0909/0909.0518.md

Holographic duality with a view toward many-body physics

John McGreevy

Center for Theoretical Physics, MIT, Cambridge, Massachusetts 02139, USA
Kavli Institute for Theoretical Physics, Santa Barbara, California 93106-4030, USA
mcgreevy at mit.edu

Abstract

These are notes based on a series of lectures given at the KITP workshop Quantum 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.

September 2009, revised May 2010# Contents

1	Introductory remarks	2
2	Motivating the correspondence	3
2.1	Counting of degrees of freedom . . . . .	9
2.2	Preview of the AdS/CFT correspondence . . . . .	10
3	When is the gravity theory classical?	12
3.1	Large $n$ vector models . . . . .	13
3.2	't Hooft counting . . . . .	13
3.3	$N$-counting of correlation functions . . . . .	19
3.4	Simple generalizations . . . . .	22
4	Vacuum CFT correlators from fields in $AdS$	23
4.1	Wave equation near the boundary and dimensions of operators . . . . .	26
4.2	Solutions of the $AdS$ wave equation and real-time issues . . . . .	30
4.3	Bulk-to-boundary propagator in momentum space . . . . .	32
4.4	The response of the system to an arbitrary source . . . . .	34
4.5	A useful visualization . . . . .	36
4.6	$n$-point functions . . . . .	36
4.7	Which scaling dimensions are attainable? . . . . .	38
4.8	Geometric optics limit . . . . .	39
4.9	Comment on the physics of the warp factor . . . . .	41
5	Finite temperature and density	42
5.1	Interjection on expectations for CFT at finite temperature . . . . .	42
5.2	Back to the gravity dual . . . . .	43
5.3	Finite density . . . . .	47
6	Hydrodynamics and response functions	49
6.1	Linear response, transport coefficients . . . . .	49

6.2	Holographic calculation of transport coefficients . . . . .	50
7	Concluding remarks	53
7.1	Remarks on other observables . . . . .	53
7.2	Remarks on the role of supersymmetry . . . . .	54
7.3	Lessons for how to use AdS/CFT . . . . .	55

1 Introductory remarks

My task in these lectures is to engender some understanding of the following

Bold 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 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 is true, 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 over-represented 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, 3, 4, 2, 5, 6, 7], has been liberally borrowed to construct these notes. In addition, some of the tex 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. In particular I wish to thank Christiana Athanasiou, Francesco D’Eramo, Tom Faulkner, Tarun Grover, Wing-Ko Ho, Vijay Kumar,Tongyan Lin, Daniel Park, and Brian Swingle; specific figure credits appear in the margins. The discussion in section 6.2 follows a talk given by Nabil Iqbal. I’m grateful to Sean Hartnoll, Joe Polchinski, and Subir Sachdev for helpful discussions and for giving me the opportunity to inflict my perspective on their workshop participants, those participants for their many lively questions, and to Pavel Kovtun for teaching me many of the ideas discussed herein. Thanks also to Brian Swingle and T. Senthil for being the first victims of many of these explanations, and to Koushik Balasubramanian, Dan Freedman, Hong Liu, Kostas Skenderis and Erik Tonni for comments on the draft. The title is adapted from [9]. The selection of references was made based on perceived pedagogical value and personal bias.

2 Motivating the correspondence

To understand what one might mean by a more precise version of the Bold 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’s 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 the 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:

Theorem [Weinberg-Witten[10]]: A QFT with a Poincaré covariant conserved stress tensor $T^{\mu\nu}$ forbids massless particles of spin $j > 1$ which carry momentum (i.e. with $P^\mu = \int d^D x T^{0\mu} \neq 0$ ).

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: $T^{\mu\nu} \propto \frac{\delta S}{\delta g_{\mu\nu}} = 0$ . (Alternatively, the‘matter stress tensor,’ which doesn’t 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 needn’t 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 you have in a volume $V$ (bounded by an area $A$ ) a configuration with entropy $S > S_{BH} = \frac{A}{4G_N}$ (where $S_{BH}$ is the entropy of the biggest black hole fittable in $V$ ), but which has less energy. Then by throwing in more stuff (as arbitrarily non-adiabatically as necessary, i.e. you can increase the entropy), since stuff that carries entropy also carries energy¹, 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 you can’t 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 $N_s \sim e^V$ (maximum entropy = $\ln N_s$ ). 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

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¹ Matthew Fisher raises the point that there are systems (ones with topological order) where it is possible to create an information-carrying excitation which doesn’t change the energy. I’m not sure exactly how to defend Bekenstein’s argument from this. I think an important point must be that the effects of such excitations are not completely local (which is why they would be good for quantum computing). A related issue about which more work has been done is the species problem: if there are many species of fields in the bulk, information can be carried by the species label, without any cost in energy. There are two points which save Bekenstein from this: (1) if there are a large number of species of fields, their fluctuations renormalize the Newton constant (to make gravity weaker), and weaken the bound. (2) Being able to encode information in the species label implies that there is some continuous global symmetry. It is believed that theories of quantum gravity do not have continuous global symmetries (roughly because virtual black holes can eat the charge and therefore generate symmetry-breaking operators in the effective action, see e.g. page 12 of [12]).thing is actually true, and the AdS/CFT correspondence [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 don't 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) Don't 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

$$S_{CS}\sim {\int }_M\text{tr}A\wedge dA+\dots S\_\{CS\}\; \backslash sim\; \backslash int\_M\; \backslash text\{tr\}\; A\; \backslash wedge\; dA\; +\; \backslash dots$$

(where the dots is extra stuff to make the nonabelian case gauge invariant); note that there's 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:

$$u{\mathrm{\partial }}_{u}g=\beta (g(u))\mathrm{.}u\; \backslash partial\_u\; g\; =\; \backslash beta(g(u))\; .$$

The beta function is determined by the coupling constant evaluated at the energy scale $u$ , and we don't 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 [23]), but it is a good hint.To summarize, we have three hints for interpreting the Bold Assertion:

1. 1. The Weinberg-Witten theorem suggests that the graviton lives on a different space than the QFT in question.
2. 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. 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 non-overlapping regimes of validity.²

Next we will make a simplifying assumption in an effort to find concrete examples. The simplest case of an RG flow is when $\beta = 0$ and the system is self-similar. In a Lorentz invariant theory (which we also assume for simplicity), this means that the following scale transformation $x^\mu \rightarrow \lambda x^\mu$ ( $\mu = 0, 1, 2, \dots, d-1$ ) 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 $u \rightarrow \frac{u}{\lambda}$ . The most general $(d+1)$ -dimensional metric (one extra dimension) with this symmetry and Poincaré invariance is of the following form:

$$d{s}^2={\left(\frac{\tilde{u}}{\tilde{L}}\right)}^2{\eta }_{\mu \nu }d{x}^{\mu }d{x}^{\nu }+\frac{d{\tilde{u}}^2}{{\tilde{u}}^2}{L}^2\mathrm{.}ds^2\; =\; \backslash left(\backslash frac\{\backslash tilde\{u\}\}\{\backslash tilde\{L\}\}\backslash right)^2\; \backslash eta\_\{\backslash mu\backslash nu\}\; dx^\backslash mu\; dx^\backslash nu\; +\; \backslash frac\{d\backslash tilde\{u\}^2\}\{\backslash tilde\{u\}^2\}\; L^2\; .$$

We can bring it into a more familiar form by a change of coordinates, $\tilde{u} = \frac{\tilde{L}}{L} u$ :

$$d{s}^2={\left(\frac{u}{L}\right)}^2{\eta }_{\mu \nu }d{x}^{\mu }d{x}^{\nu }+\frac{d{u}^2}{u^2}{L}^2\mathrm{.}ds^2\; =\; \backslash left(\backslash frac\{u\}\{L\}\backslash right)^2\; \backslash eta\_\{\backslash mu\backslash nu\}\; dx^\backslash mu\; dx^\backslash nu\; +\; \backslash frac\{du^2\}\{u^2\}\; L^2\; .$$

This is $AdS_{d+1}$ ³. It is a family of copies of Minkowski space, parametrized by $u$ , whose size varies with $u$ (see Fig. 1). The parameter $L$ is called the ‘AdS radius’ and it has dimensions

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² The criterion ‘different’ may require some effort to check. This Principle is sometimes also called ‘Conservation of Difficulty’.

³It turns out that this metric also has conformal invariance. So scale and Poincaré symmetry implies conformal invariance, at least when there is a gravity dual. This is believed to be true more generally [14],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 parameter $z$ . 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.

of length. Although this is a dimensionful parameter, a scale transformation $x^\mu \rightarrow \lambda x^\mu$ can be absorbed by rescaling the radial coordinate $u \rightarrow u/\lambda$ (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 $z \equiv \frac{L^2}{u}$ , in which the metric takes the form

$$d{s}^2={\left(\frac{L}{z}\right)}^2({\eta }_{\mu \nu }d{x}^{\mu }d{x}^{\nu }+d{z}^2)\mathrm{.}(2.1)ds^2\; =\; \backslash left(\backslash frac\{L\}\{z\}\backslash right)^2\; (\backslash eta\_\{\backslash mu\backslash nu\}\; dx^\backslash mu\; dx^\backslash nu\; +\; dz^2)\; \backslash quad\; .\; \backslash quad\; (2.1)$$

These coordinates are better because fewer symbols are required to write the metric. $z$ will map to the length scale in the dual theory.

