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

We calculate the fusion rules among ${\mathbb{Z}}_2$-twisted modules $L_{{\mathfrak{s}\mathfrak{l}}_2}(\mathrm{\ell },0)$ 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 $k=-1\mathrm{/}2$, $A_2^{(1)}$ at boundary admissible level $k=-3\mathrm{/}2$, and ${\mathrm{B}\mathrm{P}}^k$-algebra with special value $k=-9\mathrm{/}4$. 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.