散射振幅中的正奇点与体积
Positive Singularities and Volumes in Scattering Amplitudes
AI总结：
本论文围绕正几何，从对偶体积表示、圈级振幅奇点结构、朗道奇点与簇代数的猜想关系三方面展开研究，确立正几何为散射振幅的分析与几何组织提供统一语言。
AI中文摘要：
近期进展表明，某些量子场论中的散射振幅可通过正几何（positive geometries）实现几何表述。在该框架下，振幅由微分形式编码，其边界结构反映定域性、幺正性、因子化等物理原理。本论文对正几何给出了自包含的介绍，尤其侧重振幅体（Amplituhedron）——其通过典范形式描述平面最大超对称杨-米尔斯理论的振幅。我们发展了三个与正性、体积及奇点相关的方向：其一，通过对偶体积表示研究典范形式的正性性质，对于多面体，典范函数计算对偶多面体的体积；我们将该图像扩展至非线性正几何，揭示了非负超越测度及其与完全单调性的关联。其二，通过奇点结构研究圈级振幅，结合振幅体几何与朗道分析，我们对带拉格朗日插入的威尔逊圈的主导奇点进行分类，并约束了可能的奇异轨迹。其三，我们考察朗道奇点、正性与簇代数（cluster algebras）之间的猜想关系，利用动量-扭量（momentum-twistor）和格拉斯曼流形（Grassmannian）方法，我们识别了跨圈阶的递归结构，证明了若干无限族的情形，并提出了通向一般猜想的策略。总体而言，本论文确立了正几何为散射振幅的分析与几何组织提供了统一语言，涵盖从典范形式、体积到圈积分后出现的奇点等内容。
英文摘要：
Recent advances have revealed that scattering amplitudes in certain quantum field theories admit a geometric formulation in terms of positive geometries. In this framework, amplitudes are encoded by differential forms whose boundary structure reflects physical principles such as locality, unitarity, and factorization. This thesis gives a self-contained introduction to positive geometries, with particular emphasis on the Amplituhedron, which describes amplitudes in planar maximally supersymmetric Yang--Mills theory through its canonical form. We develop three related directions connecting positivity, volumes, and singularities. First, we study positivity properties of canonical forms through dual volume representations. For polytopes, canonical functions compute volumes of dual polytopes; we extend this picture to nonlinear positive geometries, uncovering non-negative transcendental measures and links with complete monotonicity. Second, we investigate loop-level amplitudes via their singularity structure. Combining Amplituhedron geometry with Landau analysis, we classify leading singularities of the Wilson loop with Lagrangian insertion and constrain the possible singular loci. Third, we examine conjectural relations between Landau singularities, positivity, and cluster algebras. Using momentum-twistor and Grassmannian methods, we identify recursive structures across loop orders, prove several infinite families of cases, and propose a strategy toward the general conjectures. Overall, the thesis develops the perspective that positive geometry provides a unifying language for the analytic and geometric organization of scattering amplitudes, from canonical forms and volumes to the singularities that emerge after loop integration.
