Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 22 / 5 / b / Solution

By definition $H_A^0=\ker d_A$ consists of parallel adjoint sections. Evaluation at one point identifies it with the subspace of $\mathfrak{su}(2)$ fixed by the full holonomy group. For holonomy equal to a maximal circle, write its elements as $\mathrm{diag}(\lambda ,{\lambda }^{-1})$. The fixed Lie algebra isThe off-diagonal real plane rotates with weight two and has no nonzero vector fixed by the whole circle. A parallel section is uniquely determined by its value, and every invariant value gives a globally well-defined parallel section by parallel transport. Therefore ${\dim }_{\mathbb{R}}{H}_A^0=1$. The full-circle holonomy hypothesis is essential: central holonomy would fix all of $\mathfrak{su}(2)$.