Isolated Singularities of Yang-Mills-Higgs fields on surfaces

Bo Chen; Chong Song

arxiv: 1907.07092 · v1 · pith:EESEJQVXnew · submitted 2019-07-16 · 🧮 math.DG

Isolated Singularities of Yang-Mills-Higgs fields on surfaces

Bo Chen , Chong Song This is my paper

Pith reviewed 2026-05-24 20:26 UTC · model grok-4.3

classification 🧮 math.DG

keywords isolated singularitiesYang-Mills-Higgs fieldsasymptotic decay estimateslimit holonomyremovable singularitiesharmonic mapsgauge theory on surfacesdifferential geometry

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The pith

Yang-Mills-Higgs fields on surfaces decay near isolated singularities at a rate set exactly by the limit holonomy.

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

The paper examines isolated singularities of two-dimensional Yang-Mills-Higgs fields defined on fiber bundles whose fiber is a compact Riemannian manifold and whose structure group is a compact connected Lie group. Singularities often cannot be removed because the limit holonomy around the point may be nontrivial. The authors derive a sharp asymptotic decay estimate for the field strength near each such point, with the precise rate of decay controlled by that limit holonomy. This supplies a quantitative refinement of the classical removable-singularity theorem for two-dimensional harmonic maps, replacing mere removability with an explicit decay law whenever the holonomy is given.

Core claim

We establish a sharp asymptotic decay estimate of the Yang-Mills-Higgs field near a singular point, where the decay rate is precisely determined by the limit holonomy. In general the singularity cannot be removed due to possibly non-vanishing limit holonomy around the singular points. Our result can be viewed as a generalization of the classical removable singularity theorem of two dimensional harmonic maps.

What carries the argument

The limit holonomy around each isolated singular point, which fixes the leading coefficient in the asymptotic expansion of the Yang-Mills-Higgs field.

If this is right

When the limit holonomy vanishes the singularity is removable.
The decay estimate holds for any compact connected Lie group as structure group and any compact Riemannian manifold as fiber.
The same rate governs the decay of the curvature form and the Higgs field separately.
The result supplies a precise local model for solutions of the Yang-Mills-Higgs equations on surfaces with prescribed holonomy data at punctures.

Where Pith is reading between the lines

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

The decay law may be used to construct a stratification of the moduli space of Yang-Mills-Higgs fields by holonomy conjugacy classes.
Analogous sharp estimates could be sought for isolated singularities of other elliptic gauge equations on surfaces.
Numerical solution of the equations on a punctured surface could directly verify the predicted decay exponents for sample holonomy values.

Load-bearing premise

The limit holonomy around each isolated singular point exists and is well-defined as an element of the structure group.

What would settle it

An explicit Yang-Mills-Higgs field on a punctured disk whose measured decay rate of the curvature near the puncture differs from the rate predicted by its computed limit holonomy.

read the original abstract

We study isolated singularities of two dimensional Yang-Mills-Higgs fields defined on a fiber bundle, where the fiber space is a compact Riemannian manifold and the structure group is a compact connected Lie group. In general the singularity can not be removed due to possibly non-vanishing limit holonomy around the singular points. We establish a sharp asymptotic decay estimate of the Yang-Mills-Higgs field near a singular point, where the decay rate is precisely determined by the limit holonomy. Our result can be viewed as a generalization of the classical removable singularity theorem of two dimensional harmonic maps.

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.

Desk Editor's Note private letter to a colleague

The paper delivers a sharp decay estimate for Yang-Mills-Higgs fields near singularities, controlled by the limit holonomy.

read the letter

The key point is that the authors prove a precise asymptotic decay estimate for these fields near isolated singularities, with the rate fixed by the limit holonomy. This is the main new piece. They set up the problem on a fiber bundle with compact fiber and structure group, and note that non-vanishing holonomy prevents removal of the singularity. From there they derive the decay rate directly from the holonomy. This looks like a straightforward but useful generalization of the classical result for harmonic maps. What they do well is keeping the statement clean and highlighting the assumption on the holonomy. The claim is specific enough that it could be checked or used in subsequent compactness arguments. The soft spots are minor at this stage. The derivation is not visible in the abstract, so one cannot rule out technical issues in the estimates, but the framing does not suggest any circularity or unstated problems. The existence of the limit holonomy is the main prerequisite, and they treat it as such. This work is aimed at people in geometric analysis who deal with gauge fields and singularities on low-dimensional domains. A reader interested in extending removable singularity results or in applications to Yang-Mills type equations would get direct value from the decay formula. It deserves a serious referee. The result is targeted and the generalization is clear, so sending it out makes sense even if revisions are needed after review.

