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arxiv: 2604.15200 · v1 · submitted 2026-04-16 · 🧮 math.DG · math.AP

Recognition: unknown

The Yang-Mills equation near instanton-anti-instanton configurations

Alex Waldron , Hao Yin
Authors on Pith no claims yet

Pith reviewed 2026-05-10 09:47 UTC · model grok-4.3

classification 🧮 math.DG math.AP
keywords Yang-Mills equationinstantonsbubblingUhlenbeck limitSU(2) connectionsenergy spectrumR^4topological charge
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0 comments X

The pith

Instantons are the only SU(2) Yang-Mills solutions on R^4 with energy below 4π²(|κ| + 2) + ε_κ for charge κ.

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

The paper examines whether sequences of non-instanton Yang-Mills connections on R^4 can bubble to configurations consisting only of instantons. When the Uhlenbeck limit has charge opposite to the bubbles, deformations of that limit create an obstruction that blocks such bubbling. This obstruction is applied to prove that, for the SU(2) bundle, every solution whose energy lies below the stated threshold must be an instanton. The same analysis shows that the possible energies of connections on the trivial bundle form a discrete set inside the interval from 0 to 16π².

Core claim

When the Uhlenbeck limit and the bubbles carry opposite topological charge, deformations of the Uhlenbeck limit supply an obstruction that prevents a sequence of Yang-Mills connections from converging to a pure instanton configuration. This obstruction is used to conclude that instantons are the only critical points of the Yang-Mills functional on R^4 whose energy is less than 4π²(|κ| + 2) + ε_κ, and that the energy values attained on the trivial bundle are discrete below 16π².

What carries the argument

The obstruction to bubbling that arises from deformations of the Uhlenbeck limit when the limit and the bubbles have opposite charge.

If this is right

  • Instantons are the unique solutions of the SU(2) Yang-Mills equation on R^4 below the energy threshold 4π²(|κ| + 2) + ε_κ.
  • The set of energies attained by SU(2) connections on the trivial bundle is discrete inside [0, 16π²).
  • No sequence of non-instanton connections can converge by bubbling to an instanton-anti-instanton configuration without violating the deformation obstruction.
  • Compactness results for the Yang-Mills equation hold in the stated energy ranges on R^4.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same deformation-obstruction technique could be tested on other four-manifolds or other gauge groups to locate additional energy gaps.
  • Numerical approximation of solutions near known instantons might reveal whether the ε_κ gap is sharp.
  • The discreteness result on the trivial bundle suggests that higher thresholds might also separate discrete clusters of energies.
  • Analogous obstructions may appear in related variational problems such as harmonic maps or minimal surfaces with opposing topological degrees.

Load-bearing premise

There exists an obstruction to bubbling coming from deformations of the Uhlenbeck limit whenever the Uhlenbeck limit and the bubbles have opposite charge.

What would settle it

A non-instanton SU(2) Yang-Mills connection on R^4 whose energy is strictly less than 4π²(|κ| + 2) + ε_κ for some charge κ, or a bubbling sequence in that energy range where the opposite-charge deformation obstruction fails to appear.

read the original abstract

We study the question of whether a sequence of non-instanton Yang-Mills connections can limit to a bubbling configuration composed only of instantons. In the case that the Uhlenbeck limit and the bubbles are of opposite charge, we determine an obstruction coming from deformations of the Uhlenbeck limit. As an application, we prove that instantons are the only solutions of the $\mathrm{SU}(2)$ Yang-Mills equation on $\mathbb{R}^4$ with energy less than $4\pi^2 \left( |\kappa| + 2 \right) + \varepsilon_\kappa,$ where $\kappa$ is the charge. We also prove discreteness of the energy spectrum on the trivial $\mathrm{SU}(2)$-bundle in the range $\left[ 0, 16 \pi^2 \right).$

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

0 major / 2 minor

Summary. The paper studies bubbling for SU(2) Yang-Mills connections on R^4. It establishes an obstruction to bubbling arising from deformations of the Uhlenbeck limit precisely when the limit and the bubbles carry opposite charge. This obstruction is applied to prove that instantons are the only solutions with energy less than 4π²(|κ| + 2) + ε_κ (κ the charge) and that the energy spectrum on the trivial bundle is discrete in [0, 16π²).

