Yang-Mills Theory and Large N Limits.

这项研究涉及量子场论（QFT），这是一个数学框架，用于研究具有许多随机波动变量的系统。它是微观物理学的语言，也是相变物理学的语言（例如。水的沸腾），其中随机性是量子的或热的。值得注意的是，QFT的思想也被用来解决纯数学中的几个突出问题。QFT方法也被用来分析金融市场中的湍流和波动。杨米尔斯或规范理论概括了麦克斯韦的电动力学，它描述了光和电磁学。量子色动力学（QCD，N = 3色夸克的规范理论）描述夸克和胶子之间的强作用力。然而，所有强相互作用的亚原子粒子（强子，例如。质子、中子）是夸克和胶子的束缚态。一个突出的数学挑战是从QCD推导出束缚态理论。有两个主要的困难：规范不变性和无穷大。虽然这个理论是用夸克和胶子写成的，但它们不是规范不变的。这伴随着神秘的禁闭现象：不可能从强子中移除夸克或胶子。无穷大是由于缺乏一个数学框架（重正化）来处理无限多个自由度之间的短距离相互作用的束缚态。杨-米尔斯理论是公认的与几何、代数和分析有着深刻联系的一个困难而重要的问题。它出现在克莱研究所的百万美元数学问题清单上。大N极限是Yang-Mills理论的一个有希望的近似。这是一个经典极限：规范不变观测量的涨落很小。虽然普朗克常数并不小，但经典力学的技术是适用的。我们的目标是理解数学结构，并近似或精确地解决一些大N规范理论，如球QCD和圆柱上的QCD。前一种模型的强子包含一个重夸克，周围是轻夸克。胶子不是动力学的，但贡献了一个库仑势，就像原子一样。由于无穷大，能量在下面是无限的。我们的目标是重整化的理论，以便研究束缚态的光谱。这个模型的有趣之处在于它被证明是渐近自由的，即相互作用在短距离内从自由粒子的相互作用中偏离，就像QCD一样。在圆柱上的QCD中，我们假设空间是一个圆。夸克有无限多个自由度，胶子缺乏局部动力学，但有围绕圆的整体自由度。这是QCD的最简单版本，具有动态规范场，尚未解决。为了更好地理解如何在大N极限下求解汉密尔顿方程，我们的目标是用规范不变变量来重新表述理论，并研究它们的泊松代数。我们还研究了局域胶子动力学的矩阵模型：具有几个动力学厄米矩阵的量子系统，但基本独立的可观测量。我们开发的方法来确定大N多矩阵模型的相关性。对于欧几里得模型，它是一种基于作用和测度对称性的方法，满足非交换李代数。对于Hamilton矩阵模型，我们从自旋链和可积系统出发，研究了在大N极限下四维O（N）标量场理论中相互作用紫外不动点的存在性。这在粒子物理学的希格斯部分是有趣的，希格斯部分基于具有四次相互作用的大质量标量场，但遭受糟糕的UV行为和自然性问题。UV固定点的线将确保良好的高能量行为和自然性，通过在质量设置为零时导致对称性（尺度不变性）。

The research concerns quantum field theory(QFT), a mathematical framework for systems with many variables subject to random fluctuations. It is the language of microscopic physics as well as the physics of phase transitions(eg. boiling of water), where the randomness is either quantum or thermal. Remarkably, ideas from QFT have also been used to solve several outstanding problems in pure mathematics. QFT methods are also used to analyse turbulent fluid flow and fluctuations in financial markets.Yang-Mills or gauge theories generalise Maxwell's electrodynamics, which describes light and electromagnetism. Quantum Chromodynamics (QCD, a gauge theory with N=3 colours of quarks) describes strong forces between quarks and gluons. However, all strongly interacting sub-atomic particles (hadrons, eg. proton, neutron) are bound states of quarks and gluons. An outstanding mathematical challenge is to deduce from QCD a theory of the bound states. There are two main difficulties: gauge invariance and infinities. Though the theory is written in terms of quarks and gluons, these are not gauge invariant. This is accompanied by the mysterious phenomenon of confinement: the impossibility to remove a quark or gluon from a hadron. Infinities are due to the lack of a mathematical framework (renormalisation) to treat the short distance interactions between infinitely many degrees of freedom in bound states. We develop methods to make progress with these two mathematical difficulties.Yang-Mills theory is acknowledged as a difficult and important problem with deep relations to geometry, algebra and analysis. It appears on the Clay Institute's list of million dollar maths problems. The large-N limit is a promising approximation for Yang-Mills theory. It is a classical limit: fluctuations in gauge invariant observables are small. Though Planck's constant is not assumed small, techniques of classical mechanics are applicable.We aim to understand mathematical structures and solve, either approximately or exactly, some large-N gauge theories such as spherical QCD and QCD on a cylinder. The former models hadrons containing a heavy quark surrounded by light quarks. Gluons are not dynamical, but contribute a coulomb potential, as in an atom. The energy is unbounded below due to infinities. We aim to renormalise the theory so as to study the spectrum of bound states. What makes this model very interesting is that it is conjectured to be asymptotically free, i.e. interactions deviate logarithmically from that of free particles at short distances, as in QCD. In QCD on a cylinder, we assume space is a circle. While quarks have infinitely many degrees of freedom, gluons lack local dynamics, but have global degrees of freedom around the circle. This is the simplest version of QCD, with dynamical gauge fields, that has not yet been solved. We aim to reformulate the theory in terms of gauge invariant variables and study their Poisson algebra in order to better understand how to solve Hamilton's equations in the large-N limit.We also study matrix models of local gluon dynamics: quantum systems with several dynamical hermitian matrices, but basis independent observables. We develop methods to determine the correlations of large-N multi-matrix models. For Euclidean models, it is a method based on symmetries of both action and measure, satisfying a non-abelian Lie algebra. For hamiltonian matrix models, we investigate insights from spin chains and integrable systems.Finally, we study the existence of a line of interacting ultraviolet fixed points in four dimensional O(N) scalar field theory in the large-N limit. This is of interest in the Higgs sector of particle physics, which is based on massive scalar fields with a quartic interaction, but suffers from bad UV behavior and a naturalness problem. A line of UV fixed points would ensure good high energy behavior and naturalness, by leading to a symmetry (scale invariance) when masses are set to zero.