This paper is about the orbifold theory of affine and parafermion vertex operator algebras. It is known that the parafermion vertex operator algebra K(sl2,k) associated to the integrable highest weight modules for the affine Kac—Moody algebra $$A_1^{(1)}$$ is the building block of the general parafermion vertex operator K( $$K(\mathfrak{g},k)$$ ,k) for any finite-dimensional simple Lie algebra $$\mathfrak{g}$$ and any positive integer k. We first classify the irreducible modules of ℤ2-orbifold of the simple affine vertex operator algebra of type $$A_1^{(1)}$$ and determine their fusion rules. Then we study the representations of the ℤ2-orbifold of the parafermion vertex operator algebra K(sl2, k). The quantum dimensions, and more technically, fusion rules for the ℤ2-orbifold of the parafermion vertex operator algebra K(sl2, k) are completely determined.

中文翻译：

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仿射和准费米子顶点算子代数的ℤ2-orbifolds的融合规则

这篇论文是关于仿射和准费米子顶点算子代数的orbifold理论。已知与仿射 Kac-Moody 代数 $$A_1^{(1)}$$ 的可积最高权重模块相关联的副费米子顶点算子代数 K(sl2,k) 是通用副费米子顶点算子的构建块K( $$K(\mathfrak{g},k)$$ ,k) 对于任何有限维简单李代数 $$\mathfrak{g}$$ 和任何正整数 k。我们首先对$$A_1^{(1)}$$类型的简单仿射顶点算子代数的ℤ2-orbifold的不可约模进行分类，并确定它们的融合规则。然后我们研究了准费米子顶点算子代数K(sl2, k)的ℤ2-orbifold的表示。准费米子顶点算子代数 K(sl2,