Ramanujan Identities and Quasi-Modularity in Gromov-Witten Theory

Abstract

We prove that the ancestor Gromov-Witten correlation functions of one-dimensional compact Calabi-Yau orbifolds are quasi-modular forms. This includes the pillowcase orbifold which can not yet be handled by using Milanov-Ruan's B-model technique. We first show that genus zero modularity is obtained from the phenomenon that the system of WDVV equations is essentially equivalent to the set of Ramanujan identities satisfied by the generators of the ring of quasi-modular forms for a certain modular group associated to the orbifold curve. Higher genus modularity then follows by using tautological relations.

Version 3 (journal version): typos corrected, shortened. See version 2 for complete discussions. Mathematica Notebook files as an aid in doing computations are available upon request

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Cited by in corpus (8)

• LG/CY Correspondence for Elliptic Orbifold Curves via Modularity
• Local Calabi-Yau manifolds of type \tilde{A} via SYZ mirror symmetry
• WDVV equations and invariant bi-Hamiltonian formalism
• Genus Two Quasi-Siegel Modular Forms and Gromov-Witten Theory of Toric Calabi-Yau Threefolds
• Mirror Symmetry for Plane Cubics Revisited
• Higher Genus FJRW Invariants of a Fermat Cubic
• Remarks on Symplectic Geometry
• Counting of Holomorphic Orbi-spheres in ${\mathbb{P}}_{2,2,2,2}^1$ and Determinant Equation