Constraints on extremal self-dual CFTs

TL;DR

This paper explores the implications of modular differential equations on extremal self-dual conformal field theories, suggesting their non-existence at high central charges based on algebraic constraints.

Contribution

It proposes a new link between modular differential equations and the structure of Zhu's C2 quotient space, providing evidence that certain extremal self-dual CFTs cannot exist for large k.

Findings

Modular differential equations imply a specific vector vanishes in Zhu's C2 space.

Evidence suggests extremal self-dual CFTs at c=24k do not exist for k≥42.

Numerous examples support the proposed connection.

Abstract

We argue that the existence of a modular differential equation implies that a certain vector vanishes in Zhu's C2 quotient space, and we check this assertion in numerous examples. If this connection is true in general, it would imply that the recently conjectured extremal self-dual conformal field theories at c=24 k cannot exist for k\geq 42.