## July 22, 2007

#### Posted by Urs Schreiber

The last session of Recent Developments in QFT in Leipzig was general discussion, which happened to be quite interesting for various reasons.

As you know, work in physics from time to time leads to interesting conjectures in mathematics. Usually what happens is that first experimental physicists measure something which leaves everyone more or less puzzled. After a while theoretical physicists are beginning to understand what’s going on and are growing fond of some idea which they then regard as established textbook fact. Then mathematical physicists come along and try to find a precise statement of this idea. Ideally this results in a well defined conjecture. Then finally mathematicians try to prove this. Or roughly like this. It’s an academic food chain.

For this food chain to work, it’s important that the end product at every step is digestible by the species in the next layer.

Recently, there was a rather remarkable example where this didn’t work:

Many theoretical physicists are very fond of something called AdS/CFT duality. It’s an example, or a realization, of what has been called the holographic principle, which is an idea going back to t’Hooft. I had recently mentioned that in Extended QFT and Cohomology II: Sections, States, Twists and Holography and Why Theoretical Physics is Hard…. But if you want to learn more, you should better look at something like this Introduction to the AdS/CFT correspondence.

But beware: that’s clearly within the second layer of the food chain, not any higher yet.

In fact, it is quite hard to track down more than less precise statements of what this conjectural correspondence actually says in detail. Still, the excitement about it is comparatively large.

Now, a while ago, there was an attempt by a mathematical physicist to process this conjecture from the second to the third and forth layer of the food chain:

K.-H. Rehren
Algebraic Duality
hep-th/9905179.

But something strange happened: people in the second layer complained that this work did in fact not actually process the output which they produced, Maldacena’s AdS/CFT conjecture, but instead something more or less completely different!

See Jacques Distler’s entry on Rehren Duality.

Now, it so happened that yesterday at the conference which I am attending here, Hanno Gottschalk had given a talk based on Rehren’s work:

Hanno Gottschalk
AdS/CFT correspondence: lessons from constructive field theory

Abstract: Following ideas of Dütsch and Rehren, we give a proof of the equivalence of the two prescriptions of the conformal boundary field in AdS/CFT (“duality”). We use rigorous Euclidean path integrals to define functional delta functions of boundary values and restrictions to the conformal boundary. Both, in the interacting and non-interacting case, divergences arise that have not been observed so far. These divergences can, however, be removed without doing damage to reflection positivity and symmetry. Interactions with IR cut-off are taken into account in D=2 dimensional Euclidean AdS. We comment on some intriguing novel features of the IR-problem which comes from the fact that boundary sources in AdS/CFT lead to shifts in the bulk interaction that create an infinite amount of energy. This talk is based on joint work with Horst Thaler, math-ph/0611006.

As you know, I missed that talk. But I was told that after the talk a couple of string theorists at the conference raised those objections again: whatever this work is about, it is not really about what they call AdS/CFT duality.

Now, to some extent this is of course not a surprise: AdS/CFT duality is supposed to related a quantum field theory called $N=4$ super Yang-Mills to closed quantum string field theory. This means that in its full form, this is extremely far indeed from anything remotely tractable by precise mathematics.

So nobody should expect mathematical physicists to come up with an axiomatization of anything but a simplified toy model version of the conjecture.

But, and that’s important, this is not what the complaints are about: the complaint which is being voiced is that “Rehren duality” is not even roughly the same idea as string theorist’s AdS/CFT.

Whatever the reason for this disagreement (personally, I think there are problems with both layers of the food chain involved here) it is good that people are talking about it. Communication is important.

In our dicussion session, Matthias Blau played the role of the spokesman of the critics. His main emphasis was on this important fact:

The original AdS/CFT duality conjecture is not a statement which relates a quantum field theory in higher dimensions to whatever quantum field theory (or whatever) is obtained as the restriction of the former one to some boundary of its paremeter space.

It’s more indirect than that. The way I would phrase it is this:

The states of the higher dimensional QFT encode the correlators of a lower dimensional theory on the boundary.

