Coulomb branches for quaternionic representations

Abstract

I describe the \emph{Chiral rings} $R_{3,4}$ for $3$D, $N=4$ supersymmetric $G$-gauge theory and matter fields in quaternionic representations $E$: first, by a topological tweak of the construction of arxiv:1601.03586, and second, more explicitly, by Weyl group descent from the maximal torus. A topological obstruction is $w_4(E)$ modulo squares, for $R_3$; a secondary obstruction, from $\eta\cdot E$, may appear for $R_4$. Flatness over the Toda bases allows their calculation by reduction to $\mathrm{SU}_2$. For some representations, an Abelianization formula describes the $R$ in terms of the maximal torus and the Weyl group. This provides an alternative to a recent attempt arxiv:2201.09475.