added pointer to:

- Katrin Wendland,
*Snapshots of Conformal Field Theory*, in: Mathematical Aspects of Quantum Field Theories. Mathematical Physics Studies. Springer 2015 (arXiv:1404.3108, doi:10.1007/978-3-319-09949-1_4)

added pointer to the original:

- Alexander Belavin, Alexander Polyakov, Alexander Zamolodchikov,
*Infinite conformal symmetry in two–dimensional quantum field theory*, Nuclear Physics B Volume 241, Issue 2, 23 July 1984, Pages 333-380 (doi:10.1016/0550-3213(84)90052-X)

added pointer to

- James E. Tener,
*Representation theory in chiral conformal field theory: from fields to observables*(arXiv:1810.08168)

(here and in related entries)

]]>I added

- Clay Cordova, Thomas T. Dumitrescu, Kenneth Intriligator,
*Deformations of Superconformal Theories*, (arXiv:1602.01217)

to SCFT. Jacques Distler mentioned it as a source for the claim

]]>a general feature of ((1,0) or (2,0)) SCFTs in 6 dimensions (and $\mathcal{N}=3$ SCFTs in 4 dimensions): they have no relevant or marginal supersymmetry-preserving deformations.

Hey Zoran: it looks like in #28 you are replying to #25. Notice that this dates from over a year back! If you want Tim to see you message, you’d better email him. :-)

]]>Just go to the web page of Oberwolfach http://www.mfo.de and you can find pdfs of the last few years of online reports.

]]>I noticed that the references on the FQFT-perspective on 2dCFT were missing at at conformal field theory. So I added in some, in a new subsection References–Formulation by functors on conformal cobordisms.

]]>The workshop reports are not freely available, are they?

The Oberwolfach workshops reports are usually made freely available online, as far as I am aware.

]]>So they will be at the workshop “Geometry, Quantum Fields, and Strings: Categorial Aspects” next week in Oberwolfach?

The workshop reports are not freely available, are they?

(I hope you get well soon).

]]>Today's arxiv reference form Igor Kříž and collaborators at vertex operator algebra. Probably Urs will hear more from Prof. Kříž at Oberwolfach next week (I was scheduled to go and to my regret do not have physical strength to travel that far without health risk).

]]>Right Tim, this is the canonical kind of statement I learned from (first from Ginsparg’s review when first studying the subject in early 1990s and then from that book when it appeared in 1997 to my delight). And it makes “nevertheless” out of place here.

]]>I am just saying that 2d CFT is better understood than higher dimensional CFT.

Well this is what I agree, however the original statement explictly claimed that the problem/difference is in the problems with **rigour** of defining QFT. In 2d the rigour has been avoided by alternative combinatorial treatment (which is often non-rigourous) of the classification problem of a consistent class (rather than the definition itself), but it works, at least in rational case.

Statements like “more interesting in 2 dim than” or “better understood in 2 dim” or “conformal symmetries are more complicated in 2 dim” etc. have a tendency to be subjective.

The foreword of the Francesco/Mathieu/Sénéchal book does a good job, in my opinion, to dodge this difficulty:

“In d spatial dimensions, there are 1/2(d+1)(d+2) parameters needed to specify a conformal transformation. The consequence of this finiteness is that conformal invariance can say relatively little about the form of correlations, in fact just slightly more than rotation or scale invariance. The exception is in two dimensions, where the above formula gives only the number of parameters specifying conformal transformations that are everywhere well-defined, while there is an infinity variety of local transformations, namely the locally analytic functions. In two dimensions the conformal symmetry is so powerful as to allow…”

(the authors write next “an exact solution of the problem”, which is suboptimal for various reasons, one being that the “problem” isn’t defined yet :-)

]]>Zoran,

maybe there is a misunderstanding here. I am not talking about Feynman integrals etc. I am just saying that 2d CFT is better understood than higher dimensional CFT. I don’t think this is controversial. In fact, I think you make the same point.

]]>It is easy to be an admiral after the battle. If you think you can do the renormalization in 2d etc. hence that the difficulties with **rigour** prevent you to extend than go on. Most of the breakthrough is due to BPZ revolution in 1984 which had nothing to do with rigour in 2d as opposed to “rigour in handling higher dimensional QFT”, but with combinatorial handling of constraints performed in dimension 2 due infinite Virasoro symmetry at physicists’ level of rigour.

Still one can not systematically do the Feynman integral and renormalization etc. in 2d. One **avoids** this by doing bootstrap or something else, instead of defining the analytic Feynman integral one replaces it with comgbinatorial device defined ad hoc with help of highly symmetric situation. Similarly one can do something for TQFTs in higher dimension here and there. But no progress in true general QFT.

Bootstrap or not, very few QFTs in higher dimensions are rigorously understood.

2d CFT has to a large extent been constructed and classified. This is far from true in higher dimensions. Independend of which formalism you use.

]]>You also wrote:

due to the general problems with rigorously handling higher dimensional QFTs

The fact is that the bootstrap program in any number of dimension does not care about the definition of Feynman integral or renormalization or anything of the sort. It is not computing the amplitudes; it just looks at consistent systems of correlation functions which satisfy all the constraints. It is a classification program out of knowing the symmetries and axioms, not a computational program from a path integral and an action. So by definition it is irrelevant for this program weather people know how to regularize an integral in some number of dimensions.

]]>A manifest symmetry is never a factor which complicates a structure. It is just a statement of a manifest better order in a structure – knowing a symmetry makes it easier to analyse the structure.

]]>Well, I added the “nevertheless” after somebody had changed the entry to saying that the 2-d case is the *least* understood one, because the conformal group in 2d is so much richer. That made me think that one should point out that *even though* the structure is rich, it has been fully understood.

But feel free to reword it as you deem appropriate.

]]>I have further added a remark on how nevertheless the case of d=2 CFT is the best understood case

**Nevertheless** ???? The infinite-dimensional conformal group is generally recognized as a lucky constraint which has enabled in 1984 to make a breakthrough in 2d case of the Polyakov’s 1971 bootstrap program which has been originally formulated in all dimensions. More symmetries easier problem in mathematical physics.

This blunder should be corrected in the entry, in my opinion.

]]>completed reformatting from beamer. still links to be added; another day.

]]>a few lines added to the notes. we’ll meet OPEs tomorrow :)

]]>Thanks! both for the reference and for the TOC.

There’s a lot of editing to do there apart from reformatting: adding links to nLab pages! I’ll do at the end of the reformatting, but if anyone wills to add a few links while reading the cleaned up part..

]]>Thanks, Domenico, nice notes.

I added a link to them in the References-section at vertex operator algebra.

I also took the liberty of adding a TOC to your page. Hope you don’t mind.

]]>