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

The Hodge star operator is defined byOn two-forms in oriented Euclidean four-space, ${\star }^2=1$ and the wedge product is symmetric. ThereforeFor an anti-self-dual curvature $F=-\star F$, the Yang-Mills instanton action becomes purely topological. Withthe standard positive-action convention gives

Let $D_{\mu }={\partial }_{\mu }+A_{\mu }$. A Lax pair with spectral parameter $\lambda$ isThe coefficients of $[L,M]=0$ at orders $1,\lambda ,{\lambda }^2$ are respectivelywhich are precisely the anti-self-dual Yang-Mills equations.

The equation $F_{wz}=0$ says that the partial connection $D_w,D_z$ is flat. Its compatibility condition therefore guarantees a local $G$-valued solution $g$ ofAfter the associated gauge transformation, both transformed components vanish:

In this gauge, $F_{\bar{w}\bar{z}}=0$ says that the remaining partial connection is flat. Hence locallyorThe remaining curvature equation then becomes

For Maxwell theory the group is Abelian, so write $J=e^{\chi }$. The reduced equation loses its commutators and becomesSince the Euclidean Laplacian isthe anti-self-dual Maxwell equations in this gauge are equivalent to