Fusion rules for ℤ2-orbifolds of affine and parafermion vertex operator algebras

The structure of parafermion vertex operator algebras $K(osp(1|2n),k)$

“…We see that the building blocks of parafermion vertex operator algebras K(osp(1|2n), k) are K(osp(1|2), k) and K(sl 2 , 2k). The structural and representation theory for K(sl 2 , k) are studied in [9], [13], [5], [20], [21] etc., and the representation theory for K(osp(1|2), k) are studied in [6]. These may shed light on the study of the representation theory for rational parafermion vertex operator algebras K(osp(1|2n), k).…”

“…Also in [7], Creutzig and Linshaw proved the rationality of the affine vertex operator superalgebra L ĝ(k, 0) with k being a positive integer. The structural and representation theory of the rational parafermion vertex operator algebras associated to the integrable highest weight modules of affine Kac-Moody Lie algebras and their orbifolds have been fully studied (see [1,3,4,8,9,11,12,13,14,20,21,24,27] etc.) In this paper, we turn our attention to the rational parafermion vertex operator algebras associated to the affine vertex superalgebras.…”

“…The generator result shows that the parafermion vertex operator algebra K(osp(1|2), k) associated to osp(1|2) together with K(sl 2 , 2k) associated to sl 2 are building blocks of K(osp(1|2n), k). The structural and representation theories for K(sl 2 , k) are studied in [9], [13], [5], [20], [21] etc. And the representation theory for K(osp(1|2), k) are studied in [6].…”

Fusion rules for the orbifold vertex operator algebras $L_{\widehat{\frak{sl}_2}}(k,0)^{K}$

“…It is very natrual and desirable to study the structure and representations of the orbifold vertex operator algebra L sl 2 (k, 0) G . If |G| = 2, the irreducible modules and fusion rules of L sl 2 (k, 0) G are characterized in [JW20]. If G is the Klein subgroup K of Aut(L sl 2 (k, 0)) generated by involutions σ 1 and σ 2 , which are defined by σ 1 (h(−1)½) = h(−1)½, σ 1 (e(−1)½) = −e(−1)½, σ 1 (f (−1)½) = −f (−1)½, and σ 2 (h(−1)½) = −h(−1)½, σ 2 (e(−1)½) = f (−1)½, σ 2 (f (−1)½) = e(−1)½, respectively, the irreducible modules of L sl 2 (k, 0) K are classified in [JWa21], where {e, f, h} is a standard basis of sl 2 (C).…”

“…Then we can decompose the irreducibule modules of L sl 2 (k, 0) K into direct sum of irreducible modules of V + Zγ ⊗K + 0 . By using the fusion rules obtained in [Ab01] and [JW20] we get the desired fusion products, where V + Zγ and K + 0 are the orbifold subalgebras of lattice vertex operator algbera and parafermion vertex operator algbera with respect to the involution σ 2 and the Heisenberg Lie algebra Ch, respectively. The case that k is an odd integer is easier to deal with.…”