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 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. © 2021 Jagellonian University. All rights reserved.