Orbifold Cohomology of Torus Quotients

We introduce the inertial cohomology ring NH T (Y) of a stably almost complex manifold carrying an action of a torus T . We show that in the case that Y has a locally free action by T , the inertial cohomology ring is isomorphic to the Chen-Ruan orbifold cohomology ring H∗CR(Y/T ) (as defined in [Chen-Ruan]) of the quotient orbifold Y/T . For Y a compact Hamiltonian T -space, we extend to orbifold cohomology two techniques that are standard in ordinary cohomology. We show that NH T (Y) has a natural ring surjection onto H∗CR(Y//T ), where Y//T is the symplectic reduction of Y by T at a regular value of the moment map. We extend to NH T (Y) the graphical GKM calculus (as detailed in e.g. [Harada-Henriques-Holm]), and the kernel computations of [Tolman-Weitsman, Goldin]. 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 Q coefficients, in [Borisov-Chen-Smith]); we reproduce their results over Q for all symplectic toric orbifolds obtained by reduction by a connected torus (though with different computational methods), and extend them to Z coefficients in certain cases, including weighted projective spaces. CONTENTS