Quantum gravity in AdS/CFT is inevitably stringy

One of the favorite lies emitted by the coalition of the dishonest, deluded, and brainwashed anti-stringy demagogues and imbeciles is the claim that the AdS/CFT correspondence doesn't really "need" string theory. See this question to check that this myth is alive and kicking.



Leonard got it right – and he got the well-deserved reward, too.

The truth is that everything in the AdS/CFT has to be stringy for both sides of the duality to work and for their equivalence to hold. After all, one of the sides in AdS/CFT is a quantum theory of gravity in an Anti de Sitter space and string/M-theory is the only known consistent quantum theory of gravity – and most likely, the only mathematically possible theory of quantum gravity.




In this blog post, let me mention some of the aspects of the unavoidable "stringiness" of the theories participating in the AdS/CFT pairing. First, the AdS/CFT duality began with the 1997 paper by Maldacena that is approaching 10,000 citations now.




Enriched by some experience with black hole entropy calculations in string theory that had seemed to work "unexpectedly well", better than expected, Juan almost immediately declared that the gravitational theory is equivalent to a non-gravitational, lower-dimensional one. To prove that, he needed to take the branes in the full string/M-theory and to construct the decoupling limit.

The most famous example deals with a stack of \(N\) D3-branes in type IIB string theory. They have 3+1 dimensions in their world volume and 6 transverse spatial dimensions. Yes, the total number of dimensions in type IIB string theory is and has to be 10. The correct, stringy total spacetime dimension may be extracted from the CFTs in the most famous examples of the AdS/CFT pairs – one hint that the CFTs really know about the whole string/M-theory.



The key observation that allowed Maldacena to complete his first, somewhat heuristic proof of the equivalence is the fact that since the mid 1990s, string theory has offered you (at least) two ways to describe the dynamics (degrees of freedom and their time evolution) of this stack. One of them is the perturbative D-brane calculus. The objects embedded in the surrounding flat space are viewed as loci where fundamental strings of string theory are allowed to end. These open strings attached to these D-branes may be excited and their interactions may be computed via the usual rules of perturbative string theory (the world sheets are allowed to have boundaries because we deal with open strings and some of the boundary conditions are Dirichlet boundary conditions because there are D-branes).



The other description of the system boils down to the observation that a large enough stack of D-branes has a pretty large mass and behaves like a gravitational source – really a higher-dimensional counterpart of a black hole, a black \(p\)-brane – that is curving the space around it.

You could think that all the open strings as well as all the wiggles on the curved gravitational field have to be incorporated to describe what may happen to these objects, the stack of branes. However, Maldacena was sensible, original, and ingenious enough to realize that this would be a double counting. In fact, at weak coupling, the open strings are "the whole story". Similarly, at strong coupling, the objects really look like black holes without any visible strings attached, so one only needs the gravitational degrees of freedom.

Andy Strominger would be telling me about his lunch conversations with Joe Polchinski in Santa Barbara sometime in 1994 or so. Strominger would talk about black \(p\)-branes for half of the lunch; Polchinski would discuss something else, the emergent D-branes, for the other half of the lunch. Only when the year 1995 arrived, they have realized that they were talking about the same physical object! ;-)



Not everyone may be hired as a drunk homeless man in Prague! The same holds for cops in Prague. The main railway station became a concert hall, too. Those who can play Ježek's Bugatti Step impress me a lot.

A funny thing is that these two descriptions (open strings attached do D-branes; closed strings with a curved gravitational field) may be used for any value of the coupling, so they could be equivalent. Maldacena has presented a "decoupling" argument:

Consider all processes around this object (stack of branes) that occur very slowly in the Schwarzschild time (that's equivalent to low energy).

The open-string description of the D-branes at low energies reduces to the lowest-energy excitations of the open strings. In the case of D3-branes, the relevant description is Yang-Mills theory, the dynamics describing the interactions of the massless states of the open strings.

In the gravitational, closed-string description, low-energy or low-frequency processes are those that occur close to the event horizon of the black \(p\)-brane. You know that everything that happens to a guy right before he falls through the black hole event horizon looks increasingly slowed doowwwnnnn... to the observer at infinity.

The equivalence between the two, open-string-based and closed-string-based, pictures survives in the low-energy limit as well. Consequently, the \(SU(N)\) Yang-Mills theory in \(p+1\) dimensions is equivalent to the gravity in the near-horizon geometry of the black \(p\)-brane geometry – and this near-horizon geometry happens to be the \(AdS_{p+2}\) space in the case of extremal \(p\)-branes.