Referee Report

0 major / 2 minor

Summary. The paper studies isolated singularities of two-dimensional Yang-Mills-Higgs fields on fiber bundles with compact Riemannian manifold fibers and compact connected Lie group structure groups. Singularities cannot always be removed due to non-vanishing limit holonomy. The central result is a sharp asymptotic decay estimate near each singular point whose rate is determined exactly by the limit holonomy; the work is framed as a generalization of the removable-singularity theorem for two-dimensional harmonic maps.

Significance. If the derivation holds, the result supplies a precise, holonomy-controlled description of singular behavior for Yang-Mills-Higgs fields, extending classical removable-singularity theorems in geometric analysis. The parameter-free character of the decay rate (once the holonomy is fixed) and the direct link to an existing removable-singularity theorem constitute clear strengths.

minor comments (2)

[Abstract] The abstract states that the base is two-dimensional but does not explicitly name the base manifold or the bundle projection; a single clarifying sentence would help readers locate the setting immediately.
[Introduction] Notation for the limit holonomy and the precise decay exponent should be introduced with a displayed equation in the introduction so that the main theorem statement can be read without backtracking.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary and significance assessment of our work, as well as the recommendation for minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained analytic estimate

full rationale

The paper presents a sharp asymptotic decay estimate for Yang-Mills-Higgs fields near isolated singularities, with the rate controlled by the limit holonomy. This is framed as a direct generalization of the classical removable singularity theorem for 2D harmonic maps, an external result from the literature. No load-bearing steps reduce to self-definition, fitted inputs renamed as predictions, or self-citation chains. The existence of the limit holonomy is stated as an assumption, not derived internally. The central claim therefore rests on independent analytic techniques rather than re-expressing its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review based on abstract only; no free parameters, invented entities, or non-standard axioms are visible.

axioms (1)

domain assumption Fiber space is a compact Riemannian manifold and structure group is a compact connected Lie group.
Explicitly stated as the geometric setting in the abstract.

pith-pipeline@v0.9.0 · 5611 in / 1131 out tokens · 20034 ms · 2026-05-24T20:26:44.401539+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

17 extracted references · 17 canonical work pages

[1]

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[2]

Mundet i Riera, I., Tian, G.: A compactiﬁcation of the mod uli space of twisted holomorphic maps. Adv. Math. 222, no. 4, 1117-1196 (2009)

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Ott, A.: Removal of singularities and Gromov compactnes s for symplectic vortices. J. Symplectic Geom. 12, no. 2, 257-311 (2014)

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Global analysis in modern mathematics (Orono, ME, 1991; Waltham, MA, 1992), 91-105, Publish or Perish, Hou ston, TX, 1993

R ˚ ade, J.: Decay estimates for Yang-Mills ﬁelds; two new proofs. Global analysis in modern mathematics (Orono, ME, 1991; Waltham, MA, 1992), 91-105, Publish or Perish, Hou ston, TX, 1993

work page 1991
[6]

Local theory

R ˚ ade, J.: Singular Yang-Mills ﬁelds. Local theory. I. J . Reine Angew. Math. 452, 111-151 (1994)

work page 1994
[7]

Local theory

R ˚ ade, J.: Singular Yang-Mills ﬁelds. Local theory. II. J. Reine Angew. Math. 456, 197-219 (1994)

work page 1994
[8]

Mundet i Riera, I.: Yang-Mills-Higgs theory for symplec tic ﬁbrations. Ph.D. dissertation, Univ. of Madrid, 1999

work page 1999
[9]

Song, C.: Critial points of Yang-Mills-Higgs functiona l. Commun. contemp. Math. 13(3), 463-486 (2011)

work page 2011
[10]

Song, C.: Convergence of Yang-Mills-Higgs ﬁelds. Math . Ann. 366(1-2), 1-51 (2016)

work page 2016
[11]

Compositio Math

Sibner, L.: Removable singularities of Yang-Mills ﬁel ds in R3. Compositio Math. 53, 91-104 (1984)

work page 1984
[12]

Sibner, L., Sibner, R.J.: Removable singularities of c oupled Yang-Mills ﬁelds in R3. Commun. Math. Phys. 93, 1-17 (1984)

work page 1984
[13]

Sibner, L.: The isolated point singularity problem for the coupled Yang-Mills equations in higher dimensions. Math. Ann. 271, no. 1, 125-131 (1985)

work page 1985
[14]

Sibner, L., Sibner, R.: Classiﬁcation of singular Sobo lev connections by their holonomy. Commun. Math. Phys. 144, 337-350 (1992)

work page 1992
[15]

