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We compute, with Symplectic Field Theory techniques, the Gromov-Witten theoryof the complex projective line with orbifold points. A natural subclass ofthese orbifolds, the ones with polynomial quantum cohomology, gives rise to afamily of (polynomial) Frobenius manifolds and integrable systems ofHamiltonian PDEs, which extend the (dispersionless) bigraded Toda hierarchy. Wethen define a Frobenius structure on the spaces of polynomials in three complexvariables of the form F(x,y,z)= -xyz+P_1(x)+P_2(y)+P_3(z) which contains asspecial cases the ones constructed on the space of Laurent polynomials. Weprove a mirror theorem stating that these Frobenius structures are isomorphicto the ones found before for polynomial P1-orbifolds. Finally we link rationalSymplectic Field Theory of Seifert fibrations over S^2 and three singularfibers with orbifold Gromov-Witten invariants of the base, extending a knownresult valid in the smooth case.

Gromov-Witten theory of orbicurves, the space of tri-polynomials and Symplectic Field Theory of Seifert fibrations