I encounter recently admissible affine Lie algebras when visiting some physics problems. I am reading Adamovic's A construction of admissible $A_1^{(1)}$-modules of level $−4/3$.
In section 3, it is claimed that there is a unique maximal ideal generated by a singular vector (here I translated to the $J^\pm, J^3$ basis, $T$ is the stress tensor/Virasoro element, and $(..)$ means normal-ordered product) $$ v^+_\text{sing} = (J^+ T) + \frac{1}{6} \partial^2 J^+ - (J^3 \partial J^+) + (\partial J^3 J^+) \ . $$
By direct computation, I also observe that
there are two more vectors $v_\text{sing}^3, v_\text{sing}^-$, that together with $v^+_\text{sing}$ form an $\operatorname{SU}(2)$ triplet.
All three vectors comes from a next level state ($K$ being the Killing form) $$ \chi = (TT) - \frac{1}{4} K_{ab}(\partial^2 J^a J^b) - \frac{3}{4}K_{ab}(\partial J^a \partial J^b) \ , $$ by $v^a_\text{sing} = J^a_1 \chi$.
My question is:
Given the relation with the singular vectors, should one expect $\chi$ to lead to vanishing torus one-point function (along the line of Gaberdiel and Keller - Modular differential equations and null vectors), $$ \operatorname{tr} o(\chi) q^{L_0 - c/24} b^{J^3_0} = 0? $$
(By direct application of Zhu's recursion formula, it seems it does. For example, when $b = 1$, it gives the equation (see Arakawa and Kawasetsu - Quasi-lisse vertex algebras and modular linear differential equations) $$ (D_q^{(2)} - 15E_4) \operatorname{ch}_0 = 0 \ . $$ But I don't know how to understand in the framework of Gaberdiel and Keller - Modular differential equations and null vectors.)
