nLab
2d SCFT

Idea

Super-conformal field theory in dimension $d = 2$, locally given by a super vertex operator algebra.

For central charge 15 this is the worldsheet theory of the superstring.

May be regarded as a “2-spectral triple” (see there for more), the 2-dimensional generalization of spectral triples describing the quantum mechanics of spinning particles (super-particles).

Examples

• superstringsigma-model

  • spinning stringsuperstring

  • heterotic string, type II superstring

  • Gepner model

  • (2,1)-dimensional Euclidean field theories and tmf

Properties

Classification

See at supersymmetry – Classification – Superconformal algebra – In dimension 2.

Related concepts

• perturbative string theory vacuum

References

General

A basic but detailed exposition focusing on the super-WZW model (and the perspective of 2-spectral triples) is in Fröhlich Gawedzki 93

Other accounts include

• Lance Dixon, Paul Ginsparg, and Jeffrey Harvey, $\hat c = 1$ Superconformal field theory (pdf)

• Yasuyuki Kawahigashi, $N = 2$ Superconformal Field Theory

  and Operator Algebras_ (pdf)

Relation to 2-spectral triples

Discussion of 2d SCFTs as a higher analog of spectral triples (“2-spectral triples”, see there for more) is in terms of vertex operator algebras in

• Jürg Fröhlich, Krzysztof Gawędzki, Conformal Field Theory and Geometry of Strings, extended lecture notes for lecture given at the Mathematical Quantum Theory Conference, Vancouver, Canada, August 4-8 (arXiv:hep-th/9310187)

• Yan Soibelman, Collapsing CFTs, spaces with non-negative Ricci curvature and nc-geometry , in Hisham Sati, Urs Schreiber (eds.), Mathematical Foundations of Quantum Field and Perturbative String Theory, Proceedings of Symposia in Pure Mathematics, AMS (2001)

and in terms of conformal 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)