Orthosymplectic quivers: indices, Hilbert series, and generalised symmetries

We investigate generalised global symmetries in 3d \( \mathcal{N}=4 \) orthosymplectic quiver gauge theories. Using the superconformal index, we identify a D₈ 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 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 (O(N)^±, Spin(N), 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[SO(N)] and T[USp(2N)] theories. The analysis also readily generalises to the T_ρ[SO(N)] and T_ρ[USp(2N)] theories associated with partition ρ.