For Maxwell theory the group is Abelian, so write $J=e^\chi$. The reduced equation loses its commutators and becomes
$$
\partial_w\partial_{\bar w}\chi
+\partial_z\partial_{\bar z}\chi=0.
$$
Since the Euclidean <Laplacian> is
$$
\Delta_{\mathbb R^4}=4(\partial_w\partial_{\bar w}
+\partial_z\partial_{\bar z}),
$$
the anti-self-dual Maxwell equations in this gauge are equivalent to
$$
\boxed{\Delta_{\mathbb R^4}\chi=0}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 313 / 2 / v / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/iii/paper-313/2/v/solution