Abstract

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 R 4 × σ as the world-volume of a five-brane and show that BPS states correspond to two-branes ending on this five-brane.

Highlights

  • It is becoming increasingly clear that dualities in field theory and string theory are very strongly interrelated

  • There is one basic puzzle: The field theory results are naturally phrased in terms of a Riemann surface, and in some of the examples considered in [5] (for instance, one with an SU (3) gauge symmetry) this did not appear

  • We show that the BPS states of field theory can be best understood as the two-branes whose boundaries are self-dual strings on the Riemann surface

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Summary

Introduction

It is becoming increasingly clear that dualities in field theory and string theory are very strongly interrelated. There is one basic puzzle: The field theory results are naturally phrased in terms of a Riemann surface, and in some of the examples considered in [5] (for instance, one with an SU (3) gauge symmetry) this did not appear. Considering geodesics on the Riemann surface with the metric determined by this one-form allows one to explicitly study the spectrum of stable BPS states. The relationship between such self-dual strings and ordinary Yang-Mills field theory is the rigid analog of the duality [2] between type II and heterotic strings, and is, as we will show, a consequence of it.

K3-fibrations revisited
Geometrical Interpretation
Type IIB Perspective
Type IIA Perspective
Outlook
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