# Holographic Duality with a View Toward Many-Body Physics

Authors: John McGreevy
Journal: Advances in High Energy Physics
 (2010)

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## Abstract

These are notes based on a series of lectures given at the KITP workshopQuantum Criticality and the AdS/CFT Correspondence in July, 2009. The goal of the lectures was to introduce condensed matter physicists to the AdS/CFT correspondence. Discussion of string theory and of supersymmetry is avoided to the extent possible.


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## Body

## 1. Introductory Remarks

My task in these lectures is to engender some understanding of the followingBold Assertion:
(a)
Some ordinary quantum field theories (QFTs) are secretly quantum theories of gravity.(b)
Sometimes the gravity theory is classical, and therefore we can use it to compute interesting observables of the QFT.Part (a) is vague enough that it really just raises the following questions: “which QFTs?” and “what the heck is a quantum theory of gravity?” Part (b) begs the question “when??!”In trying to answer these questions, I have two conflicting goals: on the one hand, I want to convince you that some statement along these lines istrue, and on the other hand I want to convince you that it is interesting. These goals conflict because our best evidence for the Assertion comes with the aid of supersymmetry and complicated technology from string theory and applies to very peculiar theories which represent special cases of the correspondence, wildly overrepresented in the literature on the subject. Since most of this technology is completely irrelevant for the applications that we have in mind (which I will also not discuss explicitly except to say a few vague words at the very end), I will attempt to accomplish the first goal by way of showing that the correspondence gives sensible answers to some interesting questions. Along the way we will try to get a picture of its regime of validity.Material from other review articles, including [1–7], has been liberally borrowed to construct these notes. In addition, some of the text source and most of the figures were pillaged from lecture notes from my class at MIT during Fall 2008 [8], some of which were created by students in the class.


## 2. Motivating the Correspondence

To understand what one might mean by a more precise version of theBold Assertion above, we will follow for a little while the interesting logic of [1], which liberally uses hindsight, but does not use string theory.Here are three facts which make the Assertion seem less unreasonable.(1) First we must define what we mean by a quantum gravity (QG). As a working definition, let us say that a QG is a quantum theory with a dynamical metric. In enough dimensions, this usually means that there are local degrees of freedom. In particular, linearizing equations of motion (EoM) for a metric usually reveals a propagating mode of the metric, some spin-2 massless particle which we can call a “graviton”.So at least the assertion must mean that there is some spin-two graviton particle that is somehow a composite object made of gauge theory degrees of freedom. This statement seems to run afoul of the Weinberg-Witten no-go theorem, which says the following.Theorem 2.1 (Weinberg-Witten [9, 10]).
A QFT with a Poincaré covariant conserved stress tensorTμν forbids massless particles of spin j>1 which carry momentum (i.e., with Pμ=∫dDxT0μ≠0).You may worry that the assumption of Poincaré invariance plays an important role in the proof, but the set of QFTs to which the Bold Assertion applies includes relativistic theories.General relativity (GR) gets around this theorem because the total stress tensor (including the gravitational bit) vanishes by the metric EoM:Tμν∝δS/δgμν=0. (Alternatively, the “matter stress tensor,” which does not vanish, is not general-coordinate invariant.)Like any good no-go theorem, it is best considered a sign pointing away from wrong directions. The loophole in this case is blindingly obvious in retrospect: the graviton need not live in the same spacetime as the QFT.(2) Hint number two comes from the Holographic Principle (a good reference is [11]). This is a far-reaching consequence of black hole thermodynamics. The basic fact is that a black hole must be assigned an entropy proportional to the area of its horizon (in Planck units). On the other hand, dense matter will collapse into a black hole. The combination of these two observations leads to the following crazy thing: the maximum entropy in a region of space is the area of its boundary, in Planck units. To see this, suppose that you have in a volume V (bounded by an area A) a configuration with entropy S>SBH=A/4GN (where SBH is the entropy of the biggest black hole fittable in V), but which has less energy. Then by throwing in more stuff (as arbitrarily nonadiabatically as necessary, i.e., you can increase the entropy), since stuff that carries entropy also carries energy,1 you can make a black hole. This would violate the second law of thermodynamics, and you can use it to save the planet from the humans. This probably means that you cannot do it, and instead we conclude that the black hole is the most entropic configuration of the theory in this volume. But its entropy goes like the area! This is much smaller than the entropy of a local quantum field theory on the same space, even with some UV cutoff, which would have a number of states Ns~eV (maximum entropy =lnNs). Indeed it is smaller (when the linear dimensions are large compared to the Planck length) than that of any system with local degrees of freedom, such as a bunch of spins on a spacetime lattice.We conclude from this that a quantum theory of gravity must have a number of degrees of freedom which scales like that of a QFT in a smaller number of dimensions. This crazy thing is actually true, and the AdS/CFT correspondence [12, 13] is a precise implementation of it.Actually, we already know some examples like this in low dimensions. An alternative, more general, definition of a quantum gravity is a quantum theory where we do not need to introduce the geometry of spacetime (i.e., the metric) as input. We know two ways to accomplish this.(a)
Integrate over all metrics (fixing some asymptotic data). This is how GR works.(b)
Do not ever introduce a metric. Such a thing is generally called a topological field theory. The best-understood example is Chern-Simons gauge theory in three dimensions, where the dynamical variable is a one-form field and the action is(2.1)SCS~∫MtrA∧dA+⋯,
(where the dots are extra stuff to make the non-Abelian case gauge invariant); note that there is no metric anywhere here. With option (b) there are no local degrees of freedom. But if you put the theory on a space with boundary, there are local degrees of freedom which live on the boundary. Chern-Simons theory on some three-manifold M induces a WZW model (a 2d CFT) on the boundary of M. So this can be considered an example of the correspondence, but the examples to be discussed below are quite a bit more dramatic, because there will be dynamics in the bulk.(3) A beautiful hint as to the possible identity of the extra dimensions is this. Wilson taught us that a QFT is best thought of as being sliced up by length (or energy) scale, as a family of trajectories of the renormalization group (RG). A remarkable fact about this is that the RG equations for the behavior of the coupling constants as a function of RG scale u are local in scale: (2.2)u∂ug=β(g(u)).
The beta function is determined by the coupling constant evaluated at the energy scale u, and we do not need to know its behavior in the deep UV or IR to figure out how it's changing. This fact is basically a consequence of locality in ordinary spacetime. This opens the possibility that we can associate the extra dimensions suggested by the Holographic idea with energy scale. This notion of locality in the extra dimension actually turns out to be much weaker than what we will find in AdS/CFT (as discussed recently in [14]), but it is a good hint.To summarize, we have three hints for interpreting the Bold Assertion:(1)
The Weinberg-Witten theorem suggests that the graviton lives on a different space than the QFT in question.(2)
The holographic principle says that the theory of gravity should have a number of degrees of freedom that grows more slowly than the volume. This suggests that the quantum gravity should live in more dimensions than the QFT.(3)
The structure of the Renormalization Group suggests that we can identify one of these extra dimensions as the RG-scale.Clearly the field theory in question needs to be strongly coupled. Otherwise, we can compute and we can see that there is no large extra dimension sticking out. This is an example of the extremely useful Principle of Conservation of Evil. Different weakly coupled descriptions should have nonoverlapping regimes of validity.2Next we will make a simplifying assumption in an effort to find concrete examples. The simplest case of an RG flow is whenβ=0 and the system is self-similar. In a Lorentz invariant theory (which we also assume for simplicity), this means that the following scale transformation xμ→λxμ (μ=0,1,2,…d-1) is a symmetry. If the extra dimension coordinate u is to be thought of as an energy scale, then dimensional analysis says that u will scale under the scale transformation as u→u/λ. The most general (d+1)-dimensional metric (one extra dimension) with this symmetry and Poincaré invariance is of the following form: (2.3)ds2=(ũL̃)2ημνdxμdxν+dũ2ũ2L2.
We can bring it into a more familiar form by a change of coordinates,ũ=(L̃/L)ũ:(2.4)ds2=(uL)2ημνdxμdxν+du2u2L2.
This is AdSd+1.3 It is a family of copies of Minkowski space, parametrized by u, whose size varies with u (see Figure 1). The parameter L is called the “AdS radius” and it has dimensions of length. Although this is a dimensionful parameter, a scale transformation xμ→λxμ can be absorbed by rescaling the radial coordinate u→u/λ (by design); we will see below more explicitly how this is consistent with scale invariance of the dual theory. It is convenient to do one more change of coordinates, to z≡L2/u, in which the metric takes the form(2.5)ds2=(Lz)2(ημνdxμdxν+dz2).
These coordinates are better because fewer symbols are required to write the metric. z will map to the length scale in the dual theory.Figure 1
The extra (radial) dimension of the bulk is the resolution scale of the field theory. The left figure indicates a series of block spin transformations labelled by a parameterz. The right figure is a cartoon of AdS space, which organizes the field theory information in the same way. In this sense, the bulk picture is a hologram: excitations with different wavelengths get put in different places in the bulk image. The connection between these two pictures is pursued further in [15]. This paper contains a useful discussion of many features of the correspondence for those familiar with the real-space RG techniques developed recently from quantum information theory.So it seems that ad-dimensional conformal field theory (CFT) should be related to a theory of gravity on AdSd+1. This metric (2.5) solves the equations of motion of the following action (and many others)4:(2.6)Sbulk[g,…]=116πGN∫dd+1xg(-2Λ+ℛ+⋯).
Here, g≡|detg| makes the integral coordinate-invariant, and ℛ is the Ricci scalar curvature. The cosmological constant Λ is related by the equations of motion  (2.7)0=δSbulkδgAB⇒RAB+dL2gAB=0
to the value of the AdS radius: -2Λ=d(d-1)/L2. This form of the action (2.6) is what we would guess using Wilsonian naturalness (which in some circles is called the “Landau-Ginzburg-Wilson paradigm”): we include all the terms which respect the symmetries (in this case, this is general coordinate invariance), organized by decreasing relevantness, that is, by the number of derivatives. The Einstein-Hilbert term (the one with the Ricci scalar) is an irrelevant operator: ℛ~∂2g+(∂g)2 has dimensions of length-2, and so GN here is a lengthd-1, the Planck length: GN≡ℓpld-1≡Mpl1-d (in units where ℏ=c=1). The gravity theory is classical if L≫ℓpl. In this spirit, the … on the RHS denotes more irrelevant terms involving more powers of the curvature. Also hidden in the … are other bulk fields which vanish in the dual of the CFT vacuum (i.e., in the AdS solution).This form of the action (2.6) is indeed what comes from string theory at low energies and when the curvature (here, ℛ~1/L2) is small (compared to the string tension, 1/α'≡1/ℓs2; this is the energy scale that determines the masses of excited vibrational modes of the string), at least in cases where we are able to tell. The main role of string theory in this business (at the moment) is to provide consistent ways of filling in the dots.In a theory of gravity, the space-time metric is a dynamical variable, and we only get to specify the boundary behavior. TheAdS metric above has a boundary at z=0. This is a bit subtle. Keeping xμ fixed and moving in the z direction from a finite value of z to z=0 is actually infinite distance. However, massless particles in AdS (such as the graviton discussed above) travel along null geodesics; these reach the boundary in finite time. This means that in order to specify the future evolution of the system from some initial data, we have also to specify boundary conditions at z=0. These boundary conditions will play a crucial role in the discussion below.So we should amend our statement to say that ad-dimensional conformal field theory is related to a theory of gravity on spaces which are asymptoticallyAdSd+1. Note that this case of negative cosmological constant (CC) turns out to be much easier to understand holographically than the naively-simpler (asymptotically-flat) case of zero CC. Let us not even talk about the case of positive CC (asymptotically de Sitter).Different CFTs will correspond to such theories of gravity with different field content and different bulk actions, for example, different values of the coupling constants inSbulk. The example which is understood best is the case of the 𝒩=4 super Yang-Mills theory (SYM) in four dimensions. This is dual to maximal supergravity in AdS5 (which arises by dimensional reduction of ten-dimensional IIB supergravity on AdS5×S5). In that case, we know the precise values of many of the coefficients in the bulk action. This will not be very relevant for our discussion below. An important conceptual point is that the values of the bulk parameters which are realizable will in general be discrete.5 This discreteness is hidden by the classical limit.We will focus on the case of relativistic CFT for a while, but let me emphasize here that the name “AdS/CFT” is a very poor one: the correspondence is much more general. It can describe deformations of UV fixed points by relevant operators, and it has been extended to cases which are not even relativistic CFTs in the UV: examples include fixed points with dynamical critical exponentz≠1 [16], Galilean-invariant theories [17, 18], and theories which do more exotic things in the UV like the “duality cascade” of [19].


### 2.1. Counting of Degrees of Freedom

We can already make a check of the conjecture that a gravity theory inAdSd+1 might be dual to a QFT in d dimensions. The holographic principle tells us that the area of the boundary in Planck units is the number of degrees of freedom (dof), that is, the maximum entropy: (2.8)Areaofboundary4GN=?numberofdofofQFT≡Nd.
Is this true [20]? Yes: both sides are equal to infinity. We need to regulate our counting.Let's regulate the field theory first. There are both UV and IR divergences. We put the thing on a lattice, introducing a short-distance cut-offϵ (e.g., the lattice spacing) and we put it in a cubical box of linear size R. The total number of degrees of freedom is the number of cells (R/ϵ)d-1, times the number of degrees of freedom per lattice site, which we will call “N2”. The behavior suggested by the name we have given this number is found in well-understood examples. It is, however, clear (e.g., from the structure of known AdS vacua of string theory [21]) that other behaviors Nb are possible, and that's why I made it a funny color and put it in quotes. So Nd=(Rd-1/ϵd-1)N2.The picture we have of AdSd+1 is a collection of copies of d-dimensional Minkowski space of varying size; the boundary is the locus z→0 where they get really big. The area of the boundary is  (2.9)A=∫ℝd-1,z→0,fixedtgdd-1x=∫ℝd-1,z→0dd-1xLd-1zd-1.
As in the field theory counting, this is infinite for two reasons: from the integral over x and from the fact that z is going to zero. To regulate this integral, we integrate not to z=0 but rather cut it off at z=ϵ. We will see below a great deal more evidence for this idea that the boundary of AdS is associated with the UV behavior of the field theory, and that cutting off the geometry at z=ϵ is a UV cutoff (not identical to the lattice cutoff, but close enough for our present purposes). Given this,  (2.10)A=∫0Rdd-1xLd-1zd-1|z=ϵ=(RLϵ)d-1.
The holographic principle then  says that the maximum entropy in the bulk is  (2.11)A4GN~Ld-14GN(Rϵ)d-1.We see that the scaling with the system size agrees—both hand side go likeRd-1. So AdS/CFT is indeed an implementation of the holographic principle. We can learn more from this calcluation: In order for the prefactors of Rd-1 to agree, we need to relate the AdS radius in Planck units Ld-1/GN~(LMpl)d-1 to the number of degrees of freedom per site of the field theory:  (2.12)Ld-1GN=N2
up to numerical prefactors.


### 2.2. Preview of the AdS/CFT Correspondence

Here is the ideology:(2.13)fieldsinAdS⟷localoperatorsofCFTspinspinmassscalingdimensionΔ
In particular, for a scalar field in AdS, the formula relating the mass of the scalar field to the scaling dimension of the corresponding operator in the CFT is m2LAdS2=Δ(Δ-d), as we will show in Section 4.1.One immediate lesson from this formula is that a simple bulk theory with a small number of light fields is dual to a CFT with a hierarchy in its spectrum of operator dimensions. In particular, there need to be a small number of operators with small (e.g., of orderN0) dimensions. If you are aware of explicit examples of such theories, please let me know.6,  7 This is to be distinguished from the thus-far-intractable case where some whole tower of massive string modes in the bulk is needed.Now let us consider some observables of a QFT (we'll assume Euclidean spacetime for now), namely vacuum correlation functions of local operators in the CFT:(2.14)〈𝒪1(x1)𝒪2(x2)⋯𝒪n(xn)〉.
We can write down a generating functional Z[J] for these correlators by perturbing the action of the QFT: (2.15)ℒ(x)→ℒ(x)+∑AJA(x)𝒪A(x)≡ℒ(x)+ℒJ(x),Z[J]=〈e-∫ℒJ〉CFT,
where JA(x) are arbitrary functions (sources) and {𝒪A(x)} is some basis of local operators. The n-point function is then given by (2.16)〈∏n𝒪n(xn)〉=∏nδδJn(xn)lnZ|J=0.SinceℒJ is a UV perturbation (because it is a perturbation of the bare Lagrangian by local operators), in AdS it corresponds to a perturbation near the boundary, z→0. (Recall from the counting of degrees of freedom in Section 2.1 QFT with UV cutoff E<1/ϵ↔ AdS cutoff z>ϵ.) The perturbation J of the CFT action will be encoded in the boundary condition on bulk fields.The idea ([22, 23], often referred to as GKPW) for computing Z[J] is then schematically (2.17)Z[J]≡〈e-∫ℒJ〉CFT=ZQG[b.c.dependsonJ]︸=???~N≫1e-Sgrav|EOM,b.c.dependonJ.
The middle object is the partition function of quantum gravity. We do not have a very useful idea of what this is, except in perturbation theory and via this very equality. In a limit where this gravity theory becomes classical, however, we know quite well what we are doing, and we can do the path integral by saddle point, as indicated on the RHS of (2.17).An important point here is that even though we are claiming that the QFT path integral is dominated by a classical saddle point, this does not mean that the field theory degrees of freedom are free. How this works depends on what kind of large-N limit we take to make the gravity theory classical. This is our next subject.


## 2.1. Counting of Degrees of Freedom

We can already make a check of the conjecture that a gravity theory inAdSd+1 might be dual to a QFT in d dimensions. The holographic principle tells us that the area of the boundary in Planck units is the number of degrees of freedom (dof), that is, the maximum entropy: (2.8)Areaofboundary4GN=?numberofdofofQFT≡Nd.
Is this true [20]? Yes: both sides are equal to infinity. We need to regulate our counting.Let's regulate the field theory first. There are both UV and IR divergences. We put the thing on a lattice, introducing a short-distance cut-offϵ (e.g., the lattice spacing) and we put it in a cubical box of linear size R. The total number of degrees of freedom is the number of cells (R/ϵ)d-1, times the number of degrees of freedom per lattice site, which we will call “N2”. The behavior suggested by the name we have given this number is found in well-understood examples. It is, however, clear (e.g., from the structure of known AdS vacua of string theory [21]) that other behaviors Nb are possible, and that's why I made it a funny color and put it in quotes. So Nd=(Rd-1/ϵd-1)N2.The picture we have of AdSd+1 is a collection of copies of d-dimensional Minkowski space of varying size; the boundary is the locus z→0 where they get really big. The area of the boundary is  (2.9)A=∫ℝd-1,z→0,fixedtgdd-1x=∫ℝd-1,z→0dd-1xLd-1zd-1.
As in the field theory counting, this is infinite for two reasons: from the integral over x and from the fact that z is going to zero. To regulate this integral, we integrate not to z=0 but rather cut it off at z=ϵ. We will see below a great deal more evidence for this idea that the boundary of AdS is associated with the UV behavior of the field theory, and that cutting off the geometry at z=ϵ is a UV cutoff (not identical to the lattice cutoff, but close enough for our present purposes). Given this,  (2.10)A=∫0Rdd-1xLd-1zd-1|z=ϵ=(RLϵ)d-1.
The holographic principle then  says that the maximum entropy in the bulk is  (2.11)A4GN~Ld-14GN(Rϵ)d-1.We see that the scaling with the system size agrees—both hand side go likeRd-1. So AdS/CFT is indeed an implementation of the holographic principle. We can learn more from this calcluation: In order for the prefactors of Rd-1 to agree, we need to relate the AdS radius in Planck units Ld-1/GN~(LMpl)d-1 to the number of degrees of freedom per site of the field theory:  (2.12)Ld-1GN=N2
up to numerical prefactors.


