Hilbert series and mixed branches of $T[SU(N)]$ theory

TL;DR

This paper analyzes mixed branches of 3d $ ext{N}=4$ $T[SU(N)]$ theory by computing Hilbert series for Coulomb and Higgs branches using a restriction rule, confirmed by brane picture and mirror symmetry.

Contribution

It introduces a restriction rule for computing Hilbert series of mixed branches, connecting magnetic charges to brane positions and applying mirror symmetry for Higgs branches.

Findings

Restriction rule accurately computes Coulomb branch Hilbert series.

Results agree with alternative calculation methods.

Brane picture provides a geometric understanding of magnetic charge restrictions.

Abstract

We consider mixed branches of 3d $\mathcal{N}=4$ $T[SU(N)]$ theory. We compute the Hilbert series of the Coulomb branch part of the mixed branch from a restriction rule acting on the Hilbert series of the full Coulomb branch that will truncate the magnetic charge summation only to the subset of BPS dressed monopole operators that arise in the Coulomb branch sublocus where the mixed branch stems. This restriction can be understood directly from the type IIB brane picture by a relation between the magnetic charges of the monopoles and brane position moduli. We also apply the restriction rule to the Higgs branch part of a given mixed branch by exploiting 3d mirror symmetry. Both cases show complete agreement with the results calculated by different methods.