Vanishing Chiral Algebras and H\"ohn-Stolz Conjecture

TL;DR

This paper explores how the vanishing of chiral algebras in certain supersymmetric sigma models impacts the H"ohn-Stolz conjecture related to the Witten genus, linking quantum field theory and differential geometry.

Contribution

It demonstrates the phenomenon of chiral algebra vanishing due to instantons and discusses its implications for the H"ohn-Stolz conjecture on the Witten genus.

Findings

Chiral algebras can vanish in (0,2) supersymmetric models due to instantons.

Vanishing of chiral algebras affects the understanding of the Witten genus.

Implications for the H"ohn-Stolz conjecture are discussed.

Abstract

Given a two-dimensional quantum field theory with (0,2) supersymmetry, one can construct a chiral (or vertex) algebra. The chiral algebra of a (0,2) supersymmetric sigma model is, perturbatively, the cohomology of a sheaf of chiral differential operators on a string K\"ahler manifold. However, it vanishes in some cases when instantons are taken into account. I will discuss the implication of this phenomenon for the H\"ohn-Stolz conjecture on the Witten genus.