The rank 2 classification problem II: mapping scale-invariant solutions to SCFTs

This is the second of a series of papers outlining an approach to the classification of $\mathcal{N}{=}2$ superconformal field theories at rank 2 via a systematic analysis of their Coulomb branches, mathematically described by special K\"ahler scale invariant geometries. Here we describe how to make the translation between geometry and field theory data. We apply this strategy to the special K\"ahler geometries found in the first paper of the series where we made strong simplifying assumptions on the form of the solutions. Remarkably, we find that our bottom-up classification strategy pays off even in this simplified setup. All scale invariant solutions in the first paper of the series have an interpretation as the Coulomb branch of at least one $\mathcal{N}{=}2$ superconformal field theory, many matching what is already known. But we also predict the existence of six new rank 2 theories, including an entire new set closed under renormalization group flow. We characterise and discuss in detail each one of these new $\mathcal{N}{=}2$ superconformal field theories. To our knowledge none of them have known string theoretic or higher dimensional realisations.