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

TL;DR

This paper proves that the category of finitely-generated weight modules for certain affine vertex operator algebras at admissible levels is rigid and braided, using advanced categorical embedding techniques and algebraic reductions.

Contribution

It establishes the rigidity and braided ribbon structure of weight module categories for affine $rak{sl}_2$ at admissible levels via a novel embedding approach involving commutative algebra objects.

Findings

The category of modules is shown to be rigid and braided ribbon.

Embedding into the Drinfeld center is achieved through algebraic reduction.

Results apply to super Virasoro vertex operator superalgebras at specific central charges.

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 Adamovi\'{c}'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 $(\ell+1)(k+2) = 1$.