Generalized twisted modules associated to general automorphisms of a vertex operator algebra

TL;DR

This paper develops a new framework for constructing generalized twisted modules for vertex operator algebras associated with arbitrary automorphisms, including those of infinite order, using complex gradings and logarithmic operators.

Contribution

It introduces strongly C^{ imes}-graded and C/Z-graded generalized g-twisted modules for automorphisms of any order, expanding the scope of twisted module theory.

Findings

Constructed generalized twisted modules from automorphisms generated by exponential of weight 1 elements.

Provided examples related to screening operators on triplet W-algebras.

Demonstrated the use of logarithmic vertex operators in the twisted module framework.

Abstract

We introduce a notion of strongly C^{\times}-graded, or equivalently, C/Z-graded generalized g-twisted V-module associated to an automorphism g, not necessarily of finite order, of a vertex operator algebra. We also introduce a notion of strongly C-graded generalized g-twisted V-module if V admits an additional C-grading compatible with g. Let V=\coprod_{n\in \Z}V_{(n)} be a vertex operator algebra such that V_{(0)}=\C\one and V_{(n)}=0 for n<0 and let u be an element of V of weight 1 such that L(1)u=0. Then the exponential of 2\pi \sqrt{-1} Res_{x} Y(u, x) is an automorphism g_{u} of V. In this case, a strongly C-graded generalized g_{u}-twisted V-module is constructed from a strongly C-graded generalized V-module with a compatible action of g_{u} by modifying the vertex operator map for the generalized V-module using the exponential of the negative-power part of the vertex operator Y(u, x). In particular, we give examples of such generalized twisted modules associated to the exponentials of some screening operators on certain vertex operator algebras related to the triplet W-algebras. An important feature is that we have to work with generalized (twisted) V-modules which are doubly graded by the group C/Z or C and by generalized eigenspaces (not just eigenspaces) for L(0), and the twisted vertex operators in general involve the logarithm of the formal variable.