Operator-Field Dictionary, Witten’s Generating Functional Formula

We have already seen in section 2.3.2 that the global symmetries of the N = 4 SYM theory match the isometries of the AdS5 ×S5 background. This is an important check

for the correspondence. It also gives a hint how to further concretise the statement of the correspondence: In the N = 4 SYM theory, operators fall into multiplets of the full SU(2,2|4) superconformal group. These multiplets can be short multiplets or long multiplets, depending on whether they preserve some supersymmetry of the theory or not. Of most interest for checking the AdS/CFT correspondence are short multiplets, in particular the 1₂BPS ones, as it turns out that they are protected by supersymmetry and their correlators do not renormalise. In particular, their correlators are independent of the ’t Hooft couplingλ [212],37 _{which makes an extrapolation possible from weak ’t Hooft}

coupling (where the correlator can be calculated in the field theory) to strong ’t Hooft coupling, where we can trust the supergravity calculations on the AdS5×S5 background.

But how are they identified on the gravity side of the correspondence? There must exist a map between local38 _{gauge invariant operators on the field theory side and the excitations}

of type IIB supergravity on AdS5×S5on the field theory side. Since the latter also falls into

multiplets of the corresponding global symmetries, in particular of the SU(4)_R symmetry if one Kaluza-Klein reduces the IIB supergravity on the S5 [217, 218], one needs to show that there is a one-to-one mapping between the representations showing up in the field theory and the representations showing up in the Kaluza-Klein tower of IIB supergravity states. This would then establish an operator-field dictionary. And indeed this is the case, as described nicely e.g. in chapter 5.6 of [80], see also the references cited there. I will not try to reproduce the full spectrum here, it should suffice to say that one needs to expand the ten-dimensional supergravity fields of different spin in spherical harmonics on the five-sphere which are labeled by (amongst other quantum numbers) the scaling dimension ∆, which describes the scaling of the corresponding field theory operator under dilatations x 7→ λx. The resulting five-dimensional fields will eventually become massive (just as when reducing a theory on a circle), and these masses are in correspondence with the scaling dimensions via the following mapping (for AdSd+1):

scalars and massive spin two fields R2m2 = ∆(∆−d),

massless spin two fields R2m2 = 0, ∆ =d ,

spin 1/2 and 3/2 R_|m_|= ∆₋ d

2, (2.115)

p-form fields R2m2 = (∆₋p)(∆ +p₋d).

Note that it is the linearised fields (i.e. gravitons, dilatons etc.) which are reduced on the five-sphere to yield these representations. Upon reduces the full nonlinear theory, also the

37_{Nonrenormalisation properties of} 1

4BPS operators are also known [213–215].

interactions between these modes are obtained. We will see later in this section that these interactions are important in holographic calculations of correlation functions.

The above mapping on the basis of symmetry arguments is partly a kinematical feature of the correspondence, since the symmetries are strong enough in this case to protect the field theory from most or probably all quantum corrections. To establish a full correspondence it is however also necessary to explain how each field mode exactly encodes the information about the corresponding operator and also a possible vacuum expectation value. This explanation was given by Edward Witten in another seminal paper [219] which laid the foundations of the AdS/CFT correspondence. It is remarkable simple: For every field mode ϕ obtained after Kaluza-Klein reduction, consider the Laplace-Beltrami equation on the Euclidean version of AdS5 space (in Poincar´e coordinates)

ds2 =R2 dx i2 + dz2 z2 , i= 0,1,2,3, z = R2 r , (2.116)

where a new radial coordinate z was introduced. The reason for doing this calculation in Euclidean AdS is two-fold: Historically this route was taken in e.g. [220, 221], and one does not need to worry about boundary conditions too much, since the metric (2.116) has a boundary of topology 4 _{at z = 0 (henceforth called “the boundary”), plus a single}

point in the “interior” of Anti-de Sitter space at z = _∞. In Minkowskian signature the boundary would be four-dimensional Minkowski space, while z =_∞ will become a Killing horizon. In the Euclidean case one can just impose regularity as the boundary condition at z =_∞, but in the Minkowskian case different boundary conditions (e.g. regularity for spacelike momenta and infalling wave boundary conditions for timelike momenta) have to be imposed, complicating the situation. It is however enough to consider the Euclidean case for extracting the wanted information.

What is important for establishing the exact dictionary between the behaviour of the supergravity modes and the field theory operators is the behaviour of the solutions to the Laplace-Beltrami equation at the boundary of AdS5, which is at z = 0. We thus need to

solve

0 =2AdS5ϕ(x

µ_{, z) (2.117)}

and extract the behaviour of the solution as z → 0. It turns out [219] that any solution of eq. (2.117) in an asymptotically Anti-de Sitter space-time (i.e. a space-time which is of the form (2.116) for z → 0) has a leading and a subleading behaviour. For e.g. a scalar excitationϕ on AdS5 the asymptotic behaviour is

ϕ(xi, z)m2_{=∆(∆−4)} ≃z4−∆J_O(x) +z∆hO(x)i, (2.118)

where the leading term is interpreted as a source for the operator _O corresponding to this field (which has mass m2 _{= ∆(∆ −4)), and the subleading term is identified}

with the expectation value (the condensate) of this operator. Note that this is only the asymptotic behaviour of the eq. (2.117) and no boundary conditions in the interior

have been imposed so far, so the source and the expectation value are actually the two integration constants of eq. (2.117). Imposing a boundary condition in the interior would then relate the expectation value with the source, enabling one to calculate the vacuum expectation value _hOi|JO=0.