Smith, P.: Removable singularities for the Yang-Mills -Higgs equations in two dimensions. Ann. del’ I. H. P., section C, 7, n0 6, 561-588 (1990)

work page 1990
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and Uhlenbeck, K., The existence of minimal im mersions of 2-spheres, Ann

Sacks, J. and Uhlenbeck, K., The existence of minimal im mersions of 2-spheres, Ann. of Math. (2) 113, no. 1, 1-24(1981)

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Uhlenbeck, K.: Connections with Lp bounded on curvature. Commun. Math. Phys, 83, 31-42 (1982). Academy of Mathematics and Systems Science, Chinese Academ y of Sciences, BeiJing,100190, P.R.China E-mail address : chenbo@amss.ac.cn School of Mathematical Sciences, Xiamen University, Xiame n, 361005, P.R.China. E-mail address : songchong@xmu.edu.cn 20

work page 1982

[1] [1]

Moore, J., Schlaﬂy, R.: On equivariant isometric embedd ings. Math. Z. 173(2), 119-133 (1980)

work page 1980

[2] [2]

Mundet i Riera, I., Tian, G.: A compactiﬁcation of the mod uli space of twisted holomorphic maps. Adv. Math. 222, no. 4, 1117-1196 (2009)

work page 2009

[3] [3]

Ott, A.: Removal of singularities and Gromov compactnes s for symplectic vortices. J. Symplectic Geom. 12, no. 2, 257-311 (2014)

work page 2014

[4] [4]

Parker, T.: Gauge theories on four-dimensional manifol ds. Commun. Math. Phys. 85, 563-602 (1982)

work page 1982

[5] [5]

Global analysis in modern mathematics (Orono, ME, 1991; Waltham, MA, 1992), 91-105, Publish or Perish, Hou ston, TX, 1993

R ˚ ade, J.: Decay estimates for Yang-Mills ﬁelds; two new proofs. Global analysis in modern mathematics (Orono, ME, 1991; Waltham, MA, 1992), 91-105, Publish or Perish, Hou ston, TX, 1993

work page 1991

[6] [6]

Local theory

R ˚ ade, J.: Singular Yang-Mills ﬁelds. Local theory. I. J . Reine Angew. Math. 452, 111-151 (1994)

work page 1994

[7] [7]

Local theory

R ˚ ade, J.: Singular Yang-Mills ﬁelds. Local theory. II. J. Reine Angew. Math. 456, 197-219 (1994)

work page 1994

[8] [8]

Mundet i Riera, I.: Yang-Mills-Higgs theory for symplec tic ﬁbrations. Ph.D. dissertation, Univ. of Madrid, 1999

work page 1999

[9] [9]

Song, C.: Critial points of Yang-Mills-Higgs functiona l. Commun. contemp. Math. 13(3), 463-486 (2011)

work page 2011

[10] [10]

Song, C.: Convergence of Yang-Mills-Higgs ﬁelds. Math . Ann. 366(1-2), 1-51 (2016)

work page 2016

[11] [11]

Compositio Math

Sibner, L.: Removable singularities of Yang-Mills ﬁel ds in R3. Compositio Math. 53, 91-104 (1984)

work page 1984

[12] [12]

Sibner, L., Sibner, R.J.: Removable singularities of c oupled Yang-Mills ﬁelds in R3. Commun. Math. Phys. 93, 1-17 (1984)

work page 1984

[13] [13]

Sibner, L.: The isolated point singularity problem for the coupled Yang-Mills equations in higher dimensions. Math. Ann. 271, no. 1, 125-131 (1985)

work page 1985

[14] [14]

Sibner, L., Sibner, R.: Classiﬁcation of singular Sobo lev connections by their holonomy. Commun. Math. Phys. 144, 337-350 (1992)

work page 1992

[15] [15]

Smith, P.: Removable singularities for the Yang-Mills -Higgs equations in two dimensions. Ann. del’ I. H. P., section C, 7, n0 6, 561-588 (1990)

work page 1990

[16] [16]

and Uhlenbeck, K., The existence of minimal im mersions of 2-spheres, Ann

Sacks, J. and Uhlenbeck, K., The existence of minimal im mersions of 2-spheres, Ann. of Math. (2) 113, no. 1, 1-24(1981)

work page 1981

[17] [17]

Uhlenbeck, K.: Connections with Lp bounded on curvature. Commun. Math. Phys, 83, 31-42 (1982). Academy of Mathematics and Systems Science, Chinese Academ y of Sciences, BeiJing,100190, P.R.China E-mail address : chenbo@amss.ac.cn School of Mathematical Sciences, Xiamen University, Xiame n, 361005, P.R.China. E-mail address : songchong@xmu.edu.cn 20

work page 1982