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

Research Project

Project/Area Number	19540024
Research Category	Grant-in-Aid for Scientific Research (C)
Allocation Type	Single-year Grants
Section	一般
Research Field	Algebra
Research Institution	Kyoto University
Principal Investigator	KAWAI Toshiya Kyoto University, 数理解析研究所, 准教授 (20293970)
Project Period (FY)	2007 – 2009
Project Status	Completed (Fiscal Year 2009)
Budget Amount *help	¥3,640,000 (Direct Cost: ¥2,800,000、Indirect Cost: ¥840,000) Fiscal Year 2009: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000) Fiscal Year 2008: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000) Fiscal Year 2007: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
Keywords	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
Research Abstract	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.