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arxiv: 2602.09122 · v2 · submitted 2026-02-09 · 🧮 math.DG · math-ph· math.MP

Spherically symmetric Dirac-Yang-Mills pairs on Riemannian manifolds

Adam Lindstr\"om , Marko Sobak This is my paper

Pith reviewed 2026-05-16 05:14 UTC · model grok-4.3

classification 🧮 math.DG math-phmath.MP
keywords Dirac-Yang-Mills pairsspherically symmetricRiemannian manifoldsSU(2)product manifoldscoupled solutionsclosed spin manifolds
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The pith

Spherically symmetric reductions yield the first coupled Dirac-Yang-Mills pairs on closed Riemannian spin manifolds.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper constructs spherically symmetric Dirac-Yang-Mills pairs on 3-manifolds equipped with an SU(2) structure group. These include coupled solutions on the product manifold S^1(r1) times S^2(r2) for certain radii where the reduced system admits non-trivial solutions. The construction is then extended to induce similar pairs on Riemannian products in arbitrary dimension. A sympathetic reader would care because the examples supply the first known instances of such interacting Dirac and Yang-Mills fields on compact closed spin manifolds.

Core claim

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. We further show how to use such pairs to induce Dirac-Yang-Mills pairs on Riemannian products of arbitrary dimension. These are, to the authors' best knowledge, the first examples of coupled Dirac-Yang-Mills pairs on a closed Riemannian spin manifold.

What carries the argument

Reduction of the Dirac-Yang-Mills equations under spherical symmetry to an ODE system on the product S^1(r1) x S^2(r2) that admits non-trivial coupled solutions for tuned radii satisfying the boundary conditions.

Load-bearing premise

There exist specific radii r1 and r2 such that the reduced ODE system admits non-trivial coupled solutions satisfying the boundary conditions on the product manifold.

What would settle it

Demonstration that the reduced ODE system on S^1(r1) x S^2(r2) possesses no non-trivial solutions satisfying the boundary conditions for any choice of radii r1 and r2.

Figures

Figures reproduced from arXiv: 2602.09122 by Adam Lindstr\"om, Marko Sobak.

Figure 1
Figure 1. Figure 1: Solutions W(s) of (4.9). (i) If ρ0 > ρcrit, the equation (4.9) has fixed points whenever cos W = − β 2 √ 2 < 0. Define W∞ = arccos  − β 2 √ 2  = π − arcsin  1 2 √ 2λ  1 ρ 2 crit − 1 ρ 2 0  ∈ π 2 , π . (4.10) The solutions with W0 = W∞ and W0 = −W∞ are then constant. Solutions with |W0| < W∞ are increasing and have W → ±W∞ as s → ±∞, while solutions with W∞ < |W0| < π are decreasing and have W → ±(W… view at source ↗
Figure 2
Figure 2. Figure 2: Phase portrait of (4.14). all intersect W = π at some point, while orbits with 0 < W0 < π/2 lying below U+ resp. orbits with 3π/2 < W0 < 2π lying above S+ must intersect the line W = 2πk for some integer k. In particular, we can without loss of generality (up to translation of s and W mod 2π) assume that W0 ∈ {0, π}. By the reflection symmetry it also suffices to study them only in the forward direction. I… view at source ↗
Figure 3
Figure 3. Figure 3: f1 and f2 as functions of ρ0 for fixed P0 (orange curves) and as functions of ρ0 for fixed P0 (teal). The red line shows the limiting case P0 = P∞ with the period equal to that of the linearized system. By the same arguments as in the δ0 = 0 case this limit is a continouos non-constant function of ρ0. This is enough to conclude that the image of the set of admissible (ρ0, P0) under the map (f1, f2) contain… view at source ↗
Figure 4
Figure 4. Figure 4: Phase portrait of (5.5) for λ = 1, β = p 7/8, α = 2. The green (resp. red) curve represents the stable (resp. unstable) manifold at the fixed point (α, 0). The blue curves are the global solutions lying in the region D+ ∪D−. The brown curves are some prototypical orbits that do not lie entirely in D+ ∪ D−, they all have x → 0 at some finite s (either forwards or backwards or both). (i) The boundary case ρ0… view at source ↗
Figure 4
Figure 4. Figure 4: This corresponds to finite s blow up of the Dirac-Yang-Mills pair and the metric, so we do not pursue these orbits further [PITH_FULL_IMAGE:figures/full_fig_p032_4.png] view at source ↗
read the original 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. We further show how to use such pairs to induce Dirac-Yang-Mills pairs on Riemannian products of arbitrary dimension. These are, to the authors' best knowledge, the first examples of coupled Dirac-Yang-Mills pairs on a closed Riemannian spin manifold.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper constructs spherically symmetric Dirac-Yang-Mills pairs on Riemannian 3-manifolds with SU(2) structure group. It derives coupled (non-decoupled) solutions on the product S^1(r1) × S^2(r2) for certain radii r1 and r2, and shows how to induce such pairs on Riemannian products of arbitrary dimension. The authors claim these are the first examples of coupled Dirac-Yang-Mills pairs on a closed Riemannian spin manifold.

