标题： N=4 SYM 和 ABJM 中散射幅度的正几何结构 显示英文标题

标题： Positive Geometries for Scattering Amplitudes in N=4 SYM and ABJM

摘要： 本论文研究散射振幅的几何描述，特别关注N=4 SYM和ABJM理论中的散射振幅。 正几何领域的最新发展为我们提供了这一研究的合适框架。 特别是，我们将详细阐述幅度多面体、动量幅度多面体以及ABJM动量幅度多面体。 除了这些几何结构外，我们还将讨论ABHY关肇维多面体，它概括了双伴随标量理论中的树图散射振幅。 我们对这些正几何结构进行了详细的介绍，包括对其结构的全面讨论。 对于动量幅度多面体、ABJM动量幅度多面体和ABHY关肇维多面体，我们给出了它们边界的完整分层，这等价地阐明了树图散射振幅的奇异性结构。 值得注意的是，我们证明了ABJM动量幅度多面体的欧拉示性数等于一。 此外，我们探讨了这些及其他正几何结构之间的相互联系。 这些联系部分是通过散射方程的前推获得的。 我们开发了计算这些前推的技术，从而避免了显式求解散射方程的必要性。 超越树图层次，我们说明了正几何如何用于描述平面N=4 SYM和ABJM中的环积分。 建立了一个新的框架来研究双动量空间中的这些环几何结构。 该构造仅依赖于光锥及其交集，并且该框架同时涵盖了幅度多面体、动量幅度多面体和ABJM动量幅度多面体的环层次结构。 这进一步导致了N=4 SYM和ABJM中所有一环积分的简洁通用公式。 显示英文摘要

摘要： This thesis investigates geometric descriptions of scattering amplitudes, with a specific focus on scattering amplitudes in N=4 SYM and ABJM theory. The recent development of the field of positive geometries provides us with a suitable framework for this endeavour. In particular, we will give a detailed account of the amplituhedron, the momentum amplituhedron, and the ABJM momentum amplituhedron. Alongside these geometries, we will also discuss the ABHY associahedron, which encapsulates tree-level scattering amplitudes in bi-adjoint scalar theory. We provide a detailed introduction to these positive geometries, which includes a comprehensive discussion of their structure. For the momentum amplituhedron, ABJM momentum amplituhedron, and ABHY associahedron we give a full stratification of their boundaries, which equivalently elucidates the singularity structure of the tree-level scattering amplitudes. Notably, we show that the ABJM momentum amplituhedron has an Euler characteristic equal to one. Furthermore, we explore the interconnections between these, and other, positive geometries. These connections are in part obtained via push forwards through the scattering equations. We develop techniques to calculate these push forwards which circumvents the necessity to solve the scattering equations explicitly. Beyond tree-level, we illustrate how positive geometries can be used to describe loop integrands in planar N=4 SYM and ABJM. A new framework is established to investigate these loop geometries in the space of dual momenta. The construction relies solely on lightcones and their intersections, and the framework simultaneously encompasses the loop level structure of the amplituhedron, momentum amplituhedron, and the ABJM momentum amplituhedron. This further leads to compact general formulae for all one-loop integrands in N=4 SYM and ABJM.