$A_{2l}^{(2)}$ 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 Perše. We find that there are finitely many such modules up to isomorphism, and the $\nu$-twisted (weak) $L_l$-modules that are in category $\mathcal{O}$ for $A_{2l}^{(2)}$ are semi-simple.

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