Modular anomaly equation for Schur index of N $$ \mathcal{N} $$ = 4 super-Yang-Mills

Abstract We propose a novel modular anomaly equation for the unflavored Schur index in the N $$ \mathcal{N} $$ = 4 SU(N) super-Yang-Mills theory. The vanishing conditions overdetermine the modular ambiguity ansatz from the equation, thus together they are sufficient to recursively compute the exact Schur indices for all SU(N) gauge groups. Using the representations as MacMahon’s generalized sum-of-divisors functions and Jacobi forms, we then prove our proposal as well as elucidate a general formula conjectured by Pan and Peelaers.