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

物理学中的散射幅度问题是描述基本粒子接近、相互作用和散射时发生的事情。这个问题除了具有基本的理论重要性外，还具有实际意义：为了分析来自大型强子对撞机等粒子加速器的数据，需要对散射幅度有精确的了解。这个问题的传统方法是使用一种名为费曼图的技术。每个费曼图都是粒子之间一种可能的相互作用的示意图；为了解决散射幅度问题，人们总结了(许多)相关费曼图的贡献。大约十年前，普林斯顿高级研究所的尼玛·阿卡尼-哈米德身边的一群物理学家发起了一个项目，以更全面的方式重新定义散射幅度问题的解决方案。令人惊讶的是，他们的方法将散射幅度与我是专家的纯数学中的许多主题联系在一起，特别是簇代数和有限维代数的表示理论。我的建议有两个方面：我将解决源自物理学家方法的新的、令人兴奋的数学问题，我将积极与物理学家合作，将这些结果应用于散射幅度问题。物理学家解决散射幅度问题的方法是围绕几何对象的构造，该几何对象包含编码在其中的答案的基本元素。对于正在讨论的量子场论来说，这需要构造一系列日益复杂的多面体。该系列中的第0个多面体是缔合体，最初由James Stasheff在20世纪60年代的S中描述，但最近由于它与簇代数的联系而重新引起人们的兴趣。与合作者一起，我已经证明了该系列中的下一个多面体也与簇代数相连。我建议使用簇代数和相关的有限维代数来构造所有需要的多面体。还有一些代数变体提供了这些空间的重要的非多面体类似物，由非线性方程定义。这些代数簇被证明定义为一大类有限维代数，并且与它们的tau倾斜理论密切相关。这些变体是一个引人入胜的新研究对象，完全可以从表示论的研究中激发出来，但它们永远不会出现在我所从事的散射幅度和表示理论之间相互作用的背景下。综上所述，这个研究计划将为团簇代数和有限维代数的表示理论提供新的启示，并推动散射幅度的研究。

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.