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Advances in High Energy Physics
Volume 2010, Article ID 723105, 54 pages
http://dx.doi.org/10.1155/2010/723105
Review Article

Holographic Duality with a View Toward Many-Body Physics

1Center for Theoretical Physics, MIT, Cambridge 02139, USA
2Kavli Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106-4030, USA

Received 15 March 2010; Accepted 7 May 2010

Academic Editor: Wolfgang Mück

Copyright © 2010 John McGreevy. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

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 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 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 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 [17], 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 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.

() 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 tensor forbids massless particles of spin which carry momentum (i.e., with ).

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: . (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.

() 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 (bounded by an area ) a configuration with entropy (where is the entropy of the biggest black hole fittable in ), 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 (maximum entropy ). 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 (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 induces a WZW model (a 2d CFT) on the boundary of . 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.

() 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 are local in scale: The beta function is determined by the coupling constant evaluated at the energy scale , 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: ()The Weinberg-Witten theorem suggests that the graviton lives on a different space than the QFT in question. ()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. ()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.2

Next we will make a simplifying assumption in an effort to find concrete examples. The simplest case of an RG flow is when and the system is self-similar. In a Lorentz invariant theory (which we also assume for simplicity), this means that the following scale transformation () is a symmetry. If the extra dimension coordinate is to be thought of as an energy scale, then dimensional analysis says that will scale under the scale transformation as . The most general ()-dimensional metric (one extra dimension) with this symmetry and Poincaré invariance is of the following form: We can bring it into a more familiar form by a change of coordinates: This is .3 It is a family of copies of Minkowski space, parametrized by , whose size varies with (see Figure 1). The parameter is called the “ radius” and it has dimensions of length. Although this is a dimensionful parameter, a scale transformation can be absorbed by rescaling the radial coordinate (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 , in which the metric takes the form These coordinates are better because fewer symbols are required to write the metric. 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 parameter . 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 a -dimensional conformal field theory (CFT) should be related to a theory of gravity on . This metric (2.5) solves the equations of motion of the following action (and many others)4: Here, makes the integral coordinate-invariant, and is the Ricci scalar curvature. The cosmological constant is related by the equations of motion to the value of the AdS radius: . 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: has dimensions of length, and so here is a length, the Planck length: (in units where ). The gravity theory is classical if . 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, ) is small (compared to the string tension, ; 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 metric above has a boundary at . This is a bit subtle. Keeping fixed and moving in the direction from a finite value of to is actually infinite distance. However, massless particles in (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 . These boundary conditions will play a crucial role in the discussion below.

So we should amend our statement to say that a -dimensional conformal field theory is related to a theory of gravity on spaces which are asymptotically . 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 in . The example which is understood best is the case of the super Yang-Mills theory (SYM) in four dimensions. This is dual to maximal supergravity in (which arises by dimensional reduction of ten-dimensional IIB supergravity on ). 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 exponent [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 might be dual to a QFT in 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: 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 (e.g., the lattice spacing) and we put it in a cubical box of linear size . The total number of degrees of freedom is the number of cells , times the number of degrees of freedom per lattice site, which we will call “”. 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 vacua of string theory [21]) that other behaviors are possible, and that's why I made it a funny color and put it in quotes. So .

The picture we have of AdS is a collection of copies of -dimensional Minkowski space of varying size; the boundary is the locus where they get really big. The area of the boundary is As in the field theory counting, this is infinite for two reasons: from the integral over and from the fact that is going to zero. To regulate this integral, we integrate not to but rather cut it off at . We will see below a great deal more evidence for this idea that the boundary of is associated with the UV behavior of the field theory, and that cutting off the geometry at is a UV cutoff (not identical to the lattice cutoff, but close enough for our present purposes). Given this, The holographic principle then says that the maximum entropy in the bulk is

We see that the scaling with the system size agrees—both hand side go like . So AdS/CFT is indeed an implementation of the holographic principle. We can learn more from this calcluation: In order for the prefactors of to agree, we need to relate the radius in Planck units to the number of degrees of freedom per site of the field theory: up to numerical prefactors.