This interpretation of classical gravity solutions also yields a correspondence between the ultraviolet and infrared physics of the field theory, and the different regions of Anti-de Sitter space-time. In terms of renormalisation group flows, the procedure is normally such that one defines a theory through an action in the ultraviolet, and studies its behaviour under the renormalisation group flow, i.e. when integrating out high-energetic degrees of freedom in the Wilsonian sense [222]. The theory then can flow to an infrared fixed point (this is believed to happen for QCD), or it can be already at a fixed point (this is what happens e.g. for conformal theories like _N = 4 SYM). For the latter case of theories, it is then the interesting to perturb the theory by adding additional operators with corresponding coefficients (the sources which can for example be mass matrices or additional couplings), and investigate whether this creates a new renormalisation group flow [223]. If the system is driven by this deformation to a new fixed point, the deformed Lagrangian serves as the ultraviolet definition of a new theory with a nontrivial RG flow. This theory can then flow to another fixed point in the infrared, but the important point here is that the deformation with its nontrivial sources for certain operators was added in the ultraviolet description of the theory. Since we know that sources in the AdS/CFT context are encoded in the nonnormalisable modes of supergravity fields, i.e. that they die off slowest towards z = 0, it is natural to identify the z = 0 region of the Anti-de Sitter space as the region encoding the ultraviolet physics of the dual field theory. On the other hand, the interior part of Anti-de Sitter space, in particular the point z = _∞, should correspond to the infrared physics in the field theory since for example condensates, which in the field theory are often governed by infrared effects, are calculated after giving boundary conditions at z = _∞. Later on in section 2.3.6 we will see an example of this mechanism. In conclusion, we thus identify the region z _→0 with ultraviolet effects in the dual field theory, and the regionz → ∞with infrared effects. A renormalisation group flow of some observable from the ultraviolet to the infrared is encoded in the full z dependence of the corresponding dual supergravity field.

One might wonder why the mass squares in eq. (2.115) are not bounded by zero from below. The reason for this is a peculiar property of fields in Anti-de Sitter space-time found by Peter Breitenlohner and Daniel Z. Freedman [224, 225] (see also [226]): Since the metric (2.116) ofAdSd+1 diverges as one approaches the boundary atz = 0, less restrictive

fall-off conditions at spatial infinity are necessary for fields to have positive energy, and thus tachyonic (i.e. negative) mass squares are allowed if they are not too negative. For scalars39 _{one finds}

R2_m2 _{≥ −}d2

4 . (2.119)

39_{The Breitenlohner-Freedman bound for fields other than scalars can readily be deduced from the} unitarity bound ∆_≥0 by using eqs. (2.115).

This restriction is called theBreitenlohner-Freedman bound. There is actually a nice correspondence between the (ir)relevance of operators and the masses of the corresponding supergravity fields (see. ch. 9.1 of [80] or [219]), which I will describe for scalars: In general, deforming the field theory by a certain operator corresponds to searching for a new supergravity background with the corresponding field excited on the nonlinear level, i.e. solving the coupled equations of the new field together with the equations of motion of the old theory. In particular, scalar fields can have quite complicated potentials in the gauged supergravity theories which arise from the reduction on the five-sphere (or other internal spaces). Scalar operators withm2 _{<0 then correspond torelevant deformations}

of the field theory, creating a RG flow towards a different infrared fixed point. Massless scalars correspond to marginal deformations, which might or might not create a RG flow, depending on whether they are marginally relevant, irrelevant or exactly marginal. Finally, scalar operators with positive mass squared (i.e. ∆>4) correspond to irrelevant deformations of the dual CFT, describing a renormalisation group flow to N = 4 SYM in the infrared from some other unknown theory in the UV. The way to describe these RG flows holographically is via domain wall solutions of truncated gauged supergravities interpolating between different asymptotically Anti-de Sitter spaces (see e.g. ch. 9 of [80]). We have seen that free supergravity fields encode in their boundary behaviour both ul- traviolet physics (the sources) and infrared physics (the vacuum expectation values). For giving a detailed description of the correspondence we also need to give a recipe of how to calculate correlation functions. This was given again by Witten in [219] (and inde- pendently in [220]), by stating the following equivalence between the supergravity on-shell action and the generating functional of the dual field theory. Let Oi be

a set of operators of_N = 4 SYM theory, then, according to [219], the generating functional of Euclidean field theory correlators is identified with the on-shell type IIB supergravity action, where the solutions for the linearised fieldsϕi _{dual to the correlatorsO}

i are subject

to the boundary behaviour as in (2.118). In a formula,

* exp  − Z 4 Ji_Oi   + CM = exp ₋SIIB ϕi(Ji,_hOi(Ji)i) . (2.120)

