I review recent work of Heckman, Morrison, Vafa on a new approach to classify superconformal field theories in 6D using F-theory on non-compact Calabi-Yau threefolds. F-theory setups with shrinkable curves in their non-compact two-dimensional base are believed to yield 6D supersymmetric effective theories, that are superconformal at strong coupling. Thus, the classification of all 6D SCFTs that can arise in this way reduces to a classification of all possible configurations of shrinkable curves in the two-dimensional base so that a Calabi-Yau elliptic fibration still exists. This classification relies heavily on the method of non-Higgsable clusters. All the “minimal" bases arise as the blow-ups of C^2/Gamma orbifolds with Gamma certain discrete subgroups of U(2). Non-minimal bases are obtained by blowing up -1-curves corresponding to E-string theories and further equipping existing curve configurations with non-minimal gauge algebras. It is argued that the F-theory geometry not only encodes the gauge algebra and (conformal) matter content of these SCFTs, but also allows the study of flows between different SCFTs and dualities between them.
