In this talk we will present some analysis aspects of gauge theory in high dimension. First, we will study the completion of the space of arbitrary
smooth bundles and connections under Lp-control of their curvature. We will start from the classical theory in critical dimension (i.e. n=2p) and then move to
the super-critical dimension (i.e. n>2p), making a digression about the so called “abelian” case and thus showing an analogy between p-Yang-Mills fields on
abelian bundles and a special class of singular vector fields. In the last part, we will show how the previous analysis can be used in order to
build a Schoen-Uhlenbeck type regularity theory for Yang-Mills fields in supercritical dimension. | 
