From the construction of Chen–Ruan cohomology, it is clear that the only non-topological datum is the obstruction bundle. This phenomenon is also reflected in calculations. That is, it is fairly easy to compute Chen–Ruan cohomology so long as there is no contribution from the obstruction bundle, but when the obstruction bundle does contribute, the calculation becomes more subtle. In such situations it is necessary to develop new technology. During the last several years, many efforts have been made to perform such calculations. So far, major success has been achieved in two special cases: abelian orbifolds (such as toric varieties) and symmetric products. For both these sorts of orbifolds, we have elegant – and yet very different – solutions.

Abelian orbifolds

An orbifold is abelian if and only if each local group Gx is an abelian group. Abelian orbifolds constitute a large class of orbifolds, and include toric varieties and complete intersections of toric varieties. Such orbifolds were the first class of examples to be studied extensively. Immediately after Chen and Ruan introduced their cohomology, Poddar [123] identified the twisted sectors of toric varieties and their complete intersections. There followed a series of works on abelian orbifolds by Borisov and Mavlyutov [28], Park and Poddar [122], Jiang [74], and Borisov, Chen, and Smith [26].