## 2.2. Preview of the AdS/CFT Correspondence

Here is the ideology:(2.13)fieldsinAdS⟷localoperatorsofCFTspinspinmassscalingdimensionΔ
In particular, for a scalar field in AdS, the formula relating the mass of the scalar field to the scaling dimension of the corresponding operator in the CFT is m2LAdS2=Δ(Δ-d), as we will show in Section 4.1.One immediate lesson from this formula is that a simple bulk theory with a small number of light fields is dual to a CFT with a hierarchy in its spectrum of operator dimensions. In particular, there need to be a small number of operators with small (e.g., of orderN0) dimensions. If you are aware of explicit examples of such theories, please let me know.6,  7 This is to be distinguished from the thus-far-intractable case where some whole tower of massive string modes in the bulk is needed.Now let us consider some observables of a QFT (we'll assume Euclidean spacetime for now), namely vacuum correlation functions of local operators in the CFT:(2.14)〈𝒪1(x1)𝒪2(x2)⋯𝒪n(xn)〉.
We can write down a generating functional Z[J] for these correlators by perturbing the action of the QFT: (2.15)ℒ(x)→ℒ(x)+∑AJA(x)𝒪A(x)≡ℒ(x)+ℒJ(x),Z[J]=〈e-∫ℒJ〉CFT,
where JA(x) are arbitrary functions (sources) and {𝒪A(x)} is some basis of local operators. The n-point function is then given by (2.16)〈∏n𝒪n(xn)〉=∏nδδJn(xn)lnZ|J=0.SinceℒJ is a UV perturbation (because it is a perturbation of the bare Lagrangian by local operators), in AdS it corresponds to a perturbation near the boundary, z→0. (Recall from the counting of degrees of freedom in Section 2.1 QFT with UV cutoff E<1/ϵ↔ AdS cutoff z>ϵ.) The perturbation J of the CFT action will be encoded in the boundary condition on bulk fields.The idea ([22, 23], often referred to as GKPW) for computing Z[J] is then schematically (2.17)Z[J]≡〈e-∫ℒJ〉CFT=ZQG[b.c.dependsonJ]︸=???~N≫1e-Sgrav|EOM,b.c.dependonJ.
The middle object is the partition function of quantum gravity. We do not have a very useful idea of what this is, except in perturbation theory and via this very equality. In a limit where this gravity theory becomes classical, however, we know quite well what we are doing, and we can do the path integral by saddle point, as indicated on the RHS of (2.17).An important point here is that even though we are claiming that the QFT path integral is dominated by a classical saddle point, this does not mean that the field theory degrees of freedom are free. How this works depends on what kind of large-N limit we take to make the gravity theory classical. This is our next subject.


## 3. When Is the Gravity Theory Classical?

So we have said that some QFT path integrals are dominated by saddle points8 where the degrees of freedom near the saddle are those of a gravitational theory in extra dimensions:  (3.1)ZsomeQFTs[sources]≈e-Sbulk[boundaryconditionsatz→0]|extremumofSbulk.
The sharpness of the saddle (the size of the second derivatives of the action evaluated at the saddle) is equivalent to the classicalness of the bulk theory. In a theory of gravity, this is controlled by the Newton constant in front of the action. More precisely, in an asymptotically AdS space with AdS radius L, the theory is classical when  (3.2)Ld-1GN≡“N2”≫1.
This quantity, the AdS radius in Planck units Ld-1/GN≡(LMpl)d-1, is what we identified (using the holographic principle) as the number of degrees of freedom per site of the QFT.In the context of our current goal, it is worth spending some time talking about different kinds of large-species limits of QFTs. In particular, in the condensed matter literature, the phrase “large-enn” usually means that one promotes a two-component object to ann-component vector, with O(n)-invariant interactions. This is probably not what we need to have a simple gravity dual, for the reasons described next.


### 3.1. Largen Vector Models

A simple paradigmatic example of this vector-like large-n limit (I use a different n to distinguish it from the matrix case to be discussed next) is a QFT of n scalar fields φ⃗=(φ1,…,φn) with the following action:  (3.3)S[φ]=-12∫ddx(∂μφ⃗∂μφ⃗+m2φ⃗·φ⃗+λvn(φ⃗·φ⃗)2).
The fields φ⃗ transform in the fundamental representation of the O(n) symmetry group. Some foresight has been used to determine that the quartic coupling λv is to be held fixed in the large-n limit. An effective description (i.e., a well-defined saddle-point) can be found in terms of σ≡φ⃗·φ⃗ by standard path-integral tricks, and the effective action for σ is  (3.4)Seff[σ]=-n2[∫σ22λ+trln(-∂2+m2+σ)].
The important thing is the giant factor of n in front of the action which makes the theory of σ classical. Alternatively, the only interactions in this n vector model are “cactus” diagrams; this means that, modulo some self energy corrections, the theory is free.So we have found a description of this saddle pointwithin weakly coupled quantum field theory. The Principle of Conservation of Evil then suggests that this should not also be a simple, classical theory of gravity. Klebanov and Polyakov [24] have suggested what the (not simple) gravity dual might be.


### 3.2. 't Hooft Counting

“You can hide a lot in a large-Nmatrix."Steve ShenkerGiven some system with a few degrees of freedom, there exist many interesting large-N generalizations, many of which may admit saddle-point descriptions. It is not guaranteed that the effective degrees of freedom near the saddle (sometimes ominously called “the masterfield”) are simple field theory degrees of freedom (at least not in the same number of dimensions). If they are not, this means that such a limit is not immediately useful, but it is not necessarily more distant from the physical situation than the limit of the previous subsection. In fact, we will see dramatically below that the 't Hooft limit described here preserves more features of the interacting small-N theory than the usual vector-like limit. The remaining problem is to find a description of the masterfield, and this is precisely what is accomplished by AdS/CFT.Next we describe in detail a large-N limit (found by 't Hooft9) where the right degrees of freedom seem to be closed strings (and hence gravity). In this case, the number of degrees of freedom per point in the QFT will go like N2. Evidence from the space of string vacua suggests that there are many generalizations of this where the number of dofs per point goes like Nb for b≠2 [21]. However, a generalization of the 't Hooft limit is not yet well understood for other cases.10Consider a (any) quantum field theory withmatrix fields, Φa=1,…,N̅b=1,…,N. By matrix fields, we mean that their products appear in the Lagrangian only in the form of matrix multiplication, for example, (Φ2)ac=ΦabΦbc, which is a big restriction on the interactions. It means the interactions must be invariant under Φ→U-1ΦU; for concreteness we will take the matrix group to be U∈U(N).11 The fact that this theory has many more interaction terms than the vector model with the same number of fields (which would have a much larger O(N2) symmetry) changes the scaling of the coupling in the large N limit.In particular, consider the't Hooft limit in which N→∞ and g→0 with λ=g2N held fixed in the limit. Is the theory free in this limit? The answer turns out to be no. The loophole is that even though the coupling goes to zero, the number of modes diverges. Compared to the vector model, the quartic coupling in the matrix model g~1/N goes to zero slower than the coupling in the vector model gv≡λv/N~1/N.We will be agnostic here about whether theU(N) symmetry is gauged, but if it is not, there are many more states than we can handle using the gravity dual. The important role of the gauge symmetry for our purpose is to restrict the physical spectrum to gauge-invariant operators, like trΦk.The fields can have all kinds of spin labels and global symmetry labels, but we will just call themΦ. In fact, the location in space can also for the purposes of the discussion of this section be considered as merely a label on the field (which we are suppressing). So consider a schematic Lagrangian of the following form: (3.5)ℒ~1g2Tr((∂Φ)2+Φ2+Φ3+Φ4+⋯).
I suppose that we want Φ to be Hermitian so that this Lagrangian is real, but this will not be important for our considerations.We will now draw some diagrams which let us keep track of theN-dependence of various quantities. It is convenient to adopt the double line notation, in which oriented index lines follow conserved color flow. We denote the propagator:12(3.6)and the vertices by(3.7)Created by Brian Swingle.To see the consequences of this more concretely, let us consider some vacuum-to-vacuum diagrams (see Figures3 and 4 for illustration). We will keep track of the color structure and not worry even about how many dimensions we are in (the theory could even be zero-dimensional, such as the matrix integral which constructs the Wigner-Dyson distribution).A general diagram consists of propagators, interaction vertices, and index loops, and gives a contribution:(3.8)diagram~(λN)no.ofprop.(Nλ)no.ofint.vert.Nno.ofindexloops.
For example, the diagram in Figure 2 has 4 three-point vertices, 6 propagators, and 4 index loops, giving the final result N2λ2. In Figure 3 we have a set of planar graphs, meaning that we can draw them on a piece of paper without any lines crossing; their contributions take the general form λnN2. However, there also exist nonplanar graphs, such as the one in Figure 4, whose contributions are down by (an even number of) powers of N. One thing that is great about this expansion is that the diagrams which are harder to draw are less important.Figure 2
This diagram consists of 4 three-point vertices, 6 propagators, and 4 index loops.Planar graphs that contribute to the vacuum→ vacuum amplitude.
(a)(b)(c)Figure 4
Non-planar (but still oriented!) graph that contributes to the vacuum→ vacuum amplitude. Created by Wing-Ko Ho.We can be more precise about how the diagrams are organized. Every double-line graph specifies a triangulation of a 2-dimensional surfaceΣ. There are two ways to construct the explicit mapping.Method 1 (direct surface).
Fill in index loops with little plaquettes.Method 2 (dual surface).
(1) draw a vertex13 in every index loop and (2) draw an edge across every propagator.These constructions are illustrated in Figures5 and 6.Direct surfaces constructed from the vacuum diagram in (a) Figure3(a) and (b) Figure 4.
(a)(b)Figure 6
Dual surface constructed from the vacuum diagram in Figure3(c). Note that points at infinity are identified.IfE= number of propagators, V= number of vertices, and F= number of index loops, then the diagram gives a contribution NF-E+VλE-V. The letters refer to the “direct” triangulation of the surface in which interaction vertices are triangulation vertices. Then we interpret E as the number of edges, F as the number of faces, and V as the number of vertices in the triangulation. In the dual triangulation there are dual faces F̃, dual edges Ẽ, and dual vertices Ṽ. The relationship between the original and dual variables is E=Ẽ, V=F̃, and F=Ṽ. The exponent χ=F-E+V=F̃-Ẽ+Ṽ is the Euler character and it is a topological invariant of two-dimensional surfaces. In general it is given by χ(Σ)=2-2h-b where h is the number of handles (the genus) and b is the number of boundaries. Note that the exponent of λ, E-V or Ẽ-F̃ is not a topological invariant and depends on the triangulation (Feynman diagram).Because theN-counting is topological (depending only on χ(Σ)), we can sensibly organize the perturbation series for the effective action lnZ in terms of a sum over surface topology. Because we are computing only vacuum diagrams for the moment, the surfaces we are considering have no boundaries b=0 and are classified by their number of handles h (h=0 is the two-dimensional sphere, h=1 is the torus, and so on). We may write the effective action (the sum over connected vacuum-to-vacuum diagrams) as (3.9)lnZ=∑h=0∞N2-2h∑ℓ=0∞cℓ,hλℓ=∑h=0∞N2-2h𝔉h(λ),
where the sum over topologies is explicit.Now we can see some similarities between this expansion and perturbative string expansions.141/N plays the role of the string coupling gs, the amplitude joining and splitting of the closed strings. In the large N limit, this process is suppressed and the theory is classical. Closed string theory generically predicts gravity, with Newton's constant GN∝gs2; so this reproduces our result GN~N-2 from the holographic counting of degrees of freedom (this time, without the quotes around it).It is reasonable to ask what plays the role of the worldsheet coupling: there is a 2d QFT living on the worldsheet of the string, which describes its embeddings into the target space; this theory has a weak-coupling limit when the target-space curvatureL-2 is small, and it can be studied in perturbation theory in powers of ℓs/L, where ℓs-2 is the string tension. We can think of λ as a sort of chemical potential for edges in our triangulation. Looking back at our diagram counting we can see that if λ becomes large then diagrams with lots of edges are important. Thus large λ encourages a smoother triangulation of the worldsheet which we might interpret as fewer quantum fluctuations on the worldsheet. We expect a relation of the form λ-1~α' which encodes our intuition about large λ suppressing fluctuations. This is what is found in well-understood examples.This story is very general in the sense that all matrix models define something like a theory of two-dimensional fluctuating surfaces via these random triangulations. The connection is even more interesting when we remember all the extra labels we have been suppressing on our fieldΦ. For example, the position labeling where the field Φ sits plays the role of embedding coordinates on the worldsheet. Other indices (spin, etc.) indicate further worldsheet degrees of freedom. However, the microscopic details of the worldsheet theory are not so easily discovered. It took about fifteen years between the time when 't Hooft described the large-N perturbation series in this way and the first examples where the worldsheet dynamics were identified (these old examples are reviewed in, e.g., [25]).As a final check on the nontriviality of the theory in the 't Hooft limit, let us see if the 't Hooft coupling runs with scale. For argument let us think about the case when the matrices are gauge fields andL=-(1/gYM2)trFμνFμν. In d dimensions, the behavior through one loop is  (3.10)μ∂μgYM≡βg~4-d2gYM+b0gYM3N.
(b0 is a coefficient which depends on the matter content and vanishes for 𝒩=4 SYM.) So we find that βλ~((4-d)/2)λ+b0λ2. Thus λ can still run in the large N limit and the theory is nontrivial.


### 3.3.N-Counting of Correlation Functions

Let us now consider theN-counting for correlation functions of local gauge-invariant operators. Motivated by gauge invariance and simplicity, we will consider “single trace" operators, operators 𝒪(x) that look like (3.11)𝒪(x)=c(k,N)Tr(Φ1(x)⋯Φk(x))
and which we will abbreviate as Tr(Φk). We will keep k finite as N→∞.15 There are two little complications here. We must be careful about how we normalize the fields Φ and we must be careful about how we normalize the operator 𝒪. The normalization of the fields will continue to be such that the Lagrangian takes the form L=(1/gYM2)ℒ=(N/λ)ℒ with ℒ(Φ) containing no explicit factors of N. To fix the normalization of 𝒪 (to determine the constant c(k,N)) we will demand that when acting on the vacuum, the operator 𝒪 creates states of finite norm in the large-N limit, that is, 〈𝒪𝒪〉c~N0 where the subscript c stands for connected.To determinec(k,N) we need to know how to insert single trace operators into the t'Hooft counting. Each single-trace operator in the correlator is a new vertex which is required to be present in every contributing diagram. This vertex has k legs where k propagators can be attached and looks like a big squid. An example of such a new vertex appears in Figure 7 which corresponds to the insertion  of the operator Tr(Φ6). For the moment we do not associate any explicit factors of N with the new vertex. Let us consider the example 〈Tr(Φ4)Tr(Φ4)〉. We need to draw two four point vertices for the two single trace operators in the correlation function. How are we to connect these vertices with propagators? The dominant contribution comes from disconnected diagrams like the one shown in Figure 8. The leading disconnected diagram has four propagators and six index loops and so gives a factor λ4N2~N2. On the other hand, the leading connected diagram shown in Figure 9 has four propagators and four index loops and so only gives a contribution λ4~N0. (A way to draw the connected diagram in Figure 9 which makes the N-counting easier is shown in Figure 10 where we have deformed the two four point operator insertion vertices so that they are “ready for contraction".)Figure 7
New vertex for an operator insertion ofTr(Φk) with k=6.Disconnected diagram contributing to the correlation function〈Tr(Φ4)Tr(Φ4)〉. Created by Brian Swingle.
(a)(b)Figure 9
Connected diagram contributing to the correlation function〈Tr(Φ4)Tr(Φ4)〉.Figure 10
A redrawing of the connected diagram shown in Figure9.The fact that disconnected diagrams win in the largeN limit is general and goes by the name “large-N factorization". It says that single trace operators are basically classical objects in the large-N limit 〈𝒪𝒪〉~〈𝒪〉〈𝒪〉+𝒪(1/N2).The leading connected contribution to the correlation function is independent ofN and so 〈𝒪𝒪〉c~c2N0. Requiring that 〈𝒪𝒪〉c~N0 means that we can just set c~N0. Having fixed the normalization of 𝒪 we can now determine the N-dependence of higher-order correlation functions. For example, the leading connected diagram for 〈𝒪3〉 where 𝒪=Tr(Φ2) is just a triangle and contributes a factor λ3N-1~N-1. In fact, quite generally we have 〈𝒪n〉c~N2-n for the leading contribution.So the operators𝒪 (called glueballs in the context of QCD) create excitations of the theory that are free at large N—they interact with coupling 1/N. In QCD with N=3, quarks and gluons interact strongly, and so do their hadron composites. The role of large-N here is to make the color-neutral objects weakly interacting, in spite of the strong interactions of the constituents. So this is the sense in which the theory is classical: although the dimensions of these operators can be highly nontrivial (examples are known where they are irrational [26]), the dimensions of their products are additive at leading order in N.Finally, we should make a comment about theN-scaling of the generating functional Z=e-W=〈e-N∑AλA𝒪A〉. We have normalized the sources so that each λA is an 't Hooft-like coupling, in that it is finite as N→∞. The effective action W is the sum of connected vacuum diagrams, which at large-N is dominated by the planar diagrams. As we have shown, their contributions go like N2. This agrees with our normalization of the gravity action,  (3.12)Sbulk~Ld-1GNIdimensionless~N2Idimensionless.


### 3.4. Simple Generalizations

We can generalize the analysis performed so far without too much effort. One possibility is the addition of fields, “quarks”, in the fundamental ofU(N). We can add fermions ΔL~q̅γμDμq or bosons ΔL~|Dμq|2. Because quarks are in the fundamental of U(N), their propagator consists of only a single line. When using Feynman diagrams to triangulate surfaces we now have the possibility of surfaces with boundary. Two quark diagrams are shown in Figure 11 both of which triangulate a disk. Notice in particular the presence of only a single outer line representing the quark propagator. We can conclude that adding quarks into our theory corresponds to admitting open strings into the string theory. We can also consider "meson" operators like q̅q or q̅Φkq in addition to single trace operators. The extension of the holographic correspondence to include this case [27] has had many applications [28, 29], which are not discussed here for lack of time.A quark vacuum bubble and a quark vacuum bubble with “gluon” exchange. Created by Brian Swingle.
(a)(b)Another direction for generalization is to consider different matrix groups such asSO(N) or Sp(N). The adjoint of U(N) is just the fundamental times the antifundamental. However, the adjoint representations of SO(N) and Sp(N) are more complicated. For SO(N) the adjoint is given by the antisymmetric product of two fundamentals (vectors), and for Sp(N) the adjoint is the symmetric product of two fundamentals. In both of these cases, the lines in the double-line formalism no longer have arrows. As a consequence, the lines in the propagator for the matrix field can join directly or cross and then join as shown in Figure 12. In the string language the worldsheet can now be unoriented, an example being given by a matrix field vacuum bubble where the lines cross giving rise to the worldsheet ℝℙ2.Figure 12
Propagator for SO(N) (+) or Sp(N) (-) matrix models.Feynman graphs in AdS. We did the one with two external legs in Section4.3. Created by Francesco D’Eramo.
(a)(b)


## 3.1. Largen Vector Models

A simple paradigmatic example of this vector-like large-n limit (I use a different n to distinguish it from the matrix case to be discussed next) is a QFT of n scalar fields φ⃗=(φ1,…,φn) with the following action:  (3.3)S[φ]=-12∫ddx(∂μφ⃗∂μφ⃗+m2φ⃗·φ⃗+λvn(φ⃗·φ⃗)2).
The fields φ⃗ transform in the fundamental representation of the O(n) symmetry group. Some foresight has been used to determine that the quartic coupling λv is to be held fixed in the large-n limit. An effective description (i.e., a well-defined saddle-point) can be found in terms of σ≡φ⃗·φ⃗ by standard path-integral tricks, and the effective action for σ is  (3.4)Seff[σ]=-n2[∫σ22λ+trln(-∂2+m2+σ)].
The important thing is the giant factor of n in front of the action which makes the theory of σ classical. Alternatively, the only interactions in this n vector model are “cactus” diagrams; this means that, modulo some self energy corrections, the theory is free.So we have found a description of this saddle pointwithin weakly coupled quantum field theory. The Principle of Conservation of Evil then suggests that this should not also be a simple, classical theory of gravity. Klebanov and Polyakov [24] have suggested what the (not simple) gravity dual might be.


