Geometric constraints on the space of N=2 SCFTs III: enhanced Coulomb branches and central charges

This is the third in a series of three papers on the systematic analysis of rank 1 four dimensional $\mathcal{N}=2$ SCFTs. In the first two papers we developed and carried out a strategy for classifying and constructing physical planar rank-1 Coulomb branch geometries of $\mathcal{N}=2$ SCFTs. Here we describe general features of the Higgs and mixed branch geometries of the moduli space of these SCFTs, and use this, along with their Coulomb branch geometry, to compute their conformal and flavor central charges. We conclude with a summary of the state of the art for rank-1 $\mathcal{N}=2$ SCFTs.