Coulomb Branches, Shifted Quantum Groups, and their Applications

库仑支、移位量子群及其应用

• 批准号: 2001247
• 负责人: Oleksandr Tsymbaliuk
• 金额: $ 16.5万
• 依托单位: Yale University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2020
• 资助国家: 美国
• 起止时间: 2020-06-01 至 2020-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2001247&HistoricalAwards=false
• 关键词: Coulomb; Branches; Shifted; Quantum; Groups

This project lies at the intersection of several fields of mathematics: representation theory, classical and quantum integrable systems, mathematical physics, and enumerative algebraic geometry. While the former three branches originate from quantum physics, the last one deals with applications of purely algebraic concepts to geometry. Representation theory concerns the study of symmetries of a vector space such as three-dimensional Euclidean space (more generally, an infinite dimensional space) with additional structures. These symmetries can be often thought of as algebraic structures such as groups, Lie algebras, or associative algebras. Two cases are of particular interest: (1) the case of sufficiently many pair-wise commuting symmetries, which is a primary subject of study in integrable systems, and (2) the case when the underlying vector spaces arise via generalized cohomology theories associated with geometric moduli spaces. This project aims at resolving several open questions pertaining to those cases through the study of shifted quantum groups; the first surprising connections of those novel algebras to Toda-like quantum (difference) integrable systems and quantized Coulomb branches were discovered in the recent work of the PI. The major theme of the proposed research is the study of shifted quantum affine algebras and the corresponding new structures on the quantized Coulomb branches. The project is broken down into five parts, as follows. The first part will investigate integral forms of shifted quantum affine algebras. One objective is to show that they map surjectively onto quantized K-theoretic Coulomb branches and to describe explicitly the kernel of these maps using the shuffle approach. Another important structure to be constructed on such integral forms are coproduct homomorphisms: these will descend to the truncated counterparts, thus quantizing multiplications of the corresponding classical K-theoretic Coulomb branches. The second and the third parts of the project are aimed at the construction and study of monoidal categorification of the quantum cluster algebra structure on quantized K-theoretic Coulomb branches via shifted quantum affine algebras, and a construction of new vertex operator algebras via shifted affine Yangians of gl(n). The fourth part deals with a novel approach to Lax matrices via antidominantly shifted quantum groups. This will bring new insights into now relatively old subject of the inverse scattering method. At the same time, it will also emphasize an overlooked importance of antidominant shifts, leading to a new study of Bethe subalgebras of the quantized Coulomb branches. This work will also provide a systematic construction of Baxter Q-operators, implying functional and TQ-relations for them. The fifth part of the project aims to obtain Kazhdan-Lusztig type character formulas for finite-dimensional representations of DeConcini-Kac truncated shifted quantum affine algebras at roots of unity.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

该项目位于数学几个领域的交集：表示理论，经典和量子整合系统，数学物理学以及枚举代数几何形状。虽然前三个分支来自量子物理学，但最后一个分支介绍了纯粹代数概念到几何形状的应用。表示理论涉及对矢量空间的对称性的研究，例如三维欧几里得空间（更一般而言，是无限的尺寸空间），并具有其他结构。这些对称性通常被认为是代数结构，例如群体，谎言代数或联想代数。两种情况特别令人感兴趣：（1）足够多的成对通勤对称性的情况，这是在可集成系统中的主要研究主题，以及（2）当基础矢量空间通过与几何模态空间相关的广义共同体理论而产生的情况。该项目旨在通过研究转移的量子组解决与这些情况有关的几个开放问题。在PI的最新工作中，发现了这些新型代数与TODA样量子（差异）集成系统和量化库仑分支的第一个令人惊讶的连接。拟议的研究的主要主题是研究量子转移的代数和量化库仑分支上相应的新结构。该项目分为五个部分，如下所示。第一部分将研究移动的量子仿射代数的积分形式。一个目的是证明它们将其映射到量化的K理论库仑分支上，并使用洗牌方法明确描述这些地图的内核。在此类积分形式上要构建的另一个重要结构是共生同构：这些结构将下降到截短的对应物，从而量化相应的经典K理论库仑分支的乘法。该项目的第二部分和第三部分旨在通过移动的量子仿射代数来构建和研究量子群集代数结构的单体分类，并通过移动的gl（n）的转移偏爱Yangians构建新的Vertex操作员代数。第四部分介绍了一种通过抗主流量子组的新方法来宽松矩阵。这将使逆散射方法的现在相对较古老的主题带来新的见解。同时，它还将强调抗主导偏移的重要性，从而导致对量化库仑分支的伯特伯氏骨架的新研究。这项工作还将为百特Q-操作员提供系统的构建，这意味着它们的功能和TQ关系。该项目的第五部分旨在获取kazhdan-lusztig类型字符公式，用于在Unity的基础上截断了Deconcini-Kac截断的截断量子代数的有限维度，这奖反映了NSF的法定任务，并通过使用基础的智力效果和宽阔的范围进行评估，以评估值得评估。