Hurwitz–Hodge integrals, the E6 and D4 root systems, and the Crepant Resolution Conjecture

Let G be the group A4 or Z2×Z2. We compute the integral of λg on the Hurwitz locus of curves admitting a degree 4 cover of P1 having monodromy group G. We compute the generating functions for these integrals and write them as a trigonometric expression summed over the positive roots of the E6 and D4 root systems, respectively. As an application, we prove the Crepant Resolution Conjecture for the orbifolds [C3/A4] and [C3/(Z2×Z2)].