Chiral Algebras of (0,2) Models

TL;DR

This paper investigates the chiral algebras of (0,2) supersymmetric sigma models, revealing their complex structure and how instantons can lead to supersymmetry breaking, providing insights into the H"ohn-Stolz conjecture.

Contribution

It demonstrates how instantons can drastically deform or annihilate the chiral algebra in (0,2) models, linking physical phenomena to mathematical conjectures.

Findings

Perturbative chiral algebras are described by sheaves of chiral differential operators.

Instantons can completely annihilate the chiral algebra, causing supersymmetry breaking.

Supports the H"ohn-Stolz conjecture by relating loop space harmonic spinors to supersymmetry.

Abstract

We explore two-dimensional sigma models with (0,2) supersymmetry through their chiral algebras. Perturbatively, the chiral algebras of (0,2) models have a rich infinite-dimensional structure described by the cohomology of a sheaf of chiral differential operators. Nonperturbatively, instantons can deform this structure drastically. We show that under some conditions they even annihilate the whole algebra, thereby triggering the spontaneous breaking of supersymmetry. For a certain class of K\"ahler manifolds, this suggests that there are no harmonic spinors on their loop spaces and gives a physical proof of the H\"ohn-Stolz conjecture.