Geometrical structures in mathematical physics

数学物理中的几何结构

• 批准号: RGPIN-2018-05413
• 负责人: Shramchenko, Vasilisa
• 金额: $ 1.68万
• 依托单位: Université de Sherbrooke
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=759339
• 关键词: Geometrical; structures; mathematical; physics

The proposed research aims at establishing and strengthening connections between mathematics and quantum physics. We believe this to be an important direction as such connections can bring (as they have done in the past) new insights in both fields. In general, using techniques of one field in another might lead to new ways of looking at problems and thus to discoveries that would be less easy to make if one stays within a closed area of research. On one hand, we are planning to use our insight on one differential equation important in mathematics (Painlevé-6) to reveal similarities between its structure and the structure of some other equations arising in mathematical and quantum physics (Korteweg-de Vries and Schrödinger equations). On the other hand, we propose to study the topological recursion approach in application to quantum mechanics. Topological recursion method, having recently appeared in mathematics, suggests that information about quantum mechanical systems might be encoded in an associated mathematical object called Riemann surface. We plan to discover consequences for physics of this new approach to quantum mechanics. Another part of the proposal suggests, on the contrary, to use methods of quantum field theories to enumerate mathematical objects, namely graphs on surfaces with certain properties. This is possible due to the correspondence established in our recent work between such graphs and Feynman diagrams of one particular quantum field theory. The fourth part of the proposal concerns the relationship between mathematical structures called cluster algebras and the very special so-called BPS (Bogomolnyi-Prasad-Sommerfield) states in quantum field theories. It turned out that certain numbers in cluster algebra theory count the number of such BPS states. This fact has been discovered by both physicists and mathematicians studying the same questions but using quite different approaches. We are planning to translate results of one area into the language of the other, and discover new implications of results of each field for the other. Students working on these projects, in addition to the mathematics involved, will also learn some physics, which will give them an excellent start in research on mathematical physics.

这项研究旨在建立和加强数学与量子物理之间的联系。我们认为这是一个重要的方向，因为这种联系可以在这两个领域带来新的见解（就像他们过去所做的那样）。一般来说，在另一个领域中使用一个领域的技术可能会导致看待问题的新方法，从而导致如果一个人停留在封闭的研究领域中就不那么容易做出的发现。一方面，我们计划利用我们对数学中一个重要微分方程（Painlevé-6）的洞察力，揭示它的结构与数学和量子物理中出现的其他一些方程（Korteweg-de弗里斯和薛定谔方程）的结构之间的相似性。另一方面，我们提出研究拓扑递归方法在量子力学中的应用。最近出现在数学中的拓扑递归方法表明，有关量子力学系统的信息可能被编码在一个称为黎曼面的相关数学对象中。我们计划发现量子力学的这种新方法对物理学的影响。另一部分建议，相反，使用量子场论的方法来枚举数学对象，即具有某些属性的表面上的图形。这是可能的，因为在我们最近的工作中建立了这样的图和一个特定量子场论的费曼图之间的对应关系。第四部分的建议涉及的数学结构之间的关系称为集群代数和非常特殊的所谓的BPS（Bogomolnyi-Prasad-Sommerfield）状态在量子场论。原来，在团簇代数理论中，某些数字计算了这种BPS状态的数量。物理学家和数学家都发现了这个事实，他们研究同样的问题，但使用的方法完全不同。我们计划将一个领域的成果翻译成另一个领域的语言，并发现每个领域的成果对另一个领域的新影响。从事这些项目的学生，除了所涉及的数学之外，还将学习一些物理学，这将使他们在数学物理研究方面有一个良好的开端。