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: 2016
• 资助国家: 加拿大
• 起止时间: 2016-01-01 至 2017-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=612259
• 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.

弦理论是我们对物理学中的“一切理论”的最大希望。弦理论实际描述了宇宙的精确程度仍然完全不确定，但是其对数学的影响很明显，并且在历史上可能是无与伦比的。 我的兴趣是研究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。更确切地说，有两种主要方法是保形场理论，但从表面上看，它们看起来截然不同。有些事情在一种方法中更容易理解，而另一些事情在其他方法中也容易得多。我已经成为两者的专家。极其有价值的是它们之间的数学上精确的词典，使我们能够仔细地将构造，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，但这很难在数学上看到。我的大部分工作都围绕着理解，测试，加强该词典。 加拿大人已经在这方面出名。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）：某些代数对称性和“模块化功能”（即生活在表面上的函数）。我们对月光的最佳理解使用保形场理论来解释它。罗伯特·穆迪（Robert Moody）共同发现了现在所谓的kac-moody代数；也许最重要的认识是某些非常特殊的字符串理论的对称性。我对这两个方面都非常感兴趣。