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.

该项目位于多个数学领域的交叉点：表示论、经典和量子可积系统、数学物理和枚举代数几何。前三个分支源于量子物理学，而最后一个分支则涉及纯代数概念在几何中的应用。表示论涉及向量空间的对称性研究，例如具有附加结构的三维欧几里得空间（更一般地说，无限维空间）。这些对称性通常可以被认为是代数结构，例如群、李代数或关联代数。有两种情况特别令人感兴趣：（1）足够多的成对交换对称性的情况，这是可积系统研究的主要主题，以及（2）当基础向量空间通过与以下相关的广义上同调理论产生时的情况几何模空间。该项目旨在通过研究移位量子群来解决与这些案例相关的几个悬而未决的问题； PI 最近的工作中首次发现了这些新颖代数与 Toda 类量子（差分）可积系统和量子化库仑支的令人惊讶的联系。该研究的主要主题是研究移位量子仿射代数以及量子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。第一部分将研究移位量子仿射代数的积分形式。一个目标是表明它们满射地映射到量化 K 理论库仑分支上，并使用洗牌方法明确描述这些映射的内核。在这种积分形式上构建的另一个重要结构是余积同态：它们将下降到截断的对应部分，从而量化相应经典 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位量子仿射代数构造和研究量子簇代数结构在量化K理论库仑分支上的幺半群分类，以及通过移位仿射Yangian构造新的顶点算子代数的gl(n)。第四部分讨论了一种通过反显性移位量子群来处理 Lax 矩阵的新方法。这将为现在相对古老的逆散射方法主题带来新的见解。同时，它还将强调反显性位移被忽视的重要性，从而引发对量子化库仑分支的贝特子代数的新研究。这项工作还将提供 Baxter Q 算子的系统构建，暗示它们的函数关系和 TQ 关系。该项目的第五部分旨在获得统一根处 DeConcini-Kac 截断移位量子仿射代数的有限维表示的 Kazhdan-Lusztig 型特征公式。该奖项反映了 NSF 的法定使命，并通过使用评估结果被认为值得支持。基金会的智力价值和更广泛的影响审查标准。