Spectral flow, twisted modules and MLDE of quasi-lisse vertex algebras

We calculate the fusion rules among $\mathbb{Z}_2$-twisted modules $L_{\mathfrak{sl}_2}(\ell,0)$ at admissible levels. We derive a series MLDEs for normalized characters of ordinary twisted modules of quasi-lisse vertex algebras. Examples include affine VOAs of type $A_1^{(1)}$ at boundary admissible level, admissible level $k=-1/2$, $A^{(1)}_{2}$ at boundary admissible level $k=-3/2$, and $\mathrm{BP}^{k}$-algebra with special value $k=-9/4$. We also derive characters of some non-vacuum modules for affine VOA of type $D_4$ at non-admissible level $-2$ from spectral flow automorphism.