nLab Stolz conjecture

Idea

The Stolz conjecture (also: Stolz-Höhn conjecture) due to (Stolz 96) asserts that if $X$ is a closed manifold with String structure which furthermore admits a Riemannian metric with positive Ricci curvature, then its Witten genus vanishes.

See (Dessai) for a review.

Motivation

One part of the reasoning that motivates the conjecture is the idea that string geometry should be a “delooping” of spin geometry, and that the Witten genus is roughly like the index of a Dirac operator on loop space. Now for spin geometry the Lichnerowicz formula implies that for positive scalar curvature there are no harmonic spinors on a Riemannian manifold $X$, and hence that the index of the Dirac operator vanishes. One might then expect that there is a sensible concept of scalar curvature of smooth loop space $\mathcal{L} X$ obtained by integrating the Ricci curvature on $X$ along loops (transgression). Therefore, in this reasoning, a positive Ricci curvature of $X$ would imply a positive scalar curvature of the smooth loop space $\mathcal{L}X$, thus a vanishing of the index of the “Dirac operator on smooth loop space”, hence a vanishing of the Witten genus.

Evidence

From perturbation theory

The Witten genus is by construction the partition function of the heterotic string, in the large volume limit. One may further approximate this in perturbative quantum field theory. Working equivalently with the 2d (2,0)-superconformal QFT in perturbation theory, one may say something. A decent review is in (Yagi 10).

Related concepts

• p-curvature

• elliptic cohomology

References

The original article is

• Stephan Stolz, A conjecture concerning positive Ricci curvature and the Witten genus, Mathematische Annalen Volume 304, Number 1 (1996) (doi:10.1007/BF01446319)

Review:

• Anand Dessai, Some geometric properties of the Witten genus, in: Christian Ausoni, Kathryn Hess, Jérôme Scherer (eds.) Alpine Perspectives on Algebraic Topology, Contemporary Mathematics 504 (2009) (pdf, pdf,

  doi:10.1090/conm/504)

Discussion in the context of perturbation theory for the 2d (2,0)-superconformal QFT includes

• Junya Yagi, Vanishing Chiral Algebras and Höhn-Stolz Conjecture (arXiv:1002.0028)