一类秩为2和3的Donaldson-Thomas不变量

摘要

在这篇文章中我们报告了Donaldson-Thomas不变量的基本理论。这些不变量足发牛在Calabi-Yau三维流形上的现象。本文首先介绍了Donaldson-Thomas理论的基础知识,然后给出了一个秩为2的稳定向量丛的模空问的完整结构,从而得到了相应的Donaldson-Thomas不变量。接着我们比较了P1×Pn上向量丛的稳定性,并根据这些稳,定性结果在P1×Pn上构造了秩为3的稳定向量从。本文最后总结了Donaldson-Thomas不变量与Gromov-Witten不变量的关系和一般Donaldson-Thomas不变量的理论,这些最新的研究进展将会给我们之后的研究工作带来一些启发。

目录

文摘

英文文摘

Chapter Ⅰ Introduction

Chapter Ⅱ Preliminaries of Donaldson-Thomas invariants

§2.1 The moduli space

§2.2 Virtual moduli cycle

§2.3 Donaldson-Thomas invariants

Chapter Ⅲ On some computations of Donaldson-Thomas invariants

§3.1 Rank-2 case

§3.1.1 Chamber structures of Y=Pm×Pn

§3.1.2 Stable rank-2 bundles on Y=P1×Pn

§3.1.3 Stable rank-2 bundles on Calabi-Yau manifolds

§3.1.4 Donaldson-Thomas invariants

§3.2 Rank-3 case

§3.2.1 Stabilities of vector bundles

§3.2.2 Chamber structures of Y=P1×Pn

§3.2.3 Construction of stable rank-3 bundles on Y=P1×Pn

Chapter Ⅳ Relations with Gromov-Witten invariants

§4.1 Introduction

§4.2 Gromov-Witten theory and Donaldson-Thomas theory

§4.3 Gauge theory and string dualities

Chapter Ⅴ Generalized Donaldson-Thomas invariants

References

Acknowledgements

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