noncommutative topology, noncommutative geometry
noncommutative stable homotopy theory
genus, orientation in generalized cohomology
group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
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.
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.
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).
The original article is
Review:
Discussion in the context of perturbation theory for the 2d (2,0)-superconformal QFT includes
Last revised on November 17, 2020 at 17:42:29. See the history of this page for a list of all contributions to it.