Asymptotic behaviour and the Nahm transform of doubly periodic instantons with square integrable curvature

We study the asymptotic behaviour of doubly periodic instantons with square-integrable curvature. Then we establish an equivalence given by the Nahm transform between the doubly periodic instantons with square integrable curvature and wild harmonic bundles on the dual torus. We also introduce algebraic Nahm transforms, which describe the transformations of the underlying filtered objects.