So it seems that a $d$ -dimensional conformal field theory (CFT) should be related to a theory of gravity on $AdS_{d+1}$ . This metric (2.1) solves the equations of motion of the following action (and many others)⁴

$$S_{\text{bulk}}[g,\dots ]=\frac{1}{16\pi G_N}\int d^{d+1}x\sqrt{g}(-2\mathrm{\Lambda }+\mathcal{R}+\dots )\mathrm{.}(2.2)S\_\{\backslash text\{bulk\}\}[g,\; \backslash dots]\; =\; \backslash frac\{1\}\{16\backslash pi\; G\_N\}\; \backslash int\; d^\{d+1\}x\; \backslash sqrt\{g\}\; (-2\backslash Lambda\; +\; \backslash mathcal\{R\}\; +\; \backslash dots)\; \backslash quad\; .\; \backslash quad\; (2.2)$$

Here, $\sqrt{g} \equiv \sqrt{|\det g|}$ makes the integral coordinate-invariant, and $\mathcal{R}$ is the Ricci scalar

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but there is no proof for $d > 1 + 1$ . Without Poincaré invariance, scale invariance definitely does not imply conformal invariance; indeed there are scale-invariant metrics without Poincaré symmetry, which do not have special conformal symmetry [16].

⁴For verifying statements like this, it can be helpful to use Mathematica or some such thing.curvature. The cosmological constant $\Lambda$ is related by the equations of motion

$$0=\frac{\delta S_{\text{bulk}}}{\delta g^{AB}}⟹R_{AB}+\frac{d}{L^2}{g}_{AB}=0(2.3)0\; =\; \backslash frac\{\backslash delta\; S\_\{\backslash text\{bulk\}\}\}\{\backslash delta\; g^\{AB\}\}\; \backslash implies\; R\_\{AB\}\; +\; \backslash frac\{d\}\{L^2\}\; g\_\{AB\}\; =\; 0\; \backslash quad\; (2.3)$$

to the value of the AdS radius: $-2\Lambda = \frac{d(d-1)}{L^2}$ . This form of the action (2.2) 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, i.e. by the number of derivatives. The Einstein-Hilbert term (the one with the Ricci scalar) is an irrelevant operator: $\mathcal{R} \sim \partial^2 g + (\partial g)^2$ has dimensions of length $^{-2}$ , so $G_N$ here is a length $^{d-1}$ , the Planck length: $G_N \equiv \ell_{pl}^{d-1} \equiv M_{pl}^{1-d}$ (in units where $\hbar = c = 1$ ). The gravity theory is classical if $L \gg \ell_{pl}$ . In this spirit, the ... on the RHS denote 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 (2.2) is indeed what comes from string theory at low energies and when the curvature (here, $\mathcal{R} \sim \frac{1}{L^2}$ ) is small (compared to the string tension, $\frac{1}{\alpha'} \equiv \frac{1}{\ell_s^2}$ ; 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. The AdS metric above has a boundary at $z = 0$ . This is a bit subtle. Keeping $x^\mu$ 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 a $d$ -dimensional conformal field theory is related to a theory of gravity on spaces which are asymptotically $AdS_{d+1}$ . 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’s 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 anddifferent bulk actions, e.g. different values of the coupling constants in $S_{\text{bulk}}$ . The example which is understood best is the case of the $\mathcal{N} = 4$ super Yang-Mills theory (SYM) in four dimensions. This is dual to maximal supergravity in $AdS_5$ (which arises by dimensional reduction of ten-dimensional IIB supergravity on $AdS_5 \times S^5$ ). 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 discrete⁵. 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 exponent $z \neq 1$ [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 in $AdS_{d+1}$ 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), i.e. the maximum entropy:

$$\frac{\text{Area of boundary}}{4G_N}\text{number of dof of QFT}\equiv N_d\text{ .}\backslash frac\{\backslash text\{Area\; of\; boundary\}\}\{4G\_N\}\; \backslash stackrel\{?\}\{=\}\; \backslash text\{number\; of\; dof\; of\; QFT\}\; \backslash equiv\; N\_d\; \backslash text\{\; .\}$$

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-off $\epsilon$ (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 $(\frac{R}{\epsilon})^{d-1}$ , times the number of degrees of freedom per lattice site, which we will call ‘ $N^2$ ’. The behavior suggested by the name we have given this number is found in well-understood examples. It is, however, clear (for example from the structure of known $AdS$ vacua of string theory [21]) that other behaviors $N^b$ are possible, and that’s why I made it a funny color and put it in quotes. So $N_d = \frac{R^{d-1}}{\epsilon^{d-1}} N^2$ .

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⁵An example of this is the relationship (2.7) between the Newton constant in the bulk and the number of species in the field theory, which we will find in the next subsection.The picture we have of $\text{AdS}_{d+1}$ is a collection of copies of $d$ -dimensional Minkowski space of varying size; the boundary is the locus $z \rightarrow 0$ where they get really big. The area of the boundary is

$$A={\int }_{{\mathbb{R}}^{d-1},z\to 0,\text{ fixed }t}\sqrt{g}{d}^{d-1}x={\int }_{{\mathbb{R}}^{d-1},z\to 0}{d}^{d-1}x\frac{L^{d-1}}{z^{d-1}}\mathrm{.}(2.4)A\; =\; \backslash int\_\{\backslash mathbb\{R\}^\{d-1\},\; z\; \backslash rightarrow\; 0,\; \backslash text\{\; fixed\; \}\; t\}\; \backslash sqrt\{g\}\; d^\{d-1\}x\; =\; \backslash int\_\{\backslash mathbb\{R\}^\{d-1\},\; z\; \backslash rightarrow\; 0\}\; d^\{d-1\}x\; \backslash frac\{L^\{d-1\}\}\{z^\{d-1\}\}\; .\; \backslash quad\; (2.4)$$

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 $z = \epsilon$ . 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 $z = \epsilon$ is a UV cutoff (not identical to the lattice cutoff, but close enough for our present purposes). Given this,

$$A={\int }_0^R{d}^{d-1}x\frac{L^{d-1}}{z^{d-1}}{\mid }_{z=ϵ}={\left(\frac{RL}{ϵ}\right)}^{d-1}\mathrm{.}(2.5)A\; =\; \backslash int\_0^R\; d^\{d-1\}x\; \backslash frac\{L^\{d-1\}\}\{z^\{d-1\}\}\; \backslash Big|\_\{z=\backslash epsilon\}\; =\; \backslash left(\; \backslash frac\{RL\}\{\backslash epsilon\}\; \backslash right)^\{d-1\}\; .\; \backslash quad\; (2.5)$$

The holographic principle then says that the maximum entropy in the bulk is

$$\frac{A}{4G_N}\sim \frac{L^{d-1}}{4G_N}{\left(\frac{R}{ϵ}\right)}^{d-1}\mathrm{.}(2.6)\backslash frac\{A\}\{4G\_N\}\; \backslash sim\; \backslash frac\{L^\{d-1\}\}\{4G\_N\}\; \backslash left(\; \backslash frac\{R\}\{\backslash epsilon\}\; \backslash right)^\{d-1\}\; .\; \backslash quad\; (2.6)$$

We see that the scaling with the system size agrees – the both-hand-side goes like $R^{d-1}$ . So $\text{AdS/CFT}$ is indeed an implementation of the holographic principle. We can learn more from this calculation: In order for the prefactors of $R^{d-1}$ to agree, we need to relate the $AdS$ radius in Planck units $\frac{L^{d-1}}{G_N} \sim (LM_{pl})^{d-1}$ to the number of degrees of freedom per site of the field theory:

$$\boxed{{\displaystyle \frac{L^{d-1}}{G_N}=N^2}}(2.7)\backslash boxed\{\backslash frac\{L^\{d-1\}\}\{G\_N\}\; =\; N^2\}\; \backslash quad\; (2.7)$$

up to numerical prefactors.

2.2 Preview of the $\text{AdS/CFT}$ correspondence

Here's the ideology:

fields in $\backslash text\{AdS\}$	$\backslash longleftrightarrow$	local operators of CFT
spin		spin
mass		scaling dimension $\backslash Delta$

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 $m^2 L_{AdS}^2 = \Delta(\Delta - d)$ , as we'll 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 order $N^0$ ) dimensions. If you are aware of explicit examples of such theories, please let me know⁶⁷. This is to be distinguished from the thus-far-intractable case where some whole tower of massive string modes in the bulk are needed.

