$L$ and positive scalar curvature. This metric is invariant under the action of complex conjugation, and it gives rise to a conformally anti-self-dual metric on the bifold $\check
    S^{4}$ corresponding to $(S^{4}, \Psi_{1})$. For such a metric, the obstruction space $H^{2}_{A}$ in the deformation complex of an anti-self-dual bifold connection $A$ is trivial [@AHS Theorem 6.1], so the moduli space is zero-dimensional and consists of finitely many points, all of which are unobstructed solutions. If $A$ is such a bifold connection, consider its pull-back, $\tilde A$, on the double-cover $-\CP^{2}$, regarded as a bifold with singular locus $L$. The action of $\tilde A$ is $1/16$, and the dimension formula shows that the moduli space containing $\tilde
    A$ on this bifold has formal dimension $-1$. Since it is unobstructed, the solution must be reducible, and must therefore be an $\SO(2)$ connection, with holonomy $-1$ around the link of $L\subset
    -\CP^{2}$. There is a unique such $\SO(2)$ solution on $-\CP^{2}$ with the correct action, and it gives rise to a unique $O(2)$ connection on the original bifold.

In the case $n=3$, the bifold corresponding to $\Psi_{3}\subset
    S^{4}$ has a smooth $8$-fold cover, which is $-\CP^{2}$. The covering map is the quotient map 