nLab conformal net

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

(…)

• 3-category of fermionic 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)

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

Related concepts

• conformal structure, CFT

• conformal bootstrap

References

Fundamentals

Standard AQFT references on conformal nets include

• Roberto Longo, Conformal Subnets and Intermediate Subfactors (arXiv:0102196)

and lots of others…

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

• Arthur Bartels, Chris Douglas, Andre Henriques, Conformal nets and local field theory (arXiv:0912.5307)

• Arthur Bartels, Chris Douglas, Andre Henriques, Conformal nets I: coordinate-free nets (arXiv:1302.2604)

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

• Chris Douglas, André Henriques, Topological modular forms and conformal nets , in Branislav Jurčo, Hisham Sati, Urs Schreiber (eds.), Mathematical Foundations of Quantum Field and Perturbative String Theory

• Chris Douglas, André Henriques, Geometric string structures (pdf)

Relation to vertex operator algebras is discussed in

• Sebastiano Carpi, Yasuyuki Kawahigashi, Roberto Longo, Mihály Weiner, From vertex operator algebras to conformal nets and back, Memoirs of the American Mathematical Society, Volume 254, Number 1213 2018; (doi:10.1090/memo/1213, arXiv:1503.01260)

• Sebastiano Carpi, Operator algebras and vertex operator algebras, Contribution to the Proceedings of the 14th Marcel Grossmann Meeting - MG14 (Rome, 2015) (arXiv:1603.06742)

and for super vertex operator algebras:

• Sebastiano Carpi, Tiziano Gaudio, Robin Hillier, From vertex operator superalgebras to graded-local conformal nets and back [arXiv:2304.14263]

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:

• Bin Gui, Categorical extensions of conformal nets (arXiv:1812.04470)

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

• Yan Soibelman, Collapsing CFTs, spaces with non-negative Ricci curvature and nc-geometry , in Branislav Jurčo, Hisham Sati, Urs Schreiber (eds.), Mathematical Foundations of Quantum Field and Perturbative String Theory

and in terms of actual local nets in

• Sebastiano Carpi, Robin Hillier, Yasuyuki Kawahigashi, Roberto Longo, Spectral triples and the super-Virasoro algebra, Commun.Math.Phys.295:71-97 (2010) (arXiv:0811.4128)

Classification

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

• Yasuyuki Kawahigashi, Roberto Longo, Classification of Two-dimensional Local Conformal Nets with $c \lt 1$ and 2-cohomology Vanishing for Tensor Categories (arXiv)

• Yasuyuki Kawahigashi, Roberto Longo, Classification of Local Conformal Nets. Case $c \lt 1$ (arXiv)

Representation theory

Articles that discuss the representation theory of conformal nets include

• Sebastiano Carpi, Roberto Conti, Robin Hillier, Mihaly Weiner, Representations of Conformal Nets, Universal $C^\ast$-Algebras and K-Theory (arXiv:1202.2543)