## 3.2. 't Hooft Counting

“You can hide a lot in a large-Nmatrix."Steve ShenkerGiven some system with a few degrees of freedom, there exist many interesting large-N generalizations, many of which may admit saddle-point descriptions. It is not guaranteed that the effective degrees of freedom near the saddle (sometimes ominously called “the masterfield”) are simple field theory degrees of freedom (at least not in the same number of dimensions). If they are not, this means that such a limit is not immediately useful, but it is not necessarily more distant from the physical situation than the limit of the previous subsection. In fact, we will see dramatically below that the 't Hooft limit described here preserves more features of the interacting small-N theory than the usual vector-like limit. The remaining problem is to find a description of the masterfield, and this is precisely what is accomplished by AdS/CFT.Next we describe in detail a large-N limit (found by 't Hooft9) where the right degrees of freedom seem to be closed strings (and hence gravity). In this case, the number of degrees of freedom per point in the QFT will go like N2. Evidence from the space of string vacua suggests that there are many generalizations of this where the number of dofs per point goes like Nb for b≠2 [21]. However, a generalization of the 't Hooft limit is not yet well understood for other cases.10Consider a (any) quantum field theory withmatrix fields, Φa=1,…,N̅b=1,…,N. By matrix fields, we mean that their products appear in the Lagrangian only in the form of matrix multiplication, for example, (Φ2)ac=ΦabΦbc, which is a big restriction on the interactions. It means the interactions must be invariant under Φ→U-1ΦU; for concreteness we will take the matrix group to be U∈U(N).11 The fact that this theory has many more interaction terms than the vector model with the same number of fields (which would have a much larger O(N2) symmetry) changes the scaling of the coupling in the large N limit.In particular, consider the't Hooft limit in which N→∞ and g→0 with λ=g2N held fixed in the limit. Is the theory free in this limit? The answer turns out to be no. The loophole is that even though the coupling goes to zero, the number of modes diverges. Compared to the vector model, the quartic coupling in the matrix model g~1/N goes to zero slower than the coupling in the vector model gv≡λv/N~1/N.We will be agnostic here about whether theU(N) symmetry is gauged, but if it is not, there are many more states than we can handle using the gravity dual. The important role of the gauge symmetry for our purpose is to restrict the physical spectrum to gauge-invariant operators, like trΦk.The fields can have all kinds of spin labels and global symmetry labels, but we will just call themΦ. In fact, the location in space can also for the purposes of the discussion of this section be considered as merely a label on the field (which we are suppressing). So consider a schematic Lagrangian of the following form: (3.5)ℒ~1g2Tr((∂Φ)2+Φ2+Φ3+Φ4+⋯).
I suppose that we want Φ to be Hermitian so that this Lagrangian is real, but this will not be important for our considerations.We will now draw some diagrams which let us keep track of theN-dependence of various quantities. It is convenient to adopt the double line notation, in which oriented index lines follow conserved color flow. We denote the propagator:12(3.6)and the vertices by(3.7)Created by Brian Swingle.To see the consequences of this more concretely, let us consider some vacuum-to-vacuum diagrams (see Figures3 and 4 for illustration). We will keep track of the color structure and not worry even about how many dimensions we are in (the theory could even be zero-dimensional, such as the matrix integral which constructs the Wigner-Dyson distribution).A general diagram consists of propagators, interaction vertices, and index loops, and gives a contribution:(3.8)diagram~(λN)no.ofprop.(Nλ)no.ofint.vert.Nno.ofindexloops.
For example, the diagram in Figure 2 has 4 three-point vertices, 6 propagators, and 4 index loops, giving the final result N2λ2. In Figure 3 we have a set of planar graphs, meaning that we can draw them on a piece of paper without any lines crossing; their contributions take the general form λnN2. However, there also exist nonplanar graphs, such as the one in Figure 4, whose contributions are down by (an even number of) powers of N. One thing that is great about this expansion is that the diagrams which are harder to draw are less important.Figure 2
This diagram consists of 4 three-point vertices, 6 propagators, and 4 index loops.Planar graphs that contribute to the vacuum→ vacuum amplitude.
(a)(b)(c)Figure 4
Non-planar (but still oriented!) graph that contributes to the vacuum→ vacuum amplitude. Created by Wing-Ko Ho.We can be more precise about how the diagrams are organized. Every double-line graph specifies a triangulation of a 2-dimensional surfaceΣ. There are two ways to construct the explicit mapping.Method 1 (direct surface).
Fill in index loops with little plaquettes.Method 2 (dual surface).
(1) draw a vertex13 in every index loop and (2) draw an edge across every propagator.These constructions are illustrated in Figures5 and 6.Direct surfaces constructed from the vacuum diagram in (a) Figure3(a) and (b) Figure 4.
(a)(b)Figure 6
Dual surface constructed from the vacuum diagram in Figure3(c). Note that points at infinity are identified.IfE= number of propagators, V= number of vertices, and F= number of index loops, then the diagram gives a contribution NF-E+VλE-V. The letters refer to the “direct” triangulation of the surface in which interaction vertices are triangulation vertices. Then we interpret E as the number of edges, F as the number of faces, and V as the number of vertices in the triangulation. In the dual triangulation there are dual faces F̃, dual edges Ẽ, and dual vertices Ṽ. The relationship between the original and dual variables is E=Ẽ, V=F̃, and F=Ṽ. The exponent χ=F-E+V=F̃-Ẽ+Ṽ is the Euler character and it is a topological invariant of two-dimensional surfaces. In general it is given by χ(Σ)=2-2h-b where h is the number of handles (the genus) and b is the number of boundaries. Note that the exponent of λ, E-V or Ẽ-F̃ is not a topological invariant and depends on the triangulation (Feynman diagram).Because theN-counting is topological (depending only on χ(Σ)), we can sensibly organize the perturbation series for the effective action lnZ in terms of a sum over surface topology. Because we are computing only vacuum diagrams for the moment, the surfaces we are considering have no boundaries b=0 and are classified by their number of handles h (h=0 is the two-dimensional sphere, h=1 is the torus, and so on). We may write the effective action (the sum over connected vacuum-to-vacuum diagrams) as (3.9)lnZ=∑h=0∞N2-2h∑ℓ=0∞cℓ,hλℓ=∑h=0∞N2-2h𝔉h(λ),
where the sum over topologies is explicit.Now we can see some similarities between this expansion and perturbative string expansions.141/N plays the role of the string coupling gs, the amplitude joining and splitting of the closed strings. In the large N limit, this process is suppressed and the theory is classical. Closed string theory generically predicts gravity, with Newton's constant GN∝gs2; so this reproduces our result GN~N-2 from the holographic counting of degrees of freedom (this time, without the quotes around it).It is reasonable to ask what plays the role of the worldsheet coupling: there is a 2d QFT living on the worldsheet of the string, which describes its embeddings into the target space; this theory has a weak-coupling limit when the target-space curvatureL-2 is small, and it can be studied in perturbation theory in powers of ℓs/L, where ℓs-2 is the string tension. We can think of λ as a sort of chemical potential for edges in our triangulation. Looking back at our diagram counting we can see that if λ becomes large then diagrams with lots of edges are important. Thus large λ encourages a smoother triangulation of the worldsheet which we might interpret as fewer quantum fluctuations on the worldsheet. We expect a relation of the form λ-1~α' which encodes our intuition about large λ suppressing fluctuations. This is what is found in well-understood examples.This story is very general in the sense that all matrix models define something like a theory of two-dimensional fluctuating surfaces via these random triangulations. The connection is even more interesting when we remember all the extra labels we have been suppressing on our fieldΦ. For example, the position labeling where the field Φ sits plays the role of embedding coordinates on the worldsheet. Other indices (spin, etc.) indicate further worldsheet degrees of freedom. However, the microscopic details of the worldsheet theory are not so easily discovered. It took about fifteen years between the time when 't Hooft described the large-N perturbation series in this way and the first examples where the worldsheet dynamics were identified (these old examples are reviewed in, e.g., [25]).As a final check on the nontriviality of the theory in the 't Hooft limit, let us see if the 't Hooft coupling runs with scale. For argument let us think about the case when the matrices are gauge fields andL=-(1/gYM2)trFμνFμν. In d dimensions, the behavior through one loop is  (3.10)μ∂μgYM≡βg~4-d2gYM+b0gYM3N.
(b0 is a coefficient which depends on the matter content and vanishes for 𝒩=4 SYM.) So we find that βλ~((4-d)/2)λ+b0λ2. Thus λ can still run in the large N limit and the theory is nontrivial.


## 3.3.N-Counting of Correlation Functions

Let us now consider theN-counting for correlation functions of local gauge-invariant operators. Motivated by gauge invariance and simplicity, we will consider “single trace" operators, operators 𝒪(x) that look like (3.11)𝒪(x)=c(k,N)Tr(Φ1(x)⋯Φk(x))
and which we will abbreviate as Tr(Φk). We will keep k finite as N→∞.15 There are two little complications here. We must be careful about how we normalize the fields Φ and we must be careful about how we normalize the operator 𝒪. The normalization of the fields will continue to be such that the Lagrangian takes the form L=(1/gYM2)ℒ=(N/λ)ℒ with ℒ(Φ) containing no explicit factors of N. To fix the normalization of 𝒪 (to determine the constant c(k,N)) we will demand that when acting on the vacuum, the operator 𝒪 creates states of finite norm in the large-N limit, that is, 〈𝒪𝒪〉c~N0 where the subscript c stands for connected.To determinec(k,N) we need to know how to insert single trace operators into the t'Hooft counting. Each single-trace operator in the correlator is a new vertex which is required to be present in every contributing diagram. This vertex has k legs where k propagators can be attached and looks like a big squid. An example of such a new vertex appears in Figure 7 which corresponds to the insertion  of the operator Tr(Φ6). For the moment we do not associate any explicit factors of N with the new vertex. Let us consider the example 〈Tr(Φ4)Tr(Φ4)〉. We need to draw two four point vertices for the two single trace operators in the correlation function. How are we to connect these vertices with propagators? The dominant contribution comes from disconnected diagrams like the one shown in Figure 8. The leading disconnected diagram has four propagators and six index loops and so gives a factor λ4N2~N2. On the other hand, the leading connected diagram shown in Figure 9 has four propagators and four index loops and so only gives a contribution λ4~N0. (A way to draw the connected diagram in Figure 9 which makes the N-counting easier is shown in Figure 10 where we have deformed the two four point operator insertion vertices so that they are “ready for contraction".)Figure 7
New vertex for an operator insertion ofTr(Φk) with k=6.Disconnected diagram contributing to the correlation function〈Tr(Φ4)Tr(Φ4)〉. Created by Brian Swingle.
(a)(b)Figure 9
Connected diagram contributing to the correlation function〈Tr(Φ4)Tr(Φ4)〉.Figure 10
A redrawing of the connected diagram shown in Figure9.The fact that disconnected diagrams win in the largeN limit is general and goes by the name “large-N factorization". It says that single trace operators are basically classical objects in the large-N limit 〈𝒪𝒪〉~〈𝒪〉〈𝒪〉+𝒪(1/N2).The leading connected contribution to the correlation function is independent ofN and so 〈𝒪𝒪〉c~c2N0. Requiring that 〈𝒪𝒪〉c~N0 means that we can just set c~N0. Having fixed the normalization of 𝒪 we can now determine the N-dependence of higher-order correlation functions. For example, the leading connected diagram for 〈𝒪3〉 where 𝒪=Tr(Φ2) is just a triangle and contributes a factor λ3N-1~N-1. In fact, quite generally we have 〈𝒪n〉c~N2-n for the leading contribution.So the operators𝒪 (called glueballs in the context of QCD) create excitations of the theory that are free at large N—they interact with coupling 1/N. In QCD with N=3, quarks and gluons interact strongly, and so do their hadron composites. The role of large-N here is to make the color-neutral objects weakly interacting, in spite of the strong interactions of the constituents. So this is the sense in which the theory is classical: although the dimensions of these operators can be highly nontrivial (examples are known where they are irrational [26]), the dimensions of their products are additive at leading order in N.Finally, we should make a comment about theN-scaling of the generating functional Z=e-W=〈e-N∑AλA𝒪A〉. We have normalized the sources so that each λA is an 't Hooft-like coupling, in that it is finite as N→∞. The effective action W is the sum of connected vacuum diagrams, which at large-N is dominated by the planar diagrams. As we have shown, their contributions go like N2. This agrees with our normalization of the gravity action,  (3.12)Sbulk~Ld-1GNIdimensionless~N2Idimensionless.


## 3.4. Simple Generalizations

We can generalize the analysis performed so far without too much effort. One possibility is the addition of fields, “quarks”, in the fundamental ofU(N). We can add fermions ΔL~q̅γμDμq or bosons ΔL~|Dμq|2. Because quarks are in the fundamental of U(N), their propagator consists of only a single line. When using Feynman diagrams to triangulate surfaces we now have the possibility of surfaces with boundary. Two quark diagrams are shown in Figure 11 both of which triangulate a disk. Notice in particular the presence of only a single outer line representing the quark propagator. We can conclude that adding quarks into our theory corresponds to admitting open strings into the string theory. We can also consider "meson" operators like q̅q or q̅Φkq in addition to single trace operators. The extension of the holographic correspondence to include this case [27] has had many applications [28, 29], which are not discussed here for lack of time.A quark vacuum bubble and a quark vacuum bubble with “gluon” exchange. Created by Brian Swingle.
(a)(b)Another direction for generalization is to consider different matrix groups such asSO(N) or Sp(N). The adjoint of U(N) is just the fundamental times the antifundamental. However, the adjoint representations of SO(N) and Sp(N) are more complicated. For SO(N) the adjoint is given by the antisymmetric product of two fundamentals (vectors), and for Sp(N) the adjoint is the symmetric product of two fundamentals. In both of these cases, the lines in the double-line formalism no longer have arrows. As a consequence, the lines in the propagator for the matrix field can join directly or cross and then join as shown in Figure 12. In the string language the worldsheet can now be unoriented, an example being given by a matrix field vacuum bubble where the lines cross giving rise to the worldsheet ℝℙ2.Figure 12
Propagator for SO(N) (+) or Sp(N) (-) matrix models.Feynman graphs in AdS. We did the one with two external legs in Section4.3. Created by Francesco D’Eramo.
(a)(b)


## 4. Vacuum CFT Correlators from Fields in AdS

Our next goal is to evaluate〈e-∫ϕ0𝒪〉CFT≡e-WCFT[ϕ0], where ϕ0 is some small perturbation around some reference value associated with a CFT. You may not be interested in such a quantity in itself, but we will calculate it in a way which extends directly to more physically relevant quanitities (such as real-time thermal response functions). The general form of the AdS/CFT conjecture for the generating funcitonal is the GKPW equation [22, 23]  (4.1)〈e-∫ϕ0𝒪〉CFT=ZstringsinAdS[ϕ0].
This thing on the RHS is not yet a computationally effective object; the currently-practical version of the GKPW formula is the classical limit: (4.2)WCFT[ϕ0]=-ln〈e∫ϕ0𝒪〉CFT≃extremum“ϕ|z=ϵ~ϕ0"(N2Igrav[ϕ])+O(1N2)+O(1λ).There are many things to say about this formula.(i)
In the case of matrix theories like those described in the previous section, the classical gravity description is valid for largeN and large λ. In some examples there is only one parameter which controls the validity of the gravity description. In (4.2) we have made the N-dependence explicit: in units of the AdS radius, the Newton constant is Ld-1/GN=N2. Igrav is some dimensionless action.(ii)
We said that we are going to think ofϕ0 as a small perturbation. Let us then make a perturbative expansion in powers of ϕ0:(4.3)WCFT[ϕ0]=WCFT[0]+∫dDxϕ0(x)G1(x)+12∬dDx1dDx2ϕ0(x1)ϕ0(x2)G2(x1,x2)+⋯,
where (4.4)G1(x)=〈𝒪(x)〉=δWδϕ0(x)|ϕ0=0,G2(x)=〈𝒪(x1)𝒪(x2)〉c=δ2Wδϕ0(x1)δϕ0(x2)|ϕ0=0.
Now if there is no instability, then ϕ0 is small implies ϕ is small. For one thing, this means that we can ignore interactions of the bulk fields in computing two-point functions. For n-point functions, we will need to know terms in the bulk action of degree up to n in the fields.(i)
Anticipating divergences atz→0, we have introduced a cutoff in (4.2) (which will be a UV cutoff in the CFT) and set boundary conditions at z=ϵ. They are in quotes because they require a bit of refinement (this will happen in Section 4.1).(ii)
Equation (4.2) is written as if there is just one field in the bulk. Really there is a ϕ for every operator 𝒪 in the dual field theory. For such a pair, we will say “ϕ couples to 𝒪” at the boundary. How to match up fields in the bulk and operators in the QFT? In general this is hard and information from string theory is useful. Without specifying a definite field theory, we can say a few general things.
(1)
We can organize both hand sides into representations of the conformal group. In fact only conformal primary operators correspond to “elementary fields” in the gravity action, and their descendants correspond to derivatives of those fields. More about this loaded word “elementary” in a moment.(2)
Only “single-trace” operators (like thetrΦks of the previous section) correspond to “elementary fields” ϕ in the bulk. The excitations created by multitrace operators (like (trΦk)2) correspond to multiparticle states of ϕ (in this example, a 2-particle state). Here I should stop and emphasize that this word “elementary” is well defined because we have assumed that we have a weakly coupled theory in the bulk, and hence the Hilbert space is approximately a Fock space, organized according to the number of particles in the bulk. A well-defined notion of single-particle state depends on large-N—if N is not large, it is not true that the overlap between trΦ2trΦ2|0〉 and trΦ4|0〉 is small.16(3)
There are some simple examples of the correspondence between bulk fields and boundary operators that are determined by symmetry. The stress-energy tensorTμν is the response of a local QFT to local change in the metric, Sbdy∋∫γμνTμν.Here we are writingγμν for the metric on the boundary. In this case
(4.5)gμν⟷Tμν.Gauge fields in the bulk correspond to currents in the boundary theory:(4.6)Aμa⟷Jaμ,
that is, Sbdy∋∫AμaJaμ. We say this mostly because we can contract all the indices to make a singlet action. In the special case where the gauge field is massless, the current is conserved.(iii)
Finally, something that needs to be emphasized is that changing theLagrangian of the CFT (by changing ϕ0) is accomplished by changing the boundary condition in the bulk. The bulk equations of motion remain the same (e.g., the masses of the bulk fields do not change). This means that actually changing the bulk action corresponds to something more drastic in the boundary theory. One context in which it is useful to think about varying the bulk coupling constant is in thinking about the renormalization group. We motivated the form ∫(2Λ+ℛ+⋯) of the bulk action by Wilsonian naturalness, which is usually enforced by the RG; so this is a delicate point. For example, soon we will compute the ratio of the shear viscosity to the entropy density, η/s, for the plasma made from any CFT that has an Einstein gravity dual; the answer is always 1/4π. Each such CFT is what we usually think of as a universality class, since it will have some basin of attraction in the space of nearby QFT couplings. Here we are saying that a whole class of universality classes exhibits the same behavior.What is special about these theories from the QFT point of view? Our understanding of this “bulk universality” is obscured by our ignorance about quantum mechanics in the bulk. Physicists with what could be called a monovacuist inclination may say that what is special about them is that they exist.17 The issue, however, arises for interactions in the bulk which are quite a bit less contentious than gravity; so this seems unlikely to me to be the answer.


