Physical Mathematics: String Theory, Quantization and Geometry

物理数学：弦论、量化和几何

• 批准号: SAPIN-2018-00029
• 负责人: Bouchard, Vincent
• 金额: $ 5.25万
• 依托单位: University of Alberta
• 依托单位国家: 加拿大
• 项目类别: Subatomic Physics Envelope - Individual
• 财政年份: 2019
• 资助国家: 加拿大
• 起止时间: 2019-01-01 至 2020-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=691653
• 关键词: Physical; Mathematics; String; Theory; Quantization

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.**

物理学以许多不同的方式影响了数学的发展。众所周知，许多数学领域的发展都是为了提供一种表达物理理论的语言。但人们可能不太清楚，物理理论所需的复杂数学一致性常常会揭示数学中新的、意想不到的结构。特别是，量子场论和弦理论中的对偶性常常会在原本不相关的数学领域之间产生深刻的、令人着迷的联系。我的研究项目集中在数学和物理之间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”，它起源于解决某些特定量子场论中物理可观测量的计算。由于弦理论和量子场论的对偶性，现在很明显，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根本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化相关：通过所谓的“量子艾里结构”，以及通过“量子曲线”。我的研究计划的目标之一是阐明这两个量化过程、拓扑递归之间的联系及其在几何、结理论和模形式理论中的各种应用。特别是，我提出了这种量子艾里结构的新颖概括，这意味着代数、几何和物理学之间令人着迷的新联系，并提出了许多新的研究问题。我还建议研究是否有可能在这些量化过程上施加额外的对称性（称为超对称性）的问题。那么通过这种超对称拓扑递归计算出的对象的几何意义是什么？********在物理学中，给定理论的可观测量经常具有很强的不变性。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。这些不变性通常遵循物理一致性，但在数学上往往远非显而易见。我的研究计划的另一方面在于研究这样一个不变性要求，该要求源自我们对弦理论中特定 D 膜状态的研究。这些状态的不变性暗示了模形式理论的一种新的构造，这是相当普遍和优雅的。我打算完成这个构造，并从数学和物理角度研究其属性和后果。**