  Motivated by logarithmic conformal field theory and Gromov-Witten theory, we
introduce a notion of a twisted module of a vertex algebra under an arbitrary
(not necessarily semisimple) automorphism. Its main feature is that the twisted
fields involve the logarithm of the formal variable. We develop the theory of
such twisted modules and, in particular, derive a Borcherds identity and
commutator formula for them. We investigate in detail the examples of affine
and Heisenberg vertex algebras.
