetries.
To ensure a gauge invariant field theory action, including the source term (2.31), we have to restrict our attention to operatorsOwhich are gauge invari-ant. The localSU(N)gauge symmetry of the quantum field theory in fact has no counterpart on the supergravity side in the Maldacena limit. The parameter N is translated into the number of D3-branes on the string theory side of the correspondence. The stack of D3-branes merely accounts for the emergence of theAdS5×S5 spacetime, see section 2.1.2 on page 15. Moreover, the following arguments strictly only apply to BPS states.
Comparing the remaining symmetry groups ofN = 4SYM theory and type IIB supergravity we indeed observe a matching of symmetries. In sec-tion 2.1.1 we noted the symmetry group of the gauge theory to bePSU(2,2|4).
The bosonic subgroup of this isSU(2,2)×SU(4)R ∼= SO(2,4)×SO(6).
These are precisely the isometry groups ofAdS5 ×S5, whereSO(2,4)is the isometry group of theAdS5part, while the five-sphere is invariant under SO(6)transformations. The fermionic symmetries can be shown to coincide as well, leading to the overall symmetry groupPSU(2,2|4).
In fact the isometries ofAdS5×S5 act as the conformal group on the boundary[15]. Any gauge invariant field theory operatorOdoes transform under some representation of the conformal group. Since the boundary theory is invariant under conformal transformations, the supergravity fieldφin the source term(2.31) has to transform in the dual (conjugate) representation.
Conformal invariance of the theory restricts the fieldφfurther. For instance, in the coordinates where the boundary is located atu = 0the supergravity equations of motion for a scalarφhave two linear independent solutions at asymptotically smallu=near the boundary,
φ(~x, ) =φ0(~x)d−∆+φ1(~x)∆. (2.36) Hereddenotes the number of dimensions of the boundary, which in our case isd= 4. Generically, the value of∆for a scalar supergravity fieldφdepends on the massmφof the field[13]as
m2φ= ∆(∆−d), (2.37)
with∆>0. The second term of(2.36)vanishes at the boundary→0while the first term may diverge. The existence of a well defined boundary value φ0(~x)tells us that this function has scaling dimensiond−∆,i. e.φ0(~x) 7→
φ0(~x)/d−∆ on rescalings. From the interaction term of the conformally
invariant action(2.31)we thus see that the boundary value ofφ(~x, u)acts as the source to an operatorO(~x)of scaling dimension∆.
In summary, to identify the supergravity fieldφdual to an operatorOwe have to spot all supergravity fields transforming in the dual representation to that of the operator under consideration. The conformal weight ∆ of the operator determines the mass of the supergravity field by (2.37). The mass spectrum ofN = 2supergravity compactified onAdS5×S5has been computed[16], and therefore the field can be identified uniquely. Examples of computations of dual field–operator pairs can be founde. g.in refs. 8, 9, 13, 14.
We also identified the boundary of AdS space with four-dimensional Minkowski spacetime. This spacetime was only defined up to conformal transformations, and we will identify it from now on with the domain of the conformally invariantN = 4SYM theory. Notice that near the boundary all processes occurring in the field theory directions can be thought of as being scaled in such a way that all lengths of theAdS theory, even long distance or IR phenomena, are mapped to short scales,i. e.the UV limit on the conformal field theory side. To see this, consider the metric(2.21)in the near horizon limit. Then distancesds2CFTin the field theory, which are measured along~x appear with a warp factor relative to the distanceds2AdSinAdSspace,
ds2CFT= R2
r2 ds2AdS. (2.38)
At larger 1(IRAdS) close to the boundary, short scale phenomena of the CFT (UVCFT) match the events inAdSspace, while at smallr 1(UVAdS), far from the boundary, long scale phenomena in the CFT (IRCFT) match the AdS distances. The radial coordinate in this way sets the renormalization scale of the field theory which is holographically described by the supergravity theory. This phenomenon is called theUV/IR duality.