So, you see, it’s not the states which restrict to states. But states restrict to correlators.

But, in saying so, I am already trying to move upwards in the food chain a little. What one finds in standard literature on AdS/CFT (like this Introduction to the AdS/CFT correspondence) looks like this:

On one side of the duality, we have something like a field theory (a string field theory, really, the second quantization of a certain 2-dimensional CFT) in higher dimensions, which is encoded by something like a path integral $\int D\phi \exp(i S[\phi])$ on a “exponentiated action” functional $\phi \mapsto \exp(i S[\phi])$ on some space of “fields” on some parameter space $X$.

On an (asymptotic) boundary $\partial X$ of parameter space $X$ one may fix the value of these fields $\phi_0 := \phi|_{\partial X}$ and then consider the path integral over the space of fields with that fixed restriction, only. This yields a functional $\phi_0 \mapsto \exp(i W[\phi_0]) := \int_{\phi_{\partial X} = \phi_0} D\phi \exp(i S[\phi])$ on fields on the boundary.

Now, naively one might think that this functional $\exp(i W[\cdot])$ should now be regarded as the action functional on fields $\phi_0$ on $\partial X$ and thus define a new field theory on $\partial X$.

But that’s at least not what the original AdS/CFT correspondence is about! Rather, this correspondence states that $\exp(i W[\cdot])$ is the generating functional of correlators of some field theory on the boundary.

This means that variations $\frac{\delta}{\delta \phi_0(x_1)} \cdots \frac{\delta}{\delta \phi_0(x_n)} \exp(i W[\phi_0])|_{\phi_0 = 0} = \langle F(x_1) \cdots F(x_n) \rangle$ are the correlator of some quantum field theory on the boundary. In terms of path integrals this reads $\langle F(x_1) \cdots F(x_n) \rangle := \int D A\; F(x_1) \cdots F(x_n) \exp(i S_{\mathrm{bd}}[A]) \,.$

I am glossing over mountains of details here, for good reason. The claim is that it is this broad structure

states in higher dimensions $\leftrightarrow$ correlators in lower dimensions

which is crucial to AdS/CFT – and which is not captured by those attempted axiomatizations by Rehren and others.

I think that instead of trying to tackle the full AdS/CFT correspondence, which is about theories in 10 and 4 dimensions (5 dimensions somehow disappear under some carpet, a non-rigorous carpet, of course), mathematical physicists should first try to understand the analogous situation in 3 and 2 dimensions.

This is the situation that Witten’s recent work applies to, which John very nicely summarized in TWF 254.

And for non-gravitational versions of this, there is already a pretty good mathematical understanding. I mention this from time to time here. Last time in FFRS on Uniqueness of Conformal Field Theory.

Posted at July 22, 2007 6:15 PM UTC

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Completely out-of-the blue, and possibly stupid, but I have to ask: do you think a useful analogy of this switch from states to corelators can be drawn to Plank’s original shift from classical degrees of freedom (like wavemode amplitudes) to quantum states (the individual wavemodes) that e.g. fixed black body theory? Or the opposite shift?

Posted by: Some Guy on the Street on July 24, 2007 12:12 AM | Permalink | Reply to this

do you think a useful analogy of this switch from states to corelators can be drawn to Plank’s original shift from classical degrees of freedom (like wavemode amplitudes) to quantum states (the individual wavemodes)

Hm, my first reaction to this is a decided No.

But maybe if you give more details about why you think there might be something to this, I can give a better answer. I mean, John here at the $n$-Café has often talked about the fact that classical mechanics can be viewed as quantum mechanics over a different rig. Maybe using some magic like that one could try to see if the holographic principle somehow has a classical analog along the lines you seem to be imagining. But I really don’t know. Right now I strongly doubt it. But maybe something deep is going on here which escapes me.

