# Uhlenbeck's theorem novelty

This link provides a short introduction to the contributions of Uhlenbeck about regular gauge fixing. However, I feel quite puzzled about it and I do not understand the real novelty apported by this work. More concisely, it would be helpful for me to understand

1. which is the transcendence and novelty of the local Coulomb gauge theorem (theorem 3.2), in case it can be shortly explained, and what it is useful for.

2. which relation has this result with physics (I have heard in the web that the so-called $$\epsilon$$ plays some Planck's constant role, and that everything may be translated into quantum physics, or at least physics world).

Any suggestion for those two questions will be welcomed.

• I have to say that what is described in this note is the least of what Uhlenbeck did in her seminal work. The note describes how to use the Coulomb gauge to get compactness of self-dual connections with a uniform $L^p$ bound on the curvature, where $p > n/2$. What was truly revolutionary was her analysis of what happens when $p = n/2$, where she showed that there was a "bubbling" effect that can occur due to the scale invariance of the $L^{n/2}$ norm of curvature. The phenomenon is crucial to Donaldson's thesis. – Deane Yang Dec 10 '14 at 4:13
• @DeaneYang Thank you very much for the information. Could you tell me about some source where this 'truly revolutionary seminal work' is explained (at least roughly). And the same would be helpful about the point where this phenomenon is crucial to Donaldson's thesis. – Jjm Dec 11 '14 at 9:09
• I'm sure there are other references by now, but one is Instantons and Four-Manifolds. by Freed and Uhlenbeck. – Deane Yang Dec 11 '14 at 15:09
• What is “apported”? – Monroe Eskew Apr 19 at 17:57
• @MonroeEskew apported is intended to mean brought ( = French apporté) – Andreas Blass Apr 19 at 18:09

Denote by $$A$$ the connection and by $$F_A$$ its curvature. Then

$$dA=F_A-A\wedge A.$$

If $$A$$ is in Coulomb gauge we have an additional equation

$$d^*A=0.$$

The advantage is that the operator $$d\oplus d^*$$ is elliptic and now we have an equation of the form

$$(d\oplus d^*)A= \mbox{something}.$$

Elliptic theory allows us to convert bounds on "something'' into bounds on $$A$$. Then, the bounds on $$A$$ can be converted into compactness results using standard compactness results in Sobolev spaces.

Finding a local Coulomb gauge on a region $$D$$ is possible as long as the "energy" $$\Vert F_A\Vert_{L^2(D)}$$ is smaller than $$<\epsilon$$, where $$\epsilon$$ is related to the second Chern number of a principal $$G$$-bundle over $$S^4$$, the conformal compactification of $$\mathbb{R}^4$$. The energy of an instanton on $$S^4$$ is equal, up to a universal constant, to the second Chern number which is an integer. You can regard this as a quantization result, stating that the energy of an instanton is an integral multiple of a universal constant. (If my memory serves me right this constant is $$4\pi^2$$, give or take a factor of $$2$$.)

• To a physicist, it is strange that this gauge fixing condition is referred to as "Coulomb gauge". It is true that in 4d electrodynamics this name refers to the condition that the divergence of the vector potential $\mathbf{A}_t$ is zero. However, here $\mathbf{A}_t$ should be interpreted as the pullback to the level sets of a special coordinate $t$ of a 1-form $A$ on $M$, with the divergence taken with respect to the induced metric on each level set. The simple divergence of $A$ on $M$ is called the Lorenz gauge and seems to be the more appropriate analog. – Igor Khavkine Dec 9 '14 at 23:18
• Igor, this is in 4 dimensions, but time $t$ has been replaced by "imaginary time" so that the metric is no longer Lorentzian but is Riemannian. – Deane Yang Dec 10 '14 at 4:07
• @DeaneYang, I'm well aware, thanks. I could also rephrase my comment by saying that Uhlenbeck's "Coulomb gauge", when the "imaginary time" is replaced by "real time", becomes the Lorenz gauge, rather than the usual Coulomb gauge in electrodynamics or Yang-Mills theory. But I imagine the incongruence isn't going to go away since all of this terminology already seems to have become standard. – Igor Khavkine Dec 10 '14 at 9:53
• Do you know a reference expanding on the last paragraph? I'd be especially curious to find out more about situations where a Coulomb gauge exists despite the curvature being 'large', that is $\lvert F \rvert_{L^2(D)} > \epsilon$ or $4 \pi^2$ say. – Leo Moos Apr 19 at 17:23
• I think that in her paper on the removal of singularities you see this phenomenon better. Technically the bound on energy is required by the method. The Coulomb gauge is found by solving a nonlinear equation using the implicit function theorem. – Liviu Nicolaescu Apr 20 at 17:39