Fixed point subalgebras of lattice vertex operator algebras by an automorphism of order three (Q392584)

In this article, the authors study the representation theory of the fixed point vertex sub-algebra of an automorphism of order \(3\) of certain lattice vertex operator algebra \(V_X\), where \(X\) contains a full sublattice isometric to \(\sqrt{2}A_2^n\).NEWLINENEWLINELet \(X\) be an even lattice such that \(X \supset \sqrt{2}A_2^n\) as a full rank sublattice. It is known that \(X\) can be constructed as a overlattice of \(\sqrt{2}A_2^n\) by using a self-orthogonal Kleinian code \(\mathcal{C}\) and a self orthogonal \(\mathbb{Z}_3\) code \(\mathcal{D}\) and \(X\) is denoted by \(L_{\mathcal{C}\times \mathcal{D}}\) in the article. Let \(\tau\) be an fixed point free isometry of order \(3\) of \(\sqrt{2}A_2\). Then \(\tau\) induces an isometry of order \(3\) on the dual lattice \([(\sqrt{2}A_2)^*]^n\). If the code \(\mathcal{C}\) is \(\tau\)-invariant, then \(\tau\) also acts on \(L_{\mathcal{C}\times \mathcal{D}} \).NEWLINENEWLINEThe main result of the article is the classification of all irreducible modules of the fixed point vertex operator algebra \(V_{L_{\mathcal{C}\times \mathcal{D}}}^\tau\) when \(\mathcal{C}=0\) and when both \(\mathcal{C}\) and \(\mathcal{D}\) are self-dual and the minimal weight of \(\mathcal{C}\) is \(4\). It is also shown that \(V_{L_{\mathcal{C}\times \mathcal{D}}}^\tau\) is simple, \(C_2\)-cofinite, rational, and of CFT type in these two cases.