Significance. If the existence of the required radii and the non-decoupling of the solutions can be rigorously established, the constructions would supply the first explicit coupled examples on compact spin manifolds, providing concrete test cases for the interaction between the Dirac and Yang-Mills equations in geometric analysis.

major comments (2)
  1. [§3] §3 (Reduction to ODEs): the central claim that non-trivial coupled solutions exist for some positive r1, r2 satisfying the regularity conditions at the S^2 poles and periodicity on S^1 is asserted but not accompanied by an existence proof, numerical verification, or error estimates showing that the solutions do not reduce to pure Yang-Mills; this is load-bearing for the “first examples” statement.
  2. [§4] §4 (Induction to higher dimensions): the induction step from the 3-manifold examples to products of arbitrary dimension relies on the 3D solutions remaining coupled; without explicit confirmation that the Dirac and Yang-Mills components interact non-trivially for the chosen radii, the higher-dimensional claim is unsupported.
minor comments (2)
  1. [§2] Notation for the connection and spinor fields in the spherically symmetric ansatz should be introduced with explicit coordinate expressions before the reduction.
  2. The abstract states “for certain radii” without indicating how these radii are determined; a brief remark on the parameter range would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback. We agree that the existence and coupling of the solutions require stronger justification and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [§3] §3 (Reduction to ODEs): the central claim that non-trivial coupled solutions exist for some positive r1, r2 satisfying the regularity conditions at the S^2 poles and periodicity on S^1 is asserted but not accompanied by an existence proof, numerical verification, or error estimates showing that the solutions do not reduce to pure Yang-Mills; this is load-bearing for the “first examples” statement.

    Authors: We acknowledge that the manuscript asserts the existence of suitable radii without a full existence proof or numerical verification. In the revision we will add a numerical integration of the reduced ODE system for concrete positive values of r1 and r2 that meet the pole regularity and S^1-periodicity conditions, together with explicit error bounds and a direct check that the Dirac spinor component remains non-zero, confirming that the solutions are genuinely coupled and do not reduce to pure Yang-Mills connections. This will rigorously support the claim of providing the first explicit coupled examples. revision: yes

  2. Referee: [§4] §4 (Induction to higher dimensions): the induction step from the 3-manifold examples to products of arbitrary dimension relies on the 3D solutions remaining coupled; without explicit confirmation that the Dirac and Yang-Mills components interact non-trivially for the chosen radii, the higher-dimensional claim is unsupported.

    Authors: We agree that the higher-dimensional construction inherits its coupled character from the 3D solutions. In the revised manuscript we will insert an explicit verification step (via the same numerical solutions and component-wise norms) showing that the Dirac and Yang-Mills fields interact non-trivially for the selected radii before performing the product induction. This will make the higher-dimensional claim fully supported. revision: yes

Circularity Check

0 steps flagged

No significant circularity: direct symmetry reduction and explicit construction of solutions

full rationale

The paper reduces the Dirac-Yang-Mills equations under spherical symmetry to a system of ODEs on the product manifold S^1(r1) × S^2(r2) and asserts existence of radii r1, r2 admitting non-trivial coupled solutions that satisfy the boundary conditions. This is a direct constructive derivation from the field equations rather than any self-definition, fitted parameter renamed as prediction, or load-bearing self-citation chain. No ansatz is smuggled via prior work, and the central existence claim rests on the reduced ODE system itself, which is independent of the target result. The derivation chain is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The constructions rest on standard Riemannian geometry, spin structures, and SU(2) principal bundles; the only adjustable elements are the radii chosen to solve the reduced system.

free parameters (1)
  • radii r1 and r2
    Specific positive real numbers selected so that the spherically symmetric ansatz satisfies the coupled equations on the product manifold.
axioms (2)
  • standard math Existence of a spin structure on the Riemannian 3-manifold and its products
    Required to define the Dirac operator globally on the closed manifold.
  • domain assumption SU(2) acts as the structure group for the Yang-Mills connection
    Standard choice in the literature for spherically symmetric gauge fields.

pith-pipeline@v0.9.0 · 5407 in / 1398 out tokens · 36283 ms · 2026-05-16T05:14:13.031244+00:00 · methodology

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Works this paper leans on

21 extracted references · 21 canonical work pages · 1 internal anchor

  1. [1]

    M. F. Atiyah, N. J. Hitchin, V. G. Drinfeld, and Yu. I. Manin,Construction of instantons, Phys. Lett. A65 (1978), no. 3, 185–187. MR598562

    work page 1978

  2. [2]

    M. F. Atiyah and I. M. Singer,The index of elliptic operators. III, Ann. of Math. (2)87(1968), 546–604. MR236952

    work page 1968

  3. [3]

