Self-dual strings and N=2 supersymmetric field theory

We show how the Riemann surface \Sigma of N=2 Yang-Mills field theory arises in type II string compactifications on Calabi-Yau threefolds. The relevant local geometry is given by fibrations of ALE spaces. The 3-branes that give rise to BPS multiplets in the string descend to self-dual strings on the Riemann surface, with tension determined by a canonically fixed Seiberg-Witten differential \lambda. The existence of BPS states is essentially reduced to a geodesic problem on the Riemann surface with metric |\lambda|^2. This allows us in particular, to determine the spectrum of {\it stable} BPS states in field theory. Moreover, we identify the six-dimensional space \IR^4\times \Sigma as the world-volume of a five-brane and show that BPS states correspond to two-branes ending on this five-brane.