### 4.1. Wave Equation near the Boundary and Dimensions of Operators

The metric of AdS (in Poincaré coordinates, so that the constant-z slices are just copies of Minkowski space) is (4.7)ds2=L2dz2+dxμdxμz2≡gABdxAdxB,A=0,…,d,xA=(z,xμ).
As the simplest case to consider, let's think about a scalar field in the bulk. An action for such a scalar field suggested by Naturalness is (4.8)S=-𝔎2∫dd+1xg[gAB∂Aϕ∂Bϕ+m2ϕ2+bϕ3+⋯].
Here 𝔎 is just a normalization constant; we are assuming that the theory of ϕ is weakly coupled and one may think of 𝔎 as proportional to N2. For this metric g=|detg|=(L/z)d+1. Our immediate goal is to compute a two-point function of the operator 𝒪 to which ϕ couples, so we will ignore the interaction terms in (4.8) for a while. Since ϕ is a scalar field we can rewrite the kinetic term as (4.9)gAB∂Aϕ∂Bϕ=(∂ϕ)2=gABDAϕDBϕ,
where DA is the covariant derivative, which has the nice property that DA(gBC)=0, so we can move the Ds around the gs with impunity. By integrating by parts we can rewrite the action in a useful way:(4.10)S=-𝔎2∫dd+1x[∂A(ggABϕ∂Bϕ)-ϕ∂A(ggAB∂Bϕ)+g(m2ϕ2+⋯)]
and finally by using Stokes' theorem we can rewrite the action as (4.11)S=-𝔎2∫∂AdSddxggzBϕ∂Bϕ-𝔎2∫gϕ(-□+m2)ϕ+O(ϕ3),
where we define the scalar Laplacian □ϕ=(1/g)∂A(ggAB∂B)ϕ=DADAϕ. Note that we wrote all these covariant expressions without ever introducing Christoffel symbols.We can rewrite the boundary term more covariantly as(4.12)∫ℳgDAJA=∫∂ℳγnAJA.
The metric tensor γ is defined as (4.13)ds2|z=ϵ≡γμνdxμdxν=L2ϵ2ημνdxμdxν;
that is, it is the induced metric on the boundary surface z=ϵ. The vector nA is a unit vector normal to boundary (z=ϵ). We can find an explicit expression for it: (4.14)nA∝∂∂zgABnAnB|z=ϵ=1⇒n=1gzz∂∂z=zL∂∂z.From this discussion we have learned the following.(i)
The equation of motion for small fluctuations ofϕ is (-□+m2)ϕ̲=0.18(ii)
Ifϕ̲ solves the equation of motion, the on-shell action is just given by the boundary term.Next we will derive the promised formula relating bulk masses and operator dimensions(4.15)Δ(Δ-d)=m2L2
by studying the AdS wave equation near the boundary.Let us take advantage of translational invariance ind dimensions, xμ→xμ+aμ, to Fourier decompose the scalar field: (4.16)ϕ(z,xμ)=eikμxμfk(z),kμxμ≡-ωt+k⃗·x⃗.
In the Fourier basis, substituting (4.16) into the wave equation (-□+m2)ϕ=0 and using the fact that the metric only depends on z, the wave equation is (4.17)0=(gμνkμkν-1g∂z(ggzz∂z)+m2)fk(z)(4.18)=1L2[z2k2-zd+1∂z(z-d+1∂z)+m2L2]fk(z),
where we have used gμν=(z/L)2δμν. The solutions of (4.18) are Bessel functions; we can learn a lot without using that information. For example, look at the solutions near the boundary (i.e., z→0). In this limit we have power law solutions, which are spoiled by the z2k2 term. To see this, try using fk=zΔ in (4.18): (4.19)0=k2z2+Δ-zd+1∂z(Δz-d+Δ)+m2L2zΔ=(k2z2-Δ(Δ-d)+m2L2)zΔ,
and for z→0 we get (4.20)Δ(Δ-d)=m2L2.
The two roots of (4.20) are (4.21)Δ±=d2±(d2)2+m2L2.Comments
(i)
The solution proportional tozΔ- is bigger near z→0.(ii)
Δ+>0forallm, therefore zΔ+ decays near the boundary for any value of the mass.(iii)
Δ++Δ-=d.We want to impose boundary conditions that allow solutions. Since the leading behavior near the boundary of a generic solution isϕ~zΔ-, we impose (4.22)ϕ(x,z)|z=ϵ=ϕ0(x,ϵ)=ϵΔ-ϕ0Ren(x),
where ϕ0Ren is a renormalized source field. With this boundary condition ϕ0Ren is a finite quantity in the limit ϵ→0.Wavefunction Renormalization of𝒪 (Heuristic but useful)
Suppose that(4.23)Sbdy∋∫z=ϵddxγϵϕ0(x,ϵ)𝒪(x,ϵ)=∫ddx(Lϵ)d(ϵΔ-ϕ0Ren(x))𝒪(x,ϵ),
where we have used γ=(L/ϵ)d. Demanding this to be finite as ϵ→0 we get
(4.24)𝒪(x,ϵ)~ϵd-Δ-𝒪Ren(x)=ϵΔ+𝒪Ren(x),
where in the last line we have used Δ++Δ-=d. Therefore, the scaling dimension of 𝒪Ren is Δ+≡Δ. We will soon see that 〈𝒪(x)𝒪(0)〉~1/|x|2Δ, confirming that Δ is indeed the scaling dimension.We are solving a second-order ODE; therefore we need two conditions in order to determine a solution (for eachk). So far we have imposed one condition at the boundary of AdS.(i)
Forz→ϵ,ϕ~zΔ-ϕ0+ (terms subleading in z).In the Euclidean case (we discuss real time in the next subsection), we will also impose(ii)
ϕ regular in the interior of AdS (i.e., at z→∞).Comments onΔ
(1)
TheϵΔ- factor is independent of k and x, which is a consequence of a local QFT (this fails in exotic examples).(2)
Relevantness: If m2>0: This implies Δ≡Δ+>d, so 𝒪Δ is an irrelevant operator. This means that if you perturb the CFT by adding 𝒪Δ to the Lagrangian, then its coefficient is some mass scale to a negative power:
(4.25)ΔS=∫ddx(mass)d-Δ𝒪Δ,
where the exponent is negative; so the effects of such an operator go away in the IR, at energies E<mass. ϕ~zΔ-ϕ0is this coupling. It grows in the UV (small z). If ϕ0 is a finite perturbation, it will back-react on the metric and destroy the asymptotic AdS-ness of the geometry: extra data about the UV will be required.m2=0↔Δ=d means that 𝒪 is marginal.
Ifm2<0, then Δ<d; so 𝒪 is a relevant operator. Note that in AdS, m2<0 is ok if m2 is not too negative. Such fields with m2>-|mBF|2≡-(d/2L)2 are called “Breitenlohner-Freedman- (BF-) allowed tachyons”. The reason you might think that m2<0 is bad is that usually it means an instability of the vacuum at ϕ=0. An instability occurs when a normalizable mode grows with time without a source. But for m2<0, ϕ~zΔ- decays near the boundary (i.e., in the UV). This requires a gradient energy of order ~1/L, which can stop the field from condensing.
To see what is too negative, consider the formula for the dimension,Δ±=d/2±(d/2)2+m2L2. For m2<mBF2, the dimension becomes complex.(3)
The formula relating the mass of a bulk field and the dimension of the associated operator depends on their spin. For example, for a massive gauge field inAdS with action
(4.26)S=-∫AdS(14FμνFμν-12m2AμAμ),
the boundary behavior of the wave equation implies that Aμ~z→0zα with
(4.27)α(α-d+2)=m2L2.
For the particular case of Aμ massless this can be seen to lead to Δ(jμ)=d-1, which is the dimension of a conserved current in a CFT. Also, the fact that gμν is massless implies
(4.28)Δ(Tμν)=d,
which is required by conformal Ward identities.


### 4.2. Solutions of the AdS Wave Equation and Real-Time Issues

An approach which uses the symmetries ofAdS [23] is appealing. However, it is not always available (e.g., if there is a black hole in the spacetime). Also, it can be misleading: as in quantum field theory, scale-invariance is inevitably broken by the UV cutoff.Return to the scalar wave equation in momentum space:(4.29)0=[zd+1∂z(z-d+1∂z)-m2L2-z2k2]fk(z).
We will now stop being agnostic about the signature and confront some issues that arise for real time correlators. If k2>0, that is, kμ is spacelike (or Euclidean), the solution is  (4.30)fk(z)=AKzd/2Kν(kz)+AIzd/2Iν(kz),
where ν=Δ-d/2=(d/2)2+m2L2. In the interior of AdS (z→∞), the Bessel functions behave as  (4.31)Kν(kz)~e-kz,Iν(kz)~ekz.
So we see that the regularity in the interior uniquely fixes f̲k and hence the bulk-to-boundary propagator. Actually there is a theorem (the Graham-Lee theorem) addressing this issue for gravity fields (and not just linearly in the fluctuations); it states that if you specify a Euclidean metric on the boundary of a Euclidean AdS (which we can think of as topologically a Sd by adding the point at infinity in ℝd) modulo conformal rescaling, then the metric on the space inside of the Sd is uniquely determined. A similar result holds for gauge fields.In contrast to this, in Lorentzian signature with timelikek2, that is, for on-shell states with ω2>k⃗2, there exist two linearly independent solutions with the same leading behavior at the UV boundary. In terms of q≡ω2-k⃗2, the solutions are  (4.32)zd/2K±ν(iqz)~e±iqz(z→∞);
so these modes oscillate in the far IR region of the geometry.19 This ambiguity reflects the many possibilities for real-time Green's functions in the QFT. One useful choice is the retarded Green's function, which describes causal response of the system to a perturbation. This choice corresponds to a choice of boundary conditions at z→∞ describing stuff falling into the horizon [30], that is, moving towards larger z as time passes. There are three kinds of reasons for this prescription.20(i)
Both the retarded Green's functions and stuff falling through the horizon describe things thathappen, rather than unhappen.(ii)
You can check that this prescription gives the correct analytic structure ofGR(ω) ([30] and all the hundreds of papers that have used this prescription).(iii)
It has been derived from a holographic version of the Schwinger-Keldysh prescription [31–33].The fact that stuff goes past the horizon and does not come out is what breaks time-reversal invariance in the holographic computation ofGR. Here, the ingoing choice is ϕ(t,z)~e-iωt+iqz, since as t grows, the wavefront moves to larger z. This specifies the solution which computes the causal response to be zd/2K+ν(iqz).The same prescription, adapted to the black hole horizon, will work in the finite temperature case.One last thing we must deal with before proceeding is to define what we mean by a “normalizable” mode, or solution, when we say that we have many normalizable solutions fork2<0 with a given scaling behavior. In Euclidean space, ϕ is normalizable when S[ϕ]<∞. This is because when we are thinking about the partition function Z[ϕ]=∑ϕe-S[ϕ], modes with boundary conditions which force S[ϕ]=∞ would not contribute. In real-time, we say that ϕ is normalizable if E[ϕ]<∞ where  (4.33)E[ϕ]=∫Σdd-1xdzhTμν[ϕ]nμξν=∫x0=constantdd-1xdzhT00[ϕ],
where Σ is a given spatial slice, h is the induced metric on that slice, nμ is a normal unit vector to Σ, and ξμ is a timelike killing vector. TAB is defined as  (4.34)TAB[ϕ]=-2gδδgABSBulk[ϕ].


### 4.3. Bulk-to-Boundary Propagator in Momentum Space

We return to considering spacelikek in this section. Let us normalize our solution of the wave equation by the condition fk(z=ϵ)=1. This means that its Fourier transform is a δ-function in position space δd(x) when evaluated at z=ϵ, not at the actual boundary z=0. The solution, which we can call the “regulated bulk-to-boundary propagator”, is then  (4.35)f̲k(z)=zd/2Kν(kz)ϵd/2Kν(kϵ).
The general position space solution can be obtained by Fourier decomposition:  (4.36)ϕ̲[ϕ0](x)=∫ddkeikxf̲k(z)ϕ0(k,ϵ).
The “on-shell action” (i.e., the action evaluated on the saddle-point solution) is (using (4.11)) (4.37)S[ϕ̲]=-𝔎2∫ddxγϕ̲n·∂ϕ̲=-𝔎2∫ddx∫ddk1∫ddk2ei(k1+k2)xϕ0(k1,ϵ)ϕ0(k2,ϵ)Ld-1z-df̲k1(z)z∂zf̲k2(z)|z=ϵ=-𝔎Ld-12∫ddkϕ0(k,ϵ)ϕ0(-k,ϵ)ℱϵ(k),
and therefore21(4.38)〈𝒪(k1)𝒪(k2)〉cϵ=-δδϕ0(k1)δδϕ0(k2)S=(2π)dδd(k1+k2)ℱϵ(k1).
Here ℱϵ(k) (sometimes called the “flux factor”) is  (4.39)ℱϵ(k)=z-df̲-k(z)z∂zf̲k(z)|z=ϵ+(k⟷-k)=2ϵ-d+1∂z(zd/2Kν(kz)ϵd/2Kν(ϵz))|z=ϵ.
The small-u (near boundary) behavior of Kν(u) is (4.40)Kν(u)=u-ν(a0+a1u2+a2u4+⋯)(leadingterm)+uνlnu(b0+b1u2+b2u4+⋯)(subleadingterm),
where the coefficients of the series ai,bi depend on ν. For noninteger ν, there would be no lnu in the second line, and so we make it pink. Of course, we saw in the previous subsection (with very little work) that any solution of the bulk wave equation has this kind of form (the boundary is a regular singular point of the ODE). We could determine the as and bs recursively by the same procedure. This is just like a scattering problem in 1d quantum mechanics. The point of the Bessel function here is to choose which values of the coefficients a,b give a function which has the correct behavior at the other end, that is, at z→∞. Plugging the asymptotic expansion of the Bessel function into (4.39), (4.41)ℱϵ(k)=2ϵ-d+1∂z((kz)-ν+d/2(a0+⋯)+(kz)ν+d/2lnkz(b0+⋯)(kϵ)-ν+d/2(a0+⋯)+(kϵ)ν+d/2lnkϵ(b0+⋯))|z=ϵ=2ϵ-d[{d2-ν(1+c2(ϵ2k2)+c4(ϵ4k4)+⋯)}+{ν2b0a0(ϵk)2νIn(ϵk)(1+d2(ϵk)2+⋯)}]≡(I)+(II),
where (I) and (II) denote the first and second groups of terms of the previous line.(I) is a Laurent series inϵ with coefficients which are positive powers of k (i.e., analytic in k at k=0). These are contact terms, that is, short distance goo that we do not care about and can subtract off. We can see this by noting that  (4.42)∫ddke-ikx(ϵk)2mϵ-d=ϵ2m-d□xmδd(x)
for m>0. The ϵ2m-d factor reinforces the notion that ϵ, which is an IR cutoff in AdS, is a UV cutoff for the QFT.The interesting bit ofℱ(k), which gives the x1≠x2 behavior of the correlator (4.38), is nonanalytic in k:  (4.43)(II)=-2ν·b0a0k2νln(kϵ)·ϵ2ν-d(1+𝒪(ϵ2)),(b0a0=(-1)ν-122ννΓ(ν)2forν∈ℤ).
To get the factor of 2ν, one must expand both the numerator and the denominator in (4.41); this important subtlety was pointed out in [34].22 The Fourier transformation of the leading term of (II) is given by  (4.44)∫ddke-ikx(II)=2νΓ(Δ+)πd/2Γ(Δ+-d/2)1x2Δ+ϵ2ν-d.
Note that the AdS radius appears only through the overall normalization of the correlator (4.38), in the combination 𝔎Ld-1.Now let us deal with the pesky cutoff dependence. Sinceϵ2ν-d=ϵ-2Δ- if we let ϕ0(k,ϵ)=ϕ0Ren(k)ϵΔ- as before, the operation  (4.45)δδϕ0(k,ϵ)=ϵ-Δ-δδϕ0Ren(k)
removes the potentially divergent factor of ϵ-2Δ-. We also see that for ϵ→0, the 𝒪(ϵ2) terms vanish.If you are bothered by the infinite contact terms (I), there is a prescription to cancel them, appropriately called Holographic Renormalization [35]. Add to Sbulk the local, intrinsic boundary term: (4.46)ΔS=Sc.t.=𝔎2∫bdyddx(-Δ-Ld-1ϵ2Δ--d(ϕ0Ren(x))2)=-Δ-𝔎2L∫∂AdS,z=ϵγϕ2(z,x),
and redo the preceding calculation. Note that this does not affect the equations of motion, nor does it affect G2(x1≠x2).


### 4.4. The Response of the System to an Arbitrary Source

Next we will derive a very important formula for the response of the system to an arbitrary source. The preceding business with the on-shell action is cumbersome, and is inconvenient for computing real-time Green's functions. The following result [36–40] circumvents this confusion.The solution of the equations of motion, satisfying the conditions we want in the interior of the geometry, behaves near the boundary as(4.47)ϕ(z,x)≈(zL)Δ-ϕ0(x)(1+𝒪(z2))+(zL)Δ+ϕ1(x)(1+𝒪(z2));
this formula defines the coefficient ϕ1 of the subleading behavior of the solution. First we recall some facts from classical mechanics. Consider the dynamics of a particle in one dimension, with action S[x]=∫titfdtL(x(t),ẋ(t)). The variation of the action with respect to the initial value of the coordinate is the momentum:  (4.48)δSδx(ti)=Π(ti),Π(t)≡∂L∂ẋ.
Thinking of the radial direction of AdS as time, the following is a mild generalization of (4.48):  (4.49)〈𝒪(x)〉=δW[ϕ0]δϕ0(x)=limz→0(zL)Δ-Π(z,x)|finite,
where Π≡∂ℒ/∂(∂zϕ) is the bulk field-momentum, with z thought of as time. There are two minor subtleties. (1)
The factor ofz-Δ arises because ϕ0 differs from the boundary value of ϕ by a factor: ϕ~zΔ-ϕ0, so ∂/∂ϕ0=z-Δ-(∂/∂ϕ(z=ϵ)).(2)
Π itself in general (for m≠0) has a term proportional to the source ϕ0, which diverges near the boundary; this is related to the contact terms in the previous description. Do not include these terms in (4.49). Finally, then, we can evaluate the field momentum in the solution (4.47) and find23(4.50)〈𝒪(x)〉=𝔎2Δ-dLϕ1(x).
This is an extremely important formula. We already knew that the leading behavior of the solution encoded the source in the dual theory, that is, the perturbation of the action of the QFT. Now we see that the coefficient of the subleading falloff encodes the response [36]. It tells us how the state of the QFT changes as a result of the perturbation.24This formula applies in the real-time case [41]. For example, to describe linear response, δ〈𝒪〉=δϕ0G+O(δϕ0)2), then (4.50) says that  (4.51)G(ω,k)=𝔎2Δ-dLϕ1(ω,k)ϕ0(ω,k).
Which kind of Green's function we get depends on the boundary conditions we impose in the interior.


### 4.5. A Useful Visualization

We are doing classical field theory in the bulk, that is, solving a boundary value problem. We can describe the expansion about an extremum of classical action in powers ofϕ0 in terms of tree level Feynman graphs. External legs of the graphs correspond to the wavefunctions of asymptotic states. In the usual example of QFT in flat space, these are plane waves. In the expansion we are setting up in AdS, the external legs of graphs are associated with the boundary behavior of ϕ (bulk-to-boundary propagators). These diagrams are sometimes called “Witten diagrams,” after [23].


### 4.6.n-Point Functions

Next we will briefly talk about connected correlation functions of three or more operators. Unlike two-point functions, such observables are sensitive to the details of the bulk interactions, and we need to make choices. For definiteness, we will consider the three-point functions of scalar operators dual to scalar fields with the following bulk action:(4.52)S=12∫dd+1xg[∑i=13((∂ϕi)2+mi2ϕi2)+bϕ1ϕ2ϕ3].
The discussion can easily be extended to n-point functions with n>3.The equations of motion are(4.53)(□-m12)ϕ1(z,x)=bϕ2ϕ3
and its permutations. We solve this perturbatively in the sources, ϕ0i: (4.54)ϕ̲1(z,x)=∫ddx1KΔ1(z,x;x1)ϕ01(x1)+b∫ddx'dz'gGΔ1(z,x;z',x')∫ddx1∫ddx2KΔ2(z',x';x1)ϕ02(x1)KΔ3(z',x';x2)ϕ03(x2)+𝒪(b2ϕ03),
with similar expressions for ϕ̲2,3. We need to define the quantities K,G appearing in (4.54). KΔ is the bulk-to-boundary propagator for a bulk field dual to an operator of dimension Δ. We determined this in the previous subsection: it is defined to be the solution to the homogeneous scalar wave equation (□-m2)KΔ(z,x;x′)=0 which approaches a delta function at the boundary,25KΔ(z,x;x')→z→0zΔ-δ(x,x'). So K is given by (4.36) with ϕ0(k)=e-ikx'. GΔi(z,x;z'x') is the bulk-to-bulk propagator, which is the normalizable solution to the wave equation with a source  (4.55)(□-mi2)GΔi(z,x;z',x')=1gδ(z-z')δd(x-x')
(so that (□-mi2)∫gGJ=J for a source J).The first and second terms in (4.54) are summarized by the Witten diagrams in Figures 14 and 15. A typical higher-order diagram would look something like Figure 16. This result can be inserted into our master formula (4.50) to find G3.Figure 14
Witten diagram 1, created by Daniel Park.Figure 15
Witten diagram 2, created by Daniel Park.Figure 16
Witten diagram 3, created by Daniel Park.Figure 17
The curveΣ⊂AdS connecting the two-points in C. The arrows on the curve indicate the orientation, created by Vijay Kumar.Figure 18
The geometry of the near-extremalAdSd+1 charged black hole for T≪μ.


