Constraints on extremal self-dual CFTs

TL;DR

This work links modular differential equations to a vacuum-null-vector condition in Zhu's $C_2$ quotient for rational CFTs, and tests the idea across minimal models, WZW theories, the Monster, and self-dual $c=24$ theories. It argues that the minimal differential equation order equals the chiral algebra size $s$, with $L_{-2}^s\Omega\in O_{[2]}$, and uses this to predict null vectors at low levels for extremal $c=24k$ theories. Applying the method to extremal candidates shows that for $k\ge 42$ the required null-vector would appear below the expected level, signaling inconsistency, though low-$k$ genus-two checks do not fully rule out all scenarios. The results constrain the landscape of extremal self-dual CFTs and point to possible modifications of the proposed $AdS_3$ dual chiral theory at large $k$, while preserving some consistency in smaller cases.

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.