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

2008/08/19 by Paolo Rossi, Rossi, Paolo · 4 citations
Mathematics · #Algebraic Geometry and Number Theory #Advanced Algebra and Geometry #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.0808.2626

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+P₁(x)+P₂(y)+P₃(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 S² and three singular fibers with orbifold Gromov-Witten invariants of the base, extending a known result valid in the smooth case.

Cited by

• Dubrovin duality for open Hurwitz flat F-manifolds
• Gromov--Witten invariants for mirror orbifolds of simple elliptic singularities
• Primitive Forms for Affine Cusp Polynomials
• Equivariant Gromov-Witten theory of one dimensional stacks

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