Argyres-Seiberg duality and the Higgs branch

TL;DR

The paper proves that the Higgs branches of the Argyres–Seiberg dual pair—the $ ext{SU}(3)$ theory with six quarks and the $ ext{SU}(2)$ theory coupled to the $ ext{SCFT}[E_6]$—coincide as hyperkähler cones. By modeling the SU(3) side with gauge-invariant operators $M,B, ilde B$ and the exceptional side with the minimal nilpotent orbit $ ext{O}_{ ext{min}}(E_6)$ realized through $X,Y,Z$ and quarks $v, ilde v$, the authors establish a precise operator dictionary that matches Poisson brackets, conjugation, and Joseph-like constraints. They map $ ext{hat}M^i{}_j o X^i{}_j$, $ ext{tr}M o -3(v ilde v)$, and $B^{ijk}=i(Y^{ijk}v)$, $ ilde B_{ijk}=i(Y_{ijk} ilde v)$, yielding consistent algebraic relations and constraints; the Higgs branches are further supported by numerical agreement of Kähler potentials. This provides strong evidence for the duality, with potential extensions to other E-type SCFTs and a direction toward a string-theoretic embedding.

Abstract

We demonstrate the agreement between the Higgs branches of two N=2 theories proposed by Argyres and Seiberg to be S-dual, namely the SU(3) gauge theory with six quarks, and the SU(2) gauge theory with one pair of quarks coupled to the superconformal theory with E_6 flavor symmetry. In mathematical terms, we demonstrate the equivalence between a hyperkaehler quotient of a linear space and another hyperkaehler quotient involving the minimal nilpotent orbit of E_6, modulo the identification of the twistor lines.