Now let's consider some observables of a QFT (we'll assume Euclidean spacetime for now), namely vacuum correlation functions of local operators in the CFT:

$$⟨{\mathcal{O}}_1(x_1){\mathcal{O}}_2(x_2)\cdots {\mathcal{O}}_n(x_n)⟩\mathrm{.}\backslash langle\; \backslash mathcal\{O\}\_1(x\_1)\; \backslash mathcal\{O\}\_2(x\_2)\; \backslash cdots\; \backslash mathcal\{O\}\_n(x\_n)\; \backslash rangle\; .$$

We can write down a generating functional $Z[J]$ for these correlators by perturbing the action of the QFT:

where $J_A(x)$ are arbitrary functions (sources) and ${\mathcal{O}_A(x)}$ is some basis of local operators. The $n$ -point function is then given by:

$$⟨\prod _n{\mathcal{O}}_n(x_n)⟩=\prod _n\frac{\delta }{\delta J_n(x_n)}\ln Z{\mid }_{J=0}\mathrm{.}\backslash langle\; \backslash prod\_n\; \backslash mathcal\{O\}\_n(x\_n)\; \backslash rangle\; =\; \backslash prod\_n\; \backslash frac\{\backslash delta\}\{\backslash delta\; J\_n(x\_n)\}\; \backslash ln\; Z\; \backslash Big|\_\{J=0\}\; .$$

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⁶Rational CFTs in two dimensions don't count because they fail our other criterion for a simple gravity dual: in the case of a 2d CFT, the central charge of the Virasoro algebra, $c$ , is a good measure of ' $N^{2'}$ , the number of degrees of freedom per point. But rational CFTs have $c$ of order unity, and therefore can only be dual to very quantum mechanical theories of gravity. But this is the right idea. Joe Polchinski has referred to the general strategy being applied here as "the Bootstrap for condensed matter physics". The connection with the bootstrap in its CFT incarnation [22] is made quite direct in [23].

⁷ Eva Silverstein and Shamit Kachru have emphasized that this special property of these field theories is a version of the 'cosmological constant problem,' i.e. it is dual to the specialness of having a small cosmological constant in the bulk. At least in the absence of supersymmetry, there is some tuning that needs to be done in the landscape of string vacua to choose these vacua with a small vacuum energy, and hence a large AdS radius. Here is a joke about this: when experimentalists look at some material and see lots of complicated crossovers, they will tend to throw it away; if they see instead some simple beautiful power laws, as would happen in a system with few low-dimension operators, they will keep it. Perhaps these selection effects are dual to each other.Since $\mathcal{L}_J$ 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, $z \rightarrow 0$ . (Recall from the counting of degrees of freedom in section 2.1 that QFT with UV cutoff $E < 1/\epsilon \longleftrightarrow$ AdS cutoff $z > \epsilon$ .) The perturbation $J$ of the CFT action will be encoded in the boundary condition on bulk fields.

The idea ([24, 25], often referred to as GKPW) for computing $Z[J]$ is then, schematically:

$$Z[J]\equiv ⟨e^{-\int {\mathcal{L}}_J}{⟩}_{CFT}=\underset{=\text{???}}{\underbrace{Z_{\text{QG}}[\text{b.c. depends on }J]}}{e}^{-S_{\text{grav}}}{\mid }_{\text{EOM, b.c. depend on }J}\mathrm{.}(2.8)Z[J]\; \backslash equiv\; \backslash langle\; e^\{-\backslash int\; \backslash mathcal\{L\}\_J\}\; \backslash rangle\_\{CFT\}\; =\; \backslash underbrace\{Z\_\{\backslash text\{QG\}\}[\backslash text\{b.c.\; depends\; on\; \}\; J]\}\_\{=\backslash text\{???\}\}\; \backslash underset\{N\; \backslash gg\; 1\}\{\backslash sim\}\; e^\{-S\_\{\backslash text\{grav\}\}\}\; \backslash Big|\_\{\backslash text\{EOM,\; b.c.\; depend\; on\; \}\; J\}\; .\; \backslash quad\; (2.8)$$

The middle object is the partition function of quantum gravity. We don't 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're doing, and we can do the path integral by saddle point, as indicated on the RHS of (2.8).

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've 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:

$$Z_{\text{some }QFTs}[\text{sources}]\approx e^{-S_{\text{bulk}}[\text{boundary conditions at }z\to 0]}{\mid }_{\text{extremum of }{S}_{\text{bulk}}}\mathrm{.}(3.9)Z\_\{\backslash text\{some\; \}\; QFTs\}[\backslash text\{sources\}]\; \backslash approx\; e^\{-S\_\{\backslash text\{bulk\}\}[\backslash text\{boundary\; conditions\; at\; \}\; z\; \backslash rightarrow\; 0]\}\; \backslash Big|\_\{\backslash text\{extremum\; of\; \}\; S\_\{\backslash text\{bulk\}\}\}\; .\; \backslash quad\; (3.9)$$

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

$$\frac{L^{d-1}}{G_N}{\equiv }^{\mathrm{\prime }}{N}^{2^{\mathrm{\prime }}}\gg 1.(3.10)\backslash frac\{L^\{d-1\}\}\{G\_N\}\; \backslash equiv\; \text{'}N^\{2\text{'}\}\; \backslash gg\; 1.\; \backslash quad\; (3.10)$$

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⁸Note that I am not saying here that the configuration of the elementary fields in the path integral necessarily have some simple description at the saddle point. Thanks to Larry Yaffe for emphasizing this point.This quantity, the $AdS$ radius in Planck units $\frac{L^{d-1}}{G_N} \equiv (LM_{pl})^{d-1}$ , 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 an $n$ -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 Large $n$ 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 $\vec{\varphi} = (\varphi_1, \dots, \varphi_n)$ with the following action

$$S[\phi ]=-\frac{1}{2}\int d^{d}x({\mathrm{\partial }}_{\mu }\vec{\phi }{\mathrm{\partial }}^{\mu }\vec{\phi }+m^2\vec{\phi }\cdot \vec{\phi }+\frac{{\lambda }_v}{n}(\vec{\phi }\cdot \vec{\phi }{)}^2)\mathrm{.}(3.11)S[\backslash varphi]\; =\; -\backslash frac\{1\}\{2\}\; \backslash int\; d^d\; x\; \backslash left(\; \backslash partial\_\backslash mu\; \backslash vec\{\backslash varphi\}\; \backslash partial^\backslash mu\; \backslash vec\{\backslash varphi\}\; +\; m^2\; \backslash vec\{\backslash varphi\}\; \backslash cdot\; \backslash vec\{\backslash varphi\}\; +\; \backslash frac\{\backslash lambda\_v\}\{n\}\; (\backslash vec\{\backslash varphi\}\; \backslash cdot\; \backslash vec\{\backslash varphi\})^2\; \backslash right)\; .\; \backslash quad\; (3.11)$$

The fields $\vec{\varphi}$ transform in the fundamental representation of the $O(n)$ symmetry group. Some foresight has been used to determine that the quartic coupling $\lambda_v$ 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 $\sigma \equiv \vec{\varphi} \cdot \vec{\varphi}$ by standard path-integral tricks, and the effective action for $\sigma$ is

$$S_{\text{eff}}[\sigma ]=-\frac{n}{2}[\int \frac{{\sigma }^2}{2\lambda }+\text{tr}\ln (-{\mathrm{\partial }}^2+m^2+\sigma )]\mathrm{.}(3.12)S\_\{\backslash text\{eff\}\}[\backslash sigma]\; =\; -\backslash frac\{n\}\{2\}\; \backslash left[\; \backslash int\; \backslash frac\{\backslash sigma^2\}\{2\backslash lambda\}\; +\; \backslash text\{tr\}\; \backslash ln\; (-\backslash partial^2\; +\; m^2\; +\; \backslash sigma)\; \backslash right]\; .\; \backslash quad\; (3.12)$$

The important thing is the giant factor of $n$ in front of the action which makes the theory of $\sigma$ 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’ve found a description of this saddle point within 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 [32] have suggested what the (not simple) gravity dual might be.

3.2 ’t Hooft counting

“You can hide a lot in a large- $N$ matrix.”

– 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 isn’t immediately useful, but it isn’t 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’s 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 $N^2$ . Evidence from the space of string vacua suggests that there are many generalizations of this where the number of dofs per point goes like $N^b$ for $b \neq 2$ [21]. However, a generalization of the ’t Hooft limit is not yet well-understood for other cases¹⁰.

