nLab elliptic cohomology -- references

Elliptic cohomology

General

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)

Surveys:

Textbook accounts:

Equivariant elliptic cohomology

Relation to Kac-Weyl characters of loop group representations

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

Via derived $E_\infty$-geometry

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

Elliptic genera

General

The general concept of elliptic genus originates with:

Early development:

Review:

• 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)

For the Ochanine genus:

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:

• Anand Dessai, The Witten genus and $S^3$-actions on manifolds, 1994 (pdf, pdf)

Twisted elliptic genera

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

Elliptic genera as super $p$-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:

Formulations

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 $X$ are the generalized cohomology of $X$ 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 $\mathbb{Z}/24$ $\simeq$ $tmf^{-3}(\ast) = \pi_3(tmf)$ $\simeq$ $\pi_3(\mathbb{S})$ (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:

• J. A. Minahan, D. Nemeschansky, Cumrun Vafa, N. P. Warner, E-Strings and $N=4$ Topological Yang-Mills Theories, Nucl. Phys. B527 (1998) 581-623 (arXiv:hep-th/9802168)

• Wenhe Cai, Min-xin Huang, Kaiwen Sun, On the Elliptic Genus of Three E-strings and Heterotic Strings, J. High Energ. Phys. 2015, 79 (2015). (arXiv:1411.2801, doi:10.1007/JHEP01(2015)079)

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