论文标题
$ \ mathfrak的顶点代数的普通模块{osp} _ {1 | 2n} $
Ordinary modules for vertex algebras of $\mathfrak{osp}_{1|2n}$
论文作者
论文摘要
我们表明，仿射顶点superalgebra $ v^k（\ mathfrak {osp} _ {1 | 2n}）$在equivariant $ \ mathcal $ \ mathcal w $ -algebra中嵌入$ k $ $ k $ $ k $ $ \ algebra，$ \ m mathfrak {sp} _ {2n} _ {2n} _ {2n} $ 4n $ 4n $ 4n $ 4n $ 4n $ 4n $ 4n。这有两种推论：（1）提供了一个新的证明，即对于通用$ k $，coset $ \ text {com}（v^k（\ mathfrak {sp} _ {2n}），v^k（\ mathfrak {osp} _ {osp} _ {1 | 2n}） w^\ ell（\ Mathfrak {sp} _ {2n}）$ for $ \ ell = - （n+1）+\ frac {k+n+n+1} {2k+2n+1} $，（2）我们获得了普通$ v^k的分解$ v^k（\ mathfrak {sp} _ {2n}）\ otimes \ mathcal w^\ ell（\ mathfrak {sp} _ {2n}）$ - 模块。接下来，如果$ k $是一个可接受的水平，而$ \ ell $是$ \ mathfrak {sp} _ {2n} $的不可替代水平$ l_k（\ Mathfrak {sp} _ {2n}）\ otimes {\ Mathcal W} _ {\ ell}（\ Mathfrak {sp} _ {2n}）$。使用顶点超级级扩展理论，我们证明了普通$ L_K（\ Mathfrak {osp} _ {1 | 2n}）$的类别 - 模块是一个半密度，刚性顶点tensor superCategory，只有许多与等量的简单对象，只有许多有限的简单对象。它等效于$ \ Mathcal w_ \ ell（\ Mathfrak {sp} _ {2n}）$ - 模块的某个子类别。 Ramond Twisted模块类别也有类似的结果。由于罗伯特·麦克雷（Robert McRae）的最新定理，我们得到了推论，即普通$ l_k（\ mathfrak {sp} _ {2n}）$ - 模块是刚性的。
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.
