Geometric constraints on the space of N=2 SCFTs II: Construction of special K\"ahler geometries and RG flows

This is the second in a series of three papers on systematic analysis of rank 1 Coulomb branch geometries of four dimensional $\mathcal{N}$=2 SCFTs. In the first paper we developed a strategy for classifying physical rank-1 CB geometries of $\mathcal{N}$=2 SCFTs. Here we show how to carry out this strategy computationally to construct the Seiberg-Witten curves and one-forms for all the rank-1 SCFTs. Explicit expressions are given for all cases, with the exception of the $N_f$=4 SU(2) gauge theory and the En SCFTs which were previously constructed. Our classification includes all known rank-1 theories plus a new one with an abelian flavor group, plus nine additional theories whose existence is more speculative. Four of those, reported in our first paper, depend on the assumption of new frozen rank-1 SCFTs. Here we also also show that the assumption of the existence of certain rank-0 $\mathcal{N}$=2 SCFTs leads to five additional consistent rank-1 CB geometries.