Instantons on multi-Taub-NUT Spaces III: Down Transform, Completeness,
and Isometry

The index bundle of a family of Dirac operators associated to an instanton on a multi-Taub-NUT space forms a bow representation. We prove that the gauge equivalence classes of solutions of this bow representation are in one-to-one correspondence with the instantons. We also prove that this correspondence establishes an isometry of the bow and instanton moduli spaces.