Gromov-Witten theory and cycle-valued modular forms

Todor Milanov; Yongbin Ruan; Yefeng Shen

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arXiv · 1206.3879

Gromov-Witten theory and cycle-valued modular forms

Todor Milanov·Yongbin Ruan·Yefeng Shen

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Abstract

In this paper, we proved generating functions of Gromov-Witten cycles of the elliptic orbifold lines with weights (3,3,3), (4,4,2), and (6,3,2) are cycle-valued quasi-modular forms. This is a generalization of Milanov and Ruan's work on cycle-valued level. First we construct a global cohomology field theory (CohFT) for simple elliptic singularities (modulo an extension problem) and prove its modularity. Then, we apply Teleman's reconstruction theorem to prove mirror theorems on cycled-valued level and match it with a CohFT from Gromov-Witten theory of a corresponding orbifold.This solves the extension property as well as inducing the modularity for a Gromov-Witten CohFT.

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Todor Milanov, Yongbin Ruan, Yefeng Shen. 2012-06-18. Gromov-Witten theory and cycle-valued modular forms. https://arxiv.org/abs/1206.3879

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