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“万物理论”的最大希望。弦理论对宇宙的实际描述有多准确仍是完全不确定的，但它对数学的影响是显而易见的，可能是历史上无与伦比的。 我的兴趣是学习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价的理论，称为共形场理论，因为它在数学上更容易理解。更准确地说，共形场理论有两种主要的方法，但表面上看，它们看起来非常不同。有些事情在一种方法中更容易理解，而另一些事情在另一种方法中更容易理解。我已经成为这两方面的专家。最有价值的将是它们之间的一本数学上精确的词典，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，但从数学上很难看出这一点。我的大部分工作都围绕着理解、测试和加强这本词典。 加拿大人已经在这方面发挥了突出作用。例如，约翰·麦凯发现了两个看似不相关的领域之间的桥梁(称为月光)：某些代数对称和‘模函数’(即存在于曲面上的函数)。我们对月光的最好理解是用共形场理论来解释它。罗伯特·穆迪与人共同发现了现在被称为Kac-Moody代数的东西；或许其中最重要的实现是某些非常特殊的弦理论的对称性。我对这两个方面都非常感兴趣。