Killing vectors in self-dual, Euclidean Einstein spaces

Source abstract

Using the formalism of complex ℋ-spaces, we show that all real, Euclidean self-dual spaces that admit (at least) one Killing vector may be gauged so that only two distinct types of Killing vectors appear; in Kähler coordinates these are the generators of a translational or a rotational symmetry. We give explicit forms both for the Killing vectors and for the constraint on the Kähler potential function Ω which allows for such a Killing vector. In the translational case we show how all such spaces are determined by the general solution of the three-dimensional, flat Laplace’s equation and how these are related to the multi-Taub–NUT metrics of Gibbons and Hawking. In the rotational case we simplify the equation determining Ω, but this is not sufficient to obtain the general solution.