On Some Geometrical Properties of Gauge Theories

Gauge theories have become the universal language of fundamental interactions. To this discovery, Martinus J.G. Veltman played a major role. In this short note, dedicated to his memory, we try to understand some of their geometrical properties. We show that a d-dimensional $\mathrm{SU}(N)$ Yang– Mills theory can be formulated on a ($d + 2$)-dimensional space, with the extra two dimensions forming a surface with non-commutative geometry. The non-commutativity parameter is proportional to $1/N$ and the equivalence is valid to any order in $1/N$. We study explicitly the case of the sphere and the torus.