On the BPS spectrum at the root of the Higgs branch

Abstract

We study the BPS spectrum and walls of marginal stability of the $\mathcal{N} = 2$ supersymmetric theory in four dimensions with gauge group SU(n) and n ≤ N _f < 2n fundamental flavours at the root of the Higgs branch considering only magnetic charges (. . . , 0, 0, −1, 1, 0, 0, . . . ), corresponding to the lightest towers of dyons at weak coupling. The strong-coupling spectrum of this theory was conjectured in hep-th/9902134 to coincide with that of the two-dimensional supersymmetric $\mathbb{C}{\mathbb{P}^{{2n - {N_f} - 1}}}$ sigma model. Using the Kontsevich-Soibelman wall-crossing formula, we start with the conjectured strong-coupling spectrum and extrapolate it to the weak coupling; we restrict our attention to the special case of ${\mathbb{Z}_n}$-symmetric masses, where we find the walls of marginal stability explicitly. In the weak-coupling regime, our results precisely agree with the semiclassical analysis of hep- th/9902134 for any values of the complex masses: in addition to the usual dyons, quarks, and W bosons, if the masses obey a particular inequality, the resulting weak-coupling spectrum includes a tower of bound states consisting of a dyon and one or more quarks. For ${\mathbb{Z}_n}$-symmetric masses, there are bound states with one quark for odd n and no bound states for even n.