Coulomb Branches, Shifted Quantum Groups, and their Applications

这个项目是在数学的几个领域的交叉点：表示论，经典和量子可积系统，数学物理学和枚举代数几何。虽然前三个分支起源于量子物理学，但最后一个分支涉及纯粹代数概念在几何学中的应用。表示论涉及研究向量空间的对称性，例如三维欧几里得空间（更一般地说，无限维空间）。这些对称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）有足够多的成对交换对称的情况，这是可积系统的主要研究课题;（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的情况。该项目旨在通过研究移位量子群来解决与这些情况有关的几个悬而未决的问题; PI最近的工作中发现了这些新代数与Toda类量子（差分）可积系统和量化库仑分支的第一个令人惊讶的联系。本论文的主要研究方向是移位量子仿射代数及其量子化库仑分支上的新结构。该项目分为以下五个部分。第一部分研究移位量子仿射代数的积分形式。一个目标是表明，他们满射映射到量化的K-理论库仑分支，并明确描述这些地图的内核使用洗牌的方法。另一个重要的结构是在这样的积分形式上构造的余积同态：这些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的主要工作是利用移位量子仿射代数构造和研究量子K理论库仑分支上量子团簇代数结构的monoidal分类，以及利用gl（n）的移位仿射Yangians构造新的顶点算子代数。第四部分讨论了利用量子群的反支配位移构造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，它也将强调一个被忽视的重要性的反优势位移，导致一个新的研究Bethe子代数的量化库仑分支。这项工作也将提供一个系统的巴克斯特Q-运营商的建设，这意味着他们的功能和TQ关系。该项目的第五部分旨在获得DeConcini-Kac截断移位量子仿射代数在单位根的有限维表示的Kazhdan-Lusztig型特征公式。该奖项反映了NSF的法定使命，并通过使用基金会的智力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.