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

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