Ramanujan Identities and Quasi-Modularity in Gromov-Witten Theory

TL;DR

This paper proves that Gromov-Witten correlation functions for one-dimensional Calabi-Yau orbifolds are quasi-modular forms, extending modularity results to cases not accessible by previous B-model techniques, and establishes genus zero and higher genus modularity.

Contribution

It demonstrates that the ancestor Gromov-Witten functions are quasi-modular forms and connects WDVV equations with Ramanujan identities to establish modularity.

Findings

Genus zero modularity from WDVV and Ramanujan identities

Higher genus modularity via tautological relations

Includes cases not handled by previous B-model methods

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.