Let be a smooth proper Deligne–Mumford stack over . One can define twisted orbifold Gromov–Witten invariants of by considering multiplicative invertible characteristic classes of various bundles on the moduli spaces of stable maps , 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 a paper with Givental about the quantum -theory of a complex compact manifold .
Valentin Tonita. "Twisted orbifold Gromov–Witten invariants." Nagoya Math. J. 213 141 - 187, March 2014. https://doi.org/10.1215/00277630-2393950