nLab
Stolz conjecture

Context

Index theory

Riemannian geometry

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

String theory

Contents

Idea

The Stolz conjecture (also: Stolz-Höhn conjecture) due to (Stolz 96) asserts that if XX 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 XX, 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 X\mathcal{L} X obtained by integrating the Ricci curvature on XX along loops (transgression). Therefore in this reasoning a positive Ricci curvature of XX would imply a positive scalar curvature of the smooth loop space X\mathcal{L}X, thus a vanishing of the index of the “Dirac operator on 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 perturbation theory. Working equivalently with the 2d (2,0)-superconformal QFT in perturbation theory, one may say something. A decent review is in (Yagi 10).

References

The original article is

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

Reviews include

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

Last revised on April 21, 2018 at 07:47:37. See the history of this page for a list of all contributions to it.