Special and general types
Critical string models
The original Witten genus is an elliptic genus evaluated on the Tate curve, hence a genus for string structure-cobordisms with values in topological modular forms.
Witten obtained this originally as the partition function of the heterotic string in the perturbation theory about constant worldsheet configurations. This is what the Tate curve expresses.
The universal elliptic genus is a morphism from the complexified cobordism ring. Edward Witten argued that the value of the elliptic genus on can be understood as the -equivariant index of a Dirac operator on a loop space .
As such it can be understood as the partition function of an sigma-model SCFT (“the heterotic string”). Formalizations of this construction exist both in AQFT-type (Costello) and in FQFT-type quantum field theory (Stolz-Teichner)
This can be refined to a morphism of ring spectra (Ando-Hopkins-Rezk 10)
\sigma : MString \to tmf
from the Thom spectrum of String bordism to the tmf-spectrum, also called the sigma-orientation, the string orientation of tmf.
The original reference on the string theoretic analytic interpretation of elliptic genera and on the Witten genus is
Elliptic Genera And Quantum Field Theory , Commun.Math.Phys. 109 525 (1987) (Euclid)
The Index Of The Dirac Operator In Loop Space Proc. of Conf. on Elliptic Curves and Modular Forms in Algebraic Topology Princeton (1986).
Rigorous formulations of the relation then appeared in
A simpler proof and many further cases were then discussed by Kefeng Liu.
That it takes a string structure to make the elliptic genus land in modular forms was noticed in
- Don Zagier, Note on the Landweber-Stong elliptic genus 1986 (pdf)
Surveys are in
Gerald Höhn, Complex elliptic genera and -equivariant cobordism theory (pdf)
Anand Dessai, Some geometric properties of the Witten genus (pdf)
Further discussion of the relation to quantum anomalies and the Green-Schwarz mechanism (string structure, string^c structure) is in
Wolfgang Lerche, B. Nilsson, A. Schellekens, N. Warner, Anomaly cancelling terms from the elliptic genus (1987) (pdf)
Qingtao Chen, Fei Han, Weiping Zhang, Generalized Witten Genus and Vanishing Theorems, JDG, 88 (2011) 1-39 (arXiv:1003.2325)
Discussion of the Witten genus via a KO-valued Chern-character on elliptic cohomology is in
- Haynes Miller, The elliptic character and the Witten genus, Contemporary mathematics, volume 96, 1989 (pdf)
The refinement of the Witten genus from values in modular forms to topological modular forms and further to a morphism of E-∞ rings, hence to the string orientation of tmf is due to
Michael Hopkins, Topological modular forms, the Witten Genus, and the theorem of the cube, Proceedings of the International Congress of Mathematics, Zürich 1994 (pdf)
Michael Hopkins, Algebraic topology and modular forms, Proceedings of the ICM, Beijing 2002, vol. 1, 283–309 (arXiv:math/0212397)
Matthew Ando, Michael Hopkins, Charles Rezk, Multiplicative orientations of KO-theory and the spectrum of topological modular forms, 2010 (pdf)
see also remark 1.4 of
- Paul Goerss, Topological modular forms (after Hopkins, Miller and Lurie) (pdf).
and for more on the sigma-orientation see
Discussion of the relation to vertex operator algebras is in
- Chongying Dong, Elliptic Genus and Vertex Operator Algebras, (pdf)
Other references include
Stephan Stolz, A conjecture concerning positive Ricci curvature and the Witten genus, Mathematische Annalen Volume 304, Number 1 (1996),
Matthew Ando, Maria Basterra, The Witten genus and equivariant elliptic cohomology (arXiv:0008192)
Constructions of the sigma-model QFT that is supposed to give the Witten genus are proposed
in terms of chiral differential operators in
in terms of vertex operator algebra in
in terms of FQFT in
and in terms of factorization algebra in
The twisted Witten genus in the present of background gauge field hence for a twisted string structure/string^c structure was considered in
For the moment see the references at string^c structure for more on this.