Asymptotic Hodge Theory and Instantons

渐近霍奇理论和瞬子

• 批准号: 1005761
• 负责人: Mark Stern
• 金额: $ 17万
• 依托单位: Duke University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-09-01 至 2014-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1005761&HistoricalAwards=false
• 关键词: Asymptotic; Hodge; Theory; Instantons

AbstractAward: DMS-1005761Principal Investigator: Mark A. SternThe PI's proposed research focuses on several problems related to the existence and geometry of connections minimizing the Yang-Mills and related energy functionals of connections on vector bundles over manifolds of special holonomy and on smooth projective varieties. In the first project the PI, in joint work with Bianca Santoro, will study critical points and parabolic flows of the energy, E', given by the L2 norm of the (0,2)component of the curvature tensor. In many special geometries where the de Rham cohomology is all of type (p,p), they will attempt to prove holomorphicity of connections that are stable critical points of E'. They will also study obstructions to the long time existence of solutions to the gradient flow for E'. The primary goal of this project is to improve our understanding of when vector bundles admit holomorphic structures. In a related project, joint with Benoit Charbonneau, the PI will use techniques from geometric quantization to study refined Hodge structures on isomorphism classes of vector bundles on smooth projective varieties. They will study the large k asymptotics of the projection operator onto the kernel of a generalized Diracoperator associated to a bundle F tensored by the kth power of a polarizing line bundle.They have used these asymptotics to define a refined Hodge filtration on bundles and will attempt to exploit the asymptotics further to prove that the Chern character mapping from bundles to de Rham cohomology (equipped with its usual Hodge structure) respects these new Hodge filtrations. Connection analogs of Donaldson's balanced metric conditions arise in this study. The PI will seek corresponding analogs of Gieseker stability. In joint work with Savdeep Sethi, the PI will study the moduli space of instantons on RxT^3, interpolating between flat connections on T^3, with unequal Chern Simons invariants.The physics of the strong, weak, and electromagnetic forces is based on gauge theory, the study of differential calculus on vector bundles. The primary objects in these gauge theories are connections, which may be viewed as rules for defining differentiation. The primary energy guiding the dynamics in these physical theories is the Yang-Mills energy, and the fields dominating the physics are those that minimize the Yang-Mills energy. In dimensions greater than 4, the existence and structure of these optimal connections are poorly understood. In 4-dimensions, optimal connections have been a powerful tool for answering questions in geometry and topology. In higher dimensions they can potentially provide extremely important techniques for attacking questions in differential and algebraic geometry. In particular, they may give techniques for transforming questions involving the solutions of complex partial differential equations to dramatically simpler algebraic problems. As the current theories of quantum gravity all require an understanding of physics in dimensions greater than 4, there is strong interplay between the physical and geometric study of optimal connections. In fact, the antecedents to this project arose from questions posed to the PI by physicists; the answer to these questions then led to new results in geometry.

摘要/ abstract摘要：项目负责人：Mark A. stern该项目主要研究特殊完整流形和光滑射影变体上向量束上连接的Yang-Mills及其相关能量泛函的存在性和几何性问题。在第一个项目中，PI将与Bianca Santoro合作，研究能量E‘的临界点和抛物线流，E’由曲率张量（0,2）分量的L2范数给出。在许多de Rham上同调都是（p,p）型的特殊几何中，他们试图证明作为E'稳定临界点的连接的全纯性。他们还将研究E'梯度流解长期存在的障碍。这个项目的主要目标是提高我们对向量束何时承认全纯结构的理解。在与Benoit Charbonneau合作的相关项目中，PI将使用几何量化技术研究光滑射影变体上向量束同构类上的精细Hodge结构。他们将研究广义dirac算子核上的投影算子的大k渐近性，该广义dirac算子与由偏振线束的k次幂组成的束F相关。他们已经使用这些渐近定义了束上的一个改进的Hodge过滤，并将尝试进一步利用渐近来证明从束到de Rham上同调的chen字符映射（配备了其通常的Hodge结构）尊重这些新的Hodge过滤。在本研究中出现了唐纳森平衡度量条件的连接类比。PI将寻找Gieseker稳定性的相应类似物。在与Savdeep Sethi的合作中，PI将研究RxT^3上的实例的模空间，在T^3上具有不相等的chen Simons不变量的平连接之间进行插值。强、弱和电磁力的物理学是建立在规范理论的基础上的，这是对矢量束的微分微积分的研究。这些规范理论的主要对象是连接，这些连接可以被视为定义微分的规则。在这些物理理论中，指导动力学的主要能量是杨-米尔斯能，而控制物理学的场是那些使杨-米尔斯能最小化的场。在大于4的维度中，人们对这些最佳连接的存在和结构知之甚少。在四维空间中，最优连接是回答几何和拓扑问题的有力工具。在更高的维度，它们可以潜在地为解决微分几何和代数几何中的问题提供极其重要的技术。特别是，他们可能会给出将涉及复杂偏微分方程解的问题转化为简单得多的代数问题的技术。由于目前的量子引力理论都需要理解大于4维的物理，因此在最佳连接的物理和几何研究之间存在很强的相互作用。事实上，这个项目的前身是物理学家向PI提出的问题；这些问题的答案引出了几何学的新结果。