Monopole moduli spaces and manifolds with corners

单极模空间和带角流形

• 批准号: EP/K036696/1
• 负责人: Michael Singer
• 金额: $ 45.13万
• 依托单位: University College London
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2014
• 资助国家: 英国
• 起止时间: 2014 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FK036696%2F1
• 关键词: Monopole; moduli; spaces; manifolds; corners

Non-abelian magnetic monopoles are particle-like solutions of a differential equation that live on ordinary three-dimensional space. It is known that some of the solutions correspond to widely separated particles, but that as the separations decrease, the particles lose their individual identity and can no longer be distinguished. These monopoles are part of a physical theory (Yang--Mills--Higgs theory) which cannot be solved exactly but which predicts that monopoles will interact non-trivially with each other when they move. When they move at low speeds, their motion is well approximated by geodesics on a certain curved space, much as particles in Einstein's theory of general relativity move under the influence of gravity. This project will study the distance function (metric) underlying this curved space M, and will study in particular how it looks at large distances.By undertaking a detailed study of the metric, we shall be also be able to study natural differential equations which are defined on the space M. This is of interest in the study of differential equations generally, since the large-scale structure of M is rather complicated, and can only be built up recursively. The behaviour of differential equations on M is important since they are necessary for the understanding of the quantum theory of the original Yang--Mills--Higgs theory. The important ingredient here again is the large-scale structure of M and its metric.This research will be important not only for understanding monopoles and features of the Yang--Mills--Higgs theory of which they are a part, but also for the development of a suite of sophisticated tools for understanding differential equations in this type of setting. These tools will be useful in similar problems and can be expected to have an importance beyond their applications to the theory of monopoles.

非亚伯磁性单极管是属于普通三维空间的微分方程的粒子样溶液。众所周知，某些溶液对应于广泛分离的颗粒，但是随着分离的减少，颗粒会失去其个体身份，无法再区分。这些单孔是物理理论（Yang-Mills--higgs理论）的一部分，该理论无法准确解决，但它预测单极会在移动时彼此之间的非琐事相互作用。当它们以低速移动时，它们的运动被一定弯曲空间上的大地学近似，就像爱因斯坦在重力影响下的一般相对性移动理论中一样。该项目将研究该弯曲空间M的距离函数（度量），并将特别研究它在大距离上的外观。通过对度量的详细研究，我们还将能够研究自然微分方程在空间M上定义。这通常是对微分方程的研究，因为M的大规模结构相当复杂，并且只能递归地构建。微分方程在M上的行为很重要，因为它们对于理解原始Yangs-Mills--higgs理论的量子理论是必不可少的。这里的重要成分再次是M及其度量的大规模结构。这项研究不仅对于理解Yang-Mills的单孔和特征 - higgs- higgs理论很重要，而且对它们是其中的一部分也很重要，而且对于对开发一套复杂工具，以理解这种类型的环境中的微分方程。这些工具在类似的问题中将很有用，并且可以预期在其对单极理论中的应用之外具有重要意义。

项目成果

Full asymptotics and Laurent series of layer potentials for Laplace's equation on the half-space

半空间上拉普拉斯方程的完全渐近和层势的洛朗级数

• DOI: 10.1002/mana.201800145
• 发表时间: 2019
• 期刊: Mathematische Nachrichten
• 影响因子: 1
• 作者: Fritzsch K

Partial compactification of monopoles and metric asymptotics

单极子的部分紧化和度量渐近

• DOI: 10.48550/arxiv.1512.02979
• 发表时间: 2015
• 期刊: arXiv e-prints
• 作者: Kottke Chris

Monopoles and the Sen Conjecture: Part I

单极子和森猜想：第一部分

• DOI: 10.48550/arxiv.1811.00601
• 发表时间: 2018
• 作者: Fritzsch K