On quasi-lisse affine vertex algebras $L_{-1-h\mathrm{/}6}(\mathfrak{g})$ from rank-one Argyres-Douglas theories

We study certain simple affine vertex algebras $L_{-1-h\mathrm{/}6}(\mathfrak{g})$ from rank-one Argyres-Douglas theories, where $\mathfrak{g}$ is a non-simply-laced simple Lie algebra and $h$ is its Coxeter number. For $L_{-3}(F_4)$, we determine the maximal ideal of $V^{-3}(F_4)$, Zhu’s algebra, and the associated variety. We also classify the irreducible highest-weight $L_{-3}(F_4)$-modules and determine the characters of its irreducible ordinary modules. The associated variety is ${\bar{\mathbb{O}}}_{{\tilde{A}}_1}$, and the level $-3$ is totally collapsing for the orbit ${\mathbb{O}}_{{\tilde{A}}_1}$. The Poisson coinvariants of the $C_2$-algebra of $L_{-3}(F_4)$ are isomorphic to $\mathbb{C}[\mathrm{\ell }]\mathrm{/}({\mathrm{\ell }}^5)$, where $\mathrm{\ell }$ is induced by the conformal vector. Consequently, the characters of ordinary $L_{-3}(F_4)$-modules satisfy a fifth-order modular linear differential equation. We also give modular linear differential equations of orders five and eleven satisfied by the characters of ordinary modules for $L_{-2}(B_3)$ and $L_{-2}(G_2)$ respectively.