Posted by: Urs Schreiber on July 24, 2007 8:59 AM | Permalink | Reply to this

This might be a rephrasing of what you’re saying, but my thinking about what AdS/CFT does has always been not so much correlators $\leftrightarrow$ states as fields (or operators) $\leftrightarrow$ sources. In some sense, AdS/CFT is talking about Legendre transforms, taking an ordinary theory of some fields $\phi_i$ and rewriting it in terms of the sources $J_i$ that appear in terms $J_i \phi_i$ in the partition function. (Since the boundary values of the 5D fields are the values of these sources.) Now, in field theory the sources look like a different sort of object from a local propagating quantum field, but the magic is that somehow the physics in terms of the sources is organized scale-by-scale so as to be described by local propagating fields in five dimensions where scale is the fifth dimension.

Does an intuition like this fit into the sort of picture you want to build? I’ve always thought that the RG interpretation of AdS/CFT is very nice, and it would be interesting if someday it could be made completely explicit in some relatively simple example.

Posted by: Matt Reece on July 24, 2007 9:09 PM | Permalink | Reply to this

This might be a rephrasing of what you’re saying, but my thinking about what AdS/CFT does has always been not so much correlators $\leftrightarrow$ states as fields (or operators) $\leftrightarrow$ sources.

Yes, I think that’s supposed to mean the same thing. Under the following dictionary:

The path integral of a $d$-dimensional QFT is something which assigns a vector “space of states” to each $(d-1)$-dimensional piece of parameter space and a linear map between two such for every $d$-dimensional piece cobounding these.

In particular, for a space $X$ with boundary $\partial X$ considered as “outgoing” (and hence with the empty set considered as ingoing) $\{\} \stackrel{X}{\to} \partial X$ we get a map $Z(X) : \mathbb{C} \to Z(\partial X) \,.$ Here $H = Z(\partial X)$ is the vector space of states.

Hence $Z(X)$ in fact produces one particular state in $H$.

In the path integral language, this corresponds to the following:

$H$ is a space of sections of some bundle over the space of field configurations $\mathrm{conf} := \mathrm{Maps}(\partial X, \mathrm{space}\; \mathrm{of} \; \mathrm{field} \; \mathrm{values})$ on the boundary. Assume this bundle to be a trivial line bundle. Then $H$ is just a space of functions on $\mathrm{conf}$.

Now, the path integral produces such a function by the assignment $\phi_0 \mapsto \int_{\phi|_{\partial X} = \phi_0} \exp(i S[\phi])$ for $\phi_0$ a field configuration on the boundary.

That’s the function (“wave function”) called $\exp(i W[\cdot])$ in the above entry.

It is a state of the QFT in that it assigns a probability amplitude to each field configuration on a given boundary of parameter space.

To remind us of that, we should write $\exp(i W[\cdot]) := \Psi \,.$

The statement then is that the derivatives (functional variations) $\partial \psi$ of $\Psi$ are correlators of a QFT defined only on the boundary.

And if a correlator is expressed as the variation of some quantity after a veriable $\phi_0$, we say that $\phi_0$ is a “source”.

But $\phi_0$ was a configuration of the original theory.

So I’d say that under the correspondence we have

states $\leftrightarrow$ (generating functionals for) correlators

and

configurations $\leftrightarrow$ sources.

Would you agree with that?

(I notice that sometimes people in physics use the word “state” also for what here I call “configuration”. But one shouldn’t, I believe. Even classically, a configuration is not a state, only a configuration and a momentum is.)

Posted by: Urs Schreiber on July 25, 2007 11:05 AM | Permalink | Reply to this

The problem as I see it is that ads/cft is actually a very precise statement, albeit maybe not formulated in the right language for some. If you want to perform a certain calculation, there is no ambiguity in how to do it. The suggested “formalization” of ads/cft gives a different, inequivalent ways of performing the same calculations, one that doesn’t make much sense, but in any event it is different from how you suppose to do calculations in ads/cft. As such it is not making ads/cft precise, it is making a different conjecture altogether. It is also relevant that such conjecture would not pass some very basic consistency checks (e.g, locality and unitarity of the boundary theory).

Also, I wish that people that want to make that independent conjecture would find their own term for the suggested duality. There are students there that are studying the subject, and I can tell you it is very confusing for some of them to have this amount of noise out there.