### 4.7. Which Scaling Dimensions Are Attainable?

Let us think about whichΔ are attainable by varying m.26Δ+≥d/2 is the smallest dimension we have obtained so far, but the bound from unitarity is lower: Δ+≥(d-2)/2. There is a range of values for which we have a choice about which of zΔ± is the source and which is the response. This comes about as follows.ϕ is normalizable if S[ϕ]<∞. With the action we have been using  (4.56)SBulk=∫ϵdd+1xg(gAB∂Aϕ∂Bϕ+m2ϕ2),
with g=z-d-1, our boundary conditions demand that ϕ~zΔ(1+𝒪(z2)) with Δ=Δ+ or Δ-,  (4.57)gzz(∂zϕ)2=(z∂zϕ)2~Δ2z2Δ
and hence,  (4.58)gAB∂Aϕ∂Bϕ+m2ϕ2≃Δ2z2Δ+k2z2Δ+2+m2z2=(Δ2+m2)z2Δ(1+𝒪(z2))
in the limit z→0. Since for Δ=Δ±, Δ2+m2=-dΔ≠0,  (4.59)SBulk[zΔ]~∫ϵdzz-d-1(-dΔ)z2Δ(1+𝒪(z2))∝12Δ-dϵ2Δ-d.
We emphasize that we have only specified the boundary behavior of ϕ, and it is not assumed that ϕ satisfies the equation of motion. We see that  (4.60)SBulk[zΔ]<∞⇔Δ>d2.
This does not saturate the lower bound from unitarity on the dimension of a scalar operator, which is Δ>(d-2)/2; the bound coincides with the engineering dimension of a free field.We can change which fall-off is the source by adding a boundary term to the action (4.56) [37]:  (4.61)SBulkKW=∫ϵdd+1xgϕ(-□+m2)ϕ=SBulk-∫∂AdSγϕn·∂zϕ.
For this action we see that (4.62)SBulkKW[ϕ~zΔ(1+𝒪(z2))]=∫ϵdzz-d-1zΔ(1+𝒪(z2))[(-Δ(Δ-d)+m2)zΔ(1+𝒪(z2))+k2z2Δ+2]~∫ϵdzz-d-1+2Δ+2~ϵ2Δ-d+2<∞
is equivalent to  (4.63)Δ≥d-22
which is exactly the unitary bound. We see that in this case both Δ± give normalizable modes for ν≤1. Note that it is actually Δ- that gives the value which saturates the unitarity bound, that is, when  (4.64)Δ-=(d2-(d2)2+m2)m2=1-d2/4=d-22.
The coefficient of zΔ+ would be the source in this case.We have found a description of a different boundary CFT from the same bulk action, which we have obtained by adding a boundary term to the action. The effect of the new boundary term is to lead us to impose Neumann boundary conditions onϕ, rather than Dirichlet conditions. The procedure of interchanging the two is similar to a Legendre transformation.


### 4.8. Geometric Optics Limit

When the dimension of our operator𝒪 is very large, the mass of the bulk field is large in units of the AdS radius:  (4.65)m2L2=Δ(Δ-d)~Δ2≫1⇒mL≫1.
This means that the path integral which computes the solution of the bulk wave equation has a sharply peaked saddle point, associated with geodesics in the bulk. That is, we can think of the solution of the bulk wave equation from a first-quantized point of view, in terms of particles of mass m in the bulk. For convenience, consider the case of a complex operator 𝒪, so that the worldlines of these particles are oriented. Then (4.66)〈𝒪(+a)𝒪̅(-a)〉=Z[±a]~m≫L-1exp(-S[z̲]).
(𝒪̅ is the complex conjugate operator.) The middle expression is the Feynman path integral Z[±a]=∫[dz(τ)]exp(-S[z]); the action for a point particle of mass m whose world-line is Σ is given by S[z]=m∫Σds. In the limit of large m, we have (4.67)Z[±a]~exp(-S[z̲]),
where we have used the saddle point approximation; z̲ is the geodesic connecting the points ±a on the boundary.We now computeZ[±a] in the saddle point approximation. The metric restricted to Σ is given by ds2∣Σ=(L2/z2)(1+z'2)dτ2, where z':=dz/dτ. This implies that the action is (4.68)S[z]=∫dτLz1+z'2.
The geodesic can be computed by noting that the action S[z] does not depend on τ explicitly. This implies, we have a conserved quantity: (4.69)h=z′∂ℒ∂z′-ℒ=Lz11+z'2⇒z'2=(L2h2-z2)1z2.
The above is a first-order differential equation with solution τ=zmax2-z2, with z(τ=0)=zmax=L/h=a. This is the equation of a semicircle. Substituting the solution back into the action gives (4.70)S[z̲]=∫dτLz̲1+z̲'2=2L∫ϵa‍dza2-z̲2az̲=2Llog2aϵ+(terms→0asϵ→0).
You might think that since we are computing this in a conformal field theory, the only scale in the problem is a and therefore the path integral should be independent of a. This argument fails in the case at hand because there are two scales: a and ϵ, the UV cutoff. The scale transformation is anomalous and this is manifested in the ϵ dependence of S[z̲]: (4.71)〈𝒪(+a)𝒪¯(-a)〉=Z[±a]~m≫L-1exp(-S[z̲])~1a2mL.
This is exactly what we expect for the two-point function, since m2L2=Δ(Δ-d)≈Δ2 in the large Δ limit. Without this anomaly, the two-point function of the operator 𝒪 would be independent of a forcing Δ=0, which is impossible in a unitary CFT. This is similar to the anomaly that gives a nonzero scaling dimension of k2/4 to the operator exp(ikX) for the 2d free boson CFT which has classical scaling dimension zero. Graham and Witten [42] showed that such a scale anomaly exists for surface observables for any even k. This relation between correlators of large Δ observables and geodesics in the bulk theory offers a probe of the bulk spacetime [43, 44].


### 4.9. Comment on the Physics of the Warp Factor

Recall that the bulk radial coordinate behaves like a spectrograph separating the theory into energy scales.z=0 is associated with the UV while z→∞ describes the IR.One way to think about the implementation of this relationship between the radial coordinate of the bulk geometry and the field theory energy scale is as follows. The geometry dual to a more general Lorentz-invariant state is(4.72)ds2=w2(z)(-dt2+dx⃗2+dz2),
where w(z) is called the “warp factor”. The warp factor for AdS is simply L/z. The coordinates t,x⃗ in (4.72) are the field theory time and space coordinates. This means that the size of a field theory object is related to the size of its holographic image by the relation: (4.73)(size)FT=1w(z)(propersize)
and similarly the energy of a field theory object is related to the proper energy of the corresponding bulk excitation by (4.74)EFT~i∂t=w(z)Eproper.The fact that AdS goes forever in thez direction is related to the fact that the dual theory is a conformal field theory with degrees of freedom at all energies. We can interpret the existence of modes at arbitrarily low energy as the statement that the warp factor has a zero at z=∞. More precisely, there are 𝒪(N2) degrees of freedom at every energy scale in the CFT. One concrete manifestation of this is the fact that we found a continuum of solutions of (-□+m2)ϕ=0—in particular, for any kμ we could match the required boundary  condition in the IR region, z→∞.In the gravity dual of a QFT with an energy gap, the warp factor has aminimum. This will impose a boundary condition on the bulk wave equation at the IR end of the geometry. The problem becomes like QM of a 1d particle trapped at both ends and therefore has a discrete spectrum.There exist gravity solutions [19, 45, 46] which are dual to field theories with logarithmically running couplings which become strong in the IR, leading to a gap to almost all excitations (except some order N0 number of Goldstone bosons), as in QCD.


## 4.1. Wave Equation near the Boundary and Dimensions of Operators

The metric of AdS (in Poincaré coordinates, so that the constant-z slices are just copies of Minkowski space) is (4.7)ds2=L2dz2+dxμdxμz2≡gABdxAdxB,A=0,…,d,xA=(z,xμ).
As the simplest case to consider, let's think about a scalar field in the bulk. An action for such a scalar field suggested by Naturalness is (4.8)S=-𝔎2∫dd+1xg[gAB∂Aϕ∂Bϕ+m2ϕ2+bϕ3+⋯].
Here 𝔎 is just a normalization constant; we are assuming that the theory of ϕ is weakly coupled and one may think of 𝔎 as proportional to N2. For this metric g=|detg|=(L/z)d+1. Our immediate goal is to compute a two-point function of the operator 𝒪 to which ϕ couples, so we will ignore the interaction terms in (4.8) for a while. Since ϕ is a scalar field we can rewrite the kinetic term as (4.9)gAB∂Aϕ∂Bϕ=(∂ϕ)2=gABDAϕDBϕ,
where DA is the covariant derivative, which has the nice property that DA(gBC)=0, so we can move the Ds around the gs with impunity. By integrating by parts we can rewrite the action in a useful way:(4.10)S=-𝔎2∫dd+1x[∂A(ggABϕ∂Bϕ)-ϕ∂A(ggAB∂Bϕ)+g(m2ϕ2+⋯)]
and finally by using Stokes' theorem we can rewrite the action as (4.11)S=-𝔎2∫∂AdSddxggzBϕ∂Bϕ-𝔎2∫gϕ(-□+m2)ϕ+O(ϕ3),
where we define the scalar Laplacian □ϕ=(1/g)∂A(ggAB∂B)ϕ=DADAϕ. Note that we wrote all these covariant expressions without ever introducing Christoffel symbols.We can rewrite the boundary term more covariantly as(4.12)∫ℳgDAJA=∫∂ℳγnAJA.
The metric tensor γ is defined as (4.13)ds2|z=ϵ≡γμνdxμdxν=L2ϵ2ημνdxμdxν;
that is, it is the induced metric on the boundary surface z=ϵ. The vector nA is a unit vector normal to boundary (z=ϵ). We can find an explicit expression for it: (4.14)nA∝∂∂zgABnAnB|z=ϵ=1⇒n=1gzz∂∂z=zL∂∂z.From this discussion we have learned the following.(i)
The equation of motion for small fluctuations ofϕ is (-□+m2)ϕ̲=0.18(ii)
Ifϕ̲ solves the equation of motion, the on-shell action is just given by the boundary term.Next we will derive the promised formula relating bulk masses and operator dimensions(4.15)Δ(Δ-d)=m2L2
by studying the AdS wave equation near the boundary.Let us take advantage of translational invariance ind dimensions, xμ→xμ+aμ, to Fourier decompose the scalar field: (4.16)ϕ(z,xμ)=eikμxμfk(z),kμxμ≡-ωt+k⃗·x⃗.
In the Fourier basis, substituting (4.16) into the wave equation (-□+m2)ϕ=0 and using the fact that the metric only depends on z, the wave equation is (4.17)0=(gμνkμkν-1g∂z(ggzz∂z)+m2)fk(z)(4.18)=1L2[z2k2-zd+1∂z(z-d+1∂z)+m2L2]fk(z),
where we have used gμν=(z/L)2δμν. The solutions of (4.18) are Bessel functions; we can learn a lot without using that information. For example, look at the solutions near the boundary (i.e., z→0). In this limit we have power law solutions, which are spoiled by the z2k2 term. To see this, try using fk=zΔ in (4.18): (4.19)0=k2z2+Δ-zd+1∂z(Δz-d+Δ)+m2L2zΔ=(k2z2-Δ(Δ-d)+m2L2)zΔ,
and for z→0 we get (4.20)Δ(Δ-d)=m2L2.
The two roots of (4.20) are (4.21)Δ±=d2±(d2)2+m2L2.Comments
(i)
The solution proportional tozΔ- is bigger near z→0.(ii)
Δ+>0forallm, therefore zΔ+ decays near the boundary for any value of the mass.(iii)
Δ++Δ-=d.We want to impose boundary conditions that allow solutions. Since the leading behavior near the boundary of a generic solution isϕ~zΔ-, we impose (4.22)ϕ(x,z)|z=ϵ=ϕ0(x,ϵ)=ϵΔ-ϕ0Ren(x),
where ϕ0Ren is a renormalized source field. With this boundary condition ϕ0Ren is a finite quantity in the limit ϵ→0.Wavefunction Renormalization of𝒪 (Heuristic but useful)
Suppose that(4.23)Sbdy∋∫z=ϵddxγϵϕ0(x,ϵ)𝒪(x,ϵ)=∫ddx(Lϵ)d(ϵΔ-ϕ0Ren(x))𝒪(x,ϵ),
where we have used γ=(L/ϵ)d. Demanding this to be finite as ϵ→0 we get
(4.24)𝒪(x,ϵ)~ϵd-Δ-𝒪Ren(x)=ϵΔ+𝒪Ren(x),
where in the last line we have used Δ++Δ-=d. Therefore, the scaling dimension of 𝒪Ren is Δ+≡Δ. We will soon see that 〈𝒪(x)𝒪(0)〉~1/|x|2Δ, confirming that Δ is indeed the scaling dimension.We are solving a second-order ODE; therefore we need two conditions in order to determine a solution (for eachk). So far we have imposed one condition at the boundary of AdS.(i)
Forz→ϵ,ϕ~zΔ-ϕ0+ (terms subleading in z).In the Euclidean case (we discuss real time in the next subsection), we will also impose(ii)
ϕ regular in the interior of AdS (i.e., at z→∞).Comments onΔ
(1)
TheϵΔ- factor is independent of k and x, which is a consequence of a local QFT (this fails in exotic examples).(2)
Relevantness: If m2>0: This implies Δ≡Δ+>d, so 𝒪Δ is an irrelevant operator. This means that if you perturb the CFT by adding 𝒪Δ to the Lagrangian, then its coefficient is some mass scale to a negative power:
(4.25)ΔS=∫ddx(mass)d-Δ𝒪Δ,
where the exponent is negative; so the effects of such an operator go away in the IR, at energies E<mass. ϕ~zΔ-ϕ0is this coupling. It grows in the UV (small z). If ϕ0 is a finite perturbation, it will back-react on the metric and destroy the asymptotic AdS-ness of the geometry: extra data about the UV will be required.m2=0↔Δ=d means that 𝒪 is marginal.
Ifm2<0, then Δ<d; so 𝒪 is a relevant operator. Note that in AdS, m2<0 is ok if m2 is not too negative. Such fields with m2>-|mBF|2≡-(d/2L)2 are called “Breitenlohner-Freedman- (BF-) allowed tachyons”. The reason you might think that m2<0 is bad is that usually it means an instability of the vacuum at ϕ=0. An instability occurs when a normalizable mode grows with time without a source. But for m2<0, ϕ~zΔ- decays near the boundary (i.e., in the UV). This requires a gradient energy of order ~1/L, which can stop the field from condensing.
To see what is too negative, consider the formula for the dimension,Δ±=d/2±(d/2)2+m2L2. For m2<mBF2, the dimension becomes complex.(3)
The formula relating the mass of a bulk field and the dimension of the associated operator depends on their spin. For example, for a massive gauge field inAdS with action
(4.26)S=-∫AdS(14FμνFμν-12m2AμAμ),
the boundary behavior of the wave equation implies that Aμ~z→0zα with
(4.27)α(α-d+2)=m2L2.
For the particular case of Aμ massless this can be seen to lead to Δ(jμ)=d-1, which is the dimension of a conserved current in a CFT. Also, the fact that gμν is massless implies
(4.28)Δ(Tμν)=d,
which is required by conformal Ward identities.


## 4.2. Solutions of the AdS Wave Equation and Real-Time Issues

An approach which uses the symmetries ofAdS [23] is appealing. However, it is not always available (e.g., if there is a black hole in the spacetime). Also, it can be misleading: as in quantum field theory, scale-invariance is inevitably broken by the UV cutoff.Return to the scalar wave equation in momentum space:(4.29)0=[zd+1∂z(z-d+1∂z)-m2L2-z2k2]fk(z).
We will now stop being agnostic about the signature and confront some issues that arise for real time correlators. If k2>0, that is, kμ is spacelike (or Euclidean), the solution is  (4.30)fk(z)=AKzd/2Kν(kz)+AIzd/2Iν(kz),
where ν=Δ-d/2=(d/2)2+m2L2. In the interior of AdS (z→∞), the Bessel functions behave as  (4.31)Kν(kz)~e-kz,Iν(kz)~ekz.
So we see that the regularity in the interior uniquely fixes f̲k and hence the bulk-to-boundary propagator. Actually there is a theorem (the Graham-Lee theorem) addressing this issue for gravity fields (and not just linearly in the fluctuations); it states that if you specify a Euclidean metric on the boundary of a Euclidean AdS (which we can think of as topologically a Sd by adding the point at infinity in ℝd) modulo conformal rescaling, then the metric on the space inside of the Sd is uniquely determined. A similar result holds for gauge fields.In contrast to this, in Lorentzian signature with timelikek2, that is, for on-shell states with ω2>k⃗2, there exist two linearly independent solutions with the same leading behavior at the UV boundary. In terms of q≡ω2-k⃗2, the solutions are  (4.32)zd/2K±ν(iqz)~e±iqz(z→∞);
so these modes oscillate in the far IR region of the geometry.19 This ambiguity reflects the many possibilities for real-time Green's functions in the QFT. One useful choice is the retarded Green's function, which describes causal response of the system to a perturbation. This choice corresponds to a choice of boundary conditions at z→∞ describing stuff falling into the horizon [30], that is, moving towards larger z as time passes. There are three kinds of reasons for this prescription.20(i)
Both the retarded Green's functions and stuff falling through the horizon describe things thathappen, rather than unhappen.(ii)
You can check that this prescription gives the correct analytic structure ofGR(ω) ([30] and all the hundreds of papers that have used this prescription).(iii)
It has been derived from a holographic version of the Schwinger-Keldysh prescription [31–33].The fact that stuff goes past the horizon and does not come out is what breaks time-reversal invariance in the holographic computation ofGR. Here, the ingoing choice is ϕ(t,z)~e-iωt+iqz, since as t grows, the wavefront moves to larger z. This specifies the solution which computes the causal response to be zd/2K+ν(iqz).The same prescription, adapted to the black hole horizon, will work in the finite temperature case.One last thing we must deal with before proceeding is to define what we mean by a “normalizable” mode, or solution, when we say that we have many normalizable solutions fork2<0 with a given scaling behavior. In Euclidean space, ϕ is normalizable when S[ϕ]<∞. This is because when we are thinking about the partition function Z[ϕ]=∑ϕe-S[ϕ], modes with boundary conditions which force S[ϕ]=∞ would not contribute. In real-time, we say that ϕ is normalizable if E[ϕ]<∞ where  (4.33)E[ϕ]=∫Σdd-1xdzhTμν[ϕ]nμξν=∫x0=constantdd-1xdzhT00[ϕ],
where Σ is a given spatial slice, h is the induced metric on that slice, nμ is a normal unit vector to Σ, and ξμ is a timelike killing vector. TAB is defined as  (4.34)TAB[ϕ]=-2gδδgABSBulk[ϕ].


## 4.3. Bulk-to-Boundary Propagator in Momentum Space

We return to considering spacelikek in this section. Let us normalize our solution of the wave equation by the condition fk(z=ϵ)=1. This means that its Fourier transform is a δ-function in position space δd(x) when evaluated at z=ϵ, not at the actual boundary z=0. The solution, which we can call the “regulated bulk-to-boundary propagator”, is then  (4.35)f̲k(z)=zd/2Kν(kz)ϵd/2Kν(kϵ).
The general position space solution can be obtained by Fourier decomposition:  (4.36)ϕ̲[ϕ0](x)=∫ddkeikxf̲k(z)ϕ0(k,ϵ).
The “on-shell action” (i.e., the action evaluated on the saddle-point solution) is (using (4.11)) (4.37)S[ϕ̲]=-𝔎2∫ddxγϕ̲n·∂ϕ̲=-𝔎2∫ddx∫ddk1∫ddk2ei(k1+k2)xϕ0(k1,ϵ)ϕ0(k2,ϵ)Ld-1z-df̲k1(z)z∂zf̲k2(z)|z=ϵ=-𝔎Ld-12∫ddkϕ0(k,ϵ)ϕ0(-k,ϵ)ℱϵ(k),
and therefore21(4.38)〈𝒪(k1)𝒪(k2)〉cϵ=-δδϕ0(k1)δδϕ0(k2)S=(2π)dδd(k1+k2)ℱϵ(k1).
Here ℱϵ(k) (sometimes called the “flux factor”) is  (4.39)ℱϵ(k)=z-df̲-k(z)z∂zf̲k(z)|z=ϵ+(k⟷-k)=2ϵ-d+1∂z(zd/2Kν(kz)ϵd/2Kν(ϵz))|z=ϵ.
The small-u (near boundary) behavior of Kν(u) is (4.40)Kν(u)=u-ν(a0+a1u2+a2u4+⋯)(leadingterm)+uνlnu(b0+b1u2+b2u4+⋯)(subleadingterm),
where the coefficients of the series ai,bi depend on ν. For noninteger ν, there would be no lnu in the second line, and so we make it pink. Of course, we saw in the previous subsection (with very little work) that any solution of the bulk wave equation has this kind of form (the boundary is a regular singular point of the ODE). We could determine the as and bs recursively by the same procedure. This is just like a scattering problem in 1d quantum mechanics. The point of the Bessel function here is to choose which values of the coefficients a,b give a function which has the correct behavior at the other end, that is, at z→∞. Plugging the asymptotic expansion of the Bessel function into (4.39), (4.41)ℱϵ(k)=2ϵ-d+1∂z((kz)-ν+d/2(a0+⋯)+(kz)ν+d/2lnkz(b0+⋯)(kϵ)-ν+d/2(a0+⋯)+(kϵ)ν+d/2lnkϵ(b0+⋯))|z=ϵ=2ϵ-d[{d2-ν(1+c2(ϵ2k2)+c4(ϵ4k4)+⋯)}+{ν2b0a0(ϵk)2νIn(ϵk)(1+d2(ϵk)2+⋯)}]≡(I)+(II),
where (I) and (II) denote the first and second groups of terms of the previous line.(I) is a Laurent series inϵ with coefficients which are positive powers of k (i.e., analytic in k at k=0). These are contact terms, that is, short distance goo that we do not care about and can subtract off. We can see this by noting that  (4.42)∫ddke-ikx(ϵk)2mϵ-d=ϵ2m-d□xmδd(x)
for m>0. The ϵ2m-d factor reinforces the notion that ϵ, which is an IR cutoff in AdS, is a UV cutoff for the QFT.The interesting bit ofℱ(k), which gives the x1≠x2 behavior of the correlator (4.38), is nonanalytic in k:  (4.43)(II)=-2ν·b0a0k2νln(kϵ)·ϵ2ν-d(1+𝒪(ϵ2)),(b0a0=(-1)ν-122ννΓ(ν)2forν∈ℤ).
To get the factor of 2ν, one must expand both the numerator and the denominator in (4.41); this important subtlety was pointed out in [34].22 The Fourier transformation of the leading term of (II) is given by  (4.44)∫ddke-ikx(II)=2νΓ(Δ+)πd/2Γ(Δ+-d/2)1x2Δ+ϵ2ν-d.
Note that the AdS radius appears only through the overall normalization of the correlator (4.38), in the combination 𝔎Ld-1.Now let us deal with the pesky cutoff dependence. Sinceϵ2ν-d=ϵ-2Δ- if we let ϕ0(k,ϵ)=ϕ0Ren(k)ϵΔ- as before, the operation  (4.45)δδϕ0(k,ϵ)=ϵ-Δ-δδϕ0Ren(k)
removes the potentially divergent factor of ϵ-2Δ-. We also see that for ϵ→0, the 𝒪(ϵ2) terms vanish.If you are bothered by the infinite contact terms (I), there is a prescription to cancel them, appropriately called Holographic Renormalization [35]. Add to Sbulk the local, intrinsic boundary term: (4.46)ΔS=Sc.t.=𝔎2∫bdyddx(-Δ-Ld-1ϵ2Δ--d(ϕ0Ren(x))2)=-Δ-𝔎2L∫∂AdS,z=ϵγϕ2(z,x),
and redo the preceding calculation. Note that this does not affect the equations of motion, nor does it affect G2(x1≠x2).


