Conformal field theory and the nature of symmetry

弦理论是物理学中“万有理论”的最大希望。弦理论对未来的物理学有多大意义还不确定，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。我学习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。它是量子场论的一个特别友好的例子，量子场论是现代物理学的主要语言。根据菲尔兹奖得主威滕的说法，这是数学最终与量子场论达成协议的世纪。预计这对数学的意义至少会和18、19世纪的微积分一样重大。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，比如对称性，在多大程度上被破坏了。我想知道这种破坏有多严重。虽然我们正面临着这些新的思想和例子，但经典的观念仍然渗透在今天的理论中。这是否暗示经典观念是真正的基础，并将继续相关，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允许的一些可能性范围，并自己判断经典概念似乎应该得到的作用。我发现，未来的变化将是深远的。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在两个看似无关的领域之间发现了一座桥梁（称为Moonshine）：某些经典对称性和“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插入它们。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；也许它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。我将继承他们的遗产的想法激励着我。

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.