Conformal symmetries of self-dual hyperbolic monopole metrics

TL;DR

This paper characterizes the conformal automorphism groups of self-dual hyperbolic monopole metrics, revealing their structure as lifts of hyperbolic isometries and detailing the automorphism group for the case n=2.

Contribution

It determines the conformal automorphism groups of LeBrun and Poon's self-dual metrics, showing their relation to hyperbolic isometries and explicitly describing the automorphism group for n=2.

Findings

For n>2, automorphisms lift hyperbolic isometries preserving monopole points.

For n=2, automorphisms form a subgroup of index 2, with a semi-direct product structure.

Automorphism group for n=2 is (U(1) × U(1)) × D_4.

Abstract

We determine the group of conformal automorphisms of the self-dual metrics on n#CP^2 due to LeBrun for n>2, and Poon for n=2. These metrics arise from an ansatz involving a circle bundle over hyperbolic three-space H^3 minus a finite number of points, called monopole points. We show that for n>2 connected sums, any conformal automorphism is a lift of an isometry of 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 is a semi-direct product (U(1) \times U(1)) \times D_4, the dihedral group of order 8.