nLab
2d SCFT

Contents

Context

Quantum field theory

String theory

Super-Geometry

Contents

Idea

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).

Examples

Properties

Classification

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

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

A hint for a relation to tmf, vaguely in line with the lift of the Witten genus to the string orientation of tmf:

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

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 – originates with:

Review in:

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:

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 functorial QFT

Tentative formulation via functorial quantum field theory ((2,1)-dimensional Euclidean field theories and tmf):

Via conformal nets

Tentative formulation via conformal nets:

Occurrences in string theory

H-string elliptic genus

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

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 November 17, 2020 at 10:09:59. See the history of this page for a list of all contributions to it.