ADHM Construction of Instantons on the Torus

TL;DR

This paper extends the ADHM construction to instantons on torus-structured spaces, relating the problem to Weyl operators and Nahm transformation, and provides bounds on instanton parameters in specific cases.

Contribution

It develops a method to construct instantons on T^n x R^{4-n} using an infinite-dimensional ADHM algebra and relates it to Weyl operators on the dual torus, including explicit bounds.

Findings

Constructed Weyl operators for one-instantons on T^n x R^{4-n}

Related instanton construction to a variant of the Nahm transformation

Found an upper bound for the scale parameter in the T^2 x R^2 case

Abstract

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.