Anti-self-dual Yang-Mills equations in temporal gauge (ASDYM flow)

With orientation $d{x}^1\wedge d{x}^2\wedge d{x}^3\wedge d\tau$ and $A_{\tau }=0$, one anti-self-duality convention becomes

For three Lie-algebra-valued functions of one variable, the Nahm equations areThey arise by taking a solution of the ASDYM equations with orientation $d{x}^1\wedge d{x}^2\wedge d{x}^3\wedge dt$, imposing independence of $x^1,x^2,x^3$, and setting $A_t=0$ by a local Yang-Mills gauge transformation. Then $F_{ab}=[A_a,A_b]$, $F_{at}=-{\dot{A}}_a$, and anti-self-duality of gauge curvature gives the displayed equations. The opposite orientation reverses their sign.

Put $P=A_1+i{A}_2$, $Q=A_1-i{A}_2$ and $R=A_3$, and defineDirect expansion gives $[L,M]=-i[P,R]+i[P,Q]\lambda +i[R,Q]{\lambda }^2$. The Nahm equations imply $\dot{P}=-i[P,R]$, $2\dot{R}=i[P,Q]$ and $-\dot{Q}=i[R,Q]$, so $\dot{L}=[L,M]$. In any finite-dimensional matrix representation, cyclicity of the matrix trace givesThus every coefficient of the degree-at-most-$2p$ matrix trace polynomial is conserved. At projective infinity the polynomial is interpreted as a section of $\mathcal{O}(2p)$, not a globally holomorphic scalar function on the complex projective line.