Elliptic cohomology and the enumerative geometry of elliptic Calabi-Yau manifolds : Towards the understanding of string dualities

椭圆上同调和椭圆卡拉比-丘流形的枚举几何：理解弦对偶性

• 批准号: 19540024
• 负责人: KAWAI Toshiya
• 金额: $ 2.33万
• 依托单位: Kyoto University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2007
• 资助国家: 日本
• 起止时间: 2007 至 2009
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-19540024/
• 关键词: 数理物理; 代数幾何; 素粒子論; カラビーヤウ多様体; 弦理論; ボーチャーズ積; 楕円種数; 双対性; カラビ-ヤウ多様体

There are mysterious dualities among superstring theories, unified theories including quantum gravity. To make quantative investigation of them we need to relate enumerative geometry of internal Calabi-Yau manifolds with the BPS state counting. The aim of the present research has been to make progress in this direction with the aid of elliptic cohomology and an analogy to Borcherds products by focusing our attention to the cases of elliptic Calabi-Yau threefolds. We have systematically constructed the relevant Borcherds-like products and investigated their functional properties and the consistency with the expectation from enumerative geometry. Unfortunately, the research was not fully completed within the given period to the level of writing up an article despite all the encouraging results we obtained so far. However, we were able to publish loosely related or technical results to be used for the main purpose of this research.

在超弦理论和包括量子引力在内的统一理论中，存在着神秘的对偶性。为了对它们进行定量研究，我们需要把内Calabi-Yau流形的计数几何与BPS态计数联系起来。本研究的目的是在这个方向上取得进展的帮助下，椭圆上同调和类比Borcherds产品通过集中我们的注意力的情况下，椭圆Calabi-Yau三倍。我们系统地构造了相应的Borcherds类积，并研究了它们的函数性质和与计数几何期望的一致性。不幸的是，尽管我们迄今为止取得了令人鼓舞的成果，但这项研究并没有在给定的时间内完全完成，没有达到撰写一篇文章的水平。然而，我们能够发布松散相关或技术结果，用于本研究的主要目的。

项目成果

Abelian voritices on noadal and cuspidal curves

节点曲线和尖端曲线上的阿贝尔涡流

• 发表时间: 2009
• 期刊: Journal of High Energy Physics 11
• 作者: R.Inoue;O.Iyama;B.Keller;A.Kuniba;T..Nakanishi;Toshiya Kawai
• 通讯作者: Toshiya Kawai