Modular Differential Equations with Movable Poles and Admissible RCFT Characters

TL;DR

This paper explores modular linear differential equations with movable poles for classifying rational conformal field theory characters, extending previous work to more general cases with multiple poles and establishing new relations and solutions.

Contribution

It introduces a parametrization of MLDEs with arbitrary poles, derives constraints for admissible solutions, and demonstrates the existence of genuine CFTs in these broader settings.

Findings

Parameter count for general MLDEs with poles

Recursion relations between solutions with different poles

Existence of genuine CFTs corresponding to new cases

Abstract

Studies of modular linear differential equations (MLDE) for the classification of rational CFT characters have been limited to the case where the coefficient functions (in monic form) have no poles, or poles at special points of moduli space. Here we initiate an exploration of the vast territory of MLDEs with two characters and any number of poles at arbitrary points of moduli space. We show how to parametrise the most general equation precisely and count its parameters. Eliminating logarithmic singularities at all the poles provides constraint equations for the accessory parameters. By taking suitable limits, we find recursion relations between solutions for different numbers of poles. The cases of one and two movable poles are examined in detail and compared with predictions based on quasi-characters to find complete agreement. We also comment on the limit of coincident poles. Finally we show that there exist genuine CFT corresponding to many of the newly-studied cases. We emphasise that the modular data is an output, rather than an input, of our approach.