非微擾現(xiàn)象是量子理論中經(jīng)常出現(xiàn)的現(xiàn)象,但是對于一般的情況,我們至今也沒有有效的研究方法�；谕�?fù)湎液土孔恿W(xué)的對應(yīng),我們對一類相對論可積系統(tǒng)提出了一個嚴(yán)格的非微擾量子化條件。由于Nekrasov-Shatashvili（NS）之前的研究,我們把這個量子化方法命名為NS量子化方案。除此之外,還有一套量子化方案是直接考慮粒子對應(yīng)的哈密頓量算子的譜問題,則這個算子的譜行列式的零點(diǎn)就給出了量子化條件,一般的來講,這個“零點(diǎn)”應(yīng)為除子。這一套方案首先由Grassi-Hatsuda-Marino（GHM）提出,我們把它稱為GHM量子化方案。即便是考慮的同一套系統(tǒng),這兩套方案卻看起來十分不同。通過引入ri場,我們論證了這里實(shí)際存在很多個譜行列式,他們以相位ri進(jìn)行區(qū)分,而這些不同的譜行列式給出的除子的交點(diǎn)剛好給出NS量子化方案的量子化條件。我們詳細(xì)討論了這個等價(jià)性,并且提出,二者等價(jià)需要一組等式恒成立。這一等式實(shí)際上就是拉開方程的一個特殊情況。我們又對拉開方程進(jìn)行了詳細(xì)研究,我們發(fā)現(xiàn)它在模變換下是不變的。在我們考慮的模型中,這一性質(zhì)預(yù)測了所有的ri場,在我們的驗(yàn)證精度內(nèi),僅僅通過拓?fù)湎业奈_信息和一... 

【文章來源】：中國科學(xué)技術(shù)大學(xué)安徽省 211工程院校 985工程院校

【文章頁數(shù)】：126 頁

【學(xué)位級別】：博士

【文章目錄】：
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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