Ribbon categories of weight modules for affine ${\mathfrak{sl}}_2$ at admissible levels

Abstract

We show that the braided tensor category of finitely-generated weight modules for the simple affine vertex operator algebra $L_k({\mathfrak{sl}}_2)$ of ${\mathfrak{sl}}_2$ at any admissible level $k$ is rigid and hence a braided ribbon category. The proof uses a recent result of the first two authors with Shimizu and Yadav on embedding a braided Grothendieck-Verdier category $\mathcal{C}$ into the Drinfeld center of the category of modules for a suitable commutative algebra $A$ in $\mathcal{C}$, in situations where the braided tensor category of local $A$-modules is rigid. Here, the commutative algebra $A$ is Adamović's inverse quantum Hamiltonian reduction of $L_k({\mathfrak{sl}}_2)$, which is the simple rational Virasoro vertex operator algebra at central charge $1-\frac{6(k+1{)}^2}{k+2}$ tensored with a half-lattice conformal vertex algebra. As a corollary, we also show that the category of finitely-generated weight modules for the $N=2$ super Virasoro vertex operator superalgebra at central charge $-6\ell -3$ is rigid for $\ell$ such that $(\ell +1)(k+2)=1$.

34 pages