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。该项目融合了几何和量子场论的技术，以揭示代数奇点的隐藏复杂性，提取新的变量的枚举不变量，并在表示论中构建新的对偶。将研究五个问题。第一个目的是构造新的代数不变量的模空间和辛奇点。这些不变量的建立和计算需要非交换几何中移位辛结构形式的理论发展。 在第二个项目中，我们提出了一种新的方法，通过对Calabi-Yau型微分分次范畴中对象的模进行移位量子化来构造motivic枚举不变量。第三个项目分析了非对易Hodge理论控制Landau-Ginzburg模型变形的方式，并提出了一个非常一般的无障碍定理。 第四个项目给出了一个策略，重新制定和证明经典的极限朗兰兹对偶作为一个纯粹的拓扑对偶的反常层。最后一个项目将建立分歧和扭曲的非阿贝尔霍奇对应的函子性。这些问题的解决将巩固和消除几何，辛拓扑和场论中几个现有的量子化方案的神秘性。该项目为理解代数簇的基本结构奠定了基础，适合在广泛的应用中实用。除了代数拓扑的自然应用外，所提出的工作将直接与范畴论、几何表示论、可积系统理论、弦理论、规范理论、量子引力和宇宙学中的深层问题相关。该项目概述了具体的跨学科应用的物理镜像对称，重整化群流，规范理论的量子化。该项目还旨在组织一个集中的努力，加强和建立一个新的几何武器库的技术适用于理论的代数循环，辛拓扑结构和高能物理。这将通过培训一批年轻的研究人员和数学和物理学研究生，以及通过编制关于派生辛几何的课程和关于非交换霍奇理论的课程来实现。具体的研究机会的接口几何和弦理论的研究生和博士后进行了讨论。拟议的工作将通过多学科会议上的演讲、研究研讨会和同行评审的科学期刊上的出版物进行传播。