12-08 Notes

We consider the Yang-Mills equations in four dimensions.

Let \((X,g)\) be a Riemannian \(4\)-manifold. Let \(\mathcal{A}\) be the space of connections for some principal bundle over \(X\). For \(A\in \mathcal{A}\), we have two quantities: the Yang-Mills energy \[\mathcal{E}(A) = -\int_X \mathrm{tr}\left(F_A \wedge *F_A\right) = \|F_A\|^2_{L^2}\] and a characteristic number independent of \(A\) (if \(X\) is closed) \[c(A) = -\int_X \mathrm{tr}\left(F_A \wedge F_A\right).\]

Here, \(*\) is the Hodge star operator \(*:\Omega^2(X)\rightarrow \Omega^2(X)\) induced by the metric \(g\). At a point \(x \in X\), we have \(*:\Lambda^2 T^*_x X \rightarrow \Lambda^2 T_x X\). We have \(*^2 = 1\), so we get a splitting into \(\pm 1\)-eigenspaces of \(*\): \[\Lambda^2 = \Lambda^2_+ \oplus \Lambda^2_-.\] We call the former self-dual (SD) and the latter anti-self-dual (ASD). If \(e^1\), \(e^2\), \(e^3\), \(e^4\) is an orthonormal frame for \(T^* X\), then \[\begin{aligned} e^1\wedge e^2 \pm e^3 \wedge e^4 \\ e^1\wedge e^3 \mp e^2 \wedge e^4 \\ e^1\wedge e^4 \pm e^2 \wedge e^3 \end{aligned}\] form a basis for \(\Lambda^2_{\pm}\) (i.e. are self-dual and anti-self-dual respectively).

Applying this decomposition to the curvature, we have \(F_A = F_A^+ + F_A^-\). Because \(\Lambda^2_+\) and \(\Lambda^2_-\) are orthogonal, and thus \(F_A^+\) and \(F_A^-\) are pointwise orthogonal, we have \[\mathcal{E}(A) = \|F_A^+\|^2_{L^2} + \|F_A^-\|^2_{L^2}\] and in addition we get \[c(A) = \|F_A^+\|^2_{L^2} - \|F_A^-\|^2_{L^2}\] by examination. Thus we have

Lemma 1. \(\mathcal{E}(A) \geq |c(A)|\) with equality if and only if \(F_A\) is purely self-dual (if \(c(A) \geq 0\)) or purely anti-self-dual (if \(c(A) \leq 0\)).

From this we see that purely SD or ASD solutions are Yang-Mills (YM) solutions:

Lemma 2. If \(F_A\) is purely self-dual or purely anti-self-dual, then it is Yang-Mills.

Proof. Method 1: If \(F_A = \mp *F_A\) then \[d_A^* F_A = -*d_A*F_A = \mp *d_AF_A = 0\] (by Bianchi), so \(F_A\) is a Yang-Mills solution. Method 2: If \(F_A\) is purely self-dual or purely anti-self-dual, then by the above Lemma, \(A\) is a minimal point for \(\mathcal{E}(A)\), and thus is a critical point of \(\mathcal{E}(A)\), i.e. a Yang-Mills solution. ◻