Posted by: Moshe on July 24, 2007 11:46 PM | Permalink | Reply to this

The problem as I see it is that ads/cft is actually a very precise statement, albeit maybe not formulated in the right language for some.

The conjecture involves a quantum field theory and even string field theory. People who would like to make it “precise” in the mathematical sense will have to first find axiomatics for these ingredients, or at least for some aspects of them.

There are essentially two approaches to axiomatize QFT: a) the Atiyah-Segal functorial description (“Schrödinger picture”) and b) the Haag-Kastler “AQFT” formulation (Heisenberg picture).

(I am not aware of any attempt to analogously put string field theory in a solid form. Though I allow myself to have a hunch for how it should proceed in the context of a).)

Rehren went ahead and tried to see how far he can get with using b) to attack AdS/CFT.

Given the state of development of AQFT, one can clearly not expect this to capture all of the correspondence. But it is not out of the question, I think, that some version of its main aspect could be conceived this way.

The suggested “formalization” of ads/cft gives a different, inequivalent ways of performing the same calculations,

My impression was that what Rehren, Gottschalk and others do is in fact another kind of calculation, not just a different way to do the same calculation.

it is making a different conjecture altogether

Yes, it seems so. On the other hand, at least it provides a proof of that conjecture! :-)

I wish that people that want to make that independent conjecture would find their own term for the suggested duality

Yes. On the other hand, the ingredients “AdS” and “CFT” appear in both cases. So maybe the term “AdS/CFT correspondence” is by itself not optimal.

Posted by: Urs Schreiber on July 25, 2007 10:17 AM | Permalink | Reply to this

Urs, a few points to your comment:

1. The calculations done in ads/cft are all for on-shell quantities, so I don’t see any need for string field theory.

2. I am all for trying to formalize and making precise the correspondence. However, the attempt you refer to explicitly contradicts ads/cft. For example if you waste your time trying to calculate correlation functions that way, you’ll get a different answer from that of ads/cft.

Again, everyone is free to do whatever they find interesting, I just witnessed a lot of confusion resulting from them insisting using the pre-exisiting name for their different conjecture.

3. As for their correspondence, as discussed in Jacques’ piece, their correlation functions do not obey the basic conventional axioms of QFT. Yes, you can always redefine those axioms, and then get your desired proof, in the process completely mutilating the objects you are interested in, but I assume we don’t just want to play with words, we want to actually understand something.

Posted by: Moshe on July 25, 2007 6:27 PM | Permalink | Reply to this

Moshe,

concerning your point 2, I am not sure what it is your are replying to exactly. If by “that way” you mean Rehren’s approach, we already perfectly agree that this is not string theorist’s AdS/CFT. That was the point of my entry.

Posted by: Urs Schreiber on July 25, 2007 6:40 PM | Permalink | Reply to this

Yeah Urs, I think we agree. I am just generally frustrated by the amount of misinformation students of string theory have to overcome these days, this is just a drop in the bucket. Anyhow, thanks for making the correct statement the point of your entry.

Posted by: Moshe on July 25, 2007 8:45 PM | Permalink | Reply to this

Also, point 2 was in reply to your “My impression was that what Rehren, Gottschalk and others do is in fact another kind of calculation, not just a different way to do the same calculation”… Even if true it doesn’t matter, it is clear that when calculating the same quantities the answers are going to be different. But this is beating a dead horse, we already agree on that point I believe, just wanted to clarify myself.

Posted by: Moshe on July 25, 2007 8:55 PM | Permalink | Reply to this

Also, point 2 was in reply to your “My impression was that what Rehren, Gottschalk and others do is in fact another kind of calculation, not just a different way to do the same calculation”…

Oh, I see. That wasn’t phrased very clearly by me. I really mean: Rehren’s “algebraic duality” is pretty much entirely something different from Maldacena’s duality.

My sentence above was to say that it’s not just a different way to attack the same question, but a different question altogether.

Okay, so we agree.

Posted by: Urs Schreiber on July 26, 2007 9:53 AM | Permalink | Reply to this
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