Nahm Transform and Spectral Curves¶for Doubly-Periodic Instantons

We present the Nahm transform of the doubly-periodic instantons (i.e. instantons on T²×&R;²), converting them into certain meromorphic solutions of Hitchin's equations over an elliptic curve. We then show how to construct a triple consisting of an algebraic curve plus a line bundle with connection over it from a doubly-periodic instanton, and that such data coincides with the Hitchin spectral data associated with the Nahm transformed Higgs bundle.