Casimir connections, Yangians and quantum loop algebras

卡西米尔连接、Yangians 和量子环代数

• 批准号: 1206305
• 负责人: Valerio Toledano Laredo
• 金额: $ 14.14万
• 依托单位: Northeastern University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-08-01 至 2015-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1206305&HistoricalAwards=false
• 关键词: Casimir; connections; Yangians; quantum; loop

Quantum groups were discovered in the mid--eighties as symmetries of 1 and 2-dimensional Statistical Mechanical models. After a period of intense development, they reemerged more recently as the symmetries of 4-dimensional supersymmetric quantum gauge theories, through the work of Nekrasov--Shatashvilii, and as the constraints governing the enumerative geometry of Nakajima quiver varieties, through the work of Maulik-Okounkov. The present proposal bears upon two such infinite-dimensional quantum groups which are associated to a complex semisimple Lie algebra: the Yangian and the quantum loop algebra. An important component of the proposal is joint between the PI and S. Gautam, and builds upon a precise link they recently discovered between these quantum groups. It endeavors on the one hand to promote the above link to an equivalence between categories of finite-dimensional representations and, on the other, to use this equivalence to compute the monodromy the trigonometric Casimir connection of the Yangian in terms of the quantum Weyl group operators of the quantum loop algebra, in a way reminiscent of the Kohno--Drinfeld theorem. The proposal has potential implications for the representation theory of Yangians and quantum loop algebras, quantum integrable systems and enumerative geometry.Quantum groups are deformations of the most basic symmetries of Nature. They were discovered in the mid-eighties as symmetries of 1 and 2-dimensional statistical mechanical models and, amazingly, reemerged recently as the symmetries of 4-dimensional quantum gauge theories, as well as the constraints of a class of enumerative problems in geometry. The mathematical study of their intrinsic and extrinsic structures is often key in understanding, and solving, the physical and mathematical systems they govern, since the presence of these symmetries greatly constrains these systems. The goal of this project is to better understand the relationship between two such classes of infinite-dimensional quantum groups, and to use this relation to describe the evolution of the systems governed by the first as some of its physical parameters (masses for example) are tuned, in terms of the second.

量子群是在80年代中期作为1维和2维统计力学模型的对称性被发现的。经过一段时间的激烈发展，他们重新出现最近作为四维超对称量子规范理论的对称性，通过工作的Nekrasov-Shatashvilii，并作为约束管理枚举几何的Nakajima的变种，通过工作的Maulik-Okounkov。目前的建议承担两个这样的无限维量子群，这是一个复杂的半单李代数：杨和量子回路代数。该提案的一个重要组成部分是PI和S之间的联合。Gautam，并建立在他们最近发现的这些量子群之间的精确联系之上。它一方面努力促进上述联系之间的等价类别的有限维表示，另一方面，使用这种等价计算单值三角卡西米尔连接的杨在量子Weyl群运营商的量子圈代数，在某种程度上让人想起Kohno-德林费尔德定理。这个提议对杨格群的表示理论、量子圈代数、量子可积系统和计数几何都有潜在的意义。量子群是自然界最基本对称性的变形。它们在80年代中期作为1维和2维统计力学模型的对称性被发现，令人惊讶的是，它们最近作为4维量子规范理论的对称性以及一类几何计数问题的约束重新出现。对它们的内在和外在结构的数学研究通常是理解和解决它们所支配的物理和数学系统的关键，因为这些对称性的存在极大地限制了这些系统。该项目的目标是更好地理解两类无限维量子群之间的关系，并使用这种关系来描述由第一类量子群控制的系统的演化，因为它的一些物理参数（例如质量）是根据第二类量子群调整的。