Asymptotic behaviour and the moduli space of doubly-periodic instantons

Abstract

We study doubly-periodic instantons, i.e. instantons on the product of a 1-dimensional complex torus T with a complex line ³, with quadratic curvature decay. We determine the asymptotic behaviour of these instantons, constructing new asymptotic invariants. We show that the underlying holomorphic bundle extends to T2À1. The converse statement is also true, namely a holomorphic bundle on T2À1 which is flat on the torus at infinity, and satisfies a stability condition, comes from a doubly-periodic instanton. Finally, we study the hyperkähler geometry of the moduli space of doubly-periodic instantons, and prove that the Nahm transform previously defined by the second author is a hyperkähler isometry with the moduli space of certain meromorphic Higgs bundles on the dual torus.