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A conformal net is a local net of observables that describes (2-dimensional) conformal field theory in the framework of AQFT.
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Subject to some conditions a vertex operator algebra may be turned into a conformal net and vice versa (Carpi-Kawahigahshi-Longo-Weiner 15).
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).
Standard AQFT references on conformal nets include
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.
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 (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)
Relation to 2-spectral triples is discussed in terms of vertex operator algebras
and in terms of actual local nets in
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)
Articles that discuss the representation theory of conformal nets include