This is a research announcement on a theory of Gromov-Witten invariants and quantum cohomology of symplectic or projective orbifolds. Our project started in the summer of 98 where our original motivation was to study the quantum cohomology under singular flops in complex dimension three. In this setting, we allow our three-fold to have terminal singularities which can be deformed into a symplectic orbifold. We spent the second half of 98 and most of spring of 99 to develop the foundation of Gromov-Witten invariants over orbifolds, including the key conceptual ingredient — the notion of good map. In the April of 99, we were lucky to meet R. Dijkgraaf who explained to us the orbifold string theory and the role of twisted sectors. The twisted sector provides the precise topological framework for our orbifold quantum cohomology. Our theory of orbifold quantum cohomology was virtually completed in the summer of 99. Here, we give an overlook of the foundation of orbifold quantum cohomology while we leave its applications in other fields such as birational geometry for future research. The first part of our work has already appeared in [CR1]. The second part [CR2] will appear shortly. We would like to thank R. Dijkgraaf for bringing orbifold string theory to our attention at the critical moment of our project. We are also benefited from many discussions with E. Zaslow. Finally, the second author would like to thank E. Witten for many stimulating discussions about orbifold string theory.
Available at: http://works.bepress.com/weiminchen_chen/19/