Quantum Invariants, Enhanced Moduli, and Integrable Systems

该奖项支持代数几何领域的研究。这项研究为理解基本问题的基本结构奠定了基础，这种方式适合在广泛的应用中使用。这项工作的结果旨在与可积系统理论、弦理论、规范理论和拓扑绝缘体的研究中的问题直接相关。此外，该研究项目旨在对场理论和量子化有具体的应用。该项目还旨在建立一个新的几何技术武器库，适用于代数循环理论、辛拓扑和高能物理。这将通过为数学和物理学的研究生和博士后助理提供几何和弦论界面的研究机会，以及通过开发关于形式局部化和移动量子化以及关于Calabi-Yau可积系统和更高的Donaldson-Thomas理论的课程来实现。该项目融合了衍生几何和量子场论的思想，以揭开模问题的隐藏复杂性，提取新的变种的计数不变量，并研究完全可积系统。这些问题的解决将巩固和揭开几何学、辛拓扑和场论中现有的几个量子化方案的神秘面纱。将研究三个方向。第一个目的是刻画那些允许实现为势的临界轨迹的模空间。这种刻画要求发展派生几何中移位辛和泊松结构的形式主义和各向同性叶理理论。在第二个项目中，我们将研究一种新的方法，通过建立显式的拉格朗日叶和计算相关的势函数来构建非阿贝尔上同调的基序取向数据。该项目旨在构建更高的Chern-Simons泛函，并为从高维量子场论中提取计数不变量奠定了基础。最后一个项目是寻找实现ADE结构群的驯服或野生亚纯Hitchin颤动的Calabi-Yau可积系统。

This award supports research in the field of algebraic geometry. The research sets the stage for understanding the basic structure of fundamental problems in a way suitable for pragmatic use in a broad spectrum of applications. The results of the work are intended to be immediately relevant to questions in the theory of integrable systems, string theory, gauge theory, and the study of topological insulators. In addition, the research project aims to have concrete applications to field theory and quantization. The project also aims to build a new arsenal of geometric techniques applicable to the theory of algebraic cycles, symplectic topology, and high energy physics. This will be achieved by providing research opportunities on the interface of geometry and string theory for graduate students and postdoctoral associates in mathematics and physics, and by development of courses on formal localization and shifted quantization and on Calabi-Yau integrable systems and higher Donaldson-Thomas theory. The project integrates ideas from derived geometry and quantum field theory to unravel the hidden complexity of moduli problems, to extract new enumerative invariants of varieties, and to study completely integrable systems. The resolution of these questions will consolidate and demystify several existing quantization schemes in geometry, symplectic topology, and field theory. Three directions will be studied. The first aims to characterize those moduli spaces that admit a realization as the critical locus of a potential. The characterization requires the development of the formalism of shifted symplectic and Poisson structures in derived geometry and the theory of isotropic foliations. In the second project a new method will be investigated for constructing motivic orientation data on non-abelian cohomology by building explicit Lagrangian foliations and computing the associated potential functions. The project aims to construct higher Chern-Simons functionals, and sets the stage for extracting enumerative invariants from higher dimensional quantum field theory. The final project searches for Calabi-Yau integrable systems that realize the tame or wild meromorphic Hitchin fibrations for ADE structure groups.