Consider a (any) quantum field theory with matrix fields, $\Phi_{a=1,\dots,N}^{b=1,\dots,N}$ . By matrix fields, we mean that their products appear in the Lagrangian only in the form of matrix multiplication, e.g. $(\Phi^2)_a^c = \Phi_a^b \Phi_b^c$ , which is a big restriction on the interactions. It means the interactions must be invariant under $\Phi \rightarrow U^{-1} \Phi U$ ; for concreteness we’ll take the matrix group to be $U \in U(N)$ 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 $O(N^2)$ symmetry) changes the scaling of the coupling in the large $N$ limit.

In particular, consider the ’t Hooft limit in which $N \rightarrow \infty$ and $g \rightarrow 0$ with $\lambda = g^2 N$ 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 $g \sim 1/\sqrt{N}$ goes to

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⁹The standard pedagogical source for this material is [29], available from the KEK KISS server.

¹⁰Recently, there has been an explosion of literature on a case where the number of degrees of freedom per point should go like $N^{3/2}$ [30].

¹¹Note that the important distinction between these models and those of the previous subsection is not the difference in groups ( $U(N)$ vs $O(N)$ ), but rather the difference in representation in which the fields transform: here the fields transform in the adjoint representation rather than the fundamental.zero slower than the coupling in the vector model $g_v \equiv \lambda_v/N \sim 1/N$ .

We will be agnostic here about whether the $U(N)$ 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 $\text{tr}\Phi^k$ .

The fields can have all kinds of spin labels and global symmetry labels, but we will just call them $\Phi$ . 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 form:

$$\mathcal{L}\sim \frac{1}{g^2}\text{Tr}((\mathrm{\partial }\mathrm{\Phi }{)}^2+{\mathrm{\Phi }}^2+{\mathrm{\Phi }}^3+{\mathrm{\Phi }}^4+\dots )\mathrm{.}\backslash mathcal\{L\}\; \backslash sim\; \backslash frac\{1\}\{g^2\}\; \backslash text\{Tr\}\; ((\backslash partial\backslash Phi)^2\; +\; \backslash Phi^2\; +\; \backslash Phi^3\; +\; \backslash Phi^4\; +\; \backslash dots)\; .$$

I suppose we want $\Phi$ 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 the $N$ -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 by¹²:

$$⟨{\mathrm{\Phi }}_b^a{\mathrm{\Phi }}_c^d⟩\propto g^2{\delta }_c^a{\delta }_b^d\equiv g^2\begin{array}{c}\text{a}\\ \text{b}\end{array}\begin{array}{c}\text{c}\\ \text{d}\end{array}\backslash langle\; \backslash Phi\_b^a\; \backslash Phi\_c^d\; \backslash rangle\; \backslash propto\; g^2\; \backslash delta\_c^a\; \backslash delta\_b^d\; \backslash equiv\; g^2\; \backslash begin\{array\}\{c\}\; \backslash text\{a\}\; \backslash \backslash \; \backslash text\{b\}\; \backslash end\{array\}\; \backslash begin\{array\}\{c\}\; \backslash text\{c\}\; \backslash \backslash \; \backslash text\{d\}\; \backslash end\{array\}$$

and the vertices by:

$\propto g^{-2}$ $\propto g^{-2}$

[Brian Swingle]

To see the consequences of this more concretely, let's consider some vacuum-to-vacuum diagrams (see Fig. 3 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

$$\text{diagram}\sim {\left(\frac{\lambda }{N}\right)}^{\text{no. of prop.}}{\left(\frac{N}{\lambda }\right)}^{\text{no. of int. vert.}}{N}^{\text{no. of index loops}}\mathrm{.}(3.13)\backslash text\{diagram\}\; \backslash sim\; \backslash left(\backslash frac\{\backslash lambda\}\{N\}\backslash right)^\{\backslash text\{no.\; of\; prop.\}\}\; \backslash left(\backslash frac\{N\}\{\backslash lambda\}\backslash right)^\{\backslash text\{no.\; of\; int.\; vert.\}\}\; N^\{\backslash text\{no.\; of\; index\; loops\}\}\; .\; \backslash quad\; (3.13)$$

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¹²Had we been considering $SU(N)$ , the result would be $\langle \Phi_b^a \Phi_c^d \rangle \propto \delta_c^a \delta_b^d - \delta_b^a \delta_c^d / N^2 = \begin{array}{c} \text{two parallel lines with arrows pointing right} \ \text{minus} \ \text{two parallel lines with arrows pointing left} \end{array} = \begin{array}{c} \text{two parallel lines with arrows pointing right} \ \text{minus} \ \text{two parallel lines with arrows pointing left} \end{array} .$
This difference can be ignored at leading order in the $1/N$ expansion.For example, the diagram in Fig. 2 has 4 three point vertices, 6 propagators, and 4 index loops

Figure 2: This diagram consists of 4 three point vertices, 6 propagators, and 4 index loops

can draw them on a piece of paper without any lines crossing; their contributions take the general form $\lambda^n N^2$ . However, there also exist non-planar graphs, such as the one in Fig. 4, whose contributions are down by (an even number of) powers of $N$ . One thing that's great about this expansion is that the diagrams which are harder to draw are less important.

Figure 3: planar graphs that contribute to the vacuum $\rightarrow$ vacuum amplitude.

Figure 4: Non-planar (but still oriented!) graph that contributes to the vacuum $\rightarrow$ vacuum amplitude.

We can be more precise about how the diagrams are organized. Every double-line graph specifies a triangulation of a 2-dimensional surface $\Sigma$ . 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 Fig. 5 and 6.

Figure 5: Direct surfaces constructed from the vacuum diagram in (a) Fig. 3a and (b) Fig. 4.

Figure 6: Dual surface constructed from the vacuum diagram in Fig. 3c. Note that points at infinity are identified.

If $E$ = number of propagators, $V$ = number of vertices, and $F$ = number of index loops, then the diagram gives a contribution $N^{F-E+V} \lambda^{E-V}$ . 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 $\tilde{F}$ , dual edges $\tilde{E}$ , and dual vertices $\tilde{V}$ . The relationship between the original and dual variables is $E = \tilde{E}$ , $V = \tilde{F}$ , and $F = \tilde{V}$ . The exponent $\chi = F - E + V = \tilde{F} - \tilde{E} + \tilde{V}$ is the Euler character and it is a topological invariant of two dimensional surfaces. In general it is given by $\chi(\Sigma) = 2 - 2h - b$ where $h$ is the number of handles (the genus) and $b$ is the number of boundaries. Note that the exponent of $\lambda$ , $E - V$ or $\tilde{E} - \tilde{F}$ is not a topological invariant and depends on the triangulation (Feynman diagram).

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¹³Please don’t be confused by multiple uses of the word ‘vertex’. There are interaction vertices of various kinds in the Feynman diagrams and these correspond to vertices in the triangulation only in the first formulation.Because the $N$ -counting is topological (depending only on $\chi(\Sigma)$ ) we can sensibly organize the perturbation series for the effective action $\ln Z$ in terms of a sum over surface topology. Because we're computing only vacuum diagrams for the moment, the surfaces we're 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

$$\ln Z=\sum _{h=0}^{\mathrm{\infty }}{N}^{2-2h}\sum _{\mathrm{\ell }=0}^{\mathrm{\infty }}{c}_{\mathrm{\ell },h}{\lambda }^{\mathrm{\ell }}=\sum _{h=0}^{\mathrm{\infty }}{N}^{2-2h}{\mathfrak{F}}_h(\lambda )(3.14)\backslash ln\; Z\; =\; \backslash sum\_\{h=0\}^\{\backslash infty\}\; N^\{2-2h\}\; \backslash sum\_\{\backslash ell=0\}^\{\backslash infty\}\; c\_\{\backslash ell,h\}\; \backslash lambda^\{\backslash ell\}\; =\; \backslash sum\_\{h=0\}^\{\backslash infty\}\; N^\{2-2h\}\; \backslash mathfrak\{F\}\_h(\backslash lambda)\; \backslash quad\; (3.14)$$

where the sum over topologies is explicit.

Now we can see some similarities between this expansion and perturbative string expansions¹⁴. $1/N$ plays the role of the string coupling $g_s$ , 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 $G_N \propto g_s^2$ , so this reproduces our result $G_N \sim N^{-2}$ 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 curvature $L^{-2}$ is small, and it can be studied in perturbation theory in powers of $\frac{\ell_s}{L}$ , where $\ell_s^{-2}$ is the string tension. We can think of $\lambda$ as a sort of chemical potential for edges in our triangulation. Looking back at our diagram counting we can see that if $\lambda$ becomes large then diagrams with lots of edges are important. Thus large $\lambda$ 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 $\lambda^{-1} \sim \alpha'$ which encodes our intuition about large $\lambda$ 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've been suppressing on our field $\Phi$ . For example, the position labeling where the field $\Phi$ sits plays the role of embedding coordinates on the worldsheet. Other indices (spin, etc.) indicate further worldsheet degrees

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¹⁴The following two paragraphs may be skipped by the reader who doesn't want to hear about string theory.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. [31]).

