Coulomb Branches, Shifted Quantum Groups, and their Applications

该项目位于数学几个领域的交叉点：表示理论、经典和量子可积系统、数学物理和枚举代数几何。前三个分支起源于量子物理学，而后一个分支则涉及纯代数概念在几何中的应用。表征理论关注具有附加结构的向量空间的对称性研究，例如三维欧几里得空间（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，如群、李代数或结合代数。有两种情况特别令人感兴趣：(1)足够多的对交换对称的情况，这是可积系统研究的一个主要课题；(2)当潜在的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的情况。该项目旨在通过研究位移量子群来解决与这些情况有关的几个悬而未决的问题；这些新代数与toda类量子（差分）可积系统和量子化库仑分支的第一个令人惊讶的联系是在PI最近的工作中发现的。本课题主要研究移量子仿射代数及其在量子化库仑分支上的新结构。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。一个目标是证明它们映射到量子化的k -理论库仑分支上，并使用shuffle方法显式地描述这些映射的核。在这种积分形式上构造的另一个重要结构是协积同态：它们将下降到截断的对应物，从而量化相应的经典k理论库仑分支的乘法。第二部分和第三部分是通过位移量子仿射代数构造和研究量子化k -理论库仑分支上量子簇代数结构的一元分类，以及通过gl(n)的位移仿射yangian构造新的顶点算子代数。第四部分讨论了一种利用反优势位移量子群求解Lax矩阵的新方法。这将为逆散射法这一相对陈旧的课题带来新的见解。同时，它还将强调被忽视的反优势位移的重要性，从而导致对量子化库仑分支的贝特子代数的新研究。本工作还将提供Baxter q -算子的系统构建，暗示它们的功能关系和tq关系。项目的第五部分旨在获得单位根处DeConcini-Kac截断位移量子仿射代数的有限维表示的Kazhdan-Lusztig型特征公式。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.