Multidimensional Hypergeometric Function Associated with Riemann Surfaces and Dynamical Quantum Groups

主要研究人员：Alexander Varchenko摘要：该项目的目标是求解任意亏格的Riemann曲面上的共形块的微分方程组，量子化共形块的微分方程组，发展量化差分方程解的模性质，构造动态量子群的表示理论，形成量子化共形块性质的代数结构。特殊函数理论是求解具有重要应用意义的微分方程组的有用工具。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著的超几何微分方程。超几何微分方程与仿射李代数的表示理论相联系。高斯超几何函数的单调性质导致了量子群理论的产生。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，即把超几何微分方程看作黎曼球面上共形块的微分方程的特例，把高斯超几何函数看作黎曼球上的特殊共形块。于是，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，需要新的代数结构。也就是说，黎曼球面上共形块性质的基本代数结构是标准量子群结构。环面上共形块性质的基本代数结构是最近发现的动态量子群结构。可以预期，要描述给定亏格的Riemann曲面上的共形块的性质，将需要对应于量子群论的这个亏格变体。这个项目的目的是发展与任意亏格的Riemann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.