Flavored modular differential equations sometimes arise from null states or their descendants in a chiral algebra with continuous flavor symmetry. In this paper we focus on Kac-Moody algebras ${\widehat{\mathfrak{g}}}_k$ that contain a level-four null state $|{\mathcal{N}}_T⟩$ which implements the nilpotency of the Sugawara stress tensor. We study the properties of the corresponding flavored modular differential equations, and show that the equations exhibit almost covariance under modular $S$-transformation, connecting null states and their descendants at different levels. The modular property of the equations fixes the structure of $\mathfrak{g}$ and the level $k$, as well as the flavored characters of all the highest weight representations. Shift property of the equations can generate nonvacuum characters starting from the vacuum character.

• Received 9 August 2023
• Accepted 3 October 2023

DOI:https://doi.org/10.1103/PhysRevD.108.085027

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Published by the American Physical Society