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We study finite action anti-self-dual Yang-Mills connections on themulti-Taub-NUT space. We establish the curvature and the harmonic spinors decayrates and compute the index of the associated Dirac operator.  This is the first in a series of papers proving the completeness of the bowconstruction of instantons on multi-Taub-NUT spaces and exploring it in detail.

Instantons on multi-Taub-NUT Spaces I: Asymptotic Form and Index Theorem