Monodromy Theorems, Affine Quantum Groups, and Meromorphic Tensor Categories

量子群是自然界中最基本对称性的变形。它们是在80年代中期被发现的，它们是一维和二维统计力学模型的对称性，例如描述薄冰层。令人惊讶的是，量子群最近作为四维规范理论的对称性出现，该理论描述了诸如夸克等基本粒子的相互作用，以及它们的五维和六维推广，以及几何中一类计数问题的约束。辫子群是另一类普遍存在的对称，它出现在纽结的数学研究、三维形状的分析、被约束为在二维中运动的基本粒子的统计以及量子计算中。因为对称性的存在极大地限制了这些系统，所以对它们对称组的内在和外在结构的数学研究是理解它们的物理和数学性质的关键。这一研究项目将使人们对平面、球面和环面对应的量子群之间的关系有更深入的了解。传统上，它们被认为处于日益复杂的阶梯中的不同阶梯上。正在进行的研究正在揭示一个惊人的事实，即这些量子群实际上是等价的。这种等价性本身很有趣，但它们也允许精确描述辫子群在不同的数学和物理背景下的外观。这个研究项目集中在与一维代数群相关的无限维量子群：延安、量子循环代数和椭圆曲线。该项目建立在延安和量子循环代数的有限维表示的等价性基础上。Pi和合作者将把有限维代数的这种等价推广到无限维量子循环代数和椭圆量子群之间的等价，从而阐明椭圆量子群的结构，后者在类型A之外仍然鲜为人知。这些等价可以被认为是仿射李代数和有限维量子群之间Kazhdan-Lusztig等价的q变形亚纯版本。这个项目的一个关键方面是研究这些仿射量子群表示的亚纯辫子范畴，这为这类范畴提供了新的例子。利用新导出的等价性，有理仿射、三角和椭圆Casimir连接的单调性将以量子Weyl群算符的形式计算，其方式类似于Kohno-Drinfeld量子群。这项研究将应用于延安、量子循环代数和椭圆量子群、量子可积系统和计数几何中的问题。

Quantum groups are deformations of the most basic symmetries in nature. They were discovered in the mid-eighties as symmetries of one- and two-dimensional statistical mechanical models describing, for example, thin layers of ice. Amazingly, quantum groups have recently arisen as the symmetries of four-dimensional gauge theories, which describe the interaction of elementary particles such as quarks, and their five- and six-dimensional generalizations, as well as the constraints of a class of counting problems in geometry. Braid groups are another pervasive class of symmetries that arise in the mathematical study of knots, the analysis of three-dimensional shapes, the statistics of elementary particles constrained to move in two dimensions, and quantum computing. Because the presence of symmetries greatly constrains these systems, the mathematical study of the intrinsic and extrinsic structures of their groups of symmetries is key in understanding their physical and mathematical properties. This research project will lead to a deeper understanding of the relationship between the quantum groups corresponding to the plane, the sphere, and the torus. These have been traditionally thought to sit on distinct rungs in a ladder of increasing complexity. Research in progress is uncovering the striking fact that these quantum groups are, in fact, equivalent. Such equivalences are interesting in themselves, but they also allow precise characterization of the appearance of braid groups in disparate mathematical and physical contexts. This research project centers on infinite-dimensional quantum groups associated to one-dimensional algebraic groups: Yangians, quantum loop algebras, and elliptic curves. The project builds on the equivalence of finite-dimensional representations of Yangians and quantum loop algebras. The PI and collaborators will promote this equivalence of finite-dimensional algebras to an equivalence between infinite-dimensional quantum loop algebras and elliptic quantum groups, thereby elucidating the structure of the latter, which is still little understood outside type A. These equivalences may be thought of as q-deformed, meromorphic versions of the Kazhdan-Lusztig equivalence between affine Lie algebras and finite-dimensional quantum groups. A key aspect of this project is the study of the meromorphic braided categories of representations of these affine quantum groups, which provides new examples of such categories. Using the newly derived equivalence, the monodromy of rational affine, trigonometric, and elliptic Casimir connections will be computed in terms of quantum Weyl group operators in a way reminiscent of the Kohno-Drinfeld quantum group. This research will have applications to Yangians, quantum loop algebras and elliptic quantum groups, quantum integrable systems, and questions in enumerative geometry.