$D_k$ Gravitational Instantons as Superpositions of Atiyah-Hitchin and Taub-NUT Geometries

Abstract

We obtain $D_k$ ALF gravitational instantons by a gluing construction which captures, in a precise and explicit fashion, their interpretation as non-linear superpositions of the moduli space of centred $SU(2)$ monopoles, equipped with the Atiyah-Hitchin metric, and $k$ copies of the Taub-NUT manifold. The construction proceeds from a finite set of points in euclidean space, reflection symmetric about the origin, and depends on an adiabatic parameter which is incorporated into the geometry as a fifth dimension. Using a formulation in terms of hyperKähler triples on manifolds with boundaries, we show that the constituent Atiyah-Hitchin and Taub-NUT geometries arise as boundary components of the 5-dimensional geometry as the adiabatic parameter is taken to zero.

44 pages, 6 figures, to appear in a special issue of The Quarterly Journal of Mathematics dedicated to Sir Michael Atiyah; presentation improved and typos corrected