Gromov-Witten theory of elliptic orbifold P^1 and quasi-modular forms

Abstract

In this paper we prove that the GW invariants of the elliptic orbifold lines with weights (3,3,3), (4,4,2), and (6,3,2) are quasi-modular forms. Our method is based on Givental's higher genus reconstruction formalism applied to the settings of Saito's Frobenius structures for simple elliptic singularities. Our results are part of a larger project whose goal is to prove the Landau-Ginzburg/Calabi-Yau correspondence for simple elliptic singularities. The correspondence describes a relation between Gromov-Witten theory (of a certain hypersurface) and Fan-Jarvis-Ruan-Witten theory (of a certain Landau-Ginzburg potential). Roughly, the main statement is that the Saito's Frobenius manifold for simple elliptic singularities has some special points such that locally near these points the Frobenius structure governs one of the two theories. The local part of the correspondence is established in a companion article by M. Krawitz and Y. Shen, while here we describe the global picture.

arXiv comment49 pages

References in corpus (6)

• Direct Integration of the Topological String
• FJRW rings and Landau-Ginzburg Mirror Symmetry
• The Witten equation and its virtual fundamental cycle
• Quantum BCOV theory on Calabi-Yau manifolds and the higher genus B-model
• Gromov--Witten invariants for mirror orbifolds of simple elliptic singularities
• Simple Elliptic Singularities: a note on their G-function

Cited by in corpus (15)

• The Omega deformed B-model for rigid N=2 theories
• Primitive forms via polyvector fields
• Landau-Ginzburg/Calabi-Yau Correspondence of all Genera for Elliptic Orbifold $p^1$
• LG/CY Correspondence for Elliptic Orbifold Curves via Modularity
• Ramanujan Identities and Quasi-Modularity in Gromov-Witten Theory
• Gromov--Witten invariants for mirror orbifolds of simple elliptic singularities
• Local Calabi-Yau manifolds of type \tilde{A} via SYZ mirror symmetry
• Localized mirror functor for Lagrangian immersions, and homological mirror symmetry for P^1_{a,b,c}
• Simple Elliptic Singularities: a note on their G-function
• Gromov-Witten theory and cycle-valued modular forms
• A global mirror symmetry framework for the Landau-Ginzburg/Calabi-Yau correspondence
• The modular group for the total ancestor potential of Fermat simple elliptic singularities
• On quantum cohomology ring of elliptic ${\mathbb{P}}^1$ orbifolds
• From primitive form to mirror symmetry
• Counting of Holomorphic Orbi-spheres in ${\mathbb{P}}_{2,2,2,2}^1$ and Determinant Equation