$A_{2l}^{(2)}$ at level $-l-\frac{1}{2}$

Abstract

Let $L_{l}=L(\mathfrak{sl}_{2l+1},-l-\frac{1}{2})$ be the simple vertex operator algebra based on the affine Lie algebra $\widehat{\mathfrak{sl}}_{2l+1}$ at boundary admissible level $-l-\frac{1}{2}$. We consider a lift $\nu$ of the Dynkin diagram involution of $A_{2l}=\mathfrak{sl}_{2l+1}$ to an involution of $L_{l}$. The $\nu$-twisted $L_l$-modules are $A_{2l}^{(2)}$-modules of level $-l-\frac{1}{2}$ with an anti-homogeneous realization. We classify simple $\nu$-twisted highest-weight (weak) $L_l$-modules using twisted Zhu algebras and singular vectors for $\widehat{\mathfrak{sl}}_{2l+1}$ at level $-l-\frac{1}{2}$ obtained by Per\v{s}e. We find that there are finitely many such modules up to isomorphism, and the $\nu$-twisted (weak) $L_l$-modules that are in category $\mathscr{O}$ for $A_{2l}^{(2)}$ are semi-simple.