Conformal compactification and finite-action instantons

Intuition Beginner

Four-dimensional Euclidean space is noncompact: it has directions that run off forever. Conformal compactification adds one point at infinity, turning the space into a round sphere. This is like drawing an infinite flat map on a globe with one extra point representing every far-away direction.

Instantons are unusually compatible with this move. Their action stays finite, and the four-dimensional Yang-Mills action does not change under conformal rescaling. That means an instanton on flat space can often be studied as a smooth field on the sphere, with the far-away point treated as an ordinary point.

The hard theorem says the far-away point is not hiding a defect. If the action is finite and the field satisfies the right equation, a suitable gauge makes the field smooth there.

Visual Beginner

The marked point is not a boundary. It is a new point added to the space. The theorem says a finite-action instanton can be described smoothly across that point after choosing a good gauge.

Worked example Beginner

Start with the simplest instanton bump centered at the origin. Near the center the field is strong. Far away the field fades rapidly, so the total action remains eight times pi squared.

Now add one point representing all far-away directions. The old flat space becomes a sphere missing that one point. Since the field fades fast enough, the added point should not carry extra action. Uhlenbeck's theorem turns that expectation into a precise statement: after changing gauge, the field extends smoothly across the added point.

What this tells us: the one-instanton on flat space is not merely a field on an open space. It is the same geometric object as a charge-one connection on the sphere, written in a chart that omits one point.

Check your understanding Beginner

Formal definition Intermediate+

The conformal compactification of ${\mathbb{R}}^4$ is the one-point compactification $$ S^4=\mathbb{R}^4\cup{\infty}, $$ equipped with the conformal class of the round metric under stereographic projection. In coordinates, the round metric pulled back to ${\mathbb{R}}^4$ is conformal to the Euclidean metric: $$ g_{S^4}=\Omega(x)^2 g_{\mathrm{euc}},\qquad \Omega(x)=\frac{2}{1+|x|^2}. $$

For a connection $A$ on a principal $G$-bundle over a four-manifold, the Yang-Mills action is $$ \operatorname{YM}(A)=\frac12\int |F_A|^2,d\operatorname{vol}. $$ If $g$ is replaced by $\hat{g}=e^{2f}g$, then the pointwise norm of a two-form scales by $e^{-4f}$ while the volume form scales by $e^{4f}$. Hence the action is conformally invariant in dimension four. The same calculation gives conformal invariance of the ASD equation because the Hodge star on two-forms is unchanged 03.07.06.

An isolated removable singularity for a Yang-Mills connection is a point $p$ such that a smooth connection is given on $B^4\setminus \{p\}$, has finite action near $p$, and is gauge-equivalent on $B^4\setminus \{p\}$ to a connection extending smoothly over $B^4$.

Uhlenbeck's theorem states that a Yang-Mills connection on a punctured four-ball with sufficiently controlled finite curvature energy has such a removable singularity after gauge transformation ^{[Uhlenbeck 1982]}. Applied at the point $\infty \in S^4$, this says a finite-action ASD instanton on ${\mathbb{R}}^4$ extends, after gauge, to a smooth connection on a bundle over $S^4$.

Key theorem with proof Intermediate+

Theorem (finite-action instantons extend over infinity). Let $A$ be a smooth finite-action ASD connection on a principal $\mathrm{SU}(2)$-bundle over ${\mathbb{R}}^4$. After conformal compactification and a gauge transformation near $\infty$, $A$ extends to a smooth ASD connection on a principal $\mathrm{SU}(2)$-bundle over $S^4$.

Proof. Stereographic projection identifies ${\mathbb{R}}^4$ with $S^4\setminus \{\infty \}$ and identifies the Euclidean conformal class with the round conformal class. Since the Yang-Mills action and the ASD equation are conformally invariant in dimension four, $A$ becomes a finite-action ASD connection on $S^4\setminus \{\infty \}$.

The point $\infty$ is an isolated puncture. Uhlenbeck's removable-singularity theorem applies to finite-action Yang-Mills connections on punctured four-balls: after choosing a suitable gauge near the puncture, the connection extends smoothly across the missing point. An ASD connection is Yang-Mills by 03.07.06, so the theorem applies to the compactified connection near $\infty$.

