Integrability, Jacobians and Calabi-Yau threefolds

The integrable systems associated with Seiberg-Witten geometry are considered both from the Hitchin-Donagi-Witten gauge model and in terms of intermediate Jacobians of Calabi-Yau threefolds. Dual pairs and enhancement of gauge symmetries are discussed on the basis of a map from the Donagi-Witten “moduli” into the moduli of complex structures of the Calabi-Yau threefold. Based on a lecture given by C. Gomez at the “VIII Regional Meeting on Mathematical Physics”, October 1995, Iran. Looking for a unified point of view to describe the great amount of exciting new results appearing in the last few months in the context of string theory can be considered at this moment premature and maybe a pure question of taste. Nevertheless, in this lecture we would like to present a conjectural unified approach supported by some evidence [1] and based on the mathematical structure of integrable systems [2, 3, 4]. 1 Integrability and Seiberg-Witten geometry A very elegant way to describe Seiberg-Witten geometry [5, 6] for N = 2 non abelian gauge theories, is [3] by means of families of complex tori [4]. Let π : X → U (1) with fiberXu (u ∈ U) a complex tori of dimension g. We will assume thatX is a symplectic manifold possessing a closed non degenerate holomorphic two form w and that dim U is equal to g. Given a basis γi(u) i = 1, .., g of H1(Xu, Z), we define “special coordinates” ai(u) = ∮