Uncoupled Dirac-Yang-Mills Pairs on Closed Riemannian Spin Manifolds

TL;DR

This paper investigates uncoupled solutions to Dirac-Yang-Mills equations on closed spin manifolds, classifying such solutions, analyzing their properties, and constructing explicit examples, especially in four dimensions.

Contribution

It provides a classification of connection forms with vanishing Dirac current on harmonic spinors and constructs explicit uncoupled solutions using advanced geometric methods.

Findings

Open and dense subset of connections satisfy the vanishing Dirac current condition.

Existence of uncoupled solutions in dimension 4 established via index theorem.

Explicit constructions of solutions on manifolds with special spinors.

Abstract

We study the Dirac-Yang-Mills equations on closed spin manifolds with a focus on uncoupled solutions, i.e. solutions for which the connection form satisfies the Yang-Mills equation. Such solutions require the Dirac current, a quadratic form on the spinor bundle, to vanish. We study the condition that this current vanishes on all harmonic spinors using perturbation theory and obtain a classification of the connection forms for which this holds, which we show contains an open and dense subset of connections. This has several implications for the generic dimension of the kernel of the Dirac operator. We further establish existence results for uncoupled solutions, in particular in dimension $4$ using the index theorem. Finally we generalize a construction method for twisted harmonic spinors to construct explicit uncoupled solutions on $4$-manifolds admitting twistor spinors and on spin manifolds of any dimension admitting parallel spinors.