Interactions between representation theory, algebraic geometry, and physics

表示论、代数几何和物理学之间的相互作用

• 批准号: RGPIN-2022-03135
• 负责人: Weekes, Alexander
• 金额: $ 1.89万
• 依托单位: University of Saskatchewan
• 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=759165
• 关键词: Interactions; between; representation; theory; algebraic

Representation theory is an area of mathematics which deals with the study of symmetry, and is fundamental to many areas of scientific study. For example, the electron orbitals of a hydrogen atom are classified in part through the representation theory associated to the symmetries of a sphere. Representation theory also has diverse theoretical applications within mathematics itself. Some of the most modern developments in this field fall under the umbrella of higher representation theory. Here, problems are recast in geometric or abstract terms, allowing for the application of novel techniques. This often takes place in the framework of algebraic geometry: the study of solutions of systems of polynomial equations. The interplay between higher representation theory and algebraic geometry is extremely rich, and has led to the resolution of challenging problems such as the Kazhdan-Lusztig Conjectures. Less understood, but perhaps equally as rich, are emerging relations between higher representation theory and theoretical physics. Many of the spaces and concepts that are central to higher representation theory also appear in quantum field theory, and the perspective of physics opens the door to new proofs, techniques, and intuition. A prime example is Kapustin and Witten's physical interpretation of the Geometric Langlands program in terms of super Yang-Mills theory, which has inspired many mathematicians. The proposed research focuses on the interactions between higher representation theory, geometry, and physics. It will address natural questions which arise at the intersection of these fields, such as: - What problems and structures in higher representation theory can we understand using ideas from physics, and vice versa? - What are the special algebro-geometric properties of spaces arising in higher representation theory, and how do they relate to physical constructions? - How can we further expand the connections between higher representation theory and physics, and foster interdisciplinary collaboration? A primary focus of this proposal is Coulomb branches, which are spaces arising in quantum field theory. They were very recently given a rigourous mathematical definition by Braverman, Finkelberg, and Nakajima. Subsequent work has established ties between Coulomb branches and several areas of mathematics, including representation theory, algebraic geometry, number theory, quantum groups, and integrable systems. The proposed research aims to develop and understand these connections, and apply Coulomb branch techniques to resolve important mathematical problems in these fields. The results of this research will be of interest to mathematicians and to physicists, and contribute foundational results to a developing area of research. This research program offers opportunities for training at all levels. It will expose students and postdocs to a very active field of study, and foster the development of new research expertise in Saskatchewan.

表示论是一个研究对称性的数学领域，是许多科学研究领域的基础。例如，氢原子的电子轨道部分地通过与球体对称性相关的表示论来分类。表示论在数学本身也有不同的理论应用。在这个领域的一些最现代的发展属于更高的代表性理论的保护伞。在这里，问题被重新塑造成几何或抽象的术语，允许新技术的应用。这通常发生在代数几何的框架内：研究多项式方程组的解。高等表示理论和代数几何之间的相互作用是非常丰富的，并导致解决了具有挑战性的问题，如Kazhdan-Lusztig猜想。更高级表示论和理论物理学之间的新兴关系不太为人所知，但也许同样丰富。许多空间和概念是中央更高的表示理论也出现在量子场论，物理学的角度打开了大门，新的证明，技术和直觉。一个最好的例子是卡普斯廷和维滕根据超级杨-米尔斯理论对几何朗兰兹纲领的物理解释，这启发了许多数学家。拟议的研究重点是高等表示论、几何和物理之间的相互作用。它将解决自然问题出现在这些领域的交叉点，如：-什么问题和结构，在更高的代表性理论，我们可以理解使用的想法从物理学，反之亦然？- 在高等表示论中出现的空间的特殊代数几何性质是什么？它们与物理构造有什么关系？- 我们如何进一步扩大高等表征理论与物理学之间的联系，促进跨学科合作？这个建议的主要焦点是库仑分支，这是量子场论中出现的空间。最近布雷弗曼、芬克尔伯格和中岛给出了严格的数学定义。随后的工作建立了库仑分支和数学的几个领域之间的联系，包括表示论，代数几何，数论，量子群和可积系统。本研究的目的是发展和理解这些联系，并应用库仑分支技术来解决这些领域中的重要数学问题。 这项研究的结果将感兴趣的数学家和物理学家，并有助于基础性的结果，一个发展中的研究领域。该研究计划为各级培训提供了机会。它将使学生和博士后接触到一个非常活跃的研究领域，并促进萨斯喀彻温省新的研究专业知识的发展。