nLab Dirac-Ramond operator



Higher spin geometry

String theory



The 0-mode of the supercharge in a 2d SCFT (an operator in a (sheaf of) super vertex operator algebra) behaves like a higher dimensional analog of the operator DD in a spectral triple, hence like the supercharge in supersymmetric quantum mechanics (see the references there).

Specifically for a sigma-model 2d SCFT induced from some target space geometry – such as the worldsheet-quantum field theory of a superstring propagating on that target spacetimes – the Dirac-Ramond operator is a higher analogue of a Dirac operator on that target spacetime (roughly like what one would expect of a Dirac operator on a smooth loop space). This is called the Dirac-Ramond operator (Ramond 71).

The index of the large volume limit of the Dirac-Ramond operator is what is now known as the Witten genus (but in fact the original article Alvarez-Killingback-Mangano-Windey 87 appeared independently and almost in parallel of Witten’s discussion).

partition functions in quantum field theory as indices/genera/orientations in generalized cohomology theory:

ddpartition function in dd-dimensional QFTsuperchargeindex in cohomology theorygenuslogarithmic coefficients of Hirzebruch series
0push-forward in ordinary cohomology: integration of differential formsorientation
1spinning particleDirac operatorKO-theory indexA-hat genusBernoulli numbersAtiyah-Bott-Shapiro orientation MSpinKOM Spin \to KO
endpoint of 2d Poisson-Chern-Simons theory stringSpin^c Dirac operator twisted by prequantum line bundlespace of quantum states of boundary phase space/Poisson manifoldTodd genusBernoulli numbersAtiyah-Bott-Shapiro orientation MSpin cKUM Spin^c \to KU
endpoint of type II superstringSpin^c Dirac operator twisted by Chan-Paton gauge fieldD-brane chargeTodd genusBernoulli numbersAtiyah-Bott-Shapiro orientation MSpin cKUM Spin^c \to KU
2type II superstringDirac-Ramond operatorsuperstring partition function in NS-R sectorOchanine elliptic genusSO orientation of elliptic cohomology
heterotic superstringDirac-Ramond operatorsuperstring partition functionWitten genusEisenstein seriesstring orientation of tmf
self-dual stringM5-brane charge
3w4-orientation of EO(2)-theory



The original article, in the context of the superstring of string theory

The Dirac-Ramond operator originates with the early beginning of superstring models, when they were still called spinning strings – see there for more references.

The concept gained more attention in pure mathematics when it was found that the large volume limit of its index, when properly construed, is a universal elliptic genus, now known as the Witten genus. See there for more references.

Articles that explicitly consider the Dirac-Ramond operator in this context:

  • Orlando Alvarez, T. P. Killingback, Michelangelo Mangano, and Paul Windey, The Dirac-Ramond operator in string theory and loop space index theorems, Nuclear Phys. B Proc. Suppl., 1A:189–215, 1987. Nonperturbative methods in field theory (Irvine, CA, 1987)., also: Comm. Math. Phys. Volume 111, Number 1 (1987), 1-160 (Ecudid)

  • Gregory Landweber, Dirac operators on loop space PhD thesis (Harvard 1999) (pdf)

  • Orlando Alvarez, Paul Windey, Analytic index for a family of Dirac-Ramond operators, Proc. Natl. Acad. Sci. USA, 107(11):4845–4850, 2010

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 January 11, 2021 at 09:01:32. See the history of this page for a list of all contributions to it.