ADHM construction of instantons on the torus Original Research Article

We apply the ADHM instanton construction to SU(2) gauge theory on Tn×R4−n for n=1,2,3,4. To do this we regard instantons on Tn×R4−n as periodic (modulo gauge transformations) instantons on R4. Since the R4 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 T̃n. We construct the Weyl operator corresponding to the one-instantons on Tn×R4−n. In order to derive the self-dual potential on Tn×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., T2×R2) we essentially have an Aharonov–Bohm problem on T̃2. In the one-instanton sector we find that the scale parameter, λ, is bounded above, λ2Ṽ<4π, Ṽ being the volume of the dual torus T̃2.