= 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 $\operatorname{diag}(\lambda,\lambda^{-1})$. The fixed Lie algebra is
$$
\left\{\begin{pmatrix}it&0\\0&-it\end{pmatrix}:t\in\mathbb R\right\}.
$$
The 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 \b[$\dim_{\mathbb R}H_A^0=1$]. The full-circle holonomy hypothesis is essential: central holonomy would fix all of $\mathfrak{su}(2)$.
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