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

We introduce a notion of strongly C×-graded, or equivalently, C/Zgraded 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 = ∐ n∈Z V(n) be a vertex operator algebra such that V(0) = C1 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 2π √ −1 ResxY (u, x) is an automorphism gu of V . In this case, a strongly C-graded generalized gu-twisted V -module is constructed from a strongly C-graded generalized V -module with a compatible action of gu 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.