Mathematics of Conformal Field Theory

弦理论是我们对物理学中“万有理论”的最大希望。弦理论实际上如何准确地描述宇宙仍然完全不确定，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。我的兴趣是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。更准确地说，共形场论有两种主要方法，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。我已成为这两方面的专家。极其有价值的是它们之间有一本数学上精确的字典，使我们能够小心地将结构、见解、定理从一个传递到另一个。物理学说这应该是可能的，但这在数学上很难看到。我的大部分工作都是围绕着理解、测试、强化这本词典。加拿大人在这方面已经发挥了突出作用。例如，约翰·麦凯发现了两个看似无关的领域之间的一座桥梁（称为 Moonshine）：某些代数对称性和“模函数”（即存在于表面上的函数）。我们对月光的最佳理解是使用共形场理论来解释它。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。我对这两个方面都非常感兴趣。

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.