Quantized Lagrangian submanifolds of moduli spaces and representation theory

模空间的量化拉格朗日子流形和表示理论

• 批准号: 2302624
• 负责人: Gus Schrader
• 金额: $ 28万
• 依托单位: Northwestern University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-06-01 至 2026-05-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2302624&HistoricalAwards=false
• 关键词: Quantized; Lagrangian; submanifolds; moduli; spaces

Despite its relatively recent development, the theory of cluster algebras has proven to be a powerful and versatile tool across a broad range of areas of modern mathematics and physics, and has been instrumental in building bridges between these disciplines. This project focuses on exploring new structures in representation theory, quantum topology and enumerative geometry from the cluster-algebraic perspective. In many problems in these areas, identifying an underlying cluster structure reveals hidden combinatorial structures and symmetries, thereby leading to explicit, constructive proofs of deep results. This research program will closely involve early career researchers, with plans to disseminate both the necessary background ideas and cutting edge results from the project through the organization of mini-schools aimed at graduate students and postdocs in adjacent areas of research.More specifically, this project focuses on the quantum geometry of moduli spaces of local systems on surfaces, and the problem of quantizing Lagrangian submanifolds of these symplectic moduli spaces. Constructing such a quantization amounts to producing a canonical vector in the Hilbert space associated to the surface, and in accordance with the philosophy of topological quantum field theory, these quantized Lagrangians are closely related to the geometry of three-manifolds. The PI will systematically study this quantization problem, developing along the way new structures on the underlying moduli spaces of local systems based on their connection with representation theory. New directions to be explored include the construction of integrable systems providing higher Teichmueller-theoretic analogs of the classical Fenchel-Nielsen Hamiltonians on Teichmueller spaces, as well understanding the behavior of the cluster structure for moduli spaces of local systems with non-generic monodromy data at punctures, which is intimately connected with the theory of double affine Hecke algebras and their higher genus analogs.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

尽管簇代数理论的发展相对较新，但它已被证明是跨越现代数学和物理学广泛领域的强大且多功能的工具，并且有助于在这些学科之间建立桥梁。该项目重点从簇代数的角度探索表示论、量子拓扑和枚举几何中的新结构。在这些领域的许多问题中，识别潜在的簇结构揭示了隐藏的组合结构和对称性，从而为深层结果提供了明确的、建设性的证明。该研究计划将密切涉及早期职业研究人员，并计划通过组织针对相邻研究领域的研究生和博士后的迷你学校来传播该项目必要的背景思想和前沿成果。更具体地说，该项目重点研究表面局部系统模空间的量子几何，以及量化这些辛模空间的拉格朗日子流形的问题。构建这样的量子化相当于在与表面相关的希尔伯特空间中产生一个规范向量，并且根据拓扑量子场论的哲学，这些量子化的拉格朗日量与三流形的几何形状密切相关。 PI 将系统地研究这个量化问题，并根据局部系统与表示论的联系，在其基础模空间上开发新的结构。待探索的新方向包括构建可积系统，提供 Teichmueller 空间上经典 Fenchel-Nielsen 哈密顿量的更高 Teichmueller 理论模拟，以及理解具有非通用单向数据的局部系统模空间的簇结构的行为。穿刺，这与双仿射赫克代数及其更高的属类比理论密切相关。该奖项反映了 NSF 的法定使命，并具有通过使用基金会的智力优点和更广泛的影响审查标准进行评估，被认为值得支持。