  Let $\ix$ be a smooth Deligne-Mumford stack over the complex numbers. One can
define twisted orbifold Gromov-Witten invariants of $\ix$ by considering
multiplicative invertible characteristic classes of various bundles on the
moduli spaces of stable maps $\ix_{g,n,d}$, cupping them with evaluation and
cotangent line classes and then integrating against the virtual fundamental
class. These are more general than the twisted invariants introduced by Tseng.
We express the generating series of the twisted invariants in terms of the
generating series of the untwisted ones. We derive the corollaries which are
used in the work of Givental-Tonita on the quantum K-theory of a complex
manifold X.
