ADHM construction of instantons on the torus

We apply the ADHM instanton construction to SU(2) gauge theory on T-n x R4-n for n = 1, 2, 3, 4. To do this we regard instantons an T-n x R4-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 (T) over tilde (n). We construct the Weyl operator corresponding to the one-instantons on Tn x R4-n. In order to derive the self-dual potential on T-n x R4-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 (T) over tilde (2). In the one-instanton sector we find that the scale parameter, lambda, is bounded above, lambda (2)(V) over tilde < 4<pi>, (V) over tilde being the volume of the dual torus (T) over tilde (2). (C) 2001 Elsevier Science B.V, All rights reserved.