The extended connection remains ASD because the equation $F_A^{+}=0$ is local and the curvature of the extended connection agrees with the old curvature away from $\infty$. Smoothness lets the equation extend to the added point by continuity. Therefore the compactified connection is a smooth ASD connection over all of $S^4$. $\square$

Bridge. Conformal compactification builds toward 03.07.09 by replacing finite-action instantons on open Euclidean space with smooth points of a compact moduli problem, and it appears again in 03.07.20 when bubbling on cylinders is modeled by rescaled instantons on $S^4$. The foundational reason is the four-dimensional conformal invariance of the action; this is exactly what identifies energy concentration with sphere bubbles. Putting these together, the compactification theorem generalises removable-singularity theorems from functions to connections, and the bridge is dual to the local gauge-fixing estimates behind Uhlenbeck compactness.

Exercises Intermediate+

Advanced results Master

The analytic core of Uhlenbeck's theorem is gauge fixing under a small $L^2$ curvature hypothesis. On a four-ball, if $\Vert F_A{\Vert }_{L^2}$ is sufficiently small, there is a gauge in which the connection form satisfies a Coulomb condition $$ d^*a=0 $$ and a Sobolev estimate $$ |a|{W^{1,2}}\leq C|F_A|{L^2}. $$ This converts a gauge-invariant curvature bound into an elliptic estimate for a local representative of the connection ^{[Uhlenbeck Lp 1982]}.

The removable-singularity theorem strengthens this estimate. For a Yang-Mills connection on $B^4\setminus \{0\}$ with finite $L^2$ curvature, one first chooses radii on which the curvature energy is small, then patches Coulomb gauges across annuli. The Yang-Mills equation becomes an elliptic system in Coulomb gauge. Elliptic regularity upgrades the Sobolev extension to smoothness. The theorem is false in this form without the gauge transformation because connection one-forms are not gauge-invariant objects.

The conformal compactification point is the first instance of the bubble compactness mechanism. In a sequence of ASD connections, energy may concentrate at finitely many points. Around each concentration point, rescaling produces a finite-action ASD connection on ${\mathbb{R}}^4$; this unit turns that local limit into an instanton on $S^4$. The quantized amount of lost energy is therefore the action of a smooth sphere instanton, an integer multiple of $8{\pi }^2$ in the $\mathrm{SU}(2)$ normalization.

For moduli spaces, compactification by ideal instantons attaches lower-charge smooth instantons plus unordered points carrying bubble charge. The Uhlenbeck compactification of the charge-$k$ moduli space on $S^4$ has boundary strata modeled by charge-$k-\ell$ instantons together with $\ell$ concentration points. Donaldson theory uses this compactness to control intersection numbers on moduli spaces and to define invariants despite noncompactness.

Synthesis. The foundational reason compactification works is the equality between the curvature degree and half the dimension. This is exactly the same equality that makes ASD a first-order Yang-Mills equation in 03.07.06. The central insight identifies the point at infinity with an isolated removable singularity, and the gauge-fixing theorem is dual to the geometric compactification of instanton moduli. Putting these together, Uhlenbeck's removable-singularity theorem supplies the bridge from explicit Euclidean instantons to compact four-manifold gauge theory.

Full proof set Master

Proposition 1 (conformal invariance of the Yang-Mills action in dimension four). Let $A$ be a connection on a principal bundle over an oriented four-manifold. If $\hat{g}=e^{2f}g$, then ${\mathrm{YM}}_{\hat{g}}(A)={\mathrm{YM}}_g(A)$.

Proof. The curvature $F_A$ is a two-form with values in $\mathrm{ad}P$, independent of the Riemannian metric. Under $\hat{g}=e^{2f}g$, the pointwise squared norm of an ordinary two-form scales as $e^{-4f}$; the same scaling holds for adjoint-valued two-forms after pairing the adjoint factor with a fixed invariant inner product. The volume form scales as $e^{4f}$. Therefore $$ |F_A|{\hat g}^2,d\operatorname{vol}{\hat g} =e^{-4f}|F_A|_g^2,e^{4f}d\operatorname{vol}_g =|F_A|_g^2,d\operatorname{vol}_g. $$ Integrating and multiplying by the common factor $\frac{1}{2}$ gives the result. $\square$

Proposition 2 (stereographic compactification is conformal). The round metric on $S^4$ pulls back under stereographic projection to $\Omega (x{)}^2{g}_{\mathrm{euc}}$ on ${\mathbb{R}}^4$, where $\Omega (x)=2/(1+|x{|}^2)$.

