Physical Mathematics: String Theory, Quantization and Geometry

物理学在许多方面影响了数学的发展。众所周知，许多数学领域的发展都是为了提供一种语言来表述物理理论。但也许不太为人所知的是，物理理论所要求的复杂的数学一致性往往会揭示数学中新的、意想不到的结构。特别是，量子场论和弦论中的对偶性常常在先验无关的数学领域之间产生深刻而迷人的联系。我的研究计划侧重于数学和物理之间的这种有趣的相互作用，有时被称为“物理数学”。这种相互作用的一个特殊例子是所谓的“拓扑递归”，它起源于某些特定量子场论中物理观测量计算的解决方案。由于弦论和量子场论中的对偶性，现在很清楚，这种递归结构是几何学许多领域的统一主题。这种结构普遍存在的数学原因似乎在于量子化过程。事实上，这种递归结构似乎以两种不同的方式与量子化相联系：通过所谓的“量子艾里结构”和通过“量子曲线”。我的研究计划的目标之一是阐明这两个量子化过程之间的联系，拓扑递归，以及它在几何，纽结理论和模形式理论中的各种应用。特别是，我提出了一个新的推广这种量子艾里结构，这意味着迷人的代数，几何和物理之间的新的联系，并开辟了许多新的研究问题。我还建议研究是否有可能在这些量子化过程中强加一种额外的对称性，即所谓的超对称性。那么，由这种超对称拓扑递归计算出的物体的几何意义是什么呢？在物理学中，通常情况下，给定理论的可观测量具有很强的不变性。例如，它们不应该依赖于用于描述物理现象的坐标系的选择。这些不变性通常来自物理一致性，但在数学上往往远非显而易见。我的研究计划的另一个方面是研究一个这样的不变性要求，它产生于我们对弦理论中特定D-膜态的研究。这些状态的不变性表明模形式理论中有一种新的结构，它非常通用且优雅。我打算完成这个构造，并从数学和物理两方面研究它的性质和结果。

Physics has influenced the development of mathematics in many different ways. It is well known that many areas of mathematics have been developed to provide a language to formulate physical theories. But it is perhaps not as well known that the intricate mathematical consistency required of physical theories often uncover new, unexpected structures in mathematics. In particular, dualities in quantum field theory and string theory often give rise to deep, fascinating connections between areas of mathematics that are a priori unrelated. My research program focuses on this intriguing interaction between mathematics and physics, sometimes known as "physical mathematics".******One particular example of such interaction is the so-called "topological recursion", which originated as a solution to the calculation of physical observables in some particular quantum field theory. Because of dualities in string theory and quantum field theory, it is now clear that this recursive structure is a unifying theme in many areas of geometry. The underlying mathematical reason for the ubiquity of this structure appears to lie in the process of quantization. In fact, this recursive structure appears to be connected to quantization in two different ways: through so-called "quantum Airy structures", and via "quantum curves". One of the goals of my research program is to shed light on the connections between these two quantization processes, the topological recursion, and its various applications in geometry, knot theory, and the theory of modular forms. In particular, I propose a novel generalization of this quantum Airy structures, which implies fascinating new connections between algebra, geometry and physics, and opens up many new research questions. I also propose to study the question of whether it is possible impose an extra symmetry, known as supersymmetry, on these quantization processes. What would then be the geometric meaning of the objects calculated by such a supersymmetric topological recursion?******In physics it is often the case that observables of a given theory have strong invariance properties. For instance, they should not depend on a choice of coordinate system used to describe a physical phenomenon. Those invariance properties generally follow from physical consistency, but are often far from obvious mathematically. Another aspect of my research program consists in studying one such invariance requirement that arose from our study of particular D-brane states in string theory. Invariance of these states suggests a new construction in the theory of modular forms, which is quite general and elegant. I intend to complete this construction and study its properties and consequences, both mathematically and physically.**