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

TL;DR

This paper classifies irreducible modules of orbifold vertex algebras derived from lattice automorphisms, calculates their quantum dimensions and fusion products, with detailed examples involving root lattices and diagram automorphisms.

Contribution

It provides explicit calculations of quantum dimensions and fusion products for orbifold modules associated with involutive lattice automorphisms, extending previous classifications.

Findings

Classified irreducible modules of V_Q^s for order-two automorphisms.

Calculated quantum dimensions of these modules.

Determined fusion products for specific lattice automorphisms.

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.