Proof. View $S^4\subset {\mathbb{R}}^5$ and project from the north pole to the equatorial copy of ${\mathbb{R}}^4$. The inverse stereographic map is $$ x\longmapsto \left(\frac{2x}{1+|x|^2},\frac{|x|^2-1}{1+|x|^2}\right). $$ Differentiating and taking the Euclidean inner product in ${\mathbb{R}}^5$ gives $$ ds_{S^4}^2=\frac{4}{(1+|x|^2)^2},ds_{\mathbb{R}^4}^2. $$ Thus the pullback is conformal to the Euclidean metric with factor $\Omega (x{)}^2$. $\square$

Proposition 3 (the compactified extension remains ASD). If an ASD connection on $S^4\setminus \{\infty \}$ extends smoothly over $\infty$, then the extension is ASD on all of $S^4$.

Proof. Let $\tilde{A}$ be the smooth extension. On $S^4\setminus \{\infty \}$, its curvature agrees with the original curvature, so $F_{\tilde{A}}^{+}=0$ there. The self-dual projection $F_{\tilde{A}}^{+}$ is a smooth adjoint-valued two-form on all of $S^4$. A smooth section that vanishes on the dense open subset $S^4\setminus \{\infty \}$ vanishes everywhere. Hence $F_{\tilde{A}}^{+}=0$ at $\infty$ as well. $\square$

Connections Master

• ASD equation 03.07.06. The conformal invariance of the Hodge star on two-forms is the shared reason both ASD and Yang-Mills action survive compactification.

• BPST instanton and Bogomolny bound 03.07.07. The BPST formula on ${\mathbb{R}}^4$ becomes a smooth charge-one connection on $S^4$ after the removable-singularity theorem is applied at infinity.

• ASD moduli space 03.07.09. Instanton moduli on $S^4$ use this theorem to treat finite-action Euclidean instantons as points in a compact four-manifold problem.

• Uhlenbeck compactness for ASD equations on cylinders 03.07.20. Bubble formation in Floer trajectories is modeled by rescaled finite-action instantons, then compactified to smooth sphere instantons by the theorem in this unit.

Historical & philosophical context Master

Atiyah's 1979 Pisa lectures placed Yang-Mills instantons on $S^4$ at the center of the geometric theory, using conformal compactification to pass between explicit Euclidean formulas and compact four-manifold geometry ^{[Atiyah 1979]}. This framing made the BPST solution a geometric object rather than only a formula in coordinates.

Uhlenbeck's 1982 papers supplied the analytic engine. The first established removable singularities for Yang-Mills fields, and the second developed gauge choices under $L^p$ curvature bounds ^{[Uhlenbeck 1982]}. Freed-Uhlenbeck and Donaldson-Kronheimer then made these estimates part of the standard compactness package for instanton moduli spaces ^{[Donaldson-Kronheimer 1990]}.

Bibliography Master

@article{Uhlenbeck1982Removable,
  author = {Uhlenbeck, Karen K.},
  title = {Removable singularities in Yang-Mills fields},
  journal = {Communications in Mathematical Physics},
  volume = {83},
  pages = {11--29},
  year = {1982}
}

@article{Uhlenbeck1982Lp,
  author = {Uhlenbeck, Karen K.},
  title = {Connections with {$L^p$} bounds on curvature},
  journal = {Communications in Mathematical Physics},
  volume = {83},
  pages = {31--42},
  year = {1982}
}

@book{Atiyah1979YangMillsCompactification,
  author = {Atiyah, Michael F.},
  title = {Geometry of Yang-Mills Fields},
  publisher = {Scuola Normale Superiore},
  address = {Pisa},
  year = {1979}
}

@book{DonaldsonKronheimer1990Compactness,
  author = {Donaldson, Simon K. and Kronheimer, Peter B.},
  title = {The Geometry of Four-Manifolds},
  publisher = {Oxford University Press},
  year = {1990}
}

@book{FreedUhlenbeck1991Instantons,
  author = {Freed, Daniel S. and Uhlenbeck, Karen K.},
  title = {Instantons and Four-Manifolds},
  series = {Mathematical Sciences Research Institute Publications},
  volume = {1},
  publisher = {Springer},
  year = {1991}
}