2.2. Preview of the AdS/CFT Correspondence

Here is the ideology: 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 , 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 order ) 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: We can write down a generating functional for these correlators by perturbing the action of the QFT: where are arbitrary functions (sources) and is some basis of local operators. The -point function is then given by

Since is a perturbation (because it is a perturbation of the bare Lagrangian by local operators), in AdS it corresponds to a perturbation near the boundary, . (Recall from the counting of degrees of freedom in Section 2.1 QFT with UV cutoff AdS cutoff .) The perturbation 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 is then schematically 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- 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: 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 space with radius , the theory is classical when This quantity, the radius in Planck units , 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 -component vector, with -invariant interactions. This is probably not what we need to have a simple gravity dual, for the reasons described next.

3.1. Large Vector Models

A simple paradigmatic example of this vector-like large- limit (I use a different to distinguish it from the matrix case to be discussed next) is a QFT of scalar fields with the following action: The fields transform in the fundamental representation of the symmetry group. Some foresight has been used to determine that the quartic coupling is to be held fixed in the large- limit. An effective description (i.e., a well-defined saddle-point) can be found in terms of by standard path-integral tricks, and the effective action for is The important thing is the giant factor of in front of the action which makes the theory of classical. Alternatively, the only interactions in this 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 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 [24] have suggested what the (not simple) gravity dual might be.

3.2. 't Hooft Counting

“You can hide a lot in a large- matrix."

Steve Shenker

Given some system with a few degrees of freedom, there exist many interesting large- 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- 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- 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 . Evidence from the space of string vacua suggests that there are many generalizations of this where the number of dofs per point goes like for [21]. However, a generalization of the 't Hooft limit is not yet well understood for other cases.10

Consider a (any) quantum field theory with matrix fields, . By matrix fields, we mean that their products appear in the Lagrangian only in the form of matrix multiplication, for example, , which is a big restriction on the interactions. It means the interactions must be invariant under ; for concreteness we will take the matrix group to be .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 symmetry) changes the scaling of the coupling in the large limit.

In particular, consider the 't Hooft limit in which and with 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 goes to zero slower than the coupling in the vector model .

We will be agnostic here about whether the 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 .

The fields can have all kinds of spin labels and global symmetry labels, but we will just call them . 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: I suppose that we want 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 -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

and the vertices by

Created by Brian Swingle.

To see the consequences of this more concretely, let us consider some vacuum-to-vacuum diagrams (see Figures 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: For example, the diagram in Figure 2 has three-point vertices, propagators, and index loops, giving the final result . 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 . However, there also exist nonplanar graphs, such as the one in Figure 4, whose contributions are down by (an even number of) powers of . 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.
Figure 3: Planar graphs that contribute to the vacuum vacuum amplitude.
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 surface . There are two ways to construct the explicit mapping.

Method 1 (direct surface). Fill in index loops with little plaquettes.

Method 2 (dual surface). () draw a vertex13 in every index loop and () draw an edge across every propagator.

These constructions are illustrated in Figures 5 and 6.

Figure 5: Direct surfaces constructed from the vacuum diagram in (a) Figure 3(a) and (b) Figure 4.
Figure 6: Dual surface constructed from the vacuum diagram in Figure 3(c). Note that points at infinity are identified.

If number of propagators, number of vertices, and number of index loops, then the diagram gives a contribution . The letters refer to the “direct” triangulation of the surface in which interaction vertices are triangulation vertices. Then we interpret as the number of edges, as the number of faces, and as the number of vertices in the triangulation. In the dual triangulation there are dual faces , dual edges , and dual vertices . The relationship between the original and dual variables is , , and . The exponent is the Euler character and it is a topological invariant of two-dimensional surfaces. In general it is given by where is the number of handles (the genus) and is the number of boundaries. Note that the exponent of , or is not a topological invariant and depends on the triangulation (Feynman diagram).

Because the -counting is topological (depending only on ), we can sensibly organize the perturbation series for the effective action 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 and are classified by their number of handles ( is the two-dimensional sphere, is the torus, and so on). We may write the effective action (the sum over connected vacuum-to-vacuum diagrams) as where the sum over topologies is explicit.

Now we can see some similarities between this expansion and perturbative string expansions.14 plays the role of the string coupling , the amplitude joining and splitting of the closed strings. In the large limit, this process is suppressed and the theory is classical. Closed string theory generically predicts gravity, with Newton's constant ; so this reproduces our result 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 is small, and it can be studied in perturbation theory in powers of , where is the string tension. We can think of as a sort of chemical potential for edges in our triangulation. Looking back at our diagram counting we can see that if becomes large then diagrams with lots of edges are important. Thus large 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 which encodes our intuition about large 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 field . For example, the position labeling where the field 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- 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 and . In dimensions, the behavior through one loop is ( is a coefficient which depends on the matter content and vanishes for SYM.) So we find that . Thus can still run in the large limit and the theory is nontrivial.

