Twisted modules for vertex algebras associated with even lattices (Q1327021)

Let \(L\) be an even lattice and \(V_ L\) be the associated vertex algebra. The author has proved [see J. Algebra 161, 245-265 (1993; reviewed above)] that every irreducible module for \(V_ L\) is isomorphic to the untwisted Fock space associated with some coset of \(L\) in its \(\mathbb{Z}\)- dual lattice. In the book ``Vertex operator algebras and the Monster'' [see \textit{I. Frenkel}, \textit{J. Lepowsky}, and \textit{A. Meurman}, Academic Press (1988; Zbl 0674.17001)], a family of \(\theta\)-twisted \(V_ L\) modules has been constructed for the involutory automorphisms \(\theta\) of \(V_ L\) of order 2. In this paper it has been proved that for positive definite \(L\), any irreducible \(\theta\)-twisted module for \(V_ L\) is isomorphic to one of these known modules. In particular, if \(L\) is the Leech lattice, it follows that there is only one \(\theta\)-twisted module up to isomorphism.