Here the chosen vacuum expectation values _hOii|Ji₌₀ on the right-hand side define the

vacuum in which the generating functional on the left-hand side is taken, while the non- normaliseable mode of the linearised fields (2.118) provides the dependence on the source

J. Possible additional boundary conditions imposed on the linearised field equations relate the normaliseable behaviour, i.e. the expectation value, with the source for the operator. The action on the right-hand side is the type IIB supergravity action after Kaluza-Klein reduction on the S5. The correct definition of the on-shell action still needs the addition of surface terms, such that the theory admits a well-defined semi-classical approximation, a process called “holographic renormalisation”. I will not dwell on this at this point, but come back to it in the appropriate places in the next chapters. See however [227] for a good review of this extensive topic and [228,229] for a systematic treatment of the simpler

(a) (b) (c) (d) (e) AdS

boundary AdS

Figure 2.6: Several Witten diagrams describing a) the vacuum, b,c) two- and three-point functions and d,e) contributions to a four-point function (figure taken from [80]).

and thus more transparent situation in 1+1 dimensions. In any case the formula (2.120) now allows to calculate Euclidean correlators by differentiating with respect to the sources as usual. Note that the on-shell type IIB supergravity action on the right-hand side is the full action, including interaction terms between the linearised fields. These interactions are important, since they encode the nontrivial information about the field theory inter- actions between the operators. Since, as explained in section 2.3.3, the five-dimensional gravitational coupling (i.e. the relevant gravitational coupling after reduction on the S5)

κ2

5 ∼Nc−2is small in the largeNclimit, it is possible to set up a perturbation theory scheme

on the gravity side by expanding the supergravity action in κ5. The on-shell supergravity

action can then be evaluated in perturbation theory to the required order by the following prescription: Connect the sources Ji _{located at the boundary of Euclidean AdS with the}

interior by “bulk-to-boundary propagators”, and connect the end points of the bulk-to- boundary propagators with “bulk-to-bulk” propagators. If necessary, loops in the interior of Euclidean AdS can be build up by bulk-to-bulk propagators. Include integrations over Euclidean AdS space for each internal interaction point. At the end, do a summation over all diagrams to the wanted order in κ5. These diagrams are called “Witten diagrams”,

depicted in figure 2.6. I will however not dwell longer on the actual calculation of corre- lators, since in this thesis we will mostly work with one-point functions. The interested reader is referred to the reviews [79,80,200,201] or the classics [212,219–221,230–237]. The classical checks of the AdS/CFT correspondence via the calculation of correlators of 1₂BPS operators which are protected by supersymmetry and thus do not depend on the ’t Hooft coupling can also be found in these references.

Some additional words have to be said about the relation between Minkowskian and Eu- clidean signature in the AdS/CFT correspondence. It has been observed by Dam T. Son and Andrei Starinets in [238] that Witten’s formula for the generating functional (2.120) does not apply to the case of Minkowskian signature. In particular, it would yield real boundary propagators for timelike momenta, while e.g. the retarded/advanced propaga- tors are complex. Son and Starinets proposed a recipe which correctly provides the right boundary propagators by imposing corresponding boundary conditions at the z =∞hori- zon of the Poincar´e patch of AdS5. However, since their approach is ad hoc and does not

to such a generalisation was made in [239], where it was shown that holographically im- plementing the Schwinger-Keldysh formalism with its doubling of the degrees of freedom on the contour needs the full global Anti-de Sitter manifold rather than just the Poincar´e patch (which covers only half of the manifold, see e.g. [204]). Physically speaking, the second copy of N = 4 in the Schwinger-Keldysh formalism lives on the other boundary of Anti-de Sitter space which bounds the second Poincar´e patch. In this way [239] it was possible to obtain the Schwinger-Keldysh propagator by functional variation of a corre- sponding boundary action. The upshot is that the relevant on-shell gravity action is a sum over action functionals, one with Minkowskian signature for each real-time segment of the contour, and a Wick-rotated one for each imaginary-time segment. Recently Kostas Skenderis and Balt C. van Rees gave an in-depth analysis of this procedure for general contours [240, 241], including holographic renormalisation and several examples.

In conclusion, we have seen that and how the AdS/CFT correspondence can be formulated on the levels of operators and fields, correlation functions and generating functionals. It is possible to incorporate vacua with nontrivial expectation values and to describe different field theory perturbations and holographic renormalisation group flows. This is already a rather concrete framework which can be used to analyse the dynamics of N = 4 su- persymmetric Yang-Mills theory at strong coupling. In the remaining two sections of this chapter I will describe two generalisations of the AdS/CFT correspondence to include fi- nite temperature as well as matter fields in the fundamental representation of the gauge group (recall that all fields in _N = 4 SYM where adjoints). These two generalisations will be most important for the understanding of the research work of the author presented in chapters 3 and 4.