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I began to add a definition of conformal field theory using the Wightman resp. Osterwalder-Schrader axiomatic approach. My intention is to define and explain the most common concepts that appear again and again in the physics literature, but are rarely defined, like “primary field” or “operator product expansion”.
(I remember that I asked myself, when I first saw an operator product expansion, if the existence of one is an axiom or a theorem, I don’t remember reading or hearing an answer of that until I looked in the book by Schottenloher).
Tim,
thanks a million. Very nice that you put energy into this!
While I have not time to join you in your efforts here right now, I just restructred the section outline slightly, in order to remind us that eventually the discussion that you put in (which is about CFT or even “rational” CFT on the plane) needs to be accompanied by more dicussion on the definition of CFT on surfaces of arbitrary genus.
I remember that I asked myself, when I first saw an operator product expansion, if the existence of one is an axiom or a theorem,
Good point. The literature seems to be divided into those who assume that “vertex operator algebra” is a definition of (chiral) CFT, those who use AFT-style definitions, and, finally, those who don’t use any real definition, or maybe some path integral heuristics.
some path integral heuristics
this actually leads to OPEs. I have some never cleaned up notes on this, for an old talk “Vertex algebras avant Borcherds” I gave in Milan a few years ago. now I see my nLab area is a good place for letting them free. Just give me a day to reformat them, within 24 hours from now they’ll be there :)
some path integral heuristics
this actually leads to OPEs.
Oh for sure they do. But it’s a bit heuristic. That’s all I meant to say.
Just give me a day to reformat them, within 24 hours from now they’ll be there :)
Very nice. Thanks!!
But it’s a bit heuristic.
much more than a bit.. :-)
Dmitriy Drichel had kindly added some comments to conformal field theory on how the conformal group is most interesting in $d = 2$. I have further added a remark on how nevertheless the case of d=2 CFT is the best understood case, as far as really the QFTs go.
Much more should be said here eventually, of course.
Urs said:
eventually the discussion … needs to be accompanied by more dicussion on the definition of CFT on surfaces of arbitrary genus.
Ok, but that’s a topic that I do not know much about (but are willing to learn :-)
Domenico said:
now I see my nLab area is a good place for letting them free.
Yes, definitly! Maybe OPE should then get their own page. Is there a canonical way to find someone’s nLab area?
Dmitriy Drichel had kindly added some comments to conformal field theory on how the conformal group is most interesting in d=2.
The book by Schottenloher has a nice discussion of this and an explanation how and why the physics terminology is confusing for mathematicians. Eventually I would like to discuss some aspects of this on a conformal group page, but everyone should feel free to beat me to it :-)
Here are the notes from my talk (first 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.
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..
a few lines added to the notes. we’ll meet OPEs tomorrow :)
completed reformatting from beamer. still links to be added; another day.
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.
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.
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.
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.
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.
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.
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.
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 :-)
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.
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.
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).
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).
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.
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.
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.
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. :-)
I added
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.
added pointer to
(here and in related entries)
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