The Thinker Musings

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.

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

Finn Larsen, Partition functions and elliptic genera from supergravity

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

In

Taejin Lee, Topological Ward identity and AdS/CFT correspondence

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

AdS/CFT vs CS/WZW

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.

Muddy waters

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.

Re: Link underline behavior

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.