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.

这个项目涉及几个数学领域的交叉：表象理论、经典和量子可积系统、数学物理和计数代数几何。前三个分支起源于量子物理，后一个分支处理纯代数概念在几何学中的应用。表示论主要研究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)等向量空间的对称性。这些对称性通常可以被认为是代数结构，如群、李代数或结合代数。有两种情况特别令人感兴趣：(1)足够多的成对交换对称的情况，这是可积系统中的一个主要研究主题；(2)当基础向量空间通过与几何模空间相关的广义上同调理论出现时。这个项目的目的是通过对移位量子群的研究来解决与这些情况相关的几个公开问题；在PI最近的工作中发现了这些新的代数与类Toda量子(差)可积系统和量子化库仑分支的第一个令人惊讶的联系。这项研究的主要主题是研究移位量子仿射代数以及相应的量子化库仑分支上的新结构。该项目分为五个部分，具体如下。第一部分研究移位量子仿射代数的积分形式。一个目的是证明它们满射地映射到量子化的K-理论库仑分支上，并用随机方法显式地描述这些映射的核。建立在这种积分形式上的另一个重要结构是余积同态：这些同态将下降到截断的对应形式，从而量化对应的经典K-理论库仑分支的乘法。第二部分和第三部分是利用移位量子仿射代数构造和研究量子化K-理论库仑分支上的量子簇代数结构的单态范畴，以及利用gl(N)的移位仿射延安构造新的顶点算子代数。第四部分讨论了一种利用反支配移位量子群构造Lax矩阵的新方法。这将为现在相对古老的逆散射方法带来新的见解。同时，它还将强调被忽视的反支配移位的重要性，从而导致对量子化库仑分支的Bethe子代数的新的研究。这项工作还将提供Baxter Q-算子的系统构造，为它们隐含泛函关系和TQ-关系。该项目的第五部分旨在获得用于单位根的DeConcini-Kac截断移位量子仿射代数的有限维表示的Kazhdan-Lusztig型特征标公式。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。