Conformal symmetries of self-dual hyperbolic monopole metrics

Abstract

We determine the group of conformal automorphisms of the self-dual metrics on $n \# \mathbb{CP}^{2}$ due to LeBrun for $n \geq 3$, and Poon for $n = 2$. These metrics arise from an ansatz involving a circle bundle over hyperbolic three-space $\mathcal{H}^{3}$ minus a finite number of points, called monopole points. We show that for $n \geq 3$, any conformal automorphism is a lift of an isometry of $\mathcal{H}^{3}$ which preserves the set of monopole points. Furthermore, we prove that for $n = 2$, such lifts form a subgroup of index $2$ in the full automorphism group, which we show to be a semi-direct product $(\mathrm{U}(1) \times \mathrm{U}(1)) \rtimes \mathrm{D}_{4}$, where $\mathrm{D}_{4}$ is the dihedral group of order $8$.