## 4.4. The Response of the System to an Arbitrary Source

Next we will derive a very important formula for the response of the system to an arbitrary source. The preceding business with the on-shell action is cumbersome, and is inconvenient for computing real-time Green's functions. The following result [36–40] circumvents this confusion.The solution of the equations of motion, satisfying the conditions we want in the interior of the geometry, behaves near the boundary as(4.47)ϕ(z,x)≈(zL)Δ-ϕ0(x)(1+𝒪(z2))+(zL)Δ+ϕ1(x)(1+𝒪(z2));
this formula defines the coefficient ϕ1 of the subleading behavior of the solution. First we recall some facts from classical mechanics. Consider the dynamics of a particle in one dimension, with action S[x]=∫titfdtL(x(t),ẋ(t)). The variation of the action with respect to the initial value of the coordinate is the momentum:  (4.48)δSδx(ti)=Π(ti),Π(t)≡∂L∂ẋ.
Thinking of the radial direction of AdS as time, the following is a mild generalization of (4.48):  (4.49)〈𝒪(x)〉=δW[ϕ0]δϕ0(x)=limz→0(zL)Δ-Π(z,x)|finite,
where Π≡∂ℒ/∂(∂zϕ) is the bulk field-momentum, with z thought of as time. There are two minor subtleties. (1)
The factor ofz-Δ arises because ϕ0 differs from the boundary value of ϕ by a factor: ϕ~zΔ-ϕ0, so ∂/∂ϕ0=z-Δ-(∂/∂ϕ(z=ϵ)).(2)
Π itself in general (for m≠0) has a term proportional to the source ϕ0, which diverges near the boundary; this is related to the contact terms in the previous description. Do not include these terms in (4.49). Finally, then, we can evaluate the field momentum in the solution (4.47) and find23(4.50)〈𝒪(x)〉=𝔎2Δ-dLϕ1(x).
This is an extremely important formula. We already knew that the leading behavior of the solution encoded the source in the dual theory, that is, the perturbation of the action of the QFT. Now we see that the coefficient of the subleading falloff encodes the response [36]. It tells us how the state of the QFT changes as a result of the perturbation.24This formula applies in the real-time case [41]. For example, to describe linear response, δ〈𝒪〉=δϕ0G+O(δϕ0)2), then (4.50) says that  (4.51)G(ω,k)=𝔎2Δ-dLϕ1(ω,k)ϕ0(ω,k).
Which kind of Green's function we get depends on the boundary conditions we impose in the interior.


## 4.5. A Useful Visualization

We are doing classical field theory in the bulk, that is, solving a boundary value problem. We can describe the expansion about an extremum of classical action in powers ofϕ0 in terms of tree level Feynman graphs. External legs of the graphs correspond to the wavefunctions of asymptotic states. In the usual example of QFT in flat space, these are plane waves. In the expansion we are setting up in AdS, the external legs of graphs are associated with the boundary behavior of ϕ (bulk-to-boundary propagators). These diagrams are sometimes called “Witten diagrams,” after [23].


## 4.6.n-Point Functions

Next we will briefly talk about connected correlation functions of three or more operators. Unlike two-point functions, such observables are sensitive to the details of the bulk interactions, and we need to make choices. For definiteness, we will consider the three-point functions of scalar operators dual to scalar fields with the following bulk action:(4.52)S=12∫dd+1xg[∑i=13((∂ϕi)2+mi2ϕi2)+bϕ1ϕ2ϕ3].
The discussion can easily be extended to n-point functions with n>3.The equations of motion are(4.53)(□-m12)ϕ1(z,x)=bϕ2ϕ3
and its permutations. We solve this perturbatively in the sources, ϕ0i: (4.54)ϕ̲1(z,x)=∫ddx1KΔ1(z,x;x1)ϕ01(x1)+b∫ddx'dz'gGΔ1(z,x;z',x')∫ddx1∫ddx2KΔ2(z',x';x1)ϕ02(x1)KΔ3(z',x';x2)ϕ03(x2)+𝒪(b2ϕ03),
with similar expressions for ϕ̲2,3. We need to define the quantities K,G appearing in (4.54). KΔ is the bulk-to-boundary propagator for a bulk field dual to an operator of dimension Δ. We determined this in the previous subsection: it is defined to be the solution to the homogeneous scalar wave equation (□-m2)KΔ(z,x;x′)=0 which approaches a delta function at the boundary,25KΔ(z,x;x')→z→0zΔ-δ(x,x'). So K is given by (4.36) with ϕ0(k)=e-ikx'. GΔi(z,x;z'x') is the bulk-to-bulk propagator, which is the normalizable solution to the wave equation with a source  (4.55)(□-mi2)GΔi(z,x;z',x')=1gδ(z-z')δd(x-x')
(so that (□-mi2)∫gGJ=J for a source J).The first and second terms in (4.54) are summarized by the Witten diagrams in Figures 14 and 15. A typical higher-order diagram would look something like Figure 16. This result can be inserted into our master formula (4.50) to find G3.Figure 14
Witten diagram 1, created by Daniel Park.Figure 15
Witten diagram 2, created by Daniel Park.Figure 16
Witten diagram 3, created by Daniel Park.Figure 17
The curveΣ⊂AdS connecting the two-points in C. The arrows on the curve indicate the orientation, created by Vijay Kumar.Figure 18
The geometry of the near-extremalAdSd+1 charged black hole for T≪μ.


## 4.7. Which Scaling Dimensions Are Attainable?

Let us think about whichΔ are attainable by varying m.26Δ+≥d/2 is the smallest dimension we have obtained so far, but the bound from unitarity is lower: Δ+≥(d-2)/2. There is a range of values for which we have a choice about which of zΔ± is the source and which is the response. This comes about as follows.ϕ is normalizable if S[ϕ]<∞. With the action we have been using  (4.56)SBulk=∫ϵdd+1xg(gAB∂Aϕ∂Bϕ+m2ϕ2),
with g=z-d-1, our boundary conditions demand that ϕ~zΔ(1+𝒪(z2)) with Δ=Δ+ or Δ-,  (4.57)gzz(∂zϕ)2=(z∂zϕ)2~Δ2z2Δ
and hence,  (4.58)gAB∂Aϕ∂Bϕ+m2ϕ2≃Δ2z2Δ+k2z2Δ+2+m2z2=(Δ2+m2)z2Δ(1+𝒪(z2))
in the limit z→0. Since for Δ=Δ±, Δ2+m2=-dΔ≠0,  (4.59)SBulk[zΔ]~∫ϵdzz-d-1(-dΔ)z2Δ(1+𝒪(z2))∝12Δ-dϵ2Δ-d.
We emphasize that we have only specified the boundary behavior of ϕ, and it is not assumed that ϕ satisfies the equation of motion. We see that  (4.60)SBulk[zΔ]<∞⇔Δ>d2.
This does not saturate the lower bound from unitarity on the dimension of a scalar operator, which is Δ>(d-2)/2; the bound coincides with the engineering dimension of a free field.We can change which fall-off is the source by adding a boundary term to the action (4.56) [37]:  (4.61)SBulkKW=∫ϵdd+1xgϕ(-□+m2)ϕ=SBulk-∫∂AdSγϕn·∂zϕ.
For this action we see that (4.62)SBulkKW[ϕ~zΔ(1+𝒪(z2))]=∫ϵdzz-d-1zΔ(1+𝒪(z2))[(-Δ(Δ-d)+m2)zΔ(1+𝒪(z2))+k2z2Δ+2]~∫ϵdzz-d-1+2Δ+2~ϵ2Δ-d+2<∞
is equivalent to  (4.63)Δ≥d-22
which is exactly the unitary bound. We see that in this case both Δ± give normalizable modes for ν≤1. Note that it is actually Δ- that gives the value which saturates the unitarity bound, that is, when  (4.64)Δ-=(d2-(d2)2+m2)m2=1-d2/4=d-22.
The coefficient of zΔ+ would be the source in this case.We have found a description of a different boundary CFT from the same bulk action, which we have obtained by adding a boundary term to the action. The effect of the new boundary term is to lead us to impose Neumann boundary conditions onϕ, rather than Dirichlet conditions. The procedure of interchanging the two is similar to a Legendre transformation.


## 4.8. Geometric Optics Limit

When the dimension of our operator𝒪 is very large, the mass of the bulk field is large in units of the AdS radius:  (4.65)m2L2=Δ(Δ-d)~Δ2≫1⇒mL≫1.
This means that the path integral which computes the solution of the bulk wave equation has a sharply peaked saddle point, associated with geodesics in the bulk. That is, we can think of the solution of the bulk wave equation from a first-quantized point of view, in terms of particles of mass m in the bulk. For convenience, consider the case of a complex operator 𝒪, so that the worldlines of these particles are oriented. Then (4.66)〈𝒪(+a)𝒪̅(-a)〉=Z[±a]~m≫L-1exp(-S[z̲]).
(𝒪̅ is the complex conjugate operator.) The middle expression is the Feynman path integral Z[±a]=∫[dz(τ)]exp(-S[z]); the action for a point particle of mass m whose world-line is Σ is given by S[z]=m∫Σds. In the limit of large m, we have (4.67)Z[±a]~exp(-S[z̲]),
where we have used the saddle point approximation; z̲ is the geodesic connecting the points ±a on the boundary.We now computeZ[±a] in the saddle point approximation. The metric restricted to Σ is given by ds2∣Σ=(L2/z2)(1+z'2)dτ2, where z':=dz/dτ. This implies that the action is (4.68)S[z]=∫dτLz1+z'2.
The geodesic can be computed by noting that the action S[z] does not depend on τ explicitly. This implies, we have a conserved quantity: (4.69)h=z′∂ℒ∂z′-ℒ=Lz11+z'2⇒z'2=(L2h2-z2)1z2.
The above is a first-order differential equation with solution τ=zmax2-z2, with z(τ=0)=zmax=L/h=a. This is the equation of a semicircle. Substituting the solution back into the action gives (4.70)S[z̲]=∫dτLz̲1+z̲'2=2L∫ϵa‍dza2-z̲2az̲=2Llog2aϵ+(terms→0asϵ→0).
You might think that since we are computing this in a conformal field theory, the only scale in the problem is a and therefore the path integral should be independent of a. This argument fails in the case at hand because there are two scales: a and ϵ, the UV cutoff. The scale transformation is anomalous and this is manifested in the ϵ dependence of S[z̲]: (4.71)〈𝒪(+a)𝒪¯(-a)〉=Z[±a]~m≫L-1exp(-S[z̲])~1a2mL.
This is exactly what we expect for the two-point function, since m2L2=Δ(Δ-d)≈Δ2 in the large Δ limit. Without this anomaly, the two-point function of the operator 𝒪 would be independent of a forcing Δ=0, which is impossible in a unitary CFT. This is similar to the anomaly that gives a nonzero scaling dimension of k2/4 to the operator exp(ikX) for the 2d free boson CFT which has classical scaling dimension zero. Graham and Witten [42] showed that such a scale anomaly exists for surface observables for any even k. This relation between correlators of large Δ observables and geodesics in the bulk theory offers a probe of the bulk spacetime [43, 44].


## 4.9. Comment on the Physics of the Warp Factor

Recall that the bulk radial coordinate behaves like a spectrograph separating the theory into energy scales.z=0 is associated with the UV while z→∞ describes the IR.One way to think about the implementation of this relationship between the radial coordinate of the bulk geometry and the field theory energy scale is as follows. The geometry dual to a more general Lorentz-invariant state is(4.72)ds2=w2(z)(-dt2+dx⃗2+dz2),
where w(z) is called the “warp factor”. The warp factor for AdS is simply L/z. The coordinates t,x⃗ in (4.72) are the field theory time and space coordinates. This means that the size of a field theory object is related to the size of its holographic image by the relation: (4.73)(size)FT=1w(z)(propersize)
and similarly the energy of a field theory object is related to the proper energy of the corresponding bulk excitation by (4.74)EFT~i∂t=w(z)Eproper.The fact that AdS goes forever in thez direction is related to the fact that the dual theory is a conformal field theory with degrees of freedom at all energies. We can interpret the existence of modes at arbitrarily low energy as the statement that the warp factor has a zero at z=∞. More precisely, there are 𝒪(N2) degrees of freedom at every energy scale in the CFT. One concrete manifestation of this is the fact that we found a continuum of solutions of (-□+m2)ϕ=0—in particular, for any kμ we could match the required boundary  condition in the IR region, z→∞.In the gravity dual of a QFT with an energy gap, the warp factor has aminimum. This will impose a boundary condition on the bulk wave equation at the IR end of the geometry. The problem becomes like QM of a 1d particle trapped at both ends and therefore has a discrete spectrum.There exist gravity solutions [19, 45, 46] which are dual to field theories with logarithmically running couplings which become strong in the IR, leading to a gap to almost all excitations (except some order N0 number of Goldstone bosons), as in QCD.


## 5. Finite Temperature and Density

AdS is scale invariant. It is a solution dual to the vacuum of a CFT. The correspondence we have developed can also describe systems which are not scale invariant. This includes QFTs with a scale in the Lagrangian, or where a scale is generated quantum mechanically. A simpler thing to describe is the dual of a CFT in an ensemble which introduces a scale. A saddle point of the bulk path integral describing such a state would correspond to a geometry which approaches AdS near the boundary, and solves the same equations of motion: (5.1)0=RAB+dL2gAB∝δSbulkδgAB,
but does something else in the interior.


### 5.1. Interjection on Expectations for CFT at Finite Temperature

In particular we mean ad dimensional relativistic CFT. The partition function is (5.2)Z(τ)=tre-H/T=e-F/T
with free energy F, on a space with geometry Sth1×Σd-1 where the Sth1 has radius 1/T,τ~τ+1/T and Σd-1 is some (d-1)-manifold. We can give Σd-1 finite volume as an IR regulator. The temperature is a deformation of the IR physics (modes with ω≫T=EKK do not notice).For largeV=Vol(Σd-1), then (5.3)F=-cVTd
which follows from extensivity of F and dimensional analysis. This is the familiar Stefan-Boltzmann behavior for blackbody radiation. The pressure is P=-∂VF. Note that in a relativistic theory, just putting it at finite temperature is enough to cause stuff to be present, because of the existence of antiparticles. It is the physics of this collection of CFT stuff that we would like to understand.The prefactorc in (5.3) is a measure the number of degrees-of-freedom-per-site, that is, the number of species of fields in the CFT, which we called “N2” above.


### 5.2. Back to the Gravity Dual

The desired object goes by many names, such as planar black hole, Poincaré black hole, and black brane…. Let us just call it a black hole in AdSd+1. The metric is (5.4)ds2=L2z2(-fdt2+dx⃗2+dz2f),f=1-zdzHd.
We again put the x⃗ coordinates on a finite volume space, for example, in box of volume V,x~x+V1/d. Notice that if we set the emblackening factor f=1, we recover the Poincaré AdS metric, and in fact f approaches 1 as z→0, demonstrating that this is an IR deformation.This metric solves Einstein's equations with a cosmological constantΛ=-d(d-1)/2L2 and asymptotes to Poincaré AdS. It has a horizon at z=zH, where the emblackening factor f vanishes linearly. This means that events at z>zH cannot influence the boundary near z=0.The fact that the horizon is actually translation invariant (i.e., it is a copy ofℝd-1, rather than a sphere) leads some people to call this a “black brane”.In general, horizons describe thermally mixed states. Here we can see the connection more directly: this solution (5.4) is the extremum of the Euclidean gravity action dual to the QFT path integral with thermal boundary conditions. Recall that the boundary behavior of the bulk metric acts as a source for the boundary stress tensor—changing the boundary behavior of the bulk metric is the same as changing the metric of the space on which we put the boundary theory. This is to say that if (5.5)dsbulk2≈z→0dz2z2+L2z2gμν(0)dxμdxν,
then, up to a factor, the boundary metric is gμν(0). This includes any periodic identifications we might make on the geometry, such as making the Euclidean time periodic.A study of the geometry near the horizon will give a relationship between the temperature and the parameterzH appearing in the metric (5.4). The near-horizon (z~zH) metric is (5.6)ds2~-κ2ρ2dt2+dρ2+L2zH2dx⃗2,
where ρ2=(2/κ)(L/zH2)(zH-z)+𝒪(zH-z)2, and κ≡|f'(zH)|/2=d/2zH is called the “surface gravity”. If we analytically continue this geometry to Euclidean time, t→iτ, it becomes (5.7)ds2~κ2ρ2dτ2+dρ2+L2zH2dx⃗2
which looks like ℝd-1 times a Euclidean plane in polar coordinates ρ,κτ. There is a deficit angle in this plane unless τ is periodic according to (5.8)κτ≃κτ+2π.
A deficit angle would mean nonzero Ricci scalar curvature, which would mean that the geometry is not a saddle point of our bulk path integral. Therefore we identify the temperature as T=κ/(2π)=d/(4πzH).Meanwhile the area of the horizon (the set of points withz=zH,t fixed) is (5.9)A=∫z=zH,fixedtgdd-1x=(LzH)d-1V.
Therefore the entropy is (5.10)S=A4GN=Ld-14GNVzHd-1=N22π(πT)d-1V=π22N2VTd-1.
Here I have used the relation Ld-1/4GN=N2/2π. (The factor of 2π is the correct factor for the case of the duality involving the 𝒩=4 SYM, and so this factor of N2 is not red.) The entropy density is (5.11)sBH=SBHV=aBH4GN,
where aBH≡A/V is the “area density” (area per unit volume!) of the black hole.We would like to identify this entropy with that of the CFT at finite temperature. The power ofT is certainly consistent with the constraints of scale invariance. It is not a coincidence that the number of degrees-of-freedom-per-site Ld-1/GN appears in the prefactor. In the concrete example of 𝒩=4 SYM in d=3+1, we can compute this prefactor in the weak-coupling limit λ→0 and the answer is [47, 48] (5.12)F(λ=0)=34F(λ=∞);
the effect of strong coupling is to reduce the effective number of degrees-of-freedom-per-site by an order-one factor. Similar behavior is also seen in lattice simulations of QCD [49].To support the claim that this metric describes the saddle point of the partition sum of a CFT in thermal equilibrium, consider again the partition function:(5.13)ZCFT≡e-βF=e-Sg[g̲],
where g̲ is the Euclidean saddle-point metric.27Sg is the on-shell gravity action for the black hole solution and is equal to (5.14)Sg=SEH+SGH+SctSEH=-116πGN∫dd+1xg(R+d(d-1)L2)
is just the usual bulk gravity action. In addition to this there are two boundary terms. The “Gibbons-Hawking” term is an extrinsic boundary term which affects which boundary conditions we impose on the metric, in the same way that the ∫∂AdSϕn·∂ϕ term in the scalar case changed the scalar boundary conditions from Neumann to Dirichlet. Its role is to guarantee that when we impose Dirichlet boundary conditions on the metric by specifying gμν(0), the action evaluated on a solution is stationary under an arbitrary variation of the metric satisfying that boundary condition.28 Its specific form is: (5.15)SGH=-2∫∂MddxγΘ,
where Θ is the extrinsic curvature of the boundary (5.16)Θ≡γμν∇μnν=nz2γμν∂zγμν,
where nA is an outward-pointing unit normal to the boundary z=ϵ, and we have defined γ by (5.17)ds2≈z→0L2dz2z2+γμνdxμdxν.Finally,(5.18)Sct=∫∂Mddxγ2(d-1)L+⋯
is a local, intrinsic boundary counter-term, as we need to subtract some divergences as z→0, just like in the calculation of vacuum correlation functions of local operators. The … are terms proportional to the intrinsic curvature of the boundary metric gμν(0), which we have taken to be flat. See [50–52] for more details.Thus by plugging in the AdS planar black hole solution (the saddle point) we obtain the free energy. Specializing tod=3+1 and using the 𝒩=4 SYM normalization for the rest of this section, we find (5.19)-FV=L216πGN1zH4=π28N2T4.
This is consistent with the Bekenstein-Hawking entropy calculation above: (5.20)SBH=-∂TF.
We can also check the first law of thermodynamics:  (5.21)dE+PdV=TBHdSBH+ΩdJ+ΦdQ,
where the pressure is P=-∂VF. This is actually a strong check (at least on the correctness of our numerical factors), because it relates horizon quantities such as T,S to global quantities such as the free energy F. The red terms on the RHS of (5.21) would be present if we were studying a black hole with angular momentum J, or with charge Q as in the next subsection.We can also compute the expectation value of the field theory stress tensorTμν. We just use the usual GKPW prescription, using the fact that Tμν couples to the induced metric on the boundary γμν. The energy is E=-VγTtt and the pressure is P=VγTxx. The answers agree with the form we found from thermodynamics:  (5.22)(EV)=N2π2832zH4,P=N2π2812zH4,
which satisfies E=3P, so Tμμ=0 as required in a 3 + 1-dimensional relativistic CFT. One can do the same for the black hole with a spherical horizon (in “global coordinates”) which describes the field theory on a sphere. The answer is E=(N2π2/8)(3/(2zH4)+3L2/(8G)). The limit zH→∞ describes the zero-temperature vacuum. In this limit E is nonzero, and in fact matches the calculation for the zero-point (Casimir) energy of the field theory on a sphere of radius L. The fact that this quantity matches precisely the field theory result at weak coupling [50–52] (unlike, e.g., the free energy which depends on the 't Hooft coupling) is because it arises from an anomaly.


