Needles in a haystack: An algorithmic approach to the classification of 4d $\mathcal{N}=2$ SCFTs

There is a well-known map from 4d $\mathcal{N}=2$ superconformal field theories (SCFTs) to 2d vertex operator algebras (VOAs). The 4d Schur index corresponds to the VOA vacuum character, and must be a solution with integral coefficients of a modular differential equation. This suggests a classification program for 4d $\mathcal{N}=2$ SCFTs that starts with modular differential equations and proceeds by imposing all known constraints that follow from the 4d $\to$ 2d map. This program becomes fully algorithmic once one specifies the $\mathrm{\textit{order}}$ of the modular differential equation and the $\mathrm{\textit{rank}}$ (complex dimension of the Coulomb branch) of the $\mathcal{N}=2$ theory. As a proof of concept, we apply the algorithm to the study of rank-two $\mathcal{N}=2$ SCFTs whose Schur indices satisfy a fourth-order untwisted modular differential equation. Scanning over a large number of putative cases, only 15 satisfy all of the constraints imposed by our algorithm, six of which correspond to known 4d SCFTs. More sophisticated constraints can be used to argue against the existence of the remaining nine cases. Altogether, this indicates that our knowledge of such rank-two SCFTs is surprisingly complete.