This [link](http://math.mit.edu/%7Ebottman/docs/Uhlenbeck_1.pdf) 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.



Uhlenbeck's theorem novelty
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 $2$.)


