Abstract

Coset constructions of ${{\mathcal{W}}}$-algebras have many applications and were recently given for principal ${{\mathcal{W}}}$-algebras of $A$, $D$, and $E$ types by Arakawa together with the 1st and 3rd authors. In this paper, we give coset constructions of the large and small $N=4$ superconformal algebras, which are the minimal ${{\mathcal{W}}}$-algebras of ${{\mathfrak{d}}}(2,1;a)$ and ${{\mathfrak{p}}}{{\mathfrak{s}}}{{\mathfrak{l}}}(2|2)$, respectively. From these realizations, one finds a remarkable connection between the large $N=4$ algebra and the diagonal coset $C^{k_1, k_2} = \textrm{Com}(V^{k_1+k_2}({{\mathfrak{s}}}{{\mathfrak{l}}}_2), V^{k_1}({{\mathfrak{s}}}{{\mathfrak{l}}}_2) \otimes V^{k_2}({{\mathfrak{s}}}{{\mathfrak{l}}}_2))$, namely, as two-parameter vertex algebras, $C^{k_1, k_2}$ coincides with the coset of the large $N=4$ algebra by its affine subalgebra. We also show that at special points in the parameter space, the simple quotients of these cosets are isomorphic to various ${{\mathcal{W}}}$-algebras. As a corollary, we give new examples of strongly rational principal ${{\mathcal{W}}}$-algebras of type $C$ at degenerate admissible levels.

中文翻译：

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N = 4 个超共形代数和对角陪集

摘要

${{\mathcal{W}}}$-代数的陪集结构有很多应用，最近给出了 $A$、$D$ 和 $E 的主要 ${{\mathcal{W}}}$-代数$ 类型由 Arakawa 与第一和第三作者一起。在本文中，我们给出了大小 $N=4$ 超共形代数的陪集构造，它们是 ${{\mathfrak{d}}}( 2,1;a)$ 和 ${{\mathfrak{p}}}{{\mathfrak{s}}}{{\mathfrak{l}}}(2|2)$。从这些实现中，我们发现大型 $N=4$ 代数和对角陪集 $C^{k_1, k_2} = \textrm{Com}(V^{k_1+k_2}({{\mathfrak{ s}}}{{\mathfrak{l}}}_2), V^{k_1}({{\mathfrak{s}}}{{\mathfrak{l}}}_2) \otimes V^{k_2}( {{\mathfrak{s}}}{{\mathfrak{l}}}_2))$，即作为二参数顶点代数，$C^{k_1，k_2}$ 通过其仿射子代数与大 $N=4$ 代数的陪集一致。我们还表明，在参数空间的特殊点，这些陪集的简单商同构于各种 ${{\mathcal{W}}}$-代数。作为推论，我们给出了在退化可接受水平上的 $C$ 类型的强理性主 ${{\mathcal{W}}}$-代数的新例子。