3.3. -Counting of Correlation Functions

Let us now consider the -counting for correlation functions of local gauge-invariant operators. Motivated by gauge invariance and simplicity, we will consider “single trace" operators, operators that look like and which we will abbreviate as . We will keep finite as .15 There are two little complications here. We must be careful about how we normalize the fields and we must be careful about how we normalize the operator . The normalization of the fields will continue to be such that the Lagrangian takes the form with containing no explicit factors of . To fix the normalization of (to determine the constant ) we will demand that when acting on the vacuum, the operator creates states of finite norm in the large- limit, that is, where the subscript stands for connected.

To determine 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 legs where 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 . For the moment we do not associate any explicit factors of with the new vertex. Let us consider the example . 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 . 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 . (A way to draw the connected diagram in Figure 9 which makes the -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 of with .
Figure 8: Disconnected diagram contributing to the correlation function . Created by Brian Swingle.
Figure 9: Connected diagram contributing to the correlation function .
Figure 10: A redrawing of the connected diagram shown in Figure 9.

The fact that disconnected diagrams win in the large limit is general and goes by the name “large- factorization". It says that single trace operators are basically classical objects in the large- limit .

The leading connected contribution to the correlation function is independent of and so . Requiring that means that we can just set . Having fixed the normalization of we can now determine the -dependence of higher-order correlation functions. For example, the leading connected diagram for where is just a triangle and contributes a factor . In fact, quite generally we have for the leading contribution.

So the operators (called glueballs in the context of QCD) create excitations of the theory that are free at large —they interact with coupling . In QCD with , quarks and gluons interact strongly, and so do their hadron composites. The role of large- 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 .

Finally, we should make a comment about the -scaling of the generating functional . We have normalized the sources so that each is an 't Hooft-like coupling, in that it is finite as . The effective action is the sum of connected vacuum diagrams, which at large- is dominated by the planar diagrams. As we have shown, their contributions go like . This agrees with our normalization of the gravity action,

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 . We can add fermions or bosons . Because quarks are in the fundamental of , 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 or 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.

Figure 11: A quark vacuum bubble and a quark vacuum bubble with “gluon” exchange. Created by Brian Swingle.

Another direction for generalization is to consider different matrix groups such as or . The adjoint of is just the fundamental times the antifundamental. However, the adjoint representations of and are more complicated. For the adjoint is given by the antisymmetric product of two fundamentals (vectors), and for 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 .

Figure 12: Propagator for SO() () or Sp() () matrix models.
Figure 13: Feynman graphs in AdS. We did the one with two external legs in Section 4.3. Created by Francesco D’Eramo.

4. Vacuum CFT Correlators from Fields in AdS

Our next goal is to evaluate , where 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] This thing on the RHS is not yet a computationally effective object; the currently-practical version of the GKPW formula is the classical limit:

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 large and large . In some examples there is only one parameter which controls the validity of the gravity description. In (4.2) we have made the -dependence explicit: in units of the radius, the Newton constant is . is some dimensionless action.(ii)We said that we are going to think of as a small perturbation. Let us then make a perturbative expansion in powers of : where Now if there is no instability, then is small implies is small. For one thing, this means that we can ignore interactions of the bulk fields in computing two-point functions. For -point functions, we will need to know terms in the bulk action of degree up to in the fields.(i)Anticipating divergences at , we have introduced a cutoff in (4.2) (which will be a UV cutoff in the CFT) and set boundary conditions at . 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 for every operator in the dual field theory. For such a pair, we will say “ couples to ” 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. ()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.()Only “single-trace” operators (like the s of the previous section) correspond to “elementary fields” in the bulk. The excitations created by multitrace operators (like ) correspond to multiparticle states of (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-—if is not large, it is not true that the overlap between and is small.16()There are some simple examples of the correspondence between bulk fields and boundary operators that are determined by symmetry. The stress-energy tensor is the response of a local QFT to local change in the metric, .

