Orthosymplectic quivers: indices, Hilbert series, and generalised symmetries

A bstract We investigate generalised global symmetries in 3d $$ \mathcal{N}=4 $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> <mml:mo>=</mml:mo> <mml:mn>4</mml:mn> </mml:math> orthosymplectic quiver gauge theories. Using the superconformal index, we identify a D 8 categorical symmetry web in a class of theories featuring $$ \mathfrak{so}(2N)\times \mathfrak{usp}(2N) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>so</mml:mi> <mml:mfenced> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>N</mml:mi> </mml:mrow> </mml:mfenced> <mml:mo>×</mml:mo> <mml:mi>usp</mml:mi> <mml:mfenced> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>N</mml:mi> </mml:mrow> </mml:mfenced> </mml:math> 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(2 N )] theories. The analysis also readily generalises to the T ρ [SO( N )] and T ρ [USp(2 N )] theories associated with partition ρ .