### 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

## 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.