### 5.3. Finite Density

Suppose that the boundary theory has a conservedU(1) symmetry, with current Jμ. This means that there should be a massless gauge field Aμ in the bulk. Naturalness suggests the action  (5.23)ΔSbulk=-14gF2∫dd+1xgFμνFμν,Fμν≡∂μAν-∂νAμ.
The Maxwell equation near the boundary implies the behavior  (5.24)A≈z→0A(0)(x)+zd-2A(1)(x),
and in particular for the zeromode of the time component, At≈z→0μ+zd-2ρ. Using the result (4.50), this is the statement that the charge density ρ of the boundary theory is related to the electric field in the bulk:  (5.25)ΠAt=∂ℒ∂(∂zAt)=Ez=A(1)=ρ;
the subleading behavior encodes the field momentum.A black hole which describes the equilibrium configuration of the field theory at finite density has the same form of the metric as above:(5.26)ds2=L2z2(-fdt2+dx⃗2+dz2f),
but with a different emblackening factor f=1-Mzd+Qz2d-2, and a nonzero gauge field  (5.27)A=dt(μ+ρzd-2).M,Q,ρ can be written in terms of gF,μ,T; see, for example, [2].I will restrict myself to two comments about this solution. First, in the grand canonical ensemble,μ is fixed, and we should think of the z0 term in At as the source, and the coefficient of zd-2, namely, ρ as the response. Changing the boundary conditions on the gauge field can be accomplished as described in Section 4.7, by adding a term of the form ∫∂AdSnμAνFμν. In this case, the trick is the Legendre transformation which takes us to the canonical ensemble, where the ρ-term is the  source and μ is the response.Secondly, this geometry is interesting. AtT≪μ it describes an example of a “holographic RG flow” between RG fixed points. In particular, the function f now has multiple zeros at real, positive values of z. Let us call the one closest to the boundary z0; this represents a horizon; there is another zero z1 at a larger value of z, in the inaccessible region behind the horizon. These zeros collide z1→z0 when we take T→0, in which case f has a double-zero, and the black hole is said to be extremal. In this limit, the geometry interpolates between AdSd+1 near the boundary z~0, and AdS2×ℝd-1 near the horizon. The presence of an AdS factor means that this IR region of the geometry is scale invariant. The original scale invariance of the d-dimensional CFT is broken by the chemical potential; so this is an emergent scale invariance, like the kind that one might see in a real system. The IR geometry is dual to some nonrelativistic scale-invariant theory. At finite T≪μ, the IR geometry is a black hole in AdS2×ℝd-1, related to this CFT at finite temperature.If we put a charged boson in this geometry, we find the holographic superconductor phenomenon ([53–55], for a review see [56]). If we put a charged spinor in this geometry, we can study the two-point functions of the dual fermionic operator, and in particular its spectral density~ImGR. This study was initiated in [57] and developed by [58–60], and one finds Fermi surfaces, which sometimes describe nonFermi liquids. For some values of the bulk parameters, the fermion spectral density is that of a “marginal Fermi liquid” introduced [61] to model the strange metal phase of high-Tc superconductors [59].


## 5.1. Interjection on Expectations for CFT at Finite Temperature

In particular we mean ad dimensional relativistic CFT. The partition function is (5.2)Z(τ)=tre-H/T=e-F/T
with free energy F, on a space with geometry Sth1×Σd-1 where the Sth1 has radius 1/T,τ~τ+1/T and Σd-1 is some (d-1)-manifold. We can give Σd-1 finite volume as an IR regulator. The temperature is a deformation of the IR physics (modes with ω≫T=EKK do not notice).For largeV=Vol(Σd-1), then (5.3)F=-cVTd
which follows from extensivity of F and dimensional analysis. This is the familiar Stefan-Boltzmann behavior for blackbody radiation. The pressure is P=-∂VF. Note that in a relativistic theory, just putting it at finite temperature is enough to cause stuff to be present, because of the existence of antiparticles. It is the physics of this collection of CFT stuff that we would like to understand.The prefactorc in (5.3) is a measure the number of degrees-of-freedom-per-site, that is, the number of species of fields in the CFT, which we called “N2” above.


## 5.2. Back to the Gravity Dual

The desired object goes by many names, such as planar black hole, Poincaré black hole, and black brane…. Let us just call it a black hole in AdSd+1. The metric is (5.4)ds2=L2z2(-fdt2+dx⃗2+dz2f),f=1-zdzHd.
We again put the x⃗ coordinates on a finite volume space, for example, in box of volume V,x~x+V1/d. Notice that if we set the emblackening factor f=1, we recover the Poincaré AdS metric, and in fact f approaches 1 as z→0, demonstrating that this is an IR deformation.This metric solves Einstein's equations with a cosmological constantΛ=-d(d-1)/2L2 and asymptotes to Poincaré AdS. It has a horizon at z=zH, where the emblackening factor f vanishes linearly. This means that events at z>zH cannot influence the boundary near z=0.The fact that the horizon is actually translation invariant (i.e., it is a copy ofℝd-1, rather than a sphere) leads some people to call this a “black brane”.In general, horizons describe thermally mixed states. Here we can see the connection more directly: this solution (5.4) is the extremum of the Euclidean gravity action dual to the QFT path integral with thermal boundary conditions. Recall that the boundary behavior of the bulk metric acts as a source for the boundary stress tensor—changing the boundary behavior of the bulk metric is the same as changing the metric of the space on which we put the boundary theory. This is to say that if (5.5)dsbulk2≈z→0dz2z2+L2z2gμν(0)dxμdxν,
then, up to a factor, the boundary metric is gμν(0). This includes any periodic identifications we might make on the geometry, such as making the Euclidean time periodic.A study of the geometry near the horizon will give a relationship between the temperature and the parameterzH appearing in the metric (5.4). The near-horizon (z~zH) metric is (5.6)ds2~-κ2ρ2dt2+dρ2+L2zH2dx⃗2,
where ρ2=(2/κ)(L/zH2)(zH-z)+𝒪(zH-z)2, and κ≡|f'(zH)|/2=d/2zH is called the “surface gravity”. If we analytically continue this geometry to Euclidean time, t→iτ, it becomes (5.7)ds2~κ2ρ2dτ2+dρ2+L2zH2dx⃗2
which looks like ℝd-1 times a Euclidean plane in polar coordinates ρ,κτ. There is a deficit angle in this plane unless τ is periodic according to (5.8)κτ≃κτ+2π.
A deficit angle would mean nonzero Ricci scalar curvature, which would mean that the geometry is not a saddle point of our bulk path integral. Therefore we identify the temperature as T=κ/(2π)=d/(4πzH).Meanwhile the area of the horizon (the set of points withz=zH,t fixed) is (5.9)A=∫z=zH,fixedtgdd-1x=(LzH)d-1V.
Therefore the entropy is (5.10)S=A4GN=Ld-14GNVzHd-1=N22π(πT)d-1V=π22N2VTd-1.
Here I have used the relation Ld-1/4GN=N2/2π. (The factor of 2π is the correct factor for the case of the duality involving the 𝒩=4 SYM, and so this factor of N2 is not red.) The entropy density is (5.11)sBH=SBHV=aBH4GN,
where aBH≡A/V is the “area density” (area per unit volume!) of the black hole.We would like to identify this entropy with that of the CFT at finite temperature. The power ofT is certainly consistent with the constraints of scale invariance. It is not a coincidence that the number of degrees-of-freedom-per-site Ld-1/GN appears in the prefactor. In the concrete example of 𝒩=4 SYM in d=3+1, we can compute this prefactor in the weak-coupling limit λ→0 and the answer is [47, 48] (5.12)F(λ=0)=34F(λ=∞);
the effect of strong coupling is to reduce the effective number of degrees-of-freedom-per-site by an order-one factor. Similar behavior is also seen in lattice simulations of QCD [49].To support the claim that this metric describes the saddle point of the partition sum of a CFT in thermal equilibrium, consider again the partition function:(5.13)ZCFT≡e-βF=e-Sg[g̲],
where g̲ is the Euclidean saddle-point metric.27Sg is the on-shell gravity action for the black hole solution and is equal to (5.14)Sg=SEH+SGH+SctSEH=-116πGN∫dd+1xg(R+d(d-1)L2)
is just the usual bulk gravity action. In addition to this there are two boundary terms. The “Gibbons-Hawking” term is an extrinsic boundary term which affects which boundary conditions we impose on the metric, in the same way that the ∫∂AdSϕn·∂ϕ term in the scalar case changed the scalar boundary conditions from Neumann to Dirichlet. Its role is to guarantee that when we impose Dirichlet boundary conditions on the metric by specifying gμν(0), the action evaluated on a solution is stationary under an arbitrary variation of the metric satisfying that boundary condition.28 Its specific form is: (5.15)SGH=-2∫∂MddxγΘ,
where Θ is the extrinsic curvature of the boundary (5.16)Θ≡γμν∇μnν=nz2γμν∂zγμν,
where nA is an outward-pointing unit normal to the boundary z=ϵ, and we have defined γ by (5.17)ds2≈z→0L2dz2z2+γμνdxμdxν.Finally,(5.18)Sct=∫∂Mddxγ2(d-1)L+⋯
is a local, intrinsic boundary counter-term, as we need to subtract some divergences as z→0, just like in the calculation of vacuum correlation functions of local operators. The … are terms proportional to the intrinsic curvature of the boundary metric gμν(0), which we have taken to be flat. See [50–52] for more details.Thus by plugging in the AdS planar black hole solution (the saddle point) we obtain the free energy. Specializing tod=3+1 and using the 𝒩=4 SYM normalization for the rest of this section, we find (5.19)-FV=L216πGN1zH4=π28N2T4.
This is consistent with the Bekenstein-Hawking entropy calculation above: (5.20)SBH=-∂TF.
We can also check the first law of thermodynamics:  (5.21)dE+PdV=TBHdSBH+ΩdJ+ΦdQ,
where the pressure is P=-∂VF. This is actually a strong check (at least on the correctness of our numerical factors), because it relates horizon quantities such as T,S to global quantities such as the free energy F. The red terms on the RHS of (5.21) would be present if we were studying a black hole with angular momentum J, or with charge Q as in the next subsection.We can also compute the expectation value of the field theory stress tensorTμν. We just use the usual GKPW prescription, using the fact that Tμν couples to the induced metric on the boundary γμν. The energy is E=-VγTtt and the pressure is P=VγTxx. The answers agree with the form we found from thermodynamics:  (5.22)(EV)=N2π2832zH4,P=N2π2812zH4,
which satisfies E=3P, so Tμμ=0 as required in a 3 + 1-dimensional relativistic CFT. One can do the same for the black hole with a spherical horizon (in “global coordinates”) which describes the field theory on a sphere. The answer is E=(N2π2/8)(3/(2zH4)+3L2/(8G)). The limit zH→∞ describes the zero-temperature vacuum. In this limit E is nonzero, and in fact matches the calculation for the zero-point (Casimir) energy of the field theory on a sphere of radius L. The fact that this quantity matches precisely the field theory result at weak coupling [50–52] (unlike, e.g., the free energy which depends on the 't Hooft coupling) is because it arises from an anomaly.


## 5.3. Finite Density

Suppose that the boundary theory has a conservedU(1) symmetry, with current Jμ. This means that there should be a massless gauge field Aμ in the bulk. Naturalness suggests the action  (5.23)ΔSbulk=-14gF2∫dd+1xgFμνFμν,Fμν≡∂μAν-∂νAμ.
The Maxwell equation near the boundary implies the behavior  (5.24)A≈z→0A(0)(x)+zd-2A(1)(x),
and in particular for the zeromode of the time component, At≈z→0μ+zd-2ρ. Using the result (4.50), this is the statement that the charge density ρ of the boundary theory is related to the electric field in the bulk:  (5.25)ΠAt=∂ℒ∂(∂zAt)=Ez=A(1)=ρ;
the subleading behavior encodes the field momentum.A black hole which describes the equilibrium configuration of the field theory at finite density has the same form of the metric as above:(5.26)ds2=L2z2(-fdt2+dx⃗2+dz2f),
but with a different emblackening factor f=1-Mzd+Qz2d-2, and a nonzero gauge field  (5.27)A=dt(μ+ρzd-2).M,Q,ρ can be written in terms of gF,μ,T; see, for example, [2].I will restrict myself to two comments about this solution. First, in the grand canonical ensemble,μ is fixed, and we should think of the z0 term in At as the source, and the coefficient of zd-2, namely, ρ as the response. Changing the boundary conditions on the gauge field can be accomplished as described in Section 4.7, by adding a term of the form ∫∂AdSnμAνFμν. In this case, the trick is the Legendre transformation which takes us to the canonical ensemble, where the ρ-term is the  source and μ is the response.Secondly, this geometry is interesting. AtT≪μ it describes an example of a “holographic RG flow” between RG fixed points. In particular, the function f now has multiple zeros at real, positive values of z. Let us call the one closest to the boundary z0; this represents a horizon; there is another zero z1 at a larger value of z, in the inaccessible region behind the horizon. These zeros collide z1→z0 when we take T→0, in which case f has a double-zero, and the black hole is said to be extremal. In this limit, the geometry interpolates between AdSd+1 near the boundary z~0, and AdS2×ℝd-1 near the horizon. The presence of an AdS factor means that this IR region of the geometry is scale invariant. The original scale invariance of the d-dimensional CFT is broken by the chemical potential; so this is an emergent scale invariance, like the kind that one might see in a real system. The IR geometry is dual to some nonrelativistic scale-invariant theory. At finite T≪μ, the IR geometry is a black hole in AdS2×ℝd-1, related to this CFT at finite temperature.If we put a charged boson in this geometry, we find the holographic superconductor phenomenon ([53–55], for a review see [56]). If we put a charged spinor in this geometry, we can study the two-point functions of the dual fermionic operator, and in particular its spectral density~ImGR. This study was initiated in [57] and developed by [58–60], and one finds Fermi surfaces, which sometimes describe nonFermi liquids. For some values of the bulk parameters, the fermion spectral density is that of a “marginal Fermi liquid” introduced [61] to model the strange metal phase of high-Tc superconductors [59].


## 6. Hydrodynamics and Response Functions

So far we have given evidence that the black hole thermodynamics of the AdS black hole solution is dual to the thermal ensemble of some strongly coupled CFT onℝd-1 at large N and large 't Hooft coupling λ. Thus the gauge theory provides the microstates that are being coarse grained by the Bekenstein-Hawking entropy of the black hole SBH. The static black hole describes the field theory in thermodynamic equilibrium.A few comments on this observation are as follows.(i)
Perturbing the equilibrium of theboundary theory with a kick will result in thermalization—relaxation back to equilibrium.(ii)
In thebulk the response to such a kick is for the energy of the kick to fall into the black hole.The above two statements are related by the duality [62]. In the long wavelength and small frequency limit both are consistent with the hydrodynamics of a relativistic CFT. Additionally the duality allows one to compute various transport coefficients of the gauge theory at large λ, such as the shear viscosity, and conductivity as we discuss next. We emphasize that these calculations are done at leading order in this mean-field-theory-like description, but they include dissipation.


### 6.1. Linear Response and Transport Coefficients

As a demonstration of the real-time prescription [30], in the following subsection we will derive the viscosity of a large class of strongly interacting plasmas, made from CFT at finite temperature. First, we quickly recall the formalism of linear response to establish notation.To see how our system responds to our poking at it, consider the following small perturbation of the field theory action:(6.1)ΔSQFT=∫ddx𝒪ϕ0,ϕ0small.
The response is  (6.2)δ〈𝒪〉CFT,T=ϕ0→0-GR(ω,k)ϕ0(ω,k).
For simplicity, we are just asking about the diagonal response, perturbing with the operator 𝒪, and measuring the operator 𝒪: GR≡G𝒪𝒪R. The subscript on the LHS indicates that we are computing thermal averages at temperature T. In the long-wavelength, low-frequency limit, on very general grounds, this will reduce to the Kubo formula:  (6.3)δ〈𝒪〉CFT,T→k→0,ω→0iωχϕ0.
For example, in the case where 𝒪=jμ is a conserved current, ϕ0=Aμ is the boundary behavior of a bulk gauge field, and the transport coefficient is the conductivity: (6.4)δ〈J⃗〉CFT,T→k→0,ω→0iωχ(j)A⃗=σE⃗(Ohm'slaw).
In the case where 𝒪=Tyx(kz), the source is the boundary value of a metric perturbation δgxy, and the transport coefficient is the shear viscosity, χ(Tyx)=η. Do not forget that the order of limits here matters: k must be taken to zero before ω to get the transport coefficient.