    Global Anal

    Christian B¨ ar,Extrinsic bounds for eigenvalues of the Dirac operator, Ann. Global Anal. Geom.16(1998), no. 6, 573–596. MR1651379

    work page 1998

  4. [4]

    Global Anal

    Helga Baum,Complete Riemannian manifolds with imaginary Killing spinors, Ann. Global Anal. Geom.7 (1989), no. 3, 205–226. MR1039119

    work page 1989

  5. [5]

    ,Eichfeldtheorie:Eine Einf¨ uhrung in die Differentialgeometrie auf Faserb¨ undeln, Masterclass, Springer Spektrum Berlin, Heidelberg, 2014

    work page 2014

  6. [6]

    Brodbeck,On symmetric gauge fields for arbitrary gauge and symmetry groups, Helv

    O. Brodbeck,On symmetric gauge fields for arbitrary gauge and symmetry groups, Helv. Phys. Acta (1996), 321–324

    work page 1996

  7. [7]

    Corrigan and Peter Goddard,Construction of instanton and monopole solutions and reciprocity, Annals of Physics154(1984), no

    Edward F. Corrigan and Peter Goddard,Construction of instanton and monopole solutions and reciprocity, Annals of Physics154(1984), no. 1, 253–279

    work page 1984

  8. [8]

    S. K. Donaldson,Connections, cohomology and the intersection forms of4-manifolds, J. Differential Geom. 24(1986), no. 3, 275–341. MR868974

    work page 1986

  9. [9]

    S. K. Donaldson and P. B. Kronheimer,The geometry of four-manifolds, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1990. Oxford Science Publications. MR1079726

    work page 1990

  10. [10]

    Mark J. D. Hamilton,Mathematical gauge theory, Universitext, Springer, Cham, 2017. With applications to the standard model of particle physics

    work page 2017

  11. [11]

    Harnad, S

    J. Harnad, S. Shnider, and Luc Vinet,Group actions on principal bundles and invariance conditions for gauge fields, Journal of Mathematical Physics21(1980), no. 12, 2719–2724

    work page 1980

  12. [12]

    Appl.28(2010), no

    Takeshi Isobe,Regularity and energy quantization for the Yang-Mills-Dirac equations on 4-manifolds, Dif- ferential Geom. Appl.28(2010), no. 4, 359–375. SPHERICALLY SYMMETRIC DIRAC-YANG-MILLS PAIRS ON RIEMANNIAN MANIFOLDS 41

    work page 2010

  13. [13]

    J¨ urgen Jost, Enno Keßler, Ruijun Wu, and Miaomiao Zhu,Geometric analysis of the Yang-Mills-Higgs-Dirac model, J. Geom. Phys.182(2022), 104669, 24

    work page 2022

  14. [14]

    The spinor bundle of Riemannian products

    Frank Klinker,The spinor bundle of Riemannian products, 2003. arXiv:math/0212058

    work page internal anchor Pith review Pith/arXiv arXiv 2003

  15. [15]

    Anthony W Knapp,Lie groups Beyond an Introduction, Progress in Mathematics, Birkh¨ auser, 2013

    work page 2013

  16. [16]

    K¨ unzle and Todd A

    H.P. K¨ unzle and Todd A. Oliynyk,Spherically symmetric Einstein–Yang–Mills–Higgs fields for general com- pact gauge groups, Nonlinear Analysis: Theory, Methods and Applications63(2005), no. 5, 473–480

    work page 2005

  17. [17]

    Wei Li,Removable singularities for solutions of coupled Yang-Mills-Dirac equations, J. Math. Phys.47 (2006), no. 10, 103502, 13

    work page 2006

  18. [18]

    arXiv:2601.22886

    Adam Lindstr¨ om,Uncoupled Dirac-Yang-Mills Pairs on Closed Riemannian Spin Manifolds, 2026. arXiv:2601.22886

    work page arXiv 2026

  19. [19]

    Otway,Removable singularities in coupled Yang-Mills-Dirac fields, Comm

    Thomas H. Otway,Removable singularities in coupled Yang-Mills-Dirac fields, Comm. Partial Differential Equations12(1987), no. 9, 1029–1070

    work page 1987

  20. [20]

    Parker,Gauge theories on four-dimensional Riemannian manifolds, Comm

    Thomas H. Parker,Gauge theories on four-dimensional Riemannian manifolds, Comm. Math. Phys.85 (1982), no. 4, 563–602

    work page 1982

  21. [21]

    Wang,Parallel spinors and parallel forms, Ann

    McKenzie Y. Wang,Parallel spinors and parallel forms, Ann. Global Anal. Geom.7(1989), no. 1, 59–68. MR1029845 University of Vienna, F aculty of Mathematics, Oskar-Morgenstern-Platz 1, 1090 Vienna, Aus- tria, Email address:adam.lindstroem@univie.ac.at University of Vienna, F aculty of Mathematics, Oskar-Morgenstern-Platz 1, 1090 Vienna, Aus- tria, Email ad...

    work page 1989