Self-Dual Strings and N=2 Supersymmetric Field Theory

We show how the Riemann surface Σ 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 λ. The existence of BPS states is essentially reduced to a geodesic problem on the Riemann surface with metric |λ|2. This allows us, in particular, to determine the spectrum of stable BPS states in field theory. Moreover, we identify the six-dimensional space IR × Σ as the world-volume of a five-brane and show that BPS states correspond to two-branes ending on this five-brane.