nLab conformal net

Contents

Context

AQFT

algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)

Introduction

Concepts

field theory:

Lagrangian field theory

quantization

quantum mechanical system, quantum probability

free field quantization

gauge theories

interacting field quantization

renormalization

Theorems

States and observables

Operator algebra

Local QFT

Perturbative QFT

Physics

physics, mathematical physics, philosophy of physics

Surveys, textbooks and lecture notes


theory (physics), model (physics)

experiment, measurement, computable physics

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.

Definition

(…)

Properties

3-Category of conformal nets

(…)

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

(DH, section 4)

Fer n(I)Cl(L 2(I, nS)) 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

and for super vertex operator algebras:

See also

  • 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

Classification

Representation theory

Articles that discuss the representation theory of conformal nets include

Last revised on April 28, 2023 at 09:42:08. See the history of this page for a list of all contributions to it.