Degeneration and gluing of Kuranishi structures in Gromov-Witten theory   and the degeneration/gluing axioms for open Gromov-Witten invariants under a   symplectic cut
  We study the degeneration and the gluing of Kuranishi structures in Gromov-Witten theory under a symplectic cut. This leads us to a degeneration axiom and a gluing axiom for open Gromov-Witten invariants. They provide then a route to the construction of virtual fundamental chains via specialization. Comments on the equivalence of the degeneration formula of closed Gromov-Witten invariants by Li-Ruan/Li versus Ionel-Parker are given in the Appendix. 
Chien-Hao Liu and Shing-Tung Yau