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

Abstract: This paper is about the orbifold theory of affine and parafermion vertex operator algebras. It is known that the parafermion vertex operator algebra $K(sl_2,k)$ 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 $K(\mathfrak{g},k)$ for any finite dimensional simple Lie algebra $\mathfrak{g}$ and any positive integer $k$. We first classify the irreducible modules of $\mathbb{Z}{2}$-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 $\mathbb{Z}{2}$-orbifold of the parafermion vertex operator algebra $K(sl_2,k)$, we give the quantum dimensions, and more technically, fusion rules for the $\mathbb{Z}_{2}$-orbifold of the parafermion vertex operator algebra $K(sl_2,k)$ are completely determined.