nLab
Witten genus

Idea

The Witten genus is a genus for string structure-cobordisms with values in topological modular forms.

The universal elliptic genus is a morphism ${\varphi }_2:{\Omega }_{•}^{\mathrm{SO}}\otimes ℂ\to M_{•}({\Gamma }_1(2))\simeq ℂ[\delta ,ϵ]$ from the complexified cobordism ring. Edward Witten argued that the value of the elliptic genus on $X$ can be understood as the $S^1$-equivariant index of a Dirac operator on a loop space $ℒX$.

As such it can be understood as the partition function of an $N=1$ $d=2$ 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)

$$\sigma :\mathrm{MString}\to \mathrm{tmf}$$\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.

References

The original reference on the Witten genus is

• Edward Witten,

  • 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).

Surveys are in

• Gerald Höhn, Complex elliptic genera and $S^1$-equivariant cobordism theory (pdf)

• Anand Dessai, Some geometric properties of the Witten genus (pdf)

The discussion of the refinement of the Witten genus to a morphism of ring spectra, to the string orientation of tmf is in

• Matthew Ando, Mike Hopkins, Charles Rezk, Multiplicative orientations of KO-theory and the spectrum of topological modular forms (pdf)

see also remark 1.4 of

• Paul Goerss, Topological modular forms (after Hopkins, Miller and Lurie) (pdf).

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

• V. Gorbounov, F. Malikov and V. Schechtman, Gerbes of chiral differential operators, Math. Res. Lett. 7(1), 55–66 (2000) (arXiv:math/9906117, arXiv:math/0003170, arXiv:math/0005201)

in terms of vertex operator algebra in

• Pokman Cheung, The Witten genus and vertex algebras (arXiv:0811.1418)

in terms of FQFT in

• Stephan Stolz, Peter Teichner, Supersymmetric field theories and generalized cohomology in Mathematical Foundations of Quantum Field and Perturbative String Theory (arXiv:1108.0189)

and in terms of factorization algebra in

• Kevin Costello, A geometric construction of the Witten genus, II (arXiv:1112.0816)