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

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