The Chen-Ruan Cohomology of Weighted Projective Spaces

Abstract

Chen and Ruan [6] defined a very interesting cohomology theory for orbifolds, which is now called Chen-Ruan cohomology. The primary objective of this paper is to compute the Chen-Ruan cohomology rings of the weighted projective spaces, a class of important spaces in physics. The classical tools (Chen-Ruan cohomology, toric varieties, the localization technique) which have been proved to be successful are used to study the orbifold cohomology of weighted projective spaces. Given a weighted projective space ${\bf P}^{n}_{q_{0}, >..., q_{n}}$, we determine all of its twisted sectors and the corresponding degree shifting numbers, and we calculate the orbifold cohomology group of ${\bf P}^{n}_{q_{0}, ..., q_{n}}$. For a general reduced weighted projective space, we give a formula to compute the 3-point function which is the key in the definition of Chen-Ruan cohomology ring. Finally we concretely calculate the Chen-Ruan cohomology ring of weighted projective space ${\bf P}^{5}_{1,2,2,3,3,3}$.