Geometry and Dynamics of Topological Solitons

场论是解释基础物理学的自然数学语言，从粒子物理学到凝聚态，再到引力和宇宙学。在许多场论中，动力学场（一个从时空到某个流形的映射）可以像一个高维结一样把自己包裹起来。这样的场将大量能量捕获在一个光滑的、空间局部化的块中，称为拓扑孤子，它可以以非常类似粒子的方式四处移动并与其他孤子相互作用。孤立子是基本粒子（例如磁单极子，质子，中子）的天然候选者，但它们也可以模拟凝聚态物理学（例如超导体中的涡旋）和宇宙学（例如宇宙弦）中的大尺度结构。有一个关于孤子动力学的美丽几何理论，它用静态多孤子的模空间来描述它们的运动，静态多孤子本身就是一个数学上丰富而迷人的对象。理论中的关键挑战是在没有明确公式的情况下计算或理解孤子模空间上的正则度量。本项目的目的是在规范理论的特定背景下研究拓扑孤子的几何结构，并将其与动力学联系起来。涡旋模空间在某些易处理极限下的正则度量的计算.在大于4维的空间上构造瞬子，这是一个纯数学工程，预计不会有任何技术应用。

Field theory is the natural mathematical language to explain fundamental physics, from particle physics, to condensed matter, to gravitation and cosmology. In many field theories it is possible for the dynamical field, a map from spacetime into some manifold, to wrap itself up rather like a high-dimensional knot. Such a field traps large amounts of energy in a smooth, spatially localized lump, called a topological soliton, which can move around and interact with other solitons in remarkably particle-like fashion. Solitons are natural candidates for fundamental particles (e.g. magnetic monopoles, protons, neutrons), but they also model large scale structures in condensed matter physics (e.g. vortices in superconductors) and cosmology (e.g. cosmic strings). There is a beautiful geometric theory of the dynamics of solitons which describes their motion in terms of the moduli space of static multisolitons, a mathematically rich and fascinating object in its own right. The key challenge in the theory is to compute, or understand in the absence of explicit formulae, the canonical metric on the soliton moduli space. The aim of this project is to study the geometry of topological solitons and connect it to their dynamics, in the specific context of gauge theory.Specific goals are:1. The compuation of the canonical metric on vortex moduli spaces in certain tractable limits.2. The construction of instantons on spaces of dimension higher than 4.This is a pure mathematical project, and is not expected to have any technological applications.