Twisted modules associated to general automorphisms of a vertex operator algebra

We introduce a notion of strongly C×-graded generalized g-twisted V -module for an automorphism g, not necessarily of finite order, of a vertex operator algebra. 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, ResxY (u, x) has only real eigenvalues, and the sizes of the Jordan blocks of ResxY (u, x) on V(n) for n ∈ Z are bounded. Then the exponential of 2πiResxY (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 using the exponential of the negative-power part of the vertex operator Y (u, x). An important feature is that we have to work with generalized (twisted) V -modules which are doubly graded by the group 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.