As a final check on the non-triviality of the theory in the 't Hooft limit, let's see if the 't Hooft coupling runs with scale. For argument let's think about the case when the matrices are gauge fields and $L = -\frac{1}{g_{YM}^2} \text{tr} F_{\mu\nu} F^{\mu\nu}$ . In $d$ dimensions, the behavior through one loop is

$$\mu {\mathrm{\partial }}_{\mu }{g}_{YM}\equiv {\beta }_g\sim \frac{4-d}{2}{g}_{YM}+b_0{g}_{YM}^{3}N\mathrm{.}(3.15)\backslash mu\; \backslash partial\_\backslash mu\; g\_\{YM\}\; \backslash equiv\; \backslash beta\_g\; \backslash sim\; \backslash frac\{4-d\}\{2\}\; g\_\{YM\}\; +\; b\_0\; g\_\{YM\}^3\; N\; .\; \backslash quad\; (3.15)$$

( $b_0$ is a coefficient which depends on the matter content, and vanishes for $\mathcal{N} = 4$ SYM.) So we find that $\beta_\lambda \sim \frac{4-d}{2} \lambda + b_0 \lambda^2$ . Thus $\lambda$ can still run in the large $N$ limit and the theory is non-trivial.

3.3 $N$ -counting of correlation functions

Let's now consider the $N$ -counting for correlation functions of local gauge-invariant operators. Motivated by gauge invariance and simplicity, we will consider “single trace” operators, operators $\mathcal{O}(x)$ that look like

$$\mathcal{O}(x)=c(k,N)\text{Tr}({\mathrm{\Phi }}_1(x)\dots {\mathrm{\Phi }}_k(x))(3.16)\backslash mathcal\{O\}(x)\; =\; c(k,\; N)\; \backslash text\{Tr\}(\backslash Phi\_1(x)\; \backslash dots\; \backslash Phi\_k(x))\; \backslash quad\; (3.16)$$

and which we will abbreviate as $\text{Tr}(\Phi^k)$ . We will keep $k$ finite as $N \rightarrow \infty$ ¹⁵. There are two little complications here. We must be careful about how we normalize the fields $\Phi$ and we must be careful about how we normalize the operator $\mathcal{O}$ . The normalization of the fields will continue to be such that the Lagrangian takes the form $L = \frac{1}{g_{YM}^2} \mathcal{L} = \frac{N}{\lambda} \mathcal{L}$ with $\mathcal{L}(\Phi)$ containing no explicit factors of $N$ . To fix the normalization of $\mathcal{O}$ (to determine the constant $c(k, N)$ ) we will demand that when acting on the vacuum, the operator $\mathcal{O}$ creates states of finite norm in the large- $N$ limit, i.e. $\langle \mathcal{O} \mathcal{O} \rangle_c \sim N^0$ where the subscript $c$ stands for connected.

To determine $c(k, N)$ 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

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¹⁵From the point of view of the worldsheet, these operators create closed-string excitations, such as the graviton.Figure 7: New vertex for an operator insertion of $\text{Tr}(\Phi^k)$ with $k = 6$

be attached and looks like a big squid. An example of such a new vertex appears in Fig. 7 which corresponds to the insertion of the operator $\text{Tr}(\Phi^6)$ . For the moment we don't associate any explicit factors of $N$ with the new vertex. Let's consider the example $\langle \text{Tr}(\Phi^4)\text{Tr}(\Phi^4) \rangle$ . 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 Fig. 8. The leading disconnected diagram has four propagators and six index loops and so gives a factor $\lambda^4 N^2 \sim N^2$ . On the other hand, the leading connected diagram shown in Fig. 9 has four propagators and four index loops and so only gives a contribution $\lambda^4 \sim N^0$ . (A way to draw the connected diagram in Fig. 9 which makes the $N$ -counting easier is shown in Fig. 10 where we have deformed the two four point operator insertion vertices so that they are “ready for contraction”.)

The fact that disconnected diagrams win in the large $N$ 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 $\langle \mathcal{O}\mathcal{O} \rangle \sim \langle \mathcal{O} \rangle \langle \mathcal{O} \rangle + O(1/N^2)$ .

Figure 8: Disconnected diagram contributing to the correlation function $\langle \text{Tr}(\Phi^4)\text{Tr}(\Phi^4) \rangle$

The leading connected contribution to the correlation function is independent of $N$ and so $\langle \mathcal{O}\mathcal{O} \rangle_c \sim c^2 N^0$ . Requiring that $\langle \mathcal{O}\mathcal{O} \rangle_c \sim N^0$ means we can just set $c \sim N^0$ . Having fixed the normalization of $\mathcal{O}$ we can now determine the $N$ -dependence of higher-order correlation functions. For example, the leading connected diagram for $\langle \mathcal{O}^3 \rangle$ where $\mathcal{O} = \text{Tr}(\Phi^2)$ is just a triangle and contributes a factor $\lambda^3 N^{-1} \sim N^{-1}$ . In fact, quite generally we haveFigure 9: Connected diagram contributing to the correlation function $\langle \text{Tr}(\Phi^4) \text{Tr}(\Phi^4) \rangle$

Figure 10: A redrawing of the connected diagram shown in Fig. 9

$\langle \mathcal{O}^n \rangle_c \sim N^{2-n}$ for the leading contribution.

So the operators $\mathcal{O}$ (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 [28]), the dimensions of their products are additive at leading order in $N$ .

Finally, we should make a comment about the $N$ -scaling of the generating functional $Z = e^{-W} = \langle e^{-N \sum_A \lambda_A \mathcal{O}^A} \rangle$ . We've normalized the sources so that each $\lambda_A$ is an 't Hooft-like coupling, in that it is finite as $N \rightarrow \infty$ . The effective action $W$ is the sum of connected vacuum diagrams, which at large- $N$ is dominated by the planar diagrams. As we've shown, their contributions go like $N^2$ . This agrees with our normalization of the gravity action,

$$S_{\text{bulk}}\sim \frac{L^{d-1}}{G_N}{I}_{\text{dimensionless}}\sim N^2{I}_{\text{dimensionless}}\mathrm{.}(3.17)S\_\{\backslash text\{bulk\}\}\; \backslash sim\; \backslash frac\{L^\{d-1\}\}\{G\_N\}\; I\_\{\backslash text\{dimensionless\}\}\; \backslash sim\; N^2\; I\_\{\backslash text\{dimensionless\}\}\; \backslash quad\; .\; \backslash quad\; (3.17)$$### 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 of $U(N)$ . We can add fermions $\Delta L \sim \bar{q}\gamma^\mu D_\mu q$ or bosons $\Delta L \sim |D_\mu q|^2$ . 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 Fig. 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 $\bar{q}q$ or $\bar{q}\Phi^k q$ in addition to single trace operators. The extension of the holographic correspondence to include this case [33] has had many applications [34], which are not discussed here for lack of time.

Figure 11: A quark vacuum bubble and a quark vacuum bubble with "gluon" exchange

Another direction for generalization is to consider different matrix groups such as $SO(N)$ or $Sp(N)$ . The adjoint of $U(N)$ is just the fundamental times the anti-fundamental. However, the adjoint representations of $SO(N)$ and $Sp(N)$ are more complicated. For $SO(N)$ the adjoint is given by the anti-symmetric 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 Fig. 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 $\mathbb{RP}^2$ .Figure 12: Propagator for $\text{SO}(N)$ (+) or $\text{Sp}(N)$ (−) matrix models

4 Vacuum CFT correlators from fields in $AdS$

Our next goal is to evaluate $\langle e^{-\int \phi_0 \mathcal{O}} \rangle_{CFT} \equiv e^{-W_{CFT}[\phi_0]}$ , where $\phi_0$ 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 quantities (such as real-time thermal response functions). The general form of the AdS/CFT conjecture for the generating functional is the GKPW equation [24, 25]

$$⟨e^{-\int {\varphi }_0\mathcal{O}}{⟩}_{CFT}=Z_{\text{strings in AdS}}[{\varphi }_0]\mathrm{.}(4.18)\backslash langle\; e^\{-\backslash int\; \backslash phi\_0\; \backslash mathcal\{O\}\}\; \backslash rangle\_\{CFT\}\; =\; Z\_\{\backslash text\{strings\; in\; AdS\}\}[\backslash phi\_0]\; .\; \backslash quad\; (4.18)$$

This thing on the RHS is not yet a computationally effective object; the currently-practical version of the GKPW formula is the classical limit:

$$W_{CFT}[{\varphi }_0]=-\ln ⟨e^{\int {\varphi }_0\mathcal{O}}{⟩}_{CFT}\simeq {\text{extremum}}_{\varphi {\mathrm{\mid }}_{z=ϵ}\sim {\varphi }_0}(N^2{I}_{grav}[\varphi ])+O\left(\frac{1}{N^2}\right)+O\left(\frac{1}{\sqrt{\lambda }}\right)(4.19)W\_\{CFT\}[\backslash phi\_0]\; =\; -\backslash ln\; \backslash langle\; e^\{\backslash int\; \backslash phi\_0\; \backslash mathcal\{O\}\}\; \backslash rangle\_\{CFT\}\; \backslash simeq\; \backslash text\{extremum\}\_\{\backslash phi|\_\{z=\backslash epsilon\}\; \backslash sim\; \backslash phi\_0\}\; \backslash left(\; N^2\; I\_\{grav\}[\backslash phi]\; \backslash right)\; +\; O\backslash left(\backslash frac\{1\}\{N^2\}\backslash right)\; +\; O\backslash left(\backslash frac\{1\}\{\backslash sqrt\{\backslash lambda\}\}\backslash right)\; \backslash quad\; (4.19)$$

There are many things to say about this formula.

