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

Adam Lindström

TL;DR

The paper analyzes the Dirac-Yang-Mills system on closed spin manifolds, focusing on uncoupled solutions where the connection satisfies the Yang-Mills equation and the Dirac current vanishes. It develops a variational criterion for current vanishing, proves that decoupling is generic via perturbation minimality, and uses index theory to establish existence of uncoupled Dirac-Yang-Mills pairs in several settings, including dimensions where $\hat{A}[M]$ or $\mathrm{ind}(\slashed{D}^+_E)$ are nonzero. It extends construction methods for twisted harmonic spinors to yield explicit uncoupled solutions on Calabi–Yau manifolds and in any dimension admitting parallel spinors, and provides concrete examples on $\mathbb{S}^4$ and $\mathbb{T}^4$ via twistor spinors. These results give new insight into kernel dimensions of twisted Dirac operators, produce broad classes of uncoupled solutions, and connect geometric analysis with gauge theory through conformal invariance and stress-energy considerations.

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.

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

TL;DR

The paper analyzes the Dirac-Yang-Mills system on closed spin manifolds, focusing on uncoupled solutions where the connection satisfies the Yang-Mills equation and the Dirac current vanishes. It develops a variational criterion for current vanishing, proves that decoupling is generic via perturbation minimality, and uses index theory to establish existence of uncoupled Dirac-Yang-Mills pairs in several settings, including dimensions where or are nonzero. It extends construction methods for twisted harmonic spinors to yield explicit uncoupled solutions on Calabi–Yau manifolds and in any dimension admitting parallel spinors, and provides concrete examples on and via twistor spinors. These results give new insight into kernel dimensions of twisted Dirac operators, produce broad classes of uncoupled solutions, and connect geometric analysis with gauge theory through conformal invariance and stress-energy considerations.

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 using the index theorem. Finally we generalize a construction method for twisted harmonic spinors to construct explicit uncoupled solutions on -manifolds admitting twistor spinors and on spin manifolds of any dimension admitting parallel spinors.
Paper Structure (19 sections, 173 equations)

This paper contains 19 sections, 173 equations.

Table of Contents

  1. Introduction
  2. Setting
  3. Main Results
  4. Organization
  5. Conventions and notations
  6. Differential forms
  7. Principal bundles
  8. Spinors and the Dirac sector
  9. The Euler Lagrange Equations
  10. Variations with respect to the metric, Conformal invariance and the stress-energy tensor
  11. Conformal Invariance
  12. The stress-energy tensor
  13. Vanishing of the Current & Characterisation of Decoupling Connections
  14. Implications of non-vanishing current
  15. Existence Results via Index Theory
  16. ...and 4 more sections

Theorems & Definitions (24)

  • proof
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  • proof : Proof of Theorem \ref{['thm: decoupling classification']}
  • proof
  • proof
  • ...and 14 more