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

超弦理论和包括量子引力在内的统一理论之间存在着神秘的二元性。为了对它们进行定量研究，我们需要将内部Calabi-Yau流形的计数几何与BPS态计数联系起来。本研究的目的是借助于椭圆上同调，并通过将注意力集中在椭圆Calabi-Yau三重情形来类比Borcherds积，从而在这一方向上取得进展。我们系统地构造了相关的Borcherds型乘积，并研究了它们的功能性质及其与计数几何期望的一致性。不幸的是，尽管到目前为止我们取得了所有令人鼓舞的结果，但研究并未在给定的时间内完全完成，达到撰写一篇文章的水平。然而，我们能够发布松散的相关或技术结果，以用于这项研究的主要目的。

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.