Modular differential equations and null vectors

TL;DR

This paper demonstrates that modular differential equations in rational conformal field theories originate from null vectors in the vacuum Verma module, impacting the understanding of extremal self-dual theories at specific central charges.

Contribution

It establishes a direct link between modular differential equations and null vectors, providing new insights into the structure of rational conformal field theories.

Findings

Modular differential equations derive from null vectors in the vacuum module.

Implications for the consistency of extremal self-dual conformal field theories at c=24k.

Clarifies the algebraic origin of differential equations in RCFTs.

Abstract

We show that every modular differential equation of a rational conformal field theory comes from a null vector in the vacuum Verma module. We also comment on the implications of this result for the consistency of the extremal self-dual conformal field theories at c=24 k.