Mutation Symmetries in BPS Quiver Theories: Building the BPS Spectra

TL;DR

This paper explores the symmetries of BPS quiver mutations in 4D N=2 supersymmetric theories with ADE gauge groups, revealing how these symmetries determine BPS spectra at strong and weak couplings.

Contribution

It introduces the isotropy groups of quiver mutations for ADE gauge symmetries and connects these to Coxeter groups, providing new methods to analyze BPS spectra.

Findings

Strong coupling BPS chambers correspond to finite groupoid orbits.

Isotropy symmetry groups are isomorphic to dihedral groups Dih_{2h_{G}}.

Weak coupling mutation groups relate to infinite Coxeter groups, such as I_{2}^{inite}.

Abstract

We study the basic features of BPS quiver mutations in 4D $\mathcal{N}=2$ supersymmetric quantum field theory with $G=ADE$ gauge symmetries.\ We show, for these gauge symmetries, that there is an isotropy group $\mathcal{G}_{Mut}^{G}$ associated to a set of quiver mutations capturing information about the BPS spectra. In the strong coupling limit, it is shown that BPS chambers correspond to finite and closed groupoid orbits with an isotropy symmetry group $\mathcal{G}_{strong}^{G}$ isomorphic to the discrete dihedral groups $Dih_{2h_{G}}$ contained in Coxeter$(G) $ with $% h_{G}$ the Coxeter number of G. These isotropy symmetries allow to determine the BPS spectrum of the strong coupling chamber; and give another way to count the total number of BPS and anti-BPS states of $\mathcal{N}=2$ gauge theories. We also build the matrix realization of these mutation groups $% \mathcal{G}_{strong}^{G}$ from which we read directly the electric-magnetic charges of the BPS and anti-BPS states of $\mathcal{N}=2$ QFT$_{4}$ as well as their matrix intersections. We study as well the quiver mutation symmetries in the weak coupling limit and give their links with infinite Coxeter groups. We show amongst others that $\mathcal{G}_{weak}^{su_{2}}$ is contained in ${GL}({2,}\mathbb{Z}) $; and isomorphic to the infinite Coxeter ${I_{2}^{\infty}}$. Other issues such as building $\mathcal{G}%_{weak}^{so_{4}}$ and $\mathcal{G}_{weak}^{su_{3}}$ are also studied.