Contact invariants of Q-Gorenstein toric contact manifolds, the Ehrhart polynomial and Chen-Ruan cohomology

Q-Gorenstein toric contact manifolds provide an interesting class of examples of contact manifolds with torsion first Chern class. They are completely determined by certain rational convex polytopes, called toric diagrams, and arise both as links of toric isolated singularities and as prequantizations of monotone toric symplectic orbifolds. In this paper we show how the cylindrical contact homology invariants of a Q-Gorenstein toric contact manifold are related to (i) the Ehrhart (quasi-)polynomial of its toric diagram; (ii) the Chen-Ruan cohomology of any crepant toric orbifold resolution of its corresponding toric isolated singularity; (iii) the Chen-Ruan cohomology of any monotone toric symplectic orbifold base that gives rise to it through prequantization.