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 {\it stable} BPS states in field theory. Moreover, we identify the six-dimensional space $\IR^4\times Σ$ as the world-volume of a five-brane and show that BPS states correspond to two-branes ending on this five-brane.

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