nLab elliptic cohomology -- references

Elliptic cohomology


The concept of elliptic cohomology originates around:

and in the universal guise of topological modular forms in:

  • Michael Hopkins, Algebraic topology and modular forms in Proceedings of the International Congress of Mathematicians, Vol. I (Beijing, 2002), pages 291–317, Beijing, 2002. Higher Ed. Press (arXiv:math/0212397)


Textbook accounts:

Equivariant elliptic cohomology

On equivariant elliptic cohomology and positive energy representations of loop groups:

Relation to Kac-Weyl characters of loop group representations

The case of twisted ad-equivariant Tate K-theory:

See also:

Via derived E E_\infty-geometry

Formulation of (equivariant) elliptic cohomology in derived algebraic geometry/E-∞ geometry (derived elliptic curves):

Elliptic genera


The general concept of elliptic genus originates with:

Early development:


  • Peter Landweber, Elliptic genera: An introductory overview In: P. Landweber (eds.) Elliptic Curves and Modular Forms in Algebraic Topology, Lecture Notes in Mathematics, vol 1326. Springer (1988) (doi:10.1007/BFb0078036)

  • Kefeng Liu, Modular forms and topology, Proc. of the AMS Conference on the Monster and Related Topics, Contemporary Math. (1996) (pdf, pdf, doi:10.1090/conm/193)

  • Serge Ochanine, What is… an elliptic genus?, Notices of the AMS, volume 56, number 6 (2009) (pdf)

The Stolz conjecture on the Witten genus:

The Jacobi form-property of the Witten genus:

  • Matthew Ando, Christopher French, Nora Ganter, The Jacobi orientation and the two-variable elliptic genus, Algebraic and Geometric Topology 8 (2008) p. 493-539 (pdf)

The identification of elliptic genera, via fiber integration/Pontrjagin-Thom collapse, as complex orientations of elliptic cohomology (sigma-orientation/string-orientation of tmf/spin-orientation of Tate K-theory):

For the Ochanine genus:

Equivariant elliptic genera

Genera in equivariant elliptic cohomology and the rigidity theorem for equivariant elliptic genera:

The statement, with a string theory-motivated plausibility argument, is due to Witten 87.

The first proof was given in:

Reviewed in:

  • Raoul Bott, On the Fixed Point Formula and the Rigidity Theorems of Witten, Lectures at Cargése 1987. In: ’t Hooft G., Jaffe A., Mack G., Mitter P.K., Stora R. (eds) Nonperturbative Quantum Field Theory. NATO ASI Series (Series B: Physics), vol 185. Springer (1988) (doi:10.1007/978-1-4613-0729-7_2)

Further proofs and constructions:

On manifolds with SU(2)-action:

Twisted elliptic genera

Discussion of elliptic genera twisted by a gauge bundle, i.e. for string^c structure):

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:


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 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):

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:

Speculations 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 July 3, 2022 at 18:05:42. See the history of this page for a list of all contributions to it.