• • In the case of matrix theories like those described in the previous section, the classical gravity description is valid for large $N$ and large $\lambda$ . In some examples there is only one parameter which controls the validity of the gravity description. In (4.19) we've made the $N$ -dependence explicit: in units of the $AdS$ radius, the Newton constant is $\frac{L^{d-1}}{G_N} = N^2$ . $I_{grav}$ is some dimensionless action.
• • We said that we are going to think of $\phi_0$ as a small perturbation. Let us then make a perturbative expansion in powers of $\phi_0$ :

$$W_{CFT}[{\varphi }_0]=W_{CFT}[0]+\int d^{D}x{\varphi }_0(x)G_1(x)+\frac{1}{2}\int \int d^D{x}_1{d}^D{x}_2{\varphi }_0(x_1){\varphi }_0(x_2)G_2(x_1,x_2)+\dots (4.20)W\_\{CFT\}[\backslash phi\_0]\; =\; W\_\{CFT\}[0]\; +\; \backslash int\; d^D\; x\; \backslash phi\_0(x)\; G\_1(x)\; +\; \backslash frac\{1\}\{2\}\; \backslash int\; \backslash int\; d^D\; x\_1\; d^D\; x\_2\; \backslash phi\_0(x\_1)\; \backslash phi\_0(x\_2)\; G\_2(x\_1,\; x\_2)\; +\; \backslash dots\; \backslash quad\; (4.20)$$

where

$$G_1(x)=⟨\mathcal{O}(x)⟩=\frac{\delta W}{\delta {\varphi }_0(x)}{\mid }_{{\varphi }_0=0},(4.21)G\_1(x)\; =\; \backslash langle\; \backslash mathcal\{O\}(x)\; \backslash rangle\; =\; \backslash frac\{\backslash delta\; W\}\{\backslash delta\; \backslash phi\_0(x)\}\; \backslash Big|\_\{\backslash phi\_0=0\},\; \backslash quad\; (4.21)$$

$$G_2(x)=⟨\mathcal{O}(x_1)\mathcal{O}(x_2){⟩}_c=\frac{{\delta }^{2}W}{\delta {\varphi }_0(x_1)\delta {\varphi }_0(x_2)}{\mid }_{{\varphi }_0=0}\mathrm{.}(4.22)G\_2(x)\; =\; \backslash langle\; \backslash mathcal\{O\}(x\_1)\; \backslash mathcal\{O\}(x\_2)\; \backslash rangle\_c\; =\; \backslash frac\{\backslash delta^2\; W\}\{\backslash delta\; \backslash phi\_0(x\_1)\; \backslash delta\; \backslash phi\_0(x\_2)\}\; \backslash Big|\_\{\backslash phi\_0=0\}.\; \backslash quad\; (4.22)$$Now if there is no instability, then $\phi_0$ is small implies $\phi$ 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.

• • Anticipating divergences at $z \rightarrow 0$ , we have introduced a cutoff in (4.19) (which will be a UV cutoff in the CFT) and set boundary conditions at $z = \epsilon$ . They are in quotes because they require a bit of refinement (this will happen in subsection 4.1).
• • Eqn (4.19) is written as if there is just one field in the bulk. Really there is a $\phi$ for every operator $\mathcal{O}$ in the dual field theory. For such a pair, we'll say ' $\phi$ couples to $\mathcal{O}$ ' 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. 1. We can organize the both hand side 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. 2. Only 'single-trace' operators (like the $\text{tr}\Phi^k$ s of the previous section) correspond to 'elementary fields' $\phi$ in the bulk. The excitations created by multi-trace operators (like $(\text{tr}\Phi^k)^2$ ) correspond to multi-particle states of $\phi$ (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's not true that the overlap between $\text{tr}\Phi^2\text{tr}\Phi^2|0\rangle$ and $\text{tr}\Phi^4|0\rangle$ is small¹⁶.
  3. 3. There are some simple examples of the correspondence between bulk fields and boundary operators that are determined by symmetry. The stress-energy tensor $T_{\mu\nu}$ is the response of a local QFT to local change in the metric, $S_{bdy} \ni \int \gamma_{\mu\nu} T^{\mu\nu}$ .

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¹⁶It is clear that the 't Hooft limit is not the only way to achieve such a situation, but I am using the language specific to it because it is the one I understand.Here we are writing $\gamma_{\mu\nu}$ for the metric on the boundary. In this case

$$g_{\mu \nu }\leftrightarrow T_{\mu \nu }\mathrm{.}(4.23)g\_\{\backslash mu\backslash nu\}\; \backslash leftrightarrow\; T\_\{\backslash mu\backslash nu\}\; \backslash quad\; .\; \backslash quad\; (4.23)$$

Gauge fields in the bulk correspond to currents in the boundary theory:

$$A_{\mu }^a\leftrightarrow J_a^{\mu }(4.24)A\_\backslash mu^a\; \backslash leftrightarrow\; J\_a^\backslash mu\; \backslash quad\; (4.24)$$

i.e. $S_{bdy} \ni \int A_\mu^a J_a^\mu$ . 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.

• • Finally, something that needs to be emphasized is that changing the Lagrangian of the CFT (by changing $\phi_0$ ) 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 don't 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 $\int (2\Lambda + \mathcal{R} + \dots)$ 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, $\frac{\eta}{s}$ , for the plasma made from any CFT that has an Einstein gravity dual; the answer is always $\frac{1}{4\pi}$ . 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's 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 monovacuitist inclination may say that what's special about them is that they exist¹⁷. 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.

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¹⁷ Monovacuitist (n): One who believes that a theory of quantum gravity should have a unique groundstate (in spite of the fact that we know many examples of much simpler systems which have many groundstates, and in spite of all the evidence to the contrary (e.g. [26, 27])).## 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

$$d{s}^2=L^2\frac{d{z}^2+d{x}^{\mu }d{x}_{\mu }}{z^2}\equiv g_{AB}d{x}^{A}d{x}^{B}A=0,\dots ,d,x^A=(z,x^{\mu })\mathrm{.}(4.25)ds^2\; =\; L^2\; \backslash frac\{dz^2\; +\; dx^\backslash mu\; dx\_\backslash mu\}\{z^2\}\; \backslash equiv\; g\_\{AB\}\; dx^A\; dx^B\; \backslash quad\; A\; =\; 0,\; \backslash dots,\; d,\; \backslash quad\; x^A\; =\; (z,\; x^\backslash mu)\; .\; \backslash quad\; (4.25)$$

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

$$S=-\frac{\mathfrak{K}}{2}\int d^{d+1}x\sqrt{g}[g^{AB}{\mathrm{\partial }}_A\varphi {\mathrm{\partial }}_B\varphi +m^2{\varphi }^2+b{\varphi }^3+\dots ]\mathrm{.}(4.26)S\; =\; -\backslash frac\{\backslash mathfrak\{K\}\}\{2\}\; \backslash int\; d^\{d+1\}x\; \backslash sqrt\{g\}\; [g^\{AB\}\; \backslash partial\_A\; \backslash phi\; \backslash partial\_B\; \backslash phi\; +\; m^2\; \backslash phi^2\; +\; b\backslash phi^3\; +\; \backslash dots]\; .\; \backslash quad\; (4.26)$$

