  We compute, with Symplectic Field Theory techniques, the Gromov-Witten theory
of the complex projective line with orbifold points. A natural subclass of
these orbifolds, the ones with polynomial quantum cohomology, gives rise to a
family of (polynomial) Frobenius manifolds and integrable systems of
Hamiltonian PDEs, which extend the (dispersionless) bigraded Toda hierarchy. We
then define a Frobenius structure on the spaces of polynomials in three complex
variables of the form F(x,y,z)= -xyz+P_1(x)+P_2(y)+P_3(z) which contains as
special cases the ones constructed on the space of Laurent polynomials. We
prove a mirror theorem stating that these Frobenius structures are isomorphic
to the ones found before for polynomial P1-orbifolds. Finally we link rational
Symplectic Field Theory of Seifert fibrations over S^2 and three singular
fibers with orbifold Gromov-Witten invariants of the base, extending a known
result valid in the smooth case.
