Boundaries, Vermas and factorisation

We revisit the factorisation of supersymmetric partition functions 3d $\mathcal{N}=4$ gauge theories. The building blocks are hemisphere a class UV $\mathcal{N}=(2,2)$ boundary conditions that mimic presence isolated vacua at infinity in real mass and FI parameters. These can be unambiguously defined computed using localisation. show certain limits these coincide with characters lowest weight Verma modules over quantised Higgs Coulomb branch chiral rings. This leads to expressions for superconformal index, twisted index $S^3$ function terms such characters. On way we uncover new connections between 't Hooft anomalies, weights modules.