拓扑弦和量子力学

发布时间：2022-01-05 04:30

　　非微扰现象是量子理论中经常出现的现象,但是对于一般的情况,我们至今也没有有效的研究方法。基于拓扑弦和量子力学的对应,我们对一类相对论可积系统提出了一个严格的非微扰量子化条件。由于Nekrasov-Shatashvili（NS）之前的研究,我们把这个量子化方法命名为NS量子化方案。除此之外,还有一套量子化方案是直接考虑粒子对应的哈密顿量算子的谱问题,则这个算子的谱行列式的零点就给出了量子化条件,一般的来讲,这个“零点”应为除子。这一套方案首先由Grassi-Hatsuda-Marino（GHM）提出,我们把它称为GHM量子化方案。即便是考虑的同一套系统,这两套方案却看起来十分不同。通过引入ri场,我们论证了这里实际存在很多个谱行列式,他们以相位ri进行区分,而这些不同的谱行列式给出的除子的交点刚好给出NS量子化方案的量子化条件。我们详细讨论了这个等价性,并且提出,二者等价需要一组等式恒成立。这一等式实际上就是拉开方程的一个特殊情况。我们又对拉开方程进行了详细研究,我们发现它在模变换下是不变的。在我们考虑的模型中,这一性质预测了所有的ri场,在我们的验证精度内,仅仅通过拓扑弦的微扰信息和一... 

【文章来源】：中国科学技术大学安徽省 211工程院校 985工程院校

【文章页数】：126 页

【学位级别】：博士

【文章目录】：
摘要
ABSTRACT
Chapter 1 Introduction
Chapter 2 Introduction to Topological String Theory
    2.0
        2.0.1 N＝(2,2),d＝2 supersymmetric non-linear sigma model
        2.0.2 R-symmetry Anomaly
        2.0.3 Topological Twist
        2.0.4 Topological String Theory
        2.0.5 Holomorphic Anomaly
        2.0.6 Physical Interpretation
        2.0.7 Toric Calabi-Yau and Local Mirror Symmetry
Chapter 3 Quantization Conditions
    3.1 Nekrasov-Shatashvili Quantization Scheme
        3.1.1 Bethe/Gauge Correspondence
        3.1.2 Exact Nekrasov-Shatashvili Quantization Conditions
        3.1.3 Pole Cancellation
        3.1.4 Derivation from Lockhart-Vafa Partition Function
    3.2 Grassi-Hatsuda-Marino Quantization Scheme
        3.2.1 Review on GHM Conjecture
        3.2.2 GHM Conjecture at Rational Planck Constant
        3.2.3 Generalized GHM Conjecture
    3.3 Equivalence between the Two Quantization Schemes
        3.3.1 Generic Planck Constant h
        3.3.2 Proof at h＝2π/k
        3.3.3 Comments on the Equivalence
Chapter 4 K-theoretic blowup equations
    4.0
        4.0.1 K-theoretic blowup equations
        4.0.2 Vanishing blowup equations
        4.0.3 Unity blowup equations
        4.0.4 Solving the r fields
        4.0.5 Reflective property of the r fields
        4.0.6 Constrains on refined BPS invariants
        4.0.7 Blowup equations and (Siegel) modular forms
        4.0.8 Non-perturbative formulation
    4.1 Examples
        4.1.1 Genus zero examples
        4.1.2 Local P~2
        4.1.3 Local Hirzebruch surfaces
        4.1.4 Local B_3
    4.2 Blowup equations at generic points of moduli space
        4.2.1 Modular transformation
        4.2.2 Conifold point
        4.2.3 Orbifold point
    4.3 A non-toric case: local half K3
        4.3.1 Local half K3 and E-string theory
        4.3.2 Refined partition function of E-strings
        4.3.3 Vanishing blowup equation and Jacobi forms
    4.4 Solving refined BPS invariants from blowup equations
        4.4.1 Counting the equations
        4.4.2 Proof for resolved conifold
        4.4.3 A test for local P~2
Chapter 5 Conclusion
References
Thanks
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