Self-Dual Strings and N=2 Supersymmetric Field Theory

TL;DR

This paper explores the geometric origin of N=2 supersymmetric Yang-Mills theory via string compactifications, revealing a dual formulation with self-dual strings on a Riemann surface and analyzing BPS states as geodesics.

Contribution

It introduces a geometric framework linking string theory, Riemann surfaces, and BPS states, providing a novel dual perspective on N=2 Yang-Mills theory.

Findings

Self-dual strings on Riemann surface encode BPS states.

Spectrum of stable BPS states determined by geodesic analysis.

Identification of BPS states with two-branes ending on a five-brane.

Abstract

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 {\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.