## July 10, 2008

### Talk on AQFT from FQFT and Applications

#### Posted by Urs Schreiber

Yesterday it was my turn again in our “internal seminar” at HIM: I had been asked to talk about the ideas on AQFT from $n$-functorial QFT (arXiv, blog).

The notes for the talk

AQFT from FQFT and Applications
(pdf)

dwell, after some motivation and a quick tour through the main theorem, a bit more on applications than the currently available article on the arXiv does. In particular, there was some further progress on my part with understanding the questions concerning lattice models. This owes a lot to very helpful discussion I had with Pasquale Zito who educated me more about his thesis work and especially about hard-to-find work (still have to try to track down some of it) by A. Ocneanu on asymptotic inclusion of von Neumann algebra subfactors.

All experts I talked to so far assure me that there should be nice constructions of AQFT nets from continuum limits of lattice models. But no literature at all seems to exist. Over on his blog, Alain Connes once said, in a closely related context, that

It is not really nicely spelled out anywhere

Local nets of algebra versus vertex operator algebras

Most everywhere one looks in the context of these investigations, one sees a curious lack of communication across borders of supposedly closely related fields. For instance, there has been every indication that local conformal AQFT nets in 2-dimensions, originally conceived in

Jürg Fröhlich and Fabrizio Gabbiani
Operator algebras and conformal field theory
Comm. Math. Phys. Volume 155, Number 3 (1993), 569-640

are equivalent to vertex operator algebras: both completely characterize the local (“chiral”) behaviour of conformal field theories and have been used to classify chiral CFTs. On the AQFT side of life for instance in

Yasuyuki Kawahigashi
Classification of operator algebraic conformal field theories in dimensions one and two
arXiv:math-ph/0308029 .

First attempts to make the equivalence between conformal nets and vertex operator algebras explcit and precise are given in

Yasuyuki Kawahigashi, Roberto Longo
Local conformal nets arising from framed vertex operator algebras
arXiv:math/0407263 .

The introduction starts with

We have two mathematically rigorous approaches to study chiral conformal field theory using infinite dimensional algebraic systems. One is algebraic quantum field theory where we study local conformal nets of von Neumann algebras (factors) on the circle, and the other is theory of vertex operator algebras. One local conformal net of factors corresponds to one vertex operator algebra, at least conceptually, and each describes one chiral conformal field theory. Since these two mathematical theories are supposed to study the same physical objects, it is natural that the two theories have much in common. For example, both theories have mathematical objects corresponding to the affine Lie algebras and the Virasoro algebra, and also, both have simple current extension, the coset construction, and the orbifold construction as constructions of a new object from a given object. However, the interactions between the two theories have been relatively small, and different people have studied different aspects of the two approaches from different motivations.

The authors of that article then go on to construct conformal nets from “framed” vertex operator algebras and in particular build the AQFT net corresponding to the moonshine vertex operator algebra

Posted at July 10, 2008 1:24 PM UTC

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### Re: Talk on AQFT from FQFT and Applications

Hi Urs,

I’m a bit confused by that sentence,

“…both have been used to classify chiral CFTs. On the AQFT side of life for instance…”

I can’t seem to parse it correctly. Also, did the blogging software cut off your post somehow? There’s a full stop missing at the end of the post, but I get the feeling you were going to go on to say more.

Posted by: Bruce Bartlett on July 10, 2008 5:28 PM | Permalink | Reply to this

### Re: Talk on AQFT from FQFT and Applications

Hi Bruce,

concerning the classification:

One can try to classify vertex operator algebras and one can try to classify 2-d conformal nets of algebras. In both cases one is classifying what is called chiral 2-dimensional conformal field theories.

Much work on the classification of conformal 2d nets has been done by Kawahigashi.

Concerning the end of the entry:

I did originally intend to say more about the asymptotic inclusion business that is mentioned in the last sections of the linked pdf. But then I ran out of time and posted anyway. One thing I can add easily, though, is the full stop. :-)

Posted by: Urs Schreiber on July 10, 2008 8:02 PM | Permalink | Reply to this

### Re: Talk on AQFT from FQFT and Applications

I have now posted the replacement which contains the above examples.

Posted by: Urs Schreiber on September 10, 2008 2:24 PM | Permalink | Reply to this
Read the post Planar Algebras, TFTs with Defects
Weblog: The n-Category Café
Excerpt: I am in Vienna at the ESi attending a few days of the program Operator algebras and CFT. This morning we had a nice talk by Dietmar Bisch on Dietmar Bisch, Paramita Das, Shamindra Kumar Ghosh The planar algebra...
Tracked: September 11, 2008 3:52 PM

### Re: Talk on AQFT from FQFT and Applications

On the relation between vertex operator algebras and conformal nets:

For Monday Sept 15 there is a talk announced here by Sebastiano Carpi titled “From vertex operator algebras to conformal nets”.

I will have to miss this talk and haven’t managed to meet Carpi so far. But from somebody else who has I learned the following:

apparently Carpi and collaborators manage to handle the obvious approach to turning a VOA into a conformal net:

define the algebra of the net assigned to a region to be all possible exponentiated smearings of the VOA fields.

Apparently they can show that without the exponentiation (i.e. using the net of unbounded operators obtained this way) the result is indeed a local net, but there is an unsolved technical subtlety in determining whether this property survives the exponentiation.

I am hoping to be able to find out more details. There does not seem to be anything written up yet on this. Or does anyone know?

Posted by: Urs Schreiber on September 11, 2008 4:29 PM | Permalink | Reply to this

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