Modular anomaly equation for Schur index of N=4 super-Yang-Mills

We propose a novel modular anomaly equation for the unflavored Schur index in the 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.

Automated summary

A novel modular anomaly equation is proposed for calculating the unflavored Schur index in the N = 4 SU(N) super-Yang-Mills theory. The equation overdetermines the modular ambiguity ansatz with vanishing conditions, enabling recursive computation of the exact Schur indices for all SU(N) gauge groups. Using MacMahon's generalized sum-of-divisors functions and Jacobi forms as representations, our proposal is proven, along with an elucidation of a general formula conjectured by Pan and Peelaers.