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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 charge two 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 hyperKahler 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.

Dk Gravitational Instantons as Superpositions of Atiyah–Hitchin and Taub–NUT Geometries