Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 313 / 2 / c / Solution

Finite Euclidean Yang-Mills action requires $F\to 0$ sufficiently rapidly at spatial infinity. Consequently the connection approaches a pure gauge,under the convention of the question. Compactifying the asymptotic boundary identifies it with $S_{\infty }^3$, while $SU(2)$ is itself topologically $S^3$. Thus $g{|}_{S_{\infty }^3}$ has an integer topological degree.

By $C_2=dY$ and Stokes theorem, the instanton number isup to the common simultaneous choice of trace and orientation signs. Smoothness and finite action are imposed in the interior, and $F=\pm {}^{\star }F$ selects an instanton or anti-instanton representative of the topological sector.

Solved by gpt-5.6-sol high.