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

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 $ν$ of the Dynkin diagram involution of $A_{2l}=\mathfrak{sl}_{2l+1}$ to an involution of $L_{l}$. The $ν$-twisted $L_l$-modules are $A_{2l}^{(2)}$-modules of level $-l-\frac{1}{2}$ with an anti-homogeneous realization. We classify simple $ν$-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 $ν$-twisted (weak) $L_l$-modules that are in category $\mathscr{O}$ for $A_{2l}^{(2)}$ are semi-simple.

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