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Abstract The theory of topological modular forms (TMF) predicts that elliptic genera of physical theories satisfy a certain divisibility property, determined by the theory’s gravitational anomaly. In this note we verify this prediction in Duncan’s supermoonshine module, as well as in tensor products and orbifolds thereof. Along the way we develop machinery for computing the elliptic genera of general alternating orbifolds and discuss the relation of this construction to the elusive “periodicity class” of TMF. 

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 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 $$ \mathcal{N} $$ N = 2 SCFTs that starts with modular differential equations and proceeds by imposing all known constraints that follow from the 4d → 2d map. This program becomes fully algorithmic once one specifies the order of the modular differential equation and the rank (complex dimension of the Coulomb branch) of the $$ \mathcal{N} $$ N = 2 theory. As a proof of concept, we apply the algorithm to the study of rank-two $$ \mathcal{N} $$ 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.