Contents

# Contents

## Idea

A conformal net is a local net of observables that describes (2-dimensional) conformal field theory in the framework of algebraic quantum field theory.

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## Properties

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### Relation to vertex operator algebras

Subject to some conditions a vertex operator algebra may be turned into a conformal net and vice versa (Carpi-Kawahigahshi-Longo-Weiner 15, Carpi 16) also (Gui 18).

### Relation to spectral triples

Superconformal nets encoding 2d SCFTs may be related to local nets of spectral triples (CHKL 09) (an incarnation of “2-spectral triples”, see there for more).

## Examples

### Free fermion

$Fer_n(I) \coloneqq Cl(L^2(I, \mathbb{R}^n \otimes S))$

## References

### Fundamentals

Standard AQFT references on conformal nets include

and lots of others…

A somewhat novel take on conformal nets is presented and studied in

where conformal nets are arranged into a tricategory with symmetric monoidal structure that is a delooping of the bicategory of von Neumann algebras and bimodules between these. Among other things, this work connects the AQFT notion of conformal nets with the FQFT notion of cobordism representations.

More on this in

Relation to vertex operator algebras is discussed in

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

Relation of the corresponding ribbon categories:

Relation to 2-spectral triples is discussed in terms of vertex operator algebras

and in terms of actual local nets in

### Representation theory

Articles that discuss the representation theory of conformal nets include

Last revised on December 12, 2018 at 12:07:05. See the history of this page for a list of all contributions to it.