Abelianization of Connections in Two and Three Dimensions

二维和三维连接的阿贝尔化

• 批准号: 1711692
• 负责人: David Ben-Zvi
• 金额: $ 33.42万
• 依托单位: University of Texas at Austin
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2017
• 资助国家: 美国
• 起止时间: 2017-07-01 至 2021-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1711692&HistoricalAwards=false
• 关键词: Abelianization; Connections; Two; Three; Dimensions

The PI studies problems of geometry using methods imported from particle physics. In joint work with his collaborators, he has recently developed a new geometric technique of "abelianization" -- so called because it reduces nonabelian problems (involving operations for which the order in which we do the operations matters) to simpler abelian ones (where the order does not matter). The PI, together with his collaborators and graduate students, will work on several new applications of abelianization. One application is a new approach to solving certain differential equations, including the Schrodinger equation which governs the physics of some quantum systems. A second application is a new way of measuring the topology of 3-dimensional spaces. The results of this work will be disseminated broadly both in the mathematics and high-energy physics communities, helping to bring these two areas closer together. The work will also contribute to the training of graduate students in both fields.The PI's recent joint work with collaborators introduced a new ingredient to the theory of flat connections: a way of "abelianizing" flat connections on a rank N complex vector bundle over a surface, replacing them by almost-flat connections on a line bundle over an N-fold branched covering surface. The full scope of this new theory is not yet known: it appears that there are many more uses of abelianization yet to be discovered. The PI aims to develop some of these. First, he will study a family of special connections on surfaces called "opers," which can be abelianized in a canonical way. On the one hand, this is a warmup for the abelianization of the twistor lines in the Hitchin system. On the other hand, it gives a new way of understanding the locus of opers and thus a new perspective on many related issues, from the classical theory of linear scalar differential operators to nonperturbative extensions of topological string theory. Second, he will consider abelianization on a 3-manifold instead of a surface. One immediate application is the development of new formulas for classical complex Chern-Simons invariants. Third, the PI aims to develop a new relation between abelianization and Floer theory on cotangent bundles.

PI使用从粒子物理学导入的方法研究几何问题。在与他的合作者的联合合作中，他最近开发了一种新的“ Abelianization”的几何技术 - 之所以如此，是因为它减少了非阿贝尔问题（涉及我们执行操作重要的顺序）到更简单的Abelian（命令无关紧要）。 PI与他的合作者和研究生一起，将研究几种新的Abelianization应用程序。一种应用是一种解决某些微分方程的新方法，包括控制某些量子系统物理的Schrodinger方程。第二个应用是一种测量3维空间拓扑的新方法。这项工作的结果将在数学和高能物理社区中广泛传播，从而使这两个领域更加紧密地结合在一起。这项工作还将有助于对两个领域的研究生进行培训。PI最近与合作者的联合合作引入了对平坦联系理论的新成分：一种在表面上的平台N复合矢量捆绑包上“ Abelianization”平面连接的方法，将它们替换为N-Flat连接在N-Flat bunndle的n-Flat Bundled bunderd覆盖封面的表面上。该新理论的全部范围尚不清楚：似乎还有更多的Abelianization使用。 PI旨在开发其中一些。 首先，他将研究一个称为“ Opers”的表面上的特殊连接家族，可以以规范的方式进行ABELIAN。 一方面，这是Hitchin系统中扭曲线的Abelianization的热身。另一方面，它给出了一种新的理解操作源的方式，因此从线性标量差分运算符的经典理论到拓扑字符串理论的非扰动扩展。其次，他将考虑在3个manifold而不是表面上的Abelianization。一个即时应用是为古典复​​杂的Chern-Simons开发新公式。第三，PI旨在建立Abelianization与Cotangengent束的浮动理论之间的新关系。