nLab 2d SCFT



Quantum field theory

String theory




Super-conformal field theory in dimension d=2d = 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).




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



A basic but detailed exposition focusing on the super-WZW model (and the perspective of 2-spectral triples) is in Fröhlich & Gawedzki (1993).

Textbook account:

Other accounts:

Constructing 2d SCFTs from error-correcting codes and a hint for a relation to tmf, vaguely in line with the lift of the Witten genus to the string orientation of tmf:

further on the resulting elliptic genera:

  • Kohki Kawabata, Shinichiro Yahagi, Elliptic genera from classical error-correcting codes [arXiv:2308.12592]

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

and in terms of conformal nets in

D=2D=2 CFT as functorial field theory

Discussion of D=2 conformal field theory as a functorial field theory, namely as a monoidal functor from a 2d conformal cobordism category to Hilbert spaces:

  • Graeme Segal, The definition of conformal field theory, in: K. Bleuler, M. Werner (eds.), Differential geometrical methods in theoretical physics (Proceedings of Research Workshop, Como 1987), NATO Adv. Sci. Inst., Ser. C: Math. Phys. Sci. 250 Kluwer Acad. Publ., Dordrecht (1988) 165-171 [[doi:10.1007/978-94-015-7809-7]]

and including discussion of modular functors:

  • Graeme Segal, Two-dimensional conformal field theories and modular functors, in: Proceedings of the IXth International Congress on Mathematical Physics, Swansea, 1988, Hilger, Bristol (1989) 22-37.

  • Graeme Segal, The definition of conformal field theory, in: Ulrike Tillmann (ed.), Topology, geometry and quantum field theory , London Math. Soc. Lect. Note Ser. 308, Cambridge University Press (2004) 421-577 [[doi:10.1017/CBO9780511526398.019, pdf, pdf]]

General construction for the case of rational 2d conformal field theory is given by the

See also:

A different but closely analogous development for chiral 2d CFT (vertex operator algebras, see there for more):

Discussion of the case of Liouville theory:

Early suggestions to refine this to an extended 2-functorial construction:

A step towards generalization to 2d super-conformal field theory:

Discussion of 2-functorial chiral 2d CFT:

Elliptic genera as super pp-brane partition functions

The interpretation of elliptic genera (especially the Witten genus) as the partition function of a 2d superconformal field theory (or Landau-Ginzburg model) – and especially of the heterotic string (“H-string”) or type II superstring worldsheet theory has precursors in

and then strictly originates with:

Review in:

With emphasis on orbifold CFTs:


Via super vertex operator algebra

Formulation via super vertex operator algebras:

and for the topologically twisted 2d (2,0)-superconformal QFT (the heterotic string with enhanced supersymmetry) via sheaves of vertex operator algebras in

based on chiral differential operators:

In relation to error-correcting codes:

  • Kohki Kawabata, Shinichiro Yahagi, Elliptic genera from classical error-correcting codes [[arXiv:2308.12592]]
Via Dirac-Ramond operators on free loop space

Tentative interpretation as indices of Dirac-Ramond operators as would-be Dirac operators on smooth loop space:

Via conformal nets

Tentative formulation via conformal nets:

Conjectural interpretation in tmf-cohomology

The resulting suggestion that, roughly, deformation-classes (concordance classes) of 2d SCFTs with target space XX are the generalized cohomology of XX with coefficients in the spectrum of topological modular forms (tmf):

and the more explicit suggestion that, under this identification, the Chern-Dold character from tmf to modular forms, sends a 2d SCFT to its partition function/elliptic genus/supersymmetric index:

This perspective is also picked up in Gukov, Pei, Putrov & Vafa 18.

Discussion of the 2d SCFTs (namely supersymmetric SU(2)-WZW-models) conjecturally corresponding, under this conjectural identification, to the elements of /24\mathbb{Z}/24 \simeq tmf 3(*)=π 3(tmf) tmf^{-3}(\ast) = \pi_3(tmf) \simeq π 3(𝕊)\pi_3(\mathbb{S}) (the third stable homotopy group of spheres):

Discussion properly via (2,1)-dimensional Euclidean field theory:

See also:

Occurrences in string theory

H-string elliptic genus

Further on the elliptic genus of the heterotic string being the Witten genus:

The interpretation of equivariant elliptic genera as partition functions of parametrized WZW models in heterotic string theory:

Proposals on physics aspects of lifting the Witten genus to topological modular forms:

M5-brane elliptic genus

On the M5-brane elliptic genus:

A 2d SCFT argued to describe the KK-compactification of the M5-brane on a 4-manifold (specifically: a complex surface) originates with

Discussion of the resulting elliptic genus (2d SCFT partition function) originates with:

Further discussion in:

M-string elliptic genus

On the elliptic genus of M-strings inside M5-branes:

E-string elliptic genus

On the elliptic genus of E-strings as wrapped M5-branes:

On the elliptic genus of E-strings as M2-branes ending on M5-branes:

Last revised on August 25, 2023 at 07:21:03. See the history of this page for a list of all contributions to it.