## June 1, 2008

### Not Forgotten

The comment sections of all my old posts remain open for business. You never know when some old topic will prove relevant to someone.

Lately, Urs Schreiber took an interest in my old post on Rehren’s Algebraic Holography. The ensuing discussion eventually focussed on one of the questionable aspects of Rehren’s proposal that I didn’t emphasize in my original post.

As you’ll recall, in AdS/CFT, the boundary values of the bulk fields are sources for the correlation functions of the boundary CFT. In Rehren’s formulation, he wants them to be the fields of the boundary theory themselves, and he claims that the two formulations are related by a functional Fourier transform in the bulk theory. Alas, that’s wrong, and there is no bulk QFT whose boundary values are suitable candidates for the fields of the boundary theory.

All of this is kinda obvious to anyone with a passing acquaintance with AdS/CFT. But, since the arXivs continue to receive papers which cite this notion of Algebraic Holography as an established fact, it’s probably worth dragging this point up “above the fold.” This way, someone, somewhere, will be saved the need to wade through the original papers.

Posted by distler at June 1, 2008 10:59 AM

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### Re: Not Forgotten

In any case, I think it is, in general, a good idea to try to understand something as supposedly fundamental as AdS/CFT from a more formal QFT point of view, if possible (i.e. if it is within reach).

Of course care has to be exercise about the role of rigour, as you very well emphasized. So it would be good to look at special cases where a working rigorous description is already available:

there is already a pretty good, if also still not complete, formal understanding of the participating components and the “duality” for the simple but important case

$rational 2d CFT (in particular WZW) \leftrightarrow rational 3d TFT (in particular Chern-Simons)$

My understanding is that CS/WZW is to be regarded as a special case of AdS/CFT. Just for my own benefit, I give the following links:

From

I learn that $CS_3/WZW_2$ is supposed to capture the “asymptotic behaviour” of $AdS_3/CFT_2$.

In

the topological Ward identity for CS is interpreted as realizing AdS/CFT duality.

In

Banados, Schwimmer, Theisen, Chern-Simons gravity and holographic anomalies

theories of Chern-Simons gravity, which generalize 3d Chern-Simons theory to higher odd dimensions, are related to their holographic CFT duals.

The point of these random links for the moment just being to emphasize that for an axiomatic/formal/rigorous discussion of AdS/CFT the ordinary CS/WZW relation seem to be a good starting point.

I have to run now. But I’d be grateful for whatever comment and advise you might have

Posted by: Urs Schreiber on June 2, 2008 10:41 AM | Permalink | Reply to this

There are a couple of — accidentally similar — things that are probably best kept distinct.

1. 3D TQFT on a manifold with boundary yields a Hilbert space. Canonically quantizing a 3D TQFT on $\Sigma\times\mathbb{R}$ requires choosing a polarization, which amounts to choosing a conformal structure on $\Sigma$. The aforementioned Hilbert space is the space of conformal blocks of the 2D CFT. For CS/WZW, heuristically, what’s going on is that the bulk CS theory is not invariant under gauge transformations that fail to go to the identity at the boundary. So there are actual local degrees of freedom (the “$g$” of the WZW model) living at the boundary. Note, in particular, that the stress tensor of the WZW model (which, among other things, measures the response to a change in polarization) is obtained by the Sugawara construction from the currents (built out of $g$).
2. Pure gravity in $\text{AdS}_3$ is, formally, a Chern-Simons theory. The boundary theory is, again, a $\text{CFT}_2$. But, unlike the previous case, the stress tensor of the CFT is not quadratic in the currents. Rather, the generators of global conformal transformation of the boundary theory are linear in the currents. In AdS/CFT, the stress tensor of the boundary CFT is sourced by the metric of the bulk theory. The latter just happens to be expressible as a CS gauge field, in the present formulation.

So, even though these looked similar (a WZW model on one side, a CFT on the other), they are very different.

Posted by: Jacques Distler on June 2, 2008 11:20 AM | Permalink | PGP Sig | Reply to this

### Re: Not Forgotten

I have to say, from all the misleading technical claims one encounters in the literature, this is one of the more annoying one, this series of papers really does cause lots of confusion, hopefully this post and other rectify the situation a bit.

