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

点击翻译按钮获取中文摘要

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.