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: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=758341
• 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中“万有理论”的最大希望。弦理论与未来物理学的相关性仍不确定，但它对数学的影响是显而易见的，而且可能是历史上无与伦比的。我研究受弦理论启发的数学。我使用的不是弦理论本身，而是一种或多或少等效的理论，称为共形场论，因为它在数学上更容易理解。这是现代物理学的主要语言——量子场论的一个特别友好的例子。根据菲尔兹奖获得者维滕的说法，这是数学最终接受量子场论的世纪。预计这对于数学来说至少会像 18 世纪和 19 世纪微积分的出现一样重要。我正在探索随着我们更好地理解共形场论，我们的一些经典概念（例如对称性）在多大程度上被破坏。我想知道这种破坏有多重要。尽管我们面临着这些新的想法和例子，经典的观念仍然渗透到今天的理论中。这是否暗示经典概念确实是基础性的并将继续具有相关性，或者我们最终会将此视为一种反动偏见、一种令人尴尬的偏见？我将列出共形场论中允许的一些可能性范围，并亲自判断经典概念似乎应有的作用。我发现即将发生的变化将是深远的。加拿大数学家已经在这一不断发展的理论的各个方面发挥了突出作用。例如，约翰·麦凯在两个看似无关的领域之间发现了一座桥梁（称为月光）：某些经典对称性和“模函数”（即存在于表面上的函数）。我们对 Moonshine 的最佳理解是使用共形场理论对它们进行插值。罗伯特·穆迪 (Robert Moody) 与他人共同发现了现在所谓的 Kac-Moody 代数；也许他们最重要的实现是某些非常特殊（并且非常经典）的共形场论的对称性。我对这两方面都非常感兴趣。我继承他们的遗产的想法激励着我。