Here $\mathfrak{K}$ is just a normalization constant; we are assuming that the theory of $\phi$ is weakly coupled and one may think of $\mathfrak{K}$ as proportional to $N^2$ . For this metric $\sqrt{g} = \sqrt{|\det g|} = (\frac{L}{z})^{d+1}$ . Our immediate goal is to compute a two-point function of the operator $\mathcal{O}$ to which $\phi$ couples, so we will ignore the interaction terms in (4.26) for a while. Since $\phi$ is a scalar field we can rewrite the kinetic term as

$$g^{AB}{\mathrm{\partial }}_A\varphi {\mathrm{\partial }}_B\varphi =(\mathrm{\partial }\varphi {)}^2=g^{AB}{D}_A\varphi D_B\varphi (4.27)g^\{AB\}\; \backslash partial\_A\; \backslash phi\; \backslash partial\_B\; \backslash phi\; =\; (\backslash partial\; \backslash phi)^2\; =\; g^\{AB\}\; D\_A\; \backslash phi\; D\_B\; \backslash phi\; \backslash quad\; (4.27)$$

where $D_A$ is the covariant derivative, which has the nice property that $D_A(g_{BC}) = 0$ , so we can move the $D$ s around the $g$ s with impunity. By integrating by parts we can rewrite the action in a useful way:

$$S=-\frac{\mathfrak{K}}{2}\int d^{d+1}x[{\mathrm{\partial }}_A(\sqrt{g}{g}^{AB}\varphi {\mathrm{\partial }}_B\varphi )-\varphi {\mathrm{\partial }}_A(\sqrt{g}{g}^{AB}{\mathrm{\partial }}_B\varphi )+\sqrt{g}(m^2{\varphi }^2+\dots )](4.28)S\; =\; -\backslash frac\{\backslash mathfrak\{K\}\}\{2\}\; \backslash int\; d^\{d+1\}x\; [\backslash partial\_A\; (\backslash sqrt\{g\}\; g^\{AB\}\; \backslash phi\; \backslash partial\_B\; \backslash phi)\; -\; \backslash phi\; \backslash partial\_A\; (\backslash sqrt\{g\}\; g^\{AB\}\; \backslash partial\_B\; \backslash phi)\; +\; \backslash sqrt\{g\}\; (m^2\; \backslash phi^2\; +\; \backslash dots)]\; \backslash quad\; (4.28)$$

and finally by using Stokes' theorem we can rewrite the action as

$$S=-\frac{\mathfrak{K}}{2}{\int }_{\mathrm{\partial }AdS}{d}^{d}x\sqrt{g}{g}^{zB}\varphi {\mathrm{\partial }}_B\varphi -\frac{\mathfrak{K}}{2}\int \sqrt{g}\varphi (-\mathrm{\square }+m^2)\varphi +O({\varphi }^3)(4.29)S\; =\; -\backslash frac\{\backslash mathfrak\{K\}\}\{2\}\; \backslash int\_\{\backslash partial\; AdS\}\; d^d\; x\; \backslash sqrt\{g\}\; g^\{zB\}\; \backslash phi\; \backslash partial\_B\; \backslash phi\; -\; \backslash frac\{\backslash mathfrak\{K\}\}\{2\}\; \backslash int\; \backslash sqrt\{g\}\; \backslash phi\; (-\backslash square\; +\; m^2)\; \backslash phi\; +\; O(\backslash phi^3)\; \backslash quad\; (4.29)$$

where we define the scalar Laplacian $\square \phi = \frac{1}{\sqrt{g}} \partial_A (\sqrt{g} g^{AB} \partial_B \phi) = D^A D_A \phi$ . Note that we wrote all these covariant expressions without ever introducing Christoffel symbols.

We can rewrite the boundary term more covariantly as

$${\int }_{\mathcal{M}}\sqrt{g}{D}_A{J}^A={\int }_{\mathrm{\partial }\mathcal{M}}\sqrt{\gamma }{n}_A{J}^A\mathrm{.}(4.30)\backslash int\_\{\backslash mathcal\{M\}\}\; \backslash sqrt\{g\}\; D\_A\; J^A\; =\; \backslash int\_\{\backslash partial\; \backslash mathcal\{M\}\}\; \backslash sqrt\{\backslash gamma\}\; n\_A\; J^A\; .\; \backslash quad\; (4.30)$$

The metric tensor $\gamma$ is defined as

$$d{s}^2{\mathrm{\mid }}_{z=ϵ}\equiv {\gamma }_{\mu \nu }d{x}^{\mu }d{x}^{\nu }=\frac{L^2}{{ϵ}^2}{\eta }_{\mu \nu }d{x}^{\mu }d{x}^{\nu }(4.31)ds^2|\_\{z=\backslash epsilon\}\; \backslash equiv\; \backslash gamma\_\{\backslash mu\backslash nu\}\; dx^\backslash mu\; dx^\backslash nu\; =\; \backslash frac\{L^2\}\{\backslash epsilon^2\}\; \backslash eta\_\{\backslash mu\backslash nu\}\; dx^\backslash mu\; dx^\backslash nu\; \backslash quad\; (4.31)$$i.e. it is the induced metric on the boundary surface $z = \epsilon$ . The vector $n_A$ is a unit vector normal to boundary ( $z = \epsilon$ ). We can find an explicit expression for it

$$n_A\propto \frac{\mathrm{\partial }}{\mathrm{\partial }z}{g}_{AB}{n}^A{n}^B{\mathrm{\mid }}_{z=ϵ}=1\Rightarrow n=\frac{1}{\sqrt{g_{zz}}}\frac{\mathrm{\partial }}{\mathrm{\partial }z}=\frac{z}{L}\frac{\mathrm{\partial }}{\mathrm{\partial }z}\mathrm{.}(4.32)n\_A\; \backslash propto\; \backslash frac\{\backslash partial\}\{\backslash partial\; z\}\; \backslash quad\; g\_\{AB\}\; n^A\; n^B|\_\{z=\backslash epsilon\}\; =\; 1\; \backslash quad\; \backslash Rightarrow\; \backslash quad\; n\; =\; \backslash frac\{1\}\{\backslash sqrt\{g\_\{zz\}\}\}\; \backslash frac\{\backslash partial\}\{\backslash partial\; z\}\; =\; \backslash frac\{z\}\{L\}\; \backslash frac\{\backslash partial\}\{\backslash partial\; z\}\; .\; \backslash quad\; (4.32)$$

From this discussion we have learned the following:

• • the equation of motion for small fluctuations of $\phi$ is $(-\square + m^2)\underline{\phi} = 0$ ¹⁸.
• • If $\underline{\phi}$ 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

$$\mathrm{\Delta }(\mathrm{\Delta }-d)=m^2{L}^2\backslash Delta(\backslash Delta\; -\; d)\; =\; m^2\; L^2$$

by studying the AdS wave equation near the boundary.

Let's take advantage of translational invariance in $d$ dimensions, $x^\mu \rightarrow x^\mu + a^\mu$ , to Fourier decompose the scalar field:

$$\varphi (z,x^{\mu })=e^{i{k}_{\mu }{x}^{\mu }}{f}_k(z),k_{\mu }{x}^{\mu }\equiv -\omega t+\vec{k}\cdot \vec{x}\mathrm{.}(4.33)\backslash phi(z,\; x^\backslash mu)\; =\; e^\{ik\_\backslash mu\; x^\backslash mu\}\; f\_k(z),\; \backslash quad\; k\_\backslash mu\; x^\backslash mu\; \backslash equiv\; -\backslash omega\; t\; +\; \backslash vec\{k\}\; \backslash cdot\; \backslash vec\{x\}\; .\; \backslash quad\; (4.33)$$

In the Fourier basis, substituting (4.33) into the wave equation $(-\square + m^2)\phi = 0$ and using the fact that the metric only depends on $z$ , the wave equation is:

$$0=(g^{\mu \nu }{k}_{\mu }{k}_{\nu }-\frac{1}{\sqrt{g}}{\mathrm{\partial }}_z(\sqrt{g}{g}^{zz}{\mathrm{\partial }}_z)+m^2)f_k(z)(4.34)0\; =\; (g^\{\backslash mu\backslash nu\}\; k\_\backslash mu\; k\_\backslash nu\; -\; \backslash frac\{1\}\{\backslash sqrt\{g\}\}\; \backslash partial\_z\; (\backslash sqrt\{g\}\; g^\{zz\}\; \backslash partial\_z)\; +\; m^2)\; f\_k(z)\; \backslash quad\; (4.34)$$

$$=\frac{1}{L^2}[z^2{k}^2-z^{d+1}{\mathrm{\partial }}_z(z^{-d+1}{\mathrm{\partial }}_z)+m^2{L}^2]f_k(z),(4.35)=\; \backslash frac\{1\}\{L^2\}\; [z^2\; k^2\; -\; z^\{d+1\}\; \backslash partial\_z\; (z^\{-d+1\}\; \backslash partial\_z)\; +\; m^2\; L^2]\; f\_k(z),\; \backslash quad\; (4.35)$$

where we have used $g^{\mu\nu} = (z/L)^2 \delta^{\mu\nu}$ . The solutions of (4.35) are Bessel functions; we can learn a lot without using that information. For example, look at the solutions near the boundary (i.e. $z \rightarrow 0$ ). In this limit we have power law solutions, which are spoiled by the $z^2 k^2$ term. To see this, try using $f_k = z^\Delta$ in (4.35):

$$0=k^2{z}^{2+\mathrm{\Delta }}-z^{d+1}{\mathrm{\partial }}_z(\mathrm{\Delta }{z}^{-d+\mathrm{\Delta }})+m^2{L}^2{z}^{\mathrm{\Delta }}(4.36)0\; =\; k^2\; z^\{2+\backslash Delta\}\; -\; z^\{d+1\}\; \backslash partial\_z\; (\backslash Delta\; z^\{-d+\backslash Delta\})\; +\; m^2\; L^2\; z^\backslash Delta\; \backslash quad\; (4.36)$$

$$=(k^2{z}^2-\mathrm{\Delta }(\mathrm{\Delta }-d)+m^2{L}^2)z^{\mathrm{\Delta }},(4.37)=\; (k^2\; z^2\; -\; \backslash Delta(\backslash Delta\; -\; d)\; +\; m^2\; L^2)\; z^\backslash Delta,\; \backslash quad\; (4.37)$$

