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 mechnanics of spinning particles (super-particles).

Examples

• superstring sigma-model

  • spinning string superstring

  • 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

Table of branes appearing in supergravity/string theory (for classification see at brane scan).

brane	in supergravity	charged under gauge field	has worldvolume theory
black brane	supergravity	higher gauge field	SCFT
D-brane	type II	RR-field	super Yang-Mills theory
$(D = 2n)$	type IIA	$\,$	$\,$
D0-brane	$\,$	$\,$	BFSS matrix model
D2-brane	$\,$	$\,$	$\,$
D4-brane	$\,$	$\,$	D=5 super Yang-Mills theory with Khovanov homology observables
D6-brane	$\,$	$\,$	D=7 super Yang-Mills theory
D8-brane	$\,$	$\,$
$(D = 2n+1)$	type IIB	$\,$	$\,$
D(-1)-brane	$\,$	$\,$	$\,$
D1-brane	$\,$	$\,$	2d CFT with BH entropy
D3-brane	$\,$	$\,$	N=4 D=4 super Yang-Mills theory
D5-brane	$\,$	$\,$	$\,$
D7-brane	$\,$	$\,$	$\,$
D9-brane	$\,$	$\,$	$\,$
(p,q)-string	$\,$	$\,$	$\,$
(D25-brane)	(bosonic string theory)
NS-brane	type I, II, heterotic	circle n-connection	$\,$
string	$\,$	B2-field	2d SCFT
NS5-brane	$\,$	B6-field	little string theory
D-brane for topological string			$\,$
A-brane			$\,$
B-brane			$\,$
M-brane	11D SuGra/M-theory	circle n-connection	$\,$
M2-brane	$\,$	C3-field	ABJM theory, BLG model
M5-brane	$\,$	C6-field	6d (2,0)-superconformal QFT
M9-brane/O9-plane			heterotic string theory
M-wave
topological M2-brane	topological M-theory	C3-field on G2-manifold
topological M5-brane	$\,$	C6-field on G2-manifold
solitons on M5-brane	6d (2,0)-superconformal QFT
self-dual string		self-dual B-field
3-brane in 6d

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)