Ribbon categories of weight modules for affine $\mathfrak{sl}_2$ at admissible levels
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2024
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| author | Creutzig, Thomas McRae, Robert Yang, Jinwei |
| author_facet | Creutzig, Thomas McRae, Robert Yang, Jinwei |
| contents | 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 $(\ell+1)(k+2) = 1$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_11386 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Ribbon categories of weight modules for affine $\mathfrak{sl}_2$ at admissible levels Creutzig, Thomas McRae, Robert Yang, Jinwei Quantum Algebra High Energy Physics - Theory Mathematical Physics Representation Theory 17B67, 17B69, 18M15, 81R10, 81T40 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 $(\ell+1)(k+2) = 1$. |
| title | Ribbon categories of weight modules for affine $\mathfrak{sl}_2$ at admissible levels |
| topic | Quantum Algebra High Energy Physics - Theory Mathematical Physics Representation Theory 17B67, 17B69, 18M15, 81R10, 81T40 |
| url | https://arxiv.org/abs/2411.11386 |