Flat Connections, Irregular Singularities and Quantum Groups

这一建议源于Pi最近对Kohno-Drinfeld定理的推广，根据该定理，Casimir联络的单调性由相应量子群的量子Weyl群算子来描述。Casimir联络是单李代数的根超平面上具有对数奇点的平坦联络。这一结果是PI和De Concini独立猜测的。第一个项目与B.Feigin和E.Frenkel合作，试图通过展示Wess-Zumino-Witten模型中的新场来建立Casimir联系和共形场理论之间的精确联系，这些场的关联函数满足相应的微分方程组，从而将具有不规则奇点的联系纳入该模型。除了共形场论之外，这个项目还有许多重要的潜在应用：单李代数的结构和表示理论(论元子代数的移位的量子化，不可约有限维表示的新的基的构造)和统计力学(构造推广高丁模型的新的可积模型)。第二个项目的目的是在单位根上推广圆周率的单行定理。这一推广可以为广义辫子张量范畴的研究提供动力，广义辫子张量范畴的基本辫子群是广义辫子群，其定义隐含在拟Coxeter代数的公理中。最后一个与R.Rouquier合作的项目是关于动态图上同调的，这是PI为了控制拟Coxeter代数的形变而引入的。它的目的是通过计算Coxeter群和半单李代数的包络代数来得到对这种上同调的更好的组合和拓扑理解。量子群是自然界最基本的对称群的变形。它们在80年代中期作为某些量子力学系统的对称性被发现，此后出现在数学和物理的广泛领域，如弦论、量子统计力学、有限对称群的研究、拓扑学和组合学。它们令人困惑的方面之一是它们处理和解决这些领域中长期存在的问题的不可思议的能力，这些问题通常是在没有吸引量子群的情况下制定的。PI一直在寻求这样的研究途径，通过使用量子群来描述复域中某些微分方程组的解的分支行为。这项提议旨在进一步扩大我们对这一耐人寻味的现象的理解。该项目对弦理论和表示理论具有重要的潜在应用价值。预计它还将导致编织张量范畴的广泛推广，这些范畴在目前的伪装下，已被广泛用于计算机科学、逻辑和最近的量子计算。国际和平研究所曾多次向年轻研究人员和研究生的听众介绍他的最新成果，并总是找到非常容易接受的听众。如上所述，由它们产生的进一步发展很可能会吸引这些年轻的数学家。

This proposal stems from the PI's recent generalisation of the Kohno-Drinfeld theorem according to which the monodromy of the Casimir connection, a flat connection with logarithmic singularities on the root hyperplanes of a simple Lie algebra, is described by the quantum Weyl group operators of the corresponding quantum group. This result had been conjectured independently by the PI and De Concini. The first project, in collaboration with B. Feigin and E. Frenkel, endeavours to establish a precise link between the Casimir connection and Conformal Field Theory by exhibiting new fields in the Wess-Zumino-Witten model whose correlation functions satisfy the corresponding system of differential equations, thus incorporating connections with irregular singularities into this model. This project has a number of important potential applications beyond those to Conformal Field Theory: to the structure and representation theory of simple Lie algebras (quantisation of the shift of argument subalgebra, construction of new basis of irreducible finite dimensional representations) and to Statistical Mechanics (construction of new integrable models generalising the Gaudin model). The second project aims at extending the PI's monodromy theorem at roots of unity. This extension could provide the impetus for the study of generalised braided tensor categories, that is tensor categories whose underlying braid group is a generalised one, and whose definition is implicit in the axioms of a quasi-Coxeter algebra. The last project, in collaboration with R. Rouquier, is concerned with Dynkin diagram cohomology, which was introduced by the PI to control deformations of quasi-Coxeter algebras. Its aim is to arrive at a better combinatorial and topological understanding of this cohomology by computing it for Coxeter groups and the enveloping algebras of semi-simple Lie algebras.Quantum groups are deformations of the most basic symmetry groups of Nature. They were discovered in the mid-eighties as symmetries of certain Quantum mechanical systems and have since appeared in a wide range of fields in Mathematics and Physics such as String Theory, Quantum Statistical Mechanics, the study of finite symmetry groups, Topology and Combinatorics. One of their bewildering aspects is their uncanny ability to address, and solve, long standing problems in these fields which are often formulated without appealling to quantum groups. The PI has been pursuing one such avenue of investigation, by using quantum groups to describe the branching behaviour of solutions of certain systems of differential equations in the complex domain. This proposal aims at further broadening our understanding of this intriguing phenomenon. This project has important potential applications to String Theory and Representation Theory. It is also expected that it will lead to a wide generalisation of Braided Tensor Categories which, in their present guise, have been extensively used in Computer Science, Logic and, more recently, Quantum Computing. The PI has, on several occasions, given lectures to audiences of young researchers and graduate students about his recent results and always found an extremely receptive audience. It is likely that the further developments stemming from them which are presented above will attract such young mathematicians.