Quantum dimensions and fusion products for irreducible V_Q^s-modules, where s is an isometry of Q with s^2=1

Abstract

Every isometry s of a positive-definite even lattice Q can be lifted to an automorphism of the lattice vertex algebra V_Q. An important problem in vertex algebra theory and conformal field theory is to classify the representations of the s-invariant subalgebra V_Q^s of V_Q, known as an orbifold. In the case when s is an isometry of Q of order two, we have classified the irreducible modules of the orbifold vertex algebra V_Q^s and identified them as submodules of twisted or untwisted V_Q-modules in [Bavalov-Elsinger]. Here we calculate their quantum dimensions and fusion products. The examples where Q is the orthogonal direct sum of two copies of the A_2 root lattice and s is the 2-cycle permutation as well as where Q is the A_n root latice and s is a Dynkin diagram automorphism are presented in detail.