Modular differential equations and null vectors

TL;DR

The paper proves that modular differential equations in rational CFTs originate from null-vectors in the vacuum Verma module, via the Zhu algebra and the $O_q(V)$ construction. It shows how such null-vectors can be reconstructed from the differential equations and discusses conditions under which they imply null-vectors at weight $2s$, with implications for extremal self-dual theories at $c=24k$. The Monster theory and tensor products are analyzed to illustrate counterexamples where holomorphic holomorphicity of coefficients fails, emphasizing the role of additional low-level null-vectors. Under holomorphicity, the extremal self-dual CFTs face potential contradictions for large $k$ (notably $k\, extgeq 42$), suggesting these theories may be inconsistent. Overall, the work connects modular constraints to deep structural features of vertex operator algebras and their representations, providing a framework to assess the viability of extremal theories in AdS$_3$/CFT$_2$ contexts.

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.