### 6.2. Holographic Calculation of Transport Coefficients

Now we discuss the bulk calculation of these quantities. We will follow the discussion of [40], but we should emphasize that this calculation has a long history [63–66]. We can consider a very general bulk metric: (6.5)ds2=gttdt2+gzzdz2+gijdxidxj,
which satisfies two conditions.(i)
Nearz→0, it approaches AdS, or some other asymptotics for which we know how to construct a holographic GKPW-like prescription. Other examples include systems which in the UV are described by nonrelativistic CFTs, as described in [16–18].(ii)
The geometry has a horizon at somez=zH, near which the metric coefficients take the form gtt≈2κ(zH-z),gzz≈1/2κ(zH-z). Such a thing is called a Rindler horizon, and means that the Euclidean time coordinate must have period 1/T=2π/κ.T is therefore the temperature at which we have put the field theory.In this spacetime, consider the bulk action:(6.6)S=-12∫dd+1xg∂Aϕ∂Aϕq(z).
Several points are noteworthy here. First, the quantity q(z) is some effective coupling of the mode, which can depend on the radial direction; such dependence can arise, for example from the profiles of some background fields to which the mode is coupled. In addition, we have assumed that the field ϕ is massless and does not mix with other modes (at least in the kinematical regime of interest); this will be important for the calculation below. An important example of such a ϕ is precisely the metric fluctuation δgxy(kz) which computes the shear viscosity. In the case of Einstein gravity, the coefficient is 1/qEinstein=1/16πGN. For other theories of gravity with higher-curvature corrections, q can take some other forms [40]. We will continue to leave q general and refer to the operator to which ϕ couples as 𝒪.Recall that the general formula for the expectation value of an operator in terms of the behavior of its dual bulk field is(6.7)〈𝒪(xμ)〉QFT=limz→0Πϕ(z,xμ),
where Πϕ≡∂ℒbulk/∂(∂zϕ) is the field momentum (with the radial direction thought of as time). We have specialized the formula (4.50) to the case of a massless field. The fact that (6.7), evaluated on the infalling solution for ϕ, correctly computes the retarded response was demonstrated in [41]. Equation (6.7) means that the transport coefficient is (6.8)χ=limω→0limk→0limz→0(Πϕ(z,kμ)iωϕ(z,kμ)).
We will calculate this in two steps. First we find its value at the black hole horizon, and then we propagate it to the boundary using the equation of motion.By assumption (2) about the metric, the horizon at z=zH is a regular singular point of the wave equation, near which solutions behave as  (6.9)ϕ(z)≈(zH-z)±iω/4πT
(the exponents are determined by plugging in a power-law ansatz into the wave equation and Taylor expanding). Since the time-dependence of the solution is of the form e-iωt (recall (4.16)), these two solutions describe waves which fall into (−) or come out of (+) the black hole horizon as time passes. To compute the retarded Green's function, we pick the ingoing solution [30]. This says that near the horizon, the field momentum is  (6.10)Πϕ(z,k)=|g|q(z)gzz∂zϕ≈z→zH1q(zH)|ggzzgtt|iωϕ(z,k)|z=zH.
The outgoing solution would give a minus sign in front here.To propagate this to the boundary, we use the bulk equations of motion, which relate∂zΠϕ~∂z2ϕ to ϕ. It is not hard to show that in the limit k→0,ω→0,  (6.11)∂zΠϕ(z,k→0,ω→0)=0.
A similar statement holds for the denominator of Green's function, ωϕ. This means that  (6.12)Πωϕ|z=0=Πωϕ|z=zH,
from which we learn that  (6.13)χ=1q(zH)|ggzzgtt|.
Here it was important that the bulk field was massless; this fails, for example, for the mode which computes the bulk viscosity.Let us apply this discussion to the case whereχ is the shear viscosity η, defined in the previous subsection. The shear viscosity is dimensionful (it comes in some units called “poise”); a dimensionless measure of the quality of a liquid is its ratio with the entropy density, which is also something we know how to compute. The entropy density of our system is related to the “area-density” aBH of the black hole:  (6.14)s=aBH4GN=14GN|ggzzgtt|.
We see therefore that  (6.15)ηs=1q(zH).
In the special case of Einstein gravity in the bulk this gives the celebrated KSS value [65]:  (6.16)ηs=14π.
This value is much smaller than that of common liquids. The substances which come the closest [67] are cold atoms at unitarity (η/s~1/2 [68, 69]) and the fireball at RHIC (η/s~0.16 [70, 71]). This computation of the shear viscosity of a strongly interacting plasma seems to have been quite valuable to people trying to interpret the experiments at RHIC.29It seems not to be a lower bound. For example, in some particular higher-curvature gravity theory called Gauss-Bonnet gravity, where the black-hole solution is known, the parameterq is related to a coefficient of a higher-derivative term λGB, and one finds [72, 73]  (6.17)ηs=1q(zH)=14π>1625.
The inequality on the right arises from demanding causality of the boundary theory, which fails if λGB is too large. It is not clear that GB gravity has a sensible UV completion, but other theories where the KSS value is violated by a small amount do [74].A nice thing about the ratioη/s is that the number of degrees of freedom (‘N2) cancels out. Attempts to find other such observables, for example, related to charge transport, include [75].For lack of time, I have spoken here only about the extreme long-wavelength limit of the response functions. The frequency and momentum dependence is also very revealing; see, for example, [76, 77].


## 6.1. Linear Response and Transport Coefficients

As a demonstration of the real-time prescription [30], in the following subsection we will derive the viscosity of a large class of strongly interacting plasmas, made from CFT at finite temperature. First, we quickly recall the formalism of linear response to establish notation.To see how our system responds to our poking at it, consider the following small perturbation of the field theory action:(6.1)ΔSQFT=∫ddx𝒪ϕ0,ϕ0small.
The response is  (6.2)δ〈𝒪〉CFT,T=ϕ0→0-GR(ω,k)ϕ0(ω,k).
For simplicity, we are just asking about the diagonal response, perturbing with the operator 𝒪, and measuring the operator 𝒪: GR≡G𝒪𝒪R. The subscript on the LHS indicates that we are computing thermal averages at temperature T. In the long-wavelength, low-frequency limit, on very general grounds, this will reduce to the Kubo formula:  (6.3)δ〈𝒪〉CFT,T→k→0,ω→0iωχϕ0.
For example, in the case where 𝒪=jμ is a conserved current, ϕ0=Aμ is the boundary behavior of a bulk gauge field, and the transport coefficient is the conductivity: (6.4)δ〈J⃗〉CFT,T→k→0,ω→0iωχ(j)A⃗=σE⃗(Ohm'slaw).
In the case where 𝒪=Tyx(kz), the source is the boundary value of a metric perturbation δgxy, and the transport coefficient is the shear viscosity, χ(Tyx)=η. Do not forget that the order of limits here matters: k must be taken to zero before ω to get the transport coefficient.


## 6.2. Holographic Calculation of Transport Coefficients

Now we discuss the bulk calculation of these quantities. We will follow the discussion of [40], but we should emphasize that this calculation has a long history [63–66]. We can consider a very general bulk metric: (6.5)ds2=gttdt2+gzzdz2+gijdxidxj,
which satisfies two conditions.(i)
Nearz→0, it approaches AdS, or some other asymptotics for which we know how to construct a holographic GKPW-like prescription. Other examples include systems which in the UV are described by nonrelativistic CFTs, as described in [16–18].(ii)
The geometry has a horizon at somez=zH, near which the metric coefficients take the form gtt≈2κ(zH-z),gzz≈1/2κ(zH-z). Such a thing is called a Rindler horizon, and means that the Euclidean time coordinate must have period 1/T=2π/κ.T is therefore the temperature at which we have put the field theory.In this spacetime, consider the bulk action:(6.6)S=-12∫dd+1xg∂Aϕ∂Aϕq(z).
Several points are noteworthy here. First, the quantity q(z) is some effective coupling of the mode, which can depend on the radial direction; such dependence can arise, for example from the profiles of some background fields to which the mode is coupled. In addition, we have assumed that the field ϕ is massless and does not mix with other modes (at least in the kinematical regime of interest); this will be important for the calculation below. An important example of such a ϕ is precisely the metric fluctuation δgxy(kz) which computes the shear viscosity. In the case of Einstein gravity, the coefficient is 1/qEinstein=1/16πGN. For other theories of gravity with higher-curvature corrections, q can take some other forms [40]. We will continue to leave q general and refer to the operator to which ϕ couples as 𝒪.Recall that the general formula for the expectation value of an operator in terms of the behavior of its dual bulk field is(6.7)〈𝒪(xμ)〉QFT=limz→0Πϕ(z,xμ),
where Πϕ≡∂ℒbulk/∂(∂zϕ) is the field momentum (with the radial direction thought of as time). We have specialized the formula (4.50) to the case of a massless field. The fact that (6.7), evaluated on the infalling solution for ϕ, correctly computes the retarded response was demonstrated in [41]. Equation (6.7) means that the transport coefficient is (6.8)χ=limω→0limk→0limz→0(Πϕ(z,kμ)iωϕ(z,kμ)).
We will calculate this in two steps. First we find its value at the black hole horizon, and then we propagate it to the boundary using the equation of motion.By assumption (2) about the metric, the horizon at z=zH is a regular singular point of the wave equation, near which solutions behave as  (6.9)ϕ(z)≈(zH-z)±iω/4πT
(the exponents are determined by plugging in a power-law ansatz into the wave equation and Taylor expanding). Since the time-dependence of the solution is of the form e-iωt (recall (4.16)), these two solutions describe waves which fall into (−) or come out of (+) the black hole horizon as time passes. To compute the retarded Green's function, we pick the ingoing solution [30]. This says that near the horizon, the field momentum is  (6.10)Πϕ(z,k)=|g|q(z)gzz∂zϕ≈z→zH1q(zH)|ggzzgtt|iωϕ(z,k)|z=zH.
The outgoing solution would give a minus sign in front here.To propagate this to the boundary, we use the bulk equations of motion, which relate∂zΠϕ~∂z2ϕ to ϕ. It is not hard to show that in the limit k→0,ω→0,  (6.11)∂zΠϕ(z,k→0,ω→0)=0.
A similar statement holds for the denominator of Green's function, ωϕ. This means that  (6.12)Πωϕ|z=0=Πωϕ|z=zH,
from which we learn that  (6.13)χ=1q(zH)|ggzzgtt|.
Here it was important that the bulk field was massless; this fails, for example, for the mode which computes the bulk viscosity.Let us apply this discussion to the case whereχ is the shear viscosity η, defined in the previous subsection. The shear viscosity is dimensionful (it comes in some units called “poise”); a dimensionless measure of the quality of a liquid is its ratio with the entropy density, which is also something we know how to compute. The entropy density of our system is related to the “area-density” aBH of the black hole:  (6.14)s=aBH4GN=14GN|ggzzgtt|.
We see therefore that  (6.15)ηs=1q(zH).
In the special case of Einstein gravity in the bulk this gives the celebrated KSS value [65]:  (6.16)ηs=14π.
This value is much smaller than that of common liquids. The substances which come the closest [67] are cold atoms at unitarity (η/s~1/2 [68, 69]) and the fireball at RHIC (η/s~0.16 [70, 71]). This computation of the shear viscosity of a strongly interacting plasma seems to have been quite valuable to people trying to interpret the experiments at RHIC.29It seems not to be a lower bound. For example, in some particular higher-curvature gravity theory called Gauss-Bonnet gravity, where the black-hole solution is known, the parameterq is related to a coefficient of a higher-derivative term λGB, and one finds [72, 73]  (6.17)ηs=1q(zH)=14π>1625.
The inequality on the right arises from demanding causality of the boundary theory, which fails if λGB is too large. It is not clear that GB gravity has a sensible UV completion, but other theories where the KSS value is violated by a small amount do [74].A nice thing about the ratioη/s is that the number of degrees of freedom (‘N2) cancels out. Attempts to find other such observables, for example, related to charge transport, include [75].For lack of time, I have spoken here only about the extreme long-wavelength limit of the response functions. The frequency and momentum dependence is also very revealing; see, for example, [76, 77].


## 7. Concluding Remarks

### 7.1. Remarks on Other Observables

Besides correlation functions and thermodynamic potentials, a number of other observables of interest can be computed easily using the correspondence. Some, like expectation values of Wilson loops, are relatively specific to gauge theories.A very ubiquitous observable, which is notoriously hard to compute otherwise, is the entanglement entropy. If we divide the hilbert space of the QFT into(7.1)ℋ=ℋA⊕ℋB
and declare ourselves ignorant of ℋB, we have an ensemble described by the density matrix:  (7.2)ρA=trHBρ,
where ρ is the density matrix of the whole system, which let's take to be the one associated with the ground state ρ=|Ω〉〈Ω|. Then the entanglement entropy between A and B is the von Neumann entropy of ρA:  (7.3)S=-trAρAlnρA.In the special case where the subdivision (7.1) is done by cutting out a region of space B bounded by some d-2-dimensional surface Σ, there exists a proposal for how to calculate the associated entanglement entropy SΣ using the holographic dual [78] (for a recent review, see [79]). The idea is extremely simple: just find the surface M (d-1-dimensional), ending on the boundary at Σ, which extremizes its area. The formula for SΣ is then  (7.4)SΣ=extremum∂M=Σarea(M)4GN,
very reminiscent of the Bekenstein formula for the entropy of a black hole. This formula passes many checks. For example, it gives the correct universal behavior S~(c/3)lnL (L is the length of the region B) in a 1+1-dimensional CFT of central charge c. Like the Casimir energy, this match between weak and strong coupling is precise because it is determined by the conformal anomaly. In higher dimensions, the holographic prescription gives a prediction for which terms in the expansion of S(L) in powers of L are universal. There even exists a heuristic derivation [80].30


### 7.2. Remarks on the Role of Supersymmetry

Supersymmetry has played important roles in the historical development of the AdS/CFT correspondence.(i)
It constrains the form of the interactions, meaning that there are fewer possible candidates for the dual (e.g., the maximalAdS supergravity theory in five dimensions is unique).(ii)
A supersymmetric theory has many observables which are independent of the coupling. These so-called BPS quantities allow for many quantitative checks of a proposed dual pair.(iii)
Supersymmetry can stabilize aline of exact fixed points (e.g., in the 𝒩=4 SYM), rendering the coupling constant a dimensionless parameter which interpolates between the weakly coupled field theory description and the gravity regime.However, it has played no role in our discussion. Some people believe that supersymmetry may be necessary for the construction of a consistent theory of quantum gravity. But it seems more likely to me that the formulation of specific examples of the duality without supersymmetry is a (perhaps hard) technical problem, not one of principle.


### 7.3. Lessons for How to Use AdS/CFT

Critical exponents depend on “Landscape Issues”. By this I mean just that they depend on the values of the couplings in the bulk action (in particular, the masses of bulk fields), which are specified only by some UV completion, that is, by available string vacua. For each possible bulk coupling, it is very much an open question which values arise in a consistent theory of quantum gravity. This situation—that the critical exponents are UV-sensitive quantities—is a rather unfamiliar one!At least in examples we know, thermodynamics is not very sensitive to strong coupling. In both the𝒩=4 SYM, and from lattice QCD, we find for, for example, the free energy a relation of the form  (7.5)Fstrong~34Fweak.Real-time dynamics and transportare very sensitive to the strength of the interactions. For example,  (7.6)(ηs)weak~1g4Ing≫(ηs)strong~14π.
Not only are these observables sensitive to strong coupling, but they are very natural things to compute using the holographic technology. In particular, although it is a classical description, it automatically includes dissipation. Ordinary techniques seem to require the existence of a description of what is being transported in terms of quasiparticles, so that the Boltzmann equation can be used. Since we know that such a description need not exist, this is a very good opportunity for the machinery described above to be useful.I would like to close with a final optimistic philosophical comment. The following gedanken experiment was proposed by Weisskopf [81]: take a bunch of theoretical physicists and lock them away from birth so that they are never exposed to any substance in the liquid phase. Will they predict the existence of the liquid phase from first principles? Weisskopf thinks not, because its existence depends on the fact that the constituents interact strongly with each other. The same statement applies to any state of matter which depends for its existence on strong interactions, such as confinement, and fractional quantum hall phases.31,  32I think it is a defensible claim [82] that if we did not know the IR physics of QCD (e.g., if we did not happen to be made out of color-neutral boundstates of quarks and gluons) before the discovery of AdS/CFT, we would have predicted color confinement by finding its dual geometry [19, 45, 46]. Our ability to imagine the possible behavior of a bunch of stuff has been limited by our dependence on our weak coupling tools and on experimenters to actually assemble the stuff. It is exciting that we now have another tool, which allows us to ask these questions in a way which involves such simple geometrical pictures. Perhaps there are even states of matter that we can describe this way which have already been seen, but which have not yet been understood.


## 7.1. Remarks on Other Observables

Besides correlation functions and thermodynamic potentials, a number of other observables of interest can be computed easily using the correspondence. Some, like expectation values of Wilson loops, are relatively specific to gauge theories.A very ubiquitous observable, which is notoriously hard to compute otherwise, is the entanglement entropy. If we divide the hilbert space of the QFT into(7.1)ℋ=ℋA⊕ℋB
and declare ourselves ignorant of ℋB, we have an ensemble described by the density matrix:  (7.2)ρA=trHBρ,
where ρ is the density matrix of the whole system, which let's take to be the one associated with the ground state ρ=|Ω〉〈Ω|. Then the entanglement entropy between A and B is the von Neumann entropy of ρA:  (7.3)S=-trAρAlnρA.In the special case where the subdivision (7.1) is done by cutting out a region of space B bounded by some d-2-dimensional surface Σ, there exists a proposal for how to calculate the associated entanglement entropy SΣ using the holographic dual [78] (for a recent review, see [79]). The idea is extremely simple: just find the surface M (d-1-dimensional), ending on the boundary at Σ, which extremizes its area. The formula for SΣ is then  (7.4)SΣ=extremum∂M=Σarea(M)4GN,
very reminiscent of the Bekenstein formula for the entropy of a black hole. This formula passes many checks. For example, it gives the correct universal behavior S~(c/3)lnL (L is the length of the region B) in a 1+1-dimensional CFT of central charge c. Like the Casimir energy, this match between weak and strong coupling is precise because it is determined by the conformal anomaly. In higher dimensions, the holographic prescription gives a prediction for which terms in the expansion of S(L) in powers of L are universal. There even exists a heuristic derivation [80].30


## 7.2. Remarks on the Role of Supersymmetry

Supersymmetry has played important roles in the historical development of the AdS/CFT correspondence.(i)
It constrains the form of the interactions, meaning that there are fewer possible candidates for the dual (e.g., the maximalAdS supergravity theory in five dimensions is unique).(ii)
A supersymmetric theory has many observables which are independent of the coupling. These so-called BPS quantities allow for many quantitative checks of a proposed dual pair.(iii)
Supersymmetry can stabilize aline of exact fixed points (e.g., in the 𝒩=4 SYM), rendering the coupling constant a dimensionless parameter which interpolates between the weakly coupled field theory description and the gravity regime.However, it has played no role in our discussion. Some people believe that supersymmetry may be necessary for the construction of a consistent theory of quantum gravity. But it seems more likely to me that the formulation of specific examples of the duality without supersymmetry is a (perhaps hard) technical problem, not one of principle.


## 7.3. Lessons for How to Use AdS/CFT

Critical exponents depend on “Landscape Issues”. By this I mean just that they depend on the values of the couplings in the bulk action (in particular, the masses of bulk fields), which are specified only by some UV completion, that is, by available string vacua. For each possible bulk coupling, it is very much an open question which values arise in a consistent theory of quantum gravity. This situation—that the critical exponents are UV-sensitive quantities—is a rather unfamiliar one!At least in examples we know, thermodynamics is not very sensitive to strong coupling. In both the𝒩=4 SYM, and from lattice QCD, we find for, for example, the free energy a relation of the form  (7.5)Fstrong~34Fweak.Real-time dynamics and transportare very sensitive to the strength of the interactions. For example,  (7.6)(ηs)weak~1g4Ing≫(ηs)strong~14π.
Not only are these observables sensitive to strong coupling, but they are very natural things to compute using the holographic technology. In particular, although it is a classical description, it automatically includes dissipation. Ordinary techniques seem to require the existence of a description of what is being transported in terms of quasiparticles, so that the Boltzmann equation can be used. Since we know that such a description need not exist, this is a very good opportunity for the machinery described above to be useful.I would like to close with a final optimistic philosophical comment. The following gedanken experiment was proposed by Weisskopf [81]: take a bunch of theoretical physicists and lock them away from birth so that they are never exposed to any substance in the liquid phase. Will they predict the existence of the liquid phase from first principles? Weisskopf thinks not, because its existence depends on the fact that the constituents interact strongly with each other. The same statement applies to any state of matter which depends for its existence on strong interactions, such as confinement, and fractional quantum hall phases.31,  32I think it is a defensible claim [82] that if we did not know the IR physics of QCD (e.g., if we did not happen to be made out of color-neutral boundstates of quarks and gluons) before the discovery of AdS/CFT, we would have predicted color confinement by finding its dual geometry [19, 45, 46]. Our ability to imagine the possible behavior of a bunch of stuff has been limited by our dependence on our weak coupling tools and on experimenters to actually assemble the stuff. It is exciting that we now have another tool, which allows us to ask these questions in a way which involves such simple geometrical pictures. Perhaps there are even states of matter that we can describe this way which have already been seen, but which have not yet been understood.



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*Source: 723105-2010-08-05.xml*