On the Haagerup Vertex Operator Algebras

Abstract

Three constructions of vertex operator algebras realizing the Haagerup center were announced independently and almost simultaneously. The relations between these constructions are studied in this paper. The generators are expressed in a common lattice basis for $E_6\oplus A_2$. It is proved that the constructions of Riesen and Huang give isomorphic vertex operator algebras, and that this algebra and the algebra constructed by Jiao have isomorphic fixed point subalgebras under automorphisms of order two. The isomorphisms are given explicitly on generators. The common fixed point subalgebra contains the parafermion vertex operator algebras $K(\mathfrak{sl}_2,3)$ and $K(\mathfrak{sl}_2,13)$. A general rationality criterion for unitary vertex operator algebras is given in terms of positive linear combinations of conformal vectors and is applied to these constructions. The two vertex operator algebras are simple current extensions of the common fixed point subalgebra and have equivalent module categories as ribbon categories. The equivalence follows from decomposition with respect to a nondegenerate pointed subcategory with four simple objects in the module category of the common fixed point subalgebra. It is also proved that the conformal nets associated with these unitary vertex operator algebras are completely rational.