Intuitive

Concrete

Abstract

The Yang-Mills equations can be expressed with the Hodge star operator as \begin{equation} 0=d_A F=d_a *F \quad F=dA+A\wedge A \end{equation} where $d_A$ is the gauge-covariant extension of the exterior derivative. The gauge field $A$ is a one-form \begin{equation} A(x)=A_{\mu}^a(x)t^adx^{\mu} \end{equation} with the values on the Lie algebra of a compact simple Lie group $G$. The curvature is a two-form \begin{equation} \begin{gathered} F=dA+A\wedge A \\ F=F_{\mu\nu}^at^adx^{\mu}\wedge dx^{\nu}\\ F=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}+f^{abc}A_{\mu}^bA_{\nu}^c \end{gathered} \end{equation}

• For a nice description see Mass in Quantum Yang-Mills Theory by L. D. Faddeev

Why is it interesting?

In physics, of course, Maxwell's equations of electromagnetism are linear partial differential equations. Their counterparts, the famous Yang-Mills equations, are non-linear equations which are supposed to govern the forces involved in the structure of matter. The equations are non-linear, because the Yang-Mills equations are essentially matrix versions of Maxwell's equations, and the fact that matrices do not commute is what produces the non-linear term in the equations.

The Aleph of Space by Luciano Boi