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This paper is about the orbifold theory of affine and parafermion vertexoperator algebras. It is known that the parafermion vertex operator algebra$K(sl_2,k)$ associated to the integrable highest weight modules for the affineKac-Moody algebra $A_1^{(1)}$ is the building block of the general parafermionvertex operator $K(\mathfrak{g},k)$ for any finite dimensional simple Liealgebra $\mathfrak{g}$ and any positive integer $k$. We first classify theirreducible modules of $\mathbb{Z}_{2}$-orbifold of the simple affine vertexoperator algebra of type $A_1^{(1)}$ and determine their fusion rules. Then westudy the representations of the $\mathbb{Z}_{2}$-orbifold of the parafermionvertex operator algebra $K(sl_2,k)$, we give the quantum dimensions, and moretechnically, fusion rules for the $\mathbb{Z}_{2}$-orbifold of the parafermionvertex operator algebra $K(sl_2,k)$ are completely determined.

Fusion rules for $\mathbb{Z}_{2}$-orbifolds of affine and parafermion vertex operator algebras