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: 2019
• 资助国家: 加拿大
• 起止时间: 2019-01-01 至 2020-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=681407
• 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。弦理论与未来的物理学有多大的关联还不确定，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。我研究数学的灵感来自弦理论。而不是弦理论本身，我用一个或多或少等价的理论，叫做共形场论，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。根据菲尔兹奖得主维滕的说法，这是数学最终与量子场论达成一致的世纪。预计这对数学的意义至少与18世纪和19世纪微积分的出现一样重要。我正在探索当我们更好地理解共形场论时，我们的一些经典概念（如对称性）被打乱的程度。我想知道这次破坏有多严重。尽管我们面对这些新的观点和例子，但经典的概念仍然渗透到今天的理论中。这是否暗示着经典概念是真正的基础，并将继续具有相关性，或者我们最终会将其视为一种反动的偏见，一种令人尴尬的偏见？我将描绘出共形场论中所允许的一些可能性的范围，并自己判断经典概念似乎应该扮演的角色。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个不断发展的理论的各个方面发挥了突出作用。例如，John McKay发现了两个看似无关的领域之间的桥梁（称为月光）：某些经典对称性和“模函数”（即生活在表面上的函数）。我们对月光的最好理解是使用共形场论来插值它们。罗伯特·穆迪共同发现了现在被称为卡茨-穆迪代数;也许他们最重要的实现是作为某些非常特殊（和非常经典）的共形场论的对称性。我对这两个方面都很感兴趣。一想到我在继承他们的遗产，我就受到鼓舞。