Gauge theoretic moduli spaces

规范理论模空间

• 批准号: RGPIN-2019-04375
• 负责人: Charbonneau, Benoit
• 金额: $ 1.53万
• 依托单位: University of Waterloo
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2021
• 资助国家: 加拿大
• 起止时间: 2021-01-01 至 2022-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=737592
• 关键词: Gauge; theoretic; moduli; spaces

From a mathematical perspective, gauge theory is the study of objects called vector bundles and structures on them (e.g., connections, Higgs fields, etc) satisfying differential equations derived from physics. Many solutions for such differential equations may be considered equivalent, and mathematicians and physicists call the parameter space of inequivalent solutions the moduli space. Dr. Charbonneau's research program is mostly concerned with studying the geometry and topology of moduli spaces of instantons and monopoles, and in assembling tools for their study. For monopoles on Euclidean spaces, with periodic conditions or not, a tool called the Nahm transform heuristically allows for a computation of the moduli space. This heuristic has been proven to work in some settings, but Dr. Charbonneau, through this grant, aims to improve the conditions under which this tool can be used. Through it, he intends to settle longstanding conjectures about the moduli spaces of monopoles, some of them coming from physics, others from mathematics. With his earlier successes in establishing the Nahm transform in other settings, Dr. Charbonneau is well established to settle those cases. A current theme of interest is the study of instantons on spaces of high dimensions. These spaces feature interesting geometries we call "special holonomy." There is a celebrated research program proposed by Donaldson and Thomas relating the gauge theory on those spaces to the sub-geometry of special objects in these spaces called calibrated sub-manifolds. At the moment, the international mathematical community has a deeper understanding on the calibrated sub-geometry side than on the gauge theory side and Dr. Charbonneau's work during the tenure on this grant helps advance knowledge on the gauge theory side. Another aspect of Dr. Charbonneau's research program is interdisciplinary, where his geometric expertise facilitates the research of soft-matter scientist who study glass transition.

从数学角度来看，规范理论是对称为矢量丛的对象及其结构（例如，连接、希格斯场等）的研究，满足从物理学中导出的微分方程。这种微分方程的许多解可以被认为是等价的，数学家和物理学家将不等价解的参数空间称为模空间。 Charbonneau 博士的研究项目主要涉及研究瞬子和单极子模空间的几何和拓扑，以及为他们的研究组装工具。 对于欧几里得空间上的单极子，无论是否具有周期性条件，一种称为纳姆变换的工具启发式地允许计算模空间。这种启发式方法已被证明在某些情况下有效，但 Charbonneau 博士希望通过这笔资助改善使用该工具的条件。通过它，他打算解决长期以来关于单极子模空间的猜想，其中一些来自物理学，另一些来自数学。凭借早期在其他环境中建立 Nahm 变换的成功，Charbonneau 博士已经做好了解决这些案例的准备。当前感兴趣的主题是高维空间瞬子的研究。这些空间具有有趣的几何形状，我们称之为“特殊的完整”。唐纳森和托马斯提出了一个著名的研究计划，将这些空间的规范理论与这些空间中特殊物体的子几何结构（称为校准子流形）联系起来。目前，国际数学界对校准子几何方面的理解比对规范理论方面的理解更深，Charbonneau 博士在这项资助任职期间的工作有助于推进规范理论方面的知识。 Charbonneau 博士研究项目的另一个方面是跨学科的，他的几何专业知识促进了研究玻璃转变的软物质科学家的研究。