Enhanced moduli, Hodge theory, and quantization

增强模、Hodge 理论和量化

• 批准号: 1302242
• 负责人: Tony Pantev
• 金额: $ 30.56万
• 依托单位: University of Pennsylvania
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-07-01 至 2017-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1302242&HistoricalAwards=false
• 关键词: Enhanced; moduli; Hodge; theory; quantization

This is a research in the field of algebraic geometry. The project fuses techniques from geometry and quantum field theory to unravel the hidden complexity of algebraic singularities, to extract new enumerative invariants of varieties, and to construct novel dualities in representation theory. Five problems will be studied. The first one aims to construct new algebraic invariants of moduli spaces and of symplectic singularities. The building and computation of these invariants requires a theoretical development of the formalism of shifted symplectic structures in non-commutative geometry. In the second project a new method is proposed for constructing motivic enumerative invariants by shifted quantization of the moduli of objects in differential graded categories of Calabi-Yau type. The third project analyzes the way in which non-commutative Hodge theory controls the deformations of Landau-Ginzburg models and proposes a very general unobstructedness theorem. The fourth project gives a strategy for reformulating and proving the classical limit Langlands duality as a purely topological duality for perverse sheaves. The last project will establish the functoriality of the ramified and the twisted non-abelian Hodge correspondences.The resolution of these questions will consolidate and demystify several existing quantization schemes in geometry, symplectic topology, and field theory. The project sets the stage for understanding the basic structure of algebraic varieties in a way suitable for pragmatic use in a broad spectrum of applications. Aside from the natural applications to algebraic topology, the work proposed will be immediately relevant to deep questions in category theory, geometric representation theory, the theory of integrable systems, string theory, gauge theory, quantum gravity and cosmology. The project outlines concrete cross-discipline applications to the physics of mirror symmetry, renormalization group flow, and the quantization of gauge theories. This project also aims to organize a concentrated effort on enhancing and building a new geometric arsenal of techniques applicable to the theory of algebraic cycles, symplectic topology, and high energy physics. This will be achieved by training a group of young researchers, and graduate students in mathematics and physics, and by a curriculum development of a course on derived symplectic geometry, and a course on non-commutative Hodge theory. Specific research opportunities on the interface of geometry and string theory for graduate students and postdocs are discussed. The proposed work will be disseminated through talks at multidisciplinary conferences, research seminars and publications in peer reviewed scientific journals.

这是代数几何领域的一项研究。该项目融合了几何学和量子场论的技术，以揭示代数奇点隐藏的复杂性，提取新的簇的枚举不变量，并在表示论中构建新颖的对偶性。将研究五个问题。第一个旨在构造模空间和辛奇点的新代数不变量。这些不变量的构建和计算需要非交换几何中移位辛结构的形式主义的理论发展。 在第二个项目中，提出了一种通过 Calabi-Yau 类型的微分分级类别中对象模的移位量化来构造动机枚举不变量的新方法。第三个项目分析了非交换Hodge理论控制Landau-Ginzburg模型变形的方式，并提出了一个非常普遍的无阻碍定理。 第四个项目给出了重新表述和证明经典极限朗兰兹对偶性作为反常滑轮的纯粹拓扑对偶性的策略。最后一个项目将建立分支和扭曲非交换霍奇对应的函子性。这些问题的解决将巩固和揭开几何、辛拓扑和场论中现有的几种量化方案的神秘面纱。该项目为理解代数簇的基本结构奠定了基础，以适合在广泛的应用中实际使用的方式。除了代数拓扑的自然应用之外，所提出的工作还将与范畴论、几何表示论、可积系统理论、弦论、规范论、量子引力和宇宙学中的深层问题直接相关。该项目概述了镜像对称物理、重整化群流和规范理论量子化的具体跨学科应用。该项目还旨在组织集中力量，增强和建立适用于代数循环、辛拓扑和高能物理理论的新几何技术库。这将通过培训一批数学和物理学方面的年轻研究人员和研究生，以及开发派生辛几何课程和非交换霍奇理论课程来实现。讨论了研究生和博士后关于几何和弦理论接口的具体研究机会。拟议的工作将通过多学科会议的演讲、研究研讨会和同行评审的科学期刊上的出版物进行传播。