We introduce the inertial cohomology ring of a stably almost complex manifold carrying an action of a torus . We show that in the case where has a locally free action by , the inertial cohomology ring is isomorphic to the Chen-Ruan orbifold cohomology ring (as defined in [CR]) of the quotient orbifold .
For a compact Hamiltonian -space, we extend to orbifold cohomology two techniques that are standard in ordinary cohomology. We show that has a natural ring surjection onto , where is the symplectic reduction of by at a regular value of the moment map. We extend to the graphical Goresky-Kottwitz-MacPherson (GKM) calculus (as detailed in, e.g., [HHH]) and the kernel computations of [TW] and [G1], [G2].
We detail this technology in two examples: toric orbifolds and weight varieties, which are symplectic reductions of flag manifolds. The Chen-Ruan ring has been computed for toric orbifolds, with -coefficients, in [BCS]; we reproduce their results over for all symplectic toric orbifolds obtained by reduction by a connected torus (though with different computational methods) and extend them to -coefficients in certain cases, including weighted projective spaces
"Orbifold cohomology of torus quotients." Duke Math. J. 139 (1) 89 - 139, 15 July 2007. https://doi.org/10.1215/S0012-7094-07-13912-7