Combinatorics of finite-dimensional algebras, with applications to scattering amplitudes

物理学中的散射振幅问题是描述当基本粒子接近、相互作用和散射时发生的事情的问题。除了具有基本的理论重要性外，这个问题也具有实际意义：为了分析来自粒子加速器（如大型强子对撞机）的数据，需要精确了解散射振幅。 解决这个问题的传统方法是使用一种称为“费曼图”的技术。每个费曼图都是粒子之间一种可能相互作用的示意图;为了解决散射振幅问题，人们总结了（许多）相关费曼图的贡献。大约10年前，普林斯顿高等研究院（Institute for Advanced Study in Princeton）的尼玛·阿尔卡尼-哈米德（Nima Arkani-Hamed）周围的一群物理学家发起了一个项目，以更全面的方式重新设计散射振幅问题的解决方案。他们的做法，令人惊讶的是，连接散射振幅的一些议题，在纯数学，我是一个专家，特别是集群代数和表示理论的有限维代数。我的建议有两个方面：我将解决新的和令人兴奋的数学问题起源于物理学家的方法，我将积极与物理学家合作，将这些结果应用到散射振幅问题。物理学家解决散射振幅问题的方法是围绕着构造一个几何对象，这个几何对象包含了编码在其中的答案的基本元素。对于正在讨论的量子场论，这需要构造一系列越来越复杂的多面体。第0个多面体是associahedron，最初由James Stasheff在20世纪60年代描述，但最近由于它与簇代数的联系而重新引起了人们的兴趣。与合作者一起，我已经证明了系列中的下一个多面体也连接到一个簇代数。我建议使用簇代数，以及相关的有限维代数，来构造所有需要的多面体。也有代数簇提供了重要的非多面体类似物的这些空间，定义的非线性方程。这些代数簇原来是定义为一个广泛的一类有限维代数，并密切相关的τ-倾斜理论。这些变体是一个令人着迷的新研究对象，可以完全从表征理论的研究中得到启发，但它们绝不会在我所从事的散射振幅和表征理论之间的相互作用的背景之外出现。 总之，这项研究计划将为集团代数和有限维代数的表示理论提供新的启发，并推动散射振幅的研究。

The scattering amplitudes problem in physics is the problem of describing what happens when elementary particles approach, interact, and scatter. As well as being of fundamental theoretical importance, this problem is also of practical significance: a precise knowledge of scattering amplitudes is needed in order to analyze data from particle accelerators such as the Large Hadron Collider.  The traditional approach to this problem uses a technique called "Feynman diagrams." Each Feynman diagram is a schematic representation of one possible interaction of the particles; in order to solve the scattering amplitudes problem, one sums up contributions from the (many) relevant Feynman diagrams. Some ten years ago, a group of physicists around Nima Arkani-Hamed of the Institute for Advanced Study in Princeton initiated a program to recast the solution to the scattering amplitudes problem in a more holistic way. Their approach has, surprisingly, connected scattering amplitudes to a number of topics in pure mathematics in which I am an expert, notably cluster algebras and the representation theory of finite dimensional algebras. My proposal is two-fold: I will address the new and exciting mathematical questions originating from the physicsts' approach, and I will collaborate actively with physicists to apply these results to the scattering amplitudes problem. The physicists' approach to the scattering amplitudes problem revolves around the construction of a geometrical object which contains the essential element of the answer encoded within it. For the quantum field theory under discussion, this requires the construction of a series of polyhedra of increasing complexity. The 0-th polyhedron in the series is the associahedron, originally described by James Stasheff in the 1960's, but seeing renewed interest recently because of its connection to cluster algebras. Together with collaborators, I have shown that the next polytope in the series is also connected to a cluster algebra. I propose to use cluster algebras, and related finite dimensional algebras, to construct all the needed polyhedra. There are also algebraic varieties which provide important non-polyhedral analogues of these spaces, defined by non-linear equations. These algebraic varieties turn out to be defined for a broad class of finite-dimensional algebras, and to be closely related to their tau-tilting theory. These varieties are a fascinating new object of study which can be motivated entirely from within the study of representation theory, but they would never have arisen outside of the context of the interplay between scattering amplitudes and representation theory in which I am engaged.  To summarize, this research program will shed new light on cluster algebras and representation theory of finite dimensional algebras, as well as driving forward research in scattering amplitudes.