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

Factorisation of 3d $\mathcal{N}=4$ Twisted Indices and the Geometry of Vortex Moduli Space