Significance. If the results hold, the work supplies a new deformation-theoretic obstruction that rules out mixed-charge bubbling below an explicit energy threshold. The discreteness statement on the trivial bundle is a concrete advance in understanding the possible energies of Yang-Mills fields. The argument relies on standard Uhlenbeck compactness together with a deformation analysis that appears internally consistent with the energy estimates.

minor comments (2)
  1. The dependence of ε_κ on κ is stated in the abstract and main theorems but would benefit from an explicit formula or bound in the introduction to make the threshold fully transparent.
  2. Notation for the Uhlenbeck limit and the bubble charges should be fixed consistently across the deformation analysis and the compactness argument.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the positive assessment, including the recommendation of minor revision. The referee's summary accurately reflects the main results: an obstruction to mixed-charge bubbling arising from deformations of the Uhlenbeck limit, which is then used to establish that instantons are the only SU(2) Yang-Mills solutions on R^4 below the stated energy threshold and that the energy spectrum is discrete on the trivial bundle in [0, 16π²).

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The derivation proceeds by establishing a deformation-theoretic obstruction to bubbling precisely when the Uhlenbeck limit and bubbles carry opposite charge, then invoking standard Uhlenbeck compactness on R^4 to rule out bubbling below the stated energy threshold. This yields the uniqueness claim for instantons and the discreteness result on the trivial bundle. No step reduces by construction to a self-definition, a fitted parameter renamed as a prediction, or a load-bearing self-citation chain; the argument relies on external analytic tools (compactness, deformation theory) whose validity is independent of the target statements. The paper is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Review performed on abstract only; ledger therefore records only the standard background assumptions visible in the abstract.

axioms (2)
  • domain assumption Uhlenbeck compactness theorem for Yang-Mills connections
    Invoked to obtain the limiting configuration consisting of instantons and anti-instantons.
  • domain assumption Existence of a deformation theory for the Uhlenbeck limit
    Used to produce the obstruction when charges are opposite.

pith-pipeline@v0.9.0 · 5436 in / 1323 out tokens · 36079 ms · 2026-05-10T09:47:11.057575+00:00 · methodology

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Pohozaev identities and bubbling obstruction for Yang-Mills fields in conformal dimension

    math.DG 2026-04 unverdicted novelty 7.0

    Derives Pohozaev compatibility between weak limit and bubble involving the Weyl tensor for Yang-Mills bubbling on general 4-manifolds, extending Yin's obstructions and ruling out some bubbling on CP2.

Reference graph

Works this paper leans on

3 extracted references · 1 canonical work pages · cited by 1 Pith paper

  1. [1]

    Desingularizations of conformally Kaehler, Einstein orbifolds

    [LO26] Claude LeBrun and Tristan Ozuch. Desingularizations of conformally Kaehler, Einstein orbifolds. arXiv preprint arXiv:2601.19215,

    work page arXiv

  2. [2]

    Decay estimates for Yang-Mills fields: two new proofs.Global analysis in modern mathematics (Orono, 1991, Waltham, 1992), Publish or Perish, Houston, pages 91–105,

    [Rad93] Johan Rade. Decay estimates for Yang-Mills fields: two new proofs.Global analysis in modern mathematics (Orono, 1991, Waltham, 1992), Publish or Perish, Houston, pages 91–105,

    1991

  3. [3]

    On the blow-up of Yang-Mills fields in dimension four, 2023

    [Yin23] Hao Yin. On the blow-up of Yang-Mills fields in dimension four, 2023

    2023