Orthosymplectic Quivers: Indices, Hilbert Series, and Generalised Symmetries

TL;DR

This paper explores the structure of generalised symmetries in 3d $ abla=4$ orthosymplectic quiver gauge theories, improving computational methods for Coulomb branch Hilbert series and analyzing symmetry mappings under mirror symmetry.

Contribution

It introduces an improved prescription for computing Coulomb branch Hilbert series that includes discrete zero-form symmetries and background fluxes, and applies this to orthosymplectic quivers and mirror symmetry analysis.

Findings

Identified a $D_8$ categorical symmetry web in specific theories.

Extended Hilbert series computation methods to include discrete symmetries.

Verified the methods through examples and symmetry mapping under mirror symmetry.

Abstract

We investigate generalised global symmetries in 3d $\mathcal{N}=4$ orthosymplectic quiver gauge theories. Using the superconformal index, we identify a $D_8$ categorical symmetry web in a class of theories featuring $\mathfrak{so}(2N) \times \mathfrak{usp}(2N)$ gauge algebra (at zero Chern-Simons levels) and $n$ bifundamental half-hypermultiplets, analogous to ABJ-type models. As a distinct contribution, we improve the prescription, previously studied in the literature, for computing Coulomb branch Hilbert series of $\mathrm{SO}(N)$ gauge theories with $N_f$ vector hypermultiplets. Our improved prescription extends these methods by incorporating fugacities for discrete zero-form symmetries - specifically charge conjugation and magnetic symmetries - and properly treating background magnetic fluxes for the flavour symmetry. This refinement enables calculations for various global forms ($\mathrm{O}(N)^\pm, \mathrm{Spin}(N), \mathrm{Pin}(N)$) and ensures consistency with the Coulomb branch limit of the superconformal index and known dualities. The proper treatment of fluxes is particularly essential for analysing orthosymplectic quivers where such a flavour symmetry is gauged. We verify our methods through several examples, including an analysis of the mapping of discrete symmetries under mirror symmetry for $T[\mathrm{SO}(N)]$ and $T[\mathrm{USp}(2N)]$ theories. The analysis also readily generalises to the $T_\rho[\mathrm{SO}(N)]$ and $T_\rho[\mathrm{USp}(2N)]$ theories associated with partition $\rho$.