Ramanujan identities and quasi-modularity in Gromov–Witten theory
Ramanujan identities and quasi-modularity in Gromov–Witten theory
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格罗莫夫-维滕理论中的拉马努金恒等式和拟模性
DOI：
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发表时间：
2014
期刊：
影响因子：
--
通讯作者：
Jie Zhou
中科院分区：
文献类型：
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作者：
Yefeng Shen;Jie Zhou
一个著名的民间猜想断言紧Calabi-Yau轨道的Gromov-Witten不变量的生成函数是拟模形式或它们的推广。到目前为止，证明这一猜想的主要策略是使用镜像对称和b模型技术。不幸的是，并不是每个有趣的Calabi-Yau轨道都有已知的更高属b型镜像。一个著名的例子是$\mathbb{P}^1_{2,2,2,2}$，椭圆的轨道曲线有四个$\mathbb{Z}_{2}$ -轨道点。
在本文中，我们将介绍一种纯a模型方法。在此方法中，我们从一个惊人的现象得到了属零模性，即WDVV方程组等价于某些模群的拟模形式环的生成子所满足的Ramanujan恒等式集。然后利用同义关系证明了高属模性。这种方法适用于所有一维紧凑的Calabi-Yau轨道，包括$\mathbb{P}^1_{2,2,2,2}$，它还不能被Milanov-Ruan的b模型技术处理\cite{Milanov:2011}。
A celebrated folklore conjecture asserts that the generating functions of Gromov-Witten invariants of compact Calabi-Yau orbifolds are quasi-modular forms or their generalizations. So far, the main strategy in proving this conjecture is to use mirror symmetry and B-model techniques. Unfortunately, not every interesting Calabi-Yau orbifold has a known higher genus B-model mirror. A well-known example is $\mathbb{P}^1_{2,2,2,2}$, the elliptic orbifold curve with four $\mathbb{Z}_{2}$-orbifold points.
In this article, we introduce a purely A-model approach. In this approach, we show that genus zero modularity is obtained from a surprising phenomenon that the system of WDVV equations is equivalent to the set of Ramanujan identities satisfied by the generators of the ring of quasi-modular forms for certain modular group. Higher genus modularity is then proved by using tautological relations. This approach works perfectly for all one-dimensional compact Calabi-Yau orbifolds, including $\mathbb{P}^1_{2,2,2,2}$ which can not yet be handled by Milanov-Ruan's B-model technique \cite{Milanov:2011}.
Orbifold GW 理论作为 HurwitzâFrobenius 子流形
影响因子：
1.5
作者：
A. Basalaev
通讯作者：
A. Basalaev
Noether-Lefschetz 理论和 Yau-Zaslow 猜想
DOI：
10.1090/s0894-0347-2010-00672-8
发表时间：
2010
期刊：
arXiv: Algebraic Geometry
影响因子：
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作者：
A. Klemm;D. Maulik;R. Pandharipande;E. Scheidegger
通讯作者：
E. Scheidegger