The age grading and the Chen-Ruan cup product

We prove that the obstruction bundle used to define the cup-product in Chen–Ruan cohomology has a simple representation-theoretic description. The key quantities in this description are the age gradings or degree-shifting numbers of the representations of the local groups of the orbifold. We obtain a Künneth Theorem for Chen–Ruan cohomology as a direct consequence of an elementary property of the age grading, and explain how several other results, including associativity of the cup product, can be proved in a similar way.