Discrete Gauging of 5d $\mathcal{N}=1$ $E_n$ SCFTs

Abstract

We study the gauging of discrete symmetries of the 5d rank-one $E_n$ SCFTs that fix the Coulomb branch coordinate but act nontrivially on the Higgs branch chiral ring. Using generalized toric polygons and $(p,q)$-brane webs, we identify symmetries of the moduli space that extend to symmetries of the full SCFT. In many cases they are non-anomalous, and we analyze the moduli space of the gauged theories. We realize the resulting Higgs branch quotients as Coulomb branches of wreathed magnetic quivers and compute their Hilbert series; the flavor algebras extracted from moment maps agree with the fixed-point subalgebras of $\mathfrak{e}_n$. We also lift the discrete actions to the Seiberg--Witten geometry and recover the expected restriction on the mass deformations. For $\mathbb{Z}_2$-wreathings that exchange adjacent gauge nodes, we find that the monopole formula requires a flux-dependent sign and propose a corrected prescription. Finally, we conjecture a higher-rank extension and verify it for the rank-two $E_6/\mathbb{Z}_2$ Higgs branch.