  We apply the ADHM instanton construction to SU(2) gauge theory on T^n x
R^(4-n)for n=1,2,3,4. To do this we regard instantons on T^n x R^(4-n) as
periodic (modulo gauge transformations) instantons on R^4. Since the R^4
topological charge of such instantons is infinite the ADHM algebra takes place
on an infinite dimensional linear space. The ADHM matrix M is related to a Weyl
operator (with a self-dual background) on the dual torus tilde T^n. We
construct the Weyl operator corresponding to the one-instantons on T^n x
R^(4-n). In order to derive the self-dual potential on T^n x R^(4-n) it is
necessary to solve a specific Weyl equation. This is a variant of the Nahm
transformation. In the case n=2 (i.e. T^2 x R^2) we essentially have an
Aharonov Bohm problem on tilde T^2. In the one-instanton sector we find that
the scale parameter, lambda, is bounded above, (lambda)^2 tv<4 pi, tv being the
volume of the dual torus tilde T^2.
