cohomology

# Contents

## Idea

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

The universal elliptic genus is a morphism ${\varphi }_{2}:{\Omega }_{•}^{\mathrm{SO}}\otimes ℂ\to {M}_{•}\left({\Gamma }_{1}\left(2\right)\right)\simeq ℂ\left[\delta ,ϵ\right]$ 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

• 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

• 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

in terms of vertex operator algebra in

in terms of FQFT in

and in terms of factorization algebra in

Revised on May 14, 2013 20:19:57 by Urs Schreiber (131.174.41.96)