't Hooft tensor for generic gauge group

TL;DR

This paper generalizes the 't Hooft tensor to arbitrary gauge groups, providing a framework to classify magnetic monopoles via topological methods, which is crucial for understanding color confinement in gauge theories.

Contribution

It derives the 't Hooft tensor for any gauge group, extending previous results limited to specific cases, and clarifies the topological classification of monopoles.

Findings

Derived the 't Hooft tensor for generic gauge groups

Established the correspondence between magnetic charges and second homotopy classes

Implications for understanding color confinement

Abstract

We study monopoles in gauge theories with generic gauge group. Magnetic charges are in one-to-one correspondence with the second homotopy classes at spatial infinity (${\Pi}_2$), which are therefore identified by the 't Hooft tensor. We determine the 't Hooft tensor in the general case. These issues are relevant to the understanding of Color Confinement.