  This paper establishes that the Nahm transform sending spatially periodic
instantons (instantons on the product of the real line and a three-torus) to
singular monopoles on the dual three-torus is indeed a bijection as suggested
by the heuristic. In the process, we show how the Nahm transform intertwines to
a Fourier-Mukai transform via Kobayashi-Hitchin correspondences. We also prove
existence and non-existence results.
