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Spherically symmetric Dirac-Yang-Mills pairs on Riemannian 3-manifolds

Adam Lindström, Marko Sobak

TL;DR

This work constructs spherically symmetric Dirac-Yang-Mills pairs with structure group $SU(2)$ on 3-manifolds of the form $M = N\times\mathbb{S}^2$ by developing an invariant bundle framework and an invariant spinor/gauge ansatz. The authors reduce the DYM equations to a tractable system of ordinary differential equations using a polar decomposition and a detailed invariant-spinor analysis, yielding explicit global solutions and a rich family of coupled, periodically-reducing configurations. They identify regimes with constant spinor-norm $\rho$ (and two distinct regimes $\delta_0=0$ and $\delta_0>0$), obtaining explicit solutions, singular behavior, and conditions under which solutions descend to closed manifolds like $\mathbb{S}^1\times\mathbb{S}^2$. The results provide the first known examples of coupled DYM pairs on closed Riemannian spin manifolds and demonstrate the feasibility of generating entire families of periodic DYM configurations by a carefully chosen spherically symmetric ansatz. The constructions have potential implications for geometric models of gauge-fermion interactions and offer a concrete setting to explore variational and topological aspects of DYM systems on curved spaces.

Abstract

In this paper we construct examples of spherically symmetric Dirac-Yang-Mills pairs on Riemannian 3-manifolds with the structure group SU(2). This approach yields coupled solutions (i.e. the connection is not a Yang-Mills connection) and among them are solutions on S^1(r_1) x S^2(r_2) for certain radii r_1 and r_2. These are, to the authors' best knowledge, the first examples of coupled Dirac-Yang-Mills pairs on a closed Riemannian spin manifold.

Spherically symmetric Dirac-Yang-Mills pairs on Riemannian 3-manifolds

TL;DR

This work constructs spherically symmetric Dirac-Yang-Mills pairs with structure group on 3-manifolds of the form by developing an invariant bundle framework and an invariant spinor/gauge ansatz. The authors reduce the DYM equations to a tractable system of ordinary differential equations using a polar decomposition and a detailed invariant-spinor analysis, yielding explicit global solutions and a rich family of coupled, periodically-reducing configurations. They identify regimes with constant spinor-norm (and two distinct regimes and ), obtaining explicit solutions, singular behavior, and conditions under which solutions descend to closed manifolds like . The results provide the first known examples of coupled DYM pairs on closed Riemannian spin manifolds and demonstrate the feasibility of generating entire families of periodic DYM configurations by a carefully chosen spherically symmetric ansatz. The constructions have potential implications for geometric models of gauge-fermion interactions and offer a concrete setting to explore variational and topological aspects of DYM systems on curved spaces.

Abstract

In this paper we construct examples of spherically symmetric Dirac-Yang-Mills pairs on Riemannian 3-manifolds with the structure group SU(2). This approach yields coupled solutions (i.e. the connection is not a Yang-Mills connection) and among them are solutions on S^1(r_1) x S^2(r_2) for certain radii r_1 and r_2. These are, to the authors' best knowledge, the first examples of coupled Dirac-Yang-Mills pairs on a closed Riemannian spin manifold.
Paper Structure (26 sections, 7 theorems, 171 equations, 3 figures, 2 tables)

This paper contains 26 sections, 7 theorems, 171 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. SU(2)-ansatz in spherical symmetry
  3. General theory of invariant bundles, connections, and sections
  4. Some important matrices and maps
  5. Base manifold
  6. Invariant principal bundles
  7. Invariant connections
  8. Spin structure
  9. Invariant twisted spinors
  10. Dirac-Yang-Mills equations
  11. Summary of the ansatz
  12. The ODE system
  13. Invariances, constraints, and initial data
  14. Polar form
  15. Solutions with constant rho
  16. ...and 11 more sections

Key Result

Theorem 1.1

Let $N = \mathbb{R}$ or $N = \mathbb{S}^1$. Then in both cases, there exist families of spherically symmetric Riemannian metrics $g$ on $N\times \mathbb{S}^2$ admitting globally defined coupled spherically symmetric Dirac-Yang-Mills pairs with structure group $\mathbf{SU}(2)$.

Figures (3)

  • Figure 1: Solutions $W(s)$ of \ref{['eq-W-delta=0-rho<1']}.
  • Figure 2: Phase portrait of \ref{['eq-P-W-system-delta0>0-rho0<1']}.
  • Figure 4: Phase portrait of \ref{['eq-autonomous-kappa-pos']} for $\lambda=1$, $\beta=\sqrt{7/8}$, $\alpha=2$. The green (resp. red) curve represents the stable (resp. unstable) manifold at the fixed point $(\alpha,0)$. The blue curves are the global solutions lying in the region $D_+ \cup D_-$. The brown curves are some prototypical orbits that do not lie entirely in $D_+ \cup D_-$, they all have $x \to 0$ at some finite $s$ (either forwards or backwards or both).

Theorems & Definitions (16)

  • Theorem 1.1
  • Remark 2.1
  • Remark 2.2
  • Remark 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3: Polar form
  • Remark 3.4
  • Proposition 4.1: The case $\delta_0 = 0$
  • Remark 4.2
  • ...and 6 more