Unlocking the Wronskian Tower: A Simplification of the Holomorphic Modular Bootstrap

Characters of rational conformal field theories solve modular linear differential equations labelled by their order and the Wronskian index $\mathrm{\ell }$. Direct classification of admissible solutions by solving MLDEs becomes increasingly difficult at higher $\mathrm{\ell }$ – where movable poles and accessory parameters appear. In this work we introduce differential operators that relate higher-$\mathrm{\ell }$ solutions to lower-$\mathrm{\ell }$ ones while preserving modular covariance and integrality of the $q$-series. In rank two, this generates all allowed Wronskian sectors from the Mathur–Mukhi–Sen equation. In rank three and higher, it reduces the construction of higher-$\mathrm{\ell }$ quasi-characters to simpler equations with lower $\mathrm{\ell }$. This gives an efficient new route for organising candidate RCFT characters, and more generally quasi-characters, across the Wronskian tower. As an application, we apply our construction to prove a previously conjectured property on the signs of $\mathrm{\ell }=2$ quasi-characters in rank 2.