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

点击翻译按钮获取中文摘要

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.