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We compute the Coulomb branch partition function of the 4d $\mathcal{N}=2$vector multiplet on closed simply-connected quasi-toric manifolds $B$. Thisincludes a large class of theories, localising to either instantons oranti-instantons at the torus fixed points (including Donaldson-Witten andPestun-like theories as examples). The main difficulty is to obtain fluxcontributions from the localisation procedure. We achieve this by taking adetour via the 5d $\mathcal{N}=1$ vector multiplet on closed simply-connectedtoric Sasaki-manifolds $M$ which are principal $S^1$-bundles over $B$. Theperturbative partition function can be expressed as a product over slices ofthe toric cone. By taking finite quotients $M/\mathbb{Z}_h$ along the $S^1$,the locus picks up non-trivial flat connections which, in the limit$h\to\infty$, provide the sought-after fluxes on $B$. We compute the one-looppartition functions around each topological sector on $M/\mathbb{Z}_h$ and $B$explicitly, and then factorise them into contributions from the torus fixedpoints. This enables us to also write down the conjectured instanton part ofthe partition function on $B$.

From 5d Flat Connections to 4d Fluxes (the Art of Slicing the Cone)