Conformal field theory and the nature of symmetry

共形场论和对称性的本质

• 批准号: RGPIN-2019-06049
• 负责人: Gannon, Terry
• 金额: $ 1.89万
• 依托单位: University of Alberta
• 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=738287
• 关键词: Conformal; field; theory; nature; symmetry

String theory is our best hope for a `theory of everything' in physics. How relevant string theory will be to the physics of tomorrow is still uncertain, but its impact on mathematics is clear, and probably unparalleled in history. I study the mathematics inspired by string theory. Instead of string theory itself, I work with a more or less equivalent theory called conformal field theory, because it's much more accessible mathematically. It is an especially friendly example of a quantum field theory, the main language of modern physics. According to Fields' medalist Witten, this is the century when mathematics finally comes to terms with quantum field theory. It is expected that this will be at least as significant for mathematics, as was the coming to terms with calculus in the 18th and 19th centuries. I am exploring the extent to which some of our classical notions, like symmetry, are being disrupted as we understand conformal field theory better. I want to know how significant is this disruption. Though we are being confronted by these new ideas and examples, classical notions still permeate today's theory. Is this a hint that the classical notions are truly foundational and will continue to be relevant, or will we eventually see this as a reactionary bias, an embarrassing prejudice? I will map out some of the range of possibilities allowed in conformal field theory, and judge for myself the role the classical notions appear to deserve. What I am finding is that the coming changes will be profound. Canadian mathematicians have already featured prominently in aspects of this evolving theory. For example, John McKay discovered a bridge (called Moonshine) between two seemingly unrelated areas: certain classical symmetries, and `modular functions' (i.e. functions which live on surfaces). Our best understanding of Moonshine interpolates them using conformal field theory. Robert Moody co-discovered what are now called Kac-Moody algebras; perhaps their most important realization is as symmetries of certain very special (and very classical) conformal field theories. I am very interested in both of these aspects. The thought that I am continuing their legacy inspires me.

弦理论是我们对物理学的“一切理论”的最大希望。弦理论与明天的物理学的相关理论是多么相关，但其对数学的影响很明显，并且在历史上可能是无与伦比的。我研究了受字符串理论启发的数学。我不是弦理论本身，而是与一个或多或少的等效理论一起工作，称为保形场理论，因为它在数学上更容易访问。这是量子场理论的一个特别友好的例子，这是现代物理学的主要语言。根据Fields的奖牌获得者Witten的说法，这是数学最终与量子场理论相关的世纪。可以预期，这对于数学至少将同样重要，而在18世纪和19世纪与微积分也是如此。我正在探索我们的某些古典概念（例如对称性）的程度，因为我们更好地理解保形田地理论。我想知道这种破坏有多重要。尽管我们正面临这些新想法和例子，但古典概念仍然渗透到当今的理论。这是否暗示古典概念是真正的基础并将继续相关的，还是我们最终将其视为反动偏见，令人尴尬的偏见？我将绘制在保形领域理论中允许的一些可能性，并亲自判断古典概念似乎应得的作用。我发现，即将到来的变化将是深刻的。加拿大数学家已经在这种不断发展的理论的方面出名。例如，约翰·麦凯（John McKay）在两个看似无关的区域之间发现了一座桥（称为月光）：某些经典对称性和“模块化功能”（即生活在表面上的功能）。我们对月光的最佳理解使用保形场理论可以介入它们。罗伯特·穆迪（Robert Moody）共同发现了现在所谓的kac-moody代数；他们最重要的认识也许是某些非常特殊（且非常经典）的形状理论的对称性。我对这两个方面都非常感兴趣。我继续他们的遗产的想法激发了我的启发。