Equivariant cohomology for Hamiltonian torus actions on symplectic orbifolds

TL;DR

This paper extends equivariant cohomology techniques to Hamiltonian torus actions on symplectic orbifolds, providing new injectivity results and a combinatorial method for computing Chen-Ruan cohomology.

Contribution

It generalizes the GKM theorem to symplectic orbifolds and introduces a combinatorial approach to compute equivariant Chen-Ruan cohomology.

Findings

Proved an injectivity theorem for Hamiltonian torus actions on orbifolds.

Generalized GKM theorem to the orbifold setting.

Developed a combinatorial method for computing Chen-Ruan cohomology.

Abstract

In this paper, we study Hamiltonian R-actions on symplectic orbifolds [M/S], where R and S are tori. We prove an injectivity theorem and generalize Tolman-Weitsman's proof of the GKM theorem in this setting. The main example is the symplectic reduction X//S of a Hamiltonian T-manifold X by a subtorus S of T. This includes the class of symplectic toric orbifolds. We define the equivariant Chen-Ruan cohomology ring and use the above results to establish a combinatorial method of computing this equivariant Chen-Ruan cohomology in terms of orbifold fixed point data.