Multidimensional Hypergeometric Function Associated with Riemann Surfaces and Dynamical Quantum Groups

题目：“与黎曼曲面和动态量子群相关的多维超几何函数” 首席研究员：Alexander Varchenko 摘要：该项目的目标是求解任意亏格黎曼曲面上的共形块的微分方程，量化共形块的微分方程并开发量化差分方程解的模性质，构建动态量子群的表示理论，形成动态量子群性质的代数结构量化共形块。特殊函数理论是求解对应用很重要的微分方程的有用工具。数学物理的新问题需要新的特殊函数，并提出了研究特殊函数的新方法。 高斯超几何函数是经典特殊函数的一个例子。 高斯超几何函数满足显着的超几何微分方程。 超几何微分方程与仿射李代数的表示论相联系。 高斯超几何函数的单性性质引出了量子群理论。 现代场论和统计力学对高斯超几何函数提出了一种新的观点，即将超几何微分方程视为黎曼球上共形块微分方程的特例，将高斯超几何函数视为黎曼球上特殊共形块。 因此，出现了一类新的特殊函数——黎曼曲面上的共形块。为了描述共形块的代数性质，我们需要新的代数结构。即，黎曼球上共形块性质的代数结构是标准量子群结构。环面上共形块特性的代数结构是最近发现的动态量子群结构。人们可以预期，为了描述给定属的黎曼表面上的共形块的性质，我们将需要与量子群论的该属变体相对应的。 该项目的目的是发展与任意属的黎曼曲面相关的特殊函数理论，并研究与这些特殊函数相关的代数结构。

Title: "Multidimensional Hypergeometric Functions Associated with Riemann Surfaces and Dynamical Quantum Groups" Principal Investigator: Alexander Varchenko Abstract: The goal of the project is to solve differential equations for conformal blocks on Riemann surfaces of an arbitrary genus, to quantize the differential equations for conformal blocks and develop modular properties of solutions of the quantized difference equations, to construct the representation theory of dynamical quantum groups, which form an algebraic structure underlying the properties of the quantized conformal blocks.The theory of special functions is a useful tool to solve differential equations important for applications. New problems of mathematical physics require new special functions and suggest new approaches to studying special functions. The Gauss hypergeometric function is an example of a classical special function. The Gauss hypergeometric function satisfies the remarkable hypergeometric differential equation. The hypergeometric differential equation is connected with representation theory of affine Lie algebras. The monodromy properties of the Gauss hypergeometric function lead to the theory of quantum groups. Modern field theory and statistical mechanics suggest a new point of view on the Gauss hypergeometric function, namely, one considers the hypergeometric differential equation as a special example of the differential equation for conformal blocks on the Riemann sphere and the Gauss hypergeometric function as a special conformal block on the Riemann sphere. Thus, a new class of special functions arises - the conformal blocks on Riemann surfaces. To describe algebraic properties of conformal blocks one will need new algebraic structures. Namely, the algebraic structure underlying the properties of conformal blocks on the Riemann sphere is the standard quantum group structure. The algebraic structure underlying the properties of conformal blo cks on the torus is the recently discovered dynamical quantum group structure. One can expect that to describe properties of conformal blocks on Riemann surfaces of a given genus, one will need the corresponding to this genus variant of the quantum group theory. The aim of this project is to develop the theory of special functions associated with Riemann surfaces of an arbitrary genus and study algebraic structures associated with these special functions.