Finite Euclidean Yang-Mills action requires $F\to0$ sufficiently rapidly at spatial infinity. Consequently the connection approaches a pure gauge,
$$
A\longrightarrow-dg\,g^{-1}
$$
under the convention of the question. Compactifying the asymptotic boundary identifies it with $S^3_\infty$, while $SU(2)$ is itself topologically $S^3$. Thus $g|_{S^3_\infty}$ has an integer <topological degree>.

By $C_2=dY$ and <Stokes theorem>, the <instanton number> is
$$
k=\int_{\mathbb R^4}C_2
=\int_{S^3_\infty}Y(-dg\,g^{-1})
=\boxed{\frac1{24\pi^2}
\int_{S^3_\infty}\operatorname{Tr}[(dg\,g^{-1})^3]}
=\deg(g),
$$
up 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.

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/iii/paper-313/2/c/solution