Asymptotic behaviour and the moduli space of doubly-periodic instantons

We study doubly-periodic instantons, i.e. instantons on the product of a1-dimensional complex torus T with a complex line C, with quadratic curvaturedecay. We determine the asymptotic behaviour of these instantons, constructingnew asymptotic invariants. We show that the underlying holomorphic bundleextends to TxP1. The converse statement is also true, namely a holomorphicbundle on TxP1 which is flat on the torus at infinity, and satisfies astability condition, comes from a doubly-periodic instanton. Finally, we studythe hyperkahler geometry of the moduli space of doubly-periodic instantons, andprove that the Nahm transform previously defined by the second author is ahyperkahler isometry with the moduli space of certain meromorphic Higgs bundleson the dual torus.