Gauge theories and non-commutative geometry

TL;DR

This paper demonstrates how a classical SU(N) Yang-Mills theory in d dimensions can be reformulated in d+2 dimensions with a non-commutative geometric surface, providing explicit proofs for torus and sphere cases.

Contribution

It introduces a novel formulation of Yang-Mills theories in higher dimensions with non-commutative geometry, explicitly proven for torus and sphere surfaces.

Findings

Yang-Mills theory reformulated in higher dimensions

Explicit proofs for torus and sphere cases

Connection between gauge theories and non-commutative geometry

Abstract

It is shown that a $d$-dimensional classical SU(N) Yang-Mills theory can be formulated in a $d+2$-dimensional space, with the extra two dimensions forming a surface with non-commutative geometry. In this paper we present an explicit proof for the case of the torus and the sphere.