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. This gives, effectively, a dual formulation of Yang-Mills theory in which gauge bosons and monopoles are treated on equal footing, and represents the rigid analog of type II/heterotic string duality. 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 easily determine the spectrum of stable BPS states in field theory. Moreover, we identify the six-dimensional space R(4) X Sigma as the world-volume of a five-brane and show that BPS states correspond to two-branes ending on this five-brane.