The irreducible weak modules for the fixed point subalgebra of the vertex algebra associated to a non-degenerate even lattice by an automorphism of order 2 (Part 1)

Abstract

Let VL be the vertex algebra associated to a non-degenerate even lattice L, θ the automorphism of VL induced from the −1-isometry of L, and VL+ the fixed point subalgebra of VL under the action of θ. In this series of papers we classify the irreducible weak VL+-modules and show that any irreducible weak VL+-module is isomorphic to a weak submodule of some irreducible weak VL-module or to a submodule of some irreducible θ-twisted VL-module.In this paper (Part 1), we show that when the rank of L is 1, every non-zero weak VL+-module contains a non-zero M(1)+-module, where M(1)+ is the fixed point subalgebra of the Heisenberg vertex operator algebra M(1) under the action of θ.