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

Abstract

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.

arXiv comment(fixed a mistake in computing flat coordinates for tri-polynomials)

References in corpus (5)

• A New Cohomology Theory for Orbifold
• Gromov-Witten invariants of target curves via Symplectic Field Theory
• Weighted Projective Lines Associated to Regular Systems of Weights of Dual Type
• The spaces of Laurent polynomials, ${\mathbb{P}}^1$-orbifolds, and integrable hierarchies
• Realizability and exceptionality of candidate surface branched covers: methods and results

Cited by in corpus (6)

• Equivariant Gromov-Witten theory of one dimensional stacks
• Integrable hierarchies and the mirror model of local CP1
• Gromov--Witten invariants for mirror orbifolds of simple elliptic singularities
• Primitive Forms for Affine Cusp Polynomials
• A Uniqueness Theorem for Frobenius Manifolds and Gromov--Witten Theory for Orbifold Projective Lines
• On Frobenius Manifolds from Gromov--Witten Theory of Orbifold Projective Lines with $r$ orbifold points