Spectral Flow, Twisted Modules, and MLDE of Quasi-Lisse Vertex Algebras

Abstract

We calculate the dimensions of intertwining operators among highest weight modules and their contragredient modules of $L_k({\mathfrak{sl}}_2)$ at the admissible level. We derive a series of ${\Gamma }^0(2)$-MLDEs satisfied by normalized characters of some ordinary twisted modules of some quasi-lisse vertex algebras. Examples include simple affine vertex algebras $L_{-2+2/u}({\mathfrak{sl}}_2)$ ($u=3,5,7$), $L_{-1/2}({\mathfrak{sl}}_2)$, $L_{-3/2}({\mathfrak{sl}}_3)$, $L_{-2}({\mathfrak{so}}_8)$, and the Bershadsky–Polyakov algebra ${\mathrm{BP}}^{-9/4}$. We also derive the full characters of some nonvacuum modules of $L_{-2}({\mathfrak{so}}_8)$ using the spectral flow automorphism and SCFT/VOA correspondence.