Nahm Transform for Integrable Connections on the Riemann Sphere

The topic treated in this book fits into the broader program of a study of properties of instantons (i.e. self-dual solutions of Yang-Mills field equations) on 4-manifolds. In this context, the Nahm transform is relating solutions on two different manifolds that are dual to each other. Its properties are in many aspects similar to properties of the standard Fourier transform. Dual manifolds are constructed as the quotient of X=R4 / Λ of R4 by a close additive subgroup Λ, and as the quotient X*=(R4)*/ Λ*, where Λ* is the dual subgroup. Self-dual solutions of the Yang-Mills equations on R4 invariant with respect to Λ form the space, which is an analogue of the space of real functions for the standard Fourier transform. The same definition works for the dual version. The Nahm transform then maps solutions on X to solutions on X*. A lot of special cases have already been studied and are well understood. The author treats here the case where X and X* are projective spaces of dimension 1. In this situation, it is necessary to allow solutions with singularities; the transform then works under suitable conditions on the singularity behaviour of solutions.

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