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. 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 | λ| 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 R⁴ × σ as the world-volume of a five-brane and show that BPS states correspond to two-branes ending on this five-brane.