  We study the twisted indices of $\mathcal{N}=4$ supersymmetric gauge theories
in three dimensions on spatial $S^{2}$ with an angular momentum refinement. We
demonstrate factorisation of the index into holomorphic blocks for the
$T[SU(N)]$ theory in the presence of generic fluxes and fugacities. We also
investigate the relation between the twisted index, Hilbert series and the
moduli space of vortices. In particular, we show that each holomorphic block
coincides with a generating function for the $\chi_{t}$ genera of the moduli
spaces of "local" vortices. The twisted index itself coincides with a
corresponding generating function for the $\chi_{t}$ genera of moduli spaces of
"global" vortices in agreement with a proposal of Bullimore et. al. We
generalise this geometric interpretation of the twisted index to include fluxes
and Chern-Simons levels. For the $T[SU(N)]$ theory, the relevant moduli spaces
are the local and global versions of Laumon space respectively and we
demonstrate the proposed agreements explicitly using results from the
mathematical literature. Finally, we exhibit a precise relation between the
Coulomb branch Hilbert series and the Poincar\'e polynomials of the
corresponding vortex moduli spaces.

High Energy Physics - Theory