Ordinary modules for vertex algebras of $\mathfrak{osp}_{1|2n}$
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| Format: | Preprint |
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2022
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| author | Creutzig, Thomas Genra, Naoki Linshaw, Andrew |
| author_facet | Creutzig, Thomas Genra, Naoki Linshaw, Andrew |
| contents | We show that the affine vertex superalgebra $V^k(\mathfrak{osp}_{1|2n})$ at generic level $k$ embeds in the equivariant $\mathcal W$-algebra of $\mathfrak{sp}_{2n}$ times $4n$ free fermions. This has two corollaries: (1) it provides a new proof that for generic $k$, the coset $\text{Com}(V^k(\mathfrak{sp}_{2n}), V^k(\mathfrak{osp}_{1|2n}))$ is isomorphic to $\mathcal W^\ell(\mathfrak{sp}_{2n})$ for $\ell = -(n+1) + \frac{k+n+1}{2k+2n+1}$, and (2) we obtain the decomposition of ordinary $V^k(\mathfrak{osp}_{1|2n})$-modules into $V^k(\mathfrak{sp}_{2n}) \otimes \mathcal W^\ell(\mathfrak{sp}_{2n})$-modules. Next, if $k$ is an admissible level and $\ell$ is a non-degenerate admissible level for $\mathfrak{sp}_{2n}$, we show that the simple algebra $L_k(\mathfrak{osp}_{1|2n})$ is an extension of the simple subalgebra $L_k(\mathfrak{sp}_{2n}) \otimes {\mathcal W}_{\ell}(\mathfrak{sp}_{2n})$. Using the theory of vertex superalgebra extensions, we prove that the category of ordinary $L_k(\mathfrak{osp}_{1|2n})$-modules is a semisimple, rigid vertex tensor supercategory with only finitely many inequivalent simple objects. It is equivalent to a certain subcategory of $\mathcal W_\ell(\mathfrak{sp}_{2n})$-modules. A similar result also holds for the category of Ramond twisted modules. Due to a recent theorem of Robert McRae, we get as a corollary that categories of ordinary $L_k(\mathfrak{sp}_{2n})$-modules are rigid. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2203_08188 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Ordinary modules for vertex algebras of $\mathfrak{osp}_{1|2n}$ Creutzig, Thomas Genra, Naoki Linshaw, Andrew Representation Theory High Energy Physics - Theory Quantum Algebra We show that the affine vertex superalgebra $V^k(\mathfrak{osp}_{1|2n})$ at generic level $k$ embeds in the equivariant $\mathcal W$-algebra of $\mathfrak{sp}_{2n}$ times $4n$ free fermions. This has two corollaries: (1) it provides a new proof that for generic $k$, the coset $\text{Com}(V^k(\mathfrak{sp}_{2n}), V^k(\mathfrak{osp}_{1|2n}))$ is isomorphic to $\mathcal W^\ell(\mathfrak{sp}_{2n})$ for $\ell = -(n+1) + \frac{k+n+1}{2k+2n+1}$, and (2) we obtain the decomposition of ordinary $V^k(\mathfrak{osp}_{1|2n})$-modules into $V^k(\mathfrak{sp}_{2n}) \otimes \mathcal W^\ell(\mathfrak{sp}_{2n})$-modules. Next, if $k$ is an admissible level and $\ell$ is a non-degenerate admissible level for $\mathfrak{sp}_{2n}$, we show that the simple algebra $L_k(\mathfrak{osp}_{1|2n})$ is an extension of the simple subalgebra $L_k(\mathfrak{sp}_{2n}) \otimes {\mathcal W}_{\ell}(\mathfrak{sp}_{2n})$. Using the theory of vertex superalgebra extensions, we prove that the category of ordinary $L_k(\mathfrak{osp}_{1|2n})$-modules is a semisimple, rigid vertex tensor supercategory with only finitely many inequivalent simple objects. It is equivalent to a certain subcategory of $\mathcal W_\ell(\mathfrak{sp}_{2n})$-modules. A similar result also holds for the category of Ramond twisted modules. Due to a recent theorem of Robert McRae, we get as a corollary that categories of ordinary $L_k(\mathfrak{sp}_{2n})$-modules are rigid. |
| title | Ordinary modules for vertex algebras of $\mathfrak{osp}_{1|2n}$ |
| topic | Representation Theory High Energy Physics - Theory Quantum Algebra |
| url | https://arxiv.org/abs/2203.08188 |