2-cyclic permutations of lattice vertex operator algebras (Q2809177)

The focus of this paper is on vertex operator algebras of the form \((V_{\mathbb{Z}\alpha}\otimes V_{\mathbb{Z}\alpha})^{\mathbb{Z}_2}\), that is, \(2\)-cyclic permutation orbifold models for rank one lattice vertex operator algebras. Here, \(V_{\mathbb{Z}\alpha}\) is a rank one lattice vertex operator algebra with \(\langle \alpha ,\alpha\rangle =2k\) (\(k\) a positive integer) and \((V_{\mathbb{Z}\alpha}\otimes V_{\mathbb{Z}\alpha})^{\mathbb{Z}_2} = \{v\in V_{\mathbb{Z}\alpha}\otimes V_{\mathbb{Z}\alpha} \mid g(v)=v \text{ for any }g\in \mathbb{Z}_2\}\). The main results in this paper for such vertex operator algebras are (i) proving their rationality, (ii) classifying their irreducible modules, (iii) calculating the quantum dimensions, and (iii) obtaining the fusion rules. These are performed in Sections 4, 5, 6, and 7 of the paper, respectively. Additionally, beyond the introduction, Section 2 provides preliminary material while Section 3 details the vertex operator algebras \(V_{\mathbb{Z}\alpha}\) and \(V_{\mathbb{Z}\alpha}^+\).NEWLINENEWLINEThe proof of rationality rests on showing that \((V_{\mathbb{Z}\alpha}\otimes V_{\mathbb{Z}\alpha})^{\mathbb{Z}_2}\) is isomorphic to a simple current extension of a rational, \(C_2\)-cofinite vertex operator algebra of CFT-type, and thus rational. This isomorphism also proves useful in the classification of the irreducible modules, quantum dimensions, and fusion rules, as known fusion rules of irreducible modules in the decomposition of the vertex operator algebra are exploited in producing these nice results.NEWLINENEWLINEIn later works, the authors extend these results to higher rank lattice vertex operator algebras [J.\ Algebra 476, 1--25 (2017; Zbl 1395.17066)] and also consider the \(3\)-permutation orbifold of a rank one lattice vertex operator algebra [J.\ Pure Appl.\ Algebra 222, No.\ 6, 1316--1336 (2018; Zbl 1395.17067)].