Quantum dimensions and fusion products for irreducible VQσ-modules with σ2 = 1

Abstract

ABSTRACTEvery isometry σ of a positive-definite even lattice Q can be lifted to an automorphism of the lattice vertex algebra VQ. An important problem in vertex algebra theory and conformal field theory is to classify the representations of the σ-invariant subalgebra of VQ, known as an orbifold. In the case when σ is an isometry of Q of order two, we have classified the irreducible modules of the orbifold vertex algebra and identified them as submodules of twisted or untwisted VQ-modules in [6]. Here we calculate their quantum dimensions and fusion products.The examples where Q is the orthogonal direct sum of two copies of the A2 root lattice and σ is the 2-cycle permutation as well as where Q is the An root lattice and σ is a Dynkin diagram automorphism are presented in detail.