Multidimensional Hypergeometric Functions and Dynamical Quantum Groups

主要研究者：亚历山大瓦尔琴科，英国查佩尔山DMS-0244579摘要：该项目的目标是求解任意亏格的黎曼曲面上的共形块的微分方程，对共形块的微分方程进行离散化，描述方程解的模性质，构造动力学量子群的表示理论，这是一种基于共形块性质的代数结构。自世纪以来，特殊函数一直是数学研究的对象。高斯超几何函数就是一个例子。特殊函数理论是求解微分方程的一个有用工具，具有重要的应用价值。 新的数学物理问题需要新的特殊函数，并提出了研究特殊函数的新方法。 共形场论起源于临界温度下的表面物理学，以及粒子相互作用的统一模型弦理论。本文的研究对象是共形场论的数学结构。该理论明确地将一个称为共形块的复向量空间的数学对象与模拟物理系统的方程相关联。

PI: Alexander Varchenko, UNC Chapel Hill DMS-0244579ABSTRACT: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, to describe the modular properties of solutions of the equations, to construct the representation theory of dynamical quantum groups, which is an algebraic structure underlying the properties of the conformal blocks.Special functions have been the objects of mathematical study since the seventeenth century. The Gauss hypergeometric function is an example. 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. Conformal field theory finds its origin in the physics of surface at critical temperature, and string theory, a unified model of particle interactions. The object of this research is the mathematical structure of conformal field theory. This theory explicitly associates a mathematical object called a complex vector space of conformal blocks with equations that model physical systems.