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

For the curvature $F=dA+A\wedge A$, define the Second Chern formUsing the graded cyclicity of the trace and $d^2=0$,whileExpanding $\mathrm{Tr}(F\wedge F)$ gives the same two terms with coefficient two on $dA\wedge A\wedge A$; the quartic term has vanishing trace by graded cyclicity. Thereforeso $\boxed{\alpha =2/3}$. This is the Chern-Simons 3-form.

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SetThe right Maurer-Cartan equation is $d\omega =\omega \wedge \omega$. Direct substitution givesso trace invariance proves ${\tilde{C}}_2=C_2$.

For the Chern-Simons 3-form, expansion and graded cyclicity giveSince cyclicity also givesthe required two-form may be chosen asThus

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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.

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