The Eynard-Orantin recursion and equivariant mirror symmetry for the projective line

We study the equivariantly perturbed mirror Landau-Ginzburg model of P-1. We show that the Eynard-Orantin recursion on this model encodes all-genus, all-descendants equivariant Gromov-Witten invariants of P-1. The nonequivariant limit of this result is the Norbury-Scott conjecture, while by taking large radius limit we recover the Bouchard-Marino conjecture on simple Hurwitz numbers.