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 2)

Abstract

Let 𝑉_𝐿 be the vertex algebra associated to a non-degenerate even lattice 𝐿, 𝜃 the automorphism of 𝑉_𝐿 induced from the −1 symmetry of 𝐿, and 𝑉_𝐿⁺ the fixed point subalgebra of 𝑉_𝐿 under the action of 𝜃. In this series of papers, we classify the irreducible weak 𝑉_𝐿⁺-modules and show that any irreducible weak 𝑉_𝐿⁺-module is isomorphic to a weak submodule of some irreducible weak 𝑉_𝐿-module or to a submodule of some irreducible 𝜃-twisted 𝑉_𝐿-module. Let 𝑀(1)⁺ be the fixed point subalgebra of the Heisenberg vertex operator algebra 𝑀(1) under the action of 𝜃. In this paper (Part 2), we show that there exists an irreducible 𝑀(1)⁺-submodule in any non-zero weak 𝑉_𝐿⁺-module and we compute extension groups for 𝑀(1)⁺.