Here we are writing for the metric on the boundary. In this case

Gauge fields in the bulk correspond to currents in the boundary theory: that is, . 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 the Lagrangian of the CFT (by changing ) 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 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, , for the plasma made from any CFT that has an Einstein gravity dual; the answer is always . 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- slices are just copies of Minkowski space) is 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 Here is just a normalization constant; we are assuming that the theory of is weakly coupled and one may think of as proportional to . For this metric . Our immediate goal is to compute a two-point function of the operator to which couples, so we will ignore the interaction terms in (4.8) for a while. Since is a scalar field we can rewrite the kinetic term as where is the covariant derivative, which has the nice property that , so we can move the s around the s with impunity. By integrating by parts we can rewrite the action in a useful way: and finally by using Stokes' theorem we can rewrite the action as where we define the scalar Laplacian . Note that we wrote all these covariant expressions without ever introducing Christoffel symbols.

We can rewrite the boundary term more covariantly as The metric tensor is defined as that is, it is the induced metric on the boundary surface . The vector is a unit vector normal to boundary (). We can find an explicit expression for it:

From this discussion we have learned the following. (i)The equation of motion for small fluctuations of is .18(ii)If 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 by studying the AdS wave equation near the boundary.

Let us take advantage of translational invariance in dimensions, , to Fourier decompose the scalar field: In the Fourier basis, substituting (4.16) into the wave equation and using the fact that the metric only depends on , the wave equation is where we have used . 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., ). In this limit we have power law solutions, which are spoiled by the term. To see this, try using in (4.18): and for we get The two roots of (4.20) are

Comments
(i)The solution proportional to is bigger near . (ii), therefore decays near the boundary for any value of the mass. (iii).

We want to impose boundary conditions that allow solutions. Since the leading behavior near the boundary of a generic solution is , we impose where is a renormalized source field. With this boundary condition is a finite quantity in the limit .

Wavefunction Renormalization of (Heuristic but useful)
Suppose that where we have used . Demanding this to be finite as we get where in the last line we have used . Therefore, the scaling dimension of is . We will soon see that , confirming that 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 ). So far we have imposed one condition at the boundary of AdS.(i)For (terms subleading in ).

In the Euclidean case (we discuss real time in the next subsection), we will also impose(ii) regular in the interior of AdS (i.e., at ).

Comments on
()The factor is independent of and , which is a consequence of a local QFT (this fails in exotic examples). ()Relevantness: If : This implies , so is an irrelevant operator. This means that if you perturb the CFT by adding to the Lagrangian, then its coefficient is some mass scale to a negative power: where the exponent is negative; so the effects of such an operator go away in the IR, at energies . is this coupling. It grows in the UV (small ). If 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. means that is marginal.If , then ; so is a relevant operator. Note that in , is ok if is not too negative. Such fields with are called “Breitenlohner-Freedman- (BF-) allowed tachyons”. The reason you might think that is bad is that usually it means an instability of the vacuum at . An instability occurs when a normalizable mode grows with time without a source. But for , decays near the boundary (i.e., in the UV). This requires a gradient energy of order , which can stop the field from condensing.
To see what is too negative, consider the formula for the dimension, . For , the dimension becomes complex.()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 with action the boundary behavior of the wave equation implies that with For the particular case of massless this can be seen to lead to which is the dimension of a conserved current in a CFT. Also, the fact that is massless implies 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 of [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: We will now stop being agnostic about the signature and confront some issues that arise for real time correlators. If , that is, is spacelike (or Euclidean), the solution is where . In the interior of (), the Bessel functions behave as So we see that the regularity in the interior uniquely fixes 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 by adding the point at infinity in ) modulo conformal rescaling, then the metric on the space inside of the is uniquely determined. A similar result holds for gauge fields.

In contrast to this, in Lorentzian signature with timelike , that is, for on-shell states with , there exist two linearly independent solutions with the same leading behavior at the UV boundary. In terms of , the solutions are 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 describing stuff falling into the horizon [30], that is, moving towards larger 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 that happen, rather than unhappen.(ii)You can check that this prescription gives the correct analytic structure of ([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 [3133].

The fact that stuff goes past the horizon and does not come out is what breaks time-reversal invariance in the holographic computation of . Here, the ingoing choice is , since as grows, the wavefront moves to larger . This specifies the solution which computes the causal response to be .

