Orbifolds of lattice vertex algebras under an isometry of order two

Every isometry σ of a positive-definite even lattice Q   can be lifted to an automorphism of the lattice vertex algebra VQVQ. An important problem in vertex algebra theory and conformal field theory is to classify the representations of the σ  -invariant subalgebra VQσ of VQVQ, known as an orbifold. In the case when σ is an isometry of Q   of order two, we classify the irreducible modules of the orbifold vertex algebra VQσ and identify them as submodules of twisted or untwisted VQVQ-modules. The examples where Q is a root lattice and σ is a Dynkin diagram automorphism are presented in detail.