Self-dual strings and N = 2 supersymmetric field theory

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

Nonsupersymmetric flux vacua and perturbed 𝒩 = 2 systems

“…Compactifying on such a Calabi-Yau leaves N = 2 supersymmetry unbroken in four dimensions. Important roles in these Calabi-Yau's are played by the underlying one-dimensional complex curve in the x, y-plane defined by F (x, y) = 0 [25,45]. In most of our examples this curve is smooth, and we will refer to it as the Riemann surface Σ.…”

“…First of all, the holomorphic 3-form Ω of M Σ , which is given e.g. for x, y ∈ C by (3.2), is easily seen to reduce to a meromorphic 1-form λ = y dx on the Riemann surface in this case [25,45]. The special coordinates (2.1) parametrizing complex structure moduli are 4) and the Kähler potential (2.8) is given by…”

“…The holomorphic 3-form on M SW is Ω SW = du u ∧ dx ∧ dy y and reduces to [25] λ SW = x dy y (3.16) on the Riemann surface Σ SW .…”

“…The first step is achieved by the standard geometric engineering of N = 2 gauge theories by IIB on noncompact Calabi-Yau manifolds [25,26]. It is well known that the moduli space of the Calabi-Yau compactification encodes the geometry of the Coulomb branch of the gauge theory and that the Seiberg-Witten solution can be rederived by the complex geometry of the Calabi-Yau.…”

Mirror symmetry and aG2flop

“…As discussed in [30] this gives rise to a type IIA description involving NS 5-brane wrapping the Riemann surface Σ given by F (u, v) = 0 and filling R 4 . Without the extra D5 brane, this was used in [26] to derive the Seiberg-Witten vacuum geometry of N = 2 systems, where Σ was identified with the Seiberg-Witten Riemann surface. The only additional ingredient here is the fact that we also have an extra D5 brane, filling the z-plane.…”