Local Calabi-Yau，量子曲线与Refined拓扑弦

摘要

Calabi-Yau流形上的拓扑弦是数学物理中最为丰硕的领域之一。在数学上，其导出的镜像对称联系起Calabi-Yau上的辛几何与代数几何。本文研究拓扑弦理论中的若干问题，分两部分。第一部分证明了Haghighat-L ockhart-Vafa猜想在n=3时的情况，即在E8×E8杂化弦理论中，三对E-strings构型等价于三个杂化弦构型。在数学上这涉及到local half K3 Calabi-Yau threefold的性质以及E8 Weyl-invariantJacobi形式的新的非平凡恒等式。第二部分研究一般local Calabi-Yau的镜曲线的严格量子化，建立了exact Nekrasov-Shatashivili量子化与Grassi-Hatsuda-Mari(n)o猜想之间的等价性条件。对于一条亏格g的镜曲线，exact Nekrasov-Shatashivili量子化给出g个量子化条件，可以从非微扰拓扑弦的Lockhart-Vafa配分函数得到。而Grassi-Hatsuda-Mari(n)o猜想联系起拓扑弦与谱理论，其中曲线量子化由一个量子Riemann theta函数为零得到。本文我们发现至少存在g个不等价的量子Riemann theta函数，使得全部不等价量子Riemann theta函数的Theta除子之交恰好与g个exact Nekrasov-Shatashivili量子化条件的谱重合。此两种量子化途径的等价对local Calabi-Yau的refined Gopakumar-Vafa不变量给出无穷多约束。

目录

声明

摘要

ABSTRACT

Table of Contents

List of Tables

List of Figures

Chapter 1 Introduction

Chapter 2 Basics of Toric Calabi-Yau and Topological String

2．1 Toric Calabi-Yau and Local Mirror Symmetry

2．2 Topological String

Chapter 3 Known results for E-strings and heterotic strings

3．1 Background

3．2 Orbifold formula for n H-strings

3．3 Results from topological strings on half K3 surface

3．4 Domain wall method for two E-strings

Chapter 4 Elliptic genus of three E-strings and three H-strings

4．1 E-string holomorphic anomaly

4．2 M9 domain walls

4．3 Elliptic genus of three E-strings

4．4 Elliptic genus of three H-strings

4．5 Discussion of results

Chapter 5 Nekrasov-Shatashvili Quantization Scheme

5．1 Bethe／Gauge Correspondence

5．2 Exact Nekrasov-Shatashvili Quantization Conditions

5．3 Pole Cancellation

5．4 Derivation from Lockhart-Vafa Partition Function

5．5 Remarks on B Field

Chapter 6 Grassi-Hatsuda-Mari(n)o Quantization Scheme

6．1 Review on GHM Conjecture

6．2 GHM Conjecture at Rational Planck Constant

6．3 Generalized GHM Conjecture

Chapter 7 Equivalence between the Two Quantization Schemes

7．1 Generic Planck Constant

7．2 Proof at Special Planck Constants

7．3 Remarks on the Equivalence

Chapter 8 Examples

8．1 Local Del Pezzo Surfaces

8．2 Resolved C3／Z5 Orbifold

8．3 SU(N)Geometries

8．4 SU(3)Geometries with m=0，1，2

8．4．1 SU(3)Geometries with m=0

8．4．2 SU(3)Geometries with m=1

8．4．3 SU(3) Geometries with m=2

Chapter 9 Conclusion

References

Appendix

Thanks

在读期间发表的学术论文与取得的研究成果