Great. It worked, tests could have been done for the supergravity part of it (correlators of the conformal field theory, or CFT for short, corresponded to scattering in the anti de Sitter i.e. AdS space: Gubser, Klebanov, Polyakov and Witten), and so on. But the derivation used all of string theory and people could continue with tests of other features of string theory beyond the supergravity.

All of the tests have worked so far; the seemingly "non-gravitational" CFT on the boundary was shown to contain all the stringy objects and phenomena that were predicted by string theory. It quickly became clear that by analyzing the emergent gravitational description of the non-gravitational CFTs, one could have discovered all of string theory "from a different angle".

Let me mention several characteristic features of string theory that were found in the seemingly non-gravitational CFTs.

First, Witten has found the wrapped branes of string theory in the CFT: they were given by nothing else than the baryonic operators. Recall that in a proton, the three quarks have different colors so their fields are really contracted against the antisymmetric \(\varepsilon_{abc}\) symbol with three color indices. Operators similar to this "creation operator for the proton" exist in \(SU(N)\) gauge theories (there are \(N\) indices in the antisymmetric tensor) and one may ask what they correspond to in the gravitational language. They are branes wrapped on various cycles of the compact space in the \(AdS_{p+2}\times{\mathcal M}\) manifold. The paper has nearly 500 citations so you may see lots of extra extensions of Witten's observations.

So the CFT contains wrapped branes. But does it also contain strings? You bet.

The clearest demonstration of the presence of excited, vibrating type IIB strings in the gauge theory was the 2002 paper by Berenstein-Maldacena-Nastase (BMN) who found these strings in a limit of the AdS space, the Penrose \(pp\)-wave limit of the geometry. The gauge theory has three complex scalars \(A,B,Z\) and one may write operators of the type\[

{\rm Tr}(ZZZZZZ {\bf A} ZZZZ {\bf B} ZZZZZ)

\] and so on. It is a trace of a long product so the operator looks like a closed string. The general "string bits" in this closed string are given by the complex scalar \(Z\) but there may be impurities such as \(A,B\) – I wrote them in the boldface purely for the sake of clarity. BMN calculated the matrix elements of the Hamiltonian between the states "created" by these trace-like operators and they verified that the Hamiltonian coincides with the Hamiltonian for vibrating type IIB strings in the \(pp\)-wave curved limit of the AdS space (which admits a description using free but massive fields in the light-cone gauge).



A string of beads resembles the way how closed strings appear in the AdS/CFT correspondence. The strings between the beads are pretty much the QCD flux tubes. Each bead carries two "hands" which correspond to the two indices of the adjoint fields.

Various people have verified that the BMN picture of the closed type IIB string (the BMN paper is approaching 1,500 citations) also contain the right stringy interactions, see e.g. this paper of ours. All of this works perfectly in the \(pp\)-wave limit but it's clear that the strings continue to exist outside the limit and their behavior and interactions gets deformed to exactly what you would expect in the general AdS space.

(BMN also presented a massive BFSS-like DLCQ model for the \(pp\)-wave background of string theory and I am convinced it is right, too. Lots of interesting maths follow from that.)

It's not just excited vibrating strings and wrapped branes you may find in the CFT. You also find the characteristically stringy shapes of the compactification space. For example, in 1999, Klebanov and Witten started with the analysis of massive deformations of the CFT. They found out that the dual gravitational side involves various orbifolds and conifolds and they may be deformed (and they actually do deform) much like the conifold – a special case of a Calabi-Yau manifold – in string theory. So not only the supersymmetric AdS/CFT pairs have the right total spacetime dimension on the gravitational side, \(d=10\) or \(d=11\) – the latter in the known M-theory examples of AdS/CFT that are known as well. The allowed shapes of the compactified "extra" dimensions agree with what string theorists have known to be possible for quite some time, too.

I could continue for quite some time. You really find everything you have a reason to look for. Whenever some stringy objects or processes can be seen to leave the traces in the strong (gravitational) coupling limit of the CFT, you will find them. This diverse agreement makes us almost certain that even the aspects we can't fully calculate yet almost certainly work properly, too. But I also want to mention several "disclaimers" concerning the claim that the stringiness is absolutely paramount in AdS/CFT.

