Gromov-Witten theory of orbicurves, the space of tri-polynomials and symplectic field theory of Seifert fibrations

We compute, with symplectic field theory (SFT) techniques, the Gromov-Witten theory of $${\mathbb{P}^1_{\alpha_1,\ldots,\alpha_a}}$$, i.e., the complex projective line with a 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 (Carlet, The extended bigraded toda hierarchy. arXiv preprint arXiv:math-ph/0604024). We then define a Frobenius structure on the spaces of polynomials in three complex variables of the form F(x, y, z) = −xyz + P1(x) + P2(y) + P3(z) which contains as special cases the ones constructed on the space of Laurent polynomials (Dubrovin, Geometry of 2D topologica field theories. Integrable systems and quantum groups, Springer Lecture Notes in Mathematics 1620:120–348, 1996; Milanov and Tseng, The space of Laurent polynomials, $${\mathbb{P}^1}$$-orbifolds, and integrable hierarchies. preprint arXiv:math/0607012v3 [math.AG]). We prove a mirror theorem stating that these Frobenius structures are isomorphic to the ones found before for polynomial $${\mathbb{P}^1}$$-orbifolds. Finally we link rational SFT of Seifert fibrations over $${\mathbb{P}^1_{a,b,c}}$$ with orbifold Gromov-Witten invariants of the base, extending a known result (Bourgeois, A Morse-Bott approach to contact homology. Ph.D. dissertation, Stanford University, 2002) valid in the smooth case.