Multidimensional Hypergeometric Function Associated with Riemann Surfaces and Dynamical Quantum Groups

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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.