ADHM Construction of Instantons on the Torus

Details ADHM Construction of Instantons on the Torus ADHM Construction of Instantons on the Torus

2000-05-24

arxiv.org/abs/hep-th/0005221v3

Nucl.Phys. B596 (2001) 387-414

Physics

High Energy Physics

High Energy Physics - Theory

35 pages, LATeX. New section on Nahm transform included, presentation improved, reference added, to appear in Nuclear Physics

Scientific paper

10.1016/S0550-3213(00)00704-5

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.

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