Multidimensional Hypergeometric Functions and Dynamical Quantum Groups

University Of North Carolina At Chapel Hill, Chapel Hill NC

Investigators

Abstract

PI: Alexander Varchenko, UNC Chapel Hill DMS-0244579 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, 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.