Mathematics of Conformal Field Theory

共形场论数学

• 批准号: RGPIN-2014-06494
• 负责人: Gannon, Terry
• 金额: $ 1.68万
• 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: 2018
• 资助国家: 加拿大
• 起止时间: 2018-01-01 至 2019-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=651496
• 关键词: Mathematics; Conformal; Field; Theory

String theory is our best hope for a `theory-of-everything' in physics. How accurately string theory actually describes the universe is still completely uncertain, but its impact on math is clear, and probably unparalleled in history.**My interest is in studying 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. More precisely, there are two main approaches to conformal field theory, but superficially they look very different. Some things are much easier to understand in one approach, other things are much easier in the other. I have become an expert in both. What would be extremely valuable would be a mathematically precise dictionary between them, allowing us to carefully pass constructions, insights, theorems from one to the other. The physics says this should be possible, but this is very hard to see mathematically. Much of my work surrounds understanding, testing, strengthening that dictionary.**Canadians have already featured prominently in aspects of this. For example, John McKay discovered a bridge (called Moonshine) between two seemingly unrelated areas: certain algebraic symmetries, and `modular functions' (i.e. functions that live on surfaces). Our best understanding of Moonshine interprets it using conformal field theory. Robert Moody co-discovered what are now called Kac-Moody algebras; perhaps there most important realisation is as symmetries of certain very special string theories. I am very interested in both these aspects.

弦理论是物理学中“万有理论”的最大希望。弦理论对宇宙的描述到底有多精确还完全不确定，但它对数学的影响是显而易见的，而且在历史上可能是无与伦比的。我的兴趣是研究受弦理论启发的数学。而不是弦理论本身，我研究的是一个或多或少等效的理论，叫做共形场论，因为它在数学上更容易理解。更准确地说，有两种主要的共形场论方法，但表面上它们看起来非常不同。有些事情用一种方法更容易理解，另一些事情用另一种方法更容易理解。我已经成为这两方面的专家。在它们之间有一本精确的数学字典，这将是极有价值的，让我们能够仔细地将结构、见解和定理从一个传递到另一个。物理学说这应该是可能的，但这很难用数学来解释。我的大部分工作都是围绕着理解、测试和加强词典。**加拿大人已经在这方面发挥了突出作用。例如，约翰·麦凯（John McKay）在两个看似无关的领域之间发现了一座桥梁（名为Moonshine）：某些代数对称性和“模函数”（即存在于表面上的函数）。我们对月光最好的理解是用共形场论来解释它。罗伯特·穆迪与人共同发现了现在被称为kac -穆迪代数的东西；也许最重要的认识是某些非常特殊的弦理论的对称性。我对这两个方面都很感兴趣。