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 for with a given scaling behavior. In Euclidean space, is normalizable when . This is because when we are thinking about the partition function , modes with boundary conditions which force would not contribute. In real-time, we say that is normalizable if where where is a given spatial slice, is the induced metric on that slice, is a normal unit vector to , and is a timelike killing vector. is defined as

4.3. Bulk-to-Boundary Propagator in Momentum Space

We return to considering spacelike in this section. Let us normalize our solution of the wave equation by the condition . This means that its Fourier transform is a -function in position space when evaluated at , not at the actual boundary . The solution, which we can call the “regulated bulk-to-boundary propagator”, is then The general position space solution can be obtained by Fourier decomposition: The “on-shell action” (i.e., the action evaluated on the saddle-point solution) is (using (4.11)) and therefore21 Here (sometimes called the “flux factor”) is The small- (near boundary) behavior of is where the coefficients of the series depend on . For noninteger , there would be no 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 s and s 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 give a function which has the correct behavior at the other end, that is, at . Plugging the asymptotic expansion of the Bessel function into (4.39), where (I) and (II) denote the first and second groups of terms of the previous line.

(I) is a Laurent series in with coefficients which are positive powers of (i.e., analytic in at ). 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 for . The factor reinforces the notion that , which is an IR cutoff in AdS, is a UV cutoff for the QFT.

The interesting bit of , which gives the behavior of the correlator (4.38), is nonanalytic in : To get the factor of , 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 Note that the radius appears only through the overall normalization of the correlator (4.38), in the combination .

Now let us deal with the pesky cutoff dependence. Since if we let as before, the operation removes the potentially divergent factor of . We also see that for , the 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 the local, intrinsic boundary term: and redo the preceding calculation. Note that this does not affect the equations of motion, nor does it affect .

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 [3640] 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 this formula defines the coefficient 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 . The variation of the action with respect to the initial value of the coordinate is the momentum: Thinking of the radial direction of as time, the following is a mild generalization of (4.48): where is the bulk field-momentum, with thought of as time. There are two minor subtleties.

()The factor of arises because differs from the boundary value of by a factor: , so .() itself in general (for ) has a term proportional to the source , 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 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.24

This formula applies in the real-time case [41]. For example, to describe linear response, , then (4.50) says that 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 of 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 (bulk-to-boundary propagators). These diagrams are sometimes called “Witten diagrams,” after [23].

4.6. -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: The discussion can easily be extended to -point functions with .

The equations of motion are and its permutations. We solve this perturbatively in the sources, : with similar expressions for . We need to define the quantities appearing in (4.54). is the bulk-to-boundary propagator for a bulk field dual to an operator of dimension . We determined this in the previous subsection: it is defined to be the solution to the homogeneous scalar wave equation which approaches a delta function at the boundary,25. So is given by (4.36) with . is the bulk-to-bulk propagator, which is the normalizable solution to the wave equation with a source (so that for a source ).

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 .

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 curve connecting the two-points in . The arrows on the curve indicate the orientation, created by Vijay Kumar.
Figure 18: The geometry of the near-extremal charged black hole for .
4.7. Which Scaling Dimensions Are Attainable?

Let us think about which are attainable by varying .26 is the smallest dimension we have obtained so far, but the bound from unitarity is lower: . There is a range of values for which we have a choice about which of is the source and which is the response. This comes about as follows.

is normalizable if . With the action we have been using with , our boundary conditions demand that with or , and hence, in the limit . Since for , , We emphasize that we have only specified the boundary behavior of , and it is not assumed that satisfies the equation of motion. We see that This does not saturate the lower bound from unitarity on the dimension of a scalar operator, which is ; 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]: For this action we see that is equivalent to which is exactly the unitary bound. We see that in this case both give normalizable modes for . Note that it is actually that gives the value which saturates the unitarity bound, that is, when The coefficient of 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 on , 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 operator is very large, the mass of the bulk field is large in units of the radius: 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 in the bulk. For convenience, consider the case of a complex operator , so that the worldlines of these particles are oriented. Then ( is the complex conjugate operator.) The middle expression is the Feynman path integral ; the action for a point particle of mass whose world-line is is given by . In the limit of large , we have where we have used the saddle point approximation; is the geodesic connecting the points on the boundary.

We now compute in the saddle point approximation. The metric restricted to is given by , where . This implies that the action is The geodesic can be computed by noting that the action does not depend on explicitly. This implies, we have a conserved quantity: The above is a first-order differential equation with solution , with . This is the equation of a semicircle. Substituting the solution back into the action gives You might think that since we are computing this in a conformal field theory, the only scale in the problem is and therefore the path integral should be independent of . This argument fails in the case at hand because there are two scales: and , the UV cutoff. The scale transformation is anomalous and this is manifested in the dependence of :