Anti-self-dual Yang-Mills equations (ASDYM equations)

The anti-self-dual Yang-Mills equations require the curvature of a connection on an oriented Euclidean four-manifold to satisfy $F=-\ast F$. In complex coordinates $(w,z)$ they can be written

Anti-self-dual gauge curvature obeys $F=-\ast F$ in oriented Euclidean four-space. In the corresponding two-spinor convention it has vanishing symmetric primed curvature spinor. A matrix-valued two-form which is both self-dual and anti-self-dual is zero, because adding the two equations gives $2F=0$.

Locally, the ASDYM equations permit a complex gauge with $A_w=A_z=0$. The trace equation then gives $A_{\bar{w}}=K_z$ and $A_{\bar{z}}=-K_w$. The remaining curvature equation is $K_{w\bar{w}}+K_{z\bar{z}}-[K_w,K_z]=0$. A complex gauge generally makes $K$ valued in the complexification of a Lie algebra; reconstructing a real connection requires its inherited reality condition. The gauge/potential construction is local and need not preserve prescribed behaviour at infinity.

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.