Advances in High Energy Physics

Volume 2010, Article ID 723105, 54 pages

http://dx.doi.org/10.1155/2010/723105

## Holographic Duality with a View Toward Many-Body Physics

^{1}Center for Theoretical Physics, MIT, Cambridge 02139, USA^{2}Kavli 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 [1–7], has been liberally borrowed to construct these notes. In addition, some of the text source and most of the figures were pillaged from lecture notes from my class at MIT during Fall 2008 [8], some of which were created by students in the class.

#### 2. Motivating the Correspondence

To understand what one might mean by a more precise version of 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.