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¹⁸We will use an underline to denote fields which solve the equations of motion.and for $z \rightarrow 0$ we get:

$$\mathrm{\Delta }(\mathrm{\Delta }-d)=m^2{L}^2\mathrm{.}(4.38)\backslash Delta(\backslash Delta\; -\; d)\; =\; m^2\; L^2\; .\; \backslash quad\; (4.38)$$

The two roots of (4.38) are

$${\mathrm{\Delta }}_{\pm }=\frac{d}{2}\pm \sqrt{{\left(\frac{d}{2}\right)}^2+m^2{L}^2}\mathrm{.}(4.39)\backslash Delta\_\{\backslash pm\}\; =\; \backslash frac\{d\}\{2\}\; \backslash pm\; \backslash sqrt\{\backslash left(\backslash frac\{d\}\{2\}\backslash right)^2\; +\; m^2\; L^2\}.\; \backslash quad\; (4.39)$$

Comments

• • The solution proportional to $z^{\Delta_-}$ is bigger near $z \rightarrow 0$ .
• • $\Delta_+ > 0 \ \forall \ m$ , therefore $z^{\Delta_+}$ decays near the boundary for any value of the mass.
• • $\Delta_+ + \Delta_- = d$ .

We want to impose boundary conditions that allow solutions. Since the leading behavior near the boundary of a generic solution is $\phi \sim z^{\Delta_-}$ , we impose

$$\varphi (x,z){\mathrm{\mid }}_{z=ϵ}={\varphi }_0(x,ϵ)={ϵ}^{{\mathrm{\Delta }}_{-}}{\varphi }_0^{Ren}(x),(4.40)\backslash phi(x,\; z)|\_\{z=\backslash epsilon\}\; =\; \backslash phi\_0(x,\; \backslash epsilon)\; =\; \backslash epsilon^\{\backslash Delta\_-\}\; \backslash phi\_0^\{Ren\}(x),\; \backslash quad\; (4.40)$$

where $\phi_0^{Ren}$ is a renormalized source field. With this boundary condition $\phi_0^{Ren}$ is a finite quantity in the limit $\epsilon \rightarrow 0$ .

Wavefunction renormalization of $\mathcal{O}$ (Heuristic but useful)

Suppose:

$$S_{bdy}\ni {\int }_{z=ϵ}{d}^{d}x\sqrt{{\gamma }_{ϵ}}{\varphi }_0(x,ϵ)\mathcal{O}(x,ϵ)(4.41)S\_\{bdy\}\; \backslash ni\; \backslash int\_\{z=\backslash epsilon\}\; d^d\; x\; \backslash sqrt\{\backslash gamma\_\backslash epsilon\}\; \backslash phi\_0(x,\; \backslash epsilon)\; \backslash mathcal\{O\}(x,\; \backslash epsilon)\; \backslash quad\; (4.41)$$

$$=\int d^{d}x{\left(\frac{L}{ϵ}\right)}^d({ϵ}^{{\mathrm{\Delta }}_{-}}{\varphi }_0^{Ren}(x))\mathcal{O}(x,ϵ),(4.42)=\; \backslash int\; d^d\; x\; \backslash left(\backslash frac\{L\}\{\backslash epsilon\}\backslash right)^d\; (\backslash epsilon^\{\backslash Delta\_-\}\; \backslash phi\_0^\{Ren\}(x))\; \backslash mathcal\{O\}(x,\; \backslash epsilon),\; \backslash quad\; (4.42)$$

where we have used $\sqrt{\gamma} = (L/\epsilon)^d$ . Demanding this to be finite as $\epsilon \rightarrow 0$ we get:

$$\mathcal{O}(x,ϵ)\sim {ϵ}^{d-{\mathrm{\Delta }}_{-}}{\mathcal{O}}^{Ren}(x)(4.43)\backslash mathcal\{O\}(x,\; \backslash epsilon)\; \backslash sim\; \backslash epsilon^\{d-\backslash Delta\_-\}\; \backslash mathcal\{O\}^\{Ren\}(x)\; \backslash quad\; (4.43)$$

$$={ϵ}^{{\mathrm{\Delta }}_{+}}{\mathcal{O}}^{Ren}(x),(4.44)=\; \backslash epsilon^\{\backslash Delta\_+\}\; \backslash mathcal\{O\}^\{Ren\}(x),\; \backslash quad\; (4.44)$$

where in the last line we have used $\Delta_+ + \Delta_- = d$ . Therefore, the scaling dimension of $\mathcal{O}^{Ren}$ is $\Delta_+ \equiv \Delta$ . We will soon see that $\langle \mathcal{O}(x) \mathcal{O}(0) \rangle \sim \frac{1}{|x|^{2\Delta}}$ , confirming that $\Delta$ is indeed the scaling dimension.

We are solving a second order ODE, therefore we need two conditions in order to determine a solution (for each $k$ ). So far we have imposed one condition at the boundary of AdS:- • For $z \rightarrow \epsilon$ , $\phi \sim z^{\Delta} \phi_0 + (\text{terms subleading in } z)$ .

In the Euclidean case (we discuss real time in the next subsection), we will also impose

• • $\phi$ regular in the interior of AdS (i.e. at $z \rightarrow \infty$ ).

Comments on $\Delta$

1. 1. The $\epsilon^{\Delta}$ - factor is independent of $k$ and $x$ , which is a consequence of a local QFT (this fails in exotic examples).
2. 2. Relevantness: If $m^2 > 0$ : This implies $\Delta \equiv \Delta_+ > d$ , so $\mathcal{O}{\Delta}$ is an irrelevant operator. This means that if you perturb the CFT by adding $\mathcal{O}{\Delta}$ to the Lagrangian, then its coefficient is some mass scale to a negative power:

$$\mathrm{\Delta }S=\int d^{d}x(\text{mass}{)}^{d-\mathrm{\Delta }}{\mathcal{O}}_{\mathrm{\Delta }},(4.45)\backslash Delta\; S\; =\; \backslash int\; d^d\; x\; (\backslash text\{mass\})^\{d-\backslash Delta\}\; \backslash mathcal\{O\}\_\{\backslash Delta\},\; \backslash quad\; (4.45)$$

where the exponent is negative, so the effects of such an operator go away in the IR, at energies $E < \text{mass}$ . $\phi \sim z^{\Delta} \phi_0$ is this coupling. It grows in the UV (small $z$ ). If $\phi_0$ 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.

$m^2 = 0 \leftrightarrow \Delta = d$ means that $\mathcal{O}$ is marginal.

If $m^2 < 0$ , then $\Delta < d$ , so $\mathcal{O}$ is a relevant operator. Note that in $AdS$ , $m^2 < 0$ is ok if $m^2$ is not too negative. Such fields with $m^2 > -|m_{BF}|^2 \equiv -(d/2L)^2$ are called “Breitenlohner-Freedman (BF)-allowed tachyons”. The reason you might think that $m^2 < 0$ is bad is that usually it means an instability of the vacuum at $\phi = 0$ . An instability occurs when a normalizable mode grows with time without a source. But for $m^2 < 0$ , $\phi \sim z^{\Delta}$ decays near the boundary (i.e. in the UV). This requires a gradient energy of order $\sim \frac{1}{L}$ , which can stop the field from condensing.

To see what’s too negative, consider the formula for the dimension, $\Delta_{\pm} = \frac{d}{2} \pm \sqrt{\left(\frac{d}{2}\right)^2 + m^2 L^2}$ .

For $m^2 < m_{BF}^2$ , the dimension becomes complex.

1. 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 in $AdS$ with action

$$S=-{\int }_{AdS}(\frac{1}{4}{F}_{\mu \nu }{F}^{\mu \nu }-\frac{1}{2}{m}^2{A}_{\mu }{A}^{\mu }),S\; =\; -\; \backslash int\_\{AdS\}\; \backslash left(\; \backslash frac\{1\}\{4\}\; F\_\{\backslash mu\backslash nu\}\; F^\{\backslash mu\backslash nu\}\; -\; \backslash frac\{1\}\{2\}\; m^2\; A\_\{\backslash mu\}\; A^\{\backslash mu\}\; \backslash right),$$