# nLab Witten genus

### Context

#### Index theory

index theory, KK-theory

noncommutative stable homotopy theory

partition function

genus, orientation in generalized cohomology

## Definitions

operator K-theory

K-homology

cohomology

# Contents

## Idea

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 ${\varphi }_{2}:{\Omega }_{•}^{\mathrm{SO}}\otimes ℂ\to {M}_{•}\left({\Gamma }_{1}\left(2\right)\right)\simeq ℂ\left[\delta ,ϵ\right]$ 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 10)

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

$d$partition function in $d$-dimensional QFTindex/genus in cohomology theory
0push-forward in ordinary cohomology: integration of differential forms
1spinning particleK-theory index
endpoint of 2d Poisson-Chern-Simons theory stringspace of quantum states of boundary phase space/Poisson manifold
endpoint of type II superstringD-brane charge
2type II superstringelliptic genus
heterotic stringWitten genus

## References

### General

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

• Clifford Taubes, ${S}^{1}$ actions and elliptic genera, Comm. Math. Phys., 122(3):455–526, 1989.

• Raoul Bott, Clifford Taubes, On the rigidity theorems of Witten, J. of the Amer. Math. Soc., 2, 1989.

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 ${S}^{1}$-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

• 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

### Twisted case

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.

Revised on November 6, 2013 11:29:16 by Urs Schreiber (145.116.129.122)