We study permutation orbifolds of the $2$-fold and $3$-fold tensor product for the Virasoro vertex algebra $\mathcal{V}_c$ of central charge $c$. In particular, we show that for all but finitely many central charges $\left(\mathcal{V}_c^{\otimes 3}\right)^{\mathbb{Z}_3}$ is a $W$-algebra of type $(2, 4, 5, 6^3 , 7, 8^3 , 9^3 , 10^2 )$. We also study orbifolds of their simple quotients and obtain new realizations of certain rational affine $W$-algebras associated to a principal nilpotent element. Further analysis of permutation orbifolds of the celebrated $(2,5)$-minimal vertex algebra $\mathcal{L}_{-\frac{22}{5}}$ is presented.

中文翻译：

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Virasoro 顶点代数和 W-代数的置换轨道

我们研究了中心电荷 $c$ 的 Virasoro 顶点代数 $\mathcal{V}_c$ 的 $2$-fold 和 $3$-fold 张量积的排列轨道。特别地，我们证明对于所有但有限的中心电荷 $\left(\mathcal{V}_c^{\otimes 3}\right)^{\mathbb{Z}_3}$ 是 $W$-代数输入 $(2, 4, 5, 6^3 , 7, 8^3 , 9^3 , 10^2 )$。我们还研究了它们的简单商的 orbifolds，并获得了与主幂零元相关联的某些有理仿射 $W$-代数的新实现。进一步分析了著名的 $(2,5)$-最小顶点代数 $\mathcal{L}_{-\frac{22}{5}}$ 的排列轨道。