First, I want to say that the AdS/CFT correspondence has been used to calculate the entropy of various black holes. The "essence" of these calculations only requires one aspect of string theory that you could "separate" from the rest of string theory and treat as a string-theory-independent aspect of AdS/CFT. To count the microstates, one needs the Cardy's formula for the number of highly excited states which is a universal trick in CFTs, regardless of their stringy or gravitational interpretation; and one needs to realize that via AdS/CFT, the master 2D CFT example of that always gets mapped to a three-dimensional BTZ black hole that may be found in the "heart" of almost any black hole whose entropy has been accurately calculated in AdS/CFT.

So the previous paragraph implies that "not all of string theory is needed for one particular calculation". However, if you want to be able to perform "all meaningful calculations", you will need a whole consistent theory – and you will need all insights and objects from string theory, too.

Second, there exist examples of AdS/CFT pairs that "marginally" look non-stringy. My favorite example is the pure 3D AdS gravity linked to a 2D CFT with the monster group symmetry by Witten. This is very interesting but I think that one may still interpret this vacuum e.g. as an "exceptional" compactification of some 27-dimensional bosonic M-theory on something like the 24-dimensional Leech lattice. Because this vacuum is non-supersymmetric, it isn't and cannot be easily linked to the usual \(d=10\) or \(d=11\) vacua of string/M-theory. But otherwise, the theory still smells "stringy". You should surely expect no confirmation of the fantasies of the people who imagine that quantum gravity is something completely different than what string theory dictates it to be.

My second example of a "potentially non-stringy AdS/CFT pair" will have a clearer resolution. There is something called "the Vasiliev theory" or "higher-spin theory" with infinitely many fields of arbitrarily high spins (like in string theory). However, the number of fields isn't growing exponentially with the mass (in string theory, it is). Nevertheless, in 2002, Klebanov and Polyakov proposed that this bizarre quantum-gravity-like theory is equivalent, via the AdS/CFT correspondence, to the critical \(O(N)\) vector model. It is a theory whose main fields don't transform in the adjoint (matrix) representation of the group like in the most notorious AdS/CFT example; instead, they transform in the fundamental (column) representation.

For years, I've been puzzled by the question whether these folks would claim that the Vasiliev theory is a consistent theory of quantum gravity unrelated to string theory. It had to be either inconsistent, or non-gravitational, or it had to be a vacuum of string theory, I thought. Needless to say, I was right. In 2012, Xi Yin, Shiraz Minwalla, and Chi-Ming Chang and Tarun Sharma who co-wrote their paper showed that the Vasiliev theory and/or the vector model may be obtained as a limit of type IIA string theory compactified on \(AdS_4\times{\mathbb C \mathbb P}^3\) which allows them to prove the equivalence with the help of some usual stringy arguments. The understanding of the origin is good enough that the authors even guess where all the terms in the Vasiliev equations come from; the right hand sides probably arise from the world sheet instantons. It just happens that this AdS/CFT pair was also linked to the ABJ theory found in another recent development, the membrane minirevolution.

For the single monstrous CFT, such a stringy construction may be not known equally explicitly but because it has worked in all other examples, one should say that it is extremely likely that all consistent examples of the AdS/CFT correspondence may be shown to be vacua of string/M-theory and all the objects and processes predicted by string/M-theory may be found in these holographic definitions of quantum gravity, too. Quantum gravity and string/M-theory are ultimately synonyma and the AdS/CFT correspondence has provided us with thousands of pieces of new circumstantial evidence that this is the case.

Incidentally, the Physics Stack Exchange question mentioned the "Rehren duality". Karl-Henning Rehren's "holography" mentioned in another answer over there is a vacuous pseudoscience that has nothing to do with the depth of the actual holography in quantum gravity. What is done in that paper is just rewriting fields \(\phi(x^0,x^1,x^2,x^3,x^4)\) as \(\phi_{x^4}(x^0,x^1,x^2,x^3)\) i.e. rewriting one of the coordinates as an "index" of the fields, and claiming that coordinates may be reduced by one (or any number, for that matter). However, nothing changes about the physical "number of dimensions" by this sleight-of-hand. See some TRF blog posts about Rehren, especially one on the temptation of rigor.

Previous articles on a similar topic:

Unity of strings (2004)
Unity of strings (2006)
Two roads from \(\NNN=8\) SUGRA to string theory (2008)
Are AdS/CFT and AdS/CMT relevant for unification? (2009)
Why unification sits at the core of string theory (2010)
Unification as a source of certainty (2011)
Unity and uniqueness of string theory became a heresy (2011)

So please, if you hear a Sh(mo)ithead talking about the AdS/CFT correspondence's not being dependent on string/M-theory again, reach for your firearm and press the trigger. Let me emphasize that this is no manslaughter, it is self-defense.