On the BPS Spectrum at the Root of the Higgs Branch

TL;DR

This paper investigates the BPS spectrum and wall-crossing phenomena in a four-dimensional $ ext{SU}(n)$ $ ext{N}=2$ supersymmetric gauge theory at the Higgs branch root, confirming conjectures and revealing new bound states across coupling regimes.

Contribution

It applies the Kontsevich--Soibelman wall-crossing formula to extend the conjectured strong-coupling spectrum to all moduli space regions, matching semiclassical results and uncovering new bound states.

Findings

Strong-coupling spectrum matches conjecture from hep-th/9902134.

Weak-coupling spectrum includes a tower of bound states under certain mass conditions.

Bound states depend on mass symmetry, with distinct patterns for odd and even n.

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 $n\le N_{f}<2n$ fundamental flavours at the root of the Higgs branch. The strong-coupling spectrum of this theory was conjectured in hep-th/9902134 to coincide with that of the two-dimensional supersymmetric $\mathbb{CP}^{2n-N_{f}-1}$ sigma model. Using the Kontsevich--Soibelman wall-crossing formula, we start with the conjectured strong-coupling spectrum and extrapolate it to all other regions of the moduli space. In the weak-coupling regime, our results precisely agree with the semiclassical analysis of hep-th/9902134: in addition to the usual dyons, quarks, and $W$ bosons, if the complex 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. In the special case of $\mathbb{Z}_{n}$-symmetric masses, there are bound states with one quark for odd $n$ and no bound states for even $n$.