Yang–Mills equations

Remarks

The equation is equivalent to $d_A^* F_A = 0$, where $d_A^*$ is the formal adjoint of $d_A$. Every connection also satisfies the Bianchi identity $d_A F_A = 0$. For compact $M$, the Yang–Mills connections are the critical points of the Yang–Mills functional $\mathrm{YM}(A) = \tfrac{1}{2} \int_M \abs{F_A}^2 \, d\mathrm{vol}_g$, where the norm uses an invariant inner product on $\mathfrak{g}$.

Example: a flat connection, with $F_A = 0$, satisfies the equations. On a four-manifold, a connection with $\star F_A = \pm F_A$, an instanton or anti-instanton, also satisfies them, because $d_A \star F_A = \pm d_A F_A = 0$ by the Bianchi identity.

Non-example: let $M = \R^4$ with the Euclidean metric and $G = U(1)$, so that $d_A = d$, and take $A = i x_1^2 \, dx_2$. Then $F_A = 2i x_1 \, dx_1 \wedge dx_2$, $\star F_A = 2i x_1 \, dx_3 \wedge dx_4$, and $d \star F_A = 2i \, dx_1 \wedge dx_3 \wedge dx_4 \ne 0$.

For $G = U(1)$ and a Lorentzian metric the equations are Maxwell's equations in vacuum.