In fact the behavior of correlation functions under renormalization group flows can be computed holographically. The UV divergences known from field theory translate into IR divergences on the gravity side. The procedure to incorporate scale dependence and renormalizen-point correlation functions is known asholographic renormalization. We will not review the procedure in detail here, but rather give an idea of the procedure and state some results. A nice overview which also addresses some subtleties can be found in ref. 17.
The correlation functions(2.35)in general suffer from IR divergences,i. e.
divergent terms at large values of the radial coordinate. Analogous to quantum field theory renormalization, they can be cured by analyzing the behavior of the field solutions near the boundary and adding appropriate counterterms Sctto the actionSwhich do not alter the equations of motion but render the resulting correlators finite.
To analyze the field behavior near the boundary it is convenient to work in coordinatesuas in(2.26)where the boundary is located atu → 0. The solution for the second order equation of motion of any fieldFcan then be
expanded in a series around u = 0. In general there are two independent solutions scaling asum andum+nnear the boundary. The general solution can be written as
F(x, u) =um
f(0)(x) +u2f(2)(x) +. . .+un
f(2n)+ ˜f(2n)lnu
+. . .
(2.39) in a well defined manner where the coefficientsf(n)(x)carry the dependence on the other coordinates. The values of m and n are determined by the mass of the supergravity field and related to the conformal dimension of the dual operator, as in the example above. The coefficientf(0) determines the boundary behavior of the two independent solutions for the equation of motion of F. Solving these equations order by order in u determines the relevant coefficientsf(k)fork <2nas functions off(0), which thereby can be used as the initial value of the first of the two linearly independent solutions.
The second parameter needed to define the full solution of the second order equation of motion toF is the coefficientf(2n), which in turn determines the remaining higher order coefficients. It then is possible to extract the divergent terms in the regularized actionSreg, which is given by the on shell action with respect to theudependence of the solution, evaluated at the cutoff1,
Sreg= Z
d4x a(0)u−ν+a(1)u−(ν+1)+. . .
u=. (2.40)
The coefficientsa(n)now are functions of the coefficientf(0), and theν >0 solely depend on the scale dimension of the operator in the conformal field theory. Defining the counterterm action as
Sct=−divergent terms fromSreg (2.41)
The renormalized action is given by Sren = lim
→0 Sreg+Sct
. (2.42)
Finding the renormalized action therefore involves a careful analysis of the equations of motion. An extremely useful result of holographic renormalization is the fact that the solutions to the equations of motion of a supergravity field F can directly be related to the source and the vacuum expectation value of the dual operator in the field theory[17, 18]. In particular holographic renormalization unveils that the modef(0)is proportional to the source of the dual operator, while the modef(2n)is proportional to the vacuum expectation value of the same operator. In general, the mode that is proportional to the source scales in a non-normalizable way withu, while the mode proportional to the vacuum expectation value is normalizable. An example is given in (2.36)for the case of a scalar field. The integersmandnare determined in terms of the supergravity field’s mass, which in turn translates to the conformal
Supergravity onAdS5×S5 4dim. N = 4CFT
boundary ofAdS5 field theory domain
isometry ofAdS5 conformal symmetry
isometry ofS5 R-symmetry
weak coupling ings strong coupling inλ
variations in radial coordinate renormalization group flow
field boundary valueφ0 source for operatorO
field massm conformal weight∆
IR normalizable mode hOiof dual operator
IR non normalizable mode source of dual operator
quantum corrections ofO(gs) corrections in1/N stringy corrections ofO(α0) corrections in1/λ
TABLE2.1: A few examples for entries of theAdS/CFT dictionary. Precise operator–field pairings can be found e. g. in refs. 8, 9, 13, 14.
dimension of the dual operator. In(2.36)the value ofφ0is proportional to the source and φ1 is proportional to the vacuum expectation value of the dual operator. We will encounter an explicit example for the source and vacuum expectation value when we introduce the prominent pair of a D-brane embedding function and its field theory dual operator in section 2.2.2.