Multidimensional Hypergeometric Functions

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This project will focus on multidimensional hypergeometric functions and their q-analogs appear as solutions to differential and difference equations in representation theory, conformal field theory, and statistical mechanics. The equations are formulated in terms of representation theory of Lie algebras and quantum groups. The hypergeometric and q-hypergeometric functions appearing in these considerations have rich and interesting mathematical structures. Studying these structures will lead to better understanding of interrelations of the above theories as well as to establishing new connections among them. A class of (q-hypergeometric functions is determined by a choice of a simple Lie algebra and a number called central charge or level. The theory of hypergeometric functions is better developed when the Lie algebra is sl2 and the level is not critical. The limit when the level tends to its critical value is the theory of the Bethe ansatz method in quantum integrable systems. The goal of the proposal is to develop the theory of (q-)hypergeometric functions for higher rank Lie algebras and the associated Bethe ansatz method with applications to representation theory, CFT, and statistical mechanics