Mirror Symmetry and Level-rank Duality for 3d $\mathcal{N}=4$ Rank 0 SCFTs

We introduce a family of 3d $\mathcal{N}=4$ superconformal field theories that have zero-dimensional Coulomb and Higgs branches and propose that the rational vertex operator algebras $W_{k-\frac{1}{2}}^{\text{min}}({\mathfrak{s}\mathfrak{p}}_{2N})$ and $L_k({\mathfrak{o}\mathfrak{s}\mathfrak{p}}_{1\mathrm{\mid }2N})$ model the modular tensor categories of line operators in their topological $A$ and $B$ twists, respectively. Our analysis indicates that the action of 3d mirror symmetry on this family of theories is related to a novel level-rank duality and leads to several conjectural $q$-series identities of independent interest.