### 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.

## 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

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