Nahm transform for integrable connections on the Riemann sphere

Abstract

In this work we are concerned with reduction of the ASD-equations to the Riemann sphere, that is integrable connections with a harmonic metric, or equivalently Higgs bundles with a Hermitian-Einstein metric. In the first chapter, we introduce the type of singularities we allow for the solutions to have, and announce the main results of the paper. In the second chapter, we study the analytical properties of the Dirac and Green's operators of our problem. In the third one, we carry out the main construction of the paper, namely that of the Nahm transform of an integrable connection with parabolic structure and with a compatible harmonic metric. In the fourth chapter, we interpret the construction from the point of view of Higgs bundles, and compute the effect of the transform on the level of topological and singularity parameters - this latter being achieved via an understanding of the behaviour of the spectral curve. Finally, in the fifth chapter we show that Nahm transform is (up to a sign) an involution.