Supersymmetric sigma-models, twistors, and the Atiyah-Hitchin metric

The Legendre transform and its generalizations, originally found in supersymmetric σ -models, are techniques that can be used to give local constructions of hyperk¨ahler metrics. We give a twistor space interpretation to the generalizations of the Legendre transform construction. The Atiyah-Hitchin metric on the moduli space of two monopoles is used as a detailed example. from a few “flat space” ingredients. We propose more complete metrics based on these.