Z 3 Symmetry and W 3 Algebra in Lattice Vertex Operator Algebras

The vertex operator algebras associated with positive definite even lattices afford a large family of known examples of vertex operator algebras. An isometry of the lattice induces an automorphism of the lattice vertex operator algebra. The subalgebra of fixed points of an automorphism is the so-called orbifold vertex operator algebra. In this paper we deal with the case where the lattice L = √ 2A2 is √ 2 times an ordinary root lattice of type A2 and the isometry τ is an element of the Weyl group of order 3. We use this algebra to study the W3 algebra of central charge 6/5. In fact, by using both coset construction and orbifold theory we construct the W3 algebra of central charge 6/5 inside VL and classify its irreducible modules. We also prove that the W3 algebra is rational and compute the characters of the irreducible modules. The vertex operator algebra VL associated with L = √ 2A2 contains three mutually orthogonal conformal vectors ω1, ω2, ω3 with central charge c = 1/2, 7/10, or 4/5 respectively [10]. The subalgebra Vir(ωi) generated by ωi is the Virasoro vertex operator algebra L(c, 0), which is the irreducible unitary highest weight module for the Virasoro algebra with central charge c and highest weight 0. The structure of VL as a module for Vir(ω 1) ⊗ Vir(ω2)⊗ Vir(ω3) was discussed in [23]. Among other things it was shown that VL contains a subalgebra of the form L(4/5, 0) ⊕ L(4/5, 3). Such a vertex operator algebra is called a 3-state Potts model. This subalgebra is contained in the subalgebra (VL) τ of fixed points of τ . There is another