High-temperature expansion of the Schur index and modularity

A bstract High-temperature (q → 1) asymptotics of 4d superconformal indices of Lagrangian theories have been recently analyzed up to exponentially suppressed corrections. Here we use RG-inspired tools to extend the analysis to the exponentially suppressed terms in the context of Schur indices of N=2 N = 2 SCFTs. In particular, our approach explains the curious patterns of logarithms (polynomials in 1 / log q) found by Dedushenko and Fluder in their numerical study of the high-temperature expansion of rank-1 theories. We also demonstrate compatibility of our results with the conjecture of Beem and Rastelli that Schur indices satisfy finite-order, possibly twisted, modular linear differential equations (MLDEs), and discuss the interplay between our approach and the MLDE approach to the high-temperature expansion. The expansions for q near roots of unity are also treated. A byproduct of our analysis is a proof (for Lagrangian theories) of rationality of the conformal dimensions of all characters of the associated VOA, that mix with the Schur index under modular transformations.