Whatever algebraic holography is supposed to be, it differs from AdS/CFT in concrete ways (e.g. calculating correlations functions etc.). Now if you are a student, you are likely to encounter one of the papers out there using the term AdS/CFT, yet giving information orthogonal to the established facts, and get yourself very confused… It is one thing to try to translate those established facts to your language, it is another to try and construct something analogous which fits better with your world view. If you are doing the latter, please use a new term.

Posted by: Moshe on June 2, 2008 11:15 AM | Permalink | Reply to this

### Muddy waters

Whatever algebraic holography is supposed to be, it differs from AdS/CFT in concrete ways (e.g. calculating correlations functions etc.).

The claim of the fallacious paper, under discussion, is that they are not different, and hence that the Algebraic Holography people really are discussing AdS/CFT.

And it’s not just students who are liable to be confused.

Posted by: Jacques Distler on June 2, 2008 11:43 AM | Permalink | PGP Sig | Reply to this

### Wikipedia

I wonder if someone could fix the Wikipedia article. Lubos created it with a typically forthright statement of the problem, but it was immediately changed to the current version.

Posted by: mitchell porter on June 4, 2008 5:05 AM | Permalink | Reply to this

### Re: Wikipedia

I wonder if someone could fix the Wikipedia article.

I would guess that’s impossible.

Posted by: Jacques Distler on June 4, 2008 8:12 AM | Permalink | PGP Sig | Reply to this

### Re: Wikipedia

I had a go. I think the changes might survive.

Posted by: mitchell porter on June 6, 2008 6:55 AM | Permalink | Reply to this

### Re: Wikipedia

I had a go. I think the changes might survive.

Good luck!

One change I would suggest: where you list the main differences between AdS/CFT and Rehren Duality, I would not bother with the AdS/universal covering space distinction. Relatively speaking, that’s a minor technicality.

The main differences are

1. In AdS/CFT, the boundary values of bulk fields are sources for operators of the boundary theory. In Rehren Duality, the boundary values of the bulk fields are the operators of the boundary theory.
2. In AdS/CFT, the bulk theory is necessarily a gravitational one. The source for the conserved stress tensor of the boundary theory is the boundary value of the bulk metric tensor. In Rehren Duality, the bulk theory is an “ordinary” (non-gravitational) QFT.
Posted by: Jacques Distler on June 6, 2008 9:34 AM | Permalink | PGP Sig | Reply to this

Sorry for off topic, but I noticed that in Safari (10.5.3), all of the links in the same paragraph get underlined when the mouse pointer is in the paragraph. Is it the expected behavior ? I think underlining a link only when the mouse pointer is on top of it is enough, but it may just be my preference.

Posted by: Anonymous on June 2, 2008 7:40 PM | Permalink | Reply to this

That’s the intended behaviour.

* Hover over a paragraph, and all the links in the paragraph get underlined.
* Links which open in a new window (e.g., the comment form) sport a distinct cursor, so you can tell, in advance, that’s going to happen.

Posted by: Jacques Distler on June 2, 2008 7:50 PM | Permalink | PGP Sig | Reply to this

Thank you very much for the explanation.

Posted by: Anonymous on June 2, 2008 9:11 PM | Permalink | Reply to this

### Re: Not Forgotten

I just heard a talk at the Max-Planck institute for math in Bonn by H. Thaler, who worked on the Dütsch-Rehren functional Fourier transformation. I haven’t looked yet at the accompanying article AdS/CFT correspondence in the Euclidean context yet, but H. Thaler tells me that their main result is:

they can make rigorous (i.e. non-perturbative) sense of the functional Fourier transform (only) for the case that the mass squared of the field is non-positive (i.e. tachyonic) and then precisely for the range that both scaling dimensions $\Delta_\pm$ are positive.

This is of course precisely what you said.

Curiously, he says they show that in these tachyonic cases both sides of the Fourier transform are well defined and equal – but both vanish identically.

Posted by: Urs Schreiber on June 30, 2008 9:40 AM | Permalink | Reply to this
Read the post The Manifold Geometries of QFT, I
Weblog: The n-Category Café
Excerpt: Some notes on the first day at "The manifold geometries of QFT" at the MPI in Bonn.
Tracked: June 30, 2008 12:21 PM

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