The Strong-Coupling Spectrum of the Seiberg-Witten Theory

TL;DR

The paper determines the exact strong-coupling BPS spectrum of N=2 SU(2) Yang–Mills by exploiting a ${f Z}_2$ symmetry of the Seiberg–Witten solution and the curve ${ t C}$ that separates strong and weak coupling. It shows that only the magnetic monopole and a unit-charge dyon survive in the strong region, while the weak region hosts the full tower of dyons $(n,1)$ and W bosons; crossing ${ t C}$ triggers controlled BPS decays that reconcile the two regimes. The analysis relies on the SW periods $(a_D,a)$, their ${ m Sp}(2,Z)$ duality properties, and the mass formula $m= ext{sqrt}(2)|a n_e - a_D n_m|$, together with a precise understanding of monodromies and the curve ${ t C}$. The results provide a non-semiclassical, global consistency check of the SW solution and pave the way for extensions to other gauge groups and matter content.

Abstract

We carefully study the global structure of the solution of the $N=2$ supersymmetric pure Yang-Mills theory with gauge group $SU(2)$ obtained by Seiberg and Witten. We exploit its ${\bf Z}_2$-symmetry and describe the curve in moduli space where BPS states can become unstable, separating the strong-coupling from the weak-coupling region. This allows us to obtain the spectrum of stable BPS states in the strong-coupling region: we prove that only the two particles responsible for the singularities of the solution (the magnetic monopole and the dyon of unit electric charge) are present in this region. Our method also permits us to very easily obtain the well-known weak-coupling spectrum, without using semi-classical methods. We discuss how the BPS states disintegrate when crossing the border from the weak to the strong-coupling region.

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