论文标题
正交类型的广义副作用
Generalized parafermions of orthogonal type
论文作者
论文摘要
仿射顶点代数$ v^k（\ mathfrak {gl} _n）\ hookrightArrow v^k（\ mathfrak {sl} _ {n+1}）$嵌入v^k（\ mathfrak {sl} _ {n+1}））$是$ \ mathfrak {sl} _2 $的parafermion代数的自然概括。它被第三作者称为广义派象的代数，并被证明是通用的两参数$ \ Mathcal {w} _ {\ infty} $ - 类型$ \ Mathcal {w}（w}（w}（2,3，\ dots）的单参数的单参数商。在本文中，我们考虑了正交类型的类似结构，即$ \ Mathcal {d}^k（n）= \ text {com}（v^k（\ mathfrak {so} _ {2n} _ {2n}），v^k（\ mathfrak {so} so} _ {so} _ {2n+1}）我们将这个代数视为两参数的单参数商，甚至是旋转$ \ MATHCAL {w} _ {\ infty} $ - 类型$ \ Mathcal {W}（2,4，\ dots）的类型的代数 - 我们对其简单的$ \ nater $ \ MathCal calbras and n andbr a} n anderge进行了分类。 $ \ MATHCAL {W} _ {\ ELL}（\ Mathfrak {so} _ {2m+1}）$和$ \ Mathcal {w} _ {\ ell}（\ Ell}（\ Mathfrak {\ Mathfrak {so} _} _ {2m} _ {2m}）作为推论，我们表明，对于可允许的级别，$ k = - （2n-2） + \ frac {1} {2} {2 n + 2 m -1）$ for $ \ wideHat {\ mathfrak {\ mathfrak {so}}} _ {2n} $ l_k（\ Mathfrak {so} _ {2n+1}）$，coset非常合理。结果，在这样一个级别上，$ l_k（\ mathfrak {so} _ {2n+1}）的普通模块的类别是一个编织的融合类别。
There is an embedding of affine vertex algebras $V^k(\mathfrak{gl}_n) \hookrightarrow V^k(\mathfrak{sl}_{n+1})$, and the coset $\mathcal{C}^k(n) = \text{Com}(V^k(\mathfrak{gl}_n), V^k(\mathfrak{sl}_{n+1}))$ is a natural generalization of the parafermion algebra of $\mathfrak{sl}_2$. It was called the algebra of generalized parafermions by the third author and was shown to arise as a one-parameter quotient of the universal two-parameter $\mathcal{W}_{\infty}$-algebra of type $\mathcal{W}(2,3,\dots)$. In this paper, we consider an analogous structure of orthogonal type, namely $\mathcal{D}^k(n) = \text{Com}(V^k(\mathfrak{so}_{2n}), V^k(\mathfrak{so}_{2n+1}))^{\mathbb{Z}_2}$. We realize this algebra as a one-parameter quotient of the two-parameter even spin $\mathcal{W}_{\infty}$-algebra of type $\mathcal{W}(2,4,\dots)$, and we classify all coincidences between its simple quotient $\mathcal{D}_k(n)$ and the algebras $\mathcal{W}_{\ell}(\mathfrak{so}_{2m+1})$ and $\mathcal{W}_{\ell}(\mathfrak{so}_{2m})^{\mathbb{Z}_2}$. As a corollary, we show that for the admissible levels $k = -(2n-2) + \frac{1}{2} (2 n + 2 m -1)$ for $\widehat{\mathfrak{so}}_{2n}$ the simple affine algebra $L_k(\mathfrak{so}_{2n})$ embeds in $L_k(\mathfrak{so}_{2n+1})$, and the coset is strongly rational. As a consequence, the category of ordinary modules of $L_k(\mathfrak{so}_{2n+1})$ at such a level is a braided fusion category.
